# Properties

 Label 768.2.d.b.385.2 Level $768$ Weight $2$ Character 768.385 Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ 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 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 385.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.385 Dual form 768.2.d.b.385.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +6.00000 q^{17} -2.00000i q^{21} -4.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} -4.00000i q^{29} +10.0000 q^{31} +4.00000 q^{33} +2.00000i q^{37} +6.00000 q^{39} +2.00000 q^{41} +8.00000i q^{43} -12.0000 q^{47} -3.00000 q^{49} +6.00000i q^{51} -12.0000i q^{53} -4.00000i q^{59} -2.00000i q^{61} +2.00000 q^{63} -4.00000i q^{67} -4.00000i q^{69} +4.00000 q^{71} +10.0000 q^{73} +5.00000i q^{75} +8.00000i q^{77} -6.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +4.00000 q^{87} -2.00000 q^{89} +12.0000i q^{91} +10.0000i q^{93} -6.00000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{7} - 2q^{9} + 12q^{17} - 8q^{23} + 10q^{25} + 20q^{31} + 8q^{33} + 12q^{39} + 4q^{41} - 24q^{47} - 6q^{49} + 4q^{63} + 8q^{71} + 20q^{73} - 12q^{79} + 2q^{81} + 8q^{87} - 4q^{89} - 12q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 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
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ − 2.00000i − 0.436436i
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ − 4.00000i − 0.742781i −0.928477 0.371391i $$-0.878881\pi$$
0.928477 0.371391i $$-0.121119\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 6.00000i 0.840168i
$$52$$ 0 0
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ − 2.00000i − 0.256074i −0.991769 0.128037i $$-0.959132\pi$$
0.991769 0.128037i $$-0.0408676\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ − 4.00000i − 0.481543i
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 5.00000i 0.577350i
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 12.0000i 1.25794i
$$92$$ 0 0
$$93$$ 10.0000i 1.03695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 4.00000i 0.398015i 0.979998 + 0.199007i $$0.0637718\pi$$
−0.979998 + 0.199007i $$0.936228\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ 2.00000i 0.191565i 0.995402 + 0.0957826i $$0.0305354\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ − 4.00000i − 0.339276i −0.985506 0.169638i $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 0 0
$$141$$ − 12.0000i − 1.01058i
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 3.00000i − 0.247436i
$$148$$ 0 0
$$149$$ 16.0000i 1.31077i 0.755295 + 0.655386i $$0.227494\pi$$
−0.755295 + 0.655386i $$0.772506\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 16.0000i 1.21646i 0.793762 + 0.608229i $$0.208120\pi$$
−0.793762 + 0.608229i $$0.791880\pi$$
$$174$$ 0 0
$$175$$ −10.0000 −0.755929
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ 12.0000i 0.896922i 0.893802 + 0.448461i $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ 14.0000i 1.04061i 0.853980 + 0.520306i $$0.174182\pi$$
−0.853980 + 0.520306i $$0.825818\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 8.00000i 0.561490i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000i 0.826114i 0.910705 + 0.413057i $$0.135539\pi$$
−0.910705 + 0.413057i $$0.864461\pi$$
$$212$$ 0 0
$$213$$ 4.00000i 0.274075i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −20.0000 −1.35769
$$218$$ 0 0
$$219$$ 10.0000i 0.675737i
$$220$$ 0 0
$$221$$ − 36.0000i − 2.42162i
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ − 10.0000i − 0.660819i −0.943838 0.330409i $$-0.892813\pi$$
0.943838 0.330409i $$-0.107187\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 6.00000i − 0.389742i
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ − 20.0000i − 1.26239i −0.775625 0.631194i $$-0.782565\pi$$
0.775625 0.631194i $$-0.217435\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ − 4.00000i − 0.248548i
$$260$$ 0 0
$$261$$ 4.00000i 0.247594i
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 2.00000i − 0.122398i
$$268$$ 0 0
$$269$$ 28.0000i 1.70719i 0.520937 + 0.853595i $$0.325583\pi$$
−0.520937 + 0.853595i $$0.674417\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ −12.0000 −0.726273
$$274$$ 0 0
$$275$$ − 20.0000i − 1.20605i
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ − 6.00000i − 0.351726i
