# Properties

 Label 8470.2.a.bk.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} -1.61803 q^{12} -0.763932 q^{13} -1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} +0.618034 q^{17} +0.381966 q^{18} -2.61803 q^{19} -1.00000 q^{20} -1.61803 q^{21} +0.472136 q^{23} +1.61803 q^{24} +1.00000 q^{25} +0.763932 q^{26} +5.47214 q^{27} +1.00000 q^{28} +4.00000 q^{29} -1.61803 q^{30} +4.47214 q^{31} -1.00000 q^{32} -0.618034 q^{34} -1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} +2.61803 q^{38} +1.23607 q^{39} +1.00000 q^{40} +6.38197 q^{41} +1.61803 q^{42} -2.38197 q^{43} +0.381966 q^{45} -0.472136 q^{46} -11.2361 q^{47} -1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -0.763932 q^{52} +2.47214 q^{53} -5.47214 q^{54} -1.00000 q^{56} +4.23607 q^{57} -4.00000 q^{58} -1.38197 q^{59} +1.61803 q^{60} -8.00000 q^{61} -4.47214 q^{62} -0.381966 q^{63} +1.00000 q^{64} +0.763932 q^{65} +5.09017 q^{67} +0.618034 q^{68} -0.763932 q^{69} +1.00000 q^{70} -6.47214 q^{71} +0.381966 q^{72} -11.3262 q^{73} +3.70820 q^{74} -1.61803 q^{75} -2.61803 q^{76} -1.23607 q^{78} -11.7082 q^{79} -1.00000 q^{80} -7.70820 q^{81} -6.38197 q^{82} -0.0901699 q^{83} -1.61803 q^{84} -0.618034 q^{85} +2.38197 q^{86} -6.47214 q^{87} +3.14590 q^{89} -0.381966 q^{90} -0.763932 q^{91} +0.472136 q^{92} -7.23607 q^{93} +11.2361 q^{94} +2.61803 q^{95} +1.61803 q^{96} +13.7984 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 $$2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 2 q^{10} - q^{12} - 6 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} - q^{17} + 3 q^{18} - 3 q^{19} - 2 q^{20} - q^{21} - 8 q^{23} + q^{24} + 2 q^{25} + 6 q^{26} + 2 q^{27} + 2 q^{28} + 8 q^{29} - q^{30} - 2 q^{32} + q^{34} - 2 q^{35} - 3 q^{36} + 6 q^{37} + 3 q^{38} - 2 q^{39} + 2 q^{40} + 15 q^{41} + q^{42} - 7 q^{43} + 3 q^{45} + 8 q^{46} - 18 q^{47} - q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{51} - 6 q^{52} - 4 q^{53} - 2 q^{54} - 2 q^{56} + 4 q^{57} - 8 q^{58} - 5 q^{59} + q^{60} - 16 q^{61} - 3 q^{63} + 2 q^{64} + 6 q^{65} - q^{67} - q^{68} - 6 q^{69} + 2 q^{70} - 4 q^{71} + 3 q^{72} - 7 q^{73} - 6 q^{74} - q^{75} - 3 q^{76} + 2 q^{78} - 10 q^{79} - 2 q^{80} - 2 q^{81} - 15 q^{82} + 11 q^{83} - q^{84} + q^{85} + 7 q^{86} - 4 q^{87} + 13 q^{89} - 3 q^{90} - 6 q^{91} - 8 q^{92} - 10 q^{93} + 18 q^{94} + 3 q^{95} + q^{96} + 3 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 + 2 * q^10 - q^12 - 6 * q^13 - 2 * q^14 + q^15 + 2 * q^16 - q^17 + 3 * q^18 - 3 * q^19 - 2 * q^20 - q^21 - 8 * q^23 + q^24 + 2 * q^25 + 6 * q^26 + 2 * q^27 + 2 * q^28 + 8 * q^29 - q^30 - 2 * q^32 + q^34 - 2 * q^35 - 3 * q^36 + 6 * q^37 + 3 * q^38 - 2 * q^39 + 2 * q^40 + 15 * q^41 + q^42 - 7 * q^43 + 3 * q^45 + 8 * q^46 - 18 * q^47 - q^48 + 2 * q^49 - 2 * q^50 - 2 * q^51 - 6 * q^52 - 4 * q^53 - 2 * q^54 - 2 * q^56 + 4 * q^57 - 8 * q^58 - 5 * q^59 + q^60 - 16 * q^61 - 3 * q^63 + 2 * q^64 + 6 * q^65 - q^67 - q^68 - 6 * q^69 + 2 * q^70 - 4 * q^71 + 3 * q^72 - 7 * q^73 - 6 * q^74 - q^75 - 3 * q^76 + 2 * q^78 - 10 * q^79 - 2 * q^80 - 2 * q^81 - 15 * q^82 + 11 * q^83 - q^84 + q^85 + 7 * q^86 - 4 * q^87 + 13 * q^89 - 3 * q^90 - 6 * q^91 - 8 * q^92 - 10 * q^93 + 18 * q^94 + 3 * q^95 + q^96 + 3 * q^97 - 2 * q^98

## 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
$$3$$ −1.61803 −0.934172 −0.467086 0.884212i $$-0.654696\pi$$
−0.467086 + 0.884212i $$0.654696\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.61803 0.660560
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −0.381966 −0.127322
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ −1.61803 −0.467086
$$13$$ −0.763932 −0.211877 −0.105938 0.994373i $$-0.533785\pi$$
−0.105938 + 0.994373i $$0.533785\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 1.61803 0.417775
$$16$$ 1.00000 0.250000
$$17$$ 0.618034 0.149895 0.0749476 0.997187i $$-0.476121\pi$$
0.0749476 + 0.997187i $$0.476121\pi$$
$$18$$ 0.381966 0.0900303
$$19$$ −2.61803 −0.600618 −0.300309 0.953842i $$-0.597090\pi$$
−0.300309 + 0.953842i $$0.597090\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −1.61803 −0.353084
$$22$$ 0 0
$$23$$ 0.472136 0.0984472 0.0492236 0.998788i $$-0.484325\pi$$
0.0492236 + 0.998788i $$0.484325\pi$$
$$24$$ 1.61803 0.330280
$$25$$ 1.00000 0.200000
$$26$$ 0.763932 0.149819
$$27$$ 5.47214 1.05311
$$28$$ 1.00000 0.188982
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ −1.61803 −0.295411
$$31$$ 4.47214 0.803219 0.401610 0.915811i $$-0.368451\pi$$
0.401610 + 0.915811i $$0.368451\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −0.618034 −0.105992
$$35$$ −1.00000 −0.169031
$$36$$ −0.381966 −0.0636610
$$37$$ −3.70820 −0.609625 −0.304812 0.952412i $$-0.598594\pi$$
−0.304812 + 0.952412i $$0.598594\pi$$
$$38$$ 2.61803 0.424701
$$39$$ 1.23607 0.197929
$$40$$ 1.00000 0.158114
$$41$$ 6.38197 0.996696 0.498348 0.866977i $$-0.333940\pi$$
0.498348 + 0.866977i $$0.333940\pi$$
$$42$$ 1.61803 0.249668
$$43$$ −2.38197 −0.363246 −0.181623 0.983368i $$-0.558135\pi$$
−0.181623 + 0.983368i $$0.558135\pi$$
$$44$$ 0 0
$$45$$ 0.381966 0.0569401
$$46$$ −0.472136 −0.0696126
