# Properties

 Label 550.2.a.f.1.1 Level $550$ Weight $2$ Character 550.1 Self dual yes Analytic conductor $4.392$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.39177211117$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +2.00000 q^{18} +5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} -2.00000 q^{36} -3.00000 q^{37} -5.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +7.00000 q^{51} +6.00000 q^{52} +1.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} +5.00000 q^{57} -5.00000 q^{58} -10.0000 q^{59} +7.00000 q^{61} +3.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} +7.00000 q^{68} +6.00000 q^{69} +7.00000 q^{71} +2.00000 q^{72} -14.0000 q^{73} +3.00000 q^{74} +5.00000 q^{76} -3.00000 q^{77} -6.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -3.00000 q^{84} +4.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} -18.0000 q^{91} +6.00000 q^{92} -3.00000 q^{93} -2.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 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 $$0.187167\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ −1.00000 −0.213201
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ −5.00000 −0.962250
$$28$$ −3.00000 −0.566947
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ −7.00000 −1.20049
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ −5.00000 −0.811107
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 3.00000 0.462910
$$43$$ −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
$$46$$ −6.00000 −0.884652
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 7.00000 0.980196
$$52$$ 6.00000 0.832050
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 5.00000 0.662266
$$58$$ −5.00000 −0.656532
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 7.00000 0.848875
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 2.00000 0.235702
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 5.00000 0.573539
$$77$$ −3.00000 −0.341882
$$78$$ −6.00000 −0.679366
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 5.00000 0.536056
$$88$$ −1.00000 −0.106600
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ −18.0000 −1.88691
$$92$$ 6.00000 0.625543
$$93$$ −3.00000 −0.311086
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −7.00000 −0.693103
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ −3.00000 −0.283473
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ −5.00000 −0.468293
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ −12.0000 −1.10940
$$118$$ 10.0000 0.920575
$$119$$ −21.0000 −1.92507
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −7.00000 −0.633750
$$123$$ 2.00000 0.180334
$$124$$ −3.00000 −0.269408
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 17.0000 1.48530 0.742648 0.669681i $$-0.233569\pi$$
0.742648 + 0.669681i $$0.233569\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ −15.0000 −1.30066
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −7.00000 −0.600245
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ −7.00000 −0.587427
$$143$$ 6.00000 0.501745
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 2.00000 0.164957
$$148$$ −3.00000 −0.246598
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ −5.00000 −0.405554
$$153$$ −14.0000 −1.13183
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ −3.00000 −0.239426 −0.119713 0.992809i $$-0.538197\pi$$
−0.119713 + 0.992809i $$0.538197\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 1.00000 0.0793052
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ −1.00000 −0.0785674
$$163$$ −19.0000 −1.48819 −0.744097 0.668071i $$-0.767120\pi$$
−0.744097 + 0.668071i $$0.767120\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −10.0000 −0.764719
$$172$$ −4.00000 −0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ −10.0000 −0.751646
$$178$$ 15.0000 1.12430
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 18.0000 1.33425
$$183$$ 7.00000 0.517455
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 3.00000 0.219971
$$187$$ 7.00000 0.511891
$$188$$ 2.00000 0.145865
$$189$$ 15.0000 1.09109
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 2.00000 0.142134
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ −2.00000 −0.140720
