# Properties

 Label 1862.2.a.f.1.1 Level $1862$ Weight $2$ Character 1862.1 Self dual yes Analytic conductor $14.868$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1862,2,Mod(1,1862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1862, 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("1862.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1862 = 2 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1862.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: $$14.8681448564$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1862.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} +4.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -5.00000 q^{29} +4.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +4.00000 q^{40} +8.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +11.0000 q^{50} -3.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -5.00000 q^{54} +8.00000 q^{55} +1.00000 q^{57} -5.00000 q^{58} -15.0000 q^{59} +4.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} +3.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +2.00000 q^{71} -2.00000 q^{72} -9.00000 q^{73} -2.00000 q^{74} +11.0000 q^{75} +1.00000 q^{76} +1.00000 q^{78} -10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} +6.00000 q^{83} -12.0000 q^{85} +4.00000 q^{86} -5.00000 q^{87} +2.00000 q^{88} -8.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## 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 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 4.00000 1.26491
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ 1.00000 0.229416
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 11.0000 2.20000
$$26$$ 1.00000 0.196116
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 4.00000 0.730297
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 1.00000 0.160128
$$40$$ 4.00000 0.632456
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −8.00000 −1.19257
$$46$$ −1.00000 −0.147442
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 11.0000 1.55563
$$51$$ −3.00000 −0.420084
$$52$$ 1.00000 0.138675
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ −5.00000 −0.656532
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 4.00000 0.516398
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 2.00000 0.246183
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 11.0000 1.27017
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 1.00000 0.111111
$$82$$ 8.00000 0.883452
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ 4.00000 0.431331
$$87$$ −5.00000 −0.536056
$$88$$ 2.00000 0.213201
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −8.00000 −0.843274
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 8.00000 0.829561
$$94$$ −8.00000 −0.825137
$$95$$ 4.00000 0.410391
$$96$$ 1.00000 0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 11.0000 1.10000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ −7.00000 −0.676716 −0.338358 0.941018i $$-0.609871\pi$$
−0.338358 + 0.941018i $$0.609871\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 8.00000 0.762770
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ −4.00000 −0.373002
$$116$$ −5.00000 −0.464238
$$117$$ −2.00000 −0.184900
$$118$$ −15.0000 −1.38086
$$119$$ 0 0
$$120$$ 4.00000 0.365148
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 8.00000 0.721336
$$124$$ 8.00000 0.718421
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ 18.0000 1.59724 0.798621 0.601834i $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 0 0
$$134$$ 3.00000 0.259161
