# Properties

 Label 1014.2.a.a.1.1 Level $1014$ Weight $2$ Character 1014.1 Self dual yes Analytic conductor $8.097$ 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] = [1014,2,Mod(1,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, 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("1014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$8.09683076496$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1014.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -9.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -11.0000 q^{37} +2.00000 q^{38} +1.00000 q^{40} +5.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -5.00000 q^{51} -1.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} +9.00000 q^{58} -8.00000 q^{59} +1.00000 q^{60} -11.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} +5.00000 q^{68} -6.00000 q^{69} -2.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +11.0000 q^{74} +4.00000 q^{75} -2.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -5.00000 q^{85} -10.0000 q^{86} +9.00000 q^{87} -2.00000 q^{88} +2.00000 q^{89} +1.00000 q^{90} +6.00000 q^{92} +4.00000 q^{93} -2.00000 q^{94} +2.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 2.00000 0.436436
$$22$$ −2.00000 −0.426401
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.00000 −0.348155
$$34$$ −5.00000 −0.857493
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ −11.0000 −1.80839 −0.904194 0.427121i $$-0.859528\pi$$
−0.904194 + 0.427121i $$0.859528\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −1.00000 −0.149071
$$46$$ −6.00000 −0.884652
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 4.00000 0.565685
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −2.00000 −0.269680
$$56$$ 2.00000 0.267261
$$57$$ 2.00000 0.264906
$$58$$ 9.00000 1.18176
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 5.00000 0.606339
$$69$$ −6.00000 −0.722315
$$70$$ −2.00000 −0.239046
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −13.0000 −1.52153 −0.760767 0.649025i $$-0.775177\pi$$
−0.760767 + 0.649025i $$0.775177\pi$$
$$74$$ 11.0000 1.27872
$$75$$ 4.00000 0.461880
$$76$$ −2.00000 −0.229416
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −5.00000 −0.552158
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −5.00000 −0.542326
$$86$$ −10.0000 −1.07833
$$87$$ 9.00000 0.964901
$$88$$ −2.00000 −0.213201
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 4.00000 0.414781
$$94$$ −2.00000 −0.206284
$$95$$ 2.00000 0.205196
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 2.00000 0.201008
$$100$$ −4.00000 −0.400000
$$101$$ −5.00000 −0.497519 −0.248759 0.968565i $$-0.580023\pi$$
−0.248759 + 0.968565i $$0.580023\pi$$
$$102$$ 5.00000 0.495074
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 1.00000 0.0971286
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 11.0000 1.04407
$$112$$ −2.00000 −0.188982
$$113$$ −3.00000 −0.282216 −0.141108 0.989994i $$-0.545067\pi$$
−0.141108 + 0.989994i $$0.545067\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ −6.00000 −0.559503
$$116$$ −9.00000 −0.835629
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ −10.0000 −0.916698
$$120$$ −1.00000 −0.0912871
$$121$$ −7.00000 −0.636364
$$122$$ 11.0000 0.995893
$$123$$ −5.00000 −0.450835
$$124$$ −4.00000 −0.359211
$$125$$ 9.00000 0.804984
$$126$$ 2.00000 0.178174
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −10.0000 −0.880451
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 4.00000 0.346844
$$134$$ −2.00000 −0.172774
$$135$$ 1.00000 0.0860663
$$136$$ −5.00000 −0.428746
$$137$$ 17.0000 1.45241 0.726204 0.687479i $$-0.241283\pi$$
0.726204 + 0.687479i $$0.241283\pi$$
$$138$$ 6.00000 0.510754
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 2.00000 0.169031
$$141$$ −2.00000 −0.168430
$$142$$ 14.0000 1.17485
