Properties

Label 1856.4.a.z.1.5
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-6.05843\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +O(q^{10})\) \(q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +18.6870 q^{11} +39.8582 q^{13} +88.4931 q^{15} +19.3391 q^{17} -2.14572 q^{19} -40.2506 q^{21} +103.945 q^{23} +99.9629 q^{25} -113.220 q^{27} -29.0000 q^{29} +213.167 q^{31} +110.254 q^{33} -102.323 q^{35} +226.054 q^{37} +235.164 q^{39} -191.267 q^{41} -34.3923 q^{43} +117.145 q^{45} +501.173 q^{47} -296.459 q^{49} +114.101 q^{51} -259.952 q^{53} +280.282 q^{55} -12.6598 q^{57} -280.736 q^{59} -81.0752 q^{61} -53.2826 q^{63} +597.824 q^{65} +799.332 q^{67} +613.278 q^{69} -575.901 q^{71} -22.5292 q^{73} +589.783 q^{75} -127.485 q^{77} -118.878 q^{79} -878.877 q^{81} -235.673 q^{83} +290.063 q^{85} -171.101 q^{87} +1153.23 q^{89} -271.917 q^{91} +1257.69 q^{93} -32.1832 q^{95} +295.976 q^{97} +145.951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 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
\(3\) 5.90002 1.13546 0.567730 0.823215i \(-0.307822\pi\)
0.567730 + 0.823215i \(0.307822\pi\)
\(4\) 0 0
\(5\) 14.9988 1.34153 0.670765 0.741670i \(-0.265966\pi\)
0.670765 + 0.741670i \(0.265966\pi\)
\(6\) 0 0
\(7\) −6.82211 −0.368359 −0.184180 0.982893i \(-0.558963\pi\)
−0.184180 + 0.982893i \(0.558963\pi\)
\(8\) 0 0
\(9\) 7.81028 0.289270
\(10\) 0 0
\(11\) 18.6870 0.512213 0.256107 0.966649i \(-0.417560\pi\)
0.256107 + 0.966649i \(0.417560\pi\)
\(12\) 0 0
\(13\) 39.8582 0.850360 0.425180 0.905109i \(-0.360211\pi\)
0.425180 + 0.905109i \(0.360211\pi\)
\(14\) 0 0
\(15\) 88.4931 1.52325
\(16\) 0 0
\(17\) 19.3391 0.275908 0.137954 0.990439i \(-0.455947\pi\)
0.137954 + 0.990439i \(0.455947\pi\)
\(18\) 0 0
\(19\) −2.14572 −0.0259086 −0.0129543 0.999916i \(-0.504124\pi\)
−0.0129543 + 0.999916i \(0.504124\pi\)
\(20\) 0 0
\(21\) −40.2506 −0.418257
\(22\) 0 0
\(23\) 103.945 0.942349 0.471174 0.882040i \(-0.343830\pi\)
0.471174 + 0.882040i \(0.343830\pi\)
\(24\) 0 0
\(25\) 99.9629 0.799703
\(26\) 0 0
\(27\) −113.220 −0.807006
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 213.167 1.23503 0.617514 0.786560i \(-0.288140\pi\)
0.617514 + 0.786560i \(0.288140\pi\)
\(32\) 0 0
\(33\) 110.254 0.581598
\(34\) 0 0
\(35\) −102.323 −0.494165
\(36\) 0 0
\(37\) 226.054 1.00441 0.502204 0.864749i \(-0.332523\pi\)
0.502204 + 0.864749i \(0.332523\pi\)
\(38\) 0 0
\(39\) 235.164 0.965550
\(40\) 0 0
\(41\) −191.267 −0.728556 −0.364278 0.931290i \(-0.618684\pi\)
−0.364278 + 0.931290i \(0.618684\pi\)
\(42\) 0 0
\(43\) −34.3923 −0.121971 −0.0609857 0.998139i \(-0.519424\pi\)
−0.0609857 + 0.998139i \(0.519424\pi\)
\(44\) 0 0
\(45\) 117.145 0.388064
\(46\) 0 0
\(47\) 501.173 1.55540 0.777698 0.628638i \(-0.216387\pi\)
0.777698 + 0.628638i \(0.216387\pi\)
\(48\) 0 0
\(49\) −296.459 −0.864312
\(50\) 0 0
\(51\) 114.101 0.313282
\(52\) 0 0
\(53\) −259.952 −0.673719 −0.336860 0.941555i \(-0.609365\pi\)
−0.336860 + 0.941555i \(0.609365\pi\)
\(54\) 0 0
\(55\) 280.282 0.687149
\(56\) 0 0
\(57\) −12.6598 −0.0294181
\(58\) 0 0
\(59\) −280.736 −0.619471 −0.309735 0.950823i \(-0.600240\pi\)
−0.309735 + 0.950823i \(0.600240\pi\)
\(60\) 0 0
\(61\) −81.0752 −0.170174 −0.0850870 0.996374i \(-0.527117\pi\)
−0.0850870 + 0.996374i \(0.527117\pi\)
\(62\) 0 0
\(63\) −53.2826 −0.106555
\(64\) 0 0
\(65\) 597.824 1.14078
\(66\) 0 0
\(67\) 799.332 1.45752 0.728761 0.684768i \(-0.240097\pi\)
0.728761 + 0.684768i \(0.240097\pi\)
\(68\) 0 0
\(69\) 613.278 1.07000
\(70\) 0 0
\(71\) −575.901 −0.962632 −0.481316 0.876547i \(-0.659841\pi\)
−0.481316 + 0.876547i \(0.659841\pi\)
\(72\) 0 0
\(73\) −22.5292 −0.0361211 −0.0180606 0.999837i \(-0.505749\pi\)
−0.0180606 + 0.999837i \(0.505749\pi\)
\(74\) 0 0
\(75\) 589.783 0.908031
\(76\) 0 0
\(77\) −127.485 −0.188678
\(78\) 0 0
\(79\) −118.878 −0.169302 −0.0846509 0.996411i \(-0.526977\pi\)
−0.0846509 + 0.996411i \(0.526977\pi\)
\(80\) 0 0
\(81\) −878.877 −1.20559
\(82\) 0 0
\(83\) −235.673 −0.311669 −0.155834 0.987783i \(-0.549807\pi\)
−0.155834 + 0.987783i \(0.549807\pi\)
\(84\) 0 0
