# Properties

 Label 5550.2.a.bd.1.1 Level $5550$ Weight $2$ Character 5550.1 Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} +6.00000 q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -8.00000 q^{43} +6.00000 q^{44} -6.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -2.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +8.00000 q^{61} +8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{72} -14.0000 q^{73} -1.00000 q^{74} +2.00000 q^{76} +24.0000 q^{77} +2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} -8.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -8.00000 q^{97} +9.00000 q^{98} +6.00000 q^{99} +O(q^{100})$$

## 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
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 6.00000 1.27920
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ −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
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −6.00000 −1.04447
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ 2.00000 0.324443
$$39$$ 2.00000 0.320256
$$40$$ 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
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ −2.00000 −0.264906
$$58$$ 6.00000 0.787839
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 24.0000 2.73505
$$78$$ 2.00000 0.226455
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −6.00000 −0.643268
$$88$$ 6.00000 0.639602
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 20.0000 1.91565 0.957826 0.287348i $$-0.0927736\pi$$
0.957826 + 0.287348i $$0.0927736\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ 4.00000 0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ −12.0000 −1.10469
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 8.00000 0.724286
$$123$$ 6.00000 0.541002
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ 8.00000 0.693688
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ −9.00000 −0.742307
$$148$$ −1.00000 −0.0821995
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 6.00000 0.485071
$$154$$ 24.0000 1.93398
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 12.0000 0.901975
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ −8.00000 −0.591377
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 36.0000 2.63258
$$188$$ −6.00000 −0.437595
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 32.0000 2.17230
$$218$$ 20.0000 1.35457
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 1.00000 0.0671156
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ −24.0000 −1.57908
$$232$$ 6.00000 0.393919
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 16.0000 1.03931
$$238$$ 24.0000 1.55569
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 25.0000 1.60706
$$243$$ −1.00000 −0.0641500
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ −4.00000 −0.254514
$$248$$ 8.00000 0.508001
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 4.00000 0.251976
$$253$$ 0 0
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 8.00000 0.498058
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ −12.0000 −0.741362
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ −6.00000 −0.367194
$$268$$ 4.00000 0.244339
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 8.00000 0.484182
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ −24.0000 −1.41668
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ −14.0000 −0.819288
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −6.00000 −0.348155
$$298$$ −12.0000 −0.695141
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 24.0000 1.36753
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 16.0000 0.904373 0.452187 0.891923i $$-0.350644\pi$$
0.452187 + 0.891923i $$0.350644\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 36.0000 2.01561
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ −20.0000 −1.10600
$$328$$ −6.00000 −0.331295
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 14.0000 0.769510 0.384755 0.923019i $$-0.374286\pi$$
0.384755 + 0.923019i $$0.374286\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −1.00000 −0.0547997
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 48.0000 2.59935
$$342$$ 2.00000 0.108148
$$343$$ 8.00000 0.431959
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 6.00000 0.319801
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ −24.0000 −1.27021
$$358$$ −12.0000 −0.634220
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −10.0000 −0.525588
$$363$$ −25.0000 −1.31216
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −8.00000 −0.418167
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ −8.00000 −0.414781
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 36.0000 1.86152
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ −12.0000 −0.618031
$$378$$ −4.00000 −0.205738
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ −8.00000 −0.406663
$$388$$ −8.00000 −0.406138
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 9.00000 0.454569
$$393$$ 12.0000 0.605320
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −16.0000 −0.797017
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 24.0000 1.19110
$$407$$ −6.00000 −0.297409
$$408$$ −6.00000 −0.297044
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ −14.0000 −0.689730
$$413$$ −48.0000 −2.36193
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 4.00000 0.195881
$$418$$ 12.0000 0.586939
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ −16.0000 −0.778868
$$423$$ −6.00000 −0.291730
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 32.0000 1.54859
$$428$$ 0 0
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −38.0000 −1.82616 −0.913082 0.407777i $$-0.866304\pi$$
−0.913082 + 0.407777i $$0.866304\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ 20.0000 0.957826
$$437$$ 0 0
$$438$$ 14.0000 0.668946
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −12.0000 −0.570782
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 12.0000 0.567581
$$448$$ 4.00000 0.188982
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ −6.00000 −0.282216
$$453$$ 16.0000 0.751746
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 26.0000 1.21490
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ −24.0000 −1.11658
$$463$$ 34.0000 1.58011 0.790057 0.613033i $$-0.210051\pi$$
0.790057 + 0.613033i $$0.210051\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ −12.0000 −0.552345
$$473$$ −48.0000 −2.20704
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 24.0000 1.10004
$$477$$ −6.00000 −0.274721
$$478$$ 24.0000 1.09773
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 8.00000 0.362143
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 36.0000 1.62136
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 4.00000 0.177471
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 1.00000 0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ −36.0000 −1.58328
$$518$$ −4.00000 −0.175750
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −32.0000 −1.39926 −0.699631 0.714504i $$-0.746652\pi$$
−0.699631 + 0.714504i $$0.746652\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ 48.0000 2.09091
$$528$$ −6.00000 −0.261116
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 8.00000 0.346844
$$533$$ 12.0000 0.519778
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 12.0000 0.517838
$$538$$ −24.0000 −1.03471
$$539$$ 54.0000 2.32594
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 10.0000 0.429141
