Properties

Label 1856.4.a.y.1.2
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.957567\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.64574 q^{3} -12.8729 q^{5} +26.0540 q^{7} -5.41713 q^{9} +O(q^{10})\) \(q-4.64574 q^{3} -12.8729 q^{5} +26.0540 q^{7} -5.41713 q^{9} +62.8274 q^{11} -22.3936 q^{13} +59.8039 q^{15} -57.9808 q^{17} -71.3143 q^{19} -121.040 q^{21} -49.5307 q^{23} +40.7104 q^{25} +150.601 q^{27} +29.0000 q^{29} +62.9198 q^{31} -291.880 q^{33} -335.389 q^{35} -119.123 q^{37} +104.035 q^{39} -414.916 q^{41} +348.009 q^{43} +69.7340 q^{45} +553.259 q^{47} +335.808 q^{49} +269.364 q^{51} +107.308 q^{53} -808.768 q^{55} +331.308 q^{57} -136.881 q^{59} +579.408 q^{61} -141.138 q^{63} +288.269 q^{65} -919.959 q^{67} +230.106 q^{69} +781.802 q^{71} -133.237 q^{73} -189.130 q^{75} +1636.90 q^{77} +868.196 q^{79} -553.392 q^{81} +83.3560 q^{83} +746.379 q^{85} -134.726 q^{87} -357.919 q^{89} -583.442 q^{91} -292.309 q^{93} +918.019 q^{95} -187.105 q^{97} -340.344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 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
\(3\) −4.64574 −0.894072 −0.447036 0.894516i \(-0.647520\pi\)
−0.447036 + 0.894516i \(0.647520\pi\)
\(4\) 0 0
\(5\) −12.8729 −1.15138 −0.575692 0.817667i \(-0.695267\pi\)
−0.575692 + 0.817667i \(0.695267\pi\)
\(6\) 0 0
\(7\) 26.0540 1.40678 0.703391 0.710804i \(-0.251669\pi\)
0.703391 + 0.710804i \(0.251669\pi\)
\(8\) 0 0
\(9\) −5.41713 −0.200635
\(10\) 0 0
\(11\) 62.8274 1.72211 0.861053 0.508515i \(-0.169805\pi\)
0.861053 + 0.508515i \(0.169805\pi\)
\(12\) 0 0
\(13\) −22.3936 −0.477759 −0.238879 0.971049i \(-0.576780\pi\)
−0.238879 + 0.971049i \(0.576780\pi\)
\(14\) 0 0
\(15\) 59.8039 1.02942
\(16\) 0 0
\(17\) −57.9808 −0.827201 −0.413601 0.910458i \(-0.635729\pi\)
−0.413601 + 0.910458i \(0.635729\pi\)
\(18\) 0 0
\(19\) −71.3143 −0.861086 −0.430543 0.902570i \(-0.641678\pi\)
−0.430543 + 0.902570i \(0.641678\pi\)
\(20\) 0 0
\(21\) −121.040 −1.25776
\(22\) 0 0
\(23\) −49.5307 −0.449037 −0.224519 0.974470i \(-0.572081\pi\)
−0.224519 + 0.974470i \(0.572081\pi\)
\(24\) 0 0
\(25\) 40.7104 0.325683
\(26\) 0 0
\(27\) 150.601 1.07345
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 62.9198 0.364540 0.182270 0.983249i \(-0.441656\pi\)
0.182270 + 0.983249i \(0.441656\pi\)
\(32\) 0 0
\(33\) −291.880 −1.53969
\(34\) 0 0
\(35\) −335.389 −1.61974
\(36\) 0 0
\(37\) −119.123 −0.529288 −0.264644 0.964346i \(-0.585254\pi\)
−0.264644 + 0.964346i \(0.585254\pi\)
\(38\) 0 0
\(39\) 104.035 0.427151
\(40\) 0 0
\(41\) −414.916 −1.58046 −0.790231 0.612810i \(-0.790039\pi\)
−0.790231 + 0.612810i \(0.790039\pi\)
\(42\) 0 0
\(43\) 348.009 1.23421 0.617103 0.786882i \(-0.288306\pi\)
0.617103 + 0.786882i \(0.288306\pi\)
\(44\) 0 0
\(45\) 69.7340 0.231007
\(46\) 0 0
\(47\) 553.259 1.71705 0.858523 0.512775i \(-0.171382\pi\)
0.858523 + 0.512775i \(0.171382\pi\)
\(48\) 0 0
\(49\) 335.808 0.979033
\(50\) 0 0
\(51\) 269.364 0.739578
\(52\) 0 0
\(53\) 107.308 0.278111 0.139055 0.990285i \(-0.455593\pi\)
0.139055 + 0.990285i \(0.455593\pi\)
\(54\) 0 0
\(55\) −808.768 −1.98280
\(56\) 0 0
\(57\) 331.308 0.769873
\(58\) 0 0
\(59\) −136.881 −0.302041 −0.151020 0.988531i \(-0.548256\pi\)
−0.151020 + 0.988531i \(0.548256\pi\)
\(60\) 0 0
\(61\) 579.408 1.21616 0.608078 0.793877i \(-0.291941\pi\)
0.608078 + 0.793877i \(0.291941\pi\)
\(62\) 0 0
\(63\) −141.138 −0.282249
\(64\) 0 0
\(65\) 288.269 0.550084
\(66\) 0 0
\(67\) −919.959 −1.67748 −0.838738 0.544535i \(-0.816706\pi\)
−0.838738 + 0.544535i \(0.816706\pi\)
\(68\) 0 0
\(69\) 230.106 0.401472
\(70\) 0 0
\(71\) 781.802 1.30680 0.653400 0.757013i \(-0.273342\pi\)
0.653400 + 0.757013i \(0.273342\pi\)
\(72\) 0 0
\(73\) −133.237 −0.213619 −0.106810 0.994279i \(-0.534064\pi\)
−0.106810 + 0.994279i \(0.534064\pi\)
\(74\) 0 0
\(75\) −189.130 −0.291185
\(76\) 0 0
\(77\) 1636.90 2.42263
\(78\) 0 0
\(79\) 868.196 1.23645 0.618225 0.786001i \(-0.287852\pi\)
0.618225 + 0.786001i \(0.287852\pi\)
\(80\) 0 0
\(81\) −553.392 −0.759111
\(82\) 0 0
\(83\) 83.3560 0.110235 0.0551175 0.998480i \(-0.482447\pi\)
0.0551175 + 0.998480i \(0.482447\pi\)
