# Properties

 Label 1215.1.d.a.1214.1 Level $1215$ Weight $1$ Character 1215.1214 Self dual yes Analytic conductor $0.606$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -15 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1215,1,Mod(1214,1215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1215, 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("1215.1214");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1215 = 3^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1215.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: $$0.606363990349$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.242137805625.3

## Embedding invariants

 Embedding label 1214.1 Root $$-1.53209$$ of defining polynomial Character $$\chi$$ $$=$$ 1215.1214

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.53209 q^{2} +1.34730 q^{4} -1.00000 q^{5} -0.532089 q^{8} +O(q^{10})$$ $$q-1.53209 q^{2} +1.34730 q^{4} -1.00000 q^{5} -0.532089 q^{8} +1.53209 q^{10} -0.532089 q^{16} +1.87939 q^{17} -1.87939 q^{19} -1.34730 q^{20} -0.347296 q^{23} +1.00000 q^{25} +1.53209 q^{31} +1.34730 q^{32} -2.87939 q^{34} +2.87939 q^{38} +0.532089 q^{40} +0.532089 q^{46} +1.00000 q^{47} +1.00000 q^{49} -1.53209 q^{50} -0.347296 q^{53} +1.53209 q^{61} -2.34730 q^{62} -1.53209 q^{64} +2.53209 q^{68} -2.53209 q^{76} +0.347296 q^{79} +0.532089 q^{80} +1.87939 q^{83} -1.87939 q^{85} -0.467911 q^{92} -1.53209 q^{94} +1.87939 q^{95} -1.53209 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{4} - 3 q^{5} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^4 - 3 * q^5 + 3 * q^8 $$3 q + 3 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{16} - 3 q^{20} + 3 q^{25} + 3 q^{32} - 3 q^{34} + 3 q^{38} - 3 q^{40} - 3 q^{46} + 3 q^{47} + 3 q^{49} - 6 q^{62} + 3 q^{68} - 3 q^{76} - 3 q^{80} - 6 q^{92}+O(q^{100})$$ 3 * q + 3 * q^4 - 3 * q^5 + 3 * q^8 + 3 * q^16 - 3 * q^20 + 3 * q^25 + 3 * q^32 - 3 * q^34 + 3 * q^38 - 3 * q^40 - 3 * q^46 + 3 * q^47 + 3 * q^49 - 6 * q^62 + 3 * q^68 - 3 * q^76 - 3 * q^80 - 6 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1215\mathbb{Z}\right)^\times$$.

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$3$$ 0 0
$$4$$ 1.34730 1.34730
$$5$$ −1.00000 −1.00000
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ −0.532089 −0.532089
$$9$$ 0 0
$$10$$ 1.53209 1.53209
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −0.532089 −0.532089
$$17$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$18$$ 0 0
$$19$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$20$$ −1.34730 −1.34730
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$24$$ 0 0
$$25$$ 1.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$32$$ 1.34730 1.34730
$$33$$ 0 0
$$34$$ −2.87939 −2.87939
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 2.87939 2.87939
$$39$$ 0 0
$$40$$ 0.532089 0.532089
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0.532089 0.532089
$$47$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$48$$ 0 0
$$49$$ 1.00000 1.00000
$$50$$ −1.53209 −1.53209
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$62$$ −2.34730 −2.34730
$$63$$ 0 0
$$64$$ −1.53209 −1.53209
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 2.53209 2.53209
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −2.53209 −2.53209
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$80$$ 0.532089 0.532089
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$84$$ 0 0
$$85$$ −1.87939 −1.87939
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −0.467911 −0.467911
$$93$$ 0 0
$$94$$ −1.53209 −1.53209
$$95$$ 1.87939 1.87939
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −1.53209 −1.53209
$$99$$ 0 0
$$100$$ 1.34730 1.34730
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0.532089 0.532089
$$107$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$108$$ 0 0
$$109$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$110$$ 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
$$115$$ 0.347296 0.347296
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 1.00000
$$122$$ −2.34730 −2.34730
$$123$$ 0 0
$$124$$ 2.06418 2.06418
$$125$$ −1.00000 −1.00000
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000 1.00000
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −1.00000 −1.00000
$$137$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$138$$ 0 0
