Properties

Label 8004.2.a.i.1.9
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.282383\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.282383 q^{5} +5.02462 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.282383 q^{5} +5.02462 q^{7} +1.00000 q^{9} +1.61549 q^{11} +0.572564 q^{13} -0.282383 q^{15} -0.464906 q^{17} +0.460906 q^{19} -5.02462 q^{21} +1.00000 q^{23} -4.92026 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.38151 q^{31} -1.61549 q^{33} +1.41887 q^{35} +4.37755 q^{37} -0.572564 q^{39} +2.95258 q^{41} -6.06819 q^{43} +0.282383 q^{45} +6.36723 q^{47} +18.2468 q^{49} +0.464906 q^{51} -5.96026 q^{53} +0.456188 q^{55} -0.460906 q^{57} +12.5554 q^{59} +7.82665 q^{61} +5.02462 q^{63} +0.161682 q^{65} -4.23458 q^{67} -1.00000 q^{69} +13.7536 q^{71} +5.71699 q^{73} +4.92026 q^{75} +8.11724 q^{77} -5.81674 q^{79} +1.00000 q^{81} +13.1920 q^{83} -0.131282 q^{85} -1.00000 q^{87} +5.33893 q^{89} +2.87692 q^{91} +1.38151 q^{93} +0.130152 q^{95} -17.5054 q^{97} +1.61549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.282383 0.126286 0.0631428 0.998005i \(-0.479888\pi\)
0.0631428 + 0.998005i \(0.479888\pi\)
\(6\) 0 0
\(7\) 5.02462 1.89913 0.949565 0.313571i \(-0.101525\pi\)
0.949565 + 0.313571i \(0.101525\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.61549 0.487089 0.243545 0.969890i \(-0.421690\pi\)
0.243545 + 0.969890i \(0.421690\pi\)
\(12\) 0 0
\(13\) 0.572564 0.158801 0.0794004 0.996843i \(-0.474699\pi\)
0.0794004 + 0.996843i \(0.474699\pi\)
\(14\) 0 0
\(15\) −0.282383 −0.0729110
\(16\) 0 0
\(17\) −0.464906 −0.112756 −0.0563781 0.998409i \(-0.517955\pi\)
−0.0563781 + 0.998409i \(0.517955\pi\)
\(18\) 0 0
\(19\) 0.460906 0.105739 0.0528695 0.998601i \(-0.483163\pi\)
0.0528695 + 0.998601i \(0.483163\pi\)
\(20\) 0 0
\(21\) −5.02462 −1.09646
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.92026 −0.984052
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.38151 −0.248127 −0.124064 0.992274i \(-0.539593\pi\)
−0.124064 + 0.992274i \(0.539593\pi\)
\(32\) 0 0
\(33\) −1.61549 −0.281221
\(34\) 0 0
\(35\) 1.41887 0.239833
\(36\) 0 0
\(37\) 4.37755 0.719665 0.359832 0.933017i \(-0.382834\pi\)
0.359832 + 0.933017i \(0.382834\pi\)
\(38\) 0 0
\(39\) −0.572564 −0.0916836
\(40\) 0 0
\(41\) 2.95258 0.461115 0.230557 0.973059i \(-0.425945\pi\)
0.230557 + 0.973059i \(0.425945\pi\)
\(42\) 0 0
\(43\) −6.06819 −0.925390 −0.462695 0.886518i \(-0.653117\pi\)
−0.462695 + 0.886518i \(0.653117\pi\)
\(44\) 0 0
\(45\) 0.282383 0.0420952
\(46\) 0 0
\(47\) 6.36723 0.928756 0.464378 0.885637i \(-0.346278\pi\)
0.464378 + 0.885637i \(0.346278\pi\)
\(48\) 0 0
\(49\) 18.2468 2.60669
\(50\) 0 0
\(51\) 0.464906 0.0650998
\(52\) 0 0
\(53\) −5.96026 −0.818705 −0.409353 0.912376i \(-0.634245\pi\)
−0.409353 + 0.912376i \(0.634245\pi\)
\(54\) 0 0
\(55\) 0.456188 0.0615124
\(56\) 0 0
\(57\) −0.460906 −0.0610484
\(58\) 0 0
\(59\) 12.5554 1.63457 0.817284 0.576235i \(-0.195479\pi\)
0.817284 + 0.576235i \(0.195479\pi\)
\(60\) 0 0
\(61\) 7.82665 1.00210 0.501050 0.865418i \(-0.332947\pi\)
0.501050 + 0.865418i \(0.332947\pi\)
\(62\) 0 0
\(63\) 5.02462 0.633043
\(64\) 0 0
\(65\) 0.161682 0.0200542
\(66\) 0 0
\(67\) −4.23458 −0.517336 −0.258668 0.965966i \(-0.583284\pi\)
−0.258668 + 0.965966i \(0.583284\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 13.7536 1.63225 0.816126 0.577873i \(-0.196117\pi\)
0.816126 + 0.577873i \(0.196117\pi\)
\(72\) 0 0
\(73\) 5.71699 0.669123 0.334561 0.942374i \(-0.391412\pi\)
0.334561 + 0.942374i \(0.391412\pi\)
\(74\) 0 0
\(75\) 4.92026 0.568143
\(76\) 0 0
\(77\) 8.11724 0.925046
\(78\) 0 0
\(79\) −5.81674 −0.654435 −0.327217 0.944949i \(-0.606111\pi\)
−0.327217 + 0.944949i \(0.606111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.1920 1.44802 0.724008 0.689792i \(-0.242298\pi\)
0.724008 + 0.689792i \(0.242298\pi\)
\(84\) 0 0
\(85\) −0.131282 −0.0142395
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 5.33893 0.565926 0.282963 0.959131i \(-0.408683\pi\)
0.282963 + 0.959131i \(0.408683\pi\)
\(90\) 0 0
