# Properties

 Label 285.2.a.a.1.1 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} +2.00000 q^{42} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -6.00000 q^{56} +1.00000 q^{57} -4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} -8.00000 q^{69} -2.00000 q^{70} +16.0000 q^{71} +3.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{84} +6.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -18.0000 q^{88} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -12.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})$$

## 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 $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 2.00000 0.534522
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.00000 0.229416
$$20$$ 1.00000 0.223607
$$21$$ −2.00000 −0.436436
$$22$$ 6.00000 1.27920
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ −6.00000 −1.04447
$$34$$ 6.00000 1.02899
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 6.00000 0.904534
$$45$$ −1.00000 −0.149071
$$46$$ 8.00000 1.17954
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 6.00000 0.809040
$$56$$ −6.00000 −0.801784
$$57$$ 1.00000 0.132453
$$58$$ −4.00000 −0.525226
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −8.00000 −0.963087
$$70$$ −2.00000 −0.239046
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 1.00000 0.115470
$$76$$ −1.00000 −0.114708
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 6.00000 0.650791
$$86$$ 2.00000 0.215666
$$87$$ 4.00000 0.428845
$$88$$ −18.0000 −1.91881
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −1.00000 −0.102598
$$96$$ −5.00000 −0.510310
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −6.00000 −0.603023
$$100$$ −1.00000 −0.100000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −6.00000 −0.572078
$$111$$ 4.00000 0.379663
$$112$$ 2.00000 0.188982
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 8.00000 0.746004
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 12.0000 1.10004
$$120$$ −3.00000 −0.273861
$$121$$ 25.0000 2.27273
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 2.00000 0.178174
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ 6.00000 0.522233
$$133$$ −2.00000 −0.173422
$$134$$ 8.00000 0.691095
$$135$$ −1.00000 −0.0860663
$$136$$ −18.0000 −1.54349
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −8.00000 −0.673722
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −14.0000 −1.15865
$$147$$ −3.00000 −0.247436
$$148$$ −4.00000 −0.328798
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 3.00000 0.243332
$$153$$ −6.00000 −0.485071
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 2.00000 0.158610
$$160$$ 5.00000 0.395285
$$161$$ 16.0000 1.26098
$$162$$ −1.00000 −0.0785674
$$163$$ −22.0000 −1.72317 −0.861586 0.507611i $$-0.830529\pi$$
−0.861586 + 0.507611i $$0.830529\pi$$
$$164$$ 0 0
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ −6.00000 −0.462910
$$169$$ −13.0000 −1.00000
$$170$$ −6.00000 −0.460179
$$171$$ 1.00000 0.0764719
$$172$$ 2.00000 0.152499
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ −2.00000 −0.151186
$$176$$ 6.00000 0.452267
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ −24.0000 −1.76930
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 36.0000 2.63258
$$188$$ 8.00000 0.583460
$$189$$ −2.00000 −0.145479
$$190$$ 1.00000 0.0725476
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 7.00000 0.505181
$$193$$ 24.0000 1.72756 0.863779 0.503871i $$-0.168091\pi$$
0.863779 + 0.503871i $$0.168091\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 3.00000 0.212132
$$201$$ −8.00000 −0.564276
$$202$$ 18.0000 1.26648
$$203$$ −8.00000 −0.561490
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ −2.00000 −0.138013
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 16.0000 1.09630
$$214$$ 12.0000 0.820303
$$215$$ 2.00000 0.136399
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 14.0000 0.946032
$$220$$ −6.00000 −0.404520
$$221$$ 0 0
$$222$$ −4.00000 −0.268462
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 1.00000 0.0666667
$$226$$ 6.00000 0.399114
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 12.0000 0.789542
$$232$$ 12.0000 0.787839
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ −12.0000 −0.781133
$$237$$ 8.00000 0.519656
$$238$$ −12.0000 −0.777844
