# Properties

 Label 1000.1.g.a.251.2 Level $1000$ Weight $1$ Character 1000.251 Analytic conductor $0.499$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -40 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1000 = 2^{3} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1000.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.499065012633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.1000000.1

## Embedding invariants

 Embedding label 251.2 Root $$-1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 1000.251 Dual form 1000.1.g.a.251.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.61803i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.61803i q^{7} +1.00000i q^{8} -1.00000 q^{9} +0.618034 q^{11} +0.618034i q^{13} +1.61803 q^{14} +1.00000 q^{16} +1.00000i q^{18} +1.61803 q^{19} -0.618034i q^{22} +0.618034i q^{23} +0.618034 q^{26} -1.61803i q^{28} -1.00000i q^{32} +1.00000 q^{36} +1.61803i q^{37} -1.61803i q^{38} -1.61803 q^{41} -0.618034 q^{44} +0.618034 q^{46} -0.618034i q^{47} -1.61803 q^{49} -0.618034i q^{52} -1.61803i q^{53} -1.61803 q^{56} +1.61803 q^{59} -1.61803i q^{63} -1.00000 q^{64} -1.00000i q^{72} +1.61803 q^{74} -1.61803 q^{76} +1.00000i q^{77} +1.00000 q^{81} +1.61803i q^{82} +0.618034i q^{88} -0.618034 q^{89} -1.00000 q^{91} -0.618034i q^{92} -0.618034 q^{94} +1.61803i q^{98} -0.618034 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{9} - 2q^{11} + 2q^{14} + 4q^{16} + 2q^{19} - 2q^{26} + 4q^{36} - 2q^{41} + 2q^{44} - 2q^{46} - 2q^{49} - 2q^{56} + 2q^{59} - 4q^{64} + 2q^{74} - 2q^{76} + 4q^{81} + 2q^{89} - 4q^{91} + 2q^{94} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$$.

 $$n$$ $$377$$ $$501$$ $$751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 1.00000i
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$8$$ 1.00000i 1.00000i
$$9$$ −1.00000 −1.00000
$$10$$ 0 0
$$11$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$12$$ 0 0
$$13$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$14$$ 1.61803 1.61803
$$15$$ 0 0
$$16$$ 1.00000 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.00000i 1.00000i
$$19$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 0.618034i − 0.618034i
$$23$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.618034 0.618034
$$27$$ 0 0
$$28$$ − 1.61803i − 1.61803i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ − 1.00000i − 1.00000i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 1.00000
$$37$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$38$$ − 1.61803i − 1.61803i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −0.618034 −0.618034
$$45$$ 0 0
$$46$$ 0.618034 0.618034
$$47$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$48$$ 0 0
$$49$$ −1.61803 −1.61803
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 0.618034i − 0.618034i
$$53$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.61803 −1.61803
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ − 1.61803i − 1.61803i
$$64$$ −1.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 1.00000i − 1.00000i
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 1.61803 1.61803
$$75$$ 0 0
$$76$$ −1.61803 −1.61803
$$77$$ 1.00000i 1.00000i
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 1.00000 1.00000
$$82$$ 1.61803i 1.61803i
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0.618034i 0.618034i
$$89$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −1.00000
$$92$$ − 0.618034i − 0.618034i
$$93$$ 0 0
$$94$$ −0.618034 −0.618034
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 1.61803i 1.61803i
$$99$$ −0.618034 −0.618034
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$104$$ −0.618034 −0.618034
$$105$$ 0 0
$$106$$ −1.61803 −1.61803
$$107$$ 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
$$111$$ 0 0
$$112$$ 1.61803i 1.61803i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 0.618034i − 0.618034i
$$118$$ − 1.61803i − 1.61803i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −0.618034 −0.618034
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −1.61803 −1.61803
$$127$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$128$$ 1.00000i 1.00000i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$132$$ 0 0
$$133$$ 2.61803i 2.61803i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0.381966i 0.381966i
$$144$$ −1.00000 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ − 1.61803i − 1.61803i
