Properties

Label 1080.4.a.o.1.1
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 1.1
Root \(7.09129\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} -30.4893 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} -30.4893 q^{7} +37.9574 q^{11} -43.3934 q^{13} +29.8404 q^{17} -28.6063 q^{19} -51.2553 q^{23} +25.0000 q^{25} -178.457 q^{29} -237.595 q^{31} +152.446 q^{35} -12.7129 q^{37} -222.776 q^{41} +464.063 q^{43} -400.138 q^{47} +586.595 q^{49} -249.541 q^{53} -189.787 q^{55} +779.434 q^{59} +609.977 q^{61} +216.967 q^{65} -172.872 q^{67} +1044.33 q^{71} -38.6474 q^{73} -1157.29 q^{77} +1194.75 q^{79} -150.860 q^{83} -149.202 q^{85} -119.500 q^{89} +1323.03 q^{91} +143.031 q^{95} +1266.01 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} + 14 q^{7}+O(q^{10}) \) 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
\(7\) −30.4893 −1.64627 −0.823133 0.567849i \(-0.807776\pi\)
−0.823133 + 0.567849i \(0.807776\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 37.9574 1.04042 0.520209 0.854039i \(-0.325854\pi\)
0.520209 + 0.854039i \(0.325854\pi\)
\(12\) 0 0
\(13\) −43.3934 −0.925781 −0.462891 0.886415i \(-0.653188\pi\)
−0.462891 + 0.886415i \(0.653188\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 29.8404 0.425727 0.212864 0.977082i \(-0.431721\pi\)
0.212864 + 0.977082i \(0.431721\pi\)
\(18\) 0 0
\(19\) −28.6063 −0.345407 −0.172703 0.984974i \(-0.555250\pi\)
−0.172703 + 0.984974i \(0.555250\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −51.2553 −0.464672 −0.232336 0.972636i \(-0.574637\pi\)
−0.232336 + 0.972636i \(0.574637\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −178.457 −1.14271 −0.571357 0.820702i \(-0.693583\pi\)
−0.571357 + 0.820702i \(0.693583\pi\)
\(30\) 0 0
\(31\) −237.595 −1.37656 −0.688280 0.725445i \(-0.741634\pi\)
−0.688280 + 0.725445i \(0.741634\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 152.446 0.736232
\(36\) 0 0
\(37\) −12.7129 −0.0564860 −0.0282430 0.999601i \(-0.508991\pi\)
−0.0282430 + 0.999601i \(0.508991\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −222.776 −0.848581 −0.424291 0.905526i \(-0.639476\pi\)
−0.424291 + 0.905526i \(0.639476\pi\)
\(42\) 0 0
\(43\) 464.063 1.64579 0.822894 0.568194i \(-0.192358\pi\)
0.822894 + 0.568194i \(0.192358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −400.138 −1.24183 −0.620916 0.783877i \(-0.713239\pi\)
−0.620916 + 0.783877i \(0.713239\pi\)
\(48\) 0 0
\(49\) 586.595 1.71019
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −249.541 −0.646738 −0.323369 0.946273i \(-0.604815\pi\)
−0.323369 + 0.946273i \(0.604815\pi\)
\(54\) 0 0
\(55\) −189.787 −0.465289
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 779.434 1.71989 0.859946 0.510385i \(-0.170497\pi\)
0.859946 + 0.510385i \(0.170497\pi\)
\(60\) 0 0
\(61\) 609.977 1.28032 0.640161 0.768241i \(-0.278868\pi\)
0.640161 + 0.768241i \(0.278868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 216.967 0.414022
\(66\) 0 0
\(67\) −172.872 −0.315218 −0.157609 0.987502i \(-0.550379\pi\)
−0.157609 + 0.987502i \(0.550379\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1044.33 1.74562 0.872809 0.488062i \(-0.162296\pi\)
0.872809 + 0.488062i \(0.162296\pi\)
\(72\) 0 0
\(73\) −38.6474 −0.0619635 −0.0309818 0.999520i \(-0.509863\pi\)
−0.0309818 + 0.999520i \(0.509863\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1157.29 −1.71280
\(78\) 0 0
\(79\) 1194.75 1.70152 0.850761 0.525553i \(-0.176141\pi\)
0.850761 + 0.525553i \(0.176141\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −150.860 −0.199507 −0.0997535 0.995012i \(-0.531805\pi\)
−0.0997535 + 0.995012i \(0.531805\pi\)
\(84\) 0 0
\(85\) −149.202 −0.190391
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −119.500 −0.142326 −0.0711628 0.997465i \(-0.522671\pi\)
−0.0711628 + 0.997465i \(0.522671\pi\)
\(90\) 0 0
\(91\) 1323.03 1.52408
