# Properties

 Label 3360.1.ft.a.1937.2 Level $3360$ Weight $1$ Character 3360.1937 Analytic conductor $1.677$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3360.ft (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## Embedding invariants

 Embedding label 1937.2 Root $$0.965926 - 0.258819i$$ of defining polynomial Character $$\chi$$ $$=$$ 3360.1937 Dual form 3360.1.ft.a.2033.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(0.258819 - 0.965926i) q^{3} +(0.965926 + 0.258819i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})$$ $$q+(0.258819 - 0.965926i) q^{3} +(0.965926 + 0.258819i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.866025 - 0.500000i) q^{9} +(-0.258819 - 0.448288i) q^{11} +(0.500000 - 0.866025i) q^{15} +(-0.965926 + 0.258819i) q^{21} +(0.866025 + 0.500000i) q^{25} +(-0.707107 + 0.707107i) q^{27} -1.93185i q^{29} +(0.866025 - 0.500000i) q^{31} +(-0.500000 + 0.133975i) q^{33} +(-0.258819 - 0.965926i) q^{35} +(-0.707107 - 0.707107i) q^{45} +(-0.500000 + 0.866025i) q^{49} +(-1.67303 - 0.448288i) q^{53} +(-0.133975 - 0.500000i) q^{55} +(-0.965926 - 1.67303i) q^{59} +1.00000i q^{63} +(-0.366025 + 1.36603i) q^{73} +(0.707107 - 0.707107i) q^{75} +(-0.258819 + 0.448288i) q^{77} +(1.50000 + 0.866025i) q^{79} +(0.500000 + 0.866025i) q^{81} +(1.22474 + 1.22474i) q^{83} +(-1.86603 - 0.500000i) q^{87} +(-0.258819 - 0.965926i) q^{93} +(1.36603 - 1.36603i) q^{97} +0.517638i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{7}+O(q^{10})$$ 8 * q - 4 * q^7 $$8 q - 4 q^{7} + 4 q^{15} - 4 q^{33} - 4 q^{49} - 8 q^{55} + 4 q^{73} + 12 q^{79} + 4 q^{81} - 8 q^{87} + 4 q^{97}+O(q^{100})$$ 8 * q - 4 * q^7 + 4 * q^15 - 4 * q^33 - 4 * q^49 - 8 * q^55 + 4 * q^73 + 12 * q^79 + 4 * q^81 - 8 * q^87 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$$.

 $$n$$ $$421$$ $$1121$$ $$1471$$ $$1921$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{4}\right)$$

## 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 0
$$3$$ 0.258819 0.965926i 0.258819 0.965926i
$$4$$ 0 0
$$5$$ 0.965926 + 0.258819i 0.965926 + 0.258819i
$$6$$ 0 0
$$7$$ −0.500000 0.866025i −0.500000 0.866025i
$$8$$ 0 0
$$9$$ −0.866025 0.500000i −0.866025 0.500000i
$$10$$ 0 0
$$11$$ −0.258819 0.448288i −0.258819 0.448288i 0.707107 0.707107i $$-0.250000\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$12$$ 0 0
$$13$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$14$$ 0 0
$$15$$ 0.500000 0.866025i 0.500000 0.866025i
$$16$$ 0 0
$$17$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$18$$ 0 0
$$19$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$20$$ 0 0
$$21$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$22$$ 0 0
$$23$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$24$$ 0 0
$$25$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$26$$ 0 0
$$27$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$28$$ 0 0
$$29$$ 1.93185i 1.93185i −0.258819 0.965926i $$-0.583333\pi$$
0.258819 0.965926i $$-0.416667\pi$$
$$30$$ 0 0
$$31$$ 0.866025 0.500000i 0.866025 0.500000i 1.00000i $$-0.5\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$32$$ 0 0
$$33$$ −0.500000 + 0.133975i −0.500000 + 0.133975i
$$34$$ 0 0
$$35$$ −0.258819 0.965926i −0.258819 0.965926i
$$36$$ 0 0
$$37$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$44$$ 0 0
$$45$$ −0.707107 0.707107i −0.707107 0.707107i
$$46$$ 0 0
$$47$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$48$$ 0 0
$$49$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.67303 0.448288i −1.67303 0.448288i −0.707107 0.707107i $$-0.750000\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$54$$ 0 0
$$55$$ −0.133975 0.500000i −0.133975 0.500000i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −0.965926 1.67303i −0.965926 1.67303i −0.707107 0.707107i $$-0.750000\pi$$
−0.258819 0.965926i $$-0.583333\pi$$
$$60$$ 0 0
$$61$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 1.00000i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i $$0.333333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$74$$ 0 0
$$75$$ 0.707107 0.707107i 0.707107 0.707107i
$$76$$ 0 0
$$77$$ −0.258819 + 0.448288i −0.258819 + 0.448288i
$$78$$ 0 0
$$79$$ 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 $$0$$
0.500000 + 0.866025i $$0.333333\pi$$
$$80$$ 0 0
$$81$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$82$$ 0 0
$$83$$ 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i $$0.0833333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −1.86603 0.500000i −1.86603 0.500000i
$$88$$ 0 0
$$89$$ 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
