# Properties

 Label 209.2.a.a.1.1 Level $209$ Weight $2$ Character 209.1 Self dual yes Analytic conductor $1.669$ 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] = [209,2,Mod(1,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("209.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.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: $$1.66887340224$$ 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 $$\chi$$ $$=$$ 209.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +1.00000 q^{19} +6.00000 q^{20} -4.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} +8.00000 q^{28} -6.00000 q^{29} -7.00000 q^{31} +1.00000 q^{33} +12.0000 q^{35} +4.00000 q^{36} -7.00000 q^{37} +2.00000 q^{39} -10.0000 q^{43} -2.00000 q^{44} +6.00000 q^{45} +4.00000 q^{48} +9.00000 q^{49} -4.00000 q^{52} +6.00000 q^{53} -3.00000 q^{55} +1.00000 q^{57} +3.00000 q^{59} +6.00000 q^{60} -10.0000 q^{61} +8.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +11.0000 q^{67} +3.00000 q^{69} +15.0000 q^{71} +8.00000 q^{73} +4.00000 q^{75} -2.00000 q^{76} -4.00000 q^{77} -16.0000 q^{79} -12.0000 q^{80} +1.00000 q^{81} +8.00000 q^{84} -6.00000 q^{87} +9.00000 q^{89} -8.00000 q^{91} -6.00000 q^{92} -7.00000 q^{93} -3.00000 q^{95} -1.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −2.00000 −0.577350
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ 4.00000 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 6.00000 1.34164
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 8.00000 1.51186
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ 12.0000 2.02837
$$36$$ 4.00000 0.666667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 6.00000 0.894427
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 6.00000 0.774597
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 8.00000 1.00791
$$64$$ −8.00000 −1.00000
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ 0 0
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ −2.00000 −0.229416
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −12.0000 −1.34164
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 8.00000 0.872872
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 9.00000 0.953998 0.476999 0.878904i $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ −6.00000 −0.625543
$$93$$ −7.00000 −0.725866
$$94$$ 0 0
$$95$$ −3.00000 −0.307794
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ −8.00000 −0.800000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 12.0000 1.17108
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 10.0000 0.962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ −16.0000 −1.51186
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ 12.0000 1.11417
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 14.0000 1.25724
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ −10.0000 −0.880451
$$130$$ 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
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ 15.0000 1.29099
$$136$$ 0 0
$$137$$ 21.0000 1.79415 0.897076 0.441877i $$-0.145687\pi$$
0.897076 + 0.441877i $$0.145687\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −24.0000 −2.02837
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 2.00000 0.167248
$$144$$ −8.00000 −0.666667
$$145$$ 18.0000 1.49482
$$146$$ 0 0
$$147$$ 9.00000 0.742307
$$148$$ 14.0000 1.15079
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 21.0000 1.68676
$$156$$ −4.00000 −0.320256
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −3.00000 −0.233550
$$166$$ 0 0
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 20.0000 1.52499
$$173$$ 24.0000 1.82469 0.912343 0.409426i $$-0.134271\pi$$
0.912343 + 0.409426i $$0.134271\pi$$
$$174$$ 0 0
$$175$$ −16.0000 −1.20949
$$176$$ 4.00000 0.301511
$$177$$ 3.00000 0.225494
$$178$$ 0 0
$$179$$ 15.0000 1.12115 0.560576 0.828103i $$-0.310580\pi$$
0.560576 + 0.828103i $$0.310580\pi$$
$$180$$ −12.0000 −0.894427
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 21.0000 1.54395
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 20.0000 1.45479
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ −6.00000 −0.429669
$$196$$ −18.0000 −1.28571
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 11.0000 0.775880
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 8.00000 0.554700
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 15.0000 1.02778
$$214$$ 0 0
