# Properties

 Label 8550.2.a.ck.1.2 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +0.636672 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +0.636672 q^{7} -1.00000 q^{8} -3.50466 q^{11} -0.141336 q^{13} -0.636672 q^{14} +1.00000 q^{16} -2.14134 q^{17} +1.00000 q^{19} +3.50466 q^{22} -4.91934 q^{23} +0.141336 q^{26} +0.636672 q^{28} -7.15066 q^{29} -7.78734 q^{31} -1.00000 q^{32} +2.14134 q^{34} +3.27334 q^{37} -1.00000 q^{38} +4.23132 q^{41} +2.49534 q^{43} -3.50466 q^{44} +4.91934 q^{46} +10.2827 q^{47} -6.59465 q^{49} -0.141336 q^{52} -8.14134 q^{53} -0.636672 q^{56} +7.15066 q^{58} +5.64600 q^{59} -6.49534 q^{61} +7.78734 q^{62} +1.00000 q^{64} +8.37266 q^{67} -2.14134 q^{68} +8.95798 q^{71} -3.69735 q^{73} -3.27334 q^{74} +1.00000 q^{76} -2.23132 q^{77} -4.17997 q^{79} -4.23132 q^{82} +9.00933 q^{83} -2.49534 q^{86} +3.50466 q^{88} +6.77801 q^{89} -0.0899847 q^{91} -4.91934 q^{92} -10.2827 q^{94} +14.5653 q^{97} +6.59465 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{4} + 4q^{7} - 3q^{8} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{4} + 4q^{7} - 3q^{8} + 8q^{13} - 4q^{14} + 3q^{16} + 2q^{17} + 3q^{19} - 8q^{26} + 4q^{28} + 8q^{29} + 4q^{31} - 3q^{32} - 2q^{34} + 14q^{37} - 3q^{38} - 2q^{41} + 18q^{43} + 14q^{47} - 3q^{49} + 8q^{52} - 16q^{53} - 4q^{56} - 8q^{58} - 2q^{59} - 30q^{61} - 4q^{62} + 3q^{64} + 2q^{67} + 2q^{68} + 8q^{71} + 10q^{73} - 14q^{74} + 3q^{76} + 8q^{77} + 2q^{82} + 6q^{83} - 18q^{86} + 14q^{89} + 6q^{91} - 14q^{94} + 10q^{97} + 3q^{98} + O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.636672 0.240639 0.120320 0.992735i $$-0.461608\pi$$
0.120320 + 0.992735i $$0.461608\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.50466 −1.05670 −0.528348 0.849028i $$-0.677188\pi$$
−0.528348 + 0.849028i $$0.677188\pi$$
$$12$$ 0 0
$$13$$ −0.141336 −0.0391996 −0.0195998 0.999808i $$-0.506239\pi$$
−0.0195998 + 0.999808i $$0.506239\pi$$
$$14$$ −0.636672 −0.170158
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.14134 −0.519350 −0.259675 0.965696i $$-0.583615\pi$$
−0.259675 + 0.965696i $$0.583615\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.50466 0.747197
$$23$$ −4.91934 −1.02575 −0.512877 0.858462i $$-0.671420\pi$$
−0.512877 + 0.858462i $$0.671420\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.141336 0.0277183
$$27$$ 0 0
$$28$$ 0.636672 0.120320
$$29$$ −7.15066 −1.32785 −0.663923 0.747801i $$-0.731110\pi$$
−0.663923 + 0.747801i $$0.731110\pi$$
$$30$$ 0 0
$$31$$ −7.78734 −1.39865 −0.699323 0.714805i $$-0.746515\pi$$
−0.699323 + 0.714805i $$0.746515\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 2.14134 0.367236
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.27334 0.538134 0.269067 0.963121i $$-0.413285\pi$$
0.269067 + 0.963121i $$0.413285\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.23132 0.660821 0.330411 0.943837i $$-0.392813\pi$$
0.330411 + 0.943837i $$0.392813\pi$$
$$42$$ 0 0
$$43$$ 2.49534 0.380535 0.190268 0.981732i $$-0.439064\pi$$
0.190268 + 0.981732i $$0.439064\pi$$
$$44$$ −3.50466 −0.528348
$$45$$ 0 0
$$46$$ 4.91934 0.725318
$$47$$ 10.2827 1.49988 0.749941 0.661505i $$-0.230082\pi$$
0.749941 + 0.661505i $$0.230082\pi$$
$$48$$ 0 0
$$49$$ −6.59465 −0.942093
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.141336 −0.0195998
$$53$$ −8.14134 −1.11830 −0.559149 0.829067i $$-0.688872\pi$$
−0.559149 + 0.829067i $$0.688872\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −0.636672 −0.0850788
$$57$$ 0 0
$$58$$ 7.15066 0.938928
$$59$$ 5.64600 0.735047 0.367523 0.930014i $$-0.380206\pi$$
0.367523 + 0.930014i $$0.380206\pi$$
$$60$$ 0 0
$$61$$ −6.49534 −0.831643 −0.415821 0.909446i $$-0.636506\pi$$
−0.415821 + 0.909446i $$0.636506\pi$$
$$62$$ 7.78734 0.988993
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.37266 1.02288 0.511441 0.859318i $$-0.329112\pi$$
0.511441 + 0.859318i $$0.329112\pi$$
$$68$$ −2.14134 −0.259675
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.95798 1.06312 0.531558 0.847022i $$-0.321607\pi$$
0.531558 + 0.847022i $$0.321607\pi$$
$$72$$ 0 0
$$73$$ −3.69735 −0.432742 −0.216371 0.976311i $$-0.569422\pi$$
−0.216371 + 0.976311i $$0.569422\pi$$
$$74$$ −3.27334 −0.380518
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −2.23132 −0.254283
$$78$$ 0 0
$$79$$ −4.17997 −0.470283 −0.235142 0.971961i $$-0.575555\pi$$
−0.235142 + 0.971961i $$0.575555\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −4.23132 −0.467271
$$83$$ 9.00933 0.988902 0.494451 0.869205i $$-0.335369\pi$$
