# Properties

 Label 1083.2.a.b.1.1 Level $1083$ Weight $2$ Character 1083.1 Self dual yes Analytic conductor $8.648$ 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] = [1083,2,Mod(1,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, 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("1083.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1083 = 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1083.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: $$8.64779853890$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1083.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} -3.00000 q^{37} -5.00000 q^{39} +12.0000 q^{41} -1.00000 q^{42} -1.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -4.00000 q^{51} +5.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -8.00000 q^{58} -10.0000 q^{59} -13.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -11.0000 q^{67} +4.00000 q^{68} -4.00000 q^{69} -6.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} +3.00000 q^{74} -5.00000 q^{75} -2.00000 q^{77} +5.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} -1.00000 q^{84} +1.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -5.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +6.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −5.00000 −1.38675 −0.693375 0.720577i $$-0.743877\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 2.00000 0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −5.00000 −1.00000
$$26$$ 5.00000 0.980581
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ −2.00000 −0.348155
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 0 0
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 5.00000 0.707107
$$51$$ −4.00000 −0.560112
$$52$$ 5.00000 0.693375
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −8.00000 −1.05045
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 1.00000 0.125988
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −11.0000 −1.34386 −0.671932 0.740613i $$-0.734535\pi$$
−0.671932 + 0.740613i $$0.734535\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 3.00000 0.353553
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 3.00000 0.348743
$$75$$ −5.00000 −0.577350
$$76$$ 0 0
$$77$$ −2.00000 −0.227921
$$78$$ 5.00000 0.566139
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −12.0000 −1.32518
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 8.00000 0.857690
$$88$$ −6.00000 −0.639602
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −5.00000 −0.524142
$$92$$ 4.00000 0.417029
$$93$$ 3.00000 0.311086
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 6.00000 0.606092
$$99$$ −2.00000 −0.201008
$$100$$ 5.00000 0.500000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 4.00000 0.396059
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ −15.0000 −1.47087
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ −1.00000 −0.0944911
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −8.00000 −0.742781
$$117$$ −5.00000 −0.462250
$$118$$ 10.0000 0.920575
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 13.0000 1.17696
$$123$$ 12.0000 1.08200
$$124$$ −3.00000 −0.269408
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 0 0
$$134$$ 11.0000 0.950255
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 6.00000 0.503509
$$143$$ 10.0000 0.836242
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 11.0000 0.910366
$$147$$ −6.00000 −0.494872
$$148$$ 3.00000 0.246598
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ 5.00000 0.400320
$$157$$ 3.00000 0.239426 0.119713 0.992809i $$-0.461803\pi$$
0.119713 + 0.992809i $$0.461803\pi$$
$$158$$ 1.00000 0.0795557
$$159$$ −4.00000 −0.317221
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ −1.00000 −0.0785674
$$163$$ 3.00000 0.234978 0.117489 0.993074i $$-0.462515\pi$$
0.117489 + 0.993074i $$0.462515\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2.00000 0.154765 0.0773823 0.997001i $$-0.475344\pi$$
0.0773823 + 0.997001i $$0.475344\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.00000 0.0762493
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ −5.00000 −0.377964
$$176$$ 2.00000 0.150756
$$177$$ −10.0000 −0.751646
