# Properties

 Label 4800.2.f.y.3649.2 Level $4800$ Weight $2$ Character 4800.3649 Analytic conductor $38.328$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.3281929702$$ 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 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4800.3649 Dual form 4800.2.f.y.3649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{13} +2.00000i q^{17} +5.00000 q^{19} -3.00000 q^{21} +6.00000i q^{23} -1.00000i q^{27} +10.0000 q^{29} +3.00000 q^{31} +2.00000i q^{33} -2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} -1.00000i q^{43} -2.00000i q^{47} -2.00000 q^{49} -2.00000 q^{51} -4.00000i q^{53} +5.00000i q^{57} +10.0000 q^{59} -7.00000 q^{61} -3.00000i q^{63} -3.00000i q^{67} -6.00000 q^{69} +8.00000 q^{71} +14.0000i q^{73} +6.00000i q^{77} +1.00000 q^{81} -6.00000i q^{83} +10.0000i q^{87} -3.00000 q^{91} +3.00000i q^{93} +17.0000i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{11} + 10q^{19} - 6q^{21} + 20q^{29} + 6q^{31} - 2q^{39} - 16q^{41} - 4q^{49} - 4q^{51} + 20q^{59} - 14q^{61} - 12q^{69} + 16q^{71} + 2q^{81} - 6q^{91} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ − 4.00000i − 0.549442i −0.961524 0.274721i $$-0.911414\pi$$
0.961524 0.274721i $$-0.0885855\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.00000i 0.662266i
$$58$$ 0 0
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 0 0
$$63$$ − 3.00000i − 0.377964i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 3.00000i − 0.366508i −0.983066 0.183254i $$-0.941337\pi$$
0.983066 0.183254i $$-0.0586631\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.00000i 0.683763i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.0000i 1.07211i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ 3.00000i 0.311086i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 17.0000i 1.72609i 0.505128 + 0.863044i $$0.331445\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 4.00000i 0.376288i 0.982141 + 0.188144i $$0.0602472\pi$$
−0.982141 + 0.188144i $$0.939753\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 15.0000i 1.30066i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 0 0
$$143$$ 2.00000i 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2.00000i − 0.164957i
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000i 1.03751i 0.854922 + 0.518756i $$0.173605\pi$$
−0.854922 + 0.518756i $$0.826395\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 0 0
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.0000i 0.751646i
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −17.0000 −1.26360 −0.631800 0.775131i $$-0.717684\pi$$
−0.631800 + 0.775131i $$0.717684\pi$$
$$182$$ 0 0
$$183$$ − 7.00000i − 0.517455i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ − 11.0000i − 0.791797i −0.918294 0.395899i $$-0.870433\pi$$
0.918294 0.395899i $$-0.129567\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ 30.0000i 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 9.00000i 0.610960i
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ − 19.0000i − 1.27233i −0.771551 0.636167i $$-0.780519\pi$$
0.771551 0.636167i $$-0.219481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000i 0.318142i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 0 0
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ − 3.00000i − 0.181568i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000i 0.180253i 0.995930 + 0.0901263i $$0.0287271\pi$$
−0.995930 + 0.0901263i $$0.971273\pi$$
$$278$$ 0 0
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 9.00000i 0.534994i 0.963559 + 0.267497i $$0.0861966\pi$$
−0.963559 + 0.267497i $$0.913803\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −17.0000 −0.996558
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2.00000i − 0.116052i
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 3.00000 0.172917
$$302$$ 0 0
$$303$$ − 12.0000i − 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ − 11.0000i − 0.621757i −0.950450 0.310878i $$-0.899377\pi$$
0.950450 0.310878i $$-0.100623\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000i 0.449325i 0.974437 + 0.224662i $$0.0721279\pi$$
