# Properties

 Label 2175.1.h.d.1826.2 Level $2175$ Weight $1$ Character 2175.1826 Self dual yes Analytic conductor $1.085$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -87 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,1,Mod(1826,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("2175.1826");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2175.h (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: $$1.08546640248$$ 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 9.1.895152515625.1

## Embedding invariants

 Embedding label 1826.2 Root $$1.87939$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1826

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} +0.347296 q^{6} -1.87939 q^{7} +0.652704 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} +0.347296 q^{6} -1.87939 q^{7} +0.652704 q^{8} +1.00000 q^{9} -1.53209 q^{11} +0.879385 q^{12} +0.347296 q^{13} +0.652704 q^{14} +0.652704 q^{16} -1.53209 q^{17} -0.347296 q^{18} +1.87939 q^{21} +0.532089 q^{22} -0.652704 q^{24} -0.120615 q^{26} -1.00000 q^{27} +1.65270 q^{28} -1.00000 q^{29} -0.879385 q^{32} +1.53209 q^{33} +0.532089 q^{34} -0.879385 q^{36} -0.347296 q^{39} +1.00000 q^{41} -0.652704 q^{42} +1.34730 q^{44} +1.87939 q^{47} -0.652704 q^{48} +2.53209 q^{49} +1.53209 q^{51} -0.305407 q^{52} +0.347296 q^{54} -1.22668 q^{56} +0.347296 q^{58} -1.87939 q^{63} -0.347296 q^{64} -0.532089 q^{66} +1.53209 q^{67} +1.34730 q^{68} +0.652704 q^{72} +2.87939 q^{77} +0.120615 q^{78} +1.00000 q^{81} -0.347296 q^{82} -1.65270 q^{84} +1.00000 q^{87} -1.00000 q^{88} -0.347296 q^{89} -0.652704 q^{91} -0.652704 q^{94} +0.879385 q^{96} -0.879385 q^{98} -1.53209 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{12} + 3 q^{14} + 3 q^{16} - 3 q^{22} - 3 q^{24} - 6 q^{26} - 3 q^{27} + 6 q^{28} - 3 q^{29} + 3 q^{32} - 3 q^{34} + 3 q^{36} + 3 q^{41} - 3 q^{42} + 3 q^{44} - 3 q^{48} + 3 q^{49} - 3 q^{52} + 3 q^{56} + 3 q^{66} + 3 q^{68} + 3 q^{72} + 3 q^{77} + 6 q^{78} + 3 q^{81} - 6 q^{84} + 3 q^{87} - 3 q^{88} - 3 q^{91} - 3 q^{94} - 3 q^{96} + 3 q^{98}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^12 + 3 * q^14 + 3 * q^16 - 3 * q^22 - 3 * q^24 - 6 * q^26 - 3 * q^27 + 6 * q^28 - 3 * q^29 + 3 * q^32 - 3 * q^34 + 3 * q^36 + 3 * q^41 - 3 * q^42 + 3 * q^44 - 3 * q^48 + 3 * q^49 - 3 * q^52 + 3 * q^56 + 3 * q^66 + 3 * q^68 + 3 * q^72 + 3 * q^77 + 6 * q^78 + 3 * q^81 - 6 * q^84 + 3 * q^87 - 3 * q^88 - 3 * q^91 - 3 * q^94 - 3 * q^96 + 3 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$3$$ −1.00000 −1.00000
$$4$$ −0.879385 −0.879385
$$5$$ 0 0
$$6$$ 0.347296 0.347296
$$7$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$8$$ 0.652704 0.652704
$$9$$ 1.00000 1.00000
$$10$$ 0 0
$$11$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$12$$ 0.879385 0.879385
$$13$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$14$$ 0.652704 0.652704
$$15$$ 0 0
$$16$$ 0.652704 0.652704
$$17$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$18$$ −0.347296 −0.347296
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 1.87939 1.87939
$$22$$ 0.532089 0.532089
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −0.652704 −0.652704
$$25$$ 0 0
$$26$$ −0.120615 −0.120615
$$27$$ −1.00000 −1.00000
$$28$$ 1.65270 1.65270
$$29$$ −1.00000 −1.00000
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ −0.879385 −0.879385
$$33$$ 1.53209 1.53209
$$34$$ 0.532089 0.532089
$$35$$ 0 0
$$36$$ −0.879385 −0.879385
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −0.347296 −0.347296
$$40$$ 0 0
$$41$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$42$$ −0.652704 −0.652704
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 1.34730 1.34730
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$48$$ −0.652704 −0.652704
$$49$$ 2.53209 2.53209
$$50$$ 0 0
$$51$$ 1.53209 1.53209
$$52$$ −0.305407 −0.305407
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0.347296 0.347296
$$55$$ 0 0
$$56$$ −1.22668 −1.22668
$$57$$ 0 0
$$58$$ 0.347296 0.347296
