# Properties

 Label 2070.2.a.r.1.1 Level $2070$ Weight $2$ Character 2070.1 Self dual yes Analytic conductor $16.529$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} -6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +6.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} +14.0000 q^{53} +4.00000 q^{55} +6.00000 q^{58} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} +6.00000 q^{68} -8.00000 q^{71} +2.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +1.00000 q^{80} -10.0000 q^{82} +16.0000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +2.00000 q^{89} -1.00000 q^{92} +8.00000 q^{94} +4.00000 q^{95} -14.0000 q^{97} -7.00000 q^{98} +O(q^{100})$$

## 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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −6.00000 −0.526235
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 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 $$0.423041\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 6.00000 0.460179
$$171$$ 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
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ −18.0000 −1.26648
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.0000 −0.698430
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ −36.0000 −2.42162
$$222$$ 0 0
$$223$$ −20.0000 −1.33930 −0.669650 0.742677i $$-0.733556\pi$$
−0.669650 + 0.742677i $$0.733556\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ −7.00000 −0.447214
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ −8.00000 −0.508001
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −6.00000 −0.372104
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 32.0000 1.97320 0.986602 0.163144i $$-0.0521635\pi$$
0.986602 + 0.163144i $$0.0521635\pi$$
$$264$$ 0 0
$$265$$ 14.0000 0.860013
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 6.00000 0.352332
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 0 0
$$325$$ −6.00000 −0.332820
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 0 0
$$334$$ 16.0000 0.875481
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ 6.00000 0.325396
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −6.00000 −0.315353
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ 6.00000 0.311925
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −18.0000 −0.932005 −0.466002 0.884783i $$-0.654306\pi$$
−0.466002 + 0.884783i $$0.654306\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ −7.00000 −0.353553
$$393$$ 0 0
$$394$$ −26.0000 −1.30986
$$395$$ −12.0000 −0.603786
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 48.0000 2.39105
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 0 0
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 16.0000 0.785409
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 14.0000 0.679900
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 0 0
$$442$$ −36.0000 −1.71235
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 2.00000 0.0948091
$$446$$ −20.0000 −0.947027
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ −1.00000 −0.0466252
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 8.00000 0.365911
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −14.0000 −0.635707
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ −7.00000 −0.316228
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ −18.0000 −0.800989
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 32.0000 1.40736
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 14.0000 0.608121
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 60.0000 2.59889
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −10.0000 −0.431131
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 0 0
$$544$$ 6.00000 0.257248
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ 8.00000 0.337160 0.168580 0.985688i $$-0.446082\pi$$
0.168580 + 0.985688i $$0.446082\pi$$
$$564$$ 0 0
$$565$$ 14.0000 0.588984
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 56.0000 2.31928
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ −48.0000 −1.94187
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 8.00000 0.320771
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 42.0000 1.66410
$$638$$ 24.0000 0.950169
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −44.0000 −1.73519 −0.867595 0.497271i $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.00000 −0.232321
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ 0 0
$$682$$ −32.0000 −1.22534
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −84.0000 −3.20015
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −60.0000 −2.27266
$$698$$ −34.0000 −1.28692
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ −24.0000 −0.897549
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −6.00000 −0.222988
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 2.00000 0.0740233
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ −18.0000 −0.659027
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000 0.359908
$$773$$ −2.00000 −0.0719350 −0.0359675 0.999353i $$-0.511451\pi$$
−0.0359675 + 0.999353i $$0.511451\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −6.00000 −0.214560
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 6.00000 0.214149
$$786$$ 0 0
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ −26.0000 −0.926212
$$789$$ 0 0
$$790$$ −12.0000 −0.426941
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −60.0000 −2.13066
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −6.00000 −0.211867
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 0 0
$$808$$ −18.0000 −0.633238
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ 0 0
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000 0.278187 0.139094 0.990279i $$-0.455581\pi$$
0.139094 + 0.990279i $$0.455581\pi$$
$$828$$ 0 0
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 16.0000 0.555368
$$831$$ 0 0
$$832$$ −6.00000 −0.208013
$$833$$ −42.0000 −1.45521
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −38.0000 −1.30957
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 14.0000 0.480762
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ −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
$$866$$ 2.00000 0.0679628
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.00000 0.0670402
$$891$$ 0 0
$$892$$ −20.0000 −0.669650
$$893$$ 32.0000 1.07084
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −10.0000 −0.333704
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ 84.0000 2.79845
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ 64.0000 2.11809
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ −2.00000 −0.0658665
$$923$$ 48.0000 1.57994
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ −20.0000 −0.657241
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ −22.0000 −0.720634
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 24.0000 0.784884
$$936$$ 0 0
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ 10.0000 0.325645
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 20.0000 0.649913 0.324956 0.945729i $$-0.394650\pi$$
0.324956 + 0.945729i $$0.394650\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −36.0000 −1.16069
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −14.0000 −0.449513
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 8.00000 0.255681
$$980$$ −7.00000 −0.223607
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −26.0000 −0.828429
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ −24.0000 −0.763542
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 0 0
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 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.r.1.1 1
3.2 odd 2 690.2.a.e.1.1 1
12.11 even 2 5520.2.a.f.1.1 1
15.2 even 4 3450.2.d.a.2899.1 2
15.8 even 4 3450.2.d.a.2899.2 2
15.14 odd 2 3450.2.a.p.1.1 1

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