Properties

Label 1080.4.a.o.1.2
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.55367\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} -2.40438 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} -2.40438 q^{7} -17.3282 q^{11} +71.2940 q^{13} -111.164 q^{17} -86.2397 q^{19} +148.266 q^{23} +25.0000 q^{25} -71.1182 q^{29} +212.038 q^{31} +12.0219 q^{35} -132.980 q^{37} +29.6830 q^{41} -126.879 q^{43} +103.317 q^{47} -337.219 q^{49} +298.464 q^{53} +86.6410 q^{55} -893.675 q^{59} +752.720 q^{61} -356.470 q^{65} +477.167 q^{67} +58.7339 q^{71} +873.851 q^{73} +41.6635 q^{77} +819.768 q^{79} +959.392 q^{83} +555.818 q^{85} +406.735 q^{89} -171.418 q^{91} +431.199 q^{95} +623.748 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} + 14 q^{7} + 4 q^{11} + 30 q^{13} + 28 q^{17} + 78 q^{19} - 182 q^{23} + 100 q^{25} - 202 q^{29} - 76 q^{31} - 70 q^{35} + 302 q^{37} - 380 q^{41} + 178 q^{43} - 114 q^{47} + 958 q^{49} + 256 q^{53} - 20 q^{55} + 204 q^{59} + 766 q^{61} - 150 q^{65} + 330 q^{67} + 1060 q^{71} + 1442 q^{73} - 216 q^{77} + 742 q^{79} + 768 q^{83} - 140 q^{85} + 400 q^{89} + 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −2.40438 −0.129824 −0.0649121 0.997891i \(-0.520677\pi\)
−0.0649121 + 0.997891i \(0.520677\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −17.3282 −0.474968 −0.237484 0.971391i \(-0.576323\pi\)
−0.237484 + 0.971391i \(0.576323\pi\)
\(12\) 0 0
\(13\) 71.2940 1.52103 0.760515 0.649320i \(-0.224946\pi\)
0.760515 + 0.649320i \(0.224946\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −111.164 −1.58595 −0.792974 0.609255i \(-0.791468\pi\)
−0.792974 + 0.609255i \(0.791468\pi\)
\(18\) 0 0
\(19\) −86.2397 −1.04130 −0.520651 0.853769i \(-0.674311\pi\)
−0.520651 + 0.853769i \(0.674311\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 148.266 1.34416 0.672080 0.740479i \(-0.265401\pi\)
0.672080 + 0.740479i \(0.265401\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −71.1182 −0.455390 −0.227695 0.973732i \(-0.573119\pi\)
−0.227695 + 0.973732i \(0.573119\pi\)
\(30\) 0 0
\(31\) 212.038 1.22849 0.614244 0.789116i \(-0.289461\pi\)
0.614244 + 0.789116i \(0.289461\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0219 0.0580591
\(36\) 0 0
\(37\) −132.980 −0.590859 −0.295430 0.955365i \(-0.595463\pi\)
−0.295430 + 0.955365i \(0.595463\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 29.6830 0.113066 0.0565330 0.998401i \(-0.481995\pi\)
0.0565330 + 0.998401i \(0.481995\pi\)
\(42\) 0 0
\(43\) −126.879 −0.449973 −0.224987 0.974362i \(-0.572234\pi\)
−0.224987 + 0.974362i \(0.572234\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 103.317 0.320647 0.160323 0.987065i \(-0.448746\pi\)
0.160323 + 0.987065i \(0.448746\pi\)
\(48\) 0 0
\(49\) −337.219 −0.983146
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 298.464 0.773531 0.386765 0.922178i \(-0.373592\pi\)
0.386765 + 0.922178i \(0.373592\pi\)
\(54\) 0 0
\(55\) 86.6410 0.212412
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −893.675 −1.97198 −0.985988 0.166817i \(-0.946651\pi\)
−0.985988 + 0.166817i \(0.946651\pi\)
\(60\) 0 0
\(61\) 752.720 1.57993 0.789966 0.613150i \(-0.210098\pi\)
0.789966 + 0.613150i \(0.210098\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −356.470 −0.680226
\(66\) 0 0
\(67\) 477.167 0.870078 0.435039 0.900412i \(-0.356735\pi\)
0.435039 + 0.900412i \(0.356735\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 58.7339 0.0981751 0.0490875 0.998794i \(-0.484369\pi\)
0.0490875 + 0.998794i \(0.484369\pi\)
\(72\) 0 0
\(73\) 873.851 1.40105 0.700524 0.713628i \(-0.252949\pi\)
0.700524 + 0.713628i \(0.252949\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 41.6635 0.0616623
\(78\) 0 0
\(79\) 819.768 1.16748 0.583741 0.811940i \(-0.301588\pi\)
0.583741 + 0.811940i \(0.301588\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 959.392 1.26876 0.634379 0.773022i \(-0.281256\pi\)
