# Properties

 Label 1734.2.b.f.577.1 Level $1734$ Weight $2$ Character 1734.577 Analytic conductor $13.846$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(577,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1734.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.b (of order $$2$$, degree $$1$$, not 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: $$13.8460597105$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 577.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1734.577 Dual form 1734.2.b.f.577.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000 q^{13} +2.00000i q^{14} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{21} -6.00000i q^{23} -1.00000i q^{24} +5.00000 q^{25} +2.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} +10.0000i q^{31} +1.00000 q^{32} -1.00000 q^{36} -8.00000i q^{37} +4.00000 q^{38} -2.00000i q^{39} +6.00000i q^{41} +2.00000 q^{42} +4.00000 q^{43} -6.00000i q^{46} +12.0000 q^{47} -1.00000i q^{48} +3.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +1.00000i q^{54} +2.00000i q^{56} -4.00000i q^{57} +12.0000 q^{59} +8.00000i q^{61} +10.0000i q^{62} -2.00000i q^{63} +1.00000 q^{64} -4.00000 q^{67} -6.00000 q^{69} -6.00000i q^{71} -1.00000 q^{72} -2.00000i q^{73} -8.00000i q^{74} -5.00000i q^{75} +4.00000 q^{76} -2.00000i q^{78} -10.0000i q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -18.0000 q^{89} +4.00000i q^{91} -6.00000i q^{92} +10.0000 q^{93} +12.0000 q^{94} -1.00000i q^{96} -14.0000i q^{97} +3.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 4 q^{13} + 2 q^{16} - 2 q^{18} + 8 q^{19} + 4 q^{21} + 10 q^{25} + 4 q^{26} + 2 q^{32} - 2 q^{36} + 8 q^{38} + 4 q^{42} + 8 q^{43} + 24 q^{47} + 6 q^{49} + 10 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 2 q^{64} - 8 q^{67} - 12 q^{69} - 2 q^{72} + 8 q^{76} + 2 q^{81} - 24 q^{83} + 4 q^{84} + 8 q^{86} - 36 q^{89} + 20 q^{93} + 24 q^{94} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 4 * q^13 + 2 * q^16 - 2 * q^18 + 8 * q^19 + 4 * q^21 + 10 * q^25 + 4 * q^26 + 2 * q^32 - 2 * q^36 + 8 * q^38 + 4 * q^42 + 8 * q^43 + 24 * q^47 + 6 * q^49 + 10 * q^50 + 4 * q^52 - 12 * q^53 + 24 * q^59 + 2 * q^64 - 8 * q^67 - 12 * q^69 - 2 * q^72 + 8 * q^76 + 2 * q^81 - 24 * q^83 + 4 * q^84 + 8 * q^86 - 36 * q^89 + 20 * q^93 + 24 * q^94 + 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$$.

