# Properties

 Label 285.2.a.c.1.1 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.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: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 285.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -1.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -12.0000 q^{56} +1.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} +4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{84} +2.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 4.00000 1.06904
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.00000 −0.229416
$$20$$ −1.00000 −0.223607
$$21$$ −4.00000 −0.872872
$$22$$ 4.00000 0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −4.00000 −0.755929
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −4.00000 −0.696311
$$34$$ 2.00000 0.342997
$$35$$ 4.00000 0.676123
$$36$$ −1.00000 −0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −2.00000 −0.320256
$$40$$ −3.00000 −0.474342
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 1.00000 0.149071
$$46$$ −4.00000 −0.589768
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ −2.00000 −0.280056
$$52$$ −2.00000 −0.277350
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.00000 0.539360
$$56$$ −12.0000 −1.60357
$$57$$ 1.00000 0.132453
$$58$$ −2.00000 −0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ −4.00000 −0.492366
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 4.00000 0.481543
$$70$$ 4.00000 0.478091
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ 1.00000 0.114708
$$77$$ 16.0000 1.82337
$$78$$ −2.00000 −0.226455
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 2.00000 0.216930
$$86$$ 8.00000 0.862662
$$87$$ 2.00000 0.214423
$$88$$ −12.0000 −1.27920
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 8.00000 0.838628
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ −1.00000 −0.102598
$$96$$ −5.00000 −0.510310
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 4.00000 0.402015
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ −4.00000 −0.390360
$$106$$ −14.0000 −1.35980
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 6.00000 0.569495
$$112$$ −4.00000 −0.377964
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ −4.00000 −0.373002
$$116$$ 2.00000 0.185695
$$117$$ 2.00000 0.184900
$$118$$ 4.00000 0.368230
$$119$$ 8.00000 0.733359
$$120$$ 3.00000 0.273861
$$121$$ 5.00000 0.454545
$$122$$ 14.0000 1.26750
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 4.00000 0.356348
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −8.00000 −0.704361
$$130$$ 2.00000 0.175412
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −4.00000 −0.346844
$$134$$ −4.00000 −0.345547
$$135$$ −1.00000 −0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ −14.0000 −1.15865
$$147$$ −9.00000 −0.742307
$$148$$ 6.00000 0.493197
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 2.00000 0.161690
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 14.0000 1.11027
$$160$$ 5.00000 0.395285
$$161$$ −16.0000 −1.26098
$$162$$ 1.00000 0.0785674
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 6.00000 0.468521
$$165$$ −4.00000 −0.311400
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 12.0000 0.925820
$$169$$ −9.00000 −0.692308
$$170$$ 2.00000 0.153393
$$171$$ −1.00000 −0.0764719
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 4.00000 0.302372
$$176$$ −4.00000 −0.301511
$$177$$ −4.00000 −0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −14.0000 −1.03491
$$184$$ 12.0000 0.884652
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 12.0000 0.875190
$$189$$ −4.00000 −0.290957
$$190$$ −1.00000 −0.0725476
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ −2.00000 −0.143223
$$196$$ −9.00000 −0.642857
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 4.00000 0.282138
$$202$$ −10.0000 −0.703598
$$203$$ −8.00000 −0.561490