\(85\) 290.063 0.370138
\(86\) 0 0
\(87\) −171.101 −0.210850
\(88\) 0 0
\(89\) 1153.23 1.37351 0.686755 0.726889i \(-0.259035\pi\)
0.686755 + 0.726889i \(0.259035\pi\)
\(90\) 0 0
\(91\) −271.917 −0.313238
\(92\) 0 0
\(93\) 1257.69 1.40232
\(94\) 0 0
\(95\) −32.1832 −0.0347571
\(96\) 0 0
\(97\) 295.976 0.309812 0.154906 0.987929i \(-0.450493\pi\)
0.154906 + 0.987929i \(0.450493\pi\)
\(98\) 0 0
\(99\) 145.951 0.148168
\(100\) 0 0
\(101\) 1163.17 1.14594 0.572971 0.819575i \(-0.305791\pi\)
0.572971 + 0.819575i \(0.305791\pi\)
\(102\) 0 0
\(103\) 790.763 0.756468 0.378234 0.925710i \(-0.376531\pi\)
0.378234 + 0.925710i \(0.376531\pi\)
\(104\) 0 0
\(105\) −603.709 −0.561104
\(106\) 0 0
\(107\) 906.764 0.819254 0.409627 0.912253i \(-0.365659\pi\)
0.409627 + 0.912253i \(0.365659\pi\)
\(108\) 0 0
\(109\) 897.545 0.788708 0.394354 0.918959i \(-0.370968\pi\)
0.394354 + 0.918959i \(0.370968\pi\)
\(110\) 0 0
\(111\) 1333.73 1.14046
\(112\) 0 0
\(113\) 871.936 0.725884 0.362942 0.931812i \(-0.381772\pi\)
0.362942 + 0.931812i \(0.381772\pi\)
\(114\) 0 0
\(115\) 1559.05 1.26419
\(116\) 0 0
\(117\) 311.304 0.245983
\(118\) 0 0
\(119\) −131.934 −0.101633
\(120\) 0 0
\(121\) −981.796 −0.737638
\(122\) 0 0
\(123\) −1128.48 −0.827247
\(124\) 0 0
\(125\) −375.526 −0.268704
\(126\) 0 0
\(127\) 1417.53 0.990433 0.495217 0.868770i \(-0.335089\pi\)
0.495217 + 0.868770i \(0.335089\pi\)
\(128\) 0 0
\(129\) −202.915 −0.138494
\(130\) 0 0
\(131\) −473.020 −0.315480 −0.157740 0.987481i \(-0.550421\pi\)
−0.157740 + 0.987481i \(0.550421\pi\)
\(132\) 0 0
\(133\) 14.6384 0.00954366
\(134\) 0 0
\(135\) −1698.16 −1.08262
\(136\) 0 0
\(137\) 1636.81 1.02075 0.510374 0.859953i \(-0.329507\pi\)
0.510374 + 0.859953i \(0.329507\pi\)
\(138\) 0 0
\(139\) −1485.21 −0.906286 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(140\) 0 0
\(141\) 2956.93 1.76609
\(142\) 0 0
\(143\) 744.831 0.435566
\(144\) 0 0
\(145\) −434.964 −0.249116
\(146\) 0 0
\(147\) −1749.11 −0.981391
\(148\) 0 0
\(149\) −1370.03 −0.753269 −0.376634 0.926362i \(-0.622919\pi\)
−0.376634 + 0.926362i \(0.622919\pi\)
\(150\) 0 0
\(151\) 948.384 0.511115 0.255557 0.966794i \(-0.417741\pi\)
0.255557 + 0.966794i \(0.417741\pi\)
\(152\) 0 0
\(153\) 151.044 0.0798117
\(154\) 0 0
\(155\) 3197.24 1.65683
\(156\) 0 0
\(157\) 1499.20 0.762098 0.381049 0.924555i \(-0.375563\pi\)
0.381049 + 0.924555i \(0.375563\pi\)
\(158\) 0 0
\(159\) −1533.72 −0.764982
\(160\) 0 0
\(161\) −709.123 −0.347123
\(162\) 0 0
\(163\) 380.266 0.182728 0.0913642 0.995818i \(-0.470877\pi\)
0.0913642 + 0.995818i \(0.470877\pi\)
\(164\) 0 0
\(165\) 1653.67 0.780231
\(166\) 0 0
\(167\) 1418.55 0.657309 0.328655 0.944450i \(-0.393405\pi\)
0.328655 + 0.944450i \(0.393405\pi\)
\(168\) 0 0
\(169\) −608.322 −0.276888
\(170\) 0 0
\(171\) −16.7587 −0.00749456
\(172\) 0 0
\(173\) −3637.40 −1.59853 −0.799266 0.600977i \(-0.794778\pi\)
−0.799266 + 0.600977i \(0.794778\pi\)
\(174\) 0 0
\(175\) −681.957 −0.294578
\(176\) 0 0
\(177\) −1656.35 −0.703384
\(178\) 0 0
\(179\) 2054.57 0.857907 0.428954 0.903326i \(-0.358882\pi\)
0.428954 + 0.903326i \(0.358882\pi\)
\(180\) 0 0
\(181\) −4423.60 −1.81660 −0.908298 0.418324i \(-0.862618\pi\)
−0.908298 + 0.418324i \(0.862618\pi\)
\(182\) 0 0
\(183\) −478.346 −0.193226
\(184\) 0 0
\(185\) 3390.53 1.34744
\(186\) 0 0
\(187\) 361.390 0.141323
\(188\) 0 0
\(189\) 772.398 0.297268
\(190\) 0 0
\(191\) −2132.86 −0.808000 −0.404000 0.914759i \(-0.632380\pi\)
−0.404000 + 0.914759i \(0.632380\pi\)
\(192\) 0 0
\(193\) 137.560 0.0513048 0.0256524 0.999671i \(-0.491834\pi\)
0.0256524 + 0.999671i \(0.491834\pi\)
\(194\) 0 0
\(195\) 3527.18 1.29531
\(196\) 0 0
\(197\) −4492.27 −1.62467 −0.812337 0.583189i \(-0.801805\pi\)
−0.812337 + 0.583189i \(0.801805\pi\)
\(198\) 0 0
\(199\) −2435.95 −0.867738 −0.433869 0.900976i \(-0.642852\pi\)
−0.433869 + 0.900976i \(0.642852\pi\)
\(200\) 0 0
\(201\) 4716.08 1.65496
\(202\) 0 0
\(203\) 197.841 0.0684026
\(204\) 0 0
\(205\) −2868.76 −0.977380
\(206\) 0 0
\(207\) 811.839 0.272593
\(208\) 0 0
\(209\) −40.0971 −0.0132707
\(210\) 0 0