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ −25.0000 −1.60706
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 48.0000 3.01773
$$254$$ −4.00000 −0.250982
$$255$$ 6.00000 0.375735
$$256$$ −17.0000 −1.06250
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 2.00000 0.124515
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 2.00000 0.123560
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ −18.0000 −1.10782
$$265$$ −2.00000 −0.122859
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ −6.00000 −0.361814
$$276$$ 8.00000 0.481543
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 16.0000 0.959616
$$279$$ 0 0
$$280$$ 6.00000 0.358569
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ 19.0000 1.11765
$$290$$ 4.00000 0.234888
$$291$$ −12.0000 −0.703452
$$292$$ −14.0000 −0.819288
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 3.00000 0.174964
$$295$$ −12.0000 −0.698667
$$296$$ 12.0000 0.697486
$$297$$ −6.00000 −0.348155
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 4.00000 0.230556
$$302$$ 16.0000 0.920697
$$303$$ −18.0000 −1.03407
$$304$$ −1.00000 −0.0573539
$$305$$ −2.00000 −0.114520
$$306$$ 6.00000 0.342997
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ −12.0000 −0.683763
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 2.00000 0.112687
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ −24.0000 −1.34374
$$320$$ −7.00000 −0.391312
$$321$$ −12.0000 −0.669775
$$322$$ −16.0000 −0.891645
$$323$$ −6.00000 −0.333849
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ −6.00000 −0.331801
$$328$$ 0 0
$$329$$ 16.0000 0.882109
$$330$$ −6.00000 −0.330289
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ −16.0000 −0.875481
$$335$$ 8.00000 0.437087
$$336$$ 2.00000 0.109109
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 13.0000 0.707107
$$339$$ −6.00000 −0.325875
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 20.0000 1.07990
$$344$$ −6.00000 −0.323498
$$345$$ 8.00000 0.430706
$$346$$ −6.00000 −0.322562
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 30.0000 1.59901
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ −16.0000 −0.849192
$$356$$ 0 0
$$357$$ 12.0000 0.635107
$$358$$ 20.0000 1.05703
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ 1.00000 0.0526316
$$362$$ −10.0000 −0.525588
$$363$$ 25.0000 1.31216
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ −2.00000 −0.104542
$$367$$ −14.0000 −0.730794 −0.365397 0.930852i $$-0.619067\pi$$
−0.365397 + 0.930852i $$0.619067\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ −36.0000 −1.86152
$$375$$ −1.00000 −0.0516398
$$376$$ −24.0000 −1.23771
$$377$$ 0 0
$$378$$ 2.00000 0.102869
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 4.00000 0.204926
$$382$$ −10.0000 −0.511645
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 3.00000 0.153093
$$385$$ −12.0000 −0.611577
$$386$$ −24.0000 −1.22157
$$387$$ −2.00000 −0.101666
$$388$$ 12.0000 0.609208
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ −9.00000 −0.454569
$$393$$ −2.00000 −0.100887
$$394$$ −6.00000 −0.302276
$$395$$ −8.00000 −0.402524
$$396$$ 6.00000 0.301511
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 20.0000 1.00251
$$399$$ −2.00000 −0.100125
$$400$$ −1.00000 −0.0500000
$$401$$ 4.00000 0.199750 0.0998752 0.995000i $$-0.468156\pi$$
0.0998752 + 0.995000i $$0.468156\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ −1.00000 −0.0496904
$$406$$ 8.00000 0.397033
$$407$$ −24.0000 −1.18964
$$408$$ −18.0000 −0.891133
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 8.00000 0.394132
$$413$$ −24.0000 −1.18096
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −16.0000 −0.783523
$$418$$ 6.00000 0.293470
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −8.00000 −0.388973
$$424$$ 6.00000 0.291386
$$425$$ −6.00000 −0.291043
$$426$$ −16.0000 −0.775203
$$427$$ −4.00000 −0.193574
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ −4.00000 −0.191785
$$436$$ 6.00000 0.287348
$$437$$ −8.00000 −0.382692
$$438$$ −14.0000 −0.668946
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 18.0000 0.858116
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 6.00000 0.283790
$$448$$ −14.0000 −0.661438
$$449$$ −8.00000 −0.377543 −0.188772 0.982021i $$-0.560451\pi$$
−0.188772 + 0.982021i $$0.560451\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ −16.0000 −0.751746
