# Properties

 Label 38.2.a.b.1.1 Level $38$ Weight $2$ Character 38.1 Self dual yes Analytic conductor $0.303$ 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] = [38,2,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 38.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: $$0.303431527681$$ 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 $$\chi$$ $$=$$ 38.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -4.00000 q^{20} -3.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} +5.00000 q^{27} +3.00000 q^{28} -5.00000 q^{29} +4.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +3.00000 q^{34} -12.0000 q^{35} -2.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +1.00000 q^{39} -4.00000 q^{40} -8.00000 q^{41} -3.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +8.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +11.0000 q^{50} -3.00000 q^{51} -1.00000 q^{52} -1.00000 q^{53} +5.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} +1.00000 q^{57} -5.00000 q^{58} +15.0000 q^{59} +4.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} +3.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -12.0000 q^{70} +2.00000 q^{71} -2.00000 q^{72} +9.00000 q^{73} -2.00000 q^{74} -11.0000 q^{75} -1.00000 q^{76} +6.00000 q^{77} +1.00000 q^{78} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -6.00000 q^{83} -3.00000 q^{84} -12.0000 q^{85} +4.00000 q^{86} +5.00000 q^{87} +2.00000 q^{88} +8.00000 q^{90} -3.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} +2.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 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 $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 4.00000 1.03280
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −1.00000 −0.229416
$$20$$ −4.00000 −0.894427
$$21$$ −3.00000 −0.654654
$$22$$ 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
$$28$$ 3.00000 0.566947
$$29$$ −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 $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 3.00000 0.514496
$$35$$ −12.0000 −2.02837
$$36$$ −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 $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ −3.00000 −0.462910
$$43$$ 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 $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 2.00000 0.285714
$$50$$ 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
$$56$$ 3.00000 0.400892
$$57$$ 1.00000 0.132453
$$58$$ −5.00000 −0.656532
$$59$$ 15.0000 1.95283 0.976417 0.215894i $$-0.0692665\pi$$
0.976417 + 0.215894i $$0.0692665\pi$$
$$60$$ 4.00000 0.516398
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ −6.00000 −0.755929
$$64$$ 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
$$70$$ −12.0000 −1.43427
$$71$$ 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 $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −11.0000 −1.27017
$$76$$ −1.00000 −0.114708
$$77$$ 6.00000 0.683763
$$78$$ 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 $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ −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
$$91$$ −3.00000 −0.314485
$$92$$ −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 $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 2.00000 0.202031
$$99$$ −4.00000 −0.402015
$$100$$ 11.0000 1.10000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 12.0000 1.17108
$$106$$ −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
$$112$$ 3.00000 0.283473
$$113$$ 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
$$119$$ 9.00000 0.825029
$$120$$ 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
$$126$$ −6.00000 −0.534522
$$127$$ 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 $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ −3.00000 −0.260133
$$134$$ 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 $$0.5\pi$$
$$140$$ −12.0000 −1.01419
$$141$$ −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
$$147$$ −2.00000 −0.164957
$$148$$ −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
$$154$$ 6.00000 0.483494
$$155$$ 32.0000 2.57030
$$156$$ 1.00000 0.0800641
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 1.00000 0.0793052
$$160$$ −4.00000 −0.316228
$$161$$ −3.00000 −0.236433