\(84\) 0 0
\(85\) 746.379 0.952425
\(86\) 0 0
\(87\) −134.726 −0.166025
\(88\) 0 0
\(89\) −357.919 −0.426284 −0.213142 0.977021i \(-0.568370\pi\)
−0.213142 + 0.977021i \(0.568370\pi\)
\(90\) 0 0
\(91\) −583.442 −0.672102
\(92\) 0 0
\(93\) −292.309 −0.325925
\(94\) 0 0
\(95\) 918.019 0.991440
\(96\) 0 0
\(97\) −187.105 −0.195852 −0.0979260 0.995194i \(-0.531221\pi\)
−0.0979260 + 0.995194i \(0.531221\pi\)
\(98\) 0 0
\(99\) −340.344 −0.345514
\(100\) 0 0
\(101\) 959.423 0.945209 0.472605 0.881275i \(-0.343314\pi\)
0.472605 + 0.881275i \(0.343314\pi\)
\(102\) 0 0
\(103\) 78.8738 0.0754531 0.0377265 0.999288i \(-0.487988\pi\)
0.0377265 + 0.999288i \(0.487988\pi\)
\(104\) 0 0
\(105\) 1558.13 1.44817
\(106\) 0 0
\(107\) −713.851 −0.644959 −0.322479 0.946577i \(-0.604516\pi\)
−0.322479 + 0.946577i \(0.604516\pi\)
\(108\) 0 0
\(109\) −536.561 −0.471497 −0.235749 0.971814i \(-0.575754\pi\)
−0.235749 + 0.971814i \(0.575754\pi\)
\(110\) 0 0
\(111\) 553.412 0.473222
\(112\) 0 0
\(113\) 1946.36 1.62034 0.810170 0.586195i \(-0.199375\pi\)
0.810170 + 0.586195i \(0.199375\pi\)
\(114\) 0 0
\(115\) 637.601 0.517014
\(116\) 0 0
\(117\) 121.309 0.0958549
\(118\) 0 0
\(119\) −1510.63 −1.16369
\(120\) 0 0
\(121\) 2616.28 1.96565
\(122\) 0 0
\(123\) 1927.59 1.41305
\(124\) 0 0
\(125\) 1085.05 0.776397
\(126\) 0 0
\(127\) −1995.14 −1.39402 −0.697009 0.717062i \(-0.745486\pi\)
−0.697009 + 0.717062i \(0.745486\pi\)
\(128\) 0 0
\(129\) −1616.76 −1.10347
\(130\) 0 0
\(131\) −1544.84 −1.03033 −0.515164 0.857092i \(-0.672269\pi\)
−0.515164 + 0.857092i \(0.672269\pi\)
\(132\) 0 0
\(133\) −1858.02 −1.21136
\(134\) 0 0
\(135\) −1938.67 −1.23596
\(136\) 0 0
\(137\) −1294.93 −0.807543 −0.403771 0.914860i \(-0.632301\pi\)
−0.403771 + 0.914860i \(0.632301\pi\)
\(138\) 0 0
\(139\) −1999.66 −1.22021 −0.610105 0.792320i \(-0.708873\pi\)
−0.610105 + 0.792320i \(0.708873\pi\)
\(140\) 0 0
\(141\) −2570.30 −1.53516
\(142\) 0 0
\(143\) −1406.93 −0.822752
\(144\) 0 0
\(145\) −373.313 −0.213807
\(146\) 0 0
\(147\) −1560.08 −0.875326
\(148\) 0 0
\(149\) −1187.63 −0.652984 −0.326492 0.945200i \(-0.605867\pi\)
−0.326492 + 0.945200i \(0.605867\pi\)
\(150\) 0 0
\(151\) −2257.61 −1.21670 −0.608350 0.793669i \(-0.708168\pi\)
−0.608350 + 0.793669i \(0.708168\pi\)
\(152\) 0 0
\(153\) 314.090 0.165965
\(154\) 0 0
\(155\) −809.958 −0.419725
\(156\) 0 0
\(157\) −1188.18 −0.603995 −0.301997 0.953309i \(-0.597653\pi\)
−0.301997 + 0.953309i \(0.597653\pi\)
\(158\) 0 0
\(159\) −498.524 −0.248651
\(160\) 0 0
\(161\) −1290.47 −0.631697
\(162\) 0 0
\(163\) 2452.33 1.17841 0.589207 0.807982i \(-0.299440\pi\)
0.589207 + 0.807982i \(0.299440\pi\)
\(164\) 0 0
\(165\) 3757.32 1.77277
\(166\) 0 0
\(167\) −2020.14 −0.936067 −0.468034 0.883711i \(-0.655037\pi\)
−0.468034 + 0.883711i \(0.655037\pi\)
\(168\) 0 0
\(169\) −1695.53 −0.771746
\(170\) 0 0
\(171\) 386.319 0.172764
\(172\) 0 0
\(173\) −2862.12 −1.25782 −0.628910 0.777478i \(-0.716499\pi\)
−0.628910 + 0.777478i \(0.716499\pi\)
\(174\) 0 0
\(175\) 1060.67 0.458165
\(176\) 0 0
\(177\) 635.914 0.270046
\(178\) 0 0
\(179\) 232.651 0.0971460 0.0485730 0.998820i \(-0.484533\pi\)
0.0485730 + 0.998820i \(0.484533\pi\)
\(180\) 0 0
\(181\) 2607.67 1.07086 0.535432 0.844578i \(-0.320149\pi\)
0.535432 + 0.844578i \(0.320149\pi\)
\(182\) 0 0
\(183\) −2691.78 −1.08733
\(184\) 0 0
\(185\) 1533.45 0.609413
\(186\) 0 0
\(187\) −3642.78 −1.42453
\(188\) 0 0
\(189\) 3923.76 1.51012
\(190\) 0 0
\(191\) 1528.90 0.579202 0.289601 0.957147i \(-0.406477\pi\)
0.289601 + 0.957147i \(0.406477\pi\)
\(192\) 0 0
\(193\) 1017.58 0.379518 0.189759 0.981831i \(-0.439229\pi\)
0.189759 + 0.981831i \(0.439229\pi\)
\(194\) 0 0
\(195\) −1339.22 −0.491815
\(196\) 0 0
\(197\) −3290.20 −1.18994 −0.594968 0.803749i \(-0.702835\pi\)
−0.594968 + 0.803749i \(0.702835\pi\)
\(198\) 0 0
\(199\) −29.9190 −0.0106578 −0.00532891 0.999986i \(-0.501696\pi\)
−0.00532891 + 0.999986i \(0.501696\pi\)
\(200\) 0 0
\(201\) 4273.89 1.49979
\(202\) 0 0
\(203\) 755.565 0.261233
\(204\) 0 0
\(205\) 5341.15 1.81972
\(206\) 0 0
\(207\) 268.314 0.0900924
\(208\) 0 0