−0.974437 + 0.224662i $$0.927872\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 10.0000i 0.556415i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 5.00000i 0.276501i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 23.0000i − 1.25289i −0.779466 0.626445i $$-0.784509\pi$$
0.779466 0.626445i $$-0.215491\pi$$
$$338$$ 0 0
$$339$$ −4.00000 −0.217250
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 15.0000i 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6.00000i − 0.317554i
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 27.0000i − 1.40939i −0.709511 0.704694i $$-0.751084\pi$$
0.709511 0.704694i $$-0.248916\pi$$
$$368$$ 0 0
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ − 29.0000i − 1.50156i −0.660551 0.750782i $$-0.729677\pi$$
0.660551 0.750782i $$-0.270323\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.0000i 0.515026i
$$378$$ 0 0
$$379$$ −25.0000 −1.28416 −0.642082 0.766636i $$-0.721929\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000i 0.0508329i
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 12.0000i 0.605320i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7.00000i − 0.351320i −0.984451 0.175660i $$-0.943794\pi$$
0.984451 0.175660i $$-0.0562059\pi$$
$$398$$ 0 0
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 3.00000i 0.149441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 4.00000i − 0.198273i
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 0 0
$$413$$ 30.0000i 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 20.0000i − 0.979404i
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ 2.00000i 0.0972433i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 21.0000i − 1.01626i
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 29.0000i 1.39365i 0.717241 + 0.696826i $$0.245405\pi$$
−0.717241 + 0.696826i $$0.754595\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.0000i 1.43509i
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 10.0000i 0.472984i
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ 0 0
$$453$$ − 7.00000i − 0.328889i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 38.0000i − 1.75843i −0.476425 0.879215i $$-0.658068\pi$$
0.476425 0.879215i $$-0.341932\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ − 2.00000i − 0.0919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4.00000i 0.183147i
$$478$$ 0 0
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ − 18.0000i − 0.819028i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 13.0000i 0.589086i 0.955638 + 0.294543i $$0.0951675\pi$$
−0.955638 + 0.294543i $$0.904833\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 20.0000i 0.900755i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −42.0000 −1.85797
$$512$$ 0 0
$$513$$ − 5.00000i − 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 4.00000i − 0.175920i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ − 31.0000i − 1.35554i −0.735276 0.677768i $$-0.762948\pi$$
0.735276 0.677768i $$-0.237052\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000i 0.261364i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 0 0
$$533$$ − 8.00000i − 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10.0000i 0.431532i
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ 0 0
$$543$$ − 17.0000i − 0.729540i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ 50.0000 2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 42.0000i − 1.77960i −0.456354 0.889799i $$-0.650845\pi$$
0.456354 0.889799i $$-0.349155\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ − 6.00000i − 0.252870i −0.991975 0.126435i $$-0.959647\pi$$
0.991975 0.126435i $$-0.0403535\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.00000i 0.125988i
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ 0 0
$$573$$ − 22.0000i − 0.919063i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 13.0000i − 0.541197i −0.962692 0.270599i $$-0.912778\pi$$
0.962692 0.270599i $$-0.0872216\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ − 8.00000i − 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 0 0
$$593$$ − 16.0000i − 0.657041i −0.944497 0.328521i $$-0.893450\pi$$
0.944497 0.328521i $$-0.106550\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 5.00000i − 0.204636i
