# Properties

 Label 87.1.d.a.86.1 Level $87$ Weight $1$ Character 87.86 Self dual yes Analytic conductor $0.043$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -87 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,1,Mod(86,87)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("87.86");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$87 = 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 87.d (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: $$0.0434186560991$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.87.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.87.1

## Embedding invariants

 Embedding label 86.1 Character $$\chi$$ $$=$$ 87.86

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -1.00000 q^{33} +1.00000 q^{34} -1.00000 q^{39} +2.00000 q^{41} +1.00000 q^{42} -1.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{54} -1.00000 q^{56} -1.00000 q^{58} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -1.00000 q^{67} +1.00000 q^{72} +1.00000 q^{75} +1.00000 q^{77} +1.00000 q^{78} +1.00000 q^{81} -2.00000 q^{82} +1.00000 q^{87} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{91} +1.00000 q^{94} -1.00000 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$59$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$3$$ 1.00000 1.00000
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ −1.00000 −1.00000
$$7$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$8$$ 1.00000 1.00000
$$9$$ 1.00000 1.00000
$$10$$ 0 0
$$11$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$14$$ 1.00000 1.00000
$$15$$ 0 0
$$16$$ −1.00000 −1.00000
$$17$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$18$$ −1.00000 −1.00000
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −1.00000
$$22$$ 1.00000 1.00000
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.00000 1.00000
$$25$$ 1.00000 1.00000
$$26$$ 1.00000 1.00000
$$27$$ 1.00000 1.00000
$$28$$ 0 0
$$29$$ 1.00000 1.00000
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −1.00000
$$34$$ 1.00000 1.00000
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −1.00000
$$40$$ 0 0
$$41$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$42$$ 1.00000 1.00000
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$48$$ −1.00000 −1.00000
$$49$$ 0 0
$$50$$ −1.00000 −1.00000
$$51$$ −1.00000 −1.00000
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ −1.00000 −1.00000
$$55$$ 0 0
$$56$$ −1.00000 −1.00000
$$57$$ 0 0
$$58$$ −1.00000 −1.00000
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −1.00000
$$64$$ 1.00000 1.00000
$$65$$ 0 0
$$66$$ 1.00000 1.00000
$$67$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 1.00000 1.00000
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 1.00000 1.00000
$$76$$ 0 0
$$77$$ 1.00000 1.00000
$$78$$ 1.00000 1.00000
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 1.00000 1.00000
$$82$$ −2.00000 −2.00000
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.00000 1.00000
$$88$$ −1.00000 −1.00000
$$89$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$90$$ 0 0
$$91$$ 1.00000 1.00000
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 1.00000 1.00000
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −1.00000
$$100$$ 0 0
$$101$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$102$$ 1.00000 1.00000
$$103$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$104$$ −1.00000 −1.00000
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 1.00000
$$113$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −1.00000
$$118$$ 0 0
$$119$$ 1.00000 1.00000
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 2.00000 2.00000
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 1.00000 1.00000
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ −1.00000 −1.00000
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1.00000 1.00000
$$135$$ 0 0
