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 $1$

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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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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

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

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

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