\(91\) 2.87692 0.301583
\(92\) 0 0
\(93\) 1.38151 0.143256
\(94\) 0 0
\(95\) 0.130152 0.0133533
\(96\) 0 0
\(97\) −17.5054 −1.77740 −0.888702 0.458486i \(-0.848392\pi\)
−0.888702 + 0.458486i \(0.848392\pi\)
\(98\) 0 0
\(99\) 1.61549 0.162363
\(100\) 0 0
\(101\) 11.9008 1.18417 0.592085 0.805876i \(-0.298305\pi\)
0.592085 + 0.805876i \(0.298305\pi\)
\(102\) 0 0
\(103\) −7.02283 −0.691980 −0.345990 0.938238i \(-0.612457\pi\)
−0.345990 + 0.938238i \(0.612457\pi\)
\(104\) 0 0
\(105\) −1.41887 −0.138467
\(106\) 0 0
\(107\) −13.4809 −1.30325 −0.651626 0.758540i \(-0.725913\pi\)
−0.651626 + 0.758540i \(0.725913\pi\)
\(108\) 0 0
\(109\) −7.29204 −0.698451 −0.349226 0.937039i \(-0.613555\pi\)
−0.349226 + 0.937039i \(0.613555\pi\)
\(110\) 0 0
\(111\) −4.37755 −0.415499
\(112\) 0 0
\(113\) −20.9016 −1.96625 −0.983127 0.182925i \(-0.941443\pi\)
−0.983127 + 0.182925i \(0.941443\pi\)
\(114\) 0 0
\(115\) 0.282383 0.0263324
\(116\) 0 0
\(117\) 0.572564 0.0529336
\(118\) 0 0
\(119\) −2.33598 −0.214139
\(120\) 0 0
\(121\) −8.39018 −0.762744
\(122\) 0 0
\(123\) −2.95258 −0.266225
\(124\) 0 0
\(125\) −2.80131 −0.250557
\(126\) 0 0
\(127\) 18.5052 1.64207 0.821037 0.570874i \(-0.193396\pi\)
0.821037 + 0.570874i \(0.193396\pi\)
\(128\) 0 0
\(129\) 6.06819 0.534274
\(130\) 0 0
\(131\) 17.6604 1.54300 0.771498 0.636232i \(-0.219508\pi\)
0.771498 + 0.636232i \(0.219508\pi\)
\(132\) 0 0
\(133\) 2.31588 0.200812
\(134\) 0 0
\(135\) −0.282383 −0.0243037
\(136\) 0 0
\(137\) −13.0006 −1.11072 −0.555358 0.831612i \(-0.687419\pi\)
−0.555358 + 0.831612i \(0.687419\pi\)
\(138\) 0 0
\(139\) −0.446628 −0.0378825 −0.0189412 0.999821i \(-0.506030\pi\)
−0.0189412 + 0.999821i \(0.506030\pi\)
\(140\) 0 0
\(141\) −6.36723 −0.536217
\(142\) 0 0
\(143\) 0.924973 0.0773501
\(144\) 0 0
\(145\) 0.282383 0.0234506
\(146\) 0 0
\(147\) −18.2468 −1.50497
\(148\) 0 0
\(149\) −2.48939 −0.203939 −0.101969 0.994788i \(-0.532514\pi\)
−0.101969 + 0.994788i \(0.532514\pi\)
\(150\) 0 0
\(151\) 6.97623 0.567717 0.283859 0.958866i \(-0.408385\pi\)
0.283859 + 0.958866i \(0.408385\pi\)
\(152\) 0 0
\(153\) −0.464906 −0.0375854
\(154\) 0 0
\(155\) −0.390116 −0.0313349
\(156\) 0 0
\(157\) 0.517289 0.0412842 0.0206421 0.999787i \(-0.493429\pi\)
0.0206421 + 0.999787i \(0.493429\pi\)
\(158\) 0 0
\(159\) 5.96026 0.472680
\(160\) 0 0
\(161\) 5.02462 0.395996
\(162\) 0 0
\(163\) 11.5780 0.906858 0.453429 0.891292i \(-0.350200\pi\)
0.453429 + 0.891292i \(0.350200\pi\)
\(164\) 0 0
\(165\) −0.456188 −0.0355142
\(166\) 0 0
\(167\) −8.90578 −0.689150 −0.344575 0.938759i \(-0.611977\pi\)
−0.344575 + 0.938759i \(0.611977\pi\)
\(168\) 0 0
\(169\) −12.6722 −0.974782
\(170\) 0 0
\(171\) 0.460906 0.0352463
\(172\) 0 0
\(173\) 18.0923 1.37553 0.687765 0.725933i \(-0.258592\pi\)
0.687765 + 0.725933i \(0.258592\pi\)
\(174\) 0 0
\(175\) −24.7225 −1.86884
\(176\) 0 0
\(177\) −12.5554 −0.943719
\(178\) 0 0
\(179\) 7.62128 0.569641 0.284820 0.958581i \(-0.408066\pi\)
0.284820 + 0.958581i \(0.408066\pi\)
\(180\) 0 0
\(181\) 20.9090 1.55415 0.777075 0.629408i \(-0.216702\pi\)
0.777075 + 0.629408i \(0.216702\pi\)
\(182\) 0 0
\(183\) −7.82665 −0.578563
\(184\) 0 0
\(185\) 1.23615 0.0908833
\(186\) 0 0
\(187\) −0.751052 −0.0549224
\(188\) 0 0
\(189\) −5.02462 −0.365488
\(190\) 0 0
\(191\) 5.10381 0.369299 0.184649 0.982804i \(-0.440885\pi\)
0.184649 + 0.982804i \(0.440885\pi\)
\(192\) 0 0
\(193\) −7.73467 −0.556754 −0.278377 0.960472i \(-0.589796\pi\)
−0.278377 + 0.960472i \(0.589796\pi\)
\(194\) 0 0
\(195\) −0.161682 −0.0115783
\(196\) 0 0
\(197\) −10.0993 −0.719545 −0.359772 0.933040i \(-0.617146\pi\)
−0.359772 + 0.933040i \(0.617146\pi\)
\(198\) 0 0
\(199\) −12.0031 −0.850880 −0.425440 0.904987i \(-0.639881\pi\)
−0.425440 + 0.904987i \(0.639881\pi\)
\(200\) 0 0
\(201\) 4.23458 0.298684
\(202\) 0 0
\(203\) 5.02462 0.352659
\(204\) 0 0
\(205\) 0.833758 0.0582322
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0.744590 0.0515043
\(210\) 0 0
\(211\) 8.53376 0.587488 0.293744 0.955884i \(-0.405099\pi\)
0.293744 + 0.955884i \(0.405099\pi\)