$$292$$ 0 0
$$293$$ 4.00000i 0.233682i 0.993151 + 0.116841i $$0.0372769\pi$$
−0.993151 + 0.116841i $$0.962723\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 24.0000i 1.38796i
$$300$$ 0 0
$$301$$ − 16.0000i − 0.922225i
$$302$$ 0 0
$$303$$ −4.00000 −0.229794
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 10.0000i 0.568880i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 28.0000i − 1.57264i −0.617822 0.786318i $$-0.711985\pi$$
0.617822 0.786318i $$-0.288015\pi$$
$$318$$ 0 0
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ − 30.0000i − 1.66410i
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ − 28.0000i − 1.53902i −0.638635 0.769510i $$-0.720501\pi$$
0.638635 0.769510i $$-0.279499\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ − 14.0000i − 0.760376i
$$340$$ 0 0
$$341$$ − 40.0000i − 2.16612i
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 36.0000i 1.93258i 0.257454 + 0.966291i $$0.417117\pi$$
−0.257454 + 0.966291i $$0.582883\pi$$
$$348$$ 0 0
$$349$$ 14.0000i 0.749403i 0.927146 + 0.374701i $$0.122255\pi$$
−0.927146 + 0.374701i $$0.877745\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 12.0000i − 0.635107i
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 24.0000i 1.24602i
$$372$$ 0 0
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ − 8.00000i − 0.410932i −0.978664 0.205466i $$-0.934129\pi$$
0.978664 0.205466i $$-0.0658711\pi$$
$$380$$ 0 0
$$381$$ 14.0000i 0.717242i
$$382$$ 0 0
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 10.0000i − 0.501886i −0.968002 0.250943i $$-0.919259\pi$$
0.968002 0.250943i $$-0.0807406\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ − 60.0000i − 2.98881i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ − 6.00000i − 0.295958i
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ 4.00000i 0.195413i 0.995215 + 0.0977064i $$0.0311506\pi$$
−0.995215 + 0.0977064i $$0.968849\pi$$
$$420$$ 0 0
$$421$$ − 2.00000i − 0.0974740i −0.998812 0.0487370i $$-0.984480\pi$$
0.998812 0.0487370i $$-0.0155196\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 30.0000 1.45521
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ − 24.0000i − 1.15873i
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −6.00000 −0.286364 −0.143182 0.989696i $$-0.545733\pi$$
−0.143182 + 0.989696i $$0.545733\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −16.0000 −0.756774
$$448$$ 0 0
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ − 8.00000i − 0.376705i
$$452$$ 0 0
$$453$$ − 10.0000i − 0.469841i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ 0 0
$$459$$ − 6.00000i − 0.280056i
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 22.0000 1.02243 0.511213 0.859454i $$-0.329196\pi$$
0.511213 + 0.859454i $$0.329196\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 0 0
$$469$$ 8.00000i 0.369406i
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 0 0
$$473$$ 32.0000 1.47136
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 0 0
$$483$$ 8.00000i 0.364013i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −18.0000 −0.815658 −0.407829 0.913058i $$-0.633714\pi$$
−0.407829 + 0.913058i $$0.633714\pi$$
$$488$$ 0 0
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ − 24.0000i − 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ 36.0000i 1.61158i 0.592200 + 0.805791i $$0.298259\pi$$
−0.592200 + 0.805791i $$0.701741\pi$$
$$500$$ 0 0
$$501$$ − 16.0000i − 0.714827i
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 23.0000i − 1.02147i
$$508$$ 0 0
$$509$$ − 28.0000i − 1.24108i −0.784176 0.620539i $$-0.786914\pi$$
0.784176 0.620539i $$-0.213086\pi$$
$$510$$ 0 0
$$511$$ −20.0000 −0.884748
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 48.0000i 2.11104i
$$518$$ 0 0
$$519$$ −16.0000 −0.702322
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ 40.0000i 1.74908i 0.484955 + 0.874539i $$0.338836\pi$$
−0.484955 + 0.874539i $$0.661164\pi$$
$$524$$ 0 0
$$525$$ − 10.0000i − 0.436436i
$$526$$ 0 0
$$527$$ 60.0000 2.61364
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 4.00000i 0.173585i
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 12.0000i 0.516877i
$$540$$ 0 0
$$541$$ − 38.0000i − 1.63375i −0.576816 0.816874i $$-0.695705\pi$$
0.576816 0.816874i $$-0.304295\pi$$
$$542$$ 0 0
$$543$$ −14.0000 −0.600798
