# Properties

 Label 3744.1.gs.c.2017.1 Level $3744$ Weight $1$ Character 3744.2017 Analytic conductor $1.868$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -4 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3744.gs (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 416) Projective image: $$D_{12}$$ Projective field: Galois closure of 12.0.469804094334435328.7

## Embedding invariants

 Embedding label 2017.1 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3744.2017 Dual form 3744.1.gs.c.865.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.366025 + 0.366025i) q^{5} +O(q^{10})$$ $$q+(-0.366025 + 0.366025i) q^{5} +(-0.866025 + 0.500000i) q^{13} +(-0.866025 - 0.500000i) q^{17} +0.732051i q^{25} +(0.866025 + 1.50000i) q^{29} +(-0.500000 + 1.86603i) q^{37} +(1.86603 + 0.500000i) q^{41} +(-0.866025 + 0.500000i) q^{49} -1.00000 q^{53} +(-0.500000 + 0.866025i) q^{61} +(0.133975 - 0.500000i) q^{65} +(-1.36603 - 1.36603i) q^{73} +(0.500000 - 0.133975i) q^{85} +(0.366025 - 1.36603i) q^{89} +(0.366025 + 1.36603i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + O(q^{10})$$ $$4 q + 2 q^{5} - 2 q^{37} + 4 q^{41} - 4 q^{53} - 2 q^{61} + 4 q^{65} - 2 q^{73} + 2 q^{85} - 2 q^{89} - 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$1$$

