# Properties

 Label 768.2.d.a.385.2 Level 768 Weight 2 Character 768.385 Analytic conductor 6.133 Analytic rank 1 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: $$1$$ 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 96) 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.a.385.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +2.00000i q^{13} -2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +4.00000 q^{33} -8.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} -2.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} -6.00000i q^{51} +10.0000i q^{53} +8.00000 q^{55} +4.00000 q^{57} +4.00000i q^{59} -6.00000i q^{61} +4.00000 q^{63} -4.00000 q^{65} +4.00000i q^{67} -16.0000 q^{71} +6.00000 q^{73} +1.00000i q^{75} +16.0000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -12.0000i q^{85} +2.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} -4.00000i q^{93} +8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{7} - 2q^{9} + O(q^{10})$$ $$2q - 8q^{7} - 2q^{9} - 4q^{15} - 12q^{17} + 2q^{25} - 8q^{31} + 8q^{33} - 4q^{39} - 4q^{41} - 16q^{47} + 18q^{49} + 16q^{55} + 8q^{57} + 8q^{63} - 8q^{65} - 32q^{71} + 12q^{73} - 8q^{79} + 2q^{81} + 4q^{87} - 20q^{89} + 16q^{95} - 28q^{97} + O(q^{100})$$

## 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
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 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 $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ − 4.00000i − 0.872872i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ − 2.00000i − 0.371391i −0.982607 0.185695i $$-0.940546\pi$$
0.982607 0.185695i $$-0.0594537\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 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
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ − 6.00000i − 0.840168i
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ 4.00000i 0.520756i 0.965507 + 0.260378i $$0.0838471\pi$$
−0.965507 + 0.260378i $$0.916153\pi$$
$$60$$ 0 0
$$61$$ − 6.00000i − 0.768221i −0.923287 0.384111i $$-0.874508\pi$$
0.923287 0.384111i $$-0.125492\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 1.00000i 0.115470i
$$76$$ 0 0
$$77$$ 16.0000i 1.82337i
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 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
$$85$$ − 12.0000i − 1.30158i
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ − 8.00000i − 0.838628i
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ − 6.00000i − 0.597022i −0.954406 0.298511i $$-0.903510\pi$$
0.954406 0.298511i $$-0.0964900\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 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
$$109$$ − 14.0000i − 1.34096i −0.741929 0.670478i $$-0.766089\pi$$
0.741929 0.670478i $$-0.233911\pi$$
$$110$$ 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
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 12.0000i 1.07331i
$$126$$ 0 0
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 4.00000i 0.349482i 0.984614 + 0.174741i $$0.0559088\pi$$
−0.984614 + 0.174741i $$0.944091\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 2.00000 0.172133
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 20.0000i 1.69638i 0.529694 + 0.848189i $$0.322307\pi$$
−0.529694 + 0.848189i $$0.677693\pi$$
$$140$$ 0 0
$$141$$ − 8.00000i − 0.673722i
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ 9.00000i 0.742307i
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ − 8.00000i − 0.642575i
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 8.00000i 0.622799i
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 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
$$181$$ − 10.0000i − 0.743294i −0.928374 0.371647i $$-0.878793\pi$$
0.928374 0.371647i $$-0.121207\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 24.0000i 1.75505i
$$188$$ 0 0
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 0 0
$$195$$ − 4.00000i − 0.286446i
$$196$$ 0 0
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 8.00000i 0.561490i
$$204$$ 0 0
$$205$$ − 4.00000i − 0.279372i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000i 1.37686i 0.725304 + 0.688428i $$0.241699\pi$$
−0.725304 + 0.688428i $$0.758301\pi$$
$$212$$ 0 0
$$213$$ − 16.0000i − 1.09630i
