Properties

Label 8470.2.a.db.1.6
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1\)
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 1.6
Root \(-1.92474\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.92474 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.92474 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.704605 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.92474 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.92474 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.704605 q^{9} +1.00000 q^{10} +1.92474 q^{12} -5.24489 q^{13} -1.00000 q^{14} +1.92474 q^{15} +1.00000 q^{16} -4.30919 q^{17} +0.704605 q^{18} -0.740633 q^{19} +1.00000 q^{20} -1.92474 q^{21} -0.856965 q^{23} +1.92474 q^{24} +1.00000 q^{25} -5.24489 q^{26} -4.41803 q^{27} -1.00000 q^{28} -8.37564 q^{29} +1.92474 q^{30} +1.10298 q^{31} +1.00000 q^{32} -4.30919 q^{34} -1.00000 q^{35} +0.704605 q^{36} -1.77921 q^{37} -0.740633 q^{38} -10.0950 q^{39} +1.00000 q^{40} -10.7200 q^{41} -1.92474 q^{42} +1.13472 q^{43} +0.704605 q^{45} -0.856965 q^{46} +8.40172 q^{47} +1.92474 q^{48} +1.00000 q^{49} +1.00000 q^{50} -8.29405 q^{51} -5.24489 q^{52} -8.80661 q^{53} -4.41803 q^{54} -1.00000 q^{56} -1.42552 q^{57} -8.37564 q^{58} -12.9838 q^{59} +1.92474 q^{60} +13.0887 q^{61} +1.10298 q^{62} -0.704605 q^{63} +1.00000 q^{64} -5.24489 q^{65} +8.93617 q^{67} -4.30919 q^{68} -1.64943 q^{69} -1.00000 q^{70} -4.50750 q^{71} +0.704605 q^{72} -3.33413 q^{73} -1.77921 q^{74} +1.92474 q^{75} -0.740633 q^{76} -10.0950 q^{78} -1.56183 q^{79} +1.00000 q^{80} -10.6173 q^{81} -10.7200 q^{82} +0.609215 q^{83} -1.92474 q^{84} -4.30919 q^{85} +1.13472 q^{86} -16.1209 q^{87} +1.46836 q^{89} +0.704605 q^{90} +5.24489 q^{91} -0.856965 q^{92} +2.12295 q^{93} +8.40172 q^{94} -0.740633 q^{95} +1.92474 q^{96} +8.41705 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (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
\(3\) 1.92474 1.11125 0.555623 0.831434i \(-0.312480\pi\)
0.555623 + 0.831434i \(0.312480\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.92474 0.785770
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0.704605 0.234868
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.92474 0.555623
\(13\) −5.24489 −1.45467 −0.727335 0.686283i \(-0.759241\pi\)
−0.727335 + 0.686283i \(0.759241\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.92474 0.496964
\(16\) 1.00000 0.250000
\(17\) −4.30919 −1.04513 −0.522566 0.852599i \(-0.675025\pi\)
−0.522566 + 0.852599i \(0.675025\pi\)
\(18\) 0.704605 0.166077
\(19\) −0.740633 −0.169913 −0.0849564 0.996385i \(-0.527075\pi\)
−0.0849564 + 0.996385i \(0.527075\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.92474 −0.420012
\(22\) 0 0
\(23\) −0.856965 −0.178689 −0.0893447 0.996001i \(-0.528477\pi\)
−0.0893447 + 0.996001i \(0.528477\pi\)
\(24\) 1.92474 0.392885
\(25\) 1.00000 0.200000
\(26\) −5.24489 −1.02861
\(27\) −4.41803 −0.850250
\(28\) −1.00000 −0.188982
\(29\) −8.37564 −1.55532 −0.777658 0.628687i \(-0.783593\pi\)
−0.777658 + 0.628687i \(0.783593\pi\)
\(30\) 1.92474 0.351407
\(31\) 1.10298 0.198101 0.0990506 0.995082i \(-0.468419\pi\)
0.0990506 + 0.995082i \(0.468419\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.30919 −0.739020
\(35\) −1.00000 −0.169031
\(36\) 0.704605 0.117434
\(37\) −1.77921 −0.292500 −0.146250 0.989248i \(-0.546720\pi\)
−0.146250 + 0.989248i \(0.546720\pi\)
\(38\) −0.740633 −0.120146
\(39\) −10.0950 −1.61650
\(40\) 1.00000 0.158114
\(41\) −10.7200 −1.67418 −0.837090 0.547065i \(-0.815745\pi\)
−0.837090 + 0.547065i \(0.815745\pi\)
\(42\) −1.92474 −0.296993
\(43\) 1.13472 0.173044 0.0865218 0.996250i \(-0.472425\pi\)
0.0865218 + 0.996250i \(0.472425\pi\)
\(44\) 0 0
\(45\) 0.704605 0.105036
\(46\) −0.856965 −0.126353
\(47\) 8.40172 1.22552 0.612758 0.790271i \(-0.290060\pi\)
0.612758 + 0.790271i \(0.290060\pi\)
\(48\) 1.92474 0.277812
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −8.29405 −1.16140
\(52\) −5.24489 −0.727335
\(53\) −8.80661 −1.20968 −0.604840 0.796347i \(-0.706763\pi\)
−0.604840 + 0.796347i \(0.706763\pi\)
\(54\) −4.41803 −0.601217
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −1.42552 −0.188815
\(58\) −8.37564 −1.09977
\(59\) −12.9838 −1.69035 −0.845175 0.534490i \(-0.820504\pi\)
−0.845175 + 0.534490i \(0.820504\pi\)
\(60\) 1.92474 0.248482
\(61\) 13.0887 1.67584 0.837920 0.545793i \(-0.183772\pi\)
0.837920 + 0.545793i \(0.183772\pi\)
\(62\) 1.10298 0.140079
\(63\) −0.704605 −0.0887719
\(64\) 1.00000 0.125000
\(65\) −5.24489 −0.650548
\(66\) 0 0
\(67\) 8.93617 1.09173 0.545864 0.837874i \(-0.316202\pi\)
0.545864 + 0.837874i \(0.316202\pi\)
\(68\) −4.30919 −0.522566
\(69\) −1.64943 −0.198568
\(70\) −1.00000 −0.119523
\(71\) −4.50750 −0.534942 −0.267471 0.963566i \(-0.586188\pi\)
−0.267471 + 0.963566i \(0.586188\pi\)
\(72\) 0.704605 0.0830385
\(73\) −3.33413 −0.390230 −0.195115 0.980780i \(-0.562508\pi\)
−0.195115 + 0.980780i \(0.562508\pi\)
