# Properties

 Label 4000.1.bo.a.2593.1 Level 4000 Weight 1 Character 4000.2593 Analytic conductor 1.996 Analytic rank 0 Dimension 8 Projective image $$D_{20}$$ CM disc. -4 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 4000.bo (of order $$20$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Projective image $$D_{20}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{20} + \cdots)$$

## Embedding invariants

 Embedding label 2593.1 Root $$-0.587785 - 0.809017i$$ Character $$\chi$$ = 4000.2593 Dual form 4000.1.bo.a.3457.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.587785 + 0.809017i) q^{9} +O(q^{10})$$ $$q+(-0.587785 + 0.809017i) q^{9} +(0.142040 + 0.896802i) q^{13} +(-0.278768 - 0.142040i) q^{17} +(1.11803 + 0.363271i) q^{29} +(-1.95106 + 0.309017i) q^{37} +(1.53884 + 1.11803i) q^{41} +1.00000i q^{49} +(-1.58779 + 0.809017i) q^{53} +(-1.53884 + 1.11803i) q^{61} +(1.76007 + 0.278768i) q^{73} +(-0.309017 - 0.951057i) q^{81} +(0.690983 + 0.951057i) q^{89} +(-0.896802 - 1.76007i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 2q^{13} - 2q^{17} - 8q^{37} - 8q^{53} + 2q^{73} + 2q^{81} + 10q^{89} + 2q^{97} + O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$$.

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$e\left(\frac{11}{20}\right)$$ $$1$$ $$1$$

## 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 −0.453990 0.891007i $$-0.650000\pi$$
0.453990 + 0.891007i $$0.350000\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ 0 0
$$9$$ −0.587785 + 0.809017i −0.587785 + 0.809017i
$$10$$ 0 0
$$11$$ 0 0 −0.587785 0.809017i $$-0.700000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$12$$ 0 0
$$13$$ 0.142040 + 0.896802i 0.142040 + 0.896802i 0.951057 + 0.309017i $$0.100000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.278768 0.142040i −0.278768 0.142040i 0.309017 0.951057i $$-0.400000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$18$$ 0 0
$$19$$ 0 0 −0.309017 0.951057i $$-0.600000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.156434 0.987688i $$-0.450000\pi$$
−0.156434 + 0.987688i $$0.550000\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.11803 + 0.363271i 1.11803 + 0.363271i 0.809017 0.587785i $$-0.200000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$30$$ 0 0
$$31$$ 0 0 0.951057 0.309017i $$-0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.95106 + 0.309017i −1.95106 + 0.309017i −0.951057 + 0.309017i $$0.900000\pi$$
−1.00000 $$1.00000\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i $$0.100000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.891007 0.453990i $$-0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$48$$ 0 0
$$49$$ 1.00000i 1.00000i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.58779 + 0.809017i −1.58779 + 0.809017i −0.587785 + 0.809017i $$0.700000\pi$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 −0.809017 0.587785i $$-0.800000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$60$$ 0 0
$$61$$ −1.53884 + 1.11803i −1.53884 + 1.11803i −0.587785 + 0.809017i $$0.700000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 0.453990 0.891007i $$-0.350000\pi$$
−0.453990 + 0.891007i $$0.650000\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 −0.951057 0.309017i $$-0.900000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$72$$ 0 0
$$73$$ 1.76007 + 0.278768i 1.76007 + 0.278768i 0.951057 0.309017i $$-0.100000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 0.309017 0.951057i $$-0.400000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$80$$ 0 0
$$81$$ −0.309017 0.951057i −0.309017 0.951057i
$$82$$ 0 0
$$83$$ 0 0 −0.891007 0.453990i $$-0.850000\pi$$
0.891007 + 0.453990i $$0.150000\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 $$0$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.896802 1.76007i −0.896802 1.76007i −0.587785 0.809017i $$-0.700000\pi$$
−0.309017 0.951057i $$-0.600000\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$102$$ 0 0
$$103$$ 0 0 −0.453990 0.891007i $$-0.650000\pi$$
0.453990 + 0.891007i $$0.350000\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$108$$ 0 0
$$109$$ 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i $$-0.700000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0.309017 + 1.95106i 0.309017 + 1.95106i 0.309017 + 0.951057i $$0.400000\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −0.809017 0.412215i −0.809017 0.412215i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −0.309017 + 0.951057i −0.309017 + 0.951057i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 −0.987688 0.156434i $$-0.950000\pi$$