$$136$$ −1.00000 −1.00000
$$137$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$138$$ 0 0
$$139$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −1.00000
$$142$$ 0 0
$$143$$ 1.00000 1.00000
$$144$$ −1.00000 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ −1.00000 −1.00000
$$151$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$152$$ 0 0
$$153$$ −1.00000 −1.00000
$$154$$ −1.00000 −1.00000
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −1.00000
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ −1.00000 −1.00000
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ −1.00000 −1.00000
$$175$$ −1.00000 −1.00000
$$176$$ 1.00000 1.00000
$$177$$ 0 0
$$178$$ 1.00000 1.00000
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$182$$ −1.00000 −1.00000
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.00000 1.00000
$$188$$ 0 0
$$189$$ −1.00000 −1.00000
$$190$$ 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 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 1.00000 1.00000
$$199$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$200$$ 1.00000 1.00000
$$201$$ −1.00000 −1.00000
$$202$$ 1.00000 1.00000
$$203$$ −1.00000 −1.00000
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.00000 −2.00000
$$207$$ 0 0
$$208$$ 1.00000 1.00000
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000 1.00000
$$217$$ 0 0
$$218$$ 1.00000 1.00000
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.00000 1.00000
$$222$$ 0 0
$$223$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$224$$ 0 0
$$225$$ 1.00000 1.00000
$$226$$ 1.00000 1.00000
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 1.00000 1.00000
$$232$$ 1.00000 1.00000
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 1.00000 1.00000
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −1.00000 −1.00000
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$242$$ 0 0
$$243$$ 1.00000 1.00000
$$244$$ 0 0
$$245$$ 0 0
$$246$$ −2.00000 −2.00000
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1.00000 1.00000
$$262$$ 1.00000 1.00000
$$263$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$264$$ −1.00000 −1.00000
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.00000 −1.00000
$$268$$ 0 0
$$269$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 1.00000 1.00000
$$273$$ 1.00000 1.00000
$$274$$ −2.00000 −2.00000
$$275$$ −1.00000 −1.00000
$$276$$ 0 0
$$277$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$278$$ 1.00000 1.00000
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 1.00000 1.00000
$$283$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −1.00000 −1.00000
$$287$$ −2.00000 −2.00000
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$301$$ 0 0
$$302$$ −2.00000 −2.00000
$$303$$ −1.00000 −1.00000
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 1.00000 1.00000
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 2.00000 2.00000
$$310$$ 0 0
$$311$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$312$$ −1.00000 −1.00000
$$313$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$318$$ 0 0
$$319$$ −1.00000 −1.00000
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −1.00000 −1.00000
$$326$$ 0 0
$$327$$ −1.00000 −1.00000
$$328$$ 2.00000 2.00000
$$329$$ 1.00000 1.00000
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 1.00000 1.00000
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ −1.00000 −1.00000
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$350$$ 1.00000 1.00000
$$351$$ −1.00000 −1.00000
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.00000 1.00000
$$358$$ 0 0
$$359$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$360$$ 0 0
$$361$$ 1.00000 1.00000
$$362$$ 1.00000 1.00000
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 2.00000 2.00000
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$374$$ −1.00000 −1.00000