0.634379 + 0.773022i \(0.281256\pi\)
\(84\) 0 0
\(85\) 555.818 0.709258
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 406.735 0.484425 0.242212 0.970223i \(-0.422127\pi\)
0.242212 + 0.970223i \(0.422127\pi\)
\(90\) 0 0
\(91\) −171.418 −0.197467
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 431.199 0.465685
\(96\) 0 0
\(97\) 623.748 0.652908 0.326454 0.945213i \(-0.394146\pi\)
0.326454 + 0.945213i \(0.394146\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 468.202 0.461266 0.230633 0.973041i \(-0.425920\pi\)
0.230633 + 0.973041i \(0.425920\pi\)
\(102\) 0 0
\(103\) −949.097 −0.907935 −0.453967 0.891018i \(-0.649992\pi\)
−0.453967 + 0.891018i \(0.649992\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 649.086 0.586444 0.293222 0.956044i \(-0.405273\pi\)
0.293222 + 0.956044i \(0.405273\pi\)
\(108\) 0 0
\(109\) 1152.44 1.01269 0.506346 0.862330i \(-0.330996\pi\)
0.506346 + 0.862330i \(0.330996\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1537.75 −1.28017 −0.640087 0.768302i \(-0.721102\pi\)
−0.640087 + 0.768302i \(0.721102\pi\)
\(114\) 0 0
\(115\) −741.331 −0.601126
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 267.279 0.205894
\(120\) 0 0
\(121\) −1030.73 −0.774405
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1950.33 1.36271 0.681353 0.731955i \(-0.261392\pi\)
0.681353 + 0.731955i \(0.261392\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1219.32 0.813222 0.406611 0.913601i \(-0.366710\pi\)
0.406611 + 0.913601i \(0.366710\pi\)
\(132\) 0 0
\(133\) 207.353 0.135186
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2632.77 −1.64185 −0.820924 0.571038i \(-0.806541\pi\)
−0.820924 + 0.571038i \(0.806541\pi\)
\(138\) 0 0
\(139\) 943.117 0.575498 0.287749 0.957706i \(-0.407093\pi\)
0.287749 + 0.957706i \(0.407093\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1235.40 −0.722441
\(144\) 0 0
\(145\) 355.591 0.203657
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 833.371 0.458204 0.229102 0.973402i \(-0.426421\pi\)
0.229102 + 0.973402i \(0.426421\pi\)
\(150\) 0 0
\(151\) 2413.13 1.30052 0.650258 0.759714i \(-0.274661\pi\)
0.650258 + 0.759714i \(0.274661\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1060.19 −0.549396
\(156\) 0 0
\(157\) 1085.41 0.551755 0.275877 0.961193i \(-0.411032\pi\)
0.275877 + 0.961193i \(0.411032\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −356.488 −0.174504
\(162\) 0 0
\(163\) 3657.89 1.75772 0.878859 0.477082i \(-0.158306\pi\)
0.878859 + 0.477082i \(0.158306\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1506.46 0.698043 0.349022 0.937115i \(-0.386514\pi\)
0.349022 + 0.937115i \(0.386514\pi\)
\(168\) 0 0
\(169\) 2885.84 1.31353
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 83.8876 0.0368662 0.0184331 0.999830i \(-0.494132\pi\)
0.0184331 + 0.999830i \(0.494132\pi\)
\(174\) 0 0
\(175\) −60.1094 −0.0259648
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4698.85 1.96206 0.981030 0.193855i \(-0.0620990\pi\)
0.981030 + 0.193855i \(0.0620990\pi\)
\(180\) 0 0
\(181\) −2160.95 −0.887413 −0.443707 0.896172i \(-0.646337\pi\)
−0.443707 + 0.896172i \(0.646337\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 664.900 0.264240
\(186\) 0 0
\(187\) 1926.26 0.753275
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 160.758 0.0609009 0.0304504 0.999536i \(-0.490306\pi\)
0.0304504 + 0.999536i \(0.490306\pi\)
\(192\) 0 0
\(193\) 3743.78 1.39628 0.698142 0.715959i \(-0.254010\pi\)
0.698142 + 0.715959i \(0.254010\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3812.48 −1.37882 −0.689411 0.724371i \(-0.742130\pi\)
−0.689411 + 0.724371i \(0.742130\pi\)
\(198\) 0 0
\(199\) −52.1965 −0.0185935 −0.00929676 0.999957i \(-0.502959\pi\)
−0.00929676 + 0.999957i \(0.502959\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 170.995 0.0591206
\(204\) 0 0
\(205\) −148.415 −0.0505646
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1494.38 0.494585