\(212\) 0 0
\(213\) −13.7536 −0.942382
\(214\) 0 0
\(215\) −1.71355 −0.116863
\(216\) 0 0
\(217\) −6.94159 −0.471226
\(218\) 0 0
\(219\) −5.71699 −0.386318
\(220\) 0 0
\(221\) −0.266188 −0.0179058
\(222\) 0 0
\(223\) −19.1107 −1.27975 −0.639874 0.768480i \(-0.721013\pi\)
−0.639874 + 0.768480i \(0.721013\pi\)
\(224\) 0 0
\(225\) −4.92026 −0.328017
\(226\) 0 0
\(227\) 12.2099 0.810402 0.405201 0.914228i \(-0.367202\pi\)
0.405201 + 0.914228i \(0.367202\pi\)
\(228\) 0 0
\(229\) −27.4713 −1.81536 −0.907678 0.419668i \(-0.862147\pi\)
−0.907678 + 0.419668i \(0.862147\pi\)
\(230\) 0 0
\(231\) −8.11724 −0.534075
\(232\) 0 0
\(233\) 21.9922 1.44076 0.720378 0.693582i \(-0.243968\pi\)
0.720378 + 0.693582i \(0.243968\pi\)
\(234\) 0 0
\(235\) 1.79800 0.117288
\(236\) 0 0
\(237\) 5.81674 0.377838
\(238\) 0 0
\(239\) 3.93422 0.254483 0.127242 0.991872i \(-0.459388\pi\)
0.127242 + 0.991872i \(0.459388\pi\)
\(240\) 0 0
\(241\) 7.44231 0.479401 0.239701 0.970847i \(-0.422951\pi\)
0.239701 + 0.970847i \(0.422951\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.15260 0.329188
\(246\) 0 0
\(247\) 0.263898 0.0167914
\(248\) 0 0
\(249\) −13.1920 −0.836012
\(250\) 0 0
\(251\) −19.5898 −1.23650 −0.618250 0.785982i \(-0.712158\pi\)
−0.618250 + 0.785982i \(0.712158\pi\)
\(252\) 0 0
\(253\) 1.61549 0.101565
\(254\) 0 0
\(255\) 0.131282 0.00822117
\(256\) 0 0
\(257\) 17.9223 1.11796 0.558982 0.829180i \(-0.311192\pi\)
0.558982 + 0.829180i \(0.311192\pi\)
\(258\) 0 0
\(259\) 21.9955 1.36674
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 10.2174 0.630034 0.315017 0.949086i \(-0.397990\pi\)
0.315017 + 0.949086i \(0.397990\pi\)
\(264\) 0 0
\(265\) −1.68308 −0.103391
\(266\) 0 0
\(267\) −5.33893 −0.326737
\(268\) 0 0
\(269\) 22.1372 1.34973 0.674865 0.737942i \(-0.264202\pi\)
0.674865 + 0.737942i \(0.264202\pi\)
\(270\) 0 0
\(271\) −32.0849 −1.94902 −0.974510 0.224345i \(-0.927976\pi\)
−0.974510 + 0.224345i \(0.927976\pi\)
\(272\) 0 0
\(273\) −2.87692 −0.174119
\(274\) 0 0
\(275\) −7.94864 −0.479321
\(276\) 0 0
\(277\) 16.0334 0.963353 0.481677 0.876349i \(-0.340028\pi\)
0.481677 + 0.876349i \(0.340028\pi\)
\(278\) 0 0
\(279\) −1.38151 −0.0827091
\(280\) 0 0
\(281\) 15.2874 0.911970 0.455985 0.889987i \(-0.349287\pi\)
0.455985 + 0.889987i \(0.349287\pi\)
\(282\) 0 0
\(283\) 14.5611 0.865566 0.432783 0.901498i \(-0.357532\pi\)
0.432783 + 0.901498i \(0.357532\pi\)
\(284\) 0 0
\(285\) −0.130152 −0.00770954
\(286\) 0 0
\(287\) 14.8356 0.875717
\(288\) 0 0
\(289\) −16.7839 −0.987286
\(290\) 0 0
\(291\) 17.5054 1.02618
\(292\) 0 0
\(293\) 26.1553 1.52801 0.764005 0.645210i \(-0.223230\pi\)
0.764005 + 0.645210i \(0.223230\pi\)
\(294\) 0 0
\(295\) 3.54542 0.206422
\(296\) 0 0
\(297\) −1.61549 −0.0937404
\(298\) 0 0
\(299\) 0.572564 0.0331122
\(300\) 0 0
\(301\) −30.4904 −1.75744
\(302\) 0 0
\(303\) −11.9008 −0.683681
\(304\) 0 0
\(305\) 2.21012 0.126551
\(306\) 0 0
\(307\) −22.8745 −1.30551 −0.652757 0.757567i \(-0.726388\pi\)
−0.652757 + 0.757567i \(0.726388\pi\)
\(308\) 0 0
\(309\) 7.02283 0.399515
\(310\) 0 0
\(311\) −6.38071 −0.361817 −0.180908 0.983500i \(-0.557904\pi\)
−0.180908 + 0.983500i \(0.557904\pi\)
\(312\) 0 0
\(313\) 31.6435 1.78859 0.894297 0.447474i \(-0.147676\pi\)
0.894297 + 0.447474i \(0.147676\pi\)
\(314\) 0 0
\(315\) 1.41887 0.0799442
\(316\) 0 0
\(317\) −1.40227 −0.0787591 −0.0393796 0.999224i \(-0.512538\pi\)
−0.0393796 + 0.999224i \(0.512538\pi\)
\(318\) 0 0
\(319\) 1.61549 0.0904502
\(320\) 0 0
\(321\) 13.4809 0.752433
\(322\) 0 0
\(323\) −0.214278 −0.0119227
\(324\) 0 0
\(325\) −2.81716 −0.156268
\(326\) 0 0
\(327\) 7.29204 0.403251
\(328\) 0 0
\(329\) 31.9929 1.76383
\(330\) 0 0
\(331\) 7.04905 0.387451 0.193725 0.981056i \(-0.437943\pi\)
0.193725 + 0.981056i \(0.437943\pi\)
\(332\) 0 0
\(333\) 4.37755 0.239888
\(334\) 0 0
\(335\) −1.19577 −0.0653321
\(336\) 0 0
\(337\) −17.3534 −0.945299 −0.472649 0.881251i \(-0.656702\pi\)
−0.472649 + 0.881251i \(0.656702\pi\)
\(338\) 0 0
\(339\) 20.9016 1.13522
\(340\) 0 0
\(341\) −2.23183 −0.120860