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 143.031 0.154471
\(96\) 0 0
\(97\) 1266.01 1.32519 0.662596 0.748977i \(-0.269455\pi\)
0.662596 + 0.748977i \(0.269455\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1762.34 1.73623 0.868114 0.496365i \(-0.165332\pi\)
0.868114 + 0.496365i \(0.165332\pi\)
\(102\) 0 0
\(103\) 746.636 0.714255 0.357127 0.934056i \(-0.383756\pi\)
0.357127 + 0.934056i \(0.383756\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −351.360 −0.317451 −0.158726 0.987323i \(-0.550739\pi\)
−0.158726 + 0.987323i \(0.550739\pi\)
\(108\) 0 0
\(109\) −755.508 −0.663895 −0.331947 0.943298i \(-0.607706\pi\)
−0.331947 + 0.943298i \(0.607706\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 787.742 0.655792 0.327896 0.944714i \(-0.393660\pi\)
0.327896 + 0.944714i \(0.393660\pi\)
\(114\) 0 0
\(115\) 256.276 0.207808
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −909.812 −0.700860
\(120\) 0 0
\(121\) 109.765 0.0824684
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1415.65 −0.989125 −0.494563 0.869142i \(-0.664672\pi\)
−0.494563 + 0.869142i \(0.664672\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 882.082 0.588304 0.294152 0.955759i \(-0.404963\pi\)
0.294152 + 0.955759i \(0.404963\pi\)
\(132\) 0 0
\(133\) 872.184 0.568631
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −503.449 −0.313960 −0.156980 0.987602i \(-0.550176\pi\)
−0.156980 + 0.987602i \(0.550176\pi\)
\(138\) 0 0
\(139\) 587.011 0.358199 0.179099 0.983831i \(-0.442682\pi\)
0.179099 + 0.983831i \(0.442682\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1647.10 −0.963199
\(144\) 0 0
\(145\) 892.287 0.511037
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2927.36 −1.60952 −0.804760 0.593600i \(-0.797706\pi\)
−0.804760 + 0.593600i \(0.797706\pi\)
\(150\) 0 0
\(151\) −841.091 −0.453291 −0.226646 0.973977i \(-0.572776\pi\)
−0.226646 + 0.973977i \(0.572776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1187.98 0.615617
\(156\) 0 0
\(157\) 1763.63 0.896518 0.448259 0.893904i \(-0.352044\pi\)
0.448259 + 0.893904i \(0.352044\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1562.74 0.764974
\(162\) 0 0
\(163\) 2522.29 1.21203 0.606016 0.795452i \(-0.292767\pi\)
0.606016 + 0.795452i \(0.292767\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2527.79 −1.17129 −0.585647 0.810566i \(-0.699159\pi\)
−0.585647 + 0.810566i \(0.699159\pi\)
\(168\) 0 0
\(169\) −314.014 −0.142929
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 389.805 0.171308 0.0856541 0.996325i \(-0.472702\pi\)
0.0856541 + 0.996325i \(0.472702\pi\)
\(174\) 0 0
\(175\) −762.232 −0.329253
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2305.67 0.962760 0.481380 0.876512i \(-0.340136\pi\)
0.481380 + 0.876512i \(0.340136\pi\)
\(180\) 0 0
\(181\) 2644.87 1.08614 0.543071 0.839687i \(-0.317262\pi\)
0.543071 + 0.839687i \(0.317262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 63.5643 0.0252613
\(186\) 0 0
\(187\) 1132.67 0.442934
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1152.77 −0.436711 −0.218356 0.975869i \(-0.570069\pi\)
−0.218356 + 0.975869i \(0.570069\pi\)
\(192\) 0 0
\(193\) −2033.47 −0.758406 −0.379203 0.925313i \(-0.623802\pi\)
−0.379203 + 0.925313i \(0.623802\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1701.78 −0.615466 −0.307733 0.951473i \(-0.599570\pi\)
−0.307733 + 0.951473i \(0.599570\pi\)
\(198\) 0 0
\(199\) 5087.64 1.81233 0.906163 0.422928i \(-0.138998\pi\)
0.906163 + 0.422928i \(0.138998\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5441.04 1.88121
\(204\) 0 0
\(205\) 1113.88 0.379497
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1085.82 −0.359367
\(210\) 0 0
\(211\) 128.178 0.0418206 0.0209103 0.999781i \(-0.493344\pi\)
0.0209103 + 0.999781i \(0.493344\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2320.31 −0.736019