$$139$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$152$$ 1.00000 1.00000
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.53209 −1.53209
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ −0.532089 −0.532089
$$159$$ 0 0
$$160$$ −1.34730 −1.34730
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −2.87939 −2.87939
$$167$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$168$$ 0 0
$$169$$ 1.00000 1.00000
$$170$$ 2.87939 2.87939
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0.184793 0.184793
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 1.34730 1.34730
$$189$$ 0 0
$$190$$ −2.87939 −2.87939
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1.34730 1.34730
$$197$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$200$$ −0.532089 −0.532089
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$212$$ −0.467911 −0.467911
$$213$$ 0 0
$$214$$ −1.53209 −1.53209
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −0.532089 −0.532089
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1.53209 −1.53209
$$227$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$228$$ 0 0
$$229$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$230$$ −0.532089 −0.532089
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$234$$ 0 0
$$235$$ −1.00000 −1.00000
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$242$$ −1.53209 −1.53209
$$243$$ 0 0
$$244$$ 2.06418 2.06418
$$245$$ −1.00000 −1.00000
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −0.815207 −0.815207
$$249$$ 0 0
$$250$$ 1.53209 1.53209
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0.347296 0.347296
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$272$$ −1.00000 −1.00000
$$273$$ 0 0
$$274$$ 0.532089 0.532089
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 1.53209 1.53209
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.53209 2.53209
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 1.53209 1.53209
$$303$$ 0 0
$$304$$ 1.00000 1.00000
$$305$$ −1.53209 −1.53209
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 2.34730 2.34730
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0.467911 0.467911
$$317$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 1.53209 1.53209
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.53209 −3.53209
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$332$$ 2.53209 2.53209
$$333$$ 0 0
$$334$$ 2.34730 2.34730
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ −1.53209 −1.53209
$$339$$ 0 0
$$340$$ −2.53209 −2.53209
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 2.34730 2.34730
$$347$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$348$$ 0 0
$$349$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 2.53209 2.53209
$$362$$ −0.532089 −0.532089
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0.184793 0.184793
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −0.532089 −0.532089
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$380$$ 2.53209 2.53209
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ −0.652704 −0.652704
$$392$$ −0.532089 −0.532089
$$393$$ 0 0
$$394$$ 2.34730 2.34730
$$395$$ −0.347296 −0.347296
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 1.53209 1.53209
$$399$$ 0 0
$$400$$ −0.532089 −0.532089
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.87939 −1.87939
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$422$$ −2.34730 −2.34730
$$423$$ 0 0
$$424$$ 0.184793 0.184793
$$425$$ 1.87939 1.87939
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1.34730 1.34730
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0.467911 0.467911
$$437$$ 0.652704 0.652704
$$438$$ 0 0
$$439$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 1.34730 1.34730
$$453$$ 0 0
$$454$$ 0.532089 0.532089
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 2.87939 2.87939
$$459$$ 0 0
$$460$$ 0.467911 0.467911
$$461$$ 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
$$466$$ −1.53209 −1.53209
$$467$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1.53209 1.53209
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −1.87939 −1.87939
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 2.87939 2.87939
$$483$$ 0 0
$$484$$ 1.34730 1.34730
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ −0.815207 −0.815207
$$489$$ 0 0
$$490$$ 1.53209 1.53209
