# Properties

 Label 1152.2.a.t.1.1 Level $1152$ Weight $2$ Character 1152.1 Self dual yes Analytic conductor $9.199$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1152.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{5} +2.00000 q^{7} +O(q^{10})$$ $$q+4.00000 q^{5} +2.00000 q^{7} -4.00000 q^{11} +2.00000 q^{13} +2.00000 q^{17} +8.00000 q^{19} -4.00000 q^{23} +11.0000 q^{25} -6.00000 q^{31} +8.00000 q^{35} -2.00000 q^{37} -6.00000 q^{41} -4.00000 q^{47} -3.00000 q^{49} -16.0000 q^{55} +4.00000 q^{59} +14.0000 q^{61} +8.00000 q^{65} +4.00000 q^{67} -12.0000 q^{71} -10.0000 q^{73} -8.00000 q^{77} +10.0000 q^{79} +12.0000 q^{83} +8.00000 q^{85} +14.0000 q^{89} +4.00000 q^{91} +32.0000 q^{95} +10.0000 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 8.00000 1.35225
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −16.0000 −2.15744
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 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
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −16.0000 −1.49201
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 0 0
$$175$$ 22.0000 1.66304
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −24.0000 −1.67623
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −32.0000 −2.21349
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ −18.0000 −1.20537 −0.602685 0.797980i $$-0.705902\pi$$
−0.602685 + 0.797980i $$0.705902\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −16.0000 −1.04372
$$236$$ 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 $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −12.0000 −0.766652
$$246$$ 0 0
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −16.0000 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −44.0000 −2.65330
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 56.0000 3.20655
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 16.0000 0.890264
$$324$$ 0 0
$$325$$ 22.0000 1.22034
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 0 0
$$355$$ −48.0000 −2.54758
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −40.0000 −2.09370
$$366$$ 0 0
$$367$$ −38.0000 −1.98358 −0.991792 0.127862i $$-0.959188\pi$$
−0.991792 + 0.127862i $$0.959188\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −32.0000 −1.64373 −0.821865 0.569683i $$-0.807066\pi$$
−0.821865 + 0.569683i $$0.807066\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 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
$$389$$ 20.0000 1.01404 0.507020 0.861934i $$-0.330747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 40.0000 2.01262
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ −12.0000 −0.597763
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 48.0000 2.35623
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 22.0000 1.06716
$$426$$ 0 0
$$427$$ 28.0000 1.35501
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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
$$437$$ −32.0000 −1.53077
$$438$$ 0 0
$$439$$ 6.00000 0.286364 0.143182 0.989696i $$-0.454267\pi$$
0.143182 + 0.989696i $$0.454267\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 56.0000 2.65465
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 16.0000 0.750092
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −4.00000 −0.185098 −0.0925490 0.995708i $$-0.529501\pi$$
−0.0925490 + 0.995708i $$0.529501\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 88.0000 4.03772
$$476$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 40.0000 1.81631
$$486$$ 0 0
$$487$$ −14.0000 −0.634401 −0.317200 0.948359i $$-0.602743\pi$$
−0.317200 + 0.948359i $$0.602743\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −44.0000 −1.96186 −0.980932 0.194354i $$-0.937739\pi$$
−0.980932 + 0.194354i $$0.937739\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 8.00000 0.354594 0.177297 0.984157i $$-0.443265\pi$$
0.177297 + 0.984157i $$0.443265\pi$$
$$510$$ 0 0
$$511$$ −20.0000 −0.884748
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −40.0000 −1.76261
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −24.0000 −1.02805
$$546$$ 0 0
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 20.0000 0.850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 28.0000 1.18640 0.593199 0.805056i $$-0.297865\pi$$
0.593199 + 0.805056i $$0.297865\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ 0 0
$$589$$ −48.0000 −1.97781
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 20.0000 0.813116
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 28.0000 1.12180
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 34.0000 1.35352 0.676759 0.736204i $$-0.263384\pi$$
0.676759 + 0.736204i $$0.263384\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 0 0
$$643$$ 48.0000 1.89294 0.946468 0.322799i $$-0.104624\pi$$
