# Properties

 Label 2850.2.a.a.1.1 Level $2850$ Weight $2$ Character 2850.1 Self dual yes Analytic conductor $22.757$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +4.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{38} -2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} -8.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +1.00000 q^{57} -6.00000 q^{58} -10.0000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -10.0000 q^{74} -1.00000 q^{76} +16.0000 q^{77} +2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +8.00000 q^{83} +4.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000 q^{92} -4.00000 q^{93} +1.00000 q^{96} -18.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 4.00000 0.852803
$$23$$ 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
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −4.00000 −0.755929
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.00000 0.696311
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 1.00000 0.132453
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ −4.00000 −0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 2.00000 0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 16.0000 1.82337
$$78$$ 2.00000 0.226455
$$79$$ −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 $$0.355313\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ −6.00000 −0.643268
$$88$$ 4.00000 0.426401
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 8.00000 0.834058
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 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
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ −4.00000 −0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 2.00000 0.180334
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 4.00000 0.346844
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ −9.00000 −0.742307
$$148$$ 10.0000 0.821995
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 2.00000 0.161690
$$154$$ −16.0000 −1.28932
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −32.0000 −2.52195
$$162$$ −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
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ −12.0000 −0.914991
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 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 $$0.476318\pi$$
$$182$$ 8.00000 0.592999
$$183$$ 10.0000 0.739221
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −8.00000 −0.585018
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 10.0000 0.703598
$$203$$ −24.0000 −1.68447
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ 8.00000 0.556038
$$208$$ 2.00000 0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 8.00000 0.548151
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ −10.0000 −0.677285
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 10.0000 0.671156
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ −6.00000 −0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 12.0000 0.779484
$$238$$ 8.00000 0.518563
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ −2.00000 −0.127257
$$248$$ −4.00000 −0.254000
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ −32.0000 −2.01182
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −12.0000 −0.747087
$$259$$ −40.0000 −2.48548
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ −6.00000 −0.367194
$$268$$ 4.00000 0.244339
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 8.00000 0.484182
$$274$$ 14.0000 0.845771
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 8.00000 0.472225
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ −2.00000 −0.117041
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 4.00000 0.232104
$$298$$ −22.0000 −1.27443
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 48.0000 2.76667
$$302$$ −4.00000 −0.230174
$$303$$ 10.0000 0.574485
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 30.0000 1.69570 0.847850 0.530236i $$-0.177897\pi$$
0.847850 + 0.530236i $$0.177897\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 32.0000 1.78329
$$323$$ −2.00000 −0.111283
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −10.0000 −0.553001
$$328$$ 2.00000 0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 10.0000 0.547997
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 1.00000 0.0540738
$$343$$ −8.00000 −0.431959
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 4.00000 0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 8.00000 0.423405
$$358$$ 0 0
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −2.00000 −0.105118
$$363$$ −5.00000 −0.262432
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ −4.00000 −0.207390
$$373$$ −30.0000 −1.55334 −0.776671 0.629907i $$-0.783093\pi$$
−0.776671 + 0.629907i $$0.783093\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ −4.00000 −0.205738
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ −12.0000 −0.609994
$$388$$ −18.0000 −0.913812
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ −9.00000 −0.454569
$$393$$ 12.0000 0.605320
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 8.00000 0.398508
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 24.0000 1.19110
$$407$$ −40.0000 −1.98273
$$408$$ 2.00000 0.0990148
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 12.0000 0.587643
$$418$$ −4.00000 −0.195646
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 40.0000 1.93574
$$428$$ 12.0000 0.580042
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −8.00000 −0.382692
$$438$$ −2.00000 −0.0955637
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −4.00000 −0.190261
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ −22.0000 −1.04056
$$448$$ −4.00000 −0.188982
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 6.00000 0.282216
$$453$$ −4.00000 −0.187936
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 16.0000 0.744387
$$463$$ 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
$$466$$ 6.00000 0.277945
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 0 0
$$473$$ 48.0000 2.20704
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ −6.00000 −0.274721
$$478$$ 8.00000 0.365911
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ −2.00000 −0.0910975
$$483$$ 32.0000 1.45605
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ 12.0000 0.540453
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 32.0000 1.43540
$$498$$ 8.00000 0.358489
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 28.0000 1.24970
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 9.00000 0.399704
$$508$$ −8.00000 −0.354943
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 40.0000 1.75750
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 8.00000 0.348485
$$528$$ 4.00000 0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −4.00000 −0.173259
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −2.00000 −0.0858282
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 8.00000 0.340503
$$553$$ 48.0000 2.04117
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ −30.0000 −1.26547
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ −4.00000 −0.167984
$$568$$ 8.00000 0.335673
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ −8.00000 −0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ −32.0000 −1.32758
$$582$$ −18.0000 −0.746124
$$583$$ 24.0000 0.993978
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 40.0000 1.65098 0.825488 0.564419i $$-0.190900\pi$$
0.825488 + 0.564419i $$0.190900\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 10.0000 0.410997
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ −16.0000 −0.654836