$$162$$ 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 $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ −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 $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 5.00000 0.379049
$$175$$ 33.0000 2.49457
$$176$$ 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 $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ −3.00000 −0.222375
$$183$$ −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
$$189$$ 15.0000 1.09109
$$190$$ 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
$$196$$ 2.00000 0.142857
$$197$$ 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 $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 11.0000 0.777817
$$201$$ −3.00000 −0.211604
$$202$$ 2.00000 0.140720
$$203$$ −15.0000 −1.05279
$$204$$ −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
$$210$$ 12.0000 0.828079
$$211$$ 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
$$217$$ −24.0000 −1.62923
$$218$$ −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 $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 3.00000 0.200446
$$225$$ −22.0000 −1.46667
$$226$$ 14.0000 0.931266
$$227$$ −17.0000 −1.12833 −0.564165 0.825662i $$-0.690802\pi$$
−0.564165 + 0.825662i $$0.690802\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 4.00000 0.263752
$$231$$ −6.00000 −0.394771
$$232$$ −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
$$238$$ 9.00000 0.583383
$$239$$ 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 $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −16.0000 −1.02640
$$244$$ 2.00000 0.128037
$$245$$ −8.00000 −0.511101
$$246$$ 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 $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ −6.00000 −0.377964
$$253$$ −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 $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −6.00000 −0.372822
$$260$$ 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
$$266$$ −3.00000 −0.183942
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ −20.0000 −1.21716
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 3.00000 0.181568
$$274$$ −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
$$280$$ −12.0000 −0.717137
$$281$$ −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 $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ 2.00000 0.118678
$$285$$ −4.00000 −0.236940
$$286$$ −2.00000 −0.118262
$$287$$ −24.0000 −1.41668
$$288$$ −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 $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ −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
$$301$$ 12.0000 0.691669
$$302$$ 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 $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 6.00000 0.341328
$$310$$ 32.0000 1.81748
$$311$$ 7.00000 0.396934 0.198467 0.980108i $$-0.436404\pi$$
0.198467 + 0.980108i $$0.436404\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 29.0000 1.63918 0.819588 0.572953i $$-0.194202\pi$$
0.819588 + 0.572953i $$0.194202\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 24.0000 1.35225
$$316$$ −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
$$322$$ −3.00000 −0.167183
$$323$$ −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
$$329$$ 24.0000 1.32316
$$330$$ 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
$$336$$ −3.00000 −0.163663
$$337$$ −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
$$343$$ −15.0000 −0.809924
$$344$$ 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 $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 33.0000 1.76392
$$351$$ −5.00000 −0.266880
$$352$$ 2.00000 0.106600
$$353$$ 9.00000 0.479022 0.239511 0.970894i $$-0.423013\pi$$
0.239511 + 0.970894i $$0.423013\pi$$
$$354$$ −15.0000 −0.797241
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ −9.00000 −0.476331
$$358$$ 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
$$364$$ −3.00000 −0.157243
$$365$$ −36.0000 −1.88433
$$366$$ −2.00000 −0.104542
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 16.0000 0.832927
$$370$$ 8.00000 0.415900
$$371$$ −3.00000 −0.155752
$$372$$ 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
$$378$$ 15.0000 0.771517
$$379$$ 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 $$-0.731257\pi$$
−0.664269 + 0.747494i $$0.731257\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −24.0000 −1.22315
$$386$$ −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
$$392$$ 2.00000 0.101015