0.494451 + 0.869205i $$0.335369\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.49534 −0.269079
$$87$$ 0 0
$$88$$ 3.50466 0.373598
$$89$$ 6.77801 0.718467 0.359234 0.933248i $$-0.383038\pi$$
0.359234 + 0.933248i $$0.383038\pi$$
$$90$$ 0 0
$$91$$ −0.0899847 −0.00943296
$$92$$ −4.91934 −0.512877
$$93$$ 0 0
$$94$$ −10.2827 −1.06058
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.5653 1.47889 0.739443 0.673219i $$-0.235089\pi$$
0.739443 + 0.673219i $$0.235089\pi$$
$$98$$ 6.59465 0.666160
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.6167 1.65342 0.826712 0.562626i $$-0.190209\pi$$
0.826712 + 0.562626i $$0.190209\pi$$
$$102$$ 0 0
$$103$$ 9.06068 0.892775 0.446388 0.894840i $$-0.352710\pi$$
0.446388 + 0.894840i $$0.352710\pi$$
$$104$$ 0.141336 0.0138591
$$105$$ 0 0
$$106$$ 8.14134 0.790756
$$107$$ −0.0899847 −0.00869915 −0.00434958 0.999991i $$-0.501385\pi$$
−0.00434958 + 0.999991i $$0.501385\pi$$
$$108$$ 0 0
$$109$$ −13.5946 −1.30213 −0.651066 0.759021i $$-0.725678\pi$$
−0.651066 + 0.759021i $$0.725678\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0.636672 0.0601598
$$113$$ −11.5233 −1.08402 −0.542011 0.840371i $$-0.682337\pi$$
−0.542011 + 0.840371i $$0.682337\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −7.15066 −0.663923
$$117$$ 0 0
$$118$$ −5.64600 −0.519756
$$119$$ −1.36333 −0.124976
$$120$$ 0 0
$$121$$ 1.28267 0.116607
$$122$$ 6.49534 0.588060
$$123$$ 0 0
$$124$$ −7.78734 −0.699323
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −3.29200 −0.292118 −0.146059 0.989276i $$-0.546659\pi$$
−0.146059 + 0.989276i $$0.546659\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −18.0187 −1.57430 −0.787149 0.616763i $$-0.788444\pi$$
−0.787149 + 0.616763i $$0.788444\pi$$
$$132$$ 0 0
$$133$$ 0.636672 0.0552064
$$134$$ −8.37266 −0.723287
$$135$$ 0 0
$$136$$ 2.14134 0.183618
$$137$$ 14.4240 1.23233 0.616163 0.787619i $$-0.288686\pi$$
0.616163 + 0.787619i $$0.288686\pi$$
$$138$$ 0 0
$$139$$ 15.4720 1.31232 0.656158 0.754624i $$-0.272181\pi$$
0.656158 + 0.754624i $$0.272181\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.95798 −0.751737
$$143$$ 0.495336 0.0414220
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.69735 0.305995
$$147$$ 0 0
$$148$$ 3.27334 0.269067
$$149$$ 17.1893 1.40820 0.704101 0.710100i $$-0.251350\pi$$
0.704101 + 0.710100i $$0.251350\pi$$
$$150$$ 0 0
$$151$$ 3.29200 0.267899 0.133950 0.990988i $$-0.457234\pi$$
0.133950 + 0.990988i $$0.457234\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ 2.23132 0.179805
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.1893 1.21224 0.606119 0.795374i $$-0.292726\pi$$
0.606119 + 0.795374i $$0.292726\pi$$
$$158$$ 4.17997 0.332541
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.13201 −0.246837
$$162$$ 0 0
$$163$$ −14.0700 −1.10205 −0.551024 0.834489i $$-0.685763\pi$$
−0.551024 + 0.834489i $$0.685763\pi$$
$$164$$ 4.23132 0.330411
$$165$$ 0 0
$$166$$ −9.00933 −0.699260
$$167$$ 14.7967 1.14500 0.572500 0.819905i $$-0.305974\pi$$
0.572500 + 0.819905i $$0.305974\pi$$
$$168$$ 0 0
$$169$$ −12.9800 −0.998463
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.49534 0.190268
$$173$$ 17.2920 1.31469 0.657343 0.753591i $$-0.271680\pi$$
0.657343 + 0.753591i $$0.271680\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.50466 −0.264174
$$177$$ 0 0
$$178$$ −6.77801 −0.508033
$$179$$ −17.7360 −1.32565 −0.662825 0.748774i $$-0.730643\pi$$
−0.662825 + 0.748774i $$0.730643\pi$$
$$180$$ 0 0
$$181$$ 6.17997 0.459354 0.229677 0.973267i $$-0.426233\pi$$
0.229677 + 0.973267i $$0.426233\pi$$
$$182$$ 0.0899847 0.00667011
$$183$$ 0 0
$$184$$ 4.91934 0.362659
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 7.50466 0.548795
$$188$$ 10.2827 0.749941
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −14.6367 −1.05907 −0.529536 0.848287i $$-0.677634\pi$$
−0.529536 + 0.848287i $$0.677634\pi$$
$$192$$ 0 0
$$193$$ 20.0187 1.44097 0.720487 0.693468i $$-0.243918\pi$$
0.720487 + 0.693468i $$0.243918\pi$$
$$194$$ −14.5653 −1.04573
$$195$$ 0 0
$$196$$ −6.59465 −0.471046
$$197$$ 9.94865 0.708812 0.354406 0.935092i $$-0.384683\pi$$
0.354406 + 0.935092i $$0.384683\pi$$
$$198$$ 0 0
$$199$$ 9.74870 0.691067 0.345534 0.938406i $$-0.387698\pi$$
0.345534 + 0.938406i $$0.387698\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −16.6167 −1.16915
$$203$$ −4.55263 −0.319532
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −9.06068 −0.631287
$$207$$ 0 0
$$208$$ −0.141336 −0.00979990
$$209$$ −3.50466 −0.242423
$$210$$ 0 0
$$211$$ −20.7580 −1.42904 −0.714521 0.699614i $$-0.753355\pi$$
−0.714521 + 0.699614i $$0.753355\pi$$
