# Properties

 Label 5082.2.a.l.1.1 Level $5082$ Weight $2$ Character 5082.1 Self dual yes Analytic conductor $40.580$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5082.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.5799743072$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5082.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{21} -1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} +1.00000 q^{42} +4.00000 q^{43} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -2.00000 q^{57} -6.00000 q^{58} -2.00000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -12.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} -2.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -1.00000 q^{84} -4.00000 q^{86} +6.00000 q^{87} -6.00000 q^{89} +2.00000 q^{91} +2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 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
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 2.00000 0.324443
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ −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
$$56$$ 1.00000 0.133631
$$57$$ −2.00000 −0.264906
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −5.00000 −0.577350
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 2.00000 0.207390
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 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
$$112$$ −1.00000 −0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 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
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −16.0000 −1.27694 −0.638470 0.769647i $$-0.720432\pi$$
−0.638470 + 0.769647i $$0.720432\pi$$
$$158$$ 8.00000 0.636446
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 4.00000 0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 5.00000 0.377964
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ 0 0
$$188$$ −6.00000 −0.437595
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 5.00000 0.353553
$$201$$ −4.00000 −0.282138
$$202$$ 6.00000 0.422159
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ −2.00000 −0.135769
$$218$$ −10.0000 −0.677285
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −5.00000 −0.333333
$$226$$ 18.0000 1.19734
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 28.0000 1.80364 0.901819 0.432113i $$-0.142232\pi$$
0.901819 + 0.432113i $$0.142232\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ −2.00000 −0.127000
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ −6.00000 −0.370681
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 4.00000 0.234082
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ −5.00000 −0.288675
$$301$$ −4.00000 −0.230556
$$302$$ 8.00000 0.460348
$$303$$ −6.00000 −0.344691
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 34.0000 1.94048 0.970241 0.242140i $$-0.0778494\pi$$
0.970241 + 0.242140i $$0.0778494\pi$$
$$308$$ 0 0
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 2.00000 0.113228
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 16.0000 0.902932
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 10.0000 0.554700
$$326$$ −20.0000 −1.10770
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 2.00000 0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000 0.108148
$$343$$ −1.00000 −0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −20.0000 −1.05118
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 2.00000 0.103695
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ −12.0000 −0.618031
$$378$$ 1.00000 0.0514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ −24.0000 −1.22795
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 4.00000 0.203331
$$388$$ 2.00000 0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ 6.00000 0.302660
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 2.00000 0.100125
$$400$$ −5.00000 −0.250000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −4.00000 −0.199254
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 40.0000 1.97787 0.988936 0.148340i $$-0.0473931\pi$$
0.988936 + 0.148340i $$0.0473931\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ −14.0000 −0.685583
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −6.00000 −0.291730
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 12.0000 0.581402
$$427$$ 2.00000 0.0967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ −4.00000 −0.191127
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ −6.00000 −0.283790
$$448$$ −1.00000 −0.0472456
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 5.00000 0.235702
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ −8.00000 −0.375873
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ −20.0000 −0.934539
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −16.0000 −0.737241
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 8.00000 0.367452
$$475$$ 10.0000 0.458831
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −28.0000 −1.27537
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 12.0000 0.538274
$$498$$ 6.00000 0.268866
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ −8.00000 −0.354943
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 2.00000 0.0878750
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 10.0000 0.437269 0.218635 0.975807i $$-0.429840\pi$$
0.218635 + 0.975807i $$0.429840\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 5.00000 0.218218
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ 0 0
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 20.0000 0.859074
$$543$$ 20.0000 0.858282
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ −2.00000 −0.0846668
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ −1.00000 −0.0419961
$$568$$ 12.0000 0.503509
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 24.0000 1.00261
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 17.0000 0.707107
