# Properties

 Label 3840.2.k.c.1921.1 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: $$x^{2} + 1$$ 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 1921.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3840.1921 Dual form 3840.2.k.c.1921.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} -1.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -10.0000i q^{29} +4.00000 q^{31} -4.00000 q^{33} +4.00000i q^{35} -10.0000i q^{37} -6.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +1.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} +2.00000i q^{53} -4.00000 q^{55} +4.00000 q^{57} -12.0000i q^{59} +10.0000i q^{61} +4.00000 q^{63} -6.00000 q^{65} +12.0000i q^{67} -10.0000 q^{73} +1.00000i q^{75} +16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -2.00000i q^{85} -10.0000 q^{87} +6.00000 q^{89} +24.0000i q^{91} -4.00000i q^{93} +4.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} + 4 q^{17} - 2 q^{25} + 8 q^{31} - 8 q^{33} - 12 q^{39} - 4 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} + 8 q^{57} + 8 q^{63} - 12 q^{65} - 20 q^{73} + 8 q^{79} + 2 q^{81} - 20 q^{87} + 12 q^{89} + 8 q^{95} - 28 q^{97} + O(q^{100})$$

## 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
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 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
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 4.00000i 0.872872i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ − 10.0000i − 1.85695i −0.371391 0.928477i $$-0.621119\pi$$
0.371391 0.928477i $$-0.378881\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 4.00000i 0.676123i
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ − 2.00000i − 0.280056i
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ 10.0000i 1.28037i 0.768221 + 0.640184i $$0.221142\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 1.00000i 0.115470i
$$76$$ 0 0
$$77$$ 16.0000i 1.82337i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ − 2.00000i − 0.216930i
$$86$$ 0 0
$$87$$ −10.0000 −1.07211
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 24.0000i 2.51588i
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 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
$$101$$ 2.00000i 0.199007i 0.995037 + 0.0995037i $$0.0317255\pi$$
−0.995037 + 0.0995037i $$0.968274\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 2.00000i 0.191565i 0.995402 + 0.0957826i $$0.0305354\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 20.0000i 1.74741i 0.486458 + 0.873704i $$0.338289\pi$$
−0.486458 + 0.873704i $$0.661711\pi$$
$$132$$ 0 0
$$133$$ − 16.0000i − 1.38738i
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ 0 0
$$139$$ − 4.00000i − 0.339276i −0.985506 0.169638i $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 0 0
$$141$$ 8.00000i 0.673722i
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ −10.0000 −0.830455
$$146$$ 0 0
$$147$$ − 9.00000i − 0.742307i
$$148$$ 0 0
$$149$$ 10.0000i 0.819232i 0.912258 + 0.409616i $$0.134337\pi$$
−0.912258 + 0.409616i $$0.865663\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ − 4.00000i − 0.321288i
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 12.0000i − 0.939913i −0.882690 0.469956i $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ 0 0
$$165$$ 4.00000i 0.311400i
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ − 12.0000i − 0.896922i −0.893802 0.448461i $$-0.851972\pi$$
0.893802 0.448461i $$-0.148028\pi$$
$$180$$ 0 0
$$181$$ 6.00000i 0.445976i 0.974821 + 0.222988i $$0.0715812\pi$$
−0.974821 + 0.222988i $$0.928419\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ − 4.00000i − 0.290957i
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 6.00000i 0.429669i
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 40.0000i 2.80745i