\(210\) 0 0
\(211\) 2781.76 0.907603 0.453802 0.891103i \(-0.350067\pi\)
0.453802 + 0.891103i \(0.350067\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 634.395 0.201234
\(216\) 0 0
\(217\) −509.819 −0.159487
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7925.29 −2.41228
\(222\) 0 0
\(223\) 1310.27 0.393462 0.196731 0.980458i \(-0.436968\pi\)
0.196731 + 0.980458i \(0.436968\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3413.82 −0.998163 −0.499082 0.866555i \(-0.666329\pi\)
−0.499082 + 0.866555i \(0.666329\pi\)
\(228\) 0 0
\(229\) −4317.15 −1.24579 −0.622894 0.782306i \(-0.714043\pi\)
−0.622894 + 0.782306i \(0.714043\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2244.87 −0.631185 −0.315593 0.948895i \(-0.602203\pi\)
−0.315593 + 0.948895i \(0.602203\pi\)
\(234\) 0 0
\(235\) −516.587 −0.143398
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −924.054 −0.250092 −0.125046 0.992151i \(-0.539908\pi\)
−0.125046 + 0.992151i \(0.539908\pi\)
\(240\) 0 0
\(241\) 5505.32 1.47149 0.735744 0.677260i \(-0.236833\pi\)
0.735744 + 0.677260i \(0.236833\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1686.09 0.439676
\(246\) 0 0
\(247\) −6148.37 −1.58385
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1654.48 0.416056 0.208028 0.978123i \(-0.433295\pi\)
0.208028 + 0.978123i \(0.433295\pi\)
\(252\) 0 0
\(253\) −2569.19 −0.638433
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5099.06 1.23763 0.618814 0.785538i \(-0.287613\pi\)
0.618814 + 0.785538i \(0.287613\pi\)
\(258\) 0 0
\(259\) 319.734 0.0767078
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4300.91 1.00838 0.504192 0.863591i \(-0.331790\pi\)
0.504192 + 0.863591i \(0.331790\pi\)
\(264\) 0 0
\(265\) −1492.32 −0.345933
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5589.55 −1.26692 −0.633459 0.773776i \(-0.718366\pi\)
−0.633459 + 0.773776i \(0.718366\pi\)
\(270\) 0 0
\(271\) 3435.85 0.770158 0.385079 0.922884i \(-0.374174\pi\)
0.385079 + 0.922884i \(0.374174\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −433.205 −0.0949936
\(276\) 0 0
\(277\) 4652.36 1.00914 0.504572 0.863369i \(-0.331650\pi\)
0.504572 + 0.863369i \(0.331650\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3173.95 0.673814 0.336907 0.941538i \(-0.390619\pi\)
0.336907 + 0.941538i \(0.390619\pi\)
\(282\) 0 0
\(283\) −4344.90 −0.912641 −0.456321 0.889815i \(-0.650833\pi\)
−0.456321 + 0.889815i \(0.650833\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −71.3691 −0.0146787
\(288\) 0 0
\(289\) 7444.33 1.51523
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3010.86 0.600329 0.300165 0.953887i \(-0.402958\pi\)
0.300165 + 0.953887i \(0.402958\pi\)
\(294\) 0 0
\(295\) 4468.37 0.881894
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10570.5 2.04451
\(300\) 0 0
\(301\) 305.065 0.0584174
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3763.60 −0.706567
\(306\) 0 0
\(307\) −1083.82 −0.201487 −0.100744 0.994912i \(-0.532122\pi\)
−0.100744 + 0.994912i \(0.532122\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7202.17 1.31318 0.656588 0.754249i \(-0.271999\pi\)
0.656588 + 0.754249i \(0.271999\pi\)
\(312\) 0 0
\(313\) 55.5588 0.0100331 0.00501656 0.999987i \(-0.498403\pi\)
0.00501656 + 0.999987i \(0.498403\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6762.76 −1.19822 −0.599108 0.800668i \(-0.704478\pi\)
−0.599108 + 0.800668i \(0.704478\pi\)
\(318\) 0 0
\(319\) 1232.35 0.216296
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9586.71 1.65145
\(324\) 0 0
\(325\) 1782.35 0.304206
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −248.414 −0.0416277
\(330\) 0 0
\(331\) −4689.11 −0.778660 −0.389330 0.921098i \(-0.627293\pi\)
−0.389330 + 0.921098i \(0.627293\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2385.84 −0.389111
\(336\) 0 0
\(337\) −11110.1 −1.79587 −0.897935 0.440128i \(-0.854933\pi\)