$$375$$ 0 0
$$376$$ −1.00000 −1.00000
$$377$$ −1.00000 −1.00000
$$378$$ 1.00000 1.00000
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −2.00000 −2.00000
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −1.00000 −1.00000
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 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
$$393$$ −1.00000 −1.00000
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$398$$ 1.00000 1.00000
$$399$$ 0 0
$$400$$ −1.00000 −1.00000
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 1.00000 1.00000
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 1.00000 1.00000
$$407$$ 0 0
$$408$$ −1.00000 −1.00000
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 2.00000 2.00000
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1.00000 −1.00000
$$418$$ 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
$$423$$ −1.00000 −1.00000
$$424$$ 0 0
$$425$$ −1.00000 −1.00000
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 1.00000 1.00000
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −1.00000 −1.00000
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −1.00000 −1.00000
$$443$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 1.00000 1.00000
$$447$$ 0 0
$$448$$ −1.00000 −1.00000
$$449$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$450$$ −1.00000 −1.00000
$$451$$ −2.00000 −2.00000
$$452$$ 0 0
$$453$$ 2.00000 2.00000
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −1.00000
$$460$$ 0 0
$$461$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$462$$ −1.00000 −1.00000
$$463$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$464$$ −1.00000 −1.00000
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$468$$ 0 0
$$469$$ 1.00000 1.00000
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 1.00000 1.00000
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ −1.00000 −1.00000
$$487$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$492$$ 0 0
$$493$$ −1.00000 −1.00000
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 1.00000 1.00000
$$503$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$504$$ −1.00000 −1.00000
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 1.00000
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.00000 1.00000
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ −1.00000 −1.00000
$$523$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −1.00000
$$526$$ −2.00000 −2.00000
$$527$$ 0 0
$$528$$ 1.00000 1.00000
$$529$$ 1.00000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.00000 −2.00000
$$534$$ 1.00000 1.00000
$$535$$ 0 0
$$536$$ −1.00000 −1.00000
$$537$$ 0 0
$$538$$ 1.00000 1.00000
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ −1.00000 −1.00000
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −1.00000 −1.00000
$$547$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 1.00000 1.00000
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 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
$$561$$ 1.00000 1.00000
$$562$$ 0 0
$$563$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −2.00000 −2.00000
$$567$$ −1.00000 −1.00000
$$568$$ 0 0
$$569$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$570$$ 0 0
$$571$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$572$$ 0 0
$$573$$ 2.00000 2.00000
$$574$$ 2.00000 2.00000
$$575$$ 0 0
$$576$$ 1.00000 1.00000
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 1.00000 1.00000
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 1.00000 1.00000
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.00000 −1.00000
$$598$$ 0 0
$$599$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$600$$ 1.00000 1.00000
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −1.00000 −1.00000
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 1.00000 1.00000
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −1.00000 −1.00000