$$204$$ 0 0
$$205$$ 2.00000i 0.139686i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ − 4.00000i − 0.275371i −0.990476 0.137686i $$-0.956034\pi$$
0.990476 0.137686i $$-0.0439664\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 10.0000i 0.675737i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ − 10.0000i − 0.660819i −0.943838 0.330409i $$-0.892813\pi$$
0.943838 0.330409i $$-0.107187\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 0 0
$$235$$ 8.00000i 0.521862i
$$236$$ 0 0
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 9.00000i − 0.574989i
$$246$$ 0 0
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 28.0000i 1.76734i 0.468106 + 0.883672i $$0.344936\pi$$
−0.468106 + 0.883672i $$0.655064\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 0 0
$$259$$ 40.0000i 2.48548i
$$260$$ 0 0
$$261$$ 10.0000i 0.618984i
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 0 0
$$269$$ 6.00000i 0.365826i 0.983129 + 0.182913i $$0.0585527\pi$$
−0.983129 + 0.182913i $$0.941447\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ 24.0000 1.45255
$$274$$ 0 0
$$275$$ 4.00000i 0.241209i
$$276$$ 0 0
$$277$$ 14.0000i 0.841178i 0.907251 + 0.420589i $$0.138177\pi$$
−0.907251 + 0.420589i $$0.861823\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 0 0
$$285$$ − 4.00000i − 0.236940i
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000i 0.820695i
$$292$$ 0 0
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 16.0000i − 0.922225i
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ − 4.00000i − 0.227552i
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ − 4.00000i − 0.225374i
$$316$$ 0 0
$$317$$ 22.0000i 1.23564i 0.786318 + 0.617822i $$0.211985\pi$$
−0.786318 + 0.617822i $$0.788015\pi$$
$$318$$ 0 0
$$319$$ −40.0000 −2.23957
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 6.00000i 0.332820i
$$326$$ 0 0
$$327$$ 2.00000 0.110600
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 28.0000i 1.53902i 0.638635 + 0.769510i $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 6.00000i 0.325875i
$$340$$ 0 0
$$341$$ − 16.0000i − 0.866449i
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ − 22.0000i − 1.17763i −0.808267 0.588817i $$-0.799594\pi$$
0.808267 0.588817i $$-0.200406\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000i 0.423405i
$$358$$ 0 0
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 10.0000i 0.523424i
$$366$$ 0 0
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ − 8.00000i − 0.415339i
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −60.0000 −3.09016
$$378$$ 0 0
$$379$$ − 12.0000i − 0.616399i −0.951322 0.308199i $$-0.900274\pi$$
0.951322 0.308199i $$-0.0997264\pi$$
$$380$$ 0 0
$$381$$ 12.0000i 0.614779i
$$382$$ 0 0
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ − 4.00000i − 0.203331i
$$388$$ 0 0
$$389$$ 26.0000i 1.31825i 0.752032 + 0.659126i $$0.229074\pi$$
−0.752032 + 0.659126i $$0.770926\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 20.0000 1.00887
$$394$$ 0 0
$$395$$ − 4.00000i − 0.201262i
$$396$$ 0 0
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ − 24.0000i − 1.19553i
$$404$$ 0 0
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ − 22.0000i − 1.08518i
$$412$$ 0 0
$$413$$ 48.0000i 2.36193i
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 12.0000i 0.586238i 0.956076 + 0.293119i $$0.0946933\pi$$
−0.956076 + 0.293119i $$0.905307\pi$$
$$420$$ 0 0
$$421$$ − 26.0000i − 1.26716i −0.773676 0.633581i $$-0.781584\pi$$
0.773676 0.633581i $$-0.218416\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ − 40.0000i − 1.93574i
$$428$$ 0 0
$$429$$ 24.0000i 1.15873i
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 10.0000i 0.479463i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 0 0
$$445$$ − 6.00000i − 0.284427i