$$212$$ −8.14134 −0.559149
$$213$$ 0 0
$$214$$ 0.0899847 0.00615123
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.95798 −0.336569
$$218$$ 13.5946 0.920746
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.302648 0.0203583
$$222$$ 0 0
$$223$$ −10.7267 −0.718310 −0.359155 0.933278i $$-0.616935\pi$$
−0.359155 + 0.933278i $$0.616935\pi$$
$$224$$ −0.636672 −0.0425394
$$225$$ 0 0
$$226$$ 11.5233 0.766520
$$227$$ −12.5526 −0.833147 −0.416574 0.909102i $$-0.636769\pi$$
−0.416574 + 0.909102i $$0.636769\pi$$
$$228$$ 0 0
$$229$$ 25.4720 1.68324 0.841618 0.540074i $$-0.181604\pi$$
0.841618 + 0.540074i $$0.181604\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 7.15066 0.469464
$$233$$ 3.11203 0.203876 0.101938 0.994791i $$-0.467496\pi$$
0.101938 + 0.994791i $$0.467496\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5.64600 0.367523
$$237$$ 0 0
$$238$$ 1.36333 0.0883714
$$239$$ 1.54330 0.0998276 0.0499138 0.998754i $$-0.484105\pi$$
0.0499138 + 0.998754i $$0.484105\pi$$
$$240$$ 0 0
$$241$$ 10.2827 0.662365 0.331183 0.943567i $$-0.392552\pi$$
0.331183 + 0.943567i $$0.392552\pi$$
$$242$$ −1.28267 −0.0824533
$$243$$ 0 0
$$244$$ −6.49534 −0.415821
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.141336 −0.00899300
$$248$$ 7.78734 0.494496
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.51399 0.158682 0.0793409 0.996848i $$-0.474718\pi$$
0.0793409 + 0.996848i $$0.474718\pi$$
$$252$$ 0 0
$$253$$ 17.2406 1.08391
$$254$$ 3.29200 0.206559
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 24.7967 1.54677 0.773387 0.633934i $$-0.218561\pi$$
0.773387 + 0.633934i $$0.218561\pi$$
$$258$$ 0 0
$$259$$ 2.08405 0.129496
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0187 1.11320
$$263$$ 22.5653 1.39144 0.695719 0.718314i $$-0.255086\pi$$
0.695719 + 0.718314i $$0.255086\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −0.636672 −0.0390369
$$267$$ 0 0
$$268$$ 8.37266 0.511441
$$269$$ 26.5653 1.61972 0.809859 0.586625i $$-0.199544\pi$$
0.809859 + 0.586625i $$0.199544\pi$$
$$270$$ 0 0
$$271$$ −24.9380 −1.51488 −0.757438 0.652907i $$-0.773549\pi$$
−0.757438 + 0.652907i $$0.773549\pi$$
$$272$$ −2.14134 −0.129838
$$273$$ 0 0
$$274$$ −14.4240 −0.871386
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.5467 1.11436 0.557181 0.830391i $$-0.311883\pi$$
0.557181 + 0.830391i $$0.311883\pi$$
$$278$$ −15.4720 −0.927947
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −24.7967 −1.47925 −0.739623 0.673022i $$-0.764996\pi$$
−0.739623 + 0.673022i $$0.764996\pi$$
$$282$$ 0 0
$$283$$ 13.5747 0.806931 0.403465 0.914995i $$-0.367806\pi$$
0.403465 + 0.914995i $$0.367806\pi$$
$$284$$ 8.95798 0.531558
$$285$$ 0 0
$$286$$ −0.495336 −0.0292898
$$287$$ 2.69396 0.159020
$$288$$ 0 0
$$289$$ −12.4147 −0.730275
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.69735 −0.216371
$$293$$ 15.6133 0.912139 0.456070 0.889944i $$-0.349257\pi$$
0.456070 + 0.889944i $$0.349257\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −3.27334 −0.190259
$$297$$ 0 0
$$298$$ −17.1893 −0.995749
$$299$$ 0.695281 0.0402091
$$300$$ 0 0
$$301$$ 1.58871 0.0915717
$$302$$ −3.29200 −0.189433
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 34.0187 1.94155 0.970774 0.239997i $$-0.0771464\pi$$
0.970774 + 0.239997i $$0.0771464\pi$$
$$308$$ −2.23132 −0.127141
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.93800 −0.506828 −0.253414 0.967358i $$-0.581553\pi$$
−0.253414 + 0.967358i $$0.581553\pi$$
$$312$$ 0 0
$$313$$ −18.4240 −1.04139 −0.520693 0.853744i $$-0.674326\pi$$
−0.520693 + 0.853744i $$0.674326\pi$$
$$314$$ −15.1893 −0.857182
$$315$$ 0 0
$$316$$ −4.17997 −0.235142
$$317$$ −33.5547 −1.88462 −0.942310 0.334742i $$-0.891351\pi$$
−0.942310 + 0.334742i $$0.891351\pi$$
$$318$$ 0 0
$$319$$ 25.0607 1.40313
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3.13201 0.174540
$$323$$ −2.14134 −0.119147
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 14.0700 0.779266
$$327$$ 0 0
$$328$$ −4.23132 −0.233636
$$329$$ 6.54669 0.360931
$$330$$ 0 0
$$331$$ 2.25130 0.123742 0.0618712 0.998084i $$-0.480293\pi$$
0.0618712 + 0.998084i $$0.480293\pi$$
$$332$$ 9.00933 0.494451
$$333$$ 0 0
$$334$$ −14.7967 −0.809638
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −21.3620 −1.16366 −0.581831 0.813309i $$-0.697664\pi$$
−0.581831 + 0.813309i $$0.697664\pi$$
$$338$$ 12.9800 0.706020
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 27.2920 1.47794
$$342$$ 0 0
$$343$$ −8.65533 −0.467344
$$344$$ −2.49534 −0.134539
$$345$$ 0 0
$$346$$ −17.2920 −0.929624