$$610$$ 0 0
$$611$$ 1.00000 1.00000
$$612$$ 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
$$616$$ 1.00000 1.00000
$$617$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$618$$ −2.00000 −2.00000
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 1.00000 1.00000
$$623$$ 1.00000 1.00000
$$624$$ 1.00000 1.00000
$$625$$ 1.00000 1.00000
$$626$$ 1.00000 1.00000
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 1.00000 1.00000
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.00000 1.00000
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$642$$ 0 0
$$643$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 1.00000 1.00000
$$649$$ 0 0
$$650$$ 1.00000 1.00000
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$654$$ 1.00000 1.00000
$$655$$ 0 0
$$656$$ −2.00000 −2.00000
$$657$$ 0 0
$$658$$ −1.00000 −1.00000
$$659$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$660$$ 0 0
$$661$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$662$$ 0 0
$$663$$ 1.00000 1.00000
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −1.00000 −1.00000
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$674$$ 0 0
$$675$$ 1.00000 1.00000
$$676$$ 0 0
$$677$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$678$$ 1.00000 1.00000
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −1.00000
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$692$$ 0 0
$$693$$ 1.00000 1.00000
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 1.00000 1.00000
$$697$$ −2.00000 −2.00000
$$698$$ −2.00000 −2.00000
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 1.00000 1.00000
$$703$$ 0 0
$$704$$ −1.00000 −1.00000
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.00000 1.00000
$$708$$ 0 0
$$709$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −1.00000 −1.00000
$$713$$ 0 0
$$714$$ −1.00000 −1.00000
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −2.00000 −2.00000
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −2.00000 −2.00000
$$722$$ −1.00000 −1.00000
$$723$$ −1.00000 −1.00000
$$724$$ 0 0
$$725$$ 1.00000 1.00000
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 1.00000 1.00000
$$729$$ 1.00000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.00000 1.00000
$$738$$ −2.00000 −2.00000
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −2.00000 −2.00000
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 1.00000 1.00000
$$753$$ −1.00000 −1.00000
$$754$$ 1.00000 1.00000
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 1.00000 1.00000
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 1.00000 1.00000
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 1.00000 1.00000
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 1.00000 1.00000
$$787$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$788$$ 0 0
$$789$$ 2.00000 2.00000
$$790$$ 0 0
$$791$$ 1.00000 1.00000
$$792$$ −1.00000 −1.00000
$$793$$ 0 0
$$794$$ −2.00000 −2.00000
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$798$$ 0 0
$$799$$ 1.00000 1.00000
$$800$$ 0 0
$$801$$ −1.00000 −1.00000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.00000 −1.00000
$$808$$ −1.00000 −1.00000
$$809$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$810$$ 0 0
$$811$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 1.00000 1.00000
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 1.00000 1.00000
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ −2.00000 −2.00000
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 2.00000 2.00000
$$825$$ −1.00000 −1.00000
$$826$$ 0 0
$$827$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ −1.00000 −1.00000
$$832$$ −1.00000 −1.00000
$$833$$ 0 0
$$834$$ 1.00000 1.00000
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$840$$ 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 0
$$848$$ 0 0
$$849$$ 2.00000 2.00000