$$446$$ 0 0
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ −38.0000 −1.79333 −0.896665 0.442709i $$-0.854018\pi$$
−0.896665 + 0.442709i $$0.854018\pi$$
$$450$$ 0 0
$$451$$ 8.00000i 0.376705i
$$452$$ 0 0
$$453$$ − 4.00000i − 0.187936i
$$454$$ 0 0
$$455$$ 24.0000 1.12514
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 2.00000i 0.0933520i
$$460$$ 0 0
$$461$$ − 18.0000i − 0.838344i −0.907907 0.419172i $$-0.862320\pi$$
0.907907 0.419172i $$-0.137680\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ − 48.0000i − 2.21643i
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 14.0000i 0.635707i
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 4.00000i 0.180517i 0.995918 + 0.0902587i $$0.0287694\pi$$
−0.995918 + 0.0902587i $$0.971231\pi$$
$$492$$ 0 0
$$493$$ − 20.0000i − 0.900755i
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 20.0000i − 0.895323i −0.894203 0.447661i $$-0.852257\pi$$
0.894203 0.447661i $$-0.147743\pi$$
$$500$$ 0 0
$$501$$ 8.00000i 0.357414i
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ 0 0
$$509$$ 6.00000i 0.265945i 0.991120 + 0.132973i $$0.0424523\pi$$
−0.991120 + 0.132973i $$0.957548\pi$$
$$510$$ 0 0
$$511$$ 40.0000 1.76950
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ − 4.00000i − 0.176261i
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ − 4.00000i − 0.174574i
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 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
$$539$$ − 36.0000i − 1.55063i
$$540$$ 0 0
$$541$$ − 14.0000i − 0.601907i −0.953639 0.300954i $$-0.902695\pi$$
0.953639 0.300954i $$-0.0973049\pi$$
$$542$$ 0 0
$$543$$ 6.00000 0.257485
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 0 0
$$549$$ − 10.0000i − 0.426790i
$$550$$ 0 0
$$551$$ 40.0000 1.70406
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 10.0000i 0.424476i
$$556$$ 0 0
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 6.00000i 0.252422i
$$566$$ 0 0
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ − 20.0000i − 0.836974i −0.908223 0.418487i $$-0.862561\pi$$
0.908223 0.418487i $$-0.137439\pi$$
$$572$$ 0 0
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 0 0
$$579$$ − 2.00000i − 0.0831172i
$$580$$ 0 0
$$581$$ − 16.0000i − 0.663792i
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 0 0
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 0 0
$$589$$ 16.0000i 0.659269i
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 8.00000i 0.327968i
$$596$$ 0 0
$$597$$ 20.0000i 0.818546i
$$598$$ 0 0
$$599$$ 48.0000 1.96123 0.980613 0.195952i $$-0.0627798\pi$$
0.980613 + 0.195952i $$0.0627798\pi$$
$$600$$ 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
$$607$$ 4.00000 0.162355 0.0811775 0.996700i $$-0.474132\pi$$
0.0811775 + 0.996700i $$0.474132\pi$$
$$608$$ 0 0
$$609$$ 40.0000 1.62088
$$610$$ 0 0
$$611$$ 48.0000i 1.94187i
$$612$$ 0 0
$$613$$ 38.0000i 1.53481i 0.641165 + 0.767403i $$0.278451\pi$$
−0.641165 + 0.767403i $$0.721549\pi$$
$$614$$ 0 0
$$615$$ 2.00000 0.0806478
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ − 20.0000i − 0.797452i
$$630$$ 0 0
$$631$$ −12.0000 −0.477712 −0.238856 0.971055i $$-0.576772\pi$$
−0.238856 + 0.971055i $$0.576772\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 12.0000i 0.476205i
$$636$$ 0 0
$$637$$ − 54.0000i − 2.13956i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ 36.0000i 1.41970i 0.704352 + 0.709851i $$0.251238\pi$$
−0.704352 + 0.709851i $$0.748762\pi$$
$$644$$ 0 0
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 16.0000i 0.627089i
$$652$$ 0 0
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ 0 0
$$655$$ 20.0000 0.781465
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ − 44.0000i − 1.71400i −0.515319 0.856998i $$-0.672327\pi$$
0.515319 0.856998i $$-0.327673\pi$$
