Properties

Label 2205.4.a.be.1.2
Level 2205
Weight 4
Character 2205.1
Self dual yes
Analytic conductor 130.099
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.23607 q^{2} +9.94427 q^{4} -5.00000 q^{5} +8.23607 q^{8} +O(q^{10})\) \(q+4.23607 q^{2} +9.94427 q^{4} -5.00000 q^{5} +8.23607 q^{8} -21.1803 q^{10} +41.5279 q^{11} -88.9706 q^{13} -44.6656 q^{16} -120.387 q^{17} +112.138 q^{19} -49.7214 q^{20} +175.915 q^{22} +115.279 q^{23} +25.0000 q^{25} -376.885 q^{26} +144.833 q^{29} +258.079 q^{31} -255.095 q^{32} -509.967 q^{34} +48.3344 q^{37} +475.023 q^{38} -41.1803 q^{40} +200.885 q^{41} -218.217 q^{43} +412.964 q^{44} +488.328 q^{46} +575.659 q^{47} +105.902 q^{50} -884.748 q^{52} +184.302 q^{53} -207.639 q^{55} +613.522 q^{58} -151.502 q^{59} +529.830 q^{61} +1093.24 q^{62} -723.276 q^{64} +444.853 q^{65} +1.28485 q^{67} -1197.16 q^{68} +61.4226 q^{71} -484.800 q^{73} +204.748 q^{74} +1115.13 q^{76} +878.257 q^{79} +223.328 q^{80} +850.964 q^{82} +491.830 q^{83} +601.935 q^{85} -924.381 q^{86} +342.026 q^{88} -415.560 q^{89} +1146.36 q^{92} +2438.53 q^{94} -560.689 q^{95} +1031.70 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 2q^{4} - 10q^{5} + 12q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 2q^{4} - 10q^{5} + 12q^{8} - 20q^{10} + 92q^{11} - 8q^{13} + 18q^{16} - 44q^{17} + 108q^{19} - 10q^{20} + 164q^{22} + 320q^{23} + 50q^{25} - 396q^{26} + 236q^{29} + 60q^{31} - 300q^{32} - 528q^{34} + 204q^{37} + 476q^{38} - 60q^{40} + 44q^{41} + 136q^{43} + 12q^{44} + 440q^{46} + 400q^{47} + 100q^{50} - 1528q^{52} - 16q^{53} - 460q^{55} + 592q^{58} - 464q^{59} + 684q^{61} + 1140q^{62} - 1214q^{64} + 40q^{65} + 736q^{67} - 1804q^{68} + 740q^{71} - 424q^{73} + 168q^{74} + 1148q^{76} - 408q^{79} - 90q^{80} + 888q^{82} + 608q^{83} + 220q^{85} - 1008q^{86} + 532q^{88} - 1332q^{89} - 480q^{92} + 2480q^{94} - 540q^{95} + 2448q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.23607 1.49768 0.748838 0.662753i \(-0.230612\pi\)
0.748838 + 0.662753i \(0.230612\pi\)
\(3\) 0 0
\(4\) 9.94427 1.24303
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 8.23607 0.363986
\(9\) 0 0
\(10\) −21.1803 −0.669781
\(11\) 41.5279 1.13828 0.569142 0.822239i \(-0.307275\pi\)
0.569142 + 0.822239i \(0.307275\pi\)
\(12\) 0 0
\(13\) −88.9706 −1.89815 −0.949077 0.315044i \(-0.897981\pi\)
−0.949077 + 0.315044i \(0.897981\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −44.6656 −0.697900
\(17\) −120.387 −1.71754 −0.858769 0.512364i \(-0.828770\pi\)
−0.858769 + 0.512364i \(0.828770\pi\)
\(18\) 0 0
\(19\) 112.138 1.35401 0.677004 0.735979i \(-0.263278\pi\)
