# Properties

 Label 38.2.a.a.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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -1.00000 q^{21} +6.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} +5.00000 q^{39} +1.00000 q^{42} +8.00000 q^{43} -6.00000 q^{44} -3.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +3.00000 q^{51} +5.00000 q^{52} -3.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} -9.00000 q^{58} +9.00000 q^{59} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +5.00000 q^{67} +3.00000 q^{68} +3.00000 q^{69} -6.00000 q^{71} +2.00000 q^{72} -7.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +1.00000 q^{76} +6.00000 q^{77} -5.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} +9.00000 q^{87} +6.00000 q^{88} -12.0000 q^{89} -5.00000 q^{91} +3.00000 q^{92} -4.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} +6.00000 q^{98} +12.0000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 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
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 6.00000 1.27920
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −5.00000 −1.00000
$$26$$ −5.00000 −0.980581
$$27$$ −5.00000 −0.962250
$$28$$ −1.00000 −0.188982
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −6.00000 −1.04447
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 5.00000 0.800641
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 5.00000 0.707107
$$51$$ 3.00000 0.420084
$$52$$ 5.00000 0.693375
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 1.00000 0.132453
$$58$$ −9.00000 −1.18176
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 2.00000 0.235702
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −5.00000 −0.577350
$$76$$ 1.00000 0.114708
$$77$$ 6.00000 0.683763
$$78$$ −5.00000 −0.566139
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 9.00000 0.964901
$$88$$ 6.00000 0.639602
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ −5.00000 −0.524142
$$92$$ 3.00000 0.312772
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 12.0000 1.20605
$$100$$ −5.00000 −0.500000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ −10.0000 −0.924500
$$118$$ −9.00000 −0.828517
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ −1.00000 −0.0867110
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ −3.00000 −0.255377
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ −30.0000 −2.50873
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ −6.00000 −0.494872
$$148$$ 2.00000 0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −6.00000 −0.485071
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 5.00000 0.400320
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 10.0000 0.795557
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 8.00000 0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ 5.00000 0.377964
$$176$$ −6.00000 −0.452267
$$177$$ 9.00000 0.676481
$$178$$ 12.0000 0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 5.00000 0.370625
$$183$$ −10.0000 −0.739221
$$184$$ −3.00000 −0.221163
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 5.00000 0.352673
$$202$$ −18.0000 −1.26648
$$203$$ −9.00000 −0.631676
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ −6.00000 −0.417029
$$208$$ 5.00000 0.346688
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ −6.00000 −0.411113
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 4.00000 0.271538
$$218$$ −11.0000 −0.745014
$$219$$ −7.00000 −0.473016
$$220$$ 0 0
$$221$$ 15.0000 1.00901
$$222$$ −2.00000 −0.134231
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 10.0000 0.666667
$$226$$ −6.00000 −0.399114
$$227$$ −15.0000 −0.995585 −0.497792 0.867296i $$-0.665856\pi$$
−0.497792 + 0.867296i $$0.665856\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ −9.00000 −0.590879
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 10.0000 0.653720
$$235$$ 0 0
$$236$$ 9.00000 0.585850
$$237$$ −10.0000 −0.649570
$$238$$ 3.00000 0.194461
$$239$$ −21.0000 −1.35838 −0.679189 0.733964i $$-0.737668\pi$$
−0.679189 + 0.733964i $$0.737668\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ −25.0000 −1.60706
$$243$$ 16.0000 1.02640
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 4.00000 0.254000
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −18.0000 −1.13165