$$660$$ 0 0
$$661$$ − 2.00000i − 0.0777910i −0.999243 0.0388955i $$-0.987616\pi$$
0.999243 0.0388955i $$-0.0123839\pi$$
$$662$$ 0 0
$$663$$ −12.0000 −0.466041
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 28.0000i 1.08254i
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 0 0
$$675$$ − 1.00000i − 0.0384900i
$$676$$ 0 0
$$677$$ − 22.0000i − 0.845529i −0.906240 0.422764i $$-0.861060\pi$$
0.906240 0.422764i $$-0.138940\pi$$
$$678$$ 0 0
$$679$$ 56.0000 2.14908
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 28.0000i − 1.07139i −0.844411 0.535695i $$-0.820050\pi$$
0.844411 0.535695i $$-0.179950\pi$$
$$684$$ 0 0
$$685$$ − 22.0000i − 0.840577i
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ − 12.0000i − 0.456502i −0.973602 0.228251i $$-0.926699\pi$$
0.973602 0.228251i $$-0.0733006\pi$$
$$692$$ 0 0
$$693$$ − 16.0000i − 0.607790i
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ 2.00000i 0.0756469i
$$700$$ 0 0
$$701$$ 6.00000i 0.226617i 0.993560 + 0.113308i $$0.0361448\pi$$
−0.993560 + 0.113308i $$0.963855\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 8.00000 0.301297
$$706$$ 0 0
$$707$$ − 8.00000i − 0.300871i
$$708$$ 0 0
$$709$$ − 10.0000i − 0.375558i −0.982211 0.187779i $$-0.939871\pi$$
0.982211 0.187779i $$-0.0601289\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 24.0000i 0.897549i
$$716$$ 0 0
$$717$$ − 16.0000i − 0.597531i
$$718$$ 0 0
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ − 18.0000i − 0.669427i
$$724$$ 0 0
$$725$$ 10.0000i 0.371391i
$$726$$ 0 0
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000i 0.295891i
$$732$$ 0 0
$$733$$ − 6.00000i − 0.221615i −0.993842 0.110808i $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ 0 0
$$735$$ −9.00000 −0.331970
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ − 52.0000i − 1.91285i −0.291977 0.956425i $$-0.594313\pi$$
0.291977 0.956425i $$-0.405687\pi$$
$$740$$ 0 0
$$741$$ − 24.0000i − 0.881662i
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 0 0
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ − 48.0000i − 1.75388i
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ 28.0000 1.02038
$$754$$ 0 0
$$755$$ − 4.00000i − 0.145575i
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 0 0
$$763$$ − 8.00000i − 0.289619i
$$764$$ 0 0
$$765$$ 2.00000i 0.0723102i
$$766$$ 0 0
$$767$$ −72.0000 −2.59977
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ − 26.0000i − 0.936367i
$$772$$ 0 0
$$773$$ 34.0000i 1.22290i 0.791285 + 0.611448i $$0.209412\pi$$
−0.791285 + 0.611448i $$0.790588\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 40.0000 1.43499
$$778$$ 0 0
$$779$$ − 8.00000i − 0.286630i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 10.0000 0.357371
$$784$$ 0 0
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 0 0
$$789$$ 8.00000i 0.284808i
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ 60.0000 2.13066
$$794$$ 0 0
$$795$$ − 2.00000i − 0.0709327i
$$796$$ 0 0
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 52.0000i 1.82597i 0.407997 + 0.912983i $$0.366228\pi$$
−0.407997 + 0.912983i $$0.633772\pi$$
$$812$$ 0 0
$$813$$ − 28.0000i − 0.982003i
$$814$$ 0 0
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ − 24.0000i − 0.838628i
$$820$$ 0 0
$$821$$ 34.0000i 1.18661i 0.804978 + 0.593304i $$0.202177\pi$$
−0.804978 + 0.593304i $$0.797823\pi$$
$$822$$ 0 0
$$823$$ 12.0000 0.418294 0.209147 0.977884i $$-0.432931\pi$$
0.209147 + 0.977884i $$0.432931\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ − 52.0000i − 1.80822i −0.427303 0.904109i $$-0.640536\pi$$
0.427303 0.904109i $$-0.359464\pi$$
$$828$$ 0 0
$$829$$ − 14.0000i − 0.486240i −0.969996 0.243120i $$-0.921829\pi$$
0.969996 0.243120i $$-0.0781709\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 8.00000i 0.276851i