$$47$$ −11.2361 −1.63895 −0.819474 0.573116i $$-0.805735\pi$$
−0.819474 + 0.573116i $$0.805735\pi$$
$$48$$ −1.61803 −0.233543
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −1.00000 −0.140028
$$52$$ −0.763932 −0.105938
$$53$$ 2.47214 0.339574 0.169787 0.985481i $$-0.445692\pi$$
0.169787 + 0.985481i $$0.445692\pi$$
$$54$$ −5.47214 −0.744663
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 4.23607 0.561081
$$58$$ −4.00000 −0.525226
$$59$$ −1.38197 −0.179917 −0.0899583 0.995946i $$-0.528673\pi$$
−0.0899583 + 0.995946i $$0.528673\pi$$
$$60$$ 1.61803 0.208887
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −4.47214 −0.567962
$$63$$ −0.381966 −0.0481232
$$64$$ 1.00000 0.125000
$$65$$ 0.763932 0.0947541
$$66$$ 0 0
$$67$$ 5.09017 0.621863 0.310932 0.950432i $$-0.399359\pi$$
0.310932 + 0.950432i $$0.399359\pi$$
$$68$$ 0.618034 0.0749476
$$69$$ −0.763932 −0.0919666
$$70$$ 1.00000 0.119523
$$71$$ −6.47214 −0.768101 −0.384051 0.923312i $$-0.625471\pi$$
−0.384051 + 0.923312i $$0.625471\pi$$
$$72$$ 0.381966 0.0450151
$$73$$ −11.3262 −1.32564 −0.662818 0.748781i $$-0.730640\pi$$
−0.662818 + 0.748781i $$0.730640\pi$$
$$74$$ 3.70820 0.431070
$$75$$ −1.61803 −0.186834
$$76$$ −2.61803 −0.300309
$$77$$ 0 0
$$78$$ −1.23607 −0.139957
$$79$$ −11.7082 −1.31728 −0.658638 0.752460i $$-0.728867\pi$$
−0.658638 + 0.752460i $$0.728867\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ −7.70820 −0.856467
$$82$$ −6.38197 −0.704770
$$83$$ −0.0901699 −0.00989744 −0.00494872 0.999988i $$-0.501575\pi$$
−0.00494872 + 0.999988i $$0.501575\pi$$
$$84$$ −1.61803 −0.176542
$$85$$ −0.618034 −0.0670352
$$86$$ 2.38197 0.256854
$$87$$ −6.47214 −0.693886
$$88$$ 0 0
$$89$$ 3.14590 0.333465 0.166732 0.986002i $$-0.446678\pi$$
0.166732 + 0.986002i $$0.446678\pi$$
$$90$$ −0.381966 −0.0402628
$$91$$ −0.763932 −0.0800818
$$92$$ 0.472136 0.0492236
$$93$$ −7.23607 −0.750345
$$94$$ 11.2361 1.15891
$$95$$ 2.61803 0.268605
$$96$$ 1.61803 0.165140
$$97$$ 13.7984 1.40101 0.700506 0.713646i $$-0.252958\pi$$
0.700506 + 0.713646i $$0.252958\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.7639 1.27006 0.635029 0.772488i $$-0.280988\pi$$
0.635029 + 0.772488i $$0.280988\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 13.7082 1.35071 0.675355 0.737493i $$-0.263991\pi$$
0.675355 + 0.737493i $$0.263991\pi$$
$$104$$ 0.763932 0.0749097
$$105$$ 1.61803 0.157904
$$106$$ −2.47214 −0.240115
$$107$$ 8.09017 0.782106 0.391053 0.920368i $$-0.372111\pi$$
0.391053 + 0.920368i $$0.372111\pi$$
$$108$$ 5.47214 0.526557
$$109$$ 10.1803 0.975100 0.487550 0.873095i $$-0.337891\pi$$
0.487550 + 0.873095i $$0.337891\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 1.00000 0.0944911
$$113$$ −10.7984 −1.01583 −0.507913 0.861409i $$-0.669583\pi$$
−0.507913 + 0.861409i $$0.669583\pi$$
$$114$$ −4.23607 −0.396744
$$115$$ −0.472136 −0.0440269
$$116$$ 4.00000 0.371391
$$117$$ 0.291796 0.0269766
$$118$$ 1.38197 0.127220
$$119$$ 0.618034 0.0566551
$$120$$ −1.61803 −0.147706
$$121$$ 0 0
$$122$$ 8.00000 0.724286
$$123$$ −10.3262 −0.931086
$$124$$ 4.47214 0.401610
$$125$$ −1.00000 −0.0894427
$$126$$ 0.381966 0.0340282
$$127$$ −8.18034 −0.725888 −0.362944 0.931811i $$-0.618228\pi$$
−0.362944 + 0.931811i $$0.618228\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 3.85410 0.339335
$$130$$ −0.763932 −0.0670013
$$131$$ 8.14590 0.711710 0.355855 0.934541i $$-0.384190\pi$$
0.355855 + 0.934541i $$0.384190\pi$$
$$132$$ 0 0
$$133$$ −2.61803 −0.227012
$$134$$ −5.09017 −0.439724
$$135$$ −5.47214 −0.470966
$$136$$ −0.618034 −0.0529960
$$137$$ −12.8541 −1.09820 −0.549100 0.835757i $$-0.685029\pi$$
−0.549100 + 0.835757i $$0.685029\pi$$
$$138$$ 0.763932 0.0650302
$$139$$ −1.52786 −0.129592 −0.0647959 0.997899i $$-0.520640\pi$$
−0.0647959 + 0.997899i $$0.520640\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 18.1803 1.53106
$$142$$ 6.47214 0.543130
$$143$$ 0 0
$$144$$ −0.381966 −0.0318305
$$145$$ −4.00000 −0.332182
$$146$$ 11.3262 0.937366
$$147$$ −1.61803 −0.133453
$$148$$ −3.70820 −0.304812
$$149$$ −7.70820 −0.631481 −0.315740 0.948846i $$-0.602253\pi$$
−0.315740 + 0.948846i $$0.602253\pi$$
$$150$$ 1.61803 0.132112
$$151$$ 20.1803 1.64225 0.821126 0.570746i $$-0.193346\pi$$
0.821126 + 0.570746i $$0.193346\pi$$
$$152$$ 2.61803 0.212351
$$153$$ −0.236068 −0.0190850
$$154$$ 0 0
$$155$$ −4.47214 −0.359211
$$156$$ 1.23607 0.0989646
$$157$$ −6.18034 −0.493245 −0.246622 0.969112i $$-0.579321\pi$$
−0.246622 + 0.969112i $$0.579321\pi$$
$$158$$ 11.7082 0.931455
$$159$$ −4.00000 −0.317221
$$160$$ 1.00000 0.0790569
$$161$$ 0.472136 0.0372095
$$162$$ 7.70820 0.605614
$$163$$ −22.6180 −1.77158 −0.885791 0.464085i $$-0.846383\pi$$
−0.885791 + 0.464085i $$0.846383\pi$$
$$164$$ 6.38197 0.498348
$$165$$ 0 0
$$166$$ 0.0901699 0.00699854
$$167$$ 10.0000 0.773823 0.386912 0.922117i $$-0.373542\pi$$
0.386912 + 0.922117i $$0.373542\pi$$
$$168$$ 1.61803 0.124834
$$169$$ −12.4164 −0.955108
$$170$$ 0.618034 0.0474010
$$171$$ 1.00000 0.0764719
$$172$$ −2.38197 −0.181623
$$173$$ 4.76393 0.362195 0.181098 0.983465i $$-0.442035\pi$$
0.181098 + 0.983465i $$0.442035\pi$$
