# Properties

 Label 4800.2.a.b.1.1 Level $4800$ Weight $2$ Character 4800.1 Self dual yes Analytic conductor $38.328$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -5.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} +3.00000 q^{13} -2.00000 q^{17} -1.00000 q^{19} +5.00000 q^{21} -2.00000 q^{23} -1.00000 q^{27} -6.00000 q^{29} +3.00000 q^{31} -6.00000 q^{33} +6.00000 q^{37} -3.00000 q^{39} +4.00000 q^{41} -11.0000 q^{43} -10.0000 q^{47} +18.0000 q^{49} +2.00000 q^{51} +8.00000 q^{53} +1.00000 q^{57} +6.00000 q^{59} -3.00000 q^{61} -5.00000 q^{63} +1.00000 q^{67} +2.00000 q^{69} -12.0000 q^{71} +10.0000 q^{73} -30.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +6.00000 q^{87} -16.0000 q^{89} -15.0000 q^{91} -3.00000 q^{93} -7.00000 q^{97} +6.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 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
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 0 0
$$63$$ −5.00000 −0.629941
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.00000 0.122169 0.0610847 0.998133i $$-0.480544\pi$$
0.0610847 + 0.998133i $$0.480544\pi$$
$$68$$ 0 0
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −30.0000 −3.41882
$$78$$ 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
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ −15.0000 −1.57243
$$92$$ 0 0
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.00000 0.277350
$$118$$ 0 0
$$119$$ 10.0000 0.916698
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ −4.00000 −0.360668
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 11.0000 0.968496
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 0 0
$$143$$ 18.0000 1.50524
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 9.00000 0.732410 0.366205 0.930534i $$-0.380657\pi$$
0.366205 + 0.930534i $$0.380657\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 10.0000 0.788110
$$162$$ 0 0
$$163$$ 7.00000 0.548282 0.274141 0.961689i $$-0.411606\pi$$
0.274141 + 0.961689i $$0.411606\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ 19.0000 1.41226 0.706129 0.708083i $$-0.250440\pi$$
0.706129 + 0.708083i $$0.250440\pi$$
$$182$$ 0 0
$$183$$ 3.00000 0.221766
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ 3.00000 0.215945 0.107972 0.994154i $$-0.465564\pi$$
0.107972 + 0.994154i $$0.465564\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ 0 0
$$201$$ −1.00000 −0.0705346
$$202$$ 0 0
$$203$$ 30.0000 2.10559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.00000 −0.139010
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ 0 0
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −15.0000 −1.01827
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ 3.00000 0.200895 0.100447 0.994942i $$-0.467973\pi$$
0.100447 + 0.994942i $$0.467973\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 3.00000 0.198246 0.0991228 0.995075i $$-0.468396\pi$$
0.0991228 + 0.995075i $$0.468396\pi$$
$$230$$ 0 0
$$231$$ 30.0000 1.97386
$$232$$ 0 0
$$233$$ 20.0000 1.31024 0.655122 0.755523i $$-0.272617\pi$$
0.655122 + 0.755523i $$0.272617\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.00000 −0.190885
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000 0.979184
$$268$$ 0 0
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 15.0000 0.907841
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −13.0000 −0.772770 −0.386385 0.922338i $$-0.626276\pi$$
−0.386385 + 0.922338i $$0.626276\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −6.00000 −0.348155
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 55.0000 3.17015
$$302$$ 0 0
$$303$$ −8.00000 −0.459588
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −13.0000 −0.741949 −0.370975 0.928643i $$-0.620976\pi$$
−0.370975 + 0.928643i $$0.620976\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ −29.0000 −1.63918 −0.819588 0.572953i $$-0.805798\pi$$
−0.819588 + 0.572953i $$0.805798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.0000 0.898650 0.449325 0.893368i $$-0.351665\pi$$
0.449325 + 0.893368i $$0.351665\pi$$
$$318$$ 0 0
$$319$$ −36.0000 −2.01561
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7.00000 −0.387101
$$328$$ 0 0
$$329$$ 50.0000 2.75659
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −23.0000 −1.25289 −0.626445 0.779466i $$-0.715491\pi$$
−0.626445 + 0.779466i $$0.715491\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ −55.0000 −2.96972
