# Properties

 Label 1650.2.a.k.1.1 Level $1650$ Weight $2$ Character 1650.1 Self dual yes Analytic conductor $13.175$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.1753163335$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1650.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000 q^{77} -6.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +24.0000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 1.00000 0.213201
$$23$$ −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
$$26$$ −6.00000 −1.17670
$$27$$ 1.00000 0.192450
$$28$$ 4.00000 0.755929
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 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
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 6.00000 0.832050
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 4.00000 0.529813
$$58$$ −6.00000 −0.787839
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ −4.00000 −0.455842
$$78$$ −6.00000 −0.679366
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 6.00000 0.643268
$$88$$ 1.00000 0.106600
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 24.0000 2.51588
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 4.00000 0.377964
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 6.00000 0.554700
$$118$$ −12.0000 −1.10469
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 14.0000 1.26750
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 12.0000 1.00702
$$143$$ −6.00000 −0.501745
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ 9.00000 0.742307
$$148$$ −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
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 4.00000 0.318223
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 12.0000 0.901975
$$178$$ −10.0000 −0.749532
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ −24.0000 −1.77900
$$183$$ −14.0000 −1.03491
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.00000 0.146254
$$188$$ 12.0000 0.875190
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −14.0000 −0.985037
$$203$$ 24.0000 1.68447
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 6.00000 0.416025
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ −12.0000 −0.822226
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 6.00000 0.402694
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ −6.00000 −0.393919
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ −4.00000 −0.259828
$$238$$ 8.00000 0.518563
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 4.00000 0.251976
$$253$$ 4.00000 0.251478
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ −4.00000 −0.247121
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ −16.0000 −0.981023
$$267$$ 10.0000 0.611990
$$268$$ −4.00000 −0.244339
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 24.0000 1.45255
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ −24.0000 −1.41668
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 6.00000 0.351123
$$293$$ −22.0000 −1.28525 −0.642627 0.766179i $$-0.722155\pi$$
−0.642627 + 0.766179i $$0.722155\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ −1.00000 −0.0580259
$$298$$ 10.0000 0.579284
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ −4.00000 −0.230174
$$303$$ 14.0000 0.804279
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 16.0000 0.891645
$$323$$ −8.00000 −0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ −6.00000 −0.331801
$$328$$ 6.00000 0.331295
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 8.00000 0.431959
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 1.00000 0.0533002
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ −8.00000 −0.423405
$$358$$ −20.0000 −1.05703
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000 0.105118
$$363$$ 1.00000 0.0524864
$$364$$ 24.0000 1.25794
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 36.0000 1.85409
$$378$$ −4.00000 −0.205738
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 12.0000 0.613973
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ −4.00000 −0.203331
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ −9.00000 −0.454569
$$393$$ 4.00000 0.201773
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ −38.0000 −1.89763 −0.948815 0.315833i $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 6.00000 0.297409
$$408$$ 2.00000 0.0990148
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ 48.0000 2.36193
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ −4.00000 −0.195881
$$418$$ 4.00000 0.195646
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 12.0000 0.583460
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 12.0000 0.581402
$$427$$ −56.0000 −2.71003
$$428$$ −4.00000 −0.193347
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ −16.0000 −0.765384
$$438$$ −6.00000 −0.286691
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 12.0000 0.570782
$$443$$ −28.0000 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −10.0000 −0.472984
$$448$$ 4.00000 0.188982
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ −2.00000 −0.0940721
$$453$$ 4.00000 0.187936
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −34.0000 −1.59045 −0.795226 0.606313i $$-0.792648\pi$$