\(216\) 0 0
\(217\) 7244.11 2.26618
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1294.88 −0.394130
\(222\) 0 0
\(223\) 2803.55 0.841882 0.420941 0.907088i \(-0.361700\pi\)
0.420941 + 0.907088i \(0.361700\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2485.68 0.726786 0.363393 0.931636i \(-0.381618\pi\)
0.363393 + 0.931636i \(0.381618\pi\)
\(228\) 0 0
\(229\) −1733.54 −0.500241 −0.250121 0.968215i \(-0.580470\pi\)
−0.250121 + 0.968215i \(0.580470\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4163.48 −1.17064 −0.585319 0.810803i \(-0.699031\pi\)
−0.585319 + 0.810803i \(0.699031\pi\)
\(234\) 0 0
\(235\) 2000.69 0.555364
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5658.52 1.53146 0.765730 0.643162i \(-0.222378\pi\)
0.765730 + 0.643162i \(0.222378\pi\)
\(240\) 0 0
\(241\) 3016.40 0.806239 0.403119 0.915147i \(-0.367926\pi\)
0.403119 + 0.915147i \(0.367926\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2932.98 −0.764821
\(246\) 0 0
\(247\) 1241.32 0.319771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6422.21 1.61501 0.807503 0.589863i \(-0.200818\pi\)
0.807503 + 0.589863i \(0.200818\pi\)
\(252\) 0 0
\(253\) −1945.52 −0.483453
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6124.07 −1.48642 −0.743208 0.669060i \(-0.766697\pi\)
−0.743208 + 0.669060i \(0.766697\pi\)
\(258\) 0 0
\(259\) 387.606 0.0929910
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3487.57 0.817692 0.408846 0.912603i \(-0.365931\pi\)
0.408846 + 0.912603i \(0.365931\pi\)
\(264\) 0 0
\(265\) 1247.71 0.289230
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2681.05 −0.607683 −0.303841 0.952723i \(-0.598269\pi\)
−0.303841 + 0.952723i \(0.598269\pi\)
\(270\) 0 0
\(271\) 6504.70 1.45805 0.729027 0.684485i \(-0.239973\pi\)
0.729027 + 0.684485i \(0.239973\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 948.935 0.208083
\(276\) 0 0
\(277\) 2919.77 0.633329 0.316664 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8232.92 −1.74781 −0.873905 0.486097i \(-0.838420\pi\)
−0.873905 + 0.486097i \(0.838420\pi\)
\(282\) 0 0
\(283\) 579.928 0.121813 0.0609066 0.998143i \(-0.480601\pi\)
0.0609066 + 0.998143i \(0.480601\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6792.29 1.39699
\(288\) 0 0
\(289\) −4022.55 −0.818756
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4788.47 0.954763 0.477382 0.878696i \(-0.341586\pi\)
0.477382 + 0.878696i \(0.341586\pi\)
\(294\) 0 0
\(295\) −3897.17 −0.769159
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2224.14 0.430185
\(300\) 0 0
\(301\) −14148.9 −2.70941
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3049.89 −0.572577
\(306\) 0 0
\(307\) −8910.36 −1.65649 −0.828243 0.560369i \(-0.810659\pi\)
−0.828243 + 0.560369i \(0.810659\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9512.00 −1.73433 −0.867164 0.498023i \(-0.834060\pi\)
−0.867164 + 0.498023i \(0.834060\pi\)
\(312\) 0 0
\(313\) −667.152 −0.120478 −0.0602391 0.998184i \(-0.519186\pi\)
−0.0602391 + 0.998184i \(0.519186\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4543.67 −0.805041 −0.402520 0.915411i \(-0.631866\pi\)
−0.402520 + 0.915411i \(0.631866\pi\)
\(318\) 0 0
\(319\) −6773.78 −1.18890
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −853.623 −0.147049
\(324\) 0 0
\(325\) −1084.83 −0.185156
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12199.9 2.04438
\(330\) 0 0
\(331\) 473.632 0.0786500 0.0393250 0.999226i \(-0.487479\pi\)
0.0393250 + 0.999226i \(0.487479\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 864.358 0.140970
\(336\) 0 0
\(337\) −968.065 −0.156480 −0.0782401 0.996935i \(-0.524930\pi\)
−0.0782401 + 0.996935i \(0.524930\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −9018.50 −1.43220
\(342\) 0 0
\(343\) −7427.05 −1.16916
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −646.162 −0.0999648 −0.0499824 0.998750i \(-0.515917\pi\)
