# Properties

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

## $q$-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 $$-0.852416\pi$$
0.894427 0.447214i $$-0.147584\pi$$
$$6$$ 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 $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ −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 $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 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 $$0.0594537\pi$$
−0.982607 + 0.185695i $$0.940546\pi$$
$$30$$ 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 $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 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 $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 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 $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 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 $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 0 0
$$61$$ 6.00000i 0.768221i 0.923287 + 0.384111i $$0.125492\pi$$
−0.923287 + 0.384111i $$0.874508\pi$$
$$62$$ 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 $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 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 $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 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 $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 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 $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ 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 $$-0.944091\pi$$
0.984614 0.174741i $$-0.0559088\pi$$
$$132$$ 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 $$-0.677693\pi$$
0.529694 0.848189i $$-0.322307\pi$$
$$140$$ 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 $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 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 $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 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 $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 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 $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 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 $$0.148028\pi$$
−0.893802 + 0.448461i $$0.851972\pi$$
$$180$$ 0 0
$$181$$ 10.0000i 0.743294i 0.928374 + 0.371647i $$0.121207\pi$$
−0.928374 + 0.371647i $$0.878793\pi$$
$$182$$ 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 $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ 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 $$-0.758301\pi$$
0.725304 0.688428i $$-0.241699\pi$$
$$212$$ 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 $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 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 $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 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 $$0.184892\pi$$
−0.835993 + 0.548740i $$0.815108\pi$$
$$270$$ 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 $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 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 $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 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 $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 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 $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 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 $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 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 $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ 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 $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 0 0
$$349$$ − 26.0000i − 1.39175i −0.718164 0.695874i $$-0.755017\pi$$
0.718164 0.695874i $$-0.244983\pi$$
$$350$$ 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 $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ 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 $$-0.744540\pi$$
0.694874 0.719132i $$-0.255460\pi$$
$$380$$ 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 $$0.275061\pi$$
−0.649303 + 0.760530i $$0.724939\pi$$
$$390$$ 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 $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ 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 $$0.162467\pi$$
−0.872546 + 0.488532i $$0.837533\pi$$
$$420$$ 0 0
$$421$$ − 38.0000i − 1.85201i −0.377515 0.926003i $$-0.623221\pi$$
0.377515 0.926003i $$-0.376779\pi$$
$$422$$ 0 0
$$423$$ 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 $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 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 $$-0.955378\pi$$
0.990190 0.139724i $$-0.0446215\pi$$
$$462$$ 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 $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 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 $$0.217688\pi$$
−0.775122 + 0.631811i $$0.782312\pi$$
$$492$$ 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 $$0.0865599\pi$$
−0.963253 + 0.268597i $$0.913440\pi$$
$$500$$ 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 $$0.130614\pi$$
−0.916987 + 0.398918i $$0.869386\pi$$
$$510$$ 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 $$-0.915499\pi$$
0.964970 0.262362i $$-0.0845013\pi$$
$$524$$ 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 $$-0.986311\pi$$
0.999075 0.0429934i $$-0.0136894\pi$$
$$542$$ 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 $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ 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 $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ 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 $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 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 $$0.372358\pi$$
−0.390339 + 0.920671i $$0.627642\pi$$
$$572$$ 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 $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 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 $$0.118421\pi$$
−0.931592 + 0.363507i $$0.881579\pi$$
$$614$$ 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 $$0.0775280\pi$$
