# Properties

 Label 136.1.j.a.123.1 Level $136$ Weight $1$ Character 136.123 Analytic conductor $0.068$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 136.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0678728417181$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.314432.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.20123648.1

## Embedding invariants

 Embedding label 123.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 136.123 Dual form 136.1.j.a.115.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +(-1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 1.00000i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +(-1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 1.00000i) q^{6} -1.00000i q^{8} -1.00000i q^{9} +(1.00000 + 1.00000i) q^{11} +(1.00000 - 1.00000i) q^{12} +1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{18} -2.00000i q^{19} +(-1.00000 + 1.00000i) q^{22} +(1.00000 + 1.00000i) q^{24} -1.00000i q^{25} +1.00000i q^{32} -2.00000 q^{33} -1.00000 q^{34} +1.00000i q^{36} +2.00000 q^{38} +(-1.00000 - 1.00000i) q^{41} +(-1.00000 - 1.00000i) q^{44} +(-1.00000 + 1.00000i) q^{48} +1.00000i q^{49} +1.00000 q^{50} +(-1.00000 - 1.00000i) q^{51} +(2.00000 + 2.00000i) q^{57} -1.00000 q^{64} -2.00000i q^{66} -1.00000i q^{68} -1.00000 q^{72} +(-1.00000 + 1.00000i) q^{73} +(1.00000 + 1.00000i) q^{75} +2.00000i q^{76} +1.00000 q^{81} +(1.00000 - 1.00000i) q^{82} +(1.00000 - 1.00000i) q^{88} +(-1.00000 - 1.00000i) q^{96} +(1.00000 - 1.00000i) q^{97} -1.00000 q^{98} +(1.00000 - 1.00000i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} - 2q^{6} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} - 2q^{6} + 2q^{11} + 2q^{12} + 2q^{16} + 2q^{18} - 2q^{22} + 2q^{24} - 4q^{33} - 2q^{34} + 4q^{38} - 2q^{41} - 2q^{44} - 2q^{48} + 2q^{50} - 2q^{51} + 4q^{57} - 2q^{64} - 2q^{72} - 2q^{73} + 2q^{75} + 2q^{81} + 2q^{82} + 2q^{88} - 2q^{96} + 2q^{97} - 2q^{98} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$e\left(\frac{3}{4}\right)$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 1.00000i
$$3$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −1.00000
$$5$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$6$$ −1.00000 1.00000i −1.00000 1.00000i
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ 1.00000i 1.00000i
$$9$$ 1.00000i 1.00000i
$$10$$ 0 0
$$11$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 1.00000i 1.00000 1.00000i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 1.00000
$$17$$ 1.00000i 1.00000i
$$18$$ 1.00000 1.00000
$$19$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$23$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$24$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$25$$ 1.00000i 1.00000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$30$$ 0 0
$$31$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$32$$ 1.00000i 1.00000i
$$33$$ −2.00000 −2.00000
$$34$$ −1.00000 −1.00000
$$35$$ 0 0
$$36$$ 1.00000i 1.00000i
$$37$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$38$$ 2.00000 2.00000
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ −1.00000 1.00000i −1.00000 1.00000i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$49$$ 1.00000i 1.00000i
$$50$$ 1.00000 1.00000
$$51$$ −1.00000 1.00000i −1.00000 1.00000i
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.00000 −1.00000
$$65$$ 0 0
$$66$$ 2.00000i 2.00000i
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 1.00000i 1.00000i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$72$$ −1.00000 −1.00000
$$73$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$76$$ 2.00000i 2.00000i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$80$$ 0 0
$$81$$ 1.00000 1.00000
$$82$$ 1.00000 1.00000i 1.00000 1.00000i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 1.00000 1.00000i 1.00000 1.00000i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 1.00000i −1.00000 1.00000i
$$97$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$98$$ −1.00000 −1.00000
$$99$$ 1.00000 1.00000i 1.00000 1.00000i
$$100$$ 1.00000i 1.00000i
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 1.00000 1.00000i 1.00000 1.00000i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$108$$ 0 0
$$109$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$114$$ −2.00000 + 2.00000i −2.00000 + 2.00000i
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000i 1.00000i
$$122$$ 0 0
$$123$$ 2.00000 2.00000
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000i 1.00000i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$132$$ 2.00000 2.00000
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 1.00000 1.00000
$$137$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000i 1.00000i
$$145$$ 0 0
$$146$$ −1.00000 1.00000i −1.00000 1.00000i
$$147$$ −1.00000 1.00000i −1.00000 1.00000i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −2.00000 −2.00000
$$153$$ 1.00000 1.00000
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 1.00000i
$$163$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$164$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$168$$ 0 0
$$169$$ 1.00000 1.00000
$$170$$ 0 0
$$171$$ −2.00000 −2.00000
$$172$$ 0 0
$$173$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 1.00000 1.00000i 1.00000 1.00000i
$$193$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$194$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$195$$ 0 0
$$196$$ 1.00000i 1.00000i
$$197$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$198$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$199$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$200$$ −1.00000 −1.00000
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2.00000 2.00000i 2.00000 2.00000i
$$210$$ 0 0
$$211$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 2.00000i 2.00000i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −1.00000
$$226$$ 1.00000 1.00000i 1.00000 1.00000i
$$227$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$228$$ −2.00000 2.00000i −2.00000 2.00000i
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$234$$ 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 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$242$$ −1.00000 −1.00000
$$243$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 2.00000i 2.00000i
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 1.00000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −1.00000 1.00000i −1.00000 1.00000i
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 2.00000i 2.00000i
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 1.00000i 1.00000i
$$273$$ 0 0
$$274$$ 2.00000i 2.00000i
$$275$$ 1.00000 1.00000i 1.00000 1.00000i
$$276$$ 0 0
$$277$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$278$$ −1.00000 1.00000i −1.00000 1.00000i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 1.00000
$$289$$ −1.00000 −1.00000
$$290$$ 0 0
$$291$$ 2.00000i 2.00000i
$$292$$ 1.00000 1.00000i 1.00000 1.00000i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 1.00000 1.00000i 1.00000 1.00000i