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −3.00000 −0.160128
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −10.0000 −0.529256
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ 0 0
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ −40.0000 −2.07670
$$372$$ 0 0
$$373$$ 25.0000 1.29445 0.647225 0.762299i $$-0.275929\pi$$
0.647225 + 0.762299i $$0.275929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −18.0000 −0.927047
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11.0000 −0.559161
$$388$$ 0 0
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ 16.0000 0.807093
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −11.0000 −0.552074 −0.276037 0.961147i $$-0.589021\pi$$
−0.276037 + 0.961147i $$0.589021\pi$$
$$398$$ 0 0
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 9.00000 0.448322
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 36.0000 1.78445
$$408$$ 0 0
$$409$$ −11.0000 −0.543915 −0.271957 0.962309i $$-0.587671\pi$$
−0.271957 + 0.962309i $$0.587671\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ −30.0000 −1.47620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 0 0
$$423$$ −10.0000 −0.486217
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.0000 0.725901
$$428$$ 0 0
$$429$$ −18.0000 −0.869048
$$430$$ 0 0
$$431$$ 22.0000 1.05970 0.529851 0.848091i $$-0.322248\pi$$
0.529851 + 0.848091i $$0.322248\pi$$
$$432$$ 0 0
$$433$$ 19.0000 0.913082 0.456541 0.889702i $$-0.349088\pi$$
0.456541 + 0.889702i $$0.349088\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.00000 0.0956730
$$438$$ 0 0
$$439$$ −5.00000 −0.238637 −0.119318 0.992856i $$-0.538071\pi$$
−0.119318 + 0.992856i $$0.538071\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 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
$$453$$ −9.00000 −0.422857
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 16.0000 0.745194 0.372597 0.927993i $$-0.378467\pi$$
0.372597 + 0.927993i $$0.378467\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.0000 0.647843 0.323921 0.946084i $$-0.394999\pi$$
0.323921 + 0.946084i $$0.394999\pi$$
$$468$$ 0 0
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ 7.00000 0.322543
$$472$$ 0 0
$$473$$ −66.0000 −3.03468
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 18.0000 0.820729
$$482$$ 0 0
$$483$$ −10.0000 −0.455016
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3.00000 −0.135943 −0.0679715 0.997687i $$-0.521653\pi$$
−0.0679715 + 0.997687i $$0.521653\pi$$
$$488$$ 0 0
$$489$$ −7.00000 −0.316551
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 60.0000 2.69137
$$498$$ 0 0
$$499$$ −17.0000 −0.761025 −0.380512 0.924776i $$-0.624252\pi$$
−0.380512 + 0.924776i $$0.624252\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 4.00000 0.177646
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −50.0000 −2.21187
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −60.0000 −2.63880
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −21.0000 −0.918266 −0.459133 0.888368i $$-0.651840\pi$$
−0.459133 + 0.888368i $$0.651840\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 2.00000 0.0863064
$$538$$ 0 0
$$539$$ 108.000 4.65189
$$540$$ 0 0
$$541$$ −17.0000 −0.730887 −0.365444 0.930834i $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ −19.0000 −0.815368
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 0 0
$$549$$ −3.00000 −0.128037
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 40.0000 1.70097
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ −33.0000 −1.39575
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −5.00000 −0.209980
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −1.00000 −0.0418487 −0.0209243 0.999781i $$-0.506661\pi$$
−0.0209243 + 0.999781i $$0.506661\pi$$
$$572$$ 0 0
$$573$$ 10.0000 0.417756
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 43.0000 1.79011 0.895057 0.445952i $$-0.147135\pi$$
0.895057 + 0.445952i $$0.147135\pi$$
$$578$$ 0 0
$$579$$ −3.00000 −0.124676
$$580$$ 0 0
$$581$$ −30.0000 −1.24461
$$582$$ 0 0
$$583$$ 48.0000 1.98796
$$584$$ 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
$$589$$ −3.00000 −0.123613
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ 4.00000 0.164260 0.0821302 0.996622i $$-0.473828\pi$$
0.0821302 + 0.996622i $$0.473828\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −21.0000 −0.859473
