# Properties

 Label 2070.2.a.e.1.1 Level $2070$ Weight $2$ Character 2070.1 Self dual yes Analytic conductor $16.529$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.5290332184$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -2.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -8.00000 q^{37} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} +2.00000 q^{55} +2.00000 q^{56} +2.00000 q^{58} -8.00000 q^{61} +1.00000 q^{64} -6.00000 q^{65} -4.00000 q^{67} +4.00000 q^{68} +2.00000 q^{70} -16.0000 q^{71} +6.00000 q^{73} +8.00000 q^{74} -4.00000 q^{77} +14.0000 q^{79} +1.00000 q^{80} -6.00000 q^{82} -14.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{88} +8.00000 q^{89} +12.0000 q^{91} -1.00000 q^{92} -6.00000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## 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
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 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
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −6.00000 −0.662589
$$83$$ −14.0000 −1.53670 −0.768350 0.640030i $$-0.778922\pi$$
−0.768350 + 0.640030i $$0.778922\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 6.00000 0.526235
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −16.0000 −1.27694 −0.638470 0.769647i $$-0.720432\pi$$
−0.638470 + 0.769647i $$0.720432\pi$$
$$158$$ −14.0000 −1.11378
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 2.00000 0.157622
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ −8.00000 −0.599625
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ 0 0
$$184$$ 1.00000 0.0737210
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 2.00000 0.139347
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −18.0000 −1.23045
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 8.00000 0.518563
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 16.0000 0.994192
$$260$$ −6.00000 −0.372104
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 2.00000 0.119523
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 2.00000 0.117444
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −12.0000 −0.690522
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 32.0000 1.81455 0.907277 0.420534i $$-0.138157\pi$$
0.907277 + 0.420534i $$0.138157\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 16.0000 0.902932
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −2.00000 −0.111456
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −6.00000 −0.332820
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ 0 0
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ 8.00000 0.423999
$$357$$ 0 0
$$358$$ −20.0000 −1.05703
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 12.0000 0.630706
$$363$$ 0 0
$$364$$ 12.0000 0.628971
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ 8.00000 0.415900
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 32.0000 1.65690 0.828449 0.560065i $$-0.189224\pi$$
0.828449 + 0.560065i $$0.189224\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ −6.00000 −0.304604
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 14.0000 0.704416
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 2.00000 0.100251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ −16.0000 −0.793091
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 0 0
$$412$$ −2.00000 −0.0985329
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −14.0000 −0.687233
$$416$$ 6.00000 0.294174
$$417$$ 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
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 16.0000 0.774294
$$428$$ 18.0000 0.870063
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 24.0000 1.14156
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ 8.00000 0.379236
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ −2.00000 −0.0944911
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ −18.0000 −0.844782
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 20.0000 0.934539
$$459$$ 0 0
$$460$$ −1.00000 −0.0466252
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ 30.0000 1.38823 0.694117 0.719862i $$-0.255795\pi$$
0.694117 + 0.719862i $$0.255795\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ −6.00000 −0.272446
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 0 0
$$490$$ 3.00000 0.135526
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000 1.43540
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 2.00000 0.0889108
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ −2.00000 −0.0881305
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −16.0000 −0.703000
$$519$$ 0 0
$$520$$ 6.00000 0.263117
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −36.0000 −1.55933
$$534$$ 0 0
$$535$$ 18.0000 0.778208
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 8.00000 0.341743
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −28.0000 −1.19068
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ 8.00000 0.337460
$$563$$ 10.0000 0.421450 0.210725 0.977545i $$-0.432418\pi$$
0.210725 + 0.977545i $$0.432418\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ 16.0000 0.672530
$$567$$ 0 0
$$568$$ 16.0000 0.671345
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −2.00000 −0.0830455
$$581$$ 28.0000 1.16164
$$582$$ 0 0
$$583$$ −12.0000 −0.496989
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ −8.00000 −0.330195 −0.165098 0.986277i $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.00000 −0.328798
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ −8.00000 −0.327968
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ −6.00000 −0.245358
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 0 0
$$604$$ 12.0000 0.488273
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −4.00000 −0.161558 −0.0807792 0.996732i $$-0.525741\pi$$
−0.0807792 + 0.996732i $$0.525741\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ −28.0000 −1.12724 −0.563619 0.826035i $$-0.690591\pi$$