$$135$$ −20.0000 −1.72133
$$136$$ −3.00000 −0.257248
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 2.00000 0.167836
$$143$$ 2.00000 0.167248
$$144$$ −2.00000 −0.166667
$$145$$ −20.0000 −1.66091
$$146$$ −9.00000 −0.744845
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 11.0000 0.898146
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 32.0000 2.57030
$$156$$ 1.00000 0.0800641
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ −1.00000 −0.0793052
$$160$$ 4.00000 0.316228
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 8.00000 0.622799
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ −12.0000 −0.920358
$$171$$ −2.00000 −0.152944
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ −15.0000 −1.12747
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ −8.00000 −0.596285
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ −1.00000 −0.0737210
$$185$$ −8.00000 −0.588172
$$186$$ 8.00000 0.586588
$$187$$ −6.00000 −0.438763
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 7.00000 0.506502 0.253251 0.967401i $$-0.418500\pi$$
0.253251 + 0.967401i $$0.418500\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 4.00000 0.286446
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 25.0000 1.77220 0.886102 0.463491i $$-0.153403\pi$$
0.886102 + 0.463491i $$0.153403\pi$$
$$200$$ 11.0000 0.777817
$$201$$ 3.00000 0.211604
$$202$$ −2.00000 −0.140720
$$203$$ 0 0
$$204$$ −3.00000 −0.210042
$$205$$ 32.0000 2.23498
$$206$$ 6.00000 0.418040
$$207$$ 2.00000 0.139010
$$208$$ 1.00000 0.0693375
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 27.0000 1.85876 0.929378 0.369129i $$-0.120344\pi$$
0.929378 + 0.369129i $$0.120344\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ 2.00000 0.137038
$$214$$ −7.00000 −0.478510
$$215$$ 16.0000 1.09119
$$216$$ −5.00000 −0.340207
$$217$$ 0 0
$$218$$ −15.0000 −1.01593
$$219$$ −9.00000 −0.608164
$$220$$ 8.00000 0.539360
$$221$$ −3.00000 −0.201802
$$222$$ −2.00000 −0.134231
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 0 0
$$225$$ −22.0000 −1.46667
$$226$$ 14.0000 0.931266
$$227$$ 17.0000 1.12833 0.564165 0.825662i $$-0.309198\pi$$
0.564165 + 0.825662i $$0.309198\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −5.00000 −0.328266
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −32.0000 −2.08745
$$236$$ −15.0000 −0.976417
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ 4.00000 0.258199
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 16.0000 1.02640
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ 1.00000 0.0636285
$$248$$ 8.00000 0.508001
$$249$$ 6.00000 0.380235
$$250$$ 24.0000 1.51789
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 18.0000 1.12942
$$255$$ −12.0000 −0.751469
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ 10.0000 0.618984
$$262$$ −12.0000 −0.741362
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 2.00000 0.123091
$$265$$ −4.00000 −0.245718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ −20.0000 −1.21716
$$271$$ −7.00000 −0.425220 −0.212610 0.977137i $$-0.568196\pi$$
−0.212610 + 0.977137i $$0.568196\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −17.0000 −1.02701
$$275$$ 22.0000 1.32665
$$276$$ −1.00000 −0.0601929
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ −16.0000 −0.957895
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 4.00000 0.236940
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ −2.00000 −0.117851
$$289$$ −8.00000 −0.470588
$$290$$ −20.0000 −1.17444
$$291$$ 2.00000 0.117242
$$292$$ −9.00000 −0.526685
