# Properties

 Label 2880.2.k.l.1441.4 Level $2880$ Weight $2$ Character 2880.1441 Analytic conductor $22.997$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1441.4 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2880.1441 Dual form 2880.2.k.l.1441.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{5} +4.73205 q^{7} +O(q^{10})$$ $$q+1.00000i q^{5} +4.73205 q^{7} +3.46410i q^{11} +3.46410i q^{13} +3.46410 q^{17} +2.00000i q^{19} +2.19615 q^{23} -1.00000 q^{25} -2.53590 q^{31} +4.73205i q^{35} +6.00000i q^{37} -9.46410 q^{41} +0.196152i q^{43} -2.19615 q^{47} +15.3923 q^{49} -10.3923i q^{53} -3.46410 q^{55} +6.00000i q^{59} -0.928203i q^{61} -3.46410 q^{65} +0.196152i q^{67} -16.3923 q^{71} +6.39230 q^{73} +16.3923i q^{77} -12.0000 q^{79} +1.26795i q^{83} +3.46410i q^{85} +12.9282 q^{89} +16.3923i q^{91} -2.00000 q^{95} +14.3923 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7}+O(q^{10})$$ 4 * q + 12 * q^7 $$4 q + 12 q^{7} - 12 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{41} + 12 q^{47} + 20 q^{49} - 24 q^{71} - 16 q^{73} - 48 q^{79} + 24 q^{89} - 8 q^{95} + 16 q^{97}+O(q^{100})$$ 4 * q + 12 * q^7 - 12 * q^23 - 4 * q^25 - 24 * q^31 - 24 * q^41 + 12 * q^47 + 20 * q^49 - 24 * q^71 - 16 * q^73 - 48 * q^79 + 24 * q^89 - 8 * q^95 + 16 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ 4.73205 1.78855 0.894274 0.447521i $$-0.147693\pi$$
0.894274 + 0.447521i $$0.147693\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 3.46410i 0.960769i 0.877058 + 0.480384i $$0.159503\pi$$
−0.877058 + 0.480384i $$0.840497\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 0 0
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.19615 0.457929 0.228965 0.973435i $$-0.426466\pi$$
0.228965 + 0.973435i $$0.426466\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −2.53590 −0.455461 −0.227730 0.973724i $$-0.573130\pi$$
−0.227730 + 0.973724i $$0.573130\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.73205i 0.799863i
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.46410 −1.47804 −0.739022 0.673681i $$-0.764712\pi$$
−0.739022 + 0.673681i $$0.764712\pi$$
$$42$$ 0 0
$$43$$ 0.196152i 0.0299130i 0.999888 + 0.0149565i $$0.00476097\pi$$
−0.999888 + 0.0149565i $$0.995239\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.19615 −0.320342 −0.160171 0.987089i $$-0.551205\pi$$
−0.160171 + 0.987089i $$0.551205\pi$$
$$48$$ 0 0
$$49$$ 15.3923 2.19890
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 10.3923i − 1.42749i −0.700404 0.713746i $$-0.746997\pi$$
0.700404 0.713746i $$-0.253003\pi$$
$$54$$ 0 0
$$55$$ −3.46410 −0.467099
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 6.00000i 0.781133i 0.920575 + 0.390567i $$0.127721\pi$$
−0.920575 + 0.390567i $$0.872279\pi$$
$$60$$ 0 0
$$61$$ − 0.928203i − 0.118844i −0.998233 0.0594221i $$-0.981074\pi$$
0.998233 0.0594221i $$-0.0189258\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.46410 −0.429669
$$66$$ 0 0
$$67$$ 0.196152i 0.0239638i 0.999928 + 0.0119819i $$0.00381405\pi$$
−0.999928 + 0.0119819i $$0.996186\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −16.3923 −1.94541 −0.972704 0.232048i $$-0.925457\pi$$
−0.972704 + 0.232048i $$0.925457\pi$$
$$72$$ 0 0
$$73$$ 6.39230 0.748163 0.374081 0.927396i $$-0.377958\pi$$
0.374081 + 0.927396i $$0.377958\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 16.3923i 1.86808i
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.26795i 0.139176i 0.997576 + 0.0695878i $$0.0221684\pi$$
−0.997576 + 0.0695878i $$0.977832\pi$$
$$84$$ 0 0
$$85$$ 3.46410i 0.375735i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.9282 1.37039 0.685193 0.728361i $$-0.259718\pi$$
0.685193 + 0.728361i $$0.259718\pi$$
$$90$$ 0 0
$$91$$ 16.3923i 1.71838i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ 14.3923 1.46132 0.730659 0.682743i $$-0.239213\pi$$