\(211\) −4150.84 −1.35429 −0.677145 0.735849i \(-0.736783\pi\)
−0.677145 + 0.735849i \(0.736783\pi\)
\(212\) 0 0
\(213\) −3397.83 −1.09303
\(214\) 0 0
\(215\) −515.842 −0.163628
\(216\) 0 0
\(217\) −1454.25 −0.454934
\(218\) 0 0
\(219\) −132.923 −0.0410141
\(220\) 0 0
\(221\) 770.823 0.234621
\(222\) 0 0
\(223\) 4405.99 1.32308 0.661540 0.749910i \(-0.269903\pi\)
0.661540 + 0.749910i \(0.269903\pi\)
\(224\) 0 0
\(225\) 780.738 0.231330
\(226\) 0 0
\(227\) −1996.06 −0.583627 −0.291813 0.956475i \(-0.594259\pi\)
−0.291813 + 0.956475i \(0.594259\pi\)
\(228\) 0 0
\(229\) −4659.18 −1.34449 −0.672243 0.740330i \(-0.734669\pi\)
−0.672243 + 0.740330i \(0.734669\pi\)
\(230\) 0 0
\(231\) −752.163 −0.214237
\(232\) 0 0
\(233\) 5821.24 1.63675 0.818374 0.574686i \(-0.194876\pi\)
0.818374 + 0.574686i \(0.194876\pi\)
\(234\) 0 0
\(235\) 7516.98 2.08661
\(236\) 0 0
\(237\) −701.384 −0.192235
\(238\) 0 0
\(239\) 6555.92 1.77434 0.887170 0.461442i \(-0.152668\pi\)
0.887170 + 0.461442i \(0.152668\pi\)
\(240\) 0 0
\(241\) 2578.14 0.689097 0.344548 0.938769i \(-0.388032\pi\)
0.344548 + 0.938769i \(0.388032\pi\)
\(242\) 0 0
\(243\) −2128.46 −0.561896
\(244\) 0 0
\(245\) −4446.52 −1.15950
\(246\) 0 0
\(247\) −85.5247 −0.0220316
\(248\) 0 0
\(249\) −1390.48 −0.353887
\(250\) 0 0
\(251\) 2659.86 0.668880 0.334440 0.942417i \(-0.391453\pi\)
0.334440 + 0.942417i \(0.391453\pi\)
\(252\) 0 0
\(253\) 1942.42 0.482683
\(254\) 0 0
\(255\) 1711.38 0.420277
\(256\) 0 0
\(257\) −6610.31 −1.60443 −0.802217 0.597032i \(-0.796346\pi\)
−0.802217 + 0.597032i \(0.796346\pi\)
\(258\) 0 0
\(259\) −1542.17 −0.369983
\(260\) 0 0
\(261\) −226.498 −0.0537160
\(262\) 0 0
\(263\) 4167.41 0.977085 0.488543 0.872540i \(-0.337529\pi\)
0.488543 + 0.872540i \(0.337529\pi\)
\(264\) 0 0
\(265\) −3898.96 −0.903815
\(266\) 0 0
\(267\) 6804.10 1.55957
\(268\) 0 0
\(269\) 5180.05 1.17410 0.587050 0.809550i \(-0.300289\pi\)
0.587050 + 0.809550i \(0.300289\pi\)
\(270\) 0 0
\(271\) −4805.98 −1.07728 −0.538639 0.842537i \(-0.681061\pi\)
−0.538639 + 0.842537i \(0.681061\pi\)
\(272\) 0 0
\(273\) −1604.32 −0.355669
\(274\) 0 0
\(275\) 1868.01 0.409618
\(276\) 0 0
\(277\) 1303.90 0.282830 0.141415 0.989950i \(-0.454835\pi\)
0.141415 + 0.989950i \(0.454835\pi\)
\(278\) 0 0
\(279\) 1664.89 0.357256
\(280\) 0 0
\(281\) −4084.01 −0.867016 −0.433508 0.901150i \(-0.642724\pi\)
−0.433508 + 0.901150i \(0.642724\pi\)
\(282\) 0 0
\(283\) 4493.32 0.943816 0.471908 0.881648i \(-0.343565\pi\)
0.471908 + 0.881648i \(0.343565\pi\)
\(284\) 0 0
\(285\) −189.882 −0.0394653
\(286\) 0 0
\(287\) 1304.84 0.268370
\(288\) 0 0
\(289\) −4539.00 −0.923875
\(290\) 0 0
\(291\) 1746.26 0.351779
\(292\) 0 0
\(293\) 2591.95 0.516803 0.258402 0.966038i \(-0.416804\pi\)
0.258402 + 0.966038i \(0.416804\pi\)
\(294\) 0 0
\(295\) −4210.70 −0.831038
\(296\) 0 0
\(297\) −2115.74 −0.413359
\(298\) 0 0
\(299\) 4143.06 0.801336
\(300\) 0 0
\(301\) 234.628 0.0449293
\(302\) 0 0
\(303\) 6862.76 1.30117
\(304\) 0 0
\(305\) −1216.03 −0.228294
\(306\) 0 0
\(307\) 2355.83 0.437962 0.218981 0.975729i \(-0.429727\pi\)
0.218981 + 0.975729i \(0.429727\pi\)
\(308\) 0 0
\(309\) 4665.52 0.858940
\(310\) 0 0
\(311\) 3087.64 0.562971 0.281486 0.959565i \(-0.409173\pi\)
0.281486 + 0.959565i \(0.409173\pi\)
\(312\) 0 0
\(313\) −1669.67 −0.301518 −0.150759 0.988571i \(-0.548172\pi\)
−0.150759 + 0.988571i \(0.548172\pi\)
\(314\) 0 0
\(315\) −799.172 −0.142947
\(316\) 0 0
\(317\) −6613.47 −1.17176 −0.585882 0.810396i \(-0.699252\pi\)
−0.585882 + 0.810396i \(0.699252\pi\)
\(318\) 0 0
\(319\) −541.923 −0.0951156
\(320\) 0 0
\(321\) 5349.93 0.930230
\(322\) 0 0
\(323\) −41.4964 −0.00714837
\(324\) 0 0
\(325\) 3984.34 0.680036
\(326\) 0 0
\(327\) 5295.54 0.895547
\(328\) 0 0
\(329\) −3419.06 −0.572945
\(330\) 0 0
\(331\) 11636.2 1.93228 0.966138 0.258027i \(-0.0830722\pi\)
0.966138 + 0.258027i \(0.0830722\pi\)
\(332\) 0 0
\(333\) 1765.55 0.290545
\(334\) 0 0
\(335\) 11989.0 1.95531
\(336\) 0 0
\(337\) −11798.2 −1.90708 −0.953542 0.301262i \(-0.902592\pi\)
−0.953542 + 0.301262i \(0.902592\pi\)
\(338\) 0 0
\(339\) 5144.44 0.824212
