# Properties

 Label 528.2.b.a.65.1 Level $528$ Weight $2$ Character 528.65 Analytic conductor $4.216$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 528.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.21610122672$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 65.1 Root $$0.500000 + 1.65831i$$ of defining polynomial Character $$\chi$$ $$=$$ 528.65 Dual form 528.2.b.a.65.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 + 1.65831i) q^{9} +O(q^{10})$$ $$q+(-0.500000 - 1.65831i) q^{3} -3.31662i q^{5} +(-2.50000 + 1.65831i) q^{9} -3.31662i q^{11} +(-5.50000 + 1.65831i) q^{15} +3.31662i q^{23} -6.00000 q^{25} +(4.00000 + 3.31662i) q^{27} -5.00000 q^{31} +(-5.50000 + 1.65831i) q^{33} -7.00000 q^{37} +(5.50000 + 8.29156i) q^{45} -6.63325i q^{47} +7.00000 q^{49} -13.2665i q^{53} -11.0000 q^{55} +3.31662i q^{59} +13.0000 q^{67} +(5.50000 - 1.65831i) q^{69} -16.5831i q^{71} +(3.00000 + 9.94987i) q^{75} +(3.50000 - 8.29156i) q^{81} +16.5831i q^{89} +(2.50000 + 8.29156i) q^{93} +17.0000 q^{97} +(5.50000 + 8.29156i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 5 q^{9} + O(q^{10})$$ $$2 q - q^{3} - 5 q^{9} - 11 q^{15} - 12 q^{25} + 8 q^{27} - 10 q^{31} - 11 q^{33} - 14 q^{37} + 11 q^{45} + 14 q^{49} - 22 q^{55} + 26 q^{67} + 11 q^{69} + 6 q^{75} + 7 q^{81} + 5 q^{93} + 34 q^{97} + 11 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −0.500000 1.65831i −0.288675 0.957427i
$$4$$ 0 0
$$5$$ 3.31662i 1.48324i −0.670820 0.741620i $$-0.734058\pi$$
0.670820 0.741620i $$-0.265942\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ −2.50000 + 1.65831i −0.833333 + 0.552771i
$$10$$ 0 0
$$11$$ 3.31662i 1.00000i
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ −5.50000 + 1.65831i −1.42009 + 0.428174i
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.31662i 0.691564i 0.938315 + 0.345782i $$0.112386\pi$$
−0.938315 + 0.345782i $$0.887614\pi$$
$$24$$ 0 0
$$25$$ −6.00000 −1.20000
$$26$$ 0 0
$$27$$ 4.00000 + 3.31662i 0.769800 + 0.638285i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ −5.50000 + 1.65831i −0.957427 + 0.288675i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 5.50000 + 8.29156i 0.819892 + 1.23603i
$$46$$ 0 0
$$47$$ 6.63325i 0.967559i −0.875190 0.483779i $$-0.839264\pi$$
0.875190 0.483779i $$-0.160736\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 13.2665i 1.82229i −0.412082 0.911147i $$-0.635198\pi$$
0.412082 0.911147i $$-0.364802\pi$$
$$54$$ 0 0
$$55$$ −11.0000 −1.48324
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.31662i 0.431788i 0.976417 + 0.215894i $$0.0692665\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 13.0000 1.58820 0.794101 0.607785i $$-0.207942\pi$$
0.794101 + 0.607785i $$0.207942\pi$$
$$68$$ 0 0
$$69$$ 5.50000 1.65831i 0.662122 0.199637i
$$70$$ 0 0
$$71$$ 16.5831i 1.96805i −0.178017 0.984027i $$-0.556968\pi$$
0.178017 0.984027i $$-0.443032\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 3.00000 + 9.94987i 0.346410 + 1.14891i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 3.50000 8.29156i 0.388889 0.921285i
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 16.5831i 1.75781i 0.476999 + 0.878904i $$0.341725\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 2.50000 + 8.29156i 0.259238 + 0.859795i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 0 0
$$99$$ 5.50000 + 8.29156i 0.552771 + 0.833333i
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 3.50000 + 11.6082i 0.332205 + 1.10180i
$$112$$ 0 0
$$113$$ 3.31662i 0.312002i −0.987757 0.156001i $$-0.950140\pi$$
0.987757 0.156001i $$-0.0498603\pi$$
$$114$$ 0 0
$$115$$ 11.0000 1.02576
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.31662i 0.296648i
