# Properties

 Label 600.1.q.b.107.3 Level 600 Weight 1 Character 600.107 Analytic conductor 0.299 Analytic rank 0 Dimension 8 Projective image $$D_{6}$$ CM discriminant -8 Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 600.q (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.299439007580$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.5400000.2

## Embedding invariants

 Embedding label 107.3 Root $$0.965926 - 0.258819i$$ of defining polynomial Character $$\chi$$ $$=$$ 600.107 Dual form 600.1.q.b.443.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(0.707107 - 0.707107i) q^{2} +(-0.965926 + 0.258819i) q^{3} -1.00000i q^{4} +(-0.500000 + 0.866025i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})$$ $$q+(0.707107 - 0.707107i) q^{2} +(-0.965926 + 0.258819i) q^{3} -1.00000i q^{4} +(-0.500000 + 0.866025i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(0.866025 - 0.500000i) q^{9} -1.73205i q^{11} +(0.258819 + 0.965926i) q^{12} -1.00000 q^{16} +(0.707107 - 0.707107i) q^{17} +(0.258819 - 0.965926i) q^{18} +1.00000i q^{19} +(-1.22474 - 1.22474i) q^{22} +(0.866025 + 0.500000i) q^{24} +(-0.707107 + 0.707107i) q^{27} +(-0.707107 + 0.707107i) q^{32} +(0.448288 + 1.67303i) q^{33} -1.00000i q^{34} +(-0.500000 - 0.866025i) q^{36} +(0.707107 + 0.707107i) q^{38} +1.73205i q^{41} -1.73205 q^{44} +(0.965926 - 0.258819i) q^{48} +1.00000i q^{49} +(-0.500000 + 0.866025i) q^{51} +1.00000i q^{54} +(-0.258819 - 0.965926i) q^{57} +1.00000i q^{64} +(1.50000 + 0.866025i) q^{66} +(1.22474 + 1.22474i) q^{67} +(-0.707107 - 0.707107i) q^{68} +(-0.965926 - 0.258819i) q^{72} +(1.22474 - 1.22474i) q^{73} +1.00000 q^{76} +(0.500000 - 0.866025i) q^{81} +(1.22474 + 1.22474i) q^{82} +(0.707107 + 0.707107i) q^{83} +(-1.22474 + 1.22474i) q^{88} -1.73205 q^{89} +(0.500000 - 0.866025i) q^{96} +(0.707107 + 0.707107i) q^{98} +(-0.866025 - 1.50000i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{6} + O(q^{10})$$ $$8q - 4q^{6} - 8q^{16} - 4q^{36} - 4q^{51} + 12q^{66} + 8q^{76} + 4q^{81} + 4q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{4}\right)$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.707107 0.707107i 0.707107 0.707107i
$$3$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$4$$ 1.00000i 1.00000i
$$5$$ 0 0
$$6$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ −0.707107 0.707107i −0.707107 0.707107i
$$9$$ 0.866025 0.500000i 0.866025 0.500000i
$$10$$ 0 0
$$11$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$12$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$13$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −1.00000
$$17$$ 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i $$-0.583333\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$18$$ 0.258819 0.965926i 0.258819 0.965926i
$$19$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.22474 1.22474i −1.22474 1.22474i
$$23$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$24$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$33$$ 0.448288 + 1.67303i 0.448288 + 1.67303i
$$34$$ 1.00000i 1.00000i
$$35$$ 0 0
$$36$$ −0.500000 0.866025i −0.500000 0.866025i
$$37$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$38$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$42$$ 0 0
$$43$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$44$$ −1.73205 −1.73205
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$48$$ 0.965926 0.258819i 0.965926 0.258819i
$$49$$ 1.00000i 1.00000i
$$50$$ 0 0
$$51$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$52$$ 0 0
$$53$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$54$$ 1.00000i 1.00000i
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.258819 0.965926i −0.258819 0.965926i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000i 1.00000i
$$65$$ 0 0
$$66$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$67$$ 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i $$0.0833333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$68$$ −0.707107 0.707107i −0.707107 0.707107i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −0.965926 0.258819i −0.965926 0.258819i
$$73$$ 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i $$-0.416667\pi$$
0.965926 0.258819i $$-0.0833333\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.00000 1.00000
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0.500000 0.866025i 0.500000 0.866025i
$$82$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$83$$ 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i $$-0.0833333\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −1.22474 + 1.22474i −1.22474 + 1.22474i
$$89$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0.500000 0.866025i 0.500000 0.866025i
$$97$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$98$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$99$$ −0.866025 1.50000i −0.866025 1.50000i
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$103$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i $$-0.916667\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$108$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i $$-0.416667\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$114$$ −0.866025 0.500000i −0.866025 0.500000i
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −2.00000
$$122$$ 0 0
$$123$$ −0.448288 1.67303i −0.448288 1.67303i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$128$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 1.67303 0.448288i 1.67303 0.448288i
$$133$$ 0 0
$$134$$ 1.73205 1.73205
$$135$$ 0 0
$$136$$ −1.00000 −1.00000
$$137$$ −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i $$-0.916667\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$138$$ 0 0
$$139$$ 1.00000i 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$145$$ 0 0
$$146$$ 1.73205i 1.73205i
$$147$$ −0.258819 0.965926i −0.258819 0.965926i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0.707107 0.707107i 0.707107 0.707107i
$$153$$ 0.258819 0.965926i 0.258819 0.965926i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −0.258819 0.965926i −0.258819 0.965926i
$$163$$ 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i $$-0.416667\pi$$
0.965926 0.258819i $$-0.0833333\pi$$