$$347$$ −11.5560 −0.620359 −0.310180 0.950678i $$-0.600389\pi$$
−0.310180 + 0.950678i $$0.600389\pi$$
$$348$$ 0 0
$$349$$ −17.1120 −0.915986 −0.457993 0.888956i $$-0.651432\pi$$
−0.457993 + 0.888956i $$0.651432\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3.50466 0.186799
$$353$$ −11.6974 −0.622587 −0.311294 0.950314i $$-0.600762\pi$$
−0.311294 + 0.950314i $$0.600762\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.77801 0.359234
$$357$$ 0 0
$$358$$ 17.7360 0.937376
$$359$$ 4.47536 0.236200 0.118100 0.993002i $$-0.462320\pi$$
0.118100 + 0.993002i $$0.462320\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −6.17997 −0.324812
$$363$$ 0 0
$$364$$ −0.0899847 −0.00471648
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 18.7453 0.978497 0.489249 0.872144i $$-0.337271\pi$$
0.489249 + 0.872144i $$0.337271\pi$$
$$368$$ −4.91934 −0.256439
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5.18336 −0.269107
$$372$$ 0 0
$$373$$ 1.69735 0.0878855 0.0439428 0.999034i $$-0.486008\pi$$
0.0439428 + 0.999034i $$0.486008\pi$$
$$374$$ −7.50466 −0.388057
$$375$$ 0 0
$$376$$ −10.2827 −0.530288
$$377$$ 1.01065 0.0520510
$$378$$ 0 0
$$379$$ 2.63667 0.135437 0.0677184 0.997704i $$-0.478428\pi$$
0.0677184 + 0.997704i $$0.478428\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 14.6367 0.748877
$$383$$ −12.4953 −0.638482 −0.319241 0.947674i $$-0.603428\pi$$
−0.319241 + 0.947674i $$0.603428\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −20.0187 −1.01892
$$387$$ 0 0
$$388$$ 14.5653 0.739443
$$389$$ 4.51399 0.228869 0.114434 0.993431i $$-0.463494\pi$$
0.114434 + 0.993431i $$0.463494\pi$$
$$390$$ 0 0
$$391$$ 10.5340 0.532726
$$392$$ 6.59465 0.333080
$$393$$ 0 0
$$394$$ −9.94865 −0.501206
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −35.6774 −1.79060 −0.895298 0.445468i $$-0.853037\pi$$
−0.895298 + 0.445468i $$0.853037\pi$$
$$398$$ −9.74870 −0.488658
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.3434 −0.766210 −0.383105 0.923705i $$-0.625145\pi$$
−0.383105 + 0.923705i $$0.625145\pi$$
$$402$$ 0 0
$$403$$ 1.10063 0.0548264
$$404$$ 16.6167 0.826712
$$405$$ 0 0
$$406$$ 4.55263 0.225943
$$407$$ −11.4720 −0.568644
$$408$$ 0 0
$$409$$ 29.3620 1.45186 0.725929 0.687770i $$-0.241410\pi$$
0.725929 + 0.687770i $$0.241410\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 9.06068 0.446388
$$413$$ 3.59465 0.176881
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0.141336 0.00692957
$$417$$ 0 0
$$418$$ 3.50466 0.171419
$$419$$ −25.1379 −1.22807 −0.614035 0.789279i $$-0.710454\pi$$
−0.614035 + 0.789279i $$0.710454\pi$$
$$420$$ 0 0
$$421$$ 14.5454 0.708898 0.354449 0.935075i $$-0.384668\pi$$
0.354449 + 0.935075i $$0.384668\pi$$
$$422$$ 20.7580 1.01049
$$423$$ 0 0
$$424$$ 8.14134 0.395378
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4.13540 −0.200126
$$428$$ −0.0899847 −0.00434958
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −19.4020 −0.934560 −0.467280 0.884109i $$-0.654766\pi$$
−0.467280 + 0.884109i $$0.654766\pi$$
$$432$$ 0 0
$$433$$ 5.50466 0.264537 0.132269 0.991214i $$-0.457774\pi$$
0.132269 + 0.991214i $$0.457774\pi$$
$$434$$ 4.95798 0.237991
$$435$$ 0 0
$$436$$ −13.5946 −0.651066
$$437$$ −4.91934 −0.235324
$$438$$ 0 0
$$439$$ 12.2500 0.584660 0.292330 0.956318i $$-0.405570\pi$$
0.292330 + 0.956318i $$0.405570\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −0.302648 −0.0143955
$$443$$ −31.6006 −1.50139 −0.750695 0.660649i $$-0.770281\pi$$
−0.750695 + 0.660649i $$0.770281\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 10.7267 0.507922
$$447$$ 0 0
$$448$$ 0.636672 0.0300799
$$449$$ 36.0187 1.69983 0.849913 0.526923i $$-0.176655\pi$$
0.849913 + 0.526923i $$0.176655\pi$$
$$450$$ 0 0
$$451$$ −14.8294 −0.698287
$$452$$ −11.5233 −0.542011
$$453$$ 0 0
$$454$$ 12.5526 0.589124
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.1413 1.03573 0.517864 0.855463i $$-0.326727\pi$$
0.517864 + 0.855463i $$0.326727\pi$$
$$458$$ −25.4720 −1.19023
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.31537 0.107837 0.0539187 0.998545i $$-0.482829\pi$$
0.0539187 + 0.998545i $$0.482829\pi$$
$$462$$ 0 0
$$463$$ 15.8387 0.736086 0.368043 0.929809i $$-0.380028\pi$$
0.368043 + 0.929809i $$0.380028\pi$$
$$464$$ −7.15066 −0.331961
$$465$$ 0 0
$$466$$ −3.11203 −0.144162
$$467$$ −23.1379 −1.07070 −0.535348 0.844631i $$-0.679820\pi$$
−0.535348 + 0.844631i $$0.679820\pi$$
$$468$$ 0 0
$$469$$ 5.33063 0.246146
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −5.64600 −0.259878
$$473$$ −8.74531 −0.402110