$$204$$ 2.00000 0.140028
$$205$$ −6.00000 −0.419058
$$206$$ 16.0000 1.11477
$$207$$ −4.00000 −0.278019
$$208$$ −2.00000 −0.138675
$$209$$ −4.00000 −0.276686
$$210$$ −4.00000 −0.276026
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 8.00000 0.545595
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ 14.0000 0.946032
$$220$$ −4.00000 −0.269680
$$221$$ 4.00000 0.269069
$$222$$ 6.00000 0.402694
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 20.0000 1.33631
$$225$$ 1.00000 0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −16.0000 −1.05272
$$232$$ 6.00000 0.393919
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −12.0000 −0.782794
$$236$$ −4.00000 −0.260378
$$237$$ −16.0000 −1.03931
$$238$$ 8.00000 0.518563
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −14.0000 −0.896258
$$245$$ 9.00000 0.574989
$$246$$ 6.00000 0.382546
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ −16.0000 −1.00591
$$254$$ −8.00000 −0.501965
$$255$$ −2.00000 −0.125245
$$256$$ −17.0000 −1.06250
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −24.0000 −1.49129
$$260$$ −2.00000 −0.124035
$$261$$ −2.00000 −0.123797
$$262$$ 4.00000 0.247121
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 12.0000 0.738549
$$265$$ −14.0000 −0.860013
$$266$$ −4.00000 −0.245256
$$267$$ 6.00000 0.367194
$$268$$ 4.00000 0.244339
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −8.00000 −0.484182
$$274$$ −6.00000 −0.362473
$$275$$ 4.00000 0.241209
$$276$$ −4.00000 −0.240772
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ −12.0000 −0.717137
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 8.00000 0.473050
$$287$$ −24.0000 −1.41668
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ 10.0000 0.586210
$$292$$ 14.0000 0.819288
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 4.00000 0.232889
$$296$$ 18.0000 1.04623
$$297$$ −4.00000 −0.232104
$$298$$ 6.00000 0.347571
$$299$$ −8.00000 −0.462652
$$300$$ 1.00000 0.0577350
$$301$$ 32.0000 1.84445
$$302$$ 0 0
$$303$$ 10.0000 0.574485
$$304$$ 1.00000 0.0573539
$$305$$ 14.0000 0.801638
$$306$$ 2.00000 0.114332
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −16.0000 −0.911685
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 4.00000 0.225374
$$316$$ −16.0000 −0.900070
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 14.0000 0.785081
$$319$$ −8.00000 −0.447914
$$320$$ 7.00000 0.391312
$$321$$ −12.0000 −0.669775
$$322$$ −16.0000 −0.891645
$$323$$ −2.00000 −0.111283
$$324$$ −1.00000 −0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ 2.00000 0.110600
$$328$$ 18.0000 0.993884
$$329$$ −48.0000 −2.64633
$$330$$ −4.00000 −0.220193
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ −8.00000 −0.437741
$$335$$ −4.00000 −0.218543
$$336$$ 4.00000 0.218218
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 10.0000 0.543125
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 8.00000 0.431959
$$344$$ −24.0000 −1.29399
$$345$$ 4.00000 0.215353
$$346$$ −6.00000 −0.322562
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 4.00000 0.213809
$$351$$ −2.00000 −0.106752
$$352$$ 20.0000 1.06600
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ −8.00000 −0.423405
$$358$$ 12.0000 0.634220
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ 1.00000 0.0526316
$$362$$ 22.0000 1.15629
$$363$$ −5.00000 −0.262432
$$364$$ −8.00000 −0.419314
$$365$$ −14.0000 −0.732793
$$366$$ −14.0000 −0.731792
$$367$$ −20.0000 −1.04399 −0.521996 0.852948i $$-0.674812\pi$$
−0.521996 + 0.852948i $$0.674812\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −6.00000 −0.312348
$$370$$ −6.00000 −0.311925
$$371$$ −56.0000 −2.90738
$$372$$ 0 0
$$373$$ 34.0000 1.76045 0.880227 0.474554i $$-0.157390\pi$$
0.880227 + 0.474554i $$0.157390\pi$$
$$374$$ 8.00000 0.413670
$$375$$ −1.00000 −0.0516398
$$376$$ 36.0000 1.85656
$$377$$ −4.00000 −0.206010
$$378$$ −4.00000 −0.205738