 $$n$$ $$1157$$ $$1159$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 0.707107
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ − 1.00000i − 0.408248i
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 2.00000i 0.534522i
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ 5.00000 1.00000
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 10.0000i 1.79605i 0.439941 + 0.898027i $$0.354999\pi$$
−0.439941 + 0.898027i $$0.645001\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 4.00000 0.648886
$$39$$ − 2.00000i − 0.320256i
$$40$$ 0 0
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ − 6.00000i − 0.884652i
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 0 0
$$56$$ 2.00000i 0.267261i
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 8.00000i 1.02430i 0.858898 + 0.512148i $$0.171150\pi$$
−0.858898 + 0.512148i $$0.828850\pi$$
$$62$$ 10.0000i 1.27000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ − 6.00000i − 0.712069i −0.934473 0.356034i $$-0.884129\pi$$
0.934473 0.356034i $$-0.115871\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ − 8.00000i − 0.929981i
$$75$$ − 5.00000i − 0.577350i
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ − 10.0000i − 1.12509i −0.826767 0.562544i $$-0.809823\pi$$
0.826767 0.562544i $$-0.190177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 4.00000i 0.419314i
$$92$$ − 6.00000i − 0.625543i
$$93$$ 10.0000 1.03695
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 5.00000 0.500000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 20.0000i 1.91565i 0.287348 + 0.957826i $$0.407226\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 2.00000i 0.188982i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ − 4.00000i − 0.374634i
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 8.00000i 0.724286i
$$123$$ 6.00000 0.541002
$$124$$ 10.0000i 0.898027i
$$125$$ 0 0
$$126$$ − 2.00000i − 0.178174i
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ − 4.00000i − 0.352180i
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 8.00000i 0.693688i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ − 8.00000i − 0.678551i −0.940687 0.339276i $$-0.889818\pi$$
0.940687 0.339276i $$-0.110182\pi$$
$$140$$ 0 0
$$141$$ − 12.0000i − 1.01058i
$$142$$ − 6.00000i − 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ − 2.00000i − 0.165521i
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 8.00000i − 0.657596i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ − 5.00000i − 0.408248i
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ − 2.00000i − 0.160128i
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 6.00000i 0.475831i
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 1.00000 0.0785674
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 6.00000i 0.468521i
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 10.0000i 0.755929i
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ −18.0000 −1.34916
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 8.00000i 0.594635i 0.954779 + 0.297318i $$0.0960920\pi$$
−0.954779 + 0.297318i $$0.903908\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 8.00000 0.591377
$$184$$ − 6.00000i − 0.442326i
$$185$$ 0 0
$$186$$ 10.0000 0.733236
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ − 14.0000i − 1.00514i
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ − 2.00000i − 0.141776i −0.997484 0.0708881i $$-0.977417\pi$$
0.997484 0.0708881i $$-0.0225833\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 4.00000i 0.282138i
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 6.00000i 0.417029i
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000i 0.550743i 0.961338 + 0.275371i $$0.0888008\pi$$
−0.961338 + 0.275371i $$0.911199\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −6.00000 −0.411113
