# Properties

 Label 1470.2.a.p.1.1 Level $1470$ Weight $2$ Character 1470.1 Self dual yes Analytic conductor $11.738$ 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] = [1470,2,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.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: $$11.7380090971$$ 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 $$\chi$$ $$=$$ 1470.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -4.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} +8.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -16.0000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +8.00000 q^{92} +2.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −1.00000 −0.149071
$$46$$ 8.00000 1.17954
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 4.00000 0.560112
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 2.00000 0.246183
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 2.00000 0.213201
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 2.00000 0.207390
$$94$$ −10.0000 −1.03142
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 1.00000 0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 20.0000 1.97066 0.985329 0.170664i $$-0.0545913\pi$$
0.985329 + 0.170664i $$0.0545913\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 8.00000 0.759326
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ −7.00000 −0.636364
$$122$$ 10.0000 0.905357
$$123$$ 2.00000 0.180334
$$124$$ 2.00000 0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ 2.00000 0.175412
$$131$$ −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
$$134$$ 2.00000 0.172774
$$135$$ −1.00000 −0.0860663
$$136$$ 4.00000 0.342997
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ −12.0000 −1.00702
$$143$$ −4.00000 −0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ −16.0000 −1.31077 −0.655386 0.755295i $$-0.727494\pi$$
−0.655386 + 0.755295i $$0.727494\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −2.00000 −0.160644
$$156$$ −2.00000 −0.160128
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 16.0000 1.27289
$$159$$ −2.00000 −0.158610
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 2.00000 0.156174
$$165$$ −2.00000 −0.155700
$$166$$ −16.0000 −1.24184
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ −4.00000 −0.300658
$$178$$ −14.0000 −1.04934
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 8.00000 0.589768
$$185$$ −8.00000 −0.588172
$$186$$ 2.00000 0.146647
$$187$$ 8.00000 0.585018
$$188$$ −10.0000 −0.729325
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 2.00000 0.142134
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 2.00000 0.141069
$$202$$ 14.0000 0.985037
$$203$$ 0 0
$$204$$ 4.00000 0.280056
$$205$$ −2.00000 −0.139686
$$206$$ 20.0000 1.39347
$$207$$ 8.00000 0.556038
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ −12.0000 −0.822226
$$214$$ 12.0000 0.820303
$$215$$ 2.00000 0.136399
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ −10.0000 −0.675737
$$220$$ −2.00000 −0.134840
$$221$$ −8.00000 −0.538138
$$222$$ 8.00000 0.536925
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −14.0000 −0.931266
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 10.0000 0.652328
$$236$$ −4.00000 −0.260378
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 2.00000 0.127000
$$249$$ −16.0000 −1.01396
$$250$$ −1.00000 −0.0632456
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −12.0000 −0.752947
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ −2.00000 −0.124515
$$259$$ 0 0
$$260$$ 2.00000 0.124035
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ −14.0000 −0.856786
$$268$$ 2.00000 0.122169
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 2.00000 0.120605
$$276$$ 8.00000 0.481543
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ −10.0000 −0.595491
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ −10.0000 −0.585206
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 8.00000 0.464991
$$297$$ 2.00000 0.116052
$$298$$ −16.0000 −0.926855
$$299$$ −16.0000 −0.925304
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ −10.0000 −0.572598
$$306$$ 4.00000 0.228665
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 20.0000 1.13776
$$310$$ −2.00000 −0.113592
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ −10.0000 −0.553849
$$327$$ −2.00000 −0.110600
$$328$$ 2.00000 0.110432
$$329$$ 0 0
$$330$$ −2.00000 −0.110096
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 8.00000 0.438397
$$334$$ −18.0000 −0.984916
$$335$$ −2.00000 −0.109272
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −14.0000 −0.760376
$$340$$ −4.00000 −0.216930
