Properties

Label 1521.4.a.x.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5054412.1
Defining polynomial: \( x^{4} - 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(5.21898\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.21898 q^{2} +19.2377 q^{4} -5.83936 q^{5} +31.3139 q^{7} +58.6495 q^{8} +O(q^{10})\) \(q+5.21898 q^{2} +19.2377 q^{4} -5.83936 q^{5} +31.3139 q^{7} +58.6495 q^{8} -30.4755 q^{10} -16.2773 q^{11} +163.426 q^{14} +152.189 q^{16} +54.0000 q^{17} -66.3500 q^{19} -112.336 q^{20} -84.9510 q^{22} +182.853 q^{23} -90.9019 q^{25} +602.408 q^{28} +164.853 q^{29} +58.9055 q^{31} +325.073 q^{32} +281.825 q^{34} -182.853 q^{35} +110.366 q^{37} -346.279 q^{38} -342.475 q^{40} +55.0357 q^{41} -113.147 q^{43} -313.139 q^{44} +954.305 q^{46} +514.089 q^{47} +637.559 q^{49} -474.415 q^{50} -242.559 q^{53} +95.0490 q^{55} +1836.54 q^{56} +860.364 q^{58} -265.036 q^{59} -468.098 q^{61} +307.426 q^{62} +479.042 q^{64} -852.919 q^{67} +1038.84 q^{68} -954.305 q^{70} +165.619 q^{71} -315.325 q^{73} +576.000 q^{74} -1276.42 q^{76} -509.706 q^{77} +479.608 q^{79} -888.684 q^{80} +287.230 q^{82} +574.235 q^{83} -315.325 q^{85} -590.512 q^{86} -954.657 q^{88} +66.7144 q^{89} +3517.68 q^{92} +2683.02 q^{94} +387.441 q^{95} +1438.25 q^{97} +3327.40 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.21898 1.84519 0.922594 0.385773i \(-0.126065\pi\)
0.922594 + 0.385773i \(0.126065\pi\)
\(3\) 0 0
\(4\) 19.2377 2.40472
\(5\) −5.83936 −0.522288 −0.261144 0.965300i \(-0.584100\pi\)
−0.261144 + 0.965300i \(0.584100\pi\)
\(6\) 0 0
\(7\) 31.3139 1.69079 0.845395 0.534141i \(-0.179365\pi\)
0.845395 + 0.534141i \(0.179365\pi\)
\(8\) 58.6495 2.59197
\(9\) 0 0
\(10\) −30.4755 −0.963719
\(11\) −16.2773 −0.446163 −0.223082 0.974800i \(-0.571612\pi\)
−0.223082 + 0.974800i \(0.571612\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 163.426 3.11983
\(15\) 0 0
\(16\) 152.189 2.37795
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) −66.3500 −0.801144 −0.400572 0.916265i \(-0.631189\pi\)
−0.400572 + 0.916265i \(0.631189\pi\)
\(20\) −112.336 −1.25595
\(21\) 0 0
\(22\) −84.9510 −0.823255
\(23\) 182.853 1.65772 0.828858 0.559459i \(-0.188991\pi\)
0.828858 + 0.559459i \(0.188991\pi\)
\(24\) 0 0
\(25\) −90.9019 −0.727215
\(26\) 0 0
\(27\) 0 0
\(28\) 602.408 4.06587
\(29\) 164.853 1.05560 0.527800 0.849369i \(-0.323017\pi\)
0.527800 + 0.849369i \(0.323017\pi\)
\(30\) 0 0
\(31\) 58.9055 0.341282 0.170641 0.985333i \(-0.445416\pi\)
0.170641 + 0.985333i \(0.445416\pi\)
\(32\) 325.073 1.79579
\(33\) 0 0
\(34\) 281.825 1.42155
\(35\) −182.853 −0.883079
\(36\) 0 0
\(37\) 110.366 0.490382 0.245191 0.969475i \(-0.421149\pi\)
0.245191 + 0.969475i \(0.421149\pi\)
\(38\) −346.279 −1.47826
\(39\) 0 0
\(40\) −342.475 −1.35375
\(41\) 55.0357 0.209637 0.104819 0.994491i \(-0.466574\pi\)
0.104819 + 0.994491i \(0.466574\pi\)
\(42\) 0 0
\(43\) −113.147 −0.401274 −0.200637 0.979666i \(-0.564301\pi\)
−0.200637 + 0.979666i \(0.564301\pi\)
\(44\) −313.139 −1.07290
\(45\) 0 0
\(46\) 954.305 3.05880
\(47\) 514.089 1.59548 0.797740 0.603001i \(-0.206029\pi\)
0.797740 + 0.603001i \(0.206029\pi\)
\(48\) 0 0
\(49\) 637.559 1.85877
\(50\) −474.415 −1.34185
\(51\) 0 0
\(52\) 0 0
\(53\) −242.559 −0.628641 −0.314321 0.949317i \(-0.601777\pi\)
−0.314321 + 0.949317i \(0.601777\pi\)
\(54\) 0 0
\(55\) 95.0490 0.233026
\(56\) 1836.54 4.38247
\(57\) 0 0
\(58\) 860.364 1.94778
\(59\) −265.036 −0.584825 −0.292413 0.956292i \(-0.594458\pi\)
−0.292413 + 0.956292i \(0.594458\pi\)
\(60\) 0 0
\(61\) −468.098 −0.982522 −0.491261 0.871013i \(-0.663464\pi\)
−0.491261 + 0.871013i \(0.663464\pi\)
\(62\) 307.426 0.629729
\(63\) 0 0
\(64\) 479.042 0.935628
\(65\) 0 0
\(66\) 0 0
\(67\) −852.919 −1.55523 −0.777617 0.628739i \(-0.783572\pi\)
−0.777617 + 0.628739i \(0.783572\pi\)
\(68\) 1038.84 1.85261
\(69\) 0 0
\(70\) −954.305 −1.62945
\(71\) 165.619 0.276836 0.138418 0.990374i \(-0.455798\pi\)
0.138418 + 0.990374i \(0.455798\pi\)
\(72\) 0 0
\(73\) −315.325 −0.505562 −0.252781 0.967524i \(-0.581345\pi\)
−0.252781 + 0.967524i \(0.581345\pi\)
\(74\) 576.000 0.904846
\(75\) 0 0
\(76\) −1276.42 −1.92653
\(77\) −509.706 −0.754368
\(78\) 0 0
\(79\) 479.608 0.683039 0.341519 0.939875i \(-0.389058\pi\)
0.341519 + 0.939875i \(0.389058\pi\)
\(80\) −888.684 −1.24197
\(81\) 0 0
\(82\) 287.230 0.386820
