Properties

 Label 660.2.a.e.1.2 Level $660$ Weight $2$ Character 660.1 Self dual yes Analytic conductor $5.270$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$660 = 2^{2} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 660.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$5.27012653340$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 660.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +4.60555 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +4.60555 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.60555 q^{13} -1.00000 q^{15} +6.60555 q^{17} -7.21110 q^{19} -4.60555 q^{21} +1.00000 q^{25} -1.00000 q^{27} +8.00000 q^{29} +9.21110 q^{31} -1.00000 q^{33} +4.60555 q^{35} -3.21110 q^{37} +4.60555 q^{39} +8.00000 q^{41} -3.39445 q^{43} +1.00000 q^{45} -5.21110 q^{47} +14.2111 q^{49} -6.60555 q^{51} +2.00000 q^{53} +1.00000 q^{55} +7.21110 q^{57} +8.00000 q^{59} +7.21110 q^{61} +4.60555 q^{63} -4.60555 q^{65} -4.00000 q^{67} -14.4222 q^{71} +0.605551 q^{73} -1.00000 q^{75} +4.60555 q^{77} -11.2111 q^{79} +1.00000 q^{81} -10.6056 q^{83} +6.60555 q^{85} -8.00000 q^{87} +6.00000 q^{89} -21.2111 q^{91} -9.21110 q^{93} -7.21110 q^{95} -12.4222 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 16 q^{29} + 4 q^{31} - 2 q^{33} + 2 q^{35} + 8 q^{37} + 2 q^{39} + 16 q^{41} - 14 q^{43} + 2 q^{45} + 4 q^{47} + 14 q^{49} - 6 q^{51} + 4 q^{53} + 2 q^{55} + 16 q^{59} + 2 q^{63} - 2 q^{65} - 8 q^{67} - 6 q^{73} - 2 q^{75} + 2 q^{77} - 8 q^{79} + 2 q^{81} - 14 q^{83} + 6 q^{85} - 16 q^{87} + 12 q^{89} - 28 q^{91} - 4 q^{93} + 4 q^{97} + 2 q^{99} + O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −4.60555 −1.27735 −0.638675 0.769477i $$-0.720517\pi$$
−0.638675 + 0.769477i $$0.720517\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 6.60555 1.60208 0.801041 0.598610i $$-0.204280\pi$$
0.801041 + 0.598610i $$0.204280\pi$$
$$18$$ 0 0
$$19$$ −7.21110 −1.65434 −0.827170 0.561951i $$-0.810051\pi$$
−0.827170 + 0.561951i $$0.810051\pi$$
$$20$$ 0 0
$$21$$ −4.60555 −1.00501
$$22$$ 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
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 9.21110 1.65436 0.827181 0.561935i $$-0.189943\pi$$
0.827181 + 0.561935i $$0.189943\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −0.174078
$$34$$ 0 0
$$35$$ 4.60555 0.778480
$$36$$ 0 0
$$37$$ −3.21110 −0.527902 −0.263951 0.964536i $$-0.585026\pi$$
−0.263951 + 0.964536i $$0.585026\pi$$
$$38$$ 0 0
$$39$$ 4.60555 0.737478
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ −3.39445 −0.517649 −0.258824 0.965924i $$-0.583335\pi$$
−0.258824 + 0.965924i $$0.583335\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −5.21110 −0.760117 −0.380059 0.924962i $$-0.624096\pi$$
−0.380059 + 0.924962i $$0.624096\pi$$
$$48$$ 0 0
$$49$$ 14.2111 2.03016
$$50$$ 0 0
$$51$$ −6.60555 −0.924962
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 7.21110 0.955134
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 7.21110 0.923287 0.461644 0.887066i $$-0.347260\pi$$
0.461644 + 0.887066i $$0.347260\pi$$
$$62$$ 0 0
$$63$$ 4.60555 0.580245
$$64$$ 0 0
$$65$$ −4.60555 −0.571248
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.4222 −1.71160 −0.855800 0.517306i $$-0.826935\pi$$
−0.855800 + 0.517306i $$0.826935\pi$$
$$72$$ 0 0
$$73$$ 0.605551 0.0708744 0.0354372 0.999372i $$-0.488718\pi$$
0.0354372 + 0.999372i $$0.488718\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 4.60555 0.524851
$$78$$ 0 0
$$79$$ −11.2111 −1.26135 −0.630674 0.776048i $$-0.717221\pi$$
−0.630674 + 0.776048i $$0.717221\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −10.6056 −1.16411 −0.582055 0.813149i $$-0.697751\pi$$
−0.582055 + 0.813149i $$0.697751\pi$$
$$84$$ 0 0
$$85$$ 6.60555 0.716473
$$86$$ 0 0
$$87$$ −8.00000 −0.857690
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −21.2111 −2.22353
