# Properties

 Label 768.2.d.f.385.1 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 385.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.385 Dual form 768.2.d.f.385.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -4.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -4.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} -4.00000 q^{15} -2.00000 q^{17} +8.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} -11.0000 q^{25} +1.00000i q^{27} +6.00000 q^{31} -4.00000 q^{33} -8.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000i q^{51} -16.0000 q^{55} +8.00000 q^{57} +4.00000i q^{59} -14.0000i q^{61} -2.00000 q^{63} -8.00000 q^{65} +4.00000i q^{67} -4.00000i q^{69} +12.0000 q^{71} +10.0000 q^{73} +11.0000i q^{75} -8.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +8.00000i q^{85} +14.0000 q^{89} -4.00000i q^{91} -6.00000i q^{93} +32.0000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} - 8q^{15} - 4q^{17} + 8q^{23} - 22q^{25} + 12q^{31} - 8q^{33} - 4q^{39} - 12q^{41} - 8q^{47} - 6q^{49} - 32q^{55} + 16q^{57} - 4q^{63} - 16q^{65} + 24q^{71} + 20q^{73} - 20q^{79} + 2q^{81} + 28q^{89} + 64q^{95} + 20q^{97} + O(q^{100})$$

## 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
$$5$$ − 4.00000i − 1.78885i −0.447214 0.894427i $$-0.647584\pi$$
0.447214 0.894427i $$-0.352416\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 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
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 8.00000i 1.83533i 0.397360 + 0.917663i $$0.369927\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ − 2.00000i − 0.436436i
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −11.0000 −2.20000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ − 8.00000i − 1.35225i
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 4.00000i 0.596285i
$$46$$ 0 0
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 2.00000i 0.280056i
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ −16.0000 −2.15744
$$56$$ 0 0
$$57$$ 8.00000 1.05963
$$58$$ 0 0
$$59$$ 4.00000i 0.520756i 0.965507 + 0.260378i $$0.0838471\pi$$
−0.965507 + 0.260378i $$0.916153\pi$$
$$60$$ 0 0
$$61$$ − 14.0000i − 1.79252i −0.443533 0.896258i $$-0.646275\pi$$
0.443533 0.896258i $$-0.353725\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ − 4.00000i − 0.481543i
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 11.0000i 1.27017i
$$76$$ 0 0
$$77$$ − 8.00000i − 0.911685i
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 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
$$85$$ 8.00000i 0.867722i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ − 4.00000i − 0.419314i
$$92$$ 0 0
$$93$$ − 6.00000i − 0.622171i
$$94$$ 0 0
$$95$$ 32.0000 3.28313
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ −8.00000 −0.780720
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ 6.00000i 0.574696i 0.957826 + 0.287348i $$0.0927736\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ − 16.0000i − 1.49201i
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 24.0000i 2.14663i
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000i 1.04844i 0.851581 + 0.524222i $$0.175644\pi$$
−0.851581 + 0.524222i $$0.824356\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ 4.00000i 0.339276i 0.985506 + 0.169638i $$0.0542598\pi$$
−0.985506 + 0.169638i $$0.945740\pi$$
$$140$$ 0 0
$$141$$ 4.00000i 0.336861i
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.00000i 0.247436i
$$148$$ 0 0
$$149$$ 12.0000i 0.983078i 0.870855 + 0.491539i $$0.163566\pi$$
−0.870855 + 0.491539i $$0.836434\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ − 24.0000i − 1.92773i