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ 0 0
$$219$$ 6.00000i 0.405442i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ − 10.0000i − 0.660819i −0.943838 0.330409i $$-0.892813\pi$$
0.943838 0.330409i $$-0.107187\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 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
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 18.0000i 1.14998i
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 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
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 12.0000 0.751469
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 8.00000i 0.497096i
$$260$$ 0 0
$$261$$ 2.00000i 0.123797i
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ 0 0
$$269$$ − 18.0000i − 1.09748i −0.835993 0.548740i $$-0.815108\pi$$
0.835993 0.548740i $$-0.184892\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 8.00000 0.484182
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 0 0
$$285$$ 8.00000i 0.473879i
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ − 14.0000i − 0.820695i
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000i 0.922225i
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 12.0000 0.687118
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ − 12.0000i − 0.682656i
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ 8.00000i 0.450749i
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 0 0
$$325$$ 2.00000i 0.110940i
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ − 28.0000i − 1.53902i −0.638635 0.769510i $$-0.720501\pi$$
0.638635 0.769510i $$-0.279499\pi$$
$$332$$ 0 0
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 2.00000i 0.108625i
$$340$$ 0 0
$$341$$ 16.0000i 0.866449i
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 20.0000i − 1.07366i −0.843692 0.536828i $$-0.819622\pi$$
0.843692 0.536828i $$-0.180378\pi$$
$$348$$ 0 0
$$349$$ 26.0000i 1.39175i 0.718164 + 0.695874i $$0.244983\pi$$
−0.718164 + 0.695874i $$0.755017\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ − 32.0000i − 1.69838i
$$356$$ 0 0
$$357$$ 24.0000i 1.27021i
$$358$$ 0 0
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 12.0000i 0.628109i
$$366$$ 0 0
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ − 40.0000i − 2.07670i
$$372$$ 0 0
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 28.0000i 1.43826i 0.694874 + 0.719132i $$0.255460\pi$$
−0.694874 + 0.719132i $$0.744540\pi$$
$$380$$ 0 0
$$381$$ 20.0000i 1.02463i
$$382$$ 0 0
$$383$$ −32.0000 −1.63512 −0.817562 0.575841i $$-0.804675\pi$$
−0.817562 + 0.575841i $$0.804675\pi$$
$$384$$ 0 0
$$385$$ −32.0000 −1.63087
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ − 30.0000i − 1.52106i −0.649303 0.760530i $$-0.724939\pi$$
0.649303 0.760530i $$-0.275061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ − 8.00000i − 0.402524i
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 0 0
$$405$$ 2.00000i 0.0993808i
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ − 18.0000i − 0.887875i
$$412$$ 0 0
$$413$$ − 16.0000i − 0.787309i
$$414$$ 0 0
$$415$$ −24.0000 −1.17811
$$416$$ 0 0
$$417$$ −20.0000 −0.979404
$$418$$ 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
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 24.0000i 1.16144i
$$428$$ 0 0
$$429$$ 8.00000i 0.386244i
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 4.00000i 0.191785i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ − 20.0000i − 0.948091i
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 8.00000i 0.376705i
$$452$$ 0 0
$$453$$ 12.0000i 0.563809i
$$454$$ 0 0
$$455$$ 16.0000 0.750092
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 0 0
$$459$$ 6.00000i 0.280056i
$$460$$ 0 0
$$461$$ 6.00000i 0.279448i 0.990190 + 0.139724i $$0.0446215\pi$$
−0.990190 + 0.139724i $$0.955378\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ 0 0
$$465$$ 8.00000 0.370991
$$466$$ 0 0
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 0 0
$$469$$ − 16.0000i − 0.738811i
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ − 10.0000i − 0.457869i
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 28.0000i − 1.27141i
$$486$$ 0 0
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ − 28.0000i − 1.26362i −0.775122 0.631811i $$-0.782312\pi$$