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 0 0
$$295$$ −60.0000 −3.49334
$$296$$ −2.00000 −0.116248
$$297$$ −10.0000 −0.580259
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 11.0000 0.635085
$$301$$ 0 0
$$302$$ 2.00000 0.115087
$$303$$ −2.00000 −0.114897
$$304$$ 1.00000 0.0573539
$$305$$ −8.00000 −0.458079
$$306$$ 6.00000 0.342997
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 6.00000 0.341328
$$310$$ 32.0000 1.81748
$$311$$ −7.00000 −0.396934 −0.198467 0.980108i $$-0.563596\pi$$
−0.198467 + 0.980108i $$0.563596\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −29.0000 −1.63918 −0.819588 0.572953i $$-0.805798\pi$$
−0.819588 + 0.572953i $$0.805798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −27.0000 −1.51647 −0.758236 0.651981i $$-0.773938\pi$$
−0.758236 + 0.651981i $$0.773938\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ −10.0000 −0.559893
$$320$$ 4.00000 0.223607
$$321$$ −7.00000 −0.390702
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 1.00000 0.0555556
$$325$$ 11.0000 0.610170
$$326$$ −16.0000 −0.886158
$$327$$ −15.0000 −0.829502
$$328$$ 8.00000 0.441726
$$329$$ 0 0
$$330$$ 8.00000 0.440386
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 4.00000 0.219199
$$334$$ 12.0000 0.656611
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −32.0000 −1.74315 −0.871576 0.490261i $$-0.836901\pi$$
−0.871576 + 0.490261i $$0.836901\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 14.0000 0.760376
$$340$$ −12.0000 −0.650791
$$341$$ 16.0000 0.866449
$$342$$ −2.00000 −0.108148
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ −4.00000 −0.215353
$$346$$ 6.00000 0.322562
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ −5.00000 −0.268028
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 2.00000 0.106600
$$353$$ −9.00000 −0.479022 −0.239511 0.970894i $$-0.576987\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$354$$ −15.0000 −0.797241
$$355$$ 8.00000 0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ −8.00000 −0.421637
$$361$$ 1.00000 0.0526316
$$362$$ −22.0000 −1.15629
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −36.0000 −1.88433
$$366$$ −2.00000 −0.104542
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −16.0000 −0.832927
$$370$$ −8.00000 −0.415900
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 24.0000 1.23935
$$376$$ −8.00000 −0.412568
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ 15.0000 0.770498 0.385249 0.922813i $$-0.374116\pi$$
0.385249 + 0.922813i $$0.374116\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 18.0000 0.922168
$$382$$ 7.00000 0.358151
$$383$$ 26.0000 1.32854 0.664269 0.747494i $$-0.268743\pi$$
0.664269 + 0.747494i $$0.268743\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ −8.00000 −0.406663
$$388$$ 2.00000 0.101535
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 8.00000 0.403034
$$395$$ −40.0000 −2.01262
$$396$$ −4.00000 −0.201008
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 25.0000 1.25314
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 3.00000 0.149626
$$403$$ 8.00000 0.398508
$$404$$ −2.00000 −0.0995037
$$405$$ 4.00000 0.198762
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ −3.00000 −0.148522
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 32.0000 1.58037
$$411$$ −17.0000 −0.838548
$$412$$ 6.00000 0.295599
$$413$$ 0 0
$$414$$ 2.00000 0.0982946
$$415$$ 24.0000 1.17811
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 2.00000 0.0978232
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ 27.0000 1.31434