0.946468 + 0.322799i $$0.104624\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 12.0000 0.469596 0.234798 0.972044i $$-0.424557\pi$$
0.234798 + 0.972044i $$0.424557\pi$$
$$654$$ 0 0
$$655$$ −48.0000 −1.87552
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 64.0000 2.48181
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −20.0000 −0.768662 −0.384331 0.923195i $$-0.625568\pi$$
−0.384331 + 0.923195i $$0.625568\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −24.0000 −0.913003 −0.456502 0.889723i $$-0.650898\pi$$
−0.456502 + 0.889723i $$0.650898\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 40.0000 1.51078 0.755390 0.655276i $$-0.227448\pi$$
0.755390 + 0.655276i $$0.227448\pi$$
$$702$$ 0 0
$$703$$ −16.0000 −0.603451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ −32.0000 −1.19673
$$716$$ 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
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 42.0000 1.55131 0.775653 0.631160i $$-0.217421\pi$$
0.775653 + 0.631160i $$0.217421\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ −48.0000 −1.75858
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 40.0000 1.45575
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ −12.0000 −0.434429
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ −66.0000 −2.37079
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −48.0000 −1.71978
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −72.0000 −2.56979
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ 28.0000 0.994309
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 40.0000 1.41157
$$804$$ 0 0
$$805$$ −32.0000 −1.12785
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 32.0000 1.12091
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.00000 0.279202 0.139601 0.990208i $$-0.455418\pi$$
0.139601 + 0.990208i $$0.455418\pi$$
$$822$$ 0 0
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ −64.0000 −2.21481
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 38.0000 1.30110 0.650548 0.759465i $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −40.0000 −1.35691
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 48.0000 1.62270
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 48.0000 1.60446
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 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
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 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
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −24.0000 −0.792550
$$918$$ 0 0
$$919$$ −6.00000 −0.197922 −0.0989609 0.995091i $$-0.531552\pi$$
−0.0989609 + 0.995091i $$0.531552\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −22.0000 −0.723356
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −32.0000 −1.04651
$$936$$ 0 0
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −28.0000 −0.912774 −0.456387 0.889781i $$-0.650857\pi$$
−0.456387 + 0.889781i $$0.650857\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ −32.0000 −1.03550
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −8.00000 −0.257529
$$966$$ 0 0
$$967$$ −2.00000 −0.0643157 −0.0321578 0.999483i $$-0.510238\pi$$
−0.0321578 + 0.999483i $$0.510238\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ −56.0000 −1.78977
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −56.0000 −1.78612 −0.893061 0.449935i $$-0.851447\pi$$
−0.893061 + 0.449935i $$0.851447\pi$$
$$984$$ 0 0
$$985$$ 32.0000 1.01960
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 56.0000 1.77532
$$996$$ 0 0
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1152.2.a.t.1.1 1
3.2 odd 2 384.2.a.a.1.1 1
4.3 odd 2 1152.2.a.s.1.1 1
8.3 odd 2 1152.2.a.a.1.1 1
8.5 even 2 1152.2.a.b.1.1 1
12.11 even 2 384.2.a.e.1.1 yes 1
15.14 odd 2 9600.2.a.bk.1.1 1
16.3 odd 4 2304.2.d.o.1153.1 2
16.5 even 4 2304.2.d.f.1153.2 2
16.11 odd 4 2304.2.d.o.1153.2 2
16.13 even 4 2304.2.d.f.1153.1 2
24.5 odd 2 384.2.a.h.1.1 yes 1
24.11 even 2 384.2.a.d.1.1 yes 1
48.5 odd 4 768.2.d.c.385.2 2
48.11 even 4 768.2.d.f.385.1 2
48.29 odd 4 768.2.d.c.385.1 2
48.35 even 4 768.2.d.f.385.2 2
60.59 even 2 9600.2.a.t.1.1 1
120.29 odd 2 9600.2.a.e.1.1 1
120.59 even 2 9600.2.a.bz.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.a.1.1 1 3.2 odd 2
384.2.a.d.1.1 yes 1 24.11 even 2
384.2.a.e.1.1 yes 1 12.11 even 2
384.2.a.h.1.1 yes 1 24.5 odd 2
768.2.d.c.385.1 2 48.29 odd 4
768.2.d.c.385.2 2 48.5 odd 4
768.2.d.f.385.1 2 48.11 even 4
768.2.d.f.385.2 2 48.35 even 4
1152.2.a.a.1.1 1 8.3 odd 2
1152.2.a.b.1.1 1 8.5 even 2
1152.2.a.s.1.1 1 4.3 odd 2
1152.2.a.t.1.1 1 1.1 even 1 trivial
2304.2.d.f.1153.1 2 16.13 even 4
2304.2.d.f.1153.2 2 16.5 even 4
2304.2.d.o.1153.1 2 16.3 odd 4
2304.2.d.o.1153.2 2 16.11 odd 4
9600.2.a.e.1.1 1 120.29 odd 2
9600.2.a.t.1.1 1 60.59 even 2
9600.2.a.bk.1.1 1 15.14 odd 2
9600.2.a.bz.1.1 1 120.59 even 2