$$393$$ −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 $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ −25.0000 −1.25314
$$399$$ 3.00000 0.150188
$$400$$ 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
$$406$$ −15.0000 −0.744438
$$407$$ −4.00000 −0.198273
$$408$$ −3.00000 −0.148522
$$409$$ −20.0000 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$410$$ 32.0000 1.58037
$$411$$ 17.0000 0.838548
$$412$$ −6.00000 −0.295599
$$413$$ 45.0000 2.21431
$$414$$ 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 $$0.5\pi$$
$$420$$ 12.0000 0.585540
$$421$$ −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
$$427$$ 6.00000 0.290360
$$428$$ −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 $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ −24.0000 −1.15204
$$435$$ −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 $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ −4.00000 −0.190476
$$442$$ −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
$$448$$ 3.00000 0.141737
$$449$$ 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
$$455$$ 12.0000 0.562569
$$456$$ 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 $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ −6.00000 −0.279145
$$463$$ 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 $$-0.514735\pi$$
−0.0462745 + 0.998929i $$0.514735\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 9.00000 0.415581
$$470$$ −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
$$476$$ 9.00000 0.412514
$$477$$ 2.00000 0.0915737
$$478$$ 15.0000 0.686084
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 4.00000 0.182574
$$481$$ 2.00000 0.0911922
$$482$$ −8.00000 −0.364390
$$483$$ 3.00000 0.136505
$$484$$ −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
$$490$$ −8.00000 −0.361403
$$491$$ −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
$$497$$ 6.00000 0.269137
$$498$$ 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 $$-0.164466\pi$$
0.869462 + 0.494000i $$0.164466\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ −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 $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 12.0000 0.531369
$$511$$ 27.0000 1.19441
$$512$$ 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
$$518$$ −6.00000 −0.263625
$$519$$ 6.00000 0.263371
$$520$$ 4.00000 0.175412
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 12.0000 0.524222
$$525$$ −33.0000 −1.44024
$$526$$ 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
$$532$$ −3.00000 −0.130066
$$533$$ 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
$$539$$ 4.00000 0.172292
$$540$$ −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
$$546$$ 3.00000 0.128388
$$547$$ 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
$$553$$ −30.0000 −1.27573
$$554$$ 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
$$560$$ −12.0000 −0.507093
$$561$$ −6.00000 −0.253320
$$562$$ −8.00000 −0.337460
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ −56.0000 −2.35594
$$566$$ −6.00000 −0.252199
$$567$$ 3.00000 0.125988
$$568$$ 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
$$574$$ −24.0000 −1.00174
$$575$$ −11.0000 −0.458732
$$576$$ −2.00000 −0.0833333
$$577$$ −37.0000 −1.54033 −0.770165 0.637845i $$-0.779826\pi$$
−0.770165 + 0.637845i $$0.779826\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 6.00000 0.249351
$$580$$ 20.0000 0.830455
$$581$$ −18.0000 −0.746766
$$582$$ 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 $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 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 $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 10.0000 0.410305
$$595$$ −36.0000 −1.47586
$$596$$ 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 $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 12.0000 0.489083
$$603$$ −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 $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 15.0000 0.607831
$$610$$ −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
$$616$$ 6.00000 0.241747
$$617$$ 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 $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 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
$$630$$ 24.0000 0.956183
$$631$$ 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
$$637$$ −2.00000 −0.0792429