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 1.00000 0.0613139
$$267$$ −12.0000 −0.734388
$$268$$ 5.00000 0.305424
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 3.00000 0.181902
$$273$$ −5.00000 −0.302614
$$274$$ 9.00000 0.543710
$$275$$ 30.0000 1.80907
$$276$$ 3.00000 0.180579
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 30.0000 1.77394
$$287$$ 0 0
$$288$$ 2.00000 0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ −7.00000 −0.409644
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 30.0000 1.74078
$$298$$ 0 0
$$299$$ 15.0000 0.867472
$$300$$ −5.00000 −0.288675
$$301$$ −8.00000 −0.461112
$$302$$ 10.0000 0.575435
$$303$$ 18.0000 1.03407
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ −5.00000 −0.283069
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −9.00000 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$318$$ 3.00000 0.168232
$$319$$ −54.0000 −3.02342
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 3.00000 0.167183
$$323$$ 3.00000 0.166924
$$324$$ 1.00000 0.0555556
$$325$$ −25.0000 −1.38675
$$326$$ −20.0000 −1.10770
$$327$$ 11.0000 0.608301
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.00000 −0.0549650 −0.0274825 0.999622i $$-0.508749\pi$$
−0.0274825 + 0.999622i $$0.508749\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ −4.00000 −0.219199
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 2.00000 0.108148
$$343$$ 13.0000 0.701934
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 9.00000 0.482451
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ −25.0000 −1.33440
$$352$$ 6.00000 0.319801
$$353$$ −15.0000 −0.798369 −0.399185 0.916871i $$-0.630707\pi$$
−0.399185 + 0.916871i $$0.630707\pi$$
$$354$$ −9.00000 −0.478345
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ −3.00000 −0.158777
$$358$$ 0 0
$$359$$ 21.0000 1.10834 0.554169 0.832404i $$-0.313036\pi$$
0.554169 + 0.832404i $$0.313036\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −2.00000 −0.105118
$$363$$ 25.0000 1.31216
$$364$$ −5.00000 −0.262071
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ −4.00000 −0.207390
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 45.0000 2.31762
$$378$$ −5.00000 −0.257172
$$379$$ −7.00000 −0.359566 −0.179783 0.983706i $$-0.557540\pi$$
−0.179783 + 0.983706i $$0.557540\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ −3.00000 −0.153493
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −16.0000 −0.813326
$$388$$ −10.0000 −0.507673
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ −11.0000 −0.551380
$$399$$ −1.00000 −0.0500626
$$400$$ −5.00000 −0.250000
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ −5.00000 −0.249377
$$403$$ −20.0000 −0.996271
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ −12.0000 −0.594818
$$408$$ −3.00000 −0.148522
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 14.0000 0.689730
$$413$$ −9.00000 −0.442861
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ −4.00000 −0.195881
$$418$$ 6.00000 0.293470
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ −5.00000 −0.243396
$$423$$ 0 0
$$424$$ 3.00000 0.145693
$$425$$ −15.0000 −0.727607
$$426$$ 6.00000 0.290701
$$427$$ 10.0000 0.483934
$$428$$ −9.00000 −0.435031
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 3.00000 0.143509
$$438$$ 7.00000 0.334473
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ −15.0000 −0.713477
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −10.0000 −0.471405
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ −10.0000 −0.469841
$$454$$ 15.0000 0.703985
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 17.0000 0.795226 0.397613 0.917553i $$-0.369839\pi$$
0.397613 + 0.917553i $$0.369839\pi$$
$$458$$ 22.0000 1.02799
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ −6.00000 −0.279145
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ −10.0000 −0.462250
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ −9.00000 −0.414259
$$473$$ −48.0000 −2.20704
$$474$$ 10.0000 0.459315
$$475$$ −5.00000 −0.229416
$$476$$ −3.00000 −0.137505
$$477$$ 6.00000 0.274721
$$478$$ 21.0000 0.960518
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ −8.00000 −0.364390
$$483$$ −3.00000 −0.136505
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ −5.00000 −0.224961
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 6.00000 0.269137
$$498$$ 6.00000 0.268866