−0.970485 + 0.241160i $$0.922472\pi$$
$$620$$ 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 $$-0.923961\pi$$
0.971603 0.236617i $$-0.0760386\pi$$
$$644$$ 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 $$0.0626866\pi$$
−0.980671 + 0.195665i $$0.937313\pi$$
$$654$$ 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 $$-0.752657\pi$$
0.712984 0.701180i $$-0.247343\pi$$
$$660$$ 0 0
$$661$$ − 14.0000i − 0.544537i −0.962221 0.272268i $$-0.912226\pi$$
0.962221 0.272268i $$-0.0877739\pi$$
$$662$$ 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 $$0.0867037\pi$$
−0.963131 + 0.269032i $$0.913296\pi$$
$$678$$ 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 $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 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 $$0.452930\pi$$
−0.147335 + 0.989087i $$0.547070\pi$$
$$692$$ 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 $$0.110402\pi$$
−0.940452 + 0.339925i $$0.889598\pi$$
$$702$$ 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 $$-0.864441\pi$$
0.910679 0.413114i $$-0.135559\pi$$
$$710$$ 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 $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 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 $$-0.880092\pi$$
0.929883 0.367856i $$-0.119908\pi$$
$$740$$ 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 $$-0.965223\pi$$
0.994038 0.109037i $$-0.0347767\pi$$
$$758$$ 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 $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 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 $$-0.931391\pi$$
0.976861 0.213877i $$-0.0686091\pi$$
$$788$$ 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 $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 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 $$-0.836410\pi$$
0.870817 0.491606i $$-0.163590\pi$$
$$812$$ 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 $$-0.850103\pi$$
0.891153 0.453703i $$-0.149897\pi$$
$$822$$ 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 $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ − 34.0000i − 1.18087i −0.807086 0.590434i $$-0.798956\pi$$
0.807086 0.590434i $$-0.201044\pi$$
$$830$$ 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 $$0.0997113\pi$$
−0.951336 + 0.308154i $$0.900289\pi$$
$$854$$ 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 $$0.0217381\pi$$
−0.997669 + 0.0682391i $$0.978262\pi$$
$$860$$ 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 $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 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 $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 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 $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 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 $$0.394302\pi$$
−0.325991 + 0.945373i $$0.605698\pi$$
$$942$$ 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 $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 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 $$0.314174\pi$$
−0.551190 + 0.834380i $$0.685826\pi$$
$$972$$ 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 $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 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.385.1 2
3.2 odd 2 2304.2.d.c.1153.2 2
4.3 odd 2 768.2.d.h.385.2 2
8.3 odd 2 768.2.d.h.385.1 2
8.5 even 2 inner 768.2.d.a.385.2 2
12.11 even 2 2304.2.d.s.1153.2 2
16.3 odd 4 192.2.a.a.1.1 1
16.5 even 4 96.2.a.a.1.1 1
16.11 odd 4 96.2.a.b.1.1 yes 1
16.13 even 4 192.2.a.c.1.1 1
24.5 odd 2 2304.2.d.c.1153.1 2
24.11 even 2 2304.2.d.s.1153.1 2
48.5 odd 4 288.2.a.c.1.1 1
48.11 even 4 288.2.a.b.1.1 1
48.29 odd 4 576.2.a.h.1.1 1
48.35 even 4 576.2.a.g.1.1 1
80.3 even 4 4800.2.f.e.3649.1 2
80.13 odd 4 4800.2.f.bh.3649.2 2
80.19 odd 4 4800.2.a.co.1.1 1
80.27 even 4 2400.2.f.r.1249.1 2
80.29 even 4 4800.2.a.f.1.1 1
80.37 odd 4 2400.2.f.a.1249.2 2
80.43 even 4 2400.2.f.r.1249.2 2
80.53 odd 4 2400.2.f.a.1249.1 2
80.59 odd 4 2400.2.a.q.1.1 1
80.67 even 4 4800.2.f.e.3649.2 2
80.69 even 4 2400.2.a.r.1.1 1
80.77 odd 4 4800.2.f.bh.3649.1 2
112.13 odd 4 9408.2.a.bj.1.1 1
112.27 even 4 4704.2.a.e.1.1 1
112.69 odd 4 4704.2.a.t.1.1 1
112.83 even 4 9408.2.a.ct.1.1 1
144.5 odd 12 2592.2.i.q.865.1 2
144.11 even 12 2592.2.i.w.1729.1 2
144.43 odd 12 2592.2.i.h.1729.1 2
144.59 even 12 2592.2.i.w.865.1 2
144.85 even 12 2592.2.i.b.865.1 2
144.101 odd 12 2592.2.i.q.1729.1 2
144.133 even 12 2592.2.i.b.1729.1 2
144.139 odd 12 2592.2.i.h.865.1 2
240.53 even 4 7200.2.f.x.6049.1 2
240.59 even 4 7200.2.a.bx.1.1 1
240.107 odd 4 7200.2.f.f.6049.1 2
240.149 odd 4 7200.2.a.e.1.1 1
240.197 even 4 7200.2.f.x.6049.2 2
240.203 odd 4 7200.2.f.f.6049.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.a.a.1.1 1 16.5 even 4
96.2.a.b.1.1 yes 1 16.11 odd 4
192.2.a.a.1.1 1 16.3 odd 4
192.2.a.c.1.1 1 16.13 even 4
288.2.a.b.1.1 1 48.11 even 4
288.2.a.c.1.1 1 48.5 odd 4
576.2.a.g.1.1 1 48.35 even 4
576.2.a.h.1.1 1 48.29 odd 4
768.2.d.a.385.1 2 1.1 even 1 trivial
768.2.d.a.385.2 2 8.5 even 2 inner
768.2.d.h.385.1 2 8.3 odd 2
768.2.d.h.385.2 2 4.3 odd 2
2304.2.d.c.1153.1 2 24.5 odd 2
2304.2.d.c.1153.2 2 3.2 odd 2
2304.2.d.s.1153.1 2 24.11 even 2
2304.2.d.s.1153.2 2 12.11 even 2
2400.2.a.q.1.1 1 80.59 odd 4
2400.2.a.r.1.1 1 80.69 even 4
2400.2.f.a.1249.1 2 80.53 odd 4
2400.2.f.a.1249.2 2 80.37 odd 4
2400.2.f.r.1249.1 2 80.27 even 4
2400.2.f.r.1249.2 2 80.43 even 4
2592.2.i.b.865.1 2 144.85 even 12
2592.2.i.b.1729.1 2 144.133 even 12
2592.2.i.h.865.1 2 144.139 odd 12
2592.2.i.h.1729.1 2 144.43 odd 12
2592.2.i.q.865.1 2 144.5 odd 12
2592.2.i.q.1729.1 2 144.101 odd 12
2592.2.i.w.865.1 2 144.59 even 12
2592.2.i.w.1729.1 2 144.11 even 12
4704.2.a.e.1.1 1 112.27 even 4
4704.2.a.t.1.1 1 112.69 odd 4
4800.2.a.f.1.1 1 80.29 even 4
4800.2.a.co.1.1 1 80.19 odd 4
4800.2.f.e.3649.1 2 80.3 even 4
4800.2.f.e.3649.2 2 80.67 even 4
4800.2.f.bh.3649.1 2 80.77 odd 4
4800.2.f.bh.3649.2 2 80.13 odd 4
7200.2.a.e.1.1 1 240.149 odd 4
7200.2.a.bx.1.1 1 240.59 even 4
7200.2.f.f.6049.1 2 240.107 odd 4
7200.2.f.f.6049.2 2 240.203 odd 4
7200.2.f.x.6049.1 2 240.53 even 4
7200.2.f.x.6049.2 2 240.197 even 4
9408.2.a.bj.1.1 1 112.13 odd 4
9408.2.a.ct.1.1 1 112.83 even 4