# Properties

 Label 2850.2.a.ba.1.1 Level $2850$ Weight $2$ Character 2850.1 Self dual yes Analytic conductor $22.757$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} -2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} +2.00000 q^{28} +1.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +1.00000 q^{57} -8.00000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +2.00000 q^{68} +8.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -4.00000 q^{74} +1.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -8.00000 q^{82} +16.0000 q^{83} +2.00000 q^{84} +6.00000 q^{86} -2.00000 q^{88} +16.0000 q^{89} +8.00000 q^{92} +8.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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 $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 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
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 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
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 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
$$46$$ 8.00000 1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −8.00000 −0.883452
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 6.00000 0.646997
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ −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
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −20.0000 −1.93347 −0.966736 0.255774i $$-0.917670\pi$$
−0.966736 + 0.255774i $$0.917670\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 2.00000 0.188982
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ −8.00000 −0.736460
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000 0.181071
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 2.00000 0.173422
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 8.00000 0.671345
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ −3.00000 −0.247436
$$148$$ −4.00000 −0.328798
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 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
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 1.00000 0.0785674
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 6.00000 0.457496
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ −8.00000 −0.601317
$$178$$ 16.0000 1.19925
$$179$$ −8.00000 −0.597948 −0.298974 0.954261i $$-0.596644\pi$$
−0.298974 + 0.954261i $$0.596644\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 8.00000 0.583460
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000 0.548151
$$214$$ −20.0000 −1.36717
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −4.00000 −0.268462
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 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 $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ −8.00000 −0.519656
$$238$$ 4.00000 0.259281
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −8.00000 −0.510061
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −16.0000 −1.00591
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 6.00000 0.373544
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 16.0000 0.979184
$$268$$ 0 0
$$269$$ 8.00000 0.487769 0.243884 0.969804i $$-0.421578\pi$$
0.243884 + 0.969804i $$0.421578\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 14.0000 0.841178 0.420589 0.907251i $$-0.361823\pi$$
0.420589 + 0.907251i $$0.361823\pi$$
$$278$$ 8.00000 0.479808
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −16.0000 −0.944450
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ 2.00000 0.117041
$$293$$ 22.0000 1.28525 0.642627 0.766179i $$-0.277845\pi$$
0.642627 + 0.766179i $$0.277845\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ −2.00000 −0.116052
$$298$$ 2.00000 0.115857
$$299$$ 0 0
$$300$$ 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
$$307$$ 24.0000 1.36975 0.684876 0.728659i $$-0.259856\pi$$
0.684876 + 0.728659i $$0.259856\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ −18.0000 −1.01742 −0.508710 0.860938i $$-0.669877\pi$$
−0.508710 + 0.860938i $$0.669877\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ 16.0000 0.891645
$$323$$ 2.00000 0.111283
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 10.0000 0.553849
$$327$$ 2.00000 0.110600
$$328$$ −8.00000 −0.441726
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 16.0000 0.878114
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000 0.0540738
$$343$$ −20.0000 −1.07990
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ 16.0000 0.847998
$$357$$ 4.00000 0.211702
$$358$$ −8.00000 −0.422813
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −6.00000 −0.315353
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −26.0000 −1.35719 −0.678594 0.734513i $$-0.737411\pi$$
−0.678594 + 0.734513i $$0.737411\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 2.00000 0.102869
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 10.0000 0.511645
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 6.00000 0.304997
$$388$$ −8.00000 −0.406138
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ −3.00000 −0.151523
$$393$$ −6.00000 −0.302660
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 2.00000 0.0990148
