# Properties

 Label 1050.2.a.p.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $0$ 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: $$0$$ 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 $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 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 $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 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 $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ −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 $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 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 $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 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 $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 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 $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 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 $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ −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 $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 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 $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 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 $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −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 $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 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 $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ −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 $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −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 $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −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 $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 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 $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 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 $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ −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 $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 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 $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 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 $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ −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 $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ −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 $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −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 $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 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 $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 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 $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ 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 $$-0.321348\pi$$
0.532246 + 0.846590i $$0.321348\pi$$
$$354$$ −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 $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ −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 $$-0.713410\pi$$
−0.621336 + 0.783544i $$0.713410\pi$$
$$374$$ 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 $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 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 $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 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 $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 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 $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 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 $$-0.272714\pi$$
0.654892 + 0.755722i $$0.272714\pi$$
$$458$$ 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 $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 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 $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ −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 $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ −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 $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −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 $$-0.173472\pi$$
0.855138 + 0.518400i $$0.173472\pi$$
$$548$$ −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 $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ −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 $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 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 $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ −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 $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 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 $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ 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 $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −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 $$-0.308701\pi$$
0.565455 + 0.824779i $$0.308701\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 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 $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 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 $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ −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 $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 40.0000 1.53732 0.768662 0.639655i $$-0.220923\pi$$
0.768662 + 0.639655i $$0.220923\pi$$
$$678$$ 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 $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ −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 $$-0.353184\pi$$
0.445055 + 0.895503i $$0.353184\pi$$
$$728$$ −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 $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −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 $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ −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 $$-0.697546\pi$$
−0.581530 + 0.813525i $$0.697546\pi$$
$$758$$ 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 $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 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 $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 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 $$-0.432931\pi$$
0.209147 + 0.977884i $$0.432931\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ −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 $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 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 $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 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 $$-0.656737\pi$$
−0.472746 + 0.881199i $$0.656737\pi$$
$$878$$ −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 $$-0.367678\pi$$
0.403832 + 0.914833i $$0.367678\pi$$
$$884$$ 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 $$-0.369547\pi$$
0.398453 + 0.917189i $$0.369547\pi$$
$$908$$ −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 $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ −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 $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −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 $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ 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 $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ −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 $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −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 $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 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 $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −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.p.1.1 1
3.2 odd 2 3150.2.a.d.1.1 1
4.3 odd 2 8400.2.a.w.1.1 1
5.2 odd 4 210.2.g.b.169.2 yes 2
5.3 odd 4 210.2.g.b.169.1 2
5.4 even 2 1050.2.a.d.1.1 1
7.6 odd 2 7350.2.a.bz.1.1 1
15.2 even 4 630.2.g.c.379.1 2
15.8 even 4 630.2.g.c.379.2 2
15.14 odd 2 3150.2.a.bk.1.1 1
20.3 even 4 1680.2.t.e.1009.1 2
20.7 even 4 1680.2.t.e.1009.2 2
20.19 odd 2 8400.2.a.bp.1.1 1
35.2 odd 12 1470.2.n.b.949.2 4
35.3 even 12 1470.2.n.f.79.2 4
35.12 even 12 1470.2.n.f.949.2 4
35.13 even 4 1470.2.g.b.589.1 2
35.17 even 12 1470.2.n.f.79.1 4
35.18 odd 12 1470.2.n.b.79.2 4
35.23 odd 12 1470.2.n.b.949.1 4
35.27 even 4 1470.2.g.b.589.2 2
35.32 odd 12 1470.2.n.b.79.1 4
35.33 even 12 1470.2.n.f.949.1 4
35.34 odd 2 7350.2.a.bk.1.1 1
60.23 odd 4 5040.2.t.h.1009.1 2
60.47 odd 4 5040.2.t.h.1009.2 2

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