# Properties

 Label 3840.2.k.p.1921.2 Level $3840$ Weight $2$ Character 3840.1921 Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1921.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3840.1921 Dual form 3840.2.k.p.1921.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} +8.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} +6.00000i q^{29} +4.00000 q^{33} +2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +1.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} -2.00000i q^{51} -6.00000i q^{53} -4.00000 q^{55} -8.00000 q^{57} -12.0000i q^{59} -14.0000i q^{61} -2.00000 q^{65} -12.0000i q^{67} +4.00000i q^{69} -2.00000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} +8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{15} - 4 q^{17} + 8 q^{23} - 2 q^{25} + 8 q^{33} + 4 q^{39} + 12 q^{41} + 24 q^{47} - 14 q^{49} - 8 q^{55} - 16 q^{57} - 4 q^{65} - 4 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89} + 16 q^{95} - 28 q^{97}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^15 - 4 * q^17 + 8 * q^23 - 2 * q^25 + 8 * q^33 + 4 * q^39 + 12 * q^41 + 24 * q^47 - 14 * q^49 - 8 * q^55 - 16 * q^57 - 4 * q^65 - 4 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 16 * q^95 - 28 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ − 1.00000i − 0.447214i
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 8.00000i 1.83533i 0.397360 + 0.917663i $$0.369927\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ − 2.00000i − 0.280056i
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ − 14.0000i − 1.79252i −0.443533 0.896258i $$-0.646275\pi$$
0.443533 0.896258i $$-0.353725\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 0 0
$$69$$ 4.00000i 0.481543i
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ − 1.00000i − 0.115470i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 2.00000i 0.216930i
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ − 14.0000i − 1.39305i −0.717532 0.696526i $$-0.754728\pi$$
0.717532 0.696526i $$-0.245272\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ − 14.0000i − 1.34096i −0.741929 0.670478i $$-0.766089\pi$$
0.741929 0.670478i $$-0.233911\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ − 4.00000i − 0.373002i
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ − 20.0000i − 1.74741i −0.486458 0.873704i $$-0.661711\pi$$
0.486458 0.873704i $$-0.338289\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ − 16.0000i − 1.35710i −0.734553 0.678551i $$-0.762608\pi$$
0.734553 0.678551i $$-0.237392\pi$$
$$140$$ 0 0
$$141$$ 12.0000i 1.01058i
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ − 7.00000i − 0.577350i
$$148$$ 0 0
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 0 0
$$165$$ − 4.00000i − 0.311400i
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ − 8.00000i − 0.611775i
$$172$$ 0 0
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 12.0000i 0.896922i 0.893802 + 0.448461i $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ 22.0000i 1.63525i 0.575753 + 0.817624i $$0.304709\pi$$
−0.575753 + 0.817624i $$0.695291\pi$$
$$182$$ 0 0
$$183$$ 14.0000 1.03491
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 0 0
$$195$$ − 2.00000i − 0.143223i
$$196$$ 0 0
$$197$$ 26.0000i 1.85242i 0.377004 + 0.926212i $$0.376954\pi$$
−0.377004 + 0.926212i $$0.623046\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ − 6.00000i − 0.419058i
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 32.0000 2.21349
$$210$$ 0 0
$$211$$ − 8.00000i − 0.550743i −0.961338 0.275371i $$-0.911199\pi$$
0.961338 0.275371i $$-0.0888008\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 2.00000i − 0.135147i
$$220$$ 0 0
$$221$$ 4.00000i 0.269069i
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 0 0
$$229$$ − 18.0000i − 1.18947i −0.803921 0.594737i $$-0.797256\pi$$
0.803921 0.594737i $$-0.202744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ − 12.0000i − 0.782794i
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 7.00000i 0.447214i
$$246$$ 0 0
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ − 12.0000i − 0.757433i −0.925513 0.378717i $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 0 0
$$253$$ − 16.0000i − 1.00591i
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ − 6.00000i − 0.371391i
$$262$$ 0 0
$$263$$ 20.0000 1.23325 0.616626 0.787256i $$-0.288499\pi$$
0.616626 + 0.787256i $$0.288499\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ − 2.00000i − 0.122398i