−0.0499824 + 0.998750i \(0.515917\pi\)
\(348\) 0 0
\(349\) 8806.56 1.35073 0.675365 0.737484i \(-0.263986\pi\)
0.675365 + 0.737484i \(0.263986\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −237.161 −0.0357586 −0.0178793 0.999840i \(-0.505691\pi\)
−0.0178793 + 0.999840i \(0.505691\pi\)
\(354\) 0 0
\(355\) −5221.64 −0.780664
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11440.6 1.68192 0.840962 0.541094i \(-0.181990\pi\)
0.840962 + 0.541094i \(0.181990\pi\)
\(360\) 0 0
\(361\) −6040.68 −0.880694
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 193.237 0.0277109
\(366\) 0 0
\(367\) 4164.30 0.592301 0.296151 0.955141i \(-0.404297\pi\)
0.296151 + 0.955141i \(0.404297\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7608.32 1.06470
\(372\) 0 0
\(373\) −2502.13 −0.347333 −0.173667 0.984804i \(-0.555562\pi\)
−0.173667 + 0.984804i \(0.555562\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7743.87 1.05790
\(378\) 0 0
\(379\) −3861.08 −0.523300 −0.261650 0.965163i \(-0.584267\pi\)
−0.261650 + 0.965163i \(0.584267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5093.32 0.679521 0.339760 0.940512i \(-0.389654\pi\)
0.339760 + 0.940512i \(0.389654\pi\)
\(384\) 0 0
\(385\) 5786.47 0.765989
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13388.1 1.74500 0.872500 0.488614i \(-0.162497\pi\)
0.872500 + 0.488614i \(0.162497\pi\)
\(390\) 0 0
\(391\) −1529.48 −0.197824
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5973.77 −0.760944
\(396\) 0 0
\(397\) −10809.2 −1.36650 −0.683249 0.730185i \(-0.739434\pi\)
−0.683249 + 0.730185i \(0.739434\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7927.12 −0.987185 −0.493593 0.869693i \(-0.664317\pi\)
−0.493593 + 0.869693i \(0.664317\pi\)
\(402\) 0 0
\(403\) 10310.1 1.27439
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −482.548 −0.0587690
\(408\) 0 0
\(409\) −5186.81 −0.627069 −0.313535 0.949577i \(-0.601513\pi\)
−0.313535 + 0.949577i \(0.601513\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −23764.4 −2.83140
\(414\) 0 0
\(415\) 754.302 0.0892223
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8835.54 −1.03018 −0.515089 0.857137i \(-0.672241\pi\)
−0.515089 + 0.857137i \(0.672241\pi\)
\(420\) 0 0
\(421\) 12713.5 1.47177 0.735886 0.677105i \(-0.236766\pi\)
0.735886 + 0.677105i \(0.236766\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 746.010 0.0851455
\(426\) 0 0
\(427\) −18597.8 −2.10775
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4171.84 0.466242 0.233121 0.972448i \(-0.425106\pi\)
0.233121 + 0.972448i \(0.425106\pi\)
\(432\) 0 0
\(433\) 13112.5 1.45531 0.727653 0.685946i \(-0.240611\pi\)
0.727653 + 0.685946i \(0.240611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1466.22 0.160501
\(438\) 0 0
\(439\) 9951.32 1.08189 0.540946 0.841057i \(-0.318066\pi\)
0.540946 + 0.841057i \(0.318066\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 724.845 0.0777391 0.0388696 0.999244i \(-0.487624\pi\)
0.0388696 + 0.999244i \(0.487624\pi\)
\(444\) 0 0
\(445\) 597.500 0.0636500
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1392.90 0.146403 0.0732015 0.997317i \(-0.476678\pi\)
0.0732015 + 0.997317i \(0.476678\pi\)
\(450\) 0 0
\(451\) −8456.02 −0.882879
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6615.16 −0.681590
\(456\) 0 0
\(457\) −8721.35 −0.892708 −0.446354 0.894856i \(-0.647278\pi\)
−0.446354 + 0.894856i \(0.647278\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −927.718 −0.0937269 −0.0468635 0.998901i \(-0.514923\pi\)
−0.0468635 + 0.998901i \(0.514923\pi\)
\(462\) 0 0
\(463\) −17034.5 −1.70985 −0.854926 0.518749i \(-0.826398\pi\)
−0.854926 + 0.518749i \(0.826398\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13675.8 −1.35512 −0.677561 0.735467i \(-0.736963\pi\)
−0.677561 + 0.735467i \(0.736963\pi\)
\(468\) 0 0
