# Properties

 Label 1950.2.a.z.1.1 Level $1950$ Weight $2$ Character 1950.1 Self dual yes Analytic conductor $15.571$ 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] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.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: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1950.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -1.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} +10.0000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -4.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} +8.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 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
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 2.00000 0.426401
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 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 $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −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
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ −1.00000 −0.138675
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ −4.00000 −0.529813
$$58$$ 4.00000 0.525226
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 4.00000 0.455842
$$78$$ −1.00000 −0.113228
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 4.00000 0.428845
$$88$$ 2.00000 0.213201
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 2.00000 0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ −1.00000 −0.0924500
$$118$$ 10.0000 0.920575
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −14.0000 −1.26750
$$123$$ −6.00000 −0.541002
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −8.00000 −0.693688
$$134$$ −16.0000 −1.38219
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ −4.00000 −0.335673
$$143$$ −2.00000 −0.167248
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ −3.00000 −0.247436
$$148$$ 6.00000 0.493197
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 2.00000 0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 10.0000 0.751646
$$178$$ 6.00000 0.449719
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −14.0000 −1.03491
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 4.00000 0.292509
$$188$$ −8.00000 −0.583460
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 2.00000 0.142134
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 4.00000 0.281439
$$203$$ 8.00000 0.561490
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −4.00000 −0.274075
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 16.0000 1.08615
$$218$$ −18.0000 −1.21911
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 6.00000 0.402694
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 4.00000 0.262613
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ −8.00000 −0.519656
$$238$$ 4.00000 0.259281
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 1.00000 0.0641500
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 4.00000 0.254514
$$248$$ 8.00000 0.508001
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 6.00000 0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ −6.00000 −0.370681
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ 6.00000 0.367194
$$268$$ −16.0000 −0.977356
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 2.00000 0.121268
$$273$$ −2.00000 −0.121046
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ −12.0000 −0.708338
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ 8.00000 0.468165
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 2.00000 0.116052
$$298$$ 12.0000 0.695141
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −16.0000 −0.920697
$$303$$ 4.00000 0.229794
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 6.00000 0.341328
$$310$$ 0 0
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −20.0000 −1.13047 −0.565233 0.824931i $$-0.691214\pi$$
−0.565233 + 0.824931i $$0.691214\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ −18.0000 −0.995402
$$328$$ −6.00000 −0.331295
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 6.00000 0.328798
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 20.0000 1.08947 0.544735 0.838608i $$-0.316630\pi$$
0.544735 + 0.838608i $$0.316630\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ −4.00000 −0.216295
$$343$$ −20.0000 −1.07990
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 4.00000 0.214423
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 2.00000 0.106600
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 4.00000 0.211702
$$358$$ −18.0000 −0.951330
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −6.00000 −0.315353
$$363$$ −7.00000 −0.367405
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 8.00000 0.414781
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ −4.00000 −0.206010
$$378$$ 2.00000 0.102869
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 6.00000 0.307389
$$382$$ −12.0000 −0.613973
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 12.0000 0.609208
$$389$$ 16.0000 0.811232 0.405616 0.914044i $$-0.367057\pi$$
0.405616 + 0.914044i $$0.367057\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ −6.00000 −0.302660
$$394$$ 10.0000 0.503793
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ −8.00000 −0.398508
$$404$$ 4.00000 0.199007
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 12.0000 0.594818
$$408$$ 2.00000 0.0990148
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ 6.00000 0.295599
$$413$$ 20.0000 0.984136
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 4.00000 0.195881
$$418$$ −8.00000 −0.391293
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ −8.00000 −0.388973
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −4.00000 −0.193801