$$268$$ 0 0
$$269$$ − 10.0000i − 0.609711i −0.952399 0.304855i $$-0.901392\pi$$
0.952399 0.304855i $$-0.0986081\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 4.00000i 0.241209i
$$276$$ 0 0
$$277$$ − 30.0000i − 1.80253i −0.433273 0.901263i $$-0.642641\pi$$
0.433273 0.901263i $$-0.357359\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 8.00000i 0.473879i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 14.0000i − 0.820695i
$$292$$ 0 0
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ − 8.00000i − 0.462652i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ −14.0000 −0.801638
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 0 0
$$309$$ 8.00000i 0.455104i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ 2.00000i 0.110940i
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ 6.00000i 0.325875i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ 0 0
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ 2.00000i 0.107058i 0.998566 + 0.0535288i $$0.0170469\pi$$
−0.998566 + 0.0535288i $$0.982953\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −45.0000 −2.36842
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 2.00000i 0.104685i
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 16.0000i 0.821865i 0.911666 + 0.410932i $$0.134797\pi$$
−0.911666 + 0.410932i $$0.865203\pi$$
$$380$$ 0 0
$$381$$ 16.0000i 0.819705i
$$382$$ 0 0
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ − 6.00000i − 0.304212i −0.988364 0.152106i $$-0.951394\pi$$
0.988364 0.152106i $$-0.0486055\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ 20.0000 1.00887
$$394$$ 0 0
$$395$$ − 8.00000i − 0.402524i
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 2.00000i 0.0986527i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 16.0000 0.783523
$$418$$ 0 0
$$419$$ − 12.0000i − 0.586238i −0.956076 0.293119i $$-0.905307\pi$$
0.956076 0.293119i $$-0.0946933\pi$$
$$420$$ 0 0
$$421$$ − 2.00000i − 0.0974740i −0.998812 0.0487370i $$-0.984480\pi$$
0.998812 0.0487370i $$-0.0155196\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ − 8.00000i − 0.386244i
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 6.00000i 0.287678i
$$436$$ 0 0
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ 28.0000i 1.33032i 0.746701 + 0.665160i $$0.231637\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$444$$ 0 0
$$445$$ 2.00000i 0.0948091i
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ − 24.0000i − 1.13012i
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 0 0
$$459$$ 2.00000i 0.0933520i
$$460$$ 0 0
$$461$$ 14.0000i 0.652045i 0.945362 + 0.326023i $$0.105709\pi$$
−0.945362 + 0.326023i $$0.894291\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ − 8.00000i − 0.367065i
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 40.0000 1.82765 0.913823 0.406112i $$-0.133116\pi$$
0.913823 + 0.406112i $$0.133116\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 14.0000i 0.635707i
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ − 12.0000i − 0.541552i −0.962642 0.270776i $$-0.912720\pi$$
0.962642 0.270776i $$-0.0872803\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 8.00000i 0.358129i 0.983837 + 0.179065i $$0.0573071\pi$$
−0.983837 + 0.179065i $$0.942693\pi$$
$$500$$ 0 0
$$501$$ 12.0000i 0.536120i
$$502$$ 0 0
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ 38.0000i 1.68432i 0.539227 + 0.842160i $$0.318716\pi$$
−0.539227 + 0.842160i $$0.681284\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 8.00000 0.353209
$$514$$ 0 0
$$515$$ − 8.00000i − 0.352522i
$$516$$ 0 0
$$517$$ − 48.0000i − 2.11104i
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 12.0000i 0.520756i
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 28.0000i 1.20605i
$$540$$ 0 0
$$541$$ − 30.0000i − 1.28980i −0.764267 0.644900i $$-0.776899\pi$$
0.764267 0.644900i $$-0.223101\pi$$
$$542$$ 0 0
$$543$$ −22.0000 −0.944110
$$544$$ 0 0
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 0 0
$$549$$ 14.0000i 0.597505i
$$550$$ 0 0
$$551$$ −48.0000 −2.04487
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.00000i 0.0848953i
$$556$$ 0 0
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ − 6.00000i − 0.252422i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ − 32.0000i − 1.33916i −0.742741 0.669579i $$-0.766474\pi$$
0.742741 0.669579i $$-0.233526\pi$$
$$572$$ 0 0
$$573$$ − 16.0000i − 0.668410i