0.987688 + 0.156434i $$0.0500000\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 0.951057 0.309017i $$-0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.76007 0.278768i 1.76007 0.278768i 0.809017 0.587785i $$-0.200000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$138$$ 0 0
$$139$$ 0 0 0.809017 0.587785i $$-0.200000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0.618034i 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0.278768 0.142040i 0.278768 0.142040i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −0.221232 + 0.221232i −0.221232 + 0.221232i −0.809017 0.587785i $$-0.800000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 0.987688 0.156434i $$-0.0500000\pi$$
−0.987688 + 0.156434i $$0.950000\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.453990 0.891007i $$-0.350000\pi$$
−0.453990 + 0.891007i $$0.650000\pi$$
$$168$$ 0 0
$$169$$ 0.166977 0.0542543i 0.166977 0.0542543i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −0.309017 0.0489435i −0.309017 0.0489435i 1.00000i $$-0.5\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.309017 0.951057i $$-0.400000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$180$$ 0 0
$$181$$ −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i $$-0.800000\pi$$
0.309017 0.951057i $$-0.400000\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 0.587785 0.809017i $$-0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$192$$ 0 0
$$193$$ −0.642040 0.642040i −0.642040 0.642040i 0.309017 0.951057i $$-0.400000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.809017 + 1.58779i 0.809017 + 1.58779i 0.809017 + 0.587785i $$0.200000\pi$$
1.00000i $$0.500000\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 −0.587785 0.809017i $$-0.700000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$212$$ 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
$$221$$ 0.0877853 0.270175i 0.0877853 0.270175i
$$222$$ 0 0
$$223$$ 0 0 0.156434 0.987688i $$-0.450000\pi$$
−0.156434 + 0.987688i $$0.550000\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 −0.987688 0.156434i $$-0.950000\pi$$
0.987688 + 0.156434i $$0.0500000\pi$$
$$228$$ 0 0
$$229$$ −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i $$-0.800000\pi$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0.412215 0.809017i 0.412215 0.809017i −0.587785 0.809017i $$-0.700000\pi$$
1.00000 $$0$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 0.809017 0.587785i $$-0.200000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$240$$ 0 0
$$241$$ 0.951057 + 0.690983i 0.951057 + 0.690983i 0.951057 0.309017i $$-0.100000\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 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
$$256$$ 0 0
$$257$$ −0.642040 + 0.642040i −0.642040 + 0.642040i −0.951057 0.309017i $$-0.900000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −0.951057 + 0.690983i −0.951057 + 0.690983i
$$262$$ 0 0
$$263$$ 0 0 0.987688 0.156434i $$-0.0500000\pi$$
−0.987688 + 0.156434i $$0.950000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0.587785 0.190983i 0.587785 0.190983i 1.00000i $$-0.5\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$270$$ 0 0
$$271$$ 0 0 −0.951057 0.309017i $$-0.900000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.0489435 0.309017i 0.0489435 0.309017i −0.951057 0.309017i $$-0.900000\pi$$
1.00000 $$0$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −0.587785 1.80902i −0.587785 1.80902i −0.587785 0.809017i $$-0.700000\pi$$
1.00000i $$-0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 −0.891007 0.453990i $$-0.850000\pi$$
0.891007 + 0.453990i $$0.150000\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −0.530249 0.729825i −0.530249 0.729825i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1.39680 1.39680i −1.39680 1.39680i −0.809017 0.587785i $$-0.800000\pi$$
−0.587785 0.809017i $$-0.700000\pi$$
$$294$$ 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 0
$$306$$ 0 0
$$307$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 −0.587785 0.809017i $$-0.700000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$312$$ 0 0
$$313$$ −0.221232 1.39680i −0.221232 1.39680i −0.809017 0.587785i $$-0.800000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.26007 + 0.642040i 1.26007 + 0.642040i 0.951057 0.309017i $$-0.100000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$318$$ 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
