# Properties

 Label 5376.2.c.bc.2689.2 Level $5376$ Weight $2$ Character 5376.2689 Analytic conductor $42.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5376.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.9275761266$$ 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 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2689.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5376.2689 Dual form 5376.2.c.bc.2689.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{33} -2.00000i q^{35} -10.0000i q^{37} +6.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +2.00000i q^{45} +1.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} +4.00000i q^{59} -6.00000i q^{61} -1.00000 q^{63} -12.0000 q^{65} -4.00000i q^{67} -8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} +1.00000i q^{75} -4.00000i q^{77} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} -6.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} + 4q^{15} + 4q^{17} - 16q^{23} + 2q^{25} + 8q^{33} + 12q^{39} + 12q^{41} + 2q^{49} - 16q^{55} - 8q^{57} - 2q^{63} - 24q^{65} - 16q^{71} - 20q^{73} + 2q^{81} - 4q^{87} + 12q^{89} + 16q^{95} - 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$$.

 $$n$$ $$1793$$ $$2815$$ $$4609$$ $$5125$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ − 2.00000i − 0.894427i −0.894427 0.447214i $$-0.852416\pi$$
0.894427 0.447214i $$-0.147584\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 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
$$15$$ 2.00000 0.516398
$$16$$ 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
$$21$$ 1.00000i 0.218218i
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 2.00000i 0.371391i 0.982607 + 0.185695i $$0.0594537\pi$$
−0.982607 + 0.185695i $$0.940546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ − 2.00000i − 0.338062i
$$36$$ 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
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 2.00000i 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000i 0.280056i
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ 4.00000i 0.520756i 0.965507 + 0.260378i $$0.0838471\pi$$
−0.965507 + 0.260378i $$0.916153\pi$$
$$60$$ 0 0
$$61$$ − 6.00000i − 0.768221i −0.923287 0.384111i $$-0.874508\pi$$
0.923287 0.384111i $$-0.125492\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ − 8.00000i − 0.963087i
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 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
$$77$$ − 4.00000i − 0.455842i
$$78$$ 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
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ − 4.00000i − 0.433861i
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ − 6.00000i − 0.628971i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 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 $$-0.968274\pi$$
0.995037 0.0995037i $$-0.0317255\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 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
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 16.0000i 1.49201i
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ − 12.0000i − 1.07331i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 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
$$133$$ 4.00000i 0.346844i
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 4.00000i 0.339276i 0.985506 + 0.169638i $$0.0542598\pi$$
−0.985506 + 0.169638i $$0.945740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ 1.00000i 0.0824786i
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 0 0
$$165$$ − 8.00000i − 0.622799i
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ − 22.0000i − 1.67263i −0.548250 0.836315i $$-0.684706\pi$$
0.548250 0.836315i $$-0.315294\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 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
$$181$$ − 18.0000i − 1.33793i −0.743294 0.668965i $$-0.766738\pi$$