$$215$$ 30.0000 2.04598
$$216$$ 0 0
$$217$$ 28.0000 1.90076
$$218$$ 0 0
$$219$$ 8.00000 0.540590
$$220$$ 6.00000 0.404520
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5.00000 0.334825 0.167412 0.985887i $$-0.446459\pi$$
0.167412 + 0.985887i $$0.446459\pi$$
$$224$$ 0 0
$$225$$ −8.00000 −0.533333
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ −18.0000 −1.16432 −0.582162 0.813073i $$-0.697793\pi$$
−0.582162 + 0.813073i $$0.697793\pi$$
$$240$$ −12.0000 −0.774597
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 20.0000 1.28037
$$245$$ −27.0000 −1.72497
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 21.0000 1.32551 0.662754 0.748837i $$-0.269387\pi$$
0.662754 + 0.748837i $$0.269387\pi$$
$$252$$ −16.0000 −1.00791
$$253$$ 3.00000 0.188608
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 28.0000 1.73984
$$260$$ 12.0000 0.744208
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ −30.0000 −1.84988 −0.924940 0.380114i $$-0.875885\pi$$
−0.924940 + 0.380114i $$0.875885\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ 0 0
$$267$$ 9.00000 0.550791
$$268$$ −22.0000 −1.34386
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ 4.00000 0.241209
$$276$$ −6.00000 −0.361158
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 0 0
$$279$$ 14.0000 0.838158
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ −30.0000 −1.78017
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −1.00000 −0.0586210
$$292$$ −16.0000 −0.936329
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ −9.00000 −0.524000
$$296$$ 0 0
$$297$$ −5.00000 −0.290129
$$298$$ 0 0
$$299$$ 6.00000 0.346989
$$300$$ −8.00000 −0.461880
$$301$$ 40.0000 2.30556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 30.0000 1.71780
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 8.00000 0.455842
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ −24.0000 −1.35225
$$316$$ 32.0000 1.80014
$$317$$ 15.0000 0.842484 0.421242 0.906948i $$-0.361594\pi$$
0.421242 + 0.906948i $$0.361594\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 24.0000 1.34164
$$321$$ −18.0000 −1.00466
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −2.00000 −0.111111
$$325$$ 8.00000 0.443760
$$326$$ 0 0
$$327$$ 2.00000 0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 0 0
$$333$$ 14.0000 0.767195
$$334$$ 0 0
$$335$$ −33.0000 −1.80298
$$336$$ −16.0000 −0.872872
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ 0 0
$$339$$ −9.00000 −0.488813
$$340$$ 0 0
$$341$$ −7.00000 −0.379071
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ −9.00000 −0.484544
$$346$$ 0 0
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 12.0000 0.643268
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ −10.0000 −0.533761
$$352$$ 0 0
$$353$$ −21.0000 −1.11772 −0.558859 0.829263i $$-0.688761\pi$$
−0.558859 + 0.829263i $$0.688761\pi$$
$$354$$ 0 0
$$355$$ −45.0000 −2.38835
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 1.00000 0.0524864
$$364$$ 16.0000 0.838628
$$365$$ −24.0000 −1.25622
$$366$$ 0 0
$$367$$ 23.0000 1.20059 0.600295 0.799779i $$-0.295050\pi$$
0.600295 + 0.799779i $$0.295050\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 14.0000 0.725866
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 2.00000 0.102463
$$382$$ 0 0
$$383$$ 33.0000 1.68622 0.843111 0.537740i $$-0.180722\pi$$
0.843111 + 0.537740i $$0.180722\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 0 0
$$387$$ 20.0000 1.01666
$$388$$ 2.00000 0.101535
$$389$$ −3.00000 −0.152106 −0.0760530 0.997104i $$-0.524232\pi$$
−0.0760530 + 0.997104i $$0.524232\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ 48.0000 2.41514
$$396$$ 4.00000 0.201008
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 16.0000 0.800000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ −14.0000 −0.697390
$$404$$ 0 0
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ −7.00000 −0.346977
$$408$$ 0 0
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ 21.0000 1.03585
$$412$$ 32.0000 1.57653
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ −24.0000 −1.17108
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 40.0000 1.93574
$$428$$ 36.0000 1.74013
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ −20.0000 −0.962250
$$433$$ −13.0000 −0.624740 −0.312370 0.949960i $$-0.601123\pi$$
−0.312370 + 0.949960i $$0.601123\pi$$
$$434$$ 0 0
$$435$$ 18.0000 0.863034
