# Properties

 Label 1050.2.a.d.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

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

## 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
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ −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
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.00000 −0.348155
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ −6.00000 −0.832050
$$53$$ −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
$$56$$ −1.00000 −0.133631
$$57$$ 6.00000 0.794719
$$58$$ −6.00000 −0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 2.00000 0.227921
$$78$$ −6.00000 −0.679366
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ −6.00000 −0.643268
$$88$$ −2.00000 −0.213201
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 8.00000 0.834058
$$93$$ 2.00000 0.207390
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 1.00000 0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −6.00000 −0.554700
$$118$$ 8.00000 0.736460
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 10.0000 0.905357
$$123$$ −2.00000 −0.180334
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ −6.00000 −0.520266
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 6.00000 0.503509
$$143$$ −12.0000 −1.00349
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ −1.00000 −0.0824786
$$148$$ −4.00000 −0.328798
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −4.00000 −0.323381
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ −4.00000 −0.304997
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 8.00000 0.601317
$$178$$ 10.0000 0.749532
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 10.0000 0.739221
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ −8.00000 −0.585018
$$188$$ −8.00000 −0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 8.00000 0.575853 0.287926 0.957653i $$-0.407034\pi$$
0.287926 + 0.957653i $$0.407034\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 6.00000 0.425329 0.212664 0.977125i $$-0.431786\pi$$
0.212664 + 0.977125i $$0.431786\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ −10.0000 −0.703598
$$203$$ 6.00000 0.421117
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 8.00000 0.556038
$$208$$ −6.00000 −0.416025
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 6.00000 0.411113
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −2.00000 −0.135769
$$218$$ 14.0000 0.948200
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ −4.00000 −0.268462
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 6.00000 0.397360
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ −6.00000 −0.393919
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ 12.0000 0.779484
$$238$$ 4.00000 0.259281
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 7.00000 0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 36.0000 2.29063
$$248$$ 2.00000 0.127000
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 16.0000 1.00591
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 6.00000 0.367884
$$267$$ 10.0000 0.611990
$$268$$ −8.00000 −0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 10.0000 0.607457 0.303728 0.952759i $$-0.401768\pi$$
0.303728 + 0.952759i $$0.401768\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 6.00000 0.363137
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 14.0000 0.839664
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 2.00000 0.118056
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000 0.819288
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ −2.00000 −0.116052
$$298$$ −10.0000 −0.579284
$$299$$ −48.0000 −2.77591
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 8.00000 0.460348
$$303$$ −10.0000 −0.574485
$$304$$ −6.00000 −0.344124
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ −8.00000 −0.445823
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 14.0000 0.774202
$$328$$ −2.00000 −0.110432
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 8.00000 0.439057
$$333$$ −4.00000 −0.219199
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 6.00000 0.324443
$$343$$ 1.00000 0.0539949
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 8.00000 0.430083
$$347$$ −36.0000 −1.93258 −0.966291 0.257454i $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ −2.00000 −0.106600
$$353$$ −20.0000 −1.06449 −0.532246 0.846590i $$-0.678652\pi$$
−0.532246 + 0.846590i $$0.678652\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 4.00000 0.211702
