# Properties

 Label 4650.2.a.p.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.1304369399$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

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

## 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 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 3.00000 0.639602
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ −1.00000 −0.176777
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ 5.00000 0.737210
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ −5.00000 −0.686803 −0.343401 0.939189i $$-0.611579\pi$$
−0.343401 + 0.939189i $$0.611579\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −1.00000 −0.132453
$$58$$ −2.00000 −0.262613
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 1.00000 0.127000
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 0 0
$$69$$ −5.00000 −0.601929
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 3.00000 0.341882
$$78$$ −4.00000 −0.452911
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −5.00000 −0.539164
$$87$$ 2.00000 0.214423
$$88$$ 3.00000 0.319801
$$89$$ 1.00000 0.106000 0.0529999 0.998595i $$-0.483122\pi$$
0.0529999 + 0.998595i $$0.483122\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −5.00000 −0.521286
$$93$$ −1.00000 −0.103695
$$94$$ −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 $$0.330506\pi$$
$$98$$ 6.00000 0.606092
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 5.00000 0.485643
$$107$$ −19.0000 −1.83680 −0.918400 0.395654i $$-0.870518\pi$$
−0.918400 + 0.395654i $$0.870518\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ −1.00000 −0.0944911
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 4.00000 0.369800
$$118$$ 6.00000 0.552345
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 2.00000 0.181071
$$123$$ −10.0000 −0.901670
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 5.00000 0.440225
$$130$$ 0 0
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 1.00000 0.0867110
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 5.00000 0.425628
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 5.00000 0.419591
$$143$$ −12.0000 −1.00349
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −7.00000 −0.579324
$$147$$ −6.00000 −0.494872
$$148$$ −4.00000 −0.328798
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ −9.00000 −0.718278 −0.359139 0.933284i $$-0.616930\pi$$
−0.359139 + 0.933284i $$0.616930\pi$$
$$158$$ −3.00000 −0.238667
$$159$$ −5.00000 −0.396526
$$160$$ 0 0
$$161$$ 5.00000 0.394055
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −2.00000 −0.155230
$$167$$ −21.0000 −1.62503 −0.812514 0.582941i $$-0.801902\pi$$
−0.812514 + 0.582941i $$0.801902\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 5.00000 0.381246
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ −6.00000 −0.450988
$$178$$ −1.00000 −0.0749532
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −2.00000 −0.147844
$$184$$ 5.00000 0.368605
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 0 0
$$188$$ 8.00000 0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ 15.0000 1.05540
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −5.00000 −0.347524
$$208$$ 4.00000 0.277350
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ 7.00000 0.481900 0.240950 0.970538i $$-0.422541\pi$$
0.240950 + 0.970538i $$0.422541\pi$$
$$212$$ −5.00000 −0.343401
$$213$$ −5.00000 −0.342594
$$214$$ 19.0000 1.29881
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 1.00000 0.0678844
$$218$$ −2.00000 −0.135457
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000 0.268462
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −15.0000 −0.997785
$$227$$ 9.00000 0.597351 0.298675 0.954355i $$-0.403455\pi$$
0.298675 + 0.954355i $$0.403455\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ −2.00000 −0.131306
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 3.00000 0.194871
$$238$$ 0 0
$$239$$ −14.0000 −0.905585 −0.452792 0.891616i $$-0.649572\pi$$
−0.452792 + 0.891616i $$0.649572\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ −4.00000 −0.254514
$$248$$ 1.00000 0.0635001