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000i 0.0680414i
$$217$$ −20.0000 −1.35769
$$218$$ 20.0000i 1.35457i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −8.00000 −0.536925
$$223$$ 28.0000 1.87502 0.937509 0.347960i $$-0.113126\pi$$
0.937509 + 0.347960i $$0.113126\pi$$
$$224$$ 2.00000i 0.133631i
$$225$$ −5.00000 −0.333333
$$226$$ − 6.00000i − 0.399114i
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 30.0000i 1.96537i 0.185296 + 0.982683i $$0.440675\pi$$
−0.185296 + 0.982683i $$0.559325\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 10.0000i 0.644157i 0.946713 + 0.322078i $$0.104381\pi$$
−0.946713 + 0.322078i $$0.895619\pi$$
$$242$$ 11.0000 0.707107
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 8.00000i 0.512148i
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 8.00000 0.509028
$$248$$ 10.0000i 0.635001i
$$249$$ 12.0000i 0.760469i
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 16.0000 0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000i 0.490511i
$$267$$ 18.0000i 1.10158i
$$268$$ −4.00000 −0.244339
$$269$$ − 24.0000i − 1.46331i −0.681677 0.731653i $$-0.738749\pi$$
0.681677 0.731653i $$-0.261251\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 4.00000 0.242091
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 4.00000i 0.240337i 0.992754 + 0.120168i $$0.0383434\pi$$
−0.992754 + 0.120168i $$0.961657\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ − 10.0000i − 0.598684i
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ − 6.00000i − 0.356034i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ 0 0
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ − 2.00000i − 0.117041i
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ 0 0
$$296$$ − 8.00000i − 0.464991i
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ − 12.0000i − 0.693978i
$$300$$ − 5.00000i − 0.288675i
$$301$$ 8.00000i 0.461112i
$$302$$ −8.00000 −0.460348
$$303$$ − 6.00000i − 0.344691i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 4.00000i 0.227552i
$$310$$ 0 0
$$311$$ − 18.0000i − 1.02069i −0.859971 0.510343i $$-0.829518\pi$$
0.859971 0.510343i $$-0.170482\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ − 10.0000i − 0.562544i
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 10.0000 0.554700
$$326$$ 20.0000i 1.10770i
$$327$$ 20.0000 1.10600
$$328$$ 6.00000i 0.331295i
$$329$$ 24.0000i 1.32316i
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 8.00000i 0.438397i
$$334$$ − 18.0000i − 0.984916i
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 20.0000i 1.07990i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 10.0000i 0.534522i
$$351$$ 2.00000i 0.106752i
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ − 12.0000i − 0.637793i
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 8.00000i 0.420471i
$$363$$ − 11.0000i − 0.577350i
$$364$$ 4.00000i 0.209657i
$$365$$ 0 0
$$366$$ 8.00000 0.418167
$$367$$ − 10.0000i − 0.521996i −0.965339 0.260998i $$-0.915948\pi$$
0.965339 0.260998i $$-0.0840516\pi$$
$$368$$ − 6.00000i − 0.312772i
$$369$$ − 6.00000i − 0.312348i
$$370$$ 0 0
$$371$$ − 12.0000i − 0.623009i
$$372$$ 10.0000 0.518476
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 28.0000i 1.43826i 0.694874 + 0.719132i $$0.255460\pi$$
−0.694874 + 0.719132i $$0.744540\pi$$
$$380$$ 0 0
$$381$$ 8.00000i 0.409852i
$$382$$ −12.0000 −0.613973
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 0 0
$$386$$ 14.0000i 0.712581i
$$387$$ −4.00000 −0.203331
$$388$$ − 14.0000i − 0.710742i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ − 12.0000i − 0.604551i
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 16.0000i − 0.803017i −0.915855 0.401508i $$-0.868486\pi$$
0.915855 0.401508i $$-0.131514\pi$$
$$398$$ − 2.00000i − 0.100251i
$$399$$ 8.00000 0.400501
$$400$$ 5.00000 0.250000
