# Properties

 Label 150.2.a.a.1.1 Level $150$ Weight $2$ Character 150.1 Self dual yes Analytic conductor $1.198$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 150.1

## $q$-expansion

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

## 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
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −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 $$0.187167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 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
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ −2.00000 −0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.00000 −0.348155
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 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
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 6.00000 0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ −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
$$91$$ 12.0000 1.25794
$$92$$ −4.00000 −0.417029
$$93$$ 8.00000 0.829561
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −8.00000 −0.796030 −0.398015 0.917379i $$-0.630301\pi$$
−0.398015 + 0.917379i $$0.630301\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ −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
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 2.00000 0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ −10.0000 −0.920575
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ −2.00000 −0.180334
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −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
$$142$$ −12.0000 −1.00702
$$143$$ 12.0000 1.00349
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 3.00000 0.247436
$$148$$ 2.00000 0.164399
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ −10.0000 −0.751646
$$178$$ 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 $$0.476318\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ −2.00000 −0.147844
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 4.00000 0.292509
$$188$$ −8.00000 −0.583460
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 8.00000 0.562878
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ −4.00000 −0.278019
$$208$$ 6.00000 0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −12.0000 −0.822226
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ −10.0000 −0.677285
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 2.00000 0.134231
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 7.00000 0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −8.00000 −0.502956
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000 1.11204
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ −8.00000 −0.488678
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000 0.121268
$$273$$ −12.0000 −0.726273
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 20.0000 1.19952
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 4.00000 0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ −4.00000 −0.234082
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ −2.00000 −0.116052
$$298$$ 20.0000 1.15857
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 8.00000 0.460348
$$303$$ 8.00000 0.459588
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −4.00000 −0.226093 −0.113047 0.993590i $$-0.536061\pi$$
−0.113047 + 0.993590i $$0.536061\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ −10.0000 −0.553001
$$328$$ −2.00000 −0.110432
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 2.00000 0.109599
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ −2.00000 −0.106600
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ −4.00000 −0.211702
$$358$$ −10.0000 −0.528516
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −2.00000 −0.105118
$$363$$ 7.00000 0.367405
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 8.00000 0.414781
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 2.00000 0.102869
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ −12.0000 −0.613973
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −4.00000 −0.203331
$$388$$ −8.00000 −0.406138
$$389$$ 20.0000 1.01404 0.507020 0.861934i $$-0.330747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 3.00000 0.151523