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 15.0000 0.943042
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −5.00000 −0.311891 −0.155946 0.987766i $$-0.549842\pi$$
−0.155946 + 0.987766i $$0.549842\pi$$
$$258$$ −5.00000 −0.311286
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 2.00000 0.123560
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ −1.00000 −0.0613139
$$267$$ 1.00000 0.0611990
$$268$$ −2.00000 −0.122169
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 9.00000 0.546711 0.273356 0.961913i $$-0.411866\pi$$
0.273356 + 0.961913i $$0.411866\pi$$
$$272$$ 0 0
$$273$$ −4.00000 −0.242091
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ −5.00000 −0.300965
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 20.0000 1.19952
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ −32.0000 −1.90896 −0.954480 0.298275i $$-0.903589\pi$$
−0.954480 + 0.298275i $$0.903589\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 10.0000 0.590281
$$288$$ −1.00000 −0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 7.00000 0.409644
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ −3.00000 −0.174078
$$298$$ −15.0000 −0.868927
$$299$$ −20.0000 −1.15663
$$300$$ 0 0
$$301$$ −5.00000 −0.288195
$$302$$ 16.0000 0.920697
$$303$$ −15.0000 −0.861727
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 9.00000 0.507899
$$315$$ 0 0
$$316$$ 3.00000 0.168763
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 5.00000 0.280386
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −19.0000 −1.06048
$$322$$ −5.00000 −0.278639
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 2.00000 0.110600
$$328$$ 10.0000 0.552158
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 2.00000 0.109764
$$333$$ −4.00000 −0.219199
$$334$$ 21.0000 1.14907
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 15.0000 0.814688
$$340$$ 0 0
$$341$$ 3.00000 0.162459
$$342$$ 1.00000 0.0540738
$$343$$ 13.0000 0.701934
$$344$$ −5.00000 −0.269582
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ 22.0000 1.18102 0.590511 0.807030i $$-0.298926\pi$$
0.590511 + 0.807030i $$0.298926\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 3.00000 0.159901
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ 1.00000 0.0529999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3.00000 0.158334 0.0791670 0.996861i $$-0.474774\pi$$
0.0791670 + 0.996861i $$0.474774\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 5.00000 0.262794
$$363$$ −2.00000 −0.104973
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −14.0000 −0.730794 −0.365397 0.930852i $$-0.619067\pi$$
−0.365397 + 0.930852i $$0.619067\pi$$
$$368$$ −5.00000 −0.260643
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 5.00000 0.259587
$$372$$ −1.00000 −0.0518476
$$373$$ 11.0000 0.569558 0.284779 0.958593i $$-0.408080\pi$$
0.284779 + 0.958593i $$0.408080\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 8.00000 0.412021
$$378$$ 1.00000 0.0514344
$$379$$ −15.0000 −0.770498 −0.385249 0.922813i $$-0.625884\pi$$
−0.385249 + 0.922813i $$0.625884\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 16.0000 0.818631
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ 5.00000 0.254164
$$388$$ 10.0000 0.507673
$$389$$ −28.0000 −1.41966 −0.709828 0.704375i $$-0.751227\pi$$
−0.709828 + 0.704375i $$0.751227\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 6.00000 0.303046
$$393$$ −2.00000 −0.100887
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ 31.0000 1.55585 0.777923 0.628360i $$-0.216273\pi$$
0.777923 + 0.628360i $$0.216273\pi$$
$$398$$ 21.0000 1.05263
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 5.00000 0.249688 0.124844 0.992176i $$-0.460157\pi$$
0.124844 + 0.992176i $$0.460157\pi$$
$$402$$ 2.00000 0.0997509
$$403$$ −4.00000 −0.199254
$$404$$ −15.0000 −0.746278
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 12.0000 0.594818
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ 6.00000 0.295241
$$414$$ 5.00000 0.245737
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ −20.0000 −0.979404
$$418$$ −3.00000 −0.146735
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ −7.00000 −0.340755