$$598$$ 0 0
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ 1.00000 0.0407231
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −32.0000 −1.28827 −0.644136 0.764911i $$-0.722783\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ 0 0
$$619$$ 19.0000 0.763674 0.381837 0.924230i $$-0.375291\pi$$
0.381837 + 0.924230i $$0.375291\pi$$
$$620$$ 0 0
$$621$$ 2.00000 0.0802572
$$622$$ 0 0
$$623$$ 80.0000 3.20513
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 6.00000 0.239617
$$628$$ 0 0
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 31.0000 1.23409 0.617045 0.786928i $$-0.288330\pi$$
0.617045 + 0.786928i $$0.288330\pi$$
$$632$$ 0 0
$$633$$ −15.0000 −0.596196
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 54.0000 2.13956
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 0 0
$$643$$ 12.0000 0.473234 0.236617 0.971603i $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 15.0000 0.587896
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ 6.00000 0.233021
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000 0.464642
$$668$$ 0 0
$$669$$ −3.00000 −0.115987
$$670$$ 0 0
$$671$$ −18.0000 −0.694882
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 35.0000 1.34318
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −3.00000 −0.114457
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 0 0
$$693$$ −30.0000 −1.13961
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 0 0
$$699$$ −20.0000 −0.756469
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 0 0
$$723$$ 7.00000 0.260333
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −13.0000 −0.482143 −0.241072 0.970507i $$-0.577499\pi$$
−0.241072 + 0.970507i $$0.577499\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 22.0000 0.813699
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.00000 0.221013
$$738$$ 0 0
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 3.00000 0.110208
$$742$$ 0 0
$$743$$ −12.0000 −0.440237 −0.220119 0.975473i $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ −40.0000 −1.46157
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ −20.0000 −0.728841
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.00000 0.109037 0.0545184 0.998513i $$-0.482638\pi$$
0.0545184 + 0.998513i $$0.482638\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −52.0000 −1.88500 −0.942499 0.334208i $$-0.891531\pi$$
−0.942499 + 0.334208i $$0.891531\pi$$
$$762$$ 0 0
$$763$$ −35.0000 −1.26709
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.0000 0.649942
$$768$$ 0 0
$$769$$ 27.0000 0.973645 0.486822 0.873501i $$-0.338156\pi$$
0.486822 + 0.873501i $$0.338156\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ −28.0000 −1.00709 −0.503545 0.863969i $$-0.667971\pi$$
−0.503545 + 0.863969i $$0.667971\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 30.0000 1.07624
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ −72.0000 −2.57636
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.00000 0.106938 0.0534692 0.998569i $$-0.482972\pi$$
0.0534692 + 0.998569i $$0.482972\pi$$
$$788$$ 0 0
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ −60.0000 −2.13335
$$792$$ 0 0
$$793$$ −9.00000 −0.319599
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −28.0000 −0.991811 −0.495905 0.868377i $$-0.665164\pi$$
−0.495905 + 0.868377i $$0.665164\pi$$
$$798$$ 0 0
$$799$$ 20.0000 0.707549
$$800$$ 0 0
$$801$$ −16.0000 −0.565332
$$802$$ 0 0
$$803$$ 60.0000 2.11735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ −24.0000 −0.843795 −0.421898 0.906644i $$-0.638636\pi$$
−0.421898 + 0.906644i $$0.638636\pi$$
$$810$$ 0 0
$$811$$ −9.00000 −0.316033 −0.158016 0.987436i $$-0.550510\pi$$
−0.158016 + 0.987436i $$0.550510\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 11.0000 0.384841
$$818$$ 0 0
$$819$$ −15.0000 −0.524142
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ −49.0000 −1.70803 −0.854016 0.520246i $$-0.825840\pi$$
−0.854016 + 0.520246i $$0.825840\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.0000 1.11275 0.556375 0.830932i $$-0.312192\pi$$
0.556375 + 0.830932i $$0.312192\pi$$
$$828$$ 0 0
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ 1.00000 0.0346896
$$832$$ 0 0
$$833$$ −36.0000 −1.24733
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −3.00000 −0.103695
$$838$$ 0 0
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −26.0000 −0.895488