$$401$$ − 18.0000i − 0.898877i −0.893311 0.449439i $$-0.851624\pi$$
0.893311 0.449439i $$-0.148376\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 20.0000i 0.996271i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ 6.00000i 0.295958i
$$412$$ −4.00000 −0.197066
$$413$$ 24.0000i 1.18096i
$$414$$ 6.00000i 0.294884i
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ − 36.0000i − 1.75872i −0.476162 0.879358i $$-0.657972\pi$$
0.476162 0.879358i $$-0.342028\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ −12.0000 −0.583460
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ −16.0000 −0.774294
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 30.0000i − 1.44505i −0.691345 0.722525i $$-0.742982\pi$$
0.691345 0.722525i $$-0.257018\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ −20.0000 −0.960031
$$435$$ 0 0
$$436$$ 20.0000i 0.957826i
$$437$$ − 24.0000i − 1.14808i
$$438$$ −2.00000 −0.0955637
$$439$$ − 26.0000i − 1.24091i −0.784241 0.620456i $$-0.786947\pi$$
0.784241 0.620456i $$-0.213053\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ 28.0000 1.32584
$$447$$ 6.00000i 0.283790i
$$448$$ 2.00000i 0.0944911i
$$449$$ 6.00000i 0.283158i 0.989927 + 0.141579i $$0.0452178\pi$$
−0.989927 + 0.141579i $$0.954782\pi$$
$$450$$ −5.00000 −0.235702
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000i 0.563188i
$$455$$ 0 0
$$456$$ − 4.00000i − 0.187317i
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 30.0000i 1.38972i
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ − 8.00000i − 0.369406i
$$470$$ 0 0
$$471$$ 10.0000i 0.460776i
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ 20.0000 0.917663
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ −12.0000 −0.548867
$$479$$ − 6.00000i − 0.274147i −0.990561 0.137073i $$-0.956230\pi$$
0.990561 0.137073i $$-0.0437697\pi$$
$$480$$ 0 0
$$481$$ − 16.0000i − 0.729537i
$$482$$ 10.0000i 0.455488i
$$483$$ − 12.0000i − 0.546019i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 10.0000i − 0.453143i −0.973995 0.226572i $$-0.927248\pi$$
0.973995 0.226572i $$-0.0727517\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 0 0
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 10.0000i 0.449013i
$$497$$ 12.0000 0.538274
$$498$$ 12.0000i 0.537733i
$$499$$ 32.0000i 1.43252i 0.697835 + 0.716258i $$0.254147\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ 0 0
$$501$$ −18.0000 −0.804181
$$502$$ 12.0000 0.535586
$$503$$ − 6.00000i − 0.267527i −0.991013 0.133763i $$-0.957294\pi$$
0.991013 0.133763i $$-0.0427062\pi$$
$$504$$ − 2.00000i − 0.0890871i
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ −8.00000 −0.354943
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000i 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ − 4.00000i − 0.176090i
$$517$$ 0 0
$$518$$ 16.0000 0.703000
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000i 0.262865i 0.991325 + 0.131432i $$0.0419576\pi$$
−0.991325 + 0.131432i $$0.958042\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 10.0000 0.436436
$$526$$ −12.0000 −0.523225
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 8.00000i 0.346844i
$$533$$ 12.0000i 0.519778i
$$534$$ 18.0000i 0.778936i
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 24.0000i − 1.03471i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16.0000i 0.687894i 0.938989 + 0.343947i $$0.111764\pi$$
−0.938989 + 0.343947i $$0.888236\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 8.00000 0.343313
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ − 8.00000i − 0.341432i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −6.00000 −0.255377
$$553$$ 20.0000 0.850487
$$554$$ 4.00000i 0.169944i
$$555$$ 0 0
$$556$$ − 8.00000i − 0.339276i
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ − 10.0000i − 0.423334i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ − 12.0000i − 0.505291i