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.36333 −0.0624880
$$477$$ 0 0
$$478$$ −1.54330 −0.0705888
$$479$$ 10.1214 0.462457 0.231228 0.972900i $$-0.425726\pi$$
0.231228 + 0.972900i $$0.425726\pi$$
$$480$$ 0 0
$$481$$ −0.462642 −0.0210946
$$482$$ −10.2827 −0.468363
$$483$$ 0 0
$$484$$ 1.28267 0.0583033
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 20.3854 0.923750 0.461875 0.886945i $$-0.347177\pi$$
0.461875 + 0.886945i $$0.347177\pi$$
$$488$$ 6.49534 0.294030
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.78734 0.351438 0.175719 0.984440i $$-0.443775\pi$$
0.175719 + 0.984440i $$0.443775\pi$$
$$492$$ 0 0
$$493$$ 15.3120 0.689617
$$494$$ 0.141336 0.00635901
$$495$$ 0 0
$$496$$ −7.78734 −0.349662
$$497$$ 5.70329 0.255828
$$498$$ 0 0
$$499$$ 4.31537 0.193182 0.0965912 0.995324i $$-0.469206\pi$$
0.0965912 + 0.995324i $$0.469206\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −2.51399 −0.112205
$$503$$ −18.5526 −0.827221 −0.413610 0.910454i $$-0.635732\pi$$
−0.413610 + 0.910454i $$0.635732\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −17.2406 −0.766440
$$507$$ 0 0
$$508$$ −3.29200 −0.146059
$$509$$ 7.73599 0.342892 0.171446 0.985194i $$-0.445156\pi$$
0.171446 + 0.985194i $$0.445156\pi$$
$$510$$ 0 0
$$511$$ −2.35400 −0.104135
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −24.7967 −1.09373
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −36.0373 −1.58492
$$518$$ −2.08405 −0.0915677
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.2080 −0.666273 −0.333136 0.942879i $$-0.608107\pi$$
−0.333136 + 0.942879i $$0.608107\pi$$
$$522$$ 0 0
$$523$$ −18.2113 −0.796327 −0.398163 0.917315i $$-0.630352\pi$$
−0.398163 + 0.917315i $$0.630352\pi$$
$$524$$ −18.0187 −0.787149
$$525$$ 0 0
$$526$$ −22.5653 −0.983896
$$527$$ 16.6753 0.726388
$$528$$ 0 0
$$529$$ 1.19995 0.0521715
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.636672 0.0276032
$$533$$ −0.598038 −0.0259039
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8.37266 −0.361644
$$537$$ 0 0
$$538$$ −26.5653 −1.14531
$$539$$ 23.1120 0.995506
$$540$$ 0 0
$$541$$ 16.5140 0.709992 0.354996 0.934868i $$-0.384482\pi$$
0.354996 + 0.934868i $$0.384482\pi$$
$$542$$ 24.9380 1.07118
$$543$$ 0 0
$$544$$ 2.14134 0.0918090
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 16.2827 0.696197 0.348098 0.937458i $$-0.386828\pi$$
0.348098 + 0.937458i $$0.386828\pi$$
$$548$$ 14.4240 0.616163
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7.15066 −0.304629
$$552$$ 0 0
$$553$$ −2.66127 −0.113169
$$554$$ −18.5467 −0.787973
$$555$$ 0 0
$$556$$ 15.4720 0.656158
$$557$$ −37.4533 −1.58695 −0.793474 0.608604i $$-0.791730\pi$$
−0.793474 + 0.608604i $$0.791730\pi$$
$$558$$ 0 0
$$559$$ −0.352681 −0.0149168
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 24.7967 1.04598
$$563$$ 29.1307 1.22771 0.613856 0.789418i $$-0.289618\pi$$
0.613856 + 0.789418i $$0.289618\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −13.5747 −0.570586
$$567$$ 0 0
$$568$$ −8.95798 −0.375868
$$569$$ −14.8480 −0.622461 −0.311231 0.950334i $$-0.600741\pi$$
−0.311231 + 0.950334i $$0.600741\pi$$
$$570$$ 0 0
$$571$$ 41.9087 1.75382 0.876912 0.480651i $$-0.159599\pi$$
0.876912 + 0.480651i $$0.159599\pi$$
$$572$$ 0.495336 0.0207110
$$573$$ 0 0
$$574$$ −2.69396 −0.112444
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 16.4427 0.684517 0.342259 0.939606i $$-0.388808\pi$$
0.342259 + 0.939606i $$0.388808\pi$$
$$578$$ 12.4147 0.516383
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5.73599 0.237969
$$582$$ 0 0
$$583$$ 28.5327 1.18170
$$584$$ 3.69735 0.152998
$$585$$ 0 0
$$586$$ −15.6133 −0.644980
$$587$$ 42.5327 1.75551 0.877755 0.479109i $$-0.159040\pi$$
0.877755 + 0.479109i $$0.159040\pi$$
$$588$$ 0 0
$$589$$ −7.78734 −0.320872
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3.27334 0.134534
$$593$$ −3.92273 −0.161087 −0.0805437 0.996751i $$-0.525666\pi$$
−0.0805437 + 0.996751i $$0.525666\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 17.1893 0.704101
$$597$$ 0 0
$$598$$ −0.695281 −0.0284322
$$599$$ 10.7594 0.439615 0.219808 0.975543i $$-0.429457\pi$$
0.219808 + 0.975543i $$0.429457\pi$$
$$600$$ 0 0
$$601$$ 25.2220 1.02883 0.514413 0.857542i $$-0.328010\pi$$
0.514413 + 0.857542i $$0.328010\pi$$
$$602$$ −1.58871 −0.0647510
$$603$$ 0 0
$$604$$ 3.29200 0.133950
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 39.2920 1.59481 0.797407 0.603442i $$-0.206205\pi$$
0.797407 + 0.603442i $$0.206205\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.45331 −0.0587947
$$612$$ 0 0