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ 0 0
$$165$$ 16.0000i 1.24560i
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ − 8.00000i − 0.611775i
$$172$$ 0 0
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 0 0
$$175$$ −22.0000 −1.66304
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ − 12.0000i − 0.896922i −0.893802 0.448461i $$-0.851972\pi$$
0.893802 0.448461i $$-0.148028\pi$$
$$180$$ 0 0
$$181$$ − 22.0000i − 1.63525i −0.575753 0.817624i $$-0.695291\pi$$
0.575753 0.817624i $$-0.304709\pi$$
$$182$$ 0 0
$$183$$ −14.0000 −1.03491
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 8.00000i 0.572892i
$$196$$ 0 0
$$197$$ − 8.00000i − 0.569976i −0.958531 0.284988i $$-0.908010\pi$$
0.958531 0.284988i $$-0.0919897\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 24.0000i 1.67623i
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 32.0000 2.21349
$$210$$ 0 0
$$211$$ 20.0000i 1.37686i 0.725304 + 0.688428i $$0.241699\pi$$
−0.725304 + 0.688428i $$0.758301\pi$$
$$212$$ 0 0
$$213$$ − 12.0000i − 0.822226i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ − 10.0000i − 0.675737i
$$220$$ 0 0
$$221$$ 4.00000i 0.269069i
$$222$$ 0 0
$$223$$ 18.0000 1.20537 0.602685 0.797980i $$-0.294098\pi$$
0.602685 + 0.797980i $$0.294098\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 16.0000i 1.04372i
$$236$$ 0 0
$$237$$ 10.0000i 0.649570i
$$238$$ 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 $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 12.0000i 0.766652i
$$246$$ 0 0
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000i 0.757433i 0.925513 + 0.378717i $$0.123635\pi$$
−0.925513 + 0.378717i $$0.876365\pi$$
$$252$$ 0 0
$$253$$ − 16.0000i − 1.00591i
$$254$$ 0 0
$$255$$ 8.00000 0.500979
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ − 4.00000i − 0.248548i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 14.0000i − 0.856786i
$$268$$ 0 0
$$269$$ − 16.0000i − 0.975537i −0.872973 0.487769i $$-0.837811\pi$$
0.872973 0.487769i $$-0.162189\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 0 0
$$273$$ −4.00000 −0.242091
$$274$$ 0 0
$$275$$ 44.0000i 2.65330i
$$276$$ 0 0
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ 0 0
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 12.0000i 0.713326i 0.934233 + 0.356663i $$0.116086\pi$$
−0.934233 + 0.356663i $$0.883914\pi$$
$$284$$ 0 0
$$285$$ − 32.0000i − 1.89552i
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 10.0000i − 0.586210i
$$292$$ 0 0
$$293$$ 16.0000i 0.934730i 0.884064 + 0.467365i $$0.154797\pi$$
−0.884064 + 0.467365i $$0.845203\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ − 8.00000i − 0.462652i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −56.0000 −3.20655
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 10.0000i 0.568880i
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 0 0
$$315$$ 8.00000i 0.450749i
$$316$$ 0 0
$$317$$ 24.0000i 1.34797i 0.738743 + 0.673987i $$0.235420\pi$$
−0.738743 + 0.673987i $$0.764580\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ 22.0000i 1.22034i
$$326$$ 0 0
$$327$$ 6.00000 0.331801
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 16.0000 0.874173
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ − 2.00000i − 0.108625i
$$340$$ 0 0
$$341$$ − 24.0000i − 1.29967i
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ −16.0000 −0.861411
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 2.00000i 0.107058i 0.998566 + 0.0535288i $$0.0170469\pi$$
−0.998566 + 0.0535288i $$0.982953\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ − 48.0000i − 2.54758i
$$356$$ 0 0
$$357$$ 4.00000i 0.211702i
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −45.0000 −2.36842
