Properties

Label 8004.2.a.e.1.2
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.47135\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.48372 q^{5} +2.18462 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.48372 q^{5} +2.18462 q^{7} +1.00000 q^{9} -4.51219 q^{11} +4.71609 q^{13} +3.48372 q^{15} -4.53301 q^{17} -1.79821 q^{19} -2.18462 q^{21} -1.00000 q^{23} +7.13634 q^{25} -1.00000 q^{27} +1.00000 q^{29} +5.46631 q^{31} +4.51219 q^{33} -7.61063 q^{35} +2.75195 q^{37} -4.71609 q^{39} +0.252352 q^{41} +5.62492 q^{43} -3.48372 q^{45} +2.83786 q^{47} -2.22742 q^{49} +4.53301 q^{51} -4.43857 q^{53} +15.7192 q^{55} +1.79821 q^{57} +4.10392 q^{59} -0.160730 q^{61} +2.18462 q^{63} -16.4296 q^{65} -3.72772 q^{67} +1.00000 q^{69} -0.134204 q^{71} +8.61415 q^{73} -7.13634 q^{75} -9.85744 q^{77} -2.48616 q^{79} +1.00000 q^{81} -4.04290 q^{83} +15.7918 q^{85} -1.00000 q^{87} +14.3980 q^{89} +10.3029 q^{91} -5.46631 q^{93} +6.26448 q^{95} -11.5595 q^{97} -4.51219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) −3.48372 −1.55797 −0.778985 0.627043i \(-0.784265\pi\)
−0.778985 + 0.627043i \(0.784265\pi\)
\(6\) 0 0
\(7\) 2.18462 0.825710 0.412855 0.910797i \(-0.364532\pi\)
0.412855 + 0.910797i \(0.364532\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.51219 −1.36048 −0.680239 0.732990i \(-0.738124\pi\)
−0.680239 + 0.732990i \(0.738124\pi\)
\(12\) 0 0
\(13\) 4.71609 1.30801 0.654004 0.756491i \(-0.273088\pi\)
0.654004 + 0.756491i \(0.273088\pi\)
\(14\) 0 0
\(15\) 3.48372 0.899494
\(16\) 0 0
\(17\) −4.53301 −1.09942 −0.549708 0.835357i \(-0.685261\pi\)
−0.549708 + 0.835357i \(0.685261\pi\)
\(18\) 0 0
\(19\) −1.79821 −0.412538 −0.206269 0.978495i \(-0.566132\pi\)
−0.206269 + 0.978495i \(0.566132\pi\)
\(20\) 0 0
\(21\) −2.18462 −0.476724
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 7.13634 1.42727
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.46631 0.981778 0.490889 0.871222i \(-0.336672\pi\)
0.490889 + 0.871222i \(0.336672\pi\)
\(32\) 0 0
\(33\) 4.51219 0.785472
\(34\) 0 0
\(35\) −7.61063 −1.28643
\(36\) 0 0
\(37\) 2.75195 0.452418 0.226209 0.974079i \(-0.427367\pi\)
0.226209 + 0.974079i \(0.427367\pi\)
\(38\) 0 0
\(39\) −4.71609 −0.755179
\(40\) 0 0
\(41\) 0.252352 0.0394108 0.0197054 0.999806i \(-0.493727\pi\)
0.0197054 + 0.999806i \(0.493727\pi\)
\(42\) 0 0
\(43\) 5.62492 0.857792 0.428896 0.903354i \(-0.358903\pi\)
0.428896 + 0.903354i \(0.358903\pi\)
\(44\) 0 0
\(45\) −3.48372 −0.519323
\(46\) 0 0
\(47\) 2.83786 0.413945 0.206972 0.978347i \(-0.433639\pi\)
0.206972 + 0.978347i \(0.433639\pi\)
\(48\) 0 0
\(49\) −2.22742 −0.318203
\(50\) 0 0
\(51\) 4.53301 0.634748
\(52\) 0 0
\(53\) −4.43857 −0.609685 −0.304842 0.952403i \(-0.598604\pi\)
−0.304842 + 0.952403i \(0.598604\pi\)
\(54\) 0 0
\(55\) 15.7192 2.11958
\(56\) 0 0
\(57\) 1.79821 0.238179
\(58\) 0 0
\(59\) 4.10392 0.534285 0.267142 0.963657i \(-0.413921\pi\)
0.267142 + 0.963657i \(0.413921\pi\)
\(60\) 0 0
\(61\) −0.160730 −0.0205793 −0.0102897 0.999947i \(-0.503275\pi\)
−0.0102897 + 0.999947i \(0.503275\pi\)
\(62\) 0 0
\(63\) 2.18462 0.275237
\(64\) 0 0
\(65\) −16.4296 −2.03784
\(66\) 0 0
\(67\) −3.72772 −0.455413 −0.227706 0.973730i \(-0.573123\pi\)
−0.227706 + 0.973730i \(0.573123\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.134204 −0.0159271 −0.00796356 0.999968i \(-0.502535\pi\)
−0.00796356 + 0.999968i \(0.502535\pi\)
\(72\) 0 0
\(73\) 8.61415 1.00821 0.504105 0.863642i \(-0.331822\pi\)
0.504105 + 0.863642i \(0.331822\pi\)
\(74\) 0 0
\(75\) −7.13634 −0.824033
\(76\) 0 0
\(77\) −9.85744 −1.12336
\(78\) 0 0
\(79\) −2.48616 −0.279715 −0.139857 0.990172i \(-0.544664\pi\)
−0.139857 + 0.990172i \(0.544664\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.04290 −0.443766 −0.221883 0.975073i \(-0.571220\pi\)
−0.221883 + 0.975073i \(0.571220\pi\)
\(84\) 0 0
\(85\) 15.7918 1.71286
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 14.3980 1.52618 0.763091 0.646291i \(-0.223681\pi\)
0.763091 + 0.646291i \(0.223681\pi\)
\(90\) 0 0