$$613$$ 9.80599 0.396060 0.198030 0.980196i $$-0.436546\pi$$
0.198030 + 0.980196i $$0.436546\pi$$
$$614$$ −34.0187 −1.37288
$$615$$ 0 0
$$616$$ 2.23132 0.0899025
$$617$$ 35.0093 1.40942 0.704711 0.709494i $$-0.251077\pi$$
0.704711 + 0.709494i $$0.251077\pi$$
$$618$$ 0 0
$$619$$ 13.4206 0.539420 0.269710 0.962942i $$-0.413072\pi$$
0.269710 + 0.962942i $$0.413072\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8.93800 0.358381
$$623$$ 4.31537 0.172891
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 18.4240 0.736371
$$627$$ 0 0
$$628$$ 15.1893 0.606119
$$629$$ −7.00933 −0.279480
$$630$$ 0 0
$$631$$ 40.5254 1.61329 0.806645 0.591036i $$-0.201281\pi$$
0.806645 + 0.591036i $$0.201281\pi$$
$$632$$ 4.17997 0.166270
$$633$$ 0 0
$$634$$ 33.5547 1.33263
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.932062 0.0369296
$$638$$ −25.0607 −0.992162
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.0700 1.02970 0.514852 0.857279i $$-0.327847\pi$$
0.514852 + 0.857279i $$0.327847\pi$$
$$642$$ 0 0
$$643$$ 30.1400 1.18861 0.594303 0.804241i $$-0.297428\pi$$
0.594303 + 0.804241i $$0.297428\pi$$
$$644$$ −3.13201 −0.123418
$$645$$ 0 0
$$646$$ 2.14134 0.0842497
$$647$$ 20.1086 0.790552 0.395276 0.918562i $$-0.370649\pi$$
0.395276 + 0.918562i $$0.370649\pi$$
$$648$$ 0 0
$$649$$ −19.7873 −0.776721
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −14.0700 −0.551024
$$653$$ 28.0373 1.09718 0.548592 0.836090i $$-0.315164\pi$$
0.548592 + 0.836090i $$0.315164\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.23132 0.165205
$$657$$ 0 0
$$658$$ −6.54669 −0.255216
$$659$$ −4.90069 −0.190904 −0.0954518 0.995434i $$-0.530430\pi$$
−0.0954518 + 0.995434i $$0.530430\pi$$
$$660$$ 0 0
$$661$$ −8.03863 −0.312667 −0.156333 0.987704i $$-0.549967\pi$$
−0.156333 + 0.987704i $$0.549967\pi$$
$$662$$ −2.25130 −0.0874991
$$663$$ 0 0
$$664$$ −9.00933 −0.349630
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 35.1766 1.36204
$$668$$ 14.7967 0.572500
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 22.7640 0.878793
$$672$$ 0 0
$$673$$ −4.82936 −0.186158 −0.0930791 0.995659i $$-0.529671\pi$$
−0.0930791 + 0.995659i $$0.529671\pi$$
$$674$$ 21.3620 0.822834
$$675$$ 0 0
$$676$$ −12.9800 −0.499232
$$677$$ −12.8094 −0.492305 −0.246152 0.969231i $$-0.579166\pi$$
−0.246152 + 0.969231i $$0.579166\pi$$
$$678$$ 0 0
$$679$$ 9.27334 0.355878
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −27.2920 −1.04506
$$683$$ −37.1307 −1.42077 −0.710383 0.703815i $$-0.751478\pi$$
−0.710383 + 0.703815i $$0.751478\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.65533 0.330462
$$687$$ 0 0
$$688$$ 2.49534 0.0951338
$$689$$ 1.15066 0.0438368
$$690$$ 0 0
$$691$$ −18.1986 −0.692308 −0.346154 0.938178i $$-0.612513\pi$$
−0.346154 + 0.938178i $$0.612513\pi$$
$$692$$ 17.2920 0.657343
$$693$$ 0 0
$$694$$ 11.5560 0.438660
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.06068 −0.343198
$$698$$ 17.1120 0.647700
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.2827 0.992683 0.496341 0.868127i $$-0.334676\pi$$
0.496341 + 0.868127i $$0.334676\pi$$
$$702$$ 0 0
$$703$$ 3.27334 0.123456
$$704$$ −3.50466 −0.132087
$$705$$ 0 0
$$706$$ 11.6974 0.440236
$$707$$ 10.5794 0.397879
$$708$$ 0 0
$$709$$ −14.9253 −0.560531 −0.280265 0.959923i $$-0.590422\pi$$
−0.280265 + 0.959923i $$0.590422\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −6.77801 −0.254017
$$713$$ 38.3086 1.43467
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −17.7360 −0.662825
$$717$$ 0 0
$$718$$ −4.47536 −0.167019
$$719$$ −32.3327 −1.20581 −0.602903 0.797814i $$-0.705989\pi$$
−0.602903 + 0.797814i $$0.705989\pi$$
$$720$$ 0 0
$$721$$ 5.76868 0.214837
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 6.17997 0.229677
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −42.0246 −1.55861 −0.779303 0.626647i $$-0.784427\pi$$
−0.779303 + 0.626647i $$0.784427\pi$$
$$728$$ 0.0899847 0.00333506
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −5.34335 −0.197631
$$732$$ 0 0
$$733$$ 26.5840 0.981903 0.490951 0.871187i $$-0.336649\pi$$
0.490951 + 0.871187i $$0.336649\pi$$
$$734$$ −18.7453 −0.691902
$$735$$ 0 0
$$736$$ 4.91934 0.181329
$$737$$ −29.3434 −1.08088
$$738$$ 0 0
$$739$$ 8.14728 0.299702 0.149851 0.988709i $$-0.452121\pi$$
0.149851 + 0.988709i $$0.452121\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 5.18336 0.190287
$$743$$ −35.8247 −1.31428 −0.657139 0.753769i $$-0.728234\pi$$
−0.657139 + 0.753769i $$0.728234\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.69735 −0.0621445
$$747$$ 0 0
