Properties

 Label 847.2.a.c.1.1 Level $847$ Weight $2$ Character 847.1 Self dual yes Analytic conductor $6.763$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

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: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{12} +4.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} -2.00000 q^{19} -6.00000 q^{20} -1.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +5.00000 q^{31} -3.00000 q^{35} +4.00000 q^{36} +11.0000 q^{37} +4.00000 q^{39} -6.00000 q^{41} -8.00000 q^{43} -6.00000 q^{45} +4.00000 q^{48} +1.00000 q^{49} +6.00000 q^{51} -8.00000 q^{52} -6.00000 q^{53} -2.00000 q^{57} -9.00000 q^{59} -6.00000 q^{60} +10.0000 q^{61} +2.00000 q^{63} -8.00000 q^{64} +12.0000 q^{65} +5.00000 q^{67} -12.0000 q^{68} +3.00000 q^{69} +9.00000 q^{71} -2.00000 q^{73} +4.00000 q^{75} +4.00000 q^{76} +10.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{84} +18.0000 q^{85} +6.00000 q^{87} -3.00000 q^{89} -4.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} -6.00000 q^{95} -1.00000 q^{97} +O(q^{100})$$

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 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −2.00000 −0.577350
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 4.00000 1.00000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −6.00000 −1.34164
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 2.00000 0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 4.00000 0.666667
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ −6.00000 −0.894427
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ −8.00000 −1.10940
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ −6.00000 −0.774597
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ −8.00000 −1.00000
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ −12.0000 −1.45521
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 12.0000 1.34164
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 18.0000 1.95237
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −3.00000 −0.317999 −0.159000 0.987279i $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −6.00000 −0.625543
$$93$$ 5.00000 0.518476
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −8.00000 −0.800000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 10.0000 0.962250
$$109$$ −20.0000 −1.91565 −0.957826 0.287348i $$-0.907226\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 11.0000 1.04407
$$112$$ −4.00000 −0.377964
$$113$$ −3.00000 −0.282216 −0.141108 0.989994i $$-0.545067\pi$$
−0.141108 + 0.989994i $$0.545067\pi$$
$$114$$ 0 0
$$115$$ 9.00000 0.839254
$$116$$ −12.0000 −1.11417
$$117$$ −8.00000 −0.739600
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −6.00000 −0.541002
$$124$$ −10.0000 −0.898027
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 0 0
$$135$$ −15.0000 −1.29099
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −8.00000 −0.666667
$$145$$ 18.0000 1.49482
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ −22.0000 −1.80839
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ 15.0000 1.20483
$$156$$ −8.00000 −0.640513
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 16.0000 1.21999
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ −9.00000 −0.676481
$$178$$ 0 0
$$179$$ −15.0000 −1.12115 −0.560576 0.828103i $$-0.689420\pi$$
−0.560576 + 0.828103i $$0.689420\pi$$
$$180$$ 12.0000 0.894427
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ 33.0000 2.42621
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −27.0000 −1.95365 −0.976826 0.214036i $$-0.931339\pi$$
−0.976826 + 0.214036i $$0.931339\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 12.0000 0.859338
$$196$$ −2.00000 −0.142857
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ −12.0000 −0.840168
$$205$$ −18.0000 −1.25717
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 16.0000 1.10940
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 9.00000 0.616670
$$214$$ 0 0
$$215$$ −24.0000 −1.63679
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ −8.00000 −0.533333
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 5.00000 0.330409 0.165205 0.986259i $$-0.447172\pi$$