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ − 40.0000i − 2.09370i
$$366$$ 0 0
$$367$$ 38.0000 1.98358 0.991792 0.127862i $$-0.0408116\pi$$
0.991792 + 0.127862i $$0.0408116\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 2.00000i − 0.103556i −0.998659 0.0517780i $$-0.983511\pi$$
0.998659 0.0517780i $$-0.0164888\pi$$
$$374$$ 0 0
$$375$$ 24.0000 1.23935
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 32.0000i 1.64373i 0.569683 + 0.821865i $$0.307066\pi$$
−0.569683 + 0.821865i $$0.692934\pi$$
$$380$$ 0 0
$$381$$ − 2.00000i − 0.102463i
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ −32.0000 −1.63087
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 20.0000i − 1.01404i −0.861934 0.507020i $$-0.830747\pi$$
0.861934 0.507020i $$-0.169253\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 40.0000i 2.01262i
$$396$$ 0 0
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ 0 0
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ − 12.0000i − 0.597763i
$$404$$ 0 0
$$405$$ − 4.00000i − 0.198762i
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ − 2.00000i − 0.0986527i
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 0 0
$$415$$ −48.0000 −2.35623
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ 20.0000i 0.977064i 0.872546 + 0.488532i $$0.162467\pi$$
−0.872546 + 0.488532i $$0.837533\pi$$
$$420$$ 0 0
$$421$$ − 38.0000i − 1.85201i −0.377515 0.926003i $$-0.623221\pi$$
0.377515 0.926003i $$-0.376779\pi$$
$$422$$ 0 0
$$423$$ 4.00000 0.194487
$$424$$ 0 0
$$425$$ 22.0000 1.06716
$$426$$ 0 0
$$427$$ − 28.0000i − 1.35501i
$$428$$ 0 0
$$429$$ 8.00000i 0.386244i
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 0 0
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ 6.00000 0.286364 0.143182 0.989696i $$-0.454267\pi$$
0.143182 + 0.989696i $$0.454267\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 0 0
$$445$$ − 56.0000i − 2.65465i
$$446$$ 0 0
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ 24.0000i 1.13012i
$$452$$ 0 0
$$453$$ − 10.0000i − 0.469841i
$$454$$ 0 0
$$455$$ −16.0000 −0.750092
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 0 0
$$459$$ − 2.00000i − 0.0933520i
$$460$$ 0 0
$$461$$ − 28.0000i − 1.30409i −0.758180 0.652045i $$-0.773911\pi$$
0.758180 0.652045i $$-0.226089\pi$$
$$462$$ 0 0
$$463$$ 26.0000 1.20832 0.604161 0.796862i $$-0.293508\pi$$
0.604161 + 0.796862i $$0.293508\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 4.00000i 0.185098i 0.995708 + 0.0925490i $$0.0295015\pi$$
−0.995708 + 0.0925490i $$0.970499\pi$$
$$468$$ 0 0
$$469$$ 8.00000i 0.369406i
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 88.0000i − 4.03772i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ − 8.00000i − 0.364013i
$$484$$ 0 0
$$485$$ − 40.0000i − 1.81631i
$$486$$ 0 0
$$487$$ −14.0000 −0.634401 −0.317200 0.948359i $$-0.602743\pi$$
−0.317200 + 0.948359i $$0.602743\pi$$
$$488$$ 0 0
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 4.00000i 0.180517i 0.995918 + 0.0902587i $$0.0287694\pi$$
−0.995918 + 0.0902587i $$0.971231\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 16.0000 0.719147
$$496$$ 0 0
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ − 16.0000i − 0.714827i
$$502$$ 0 0
$$503$$ 44.0000 1.96186 0.980932 0.194354i $$-0.0622609\pi$$
0.980932 + 0.194354i $$0.0622609\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ 8.00000i 0.354594i 0.984157 + 0.177297i $$0.0567353\pi$$
−0.984157 + 0.177297i $$0.943265\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ 40.0000i 1.76261i
$$516$$ 0 0
$$517$$ 16.0000i 0.703679i
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 0 0