\(342\) 0 0
\(343\) 56.5112 3.05132
\(344\) 0 0
\(345\) −0.282383 −0.0152030
\(346\) 0 0
\(347\) −15.4903 −0.831561 −0.415780 0.909465i \(-0.636492\pi\)
−0.415780 + 0.909465i \(0.636492\pi\)
\(348\) 0 0
\(349\) 2.84281 0.152172 0.0760861 0.997101i \(-0.475758\pi\)
0.0760861 + 0.997101i \(0.475758\pi\)
\(350\) 0 0
\(351\) −0.572564 −0.0305612
\(352\) 0 0
\(353\) 6.16169 0.327954 0.163977 0.986464i \(-0.447568\pi\)
0.163977 + 0.986464i \(0.447568\pi\)
\(354\) 0 0
\(355\) 3.88379 0.206130
\(356\) 0 0
\(357\) 2.33598 0.123633
\(358\) 0 0
\(359\) −29.4172 −1.55258 −0.776290 0.630376i \(-0.782901\pi\)
−0.776290 + 0.630376i \(0.782901\pi\)
\(360\) 0 0
\(361\) −18.7876 −0.988819
\(362\) 0 0
\(363\) 8.39018 0.440370
\(364\) 0 0
\(365\) 1.61438 0.0845006
\(366\) 0 0
\(367\) −34.7282 −1.81280 −0.906398 0.422424i \(-0.861179\pi\)
−0.906398 + 0.422424i \(0.861179\pi\)
\(368\) 0 0
\(369\) 2.95258 0.153705
\(370\) 0 0
\(371\) −29.9481 −1.55483
\(372\) 0 0
\(373\) 29.8611 1.54615 0.773075 0.634315i \(-0.218718\pi\)
0.773075 + 0.634315i \(0.218718\pi\)
\(374\) 0 0
\(375\) 2.80131 0.144659
\(376\) 0 0
\(377\) 0.572564 0.0294886
\(378\) 0 0
\(379\) −14.2483 −0.731885 −0.365943 0.930637i \(-0.619253\pi\)
−0.365943 + 0.930637i \(0.619253\pi\)
\(380\) 0 0
\(381\) −18.5052 −0.948052
\(382\) 0 0
\(383\) 13.1526 0.672065 0.336032 0.941850i \(-0.390915\pi\)
0.336032 + 0.941850i \(0.390915\pi\)
\(384\) 0 0
\(385\) 2.29217 0.116820
\(386\) 0 0
\(387\) −6.06819 −0.308463
\(388\) 0 0
\(389\) −7.65181 −0.387962 −0.193981 0.981005i \(-0.562140\pi\)
−0.193981 + 0.981005i \(0.562140\pi\)
\(390\) 0 0
\(391\) −0.464906 −0.0235113
\(392\) 0 0
\(393\) −17.6604 −0.890849
\(394\) 0 0
\(395\) −1.64255 −0.0826457
\(396\) 0 0
\(397\) 14.9480 0.750219 0.375110 0.926980i \(-0.377605\pi\)
0.375110 + 0.926980i \(0.377605\pi\)
\(398\) 0 0
\(399\) −2.31588 −0.115939
\(400\) 0 0
\(401\) −24.7956 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(402\) 0 0
\(403\) −0.791005 −0.0394028
\(404\) 0 0
\(405\) 0.282383 0.0140317
\(406\) 0 0
\(407\) 7.07190 0.350541
\(408\) 0 0
\(409\) 31.0219 1.53393 0.766967 0.641687i \(-0.221765\pi\)
0.766967 + 0.641687i \(0.221765\pi\)
\(410\) 0 0
\(411\) 13.0006 0.641272
\(412\) 0 0
\(413\) 63.0860 3.10426
\(414\) 0 0
\(415\) 3.72521 0.182863
\(416\) 0 0
\(417\) 0.446628 0.0218714
\(418\) 0 0
\(419\) −28.8412 −1.40899 −0.704493 0.709711i \(-0.748825\pi\)
−0.704493 + 0.709711i \(0.748825\pi\)
\(420\) 0 0
\(421\) 40.1017 1.95444 0.977218 0.212237i \(-0.0680749\pi\)
0.977218 + 0.212237i \(0.0680749\pi\)
\(422\) 0 0
\(423\) 6.36723 0.309585
\(424\) 0 0
\(425\) 2.28746 0.110958
\(426\) 0 0
\(427\) 39.3260 1.90312
\(428\) 0 0
\(429\) −0.924973 −0.0446581
\(430\) 0 0
\(431\) 21.6024 1.04055 0.520275 0.853999i \(-0.325829\pi\)
0.520275 + 0.853999i \(0.325829\pi\)
\(432\) 0 0
\(433\) −12.1522 −0.583999 −0.291999 0.956418i \(-0.594321\pi\)
−0.291999 + 0.956418i \(0.594321\pi\)
\(434\) 0 0
\(435\) −0.282383 −0.0135392
\(436\) 0 0
\(437\) 0.460906 0.0220481
\(438\) 0 0
\(439\) −10.2017 −0.486900 −0.243450 0.969913i \(-0.578279\pi\)
−0.243450 + 0.969913i \(0.578279\pi\)
\(440\) 0 0
\(441\) 18.2468 0.868897
\(442\) 0 0
\(443\) 12.8464 0.610353 0.305177 0.952296i \(-0.401284\pi\)
0.305177 + 0.952296i \(0.401284\pi\)
\(444\) 0 0
\(445\) 1.50763 0.0714683
\(446\) 0 0
\(447\) 2.48939 0.117744
\(448\) 0 0
\(449\) −5.82169 −0.274743 −0.137371 0.990520i \(-0.543865\pi\)
−0.137371 + 0.990520i \(0.543865\pi\)
\(450\) 0 0
\(451\) 4.76987 0.224604
\(452\) 0 0
\(453\) −6.97623 −0.327772
\(454\) 0 0
\(455\) 0.812394 0.0380856
\(456\) 0 0
\(457\) 8.03855 0.376028 0.188014 0.982166i \(-0.439795\pi\)
0.188014 + 0.982166i \(0.439795\pi\)
\(458\) 0 0
\(459\) 0.464906 0.0216999
\(460\) 0 0
\(461\) 30.0979 1.40180 0.700901 0.713259i \(-0.252782\pi\)
0.700901 + 0.713259i \(0.252782\pi\)
\(462\) 0 0
\(463\) 37.1863 1.72820 0.864098 0.503324i \(-0.167890\pi\)
0.864098 + 0.503324i \(0.167890\pi\)
\(464\) 0 0
\(465\) 0.390116 0.0180912
\(466\) 0 0
\(467\) −31.9455 −1.47826 −0.739130 0.673563i \(-0.764763\pi\)