\(91\) 10.3029 1.08004
\(92\) 0 0
\(93\) −5.46631 −0.566830
\(94\) 0 0
\(95\) 6.26448 0.642722
\(96\) 0 0
\(97\) −11.5595 −1.17369 −0.586845 0.809699i \(-0.699630\pi\)
−0.586845 + 0.809699i \(0.699630\pi\)
\(98\) 0 0
\(99\) −4.51219 −0.453493
\(100\) 0 0
\(101\) −13.1324 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(102\) 0 0
\(103\) 5.32385 0.524574 0.262287 0.964990i \(-0.415523\pi\)
0.262287 + 0.964990i \(0.415523\pi\)
\(104\) 0 0
\(105\) 7.61063 0.742721
\(106\) 0 0
\(107\) 0.141598 0.0136888 0.00684438 0.999977i \(-0.497821\pi\)
0.00684438 + 0.999977i \(0.497821\pi\)
\(108\) 0 0
\(109\) −18.2960 −1.75244 −0.876220 0.481911i \(-0.839943\pi\)
−0.876220 + 0.481911i \(0.839943\pi\)
\(110\) 0 0
\(111\) −2.75195 −0.261203
\(112\) 0 0
\(113\) 16.8255 1.58281 0.791405 0.611292i \(-0.209350\pi\)
0.791405 + 0.611292i \(0.209350\pi\)
\(114\) 0 0
\(115\) 3.48372 0.324859
\(116\) 0 0
\(117\) 4.71609 0.436003
\(118\) 0 0
\(119\) −9.90292 −0.907799
\(120\) 0 0
\(121\) 9.35990 0.850900
\(122\) 0 0
\(123\) −0.252352 −0.0227538
\(124\) 0 0
\(125\) −7.44242 −0.665670
\(126\) 0 0
\(127\) −5.43208 −0.482019 −0.241010 0.970523i \(-0.577479\pi\)
−0.241010 + 0.970523i \(0.577479\pi\)
\(128\) 0 0
\(129\) −5.62492 −0.495246
\(130\) 0 0
\(131\) −1.54649 −0.135118 −0.0675589 0.997715i \(-0.521521\pi\)
−0.0675589 + 0.997715i \(0.521521\pi\)
\(132\) 0 0
\(133\) −3.92842 −0.340637
\(134\) 0 0
\(135\) 3.48372 0.299831
\(136\) 0 0
\(137\) 6.94494 0.593346 0.296673 0.954979i \(-0.404123\pi\)
0.296673 + 0.954979i \(0.404123\pi\)
\(138\) 0 0
\(139\) 3.92270 0.332719 0.166360 0.986065i \(-0.446799\pi\)
0.166360 + 0.986065i \(0.446799\pi\)
\(140\) 0 0
\(141\) −2.83786 −0.238991
\(142\) 0 0
\(143\) −21.2799 −1.77952
\(144\) 0 0
\(145\) −3.48372 −0.289308
\(146\) 0 0
\(147\) 2.22742 0.183715
\(148\) 0 0
\(149\) 13.2396 1.08463 0.542313 0.840176i \(-0.317549\pi\)
0.542313 + 0.840176i \(0.317549\pi\)
\(150\) 0 0
\(151\) 20.4856 1.66710 0.833548 0.552447i \(-0.186306\pi\)
0.833548 + 0.552447i \(0.186306\pi\)
\(152\) 0 0
\(153\) −4.53301 −0.366472
\(154\) 0 0
\(155\) −19.0431 −1.52958
\(156\) 0 0
\(157\) −12.5555 −1.00203 −0.501017 0.865437i \(-0.667041\pi\)
−0.501017 + 0.865437i \(0.667041\pi\)
\(158\) 0 0
\(159\) 4.43857 0.352002
\(160\) 0 0
\(161\) −2.18462 −0.172172
\(162\) 0 0
\(163\) 14.2248 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(164\) 0 0
\(165\) −15.7192 −1.22374
\(166\) 0 0
\(167\) 2.36461 0.182979 0.0914896 0.995806i \(-0.470837\pi\)
0.0914896 + 0.995806i \(0.470837\pi\)
\(168\) 0 0
\(169\) 9.24152 0.710886
\(170\) 0 0
\(171\) −1.79821 −0.137513
\(172\) 0 0
\(173\) 4.08692 0.310722 0.155361 0.987858i \(-0.450346\pi\)
0.155361 + 0.987858i \(0.450346\pi\)
\(174\) 0 0
\(175\) 15.5902 1.17851
\(176\) 0 0
\(177\) −4.10392 −0.308469
\(178\) 0 0
\(179\) 4.27445 0.319487 0.159744 0.987159i \(-0.448933\pi\)
0.159744 + 0.987159i \(0.448933\pi\)
\(180\) 0 0
\(181\) 4.85121 0.360588 0.180294 0.983613i \(-0.442295\pi\)
0.180294 + 0.983613i \(0.442295\pi\)
\(182\) 0 0
\(183\) 0.160730 0.0118815
\(184\) 0 0
\(185\) −9.58703 −0.704853
\(186\) 0 0
\(187\) 20.4538 1.49573
\(188\) 0 0
\(189\) −2.18462 −0.158908
\(190\) 0 0
\(191\) 9.30171 0.673048 0.336524 0.941675i \(-0.390749\pi\)
0.336524 + 0.941675i \(0.390749\pi\)
\(192\) 0 0
\(193\) −6.40515 −0.461053 −0.230526 0.973066i \(-0.574045\pi\)
−0.230526 + 0.973066i \(0.574045\pi\)
\(194\) 0 0
\(195\) 16.4296 1.17655
\(196\) 0 0
\(197\) −24.9249 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(198\) 0 0
\(199\) −22.7349 −1.61164 −0.805818 0.592163i \(-0.798274\pi\)
−0.805818 + 0.592163i \(0.798274\pi\)
\(200\) 0 0
\(201\) 3.72772 0.262933
\(202\) 0 0
\(203\) 2.18462 0.153330
\(204\) 0 0
\(205\) −0.879125 −0.0614008
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 8.11388 0.561249
\(210\) 0 0
\(211\) −4.01682 −0.276529 −0.138265 0.990395i \(-0.544152\pi\)
−0.138265 + 0.990395i \(0.544152\pi\)
\(212\) 0 0
\(213\) 0.134204 0.00919552
\(214\) 0 0
\(215\) −19.5957 −1.33641
\(216\) 0 0
\(217\) 11.9418 0.810664
