# Properties

 Label 3200.2.f.f.449.2 Level $3200$ Weight $2$ Character 3200.449 Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.449 Dual form 3200.2.f.f.449.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} +1.00000i q^{17} +7.00000i q^{19} +4.00000i q^{21} -4.00000i q^{23} -5.00000 q^{27} -8.00000i q^{29} +4.00000 q^{31} +3.00000i q^{33} -4.00000 q^{37} +3.00000 q^{41} -8.00000 q^{43} -9.00000 q^{49} +1.00000i q^{51} -12.0000 q^{53} +7.00000i q^{57} -8.00000i q^{59} -4.00000i q^{61} -8.00000i q^{63} +9.00000 q^{67} -4.00000i q^{69} -16.0000 q^{71} +11.0000i q^{73} -12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -1.00000 q^{83} -8.00000i q^{87} -13.0000 q^{89} +4.00000 q^{93} +14.0000i q^{97} -6.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 4 q^{9} - 10 q^{27} + 8 q^{31} - 8 q^{37} + 6 q^{41} - 16 q^{43} - 18 q^{49} - 24 q^{53} + 18 q^{67} - 32 q^{71} - 24 q^{77} - 8 q^{79} + 2 q^{81} - 2 q^{83} - 26 q^{89} + 8 q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.00000i 0.242536i 0.992620 + 0.121268i $$0.0386960\pi$$
−0.992620 + 0.121268i $$0.961304\pi$$
$$18$$ 0 0
$$19$$ 7.00000i 1.60591i 0.596040 + 0.802955i $$0.296740\pi$$
−0.596040 + 0.802955i $$0.703260\pi$$
$$20$$ 0 0
$$21$$ 4.00000i 0.872872i
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ − 8.00000i − 1.48556i −0.669534 0.742781i $$-0.733506\pi$$
0.669534 0.742781i $$-0.266494\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 3.00000i 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 1.00000i 0.140028i
$$52$$ 0 0
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 7.00000i 0.927173i
$$58$$ 0 0
$$59$$ − 8.00000i − 1.04151i −0.853706 0.520756i $$-0.825650\pi$$
0.853706 0.520756i $$-0.174350\pi$$
$$60$$ 0 0
$$61$$ − 4.00000i − 0.512148i −0.966657 0.256074i $$-0.917571\pi$$
0.966657 0.256074i $$-0.0824290\pi$$
$$62$$ 0 0
$$63$$ − 8.00000i − 1.00791i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 9.00000 1.09952 0.549762 0.835321i $$-0.314718\pi$$
0.549762 + 0.835321i $$0.314718\pi$$
$$68$$ 0 0
$$69$$ − 4.00000i − 0.481543i
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 11.0000i 1.28745i 0.765256 + 0.643726i $$0.222612\pi$$
−0.765256 + 0.643726i $$0.777388\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 8.00000i − 0.857690i
$$88$$ 0 0
$$89$$ −13.0000 −1.37800 −0.688999 0.724763i $$-0.741949\pi$$
−0.688999 + 0.724763i $$0.741949\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 0 0
$$99$$ − 6.00000i − 0.603023i
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i 0.802225 + 0.597022i $$0.203650\pi$$
−0.802225 + 0.597022i $$0.796350\pi$$
$$102$$ 0 0
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.0000 1.25676 0.628379 0.777908i $$-0.283719\pi$$
0.628379 + 0.777908i $$0.283719\pi$$
$$108$$ 0 0
$$109$$ − 20.0000i − 1.91565i −0.287348 0.957826i $$-0.592774\pi$$
0.287348 0.957826i $$-0.407226\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 9.00000i 0.846649i 0.905978 + 0.423324i $$0.139137\pi$$
−0.905978 + 0.423324i $$0.860863\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 2.00000 0.181818
$$122$$ 0 0
$$123$$ 3.00000 0.270501
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ − 8.00000i − 0.698963i −0.936943 0.349482i $$-0.886358\pi$$
0.936943 0.349482i $$-0.113642\pi$$
$$132$$ 0 0
$$133$$ −28.0000 −2.42791