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ −12.0000 −0.591198
$$413$$ −16.0000 −0.787309
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ −2.00000 −0.0978232
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 8.00000 0.388973
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 4.00000 0.193574
$$428$$ −20.0000 −0.966736
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.0000 0.963366 0.481683 0.876346i $$-0.340026\pi$$
0.481683 + 0.876346i $$0.340026\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 12.0000 0.576683 0.288342 0.957528i $$-0.406896\pi$$
0.288342 + 0.957528i $$0.406896\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 8.00000 0.382692
$$438$$ 2.00000 0.0955637
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −12.0000 −0.568216
$$447$$ 2.00000 0.0945968
$$448$$ 2.00000 0.0944911
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ 2.00000 0.0940721
$$453$$ −8.00000 −0.375873
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −18.0000 −0.836531 −0.418265 0.908325i $$-0.637362\pi$$
−0.418265 + 0.908325i $$0.637362\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ −8.00000 −0.368230
$$473$$ −12.0000 −0.551761
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 10.0000 0.457869
$$478$$ 6.00000 0.274434
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 10.0000 0.455488
$$483$$ 16.0000 0.728025
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 10.0000 0.452216
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ −8.00000 −0.360668
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 16.0000 0.716977
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ −13.0000 −0.577350
$$508$$ −16.0000 −0.709885
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 6.00000 0.264135
$$517$$ −16.0000 −0.703679
$$518$$ −8.00000 −0.351500
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ −2.00000 −0.0870388
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 16.0000 0.692388
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −8.00000 −0.345225
$$538$$ 8.00000 0.344904
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ −6.00000 −0.257485
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 40.0000 1.71028 0.855138 0.518400i $$-0.173472\pi$$
0.855138 + 0.518400i $$0.173472\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 8.00000 0.340503
$$553$$ −16.0000 −0.680389
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −20.0000 −0.843649
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ 2.00000 0.0839921
$$568$$ 8.00000 0.335673
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 10.0000 0.417756
$$574$$ −16.0000 −0.667827
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ 32.0000 1.32758
$$582$$ −8.00000 −0.331611
$$583$$ −20.0000 −0.828315
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ 44.0000 1.81607 0.908037 0.418890i $$-0.137581\pi$$
0.908037 + 0.418890i $$0.137581\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ −4.00000 −0.164399
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 12.0000 0.489083
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ −44.0000 −1.78590 −0.892952 0.450151i $$-0.851370\pi$$
−0.892952 + 0.450151i $$0.851370\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000 0.0808452
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ 24.0000 0.968561
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ −14.0000 −0.561349
$$623$$ 32.0000 1.28205
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.0000 −0.719425
$$627$$ −2.00000 −0.0798723
$$628$$ 6.00000 0.239426
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 4.00000 0.158986
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ −20.0000 −0.789337
$$643$$ −46.0000 −1.81406 −0.907031 0.421063i $$-0.861657\pi$$
−0.907031 + 0.421063i $$0.861657\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ −36.0000 −1.41531 −0.707653 0.706560i $$-0.750246\pi$$
−0.707653 + 0.706560i $$0.750246\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 10.0000 0.391630
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ 2.00000 0.0780274
$$658$$ 16.0000 0.623745
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 2.00000 0.0771517
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ 16.0000 0.616297
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 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
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −4.00000 −0.151947
$$694$$ 8.00000 0.303676
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ −26.0000 −0.984115
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 4.00000 0.150435
$$708$$ −8.00000 −0.300658
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 16.0000 0.599625
$$713$$ 0 0
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −8.00000 −0.298974
$$717$$ 6.00000 0.224074
$$718$$ −30.0000 −1.11959
$$719$$ 22.0000 0.820462 0.410231 0.911982i $$-0.365448\pi$$
0.410231 + 0.911982i $$0.365448\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 1.00000 0.0372161
$$723$$ 10.0000 0.371904
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 22.0000 0.815935 0.407967 0.912996i $$-0.366238\pi$$