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 9.00000 0.747409
$$146$$ 13.0000 1.07589
$$147$$ 3.00000 0.247436
$$148$$ −11.0000 −0.904194
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ −6.00000 −0.488273 −0.244137 0.969741i $$-0.578505\pi$$
−0.244137 + 0.969741i $$0.578505\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 5.00000 0.404226
$$154$$ 4.00000 0.322329
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 1.00000 0.0793052
$$160$$ 1.00000 0.0790569
$$161$$ −12.0000 −0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 2.00000 0.155700
$$166$$ −6.00000 −0.465690
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 5.00000 0.383482
$$171$$ −2.00000 −0.152944
$$172$$ 10.0000 0.762493
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ 8.00000 0.604743
$$176$$ 2.00000 0.150756
$$177$$ 8.00000 0.601317
$$178$$ −2.00000 −0.149906
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 11.0000 0.813143
$$184$$ −6.00000 −0.442326
$$185$$ 11.0000 0.808736
$$186$$ −4.00000 −0.293294
$$187$$ 10.0000 0.731272
$$188$$ 2.00000 0.145865
$$189$$ 2.00000 0.145479
$$190$$ −2.00000 −0.145095
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −17.0000 −1.22369 −0.611843 0.790979i $$-0.709572\pi$$
−0.611843 + 0.790979i $$0.709572\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 4.00000 0.282843
$$201$$ −2.00000 −0.141069
$$202$$ 5.00000 0.351799
$$203$$ 18.0000 1.26335
$$204$$ −5.00000 −0.350070
$$205$$ −5.00000 −0.349215
$$206$$ −10.0000 −0.696733
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 2.00000 0.138013
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ 14.0000 0.959264
$$214$$ 18.0000 1.23045
$$215$$ −10.0000 −0.681994
$$216$$ 1.00000 0.0680414
$$217$$ 8.00000 0.543075
$$218$$ 2.00000 0.135457
$$219$$ 13.0000 0.878459
$$220$$ −2.00000 −0.134840
$$221$$ 0 0
$$222$$ −11.0000 −0.738272
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −4.00000 −0.266667
$$226$$ 3.00000 0.199557
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 4.00000 0.263181
$$232$$ 9.00000 0.590879
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ −2.00000 −0.130466
$$236$$ −8.00000 −0.520756
$$237$$ 4.00000 0.259828
$$238$$ 10.0000 0.648204
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 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
$$244$$ −11.0000 −0.704203
$$245$$ 3.00000 0.191663
$$246$$ 5.00000 0.318788
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ −6.00000 −0.380235
$$250$$ −9.00000 −0.569210
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 12.0000 0.754434
$$254$$ 12.0000 0.752947
$$255$$ 5.00000 0.313112
$$256$$ 1.00000 0.0625000
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 10.0000 0.622573
$$259$$ 22.0000 1.36701
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 8.00000 0.494242
$$263$$ 14.0000 0.863277 0.431638 0.902047i $$-0.357936\pi$$
0.431638 + 0.902047i $$0.357936\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 1.00000 0.0614295
$$266$$ −4.00000 −0.245256
$$267$$ −2.00000 −0.122398
$$268$$ 2.00000 0.122169
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 5.00000 0.303170
$$273$$ 0 0
$$274$$ −17.0000 −1.02701
$$275$$ −8.00000 −0.482418
$$276$$ −6.00000 −0.361158
$$277$$ −11.0000 −0.660926 −0.330463 0.943819i $$-0.607205\pi$$
−0.330463 + 0.943819i $$0.607205\pi$$
$$278$$ 12.0000 0.719712
$$279$$ −4.00000 −0.239474
$$280$$ −2.00000 −0.119523
$$281$$ 25.0000 1.49137 0.745687 0.666296i $$-0.232121\pi$$
0.745687 + 0.666296i $$0.232121\pi$$
$$282$$ 2.00000 0.119098
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ −14.0000 −0.830747
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ −10.0000 −0.590281
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ −9.00000 −0.528498
$$291$$ 2.00000 0.117242
$$292$$ −13.0000 −0.760767
$$293$$ −1.00000 −0.0584206 −0.0292103 0.999573i $$-0.509299\pi$$