$$164$$ 1.73205 1.73205
$$165$$ 0 0
$$166$$ 1.00000 1.00000
$$167$$ 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
$$171$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$172$$ 0 0
$$173$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.73205i 1.73205i
$$177$$ 0 0
$$178$$ −1.22474 + 1.22474i −1.22474 + 1.22474i
$$179$$ 1.73205 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$180$$ 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
$$187$$ −1.22474 1.22474i −1.22474 1.22474i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −0.258819 0.965926i −0.258819 0.965926i
$$193$$ −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i $$0.583333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1.00000 1.00000
$$197$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$198$$ −1.67303 0.448288i −1.67303 0.448288i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −1.50000 0.866025i −1.50000 0.866025i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.73205 1.73205
$$210$$ 0 0
$$211$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 1.00000i 1.00000i
$$215$$ 0 0
$$216$$ 1.00000 1.00000
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −0.866025 + 1.50000i −0.866025 + 1.50000i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1.00000 −1.00000
$$227$$ 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i $$-0.250000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$228$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 0.707107i $$-0.750000\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$242$$ −1.41421 + 1.41421i −1.41421 + 1.41421i
$$243$$ −0.258819 + 0.965926i −0.258819 + 0.965926i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ −1.50000 0.866025i −1.50000 0.866025i
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −0.866025 0.500000i −0.866025 0.500000i
$$250$$ 0 0
$$251$$ 1.73205i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 1.00000
$$257$$ −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i $$0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$264$$ 0.866025 1.50000i 0.866025 1.50000i
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 1.67303 0.448288i 1.67303 0.448288i
$$268$$ 1.22474 1.22474i 1.22474 1.22474i
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$273$$ 0 0
$$274$$ 1.00000i 1.00000i
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$278$$ −0.707107 0.707107i −0.707107 0.707107i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i $$0.583333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −0.258819 + 0.965926i −0.258819 + 0.965926i
$$289$$ 0 0
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1.22474 1.22474i −1.22474 1.22474i
$$293$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$294$$ −0.866025 0.500000i −0.866025 0.500000i
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 1.00000i 1.00000i
$$305$$ 0 0
$$306$$ −0.500000 0.866025i −0.500000 0.866025i
$$307$$ −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i $$-0.916667\pi$$
−0.258819 0.965926i $$-0.583333\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0.500000 0.866025i 0.500000 0.866025i
$$322$$ 0 0
$$323$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$324$$ −0.866025 0.500000i −0.866025 0.500000i
$$325$$ 0 0
$$326$$ 1.73205i 1.73205i
$$327$$ 0 0
$$328$$ 1.22474 1.22474i 1.22474 1.22474i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$332$$ 0.707107 0.707107i 0.707107 0.707107i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i $$0.0833333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$338$$ −0.707107 0.707107i −0.707107 0.707107i
$$339$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0.965926 + 0.258819i 0.965926 + 0.258819i
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i $$-0.583333\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$353$$ 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1.73205i 1.73205i
$$357$$ 0 0
$$358$$ 1.22474 1.22474i 1.22474 1.22474i
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 0 0
$$363$$ 1.93185 0.517638i 1.93185 0.517638i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$368$$ 0 0
$$369$$ 0.866025 + 1.50000i 0.866025 + 1.50000i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$374$$ −1.73205 −1.73205
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1.00000i 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$384$$ −0.866025 0.500000i −0.866025 0.500000i
$$385$$ 0 0
$$386$$ 1.73205i 1.73205i
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0.707107 0.707107i 0.707107 0.707107i
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$397$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$402$$ −1.67303 + 0.448288i −1.67303 + 0.448288i
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0.965926 0.258819i 0.965926 0.258819i
$$409$$ 1.00000i 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$410$$ 0 0
$$411$$ 0.500000 0.866025i 0.500000 0.866025i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$418$$ 1.22474 1.22474i 1.22474 1.22474i
$$419$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0.707107 0.707107i 0.707107 0.707107i
$$433$$ −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i $$0.583333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0.448288 + 1.67303i 0.448288 + 1.67303i
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$442$$ 0 0
$$443$$ 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i $$-0.0833333\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$450$$ 0 0
$$451$$ 3.00000 3.00000
$$452$$ −0.707107 + 0.707107i −0.707107 + 0.707107i
$$453$$ 0 0
$$454$$ 2.00000i 2.00000i
$$455$$ 0 0
$$456$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$457$$ −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i $$-0.916667\pi$$
−0.258819 0.965926i $$-0.583333\pi$$
$$458$$ 0 0
$$459$$ 1.00000i 1.00000i
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −2.00000 −2.00000
$$467$$ −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i $$0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0.707107 0.707107i 0.707107 0.707107i
$$483$$ 0 0
$$484$$ 2.00000i 2.00000i
$$485$$ 0 0
$$486$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$487$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$488$$ 0 0