$$436$$ −4.00000 −0.191565
$$437$$ 3.00000 0.143509
$$438$$ 0 0
$$439$$ 14.0000 0.668184 0.334092 0.942541i $$-0.391570\pi$$
0.334092 + 0.942541i $$0.391570\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ −27.0000 −1.28281 −0.641404 0.767203i $$-0.721648\pi$$
−0.641404 + 0.767203i $$0.721648\pi$$
$$444$$ 14.0000 0.664411
$$445$$ −27.0000 −1.27992
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 32.0000 1.51186
$$449$$ −27.0000 −1.27421 −0.637104 0.770778i $$-0.719868\pi$$
−0.637104 + 0.770778i $$0.719868\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 18.0000 0.846649
$$453$$ −10.0000 −0.469841
$$454$$ 0 0
$$455$$ 24.0000 1.12514
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 18.0000 0.839254
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 29.0000 1.34774 0.673872 0.738848i $$-0.264630\pi$$
0.673872 + 0.738848i $$0.264630\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 21.0000 0.973852
$$466$$ 0 0
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 8.00000 0.369800
$$469$$ −44.0000 −2.03173
$$470$$ 0 0
$$471$$ 17.0000 0.783319
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ −14.0000 −0.638345
$$482$$ 0 0
$$483$$ −12.0000 −0.546019
$$484$$ −2.00000 −0.0909091
$$485$$ 3.00000 0.136223
$$486$$ 0 0
$$487$$ −7.00000 −0.317200 −0.158600 0.987343i $$-0.550698\pi$$
−0.158600 + 0.987343i $$0.550698\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ −28.0000 −1.25724
$$497$$ −60.0000 −2.69137
$$498$$ 0 0
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ −6.00000 −0.268328
$$501$$ −18.0000 −0.804181
$$502$$ 0 0
$$503$$ 18.0000 0.802580 0.401290 0.915951i $$-0.368562\pi$$
0.401290 + 0.915951i $$0.368562\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ −4.00000 −0.177471
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ −32.0000 −1.41560
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ 48.0000 2.11513
$$516$$ 20.0000 0.880451
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 12.0000 0.524222
$$525$$ −16.0000 −0.698297
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 8.00000 0.346844
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 54.0000 2.33462
$$536$$ 0 0
$$537$$ 15.0000 0.647298
$$538$$ 0 0
$$539$$ 9.00000 0.387657
$$540$$ −30.0000 −1.29099
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ −7.00000 −0.300399
$$544$$ 0 0
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ −42.0000 −1.79415
$$549$$ 20.0000 0.853579
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 64.0000 2.72156
$$554$$ 0 0
$$555$$ 21.0000 0.891400
$$556$$ −16.0000 −0.678551
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ −20.0000 −0.845910
$$560$$ 48.0000 2.02837
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −6.00000 −0.252870 −0.126435 0.991975i $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ 0 0
$$565$$ 27.0000 1.13590
$$566$$ 0 0
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −46.0000 −1.92504 −0.962520 0.271211i $$-0.912576\pi$$
−0.962520 + 0.271211i $$0.912576\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ −15.0000 −0.626634
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 16.0000 0.666667
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ −36.0000 −1.49482
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.00000 0.248495
$$584$$ 0 0
$$585$$ 12.0000 0.496139
$$586$$ 0 0
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −18.0000 −0.742307
$$589$$ −7.00000 −0.288430
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ −28.0000 −1.15079
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ −16.0000 −0.654836
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 20.0000 0.815817 0.407909 0.913023i $$-0.366258\pi$$
0.407909 + 0.913023i $$0.366258\pi$$
$$602$$ 0 0
$$603$$ −22.0000 −0.895909
$$604$$ 20.0000 0.813788
$$605$$ −3.00000 −0.121967
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 0 0
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ −13.0000 −0.522514 −0.261257 0.965269i $$-0.584137\pi$$
−0.261257 + 0.965269i $$0.584137\pi$$
$$620$$ −42.0000 −1.68676
$$621$$ −15.0000 −0.601929
$$622$$ 0 0
$$623$$ −36.0000 −1.44231
$$624$$ 8.00000 0.320256
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 1.00000 0.0399362
$$628$$ −34.0000 −1.35675
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.00000 −0.0398094 −0.0199047 0.999802i $$-0.506336\pi$$
−0.0199047 + 0.999802i $$0.506336\pi$$