−0.795226 + 0.606313i $$0.792648\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −12.0000 −0.552345
$$473$$ 4.00000 0.183920
$$474$$ 4.00000 0.183726
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ −2.00000 −0.0915737
$$478$$ −8.00000 −0.365911
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ −10.0000 −0.455488
$$483$$ −16.0000 −0.728025
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 14.0000 0.633750
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −12.0000 −0.540453
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −48.0000 −2.15309
$$498$$ 4.00000 0.179244
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ 23.0000 1.02147
$$508$$ −12.0000 −0.532414
$$509$$ −22.0000 −0.975133 −0.487566 0.873086i $$-0.662115\pi$$
−0.487566 + 0.873086i $$0.662115\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ −12.0000 −0.527759
$$518$$ 24.0000 1.05450
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ −1.00000 −0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 16.0000 0.693688
$$533$$ −36.0000 −1.55933
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 20.0000 0.863064
$$538$$ −26.0000 −1.12094
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −2.00000 −0.0858282
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −24.0000 −1.02711
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 4.00000 0.170251
$$553$$ −16.0000 −0.680389
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 2.00000 0.0844401
$$562$$ 22.0000 0.928014
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 4.00000 0.167984
$$568$$ 12.0000 0.503509
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ −12.0000 −0.501307
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 46.0000 1.91501 0.957503 0.288425i $$-0.0931316\pi$$
0.957503 + 0.288425i $$0.0931316\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ −14.0000 −0.580319
$$583$$ 2.00000 0.0828315
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ −6.00000 −0.246598
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ −16.0000 −0.654836
$$598$$ 24.0000 0.981433
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 16.0000 0.652111
$$603$$ −4.00000 −0.162893
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ −14.0000 −0.568711
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 72.0000 2.91281
$$612$$ −2.00000 −0.0808452
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ −4.00000 −0.160385
$$623$$ 40.0000 1.60257
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ −4.00000 −0.159745
$$628$$ 10.0000 0.399043
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −4.00000 −0.158986
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 54.0000 2.13956
$$638$$ 6.00000 0.237542
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ −48.0000 −1.87123
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ −12.0000 −0.466041
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ −24.0000 −0.929284
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ −4.00000 −0.154303
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ 56.0000 2.14908
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 14.0000 0.534133
$$688$$ −4.00000 −0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 10.0000 0.380143
$$693$$ −4.00000 −0.151947
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000 0.454532
$$698$$ 6.00000 0.227103
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ −24.0000 −0.905177
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 56.0000 2.10610
$$708$$ 12.0000 0.450988
$$709$$ −18.0000 −0.676004 −0.338002 0.941145i $$-0.609751\pi$$
−0.338002 + 0.941145i $$0.609751\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ −10.0000 −0.374766
$$713$$ 0 0
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 8.00000 0.298765
$$718$$ 0 0
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 10.0000 0.371904
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −1.00000 −0.0371135
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ −14.0000 −0.517455
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 4.00000 0.147342
$$738$$ 6.00000 0.220863
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 8.00000 0.293689
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ −4.00000 −0.146352
$$748$$ 2.00000 0.0731272
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −4.00000 −0.145768
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −24.0000 −0.868858
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ 72.0000 2.59977
$$768$$ 1.00000 0.0360844
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ −10.0000 −0.359908
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ −24.0000 −0.860995
$$778$$ 30.0000 1.07555
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ −8.00000 −0.286079
$$783$$ 6.00000 0.214423
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 1.00000 0.0355335
$$793$$ −84.0000 −2.98293
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ 54.0000 1.91278 0.956389 0.292096i $$-0.0943526\pi$$
0.956389 + 0.292096i $$0.0943526\pi$$
$$798$$ −16.0000 −0.566394
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 38.0000 1.34183
$$803$$ −6.00000 −0.211735