$$92$$ 0 0
$$93$$ −9.21110 −0.955147
$$94$$ 0 0
$$95$$ −7.21110 −0.739844
$$96$$ 0 0
$$97$$ −12.4222 −1.26128 −0.630642 0.776074i $$-0.717208\pi$$
−0.630642 + 0.776074i $$0.717208\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 5.21110 0.518524 0.259262 0.965807i $$-0.416521\pi$$
0.259262 + 0.965807i $$0.416521\pi$$
$$102$$ 0 0
$$103$$ 1.21110 0.119333 0.0596667 0.998218i $$-0.480996\pi$$
0.0596667 + 0.998218i $$0.480996\pi$$
$$104$$ 0 0
$$105$$ −4.60555 −0.449456
$$106$$ 0 0
$$107$$ 15.8167 1.52905 0.764527 0.644592i $$-0.222973\pi$$
0.764527 + 0.644592i $$0.222973\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 3.21110 0.304784
$$112$$ 0 0
$$113$$ −0.788897 −0.0742132 −0.0371066 0.999311i $$-0.511814\pi$$
−0.0371066 + 0.999311i $$0.511814\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4.60555 −0.425783
$$118$$ 0 0
$$119$$ 30.4222 2.78880
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.60555 −0.763619 −0.381810 0.924241i $$-0.624699\pi$$
−0.381810 + 0.924241i $$0.624699\pi$$
$$128$$ 0 0
$$129$$ 3.39445 0.298865
$$130$$ 0 0
$$131$$ −6.78890 −0.593149 −0.296574 0.955010i $$-0.595844\pi$$
−0.296574 + 0.955010i $$0.595844\pi$$
$$132$$ 0 0
$$133$$ −33.2111 −2.87977
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 16.4222 1.39291 0.696457 0.717599i $$-0.254759\pi$$
0.696457 + 0.717599i $$0.254759\pi$$
$$140$$ 0 0
$$141$$ 5.21110 0.438854
$$142$$ 0 0
$$143$$ −4.60555 −0.385136
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ −14.2111 −1.17211
$$148$$ 0 0
$$149$$ −9.21110 −0.754603 −0.377301 0.926090i $$-0.623148\pi$$
−0.377301 + 0.926090i $$0.623148\pi$$
$$150$$ 0 0
$$151$$ −19.2111 −1.56338 −0.781689 0.623669i $$-0.785641\pi$$
−0.781689 + 0.623669i $$0.785641\pi$$
$$152$$ 0 0
$$153$$ 6.60555 0.534027
$$154$$ 0 0
$$155$$ 9.21110 0.739854
$$156$$ 0 0
$$157$$ 3.21110 0.256274 0.128137 0.991756i $$-0.459100\pi$$
0.128137 + 0.991756i $$0.459100\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −5.21110 −0.408165 −0.204083 0.978954i $$-0.565421\pi$$
−0.204083 + 0.978954i $$0.565421\pi$$
$$164$$ 0 0
$$165$$ −1.00000 −0.0778499
$$166$$ 0 0
$$167$$ −19.8167 −1.53346 −0.766729 0.641970i $$-0.778117\pi$$
−0.766729 + 0.641970i $$0.778117\pi$$
$$168$$ 0 0
$$169$$ 8.21110 0.631623
$$170$$ 0 0
$$171$$ −7.21110 −0.551447
$$172$$ 0 0
$$173$$ 0.183346 0.0139396 0.00696978 0.999976i $$-0.497781\pi$$
0.00696978 + 0.999976i $$0.497781\pi$$
$$174$$ 0 0
$$175$$ 4.60555 0.348147
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 23.2111 1.72527 0.862634 0.505829i $$-0.168813\pi$$
0.862634 + 0.505829i $$0.168813\pi$$
$$182$$ 0 0
$$183$$ −7.21110 −0.533060
$$184$$ 0 0
$$185$$ −3.21110 −0.236085
$$186$$ 0 0
$$187$$ 6.60555 0.483046
$$188$$ 0 0
$$189$$ −4.60555 −0.335005
$$190$$ 0 0
$$191$$ 18.4222 1.33298 0.666492 0.745512i $$-0.267795\pi$$
0.666492 + 0.745512i $$0.267795\pi$$
$$192$$ 0 0
$$193$$ −21.8167 −1.57040 −0.785199 0.619244i $$-0.787439\pi$$
−0.785199 + 0.619244i $$0.787439\pi$$
$$194$$ 0 0
$$195$$ 4.60555 0.329810
$$196$$ 0 0
$$197$$ −13.3944 −0.954315 −0.477157 0.878818i $$-0.658333\pi$$
−0.477157 + 0.878818i $$0.658333\pi$$
$$198$$ 0 0
$$199$$ −10.4222 −0.738811 −0.369405 0.929268i $$-0.620439\pi$$
−0.369405 + 0.929268i $$0.620439\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 36.8444 2.58597
$$204$$ 0 0
$$205$$ 8.00000 0.558744
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −7.21110 −0.498802
$$210$$ 0 0
$$211$$ −23.2111 −1.59792 −0.798959 0.601385i $$-0.794616\pi$$
−0.798959 + 0.601385i $$0.794616\pi$$
$$212$$ 0 0
$$213$$ 14.4222 0.988193
$$214$$ 0 0
$$215$$ −3.39445 −0.231499
$$216$$ 0 0
$$217$$ 42.4222 2.87981
$$218$$ 0 0
$$219$$ −0.605551 −0.0409194
$$220$$ 0 0
$$221$$ −30.4222 −2.04642
$$222$$ 0 0
$$223$$ 9.21110 0.616821 0.308411 0.951253i $$-0.400203\pi$$
