# Properties

 Label 47.1.b.a.46.2 Level $47$ Weight $1$ Character 47.46 Self dual yes Analytic conductor $0.023$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -47 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [47,1,Mod(46,47)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(47, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("47.46");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 47.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.0234560555938$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.2209.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.2209.1

## Embedding invariants

 Embedding label 46.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 47.46

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} -1.61803 q^{3} -0.618034 q^{4} -1.00000 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.61803 q^{9} +O(q^{10})$$ $$q+0.618034 q^{2} -1.61803 q^{3} -0.618034 q^{4} -1.00000 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.61803 q^{9} +1.00000 q^{12} +0.381966 q^{14} -1.61803 q^{17} +1.00000 q^{18} -1.00000 q^{21} +1.61803 q^{24} +1.00000 q^{25} -1.00000 q^{27} -0.381966 q^{28} +1.00000 q^{32} -1.00000 q^{34} -1.00000 q^{36} -1.61803 q^{37} -0.618034 q^{42} +1.00000 q^{47} -0.618034 q^{49} +0.618034 q^{50} +2.61803 q^{51} +0.618034 q^{53} -0.618034 q^{54} -0.618034 q^{56} +0.618034 q^{59} +0.618034 q^{61} +1.00000 q^{63} +0.618034 q^{64} +1.00000 q^{68} -1.61803 q^{71} -1.61803 q^{72} -1.00000 q^{74} -1.61803 q^{75} -1.61803 q^{79} +2.00000 q^{83} +0.618034 q^{84} +0.618034 q^{89} +0.618034 q^{94} -1.61803 q^{96} +0.618034 q^{97} -0.381966 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} + q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 + q^4 - 2 * q^6 - q^7 - 2 * q^8 + q^9 $$2 q - q^{2} - q^{3} + q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + q^{9} + 2 q^{12} + 3 q^{14} - q^{17} + 2 q^{18} - 2 q^{21} + q^{24} + 2 q^{25} - 2 q^{27} - 3 q^{28} + 2 q^{32} - 2 q^{34} - 2 q^{36} - q^{37} + q^{42} + 2 q^{47} + q^{49} - q^{50} + 3 q^{51} - q^{53} + q^{54} + q^{56} - q^{59} - q^{61} + 2 q^{63} - q^{64} + 2 q^{68} - q^{71} - q^{72} - 2 q^{74} - q^{75} - q^{79} + 4 q^{83} - q^{84} - q^{89} - q^{94} - q^{96} - q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^3 + q^4 - 2 * q^6 - q^7 - 2 * q^8 + q^9 + 2 * q^12 + 3 * q^14 - q^17 + 2 * q^18 - 2 * q^21 + q^24 + 2 * q^25 - 2 * q^27 - 3 * q^28 + 2 * q^32 - 2 * q^34 - 2 * q^36 - q^37 + q^42 + 2 * q^47 + q^49 - q^50 + 3 * q^51 - q^53 + q^54 + q^56 - q^59 - q^61 + 2 * q^63 - q^64 + 2 * q^68 - q^71 - q^72 - 2 * q^74 - q^75 - q^79 + 4 * q^83 - q^84 - q^89 - q^94 - q^96 - q^97 - 3 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/47\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$3$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$4$$ −0.618034 −0.618034
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ −1.00000 −1.00000
$$7$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$8$$ −1.00000 −1.00000
$$9$$ 1.61803 1.61803
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000 1.00000
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0.381966 0.381966
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$18$$ 1.00000 1.00000
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −1.00000
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.61803 1.61803
$$25$$ 1.00000 1.00000
$$26$$ 0 0
$$27$$ −1.00000 −1.00000
$$28$$ −0.381966 −0.381966
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 1.00000 1.00000
$$33$$ 0 0
$$34$$ −1.00000 −1.00000
$$35$$ 0 0
$$36$$ −1.00000 −1.00000
$$37$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ −0.618034 −0.618034
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.00000 1.00000
$$48$$ 0 0
$$49$$ −0.618034 −0.618034
$$50$$ 0.618034 0.618034
$$51$$ 2.61803 2.61803
$$52$$ 0 0
$$53$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$54$$ −0.618034 −0.618034
$$55$$ 0 0
$$56$$ −0.618034 −0.618034
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$60$$ 0 0
$$61$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$62$$ 0 0
$$63$$ 1.00000 1.00000
$$64$$ 0.618034 0.618034
