# Properties

 Label 4410.2.a.e.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1470) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{20} +2.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +8.00000 q^{37} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} -10.0000 q^{47} -1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} -4.00000 q^{59} -10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} +12.0000 q^{71} +10.0000 q^{73} -8.00000 q^{74} +16.0000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} -4.00000 q^{85} +2.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -8.00000 q^{92} +10.0000 q^{94} +6.00000 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 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 $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 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
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 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 $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −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 $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −8.00000 −0.834058
$$93$$ 0 0
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −20.0000 −1.97066 −0.985329 0.170664i $$-0.945409\pi$$
−0.985329 + 0.170664i $$0.945409\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ −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
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2.00000 0.160644
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 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
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 8.00000 0.589768
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ −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 $$0.5\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 20.0000 1.39347
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −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 $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 10.0000 0.652328
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 2.00000 0.127000
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 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 $$0.5\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.0000 −0.835170 −0.417585 0.908638i $$-0.637123\pi$$
−0.417585 + 0.908638i $$0.637123\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ −16.0000 −0.926855
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2.00000 −0.113592
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 10.0000 0.553849
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ −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 $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 22.0000 1.15629
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 0 0
$$370$$ 8.00000 0.415900
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −36.0000 −1.86401 −0.932005 0.362446i $$-0.881942\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 0 0
$$388$$ 6.00000 0.304604
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −32.0000 −1.61831
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −16.0000 −0.793091
$$408$$ 0 0
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ −20.0000 −0.985329
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 16.0000 0.785409
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 38.0000 1.82616 0.913082 0.407777i $$-0.133696\pi$$
0.913082 + 0.407777i $$0.133696\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ −28.0000 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$444$$ 0 0
$$445$$ 14.0000 0.663664
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −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
$$459$$ 0 0
$$460$$ 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
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −10.0000 −0.461266
$$471$$ 0 0
$$472$$ 4.00000 0.184115
$$473$$ 4.00000 0.183920
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8.00000 −0.365911
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 20.0000 0.910975
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ −6.00000 −0.272446
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −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
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ 20.0000 0.881305
$$516$$ 0 0
$$517$$ 20.0000 0.879599
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 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 $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 2.00000 0.0868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ −2.00000 −0.0863868
$$537$$ 0 0
$$538$$ −10.0000 −0.431131
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 14.0000 0.601351
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ 2.00000 0.0852803
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.0000 0.590554
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 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
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 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
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ 20.0000 0.821302 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 16.0000 0.655386
$$597$$ 0 0
$$598$$ 16.0000 0.654289
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ 0 0
$$601$$ 4.00000 0.163163 0.0815817 0.996667i $$-0.474003\pi$$
0.0815817 + 0.996667i $$0.474003\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ −20.0000 −0.809113
$$612$$ 0 0
$$613$$ −8.00000 −0.323117 −0.161558 0.986863i $$-0.551652\pi$$
−0.161558 + 0.986863i $$0.551652\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ 0 0
$$622$$ −20.0000 −0.801927
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 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
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 0 0
$$649$$ 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 $$-0.512460\pi$$
−0.0391330 + 0.999234i $$0.512460\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 34.0000 1.32445 0.662226 0.749304i $$-0.269612\pi$$
0.662226 + 0.749304i $$0.269612\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −18.0000 −0.696441
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2.00000 −0.0762493
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 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
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −16.0000 −0.604312 −0.302156 0.953259i $$-0.597706\pi$$
−0.302156 + 0.953259i $$0.597706\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ 14.0000 0.524672
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 2.00000 0.0747435
$$717$$ 0 0
$$718$$ −20.0000 −0.746393
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 10.0000 0.370117
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 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 $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ 0 0
$$745$$ −16.0000 −0.586195
$$746$$ 36.0000 1.31805
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −10.0000 −0.364662
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 14.0000 0.505841
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 16.0000 0.576975 0.288487 0.957484i $$-0.406848\pi$$
0.288487 + 0.957484i $$0.406848\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −18.0000 −0.647834
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 0 0
$$775$$ −2.00000 −0.0718421
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ −24.0000 −0.860442
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ 32.0000 1.14432
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ 16.0000 0.569254
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ −14.0000 −0.494357
$$803$$ −20.0000 −0.705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ −14.0000 −0.492518
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 16.0000 0.560800
$$815$$ 10.0000 0.350285
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 32.0000 1.11885
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 20.0000 0.696733
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 0 0
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 18.0000 0.622916
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ −64.0000 −2.19389
$$852$$ 0 0
$$853$$ 38.0000 1.30110 0.650548 0.759465i $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ 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 $$0.434295\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ −38.0000 −1.29129
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 26.0000 0.877457
$$879$$ 0 0
$$880$$ 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
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ 2.00000 0.0671534 0.0335767 0.999436i $$-0.489310\pi$$
0.0335767 + 0.999436i $$0.489310\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −14.0000 −0.469281
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −2.00000 −0.0668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 0 0
$$913$$ 32.0000 1.05905
$$914$$ 42.0000 1.38924
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 0 0
$$922$$ −18.0000 −0.592798
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.0000 0.458585
$$933$$ 0 0
$$934$$ 8.00000 0.261768
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 10.0000 0.326164
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ 0 0
$$943$$ −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 $$0.5\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 58.0000 1.87880 0.939402 0.342817i $$-0.111381\pi$$
0.939402 + 0.342817i $$0.111381\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ 4.00000 0.129234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −16.0000 −0.515861
$$963$$ 0 0
$$964$$ −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 $$0.396010\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 28.0000 0.894884
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 6.00000 0.191468
$$983$$ 46.0000 1.46717 0.733586 0.679597i $$-0.237845\pi$$
0.733586 + 0.679597i $$0.237845\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −10.0000 −0.317021
$$996$$ 0 0
$$997$$ 6.00000 0.190022 0.0950110 0.995476i $$-0.469711\pi$$
0.0950110 + 0.995476i $$0.469711\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ 0 0
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 4410.2.a.e.1.1 1
3.2 odd 2 1470.2.a.n.1.1 1
7.6 odd 2 4410.2.a.n.1.1 1
15.14 odd 2 7350.2.a.bh.1.1 1
21.2 odd 6 1470.2.i.g.361.1 2
21.5 even 6 1470.2.i.c.361.1 2
21.11 odd 6 1470.2.i.g.961.1 2
21.17 even 6 1470.2.i.c.961.1 2
21.20 even 2 1470.2.a.p.1.1 yes 1
105.104 even 2 7350.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1470.2.a.n.1.1 1 3.2 odd 2
1470.2.a.p.1.1 yes 1 21.20 even 2
1470.2.i.c.361.1 2 21.5 even 6
1470.2.i.c.961.1 2 21.17 even 6
1470.2.i.g.361.1 2 21.2 odd 6
1470.2.i.g.961.1 2 21.11 odd 6
4410.2.a.e.1.1 1 1.1 even 1 trivial
4410.2.a.n.1.1 1 7.6 odd 2
7350.2.a.o.1.1 1 105.104 even 2
7350.2.a.bh.1.1 1 15.14 odd 2