$$525$$ 22.0000i 0.960159i
$$526$$ 0 0
$$527$$ −12.0000 −0.522728
$$528$$ 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
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 12.0000i 0.516877i
$$540$$ 0 0
$$541$$ 14.0000i 0.601907i 0.953639 + 0.300954i $$0.0973049\pi$$
−0.953639 + 0.300954i $$0.902695\pi$$
$$542$$ 0 0
$$543$$ −22.0000 −0.944110
$$544$$ 0 0
$$545$$ 24.0000 1.02805
$$546$$ 0 0
$$547$$ 16.0000i 0.684111i 0.939680 + 0.342055i $$0.111123\pi$$
−0.939680 + 0.342055i $$0.888877\pi$$
$$548$$ 0 0
$$549$$ 14.0000i 0.597505i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −20.0000 −0.850487
$$554$$ 0 0
$$555$$ 8.00000i 0.339581i
$$556$$ 0 0
$$557$$ 28.0000i 1.18640i 0.805056 + 0.593199i $$0.202135\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ − 8.00000i − 0.336563i
$$566$$ 0 0
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 0 0
$$571$$ 4.00000i 0.167395i 0.996491 + 0.0836974i $$0.0266729\pi$$
−0.996491 + 0.0836974i $$0.973327\pi$$
$$572$$ 0 0
$$573$$ 8.00000i 0.334205i
$$574$$ 0 0
$$575$$ −44.0000 −1.83493
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 0 0
$$579$$ 2.00000i 0.0831172i
$$580$$ 0 0
$$581$$ − 24.0000i − 0.995688i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 8.00000 0.330759
$$586$$ 0 0
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ 0 0
$$589$$ 48.0000i 1.97781i
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 16.0000i 0.655936i
$$596$$ 0 0
$$597$$ − 14.0000i − 0.572982i
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ 20.0000i 0.813116i
$$606$$ 0 0
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000i 0.323645i
$$612$$ 0 0
$$613$$ 46.0000i 1.85792i 0.370177 + 0.928961i $$0.379297\pi$$
−0.370177 + 0.928961i $$0.620703\pi$$
$$614$$ 0 0
$$615$$ 24.0000 0.967773
$$616$$ 0 0
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 0 0
$$621$$ 4.00000i 0.160514i
$$622$$ 0 0
$$623$$ 28.0000 1.12180
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ − 32.0000i − 1.27796i
$$628$$ 0 0
$$629$$ 4.00000i 0.159490i
$$630$$ 0 0
$$631$$ 34.0000 1.35352 0.676759 0.736204i $$-0.263384\pi$$
0.676759 + 0.736204i $$0.263384\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ − 8.00000i − 0.317470i
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ 48.0000i 1.89294i 0.322799 + 0.946468i $$0.395376\pi$$
−0.322799 + 0.946468i $$0.604624\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ − 12.0000i − 0.470317i
$$652$$ 0 0
$$653$$ 12.0000i 0.469596i 0.972044 + 0.234798i $$0.0754429\pi$$
−0.972044 + 0.234798i $$0.924557\pi$$
$$654$$ 0 0
$$655$$ 48.0000 1.87552
$$656$$ 0 0
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 36.0000i 1.40236i 0.712984 + 0.701180i $$0.247343\pi$$
−0.712984 + 0.701180i $$0.752657\pi$$
$$660$$ 0 0
$$661$$ 22.0000i 0.855701i 0.903850 + 0.427850i $$0.140729\pi$$
−0.903850 + 0.427850i $$0.859271\pi$$
$$662$$ 0 0
$$663$$ 4.00000 0.155347
$$664$$ 0 0
$$665$$ 64.0000 2.48181
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ − 18.0000i − 0.695920i
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ − 11.0000i − 0.423390i
$$676$$ 0 0
$$677$$ 20.0000i 0.768662i 0.923195 + 0.384331i $$0.125568\pi$$
−0.923195 + 0.384331i $$0.874432\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ − 8.00000i − 0.305664i
$$686$$ 0 0
$$687$$ −14.0000 −0.534133
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 24.0000i − 0.913003i −0.889723 0.456502i $$-0.849102\pi$$
0.889723 0.456502i $$-0.150898\pi$$
$$692$$ 0 0
$$693$$ 8.00000i 0.303895i
$$694$$ 0 0
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ − 6.00000i − 0.226941i
$$700$$ 0 0
$$701$$ 40.0000i 1.51078i 0.655276 + 0.755390i $$0.272552\pi$$