−0.897935 + 0.440128i \(0.854933\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3674.23 −0.583492
\(342\) 0 0
\(343\) 1635.50 0.257460
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8190.20 −1.26707 −0.633535 0.773714i \(-0.718397\pi\)
−0.633535 + 0.773714i \(0.718397\pi\)
\(348\) 0 0
\(349\) −2804.03 −0.430075 −0.215037 0.976606i \(-0.568987\pi\)
−0.215037 + 0.976606i \(0.568987\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5285.46 −0.796931 −0.398466 0.917183i \(-0.630457\pi\)
−0.398466 + 0.917183i \(0.630457\pi\)
\(354\) 0 0
\(355\) −293.669 −0.0439052
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3078.37 0.452563 0.226281 0.974062i \(-0.427343\pi\)
0.226281 + 0.974062i \(0.427343\pi\)
\(360\) 0 0
\(361\) 578.288 0.0843109
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4369.26 −0.626568
\(366\) 0 0
\(367\) −81.5000 −0.0115920 −0.00579600 0.999983i \(-0.501845\pi\)
−0.00579600 + 0.999983i \(0.501845\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −717.619 −0.100423
\(372\) 0 0
\(373\) −11888.6 −1.65032 −0.825160 0.564899i \(-0.808915\pi\)
−0.825160 + 0.564899i \(0.808915\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5070.30 −0.692662
\(378\) 0 0
\(379\) −10663.7 −1.44527 −0.722636 0.691229i \(-0.757070\pi\)
−0.722636 + 0.691229i \(0.757070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6178.97 0.824362 0.412181 0.911102i \(-0.364767\pi\)
0.412181 + 0.911102i \(0.364767\pi\)
\(384\) 0 0
\(385\) −208.318 −0.0275762
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7948.36 −1.03598 −0.517992 0.855386i \(-0.673320\pi\)
−0.517992 + 0.855386i \(0.673320\pi\)
\(390\) 0 0
\(391\) −16481.8 −2.13177
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4098.84 −0.522114
\(396\) 0 0
\(397\) 11191.2 1.41478 0.707392 0.706822i \(-0.249872\pi\)
0.707392 + 0.706822i \(0.249872\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9357.36 1.16530 0.582649 0.812724i \(-0.302016\pi\)
0.582649 + 0.812724i \(0.302016\pi\)
\(402\) 0 0
\(403\) 15117.0 1.86857
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2304.31 0.280639
\(408\) 0 0
\(409\) −3229.59 −0.390447 −0.195224 0.980759i \(-0.562543\pi\)
−0.195224 + 0.980759i \(0.562543\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2148.73 0.256010
\(414\) 0 0
\(415\) −4796.96 −0.567406
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4305.10 0.501951 0.250976 0.967993i \(-0.419249\pi\)
0.250976 + 0.967993i \(0.419249\pi\)
\(420\) 0 0
\(421\) −4330.36 −0.501304 −0.250652 0.968077i \(-0.580645\pi\)
−0.250652 + 0.968077i \(0.580645\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2779.09 −0.317190
\(426\) 0 0
\(427\) −1809.82 −0.205113
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2774.39 0.310064 0.155032 0.987909i \(-0.450452\pi\)
0.155032 + 0.987909i \(0.450452\pi\)
\(432\) 0 0
\(433\) −16958.3 −1.88213 −0.941065 0.338225i \(-0.890173\pi\)
−0.941065 + 0.338225i \(0.890173\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12786.4 −1.39968
\(438\) 0 0
\(439\) 1334.39 0.145072 0.0725362 0.997366i \(-0.476891\pi\)
0.0725362 + 0.997366i \(0.476891\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1990.19 −0.213447 −0.106723 0.994289i \(-0.534036\pi\)
−0.106723 + 0.994289i \(0.534036\pi\)
\(444\) 0 0
\(445\) −2033.67 −0.216641
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5858.34 0.615751 0.307876 0.951427i \(-0.400382\pi\)
0.307876 + 0.951427i \(0.400382\pi\)
\(450\) 0 0
\(451\) −514.353 −0.0537027
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 857.088 0.0883097
\(456\) 0 0
\(457\) 1838.53 0.188190 0.0940952 0.995563i \(-0.470004\pi\)
0.0940952 + 0.995563i \(0.470004\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2805.80 −0.283469 −0.141735 0.989905i \(-0.545268\pi\)
−0.141735 + 0.989905i \(0.545268\pi\)
\(462\) 0 0
\(463\) −9647.34 −0.968359 −0.484179 0.874969i \(-0.660882\pi\)