\(74\) −1.77921 −0.206829
\(75\) 1.92474 0.222249
\(76\) −0.740633 −0.0849564
\(77\) 0 0
\(78\) −10.0950 −1.14304
\(79\) −1.56183 −0.175719 −0.0878597 0.996133i \(-0.528003\pi\)
−0.0878597 + 0.996133i \(0.528003\pi\)
\(80\) 1.00000 0.111803
\(81\) −10.6173 −1.17971
\(82\) −10.7200 −1.18382
\(83\) 0.609215 0.0668700 0.0334350 0.999441i \(-0.489355\pi\)
0.0334350 + 0.999441i \(0.489355\pi\)
\(84\) −1.92474 −0.210006
\(85\) −4.30919 −0.467397
\(86\) 1.13472 0.122360
\(87\) −16.1209 −1.72834
\(88\) 0 0
\(89\) 1.46836 0.155646 0.0778228 0.996967i \(-0.475203\pi\)
0.0778228 + 0.996967i \(0.475203\pi\)
\(90\) 0.704605 0.0742719
\(91\) 5.24489 0.549813
\(92\) −0.856965 −0.0893447
\(93\) 2.12295 0.220139
\(94\) 8.40172 0.866571
\(95\) −0.740633 −0.0759873
\(96\) 1.92474 0.196442
\(97\) 8.41705 0.854622 0.427311 0.904105i \(-0.359461\pi\)
0.427311 + 0.904105i \(0.359461\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.0295 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(102\) −8.29405 −0.821233
\(103\) 5.86953 0.578342 0.289171 0.957277i \(-0.406620\pi\)
0.289171 + 0.957277i \(0.406620\pi\)
\(104\) −5.24489 −0.514303
\(105\) −1.92474 −0.187835
\(106\) −8.80661 −0.855373
\(107\) 6.83162 0.660438 0.330219 0.943904i \(-0.392877\pi\)
0.330219 + 0.943904i \(0.392877\pi\)
\(108\) −4.41803 −0.425125
\(109\) −7.76784 −0.744025 −0.372012 0.928228i \(-0.621332\pi\)
−0.372012 + 0.928228i \(0.621332\pi\)
\(110\) 0 0
\(111\) −3.42451 −0.325040
\(112\) −1.00000 −0.0944911
\(113\) 4.63948 0.436446 0.218223 0.975899i \(-0.429974\pi\)
0.218223 + 0.975899i \(0.429974\pi\)
\(114\) −1.42552 −0.133512
\(115\) −0.856965 −0.0799124
\(116\) −8.37564 −0.777658
\(117\) −3.69557 −0.341656
\(118\) −12.9838 −1.19526
\(119\) 4.30919 0.395023
\(120\) 1.92474 0.175703
\(121\) 0 0
\(122\) 13.0887 1.18500
\(123\) −20.6331 −1.86043
\(124\) 1.10298 0.0990506
\(125\) 1.00000 0.0894427
\(126\) −0.704605 −0.0627712
\(127\) 9.42065 0.835947 0.417974 0.908459i \(-0.362740\pi\)
0.417974 + 0.908459i \(0.362740\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.18404 0.192294
\(130\) −5.24489 −0.460007
\(131\) −0.862825 −0.0753854 −0.0376927 0.999289i \(-0.512001\pi\)
−0.0376927 + 0.999289i \(0.512001\pi\)
\(132\) 0 0
\(133\) 0.740633 0.0642210
\(134\) 8.93617 0.771968
\(135\) −4.41803 −0.380243
\(136\) −4.30919 −0.369510
\(137\) −17.2469 −1.47350 −0.736751 0.676164i \(-0.763641\pi\)
−0.736751 + 0.676164i \(0.763641\pi\)
\(138\) −1.64943 −0.140409
\(139\) 20.4448 1.73411 0.867054 0.498213i \(-0.166010\pi\)
0.867054 + 0.498213i \(0.166010\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 16.1711 1.36185
\(142\) −4.50750 −0.378261
\(143\) 0 0
\(144\) 0.704605 0.0587171
\(145\) −8.37564 −0.695559
\(146\) −3.33413 −0.275934
\(147\) 1.92474 0.158749
\(148\) −1.77921 −0.146250
\(149\) 1.88298 0.154260 0.0771299 0.997021i \(-0.475424\pi\)
0.0771299 + 0.997021i \(0.475424\pi\)
\(150\) 1.92474 0.157154
\(151\) 5.45095 0.443592 0.221796 0.975093i \(-0.428808\pi\)
0.221796 + 0.975093i \(0.428808\pi\)
\(152\) −0.740633 −0.0600732
\(153\) −3.03628 −0.245468
\(154\) 0 0
\(155\) 1.10298 0.0885935
\(156\) −10.0950 −0.808248
\(157\) −11.9779 −0.955944 −0.477972 0.878375i \(-0.658628\pi\)
−0.477972 + 0.878375i \(0.658628\pi\)
\(158\) −1.56183 −0.124252
\(159\) −16.9504 −1.34425
\(160\) 1.00000 0.0790569
\(161\) 0.856965 0.0675383
\(162\) −10.6173 −0.834178
\(163\) −15.5543 −1.21830 −0.609152 0.793053i \(-0.708490\pi\)
−0.609152 + 0.793053i \(0.708490\pi\)
\(164\) −10.7200 −0.837090
\(165\) 0 0
\(166\) 0.609215 0.0472843
\(167\) 18.0474 1.39655 0.698274 0.715830i \(-0.253952\pi\)
0.698274 + 0.715830i \(0.253952\pi\)
\(168\) −1.92474 −0.148497
\(169\) 14.5088 1.11606
\(170\) −4.30919 −0.330500
\(171\) −0.521854 −0.0399071
\(172\) 1.13472 0.0865218
\(173\) −9.60665 −0.730380 −0.365190 0.930933i \(-0.618996\pi\)
−0.365190 + 0.930933i \(0.618996\pi\)
\(174\) −16.1209 −1.22212
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −24.9904 −1.87839
\(178\) 1.46836 0.110058
\(179\) 13.2673 0.991643 0.495822 0.868424i \(-0.334867\pi\)
0.495822 + 0.868424i \(0.334867\pi\)
\(180\) 0.704605 0.0525182
\(181\) 19.5835 1.45563 0.727815 0.685774i \(-0.240536\pi\)
0.727815 + 0.685774i \(0.240536\pi\)
\(182\) 5.24489 0.388777
\(183\) 25.1923 1.86227
\(184\) −0.856965 −0.0631763
\(185\) −1.77921 −0.130810
\(186\) 2.12295 0.155662
\(187\) 0 0
\(188\) 8.40172 0.612758
\(189\) 4.41803 0.321364
\(190\) −0.740633 −0.0537311
\(191\) −12.7555 −0.922954 −0.461477 0.887152i \(-0.652680\pi\)
−0.461477 + 0.887152i \(0.652680\pi\)
\(192\) 1.92474 0.138906
\(193\) 11.8288 0.851459 0.425729 0.904851i \(-0.360018\pi\)
0.425729 + 0.904851i \(0.360018\pi\)
\(194\) 8.41705 0.604309
\(195\) −10.0950 −0.722919
\(196\) 1.00000 0.0714286
\(197\) −21.9105 −1.56106 −0.780528 0.625121i \(-0.785050\pi\)
−0.780528 + 0.625121i \(0.785050\pi\)
\(198\) 0 0
\(199\) 10.3121 0.731006 0.365503 0.930810i \(-0.380897\pi\)