0.677004 + 0.735979i \(0.263278\pi\)
\(20\) −49.7214 −0.555902
\(21\) 0 0
\(22\) 175.915 1.70478
\(23\) 115.279 1.04510 0.522549 0.852609i \(-0.324981\pi\)
0.522549 + 0.852609i \(0.324981\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −376.885 −2.84282
\(27\) 0 0
\(28\) 0 0
\(29\) 144.833 0.927406 0.463703 0.885991i \(-0.346520\pi\)
0.463703 + 0.885991i \(0.346520\pi\)
\(30\) 0 0
\(31\) 258.079 1.49524 0.747618 0.664128i \(-0.231197\pi\)
0.747618 + 0.664128i \(0.231197\pi\)
\(32\) −255.095 −1.40922
\(33\) 0 0
\(34\) −509.967 −2.57231
\(35\) 0 0
\(36\) 0 0
\(37\) 48.3344 0.214760 0.107380 0.994218i \(-0.465754\pi\)
0.107380 + 0.994218i \(0.465754\pi\)
\(38\) 475.023 2.02787
\(39\) 0 0
\(40\) −41.1803 −0.162780
\(41\) 200.885 0.765196 0.382598 0.923915i \(-0.375029\pi\)
0.382598 + 0.923915i \(0.375029\pi\)
\(42\) 0 0
\(43\) −218.217 −0.773901 −0.386950 0.922101i \(-0.626472\pi\)
−0.386950 + 0.922101i \(0.626472\pi\)
\(44\) 412.964 1.41493
\(45\) 0 0
\(46\) 488.328 1.56522
\(47\) 575.659 1.78657 0.893283 0.449496i \(-0.148396\pi\)
0.893283 + 0.449496i \(0.148396\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 105.902 0.299535
\(51\) 0 0
\(52\) −884.748 −2.35947
\(53\) 184.302 0.477657 0.238828 0.971062i \(-0.423237\pi\)
0.238828 + 0.971062i \(0.423237\pi\)
\(54\) 0 0
\(55\) −207.639 −0.509056
\(56\) 0 0
\(57\) 0 0
\(58\) 613.522 1.38895
\(59\) −151.502 −0.334302 −0.167151 0.985931i \(-0.553457\pi\)
−0.167151 + 0.985931i \(0.553457\pi\)
\(60\) 0 0
\(61\) 529.830 1.11209 0.556047 0.831151i \(-0.312317\pi\)
0.556047 + 0.831151i \(0.312317\pi\)
\(62\) 1093.24 2.23938
\(63\) 0 0
\(64\) −723.276 −1.41265
\(65\) 444.853 0.848880
\(66\) 0 0
\(67\) 1.28485 0.00234283 0.00117142 0.999999i \(-0.499627\pi\)
0.00117142 + 0.999999i \(0.499627\pi\)
\(68\) −1197.16 −2.13496
\(69\) 0 0
\(70\) 0 0
\(71\) 61.4226 0.102669 0.0513347 0.998682i \(-0.483652\pi\)
0.0513347 + 0.998682i \(0.483652\pi\)
\(72\) 0 0
\(73\) −484.800 −0.777282 −0.388641 0.921389i \(-0.627055\pi\)
−0.388641 + 0.921389i \(0.627055\pi\)
\(74\) 204.748 0.321641
\(75\) 0 0
\(76\) 1115.13 1.68308
\(77\) 0 0
\(78\) 0 0
\(79\) 878.257 1.25078 0.625390 0.780312i \(-0.284940\pi\)
0.625390 + 0.780312i \(0.284940\pi\)
\(80\) 223.328 0.312111
\(81\) 0 0
\(82\) 850.964 1.14602
\(83\) 491.830 0.650426 0.325213 0.945641i \(-0.394564\pi\)
0.325213 + 0.945641i \(0.394564\pi\)
\(84\) 0 0
\(85\) 601.935 0.768106
\(86\) −924.381 −1.15905
\(87\) 0 0
\(88\) 342.026 0.414320
\(89\) −415.560 −0.494936 −0.247468 0.968896i \(-0.579599\pi\)
−0.247468 + 0.968896i \(0.579599\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1146.36 1.29909