−0.739130 + 0.673563i \(0.764763\pi\)
\(468\) 0 0
\(469\) −21.2772 −0.982488
\(470\) 0 0
\(471\) −0.517289 −0.0238354
\(472\) 0 0
\(473\) −9.80311 −0.450748
\(474\) 0 0
\(475\) −2.26778 −0.104053
\(476\) 0 0
\(477\) −5.96026 −0.272902
\(478\) 0 0
\(479\) −9.62267 −0.439671 −0.219835 0.975537i \(-0.570552\pi\)
−0.219835 + 0.975537i \(0.570552\pi\)
\(480\) 0 0
\(481\) 2.50643 0.114283
\(482\) 0 0
\(483\) −5.02462 −0.228628
\(484\) 0 0
\(485\) −4.94323 −0.224460
\(486\) 0 0
\(487\) 33.9992 1.54065 0.770324 0.637652i \(-0.220094\pi\)
0.770324 + 0.637652i \(0.220094\pi\)
\(488\) 0 0
\(489\) −11.5780 −0.523575
\(490\) 0 0
\(491\) −15.8256 −0.714200 −0.357100 0.934066i \(-0.616235\pi\)
−0.357100 + 0.934066i \(0.616235\pi\)
\(492\) 0 0
\(493\) −0.464906 −0.0209383
\(494\) 0 0
\(495\) 0.456188 0.0205041
\(496\) 0 0
\(497\) 69.1067 3.09986
\(498\) 0 0
\(499\) −13.2102 −0.591369 −0.295684 0.955286i \(-0.595548\pi\)
−0.295684 + 0.955286i \(0.595548\pi\)
\(500\) 0 0
\(501\) 8.90578 0.397881
\(502\) 0 0
\(503\) −23.5178 −1.04861 −0.524303 0.851532i \(-0.675674\pi\)
−0.524303 + 0.851532i \(0.675674\pi\)
\(504\) 0 0
\(505\) 3.36057 0.149544
\(506\) 0 0
\(507\) 12.6722 0.562791
\(508\) 0 0
\(509\) 16.8420 0.746508 0.373254 0.927729i \(-0.378242\pi\)
0.373254 + 0.927729i \(0.378242\pi\)
\(510\) 0 0
\(511\) 28.7257 1.27075
\(512\) 0 0
\(513\) −0.460906 −0.0203495
\(514\) 0 0
\(515\) −1.98313 −0.0873871
\(516\) 0 0
\(517\) 10.2862 0.452387
\(518\) 0 0
\(519\) −18.0923 −0.794163
\(520\) 0 0
\(521\) 38.3149 1.67861 0.839303 0.543665i \(-0.182964\pi\)
0.839303 + 0.543665i \(0.182964\pi\)
\(522\) 0 0
\(523\) −11.4290 −0.499754 −0.249877 0.968278i \(-0.580390\pi\)
−0.249877 + 0.968278i \(0.580390\pi\)
\(524\) 0 0
\(525\) 24.7225 1.07898
\(526\) 0 0
\(527\) 0.642274 0.0279779
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.5554 0.544856
\(532\) 0 0
\(533\) 1.69054 0.0732254
\(534\) 0 0
\(535\) −3.80679 −0.164582
\(536\) 0 0
\(537\) −7.62128 −0.328882
\(538\) 0 0
\(539\) 29.4776 1.26969
\(540\) 0 0
\(541\) 1.53527 0.0660065 0.0330033 0.999455i \(-0.489493\pi\)
0.0330033 + 0.999455i \(0.489493\pi\)
\(542\) 0 0
\(543\) −20.9090 −0.897289
\(544\) 0 0
\(545\) −2.05915 −0.0882043
\(546\) 0 0
\(547\) −36.7104 −1.56962 −0.784812 0.619735i \(-0.787240\pi\)
−0.784812 + 0.619735i \(0.787240\pi\)
\(548\) 0 0
\(549\) 7.82665 0.334033
\(550\) 0 0
\(551\) 0.460906 0.0196352
\(552\) 0 0
\(553\) −29.2270 −1.24286
\(554\) 0 0
\(555\) −1.23615 −0.0524715
\(556\) 0 0
\(557\) −12.7739 −0.541246 −0.270623 0.962685i \(-0.587230\pi\)
−0.270623 + 0.962685i \(0.587230\pi\)
\(558\) 0 0
\(559\) −3.47443 −0.146953
\(560\) 0 0
\(561\) 0.751052 0.0317094
\(562\) 0 0
\(563\) −16.4675 −0.694024 −0.347012 0.937861i \(-0.612804\pi\)
−0.347012 + 0.937861i \(0.612804\pi\)
\(564\) 0 0
\(565\) −5.90225 −0.248310
\(566\) 0 0
\(567\) 5.02462 0.211014
\(568\) 0 0
\(569\) 2.85212 0.119567 0.0597836 0.998211i \(-0.480959\pi\)
0.0597836 + 0.998211i \(0.480959\pi\)
\(570\) 0 0
\(571\) 19.9188 0.833574 0.416787 0.909004i \(-0.363156\pi\)
0.416787 + 0.909004i \(0.363156\pi\)
\(572\) 0 0
\(573\) −5.10381 −0.213215
\(574\) 0 0
\(575\) −4.92026 −0.205189
\(576\) 0 0
\(577\) −19.6175 −0.816689 −0.408344 0.912828i \(-0.633894\pi\)
−0.408344 + 0.912828i \(0.633894\pi\)
\(578\) 0 0
\(579\) 7.73467 0.321442
\(580\) 0 0
\(581\) 66.2851 2.74997
\(582\) 0 0
\(583\) −9.62876 −0.398783
\(584\) 0 0
\(585\) 0.161682 0.00668475
\(586\) 0 0
\(587\) −34.1057 −1.40769 −0.703846 0.710353i \(-0.748535\pi\)
−0.703846 + 0.710353i \(0.748535\pi\)
\(588\) 0 0
\(589\) −0.636747 −0.0262367
\(590\) 0 0
\(591\) 10.0993 0.415429
\(592\) 0 0
\(593\) −36.9271 −1.51641 −0.758206 0.652015i \(-0.773924\pi\)
−0.758206 + 0.652015i \(0.773924\pi\)
\(594\) 0 0
\(595\) −0.659641 −0.0270426
\(596\) 0 0
\(597\) 12.0031 0.491256
\(598\) 0 0
\(599\) 15.2603 0.623520 0.311760 0.950161i \(-0.399082\pi\)
0.311760 + 0.950161i \(0.399082\pi\)
\(600\) 0 0
\(601\) 27.7506 1.13197 0.565985 0.824415i \(-0.308496\pi\)
