# Properties

 Label 912.1.ce.b.527.1 Level $912$ Weight $1$ Character 912.527 Analytic conductor $0.455$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 912.ce (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.455147291521$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} + \cdots)$$

## Embedding invariants

 Embedding label 527.1 Root $$-0.173648 - 0.984808i$$ of defining polynomial Character $$\chi$$ $$=$$ 912.527 Dual form 912.1.ce.b.623.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(0.173648 - 0.984808i) q^{3} +(1.70574 - 0.984808i) q^{7} +(-0.939693 - 0.342020i) q^{9} +O(q^{10})$$ $$q+(0.173648 - 0.984808i) q^{3} +(1.70574 - 0.984808i) q^{7} +(-0.939693 - 0.342020i) q^{9} +(-1.26604 + 0.223238i) q^{13} +(0.500000 + 0.866025i) q^{19} +(-0.673648 - 1.85083i) q^{21} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(-0.939693 - 1.62760i) q^{31} +0.684040i q^{37} +1.28558i q^{39} +(0.439693 - 0.524005i) q^{43} +(1.43969 - 2.49362i) q^{49} +(0.939693 - 0.342020i) q^{57} +(-1.17365 + 0.984808i) q^{61} +(-1.93969 + 0.342020i) q^{63} +(0.326352 + 0.118782i) q^{67} +(-0.266044 + 1.50881i) q^{73} +1.00000 q^{75} +(0.0603074 - 0.342020i) q^{79} +(0.766044 + 0.642788i) q^{81} +(-1.93969 + 1.62760i) q^{91} +(-1.76604 + 0.642788i) q^{93} +(0.592396 + 1.62760i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + O(q^{10})$$ $$6 q - 3 q^{13} + 3 q^{19} - 3 q^{21} - 3 q^{27} - 3 q^{43} + 3 q^{49} - 6 q^{61} - 6 q^{63} + 3 q^{67} + 3 q^{73} + 6 q^{75} + 6 q^{79} - 6 q^{91} - 6 q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$e\left(\frac{7}{18}\right)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$3$$ 0.173648 0.984808i 0.173648 0.984808i
$$4$$ 0 0
$$5$$ 0 0 −0.766044 0.642788i $$-0.777778\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$6$$ 0 0
$$7$$ 1.70574 0.984808i 1.70574 0.984808i 0.766044 0.642788i $$-0.222222\pi$$
0.939693 0.342020i $$-0.111111\pi$$
$$8$$ 0 0
$$9$$ −0.939693 0.342020i −0.939693 0.342020i
$$10$$ 0 0
$$11$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$12$$ 0 0
$$13$$ −1.26604 + 0.223238i −1.26604 + 0.223238i −0.766044 0.642788i $$-0.777778\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 0.939693 0.342020i $$-0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$18$$ 0 0
$$19$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$20$$ 0 0
$$21$$ −0.673648 1.85083i −0.673648 1.85083i
$$22$$ 0 0
$$23$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$24$$ 0 0
$$25$$ 0.173648 + 0.984808i 0.173648 + 0.984808i
$$26$$ 0 0
$$27$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$28$$ 0 0
$$29$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$30$$ 0 0
$$31$$ −0.939693 1.62760i −0.939693 1.62760i −0.766044 0.642788i $$-0.777778\pi$$
−0.173648 0.984808i $$-0.555556\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.684040i 0.684040i 0.939693 + 0.342020i $$0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$38$$ 0 0
$$39$$ 1.28558i 1.28558i
$$40$$ 0 0
$$41$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$42$$ 0 0
$$43$$ 0.439693 0.524005i 0.439693 0.524005i −0.500000 0.866025i $$-0.666667\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$48$$ 0 0
$$49$$ 1.43969 2.49362i 1.43969 2.49362i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.939693 0.342020i 0.939693 0.342020i
$$58$$ 0 0
$$59$$ 0 0 0.939693 0.342020i $$-0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$60$$ 0 0
$$61$$ −1.17365 + 0.984808i −1.17365 + 0.984808i −0.173648 + 0.984808i $$0.555556\pi$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ −1.93969 + 0.342020i −1.93969 + 0.342020i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.326352 + 0.118782i 0.326352 + 0.118782i 0.500000 0.866025i $$-0.333333\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 −0.766044 0.642788i $$-0.777778\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$72$$ 0 0
$$73$$ −0.266044 + 1.50881i −0.266044 + 1.50881i 0.500000 + 0.866025i $$0.333333\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$74$$ 0 0