0.365503 + 0.930810i \(0.380897\pi\)
\(200\) 1.00000 0.0707107
\(201\) 17.1998 1.21318
\(202\) −11.0295 −0.776036
\(203\) 8.37564 0.587854
\(204\) −8.29405 −0.580700
\(205\) −10.7200 −0.748716
\(206\) 5.86953 0.408950
\(207\) −0.603822 −0.0419685
\(208\) −5.24489 −0.363667
\(209\) 0 0
\(210\) −1.92474 −0.132819
\(211\) −24.8455 −1.71043 −0.855217 0.518270i \(-0.826576\pi\)
−0.855217 + 0.518270i \(0.826576\pi\)
\(212\) −8.80661 −0.604840
\(213\) −8.67574 −0.594452
\(214\) 6.83162 0.467000
\(215\) 1.13472 0.0773875
\(216\) −4.41803 −0.300609
\(217\) −1.10298 −0.0748752
\(218\) −7.76784 −0.526105
\(219\) −6.41731 −0.433641
\(220\) 0 0
\(221\) 22.6012 1.52032
\(222\) −3.42451 −0.229838
\(223\) −20.2394 −1.35533 −0.677667 0.735369i \(-0.737009\pi\)
−0.677667 + 0.735369i \(0.737009\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0.704605 0.0469737
\(226\) 4.63948 0.308614
\(227\) 11.8090 0.783792 0.391896 0.920009i \(-0.371819\pi\)
0.391896 + 0.920009i \(0.371819\pi\)
\(228\) −1.42552 −0.0944075
\(229\) 1.67906 0.110955 0.0554776 0.998460i \(-0.482332\pi\)
0.0554776 + 0.998460i \(0.482332\pi\)
\(230\) −0.856965 −0.0565066
\(231\) 0 0
\(232\) −8.37564 −0.549887
\(233\) −15.5220 −1.01688 −0.508440 0.861097i \(-0.669778\pi\)
−0.508440 + 0.861097i \(0.669778\pi\)
\(234\) −3.69557 −0.241587
\(235\) 8.40172 0.548067
\(236\) −12.9838 −0.845175
\(237\) −3.00610 −0.195267
\(238\) 4.30919 0.279323
\(239\) −22.0037 −1.42330 −0.711652 0.702532i \(-0.752053\pi\)
−0.711652 + 0.702532i \(0.752053\pi\)
\(240\) 1.92474 0.124241
\(241\) −23.8155 −1.53409 −0.767047 0.641591i \(-0.778274\pi\)
−0.767047 + 0.641591i \(0.778274\pi\)
\(242\) 0 0
\(243\) −7.18150 −0.460693
\(244\) 13.0887 0.837920
\(245\) 1.00000 0.0638877
\(246\) −20.6331 −1.31552
\(247\) 3.88453 0.247167
\(248\) 1.10298 0.0700393
\(249\) 1.17258 0.0743091
\(250\) 1.00000 0.0632456
\(251\) 28.5083 1.79943 0.899715 0.436477i \(-0.143774\pi\)
0.899715 + 0.436477i \(0.143774\pi\)
\(252\) −0.704605 −0.0443859
\(253\) 0 0
\(254\) 9.42065 0.591104
\(255\) −8.29405 −0.519393
\(256\) 1.00000 0.0625000
\(257\) 29.6194 1.84760 0.923802 0.382870i \(-0.125064\pi\)
0.923802 + 0.382870i \(0.125064\pi\)
\(258\) 2.18404 0.135973
\(259\) 1.77921 0.110555
\(260\) −5.24489 −0.325274
\(261\) −5.90151 −0.365295
\(262\) −0.862825 −0.0533055
\(263\) 4.09184 0.252314 0.126157 0.992010i \(-0.459736\pi\)
0.126157 + 0.992010i \(0.459736\pi\)
\(264\) 0 0
\(265\) −8.80661 −0.540986
\(266\) 0.740633 0.0454111
\(267\) 2.82620 0.172960
\(268\) 8.93617 0.545864
\(269\) 2.99735 0.182752 0.0913759 0.995816i \(-0.470874\pi\)
0.0913759 + 0.995816i \(0.470874\pi\)
\(270\) −4.41803 −0.268873
\(271\) 24.7015 1.50051 0.750255 0.661148i \(-0.229930\pi\)
0.750255 + 0.661148i \(0.229930\pi\)
\(272\) −4.30919 −0.261283
\(273\) 10.0950 0.610978
\(274\) −17.2469 −1.04192
\(275\) 0 0
\(276\) −1.64943 −0.0992840
\(277\) 2.67858 0.160941 0.0804703 0.996757i \(-0.474358\pi\)
0.0804703 + 0.996757i \(0.474358\pi\)
\(278\) 20.4448 1.22620
\(279\) 0.777166 0.0465277
\(280\) −1.00000 −0.0597614
\(281\) −11.6213 −0.693271 −0.346636 0.938000i \(-0.612676\pi\)
−0.346636 + 0.938000i \(0.612676\pi\)
\(282\) 16.1711 0.962974
\(283\) 18.1362 1.07808 0.539042 0.842279i \(-0.318786\pi\)
0.539042 + 0.842279i \(0.318786\pi\)
\(284\) −4.50750 −0.267471
\(285\) −1.42552 −0.0844406
\(286\) 0 0
\(287\) 10.7200 0.632781
\(288\) 0.704605 0.0415192
\(289\) 1.56912 0.0923010
\(290\) −8.37564 −0.491834
\(291\) 16.2006 0.949695
\(292\) −3.33413 −0.195115
\(293\) −22.8700 −1.33608 −0.668041 0.744125i \(-0.732867\pi\)
−0.668041 + 0.744125i \(0.732867\pi\)
\(294\) 1.92474 0.112253
\(295\) −12.9838 −0.755947
\(296\) −1.77921 −0.103415
\(297\) 0 0
\(298\) 1.88298 0.109078
\(299\) 4.49468 0.259934
\(300\) 1.92474 0.111125
\(301\) −1.13472 −0.0654044
\(302\) 5.45095 0.313667
\(303\) −21.2290 −1.21957
\(304\) −0.740633 −0.0424782
\(305\) 13.0887 0.749458
\(306\) −3.03628 −0.173572
\(307\) −19.9326 −1.13761 −0.568806 0.822472i \(-0.692594\pi\)
−0.568806 + 0.822472i \(0.692594\pi\)
\(308\) 0 0
\(309\) 11.2973 0.642681
\(310\) 1.10298 0.0626451
\(311\) −1.28437 −0.0728299 −0.0364149 0.999337i \(-0.511594\pi\)
−0.0364149 + 0.999337i \(0.511594\pi\)
\(312\) −10.0950 −0.571518
\(313\) −13.3964 −0.757210 −0.378605 0.925558i \(-0.623596\pi\)
−0.378605 + 0.925558i \(0.623596\pi\)
\(314\) −11.9779 −0.675954
\(315\) −0.704605 −0.0397000
\(316\) −1.56183 −0.0878597
\(317\) 16.6640 0.935941 0.467971 0.883744i \(-0.344985\pi\)
0.467971 + 0.883744i \(0.344985\pi\)
\(318\) −16.9504 −0.950530
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 13.1491 0.733909
\(322\) 0.856965 0.0477568
\(323\) 3.19153 0.177581
\(324\) −10.6173 −0.589853
\(325\) −5.24489 −0.290934
\(326\) −15.5543 −0.861471
\(327\) −14.9510 −0.826795
\(328\) −10.7200 −0.591912
\(329\) −8.40172 −0.463202
\(330\) 0 0
\(331\) 11.1340 0.611979 0.305989 0.952035i \(-0.401013\pi\)
0.305989 + 0.952035i \(0.401013\pi\)