\(83\) 574.235 0.759404 0.379702 0.925109i \(-0.376027\pi\)
0.379702 + 0.925109i \(0.376027\pi\)
\(84\) 0 0
\(85\) −315.325 −0.402374
\(86\) −590.512 −0.740426
\(87\) 0 0
\(88\) −954.657 −1.15644
\(89\) 66.7144 0.0794575 0.0397287 0.999211i \(-0.487351\pi\)
0.0397287 + 0.999211i \(0.487351\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3517.68 3.98634
\(93\) 0 0
\(94\) 2683.02 2.94396
\(95\) 387.441 0.418428
\(96\) 0 0
\(97\) 1438.25 1.50549 0.752744 0.658313i \(-0.228730\pi\)
0.752744 + 0.658313i \(0.228730\pi\)
\(98\) 3327.40 3.42978
\(99\) 0 0
\(100\) −1748.75 −1.74875
\(101\) 896.264 0.882986 0.441493 0.897265i \(-0.354449\pi\)
0.441493 + 0.897265i \(0.354449\pi\)
\(102\) 0 0
\(103\) −22.2644 −0.0212988 −0.0106494 0.999943i \(-0.503390\pi\)
−0.0106494 + 0.999943i \(0.503390\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1265.91 −1.15996
\(107\) 351.441 0.317524 0.158762 0.987317i \(-0.449250\pi\)
0.158762 + 0.987317i \(0.449250\pi\)
\(108\) 0 0
\(109\) −967.008 −0.849748 −0.424874 0.905252i \(-0.639682\pi\)
−0.424874 + 0.905252i \(0.639682\pi\)
\(110\) 496.059 0.429976
\(111\) 0 0
\(112\) 4765.62 4.02061
\(113\) −48.2943 −0.0402048 −0.0201024 0.999798i \(-0.506399\pi\)
−0.0201024 + 0.999798i \(0.506399\pi\)
\(114\) 0 0
\(115\) −1067.74 −0.865805
\(116\) 3171.40 2.53842
\(117\) 0 0
\(118\) −1383.22 −1.07911
\(119\) 1690.95 1.30260
\(120\) 0 0
\(121\) −1066.05 −0.800938
\(122\) −2442.99 −1.81294
\(123\) 0 0
\(124\) 1133.21 0.820686
\(125\) 1260.73 0.902104
\(126\) 0 0
\(127\) −1763.02 −1.23183 −0.615916 0.787812i \(-0.711214\pi\)
−0.615916 + 0.787812i \(0.711214\pi\)
\(128\) −100.479 −0.0693844
\(129\) 0 0
\(130\) 0 0
\(131\) −955.970 −0.637584 −0.318792 0.947825i \(-0.603277\pi\)
−0.318792 + 0.947825i \(0.603277\pi\)
\(132\) 0 0
\(133\) −2077.68 −1.35457
\(134\) −4451.37 −2.86970
\(135\) 0 0
\(136\) 3167.07 1.99687
\(137\) 15.9215 0.00992894 0.00496447 0.999988i \(-0.498420\pi\)
0.00496447 + 0.999988i \(0.498420\pi\)
\(138\) 0 0
\(139\) 2074.26 1.26573 0.632866 0.774261i \(-0.281878\pi\)
0.632866 + 0.774261i \(0.281878\pi\)
\(140\) −3517.68 −2.12356
\(141\) 0 0
\(142\) 864.362 0.510815
\(143\) 0 0
\(144\) 0 0
\(145\) −962.635 −0.551327
\(146\) −1645.68 −0.932857
\(147\) 0 0
\(148\) 2123.20 1.17923
\(149\) −2764.08 −1.51975 −0.759873 0.650071i \(-0.774739\pi\)
−0.759873 + 0.650071i \(0.774739\pi\)
\(150\) 0 0
\(151\) 1618.46 0.872239 0.436120 0.899889i \(-0.356352\pi\)
0.436120 + 0.899889i \(0.356352\pi\)
\(152\) −3891.40 −2.07654
\(153\) 0 0
\(154\) −2660.14 −1.39195
\(155\) −343.970 −0.178247
\(156\) 0 0
\(157\) −1109.97 −0.564237 −0.282119 0.959380i \(-0.591037\pi\)
−0.282119 + 0.959380i \(0.591037\pi\)
\(158\) 2503.06 1.26034
\(159\) 0 0
\(160\) −1898.22 −0.937921
\(161\) 5725.83 2.80285
\(162\) 0 0
\(163\) −233.201 −0.112060 −0.0560299 0.998429i \(-0.517844\pi\)
−0.0560299 + 0.998429i \(0.517844\pi\)
\(164\) 1058.76 0.504119
\(165\) 0 0
\(166\) 2996.92 1.40124
\(167\) 215.405 0.0998118 0.0499059 0.998754i \(-0.484108\pi\)
0.0499059 + 0.998754i \(0.484108\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1645.68 −0.742456
\(171\) 0 0
\(172\) −2176.69 −0.964950
\(173\) 1383.15 0.607854 0.303927 0.952695i \(-0.401702\pi\)
0.303927 + 0.952695i \(0.401702\pi\)
\(174\) 0 0
\(175\) −2846.49 −1.22957
\(176\) −2477.22 −1.06095
\(177\) 0 0
\(178\) 348.181 0.146614
\(179\) −3642.79 −1.52109 −0.760545 0.649285i \(-0.775068\pi\)
−0.760545 + 0.649285i \(0.775068\pi\)
\(180\) 0 0
\(181\) 2621.97 1.07674 0.538369 0.842709i \(-0.319041\pi\)
0.538369 + 0.842709i \(0.319041\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 10724.2 4.29674
\(185\) −644.469 −0.256121
\(186\) 0 0
\(187\) −878.975 −0.343727
\(188\) 9889.91 3.83668
\(189\) 0 0
\(190\) 2022.05 0.772078
\(191\) 3419.32 1.29536 0.647679 0.761913i \(-0.275740\pi\)
0.647679 + 0.761913i \(0.275740\pi\)
\(192\) 0 0
\(193\) 1698.39 0.633435 0.316718 0.948520i \(-0.397419\pi\)
0.316718 + 0.948520i \(0.397419\pi\)
\(194\) 7506.20 2.77791
\(195\) 0 0
\(196\) 12265.2 4.46982
\(197\) −2293.72 −0.829548 −0.414774 0.909925i \(-0.636139\pi\)
−0.414774 + 0.909925i \(0.636139\pi\)
\(198\) 0 0
\(199\) 900.981 0.320949 0.160474 0.987040i \(-0.448698\pi\)
0.160474 + 0.987040i \(0.448698\pi\)
\(200\) −5331.35 −1.88492
\(201\) 0 0
\(202\) 4677.58 1.62928
\(203\) 5162.18 1.78480
\(204\) 0 0
\(205\) −321.373 −0.109491
\(206\) −116.197 −0.0393002