\(340\) 0 0
\(341\) 3983.45 0.632597
\(342\) 0 0
\(343\) 4362.46 0.686736
\(344\) 0 0
\(345\) 9198.40 1.43544
\(346\) 0 0
\(347\) 625.125 0.0967103 0.0483551 0.998830i \(-0.484602\pi\)
0.0483551 + 0.998830i \(0.484602\pi\)
\(348\) 0 0
\(349\) 1820.90 0.279286 0.139643 0.990202i \(-0.455405\pi\)
0.139643 + 0.990202i \(0.455405\pi\)
\(350\) 0 0
\(351\) −4512.74 −0.686246
\(352\) 0 0
\(353\) 5114.36 0.771133 0.385567 0.922680i \(-0.374006\pi\)
0.385567 + 0.922680i \(0.374006\pi\)
\(354\) 0 0
\(355\) −8637.80 −1.29140
\(356\) 0 0
\(357\) −778.411 −0.115400
\(358\) 0 0
\(359\) −10503.5 −1.54415 −0.772077 0.635529i \(-0.780782\pi\)
−0.772077 + 0.635529i \(0.780782\pi\)
\(360\) 0 0
\(361\) −6854.40 −0.999329
\(362\) 0 0
\(363\) −5792.62 −0.837558
\(364\) 0 0
\(365\) −337.910 −0.0484576
\(366\) 0 0
\(367\) 5825.82 0.828625 0.414313 0.910135i \(-0.364022\pi\)
0.414313 + 0.910135i \(0.364022\pi\)
\(368\) 0 0
\(369\) −1493.84 −0.210749
\(370\) 0 0
\(371\) 1773.42 0.248171
\(372\) 0 0
\(373\) −12629.5 −1.75316 −0.876579 0.481257i \(-0.840180\pi\)
−0.876579 + 0.481257i \(0.840180\pi\)
\(374\) 0 0
\(375\) −2215.61 −0.305103
\(376\) 0 0
\(377\) −1155.89 −0.157908
\(378\) 0 0
\(379\) −4152.73 −0.562826 −0.281413 0.959587i \(-0.590803\pi\)
−0.281413 + 0.959587i \(0.590803\pi\)
\(380\) 0 0
\(381\) 8363.43 1.12460
\(382\) 0 0
\(383\) 7253.54 0.967725 0.483863 0.875144i \(-0.339233\pi\)
0.483863 + 0.875144i \(0.339233\pi\)
\(384\) 0 0
\(385\) −1912.11 −0.253118
\(386\) 0 0
\(387\) −268.613 −0.0352826
\(388\) 0 0
\(389\) −6053.84 −0.789053 −0.394527 0.918884i \(-0.629091\pi\)
−0.394527 + 0.918884i \(0.629091\pi\)
\(390\) 0 0
\(391\) 2010.20 0.260001
\(392\) 0 0
\(393\) −2790.83 −0.358215
\(394\) 0 0
\(395\) −1783.03 −0.227123
\(396\) 0 0
\(397\) 12932.7 1.63495 0.817473 0.575967i \(-0.195374\pi\)
0.817473 + 0.575967i \(0.195374\pi\)
\(398\) 0 0
\(399\) 86.3666 0.0108364
\(400\) 0 0
\(401\) 12673.6 1.57828 0.789141 0.614212i \(-0.210526\pi\)
0.789141 + 0.614212i \(0.210526\pi\)
\(402\) 0 0
\(403\) 8496.44 1.05022
\(404\) 0 0
\(405\) −13182.1 −1.61734
\(406\) 0 0
\(407\) 4224.28 0.514471
\(408\) 0 0
\(409\) −13335.1 −1.61218 −0.806088 0.591795i \(-0.798419\pi\)
−0.806088 + 0.591795i \(0.798419\pi\)
\(410\) 0 0
\(411\) 9657.23 1.15902
\(412\) 0 0
\(413\) 1915.21 0.228188
\(414\) 0 0
\(415\) −3534.81 −0.418113
\(416\) 0 0
\(417\) −8762.76 −1.02905
\(418\) 0 0
\(419\) −4564.25 −0.532167 −0.266084 0.963950i \(-0.585730\pi\)
−0.266084 + 0.963950i \(0.585730\pi\)
\(420\) 0 0
\(421\) 6992.67 0.809506 0.404753 0.914426i \(-0.367358\pi\)
0.404753 + 0.914426i \(0.367358\pi\)
\(422\) 0 0
\(423\) 3914.30 0.449929
\(424\) 0 0
\(425\) 1933.20 0.220644
\(426\) 0 0
\(427\) 553.104 0.0626852
\(428\) 0 0
\(429\) 4394.52 0.494567
\(430\) 0 0
\(431\) 9630.58 1.07631 0.538154 0.842847i \(-0.319122\pi\)
0.538154 + 0.842847i \(0.319122\pi\)
\(432\) 0 0
\(433\) 13504.7 1.49883 0.749415 0.662100i \(-0.230335\pi\)
0.749415 + 0.662100i \(0.230335\pi\)
\(434\) 0 0
\(435\) −2566.30 −0.282861
\(436\) 0 0
\(437\) −223.037 −0.0244149
\(438\) 0 0
\(439\) −2381.38 −0.258899 −0.129450 0.991586i \(-0.541321\pi\)
−0.129450 + 0.991586i \(0.541321\pi\)
\(440\) 0 0
\(441\) −2315.43 −0.250019
\(442\) 0 0
\(443\) −7708.83 −0.826766 −0.413383 0.910557i \(-0.635653\pi\)
−0.413383 + 0.910557i \(0.635653\pi\)
\(444\) 0 0
\(445\) 17297.1 1.84260
\(446\) 0 0
\(447\) −8083.20 −0.855307
\(448\) 0 0
\(449\) −2065.19 −0.217065 −0.108532 0.994093i \(-0.534615\pi\)
−0.108532 + 0.994093i \(0.534615\pi\)
\(450\) 0 0
\(451\) −3574.20 −0.373176
\(452\) 0 0
\(453\) 5595.49 0.580351
\(454\) 0 0
\(455\) −4078.42 −0.420218
\(456\) 0 0
\(457\) −4959.73 −0.507673 −0.253836 0.967247i \(-0.581692\pi\)
−0.253836 + 0.967247i \(0.581692\pi\)
\(458\) 0 0
\(459\) −2189.57 −0.222659
\(460\) 0 0
\(461\) −16339.8 −1.65080 −0.825402 0.564545i \(-0.809052\pi\)
−0.825402 + 0.564545i \(0.809052\pi\)
\(462\) 0 0
\(463\) 7546.94 0.757529 0.378765 0.925493i \(-0.376349\pi\)
0.378765 + 0.925493i \(0.376349\pi\)
\(464\) 0 0
\(465\) 18863.8 1.88126
\(466\) 0 0