$$423$$ 8.00000 0.388973
$$424$$ 5.00000 0.242821
$$425$$ 0 0
$$426$$ 5.00000 0.242251
$$427$$ 2.00000 0.0967868
$$428$$ −19.0000 −0.918400
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ −1.00000 −0.0480015
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 5.00000 0.239182
$$438$$ −7.00000 −0.334473
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −29.0000 −1.37783 −0.688916 0.724841i $$-0.741913\pi$$
−0.688916 + 0.724841i $$0.741913\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −20.0000 −0.947027
$$447$$ 15.0000 0.709476
$$448$$ −1.00000 −0.0472456
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 30.0000 1.41264
$$452$$ 15.0000 0.705541
$$453$$ −16.0000 −0.751746
$$454$$ −9.00000 −0.422391
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ −30.0000 −1.40334 −0.701670 0.712502i $$-0.747562\pi$$
−0.701670 + 0.712502i $$0.747562\pi$$
$$458$$ −15.0000 −0.700904
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 16.0000 0.745194 0.372597 0.927993i $$-0.378467\pi$$
0.372597 + 0.927993i $$0.378467\pi$$
$$462$$ −3.00000 −0.139573
$$463$$ 14.0000 0.650635 0.325318 0.945605i $$-0.394529\pi$$
0.325318 + 0.945605i $$0.394529\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −21.0000 −0.972806
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 2.00000 0.0923514
$$470$$ 0 0
$$471$$ −9.00000 −0.414698
$$472$$ 6.00000 0.276172
$$473$$ −15.0000 −0.689701
$$474$$ −3.00000 −0.137795
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −5.00000 −0.228934
$$478$$ 14.0000 0.640345
$$479$$ −25.0000 −1.14228 −0.571140 0.820853i $$-0.693499\pi$$
−0.571140 + 0.820853i $$0.693499\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 12.0000 0.546585
$$483$$ 5.00000 0.227508
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 33.0000 1.48927 0.744635 0.667472i $$-0.232624\pi$$
0.744635 + 0.667472i $$0.232624\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ 0 0
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 5.00000 0.224281
$$498$$ −2.00000 −0.0896221
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ −21.0000 −0.938211
$$502$$ −12.0000 −0.535586
$$503$$ −14.0000 −0.624229 −0.312115 0.950044i $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ −15.0000 −0.666831
$$507$$ 3.00000 0.133235
$$508$$ −8.00000 −0.354943
$$509$$ 32.0000 1.41838 0.709188 0.705020i $$-0.249062\pi$$
0.709188 + 0.705020i $$0.249062\pi$$
$$510$$ 0 0
$$511$$ −7.00000 −0.309662
$$512$$ −1.00000 −0.0441942
$$513$$ −1.00000 −0.0441511
$$514$$ 5.00000 0.220541
$$515$$ 0 0
$$516$$ 5.00000 0.220113
$$517$$ −24.0000 −1.05552
$$518$$ −4.00000 −0.175750
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −19.0000 −0.830812 −0.415406 0.909636i $$-0.636360\pi$$
−0.415406 + 0.909636i $$0.636360\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ −3.00000 −0.130558
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 1.00000 0.0433555
$$533$$ −40.0000 −1.73259
$$534$$ −1.00000 −0.0432742
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ −2.00000 −0.0862261
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ 24.0000 1.03184 0.515920 0.856637i $$-0.327450\pi$$
0.515920 + 0.856637i $$0.327450\pi$$
$$542$$ −9.00000 −0.386583
$$543$$ −5.00000 −0.214571
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 5.00000 0.212814
$$553$$ −3.00000 −0.127573
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −9.00000 −0.381342 −0.190671 0.981654i $$-0.561066\pi$$
−0.190671 + 0.981654i $$0.561066\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ 20.0000 0.845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 32.0000 1.34984
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ −1.00000 −0.0419961
$$568$$ 5.00000 0.209795
$$569$$ −39.0000 −1.63497 −0.817483 0.575953i $$-0.804631\pi$$
−0.817483 + 0.575953i $$0.804631\pi$$
$$570$$ 0 0
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ −16.0000 −0.668410
$$574$$ −10.0000 −0.417392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ −2.00000 −0.0829740
$$582$$ −10.0000 −0.414513
$$583$$ 15.0000 0.621237
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −6.00000 −0.247647 −0.123823 0.992304i $$-0.539516\pi$$
