# Properties

 Label 114.2.a.b.1.1 Level $114$ Weight $2$ Character 114.1 Self dual yes Analytic conductor $0.910$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,2,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.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: $$0.910294583043$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 114.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +1.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +10.0000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -12.0000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} -4.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −2.00000 −0.320256
$$40$$ 2.00000 0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 2.00000 0.298142
$$46$$ −4.00000 −0.589768
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ 2.00000 0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ −2.00000 −0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 4.00000 0.492366
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 1.00000 0.115470
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ 4.00000 0.431331
$$87$$ 2.00000 0.214423
$$88$$ −4.00000 −0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ −4.00000 −0.414781
$$94$$ −4.00000 −0.412568
$$95$$ −2.00000 −0.205196
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ −8.00000 −0.746004
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ 5.00000 0.454545
$$122$$ 14.0000 1.26750
$$123$$ −10.0000 −0.901670
$$124$$ 4.00000 0.359211
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 4.00000 0.350823
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ −2.00000 −0.172133
$$136$$ −6.00000 −0.514496
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −6.00000 −0.496564
$$147$$ 7.00000 0.577350
$$148$$ 10.0000 0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ −2.00000 −0.160128
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 10.0000 0.793052
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 8.00000 0.622799
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −12.0000 −0.920358
$$171$$ −1.00000 −0.0764719
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −12.0000 −0.901975
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −14.0000 −1.03491
$$184$$ −4.00000 −0.294884
$$185$$ 20.0000 1.47043
$$186$$ −4.00000 −0.293294
$$187$$ 24.0000 1.75505
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ −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
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 10.0000 0.717958
$$195$$ −4.00000 −0.286446
$$196$$ −7.00000 −0.500000
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 12.0000 0.846415
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 20.0000 1.39686
$$206$$ −12.0000 −0.836080
$$207$$ −4.00000 −0.278019
$$208$$ 2.00000 0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ −8.00000 −0.548151
$$214$$ −4.00000 −0.273434
$$215$$ 8.00000 0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 6.00000 0.405442
$$220$$ −8.00000 −0.539360
$$221$$ −12.0000 −0.807207
$$222$$ −10.0000 −0.671156
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 2.00000 0.133038
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −8.00000 −0.521862
$$236$$ 12.0000 0.781133
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 14.0000 0.896258
$$245$$ −14.0000 −0.894427
$$246$$ −10.0000 −0.637577
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ −12.0000 −0.760469
$$250$$ −12.0000 −0.758947
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −12.0000 −0.752947
$$255$$ 12.0000 0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ −2.00000 −0.123797
$$262$$ 12.0000 0.741362
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 4.00000 0.246183
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −14.0000 −0.845771
$$275$$ 4.00000 0.241209
$$276$$ 4.00000 0.240772
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 2.00000 0.118470
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ −10.0000 −0.586210
$$292$$ −6.00000 −0.351123
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 24.0000 1.39733
$$296$$ 10.0000 0.581238
$$297$$ 4.00000 0.232104
$$298$$ −6.00000 −0.347571
$$299$$ −8.00000 −0.462652
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 20.0000 1.15087
$$303$$ −2.00000 −0.114897
$$304$$ −1.00000 −0.0573539
$$305$$ 28.0000 1.60328
$$306$$ −6.00000 −0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 8.00000 0.454369
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 8.00000 0.447914
$$320$$ 2.00000 0.111803
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 20.0000 1.10770
$$327$$ 6.00000 0.331801
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 8.00000 0.440386
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ −24.0000 −1.31126
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −2.00000 −0.108625
$$340$$ −12.0000 −0.650791
$$341$$ −16.0000 −0.866449
$$342$$ −1.00000 −0.0540738
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 8.00000 0.430706