$$748$$ 7.50466 0.274398
$$749$$ −0.0572907 −0.00209336
$$750$$ 0 0
$$751$$ −14.8994 −0.543686 −0.271843 0.962342i $$-0.587633\pi$$
−0.271843 + 0.962342i $$0.587633\pi$$
$$752$$ 10.2827 0.374970
$$753$$ 0 0
$$754$$ −1.01065 −0.0368056
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −47.7920 −1.73703 −0.868514 0.495664i $$-0.834925\pi$$
−0.868514 + 0.495664i $$0.834925\pi$$
$$758$$ −2.63667 −0.0957682
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.9053 1.41032 0.705158 0.709050i $$-0.250876\pi$$
0.705158 + 0.709050i $$0.250876\pi$$
$$762$$ 0 0
$$763$$ −8.65533 −0.313344
$$764$$ −14.6367 −0.529536
$$765$$ 0 0
$$766$$ 12.4953 0.451475
$$767$$ −0.797984 −0.0288135
$$768$$ 0 0
$$769$$ 8.74663 0.315412 0.157706 0.987486i $$-0.449590\pi$$
0.157706 + 0.987486i $$0.449590\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 20.0187 0.720487
$$773$$ 23.4707 0.844181 0.422090 0.906554i $$-0.361297\pi$$
0.422090 + 0.906554i $$0.361297\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −14.5653 −0.522865
$$777$$ 0 0
$$778$$ −4.51399 −0.161834
$$779$$ 4.23132 0.151603
$$780$$ 0 0
$$781$$ −31.3947 −1.12339
$$782$$ −10.5340 −0.376694
$$783$$ 0 0
$$784$$ −6.59465 −0.235523
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.26063 0.0449364 0.0224682 0.999748i $$-0.492848\pi$$
0.0224682 + 0.999748i $$0.492848\pi$$
$$788$$ 9.94865 0.354406
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7.33657 −0.260859
$$792$$ 0 0
$$793$$ 0.918026 0.0326000
$$794$$ 35.6774 1.26614
$$795$$ 0 0
$$796$$ 9.74870 0.345534
$$797$$ 38.6481 1.36898 0.684492 0.729020i $$-0.260024\pi$$
0.684492 + 0.729020i $$0.260024\pi$$
$$798$$ 0 0
$$799$$ −22.0187 −0.778964
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 15.3434 0.541793
$$803$$ 12.9580 0.457277
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.10063 −0.0387681
$$807$$ 0 0
$$808$$ −16.6167 −0.584573
$$809$$ −51.6506 −1.81594 −0.907970 0.419036i $$-0.862368\pi$$
−0.907970 + 0.419036i $$0.862368\pi$$
$$810$$ 0 0
$$811$$ 8.19269 0.287684 0.143842 0.989601i $$-0.454054\pi$$
0.143842 + 0.989601i $$0.454054\pi$$
$$812$$ −4.55263 −0.159766
$$813$$ 0 0
$$814$$ 11.4720 0.402092
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.49534 0.0873007
$$818$$ −29.3620 −1.02662
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 49.9600 1.74362 0.871809 0.489846i $$-0.162947\pi$$
0.871809 + 0.489846i $$0.162947\pi$$
$$822$$ 0 0
$$823$$ −16.8421 −0.587078 −0.293539 0.955947i $$-0.594833\pi$$
−0.293539 + 0.955947i $$0.594833\pi$$
$$824$$ −9.06068 −0.315644
$$825$$ 0 0
$$826$$ −3.59465 −0.125074
$$827$$ 41.7487 1.45174 0.725872 0.687829i $$-0.241436\pi$$
0.725872 + 0.687829i $$0.241436\pi$$
$$828$$ 0 0
$$829$$ −5.98002 −0.207695 −0.103847 0.994593i $$-0.533115\pi$$
−0.103847 + 0.994593i $$0.533115\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −0.141336 −0.00489995
$$833$$ 14.1214 0.489276
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.50466 −0.121211
$$837$$ 0 0
$$838$$ 25.1379 0.868376
$$839$$ −32.0373 −1.10605 −0.553025 0.833164i $$-0.686527\pi$$
−0.553025 + 0.833164i $$0.686527\pi$$
$$840$$ 0 0
$$841$$ 22.1320 0.763173
$$842$$ −14.5454 −0.501267
$$843$$ 0 0
$$844$$ −20.7580 −0.714521
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.816641 0.0280601
$$848$$ −8.14134 −0.279575
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −16.1027 −0.551994
$$852$$ 0 0
$$853$$ −40.8480 −1.39861 −0.699305 0.714824i $$-0.746507\pi$$
−0.699305 + 0.714824i $$0.746507\pi$$
$$854$$ 4.13540 0.141510
$$855$$ 0 0
$$856$$ 0.0899847 0.00307561
$$857$$ −28.8667 −0.986067 −0.493033 0.870010i $$-0.664112\pi$$
−0.493033 + 0.870010i $$0.664112\pi$$
$$858$$ 0 0
$$859$$ −11.5047 −0.392534 −0.196267 0.980550i $$-0.562882\pi$$
−0.196267 + 0.980550i $$0.562882\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 19.4020 0.660833
$$863$$ 31.6074 1.07593 0.537964 0.842968i $$-0.319194\pi$$
0.537964 + 0.842968i $$0.319194\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −5.50466 −0.187056
$$867$$ 0 0
$$868$$ −4.95798 −0.168285
$$869$$ 14.6494 0.496947
$$870$$ 0 0
$$871$$ −1.18336 −0.0400966
$$872$$ 13.5946 0.460373
$$873$$ 0 0
$$874$$ 4.91934 0.166399
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 12.0641 0.407375 0.203687 0.979036i $$-0.434707\pi$$
0.203687 + 0.979036i $$0.434707\pi$$
$$878$$ −12.2500 −0.413417
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22.8480 0.769769 0.384885 0.922965i $$-0.374241\pi$$
0.384885 + 0.922965i $$0.374241\pi$$
$$882$$ 0 0
$$883$$ −34.1773 −1.15016 −0.575079 0.818098i $$-0.695029\pi$$
