# Properties

 Label 2520.1.ek.b.1909.1 Level $2520$ Weight $1$ Character 2520.1909 Analytic conductor $1.258$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2520.ek (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ 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: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.3630312000.4

## Embedding invariants

 Embedding label 1909.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2520.1909 Dual form 2520.1.ek.b.829.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{10} +(1.50000 - 0.866025i) q^{11} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{20} +1.73205i q^{22} +1.00000 q^{25} +(0.500000 - 0.866025i) q^{28} -1.73205i q^{29} +(-1.50000 + 0.866025i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{35} +1.00000 q^{40} +(-1.50000 - 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-0.500000 - 0.866025i) q^{53} +(1.50000 - 0.866025i) q^{55} +(0.500000 + 0.866025i) q^{56} +(1.50000 + 0.866025i) q^{58} +(-0.500000 - 0.866025i) q^{59} -1.73205i q^{62} +1.00000 q^{64} -1.00000 q^{70} +(-1.00000 - 1.73205i) q^{73} +(1.50000 + 0.866025i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{80} +1.73205i q^{83} +(1.50000 - 0.866025i) q^{88} +1.00000 q^{97} +(-0.500000 - 0.866025i) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 2q^{5} + q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 2q^{5} + q^{7} + 2q^{8} - q^{10} + 3q^{11} - 2q^{14} - q^{16} - q^{20} + 2q^{25} + q^{28} - 3q^{31} - q^{32} + q^{35} + 2q^{40} - 3q^{44} - q^{49} - q^{50} - q^{53} + 3q^{55} + q^{56} + 3q^{58} - q^{59} + 2q^{64} - 2q^{70} - 2q^{73} + 3q^{77} - q^{79} - q^{80} + 3q^{88} + 2q^{97} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$3$$ 0 0
$$4$$ −0.500000 0.866025i −0.500000 0.866025i
$$5$$ 1.00000 1.00000
$$6$$ 0 0
$$7$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$8$$ 1.00000 1.00000
$$9$$ 0 0
$$10$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$11$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ −1.00000 −1.00000
$$15$$ 0 0
$$16$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$17$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$18$$ 0 0
$$19$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$20$$ −0.500000 0.866025i −0.500000 0.866025i
$$21$$ 0 0
$$22$$ 1.73205i 1.73205i
$$23$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$24$$ 0 0
$$25$$ 1.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0.500000 0.866025i 0.500000 0.866025i
$$29$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$30$$ 0 0
$$31$$ −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i $$0.666667\pi$$
−1.00000 $$\pi$$
$$32$$ −0.500000 0.866025i −0.500000 0.866025i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$36$$ 0 0
$$37$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 1.00000 1.00000
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −1.50000 0.866025i −1.50000 0.866025i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$48$$ 0 0
$$49$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$50$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 1.50000 0.866025i 1.50000 0.866025i
$$56$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$57$$ 0 0
$$58$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$59$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$62$$ 1.73205i 1.73205i
$$63$$ 0 0
$$64$$ 1.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ −1.00000 −1.00000
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$78$$ 0 0
$$79$$ −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i $$0.333333\pi$$
−1.00000 $$\pi$$
$$80$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 1.50000 0.866025i 1.50000 0.866025i
$$89$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$98$$ −0.500000 0.866025i −0.500000 0.866025i
$$99$$ 0 0
$$100$$ −0.500000 0.866025i −0.500000 0.866025i
$$101$$ 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$102$$ 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
$$106$$ 1.00000 1.00000
$$107$$ −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i $$0.333333\pi$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$110$$ 1.73205i 1.73205i
$$111$$ 0 0
$$112$$ −1.00000 −1.00000
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$117$$ 0 0
$$118$$ 1.00000 1.00000
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 1.73205i 1.00000 1.73205i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$125$$ 1.00000 1.00000
$$126$$ 0 0
$$127$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−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 −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0.500000 0.866025i 0.500000 0.866025i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.73205i 1.73205i
$$146$$ 2.00000 2.00000
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$150$$ 0 0
$$151$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$155$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$156$$ 0 0
$$157$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$158$$ −0.500000 0.866025i −0.500000 0.866025i
$$159$$ 0 0
$$160$$ −0.500000 0.866025i −0.500000 0.866025i
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −1.50000 0.866025i −1.50000 0.866025i
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$174$$ 0 0
$$175$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$176$$ 1.73205i 1.73205i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 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 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$192$$ 0 0
$$193$$ −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i $$0.666667\pi$$
−1.00000 $$\pi$$
$$194$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$195$$ 0 0
$$196$$ 1.00000 1.00000
$$197$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$200$$ 1.00000 1.00000
$$201$$ 0 0
$$202$$ −2.00000 −2.00000
$$203$$ 1.50000 0.866025i 1.50000 0.866025i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1.00000 1.73205i −1.00000 1.73205i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ −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
$$217$$ −1.50000 0.866025i −1.50000 0.866025i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −1.50000 0.866025i −1.50000 0.866025i
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$224$$ 0.500000 0.866025i 0.500000 0.866025i
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$228$$ 0 0
$$229$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.73205i 1.73205i
$$233$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i $$0.666667\pi$$
−1.00000 $$\pi$$
$$242$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$249$$ 0 0
$$250$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$251$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −1.50000 0.866025i −1.50000 0.866025i
$$255$$ 0 0
$$256$$ −0.500000 0.866025i −0.500000 0.866025i
$$257$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$263$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$264$$ 0 0
$$265$$ −0.500000 0.866025i −0.500000 0.866025i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 $$0$$
0.500000 + 0.866025i $$0.333333\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.50000 0.866025i 1.50000 0.866025i
$$276$$ 0 0
$$277$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 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
$$289$$ 0.500000 0.866025i 0.500000 0.866025i
$$290$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$291$$ 0 0
$$292$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$293$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$294$$ 0 0
$$295$$ −0.500000 0.866025i −0.500000 0.866025i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 1.00000 1.00000