0.565985 + 0.824415i \(0.308496\pi\)
\(602\) 0 0
\(603\) −4.23458 −0.172445
\(604\) 0 0
\(605\) −2.36925 −0.0963236
\(606\) 0 0
\(607\) 28.3706 1.15153 0.575764 0.817616i \(-0.304705\pi\)
0.575764 + 0.817616i \(0.304705\pi\)
\(608\) 0 0
\(609\) −5.02462 −0.203608
\(610\) 0 0
\(611\) 3.64565 0.147487
\(612\) 0 0
\(613\) 9.47568 0.382719 0.191360 0.981520i \(-0.438710\pi\)
0.191360 + 0.981520i \(0.438710\pi\)
\(614\) 0 0
\(615\) −0.833758 −0.0336204
\(616\) 0 0
\(617\) 11.8218 0.475930 0.237965 0.971274i \(-0.423520\pi\)
0.237965 + 0.971274i \(0.423520\pi\)
\(618\) 0 0
\(619\) −22.5351 −0.905763 −0.452881 0.891571i \(-0.649604\pi\)
−0.452881 + 0.891571i \(0.649604\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 26.8261 1.07477
\(624\) 0 0
\(625\) 23.8103 0.952410
\(626\) 0 0
\(627\) −0.744590 −0.0297360
\(628\) 0 0
\(629\) −2.03515 −0.0811467
\(630\) 0 0
\(631\) −26.6995 −1.06289 −0.531445 0.847093i \(-0.678351\pi\)
−0.531445 + 0.847093i \(0.678351\pi\)
\(632\) 0 0
\(633\) −8.53376 −0.339186
\(634\) 0 0
\(635\) 5.22557 0.207370
\(636\) 0 0
\(637\) 10.4475 0.413945
\(638\) 0 0
\(639\) 13.7536 0.544084
\(640\) 0 0
\(641\) −19.8910 −0.785647 −0.392823 0.919614i \(-0.628502\pi\)
−0.392823 + 0.919614i \(0.628502\pi\)
\(642\) 0 0
\(643\) 45.1460 1.78039 0.890193 0.455584i \(-0.150570\pi\)
0.890193 + 0.455584i \(0.150570\pi\)
\(644\) 0 0
\(645\) 1.71355 0.0674711
\(646\) 0 0
\(647\) 8.74463 0.343787 0.171893 0.985116i \(-0.445012\pi\)
0.171893 + 0.985116i \(0.445012\pi\)
\(648\) 0 0
\(649\) 20.2831 0.796181
\(650\) 0 0
\(651\) 6.94159 0.272062
\(652\) 0 0
\(653\) 17.6687 0.691431 0.345715 0.938339i \(-0.387636\pi\)
0.345715 + 0.938339i \(0.387636\pi\)
\(654\) 0 0
\(655\) 4.98700 0.194858
\(656\) 0 0
\(657\) 5.71699 0.223041
\(658\) 0 0
\(659\) −0.656514 −0.0255741 −0.0127871 0.999918i \(-0.504070\pi\)
−0.0127871 + 0.999918i \(0.504070\pi\)
\(660\) 0 0
\(661\) 14.0189 0.545271 0.272636 0.962117i \(-0.412105\pi\)
0.272636 + 0.962117i \(0.412105\pi\)
\(662\) 0 0
\(663\) 0.266188 0.0103379
\(664\) 0 0
\(665\) 0.653965 0.0253597
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 19.1107 0.738863
\(670\) 0 0
\(671\) 12.6439 0.488112
\(672\) 0 0
\(673\) 20.4264 0.787381 0.393690 0.919243i \(-0.371198\pi\)
0.393690 + 0.919243i \(0.371198\pi\)
\(674\) 0 0
\(675\) 4.92026 0.189381
\(676\) 0 0
\(677\) −8.54817 −0.328533 −0.164266 0.986416i \(-0.552526\pi\)
−0.164266 + 0.986416i \(0.552526\pi\)
\(678\) 0 0
\(679\) −87.9580 −3.37552
\(680\) 0 0
\(681\) −12.2099 −0.467886
\(682\) 0 0
\(683\) 25.7783 0.986379 0.493190 0.869922i \(-0.335831\pi\)
0.493190 + 0.869922i \(0.335831\pi\)
\(684\) 0 0
\(685\) −3.67115 −0.140267
\(686\) 0 0
\(687\) 27.4713 1.04810
\(688\) 0 0
\(689\) −3.41263 −0.130011
\(690\) 0 0
\(691\) −18.9673 −0.721552 −0.360776 0.932653i \(-0.617488\pi\)
−0.360776 + 0.932653i \(0.617488\pi\)
\(692\) 0 0
\(693\) 8.11724 0.308349
\(694\) 0 0
\(695\) −0.126120 −0.00478401
\(696\) 0 0
\(697\) −1.37267 −0.0519936
\(698\) 0 0
\(699\) −21.9922 −0.831821
\(700\) 0 0
\(701\) 7.08092 0.267443 0.133721 0.991019i \(-0.457307\pi\)
0.133721 + 0.991019i \(0.457307\pi\)
\(702\) 0 0
\(703\) 2.01764 0.0760966
\(704\) 0 0
\(705\) −1.79800 −0.0677165
\(706\) 0 0
\(707\) 59.7968 2.24889
\(708\) 0 0
\(709\) −35.2787 −1.32492 −0.662460 0.749097i \(-0.730488\pi\)
−0.662460 + 0.749097i \(0.730488\pi\)
\(710\) 0 0
\(711\) −5.81674 −0.218145
\(712\) 0 0
\(713\) −1.38151 −0.0517381
\(714\) 0 0
\(715\) 0.261197 0.00976821
\(716\) 0 0
\(717\) −3.93422 −0.146926
\(718\) 0 0
\(719\) 48.2543 1.79958 0.899791 0.436320i \(-0.143719\pi\)
0.899791 + 0.436320i \(0.143719\pi\)
\(720\) 0 0
\(721\) −35.2871 −1.31416
\(722\) 0 0
\(723\) −7.44231 −0.276782
\(724\) 0 0
\(725\) −4.92026 −0.182734
\(726\) 0 0
\(727\) −21.1997 −0.786252 −0.393126 0.919485i \(-0.628606\pi\)
−0.393126 + 0.919485i \(0.628606\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.82114 0.104343
\(732\) 0 0
\(733\) −10.6692 −0.394075 −0.197037 0.980396i \(-0.563132\pi\)
−0.197037 + 0.980396i \(0.563132\pi\)
\(734\) 0 0
