# Properties

 Label 800.2.a.m.1.1 Level $800$ Weight $2$ Character 800.1 Self dual yes Analytic conductor $6.388$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.82843 q^{3} +2.82843 q^{7} +5.00000 q^{9} +O(q^{10})$$ $$q-2.82843 q^{3} +2.82843 q^{7} +5.00000 q^{9} -5.65685 q^{11} +2.00000 q^{13} -2.00000 q^{17} -8.00000 q^{21} -2.82843 q^{23} -5.65685 q^{27} +6.00000 q^{29} +5.65685 q^{31} +16.0000 q^{33} +10.0000 q^{37} -5.65685 q^{39} +2.00000 q^{41} +8.48528 q^{43} +2.82843 q^{47} +1.00000 q^{49} +5.65685 q^{51} -6.00000 q^{53} +11.3137 q^{59} -2.00000 q^{61} +14.1421 q^{63} +2.82843 q^{67} +8.00000 q^{69} +5.65685 q^{71} +6.00000 q^{73} -16.0000 q^{77} +11.3137 q^{79} +1.00000 q^{81} -2.82843 q^{83} -16.9706 q^{87} +10.0000 q^{89} +5.65685 q^{91} -16.0000 q^{93} -2.00000 q^{97} -28.2843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{9} + O(q^{10})$$ $$2 q + 10 q^{9} + 4 q^{13} - 4 q^{17} - 16 q^{21} + 12 q^{29} + 32 q^{33} + 20 q^{37} + 4 q^{41} + 2 q^{49} - 12 q^{53} - 4 q^{61} + 16 q^{69} + 12 q^{73} - 32 q^{77} + 2 q^{81} + 20 q^{89} - 32 q^{93} - 4 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.82843 −1.63299 −0.816497 0.577350i $$-0.804087\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ 5.00000 1.66667
$$10$$ 0 0
$$11$$ −5.65685 −1.70561 −0.852803 0.522233i $$-0.825099\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −8.00000 −1.74574
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 5.65685 1.01600 0.508001 0.861357i $$-0.330385\pi$$
0.508001 + 0.861357i $$0.330385\pi$$
$$32$$ 0 0
$$33$$ 16.0000 2.78524
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ −5.65685 −0.905822
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 8.48528 1.29399 0.646997 0.762493i $$-0.276025\pi$$
0.646997 + 0.762493i $$0.276025\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 5.65685 0.792118
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 11.3137 1.47292 0.736460 0.676481i $$-0.236496\pi$$
0.736460 + 0.676481i $$0.236496\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 14.1421 1.78174
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.82843 0.345547 0.172774 0.984962i $$-0.444727\pi$$
0.172774 + 0.984962i $$0.444727\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ 11.3137 1.27289 0.636446 0.771321i $$-0.280404\pi$$
0.636446 + 0.771321i $$0.280404\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.82843 −0.310460 −0.155230 0.987878i $$-0.549612\pi$$
−0.155230 + 0.987878i $$0.549612\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −16.9706 −1.81944
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 5.65685 0.592999
$$92$$ 0 0
$$93$$ −16.0000 −1.65912
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ −28.2843 −2.84268
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −14.1421 −1.39347 −0.696733 0.717331i $$-0.745364\pi$$
−0.696733 + 0.717331i $$0.745364\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.1421 1.36717 0.683586 0.729870i $$-0.260419\pi$$
0.683586 + 0.729870i $$0.260419\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −28.2843 −2.68462
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 10.0000 0.924500
$$118$$ 0 0
$$119$$ −5.65685 −0.518563
$$120$$ 0 0
$$121$$ 21.0000 1.90909
$$122$$ 0 0
$$123$$ −5.65685 −0.510061
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.82843 0.250982 0.125491 0.992095i $$-0.459949\pi$$
0.125491 + 0.992095i $$0.459949\pi$$
$$128$$ 0 0
$$129$$ −24.0000 −2.11308
$$130$$ 0 0
$$131$$ 5.65685 0.494242 0.247121 0.968985i $$-0.420516\pi$$
0.247121 + 0.968985i $$0.420516\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 11.3137 0.959616 0.479808 0.877373i $$-0.340706\pi$$
0.479808 + 0.877373i $$0.340706\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ −11.3137 −0.946100
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.82843 −0.233285
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −16.9706 −1.38104 −0.690522 0.723311i $$-0.742619\pi$$