0.308411 + 0.951253i $$0.400203\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −15.8167 −1.04979 −0.524894 0.851168i $$-0.675895\pi$$
−0.524894 + 0.851168i $$0.675895\pi$$
$$228$$ 0 0
$$229$$ −8.42221 −0.556555 −0.278277 0.960501i $$-0.589763\pi$$
−0.278277 + 0.960501i $$0.589763\pi$$
$$230$$ 0 0
$$231$$ −4.60555 −0.303023
$$232$$ 0 0
$$233$$ −10.6056 −0.694793 −0.347396 0.937718i $$-0.612934\pi$$
−0.347396 + 0.937718i $$0.612934\pi$$
$$234$$ 0 0
$$235$$ −5.21110 −0.339935
$$236$$ 0 0
$$237$$ 11.2111 0.728239
$$238$$ 0 0
$$239$$ 6.78890 0.439137 0.219569 0.975597i $$-0.429535\pi$$
0.219569 + 0.975597i $$0.429535\pi$$
$$240$$ 0 0
$$241$$ −15.2111 −0.979833 −0.489917 0.871769i $$-0.662973\pi$$
−0.489917 + 0.871769i $$0.662973\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 14.2111 0.907914
$$246$$ 0 0
$$247$$ 33.2111 2.11317
$$248$$ 0 0
$$249$$ 10.6056 0.672100
$$250$$ 0 0
$$251$$ −10.4222 −0.657844 −0.328922 0.944357i $$-0.606685\pi$$
−0.328922 + 0.944357i $$0.606685\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −6.60555 −0.413656
$$256$$ 0 0
$$257$$ −23.2111 −1.44787 −0.723934 0.689869i $$-0.757668\pi$$
−0.723934 + 0.689869i $$0.757668\pi$$
$$258$$ 0 0
$$259$$ −14.7889 −0.918937
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 0 0
$$263$$ −17.0278 −1.04998 −0.524988 0.851109i $$-0.675930\pi$$
−0.524988 + 0.851109i $$0.675930\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ −20.4222 −1.24516 −0.622582 0.782555i $$-0.713916\pi$$
−0.622582 + 0.782555i $$0.713916\pi$$
$$270$$ 0 0
$$271$$ −4.78890 −0.290905 −0.145452 0.989365i $$-0.546464\pi$$
−0.145452 + 0.989365i $$0.546464\pi$$
$$272$$ 0 0
$$273$$ 21.2111 1.28375
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 32.2389 1.93705 0.968523 0.248925i $$-0.0800774\pi$$
0.968523 + 0.248925i $$0.0800774\pi$$
$$278$$ 0 0
$$279$$ 9.21110 0.551454
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 4.60555 0.273772 0.136886 0.990587i $$-0.456291\pi$$
0.136886 + 0.990587i $$0.456291\pi$$
$$284$$ 0 0
$$285$$ 7.21110 0.427149
$$286$$ 0 0
$$287$$ 36.8444 2.17486
$$288$$ 0 0
$$289$$ 26.6333 1.56667
$$290$$ 0 0
$$291$$ 12.4222 0.728203
$$292$$ 0 0
$$293$$ 19.8167 1.15770 0.578851 0.815434i $$-0.303501\pi$$
0.578851 + 0.815434i $$0.303501\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ −1.00000 −0.0580259
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −15.6333 −0.901089
$$302$$ 0 0
$$303$$ −5.21110 −0.299370
$$304$$ 0 0
$$305$$ 7.21110 0.412907
$$306$$ 0 0
$$307$$ 8.60555 0.491145 0.245572 0.969378i $$-0.421024\pi$$
0.245572 + 0.969378i $$0.421024\pi$$
$$308$$ 0 0
$$309$$ −1.21110 −0.0688972
$$310$$ 0 0
$$311$$ 30.4222 1.72508 0.862542 0.505985i $$-0.168871\pi$$
0.862542 + 0.505985i $$0.168871\pi$$
$$312$$ 0 0
$$313$$ 8.78890 0.496778 0.248389 0.968660i $$-0.420099\pi$$
0.248389 + 0.968660i $$0.420099\pi$$
$$314$$ 0 0
$$315$$ 4.60555 0.259493
$$316$$ 0 0
$$317$$ 11.2111 0.629678 0.314839 0.949145i $$-0.398049\pi$$
0.314839 + 0.949145i $$0.398049\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −15.8167 −0.882800
$$322$$ 0 0
$$323$$ −47.6333 −2.65039
$$324$$ 0 0
$$325$$ −4.60555 −0.255470
$$326$$ 0 0
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −6.78890 −0.373152 −0.186576 0.982441i $$-0.559739\pi$$
−0.186576 + 0.982441i $$0.559739\pi$$
$$332$$ 0 0
$$333$$ −3.21110 −0.175967
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −28.2389 −1.53827 −0.769134 0.639087i $$-0.779312\pi$$
−0.769134 + 0.639087i $$0.779312\pi$$
$$338$$ 0 0
$$339$$ 0.788897 0.0428470
$$340$$ 0 0
$$341$$ 9.21110 0.498809
$$342$$ 0 0
$$343$$ 33.2111 1.79323
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −29.0278 −1.55829 −0.779146 0.626843i $$-0.784347\pi$$
−0.779146 + 0.626843i $$0.784347\pi$$
$$348$$ 0 0
$$349$$ −8.78890 −0.470459 −0.235229 0.971940i $$-0.575584\pi$$