$$346$$ 6.00000 0.322562
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −4.00000 −0.213201
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 16.0000 0.849192
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 2.00000 0.105409
$$361$$ 1.00000 0.0526316
$$362$$ −14.0000 −0.735824
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ −14.0000 −0.731792
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 10.0000 0.520579
$$370$$ 20.0000 1.03975
$$371$$ 0 0
$$372$$ −4.00000 −0.207390
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 12.0000 0.619677
$$376$$ −4.00000 −0.206284
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −36.0000 −1.84920 −0.924598 0.380945i $$-0.875599\pi$$
−0.924598 + 0.380945i $$0.875599\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 12.0000 0.614779
$$382$$ 4.00000 0.204658
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 4.00000 0.203331
$$388$$ 10.0000 0.507673
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ 24.0000 1.21373
$$392$$ −7.00000 −0.353553
$$393$$ −12.0000 −0.605320
$$394$$ −22.0000 −1.10834
$$395$$ −8.00000 −0.402524
$$396$$ −4.00000 −0.201008
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 8.00000 0.398508
$$404$$ 2.00000 0.0995037
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 6.00000 0.297044
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 20.0000 0.987730
$$411$$ 14.0000 0.690569
$$412$$ −12.0000 −0.591198
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 24.0000 1.17811
$$416$$ 2.00000 0.0980581
$$417$$ −12.0000 −0.587643
$$418$$ 4.00000 0.195646
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 12.0000 0.584151
$$423$$ −4.00000 −0.194487
$$424$$ −10.0000 −0.485643
$$425$$ 6.00000 0.291043
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 8.00000 0.386244
$$430$$ 8.00000 0.385794
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ −6.00000 −0.287348
$$437$$ 4.00000 0.191346
$$438$$ 6.00000 0.286691
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ −7.00000 −0.333333
$$442$$ −12.0000 −0.570782
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ −12.0000 −0.568855
$$446$$ −28.0000 −1.32584
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −40.0000 −1.88353
$$452$$ 2.00000 0.0940721
$$453$$ −20.0000 −0.939682
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 6.00000 0.280056
$$460$$ −8.00000 −0.373002
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ −8.00000 −0.370991
$$466$$ −6.00000 −0.277945
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ −22.0000 −1.01371
$$472$$ 12.0000 0.552345
$$473$$ −16.0000 −0.735681
$$474$$ 4.00000 0.183726
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 12.0000 0.548867
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 20.0000 0.911922
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 20.0000 0.908153
$$486$$ −1.00000 −0.0453609
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ 14.0000 0.633750
$$489$$ −20.0000 −0.904431
$$490$$ −14.0000 −0.632456
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ 12.0000 0.540453
$$494$$ −2.00000 −0.0899843
$$495$$ −8.00000 −0.359573
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ −28.0000 −1.24970
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 16.0000 0.711287
$$507$$ 9.00000 0.399704
$$508$$ −12.0000 −0.532414
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 12.0000 0.531369
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 2.00000 0.0882162
$$515$$ −24.0000 −1.05757
$$516$$ −4.00000 −0.176090
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 4.00000 0.175412
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ −24.0000 −1.04546
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ −20.0000 −0.868744
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 6.00000 0.259645
$$535$$ −8.00000 −0.345870
$$536$$ −12.0000 −0.518321
$$537$$ −12.0000 −0.517838
$$538$$ 6.00000 0.258678
$$539$$ 28.0000 1.20605
$$540$$ −2.00000 −0.0860663
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 14.0000 0.600798
$$544$$ −6.00000 −0.257248
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ 14.0000 0.597505
$$550$$ 4.00000 0.170561
$$551$$ 2.00000 0.0852029
$$552$$ 4.00000 0.170251
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ −20.0000 −0.848953
$$556$$ 12.0000 0.508913
$$557$$ 34.0000 1.44063 0.720313 0.693649i $$-0.243998\pi$$
0.720313 + 0.693649i $$0.243998\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 10.0000 0.421825
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 4.00000 0.168281
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ −4.00000 −0.167102
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 6.00000 0.249351
$$580$$ −4.00000 −0.166091
$$581$$ 0 0
$$582$$ −10.0000 −0.414513
$$583$$ 40.0000 1.65663
$$584$$ −6.00000 −0.248282
$$585$$ 4.00000 0.165380
$$586$$ −18.0000 −0.743573
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 7.00000 0.288675
$$589$$ −4.00000 −0.164817