\(93\) 0 0
\(94\) 2438.53 2.67570
\(95\) −560.689 −0.605531
\(96\) 0 0
\(97\) 1031.70 1.07993 0.539964 0.841688i \(-0.318438\pi\)
0.539964 + 0.841688i \(0.318438\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 248.607 0.248607
\(101\) 1447.19 1.42576 0.712878 0.701288i \(-0.247391\pi\)
0.712878 + 0.701288i \(0.247391\pi\)
\(102\) 0 0
\(103\) 163.567 0.156473 0.0782364 0.996935i \(-0.475071\pi\)
0.0782364 + 0.996935i \(0.475071\pi\)
\(104\) −732.768 −0.690902
\(105\) 0 0
\(106\) 780.715 0.715375
\(107\) 129.653 0.117141 0.0585703 0.998283i \(-0.481346\pi\)
0.0585703 + 0.998283i \(0.481346\pi\)
\(108\) 0 0
\(109\) 566.681 0.497965 0.248983 0.968508i \(-0.419904\pi\)
0.248983 + 0.968508i \(0.419904\pi\)
\(110\) −879.574 −0.762401
\(111\) 0 0
\(112\) 0 0
\(113\) −809.890 −0.674230 −0.337115 0.941463i \(-0.609451\pi\)
−0.337115 + 0.941463i \(0.609451\pi\)
\(114\) 0 0
\(115\) −576.393 −0.467382
\(116\) 1440.26 1.15280
\(117\) 0 0
\(118\) −641.771 −0.500676
\(119\) 0 0
\(120\) 0 0
\(121\) 393.563 0.295690
\(122\) 2244.39 1.66556
\(123\) 0 0
\(124\) 2566.41 1.85863
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −2584.25 −1.80563 −0.902816 0.430028i \(-0.858504\pi\)
−0.902816 + 0.430028i \(0.858504\pi\)
\(128\) −1023.08 −0.706473
\(129\) 0 0
\(130\) 1884.43 1.27135
\(131\) −1421.10 −0.947804 −0.473902 0.880578i \(-0.657155\pi\)
−0.473902 + 0.880578i \(0.657155\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 5.44272 0.00350880
\(135\) 0 0
\(136\) −991.515 −0.625160
\(137\) −104.878 −0.0654037 −0.0327019 0.999465i \(-0.510411\pi\)
−0.0327019 + 0.999465i \(0.510411\pi\)
\(138\) 0 0
\(139\) 913.160 0.557217 0.278609 0.960405i \(-0.410127\pi\)
0.278609 + 0.960405i \(0.410127\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 260.190 0.153765
\(143\) −3694.76 −2.16064
\(144\) 0 0
\(145\) −724.164 −0.414749
\(146\) −2053.65 −1.16412
\(147\) 0 0
\(148\) 480.650 0.266954
\(149\) −1781.45 −0.979476 −0.489738 0.871870i \(-0.662908\pi\)
−0.489738 + 0.871870i \(0.662908\pi\)
\(150\) 0 0
\(151\) 1407.53 0.758564 0.379282 0.925281i \(-0.376171\pi\)
0.379282 + 0.925281i \(0.376171\pi\)
\(152\) 923.574 0.492841
\(153\) 0 0
\(154\) 0 0
\(155\) −1290.39 −0.668690
\(156\) 0 0
\(157\) 1598.94 0.812798 0.406399 0.913696i \(-0.366784\pi\)
0.406399 + 0.913696i \(0.366784\pi\)
\(158\) 3720.36 1.87326
\(159\) 0 0
\(160\) 1275.48 0.630220
\(161\) 0 0
\(162\) 0 0
\(163\) −204.892 −0.0984562 −0.0492281 0.998788i \(-0.515676\pi\)
−0.0492281 + 0.998788i \(0.515676\pi\)
\(164\) 1997.66 0.951165
\(165\) 0 0
\(166\) 2083.42 0.974127