$$178$$ −6.00000 −0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 5.00000 0.370625
$$183$$ −13.0000 −0.960988
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ −3.00000 −0.219971
$$187$$ 8.00000 0.585018
$$188$$ 6.00000 0.437595
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 7.00000 0.505181
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 6.00000 0.428571
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ −15.0000 −1.06066
$$201$$ −11.0000 −0.775880
$$202$$ −14.0000 −0.985037
$$203$$ 8.00000 0.561490
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 13.0000 0.905753
$$207$$ −4.00000 −0.278019
$$208$$ 5.00000 0.346688
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ 4.00000 0.274721
$$213$$ −6.00000 −0.411113
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 3.00000 0.203653
$$218$$ −2.00000 −0.135457
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ 20.0000 1.34535
$$222$$ 3.00000 0.201347
$$223$$ −9.00000 −0.602685 −0.301342 0.953516i $$-0.597435\pi$$
−0.301342 + 0.953516i $$0.597435\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ −5.00000 −0.333333
$$226$$ 2.00000 0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 24.0000 1.57568
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 5.00000 0.326860
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ −1.00000 −0.0649570
$$238$$ 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
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 1.00000 0.0641500
$$244$$ 13.0000 0.832240
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ 0 0
$$248$$ 9.00000 0.571501
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 8.00000 0.502956
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 1.00000 0.0622573
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 14.0000 0.864923
$$263$$ 26.0000 1.60323 0.801614 0.597841i $$-0.203975\pi$$
0.801614 + 0.597841i $$0.203975\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 11.0000 0.671932
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 4.00000 0.242536
$$273$$ −5.00000 −0.302614
$$274$$ −18.0000 −1.08742
$$275$$ 10.0000 0.603023
$$276$$ 4.00000 0.240772
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −5.00000 −0.299880
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ −10.0000 −0.591312
$$287$$ 12.0000 0.708338
$$288$$ −5.00000 −0.294628
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 11.0000 0.643726
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 0 0
$$296$$ −9.00000 −0.523114
$$297$$ −2.00000 −0.116052
$$298$$ 6.00000 0.347571
$$299$$ 20.0000 1.15663
$$300$$ 5.00000 0.288675
$$301$$ −1.00000 −0.0576390
$$302$$ 12.0000 0.690522
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 2.00000 0.113961
$$309$$ −13.0000 −0.739544
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ −15.0000 −0.849208
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −3.00000 −0.169300
$$315$$ 0 0
$$316$$ 1.00000 0.0562544
$$317$$ 4.00000 0.224662 0.112331 0.993671i $$-0.464168\pi$$
0.112331 + 0.993671i $$0.464168\pi$$
$$318$$ 4.00000 0.224309
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 4.00000 0.222911
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 25.0000 1.38675
$$326$$ −3.00000 −0.166155
$$327$$ 2.00000 0.110600
$$328$$ 36.0000 1.98777
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ −3.00000 −0.164399
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ −3.00000 −0.161749
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ −21.0000 −1.12410 −0.562052 0.827102i $$-0.689988\pi$$
−0.562052 + 0.827102i $$0.689988\pi$$
$$350$$ 5.00000 0.267261
$$351$$ −5.00000 −0.266880
$$352$$ 10.0000 0.533002
$$353$$ 4.00000 0.212899 0.106449 0.994318i $$-0.466052\pi$$
0.106449 + 0.994318i $$0.466052\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −4.00000 −0.211702
$$358$$ 12.0000 0.634220
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ −2.00000 −0.105118
$$363$$ −7.00000 −0.367405
$$364$$ 5.00000 0.262071
$$365$$ 0 0
$$366$$ 13.0000 0.679521
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ −3.00000 −0.155543
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −18.0000 −0.928279
$$377$$ −40.0000 −2.06010
$$378$$ −1.00000 −0.0514344
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ −24.0000 −1.22795
