Properties

 Label 6534.2.a.b.1.1 Level $6534$ Weight $2$ Character 6534.1 Self dual yes Analytic conductor $52.174$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6534 = 2 \cdot 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6534.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$52.1742526807$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6534.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -3.00000 q^{20} +6.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{35} +2.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} +10.0000 q^{43} -6.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} -4.00000 q^{50} +4.00000 q^{52} -9.00000 q^{53} -1.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} -8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +14.0000 q^{67} +3.00000 q^{70} +7.00000 q^{73} -2.00000 q^{74} -2.00000 q^{76} -8.00000 q^{79} -3.00000 q^{80} +6.00000 q^{82} -3.00000 q^{83} -10.0000 q^{86} +18.0000 q^{89} +4.00000 q^{91} +6.00000 q^{92} +6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{97} +6.00000 q^{98} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 3.00000 0.948683
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ −4.00000 −0.784465
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 3.00000 0.358569
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 0 0
$$82$$ 6.00000 0.662589
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −18.0000 −1.67851
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ 5.00000 0.449013
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 12.0000 1.05247
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ −14.0000 −1.20942
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −3.00000 −0.253546
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −18.0000 −1.49482
$$146$$ −7.00000 −0.579324
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −15.0000 −1.20483
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 3.00000 0.237171
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 3.00000 0.232845
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 10.0000 0.762493
$$173$$ 15.0000 1.14043 0.570214 0.821496i $$-0.306860\pi$$
0.570214 + 0.821496i $$0.306860\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −6.00000 −0.437595
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 1.00000 0.0717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 9.00000 0.641223 0.320612 0.947211i $$-0.396112\pi$$
0.320612 + 0.947211i $$0.396112\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 3.00000 0.211079
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 0 0
$$214$$ −9.00000 −0.615227
$$215$$ −30.0000 −2.04598
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 18.0000 1.18688
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 18.0000 1.17419
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 30.0000 1.94054 0.970269 0.242028i $$-0.0778125\pi$$
0.970269 + 0.242028i $$0.0778125\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 18.0000 1.14998
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ −5.00000 −0.317500
$$249$$ 0 0
$$250$$ −3.00000 −0.189737
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −7.00000 −0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ −12.0000 −0.744208
$$261$$ 0 0
$$262$$ 15.0000 0.926703
$$263$$ −30.0000 −1.84988 −0.924940 0.380114i $$-0.875885\pi$$
−0.924940 + 0.380114i $$0.875885\pi$$
$$264$$ 0 0
$$265$$ 27.0000 1.65860
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ 14.0000 0.855186
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 25.0000 1.51864 0.759321 0.650716i $$-0.225531\pi$$
0.759321 + 0.650716i $$0.225531\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 3.00000 0.179284
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 18.0000 1.05700
$$291$$ 0 0
$$292$$ 7.00000 0.409644
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 36.0000 2.09600
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −3.00000 −0.173785
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 10.0000 0.576390
$$302$$ 17.0000 0.978240
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 24.0000 1.37424
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 15.0000 0.851943
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −3.00000 −0.167705
$$321$$ 0 0
$$322$$ −6.00000 −0.334367