$$358$$ −10.0000 −0.528516
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −2.00000 −0.105118
$$363$$ 7.00000 0.367405
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 2.00000 0.103695
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −36.0000 −1.85409
$$378$$ 1.00000 0.0514344
$$379$$ 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
$$382$$ 18.0000 0.920960
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ −4.00000 −0.203331
$$388$$ 10.0000 0.507673
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ −32.0000 −1.61831
$$392$$ −1.00000 −0.0505076
$$393$$ 12.0000 0.605320
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −6.00000 −0.300753
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 12.0000 0.597763
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ −8.00000 −0.396545
$$408$$ −4.00000 −0.198030
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ −8.00000 −0.394132
$$413$$ −8.00000 −0.393654
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 14.0000 0.685583
$$418$$ 12.0000 0.586939
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 0 0
$$423$$ −8.00000 −0.388973
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ −10.0000 −0.483934
$$428$$ −12.0000 −0.580042
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ 2.00000 0.0963366 0.0481683 0.998839i $$-0.484662\pi$$
0.0481683 + 0.998839i $$0.484662\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ −48.0000 −2.29615
$$438$$ 14.0000 0.668946
$$439$$ −18.0000 −0.859093 −0.429547 0.903045i $$-0.641327\pi$$
−0.429547 + 0.903045i $$0.641327\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −24.0000 −1.14156
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ −10.0000 −0.472984
$$448$$ 1.00000 0.0472456
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ −6.00000 −0.282216
$$453$$ 8.00000 0.375873
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 2.00000 0.0930484
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 8.00000 0.368230
$$473$$ −8.00000 −0.367840
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ −6.00000 −0.274721
$$478$$ −26.0000 −1.18921
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ 26.0000 1.18427
$$483$$ −8.00000 −0.364013
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ −24.0000 −1.08091
$$494$$ −36.0000 −1.61972
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ −6.00000 −0.269137
$$498$$ 8.00000 0.358489
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ −23.0000 −1.02147
$$508$$ −4.00000 −0.177471
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ −1.00000 −0.0441942
$$513$$ 6.00000 0.264906
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ −16.0000 −0.703679
$$518$$ 4.00000 0.175750
$$519$$ 8.00000 0.351161
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 8.00000 0.348485
$$528$$ −2.00000 −0.0870388
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ −6.00000 −0.260133
$$533$$ −12.0000 −0.519778
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ −10.0000 −0.431532
$$538$$ −18.0000 −0.776035
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ −10.0000 −0.429537
$$543$$ −2.00000 −0.0858282
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ −6.00000 −0.256776
$$547$$ −40.0000 −1.71028 −0.855138 0.518400i $$-0.826528\pi$$
−0.855138 + 0.518400i $$0.826528\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −36.0000 −1.53365
$$552$$ 8.00000 0.340503
$$553$$ −12.0000 −0.510292
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ −30.0000 −1.26547
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 1.00000 0.0419961
$$568$$ 6.00000 0.251754
$$569$$ −46.0000 −1.92842 −0.964210 0.265139i $$-0.914582\pi$$
−0.964210 + 0.265139i $$0.914582\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 18.0000 0.751961
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 26.0000 1.08239 0.541197 0.840896i $$-0.317971\pi$$
0.541197 + 0.840896i $$0.317971\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 10.0000 0.414513
$$583$$ −12.0000 −0.496989
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ −4.00000 −0.164399
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ −6.00000 −0.245564
$$598$$ 48.0000 1.96287
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −8.00000 −0.325785
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 6.00000 0.243332
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ −4.00000 −0.161690