\(209\) −4480.49 −1.48288
\(210\) 0 0
\(211\) −2267.20 −0.739717 −0.369859 0.929088i \(-0.620594\pi\)
−0.369859 + 0.929088i \(0.620594\pi\)
\(212\) 0 0
\(213\) −3632.05 −1.16837
\(214\) 0 0
\(215\) −4479.87 −1.42105
\(216\) 0 0
\(217\) 1639.31 0.512828
\(218\) 0 0
\(219\) 618.984 0.190991
\(220\) 0 0
\(221\) 1298.40 0.395203
\(222\) 0 0
\(223\) 4945.03 1.48495 0.742474 0.669875i \(-0.233652\pi\)
0.742474 + 0.669875i \(0.233652\pi\)
\(224\) 0 0
\(225\) −220.534 −0.0653433
\(226\) 0 0
\(227\) 3559.46 1.04075 0.520374 0.853938i \(-0.325792\pi\)
0.520374 + 0.853938i \(0.325792\pi\)
\(228\) 0 0
\(229\) −6143.40 −1.77278 −0.886391 0.462937i \(-0.846796\pi\)
−0.886391 + 0.462937i \(0.846796\pi\)
\(230\) 0 0
\(231\) −7604.61 −2.16600
\(232\) 0 0
\(233\) −1087.06 −0.305648 −0.152824 0.988253i \(-0.548837\pi\)
−0.152824 + 0.988253i \(0.548837\pi\)
\(234\) 0 0
\(235\) −7122.03 −1.97698
\(236\) 0 0
\(237\) −4033.41 −1.10548
\(238\) 0 0
\(239\) 1079.00 0.292027 0.146013 0.989283i \(-0.453356\pi\)
0.146013 + 0.989283i \(0.453356\pi\)
\(240\) 0 0
\(241\) −989.224 −0.264405 −0.132202 0.991223i \(-0.542205\pi\)
−0.132202 + 0.991223i \(0.542205\pi\)
\(242\) 0 0
\(243\) −1495.33 −0.394754
\(244\) 0 0
\(245\) −4322.81 −1.12724
\(246\) 0 0
\(247\) 1596.98 0.411391
\(248\) 0 0
\(249\) −387.250 −0.0985581
\(250\) 0 0
\(251\) −900.246 −0.226386 −0.113193 0.993573i \(-0.536108\pi\)
−0.113193 + 0.993573i \(0.536108\pi\)
\(252\) 0 0
\(253\) −3111.88 −0.773290
\(254\) 0 0
\(255\) −3467.48 −0.851537
\(256\) 0 0
\(257\) −3125.18 −0.758534 −0.379267 0.925287i \(-0.623824\pi\)
−0.379267 + 0.925287i \(0.623824\pi\)
\(258\) 0 0
\(259\) −3103.62 −0.744592
\(260\) 0 0
\(261\) −157.097 −0.0372569
\(262\) 0 0
\(263\) 2814.89 0.659976 0.329988 0.943985i \(-0.392955\pi\)
0.329988 + 0.943985i \(0.392955\pi\)
\(264\) 0 0
\(265\) −1381.36 −0.320212
\(266\) 0 0
\(267\) 1662.80 0.381129
\(268\) 0 0
\(269\) −4409.28 −0.999400 −0.499700 0.866199i \(-0.666556\pi\)
−0.499700 + 0.866199i \(0.666556\pi\)
\(270\) 0 0
\(271\) −4417.97 −0.990304 −0.495152 0.868806i \(-0.664888\pi\)
−0.495152 + 0.868806i \(0.664888\pi\)
\(272\) 0 0
\(273\) 2710.52 0.600908
\(274\) 0 0
\(275\) 2557.73 0.560862
\(276\) 0 0
\(277\) 887.577 0.192525 0.0962624 0.995356i \(-0.469311\pi\)
0.0962624 + 0.995356i \(0.469311\pi\)
\(278\) 0 0
\(279\) −340.845 −0.0731393
\(280\) 0 0
\(281\) −4286.58 −0.910021 −0.455010 0.890486i \(-0.650364\pi\)
−0.455010 + 0.890486i \(0.650364\pi\)
\(282\) 0 0
\(283\) 1709.59 0.359097 0.179549 0.983749i \(-0.442536\pi\)
0.179549 + 0.983749i \(0.442536\pi\)
\(284\) 0 0
\(285\) −4264.88 −0.886419
\(286\) 0 0
\(287\) −10810.2 −2.22336
\(288\) 0 0
\(289\) −1551.22 −0.315738
\(290\) 0 0
\(291\) 869.241 0.175106
\(292\) 0 0
\(293\) −1145.01 −0.228301 −0.114151 0.993463i \(-0.536415\pi\)
−0.114151 + 0.993463i \(0.536415\pi\)
\(294\) 0 0
\(295\) 1762.05 0.347765
\(296\) 0 0
\(297\) 9461.90 1.84860
\(298\) 0 0
\(299\) 1109.17 0.214532
\(300\) 0 0
\(301\) 9067.01 1.73626
\(302\) 0 0
\(303\) −4457.23 −0.845086
\(304\) 0 0
\(305\) −7458.63 −1.40026
\(306\) 0 0
\(307\) −1079.39 −0.200665 −0.100333 0.994954i \(-0.531991\pi\)
−0.100333 + 0.994954i \(0.531991\pi\)
\(308\) 0 0
\(309\) −366.427 −0.0674605
\(310\) 0 0
\(311\) 2313.90 0.421895 0.210947 0.977497i \(-0.432345\pi\)
0.210947 + 0.977497i \(0.432345\pi\)
\(312\) 0 0
\(313\) −7653.19 −1.38206 −0.691029 0.722827i \(-0.742842\pi\)
−0.691029 + 0.722827i \(0.742842\pi\)
\(314\) 0 0
\(315\) 1816.85 0.324977
\(316\) 0 0
\(317\) 3657.23 0.647982 0.323991 0.946060i \(-0.394975\pi\)
0.323991 + 0.946060i \(0.394975\pi\)
\(318\) 0 0
\(319\) 1821.99 0.319787
\(320\) 0 0
\(321\) 3316.36 0.576640
\(322\) 0 0
\(323\) 4134.86 0.712291
\(324\) 0 0
\(325\) −911.653 −0.155598
\(326\) 0 0
\(327\) 2492.72 0.421553
\(328\) 0 0
\(329\) 14414.6 2.41551
\(330\) 0 0
\(331\) −3237.92 −0.537681 −0.268841 0.963185i \(-0.586640\pi\)
−0.268841 + 0.963185i \(0.586640\pi\)
\(332\) 0 0
\(333\) 645.303 0.106193
\(334\) 0 0
\(335\) 11842.5 1.93142
\(336\) 0 0
\(337\) −7976.89 −1.28940 −0.644702 0.764434i \(-0.723018\pi\)
−0.644702 + 0.764434i \(0.723018\pi\)