\(207\) 0 0
\(208\) 0 0
\(209\) 1080.00 0.357441
\(210\) 0 0
\(211\) 431.019 0.140628 0.0703142 0.997525i \(-0.477600\pi\)
0.0703142 + 0.997525i \(0.477600\pi\)
\(212\) −4666.28 −1.51170
\(213\) 0 0
\(214\) 1834.17 0.585892
\(215\) 660.706 0.209580
\(216\) 0 0
\(217\) 1844.56 0.577036
\(218\) −5046.79 −1.56794
\(219\) 0 0
\(220\) 1828.53 0.560361
\(221\) 0 0
\(222\) 0 0
\(223\) −4104.30 −1.23249 −0.616243 0.787556i \(-0.711346\pi\)
−0.616243 + 0.787556i \(0.711346\pi\)
\(224\) 10179.3 3.03631
\(225\) 0 0
\(226\) −252.047 −0.0741854
\(227\) −1809.11 −0.528963 −0.264482 0.964391i \(-0.585201\pi\)
−0.264482 + 0.964391i \(0.585201\pi\)
\(228\) 0 0
\(229\) −5249.33 −1.51478 −0.757392 0.652961i \(-0.773527\pi\)
−0.757392 + 0.652961i \(0.773527\pi\)
\(230\) −5572.53 −1.59757
\(231\) 0 0
\(232\) 9668.54 2.73608
\(233\) 2808.88 0.789768 0.394884 0.918731i \(-0.370785\pi\)
0.394884 + 0.918731i \(0.370785\pi\)
\(234\) 0 0
\(235\) −3001.95 −0.833300
\(236\) −5098.69 −1.40634
\(237\) 0 0
\(238\) 8825.03 2.40354
\(239\) −6712.01 −1.81659 −0.908293 0.418334i \(-0.862614\pi\)
−0.908293 + 0.418334i \(0.862614\pi\)
\(240\) 0 0
\(241\) 2519.11 0.673321 0.336661 0.941626i \(-0.390703\pi\)
0.336661 + 0.941626i \(0.390703\pi\)
\(242\) −5563.69 −1.47788
\(243\) 0 0
\(244\) −9005.15 −2.36269
\(245\) −3722.93 −0.970814
\(246\) 0 0
\(247\) 0 0
\(248\) 3454.78 0.884591
\(249\) 0 0
\(250\) 6579.71 1.66455
\(251\) −828.000 −0.208219 −0.104109 0.994566i \(-0.533199\pi\)
−0.104109 + 0.994566i \(0.533199\pi\)
\(252\) 0 0
\(253\) −2976.35 −0.739612
\(254\) −9201.16 −2.27296
\(255\) 0 0
\(256\) −4356.73 −1.06366
\(257\) 5840.76 1.41765 0.708826 0.705383i \(-0.249225\pi\)
0.708826 + 0.705383i \(0.249225\pi\)
\(258\) 0 0
\(259\) 3456.00 0.829133
\(260\) 0 0
\(261\) 0 0
\(262\) −4989.19 −1.17646
\(263\) 4064.06 0.952854 0.476427 0.879214i \(-0.341932\pi\)
0.476427 + 0.879214i \(0.341932\pi\)
\(264\) 0 0
\(265\) 1416.39 0.328332
\(266\) −10843.3 −2.49943
\(267\) 0 0
\(268\) −16408.2 −3.73990
\(269\) −1845.44 −0.418285 −0.209142 0.977885i \(-0.567067\pi\)
−0.209142 + 0.977885i \(0.567067\pi\)
\(270\) 0 0
\(271\) −2106.78 −0.472242 −0.236121 0.971724i \(-0.575876\pi\)
−0.236121 + 0.971724i \(0.575876\pi\)
\(272\) 8218.19 1.83199
\(273\) 0 0
\(274\) 83.0939 0.0183208
\(275\) 1479.64 0.324457
\(276\) 0 0
\(277\) −4781.94 −1.03725 −0.518626 0.855001i \(-0.673556\pi\)
−0.518626 + 0.855001i \(0.673556\pi\)
\(278\) 10825.5 2.33551
\(279\) 0 0
\(280\) −10724.2 −2.28891
\(281\) −5865.81 −1.24528 −0.622642 0.782507i \(-0.713941\pi\)
−0.622642 + 0.782507i \(0.713941\pi\)
\(282\) 0 0
\(283\) −6407.02 −1.34579 −0.672894 0.739739i \(-0.734949\pi\)
−0.672894 + 0.739739i \(0.734949\pi\)
\(284\) 3186.14 0.665713
\(285\) 0 0
\(286\) 0 0
\(287\) 1723.38 0.354453
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) −5023.97 −1.01730
\(291\) 0 0
\(292\) −6066.15 −1.21573
\(293\) 3010.24 0.600204 0.300102 0.953907i \(-0.402979\pi\)
0.300102 + 0.953907i \(0.402979\pi\)
\(294\) 0 0
\(295\) 1547.64 0.305447
\(296\) 6472.94 1.27105
\(297\) 0 0
\(298\) −14425.7 −2.80422
\(299\) 0 0
\(300\) 0 0
\(301\) −3543.07 −0.678470
\(302\) 8446.69 1.60944
\(303\) 0 0
\(304\) −10097.7 −1.90508
\(305\) 2733.39 0.513159
\(306\) 0 0
\(307\) −3341.84 −0.621266 −0.310633 0.950530i \(-0.600541\pi\)
−0.310633 + 0.950530i \(0.600541\pi\)
\(308\) −9805.59 −1.81404
\(309\) 0 0
\(310\) −1795.17 −0.328900
\(311\) −8755.20 −1.59634 −0.798170 0.602432i \(-0.794199\pi\)
−0.798170 + 0.602432i \(0.794199\pi\)
\(312\) 0 0
\(313\) 1948.93 0.351949 0.175974 0.984395i \(-0.443692\pi\)
0.175974 + 0.984395i \(0.443692\pi\)
\(314\) −5792.91 −1.04112
\(315\) 0 0
\(316\) 9226.57 1.64252
\(317\) 1940.43 0.343802 0.171901 0.985114i \(-0.445009\pi\)
0.171901 + 0.985114i \(0.445009\pi\)
\(318\) 0 0
\(319\) −2683.36 −0.470970
\(320\) −2797.29 −0.488667
\(321\) 0 0
\(322\) 29883.0 5.17178
\(323\) −3582.90 −0.617207
\(324\) 0 0
\(325\) 0 0
\(326\) −1217.07 −0.206771
\(327\) 0 0
\(328\) 3227.82 0.543373
\(329\) 16098.1 2.69762
\(330\) 0 0
\(331\) −7402.13 −1.22918 −0.614589 0.788848i \(-0.710678\pi\)
−0.614589 + 0.788848i \(0.710678\pi\)
\(332\) 11047.0 1.82615
\(333\) 0 0
\(334\) 1124.20 0.184171
\(335\) 4980.50 0.812280
\(336\) 0 0
\(337\) −5494.05 −0.888071 −0.444035 0.896009i \(-0.646454\pi\)
−0.444035 + 0.896009i \(0.646454\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6066.15 −0.967597