−0.235229 + 0.971940i $$0.575584\pi$$
$$350$$ 0 0
$$351$$ 4.60555 0.245826
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ −14.4222 −0.765451
$$356$$ 0 0
$$357$$ −30.4222 −1.61011
$$358$$ 0 0
$$359$$ 18.7889 0.991640 0.495820 0.868425i $$-0.334868\pi$$
0.495820 + 0.868425i $$0.334868\pi$$
$$360$$ 0 0
$$361$$ 33.0000 1.73684
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 0.605551 0.0316960
$$366$$ 0 0
$$367$$ 6.78890 0.354378 0.177189 0.984177i $$-0.443300\pi$$
0.177189 + 0.984177i $$0.443300\pi$$
$$368$$ 0 0
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 9.21110 0.478217
$$372$$ 0 0
$$373$$ 4.60555 0.238466 0.119233 0.992866i $$-0.461956\pi$$
0.119233 + 0.992866i $$0.461956\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −36.8444 −1.89758
$$378$$ 0 0
$$379$$ −18.4222 −0.946285 −0.473143 0.880986i $$-0.656880\pi$$
−0.473143 + 0.880986i $$0.656880\pi$$
$$380$$ 0 0
$$381$$ 8.60555 0.440876
$$382$$ 0 0
$$383$$ 18.7889 0.960068 0.480034 0.877250i $$-0.340624\pi$$
0.480034 + 0.877250i $$0.340624\pi$$
$$384$$ 0 0
$$385$$ 4.60555 0.234721
$$386$$ 0 0
$$387$$ −3.39445 −0.172550
$$388$$ 0 0
$$389$$ 30.8444 1.56387 0.781937 0.623358i $$-0.214232\pi$$
0.781937 + 0.623358i $$0.214232\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 6.78890 0.342455
$$394$$ 0 0
$$395$$ −11.2111 −0.564092
$$396$$ 0 0
$$397$$ 28.4222 1.42647 0.713235 0.700925i $$-0.247229\pi$$
0.713235 + 0.700925i $$0.247229\pi$$
$$398$$ 0 0
$$399$$ 33.2111 1.66263
$$400$$ 0 0
$$401$$ 20.4222 1.01984 0.509918 0.860223i $$-0.329676\pi$$
0.509918 + 0.860223i $$0.329676\pi$$
$$402$$ 0 0
$$403$$ −42.4222 −2.11320
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −3.21110 −0.159168
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 0 0
$$413$$ 36.8444 1.81299
$$414$$ 0 0
$$415$$ −10.6056 −0.520606
$$416$$ 0 0
$$417$$ −16.4222 −0.804199
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −19.2111 −0.936292 −0.468146 0.883651i $$-0.655078\pi$$
−0.468146 + 0.883651i $$0.655078\pi$$
$$422$$ 0 0
$$423$$ −5.21110 −0.253372
$$424$$ 0 0
$$425$$ 6.60555 0.320416
$$426$$ 0 0
$$427$$ 33.2111 1.60720
$$428$$ 0 0
$$429$$ 4.60555 0.222358
$$430$$ 0 0
$$431$$ −1.57779 −0.0759997 −0.0379999 0.999278i $$-0.512099\pi$$
−0.0379999 + 0.999278i $$0.512099\pi$$
$$432$$ 0 0
$$433$$ 8.42221 0.404745 0.202373 0.979309i $$-0.435135\pi$$
0.202373 + 0.979309i $$0.435135\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 14.2111 0.676719
$$442$$ 0 0
$$443$$ 25.2111 1.19782 0.598908 0.800818i $$-0.295602\pi$$
0.598908 + 0.800818i $$0.295602\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 9.21110 0.435670
$$448$$ 0 0
$$449$$ 18.8444 0.889323 0.444661 0.895699i $$-0.353324\pi$$
0.444661 + 0.895699i $$0.353324\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ 19.2111 0.902616
$$454$$ 0 0
$$455$$ −21.2111 −0.994392
$$456$$ 0 0
$$457$$ −4.60555 −0.215439 −0.107719 0.994181i $$-0.534355\pi$$
−0.107719 + 0.994181i $$0.534355\pi$$
$$458$$ 0 0
$$459$$ −6.60555 −0.308321
$$460$$ 0 0
$$461$$ −9.21110 −0.429004 −0.214502 0.976724i $$-0.568813\pi$$
−0.214502 + 0.976724i $$0.568813\pi$$
$$462$$ 0 0
$$463$$ 1.21110 0.0562847 0.0281424 0.999604i $$-0.491041\pi$$
0.0281424 + 0.999604i $$0.491041\pi$$
$$464$$ 0 0
$$465$$ −9.21110 −0.427155
$$466$$ 0 0
$$467$$ −4.00000 −0.185098 −0.0925490 0.995708i $$-0.529501\pi$$
−0.0925490 + 0.995708i $$0.529501\pi$$
$$468$$ 0 0
$$469$$ −18.4222 −0.850658
$$470$$ 0 0
$$471$$ −3.21110 −0.147960
$$472$$ 0 0
$$473$$ −3.39445 −0.156077
$$474$$ 0 0
$$475$$ −7.21110 −0.330868
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −14.7889 −0.675722 −0.337861 0.941196i $$-0.609703\pi$$
−0.337861 + 0.941196i $$0.609703\pi$$
$$480$$ 0 0
$$481$$ 14.7889 0.674316
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12.4222 −0.564063
$$486$$ 0 0