$$75$$ 1.00000 1.00000
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.0603074 0.342020i 0.0603074 0.342020i −0.939693 0.342020i $$-0.888889\pi$$
1.00000 $$0$$
$$80$$ 0 0
$$81$$ 0.766044 + 0.642788i 0.766044 + 0.642788i
$$82$$ 0 0
$$83$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$90$$ 0 0
$$91$$ −1.93969 + 1.62760i −1.93969 + 1.62760i
$$92$$ 0 0
$$93$$ −1.76604 + 0.642788i −1.76604 + 0.642788i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.592396 + 1.62760i 0.592396 + 1.62760i 0.766044 + 0.642788i $$0.222222\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 −0.173648 0.984808i $$-0.555556\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$102$$ 0 0
$$103$$ −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i $$0.444444\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$108$$ 0 0
$$109$$ 1.11334 1.32683i 1.11334 1.32683i 0.173648 0.984808i $$-0.444444\pi$$
0.939693 0.342020i $$-0.111111\pi$$
$$110$$ 0 0
$$111$$ 0.673648 + 0.118782i 0.673648 + 0.118782i
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.26604 + 0.223238i 1.26604 + 0.223238i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0.173648 + 0.984808i 0.173648 + 0.984808i 0.939693 + 0.342020i $$0.111111\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$128$$ 0 0
$$129$$ −0.439693 0.524005i −0.439693 0.524005i
$$130$$ 0 0
$$131$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$132$$ 0 0
$$133$$ 1.70574 + 0.984808i 1.70574 + 0.984808i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.766044 0.642788i $$-0.222222\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$138$$ 0 0
$$139$$ 1.26604 0.223238i 1.26604 0.223238i 0.500000 0.866025i $$-0.333333\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.20574 1.85083i −2.20574 1.85083i
$$148$$ 0 0
$$149$$ 0 0 0.173648 0.984808i $$-0.444444\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −0.266044 0.223238i −0.266044 0.223238i 0.500000 0.866025i $$-0.333333\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1.11334 0.642788i −1.11334 0.642788i −0.173648 0.984808i $$-0.555556\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.766044 0.642788i $$-0.222222\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$168$$ 0 0
$$169$$ 0.613341 0.223238i 0.613341 0.223238i
$$170$$ 0 0
$$171$$ −0.173648 0.984808i −0.173648 0.984808i
$$172$$ 0 0
$$173$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$174$$ 0 0
$$175$$ 1.26604 + 1.50881i 1.26604 + 1.50881i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$180$$ 0 0
$$181$$ 0.592396 1.62760i 0.592396 1.62760i −0.173648 0.984808i $$-0.555556\pi$$
0.766044 0.642788i $$-0.222222\pi$$
$$182$$ 0 0
$$183$$ 0.766044 + 1.32683i 0.766044 + 1.32683i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.96962i 1.96962i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −0.673648 0.118782i −0.673648 0.118782i −0.173648 0.984808i $$-0.555556\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$198$$ 0 0
$$199$$ 0.233956 0.642788i 0.233956 0.642788i −0.766044 0.642788i $$-0.777778\pi$$
1.00000 $$0$$
$$200$$ 0 0
$$201$$ 0.173648 0.300767i 0.173648 0.300767i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.76604 + 0.642788i −1.76604 + 0.642788i −0.766044 + 0.642788i $$0.777778\pi$$
−1.00000 $$1.00000\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.20574 1.85083i −3.20574 1.85083i
$$218$$ 0 0
$$219$$ 1.43969 + 0.524005i 1.43969 + 0.524005i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0.266044 + 0.223238i 0.266044 + 0.223238i 0.766044 0.642788i $$-0.222222\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$224$$ 0 0
$$225$$ 0.173648 0.984808i 0.173648 0.984808i
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 −0.766044 0.642788i $$-0.777778\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −0.326352 0.118782i −0.326352 0.118782i
$$238$$ 0 0
$$239$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$240$$ 0 0