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −6.00000 −0.267793
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ 12.0000 0.532939
$$508$$ 2.00000 0.0887357
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ −1.00000 −0.0441942
$$513$$ −5.00000 −0.220755
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 2.00000 0.0878750
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 18.0000 0.787839
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ −24.0000 −1.04645
$$527$$ −12.0000 −0.522728
$$528$$ −6.00000 −0.261116
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −18.0000 −0.781133
$$532$$ −1.00000 −0.0433555
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 2.00000 0.0858282
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 5.00000 0.213980
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ 20.0000 0.853579
$$550$$ −30.0000 −1.27920
$$551$$ 9.00000 0.383413
$$552$$ −3.00000 −0.127688
$$553$$ 10.0000 0.425243
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 40.0000 1.69182
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ −1.00000 −0.0419961
$$568$$ 6.00000 0.251754
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −30.0000 −1.25436
$$573$$ 3.00000 0.125327
$$574$$ 0 0
$$575$$ −15.0000 −0.625543
$$576$$ −2.00000 −0.0833333
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 10.0000 0.414513
$$583$$ 18.0000 0.745484
$$584$$ 7.00000 0.289662
$$585$$ 0 0
$$586$$ 21.0000 0.867502
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ −30.0000 −1.23091
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.0000 0.450200
$$598$$ −15.0000 −0.613396
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 5.00000 0.204124
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −10.0000 −0.407231
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ −15.0000 −0.601929
$$622$$ 21.0000 0.842023
$$623$$ 12.0000 0.480770
$$624$$ 5.00000 0.200160
$$625$$ 25.0000 1.00000
$$626$$ 19.0000 0.759393
$$627$$ −6.00000 −0.239617
$$628$$ −22.0000 −0.877896
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 5.00000 0.198732
$$634$$ 9.00000 0.357436
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ −30.0000 −1.18864
$$638$$ 54.0000 2.13788
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 9.00000 0.355202
$$643$$ −22.0000 −0.867595 −0.433798 0.901010i $$-0.642827\pi$$
−0.433798 + 0.901010i $$0.642827\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −54.0000 −2.11969
$$650$$ 25.0000 0.980581
$$651$$ 4.00000 0.156772
$$652$$ 20.0000 0.783260
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ −11.0000 −0.430134
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000 0.546192
$$658$$ 0 0
$$659$$ −45.0000 −1.75295 −0.876476 0.481446i $$-0.840112\pi$$
−0.876476 + 0.481446i $$0.840112\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 1.00000 0.0388661
$$663$$ 15.0000 0.582552
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 27.0000 1.04544
$$668$$ 12.0000 0.464294
$$669$$ 26.0000 1.00522
$$670$$ 0 0
$$671$$ 60.0000 2.31627
$$672$$ 1.00000 0.0385758
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ 4.00000 0.154074
$$675$$ 25.0000 0.962250
$$676$$ 12.0000 0.461538
$$677$$ −33.0000 −1.26829 −0.634147 0.773213i $$-0.718648\pi$$
−0.634147 + 0.773213i $$0.718648\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ −15.0000 −0.574801
$$682$$ −24.0000 −0.919007
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ −22.0000 −0.839352
$$688$$ 8.00000 0.304997
$$689$$ −15.0000 −0.571454
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −12.0000 −0.455842
$$694$$ −18.0000 −0.683271
$$695$$ 0 0
$$696$$ −9.00000 −0.341144
$$697$$ 0 0
$$698$$ 10.0000 0.378506
$$699$$ −6.00000 −0.226941
$$700$$ 5.00000 0.188982
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 25.0000 0.943564
$$703$$ 2.00000 0.0754314
$$704$$ −6.00000 −0.226134
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ −18.0000 −0.676960
$$708$$ 9.00000 0.338241
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 12.0000 0.449719
$$713$$ −12.0000 −0.449404
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −21.0000 −0.784259
$$718$$ −21.0000 −0.783713
$$719$$ 39.0000 1.45445 0.727227 0.686397i $$-0.240809\pi$$
0.727227 + 0.686397i $$0.240809\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ −1.00000 −0.0372161
$$723$$ 8.00000 0.297523