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −0.815207 −0.815207
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$500$$ −1.34730 −1.34730
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$504$$ 0 0
$$505$$ 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
$$514$$ 2.34730 2.34730
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 3.06418 3.06418
$$527$$ 2.87939 2.87939
$$528$$ 0 0
$$529$$ −0.879385 −0.879385
$$530$$ −0.532089 −0.532089
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −1.00000 −1.00000
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$542$$ 2.87939 2.87939
$$543$$ 0 0
$$544$$ 2.53209 2.53209
$$545$$ −0.347296 −0.347296
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ −0.467911 −0.467911
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −1.34730 −1.34730
$$557$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ −1.00000 −1.00000
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −0.347296 −0.347296
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ −3.87939 −3.87939
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −2.87939 −2.87939
$$587$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$588$$ 0 0
$$589$$ −2.87939 −2.87939
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −1.34730 −1.34730
$$605$$ −1.00000 −1.00000
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ −2.53209 −2.53209
$$609$$ 0 0
$$610$$ 2.34730 2.34730
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$618$$ 0 0
$$619$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$620$$ −2.06418 −2.06418
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$632$$ −0.184793 −0.184793
$$633$$ 0 0
$$634$$ −2.87939 −2.87939
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −1.00000
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5.41147 5.41147
$$647$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$662$$ 1.53209 1.53209
$$663$$ 0 0
$$664$$ −1.00000 −1.00000
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −2.06418 −2.06418
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 1.34730 1.34730
$$677$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.00000 1.00000
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$684$$ 0 0
$$685$$ 0.347296 0.347296
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$692$$ −2.06418 −2.06418
$$693$$ 0 0
$$694$$ −1.53209 −1.53209
$$695$$ 1.00000 1.00000
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −2.34730 −2.34730
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −1.53209 −1.53209
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −0.532089 −0.532089
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.87939 −3.87939
$$723$$ 0 0
$$724$$ 0.467911 0.467911
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ −0.467911 −0.467911
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$752$$ −0.532089 −0.532089
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.00000 1.00000
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ −2.34730 −2.34730
$$759$$ 0 0
$$760$$ −1.00000 −1.00000
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 2.34730 2.34730
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$774$$ 0 0
$$775$$ 1.53209 1.53209
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 1.00000 1.00000
$$783$$ 0 0
$$784$$ −0.532089 −0.532089
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ −2.06418 −2.06418
$$789$$ 0 0
$$790$$ 0.532089 0.532089
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −1.34730 −1.34730
$$797$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$798$$ 0 0
$$799$$ 1.87939 1.87939
$$800$$ 1.34730 1.34730
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 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
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −0.532089 −0.532089
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$828$$ 0 0
$$829$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$830$$ 2.87939 2.87939
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.87939 1.87939
$$834$$ 0 0
$$835$$ 1.53209 1.53209
$$836$$ 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
$$842$$ −2.34730 −2.34730
$$843$$ 0 0
$$844$$ 2.06418 2.06418
$$845$$ −1.00000 −1.00000
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0.184793 0.184793
$$849$$ 0 0
$$850$$ −2.87939 −2.87939
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −0.532089 −0.532089