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ −1.87939 −1.87939
$$64$$ −0.347296 −0.347296
$$65$$ 0 0
$$66$$ −0.532089 −0.532089
$$67$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$68$$ 1.34730 1.34730
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0.652704 0.652704
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.87939 2.87939
$$78$$ 0.120615 0.120615
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 1.00000 1.00000
$$82$$ −0.347296 −0.347296
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −1.65270 −1.65270
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.00000 1.00000
$$88$$ −1.00000 −1.00000
$$89$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$90$$ 0 0
$$91$$ −0.652704 −0.652704
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −0.652704 −0.652704
$$95$$ 0 0
$$96$$ 0.879385 0.879385
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −0.879385 −0.879385
$$99$$ −1.53209 −1.53209
$$100$$ 0 0
$$101$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$102$$ −0.532089 −0.532089
$$103$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$104$$ 0.226682 0.226682
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0.879385 0.879385
$$109$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.22668 −1.22668
$$113$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.879385 0.879385
$$117$$ 0.347296 0.347296
$$118$$ 0 0
$$119$$ 2.87939 2.87939
$$120$$ 0 0
$$121$$ 1.34730 1.34730
$$122$$ 0 0
$$123$$ −1.00000 −1.00000
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0.652704 0.652704
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000 1.00000
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$132$$ −1.34730 −1.34730
$$133$$ 0 0
$$134$$ −0.532089 −0.532089
$$135$$ 0 0
$$136$$ −1.00000 −1.00000
$$137$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$138$$ 0 0
$$139$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$140$$ 0 0
$$141$$ −1.87939 −1.87939
$$142$$ 0 0
$$143$$ −0.532089 −0.532089
$$144$$ 0.652704 0.652704
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.53209 −2.53209
$$148$$ 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 $$0.666667\pi$$
$$152$$ 0 0
$$153$$ −1.53209 −1.53209
$$154$$ −1.00000 −1.00000
$$155$$ 0 0
$$156$$ 0.305407 0.305407
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −0.347296 −0.347296
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ −0.879385 −0.879385
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 1.22668 1.22668
$$169$$ −0.879385 −0.879385
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ −0.347296 −0.347296
$$175$$ 0 0
$$176$$ −1.00000 −1.00000
$$177$$ 0 0
$$178$$ 0.120615 0.120615
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$182$$ 0.226682 0.226682
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.34730 2.34730
$$188$$ −1.65270 −1.65270
$$189$$ 1.87939 1.87939
$$190$$ 0 0
$$191$$ −2.00000 −2.00000 −1.00000 $$\pi$$
−1.00000 $$\pi$$
$$192$$ 0.347296 0.347296
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −2.22668 −2.22668
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0.532089 0.532089
$$199$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$200$$ 0 0
$$201$$ −1.53209 −1.53209
$$202$$ 0.120615 0.120615
$$203$$ 1.87939 1.87939
$$204$$ −1.34730 −1.34730
$$205$$ 0 0
$$206$$ 0.347296 0.347296
$$207$$ 0 0
$$208$$ 0.226682 0.226682
$$209$$ 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
$$216$$ −0.652704 −0.652704
$$217$$ 0 0
$$218$$ −0.120615 −0.120615
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −0.532089 −0.532089
$$222$$ 0 0
$$223$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$224$$ 1.65270 1.65270
$$225$$ 0 0
$$226$$ −0.652704 −0.652704
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ −2.87939 −2.87939
$$232$$ −0.652704 −0.652704
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ −0.120615 −0.120615
$$235$$ 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
$$241$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$242$$ −0.467911 −0.467911