\(469\) 5270.73 0.518933
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17614.6 1.71231
\(474\) 0 0
\(475\) −715.157 −0.0690813
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6785.36 0.647247 0.323623 0.946186i \(-0.395099\pi\)
0.323623 + 0.946186i \(0.395099\pi\)
\(480\) 0 0
\(481\) 551.654 0.0522937
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6330.04 −0.592644
\(486\) 0 0
\(487\) 6642.50 0.618070 0.309035 0.951051i \(-0.399994\pi\)
0.309035 + 0.951051i \(0.399994\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −11103.2 −1.02053 −0.510265 0.860017i \(-0.670453\pi\)
−0.510265 + 0.860017i \(0.670453\pi\)
\(492\) 0 0
\(493\) −5325.24 −0.486485
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −31840.8 −2.87375
\(498\) 0 0
\(499\) −8157.07 −0.731785 −0.365892 0.930657i \(-0.619236\pi\)
−0.365892 + 0.930657i \(0.619236\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21298.6 1.88799 0.943994 0.329962i \(-0.107036\pi\)
0.943994 + 0.329962i \(0.107036\pi\)
\(504\) 0 0
\(505\) −8811.68 −0.776465
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4303.19 −0.374726 −0.187363 0.982291i \(-0.559994\pi\)
−0.187363 + 0.982291i \(0.559994\pi\)
\(510\) 0 0
\(511\) 1178.33 0.102008
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3733.18 −0.319424
\(516\) 0 0
\(517\) −15188.2 −1.29202
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17772.1 1.49445 0.747225 0.664572i \(-0.231386\pi\)
0.747225 + 0.664572i \(0.231386\pi\)
\(522\) 0 0
\(523\) 1629.44 0.136234 0.0681170 0.997677i \(-0.478301\pi\)
0.0681170 + 0.997677i \(0.478301\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7089.94 −0.586039
\(528\) 0 0
\(529\) −9539.90 −0.784080
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9667.02 0.785601
\(534\) 0 0
\(535\) 1756.80 0.141968
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22265.6 1.77931
\(540\) 0 0
\(541\) −15199.2 −1.20788 −0.603940 0.797030i \(-0.706403\pi\)
−0.603940 + 0.797030i \(0.706403\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3777.54 0.296903
\(546\) 0 0
\(547\) 22585.0 1.76539 0.882693 0.469950i \(-0.155728\pi\)
0.882693 + 0.469950i \(0.155728\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5105.00 0.394701
\(552\) 0 0
\(553\) −36427.2 −2.80116
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24374.9 −1.85421 −0.927107 0.374798i \(-0.877712\pi\)
−0.927107 + 0.374798i \(0.877712\pi\)
\(558\) 0 0
\(559\) −20137.3 −1.52364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2009.16 −0.150401 −0.0752006 0.997168i \(-0.523960\pi\)
−0.0752006 + 0.997168i \(0.523960\pi\)
\(564\) 0 0
\(565\) −3938.71 −0.293279
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2085.35 0.153642 0.0768210 0.997045i \(-0.475523\pi\)
0.0768210 + 0.997045i \(0.475523\pi\)
\(570\) 0 0
\(571\) −15348.8 −1.12492 −0.562459 0.826825i \(-0.690145\pi\)
−0.562459 + 0.826825i \(0.690145\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1281.38 −0.0929344
\(576\) 0 0
\(577\) 10894.2 0.786020 0.393010 0.919534i \(-0.371434\pi\)
0.393010 + 0.919534i \(0.371434\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4599.62 0.328442
\(582\) 0 0
\(583\) −9471.93 −0.672877
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16008.1 −1.12560 −0.562800 0.826593i \(-0.690276\pi\)
−0.562800 + 0.826593i \(0.690276\pi\)
\(588\) 0 0
\(589\) 6796.72 0.475473
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9103.71 0.630429 0.315215 0.949020i \(-0.397923\pi\)
0.315215 + 0.949020i \(0.397923\pi\)
\(594\) 0 0
\(595\) 4549.06 0.313434
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4319.26 0.294624 0.147312 0.989090i \(-0.452938\pi\)
0.147312 + 0.989090i \(0.452938\pi\)
\(600\) 0 0
\(601\) −376.731 −0.0255693 −0.0127847 0.999918i \(-0.504070\pi\)
−0.0127847 + 0.999918i \(0.504070\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −548.827 −0.0368810