−0.690522 + 0.723311i $$0.742619\pi$$
$$152$$ 0 0
$$153$$ −10.0000 −0.808452
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ 16.9706 1.34585
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −14.1421 −1.10770 −0.553849 0.832617i $$-0.686841\pi$$
−0.553849 + 0.832617i $$0.686841\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.1421 1.09435 0.547176 0.837018i $$-0.315703\pi$$
0.547176 + 0.837018i $$0.315703\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −32.0000 −2.40527
$$178$$ 0 0
$$179$$ 11.3137 0.845626 0.422813 0.906217i $$-0.361043\pi$$
0.422813 + 0.906217i $$0.361043\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 5.65685 0.418167
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 11.3137 0.827340
$$188$$ 0 0
$$189$$ −16.0000 −1.16383
$$190$$ 0 0
$$191$$ 16.9706 1.22795 0.613973 0.789327i $$-0.289570\pi$$
0.613973 + 0.789327i $$0.289570\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −22.6274 −1.60402 −0.802008 0.597314i $$-0.796235\pi$$
−0.802008 + 0.597314i $$0.796235\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 16.9706 1.19110
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −14.1421 −0.982946
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.9706 −1.16830 −0.584151 0.811645i $$-0.698572\pi$$
−0.584151 + 0.811645i $$0.698572\pi$$
$$212$$ 0 0
$$213$$ −16.0000 −1.09630
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ 0 0
$$219$$ −16.9706 −1.14676
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 8.48528 0.568216 0.284108 0.958792i $$-0.408302\pi$$
0.284108 + 0.958792i $$0.408302\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −19.7990 −1.31411 −0.657053 0.753845i $$-0.728197\pi$$
−0.657053 + 0.753845i $$0.728197\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 45.2548 2.97755
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −32.0000 −2.07862
$$238$$ 0 0
$$239$$ −11.3137 −0.731823 −0.365911 0.930650i $$-0.619243\pi$$
−0.365911 + 0.930650i $$0.619243\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ 14.1421 0.907218
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ −5.65685 −0.357057 −0.178529 0.983935i $$-0.557134\pi$$
−0.178529 + 0.983935i $$0.557134\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 28.2843 1.75750
$$260$$ 0 0
$$261$$ 30.0000 1.85695
$$262$$ 0 0
$$263$$ 19.7990 1.22086 0.610429 0.792071i $$-0.290997\pi$$
0.610429 + 0.792071i $$0.290997\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −28.2843 −1.73097
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 16.9706 1.03089 0.515444 0.856923i $$-0.327627\pi$$
0.515444 + 0.856923i $$0.327627\pi$$
$$272$$ 0 0
$$273$$ −16.0000 −0.968364
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 28.2843 1.69334
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 8.48528 0.504398 0.252199 0.967675i $$-0.418846\pi$$
0.252199 + 0.967675i $$0.418846\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.65685 0.333914
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 5.65685 0.331611
$$292$$ 0 0
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 32.0000 1.85683
$$298$$ 0 0
$$299$$ −5.65685 −0.327144
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ 5.65685 0.324978
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.82843 0.161427 0.0807134 0.996737i $$-0.474280\pi$$
0.0807134 + 0.996737i $$0.474280\pi$$
$$308$$ 0 0
$$309$$ 40.0000 2.27552
$$310$$ 0 0
$$311$$ −28.2843 −1.60385 −0.801927 0.597422i $$-0.796192\pi$$
−0.801927 + 0.597422i $$0.796192\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ −33.9411 −1.90034
$$320$$ 0 0
$$321$$ −40.0000 −2.23258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 50.9117 2.81542
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ 5.65685 0.310929 0.155464 0.987841i $$-0.450313\pi$$
0.155464 + 0.987841i $$0.450313\pi$$
$$332$$ 0 0
$$333$$ 50.0000 2.73998
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 5.65685 0.307238