$$427$$ −28.0000 −1.35501
$$428$$ −4.00000 −0.193347
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 12.0000 0.576683 0.288342 0.957528i $$-0.406896\pi$$
0.288342 + 0.957528i $$0.406896\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 8.00000 0.382255
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −2.00000 −0.0951303
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 12.0000 0.567581
$$448$$ 2.00000 0.0944911
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 6.00000 0.282216
$$453$$ −16.0000 −0.751746
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 10.0000 0.460287
$$473$$ 8.00000 0.367840
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 2.00000 0.0915737
$$478$$ −12.0000 −0.548867
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −18.0000 −0.819878
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −22.0000 −0.992846 −0.496423 0.868081i $$-0.665354\pi$$
−0.496423 + 0.868081i $$0.665354\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 8.00000 0.360302
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ −8.00000 −0.358849
$$498$$ 12.0000 0.537733
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 18.0000 0.803379
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 6.00000 0.266207
$$509$$ −40.0000 −1.77297 −0.886484 0.462758i $$-0.846860\pi$$
−0.886484 + 0.462758i $$0.846860\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ −16.0000 −0.703679
$$518$$ 12.0000 0.527250
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 4.00000 0.175075
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 16.0000 0.696971
$$528$$ 2.00000 0.0870388
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ −8.00000 −0.346844
$$533$$ 6.00000 0.259889
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −16.0000 −0.691095
$$537$$ −18.0000 −0.776757
$$538$$ −20.0000 −0.862261
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −6.00000 −0.257485
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ −16.0000 −0.681623
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ −18.0000 −0.759284
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 2.00000 0.0839921
$$568$$ −4.00000 −0.167836
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ −12.0000 −0.501307
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 28.0000 1.16566 0.582828 0.812596i $$-0.301946\pi$$
0.582828 + 0.812596i $$0.301946\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 12.0000 0.497416
$$583$$ 4.00000 0.165663
$$584$$ 8.00000 0.331042
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 6.00000 0.246598
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ −24.0000 −0.982255
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −46.0000 −1.87638 −0.938190 0.346122i $$-0.887498\pi$$
−0.938190 + 0.346122i $$0.887498\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −16.0000 −0.651570
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 2.00000 0.0808452
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 46.0000 1.85189 0.925945 0.377658i $$-0.123271\pi$$
0.925945 + 0.377658i $$0.123271\pi$$
$$618$$ 6.00000 0.241355
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 20.0000 0.801927
$$623$$ 12.0000 0.480770
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ −8.00000 −0.319489
$$628$$ 10.0000 0.399043
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −4.00000 −0.158986
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 3.00000 0.118864
$$638$$ 8.00000 0.316723
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ −38.0000 −1.50091 −0.750455 0.660922i $$-0.770166\pi$$
−0.750455 + 0.660922i $$0.770166\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ −8.00000 −0.313304
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 8.00000 0.312110
$$658$$ −16.0000 −0.623745
$$659$$ 10.0000 0.389545 0.194772 0.980848i $$-0.437603\pi$$
0.194772 + 0.980848i $$0.437603\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ −2.00000 −0.0776736
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ −14.0000 −0.541271
$$670$$ 0 0
$$671$$ −28.0000 −1.08093
$$672$$ 2.00000 0.0771517
$$673$$ −36.0000 −1.38770 −0.693849 0.720121i $$-0.744086\pi$$
−0.693849 + 0.720121i $$0.744086\pi$$
$$674$$ 20.0000 0.770371
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −10.0000 −0.384331 −0.192166 0.981363i $$-0.561551\pi$$
−0.192166 + 0.981363i $$0.561551\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 24.0000 0.921035
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 16.0000 0.612672
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −22.0000 −0.839352
$$688$$ 4.00000 0.152499
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 4.00000 0.151947
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 4.00000 0.151620
$$697$$ −12.0000 −0.454532
$$698$$ 14.0000 0.529908
$$699$$ −14.0000 −0.529529
$$700$$ 0 0
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −24.0000 −0.905177
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −34.0000 −1.27961
$$707$$ 8.00000 0.300871
$$708$$ 10.0000 0.375823
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −18.0000 −0.672692
$$717$$ −12.0000 −0.448148
$$718$$ 32.0000 1.19423
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ −3.00000 −0.111648
$$723$$ −18.0000 −0.669427
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 34.0000 1.26099 0.630495 0.776193i $$-0.282852\pi$$
0.630495 + 0.776193i $$0.282852\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ −14.0000 −0.517455
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −32.0000 −1.17874
$$738$$ −6.00000 −0.220863
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 4.00000 0.146845
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 12.0000 0.439057