$$241$$ −1.93969 + 0.342020i −1.93969 + 0.342020i −0.939693 + 0.342020i $$0.888889\pi$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0.766044 0.642788i 0.766044 0.642788i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.826352 0.984808i −0.826352 0.984808i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$258$$ 0 0
$$259$$ 0.673648 + 1.16679i 0.673648 + 1.16679i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$270$$ 0 0
$$271$$ 1.11334 1.32683i 1.11334 1.32683i 0.173648 0.984808i $$-0.444444\pi$$
0.939693 0.342020i $$-0.111111\pi$$
$$272$$ 0 0
$$273$$ 1.26604 + 2.19285i 1.26604 + 2.19285i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
1.00000 $$0$$
$$278$$ 0 0
$$279$$ 0.326352 + 1.85083i 0.326352 + 1.85083i
$$280$$ 0 0
$$281$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$282$$ 0 0
$$283$$ −0.592396 1.62760i −0.592396 1.62760i −0.766044 0.642788i $$-0.777778\pi$$
0.173648 0.984808i $$-0.444444\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 0.766044 0.642788i 0.766044 0.642788i
$$290$$ 0 0
$$291$$ 1.70574 0.300767i 1.70574 0.300767i
$$292$$ 0 0
$$293$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0.233956 1.32683i 0.233956 1.32683i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i $$0.222222\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$308$$ 0 0
$$309$$ 1.17365 + 0.984808i 1.17365 + 0.984808i
$$310$$ 0 0
$$311$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$312$$ 0 0
$$313$$ −0.939693 0.342020i −0.939693 0.342020i −0.173648 0.984808i $$-0.555556\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −0.439693 1.20805i −0.439693 1.20805i
$$326$$ 0 0
$$327$$ −1.11334 1.32683i −1.11334 1.32683i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i $$-0.444444\pi$$
0.766044 0.642788i $$-0.222222\pi$$
$$332$$ 0 0
$$333$$ 0.233956 0.642788i 0.233956 0.642788i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0.826352 0.984808i 0.826352 0.984808i −0.173648 0.984808i $$-0.555556\pi$$
1.00000 $$0$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 3.70167i 3.70167i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$348$$ 0 0
$$349$$ 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i $$0.222222\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$350$$ 0 0
$$351$$ 0.439693 1.20805i 0.439693 1.20805i
$$352$$ 0 0
$$353$$ 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 0
$$359$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$360$$ 0 0
$$361$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$362$$ 0 0
$$363$$ 0.939693 0.342020i 0.939693 0.342020i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0.673648 0.118782i 0.673648 0.118782i 0.173648 0.984808i $$-0.444444\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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
$$379$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$380$$ 0 0
$$381$$ 1.00000 1.00000
$$382$$ 0 0
$$383$$ 0 0 0.173648 0.984808i $$-0.444444\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.592396 + 0.342020i −0.592396 + 0.342020i
$$388$$ 0 0
$$389$$ 0 0 −0.939693 0.342020i $$-0.888889\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −1.43969 + 0.524005i −1.43969 + 0.524005i −0.939693 0.342020i $$-0.888889\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$398$$ 0 0
$$399$$ 1.26604 1.50881i 1.26604 1.50881i
$$400$$ 0 0
$$401$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$402$$ 0 0
$$403$$ 1.55303 + 1.85083i 1.55303 + 1.85083i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 −0.939693 0.342020i $$-0.888889\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1.28558i 1.28558i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 1.70574 + 0.300767i 1.70574 + 0.300767i 0.939693 0.342020i $$-0.111111\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.03209 + 2.83564i −1.03209 + 2.83564i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.173648 0.984808i $$-0.555556\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$432$$ 0 0
