# Properties

 Label 4000.1.b.a.2751.2 Level $4000$ Weight $1$ Character 4000.2751 Analytic conductor $1.996$ Analytic rank $0$ Dimension $4$ Projective image $A_{5}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{5}$$ Projective field Galois closure of 5.1.1000000.2

## Embedding invariants

 Embedding label 2751.2 Root $$1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 4000.2751 Dual form 4000.1.b.a.2751.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.618034i q^{3} +1.00000i q^{7} +0.618034 q^{9} +O(q^{10})$$ $$q-0.618034i q^{3} +1.00000i q^{7} +0.618034 q^{9} +1.00000i q^{11} +0.618034 q^{13} -1.00000 q^{17} +0.618034i q^{19} +0.618034 q^{21} -1.00000i q^{27} -1.00000 q^{29} +1.61803i q^{31} +0.618034 q^{33} -0.381966i q^{39} +1.00000 q^{41} +1.00000i q^{43} +1.61803i q^{47} +0.618034i q^{51} -1.61803 q^{53} +0.381966 q^{57} -1.61803i q^{59} +0.618034 q^{61} +0.618034i q^{63} -1.61803i q^{67} +1.00000i q^{71} -1.61803 q^{73} -1.00000 q^{77} +1.00000i q^{79} +0.618034i q^{87} +0.618034i q^{91} +1.00000 q^{93} +1.61803 q^{97} +0.618034i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{9} + O(q^{10})$$ $$4q - 2q^{9} - 2q^{13} - 4q^{17} - 2q^{21} - 4q^{29} - 2q^{33} + 4q^{41} - 2q^{53} + 6q^{57} - 2q^{61} - 2q^{73} - 4q^{77} + 4q^{93} + 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$8$$ 0 0
$$9$$ 0.618034 0.618034
$$10$$ 0 0
$$11$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$12$$ 0 0
$$13$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$18$$ 0 0
$$19$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$20$$ 0 0
$$21$$ 0.618034 0.618034
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 1.00000i
$$28$$ 0 0
$$29$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$30$$ 0 0
$$31$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$32$$ 0 0
$$33$$ 0.618034 0.618034
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ − 0.381966i − 0.381966i
$$40$$ 0 0
$$41$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$42$$ 0 0
$$43$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.618034i 0.618034i
$$52$$ 0 0
$$53$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.381966 0.381966
$$58$$ 0 0
$$59$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$60$$ 0 0
$$61$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$62$$ 0 0
$$63$$ 0.618034i 0.618034i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$72$$ 0 0
$$73$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.00000 −1.00000
$$78$$ 0 0
$$79$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.618034i 0.618034i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0.618034i 0.618034i
$$92$$ 0 0
$$93$$ 1.00000 1.00000
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$98$$ 0 0
$$99$$ 0.618034i 0.618034i
$$100$$ 0 0
$$101$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$102$$ 0 0
$$103$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$108$$ 0 0
$$109$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.381966 0.381966
$$118$$ 0 0
$$119$$ − 1.00000i − 1.00000i
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ − 0.618034i − 0.618034i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0.618034 0.618034
$$130$$ 0 0
$$131$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$132$$ 0 0
$$133$$ −0.618034 −0.618034
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$138$$ 0 0
$$139$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$140$$ 0 0
$$141$$ 1.00000 1.00000
$$142$$ 0 0
$$143$$ 0.618034i 0.618034i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$150$$ 0 0
$$151$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$152$$ 0 0
$$153$$ −0.618034 −0.618034
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$158$$ 0 0
$$159$$ 1.00000i 1.00000i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ −0.618034 −0.618034
$$170$$ 0 0
$$171$$ 0.381966i 0.381966i
$$172$$ 0 0
$$173$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.00000 −1.00000
$$178$$ 0 0
$$179$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ − 0.381966i − 0.381966i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 1.00000i − 1.00000i
$$188$$ 0 0
