Properties

 Label 1045.1.k.a.417.1 Level $1045$ Weight $1$ Character 1045.417 Analytic conductor $0.522$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 209 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,1,Mod(208,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 2, 2]))

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

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

chi := DirichletCharacter("1045.208");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.k (of order $$4$$, degree $$2$$, 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: no Analytic conductor: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.26125.1

Embedding invariants

 Embedding label 417.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1045.417 Dual form 1045.1.k.a.208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{4} -1.00000i q^{5} -1.00000i q^{9} +O(q^{10})$$ $$q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{4} -1.00000i q^{5} -1.00000i q^{9} +(-1.00000 + 1.00000i) q^{10} +1.00000 q^{11} +(1.00000 - 1.00000i) q^{13} +1.00000 q^{16} +(-1.00000 + 1.00000i) q^{18} +1.00000i q^{19} +1.00000 q^{20} +(-1.00000 - 1.00000i) q^{22} +(-1.00000 + 1.00000i) q^{23} -1.00000 q^{25} -2.00000 q^{26} -2.00000i q^{29} +(-1.00000 - 1.00000i) q^{32} +1.00000 q^{36} +(1.00000 - 1.00000i) q^{38} +1.00000i q^{44} -1.00000 q^{45} +2.00000 q^{46} +(-1.00000 - 1.00000i) q^{47} +1.00000i q^{49} +(1.00000 + 1.00000i) q^{50} +(1.00000 + 1.00000i) q^{52} -1.00000i q^{55} +(-2.00000 + 2.00000i) q^{58} +1.00000i q^{64} +(-1.00000 - 1.00000i) q^{65} -1.00000 q^{76} -1.00000i q^{80} -1.00000 q^{81} +(1.00000 + 1.00000i) q^{90} +(-1.00000 - 1.00000i) q^{92} +2.00000i q^{94} +1.00000 q^{95} +(1.00000 - 1.00000i) q^{98} -1.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2}+O(q^{10})$$ 2 * q - 2 * q^2 $$2 q - 2 q^{2} - 2 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{22} - 2 q^{23} - 2 q^{25} - 4 q^{26} - 2 q^{32} + 2 q^{36} + 2 q^{38} - 2 q^{45} + 4 q^{46} - 2 q^{47} + 2 q^{50} + 2 q^{52} - 4 q^{58} - 2 q^{65} - 2 q^{76} - 2 q^{81} + 2 q^{90} - 2 q^{92} + 2 q^{95} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^10 + 2 * q^11 + 2 * q^13 + 2 * q^16 - 2 * q^18 + 2 * q^20 - 2 * q^22 - 2 * q^23 - 2 * q^25 - 4 * q^26 - 2 * q^32 + 2 * q^36 + 2 * q^38 - 2 * q^45 + 4 * q^46 - 2 * q^47 + 2 * q^50 + 2 * q^52 - 4 * q^58 - 2 * q^65 - 2 * q^76 - 2 * q^81 + 2 * q^90 - 2 * q^92 + 2 * q^95 + 2 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{4}\right)$$