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 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
$$135$$ 11.0000 13.2665i 0.946729 1.14180i
$$136$$ 0 0
$$137$$ 23.2164i 1.98351i −0.128154 0.991754i $$-0.540905\pi$$
0.128154 0.991754i $$-0.459095\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ −11.0000 + 3.31662i −0.926367 + 0.279310i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.50000 11.6082i −0.288675 0.957427i
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.5831i 1.33199i
$$156$$ 0 0
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ 0 0
$$159$$ −22.0000 + 6.63325i −1.74471 + 0.526051i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 0 0
$$165$$ 5.50000 + 18.2414i 0.428174 + 1.42009i
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.50000 1.65831i 0.413405 0.124646i
$$178$$ 0 0
$$179$$ 16.5831i 1.23948i −0.784807 0.619740i $$-0.787238\pi$$
0.784807 0.619740i $$-0.212762\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 23.2164i 1.70690i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 23.2164i 1.67988i 0.542681 + 0.839939i $$0.317409\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ −6.50000 21.5581i −0.458475 1.52059i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −5.50000 8.29156i −0.382276 0.576303i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ −27.5000 + 8.29156i −1.88427 + 0.568128i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 1.00000 0.0669650 0.0334825 0.999439i $$-0.489340\pi$$
0.0334825 + 0.999439i $$0.489340\pi$$
$$224$$ 0 0
$$225$$ 15.0000 9.94987i 1.00000 0.663325i
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 5.00000 0.330409 0.165205 0.986259i $$-0.447172\pi$$
0.165205 + 0.986259i $$0.447172\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ −22.0000 −1.43512
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ −15.5000 1.65831i −0.994325 0.106381i
$$244$$ 0 0
$$245$$ 23.2164i 1.48324i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.5831i 1.04672i −0.852112 0.523359i $$-0.824679\pi$$
0.852112 0.523359i $$-0.175321\pi$$
$$252$$ 0 0
$$253$$ 11.0000 0.691564
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 26.5330i 1.65508i 0.561405 + 0.827541i $$0.310261\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −44.0000 −2.70290
$$266$$ 0 0
$$267$$ 27.5000 8.29156i 1.68297 0.507435i
$$268$$ 0 0
$$269$$ 13.2665i 0.808873i −0.914566 0.404436i $$-0.867468\pi$$
0.914566 0.404436i $$-0.132532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 19.8997i 1.20000i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 12.5000 8.29156i 0.748355 0.496403i
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −8.50000 28.1913i −0.498279 1.65260i
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 11.0000 0.640445
$$296$$ 0 0
$$297$$ 11.0000 13.2665i 0.638285 0.769800i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ −2.00000 6.63325i −0.113776 0.377352i
$$310$$ 0 0
$$311$$ 33.1662i 1.88069i 0.340229 + 0.940343i $$0.389495\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.2164i 1.30396i −0.758236 0.651981i $$-0.773938\pi$$
0.758236 0.651981i $$-0.226062\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 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
$$331$$ −35.0000 −1.92377 −0.961887 0.273447i $$-0.911836\pi$$
−0.961887 + 0.273447i $$0.911836\pi$$
$$332$$ 0 0
$$333$$ 17.5000 11.6082i 0.958994 0.636125i
$$334$$ 0 0
$$335$$ 43.1161i 2.35569i
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ −5.50000 + 1.65831i −0.298719 + 0.0900672i
$$340$$ 0 0
$$341$$ 16.5831i 0.898027i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −5.50000 18.2414i −0.296110 0.982086i