\(735\) −5.15260 −0.190057
\(736\) 0 0
\(737\) −6.84093 −0.251989
\(738\) 0 0
\(739\) 52.9145 1.94649 0.973245 0.229770i \(-0.0737973\pi\)
0.973245 + 0.229770i \(0.0737973\pi\)
\(740\) 0 0
\(741\) −0.263898 −0.00969454
\(742\) 0 0
\(743\) 22.1119 0.811207 0.405604 0.914049i \(-0.367061\pi\)
0.405604 + 0.914049i \(0.367061\pi\)
\(744\) 0 0
\(745\) −0.702962 −0.0257545
\(746\) 0 0
\(747\) 13.1920 0.482672
\(748\) 0 0
\(749\) −67.7367 −2.47504
\(750\) 0 0
\(751\) 42.2527 1.54182 0.770912 0.636941i \(-0.219801\pi\)
0.770912 + 0.636941i \(0.219801\pi\)
\(752\) 0 0
\(753\) 19.5898 0.713893
\(754\) 0 0
\(755\) 1.96997 0.0716945
\(756\) 0 0
\(757\) 40.4263 1.46932 0.734660 0.678435i \(-0.237342\pi\)
0.734660 + 0.678435i \(0.237342\pi\)
\(758\) 0 0
\(759\) −1.61549 −0.0586387
\(760\) 0 0
\(761\) −23.7959 −0.862600 −0.431300 0.902209i \(-0.641945\pi\)
−0.431300 + 0.902209i \(0.641945\pi\)
\(762\) 0 0
\(763\) −36.6398 −1.32645
\(764\) 0 0
\(765\) −0.131282 −0.00474650
\(766\) 0 0
\(767\) 7.18875 0.259571
\(768\) 0 0
\(769\) 18.6196 0.671438 0.335719 0.941962i \(-0.391021\pi\)
0.335719 + 0.941962i \(0.391021\pi\)
\(770\) 0 0
\(771\) −17.9223 −0.645456
\(772\) 0 0
\(773\) −47.1632 −1.69634 −0.848172 0.529722i \(-0.822296\pi\)
−0.848172 + 0.529722i \(0.822296\pi\)
\(774\) 0 0
\(775\) 6.79741 0.244170
\(776\) 0 0
\(777\) −21.9955 −0.789086
\(778\) 0 0
\(779\) 1.36086 0.0487578
\(780\) 0 0
\(781\) 22.2189 0.795053
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 0.146074 0.00521359
\(786\) 0 0
\(787\) 9.70101 0.345804 0.172902 0.984939i \(-0.444686\pi\)
0.172902 + 0.984939i \(0.444686\pi\)
\(788\) 0 0
\(789\) −10.2174 −0.363750
\(790\) 0 0
\(791\) −105.022 −3.73417
\(792\) 0 0
\(793\) 4.48126 0.159134
\(794\) 0 0
\(795\) 1.68308 0.0596926
\(796\) 0 0
\(797\) 21.0633 0.746101 0.373051 0.927811i \(-0.378312\pi\)
0.373051 + 0.927811i \(0.378312\pi\)
\(798\) 0 0
\(799\) −2.96016 −0.104723
\(800\) 0 0
\(801\) 5.33893 0.188642
\(802\) 0 0
\(803\) 9.23575 0.325923
\(804\) 0 0
\(805\) 1.41887 0.0500086
\(806\) 0 0
\(807\) −22.1372 −0.779266
\(808\) 0 0
\(809\) 42.3066 1.48742 0.743710 0.668502i \(-0.233064\pi\)
0.743710 + 0.668502i \(0.233064\pi\)
\(810\) 0 0
\(811\) −11.9537 −0.419751 −0.209875 0.977728i \(-0.567306\pi\)
−0.209875 + 0.977728i \(0.567306\pi\)
\(812\) 0 0
\(813\) 32.0849 1.12527
\(814\) 0 0
\(815\) 3.26943 0.114523
\(816\) 0 0
\(817\) −2.79686 −0.0978498
\(818\) 0 0
\(819\) 2.87692 0.100528
\(820\) 0 0
\(821\) 23.3877 0.816237 0.408119 0.912929i \(-0.366185\pi\)
0.408119 + 0.912929i \(0.366185\pi\)
\(822\) 0 0
\(823\) 4.90370 0.170932 0.0854662 0.996341i \(-0.472762\pi\)
0.0854662 + 0.996341i \(0.472762\pi\)
\(824\) 0 0
\(825\) 7.94864 0.276736
\(826\) 0 0
\(827\) 38.5936 1.34203 0.671015 0.741444i \(-0.265858\pi\)
0.671015 + 0.741444i \(0.265858\pi\)
\(828\) 0 0
\(829\) 18.4487 0.640749 0.320374 0.947291i \(-0.396191\pi\)
0.320374 + 0.947291i \(0.396191\pi\)
\(830\) 0 0
\(831\) −16.0334 −0.556192
\(832\) 0 0
\(833\) −8.48307 −0.293921
\(834\) 0 0
\(835\) −2.51484 −0.0870297
\(836\) 0 0
\(837\) 1.38151 0.0477521
\(838\) 0 0
\(839\) 40.9978 1.41540 0.707701 0.706512i \(-0.249732\pi\)
0.707701 + 0.706512i \(0.249732\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −15.2874 −0.526526
\(844\) 0 0
\(845\) −3.57841 −0.123101
\(846\) 0 0
\(847\) −42.1575 −1.44855
\(848\) 0 0
\(849\) −14.5611 −0.499735
\(850\) 0 0
\(851\) 4.37755 0.150061
\(852\) 0 0
\(853\) −44.1282 −1.51092 −0.755460 0.655194i \(-0.772587\pi\)
−0.755460 + 0.655194i \(0.772587\pi\)
\(854\) 0 0
\(855\) 0.130152 0.00445110
\(856\) 0 0
\(857\) −2.99764 −0.102398 −0.0511988 0.998688i \(-0.516304\pi\)
−0.0511988 + 0.998688i \(0.516304\pi\)
\(858\) 0 0
\(859\) −32.4605 −1.10754 −0.553769 0.832670i \(-0.686811\pi\)
−0.553769 + 0.832670i \(0.686811\pi\)
\(860\) 0 0
\(861\) −14.8356 −0.505595
\(862\) 0 0
\(863\) 44.6335 1.51934 0.759671 0.650307i \(-0.225360\pi\)
0.759671 + 0.650307i \(0.225360\pi\)
\(864\) 0 0
\(865\) 5.10896 0.173710
\(866\) 0 0
\(867\) 16.7839 0.570010
\(868\) 0 0