$$724$$ 2.00000 0.0743294
$$725$$ −45.0000 −1.67126
$$726$$ −25.0000 −0.927837
$$727$$ −37.0000 −1.37225 −0.686127 0.727482i $$-0.740691\pi$$
−0.686127 + 0.727482i $$0.740691\pi$$
$$728$$ 5.00000 0.185312
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ −10.0000 −0.369611
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ −30.0000 −1.10506
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ −3.00000 −0.110133
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −23.0000 −0.842090
$$747$$ 12.0000 0.439057
$$748$$ −18.0000 −0.658145
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ −45.0000 −1.63880
$$755$$ 0 0
$$756$$ 5.00000 0.181848
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 7.00000 0.254251
$$759$$ −18.0000 −0.653359
$$760$$ 0 0
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ −2.00000 −0.0724524
$$763$$ −11.0000 −0.398227
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ −18.0000 −0.650366
$$767$$ 45.0000 1.62486
$$768$$ 1.00000 0.0360844
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 14.0000 0.503871
$$773$$ 51.0000 1.83434 0.917171 0.398493i $$-0.130467\pi$$
0.917171 + 0.398493i $$0.130467\pi$$
$$774$$ 16.0000 0.575108
$$775$$ 20.0000 0.718421
$$776$$ 10.0000 0.358979
$$777$$ −2.00000 −0.0717496
$$778$$ −18.0000 −0.645331
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −9.00000 −0.321839
$$783$$ −45.0000 −1.60817
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −31.0000 −1.10503 −0.552515 0.833503i $$-0.686332\pi$$
−0.552515 + 0.833503i $$0.686332\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ −12.0000 −0.426401
$$793$$ −50.0000 −1.77555
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ 11.0000 0.389885
$$797$$ −39.0000 −1.38145 −0.690725 0.723117i $$-0.742709\pi$$
−0.690725 + 0.723117i $$0.742709\pi$$
$$798$$ 1.00000 0.0353996
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ 42.0000 1.48215
$$804$$ 5.00000 0.176336
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ −6.00000 −0.211210
$$808$$ −18.0000 −0.633238
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ 11.0000 0.386262 0.193131 0.981173i $$-0.438136\pi$$
0.193131 + 0.981173i $$0.438136\pi$$
$$812$$ −9.00000 −0.315838
$$813$$ 11.0000 0.385787
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 3.00000 0.105021
$$817$$ 8.00000 0.279885
$$818$$ −32.0000 −1.11885
$$819$$ 10.0000 0.349428
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 9.00000 0.313911
$$823$$ 41.0000 1.42917 0.714585 0.699549i $$-0.246616\pi$$
0.714585 + 0.699549i $$0.246616\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 30.0000 1.04447
$$826$$ 9.00000 0.313150
$$827$$ 33.0000 1.14752 0.573761 0.819023i $$-0.305484\pi$$
0.573761 + 0.819023i $$0.305484\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 11.0000 0.382046 0.191023 0.981586i $$-0.438820\pi$$
0.191023 + 0.981586i $$0.438820\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 5.00000 0.173344
$$833$$ −18.0000 −0.623663
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 20.0000 0.691301
$$838$$ 12.0000 0.414533
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ −17.0000 −0.585859
$$843$$ 0 0
$$844$$ 5.00000 0.172107
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ −3.00000 −0.103020
$$849$$ −22.0000 −0.755038
$$850$$ 15.0000 0.514496
$$851$$ 6.00000 0.205677
$$852$$ −6.00000 −0.205557
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 9.00000 0.307614
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 30.0000 1.02418
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −6.00000 −0.204361
$$863$$ −18.0000 −0.612727 −0.306364 0.951915i $$-0.599112\pi$$
−0.306364 + 0.951915i $$0.599112\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ −8.00000 −0.271694
$$868$$ 4.00000 0.135769
$$869$$ 60.0000 2.03536
$$870$$ 0 0
$$871$$ 25.0000 0.847093
$$872$$ −11.0000 −0.372507
$$873$$ 20.0000 0.676897
$$874$$ −3.00000 −0.101477
$$875$$ 0 0
$$876$$ −7.00000 −0.236508
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ 28.0000 0.944954
$$879$$ −21.0000 −0.708312
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ −12.0000 −0.404061
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 15.0000 0.504505
$$885$$ 0 0
$$886$$ 18.0000 0.604722
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 26.0000 0.870544
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 15.0000 0.500835