\(332\) 0.609215 0.0334350
\(333\) −1.25364 −0.0686991
\(334\) 18.0474 0.987509
\(335\) 8.93617 0.488235
\(336\) −1.92474 −0.105003
\(337\) −6.71838 −0.365974 −0.182987 0.983115i \(-0.558577\pi\)
−0.182987 + 0.983115i \(0.558577\pi\)
\(338\) 14.5088 0.789176
\(339\) 8.92978 0.484999
\(340\) −4.30919 −0.233699
\(341\) 0 0
\(342\) −0.521854 −0.0282186
\(343\) −1.00000 −0.0539949
\(344\) 1.13472 0.0611802
\(345\) −1.64943 −0.0888023
\(346\) −9.60665 −0.516456
\(347\) −19.4103 −1.04200 −0.520999 0.853557i \(-0.674441\pi\)
−0.520999 + 0.853557i \(0.674441\pi\)
\(348\) −16.1209 −0.864170
\(349\) 26.7685 1.43289 0.716444 0.697645i \(-0.245769\pi\)
0.716444 + 0.697645i \(0.245769\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 23.1720 1.23683
\(352\) 0 0
\(353\) −32.9268 −1.75252 −0.876259 0.481841i \(-0.839968\pi\)
−0.876259 + 0.481841i \(0.839968\pi\)
\(354\) −24.9904 −1.32823
\(355\) −4.50750 −0.239233
\(356\) 1.46836 0.0778228
\(357\) 8.29405 0.438968
\(358\) 13.2673 0.701198
\(359\) 36.1167 1.90617 0.953083 0.302709i \(-0.0978912\pi\)
0.953083 + 0.302709i \(0.0978912\pi\)
\(360\) 0.704605 0.0371359
\(361\) −18.4515 −0.971130
\(362\) 19.5835 1.02929
\(363\) 0 0
\(364\) 5.24489 0.274907
\(365\) −3.33413 −0.174516
\(366\) 25.1923 1.31682
\(367\) −8.35672 −0.436217 −0.218109 0.975925i \(-0.569989\pi\)
−0.218109 + 0.975925i \(0.569989\pi\)
\(368\) −0.856965 −0.0446724
\(369\) −7.55335 −0.393212
\(370\) −1.77921 −0.0924968
\(371\) 8.80661 0.457216
\(372\) 2.12295 0.110070
\(373\) −16.0205 −0.829509 −0.414755 0.909933i \(-0.636133\pi\)
−0.414755 + 0.909933i \(0.636133\pi\)
\(374\) 0 0
\(375\) 1.92474 0.0993929
\(376\) 8.40172 0.433285
\(377\) 43.9292 2.26247
\(378\) 4.41803 0.227239
\(379\) 25.3872 1.30405 0.652027 0.758196i \(-0.273919\pi\)
0.652027 + 0.758196i \(0.273919\pi\)
\(380\) −0.740633 −0.0379937
\(381\) 18.1323 0.928943
\(382\) −12.7555 −0.652627
\(383\) −0.303318 −0.0154988 −0.00774942 0.999970i \(-0.502467\pi\)
−0.00774942 + 0.999970i \(0.502467\pi\)
\(384\) 1.92474 0.0982212
\(385\) 0 0
\(386\) 11.8288 0.602072
\(387\) 0.799532 0.0406425
\(388\) 8.41705 0.427311
\(389\) −6.74058 −0.341761 −0.170880 0.985292i \(-0.554661\pi\)
−0.170880 + 0.985292i \(0.554661\pi\)
\(390\) −10.0950 −0.511181
\(391\) 3.69282 0.186754
\(392\) 1.00000 0.0505076
\(393\) −1.66071 −0.0837717
\(394\) −21.9105 −1.10383
\(395\) −1.56183 −0.0785841
\(396\) 0 0
\(397\) −19.9230 −0.999904 −0.499952 0.866053i \(-0.666649\pi\)
−0.499952 + 0.866053i \(0.666649\pi\)
\(398\) 10.3121 0.516899
\(399\) 1.42552 0.0713654
\(400\) 1.00000 0.0500000
\(401\) 34.0886 1.70230 0.851151 0.524920i \(-0.175905\pi\)
0.851151 + 0.524920i \(0.175905\pi\)
\(402\) 17.1998 0.857846
\(403\) −5.78501 −0.288172
\(404\) −11.0295 −0.548740
\(405\) −10.6173 −0.527580
\(406\) 8.37564 0.415676
\(407\) 0 0
\(408\) −8.29405 −0.410617
\(409\) 8.09670 0.400356 0.200178 0.979760i \(-0.435848\pi\)
0.200178 + 0.979760i \(0.435848\pi\)
\(410\) −10.7200 −0.529422
\(411\) −33.1957 −1.63742
\(412\) 5.86953 0.289171
\(413\) 12.9838 0.638892
\(414\) −0.603822 −0.0296762
\(415\) 0.609215 0.0299052
\(416\) −5.24489 −0.257152
\(417\) 39.3509 1.92702
\(418\) 0 0
\(419\) −24.0686 −1.17583 −0.587915 0.808923i \(-0.700051\pi\)
−0.587915 + 0.808923i \(0.700051\pi\)
\(420\) −1.92474 −0.0939175
\(421\) 13.7871 0.671942 0.335971 0.941872i \(-0.390936\pi\)
0.335971 + 0.941872i \(0.390936\pi\)
\(422\) −24.8455 −1.20946
\(423\) 5.91989 0.287835
\(424\) −8.80661 −0.427687
\(425\) −4.30919 −0.209026
\(426\) −8.67574 −0.420341
\(427\) −13.0887 −0.633408
\(428\) 6.83162 0.330219
\(429\) 0 0
\(430\) 1.13472 0.0547212
\(431\) −8.72005 −0.420030 −0.210015 0.977698i \(-0.567351\pi\)
−0.210015 + 0.977698i \(0.567351\pi\)
\(432\) −4.41803 −0.212562
\(433\) −8.51892 −0.409393 −0.204697 0.978825i \(-0.565621\pi\)
−0.204697 + 0.978825i \(0.565621\pi\)
\(434\) −1.10298 −0.0529448
\(435\) −16.1209 −0.772937
\(436\) −7.76784 −0.372012
\(437\) 0.634696 0.0303616
\(438\) −6.41731 −0.306631
\(439\) −22.5402 −1.07579 −0.537894 0.843013i \(-0.680780\pi\)
−0.537894 + 0.843013i \(0.680780\pi\)
\(440\) 0 0
\(441\) 0.704605 0.0335526
\(442\) 22.6012 1.07503
\(443\) 37.9089 1.80110 0.900552 0.434748i \(-0.143163\pi\)
0.900552 + 0.434748i \(0.143163\pi\)
\(444\) −3.42451 −0.162520
\(445\) 1.46836 0.0696068
\(446\) −20.2394 −0.958365
\(447\) 3.62424 0.171421
\(448\) −1.00000 −0.0472456
\(449\) −32.0551 −1.51277 −0.756387 0.654125i \(-0.773037\pi\)
−0.756387 + 0.654125i \(0.773037\pi\)
\(450\) 0.704605 0.0332154
\(451\) 0 0
\(452\) 4.63948 0.218223
\(453\) 10.4916 0.492940
\(454\) 11.8090 0.554225
\(455\) 5.24489 0.245884
\(456\) −1.42552 −0.0667562
\(457\) 24.5243 1.14720 0.573600 0.819136i \(-0.305547\pi\)
0.573600 + 0.819136i \(0.305547\pi\)
\(458\) 1.67906 0.0784572
\(459\) 19.0381 0.888623
\(460\) −0.856965 −0.0399562
\(461\) −0.188328 −0.00877132 −0.00438566 0.999990i \(-0.501396\pi\)
−0.00438566 + 0.999990i \(0.501396\pi\)