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 1.00000 1.00000
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$72$$ −1.61803 −1.61803
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ −1.00000 −1.00000
$$75$$ −1.61803 −1.61803
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$84$$ 0.618034 0.618034
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0.618034 0.618034
$$95$$ 0 0
$$96$$ −1.61803 −1.61803
$$97$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$98$$ −0.381966 −0.381966
$$99$$ 0 0
$$100$$ −0.618034 −0.618034
$$101$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$102$$ 1.61803 1.61803
$$103$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0.381966 0.381966
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0.618034 0.618034
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 2.61803 2.61803
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0.381966 0.381966
$$119$$ −1.00000 −1.00000
$$120$$ 0 0
$$121$$ 1.00000 1.00000
$$122$$ 0.381966 0.381966
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0.618034 0.618034
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ −0.618034 −0.618034
$$129$$ 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
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 1.61803 1.61803
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ −1.61803 −1.61803
$$142$$ −1.00000 −1.00000
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 1.00000
$$148$$ 1.00000 1.00000
$$149$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$150$$ −1.00000 −1.00000
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ −2.61803 −2.61803
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$158$$ −1.00000 −1.00000
$$159$$ −1.00000 −1.00000
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 1.23607 1.23607
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 1.00000 1.00000
$$169$$ 1.00000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$174$$ 0 0
$$175$$ 0.618034 0.618034
$$176$$ 0 0
$$177$$ −1.00000 −1.00000
$$178$$ 0.381966 0.381966
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −1.00000
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −0.618034 −0.618034
$$189$$ −0.618034 −0.618034
$$190$$ 0 0
$$191$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$192$$ −1.00000 −1.00000
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0.381966 0.381966
$$195$$ 0 0
$$196$$ 0.381966 0.381966
$$197$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ −1.00000 −1.00000
$$201$$ 0 0
$$202$$ −1.00000 −1.00000
$$203$$ 0 0
$$204$$ −1.61803 −1.61803
$$205$$ 0 0
$$206$$ −1.00000 −1.00000
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ −0.381966 −0.381966
$$213$$ 2.61803 2.61803
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000 1.00000
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 1.61803 1.61803
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0.618034 0.618034
$$225$$ 1.61803 1.61803
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −0.381966 −0.381966
$$237$$ 2.61803 2.61803
$$238$$ −0.618034 −0.618034
$$239$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$240$$ 0 0
$$241$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$242$$ 0.618034 0.618034
$$243$$ 1.00000 1.00000
$$244$$ −0.381966 −0.381966
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −3.23607 −3.23607
$$250$$ 0 0
$$251$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$252$$ −0.618034 −0.618034
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −1.00000 −1.00000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −1.00000
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0.381966 0.381966
$$263$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.00000 −1.00000
$$268$$ 0 0
$$269$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$270$$ 0 0
$$271$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ −1.00000 −1.00000
$$283$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$284$$ 1.00000 1.00000
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.61803 1.61803
$$289$$ 1.61803 1.61803
$$290$$ 0 0
$$291$$ −1.00000 −1.00000
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0.618034 0.618034
$$295$$ 0 0