\(218\) 0 0
\(219\) −8.61415 −0.582090
\(220\) 0 0
\(221\) −21.3781 −1.43805
\(222\) 0 0
\(223\) −21.3151 −1.42737 −0.713683 0.700469i \(-0.752974\pi\)
−0.713683 + 0.700469i \(0.752974\pi\)
\(224\) 0 0
\(225\) 7.13634 0.475756
\(226\) 0 0
\(227\) −9.37961 −0.622546 −0.311273 0.950320i \(-0.600755\pi\)
−0.311273 + 0.950320i \(0.600755\pi\)
\(228\) 0 0
\(229\) 0.756117 0.0499656 0.0249828 0.999688i \(-0.492047\pi\)
0.0249828 + 0.999688i \(0.492047\pi\)
\(230\) 0 0
\(231\) 9.85744 0.648572
\(232\) 0 0
\(233\) 9.34510 0.612218 0.306109 0.951997i \(-0.400973\pi\)
0.306109 + 0.951997i \(0.400973\pi\)
\(234\) 0 0
\(235\) −9.88633 −0.644913
\(236\) 0 0
\(237\) 2.48616 0.161493
\(238\) 0 0
\(239\) −19.1832 −1.24086 −0.620429 0.784262i \(-0.713042\pi\)
−0.620429 + 0.784262i \(0.713042\pi\)
\(240\) 0 0
\(241\) 8.64995 0.557192 0.278596 0.960408i \(-0.410131\pi\)
0.278596 + 0.960408i \(0.410131\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 7.75973 0.495751
\(246\) 0 0
\(247\) −8.48053 −0.539603
\(248\) 0 0
\(249\) 4.04290 0.256208
\(250\) 0 0
\(251\) 1.74567 0.110186 0.0550928 0.998481i \(-0.482455\pi\)
0.0550928 + 0.998481i \(0.482455\pi\)
\(252\) 0 0
\(253\) 4.51219 0.283679
\(254\) 0 0
\(255\) −15.7918 −0.988918
\(256\) 0 0
\(257\) 14.6090 0.911286 0.455643 0.890163i \(-0.349409\pi\)
0.455643 + 0.890163i \(0.349409\pi\)
\(258\) 0 0
\(259\) 6.01197 0.373566
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −14.5588 −0.897732 −0.448866 0.893599i \(-0.648172\pi\)
−0.448866 + 0.893599i \(0.648172\pi\)
\(264\) 0 0
\(265\) 15.4628 0.949870
\(266\) 0 0
\(267\) −14.3980 −0.881142
\(268\) 0 0
\(269\) −5.12131 −0.312252 −0.156126 0.987737i \(-0.549901\pi\)
−0.156126 + 0.987737i \(0.549901\pi\)
\(270\) 0 0
\(271\) 17.6495 1.07213 0.536067 0.844176i \(-0.319910\pi\)
0.536067 + 0.844176i \(0.319910\pi\)
\(272\) 0 0
\(273\) −10.3029 −0.623559
\(274\) 0 0
\(275\) −32.2006 −1.94177
\(276\) 0 0
\(277\) −14.3682 −0.863300 −0.431650 0.902041i \(-0.642068\pi\)
−0.431650 + 0.902041i \(0.642068\pi\)
\(278\) 0 0
\(279\) 5.46631 0.327259
\(280\) 0 0
\(281\) 11.2915 0.673592 0.336796 0.941578i \(-0.390657\pi\)
0.336796 + 0.941578i \(0.390657\pi\)
\(282\) 0 0
\(283\) −22.8449 −1.35799 −0.678994 0.734144i \(-0.737584\pi\)
−0.678994 + 0.734144i \(0.737584\pi\)
\(284\) 0 0
\(285\) −6.26448 −0.371076
\(286\) 0 0
\(287\) 0.551294 0.0325419
\(288\) 0 0
\(289\) 3.54818 0.208717
\(290\) 0 0
\(291\) 11.5595 0.677630
\(292\) 0 0
\(293\) 0.626712 0.0366129 0.0183065 0.999832i \(-0.494173\pi\)
0.0183065 + 0.999832i \(0.494173\pi\)
\(294\) 0 0
\(295\) −14.2969 −0.832399
\(296\) 0 0
\(297\) 4.51219 0.261824
\(298\) 0 0
\(299\) −4.71609 −0.272739
\(300\) 0 0
\(301\) 12.2883 0.708287
\(302\) 0 0
\(303\) 13.1324 0.754438
\(304\) 0 0
\(305\) 0.559938 0.0320620
\(306\) 0 0
\(307\) −13.0939 −0.747308 −0.373654 0.927568i \(-0.621895\pi\)
−0.373654 + 0.927568i \(0.621895\pi\)
\(308\) 0 0
\(309\) −5.32385 −0.302863
\(310\) 0 0
\(311\) 7.40225 0.419743 0.209872 0.977729i \(-0.432695\pi\)
0.209872 + 0.977729i \(0.432695\pi\)
\(312\) 0 0
\(313\) −2.80091 −0.158317 −0.0791584 0.996862i \(-0.525223\pi\)
−0.0791584 + 0.996862i \(0.525223\pi\)
\(314\) 0 0
\(315\) −7.61063 −0.428810
\(316\) 0 0
\(317\) −0.140364 −0.00788365 −0.00394182 0.999992i \(-0.501255\pi\)
−0.00394182 + 0.999992i \(0.501255\pi\)
\(318\) 0 0
\(319\) −4.51219 −0.252634
\(320\) 0 0
\(321\) −0.141598 −0.00790321
\(322\) 0 0
\(323\) 8.15132 0.453551
\(324\) 0 0
\(325\) 33.6556 1.86688
\(326\) 0 0
\(327\) 18.2960 1.01177
\(328\) 0 0
\(329\) 6.19966 0.341798
\(330\) 0 0
\(331\) −16.8838 −0.928020 −0.464010 0.885830i \(-0.653590\pi\)
−0.464010 + 0.885830i \(0.653590\pi\)
\(332\) 0 0
\(333\) 2.75195 0.150806
\(334\) 0 0
\(335\) 12.9863 0.709519
\(336\) 0 0
\(337\) −29.7935 −1.62296 −0.811478 0.584383i \(-0.801337\pi\)
−0.811478 + 0.584383i \(0.801337\pi\)
\(338\) 0 0
\(339\) −16.8255 −0.913836
\(340\) 0 0
\(341\) −24.6651 −1.33569
\(342\) 0 0
\(343\) −20.1584 −1.08845
\(344\) 0 0
\(345\) −3.48372 −0.187557