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −1.00000 1.00000i −1.00000 1.00000i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 2.00000i 2.00000i
$$305$$ 0 0
$$306$$ 1.00000i 1.00000i
$$307$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$312$$ 0 0
$$313$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 2.00000i 2.00000i
$$322$$ 0 0
$$323$$ 2.00000 2.00000
$$324$$ −1.00000 −1.00000
$$325$$ 0 0
$$326$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$327$$ 0 0
$$328$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$338$$ 1.00000i 1.00000i
$$339$$ 2.00000 2.00000
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000i 2.00000i
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −2.00000 −2.00000
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −3.00000
$$362$$ 0 0
$$363$$ −1.00000 1.00000i −1.00000 1.00000i
$$364$$ 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
$$369$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ −1.00000 1.00000i −1.00000 1.00000i
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$385$$ 0 0
$$386$$ 1.00000 1.00000i 1.00000 1.00000i
$$387$$ 0 0
$$388$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.00000 1.00000
$$393$$ 2.00000i 2.00000i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$397$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 1.00000i 1.00000i
$$401$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 2.00000 2.00000i 2.00000 2.00000i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 2.00000i 2.00000i
$$418$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$419$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 1.00000 1.00000i 1.00000 1.00000i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.00000 1.00000
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$432$$ 0 0
$$433$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 2.00000 2.00000
$$439$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$440$$ 0 0
$$441$$ 1.00000 1.00000
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$450$$ 1.00000i 1.00000i
$$451$$ 2.00000i 2.00000i
$$452$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$453$$ 0 0
$$454$$ 1.00000 1.00000i 1.00000 1.00000i
$$455$$ 0 0
$$456$$ 2.00000 2.00000i 2.00000 2.00000i
$$457$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$467$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.00000 −2.00000
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$483$$ 0 0
$$484$$ 1.00000i 1.00000i
$$485$$ 0 0
$$486$$ −1.00000 1.00000i −1.00000 1.00000i
$$487$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$488$$ 0 0
$$489$$ −2.00000 −2.00000
$$490$$ 0 0
$$491$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$492$$ −2.00000 −2.00000
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$500$$ 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
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 1.00000i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 1.00000 1.00000i 1.00000 1.00000i
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ −2.00000 −2.00000
$$529$$ 1.00000i 1.00000i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −2.00000 2.00000i −2.00000 2.00000i
$$538$$ 0 0
$$539$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$540$$ 0 0
$$541$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −1.00000 −1.00000
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$548$$ 2.00000 2.00000
$$549$$ 0 0
$$550$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 1.00000 1.00000i 1.00000 1.00000i
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 2.00000i 2.00000i
$$562$$ −2.00000 −2.00000
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000i 1.00000i
$$577$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$578$$ 1.00000i 1.00000i
$$579$$ 2.00000 2.00000
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −2.00000 −2.00000
$$583$$ 0 0
$$584$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$588$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 1.00000 1.00000i 1.00000 1.00000i
$$601$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$608$$ 2.00000 2.00000
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −1.00000 −1.00000
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 2.00000i 2.00000i
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.00000 −1.00000
$$626$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$627$$ 4.00000i 4.00000i
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 2.00000 2.00000
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$642$$ −2.00000 −2.00000
$$643$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2.00000i 2.00000i
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 1.00000i 1.00000i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1.00000 1.00000i −1.00000 1.00000i
$$653$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1.00000 1.00000i −1.00000 1.00000i
$$657$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$658$$ 0 0
$$659$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 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
$$673$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$674$$ −1.00000 1.00000i −1.00000 1.00000i
$$675$$ 0 0
$$676$$ −1.00000 −1.00000
$$677$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$678$$ 2.00000i 2.00000i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 2.00000 2.00000
$$682$$ 0 0
$$683$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$684$$ 2.00000 2.00000
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 1.00000 1.00000i 1.00000 1.00000i
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.00000 1.00000i 1.00000 1.00000i
$$698$$ 0 0
$$699$$ 2.00000i 2.00000i
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −1.00000 1.00000i −1.00000 1.00000i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 2.00000i 2.00000i
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 3.00000i
$$723$$ 2.00000i 2.00000i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 1.00000 1.00000i 1.00000 1.00000i
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 1.00000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −1.00000 1.00000i −1.00000 1.00000i
$$739$$ 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 $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 1.00000 1.00000i 1.00000 1.00000i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ −1.00000 1.00000i −1.00000 1.00000i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$769$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −1.00000 1.00000i −1.00000 1.00000i
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.00000 + 2.00000i −2.00000 + 2.00000i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000i 1.00000i
$$785$$ 0 0
$$786$$ 2.00000 2.00000
$$787$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −1.00000 1.00000i −1.00000 1.00000i
$$793$$ 0 0
$$794$$ 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
$$800$$ 1.00000 1.00000
$$801$$ 0 0
$$802$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$803$$ −2.00000 −2.00000
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1.00000