$$423$$ 16.0000 0.777947
$$424$$ −1.00000 −0.0485643
$$425$$ −33.0000 −1.60074
$$426$$ 2.00000 0.0969003
$$427$$ 0 0
$$428$$ −7.00000 −0.338358
$$429$$ 2.00000 0.0965609
$$430$$ 16.0000 0.771589
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −20.0000 −0.958927
$$436$$ −15.0000 −0.718370
$$437$$ −1.00000 −0.0478365
$$438$$ −9.00000 −0.430037
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 0 0
$$442$$ −3.00000 −0.142695
$$443$$ −26.0000 −1.23530 −0.617649 0.786454i $$-0.711915\pi$$
−0.617649 + 0.786454i $$0.711915\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ −22.0000 −1.03709
$$451$$ 16.0000 0.753411
$$452$$ 14.0000 0.658505
$$453$$ 2.00000 0.0939682
$$454$$ 17.0000 0.797850
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ −7.00000 −0.327446 −0.163723 0.986506i $$-0.552350\pi$$
−0.163723 + 0.986506i $$0.552350\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 15.0000 0.700140
$$460$$ −4.00000 −0.186501
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ −5.00000 −0.232119
$$465$$ 32.0000 1.48396
$$466$$ −6.00000 −0.277945
$$467$$ 2.00000 0.0925490 0.0462745 0.998929i $$-0.485265\pi$$
0.0462745 + 0.998929i $$0.485265\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ −32.0000 −1.47605
$$471$$ 2.00000 0.0921551
$$472$$ −15.0000 −0.690431
$$473$$ 8.00000 0.367840
$$474$$ −10.0000 −0.459315
$$475$$ 11.0000 0.504715
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 15.0000 0.686084
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 4.00000 0.182574
$$481$$ −2.00000 −0.0911922
$$482$$ 8.00000 0.364390
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 8.00000 0.363261
$$486$$ 16.0000 0.725775
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 8.00000 0.360668
$$493$$ 15.0000 0.675566
$$494$$ 1.00000 0.0449921
$$495$$ −16.0000 −0.719147
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 6.00000 0.268866
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 12.0000 0.536120
$$502$$ −2.00000 −0.0892644
$$503$$ −39.0000 −1.73892 −0.869462 0.494000i $$-0.835534\pi$$
−0.869462 + 0.494000i $$0.835534\pi$$
$$504$$ 0 0
$$505$$ −8.00000 −0.355995
$$506$$ −2.00000 −0.0889108
$$507$$ −12.0000 −0.532939
$$508$$ 18.0000 0.798621
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ −12.0000 −0.531369
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −5.00000 −0.220755
$$514$$ −8.00000 −0.352865
$$515$$ 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
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ 10.0000 0.437688
$$523$$ −29.0000 −1.26808 −0.634041 0.773300i $$-0.718605\pi$$
−0.634041 + 0.773300i $$0.718605\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ −24.0000 −1.04546
$$528$$ 2.00000 0.0870388
$$529$$ −22.0000 −0.956522
$$530$$ −4.00000 −0.173749
$$531$$ 30.0000 1.30189
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ −28.0000 −1.21055
$$536$$ 3.00000 0.129580
$$537$$ 0 0
$$538$$ −30.0000 −1.29339
$$539$$ 0 0
$$540$$ −20.0000 −0.860663
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −7.00000 −0.300676
$$543$$ −22.0000 −0.944110
$$544$$ −3.00000 −0.128624
$$545$$ −60.0000 −2.57012
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −17.0000 −0.726204
$$549$$ 4.00000 0.170716
$$550$$ 22.0000 0.938083
$$551$$ −5.00000 −0.213007
$$552$$ −1.00000 −0.0425628
$$553$$ 0 0
$$554$$ 28.0000 1.18961
$$555$$ −8.00000 −0.339581
$$556$$ 0 0
$$557$$ 28.0000 1.18640 0.593199 0.805056i $$-0.297865\pi$$
0.593199 + 0.805056i $$0.297865\pi$$
$$558$$ −16.0000 −0.677334