−0.123823 + 0.992304i $$0.539516\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ 1.00000 0.0412043
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ −4.00000 −0.164399
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ 15.0000 0.614424
$$597$$ −21.0000 −0.859473
$$598$$ 20.0000 0.817861
$$599$$ 13.0000 0.531166 0.265583 0.964088i $$-0.414436\pi$$
0.265583 + 0.964088i $$0.414436\pi$$
$$600$$ 0 0
$$601$$ −44.0000 −1.79480 −0.897399 0.441221i $$-0.854546\pi$$
−0.897399 + 0.441221i $$0.854546\pi$$
$$602$$ 5.00000 0.203785
$$603$$ −2.00000 −0.0814463
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 15.0000 0.609333
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 32.0000 1.29458
$$612$$ 0 0
$$613$$ −24.0000 −0.969351 −0.484675 0.874694i $$-0.661062\pi$$
−0.484675 + 0.874694i $$0.661062\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −5.00000 −0.201292 −0.100646 0.994922i $$-0.532091\pi$$
−0.100646 + 0.994922i $$0.532091\pi$$
$$618$$ 0 0
$$619$$ −12.0000 −0.482321 −0.241160 0.970485i $$-0.577528\pi$$
−0.241160 + 0.970485i $$0.577528\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ −16.0000 −0.641542
$$623$$ −1.00000 −0.0400642
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 3.00000 0.119808
$$628$$ −9.00000 −0.359139
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 25.0000 0.995234 0.497617 0.867397i $$-0.334208\pi$$
0.497617 + 0.867397i $$0.334208\pi$$
$$632$$ −3.00000 −0.119334
$$633$$ 7.00000 0.278225
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −5.00000 −0.198263
$$637$$ −24.0000 −0.950915
$$638$$ 6.00000 0.237542
$$639$$ −5.00000 −0.197797
$$640$$ 0 0
$$641$$ 50.0000 1.97488 0.987441 0.157991i $$-0.0505015\pi$$
0.987441 + 0.157991i $$0.0505015\pi$$
$$642$$ 19.0000 0.749870
$$643$$ 7.00000 0.276053 0.138027 0.990429i $$-0.455924\pi$$
0.138027 + 0.990429i $$0.455924\pi$$
$$644$$ 5.00000 0.197028
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −7.00000 −0.275198 −0.137599 0.990488i $$-0.543939\pi$$
−0.137599 + 0.990488i $$0.543939\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 18.0000 0.706562
$$650$$ 0 0
$$651$$ 1.00000 0.0391931
$$652$$ 4.00000 0.156652
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 7.00000 0.273096
$$658$$ 8.00000 0.311872
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 36.0000 1.40024 0.700119 0.714026i $$-0.253130\pi$$
0.700119 + 0.714026i $$0.253130\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 0 0
$$664$$ −2.00000 −0.0776151
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ −10.0000 −0.387202
$$668$$ −21.0000 −0.812514
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ 6.00000 0.231627
$$672$$ 1.00000 0.0385758
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −13.0000 −0.499631 −0.249815 0.968294i $$-0.580370\pi$$
−0.249815 + 0.968294i $$0.580370\pi$$
$$678$$ −15.0000 −0.576072
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 9.00000 0.344881
$$682$$ −3.00000 −0.114876
$$683$$ −35.0000 −1.33924 −0.669619 0.742705i $$-0.733543\pi$$
−0.669619 + 0.742705i $$0.733543\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 15.0000 0.572286
$$688$$ 5.00000 0.190623
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −41.0000 −1.55971 −0.779857 0.625958i $$-0.784708\pi$$
−0.779857 + 0.625958i $$0.784708\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 3.00000 0.113961
$$694$$ −22.0000 −0.835109
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 0 0
$$698$$ −20.0000 −0.757011
$$699$$ 21.0000 0.794293
$$700$$ 0 0
$$701$$ −35.0000 −1.32193 −0.660966 0.750416i $$-0.729853\pi$$
−0.660966 + 0.750416i $$0.729853\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 4.00000 0.150863
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 15.0000 0.564133
$$708$$ −6.00000 −0.225494
$$709$$ 5.00000 0.187779 0.0938895 0.995583i $$-0.470070\pi$$
0.0938895 + 0.995583i $$0.470070\pi$$
$$710$$ 0 0
$$711$$ 3.00000 0.112509
$$712$$ −1.00000 −0.0374766
$$713$$ 5.00000 0.187251
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −14.0000 −0.522840
$$718$$ −3.00000 −0.111959