\(346\) 0 0
\(347\) 4.65350 0.249813 0.124906 0.992169i \(-0.460137\pi\)
0.124906 + 0.992169i \(0.460137\pi\)
\(348\) 0 0
\(349\) −22.9746 −1.22980 −0.614901 0.788604i \(-0.710804\pi\)
−0.614901 + 0.788604i \(0.710804\pi\)
\(350\) 0 0
\(351\) −4.71609 −0.251726
\(352\) 0 0
\(353\) −19.0492 −1.01389 −0.506944 0.861979i \(-0.669225\pi\)
−0.506944 + 0.861979i \(0.669225\pi\)
\(354\) 0 0
\(355\) 0.467531 0.0248140
\(356\) 0 0
\(357\) 9.90292 0.524118
\(358\) 0 0
\(359\) 24.0776 1.27077 0.635384 0.772196i \(-0.280842\pi\)
0.635384 + 0.772196i \(0.280842\pi\)
\(360\) 0 0
\(361\) −15.7664 −0.829812
\(362\) 0 0
\(363\) −9.35990 −0.491267
\(364\) 0 0
\(365\) −30.0093 −1.57076
\(366\) 0 0
\(367\) −13.3992 −0.699435 −0.349717 0.936855i \(-0.613722\pi\)
−0.349717 + 0.936855i \(0.613722\pi\)
\(368\) 0 0
\(369\) 0.252352 0.0131369
\(370\) 0 0
\(371\) −9.69660 −0.503423
\(372\) 0 0
\(373\) −3.38531 −0.175285 −0.0876424 0.996152i \(-0.527933\pi\)
−0.0876424 + 0.996152i \(0.527933\pi\)
\(374\) 0 0
\(375\) 7.44242 0.384325
\(376\) 0 0
\(377\) 4.71609 0.242891
\(378\) 0 0
\(379\) −19.3622 −0.994572 −0.497286 0.867587i \(-0.665670\pi\)
−0.497286 + 0.867587i \(0.665670\pi\)
\(380\) 0 0
\(381\) 5.43208 0.278294
\(382\) 0 0
\(383\) −12.9709 −0.662780 −0.331390 0.943494i \(-0.607518\pi\)
−0.331390 + 0.943494i \(0.607518\pi\)
\(384\) 0 0
\(385\) 34.3406 1.75016
\(386\) 0 0
\(387\) 5.62492 0.285931
\(388\) 0 0
\(389\) 24.8051 1.25767 0.628833 0.777540i \(-0.283533\pi\)
0.628833 + 0.777540i \(0.283533\pi\)
\(390\) 0 0
\(391\) 4.53301 0.229244
\(392\) 0 0
\(393\) 1.54649 0.0780103
\(394\) 0 0
\(395\) 8.66110 0.435787
\(396\) 0 0
\(397\) −21.0970 −1.05883 −0.529414 0.848364i \(-0.677588\pi\)
−0.529414 + 0.848364i \(0.677588\pi\)
\(398\) 0 0
\(399\) 3.92842 0.196667
\(400\) 0 0
\(401\) −34.5825 −1.72697 −0.863484 0.504377i \(-0.831722\pi\)
−0.863484 + 0.504377i \(0.831722\pi\)
\(402\) 0 0
\(403\) 25.7796 1.28417
\(404\) 0 0
\(405\) −3.48372 −0.173108
\(406\) 0 0
\(407\) −12.4173 −0.615504
\(408\) 0 0
\(409\) 5.30201 0.262168 0.131084 0.991371i \(-0.458154\pi\)
0.131084 + 0.991371i \(0.458154\pi\)
\(410\) 0 0
\(411\) −6.94494 −0.342569
\(412\) 0 0
\(413\) 8.96552 0.441164
\(414\) 0 0
\(415\) 14.0843 0.691373
\(416\) 0 0
\(417\) −3.92270 −0.192096
\(418\) 0 0
\(419\) −31.1134 −1.51999 −0.759995 0.649929i \(-0.774799\pi\)
−0.759995 + 0.649929i \(0.774799\pi\)
\(420\) 0 0
\(421\) −24.3469 −1.18660 −0.593298 0.804983i \(-0.702175\pi\)
−0.593298 + 0.804983i \(0.702175\pi\)
\(422\) 0 0
\(423\) 2.83786 0.137982
\(424\) 0 0
\(425\) −32.3491 −1.56916
\(426\) 0 0
\(427\) −0.351134 −0.0169926
\(428\) 0 0
\(429\) 21.2799 1.02740
\(430\) 0 0
\(431\) −37.6740 −1.81469 −0.907346 0.420384i \(-0.861895\pi\)
−0.907346 + 0.420384i \(0.861895\pi\)
\(432\) 0 0
\(433\) −33.5751 −1.61352 −0.806759 0.590881i \(-0.798780\pi\)
−0.806759 + 0.590881i \(0.798780\pi\)
\(434\) 0 0
\(435\) 3.48372 0.167032
\(436\) 0 0
\(437\) 1.79821 0.0860202
\(438\) 0 0
\(439\) 2.70114 0.128919 0.0644593 0.997920i \(-0.479468\pi\)
0.0644593 + 0.997920i \(0.479468\pi\)
\(440\) 0 0
\(441\) −2.22742 −0.106068
\(442\) 0 0
\(443\) 13.6352 0.647830 0.323915 0.946086i \(-0.395001\pi\)
0.323915 + 0.946086i \(0.395001\pi\)
\(444\) 0 0
\(445\) −50.1586 −2.37775
\(446\) 0 0
\(447\) −13.2396 −0.626209
\(448\) 0 0
\(449\) −28.6589 −1.35250 −0.676249 0.736673i \(-0.736396\pi\)
−0.676249 + 0.736673i \(0.736396\pi\)
\(450\) 0 0
\(451\) −1.13866 −0.0536175
\(452\) 0 0
\(453\) −20.4856 −0.962498
\(454\) 0 0
\(455\) −35.8924 −1.68266
\(456\) 0 0
\(457\) −15.6633 −0.732697 −0.366348 0.930478i \(-0.619392\pi\)
−0.366348 + 0.930478i \(0.619392\pi\)
\(458\) 0 0
\(459\) 4.53301 0.211583
\(460\) 0 0
\(461\) −35.8058 −1.66764 −0.833820 0.552036i \(-0.813851\pi\)
−0.833820 + 0.552036i \(0.813851\pi\)
\(462\) 0 0
\(463\) −6.42392 −0.298545 −0.149273 0.988796i \(-0.547693\pi\)
−0.149273 + 0.988796i \(0.547693\pi\)
\(464\) 0 0
\(465\) 19.0431 0.883104
\(466\) 0 0
\(467\) 13.6231 0.630401 0.315200 0.949025i \(-0.397928\pi\)