$$632$$ 0 0
$$633$$ −22.0000 −0.874421
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ −12.0000 −0.475831
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ −30.0000 −1.18678
$$640$$ 0 0
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ 0 0
$$643$$ −7.00000 −0.276053 −0.138027 0.990429i $$-0.544076\pi$$
−0.138027 + 0.990429i $$0.544076\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 30.0000 1.18125
$$646$$ 0 0
$$647$$ 21.0000 0.825595 0.412798 0.910823i $$-0.364552\pi$$
0.412798 + 0.910823i $$0.364552\pi$$
$$648$$ 0 0
$$649$$ 3.00000 0.117760
$$650$$ 0 0
$$651$$ 28.0000 1.09741
$$652$$ 8.00000 0.313304
$$653$$ 27.0000 1.05659 0.528296 0.849060i $$-0.322831\pi$$
0.528296 + 0.849060i $$0.322831\pi$$
$$654$$ 0 0
$$655$$ 18.0000 0.703318
$$656$$ 0 0
$$657$$ −16.0000 −0.624219
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 6.00000 0.233550
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.0000 0.465340
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ 36.0000 1.39288
$$669$$ 5.00000 0.193311
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 18.0000 0.692308
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −63.0000 −2.40711
$$686$$ 0 0
$$687$$ −13.0000 −0.495981
$$688$$ −40.0000 −1.52499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ −48.0000 −1.82469
$$693$$ 8.00000 0.303895
$$694$$ 0 0
$$695$$ −24.0000 −0.910372
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 32.0000 1.20949
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −7.00000 −0.264010
$$704$$ −8.00000 −0.301511
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ −6.00000 −0.225494
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 32.0000 1.20009
$$712$$ 0 0
$$713$$ −21.0000 −0.786456
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ −30.0000 −1.12115
$$717$$ −18.0000 −0.672222
$$718$$ 0 0
$$719$$ −9.00000 −0.335643 −0.167822 0.985817i $$-0.553673\pi$$
−0.167822 + 0.985817i $$0.553673\pi$$
$$720$$ 24.0000 0.894427
$$721$$ 64.0000 2.38348
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ 14.0000 0.520306
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ 35.0000 1.29808 0.649039 0.760755i $$-0.275171\pi$$
0.649039 + 0.760755i $$0.275171\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 20.0000 0.739221
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 0 0
$$735$$ −27.0000 −0.995910
$$736$$ 0 0
$$737$$ 11.0000 0.405190
$$738$$ 0 0
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ −42.0000 −1.54395
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −30.0000 −1.10059 −0.550297 0.834969i $$-0.685485\pi$$
−0.550297 + 0.834969i $$0.685485\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 72.0000 2.63082
$$750$$ 0 0
$$751$$ −1.00000 −0.0364905 −0.0182453 0.999834i $$-0.505808\pi$$
−0.0182453 + 0.999834i $$0.505808\pi$$
$$752$$ 0 0
$$753$$ 21.0000 0.765283
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ −40.0000 −1.45479
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 0 0
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ −8.00000 −0.289619
$$764$$ 30.0000 1.08536
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.00000 0.216647
$$768$$ 16.0000 0.577350
$$769$$ −28.0000 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 8.00000 0.287926
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ −28.0000 −1.00579
$$776$$ 0 0
$$777$$ 28.0000 1.00449
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 12.0000 0.429669
$$781$$ 15.0000 0.536742
$$782$$ 0 0
$$783$$ 30.0000 1.07211
$$784$$ 36.0000 1.28571
$$785$$ −51.0000 −1.82027
$$786$$ 0 0
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ −30.0000 −1.06803
$$790$$ 0 0
$$791$$ 36.0000 1.28001
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 0 0
$$795$$ −18.0000 −0.638394
$$796$$ 32.0000 1.13421
$$797$$ 51.0000 1.80651 0.903256 0.429101i $$-0.141170\pi$$
0.903256 + 0.429101i $$0.141170\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 0 0
$$803$$ 8.00000 0.282314
$$804$$ −22.0000 −0.775880
$$805$$ 36.0000 1.26883
$$806$$ 0 0
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ −24.0000 −0.843795 −0.421898 0.906644i $$-0.638636\pi$$
−0.421898 + 0.906644i $$0.638636\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −48.0000 −1.68447
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 12.0000 0.420342
$$816$$ 0 0
$$817$$ −10.0000 −0.349856
$$818$$ 0 0
$$819$$ 16.0000 0.559085