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 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$3$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$4$$ 1.00000i 1.00000i
$$5$$ 1.00000i 1.00000i
$$6$$ 0 0
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ 0 0
$$9$$ 1.00000i 1.00000i
$$10$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$11$$ 1.00000 1.00000
$$12$$ 0 0
$$13$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 1.00000
$$17$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$18$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$19$$ 1.00000i 1.00000i
$$20$$ 1.00000 1.00000
$$21$$ 0 0
$$22$$ −1.00000 1.00000i −1.00000 1.00000i
$$23$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −1.00000
$$26$$ −2.00000 −2.00000
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ −1.00000 1.00000i −1.00000 1.00000i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 1.00000
$$37$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$38$$ 1.00000 1.00000i 1.00000 1.00000i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$44$$ 1.00000i 1.00000i
$$45$$ −1.00000 −1.00000
$$46$$ 2.00000 2.00000
$$47$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 1.00000i 1.00000i
$$50$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$51$$ 0 0
$$52$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$53$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$54$$ 0 0
$$55$$ 1.00000i 1.00000i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −2.00000 + 2.00000i −2.00000 + 2.00000i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000i 1.00000i
$$65$$ −1.00000 1.00000i −1.00000 1.00000i
$$66$$ 0 0
$$67$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −1.00000 −1.00000
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 1.00000i 1.00000i
$$81$$ −1.00000 −1.00000
$$82$$ 0 0
$$83$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$91$$ 0 0
$$92$$ −1.00000 1.00000i −1.00000 1.00000i
$$93$$ 0 0
$$94$$ 2.00000i 2.00000i
$$95$$ 1.00000 1.00000
$$96$$ 0 0
$$97$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$98$$ 1.00000 1.00000i 1.00000 1.00000i
$$99$$ 1.00000i 1.00000i
$$100$$ 1.00000i 1.00000i
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$114$$ 0 0
$$115$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$116$$ 2.00000 2.00000
$$117$$ −1.00000 1.00000i −1.00000 1.00000i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000i 1.00000i
$$126$$ 0 0
$$127$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 2.00000i 2.00000i
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.00000 1.00000i 1.00000 1.00000i
$$144$$ 1.00000i 1.00000i
$$145$$ −2.00000 −2.00000
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$161$$ 0 0
$$162$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$163$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000i 1.00000i
$$170$$ 0 0
$$171$$ 1.00000 1.00000
$$172$$ 0 0
$$173$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 1.00000
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 1.00000i 1.00000i
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 1.00000 1.00000i 1.00000 1.00000i
$$189$$ 0 0
$$190$$ −1.00000 1.00000i −1.00000 1.00000i
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1.00000 −1.00000
$$197$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$198$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$208$$ 1.00000 1.00000i 1.00000 1.00000i
$$209$$ 1.00000i 1.00000i
$$210$$ 0 0
$$211$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 2.00000i 2.00000i
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 2.00000i 2.00000 2.00000i
$$219$$ 0 0
$$220$$ 1.00000 1.00000
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$224$$ 0 0
$$225$$ 1.00000i 1.00000i
$$226$$ 0 0
$$227$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$230$$ 2.00000i 2.00000i
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$234$$ 2.00000i 2.00000i
$$235$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$242$$ −1.00000 1.00000i −1.00000 1.00000i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.00000 1.00000
$$246$$ 0 0
$$247$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 1.00000 1.00000i 1.00000 1.00000i
$$251$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$254$$ 2.00000i 2.00000i
$$255$$ 0 0
$$256$$ 1.00000 1.00000
$$257$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 1.00000 1.00000i 1.00000 1.00000i
$$261$$ −2.00000 −2.00000
$$262$$ 0 0
$$263$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 2.00000i 2.00000i
$$275$$ −1.00000 −1.00000
$$276$$ 0 0
$$277$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$282$$ 0 0
$$283$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −2.00000 −2.00000
$$287$$ 0 0
$$288$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$289$$ 1.00000i 1.00000i
$$290$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.00000i 2.00000i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −2.00000 2.00000i −2.00000 2.00000i
$$303$$ 0 0
$$304$$ 1.00000i 1.00000i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$314$$ 2.00000i 2.00000i
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$318$$ 0 0
$$319$$ 2.00000i 2.00000i
$$320$$ 1.00000 1.00000
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000i 1.00000i
$$325$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$326$$ −2.00000 −2.00000
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 2.00000i 2.00000i
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$338$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −1.00000 1.00000i −1.00000 1.00000i
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −2.00000 −2.00000
$$347$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 1.00000i −1.00000 1.00000i
$$353$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −1.00000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$368$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.00000 2.00000i −2.00000 2.00000i