$$393$$ 18.0000 0.907980
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.0000 1.09863 0.549314 0.835616i $$-0.314889\pi$$
0.549314 + 0.835616i $$0.314889\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ −48.0000 −2.39105
$$404$$ −8.00000 −0.398015
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000 0.198273
$$408$$ 2.00000 0.0990148
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ −14.0000 −0.689730
$$413$$ 20.0000 0.984136
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ −8.00000 −0.388973
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 12.0000 0.581402
$$427$$ 4.00000 0.193574
$$428$$ 12.0000 0.580042
$$429$$ −12.0000 −0.579365
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ −4.00000 −0.191127
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −12.0000 −0.570782
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 20.0000 0.945968
$$448$$ 2.00000 0.0944911
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 6.00000 0.282216
$$453$$ 8.00000 0.375873
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ −10.0000 −0.460287
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 6.00000 0.274721
$$478$$ −20.0000 −0.914779
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ −22.0000 −1.00207
$$483$$ 8.00000 0.364013
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −18.0000 −0.815658 −0.407829 0.913058i $$-0.633714\pi$$
−0.407829 + 0.913058i $$0.633714\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 24.0000 1.07655
$$498$$ −4.00000 −0.179244
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 18.0000 0.803379
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ −23.0000 −1.02147
$$508$$ 2.00000 0.0887357
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ −16.0000 −0.703679
$$518$$ −4.00000 −0.175750
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −18.0000 −0.786334
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ −16.0000 −0.696971
$$528$$ −2.00000 −0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ −10.0000 −0.431532
$$538$$ 0 0
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ 8.00000 0.343629
$$543$$ −2.00000 −0.0858282
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 18.0000 0.759284
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 2.00000 0.0839921
$$568$$ −12.0000 −0.503509
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 12.0000 0.501745
$$573$$ −12.0000 −0.501307
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 32.0000 1.33218 0.666089 0.745873i $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ −8.00000 −0.331611
$$583$$ 12.0000 0.496989
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −22.0000 −0.904959
$$592$$ 2.00000 0.0821995
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ 24.0000 0.981433
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −8.00000 −0.325785
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −8.00000 −0.324978
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 2.00000 0.0808452
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −12.0000 −0.481156
$$623$$ −20.0000 −0.801283
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ −18.0000 −0.713186
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −24.0000 −0.946468 −0.473234 0.880937i $$-0.656913\pi$$
−0.473234 + 0.880937i $$0.656913\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −48.0000 −1.88707 −0.943537 0.331266i $$-0.892524\pi$$
−0.943537 + 0.331266i $$0.892524\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 16.0000 0.626608
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ −4.00000 −0.156055
$$658$$ 16.0000 0.623745
$$659$$ −50.0000 −1.94772 −0.973862 0.227142i $$-0.927062\pi$$
−0.973862 + 0.227142i $$0.927062\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 8.00000 0.310929
$$663$$ −12.0000 −0.466041
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ −26.0000 −1.00522
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 2.00000 0.0771517
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 28.0000 1.07852
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 6.00000 0.230429
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ 16.0000 0.612672