$$544$$ 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
$$549$$ 2.00000i 0.0853579i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 12.0000 0.510292
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 40.0000i 1.69485i 0.530912 + 0.847427i $$0.321850\pi$$
−0.530912 + 0.847427i $$0.678150\pi$$
$$558$$ 0 0
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 0 0
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ − 20.0000i − 0.836974i −0.908223 0.418487i $$-0.862561\pi$$
0.908223 0.418487i $$-0.137439\pi$$
$$572$$ 0 0
$$573$$ 24.0000i 1.00261i
$$574$$ 0 0
$$575$$ −20.0000 −0.834058
$$576$$ 0 0
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 0 0
$$579$$ − 2.00000i − 0.0831172i
$$580$$ 0 0
$$581$$ 24.0000i 0.995688i
$$582$$ 0 0
$$583$$ −48.0000 −1.98796
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 18.0000i 0.736691i
$$598$$ 0 0
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −2.00000 −0.0811775 −0.0405887 0.999176i $$-0.512923\pi$$
−0.0405887 + 0.999176i $$0.512923\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ 72.0000i 2.91281i
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ − 28.0000i − 1.12542i −0.826656 0.562708i $$-0.809760\pi$$
0.826656 0.562708i $$-0.190240\pi$$
$$620$$ 0 0
$$621$$ 4.00000i 0.160514i
$$622$$ 0 0
$$623$$ 4.00000 0.160257
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ − 20.0000i − 0.783862i
$$652$$ 0 0
$$653$$ 8.00000i 0.313064i 0.987673 + 0.156532i $$0.0500315\pi$$
−0.987673 + 0.156532i $$0.949969\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ − 20.0000i − 0.779089i −0.921008 0.389545i $$-0.872632\pi$$
0.921008 0.389545i $$-0.127368\pi$$
$$660$$ 0 0
$$661$$ 10.0000i 0.388955i 0.980907 + 0.194477i $$0.0623011\pi$$
−0.980907 + 0.194477i $$0.937699\pi$$
$$662$$ 0 0
$$663$$ 36.0000 1.39812
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000i 0.619522i
$$668$$ 0 0
$$669$$ − 2.00000i − 0.0773245i
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 0 0
$$675$$ − 5.00000i − 0.192450i
$$676$$ 0 0
$$677$$ 24.0000i 0.922395i 0.887298 + 0.461197i $$0.152580\pi$$
−0.887298 + 0.461197i $$0.847420\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 0 0
$$689$$ −72.0000 −2.74298
$$690$$ 0 0
$$691$$ − 16.0000i − 0.608669i −0.952565 0.304334i $$-0.901566\pi$$
0.952565 0.304334i $$-0.0984340\pi$$
$$692$$ 0 0
$$693$$ − 8.00000i − 0.303895i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ − 10.0000i − 0.378235i
$$700$$ 0 0
$$701$$ − 12.0000i − 0.453234i −0.973984 0.226617i $$-0.927233\pi$$
0.973984 0.226617i $$-0.0727665\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 8.00000i − 0.300871i
$$708$$ 0 0
$$709$$ 6.00000i 0.225335i 0.993633 + 0.112667i $$0.0359394\pi$$
−0.993633 + 0.112667i $$0.964061\pi$$
$$710$$ 0 0
$$711$$ 6.00000 0.225018
$$712$$ 0 0
$$713$$ −40.0000 −1.49801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 0 0
$$723$$ 14.0000i 0.520666i
$$724$$ 0 0
$$725$$ − 20.0000i − 0.742781i
$$726$$ 0 0
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 48.0000i 1.77534i
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 28.0000i 1.03000i 0.857191 + 0.514998i $$0.172207\pi$$
−0.857191 + 0.514998i $$0.827793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 24.0000i 0.876941i
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ 0 0
$$753$$ 20.0000 0.728841
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 0 0
$$763$$ − 4.00000i − 0.144810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 0 0
$$771$$ − 14.0000i − 0.504198i
$$772$$ 0 0
$$773$$ − 36.0000i − 1.29483i −0.762138 0.647415i $$-0.775850\pi$$
0.762138 0.647415i $$-0.224150\pi$$
$$774$$ 0 0
$$775$$ 50.0000 1.79605
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ − 16.0000i − 0.572525i
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 40.0000i − 1.42585i −0.701242 0.712923i $$-0.747371\pi$$
0.701242 0.712923i $$-0.252629\pi$$
$$788$$ 0 0
$$789$$ 8.00000i 0.284808i
$$790$$ 0 0
$$791$$ 28.0000 0.995565
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 24.0000i 0.850124i 0.905164 + 0.425062i $$0.139748\pi$$
−0.905164 + 0.425062i $$0.860252\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ 0 0
$$803$$ − 40.0000i − 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −28.0000 −0.985647
$$808$$ 0 0