0.730659 + 0.682743i $$0.239213\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 12.0000i − 1.19404i −0.802225 0.597022i $$-0.796350\pi$$
0.802225 0.597022i $$-0.203650\pi$$
$$102$$ 0 0
$$103$$ −2.19615 −0.216393 −0.108197 0.994130i $$-0.534508\pi$$
−0.108197 + 0.994130i $$0.534508\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 13.2679i − 1.28266i −0.767265 0.641331i $$-0.778383\pi$$
0.767265 0.641331i $$-0.221617\pi$$
$$108$$ 0 0
$$109$$ − 12.9282i − 1.23830i −0.785274 0.619149i $$-0.787478\pi$$
0.785274 0.619149i $$-0.212522\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.9282 1.21618 0.608092 0.793867i $$-0.291935\pi$$
0.608092 + 0.793867i $$0.291935\pi$$
$$114$$ 0 0
$$115$$ 2.19615i 0.204792i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 16.3923 1.50268
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −14.1962 −1.25970 −0.629852 0.776715i $$-0.716885\pi$$
−0.629852 + 0.776715i $$0.716885\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.3923i 0.907980i 0.891007 + 0.453990i $$0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$132$$ 0 0
$$133$$ 9.46410i 0.820642i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.928203 0.0793018 0.0396509 0.999214i $$-0.487375\pi$$
0.0396509 + 0.999214i $$0.487375\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ 9.46410 0.770178 0.385089 0.922880i $$-0.374171\pi$$
0.385089 + 0.922880i $$0.374171\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 2.53590i − 0.203688i
$$156$$ 0 0
$$157$$ 12.9282i 1.03178i 0.856654 + 0.515891i $$0.172539\pi$$
−0.856654 + 0.515891i $$0.827461\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 10.3923 0.819028
$$162$$ 0 0
$$163$$ 16.1962i 1.26858i 0.773095 + 0.634290i $$0.218708\pi$$
−0.773095 + 0.634290i $$0.781292\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.19615 −0.169943 −0.0849717 0.996383i $$-0.527080\pi$$
−0.0849717 + 0.996383i $$0.527080\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ −4.73205 −0.357709
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 7.85641i 0.587215i 0.955926 + 0.293608i $$0.0948559\pi$$
−0.955926 + 0.293608i $$0.905144\pi$$
$$180$$ 0 0
$$181$$ 6.92820i 0.514969i 0.966282 + 0.257485i $$0.0828937\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.39230 0.317816 0.158908 0.987293i $$-0.449203\pi$$
0.158908 + 0.987293i $$0.449203\pi$$
$$192$$ 0 0
$$193$$ −14.3923 −1.03598 −0.517990 0.855386i $$-0.673320\pi$$
−0.517990 + 0.855386i $$0.673320\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.3923i 0.740421i 0.928948 + 0.370211i $$0.120714\pi$$
−0.928948 + 0.370211i $$0.879286\pi$$
$$198$$ 0 0
$$199$$ 6.92820 0.491127 0.245564 0.969380i $$-0.421027\pi$$
0.245564 + 0.969380i $$0.421027\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ − 9.46410i − 0.661002i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.92820 −0.479234
$$210$$ 0 0
$$211$$ 14.3923i 0.990807i 0.868663 + 0.495404i $$0.164980\pi$$
−0.868663 + 0.495404i $$0.835020\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −0.196152 −0.0133775
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 9.12436 0.611012 0.305506 0.952190i $$-0.401174\pi$$
0.305506 + 0.952190i $$0.401174\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 12.5885i − 0.835525i −0.908556 0.417763i $$-0.862814\pi$$
0.908556 0.417763i $$-0.137186\pi$$
$$228$$ 0 0
$$229$$ 5.07180i 0.335154i 0.985859 + 0.167577i $$0.0535942\pi$$
−0.985859 + 0.167577i $$0.946406\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.3923 1.46697 0.733484 0.679706i $$-0.237893\pi$$
0.733484 + 0.679706i $$0.237893\pi$$
$$234$$ 0 0
$$235$$ − 2.19615i − 0.143261i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.7846 1.34444 0.672222 0.740349i $$-0.265340\pi$$
0.672222 + 0.740349i $$0.265340\pi$$