$$719$$ 10.0000 0.372937 0.186469 0.982461i $$-0.440296\pi$$
0.186469 + 0.982461i $$0.440296\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ −12.0000 −0.446285
$$724$$ −5.00000 −0.185824
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ −49.0000 −1.81731 −0.908655 0.417548i $$-0.862889\pi$$
−0.908655 + 0.417548i $$0.862889\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −2.00000 −0.0739221
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 14.0000 0.516749
$$735$$ 0 0
$$736$$ 5.00000 0.184302
$$737$$ 6.00000 0.221013
$$738$$ 10.0000 0.368105
$$739$$ −18.0000 −0.662141 −0.331070 0.943606i $$-0.607410\pi$$
−0.331070 + 0.943606i $$0.607410\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ −5.00000 −0.183556
$$743$$ −15.0000 −0.550297 −0.275148 0.961402i $$-0.588727\pi$$
−0.275148 + 0.961402i $$0.588727\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −11.0000 −0.402739
$$747$$ 2.00000 0.0731762
$$748$$ 0 0
$$749$$ 19.0000 0.694245
$$750$$ 0 0
$$751$$ 22.0000 0.802791 0.401396 0.915905i $$-0.368525\pi$$
0.401396 + 0.915905i $$0.368525\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 12.0000 0.437304
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ −20.0000 −0.726912 −0.363456 0.931611i $$-0.618403\pi$$
−0.363456 + 0.931611i $$0.618403\pi$$
$$758$$ 15.0000 0.544825
$$759$$ 15.0000 0.544466
$$760$$ 0 0
$$761$$ 21.0000 0.761249 0.380625 0.924730i $$-0.375709\pi$$
0.380625 + 0.924730i $$0.375709\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −2.00000 −0.0724049
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ −24.0000 −0.866590
$$768$$ 1.00000 0.0360844
$$769$$ 45.0000 1.62274 0.811371 0.584532i $$-0.198722\pi$$
0.811371 + 0.584532i $$0.198722\pi$$
$$770$$ 0 0
$$771$$ −5.00000 −0.180071
$$772$$ 10.0000 0.359908
$$773$$ 29.0000 1.04306 0.521529 0.853234i $$-0.325362\pi$$
0.521529 + 0.853234i $$0.325362\pi$$
$$774$$ −5.00000 −0.179721
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 4.00000 0.143499
$$778$$ 28.0000 1.00385
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 15.0000 0.536742
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 2.00000 0.0713376
$$787$$ −27.0000 −0.962446 −0.481223 0.876598i $$-0.659807\pi$$
−0.481223 + 0.876598i $$0.659807\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −15.0000 −0.533339
$$792$$ 3.00000 0.106600
$$793$$ −8.00000 −0.284088
$$794$$ −31.0000 −1.10015
$$795$$ 0 0
$$796$$ −21.0000 −0.744325
$$797$$ 22.0000 0.779280 0.389640 0.920967i $$-0.372599\pi$$
0.389640 + 0.920967i $$0.372599\pi$$
$$798$$ −1.00000 −0.0353996
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 1.00000 0.0353333
$$802$$ −5.00000 −0.176556
$$803$$ −21.0000 −0.741074
$$804$$ −2.00000 −0.0705346
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 2.00000 0.0704033
$$808$$ 15.0000 0.527698
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ 35.0000 1.22902 0.614508 0.788911i $$-0.289355\pi$$
0.614508 + 0.788911i $$0.289355\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 9.00000 0.315644
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ −4.00000 −0.139857
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 6.00000 0.209147 0.104573 0.994517i $$-0.466652\pi$$
0.104573 + 0.994517i $$0.466652\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −6.00000 −0.208767
$$827$$ −14.0000 −0.486828 −0.243414 0.969923i $$-0.578267\pi$$
−0.243414 + 0.969923i $$0.578267\pi$$
$$828$$ −5.00000 −0.173762
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 4.00000 0.138675
$$833$$ 0 0
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 3.00000 0.103757
$$837$$ −1.00000 −0.0345651
$$838$$ 18.0000 0.621800
$$839$$ 39.0000 1.34643 0.673215 0.739447i $$-0.264913\pi$$
0.673215 + 0.739447i $$0.264913\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 14.0000 0.482472
$$843$$ −32.0000 −1.10214
$$844$$ 7.00000 0.240950
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ 2.00000 0.0687208
$$848$$ −5.00000 −0.171701
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ −5.00000 −0.171297
$$853$$ 55.0000 1.88316 0.941582 0.336784i $$-0.109339\pi$$