−0.655276 + 0.755390i $$0.727448\pi$$
$$702$$ 0 0
$$703$$ 16.0000 0.603451
$$704$$ 0 0
$$705$$ 16.0000 0.602595
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000i 0.0751116i 0.999295 + 0.0375558i $$0.0119572\pi$$
−0.999295 + 0.0375558i $$0.988043\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 32.0000i 1.19673i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 42.0000 1.55769 0.778847 0.627214i $$-0.215805\pi$$
0.778847 + 0.627214i $$0.215805\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ − 42.0000i − 1.55131i −0.631160 0.775653i $$-0.717421\pi$$
0.631160 0.775653i $$-0.282579\pi$$
$$734$$ 0 0
$$735$$ 12.0000 0.442627
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ − 12.0000i − 0.441427i −0.975339 0.220714i $$-0.929161\pi$$
0.975339 0.220714i $$-0.0708386\pi$$
$$740$$ 0 0
$$741$$ − 16.0000i − 0.587775i
$$742$$ 0 0
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ 48.0000 1.75858
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ − 8.00000i − 0.292314i
$$750$$ 0 0
$$751$$ 22.0000 0.802791 0.401396 0.915905i $$-0.368525\pi$$
0.401396 + 0.915905i $$0.368525\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ − 40.0000i − 1.45575i
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ 12.0000i 0.434429i
$$764$$ 0 0
$$765$$ − 8.00000i − 0.289241i
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ − 18.0000i − 0.648254i
$$772$$ 0 0
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 0 0
$$775$$ −66.0000 −2.37079
$$776$$ 0 0
$$777$$ −4.00000 −0.143499
$$778$$ 0 0
$$779$$ − 48.0000i − 1.71978i
$$780$$ 0 0
$$781$$ − 48.0000i − 1.71758i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 72.0000 2.56979
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 0 0
$$789$$ 24.0000i 0.854423i
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ −28.0000 −0.994309
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ 0 0
$$803$$ − 40.0000i − 1.41157i
$$804$$ 0 0
$$805$$ − 32.0000i − 1.12785i
$$806$$ 0 0
$$807$$ −16.0000 −0.563227
$$808$$ 0 0
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ − 24.0000i − 0.842754i −0.906886 0.421377i $$-0.861547\pi$$
0.906886 0.421377i $$-0.138453\pi$$
$$812$$ 0 0
$$813$$ 14.0000i 0.491001i
$$814$$ 0 0
$$815$$ 32.0000 1.12091
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 4.00000i 0.139771i
$$820$$ 0 0
$$821$$ − 8.00000i − 0.279202i −0.990208 0.139601i $$-0.955418\pi$$
0.990208 0.139601i $$-0.0445820\pi$$
$$822$$ 0 0
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ 0 0
$$825$$ 44.0000 1.53188
$$826$$ 0 0
$$827$$ 44.0000i 1.53003i 0.644013 + 0.765015i $$0.277268\pi$$
−0.644013 + 0.765015i $$0.722732\pi$$
$$828$$ 0 0
$$829$$ 38.0000i 1.31979i 0.751356 + 0.659897i $$0.229400\pi$$
−0.751356 + 0.659897i $$0.770600\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ − 64.0000i − 2.21481i
$$836$$ 0 0
$$837$$ 6.00000i 0.207390i
$$838$$ 0 0
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ 18.0000i 0.619953i
$$844$$ 0 0
$$845$$ − 36.0000i − 1.23844i
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ − 8.00000i − 0.274236i
$$852$$ 0 0
$$853$$ 38.0000i 1.30110i 0.759465 + 0.650548i $$0.225461\pi$$
−0.759465 + 0.650548i $$0.774539\pi$$
$$854$$ 0 0
$$855$$ −32.0000 −1.09438
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 8.00000i 0.272956i 0.990643 + 0.136478i $$0.0435784\pi$$
−0.990643 + 0.136478i $$0.956422\pi$$
$$860$$ 0 0
$$861$$ 12.0000i 0.408959i
$$862$$ 0 0
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 40.0000i 1.35691i
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ 48.0000i 1.62270i
$$876$$ 0 0
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 0 0