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 26.0000 0.915243
$$808$$ −14.0000 −0.492518
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 24.0000 0.842235
$$813$$ 20.0000 0.701431
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ −16.0000 −0.559769
$$818$$ 14.0000 0.489499
$$819$$ 24.0000 0.838628
$$820$$ 0 0
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ 52.0000 1.80822 0.904109 0.427303i $$-0.140536\pi$$
0.904109 + 0.427303i $$0.140536\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 6.00000 0.208013
$$833$$ −18.0000 −0.623663
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ −22.0000 −0.757720
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 4.00000 0.137442
$$848$$ −2.00000 −0.0686803
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ −12.0000 −0.411113
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 56.0000 1.91628
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 6.00000 0.204837
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ −20.0000 −0.680808 −0.340404 0.940279i $$-0.610564\pi$$
−0.340404 + 0.940279i $$0.610564\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 6.00000 0.203186
$$873$$ 14.0000 0.473828
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −4.00000 −0.134993
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 48.0000 1.60626
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ −24.0000 −0.801337
$$898$$ 22.0000 0.734150
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ −6.00000 −0.199778
$$903$$ −16.0000 −0.532447
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −4.00000 −0.132891
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 4.00000 0.132381
$$914$$ 34.0000 1.12462
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 16.0000 0.528367
$$918$$ 2.00000 0.0660098
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ −14.0000 −0.461065
$$923$$ −72.0000 −2.36991
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 36.0000 1.17985
$$932$$ −10.0000 −0.327561
$$933$$ 4.00000 0.130954
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ −10.0000 −0.325818
$$943$$ 24.0000 0.781548
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −4.00000 −0.129914
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 8.00000 0.259281
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ −6.00000 −0.193952
$$958$$ 0 0
$$959$$ −8.00000 −0.258333
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 36.0000 1.16069
$$963$$ −4.00000 −0.128898
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 16.0000 0.514792
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −8.00000 −0.256997
$$970$$ 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
$$973$$ −16.0000 −0.512936
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ −26.0000 −0.831814 −0.415907 0.909407i $$-0.636536\pi$$
−0.415907 + 0.909407i $$0.636536\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 28.0000 0.893516
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 48.0000 1.52786
$$988$$ 24.0000 0.763542
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 48.0000 1.52247
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 4.00000 0.126618
$$999$$ −6.00000 −0.189832
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1650.2.a.k.1.1 1
3.2 odd 2 4950.2.a.bu.1.1 1
5.2 odd 4 1650.2.c.e.199.1 2
5.3 odd 4 1650.2.c.e.199.2 2
5.4 even 2 66.2.a.b.1.1 1
15.2 even 4 4950.2.c.p.199.2 2
15.8 even 4 4950.2.c.p.199.1 2
15.14 odd 2 198.2.a.a.1.1 1
20.19 odd 2 528.2.a.j.1.1 1
35.34 odd 2 3234.2.a.t.1.1 1
40.19 odd 2 2112.2.a.e.1.1 1
40.29 even 2 2112.2.a.r.1.1 1
45.4 even 6 1782.2.e.e.1189.1 2
45.14 odd 6 1782.2.e.v.1189.1 2
45.29 odd 6 1782.2.e.v.595.1 2
45.34 even 6 1782.2.e.e.595.1 2
55.4 even 10 726.2.e.g.511.1 4
55.9 even 10 726.2.e.g.565.1 4
55.14 even 10 726.2.e.g.493.1 4
55.19 odd 10 726.2.e.o.493.1 4
55.24 odd 10 726.2.e.o.565.1 4
55.29 odd 10 726.2.e.o.511.1 4
55.39 odd 10 726.2.e.o.487.1 4
55.49 even 10 726.2.e.g.487.1 4
55.54 odd 2 726.2.a.c.1.1 1
60.59 even 2 1584.2.a.f.1.1 1
105.104 even 2 9702.2.a.x.1.1 1
120.29 odd 2 6336.2.a.bw.1.1 1
120.59 even 2 6336.2.a.cj.1.1 1
165.164 even 2 2178.2.a.g.1.1 1
220.219 even 2 5808.2.a.bc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.b.1.1 1 5.4 even 2
198.2.a.a.1.1 1 15.14 odd 2
528.2.a.j.1.1 1 20.19 odd 2
726.2.a.c.1.1 1 55.54 odd 2
726.2.e.g.487.1 4 55.49 even 10
726.2.e.g.493.1 4 55.14 even 10
726.2.e.g.511.1 4 55.4 even 10
726.2.e.g.565.1 4 55.9 even 10
726.2.e.o.487.1 4 55.39 odd 10
726.2.e.o.493.1 4 55.19 odd 10
726.2.e.o.511.1 4 55.29 odd 10
726.2.e.o.565.1 4 55.24 odd 10
1584.2.a.f.1.1 1 60.59 even 2
1650.2.a.k.1.1 1 1.1 even 1 trivial
1650.2.c.e.199.1 2 5.2 odd 4
1650.2.c.e.199.2 2 5.3 odd 4
1782.2.e.e.595.1 2 45.34 even 6
1782.2.e.e.1189.1 2 45.4 even 6
1782.2.e.v.595.1 2 45.29 odd 6
1782.2.e.v.1189.1 2 45.14 odd 6
2112.2.a.e.1.1 1 40.19 odd 2
2112.2.a.r.1.1 1 40.29 even 2
2178.2.a.g.1.1 1 165.164 even 2
3234.2.a.t.1.1 1 35.34 odd 2
4950.2.a.bu.1.1 1 3.2 odd 2
4950.2.c.p.199.1 2 15.8 even 4
4950.2.c.p.199.2 2 15.2 even 4
5808.2.a.bc.1.1 1 220.219 even 2
6336.2.a.bw.1.1 1 120.29 odd 2
6336.2.a.cj.1.1 1 120.59 even 2
9702.2.a.x.1.1 1 105.104 even 2