$$174$$ 6.47214 0.490651
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 2.23607 0.168073
$$178$$ −3.14590 −0.235795
$$179$$ −8.14590 −0.608853 −0.304427 0.952536i $$-0.598465\pi$$
−0.304427 + 0.952536i $$0.598465\pi$$
$$180$$ 0.381966 0.0284701
$$181$$ −23.1246 −1.71884 −0.859419 0.511271i $$-0.829175\pi$$
−0.859419 + 0.511271i $$0.829175\pi$$
$$182$$ 0.763932 0.0566264
$$183$$ 12.9443 0.956868
$$184$$ −0.472136 −0.0348063
$$185$$ 3.70820 0.272633
$$186$$ 7.23607 0.530574
$$187$$ 0 0
$$188$$ −11.2361 −0.819474
$$189$$ 5.47214 0.398039
$$190$$ −2.61803 −0.189932
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ −1.61803 −0.116772
$$193$$ −3.52786 −0.253941 −0.126971 0.991906i $$-0.540525\pi$$
−0.126971 + 0.991906i $$0.540525\pi$$
$$194$$ −13.7984 −0.990666
$$195$$ −1.23607 −0.0885167
$$196$$ 1.00000 0.0714286
$$197$$ −5.41641 −0.385903 −0.192952 0.981208i $$-0.561806\pi$$
−0.192952 + 0.981208i $$0.561806\pi$$
$$198$$ 0 0
$$199$$ 3.23607 0.229399 0.114699 0.993400i $$-0.463410\pi$$
0.114699 + 0.993400i $$0.463410\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −8.23607 −0.580927
$$202$$ −12.7639 −0.898067
$$203$$ 4.00000 0.280745
$$204$$ −1.00000 −0.0700140
$$205$$ −6.38197 −0.445736
$$206$$ −13.7082 −0.955096
$$207$$ −0.180340 −0.0125345
$$208$$ −0.763932 −0.0529692
$$209$$ 0 0
$$210$$ −1.61803 −0.111655
$$211$$ 28.0902 1.93381 0.966904 0.255142i $$-0.0821223\pi$$
0.966904 + 0.255142i $$0.0821223\pi$$
$$212$$ 2.47214 0.169787
$$213$$ 10.4721 0.717539
$$214$$ −8.09017 −0.553033
$$215$$ 2.38197 0.162449
$$216$$ −5.47214 −0.372332
$$217$$ 4.47214 0.303588
$$218$$ −10.1803 −0.689500
$$219$$ 18.3262 1.23837
$$220$$ 0 0
$$221$$ −0.472136 −0.0317593
$$222$$ −6.00000 −0.402694
$$223$$ −22.9443 −1.53646 −0.768231 0.640173i $$-0.778863\pi$$
−0.768231 + 0.640173i $$0.778863\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −0.381966 −0.0254644
$$226$$ 10.7984 0.718297
$$227$$ −11.5623 −0.767417 −0.383709 0.923454i $$-0.625353\pi$$
−0.383709 + 0.923454i $$0.625353\pi$$
$$228$$ 4.23607 0.280540
$$229$$ 7.70820 0.509372 0.254686 0.967024i $$-0.418028\pi$$
0.254686 + 0.967024i $$0.418028\pi$$
$$230$$ 0.472136 0.0311317
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ 19.3262 1.26610 0.633052 0.774109i $$-0.281802\pi$$
0.633052 + 0.774109i $$0.281802\pi$$
$$234$$ −0.291796 −0.0190753
$$235$$ 11.2361 0.732960
$$236$$ −1.38197 −0.0899583
$$237$$ 18.9443 1.23056
$$238$$ −0.618034 −0.0400612
$$239$$ 4.29180 0.277613 0.138807 0.990320i $$-0.455673\pi$$
0.138807 + 0.990320i $$0.455673\pi$$
$$240$$ 1.61803 0.104444
$$241$$ −13.8541 −0.892421 −0.446211 0.894928i $$-0.647227\pi$$
−0.446211 + 0.894928i $$0.647227\pi$$
$$242$$ 0 0
$$243$$ −3.94427 −0.253025
$$244$$ −8.00000 −0.512148
$$245$$ −1.00000 −0.0638877
$$246$$ 10.3262 0.658377
$$247$$ 2.00000 0.127257
$$248$$ −4.47214 −0.283981
$$249$$ 0.145898 0.00924591
$$250$$ 1.00000 0.0632456
$$251$$ 17.8885 1.12911 0.564557 0.825394i $$-0.309047\pi$$
0.564557 + 0.825394i $$0.309047\pi$$
$$252$$ −0.381966 −0.0240616
$$253$$ 0 0
$$254$$ 8.18034 0.513280
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −5.43769 −0.339194 −0.169597 0.985513i $$-0.554247\pi$$
−0.169597 + 0.985513i $$0.554247\pi$$
$$258$$ −3.85410 −0.239946
$$259$$ −3.70820 −0.230417
$$260$$ 0.763932 0.0473771
$$261$$ −1.52786 −0.0945724
$$262$$ −8.14590 −0.503255
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ −2.47214 −0.151862
$$266$$ 2.61803 0.160522
$$267$$ −5.09017 −0.311513
$$268$$ 5.09017 0.310932
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 5.47214 0.333024
$$271$$ 6.29180 0.382199 0.191100 0.981571i $$-0.438795\pi$$
0.191100 + 0.981571i $$0.438795\pi$$
$$272$$ 0.618034 0.0374738
$$273$$ 1.23607 0.0748102
$$274$$ 12.8541 0.776545
$$275$$ 0 0
$$276$$ −0.763932 −0.0459833
$$277$$ −17.1246 −1.02892 −0.514459 0.857515i $$-0.672007\pi$$
−0.514459 + 0.857515i $$0.672007\pi$$
$$278$$ 1.52786 0.0916352
$$279$$ −1.70820 −0.102267
$$280$$ 1.00000 0.0597614
$$281$$ 17.7984 1.06176 0.530881 0.847446i $$-0.321861\pi$$
0.530881 + 0.847446i $$0.321861\pi$$
$$282$$ −18.1803 −1.08262
$$283$$ 9.88854 0.587813 0.293906 0.955834i $$-0.405045\pi$$
0.293906 + 0.955834i $$0.405045\pi$$
$$284$$ −6.47214 −0.384051
$$285$$ −4.23607 −0.250923
$$286$$ 0 0
$$287$$ 6.38197 0.376716
$$288$$ 0.381966 0.0225076
$$289$$ −16.6180 −0.977531
$$290$$ 4.00000 0.234888
$$291$$ −22.3262 −1.30879
$$292$$ −11.3262 −0.662818
$$293$$ 15.7082 0.917683 0.458842 0.888518i $$-0.348265\pi$$
0.458842 + 0.888518i $$0.348265\pi$$
$$294$$ 1.61803 0.0943657
$$295$$ 1.38197 0.0804612
$$296$$ 3.70820 0.215535
$$297$$ 0 0
$$298$$ 7.70820 0.446524
$$299$$ −0.360680 −0.0208586
$$300$$ −1.61803 −0.0934172
$$301$$ −2.38197 −0.137294
$$302$$ −20.1803 −1.16125
$$303$$ −20.6525 −1.18645
$$304$$ −2.61803 −0.150155
$$305$$ 8.00000 0.458079
$$306$$ 0.236068 0.0134951
$$307$$ −3.50658 −0.200131 −0.100065 0.994981i $$-0.531905\pi$$
−0.100065 + 0.994981i $$0.531905\pi$$
$$308$$ 0 0
$$309$$ −22.1803 −1.26180
$$310$$ 4.47214 0.254000
$$311$$ −1.23607 −0.0700910 −0.0350455 0.999386i $$-0.511158\pi$$