$$243$$ −1.00000 −1.00000
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0.347296 0.347296
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$252$$ 1.65270 1.65270
$$253$$ 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
$$262$$ −0.652704 −0.652704
$$263$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$264$$ 1.00000 1.00000
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0.347296 0.347296
$$268$$ −1.34730 −1.34730
$$269$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ −1.00000 −1.00000
$$273$$ 0.652704 0.652704
$$274$$ −0.347296 −0.347296
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$278$$ 0.652704 0.652704
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0.652704 0.652704
$$283$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0.184793 0.184793
$$287$$ −1.87939 −1.87939
$$288$$ −0.879385 −0.879385
$$289$$ 1.34730 1.34730
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$294$$ 0.879385 0.879385
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.53209 1.53209
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0.347296 0.347296
$$303$$ 0.347296 0.347296
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0.532089 0.532089
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ −2.53209 −2.53209
$$309$$ 1.00000 1.00000
$$310$$ 0 0
$$311$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$312$$ −0.226682 −0.226682
$$313$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$318$$ 0 0
$$319$$ 1.53209 1.53209
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −0.879385 −0.879385
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −0.347296 −0.347296
$$328$$ 0.652704 0.652704
$$329$$ −3.53209 −3.53209
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 1.22668 1.22668
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0.305407 0.305407
$$339$$ −1.87939 −1.87939
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −2.87939 −2.87939
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ −0.879385 −0.879385
$$349$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$350$$ 0 0
$$351$$ −0.347296 −0.347296
$$352$$ 1.34730 1.34730
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0.305407 0.305407
$$357$$ −2.87939 −2.87939
$$358$$ 0 0
$$359$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$360$$ 0 0
$$361$$ 1.00000 1.00000
$$362$$ −0.532089 −0.532089
$$363$$ −1.34730 −1.34730
$$364$$ 0.573978 0.573978
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 1.00000 1.00000
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$374$$ −0.815207 −0.815207
$$375$$ 0 0
$$376$$ 1.22668 1.22668
$$377$$ −0.347296 −0.347296
$$378$$ −0.652704 −0.652704
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0.694593 0.694593
$$383$$ 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
$$389$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.65270 1.65270
$$393$$ −1.87939 −1.87939
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 1.34730 1.34730
$$397$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$398$$ −0.120615 −0.120615
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0.532089 0.532089
$$403$$ 0 0
$$404$$ 0.305407 0.305407
$$405$$ 0 0
$$406$$ −0.652704 −0.652704
$$407$$ 0 0
$$408$$ 1.00000 1.00000
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −1.00000 −1.00000
$$412$$ 0.879385 0.879385
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −0.305407 −0.305407
$$417$$ 1.87939 1.87939
$$418$$ 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
$$423$$ 1.87939 1.87939
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0.532089 0.532089
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −0.652704 −0.652704
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −0.305407 −0.305407
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$440$$ 0 0
$$441$$ 2.53209 2.53209
$$442$$ 0.184793 0.184793
$$443$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −0.532089 −0.532089
$$447$$ 0 0
$$448$$ 0.652704 0.652704