−0.484179 + 0.874969i \(0.660882\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8339.39 0.826340 0.413170 0.910654i \(-0.364421\pi\)
0.413170 + 0.910654i \(0.364421\pi\)
\(468\) 0 0
\(469\) −1147.29 −0.112957
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2198.58 0.213723
\(474\) 0 0
\(475\) −2155.99 −0.208260
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2842.91 0.271181 0.135591 0.990765i \(-0.456707\pi\)
0.135591 + 0.990765i \(0.456707\pi\)
\(480\) 0 0
\(481\) −9480.68 −0.898715
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3118.74 −0.291989
\(486\) 0 0
\(487\) 55.8309 0.00519495 0.00259747 0.999997i \(-0.499173\pi\)
0.00259747 + 0.999997i \(0.499173\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10131.2 −0.931188 −0.465594 0.884998i \(-0.654159\pi\)
−0.465594 + 0.884998i \(0.654159\pi\)
\(492\) 0 0
\(493\) 7905.75 0.722225
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −141.218 −0.0127455
\(498\) 0 0
\(499\) −18760.6 −1.68305 −0.841524 0.540219i \(-0.818341\pi\)
−0.841524 + 0.540219i \(0.818341\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1263.21 −0.111975 −0.0559877 0.998431i \(-0.517831\pi\)
−0.0559877 + 0.998431i \(0.517831\pi\)
\(504\) 0 0
\(505\) −2341.01 −0.206284
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20105.1 1.75077 0.875386 0.483424i \(-0.160607\pi\)
0.875386 + 0.483424i \(0.160607\pi\)
\(510\) 0 0
\(511\) −2101.07 −0.181890
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4745.48 0.406041
\(516\) 0 0
\(517\) −1790.31 −0.152297
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1644.53 −0.138288 −0.0691440 0.997607i \(-0.522027\pi\)
−0.0691440 + 0.997607i \(0.522027\pi\)
\(522\) 0 0
\(523\) 4830.92 0.403903 0.201952 0.979395i \(-0.435272\pi\)
0.201952 + 0.979395i \(0.435272\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −23570.9 −1.94832
\(528\) 0 0
\(529\) 9815.89 0.806764
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2116.22 0.171977
\(534\) 0 0
\(535\) −3245.43 −0.262266
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5843.40 0.466963
\(540\) 0 0
\(541\) 159.541 0.0126787 0.00633936 0.999980i \(-0.497982\pi\)
0.00633936 + 0.999980i \(0.497982\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5762.18 −0.452890
\(546\) 0 0
\(547\) 3217.83 0.251526 0.125763 0.992060i \(-0.459862\pi\)
0.125763 + 0.992060i \(0.459862\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6133.21 0.474199
\(552\) 0 0
\(553\) −1971.03 −0.151567
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21394.0 −1.62745 −0.813727 0.581247i \(-0.802565\pi\)
−0.813727 + 0.581247i \(0.802565\pi\)
\(558\) 0 0
\(559\) −9045.71 −0.684423
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3427.87 −0.256603 −0.128302 0.991735i \(-0.540953\pi\)
−0.128302 + 0.991735i \(0.540953\pi\)
\(564\) 0 0
\(565\) 7688.77 0.572511
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12420.8 0.915128 0.457564 0.889177i \(-0.348722\pi\)
0.457564 + 0.889177i \(0.348722\pi\)
\(570\) 0 0
\(571\) −3954.37 −0.289816 −0.144908 0.989445i \(-0.546289\pi\)
−0.144908 + 0.989445i \(0.546289\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3706.66 0.268832
\(576\) 0 0
\(577\) −24982.7 −1.80250 −0.901250 0.433299i \(-0.857349\pi\)
−0.901250 + 0.433299i \(0.857349\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2306.74 −0.164715
\(582\) 0 0
\(583\) −5171.84 −0.367402
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7981.44 −0.561208 −0.280604 0.959824i \(-0.590535\pi\)
−0.280604 + 0.959824i \(0.590535\pi\)
\(588\) 0 0
\(589\) −18286.1 −1.27923
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2465.27 −0.170719 −0.0853597 0.996350i \(-0.527204\pi\)
−0.0853597 + 0.996350i \(0.527204\pi\)
\(594\) 0 0
\(595\) −1336.40 −0.0920788
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21413.6 1.46066 0.730331 0.683093i \(-0.239366\pi\)
0.730331 + 0.683093i \(0.239366\pi\)
\(600\) 0 0
\(601\) −307.813 −0.0208918 −0.0104459 0.999945i \(-0.503325\pi\)