−0.0350455 + 0.999386i $$0.511158\pi$$
$$312$$ −1.23607 −0.0699786
$$313$$ −12.3262 −0.696720 −0.348360 0.937361i $$-0.613261\pi$$
−0.348360 + 0.937361i $$0.613261\pi$$
$$314$$ 6.18034 0.348777
$$315$$ 0.381966 0.0215213
$$316$$ −11.7082 −0.658638
$$317$$ 17.8885 1.00472 0.502360 0.864658i $$-0.332465\pi$$
0.502360 + 0.864658i $$0.332465\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −13.0902 −0.730622
$$322$$ −0.472136 −0.0263111
$$323$$ −1.61803 −0.0900298
$$324$$ −7.70820 −0.428234
$$325$$ −0.763932 −0.0423753
$$326$$ 22.6180 1.25270
$$327$$ −16.4721 −0.910911
$$328$$ −6.38197 −0.352385
$$329$$ −11.2361 −0.619464
$$330$$ 0 0
$$331$$ −3.32624 −0.182827 −0.0914133 0.995813i $$-0.529138\pi$$
−0.0914133 + 0.995813i $$0.529138\pi$$
$$332$$ −0.0901699 −0.00494872
$$333$$ 1.41641 0.0776187
$$334$$ −10.0000 −0.547176
$$335$$ −5.09017 −0.278106
$$336$$ −1.61803 −0.0882710
$$337$$ −1.27051 −0.0692091 −0.0346045 0.999401i $$-0.511017\pi$$
−0.0346045 + 0.999401i $$0.511017\pi$$
$$338$$ 12.4164 0.675364
$$339$$ 17.4721 0.948956
$$340$$ −0.618034 −0.0335176
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 1.00000 0.0539949
$$344$$ 2.38197 0.128427
$$345$$ 0.763932 0.0411287
$$346$$ −4.76393 −0.256111
$$347$$ 2.03444 0.109215 0.0546073 0.998508i $$-0.482609\pi$$
0.0546073 + 0.998508i $$0.482609\pi$$
$$348$$ −6.47214 −0.346943
$$349$$ 1.52786 0.0817847 0.0408923 0.999164i $$-0.486980\pi$$
0.0408923 + 0.999164i $$0.486980\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −4.18034 −0.223130
$$352$$ 0 0
$$353$$ −12.0344 −0.640529 −0.320264 0.947328i $$-0.603772\pi$$
−0.320264 + 0.947328i $$0.603772\pi$$
$$354$$ −2.23607 −0.118846
$$355$$ 6.47214 0.343505
$$356$$ 3.14590 0.166732
$$357$$ −1.00000 −0.0529256
$$358$$ 8.14590 0.430524
$$359$$ 12.2918 0.648736 0.324368 0.945931i $$-0.394848\pi$$
0.324368 + 0.945931i $$0.394848\pi$$
$$360$$ −0.381966 −0.0201314
$$361$$ −12.1459 −0.639258
$$362$$ 23.1246 1.21540
$$363$$ 0 0
$$364$$ −0.763932 −0.0400409
$$365$$ 11.3262 0.592842
$$366$$ −12.9443 −0.676608
$$367$$ 28.9443 1.51088 0.755439 0.655219i $$-0.227423\pi$$
0.755439 + 0.655219i $$0.227423\pi$$
$$368$$ 0.472136 0.0246118
$$369$$ −2.43769 −0.126901
$$370$$ −3.70820 −0.192780
$$371$$ 2.47214 0.128347
$$372$$ −7.23607 −0.375173
$$373$$ 12.9443 0.670229 0.335114 0.942177i $$-0.391225\pi$$
0.335114 + 0.942177i $$0.391225\pi$$
$$374$$ 0 0
$$375$$ 1.61803 0.0835549
$$376$$ 11.2361 0.579456
$$377$$ −3.05573 −0.157378
$$378$$ −5.47214 −0.281456
$$379$$ −16.8541 −0.865737 −0.432869 0.901457i $$-0.642499\pi$$
−0.432869 + 0.901457i $$0.642499\pi$$
$$380$$ 2.61803 0.134302
$$381$$ 13.2361 0.678104
$$382$$ 2.00000 0.102329
$$383$$ 4.29180 0.219301 0.109650 0.993970i $$-0.465027\pi$$
0.109650 + 0.993970i $$0.465027\pi$$
$$384$$ 1.61803 0.0825700
$$385$$ 0 0
$$386$$ 3.52786 0.179564
$$387$$ 0.909830 0.0462493
$$388$$ 13.7984 0.700506
$$389$$ 8.00000 0.405616 0.202808 0.979219i $$-0.434993\pi$$
0.202808 + 0.979219i $$0.434993\pi$$
$$390$$ 1.23607 0.0625907
$$391$$ 0.291796 0.0147568
$$392$$ −1.00000 −0.0505076
$$393$$ −13.1803 −0.664860
$$394$$ 5.41641 0.272875
$$395$$ 11.7082 0.589104
$$396$$ 0 0
$$397$$ 32.5410 1.63319 0.816593 0.577213i $$-0.195860\pi$$
0.816593 + 0.577213i $$0.195860\pi$$
$$398$$ −3.23607 −0.162209
$$399$$ 4.23607 0.212069
$$400$$ 1.00000 0.0500000
$$401$$ 27.5066 1.37361 0.686806 0.726840i $$-0.259012\pi$$
0.686806 + 0.726840i $$0.259012\pi$$
$$402$$ 8.23607 0.410778
$$403$$ −3.41641 −0.170183
$$404$$ 12.7639 0.635029
$$405$$ 7.70820 0.383024
$$406$$ −4.00000 −0.198517
$$407$$ 0 0
$$408$$ 1.00000 0.0495074
$$409$$ 31.3050 1.54793 0.773965 0.633228i $$-0.218271\pi$$
0.773965 + 0.633228i $$0.218271\pi$$
$$410$$ 6.38197 0.315183
$$411$$ 20.7984 1.02591
$$412$$ 13.7082 0.675355
$$413$$ −1.38197 −0.0680021
$$414$$ 0.180340 0.00886322
$$415$$ 0.0901699 0.00442627
$$416$$ 0.763932 0.0374548
$$417$$ 2.47214 0.121061
$$418$$ 0 0
$$419$$ −14.7426 −0.720225 −0.360113 0.932909i $$-0.617262\pi$$
−0.360113 + 0.932909i $$0.617262\pi$$
$$420$$ 1.61803 0.0789520
$$421$$ 8.47214 0.412907 0.206453 0.978456i $$-0.433808\pi$$
0.206453 + 0.978456i $$0.433808\pi$$
$$422$$ −28.0902 −1.36741
$$423$$ 4.29180 0.208674
$$424$$ −2.47214 −0.120058
$$425$$ 0.618034 0.0299791
$$426$$ −10.4721 −0.507377
$$427$$ −8.00000 −0.387147
$$428$$ 8.09017 0.391053
$$429$$ 0 0
$$430$$ −2.38197 −0.114869
$$431$$ 19.7082 0.949311 0.474655 0.880172i $$-0.342573\pi$$
0.474655 + 0.880172i $$0.342573\pi$$
$$432$$ 5.47214 0.263278
$$433$$ 1.20163 0.0577465 0.0288732 0.999583i $$-0.490808\pi$$
0.0288732 + 0.999583i $$0.490808\pi$$
$$434$$ −4.47214 −0.214669
$$435$$ 6.47214 0.310315
$$436$$ 10.1803 0.487550
$$437$$ −1.23607 −0.0591292
$$438$$ −18.3262 −0.875662
$$439$$ 24.3607 1.16267 0.581336 0.813664i $$-0.302530\pi$$
0.581336 + 0.813664i $$0.302530\pi$$
$$440$$ 0 0
$$441$$ −0.381966 −0.0181889
$$442$$ 0.472136 0.0224572
$$443$$ −26.3820 −1.25345 −0.626723 0.779243i $$-0.715604\pi$$
−0.626723 + 0.779243i $$0.715604\pi$$
$$444$$ 6.00000 0.284747
$$445$$ −3.14590 −0.149130
$$446$$ 22.9443 1.08644