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ − 22.0000i − 0.914289i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −26.0000 −1.06950
$$592$$ 0 0
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000i 0.654836i
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 5.00000i 0.203279i
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 24.0000i − 0.970936i
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 0 0
$$615$$ 6.00000 0.241943
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ 0 0
$$619$$ 16.0000i 0.643094i 0.946894 + 0.321547i $$0.104203\pi$$
−0.946894 + 0.321547i $$0.895797\pi$$
$$620$$ 0 0
$$621$$ − 4.00000i − 0.160514i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 32.0000i 1.27796i
$$628$$ 0 0
$$629$$ − 4.00000i − 0.159490i
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 0 0
$$635$$ − 16.0000i − 0.634941i
$$636$$ 0 0
$$637$$ 14.0000i 0.554700i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ 20.0000 0.786281 0.393141 0.919478i $$-0.371389\pi$$
0.393141 + 0.919478i $$0.371389\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 20.0000i 0.779089i 0.921008 + 0.389545i $$0.127368\pi$$
−0.921008 + 0.389545i $$0.872632\pi$$
$$660$$ 0 0
$$661$$ − 26.0000i − 1.01128i −0.862744 0.505641i $$-0.831256\pi$$
0.862744 0.505641i $$-0.168744\pi$$
$$662$$ 0 0
$$663$$ −4.00000 −0.155347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24.0000i 0.929284i
$$668$$ 0 0
$$669$$ − 24.0000i − 0.927894i
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 0 0
$$675$$ 1.00000i 0.0384900i
$$676$$ 0 0
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ − 2.00000i − 0.0764161i
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 16.0000i 0.608669i 0.952565 + 0.304334i $$0.0984340\pi$$
−0.952565 + 0.304334i $$0.901566\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ − 6.00000i − 0.226941i
$$700$$ 0 0
$$701$$ − 10.0000i − 0.377695i −0.982006 0.188847i $$-0.939525\pi$$
0.982006 0.188847i $$-0.0604752\pi$$
$$702$$ 0 0
$$703$$ −16.0000 −0.603451
$$704$$ 0 0
$$705$$ 12.0000 0.451946
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 2.00000i − 0.0751116i −0.999295 0.0375558i $$-0.988043\pi$$
0.999295 0.0375558i $$-0.0119572\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000i 0.299183i
$$716$$ 0 0
$$717$$ 8.00000i 0.298765i
$$718$$ 0 0
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 0 0
$$725$$ − 6.00000i − 0.222834i
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000i 0.295891i
$$732$$ 0 0
$$733$$ − 10.0000i − 0.369358i −0.982799 0.184679i $$-0.940875\pi$$
0.982799 0.184679i $$-0.0591246\pi$$
$$734$$ 0 0
$$735$$ −7.00000 −0.258199
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ − 8.00000i − 0.294285i −0.989115 0.147142i $$-0.952992\pi$$
0.989115 0.147142i $$-0.0470076\pi$$
$$740$$ 0 0
$$741$$ 16.0000i 0.587775i
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 4.00000i 0.146352i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ − 8.00000i − 0.291150i
$$756$$ 0 0
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ 0 0
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ − 2.00000i − 0.0723102i
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 6.00000i 0.216085i
$$772$$ 0 0
$$773$$ 10.0000i 0.359675i 0.983696 + 0.179838i $$0.0575572\pi$$
−0.983696 + 0.179838i $$0.942443\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 48.0000i 1.71978i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ 0 0
$$789$$ 20.0000i 0.712019i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −28.0000 −0.994309
$$794$$ 0 0
$$795$$ − 6.00000i − 0.212798i
$$796$$ 0 0
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ 0 0
$$803$$ 8.00000i 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10.0000 0.352017
$$808$$ 0 0
$$809$$ 22.0000 0.773479 0.386739 0.922189i $$-0.373601\pi$$
0.386739 + 0.922189i $$0.373601\pi$$
$$810$$ 0 0
$$811$$ − 24.0000i − 0.842754i −0.906886 0.421377i $$-0.861547\pi$$
0.906886 0.421377i $$-0.138453\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ −20.0000 −0.700569
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 30.0000i − 1.04701i −0.852023 0.523504i $$-0.824625\pi$$
0.852023 0.523504i $$-0.175375\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ − 30.0000i − 1.04194i −0.853574 0.520972i $$-0.825570\pi$$
0.853574 0.520972i $$-0.174430\pi$$
$$830$$ 0 0
$$831$$ 30.0000 1.04069
$$832$$ 0 0
$$833$$ 14.0000 0.485071
$$834$$ 0 0
$$835$$ − 12.0000i − 0.415277i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ − 2.00000i − 0.0688837i