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 1.61803i 1.61803i
$$153$$ 0 0
$$154$$ 1.00000 1.00000
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.00000 −1.00000
$$162$$ − 1.00000i − 1.00000i
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 1.61803 1.61803
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$168$$ 0 0
$$169$$ 0.618034 0.618034
$$170$$ 0 0
$$171$$ −1.61803 −1.61803
$$172$$ 0 0
$$173$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.618034 0.618034
$$177$$ 0 0
$$178$$ 0.618034i 0.618034i
$$179$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 1.00000i 1.00000i
$$183$$ 0 0
$$184$$ −0.618034 −0.618034
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0.618034i 0.618034i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1.61803 1.61803
$$197$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$198$$ 0.618034i 0.618034i
$$199$$ 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
$$206$$ 0.618034 0.618034
$$207$$ − 0.618034i − 0.618034i
$$208$$ 0.618034i 0.618034i
$$209$$ 1.00000 1.00000
$$210$$ 0 0
$$211$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$212$$ 1.61803i 1.61803i
$$213$$ 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 0
$$222$$ 0 0
$$223$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 1.61803 1.61803
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ −0.618034 −0.618034
$$235$$ 0 0
$$236$$ −1.61803 −1.61803
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$242$$ 0.618034i 0.618034i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000i 1.00000i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$252$$ 1.61803i 1.61803i
$$253$$ 0.381966i 0.381966i
$$254$$ −2.00000 −2.00000
$$255$$ 0 0
$$256$$ 1.00000 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −2.61803 −2.61803
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 0.618034i − 0.618034i
$$263$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.61803 2.61803
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$278$$ 0.618034i 0.618034i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0.381966 0.381966
$$287$$ − 2.61803i − 2.61803i
$$288$$ 1.00000i 1.00000i
$$289$$ −1.00000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.61803 −1.61803
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −0.381966 −0.381966
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 1.61803 1.61803
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ − 1.00000i − 1.00000i
$$309$$ 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 $$0.5\pi$$
$$314$$ −2.00000 −2.00000
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1.00000i 1.00000i
$$323$$ 0 0
$$324$$ −1.00000 −1.00000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ − 1.61803i − 1.61803i
$$329$$ 1.00000 1.00000
$$330$$ 0 0
$$331$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$332$$ 0 0
$$333$$ − 1.61803i − 1.61803i
$$334$$ 1.61803 1.61803
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ − 0.618034i − 0.618034i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.61803i 1.61803i
$$343$$ − 1.00000i − 1.00000i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −1.61803 −1.61803
$$347$$ 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
$$352$$ − 0.618034i − 0.618034i
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0.618034 0.618034
$$357$$ 0 0
$$358$$ − 1.61803i − 1.61803i
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 1.61803 1.61803
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 1.00000 1.00000
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$368$$ 0.618034i 0.618034i
$$369$$ 1.61803 1.61803
$$370$$ 0 0
$$371$$ 2.61803 2.61803
$$372$$ 0 0
$$373$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0.618034 0.618034
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 1.61803i − 1.61803i
$$393$$ 0 0
$$394$$ −2.00000 −2.00000
$$395$$ 0 0
$$396$$ 0.618034 0.618034
$$397$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.00000i 1.00000i
$$408$$ 0 0
$$409$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 0.618034i − 0.618034i
$$413$$ 2.61803i 2.61803i
$$414$$ −0.618034 −0.618034
$$415$$ 0 0
$$416$$ 0.618034 0.618034
$$417$$ 0 0
$$418$$ − 1.00000i − 1.00000i
$$419$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 1.61803i 1.61803i
$$423$$ 0.618034i 0.618034i
$$424$$ 1.61803 1.61803
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.00000i 1.00000i
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 1.61803 1.61803