$$447$$ 12.4721 0.589912
$$448$$ 1.00000 0.0472456
$$449$$ 27.7426 1.30926 0.654628 0.755951i $$-0.272825\pi$$
0.654628 + 0.755951i $$0.272825\pi$$
$$450$$ 0.381966 0.0180061
$$451$$ 0 0
$$452$$ −10.7984 −0.507913
$$453$$ −32.6525 −1.53415
$$454$$ 11.5623 0.542646
$$455$$ 0.763932 0.0358137
$$456$$ −4.23607 −0.198372
$$457$$ −5.43769 −0.254365 −0.127182 0.991879i $$-0.540593\pi$$
−0.127182 + 0.991879i $$0.540593\pi$$
$$458$$ −7.70820 −0.360181
$$459$$ 3.38197 0.157857
$$460$$ −0.472136 −0.0220135
$$461$$ 39.1246 1.82221 0.911107 0.412169i $$-0.135229\pi$$
0.911107 + 0.412169i $$0.135229\pi$$
$$462$$ 0 0
$$463$$ −30.6525 −1.42454 −0.712271 0.701905i $$-0.752333\pi$$
−0.712271 + 0.701905i $$0.752333\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 7.23607 0.335565
$$466$$ −19.3262 −0.895271
$$467$$ −0.944272 −0.0436957 −0.0218478 0.999761i $$-0.506955\pi$$
−0.0218478 + 0.999761i $$0.506955\pi$$
$$468$$ 0.291796 0.0134883
$$469$$ 5.09017 0.235042
$$470$$ −11.2361 −0.518281
$$471$$ 10.0000 0.460776
$$472$$ 1.38197 0.0636101
$$473$$ 0 0
$$474$$ −18.9443 −0.870139
$$475$$ −2.61803 −0.120124
$$476$$ 0.618034 0.0283275
$$477$$ −0.944272 −0.0432352
$$478$$ −4.29180 −0.196302
$$479$$ 23.1246 1.05659 0.528295 0.849061i $$-0.322831\pi$$
0.528295 + 0.849061i $$0.322831\pi$$
$$480$$ −1.61803 −0.0738528
$$481$$ 2.83282 0.129165
$$482$$ 13.8541 0.631037
$$483$$ −0.763932 −0.0347601
$$484$$ 0 0
$$485$$ −13.7984 −0.626552
$$486$$ 3.94427 0.178916
$$487$$ −33.7771 −1.53059 −0.765293 0.643682i $$-0.777406\pi$$
−0.765293 + 0.643682i $$0.777406\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 36.5967 1.65496
$$490$$ 1.00000 0.0451754
$$491$$ 26.9230 1.21502 0.607509 0.794313i $$-0.292169\pi$$
0.607509 + 0.794313i $$0.292169\pi$$
$$492$$ −10.3262 −0.465543
$$493$$ 2.47214 0.111339
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 4.47214 0.200805
$$497$$ −6.47214 −0.290315
$$498$$ −0.145898 −0.00653785
$$499$$ −1.61803 −0.0724331 −0.0362166 0.999344i $$-0.511531\pi$$
−0.0362166 + 0.999344i $$0.511531\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −16.1803 −0.722884
$$502$$ −17.8885 −0.798405
$$503$$ −12.2918 −0.548064 −0.274032 0.961721i $$-0.588357\pi$$
−0.274032 + 0.961721i $$0.588357\pi$$
$$504$$ 0.381966 0.0170141
$$505$$ −12.7639 −0.567988
$$506$$ 0 0
$$507$$ 20.0902 0.892236
$$508$$ −8.18034 −0.362944
$$509$$ 5.81966 0.257952 0.128976 0.991648i $$-0.458831\pi$$
0.128976 + 0.991648i $$0.458831\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ −11.3262 −0.501043
$$512$$ −1.00000 −0.0441942
$$513$$ −14.3262 −0.632519
$$514$$ 5.43769 0.239846
$$515$$ −13.7082 −0.604056
$$516$$ 3.85410 0.169667
$$517$$ 0 0
$$518$$ 3.70820 0.162929
$$519$$ −7.70820 −0.338353
$$520$$ −0.763932 −0.0335006
$$521$$ 37.5623 1.64563 0.822817 0.568306i $$-0.192401\pi$$
0.822817 + 0.568306i $$0.192401\pi$$
$$522$$ 1.52786 0.0668728
$$523$$ −35.9230 −1.57080 −0.785401 0.618987i $$-0.787543\pi$$
−0.785401 + 0.618987i $$0.787543\pi$$
$$524$$ 8.14590 0.355855
$$525$$ −1.61803 −0.0706168
$$526$$ 12.0000 0.523225
$$527$$ 2.76393 0.120399
$$528$$ 0 0
$$529$$ −22.7771 −0.990308
$$530$$ 2.47214 0.107383
$$531$$ 0.527864 0.0229073
$$532$$ −2.61803 −0.113506
$$533$$ −4.87539 −0.211177
$$534$$ 5.09017 0.220273
$$535$$ −8.09017 −0.349769
$$536$$ −5.09017 −0.219862
$$537$$ 13.1803 0.568774
$$538$$ −14.0000 −0.603583
$$539$$ 0 0
$$540$$ −5.47214 −0.235483
$$541$$ 17.8197 0.766127 0.383064 0.923722i $$-0.374869\pi$$
0.383064 + 0.923722i $$0.374869\pi$$
$$542$$ −6.29180 −0.270256
$$543$$ 37.4164 1.60569
$$544$$ −0.618034 −0.0264980
$$545$$ −10.1803 −0.436078
$$546$$ −1.23607 −0.0528988
$$547$$ 26.9098 1.15058 0.575291 0.817949i $$-0.304889\pi$$
0.575291 + 0.817949i $$0.304889\pi$$
$$548$$ −12.8541 −0.549100
$$549$$ 3.05573 0.130415
$$550$$ 0 0
$$551$$ −10.4721 −0.446128
$$552$$ 0.763932 0.0325151
$$553$$ −11.7082 −0.497883
$$554$$ 17.1246 0.727555
$$555$$ −6.00000 −0.254686
$$556$$ −1.52786 −0.0647959
$$557$$ 4.36068 0.184768 0.0923840 0.995723i $$-0.470551\pi$$
0.0923840 + 0.995723i $$0.470551\pi$$
$$558$$ 1.70820 0.0723140
$$559$$ 1.81966 0.0769634
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −17.7984 −0.750779
$$563$$ 43.6869 1.84118 0.920592 0.390526i $$-0.127707\pi$$
0.920592 + 0.390526i $$0.127707\pi$$
$$564$$ 18.1803 0.765530
$$565$$ 10.7984 0.454291
$$566$$ −9.88854 −0.415646
$$567$$ −7.70820 −0.323714
$$568$$ 6.47214 0.271565
$$569$$ −19.4508 −0.815422 −0.407711 0.913111i $$-0.633673\pi$$
−0.407711 + 0.913111i $$0.633673\pi$$
$$570$$ 4.23607 0.177429
$$571$$ 26.8328 1.12292 0.561459 0.827504i $$-0.310240\pi$$
0.561459 + 0.827504i $$0.310240\pi$$
$$572$$ 0 0
$$573$$ 3.23607 0.135189
$$574$$ −6.38197 −0.266378
$$575$$ 0.472136 0.0196894
$$576$$ −0.381966 −0.0159153
$$577$$ 24.3262 1.01271 0.506357 0.862324i $$-0.330992\pi$$
0.506357 + 0.862324i $$0.330992\pi$$
$$578$$ 16.6180 0.691219
$$579$$ 5.70820 0.237225
$$580$$ −4.00000 −0.166091
$$581$$ −0.0901699 −0.00374088
$$582$$ 22.3262 0.925452
$$583$$ 0 0
$$584$$ 11.3262 0.468683
$$585$$ −0.291796 −0.0120643
$$586$$ −15.7082 −0.648900