\(341\) −958.823 −0.152267
\(342\) 0 0
\(343\) 9223.77 1.45200
\(344\) −6636.03 −1.04009
\(345\) 0 0
\(346\) 7218.62 1.12160
\(347\) −3410.76 −0.527664 −0.263832 0.964569i \(-0.584986\pi\)
−0.263832 + 0.964569i \(0.584986\pi\)
\(348\) 0 0
\(349\) −12629.8 −1.93712 −0.968562 0.248773i \(-0.919973\pi\)
−0.968562 + 0.248773i \(0.919973\pi\)
\(350\) −14855.8 −2.26878
\(351\) 0 0
\(352\) −5291.32 −0.801217
\(353\) −2981.78 −0.449586 −0.224793 0.974407i \(-0.572171\pi\)
−0.224793 + 0.974407i \(0.572171\pi\)
\(354\) 0 0
\(355\) −967.109 −0.144588
\(356\) 1283.43 0.191073
\(357\) 0 0
\(358\) −19011.7 −2.80670
\(359\) −8942.30 −1.31464 −0.657321 0.753611i \(-0.728310\pi\)
−0.657321 + 0.753611i \(0.728310\pi\)
\(360\) 0 0
\(361\) −2456.68 −0.358168
\(362\) 13684.0 1.98678
\(363\) 0 0
\(364\) 0 0
\(365\) 1841.30 0.264049
\(366\) 0 0
\(367\) −4735.26 −0.673511 −0.336756 0.941592i \(-0.609330\pi\)
−0.336756 + 0.941592i \(0.609330\pi\)
\(368\) 27828.1 3.94196
\(369\) 0 0
\(370\) −3363.47 −0.472590
\(371\) −7595.45 −1.06290
\(372\) 0 0
\(373\) −8304.01 −1.15272 −0.576361 0.817195i \(-0.695528\pi\)
−0.576361 + 0.817195i \(0.695528\pi\)
\(374\) −4587.35 −0.634241
\(375\) 0 0
\(376\) 30151.1 4.13543
\(377\) 0 0
\(378\) 0 0
\(379\) 4088.11 0.554069 0.277035 0.960860i \(-0.410648\pi\)
0.277035 + 0.960860i \(0.410648\pi\)
\(380\) 7453.50 1.00620
\(381\) 0 0
\(382\) 17845.4 2.39018
\(383\) −13951.5 −1.86132 −0.930662 0.365879i \(-0.880768\pi\)
−0.930662 + 0.365879i \(0.880768\pi\)
\(384\) 0 0
\(385\) 2976.35 0.393997
\(386\) 8863.88 1.16881
\(387\) 0 0
\(388\) 27668.7 3.62027
\(389\) 2804.26 0.365506 0.182753 0.983159i \(-0.441499\pi\)
0.182753 + 0.983159i \(0.441499\pi\)
\(390\) 0 0
\(391\) 9874.06 1.27712
\(392\) 37392.5 4.81787
\(393\) 0 0
\(394\) −11970.9 −1.53067
\(395\) −2800.60 −0.356743
\(396\) 0 0
\(397\) 6556.18 0.828830 0.414415 0.910088i \(-0.363986\pi\)
0.414415 + 0.910088i \(0.363986\pi\)
\(398\) 4702.20 0.592211
\(399\) 0 0
\(400\) −13834.2 −1.72928
\(401\) 4730.95 0.589157 0.294579 0.955627i \(-0.404821\pi\)
0.294579 + 0.955627i \(0.404821\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 17242.1 2.12333
\(405\) 0 0
\(406\) 26941.3 3.29329
\(407\) −1796.47 −0.218790
\(408\) 0 0
\(409\) −12314.4 −1.48878 −0.744389 0.667746i \(-0.767259\pi\)
−0.744389 + 0.667746i \(0.767259\pi\)
\(410\) −1677.24 −0.202032
\(411\) 0 0
\(412\) −428.316 −0.0512175
\(413\) −8299.29 −0.988817
\(414\) 0 0
\(415\) −3353.16 −0.396627
\(416\) 0 0
\(417\) 0 0
\(418\) 5636.50 0.659546
\(419\) 5499.85 0.641254 0.320627 0.947206i \(-0.396106\pi\)
0.320627 + 0.947206i \(0.396106\pi\)
\(420\) 0 0
\(421\) 12629.7 1.46208 0.731039 0.682336i \(-0.239036\pi\)
0.731039 + 0.682336i \(0.239036\pi\)
\(422\) 2249.48 0.259486
\(423\) 0 0
\(424\) −14225.9 −1.62942
\(425\) −4908.70 −0.560252
\(426\) 0 0
\(427\) −14658.0 −1.66124
\(428\) 6760.94 0.763557
\(429\) 0 0
\(430\) 3448.21 0.386715
\(431\) −7191.15 −0.803679 −0.401840 0.915710i \(-0.631629\pi\)
−0.401840 + 0.915710i \(0.631629\pi\)
\(432\) 0 0
\(433\) 6062.68 0.672873 0.336436 0.941706i \(-0.390778\pi\)
0.336436 + 0.941706i \(0.390778\pi\)
\(434\) 9626.71 1.06474
\(435\) 0 0
\(436\) −18603.0 −2.04340
\(437\) −12132.3 −1.32807
\(438\) 0 0
\(439\) 11864.3 1.28986 0.644932 0.764240i \(-0.276886\pi\)
0.644932 + 0.764240i \(0.276886\pi\)
\(440\) 5574.58 0.603995
\(441\) 0 0
\(442\) 0 0
\(443\) −10560.5 −1.13261 −0.566303 0.824197i \(-0.691627\pi\)
−0.566303 + 0.824197i \(0.691627\pi\)
\(444\) 0 0
\(445\) −389.569 −0.0414997
\(446\) −21420.3 −2.27417
\(447\) 0 0
\(448\) 15000.6 1.58195
\(449\) 12659.7 1.33062 0.665308 0.746569i \(-0.268300\pi\)
0.665308 + 0.746569i \(0.268300\pi\)
\(450\) 0 0
\(451\) −895.834 −0.0935325
\(452\) −929.072 −0.0966812
\(453\) 0 0
\(454\) −9441.69 −0.976037
\(455\) 0 0
\(456\) 0 0
\(457\) −1544.24 −0.158067 −0.0790336 0.996872i \(-0.525183\pi\)
−0.0790336 + 0.996872i \(0.525183\pi\)
\(458\) −27396.1 −2.79506
\(459\) 0 0
\(460\) −20541.0 −2.08202
\(461\) 13196.8 1.33327 0.666635 0.745384i \(-0.267734\pi\)
0.666635 + 0.745384i \(0.267734\pi\)
\(462\) 0 0
\(463\) 16309.2 1.63705 0.818524 0.574472i \(-0.194792\pi\)
0.818524 + 0.574472i \(0.194792\pi\)
\(464\) 25088.7 2.51016
\(465\) 0 0
\(466\) 14659.5 1.45727
\(467\) 14260.8 1.41308 0.706541 0.707672i \(-0.250255\pi\)