0.941582 + 0.336784i $$0.109339\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ 19.0000 0.649407
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 12.0000 0.409673
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 10.0000 0.340799
$$862$$ 0 0
$$863$$ 29.0000 0.987171 0.493586 0.869697i $$-0.335686\pi$$
0.493586 + 0.869697i $$0.335686\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ −17.0000 −0.577350
$$868$$ 1.00000 0.0339422
$$869$$ −9.00000 −0.305304
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ −2.00000 −0.0677285
$$873$$ 10.0000 0.338449
$$874$$ −5.00000 −0.169128
$$875$$ 0 0
$$876$$ 7.00000 0.236508
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 6.00000 0.202031
$$883$$ 35.0000 1.17784 0.588922 0.808190i $$-0.299553\pi$$
0.588922 + 0.808190i $$0.299553\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 29.0000 0.974274
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 4.00000 0.134231
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 20.0000 0.669650
$$893$$ −8.00000 −0.267710
$$894$$ −15.0000 −0.501675
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ −20.0000 −0.667781
$$898$$ −2.00000 −0.0667409
$$899$$ −2.00000 −0.0667037
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −30.0000 −0.998891
$$903$$ −5.00000 −0.166390
$$904$$ −15.0000 −0.498893
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ 9.00000 0.298675
$$909$$ −15.0000 −0.497519
$$910$$ 0 0
$$911$$ −46.0000 −1.52405 −0.762024 0.647549i $$-0.775794\pi$$
−0.762024 + 0.647549i $$0.775794\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ −6.00000 −0.198571
$$914$$ 30.0000 0.992312
$$915$$ 0 0
$$916$$ 15.0000 0.495614
$$917$$ 2.00000 0.0660458
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ −16.0000 −0.526932
$$923$$ −20.0000 −0.658308
$$924$$ 3.00000 0.0986928
$$925$$ 0 0
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ −2.00000 −0.0656532
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 21.0000 0.687878
$$933$$ 16.0000 0.523816
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ −2.00000 −0.0653023
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 16.0000 0.521585 0.260793 0.965395i $$-0.416016\pi$$
0.260793 + 0.965395i $$0.416016\pi$$
$$942$$ 9.00000 0.293236
$$943$$ 50.0000 1.62822
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 15.0000 0.487692
$$947$$ −46.0000 −1.49480 −0.747400 0.664375i $$-0.768698\pi$$
−0.747400 + 0.664375i $$0.768698\pi$$
$$948$$ 3.00000 0.0974355
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ 28.0000 0.907009 0.453504 0.891254i $$-0.350174\pi$$
0.453504 + 0.891254i $$0.350174\pi$$
$$954$$ 5.00000 0.161881
$$955$$ 0 0
$$956$$ −14.0000 −0.452792
$$957$$ −6.00000 −0.193952
$$958$$ 25.0000 0.807713
$$959$$ 2.00000 0.0645834
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 16.0000 0.515861
$$963$$ −19.0000 −0.612266
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ −5.00000 −0.160872
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 20.0000 0.641171
$$974$$ −4.00000 −0.128168
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ −3.00000 −0.0958804
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ −33.0000 −1.05307
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −8.00000 −0.254643
$$988$$ −4.00000 −0.127257
$$989$$ −25.0000 −0.794954
$$990$$ 0 0
$$991$$ −53.0000 −1.68360 −0.841800 0.539789i $$-0.818504\pi$$
−0.841800 + 0.539789i $$0.818504\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ 8.00000 0.253872
$$994$$ −5.00000 −0.158590
$$995$$ 0 0
$$996$$ 2.00000 0.0633724
$$997$$ −6.00000 −0.190022 −0.0950110 0.995476i $$-0.530289\pi$$
−0.0950110 + 0.995476i $$0.530289\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ −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 4650.2.a.p.1.1 1
5.2 odd 4 930.2.d.a.559.1 2
5.3 odd 4 930.2.d.a.559.2 yes 2
5.4 even 2 4650.2.a.bc.1.1 1
15.2 even 4 2790.2.d.f.559.2 2
15.8 even 4 2790.2.d.f.559.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.a.559.1 2 5.2 odd 4
930.2.d.a.559.2 yes 2 5.3 odd 4
2790.2.d.f.559.1 2 15.8 even 4
2790.2.d.f.559.2 2 15.2 even 4
4650.2.a.p.1.1 1 1.1 even 1 trivial
4650.2.a.bc.1.1 1 5.4 even 2