0.407967 + 0.912996i $$0.366238\pi$$
$$728$$ 0 0
$$729$$ 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 $$0.340576\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ −8.00000 −0.294484
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 20.0000 0.734223
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ 16.0000 0.585409
$$748$$ −4.00000 −0.146254
$$749$$ −40.0000 −1.46157
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 6.00000 0.218652
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 4.00000 0.144810
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 4.00000 0.143963
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ −8.00000 −0.286998
$$778$$ 14.0000 0.501924
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 16.0000 0.572159
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ −52.0000 −1.85360 −0.926800 0.375555i $$-0.877452\pi$$
−0.926800 + 0.375555i $$0.877452\pi$$
$$788$$ −10.0000 −0.356235
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ −2.00000 −0.0710669
$$793$$ 0 0
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 22.0000 0.779280 0.389640 0.920967i $$-0.372599\pi$$
0.389640 + 0.920967i $$0.372599\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ 16.0000 0.565332
$$802$$ −12.0000 −0.423735
$$803$$ −4.00000 −0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8.00000 0.281613
$$808$$ 2.00000 0.0703598
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ −4.00000 −0.140286
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 6.00000 0.209913
$$818$$ −18.0000 −0.629355
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ −34.0000 −1.18517 −0.592583 0.805510i $$-0.701892\pi$$
−0.592583 + 0.805510i $$0.701892\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 0 0
$$838$$ 30.0000 1.03633
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −26.0000 −0.896019
$$843$$ −20.0000 −0.688837
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ −14.0000 −0.481046
$$848$$ 10.0000 0.343401
$$849$$ 2.00000 0.0686398
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 8.00000 0.274075
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −16.0000 −0.545279
$$862$$ 20.0000 0.681203
$$863$$ 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
$$866$$ 12.0000 0.407777
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ −8.00000 −0.270759
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −8.00000 −0.270141 −0.135070 0.990836i $$-0.543126\pi$$
−0.135070 + 0.990836i $$0.543126\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 22.0000 0.742042
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ 18.0000 0.605748 0.302874 0.953031i $$-0.402054\pi$$
0.302874 + 0.953031i $$0.402054\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ −12.0000 −0.401790
$$893$$ 8.00000 0.267710
$$894$$ 2.00000 0.0668900
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 24.0000 0.800890
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 20.0000 0.666297
$$902$$ 16.0000 0.532742
$$903$$ 12.0000 0.399335
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −32.0000 −1.05905
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ −12.0000 −0.396275
$$918$$ 2.00000 0.0660098
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 24.0000 0.790827
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −18.0000 −0.591517
$$927$$ −12.0000 −0.394132
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 6.00000 0.196537
$$933$$ −14.0000 −0.458339
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 58.0000 1.89478 0.947389 0.320085i $$-0.103712\pi$$
0.947389 + 0.320085i $$0.103712\pi$$
$$938$$ 0 0
$$939$$ −18.0000 −0.587408
$$940$$ 0 0
$$941$$ 16.0000 0.521585 0.260793 0.965395i $$-0.416016\pi$$
0.260793 + 0.965395i $$0.416016\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −64.0000 −2.08413
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 32.0000 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 4.00000 0.129641
$$953$$ 18.0000 0.583077 0.291539 0.956559i $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ −30.0000 −0.969256
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −20.0000 −0.644491
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 16.0000 0.514792
$$967$$ 6.00000 0.192947 0.0964735 0.995336i $$-0.469244\pi$$
0.0964735 + 0.995336i $$0.469244\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 16.0000 0.512936
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 10.0000 0.319765
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ −30.0000 −0.957338
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ −8.00000 −0.255031
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ −16.0000 −0.506471
$$999$$ −4.00000 −0.126554
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.ba.1.1 1
3.2 odd 2 8550.2.a.o.1.1 1
5.2 odd 4 2850.2.d.n.799.2 2
5.3 odd 4 2850.2.d.n.799.1 2
5.4 even 2 570.2.a.c.1.1 1
15.14 odd 2 1710.2.a.n.1.1 1
20.19 odd 2 4560.2.a.bd.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.c.1.1 1 5.4 even 2
1710.2.a.n.1.1 1 15.14 odd 2
2850.2.a.ba.1.1 1 1.1 even 1 trivial
2850.2.d.n.799.1 2 5.3 odd 4
2850.2.d.n.799.2 2 5.2 odd 4
4560.2.a.bd.1.1 1 20.19 odd 2
8550.2.a.o.1.1 1 3.2 odd 2