$$240$$ 0 0
$$241$$ 0.392305 0.0252706 0.0126353 0.999920i $$-0.495978\pi$$
0.0126353 + 0.999920i $$0.495978\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 15.3923i 0.983378i
$$246$$ 0 0
$$247$$ −6.92820 −0.440831
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.53590i 0.538781i 0.963031 + 0.269391i $$0.0868223\pi$$
−0.963031 + 0.269391i $$0.913178\pi$$
$$252$$ 0 0
$$253$$ 7.60770i 0.478292i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 28.3923i 1.76421i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 9.80385 0.604531 0.302266 0.953224i $$-0.402257\pi$$
0.302266 + 0.953224i $$0.402257\pi$$
$$264$$ 0 0
$$265$$ 10.3923 0.638394
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 26.7846i − 1.63309i −0.577284 0.816543i $$-0.695888\pi$$
0.577284 0.816543i $$-0.304112\pi$$
$$270$$ 0 0
$$271$$ −16.3923 −0.995762 −0.497881 0.867245i $$-0.665888\pi$$
−0.497881 + 0.867245i $$0.665888\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3.46410i − 0.208893i
$$276$$ 0 0
$$277$$ 7.85641i 0.472046i 0.971748 + 0.236023i $$0.0758440\pi$$
−0.971748 + 0.236023i $$0.924156\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7.60770 0.453837 0.226919 0.973914i $$-0.427135\pi$$
0.226919 + 0.973914i $$0.427135\pi$$
$$282$$ 0 0
$$283$$ 20.5885i 1.22386i 0.790913 + 0.611928i $$0.209606\pi$$
−0.790913 + 0.611928i $$0.790394\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −44.7846 −2.64355
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 7.60770i 0.439964i
$$300$$ 0 0
$$301$$ 0.928203i 0.0535007i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0.928203 0.0531488
$$306$$ 0 0
$$307$$ − 23.8038i − 1.35856i −0.733880 0.679279i $$-0.762293\pi$$
0.733880 0.679279i $$-0.237707\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 28.3923 1.60998 0.804990 0.593288i $$-0.202171\pi$$
0.804990 + 0.593288i $$0.202171\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 1.60770i − 0.0902972i −0.998980 0.0451486i $$-0.985624\pi$$
0.998980 0.0451486i $$-0.0143761\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.92820i 0.385496i
$$324$$ 0 0
$$325$$ − 3.46410i − 0.192154i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −10.3923 −0.572946
$$330$$ 0 0
$$331$$ − 26.3923i − 1.45065i −0.688405 0.725326i $$-0.741689\pi$$
0.688405 0.725326i $$-0.258311\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −0.196152 −0.0107170
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 8.78461i − 0.475713i
$$342$$ 0 0
$$343$$ 39.7128 2.14429
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.0526i 0.539650i 0.962909 + 0.269825i $$0.0869658\pi$$
−0.962909 + 0.269825i $$0.913034\pi$$
$$348$$ 0 0
$$349$$ − 32.7846i − 1.75492i −0.479650 0.877460i $$-0.659236\pi$$
0.479650 0.877460i $$-0.340764\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −26.7846 −1.42560 −0.712800 0.701367i $$-0.752573\pi$$
−0.712800 + 0.701367i $$0.752573\pi$$
$$354$$ 0 0
$$355$$ − 16.3923i − 0.870013i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.7846 1.73031 0.865153 0.501508i $$-0.167221\pi$$
0.865153 + 0.501508i $$0.167221\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6.39230i 0.334589i
$$366$$ 0 0
$$367$$ −28.0526 −1.46433 −0.732166 0.681126i $$-0.761490\pi$$
−0.732166 + 0.681126i $$0.761490\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 49.1769i − 2.55314i
$$372$$ 0 0
$$373$$ − 19.8564i − 1.02813i −0.857753 0.514063i $$-0.828140\pi$$
0.857753 0.514063i $$-0.171860\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 2.00000i − 0.102733i −0.998680 0.0513665i $$-0.983642\pi$$
0.998680 0.0513665i $$-0.0163577\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −14.1962 −0.725390 −0.362695 0.931908i $$-0.618143\pi$$
−0.362695 + 0.931908i $$0.618143\pi$$
$$384$$ 0 0
$$385$$ −16.3923 −0.835429