$$449$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$450$$ 0 0
$$451$$ −1.53209 −1.53209
$$452$$ −1.65270 −1.65270
$$453$$ 1.00000 1.00000
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$458$$ 0 0
$$459$$ 1.53209 1.53209
$$460$$ 0 0
$$461$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$462$$ 1.00000 1.00000
$$463$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$464$$ −0.652704 −0.652704
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$468$$ −0.305407 −0.305407
$$469$$ −2.87939 −2.87939
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.53209 −2.53209
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0.652704 0.652704
$$483$$ 0 0
$$484$$ −1.18479 −1.18479
$$485$$ 0 0
$$486$$ 0.347296 0.347296
$$487$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$492$$ 0.879385 0.879385
$$493$$ 1.53209 1.53209
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0.120615 0.120615
$$503$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$504$$ −1.22668 −1.22668
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0.879385 0.879385
$$508$$ 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
$$517$$ −2.87939 −2.87939
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0.347296 0.347296
$$523$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$524$$ −1.65270 −1.65270
$$525$$ 0 0
$$526$$ −0.347296 −0.347296
$$527$$ 0 0
$$528$$ 1.00000 1.00000
$$529$$ 1.00000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0.347296 0.347296
$$534$$ −0.120615 −0.120615
$$535$$ 0 0
$$536$$ 1.00000 1.00000
$$537$$ 0 0
$$538$$ 0.532089 0.532089
$$539$$ −3.87939 −3.87939
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ −1.53209 −1.53209
$$544$$ 1.34730 1.34730
$$545$$ 0 0
$$546$$ −0.226682 −0.226682
$$547$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$548$$ −0.879385 −0.879385
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −0.532089 −0.532089
$$555$$ 0 0
$$556$$ 1.65270 1.65270
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −2.34730 −2.34730
$$562$$ 0 0
$$563$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$564$$ 1.65270 1.65270
$$565$$ 0 0
$$566$$ 0.347296 0.347296
$$567$$ −1.87939 −1.87939
$$568$$ 0 0
$$569$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$570$$ 0 0
$$571$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$572$$ 0.467911 0.467911
$$573$$ 2.00000 2.00000
$$574$$ 0.652704 0.652704
$$575$$ 0 0
$$576$$ −0.347296 −0.347296
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ −0.467911 −0.467911
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0.120615 0.120615
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 2.22668 2.22668
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ −0.532089 −0.532089
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.347296 −0.347296
$$598$$ 0 0
$$599$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 1.53209 1.53209
$$604$$ 0.879385 0.879385
$$605$$ 0 0
$$606$$ −0.120615 −0.120615
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −1.87939 −1.87939
$$610$$ 0 0
$$611$$ 0.652704 0.652704
$$612$$ 1.34730 1.34730
$$613$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 1.87939 1.87939
$$617$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$618$$ −0.347296 −0.347296
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −0.652704 −0.652704
$$623$$ 0.652704 0.652704
$$624$$ −0.226682 −0.226682
$$625$$ 0 0
$$626$$ 0.652704 0.652704
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.87939 −1.87939 −0.939693 0.342020i $$-0.888889\pi$$
−0.939693 + 0.342020i $$0.888889\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0.120615 0.120615
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.879385 0.879385
$$638$$ −0.532089 −0.532089
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$642$$ 0 0
$$643$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0.652704 0.652704
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$654$$ 0.120615 0.120615
$$655$$ 0 0
$$656$$ 0.652704 0.652704
$$657$$ 0 0
$$658$$ 1.22668 1.22668