$$820$$ 0 0
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 0 0
$$823$$ −13.0000 −0.453152 −0.226576 0.973994i $$-0.572753\pi$$
−0.226576 + 0.973994i $$0.572753\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −42.0000 −1.46048 −0.730242 0.683189i $$-0.760592\pi$$
−0.730242 + 0.683189i $$0.760592\pi$$
$$828$$ 12.0000 0.417029
$$829$$ 11.0000 0.382046 0.191023 0.981586i $$-0.438820\pi$$
0.191023 + 0.981586i $$0.438820\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ −16.0000 −0.554700
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 54.0000 1.86875
$$836$$ −2.00000 −0.0691714
$$837$$ 35.0000 1.20978
$$838$$ 0 0
$$839$$ 33.0000 1.13929 0.569643 0.821892i $$-0.307081\pi$$
0.569643 + 0.821892i $$0.307081\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −30.0000 −1.03325
$$844$$ 44.0000 1.51454
$$845$$ 27.0000 0.928828
$$846$$ 0 0
$$847$$ −4.00000 −0.137442
$$848$$ 24.0000 0.824163
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ −30.0000 −1.02778
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ 5.00000 0.170598 0.0852989 0.996355i $$-0.472815\pi$$
0.0852989 + 0.996355i $$0.472815\pi$$
$$860$$ −60.0000 −2.04598
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −72.0000 −2.44807
$$866$$ 0 0
$$867$$ −17.0000 −0.577350
$$868$$ −56.0000 −1.90076
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 22.0000 0.745442
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ −16.0000 −0.540590
$$877$$ −4.00000 −0.135070 −0.0675352 0.997717i $$-0.521513\pi$$
−0.0675352 + 0.997717i $$0.521513\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ −12.0000 −0.404520
$$881$$ 9.00000 0.303218 0.151609 0.988441i $$-0.451555\pi$$
0.151609 + 0.988441i $$0.451555\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 0 0
$$885$$ −9.00000 −0.302532
$$886$$ 0 0
$$887$$ 54.0000 1.81314 0.906571 0.422053i $$-0.138690\pi$$
0.906571 + 0.422053i $$0.138690\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −10.0000 −0.334825
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −45.0000 −1.50418
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ 42.0000 1.40078
$$900$$ 16.0000 0.533333
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 40.0000 1.33112
$$904$$ 0 0
$$905$$ 21.0000 0.698064
$$906$$ 0 0
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 30.0000 0.991769
$$916$$ 26.0000 0.859064
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ 0 0
$$923$$ 30.0000 0.987462
$$924$$ 8.00000 0.263181
$$925$$ −28.0000 −0.920634
$$926$$ 0 0
$$927$$ 32.0000 1.05102
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ −36.0000 −1.17922
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ −60.0000 −1.95180
$$946$$ 0 0
$$947$$ −3.00000 −0.0974869 −0.0487435 0.998811i $$-0.515522\pi$$
−0.0487435 + 0.998811i $$0.515522\pi$$
$$948$$ 32.0000 1.03931
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ 15.0000 0.486408
$$952$$ 0 0
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ 45.0000 1.45617
$$956$$ 36.0000 1.16432
$$957$$ −6.00000 −0.193952
$$958$$ 0 0
$$959$$ −84.0000 −2.71250
$$960$$ 24.0000 0.774597
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 36.0000 1.16008
$$964$$ −40.0000 −1.28831
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ −32.0000 −1.02640
$$973$$ −32.0000 −1.02587
$$974$$ 0 0
$$975$$ 8.00000 0.256205
$$976$$ −40.0000 −1.28037
$$977$$ −9.00000 −0.287936 −0.143968 0.989582i $$-0.545986\pi$$
−0.143968 + 0.989582i $$0.545986\pi$$
$$978$$ 0 0
$$979$$ 9.00000 0.287641
$$980$$ 54.0000 1.72497
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ −15.0000 −0.478426 −0.239213 0.970967i $$-0.576889\pi$$
−0.239213 + 0.970967i $$0.576889\pi$$
$$984$$ 0 0
$$985$$ −36.0000 −1.14706
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ −7.00000 −0.222138
$$994$$ 0 0
$$995$$ 48.0000 1.52170
$$996$$ 0 0
$$997$$ −40.0000 −1.26681 −0.633406 0.773819i $$-0.718344\pi$$
−0.633406 + 0.773819i $$0.718344\pi$$
$$998$$ 0 0
$$999$$ 35.0000 1.10735
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 209.2.a.a.1.1 1
3.2 odd 2 1881.2.a.c.1.1 1
4.3 odd 2 3344.2.a.d.1.1 1
5.4 even 2 5225.2.a.b.1.1 1
11.10 odd 2 2299.2.a.c.1.1 1
19.18 odd 2 3971.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.a.1.1 1 1.1 even 1 trivial
1881.2.a.c.1.1 1 3.2 odd 2
2299.2.a.c.1.1 1 11.10 odd 2
3344.2.a.d.1.1 1 4.3 odd 2
3971.2.a.a.1.1 1 19.18 odd 2
5225.2.a.b.1.1 1 5.4 even 2