−0.563619 + 0.826035i $$0.690591\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −32.0000 −1.28308
$$623$$ −16.0000 −0.641026
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 34.0000 1.35891
$$627$$ 0 0
$$628$$ −16.0000 −0.638470
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ −14.0000 −0.556890
$$633$$ 0 0
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 4.00000 0.158362
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −16.0000 −0.631962 −0.315981 0.948766i $$-0.602334\pi$$
−0.315981 + 0.948766i $$0.602334\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 6.00000 0.235339
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −22.0000 −0.860927 −0.430463 0.902608i $$-0.641650\pi$$
−0.430463 + 0.902608i $$0.641650\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 14.0000 0.543305
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.00000 0.0774403
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −48.0000 −1.83667 −0.918334 0.395805i $$-0.870466\pi$$
−0.918334 + 0.395805i $$0.870466\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ −8.00000 −0.303676
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 24.0000 0.909065
$$698$$ 6.00000 0.227103
$$699$$ 0 0
$$700$$ −2.00000 −0.0755929
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ 12.0000 0.451306
$$708$$ 0 0
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ −8.00000 −0.299813
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ −12.0000 −0.445976
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ 38.0000 1.40934 0.704671 0.709534i $$-0.251095\pi$$
0.704671 + 0.709534i $$0.251095\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 0 0
$$730$$ −6.00000 −0.222070
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ −32.0000 −1.17160
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 40.0000 1.45382 0.726912 0.686730i $$-0.240955\pi$$
0.726912 + 0.686730i $$0.240955\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 8.00000 0.289619
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 4.00000 0.144150
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ 10.0000 0.358517
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 4.00000 0.143040
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ −16.0000 −0.571064
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ −14.0000 −0.498098
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ 48.0000 1.70453
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −2.00000 −0.0708881
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −24.0000 −0.847469
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 4.00000 0.140372
$$813$$ 0 0
$$814$$ 16.0000 0.560800
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −38.0000 −1.32864
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 22.0000 0.765015 0.382507 0.923952i $$-0.375061\pi$$
0.382507 + 0.923952i $$0.375061\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 14.0000 0.485947
$$831$$ 0 0
$$832$$ −6.00000 −0.208013
$$833$$ −12.0000 −0.415775
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −14.0000 −0.483622
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 0 0
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −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
$$866$$ −30.0000 −1.01944
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 4.00000 0.135457
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 56.0000 1.88669 0.943344 0.331816i $$-0.107661\pi$$
0.943344 + 0.331816i $$0.107661\pi$$
$$882$$ 0 0
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −8.00000 −0.268161
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 20.0000 0.668526
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −22.0000 −0.734150
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ −12.0000 −0.399556
$$903$$ 0 0
$$904$$ 12.0000 0.399114
$$905$$ −12.0000 −0.398893
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 18.0000 0.597351
$$909$$ 0 0
$$910$$ −12.0000 −0.397796
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ −28.0000 −0.926665
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ 0 0
$$922$$ −10.0000 −0.329332
$$923$$ 96.0000 3.15988
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 36.0000 1.18303
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 26.0000 0.851658
$$933$$ 0 0
$$934$$ −30.0000 −0.981630
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 0 0
$$943$$ −6.00000 −0.195387
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 8.00000 0.259281
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 8.00000 0.258468
$$959$$ −16.0000 −0.516667
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −48.0000 −1.54758
$$963$$ 0 0
$$964$$ −14.0000 −0.450910
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 52.0000 1.67221 0.836104 0.548572i $$-0.184828\pi$$
0.836104 + 0.548572i $$0.184828\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ 6.00000 0.192648
$$971$$ 46.0000 1.47621 0.738105 0.674686i $$-0.235721\pi$$
0.738105 + 0.674686i $$0.235721\pi$$
$$972$$ 0 0
$$973$$ 8.00000 0.256468
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ −24.0000 −0.767828 −0.383914 0.923369i $$-0.625424\pi$$
−0.383914 + 0.923369i $$0.625424\pi$$
$$978$$ 0 0
$$979$$ 16.0000 0.511362
$$980$$ −3.00000 −0.0958315
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 8.00000 0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −32.0000 −1.01498
$$995$$ −2.00000 −0.0634043
$$996$$ 0 0
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 28.0000 0.886325
$$999$$ 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 2070.2.a.e.1.1 1
3.2 odd 2 690.2.a.g.1.1 1
12.11 even 2 5520.2.a.z.1.1 1
15.2 even 4 3450.2.d.d.2899.2 2
15.8 even 4 3450.2.d.d.2899.1 2
15.14 odd 2 3450.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 3.2 odd 2
2070.2.a.e.1.1 1 1.1 even 1 trivial
3450.2.a.l.1.1 1 15.14 odd 2
3450.2.d.d.2899.1 2 15.8 even 4
3450.2.d.d.2899.2 2 15.2 even 4
5520.2.a.z.1.1 1 12.11 even 2