$$748$$ 4.00000 0.146254
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 18.0000 0.655956
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.0000 1.37750 0.688749 0.724999i $$-0.258160\pi$$
0.688749 + 0.724999i $$0.258160\pi$$
$$762$$ 6.00000 0.217357
$$763$$ −36.0000 −1.30329
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −10.0000 −0.361079
$$768$$ 1.00000 0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 0 0
$$773$$ 22.0000 0.791285 0.395643 0.918405i $$-0.370522\pi$$
0.395643 + 0.918405i $$0.370522\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 12.0000 0.430775
$$777$$ 12.0000 0.430498
$$778$$ 16.0000 0.573628
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 4.00000 0.142948
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ 10.0000 0.356235
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 2.00000 0.0710669
$$793$$ 14.0000 0.497155
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 16.0000 0.564628
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ −20.0000 −0.704033
$$808$$ 4.00000 0.140720
$$809$$ −22.0000 −0.773479 −0.386739 0.922189i $$-0.626399\pi$$
−0.386739 + 0.922189i $$0.626399\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 8.00000 0.280745
$$813$$ 16.0000 0.561144
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ −16.0000 −0.559769
$$818$$ −26.0000 −0.909069
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −14.0000 −0.488306
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 20.0000 0.695889
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ −1.00000 −0.0346688
$$833$$ −6.00000 −0.207888
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 8.00000 0.276520
$$838$$ −14.0000 −0.483622
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 34.0000 1.17172
$$843$$ −18.0000 −0.619953
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ −14.0000 −0.481046
$$848$$ 2.00000 0.0686803
$$849$$ 16.0000 0.549119
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −4.00000 −0.137038
$$853$$ −18.0000 −0.616308 −0.308154 0.951336i $$-0.599711\pi$$
−0.308154 + 0.951336i $$0.599711\pi$$
$$854$$ −28.0000 −0.958140
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 34.0000 1.16142 0.580709 0.814111i $$-0.302775\pi$$
0.580709 + 0.814111i $$0.302775\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ 16.0000 0.544962
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 12.0000 0.407777
$$867$$ −13.0000 −0.441503
$$868$$ 16.0000 0.543075
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −18.0000 −0.609557
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 42.0000 1.41824 0.709120 0.705088i $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ −10.0000 −0.336909 −0.168454 0.985709i $$-0.553878\pi$$
−0.168454 + 0.985709i $$0.553878\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ 24.0000 0.807664 0.403832 0.914833i $$-0.367678\pi$$
0.403832 + 0.914833i $$0.367678\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ −14.0000 −0.468755
$$893$$ 32.0000 1.07084
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 32.0000 1.06726
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ −12.0000 −0.399556
$$903$$ 8.00000 0.266223
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 24.0000 0.794284
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ −12.0000 −0.396275
$$918$$ 2.00000 0.0660098
$$919$$ −48.0000 −1.58337 −0.791687 0.610927i $$-0.790797\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 20.0000 0.658665
$$923$$ 4.00000 0.131662
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ 6.00000 0.197172
$$927$$ 6.00000 0.197066
$$928$$ 4.00000 0.131306
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ −14.0000 −0.458585
$$933$$ 20.0000 0.654771
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 56.0000 1.82944 0.914720 0.404088i $$-0.132411\pi$$
0.914720 + 0.404088i $$0.132411\pi$$
$$938$$ −32.0000 −1.04484
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ −44.0000 −1.43436 −0.717180 0.696888i $$-0.754567\pi$$
−0.717180 + 0.696888i $$0.754567\pi$$
$$942$$ 10.0000 0.325818
$$943$$ 0 0
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 4.00000 0.129641
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 8.00000 0.258603
$$958$$ 36.0000 1.16311
$$959$$ −28.0000 −0.904167
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −6.00000 −0.193448
$$963$$ −4.00000 −0.128898
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 58.0000 1.86515 0.932577 0.360971i $$-0.117555\pi$$
0.932577 + 0.360971i $$0.117555\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 8.00000 0.256468
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ −22.0000 −0.702048
$$983$$ 56.0000 1.78612 0.893061 0.449935i $$-0.148553\pi$$
0.893061 + 0.449935i $$0.148553\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 8.00000 0.254772
$$987$$ −16.0000 −0.509286
$$988$$ 4.00000 0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −12.0000 −0.380808
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ 36.0000 1.13956
$$999$$ 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 1950.2.a.z.1.1 1
3.2 odd 2 5850.2.a.t.1.1 1
5.2 odd 4 390.2.e.c.79.2 yes 2
5.3 odd 4 390.2.e.c.79.1 2
5.4 even 2 1950.2.a.c.1.1 1
15.2 even 4 1170.2.e.a.469.1 2
15.8 even 4 1170.2.e.a.469.2 2
15.14 odd 2 5850.2.a.bj.1.1 1
20.3 even 4 3120.2.l.g.1249.1 2
20.7 even 4 3120.2.l.g.1249.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.c.79.1 2 5.3 odd 4
390.2.e.c.79.2 yes 2 5.2 odd 4
1170.2.e.a.469.1 2 15.2 even 4
1170.2.e.a.469.2 2 15.8 even 4
1950.2.a.c.1.1 1 5.4 even 2
1950.2.a.z.1.1 1 1.1 even 1 trivial
3120.2.l.g.1249.1 2 20.3 even 4
3120.2.l.g.1249.2 2 20.7 even 4
5850.2.a.t.1.1 1 3.2 odd 2
5850.2.a.bj.1.1 1 15.14 odd 2