# Properties

 Label 2850.2.a.e.1.1 Level $2850$ Weight $2$ Character 2850.1 Self dual yes Analytic conductor $22.757$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +1.00000 q^{38} +4.00000 q^{39} -8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000 q^{57} +2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -1.00000 q^{76} +4.00000 q^{77} -4.00000 q^{78} +1.00000 q^{81} +8.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -6.00000 q^{86} -2.00000 q^{88} -8.00000 q^{91} -4.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ −2.00000 −0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 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
$$33$$ −2.00000 −0.348155
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −4.00000 −0.554700
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 4.00000 0.455842
$$78$$ −4.00000 −0.452911
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.00000 0.883452
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ −4.00000 −0.417029
$$93$$ 8.00000 0.829561
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 2.00000 0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 8.00000 0.721336
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ −2.00000 −0.173422
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 3.00000 0.247436
$$148$$ −8.00000 −0.657596
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 2.00000 0.161690
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 6.00000 0.457496
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −2.00000 −0.147844
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 4.00000 0.292509
$$188$$ 12.0000 0.875190
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −24.0000 −1.72756 −0.863779 0.503871i $$-0.831909\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ −2.00000 −0.140720
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ −4.00000 −0.278019
$$208$$ −4.00000 −0.277350
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 8.00000 0.548151
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ −10.0000 −0.677285
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ −8.00000 −0.536925
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ 10.0000 0.646846 0.323423 0.946254i $$-0.395166\pi$$
0.323423 + 0.946254i $$0.395166\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 7.00000 0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −8.00000 −0.510061
$$247$$ 4.00000 0.254514
$$248$$ 8.00000 0.508001
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −8.00000 −0.502956
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 6.00000 0.373544
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000 1.11204
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 8.00000 0.484182
$$274$$ −22.0000 −1.32907
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ −16.0000 −0.944450
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ −14.0000 −0.819288
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ −2.00000 −0.116052
$$298$$ −10.0000 −0.579284
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 8.00000 0.460348
$$303$$ −2.00000 −0.114897
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 8.00000 0.445823
$$323$$ −2.00000 −0.111283
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 14.0000 0.775388
$$327$$ −10.0000 −0.553001
$$328$$ 8.00000 0.441726
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −8.00000 −0.438397
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 1.00000 0.0540738
$$343$$ −20.0000 −1.07990
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ −2.00000 −0.106600
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 0 0
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −2.00000 −0.105118
$$363$$ 7.00000 0.367405
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 8.00000 0.414781
$$373$$ 36.0000 1.86401 0.932005 0.362446i $$-0.118058\pi$$
0.932005 + 0.362446i $$0.118058\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 2.00000 0.102869
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 18.0000 0.920960
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 24.0000 1.22157
$$387$$ 6.00000 0.304997
$$388$$ 12.0000 0.609208
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 3.00000 0.151523
$$393$$ 18.0000 0.907980
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 32.0000 1.59403
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −16.0000 −0.793091
$$408$$ 2.00000 0.0990148
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ 2.00000 0.0978232
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 12.0000 0.583460
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 4.00000 0.193574
$$428$$ 12.0000 0.580042
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 4.00000 0.191346
$$438$$ −14.0000 −0.668946
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 8.00000 0.380521
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ −10.0000 −0.472984
$$448$$ 2.00000 0.0944911
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ 6.00000 0.282216
$$453$$ 8.00000 0.375873
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 6.00000 0.274721
$$478$$ −10.0000 −0.457389
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ 32.0000 1.45907
$$482$$ −2.00000 −0.0910975
$$483$$ 8.00000 0.364013
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 14.0000 0.633102
$$490$$ 0 0
$$491$$ 22.0000 0.992846 0.496423 0.868081i $$-0.334646\pi$$
0.496423 + 0.868081i $$0.334646\pi$$
$$492$$ 8.00000 0.360668
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −16.0000 −0.717698
$$498$$ −4.00000 −0.179244
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ −2.00000 −0.0892644
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ −3.00000 −0.133235
$$508$$ −8.00000 −0.354943
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ 0 0
$$511$$ −28.0000 −1.23865
$$512$$ −1.00000 −0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −6.00000 −0.264135
$$517$$ 24.0000 1.05552
$$518$$ 16.0000 0.703000
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −18.0000 −0.786334
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ −16.0000 −0.696971
$$528$$ −2.00000 −0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ 32.0000 1.38607
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −12.0000 −0.515444
$$543$$ −2.00000 −0.0858282
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ 22.0000 0.939793
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −12.0000 −0.506189
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ 2.00000 0.0839921
$$568$$ 8.00000 0.335673
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 18.0000 0.751961
$$574$$ 16.0000 0.667827
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 24.0000 0.997406
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 12.0000 0.497416
$$583$$ 12.0000 0.496989
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ −8.00000 −0.330195 −0.165098 0.986277i $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ −8.00000 −0.328798
$$593$$ 46.0000 1.88899 0.944497 0.328521i $$-0.106550\pi$$
0.944497 + 0.328521i $$0.106550\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 20.0000 0.818546
$$598$$ −16.0000 −0.654289
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ −8.00000 −0.325785