## 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 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$6$$ 0 0
$$7$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$12$$ 0 0
$$13$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.866025 0.500000i −0.866025 0.500000i 1.00000i $$-0.5\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$18$$ 0 0
$$19$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$24$$ 0 0
$$25$$ 0.732051i 0.732051i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i $$0.166667\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.500000 + 1.86603i −0.500000 + 1.86603i 1.00000i $$0.5\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.86603 + 0.500000i 1.86603 + 0.500000i 1.00000 $$0$$
0.866025 + 0.500000i $$0.166667\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$48$$ 0 0
$$49$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$60$$ 0 0
$$61$$ −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i $$0.333333\pi$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.133975 0.500000i 0.133975 0.500000i
$$66$$ 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 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$72$$ 0 0
$$73$$ −1.36603 1.36603i −1.36603 1.36603i −0.866025 0.500000i $$-0.833333\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$84$$ 0 0
$$85$$ 0.500000 0.133975i 0.500000 0.133975i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i $$-0.666667\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i $$0.166667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i $$0.666667\pi$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$108$$ 0 0
$$109$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i $$0.5\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0.866025 + 0.500000i 0.866025 + 0.500000i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −0.633975 0.633975i −0.633975 0.633975i
$$126$$ 0 0
$$127$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.500000 0.133975i 0.500000 0.133975i 1.00000i $$-0.5\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$138$$ 0 0
$$139$$ 0 0 −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 0
$$145$$ −0.866025 0.232051i −0.866025 0.232051i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0.500000 + 1.86603i 0.500000 + 1.86603i 0.500000 + 0.866025i $$0.333333\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1.73205 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 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 0
$$166$$ 0 0
$$167$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$168$$ 0 0
$$169$$ 0.500000 0.866025i 0.500000 0.866025i
$$170$$ 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 0
$$176$$ 0 0
$$177$$ 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
$$181$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −0.500000 0.866025i −0.500000 0.866025i
$$186$$ 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
$$193$$ 0.133975 0.500000i 0.133975 0.500000i −0.866025 0.500000i $$-0.833333\pi$$
1.00000 $$0$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i $$-0.666667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$198$$ 0 0
$$199$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −0.866025 + 0.500000i −0.866025 + 0.500000i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.00000 1.00000
$$222$$ 0 0
$$223$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$228$$ 0 0
$$229$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$240$$ 0 0
$$241$$ 0.500000 0.133975i 0.500000 0.133975i 1.00000i $$-0.5\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.133975 0.500000i 0.133975 0.500000i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 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
$$256$$ 0 0
$$257$$ 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i $$-0.333333\pi$$
1.00000 $$0$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$264$$ 0 0
$$265$$ 0.366025 0.366025i 0.366025 0.366025i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$270$$ 0 0
$$271$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i $$-0.333333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$282$$ 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 0
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1.86603 0.500000i 1.86603 0.500000i 0.866025 0.500000i $$-0.166667\pi$$
1.00000 $$0$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −0.133975 0.500000i −0.133975 0.500000i
$$306$$ 0 0
$$307$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.36603 1.36603i 1.36603 1.36603i 0.500000 0.866025i $$-0.333333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −0.366025 0.633975i −0.366025 0.633975i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.00000i 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$348$$ 0 0
$$349$$ 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i $$-0.666667\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 0.866025i $$-0.333333\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$360$$ 0 0
$$361$$ 0.866025 0.500000i 0.866025 0.500000i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.00000 1.00000
$$366$$ 0 0
$$367$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i $$0.5\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.50000 0.866025i −1.50000 0.866025i
$$378$$ 0 0
$$379$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i $$-0.333333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −0.500000 + 1.86603i −0.500000 + 1.86603i 1.00000i $$0.5\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0.133975 + 0.500000i 0.133975 + 0.500000i 1.00000 $$0$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$420$$ 0 0
$$421$$ 1.36603 1.36603i 1.36603 1.36603i 0.500000 0.866025i $$-0.333333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0.366025 0.633975i 0.366025 0.633975i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$432$$ 0 0
$$433$$ −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i $$-0.666667\pi$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0.366025 + 0.633975i 0.366025 + 0.633975i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.86603 + 0.500000i 1.86603 + 0.500000i 1.00000 $$0$$
0.866025 + 0.500000i $$0.166667\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −0.500000 1.86603i −0.500000 1.86603i −0.500000 0.866025i $$-0.666667\pi$$
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
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 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 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$480$$ 0 0
$$481$$ −0.500000 1.86603i −0.500000 1.86603i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.633975 0.366025i −0.633975 0.366025i
$$486$$ 0 0
$$487$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$492$$ 0 0
$$493$$ 1.73205i 1.73205i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$504$$ 0 0
$$505$$ 0.232051 0.866025i 0.232051 0.866025i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.86603 0.500000i −1.86603 0.500000i −0.866025 0.500000i $$-0.833333\pi$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.73205 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.500000 + 0.866025i −0.500000 + 0.866025i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.86603 + 0.500000i −1.86603 + 0.500000i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i $$-0.166667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0.732051i 0.732051i
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −0.133975 + 0.500000i −0.133975 + 0.500000i 0.866025 + 0.500000i $$0.166667\pi$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$564$$ 0 0
$$565$$ −0.232051 0.866025i −0.232051 0.866025i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −1.36603 1.36603i −1.36603 1.36603i −0.866025 0.500000i $$-0.833333\pi$$
−0.500000 0.866025i $$-0.666667\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i $$0.166667\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −0.500000 + 0.133975i −0.500000 + 0.133975i
$$606$$ 0 0
$$607$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 0.866025i $$-0.333333\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0.500000 + 1.86603i 0.500000 + 1.86603i 0.500000 + 0.866025i $$0.333333\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −0.267949 −0.267949
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.36603 1.36603i 1.36603 1.36603i
$$630$$ 0 0
$$631$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.500000 0.866025i 0.500000 0.866025i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i $$-0.166667\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 0.965926 0.258819i $$-0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$660$$ 0 0
$$661$$ 0.500000 1.86603i 0.500000 1.86603i 1.00000i $$-0.5\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i $$-0.833333\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$684$$ 0 0
$$685$$ −0.133975 + 0.232051i −0.133975 + 0.232051i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0.866025 0.500000i 0.866025 0.500000i
$$690$$ 0 0
$$691$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.36603 1.36603i −1.36603 1.36603i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −0.500000 + 0.133975i −0.500000 + 0.133975i −0.500000 0.866025i $$-0.666667\pi$$
1.00000i $$0.5\pi$$
$$710$$ 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
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1.09808 + 0.633975i −1.09808 + 0.633975i
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i $$-0.833333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 −0.965926 0.258819i $$-0.916667\pi$$
0.965926 + 0.258819i $$0.0833333\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 0.258819 0.965926i $$-0.416667\pi$$
−0.258819 + 0.965926i $$0.583333\pi$$
$$744$$ 0 0
$$745$$ −0.866025 0.500000i −0.866025 0.500000i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i $$-0.333333\pi$$
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
$$769$$ 1.36603 + 0.366025i 1.36603 + 0.366025i 0.866025 0.500000i $$-0.166667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i $$-0.833333\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −0.633975 + 0.633975i −0.633975 + 0.633975i
$$786$$ 0 0
$$787$$ 0 0 −0.258819 0.965926i $$-0.583333\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.00000i 1.00000i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$798$$ 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
$$809$$ −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i $$-0.833333\pi$$
1.00000i $$-0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$812$$ 0 0
$$813$$ 0 <