−0.0292103 + 0.999573i $$0.509299\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 8.00000 0.465778
$$296$$ 11.0000 0.639362
$$297$$ −2.00000 −0.116052
$$298$$ −3.00000 −0.173785
$$299$$ 0 0
$$300$$ 4.00000 0.230940
$$301$$ −20.0000 −1.15278
$$302$$ 6.00000 0.345261
$$303$$ 5.00000 0.287242
$$304$$ −2.00000 −0.114708
$$305$$ 11.0000 0.629858
$$306$$ −5.00000 −0.285831
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −10.0000 −0.568880
$$310$$ −4.00000 −0.227185
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 7.00000 0.395033
$$315$$ 2.00000 0.112687
$$316$$ −4.00000 −0.225018
$$317$$ −33.0000 −1.85346 −0.926732 0.375722i $$-0.877395\pi$$
−0.926732 + 0.375722i $$0.877395\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ −18.0000 −1.00781
$$320$$ −1.00000 −0.0559017
$$321$$ 18.0000 1.00466
$$322$$ 12.0000 0.668734
$$323$$ −10.0000 −0.556415
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 2.00000 0.110600
$$328$$ −5.00000 −0.276079
$$329$$ −4.00000 −0.220527
$$330$$ −2.00000 −0.110096
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −11.0000 −0.602796
$$334$$ −24.0000 −1.31322
$$335$$ −2.00000 −0.109272
$$336$$ 2.00000 0.109109
$$337$$ −9.00000 −0.490261 −0.245131 0.969490i $$-0.578831\pi$$
−0.245131 + 0.969490i $$0.578831\pi$$
$$338$$ 0 0
$$339$$ 3.00000 0.162938
$$340$$ −5.00000 −0.271163
$$341$$ −8.00000 −0.433224
$$342$$ 2.00000 0.108148
$$343$$ 20.0000 1.07990
$$344$$ −10.0000 −0.539164
$$345$$ 6.00000 0.323029
$$346$$ 22.0000 1.18273
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ −8.00000 −0.427618
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ 17.0000 0.904819 0.452409 0.891810i $$-0.350565\pi$$
0.452409 + 0.891810i $$0.350565\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 14.0000 0.743043
$$356$$ 2.00000 0.106000
$$357$$ 10.0000 0.529256
$$358$$ 6.00000 0.317110
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ −5.00000 −0.262794
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ 13.0000 0.680451
$$366$$ −11.0000 −0.574979
$$367$$ −2.00000 −0.104399 −0.0521996 0.998637i $$-0.516623\pi$$
−0.0521996 + 0.998637i $$0.516623\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 5.00000 0.260290
$$370$$ −11.0000 −0.571863
$$371$$ 2.00000 0.103835
$$372$$ 4.00000 0.207390
$$373$$ 9.00000 0.466002 0.233001 0.972476i $$-0.425145\pi$$
0.233001 + 0.972476i $$0.425145\pi$$
$$374$$ −10.0000 −0.517088
$$375$$ −9.00000 −0.464758
$$376$$ −2.00000 −0.103142
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 12.0000 0.614779
$$382$$ −4.00000 −0.204658
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 4.00000 0.203859
$$386$$ 17.0000 0.865277
$$387$$ 10.0000 0.508329
$$388$$ −2.00000 −0.101535
$$389$$ 19.0000 0.963338 0.481669 0.876353i $$-0.340031\pi$$
0.481669 + 0.876353i $$0.340031\pi$$
$$390$$ 0 0
$$391$$ 30.0000 1.51717
$$392$$ 3.00000 0.151523
$$393$$ 8.00000 0.403547
$$394$$ −6.00000 −0.302276
$$395$$ 4.00000 0.201262
$$396$$ 2.00000 0.100504
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ −4.00000 −0.200250
$$400$$ −4.00000 −0.200000
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 2.00000 0.0997509
$$403$$ 0 0
$$404$$ −5.00000 −0.248759
$$405$$ −1.00000 −0.0496904
$$406$$ −18.0000 −0.893325
$$407$$ −22.0000 −1.09050
$$408$$ 5.00000 0.247537
$$409$$ 23.0000 1.13728 0.568638 0.822588i $$-0.307470\pi$$
0.568638 + 0.822588i $$0.307470\pi$$
$$410$$ 5.00000 0.246932
$$411$$ −17.0000 −0.838548
$$412$$ 10.0000 0.492665
$$413$$ 16.0000 0.787309
$$414$$ −6.00000 −0.294884
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 4.00000 0.195646
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ −23.0000 −1.12095 −0.560476 0.828171i $$-0.689382\pi$$
−0.560476 + 0.828171i $$0.689382\pi$$
$$422$$ −24.0000 −1.16830
$$423$$ 2.00000 0.0972433
$$424$$ 1.00000 0.0485643
$$425$$ −20.0000 −0.970143