$$189$$ 1.00000 1.00000
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ −1.00000 −1.00000
$$202$$ 0 0
$$203$$ − 1.00000i − 1.00000i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −0.618034 −0.618034
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0.618034 0.618034
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.61803 −1.61803
$$218$$ 0 0
$$219$$ 1.00000i 1.00000i
$$220$$ 0 0
$$221$$ −0.618034 −0.618034
$$222$$ 0 0
$$223$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$228$$ 0 0
$$229$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$230$$ 0 0
$$231$$ 0.618034i 0.618034i
$$232$$ 0 0
$$233$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0.618034 0.618034
$$238$$ 0 0
$$239$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.381966i 0.381966i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −0.618034 −0.618034
$$262$$ 0 0
$$263$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$270$$ 0 0
$$271$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$272$$ 0 0
$$273$$ 0.381966 0.381966
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$278$$ 0 0
$$279$$ 1.00000i 1.00000i
$$280$$ 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
$$287$$ 1.00000i 1.00000i
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ − 1.00000i − 1.00000i
$$292$$ 0 0
$$293$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.00000 1.00000
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1.00000 −1.00000
$$302$$ 0 0
$$303$$ − 0.618034i − 0.618034i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$308$$ 0 0
$$309$$ −0.618034 −0.618034
$$310$$ 0 0
$$311$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ − 1.00000i − 1.00000i
$$320$$ 0 0
$$321$$ −0.381966 −0.381966
$$322$$ 0 0
$$323$$ − 0.618034i − 0.618034i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 0.381966i − 0.381966i
$$328$$ 0 0
$$329$$ −1.61803 −1.61803
$$330$$ 0 0
$$331$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$338$$ 0 0
$$339$$ − 0.618034i − 0.618034i
$$340$$ 0 0
$$341$$ −1.61803 −1.61803
$$342$$ 0 0
$$343$$ 1.00000i 1.00000i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$348$$ 0 0
$$349$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$350$$ 0 0
$$351$$ − 0.618034i − 0.618034i
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −0.618034 −0.618034
$$358$$ 0 0
$$359$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$360$$ 0 0
$$361$$ 0.618034 0.618034
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$368$$ 0 0
$$369$$ 0.618034 0.618034
$$370$$ 0 0
$$371$$ − 1.61803i − 1.61803i
$$372$$ 0 0
$$373$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.618034 −0.618034
$$378$$ 0 0
$$379$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.618034i 0.618034i
$$388$$ 0 0
$$389$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −0.381966 −0.381966
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$398$$ 0 0
$$399$$ 0.381966i 0.381966i
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 1.00000i 1.00000i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ − 0.381966i − 0.381966i
$$412$$ 0 0
$$413$$ 1.61803 1.61803
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0.618034 0.618034
$$418$$ 0 0
$$419$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$420$$ 0 0
$$421$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$422$$ 0 0
$$423$$ 1.00000i 1.00000i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.618034i 0.618034i
$$428$$ 0 0
$$429$$ 0.381966 0.381966
$$430$$ 0 0
$$431$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0.618034i 0.618034i
$$448$$ 0 0
$$449$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$450$$ 0 0
$$451$$ 1.00000i 1.00000i
$$452$$ 0 0
$$453$$ −1.00000 −1.00000
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 1.00000i 1.00000i
$$460$$ 0 0
$$461$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$462$$ 0 0
$$463$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 1.61803 1.61803
$$470$$ 0 0
$$471$$ − 0.381966i − 0.381966i
$$472$$ 0 0
$$473$$ −1.00000 −1.00000
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −1.00000 −1.00000
$$478$$ 0 0