0.775122 0.631811i $$-0.217688\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 64.0000 2.87079
$$498$$ 0 0
$$499$$ − 12.0000i − 0.537194i −0.963253 0.268597i $$-0.913440\pi$$
0.963253 0.268597i $$-0.0865599\pi$$
$$500$$ 0 0
$$501$$ 8.00000i 0.357414i
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ − 18.0000i − 0.797836i −0.916987 0.398918i $$-0.869386\pi$$
0.916987 0.398918i $$-0.130614\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ − 24.0000i − 1.05757i
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ 0 0
$$525$$ − 4.00000i − 0.174574i
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ − 4.00000i − 0.173585i
$$532$$ 0 0
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ − 36.0000i − 1.55063i
$$540$$ 0 0
$$541$$ 2.00000i 0.0859867i 0.999075 + 0.0429934i $$0.0136894\pi$$
−0.999075 + 0.0429934i $$0.986311\pi$$
$$542$$ 0 0
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ 28.0000 1.19939
$$546$$ 0 0
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 0 0
$$549$$ 6.00000i 0.256074i
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 0 0
$$555$$ 4.00000i 0.169791i
$$556$$ 0 0
$$557$$ 22.0000i 0.932170i 0.884740 + 0.466085i $$0.154336\pi$$
−0.884740 + 0.466085i $$0.845664\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 0 0
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 4.00000i 0.168281i
$$566$$ 0 0
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ − 44.0000i − 1.84134i −0.390339 0.920671i $$-0.627642\pi$$
0.390339 0.920671i $$-0.372358\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 0 0
$$579$$ 18.0000i 0.748054i
$$580$$ 0 0
$$581$$ − 48.0000i − 1.99138i
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ 0 0
$$585$$ 4.00000 0.165380
$$586$$ 0 0
$$587$$ 36.0000i 1.48588i 0.669359 + 0.742940i $$0.266569\pi$$
−0.669359 + 0.742940i $$0.733431\pi$$
$$588$$ 0 0
$$589$$ 16.0000i 0.659269i
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 48.0000i 1.96781i
$$596$$ 0 0
$$597$$ 4.00000i 0.163709i
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ − 10.0000i − 0.406558i
$$606$$ 0 0
$$607$$ 20.0000 0.811775 0.405887 0.913923i $$-0.366962\pi$$
0.405887 + 0.913923i $$0.366962\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ − 16.0000i − 0.647291i
$$612$$ 0 0
$$613$$ − 18.0000i − 0.727013i −0.931592 0.363507i $$-0.881579\pi$$
0.931592 0.363507i $$-0.118421\pi$$
$$614$$ 0 0
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ − 12.0000i − 0.482321i −0.970485 0.241160i $$-0.922472\pi$$
0.970485 0.241160i $$-0.0775280\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 40.0000 1.60257
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 0 0
$$635$$ 40.0000i 1.58735i
$$636$$ 0 0
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ 8.00000i 0.315000i
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 16.0000i 0.627089i
$$652$$ 0 0
$$653$$ − 10.0000i − 0.391330i −0.980671 0.195665i $$-0.937313\pi$$
0.980671 0.195665i $$-0.0626866\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 36.0000i 1.40236i 0.712984 + 0.701180i $$0.247343\pi$$
−0.712984 + 0.701180i $$0.752657\pi$$
$$660$$ 0 0
$$661$$ 14.0000i 0.544537i 0.962221 + 0.272268i $$0.0877739\pi$$
−0.962221 + 0.272268i $$0.912226\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ −32.0000 −1.24091
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 4.00000i 0.154649i
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 0 0
$$675$$ − 1.00000i − 0.0384900i
$$676$$ 0 0
$$677$$ − 14.0000i − 0.538064i −0.963131 0.269032i $$-0.913296\pi$$
0.963131 0.269032i $$-0.0867037\pi$$
$$678$$ 0 0
$$679$$ 56.0000 2.14908
$$680$$ 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
$$685$$ − 36.0000i − 1.37549i
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 0 0
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ − 52.0000i − 1.97817i −0.147335 0.989087i $$-0.547070\pi$$
0.147335 0.989087i $$-0.452930\pi$$
$$692$$ 0 0
$$693$$ − 16.0000i − 0.607790i
$$694$$ 0 0
$$695$$ −40.0000 −1.51729
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 6.00000i 0.226941i
$$700$$ 0 0
$$701$$ − 18.0000i − 0.679851i −0.940452 0.339925i $$-0.889598\pi$$