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ −8.00000 −0.430706
$$346$$ 6.00000 0.322562
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 2.00000 0.106600
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 12.0000 0.636894
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ −2.00000 −0.105703
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −19.0000 −1.00000
$$362$$ 22.0000 1.15629
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ 10.0000 0.522708
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 2.00000 0.104116
$$370$$ −8.00000 −0.415900
$$371$$ 0 0
$$372$$ 2.00000 0.103695
$$373$$ −36.0000 −1.86401 −0.932005 0.362446i $$-0.881942\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 8.00000 0.413670
$$375$$ −1.00000 −0.0516398
$$376$$ −10.0000 −0.515711
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ −2.00000 −0.101666
$$388$$ −6.00000 −0.304604
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 32.0000 1.61831
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 18.0000 0.906827
$$395$$ −16.0000 −0.805047
$$396$$ 2.00000 0.100504
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 2.00000 0.0997509
$$403$$ −4.00000 −0.199254
$$404$$ 14.0000 0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 16.0000 0.793091
$$408$$ 4.00000 0.198030
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ −2.00000 −0.0986527
$$412$$ 20.0000 0.985329
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 16.0000 0.785409
$$416$$ −2.00000 −0.0980581
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ −10.0000 −0.486217
$$424$$ −2.00000 −0.0971286
$$425$$ 4.00000 0.194029
$$426$$ −12.0000 −0.581402
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −4.00000 −0.193122
$$430$$ 2.00000 0.0964486
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −38.0000 −1.82616 −0.913082 0.407777i $$-0.866304\pi$$
−0.913082 + 0.407777i $$0.866304\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ −10.0000 −0.477818
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 14.0000 0.663664
$$446$$ −16.0000 −0.757622
$$447$$ −16.0000 −0.756774
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 4.00000 0.188353
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −42.0000 −1.96468 −0.982339 0.187112i $$-0.940087\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 4.00000 0.186704
$$460$$ −8.00000 −0.373002
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ −2.00000 −0.0927478
$$466$$ −14.0000 −0.648537
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ 10.0000 0.461266
$$471$$ 10.0000 0.460776
$$472$$ −4.00000 −0.184115
$$473$$ −4.00000 −0.183920
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ −8.00000 −0.365911
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −16.0000 −0.729537
$$482$$ 20.0000 0.910975
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 6.00000 0.272446
$$486$$ 1.00000 0.0453609
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −10.0000 −0.452216
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ −16.0000 −0.716977
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −18.0000 −0.804181
$$502$$ 20.0000 0.892644
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 16.0000 0.711287
$$507$$ −9.00000 −0.399704
$$508$$ −12.0000 −0.532414
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ −20.0000 −0.881305
$$516$$ −2.00000 −0.0880451
$$517$$ −20.0000 −0.879599
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 2.00000 0.0877058
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 2.00000 0.0870388
$$529$$ 41.0000 1.78261
$$530$$ 2.00000 0.0868744
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ −4.00000 −0.173259
$$534$$ −14.0000 −0.605839
$$535$$ −12.0000 −0.518805
$$536$$ 2.00000 0.0863868
$$537$$ −2.00000 −0.0863064
$$538$$ 10.0000 0.431131
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 14.0000 0.601351
$$543$$ 22.0000 0.944110
$$544$$ 4.00000 0.171499
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −14.0000 −0.598597 −0.299298 0.954160i $$-0.596753\pi$$
−0.299298 + 0.954160i $$0.596753\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 10.0000 0.426790
$$550$$ 2.00000 0.0852803
$$551$$ 0 0
$$552$$ 8.00000 0.340503
$$553$$ 0 0
$$554$$ −28.0000 −1.18961
$$555$$ −8.00000 −0.339581
$$556$$ 4.00000 0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 14.0000 0.590554
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ −10.0000 −0.421076
$$565$$ 14.0000 0.588984
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −18.0000 −0.748054
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −6.00000 −0.248708
$$583$$ −4.00000 −0.165663
$$584$$ −10.0000 −0.413803
$$585$$ 2.00000 0.0826898