0.706541 + 0.707672i \(0.250255\pi\)
\(468\) 0 0
\(469\) −26708.2 −2.62957
\(470\) −15667.1 −1.53760
\(471\) 0 0
\(472\) −15544.2 −1.51585
\(473\) 1841.73 0.179034
\(474\) 0 0
\(475\) 6031.34 0.582604
\(476\) 32530.0 3.13238
\(477\) 0 0
\(478\) −35029.9 −3.35194
\(479\) −18011.5 −1.71809 −0.859046 0.511899i \(-0.828942\pi\)
−0.859046 + 0.511899i \(0.828942\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 13147.2 1.24240
\(483\) 0 0
\(484\) −20508.4 −1.92603
\(485\) −8398.46 −0.786298
\(486\) 0 0
\(487\) −14043.3 −1.30670 −0.653351 0.757055i \(-0.726637\pi\)
−0.653351 + 0.757055i \(0.726637\pi\)
\(488\) −27453.7 −2.54666
\(489\) 0 0
\(490\) −19429.9 −1.79133
\(491\) 12966.1 1.19176 0.595878 0.803075i \(-0.296804\pi\)
0.595878 + 0.803075i \(0.296804\pi\)
\(492\) 0 0
\(493\) 8902.06 0.813242
\(494\) 0 0
\(495\) 0 0
\(496\) 8964.75 0.811551
\(497\) 5186.17 0.468072
\(498\) 0 0
\(499\) −10215.5 −0.916453 −0.458227 0.888835i \(-0.651515\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(500\) 24253.6 2.16930
\(501\) 0 0
\(502\) −4321.31 −0.384203
\(503\) 16632.0 1.47432 0.737161 0.675717i \(-0.236166\pi\)
0.737161 + 0.675717i \(0.236166\pi\)
\(504\) 0 0
\(505\) −5233.61 −0.461173
\(506\) −15533.5 −1.36472
\(507\) 0 0
\(508\) −33916.5 −2.96221
\(509\) −15235.3 −1.32671 −0.663354 0.748306i \(-0.730868\pi\)
−0.663354 + 0.748306i \(0.730868\pi\)
\(510\) 0 0
\(511\) −9874.06 −0.854799
\(512\) −21933.9 −1.89326
\(513\) 0 0
\(514\) 30482.8 2.61584
\(515\) 130.009 0.0111241
\(516\) 0 0
\(517\) −8367.99 −0.711845
\(518\) 18036.8 1.52991
\(519\) 0 0
\(520\) 0 0
\(521\) −2680.23 −0.225380 −0.112690 0.993630i \(-0.535947\pi\)
−0.112690 + 0.993630i \(0.535947\pi\)
\(522\) 0 0
\(523\) −2410.38 −0.201527 −0.100764 0.994910i \(-0.532129\pi\)
−0.100764 + 0.994910i \(0.532129\pi\)
\(524\) −18390.7 −1.53321
\(525\) 0 0
\(526\) 21210.2 1.75819
\(527\) 3180.90 0.262926
\(528\) 0 0
\(529\) 21268.2 1.74802
\(530\) 7392.09 0.605834
\(531\) 0 0
\(532\) −39969.8 −3.25735
\(533\) 0 0
\(534\) 0 0
\(535\) −2052.19 −0.165839
\(536\) −50023.3 −4.03111
\(537\) 0 0
\(538\) −9631.32 −0.771814
\(539\) −10377.7 −0.829315
\(540\) 0 0
\(541\) −9969.58 −0.792284 −0.396142 0.918189i \(-0.629651\pi\)
−0.396142 + 0.918189i \(0.629651\pi\)
\(542\) −10995.2 −0.871375
\(543\) 0 0
\(544\) 17554.0 1.38349
\(545\) 5646.70 0.443813
\(546\) 0 0
\(547\) 16848.8 1.31701 0.658505 0.752576i \(-0.271189\pi\)
0.658505 + 0.752576i \(0.271189\pi\)
\(548\) 306.293 0.0238763
\(549\) 0 0
\(550\) 7722.20 0.598683
\(551\) −10938.0 −0.845688
\(552\) 0 0
\(553\) 15018.4 1.15488
\(554\) −24956.8 −1.91393
\(555\) 0 0
\(556\) 39904.2 3.04373
\(557\) −3800.83 −0.289132 −0.144566 0.989495i \(-0.546179\pi\)
−0.144566 + 0.989495i \(0.546179\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −27828.1 −2.09992
\(561\) 0 0
\(562\) −30613.5 −2.29778
\(563\) 15750.2 1.17903 0.589513 0.807759i \(-0.299320\pi\)
0.589513 + 0.807759i \(0.299320\pi\)
\(564\) 0 0
\(565\) 282.007 0.0209985
\(566\) −33438.1 −2.48323
\(567\) 0 0
\(568\) 9713.48 0.717550
\(569\) −17753.2 −1.30800 −0.654002 0.756493i \(-0.726911\pi\)
−0.654002 + 0.756493i \(0.726911\pi\)
\(570\) 0 0
\(571\) 25293.1 1.85374 0.926868 0.375388i \(-0.122491\pi\)
0.926868 + 0.375388i \(0.122491\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 8994.29 0.654032
\(575\) −16621.7 −1.20552
\(576\) 0 0
\(577\) 18488.8 1.33396 0.666982 0.745073i \(-0.267586\pi\)
0.666982 + 0.745073i \(0.267586\pi\)
\(578\) −10422.3 −0.750018
\(579\) 0 0
\(580\) −18518.9 −1.32579
\(581\) 17981.5 1.28399
\(582\) 0 0
\(583\) 3948.20 0.280477
\(584\) −18493.7 −1.31040
\(585\) 0 0
\(586\) 15710.4 1.10749
\(587\) −17376.7 −1.22183 −0.610914 0.791697i \(-0.709198\pi\)
−0.610914 + 0.791697i \(0.709198\pi\)
\(588\) 0 0
\(589\) −3908.38 −0.273416
\(590\) 8077.09 0.563608
\(591\) 0 0
\(592\) 16796.5 1.16610
\(593\) 7991.09 0.553381 0.276690 0.960959i \(-0.410762\pi\)
0.276690 + 0.960959i \(0.410762\pi\)
\(594\) 0 0
\(595\) −9874.06 −0.680331
\(596\) −53174.7 −3.65456
\(597\) 0 0
\(598\) 0 0
\(599\) −10386.5 −0.708480 −0.354240 0.935154i \(-0.615260\pi\)
−0.354240 + 0.935154i \(0.615260\pi\)
\(600\) 0 0
\(601\) −9241.77 −0.627254 −0.313627 0.949546i \(-0.601544\pi\)
−0.313627 + 0.949546i \(0.601544\pi\)
\(602\) −18491.2 −1.25190
\(603\) 0 0
\(604\) 31135.4 2.09749
\(605\) 6225.04 0.418320
\(606\) 0 0