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −6.00000 −0.253320
$$562$$ −8.00000 −0.337460
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 56.0000 2.35594
$$566$$ 6.00000 0.252199
$$567$$ 0 0
$$568$$ 2.00000 0.0839181
$$569$$ 40.0000 1.67689 0.838444 0.544988i $$-0.183466\pi$$
0.838444 + 0.544988i $$0.183466\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 7.00000 0.292429
$$574$$ 0 0
$$575$$ −11.0000 −0.458732
$$576$$ −2.00000 −0.0833333
$$577$$ 37.0000 1.54033 0.770165 0.637845i $$-0.220174\pi$$
0.770165 + 0.637845i $$0.220174\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ −6.00000 −0.249351
$$580$$ −20.0000 −0.830455
$$581$$ 0 0
$$582$$ 2.00000 0.0829027
$$583$$ −2.00000 −0.0828315
$$584$$ −9.00000 −0.372423
$$585$$ −8.00000 −0.330759
$$586$$ −9.00000 −0.371787
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ −60.0000 −2.47016
$$591$$ 8.00000 0.329076
$$592$$ −2.00000 −0.0821995
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ −10.0000 −0.410305
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 25.0000 1.02318
$$598$$ −1.00000 −0.0408930
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 11.0000 0.449073
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ 0 0
$$603$$ −6.00000 −0.244339
$$604$$ 2.00000 0.0813788
$$605$$ −28.0000 −1.13836
$$606$$ −2.00000 −0.0812444
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −8.00000 −0.323911
$$611$$ −8.00000 −0.323645
$$612$$ 6.00000 0.242536
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 32.0000 1.29036
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 6.00000 0.241355
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 32.0000 1.28515
$$621$$ 5.00000 0.200643
$$622$$ −7.00000 −0.280674
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 41.0000 1.64000
$$626$$ −29.0000 −1.15907
$$627$$ 2.00000 0.0798723
$$628$$ 2.00000 0.0798087
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 27.0000 1.07315
$$634$$ −27.0000 −1.07231
$$635$$ 72.0000 2.85723
$$636$$ −1.00000 −0.0396526
$$637$$ 0 0
$$638$$ −10.0000 −0.395904
$$639$$ −4.00000 −0.158238
$$640$$ 4.00000 0.158114
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ −7.00000 −0.276268
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ 16.0000 0.629999
$$646$$ −3.00000 −0.118033
$$647$$ −23.0000 −0.904223 −0.452112 0.891961i $$-0.649329\pi$$
−0.452112 + 0.891961i $$0.649329\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −30.0000 −1.17760
$$650$$ 11.0000 0.431455
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ −15.0000 −0.586546
$$655$$ −48.0000 −1.87552
$$656$$ 8.00000 0.312348
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ 5.00000 0.194772 0.0973862 0.995247i $$-0.468952\pi$$
0.0973862 + 0.995247i $$0.468952\pi$$
$$660$$ 8.00000 0.311400
$$661$$ 23.0000 0.894596 0.447298 0.894385i $$-0.352386\pi$$
0.447298 + 0.894385i $$0.352386\pi$$
$$662$$ 17.0000 0.660724
$$663$$ −3.00000 −0.116510
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 5.00000 0.193601
$$668$$ 12.0000 0.464294
$$669$$ −14.0000 −0.541271
$$670$$ 12.0000 0.463600
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ −55.0000 −2.11695
$$676$$ −12.0000 −0.461538
$$677$$ −13.0000 −0.499631 −0.249815 0.968294i $$-0.580370\pi$$
−0.249815 + 0.968294i $$0.580370\pi$$
$$678$$ 14.0000 0.537667
$$679$$ 0 0
$$680$$ −12.0000 −0.460179
$$681$$ 17.0000 0.651441
$$682$$ 16.0000 0.612672
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −68.0000 −2.59815
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 4.00000 0.152499
$$689$$ −1.00000 −0.0380970