0.743294 0.668965i $$-0.233262\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ −20.0000 −1.47043
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ − 1.00000i − 0.0727393i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ − 12.0000i − 0.859338i
$$196$$ 0 0
$$197$$ − 10.0000i − 0.712470i −0.934396 0.356235i $$-0.884060\pi$$
0.934396 0.356235i $$-0.115940\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 2.00000i 0.140372i
$$204$$ 0 0
$$205$$ − 12.0000i − 0.838116i
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ − 20.0000i − 1.37686i −0.725304 0.688428i $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 0 0
$$213$$ − 8.00000i − 0.548151i
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 10.0000i − 0.675737i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 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
$$229$$ − 2.00000i − 0.132164i −0.997814 0.0660819i $$-0.978950\pi$$
0.997814 0.0660819i $$-0.0210498\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ − 2.00000i − 0.127775i
$$246$$ 0 0
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ − 12.0000i − 0.757433i −0.925513 0.378717i $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ − 10.0000i − 0.621370i
$$260$$ 0 0
$$261$$ − 2.00000i − 0.123797i
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ − 22.0000i − 1.34136i −0.741745 0.670682i $$-0.766002\pi$$
0.741745 0.670682i $$-0.233998\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 8.00000i 0.473879i
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 14.0000i − 0.820695i
$$292$$ 0 0
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 48.0000i 2.77591i
$$300$$ 0 0
$$301$$ − 4.00000i − 0.230556i
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ − 8.00000i − 0.455104i
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ 2.00000i 0.112687i
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 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
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 4.00000i − 0.219860i −0.993939 0.109930i $$-0.964937\pi$$
0.993939 0.109930i $$-0.0350627\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 0 0
$$339$$ − 14.0000i − 0.760376i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −16.0000 −0.861411
$$346$$ 0 0
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 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
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ 16.0000i 0.849192i
$$356$$ 0 0
$$357$$ 2.00000i 0.105851i
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 20.0000i 1.04685i
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 6.00000i 0.311504i
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ − 20.0000i − 1.02733i −0.857991 0.513665i $$-0.828287\pi$$
0.857991 0.513665i $$-0.171713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ −8.00000 −0.407718
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ − 26.0000i − 1.31825i −0.752032 0.659126i $$-0.770926\pi$$
0.752032 0.659126i $$-0.229074\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ −20.0000 −1.00887
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 2.00000i − 0.0993808i
$$406$$ 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
$$411$$ − 10.0000i − 0.493264i
$$412$$ 0 0
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 36.0000i 1.75872i 0.476162 + 0.879358i $$0.342028\pi$$
−0.476162 + 0.879358i $$0.657972\pi$$
$$420$$ 0 0
$$421$$ 6.00000i 0.292422i 0.989253 + 0.146211i $$0.0467079\pi$$
−0.989253 + 0.146211i $$0.953292\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ − 6.00000i − 0.290360i
$$428$$ 0 0
$$429$$ − 24.0000i − 1.15873i
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 4.00000i 0.191785i
$$436$$ 0 0
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 0 0
$$445$$ − 12.0000i − 0.568855i
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ − 24.0000i − 1.13012i
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ − 2.00000i − 0.0933520i