$$203$$ −15.0000 −1.05279
$$204$$ 7.00000 0.490098
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ −12.0000 −0.834058
$$208$$ 6.00000 0.416025
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ −23.0000 −1.58339 −0.791693 0.610920i $$-0.790800\pi$$
−0.791693 + 0.610920i $$0.790800\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 7.00000 0.479632
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 9.00000 0.610960
$$218$$ 10.0000 0.677285
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 42.0000 2.82523
$$222$$ 3.00000 0.201347
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ −16.0000 −1.06430
$$227$$ 2.00000 0.132745 0.0663723 0.997795i $$-0.478857\pi$$
0.0663723 + 0.997795i $$0.478857\pi$$
$$228$$ 5.00000 0.331133
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ −5.00000 −0.328266
$$233$$ −9.00000 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 10.0000 0.649570
$$238$$ 21.0000 1.36123
$$239$$ 10.0000 0.646846 0.323423 0.946254i $$-0.395166\pi$$
0.323423 + 0.946254i $$0.395166\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 16.0000 1.02640
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 30.0000 1.90885
$$248$$ 3.00000 0.190500
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 6.00000 0.377217
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ −17.0000 −1.05026
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ 15.0000 0.919709
$$267$$ −15.0000 −0.917985
$$268$$ −8.00000 −0.488678
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 7.00000 0.424437
$$273$$ −18.0000 −1.08941
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −2.00000 −0.119098
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 7.00000 0.415374
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ −6.00000 −0.354169
$$288$$ 2.00000 0.117851
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ −14.0000 −0.819288
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ −5.00000 −0.290129
$$298$$ −15.0000 −0.868927
$$299$$ 36.0000 2.08193
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ −2.00000 −0.115087
$$303$$ 2.00000 0.114897
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ 14.0000 0.800327
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 3.00000 0.169300
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 7.00000 0.393159 0.196580 0.980488i $$-0.437017\pi$$
0.196580 + 0.980488i $$0.437017\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 18.0000 1.00310
$$323$$ 35.0000 1.94745
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 19.0000 1.05231
$$327$$ −10.0000 −0.553001
$$328$$ −2.00000 −0.110432
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 6.00000 0.328798
$$334$$ 3.00000 0.164153
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 17.0000 0.926049 0.463025 0.886345i $$-0.346764\pi$$
0.463025 + 0.886345i $$0.346764\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ 10.0000 0.540738
$$343$$ 15.0000 0.809924
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 5.00000 0.268028
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ −30.0000 −1.60128
$$352$$ −1.00000 −0.0533002
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ −21.0000 −1.11144
$$358$$ 0 0
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −2.00000 −0.105118
$$363$$ 1.00000 0.0524864
$$364$$ −18.0000 −0.943456
$$365$$ 0 0
$$366$$ −7.00000 −0.365896
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 6.00000 0.312772
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ −3.00000 −0.155543
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ −7.00000 −0.361961
$$375$$ 0 0
$$376$$ −2.00000 −0.103142
$$377$$ 30.0000 1.54508
$$378$$ −15.0000 −0.771517
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ −12.0000 −0.613973
$$383$$ −34.0000 −1.73732 −0.868659 0.495410i $$-0.835018\pi$$
−0.868659 + 0.495410i $$0.835018\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −11.0000 −0.559885
$$387$$ 8.00000 0.406663
$$388$$ 12.0000 0.609208
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 42.0000 2.12403
$$392$$ −2.00000 −0.101015
$$393$$ 17.0000 0.857537
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 25.0000 1.25314