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 13.0000i 1.10265i 0.834292 + 0.551323i $$0.185877\pi$$
−0.834292 + 0.551323i $$0.814123\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −9.00000 −0.742307
$$148$$ 0 0
$$149$$ − 8.00000i − 0.655386i −0.944784 0.327693i $$-0.893729\pi$$
0.944784 0.327693i $$-0.106271\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −20.0000 −1.59617 −0.798087 0.602542i $$-0.794154\pi$$
−0.798087 + 0.602542i $$0.794154\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ −9.00000 −0.704934 −0.352467 0.935824i $$-0.614657\pi$$
−0.352467 + 0.935824i $$0.614657\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ − 14.0000i − 1.07061i
$$172$$ 0 0
$$173$$ 16.0000 1.21646 0.608229 0.793762i $$-0.291880\pi$$
0.608229 + 0.793762i $$0.291880\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 8.00000i − 0.601317i
$$178$$ 0 0
$$179$$ − 9.00000i − 0.672692i −0.941739 0.336346i $$-0.890809\pi$$
0.941739 0.336346i $$-0.109191\pi$$
$$180$$ 0 0
$$181$$ 12.0000i 0.891953i 0.895045 + 0.445976i $$0.147144\pi$$
−0.895045 + 0.445976i $$0.852856\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.00000 −0.219382
$$188$$ 0 0
$$189$$ − 20.0000i − 1.45479i
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 7.00000i 0.503871i 0.967744 + 0.251936i $$0.0810671\pi$$
−0.967744 + 0.251936i $$0.918933\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 9.00000 0.634811
$$202$$ 0 0
$$203$$ 32.0000 2.24596
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.00000i 0.556038i
$$208$$ 0 0
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 15.0000i 1.03264i 0.856395 + 0.516321i $$0.172699\pi$$
−0.856395 + 0.516321i $$0.827301\pi$$
$$212$$ 0 0
$$213$$ −16.0000 −1.09630
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 0 0
$$219$$ 11.0000i 0.743311i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ 0 0
$$229$$ 8.00000i 0.528655i 0.964433 + 0.264327i $$0.0851500\pi$$
−0.964433 + 0.264327i $$0.914850\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ 0 0
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −15.0000 −0.966235 −0.483117 0.875556i $$-0.660496\pi$$
−0.483117 + 0.875556i $$0.660496\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −1.00000 −0.0633724
$$250$$ 0 0
$$251$$ 13.0000i 0.820553i 0.911961 + 0.410276i $$0.134568\pi$$
−0.911961 + 0.410276i $$0.865432\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ − 16.0000i − 0.994192i
$$260$$ 0 0
$$261$$ 16.0000i 0.990375i
$$262$$ 0 0
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −13.0000 −0.795587
$$268$$ 0 0
$$269$$ − 16.0000i − 0.975537i −0.872973 0.487769i $$-0.837811\pi$$
0.872973 0.487769i $$-0.162189\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −19.0000 −1.12943 −0.564716 0.825285i $$-0.691014\pi$$
−0.564716 + 0.825285i $$0.691014\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000i 0.708338i
$$288$$ 0 0
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ 14.0000i 0.820695i
$$292$$ 0 0
$$293$$ −20.0000 −1.16841 −0.584206 0.811605i $$-0.698594\pi$$
−0.584206 + 0.811605i $$0.698594\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 15.0000i − 0.870388i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 32.0000i − 1.84445i
$$302$$ 0 0
$$303$$ 12.0000i 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 0 0
$$309$$ 16.0000i 0.910208i
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.00000 −0.449325 −0.224662 0.974437i $$-0.572128\pi$$