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ 12.0000 0.487065 0.243532 0.969893i $$-0.421694\pi$$
0.243532 + 0.969893i $$0.421694\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 2.00000 0.0808452
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −32.0000 −1.29141
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ −40.0000 −1.60774 −0.803868 0.594808i $$-0.797228\pi$$
−0.803868 + 0.594808i $$0.797228\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 18.0000 0.721734
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 2.00000 0.0798723
$$628$$ −18.0000 −0.718278
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 32.0000 1.26392 0.631962 0.774999i $$-0.282250\pi$$
0.631962 + 0.774999i $$0.282250\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −14.0000 −0.552106 −0.276053 0.961142i $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ −14.0000 −0.548282
$$653$$ 46.0000 1.80012 0.900060 0.435767i $$-0.143523\pi$$
0.900060 + 0.435767i $$0.143523\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ −14.0000 −0.546192
$$658$$ −24.0000 −0.935617
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 42.0000 1.63361 0.816805 0.576913i $$-0.195743\pi$$
0.816805 + 0.576913i $$0.195743\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 8.00000 0.310694
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 2.00000 0.0771517
$$673$$ −4.00000 −0.154189 −0.0770943 0.997024i $$-0.524564\pi$$
−0.0770943 + 0.997024i $$0.524564\pi$$
$$674$$ 28.0000 1.07852
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 24.0000 0.921035
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 16.0000 0.612672
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ −10.0000 −0.381524
$$688$$ 6.00000 0.228748
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 4.00000 0.151947
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ 10.0000 0.378506
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 8.00000 0.301726
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 4.00000 0.150435
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −10.0000 −0.373457
$$718$$ 10.0000 0.373197
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ −1.00000 −0.0372161
$$723$$ −2.00000 −0.0743808
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ −2.00000 −0.0739221
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ −16.0000 −0.589368
$$738$$ 8.00000 0.294484
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ −12.0000 −0.440534
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ −36.0000 −1.31805
$$747$$ −4.00000 −0.146352
$$748$$ 4.00000 0.146254
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −2.00000 −0.0728841
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 20.0000 0.724049
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 50.0000 1.80305 0.901523 0.432731i $$-0.142450\pi$$
0.901523 + 0.432731i $$0.142450\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −24.0000 −0.863779
$$773$$ −34.0000 −1.22290 −0.611448 0.791285i $$-0.709412\pi$$
−0.611448 + 0.791285i $$0.709412\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ 16.0000 0.573997
$$778$$ −30.0000 −1.07555
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ −2.00000 −0.0710669
$$793$$ −8.00000 −0.284088
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ −2.00000 −0.0707992
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −12.0000 −0.423735
$$803$$ −28.0000 −0.988099
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −32.0000 −1.12715
$$807$$ 0 0
$$808$$ −2.00000 −0.0703598
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ −12.0000 −0.420858
$$814$$ 16.0000 0.560800
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ −6.00000 −0.209913
$$818$$ −30.0000 −1.04893
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 22.0000 0.767338
$$823$$ −34.0000 −1.18517 −0.592583 0.805510i $$-0.701892\pi$$
−0.592583 + 0.805510i $$0.701892\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ −4.00000 −0.138675
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 8.00000 0.276520
$$838$$ 30.0000 1.03633
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −22.0000 −0.758170
$$843$$ −12.0000 −0.413302
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ −14.0000 −0.481046
$$848$$ 6.00000 0.206041
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 8.00000 0.274075
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ −12.0000 −0.408722
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 13.0000 0.441503
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 32.0000 1.08428
$$872$$ −10.0000 −0.338643
$$873$$ 12.0000 0.406138
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 40.0000 1.34993
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 3.00000 0.101015
$$883$$ 26.0000 0.874970 0.437485 0.899226i $$-0.355869\pi$$
0.437485 + 0.899226i $$0.355869\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ −4.00000 −0.133930
$$893$$ −12.0000 −0.401565
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 16.0000 0.532742
$$903$$ −12.0000 −0.399335
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −8.00000 −0.264761
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ −36.0000 −1.18882
$$918$$ 2.00000 0.0660098
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ −2.00000 −0.0658665
$$923$$ 32.0000 1.05329
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −6.00000 −0.197172
$$927$$ −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
$$931$$ 3.00000 0.0983210
$$932$$ 6.00000 0.196537
$$933$$ 18.0000 0.589294
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ 32.0000 1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 0 0
$$949$$ 56.0000 1.81784
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ −4.00000 −0.129641
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 10.0000 0.323423
$$957$$ 0 0
$$958$$ 10.0000 0.323085
$$959$$ 44.0000 1.42083
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −32.0000 −1.03172
$$963$$ 12.0000 0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ −58.0000 −1.86515 −0.932577 0.360971i $$-0.882445\pi$$
−0.932577 + 0.360971i $$0.882445\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −38.0000 −1.21573 −0.607864 0.794041i $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$978$$ −14.0000 −0.447671
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ −22.0000 −0.702048
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ −8.00000 −0.255031
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 4.00000 0.127257
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −12.0000 −0.380808
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 62.0000 1.96356 0.981780 0.190022i $$-0.0608559\pi$$
0.981780 + 0.190022i $$0.0608559\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 8.00000 0.253109
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 2850.2.a.e.1.1 1
3.2 odd 2 8550.2.a.bf.1.1 1
5.2 odd 4 2850.2.d.q.799.1 2
5.3 odd 4 2850.2.d.q.799.2 2
5.4 even 2 570.2.a.l.1.1 1
15.14 odd 2 1710.2.a.b.1.1 1
20.19 odd 2 4560.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.l.1.1 1 5.4 even 2
1710.2.a.b.1.1 1 15.14 odd 2
2850.2.a.e.1.1 1 1.1 even 1 trivial
2850.2.d.q.799.1 2 5.2 odd 4
2850.2.d.q.799.2 2 5.3 odd 4
4560.2.a.o.1.1 1 20.19 odd 2
8550.2.a.bf.1.1 1 3.2 odd 2