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −8.48528 −0.455514 −0.227757 0.973718i $$-0.573139\pi$$
−0.227757 + 0.973718i $$0.573139\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ −11.3137 −0.603881
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 16.0000 0.846810
$$358$$ 0 0
$$359$$ 22.6274 1.19423 0.597115 0.802156i $$-0.296314\pi$$
0.597115 + 0.802156i $$0.296314\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −59.3970 −3.11753
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −16.9706 −0.881068
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ −22.6274 −1.16229 −0.581146 0.813799i $$-0.697396\pi$$
−0.581146 + 0.813799i $$0.697396\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ −36.7696 −1.87884 −0.939418 0.342773i $$-0.888634\pi$$
−0.939418 + 0.342773i $$0.888634\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 42.4264 2.15666
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 5.65685 0.286079
$$392$$ 0 0
$$393$$ −16.0000 −0.807093
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ 11.3137 0.563576
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −56.5685 −2.80400
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ −16.9706 −0.837096
$$412$$ 0 0
$$413$$ 32.0000 1.57462
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −32.0000 −1.56705
$$418$$ 0 0
$$419$$ −11.3137 −0.552711 −0.276355 0.961056i $$-0.589127\pi$$
−0.276355 + 0.961056i $$0.589127\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 0 0
$$423$$ 14.1421 0.687614
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.65685 −0.273754
$$428$$ 0 0
$$429$$ 32.0000 1.54497
$$430$$ 0 0
$$431$$ −5.65685 −0.272481 −0.136241 0.990676i $$-0.543502\pi$$
−0.136241 + 0.990676i $$0.543502\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 22.6274 1.07995 0.539974 0.841682i $$-0.318434\pi$$
0.539974 + 0.841682i $$0.318434\pi$$
$$440$$ 0 0
$$441$$ 5.00000 0.238095
$$442$$ 0 0
$$443$$ −2.82843 −0.134383 −0.0671913 0.997740i $$-0.521404\pi$$
−0.0671913 + 0.997740i $$0.521404\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 28.2843 1.33780
$$448$$ 0 0
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ −11.3137 −0.532742
$$452$$ 0 0
$$453$$ 48.0000 2.25524
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 11.3137 0.528079
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 8.48528 0.394344 0.197172 0.980369i $$-0.436824\pi$$
0.197172 + 0.980369i $$0.436824\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.1421 0.654420 0.327210 0.944952i $$-0.393892\pi$$
0.327210 + 0.944952i $$0.393892\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −50.9117 −2.34589
$$472$$ 0 0
$$473$$ −48.0000 −2.20704
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −30.0000 −1.37361
$$478$$ 0 0
$$479$$ −33.9411 −1.55081 −0.775405 0.631464i $$-0.782454\pi$$
−0.775405 + 0.631464i $$0.782454\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 22.6274 1.02958
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −31.1127 −1.40985 −0.704925 0.709281i $$-0.749020\pi$$
−0.704925 + 0.709281i $$0.749020\pi$$
$$488$$ 0 0
$$489$$ 40.0000 1.80886
$$490$$ 0 0
$$491$$ 39.5980 1.78703 0.893516 0.449032i $$-0.148231\pi$$
0.893516 + 0.449032i $$0.148231\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ −40.0000 −1.78707
$$502$$ 0 0
$$503$$ 8.48528 0.378340 0.189170 0.981944i $$-0.439420\pi$$
0.189170 + 0.981944i $$0.439420\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 25.4558 1.13053
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 16.9706 0.750733
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 0 0
$$519$$ −5.65685 −0.248308
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 8.48528 0.371035 0.185518 0.982641i $$-0.440604\pi$$
0.185518 + 0.982641i $$0.440604\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.3137 −0.492833
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 56.5685 2.45487
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −32.0000 −1.38090