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −13.0000 −0.649189 −0.324595 0.945853i $$-0.605228\pi$$
−0.324595 + 0.945853i $$0.605228\pi$$
$$402$$ 8.00000 0.399004
$$403$$ −18.0000 −0.896644
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 15.0000 0.744438
$$407$$ −3.00000 −0.148704
$$408$$ −7.00000 −0.346552
$$409$$ −20.0000 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ −4.00000 −0.197066
$$413$$ 30.0000 1.47620
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ −20.0000 −0.979404
$$418$$ −5.00000 −0.244558
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 23.0000 1.11962
$$423$$ −4.00000 −0.194487
$$424$$ −1.00000 −0.0485643
$$425$$ 0 0
$$426$$ −7.00000 −0.339151
$$427$$ −21.0000 −1.01626
$$428$$ −8.00000 −0.386695
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ −9.00000 −0.432014
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 30.0000 1.43509
$$438$$ 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
$$441$$ −4.00000 −0.190476
$$442$$ −42.0000 −1.99774
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ 15.0000 0.709476
$$448$$ −3.00000 −0.141737
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 16.0000 0.752577
$$453$$ 2.00000 0.0939682
$$454$$ −2.00000 −0.0938647
$$455$$ 0 0
$$456$$ −5.00000 −0.234146
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −35.0000 −1.63366
$$460$$ 0 0
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 3.00000 0.139573
$$463$$ −34.0000 −1.58011 −0.790057 0.613033i $$-0.789949\pi$$
−0.790057 + 0.613033i $$0.789949\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 9.00000 0.416917
$$467$$ −23.0000 −1.06431 −0.532157 0.846646i $$-0.678618\pi$$
−0.532157 + 0.846646i $$0.678618\pi$$
$$468$$ −12.0000 −0.554700
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ −3.00000 −0.138233
$$472$$ 10.0000 0.460287
$$473$$ −4.00000 −0.183920
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −21.0000 −0.962533
$$477$$ −2.00000 −0.0915737
$$478$$ −10.0000 −0.457389
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −18.0000 −0.820729
$$482$$ 18.0000 0.819878
$$483$$ −18.0000 −0.819028
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ −7.00000 −0.316875
$$489$$ −19.0000 −0.859210
$$490$$ 0 0
$$491$$ −3.00000 −0.135388 −0.0676941 0.997706i $$-0.521564\pi$$
−0.0676941 + 0.997706i $$0.521564\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ 35.0000 1.57632
$$494$$ −30.0000 −1.34976
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ −21.0000 −0.941979
$$498$$ −6.00000 −0.268866
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −3.00000 −0.134030
$$502$$ −2.00000 −0.0892644
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ 23.0000 1.02147
$$508$$ −8.00000 −0.354943
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ 0 0
$$511$$ 42.0000 1.85797
$$512$$ −1.00000 −0.0441942
$$513$$ −25.0000 −1.10378
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 2.00000 0.0879599
$$518$$ −9.00000 −0.395437
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 17.0000 0.742648
$$525$$ 0 0
$$526$$ 9.00000 0.392419
$$527$$ −21.0000 −0.914774
$$528$$ 1.00000 0.0435194
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 20.0000 0.867926
$$532$$ −15.0000 −0.650332
$$533$$ 12.0000 0.519778
$$534$$ 15.0000 0.649113
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 20.0000 0.862261
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −23.0000 −0.988847 −0.494424 0.869221i $$-0.664621\pi$$
−0.494424 + 0.869221i $$0.664621\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 2.00000 0.0858282
$$544$$ −7.00000 −0.300123
$$545$$ 0 0
$$546$$ 18.0000 0.770329
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 12.0000 0.512615
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ 25.0000 1.06504
$$552$$ −6.00000 −0.255377
$$553$$ −30.0000 −1.27573
$$554$$ −12.0000 −0.509831
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 7.00000 0.295540
$$562$$ 18.0000 0.759284
$$563$$ 6.00000 0.252870 0.126435 0.991975i $$-0.459647\pi$$
0.126435 + 0.991975i $$0.459647\pi$$
$$564$$ 2.00000 0.0842152
$$565$$ 0 0
$$566$$ −6.00000 −0.252199
$$567$$ −3.00000 −0.125988
$$568$$ −7.00000 −0.293713