$$487$$ 5.57779 0.252754 0.126377 0.991982i $$-0.459665\pi$$
0.126377 + 0.991982i $$0.459665\pi$$
$$488$$ 0 0
$$489$$ 5.21110 0.235654
$$490$$ 0 0
$$491$$ −5.57779 −0.251722 −0.125861 0.992048i $$-0.540169\pi$$
−0.125861 + 0.992048i $$0.540169\pi$$
$$492$$ 0 0
$$493$$ 52.8444 2.37999
$$494$$ 0 0
$$495$$ 1.00000 0.0449467
$$496$$ 0 0
$$497$$ −66.4222 −2.97944
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 19.8167 0.885343
$$502$$ 0 0
$$503$$ 4.18335 0.186526 0.0932631 0.995641i $$-0.470270\pi$$
0.0932631 + 0.995641i $$0.470270\pi$$
$$504$$ 0 0
$$505$$ 5.21110 0.231891
$$506$$ 0 0
$$507$$ −8.21110 −0.364668
$$508$$ 0 0
$$509$$ −28.4222 −1.25979 −0.629896 0.776679i $$-0.716903\pi$$
−0.629896 + 0.776679i $$0.716903\pi$$
$$510$$ 0 0
$$511$$ 2.78890 0.123374
$$512$$ 0 0
$$513$$ 7.21110 0.318378
$$514$$ 0 0
$$515$$ 1.21110 0.0533676
$$516$$ 0 0
$$517$$ −5.21110 −0.229184
$$518$$ 0 0
$$519$$ −0.183346 −0.00804800
$$520$$ 0 0
$$521$$ −8.42221 −0.368984 −0.184492 0.982834i $$-0.559064\pi$$
−0.184492 + 0.982834i $$0.559064\pi$$
$$522$$ 0 0
$$523$$ 28.2389 1.23480 0.617400 0.786650i $$-0.288186\pi$$
0.617400 + 0.786650i $$0.288186\pi$$
$$524$$ 0 0
$$525$$ −4.60555 −0.201003
$$526$$ 0 0
$$527$$ 60.8444 2.65042
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ −36.8444 −1.59591
$$534$$ 0 0
$$535$$ 15.8167 0.683814
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 14.2111 0.612116
$$540$$ 0 0
$$541$$ 3.57779 0.153821 0.0769107 0.997038i $$-0.475494\pi$$
0.0769107 + 0.997038i $$0.475494\pi$$
$$542$$ 0 0
$$543$$ −23.2111 −0.996084
$$544$$ 0 0
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ −14.1833 −0.606436 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$548$$ 0 0
$$549$$ 7.21110 0.307762
$$550$$ 0 0
$$551$$ −57.6888 −2.45763
$$552$$ 0 0
$$553$$ −51.6333 −2.19567
$$554$$ 0 0
$$555$$ 3.21110 0.136304
$$556$$ 0 0
$$557$$ −6.97224 −0.295423 −0.147712 0.989030i $$-0.547191\pi$$
−0.147712 + 0.989030i $$0.547191\pi$$
$$558$$ 0 0
$$559$$ 15.6333 0.661218
$$560$$ 0 0
$$561$$ −6.60555 −0.278887
$$562$$ 0 0
$$563$$ −19.8167 −0.835172 −0.417586 0.908637i $$-0.637124\pi$$
−0.417586 + 0.908637i $$0.637124\pi$$
$$564$$ 0 0
$$565$$ −0.788897 −0.0331892
$$566$$ 0 0
$$567$$ 4.60555 0.193415
$$568$$ 0 0
$$569$$ −27.6333 −1.15845 −0.579224 0.815168i $$-0.696644\pi$$
−0.579224 + 0.815168i $$0.696644\pi$$
$$570$$ 0 0
$$571$$ 12.4222 0.519853 0.259927 0.965628i $$-0.416302\pi$$
0.259927 + 0.965628i $$0.416302\pi$$
$$572$$ 0 0
$$573$$ −18.4222 −0.769599
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 28.7889 1.19850 0.599249 0.800563i $$-0.295466\pi$$
0.599249 + 0.800563i $$0.295466\pi$$
$$578$$ 0 0
$$579$$ 21.8167 0.906669
$$580$$ 0 0
$$581$$ −48.8444 −2.02641
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ 0 0
$$585$$ −4.60555 −0.190416
$$586$$ 0 0
$$587$$ 6.42221 0.265073 0.132536 0.991178i $$-0.457688\pi$$
0.132536 + 0.991178i $$0.457688\pi$$
$$588$$ 0 0
$$589$$ −66.4222 −2.73688
$$590$$ 0 0
$$591$$ 13.3944 0.550974
$$592$$ 0 0
$$593$$ 13.0278 0.534986 0.267493 0.963560i $$-0.413805\pi$$
0.267493 + 0.963560i $$0.413805\pi$$
$$594$$ 0 0
$$595$$ 30.4222 1.24719
$$596$$ 0 0
$$597$$ 10.4222 0.426552
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −8.78890 −0.358507 −0.179253 0.983803i $$-0.557368\pi$$
−0.179253 + 0.983803i $$0.557368\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 4.97224 0.201817 0.100909 0.994896i $$-0.467825\pi$$
0.100909 + 0.994896i $$0.467825\pi$$
$$608$$ 0 0
$$609$$ −36.8444 −1.49301
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ −21.8167 −0.881166 −0.440583 0.897712i $$-0.645228\pi$$
−0.440583 + 0.897712i $$0.645228\pi$$
$$614$$ 0 0
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ −24.7889 −0.997963 −0.498982 0.866613i $$-0.666293\pi$$
−0.498982 + 0.866613i $$0.666293\pi$$
$$618$$ 0 0