$$331$$ 0 0 0.951057 0.309017i $$-0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$332$$ 0 0
$$333$$ 0.896802 1.76007i 0.896802 1.76007i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.39680 0.221232i 1.39680 0.221232i 0.587785 0.809017i $$-0.300000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$338$$ 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 0.891007 0.453990i $$-0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$348$$ 0 0
$$349$$ 1.17557i 1.17557i 0.809017 + 0.587785i $$0.200000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1.26007 + 0.642040i −1.26007 + 0.642040i −0.951057 0.309017i $$-0.900000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.809017 0.587785i $$-0.800000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$360$$ 0 0
$$361$$ −0.809017 + 0.587785i −0.809017 + 0.587785i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 0.453990 0.891007i $$-0.350000\pi$$
−0.453990 + 0.891007i $$0.650000\pi$$
$$368$$ 0 0
$$369$$ −1.80902 + 0.587785i −1.80902 + 0.587785i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1.39680 + 0.221232i 1.39680 + 0.221232i 0.809017 0.587785i $$-0.200000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.166977 + 1.05425i −0.166977 + 1.05425i
$$378$$ 0 0
$$379$$ 0 0 0.309017 0.951057i $$-0.400000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 −0.891007 0.453990i $$-0.850000\pi$$
0.891007 + 0.453990i $$0.150000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i $$-0.300000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i $$0.100000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0.951057 1.30902i 0.951057 1.30902i 1.00000i $$-0.5\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$410$$ 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 −0.309017 0.951057i $$-0.600000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$420$$ 0 0
$$421$$ 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i $$-0.700000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 0.951057 0.309017i $$-0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$432$$ 0 0
$$433$$ 0.809017 1.58779i 0.809017 1.58779i 1.00000i $$-0.5\pi$$
0.809017 0.587785i $$-0.200000\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 0.809017 0.587785i $$-0.200000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$440$$ 0 0
$$441$$ −0.809017 0.587785i −0.809017 0.587785i
$$442$$ 0 0
$$443$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.61803i 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i $$0.600000\pi$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 0.987688 0.156434i $$-0.0500000\pi$$
−0.987688 + 0.156434i $$0.950000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.453990 0.891007i $$-0.350000\pi$$
−0.453990 + 0.891007i $$0.650000\pi$$
$$468$$ 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
$$477$$ 0.278768 1.76007i 0.278768 1.76007i
$$478$$ 0 0
$$479$$ 0 0 0.309017 0.951057i $$-0.400000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$480$$ 0 0
$$481$$ −0.554254 1.70582i −0.554254 1.70582i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 −0.156434 0.987688i $$-0.550000\pi$$
0.156434 + 0.987688i $$0.450000\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 0.587785 0.809017i $$-0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$492$$ 0 0
$$493$$ −0.260074 0.260074i −0.260074 0.260074i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.453990 0.891007i $$-0.650000\pi$$
0.453990 + 0.891007i $$0.350000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0.690983 0.951057i 0.690983 0.951057i −0.309017 0.951057i $$-0.600000\pi$$
1.00000 $$0$$
$$510$$ 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
$$521$$ −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i $$0.200000\pi$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.156434 0.987688i $$-0.450000\pi$$
−0.156434 + 0.987688i $$0.550000\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −0.951057 0.309017i −0.951057 0.309017i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −0.784079 + 1.53884i −0.784079 + 1.53884i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i $$-0.600000\pi$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 0.891007 0.453990i $$-0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$548$$ 0 0
$$549$$ 1.90211i 1.90211i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i $$-0.300000\pi$$
0.809017 0.587785i $$-0.200000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.987688 0.156434i $$-0.0500000\pi$$