−0.224662 + 0.974437i $$0.572128\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 13.0000 0.725589
$$322$$ 0 0
$$323$$ −7.00000 −0.389490
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 20.0000i − 1.10600i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 13.0000i − 0.714545i −0.934000 0.357272i $$-0.883707\pi$$
0.934000 0.357272i $$-0.116293\pi$$
$$332$$ 0 0
$$333$$ 8.00000 0.438397
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 9.00000i − 0.490261i −0.969490 0.245131i $$-0.921169\pi$$
0.969490 0.245131i $$-0.0788309\pi$$
$$338$$ 0 0
$$339$$ 9.00000i 0.488813i
$$340$$ 0 0
$$341$$ 12.0000i 0.649836i
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5.00000 0.268414 0.134207 0.990953i $$-0.457151\pi$$
0.134207 + 0.990953i $$0.457151\pi$$
$$348$$ 0 0
$$349$$ − 4.00000i − 0.214115i −0.994253 0.107058i $$-0.965857\pi$$
0.994253 0.107058i $$-0.0341429\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 0 0
$$359$$ −28.0000 −1.47778 −0.738892 0.673824i $$-0.764651\pi$$
−0.738892 + 0.673824i $$0.764651\pi$$
$$360$$ 0 0
$$361$$ −30.0000 −1.57895
$$362$$ 0 0
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ − 48.0000i − 2.49204i
$$372$$ 0 0
$$373$$ −20.0000 −1.03556 −0.517780 0.855514i $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 29.0000i − 1.48963i −0.667271 0.744815i $$-0.732538\pi$$
0.667271 0.744815i $$-0.267462\pi$$
$$380$$ 0 0
$$381$$ − 12.0000i − 0.614779i
$$382$$ 0 0
$$383$$ − 4.00000i − 0.204390i −0.994764 0.102195i $$-0.967413\pi$$
0.994764 0.102195i $$-0.0325866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 16.0000 0.813326
$$388$$ 0 0
$$389$$ 12.0000i 0.608424i 0.952604 + 0.304212i $$0.0983931\pi$$
−0.952604 + 0.304212i $$0.901607\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ − 8.00000i − 0.403547i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 0 0
$$399$$ −28.0000 −1.40175
$$400$$ 0 0
$$401$$ −15.0000 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 12.0000i − 0.594818i
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ 0 0
$$411$$ 3.00000i 0.147979i
$$412$$ 0 0
$$413$$ 32.0000 1.57462
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 13.0000i 0.636613i
$$418$$ 0 0
$$419$$ 31.0000i 1.51445i 0.653155 + 0.757225i $$0.273445\pi$$
−0.653155 + 0.757225i $$0.726555\pi$$
$$420$$ 0 0
$$421$$ − 8.00000i − 0.389896i −0.980814 0.194948i $$-0.937546\pi$$
0.980814 0.194948i $$-0.0624538\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 16.0000 0.774294
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 0 0
$$433$$ 31.0000i 1.48976i 0.667196 + 0.744882i $$0.267494\pi$$
−0.667196 + 0.744882i $$0.732506\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 28.0000 1.33942
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ −19.0000 −0.902717 −0.451359 0.892343i $$-0.649060\pi$$
−0.451359 + 0.892343i $$0.649060\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 8.00000i − 0.378387i
$$448$$ 0 0
$$449$$ 23.0000 1.08544 0.542719 0.839915i $$-0.317395\pi$$
0.542719 + 0.839915i $$0.317395\pi$$
$$450$$ 0 0
$$451$$ 9.00000i 0.423793i
$$452$$ 0 0
$$453$$ 20.0000 0.939682
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.00000i 0.140334i 0.997535 + 0.0701670i $$0.0223532\pi$$
−0.997535 + 0.0701670i $$0.977647\pi$$
$$458$$ 0 0
$$459$$ − 5.00000i − 0.233380i
$$460$$ 0 0