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 1.73205i 1.73205i
$$309$$ 0 0
$$310$$ 1.73205i 1.73205i
$$311$$ 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
$$316$$ 1.00000 1.00000
$$317$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$318$$ 0 0
$$319$$ −1.50000 2.59808i −1.50000 2.59808i
$$320$$ 1.00000 1.00000
$$321$$ 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 $$0.666667\pi$$
$$332$$ 1.50000 0.866025i 1.50000 0.866025i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$338$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1.50000 + 2.59808i −1.50000 + 2.59808i
$$342$$ 0 0
$$343$$ −1.00000 −1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ −1.00000 −1.00000
$$351$$ 0 0
$$352$$ −1.50000 0.866025i −1.50000 0.866025i
$$353$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$360$$ 0 0
$$361$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.00000 1.73205i −1.00000 1.73205i
$$366$$ 0 0
$$367$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.500000 0.866025i 0.500000 0.866025i
$$372$$ 0 0
$$373$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$374$$ 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 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$384$$ 0 0
$$385$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$386$$ 1.73205i 1.73205i
$$387$$ 0 0
$$388$$ −0.500000 0.866025i −0.500000 0.866025i
$$389$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$393$$ 0 0
$$394$$ 1.00000 1.73205i 1.00000 1.73205i
$$395$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$396$$ 0 0
$$397$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$401$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 1.00000 1.73205i 1.00000 1.73205i
$$405$$ 0 0
$$406$$ 1.73205i 1.73205i
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2.00000 2.00000
$$413$$ 0.500000 0.866025i 0.500000 0.866025i
$$414$$ 0 0
$$415$$ 1.73205i 1.73205i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ −0.500000 0.866025i −0.500000 0.866025i
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1.00000 1.00000
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$434$$ 1.50000 0.866025i 1.50000 0.866025i
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 $$0$$
0.500000 + 0.866025i $$0.333333\pi$$
$$440$$ 1.50000 0.866025i 1.50000 0.866025i
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0.500000 0.866025i 0.500000 0.866025i
$$447$$ 0 0
$$448$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 1.73205i 1.73205i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 $$0$$
0.500000 + 0.866025i $$0.333333\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −0.500000 0.866025i −0.500000 0.866025i
$$473$$ 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
$$481$$ 0 0
$$482$$ 1.73205i 1.73205i
$$483$$ 0 0
$$484$$ −2.00000 −2.00000
$$485$$ 1.00000 1.00000
$$486$$ 0 0
$$487$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −0.500000 0.866025i −0.500000 0.866025i
$$491$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.73205i 1.73205i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$500$$ −0.500000 0.866025i −0.500000 0.866025i
$$501$$ 0 0
$$502$$ 0.500000 0.866025i 0.500000 0.866025i
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 1.50000 0.866025i 1.50000 0.866025i
$$509$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$510$$ 0 0
$$511$$ 1.00000 1.73205i 1.00000 1.73205i
$$512$$ 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 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$524$$ −1.00000 −1.00000
$$525$$ 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 1.00000
$$539$$ 1.73205i 1.73205i
$$540$$ 0 0
$$541$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$542$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 1.73205i 1.73205i
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1.00000 −1.00000
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1.00000 −1.00000
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i $$0.666667\pi$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$570$$ 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
$$577$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$578$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$579$$ 0 0
$$580$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$581$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$582$$ 0 0
$$583$$ −1.50000 0.866025i −1.50000 0.866025i
$$584$$ −1.00000 1.73205i −1.00000 1.73205i
$$585$$ 0 0
$$586$$ −1.50000 0.866025i −1.50000 0.866025i
$$587$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 1.00000 1.00000
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$600$$ 0 0
$$601$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$605$$ 1.00000 1.73205i 1.00000 1.73205i
$$606$$ 0 0
$$607$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$620$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 1.00000
$$626$$ 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 $$0.666667\pi$$
$$632$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$633$$ 0 0
$$634$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$635$$ 1.73205i 1.73205i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 3.00000 3.00000
$$639$$ 0 0
$$640$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$641$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$648$$ 0 0
$$649$$ −1.50000 0.866025i −1.50000 0.866025i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$654$$ 0 0
$$655$$ 0.500000 0.866025i 0.500000 0.866025i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 1.73205i 1.73205i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$674$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$675$$ 0 0
$$676$$ −0.500000 0.866025i −0.500000 0.866025i
$$677$$ −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −1.50000 2.59808i −1.50000 2.59808i
$$683$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$684$$ 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 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 2.00000 2.00000
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0.500000 0.866025i 0.500000 0.866025i
$$701$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 1.50000 0.866025i 1.50000 0.866025i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$708$$ 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 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$720$$ 0 0
$$721$$ −2.00000 −2.00000
$$722$$ −1.00000 −1.00000
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.73205i 1.73205i
$$726$$ 0 0
$$727$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 2.00000 2.00000
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$734$$ 1.00000 1.00000
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −1.00000 −1.00000
$$750$$ 0 0
$$751$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −0.500000 0.866025i −0.500000 0.866025i
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$770$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$771$$ 0 0
$$772$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$773$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$774$$ 0 0
$$775$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$776$$ 1.00000 1.00000
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −0.500000 0.866025i −0.500000 0.866025i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$788$$ 1.00000 + 1.73205i 1.00000 + 1.73205i
$$789$$ 0 0
$$790$$ −0.500000 0.866025i −0.500000 0.866025i
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −0.500000 0.866025i −0.500000 0.866025i
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.00000 1.73205i −3.00000 1.73205i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$