$$659$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$660$$ 0 0
$$661$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$662$$ 0 0
$$663$$ 0.532089 0.532089
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −1.53209 −1.53209
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −1.65270 −1.65270
$$673$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0.773318 0.773318
$$677$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$678$$ 0.652704 0.652704
$$679$$ 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
$$691$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$692$$ 0 0
$$693$$ 2.87939 2.87939
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0.652704 0.652704
$$697$$ −1.53209 −1.53209
$$698$$ 0.347296 0.347296
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0.120615 0.120615
$$703$$ 0 0
$$704$$ 0.532089 0.532089
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0.652704 0.652704
$$708$$ 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
$$712$$ −0.226682 −0.226682
$$713$$ 0 0
$$714$$ 1.00000 1.00000
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −0.347296 −0.347296
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 1.87939 1.87939
$$722$$ −0.347296 −0.347296
$$723$$ 1.87939 1.87939
$$724$$ −1.34730 −1.34730
$$725$$ 0 0
$$726$$ 0.467911 0.467911
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ −0.426022 −0.426022
$$729$$ 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
$$737$$ −2.34730 −2.34730
$$738$$ −0.347296 −0.347296
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −0.694593 −0.694593
$$747$$ 0 0
$$748$$ −2.06418 −2.06418
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 1.22668 1.22668
$$753$$ 0.347296 0.347296
$$754$$ 0.120615 0.120615
$$755$$ 0 0
$$756$$ −1.65270 −1.65270
$$757$$ 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
$$763$$ −0.652704 −0.652704
$$764$$ 1.75877 1.75877
$$765$$ 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 $$\pi$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0.120615 0.120615
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 1.00000 1.00000
$$784$$ 1.65270 1.65270
$$785$$ 0 0
$$786$$ 0.652704 0.652704
$$787$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$788$$ 0 0
$$789$$ −1.00000 −1.00000
$$790$$ 0 0
$$791$$ −3.53209 −3.53209
$$792$$ −1.00000 −1.00000
$$793$$ 0 0
$$794$$ 0.347296 0.347296
$$795$$ 0 0
$$796$$ −0.305407 −0.305407
$$797$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$798$$ 0 0
$$799$$ −2.87939 −2.87939
$$800$$ 0 0
$$801$$ −0.347296 −0.347296
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 1.34730 1.34730
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.53209 1.53209
$$808$$ −0.226682 −0.226682
$$809$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$810$$ 0 0
$$811$$ 1.53209 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$812$$ −1.65270 −1.65270
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 1.00000 1.00000
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −0.652704 −0.652704
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0.347296 0.347296
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ −0.652704 −0.652704
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ −1.53209 −1.53209
$$832$$ −0.120615 −0.120615
$$833$$ −3.87939 −3.87939
$$834$$ −0.652704 −0.652704
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$840$$ 0 0
$$841$$ 1.00000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ −0.652704 −0.652704
$$847$$ −2.53209 −2.53209
$$848$$ 0 0
$$849$$ 1.00000 1.00000
$$850$$ 0 0
$$851$$ 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 $$\pi$$
$$858$$ −0.184793 −0.184793
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 1.87939 1.87939
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0.879385 0.879385
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −1.34730 −1.34730
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0.532089 0.532089
$$872$$ 0.226682 0.226682