\(869\) −9.39691 −0.318768
\(870\) 0 0
\(871\) −2.42457 −0.0821533
\(872\) 0 0
\(873\) −17.5054 −0.592468
\(874\) 0 0
\(875\) −14.0756 −0.475841
\(876\) 0 0
\(877\) 45.6728 1.54226 0.771131 0.636676i \(-0.219691\pi\)
0.771131 + 0.636676i \(0.219691\pi\)
\(878\) 0 0
\(879\) −26.1553 −0.882197
\(880\) 0 0
\(881\) −22.0982 −0.744507 −0.372253 0.928131i \(-0.621415\pi\)
−0.372253 + 0.928131i \(0.621415\pi\)
\(882\) 0 0
\(883\) −20.1538 −0.678228 −0.339114 0.940745i \(-0.610127\pi\)
−0.339114 + 0.940745i \(0.610127\pi\)
\(884\) 0 0
\(885\) −3.54542 −0.119178
\(886\) 0 0
\(887\) −48.9783 −1.64453 −0.822265 0.569105i \(-0.807290\pi\)
−0.822265 + 0.569105i \(0.807290\pi\)
\(888\) 0 0
\(889\) 92.9819 3.11851
\(890\) 0 0
\(891\) 1.61549 0.0541210
\(892\) 0 0
\(893\) 2.93469 0.0982057
\(894\) 0 0
\(895\) 2.15212 0.0719375
\(896\) 0 0
\(897\) −0.572564 −0.0191174
\(898\) 0 0
\(899\) −1.38151 −0.0460761
\(900\) 0 0
\(901\) 2.77096 0.0923141
\(902\) 0 0
\(903\) 30.4904 1.01466
\(904\) 0 0
\(905\) 5.90434 0.196267
\(906\) 0 0
\(907\) −35.2718 −1.17118 −0.585590 0.810607i \(-0.699137\pi\)
−0.585590 + 0.810607i \(0.699137\pi\)
\(908\) 0 0
\(909\) 11.9008 0.394723
\(910\) 0 0
\(911\) −29.5058 −0.977571 −0.488785 0.872404i \(-0.662560\pi\)
−0.488785 + 0.872404i \(0.662560\pi\)
\(912\) 0 0
\(913\) 21.3117 0.705313
\(914\) 0 0
\(915\) −2.21012 −0.0730642
\(916\) 0 0
\(917\) 88.7368 2.93035
\(918\) 0 0
\(919\) −3.33449 −0.109995 −0.0549974 0.998486i \(-0.517515\pi\)
−0.0549974 + 0.998486i \(0.517515\pi\)
\(920\) 0 0
\(921\) 22.8745 0.753739
\(922\) 0 0
\(923\) 7.87482 0.259203
\(924\) 0 0
\(925\) −21.5387 −0.708188
\(926\) 0 0
\(927\) −7.02283 −0.230660
\(928\) 0 0
\(929\) 2.51004 0.0823518 0.0411759 0.999152i \(-0.486890\pi\)
0.0411759 + 0.999152i \(0.486890\pi\)
\(930\) 0 0
\(931\) 8.41007 0.275629
\(932\) 0 0
\(933\) 6.38071 0.208895
\(934\) 0 0
\(935\) −0.212084 −0.00693590
\(936\) 0 0
\(937\) −33.9088 −1.10775 −0.553877 0.832599i \(-0.686852\pi\)
−0.553877 + 0.832599i \(0.686852\pi\)
\(938\) 0 0
\(939\) −31.6435 −1.03265
\(940\) 0 0
\(941\) −7.53677 −0.245692 −0.122846 0.992426i \(-0.539202\pi\)
−0.122846 + 0.992426i \(0.539202\pi\)
\(942\) 0 0
\(943\) 2.95258 0.0961491
\(944\) 0 0
\(945\) −1.41887 −0.0461558
\(946\) 0 0
\(947\) −3.77706 −0.122738 −0.0613690 0.998115i \(-0.519547\pi\)
−0.0613690 + 0.998115i \(0.519547\pi\)
\(948\) 0 0
\(949\) 3.27334 0.106257
\(950\) 0 0
\(951\) 1.40227 0.0454716
\(952\) 0 0
\(953\) −50.9850 −1.65157 −0.825783 0.563988i \(-0.809266\pi\)
−0.825783 + 0.563988i \(0.809266\pi\)
\(954\) 0 0
\(955\) 1.44123 0.0466371
\(956\) 0 0
\(957\) −1.61549 −0.0522215
\(958\) 0 0
\(959\) −65.3231 −2.10939
\(960\) 0 0
\(961\) −29.0914 −0.938433
\(962\) 0 0
\(963\) −13.4809 −0.434417
\(964\) 0 0
\(965\) −2.18414 −0.0703100
\(966\) 0 0
\(967\) −16.7322 −0.538070 −0.269035 0.963130i \(-0.586705\pi\)
−0.269035 + 0.963130i \(0.586705\pi\)
\(968\) 0 0
\(969\) 0.214278 0.00688359
\(970\) 0 0
\(971\) −14.4629 −0.464136 −0.232068 0.972700i \(-0.574549\pi\)
−0.232068 + 0.972700i \(0.574549\pi\)
\(972\) 0 0
\(973\) −2.24414 −0.0719437
\(974\) 0 0
\(975\) 2.81716 0.0902215
\(976\) 0 0
\(977\) 12.2710 0.392583 0.196292 0.980546i \(-0.437110\pi\)
0.196292 + 0.980546i \(0.437110\pi\)
\(978\) 0 0
\(979\) 8.62501 0.275656
\(980\) 0 0
\(981\) −7.29204 −0.232817
\(982\) 0 0
\(983\) −26.6017 −0.848461 −0.424231 0.905554i \(-0.639455\pi\)
−0.424231 + 0.905554i \(0.639455\pi\)
\(984\) 0 0
\(985\) −2.85187 −0.0908681
\(986\) 0 0
\(987\) −31.9929 −1.01835
\(988\) 0 0
\(989\) −6.06819 −0.192957
\(990\) 0 0
\(991\) −52.3584 −1.66322 −0.831610 0.555360i \(-0.812581\pi\)
−0.831610 + 0.555360i \(0.812581\pi\)
\(992\) 0 0
\(993\) −7.04905 −0.223695
\(994\) 0 0
\(995\) −3.38949 −0.107454
\(996\) 0 0
\(997\) −55.1603 −1.74694 −0.873472 0.486875i \(-0.838137\pi\)
−0.873472 + 0.486875i \(0.838137\pi\)
\(998\) 0 0
\(999\) −4.37755 −0.138500
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 8004.2.a.i.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.9 16 1.1 even 1 trivial