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 14.7846i − 0.749609i −0.927104 0.374805i $$-0.877710\pi$$
0.927104 0.374805i $$-0.122290\pi$$
$$390$$ 0 0
$$391$$ 7.60770 0.384738
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 12.0000i − 0.603786i
$$396$$ 0 0
$$397$$ − 20.5359i − 1.03067i −0.856990 0.515334i $$-0.827668\pi$$
0.856990 0.515334i $$-0.172332\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 31.8564 1.59083 0.795417 0.606063i $$-0.207252\pi$$
0.795417 + 0.606063i $$0.207252\pi$$
$$402$$ 0 0
$$403$$ − 8.78461i − 0.437593i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.7846 −1.03025
$$408$$ 0 0
$$409$$ 24.3923 1.20612 0.603061 0.797695i $$-0.293948\pi$$
0.603061 + 0.797695i $$0.293948\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 28.3923i 1.39709i
$$414$$ 0 0
$$415$$ −1.26795 −0.0622412
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.9282i 0.631584i 0.948828 + 0.315792i $$0.102270\pi$$
−0.948828 + 0.315792i $$0.897730\pi$$
$$420$$ 0 0
$$421$$ 6.00000i 0.292422i 0.989253 + 0.146211i $$0.0467079\pi$$
−0.989253 + 0.146211i $$0.953292\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3.46410 −0.168034
$$426$$ 0 0
$$427$$ − 4.39230i − 0.212559i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7.60770 −0.366450 −0.183225 0.983071i $$-0.558654\pi$$
−0.183225 + 0.983071i $$0.558654\pi$$
$$432$$ 0 0
$$433$$ 5.60770 0.269489 0.134744 0.990880i $$-0.456979\pi$$
0.134744 + 0.990880i $$0.456979\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.39230i 0.210112i
$$438$$ 0 0
$$439$$ 5.07180 0.242064 0.121032 0.992649i $$-0.461380\pi$$
0.121032 + 0.992649i $$0.461380\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 17.6603i − 0.839064i −0.907741 0.419532i $$-0.862194\pi$$
0.907741 0.419532i $$-0.137806\pi$$
$$444$$ 0 0
$$445$$ 12.9282i 0.612856i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.46410 0.446639 0.223319 0.974745i $$-0.428311\pi$$
0.223319 + 0.974745i $$0.428311\pi$$
$$450$$ 0 0
$$451$$ − 32.7846i − 1.54377i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −16.3923 −0.768483
$$456$$ 0 0
$$457$$ −18.7846 −0.878707 −0.439353 0.898314i $$-0.644792\pi$$
−0.439353 + 0.898314i $$0.644792\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 12.0000i − 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ 0 0
$$463$$ −26.1962 −1.21744 −0.608719 0.793386i $$-0.708316\pi$$
−0.608719 + 0.793386i $$0.708316\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.9808i 1.34107i 0.741878 + 0.670535i $$0.233935\pi$$
−0.741878 + 0.670535i $$0.766065\pi$$
$$468$$ 0 0
$$469$$ 0.928203i 0.0428604i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −0.679492 −0.0312431
$$474$$ 0 0
$$475$$ − 2.00000i − 0.0917663i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −20.7846 −0.947697
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 14.3923i 0.653521i
$$486$$ 0 0
$$487$$ 14.8756 0.674080 0.337040 0.941490i $$-0.390574\pi$$
0.337040 + 0.941490i $$0.390574\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.60770i 0.0725543i 0.999342 + 0.0362771i $$0.0115499\pi$$
−0.999342 + 0.0362771i $$0.988450\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −77.5692 −3.47946
$$498$$ 0 0
$$499$$ − 43.5692i − 1.95043i −0.221268 0.975213i $$-0.571020\pi$$
0.221268 0.975213i $$-0.428980\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2.19615 0.0979216 0.0489608 0.998801i $$-0.484409\pi$$
0.0489608 + 0.998801i $$0.484409\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 32.7846i − 1.45315i −0.687086 0.726576i $$-0.741110\pi$$
0.687086 0.726576i $$-0.258890\pi$$
$$510$$ 0 0
$$511$$ 30.2487 1.33812
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 2.19615i − 0.0967740i
$$516$$ 0 0
$$517$$ − 7.60770i − 0.334586i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.14359 −0.181534 −0.0907671 0.995872i $$-0.528932\pi$$