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 16.0000 0.887520
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ −3.00000 −0.164646
$$333$$ 0 0
$$334$$ 6.00000 0.328305
$$335$$ −42.0000 −2.29471
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ −15.0000 −0.806405
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ −9.00000 −0.475665
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 16.0000 0.840941
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ −21.0000 −1.09919
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 6.00000 0.311925
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5.00000 0.254493
$$387$$ 0 0
$$388$$ −1.00000 −0.0507673
$$389$$ 21.0000 1.06474 0.532371 0.846511i $$-0.321301\pi$$
0.532371 + 0.846511i $$0.321301\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ −9.00000 −0.453413
$$395$$ 24.0000 1.20757
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 7.00000 0.350878
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 20.0000 0.996271
$$404$$ −3.00000 −0.149256
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −23.0000 −1.13728 −0.568638 0.822588i $$-0.692530\pi$$
−0.568638 + 0.822588i $$0.692530\pi$$
$$410$$ −18.0000 −0.888957
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 9.00000 0.441793
$$416$$ −4.00000 −0.196116
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ −22.0000 −1.07094
$$423$$ 0 0
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 9.00000 0.435031
$$429$$ 0 0
$$430$$ 30.0000 1.44673
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −12.0000 −0.574038
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −54.0000 −2.55985
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ −18.0000 −0.839254
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 0 0
$$463$$ −13.0000 −0.604161 −0.302081 0.953282i $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 0 0
$$469$$ 14.0000 0.646460
$$470$$ −18.0000 −0.830278
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −30.0000 −1.37217
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.00000 0.136223
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 0 0
$$490$$ −18.0000 −0.813157
$$491$$ 39.0000 1.76005 0.880023 0.474932i $$-0.157527\pi$$
0.880023 + 0.474932i $$0.157527\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −18.0000 −0.802580 −0.401290 0.915951i $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 0 0
$$505$$ 9.00000 0.400495
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 7.00000 0.310575
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ 12.0000 0.528783
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 0 0
$$520$$ 12.0000 0.526235
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −15.0000 −0.655278
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −27.0000 −1.17281
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ −27.0000 −1.16731
$$536$$ −14.0000 −0.604708
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −20.0000 −0.859867 −0.429934 0.902861i $$-0.641463\pi$$
−0.429934 + 0.902861i $$0.641463\pi$$
$$542$$ −25.0000 −1.07384
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 27.0000 1.14403 0.572013 0.820244i $$-0.306163\pi$$
0.572013 + 0.820244i $$0.306163\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ −3.00000 −0.126773
$$561$$ 0 0
$$562$$ −24.0000 −1.01238
$$563$$ 3.00000 0.126435 0.0632175 0.998000i $$-0.479864\pi$$
0.0632175 + 0.998000i $$0.479864\pi$$
$$564$$ 0 0
$$565$$ −18.0000 −0.757266
$$566$$ 14.0000 0.588464
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 24.0000 1.00087
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 0 0
$$580$$ −18.0000 −0.747409
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 3.00000 0.123823 0.0619116 0.998082i $$-0.480280\pi$$
0.0619116 + 0.998082i $$0.480280\pi$$
$$588$$ 0 0
$$589$$ −10.0000 −0.412043
$$590$$ −36.0000 −1.48210
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3.00000 0.122885
$$597$$ 0 0
$$598$$ −24.0000 −0.981433
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ −35.0000 −1.42768 −0.713840 0.700309i $$-0.753046\pi$$
−0.713840 + 0.700309i $$0.753046\pi$$
$$602$$ −10.0000 −0.407570
$$603$$ 0 0
$$604$$ −17.0000 −0.691720
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ −24.0000 −0.971732
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ −15.0000 −0.602414