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ 3.00000i 0.122169i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000i 0.324710i 0.986732 + 0.162355i $$0.0519090\pi$$
−0.986732 + 0.162355i $$0.948091\pi$$
$$608$$ 0 0
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ 2.00000 0.0809113
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ 0 0
$$619$$ 25.0000 1.00483 0.502417 0.864625i $$-0.332444\pi$$
0.502417 + 0.864625i $$0.332444\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 10.0000i 0.399362i
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 23.0000 0.915616 0.457808 0.889051i $$-0.348635\pi$$
0.457808 + 0.889051i $$0.348635\pi$$
$$632$$ 0 0
$$633$$ − 13.0000i − 0.516704i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 0 0
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 14.0000i − 0.546192i
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ 0 0
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 60.0000i 2.32321i
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ − 6.00000i − 0.231283i −0.993291 0.115642i $$-0.963108\pi$$
0.993291 0.115642i $$-0.0368924\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 0 0
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 15.0000i − 0.572286i
$$688$$ 0 0
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 16.0000i − 0.606043i
$$698$$ 0 0
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 36.0000i − 1.35392i
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 18.0000i 0.674105i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.0000i 0.746914i
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ − 23.0000i − 0.855379i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.0000i 1.59478i 0.603463 + 0.797391i $$0.293787\pi$$
−0.603463 + 0.797391i $$0.706213\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ 0 0
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 6.00000i − 0.221013i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −5.00000 −0.183680
$$742$$ 0 0
$$743$$ − 4.00000i − 0.146746i −0.997305 0.0733729i $$-0.976624\pi$$
0.997305 0.0733729i $$-0.0233763\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 12.0000i 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 23.0000i 0.835949i 0.908459 + 0.417975i $$0.137260\pi$$
−0.908459 + 0.417975i $$0.862740\pi$$
$$758$$ 0 0
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ 15.0000i 0.543036i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 10.0000i 0.361079i
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 0 0
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.00000i 0.215249i
$$778$$ 0 0
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 16.0000 0.572525
$$782$$ 0 0
$$783$$ − 10.0000i − 0.357371i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ − 7.00000i − 0.248577i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 52.0000i − 1.84193i −0.389640 0.920967i $$-0.627401\pi$$
0.389640 0.920967i $$-0.372599\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 28.0000i 0.988099i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10.0000i − 0.352017i
$$808$$ 0 0
$$809$$ 20.0000 0.703163 0.351581 0.936157i $$-0.385644\pi$$
0.351581 + 0.936157i $$0.385644\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ 0 0
$$813$$ 8.00000i 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 5.00000i − 0.174928i
$$818$$ 0 0
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 41.0000i 1.42917i 0.699549 + 0.714585i $$0.253384\pi$$
−0.699549 + 0.714585i $$0.746616\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ −3.00000 −0.104069
$$832$$ 0 0
$$833$$ − 4.00000i − 0.138592i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 3.00000i − 0.103695i
$$838$$ 0 0
$$839$$ 10.0000 0.345238 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ − 18.0000i − 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 21.0000i − 0.721569i
$$848$$ 0 0
$$849$$ −9.00000 −0.308879
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 51.0000i 1.74621i 0.487535 + 0.873103i $$0.337896\pi$$
−0.487535 + 0.873103i $$0.662104\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 28.0000i − 0.956462i −0.878234 0.478231i $$-0.841278\pi$$
0.878234 0.478231i $$-0.158722\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ 16.0000i 0.544646i 0.962206 + 0.272323i $$0.0877920\pi$$