$$538$$ 0 0
$$539$$ −5.65685 −0.243658
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ −39.5980 −1.69931
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −42.4264 −1.81402 −0.907011 0.421107i $$-0.861642\pi$$
−0.907011 + 0.421107i $$0.861642\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 0 0
$$559$$ 16.9706 0.717778
$$560$$ 0 0
$$561$$ −32.0000 −1.35104
$$562$$ 0 0
$$563$$ 19.7990 0.834428 0.417214 0.908808i $$-0.363007\pi$$
0.417214 + 0.908808i $$0.363007\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.82843 0.118783
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ 28.2843 1.18366 0.591830 0.806063i $$-0.298406\pi$$
0.591830 + 0.806063i $$0.298406\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −2.00523
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 0 0
$$579$$ 50.9117 2.11582
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ 33.9411 1.40570
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 25.4558 1.05068 0.525338 0.850894i $$-0.323939\pi$$
0.525338 + 0.850894i $$0.323939\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 16.9706 0.698076
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 64.0000 2.61935
$$598$$ 0 0
$$599$$ 11.3137 0.462266 0.231133 0.972922i $$-0.425757\pi$$
0.231133 + 0.972922i $$0.425757\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 14.1421 0.575912
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.82843 0.114802 0.0574012 0.998351i $$-0.481719\pi$$
0.0574012 + 0.998351i $$0.481719\pi$$
$$608$$ 0 0
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ 5.65685 0.228852
$$612$$ 0 0
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ 0 0
$$619$$ −45.2548 −1.81895 −0.909473 0.415764i $$-0.863514\pi$$
−0.909473 + 0.415764i $$0.863514\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 0 0
$$623$$ 28.2843 1.13319
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −16.9706 −0.675587 −0.337794 0.941220i $$-0.609681\pi$$
−0.337794 + 0.941220i $$0.609681\pi$$
$$632$$ 0 0
$$633$$ 48.0000 1.90783
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ 28.2843 1.11891
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 0 0
$$643$$ 31.1127 1.22697 0.613483 0.789708i $$-0.289768\pi$$
0.613483 + 0.789708i $$0.289768\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 14.1421 0.555985 0.277992 0.960583i $$-0.410331\pi$$
0.277992 + 0.960583i $$0.410331\pi$$
$$648$$ 0 0
$$649$$ −64.0000 −2.51222
$$650$$ 0 0
$$651$$ −45.2548 −1.77368
$$652$$ 0 0
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 30.0000 1.17041
$$658$$ 0 0
$$659$$ −33.9411 −1.32216 −0.661079 0.750316i $$-0.729901\pi$$
−0.661079 + 0.750316i $$0.729901\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ 11.3137 0.439388
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −16.9706 −0.657103
$$668$$ 0 0
$$669$$ −24.0000 −0.927894
$$670$$ 0 0
$$671$$ 11.3137 0.436761
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ 0 0
$$679$$ −5.65685 −0.217090
$$680$$ 0 0
$$681$$ 56.0000 2.14592
$$682$$ 0 0
$$683$$ −14.1421 −0.541134 −0.270567 0.962701i $$-0.587211\pi$$
−0.270567 + 0.962701i $$0.587211\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −39.5980 −1.51076
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −28.2843 −1.07598 −0.537992 0.842950i $$-0.680817\pi$$
−0.537992 + 0.842950i $$0.680817\pi$$
$$692$$ 0 0
$$693$$ −80.0000 −3.03895
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ 28.2843 1.06981
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5.65685 −0.212748
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 56.5685 2.12149
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 32.0000 1.19506
$$718$$ 0 0
$$719$$ −11.3137 −0.421930 −0.210965 0.977494i $$-0.567661\pi$$
−0.210965 + 0.977494i $$0.567661\pi$$
$$720$$ 0 0
$$721$$ −40.0000 −1.48968
$$722$$ 0 0
$$723$$ −73.5391 −2.73495
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 25.4558 0.944105 0.472052 0.881570i $$-0.343513\pi$$