\(606\) 0 0
\(607\) −15923.7 −1.06478 −0.532390 0.846499i \(-0.678706\pi\)
−0.532390 + 0.846499i \(0.678706\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17363.3 1.14966
\(612\) 0 0
\(613\) −2458.04 −0.161957 −0.0809784 0.996716i \(-0.525804\pi\)
−0.0809784 + 0.996716i \(0.525804\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9810.85 0.640146 0.320073 0.947393i \(-0.396293\pi\)
0.320073 + 0.947393i \(0.396293\pi\)
\(618\) 0 0
\(619\) −10185.4 −0.661368 −0.330684 0.943742i \(-0.607279\pi\)
−0.330684 + 0.943742i \(0.607279\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3643.47 0.234306
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −379.357 −0.0240476
\(630\) 0 0
\(631\) 24370.2 1.53750 0.768750 0.639549i \(-0.220879\pi\)
0.768750 + 0.639549i \(0.220879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7078.27 0.442350
\(636\) 0 0
\(637\) −25454.4 −1.58326
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26828.4 1.65313 0.826566 0.562840i \(-0.190291\pi\)
0.826566 + 0.562840i \(0.190291\pi\)
\(642\) 0 0
\(643\) 28928.0 1.77420 0.887100 0.461577i \(-0.152716\pi\)
0.887100 + 0.461577i \(0.152716\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5640.22 0.342720 0.171360 0.985208i \(-0.445184\pi\)
0.171360 + 0.985208i \(0.445184\pi\)
\(648\) 0 0
\(649\) 29585.3 1.78941
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18973.7 −1.13705 −0.568527 0.822664i \(-0.692487\pi\)
−0.568527 + 0.822664i \(0.692487\pi\)
\(654\) 0 0
\(655\) −4410.41 −0.263098
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12979.7 −0.767249 −0.383625 0.923489i \(-0.625324\pi\)
−0.383625 + 0.923489i \(0.625324\pi\)
\(660\) 0 0
\(661\) −5073.66 −0.298551 −0.149276 0.988796i \(-0.547694\pi\)
−0.149276 + 0.988796i \(0.547694\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4360.92 −0.254300
\(666\) 0 0
\(667\) 9146.88 0.530987
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 23153.2 1.33207
\(672\) 0 0
\(673\) 6222.04 0.356377 0.178189 0.983996i \(-0.442976\pi\)
0.178189 + 0.983996i \(0.442976\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24566.3 −1.39462 −0.697311 0.716769i \(-0.745620\pi\)
−0.697311 + 0.716769i \(0.745620\pi\)
\(678\) 0 0
\(679\) −38599.7 −2.18162
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5341.08 0.299225 0.149613 0.988745i \(-0.452197\pi\)
0.149613 + 0.988745i \(0.452197\pi\)
\(684\) 0 0
\(685\) 2517.25 0.140407
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10828.4 0.598738
\(690\) 0 0
\(691\) −31206.7 −1.71803 −0.859016 0.511948i \(-0.828924\pi\)
−0.859016 + 0.511948i \(0.828924\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2935.06 −0.160191
\(696\) 0 0
\(697\) −6647.74 −0.361264
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26450.4 −1.42513 −0.712567 0.701604i \(-0.752467\pi\)
−0.712567 + 0.701604i \(0.752467\pi\)
\(702\) 0 0
\(703\) 363.668 0.0195107
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −53732.3 −2.85829
\(708\) 0 0
\(709\) 9284.18 0.491784 0.245892 0.969297i \(-0.420919\pi\)
0.245892 + 0.969297i \(0.420919\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12178.0 0.639649
\(714\) 0 0
\(715\) 8235.50 0.430756
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2340.29 0.121388 0.0606941 0.998156i \(-0.480669\pi\)
0.0606941 + 0.998156i \(0.480669\pi\)
\(720\) 0 0
\(721\) −22764.4 −1.17585
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4461.44 −0.228543
\(726\) 0 0
\(727\) 38028.0 1.94000 0.970000 0.243104i \(-0.0781657\pi\)
0.970000 + 0.243104i \(0.0781657\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 13847.8 0.700657
\(732\) 0 0
\(733\) 19036.1 0.959229 0.479614 0.877479i \(-0.340777\pi\)
0.479614 + 0.877479i \(0.340777\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6561.76 −0.327959
\(738\) 0 0
\(739\) −14047.6 −0.699253 −0.349626 0.936889i \(-0.613691\pi\)