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 27.0000 1.12991 0.564957 0.825120i $$-0.308893\pi$$
0.564957 + 0.825120i $$0.308893\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 12.0000 0.501307
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ −12.0000 −0.497416
$$583$$ 1.00000 0.0414158
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 27.0000 1.11441 0.557205 0.830375i $$-0.311874\pi$$
0.557205 + 0.830375i $$0.311874\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ −15.0000 −0.618064
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ −3.00000 −0.123299
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ 15.0000 0.614424
$$597$$ −25.0000 −1.02318
$$598$$ −36.0000 −1.47215
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ 16.0000 0.651570
$$604$$ 2.00000 0.0813788
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 47.0000 1.90767 0.953836 0.300329i $$-0.0970966\pi$$
0.953836 + 0.300329i $$0.0970966\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ −15.0000 −0.607831
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ −14.0000 −0.565916
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ −8.00000 −0.322068 −0.161034 0.986949i $$-0.551483\pi$$
−0.161034 + 0.986949i $$0.551483\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 3.00000 0.120289
$$623$$ 45.0000 1.80289
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 5.00000 0.199681
$$628$$ −3.00000 −0.119713
$$629$$ −21.0000 −0.837325
$$630$$ 0 0
$$631$$ −33.0000 −1.31371 −0.656855 0.754017i $$-0.728113\pi$$
−0.656855 + 0.754017i $$0.728113\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ −23.0000 −0.914168
$$634$$ −7.00000 −0.278006
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ 12.0000 0.475457
$$638$$ −5.00000 −0.197952
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 8.00000 0.315735
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ 0 0
$$646$$ −35.0000 −1.37706
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −10.0000 −0.392534
$$650$$ 0 0
$$651$$ 9.00000 0.352738
$$652$$ −19.0000 −0.744097
$$653$$ 31.0000 1.21312 0.606562 0.795036i $$-0.292548\pi$$
0.606562 + 0.795036i $$0.292548\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 28.0000 1.09238
$$658$$ 6.00000 0.233904
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 42.0000 1.63114
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 30.0000 1.16160
$$668$$ −3.00000 −0.116073
$$669$$ 6.00000 0.231973
$$670$$ 0 0
$$671$$ 7.00000 0.270232
$$672$$ 3.00000 0.115728
$$673$$ −29.0000 −1.11787 −0.558934 0.829212i $$-0.688789\pi$$
−0.558934 + 0.829212i $$0.688789\pi$$
$$674$$ −17.0000 −0.654816
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −28.0000 −1.07613 −0.538064 0.842904i $$-0.680844\pi$$
−0.538064 + 0.842904i $$0.680844\pi$$
$$678$$ −16.0000 −0.614476
$$679$$ −36.0000 −1.38155
$$680$$ 0 0
$$681$$ 2.00000 0.0766402
$$682$$ 3.00000 0.114876
$$683$$ 31.0000 1.18618 0.593091 0.805135i $$-0.297907\pi$$
0.593091 + 0.805135i $$0.297907\pi$$
$$684$$ −10.0000 −0.382360
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 10.0000 0.381524
$$688$$ −4.00000 −0.152499
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −38.0000 −1.44559 −0.722794 0.691063i $$-0.757142\pi$$
−0.722794 + 0.691063i $$0.757142\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 6.00000 0.227921
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ −5.00000 −0.189525
$$697$$ 14.0000 0.530288
$$698$$ −30.0000 −1.13552
$$699$$ −9.00000 −0.340411
$$700$$ 0 0
$$701$$ 7.00000 0.264386 0.132193 0.991224i $$-0.457798\pi$$
0.132193 + 0.991224i $$0.457798\pi$$
$$702$$ 30.0000 1.13228
$$703$$ −15.0000 −0.565736
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ −6.00000 −0.225653
$$708$$ −10.0000 −0.375823
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −20.0000 −0.750059
$$712$$ 15.0000 0.562149
$$713$$ −18.0000 −0.674105
$$714$$ 21.0000 0.785905
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 10.0000 0.373457
$$718$$ 20.0000 0.746393
$$719$$ 25.0000 0.932343 0.466171 0.884694i $$-0.345633\pi$$
0.466171 + 0.884694i $$0.345633\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ −6.00000 −0.223297
$$723$$ −18.0000 −0.669427