$$296$$ 1.61803 1.61803
$$297$$ 0 0
$$298$$ −1.00000 −1.00000
$$299$$ 0 0
$$300$$ 1.00000 1.00000
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.61803 2.61803
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −1.61803 −1.61803
$$307$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$308$$ 0 0
$$309$$ 2.61803 2.61803
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0.381966 0.381966
$$315$$ 0 0
$$316$$ 1.00000 1.00000
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ −0.618034 −0.618034
$$319$$ 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
$$329$$ 0.618034 0.618034
$$330$$ 0 0
$$331$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$332$$ −1.23607 −1.23607
$$333$$ −2.61803 −2.61803
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$338$$ 0.618034 0.618034
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0.381966 0.381966
$$347$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0.381966 0.381966
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$354$$ −0.618034 −0.618034
$$355$$ 0 0
$$356$$ −0.381966 −0.381966
$$357$$ 1.61803 1.61803
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 1.00000 1.00000
$$362$$ 0 0
$$363$$ −1.61803 −1.61803
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −0.618034 −0.618034
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.381966 0.381966
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −1.00000 −1.00000
$$377$$ 0 0
$$378$$ −0.381966 −0.381966
$$379$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1.23607 1.23607
$$383$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$384$$ 1.00000 1.00000
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −0.381966 −0.381966
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0.618034 0.618034
$$393$$ −1.00000 −1.00000
$$394$$ 1.23607 1.23607
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$398$$ 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
$$404$$ 1.00000 1.00000
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −2.61803 −2.61803
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 1.00000 1.00000
$$413$$ 0.381966 0.381966
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 1.61803 1.61803
$$424$$ −0.618034 −0.618034
$$425$$ −1.61803 −1.61803
$$426$$ 1.61803 1.61803
$$427$$ 0.381966 0.381966
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$440$$ 0 0
$$441$$ −1.00000 −1.00000
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −1.61803 −1.61803
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 2.61803 2.61803
$$448$$ 0.381966 0.381966
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 1.00000 1.00000
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$458$$ 0 0
$$459$$ 1.61803 1.61803
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −1.00000 −1.00000
$$472$$ −0.618034 −0.618034
$$473$$ 0 0
$$474$$ 1.61803 1.61803
$$475$$ 0 0
$$476$$ 0.618034 0.618034
$$477$$ 1.00000 1.00000
$$478$$ −1.00000 −1.00000
$$479$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −1.00000 −1.00000
$$483$$ 0 0
$$484$$ −0.618034 −0.618034
$$485$$ 0 0
$$486$$ 0.618034 0.618034
$$487$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$488$$ −0.618034 −0.618034
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.00000 −1.00000
$$498$$ −2.00000 −2.00000
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −1.00000 −1.00000
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ −1.00000 −1.00000
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.61803 −1.61803
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −0.618034 −0.618034
$$519$$ −1.00000 −1.00000
$$520$$ 0 0
$$521$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$522$$ 0 0
$$523$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$524$$ −0.381966 −0.381966
$$525$$ −1.00000 −1.00000
$$526$$ 0.381966 0.381966
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 1.00000
$$530$$ 0 0
$$531$$ 1.00000 1.00000
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −0.618034 −0.618034
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 1.23607 1.23607
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$542$$ −1.00000 −1.00000
$$543$$ 0 0
$$544$$ −1.61803 −1.61803