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 10.0000 0.381524
$$688$$ −4.00000 −0.152499
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 4.00000 0.151947
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ −10.0000 −0.378506
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 32.0000 1.20862 0.604312 0.796748i $$-0.293448\pi$$
0.604312 + 0.796748i $$0.293448\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ −16.0000 −0.601742
$$708$$ −10.0000 −0.375823
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 32.0000 1.19841
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ −20.0000 −0.746914
$$718$$ 0 0
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ −28.0000 −1.04277
$$722$$ 19.0000 0.707107
$$723$$ −22.0000 −0.818189
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ −2.00000 −0.0739221
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ −2.00000 −0.0738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ −16.0000 −0.589368
$$738$$ −2.00000 −0.0736210
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ −4.00000 −0.146352
$$748$$ 4.00000 0.146254
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 18.0000 0.655956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 2.00000 0.0724524
$$763$$ 20.0000 0.724049
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 60.0000 2.16647
$$768$$ −1.00000 −0.0360844
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −4.00000 −0.143963
$$773$$ −54.0000 −1.94225 −0.971123 0.238581i $$-0.923318\pi$$
−0.971123 + 0.238581i $$0.923318\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ −4.00000 −0.143499
$$778$$ −20.0000 −0.717035
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ −2.00000 −0.0710669
$$793$$ 12.0000 0.426132
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ −22.0000 −0.776847
$$803$$ −8.00000 −0.282314
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 0 0
$$808$$ 8.00000 0.281439
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ 10.0000 0.349642
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 6.00000 0.209147 0.104573 0.994517i $$-0.466652\pi$$
0.104573 + 0.994517i $$0.466652\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ −20.0000 −0.695889
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 6.00000 0.208013
$$833$$ −6.00000 −0.207888
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 10.0000 0.345444
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −22.0000 −0.758170
$$843$$ 18.0000 0.619953
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ −14.0000 −0.481046
$$848$$ 6.00000 0.206041
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ −12.0000 −0.411113
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 12.0000 0.409673
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ −32.0000 −1.08992
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ 13.0000 0.441503
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ −10.0000 −0.338643
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 26.0000 0.870544
$$893$$ 0 0
$$894$$ −20.0000 −0.668900
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 24.0000 0.801337
$$898$$ −30.0000 −1.00111
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ −4.00000 −0.133185
$$903$$ 8.00000 0.266223
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −28.0000 −0.929213
$$909$$ −8.00000 −0.265343
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −32.0000 −1.05847
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −36.0000 −1.18882
$$918$$ 2.00000 0.0660098
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ −12.0000 −0.395199
$$923$$ 72.0000 2.36991
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −6.00000 −0.197172
$$927$$ −14.0000 −0.459820
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −14.0000 −0.458585
$$933$$ −12.0000 −0.392862
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 4.00000 0.130535
$$940$$ 0 0
$$941$$ −28.0000 −0.912774 −0.456387 0.889781i $$-0.650857\pi$$
−0.456387 + 0.889781i $$0.650857\pi$$
$$942$$ 22.0000 0.716799
$$943$$ −8.00000 −0.260516
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ −4.00000 −0.129641
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ 0 0