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.00000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$878$$ −0.532089 −0.532089
$$879$$ 0.347296 0.347296
$$880$$ 0 0
$$881$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$882$$ −0.879385 −0.879385
$$883$$ 2.00000 2.00000 1.00000 $$0$$
1.00000 $$0$$
$$884$$ 0.467911 0.467911
$$885$$ 0 0
$$886$$ 0.120615 0.120615
$$887$$ 1.87939 1.87939 0.939693 0.342020i $$-0.111111\pi$$
0.939693 + 0.342020i $$0.111111\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.53209 −1.53209
$$892$$ −1.34730 −1.34730
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.87939 −1.87939
$$897$$ 0 0
$$898$$ −0.652704 −0.652704
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0.532089 0.532089
$$903$$ 0 0
$$904$$ 1.22668 1.22668
$$905$$ 0 0
$$906$$ −0.347296 −0.347296
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ −0.347296 −0.347296
$$910$$ 0 0
$$911$$ −0.347296 −0.347296 −0.173648 0.984808i $$-0.555556\pi$$
−0.173648 + 0.984808i $$0.555556\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0.652704 0.652704
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3.53209 −3.53209
$$918$$ −0.532089 −0.532089
$$919$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −0.347296 −0.347296
$$923$$ 0 0
$$924$$ 2.53209 2.53209
$$925$$ 0 0
$$926$$ −0.120615 −0.120615
$$927$$ −1.00000 −1.00000
$$928$$ 0.879385 0.879385
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.87939 −1.87939
$$934$$ −0.347296 −0.347296
$$935$$ 0 0
$$936$$ 0.226682 0.226682
$$937$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$938$$ 1.00000 1.00000
$$939$$ 1.87939 1.87939
$$940$$ 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
$$947$$ −1.53209 −1.53209 −0.766044 0.642788i $$-0.777778\pi$$
−0.766044 + 0.642788i $$0.777778\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0.347296 0.347296
$$952$$ 1.87939 1.87939
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −1.53209 −1.53209
$$958$$ −0.347296 −0.347296
$$959$$ −1.87939 −1.87939
$$960$$ 0 0
$$961$$ 1.00000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 1.65270 1.65270
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0.879385 0.879385
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$972$$ 0.879385 0.879385
$$973$$ 3.53209 3.53209
$$974$$ −0.694593 −0.694593
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0.532089 0.532089
$$980$$ 0 0
$$981$$ 0.347296 0.347296
$$982$$ −0.347296 −0.347296
$$983$$ 1.00000 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$984$$ −0.652704 −0.652704
$$985$$ 0 0
$$986$$ −0.532089 −0.532089
$$987$$ 3.53209 3.53209
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0.347296 0.347296 0.173648 0.984808i $$-0.444444\pi$$
0.173648 + 0.984808i $$0.444444\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ −0.120615 −0.120615
$$999$$ 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 2175.1.h.d.1826.2 yes 3
3.2 odd 2 2175.1.h.e.1826.2 yes 3
5.2 odd 4 2175.1.b.c.2174.3 6
5.3 odd 4 2175.1.b.c.2174.4 6
5.4 even 2 2175.1.h.f.1826.2 yes 3
15.2 even 4 2175.1.b.d.2174.4 6
15.8 even 4 2175.1.b.d.2174.3 6
15.14 odd 2 2175.1.h.c.1826.2 3
29.28 even 2 2175.1.h.e.1826.2 yes 3
87.86 odd 2 CM 2175.1.h.d.1826.2 yes 3
145.28 odd 4 2175.1.b.d.2174.3 6
145.57 odd 4 2175.1.b.d.2174.4 6
145.144 even 2 2175.1.h.c.1826.2 3
435.173 even 4 2175.1.b.c.2174.4 6
435.347 even 4 2175.1.b.c.2174.3 6
435.434 odd 2 2175.1.h.f.1826.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.1.b.c.2174.3 6 5.2 odd 4
2175.1.b.c.2174.3 6 435.347 even 4
2175.1.b.c.2174.4 6 5.3 odd 4
2175.1.b.c.2174.4 6 435.173 even 4
2175.1.b.d.2174.3 6 15.8 even 4
2175.1.b.d.2174.3 6 145.28 odd 4
2175.1.b.d.2174.4 6 15.2 even 4
2175.1.b.d.2174.4 6 145.57 odd 4
2175.1.h.c.1826.2 3 15.14 odd 2
2175.1.h.c.1826.2 3 145.144 even 2
2175.1.h.d.1826.2 yes 3 1.1 even 1 trivial
2175.1.h.d.1826.2 yes 3 87.86 odd 2 CM
2175.1.h.e.1826.2 yes 3 3.2 odd 2
2175.1.h.e.1826.2 yes 3 29.28 even 2
2175.1.h.f.1826.2 yes 3 5.4 even 2
2175.1.h.f.1826.2 yes 3 435.434 odd 2