$$619$$ −19.6333 −0.789129 −0.394565 0.918868i $$-0.629105\pi$$
−0.394565 + 0.918868i $$0.629105\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 27.6333 1.10711
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 7.21110 0.287984
$$628$$ 0 0
$$629$$ −21.2111 −0.845742
$$630$$ 0 0
$$631$$ 36.8444 1.46675 0.733376 0.679823i $$-0.237943\pi$$
0.733376 + 0.679823i $$0.237943\pi$$
$$632$$ 0 0
$$633$$ 23.2111 0.922559
$$634$$ 0 0
$$635$$ −8.60555 −0.341501
$$636$$ 0 0
$$637$$ −65.4500 −2.59322
$$638$$ 0 0
$$639$$ −14.4222 −0.570534
$$640$$ 0 0
$$641$$ 18.8444 0.744309 0.372155 0.928171i $$-0.378619\pi$$
0.372155 + 0.928171i $$0.378619\pi$$
$$642$$ 0 0
$$643$$ 0.366692 0.0144609 0.00723047 0.999974i $$-0.497698\pi$$
0.00723047 + 0.999974i $$0.497698\pi$$
$$644$$ 0 0
$$645$$ 3.39445 0.133656
$$646$$ 0 0
$$647$$ 45.2111 1.77743 0.888716 0.458458i $$-0.151598\pi$$
0.888716 + 0.458458i $$0.151598\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ −42.4222 −1.66266
$$652$$ 0 0
$$653$$ 4.78890 0.187404 0.0937020 0.995600i $$-0.470130\pi$$
0.0937020 + 0.995600i $$0.470130\pi$$
$$654$$ 0 0
$$655$$ −6.78890 −0.265264
$$656$$ 0 0
$$657$$ 0.605551 0.0236248
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 45.2666 1.76067 0.880334 0.474355i $$-0.157319\pi$$
0.880334 + 0.474355i $$0.157319\pi$$
$$662$$ 0 0
$$663$$ 30.4222 1.18150
$$664$$ 0 0
$$665$$ −33.2111 −1.28787
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −9.21110 −0.356122
$$670$$ 0 0
$$671$$ 7.21110 0.278382
$$672$$ 0 0
$$673$$ −11.3944 −0.439224 −0.219612 0.975587i $$-0.570479\pi$$
−0.219612 + 0.975587i $$0.570479\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 5.39445 0.207326 0.103663 0.994613i $$-0.466944\pi$$
0.103663 + 0.994613i $$0.466944\pi$$
$$678$$ 0 0
$$679$$ −57.2111 −2.19556
$$680$$ 0 0
$$681$$ 15.8167 0.606095
$$682$$ 0 0
$$683$$ −30.7889 −1.17810 −0.589052 0.808095i $$-0.700499\pi$$
−0.589052 + 0.808095i $$0.700499\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ 8.42221 0.321327
$$688$$ 0 0
$$689$$ −9.21110 −0.350915
$$690$$ 0 0
$$691$$ −13.5778 −0.516524 −0.258262 0.966075i $$-0.583150\pi$$
−0.258262 + 0.966075i $$0.583150\pi$$
$$692$$ 0 0
$$693$$ 4.60555 0.174950
$$694$$ 0 0
$$695$$ 16.4222 0.622930
$$696$$ 0 0
$$697$$ 52.8444 2.00162
$$698$$ 0 0
$$699$$ 10.6056 0.401139
$$700$$ 0 0
$$701$$ 46.4222 1.75334 0.876671 0.481090i $$-0.159759\pi$$
0.876671 + 0.481090i $$0.159759\pi$$
$$702$$ 0 0
$$703$$ 23.1556 0.873330
$$704$$ 0 0
$$705$$ 5.21110 0.196261
$$706$$ 0 0
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 5.63331 0.211563 0.105782 0.994389i $$-0.466266\pi$$
0.105782 + 0.994389i $$0.466266\pi$$
$$710$$ 0 0
$$711$$ −11.2111 −0.420449
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −4.60555 −0.172238
$$716$$ 0 0
$$717$$ −6.78890 −0.253536
$$718$$ 0 0
$$719$$ −26.4222 −0.985382 −0.492691 0.870204i $$-0.663987\pi$$
−0.492691 + 0.870204i $$0.663987\pi$$
$$720$$ 0 0
$$721$$ 5.57779 0.207728
$$722$$ 0 0
$$723$$ 15.2111 0.565707
$$724$$ 0 0
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ −28.8444 −1.06978 −0.534890 0.844922i $$-0.679647\pi$$
−0.534890 + 0.844922i $$0.679647\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −22.4222 −0.829315
$$732$$ 0 0
$$733$$ −7.39445 −0.273120 −0.136560 0.990632i $$-0.543605\pi$$
−0.136560 + 0.990632i $$0.543605\pi$$
$$734$$ 0 0
$$735$$ −14.2111 −0.524184
$$736$$ 0 0
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 41.2666 1.51802 0.759008 0.651081i $$-0.225684\pi$$
0.759008 + 0.651081i $$0.225684\pi$$
$$740$$ 0 0
$$741$$ −33.2111 −1.22004
$$742$$ 0 0
$$743$$ 37.3944 1.37187 0.685935 0.727663i $$-0.259394\pi$$
0.685935 + 0.727663i $$0.259394\pi$$
$$744$$ 0 0
$$745$$ −9.21110 −0.337469
$$746$$ 0 0
$$747$$ −10.6056 −0.388037
$$748$$ 0 0
$$749$$ 72.8444 2.66168