$$638$$ −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 $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 16.0000 0.629999
$$646$$ −3.00000 −0.118033
$$647$$ 23.0000 0.904223 0.452112 0.891961i $$-0.350671\pi$$
0.452112 + 0.891961i $$0.350671\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 30.0000 1.17760
$$650$$ −11.0000 −0.431455
$$651$$ 24.0000 0.940634
$$652$$ −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
$$658$$ 24.0000 0.935617
$$659$$ 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 $$-0.647614\pi$$
−0.447298 + 0.894385i $$0.647614\pi$$
$$662$$ 17.0000 0.660724
$$663$$ 3.00000 0.116510
$$664$$ −6.00000 −0.232845
$$665$$ 12.0000 0.465340
$$666$$ 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
$$672$$ −3.00000 −0.115728
$$673$$ 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 $$-0.419630\pi$$
0.249815 + 0.968294i $$0.419630\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ −6.00000 −0.230259
$$680$$ −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
$$686$$ −15.0000 −0.572703
$$687$$ 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 $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ −12.0000 −0.455842
$$694$$ −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
$$700$$ 33.0000 1.24728
$$701$$ −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
$$707$$ 6.00000 0.225653
$$708$$ −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
$$714$$ −9.00000 −0.336817
$$715$$ 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 $$-0.529721\pi$$
−0.0932343 + 0.995644i $$0.529721\pi$$
$$720$$ 8.00000 0.298142
$$721$$ −18.0000 −0.670355
$$722$$ 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 $$-0.602088\pi$$
−0.315248 + 0.949009i $$0.602088\pi$$
$$728$$ −3.00000 −0.111187
$$729$$ 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 $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 8.00000 0.295084
$$736$$ −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
$$742$$ −3.00000 −0.110133
$$743$$ −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
$$749$$ −21.0000 −0.767323
$$750$$ 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
$$756$$ 15.0000 0.545545
$$757$$ −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 $$-0.337225\pi$$
0.489375 + 0.872074i $$0.337225\pi$$
$$762$$ −18.0000 −0.652071
$$763$$ −45.0000 −1.62911
$$764$$ 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 $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ −24.0000 −0.864900
$$771$$ −8.00000 −0.288113
$$772$$ −6.00000 −0.215945
$$773$$ 9.00000 0.323708 0.161854 0.986815i $$-0.448253\pi$$
0.161854 + 0.986815i $$0.448253\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ −88.0000 −3.16105
$$776$$ −2.00000 −0.0717958
$$777$$ 6.00000 0.215249
$$778$$ −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
$$784$$ 2.00000 0.0714286
$$785$$ 8.00000 0.285532
$$786$$ −12.0000 −0.428026
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ 8.00000 0.284988
$$789$$ −24.0000 −0.854423
$$790$$ 40.0000 1.42314
$$791$$ 42.0000 1.49335
$$792$$ −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 $$-0.483079\pi$$
0.0531327 + 0.998587i $$0.483079\pi$$
$$798$$ 3.00000 0.106199
$$799$$ 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
$$805$$ 12.0000 0.422944
$$806$$ 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 $$-0.516774\pi$$
−0.0526721 + 0.998612i $$0.516774\pi$$
$$812$$ −15.0000 −0.526397
$$813$$ −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
$$819$$ 6.00000 0.209657
$$820$$ 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
$$826$$ 45.0000 1.56575
$$827$$ 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 $$-0.583883\pi$$
−0.260486 + 0.965478i $$0.583883\pi$$
$$830$$ 24.0000 0.833052
$$831$$ −28.0000 −0.971309
$$832$$ −1.00000 −0.0346688
$$833$$ 6.00000 0.207888
$$834$$ 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 $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 12.0000 0.414039
$$841$$ −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
$$847$$ −21.0000 −0.721569
$$848$$ −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 $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 6.00000 0.205316
$$855$$ −8.00000 −0.273594
$$856$$ −7.00000 −0.239255
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ −16.0000 −0.545595
$$861$$ 24.0000 0.817918