\(467\) −2409.12 −0.238717 −0.119358 0.992851i \(-0.538084\pi\)
−0.119358 + 0.992851i \(0.538084\pi\)
\(468\) 0 0
\(469\) −5453.13 −0.536891
\(470\) 0 0
\(471\) 8845.33 0.865332
\(472\) 0 0
\(473\) −642.689 −0.0624754
\(474\) 0 0
\(475\) −214.493 −0.0207192
\(476\) 0 0
\(477\) −2030.30 −0.194887
\(478\) 0 0
\(479\) 17370.8 1.65698 0.828488 0.560007i \(-0.189202\pi\)
0.828488 + 0.560007i \(0.189202\pi\)
\(480\) 0 0
\(481\) 9010.12 0.854108
\(482\) 0 0
\(483\) −4183.84 −0.394144
\(484\) 0 0
\(485\) 4439.27 0.415622
\(486\) 0 0
\(487\) −6367.45 −0.592478 −0.296239 0.955114i \(-0.595732\pi\)
−0.296239 + 0.955114i \(0.595732\pi\)
\(488\) 0 0
\(489\) 2243.58 0.207481
\(490\) 0 0
\(491\) 6282.71 0.577463 0.288732 0.957410i \(-0.406766\pi\)
0.288732 + 0.957410i \(0.406766\pi\)
\(492\) 0 0
\(493\) −560.835 −0.0512347
\(494\) 0 0
\(495\) 2189.08 0.198771
\(496\) 0 0
\(497\) 3928.86 0.354594
\(498\) 0 0
\(499\) −7147.33 −0.641200 −0.320600 0.947215i \(-0.603884\pi\)
−0.320600 + 0.947215i \(0.603884\pi\)
\(500\) 0 0
\(501\) 8369.48 0.746349
\(502\) 0 0
\(503\) −14496.5 −1.28502 −0.642512 0.766276i \(-0.722108\pi\)
−0.642512 + 0.766276i \(0.722108\pi\)
\(504\) 0 0
\(505\) 17446.2 1.53732
\(506\) 0 0
\(507\) −3589.11 −0.314395
\(508\) 0 0
\(509\) 4185.51 0.364478 0.182239 0.983254i \(-0.441666\pi\)
0.182239 + 0.983254i \(0.441666\pi\)
\(510\) 0 0
\(511\) 153.697 0.0133055
\(512\) 0 0
\(513\) 242.938 0.0209084
\(514\) 0 0
\(515\) 11860.5 1.01483
\(516\) 0 0
\(517\) 9365.43 0.796695
\(518\) 0 0
\(519\) −21460.7 −1.81507
\(520\) 0 0
\(521\) −12392.3 −1.04206 −0.521032 0.853537i \(-0.674453\pi\)
−0.521032 + 0.853537i \(0.674453\pi\)
\(522\) 0 0
\(523\) −18806.4 −1.57237 −0.786184 0.617993i \(-0.787946\pi\)
−0.786184 + 0.617993i \(0.787946\pi\)
\(524\) 0 0
\(525\) −4023.56 −0.334481
\(526\) 0 0
\(527\) 4122.46 0.340753
\(528\) 0 0
\(529\) −1362.45 −0.111979
\(530\) 0 0
\(531\) −2192.63 −0.179194
\(532\) 0 0
\(533\) −7623.54 −0.619535
\(534\) 0 0
\(535\) 13600.3 1.09905
\(536\) 0 0
\(537\) 12122.0 0.974120
\(538\) 0 0
\(539\) −5539.93 −0.442712
\(540\) 0 0
\(541\) −5080.90 −0.403780 −0.201890 0.979408i \(-0.564708\pi\)
−0.201890 + 0.979408i \(0.564708\pi\)
\(542\) 0 0
\(543\) −26099.4 −2.06267
\(544\) 0 0
\(545\) 13462.1 1.05808
\(546\) 0 0
\(547\) −20578.0 −1.60850 −0.804252 0.594288i \(-0.797434\pi\)
−0.804252 + 0.594288i \(0.797434\pi\)
\(548\) 0 0
\(549\) −633.220 −0.0492262
\(550\) 0 0
\(551\) 62.2260 0.00481110
\(552\) 0 0
\(553\) 811.000 0.0623639
\(554\) 0 0
\(555\) 20004.2 1.52997
\(556\) 0 0
\(557\) 3440.76 0.261741 0.130871 0.991399i \(-0.458223\pi\)
0.130871 + 0.991399i \(0.458223\pi\)
\(558\) 0 0
\(559\) −1370.82 −0.103720
\(560\) 0 0
\(561\) 2132.21 0.160467
\(562\) 0 0
\(563\) −2647.62 −0.198195 −0.0990975 0.995078i \(-0.531596\pi\)
−0.0990975 + 0.995078i \(0.531596\pi\)
\(564\) 0 0
\(565\) 13078.0 0.973795
\(566\) 0 0
\(567\) 5995.79 0.444091
\(568\) 0 0
\(569\) −24385.8 −1.79667 −0.898334 0.439313i \(-0.855222\pi\)
−0.898334 + 0.439313i \(0.855222\pi\)
\(570\) 0 0
\(571\) −15785.4 −1.15692 −0.578459 0.815712i \(-0.696346\pi\)
−0.578459 + 0.815712i \(0.696346\pi\)
\(572\) 0 0
\(573\) −12583.9 −0.917452
\(574\) 0 0
\(575\) 10390.6 0.753599
\(576\) 0 0
\(577\) −4415.15 −0.318553 −0.159277 0.987234i \(-0.550916\pi\)
−0.159277 + 0.987234i \(0.550916\pi\)
\(578\) 0 0
\(579\) 811.610 0.0582545
\(580\) 0 0
\(581\) 1607.79 0.114806
\(582\) 0 0
\(583\) −4857.72 −0.345088
\(584\) 0 0
\(585\) 4669.17 0.329994
\(586\) 0 0
\(587\) −13595.4 −0.955950 −0.477975 0.878373i \(-0.658629\pi\)
−0.477975 + 0.878373i \(0.658629\pi\)
\(588\) 0 0
\(589\) −457.397 −0.0319978
\(590\) 0 0
\(591\) −26504.5 −1.84475
\(592\) 0 0
\(593\) 23514.8 1.62839 0.814195 0.580591i \(-0.197178\pi\)
0.814195 + 0.580591i \(0.197178\pi\)
\(594\) 0 0
\(595\) −1978.84 −0.136344
\(596\) 0 0
\(597\) −14372.2 −0.985282
\(598\) 0 0
\(599\) −21041.9 −1.43531 −0.717654 0.696400i \(-0.754784\pi\)
−0.717654 + 0.696400i \(0.754784\pi\)
\(600\) 0 0
\(601\) 5250.69 0.356373 0.178186 0.983997i \(-0.442977\pi\)
0.178186 + 0.983997i \(0.442977\pi\)
\(602\) 0 0
\(603\) 6243.01 0.421617