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ 19.0000 0.967075
$$387$$ −1.00000 −0.0508329
$$388$$ 2.00000 0.101535
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ −18.0000 −0.909137
$$393$$ −14.0000 −0.706207
$$394$$ 26.0000 1.30986
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ −21.0000 −1.05263
$$399$$ 0 0
$$400$$ 5.00000 0.250000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 11.0000 0.548630
$$403$$ −15.0000 −0.747203
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 6.00000 0.297409
$$408$$ −12.0000 −0.594089
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 13.0000 0.640464
$$413$$ −10.0000 −0.492068
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 25.0000 1.22573
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −15.0000 −0.730189
$$423$$ −6.00000 −0.291730
$$424$$ −12.0000 −0.582772
$$425$$ 20.0000 0.970143
$$426$$ 6.00000 0.290701
$$427$$ −13.0000 −0.629114
$$428$$ −18.0000 −0.870063
$$429$$ 10.0000 0.482805
$$430$$ 0 0
$$431$$ 10.0000 0.481683 0.240842 0.970564i $$-0.422577\pi$$
0.240842 + 0.970564i $$0.422577\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 9.00000 0.432512 0.216256 0.976337i $$-0.430615\pi$$
0.216256 + 0.976337i $$0.430615\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 11.0000 0.525600
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ −20.0000 −0.951303
$$443$$ −32.0000 −1.52037 −0.760183 0.649709i $$-0.774891\pi$$
−0.760183 + 0.649709i $$0.774891\pi$$
$$444$$ 3.00000 0.142374
$$445$$ 0 0
$$446$$ 9.00000 0.426162
$$447$$ −6.00000 −0.283790
$$448$$ 7.00000 0.330719
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 5.00000 0.235702
$$451$$ −24.0000 −1.13012
$$452$$ 2.00000 0.0940721
$$453$$ −12.0000 −0.563809
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ −13.0000 −0.607450
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 2.00000 0.0930484
$$463$$ 23.0000 1.06890 0.534450 0.845200i $$-0.320519\pi$$
0.534450 + 0.845200i $$0.320519\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 5.00000 0.231125
$$469$$ −11.0000 −0.507933
$$470$$ 0 0
$$471$$ 3.00000 0.138233
$$472$$ −30.0000 −1.38086
$$473$$ 2.00000 0.0919601
$$474$$ 1.00000 0.0459315
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ −4.00000 −0.183147
$$478$$ 12.0000 0.548867
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ 0 0
$$481$$ 15.0000 0.683941
$$482$$ −7.00000 −0.318841
$$483$$ −4.00000 −0.182006
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ −39.0000 −1.76545
$$489$$ 3.00000 0.135665
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ −32.0000 −1.44121
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ 29.0000 1.29822 0.649109 0.760695i $$-0.275142\pi$$
0.649109 + 0.760695i $$0.275142\pi$$
$$500$$ 0 0
$$501$$ 2.00000 0.0893534
$$502$$ 2.00000 0.0892644
$$503$$ 40.0000 1.78351 0.891756 0.452517i $$-0.149474\pi$$
0.891756 + 0.452517i $$0.149474\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ −8.00000 −0.355643
$$507$$ 12.0000 0.532939
$$508$$ −8.00000 −0.354943
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −11.0000 −0.486611
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ −20.0000 −0.882162
$$515$$ 0 0
$$516$$ 1.00000 0.0440225
$$517$$ 12.0000 0.527759
$$518$$ 3.00000 0.131812
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ −8.00000 −0.350150
$$523$$ −29.0000 −1.26808 −0.634041 0.773300i $$-0.718605\pi$$
−0.634041 + 0.773300i $$0.718605\pi$$
$$524$$ 14.0000 0.611593
$$525$$ −5.00000 −0.218218
$$526$$ −26.0000 −1.13365
$$527$$ −12.0000 −0.522728
$$528$$ 2.00000 0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 0 0
$$533$$ −60.0000 −2.59889
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ −33.0000 −1.42538
$$537$$ −12.0000 −0.517838
$$538$$ 10.0000 0.431131
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ 0 0
$$543$$ 2.00000 0.0858282
$$544$$ 20.0000 0.857493
$$545$$ 0 0
$$546$$ 5.00000 0.213980
$$547$$ 29.0000 1.23995 0.619975 0.784621i $$-0.287143\pi$$
0.619975 + 0.784621i $$0.287143\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ −13.0000 −0.554826
$$550$$ −10.0000 −0.426401
$$551$$ 0 0
$$552$$ −12.0000 −0.510754
$$553$$ −1.00000 −0.0425243