−0.0104459 + 0.999945i \(0.503325\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5153.67 0.346325
\(606\) 0 0
\(607\) 3953.10 0.264335 0.132168 0.991227i \(-0.457806\pi\)
0.132168 + 0.991227i \(0.457806\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7365.92 0.487714
\(612\) 0 0
\(613\) −8356.90 −0.550623 −0.275311 0.961355i \(-0.588781\pi\)
−0.275311 + 0.961355i \(0.588781\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25301.3 −1.65088 −0.825439 0.564491i \(-0.809072\pi\)
−0.825439 + 0.564491i \(0.809072\pi\)
\(618\) 0 0
\(619\) 1542.86 0.100182 0.0500912 0.998745i \(-0.484049\pi\)
0.0500912 + 0.998745i \(0.484049\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −977.944 −0.0628900
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14782.5 0.937072
\(630\) 0 0
\(631\) 12888.9 0.813152 0.406576 0.913617i \(-0.366723\pi\)
0.406576 + 0.913617i \(0.366723\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9751.64 −0.609420
\(636\) 0 0
\(637\) −24041.7 −1.49539
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4590.77 0.282878 0.141439 0.989947i \(-0.454827\pi\)
0.141439 + 0.989947i \(0.454827\pi\)
\(642\) 0 0
\(643\) −12085.3 −0.741209 −0.370605 0.928791i \(-0.620850\pi\)
−0.370605 + 0.928791i \(0.620850\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6233.43 0.378766 0.189383 0.981903i \(-0.439351\pi\)
0.189383 + 0.981903i \(0.439351\pi\)
\(648\) 0 0
\(649\) 15485.8 0.936625
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24307.8 1.45672 0.728361 0.685194i \(-0.240282\pi\)
0.728361 + 0.685194i \(0.240282\pi\)
\(654\) 0 0
\(655\) −6096.58 −0.363684
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1445.94 −0.0854719 −0.0427359 0.999086i \(-0.513607\pi\)
−0.0427359 + 0.999086i \(0.513607\pi\)
\(660\) 0 0
\(661\) −11159.3 −0.656654 −0.328327 0.944564i \(-0.606485\pi\)
−0.328327 + 0.944564i \(0.606485\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1036.76 −0.0604571
\(666\) 0 0
\(667\) −10544.4 −0.612117
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −13043.3 −0.750418
\(672\) 0 0
\(673\) −4473.63 −0.256234 −0.128117 0.991759i \(-0.540893\pi\)
−0.128117 + 0.991759i \(0.540893\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19997.2 −1.13523 −0.567617 0.823292i \(-0.692135\pi\)
−0.567617 + 0.823292i \(0.692135\pi\)
\(678\) 0 0
\(679\) −1499.73 −0.0847632
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21261.5 1.19114 0.595571 0.803303i \(-0.296926\pi\)
0.595571 + 0.803303i \(0.296926\pi\)
\(684\) 0 0
\(685\) 13163.9 0.734256
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21278.7 1.17656
\(690\) 0 0
\(691\) 31303.9 1.72338 0.861691 0.507434i \(-0.169406\pi\)
0.861691 + 0.507434i \(0.169406\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4715.59 −0.257370
\(696\) 0 0
\(697\) −3299.67 −0.179317
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30288.1 −1.63190 −0.815952 0.578120i \(-0.803786\pi\)
−0.815952 + 0.578120i \(0.803786\pi\)
\(702\) 0 0
\(703\) 11468.2 0.615263
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1125.73 −0.0598835
\(708\) 0 0
\(709\) 26181.8 1.38685 0.693426 0.720528i \(-0.256101\pi\)
0.693426 + 0.720528i \(0.256101\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 31438.1 1.65128
\(714\) 0 0
\(715\) 6176.98 0.323085
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32762.6 1.69936 0.849679 0.527301i \(-0.176796\pi\)
0.849679 + 0.527301i \(0.176796\pi\)
\(720\) 0 0
\(721\) 2281.99 0.117872
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1777.95 −0.0910780
\(726\) 0 0
\(727\) 19439.8 0.991721 0.495861 0.868402i \(-0.334853\pi\)
0.495861 + 0.868402i \(0.334853\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14104.3 0.713634
\(732\) 0 0
\(733\) −16886.4 −0.850906 −0.425453 0.904981i \(-0.639885\pi\)
−0.425453 + 0.904981i \(0.639885\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8268.45 −0.413259