$$621$$ 0 0
$$622$$ −6.00000 −0.240578
$$623$$ 18.0000 0.721155
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 19.0000 0.759393
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ −3.00000 −0.119145
$$635$$ −21.0000 −0.833360
$$636$$ 0 0
$$637$$ −24.0000 −0.950915
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −16.0000 −0.627572
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −39.0000 −1.52619 −0.763094 0.646288i $$-0.776321\pi$$
−0.763094 + 0.646288i $$0.776321\pi$$
$$654$$ 0 0
$$655$$ 45.0000 1.75830
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ 6.00000 0.233904
$$659$$ −21.0000 −0.818044 −0.409022 0.912525i $$-0.634130\pi$$
−0.409022 + 0.912525i $$0.634130\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 0 0
$$664$$ 3.00000 0.116423
$$665$$ 6.00000 0.232670
$$666$$ 0 0
$$667$$ 36.0000 1.39393
$$668$$ −6.00000 −0.232147
$$669$$ 0 0
$$670$$ 42.0000 1.62260
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 13.0000 0.496342
$$687$$ 0 0
$$688$$ 10.0000 0.381246
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 15.0000 0.570214
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ −12.0000 −0.455186
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ 9.00000 0.339925 0.169963 0.985451i $$-0.445635\pi$$
0.169963 + 0.985451i $$0.445635\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ −3.00000 −0.112827
$$708$$ 0 0
$$709$$ 44.0000 1.65245 0.826227 0.563337i $$-0.190483\pi$$
0.826227 + 0.563337i $$0.190483\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9.00000 0.336346
$$717$$ 0 0
$$718$$ −18.0000 −0.671754
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ −16.0000 −0.594635
$$725$$ 24.0000 0.891338
$$726$$ 0 0
$$727$$ −1.00000 −0.0370879 −0.0185440 0.999828i $$-0.505903\pi$$
−0.0185440 + 0.999828i $$0.505903\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 0 0
$$730$$ 21.0000 0.777245
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ −17.0000 −0.627481
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 0 0
$$742$$ 9.00000 0.330400
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ −9.00000 −0.329734
$$746$$ 32.0000 1.17160
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ 41.0000 1.49611 0.748056 0.663636i $$-0.230988\pi$$
0.748056 + 0.663636i $$0.230988\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ −24.0000 −0.874028
$$755$$ 51.0000 1.85608
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −48.0000 −1.73318
$$768$$ 0 0
$$769$$ 31.0000 1.11789 0.558944 0.829205i $$-0.311207\pi$$
0.558944 + 0.829205i $$0.311207\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −5.00000 −0.179954
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 1.00000 0.0358979
$$777$$ 0 0
$$778$$ −21.0000 −0.752886
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 12.0000 0.428298
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 9.00000 0.320612
$$789$$ 0 0
$$790$$ −24.0000 −0.853882
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −32.0000 −1.13635
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ −7.00000 −0.248108
$$797$$ −39.0000 −1.38145 −0.690725 0.723117i $$-0.742709\pi$$
−0.690725 + 0.723117i $$0.742709\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −4.00000 −0.141421
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −18.0000 −0.634417
$$806$$ −20.0000 −0.704470
$$807$$ 0 0
$$808$$ 3.00000 0.105540
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 6.00000 0.210559
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −60.0000 −2.10171
$$816$$ 0 0
$$817$$ −20.0000 −0.699711
$$818$$ 23.0000 0.804176
$$819$$ 0 0
$$820$$ 18.0000 0.628587
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ −9.00000 −0.312395
$$831$$ 0 0
$$832$$ 4.00000 0.138675
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 18.0000 0.622916
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −8.00000 −0.275698
$$843$$ 0 0
$$844$$ 22.0000 0.757271
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −9.00000 −0.309061
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ −30.0000 −1.02299
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 0 0
$$865$$ −45.0000 −1.53005
$$866$$ −29.0000 −0.985460
$$867$$ 0 0
$$868$$ 5.00000 0.169711
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 56.0000 1.89749
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ −19.0000 −0.641219