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 2.00000
$$447$$ 0 0
$$448$$ − 1.61803i − 1.61803i
$$449$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$450$$ 0 0
$$451$$ −1.00000 −1.00000
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0.618034i 0.618034i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 1.61803i 1.61803i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.61803i 1.61803i
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ −1.00000 −1.00000
$$482$$ − 0.618034i − 0.618034i
$$483$$ 0 0
$$484$$ 0.618034 0.618034
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 1.00000 1.00000
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 2.00000i − 2.00000i
$$503$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$504$$ 1.61803 1.61803
$$505$$ 0 0
$$506$$ 0.381966 0.381966
$$507$$ 0 0
$$508$$ 2.00000i 2.00000i
$$509$$ 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
$$517$$ − 0.381966i − 0.381966i
$$518$$ 2.61803i 2.61803i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ −0.618034 −0.618034
$$525$$ 0 0
$$526$$ −1.61803 −1.61803
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 0.618034 0.618034
$$530$$ 0 0
$$531$$ −1.61803 −1.61803
$$532$$ − 2.61803i − 2.61803i
$$533$$ − 1.00000i − 1.00000i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.00000 −1.00000
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 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
$$554$$ −0.618034 −0.618034
$$555$$ 0 0
$$556$$ 0.618034 0.618034
$$557$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 0.618034i − 0.618034i
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.61803i 1.61803i
$$568$$ 0 0
$$569$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$570$$ 0 0
$$571$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$572$$ − 0.381966i − 0.381966i
$$573$$ 0 0
$$574$$ −2.61803 −2.61803
$$575$$ 0 0
$$576$$ 1.00000 1.00000
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 1.00000i 1.00000i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 1.00000i − 1.00000i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0.618034 0.618034
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.61803i 1.61803i
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0.381966i 0.381966i
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$608$$ − 1.61803i − 1.61803i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0.381966 0.381966
$$612$$ 0 0
$$613$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −1.00000 −1.00000
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 1.00000i − 1.00000i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 2.00000i 2.00000i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −0.618034 −0.618034
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 1.00000i − 1.00000i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 1.00000 1.00000
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$648$$ 1.00000i 1.00000i
$$649$$ 1.00000 1.00000
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1.61803 −1.61803
$$657$$ 0 0
$$658$$ − 1.00000i − 1.00000i
$$659$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ − 2.00000i − 2.00000i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −1.61803 −1.61803
$$667$$ 0 0
$$668$$ − 1.61803i − 1.61803i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −0.618034 −0.618034
$$677$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 1.61803 1.61803
$$685$$ 0 0
$$686$$ −1.00000 −1.00000
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1.00000 1.00000
$$690$$ 0 0
$$691$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$692$$ 1.61803i 1.61803i
$$693$$ − 1.00000i − 1.00000i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 2.61803i 2.61803i
$$704$$ −0.618034 −0.618034
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 0.618034i − 0.618034i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −1.61803 −1.61803
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −1.00000 −1.00000
$$722$$ − 1.61803i − 1.61803i
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$728$$ − 1.00000i − 1.00000i
$$729$$ −1.00000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$734$$ −2.00000 −2.00000
$$735$$ 0 0
$$736$$ 0.618034 0.618034
$$737$$ 0 0
$$738$$ − 1.61803i − 1.61803i
$$739$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 2.61803i − 2.61803i
$$743$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.61803 −1.61803