−0.0907671 + 0.995872i $$0.528932\pi$$
$$522$$ 0 0
$$523$$ − 36.9808i − 1.61706i −0.588458 0.808528i $$-0.700265\pi$$
0.588458 0.808528i $$-0.299735\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.78461 −0.382664
$$528$$ 0 0
$$529$$ −18.1769 −0.790301
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 32.7846i − 1.42006i
$$534$$ 0 0
$$535$$ 13.2679 0.573623
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 53.3205i 2.29668i
$$540$$ 0 0
$$541$$ 39.7128i 1.70739i 0.520776 + 0.853694i $$0.325643\pi$$
−0.520776 + 0.853694i $$0.674357\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 12.9282 0.553783
$$546$$ 0 0
$$547$$ − 12.1962i − 0.521470i −0.965410 0.260735i $$-0.916035\pi$$
0.965410 0.260735i $$-0.0839649\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −56.7846 −2.41473
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 26.7846i 1.13490i 0.823408 + 0.567450i $$0.192070\pi$$
−0.823408 + 0.567450i $$0.807930\pi$$
$$558$$ 0 0
$$559$$ −0.679492 −0.0287394
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 15.8038i 0.666053i 0.942918 + 0.333026i $$0.108070\pi$$
−0.942918 + 0.333026i $$0.891930\pi$$
$$564$$ 0 0
$$565$$ 12.9282i 0.543894i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.24871 −0.261960 −0.130980 0.991385i $$-0.541812\pi$$
−0.130980 + 0.991385i $$0.541812\pi$$
$$570$$ 0 0
$$571$$ 9.60770i 0.402070i 0.979584 + 0.201035i $$0.0644304\pi$$
−0.979584 + 0.201035i $$0.935570\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.19615 −0.0915859
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 6.00000i 0.248922i
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 31.5167i − 1.30083i −0.759578 0.650416i $$-0.774595\pi$$
0.759578 0.650416i $$-0.225405\pi$$
$$588$$ 0 0
$$589$$ − 5.07180i − 0.208980i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −12.9282 −0.530898 −0.265449 0.964125i $$-0.585520\pi$$
−0.265449 + 0.964125i $$0.585520\pi$$
$$594$$ 0 0
$$595$$ 16.3923i 0.672019i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 0.392305 0.0160024 0.00800122 0.999968i $$-0.497453\pi$$
0.00800122 + 0.999968i $$0.497453\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 1.00000i − 0.0406558i
$$606$$ 0 0
$$607$$ 2.19615 0.0891391 0.0445695 0.999006i $$-0.485808\pi$$
0.0445695 + 0.999006i $$0.485808\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 7.60770i − 0.307774i
$$612$$ 0 0
$$613$$ − 34.3923i − 1.38909i −0.719448 0.694546i $$-0.755605\pi$$
0.719448 0.694546i $$-0.244395\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −34.3923 −1.38458 −0.692291 0.721618i $$-0.743399\pi$$
−0.692291 + 0.721618i $$0.743399\pi$$
$$618$$ 0 0
$$619$$ 34.7846i 1.39811i 0.715067 + 0.699056i $$0.246396\pi$$
−0.715067 + 0.699056i $$0.753604\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 61.1769 2.45100
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 20.7846i 0.828737i
$$630$$ 0 0
$$631$$ 14.5359 0.578665 0.289332 0.957229i $$-0.406567\pi$$
0.289332 + 0.957229i $$0.406567\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 14.1962i − 0.563357i
$$636$$ 0 0
$$637$$ 53.3205i 2.11264i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −16.3923 −0.647457 −0.323729 0.946150i $$-0.604936\pi$$
−0.323729 + 0.946150i $$0.604936\pi$$
$$642$$ 0 0
$$643$$ 20.5885i 0.811929i 0.913889 + 0.405965i $$0.133064\pi$$
−0.913889 + 0.405965i $$0.866936\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 5.41154 0.212750 0.106375 0.994326i $$-0.466076\pi$$
0.106375 + 0.994326i $$0.466076\pi$$
$$648$$ 0 0
$$649$$ −20.7846 −0.815867
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 43.1769i − 1.68964i −0.535048 0.844822i $$-0.679707\pi$$
0.535048 0.844822i $$-0.320293\pi$$
$$654$$ 0 0
$$655$$ −10.3923 −0.406061
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 28.6410i − 1.11570i −0.829943 0.557848i $$-0.811627\pi$$