$$958$$ −20.0000 −0.646171
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −12.0000 −0.386896
$$963$$ 12.0000 0.386695
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ −38.0000 −1.22200 −0.610999 0.791632i $$-0.709232\pi$$
−0.610999 + 0.791632i $$0.709232\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 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
$$973$$ −40.0000 −1.28234
$$974$$ 18.0000 0.576757
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 16.0000 0.511624
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 18.0000 0.574403
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 8.00000 0.253872
$$994$$ −24.0000 −0.761234
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 150.2.a.a.1.1 1
3.2 odd 2 450.2.a.f.1.1 1
4.3 odd 2 1200.2.a.m.1.1 1
5.2 odd 4 30.2.c.a.19.1 2
5.3 odd 4 30.2.c.a.19.2 yes 2
5.4 even 2 150.2.a.c.1.1 1
7.6 odd 2 7350.2.a.bg.1.1 1
8.3 odd 2 4800.2.a.m.1.1 1
8.5 even 2 4800.2.a.cg.1.1 1
12.11 even 2 3600.2.a.o.1.1 1
15.2 even 4 90.2.c.a.19.2 2
15.8 even 4 90.2.c.a.19.1 2
15.14 odd 2 450.2.a.b.1.1 1
20.3 even 4 240.2.f.a.49.2 2
20.7 even 4 240.2.f.a.49.1 2
20.19 odd 2 1200.2.a.g.1.1 1
35.2 odd 12 1470.2.n.h.949.1 4
35.3 even 12 1470.2.n.a.79.1 4
35.12 even 12 1470.2.n.a.949.1 4
35.13 even 4 1470.2.g.g.589.2 2
35.17 even 12 1470.2.n.a.79.2 4
35.18 odd 12 1470.2.n.h.79.1 4
35.23 odd 12 1470.2.n.h.949.2 4
35.27 even 4 1470.2.g.g.589.1 2
35.32 odd 12 1470.2.n.h.79.2 4
35.33 even 12 1470.2.n.a.949.2 4
35.34 odd 2 7350.2.a.cc.1.1 1
40.3 even 4 960.2.f.i.769.1 2
40.13 odd 4 960.2.f.h.769.2 2
40.19 odd 2 4800.2.a.cj.1.1 1
40.27 even 4 960.2.f.i.769.2 2
40.29 even 2 4800.2.a.l.1.1 1
40.37 odd 4 960.2.f.h.769.1 2
45.2 even 12 810.2.i.b.109.2 4
45.7 odd 12 810.2.i.e.109.1 4
45.13 odd 12 810.2.i.e.379.1 4
45.22 odd 12 810.2.i.e.379.2 4
45.23 even 12 810.2.i.b.379.2 4
45.32 even 12 810.2.i.b.379.1 4
45.38 even 12 810.2.i.b.109.1 4
45.43 odd 12 810.2.i.e.109.2 4
60.23 odd 4 720.2.f.f.289.1 2
60.47 odd 4 720.2.f.f.289.2 2
60.59 even 2 3600.2.a.bg.1.1 1
80.3 even 4 3840.2.d.j.2689.2 2
80.13 odd 4 3840.2.d.y.2689.2 2
80.27 even 4 3840.2.d.j.2689.1 2
80.37 odd 4 3840.2.d.y.2689.1 2
80.43 even 4 3840.2.d.x.2689.1 2
80.53 odd 4 3840.2.d.g.2689.1 2
80.67 even 4 3840.2.d.x.2689.2 2
80.77 odd 4 3840.2.d.g.2689.2 2
120.53 even 4 2880.2.f.e.1729.2 2
120.77 even 4 2880.2.f.e.1729.1 2
120.83 odd 4 2880.2.f.c.1729.2 2
120.107 odd 4 2880.2.f.c.1729.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.c.a.19.1 2 5.2 odd 4
30.2.c.a.19.2 yes 2 5.3 odd 4
90.2.c.a.19.1 2 15.8 even 4
90.2.c.a.19.2 2 15.2 even 4
150.2.a.a.1.1 1 1.1 even 1 trivial
150.2.a.c.1.1 1 5.4 even 2
240.2.f.a.49.1 2 20.7 even 4
240.2.f.a.49.2 2 20.3 even 4
450.2.a.b.1.1 1 15.14 odd 2
450.2.a.f.1.1 1 3.2 odd 2
720.2.f.f.289.1 2 60.23 odd 4
720.2.f.f.289.2 2 60.47 odd 4
810.2.i.b.109.1 4 45.38 even 12
810.2.i.b.109.2 4 45.2 even 12
810.2.i.b.379.1 4 45.32 even 12
810.2.i.b.379.2 4 45.23 even 12
810.2.i.e.109.1 4 45.7 odd 12
810.2.i.e.109.2 4 45.43 odd 12
810.2.i.e.379.1 4 45.13 odd 12
810.2.i.e.379.2 4 45.22 odd 12
960.2.f.h.769.1 2 40.37 odd 4
960.2.f.h.769.2 2 40.13 odd 4
960.2.f.i.769.1 2 40.3 even 4
960.2.f.i.769.2 2 40.27 even 4
1200.2.a.g.1.1 1 20.19 odd 2
1200.2.a.m.1.1 1 4.3 odd 2
1470.2.g.g.589.1 2 35.27 even 4
1470.2.g.g.589.2 2 35.13 even 4
1470.2.n.a.79.1 4 35.3 even 12
1470.2.n.a.79.2 4 35.17 even 12
1470.2.n.a.949.1 4 35.12 even 12
1470.2.n.a.949.2 4 35.33 even 12
1470.2.n.h.79.1 4 35.18 odd 12
1470.2.n.h.79.2 4 35.32 odd 12
1470.2.n.h.949.1 4 35.2 odd 12
1470.2.n.h.949.2 4 35.23 odd 12
2880.2.f.c.1729.1 2 120.107 odd 4
2880.2.f.c.1729.2 2 120.83 odd 4
2880.2.f.e.1729.1 2 120.77 even 4
2880.2.f.e.1729.2 2 120.53 even 4
3600.2.a.o.1.1 1 12.11 even 2
3600.2.a.bg.1.1 1 60.59 even 2
3840.2.d.g.2689.1 2 80.53 odd 4
3840.2.d.g.2689.2 2 80.77 odd 4
3840.2.d.j.2689.1 2 80.27 even 4
3840.2.d.j.2689.2 2 80.3 even 4
3840.2.d.x.2689.1 2 80.43 even 4
3840.2.d.x.2689.2 2 80.67 even 4
3840.2.d.y.2689.1 2 80.37 odd 4
3840.2.d.y.2689.2 2 80.13 odd 4
4800.2.a.l.1.1 1 40.29 even 2
4800.2.a.m.1.1 1 8.3 odd 2
4800.2.a.cg.1.1 1 8.5 even 2
4800.2.a.cj.1.1 1 40.19 odd 2
7350.2.a.bg.1.1 1 7.6 odd 2
7350.2.a.cc.1.1 1 35.34 odd 2