\(338\) 0 0
\(339\) −9042.29 −1.44870
\(340\) 0 0
\(341\) 3953.09 0.627777
\(342\) 0 0
\(343\) −187.371 −0.0294959
\(344\) 0 0
\(345\) −2962.13 −0.462248
\(346\) 0 0
\(347\) 8355.93 1.29271 0.646354 0.763037i \(-0.276293\pi\)
0.646354 + 0.763037i \(0.276293\pi\)
\(348\) 0 0
\(349\) −5544.56 −0.850411 −0.425205 0.905097i \(-0.639798\pi\)
−0.425205 + 0.905097i \(0.639798\pi\)
\(350\) 0 0
\(351\) −3372.51 −0.512852
\(352\) 0 0
\(353\) −1682.79 −0.253727 −0.126864 0.991920i \(-0.540491\pi\)
−0.126864 + 0.991920i \(0.540491\pi\)
\(354\) 0 0
\(355\) −10064.0 −1.50463
\(356\) 0 0
\(357\) 7017.99 1.04042
\(358\) 0 0
\(359\) 7143.13 1.05014 0.525069 0.851059i \(-0.324039\pi\)
0.525069 + 0.851059i \(0.324039\pi\)
\(360\) 0 0
\(361\) −1773.27 −0.258531
\(362\) 0 0
\(363\) −12154.6 −1.75743
\(364\) 0 0
\(365\) 1715.14 0.245958
\(366\) 0 0
\(367\) −4456.16 −0.633814 −0.316907 0.948457i \(-0.602644\pi\)
−0.316907 + 0.948457i \(0.602644\pi\)
\(368\) 0 0
\(369\) 2247.65 0.317095
\(370\) 0 0
\(371\) 2795.79 0.391241
\(372\) 0 0
\(373\) −2508.28 −0.348187 −0.174094 0.984729i \(-0.555700\pi\)
−0.174094 + 0.984729i \(0.555700\pi\)
\(374\) 0 0
\(375\) −5040.85 −0.694155
\(376\) 0 0
\(377\) −649.414 −0.0887176
\(378\) 0 0
\(379\) 12733.9 1.72585 0.862926 0.505331i \(-0.168629\pi\)
0.862926 + 0.505331i \(0.168629\pi\)
\(380\) 0 0
\(381\) 9268.91 1.24635
\(382\) 0 0
\(383\) −1027.19 −0.137042 −0.0685209 0.997650i \(-0.521828\pi\)
−0.0685209 + 0.997650i \(0.521828\pi\)
\(384\) 0 0
\(385\) −21071.6 −2.78937
\(386\) 0 0
\(387\) −1885.21 −0.247624
\(388\) 0 0
\(389\) −5153.35 −0.671684 −0.335842 0.941918i \(-0.609021\pi\)
−0.335842 + 0.941918i \(0.609021\pi\)
\(390\) 0 0
\(391\) 2871.83 0.371444
\(392\) 0 0
\(393\) 7176.90 0.921187
\(394\) 0 0
\(395\) −11176.2 −1.42363
\(396\) 0 0
\(397\) −6250.95 −0.790242 −0.395121 0.918629i \(-0.629297\pi\)
−0.395121 + 0.918629i \(0.629297\pi\)
\(398\) 0 0
\(399\) 8631.87 1.08304
\(400\) 0 0
\(401\) 11083.8 1.38029 0.690145 0.723671i \(-0.257547\pi\)
0.690145 + 0.723671i \(0.257547\pi\)
\(402\) 0 0
\(403\) −1409.00 −0.174162
\(404\) 0 0
\(405\) 7123.74 0.874028
\(406\) 0 0
\(407\) −7484.17 −0.911490
\(408\) 0 0
\(409\) −3375.43 −0.408079 −0.204039 0.978963i \(-0.565407\pi\)
−0.204039 + 0.978963i \(0.565407\pi\)
\(410\) 0 0
\(411\) 6015.91 0.722002
\(412\) 0 0
\(413\) −3566.29 −0.424905
\(414\) 0 0
\(415\) −1073.03 −0.126923
\(416\) 0 0
\(417\) 9289.91 1.09096
\(418\) 0 0
\(419\) −6486.50 −0.756292 −0.378146 0.925746i \(-0.623438\pi\)
−0.378146 + 0.925746i \(0.623438\pi\)
\(420\) 0 0
\(421\) 10938.6 1.26631 0.633153 0.774026i \(-0.281760\pi\)
0.633153 + 0.774026i \(0.281760\pi\)
\(422\) 0 0
\(423\) −2997.08 −0.344499
\(424\) 0 0
\(425\) −2360.42 −0.269406
\(426\) 0 0
\(427\) 15095.9 1.71087
\(428\) 0 0
\(429\) 6536.23 0.735600
\(430\) 0 0
\(431\) 4124.34 0.460934 0.230467 0.973080i \(-0.425975\pi\)
0.230467 + 0.973080i \(0.425975\pi\)
\(432\) 0 0
\(433\) −1561.41 −0.173295 −0.0866473 0.996239i \(-0.527615\pi\)
−0.0866473 + 0.996239i \(0.527615\pi\)
\(434\) 0 0
\(435\) 1734.31 0.191158
\(436\) 0 0
\(437\) 3532.25 0.386660
\(438\) 0 0
\(439\) −15712.7 −1.70826 −0.854130 0.520060i \(-0.825909\pi\)
−0.854130 + 0.520060i \(0.825909\pi\)
\(440\) 0 0
\(441\) −1819.12 −0.196428
\(442\) 0 0
\(443\) −12763.5 −1.36888 −0.684439 0.729070i \(-0.739953\pi\)
−0.684439 + 0.729070i \(0.739953\pi\)
\(444\) 0 0
\(445\) 4607.44 0.490817
\(446\) 0 0
\(447\) 5517.42 0.583815
\(448\) 0 0
\(449\) 3117.88 0.327710 0.163855 0.986484i \(-0.447607\pi\)
0.163855 + 0.986484i \(0.447607\pi\)
\(450\) 0 0
\(451\) −26068.1 −2.72172
\(452\) 0 0
\(453\) 10488.3 1.08782
\(454\) 0 0
\(455\) 7510.56 0.773847
\(456\) 0 0
\(457\) −8479.41 −0.867943 −0.433972 0.900927i \(-0.642888\pi\)
−0.433972 + 0.900927i \(0.642888\pi\)
\(458\) 0 0
\(459\) −8732.00 −0.887962
\(460\) 0 0
\(461\) −9253.32 −0.934859 −0.467430 0.884030i \(-0.654820\pi\)
−0.467430 + 0.884030i \(0.654820\pi\)
\(462\) 0 0
\(463\) 521.395 0.0523354 0.0261677 0.999658i \(-0.491670\pi\)
0.0261677 + 0.999658i \(0.491670\pi\)
\(464\) 0 0
\(465\) 3762.85 0.375265
\(466\) 0 0