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 7.00000 0.234772
$$890$$ 54.0000 1.81008
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ −27.0000 −0.902510
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 48.0000 1.59557
$$906$$ 0 0
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 12.0000 0.397796
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ −15.0000 −0.495344
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 18.0000 0.593442
$$921$$ 0 0
$$922$$ 21.0000 0.691598
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 13.0000 0.427207
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ 12.0000 0.393707 0.196854 0.980433i $$-0.436928\pi$$
0.196854 + 0.980433i $$0.436928\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 18.0000 0.589610
$$933$$ 0 0
$$934$$ −27.0000 −0.883467
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −11.0000 −0.359354 −0.179677 0.983726i $$-0.557505\pi$$
−0.179677 + 0.983726i $$0.557505\pi$$
$$938$$ −14.0000 −0.457116
$$939$$ 0 0
$$940$$ 18.0000 0.587095
$$941$$ −33.0000 −1.07577 −0.537885 0.843018i $$-0.680776\pi$$
−0.537885 + 0.843018i $$0.680776\pi$$
$$942$$ 0 0
$$943$$ −36.0000 −1.17232
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −51.0000 −1.65728 −0.828639 0.559784i $$-0.810884\pi$$
−0.828639 + 0.559784i $$0.810884\pi$$
$$948$$ 0 0
$$949$$ 28.0000 0.908918
$$950$$ 8.00000 0.259554
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ −36.0000 −1.16493
$$956$$ 30.0000 0.970269
$$957$$ 0 0
$$958$$ −6.00000 −0.193851
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −8.00000 −0.257930
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 15.0000 0.482867
$$966$$ 0 0
$$967$$ −23.0000 −0.739630 −0.369815 0.929105i $$-0.620579\pi$$
−0.369815 + 0.929105i $$0.620579\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −3.00000 −0.0963242
$$971$$ 27.0000 0.866471 0.433236 0.901281i $$-0.357372\pi$$
0.433236 + 0.901281i $$0.357372\pi$$
$$972$$ 0 0
$$973$$ 4.00000 0.128234
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 18.0000 0.574989
$$981$$ 0 0
$$982$$ −39.0000 −1.24454
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ −27.0000 −0.860292
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 60.0000 1.90789
$$990$$ 0 0
$$991$$ 47.0000 1.49300 0.746502 0.665383i $$-0.231732\pi$$
0.746502 + 0.665383i $$0.231732\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 21.0000 0.665745
$$996$$ 0 0
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ 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 6534.2.a.b.1.1 1
3.2 odd 2 6534.2.a.bc.1.1 1
11.10 odd 2 54.2.a.b.1.1 yes 1
33.32 even 2 54.2.a.a.1.1 1
44.43 even 2 432.2.a.b.1.1 1
55.32 even 4 1350.2.c.k.649.2 2
55.43 even 4 1350.2.c.k.649.1 2
55.54 odd 2 1350.2.a.h.1.1 1
77.76 even 2 2646.2.a.bd.1.1 1
88.21 odd 2 1728.2.a.y.1.1 1
88.43 even 2 1728.2.a.z.1.1 1
99.32 even 6 162.2.c.c.55.1 2
99.43 odd 6 162.2.c.b.109.1 2
99.65 even 6 162.2.c.c.109.1 2
99.76 odd 6 162.2.c.b.55.1 2
132.131 odd 2 432.2.a.g.1.1 1
143.142 odd 2 9126.2.a.r.1.1 1
165.32 odd 4 1350.2.c.b.649.1 2
165.98 odd 4 1350.2.c.b.649.2 2
165.164 even 2 1350.2.a.r.1.1 1
231.230 odd 2 2646.2.a.a.1.1 1
264.131 odd 2 1728.2.a.d.1.1 1
264.197 even 2 1728.2.a.c.1.1 1
396.43 even 6 1296.2.i.o.433.1 2
396.131 odd 6 1296.2.i.c.865.1 2
396.175 even 6 1296.2.i.o.865.1 2
396.263 odd 6 1296.2.i.c.433.1 2
429.428 even 2 9126.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 1 33.32 even 2
54.2.a.b.1.1 yes 1 11.10 odd 2
162.2.c.b.55.1 2 99.76 odd 6
162.2.c.b.109.1 2 99.43 odd 6
162.2.c.c.55.1 2 99.32 even 6
162.2.c.c.109.1 2 99.65 even 6
432.2.a.b.1.1 1 44.43 even 2
432.2.a.g.1.1 1 132.131 odd 2
1296.2.i.c.433.1 2 396.263 odd 6
1296.2.i.c.865.1 2 396.131 odd 6
1296.2.i.o.433.1 2 396.43 even 6
1296.2.i.o.865.1 2 396.175 even 6
1350.2.a.h.1.1 1 55.54 odd 2
1350.2.a.r.1.1 1 165.164 even 2
1350.2.c.b.649.1 2 165.32 odd 4
1350.2.c.b.649.2 2 165.98 odd 4
1350.2.c.k.649.1 2 55.43 even 4
1350.2.c.k.649.2 2 55.32 even 4
1728.2.a.c.1.1 1 264.197 even 2
1728.2.a.d.1.1 1 264.131 odd 2
1728.2.a.y.1.1 1 88.21 odd 2
1728.2.a.z.1.1 1 88.43 even 2
2646.2.a.a.1.1 1 231.230 odd 2
2646.2.a.bd.1.1 1 77.76 even 2
6534.2.a.b.1.1 1 1.1 even 1 trivial
6534.2.a.bc.1.1 1 3.2 odd 2
9126.2.a.r.1.1 1 143.142 odd 2
9126.2.a.u.1.1 1 429.428 even 2