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 10.4222 0.379806
$$754$$ 0 0
$$755$$ −19.2111 −0.699164
$$756$$ 0 0
$$757$$ −12.7889 −0.464820 −0.232410 0.972618i $$-0.574661\pi$$
−0.232410 + 0.972618i $$0.574661\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 27.6333 1.00171 0.500853 0.865532i $$-0.333020\pi$$
0.500853 + 0.865532i $$0.333020\pi$$
$$762$$ 0 0
$$763$$ −46.0555 −1.66732
$$764$$ 0 0
$$765$$ 6.60555 0.238824
$$766$$ 0 0
$$767$$ −36.8444 −1.33037
$$768$$ 0 0
$$769$$ −43.2111 −1.55823 −0.779116 0.626880i $$-0.784332\pi$$
−0.779116 + 0.626880i $$0.784332\pi$$
$$770$$ 0 0
$$771$$ 23.2111 0.835927
$$772$$ 0 0
$$773$$ −48.0555 −1.72844 −0.864218 0.503117i $$-0.832186\pi$$
−0.864218 + 0.503117i $$0.832186\pi$$
$$774$$ 0 0
$$775$$ 9.21110 0.330873
$$776$$ 0 0
$$777$$ 14.7889 0.530549
$$778$$ 0 0
$$779$$ −57.6888 −2.06692
$$780$$ 0 0
$$781$$ −14.4222 −0.516067
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 3.21110 0.114609
$$786$$ 0 0
$$787$$ 39.0278 1.39119 0.695595 0.718434i $$-0.255141\pi$$
0.695595 + 0.718434i $$0.255141\pi$$
$$788$$ 0 0
$$789$$ 17.0278 0.606204
$$790$$ 0 0
$$791$$ −3.63331 −0.129186
$$792$$ 0 0
$$793$$ −33.2111 −1.17936
$$794$$ 0 0
$$795$$ −2.00000 −0.0709327
$$796$$ 0 0
$$797$$ 40.0555 1.41884 0.709420 0.704786i $$-0.248957\pi$$
0.709420 + 0.704786i $$0.248957\pi$$
$$798$$ 0 0
$$799$$ −34.4222 −1.21777
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 0.605551 0.0213694
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.4222 0.718896
$$808$$ 0 0
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ −7.57779 −0.266092 −0.133046 0.991110i $$-0.542476\pi$$
−0.133046 + 0.991110i $$0.542476\pi$$
$$812$$ 0 0
$$813$$ 4.78890 0.167954
$$814$$ 0 0
$$815$$ −5.21110 −0.182537
$$816$$ 0 0
$$817$$ 24.4777 0.856367
$$818$$ 0 0
$$819$$ −21.2111 −0.741176
$$820$$ 0 0
$$821$$ −12.8444 −0.448273 −0.224137 0.974558i $$-0.571956\pi$$
−0.224137 + 0.974558i $$0.571956\pi$$
$$822$$ 0 0
$$823$$ 41.2111 1.43653 0.718264 0.695770i $$-0.244937\pi$$
0.718264 + 0.695770i $$0.244937\pi$$
$$824$$ 0 0
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ −2.60555 −0.0906039 −0.0453019 0.998973i $$-0.514425\pi$$
−0.0453019 + 0.998973i $$0.514425\pi$$
$$828$$ 0 0
$$829$$ 6.36669 0.221124 0.110562 0.993869i $$-0.464735\pi$$
0.110562 + 0.993869i $$0.464735\pi$$
$$830$$ 0 0
$$831$$ −32.2389 −1.11835
$$832$$ 0 0
$$833$$ 93.8722 3.25248
$$834$$ 0 0
$$835$$ −19.8167 −0.685784
$$836$$ 0 0
$$837$$ −9.21110 −0.318382
$$838$$ 0 0
$$839$$ 30.4222 1.05029 0.525146 0.851012i $$-0.324011\pi$$
0.525146 + 0.851012i $$0.324011\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 8.21110 0.282471
$$846$$ 0 0
$$847$$ 4.60555 0.158249
$$848$$ 0 0
$$849$$ −4.60555 −0.158062
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −40.2389 −1.37775 −0.688876 0.724879i $$-0.741896\pi$$
−0.688876 + 0.724879i $$0.741896\pi$$
$$854$$ 0 0
$$855$$ −7.21110 −0.246615
$$856$$ 0 0
$$857$$ 25.0278 0.854932 0.427466 0.904031i $$-0.359406\pi$$
0.427466 + 0.904031i $$0.359406\pi$$
$$858$$ 0 0
$$859$$ 57.2111 1.95202 0.976009 0.217731i $$-0.0698656\pi$$
0.976009 + 0.217731i $$0.0698656\pi$$
$$860$$ 0 0
$$861$$ −36.8444 −1.25565
$$862$$ 0 0
$$863$$ 39.6333 1.34913 0.674567 0.738214i $$-0.264330\pi$$
0.674567 + 0.738214i $$0.264330\pi$$
$$864$$ 0 0
$$865$$ 0.183346 0.00623396
$$866$$ 0 0
$$867$$ −26.6333 −0.904515
$$868$$ 0 0
$$869$$ −11.2111 −0.380311
$$870$$ 0 0
$$871$$ 18.4222 0.624213
$$872$$ 0 0
$$873$$ −12.4222 −0.420428
$$874$$ 0 0
$$875$$ 4.60555 0.155696
$$876$$ 0 0
$$877$$ −24.2389 −0.818488 −0.409244 0.912425i $$-0.634208\pi$$
−0.409244 + 0.912425i $$0.634208\pi$$
$$878$$ 0 0
$$879$$ −19.8167 −0.668399
$$880$$ 0 0
$$881$$ 26.8444 0.904411 0.452206 0.891914i $$-0.350637\pi$$
0.452206 + 0.891914i $$0.350637\pi$$
$$882$$ 0 0
$$883$$ 6.42221 0.216124 0.108062 0.994144i $$-0.465535\pi$$