$$461$$ 8.00000i 0.372597i 0.982493 + 0.186299i $$0.0596492\pi$$
−0.982493 + 0.186299i $$0.940351\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 40.0000 1.85098 0.925490 0.378773i $$-0.123654\pi$$
0.925490 + 0.378773i $$0.123654\pi$$
$$468$$ 0 0
$$469$$ 36.0000i 1.66233i
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ − 24.0000i − 1.10352i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 24.0000 1.09888
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 16.0000 0.728025
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ 0 0
$$489$$ −9.00000 −0.406994
$$490$$ 0 0
$$491$$ 24.0000i 1.08310i 0.840667 + 0.541552i $$0.182163\pi$$
−0.840667 + 0.541552i $$0.817837\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 64.0000i − 2.87079i
$$498$$ 0 0
$$499$$ 8.00000i 0.358129i 0.983837 + 0.179065i $$0.0573071\pi$$
−0.983837 + 0.179065i $$0.942693\pi$$
$$500$$ 0 0
$$501$$ 16.0000i 0.714827i
$$502$$ 0 0
$$503$$ 8.00000i 0.356702i 0.983967 + 0.178351i $$0.0570763\pi$$
−0.983967 + 0.178351i $$0.942924\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 0 0
$$509$$ 12.0000i 0.531891i 0.963988 + 0.265945i $$0.0856841\pi$$
−0.963988 + 0.265945i $$0.914316\pi$$
$$510$$ 0 0
$$511$$ −44.0000 −1.94645
$$512$$ 0 0
$$513$$ − 35.0000i − 1.54529i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.0000 0.702322
$$520$$ 0 0
$$521$$ 3.00000 0.131432 0.0657162 0.997838i $$-0.479067\pi$$
0.0657162 + 0.997838i $$0.479067\pi$$
$$522$$ 0 0
$$523$$ −13.0000 −0.568450 −0.284225 0.958758i $$-0.591736\pi$$
−0.284225 + 0.958758i $$0.591736\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000i 0.174243i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 16.0000i 0.694341i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 9.00000i − 0.388379i
$$538$$ 0 0
$$539$$ − 27.0000i − 1.16297i
$$540$$ 0 0
$$541$$ − 32.0000i − 1.37579i −0.725811 0.687894i $$-0.758536\pi$$
0.725811 0.687894i $$-0.241464\pi$$
$$542$$ 0 0
$$543$$ 12.0000i 0.514969i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.00000 −0.0427569 −0.0213785 0.999771i $$-0.506805\pi$$
−0.0213785 + 0.999771i $$0.506805\pi$$
$$548$$ 0 0
$$549$$ 8.00000i 0.341432i
$$550$$ 0 0
$$551$$ 56.0000 2.38568
$$552$$ 0 0
$$553$$ − 16.0000i − 0.680389i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −3.00000 −0.126660
$$562$$ 0 0
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ 37.0000 1.55112 0.775560 0.631273i $$-0.217467\pi$$
0.775560 + 0.631273i $$0.217467\pi$$
$$570$$ 0 0
$$571$$ − 8.00000i − 0.334790i −0.985890 0.167395i $$-0.946465\pi$$
0.985890 0.167395i $$-0.0535355\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 31.0000i 1.29055i 0.763952 + 0.645273i $$0.223257\pi$$
−0.763952 + 0.645273i $$0.776743\pi$$
$$578$$ 0 0
$$579$$ 7.00000i 0.290910i
$$580$$ 0 0
$$581$$ − 4.00000i − 0.165948i
$$582$$ 0 0
$$583$$ − 36.0000i − 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5.00000 −0.206372 −0.103186 0.994662i $$-0.532904\pi$$
−0.103186 + 0.994662i $$0.532904\pi$$
$$588$$ 0 0
$$589$$ 28.0000i 1.15372i
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 0 0
$$593$$ − 23.0000i − 0.944497i −0.881466 0.472248i $$-0.843443\pi$$
0.881466 0.472248i $$-0.156557\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 0 0
$$603$$ −18.0000 −0.733017
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 0 0
$$609$$ 32.0000 1.29671