$$479$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 1.00000 1.00000
$$490$$ 0 0
$$491$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$492$$ 0 0
$$493$$ 1.00000 1.00000
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.00000 −1.00000
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0.381966i 0.381966i
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ − 1.61803i − 1.61803i
$$512$$ 0 0
$$513$$ 0.618034 0.618034
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.61803 −1.61803
$$518$$ 0 0
$$519$$ − 0.618034i − 0.618034i
$$520$$ 0 0
$$521$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1.61803i − 1.61803i
$$528$$ 0 0
$$529$$ 1.00000 1.00000
$$530$$ 0 0
$$531$$ − 1.00000i − 1.00000i
$$532$$ 0 0
$$533$$ 0.618034 0.618034
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −0.618034 −0.618034
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.61803i 1.61803i 0.587785 + 0.809017i $$0.300000\pi$$
−0.587785 + 0.809017i $$0.700000\pi$$
$$548$$ 0 0
$$549$$ 0.381966 0.381966
$$550$$ 0 0
$$551$$ − 0.618034i − 0.618034i
$$552$$ 0 0
$$553$$ −1.00000 −1.00000
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0.618034i 0.618034i
$$560$$ 0 0
$$561$$ −0.618034 −0.618034
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$578$$ 0 0
$$579$$ − 1.00000i − 1.00000i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 1.61803i − 1.61803i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ −1.00000 −1.00000
$$590$$ 0 0
$$591$$ − 0.618034i − 0.618034i
$$592$$ 0 0
$$593$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$602$$ 0 0
$$603$$ − 1.00000i − 1.00000i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −0.618034 −0.618034
$$610$$ 0 0
$$611$$ 1.00000i 1.00000i
$$612$$ 0 0
$$613$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$618$$ 0 0
$$619$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0.381966i 0.381966i
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0.618034i 0.618034i
$$640$$ 0 0
$$641$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$642$$ 0 0
$$643$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$648$$ 0 0
$$649$$ 1.61803 1.61803
$$650$$ 0 0
$$651$$ 1.00000i 1.00000i
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −1.00000 −1.00000
$$658$$ 0 0
$$659$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 0 0
$$663$$ 0.381966i 0.381966i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −1.00000 −1.00000
$$670$$ 0 0
$$671$$ 0.618034i 0.618034i
$$672$$ 0 0
$$673$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 1.61803i 1.61803i
$$680$$ 0 0
$$681$$ −1.00000 −1.00000
$$682$$ 0 0
$$683$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 1.00000i − 1.00000i
$$688$$ 0 0
$$689$$ −1.00000 −1.00000
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ −0.618034 −0.618034
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.00000 −1.00000
$$698$$ 0 0
$$699$$ 0.618034i 0.618034i
$$700$$ 0 0
$$701$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.00000i 1.00000i
$$708$$ 0 0
$$709$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$710$$ 0 0
$$711$$ 0.618034i 0.618034i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −1.00000 −1.00000
$$718$$ 0 0
$$719$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$720$$ 0 0
$$721$$ 1.00000 1.00000
$$722$$ 0 0
$$723$$ 0.618034i 0.618034i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ −0.618034 −0.618034
$$730$$ 0 0
$$731$$ − 1.00000i − 1.00000i
$$732$$ 0 0
$$733$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.61803 1.61803
$$738$$ 0 0
$$739$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$740$$ 0 0
$$741$$ 0.236068 0.236068
$$742$$ 0 0
$$743$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0.618034 0.618034
$$750$$ 0 0
$$751$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$752$$ 0 0
$$753$$ 0.381966 0.381966
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0.618034i 0.618034i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.00000i − 1.00000i
$$768$$ 0 0
$$769$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$770$$ 0 0
$$771$$ − 0.618034i − 0.618034i
$$772$$ 0 0
$$773$$ 0 0 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