$$690$$ −4.00000 −0.152277
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −2.00000 −0.0759190
$$695$$ 0 0
$$696$$ −5.00000 −0.189525
$$697$$ −24.0000 −0.909065
$$698$$ −10.0000 −0.378506
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −28.0000 −1.05755 −0.528773 0.848763i $$-0.677348\pi$$
−0.528773 + 0.848763i $$0.677348\pi$$
$$702$$ −5.00000 −0.188713
$$703$$ −2.00000 −0.0754314
$$704$$ 2.00000 0.0753778
$$705$$ −32.0000 −1.20519
$$706$$ −9.00000 −0.338719
$$707$$ 0 0
$$708$$ −15.0000 −0.563735
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 20.0000 0.750059
$$712$$ 0 0
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 0 0
$$717$$ 15.0000 0.560185
$$718$$ −15.0000 −0.559795
$$719$$ 5.00000 0.186469 0.0932343 0.995644i $$-0.470279\pi$$
0.0932343 + 0.995644i $$0.470279\pi$$
$$720$$ −8.00000 −0.298142
$$721$$ 0 0
$$722$$ 1.00000 0.0372161
$$723$$ 8.00000 0.297523
$$724$$ −22.0000 −0.817624
$$725$$ −55.0000 −2.04265
$$726$$ −7.00000 −0.259794
$$727$$ 17.0000 0.630495 0.315248 0.949009i $$-0.397912\pi$$
0.315248 + 0.949009i $$0.397912\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ −36.0000 −1.33242
$$731$$ −12.0000 −0.443836
$$732$$ −2.00000 −0.0739221
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 6.00000 0.221013
$$738$$ −16.0000 −0.588968
$$739$$ −40.0000 −1.47142 −0.735712 0.677295i $$-0.763152\pi$$
−0.735712 + 0.677295i $$0.763152\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 1.00000 0.0367359
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 29.0000 1.06177
$$747$$ −12.0000 −0.439057
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 24.0000 0.876356
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −2.00000 −0.0728841
$$754$$ −5.00000 −0.182089
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 15.0000 0.544825
$$759$$ −2.00000 −0.0725954
$$760$$ 4.00000 0.145095
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 18.0000 0.652071
$$763$$ 0 0
$$764$$ 7.00000 0.253251
$$765$$ 24.0000 0.867722
$$766$$ 26.0000 0.939418
$$767$$ −15.0000 −0.541619
$$768$$ 1.00000 0.0360844
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ −8.00000 −0.288113
$$772$$ −6.00000 −0.215945
$$773$$ −9.00000 −0.323708 −0.161854 0.986815i $$-0.551747\pi$$
−0.161854 + 0.986815i $$0.551747\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 88.0000 3.16105
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ 8.00000 0.286630
$$780$$ 4.00000 0.143223
$$781$$ 4.00000 0.143131
$$782$$ 3.00000 0.107280
$$783$$ 25.0000 0.893427
$$784$$ 0 0
$$785$$ 8.00000 0.285532
$$786$$ −12.0000 −0.428026
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 8.00000 0.284988
$$789$$ 24.0000 0.854423
$$790$$ −40.0000 −1.42314
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ −2.00000 −0.0710221
$$794$$ −8.00000 −0.283909
$$795$$ −4.00000 −0.141865
$$796$$ 25.0000 0.886102
$$797$$ −3.00000 −0.106265 −0.0531327 0.998587i $$-0.516921\pi$$
−0.0531327 + 0.998587i $$0.516921\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 11.0000 0.388909
$$801$$ 0 0
$$802$$ −8.00000 −0.282490
$$803$$ −18.0000 −0.635206
$$804$$ 3.00000 0.105802
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ −30.0000 −1.05605
$$808$$ −2.00000 −0.0703598
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 4.00000 0.140546
$$811$$ 3.00000 0.105344 0.0526721 0.998612i $$-0.483226\pi$$
0.0526721 + 0.998612i $$0.483226\pi$$
$$812$$ 0 0
$$813$$ −7.00000 −0.245501
$$814$$ −4.00000 −0.140200
$$815$$ −64.0000 −2.24182
$$816$$ −3.00000 −0.105021
$$817$$ 4.00000 0.139942