\(167\) −1165.94 −0.540259 −0.270129 0.962824i \(-0.587067\pi\)
−0.270129 + 0.962824i \(0.587067\pi\)
\(168\) 0 0
\(169\) 5718.76 2.60299
\(170\) 2549.84 1.15037
\(171\) 0 0
\(172\) −2170.01 −0.961985
\(173\) −2538.00 −1.11538 −0.557690 0.830049i \(-0.688312\pi\)
−0.557690 + 0.830049i \(0.688312\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1854.87 −0.794409
\(177\) 0 0
\(178\) −1760.34 −0.741254
\(179\) −392.255 −0.163791 −0.0818954 0.996641i \(-0.526097\pi\)
−0.0818954 + 0.996641i \(0.526097\pi\)
\(180\) 0 0
\(181\) 2978.08 1.22298 0.611489 0.791253i \(-0.290571\pi\)
0.611489 + 0.791253i \(0.290571\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 949.443 0.380401
\(185\) −241.672 −0.0960436
\(186\) 0 0
\(187\) −4999.41 −1.95504
\(188\) 5724.51 2.22076
\(189\) 0 0
\(190\) −2375.12 −0.906890
\(191\) 1097.37 0.415722 0.207861 0.978158i \(-0.433350\pi\)
0.207861 + 0.978158i \(0.433350\pi\)
\(192\) 0 0
\(193\) 3500.31 1.30548 0.652740 0.757582i \(-0.273619\pi\)
0.652740 + 0.757582i \(0.273619\pi\)
\(194\) 4370.34 1.61738
\(195\) 0 0
\(196\) 0 0
\(197\) −1573.96 −0.569237 −0.284618 0.958641i \(-0.591867\pi\)
−0.284618 + 0.958641i \(0.591867\pi\)
\(198\) 0 0
\(199\) 3396.62 1.20995 0.604976 0.796244i \(-0.293183\pi\)
0.604976 + 0.796244i \(0.293183\pi\)
\(200\) 205.902 0.0727972
\(201\) 0 0
\(202\) 6130.42 2.13532
\(203\) 0 0
\(204\) 0 0
\(205\) −1004.43 −0.342206
\(206\) 692.879 0.234346
\(207\) 0 0
\(208\) 3973.93 1.32472
\(209\) 4656.84 1.54125
\(210\) 0 0
\(211\) 3337.81 1.08903 0.544513 0.838753i \(-0.316715\pi\)
0.544513 + 0.838753i \(0.316715\pi\)
\(212\) 1832.75 0.593744
\(213\) 0 0
\(214\) 549.220 0.175439
\(215\) 1091.08 0.346099
\(216\) 0 0
\(217\) 0 0
\(218\) 2400.50 0.745791
\(219\) 0 0
\(220\) −2064.82 −0.632774
\(221\) 10710.9 3.26015
\(222\) 0 0
\(223\) −127.328 −0.0382356 −0.0191178 0.999817i \(-0.506086\pi\)
−0.0191178 + 0.999817i \(0.506086\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −3430.75 −1.00978
\(227\) 3844.12 1.12398 0.561990 0.827144i \(-0.310036\pi\)
0.561990 + 0.827144i \(0.310036\pi\)
\(228\) 0 0
\(229\) −2536.95 −0.732080 −0.366040 0.930599i \(-0.619287\pi\)
−0.366040 + 0.930599i \(0.619287\pi\)
\(230\) −2441.64 −0.699987
\(231\) 0 0
\(232\) 1192.85 0.337563
\(233\) −3987.44 −1.12114 −0.560570 0.828107i \(-0.689418\pi\)
−0.560570 + 0.828107i \(0.689418\pi\)
\(234\) 0 0
\(235\) −2878.30 −0.798976
\(236\) −1506.57 −0.415549
\(237\) 0 0
\(238\) 0 0
\(239\) 3367.18 0.911317 0.455659 0.890155i \(-0.349404\pi\)
0.455659 + 0.890155i \(0.349404\pi\)
\(240\) 0 0
\(241\) 939.551 0.251128 0.125564 0.992086i \(-0.459926\pi\)