0.315200 + 0.949025i \(0.397928\pi\)
\(468\) 0 0
\(469\) −8.14365 −0.376039
\(470\) 0 0
\(471\) 12.5555 0.578525
\(472\) 0 0
\(473\) −25.3807 −1.16701
\(474\) 0 0
\(475\) −12.8327 −0.588803
\(476\) 0 0
\(477\) −4.43857 −0.203228
\(478\) 0 0
\(479\) −28.5896 −1.30629 −0.653146 0.757232i \(-0.726551\pi\)
−0.653146 + 0.757232i \(0.726551\pi\)
\(480\) 0 0
\(481\) 12.9784 0.591766
\(482\) 0 0
\(483\) 2.18462 0.0994038
\(484\) 0 0
\(485\) 40.2701 1.82857
\(486\) 0 0
\(487\) 17.1517 0.777216 0.388608 0.921403i \(-0.372956\pi\)
0.388608 + 0.921403i \(0.372956\pi\)
\(488\) 0 0
\(489\) −14.2248 −0.643268
\(490\) 0 0
\(491\) −2.67396 −0.120674 −0.0603372 0.998178i \(-0.519218\pi\)
−0.0603372 + 0.998178i \(0.519218\pi\)
\(492\) 0 0
\(493\) −4.53301 −0.204157
\(494\) 0 0
\(495\) 15.7192 0.706528
\(496\) 0 0
\(497\) −0.293186 −0.0131512
\(498\) 0 0
\(499\) −0.493064 −0.0220726 −0.0110363 0.999939i \(-0.503513\pi\)
−0.0110363 + 0.999939i \(0.503513\pi\)
\(500\) 0 0
\(501\) −2.36461 −0.105643
\(502\) 0 0
\(503\) 19.3765 0.863955 0.431977 0.901884i \(-0.357816\pi\)
0.431977 + 0.901884i \(0.357816\pi\)
\(504\) 0 0
\(505\) 45.7498 2.03584
\(506\) 0 0
\(507\) −9.24152 −0.410430
\(508\) 0 0
\(509\) 3.02087 0.133898 0.0669488 0.997756i \(-0.478674\pi\)
0.0669488 + 0.997756i \(0.478674\pi\)
\(510\) 0 0
\(511\) 18.8187 0.832489
\(512\) 0 0
\(513\) 1.79821 0.0793930
\(514\) 0 0
\(515\) −18.5468 −0.817271
\(516\) 0 0
\(517\) −12.8050 −0.563162
\(518\) 0 0
\(519\) −4.08692 −0.179396
\(520\) 0 0
\(521\) 41.5227 1.81914 0.909571 0.415548i \(-0.136410\pi\)
0.909571 + 0.415548i \(0.136410\pi\)
\(522\) 0 0
\(523\) −25.1615 −1.10024 −0.550118 0.835087i \(-0.685417\pi\)
−0.550118 + 0.835087i \(0.685417\pi\)
\(524\) 0 0
\(525\) −15.5902 −0.680413
\(526\) 0 0
\(527\) −24.7788 −1.07938
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.10392 0.178095
\(532\) 0 0
\(533\) 1.19012 0.0515496
\(534\) 0 0
\(535\) −0.493287 −0.0213267
\(536\) 0 0
\(537\) −4.27445 −0.184456
\(538\) 0 0
\(539\) 10.0506 0.432908
\(540\) 0 0
\(541\) 25.1609 1.08175 0.540876 0.841102i \(-0.318093\pi\)
0.540876 + 0.841102i \(0.318093\pi\)
\(542\) 0 0
\(543\) −4.85121 −0.208185
\(544\) 0 0
\(545\) 63.7383 2.73025
\(546\) 0 0
\(547\) −8.28515 −0.354247 −0.177124 0.984189i \(-0.556679\pi\)
−0.177124 + 0.984189i \(0.556679\pi\)
\(548\) 0 0
\(549\) −0.160730 −0.00685978
\(550\) 0 0
\(551\) −1.79821 −0.0766064
\(552\) 0 0
\(553\) −5.43132 −0.230963
\(554\) 0 0
\(555\) 9.58703 0.406947
\(556\) 0 0
\(557\) −15.4270 −0.653663 −0.326832 0.945083i \(-0.605981\pi\)
−0.326832 + 0.945083i \(0.605981\pi\)
\(558\) 0 0
\(559\) 26.5276 1.12200
\(560\) 0 0
\(561\) −20.4538 −0.863561
\(562\) 0 0
\(563\) 21.7039 0.914711 0.457356 0.889284i \(-0.348797\pi\)
0.457356 + 0.889284i \(0.348797\pi\)
\(564\) 0 0
\(565\) −58.6154 −2.46597
\(566\) 0 0
\(567\) 2.18462 0.0917455
\(568\) 0 0
\(569\) 29.9165 1.25417 0.627083 0.778952i \(-0.284249\pi\)
0.627083 + 0.778952i \(0.284249\pi\)
\(570\) 0 0
\(571\) 9.36487 0.391908 0.195954 0.980613i \(-0.437220\pi\)
0.195954 + 0.980613i \(0.437220\pi\)
\(572\) 0 0
\(573\) −9.30171 −0.388584
\(574\) 0 0
\(575\) −7.13634 −0.297606
\(576\) 0 0
\(577\) 10.0273 0.417444 0.208722 0.977975i \(-0.433070\pi\)
0.208722 + 0.977975i \(0.433070\pi\)
\(578\) 0 0
\(579\) 6.40515 0.266189
\(580\) 0 0
\(581\) −8.83221 −0.366422
\(582\) 0 0
\(583\) 20.0277 0.829462
\(584\) 0 0
\(585\) −16.4296 −0.679279
\(586\) 0 0
\(587\) −9.59203 −0.395905 −0.197953 0.980212i \(-0.563429\pi\)
−0.197953 + 0.980212i \(0.563429\pi\)
\(588\) 0 0
\(589\) −9.82959 −0.405021
\(590\) 0 0
\(591\) 24.9249 1.02527
\(592\) 0 0
\(593\) −18.0393 −0.740784 −0.370392 0.928876i \(-0.620777\pi\)
−0.370392 + 0.928876i \(0.620777\pi\)
\(594\) 0 0
\(595\) 34.4990 1.41432
\(596\) 0 0
\(597\) 22.7349 0.930479
\(598\) 0 0
\(599\) 21.4214 0.875253 0.437627 0.899157i \(-0.355819\pi\)
0.437627 + 0.899157i \(0.355819\pi\)
\(600\) 0 0
\(601\) −0.260657 −0.0106324 −0.00531620 0.999986i \(-0.501692\pi\)
−0.00531620 + 0.999986i \(0.501692\pi\)