$$818$$ 20.0000 0.699284
$$819$$ 0 0
$$820$$ 32.0000 1.11749
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ −17.0000 −0.592943
$$823$$ 29.0000 1.01088 0.505438 0.862863i $$-0.331331\pi$$
0.505438 + 0.862863i $$0.331331\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 22.0000 0.765942
$$826$$ 0 0
$$827$$ 23.0000 0.799788 0.399894 0.916561i $$-0.369047\pi$$
0.399894 + 0.916561i $$0.369047\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ 15.0000 0.520972 0.260486 0.965478i $$-0.416117\pi$$
0.260486 + 0.965478i $$0.416117\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 28.0000 0.971309
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 48.0000 1.66111
$$836$$ 2.00000 0.0691714
$$837$$ −40.0000 −1.38260
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −13.0000 −0.448010
$$843$$ −8.00000 −0.275535
$$844$$ 27.0000 0.929378
$$845$$ −48.0000 −1.65125
$$846$$ 16.0000 0.550091
$$847$$ 0 0
$$848$$ −1.00000 −0.0343401
$$849$$ 6.00000 0.205919
$$850$$ −33.0000 −1.13189
$$851$$ 2.00000 0.0685591
$$852$$ 2.00000 0.0685189
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ −7.00000 −0.239255
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 16.0000 0.545595
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 24.0000 0.816024
$$866$$ −14.0000 −0.475739
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ −20.0000 −0.678454
$$870$$ −20.0000 −0.678064
$$871$$ 3.00000 0.101651
$$872$$ −15.0000 −0.507964
$$873$$ −4.00000 −0.135379
$$874$$ −1.00000 −0.0338255
$$875$$ 0 0
$$876$$ −9.00000 −0.304082
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ −9.00000 −0.303562
$$880$$ 8.00000 0.269680
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 34.0000 1.14419 0.572096 0.820187i $$-0.306131\pi$$
0.572096 + 0.820187i $$0.306131\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ −60.0000 −2.01688
$$886$$ −26.0000 −0.873487
$$887$$ 2.00000 0.0671534 0.0335767 0.999436i $$-0.489310\pi$$
0.0335767 + 0.999436i $$0.489310\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ −14.0000 −0.468755
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.00000 −0.0333890
$$898$$ 10.0000 0.333704
$$899$$ −40.0000 −1.33407
$$900$$ −22.0000 −0.733333
$$901$$ 3.00000 0.0999445
$$902$$ 16.0000 0.532742
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ −88.0000 −2.92522
$$906$$ 2.00000 0.0664455
$$907$$ 53.0000 1.75984 0.879918 0.475125i $$-0.157597\pi$$
0.879918 + 0.475125i $$0.157597\pi$$
$$908$$ 17.0000 0.564165
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 12.0000 0.397142
$$914$$ −7.00000 −0.231539
$$915$$ −8.00000 −0.264472
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 15.0000 0.495074
$$919$$ 5.00000 0.164935 0.0824674 0.996594i $$-0.473720\pi$$
0.0824674 + 0.996594i $$0.473720\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 12.0000 0.395413
$$922$$ 28.0000 0.922131
$$923$$ 2.00000 0.0658308
$$924$$ 0 0
$$925$$ −22.0000 −0.723356
$$926$$ 4.00000 0.131448
$$927$$ −12.0000 −0.394132
$$928$$ −5.00000 −0.164133
$$929$$ 55.0000 1.80449 0.902246 0.431222i $$-0.141918\pi$$
0.902246 + 0.431222i $$0.141918\pi$$
$$930$$ 32.0000 1.04932
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ −7.00000 −0.229170
$$934$$ 2.00000 0.0654420
$$935$$ −24.0000 −0.784884
$$936$$ −2.00000 −0.0653720
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ −29.0000 −0.946379
$$940$$ −32.0000 −1.04372
$$941$$ −7.00000 −0.228193 −0.114097 0.993470i $$-0.536397\pi$$
−0.114097 + 0.993470i $$0.536397\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −8.00000 −0.260516