$$587$$ −10.6180 −0.438253 −0.219127 0.975696i $$-0.570321\pi$$
−0.219127 + 0.975696i $$0.570321\pi$$
$$588$$ −1.61803 −0.0667266
$$589$$ −11.7082 −0.482428
$$590$$ −1.38197 −0.0568946
$$591$$ 8.76393 0.360500
$$592$$ −3.70820 −0.152406
$$593$$ −12.6180 −0.518161 −0.259080 0.965856i $$-0.583419\pi$$
−0.259080 + 0.965856i $$0.583419\pi$$
$$594$$ 0 0
$$595$$ −0.618034 −0.0253369
$$596$$ −7.70820 −0.315740
$$597$$ −5.23607 −0.214298
$$598$$ 0.360680 0.0147493
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 1.61803 0.0660560
$$601$$ 6.56231 0.267682 0.133841 0.991003i $$-0.457269\pi$$
0.133841 + 0.991003i $$0.457269\pi$$
$$602$$ 2.38197 0.0970817
$$603$$ −1.94427 −0.0791769
$$604$$ 20.1803 0.821126
$$605$$ 0 0
$$606$$ 20.6525 0.838949
$$607$$ −34.0000 −1.38002 −0.690009 0.723801i $$-0.742393\pi$$
−0.690009 + 0.723801i $$0.742393\pi$$
$$608$$ 2.61803 0.106175
$$609$$ −6.47214 −0.262264
$$610$$ −8.00000 −0.323911
$$611$$ 8.58359 0.347255
$$612$$ −0.236068 −0.00954248
$$613$$ 48.0689 1.94148 0.970742 0.240125i $$-0.0771885\pi$$
0.970742 + 0.240125i $$0.0771885\pi$$
$$614$$ 3.50658 0.141514
$$615$$ 10.3262 0.416394
$$616$$ 0 0
$$617$$ 13.2016 0.531477 0.265739 0.964045i $$-0.414384\pi$$
0.265739 + 0.964045i $$0.414384\pi$$
$$618$$ 22.1803 0.892224
$$619$$ 20.4377 0.821460 0.410730 0.911757i $$-0.365274\pi$$
0.410730 + 0.911757i $$0.365274\pi$$
$$620$$ −4.47214 −0.179605
$$621$$ 2.58359 0.103676
$$622$$ 1.23607 0.0495618
$$623$$ 3.14590 0.126038
$$624$$ 1.23607 0.0494823
$$625$$ 1.00000 0.0400000
$$626$$ 12.3262 0.492656
$$627$$ 0 0
$$628$$ −6.18034 −0.246622
$$629$$ −2.29180 −0.0913799
$$630$$ −0.381966 −0.0152179
$$631$$ −20.7639 −0.826599 −0.413300 0.910595i $$-0.635624\pi$$
−0.413300 + 0.910595i $$0.635624\pi$$
$$632$$ 11.7082 0.465727
$$633$$ −45.4508 −1.80651
$$634$$ −17.8885 −0.710445
$$635$$ 8.18034 0.324627
$$636$$ −4.00000 −0.158610
$$637$$ −0.763932 −0.0302681
$$638$$ 0 0
$$639$$ 2.47214 0.0977962
$$640$$ 1.00000 0.0395285
$$641$$ 24.6738 0.974555 0.487278 0.873247i $$-0.337990\pi$$
0.487278 + 0.873247i $$0.337990\pi$$
$$642$$ 13.0902 0.516628
$$643$$ 6.56231 0.258792 0.129396 0.991593i $$-0.458696\pi$$
0.129396 + 0.991593i $$0.458696\pi$$
$$644$$ 0.472136 0.0186048
$$645$$ −3.85410 −0.151755
$$646$$ 1.61803 0.0636607
$$647$$ −7.88854 −0.310131 −0.155065 0.987904i $$-0.549559\pi$$
−0.155065 + 0.987904i $$0.549559\pi$$
$$648$$ 7.70820 0.302807
$$649$$ 0 0
$$650$$ 0.763932 0.0299639
$$651$$ −7.23607 −0.283604
$$652$$ −22.6180 −0.885791
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 16.4721 0.644111
$$655$$ −8.14590 −0.318287
$$656$$ 6.38197 0.249174
$$657$$ 4.32624 0.168783
$$658$$ 11.2361 0.438028
$$659$$ 21.0902 0.821556 0.410778 0.911735i $$-0.365257\pi$$
0.410778 + 0.911735i $$0.365257\pi$$
$$660$$ 0 0
$$661$$ 37.7771 1.46936 0.734679 0.678415i $$-0.237333\pi$$
0.734679 + 0.678415i $$0.237333\pi$$
$$662$$ 3.32624 0.129278
$$663$$ 0.763932 0.0296687
$$664$$ 0.0901699 0.00349927
$$665$$ 2.61803 0.101523
$$666$$ −1.41641 −0.0548847
$$667$$ 1.88854 0.0731247
$$668$$ 10.0000 0.386912
$$669$$ 37.1246 1.43532
$$670$$ 5.09017 0.196650
$$671$$ 0 0
$$672$$ 1.61803 0.0624170
$$673$$ 13.8541 0.534036 0.267018 0.963691i $$-0.413962\pi$$
0.267018 + 0.963691i $$0.413962\pi$$
$$674$$ 1.27051 0.0489382
$$675$$ 5.47214 0.210623
$$676$$ −12.4164 −0.477554
$$677$$ −10.9443 −0.420623 −0.210311 0.977634i $$-0.567448\pi$$
−0.210311 + 0.977634i $$0.567448\pi$$
$$678$$ −17.4721 −0.671013
$$679$$ 13.7984 0.529533
$$680$$ 0.618034 0.0237005
$$681$$ 18.7082 0.716900
$$682$$ 0 0
$$683$$ −45.3050 −1.73355 −0.866773 0.498703i $$-0.833810\pi$$
−0.866773 + 0.498703i $$0.833810\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 12.8541 0.491130
$$686$$ −1.00000 −0.0381802
$$687$$ −12.4721 −0.475842
$$688$$ −2.38197 −0.0908116
$$689$$ −1.88854 −0.0719478
$$690$$ −0.763932 −0.0290824
$$691$$ −3.20163 −0.121796 −0.0608978 0.998144i $$-0.519396\pi$$
−0.0608978 + 0.998144i $$0.519396\pi$$
$$692$$ 4.76393 0.181098
$$693$$ 0 0
$$694$$ −2.03444 −0.0772264
$$695$$ 1.52786 0.0579552
$$696$$ 6.47214 0.245326
$$697$$ 3.94427 0.149400
$$698$$ −1.52786 −0.0578305
$$699$$ −31.2705 −1.18276
$$700$$ 1.00000 0.0377964
$$701$$ −4.18034 −0.157889 −0.0789446 0.996879i $$-0.525155\pi$$
−0.0789446 + 0.996879i $$0.525155\pi$$
$$702$$ 4.18034 0.157777
$$703$$ 9.70820 0.366152
$$704$$ 0 0
$$705$$ −18.1803 −0.684711
$$706$$ 12.0344 0.452922
$$707$$ 12.7639 0.480037
$$708$$ 2.23607 0.0840366
$$709$$ 32.7639 1.23048 0.615238 0.788342i $$-0.289060\pi$$
0.615238 + 0.788342i $$0.289060\pi$$
$$710$$ −6.47214 −0.242895
$$711$$ 4.47214 0.167718
$$712$$ −3.14590 −0.117898
$$713$$ 2.11146 0.0790747
$$714$$ 1.00000 0.0374241
$$715$$ 0 0
$$716$$ −8.14590 −0.304427
$$717$$ −6.94427 −0.259339
$$718$$ −12.2918 −0.458726
$$719$$ 5.63932 0.210311 0.105156 0.994456i $$-0.466466\pi$$
0.105156 + 0.994456i $$0.466466\pi$$
$$720$$ 0.381966 0.0142350
$$721$$ 13.7082 0.510520
$$722$$ 12.1459 0.452024
$$723$$ 22.4164 0.833675
$$724$$ −23.1246 −0.859419
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ −25.3050 −0.938509 −0.469254 0.883063i $$-0.655477\pi$$