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −6.00000 −0.280056
$$460$$ −8.00000 −0.373002
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ −12.0000 −0.558291
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ −8.00000 −0.369012
$$471$$ −2.00000 −0.0921551
$$472$$ 36.0000 1.65703
$$473$$ 12.0000 0.551761
$$474$$ −8.00000 −0.367452
$$475$$ 1.00000 0.0458831
$$476$$ −12.0000 −0.550019
$$477$$ 2.00000 0.0915737
$$478$$ −6.00000 −0.274434
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 5.00000 0.228218
$$481$$ 0 0
$$482$$ −18.0000 −0.819878
$$483$$ 16.0000 0.728025
$$484$$ −25.0000 −1.13636
$$485$$ 12.0000 0.544892
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −22.0000 −0.994874
$$490$$ −3.00000 −0.135526
$$491$$ −26.0000 −1.17336 −0.586682 0.809818i $$-0.699566\pi$$
−0.586682 + 0.809818i $$0.699566\pi$$
$$492$$ 0 0
$$493$$ −24.0000 −1.08091
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 16.0000 0.714827
$$502$$ 6.00000 0.267793
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 18.0000 0.800989
$$506$$ −48.0000 −2.13386
$$507$$ −13.0000 −0.577350
$$508$$ −4.00000 −0.177471
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ −28.0000 −1.23865
$$512$$ 11.0000 0.486136
$$513$$ 1.00000 0.0441511
$$514$$ 22.0000 0.970378
$$515$$ 8.00000 0.352522
$$516$$ 2.00000 0.0880451
$$517$$ 48.0000 2.11104
$$518$$ 8.00000 0.351500
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 2.00000 0.0873704
$$525$$ −2.00000 −0.0872872
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 6.00000 0.261116
$$529$$ 41.0000 1.78261
$$530$$ 2.00000 0.0868744
$$531$$ 12.0000 0.520756
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ −24.0000 −1.03664
$$537$$ −20.0000 −0.863064
$$538$$ −12.0000 −0.517357
$$539$$ 18.0000 0.775315
$$540$$ 1.00000 0.0430331
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 20.0000 0.859074
$$543$$ 10.0000 0.429141
$$544$$ 30.0000 1.28624
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 2.00000 0.0853579
$$550$$ 6.00000 0.255841
$$551$$ 4.00000 0.170406
$$552$$ −24.0000 −1.02151
$$553$$ −16.0000 −0.680389
$$554$$ 2.00000 0.0849719
$$555$$ −4.00000 −0.169791
$$556$$ 16.0000 0.678551
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 36.0000 1.51992
$$562$$ 4.00000 0.168730
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 6.00000 0.252422
$$566$$ 14.0000 0.588464
$$567$$ −2.00000 −0.0839921
$$568$$ 48.0000 2.01404
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 1.00000 0.0418854
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 0 0
$$573$$ 10.0000 0.417756
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 7.00000 0.291667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 24.0000 0.997406
$$580$$ 4.00000 0.166091
$$581$$ 0 0
$$582$$ 12.0000 0.497416
$$583$$ −12.0000 −0.496989
$$584$$ 42.0000 1.73797
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ −20.0000 −0.825488 −0.412744 0.910847i $$-0.635430\pi$$
−0.412744 + 0.910847i $$0.635430\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ 12.0000 0.494032
$$591$$ 6.00000 0.246807
$$592$$ −4.00000 −0.164399
$$593$$ −10.0000 −0.410651 −0.205325 0.978694i $$-0.565825\pi$$
−0.205325 + 0.978694i $$0.565825\pi$$
$$594$$ 6.00000 0.246183
$$595$$ −12.0000 −0.491952
$$596$$ −6.00000 −0.245770
$$597$$ −20.0000 −0.818546
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 3.00000 0.122474
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ −8.00000 −0.325785
$$604$$ 16.0000 0.651031
$$605$$ −25.0000 −1.01639
$$606$$ 18.0000 0.731200
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ −8.00000 −0.324176
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 8.00000 0.321807
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −34.0000 −1.36328
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −10.0000 −0.399680
$$627$$ −6.00000 −0.239617
$$628$$ 2.00000 0.0798087
$$629$$ −24.0000 −0.956943
$$630$$ −2.00000 −0.0796819
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 24.0000 0.954669
$$633$$ −20.0000 −0.794929
$$634$$ 6.00000 0.238290
$$635$$ −4.00000 −0.158735
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ 24.0000 0.950169
$$639$$ 16.0000 0.632950
$$640$$ −3.00000 −0.118585
$$641$$ 40.0000 1.57991 0.789953 0.613168i $$-0.210105\pi$$
0.789953 + 0.613168i $$0.210105\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 2.00000 0.0787499