0.125564 + 0.992086i \(0.459926\pi\)
\(242\) 1667.16 0.442848
\(243\) 0 0
\(244\) 5268.77 1.38237
\(245\) 0 0
\(246\) 0 0
\(247\) −9976.96 −2.57012
\(248\) 2125.56 0.544246
\(249\) 0 0
\(250\) −529.508 −0.133956
\(251\) −1403.96 −0.353056 −0.176528 0.984296i \(-0.556487\pi\)
−0.176528 + 0.984296i \(0.556487\pi\)
\(252\) 0 0
\(253\) 4787.28 1.18962
\(254\) −10947.1 −2.70425
\(255\) 0 0
\(256\) 1452.36 0.354579
\(257\) −1964.86 −0.476905 −0.238453 0.971154i \(-0.576640\pi\)
−0.238453 + 0.971154i \(0.576640\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4423.74 1.05519
\(261\) 0 0
\(262\) −6019.89 −1.41950
\(263\) 393.821 0.0923347 0.0461673 0.998934i \(-0.485299\pi\)
0.0461673 + 0.998934i \(0.485299\pi\)
\(264\) 0 0
\(265\) −921.509 −0.213615
\(266\) 0 0
\(267\) 0 0
\(268\) 12.7769 0.00291222
\(269\) −1877.03 −0.425444 −0.212722 0.977113i \(-0.568233\pi\)
−0.212722 + 0.977113i \(0.568233\pi\)
\(270\) 0 0
\(271\) 689.909 0.154646 0.0773228 0.997006i \(-0.475363\pi\)
0.0773228 + 0.997006i \(0.475363\pi\)
\(272\) 5377.16 1.19867
\(273\) 0 0
\(274\) −444.269 −0.0979536
\(275\) 1038.20 0.227657
\(276\) 0 0
\(277\) 6289.13 1.36418 0.682088 0.731270i \(-0.261072\pi\)
0.682088 + 0.731270i \(0.261072\pi\)
\(278\) 3868.21 0.834531
\(279\) 0 0
\(280\) 0 0
\(281\) 1954.87 0.415010 0.207505 0.978234i \(-0.433466\pi\)
0.207505 + 0.978234i \(0.433466\pi\)
\(282\) 0 0
\(283\) −5033.96 −1.05738 −0.528688 0.848816i \(-0.677316\pi\)
−0.528688 + 0.848816i \(0.677316\pi\)
\(284\) 610.803 0.127621
\(285\) 0 0
\(286\) −15651.2 −3.23594
\(287\) 0 0
\(288\) 0 0
\(289\) 9580.03 1.94993
\(290\) −3067.61 −0.621159
\(291\) 0 0
\(292\) −4820.99 −0.966188
\(293\) 6369.12 1.26993 0.634963 0.772543i \(-0.281015\pi\)
0.634963 + 0.772543i \(0.281015\pi\)
\(294\) 0 0
\(295\) 757.508 0.149504
\(296\) 398.085 0.0781697
\(297\) 0 0
\(298\) −7546.34 −1.46694
\(299\) −10256.4 −1.98376
\(300\) 0 0
\(301\) 0 0
\(302\) 5962.39 1.13608
\(303\) 0 0
\(304\) −5008.70 −0.944963
\(305\) −2649.15 −0.497344
\(306\) 0 0
\(307\) 6619.83 1.23066 0.615332 0.788268i \(-0.289022\pi\)
0.615332 + 0.788268i \(0.289022\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5466.20 −1.00148
\(311\) −9909.22 −1.80675 −0.903377 0.428848i \(-0.858920\pi\)
−0.903377 + 0.428848i \(0.858920\pi\)
\(312\) 0 0
\(313\) 422.336 0.0762678 0.0381339 0.999273i \(-0.487859\pi\)
0.0381339 + 0.999273i \(0.487859\pi\)
\(314\) 6773.22 1.21731
\(315\) 0 0
\(316\) 8733.63 1.55476
\(317\) 4902.78 0.868668 0.434334 0.900752i \(-0.356984\pi\)
0.434334 + 0.900752i \(0.356984\pi\)
\(318\) 0 0
\(319\) 6014.60 1.05565
\(320\) 3616.38 0.631755