\(738\) 0 0
\(739\) 9048.46 0.450410 0.225205 0.974311i \(-0.427695\pi\)
0.225205 + 0.974311i \(0.427695\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22272.8 −1.09975 −0.549873 0.835249i \(-0.685324\pi\)
−0.549873 + 0.835249i \(0.685324\pi\)
\(744\) 0 0
\(745\) −4166.85 −0.204915
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1560.65 −0.0761346
\(750\) 0 0
\(751\) 2223.89 0.108057 0.0540284 0.998539i \(-0.482794\pi\)
0.0540284 + 0.998539i \(0.482794\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12065.7 −0.581608
\(756\) 0 0
\(757\) 21253.7 1.02045 0.510225 0.860041i \(-0.329562\pi\)
0.510225 + 0.860041i \(0.329562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36585.2 1.74272 0.871362 0.490640i \(-0.163237\pi\)
0.871362 + 0.490640i \(0.163237\pi\)
\(762\) 0 0
\(763\) −2770.89 −0.131472
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −63713.7 −2.99944
\(768\) 0 0
\(769\) 14952.5 0.701171 0.350585 0.936531i \(-0.385983\pi\)
0.350585 + 0.936531i \(0.385983\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −41504.0 −1.93117 −0.965586 0.260082i \(-0.916250\pi\)
−0.965586 + 0.260082i \(0.916250\pi\)
\(774\) 0 0
\(775\) 5300.94 0.245697
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2559.85 −0.117736
\(780\) 0 0
\(781\) −1017.75 −0.0466300
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5427.07 −0.246752
\(786\) 0 0
\(787\) −34867.0 −1.57926 −0.789628 0.613585i \(-0.789727\pi\)
−0.789628 + 0.613585i \(0.789727\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3697.34 0.166198
\(792\) 0 0
\(793\) 53664.4 2.40313
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14744.4 0.655299 0.327649 0.944799i \(-0.393743\pi\)
0.327649 + 0.944799i \(0.393743\pi\)
\(798\) 0 0
\(799\) −11485.1 −0.508529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −15142.3 −0.665454
\(804\) 0 0
\(805\) 1782.44 0.0780407
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 26179.5 1.13773 0.568864 0.822431i \(-0.307383\pi\)
0.568864 + 0.822431i \(0.307383\pi\)
\(810\) 0 0
\(811\) −1162.53 −0.0503354 −0.0251677 0.999683i \(-0.508012\pi\)
−0.0251677 + 0.999683i \(0.508012\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18289.4 −0.786075
\(816\) 0 0
\(817\) 10942.0 0.468558
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35989.7 1.52990 0.764951 0.644088i \(-0.222763\pi\)
0.764951 + 0.644088i \(0.222763\pi\)
\(822\) 0 0
\(823\) 36003.6 1.52492 0.762459 0.647036i \(-0.223992\pi\)
0.762459 + 0.647036i \(0.223992\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22096.7 −0.929114 −0.464557 0.885543i \(-0.653786\pi\)
−0.464557 + 0.885543i \(0.653786\pi\)
\(828\) 0 0
\(829\) 14.9350 0.000625710 0 0.000312855 1.00000i \(-0.499900\pi\)
0.000312855 1.00000i \(0.499900\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 37486.5 1.55922
\(834\) 0 0
\(835\) −7532.29 −0.312174
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −41995.0 −1.72804 −0.864022 0.503453i \(-0.832063\pi\)
−0.864022 + 0.503453i \(0.832063\pi\)
\(840\) 0 0
\(841\) −19331.2 −0.792620
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −14429.2 −0.587430
\(846\) 0 0
\(847\) 2478.27 0.100537
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −19716.5 −0.794209
\(852\) 0 0
\(853\) 9809.11 0.393737 0.196868 0.980430i \(-0.436923\pi\)
0.196868 + 0.980430i \(0.436923\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31063.8 1.23818 0.619089 0.785321i \(-0.287502\pi\)
0.619089 + 0.785321i \(0.287502\pi\)
\(858\) 0 0
\(859\) −5957.06 −0.236615 −0.118307 0.992977i \(-0.537747\pi\)
−0.118307 + 0.992977i \(0.537747\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31548.8 1.24442 0.622210 0.782850i \(-0.286235\pi\)
0.622210 + 0.782850i \(0.286235\pi\)
\(864\) 0 0
\(865\) −419.438 −0.0164871
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −14205.1 −0.554517
\(870\) 0 0
\(871\) 34019.2 1.32342