\(607\) 18921.5 1.26524 0.632619 0.774463i \(-0.281980\pi\)
0.632619 + 0.774463i \(0.281980\pi\)
\(608\) −21568.6 −1.43869
\(609\) 0 0
\(610\) 14265.5 0.946875
\(611\) 0 0
\(612\) 0 0
\(613\) −17138.1 −1.12921 −0.564603 0.825363i \(-0.690971\pi\)
−0.564603 + 0.825363i \(0.690971\pi\)
\(614\) −17441.0 −1.14635
\(615\) 0 0
\(616\) −29894.0 −1.95530
\(617\) 18825.8 1.22836 0.614180 0.789166i \(-0.289487\pi\)
0.614180 + 0.789166i \(0.289487\pi\)
\(618\) 0 0
\(619\) −1392.83 −0.0904404 −0.0452202 0.998977i \(-0.514399\pi\)
−0.0452202 + 0.998977i \(0.514399\pi\)
\(620\) −6617.21 −0.428635
\(621\) 0 0
\(622\) −45693.2 −2.94555
\(623\) 2089.09 0.134346
\(624\) 0 0
\(625\) 4000.90 0.256057
\(626\) 10171.4 0.649412
\(627\) 0 0
\(628\) −21353.3 −1.35683
\(629\) 5959.79 0.377794
\(630\) 0 0
\(631\) −25488.4 −1.60804 −0.804022 0.594599i \(-0.797311\pi\)
−0.804022 + 0.594599i \(0.797311\pi\)
\(632\) 28128.8 1.77041
\(633\) 0 0
\(634\) 10127.1 0.634379
\(635\) 10294.9 0.643371
\(636\) 0 0
\(637\) 0 0
\(638\) −14004.4 −0.869028
\(639\) 0 0
\(640\) 586.735 0.0362387
\(641\) 6066.41 0.373805 0.186902 0.982379i \(-0.440155\pi\)
0.186902 + 0.982379i \(0.440155\pi\)
\(642\) 0 0
\(643\) 1598.78 0.0980554 0.0490277 0.998797i \(-0.484388\pi\)
0.0490277 + 0.998797i \(0.484388\pi\)
\(644\) 110152. 6.74006
\(645\) 0 0
\(646\) −18699.1 −1.13886
\(647\) −23067.2 −1.40164 −0.700822 0.713336i \(-0.747183\pi\)
−0.700822 + 0.713336i \(0.747183\pi\)
\(648\) 0 0
\(649\) 4314.07 0.260928
\(650\) 0 0
\(651\) 0 0
\(652\) −4486.26 −0.269472
\(653\) 23743.9 1.42293 0.711463 0.702723i \(-0.248033\pi\)
0.711463 + 0.702723i \(0.248033\pi\)
\(654\) 0 0
\(655\) 5582.25 0.333002
\(656\) 8375.81 0.498507
\(657\) 0 0
\(658\) 84015.7 4.97762
\(659\) 7497.65 0.443197 0.221598 0.975138i \(-0.428873\pi\)
0.221598 + 0.975138i \(0.428873\pi\)
\(660\) 0 0
\(661\) −1255.26 −0.0738638 −0.0369319 0.999318i \(-0.511758\pi\)
−0.0369319 + 0.999318i \(0.511758\pi\)
\(662\) −38631.5 −2.26806
\(663\) 0 0
\(664\) 33678.6 1.96835
\(665\) 12132.3 0.707474
\(666\) 0 0
\(667\) 30143.8 1.74989
\(668\) 4143.91 0.240019
\(669\) 0 0
\(670\) 25993.1 1.49881
\(671\) 7619.38 0.438365
\(672\) 0 0
\(673\) 1505.97 0.0862569 0.0431284 0.999070i \(-0.486268\pi\)
0.0431284 + 0.999070i \(0.486268\pi\)
\(674\) −28673.3 −1.63866
\(675\) 0 0
\(676\) 0 0
\(677\) 16201.4 0.919751 0.459876 0.887983i \(-0.347894\pi\)
0.459876 + 0.887983i \(0.347894\pi\)
\(678\) 0 0
\(679\) 45037.2 2.54546
\(680\) −18493.7 −1.04294
\(681\) 0 0
\(682\) −5004.08 −0.280962
\(683\) −29090.4 −1.62974 −0.814870 0.579644i \(-0.803192\pi\)
−0.814870 + 0.579644i \(0.803192\pi\)
\(684\) 0 0
\(685\) −92.9712 −0.00518576
\(686\) 48138.7 2.67922
\(687\) 0 0
\(688\) −17219.7 −0.954208
\(689\) 0 0
\(690\) 0 0
\(691\) 940.952 0.0518025 0.0259012 0.999665i \(-0.491754\pi\)
0.0259012 + 0.999665i \(0.491754\pi\)
\(692\) 26608.6 1.46172
\(693\) 0 0
\(694\) −17800.7 −0.973639
\(695\) −12112.4 −0.661077
\(696\) 0 0
\(697\) 2971.93 0.161506
\(698\) −65914.5 −3.57436
\(699\) 0 0
\(700\) −54760.1 −2.95676
\(701\) 30713.9 1.65485 0.827424 0.561578i \(-0.189805\pi\)
0.827424 + 0.561578i \(0.189805\pi\)
\(702\) 0 0
\(703\) −7322.81 −0.392866
\(704\) −7797.51 −0.417443
\(705\) 0 0
\(706\) −15561.8 −0.829571
\(707\) 28065.5 1.49294
\(708\) 0 0
\(709\) 25640.5 1.35818 0.679089 0.734056i \(-0.262375\pi\)
0.679089 + 0.734056i \(0.262375\pi\)
\(710\) −5047.32 −0.266792
\(711\) 0 0
\(712\) 3912.77 0.205951
\(713\) 10771.0 0.565748
\(714\) 0 0
\(715\) 0 0
\(716\) −70079.1 −3.65779
\(717\) 0 0
\(718\) −46669.7 −2.42576
\(719\) −20842.5 −1.08108 −0.540538 0.841319i \(-0.681779\pi\)
−0.540538 + 0.841319i \(0.681779\pi\)
\(720\) 0 0
\(721\) −697.183 −0.0360117
\(722\) −12821.3 −0.660888
\(723\) 0 0
\(724\) 50440.8 2.58925
\(725\) −14985.4 −0.767649
\(726\) 0 0
\(727\) −263.608 −0.0134480 −0.00672398 0.999977i \(-0.502140\pi\)
−0.00672398 + 0.999977i \(0.502140\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9609.69 0.487220
\(731\) −6109.94 −0.309144
\(732\) 0 0
\(733\) 21835.5 1.10029 0.550146 0.835069i \(-0.314572\pi\)
0.550146 + 0.835069i \(0.314572\pi\)
\(734\) −24713.2 −1.24275
\(735\) 0 0
\(736\) 59440.6 2.97692
\(737\) 13883.2 0.693888
\(738\) 0 0
\(739\) −19536.4 −0.972476 −0.486238 0.873826i \(-0.661631\pi\)
−0.486238 + 0.873826i \(0.661631\pi\)
\(740\) −12398.1 −0.615897
\(741\) 0 0
\(742\) −39640.5 −1.96125