\(604\) 0 0
\(605\) −14725.7 −0.989563
\(606\) 0 0
\(607\) 19758.4 1.32120 0.660599 0.750739i \(-0.270302\pi\)
0.660599 + 0.750739i \(0.270302\pi\)
\(608\) 0 0
\(609\) 1167.27 0.0776684
\(610\) 0 0
\(611\) 19975.9 1.32265
\(612\) 0 0
\(613\) 16105.3 1.06115 0.530575 0.847638i \(-0.321976\pi\)
0.530575 + 0.847638i \(0.321976\pi\)
\(614\) 0 0
\(615\) −16925.8 −1.10978
\(616\) 0 0
\(617\) −8320.76 −0.542919 −0.271460 0.962450i \(-0.587506\pi\)
−0.271460 + 0.962450i \(0.587506\pi\)
\(618\) 0 0
\(619\) 2802.58 0.181979 0.0909896 0.995852i \(-0.470997\pi\)
0.0909896 + 0.995852i \(0.470997\pi\)
\(620\) 0 0
\(621\) −11768.6 −0.760481
\(622\) 0 0
\(623\) −7867.47 −0.505945
\(624\) 0 0
\(625\) −18127.8 −1.16018
\(626\) 0 0
\(627\) −236.574 −0.0150684
\(628\) 0 0
\(629\) 4371.69 0.277124
\(630\) 0 0
\(631\) 17958.7 1.13300 0.566502 0.824060i \(-0.308296\pi\)
0.566502 + 0.824060i \(0.308296\pi\)
\(632\) 0 0
\(633\) −24490.0 −1.53774
\(634\) 0 0
\(635\) 21261.1 1.32870
\(636\) 0 0
\(637\) −11816.3 −0.734976
\(638\) 0 0
\(639\) −4497.95 −0.278460
\(640\) 0 0
\(641\) 11480.6 0.707417 0.353709 0.935356i \(-0.384920\pi\)
0.353709 + 0.935356i \(0.384920\pi\)
\(642\) 0 0
\(643\) −15144.0 −0.928802 −0.464401 0.885625i \(-0.653730\pi\)
−0.464401 + 0.885625i \(0.653730\pi\)
\(644\) 0 0
\(645\) −3043.48 −0.185793
\(646\) 0 0
\(647\) 13334.1 0.810227 0.405113 0.914266i \(-0.367232\pi\)
0.405113 + 0.914266i \(0.367232\pi\)
\(648\) 0 0
\(649\) −5246.12 −0.317301
\(650\) 0 0
\(651\) −8580.08 −0.516559
\(652\) 0 0
\(653\) 16543.2 0.991402 0.495701 0.868493i \(-0.334911\pi\)
0.495701 + 0.868493i \(0.334911\pi\)
\(654\) 0 0
\(655\) −7094.71 −0.423227
\(656\) 0 0
\(657\) −175.959 −0.0104487
\(658\) 0 0
\(659\) 9171.83 0.542161 0.271080 0.962557i \(-0.412619\pi\)
0.271080 + 0.962557i \(0.412619\pi\)
\(660\) 0 0
\(661\) −16761.6 −0.986312 −0.493156 0.869941i \(-0.664157\pi\)
−0.493156 + 0.869941i \(0.664157\pi\)
\(662\) 0 0
\(663\) 4547.88 0.266403
\(664\) 0 0
\(665\) 219.557 0.0128031
\(666\) 0 0
\(667\) −3014.40 −0.174990
\(668\) 0 0
\(669\) 25995.4 1.50230
\(670\) 0 0
\(671\) −1515.05 −0.0871654
\(672\) 0 0
\(673\) 640.780 0.0367017 0.0183508 0.999832i \(-0.494158\pi\)
0.0183508 + 0.999832i \(0.494158\pi\)
\(674\) 0 0
\(675\) −11317.8 −0.645365
\(676\) 0 0
\(677\) 30880.1 1.75305 0.876527 0.481353i \(-0.159855\pi\)
0.876527 + 0.481353i \(0.159855\pi\)
\(678\) 0 0
\(679\) −2019.18 −0.114122
\(680\) 0 0
\(681\) −11776.8 −0.662685
\(682\) 0 0
\(683\) 20491.4 1.14799 0.573997 0.818857i \(-0.305392\pi\)
0.573997 + 0.818857i \(0.305392\pi\)
\(684\) 0 0
\(685\) 24550.2 1.36936
\(686\) 0 0
\(687\) −27489.3 −1.52661
\(688\) 0 0
\(689\) −10361.2 −0.572904
\(690\) 0 0
\(691\) −20654.5 −1.13710 −0.568549 0.822649i \(-0.692495\pi\)
−0.568549 + 0.822649i \(0.692495\pi\)
\(692\) 0 0
\(693\) −995.691 −0.0545789
\(694\) 0 0
\(695\) −22276.3 −1.21581
\(696\) 0 0
\(697\) −3698.93 −0.201014
\(698\) 0 0
\(699\) 34345.5 1.85846
\(700\) 0 0
\(701\) −11890.2 −0.640636 −0.320318 0.947310i \(-0.603790\pi\)
−0.320318 + 0.947310i \(0.603790\pi\)
\(702\) 0 0
\(703\) −485.050 −0.0260228
\(704\) 0 0
\(705\) 44350.4 2.36926
\(706\) 0 0
\(707\) −7935.30 −0.422118
\(708\) 0 0
\(709\) −10009.6 −0.530209 −0.265105 0.964220i \(-0.585406\pi\)
−0.265105 + 0.964220i \(0.585406\pi\)
\(710\) 0 0
\(711\) −928.472 −0.0489739
\(712\) 0 0
\(713\) 22157.6 1.16383
\(714\) 0 0
\(715\) 11171.5 0.584324
\(716\) 0 0
\(717\) 38680.1 2.01469
\(718\) 0 0
\(719\) −25825.8 −1.33956 −0.669778 0.742561i \(-0.733611\pi\)
−0.669778 + 0.742561i \(0.733611\pi\)
\(720\) 0 0
\(721\) −5394.67 −0.278652
\(722\) 0 0
\(723\) 15211.1 0.782442
\(724\) 0 0
\(725\) −2898.92 −0.148501
\(726\) 0 0
\(727\) −22228.4 −1.13398 −0.566991 0.823724i \(-0.691893\pi\)
−0.566991 + 0.823724i \(0.691893\pi\)
\(728\) 0 0
\(729\) 11171.7 0.567582
\(730\) 0 0
\(731\) −665.117 −0.0336528
\(732\) 0 0
\(733\) −13564.3 −0.683506 −0.341753 0.939790i \(-0.611021\pi\)
−0.341753 + 0.939790i \(0.611021\pi\)
\(734\) 0 0
\(735\) −26234.5 −1.31657
\(736\) 0 0
\(737\) 14937.1 0.746562
\(738\) 0 0