0.165205 + 0.986259i $$0.447172\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 18.0000 1.17170
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 12.0000 0.774597
$$241$$ 28.0000 1.80364 0.901819 0.432113i $$-0.142232\pi$$
0.901819 + 0.432113i $$0.142232\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ −20.0000 −1.28037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 18.0000 1.12720
$$256$$ 16.0000 1.00000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ −11.0000 −0.683507
$$260$$ −24.0000 −1.48842
$$261$$ −12.0000 −0.742781
$$262$$ 0 0
$$263$$ 30.0000 1.84988 0.924940 0.380114i $$-0.124115\pi$$
0.924940 + 0.380114i $$0.124115\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ 0 0
$$267$$ −3.00000 −0.183597
$$268$$ −10.0000 −0.610847
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 24.0000 1.45521
$$273$$ −4.00000 −0.242091
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0 0
$$283$$ −32.0000 −1.90220 −0.951101 0.308879i $$-0.900046\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ −18.0000 −1.06810
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −1.00000 −0.0586210
$$292$$ 4.00000 0.234082
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 0 0
$$295$$ −27.0000 −1.57200
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ −8.00000 −0.461880
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 12.0000 0.689382
$$304$$ −8.00000 −0.458831
$$305$$ 30.0000 1.71780
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 0 0
$$315$$ 6.00000 0.338062
$$316$$ −20.0000 −1.12509
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −24.0000 −1.34164
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ −2.00000 −0.111111
$$325$$ 16.0000 0.887520
$$326$$ 0 0
$$327$$ −20.0000 −1.10600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.00000 −0.0549650 −0.0274825 0.999622i $$-0.508749\pi$$
−0.0274825 + 0.999622i $$0.508749\pi$$
$$332$$ 24.0000 1.31717
$$333$$ −22.0000 −1.20559
$$334$$ 0 0
$$335$$ 15.0000 0.819538
$$336$$ −4.00000 −0.218218
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ −3.00000 −0.162938
$$340$$ −36.0000 −1.95237
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 9.00000 0.484544
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −20.0000 −1.06752
$$352$$ 0 0
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 27.0000 1.43301
$$356$$ 6.00000 0.317999
$$357$$ −6.00000 −0.317554
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 8.00000 0.419314
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ −10.0000 −0.518476
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 11.0000 0.565032 0.282516 0.959263i $$-0.408831\pi$$
0.282516 + 0.959263i $$0.408831\pi$$
$$380$$ 12.0000 0.615587
$$381$$ −2.00000 −0.102463
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 16.0000 0.813326
$$388$$ 2.00000 0.101535
$$389$$ 33.0000 1.67317 0.836583 0.547840i $$-0.184550\pi$$
0.836583 + 0.547840i $$0.184550\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ 30.0000 1.50946
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 16.0000 0.800000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 20.0000 0.996271
$$404$$ −24.0000 −1.19404
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −3.00000 −0.147979
$$412$$ 8.00000 0.394132
$$413$$ 9.00000 0.442861
$$414$$ 0 0
$$415$$ −36.0000 −1.76717
$$416$$ 0 0
$$417$$ −14.0000 −0.685583
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 6.00000 0.292770
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 24.0000 1.16417
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ −20.0000 −0.962250
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 18.0000 0.863034
$$436$$ 40.0000 1.91565
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ 9.00000 0.427603 0.213801 0.976877i $$-0.431415\pi$$
0.213801 + 0.976877i $$0.431415\pi$$
$$444$$ −22.0000 −1.04407
$$445$$ −9.00000 −0.426641
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 8.00000 0.377964
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 10.0000 0.469841
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 0 0
$$459$$ −30.0000 −1.40028
$$460$$ −18.0000 −0.839254
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ 24.0000 1.11417
$$465$$ 15.0000 0.695608