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 32.0000 1.63512 0.817562 0.575841i $$-0.195325\pi$$
0.817562 + 0.575841i $$0.195325\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 16.0000 0.815436
$$386$$ 22.0000 1.11977
$$387$$ 8.00000 0.406663
$$388$$ 10.0000 0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ −8.00000 −0.404577
$$392$$ −27.0000 −1.36371
$$393$$ −4.00000 −0.201773
$$394$$ 22.0000 1.10834
$$395$$ 16.0000 0.805047
$$396$$ −4.00000 −0.201008
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 1.00000 0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ −24.0000 −1.18964
$$408$$ 6.00000 0.297044
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 6.00000 0.295958
$$412$$ −16.0000 −0.788263
$$413$$ 16.0000 0.787309
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ 4.00000 0.195881
$$418$$ −4.00000 −0.195646
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 4.00000 0.195180
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 12.0000 0.584151
$$423$$ −12.0000 −0.583460
$$424$$ 42.0000 2.03970
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 56.0000 2.71003
$$428$$ −12.0000 −0.580042
$$429$$ −8.00000 −0.386244
$$430$$ 8.00000 0.385794
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 2.00000 0.0957826
$$437$$ 4.00000 0.191346
$$438$$ 14.0000 0.668946
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ −12.0000 −0.572078
$$441$$ 9.00000 0.428571
$$442$$ 4.00000 0.190261
$$443$$ 40.0000 1.90046 0.950229 0.311553i $$-0.100849\pi$$
0.950229 + 0.311553i $$0.100849\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ −6.00000 −0.284427
$$446$$ −24.0000 −1.13643
$$447$$ −6.00000 −0.283790
$$448$$ 28.0000 1.32288
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −24.0000 −1.13012
$$452$$ 10.0000 0.470360
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 8.00000 0.375046
$$456$$ −3.00000 −0.140488
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ −2.00000 −0.0933520
$$460$$ 4.00000 0.186501
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ −16.0000 −0.744387
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −16.0000 −0.738811
$$470$$ −12.0000 −0.553519
$$471$$ 18.0000 0.829396
$$472$$ −12.0000 −0.552345
$$473$$ 32.0000 1.47136
$$474$$ −16.0000 −0.734904
$$475$$ −1.00000 −0.0458831
$$476$$ −8.00000 −0.366679
$$477$$ −14.0000 −0.641016
$$478$$ −24.0000 −1.09773
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ −5.00000 −0.228218
$$481$$ −12.0000 −0.547153
$$482$$ −22.0000 −1.00207
$$483$$ 16.0000 0.728025
$$484$$ −5.00000 −0.227273
$$485$$ −10.0000 −0.454077
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −42.0000 −1.90125
$$489$$ 0 0
$$490$$ 9.00000 0.406579
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −4.00000 −0.180151
$$494$$ −2.00000 −0.0899843
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 8.00000 0.357414
$$502$$ 28.0000 1.24970
$$503$$ −20.0000 −0.891756 −0.445878 0.895094i $$-0.647108\pi$$
−0.445878 + 0.895094i $$0.647108\pi$$
$$504$$ −12.0000 −0.534522
$$505$$ −10.0000 −0.444994
$$506$$ −16.0000 −0.711287
$$507$$ 9.00000 0.399704
$$508$$ 8.00000 0.354943
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ −56.0000 −2.47729
$$512$$ −11.0000 −0.486136
$$513$$ 1.00000 0.0441511
$$514$$ −2.00000 −0.0882162
$$515$$ 16.0000 0.705044
$$516$$ 8.00000 0.352180
$$517$$ −48.0000 −2.11104
$$518$$ −24.0000 −1.05450
$$519$$ 6.00000 0.263371
$$520$$ −6.00000 −0.263117
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ −4.00000 −0.174574
$$526$$ −4.00000 −0.174408
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ −14.0000 −0.608121
$$531$$ 4.00000 0.173585
$$532$$ 4.00000 0.173422
$$533$$ −12.0000 −0.519778
$$534$$ 6.00000 0.259645
$$535$$ 12.0000 0.518805
$$536$$ 12.0000 0.518321
$$537$$ −12.0000 −0.517838
$$538$$ −26.0000 −1.12094
$$539$$ 36.0000 1.55063
$$540$$ 1.00000 0.0430331