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −5.00000 −0.212047
$$557$$ −38.0000 −1.61011 −0.805056 0.593199i $$-0.797865\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ −3.00000 −0.127000
$$559$$ 5.00000 0.211477
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ −8.00000 −0.337460
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −18.0000 −0.755263
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 33.0000 1.38101 0.690504 0.723329i $$-0.257389\pi$$
0.690504 + 0.723329i $$0.257389\pi$$
$$572$$ −10.0000 −0.418121
$$573$$ 24.0000 1.00261
$$574$$ −12.0000 −0.500870
$$575$$ 20.0000 0.834058
$$576$$ 7.00000 0.291667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −19.0000 −0.789613
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 2.00000 0.0829027
$$583$$ 8.00000 0.331326
$$584$$ −33.0000 −1.36555
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ −26.0000 −1.07313 −0.536567 0.843857i $$-0.680279\pi$$
−0.536567 + 0.843857i $$0.680279\pi$$
$$588$$ 6.00000 0.247436
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −26.0000 −1.06950
$$592$$ 3.00000 0.123299
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 21.0000 0.859473
$$598$$ −20.0000 −0.817861
$$599$$ −18.0000 −0.735460 −0.367730 0.929933i $$-0.619865\pi$$
−0.367730 + 0.929933i $$0.619865\pi$$
$$600$$ −15.0000 −0.612372
$$601$$ 21.0000 0.856608 0.428304 0.903635i $$-0.359111\pi$$
0.428304 + 0.903635i $$0.359111\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ −11.0000 −0.447955
$$604$$ 12.0000 0.488273
$$605$$ 0 0
$$606$$ −14.0000 −0.568711
$$607$$ 43.0000 1.74532 0.872658 0.488332i $$-0.162394\pi$$
0.872658 + 0.488332i $$0.162394\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 30.0000 1.21367
$$612$$ 4.00000 0.161690
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 13.0000 0.522937
$$619$$ 25.0000 1.00483 0.502417 0.864625i $$-0.332444\pi$$
0.502417 + 0.864625i $$0.332444\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ −18.0000 −0.721734
$$623$$ 6.00000 0.240385
$$624$$ 5.00000 0.200160
$$625$$ 25.0000 1.00000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ −3.00000 −0.119713
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −15.0000 −0.597141 −0.298570 0.954388i $$-0.596510\pi$$
−0.298570 + 0.954388i $$0.596510\pi$$
$$632$$ −3.00000 −0.119334
$$633$$ 15.0000 0.596196
$$634$$ −4.00000 −0.158860
$$635$$ 0 0
$$636$$ 4.00000 0.158610
$$637$$ 30.0000 1.18864
$$638$$ 16.0000 0.633446
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 20.0000 0.785069
$$650$$ −25.0000 −0.980581
$$651$$ 3.00000 0.117579
$$652$$ −3.00000 −0.117489
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −12.0000 −0.468521
$$657$$ −11.0000 −0.429151
$$658$$ 6.00000 0.233904
$$659$$ −34.0000 −1.32445 −0.662226 0.749304i $$-0.730388\pi$$
−0.662226 + 0.749304i $$0.730388\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ −25.0000 −0.971653
$$663$$ 20.0000 0.776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ −32.0000 −1.23904
$$668$$ −2.00000 −0.0773823
$$669$$ −9.00000 −0.347960
$$670$$ 0 0
$$671$$ 26.0000 1.00372
$$672$$ −5.00000 −0.192879
$$673$$ 33.0000 1.27206 0.636028 0.771666i $$-0.280576\pi$$
0.636028 + 0.771666i $$0.280576\pi$$
$$674$$ 13.0000 0.500741
$$675$$ −5.00000 −0.192450
$$676$$ −12.0000 −0.461538
$$677$$ −34.0000 −1.30673 −0.653363 0.757045i $$-0.726642\pi$$
−0.653363 + 0.757045i $$0.726642\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ 6.00000 0.229752
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13.0000 0.496342
$$687$$ 13.0000 0.495981
$$688$$ 1.00000 0.0381246
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ −2.00000 −0.0759737
$$694$$ 16.0000 0.607352
$$695$$ 0 0
$$696$$ 24.0000 0.909718
$$697$$ −48.0000 −1.81813
$$698$$ 21.0000 0.794862
$$699$$ −6.00000 −0.226941
$$700$$ 5.00000 0.188982
$$701$$ 20.0000 0.755390 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 5.00000 0.188713
$$703$$ 0 0
$$704$$ −14.0000 −0.527645
$$705$$ 0 0
$$706$$ −4.00000 −0.150542
$$707$$ 14.0000 0.526524
$$708$$ 10.0000 0.375823
$$709$$ −39.0000 −1.46468 −0.732338 0.680941i $$-0.761571\pi$$
−0.732338 + 0.680941i $$0.761571\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 18.0000 0.674579
$$713$$ −12.0000 −0.449404