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 4.00000 0.161558 0.0807792 0.996732i $$-0.474259\pi$$
0.0807792 + 0.996732i $$0.474259\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 0 0
$$619$$ 40.0000i 1.60774i 0.594808 + 0.803868i $$0.297228\pi$$
−0.594808 + 0.803868i $$0.702772\pi$$
$$620$$ 0 0
$$621$$ 20.0000i 0.802572i
$$622$$ 0 0
$$623$$ − 52.0000i − 2.08334i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −21.0000 −0.838659
$$628$$ 0 0
$$629$$ − 4.00000i − 0.159490i
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ 15.0000i 0.596196i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 32.0000 1.26590
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ 0 0
$$643$$ 24.0000 0.946468 0.473234 0.880937i $$-0.343087\pi$$
0.473234 + 0.880937i $$0.343087\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 16.0000i 0.627089i
$$652$$ 0 0
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 22.0000i − 0.858302i
$$658$$ 0 0
$$659$$ 1.00000i 0.0389545i 0.999810 + 0.0194772i $$0.00620019\pi$$
−0.999810 + 0.0194772i $$0.993800\pi$$
$$660$$ 0 0
$$661$$ − 40.0000i − 1.55582i −0.628376 0.777910i $$-0.716280\pi$$
0.628376 0.777910i $$-0.283720\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −32.0000 −1.23904
$$668$$ 0 0
$$669$$ 8.00000i 0.309298i
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ 0 0
$$679$$ −56.0000 −2.14908
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −13.0000 −0.497431 −0.248716 0.968577i $$-0.580008\pi$$
−0.248716 + 0.968577i $$0.580008\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8.00000i 0.305219i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 9.00000i 0.342376i 0.985238 + 0.171188i $$0.0547606\pi$$
−0.985238 + 0.171188i $$0.945239\pi$$
$$692$$ 0 0
$$693$$ 24.0000 0.911685
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3.00000i 0.113633i
$$698$$ 0 0
$$699$$ 10.0000i 0.378235i
$$700$$ 0 0
$$701$$ 16.0000i 0.604312i 0.953259 + 0.302156i $$0.0977063\pi$$
−0.953259 + 0.302156i $$0.902294\pi$$
$$702$$ 0 0
$$703$$ − 28.0000i − 1.05604i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −48.0000 −1.80523
$$708$$ 0 0
$$709$$ 40.0000i 1.50223i 0.660171 + 0.751116i $$0.270484\pi$$
−0.660171 + 0.751116i $$0.729516\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ −64.0000 −2.38348
$$722$$ 0 0
$$723$$ −15.0000 −0.557856
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 40.0000i 1.48352i 0.670667 + 0.741759i $$0.266008\pi$$
−0.670667 + 0.741759i $$0.733992\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ − 8.00000i − 0.295891i
$$732$$ 0 0
$$733$$ −32.0000 −1.18195 −0.590973 0.806691i $$-0.701256\pi$$
−0.590973 + 0.806691i $$0.701256\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 27.0000i 0.994558i
$$738$$ 0 0
$$739$$ − 24.0000i − 0.882854i −0.897297 0.441427i $$-0.854472\pi$$
0.897297 0.441427i $$-0.145528\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 2.00000 0.0731762
$$748$$ 0 0
$$749$$ 52.0000i 1.90004i
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 13.0000i 0.473746i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −32.0000 −1.16306 −0.581530 0.813525i $$-0.697546\pi$$
−0.581530 + 0.813525i $$0.697546\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 27.0000 0.978749 0.489375 0.872074i $$-0.337225\pi$$
0.489375 + 0.872074i $$0.337225\pi$$
$$762$$ 0 0
$$763$$ 80.0000 2.89619