$$460$$ 0 0
$$461$$ − 22.0000i − 1.02464i −0.858794 0.512321i $$-0.828786\pi$$
0.858794 0.512321i $$-0.171214\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 0 0
$$469$$ − 4.00000i − 0.184703i
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 4.00000i 0.183533i
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 0 0
$$483$$ − 8.00000i − 0.364013i
$$484$$ 0 0
$$485$$ 28.0000i 1.27141i
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ 8.00000 0.359573
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ 44.0000i 1.96971i 0.173379 + 0.984855i $$0.444532\pi$$
−0.173379 + 0.984855i $$0.555468\pi$$
$$500$$ 0 0
$$501$$ 8.00000i 0.357414i
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ − 23.0000i − 1.02147i
$$508$$ 0 0
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 16.0000i 0.705044i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ 1.00000i 0.0436436i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ − 4.00000i − 0.173585i
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ 0 0
$$535$$ 24.0000 1.03761
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ − 4.00000i − 0.172292i
$$540$$ 0 0
$$541$$ − 30.0000i − 1.28980i −0.764267 0.644900i $$-0.776899\pi$$
0.764267 0.644900i $$-0.223101\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 0 0
$$549$$ 6.00000i 0.256074i
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ − 20.0000i − 0.848953i
$$556$$ 0 0
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ − 44.0000i − 1.85438i −0.374593 0.927189i $$-0.622217\pi$$
0.374593 0.927189i $$-0.377783\pi$$
$$564$$ 0 0
$$565$$ 28.0000i 1.17797i
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 12.0000i 0.502184i 0.967963 + 0.251092i $$0.0807897\pi$$
−0.967963 + 0.251092i $$0.919210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 0 0
$$579$$ 2.00000i 0.0831172i
$$580$$ 0 0
$$581$$ 4.00000i 0.165948i
$$582$$ 0 0
$$583$$ 24.0000 0.993978
$$584$$ 0 0
$$585$$ 12.0000 0.496139
$$586$$ 0 0
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ − 4.00000i − 0.163984i
$$596$$ 0 0
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 10.0000i 0.406558i
$$606$$ 0 0
$$607$$ 48.0000 1.94826 0.974130 0.225989i $$-0.0725612\pi$$
0.974130 + 0.225989i $$0.0725612\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 42.0000i − 1.69636i −0.529705 0.848182i $$-0.677697\pi$$
0.529705 0.848182i $$-0.322303\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ − 44.0000i − 1.76851i −0.467005 0.884255i $$-0.654667\pi$$
0.467005 0.884255i $$-0.345333\pi$$
$$620$$ 0 0
$$621$$ 8.00000i 0.321029i
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000i 0.638978i
$$628$$ 0 0
$$629$$ − 20.0000i − 0.797452i
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ − 8.00000i − 0.315000i
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 40.0000 1.56293
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 28.0000i 1.09073i 0.838200 + 0.545363i $$0.183608\pi$$
−0.838200 + 0.545363i $$0.816392\pi$$
$$660$$ 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
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ − 16.0000i − 0.619522i
$$668$$ 0 0
$$669$$ − 16.0000i − 0.618596i
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 0 0
$$675$$ − 1.00000i − 0.0384900i
$$676$$ 0 0
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 20.0000i 0.764161i
$$686$$ 0 0
$$687$$ 2.00000 0.0763048
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 4.00000i 0.152167i 0.997101 + 0.0760836i $$0.0242416\pi$$
−0.997101 + 0.0760836i $$0.975758\pi$$
$$692$$ 0 0
$$693$$ 4.00000i 0.151947i
$$694$$ 0 0
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 22.0000i 0.832116i
$$700$$ 0 0
$$701$$ 2.00000i 0.0755390i 0.999286 + 0.0377695i $$0.0120253\pi$$
−0.999286 + 0.0377695i $$0.987975\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 2.00000i − 0.0752177i
$$708$$ 0 0
$$709$$ − 10.0000i − 0.375558i −0.982211 0.187779i $$-0.939871\pi$$