$$898$$ 18.0000 0.600668
$$899$$ −36.0000 −1.20067
$$900$$ 10.0000 0.333333
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ −15.0000 −0.497792
$$909$$ −36.0000 −1.19404
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 36.0000 1.19143
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ 15.0000 0.495074
$$919$$ −7.00000 −0.230909 −0.115454 0.993313i $$-0.536832\pi$$
−0.115454 + 0.993313i $$0.536832\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 12.0000 0.395199
$$923$$ −30.0000 −0.987462
$$924$$ 6.00000 0.197386
$$925$$ −10.0000 −0.328798
$$926$$ 4.00000 0.131448
$$927$$ −28.0000 −0.919641
$$928$$ −9.00000 −0.295439
$$929$$ 33.0000 1.08269 0.541347 0.840799i $$-0.317914\pi$$
0.541347 + 0.840799i $$0.317914\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ −6.00000 −0.196537
$$933$$ −21.0000 −0.687509
$$934$$ −18.0000 −0.588978
$$935$$ 0 0
$$936$$ 10.0000 0.326860
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ 5.00000 0.163256
$$939$$ −19.0000 −0.620042
$$940$$ 0 0
$$941$$ 21.0000 0.684580 0.342290 0.939594i $$-0.388797\pi$$
0.342290 + 0.939594i $$0.388797\pi$$
$$942$$ 22.0000 0.716799
$$943$$ 0 0
$$944$$ 9.00000 0.292925
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ −35.0000 −1.13615
$$950$$ 5.00000 0.162221
$$951$$ −9.00000 −0.291845
$$952$$ 3.00000 0.0972306
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −21.0000 −0.679189
$$957$$ −54.0000 −1.74557
$$958$$ −36.0000 −1.16311
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −10.0000 −0.322413
$$963$$ 18.0000 0.580042
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −25.0000 −0.803530
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 4.00000 0.128234
$$974$$ −2.00000 −0.0640841
$$975$$ −25.0000 −0.800641
$$976$$ −10.0000 −0.320092
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 72.0000 2.30113
$$980$$ 0 0
$$981$$ −22.0000 −0.702406
$$982$$ 36.0000 1.14881
$$983$$ −30.0000 −0.956851 −0.478426 0.878128i $$-0.658792\pi$$
−0.478426 + 0.878128i $$0.658792\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −27.0000 −0.859855
$$987$$ 0 0
$$988$$ 5.00000 0.159071
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −1.00000 −0.0317340
$$994$$ −6.00000 −0.190308
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 4.00000 0.126618
$$999$$ −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.a.1.1 1
3.2 odd 2 342.2.a.e.1.1 1
4.3 odd 2 304.2.a.c.1.1 1
5.2 odd 4 950.2.b.b.799.1 2
5.3 odd 4 950.2.b.b.799.2 2
5.4 even 2 950.2.a.d.1.1 1
7.6 odd 2 1862.2.a.b.1.1 1
8.3 odd 2 1216.2.a.m.1.1 1
8.5 even 2 1216.2.a.e.1.1 1
11.10 odd 2 4598.2.a.p.1.1 1
12.11 even 2 2736.2.a.n.1.1 1
13.12 even 2 6422.2.a.h.1.1 1
15.14 odd 2 8550.2.a.m.1.1 1
19.2 odd 18 722.2.e.e.99.1 6
19.3 odd 18 722.2.e.e.389.1 6
19.4 even 9 722.2.e.f.415.1 6
19.5 even 9 722.2.e.f.595.1 6
19.6 even 9 722.2.e.f.245.1 6
19.7 even 3 722.2.c.e.429.1 2
19.8 odd 6 722.2.c.c.653.1 2
19.9 even 9 722.2.e.f.423.1 6
19.10 odd 18 722.2.e.e.423.1 6
19.11 even 3 722.2.c.e.653.1 2
19.12 odd 6 722.2.c.c.429.1 2
19.13 odd 18 722.2.e.e.245.1 6
19.14 odd 18 722.2.e.e.595.1 6
19.15 odd 18 722.2.e.e.415.1 6
19.16 even 9 722.2.e.f.389.1 6
19.17 even 9 722.2.e.f.99.1 6
19.18 odd 2 722.2.a.e.1.1 1
20.19 odd 2 7600.2.a.n.1.1 1
57.56 even 2 6498.2.a.f.1.1 1
76.75 even 2 5776.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 1 1.1 even 1 trivial
304.2.a.c.1.1 1 4.3 odd 2
342.2.a.e.1.1 1 3.2 odd 2
722.2.a.e.1.1 1 19.18 odd 2
722.2.c.c.429.1 2 19.12 odd 6
722.2.c.c.653.1 2 19.8 odd 6
722.2.c.e.429.1 2 19.7 even 3
722.2.c.e.653.1 2 19.11 even 3
722.2.e.e.99.1 6 19.2 odd 18
722.2.e.e.245.1 6 19.13 odd 18
722.2.e.e.389.1 6 19.3 odd 18
722.2.e.e.415.1 6 19.15 odd 18
722.2.e.e.423.1 6 19.10 odd 18
722.2.e.e.595.1 6 19.14 odd 18
722.2.e.f.99.1 6 19.17 even 9
722.2.e.f.245.1 6 19.6 even 9
722.2.e.f.389.1 6 19.16 even 9
722.2.e.f.415.1 6 19.4 even 9
722.2.e.f.423.1 6 19.9 even 9
722.2.e.f.595.1 6 19.5 even 9
950.2.a.d.1.1 1 5.4 even 2
950.2.b.b.799.1 2 5.2 odd 4
950.2.b.b.799.2 2 5.3 odd 4
1216.2.a.e.1.1 1 8.5 even 2
1216.2.a.m.1.1 1 8.3 odd 2
1862.2.a.b.1.1 1 7.6 odd 2
2736.2.a.n.1.1 1 12.11 even 2
4598.2.a.p.1.1 1 11.10 odd 2
5776.2.a.m.1.1 1 76.75 even 2
6422.2.a.h.1.1 1 13.12 even 2
6498.2.a.f.1.1 1 57.56 even 2
7600.2.a.n.1.1 1 20.19 odd 2
8550.2.a.m.1.1 1 15.14 odd 2