\(743\) 30353.1 1.49872 0.749360 0.662163i \(-0.230361\pi\)
0.749360 + 0.662163i \(0.230361\pi\)
\(744\) 0 0
\(745\) 16140.5 0.793746
\(746\) −43338.4 −2.12699
\(747\) 0 0
\(748\) −16909.5 −0.826567
\(749\) 11005.0 0.536867
\(750\) 0 0
\(751\) −3904.20 −0.189702 −0.0948510 0.995491i \(-0.530237\pi\)
−0.0948510 + 0.995491i \(0.530237\pi\)
\(752\) 78238.5 3.79397
\(753\) 0 0
\(754\) 0 0
\(755\) −9450.74 −0.455560
\(756\) 0 0
\(757\) −2900.52 −0.139262 −0.0696308 0.997573i \(-0.522182\pi\)
−0.0696308 + 0.997573i \(0.522182\pi\)
\(758\) 21335.8 1.02236
\(759\) 0 0
\(760\) 22723.3 1.08455
\(761\) 33518.7 1.59665 0.798325 0.602227i \(-0.205720\pi\)
0.798325 + 0.602227i \(0.205720\pi\)
\(762\) 0 0
\(763\) −30280.8 −1.43675
\(764\) 65780.0 3.11497
\(765\) 0 0
\(766\) −72812.5 −3.43449
\(767\) 0 0
\(768\) 0 0
\(769\) −552.769 −0.0259211 −0.0129606 0.999916i \(-0.504126\pi\)
−0.0129606 + 0.999916i \(0.504126\pi\)
\(770\) 15533.5 0.726999
\(771\) 0 0
\(772\) 32673.3 1.52323
\(773\) −36498.8 −1.69828 −0.849141 0.528166i \(-0.822880\pi\)
−0.849141 + 0.528166i \(0.822880\pi\)
\(774\) 0 0
\(775\) −5354.62 −0.248185
\(776\) 84352.8 3.90218
\(777\) 0 0
\(778\) 14635.4 0.674427
\(779\) −3651.62 −0.167950
\(780\) 0 0
\(781\) −2695.83 −0.123514
\(782\) 51532.5 2.35652
\(783\) 0 0
\(784\) 97029.2 4.42006
\(785\) 6481.51 0.294694
\(786\) 0 0
\(787\) 4284.79 0.194074 0.0970371 0.995281i \(-0.469063\pi\)
0.0970371 + 0.995281i \(0.469063\pi\)
\(788\) −44126.0 −1.99483
\(789\) 0 0
\(790\) −14616.3 −0.658258
\(791\) −1512.28 −0.0679779
\(792\) 0 0
\(793\) 0 0
\(794\) 34216.6 1.52935
\(795\) 0 0
\(796\) 17332.8 0.771792
\(797\) 29538.5 1.31281 0.656405 0.754409i \(-0.272076\pi\)
0.656405 + 0.754409i \(0.272076\pi\)
\(798\) 0 0
\(799\) 27760.8 1.22917
\(800\) −29549.8 −1.30593
\(801\) 0 0
\(802\) 24690.7 1.08711
\(803\) 5132.65 0.225563
\(804\) 0 0
\(805\) −33435.2 −1.46389
\(806\) 0 0
\(807\) 0 0
\(808\) 52565.5 2.28867
\(809\) 895.586 0.0389211 0.0194605 0.999811i \(-0.493805\pi\)
0.0194605 + 0.999811i \(0.493805\pi\)
\(810\) 0 0
\(811\) −20139.7 −0.872011 −0.436006 0.899944i \(-0.643607\pi\)
−0.436006 + 0.899944i \(0.643607\pi\)
\(812\) 99308.7 4.29194
\(813\) 0 0
\(814\) −9375.73 −0.403709
\(815\) 1361.75 0.0585274
\(816\) 0 0
\(817\) 7507.31 0.321478
\(818\) −64268.8 −2.74707
\(819\) 0 0
\(820\) −6182.49 −0.263295
\(821\) −17263.2 −0.733848 −0.366924 0.930251i \(-0.619589\pi\)
−0.366924 + 0.930251i \(0.619589\pi\)
\(822\) 0 0
\(823\) 12114.5 0.513104 0.256552 0.966530i \(-0.417413\pi\)
0.256552 + 0.966530i \(0.417413\pi\)
\(824\) −1305.79 −0.0552057
\(825\) 0 0
\(826\) −43313.8 −1.82455
\(827\) 31450.9 1.32243 0.661217 0.750194i \(-0.270040\pi\)
0.661217 + 0.750194i \(0.270040\pi\)
\(828\) 0 0
\(829\) 13760.4 0.576499 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(830\) −17500.1 −0.731852
\(831\) 0 0
\(832\) 0 0
\(833\) 34428.2 1.43201
\(834\) 0 0
\(835\) −1257.83 −0.0521305
\(836\) 20776.8 0.859545
\(837\) 0 0
\(838\) 28703.6 1.18323
\(839\) −9846.21 −0.405160 −0.202580 0.979266i \(-0.564933\pi\)
−0.202580 + 0.979266i \(0.564933\pi\)
\(840\) 0 0
\(841\) 2787.47 0.114292
\(842\) 65914.2 2.69781
\(843\) 0 0
\(844\) 8291.83 0.338171
\(845\) 0 0
\(846\) 0 0
\(847\) −33382.1 −1.35422
\(848\) −36914.7 −1.49488
\(849\) 0 0
\(850\) −25618.4 −1.03377
\(851\) 20180.8 0.812914
\(852\) 0 0
\(853\) 27574.5 1.10684 0.553420 0.832903i \(-0.313323\pi\)
0.553420 + 0.832903i \(0.313323\pi\)
\(854\) −76499.6 −3.06530
\(855\) 0 0
\(856\) 20611.9 0.823013
\(857\) 8046.95 0.320745 0.160373 0.987057i \(-0.448730\pi\)
0.160373 + 0.987057i \(0.448730\pi\)
\(858\) 0 0
\(859\) 2898.13 0.115114 0.0575570 0.998342i \(-0.481669\pi\)
0.0575570 + 0.998342i \(0.481669\pi\)
\(860\) 12710.5 0.503982
\(861\) 0 0
\(862\) −37530.5 −1.48294
\(863\) −4961.16 −0.195689 −0.0978447 0.995202i \(-0.531195\pi\)
−0.0978447 + 0.995202i \(0.531195\pi\)
\(864\) 0 0
\(865\) −8076.69 −0.317475
\(866\) 31641.0 1.24158
\(867\) 0 0
\(868\) 35485.1 1.38761
\(869\) −7806.72 −0.304747
\(870\) 0 0
\(871\) 0 0
\(872\) −56714.5 −2.20252
\(873\) 0 0
\(874\) −63318.2 −2.45054
\(875\) 39478.3 1.52527
\(876\) 0 0
\(877\) 1386.66 0.0533913 0.0266957 0.999644i \(-0.491502\pi\)
0.0266957 + 0.999644i \(0.491502\pi\)
\(878\) 61919.3 2.38004
\(879\) 0 0
\(880\) 14465.4 0.554123
\(881\) −9030.36 −0.345335 −0.172668 0.984980i \(-0.555239\pi\)