\(467\) −4337.35 −0.429783 −0.214891 0.976638i \(-0.568940\pi\)
−0.214891 + 0.976638i \(0.568940\pi\)
\(468\) 0 0
\(469\) −23968.6 −2.35984
\(470\) 0 0
\(471\) 5519.98 0.540015
\(472\) 0 0
\(473\) 21864.5 2.12544
\(474\) 0 0
\(475\) −2903.24 −0.280441
\(476\) 0 0
\(477\) −581.300 −0.0557986
\(478\) 0 0
\(479\) −11258.1 −1.07390 −0.536949 0.843615i \(-0.680423\pi\)
−0.536949 + 0.843615i \(0.680423\pi\)
\(480\) 0 0
\(481\) 2667.58 0.252872
\(482\) 0 0
\(483\) 5995.18 0.564783
\(484\) 0 0
\(485\) 2408.58 0.225501
\(486\) 0 0
\(487\) 4353.54 0.405088 0.202544 0.979273i \(-0.435079\pi\)
0.202544 + 0.979273i \(0.435079\pi\)
\(488\) 0 0
\(489\) −11392.9 −1.05359
\(490\) 0 0
\(491\) 8458.77 0.777472 0.388736 0.921349i \(-0.372912\pi\)
0.388736 + 0.921349i \(0.372912\pi\)
\(492\) 0 0
\(493\) −1681.44 −0.153607
\(494\) 0 0
\(495\) 4381.20 0.397819
\(496\) 0 0
\(497\) 20369.0 1.83838
\(498\) 0 0
\(499\) −14850.9 −1.33230 −0.666149 0.745819i \(-0.732058\pi\)
−0.666149 + 0.745819i \(0.732058\pi\)
\(500\) 0 0
\(501\) 9385.04 0.836912
\(502\) 0 0
\(503\) −1686.45 −0.149493 −0.0747464 0.997203i \(-0.523815\pi\)
−0.0747464 + 0.997203i \(0.523815\pi\)
\(504\) 0 0
\(505\) −12350.5 −1.08830
\(506\) 0 0
\(507\) 7876.97 0.689997
\(508\) 0 0
\(509\) 11113.5 0.967773 0.483887 0.875131i \(-0.339225\pi\)
0.483887 + 0.875131i \(0.339225\pi\)
\(510\) 0 0
\(511\) −3471.35 −0.300516
\(512\) 0 0
\(513\) −10740.0 −0.924336
\(514\) 0 0
\(515\) −1015.33 −0.0868754
\(516\) 0 0
\(517\) 34759.8 2.95694
\(518\) 0 0
\(519\) 13296.7 1.12458
\(520\) 0 0
\(521\) −15931.1 −1.33964 −0.669820 0.742523i \(-0.733629\pi\)
−0.669820 + 0.742523i \(0.733629\pi\)
\(522\) 0 0
\(523\) 7960.43 0.665555 0.332778 0.943005i \(-0.392014\pi\)
0.332778 + 0.943005i \(0.392014\pi\)
\(524\) 0 0
\(525\) −4927.58 −0.409633
\(526\) 0 0
\(527\) −3648.14 −0.301548
\(528\) 0 0
\(529\) −9713.71 −0.798366
\(530\) 0 0
\(531\) 741.503 0.0605998
\(532\) 0 0
\(533\) 9291.45 0.755079
\(534\) 0 0
\(535\) 9189.30 0.742594
\(536\) 0 0
\(537\) −1080.83 −0.0868555
\(538\) 0 0
\(539\) 21098.0 1.68600
\(540\) 0 0
\(541\) −13818.9 −1.09819 −0.549096 0.835760i \(-0.685028\pi\)
−0.549096 + 0.835760i \(0.685028\pi\)
\(542\) 0 0
\(543\) −12114.5 −0.957430
\(544\) 0 0
\(545\) 6907.07 0.542874
\(546\) 0 0
\(547\) −22093.3 −1.72695 −0.863474 0.504393i \(-0.831716\pi\)
−0.863474 + 0.504393i \(0.831716\pi\)
\(548\) 0 0
\(549\) −3138.73 −0.244003
\(550\) 0 0
\(551\) −2068.12 −0.159900
\(552\) 0 0
\(553\) 22619.9 1.73942
\(554\) 0 0
\(555\) −7124.00 −0.544859
\(556\) 0 0
\(557\) 3110.34 0.236606 0.118303 0.992978i \(-0.462255\pi\)
0.118303 + 0.992978i \(0.462255\pi\)
\(558\) 0 0
\(559\) −7793.17 −0.589653
\(560\) 0 0
\(561\) 16923.4 1.27363
\(562\) 0 0
\(563\) 13284.6 0.994455 0.497227 0.867620i \(-0.334351\pi\)
0.497227 + 0.867620i \(0.334351\pi\)
\(564\) 0 0
\(565\) −25055.2 −1.86563
\(566\) 0 0
\(567\) −14418.1 −1.06790
\(568\) 0 0
\(569\) −6809.18 −0.501680 −0.250840 0.968029i \(-0.580707\pi\)
−0.250840 + 0.968029i \(0.580707\pi\)
\(570\) 0 0
\(571\) 13471.8 0.987352 0.493676 0.869646i \(-0.335653\pi\)
0.493676 + 0.869646i \(0.335653\pi\)
\(572\) 0 0
\(573\) −7102.88 −0.517849
\(574\) 0 0
\(575\) −2016.41 −0.146244
\(576\) 0 0
\(577\) 5331.06 0.384636 0.192318 0.981333i \(-0.438399\pi\)
0.192318 + 0.981333i \(0.438399\pi\)
\(578\) 0 0
\(579\) −4727.41 −0.339317
\(580\) 0 0
\(581\) 2171.75 0.155077
\(582\) 0 0
\(583\) 6741.87 0.478936
\(584\) 0 0
\(585\) −1561.59 −0.110366
\(586\) 0 0
\(587\) −3333.96 −0.234425 −0.117212 0.993107i \(-0.537396\pi\)
−0.117212 + 0.993107i \(0.537396\pi\)
\(588\) 0 0
\(589\) −4487.09 −0.313900
\(590\) 0 0
\(591\) 15285.4 1.06389
\(592\) 0 0
\(593\) −23405.3 −1.62081 −0.810404 0.585872i \(-0.800752\pi\)
−0.810404 + 0.585872i \(0.800752\pi\)
\(594\) 0 0
\(595\) 19446.1 1.33985
\(596\) 0 0
\(597\) 138.996 0.00952886
\(598\) 0 0
\(599\) −10352.1 −0.706133 −0.353067 0.935598i \(-0.614861\pi\)
−0.353067 + 0.935598i \(0.614861\pi\)
\(600\) 0 0
\(601\) 15171.1 1.02969 0.514843 0.857285i \(-0.327850\pi\)
0.514843 + 0.857285i \(0.327850\pi\)
\(602\) 0 0
\(603\) 4983.54 0.336560