$$779$$ 0.618034i 0.618034i
$$780$$ 0 0
$$781$$ −1.00000 −1.00000
$$782$$ 0 0
$$783$$ 1.00000i 1.00000i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$788$$ 0 0
$$789$$ 1.00000 1.00000
$$790$$ 0 0
$$791$$ 1.00000i 1.00000i
$$792$$ 0 0
$$793$$ 0.381966 0.381966
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$798$$ 0 0
$$799$$ − 1.61803i − 1.61803i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 1.61803i − 1.61803i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0.381966i 0.381966i
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0.381966 0.381966
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −0.618034 −0.618034
$$818$$ 0 0
$$819$$ 0.381966i 0.381966i
$$820$$ 0 0
$$821$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$822$$ 0 0
$$823$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$828$$ 0 0
$$829$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$830$$ 0 0
$$831$$ 0.618034i 0.618034i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.61803 1.61803
$$838$$ 0 0
$$839$$ − 0.618034i − 0.618034i −0.951057 0.309017i $$-0.900000\pi$$
0.951057 0.309017i $$-0.100000\pi$$
$$840$$ 0 0
$$841$$ 0 0
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0.618034 0.618034
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −1.00000 −1.00000
$$870$$ 0 0
$$871$$ − 1.00000i − 1.00000i
$$872$$ 0 0
$$873$$ 1.00000 1.00000
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$878$$ 0 0
$$879$$ 0.618034i 0.618034i
$$880$$ 0 0
$$881$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$882$$ 0 0
$$883$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 2.00000i − 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −1.00000 −1.00000
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 1.61803i − 1.61803i
$$900$$ 0 0
$$901$$ 1.61803 1.61803
$$902$$ 0 0
$$903$$ 0.618034i 0.618034i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0.618034i 0.618034i 0.951057 + 0.309017i $$0.100000\pi$$
−0.951057 + 0.309017i $$0.900000\pi$$
$$908$$ 0 0
$$909$$ 0.618034 0.618034
$$910$$ 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
$$917$$ 0.618034 0.618034
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0.381966 0.381966
$$922$$ 0 0
$$923$$ 0.618034i 0.618034i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 0.618034i − 0.618034i
$$928$$ 0 0
$$929$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.00000 −1.00000
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$948$$ 0 0
$$949$$ −1.00000 −1.00000
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −0.618034 −0.618034
$$958$$ 0 0
$$959$$ 0.618034i 0.618034i
$$960$$ 0 0
$$961$$ −1.61803 −1.61803
$$962$$ 0 0
$$963$$ − 0.381966i − 0.381966i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.61803i − 1.61803i −0.587785 0.809017i $$-0.700000\pi$$
0.587785 0.809017i $$-0.300000\pi$$
$$968$$ 0 0
$$969$$ −0.381966 −0.381966
$$970$$ 0 0
$$971$$ − 1.00000i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$972$$ 0 0
$$973$$ −1.00000 −1.00000
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0.381966 0.381966
$$982$$ 0 0
$$983$$ 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
$$987$$ 1.00000i 1.00000i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 1.00000i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$992$$ 0 0
$$993$$ −1.00000 −1.00000
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −0.618034 −0.618034 −0.309017 0.951057i $$-0.600000\pi$$
−0.309017 + 0.951057i $$0.600000\pi$$
$$998$$ 0 0
$$999$$ 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 4000.1.b.a.2751.2 4
4.3 odd 2 inner 4000.1.b.a.2751.3 yes 4
5.2 odd 4 4000.1.h.b.3999.2 4
5.3 odd 4 4000.1.h.a.3999.4 4
5.4 even 2 4000.1.b.b.2751.3 yes 4
20.3 even 4 4000.1.h.b.3999.1 4
20.7 even 4 4000.1.h.a.3999.3 4
20.19 odd 2 4000.1.b.b.2751.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4000.1.b.a.2751.2 4 1.1 even 1 trivial
4000.1.b.a.2751.3 yes 4 4.3 odd 2 inner
4000.1.b.b.2751.2 yes 4 20.19 odd 2
4000.1.b.b.2751.3 yes 4 5.4 even 2
4000.1.h.a.3999.3 4 20.7 even 4
4000.1.h.a.3999.4 4 5.3 odd 4
4000.1.h.b.3999.1 4 20.3 even 4
4000.1.h.b.3999.2 4 5.2 odd 4