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 36.4829i 1.94179i 0.239511 + 0.970894i $$0.423013\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$354$$ 0 0
$$355$$ −55.0000 −2.91910
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 5.50000 + 18.2414i 0.288675 + 0.957427i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.0000 1.93138 0.965692 0.259690i $$-0.0836203\pi$$
0.965692 + 0.259690i $$0.0836203\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 5.50000 1.65831i 0.284019 0.0856349i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.31662i 0.169472i 0.996403 + 0.0847358i $$0.0270046\pi$$
−0.996403 + 0.0847358i $$0.972995\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 36.4829i 1.84976i 0.380265 + 0.924878i $$0.375833\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.5330i 1.32499i 0.749064 + 0.662497i $$0.230503\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −27.5000 11.6082i −1.36649 0.576815i
$$406$$ 0 0
$$407$$ 23.2164i 1.15079i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −38.5000 + 11.6082i −1.89906 + 0.572590i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 33.1662i 1.62028i 0.586238 + 0.810139i $$0.300608\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 11.0000 + 16.5831i 0.534838 + 0.806299i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −17.5000 + 11.6082i −0.833333 + 0.552771i
$$442$$ 0 0
$$443$$ 36.4829i 1.73335i −0.498870 0.866677i $$-0.666252\pi$$
0.498870 0.866677i $$-0.333748\pi$$
$$444$$ 0 0
$$445$$ 55.0000 2.60725
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 16.5831i 0.782606i 0.920262 + 0.391303i $$0.127976\pi$$
−0.920262 + 0.391303i $$0.872024\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 31.0000 1.44069 0.720346 0.693615i $$-0.243983\pi$$
0.720346 + 0.693615i $$0.243983\pi$$
$$464$$ 0 0
$$465$$ 27.5000 8.29156i 1.27528 0.384512i
$$466$$ 0 0
$$467$$ 43.1161i 1.99518i 0.0694117 + 0.997588i $$0.477888\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −11.5000 38.1412i −0.529892 1.75745i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 22.0000 + 33.1662i 1.00731 + 1.51858i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 56.3826i 2.56020i
$$486$$ 0 0
$$487$$ 43.0000 1.94852 0.974258 0.225436i $$-0.0723806\pi$$
0.974258 + 0.225436i $$0.0723806\pi$$
$$488$$ 0 0
$$489$$ −8.00000 26.5330i −0.361773 1.19986i
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 27.5000 18.2414i 1.23603 0.819892i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −6.50000 21.5581i −0.288675 0.957427i
$$508$$ 0 0
$$509$$ 3.31662i 0.147007i −0.997295 0.0735034i $$-0.976582\pi$$
0.997295 0.0735034i $$-0.0234180\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 13.2665i 0.584592i
$$516$$ 0 0
$$517$$ −22.0000 −0.967559
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 43.1161i 1.88895i −0.328581 0.944476i $$-0.606570\pi$$
0.328581 0.944476i $$-0.393430\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 12.0000 0.521739
$$530$$ 0 0
$$531$$ −5.50000 8.29156i −0.238680 0.359823i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −27.5000 + 8.29156i −1.18671 + 0.357807i
$$538$$ 0 0
$$539$$ 23.2164i 1.00000i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 12.5000 + 41.4578i 0.536426 + 1.77912i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 38.5000 11.6082i 1.63423 0.492740i
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ −11.0000 −0.462773
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 38.5000 11.6082i 1.60836 0.484939i
$$574$$ 0 0
$$575$$ 19.8997i 0.829877i
$$576$$ 0 0
$$577$$ 47.0000 1.95664 0.978318 0.207109i $$-0.0664056\pi$$
0.978318 + 0.207109i $$0.0664056\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −44.0000 −1.82229