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −22.0000 −0.944110
$$544$$ 10.0000 0.428746
$$545$$ −2.00000 −0.0856706
$$546$$ −8.00000 −0.342368
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 14.0000 0.597505
$$550$$ 4.00000 0.170561
$$551$$ 2.00000 0.0852029
$$552$$ −12.0000 −0.510754
$$553$$ 64.0000 2.72156
$$554$$ 22.0000 0.934690
$$555$$ 6.00000 0.254686
$$556$$ 4.00000 0.169638
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ −4.00000 −0.169031
$$561$$ −8.00000 −0.337760
$$562$$ 10.0000 0.421825
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ −10.0000 −0.420703
$$566$$ −16.0000 −0.672530
$$567$$ 4.00000 0.167984
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 1.00000 0.0418854
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −22.0000 −0.914289
$$580$$ 2.00000 0.0830455
$$581$$ 0 0
$$582$$ 10.0000 0.414513
$$583$$ −56.0000 −2.31928
$$584$$ 42.0000 1.73797
$$585$$ 2.00000 0.0826898
$$586$$ 2.00000 0.0826192
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ −22.0000 −0.904959
$$592$$ 6.00000 0.246598
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 8.00000 0.327968
$$596$$ −6.00000 −0.245770
$$597$$ 0 0
$$598$$ −8.00000 −0.327144
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 3.00000 0.122474
$$601$$ 18.0000 0.734235 0.367118 0.930175i $$-0.380345\pi$$
0.367118 + 0.930175i $$0.380345\pi$$
$$602$$ 32.0000 1.30422
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 10.0000 0.406222
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 8.00000 0.324176
$$610$$ 14.0000 0.566843
$$611$$ −24.0000 −0.970936
$$612$$ −2.00000 −0.0808452
$$613$$ −42.0000 −1.69636 −0.848182 0.529705i $$-0.822303\pi$$
−0.848182 + 0.529705i $$0.822303\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 6.00000 0.241943
$$616$$ −48.0000 −1.93398
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 2.00000 0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ 4.00000 0.159745
$$628$$ 18.0000 0.718278
$$629$$ −12.0000 −0.478471
$$630$$ 4.00000 0.159364
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −48.0000 −1.90934
$$633$$ −12.0000 −0.476957
$$634$$ 2.00000 0.0794301
$$635$$ −8.00000 −0.317470
$$636$$ −14.0000 −0.555136
$$637$$ 18.0000 0.713186
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ −3.00000 −0.118585
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 16.0000 0.630488
$$645$$ −8.00000 −0.315000
$$646$$ −2.00000 −0.0786889
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 16.0000 0.628055
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 4.00000 0.156293
$$656$$ 6.00000 0.234261
$$657$$ −14.0000 −0.546192
$$658$$ −48.0000 −1.87123
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 4.00000 0.155700
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ −4.00000 −0.155347
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ −6.00000 −0.232495
$$667$$ 8.00000 0.309761
$$668$$ 8.00000 0.309529
$$669$$ 24.0000 0.927894
$$670$$ −4.00000 −0.154533
$$671$$ 56.0000 2.16186
$$672$$ −20.0000 −0.771517
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 22.0000 0.847408
$$675$$ −1.00000 −0.0384900
$$676$$ 9.00000 0.346154
$$677$$ −14.0000 −0.538064 −0.269032 0.963131i $$-0.586704\pi$$
−0.269032 + 0.963131i $$0.586704\pi$$
$$678$$ 10.0000 0.384048
$$679$$ −40.0000 −1.53506
$$680$$ −6.00000 −0.230089
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ −6.00000 −0.229248
$$686$$ 8.00000 0.305441
$$687$$ 26.0000 0.991962
$$688$$ −8.00000 −0.304997
$$689$$ −28.0000 −1.06672
$$690$$ 4.00000 0.152277
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 16.0000 0.607790
$$694$$ 32.0000 1.21470
$$695$$ −4.00000 −0.151729
$$696$$ −6.00000 −0.227429
$$697$$ −12.0000 −0.454532
$$698$$ 30.0000 1.13552
$$699$$ −18.0000 −0.680823
$$700$$ −4.00000 −0.151186
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 6.00000 0.226294
$$704$$ 28.0000 1.05529
$$705$$ 12.0000 0.451946