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 39.0000 1.40638 0.703188 0.711004i $$-0.251759\pi$$
0.703188 + 0.711004i $$0.251759\pi$$
$$770$$ 0 0
$$771$$ 18.0000i 0.648254i
$$772$$ 0 0
$$773$$ 8.00000 0.287740 0.143870 0.989597i $$-0.454045\pi$$
0.143870 + 0.989597i $$0.454045\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 16.0000i − 0.573997i
$$778$$ 0 0
$$779$$ 21.0000i 0.752403i
$$780$$ 0 0
$$781$$ − 48.0000i − 1.71758i
$$782$$ 0 0
$$783$$ 40.0000i 1.42948i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ 0 0
$$789$$ − 4.00000i − 0.142404i
$$790$$ 0 0
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 26.0000 0.918665
$$802$$ 0 0
$$803$$ −33.0000 −1.16454
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 16.0000i − 0.563227i
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ − 8.00000i − 0.280918i −0.990086 0.140459i $$-0.955142\pi$$
0.990086 0.140459i $$-0.0448578\pi$$
$$812$$ 0 0
$$813$$ 4.00000 0.140286
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 56.0000i − 1.95919i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 36.0000i − 1.25641i −0.778048 0.628204i $$-0.783790\pi$$
0.778048 0.628204i $$-0.216210\pi$$
$$822$$ 0 0
$$823$$ 24.0000i 0.836587i 0.908312 + 0.418294i $$0.137372\pi$$
−0.908312 + 0.418294i $$0.862628\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 51.0000 1.77344 0.886722 0.462303i $$-0.152977\pi$$
0.886722 + 0.462303i $$0.152977\pi$$
$$828$$ 0 0
$$829$$ 32.0000i 1.11141i 0.831381 + 0.555703i $$0.187551\pi$$
−0.831381 + 0.555703i $$0.812449\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ − 9.00000i − 0.311832i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 0 0
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −35.0000 −1.20690
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.00000i 0.274883i
$$848$$ 0 0
$$849$$ −19.0000 −0.652078
$$850$$ 0 0
$$851$$ 16.0000i 0.548473i
$$852$$ 0 0
$$853$$ 28.0000 0.958702 0.479351 0.877623i $$-0.340872\pi$$
0.479351 + 0.877623i $$0.340872\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.0000i 0.922302i 0.887322 + 0.461151i $$0.152563\pi$$
−0.887322 + 0.461151i $$0.847437\pi$$
$$858$$ 0 0
$$859$$ 5.00000i 0.170598i 0.996355 + 0.0852989i $$0.0271845\pi$$
−0.996355 + 0.0852989i $$0.972815\pi$$
$$860$$ 0 0
$$861$$ 12.0000i 0.408959i
$$862$$ 0 0
$$863$$ 56.0000i 1.90626i 0.302558 + 0.953131i $$0.402160\pi$$
−0.302558 + 0.953131i $$0.597840\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ − 12.0000i − 0.407072i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 28.0000i − 0.947656i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ 0 0
$$879$$ −20.0000 −0.674583
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 25.0000 0.841317 0.420658 0.907219i $$-0.361799\pi$$
0.420658 + 0.907219i $$0.361799\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ 0 0
$$889$$ 48.0000 1.60987
$$890$$ 0 0
$$891$$ 3.00000i 0.100504i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 32.0000i − 1.06726i
$$900$$ 0 0
$$901$$ − 12.0000i − 0.399778i
$$902$$ 0 0
$$903$$ − 32.0000i − 1.06489i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 40.0000 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$908$$ 0 0
$$909$$ − 24.0000i − 0.796030i
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ − 3.00000i − 0.0992855i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 32.0000 1.05673