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.63325i 0.273784i −0.990586 0.136892i $$-0.956289\pi$$
0.990586 0.136892i $$-0.0437113\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10.0000 + 33.1662i 0.409273 + 1.35740i
$$598$$ 0 0
$$599$$ 33.1662i 1.35514i 0.735460 + 0.677568i $$0.236966\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −32.5000 + 21.5581i −1.32350 + 0.877912i
$$604$$ 0 0
$$605$$ 36.4829i 1.48324i
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.5330i 1.06818i 0.845428 + 0.534089i $$0.179345\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 1.00000 0.0401934 0.0200967 0.999798i $$-0.493603\pi$$
0.0200967 + 0.999798i $$0.493603\pi$$
$$620$$ 0 0
$$621$$ −11.0000 + 13.2665i −0.441415 + 0.532366i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 27.5000 + 41.4578i 1.08788 + 1.64005i
$$640$$ 0 0
$$641$$ 23.2164i 0.916992i −0.888697 0.458496i $$-0.848388\pi$$
0.888697 0.458496i $$-0.151612\pi$$
$$642$$ 0 0
$$643$$ −41.0000 −1.61688 −0.808441 0.588577i $$-0.799688\pi$$
−0.808441 + 0.588577i $$0.799688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 43.1161i 1.69507i 0.530740 + 0.847535i $$0.321914\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ 11.0000 0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3.31662i 0.129790i −0.997892 0.0648948i $$-0.979329\pi$$
0.997892 0.0648948i $$-0.0206712\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −0.500000 1.65831i −0.0193311 0.0641141i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −24.0000 19.8997i −0.923760 0.765942i
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 46.4327i 1.77670i −0.459167 0.888350i $$-0.651852\pi$$
0.459167 0.888350i $$-0.348148\pi$$
$$684$$ 0 0
$$685$$ −77.0000 −2.94202
$$686$$ 0 0
$$687$$ −2.50000 8.29156i −0.0953809 0.316343i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −17.0000 −0.646710 −0.323355 0.946278i $$-0.604811\pi$$
−0.323355 + 0.946278i $$0.604811\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 11.0000 + 36.4829i 0.414284 + 1.37402i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −19.0000 −0.713560 −0.356780 0.934188i $$-0.616125\pi$$
−0.356780 + 0.934188i $$0.616125\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 16.5831i 0.621043i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 16.5831i 0.618446i −0.950990 0.309223i $$-0.899931\pi$$
0.950990 0.309223i $$-0.100069\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −53.0000 −1.96566 −0.982831 0.184510i $$-0.940930\pi$$
−0.982831 + 0.184510i $$0.940930\pi$$
$$728$$ 0 0
$$729$$ 5.00000 + 26.5330i 0.185185 + 0.982704i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ −38.5000 + 11.6082i −1.42009 + 0.428174i
$$736$$ 0 0
$$737$$ 43.1161i 1.58820i
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 0 0
$$753$$ −27.5000 + 8.29156i −1.00216 + 0.302161i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ −5.50000 18.2414i −0.199637 0.662122i
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 44.0000 13.2665i 1.58462 0.477781i
$$772$$ 0 0
$$773$$ 13.2665i 0.477163i −0.971123 0.238581i $$-0.923318\pi$$
0.971123 0.238581i $$-0.0766824\pi$$
$$774$$ 0 0
$$775$$ 30.0000 1.07763
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −55.0000 −1.96805
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 76.2824i 2.72263i
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 22.0000 + 72.9657i 0.780260 + 2.58783i
$$796$$ 0 0
$$797$$ 56.3826i 1.99717i 0.0531327 + 0.998587i $$0.483079\pi$$
−0.0531327 + 0.998587i $$0.516921\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −27.5000 41.4578i −0.971665 1.46484i
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −22.0000 + 6.63325i −0.774437 + 0.233501i
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000