0.940452 0.339925i $$-0.110402\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 16.0000 0.602595
$$706$$ 0 0
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ 22.0000i 0.826227i 0.910679 + 0.413114i $$0.135559\pi$$
−0.910679 + 0.413114i $$0.864441\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 16.0000i 0.598366i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 0 0
$$723$$ − 14.0000i − 0.520666i
$$724$$ 0 0
$$725$$ − 2.00000i − 0.0742781i
$$726$$ 0 0
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000i 0.887672i
$$732$$ 0 0
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ 0 0
$$735$$ −18.0000 −0.663940
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 20.0000i 0.735712i 0.929883 + 0.367856i $$0.119908\pi$$
−0.929883 + 0.367856i $$0.880092\pi$$
$$740$$ 0 0
$$741$$ 8.00000i 0.293887i
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ 0 0
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ − 16.0000i − 0.584627i
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 24.0000i 0.873449i
$$756$$ 0 0
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 56.0000i 2.02734i
$$764$$ 0 0
$$765$$ 12.0000i 0.433861i
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 2.00000i 0.0720282i
$$772$$ 0 0
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ −8.00000 −0.286998
$$778$$ 0 0
$$779$$ 8.00000i 0.286630i
$$780$$ 0 0
$$781$$ 64.0000i 2.29010i
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ −20.0000 −0.713831
$$786$$ 0 0
$$787$$ 12.0000i 0.427754i 0.976861 + 0.213877i $$0.0686091\pi$$
−0.976861 + 0.213877i $$0.931391\pi$$
$$788$$ 0 0
$$789$$ − 24.0000i − 0.854423i
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 0 0
$$795$$ − 20.0000i − 0.709327i
$$796$$ 0 0
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ − 24.0000i − 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ 0 0
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 0 0
$$811$$ 28.0000i 0.983213i 0.870817 + 0.491606i $$0.163590\pi$$
−0.870817 + 0.491606i $$0.836410\pi$$
$$812$$ 0 0
$$813$$ − 12.0000i − 0.420858i
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 8.00000i 0.279543i
$$820$$ 0 0
$$821$$ 26.0000i 0.907406i 0.891153 + 0.453703i $$0.149897\pi$$
−0.891153 + 0.453703i $$0.850103\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ 20.0000i 0.695468i 0.937593 + 0.347734i $$0.113049\pi$$
−0.937593 + 0.347734i $$0.886951\pi$$
$$828$$ 0 0
$$829$$ 34.0000i 1.18087i 0.807086 + 0.590434i $$0.201044\pi$$
−0.807086 + 0.590434i $$0.798956\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 0 0
$$833$$ −54.0000 −1.87099
$$834$$ 0 0
$$835$$ 16.0000i 0.553703i
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 0 0
$$839$$ 32.0000 1.10476 0.552381 0.833592i $$-0.313719\pi$$
0.552381 + 0.833592i $$0.313719\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ 6.00000i 0.206651i
$$844$$ 0 0
$$845$$ 18.0000i 0.619219i
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 0 0
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 18.0000i − 0.616308i −0.951336 0.308154i $$-0.900289\pi$$
0.951336 0.308154i $$-0.0997113\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ − 4.00000i − 0.136478i −0.997669 0.0682391i $$-0.978262\pi$$
0.997669 0.0682391i $$-0.0217381\pi$$
$$860$$ 0 0
$$861$$ 8.00000i 0.272639i
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 16.0000i 0.542763i
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ − 48.0000i − 1.62270i
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 0 0
$$885$$ − 8.00000i − 0.268917i
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −80.0000 −2.68311
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 8.00000i 0.266815i
$$900$$ 0 0
$$901$$ − 60.0000i − 1.99889i
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 0 0
$$905$$ 20.0000 0.664822
$$906$$ 0 0
$$907$$ − 20.0000i − 0.664089i −0.943264 0.332045i $$-0.892262\pi$$
0.943264 0.332045i $$-0.107738\pi$$
$$908$$ 0 0
$$909$$ 6.00000i 0.199007i
$$910$$ 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
$$915$$ 12.0000i 0.396708i
$$916$$ 0 0
$$917$$ − 16.0000i − 0.528367i
$$918$$ 0 0
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ − 32.0000i − 1.05329i
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ − 36.0000i − 1.17985i