$$93$$ −0.258819 0.965926i −0.258819 0.965926i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.36603 1.36603i 1.36603 1.36603i 0.500000 0.866025i $$-0.333333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$98$$ 0 0
$$99$$ 0.517638i 0.517638i
$$100$$ 0 0
$$101$$ 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i $$-0.416667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$102$$ 0 0
$$103$$ −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −1.00000
$$106$$ 0 0
$$107$$ −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i $$-0.750000\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$108$$ 0 0
$$109$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.366025 0.633975i 0.366025 0.633975i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$126$$ 0 0
$$127$$ 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i $$0.166667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.448288 0.258819i −0.448288 0.258819i 0.258819 0.965926i $$-0.416667\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$136$$ 0 0
$$137$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0.500000 1.86603i 0.500000 1.86603i
$$146$$ 0 0
$$147$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$148$$ 0 0
$$149$$ 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i $$-0.0833333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$150$$ 0 0
$$151$$ 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 $$0$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0.965926 0.258819i 0.965926 0.258819i
$$156$$ 0 0
$$157$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$158$$ 0 0
$$159$$ −0.866025 + 1.50000i −0.866025 + 1.50000i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$164$$ 0 0
$$165$$ −0.517638 −0.517638
$$166$$ 0 0
$$167$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$168$$ 0 0
$$169$$ 1.00000i 1.00000i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$174$$ 0 0
$$175$$ 1.00000i 1.00000i
$$176$$ 0 0
$$177$$ −1.86603 + 0.500000i −1.86603 + 0.500000i
$$178$$ 0 0
$$179$$ −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i $$-0.916667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$180$$ 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
$$189$$ 0.965926 + 0.258819i 0.965926 + 0.258819i
$$190$$ 0 0
$$191$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$192$$ 0 0
$$193$$ −1.86603 0.500000i −1.86603 0.500000i −0.866025 0.500000i $$-0.833333\pi$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$198$$ 0 0
$$199$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1.67303 + 0.965926i −1.67303 + 0.965926i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.866025 0.500000i −0.866025 0.500000i
$$218$$ 0 0
$$219$$ 1.22474 + 0.707107i 1.22474 + 0.707107i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i $$-0.333333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$224$$ 0 0
$$225$$ −0.500000 0.866025i −0.500000 0.866025i
$$226$$ 0 0
$$227$$ 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i $$-0.250000\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$228$$ 0 0
$$229$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$230$$ 0 0
$$231$$ 0.366025 + 0.366025i 0.366025 + 0.366025i
$$232$$ 0 0
$$233$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1.22474 1.22474i 1.22474 1.22474i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 0.866025 0.500000i 0.866025 0.500000i 1.00000i $$-0.5\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$242$$ 0 0
$$243$$ 0.965926 0.258819i 0.965926 0.258819i
$$244$$ 0 0
$$245$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 1.50000 0.866025i 1.50000 0.866025i
$$250$$ 0 0
$$251$$ 0.517638i 0.517638i −0.965926 0.258819i $$-0.916667\pi$$
0.965926 0.258819i $$-0.0833333\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −0.965926 + 1.67303i −0.965926 + 1.67303i
$$262$$ 0 0
$$263$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$264$$ 0 0
$$265$$ −1.50000 0.866025i −1.50000 0.866025i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −0.258819 0.448288i −0.258819 0.448288i 0.707107 0.707107i $$-0.250000\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$270$$ 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
$$275$$ 0.517638i 0.517638i
$$276$$ 0 0
$$277$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$278$$ 0 0
$$279$$ −1.00000 −1.00000
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$290$$ 0 0
$$291$$ −0.965926 1.67303i −0.965926 1.67303i
$$292$$ 0 0
$$293$$ 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i $$-0.0833333\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$294$$ 0 0
$$295$$ −0.500000 1.86603i −0.500000 1.86603i
$$296$$ 0 0