$$862$$ −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
$$868$$ −24.0000 −0.814613
$$869$$ −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
$$875$$ −72.0000 −2.43404
$$876$$ −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 $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ 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 $$-0.510690\pi$$
−0.0335767 + 0.999436i $$0.510690\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 54.0000 1.81110
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 14.0000 0.468755
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ −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
$$903$$ −12.0000 −0.399335
$$904$$ 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
$$910$$ 12.0000 0.397796
$$911$$ 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
$$917$$ 36.0000 1.18882
$$918$$ 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
$$924$$ −6.00000 −0.197386
$$925$$ −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 $$-0.858082\pi$$
−0.902246 + 0.431222i $$0.858082\pi$$
$$930$$ −32.0000 −1.04932
$$931$$ −2.00000 −0.0655474
$$932$$ −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 $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ 9.00000 0.293860
$$939$$ −29.0000 −0.946379
$$940$$ −32.0000 −1.04372
$$941$$ 7.00000 0.228193 0.114097 0.993470i $$-0.463603\pi$$
0.114097 + 0.993470i $$0.463603\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ 8.00000 0.260516
$$944$$ 15.0000 0.488208
$$945$$ −60.0000 −1.95180
$$946$$ 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
$$952$$ 9.00000 0.291692
$$953$$ −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
$$959$$ −51.0000 −1.64688
$$960$$ 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
$$966$$ 3.00000 0.0965234
$$967$$ 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 $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ −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
$$980$$ −8.00000 −0.255551
$$981$$ 30.0000 0.957826
$$982$$ −28.0000 −0.893516
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 8.00000 0.255031
$$985$$ −32.0000 −1.01960
$$986$$ −15.0000 −0.477697
$$987$$ −24.0000 −0.763928
$$988$$ 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
$$994$$ 6.00000 0.190308
$$995$$ 100.000 3.17021
$$996$$ 6.00000 0.190117
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ 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 38.2.a.b.1.1 1
3.2 odd 2 342.2.a.d.1.1 1
4.3 odd 2 304.2.a.d.1.1 1
5.2 odd 4 950.2.b.c.799.2 2
5.3 odd 4 950.2.b.c.799.1 2
5.4 even 2 950.2.a.b.1.1 1
7.6 odd 2 1862.2.a.f.1.1 1
8.3 odd 2 1216.2.a.g.1.1 1
8.5 even 2 1216.2.a.n.1.1 1
11.10 odd 2 4598.2.a.a.1.1 1
12.11 even 2 2736.2.a.w.1.1 1
13.12 even 2 6422.2.a.b.1.1 1
15.14 odd 2 8550.2.a.u.1.1 1
19.2 odd 18 722.2.e.d.99.1 6
19.3 odd 18 722.2.e.d.389.1 6
19.4 even 9 722.2.e.c.415.1 6
19.5 even 9 722.2.e.c.595.1 6
19.6 even 9 722.2.e.c.245.1 6
19.7 even 3 722.2.c.d.429.1 2
19.8 odd 6 722.2.c.f.653.1 2
19.9 even 9 722.2.e.c.423.1 6
19.10 odd 18 722.2.e.d.423.1 6
19.11 even 3 722.2.c.d.653.1 2
19.12 odd 6 722.2.c.f.429.1 2
19.13 odd 18 722.2.e.d.245.1 6
19.14 odd 18 722.2.e.d.595.1 6
19.15 odd 18 722.2.e.d.415.1 6
19.16 even 9 722.2.e.c.389.1 6
19.17 even 9 722.2.e.c.99.1 6
19.18 odd 2 722.2.a.b.1.1 1
20.19 odd 2 7600.2.a.h.1.1 1
57.56 even 2 6498.2.a.y.1.1 1
76.75 even 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 1.1 even 1 trivial
304.2.a.d.1.1 1 4.3 odd 2
342.2.a.d.1.1 1 3.2 odd 2
722.2.a.b.1.1 1 19.18 odd 2
722.2.c.d.429.1 2 19.7 even 3
722.2.c.d.653.1 2 19.11 even 3
722.2.c.f.429.1 2 19.12 odd 6
722.2.c.f.653.1 2 19.8 odd 6
722.2.e.c.99.1 6 19.17 even 9
722.2.e.c.245.1 6 19.6 even 9
722.2.e.c.389.1 6 19.16 even 9
722.2.e.c.415.1 6 19.4 even 9
722.2.e.c.423.1 6 19.9 even 9
722.2.e.c.595.1 6 19.5 even 9
722.2.e.d.99.1 6 19.2 odd 18
722.2.e.d.245.1 6 19.13 odd 18
722.2.e.d.389.1 6 19.3 odd 18
722.2.e.d.415.1 6 19.15 odd 18
722.2.e.d.423.1 6 19.10 odd 18
722.2.e.d.595.1 6 19.14 odd 18
950.2.a.b.1.1 1 5.4 even 2
950.2.b.c.799.1 2 5.3 odd 4
950.2.b.c.799.2 2 5.2 odd 4
1216.2.a.g.1.1 1 8.3 odd 2
1216.2.a.n.1.1 1 8.5 even 2
1862.2.a.f.1.1 1 7.6 odd 2
2736.2.a.w.1.1 1 12.11 even 2
4598.2.a.a.1.1 1 11.10 odd 2
5776.2.a.d.1.1 1 76.75 even 2
6422.2.a.b.1.1 1 13.12 even 2
6498.2.a.y.1.1 1 57.56 even 2
7600.2.a.h.1.1 1 20.19 odd 2
8550.2.a.u.1.1 1 15.14 odd 2