$$613$$ −28.0000 −1.13091 −0.565455 0.824779i $$-0.691299\pi$$
−0.565455 + 0.824779i $$0.691299\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −24.0000 −0.962312
$$623$$ −10.0000 −0.400642
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 12.0000 0.479234
$$628$$ 22.0000 0.877896
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −6.00000 −0.237729
$$638$$ −12.0000 −0.475085
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ −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
$$651$$ 2.00000 0.0783862
$$652$$ 4.00000 0.156652
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 14.0000 0.546192
$$658$$ 8.00000 0.311872
$$659$$ −26.0000 −1.01282 −0.506408 0.862294i $$-0.669027\pi$$
−0.506408 + 0.862294i $$0.669027\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ 8.00000 0.310929
$$663$$ −24.0000 −0.932083
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 48.0000 1.85857
$$668$$ 12.0000 0.464294
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −20.0000 −0.772091
$$672$$ 1.00000 0.0385758
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −40.0000 −1.53732 −0.768662 0.639655i $$-0.779077\pi$$
−0.768662 + 0.639655i $$0.779077\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ −8.00000 −0.306561
$$682$$ 4.00000 0.153168
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ −14.0000 −0.534133
$$688$$ −4.00000 −0.152499
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −2.00000 −0.0760836 −0.0380418 0.999276i $$-0.512112\pi$$
−0.0380418 + 0.999276i $$0.512112\pi$$
$$692$$ −8.00000 −0.304114
$$693$$ 2.00000 0.0759737
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ −8.00000 −0.303022
$$698$$ 26.0000 0.984115
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ 24.0000 0.905177
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ 10.0000 0.376089
$$708$$ 8.00000 0.300658
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 10.0000 0.374766
$$713$$ −16.0000 −0.599205
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ −26.0000 −0.970988
$$718$$ −6.00000 −0.223918
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ −17.0000 −0.632674
$$723$$ 26.0000 0.966950
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ −24.0000 −0.890111 −0.445055 0.895503i $$-0.646816\pi$$
−0.445055 + 0.895503i $$0.646816\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 10.0000 0.369611
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ −16.0000 −0.589368
$$738$$ −2.00000 −0.0736210
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ −36.0000 −1.32249
$$742$$ 6.00000 0.220267
$$743$$ 48.0000 1.76095 0.880475 0.474093i $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 0 0
$$746$$ −24.0000 −0.878702
$$747$$ 8.00000 0.292705
$$748$$ −8.00000 −0.292509
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 36.0000 1.31104
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 32.0000 1.16306 0.581530 0.813525i $$-0.302454\pi$$
0.581530 + 0.813525i $$0.302454\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ −14.0000 −0.506834
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ 48.0000 1.73318
$$768$$ −1.00000 −0.0360844
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 4.00000 0.143499
$$778$$ −2.00000 −0.0717035
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ 32.0000 1.14432
$$783$$ −6.00000 −0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ −2.00000 −0.0710669
$$793$$ 60.0000 2.13066
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 6.00000 0.212664
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ −6.00000 −0.212398
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 14.0000 0.494357
$$803$$ 28.0000 0.988099
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −12.0000 −0.422682
$$807$$ −18.0000 −0.633630
$$808$$ −10.0000 −0.351799
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ 6.00000 0.210559
$$813$$ −10.0000 −0.350715
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 24.0000 0.839654
$$818$$ −26.0000 −0.909069
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 28.0000 0.971309
$$832$$ −6.00000 −0.208013
$$833$$ −4.00000 −0.138592
$$834$$ −14.0000 −0.484780
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ 2.00000 0.0691301
$$838$$ −20.0000 −0.690889
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −34.0000 −1.17172
$$843$$ −30.0000 −1.03325
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ −7.00000 −0.240523
$$848$$ −6.00000 −0.206041
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 6.00000 0.205557