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 1.00000 1.00000
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1.00000 −1.00000
$$554$$ −1.00000 −1.00000
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 1.00000 1.00000
$$565$$ 0 0
$$566$$ 0.381966 0.381966
$$567$$ 0 0
$$568$$ 1.61803 1.61803
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$572$$ 0 0
$$573$$ −3.23607 −3.23607
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 1.00000
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 1.00000 1.00000
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1.23607 1.23607
$$582$$ −0.618034 −0.618034
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ −0.618034 −0.618034
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −3.23607 −3.23607
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.00000 1.00000
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 1.61803 1.61803
$$601$$ 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
$$606$$ 1.61803 1.61803
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 1.61803 1.61803
$$613$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$614$$ −1.00000 −1.00000
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$618$$ 1.61803 1.61803
$$619$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0.381966 0.381966
$$624$$ 0 0
$$625$$ 1.00000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −0.381966 −0.381966
$$629$$ 2.61803 2.61803
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 1.61803 1.61803
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0.618034 0.618034
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2.61803 −2.61803
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0.381966 0.381966
$$659$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$660$$ 0 0
$$661$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$662$$ 0.381966 0.381966
$$663$$ 0 0
$$664$$ −2.00000 −2.00000
$$665$$ 0 0
$$666$$ −1.61803 −1.61803
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −1.00000 −1.00000
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0.381966 0.381966
$$675$$ −1.00000 −1.00000
$$676$$ −0.618034 −0.618034
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0.381966 0.381966
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −0.618034 −0.618034
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ −0.381966 −0.381966
$$693$$ 0 0
$$694$$ −1.00000 −1.00000
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −0.381966 −0.381966
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0.381966 0.381966
$$707$$ −1.00000 −1.00000
$$708$$ 0.618034 0.618034
$$709$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$710$$ 0 0
$$711$$ −2.61803 −2.61803
$$712$$ −0.618034 −0.618034
$$713$$ 0 0
$$714$$ 1.00000 1.00000
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 2.61803 2.61803
$$718$$ 0 0
$$719$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$720$$ 0 0
$$721$$ −1.00000 −1.00000
$$722$$ 0.618034 0.618034
$$723$$ 2.61803 2.61803
$$724$$ 0 0
$$725$$ 0 0
$$726$$ −1.00000 −1.00000
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ −1.61803 −1.61803
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0.618034 0.618034
$$733$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.236068 0.236068
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 3.23607 3.23607
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 2.61803 2.61803
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0.381966 0.381966
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0.381966 0.381966
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −1.23607 −1.23607
$$765$$ 0 0
$$766$$ −1.00000 −1.00000
$$767$$ 0 0
$$768$$ 1.61803 1.61803
$$769$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −0.618034 −0.618034
$$777$$ 1.61803 1.61803
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −0.618034 −0.618034
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ −1.23607 −1.23607
$$789$$ −1.00000 −1.00000
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −1.00000 −1.00000
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ −1.61803 −1.61803
$$800$$ 1.00000 1.00000
$$801$$ 1.00000 1.00000
$$802$$ −1.00000 −1.00000