$$850$$ 1.00000 1.00000
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ −1.00000 −1.00000
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −2.00000
$$862$$ 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
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 1.00000 1.00000
$$872$$ −1.00000 −1.00000
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$878$$ 1.00000 1.00000
$$879$$ −1.00000 −1.00000
$$880$$ 0 0
$$881$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$882$$ 0 0
$$883$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 1.00000 1.00000
$$887$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.00000 −1.00000
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 1.00000
$$897$$ 0 0
$$898$$ 1.00000 1.00000
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 2.00000 2.00000
$$903$$ 0 0
$$904$$ −1.00000 −1.00000
$$905$$ 0 0
$$906$$ −2.00000 −2.00000
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ −1.00000 −1.00000
$$910$$ 0 0
$$911$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 1.00000 1.00000
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.00000 1.00000
$$918$$ 1.00000 1.00000
$$919$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −2.00000 −2.00000
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 1.00000 1.00000
$$927$$ 2.00000 2.00000
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.00000 −1.00000
$$934$$ −2.00000 −2.00000
$$935$$ 0 0
$$936$$ −1.00000 −1.00000
$$937$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$938$$ −1.00000 −1.00000
$$939$$ −1.00000 −1.00000
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −1.00000 −1.00000
$$952$$ 1.00000 1.00000
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −1.00000 −1.00000
$$958$$ −2.00000 −2.00000
$$959$$ −2.00000 −2.00000
$$960$$ 0 0
$$961$$ 1.00000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$972$$ 0 0
$$973$$ 1.00000 1.00000
$$974$$ −2.00000 −2.00000
$$975$$ −1.00000 −1.00000
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 1.00000 1.00000
$$980$$ 0 0
$$981$$ −1.00000 −1.00000
$$982$$ −2.00000 −2.00000
$$983$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$984$$ 2.00000 2.00000
$$985$$ 0 0
$$986$$ 1.00000 1.00000
$$987$$ 1.00000 1.00000
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 1.00000 1.00000
$$999$$ 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 87.1.d.a.86.1 1
3.2 odd 2 87.1.d.b.86.1 yes 1
4.3 odd 2 1392.1.i.a.1217.1 1
5.2 odd 4 2175.1.b.a.2174.1 2
5.3 odd 4 2175.1.b.a.2174.2 2
5.4 even 2 2175.1.h.b.1826.1 1
9.2 odd 6 2349.1.h.a.782.1 2
9.4 even 3 2349.1.h.b.1565.1 2
9.5 odd 6 2349.1.h.a.1565.1 2
9.7 even 3 2349.1.h.b.782.1 2
12.11 even 2 1392.1.i.b.1217.1 1
15.2 even 4 2175.1.b.b.2174.2 2
15.8 even 4 2175.1.b.b.2174.1 2
15.14 odd 2 2175.1.h.a.1826.1 1
29.2 odd 28 2523.1.j.b.605.1 12
29.3 odd 28 2523.1.j.b.1412.1 12
29.4 even 14 2523.1.h.a.2333.1 6
29.5 even 14 2523.1.h.a.236.1 6
29.6 even 14 2523.1.h.a.1037.1 6
29.7 even 7 2523.1.h.b.1952.1 6
29.8 odd 28 2523.1.j.b.1415.1 12
29.9 even 14 2523.1.h.a.1949.1 6
29.10 odd 28 2523.1.j.b.1031.1 12
29.11 odd 28 2523.1.j.b.1619.1 12
29.12 odd 4 2523.1.b.b.842.1 2
29.13 even 14 2523.1.h.a.1745.1 6
29.14 odd 28 2523.1.j.b.2327.2 12
29.15 odd 28 2523.1.j.b.2327.1 12
29.16 even 7 2523.1.h.b.1745.1 6
29.17 odd 4 2523.1.b.b.842.2 2
29.18 odd 28 2523.1.j.b.1619.2 12
29.19 odd 28 2523.1.j.b.1031.2 12
29.20 even 7 2523.1.h.b.1949.1 6
29.21 odd 28 2523.1.j.b.1415.2 12
29.22 even 14 2523.1.h.a.1952.1 6
29.23 even 7 2523.1.h.b.1037.1 6
29.24 even 7 2523.1.h.b.236.1 6
29.25 even 7 2523.1.h.b.2333.1 6
29.26 odd 28 2523.1.j.b.1412.2 12
29.27 odd 28 2523.1.j.b.605.2 12
29.28 even 2 87.1.d.b.86.1 yes 1
87.2 even 28 2523.1.j.b.605.2 12
87.5 odd 14 2523.1.h.b.236.1 6
87.8 even 28 2523.1.j.b.1415.2 12
87.11 even 28 2523.1.j.b.1619.2 12
87.14 even 28 2523.1.j.b.2327.1 12