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −12.0000 −0.448148
$$718$$ 20.0000 0.746393
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −13.0000 −0.484145
$$722$$ 0 0
$$723$$ 7.00000 0.260333
$$724$$ −2.00000 −0.0743294
$$725$$ −40.0000 −1.48556
$$726$$ 7.00000 0.259794
$$727$$ −47.0000 −1.74313 −0.871567 0.490277i $$-0.836896\pi$$
−0.871567 + 0.490277i $$0.836896\pi$$
$$728$$ −15.0000 −0.555937
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 13.0000 0.480494
$$733$$ −50.0000 −1.84679 −0.923396 0.383849i $$-0.874598\pi$$
−0.923396 + 0.383849i $$0.874598\pi$$
$$734$$ −7.00000 −0.258375
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 22.0000 0.810380
$$738$$ −12.0000 −0.441726
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ 50.0000 1.83432 0.917161 0.398517i $$-0.130475\pi$$
0.917161 + 0.398517i $$0.130475\pi$$
$$744$$ 9.00000 0.329956
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ −45.0000 −1.64207 −0.821037 0.570875i $$-0.806604\pi$$
−0.821037 + 0.570875i $$0.806604\pi$$
$$752$$ 6.00000 0.218797
$$753$$ −2.00000 −0.0728841
$$754$$ 40.0000 1.45671
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 13.0000 0.472493 0.236247 0.971693i $$-0.424083\pi$$
0.236247 + 0.971693i $$0.424083\pi$$
$$758$$ −5.00000 −0.181608
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 2.00000 0.0724049
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ 14.0000 0.505841
$$767$$ 50.0000 1.80540
$$768$$ −17.0000 −0.613435
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 0 0
$$771$$ 20.0000 0.720282
$$772$$ 19.0000 0.683825
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ −15.0000 −0.538816
$$776$$ −6.00000 −0.215387
$$777$$ −3.00000 −0.107624
$$778$$ 24.0000 0.860442
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ −16.0000 −0.572159
$$783$$ 8.00000 0.285897
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ −25.0000 −0.891154 −0.445577 0.895244i $$-0.647001\pi$$
−0.445577 + 0.895244i $$0.647001\pi$$
$$788$$ 26.0000 0.926212
$$789$$ 26.0000 0.925625
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ −6.00000 −0.213201
$$793$$ 65.0000 2.30822
$$794$$ −7.00000 −0.248421
$$795$$ 0 0
$$796$$ −21.0000 −0.744325
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 25.0000 0.883883
$$801$$ 6.00000 0.212000
$$802$$ −6.00000 −0.211867
$$803$$ 22.0000 0.776363
$$804$$ 11.0000 0.387940
$$805$$ 0 0
$$806$$ 15.0000 0.528352
$$807$$ −10.0000 −0.352017
$$808$$ 42.0000 1.47755
$$809$$ −16.0000 −0.562530 −0.281265 0.959630i $$-0.590754\pi$$
−0.281265 + 0.959630i $$0.590754\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 0 0
$$818$$ −14.0000 −0.489499
$$819$$ −5.00000 −0.174714
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ −39.0000 −1.35863
$$825$$ 10.0000 0.348155
$$826$$ 10.0000 0.347945
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ −35.0000 −1.21341
$$833$$ 24.0000 0.831551
$$834$$ −5.00000 −0.173136
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.00000 0.103695
$$838$$ 14.0000 0.483622
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ −22.0000 −0.758170
$$843$$ 8.00000 0.275535
$$844$$ −15.0000 −0.516321
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ −7.00000 −0.240523
$$848$$ 4.00000 0.137361
$$849$$ −4.00000 −0.137280
$$850$$ −20.0000 −0.685994
$$851$$ 12.0000 0.411355
$$852$$ 6.00000 0.205557
$$853$$ −5.00000 −0.171197 −0.0855984 0.996330i $$-0.527280\pi$$
−0.0855984 + 0.996330i $$0.527280\pi$$
$$854$$ 13.0000 0.444851
$$855$$ 0 0
$$856$$ 54.0000 1.84568
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ −10.0000 −0.341394
$$859$$ 15.0000 0.511793 0.255897 0.966704i $$-0.417629\pi$$
0.255897 + 0.966704i $$0.417629\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ −10.0000 −0.340601
$$863$$ −10.0000 −0.340404 −0.170202 0.985409i $$-0.554442\pi$$
−0.170202 + 0.985409i $$0.554442\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −9.00000 −0.305832
$$867$$ −1.00000 −0.0339618
$$868$$ −3.00000 −0.101827
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ 55.0000 1.86360
$$872$$ 6.00000 0.203186
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 11.0000 0.371656
$$877$$ −15.0000 −0.506514 −0.253257 0.967399i $$-0.581502\pi$$