$$586$$ −30.0000 −1.23929
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ 18.0000 0.740421
$$592$$ 8.00000 0.328798
$$593$$ 20.0000 0.821302 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −16.0000 −0.655386
$$597$$ −10.0000 −0.409273
$$598$$ −16.0000 −0.654289
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ 2.00000 0.0814463
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 14.0000 0.568711
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ 20.0000 0.809113
$$612$$ 4.00000 0.161690
$$613$$ −8.00000 −0.323117 −0.161558 0.986863i $$-0.551652\pi$$
−0.161558 + 0.986863i $$0.551652\pi$$
$$614$$ 20.0000 0.807134
$$615$$ −2.00000 −0.0806478
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 20.0000 0.804518
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 8.00000 0.321029
$$622$$ 20.0000 0.801927
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 32.0000 1.27592
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 16.0000 0.636446
$$633$$ −4.00000 −0.158986
$$634$$ 6.00000 0.238290
$$635$$ 12.0000 0.476205
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ −1.00000 −0.0395285
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ 2.00000 0.0787499
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −8.00000 −0.314027
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ 2.00000 0.0782660 0.0391330 0.999234i $$-0.487540\pi$$
0.0391330 + 0.999234i $$0.487540\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 12.0000 0.468879
$$656$$ 2.00000 0.0780869
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ −34.0000 −1.32445 −0.662226 0.749304i $$-0.730388\pi$$
−0.662226 + 0.749304i $$0.730388\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ −8.00000 −0.310694
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ −18.0000 −0.696441
$$669$$ −16.0000 −0.618596
$$670$$ −2.00000 −0.0772667
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 1.00000 0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 0 0
$$680$$ −4.00000 −0.153393
$$681$$ 12.0000 0.459841
$$682$$ 4.00000 0.153168
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ −2.00000 −0.0762493
$$689$$ 4.00000 0.152388
$$690$$ −8.00000 −0.304555
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 8.00000 0.303022
$$698$$ −10.0000 −0.378506
$$699$$ −14.0000 −0.529529
$$700$$ 0 0
$$701$$ 16.0000 0.604312 0.302156 0.953259i $$-0.402294\pi$$
0.302156 + 0.953259i $$0.402294\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 10.0000 0.376622
$$706$$ −24.0000 −0.903252
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 16.0000 0.600047
$$712$$ −14.0000 −0.524672
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −2.00000 −0.0747435
$$717$$ −8.00000 −0.298765
$$718$$ −20.0000 −0.746393
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ −19.0000 −0.707107
$$723$$ 20.0000 0.743808
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 10.0000 0.370117
$$731$$ −8.00000 −0.295891
$$732$$ 10.0000 0.369611
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 4.00000 0.147342
$$738$$ 2.00000 0.0736210
$$739$$ 48.0000 1.76571 0.882854 0.469647i $$-0.155619\pi$$
0.882854 + 0.469647i $$0.155619\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 16.0000 0.586195
$$746$$ −36.0000 −1.31805
$$747$$ −16.0000 −0.585409
$$748$$ 8.00000 0.292509
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −10.0000 −0.364662
$$753$$ 20.0000 0.728841
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −20.0000 −0.726912 −0.363456 0.931611i $$-0.618403\pi$$
−0.363456 + 0.931611i $$0.618403\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −4.00000 −0.144620
$$766$$ −14.0000 −0.505841
$$767$$ 8.00000 0.288863
$$768$$ 1.00000 0.0360844
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ −18.0000 −0.647834
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 2.00000 0.0718421
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ −24.0000 −0.860442
$$779$$ 0 0
$$780$$ 2.00000 0.0716115
$$781$$ −24.0000 −0.858788
$$782$$ 32.0000 1.14432
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −10.0000 −0.356915
$$786$$ −12.0000 −0.428026
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ −16.0000 −0.569254
$$791$$ 0 0
$$792$$ 2.00000 0.0710669
$$793$$ −20.0000 −0.710221
$$794$$ −14.0000 −0.496841
$$795$$ 2.00000 0.0709327
$$796$$ −10.0000 −0.354441
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ 1.00000 0.0353553
$$801$$ −14.0000 −0.494666
$$802$$ −14.0000 −0.494357
$$803$$ −20.0000 −0.705785
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 10.0000 0.352017
$$808$$ 14.0000 0.492518
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 14.0000 0.491001