0.829943 0.557848i $$-0.188373\pi$$
$$660$$ 0 0
$$661$$ − 47.5692i − 1.85023i −0.379689 0.925114i $$-0.623969\pi$$
0.379689 0.925114i $$-0.376031\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −9.46410 −0.367002
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.21539 0.124129
$$672$$ 0 0
$$673$$ 23.1769 0.893404 0.446702 0.894683i $$-0.352598\pi$$
0.446702 + 0.894683i $$0.352598\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 19.1769i 0.737029i 0.929622 + 0.368514i $$0.120133\pi$$
−0.929622 + 0.368514i $$0.879867\pi$$
$$678$$ 0 0
$$679$$ 68.1051 2.61363
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5.66025i 0.216584i 0.994119 + 0.108292i $$0.0345381\pi$$
−0.994119 + 0.108292i $$0.965462\pi$$
$$684$$ 0 0
$$685$$ 0.928203i 0.0354648i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ − 26.3923i − 1.00401i −0.864865 0.502005i $$-0.832596\pi$$
0.864865 0.502005i $$-0.167404\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 10.0000 0.379322
$$696$$ 0 0
$$697$$ −32.7846 −1.24181
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.7846i 1.01164i 0.862639 + 0.505820i $$0.168810\pi$$
−0.862639 + 0.505820i $$0.831190\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 56.7846i − 2.13561i
$$708$$ 0 0
$$709$$ − 37.8564i − 1.42173i −0.703330 0.710864i $$-0.748304\pi$$
0.703330 0.710864i $$-0.251696\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −5.56922 −0.208569
$$714$$ 0 0
$$715$$ − 12.0000i − 0.448775i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −3.21539 −0.119914 −0.0599569 0.998201i $$-0.519096\pi$$
−0.0599569 + 0.998201i $$0.519096\pi$$
$$720$$ 0 0
$$721$$ −10.3923 −0.387030
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −4.05256 −0.150301 −0.0751505 0.997172i $$-0.523944\pi$$
−0.0751505 + 0.997172i $$0.523944\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0.679492i 0.0251319i
$$732$$ 0 0
$$733$$ 2.78461i 0.102852i 0.998677 + 0.0514260i $$0.0163766\pi$$
−0.998677 + 0.0514260i $$0.983623\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −0.679492 −0.0250294
$$738$$ 0 0
$$739$$ − 2.00000i − 0.0735712i −0.999323 0.0367856i $$-0.988288\pi$$
0.999323 0.0367856i $$-0.0117119\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5.41154 −0.198530 −0.0992651 0.995061i $$-0.531649\pi$$
−0.0992651 + 0.995061i $$0.531649\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 62.7846i − 2.29410i
$$750$$ 0 0
$$751$$ −19.6077 −0.715495 −0.357747 0.933818i $$-0.616455\pi$$
−0.357747 + 0.933818i $$0.616455\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 9.46410i 0.344434i
$$756$$ 0 0
$$757$$ 36.9282i 1.34218i 0.741377 + 0.671089i $$0.234173\pi$$
−0.741377 + 0.671089i $$0.765827\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.7128 −1.22209 −0.611044 0.791596i $$-0.709250\pi$$
−0.611044 + 0.791596i $$0.709250\pi$$
$$762$$ 0 0
$$763$$ − 61.1769i − 2.21475i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −20.7846 −0.750489
$$768$$ 0 0
$$769$$ 34.7846 1.25437 0.627183 0.778872i $$-0.284208\pi$$
0.627183 + 0.778872i $$0.284208\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 25.6077i 0.921045i 0.887648 + 0.460522i $$0.152338\pi$$
−0.887648 + 0.460522i $$0.847662\pi$$
$$774$$ 0 0
$$775$$ 2.53590 0.0910922
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 18.9282i − 0.678173i
$$780$$ 0 0
$$781$$ − 56.7846i − 2.03191i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −12.9282 −0.461427
$$786$$ 0 0
$$787$$ 16.1962i 0.577330i 0.957430 + 0.288665i $$0.0932115\pi$$
−0.957430 + 0.288665i $$0.906789\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 61.1769 2.17520
$$792$$ 0 0
$$793$$ 3.21539 0.114182
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 22.3923i 0.793176i 0.917997 + 0.396588i $$0.129806\pi$$