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ −2.00000 −0.0829027
$$583$$ 0 0
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 2.00000 0.0821995
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −10.0000 −0.409273
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 5.00000 0.204124
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −4.00000 −0.162893
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −34.0000 −1.37213
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ −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
$$621$$ 0 0
$$622$$ −6.00000 −0.240578
$$623$$ 6.00000 0.240385
$$624$$ −2.00000 −0.0800641
$$625$$ 25.0000 1.00000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ −16.0000 −0.638470
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −20.0000 −0.794929
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −10.0000 −0.392232
$$651$$ −2.00000 −0.0783862
$$652$$ 20.0000 0.783260
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 4.00000 0.156055
$$658$$ −6.00000 −0.233904
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 44.0000 1.71140 0.855701 0.517471i $$-0.173126\pi$$
0.855701 + 0.517471i $$0.173126\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 1.00000 0.0385758
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 14.0000 0.539260
$$675$$ −5.00000 −0.192450
$$676$$ −9.00000 −0.346154
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 18.0000 0.691286
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 20.0000 0.763048
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 0 0
$$698$$ −10.0000 −0.378506
$$699$$ −18.0000 −0.680823
$$700$$ 5.00000 0.188982
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 15.0000 0.558242
$$723$$ 28.0000 1.04133
$$724$$ 20.0000 0.743294
$$725$$ −30.0000 −1.11417
$$726$$ 0 0
$$727$$ −46.0000 −1.70605 −0.853023 0.521874i $$-0.825233\pi$$
−0.853023 + 0.521874i $$0.825233\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −2.00000 −0.0739221
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ −6.00000 −0.220267
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 0 0
$$746$$ 2.00000 0.0732252
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −10.0000 −0.362024
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 16.0000 0.576975 0.288487 0.957484i $$-0.406848\pi$$
0.288487 + 0.957484i $$0.406848\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −2.00000 −0.0719816
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −10.0000 −0.359211
$$776$$ −2.00000 −0.0717958
$$777$$ −2.00000 −0.0717496
$$778$$ −6.00000 −0.215110
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ −50.0000 −1.78231 −0.891154 0.453701i $$-0.850103\pi$$
−0.891154 + 0.453701i $$0.850103\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 36.0000 1.27519 0.637593 0.770374i $$-0.279930\pi$$
0.637593 + 0.770374i $$0.279930\pi$$
$$798$$ −2.00000 −0.0707992
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ −6.00000 −0.211867
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ 10.0000 0.351147 0.175574 0.984466i $$-0.443822\pi$$
0.175574 + 0.984466i $$0.443822\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ −20.0000 −0.701431
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ −40.0000 −1.39857
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 18.0000 0.627822
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ 14.0000 0.484780
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.00000 0.0691301
$$838$$ 12.0000 0.414533
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 6.00000 0.206651
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −12.0000 −0.411113
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −36.0000 −1.22974 −0.614868 0.788630i $$-0.710791\pi$$
−0.614868 + 0.788630i $$0.710791\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 22.0000 0.747590
$$867$$ −17.0000 −0.577350
$$868$$ −2.00000 −0.0678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −10.0000 −0.338643
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 12.0000 0.401565
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ 12.0000 0.400222
$$900$$ −5.00000 −0.166667
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −4.00000 −0.133112
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 0 0
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 34.0000 1.12034
$$922$$ 6.00000 0.197599
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −8.00000 −0.262896
$$927$$ 14.0000 0.459820
$$928$$ −6.00000 −0.196960
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ −18.0000 −0.589610
$$933$$ 6.00000 0.196431
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 52.0000 1.69877 0.849383 0.527777i $$-0.176974\pi$$
0.849383 + 0.527777i $$0.176974\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 16.0000 0.521308
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ −8.00000 −0.259691
$$950$$ −10.0000 −0.324443
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 4.00000 0.128965
$$963$$ 0 0
$$964$$ 28.0000 0.901819
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 14.0000 0.448819
$$974$$ 4.00000 0.128168
$$975$$ 10.0000 0.320256
$$976$$ −2.00000 −0.0640184
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ −12.0000 −0.382935
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 4.00000 0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ −4.00000 −0.126936
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 5082.2.a.l.1.1 1
11.10 odd 2 462.2.a.f.1.1 1
33.32 even 2 1386.2.a.c.1.1 1
44.43 even 2 3696.2.a.i.1.1 1
77.76 even 2 3234.2.a.q.1.1 1
231.230 odd 2 9702.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.f.1.1 1 11.10 odd 2
1386.2.a.c.1.1 1 33.32 even 2
3234.2.a.q.1.1 1 77.76 even 2
3696.2.a.i.1.1 1 44.43 even 2
5082.2.a.l.1.1 1 1.1 even 1 trivial
9702.2.a.n.1.1 1 231.230 odd 2