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ − 0.618034i − 0.618034i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$758$$ 0.618034i 0.618034i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −1.61803 −1.61803
$$767$$ 1.00000i 1.00000i
$$768$$ 0 0
$$769$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.61803 −2.61803
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −1.61803 −1.61803
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 2.00000i 2.00000i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 0.618034i − 0.618034i
$$793$$ 0 0
$$794$$ 1.61803 1.61803
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0.618034 0.618034
$$802$$ 1.61803i 1.61803i
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$810$$ 0 0
$$811$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 1.00000 1.00000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ − 1.61803i − 1.61803i
$$819$$ 1.00000 1.00000
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$824$$ −0.618034 −0.618034
$$825$$ 0 0
$$826$$ 2.61803 2.61803
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0.618034i 0.618034i
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 0.618034i − 0.618034i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1.00000 −1.00000
$$837$$ 0 0
$$838$$ 2.00000i 2.00000i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 1.00000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 1.61803 1.61803
$$845$$ 0 0
$$846$$ 0.618034 0.618034
$$847$$ − 1.00000i − 1.00000i
$$848$$ − 1.61803i − 1.61803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.00000 −1.00000
$$852$$ 0 0
$$853$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 1.00000 1.00000
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$882$$ − 1.61803i − 1.61803i
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$888$$ 0 0
$$889$$ 3.23607 3.23607
$$890$$ 0 0
$$891$$ 0.618034 0.618034
$$892$$ − 2.00000i − 2.00000i
$$893$$ − 1.00000i − 1.00000i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.61803 −1.61803
$$897$$ 0 0
$$898$$ − 1.61803i − 1.61803i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 1.00000i 1.00000i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.00000i 1.00000i
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 2.00000 2.00000
$$927$$ − 0.618034i − 0.618034i
$$928$$ 0 0
$$929$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$930$$ 0 0
$$931$$ −2.61803 −2.61803
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0.618034 0.618034
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ − 1.00000i − 1.00000i
$$944$$ 1.61803 1.61803
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 1.61803 1.61803
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 1.00000
$$962$$ 1.00000i 1.00000i
$$963$$ 0 0
$$964$$ −0.618034 −0.618034
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$968$$ − 0.618034i − 0.618034i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$972$$ 0 0
$$973$$ − 1.00000i − 1.00000i
$$974$$ −0.618034 −0.618034
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ −0.381966 −0.381966
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1.61803i 1.61803i
$$983$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 1.00000i − 1.00000i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$998$$ 0.618034i 0.618034i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1000.1.g.a.251.2 4
4.3 odd 2 4000.1.g.a.751.1 4
5.2 odd 4 1000.1.e.b.499.1 2
5.3 odd 4 1000.1.e.a.499.2 2
5.4 even 2 inner 1000.1.g.a.251.3 yes 4
8.3 odd 2 inner 1000.1.g.a.251.3 yes 4
8.5 even 2 4000.1.g.a.751.4 4
20.3 even 4 4000.1.e.a.1999.1 2
20.7 even 4 4000.1.e.b.1999.2 2
20.19 odd 2 4000.1.g.a.751.4 4
40.3 even 4 1000.1.e.b.499.1 2
40.13 odd 4 4000.1.e.b.1999.2 2
40.19 odd 2 CM 1000.1.g.a.251.2 4
40.27 even 4 1000.1.e.a.499.2 2
40.29 even 2 4000.1.g.a.751.1 4
40.37 odd 4 4000.1.e.a.1999.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1000.1.e.a.499.2 2 5.3 odd 4
1000.1.e.a.499.2 2 40.27 even 4
1000.1.e.b.499.1 2 5.2 odd 4
1000.1.e.b.499.1 2 40.3 even 4
1000.1.g.a.251.2 4 1.1 even 1 trivial
1000.1.g.a.251.2 4 40.19 odd 2 CM
1000.1.g.a.251.3 yes 4 5.4 even 2 inner
1000.1.g.a.251.3 yes 4 8.3 odd 2 inner
4000.1.e.a.1999.1 2 20.3 even 4
4000.1.e.a.1999.1 2 40.37 odd 4
4000.1.e.b.1999.2 2 20.7 even 4
4000.1.e.b.1999.2 2 40.13 odd 4
4000.1.g.a.751.1 4 4.3 odd 2
4000.1.g.a.751.1 4 40.29 even 2
4000.1.g.a.751.4 4 8.5 even 2
4000.1.g.a.751.4 4 20.19 odd 2