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 16.0000 0.739600
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ 12.0000 0.550019
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ 44.0000 2.00623
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −3.00000 −0.136223
$$486$$ 0 0
$$487$$ 11.0000 0.498458 0.249229 0.968445i $$-0.419823\pi$$
0.249229 + 0.968445i $$0.419823\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ −9.00000 −0.403705
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 6.00000 0.268328
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 36.0000 1.60198
$$506$$ 0 0
$$507$$ 3.00000 0.133235
$$508$$ 4.00000 0.177471
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ 10.0000 0.441511
$$514$$ 0 0
$$515$$ −12.0000 −0.528783
$$516$$ 16.0000 0.704361
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.00000 0.131432 0.0657162 0.997838i $$-0.479067\pi$$
0.0657162 + 0.997838i $$0.479067\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ 30.0000 1.30682
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 18.0000 0.781133
$$532$$ −4.00000 −0.173422
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ 0 0
$$537$$ −15.0000 −0.647298
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 30.0000 1.29099
$$541$$ 16.0000 0.687894 0.343947 0.938989i $$-0.388236\pi$$
0.343947 + 0.938989i $$0.388236\pi$$
$$542$$ 0 0
$$543$$ −7.00000 −0.300399
$$544$$ 0 0
$$545$$ −60.0000 −2.57012
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −20.0000 −0.853579
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 0 0
$$555$$ 33.0000 1.40077
$$556$$ 28.0000 1.18746
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ −32.0000 −1.35346
$$560$$ −12.0000 −0.507093
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ −9.00000 −0.378633
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −27.0000 −1.12794
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 16.0000 0.666667
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ −36.0000 −1.49482
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −24.0000 −0.992278
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ −10.0000 −0.412043
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 44.0000 1.80839
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ −18.0000 −0.737928
$$596$$ −12.0000 −0.491539
$$597$$ −16.0000 −0.654836
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 0 0
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 24.0000 0.970143
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ −19.0000 −0.763674 −0.381837 0.924230i $$-0.624709\pi$$
−0.381837 + 0.924230i $$0.624709\pi$$
$$620$$ −30.0000 −1.20483
$$621$$ −15.0000 −0.601929
$$622$$ 0 0
$$623$$ 3.00000 0.120192
$$624$$ 16.0000 0.640513
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 26.0000 1.03751
$$629$$ 66.0000 2.63159
$$630$$ 0 0
$$631$$ 11.0000 0.437903 0.218952 0.975736i $$-0.429736\pi$$
0.218952 + 0.975736i $$0.429736\pi$$
$$632$$ 0 0
$$633$$ −14.0000 −0.556450
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 12.0000 0.475831
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ −18.0000 −0.712069
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ −49.0000 −1.93237 −0.966186 0.257847i $$-0.916987\pi$$
−0.966186 + 0.257847i $$0.916987\pi$$
$$644$$ 6.00000 0.236433
$$645$$ −24.0000 −0.944999
$$646$$ 0 0
$$647$$ −33.0000 −1.29736 −0.648682 0.761060i $$-0.724679\pi$$
−0.648682 + 0.761060i $$0.724679\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −5.00000 −0.195965
$$652$$ −40.0000 −1.56652
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 18.0000 0.703318
$$656$$ −24.0000 −0.937043
$$657$$ 4.00000 0.156055
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ −49.0000 −1.90588 −0.952940 0.303160i $$-0.901958\pi$$
−0.952940 + 0.303160i $$0.901958\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ 6.00000 0.232670
$$666$$ 0 0
$$667$$ 18.0000 0.696963
$$668$$ 12.0000 0.464294
$$669$$ −19.0000 −0.734582
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 28.0000 1.07932 0.539660 0.841883i $$-0.318553\pi$$
0.539660 + 0.841883i $$0.318553\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ −6.00000 −0.230769