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −3.23607 −3.23607
$$808$$ 1.61803 1.61803
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$812$$ 0 0
$$813$$ 2.61803 2.61803
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$824$$ 1.61803 1.61803
$$825$$ 0 0
$$826$$ 0.236068 0.236068
$$827$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 2.61803 2.61803
$$832$$ 0 0
$$833$$ 1.00000 1.00000
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 1.00000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 1.00000 1.00000
$$847$$ 0.618034 0.618034
$$848$$ 0 0
$$849$$ −1.00000 −1.00000
$$850$$ −1.00000 −1.00000
$$851$$ 0 0
$$852$$ −1.61803 −1.61803
$$853$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$854$$ 0.236068 0.236068
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0.381966 0.381966
$$863$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$864$$ −1.00000 −1.00000
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −2.61803 −2.61803
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 1.00000 1.00000
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 1.23607 1.23607
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ −0.618034 −0.618034
$$883$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ −2.61803 −2.61803
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 1.61803 1.61803
$$895$$ 0 0
$$896$$ −0.381966 −0.381966
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −1.00000 −1.00000
$$901$$ −1.00000 −1.00000
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$908$$ 0 0
$$909$$ −2.61803 −2.61803
$$910$$ 0 0
$$911$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −1.00000 −1.00000
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0.381966 0.381966
$$918$$ 1.00000 1.00000
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 2.61803 2.61803
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −1.61803 −1.61803
$$926$$ 0 0
$$927$$ −2.61803 −2.61803
$$928$$ 0 0
$$929$$ −1.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$942$$ −0.618034 −0.618034
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$948$$ −1.61803 −1.61803
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.00000 1.00000
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0.618034 0.618034
$$955$$ 0 0
$$956$$ 1.00000 1.00000
$$957$$ 0 0
$$958$$ 0.381966 0.381966
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 1.00000 1.00000
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$968$$ −1.00000 −1.00000
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ −0.618034 −0.618034
$$973$$ 0 0
$$974$$ 1.23607 1.23607
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0.381966 0.381966
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −1.00000 −1.00000
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0.618034 0.618034 0.309017 0.951057i $$-0.400000\pi$$
0.309017 + 0.951057i $$0.400000\pi$$
$$992$$ 0 0
$$993$$ −1.00000 −1.00000
$$994$$ −0.618034 −0.618034
$$995$$ 0 0
$$996$$ 2.00000 2.00000
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 0 0
$$999$$ 1.61803 1.61803
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 47.1.b.a.46.2 2
3.2 odd 2 423.1.d.a.46.1 2
4.3 odd 2 752.1.g.a.657.2 2
5.2 odd 4 1175.1.b.b.1174.3 4
5.3 odd 4 1175.1.b.b.1174.2 4
5.4 even 2 1175.1.d.c.751.1 2
7.2 even 3 2303.1.f.c.704.1 4
7.3 odd 6 2303.1.f.b.422.1 4
7.4 even 3 2303.1.f.c.422.1 4
7.5 odd 6 2303.1.f.b.704.1 4
7.6 odd 2 2303.1.d.c.2255.2 2
8.3 odd 2 3008.1.g.a.1409.1 2
8.5 even 2 3008.1.g.b.1409.2 2
9.2 odd 6 3807.1.f.a.3430.2 4
9.4 even 3 3807.1.f.b.2161.1 4
9.5 odd 6 3807.1.f.a.2161.2 4
9.7 even 3 3807.1.f.b.3430.1 4
47.2 even 23 2209.1.d.a.1124.2 44
47.3 even 23 2209.1.d.a.1730.2 44
47.4 even 23 2209.1.d.a.172.2 44
47.5 odd 46 2209.1.d.a.116.1 44
47.6 even 23 2209.1.d.a.1609.2 44
47.7 even 23 2209.1.d.a.280.1 44
47.8 even 23 2209.1.d.a.1064.2 44
47.9 even 23 2209.1.d.a.295.1 44
47.10 odd 46 2209.1.d.a.2156.1 44
47.11 odd 46 2209.1.d.a.67.1 44
47.12 even 23 2209.1.d.a.655.2 44
47.13 odd 46 2209.1.d.a.1335.1 44
47.14 even 23 2209.1.d.a.1167.1 44
47.15 odd 46 2209.1.d.a.339.2 44
47.16 even 23 2209.1.d.a.1342.2 44
47.17 even 23 2209.1.d.a.1121.1 44