0.472052 + 0.881570i $$0.343513\pi$$
$$728$$ 0 0
$$729$$ −43.0000 −1.59259
$$730$$ 0 0
$$731$$ −16.9706 −0.627679
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ −11.3137 −0.416181 −0.208091 0.978110i $$-0.566725\pi$$
−0.208091 + 0.978110i $$0.566725\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −14.1421 −0.518825 −0.259412 0.965767i $$-0.583529\pi$$
−0.259412 + 0.965767i $$0.583529\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −14.1421 −0.517434
$$748$$ 0 0
$$749$$ 40.0000 1.46157
$$750$$ 0 0
$$751$$ −16.9706 −0.619265 −0.309632 0.950856i $$-0.600206\pi$$
−0.309632 + 0.950856i $$0.600206\pi$$
$$752$$ 0 0
$$753$$ 16.0000 0.583072
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ −45.2548 −1.64265
$$760$$ 0 0
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 0 0
$$763$$ −50.9117 −1.84313
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 22.6274 0.817029
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ −39.5980 −1.42609
$$772$$ 0 0
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −80.0000 −2.86998
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ −33.9411 −1.21296
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −8.48528 −0.302468 −0.151234 0.988498i $$-0.548325\pi$$
−0.151234 + 0.988498i $$0.548325\pi$$
$$788$$ 0 0
$$789$$ −56.0000 −1.99365
$$790$$ 0 0
$$791$$ −5.65685 −0.201135
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ −5.65685 −0.200125
$$800$$ 0 0
$$801$$ 50.0000 1.76666
$$802$$ 0 0
$$803$$ −33.9411 −1.19776
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 50.9117 1.79218
$$808$$ 0 0
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ −5.65685 −0.198639 −0.0993195 0.995056i $$-0.531667\pi$$
−0.0993195 + 0.995056i $$0.531667\pi$$
$$812$$ 0 0
$$813$$ −48.0000 −1.68343
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 28.2843 0.988332
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −25.4558 −0.887335 −0.443667 0.896191i $$-0.646323\pi$$
−0.443667 + 0.896191i $$0.646323\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −31.1127 −1.08189 −0.540947 0.841057i $$-0.681934\pi$$
−0.540947 + 0.841057i $$0.681934\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −28.2843 −0.981170
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −32.0000 −1.10608
$$838$$ 0 0
$$839$$ 11.3137 0.390593 0.195296 0.980744i $$-0.437433\pi$$
0.195296 + 0.980744i $$0.437433\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 84.8528 2.92249
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 59.3970 2.04090
$$848$$ 0 0
$$849$$ −24.0000 −0.823678
$$850$$ 0 0
$$851$$ −28.2843 −0.969572
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ 0 0
$$859$$ 33.9411 1.15806 0.579028 0.815308i $$-0.303432\pi$$
0.579028 + 0.815308i $$0.303432\pi$$
$$860$$ 0 0
$$861$$ −16.0000 −0.545279
$$862$$ 0 0
$$863$$ 42.4264 1.44421 0.722106 0.691783i $$-0.243174\pi$$
0.722106 + 0.691783i $$0.243174\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 36.7696 1.24876
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 5.65685 0.191675
$$872$$ 0 0
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ −28.2843 −0.954005
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ −2.82843 −0.0951842 −0.0475921 0.998867i $$-0.515155\pi$$
−0.0475921 + 0.998867i $$0.515155\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 2.82843 0.0949693 0.0474846 0.998872i $$-0.484879\pi$$
0.0474846 + 0.998872i $$0.484879\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −5.65685 −0.189512
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 16.0000 0.534224
$$898$$ 0 0
$$899$$ 33.9411 1.13200
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ −67.8823 −2.25898
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 25.4558 0.845247 0.422624 0.906305i $$-0.361109\pi$$
0.422624 + 0.906305i $$0.361109\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 5.65685 0.187420 0.0937100 0.995600i $$-0.470127\pi$$