$$565$$ 0 0
$$566$$ − 16.0000i − 0.672530i
$$567$$ 2.00000i 0.0839921i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 20.0000i 0.836974i 0.908223 + 0.418487i $$0.137439\pi$$
−0.908223 + 0.418487i $$0.862561\pi$$
$$572$$ 0 0
$$573$$ 12.0000i 0.501307i
$$574$$ −12.0000 −0.500870
$$575$$ − 30.0000i − 1.25109i
$$576$$ −1.00000 −0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ − 24.0000i − 0.995688i
$$582$$ −14.0000 −0.580319
$$583$$ 0 0
$$584$$ − 2.00000i − 0.0827606i
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 40.0000i 1.64817i
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ − 8.00000i − 0.328798i
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −2.00000 −0.0818546
$$598$$ − 12.0000i − 0.490716i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ − 5.00000i − 0.204124i
$$601$$ − 22.0000i − 0.897399i −0.893683 0.448699i $$-0.851887\pi$$
0.893683 0.448699i $$-0.148113\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ − 6.00000i − 0.243733i
$$607$$ 10.0000i 0.405887i 0.979190 + 0.202944i $$0.0650509\pi$$
−0.979190 + 0.202944i $$0.934949\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ − 28.0000i − 1.12542i −0.826656 0.562708i $$-0.809760\pi$$
0.826656 0.562708i $$-0.190240\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ − 18.0000i − 0.721734i
$$623$$ − 36.0000i − 1.44231i
$$624$$ − 2.00000i − 0.0800641i
$$625$$ 25.0000 1.00000
$$626$$ 14.0000i 0.559553i
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 8.00000 0.317971
$$634$$ 12.0000i 0.476581i
$$635$$ 0 0
$$636$$ 6.00000i 0.237915i
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 6.00000i 0.237356i
$$640$$ 0 0
$$641$$ − 30.0000i − 1.18493i −0.805597 0.592464i $$-0.798155\pi$$
0.805597 0.592464i $$-0.201845\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 10.0000 0.392232
$$651$$ 20.0000i 0.783862i
$$652$$ 20.0000i 0.783260i
$$653$$ − 24.0000i − 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 20.0000 0.782062
$$655$$ 0 0
$$656$$ 6.00000i 0.234261i
$$657$$ 2.00000i 0.0780274i
$$658$$ 24.0000i 0.935617i
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 8.00000i 0.309994i
$$667$$ 0 0
$$668$$ − 18.0000i − 0.696441i
$$669$$ − 28.0000i − 1.08254i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ 22.0000i 0.847408i
$$675$$ 5.00000i 0.192450i
$$676$$ −9.00000 −0.346154
$$677$$ 12.0000i 0.461197i 0.973049 + 0.230599i $$0.0740685\pi$$
−0.973049 + 0.230599i $$0.925932\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 28.0000 1.07454
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 20.0000i 0.763604i
$$687$$ 14.0000i 0.534133i
$$688$$ 4.00000 0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 32.0000i 1.21734i 0.793424 + 0.608669i $$0.208296\pi$$
−0.793424 + 0.608669i $$0.791704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ − 12.0000i − 0.455514i
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −26.0000 −0.984115
$$699$$ 30.0000 1.13470
$$700$$ 10.0000i 0.377964i
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 32.0000i − 1.20690i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 12.0000i 0.451306i
$$708$$ − 12.0000i − 0.450988i
$$709$$ − 20.0000i − 0.751116i −0.926799 0.375558i $$-0.877451\pi$$
0.926799 0.375558i $$-0.122549\pi$$
$$710$$ 0 0
$$711$$ 10.0000i 0.375029i
$$712$$ −18.0000 −0.674579
$$713$$ 60.0000 2.24702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ − 6.00000i − 0.223762i −0.993722 0.111881i $$-0.964312\pi$$
0.993722 0.111881i $$-0.0356876\pi$$
$$720$$ 0 0
$$721$$ − 8.00000i − 0.297936i
$$722$$ −3.00000 −0.111648
$$723$$ 10.0000 0.371904
$$724$$ 8.00000i 0.297318i
$$725$$ 0 0
$$726$$ − 11.0000i − 0.408248i
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 4.00000i 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 8.00000 0.295689