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ 7.00000 0.230658
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 32.0000i − 1.05102i
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ − 63.0000i − 2.06474i
$$932$$ 0 0
$$933$$ 12.0000 0.392862
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5.00000i 0.163343i 0.996659 + 0.0816714i $$0.0260258\pi$$
−0.996659 + 0.0816714i $$0.973974\pi$$
$$938$$ 0 0
$$939$$ 6.00000i 0.195803i
$$940$$ 0 0
$$941$$ 48.0000i 1.56476i 0.622804 + 0.782378i $$0.285993\pi$$
−0.622804 + 0.782378i $$0.714007\pi$$
$$942$$ 0 0
$$943$$ − 12.0000i − 0.390774i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −8.00000 −0.259418
$$952$$ 0 0
$$953$$ − 27.0000i − 0.874616i −0.899312 0.437308i $$-0.855932\pi$$
0.899312 0.437308i $$-0.144068\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 24.0000 0.775810
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −26.0000 −0.837838
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 0 0
$$969$$ −7.00000 −0.224872
$$970$$ 0 0
$$971$$ − 45.0000i − 1.44412i −0.691831 0.722059i $$-0.743196\pi$$
0.691831 0.722059i $$-0.256804\pi$$
$$972$$ 0 0
$$973$$ −52.0000 −1.66704
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 31.0000i − 0.991778i −0.868386 0.495889i $$-0.834842\pi$$
0.868386 0.495889i $$-0.165158\pi$$
$$978$$ 0 0
$$979$$ − 39.0000i − 1.24645i
$$980$$ 0 0
$$981$$ 40.0000i 1.27710i
$$982$$ 0 0
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.0000i 1.01754i
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 0 0
$$993$$ − 13.0000i − 0.412543i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −24.0000 −0.760088 −0.380044 0.924968i $$-0.624091\pi$$
−0.380044 + 0.924968i $$0.624091\pi$$
$$998$$ 0 0
$$999$$ 20.0000 0.632772
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 3200.2.f.f.449.2 2
4.3 odd 2 3200.2.f.a.449.1 2
5.2 odd 4 3200.2.d.a.1601.1 2
5.3 odd 4 3200.2.d.h.1601.2 yes 2
5.4 even 2 3200.2.f.b.449.1 2
8.3 odd 2 3200.2.f.e.449.1 2
8.5 even 2 3200.2.f.b.449.2 2
20.3 even 4 3200.2.d.b.1601.1 yes 2
20.7 even 4 3200.2.d.g.1601.2 yes 2
20.19 odd 2 3200.2.f.e.449.2 2
40.3 even 4 3200.2.d.b.1601.2 yes 2
40.13 odd 4 3200.2.d.h.1601.1 yes 2
40.19 odd 2 3200.2.f.a.449.2 2
40.27 even 4 3200.2.d.g.1601.1 yes 2
40.29 even 2 inner 3200.2.f.f.449.1 2
40.37 odd 4 3200.2.d.a.1601.2 yes 2
80.3 even 4 6400.2.a.v.1.1 1
80.13 odd 4 6400.2.a.c.1.1 1
80.27 even 4 6400.2.a.q.1.1 1
80.37 odd 4 6400.2.a.h.1.1 1
80.43 even 4 6400.2.a.i.1.1 1
80.53 odd 4 6400.2.a.p.1.1 1
80.67 even 4 6400.2.a.b.1.1 1
80.77 odd 4 6400.2.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.a.1601.1 2 5.2 odd 4
3200.2.d.a.1601.2 yes 2 40.37 odd 4
3200.2.d.b.1601.1 yes 2 20.3 even 4
3200.2.d.b.1601.2 yes 2 40.3 even 4
3200.2.d.g.1601.1 yes 2 40.27 even 4
3200.2.d.g.1601.2 yes 2 20.7 even 4
3200.2.d.h.1601.1 yes 2 40.13 odd 4
3200.2.d.h.1601.2 yes 2 5.3 odd 4
3200.2.f.a.449.1 2 4.3 odd 2
3200.2.f.a.449.2 2 40.19 odd 2
3200.2.f.b.449.1 2 5.4 even 2
3200.2.f.b.449.2 2 8.5 even 2
3200.2.f.e.449.1 2 8.3 odd 2
3200.2.f.e.449.2 2 20.19 odd 2
3200.2.f.f.449.1 2 40.29 even 2 inner
3200.2.f.f.449.2 2 1.1 even 1 trivial
6400.2.a.b.1.1 1 80.67 even 4
6400.2.a.c.1.1 1 80.13 odd 4
6400.2.a.h.1.1 1 80.37 odd 4
6400.2.a.i.1.1 1 80.43 even 4
6400.2.a.p.1.1 1 80.53 odd 4
6400.2.a.q.1.1 1 80.27 even 4
6400.2.a.v.1.1 1 80.3 even 4
6400.2.a.w.1.1 1 80.77 odd 4