\(321\) 0 0
\(322\) 0 0
\(323\) −13499.9 −2.32556
\(324\) 0 0
\(325\) −2224.26 −0.379631
\(326\) −867.935 −0.147455
\(327\) 0 0
\(328\) 1654.51 0.278521
\(329\) 0 0
\(330\) 0 0
\(331\) −5281.74 −0.877071 −0.438535 0.898714i \(-0.644503\pi\)
−0.438535 + 0.898714i \(0.644503\pi\)
\(332\) 4890.89 0.808501
\(333\) 0 0
\(334\) −4939.01 −0.809133
\(335\) −6.42426 −0.00104775
\(336\) 0 0
\(337\) 4459.60 0.720860 0.360430 0.932786i \(-0.382630\pi\)
0.360430 + 0.932786i \(0.382630\pi\)
\(338\) 24225.1 3.89843
\(339\) 0 0
\(340\) 5985.80 0.954782
\(341\) 10717.5 1.70200
\(342\) 0 0
\(343\) 0 0
\(344\) −1797.25 −0.281689
\(345\) 0 0
\(346\) −10751.2 −1.67048
\(347\) −5261.97 −0.814056 −0.407028 0.913416i \(-0.633435\pi\)
−0.407028 + 0.913416i \(0.633435\pi\)
\(348\) 0 0
\(349\) −960.325 −0.147292 −0.0736461 0.997284i \(-0.523464\pi\)
−0.0736461 + 0.997284i \(0.523464\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −10593.6 −1.60409
\(353\) −8925.80 −1.34581 −0.672907 0.739727i \(-0.734955\pi\)
−0.672907 + 0.739727i \(0.734955\pi\)
\(354\) 0 0
\(355\) −307.113 −0.0459151
\(356\) −4132.45 −0.615222
\(357\) 0 0
\(358\) −1661.62 −0.245306
\(359\) 3056.27 0.449314 0.224657 0.974438i \(-0.427874\pi\)
0.224657 + 0.974438i \(0.427874\pi\)
\(360\) 0 0
\(361\) 5715.88 0.833340
\(362\) 12615.4 1.83162
\(363\) 0 0
\(364\) 0 0
\(365\) 2424.00 0.347611
\(366\) 0 0
\(367\) 1813.52 0.257943 0.128971 0.991648i \(-0.458832\pi\)
0.128971 + 0.991648i \(0.458832\pi\)
\(368\) −5148.99 −0.729375
\(369\) 0 0
\(370\) −1023.74 −0.143842
\(371\) 0 0
\(372\) 0 0
\(373\) −4517.48 −0.627094 −0.313547 0.949573i \(-0.601517\pi\)
−0.313547 + 0.949573i \(0.601517\pi\)
\(374\) −21177.9 −2.92802
\(375\) 0 0
\(376\) 4741.17 0.650285
\(377\) −12885.9 −1.76036
\(378\) 0 0
\(379\) −4931.24 −0.668340 −0.334170 0.942513i \(-0.608456\pi\)
−0.334170 + 0.942513i \(0.608456\pi\)
\(380\) −5575.64 −0.752696
\(381\) 0 0
\(382\) 4648.53 0.622617
\(383\) −1482.37 −0.197770 −0.0988849 0.995099i \(-0.531528\pi\)
−0.0988849 + 0.995099i \(0.531528\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14827.5 1.95519
\(387\) 0 0
\(388\) 10259.5 1.34239
\(389\) 5448.98 0.710217 0.355109 0.934825i \(-0.384444\pi\)
0.355109 + 0.934825i \(0.384444\pi\)
\(390\) 0 0
\(391\) −13878.0 −1.79500
\(392\) 0 0
\(393\) 0 0
\(394\) −6667.38 −0.852532
\(395\) −4391.28 −0.559366
\(396\) 0 0
\(397\) 13675.9 1.72891 0.864453 0.502713i \(-0.167665\pi\)
0.864453 + 0.502713i \(0.167665\pi\)
\(398\) 14388.3 1.81212
\(399\) 0 0
\(400\) −1116.64 −0.139580
\(401\) −14109.9 −1.75714 −0.878570 0.477613i \(-0.841502\pi\)
−0.878570 + 0.477613i \(0.841502\pi\)