−0.172668 + 0.984980i \(0.555239\pi\)
\(882\) 0 0
\(883\) 15512.7 0.591216 0.295608 0.955309i \(-0.404478\pi\)
0.295608 + 0.955309i \(0.404478\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −55115.0 −2.08987
\(887\) −7431.21 −0.281303 −0.140651 0.990059i \(-0.544920\pi\)
−0.140651 + 0.990059i \(0.544920\pi\)
\(888\) 0 0
\(889\) −55207.0 −2.08277
\(890\) −2033.15 −0.0765747
\(891\) 0 0
\(892\) −78957.5 −2.96378
\(893\) −34109.8 −1.27821
\(894\) 0 0
\(895\) 21271.6 0.794447
\(896\) −3146.40 −0.117315
\(897\) 0 0
\(898\) 66070.5 2.45524
\(899\) 9710.74 0.360257
\(900\) 0 0
\(901\) −13098.2 −0.484310
\(902\) −4675.34 −0.172585
\(903\) 0 0
\(904\) −2832.44 −0.104210
\(905\) −15310.6 −0.562367
\(906\) 0 0
\(907\) 10550.2 0.386234 0.193117 0.981176i \(-0.438140\pi\)
0.193117 + 0.981176i \(0.438140\pi\)
\(908\) −34803.1 −1.27201
\(909\) 0 0
\(910\) 0 0
\(911\) −35703.3 −1.29847 −0.649234 0.760589i \(-0.724910\pi\)
−0.649234 + 0.760589i \(0.724910\pi\)
\(912\) 0 0
\(913\) −9347.01 −0.338818
\(914\) −8059.38 −0.291664
\(915\) 0 0
\(916\) −100985. −3.64263
\(917\) −29935.1 −1.07802
\(918\) 0 0
\(919\) −42896.6 −1.53975 −0.769873 0.638197i \(-0.779681\pi\)
−0.769873 + 0.638197i \(0.779681\pi\)
\(920\) −62622.6 −2.24414
\(921\) 0 0
\(922\) 68874.0 2.46013
\(923\) 0 0
\(924\) 0 0
\(925\) −10032.5 −0.356613
\(926\) 85117.5 3.02066
\(927\) 0 0
\(928\) 53589.3 1.89564
\(929\) −10366.7 −0.366114 −0.183057 0.983102i \(-0.558599\pi\)
−0.183057 + 0.983102i \(0.558599\pi\)
\(930\) 0 0
\(931\) −42302.0 −1.48914
\(932\) 54036.6 1.89917
\(933\) 0 0
\(934\) 74426.6 2.60740
\(935\) 5132.65 0.179525
\(936\) 0 0
\(937\) 20289.8 0.707405 0.353702 0.935358i \(-0.384923\pi\)
0.353702 + 0.935358i \(0.384923\pi\)
\(938\) −139390. −4.85206
\(939\) 0 0
\(940\) −57750.7 −2.00385
\(941\) 37089.1 1.28488 0.642438 0.766337i \(-0.277923\pi\)
0.642438 + 0.766337i \(0.277923\pi\)
\(942\) 0 0
\(943\) 10063.4 0.347519
\(944\) −40335.4 −1.39068
\(945\) 0 0
\(946\) 9611.96 0.330351
\(947\) 23458.2 0.804952 0.402476 0.915430i \(-0.368150\pi\)
0.402476 + 0.915430i \(0.368150\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 31477.5 1.07501
\(951\) 0 0
\(952\) 99173.4 3.37629
\(953\) −34695.5 −1.17933 −0.589663 0.807649i \(-0.700740\pi\)
−0.589663 + 0.807649i \(0.700740\pi\)
\(954\) 0 0
\(955\) −19966.6 −0.676550
\(956\) −129124. −4.36838
\(957\) 0 0
\(958\) −94001.6 −3.17020
\(959\) 498.563 0.0167878
\(960\) 0 0
\(961\) −26321.1 −0.883527
\(962\) 0 0
\(963\) 0 0
\(964\) 48462.1 1.61915
\(965\) −9917.53 −0.330836
\(966\) 0 0
\(967\) −6289.66 −0.209164 −0.104582 0.994516i \(-0.533351\pi\)
−0.104582 + 0.994516i \(0.533351\pi\)
\(968\) −62523.3 −2.07601
\(969\) 0 0
\(970\) −43831.4 −1.45087
\(971\) 20185.9 0.667145 0.333573 0.942724i \(-0.391746\pi\)
0.333573 + 0.942724i \(0.391746\pi\)
\(972\) 0 0
\(973\) 64953.2 2.14009
\(974\) −73291.8 −2.41111
\(975\) 0 0
\(976\) −71239.2 −2.33639
\(977\) 44244.0 1.44881 0.724406 0.689373i \(-0.242114\pi\)
0.724406 + 0.689373i \(0.242114\pi\)
\(978\) 0 0
\(979\) −1085.93 −0.0354510
\(980\) −71620.8 −2.33453
\(981\) 0 0
\(982\) 67669.9 2.19901
\(983\) 8835.11 0.286670 0.143335 0.989674i \(-0.454217\pi\)
0.143335 + 0.989674i \(0.454217\pi\)
\(984\) 0 0
\(985\) 13393.9 0.433263
\(986\) 46459.6 1.50058
\(987\) 0 0
\(988\) 0 0
\(989\) −20689.3 −0.665198
\(990\) 0 0
\(991\) 34915.1 1.11919 0.559594 0.828767i \(-0.310957\pi\)
0.559594 + 0.828767i \(0.310957\pi\)
\(992\) 19148.6 0.612872
\(993\) 0 0
\(994\) 27066.5 0.863680
\(995\) −5261.15 −0.167628
\(996\) 0 0
\(997\) 37962.8 1.20591 0.602956 0.797774i \(-0.293989\pi\)
0.602956 + 0.797774i \(0.293989\pi\)
\(998\) −53314.6 −1.69103
\(999\) 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 1521.4.a.x.1.4 4
3.2 odd 2 507.4.a.j.1.1 4
13.5 odd 4 117.4.b.d.64.1 4
13.8 odd 4 117.4.b.d.64.4 4
13.12 even 2 inner 1521.4.a.x.1.1 4
39.5 even 4 39.4.b.a.25.4 yes 4
39.8 even 4 39.4.b.a.25.1 4
39.38 odd 2 507.4.a.j.1.4 4
156.47 odd 4 624.4.c.e.337.3 4
156.83 odd 4 624.4.c.e.337.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.a.25.1 4 39.8 even 4
39.4.b.a.25.4 yes 4 39.5 even 4
117.4.b.d.64.1 4 13.5 odd 4
117.4.b.d.64.4 4 13.8 odd 4
507.4.a.j.1.1 4 3.2 odd 2
507.4.a.j.1.4 4 39.38 odd 2
624.4.c.e.337.2 4 156.83 odd 4
624.4.c.e.337.3 4 156.47 odd 4
1521.4.a.x.1.1 4 13.12 even 2 inner
1521.4.a.x.1.4 4 1.1 even 1 trivial