# Properties

 Label 4560.2.a.r.1.1 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} +2.00000 q^{35} -4.00000 q^{37} -4.00000 q^{41} +6.00000 q^{43} -1.00000 q^{45} +12.0000 q^{47} -3.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -6.00000 q^{55} +1.00000 q^{57} +4.00000 q^{59} +2.00000 q^{61} -2.00000 q^{63} +8.00000 q^{67} -4.00000 q^{69} +6.00000 q^{73} +1.00000 q^{75} -12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -2.00000 q^{85} -8.00000 q^{87} -4.00000 q^{89} +8.00000 q^{93} -1.00000 q^{95} +12.0000 q^{97} +6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −6.00000 −0.809040
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ −8.00000 −0.857690
$$88$$ 0 0
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ −4.00000 −0.360668
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −22.0000 −1.92215 −0.961074 0.276289i $$-0.910895\pi$$
−0.961074 + 0.276289i $$0.910895\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ 0 0
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ −6.00000 −0.467099
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 22.0000 1.59186 0.795932 0.605386i $$-0.206981\pi$$
0.795932 + 0.605386i $$0.206981\pi$$
$$192$$ 0 0
$$193$$ −8.00000 −0.575853 −0.287926 0.957653i $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 16.0000 1.12298
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −6.00000 −0.409197
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 14.0000 0.883672 0.441836 0.897096i $$-0.354327\pi$$
0.441836 + 0.897096i $$0.354327\pi$$
$$252$$ 0 0
$$253$$ −24.0000 −1.50887
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −8.00000 −0.495188
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ −4.00000 −0.244796
$$268$$ 0 0
$$269$$ 8.00000 0.487769 0.243884 0.969804i $$-0.421578\pi$$
0.243884 + 0.969804i $$0.421578\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ 0 0
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 6.00000 0.348155
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ −10.0000 −0.574485
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ 2.00000 0.112687
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ −48.0000 −2.68748
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −14.0000 −0.774202
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 0 0
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 48.0000 2.59935
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ 0 0
$$347$$ −36.0000 −1.93258 −0.966291 0.257454i $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 0 0
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 25.0000 1.31216
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −24.0000 −1.24267 −0.621336 0.783544i $$-0.713410\pi$$
−0.621336 + 0.783544i $$0.713410\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 0 0
$$387$$ 6.00000 0.304997
$$388$$ 0 0
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ −22.0000 −1.10975
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 4.00000 0.189618
$$446$$ 0 0
$$447$$ 22.0000 1.04056
$$448$$ 0 0
$$449$$ −4.00000 −0.188772 −0.0943858 0.995536i $$-0.530089\pi$$
−0.0943858 + 0.995536i $$0.530089\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −14.0000 −0.654892 −0.327446 0.944870i $$-0.606188\pi$$
−0.327446 + 0.944870i $$0.606188\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ 26.0000 1.20832 0.604161 0.796862i $$-0.293508\pi$$
0.604161 + 0.796862i $$0.293508\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 36.0000 1.65528
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 8.00000 0.364013
$$484$$ 0 0
$$485$$ −12.0000 −0.544892
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ 34.0000 1.53440 0.767199 0.641409i $$-0.221650\pi$$
0.767199 + 0.641409i $$0.221650\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ 0 0
$$495$$ −6.00000 −0.269680
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 0 0
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 0 0
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 72.0000 3.16656
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 24.0000 1.05146 0.525730 0.850652i $$-0.323792\pi$$
0.525730 + 0.850652i $$0.323792\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ −2.00000 −0.0872872
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ 2.00000 0.0858282
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ −40.0000 −1.71028 −0.855138 0.518400i $$-0.826528\pi$$
−0.855138 + 0.518400i $$0.826528\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 0 0
$$555$$ 4.00000 0.169791
$$556$$ 0 0
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −14.0000 −0.588984
$$566$$ 0 0
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ −20.0000 −0.838444 −0.419222 0.907884i $$-0.637697\pi$$
−0.419222 + 0.907884i $$0.637697\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ 22.0000 0.919063