$$426$$ −14.0000 −0.678302
$$427$$ 22.0000 1.06465
$$428$$ −18.0000 −0.870063
$$429$$ 0 0
$$430$$ 10.0000 0.482243
$$431$$ −2.00000 −0.0963366 −0.0481683 0.998839i $$-0.515338\pi$$
−0.0481683 + 0.998839i $$0.515338\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −21.0000 −1.00920 −0.504598 0.863355i $$-0.668359\pi$$
−0.504598 + 0.863355i $$0.668359\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ −9.00000 −0.431517
$$436$$ −2.00000 −0.0957826
$$437$$ −12.0000 −0.574038
$$438$$ −13.0000 −0.621164
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 11.0000 0.522037
$$445$$ −2.00000 −0.0948091
$$446$$ 16.0000 0.757622
$$447$$ −3.00000 −0.141895
$$448$$ −2.00000 −0.0944911
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 4.00000 0.188562
$$451$$ 10.0000 0.470882
$$452$$ −3.00000 −0.141108
$$453$$ 6.00000 0.281905
$$454$$ −14.0000 −0.657053
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 3.00000 0.140334 0.0701670 0.997535i $$-0.477647\pi$$
0.0701670 + 0.997535i $$0.477647\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −5.00000 −0.233380
$$460$$ −6.00000 −0.279751
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ −4.00000 −0.185496
$$466$$ 6.00000 0.277945
$$467$$ −22.0000 −1.01804 −0.509019 0.860755i $$-0.669992\pi$$
−0.509019 + 0.860755i $$0.669992\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 2.00000 0.0922531
$$471$$ 7.00000 0.322543
$$472$$ 8.00000 0.368230
$$473$$ 20.0000 0.919601
$$474$$ −4.00000 −0.183726
$$475$$ 8.00000 0.367065
$$476$$ −10.0000 −0.458349
$$477$$ −1.00000 −0.0457869
$$478$$ 6.00000 0.274434
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −7.00000 −0.318841
$$483$$ 12.0000 0.546019
$$484$$ −7.00000 −0.318182
$$485$$ 2.00000 0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 11.0000 0.497947
$$489$$ −20.0000 −0.904431
$$490$$ −3.00000 −0.135526
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ −5.00000 −0.225417
$$493$$ −45.0000 −2.02670
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ −4.00000 −0.179605
$$497$$ 28.0000 1.25597
$$498$$ 6.00000 0.268866
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 9.00000 0.402492
$$501$$ −24.0000 −1.07224
$$502$$ −4.00000 −0.178529
$$503$$ −14.0000 −0.624229 −0.312115 0.950044i $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 5.00000 0.222497
$$506$$ −12.0000 −0.533465
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ −5.00000 −0.221404
$$511$$ 26.0000 1.15017
$$512$$ −1.00000 −0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ 3.00000 0.132324
$$515$$ −10.0000 −0.440653
$$516$$ −10.0000 −0.440225
$$517$$ 4.00000 0.175920
$$518$$ −22.0000 −0.966625
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ 25.0000 1.09527 0.547635 0.836717i $$-0.315528\pi$$
0.547635 + 0.836717i $$0.315528\pi$$
$$522$$ 9.00000 0.393919
$$523$$ −38.0000 −1.66162 −0.830812 0.556553i $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ −8.00000 −0.349149
$$526$$ −14.0000 −0.610429
$$527$$ −20.0000 −0.871214
$$528$$ −2.00000 −0.0870388
$$529$$ 13.0000 0.565217
$$530$$ −1.00000 −0.0434372
$$531$$ −8.00000 −0.347170
$$532$$ 4.00000 0.173422
$$533$$ 0 0
$$534$$ 2.00000 0.0865485
$$535$$ 18.0000 0.778208
$$536$$ −2.00000 −0.0863868
$$537$$ 6.00000 0.258919
$$538$$ 14.0000 0.603583
$$539$$ −6.00000 −0.258438
$$540$$ 1.00000 0.0430331
$$541$$ −7.00000 −0.300954 −0.150477 0.988614i $$-0.548081\pi$$
−0.150477 + 0.988614i $$0.548081\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ −5.00000 −0.214571
$$544$$ −5.00000 −0.214373
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 17.0000 0.726204
$$549$$ −11.0000 −0.469469
$$550$$ 8.00000 0.341121
$$551$$ 18.0000 0.766826
$$552$$ 6.00000 0.255377
$$553$$ 8.00000 0.340195
$$554$$ 11.0000 0.467345
$$555$$ −11.0000 −0.466924
$$556$$ −12.0000 −0.508913
$$557$$ −9.00000 −0.381342 −0.190671 0.981654i $$-0.561066\pi$$