0.982211 0.187779i $$-0.0601289\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 48.0000i 1.79510i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 0 0
$$725$$ 2.00000i 0.0742781i
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 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
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 12.0000i 0.441427i 0.975339 + 0.220714i $$0.0708386\pi$$
−0.975339 + 0.220714i $$0.929161\pi$$
$$740$$ 0 0
$$741$$ 24.0000i 0.881662i
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ − 16.0000i − 0.582300i
$$756$$ 0 0
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ 0 0
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ 0 0
$$765$$ 4.00000i 0.144620i
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ − 30.0000i − 1.08042i
$$772$$ 0 0
$$773$$ − 2.00000i − 0.0719350i −0.999353 0.0359675i $$-0.988549\pi$$
0.999353 0.0359675i $$-0.0114513\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 10.0000 0.358748
$$778$$ 0 0
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ 32.0000i 1.14505i
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 20.0000 0.713831
$$786$$ 0 0
$$787$$ 36.0000i 1.28326i 0.767014 + 0.641631i $$0.221742\pi$$
−0.767014 + 0.641631i $$0.778258\pi$$
$$788$$ 0 0
$$789$$ 24.0000i 0.854423i
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ −36.0000 −1.27840
$$794$$ 0 0
$$795$$ 12.0000i 0.425596i
$$796$$ 0 0
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 16.0000i 0.563926i
$$806$$ 0 0
$$807$$ 22.0000 0.774437
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ − 44.0000i − 1.54505i −0.634985 0.772524i $$-0.718994\pi$$
0.634985 0.772524i $$-0.281006\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −40.0000 −1.40114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ 6.00000i 0.209657i
$$820$$ 0 0
$$821$$ 38.0000i 1.32621i 0.748527 + 0.663105i $$0.230762\pi$$
−0.748527 + 0.663105i $$0.769238\pi$$
$$822$$ 0 0
$$823$$ 56.0000 1.95204 0.976019 0.217687i $$-0.0698512\pi$$
0.976019 + 0.217687i $$0.0698512\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 0 0
$$829$$ 26.0000i 0.903017i 0.892267 + 0.451509i $$0.149114\pi$$
−0.892267 + 0.451509i $$0.850886\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ − 16.0000i − 0.553703i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −56.0000 −1.93333 −0.966667 0.256036i $$-0.917584\pi$$
−0.966667 + 0.256036i $$0.917584\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ − 26.0000i − 0.895488i
$$844$$ 0 0
$$845$$ 46.0000i 1.58245i
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 80.0000i 2.74236i
$$852$$ 0 0
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 20.0000i 0.682391i 0.939992 + 0.341196i $$0.110832\pi$$
−0.939992 + 0.341196i $$0.889168\pi$$
$$860$$ 0 0
$$861$$ 6.00000i 0.204479i
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ −44.0000 −1.49604
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ − 12.0000i − 0.405674i
$$876$$ 0 0
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 0 0
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 0 0
$$885$$ 8.00000i 0.268917i
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ −48.0000 −1.60267
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000i 0.399778i
$$902$$ 0 0
$$903$$ 4.00000 0.133112
$$904$$ 0 0
$$905$$ −36.0000 −1.19668
$$906$$ 0 0
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 0 0
$$909$$ 2.00000i 0.0663358i
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ − 12.0000i − 0.396708i
$$916$$ 0 0
$$917$$ 20.0000i 0.660458i
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 0 0
$$923$$ 48.0000i 1.57994i
$$924$$ 0 0
$$925$$ − 10.0000i − 0.328798i
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 4.00000i 0.131095i
$$932$$ 0 0
$$933$$ 8.00000i 0.261908i
$$934$$ 0 0
$$935$$ −16.0000 −0.523256
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ − 10.0000i − 0.326338i