$$433$$ −1.26604 1.50881i −1.26604 1.50881i −0.766044 0.642788i $$-0.777778\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −1.43969 + 0.524005i −1.43969 + 0.524005i −0.939693 0.342020i $$-0.888889\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$440$$ 0 0
$$441$$ −2.20574 + 1.85083i −2.20574 + 1.85083i
$$442$$ 0 0
$$443$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −0.173648 + 0.984808i −0.173648 + 0.984808i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 −0.766044 0.642788i $$-0.777778\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$462$$ 0 0
$$463$$ −1.11334 + 0.642788i −1.11334 + 0.642788i −0.939693 0.342020i $$-0.888889\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$468$$ 0 0
$$469$$ 0.673648 0.118782i 0.673648 0.118782i
$$470$$ 0 0
$$471$$ −0.266044 + 0.223238i −0.266044 + 0.223238i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −0.766044 + 0.642788i −0.766044 + 0.642788i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$480$$ 0 0
$$481$$ −0.152704 0.866025i −0.152704 0.866025i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ −0.826352 + 0.984808i −0.826352 + 0.984808i
$$490$$ 0 0
$$491$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.826352 0.984808i 0.826352 0.984808i −0.173648 0.984808i $$-0.555556\pi$$
1.00000 $$0$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.342020 0.939693i $$-0.388889\pi$$
−0.342020 + 0.939693i $$0.611111\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −0.113341 0.642788i −0.113341 0.642788i
$$508$$ 0 0
$$509$$ 0 0 −0.642788 0.766044i $$-0.722222\pi$$
0.642788 + 0.766044i $$0.277778\pi$$
$$510$$ 0 0
$$511$$ 1.03209 + 2.83564i 1.03209 + 2.83564i
$$512$$ 0 0
$$513$$ −1.00000 −1.00000
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$522$$ 0 0
$$523$$ −1.76604 0.642788i −1.76604 0.642788i −0.766044 0.642788i $$-0.777778\pi$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 1.70574 0.984808i 1.70574 0.984808i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.173648 + 0.984808i −0.173648 + 0.984808i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.326352 + 0.118782i 0.326352 + 0.118782i 0.500000 0.866025i $$-0.333333\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$542$$ 0 0
$$543$$ −1.50000 0.866025i −1.50000 0.866025i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0.266044 0.223238i 0.266044 0.223238i −0.500000 0.866025i $$-0.666667\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$548$$ 0 0
$$549$$ 1.43969 0.524005i 1.43969 0.524005i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −0.233956 0.642788i −0.233956 0.642788i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 −0.173648 0.984808i $$-0.555556\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$558$$ 0 0
$$559$$ −0.439693 + 0.761570i −0.439693 + 0.761570i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.93969 + 0.342020i 1.93969 + 0.342020i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0.684040i 0.684040i 0.939693 + 0.342020i $$0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
−1.00000 $$\pi$$
$$578$$ 0 0
$$579$$ −0.233956 + 0.642788i −0.233956 + 0.642788i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.342020 0.939693i $$-0.611111\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$588$$ 0 0
$$589$$ 0.939693 1.62760i 0.939693 1.62760i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 0.766044 0.642788i $$-0.222222\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.592396 0.342020i −0.592396 0.342020i
$$598$$ 0 0
$$599$$ 0 0 −0.939693 0.342020i $$-0.888889\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$600$$ 0 0
$$601$$ 1.11334 0.642788i 1.11334 0.642788i 0.173648 0.984808i $$-0.444444\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$602$$ 0 0
$$603$$ −0.266044 0.223238i −0.266044 0.223238i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0.766044 + 0.642788i 0.766044 + 0.642788i 0.939693 0.342020i $$-0.111111\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 −0.939693 0.342020i $$-0.888889\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$618$$ 0 0