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 1.00000 0.0383765
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ −9.00000 −0.343872
$$686$$ 0 0
$$687$$ 5.00000 0.190762
$$688$$ −32.0000 −1.21999
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 35.0000 1.33146 0.665731 0.746191i $$-0.268120\pi$$
0.665731 + 0.746191i $$0.268120\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −42.0000 −1.59315
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 8.00000 0.302372
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −22.0000 −0.829746
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −12.0000 −0.451306
$$708$$ 18.0000 0.676481
$$709$$ −1.00000 −0.0375558 −0.0187779 0.999824i $$-0.505978\pi$$
−0.0187779 + 0.999824i $$0.505978\pi$$
$$710$$ 0 0
$$711$$ −20.0000 −0.750059
$$712$$ 0 0
$$713$$ 15.0000 0.561754
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 30.0000 1.12115
$$717$$ 12.0000 0.448148
$$718$$ 0 0
$$719$$ −39.0000 −1.45445 −0.727227 0.686397i $$-0.759191\pi$$
−0.727227 + 0.686397i $$0.759191\pi$$
$$720$$ −24.0000 −0.894427
$$721$$ 4.00000 0.148968
$$722$$ 0 0
$$723$$ 28.0000 1.04133
$$724$$ 14.0000 0.520306
$$725$$ 24.0000 0.891338
$$726$$ 0 0
$$727$$ 17.0000 0.630495 0.315248 0.949009i $$-0.397912\pi$$
0.315248 + 0.949009i $$0.397912\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ −20.0000 −0.739221
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 34.0000 1.25071 0.625355 0.780340i $$-0.284954\pi$$
0.625355 + 0.780340i $$0.284954\pi$$
$$740$$ −66.0000 −2.42621
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 0 0
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ −31.0000 −1.13121 −0.565603 0.824678i $$-0.691357\pi$$
−0.565603 + 0.824678i $$0.691357\pi$$
$$752$$ 0 0
$$753$$ −9.00000 −0.327978
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ −10.0000 −0.363696
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 0 0
$$763$$ 20.0000 0.724049
$$764$$ 54.0000 1.95365
$$765$$ −36.0000 −1.30158
$$766$$ 0 0
$$767$$ −36.0000 −1.29988
$$768$$ 16.0000 0.577350
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 28.0000 1.00774
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ −11.0000 −0.394623
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ −24.0000 −0.859338
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −30.0000 −1.07211
$$784$$ 4.00000 0.142857
$$785$$ −39.0000 −1.39197
$$786$$ 0 0
$$787$$ −50.0000 −1.78231 −0.891154 0.453701i $$-0.850103\pi$$
−0.891154 + 0.453701i $$0.850103\pi$$
$$788$$ 36.0000 1.28245
$$789$$ 30.0000 1.06803
$$790$$ 0 0
$$791$$ 3.00000 0.106668
$$792$$ 0 0
$$793$$ 40.0000 1.42044
$$794$$ 0 0
$$795$$ −18.0000 −0.638394
$$796$$ 32.0000 1.13421
$$797$$ −21.0000 −0.743858 −0.371929 0.928261i $$-0.621304\pi$$
−0.371929 + 0.928261i $$0.621304\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ −10.0000 −0.352673
$$805$$ −9.00000 −0.317208
$$806$$ 0 0
$$807$$ 30.0000 1.05605
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ 60.0000 2.10171
$$816$$ 24.0000 0.840168
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ 8.00000 0.279543
$$820$$ 36.0000 1.25717
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ 23.0000 0.801730 0.400865 0.916137i $$-0.368710\pi$$
0.400865 + 0.916137i $$0.368710\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ −32.0000 −1.10940
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ −18.0000 −0.622916
$$836$$ 0 0
$$837$$ −25.0000 −0.864126
$$838$$ 0 0
$$839$$ −15.0000 −0.517858 −0.258929 0.965896i $$-0.583369\pi$$
−0.258929 + 0.965896i $$0.583369\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −12.0000 −0.413302
$$844$$ 28.0000 0.963800
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −24.0000 −0.824163
$$849$$ −32.0000 −1.09824
$$850$$ 0 0
$$851$$ 33.0000 1.13123
$$852$$ −18.0000 −0.616670
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ 12.0000 0.410391
$$856$$ 0 0
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 0 0
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 48.0000 1.63679
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ −12.0000 −0.408485 −0.204242 0.978920i $$-0.565473\pi$$
−0.204242 + 0.978920i $$0.565473\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000 0.645274