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 1.00000i 1.00000i
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −2.00000
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 1.00000 1.00000
$$397$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −1.00000
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000i 1.00000i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 2.00000i 2.00000i
$$415$$ 0 0
$$416$$ −2.00000 −2.00000
$$417$$ 0 0
$$418$$ 1.00000 1.00000i 1.00000 1.00000i
$$419$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$423$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −2.00000
$$437$$ −1.00000 1.00000i −1.00000 1.00000i
$$438$$ 0 0
$$439$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$440$$ 0 0
$$441$$ 1.00000 1.00000
$$442$$ 0 0
$$443$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 1.00000 1.00000i 1.00000 1.00000i
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 2.00000i 2.00000i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$458$$ −2.00000 + 2.00000i −2.00000 + 2.00000i
$$459$$ 0 0
$$460$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$464$$ 2.00000i 2.00000i
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$468$$ 1.00000 1.00000i 1.00000 1.00000i
$$469$$ 0 0
$$470$$ 2.00000 2.00000
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000i 1.00000i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$483$$ 0 0
$$484$$ 1.00000i 1.00000i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −1.00000 1.00000i −1.00000 1.00000i
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 2.00000i 2.00000i
$$495$$ −1.00000 −1.00000
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.00000i 2.00000i 1.00000i $$-0.5\pi$$
1.00000i $$-0.5\pi$$
$$500$$ −1.00000 −1.00000
$$501$$ 0 0
$$502$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$503$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.00000 2.00000
$$507$$ 0 0
$$508$$ 1.00000 1.00000i 1.00000 1.00000i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 1.00000i −1.00000 1.00000i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.00000 1.00000i −1.00000 1.00000i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$523$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000i 1.00000i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 1.00000 1.00000i 1.00000 1.00000i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1.00000i 1.00000i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.00000 2.00000
$$546$$ 0 0
$$547$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$548$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$549$$ 0 0
$$550$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$551$$ 2.00000 2.00000
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2.00000 2.00000i −2.00000 2.00000i
$$563$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.00000 1.00000i 1.00000 1.00000i
$$576$$ 1.00000 1.00000
$$577$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$578$$ 1.00000 1.00000i 1.00000 1.00000i
$$579$$ 0 0
$$580$$ 2.00000i 2.00000i
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$586$$ 2.00000 2.00000
$$587$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 2.00000 2.00000i 2.00000 2.00000i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 2.00000i 2.00000i
$$605$$ 1.00000i 1.00000i
$$606$$ 0 0
$$607$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$608$$ 1.00000 1.00000i 1.00000 1.00000i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.00000 −2.00000
$$612$$ 0 0
$$613$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$614$$ 2.00000i 2.00000i
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 1.00000
$$626$$ −2.00000 −2.00000
$$627$$ 0 0
$$628$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$636$$ 0 0
$$637$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$638$$ −2.00000 + 2.00000i −2.00000 + 2.00000i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 2.00000 2.00000
$$651$$ 0 0
$$652$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$653$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.00000 + 2.00000i 2.00000 + 2.00000i
$$668$$ −1.00000 + 1.00000i −1.00000 + 1.00000i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 1.00000 1.00000i 1.00000 1.00000i 1.00000i $$-0.5\pi$$
1.00000 $$0$$
$$674$$ 2.00000i 2.00000i
$$675$$ 0 0
$$676$$ 1.00000 1.00000
$$677$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$684$$ 1.00000i 1.00000i
$$685$$ 1.00000 1.00000i 1.00000 1.00000i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 1.00000i 1.00000i
$$705$$ 0 0
$$706$$ 2.00000 2.00000
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000i 2.00000i 1.00000i $$0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −1.00000 1.00000i −1.00000 1.00000i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ −1.00000 −1.00000
$$721$$ 0 0
$$722$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 2.00000i 2.00000i
$$726$$ 0 0
$$727$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 1.00000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$734$$ 2.00000i 2.00000i
$$735$$ 0 0
$$736$$ 2.00000 2.00000
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i $$0.5\pi$$
−1.00000 $$\pi$$
$$744$$ 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 1.00000i −1.00000 1.00000i
$$753$$ 0 0
$$754$$ 4.00000i 4.00000i
$$755$$ 2.00000i 2.00000i
$$756$$ 0 0
$$757$$ −1.00000 1.00000i −1.00000 1.00000i 1.00000i $$-0.5\pi$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$773$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000i 1.00000i
$$785$$ 1.00000 1.00000i 1.00000 1.00000i
$$786$$ 0 0
$$787$$ 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 $$0$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 2.00000i 2.00000i
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 + 1.00000i 1.00000 + 1.00000i
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0