$$944$$ −15.0000 −0.488208
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ −9.00000 −0.292152
$$950$$ 11.0000 0.356887
$$951$$ −27.0000 −0.875535
$$952$$ 0 0
$$953$$ −46.0000 −1.49009 −0.745043 0.667016i $$-0.767571\pi$$
−0.745043 + 0.667016i $$0.767571\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 28.0000 0.906059
$$956$$ 15.0000 0.485135
$$957$$ −10.0000 −0.323254
$$958$$ 20.0000 0.646171
$$959$$ 0 0
$$960$$ 4.00000 0.129099
$$961$$ 33.0000 1.06452
$$962$$ −2.00000 −0.0644826
$$963$$ 14.0000 0.451144
$$964$$ 8.00000 0.257663
$$965$$ −24.0000 −0.772587
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ −3.00000 −0.0963739
$$970$$ 8.00000 0.256865
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 0 0
$$974$$ −2.00000 −0.0640841
$$975$$ 11.0000 0.352282
$$976$$ −2.00000 −0.0640184
$$977$$ 8.00000 0.255943 0.127971 0.991778i $$-0.459153\pi$$
0.127971 + 0.991778i $$0.459153\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 30.0000 0.957826
$$982$$ −28.0000 −0.893516
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 32.0000 1.01960
$$986$$ 15.0000 0.477697
$$987$$ 0 0
$$988$$ 1.00000 0.0318142
$$989$$ −4.00000 −0.127193
$$990$$ −16.0000 −0.508513
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 17.0000 0.539479
$$994$$ 0 0
$$995$$ 100.000 3.17021
$$996$$ 6.00000 0.190117
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 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 1862.2.a.f.1.1 1
7.6 odd 2 38.2.a.b.1.1 1
21.20 even 2 342.2.a.d.1.1 1
28.27 even 2 304.2.a.d.1.1 1
35.13 even 4 950.2.b.c.799.1 2
35.27 even 4 950.2.b.c.799.2 2
35.34 odd 2 950.2.a.b.1.1 1
56.13 odd 2 1216.2.a.n.1.1 1
56.27 even 2 1216.2.a.g.1.1 1
77.76 even 2 4598.2.a.a.1.1 1
84.83 odd 2 2736.2.a.w.1.1 1
91.90 odd 2 6422.2.a.b.1.1 1
105.104 even 2 8550.2.a.u.1.1 1
133.6 odd 18 722.2.e.c.245.1 6
133.13 even 18 722.2.e.d.245.1 6
133.27 even 6 722.2.c.f.653.1 2
133.34 even 18 722.2.e.d.415.1 6
133.41 even 18 722.2.e.d.389.1 6
133.48 even 18 722.2.e.d.423.1 6
133.55 odd 18 722.2.e.c.99.1 6
133.62 odd 18 722.2.e.c.595.1 6
133.69 even 6 722.2.c.f.429.1 2
133.83 odd 6 722.2.c.d.429.1 2
133.90 even 18 722.2.e.d.595.1 6
133.97 even 18 722.2.e.d.99.1 6
133.104 odd 18 722.2.e.c.423.1 6
133.111 odd 18 722.2.e.c.389.1 6
133.118 odd 18 722.2.e.c.415.1 6
133.125 odd 6 722.2.c.d.653.1 2
133.132 even 2 722.2.a.b.1.1 1
140.139 even 2 7600.2.a.h.1.1 1
399.398 odd 2 6498.2.a.y.1.1 1
532.531 odd 2 5776.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 7.6 odd 2
304.2.a.d.1.1 1 28.27 even 2
342.2.a.d.1.1 1 21.20 even 2
722.2.a.b.1.1 1 133.132 even 2
722.2.c.d.429.1 2 133.83 odd 6
722.2.c.d.653.1 2 133.125 odd 6
722.2.c.f.429.1 2 133.69 even 6
722.2.c.f.653.1 2 133.27 even 6
722.2.e.c.99.1 6 133.55 odd 18
722.2.e.c.245.1 6 133.6 odd 18
722.2.e.c.389.1 6 133.111 odd 18
722.2.e.c.415.1 6 133.118 odd 18
722.2.e.c.423.1 6 133.104 odd 18
722.2.e.c.595.1 6 133.62 odd 18
722.2.e.d.99.1 6 133.97 even 18
722.2.e.d.245.1 6 133.13 even 18
722.2.e.d.389.1 6 133.41 even 18
722.2.e.d.415.1 6 133.34 even 18
722.2.e.d.423.1 6 133.48 even 18
722.2.e.d.595.1 6 133.90 even 18
950.2.a.b.1.1 1 35.34 odd 2
950.2.b.c.799.1 2 35.13 even 4
950.2.b.c.799.2 2 35.27 even 4
1216.2.a.g.1.1 1 56.27 even 2
1216.2.a.n.1.1 1 56.13 odd 2
1862.2.a.f.1.1 1 1.1 even 1 trivial
2736.2.a.w.1.1 1 84.83 odd 2
4598.2.a.a.1.1 1 77.76 even 2
5776.2.a.d.1.1 1 532.531 odd 2
6422.2.a.b.1.1 1 91.90 odd 2
6498.2.a.y.1.1 1 399.398 odd 2
7600.2.a.h.1.1 1 140.139 even 2
8550.2.a.u.1.1 1 105.104 even 2