−0.190671 + 0.981654i $$0.561066\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ −10.0000 −0.422200
$$562$$ −25.0000 −1.05456
$$563$$ 40.0000 1.68580 0.842900 0.538071i $$-0.180847\pi$$
0.842900 + 0.538071i $$0.180847\pi$$
$$564$$ −2.00000 −0.0842152
$$565$$ 3.00000 0.126211
$$566$$ −26.0000 −1.09286
$$567$$ −2.00000 −0.0839921
$$568$$ 14.0000 0.587427
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ −2.00000 −0.0836974 −0.0418487 0.999124i $$-0.513325\pi$$
−0.0418487 + 0.999124i $$0.513325\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ 10.0000 0.417392
$$575$$ −24.0000 −1.00087
$$576$$ 1.00000 0.0416667
$$577$$ 27.0000 1.12402 0.562012 0.827129i $$-0.310027\pi$$
0.562012 + 0.827129i $$0.310027\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 17.0000 0.706496
$$580$$ 9.00000 0.373705
$$581$$ −12.0000 −0.497844
$$582$$ −2.00000 −0.0829027
$$583$$ −2.00000 −0.0828315
$$584$$ 13.0000 0.537944
$$585$$ 0 0
$$586$$ 1.00000 0.0413096
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 8.00000 0.329634
$$590$$ −8.00000 −0.329355
$$591$$ −6.00000 −0.246807
$$592$$ −11.0000 −0.452097
$$593$$ −39.0000 −1.60154 −0.800769 0.598973i $$-0.795576\pi$$
−0.800769 + 0.598973i $$0.795576\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 10.0000 0.409960
$$596$$ 3.00000 0.122885
$$597$$ −10.0000 −0.409273
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ 11.0000 0.448699 0.224350 0.974509i $$-0.427974\pi$$
0.224350 + 0.974509i $$0.427974\pi$$
$$602$$ 20.0000 0.815139
$$603$$ 2.00000 0.0814463
$$604$$ −6.00000 −0.244137
$$605$$ 7.00000 0.284590
$$606$$ −5.00000 −0.203111
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −18.0000 −0.729397
$$610$$ −11.0000 −0.445377
$$611$$ 0 0
$$612$$ 5.00000 0.202113
$$613$$ 13.0000 0.525065 0.262533 0.964923i $$-0.415442\pi$$
0.262533 + 0.964923i $$0.415442\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 5.00000 0.201619
$$616$$ 4.00000 0.161165
$$617$$ −15.0000 −0.603877 −0.301939 0.953327i $$-0.597634\pi$$
−0.301939 + 0.953327i $$0.597634\pi$$
$$618$$ 10.0000 0.402259
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 4.00000 0.160644
$$621$$ −6.00000 −0.240772
$$622$$ −6.00000 −0.240578
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ −6.00000 −0.239808
$$627$$ 4.00000 0.159745
$$628$$ −7.00000 −0.279330
$$629$$ −55.0000 −2.19299
$$630$$ −2.00000 −0.0796819
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −24.0000 −0.953914
$$634$$ 33.0000 1.31060
$$635$$ 12.0000 0.476205
$$636$$ 1.00000 0.0396526
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ −14.0000 −0.553831
$$640$$ 1.00000 0.0395285
$$641$$ 5.00000 0.197488 0.0987441 0.995113i $$-0.468517\pi$$
0.0987441 + 0.995113i $$0.468517\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 10.0000 0.393750
$$646$$ 10.0000 0.393445
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 20.0000 0.783260
$$653$$ −22.0000 −0.860927 −0.430463 0.902608i $$-0.641650\pi$$
−0.430463 + 0.902608i $$0.641650\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 8.00000 0.312586
$$656$$ 5.00000 0.195217
$$657$$ −13.0000 −0.507178
$$658$$ 4.00000 0.155936
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ −4.00000 −0.155113
$$666$$ 11.0000 0.426241
$$667$$ −54.0000 −2.09089
$$668$$ 24.0000 0.928588
$$669$$ 16.0000 0.618596
$$670$$ 2.00000 0.0772667
$$671$$ −22.0000 −0.849301
$$672$$ −2.00000 −0.0771517
$$673$$ 43.0000 1.65753 0.828764 0.559598i $$-0.189045\pi$$
0.828764 + 0.559598i $$0.189045\pi$$
$$674$$ 9.00000 0.346667
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ −3.00000 −0.115214
$$679$$ 4.00000 0.153506
$$680$$ 5.00000 0.191741
$$681$$ −14.0000 −0.536481
$$682$$ 8.00000 0.306336
$$683$$ −40.0000 −1.53056 −0.765279 0.643699i $$-0.777399\pi$$