$$297$$ 0.500000 + 0.133975i 0.500000 + 0.133975i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −0.366025 1.36603i −0.366025 1.36603i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$308$$ 0 0
$$309$$ 1.41421i 1.41421i
$$310$$ 0 0
$$311$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$312$$ 0 0
$$313$$ 1.86603 0.500000i 1.86603 0.500000i 0.866025 0.500000i $$-0.166667\pi$$
1.00000 $$0$$
$$314$$ 0 0
$$315$$ −0.258819 + 0.965926i −0.258819 + 0.965926i
$$316$$ 0 0
$$317$$ −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i $$-0.750000\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$318$$ 0 0
$$319$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$320$$ 0 0
$$321$$ 1.00000i 1.00000i
$$322$$ 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 $$0.666667\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i $$-0.166667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −0.448288 0.258819i −0.448288 0.258819i
$$342$$ 0 0
$$343$$ 1.00000 1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$354$$ 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
$$363$$ −0.517638 0.517638i −0.517638 0.517638i
$$364$$ 0 0
$$365$$ −0.707107 + 1.22474i −0.707107 + 1.22474i
$$366$$ 0 0
$$367$$ 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 0.866025i $$-0.333333\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.448288 + 1.67303i 0.448288 + 1.67303i
$$372$$ 0 0
$$373$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$374$$ 0 0
$$375$$ 0.866025 0.500000i 0.866025 0.500000i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 1.67303 0.965926i 1.67303 0.965926i
$$382$$ 0 0
$$383$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$384$$ 0 0
$$385$$ −0.366025 + 0.366025i −0.366025 + 0.366025i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i $$-0.916667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −0.366025 + 0.366025i −0.366025 + 0.366025i
$$394$$ 0 0
$$395$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$396$$ 0 0
$$397$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$406$$ 0 0
$$407$$ 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
$$412$$ 0 0
$$413$$ −0.965926 + 1.67303i −0.965926 + 1.67303i
$$414$$ 0 0
$$415$$ 0.866025 + 1.50000i 0.866025 + 1.50000i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 1.41421 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$432$$ 0 0
$$433$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ −1.67303 0.965926i −1.67303 0.965926i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0.866025 1.50000i 0.866025 1.50000i 1.00000i $$-0.5\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$440$$ 0 0
$$441$$ 0.866025 0.500000i 0.866025 0.500000i
$$442$$ 0 0
$$443$$ 0.448288 + 1.67303i 0.448288 + 1.67303i 0.707107 + 0.707107i $$0.250000\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1.00000 1.00000i 1.00000 1.00000i
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0.965926 0.258819i 0.965926 0.258819i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.500000 + 0.133975i −0.500000 + 0.133975i −0.500000 0.866025i $$-0.666667\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1.41421i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$462$$ 0 0
$$463$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 1.00000i 1.00000i
$$466$$ 0 0
$$467$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$468$$ 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
$$477$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$478$$ 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
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.67303 0.965926i 1.67303 0.965926i
$$486$$ 0 0
$$487$$ −0.133975 + 0.500000i −0.133975 + 0.500000i 0.866025 + 0.500000i $$0.166667\pi$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.93185 1.93185 0.965926 0.258819i $$-0.0833333\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −0.133975 + 0.500000i −0.133975 + 0.500000i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$504$$ 0 0
$$505$$ 1.36603 0.366025i 1.36603 0.366025i
$$506$$ 0 0
$$507$$ 0.965926 + 0.258819i 0.965926 + 0.258819i
$$508$$ 0 0
$$509$$ −0.258819 + 0.448288i −0.258819 + 0.448288i −0.965926 0.258819i $$-0.916667\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$510$$ 0 0
$$511$$ 1.36603 0.366025i 1.36603 0.366025i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1.41421 −1.41421
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$524$$ 0 0
$$525$$ −0.965926 0.258819i −0.965926 0.258819i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$530$$ 0 0
$$531$$ 1.93185i 1.93185i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −1.00000 −1.00000
$$536$$ 0 0
$$537$$ 0.366025 + 1.36603i 0.366025 + 1.36603i