$$706$$ −14.0000 −0.526897
$$707$$ −40.0000 −1.50435
$$708$$ 4.00000 0.150329
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 18.0000 0.674579
$$713$$ 0 0
$$714$$ −8.00000 −0.299392
$$715$$ 8.00000 0.299183
$$716$$ −12.0000 −0.448461
$$717$$ 24.0000 0.896296
$$718$$ −8.00000 −0.298557
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 64.0000 2.38348
$$722$$ 1.00000 0.0372161
$$723$$ 22.0000 0.818189
$$724$$ −22.0000 −0.817624
$$725$$ −2.00000 −0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ 4.00000 0.148352 0.0741759 0.997245i $$-0.476367\pi$$
0.0741759 + 0.997245i $$0.476367\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 1.00000 0.0370370
$$730$$ −14.0000 −0.518163
$$731$$ 16.0000 0.591781
$$732$$ 14.0000 0.517455
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ −20.0000 −0.738213
$$735$$ −9.00000 −0.331970
$$736$$ −20.0000 −0.737210
$$737$$ −16.0000 −0.589368
$$738$$ −6.00000 −0.220863
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 2.00000 0.0734718
$$742$$ −56.0000 −2.05582
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 34.0000 1.24483
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 48.0000 1.75388
$$750$$ −1.00000 −0.0365148
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −28.0000 −1.02038
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 16.0000 0.580763
$$760$$ 3.00000 0.108821
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −8.00000 −0.289619
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 32.0000 1.15621
$$767$$ 8.00000 0.288863
$$768$$ 17.0000 0.613435
$$769$$ 50.0000 1.80305 0.901523 0.432731i $$-0.142450\pi$$
0.901523 + 0.432731i $$0.142450\pi$$
$$770$$ 16.0000 0.576600
$$771$$ 2.00000 0.0720282
$$772$$ −22.0000 −0.791797
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 30.0000 1.07694
$$777$$ 24.0000 0.860995
$$778$$ 6.00000 0.215110
$$779$$ 6.00000 0.214972
$$780$$ 2.00000 0.0716115
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ 2.00000 0.0714742
$$784$$ −9.00000 −0.321429
$$785$$ −18.0000 −0.642448
$$786$$ −4.00000 −0.142675
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 4.00000 0.142404
$$790$$ 16.0000 0.569254
$$791$$ −40.0000 −1.42224
$$792$$ −12.0000 −0.426401
$$793$$ 28.0000 0.994309
$$794$$ −2.00000 −0.0709773
$$795$$ 14.0000 0.496529
$$796$$ 0 0
$$797$$ −38.0000 −1.34603 −0.673015 0.739629i $$-0.735001\pi$$
−0.673015 + 0.739629i $$0.735001\pi$$
$$798$$ 4.00000 0.141598
$$799$$ −24.0000 −0.849059
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ −6.00000 −0.211867
$$803$$ −56.0000 −1.97620
$$804$$ −4.00000 −0.141069
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ 26.0000 0.915243
$$808$$ 30.0000 1.05540
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 8.00000 0.280745
$$813$$ −24.0000 −0.841717
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ −8.00000 −0.279885
$$818$$ 10.0000 0.349642
$$819$$ 8.00000 0.279543
$$820$$ 6.00000 0.209529
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ −48.0000 −1.67216
$$825$$ −4.00000 −0.139262
$$826$$ 16.0000 0.556711
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 14.0000 0.485363
$$833$$ 18.0000 0.623663
$$834$$ 4.00000 0.138509
$$835$$ −8.00000 −0.276851
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ 28.0000 0.967244
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 12.0000 0.414039
$$841$$ −25.0000 −0.862069
$$842$$ 14.0000 0.482472
$$843$$ −10.0000 −0.344418
$$844$$ −12.0000 −0.413057
$$845$$ −9.00000 −0.309609
$$846$$ −12.0000 −0.412568
$$847$$ 20.0000 0.687208
$$848$$ 14.0000 0.480762
$$849$$ 16.0000 0.549119
$$850$$ 2.00000 0.0685994
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 56.0000 1.91628
$$855$$ −1.00000 −0.0341993
$$856$$ −36.0000 −1.23045
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 24.0000 0.817918
$$862$$ 32.0000 1.08992
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −6.00000 −0.204006