$$590$$ 24.0000 0.988064
$$591$$ 22.0000 0.904959
$$592$$ 10.0000 0.410997
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 16.0000 0.654836
$$598$$ −8.00000 −0.327144
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ 20.0000 0.813788
$$605$$ 10.0000 0.406558
$$606$$ −2.00000 −0.0812444
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 28.0000 1.13369
$$611$$ −8.00000 −0.323645
$$612$$ −6.00000 −0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ −20.0000 −0.806478
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ 12.0000 0.482711
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 4.00000 0.160514
$$622$$ 4.00000 0.160385
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ −22.0000 −0.879297
$$627$$ −4.00000 −0.159745
$$628$$ 22.0000 0.877896
$$629$$ −60.0000 −2.39236
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ −12.0000 −0.476957
$$634$$ 6.00000 0.238290
$$635$$ −24.0000 −0.952411
$$636$$ 10.0000 0.396526
$$637$$ −14.0000 −0.554700
$$638$$ 8.00000 0.316723
$$639$$ 8.00000 0.316475
$$640$$ 2.00000 0.0790569
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 6.00000 0.236067
$$647$$ −4.00000 −0.157256 −0.0786281 0.996904i $$-0.525054\pi$$
−0.0786281 + 0.996904i $$0.525054\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 24.0000 0.937758
$$656$$ 10.0000 0.390434
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 8.00000 0.311400
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 12.0000 0.466041
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 8.00000 0.309761
$$668$$ 0 0
$$669$$ 28.0000 1.08254
$$670$$ −24.0000 −0.927201
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 1.00000 0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 0 0
$$680$$ −12.0000 −0.460179
$$681$$ −28.0000 −1.07296
$$682$$ −16.0000 −0.612672
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ −28.0000 −1.06983
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 4.00000 0.152499
$$689$$ −20.0000 −0.761939
$$690$$ 8.00000 0.304555
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ 24.0000 0.910372
$$696$$ 2.00000 0.0758098
$$697$$ −60.0000 −2.27266
$$698$$ −26.0000 −0.984115
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −10.0000 −0.377157
$$704$$ −4.00000 −0.150756
$$705$$ 8.00000 0.301297
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ −12.0000 −0.450988
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 16.0000 0.600469
$$711$$ −4.00000 −0.150012
$$712$$ −6.00000 −0.224860
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ 12.0000 0.448461
$$717$$ −12.0000 −0.448148
$$718$$ 12.0000 0.447836
$$719$$ 28.0000 1.04422 0.522112 0.852877i $$-0.325144\pi$$
0.522112 + 0.852877i $$0.325144\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 0 0
$$722$$ 1.00000 0.0372161
$$723$$ −10.0000 −0.371904
$$724$$ −14.0000 −0.520306
$$725$$ 2.00000 0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −12.0000 −0.444140
$$731$$ −24.0000 −0.887672
$$732$$ −14.0000 −0.517455
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 14.0000 0.516398
$$736$$ −4.00000 −0.147442
$$737$$ 48.0000 1.76810
$$738$$ 10.0000 0.368105
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 20.0000 0.735215
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ −12.0000 −0.439646
$$746$$ 26.0000 0.951928
$$747$$ 12.0000 0.439057
$$748$$ 24.0000 0.877527
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 28.0000 1.02038
$$754$$ −4.00000 −0.145671
$$755$$ 40.0000 1.45575
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −36.0000 −1.30758
$$759$$ −16.0000 −0.580763
$$760$$ −2.00000 −0.0725476
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 12.0000 0.434714
$$763$$ 0 0
$$764$$ 4.00000 0.144715
$$765$$ −12.0000 −0.433861
$$766$$ 16.0000 0.578103
$$767$$ 24.0000 0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ −6.00000 −0.215945
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ 4.00000 0.143777
$$775$$ −4.00000 −0.143684
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ −10.0000 −0.358287
$$780$$ −4.00000 −0.143223
$$781$$ −32.0000 −1.14505
$$782$$ 24.0000 0.858238
$$783$$ 2.00000 0.0714742
$$784$$ −7.00000 −0.250000
$$785$$ 44.0000 1.57043
$$786$$ −12.0000 −0.428026
$$787$$ −20.0000 −0.712923 −0.356462 0.934310i $$-0.616017\pi$$
−0.356462 + 0.934310i $$0.616017\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 12.0000 0.427211
$$790$$ −8.00000 −0.284627
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ 28.0000 0.994309
$$794$$ −10.0000 −0.354887
$$795$$ 20.0000 0.709327
$$796$$ −16.0000 −0.567105
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ −14.0000 −0.494357
$$803$$ 24.0000 0.846942
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ −6.00000 −0.211210
$$808$$ 2.00000 0.0703598
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ −40.0000 −1.40200
$$815$$ 40.0000 1.40114
$$816$$ 6.00000 0.210042
$$817$$ −4.00000 −0.139942
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 20.0000 0.698430