\(604\) 0 0
\(605\) −33679.0 −2.26322
\(606\) 0 0
\(607\) −20823.0 −1.39239 −0.696193 0.717854i \(-0.745124\pi\)
−0.696193 + 0.717854i \(0.745124\pi\)
\(608\) 0 0
\(609\) −3510.15 −0.233561
\(610\) 0 0
\(611\) −12389.5 −0.820334
\(612\) 0 0
\(613\) 19071.6 1.25660 0.628299 0.777972i \(-0.283751\pi\)
0.628299 + 0.777972i \(0.283751\pi\)
\(614\) 0 0
\(615\) −24813.6 −1.62696
\(616\) 0 0
\(617\) 15200.4 0.991806 0.495903 0.868378i \(-0.334837\pi\)
0.495903 + 0.868378i \(0.334837\pi\)
\(618\) 0 0
\(619\) −4358.76 −0.283026 −0.141513 0.989936i \(-0.545197\pi\)
−0.141513 + 0.989936i \(0.545197\pi\)
\(620\) 0 0
\(621\) −7459.39 −0.482021
\(622\) 0 0
\(623\) −9325.20 −0.599689
\(624\) 0 0
\(625\) −19056.5 −1.21961
\(626\) 0 0
\(627\) 20815.2 1.32580
\(628\) 0 0
\(629\) 6906.83 0.437827
\(630\) 0 0
\(631\) −2580.66 −0.162812 −0.0814062 0.996681i \(-0.525941\pi\)
−0.0814062 + 0.996681i \(0.525941\pi\)
\(632\) 0 0
\(633\) 10532.8 0.661361
\(634\) 0 0
\(635\) 25683.2 1.60505
\(636\) 0 0
\(637\) −7519.96 −0.467742
\(638\) 0 0
\(639\) −4235.12 −0.262189
\(640\) 0 0
\(641\) 19858.9 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(642\) 0 0
\(643\) −17371.9 −1.06545 −0.532723 0.846290i \(-0.678831\pi\)
−0.532723 + 0.846290i \(0.678831\pi\)
\(644\) 0 0
\(645\) 20812.3 1.27052
\(646\) 0 0
\(647\) 3275.07 0.199005 0.0995024 0.995037i \(-0.468275\pi\)
0.0995024 + 0.995037i \(0.468275\pi\)
\(648\) 0 0
\(649\) −8599.88 −0.520146
\(650\) 0 0
\(651\) −7615.80 −0.458505
\(652\) 0 0
\(653\) −20726.5 −1.24210 −0.621049 0.783772i \(-0.713293\pi\)
−0.621049 + 0.783772i \(0.713293\pi\)
\(654\) 0 0
\(655\) 19886.4 1.18630
\(656\) 0 0
\(657\) 721.762 0.0428594
\(658\) 0 0
\(659\) 18404.6 1.08792 0.543960 0.839111i \(-0.316924\pi\)
0.543960 + 0.839111i \(0.316924\pi\)
\(660\) 0 0
\(661\) 7146.10 0.420501 0.210250 0.977648i \(-0.432572\pi\)
0.210250 + 0.977648i \(0.432572\pi\)
\(662\) 0 0
\(663\) −6032.02 −0.353340
\(664\) 0 0
\(665\) 23918.0 1.39474
\(666\) 0 0
\(667\) −1436.39 −0.0833841
\(668\) 0 0
\(669\) −22973.3 −1.32765
\(670\) 0 0
\(671\) 36402.7 2.09435
\(672\) 0 0
\(673\) −26819.0 −1.53610 −0.768051 0.640389i \(-0.778773\pi\)
−0.768051 + 0.640389i \(0.778773\pi\)
\(674\) 0 0
\(675\) 6131.05 0.349606
\(676\) 0 0
\(677\) 20093.1 1.14068 0.570340 0.821408i \(-0.306811\pi\)
0.570340 + 0.821408i \(0.306811\pi\)
\(678\) 0 0
\(679\) −4874.82 −0.275521
\(680\) 0 0
\(681\) −16536.3 −0.930505
\(682\) 0 0
\(683\) 6876.22 0.385229 0.192614 0.981275i \(-0.438303\pi\)
0.192614 + 0.981275i \(0.438303\pi\)
\(684\) 0 0
\(685\) 16669.5 0.929791
\(686\) 0 0
\(687\) 28540.6 1.58500
\(688\) 0 0
\(689\) −2403.01 −0.132870
\(690\) 0 0
\(691\) −17332.7 −0.954224 −0.477112 0.878842i \(-0.658316\pi\)
−0.477112 + 0.878842i \(0.658316\pi\)
\(692\) 0 0
\(693\) −8867.31 −0.486063
\(694\) 0 0
\(695\) 25741.4 1.40493
\(696\) 0 0
\(697\) 24057.1 1.30736
\(698\) 0 0
\(699\) 5050.21 0.273271
\(700\) 0 0
\(701\) −11127.5 −0.599545 −0.299772 0.954011i \(-0.596911\pi\)
−0.299772 + 0.954011i \(0.596911\pi\)
\(702\) 0 0
\(703\) 8495.15 0.455762
\(704\) 0 0
\(705\) 33087.1 1.76756
\(706\) 0 0
\(707\) 24996.8 1.32970
\(708\) 0 0
\(709\) −17432.0 −0.923374 −0.461687 0.887043i \(-0.652756\pi\)
−0.461687 + 0.887043i \(0.652756\pi\)
\(710\) 0 0
\(711\) −4703.13 −0.248075
\(712\) 0 0
\(713\) −3116.46 −0.163692
\(714\) 0 0
\(715\) 18111.2 0.947302
\(716\) 0 0
\(717\) −5012.73 −0.261093
\(718\) 0 0
\(719\) 21082.7 1.09353 0.546767 0.837285i \(-0.315858\pi\)
0.546767 + 0.837285i \(0.315858\pi\)
\(720\) 0 0
\(721\) 2054.97 0.106146
\(722\) 0 0
\(723\) 4595.68 0.236397
\(724\) 0 0
\(725\) 1180.60 0.0604779
\(726\) 0 0
\(727\) −25839.1 −1.31818 −0.659091 0.752063i \(-0.729059\pi\)
−0.659091 + 0.752063i \(0.729059\pi\)
\(728\) 0 0
\(729\) 21888.5 1.11205
\(730\) 0 0
\(731\) −20177.9 −1.02094
\(732\) 0 0
\(733\) −1278.54 −0.0644256 −0.0322128 0.999481i \(-0.510255\pi\)
−0.0322128 + 0.999481i \(0.510255\pi\)
\(734\) 0 0
\(735\) 20082.6 1.00784
\(736\) 0 0
\(737\) −57798.6 −2.88879
\(738\) 0 0
\(739\) −4224.54 −0.210287 −0.105144 0.994457i \(-0.533530\pi\)