−0.765279 + 0.643699i $$0.777399\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −17.0000 −0.649537
$$686$$ −20.0000 −0.763604
$$687$$ −10.0000 −0.381524
$$688$$ 10.0000 0.381246
$$689$$ 0 0
$$690$$ −6.00000 −0.228416
$$691$$ −2.00000 −0.0760836 −0.0380418 0.999276i $$-0.512112\pi$$
−0.0380418 + 0.999276i $$0.512112\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ −4.00000 −0.151947
$$694$$ −6.00000 −0.227757
$$695$$ 12.0000 0.455186
$$696$$ −9.00000 −0.341144
$$697$$ 25.0000 0.946943
$$698$$ −6.00000 −0.227103
$$699$$ 6.00000 0.226941
$$700$$ 8.00000 0.302372
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 22.0000 0.829746
$$704$$ 2.00000 0.0753778
$$705$$ 2.00000 0.0753244
$$706$$ −17.0000 −0.639803
$$707$$ 10.0000 0.376089
$$708$$ 8.00000 0.300658
$$709$$ −15.0000 −0.563337 −0.281668 0.959512i $$-0.590888\pi$$
−0.281668 + 0.959512i $$0.590888\pi$$
$$710$$ −14.0000 −0.525411
$$711$$ −4.00000 −0.150012
$$712$$ −2.00000 −0.0749532
$$713$$ −24.0000 −0.898807
$$714$$ −10.0000 −0.374241
$$715$$ 0 0
$$716$$ −6.00000 −0.224231
$$717$$ 6.00000 0.224074
$$718$$ 30.0000 1.11959
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −20.0000 −0.744839
$$722$$ 15.0000 0.558242
$$723$$ −7.00000 −0.260333
$$724$$ 5.00000 0.185824
$$725$$ 36.0000 1.33701
$$726$$ −7.00000 −0.259794
$$727$$ 2.00000 0.0741759 0.0370879 0.999312i $$-0.488192\pi$$
0.0370879 + 0.999312i $$0.488192\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −13.0000 −0.481152
$$731$$ 50.0000 1.84932
$$732$$ 11.0000 0.406572
$$733$$ 13.0000 0.480166 0.240083 0.970752i $$-0.422825\pi$$
0.240083 + 0.970752i $$0.422825\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ −3.00000 −0.110657
$$736$$ −6.00000 −0.221163
$$737$$ 4.00000 0.147342
$$738$$ −5.00000 −0.184053
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 11.0000 0.404368
$$741$$ 0 0
$$742$$ −2.00000 −0.0734223
$$743$$ −12.0000 −0.440237 −0.220119 0.975473i $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ −3.00000 −0.109911
$$746$$ −9.00000 −0.329513
$$747$$ 6.00000 0.219529
$$748$$ 10.0000 0.365636
$$749$$ 36.0000 1.31541
$$750$$ 9.00000 0.328634
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 2.00000 0.0729325
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ 6.00000 0.218362
$$756$$ 2.00000 0.0727393
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ −12.0000 −0.435572
$$760$$ −2.00000 −0.0725476
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 4.00000 0.144810
$$764$$ 4.00000 0.144715
$$765$$ −5.00000 −0.180775
$$766$$ −24.0000 −0.867155
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ −4.00000 −0.144150
$$771$$ 3.00000 0.108042
$$772$$ −17.0000 −0.611843
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ −10.0000 −0.359443
$$775$$ 16.0000 0.574737
$$776$$ 2.00000 0.0717958
$$777$$ −22.0000 −0.789246
$$778$$ −19.0000 −0.681183
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ −28.0000 −1.00192
$$782$$ −30.0000 −1.07280
$$783$$ 9.00000 0.321634
$$784$$ −3.00000 −0.107143
$$785$$ 7.00000 0.249841
$$786$$ −8.00000 −0.285351
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −14.0000 −0.498413
$$790$$ −4.00000 −0.142314
$$791$$ 6.00000 0.213335
$$792$$ −2.00000 −0.0710669
$$793$$ 0 0
$$794$$ 18.0000 0.638796
$$795$$ −1.00000 −0.0354663
$$796$$ 10.0000 0.354441
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 10.0000 0.353775
$$800$$ 4.00000 0.141421
$$801$$ 2.00000 0.0706665
$$802$$ 27.0000 0.953403
$$803$$ −26.0000 −0.917520
$$804$$ −2.00000 −0.0705346
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 14.0000 0.492823
$$808$$ 5.00000 0.175899
$$809$$ 5.00000 0.175791 0.0878953 0.996130i $$-0.471986\pi$$
0.0878953 + 0.996130i $$0.471986\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −36.0000 −1.26413 −0.632065 0.774915i $$-0.717793\pi$$
−0.632065 + 0.774915i $$0.717793\pi$$
$$812$$ 18.0000 0.631676