$$844$$ 0 0
$$845$$ − 9.00000i − 0.309609i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 8.00000i 0.274236i
$$852$$ 0 0
$$853$$ − 30.0000i − 1.02718i −0.858036 0.513590i $$-0.828315\pi$$
0.858036 0.513590i $$-0.171685\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ 32.0000i 1.09183i 0.837842 + 0.545913i $$0.183817\pi$$
−0.837842 + 0.545913i $$0.816183\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4.00000 −0.136162 −0.0680808 0.997680i $$-0.521688\pi$$
−0.0680808 + 0.997680i $$0.521688\pi$$
$$864$$ 0 0
$$865$$ 14.0000 0.476014
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ − 32.0000i − 1.08553i
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 0 0
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ 0 0
$$885$$ − 12.0000i − 0.403376i
$$886$$ 0 0
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 96.0000i 3.21252i
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000i 0.399778i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 22.0000 0.731305
$$906$$ 0 0
$$907$$ 36.0000i 1.19536i 0.801735 + 0.597680i $$0.203911\pi$$
−0.801735 + 0.597680i $$0.796089\pi$$
$$908$$ 0 0
$$909$$ 14.0000i 0.464351i
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ − 14.0000i − 0.462826i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ − 56.0000i − 1.83533i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ 14.0000i 0.456873i
$$940$$ 0 0
$$941$$ − 34.0000i − 1.10837i −0.832394 0.554184i $$-0.813030\pi$$
0.832394 0.554184i $$-0.186970\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 0 0
$$949$$ 4.00000i 0.129845i
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ 0 0
$$955$$ 16.0000i 0.517748i
$$956$$ 0 0
$$957$$ 24.0000i 0.775810i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 22.0000i 0.708205i
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 16.0000 0.513994
$$970$$ 0 0
$$971$$ − 28.0000i − 0.898563i −0.893390 0.449281i $$-0.851680\pi$$
0.893390 0.449281i $$-0.148320\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 8.00000i 0.255681i
$$980$$ 0 0
$$981$$ 14.0000i 0.446986i
$$982$$ 0 0
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ 26.0000 0.828429
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 16.0000i − 0.508770i
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 16.0000i − 0.507234i
$$996$$ 0 0
$$997$$ − 6.00000i − 0.190022i −0.995476 0.0950110i $$-0.969711\pi$$
0.995476 0.0950110i $$-0.0302886\pi$$
$$998$$ 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 3840.2.k.p.1921.2 2
4.3 odd 2 3840.2.k.k.1921.1 2
8.3 odd 2 3840.2.k.k.1921.2 2
8.5 even 2 inner 3840.2.k.p.1921.1 2
16.3 odd 4 480.2.a.e.1.1 yes 1
16.5 even 4 960.2.a.o.1.1 1
16.11 odd 4 960.2.a.f.1.1 1
16.13 even 4 480.2.a.b.1.1 1
48.5 odd 4 2880.2.a.i.1.1 1
48.11 even 4 2880.2.a.j.1.1 1
48.29 odd 4 1440.2.a.k.1.1 1
48.35 even 4 1440.2.a.j.1.1 1
80.3 even 4 2400.2.f.n.1249.2 2
80.13 odd 4 2400.2.f.e.1249.1 2
80.19 odd 4 2400.2.a.j.1.1 1
80.27 even 4 4800.2.f.j.3649.2 2
80.29 even 4 2400.2.a.y.1.1 1
80.37 odd 4 4800.2.f.ba.3649.1 2
80.43 even 4 4800.2.f.j.3649.1 2
80.53 odd 4 4800.2.f.ba.3649.2 2
80.59 odd 4 4800.2.a.ca.1.1 1
80.67 even 4 2400.2.f.n.1249.1 2
80.69 even 4 4800.2.a.u.1.1 1
80.77 odd 4 2400.2.f.e.1249.2 2
240.29 odd 4 7200.2.a.bg.1.1 1
240.77 even 4 7200.2.f.bb.6049.1 2
240.83 odd 4 7200.2.f.b.6049.2 2
240.173 even 4 7200.2.f.bb.6049.2 2
240.179 even 4 7200.2.a.u.1.1 1
240.227 odd 4 7200.2.f.b.6049.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.a.b.1.1 1 16.13 even 4
480.2.a.e.1.1 yes 1 16.3 odd 4
960.2.a.f.1.1 1 16.11 odd 4
960.2.a.o.1.1 1 16.5 even 4
1440.2.a.j.1.1 1 48.35 even 4
1440.2.a.k.1.1 1 48.29 odd 4
2400.2.a.j.1.1 1 80.19 odd 4
2400.2.a.y.1.1 1 80.29 even 4
2400.2.f.e.1249.1 2 80.13 odd 4
2400.2.f.e.1249.2 2 80.77 odd 4
2400.2.f.n.1249.1 2 80.67 even 4
2400.2.f.n.1249.2 2 80.3 even 4
2880.2.a.i.1.1 1 48.5 odd 4
2880.2.a.j.1.1 1 48.11 even 4
3840.2.k.k.1921.1 2 4.3 odd 2
3840.2.k.k.1921.2 2 8.3 odd 2
3840.2.k.p.1921.1 2 8.5 even 2 inner
3840.2.k.p.1921.2 2 1.1 even 1 trivial
4800.2.a.u.1.1 1 80.69 even 4
4800.2.a.ca.1.1 1 80.59 odd 4
4800.2.f.j.3649.1 2 80.43 even 4
4800.2.f.j.3649.2 2 80.27 even 4
4800.2.f.ba.3649.1 2 80.37 odd 4
4800.2.f.ba.3649.2 2 80.53 odd 4
7200.2.a.u.1.1 1 240.179 even 4
7200.2.a.bg.1.1 1 240.29 odd 4
7200.2.f.b.6049.1 2 240.227 odd 4
7200.2.f.b.6049.2 2 240.83 odd 4
7200.2.f.bb.6049.1 2 240.77 even 4
7200.2.f.bb.6049.2 2 240.173 even 4