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ − 10.0000i − 0.369107i
$$735$$ 0 0
$$736$$ − 6.00000i − 0.221163i
$$737$$ 0 0
$$738$$ − 6.00000i − 0.220863i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ − 8.00000i − 0.293887i
$$742$$ − 12.0000i − 0.440534i
$$743$$ − 42.0000i − 1.54083i −0.637542 0.770415i $$-0.720049\pi$$
0.637542 0.770415i $$-0.279951\pi$$
$$744$$ 10.0000 0.366618
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 2.00000i − 0.0729810i −0.999334 0.0364905i $$-0.988382\pi$$
0.999334 0.0364905i $$-0.0116179\pi$$
$$752$$ 12.0000 0.437595
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ −40.0000 −1.44810
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 24.0000 0.866590
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 18.0000i 0.648254i
$$772$$ 14.0000i 0.503871i
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 50.0000i 1.79605i
$$776$$ − 14.0000i − 0.502571i
$$777$$ − 16.0000i − 0.573997i
$$778$$ 30.0000 1.07555
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ 12.0000i 0.427211i
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 16.0000i 0.568177i
$$794$$ − 16.0000i − 0.567819i
$$795$$ 0 0
$$796$$ − 2.00000i − 0.0708881i
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 8.00000 0.283197
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ 18.0000 0.635999
$$802$$ − 18.0000i − 0.635602i
$$803$$ 0 0
$$804$$ 4.00000i 0.141069i
$$805$$ 0 0
$$806$$ 20.0000i 0.704470i
$$807$$ −24.0000 −0.844840
$$808$$ 6.00000 0.211079
$$809$$ 6.00000i 0.210949i 0.994422 + 0.105474i $$0.0336361\pi$$
−0.994422 + 0.105474i $$0.966364\pi$$
$$810$$ 0 0
$$811$$ 4.00000i 0.140459i 0.997531 + 0.0702295i $$0.0223732\pi$$
−0.997531 + 0.0702295i $$0.977627\pi$$
$$812$$ 0 0
$$813$$ 16.0000i 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 38.0000 1.32864
$$819$$ − 4.00000i − 0.139771i
$$820$$ 0 0
$$821$$ 36.0000i 1.25641i 0.778048 + 0.628204i $$0.216210\pi$$
−0.778048 + 0.628204i $$0.783790\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ − 34.0000i − 1.18517i −0.805510 0.592583i $$-0.798108\pi$$
0.805510 0.592583i $$-0.201892\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 24.0000i 0.835067i
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ 4.00000 0.138758
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ − 36.0000i − 1.24360i
$$839$$ − 18.0000i − 0.621429i −0.950503 0.310715i $$-0.899432\pi$$
0.950503 0.310715i $$-0.100568\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ −22.0000 −0.758170
$$843$$ 30.0000i 1.03325i
$$844$$ 8.00000i 0.275371i
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 22.0000i 0.755929i
$$848$$ −6.00000 −0.206041
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ −6.00000 −0.205557
$$853$$ 28.0000i 0.958702i 0.877623 + 0.479351i $$0.159128\pi$$
−0.877623 + 0.479351i $$0.840872\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.00000i 0.204956i 0.994735 + 0.102478i $$0.0326771\pi$$
−0.994735 + 0.102478i $$0.967323\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 12.0000i 0.408959i
$$862$$ − 30.0000i − 1.02180i
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ −20.0000 −0.678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 20.0000i 0.677285i
$$873$$ 14.0000i 0.473828i
$$874$$ − 24.0000i − 0.811812i
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ − 4.00000i − 0.135070i −0.997717 0.0675352i $$-0.978487\pi$$
0.997717 0.0675352i $$-0.0215135\pi$$
$$878$$ − 26.0000i − 0.877457i
$$879$$ − 18.0000i − 0.607125i
$$880$$ 0 0
$$881$$ 54.0000i 1.81931i 0.415369 + 0.909653i $$0.363653\pi$$
−0.415369 + 0.909653i $$0.636347\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 18.0000i 0.604381i 0.953248 + 0.302190i $$0.0977178\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ − 16.0000i − 0.536623i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 28.0000 0.937509