$$538$$ 0 0
$$539$$ 0.517638 0.517638
$$540$$ 0 0
$$541$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 1.73205i 1.73205i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0.448288 1.67303i 0.448288 1.67303i −0.258819 0.965926i $$-0.583333\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i $$-0.750000\pi$$
0.965926 0.258819i $$-0.0833333\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0.500000 0.866025i 0.500000 0.866025i
$$568$$ 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
$$577$$ −1.86603 0.500000i −1.86603 0.500000i −0.866025 0.500000i $$-0.833333\pi$$
−1.00000 $$\pi$$
$$578$$ 0 0
$$579$$ −0.965926 + 1.67303i −0.965926 + 1.67303i
$$580$$ 0 0
$$581$$ 0.448288 1.67303i 0.448288 1.67303i
$$582$$ 0 0
$$583$$ 0.232051 + 0.866025i 0.232051 + 0.866025i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i $$-0.416667\pi$$
0.965926 0.258819i $$-0.0833333\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$594$$ 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 0
$$605$$ 0.517638 0.517638i 0.517638 0.517638i
$$606$$ 0 0
$$607$$ −0.500000 1.86603i −0.500000 1.86603i −0.500000 0.866025i $$-0.666667\pi$$
1.00000i $$-0.5\pi$$
$$608$$ 0 0
$$609$$ 0.500000 + 1.86603i 0.500000 + 1.86603i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$618$$ 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
$$625$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0.965926 + 1.67303i 0.965926 + 1.67303i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$642$$ 0 0
$$643$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$648$$ 0 0
$$649$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$650$$ 0 0
$$651$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$652$$ 0 0
$$653$$ 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i $$0.0833333\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$654$$ 0 0
$$655$$ −0.366025 0.366025i −0.366025 0.366025i
$$656$$ 0 0
$$657$$ 1.00000 1.00000i 1.00000 1.00000i
$$658$$ 0 0
$$659$$ 1.41421i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$660$$ 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
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −0.448288 + 0.258819i −0.448288 + 0.258819i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i $$0.666667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$674$$ 0 0
$$675$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$676$$ 0 0
$$677$$ −0.448288 1.67303i −0.448288 1.67303i −0.707107 0.707107i $$-0.750000\pi$$
0.258819 0.965926i $$-0.416667\pi$$
$$678$$ 0 0
$$679$$ −1.86603 0.500000i −1.86603 0.500000i
$$680$$ 0 0
$$681$$ 0.500000 0.866025i 0.500000 0.866025i
$$682$$ 0 0
$$683$$ −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i $$-0.583333\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 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
$$693$$ 0.448288 0.258819i 0.448288 0.258819i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −0.517638 −0.517638 −0.258819 0.965926i $$-0.583333\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.22474 0.707107i −1.22474 0.707107i
$$708$$ 0 0
$$709$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$710$$ 0 0
$$711$$ −0.866025 1.50000i −0.866025 1.50000i
$$712$$ 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
$$721$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$722$$ 0 0
$$723$$ −0.258819 0.965926i −0.258819 0.965926i
$$724$$ 0 0
$$725$$ 0.965926 1.67303i 0.965926 1.67303i
$$726$$ 0 0
$$727$$ −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$728$$ 0 0
$$729$$ 1.00000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$734$$ 0 0
$$735$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$744$$ 0 0
$$745$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$746$$ 0 0
$$747$$ −0.448288 1.67303i −0.448288 1.67303i
$$748$$ 0 0
$$749$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$750$$ 0 0
$$751$$ 0.866025 1.50000i 0.866025 1.50000i 1.00000i $$-0.5\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$752$$ 0 0
$$753$$ −0.500000 0.133975i −0.500000 0.133975i
$$754$$ 0 0
$$755$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$756$$ 0 0
$$757$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$774$$ 0 0
$$775$$ 1.00000 1.00000
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 1.36603 + 1.36603i 1.36603 + 1.36603i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −1.22474 + 1.22474i −1.22474 + 1.22474i
$$796$$ 0 0
$$797$$ −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i $$0.583333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0.707107 0.189469i 0.707107 0.189469i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0