−0.469254 + 0.883063i $$0.655477\pi$$
$$728$$ 0.763932 0.0283132
$$729$$ 29.5066 1.09284
$$730$$ −11.3262 −0.419203
$$731$$ −1.47214 −0.0544489
$$732$$ 12.9443 0.478434
$$733$$ −19.2361 −0.710500 −0.355250 0.934771i $$-0.615604\pi$$
−0.355250 + 0.934771i $$0.615604\pi$$
$$734$$ −28.9443 −1.06835
$$735$$ 1.61803 0.0596821
$$736$$ −0.472136 −0.0174032
$$737$$ 0 0
$$738$$ 2.43769 0.0897328
$$739$$ −26.9787 −0.992428 −0.496214 0.868200i $$-0.665277\pi$$
−0.496214 + 0.868200i $$0.665277\pi$$
$$740$$ 3.70820 0.136316
$$741$$ −3.23607 −0.118880
$$742$$ −2.47214 −0.0907550
$$743$$ −22.0000 −0.807102 −0.403551 0.914957i $$-0.632224\pi$$
−0.403551 + 0.914957i $$0.632224\pi$$
$$744$$ 7.23607 0.265287
$$745$$ 7.70820 0.282407
$$746$$ −12.9443 −0.473923
$$747$$ 0.0344419 0.00126016
$$748$$ 0 0
$$749$$ 8.09017 0.295608
$$750$$ −1.61803 −0.0590822
$$751$$ −49.4164 −1.80323 −0.901615 0.432539i $$-0.857618\pi$$
−0.901615 + 0.432539i $$0.857618\pi$$
$$752$$ −11.2361 −0.409737
$$753$$ −28.9443 −1.05479
$$754$$ 3.05573 0.111283
$$755$$ −20.1803 −0.734438
$$756$$ 5.47214 0.199020
$$757$$ 26.0689 0.947490 0.473745 0.880662i $$-0.342902\pi$$
0.473745 + 0.880662i $$0.342902\pi$$
$$758$$ 16.8541 0.612169
$$759$$ 0 0
$$760$$ −2.61803 −0.0949661
$$761$$ 6.72949 0.243944 0.121972 0.992534i $$-0.461078\pi$$
0.121972 + 0.992534i $$0.461078\pi$$
$$762$$ −13.2361 −0.479492
$$763$$ 10.1803 0.368553
$$764$$ −2.00000 −0.0723575
$$765$$ 0.236068 0.00853506
$$766$$ −4.29180 −0.155069
$$767$$ 1.05573 0.0381201
$$768$$ −1.61803 −0.0583858
$$769$$ 25.4164 0.916539 0.458270 0.888813i $$-0.348469\pi$$
0.458270 + 0.888813i $$0.348469\pi$$
$$770$$ 0 0
$$771$$ 8.79837 0.316866
$$772$$ −3.52786 −0.126971
$$773$$ 36.3607 1.30780 0.653901 0.756580i $$-0.273131\pi$$
0.653901 + 0.756580i $$0.273131\pi$$
$$774$$ −0.909830 −0.0327032
$$775$$ 4.47214 0.160644
$$776$$ −13.7984 −0.495333
$$777$$ 6.00000 0.215249
$$778$$ −8.00000 −0.286814
$$779$$ −16.7082 −0.598634
$$780$$ −1.23607 −0.0442583
$$781$$ 0 0
$$782$$ −0.291796 −0.0104346
$$783$$ 21.8885 0.782233
$$784$$ 1.00000 0.0357143
$$785$$ 6.18034 0.220586
$$786$$ 13.1803 0.470127
$$787$$ 14.0344 0.500274 0.250137 0.968210i $$-0.419524\pi$$
0.250137 + 0.968210i $$0.419524\pi$$
$$788$$ −5.41641 −0.192952
$$789$$ 19.4164 0.691242
$$790$$ −11.7082 −0.416559
$$791$$ −10.7984 −0.383946
$$792$$ 0 0
$$793$$ 6.11146 0.217024
$$794$$ −32.5410 −1.15484
$$795$$ 4.00000 0.141865
$$796$$ 3.23607 0.114699
$$797$$ 34.4721 1.22107 0.610533 0.791991i $$-0.290955\pi$$
0.610533 + 0.791991i $$0.290955\pi$$
$$798$$ −4.23607 −0.149955
$$799$$ −6.94427 −0.245671
$$800$$ −1.00000 −0.0353553
$$801$$ −1.20163 −0.0424574
$$802$$ −27.5066 −0.971291
$$803$$ 0 0
$$804$$ −8.23607 −0.290464
$$805$$ −0.472136 −0.0166406
$$806$$ 3.41641 0.120338
$$807$$ −22.6525 −0.797405
$$808$$ −12.7639 −0.449034
$$809$$ 26.4377 0.929500 0.464750 0.885442i $$-0.346144\pi$$
0.464750 + 0.885442i $$0.346144\pi$$
$$810$$ −7.70820 −0.270839
$$811$$ 49.7984 1.74866 0.874329 0.485334i $$-0.161302\pi$$
0.874329 + 0.485334i $$0.161302\pi$$
$$812$$ 4.00000 0.140372
$$813$$ −10.1803 −0.357040
$$814$$ 0 0
$$815$$ 22.6180 0.792275
$$816$$ −1.00000 −0.0350070
$$817$$ 6.23607 0.218172
$$818$$ −31.3050 −1.09455
$$819$$ 0.291796 0.0101962
$$820$$ −6.38197 −0.222868
$$821$$ 49.4853 1.72705 0.863524 0.504307i $$-0.168252\pi$$
0.863524 + 0.504307i $$0.168252\pi$$
$$822$$ −20.7984 −0.725427
$$823$$ 51.4853 1.79466 0.897332 0.441356i $$-0.145502\pi$$
0.897332 + 0.441356i $$0.145502\pi$$
$$824$$ −13.7082 −0.477548
$$825$$ 0 0
$$826$$ 1.38197 0.0480847
$$827$$ 25.7426 0.895160 0.447580 0.894244i $$-0.352286\pi$$
0.447580 + 0.894244i $$0.352286\pi$$
$$828$$ −0.180340 −0.00626724
$$829$$ 2.94427 0.102259 0.0511294 0.998692i $$-0.483718\pi$$
0.0511294 + 0.998692i $$0.483718\pi$$
$$830$$ −0.0901699 −0.00312984
$$831$$ 27.7082 0.961187
$$832$$ −0.763932 −0.0264846
$$833$$ 0.618034 0.0214136
$$834$$ −2.47214 −0.0856031
$$835$$ −10.0000 −0.346064
$$836$$ 0 0
$$837$$ 24.4721 0.845881
$$838$$ 14.7426 0.509276
$$839$$ −10.6525 −0.367764 −0.183882 0.982948i $$-0.558866\pi$$
−0.183882 + 0.982948i $$0.558866\pi$$
$$840$$ −1.61803 −0.0558275
$$841$$ −13.0000 −0.448276
$$842$$ −8.47214 −0.291969
$$843$$ −28.7984 −0.991869
$$844$$ 28.0902 0.966904
$$845$$ 12.4164 0.427137
$$846$$ −4.29180 −0.147555
$$847$$ 0 0
$$848$$ 2.47214 0.0848935
$$849$$ −16.0000 −0.549119
$$850$$ −0.618034 −0.0211984
$$851$$ −1.75078 −0.0600158
$$852$$ 10.4721 0.358769
$$853$$ −36.2918 −1.24261 −0.621304 0.783570i $$-0.713397\pi$$
−0.621304 + 0.783570i $$0.713397\pi$$
$$854$$ 8.00000 0.273754
$$855$$ −1.00000 −0.0341993
$$856$$ −8.09017 −0.276516
$$857$$ 47.6869 1.62895 0.814477 0.580196i $$-0.197024\pi$$
0.814477 + 0.580196i $$0.197024\pi$$
$$858$$ 0 0
$$859$$ 43.8541 1.49628 0.748141 0.663539i $$-0.230947\pi$$
0.748141 + 0.663539i $$0.230947\pi$$
$$860$$ 2.38197 0.0812244
$$861$$ −10.3262 −0.351917
$$862$$ −19.7082 −0.671264
$$863$$ 40.2492 1.37010 0.685050 0.728496i $$-0.259780\pi$$
0.685050 + 0.728496i $$0.259780\pi$$
$$864$$ −5.47214 −0.186166