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ −64.0000 −2.72156
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ −36.0000 −1.51992
$$562$$ −30.0000 −1.26547
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 4.00000 0.167984
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −20.0000 −0.832611 −0.416305 0.909225i $$-0.636675\pi$$
−0.416305 + 0.909225i $$0.636675\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −16.0000 −0.664937
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ 8.00000 0.331611
$$583$$ −36.0000 −1.49097
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ −1.00000 −0.0410997
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ −32.0000 −1.30422
$$603$$ 4.00000 0.162893
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 6.00000 0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 24.0000 0.966988
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 14.0000 0.563163
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 16.0000 0.639489
$$627$$ −12.0000 −0.479234
$$628$$ −14.0000 −0.558661
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 16.0000 0.635943
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −18.0000 −0.713186
$$638$$ 36.0000 1.42525
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −72.0000 −2.82625
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ 16.0000 0.626608
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −20.0000 −0.782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −14.0000 −0.546192
$$658$$ −24.0000 −0.935617
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 20.0000 0.777910 0.388955 0.921257i $$-0.372836\pi$$
0.388955 + 0.921257i $$0.372836\pi$$
$$662$$ 14.0000 0.544125
$$663$$ 12.0000 0.466041
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 48.0000 1.85302
$$672$$ −4.00000 −0.154303
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 10.0000 0.385186
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 6.00000 0.230429
$$679$$ −32.0000 −1.22805
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 48.0000 1.83801
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ −26.0000 −0.991962
$$688$$ −8.00000 −0.304997
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 24.0000 0.911685
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ −36.0000 −1.36360
$$698$$ 26.0000 0.984115
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −2.00000 −0.0754314
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ 12.0000 0.450988
$$709$$ 44.0000 1.65245 0.826227 0.563337i $$-0.190483\pi$$
0.826227 + 0.563337i $$0.190483\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ −24.0000 −0.896296
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −56.0000 −2.08555
$$722$$ −15.0000 −0.558242
$$723$$ −2.00000 −0.0743808
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ −25.0000 −0.927837
$$727$$ −38.0000 −1.40934 −0.704671 0.709534i $$-0.748905\pi$$
−0.704671 + 0.709534i $$0.748905\pi$$
$$728$$ −8.00000 −0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ −8.00000 −0.295689
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.0000 0.884051
$$738$$ −6.00000 −0.220863
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ −24.0000 −0.881068
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 12.0000 0.439057
$$748$$ 36.0000 1.31629
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 80.0000 2.89619
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 24.0000 0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ 16.0000 0.575853
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ 4.00000 0.143499
$$778$$ −30.0000 −1.07555
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 6.00000 0.213201
$$793$$ −16.0000 −0.568177
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 18.0000 0.635602
$$803$$ −84.0000 −2.96430
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 24.0000 0.844840
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 24.0000 0.842235
$$813$$ −8.00000 −0.280572
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ −16.0000 −0.559769
$$818$$ −10.0000 −0.349642
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −28.0000 −0.972480 −0.486240 0.873825i $$-0.661632\pi$$
−0.486240 + 0.873825i $$0.661632\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ −2.00000 −0.0693375
$$833$$ 54.0000 1.87099
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ −8.00000 −0.276520
$$838$$ 6.00000 0.207267
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 8.00000 0.275698
$$843$$ 30.0000 1.03325
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 100.000 3.43604
$$848$$ −6.00000 −0.206041
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 32.0000 1.09502
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 12.0000 0.409673
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ −30.0000 −1.02121 −0.510606 0.859815i $$-0.670579\pi$$
−0.510606 + 0.859815i $$0.670579\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −38.0000 −1.29129
$$867$$ −19.0000 −0.645274
$$868$$ 32.0000 1.08615
$$869$$ −96.0000 −3.25658
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 20.0000 0.677285
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 20.0000 0.674967
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ 1.00000 0.0335578
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 4.00000 0.133930
$$893$$ −12.0000 −0.401565
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ −36.0000 −1.19867
$$903$$ 32.0000 1.06489
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 40.0000 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 72.0000 2.38285
$$914$$ −8.00000 −0.264616
$$915$$ 0 0
$$916$$ 26.0000 0.859064
$$917$$ −48.0000 −1.58510
$$918$$ −6.00000 −0.198030
$$919$$ −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
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ −24.0000 −0.789542
$$925$$ 0 0
$$926$$ 34.0000 1.11731
$$927$$ −14.0000 −0.459820
$$928$$ 6.00000 0.196960
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 24.0000 0.786146
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ 14.0000 0.456145
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 24.0000 0.777844
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ −36.0000 −1.16371
$$958$$ −24.0000 −0.775405
$$959$$ 48.0000 1.55000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 2.00000 0.0644826
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46.0000 1.47926 0.739630 0.673014i $$-0.235000\pi$$
0.739630 + 0.673014i $$0.235000\pi$$
$$968$$ 25.0000 0.803530
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ −54.0000 −1.73294 −0.866471 0.499227i $$-0.833617\pi$$
−0.866471 + 0.499227i $$0.833617\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ −6.00000 −0.191468
$$983$$ 30.0000 0.956851 0.478426 0.878128i $$-0.341208\pi$$
0.478426 + 0.878128i $$0.341208\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 24.0000 0.763928
$$988$$ −4.00000 −0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −14.0000 −0.444277
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 14.0000 0.443162
$$999$$ 1.00000 0.0316386
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 5550.2.a.bd.1.1 1
5.4 even 2 1110.2.a.h.1.1 1
15.14 odd 2 3330.2.a.m.1.1 1
20.19 odd 2 8880.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.h.1.1 1 5.4 even 2
3330.2.a.m.1.1 1 15.14 odd 2
5550.2.a.bd.1.1 1 1.1 even 1 trivial
8880.2.a.n.1.1 1 20.19 odd 2