−0.349626 + 0.936889i \(0.613691\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1692.66 0.0835768 0.0417884 0.999126i \(-0.486694\pi\)
0.0417884 + 0.999126i \(0.486694\pi\)
\(744\) 0 0
\(745\) 14636.8 0.719800
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10712.7 0.522609
\(750\) 0 0
\(751\) 34968.0 1.69907 0.849535 0.527533i \(-0.176883\pi\)
0.849535 + 0.527533i \(0.176883\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4205.45 0.202718
\(756\) 0 0
\(757\) 37589.3 1.80477 0.902383 0.430936i \(-0.141816\pi\)
0.902383 + 0.430936i \(0.141816\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32525.0 1.54932 0.774659 0.632380i \(-0.217922\pi\)
0.774659 + 0.632380i \(0.217922\pi\)
\(762\) 0 0
\(763\) 23034.9 1.09295
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33822.3 −1.59224
\(768\) 0 0
\(769\) 10695.3 0.501537 0.250768 0.968047i \(-0.419317\pi\)
0.250768 + 0.968047i \(0.419317\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18685.3 0.869421 0.434711 0.900570i \(-0.356851\pi\)
0.434711 + 0.900570i \(0.356851\pi\)
\(774\) 0 0
\(775\) −5939.88 −0.275312
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6372.80 0.293106
\(780\) 0 0
\(781\) 39640.0 1.81617
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8818.17 −0.400935
\(786\) 0 0
\(787\) 33119.1 1.50009 0.750044 0.661388i \(-0.230032\pi\)
0.750044 + 0.661388i \(0.230032\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24017.7 −1.07961
\(792\) 0 0
\(793\) −26469.0 −1.18530
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34755.1 −1.54465 −0.772327 0.635225i \(-0.780907\pi\)
−0.772327 + 0.635225i \(0.780907\pi\)
\(798\) 0 0
\(799\) −11940.3 −0.528681
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1466.96 −0.0644679
\(804\) 0 0
\(805\) −7813.68 −0.342107
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16938.8 0.736137 0.368069 0.929799i \(-0.380019\pi\)
0.368069 + 0.929799i \(0.380019\pi\)
\(810\) 0 0
\(811\) 5733.26 0.248239 0.124120 0.992267i \(-0.460389\pi\)
0.124120 + 0.992267i \(0.460389\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12611.5 −0.542037
\(816\) 0 0
\(817\) −13275.1 −0.568466
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30811.9 1.30980 0.654899 0.755717i \(-0.272711\pi\)
0.654899 + 0.755717i \(0.272711\pi\)
\(822\) 0 0
\(823\) −15591.8 −0.660384 −0.330192 0.943914i \(-0.607114\pi\)
−0.330192 + 0.943914i \(0.607114\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22010.3 −0.925480 −0.462740 0.886494i \(-0.653134\pi\)
−0.462740 + 0.886494i \(0.653134\pi\)
\(828\) 0 0
\(829\) 43144.8 1.80757 0.903787 0.427982i \(-0.140775\pi\)
0.903787 + 0.427982i \(0.140775\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 17504.3 0.728075
\(834\) 0 0
\(835\) 12638.9 0.523819
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20108.6 0.827444 0.413722 0.910403i \(-0.364228\pi\)
0.413722 + 0.910403i \(0.364228\pi\)
\(840\) 0 0
\(841\) 7458.04 0.305795
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1570.07 0.0639196
\(846\) 0 0
\(847\) −3346.67 −0.135765
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 651.601 0.0262475
\(852\) 0 0
\(853\) −1628.47 −0.0653666 −0.0326833 0.999466i \(-0.510405\pi\)
−0.0326833 + 0.999466i \(0.510405\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16965.0 0.676212 0.338106 0.941108i \(-0.390214\pi\)
0.338106 + 0.941108i \(0.390214\pi\)
\(858\) 0 0
\(859\) −7815.73 −0.310442 −0.155221 0.987880i \(-0.549609\pi\)
−0.155221 + 0.987880i \(0.549609\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23398.8 −0.922948 −0.461474 0.887154i \(-0.652679\pi\)
−0.461474 + 0.887154i \(0.652679\pi\)
\(864\) 0 0
\(865\) −1949.03 −0.0766114
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 45349.7 1.77029
\(870\) 0 0
\(871\) 7501.49 0.291823
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3811.16 0.147246