0.0937100 + 0.995600i $$0.470127\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.0000 0.528367
$$918$$ 0 0
$$919$$ 33.9411 1.11961 0.559807 0.828623i $$-0.310875\pi$$
0.559807 + 0.828623i $$0.310875\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 0 0
$$923$$ 11.3137 0.372395
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −70.7107 −2.32244
$$928$$ 0 0
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 80.0000 2.61908
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ −16.9706 −0.553813
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 0 0
$$943$$ −5.65685 −0.184213
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −42.4264 −1.37867 −0.689336 0.724441i $$-0.742098\pi$$
−0.689336 + 0.724441i $$0.742098\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 84.8528 2.75154
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 96.0000 3.10324
$$958$$ 0 0
$$959$$ 16.9706 0.548008
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ 70.7107 2.27862
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −42.4264 −1.36434 −0.682171 0.731193i $$-0.738964\pi$$
−0.682171 + 0.731193i $$0.738964\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 28.2843 0.907685 0.453843 0.891082i $$-0.350053\pi$$
0.453843 + 0.891082i $$0.350053\pi$$
$$972$$ 0 0
$$973$$ 32.0000 1.02587
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −56.5685 −1.80794
$$980$$ 0 0
$$981$$ −90.0000 −2.87348
$$982$$ 0 0
$$983$$ −2.82843 −0.0902128 −0.0451064 0.998982i $$-0.514363\pi$$
−0.0451064 + 0.998982i $$0.514363\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −22.6274 −0.720239
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −16.9706 −0.539088 −0.269544 0.962988i $$-0.586873\pi$$
−0.269544 + 0.962988i $$0.586873\pi$$
$$992$$ 0 0
$$993$$ −16.0000 −0.507745
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 0 0
$$999$$ −56.5685 −1.78975
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 800.2.a.m.1.1 2
3.2 odd 2 7200.2.a.cm.1.2 2
4.3 odd 2 inner 800.2.a.m.1.2 2
5.2 odd 4 800.2.c.f.449.3 4
5.3 odd 4 800.2.c.f.449.1 4
5.4 even 2 160.2.a.c.1.2 yes 2
8.3 odd 2 1600.2.a.bc.1.1 2
8.5 even 2 1600.2.a.bc.1.2 2
12.11 even 2 7200.2.a.cm.1.1 2
15.2 even 4 7200.2.f.bh.6049.4 4
15.8 even 4 7200.2.f.bh.6049.2 4
15.14 odd 2 1440.2.a.o.1.1 2
20.3 even 4 800.2.c.f.449.4 4
20.7 even 4 800.2.c.f.449.2 4
20.19 odd 2 160.2.a.c.1.1 2
35.34 odd 2 7840.2.a.bf.1.1 2
40.3 even 4 1600.2.c.n.449.1 4
40.13 odd 4 1600.2.c.n.449.4 4
40.19 odd 2 320.2.a.g.1.2 2
40.27 even 4 1600.2.c.n.449.3 4
40.29 even 2 320.2.a.g.1.1 2
40.37 odd 4 1600.2.c.n.449.2 4
60.23 odd 4 7200.2.f.bh.6049.3 4
60.47 odd 4 7200.2.f.bh.6049.1 4
60.59 even 2 1440.2.a.o.1.2 2
80.19 odd 4 1280.2.d.l.641.1 4
80.29 even 4 1280.2.d.l.641.3 4
80.59 odd 4 1280.2.d.l.641.4 4
80.69 even 4 1280.2.d.l.641.2 4
120.29 odd 2 2880.2.a.bk.1.1 2
120.59 even 2 2880.2.a.bk.1.2 2
140.139 even 2 7840.2.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.c.1.1 2 20.19 odd 2
160.2.a.c.1.2 yes 2 5.4 even 2
320.2.a.g.1.1 2 40.29 even 2
320.2.a.g.1.2 2 40.19 odd 2
800.2.a.m.1.1 2 1.1 even 1 trivial
800.2.a.m.1.2 2 4.3 odd 2 inner
800.2.c.f.449.1 4 5.3 odd 4
800.2.c.f.449.2 4 20.7 even 4
800.2.c.f.449.3 4 5.2 odd 4
800.2.c.f.449.4 4 20.3 even 4
1280.2.d.l.641.1 4 80.19 odd 4
1280.2.d.l.641.2 4 80.69 even 4
1280.2.d.l.641.3 4 80.29 even 4
1280.2.d.l.641.4 4 80.59 odd 4
1440.2.a.o.1.1 2 15.14 odd 2
1440.2.a.o.1.2 2 60.59 even 2
1600.2.a.bc.1.1 2 8.3 odd 2
1600.2.a.bc.1.2 2 8.5 even 2
1600.2.c.n.449.1 4 40.3 even 4
1600.2.c.n.449.2 4 40.37 odd 4
1600.2.c.n.449.3 4 40.27 even 4
1600.2.c.n.449.4 4 40.13 odd 4
2880.2.a.bk.1.1 2 120.29 odd 2
2880.2.a.bk.1.2 2 120.59 even 2
7200.2.a.cm.1.1 2 12.11 even 2
7200.2.a.cm.1.2 2 3.2 odd 2
7200.2.f.bh.6049.1 4 60.47 odd 4
7200.2.f.bh.6049.2 4 15.8 even 4
7200.2.f.bh.6049.3 4 60.23 odd 4
7200.2.f.bh.6049.4 4 15.2 even 4
7840.2.a.bf.1.1 2 35.34 odd 2
7840.2.a.bf.1.2 2 140.139 even 2