\(462\) 0 0
\(463\) −9.40956 −0.437299 −0.218650 0.975803i \(-0.570165\pi\)
−0.218650 + 0.975803i \(0.570165\pi\)
\(464\) −8.37564 −0.388829
\(465\) 2.12295 0.0984492
\(466\) −15.5220 −0.719043
\(467\) 1.84720 0.0854781 0.0427390 0.999086i \(-0.486392\pi\)
0.0427390 + 0.999086i \(0.486392\pi\)
\(468\) −3.69557 −0.170828
\(469\) −8.93617 −0.412634
\(470\) 8.40172 0.387542
\(471\) −23.0544 −1.06229
\(472\) −12.9838 −0.597629
\(473\) 0 0
\(474\) −3.00610 −0.138075
\(475\) −0.740633 −0.0339826
\(476\) 4.30919 0.197511
\(477\) −6.20518 −0.284116
\(478\) −22.0037 −1.00643
\(479\) 33.3506 1.52383 0.761913 0.647679i \(-0.224260\pi\)
0.761913 + 0.647679i \(0.224260\pi\)
\(480\) 1.92474 0.0878517
\(481\) 9.33176 0.425491
\(482\) −23.8155 −1.08477
\(483\) 1.64943 0.0750517
\(484\) 0 0
\(485\) 8.41705 0.382198
\(486\) −7.18150 −0.325759
\(487\) 14.1136 0.639550 0.319775 0.947493i \(-0.396393\pi\)
0.319775 + 0.947493i \(0.396393\pi\)
\(488\) 13.0887 0.592499
\(489\) −29.9378 −1.35384
\(490\) 1.00000 0.0451754
\(491\) −8.40881 −0.379484 −0.189742 0.981834i \(-0.560765\pi\)
−0.189742 + 0.981834i \(0.560765\pi\)
\(492\) −20.6331 −0.930213
\(493\) 36.0922 1.62551
\(494\) 3.88453 0.174773
\(495\) 0 0
\(496\) 1.10298 0.0495253
\(497\) 4.50750 0.202189
\(498\) 1.17258 0.0525444
\(499\) −22.8087 −1.02106 −0.510529 0.859860i \(-0.670550\pi\)
−0.510529 + 0.859860i \(0.670550\pi\)
\(500\) 1.00000 0.0447214
\(501\) 34.7364 1.55191
\(502\) 28.5083 1.27239
\(503\) −22.6508 −1.00995 −0.504975 0.863134i \(-0.668498\pi\)
−0.504975 + 0.863134i \(0.668498\pi\)
\(504\) −0.704605 −0.0313856
\(505\) −11.0295 −0.490808
\(506\) 0 0
\(507\) 27.9256 1.24022
\(508\) 9.42065 0.417974
\(509\) −15.0533 −0.667224 −0.333612 0.942710i \(-0.608268\pi\)
−0.333612 + 0.942710i \(0.608268\pi\)
\(510\) −8.29405 −0.367267
\(511\) 3.33413 0.147493
\(512\) 1.00000 0.0441942
\(513\) 3.27214 0.144468
\(514\) 29.6194 1.30645
\(515\) 5.86953 0.258643
\(516\) 2.18404 0.0961471
\(517\) 0 0
\(518\) 1.77921 0.0781740
\(519\) −18.4902 −0.811632
\(520\) −5.24489 −0.230003
\(521\) −7.47977 −0.327695 −0.163847 0.986486i \(-0.552390\pi\)
−0.163847 + 0.986486i \(0.552390\pi\)
\(522\) −5.90151 −0.258302
\(523\) −5.14535 −0.224990 −0.112495 0.993652i \(-0.535884\pi\)
−0.112495 + 0.993652i \(0.535884\pi\)
\(524\) −0.862825 −0.0376927
\(525\) −1.92474 −0.0840023
\(526\) 4.09184 0.178413
\(527\) −4.75295 −0.207042
\(528\) 0 0
\(529\) −22.2656 −0.968070
\(530\) −8.80661 −0.382535
\(531\) −9.14846 −0.397010
\(532\) 0.740633 0.0321105
\(533\) 56.2251 2.43538
\(534\) 2.82620 0.122302
\(535\) 6.83162 0.295357
\(536\) 8.93617 0.385984
\(537\) 25.5360 1.10196
\(538\) 2.99735 0.129225
\(539\) 0 0
\(540\) −4.41803 −0.190122
\(541\) −1.92055 −0.0825709 −0.0412855 0.999147i \(-0.513145\pi\)
−0.0412855 + 0.999147i \(0.513145\pi\)
\(542\) 24.7015 1.06102
\(543\) 37.6930 1.61756
\(544\) −4.30919 −0.184755
\(545\) −7.76784 −0.332738
\(546\) 10.0950 0.432027
\(547\) 0.260956 0.0111577 0.00557884 0.999984i \(-0.498224\pi\)
0.00557884 + 0.999984i \(0.498224\pi\)
\(548\) −17.2469 −0.736751
\(549\) 9.22238 0.393602
\(550\) 0 0
\(551\) 6.20327 0.264268
\(552\) −1.64943 −0.0702044
\(553\) 1.56183 0.0664157
\(554\) 2.67858 0.113802
\(555\) −3.42451 −0.145362
\(556\) 20.4448 0.867054
\(557\) −6.31867 −0.267731 −0.133865 0.991000i \(-0.542739\pi\)
−0.133865 + 0.991000i \(0.542739\pi\)
\(558\) 0.777166 0.0329000
\(559\) −5.95149 −0.251721
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −11.6213 −0.490217
\(563\) 0.958697 0.0404043 0.0202021 0.999796i \(-0.493569\pi\)
0.0202021 + 0.999796i \(0.493569\pi\)
\(564\) 16.1711 0.680925
\(565\) 4.63948 0.195185
\(566\) 18.1362 0.762321
\(567\) 10.6173 0.445887
\(568\) −4.50750 −0.189130
\(569\) −7.11799 −0.298402 −0.149201 0.988807i \(-0.547670\pi\)
−0.149201 + 0.988807i \(0.547670\pi\)
\(570\) −1.42552 −0.0597085
\(571\) −27.1751 −1.13724 −0.568621 0.822599i \(-0.692523\pi\)
−0.568621 + 0.822599i \(0.692523\pi\)
\(572\) 0 0
\(573\) −24.5509 −1.02563
\(574\) 10.7200 0.447443
\(575\) −0.856965 −0.0357379
\(576\) 0.704605 0.0293585
\(577\) −32.3702 −1.34759 −0.673795 0.738918i \(-0.735337\pi\)
−0.673795 + 0.738918i \(0.735337\pi\)
\(578\) 1.56912 0.0652667
\(579\) 22.7674 0.946181
\(580\) −8.37564 −0.347779
\(581\) −0.609215 −0.0252745
\(582\) 16.2006 0.671536
\(583\) 0 0
\(584\) −3.33413 −0.137967
\(585\) −3.69557 −0.152793
\(586\) −22.8700 −0.944753
\(587\) −8.04888 −0.332213 −0.166106 0.986108i \(-0.553120\pi\)
−0.166106 + 0.986108i \(0.553120\pi\)
\(588\) 1.92474 0.0793747
\(589\) −0.816904 −0.0336599
\(590\) −12.9838 −0.534535
\(591\) −42.1718 −1.73472
\(592\) −1.77921 −0.0731251
\(593\) −39.8151 −1.63501 −0.817506 0.575920i \(-0.804644\pi\)
−0.817506 + 0.575920i \(0.804644\pi\)
\(594\) 0 0
\(595\) 4.30919 0.176660
\(596\) 1.88298 0.0771299
\(597\) 19.8481 0.812327
\(598\) 4.49468 0.183801
\(599\) 30.7925 1.25815 0.629074 0.777346i \(-0.283434\pi\)