\(739\) 5379.12 0.267759 0.133880 0.990998i \(-0.457256\pi\)
0.133880 + 0.990998i \(0.457256\pi\)
\(740\) 0 0
\(741\) −504.598 −0.0250160
\(742\) 0 0
\(743\) 17556.8 0.866885 0.433442 0.901181i \(-0.357299\pi\)
0.433442 + 0.901181i \(0.357299\pi\)
\(744\) 0 0
\(745\) −20548.7 −1.01053
\(746\) 0 0
\(747\) −1840.67 −0.0901562
\(748\) 0 0
\(749\) −6186.04 −0.301780
\(750\) 0 0
\(751\) −21609.7 −1.05000 −0.525000 0.851102i \(-0.675935\pi\)
−0.525000 + 0.851102i \(0.675935\pi\)
\(752\) 0 0
\(753\) 15693.2 0.759486
\(754\) 0 0
\(755\) 14224.6 0.685676
\(756\) 0 0
\(757\) −25756.2 −1.23662 −0.618311 0.785933i \(-0.712183\pi\)
−0.618311 + 0.785933i \(0.712183\pi\)
\(758\) 0 0
\(759\) 11460.3 0.548068
\(760\) 0 0
\(761\) 11264.3 0.536572 0.268286 0.963339i \(-0.413543\pi\)
0.268286 + 0.963339i \(0.413543\pi\)
\(762\) 0 0
\(763\) −6123.15 −0.290528
\(764\) 0 0
\(765\) 2265.47 0.107070
\(766\) 0 0
\(767\) −11189.7 −0.526773
\(768\) 0 0
\(769\) −7851.65 −0.368189 −0.184095 0.982909i \(-0.558935\pi\)
−0.184095 + 0.982909i \(0.558935\pi\)
\(770\) 0 0
\(771\) −39001.0 −1.82177
\(772\) 0 0
\(773\) −1339.59 −0.0623308 −0.0311654 0.999514i \(-0.509922\pi\)
−0.0311654 + 0.999514i \(0.509922\pi\)
\(774\) 0 0
\(775\) 21308.7 0.987655
\(776\) 0 0
\(777\) −9098.81 −0.420101
\(778\) 0 0
\(779\) 410.405 0.0188758
\(780\) 0 0
\(781\) −10761.9 −0.493073
\(782\) 0 0
\(783\) 3283.37 0.149857
\(784\) 0 0
\(785\) 22486.2 1.02238
\(786\) 0 0
\(787\) 4123.15 0.186753 0.0933764 0.995631i \(-0.470234\pi\)
0.0933764 + 0.995631i \(0.470234\pi\)
\(788\) 0 0
\(789\) 24587.8 1.10944
\(790\) 0 0
\(791\) −5948.44 −0.267386
\(792\) 0 0
\(793\) −3231.51 −0.144709
\(794\) 0 0
\(795\) −23003.9 −1.02625
\(796\) 0 0
\(797\) −27748.5 −1.23325 −0.616626 0.787257i \(-0.711501\pi\)
−0.616626 + 0.787257i \(0.711501\pi\)
\(798\) 0 0
\(799\) 9692.26 0.429146
\(800\) 0 0
\(801\) 9007.06 0.397315
\(802\) 0 0
\(803\) −421.003 −0.0185017
\(804\) 0 0
\(805\) −10636.0 −0.465676
\(806\) 0 0
\(807\) 30562.4 1.33314
\(808\) 0 0
\(809\) −39614.9 −1.72162 −0.860808 0.508930i \(-0.830041\pi\)
−0.860808 + 0.508930i \(0.830041\pi\)
\(810\) 0 0
\(811\) 15565.2 0.673944 0.336972 0.941515i \(-0.390597\pi\)
0.336972 + 0.941515i \(0.390597\pi\)
\(812\) 0 0
\(813\) −28355.4 −1.22321
\(814\) 0 0
\(815\) 5703.52 0.245136
\(816\) 0 0
\(817\) 73.7963 0.00316011
\(818\) 0 0
\(819\) −2123.75 −0.0906102
\(820\) 0 0
\(821\) −42319.2 −1.79897 −0.899483 0.436955i \(-0.856057\pi\)
−0.899483 + 0.436955i \(0.856057\pi\)
\(822\) 0 0
\(823\) −623.931 −0.0264263 −0.0132132 0.999913i \(-0.504206\pi\)
−0.0132132 + 0.999913i \(0.504206\pi\)
\(824\) 0 0
\(825\) 11021.3 0.465105
\(826\) 0 0
\(827\) −11493.7 −0.483283 −0.241642 0.970366i \(-0.577686\pi\)
−0.241642 + 0.970366i \(0.577686\pi\)
\(828\) 0 0
\(829\) 13934.2 0.583783 0.291892 0.956451i \(-0.405715\pi\)
0.291892 + 0.956451i \(0.405715\pi\)
\(830\) 0 0
\(831\) 7693.05 0.321142
\(832\) 0 0
\(833\) −5733.26 −0.238470
\(834\) 0 0
\(835\) 21276.5 0.881800
\(836\) 0 0
\(837\) −24134.7 −0.996675
\(838\) 0 0
\(839\) 19231.4 0.791351 0.395675 0.918390i \(-0.370511\pi\)
0.395675 + 0.918390i \(0.370511\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −24095.7 −0.984462
\(844\) 0 0
\(845\) −9124.07 −0.371453
\(846\) 0 0
\(847\) 6697.92 0.271716
\(848\) 0 0
\(849\) 26510.7 1.07167
\(850\) 0 0
\(851\) 23497.2 0.946502
\(852\) 0 0
\(853\) −31388.4 −1.25993 −0.629964 0.776624i \(-0.716930\pi\)
−0.629964 + 0.776624i \(0.716930\pi\)
\(854\) 0 0
\(855\) −251.360 −0.0100542
\(856\) 0 0
\(857\) −18862.4 −0.751839 −0.375920 0.926652i \(-0.622673\pi\)
−0.375920 + 0.926652i \(0.622673\pi\)
\(858\) 0 0
\(859\) −41839.3 −1.66186 −0.830931 0.556375i \(-0.812192\pi\)
−0.830931 + 0.556375i \(0.812192\pi\)
\(860\) 0 0
\(861\) 7698.59 0.304724
\(862\) 0 0
\(863\) −22898.2 −0.903204 −0.451602 0.892220i \(-0.649147\pi\)
−0.451602 + 0.892220i \(0.649147\pi\)
\(864\) 0 0
\(865\) −54556.4 −2.14448
\(866\) 0 0
\(867\) −26780.2 −1.04902
\(868\) 0 0
\(869\) −2221.48 −0.0867186
\(870\) 0 0
\(871\) 31860.0 1.23942
\(872\) 0 0
\(873\) 2311.65 0.0896192
\(874\) 0 0