−0.105144 + 0.994457i \(0.533530\pi\)
\(740\) 0 0
\(741\) −7419.17 −0.367814
\(742\) 0 0
\(743\) −17992.3 −0.888390 −0.444195 0.895930i \(-0.646510\pi\)
−0.444195 + 0.895930i \(0.646510\pi\)
\(744\) 0 0
\(745\) 15288.2 0.751835
\(746\) 0 0
\(747\) −451.550 −0.0221170
\(748\) 0 0
\(749\) −18598.6 −0.907316
\(750\) 0 0
\(751\) −10082.4 −0.489895 −0.244948 0.969536i \(-0.578771\pi\)
−0.244948 + 0.969536i \(0.578771\pi\)
\(752\) 0 0
\(753\) 4182.30 0.202406
\(754\) 0 0
\(755\) 29061.9 1.40089
\(756\) 0 0
\(757\) −10806.5 −0.518851 −0.259425 0.965763i \(-0.583533\pi\)
−0.259425 + 0.965763i \(0.583533\pi\)
\(758\) 0 0
\(759\) 14457.0 0.691377
\(760\) 0 0
\(761\) −30710.4 −1.46288 −0.731439 0.681907i \(-0.761151\pi\)
−0.731439 + 0.681907i \(0.761151\pi\)
\(762\) 0 0
\(763\) −13979.5 −0.663293
\(764\) 0 0
\(765\) −4043.23 −0.191089
\(766\) 0 0
\(767\) 3065.26 0.144303
\(768\) 0 0
\(769\) 10757.1 0.504436 0.252218 0.967670i \(-0.418840\pi\)
0.252218 + 0.967670i \(0.418840\pi\)
\(770\) 0 0
\(771\) 14518.8 0.678185
\(772\) 0 0
\(773\) 18077.2 0.841125 0.420563 0.907263i \(-0.361833\pi\)
0.420563 + 0.907263i \(0.361833\pi\)
\(774\) 0 0
\(775\) 2561.49 0.118725
\(776\) 0 0
\(777\) 14418.6 0.665719
\(778\) 0 0
\(779\) 29589.4 1.36091
\(780\) 0 0
\(781\) 49118.6 2.25045
\(782\) 0 0
\(783\) 4367.44 0.199335
\(784\) 0 0
\(785\) 15295.3 0.695430
\(786\) 0 0
\(787\) 31543.7 1.42873 0.714366 0.699773i \(-0.246715\pi\)
0.714366 + 0.699773i \(0.246715\pi\)
\(788\) 0 0
\(789\) −13077.3 −0.590066
\(790\) 0 0
\(791\) 50710.4 2.27946
\(792\) 0 0
\(793\) −12975.0 −0.581030
\(794\) 0 0
\(795\) 6417.42 0.286293
\(796\) 0 0
\(797\) −18280.9 −0.812477 −0.406239 0.913767i \(-0.633160\pi\)
−0.406239 + 0.913767i \(0.633160\pi\)
\(798\) 0 0
\(799\) −32078.4 −1.42034
\(800\) 0 0
\(801\) 1938.89 0.0855273
\(802\) 0 0
\(803\) −8370.93 −0.367875
\(804\) 0 0
\(805\) 16612.0 0.727326
\(806\) 0 0
\(807\) 20484.4 0.893536
\(808\) 0 0
\(809\) 21776.3 0.946372 0.473186 0.880963i \(-0.343104\pi\)
0.473186 + 0.880963i \(0.343104\pi\)
\(810\) 0 0
\(811\) −17035.7 −0.737612 −0.368806 0.929506i \(-0.620233\pi\)
−0.368806 + 0.929506i \(0.620233\pi\)
\(812\) 0 0
\(813\) 20524.7 0.885404
\(814\) 0 0
\(815\) −31568.5 −1.35681
\(816\) 0 0
\(817\) −24818.0 −1.06276
\(818\) 0 0
\(819\) 3160.58 0.134847
\(820\) 0 0
\(821\) 45120.7 1.91805 0.959027 0.283314i \(-0.0914339\pi\)
0.959027 + 0.283314i \(0.0914339\pi\)
\(822\) 0 0
\(823\) 22261.7 0.942883 0.471442 0.881897i \(-0.343734\pi\)
0.471442 + 0.881897i \(0.343734\pi\)
\(824\) 0 0
\(825\) −11882.5 −0.501451
\(826\) 0 0
\(827\) −10280.4 −0.432266 −0.216133 0.976364i \(-0.569345\pi\)
−0.216133 + 0.976364i \(0.569345\pi\)
\(828\) 0 0
\(829\) −28509.2 −1.19441 −0.597206 0.802088i \(-0.703722\pi\)
−0.597206 + 0.802088i \(0.703722\pi\)
\(830\) 0 0
\(831\) −4123.45 −0.172131
\(832\) 0 0
\(833\) −19470.4 −0.809857
\(834\) 0 0
\(835\) 26005.0 1.07777
\(836\) 0 0
\(837\) 9475.82 0.391317
\(838\) 0 0
\(839\) 4746.97 0.195332 0.0976661 0.995219i \(-0.468862\pi\)
0.0976661 + 0.995219i \(0.468862\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 19914.3 0.813625
\(844\) 0 0
\(845\) 21826.3 0.888576
\(846\) 0 0
\(847\) 68164.5 2.76524
\(848\) 0 0
\(849\) −7942.30 −0.321059
\(850\) 0 0
\(851\) 5900.22 0.237670
\(852\) 0 0
\(853\) −35313.3 −1.41747 −0.708736 0.705473i \(-0.750734\pi\)
−0.708736 + 0.705473i \(0.750734\pi\)
\(854\) 0 0
\(855\) −4973.03 −0.198917
\(856\) 0 0
\(857\) 32142.8 1.28119 0.640594 0.767880i \(-0.278688\pi\)
0.640594 + 0.767880i \(0.278688\pi\)
\(858\) 0 0
\(859\) −37568.2 −1.49221 −0.746107 0.665826i \(-0.768079\pi\)
−0.746107 + 0.665826i \(0.768079\pi\)
\(860\) 0 0
\(861\) 50221.3 1.98785
\(862\) 0 0
\(863\) 3416.95 0.134779 0.0673896 0.997727i \(-0.478533\pi\)
0.0673896 + 0.997727i \(0.478533\pi\)
\(864\) 0 0
\(865\) 36843.6 1.44823
\(866\) 0 0
\(867\) 7206.57 0.282293
\(868\) 0 0
\(869\) 54546.5 2.12930
\(870\) 0 0
\(871\) 20601.2 0.801429
\(872\) 0 0
\(873\) 1013.57 0.0392947
\(874\) 0 0
\(875\) 28269.8 1.09222
\(876\) 0 0
\(877\) 11891.0 0.457847 0.228923 0.973444i \(-0.426479\pi\)
0.228923 + 0.973444i \(0.426479\pi\)