$$940$$ 0 0
$$941$$ 26.0000i 0.847576i 0.905761 + 0.423788i $$0.139300\pi$$
−0.905761 + 0.423788i $$0.860700\pi$$
$$942$$ 0 0
$$943$$ −48.0000 −1.56310
$$944$$ 0 0
$$945$$ −2.00000 −0.0650600
$$946$$ 0 0
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ 0 0
$$949$$ 60.0000i 1.94768i
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8.00000i 0.258603i
$$958$$ 0 0
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ − 4.00000i − 0.128765i
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ − 12.0000i − 0.385098i −0.981287 0.192549i $$-0.938325\pi$$
0.981287 0.192549i $$-0.0616755\pi$$
$$972$$ 0 0
$$973$$ 4.00000i 0.128234i
$$974$$ 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
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −20.0000 −0.637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.0000i 1.01754i
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 16.0000i 0.507234i
$$996$$ 0 0
$$997$$ 14.0000i 0.443384i 0.975117 + 0.221692i $$0.0711580\pi$$
−0.975117 + 0.221692i $$0.928842\pi$$
$$998$$ 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 5376.2.c.bc.2689.2 2
4.3 odd 2 5376.2.c.e.2689.1 2
8.3 odd 2 5376.2.c.e.2689.2 2
8.5 even 2 inner 5376.2.c.bc.2689.1 2
16.3 odd 4 336.2.a.d.1.1 1
16.5 even 4 1344.2.a.q.1.1 1
16.11 odd 4 1344.2.a.i.1.1 1
16.13 even 4 42.2.a.a.1.1 1
48.5 odd 4 4032.2.a.e.1.1 1
48.11 even 4 4032.2.a.m.1.1 1
48.29 odd 4 126.2.a.a.1.1 1
48.35 even 4 1008.2.a.j.1.1 1
80.13 odd 4 1050.2.g.a.799.1 2
80.19 odd 4 8400.2.a.k.1.1 1
80.29 even 4 1050.2.a.i.1.1 1
80.77 odd 4 1050.2.g.a.799.2 2
112.3 even 12 2352.2.q.n.961.1 2
112.13 odd 4 294.2.a.g.1.1 1
112.19 even 12 2352.2.q.n.1537.1 2
112.27 even 4 9408.2.a.bw.1.1 1
112.45 odd 12 294.2.e.a.79.1 2
112.51 odd 12 2352.2.q.i.1537.1 2
112.61 odd 12 294.2.e.a.67.1 2
112.67 odd 12 2352.2.q.i.961.1 2
112.69 odd 4 9408.2.a.n.1.1 1
112.83 even 4 2352.2.a.l.1.1 1
112.93 even 12 294.2.e.c.67.1 2
112.109 even 12 294.2.e.c.79.1 2
144.13 even 12 1134.2.f.g.379.1 2
144.29 odd 12 1134.2.f.j.757.1 2
144.61 even 12 1134.2.f.g.757.1 2
144.77 odd 12 1134.2.f.j.379.1 2
176.109 odd 4 5082.2.a.d.1.1 1
208.77 even 4 7098.2.a.f.1.1 1
240.29 odd 4 3150.2.a.bo.1.1 1
240.77 even 4 3150.2.g.r.2899.1 2
240.173 even 4 3150.2.g.r.2899.2 2
336.83 odd 4 7056.2.a.k.1.1 1
336.125 even 4 882.2.a.b.1.1 1
336.173 even 12 882.2.g.j.361.1 2
336.221 odd 12 882.2.g.h.667.1 2
336.269 even 12 882.2.g.j.667.1 2
336.317 odd 12 882.2.g.h.361.1 2
560.349 odd 4 7350.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 16.13 even 4
126.2.a.a.1.1 1 48.29 odd 4
294.2.a.g.1.1 1 112.13 odd 4
294.2.e.a.67.1 2 112.61 odd 12
294.2.e.a.79.1 2 112.45 odd 12
294.2.e.c.67.1 2 112.93 even 12
294.2.e.c.79.1 2 112.109 even 12
336.2.a.d.1.1 1 16.3 odd 4
882.2.a.b.1.1 1 336.125 even 4
882.2.g.h.361.1 2 336.317 odd 12
882.2.g.h.667.1 2 336.221 odd 12
882.2.g.j.361.1 2 336.173 even 12
882.2.g.j.667.1 2 336.269 even 12
1008.2.a.j.1.1 1 48.35 even 4
1050.2.a.i.1.1 1 80.29 even 4
1050.2.g.a.799.1 2 80.13 odd 4
1050.2.g.a.799.2 2 80.77 odd 4
1134.2.f.g.379.1 2 144.13 even 12
1134.2.f.g.757.1 2 144.61 even 12
1134.2.f.j.379.1 2 144.77 odd 12
1134.2.f.j.757.1 2 144.29 odd 12
1344.2.a.i.1.1 1 16.11 odd 4
1344.2.a.q.1.1 1 16.5 even 4
2352.2.a.l.1.1 1 112.83 even 4
2352.2.q.i.961.1 2 112.67 odd 12
2352.2.q.i.1537.1 2 112.51 odd 12
2352.2.q.n.961.1 2 112.3 even 12
2352.2.q.n.1537.1 2 112.19 even 12
3150.2.a.bo.1.1 1 240.29 odd 4
3150.2.g.r.2899.1 2 240.77 even 4
3150.2.g.r.2899.2 2 240.173 even 4
4032.2.a.e.1.1 1 48.5 odd 4
4032.2.a.m.1.1 1 48.11 even 4
5082.2.a.d.1.1 1 176.109 odd 4
5376.2.c.e.2689.1 2 4.3 odd 2
5376.2.c.e.2689.2 2 8.3 odd 2
5376.2.c.bc.2689.1 2 8.5 even 2 inner
5376.2.c.bc.2689.2 2 1.1 even 1 trivial
7056.2.a.k.1.1 1 336.83 odd 4
7098.2.a.f.1.1 1 208.77 even 4
7350.2.a.f.1.1 1 560.349 odd 4
8400.2.a.k.1.1 1 80.19 odd 4
9408.2.a.n.1.1 1 112.69 odd 4
9408.2.a.bw.1.1 1 112.27 even 4