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −1.00000 −0.0371135
$$727$$ 22.0000 0.815935 0.407967 0.912996i $$-0.366238\pi$$
0.407967 + 0.912996i $$0.366238\pi$$
$$728$$ 18.0000 0.667124
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −28.0000 −1.03562
$$732$$ 7.00000 0.258727
$$733$$ −24.0000 −0.886460 −0.443230 0.896408i $$-0.646168\pi$$
−0.443230 + 0.896408i $$0.646168\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ −8.00000 −0.294684
$$738$$ 4.00000 0.147242
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 30.0000 1.10208
$$742$$ 3.00000 0.110133
$$743$$ 21.0000 0.770415 0.385208 0.922830i $$-0.374130\pi$$
0.385208 + 0.922830i $$0.374130\pi$$
$$744$$ 3.00000 0.109985
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ −12.0000 −0.439057
$$748$$ 7.00000 0.255945
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 17.0000 0.620339 0.310169 0.950681i $$-0.399614\pi$$
0.310169 + 0.950681i $$0.399614\pi$$
$$752$$ 2.00000 0.0729325
$$753$$ 2.00000 0.0728841
$$754$$ −30.0000 −1.09254
$$755$$ 0 0
$$756$$ 15.0000 0.545545
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 30.0000 1.08965
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 30.0000 1.08607
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 34.0000 1.22847
$$767$$ −60.0000 −2.16647
$$768$$ 1.00000 0.0360844
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 11.0000 0.395899
$$773$$ −19.0000 −0.683383 −0.341691 0.939812i $$-0.611000\pi$$
−0.341691 + 0.939812i $$0.611000\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ 9.00000 0.322873
$$778$$ 30.0000 1.07555
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 7.00000 0.250480
$$782$$ −42.0000 −1.50192
$$783$$ −25.0000 −0.893427
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ −17.0000 −0.606370
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 12.0000 0.427482
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 2.00000 0.0710669
$$793$$ 42.0000 1.49146
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −25.0000 −0.886102
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 15.0000 0.530994
$$799$$ 14.0000 0.495284
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 13.0000 0.459046
$$803$$ −14.0000 −0.494049
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 18.0000 0.634023
$$807$$ −20.0000 −0.704033
$$808$$ −2.00000 −0.0703598
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 7.00000 0.245803 0.122902 0.992419i $$-0.460780\pi$$
0.122902 + 0.992419i $$0.460780\pi$$
$$812$$ −15.0000 −0.526397
$$813$$ −8.00000 −0.280572
$$814$$ 3.00000 0.105150
$$815$$ 0 0
$$816$$ 7.00000 0.245049
$$817$$ −20.0000 −0.699711
$$818$$ 20.0000 0.699284
$$819$$ 36.0000 1.25794
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −30.0000 −1.04383
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.416275
$$832$$ 6.00000 0.208013
$$833$$ 14.0000 0.485071
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 5.00000 0.172929
$$837$$ 15.0000 0.518476
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −32.0000 −1.10279
$$843$$ −18.0000 −0.619953
$$844$$ −23.0000 −0.791693
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ −3.00000 −0.103081
$$848$$ 1.00000 0.0343401
$$849$$ 6.00000 0.205919
$$850$$ 0 0
$$851$$ −18.0000 −0.617032
$$852$$ 7.00000 0.239816
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 21.0000 0.718605
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ 7.00000 0.239115 0.119558 0.992827i $$-0.461852\pi$$
0.119558 + 0.992827i $$0.461852\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 30.0000 1.02359 0.511793 0.859109i $$-0.328981\pi$$
0.511793 + 0.859109i $$0.328981\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 8.00000 0.272481
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 32.0000 1.08678
$$868$$ 9.00000 0.305480
$$869$$ 10.0000 0.339227
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 10.0000 0.338643
$$873$$ −24.0000 −0.812277
$$874$$ −30.0000 −1.01477
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 4.00000 0.134687
$$883$$ −9.00000 −0.302874 −0.151437 0.988467i $$-0.548390\pi$$
−0.151437 + 0.988467i $$0.548390\pi$$
$$884$$ 42.0000 1.41261
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 3.00000 0.100673