−0.253257 + 0.967399i $$0.581502\pi$$
$$878$$ 35.0000 1.18119
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ −28.0000 −0.943344 −0.471672 0.881774i $$-0.656349\pi$$
−0.471672 + 0.881774i $$0.656349\pi$$
$$882$$ 6.00000 0.202031
$$883$$ −1.00000 −0.0336527 −0.0168263 0.999858i $$-0.505356\pi$$
−0.0168263 + 0.999858i $$0.505356\pi$$
$$884$$ −20.0000 −0.672673
$$885$$ 0 0
$$886$$ 32.0000 1.07506
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ −9.00000 −0.302020
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 9.00000 0.301342
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 20.0000 0.667781
$$898$$ −12.0000 −0.400445
$$899$$ 24.0000 0.800445
$$900$$ 5.00000 0.166667
$$901$$ 16.0000 0.533037
$$902$$ 24.0000 0.799113
$$903$$ −1.00000 −0.0332779
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 12.0000 0.398673
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 11.0000 0.363848
$$915$$ 0 0
$$916$$ −13.0000 −0.429532
$$917$$ −14.0000 −0.462321
$$918$$ 4.00000 0.132020
$$919$$ −25.0000 −0.824674 −0.412337 0.911031i $$-0.635287\pi$$
−0.412337 + 0.911031i $$0.635287\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ −30.0000 −0.987997
$$923$$ 30.0000 0.987462
$$924$$ 2.00000 0.0657952
$$925$$ 15.0000 0.493197
$$926$$ −23.0000 −0.755827
$$927$$ −13.0000 −0.426976
$$928$$ −40.0000 −1.31306
$$929$$ 28.0000 0.918650 0.459325 0.888268i $$-0.348091\pi$$
0.459325 + 0.888268i $$0.348091\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 18.0000 0.589294
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ −15.0000 −0.490290
$$937$$ −9.00000 −0.294017 −0.147009 0.989135i $$-0.546964\pi$$
−0.147009 + 0.989135i $$0.546964\pi$$
$$938$$ 11.0000 0.359163
$$939$$ 22.0000 0.717943
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ −3.00000 −0.0977453
$$943$$ −48.0000 −1.56310
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ −2.00000 −0.0650256
$$947$$ 50.0000 1.62478 0.812391 0.583113i $$-0.198166\pi$$
0.812391 + 0.583113i $$0.198166\pi$$
$$948$$ 1.00000 0.0324785
$$949$$ 55.0000 1.78538
$$950$$ 0 0
$$951$$ 4.00000 0.129709
$$952$$ −12.0000 −0.388922
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 4.00000 0.129505
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ −16.0000 −0.517207
$$958$$ −14.0000 −0.452319
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −15.0000 −0.483619
$$963$$ 18.0000 0.580042
$$964$$ −7.00000 −0.225455
$$965$$ 0 0
$$966$$ 4.00000 0.128698
$$967$$ −29.0000 −0.932577 −0.466289 0.884633i $$-0.654409\pi$$
−0.466289 + 0.884633i $$0.654409\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 5.00000 0.160293
$$974$$ −16.0000 −0.512673
$$975$$ 25.0000 0.800641
$$976$$ 13.0000 0.416120
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ −3.00000 −0.0959294
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ −6.00000 −0.191468
$$983$$ −10.0000 −0.318950 −0.159475 0.987202i $$-0.550980\pi$$
−0.159475 + 0.987202i $$0.550980\pi$$
$$984$$ 36.0000 1.14764
$$985$$ 0 0
$$986$$ 32.0000 1.01909
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −47.0000 −1.49300 −0.746502 0.665383i $$-0.768268\pi$$
−0.746502 + 0.665383i $$0.768268\pi$$
$$992$$ −15.0000 −0.476250
$$993$$ 25.0000 0.793351
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 7.00000 0.221692 0.110846 0.993838i $$-0.464644\pi$$
0.110846 + 0.993838i $$0.464644\pi$$
$$998$$ −29.0000 −0.917979
$$999$$ −3.00000 −0.0949158
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 1083.2.a.b.1.1 1
3.2 odd 2 3249.2.a.f.1.1 1
19.8 odd 6 57.2.e.a.7.1 2
19.12 odd 6 57.2.e.a.49.1 yes 2
19.18 odd 2 1083.2.a.c.1.1 1
57.8 even 6 171.2.f.a.64.1 2
57.50 even 6 171.2.f.a.163.1 2
57.56 even 2 3249.2.a.c.1.1 1
76.27 even 6 912.2.q.a.577.1 2
76.31 even 6 912.2.q.a.49.1 2
228.107 odd 6 2736.2.s.j.1873.1 2
228.179 odd 6 2736.2.s.j.577.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.a.7.1 2 19.8 odd 6
57.2.e.a.49.1 yes 2 19.12 odd 6
171.2.f.a.64.1 2 57.8 even 6
171.2.f.a.163.1 2 57.50 even 6
912.2.q.a.49.1 2 76.31 even 6
912.2.q.a.577.1 2 76.27 even 6
1083.2.a.b.1.1 1 1.1 even 1 trivial
1083.2.a.c.1.1 1 19.18 odd 2
2736.2.s.j.577.1 2 228.179 odd 6
2736.2.s.j.1873.1 2 228.107 odd 6
3249.2.a.c.1.1 1 57.56 even 2
3249.2.a.f.1.1 1 3.2 odd 2