$$646$$ 6.00000 0.236067
$$647$$ 20.0000 0.786281 0.393141 0.919478i $$-0.371389\pi$$
0.393141 + 0.919478i $$0.371389\pi$$
$$648$$ 3.00000 0.117851
$$649$$ −72.0000 −2.82625
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 22.0000 0.861586
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 2.00000 0.0781465
$$656$$ 0 0
$$657$$ 14.0000 0.546192
$$658$$ −16.0000 −0.623745
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ −6.00000 −0.233550
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ −4.00000 −0.154997
$$667$$ −32.0000 −1.23904
$$668$$ −16.0000 −0.619059
$$669$$ 8.00000 0.309298
$$670$$ −8.00000 −0.309067
$$671$$ −12.0000 −0.463255
$$672$$ 10.0000 0.385758
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 12.0000 0.462223
$$675$$ 1.00000 0.0384900
$$676$$ 13.0000 0.500000
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 24.0000 0.921035
$$680$$ 18.0000 0.690268
$$681$$ −28.0000 −1.07296
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 10.0000 0.382080
$$686$$ −20.0000 −0.763604
$$687$$ 2.00000 0.0763048
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ −8.00000 −0.304555
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 12.0000 0.455842
$$694$$ −8.00000 −0.303676
$$695$$ 16.0000 0.606915
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ 2.00000 0.0757011
$$699$$ 22.0000 0.832116
$$700$$ 2.00000 0.0755929
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ −42.0000 −1.58293
$$705$$ 8.00000 0.301297
$$706$$ 34.0000 1.27961
$$707$$ 36.0000 1.35392
$$708$$ −12.0000 −0.450988
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 6.00000 0.224074
$$718$$ −10.0000 −0.373197
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 16.0000 0.595871
$$722$$ −1.00000 −0.0372161
$$723$$ 18.0000 0.669427
$$724$$ −10.0000 −0.371647
$$725$$ 4.00000 0.148556
$$726$$ −25.0000 −0.927837
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 14.0000 0.518163
$$731$$ 12.0000 0.443836
$$732$$ −2.00000 −0.0739221
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 14.0000 0.516749
$$735$$ 3.00000 0.110657
$$736$$ 40.0000 1.47442
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 0 0
$$748$$ −36.0000 −1.31629
$$749$$ 24.0000 0.876941
$$750$$ 1.00000 0.0365148
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 8.00000 0.291730
$$753$$ −6.00000 −0.218652
$$754$$ 0 0
$$755$$ 16.0000 0.582300
$$756$$ 2.00000 0.0727393
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 48.0000 1.74229
$$760$$ −3.00000 −0.108821
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 12.0000 0.434429
$$764$$ −10.0000 −0.361787
$$765$$ 6.00000 0.216930
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ −17.0000 −0.613435
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 12.0000 0.432450
$$771$$ −22.0000 −0.792311
$$772$$ −24.0000 −0.863779
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ −36.0000 −1.29232
$$777$$ −8.00000 −0.286998
$$778$$ 14.0000 0.501924
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −96.0000 −3.43515
$$782$$ −48.0000 −1.71648
$$783$$ 4.00000 0.142948
$$784$$ 3.00000 0.107143
$$785$$ 2.00000 0.0713831
$$786$$ 2.00000 0.0713376
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −12.0000 −0.427211
$$790$$ 8.00000 0.284627
$$791$$ 12.0000 0.426671
$$792$$ −18.0000 −0.639602
$$793$$ 0 0
$$794$$ −6.00000 −0.212932
$$795$$ −2.00000 −0.0709327
$$796$$ 20.0000 0.708881
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ 48.0000 1.69812
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ −4.00000 −0.141245
$$803$$ −84.0000 −2.96430
$$804$$ 8.00000 0.282138
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ 12.0000 0.422420
$$808$$ −54.0000 −1.89971
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 8.00000 0.280745
$$813$$ −20.0000 −0.701431
$$814$$ 24.0000 0.841200
$$815$$ 22.0000 0.770626
$$816$$ 6.00000 0.210042
$$817$$ −2.00000 −0.0699711
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 10.0000 0.348790
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ −6.00000 −0.208893
$$826$$ 24.0000 0.835067
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 16.0000 0.554035
$$835$$ −16.0000 −0.553703
$$836$$ 6.00000 0.207514
$$837$$ 0 0
$$838$$ 30.0000 1.03633
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 6.00000 0.207020
$$841$$ −13.0000 −0.448276