$$865$$ −4.76393 −0.161979
$$866$$ −1.20163 −0.0408329
$$867$$ 26.8885 0.913183
$$868$$ 4.47214 0.151794
$$869$$ 0 0
$$870$$ −6.47214 −0.219426
$$871$$ −3.88854 −0.131758
$$872$$ −10.1803 −0.344750
$$873$$ −5.27051 −0.178380
$$874$$ 1.23607 0.0418106
$$875$$ −1.00000 −0.0338062
$$876$$ 18.3262 0.619186
$$877$$ −21.1246 −0.713327 −0.356664 0.934233i $$-0.616086\pi$$
−0.356664 + 0.934233i $$0.616086\pi$$
$$878$$ −24.3607 −0.822133
$$879$$ −25.4164 −0.857274
$$880$$ 0 0
$$881$$ 34.3394 1.15692 0.578462 0.815709i $$-0.303653\pi$$
0.578462 + 0.815709i $$0.303653\pi$$
$$882$$ 0.381966 0.0128615
$$883$$ 34.3820 1.15705 0.578523 0.815666i $$-0.303629\pi$$
0.578523 + 0.815666i $$0.303629\pi$$
$$884$$ −0.472136 −0.0158797
$$885$$ −2.23607 −0.0751646
$$886$$ 26.3820 0.886319
$$887$$ −22.0000 −0.738688 −0.369344 0.929293i $$-0.620418\pi$$
−0.369344 + 0.929293i $$0.620418\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −8.18034 −0.274360
$$890$$ 3.14590 0.105451
$$891$$ 0 0
$$892$$ −22.9443 −0.768231
$$893$$ 29.4164 0.984383
$$894$$ −12.4721 −0.417131
$$895$$ 8.14590 0.272287
$$896$$ −1.00000 −0.0334077
$$897$$ 0.583592 0.0194856
$$898$$ −27.7426 −0.925784
$$899$$ 17.8885 0.596616
$$900$$ −0.381966 −0.0127322
$$901$$ 1.52786 0.0509005
$$902$$ 0 0
$$903$$ 3.85410 0.128256
$$904$$ 10.7984 0.359149
$$905$$ 23.1246 0.768688
$$906$$ 32.6525 1.08481
$$907$$ 20.7984 0.690599 0.345299 0.938493i $$-0.387777\pi$$
0.345299 + 0.938493i $$0.387777\pi$$
$$908$$ −11.5623 −0.383709
$$909$$ −4.87539 −0.161706
$$910$$ −0.763932 −0.0253241
$$911$$ −26.0689 −0.863701 −0.431850 0.901945i $$-0.642139\pi$$
−0.431850 + 0.901945i $$0.642139\pi$$
$$912$$ 4.23607 0.140270
$$913$$ 0 0
$$914$$ 5.43769 0.179863
$$915$$ −12.9443 −0.427924
$$916$$ 7.70820 0.254686
$$917$$ 8.14590 0.269001
$$918$$ −3.38197 −0.111622
$$919$$ −21.2361 −0.700513 −0.350257 0.936654i $$-0.613906\pi$$
−0.350257 + 0.936654i $$0.613906\pi$$
$$920$$ 0.472136 0.0155659
$$921$$ 5.67376 0.186957
$$922$$ −39.1246 −1.28850
$$923$$ 4.94427 0.162743
$$924$$ 0 0
$$925$$ −3.70820 −0.121925
$$926$$ 30.6525 1.00730
$$927$$ −5.23607 −0.171975
$$928$$ −4.00000 −0.131306
$$929$$ 19.6738 0.645475 0.322738 0.946488i $$-0.395397\pi$$
0.322738 + 0.946488i $$0.395397\pi$$
$$930$$ −7.23607 −0.237280
$$931$$ −2.61803 −0.0858026
$$932$$ 19.3262 0.633052
$$933$$ 2.00000 0.0654771
$$934$$ 0.944272 0.0308975
$$935$$ 0 0
$$936$$ −0.291796 −0.00953765
$$937$$ −28.3951 −0.927628 −0.463814 0.885933i $$-0.653519\pi$$
−0.463814 + 0.885933i $$0.653519\pi$$
$$938$$ −5.09017 −0.166200
$$939$$ 19.9443 0.650857
$$940$$ 11.2361 0.366480
$$941$$ 53.7771 1.75308 0.876541 0.481326i $$-0.159845\pi$$
0.876541 + 0.481326i $$0.159845\pi$$
$$942$$ −10.0000 −0.325818
$$943$$ 3.01316 0.0981218
$$944$$ −1.38197 −0.0449792
$$945$$ −5.47214 −0.178009
$$946$$ 0 0
$$947$$ 51.3394 1.66831 0.834153 0.551533i $$-0.185957\pi$$
0.834153 + 0.551533i $$0.185957\pi$$
$$948$$ 18.9443 0.615281
$$949$$ 8.65248 0.280871
$$950$$ 2.61803 0.0849402
$$951$$ −28.9443 −0.938582
$$952$$ −0.618034 −0.0200306
$$953$$ 27.1459 0.879342 0.439671 0.898159i $$-0.355095\pi$$
0.439671 + 0.898159i $$0.355095\pi$$
$$954$$ 0.944272 0.0305719
$$955$$ 2.00000 0.0647185
$$956$$ 4.29180 0.138807
$$957$$ 0 0
$$958$$ −23.1246 −0.747122
$$959$$ −12.8541 −0.415081
$$960$$ 1.61803 0.0522218
$$961$$ −11.0000 −0.354839
$$962$$ −2.83282 −0.0913336
$$963$$ −3.09017 −0.0995793
$$964$$ −13.8541 −0.446211
$$965$$ 3.52786 0.113566
$$966$$ 0.763932 0.0245791
$$967$$ −26.0000 −0.836104 −0.418052 0.908423i $$-0.637287\pi$$
−0.418052 + 0.908423i $$0.637287\pi$$
$$968$$ 0 0
$$969$$ 2.61803 0.0841034
$$970$$ 13.7984 0.443039
$$971$$ −21.3050 −0.683708 −0.341854 0.939753i $$-0.611055\pi$$
−0.341854 + 0.939753i $$0.611055\pi$$
$$972$$ −3.94427 −0.126513
$$973$$ −1.52786 −0.0489811
$$974$$ 33.7771 1.08229
$$975$$ 1.23607 0.0395859
$$976$$ −8.00000 −0.256074
$$977$$ 31.5279 1.00867 0.504333 0.863509i $$-0.331738\pi$$
0.504333 + 0.863509i $$0.331738\pi$$
$$978$$ −36.5967 −1.17023
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −3.88854 −0.124152
$$982$$ −26.9230 −0.859147
$$983$$ 31.2361 0.996276 0.498138 0.867098i $$-0.334017\pi$$
0.498138 + 0.867098i $$0.334017\pi$$
$$984$$ 10.3262 0.329188
$$985$$ 5.41641 0.172581
$$986$$ −2.47214 −0.0787288
$$987$$ 18.1803 0.578687
$$988$$ 2.00000 0.0636285
$$989$$ −1.12461 −0.0357606
$$990$$ 0 0
$$991$$ −61.2361 −1.94523 −0.972614 0.232427i $$-0.925333\pi$$
−0.972614 + 0.232427i $$0.925333\pi$$
$$992$$ −4.47214 −0.141990
$$993$$ 5.38197 0.170792
$$994$$ 6.47214 0.205284
$$995$$ −3.23607 −0.102590
$$996$$ 0.145898 0.00462296
$$997$$ 12.8754 0.407768 0.203884 0.978995i $$-0.434644\pi$$
0.203884 + 0.978995i $$0.434644\pi$$
$$998$$ 1.61803 0.0512180
$$999$$ −20.2918 −0.642004
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 8470.2.a.bk.1.1 2
11.5 even 5 770.2.n.c.421.1 4
11.9 even 5 770.2.n.c.631.1 yes 4
11.10 odd 2 8470.2.a.bx.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.c.421.1 4 11.5 even 5
770.2.n.c.631.1 yes 4 11.9 even 5
8470.2.a.bk.1.1 2 1.1 even 1 trivial
8470.2.a.bx.1.1 2 11.10 odd 2