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 0 0
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 0 0
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ 0 0
$$597$$ −20.0000 −0.818546
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ −25.0000 −1.01639
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ 16.0000 0.648353
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 0 0
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 6.00000 0.239617
$$628$$ 0 0
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 28.0000 1.11290
$$634$$ 0 0
$$635$$ −12.0000 −0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ −6.00000 −0.236250
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ 0 0
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ 22.0000 0.859611
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ 32.0000 1.23904
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −28.0000 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ −12.0000 −0.455842
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 0 0
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 20.0000 0.752177
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 18.0000 0.672222
$$718$$ 0 0
$$719$$ −14.0000 −0.522112 −0.261056 0.965324i $$-0.584071\pi$$
−0.261056 + 0.965324i $$0.584071\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −22.0000 −0.818189
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 14.0000 0.510188
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 0 0
$$759$$ −24.0000 −0.871145
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ 28.0000 1.01367
$$764$$ 0 0
$$765$$ −2.00000 −0.0723102
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ 8.00000 0.286998
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ −6.00000 −0.214149
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −28.0000 −0.995565
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ −4.00000 −0.141333
$$802$$ 0 0
$$803$$ 36.0000 1.27041
$$804$$ 0 0
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ 8.00000 0.281613
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ −20.0000 −0.701431
$$814$$ 0 0
$$815$$ 6.00000 0.210171
$$816$$ 0 0
$$817$$ 6.00000 0.209913
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 0 0
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ 0 0
$$825$$ 6.00000 0.208893
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 0 0
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 16.0000 0.551069
$$844$$ 0 0
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ −50.0000 −1.71802
$$848$$ 0 0
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 0 0
$$879$$ 10.0000 0.337292
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ −14.0000 −0.471138 −0.235569 0.971858i $$-0.575695\pi$$
−0.235569 + 0.971858i $$0.575695\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 0 0
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −64.0000 −2.13452
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 0 0
$$913$$ −24.0000 −0.794284
$$914$$ 0 0
$$915$$ −2.00000 −0.0661180
$$916$$ 0 0
$$917$$ 44.0000 1.45301
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 8.00000 0.263609
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 0 0
$$933$$ 6.00000 0.196431
$$934$$ 0 0
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ −34.0000 −1.11073 −0.555366 0.831606i $$-0.687422\pi$$
−0.555366 + 0.831606i $$0.687422\pi$$
$$938$$ 0 0
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ −40.0000 −1.30396 −0.651981 0.758235i $$-0.726062\pi$$
−0.651981 + 0.758235i $$0.726062\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ −44.0000 −1.42981 −0.714904 0.699223i $$-0.753530\pi$$
−0.714904 + 0.699223i $$0.753530\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 0 0
$$955$$ −22.0000 −0.711903
$$956$$ 0 0
$$957$$ −48.0000 −1.55162
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ 8.00000 0.257529
$$966$$ 0 0
$$967$$ 18.0000 0.578841 0.289420 0.957202i $$-0.406537\pi$$
0.289420 + 0.957202i $$0.406537\pi$$
$$968$$ 0 0
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ −14.0000 −0.446077
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 20.0000 0.634043
$$996$$ 0 0
$$997$$ −58.0000 −1.83688 −0.918439 0.395562i $$-0.870550\pi$$
−0.918439 + 0.395562i $$0.870550\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 4560.2.a.r.1.1 1
4.3 odd 2 570.2.a.b.1.1 1
12.11 even 2 1710.2.a.s.1.1 1
20.3 even 4 2850.2.d.j.799.2 2
20.7 even 4 2850.2.d.j.799.1 2
20.19 odd 2 2850.2.a.y.1.1 1
60.59 even 2 8550.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.b.1.1 1 4.3 odd 2
1710.2.a.s.1.1 1 12.11 even 2
2850.2.a.y.1.1 1 20.19 odd 2
2850.2.d.j.799.1 2 20.7 even 4
2850.2.d.j.799.2 2 20.3 even 4
4560.2.a.r.1.1 1 1.1 even 1 trivial
8550.2.a.g.1.1 1 60.59 even 2