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 6.00000 0.200895
$$893$$ 10.0000 0.334637
$$894$$ −15.0000 −0.501675
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 36.0000 1.20201
$$898$$ 30.0000 1.00111
$$899$$ −15.0000 −0.500278
$$900$$ 0 0
$$901$$ 7.00000 0.233204
$$902$$ −2.00000 −0.0665927
$$903$$ 12.0000 0.399335
$$904$$ −16.0000 −0.532152
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ 57.0000 1.89265 0.946327 0.323211i $$-0.104762\pi$$
0.946327 + 0.323211i $$0.104762\pi$$
$$908$$ 2.00000 0.0663723
$$909$$ −4.00000 −0.132672
$$910$$ 0 0
$$911$$ 27.0000 0.894550 0.447275 0.894397i $$-0.352395\pi$$
0.447275 + 0.894397i $$0.352395\pi$$
$$912$$ 5.00000 0.165567
$$913$$ 6.00000 0.198571
$$914$$ 3.00000 0.0992312
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ −51.0000 −1.68417
$$918$$ 35.0000 1.15517
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ −27.0000 −0.889198
$$923$$ 42.0000 1.38245
$$924$$ −3.00000 −0.0986928
$$925$$ 0 0
$$926$$ 34.0000 1.11731
$$927$$ 8.00000 0.262754
$$928$$ −5.00000 −0.164133
$$929$$ 35.0000 1.14831 0.574156 0.818746i $$-0.305330\pi$$
0.574156 + 0.818746i $$0.305330\pi$$
$$930$$ 0 0
$$931$$ 10.0000 0.327737
$$932$$ −9.00000 −0.294805
$$933$$ −3.00000 −0.0982156
$$934$$ 23.0000 0.752583
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ −24.0000 −0.783628
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 17.0000 0.554184 0.277092 0.960843i $$-0.410629\pi$$
0.277092 + 0.960843i $$0.410629\pi$$
$$942$$ 3.00000 0.0977453
$$943$$ 12.0000 0.390774
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 10.0000 0.324785
$$949$$ −84.0000 −2.72676
$$950$$ 0 0
$$951$$ 7.00000 0.226991
$$952$$ 21.0000 0.680614
$$953$$ −39.0000 −1.26333 −0.631667 0.775240i $$-0.717629\pi$$
−0.631667 + 0.775240i $$0.717629\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ 10.0000 0.323423
$$957$$ 5.00000 0.161627
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 18.0000 0.580343
$$963$$ 16.0000 0.515593
$$964$$ −18.0000 −0.579741
$$965$$ 0 0
$$966$$ 18.0000 0.579141
$$967$$ 27.0000 0.868261 0.434131 0.900850i $$-0.357056\pi$$
0.434131 + 0.900850i $$0.357056\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 35.0000 1.12436
$$970$$ 0 0
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 60.0000 1.92351
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 19.0000 0.607553
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ 3.00000 0.0957338
$$983$$ −54.0000 −1.72233 −0.861166 0.508323i $$-0.830265\pi$$
−0.861166 + 0.508323i $$0.830265\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ −35.0000 −1.11463
$$987$$ −6.00000 −0.190982
$$988$$ 30.0000 0.954427
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ −28.0000 −0.888553
$$994$$ 21.0000 0.666080
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ 32.0000 1.01345 0.506725 0.862108i $$-0.330856\pi$$
0.506725 + 0.862108i $$0.330856\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ 15.0000 0.474579
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 550.2.a.f.1.1 1
3.2 odd 2 4950.2.a.bc.1.1 1
4.3 odd 2 4400.2.a.l.1.1 1
5.2 odd 4 550.2.b.a.199.1 2
5.3 odd 4 550.2.b.a.199.2 2
5.4 even 2 110.2.a.b.1.1 1
11.10 odd 2 6050.2.a.bj.1.1 1
15.2 even 4 4950.2.c.m.199.2 2
15.8 even 4 4950.2.c.m.199.1 2
15.14 odd 2 990.2.a.d.1.1 1
20.3 even 4 4400.2.b.i.4049.1 2
20.7 even 4 4400.2.b.i.4049.2 2
20.19 odd 2 880.2.a.i.1.1 1
35.34 odd 2 5390.2.a.bf.1.1 1
40.19 odd 2 3520.2.a.h.1.1 1
40.29 even 2 3520.2.a.y.1.1 1
55.54 odd 2 1210.2.a.b.1.1 1
60.59 even 2 7920.2.a.d.1.1 1
220.219 even 2 9680.2.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.b.1.1 1 5.4 even 2
550.2.a.f.1.1 1 1.1 even 1 trivial
550.2.b.a.199.1 2 5.2 odd 4
550.2.b.a.199.2 2 5.3 odd 4
880.2.a.i.1.1 1 20.19 odd 2
990.2.a.d.1.1 1 15.14 odd 2
1210.2.a.b.1.1 1 55.54 odd 2
3520.2.a.h.1.1 1 40.19 odd 2
3520.2.a.y.1.1 1 40.29 even 2
4400.2.a.l.1.1 1 4.3 odd 2
4400.2.b.i.4049.1 2 20.3 even 4
4400.2.b.i.4049.2 2 20.7 even 4
4950.2.a.bc.1.1 1 3.2 odd 2
4950.2.c.m.199.1 2 15.8 even 4
4950.2.c.m.199.2 2 15.2 even 4
5390.2.a.bf.1.1 1 35.34 odd 2
6050.2.a.bj.1.1 1 11.10 odd 2
7920.2.a.d.1.1 1 60.59 even 2
9680.2.a.x.1.1 1 220.219 even 2