$$857$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$858$$ 0 0
$$859$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$864$$ 0 0
$$865$$ 1.53209 1.53209
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −0.184793 −0.184793
$$873$$ 0 0
$$874$$ −1.00000 −1.00000
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 2.87939 2.87939
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0.532089 0.532089
$$887$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −1.87939 −1.87939
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −0.652704 −0.652704
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −0.532089 −0.532089
$$905$$ −0.347296 −0.347296
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ −0.467911 −0.467911
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −2.53209 −2.53209
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$920$$ −0.184793 −0.184793
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ −1.87939 −1.87939
$$932$$ 1.34730 1.34730
$$933$$ 0 0
$$934$$ −2.87939 −2.87939
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −1.34730 −1.34730
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 2.87939 2.87939
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.34730 1.34730
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −2.53209 −2.53209
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ −0.532089 −0.532089
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −0.815207 −0.815207
$$977$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −1.34730 −1.34730
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$984$$ 0 0
$$985$$ 1.53209 1.53209
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$992$$ 2.06418 2.06418
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1.00000 1.00000
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ −2.34730 −2.34730
$$999$$ 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 1215.1.d.a.1214.1 3
3.2 odd 2 1215.1.d.b.1214.3 yes 3
5.4 even 2 1215.1.d.b.1214.3 yes 3
9.2 odd 6 1215.1.h.a.404.1 6
9.4 even 3 1215.1.h.b.809.3 6
9.5 odd 6 1215.1.h.a.809.1 6
9.7 even 3 1215.1.h.b.404.3 6
15.14 odd 2 CM 1215.1.d.a.1214.1 3
27.2 odd 18 3645.1.n.c.1619.1 6
27.4 even 9 3645.1.n.a.3239.1 6
27.5 odd 18 3645.1.n.b.809.1 6
27.7 even 9 3645.1.n.a.404.1 6
27.11 odd 18 3645.1.n.b.2834.1 6
27.13 even 9 3645.1.n.g.2024.1 6
27.14 odd 18 3645.1.n.c.2024.1 6
27.16 even 9 3645.1.n.f.2834.1 6
27.20 odd 18 3645.1.n.h.404.1 6
27.22 even 9 3645.1.n.f.809.1 6
27.23 odd 18 3645.1.n.h.3239.1 6
27.25 even 9 3645.1.n.g.1619.1 6
45.4 even 6 1215.1.h.a.809.1 6
45.14 odd 6 1215.1.h.b.809.3 6
45.29 odd 6 1215.1.h.b.404.3 6
45.34 even 6 1215.1.h.a.404.1 6
135.4 even 18 3645.1.n.h.3239.1 6
135.14 odd 18 3645.1.n.g.2024.1 6
135.29 odd 18 3645.1.n.g.1619.1 6
135.34 even 18 3645.1.n.h.404.1 6
135.49 even 18 3645.1.n.b.809.1 6
135.59 odd 18 3645.1.n.f.809.1 6
135.74 odd 18 3645.1.n.a.404.1 6
135.79 even 18 3645.1.n.c.1619.1 6
135.94 even 18 3645.1.n.c.2024.1 6
135.104 odd 18 3645.1.n.a.3239.1 6
135.119 odd 18 3645.1.n.f.2834.1 6
135.124 even 18 3645.1.n.b.2834.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1215.1.d.a.1214.1 3 1.1 even 1 trivial
1215.1.d.a.1214.1 3 15.14 odd 2 CM
1215.1.d.b.1214.3 yes 3 3.2 odd 2
1215.1.d.b.1214.3 yes 3 5.4 even 2
1215.1.h.a.404.1 6 9.2 odd 6
1215.1.h.a.404.1 6 45.34 even 6
1215.1.h.a.809.1 6 9.5 odd 6
1215.1.h.a.809.1 6 45.4 even 6
1215.1.h.b.404.3 6 9.7 even 3
1215.1.h.b.404.3 6 45.29 odd 6
1215.1.h.b.809.3 6 9.4 even 3
1215.1.h.b.809.3 6 45.14 odd 6
3645.1.n.a.404.1 6 27.7 even 9
3645.1.n.a.404.1 6 135.74 odd 18
3645.1.n.a.3239.1 6 27.4 even 9
3645.1.n.a.3239.1 6 135.104 odd 18
3645.1.n.b.809.1 6 27.5 odd 18
3645.1.n.b.809.1 6 135.49 even 18
3645.1.n.b.2834.1 6 27.11 odd 18
3645.1.n.b.2834.1 6 135.124 even 18
3645.1.n.c.1619.1 6 27.2 odd 18
3645.1.n.c.1619.1 6 135.79 even 18
3645.1.n.c.2024.1 6 27.14 odd 18
3645.1.n.c.2024.1 6 135.94 even 18
3645.1.n.f.809.1 6 27.22 even 9
3645.1.n.f.809.1 6 135.59 odd 18
3645.1.n.f.2834.1 6 27.16 even 9
3645.1.n.f.2834.1 6 135.119 odd 18
3645.1.n.g.1619.1 6 27.25 even 9
3645.1.n.g.1619.1 6 135.29 odd 18
3645.1.n.g.2024.1 6 27.13 even 9
3645.1.n.g.2024.1 6 135.14 odd 18
3645.1.n.h.404.1 6 27.20 odd 18
3645.1.n.h.404.1 6 135.34 even 18
3645.1.n.h.3239.1 6 27.23 odd 18
3645.1.n.h.3239.1 6 135.4 even 18