0.629074 + 0.777346i \(0.283434\pi\)
\(600\) 1.92474 0.0785770
\(601\) 33.3140 1.35891 0.679454 0.733718i \(-0.262217\pi\)
0.679454 + 0.733718i \(0.262217\pi\)
\(602\) −1.13472 −0.0462479
\(603\) 6.29647 0.256412
\(604\) 5.45095 0.221796
\(605\) 0 0
\(606\) −21.2290 −0.862367
\(607\) −16.8748 −0.684927 −0.342463 0.939531i \(-0.611261\pi\)
−0.342463 + 0.939531i \(0.611261\pi\)
\(608\) −0.740633 −0.0300366
\(609\) 16.1209 0.653251
\(610\) 13.0887 0.529947
\(611\) −44.0660 −1.78272
\(612\) −3.03628 −0.122734
\(613\) 26.7574 1.08072 0.540360 0.841434i \(-0.318288\pi\)
0.540360 + 0.841434i \(0.318288\pi\)
\(614\) −19.9326 −0.804413
\(615\) −20.6331 −0.832008
\(616\) 0 0
\(617\) 32.5387 1.30996 0.654979 0.755647i \(-0.272677\pi\)
0.654979 + 0.755647i \(0.272677\pi\)
\(618\) 11.2973 0.454444
\(619\) −13.9252 −0.559701 −0.279850 0.960044i \(-0.590285\pi\)
−0.279850 + 0.960044i \(0.590285\pi\)
\(620\) 1.10298 0.0442968
\(621\) 3.78609 0.151931
\(622\) −1.28437 −0.0514985
\(623\) −1.46836 −0.0588285
\(624\) −10.0950 −0.404124
\(625\) 1.00000 0.0400000
\(626\) −13.3964 −0.535429
\(627\) 0 0
\(628\) −11.9779 −0.477972
\(629\) 7.66696 0.305702
\(630\) −0.704605 −0.0280721
\(631\) −12.8350 −0.510954 −0.255477 0.966815i \(-0.582232\pi\)
−0.255477 + 0.966815i \(0.582232\pi\)
\(632\) −1.56183 −0.0621262
\(633\) −47.8210 −1.90071
\(634\) 16.6640 0.661810
\(635\) 9.42065 0.373847
\(636\) −16.9504 −0.672126
\(637\) −5.24489 −0.207810
\(638\) 0 0
\(639\) −3.17601 −0.125641
\(640\) 1.00000 0.0395285
\(641\) −48.5629 −1.91812 −0.959059 0.283207i \(-0.908602\pi\)
−0.959059 + 0.283207i \(0.908602\pi\)
\(642\) 13.1491 0.518952
\(643\) −14.1751 −0.559009 −0.279505 0.960144i \(-0.590170\pi\)
−0.279505 + 0.960144i \(0.590170\pi\)
\(644\) 0.856965 0.0337691
\(645\) 2.18404 0.0859966
\(646\) 3.19153 0.125569
\(647\) −24.7982 −0.974917 −0.487459 0.873146i \(-0.662076\pi\)
−0.487459 + 0.873146i \(0.662076\pi\)
\(648\) −10.6173 −0.417089
\(649\) 0 0
\(650\) −5.24489 −0.205721
\(651\) −2.12295 −0.0832048
\(652\) −15.5543 −0.609152
\(653\) −29.0557 −1.13704 −0.568519 0.822670i \(-0.692483\pi\)
−0.568519 + 0.822670i \(0.692483\pi\)
\(654\) −14.9510 −0.584632
\(655\) −0.862825 −0.0337134
\(656\) −10.7200 −0.418545
\(657\) −2.34924 −0.0916526
\(658\) −8.40172 −0.327533
\(659\) 44.0463 1.71580 0.857899 0.513818i \(-0.171769\pi\)
0.857899 + 0.513818i \(0.171769\pi\)
\(660\) 0 0
\(661\) −0.575059 −0.0223672 −0.0111836 0.999937i \(-0.503560\pi\)
−0.0111836 + 0.999937i \(0.503560\pi\)
\(662\) 11.1340 0.432734
\(663\) 43.5013 1.68945
\(664\) 0.609215 0.0236421
\(665\) 0.740633 0.0287205
\(666\) −1.25364 −0.0485776
\(667\) 7.17762 0.277919
\(668\) 18.0474 0.698274
\(669\) −38.9556 −1.50611
\(670\) 8.93617 0.345234
\(671\) 0 0
\(672\) −1.92474 −0.0742483
\(673\) −27.7568 −1.06995 −0.534974 0.844869i \(-0.679679\pi\)
−0.534974 + 0.844869i \(0.679679\pi\)
\(674\) −6.71838 −0.258782
\(675\) −4.41803 −0.170050
\(676\) 14.5088 0.558032
\(677\) 51.1859 1.96723 0.983617 0.180272i \(-0.0576979\pi\)
0.983617 + 0.180272i \(0.0576979\pi\)
\(678\) 8.92978 0.342946
\(679\) −8.41705 −0.323017
\(680\) −4.30919 −0.165250
\(681\) 22.7292 0.870986
\(682\) 0 0
\(683\) −1.55522 −0.0595090 −0.0297545 0.999557i \(-0.509473\pi\)
−0.0297545 + 0.999557i \(0.509473\pi\)
\(684\) −0.521854 −0.0199536
\(685\) −17.2469 −0.658970
\(686\) −1.00000 −0.0381802
\(687\) 3.23174 0.123299
\(688\) 1.13472 0.0432609
\(689\) 46.1896 1.75969
\(690\) −1.64943 −0.0627927
\(691\) 4.60492 0.175179 0.0875896 0.996157i \(-0.472084\pi\)
0.0875896 + 0.996157i \(0.472084\pi\)
\(692\) −9.60665 −0.365190
\(693\) 0 0
\(694\) −19.4103 −0.736804
\(695\) 20.4448 0.775517
\(696\) −16.1209 −0.611060
\(697\) 46.1944 1.74974
\(698\) 26.7685 1.01320
\(699\) −29.8757 −1.13000
\(700\) −1.00000 −0.0377964
\(701\) −2.91484 −0.110092 −0.0550460 0.998484i \(-0.517531\pi\)
−0.0550460 + 0.998484i \(0.517531\pi\)
\(702\) 23.1720 0.874573
\(703\) 1.31774 0.0496996
\(704\) 0 0
\(705\) 16.1711 0.609038
\(706\) −32.9268 −1.23922
\(707\) 11.0295 0.414809
\(708\) −24.9904 −0.939197
\(709\) 3.62357 0.136086 0.0680430 0.997682i \(-0.478324\pi\)
0.0680430 + 0.997682i \(0.478324\pi\)
\(710\) −4.50750 −0.169163
\(711\) −1.10047 −0.0412709
\(712\) 1.46836 0.0550290
\(713\) −0.945215 −0.0353986
\(714\) 8.29405 0.310397
\(715\) 0 0
\(716\) 13.2673 0.495822
\(717\) −42.3514 −1.58164
\(718\) 36.1167 1.34786
\(719\) −8.72290 −0.325309 −0.162655 0.986683i \(-0.552006\pi\)
−0.162655 + 0.986683i \(0.552006\pi\)
\(720\) 0.704605 0.0262591
\(721\) −5.86953 −0.218593
\(722\) −18.4515 −0.686692
\(723\) −45.8386 −1.70476
\(724\) 19.5835 0.727815
\(725\) −8.37564 −0.311063
\(726\) 0 0
\(727\) 37.3206 1.38415 0.692073 0.721828i \(-0.256698\pi\)
0.692073 + 0.721828i \(0.256698\pi\)
\(728\) 5.24489 0.194388
\(729\) 18.0296 0.667761
\(730\) −3.33413 −0.123402
\(731\) −4.88974 −0.180854
\(732\) 25.1923 0.931136
\(733\) 8.80262 0.325132 0.162566 0.986698i \(-0.448023\pi\)
