# Properties

 Label 768.2.d.f.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.f.385.1

## $q$-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 $$0.352416\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 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 $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 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 $$-0.630073\pi$$
0.397360 0.917663i $$-0.369927\pi$$
$$20$$ 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 $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 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 $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ 14.0000i 1.79252i 0.443533 + 0.896258i $$0.353725\pi$$
−0.443533 + 0.896258i $$0.646275\pi$$
$$62$$ 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 $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 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 $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 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 $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ − 6.00000i − 0.574696i −0.957826 0.287348i $$-0.907226\pi$$
0.957826 0.287348i $$-0.0927736\pi$$
$$110$$ 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 $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 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 $$-0.945740\pi$$
0.985506 0.169638i $$-0.0542598\pi$$
$$140$$ 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 $$-0.836434\pi$$
0.870855 0.491539i $$-0.163566\pi$$
$$150$$ 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 $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 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 $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 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 $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 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 $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ 22.0000i 1.63525i 0.575753 + 0.817624i $$0.304709\pi$$
−0.575753 + 0.817624i $$0.695291\pi$$
$$182$$ 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 $$0.0919897\pi$$
−0.958531 + 0.284988i $$0.908010\pi$$
$$198$$ 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 $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 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 $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ 14.0000i 0.925146i 0.886581 + 0.462573i $$0.153074\pi$$
−0.886581 + 0.462573i $$0.846926\pi$$
$$230$$ 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 $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 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 $$0.162189\pi$$
−0.872973 + 0.487769i $$0.837811\pi$$
$$270$$ 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 $$0.0576915\pi$$
−0.983620 + 0.180253i $$0.942309\pi$$
$$278$$ 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 $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ 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 $$-0.845203\pi$$
0.884064 0.467365i $$-0.154797\pi$$
$$294$$ 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 $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 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 $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ 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 $$0.185239\pi$$
−0.835395 + 0.549650i $$0.814761\pi$$
$$332$$ 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 $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ − 2.00000i − 0.107058i −0.998566 0.0535288i $$-0.982953\pi$$
0.998566 0.0535288i $$-0.0170469\pi$$
$$350$$ 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 $$0.0164888\pi$$
−0.998659 + 0.0517780i $$0.983511\pi$$
$$374$$ 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 $$-0.692934\pi$$
0.569683 0.821865i $$-0.307066\pi$$
$$380$$ 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 $$0.169253\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 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 $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 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 $$-0.837533\pi$$
0.872546 0.488532i $$-0.162467\pi$$
$$420$$ 0 0
$$421$$ 38.0000i 1.85201i 0.377515 + 0.926003i $$0.376779\pi$$
−0.377515 + 0.926003i $$0.623221\pi$$
$$422$$ 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 $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ 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 $$0.226089\pi$$
−0.758180 + 0.652045i $$0.773911\pi$$
$$462$$ 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 $$-0.970499\pi$$
0.995708 0.0925490i $$-0.0295015\pi$$
$$468$$ 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 $$-0.971231\pi$$
0.995918 0.0902587i $$-0.0287694\pi$$
$$492$$ 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 $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 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 $$-0.943265\pi$$
0.984157 0.177297i $$-0.0567353\pi$$
$$510$$ 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 $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ 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 $$-0.902695\pi$$
0.953639 0.300954i $$-0.0973049\pi$$
$$542$$ 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 $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ 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 $$-0.797865\pi$$
0.805056 0.593199i $$-0.202135\pi$$
$$558$$ 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 $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 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 $$-0.973327\pi$$
0.996491 0.0836974i $$-0.0266729\pi$$
$$572$$ 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 $$-0.973694\pi$$
0.996587 0.0825488i $$-0.0263060\pi$$
$$588$$ 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 $$-0.620703\pi$$
0.370177 0.928961i $$-0.379297\pi$$
$$614$$ 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 $$0.0256155\pi$$
−0.996764 + 0.0803868i $$0.974384\pi$$
$$620$$ 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 $$-0.604624\pi$$
0.322799 0.946468i $$-0.395376\pi$$
$$644$$ 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 $$-0.924557\pi$$
0.972044 0.234798i $$-0.0754429\pi$$
$$654$$ 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 $$-0.752657\pi$$
0.712984 0.701180i $$-0.247343\pi$$
$$660$$ 0 0
$$661$$ − 22.0000i − 0.855701i −0.903850 0.427850i $$-0.859271\pi$$
0.903850 0.427850i $$-0.140729\pi$$
$$662$$ 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 $$-0.874432\pi$$
0.923195 0.384331i $$-0.125568\pi$$
$$678$$ 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 $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 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 $$0.150898\pi$$
−0.889723 + 0.456502i $$0.849102\pi$$
$$692$$ 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 $$-0.727448\pi$$
0.655276 0.755390i $$-0.272552\pi$$
$$702$$ 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 $$-0.988043\pi$$
0.999295 0.0375558i $$-0.0119572\pi$$
$$710$$ 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 $$0.282579\pi$$
−0.631160 + 0.775653i $$0.717421\pi$$
$$734$$ 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 $$0.0708386\pi$$
−0.975339 + 0.220714i $$0.929161\pi$$
$$740$$ 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 $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 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 $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 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 $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ 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 $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 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 $$0.138453\pi$$
−0.906886 + 0.421377i $$0.861547\pi$$
$$812$$ 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 $$0.0445820\pi$$
−0.990208 + 0.139601i $$0.955418\pi$$
$$822$$ 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 $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 0 0
$$829$$ − 38.0000i − 1.31979i −0.751356 0.659897i $$-0.770600\pi$$
0.751356 0.659897i $$-0.229400\pi$$
$$830$$ 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 $$-0.774539\pi$$
0.759465 0.650548i $$-0.225461\pi$$
$$854$$ 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 $$-0.956422\pi$$
0.990643 0.136478i $$-0.0435784\pi$$
$$860$$ 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 $$-0.805372\pi$$
0.818821 0.574049i $$-0.194628\pi$$
$$878$$ 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 $$-0.764982\pi$$
0.739594 0.673054i $$-0.235018\pi$$
$$884$$ 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 $$0.0424025\pi$$
−0.991140 + 0.132818i $$0.957597\pi$$
$$908$$ 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 $$0.150857\pi$$
−0.889781 + 0.456387i $$0.849143\pi$$
$$942$$ 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 $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 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 $$-0.979556\pi$$
0.997938 0.0641831i $$-0.0204442\pi$$
$$972$$ 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 $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ 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.385.2 2
3.2 odd 2 2304.2.d.o.1153.1 2
4.3 odd 2 768.2.d.c.385.1 2
8.3 odd 2 768.2.d.c.385.2 2
8.5 even 2 inner 768.2.d.f.385.1 2
12.11 even 2 2304.2.d.f.1153.1 2
16.3 odd 4 384.2.a.h.1.1 yes 1
16.5 even 4 384.2.a.e.1.1 yes 1
16.11 odd 4 384.2.a.a.1.1 1
16.13 even 4 384.2.a.d.1.1 yes 1
24.5 odd 2 2304.2.d.o.1153.2 2
24.11 even 2 2304.2.d.f.1153.2 2
48.5 odd 4 1152.2.a.s.1.1 1
48.11 even 4 1152.2.a.t.1.1 1
48.29 odd 4 1152.2.a.a.1.1 1
48.35 even 4 1152.2.a.b.1.1 1
80.19 odd 4 9600.2.a.e.1.1 1
80.29 even 4 9600.2.a.bz.1.1 1
80.59 odd 4 9600.2.a.bk.1.1 1
80.69 even 4 9600.2.a.t.1.1 1

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