−0.987688 + 0.156434i $$0.950000\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i $$-0.300000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$570$$ 0 0
$$571$$ 0 0 −0.951057 0.309017i $$-0.900000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −0.221232 + 1.39680i −0.221232 + 1.39680i 0.587785 + 0.809017i $$0.300000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$578$$ 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 −0.156434 0.987688i $$-0.550000\pi$$
0.156434 + 0.987688i $$0.450000\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0.221232 + 0.221232i 0.221232 + 0.221232i 0.809017 0.587785i $$-0.200000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −1.90211 −1.90211 −0.951057 0.309017i $$-0.900000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0.0489435 + 0.309017i 0.0489435 + 0.309017i 1.00000 $$0$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.58779 + 0.809017i 1.58779 + 0.809017i 1.00000 $$0$$
0.587785 + 0.809017i $$0.300000\pi$$
$$618$$ 0 0
$$619$$ 0 0 −0.309017 0.951057i $$-0.600000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0.587785 + 0.190983i 0.587785 + 0.190983i
$$630$$ 0 0
$$631$$ 0 0 0.951057 0.309017i $$-0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −0.896802 + 0.142040i −0.896802 + 0.142040i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 0.587785 0.809017i $$-0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$642$$ 0 0
$$643$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.891007 0.453990i $$-0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.76007 0.896802i 1.76007 0.896802i 0.809017 0.587785i $$-0.200000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −1.26007 + 1.26007i −1.26007 + 1.26007i
$$658$$ 0 0
$$659$$ 0 0 −0.809017 0.587785i $$-0.800000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$660$$ 0 0
$$661$$ 0 0 −0.587785 0.809017i $$-0.700000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$662$$ 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
$$673$$ −1.95106 0.309017i −1.95106 0.309017i −0.951057 0.309017i $$-0.900000\pi$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0.221232 1.39680i 0.221232 1.39680i −0.587785 0.809017i $$-0.700000\pi$$
0.809017 0.587785i $$-0.200000\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.891007 0.453990i $$-0.850000\pi$$
0.891007 + 0.453990i $$0.150000\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −0.951057 1.30902i −0.951057 1.30902i
$$690$$ 0 0
$$691$$ 0 0 0.587785 0.809017i $$-0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −0.270175 0.530249i −0.270175 0.530249i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.90211 1.90211 0.951057 0.309017i $$-0.100000\pi$$
0.951057 + 0.309017i $$0.100000\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i $$0.600000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$710$$ 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 −0.309017 0.951057i $$-0.600000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 −0.987688 0.156434i $$-0.950000\pi$$
0.987688 + 0.156434i $$0.0500000\pi$$
$$728$$ 0 0
$$729$$ 0.951057 + 0.309017i 0.951057 + 0.309017i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.642040 1.26007i 0.642040 1.26007i −0.309017 0.951057i $$-0.600000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 0.809017 0.587785i $$-0.200000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$740$$ 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
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.26007 1.26007i 1.26007 1.26007i 0.309017 0.951057i $$-0.400000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i $$-0.800000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 −0.309017 0.951057i $$-0.600000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0.896802 + 0.142040i 0.896802 + 0.142040i 0.587785 0.809017i $$-0.300000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$774$$ 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 0
$$786$$ 0 0
$$787$$ 0 0 −0.156434 0.987688i $$-0.550000\pi$$
0.156434 + 0.987688i $$0.450000\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −1.22123 1.22123i −1.22123 1.22123i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.412215 + 0.809017i 0.412215 + 0.809017i 1.00000 $$0$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −1.17557 −1.17557
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i $$0.400000\pi$$
−1.00000 $$1.00000\pi$$
$$810$$ 0 0
$$811$$ 0 0 −0.587785 0.809017i $$-0.700000\pi$$
0.587785 + 0.809017i $$0.300000\pi$$
$$812$$ 0 0
$$813$$ 0 0