$$813$$ −8.00000 −0.280572
$$814$$ 22.0000 0.771100
$$815$$ −20.0000 −0.700569
$$816$$ −5.00000 −0.175035
$$817$$ −20.0000 −0.699711
$$818$$ −23.0000 −0.804176
$$819$$ 0 0
$$820$$ −5.00000 −0.174608
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 17.0000 0.592943
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 8.00000 0.278524
$$826$$ −16.0000 −0.556711
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 6.00000 0.208514
$$829$$ −35.0000 −1.21560 −0.607800 0.794090i $$-0.707948\pi$$
−0.607800 + 0.794090i $$0.707948\pi$$
$$830$$ 6.00000 0.208263
$$831$$ 11.0000 0.381586
$$832$$ 0 0
$$833$$ −15.0000 −0.519719
$$834$$ −12.0000 −0.415526
$$835$$ −24.0000 −0.830554
$$836$$ −4.00000 −0.138343
$$837$$ 4.00000 0.138260
$$838$$ 32.0000 1.10542
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 52.0000 1.79310
$$842$$ 23.0000 0.792632
$$843$$ −25.0000 −0.861046
$$844$$ 24.0000 0.826114
$$845$$ 0 0
$$846$$ −2.00000 −0.0687614
$$847$$ 14.0000 0.481046
$$848$$ −1.00000 −0.0343401
$$849$$ −26.0000 −0.892318
$$850$$ 20.0000 0.685994
$$851$$ −66.0000 −2.26245
$$852$$ 14.0000 0.479632
$$853$$ 49.0000 1.67773 0.838864 0.544341i $$-0.183220\pi$$
0.838864 + 0.544341i $$0.183220\pi$$
$$854$$ −22.0000 −0.752825
$$855$$ 2.00000 0.0683986
$$856$$ 18.0000 0.615227
$$857$$ 45.0000 1.53717 0.768585 0.639747i $$-0.220961\pi$$
0.768585 + 0.639747i $$0.220961\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ −10.0000 −0.340997
$$861$$ 10.0000 0.340799
$$862$$ 2.00000 0.0681203
$$863$$ 46.0000 1.56586 0.782929 0.622111i $$-0.213725\pi$$
0.782929 + 0.622111i $$0.213725\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 22.0000 0.748022
$$866$$ 21.0000 0.713609
$$867$$ −8.00000 −0.271694
$$868$$ 8.00000 0.271538
$$869$$ −8.00000 −0.271381
$$870$$ 9.00000 0.305129
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ −2.00000 −0.0676897
$$874$$ 12.0000 0.405906
$$875$$ −18.0000 −0.608511
$$876$$ 13.0000 0.439229
$$877$$ 37.0000 1.24940 0.624701 0.780864i $$-0.285221\pi$$
0.624701 + 0.780864i $$0.285221\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ 1.00000 0.0337292
$$880$$ −2.00000 −0.0674200
$$881$$ 17.0000 0.572745 0.286372 0.958118i $$-0.407551\pi$$
0.286372 + 0.958118i $$0.407551\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ −20.0000 −0.671913
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −11.0000 −0.369136
$$889$$ 24.0000 0.804934
$$890$$ 2.00000 0.0670402
$$891$$ 2.00000 0.0670025
$$892$$ −16.0000 −0.535720
$$893$$ −4.00000 −0.133855
$$894$$ 3.00000 0.100335
$$895$$ 6.00000 0.200558
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 36.0000 1.20067
$$900$$ −4.00000 −0.133333
$$901$$ −5.00000 −0.166574
$$902$$ −10.0000 −0.332964
$$903$$ 20.0000 0.665558
$$904$$ 3.00000 0.0997785
$$905$$ −5.00000 −0.166206
$$906$$ −6.00000 −0.199337
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ 14.0000 0.464606
$$909$$ −5.00000 −0.165840
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 12.0000 0.397142
$$914$$ −3.00000 −0.0992312
$$915$$ −11.0000 −0.363649
$$916$$ 10.0000 0.330409
$$917$$ 16.0000 0.528367
$$918$$ 5.00000 0.165025
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 14.0000 0.461316
$$922$$ −3.00000 −0.0987997
$$923$$ 0 0
$$924$$ 4.00000 0.131590
$$925$$ 44.0000 1.44671
$$926$$ 14.0000 0.460069
$$927$$ 10.0000 0.328443
$$928$$ 9.00000 0.295439
$$929$$ −23.0000 −0.754606 −0.377303 0.926090i $$-0.623148\pi$$
−0.377303 + 0.926090i $$0.623148\pi$$
$$930$$ 4.00000 0.131165
$$931$$ 6.00000 0.196642
$$932$$ −6.00000 −0.196537
$$933$$ −6.00000 −0.196431
$$934$$ 22.0000 0.719862
$$935$$ −10.0000 −0.327035
$$936$$ 0 0
$$937$$ −1.00000 −0.0326686 −0.0163343 0.999867i $$-0.505200\pi$$
−0.0163343 + 0.999867i $$0.505200\pi$$
$$938$$ 4.00000 0.130605
$$939$$ −6.00000 −0.195803
$$940$$ −2.00000 −0.0652328