\(875\) 2561.88 0.0989797
\(876\) 0 0
\(877\) 15485.0 0.596226 0.298113 0.954531i \(-0.403643\pi\)
0.298113 + 0.954531i \(0.403643\pi\)
\(878\) 0 0
\(879\) 15292.6 0.586809
\(880\) 0 0
\(881\) −28152.8 −1.07661 −0.538305 0.842750i \(-0.680935\pi\)
−0.538305 + 0.842750i \(0.680935\pi\)
\(882\) 0 0
\(883\) 40543.4 1.54518 0.772590 0.634906i \(-0.218961\pi\)
0.772590 + 0.634906i \(0.218961\pi\)
\(884\) 0 0
\(885\) −24843.2 −0.943611
\(886\) 0 0
\(887\) −45338.8 −1.71627 −0.858133 0.513428i \(-0.828375\pi\)
−0.858133 + 0.513428i \(0.828375\pi\)
\(888\) 0 0
\(889\) −9670.51 −0.364835
\(890\) 0 0
\(891\) −16423.6 −0.617520
\(892\) 0 0
\(893\) −1075.38 −0.0402981
\(894\) 0 0
\(895\) 30815.9 1.15091
\(896\) 0 0
\(897\) 24444.2 0.909885
\(898\) 0 0
\(899\) −6181.83 −0.229339
\(900\) 0 0
\(901\) −5027.24 −0.185884
\(902\) 0 0
\(903\) 1384.31 0.0510154
\(904\) 0 0
\(905\) −66348.6 −2.43702
\(906\) 0 0
\(907\) −31106.8 −1.13879 −0.569396 0.822063i \(-0.692823\pi\)
−0.569396 + 0.822063i \(0.692823\pi\)
\(908\) 0 0
\(909\) 9084.72 0.331486
\(910\) 0 0
\(911\) 36007.5 1.30953 0.654765 0.755833i \(-0.272768\pi\)
0.654765 + 0.755833i \(0.272768\pi\)
\(912\) 0 0
\(913\) −4404.03 −0.159641
\(914\) 0 0
\(915\) −7174.59 −0.259218
\(916\) 0 0
\(917\) 3226.99 0.116210
\(918\) 0 0
\(919\) −47066.5 −1.68942 −0.844711 0.535222i \(-0.820228\pi\)
−0.844711 + 0.535222i \(0.820228\pi\)
\(920\) 0 0
\(921\) 13899.4 0.497288
\(922\) 0 0
\(923\) −22954.4 −0.818584
\(924\) 0 0
\(925\) 22597.0 0.803228
\(926\) 0 0
\(927\) 6176.08 0.218823
\(928\) 0 0
\(929\) −25359.2 −0.895596 −0.447798 0.894135i \(-0.647792\pi\)
−0.447798 + 0.894135i \(0.647792\pi\)
\(930\) 0 0
\(931\) 636.119 0.0223931
\(932\) 0 0
\(933\) 18217.1 0.639231
\(934\) 0 0
\(935\) 5420.41 0.189590
\(936\) 0 0
\(937\) 12778.7 0.445530 0.222765 0.974872i \(-0.428492\pi\)
0.222765 + 0.974872i \(0.428492\pi\)
\(938\) 0 0
\(939\) −9851.06 −0.342361
\(940\) 0 0
\(941\) 29785.8 1.03187 0.515935 0.856628i \(-0.327445\pi\)
0.515935 + 0.856628i \(0.327445\pi\)
\(942\) 0 0
\(943\) −19881.2 −0.686554
\(944\) 0 0
\(945\) 11585.0 0.398794
\(946\) 0 0
\(947\) 3498.58 0.120051 0.0600256 0.998197i \(-0.480882\pi\)
0.0600256 + 0.998197i \(0.480882\pi\)
\(948\) 0 0
\(949\) −897.974 −0.0307160
\(950\) 0 0
\(951\) −39019.6 −1.33049
\(952\) 0 0
\(953\) −18608.5 −0.632516 −0.316258 0.948673i \(-0.602427\pi\)
−0.316258 + 0.948673i \(0.602427\pi\)
\(954\) 0 0
\(955\) −31990.2 −1.08396
\(956\) 0 0
\(957\) −3197.36 −0.108000
\(958\) 0 0
\(959\) −11166.5 −0.376002
\(960\) 0 0
\(961\) 15649.0 0.525293
\(962\) 0 0
\(963\) 7082.08 0.236985
\(964\) 0 0
\(965\) 2063.24 0.0688269
\(966\) 0 0
\(967\) 38886.4 1.29318 0.646589 0.762838i \(-0.276195\pi\)
0.646589 + 0.762838i \(0.276195\pi\)
\(968\) 0 0
\(969\) −244.830 −0.00811669
\(970\) 0 0
\(971\) −30724.6 −1.01545 −0.507724 0.861520i \(-0.669513\pi\)
−0.507724 + 0.861520i \(0.669513\pi\)
\(972\) 0 0
\(973\) 10132.2 0.333839
\(974\) 0 0
\(975\) 23507.7 0.772153
\(976\) 0 0
\(977\) −17594.0 −0.576133 −0.288067 0.957610i \(-0.593012\pi\)
−0.288067 + 0.957610i \(0.593012\pi\)
\(978\) 0 0
\(979\) 21550.5 0.703530
\(980\) 0 0
\(981\) 7010.08 0.228149
\(982\) 0 0
\(983\) −26914.6 −0.873290 −0.436645 0.899634i \(-0.643833\pi\)
−0.436645 + 0.899634i \(0.643833\pi\)
\(984\) 0 0
\(985\) −67378.4 −2.17955
\(986\) 0 0
\(987\) −20172.5 −0.650556
\(988\) 0 0
\(989\) −3574.90 −0.114940
\(990\) 0 0
\(991\) 47353.1 1.51788 0.758940 0.651160i \(-0.225717\pi\)
0.758940 + 0.651160i \(0.225717\pi\)
\(992\) 0 0
\(993\) 68653.8 2.19402
\(994\) 0 0
\(995\) −36536.2 −1.16410
\(996\) 0 0
\(997\) 20910.2 0.664227 0.332113 0.943240i \(-0.392238\pi\)
0.332113 + 0.943240i \(0.392238\pi\)
\(998\) 0 0
\(999\) −25593.8 −0.810563
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.z.1.5 5
4.3 odd 2 1856.4.a.ba.1.1 5
8.3 odd 2 464.4.a.m.1.5 5
8.5 even 2 232.4.a.d.1.1 5
24.5 odd 2 2088.4.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.d.1.1 5 8.5 even 2
464.4.a.m.1.5 5 8.3 odd 2
1856.4.a.z.1.5 5 1.1 even 1 trivial
1856.4.a.ba.1.1 5 4.3 odd 2
2088.4.a.f.1.5 5 24.5 odd 2