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 300.547 0.0116118
\(876\) 0 0
\(877\) 44007.7 1.69445 0.847226 0.531233i \(-0.178271\pi\)
0.847226 + 0.531233i \(0.178271\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8635.87 −0.330250 −0.165125 0.986273i \(-0.552803\pi\)
−0.165125 + 0.986273i \(0.552803\pi\)
\(882\) 0 0
\(883\) −25587.5 −0.975184 −0.487592 0.873072i \(-0.662125\pi\)
−0.487592 + 0.873072i \(0.662125\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22534.0 −0.853007 −0.426503 0.904486i \(-0.640255\pi\)
−0.426503 + 0.904486i \(0.640255\pi\)
\(888\) 0 0
\(889\) −4689.32 −0.176912
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8910.07 −0.333890
\(894\) 0 0
\(895\) −23494.3 −0.877460
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −15079.7 −0.559441
\(900\) 0 0
\(901\) −33178.3 −1.22678
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10804.7 0.396863
\(906\) 0 0
\(907\) 44792.3 1.63980 0.819902 0.572504i \(-0.194028\pi\)
0.819902 + 0.572504i \(0.194028\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 44378.6 1.61397 0.806986 0.590571i \(-0.201097\pi\)
0.806986 + 0.590571i \(0.201097\pi\)
\(912\) 0 0
\(913\) −16624.5 −0.602620
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2931.69 −0.105576
\(918\) 0 0
\(919\) −19542.4 −0.701462 −0.350731 0.936476i \(-0.614067\pi\)
−0.350731 + 0.936476i \(0.614067\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4187.37 0.149327
\(924\) 0 0
\(925\) −3324.50 −0.118172
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23231.5 0.820453 0.410227 0.911984i \(-0.365450\pi\)
0.410227 + 0.911984i \(0.365450\pi\)
\(930\) 0 0
\(931\) 29081.7 1.02375
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9631.32 −0.336875
\(936\) 0 0
\(937\) −14026.4 −0.489032 −0.244516 0.969645i \(-0.578629\pi\)
−0.244516 + 0.969645i \(0.578629\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 986.880 0.0341885 0.0170943 0.999854i \(-0.494558\pi\)
0.0170943 + 0.999854i \(0.494558\pi\)
\(942\) 0 0
\(943\) 4400.99 0.151979
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34679.9 −1.19002 −0.595008 0.803720i \(-0.702851\pi\)
−0.595008 + 0.803720i \(0.702851\pi\)
\(948\) 0 0
\(949\) 62300.4 2.13104
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 52384.1 1.78058 0.890288 0.455398i \(-0.150503\pi\)
0.890288 + 0.455398i \(0.150503\pi\)
\(954\) 0 0
\(955\) −803.792 −0.0272357
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6330.18 0.213151
\(960\) 0 0
\(961\) 15169.0 0.509181
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18718.9 −0.624437
\(966\) 0 0
\(967\) 29842.4 0.992417 0.496208 0.868203i \(-0.334725\pi\)
0.496208 + 0.868203i \(0.334725\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21134.5 −0.698494 −0.349247 0.937031i \(-0.613563\pi\)
−0.349247 + 0.937031i \(0.613563\pi\)
\(972\) 0 0
\(973\) −2267.61 −0.0747135
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11583.6 0.379318 0.189659 0.981850i \(-0.439262\pi\)
0.189659 + 0.981850i \(0.439262\pi\)
\(978\) 0 0
\(979\) −7047.98 −0.230086
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 54382.8 1.76454 0.882269 0.470745i \(-0.156015\pi\)
0.882269 + 0.470745i \(0.156015\pi\)
\(984\) 0 0
\(985\) 19062.4 0.616628
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18811.9 −0.604836
\(990\) 0 0
\(991\) 8319.58 0.266680 0.133340 0.991070i \(-0.457430\pi\)
0.133340 + 0.991070i \(0.457430\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 260.982 0.00831527
\(996\) 0 0
\(997\) 33697.8 1.07043 0.535216 0.844715i \(-0.320230\pi\)
0.535216 + 0.844715i \(0.320230\pi\)
\(998\) 0 0
\(999\) 0 0
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 1080.4.a.o.1.2 4
3.2 odd 2 1080.4.a.p.1.2 yes 4
4.3 odd 2 2160.4.a.bu.1.3 4
12.11 even 2 2160.4.a.bv.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.2 4 1.1 even 1 trivial
1080.4.a.p.1.2 yes 4 3.2 odd 2
2160.4.a.bu.1.3 4 4.3 odd 2
2160.4.a.bv.1.3 4 12.11 even 2