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −24.0000 −0.819824 −0.409912 0.912125i $$-0.634441\pi$$
−0.409912 + 0.912125i $$0.634441\pi$$
$$858$$ −12.0000 −0.409673
$$859$$ −10.0000 −0.341196 −0.170598 0.985341i $$-0.554570\pi$$
−0.170598 + 0.985341i $$0.554570\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −0.0681598
$$862$$ −2.00000 −0.0681203
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ 1.00000 0.0339618
$$868$$ −2.00000 −0.0678844
$$869$$ −24.0000 −0.814144
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 14.0000 0.474100
$$873$$ 10.0000 0.338449
$$874$$ 48.0000 1.62362
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ 28.0000 0.945493 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$878$$ 18.0000 0.607471
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 8.00000 0.267860
$$893$$ 48.0000 1.60626
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 48.0000 1.60267
$$898$$ 6.00000 0.200223
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ −4.00000 −0.133185
$$903$$ 4.00000 0.133112
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 6.00000 0.198680
$$913$$ 16.0000 0.529523
$$914$$ 28.0000 0.926158
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ −12.0000 −0.396275
$$918$$ −4.00000 −0.132020
$$919$$ 60.0000 1.97922 0.989609 0.143787i $$-0.0459280\pi$$
0.989609 + 0.143787i $$0.0459280\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 18.0000 0.592798
$$923$$ 36.0000 1.18495
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ −8.00000 −0.262754
$$928$$ −6.00000 −0.196960
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 10.0000 0.327561
$$933$$ −24.0000 −0.785725
$$934$$ −24.0000 −0.785304
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ 8.00000 0.261209
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 22.0000 0.716799
$$943$$ 16.0000 0.521032
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 12.0000 0.389742
$$949$$ −84.0000 −2.72676
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 4.00000 0.129641
$$953$$ 10.0000 0.323932 0.161966 0.986796i $$-0.448217\pi$$
0.161966 + 0.986796i $$0.448217\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 26.0000 0.840900
$$957$$ −12.0000 −0.387905
$$958$$ −8.00000 −0.258468
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −24.0000 −0.773791
$$963$$ −12.0000 −0.386695
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ 7.00000 0.224989
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −14.0000 −0.448819
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 30.0000 0.957338
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 8.00000 0.254643
$$988$$ 36.0000 1.14531
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 8.00000 0.253872
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 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 1050.2.a.d.1.1 1
3.2 odd 2 3150.2.a.bk.1.1 1
4.3 odd 2 8400.2.a.bp.1.1 1
5.2 odd 4 210.2.g.b.169.1 2
5.3 odd 4 210.2.g.b.169.2 yes 2
5.4 even 2 1050.2.a.p.1.1 1
7.6 odd 2 7350.2.a.bk.1.1 1
15.2 even 4 630.2.g.c.379.2 2
15.8 even 4 630.2.g.c.379.1 2
15.14 odd 2 3150.2.a.d.1.1 1
20.3 even 4 1680.2.t.e.1009.2 2
20.7 even 4 1680.2.t.e.1009.1 2
20.19 odd 2 8400.2.a.w.1.1 1
35.2 odd 12 1470.2.n.b.949.1 4
35.3 even 12 1470.2.n.f.79.1 4
35.12 even 12 1470.2.n.f.949.1 4
35.13 even 4 1470.2.g.b.589.2 2
35.17 even 12 1470.2.n.f.79.2 4
35.18 odd 12 1470.2.n.b.79.1 4
35.23 odd 12 1470.2.n.b.949.2 4
35.27 even 4 1470.2.g.b.589.1 2
35.32 odd 12 1470.2.n.b.79.2 4
35.33 even 12 1470.2.n.f.949.2 4
35.34 odd 2 7350.2.a.bz.1.1 1
60.23 odd 4 5040.2.t.h.1009.2 2
60.47 odd 4 5040.2.t.h.1009.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.b.169.1 2 5.2 odd 4
210.2.g.b.169.2 yes 2 5.3 odd 4
630.2.g.c.379.1 2 15.8 even 4
630.2.g.c.379.2 2 15.2 even 4
1050.2.a.d.1.1 1 1.1 even 1 trivial
1050.2.a.p.1.1 1 5.4 even 2
1470.2.g.b.589.1 2 35.27 even 4
1470.2.g.b.589.2 2 35.13 even 4
1470.2.n.b.79.1 4 35.18 odd 12
1470.2.n.b.79.2 4 35.32 odd 12
1470.2.n.b.949.1 4 35.2 odd 12
1470.2.n.b.949.2 4 35.23 odd 12
1470.2.n.f.79.1 4 35.3 even 12
1470.2.n.f.79.2 4 35.17 even 12
1470.2.n.f.949.1 4 35.12 even 12
1470.2.n.f.949.2 4 35.33 even 12
1680.2.t.e.1009.1 2 20.7 even 4
1680.2.t.e.1009.2 2 20.3 even 4
3150.2.a.d.1.1 1 15.14 odd 2
3150.2.a.bk.1.1 1 3.2 odd 2
5040.2.t.h.1009.1 2 60.47 odd 4
5040.2.t.h.1009.2 2 60.23 odd 4
7350.2.a.bk.1.1 1 7.6 odd 2
7350.2.a.bz.1.1 1 35.34 odd 2
8400.2.a.w.1.1 1 20.19 odd 2
8400.2.a.bp.1.1 1 4.3 odd 2