0.108062 + 0.994144i $$0.465535\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ 13.0278 0.437429 0.218715 0.975789i $$-0.429814\pi$$
0.218715 + 0.975789i $$0.429814\pi$$
$$888$$ 0 0
$$889$$ −39.6333 −1.32926
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 37.5778 1.25749
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 73.6888 2.45766
$$900$$ 0 0
$$901$$ 13.2111 0.440126
$$902$$ 0 0
$$903$$ 15.6333 0.520244
$$904$$ 0 0
$$905$$ 23.2111 0.771563
$$906$$ 0 0
$$907$$ −34.0555 −1.13079 −0.565397 0.824819i $$-0.691277\pi$$
−0.565397 + 0.824819i $$0.691277\pi$$
$$908$$ 0 0
$$909$$ 5.21110 0.172841
$$910$$ 0 0
$$911$$ 18.4222 0.610355 0.305177 0.952296i $$-0.401284\pi$$
0.305177 + 0.952296i $$0.401284\pi$$
$$912$$ 0 0
$$913$$ −10.6056 −0.350993
$$914$$ 0 0
$$915$$ −7.21110 −0.238392
$$916$$ 0 0
$$917$$ −31.2666 −1.03251
$$918$$ 0 0
$$919$$ −26.0000 −0.857661 −0.428830 0.903385i $$-0.641074\pi$$
−0.428830 + 0.903385i $$0.641074\pi$$
$$920$$ 0 0
$$921$$ −8.60555 −0.283563
$$922$$ 0 0
$$923$$ 66.4222 2.18631
$$924$$ 0 0
$$925$$ −3.21110 −0.105580
$$926$$ 0 0
$$927$$ 1.21110 0.0397778
$$928$$ 0 0
$$929$$ 26.8444 0.880737 0.440368 0.897817i $$-0.354848\pi$$
0.440368 + 0.897817i $$0.354848\pi$$
$$930$$ 0 0
$$931$$ −102.478 −3.35857
$$932$$ 0 0
$$933$$ −30.4222 −0.995978
$$934$$ 0 0
$$935$$ 6.60555 0.216025
$$936$$ 0 0
$$937$$ 20.9722 0.685133 0.342567 0.939494i $$-0.388704\pi$$
0.342567 + 0.939494i $$0.388704\pi$$
$$938$$ 0 0
$$939$$ −8.78890 −0.286815
$$940$$ 0 0
$$941$$ 53.2111 1.73463 0.867316 0.497758i $$-0.165843\pi$$
0.867316 + 0.497758i $$0.165843\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −4.60555 −0.149819
$$946$$ 0 0
$$947$$ −3.63331 −0.118067 −0.0590333 0.998256i $$-0.518802\pi$$
−0.0590333 + 0.998256i $$0.518802\pi$$
$$948$$ 0 0
$$949$$ −2.78890 −0.0905314
$$950$$ 0 0
$$951$$ −11.2111 −0.363545
$$952$$ 0 0
$$953$$ −35.4500 −1.14834 −0.574168 0.818737i $$-0.694675\pi$$
−0.574168 + 0.818737i $$0.694675\pi$$
$$954$$ 0 0
$$955$$ 18.4222 0.596129
$$956$$ 0 0
$$957$$ −8.00000 −0.258603
$$958$$ 0 0
$$959$$ −27.6333 −0.892326
$$960$$ 0 0
$$961$$ 53.8444 1.73692
$$962$$ 0 0
$$963$$ 15.8167 0.509685
$$964$$ 0 0
$$965$$ −21.8167 −0.702303
$$966$$ 0 0
$$967$$ 25.8167 0.830208 0.415104 0.909774i $$-0.363745\pi$$
0.415104 + 0.909774i $$0.363745\pi$$
$$968$$ 0 0
$$969$$ 47.6333 1.53020
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ 75.6333 2.42469
$$974$$ 0 0
$$975$$ 4.60555 0.147496
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 0 0
$$983$$ 52.4777 1.67378 0.836890 0.547372i $$-0.184372\pi$$
0.836890 + 0.547372i $$0.184372\pi$$
$$984$$ 0 0
$$985$$ −13.3944 −0.426783
$$986$$ 0 0
$$987$$ 24.0000 0.763928
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −35.6333 −1.13193 −0.565965 0.824430i $$-0.691496\pi$$
−0.565965 + 0.824430i $$0.691496\pi$$
$$992$$ 0 0
$$993$$ 6.78890 0.215439
$$994$$ 0 0
$$995$$ −10.4222 −0.330406
$$996$$ 0 0
$$997$$ 15.0278 0.475934 0.237967 0.971273i $$-0.423519\pi$$
0.237967 + 0.971273i $$0.423519\pi$$
$$998$$ 0 0
$$999$$ 3.21110 0.101595
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 660.2.a.e.1.2 2
3.2 odd 2 1980.2.a.h.1.2 2
4.3 odd 2 2640.2.a.bc.1.1 2
5.2 odd 4 3300.2.c.l.1849.4 4
5.3 odd 4 3300.2.c.l.1849.1 4
5.4 even 2 3300.2.a.w.1.1 2
11.10 odd 2 7260.2.a.w.1.1 2
12.11 even 2 7920.2.a.bo.1.1 2
15.2 even 4 9900.2.c.q.5149.4 4
15.8 even 4 9900.2.c.q.5149.1 4
15.14 odd 2 9900.2.a.bl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
660.2.a.e.1.2 2 1.1 even 1 trivial
1980.2.a.h.1.2 2 3.2 odd 2
2640.2.a.bc.1.1 2 4.3 odd 2
3300.2.a.w.1.1 2 5.4 even 2
3300.2.c.l.1849.1 4 5.3 odd 4
3300.2.c.l.1849.4 4 5.2 odd 4
7260.2.a.w.1.1 2 11.10 odd 2
7920.2.a.bo.1.1 2 12.11 even 2
9900.2.a.bl.1.1 2 15.14 odd 2
9900.2.c.q.5149.1 4 15.8 even 4
9900.2.c.q.5149.4 4 15.2 even 4