−0.575079 + 0.818098i $$0.695029\pi$$
$$884$$ 0.302648 0.0101792
$$885$$ 0 0
$$886$$ 31.6006 1.06164
$$887$$ −26.6167 −0.893701 −0.446851 0.894609i $$-0.647454\pi$$
−0.446851 + 0.894609i $$0.647454\pi$$
$$888$$ 0 0
$$889$$ −2.09592 −0.0702950
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −10.7267 −0.359155
$$893$$ 10.2827 0.344097
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −0.636672 −0.0212697
$$897$$ 0 0
$$898$$ −36.0187 −1.20196
$$899$$ 55.6846 1.85719
$$900$$ 0 0
$$901$$ 17.4333 0.580789
$$902$$ 14.8294 0.493764
$$903$$ 0 0
$$904$$ 11.5233 0.383260
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 13.1434 0.436420 0.218210 0.975902i $$-0.429978\pi$$
0.218210 + 0.975902i $$0.429978\pi$$
$$908$$ −12.5526 −0.416574
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 32.9253 1.09086 0.545432 0.838155i $$-0.316366\pi$$
0.545432 + 0.838155i $$0.316366\pi$$
$$912$$ 0 0
$$913$$ −31.5747 −1.04497
$$914$$ −22.1413 −0.732370
$$915$$ 0 0
$$916$$ 25.4720 0.841618
$$917$$ −11.4720 −0.378838
$$918$$ 0 0
$$919$$ −24.1273 −0.795886 −0.397943 0.917410i $$-0.630276\pi$$
−0.397943 + 0.917410i $$0.630276\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −2.31537 −0.0762525
$$923$$ −1.26609 −0.0416737
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −15.8387 −0.520492
$$927$$ 0 0
$$928$$ 7.15066 0.234732
$$929$$ 15.9359 0.522841 0.261420 0.965225i $$-0.415809\pi$$
0.261420 + 0.965225i $$0.415809\pi$$
$$930$$ 0 0
$$931$$ −6.59465 −0.216131
$$932$$ 3.11203 0.101938
$$933$$ 0 0
$$934$$ 23.1379 0.757097
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.5547 0.769498 0.384749 0.923021i $$-0.374288\pi$$
0.384749 + 0.923021i $$0.374288\pi$$
$$938$$ −5.33063 −0.174051
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 25.6974 0.837710 0.418855 0.908053i $$-0.362432\pi$$
0.418855 + 0.908053i $$0.362432\pi$$
$$942$$ 0 0
$$943$$ −20.8153 −0.677840
$$944$$ 5.64600 0.183762
$$945$$ 0 0
$$946$$ 8.74531 0.284335
$$947$$ 16.3013 0.529722 0.264861 0.964287i $$-0.414674\pi$$
0.264861 + 0.964287i $$0.414674\pi$$
$$948$$ 0 0
$$949$$ 0.522569 0.0169633
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.36333 0.0441857
$$953$$ −10.2754 −0.332853 −0.166427 0.986054i $$-0.553223\pi$$
−0.166427 + 0.986054i $$0.553223\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 1.54330 0.0499138
$$957$$ 0 0
$$958$$ −10.1214 −0.327006
$$959$$ 9.18336 0.296546
$$960$$ 0 0
$$961$$ 29.6426 0.956213
$$962$$ 0.462642 0.0149162
$$963$$ 0 0
$$964$$ 10.2827 0.331183
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −53.2920 −1.71376 −0.856878 0.515520i $$-0.827599\pi$$
−0.856878 + 0.515520i $$0.827599\pi$$
$$968$$ −1.28267 −0.0412266
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −2.54669 −0.0817271 −0.0408635 0.999165i $$-0.513011\pi$$
−0.0408635 + 0.999165i $$0.513011\pi$$
$$972$$ 0 0
$$973$$ 9.85057 0.315795
$$974$$ −20.3854 −0.653190
$$975$$ 0 0
$$976$$ −6.49534 −0.207911
$$977$$ 49.2966 1.57714 0.788569 0.614946i $$-0.210822\pi$$
0.788569 + 0.614946i $$0.210822\pi$$
$$978$$ 0 0
$$979$$ −23.7546 −0.759202
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −7.78734 −0.248504
$$983$$ 36.7453 1.17199 0.585997 0.810313i $$-0.300703\pi$$
0.585997 + 0.810313i $$0.300703\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −15.3120 −0.487633
$$987$$ 0 0
$$988$$ −0.141336 −0.00449650
$$989$$ −12.2754 −0.390335
$$990$$ 0 0
$$991$$ −35.5674 −1.12984 −0.564918 0.825147i $$-0.691092\pi$$
−0.564918 + 0.825147i $$0.691092\pi$$
$$992$$ 7.78734 0.247248
$$993$$ 0 0
$$994$$ −5.70329 −0.180897
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −48.4299 −1.53379 −0.766896 0.641771i $$-0.778200\pi$$
−0.766896 + 0.641771i $$0.778200\pi$$
$$998$$ −4.31537 −0.136601
$$999$$ 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 8550.2.a.ck.1.2 3
3.2 odd 2 950.2.a.n.1.2 3
5.2 odd 4 1710.2.d.d.1369.2 6
5.3 odd 4 1710.2.d.d.1369.5 6
5.4 even 2 8550.2.a.cl.1.2 3
12.11 even 2 7600.2.a.bi.1.2 3
15.2 even 4 190.2.b.b.39.5 yes 6
15.8 even 4 190.2.b.b.39.2 6
15.14 odd 2 950.2.a.i.1.2 3
60.23 odd 4 1520.2.d.j.609.3 6
60.47 odd 4 1520.2.d.j.609.4 6
60.59 even 2 7600.2.a.cd.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.2 6 15.8 even 4
190.2.b.b.39.5 yes 6 15.2 even 4
950.2.a.i.1.2 3 15.14 odd 2
950.2.a.n.1.2 3 3.2 odd 2
1520.2.d.j.609.3 6 60.23 odd 4
1520.2.d.j.609.4 6 60.47 odd 4
1710.2.d.d.1369.2 6 5.2 odd 4
1710.2.d.d.1369.5 6 5.3 odd 4
7600.2.a.bi.1.2 3 12.11 even 2
7600.2.a.cd.1.2 3 60.59 even 2
8550.2.a.ck.1.2 3 1.1 even 1 trivial
8550.2.a.cl.1.2 3 5.4 even 2