$$866$$ 6.00000 0.203888
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 64.0000 2.17105
$$870$$ 2.00000 0.0678064
$$871$$ −8.00000 −0.271070
$$872$$ 6.00000 0.203186
$$873$$ −10.0000 −0.338449
$$874$$ 4.00000 0.135302
$$875$$ 4.00000 0.135225
$$876$$ −14.0000 −0.473016
$$877$$ 42.0000 1.41824 0.709120 0.705088i $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ 8.00000 0.269987
$$879$$ −2.00000 −0.0674583
$$880$$ −4.00000 −0.134840
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ −4.00000 −0.134459
$$886$$ 40.0000 1.34383
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ −18.0000 −0.604040
$$889$$ −32.0000 −1.07325
$$890$$ −6.00000 −0.201120
$$891$$ 4.00000 0.134005
$$892$$ 24.0000 0.803579
$$893$$ 12.0000 0.401565
$$894$$ −6.00000 −0.200670
$$895$$ 12.0000 0.401116
$$896$$ −12.0000 −0.400892
$$897$$ 8.00000 0.267112
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −28.0000 −0.932815
$$902$$ −24.0000 −0.799113
$$903$$ −32.0000 −1.06489
$$904$$ 30.0000 0.997785
$$905$$ 22.0000 0.731305
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −10.0000 −0.331679
$$910$$ 8.00000 0.265197
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ −14.0000 −0.462826
$$916$$ 26.0000 0.859064
$$917$$ 16.0000 0.528367
$$918$$ −2.00000 −0.0660098
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 12.0000 0.395413
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 16.0000 0.526361
$$925$$ −6.00000 −0.197279
$$926$$ −4.00000 −0.131448
$$927$$ 16.0000 0.525509
$$928$$ −10.0000 −0.328266
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ −9.00000 −0.294963
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 8.00000 0.261628
$$936$$ −6.00000 −0.196116
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ −10.0000 −0.326338
$$940$$ 12.0000 0.391397
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 24.0000 0.781548
$$944$$ −4.00000 −0.130189
$$945$$ −4.00000 −0.130120
$$946$$ 32.0000 1.04041
$$947$$ −40.0000 −1.29983 −0.649913 0.760009i $$-0.725195\pi$$
−0.649913 + 0.760009i $$0.725195\pi$$
$$948$$ 16.0000 0.519656
$$949$$ −28.0000 −0.908918
$$950$$ −1.00000 −0.0324443
$$951$$ −2.00000 −0.0648544
$$952$$ −24.0000 −0.777844
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 8.00000 0.258603
$$958$$ −16.0000 −0.516937
$$959$$ −24.0000 −0.775000
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ −12.0000 −0.386896
$$963$$ 12.0000 0.386695
$$964$$ 22.0000 0.708572
$$965$$ 22.0000 0.708205
$$966$$ 16.0000 0.514792
$$967$$ −20.0000 −0.643157 −0.321578 0.946883i $$-0.604213\pi$$
−0.321578 + 0.946883i $$0.604213\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 2.00000 0.0642493
$$970$$ −10.0000 −0.321081
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ 8.00000 0.256337
$$975$$ −2.00000 −0.0640513
$$976$$ −14.0000 −0.448129
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ −9.00000 −0.287494
$$981$$ −2.00000 −0.0638551
$$982$$ 12.0000 0.382935
$$983$$ −56.0000 −1.78612 −0.893061 0.449935i $$-0.851447\pi$$
−0.893061 + 0.449935i $$0.851447\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ 22.0000 0.700978
$$986$$ −4.00000 −0.127386
$$987$$ 48.0000 1.52786
$$988$$ 2.00000 0.0636285
$$989$$ −32.0000 −1.01754
$$990$$ 4.00000 0.127128
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 20.0000 0.633089
$$999$$ 6.00000 0.189832
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 285.2.a.c.1.1 1
3.2 odd 2 855.2.a.a.1.1 1
4.3 odd 2 4560.2.a.w.1.1 1
5.2 odd 4 1425.2.c.f.799.2 2
5.3 odd 4 1425.2.c.f.799.1 2
5.4 even 2 1425.2.a.c.1.1 1
15.14 odd 2 4275.2.a.j.1.1 1
19.18 odd 2 5415.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.c.1.1 1 1.1 even 1 trivial
855.2.a.a.1.1 1 3.2 odd 2
1425.2.a.c.1.1 1 5.4 even 2
1425.2.c.f.799.1 2 5.3 odd 4
1425.2.c.f.799.2 2 5.2 odd 4
4275.2.a.j.1.1 1 15.14 odd 2
4560.2.a.w.1.1 1 4.3 odd 2
5415.2.a.e.1.1 1 19.18 odd 2