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ 32.0000i 1.12367i 0.827249 + 0.561836i $$0.189905\pi$$
−0.827249 + 0.561836i $$0.810095\pi$$
$$812$$ 0 0
$$813$$ − 2.00000i − 0.0701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ − 12.0000i − 0.419314i
$$820$$ 0 0
$$821$$ − 4.00000i − 0.139601i −0.997561 0.0698005i $$-0.977764\pi$$
0.997561 0.0698005i $$-0.0222363\pi$$
$$822$$ 0 0
$$823$$ 22.0000 0.766872 0.383436 0.923567i $$-0.374741\pi$$
0.383436 + 0.923567i $$0.374741\pi$$
$$824$$ 0 0
$$825$$ 20.0000 0.696311
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ 34.0000i 1.18087i 0.807086 + 0.590434i $$0.201044\pi$$
−0.807086 + 0.590434i $$0.798956\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 10.0000i − 0.345651i
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 13.0000 0.448276
$$842$$ 0 0
$$843$$ 14.0000i 0.482186i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ − 8.00000i − 0.274236i
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 16.0000i 0.545913i 0.962026 + 0.272956i $$0.0880015\pi$$
−0.962026 + 0.272956i $$0.911998\pi$$
$$860$$ 0 0
$$861$$ − 4.00000i − 0.136320i
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 24.0000i 0.814144i
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 0 0
$$879$$ −4.00000 −0.134917
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 48.0000i 1.61533i 0.589643 + 0.807664i $$0.299269\pi$$
−0.589643 + 0.807664i $$0.700731\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ −28.0000 −0.939090
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −24.0000 −0.801337
$$898$$ 0 0
$$899$$ − 40.0000i − 1.33407i
$$900$$ 0 0
$$901$$ − 72.0000i − 2.39867i
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ − 4.00000i − 0.132672i
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 24.0000i 0.792550i
$$918$$ 0 0
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ −10.0000 −0.328443
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ 0 0
$$939$$ 26.0000i 0.848478i
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ − 60.0000i − 1.94768i
$$950$$ 0 0
$$951$$ 28.0000 0.907962
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 16.0000i − 0.517207i
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 34.0000 1.09337 0.546683 0.837340i $$-0.315890\pi$$
0.546683 + 0.837340i $$0.315890\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 4.00000i 0.128366i 0.997938 + 0.0641831i $$0.0204442\pi$$
−0.997938 + 0.0641831i $$0.979556\pi$$
$$972$$ 0 0
$$973$$ 8.00000i 0.256468i
$$974$$ 0 0
$$975$$ 30.0000 0.960769
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 8.00000i 0.255681i
$$980$$ 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
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24.0000i 0.763928i
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ 26.0000 0.825917 0.412959 0.910750i $$-0.364495\pi$$
0.412959 + 0.910750i $$0.364495\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 6.00000i − 0.190022i −0.995476 0.0950110i $$-0.969711\pi$$
0.995476 0.0950110i $$-0.0302886\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 768.2.d.b.385.2 2
3.2 odd 2 2304.2.d.d.1153.2 2
4.3 odd 2 768.2.d.g.385.1 2
8.3 odd 2 768.2.d.g.385.2 2
8.5 even 2 inner 768.2.d.b.385.1 2
12.11 even 2 2304.2.d.m.1153.1 2
16.3 odd 4 384.2.a.f.1.1 yes 1
16.5 even 4 384.2.a.g.1.1 yes 1
16.11 odd 4 384.2.a.b.1.1 1
16.13 even 4 384.2.a.c.1.1 yes 1
24.5 odd 2 2304.2.d.d.1153.1 2
24.11 even 2 2304.2.d.m.1153.2 2
48.5 odd 4 1152.2.a.k.1.1 1
48.11 even 4 1152.2.a.j.1.1 1
48.29 odd 4 1152.2.a.l.1.1 1
48.35 even 4 1152.2.a.i.1.1 1
80.19 odd 4 9600.2.a.w.1.1 1
80.29 even 4 9600.2.a.bh.1.1 1
80.59 odd 4 9600.2.a.bw.1.1 1
80.69 even 4 9600.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.b.1.1 1 16.11 odd 4
384.2.a.c.1.1 yes 1 16.13 even 4
384.2.a.f.1.1 yes 1 16.3 odd 4
384.2.a.g.1.1 yes 1 16.5 even 4
768.2.d.b.385.1 2 8.5 even 2 inner
768.2.d.b.385.2 2 1.1 even 1 trivial
768.2.d.g.385.1 2 4.3 odd 2
768.2.d.g.385.2 2 8.3 odd 2
1152.2.a.i.1.1 1 48.35 even 4
1152.2.a.j.1.1 1 48.11 even 4
1152.2.a.k.1.1 1 48.5 odd 4
1152.2.a.l.1.1 1 48.29 odd 4
2304.2.d.d.1153.1 2 24.5 odd 2
2304.2.d.d.1153.2 2 3.2 odd 2
2304.2.d.m.1153.1 2 12.11 even 2
2304.2.d.m.1153.2 2 24.11 even 2
9600.2.a.h.1.1 1 80.69 even 4
9600.2.a.w.1.1 1 80.19 odd 4
9600.2.a.bh.1.1 1 80.29 even 4
9600.2.a.bw.1.1 1 80.59 odd 4