$$489$$ −0.866025 + 1.50000i −0.866025 + 1.50000i
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ −1.67303 + 0.448288i −1.67303 + 0.448288i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$499$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$503$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0.258819 + 0.965926i 0.258819 + 0.965926i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0.707107 0.707107i 0.707107 0.707107i
$$513$$ −0.707107 0.707107i −0.707107 0.707107i
$$514$$ 2.00000i 2.00000i
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.73205i 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$522$$ 0 0
$$523$$ −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i $$0.583333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ −0.448288 1.67303i −0.448288 1.67303i
$$529$$ 1.00000i 1.00000i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0.866025 1.50000i 0.866025 1.50000i
$$535$$ 0 0
$$536$$ 1.73205i 1.73205i
$$537$$ −1.67303 + 0.448288i −1.67303 + 0.448288i
$$538$$ 0 0
$$539$$ 1.73205 1.73205
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 1.00000i 1.00000i
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i $$0.0833333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$548$$ 0.707107 + 0.707107i 0.707107 + 0.707107i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −1.00000 −1.00000
$$557$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 1.50000 + 0.866025i 1.50000 + 0.866025i
$$562$$ 0 0
$$563$$ −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 0.707107i $$-0.750000\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.73205i 1.73205i
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.73205 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$570$$ 0 0
$$571$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0.500000 + 0.866025i 0.500000 + 0.866025i
$$577$$ −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i $$-0.916667\pi$$
−0.258819 0.965926i $$-0.583333\pi$$
$$578$$ 0 0
$$579$$ 0.866025 1.50000i 0.866025 1.50000i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −1.73205 −1.73205
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i $$-0.583333\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$588$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i $$-0.0833333\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$594$$ 1.73205 1.73205
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$602$$ 0 0
$$603$$ 1.67303 + 0.448288i 1.67303 + 0.448288i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$608$$ −0.707107 0.707107i −0.707107 0.707107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −0.965926 0.258819i −0.965926 0.258819i
$$613$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$614$$ −1.73205 −1.73205
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i $$-0.250000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$618$$ 0 0
$$619$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −1.67303 + 0.448288i −1.67303 + 0.448288i
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0.965926 0.258819i 0.965926 0.258819i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ −0.258819 0.965926i −0.258819 0.965926i
$$643$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1.00000 1.00000
$$647$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$648$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1.22474 1.22474i −1.22474 1.22474i
$$653$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 1.73205i 1.73205i
$$657$$ 0.448288 1.67303i 0.448288 1.67303i
$$658$$ 0 0
$$659$$ −1.73205 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0.707107 0.707107i 0.707107 0.707107i
$$663$$ 0 0
$$664$$ 1.00000i 1.00000i
$$665$$ 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 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$674$$ 1.73205 1.73205
$$675$$ 0 0
$$676$$ −1.00000 −1.00000
$$677$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$678$$ 0.965926 0.258819i 0.965926 0.258819i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.00000 + 1.73205i −1.00000 + 1.73205i
$$682$$ 0 0
$$683$$ −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i $$-0.416667\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$684$$ 0.866025 0.500000i 0.866025 0.500000i
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 1.00000i 1.00000i
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$698$$ 0 0
$$699$$ 1.73205 + 1.00000i 1.73205 + 1.00000i
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 1.73205 1.73205
$$705$$ 0 0
$$706$$ 2.00000 2.00000
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.73205i 1.73205i
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 1.00000 1.73205i 1.00000 1.73205i
$$727$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$728$$ 0 0
$$729$$ 1.00000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.12132 2.12132i 2.12132 2.12132i
$$738$$ 1.67303 + 0.448288i 1.67303 + 0.448288i
$$739$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0.965926 + 0.258819i 0.965926 + 0.258819i
$$748$$ −1.22474 + 1.22474i −1.22474 + 1.22474i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ −0.448288 1.67303i −0.448288 1.67303i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$758$$ −0.707107 0.707107i −0.707107 0.707107i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.73205i 1.73205i 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
$$768$$ −0.965926 + 0.258819i −0.965926 + 0.258819i
$$769$$ 1.00000i 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$770$$ 0 0
$$771$$ 1.00000 1.73205i 1.00000 1.73205i
$$772$$ 1.22474 + 1.22474i 1.22474 + 1.22474i
$$773$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.73205 −1.73205
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000i 1.00000i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −0.448288 + 1.67303i −0.448288 + 1.67303i
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −1.50000 + 0.866025i −1.50000 + 0.866025i
$$802$$ −1.22474 1.22474i −1.22474 1.22474i
$$803$$ −2.12132 2.12132i −2.12132 2.12132i
$$804$$ −0.866025 + 1.50000i −0.866025 + 1.50000i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0