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ 40.0000i 1.34611i 0.739594 + 0.673054i $$0.235018\pi$$
−0.739594 + 0.673054i $$0.764982\pi$$
$$884$$ 0 0
$$885$$ − 16.0000i − 0.537834i
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ −48.0000 −1.60446
$$896$$ 0 0
$$897$$ −8.00000 −0.267112
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −88.0000 −2.92522
$$906$$ 0 0
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 56.0000i 1.85130i
$$916$$ 0 0
$$917$$ 24.0000i 0.792550i
$$918$$ 0 0
$$919$$ −6.00000 −0.197922 −0.0989609 0.995091i $$-0.531552\pi$$
−0.0989609 + 0.995091i $$0.531552\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 22.0000i 0.723356i
$$926$$ 0 0
$$927$$ 10.0000 0.328443
$$928$$ 0 0
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ − 24.0000i − 0.786568i
$$932$$ 0 0
$$933$$ 16.0000i 0.523816i
$$934$$ 0 0
$$935$$ 32.0000 1.04651
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ − 26.0000i − 0.848478i
$$940$$ 0 0
$$941$$ − 28.0000i − 0.912774i −0.889781 0.456387i $$-0.849143\pi$$
0.889781 0.456387i $$-0.150857\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ 52.0000i 1.68977i 0.534946 + 0.844886i $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ 0 0
$$949$$ − 20.0000i − 0.649227i
$$950$$ 0 0
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ 32.0000i 1.03550i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 0 0
$$965$$ 8.00000i 0.257529i
$$966$$ 0 0
$$967$$ −2.00000 −0.0643157 −0.0321578 0.999483i $$-0.510238\pi$$
−0.0321578 + 0.999483i $$0.510238\pi$$
$$968$$ 0 0
$$969$$ −16.0000 −0.513994
$$970$$ 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
$$975$$ 22.0000 0.704564
$$976$$ 0 0
$$977$$ 6.00000 0.191957 0.0959785 0.995383i $$-0.469402\pi$$
0.0959785 + 0.995383i $$0.469402\pi$$
$$978$$ 0 0
$$979$$ − 56.0000i − 1.78977i
$$980$$ 0 0
$$981$$ − 6.00000i − 0.191565i
$$982$$ 0 0
$$983$$ 56.0000 1.78612 0.893061 0.449935i $$-0.148553\pi$$
0.893061 + 0.449935i $$0.148553\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ 0 0
$$987$$ 8.00000i 0.254643i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −10.0000 −0.317660 −0.158830 0.987306i $$-0.550772\pi$$
−0.158830 + 0.987306i $$0.550772\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ − 56.0000i − 1.77532i
$$996$$ 0 0
$$997$$ 54.0000i 1.71020i 0.518465 + 0.855099i $$0.326503\pi$$
−0.518465 + 0.855099i $$0.673497\pi$$
$$998$$ 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.f.385.1 2
3.2 odd 2 2304.2.d.o.1153.2 2
4.3 odd 2 768.2.d.c.385.2 2
8.3 odd 2 768.2.d.c.385.1 2
8.5 even 2 inner 768.2.d.f.385.2 2
12.11 even 2 2304.2.d.f.1153.2 2
16.3 odd 4 384.2.a.a.1.1 1
16.5 even 4 384.2.a.d.1.1 yes 1
16.11 odd 4 384.2.a.h.1.1 yes 1
16.13 even 4 384.2.a.e.1.1 yes 1
24.5 odd 2 2304.2.d.o.1153.1 2
24.11 even 2 2304.2.d.f.1153.1 2
48.5 odd 4 1152.2.a.a.1.1 1
48.11 even 4 1152.2.a.b.1.1 1
48.29 odd 4 1152.2.a.s.1.1 1
48.35 even 4 1152.2.a.t.1.1 1
80.19 odd 4 9600.2.a.bk.1.1 1
80.29 even 4 9600.2.a.t.1.1 1
80.59 odd 4 9600.2.a.e.1.1 1
80.69 even 4 9600.2.a.bz.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.a.1.1 1 16.3 odd 4
384.2.a.d.1.1 yes 1 16.5 even 4
384.2.a.e.1.1 yes 1 16.13 even 4
384.2.a.h.1.1 yes 1 16.11 odd 4
768.2.d.c.385.1 2 8.3 odd 2
768.2.d.c.385.2 2 4.3 odd 2
768.2.d.f.385.1 2 1.1 even 1 trivial
768.2.d.f.385.2 2 8.5 even 2 inner
1152.2.a.a.1.1 1 48.5 odd 4
1152.2.a.b.1.1 1 48.11 even 4
1152.2.a.s.1.1 1 48.29 odd 4
1152.2.a.t.1.1 1 48.35 even 4
2304.2.d.f.1153.1 2 24.11 even 2
2304.2.d.f.1153.2 2 12.11 even 2
2304.2.d.o.1153.1 2 24.5 odd 2
2304.2.d.o.1153.2 2 3.2 odd 2
9600.2.a.e.1.1 1 80.59 odd 4
9600.2.a.t.1.1 1 80.29 even 4
9600.2.a.bk.1.1 1 80.19 odd 4
9600.2.a.bz.1.1 1 80.69 even 4