\(602\) 0 0
\(603\) −3.72772 −0.151804
\(604\) 0 0
\(605\) −32.6073 −1.32568
\(606\) 0 0
\(607\) 23.0460 0.935410 0.467705 0.883885i \(-0.345081\pi\)
0.467705 + 0.883885i \(0.345081\pi\)
\(608\) 0 0
\(609\) −2.18462 −0.0885254
\(610\) 0 0
\(611\) 13.3836 0.541443
\(612\) 0 0
\(613\) 27.8998 1.12686 0.563430 0.826164i \(-0.309481\pi\)
0.563430 + 0.826164i \(0.309481\pi\)
\(614\) 0 0
\(615\) 0.879125 0.0354497
\(616\) 0 0
\(617\) −5.37316 −0.216315 −0.108158 0.994134i \(-0.534495\pi\)
−0.108158 + 0.994134i \(0.534495\pi\)
\(618\) 0 0
\(619\) −5.32142 −0.213886 −0.106943 0.994265i \(-0.534106\pi\)
−0.106943 + 0.994265i \(0.534106\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 31.4542 1.26018
\(624\) 0 0
\(625\) −9.75436 −0.390174
\(626\) 0 0
\(627\) −8.11388 −0.324037
\(628\) 0 0
\(629\) −12.4746 −0.497395
\(630\) 0 0
\(631\) 5.48318 0.218282 0.109141 0.994026i \(-0.465190\pi\)
0.109141 + 0.994026i \(0.465190\pi\)
\(632\) 0 0
\(633\) 4.01682 0.159654
\(634\) 0 0
\(635\) 18.9239 0.750971
\(636\) 0 0
\(637\) −10.5047 −0.416212
\(638\) 0 0
\(639\) −0.134204 −0.00530904
\(640\) 0 0
\(641\) 2.58562 0.102126 0.0510629 0.998695i \(-0.483739\pi\)
0.0510629 + 0.998695i \(0.483739\pi\)
\(642\) 0 0
\(643\) −24.6027 −0.970234 −0.485117 0.874449i \(-0.661223\pi\)
−0.485117 + 0.874449i \(0.661223\pi\)
\(644\) 0 0
\(645\) 19.5957 0.771578
\(646\) 0 0
\(647\) 28.1739 1.10763 0.553815 0.832640i \(-0.313172\pi\)
0.553815 + 0.832640i \(0.313172\pi\)
\(648\) 0 0
\(649\) −18.5177 −0.726883
\(650\) 0 0
\(651\) −11.9418 −0.468037
\(652\) 0 0
\(653\) −29.3417 −1.14823 −0.574115 0.818774i \(-0.694654\pi\)
−0.574115 + 0.818774i \(0.694654\pi\)
\(654\) 0 0
\(655\) 5.38756 0.210509
\(656\) 0 0
\(657\) 8.61415 0.336070
\(658\) 0 0
\(659\) 2.83816 0.110559 0.0552794 0.998471i \(-0.482395\pi\)
0.0552794 + 0.998471i \(0.482395\pi\)
\(660\) 0 0
\(661\) −50.0625 −1.94720 −0.973602 0.228250i \(-0.926700\pi\)
−0.973602 + 0.228250i \(0.926700\pi\)
\(662\) 0 0
\(663\) 21.3781 0.830256
\(664\) 0 0
\(665\) 13.6855 0.530702
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 21.3151 0.824090
\(670\) 0 0
\(671\) 0.725244 0.0279977
\(672\) 0 0
\(673\) −19.6293 −0.756652 −0.378326 0.925672i \(-0.623500\pi\)
−0.378326 + 0.925672i \(0.623500\pi\)
\(674\) 0 0
\(675\) −7.13634 −0.274678
\(676\) 0 0
\(677\) 26.6125 1.02280 0.511400 0.859343i \(-0.329127\pi\)
0.511400 + 0.859343i \(0.329127\pi\)
\(678\) 0 0
\(679\) −25.2532 −0.969128
\(680\) 0 0
\(681\) 9.37961 0.359427
\(682\) 0 0
\(683\) −25.4576 −0.974109 −0.487055 0.873372i \(-0.661929\pi\)
−0.487055 + 0.873372i \(0.661929\pi\)
\(684\) 0 0
\(685\) −24.1943 −0.924415
\(686\) 0 0
\(687\) −0.756117 −0.0288477
\(688\) 0 0
\(689\) −20.9327 −0.797473
\(690\) 0 0
\(691\) 37.4686 1.42537 0.712687 0.701482i \(-0.247478\pi\)
0.712687 + 0.701482i \(0.247478\pi\)
\(692\) 0 0
\(693\) −9.85744 −0.374453
\(694\) 0 0
\(695\) −13.6656 −0.518366
\(696\) 0 0
\(697\) −1.14391 −0.0433289
\(698\) 0 0
\(699\) −9.34510 −0.353464
\(700\) 0 0
\(701\) −42.9212 −1.62111 −0.810556 0.585661i \(-0.800835\pi\)
−0.810556 + 0.585661i \(0.800835\pi\)
\(702\) 0 0
\(703\) −4.94859 −0.186640
\(704\) 0 0
\(705\) 9.88633 0.372341
\(706\) 0 0
\(707\) −28.6894 −1.07898
\(708\) 0 0
\(709\) −45.7792 −1.71927 −0.859637 0.510905i \(-0.829310\pi\)
−0.859637 + 0.510905i \(0.829310\pi\)
\(710\) 0 0
\(711\) −2.48616 −0.0932383
\(712\) 0 0
\(713\) −5.46631 −0.204715
\(714\) 0 0
\(715\) 74.1334 2.77243
\(716\) 0 0
\(717\) 19.1832 0.716410
\(718\) 0 0
\(719\) 34.2632 1.27780 0.638901 0.769289i \(-0.279389\pi\)
0.638901 + 0.769289i \(0.279389\pi\)
\(720\) 0 0
\(721\) 11.6306 0.433146
\(722\) 0 0
\(723\) −8.64995 −0.321695
\(724\) 0 0
\(725\) 7.13634 0.265037
\(726\) 0 0
\(727\) −12.9670 −0.480918 −0.240459 0.970659i \(-0.577298\pi\)
−0.240459 + 0.970659i \(0.577298\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −25.4978 −0.943070
\(732\) 0 0
\(733\) 0.810299 0.0299291 0.0149645 0.999888i \(-0.495236\pi\)
0.0149645 + 0.999888i \(0.495236\pi\)
\(734\) 0 0