$$868$$ 10.0000 0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 4.00000 0.135147
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ −9.00000 −0.303218 −0.151609 0.988441i $$-0.548445\pi$$
−0.151609 + 0.988441i $$0.548445\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −48.0000 −1.61441
$$885$$ −27.0000 −0.907595
$$886$$ 0 0
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 38.0000 1.27233
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −45.0000 −1.50418
$$896$$ 0 0
$$897$$ 12.0000 0.400668
$$898$$ 0 0
$$899$$ 30.0000 1.00056
$$900$$ 16.0000 0.533333
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −21.0000 −0.698064
$$906$$ 0 0
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ −24.0000 −0.796030
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 30.0000 0.991769
$$916$$ −10.0000 −0.330409
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 36.0000 1.18495
$$924$$ 0 0
$$925$$ 44.0000 1.44671
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ 0 0
$$939$$ −19.0000 −0.620042
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ −36.0000 −1.17170
$$945$$ 15.0000 0.487950
$$946$$ 0 0
$$947$$ −27.0000 −0.877382 −0.438691 0.898638i $$-0.644558\pi$$
−0.438691 + 0.898638i $$0.644558\pi$$
$$948$$ −20.0000 −0.649570
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 9.00000 0.291845
$$952$$ 0 0
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ 0 0
$$955$$ −81.0000 −2.62110
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ −24.0000 −0.774597
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ −56.0000 −1.80364
$$965$$ −42.0000 −1.35203
$$966$$ 0 0
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 0 0
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ −39.0000 −1.25157 −0.625785 0.779996i $$-0.715221\pi$$
−0.625785 + 0.779996i $$0.715221\pi$$
$$972$$ −32.0000 −1.02640
$$973$$ 14.0000 0.448819
$$974$$ 0 0
$$975$$ 16.0000 0.512410
$$976$$ 40.0000 1.28037
$$977$$ 9.00000 0.287936 0.143968 0.989582i $$-0.454014\pi$$
0.143968 + 0.989582i $$0.454014\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −6.00000 −0.191663
$$981$$ 40.0000 1.27710
$$982$$ 0 0
$$983$$ 33.0000 1.05254 0.526268 0.850319i $$-0.323591\pi$$
0.526268 + 0.850319i $$0.323591\pi$$
$$984$$ 0 0
$$985$$ −54.0000 −1.72058
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 16.0000 0.509028
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ −1.00000 −0.0317340
$$994$$ 0 0
$$995$$ −48.0000 −1.52170
$$996$$ 24.0000 0.760469
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ 0 0
$$999$$ −55.0000 −1.74012
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 847.2.a.c.1.1 1
3.2 odd 2 7623.2.a.i.1.1 1
7.6 odd 2 5929.2.a.d.1.1 1
11.2 odd 10 847.2.f.f.323.1 4
11.3 even 5 847.2.f.g.372.1 4
11.4 even 5 847.2.f.g.148.1 4
11.5 even 5 847.2.f.g.729.1 4
11.6 odd 10 847.2.f.f.729.1 4
11.7 odd 10 847.2.f.f.148.1 4
11.8 odd 10 847.2.f.f.372.1 4
11.9 even 5 847.2.f.g.323.1 4
11.10 odd 2 77.2.a.b.1.1 1
33.32 even 2 693.2.a.b.1.1 1
44.43 even 2 1232.2.a.d.1.1 1
55.32 even 4 1925.2.b.g.1849.1 2
55.43 even 4 1925.2.b.g.1849.2 2
55.54 odd 2 1925.2.a.f.1.1 1
77.10 even 6 539.2.e.e.177.1 2
77.32 odd 6 539.2.e.d.177.1 2
77.54 even 6 539.2.e.e.67.1 2
77.65 odd 6 539.2.e.d.67.1 2
77.76 even 2 539.2.a.b.1.1 1
88.21 odd 2 4928.2.a.i.1.1 1
88.43 even 2 4928.2.a.x.1.1 1
231.230 odd 2 4851.2.a.k.1.1 1
308.307 odd 2 8624.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.b.1.1 1 11.10 odd 2
539.2.a.b.1.1 1 77.76 even 2
539.2.e.d.67.1 2 77.65 odd 6
539.2.e.d.177.1 2 77.32 odd 6
539.2.e.e.67.1 2 77.54 even 6
539.2.e.e.177.1 2 77.10 even 6
693.2.a.b.1.1 1 33.32 even 2
847.2.a.c.1.1 1 1.1 even 1 trivial
847.2.f.f.148.1 4 11.7 odd 10
847.2.f.f.323.1 4 11.2 odd 10
847.2.f.f.372.1 4 11.8 odd 10
847.2.f.f.729.1 4 11.6 odd 10
847.2.f.g.148.1 4 11.4 even 5
847.2.f.g.323.1 4 11.9 even 5
847.2.f.g.372.1 4 11.3 even 5
847.2.f.g.729.1 4 11.5 even 5
1232.2.a.d.1.1 1 44.43 even 2
1925.2.a.f.1.1 1 55.54 odd 2
1925.2.b.g.1849.1 2 55.32 even 4
1925.2.b.g.1849.2 2 55.43 even 4
4851.2.a.k.1.1 1 231.230 odd 2
4928.2.a.i.1.1 1 88.21 odd 2
4928.2.a.x.1.1 1 88.43 even 2
5929.2.a.d.1.1 1 7.6 odd 2
7623.2.a.i.1.1 1 3.2 odd 2
8624.2.a.s.1.1 1 308.307 odd 2