\(878\) 0 0
\(879\) 5319.42 0.204118
\(880\) 0 0
\(881\) 20042.7 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(882\) 0 0
\(883\) 18042.8 0.687641 0.343821 0.939035i \(-0.388279\pi\)
0.343821 + 0.939035i \(0.388279\pi\)
\(884\) 0 0
\(885\) −8186.02 −0.310927
\(886\) 0 0
\(887\) 2247.91 0.0850929 0.0425464 0.999094i \(-0.486453\pi\)
0.0425464 + 0.999094i \(0.486453\pi\)
\(888\) 0 0
\(889\) −51981.4 −1.96108
\(890\) 0 0
\(891\) −34768.2 −1.30727
\(892\) 0 0
\(893\) −39455.3 −1.47852
\(894\) 0 0
\(895\) −2994.88 −0.111852
\(896\) 0 0
\(897\) −5152.91 −0.191807
\(898\) 0 0
\(899\) 1824.68 0.0676934
\(900\) 0 0
\(901\) −6221.80 −0.230053
\(902\) 0 0
\(903\) −42123.0 −1.55234
\(904\) 0 0
\(905\) −33568.1 −1.23298
\(906\) 0 0
\(907\) 1798.34 0.0658354 0.0329177 0.999458i \(-0.489520\pi\)
0.0329177 + 0.999458i \(0.489520\pi\)
\(908\) 0 0
\(909\) −5197.32 −0.189642
\(910\) 0 0
\(911\) 4602.02 0.167367 0.0836837 0.996492i \(-0.473331\pi\)
0.0836837 + 0.996492i \(0.473331\pi\)
\(912\) 0 0
\(913\) 5237.04 0.189836
\(914\) 0 0
\(915\) 34650.8 1.25194
\(916\) 0 0
\(917\) −40249.1 −1.44944
\(918\) 0 0
\(919\) 20622.0 0.740215 0.370107 0.928989i \(-0.379321\pi\)
0.370107 + 0.928989i \(0.379321\pi\)
\(920\) 0 0
\(921\) 5014.58 0.179409
\(922\) 0 0
\(923\) −17507.4 −0.624335
\(924\) 0 0
\(925\) −4849.53 −0.172380
\(926\) 0 0
\(927\) −427.270 −0.0151385
\(928\) 0 0
\(929\) −24061.8 −0.849777 −0.424889 0.905246i \(-0.639687\pi\)
−0.424889 + 0.905246i \(0.639687\pi\)
\(930\) 0 0
\(931\) −23947.9 −0.843032
\(932\) 0 0
\(933\) −10749.8 −0.377204
\(934\) 0 0
\(935\) 46893.0 1.64018
\(936\) 0 0
\(937\) −6581.96 −0.229481 −0.114740 0.993396i \(-0.536604\pi\)
−0.114740 + 0.993396i \(0.536604\pi\)
\(938\) 0 0
\(939\) 35554.7 1.23566
\(940\) 0 0
\(941\) −23579.7 −0.816873 −0.408436 0.912787i \(-0.633926\pi\)
−0.408436 + 0.912787i \(0.633926\pi\)
\(942\) 0 0
\(943\) 20551.0 0.709686
\(944\) 0 0
\(945\) −50510.0 −1.73872
\(946\) 0 0
\(947\) 36368.6 1.24796 0.623982 0.781439i \(-0.285514\pi\)
0.623982 + 0.781439i \(0.285514\pi\)
\(948\) 0 0
\(949\) 2983.65 0.102058
\(950\) 0 0
\(951\) −16990.5 −0.579343
\(952\) 0 0
\(953\) 44404.3 1.50933 0.754667 0.656108i \(-0.227798\pi\)
0.754667 + 0.656108i \(0.227798\pi\)
\(954\) 0 0
\(955\) −19681.4 −0.666884
\(956\) 0 0
\(957\) −8464.51 −0.285913
\(958\) 0 0
\(959\) −33738.1 −1.13604
\(960\) 0 0
\(961\) −25832.1 −0.867111
\(962\) 0 0
\(963\) 3867.02 0.129401
\(964\) 0 0
\(965\) −13099.2 −0.436971
\(966\) 0 0
\(967\) 21928.3 0.729230 0.364615 0.931158i \(-0.381201\pi\)
0.364615 + 0.931158i \(0.381201\pi\)
\(968\) 0 0
\(969\) −19209.5 −0.636840
\(970\) 0 0
\(971\) 6352.95 0.209965 0.104982 0.994474i \(-0.466521\pi\)
0.104982 + 0.994474i \(0.466521\pi\)
\(972\) 0 0
\(973\) −52099.1 −1.71657
\(974\) 0 0
\(975\) 4235.30 0.139116
\(976\) 0 0
\(977\) 12792.8 0.418913 0.209456 0.977818i \(-0.432831\pi\)
0.209456 + 0.977818i \(0.432831\pi\)
\(978\) 0 0
\(979\) −22487.1 −0.734107
\(980\) 0 0
\(981\) 2906.62 0.0945986
\(982\) 0 0
\(983\) 7536.73 0.244542 0.122271 0.992497i \(-0.460982\pi\)
0.122271 + 0.992497i \(0.460982\pi\)
\(984\) 0 0
\(985\) 42354.3 1.37007
\(986\) 0 0
\(987\) −66966.4 −2.15964
\(988\) 0 0
\(989\) −17237.1 −0.554205
\(990\) 0 0
\(991\) −28088.3 −0.900357 −0.450179 0.892938i \(-0.648640\pi\)
−0.450179 + 0.892938i \(0.648640\pi\)
\(992\) 0 0
\(993\) 15042.5 0.480726
\(994\) 0 0
\(995\) 385.144 0.0122712
\(996\) 0 0
\(997\) 44951.9 1.42793 0.713963 0.700184i \(-0.246899\pi\)
0.713963 + 0.700184i \(0.246899\pi\)
\(998\) 0 0
\(999\) −17940.0 −0.568166
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 1856.4.a.y.1.2 5
4.3 odd 2 1856.4.a.bb.1.4 5
8.3 odd 2 464.4.a.l.1.2 5
8.5 even 2 29.4.a.b.1.2 5
24.5 odd 2 261.4.a.f.1.4 5
40.29 even 2 725.4.a.c.1.4 5
56.13 odd 2 1421.4.a.e.1.2 5
232.173 even 2 841.4.a.b.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.2 5 8.5 even 2
261.4.a.f.1.4 5 24.5 odd 2
464.4.a.l.1.2 5 8.3 odd 2
725.4.a.c.1.4 5 40.29 even 2
841.4.a.b.1.4 5 232.173 even 2
1421.4.a.e.1.2 5 56.13 odd 2
1856.4.a.y.1.2 5 1.1 even 1 trivial
1856.4.a.bb.1.4 5 4.3 odd 2