−0.917997 + 0.396588i $$0.870194\pi$$
$$798$$ 0 0
$$799$$ −7.60770 −0.269141
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 22.1436i 0.781430i
$$804$$ 0 0
$$805$$ 10.3923i 0.366281i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 47.5692 1.67244 0.836222 0.548391i $$-0.184759\pi$$
0.836222 + 0.548391i $$0.184759\pi$$
$$810$$ 0 0
$$811$$ 17.6077i 0.618290i 0.951015 + 0.309145i $$0.100043\pi$$
−0.951015 + 0.309145i $$0.899957\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −16.1962 −0.567326
$$816$$ 0 0
$$817$$ −0.392305 −0.0137250
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 9.21539i − 0.321619i −0.986985 0.160810i $$-0.948589\pi$$
0.986985 0.160810i $$-0.0514105\pi$$
$$822$$ 0 0
$$823$$ −48.8372 −1.70236 −0.851178 0.524877i $$-0.824111\pi$$
−0.851178 + 0.524877i $$0.824111\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.26795i 0.0440909i 0.999757 + 0.0220455i $$0.00701786\pi$$
−0.999757 + 0.0220455i $$0.992982\pi$$
$$828$$ 0 0
$$829$$ 9.21539i 0.320064i 0.987112 + 0.160032i $$0.0511597\pi$$
−0.987112 + 0.160032i $$0.948840\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 53.3205 1.84745
$$834$$ 0 0
$$835$$ − 2.19615i − 0.0760010i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 32.7846 1.13185 0.565925 0.824457i $$-0.308519\pi$$
0.565925 + 0.824457i $$0.308519\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.00000i 0.0344010i
$$846$$ 0 0
$$847$$ −4.73205 −0.162595
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 13.1769i 0.451699i
$$852$$ 0 0
$$853$$ 3.46410i 0.118609i 0.998240 + 0.0593043i $$0.0188882\pi$$
−0.998240 + 0.0593043i $$0.981112\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 43.8564 1.49811 0.749053 0.662510i $$-0.230509\pi$$
0.749053 + 0.662510i $$0.230509\pi$$
$$858$$ 0 0
$$859$$ − 31.5692i − 1.07713i −0.842585 0.538564i $$-0.818967\pi$$
0.842585 0.538564i $$-0.181033\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 10.9808 0.373789 0.186895 0.982380i $$-0.440158\pi$$
0.186895 + 0.982380i $$0.440158\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 41.5692i − 1.41014i
$$870$$ 0 0
$$871$$ −0.679492 −0.0230237
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 4.73205i − 0.159973i
$$876$$ 0 0
$$877$$ − 19.8564i − 0.670503i −0.942129 0.335252i $$-0.891179\pi$$
0.942129 0.335252i $$-0.108821\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 28.3923 0.956561 0.478281 0.878207i $$-0.341260\pi$$
0.478281 + 0.878207i $$0.341260\pi$$
$$882$$ 0 0
$$883$$ 16.1962i 0.545044i 0.962150 + 0.272522i $$0.0878577\pi$$
−0.962150 + 0.272522i $$0.912142\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −51.3731 −1.72494 −0.862469 0.506109i $$-0.831083\pi$$
−0.862469 + 0.506109i $$0.831083\pi$$
$$888$$ 0 0
$$889$$ −67.1769 −2.25304
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 4.39230i − 0.146983i
$$894$$ 0 0
$$895$$ −7.85641 −0.262611
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ − 36.0000i − 1.19933i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6.92820 −0.230301
$$906$$ 0 0
$$907$$ − 32.5885i − 1.08208i −0.840996 0.541041i $$-0.818030\pi$$
0.840996 0.541041i $$-0.181970\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −25.1769 −0.834148 −0.417074 0.908872i $$-0.636944\pi$$
−0.417074 + 0.908872i $$0.636944\pi$$
$$912$$ 0 0
$$913$$ −4.39230 −0.145364
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 49.1769i 1.62396i
$$918$$ 0 0
$$919$$ −15.7128 −0.518318 −0.259159 0.965835i $$-0.583445\pi$$
−0.259159 + 0.965835i $$0.583445\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 56.7846i − 1.86909i
$$924$$ 0 0
$$925$$ − 6.00000i − 0.197279i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −28.3923 −0.931521 −0.465761 0.884911i $$-0.654219\pi$$
−0.465761 + 0.884911i $$0.654219\pi$$
$$930$$ 0 0