−0.962206 + 0.272323i $$0.912208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ − 17.0000i − 0.575363i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 27.0000i − 0.911725i −0.890050 0.455863i $$-0.849331\pi$$
0.890050 0.455863i $$-0.150669\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ 0 0
$$883$$ − 41.0000i − 1.37976i −0.723924 0.689880i $$-0.757663\pi$$
0.723924 0.689880i $$-0.242337\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 18.0000i 0.604381i 0.953248 + 0.302190i $$0.0977178\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ − 10.0000i − 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ 0 0
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 0 0
$$903$$ 3.00000i 0.0998337i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 0 0
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ 58.0000 1.92163 0.960813 0.277198i $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 36.0000i 1.18882i
$$918$$ 0 0
$$919$$ 55.0000 1.81428 0.907141 0.420826i $$-0.138260\pi$$
0.907141 + 0.420826i $$0.138260\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 0 0
$$923$$ 8.00000i 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.00000i 0.131377i
$$928$$ 0 0
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ 0 0
$$933$$ 18.0000i 0.589294i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 33.0000i − 1.07806i −0.842286 0.539032i $$-0.818790\pi$$
0.842286 0.539032i $$-0.181210\pi$$
$$938$$ 0 0
$$939$$ 11.0000 0.358971
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ − 48.0000i − 1.56310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 18.0000i − 0.584921i −0.956278 0.292461i $$-0.905526\pi$$
0.956278 0.292461i $$-0.0944741\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ −8.00000 −0.259418
$$952$$ 0 0
$$953$$ − 56.0000i − 1.81402i −0.421111 0.907009i $$-0.638360\pi$$
0.421111 0.907009i $$-0.361640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 20.0000i 0.646508i
$$958$$ 0 0
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 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
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 0 0
$$973$$ − 60.0000i − 1.92351i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −5.00000 −0.159638
$$982$$ 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
$$987$$ 6.00000i 0.190982i
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ 12.0000i 0.380808i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4800.2.f.y.3649.2 2
4.3 odd 2 4800.2.f.l.3649.1 2
5.2 odd 4 4800.2.a.br.1.1 1
5.3 odd 4 4800.2.a.be.1.1 1
5.4 even 2 inner 4800.2.f.y.3649.1 2
8.3 odd 2 75.2.b.a.49.2 2
8.5 even 2 1200.2.f.d.49.1 2
20.3 even 4 4800.2.a.bq.1.1 1
20.7 even 4 4800.2.a.bb.1.1 1
20.19 odd 2 4800.2.f.l.3649.2 2
24.5 odd 2 3600.2.f.p.2449.2 2
24.11 even 2 225.2.b.a.199.1 2
40.3 even 4 75.2.a.c.1.1 yes 1
40.13 odd 4 1200.2.a.p.1.1 1
40.19 odd 2 75.2.b.a.49.1 2
40.27 even 4 75.2.a.a.1.1 1
40.29 even 2 1200.2.f.d.49.2 2
40.37 odd 4 1200.2.a.c.1.1 1
120.29 odd 2 3600.2.f.p.2449.1 2
120.53 even 4 3600.2.a.bk.1.1 1
120.59 even 2 225.2.b.a.199.2 2
120.77 even 4 3600.2.a.j.1.1 1
120.83 odd 4 225.2.a.a.1.1 1
120.107 odd 4 225.2.a.e.1.1 1
280.27 odd 4 3675.2.a.b.1.1 1
280.83 odd 4 3675.2.a.q.1.1 1
440.43 odd 4 9075.2.a.a.1.1 1
440.307 odd 4 9075.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 40.27 even 4
75.2.a.c.1.1 yes 1 40.3 even 4
75.2.b.a.49.1 2 40.19 odd 2
75.2.b.a.49.2 2 8.3 odd 2
225.2.a.a.1.1 1 120.83 odd 4
225.2.a.e.1.1 1 120.107 odd 4
225.2.b.a.199.1 2 24.11 even 2
225.2.b.a.199.2 2 120.59 even 2
1200.2.a.c.1.1 1 40.37 odd 4
1200.2.a.p.1.1 1 40.13 odd 4
1200.2.f.d.49.1 2 8.5 even 2
1200.2.f.d.49.2 2 40.29 even 2
3600.2.a.j.1.1 1 120.77 even 4
3600.2.a.bk.1.1 1 120.53 even 4
3600.2.f.p.2449.1 2 120.29 odd 2
3600.2.f.p.2449.2 2 24.5 odd 2
3675.2.a.b.1.1 1 280.27 odd 4
3675.2.a.q.1.1 1 280.83 odd 4
4800.2.a.bb.1.1 1 20.7 even 4
4800.2.a.be.1.1 1 5.3 odd 4
4800.2.a.bq.1.1 1 20.3 even 4
4800.2.a.br.1.1 1 5.2 odd 4
4800.2.f.l.3649.1 2 4.3 odd 2
4800.2.f.l.3649.2 2 20.19 odd 2
4800.2.f.y.3649.1 2 5.4 even 2 inner
4800.2.f.y.3649.2 2 1.1 even 1 trivial
9075.2.a.a.1.1 1 440.43 odd 4
9075.2.a.s.1.1 1 440.307 odd 4