0.162566 + 0.986698i \(0.448023\pi\)
\(734\) −8.35672 −0.308452
\(735\) 1.92474 0.0709949
\(736\) −0.856965 −0.0315881
\(737\) 0 0
\(738\) −7.55335 −0.278043
\(739\) 50.6335 1.86258 0.931292 0.364274i \(-0.118683\pi\)
0.931292 + 0.364274i \(0.118683\pi\)
\(740\) −1.77921 −0.0654051
\(741\) 7.47670 0.274663
\(742\) 8.80661 0.323301
\(743\) −50.4494 −1.85081 −0.925404 0.378983i \(-0.876274\pi\)
−0.925404 + 0.378983i \(0.876274\pi\)
\(744\) 2.12295 0.0778310
\(745\) 1.88298 0.0689871
\(746\) −16.0205 −0.586552
\(747\) 0.429256 0.0157057
\(748\) 0 0
\(749\) −6.83162 −0.249622
\(750\) 1.92474 0.0702814
\(751\) −23.9655 −0.874512 −0.437256 0.899337i \(-0.644050\pi\)
−0.437256 + 0.899337i \(0.644050\pi\)
\(752\) 8.40172 0.306379
\(753\) 54.8710 1.99961
\(754\) 43.9292 1.59981
\(755\) 5.45095 0.198380
\(756\) 4.41803 0.160682
\(757\) −4.87670 −0.177247 −0.0886233 0.996065i \(-0.528247\pi\)
−0.0886233 + 0.996065i \(0.528247\pi\)
\(758\) 25.3872 0.922105
\(759\) 0 0
\(760\) −0.740633 −0.0268656
\(761\) −44.6872 −1.61991 −0.809955 0.586492i \(-0.800509\pi\)
−0.809955 + 0.586492i \(0.800509\pi\)
\(762\) 18.1323 0.656862
\(763\) 7.76784 0.281215
\(764\) −12.7555 −0.461477
\(765\) −3.03628 −0.109777
\(766\) −0.303318 −0.0109593
\(767\) 68.0986 2.45890
\(768\) 1.92474 0.0694529
\(769\) 52.9611 1.90983 0.954914 0.296884i \(-0.0959474\pi\)
0.954914 + 0.296884i \(0.0959474\pi\)
\(770\) 0 0
\(771\) 57.0094 2.05314
\(772\) 11.8288 0.425729
\(773\) 38.4372 1.38249 0.691246 0.722620i \(-0.257062\pi\)
0.691246 + 0.722620i \(0.257062\pi\)
\(774\) 0.799532 0.0287386
\(775\) 1.10298 0.0396202
\(776\) 8.41705 0.302154
\(777\) 3.42451 0.122854
\(778\) −6.74058 −0.241661
\(779\) 7.93957 0.284465
\(780\) −10.0950 −0.361459
\(781\) 0 0
\(782\) 3.69282 0.132055
\(783\) 37.0038 1.32241
\(784\) 1.00000 0.0357143
\(785\) −11.9779 −0.427511
\(786\) −1.66071 −0.0592356
\(787\) −11.9075 −0.424458 −0.212229 0.977220i \(-0.568072\pi\)
−0.212229 + 0.977220i \(0.568072\pi\)
\(788\) −21.9105 −0.780528
\(789\) 7.87570 0.280383
\(790\) −1.56183 −0.0555673
\(791\) −4.63948 −0.164961
\(792\) 0 0
\(793\) −68.6489 −2.43779
\(794\) −19.9230 −0.707039
\(795\) −16.9504 −0.601168
\(796\) 10.3121 0.365503
\(797\) −2.09798 −0.0743142 −0.0371571 0.999309i \(-0.511830\pi\)
−0.0371571 + 0.999309i \(0.511830\pi\)
\(798\) 1.42552 0.0504629
\(799\) −36.2046 −1.28083
\(800\) 1.00000 0.0353553
\(801\) 1.03461 0.0365562
\(802\) 34.0886 1.20371
\(803\) 0 0
\(804\) 17.1998 0.606589
\(805\) 0.856965 0.0302040
\(806\) −5.78501 −0.203768
\(807\) 5.76911 0.203082
\(808\) −11.0295 −0.388018
\(809\) −12.0315 −0.423005 −0.211503 0.977377i \(-0.567836\pi\)
−0.211503 + 0.977377i \(0.567836\pi\)
\(810\) −10.6173 −0.373056
\(811\) −8.52991 −0.299526 −0.149763 0.988722i \(-0.547851\pi\)
−0.149763 + 0.988722i \(0.547851\pi\)
\(812\) 8.37564 0.293927
\(813\) 47.5439 1.66744
\(814\) 0 0
\(815\) −15.5543 −0.544842
\(816\) −8.29405 −0.290350
\(817\) −0.840413 −0.0294023
\(818\) 8.09670 0.283094
\(819\) 3.69557 0.129134
\(820\) −10.7200 −0.374358
\(821\) 49.8308 1.73911 0.869554 0.493839i \(-0.164407\pi\)
0.869554 + 0.493839i \(0.164407\pi\)
\(822\) −33.1957 −1.15783
\(823\) −14.8465 −0.517516 −0.258758 0.965942i \(-0.583313\pi\)
−0.258758 + 0.965942i \(0.583313\pi\)
\(824\) 5.86953 0.204475
\(825\) 0 0
\(826\) 12.9838 0.451765
\(827\) −1.52007 −0.0528580 −0.0264290 0.999651i \(-0.508414\pi\)
−0.0264290 + 0.999651i \(0.508414\pi\)
\(828\) −0.603822 −0.0209842
\(829\) −10.4514 −0.362993 −0.181497 0.983392i \(-0.558094\pi\)
−0.181497 + 0.983392i \(0.558094\pi\)
\(830\) 0.609215 0.0211462
\(831\) 5.15556 0.178845
\(832\) −5.24489 −0.181834
\(833\) −4.30919 −0.149305
\(834\) 39.3509 1.36261
\(835\) 18.0474 0.624555
\(836\) 0 0
\(837\) −4.87300 −0.168435
\(838\) −24.0686 −0.831437
\(839\) −39.9761 −1.38013 −0.690064 0.723748i \(-0.742418\pi\)
−0.690064 + 0.723748i \(0.742418\pi\)
\(840\) −1.92474 −0.0664097
\(841\) 41.1513 1.41901
\(842\) 13.7871 0.475135
\(843\) −22.3680 −0.770395
\(844\) −24.8455 −0.855217
\(845\) 14.5088 0.499119
\(846\) 5.91989 0.203530
\(847\) 0 0
\(848\) −8.80661 −0.302420
\(849\) 34.9074 1.19802
\(850\) −4.30919 −0.147804
\(851\) 1.52472 0.0522668
\(852\) −8.67574 −0.297226
\(853\) −22.2228 −0.760893 −0.380446 0.924803i \(-0.624230\pi\)
−0.380446 + 0.924803i \(0.624230\pi\)
\(854\) −13.0887 −0.447887
\(855\) −0.521854 −0.0178470
\(856\) 6.83162 0.233500
\(857\) −15.8282 −0.540681 −0.270340 0.962765i \(-0.587136\pi\)
−0.270340 + 0.962765i \(0.587136\pi\)
\(858\) 0 0
\(859\) −20.7983 −0.709628 −0.354814 0.934937i \(-0.615456\pi\)
−0.354814 + 0.934937i \(0.615456\pi\)
\(860\) 1.13472 0.0386937
\(861\) 20.6331 0.703175
\(862\) −8.72005 −0.297006
\(863\) −18.8317 −0.641040 −0.320520 0.947242i \(-0.603858\pi\)
−0.320520 + 0.947242i \(0.603858\pi\)
\(864\) −4.41803 −0.150304
\(865\) −9.60665 −0.326636
\(866\) −8.51892 −0.289485
\(867\) 3.02013 0.102569
\(868\) −1.10298 −0.0374376
\(869\) 0 0