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ −7.00000 −0.228072
$$943$$ 30.0000 0.976934
$$944$$ −8.00000 −0.260378
$$945$$ −2.00000 −0.0650600
$$946$$ −20.0000 −0.650256
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 4.00000 0.129914
$$949$$ 0 0
$$950$$ −8.00000 −0.259554
$$951$$ 33.0000 1.07010
$$952$$ 10.0000 0.324102
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 1.00000 0.0323762
$$955$$ −4.00000 −0.129437
$$956$$ −6.00000 −0.194054
$$957$$ 18.0000 0.581857
$$958$$ −32.0000 −1.03387
$$959$$ −34.0000 −1.09792
$$960$$ 1.00000 0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 7.00000 0.225455
$$965$$ 17.0000 0.547249
$$966$$ −12.0000 −0.386094
$$967$$ −50.0000 −1.60789 −0.803946 0.594703i $$-0.797270\pi$$
−0.803946 + 0.594703i $$0.797270\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 10.0000 0.321246
$$970$$ −2.00000 −0.0642161
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 24.0000 0.769405
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −11.0000 −0.352101
$$977$$ 21.0000 0.671850 0.335925 0.941889i $$-0.390951\pi$$
0.335925 + 0.941889i $$0.390951\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 4.00000 0.127841
$$980$$ 3.00000 0.0958315
$$981$$ −2.00000 −0.0638551
$$982$$ 30.0000 0.957338
$$983$$ 60.0000 1.91370 0.956851 0.290578i $$-0.0938475\pi$$
0.956851 + 0.290578i $$0.0938475\pi$$
$$984$$ 5.00000 0.159394
$$985$$ −6.00000 −0.191176
$$986$$ 45.0000 1.43309
$$987$$ 4.00000 0.127321
$$988$$ 0 0
$$989$$ 60.0000 1.90789
$$990$$ 2.00000 0.0635642
$$991$$ 18.0000 0.571789 0.285894 0.958261i $$-0.407709\pi$$
0.285894 + 0.958261i $$0.407709\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 28.0000 0.888553
$$994$$ −28.0000 −0.888106
$$995$$ −10.0000 −0.317021
$$996$$ −6.00000 −0.190117
$$997$$ −23.0000 −0.728417 −0.364209 0.931317i $$-0.618661\pi$$
−0.364209 + 0.931317i $$0.618661\pi$$
$$998$$ 0 0
$$999$$ 11.0000 0.348025
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 1014.2.a.a.1.1 1
3.2 odd 2 3042.2.a.m.1.1 1
4.3 odd 2 8112.2.a.x.1.1 1
13.2 odd 12 1014.2.i.e.823.1 4
13.3 even 3 78.2.e.b.61.1 yes 2
13.4 even 6 1014.2.e.d.991.1 2
13.5 odd 4 1014.2.b.a.337.2 2
13.6 odd 12 1014.2.i.e.361.2 4
13.7 odd 12 1014.2.i.e.361.1 4
13.8 odd 4 1014.2.b.a.337.1 2
13.9 even 3 78.2.e.b.55.1 2
13.10 even 6 1014.2.e.d.529.1 2
13.11 odd 12 1014.2.i.e.823.2 4
13.12 even 2 1014.2.a.e.1.1 1
39.5 even 4 3042.2.b.d.1351.1 2
39.8 even 4 3042.2.b.d.1351.2 2
39.29 odd 6 234.2.h.b.217.1 2
39.35 odd 6 234.2.h.b.55.1 2
39.38 odd 2 3042.2.a.d.1.1 1
52.3 odd 6 624.2.q.b.529.1 2
52.35 odd 6 624.2.q.b.289.1 2
52.51 odd 2 8112.2.a.bb.1.1 1
65.3 odd 12 1950.2.z.b.1699.1 4
65.9 even 6 1950.2.i.b.601.1 2
65.22 odd 12 1950.2.z.b.1849.1 4
65.29 even 6 1950.2.i.b.451.1 2
65.42 odd 12 1950.2.z.b.1699.2 4
65.48 odd 12 1950.2.z.b.1849.2 4
156.35 even 6 1872.2.t.i.289.1 2
156.107 even 6 1872.2.t.i.1153.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.b.55.1 2 13.9 even 3
78.2.e.b.61.1 yes 2 13.3 even 3
234.2.h.b.55.1 2 39.35 odd 6
234.2.h.b.217.1 2 39.29 odd 6
624.2.q.b.289.1 2 52.35 odd 6
624.2.q.b.529.1 2 52.3 odd 6
1014.2.a.a.1.1 1 1.1 even 1 trivial
1014.2.a.e.1.1 1 13.12 even 2
1014.2.b.a.337.1 2 13.8 odd 4
1014.2.b.a.337.2 2 13.5 odd 4
1014.2.e.d.529.1 2 13.10 even 6
1014.2.e.d.991.1 2 13.4 even 6
1014.2.i.e.361.1 4 13.7 odd 12
1014.2.i.e.361.2 4 13.6 odd 12
1014.2.i.e.823.1 4 13.2 odd 12
1014.2.i.e.823.2 4 13.11 odd 12
1872.2.t.i.289.1 2 156.35 even 6
1872.2.t.i.1153.1 2 156.107 even 6
1950.2.i.b.451.1 2 65.29 even 6
1950.2.i.b.601.1 2 65.9 even 6
1950.2.z.b.1699.1 4 65.3 odd 12
1950.2.z.b.1699.2 4 65.42 odd 12
1950.2.z.b.1849.1 4 65.22 odd 12
1950.2.z.b.1849.2 4 65.48 odd 12
3042.2.a.d.1.1 1 39.38 odd 2
3042.2.a.m.1.1 1 3.2 odd 2
3042.2.b.d.1351.1 2 39.5 even 4
3042.2.b.d.1351.2 2 39.8 even 4
8112.2.a.x.1.1 1 4.3 odd 2
8112.2.a.bb.1.1 1 52.51 odd 2