\(735\) −7.75973 −0.286222
\(736\) 0 0
\(737\) 16.8202 0.619579
\(738\) 0 0
\(739\) 15.4395 0.567951 0.283975 0.958832i \(-0.408347\pi\)
0.283975 + 0.958832i \(0.408347\pi\)
\(740\) 0 0
\(741\) 8.48053 0.311540
\(742\) 0 0
\(743\) −1.62739 −0.0597033 −0.0298516 0.999554i \(-0.509503\pi\)
−0.0298516 + 0.999554i \(0.509503\pi\)
\(744\) 0 0
\(745\) −46.1230 −1.68981
\(746\) 0 0
\(747\) −4.04290 −0.147922
\(748\) 0 0
\(749\) 0.309337 0.0113029
\(750\) 0 0
\(751\) −3.93844 −0.143716 −0.0718579 0.997415i \(-0.522893\pi\)
−0.0718579 + 0.997415i \(0.522893\pi\)
\(752\) 0 0
\(753\) −1.74567 −0.0636157
\(754\) 0 0
\(755\) −71.3663 −2.59728
\(756\) 0 0
\(757\) −17.9907 −0.653884 −0.326942 0.945044i \(-0.606018\pi\)
−0.326942 + 0.945044i \(0.606018\pi\)
\(758\) 0 0
\(759\) −4.51219 −0.163782
\(760\) 0 0
\(761\) 38.2900 1.38801 0.694006 0.719969i \(-0.255844\pi\)
0.694006 + 0.719969i \(0.255844\pi\)
\(762\) 0 0
\(763\) −39.9699 −1.44701
\(764\) 0 0
\(765\) 15.7918 0.570952
\(766\) 0 0
\(767\) 19.3545 0.698849
\(768\) 0 0
\(769\) −19.3218 −0.696761 −0.348381 0.937353i \(-0.613268\pi\)
−0.348381 + 0.937353i \(0.613268\pi\)
\(770\) 0 0
\(771\) −14.6090 −0.526131
\(772\) 0 0
\(773\) 10.8833 0.391443 0.195722 0.980659i \(-0.437295\pi\)
0.195722 + 0.980659i \(0.437295\pi\)
\(774\) 0 0
\(775\) 39.0095 1.40126
\(776\) 0 0
\(777\) −6.01197 −0.215678
\(778\) 0 0
\(779\) −0.453783 −0.0162584
\(780\) 0 0
\(781\) 0.605556 0.0216685
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 43.7397 1.56114
\(786\) 0 0
\(787\) 41.9622 1.49579 0.747895 0.663817i \(-0.231065\pi\)
0.747895 + 0.663817i \(0.231065\pi\)
\(788\) 0 0
\(789\) 14.5588 0.518306
\(790\) 0 0
\(791\) 36.7574 1.30694
\(792\) 0 0
\(793\) −0.758016 −0.0269179
\(794\) 0 0
\(795\) −15.4628 −0.548408
\(796\) 0 0
\(797\) 11.7595 0.416542 0.208271 0.978071i \(-0.433216\pi\)
0.208271 + 0.978071i \(0.433216\pi\)
\(798\) 0 0
\(799\) −12.8641 −0.455098
\(800\) 0 0
\(801\) 14.3980 0.508727
\(802\) 0 0
\(803\) −38.8687 −1.37165
\(804\) 0 0
\(805\) 7.61063 0.268239
\(806\) 0 0
\(807\) 5.12131 0.180279
\(808\) 0 0
\(809\) −26.1473 −0.919290 −0.459645 0.888103i \(-0.652023\pi\)
−0.459645 + 0.888103i \(0.652023\pi\)
\(810\) 0 0
\(811\) −15.6688 −0.550207 −0.275104 0.961415i \(-0.588712\pi\)
−0.275104 + 0.961415i \(0.588712\pi\)
\(812\) 0 0
\(813\) −17.6495 −0.618996
\(814\) 0 0
\(815\) −49.5553 −1.73585
\(816\) 0 0
\(817\) −10.1148 −0.353872
\(818\) 0 0
\(819\) 10.3029 0.360012
\(820\) 0 0
\(821\) 28.6999 1.00163 0.500816 0.865553i \(-0.333033\pi\)
0.500816 + 0.865553i \(0.333033\pi\)
\(822\) 0 0
\(823\) −37.7360 −1.31539 −0.657697 0.753283i \(-0.728469\pi\)
−0.657697 + 0.753283i \(0.728469\pi\)
\(824\) 0 0
\(825\) 32.2006 1.12108
\(826\) 0 0
\(827\) 15.3602 0.534125 0.267062 0.963679i \(-0.413947\pi\)
0.267062 + 0.963679i \(0.413947\pi\)
\(828\) 0 0
\(829\) 50.3499 1.74872 0.874362 0.485274i \(-0.161280\pi\)
0.874362 + 0.485274i \(0.161280\pi\)
\(830\) 0 0
\(831\) 14.3682 0.498427
\(832\) 0 0
\(833\) 10.0969 0.349838
\(834\) 0 0
\(835\) −8.23766 −0.285076
\(836\) 0 0
\(837\) −5.46631 −0.188943
\(838\) 0 0
\(839\) 4.23121 0.146078 0.0730388 0.997329i \(-0.476730\pi\)
0.0730388 + 0.997329i \(0.476730\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −11.2915 −0.388899
\(844\) 0 0
\(845\) −32.1949 −1.10754
\(846\) 0 0
\(847\) 20.4479 0.702597
\(848\) 0 0
\(849\) 22.8449 0.784035
\(850\) 0 0
\(851\) −2.75195 −0.0943356
\(852\) 0 0
\(853\) 28.5417 0.977249 0.488625 0.872494i \(-0.337499\pi\)
0.488625 + 0.872494i \(0.337499\pi\)
\(854\) 0 0
\(855\) 6.26448 0.214241
\(856\) 0 0
\(857\) −37.6372 −1.28566 −0.642831 0.766008i \(-0.722240\pi\)
−0.642831 + 0.766008i \(0.722240\pi\)
\(858\) 0 0
\(859\) 27.4774 0.937518 0.468759 0.883326i \(-0.344701\pi\)
0.468759 + 0.883326i \(0.344701\pi\)
\(860\) 0 0
\(861\) −0.551294 −0.0187881
\(862\) 0 0
\(863\) 25.6281 0.872390 0.436195 0.899852i \(-0.356326\pi\)
0.436195 + 0.899852i \(0.356326\pi\)
\(864\) 0 0
\(865\) −14.2377 −0.484096
\(866\) 0 0
\(867\) −3.54818 −0.120503
\(868\) 0 0