\(870\) −16.1209 −0.546549
\(871\) −46.8692 −1.58810
\(872\) −7.76784 −0.263052
\(873\) 5.93069 0.200724
\(874\) 0.634696 0.0214689
\(875\) −1.00000 −0.0338062
\(876\) −6.41731 −0.216821
\(877\) −16.6176 −0.561138 −0.280569 0.959834i \(-0.590523\pi\)
−0.280569 + 0.959834i \(0.590523\pi\)
\(878\) −22.5402 −0.760697
\(879\) −44.0188 −1.48472
\(880\) 0 0
\(881\) 21.6943 0.730901 0.365450 0.930831i \(-0.380915\pi\)
0.365450 + 0.930831i \(0.380915\pi\)
\(882\) 0.704605 0.0237253
\(883\) 32.8896 1.10682 0.553412 0.832907i \(-0.313325\pi\)
0.553412 + 0.832907i \(0.313325\pi\)
\(884\) 22.6012 0.760161
\(885\) −24.9904 −0.840044
\(886\) 37.9089 1.27357
\(887\) 4.02388 0.135109 0.0675544 0.997716i \(-0.478480\pi\)
0.0675544 + 0.997716i \(0.478480\pi\)
\(888\) −3.42451 −0.114919
\(889\) −9.42065 −0.315958
\(890\) 1.46836 0.0492194
\(891\) 0 0
\(892\) −20.2394 −0.677667
\(893\) −6.22259 −0.208231
\(894\) 3.62424 0.121213
\(895\) 13.2673 0.443476
\(896\) −1.00000 −0.0334077
\(897\) 8.65107 0.288851
\(898\) −32.0551 −1.06969
\(899\) −9.23816 −0.308110
\(900\) 0.704605 0.0234868
\(901\) 37.9493 1.26428
\(902\) 0 0
\(903\) −2.18404 −0.0726804
\(904\) 4.63948 0.154307
\(905\) 19.5835 0.650977
\(906\) 10.4916 0.348561
\(907\) 20.4539 0.679162 0.339581 0.940577i \(-0.389715\pi\)
0.339581 + 0.940577i \(0.389715\pi\)
\(908\) 11.8090 0.391896
\(909\) −7.77147 −0.257763
\(910\) 5.24489 0.173866
\(911\) 27.9283 0.925305 0.462653 0.886540i \(-0.346898\pi\)
0.462653 + 0.886540i \(0.346898\pi\)
\(912\) −1.42552 −0.0472037
\(913\) 0 0
\(914\) 24.5243 0.811192
\(915\) 25.1923 0.832833
\(916\) 1.67906 0.0554776
\(917\) 0.862825 0.0284930
\(918\) 19.0381 0.628352
\(919\) −13.3202 −0.439394 −0.219697 0.975568i \(-0.570507\pi\)
−0.219697 + 0.975568i \(0.570507\pi\)
\(920\) −0.856965 −0.0282533
\(921\) −38.3649 −1.26417
\(922\) −0.188328 −0.00620226
\(923\) 23.6413 0.778163
\(924\) 0 0
\(925\) −1.77921 −0.0585001
\(926\) −9.40956 −0.309217
\(927\) 4.13570 0.135834
\(928\) −8.37564 −0.274944
\(929\) 39.2113 1.28648 0.643240 0.765664i \(-0.277590\pi\)
0.643240 + 0.765664i \(0.277590\pi\)
\(930\) 2.12295 0.0696141
\(931\) −0.740633 −0.0242733
\(932\) −15.5220 −0.508440
\(933\) −2.47207 −0.0809319
\(934\) 1.84720 0.0604421
\(935\) 0 0
\(936\) −3.69557 −0.120794
\(937\) −21.7403 −0.710225 −0.355112 0.934824i \(-0.615557\pi\)
−0.355112 + 0.934824i \(0.615557\pi\)
\(938\) −8.93617 −0.291776
\(939\) −25.7846 −0.841447
\(940\) 8.40172 0.274034
\(941\) 16.4125 0.535033 0.267516 0.963553i \(-0.413797\pi\)
0.267516 + 0.963553i \(0.413797\pi\)
\(942\) −23.0544 −0.751152
\(943\) 9.18665 0.299158
\(944\) −12.9838 −0.422587
\(945\) 4.41803 0.143718
\(946\) 0 0
\(947\) −45.1297 −1.46652 −0.733259 0.679949i \(-0.762002\pi\)
−0.733259 + 0.679949i \(0.762002\pi\)
\(948\) −3.00610 −0.0976337
\(949\) 17.4871 0.567655
\(950\) −0.740633 −0.0240293
\(951\) 32.0737 1.04006
\(952\) 4.30919 0.139662
\(953\) −46.8181 −1.51659 −0.758294 0.651913i \(-0.773967\pi\)
−0.758294 + 0.651913i \(0.773967\pi\)
\(954\) −6.20518 −0.200900
\(955\) −12.7555 −0.412758
\(956\) −22.0037 −0.711652
\(957\) 0 0
\(958\) 33.3506 1.07751
\(959\) 17.2469 0.556931
\(960\) 1.92474 0.0621206
\(961\) −29.7834 −0.960756
\(962\) 9.33176 0.300868
\(963\) 4.81359 0.155116
\(964\) −23.8155 −0.767047
\(965\) 11.8288 0.380784
\(966\) 1.64943 0.0530695
\(967\) −57.9645 −1.86401 −0.932007 0.362441i \(-0.881943\pi\)
−0.932007 + 0.362441i \(0.881943\pi\)
\(968\) 0 0
\(969\) 6.14284 0.197337
\(970\) 8.41705 0.270255
\(971\) −10.6264 −0.341019 −0.170509 0.985356i \(-0.554541\pi\)
−0.170509 + 0.985356i \(0.554541\pi\)
\(972\) −7.18150 −0.230347
\(973\) −20.4448 −0.655432
\(974\) 14.1136 0.452230
\(975\) −10.0950 −0.323299
\(976\) 13.0887 0.418960
\(977\) −33.6202 −1.07561 −0.537803 0.843071i \(-0.680745\pi\)
−0.537803 + 0.843071i \(0.680745\pi\)
\(978\) −29.9378 −0.957307
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −5.47326 −0.174748
\(982\) −8.40881 −0.268336
\(983\) 38.3652 1.22366 0.611830 0.790989i \(-0.290434\pi\)
0.611830 + 0.790989i \(0.290434\pi\)
\(984\) −20.6331 −0.657760
\(985\) −21.9105 −0.698125
\(986\) 36.0922 1.14941
\(987\) −16.1711 −0.514731
\(988\) 3.88453 0.123583
\(989\) −0.972418 −0.0309211
\(990\) 0 0
\(991\) 55.5999 1.76619 0.883094 0.469196i \(-0.155456\pi\)
0.883094 + 0.469196i \(0.155456\pi\)
\(992\) 1.10298 0.0350197
\(993\) 21.4300 0.680059
\(994\) 4.50750 0.142969
\(995\) 10.3121 0.326916
\(996\) 1.17258 0.0371545
\(997\) 22.9698 0.727462 0.363731 0.931504i \(-0.381503\pi\)
0.363731 + 0.931504i \(0.381503\pi\)
\(998\) −22.8087 −0.721997
\(999\) 7.86060 0.248698
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.db.1.6 6
11.5 even 5 770.2.n.g.421.3 12
11.9 even 5 770.2.n.g.631.3 yes 12
11.10 odd 2 8470.2.a.cv.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.421.3 12 11.5 even 5
770.2.n.g.631.3 yes 12 11.9 even 5
8470.2.a.cv.1.6 6 11.10 odd 2
8470.2.a.db.1.6 6 1.1 even 1 trivial