87.17 even 4 2523.1.b.b.842.1 2
87.20 odd 14 2523.1.h.a.1949.1 6
87.23 odd 14 2523.1.h.a.1037.1 6
87.26 even 28 2523.1.j.b.1412.1 12
87.32 even 28 2523.1.j.b.1412.2 12
87.35 odd 14 2523.1.h.b.1037.1 6
87.38 odd 14 2523.1.h.b.1949.1 6
87.41 even 4 2523.1.b.b.842.2 2
87.44 even 28 2523.1.j.b.2327.2 12
87.47 even 28 2523.1.j.b.1619.1 12
87.50 even 28 2523.1.j.b.1415.1 12
87.53 odd 14 2523.1.h.a.236.1 6
87.56 even 28 2523.1.j.b.605.1 12
87.62 odd 14 2523.1.h.b.2333.1 6
87.65 odd 14 2523.1.h.a.1952.1 6
87.68 even 28 2523.1.j.b.1031.2 12
87.71 odd 14 2523.1.h.b.1745.1 6
87.74 odd 14 2523.1.h.a.1745.1 6
87.77 even 28 2523.1.j.b.1031.1 12
87.80 odd 14 2523.1.h.b.1952.1 6
87.83 odd 14 2523.1.h.a.2333.1 6
87.86 odd 2 CM 87.1.d.a.86.1 1
116.115 odd 2 1392.1.i.b.1217.1 1
145.28 odd 4 2175.1.b.b.2174.1 2
145.57 odd 4 2175.1.b.b.2174.2 2
145.144 even 2 2175.1.h.a.1826.1 1
261.86 odd 6 2349.1.h.b.1565.1 2
261.115 even 6 2349.1.h.a.782.1 2
261.173 odd 6 2349.1.h.b.782.1 2
261.202 even 6 2349.1.h.a.1565.1 2
348.347 even 2 1392.1.i.a.1217.1 1
435.173 even 4 2175.1.b.a.2174.2 2
435.347 even 4 2175.1.b.a.2174.1 2
435.434 odd 2 2175.1.h.b.1826.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
87.1.d.a.86.1 1 1.1 even 1 trivial
87.1.d.a.86.1 1 87.86 odd 2 CM
87.1.d.b.86.1 yes 1 3.2 odd 2
87.1.d.b.86.1 yes 1 29.28 even 2
1392.1.i.a.1217.1 1 4.3 odd 2
1392.1.i.a.1217.1 1 348.347 even 2
1392.1.i.b.1217.1 1 12.11 even 2
1392.1.i.b.1217.1 1 116.115 odd 2
2175.1.b.a.2174.1 2 5.2 odd 4
2175.1.b.a.2174.1 2 435.347 even 4
2175.1.b.a.2174.2 2 5.3 odd 4
2175.1.b.a.2174.2 2 435.173 even 4
2175.1.b.b.2174.1 2 15.8 even 4
2175.1.b.b.2174.1 2 145.28 odd 4
2175.1.b.b.2174.2 2 15.2 even 4
2175.1.b.b.2174.2 2 145.57 odd 4
2175.1.h.a.1826.1 1 15.14 odd 2
2175.1.h.a.1826.1 1 145.144 even 2
2175.1.h.b.1826.1 1 5.4 even 2
2175.1.h.b.1826.1 1 435.434 odd 2
2349.1.h.a.782.1 2 9.2 odd 6
2349.1.h.a.782.1 2 261.115 even 6
2349.1.h.a.1565.1 2 9.5 odd 6
2349.1.h.a.1565.1 2 261.202 even 6
2349.1.h.b.782.1 2 9.7 even 3
2349.1.h.b.782.1 2 261.173 odd 6
2349.1.h.b.1565.1 2 9.4 even 3
2349.1.h.b.1565.1 2 261.86 odd 6
2523.1.b.b.842.1 2 29.12 odd 4
2523.1.b.b.842.1 2 87.17 even 4
2523.1.b.b.842.2 2 29.17 odd 4
2523.1.b.b.842.2 2 87.41 even 4
2523.1.h.a.236.1 6 29.5 even 14
2523.1.h.a.236.1 6 87.53 odd 14
2523.1.h.a.1037.1 6 29.6 even 14
2523.1.h.a.1037.1 6 87.23 odd 14
2523.1.h.a.1745.1 6 29.13 even 14
2523.1.h.a.1745.1 6 87.74 odd 14
2523.1.h.a.1949.1 6 29.9 even 14
2523.1.h.a.1949.1 6 87.20 odd 14
2523.1.h.a.1952.1 6 29.22 even 14
2523.1.h.a.1952.1 6 87.65 odd 14
2523.1.h.a.2333.1 6 29.4 even 14
2523.1.h.a.2333.1 6 87.83 odd 14
2523.1.h.b.236.1 6 29.24 even 7
2523.1.h.b.236.1 6 87.5 odd 14
2523.1.h.b.1037.1 6 29.23 even 7
2523.1.h.b.1037.1 6 87.35 odd 14
2523.1.h.b.1745.1 6 29.16 even 7
2523.1.h.b.1745.1 6 87.71 odd 14
2523.1.h.b.1949.1 6 29.20 even 7
2523.1.h.b.1949.1 6 87.38 odd 14
2523.1.h.b.1952.1 6 29.7 even 7
2523.1.h.b.1952.1 6 87.80 odd 14
2523.1.h.b.2333.1 6 29.25 even 7
2523.1.h.b.2333.1 6 87.62 odd 14
2523.1.j.b.605.1 12 29.2 odd 28
2523.1.j.b.605.1 12 87.56 even 28
2523.1.j.b.605.2 12 29.27 odd 28
2523.1.j.b.605.2 12 87.2 even 28
2523.1.j.b.1031.1 12 29.10 odd 28
2523.1.j.b.1031.1 12 87.77 even 28
2523.1.j.b.1031.2 12 29.19 odd 28
2523.1.j.b.1031.2 12 87.68 even 28
2523.1.j.b.1412.1 12 29.3 odd 28
2523.1.j.b.1412.1 12 87.26 even 28
2523.1.j.b.1412.2 12 29.26 odd 28
2523.1.j.b.1412.2 12 87.32 even 28
2523.1.j.b.1415.1 12 29.8 odd 28
2523.1.j.b.1415.1 12 87.50 even 28
2523.1.j.b.1415.2 12 29.21 odd 28
2523.1.j.b.1415.2 12 87.8 even 28
2523.1.j.b.1619.1 12 29.11 odd 28
2523.1.j.b.1619.1 12 87.47 even 28
2523.1.j.b.1619.2 12 29.18 odd 28
2523.1.j.b.1619.2 12 87.11 even 28
2523.1.j.b.2327.1 12 29.15 odd 28
2523.1.j.b.2327.1 12 87.14 even 28
2523.1.j.b.2327.2 12 29.14 odd 28
2523.1.j.b.2327.2 12 87.44 even 28