$$842$$ −6.00000 −0.206774
$$843$$ −4.00000 −0.137767
$$844$$ 20.0000 0.688428
$$845$$ 13.0000 0.447214
$$846$$ 8.00000 0.275046
$$847$$ −50.0000 −1.71802
$$848$$ −2.00000 −0.0686803
$$849$$ −14.0000 −0.480479
$$850$$ 6.00000 0.205798
$$851$$ −32.0000 −1.09695
$$852$$ −16.0000 −0.548151
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 4.00000 0.136877
$$855$$ −1.00000 −0.0341993
$$856$$ −36.0000 −1.23045
$$857$$ 46.0000 1.57133 0.785665 0.618652i $$-0.212321\pi$$
0.785665 + 0.618652i $$0.212321\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ −2.00000 −0.0681994
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −6.00000 −0.204006
$$866$$ 16.0000 0.543702
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 4.00000 0.135613
$$871$$ 0 0
$$872$$ −18.0000 −0.609557
$$873$$ −12.0000 −0.406138
$$874$$ 8.00000 0.270604
$$875$$ 2.00000 0.0676123
$$876$$ −14.0000 −0.473016
$$877$$ −24.0000 −0.810422 −0.405211 0.914223i $$-0.632802\pi$$
−0.405211 + 0.914223i $$0.632802\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 30.0000 1.01187
$$880$$ −6.00000 −0.202260
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 3.00000 0.101015
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ 12.0000 0.403148
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 12.0000 0.402694
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ −8.00000 −0.267860
$$893$$ −8.00000 −0.267710
$$894$$ −6.00000 −0.200670
$$895$$ 20.0000 0.668526
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 8.00000 0.266963
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 4.00000 0.133112
$$904$$ −18.0000 −0.598671
$$905$$ −10.0000 −0.332411
$$906$$ 16.0000 0.531564
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 28.0000 0.929213
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ −2.00000 −0.0661180
$$916$$ −2.00000 −0.0660819
$$917$$ 4.00000 0.132092
$$918$$ 6.00000 0.198030
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 24.0000 0.791257
$$921$$ −8.00000 −0.263609
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ −12.0000 −0.394771
$$925$$ 4.00000 0.131519
$$926$$ 14.0000 0.460069
$$927$$ −8.00000 −0.262754
$$928$$ −20.0000 −0.656532
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ −22.0000 −0.720634
$$933$$ 34.0000 1.11311
$$934$$ −12.0000 −0.392652
$$935$$ −36.0000 −1.17733
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 10.0000 0.326338
$$940$$ −8.00000 −0.260931
$$941$$ 44.0000 1.43436 0.717180 0.696888i $$-0.245433\pi$$
0.717180 + 0.696888i $$0.245433\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 2.00000 0.0650600
$$946$$ −12.0000 −0.390154
$$947$$ 48.0000 1.55979 0.779895 0.625910i $$-0.215272\pi$$
0.779895 + 0.625910i $$0.215272\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ −1.00000 −0.0324443
$$951$$ −6.00000 −0.194563
$$952$$ 36.0000 1.16677
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ −10.0000 −0.323592
$$956$$ −6.00000 −0.194054
$$957$$ −24.0000 −0.775810
$$958$$ −18.0000 −0.581554
$$959$$ 20.0000 0.645834
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ −18.0000 −0.579741
$$965$$ −24.0000 −0.772587
$$966$$ −16.0000 −0.514792
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 75.0000 2.41059
$$969$$ −6.00000 −0.192748
$$970$$ −12.0000 −0.385297
$$971$$ −8.00000 −0.256732 −0.128366 0.991727i $$-0.540973\pi$$
−0.128366 + 0.991727i $$0.540973\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 32.0000 1.02587
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 22.0000 0.703482
$$979$$ 0 0
$$980$$ −3.00000 −0.0958315
$$981$$ −6.00000 −0.191565
$$982$$ 26.0000 0.829693
$$983$$ 8.00000 0.255160 0.127580 0.991828i $$-0.459279\pi$$
0.127580 + 0.991828i $$0.459279\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 24.0000 0.764316
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ −6.00000 −0.190693
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 32.0000 1.01498
$$995$$ 20.0000 0.634043
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 24.0000 0.759707
$$999$$ 4.00000 0.126554
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.a.1.1 1
3.2 odd 2 855.2.a.c.1.1 1
4.3 odd 2 4560.2.a.h.1.1 1
5.2 odd 4 1425.2.c.c.799.1 2
5.3 odd 4 1425.2.c.c.799.2 2
5.4 even 2 1425.2.a.g.1.1 1
15.14 odd 2 4275.2.a.h.1.1 1
19.18 odd 2 5415.2.a.h.1.1 1

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