$$814$$ 16.0000 0.560800
$$815$$ 10.0000 0.350285
$$816$$ 4.00000 0.140028
$$817$$ 0 0
$$818$$ 32.0000 1.11885
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 20.0000 0.696733
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ 8.00000 0.278187 0.139094 0.990279i $$-0.455581\pi$$
0.139094 + 0.990279i $$0.455581\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 16.0000 0.555368
$$831$$ −28.0000 −0.971309
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ 4.00000 0.138509
$$835$$ 18.0000 0.622916
$$836$$ 0 0
$$837$$ 2.00000 0.0691301
$$838$$ −36.0000 −1.24360
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 38.0000 1.30957
$$843$$ 14.0000 0.482186
$$844$$ −4.00000 −0.137686
$$845$$ 9.00000 0.309609
$$846$$ −10.0000 −0.343807
$$847$$ 0 0
$$848$$ −2.00000 −0.0686803
$$849$$ −4.00000 −0.137280
$$850$$ 4.00000 0.137199
$$851$$ 64.0000 2.19389
$$852$$ −12.0000 −0.411113
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −6.00000 −0.204006
$$866$$ −38.0000 −1.29129
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ −2.00000 −0.0677285
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 26.0000 0.877457
$$879$$ −30.0000 −1.01187
$$880$$ −2.00000 −0.0674200
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ 34.0000 1.14419 0.572096 0.820187i $$-0.306131\pi$$
0.572096 + 0.820187i $$0.306131\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 4.00000 0.134459
$$886$$ 28.0000 0.940678
$$887$$ 2.00000 0.0671534 0.0335767 0.999436i $$-0.489310\pi$$
0.0335767 + 0.999436i $$0.489310\pi$$
$$888$$ 8.00000 0.268462
$$889$$ 0 0
$$890$$ 14.0000 0.469281
$$891$$ 2.00000 0.0670025
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ −16.0000 −0.535120
$$895$$ 2.00000 0.0668526
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 6.00000 0.200223
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ −8.00000 −0.266519
$$902$$ 4.00000 0.133185
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ −22.0000 −0.731305
$$906$$ 0 0
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −32.0000 −1.05905
$$914$$ −42.0000 −1.38924
$$915$$ −10.0000 −0.330590
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 4.00000 0.132020
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 20.0000 0.659022
$$922$$ 18.0000 0.592798
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 16.0000 0.525793
$$927$$ 20.0000 0.656886
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ −2.00000 −0.0655826
$$931$$ 0 0
$$932$$ −14.0000 −0.458585
$$933$$ 20.0000 0.654771
$$934$$ −8.00000 −0.261768
$$935$$ −8.00000 −0.261628
$$936$$ −2.00000 −0.0653720
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ 10.0000 0.326164
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ 10.0000 0.325818
$$943$$ 16.0000 0.521032
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ −58.0000 −1.87880 −0.939402 0.342817i $$-0.888619\pi$$
−0.939402 + 0.342817i $$0.888619\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ −4.00000 −0.129234
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ −27.0000 −0.870968
$$962$$ −16.0000 −0.515861
$$963$$ 12.0000 0.386695
$$964$$ 20.0000 0.644157
$$965$$ 18.0000 0.579441
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 6.00000 0.192648
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 28.0000 0.897178
$$975$$ −2.00000 −0.0640513
$$976$$ 10.0000 0.320092
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ −10.0000 −0.319765
$$979$$ −28.0000 −0.894884
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 6.00000 0.191468
$$983$$ 46.0000 1.46717 0.733586 0.679597i $$-0.237845\pi$$
0.733586 + 0.679597i $$0.237845\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ −2.00000 −0.0635642
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 10.0000 0.317021
$$996$$ −16.0000 −0.506979
$$997$$ −6.00000 −0.190022 −0.0950110 0.995476i $$-0.530289\pi$$
−0.0950110 + 0.995476i $$0.530289\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 8.00000 0.253109
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 1470.2.a.p.1.1 yes 1
3.2 odd 2 4410.2.a.n.1.1 1
5.4 even 2 7350.2.a.o.1.1 1
7.2 even 3 1470.2.i.c.361.1 2
7.3 odd 6 1470.2.i.g.961.1 2
7.4 even 3 1470.2.i.c.961.1 2
7.5 odd 6 1470.2.i.g.361.1 2
7.6 odd 2 1470.2.a.n.1.1 1
21.20 even 2 4410.2.a.e.1.1 1
35.34 odd 2 7350.2.a.bh.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1470.2.a.n.1.1 1 7.6 odd 2
1470.2.a.p.1.1 yes 1 1.1 even 1 trivial
1470.2.i.c.361.1 2 7.2 even 3
1470.2.i.c.961.1 2 7.4 even 3
1470.2.i.g.361.1 2 7.5 odd 6
1470.2.i.g.961.1 2 7.3 odd 6
4410.2.a.e.1.1 1 21.20 even 2
4410.2.a.n.1.1 1 3.2 odd 2
7350.2.a.o.1.1 1 5.4 even 2
7350.2.a.bh.1.1 1 35.34 odd 2