\(876\) 0 0
\(877\) −2187.35 −0.0842209 −0.0421104 0.999113i \(-0.513408\pi\)
−0.0421104 + 0.999113i \(0.513408\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −32719.9 −1.25126 −0.625631 0.780119i \(-0.715158\pi\)
−0.625631 + 0.780119i \(0.715158\pi\)
\(882\) 0 0
\(883\) 30656.1 1.16836 0.584178 0.811625i \(-0.301417\pi\)
0.584178 + 0.811625i \(0.301417\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3922.07 −0.148467 −0.0742335 0.997241i \(-0.523651\pi\)
−0.0742335 + 0.997241i \(0.523651\pi\)
\(888\) 0 0
\(889\) 43162.2 1.62836
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11446.4 0.428937
\(894\) 0 0
\(895\) −11528.4 −0.430559
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 42400.6 1.57302
\(900\) 0 0
\(901\) −7446.41 −0.275334
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13224.3 −0.485737
\(906\) 0 0
\(907\) −9849.07 −0.360566 −0.180283 0.983615i \(-0.557701\pi\)
−0.180283 + 0.983615i \(0.557701\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38738.8 −1.40886 −0.704431 0.709773i \(-0.748798\pi\)
−0.704431 + 0.709773i \(0.748798\pi\)
\(912\) 0 0
\(913\) −5726.27 −0.207571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −26894.0 −0.968505
\(918\) 0 0
\(919\) 13655.6 0.490161 0.245080 0.969503i \(-0.421186\pi\)
0.245080 + 0.969503i \(0.421186\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −45316.9 −1.61606
\(924\) 0 0
\(925\) −317.822 −0.0112972
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39109.3 1.38120 0.690600 0.723237i \(-0.257347\pi\)
0.690600 + 0.723237i \(0.257347\pi\)
\(930\) 0 0
\(931\) −16780.3 −0.590711
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5663.33 −0.198086
\(936\) 0 0
\(937\) −20164.6 −0.703041 −0.351521 0.936180i \(-0.614335\pi\)
−0.351521 + 0.936180i \(0.614335\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24512.7 0.849195 0.424597 0.905382i \(-0.360416\pi\)
0.424597 + 0.905382i \(0.360416\pi\)
\(942\) 0 0
\(943\) 11418.5 0.394312
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49278.9 −1.69097 −0.845485 0.533999i \(-0.820689\pi\)
−0.845485 + 0.533999i \(0.820689\pi\)
\(948\) 0 0
\(949\) 1677.04 0.0573647
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44093.4 1.49877 0.749384 0.662136i \(-0.230350\pi\)
0.749384 + 0.662136i \(0.230350\pi\)
\(954\) 0 0
\(955\) 5763.87 0.195303
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15349.8 0.516862
\(960\) 0 0
\(961\) 26660.5 0.894919
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 10167.4 0.339170
\(966\) 0 0
\(967\) −7183.12 −0.238876 −0.119438 0.992842i \(-0.538109\pi\)
−0.119438 + 0.992842i \(0.538109\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −37526.1 −1.24024 −0.620119 0.784508i \(-0.712916\pi\)
−0.620119 + 0.784508i \(0.712916\pi\)
\(972\) 0 0
\(973\) −17897.5 −0.589691
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39517.3 1.29403 0.647017 0.762476i \(-0.276016\pi\)
0.647017 + 0.762476i \(0.276016\pi\)
\(978\) 0 0
\(979\) −4535.91 −0.148078
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −55997.2 −1.81692 −0.908461 0.417971i \(-0.862742\pi\)
−0.908461 + 0.417971i \(0.862742\pi\)
\(984\) 0 0
\(985\) 8508.90 0.275245
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23785.7 −0.764752
\(990\) 0 0
\(991\) 18626.7 0.597069 0.298535 0.954399i \(-0.403502\pi\)
0.298535 + 0.954399i \(0.403502\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −25438.2 −0.810497
\(996\) 0 0
\(997\) −1966.50 −0.0624671 −0.0312335 0.999512i \(-0.509944\pi\)
−0.0312335 + 0.999512i \(0.509944\pi\)
\(998\) 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.1.1 4
3.2 odd 2 1080.4.a.p.1.1 yes 4
4.3 odd 2 2160.4.a.bu.1.4 4
12.11 even 2 2160.4.a.bv.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.1 4 1.1 even 1 trivial
1080.4.a.p.1.1 yes 4 3.2 odd 2
2160.4.a.bu.1.4 4 4.3 odd 2
2160.4.a.bv.1.4 4 12.11 even 2