$$619$$ −1.70574 0.984808i −1.70574 0.984808i −0.939693 0.342020i $$-0.888889\pi$$
−0.766044 0.642788i $$-0.777778\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −0.939693 + 0.342020i −0.939693 + 0.342020i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −0.439693 0.524005i −0.439693 0.524005i 0.500000 0.866025i $$-0.333333\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$632$$ 0 0
$$633$$ 0.326352 + 1.85083i 0.326352 + 1.85083i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.26604 + 3.47843i −1.26604 + 3.47843i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 0.642788 0.766044i $$-0.277778\pi$$
−0.642788 + 0.766044i $$0.722222\pi$$
$$642$$ 0 0
$$643$$ −1.93969 0.342020i −1.93969 0.342020i −0.939693 0.342020i $$-0.888889\pi$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2.37939 + 2.83564i −2.37939 + 2.83564i
$$652$$ 0 0
$$653$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0.766044 1.32683i 0.766044 1.32683i
$$658$$ 0 0
$$659$$ 0 0 −0.173648 0.984808i $$-0.555556\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$660$$ 0 0
$$661$$ 0 0 0.766044 0.642788i $$-0.222222\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0.266044 0.223238i 0.266044 0.223238i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1.70574 + 0.984808i 1.70574 + 0.984808i 0.939693 + 0.342020i $$0.111111\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$674$$ 0 0
$$675$$ −0.939693 0.342020i −0.939693 0.342020i
$$676$$ 0 0
$$677$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$678$$ 0 0
$$679$$ 2.61334 + 2.19285i 2.61334 + 2.19285i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −0.326352 + 1.85083i −0.326352 + 1.85083i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 0.939693 0.342020i $$-0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$702$$ 0 0
$$703$$ −0.592396 + 0.342020i −0.592396 + 0.342020i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −0.266044 1.50881i −0.266044 1.50881i −0.766044 0.642788i $$-0.777778\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$710$$ 0 0
$$711$$ −0.173648 + 0.300767i −0.173648 + 0.300767i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 −0.984808 0.173648i $$-0.944444\pi$$
0.984808 + 0.173648i $$0.0555556\pi$$
$$720$$ 0 0
$$721$$ 3.01763i 3.01763i
$$722$$ 0 0
$$723$$ 1.96962i 1.96962i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −0.826352 + 0.984808i −0.826352 + 0.984808i 0.173648 + 0.984808i $$0.444444\pi$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ −0.500000 0.866025i −0.500000 0.866025i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i $$0.333333\pi$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0.673648 + 1.85083i 0.673648 + 1.85083i 0.500000 + 0.866025i $$0.333333\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$740$$ 0 0
$$741$$ −1.11334 + 0.642788i −1.11334 + 0.642788i
$$742$$ 0 0
$$743$$ 0 0 0.939693 0.342020i $$-0.111111\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1.43969 0.524005i −1.43969 0.524005i −0.500000 0.866025i $$-0.666667\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −0.326352 + 1.85083i −0.326352 + 1.85083i 0.173648 + 0.984808i $$0.444444\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0.592396 3.35965i 0.592396 3.35965i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1.43969 + 0.524005i 1.43969 + 0.524005i 0.939693 0.342020i $$-0.111111\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 0.984808 0.173648i $$-0.0555556\pi$$
−0.984808 + 0.173648i $$0.944444\pi$$
$$774$$ 0 0
$$775$$ 1.43969 1.20805i 1.43969 1.20805i
$$776$$ 0 0
$$777$$ 1.26604 0.460802i 1.26604 0.460802i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0.766044 1.32683i 0.766044 1.32683i −0.173648 0.984808i $$-0.555556\pi$$
0.939693 0.342020i $$-0.111111\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.26604 1.50881i 1.26604 1.50881i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i