47.18 even 23 2209.1.d.a.1979.1 44
47.19 odd 46 2209.1.d.a.438.1 44
47.20 odd 46 2209.1.d.a.2138.1 44
47.21 even 23 2209.1.d.a.1580.1 44
47.22 odd 46 2209.1.d.a.2007.2 44
47.23 odd 46 2209.1.d.a.1586.2 44
47.24 even 23 2209.1.d.a.1586.2 44
47.25 even 23 2209.1.d.a.2007.2 44
47.26 odd 46 2209.1.d.a.1580.1 44
47.27 even 23 2209.1.d.a.2138.1 44
47.28 even 23 2209.1.d.a.438.1 44
47.29 odd 46 2209.1.d.a.1979.1 44
47.30 odd 46 2209.1.d.a.1121.1 44
47.31 odd 46 2209.1.d.a.1342.2 44
47.32 even 23 2209.1.d.a.339.2 44
47.33 odd 46 2209.1.d.a.1167.1 44
47.34 even 23 2209.1.d.a.1335.1 44
47.35 odd 46 2209.1.d.a.655.2 44
47.36 even 23 2209.1.d.a.67.1 44
47.37 even 23 2209.1.d.a.2156.1 44
47.38 odd 46 2209.1.d.a.295.1 44
47.39 odd 46 2209.1.d.a.1064.2 44
47.40 odd 46 2209.1.d.a.280.1 44
47.41 odd 46 2209.1.d.a.1609.2 44
47.42 even 23 2209.1.d.a.116.1 44
47.43 odd 46 2209.1.d.a.172.2 44
47.44 odd 46 2209.1.d.a.1730.2 44
47.45 odd 46 2209.1.d.a.1124.2 44
47.46 odd 2 CM 47.1.b.a.46.2 2
141.140 even 2 423.1.d.a.46.1 2
188.187 even 2 752.1.g.a.657.2 2
235.93 even 4 1175.1.b.b.1174.2 4
235.187 even 4 1175.1.b.b.1174.3 4
235.234 odd 2 1175.1.d.c.751.1 2
329.46 odd 6 2303.1.f.c.422.1 4
329.93 odd 6 2303.1.f.c.704.1 4
329.187 even 6 2303.1.f.b.704.1 4
329.234 even 6 2303.1.f.b.422.1 4
329.328 even 2 2303.1.d.c.2255.2 2
376.93 odd 2 3008.1.g.b.1409.2 2
376.187 even 2 3008.1.g.a.1409.1 2
423.140 even 6 3807.1.f.a.2161.2 4
423.187 odd 6 3807.1.f.b.3430.1 4
423.281 even 6 3807.1.f.a.3430.2 4
423.328 odd 6 3807.1.f.b.2161.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
47.1.b.a.46.2 2 1.1 even 1 trivial
47.1.b.a.46.2 2 47.46 odd 2 CM
423.1.d.a.46.1 2 3.2 odd 2
423.1.d.a.46.1 2 141.140 even 2
752.1.g.a.657.2 2 4.3 odd 2
752.1.g.a.657.2 2 188.187 even 2
1175.1.b.b.1174.2 4 5.3 odd 4
1175.1.b.b.1174.2 4 235.93 even 4
1175.1.b.b.1174.3 4 5.2 odd 4
1175.1.b.b.1174.3 4 235.187 even 4
1175.1.d.c.751.1 2 5.4 even 2
1175.1.d.c.751.1 2 235.234 odd 2
2209.1.d.a.67.1 44 47.11 odd 46
2209.1.d.a.67.1 44 47.36 even 23
2209.1.d.a.116.1 44 47.5 odd 46
2209.1.d.a.116.1 44 47.42 even 23
2209.1.d.a.172.2 44 47.4 even 23
2209.1.d.a.172.2 44 47.43 odd 46
2209.1.d.a.280.1 44 47.7 even 23
2209.1.d.a.280.1 44 47.40 odd 46
2209.1.d.a.295.1 44 47.9 even 23
2209.1.d.a.295.1 44 47.38 odd 46
2209.1.d.a.339.2 44 47.15 odd 46
2209.1.d.a.339.2 44 47.32 even 23
2209.1.d.a.438.1 44 47.19 odd 46
2209.1.d.a.438.1 44 47.28 even 23
2209.1.d.a.655.2 44 47.12 even 23
2209.1.d.a.655.2 44 47.35 odd 46
2209.1.d.a.1064.2 44 47.8 even 23
2209.1.d.a.1064.2 44 47.39 odd 46
2209.1.d.a.1121.1 44 47.17 even 23
2209.1.d.a.1121.1 44 47.30 odd 46
2209.1.d.a.1124.2 44 47.2 even 23
2209.1.d.a.1124.2 44 47.45 odd 46
2209.1.d.a.1167.1 44 47.14 even 23
2209.1.d.a.1167.1 44 47.33 odd 46
2209.1.d.a.1335.1 44 47.13 odd 46
2209.1.d.a.1335.1 44 47.34 even 23
2209.1.d.a.1342.2 44 47.16 even 23
2209.1.d.a.1342.2 44 47.31 odd 46
2209.1.d.a.1580.1 44 47.21 even 23
2209.1.d.a.1580.1 44 47.26 odd 46
2209.1.d.a.1586.2 44 47.23 odd 46
2209.1.d.a.1586.2 44 47.24 even 23
2209.1.d.a.1609.2 44 47.6 even 23
2209.1.d.a.1609.2 44 47.41 odd 46
2209.1.d.a.1730.2 44 47.3 even 23
2209.1.d.a.1730.2 44 47.44 odd 46
2209.1.d.a.1979.1 44 47.18 even 23
2209.1.d.a.1979.1 44 47.29 odd 46
2209.1.d.a.2007.2 44 47.22 odd 46
2209.1.d.a.2007.2 44 47.25 even 23
2209.1.d.a.2138.1 44 47.20 odd 46
2209.1.d.a.2138.1 44 47.27 even 23
2209.1.d.a.2156.1 44 47.10 odd 46
2209.1.d.a.2156.1 44 47.37 even 23
2303.1.d.c.2255.2 2 7.6 odd 2
2303.1.d.c.2255.2 2 329.328 even 2
2303.1.f.b.422.1 4 7.3 odd 6
2303.1.f.b.422.1 4 329.234 even 6
2303.1.f.b.704.1 4 7.5 odd 6
2303.1.f.b.704.1 4 329.187 even 6
2303.1.f.c.422.1 4 7.4 even 3
2303.1.f.c.422.1 4 329.46 odd 6
2303.1.f.c.704.1 4 7.2 even 3
2303.1.f.c.704.1 4 329.93 odd 6
3008.1.g.a.1409.1 2 8.3 odd 2
3008.1.g.a.1409.1 2 376.187 even 2
3008.1.g.b.1409.2 2 8.5 even 2
3008.1.g.b.1409.2 2 376.93 odd 2
3807.1.f.a.2161.2 4 9.5 odd 6
3807.1.f.a.2161.2 4 423.140 even 6
3807.1.f.a.3430.2 4 9.2 odd 6
3807.1.f.a.3430.2 4 423.281 even 6
3807.1.f.b.2161.1 4 9.4 even 3
3807.1.f.b.2161.1 4 423.328 odd 6
3807.1.f.b.3430.1 4 9.7 even 3
3807.1.f.b.3430.1 4 423.187 odd 6