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 14.0000 0.488306
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 42.0000 1.45872 0.729360 0.684130i $$-0.239818\pi$$
0.729360 + 0.684130i $$0.239818\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 26.0000 0.901930
$$832$$ 2.00000 0.0693375
$$833$$ 42.0000 1.45521
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ −4.00000 −0.138260
$$838$$ −12.0000 −0.414533
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000 0.896019
$$843$$ −10.0000 −0.344418
$$844$$ 12.0000 0.413057
$$845$$ −18.0000 −0.619219
$$846$$ −4.00000 −0.137523
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ −12.0000 −0.411839
$$850$$ 6.00000 0.205798
$$851$$ −40.0000 −1.37118
$$852$$ −8.00000 −0.274075
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ −2.00000 −0.0683986
$$856$$ −4.00000 −0.136717
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 8.00000 0.273115
$$859$$ 52.0000 1.77422 0.887109 0.461561i $$-0.152710\pi$$
0.887109 + 0.461561i $$0.152710\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 12.0000 0.408012
$$866$$ 26.0000 0.883516
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 4.00000 0.135613
$$871$$ −24.0000 −0.813209
$$872$$ −6.00000 −0.203186
$$873$$ 10.0000 0.338449
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ −4.00000 −0.134993
$$879$$ 18.0000 0.607125
$$880$$ −8.00000 −0.269680
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ −24.0000 −0.806751
$$886$$ −20.0000 −0.671913
$$887$$ −32.0000 −1.07445 −0.537227 0.843437i $$-0.680528\pi$$
−0.537227 + 0.843437i $$0.680528\pi$$
$$888$$ −10.0000 −0.335578
$$889$$ 0 0
$$890$$ −12.0000 −0.402241
$$891$$ −4.00000 −0.134005
$$892$$ −28.0000 −0.937509
$$893$$ 4.00000 0.133855
$$894$$ 6.00000 0.200670
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 34.0000 1.13459
$$899$$ −8.00000 −0.266815
$$900$$ −1.00000 −0.0333333
$$901$$ 60.0000 1.99889
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ −28.0000 −0.930751
$$906$$ −20.0000 −0.664455
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 28.0000 0.929213
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −48.0000 −1.58857
$$914$$ 26.0000 0.860004
$$915$$ −28.0000 −0.925651
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 6.00000 0.198030
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 12.0000 0.395413
$$922$$ 2.00000 0.0658665
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −8.00000 −0.262896
$$927$$ −12.0000 −0.394132
$$928$$ −2.00000 −0.0656532
$$929$$ −22.0000 −0.721797 −0.360898 0.932605i $$-0.617530\pi$$
−0.360898 + 0.932605i $$0.617530\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ 7.00000 0.229416
$$932$$ −6.00000 −0.196537
$$933$$ −4.00000 −0.130954
$$934$$ −20.0000 −0.654420
$$935$$ 48.0000 1.56977
$$936$$ 2.00000 0.0653720
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 22.0000 0.717943
$$940$$ −8.00000 −0.260931
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ −22.0000 −0.716799
$$943$$ −40.0000 −1.30258
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 4.00000 0.129914
$$949$$ −12.0000 −0.389536
$$950$$ 1.00000 0.0324443
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 8.00000 0.258874
$$956$$ 12.0000 0.388108
$$957$$ −8.00000 −0.258603
$$958$$ 4.00000 0.129234
$$959$$ 0 0
$$960$$ −2.00000 −0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ 20.0000 0.644826
$$963$$ −4.00000 −0.128898
$$964$$ 10.0000 0.322078
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ 5.00000 0.160706
$$969$$ −6.00000 −0.192748
$$970$$ 20.0000 0.642161
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 4.00000 0.128168
$$975$$ 2.00000 0.0640513
$$976$$ 14.0000 0.448129
$$977$$ −46.0000 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 24.0000 0.767043
$$980$$ −14.0000 −0.447214
$$981$$ −6.00000 −0.191565
$$982$$ 20.0000 0.638226
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ −44.0000 −1.40196
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ −16.0000 −0.508770
$$990$$ −8.00000 −0.254257
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ −32.0000 −1.01447
$$996$$ −12.0000 −0.380235
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ −10.0000 −0.316386
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 114.2.a.b.1.1 1
3.2 odd 2 342.2.a.b.1.1 1
4.3 odd 2 912.2.a.k.1.1 1
5.2 odd 4 2850.2.d.b.799.2 2
5.3 odd 4 2850.2.d.b.799.1 2
5.4 even 2 2850.2.a.j.1.1 1
7.6 odd 2 5586.2.a.y.1.1 1
8.3 odd 2 3648.2.a.c.1.1 1
8.5 even 2 3648.2.a.x.1.1 1
12.11 even 2 2736.2.a.d.1.1 1
15.14 odd 2 8550.2.a.ba.1.1 1
19.18 odd 2 2166.2.a.d.1.1 1
57.56 even 2 6498.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.b.1.1 1 1.1 even 1 trivial
342.2.a.b.1.1 1 3.2 odd 2
912.2.a.k.1.1 1 4.3 odd 2
2166.2.a.d.1.1 1 19.18 odd 2
2736.2.a.d.1.1 1 12.11 even 2
2850.2.a.j.1.1 1 5.4 even 2
2850.2.d.b.799.1 2 5.3 odd 4
2850.2.d.b.799.2 2 5.2 odd 4
3648.2.a.c.1.1 1 8.3 odd 2
3648.2.a.x.1.1 1 8.5 even 2
5586.2.a.y.1.1 1 7.6 odd 2
6498.2.a.p.1.1 1 57.56 even 2
8550.2.a.ba.1.1 1 15.14 odd 2