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −125.000 −4.29505
$$848$$ 0 0
$$849$$ 13.0000 0.446159
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −55.0000 −1.88316 −0.941582 0.336784i $$-0.890661\pi$$
−0.941582 + 0.336784i $$0.890661\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −52.0000 −1.77629 −0.888143 0.459567i $$-0.848005\pi$$
−0.888143 + 0.459567i $$0.848005\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 20.0000 0.681598
$$862$$ 0 0
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ −7.00000 −0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17.0000 0.574049 0.287025 0.957923i $$-0.407334\pi$$
0.287025 + 0.957923i $$0.407334\pi$$
$$878$$ 0 0
$$879$$ −2.00000 −0.0674583
$$880$$ 0 0
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ 0 0
$$883$$ −19.0000 −0.639401 −0.319700 0.947519i $$-0.603582\pi$$
−0.319700 + 0.947519i $$0.603582\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 0 0
$$889$$ −40.0000 −1.34156
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 0 0
$$893$$ 10.0000 0.334637
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ −18.0000 −0.600334
$$900$$ 0 0
$$901$$ −16.0000 −0.533037
$$902$$ 0 0
$$903$$ −55.0000 −1.83029
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 0 0
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ −14.0000 −0.463841 −0.231920 0.972735i $$-0.574501\pi$$
−0.231920 + 0.972735i $$0.574501\pi$$
$$912$$ 0 0
$$913$$ 36.0000 1.19143
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 80.0000 2.64183
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ 13.0000 0.428365
$$922$$ 0 0
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 0 0
$$933$$ 14.0000 0.458339
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.0000 0.751377 0.375689 0.926746i $$-0.377406\pi$$
0.375689 + 0.926746i $$0.377406\pi$$
$$938$$ 0 0
$$939$$ 29.0000 0.946379
$$940$$ 0 0
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000 0.0649913 0.0324956 0.999472i $$-0.489654\pi$$
0.0324956 + 0.999472i $$0.489654\pi$$
$$948$$ 0 0
$$949$$ 30.0000 0.973841
$$950$$ 0 0
$$951$$ −16.0000 −0.518836
$$952$$ 0 0
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 36.0000 1.16371
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 8.00000 0.257796
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ 0 0
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −46.0000 −1.47621 −0.738105 0.674686i $$-0.764279\pi$$
−0.738105 + 0.674686i $$0.764279\pi$$
$$972$$ 0 0
$$973$$ −60.0000 −1.92351
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 10.0000 0.319928 0.159964 0.987123i $$-0.448862\pi$$
0.159964 + 0.987123i $$0.448862\pi$$
$$978$$ 0 0
$$979$$ −96.0000 −3.06817
$$980$$ 0 0
$$981$$ 7.00000 0.223493
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −50.0000 −1.59152
$$988$$ 0 0
$$989$$ 22.0000 0.699559
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 4800.2.a.b.1.1 1
4.3 odd 2 4800.2.a.cs.1.1 1
5.2 odd 4 4800.2.f.bj.3649.2 2
5.3 odd 4 4800.2.f.bj.3649.1 2
5.4 even 2 4800.2.a.ct.1.1 1
8.3 odd 2 1200.2.a.i.1.1 1
8.5 even 2 600.2.a.f.1.1 yes 1
20.3 even 4 4800.2.f.a.3649.2 2
20.7 even 4 4800.2.f.a.3649.1 2
20.19 odd 2 4800.2.a.a.1.1 1
24.5 odd 2 1800.2.a.a.1.1 1
24.11 even 2 3600.2.a.bq.1.1 1
40.3 even 4 1200.2.f.i.49.1 2
40.13 odd 4 600.2.f.a.49.2 2
40.19 odd 2 1200.2.a.j.1.1 1
40.27 even 4 1200.2.f.i.49.2 2
40.29 even 2 600.2.a.e.1.1 1
40.37 odd 4 600.2.f.a.49.1 2
120.29 odd 2 1800.2.a.x.1.1 1
120.53 even 4 1800.2.f.k.649.2 2
120.59 even 2 3600.2.a.a.1.1 1
120.77 even 4 1800.2.f.k.649.1 2
120.83 odd 4 3600.2.f.b.2449.1 2
120.107 odd 4 3600.2.f.b.2449.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.e.1.1 1 40.29 even 2
600.2.a.f.1.1 yes 1 8.5 even 2
600.2.f.a.49.1 2 40.37 odd 4
600.2.f.a.49.2 2 40.13 odd 4
1200.2.a.i.1.1 1 8.3 odd 2
1200.2.a.j.1.1 1 40.19 odd 2
1200.2.f.i.49.1 2 40.3 even 4
1200.2.f.i.49.2 2 40.27 even 4
1800.2.a.a.1.1 1 24.5 odd 2
1800.2.a.x.1.1 1 120.29 odd 2
1800.2.f.k.649.1 2 120.77 even 4
1800.2.f.k.649.2 2 120.53 even 4
3600.2.a.a.1.1 1 120.59 even 2
3600.2.a.bq.1.1 1 24.11 even 2
3600.2.f.b.2449.1 2 120.83 odd 4
3600.2.f.b.2449.2 2 120.107 odd 4
4800.2.a.a.1.1 1 20.19 odd 2
4800.2.a.b.1.1 1 1.1 even 1 trivial
4800.2.a.cs.1.1 1 4.3 odd 2
4800.2.a.ct.1.1 1 5.4 even 2
4800.2.f.a.3649.1 2 20.7 even 4
4800.2.f.a.3649.2 2 20.3 even 4
4800.2.f.bj.3649.1 2 5.3 odd 4
4800.2.f.bj.3649.2 2 5.2 odd 4