$$598$$ −16.0000 −0.654289
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ −48.0000 −1.95633
$$603$$ 4.00000 0.162893
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000 0.0808452
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −24.0000 −0.962312
$$623$$ −24.0000 −0.961540
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −30.0000 −1.19904
$$627$$ −4.00000 −0.159745
$$628$$ 6.00000 0.239426
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 12.0000 0.476957
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 18.0000 0.713186
$$638$$ 24.0000 0.950169
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −10.0000 −0.394976 −0.197488 0.980305i $$-0.563278\pi$$
−0.197488 + 0.980305i $$0.563278\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ −32.0000 −1.26098
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 4.00000 0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 12.0000 0.466393
$$663$$ −4.00000 −0.155347
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 48.0000 1.85857
$$668$$ 16.0000 0.619059
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ −4.00000 −0.154303
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 72.0000 2.76311
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 16.0000 0.612672
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ −22.0000 −0.839352
$$688$$ −12.0000 −0.457496
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 16.0000 0.607790
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ −4.00000 −0.151511
$$698$$ 34.0000 1.28692
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −10.0000 −0.377157
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 40.0000 1.50435
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ −6.00000 −0.224860
$$713$$ 32.0000 1.19841
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.00000 0.298765
$$718$$ −32.0000 −1.19423
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 64.0000 2.38348
$$722$$ −1.00000 −0.0372161
$$723$$ −2.00000 −0.0743808
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ −12.0000 −0.445055 −0.222528 0.974926i $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 10.0000 0.369611
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −12.0000 −0.442928
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ −16.0000 −0.589368
$$738$$ 2.00000 0.0736210
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ −24.0000 −0.881068
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 30.0000 1.09838
$$747$$ 8.00000 0.292705
$$748$$ −8.00000 −0.292509
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 52.0000 1.89751 0.948753 0.316017i $$-0.102346\pi$$
0.948753 + 0.316017i $$0.102346\pi$$
$$752$$ 0 0
$$753$$ 28.0000 1.02038
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ −40.0000 −1.44810
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −18.0000 −0.647834
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ 40.0000 1.43499
$$778$$ −6.00000 −0.215110
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ −16.0000 −0.572159
$$783$$ −6.00000 −0.214423
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ −52.0000 −1.85360 −0.926800 0.375555i $$-0.877452\pi$$
−0.926800 + 0.375555i $$0.877452\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 4.00000 0.142134
$$793$$ −20.0000 −0.710221
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 18.0000 0.635602
$$803$$ 8.00000 0.282314
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ −6.00000 −0.211210
$$808$$ 10.0000 0.351799
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 24.0000 0.841717
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 12.0000 0.419827
$$818$$ 6.00000 0.209785
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ −14.0000 −0.488306
$$823$$ −52.0000 −1.81261 −0.906303 0.422628i $$-0.861108\pi$$
−0.906303 + 0.422628i $$0.861108\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −52.0000 −1.80822 −0.904109 0.427303i $$-0.859464\pi$$
−0.904109 + 0.427303i $$0.859464\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 2.00000 0.0693375
$$833$$ 18.0000 0.623663
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ −4.00000 −0.138260
$$838$$ −12.0000 −0.414533
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 14.0000 0.482472
$$843$$ −30.0000 −1.03325
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −20.0000 −0.687208
$$848$$ −6.00000 −0.206041
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 80.0000 2.74236
$$852$$ 8.00000 0.274075
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ 13.0000 0.441503
$$868$$ −16.0000 −0.543075
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −10.0000 −0.338643
$$873$$ −18.0000 −0.609208
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −6.00000 −0.202606 −0.101303 0.994856i $$-0.532301\pi$$
−0.101303 + 0.994856i $$0.532301\pi$$
$$878$$ 4.00000 0.134993
$$879$$ −10.0000 −0.337292
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 22.0000 0.735790
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ −16.0000 −0.534224
$$898$$ −6.00000 −0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ −8.00000 −0.266371
$$903$$ −48.0000 −1.59734
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 4.00000 0.132891
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −32.0000 −1.05905
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ 48.0000 1.58510
$$918$$ 2.00000 0.0660098
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ −14.0000 −0.461065
$$923$$ −16.0000 −0.526646
$$924$$ −16.0000 −0.526361
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ −16.0000 −0.525509
$$928$$ −6.00000 −0.196960
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −9.00000 −0.294963
$$932$$ −6.00000 −0.196537
$$933$$ −24.0000 −0.785725
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −30.0000 −0.979013
$$940$$ 0 0
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 40.0000 1.29983 0.649913 0.760009i $$-0.274805\pi$$
0.649913 + 0.760009i $$0.274805\pi$$
$$948$$ 12.0000 0.389742
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 8.00000 0.259281
$$953$$ −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
$$956$$ −8.00000 −0.258738
$$957$$ 24.0000 0.775810
$$958$$ 16.0000 0.516937
$$959$$ 56.0000 1.80833
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −20.0000 −0.644826
$$963$$ 12.0000 0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −32.0000 −1.02958
$$967$$ −52.0000 −1.67221 −0.836104 0.548572i $$-0.815172\pi$$
−0.836104 + 0.548572i $$0.815172\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 2.00000 0.0642493
$$970$$ 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
$$973$$ 48.0000 1.53881
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −50.0000 −1.59964 −0.799821 0.600239i $$-0.795072\pi$$
−0.799821 + 0.600239i $$0.795072\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ −28.0000 −0.893516
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ −96.0000 −3.05262
$$990$$ 0 0
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 12.0000 0.380808
$$994$$ −32.0000 −1.01498
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 4.00000 0.126618
$$999$$ −10.0000 −0.316386
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2850.2.a.a.1.1 1
3.2 odd 2 8550.2.a.t.1.1 1
5.2 odd 4 2850.2.d.k.799.1 2
5.3 odd 4 2850.2.d.k.799.2 2
5.4 even 2 570.2.a.m.1.1 1
15.14 odd 2 1710.2.a.f.1.1 1
20.19 odd 2 4560.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.m.1.1 1 5.4 even 2
1710.2.a.f.1.1 1 15.14 odd 2
2850.2.a.a.1.1 1 1.1 even 1 trivial
2850.2.d.k.799.1 2 5.2 odd 4
2850.2.d.k.799.2 2 5.3 odd 4
4560.2.a.k.1.1 1 20.19 odd 2
8550.2.a.t.1.1 1 3.2 odd 2