$$893$$ 48.0000 1.60626
$$894$$ 6.00000i 0.200670i
$$895$$ 0 0
$$896$$ 2.00000i 0.0668153i
$$897$$ −12.0000 −0.400668
$$898$$ 6.00000i 0.200223i
$$899$$ 0 0
$$900$$ −5.00000 −0.166667
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ − 6.00000i − 0.199557i
$$905$$ 0 0
$$906$$ 8.00000i 0.265782i
$$907$$ − 40.0000i − 1.32818i −0.747653 0.664089i $$-0.768820\pi$$
0.747653 0.664089i $$-0.231180\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 6.00000i 0.198789i 0.995048 + 0.0993944i $$0.0316906\pi$$
−0.995048 + 0.0993944i $$0.968309\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ − 20.0000i − 0.659022i
$$922$$ −30.0000 −0.987997
$$923$$ − 12.0000i − 0.394985i
$$924$$ 0 0
$$925$$ − 40.0000i − 1.31519i
$$926$$ 20.0000 0.657241
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ − 18.0000i − 0.590561i −0.955411 0.295280i $$-0.904587\pi$$
0.955411 0.295280i $$-0.0954131\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 30.0000i 0.982683i
$$933$$ −18.0000 −0.589294
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 12.0000i 0.391189i 0.980685 + 0.195594i $$0.0626636\pi$$
−0.980685 + 0.195594i $$0.937336\pi$$
$$942$$ 10.0000i 0.325818i
$$943$$ 36.0000 1.17232
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ − 4.00000i − 0.129845i
$$950$$ 20.0000 0.648886
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ − 6.00000i − 0.193851i
$$959$$ − 12.0000i − 0.387500i
$$960$$ 0 0
$$961$$ −69.0000 −2.22581
$$962$$ − 16.0000i − 0.515861i
$$963$$ 0 0
$$964$$ 10.0000i 0.322078i
$$965$$ 0 0
$$966$$ − 12.0000i − 0.386094i
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 16.0000 0.512936
$$974$$ − 10.0000i − 0.320421i
$$975$$ − 10.0000i − 0.320256i
$$976$$ 8.00000i 0.256074i
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ 0 0
$$981$$ − 20.0000i − 0.638551i
$$982$$ 36.0000 1.14881
$$983$$ − 6.00000i − 0.191370i −0.995412 0.0956851i $$-0.969496\pi$$
0.995412 0.0956851i $$-0.0305042\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24.0000 0.763928
$$988$$ 8.00000 0.254514
$$989$$ − 24.0000i − 0.763156i
$$990$$ 0 0
$$991$$ − 2.00000i − 0.0635321i −0.999495 0.0317660i $$-0.989887\pi$$
0.999495 0.0317660i $$-0.0101131\pi$$
$$992$$ 10.0000i 0.317500i
$$993$$ 20.0000i 0.634681i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 12.0000i 0.380235i
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 32.0000i 1.01294i
$$999$$ 8.00000 0.253109
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 1734.2.b.f.577.1 2
17.2 even 8 1734.2.f.b.829.1 4
17.4 even 4 1734.2.a.b.1.1 1
17.8 even 8 1734.2.f.b.1483.1 4
17.9 even 8 1734.2.f.b.1483.2 4
17.13 even 4 102.2.a.b.1.1 1
17.15 even 8 1734.2.f.b.829.2 4
17.16 even 2 inner 1734.2.b.f.577.2 2
51.38 odd 4 5202.2.a.j.1.1 1
51.47 odd 4 306.2.a.c.1.1 1
68.47 odd 4 816.2.a.d.1.1 1
85.13 odd 4 2550.2.d.g.2449.2 2
85.47 odd 4 2550.2.d.g.2449.1 2
85.64 even 4 2550.2.a.u.1.1 1
119.13 odd 4 4998.2.a.d.1.1 1
136.13 even 4 3264.2.a.i.1.1 1
136.115 odd 4 3264.2.a.w.1.1 1
204.47 even 4 2448.2.a.i.1.1 1
255.149 odd 4 7650.2.a.j.1.1 1
408.149 odd 4 9792.2.a.bg.1.1 1
408.251 even 4 9792.2.a.ba.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.b.1.1 1 17.13 even 4
306.2.a.c.1.1 1 51.47 odd 4
816.2.a.d.1.1 1 68.47 odd 4
1734.2.a.b.1.1 1 17.4 even 4
1734.2.b.f.577.1 2 1.1 even 1 trivial
1734.2.b.f.577.2 2 17.16 even 2 inner
1734.2.f.b.829.1 4 17.2 even 8
1734.2.f.b.829.2 4 17.15 even 8
1734.2.f.b.1483.1 4 17.8 even 8
1734.2.f.b.1483.2 4 17.9 even 8
2448.2.a.i.1.1 1 204.47 even 4
2550.2.a.u.1.1 1 85.64 even 4
2550.2.d.g.2449.1 2 85.47 odd 4
2550.2.d.g.2449.2 2 85.13 odd 4
3264.2.a.i.1.1 1 136.13 even 4
3264.2.a.w.1.1 1 136.115 odd 4
4998.2.a.d.1.1 1 119.13 odd 4
5202.2.a.j.1.1 1 51.38 odd 4
7650.2.a.j.1.1 1 255.149 odd 4
9792.2.a.ba.1.1 1 408.251 even 4
9792.2.a.bg.1.1 1 408.149 odd 4