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ −48.0000 −1.56977
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ − 10.0000i − 0.326338i
$$940$$ 0 0
$$941$$ − 58.0000i − 1.89075i −0.325991 0.945373i $$-0.605698\pi$$
0.325991 0.945373i $$-0.394302\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −8.00000 −0.260240
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ 12.0000i 0.389536i
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ 0 0
$$959$$ 72.0000 2.32500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ 0 0
$$965$$ 36.0000i 1.15888i
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ − 52.0000i − 1.66876i −0.551190 0.834380i $$-0.685826\pi$$
0.551190 0.834380i $$-0.314174\pi$$
$$972$$ 0 0
$$973$$ − 80.0000i − 2.56468i
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −22.0000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$978$$ 0 0
$$979$$ 40.0000i 1.27841i
$$980$$ 0 0
$$981$$ 14.0000i 0.446986i
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 44.0000 1.40196
$$986$$ 0 0
$$987$$ 32.0000i 1.01857i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 8.00000i 0.253617i
$$996$$ 0 0
$$997$$ − 18.0000i − 0.570066i −0.958518 0.285033i $$-0.907995\pi$$
0.958518 0.285033i $$-0.0920045\pi$$
$$998$$ 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.a.385.2 2
3.2 odd 2 2304.2.d.c.1153.1 2
4.3 odd 2 768.2.d.h.385.1 2
8.3 odd 2 768.2.d.h.385.2 2
8.5 even 2 inner 768.2.d.a.385.1 2
12.11 even 2 2304.2.d.s.1153.1 2
16.3 odd 4 96.2.a.b.1.1 yes 1
16.5 even 4 192.2.a.c.1.1 1
16.11 odd 4 192.2.a.a.1.1 1
16.13 even 4 96.2.a.a.1.1 1
24.5 odd 2 2304.2.d.c.1153.2 2
24.11 even 2 2304.2.d.s.1153.2 2
48.5 odd 4 576.2.a.h.1.1 1
48.11 even 4 576.2.a.g.1.1 1
48.29 odd 4 288.2.a.c.1.1 1
48.35 even 4 288.2.a.b.1.1 1
80.3 even 4 2400.2.f.r.1249.2 2
80.13 odd 4 2400.2.f.a.1249.1 2
80.19 odd 4 2400.2.a.q.1.1 1
80.27 even 4 4800.2.f.e.3649.2 2
80.29 even 4 2400.2.a.r.1.1 1
80.37 odd 4 4800.2.f.bh.3649.1 2
80.43 even 4 4800.2.f.e.3649.1 2
80.53 odd 4 4800.2.f.bh.3649.2 2
80.59 odd 4 4800.2.a.co.1.1 1
80.67 even 4 2400.2.f.r.1249.1 2
80.69 even 4 4800.2.a.f.1.1 1
80.77 odd 4 2400.2.f.a.1249.2 2
112.13 odd 4 4704.2.a.t.1.1 1
112.27 even 4 9408.2.a.ct.1.1 1
112.69 odd 4 9408.2.a.bj.1.1 1
112.83 even 4 4704.2.a.e.1.1 1
144.13 even 12 2592.2.i.b.865.1 2
144.29 odd 12 2592.2.i.q.1729.1 2
144.61 even 12 2592.2.i.b.1729.1 2
144.67 odd 12 2592.2.i.h.865.1 2
144.77 odd 12 2592.2.i.q.865.1 2
144.83 even 12 2592.2.i.w.1729.1 2
144.115 odd 12 2592.2.i.h.1729.1 2
144.131 even 12 2592.2.i.w.865.1 2
240.29 odd 4 7200.2.a.e.1.1 1
240.77 even 4 7200.2.f.x.6049.2 2
240.83 odd 4 7200.2.f.f.6049.2 2
240.173 even 4 7200.2.f.x.6049.1 2
240.179 even 4 7200.2.a.bx.1.1 1
240.227 odd 4 7200.2.f.f.6049.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.a.a.1.1 1 16.13 even 4
96.2.a.b.1.1 yes 1 16.3 odd 4
192.2.a.a.1.1 1 16.11 odd 4
192.2.a.c.1.1 1 16.5 even 4
288.2.a.b.1.1 1 48.35 even 4
288.2.a.c.1.1 1 48.29 odd 4
576.2.a.g.1.1 1 48.11 even 4
576.2.a.h.1.1 1 48.5 odd 4
768.2.d.a.385.1 2 8.5 even 2 inner
768.2.d.a.385.2 2 1.1 even 1 trivial
768.2.d.h.385.1 2 4.3 odd 2
768.2.d.h.385.2 2 8.3 odd 2
2304.2.d.c.1153.1 2 3.2 odd 2
2304.2.d.c.1153.2 2 24.5 odd 2
2304.2.d.s.1153.1 2 12.11 even 2
2304.2.d.s.1153.2 2 24.11 even 2
2400.2.a.q.1.1 1 80.19 odd 4
2400.2.a.r.1.1 1 80.29 even 4
2400.2.f.a.1249.1 2 80.13 odd 4
2400.2.f.a.1249.2 2 80.77 odd 4
2400.2.f.r.1249.1 2 80.67 even 4
2400.2.f.r.1249.2 2 80.3 even 4
2592.2.i.b.865.1 2 144.13 even 12
2592.2.i.b.1729.1 2 144.61 even 12
2592.2.i.h.865.1 2 144.67 odd 12
2592.2.i.h.1729.1 2 144.115 odd 12
2592.2.i.q.865.1 2 144.77 odd 12
2592.2.i.q.1729.1 2 144.29 odd 12
2592.2.i.w.865.1 2 144.131 even 12
2592.2.i.w.1729.1 2 144.83 even 12
4704.2.a.e.1.1 1 112.83 even 4
4704.2.a.t.1.1 1 112.13 odd 4
4800.2.a.f.1.1 1 80.69 even 4
4800.2.a.co.1.1 1 80.59 odd 4
4800.2.f.e.3649.1 2 80.43 even 4
4800.2.f.e.3649.2 2 80.27 even 4
4800.2.f.bh.3649.1 2 80.37 odd 4
4800.2.f.bh.3649.2 2 80.53 odd 4
7200.2.a.e.1.1 1 240.29 odd 4
7200.2.a.bx.1.1 1 240.179 even 4
7200.2.f.f.6049.1 2 240.227 odd 4
7200.2.f.f.6049.2 2 240.83 odd 4
7200.2.f.x.6049.1 2 240.173 even 4
7200.2.f.x.6049.2 2 240.77 even 4
9408.2.a.bj.1.1 1 112.69 odd 4
9408.2.a.ct.1.1 1 112.27 even 4