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −71.0000 −2.44828
$$842$$ 0 0
$$843$$ − 22.0000i − 0.757720i
$$844$$ 0 0
$$845$$ 23.0000i 0.791224i
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ 20.0000i 0.682391i 0.939992 + 0.341196i $$0.110832\pi$$
−0.939992 + 0.341196i $$0.889168\pi$$
$$860$$ 0 0
$$861$$ − 8.00000i − 0.272639i
$$862$$ 0 0
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ − 16.0000i − 0.542763i
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ − 4.00000i − 0.135225i
$$876$$ 0 0
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 0 0
$$885$$ 12.0000i 0.403376i
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 48.0000 1.60987
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 40.0000i − 1.33407i
$$900$$ 0 0
$$901$$ 4.00000i 0.133259i
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ 0 0
$$907$$ 20.0000i 0.664089i 0.943264 + 0.332045i $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 0 0
$$909$$ − 2.00000i − 0.0663358i
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ − 10.0000i − 0.330590i
$$916$$ 0 0
$$917$$ − 80.0000i − 2.64183i
$$918$$ 0 0
$$919$$ 4.00000 0.131948 0.0659739 0.997821i $$-0.478985\pi$$
0.0659739 + 0.997821i $$0.478985\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −54.0000 −1.77168 −0.885841 0.463988i $$-0.846418\pi$$
−0.885841 + 0.463988i $$0.846418\pi$$
$$930$$ 0 0
$$931$$ 36.0000i 1.17985i
$$932$$ 0 0
$$933$$ − 8.00000i − 0.261908i
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ − 6.00000i − 0.195803i
$$940$$ 0 0
$$941$$ 46.0000i 1.49956i 0.661689 + 0.749779i $$0.269840\pi$$
−0.661689 + 0.749779i $$0.730160\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ 60.0000i 1.94768i
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ 0 0
$$953$$ −10.0000 −0.323932 −0.161966 0.986796i $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ 0 0
$$955$$ 16.0000i 0.517748i
$$956$$ 0 0
$$957$$ 40.0000i 1.29302i
$$958$$ 0 0
$$959$$ −88.0000 −2.84167
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ − 2.00000i − 0.0643823i
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ − 36.0000i − 1.15529i −0.816286 0.577647i $$-0.803971\pi$$
0.816286 0.577647i $$-0.196029\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ 0 0
$$975$$ 6.00000 0.192154
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ − 24.0000i − 0.767043i
$$980$$ 0 0
$$981$$ − 2.00000i − 0.0638551i
$$982$$ 0 0
$$983$$ −8.00000 −0.255160 −0.127580 0.991828i $$-0.540721\pi$$
−0.127580 + 0.991828i $$0.540721\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 0 0
$$987$$ − 32.0000i − 1.01857i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 20.0000i 0.634043i
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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.c.1921.1 2
4.3 odd 2 3840.2.k.bb.1921.2 2
8.3 odd 2 3840.2.k.bb.1921.1 2
8.5 even 2 inner 3840.2.k.c.1921.2 2
16.3 odd 4 480.2.a.a.1.1 1
16.5 even 4 960.2.a.h.1.1 1
16.11 odd 4 960.2.a.m.1.1 1
16.13 even 4 480.2.a.f.1.1 yes 1
48.5 odd 4 2880.2.a.p.1.1 1
48.11 even 4 2880.2.a.c.1.1 1
48.29 odd 4 1440.2.a.n.1.1 1
48.35 even 4 1440.2.a.g.1.1 1
80.3 even 4 2400.2.f.o.1249.1 2
80.13 odd 4 2400.2.f.d.1249.2 2
80.19 odd 4 2400.2.a.bh.1.1 1
80.27 even 4 4800.2.f.h.3649.1 2
80.29 even 4 2400.2.a.a.1.1 1
80.37 odd 4 4800.2.f.bb.3649.2 2
80.43 even 4 4800.2.f.h.3649.2 2
80.53 odd 4 4800.2.f.bb.3649.1 2
80.59 odd 4 4800.2.a.bg.1.1 1
80.67 even 4 2400.2.f.o.1249.2 2
80.69 even 4 4800.2.a.bo.1.1 1
80.77 odd 4 2400.2.f.d.1249.1 2
240.29 odd 4 7200.2.a.d.1.1 1
240.77 even 4 7200.2.f.ba.6049.2 2
240.83 odd 4 7200.2.f.c.6049.2 2
240.173 even 4 7200.2.f.ba.6049.1 2
240.179 even 4 7200.2.a.bw.1.1 1
240.227 odd 4 7200.2.f.c.6049.1 2

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