\(869\) 11.2180 0.380546
\(870\) 0 0
\(871\) −17.5802 −0.595684
\(872\) 0 0
\(873\) −11.5595 −0.391230
\(874\) 0 0
\(875\) −16.2589 −0.549650
\(876\) 0 0
\(877\) −9.15603 −0.309177 −0.154589 0.987979i \(-0.549405\pi\)
−0.154589 + 0.987979i \(0.549405\pi\)
\(878\) 0 0
\(879\) −0.626712 −0.0211385
\(880\) 0 0
\(881\) 26.1589 0.881315 0.440657 0.897675i \(-0.354745\pi\)
0.440657 + 0.897675i \(0.354745\pi\)
\(882\) 0 0
\(883\) 39.4335 1.32704 0.663522 0.748157i \(-0.269061\pi\)
0.663522 + 0.748157i \(0.269061\pi\)
\(884\) 0 0
\(885\) 14.2969 0.480586
\(886\) 0 0
\(887\) −13.8289 −0.464329 −0.232164 0.972677i \(-0.574581\pi\)
−0.232164 + 0.972677i \(0.574581\pi\)
\(888\) 0 0
\(889\) −11.8671 −0.398008
\(890\) 0 0
\(891\) −4.51219 −0.151164
\(892\) 0 0
\(893\) −5.10308 −0.170768
\(894\) 0 0
\(895\) −14.8910 −0.497752
\(896\) 0 0
\(897\) 4.71609 0.157466
\(898\) 0 0
\(899\) 5.46631 0.182312
\(900\) 0 0
\(901\) 20.1201 0.670297
\(902\) 0 0
\(903\) −12.2883 −0.408930
\(904\) 0 0
\(905\) −16.9003 −0.561785
\(906\) 0 0
\(907\) −11.8379 −0.393070 −0.196535 0.980497i \(-0.562969\pi\)
−0.196535 + 0.980497i \(0.562969\pi\)
\(908\) 0 0
\(909\) −13.1324 −0.435575
\(910\) 0 0
\(911\) 42.2595 1.40012 0.700060 0.714084i \(-0.253157\pi\)
0.700060 + 0.714084i \(0.253157\pi\)
\(912\) 0 0
\(913\) 18.2423 0.603733
\(914\) 0 0
\(915\) −0.559938 −0.0185110
\(916\) 0 0
\(917\) −3.37851 −0.111568
\(918\) 0 0
\(919\) 34.9520 1.15296 0.576480 0.817111i \(-0.304426\pi\)
0.576480 + 0.817111i \(0.304426\pi\)
\(920\) 0 0
\(921\) 13.0939 0.431458
\(922\) 0 0
\(923\) −0.632920 −0.0208328
\(924\) 0 0
\(925\) 19.6388 0.645721
\(926\) 0 0
\(927\) 5.32385 0.174858
\(928\) 0 0
\(929\) −2.39407 −0.0785470 −0.0392735 0.999228i \(-0.512504\pi\)
−0.0392735 + 0.999228i \(0.512504\pi\)
\(930\) 0 0
\(931\) 4.00538 0.131271
\(932\) 0 0
\(933\) −7.40225 −0.242339
\(934\) 0 0
\(935\) −71.2555 −2.33030
\(936\) 0 0
\(937\) −2.10672 −0.0688235 −0.0344118 0.999408i \(-0.510956\pi\)
−0.0344118 + 0.999408i \(0.510956\pi\)
\(938\) 0 0
\(939\) 2.80091 0.0914042
\(940\) 0 0
\(941\) −24.4828 −0.798116 −0.399058 0.916926i \(-0.630663\pi\)
−0.399058 + 0.916926i \(0.630663\pi\)
\(942\) 0 0
\(943\) −0.252352 −0.00821771
\(944\) 0 0
\(945\) 7.61063 0.247574
\(946\) 0 0
\(947\) 23.8006 0.773416 0.386708 0.922202i \(-0.373612\pi\)
0.386708 + 0.922202i \(0.373612\pi\)
\(948\) 0 0
\(949\) 40.6251 1.31875
\(950\) 0 0
\(951\) 0.140364 0.00455162
\(952\) 0 0
\(953\) −8.22801 −0.266531 −0.133266 0.991080i \(-0.542546\pi\)
−0.133266 + 0.991080i \(0.542546\pi\)
\(954\) 0 0
\(955\) −32.4046 −1.04859
\(956\) 0 0
\(957\) 4.51219 0.145859
\(958\) 0 0
\(959\) 15.1721 0.489932
\(960\) 0 0
\(961\) −1.11944 −0.0361110
\(962\) 0 0
\(963\) 0.141598 0.00456292
\(964\) 0 0
\(965\) 22.3138 0.718306
\(966\) 0 0
\(967\) −35.1102 −1.12907 −0.564534 0.825410i \(-0.690944\pi\)
−0.564534 + 0.825410i \(0.690944\pi\)
\(968\) 0 0
\(969\) −8.15132 −0.261858
\(970\) 0 0
\(971\) −35.5969 −1.14236 −0.571179 0.820825i \(-0.693514\pi\)
−0.571179 + 0.820825i \(0.693514\pi\)
\(972\) 0 0
\(973\) 8.56963 0.274730
\(974\) 0 0
\(975\) −33.6556 −1.07784
\(976\) 0 0
\(977\) −1.67797 −0.0536829 −0.0268414 0.999640i \(-0.508545\pi\)
−0.0268414 + 0.999640i \(0.508545\pi\)
\(978\) 0 0
\(979\) −64.9665 −2.07634
\(980\) 0 0
\(981\) −18.2960 −0.584147
\(982\) 0 0
\(983\) −4.78120 −0.152497 −0.0762484 0.997089i \(-0.524294\pi\)
−0.0762484 + 0.997089i \(0.524294\pi\)
\(984\) 0 0
\(985\) 86.8315 2.76668
\(986\) 0 0
\(987\) −6.19966 −0.197337
\(988\) 0 0
\(989\) −5.62492 −0.178862
\(990\) 0 0
\(991\) −14.8296 −0.471076 −0.235538 0.971865i \(-0.575685\pi\)
−0.235538 + 0.971865i \(0.575685\pi\)
\(992\) 0 0
\(993\) 16.8838 0.535793
\(994\) 0 0
\(995\) 79.2023 2.51088
\(996\) 0 0
\(997\) 23.0282 0.729309 0.364655 0.931143i \(-0.381187\pi\)
0.364655 + 0.931143i \(0.381187\pi\)
\(998\) 0 0
\(999\) −2.75195 −0.0870678
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.e.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.2 9 1.1 even 1 trivial