$$931$$ 30.7846i 1.00892i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ −30.3923 −0.992873 −0.496437 0.868073i $$-0.665359\pi$$
−0.496437 + 0.868073i $$0.665359\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 20.7846i − 0.677559i −0.940866 0.338779i $$-0.889986\pi$$
0.940866 0.338779i $$-0.110014\pi$$
$$942$$ 0 0
$$943$$ −20.7846 −0.676840
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 56.1962i − 1.82613i −0.407815 0.913065i $$-0.633709\pi$$
0.407815 0.913065i $$-0.366291\pi$$
$$948$$ 0 0
$$949$$ 22.1436i 0.718811i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −11.0718 −0.358651 −0.179325 0.983790i $$-0.557391\pi$$
−0.179325 + 0.983790i $$0.557391\pi$$
$$954$$ 0 0
$$955$$ 4.39230i 0.142132i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 4.39230 0.141835
$$960$$ 0 0
$$961$$ −24.5692 −0.792555
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 14.3923i − 0.463305i
$$966$$ 0 0
$$967$$ 2.19615 0.0706235 0.0353118 0.999376i $$-0.488758\pi$$
0.0353118 + 0.999376i $$0.488758\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 57.0333i 1.83029i 0.403129 + 0.915143i $$0.367923\pi$$
−0.403129 + 0.915143i $$0.632077\pi$$
$$972$$ 0 0
$$973$$ − 47.3205i − 1.51703i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 29.3205 0.938046 0.469023 0.883186i $$-0.344606\pi$$
0.469023 + 0.883186i $$0.344606\pi$$
$$978$$ 0 0
$$979$$ 44.7846i 1.43132i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 42.5885 1.35836 0.679180 0.733971i $$-0.262335\pi$$
0.679180 + 0.733971i $$0.262335\pi$$
$$984$$ 0 0
$$985$$ −10.3923 −0.331126
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0.430781i 0.0136980i
$$990$$ 0 0
$$991$$ −32.1051 −1.01985 −0.509926 0.860218i $$-0.670327\pi$$
−0.509926 + 0.860218i $$0.670327\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 6.92820i 0.219639i
$$996$$ 0 0
$$997$$ − 46.3923i − 1.46926i −0.678468 0.734630i $$-0.737356\pi$$
0.678468 0.734630i $$-0.262644\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 2880.2.k.l.1441.4 4
3.2 odd 2 320.2.d.b.161.1 yes 4
4.3 odd 2 2880.2.k.e.1441.3 4
8.3 odd 2 2880.2.k.e.1441.1 4
8.5 even 2 inner 2880.2.k.l.1441.2 4
12.11 even 2 320.2.d.a.161.4 yes 4
15.2 even 4 1600.2.f.d.1249.2 4
15.8 even 4 1600.2.f.h.1249.3 4
15.14 odd 2 1600.2.d.b.801.4 4
24.5 odd 2 320.2.d.b.161.4 yes 4
24.11 even 2 320.2.d.a.161.1 4
48.5 odd 4 1280.2.a.c.1.1 2
48.11 even 4 1280.2.a.p.1.2 2
48.29 odd 4 1280.2.a.m.1.2 2
48.35 even 4 1280.2.a.b.1.1 2
60.23 odd 4 1600.2.f.e.1249.2 4
60.47 odd 4 1600.2.f.i.1249.3 4
60.59 even 2 1600.2.d.h.801.1 4
120.29 odd 2 1600.2.d.b.801.1 4
120.53 even 4 1600.2.f.d.1249.1 4
120.59 even 2 1600.2.d.h.801.4 4
120.77 even 4 1600.2.f.h.1249.4 4
120.83 odd 4 1600.2.f.i.1249.4 4
120.107 odd 4 1600.2.f.e.1249.1 4
240.29 odd 4 6400.2.a.bf.1.1 2
240.59 even 4 6400.2.a.y.1.1 2
240.149 odd 4 6400.2.a.ck.1.2 2
240.179 even 4 6400.2.a.cd.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.1 4 24.11 even 2
320.2.d.a.161.4 yes 4 12.11 even 2
320.2.d.b.161.1 yes 4 3.2 odd 2
320.2.d.b.161.4 yes 4 24.5 odd 2
1280.2.a.b.1.1 2 48.35 even 4
1280.2.a.c.1.1 2 48.5 odd 4
1280.2.a.m.1.2 2 48.29 odd 4
1280.2.a.p.1.2 2 48.11 even 4
1600.2.d.b.801.1 4 120.29 odd 2
1600.2.d.b.801.4 4 15.14 odd 2
1600.2.d.h.801.1 4 60.59 even 2
1600.2.d.h.801.4 4 120.59 even 2
1600.2.f.d.1249.1 4 120.53 even 4
1600.2.f.d.1249.2 4 15.2 even 4
1600.2.f.e.1249.1 4 120.107 odd 4
1600.2.f.e.1249.2 4 60.23 odd 4
1600.2.f.h.1249.3 4 15.8 even 4
1600.2.f.h.1249.4 4 120.77 even 4
1600.2.f.i.1249.3 4 60.47 odd 4
1600.2.f.i.1249.4 4 120.83 odd 4
2880.2.k.e.1441.1 4 8.3 odd 2
2880.2.k.e.1441.3 4 4.3 odd 2
2880.2.k.l.1441.2 4 8.5 even 2 inner
2880.2.k.l.1441.4 4 1.1 even 1 trivial
6400.2.a.y.1.1 2 240.59 even 4
6400.2.a.bf.1.1 2 240.29 odd 4
6400.2.a.cd.1.2 2 240.179 even 4
6400.2.a.ck.1.2 2 240.149 odd 4