\(402\) 0 0
\(403\) −22961.4 −2.83819
\(404\) 14391.3 1.77226
\(405\) 0 0
\(406\) 0 0
\(407\) 2007.22 0.244458
\(408\) 0 0
\(409\) 13995.6 1.69203 0.846015 0.533159i \(-0.178995\pi\)
0.846015 + 0.533159i \(0.178995\pi\)
\(410\) −4254.82 −0.512514
\(411\) 0 0
\(412\) 1626.55 0.194501
\(413\) 0 0
\(414\) 0 0
\(415\) −2459.15 −0.290879
\(416\) 22696.0 2.67491
\(417\) 0 0
\(418\) 19726.7 2.30829
\(419\) −9840.61 −1.14736 −0.573682 0.819078i \(-0.694485\pi\)
−0.573682 + 0.819078i \(0.694485\pi\)
\(420\) 0 0
\(421\) −12660.5 −1.46564 −0.732822 0.680420i \(-0.761797\pi\)
−0.732822 + 0.680420i \(0.761797\pi\)
\(422\) 14139.2 1.63101
\(423\) 0 0
\(424\) 1517.92 0.173860
\(425\) −3009.67 −0.343507
\(426\) 0 0
\(427\) 0 0
\(428\) 1289.31 0.145610
\(429\) 0 0
\(430\) 4621.90 0.518344
\(431\) 4578.91 0.511736 0.255868 0.966712i \(-0.417639\pi\)
0.255868 + 0.966712i \(0.417639\pi\)
\(432\) 0 0
\(433\) 3279.88 0.364020 0.182010 0.983297i \(-0.441740\pi\)
0.182010 + 0.983297i \(0.441740\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5635.23 0.618988
\(437\) 12927.1 1.41507
\(438\) 0 0
\(439\) 427.807 0.0465105 0.0232552 0.999730i \(-0.492597\pi\)
0.0232552 + 0.999730i \(0.492597\pi\)
\(440\) −1710.13 −0.185289
\(441\) 0 0
\(442\) 45372.1 4.88265
\(443\) −15441.2 −1.65605 −0.828027 0.560688i \(-0.810537\pi\)
−0.828027 + 0.560688i \(0.810537\pi\)
\(444\) 0 0
\(445\) 2077.80 0.221342
\(446\) −539.371 −0.0572645
\(447\) 0 0
\(448\) 0 0
\(449\) −9382.02 −0.986113 −0.493057 0.869997i \(-0.664120\pi\)
−0.493057 + 0.869997i \(0.664120\pi\)
\(450\) 0 0
\(451\) 8342.34 0.871010
\(452\) −8053.77 −0.838091
\(453\) 0 0
\(454\) 16284.0 1.68336
\(455\) 0 0
\(456\) 0 0
\(457\) 13570.4 1.38905 0.694524 0.719469i \(-0.255615\pi\)
0.694524 + 0.719469i \(0.255615\pi\)
\(458\) −10746.7 −1.09642
\(459\) 0 0
\(460\) −5731.81 −0.580972
\(461\) 1251.88 0.126477 0.0632386 0.997998i \(-0.479857\pi\)
0.0632386 + 0.997998i \(0.479857\pi\)
\(462\) 0 0
\(463\) 7934.36 0.796417 0.398209 0.917295i \(-0.369632\pi\)
0.398209 + 0.917295i \(0.369632\pi\)
\(464\) −6469.05 −0.647237
\(465\) 0 0
\(466\) −16891.1 −1.67911
\(467\) 7583.76 0.751466 0.375733 0.926728i \(-0.377391\pi\)
0.375733 + 0.926728i \(0.377391\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −12192.7 −1.19661
\(471\) 0 0
\(472\) −1247.78 −0.121681
\(473\) −9062.07 −0.880919
\(474\) 0 0
\(475\) 2803.44 0.270802
\(476\) 0 0
\(477\) 0 0
\(478\) 14263.6 1.36486
\(479\) −5829.34 −0.556053 −0.278027 0.960573i \(-0.589680\pi\)
−0.278027 + 0.960573i \(0.589680\pi\)
\(480\) 0 0
\(481\) −4300.34 −0.407648
\(482\) 3980.00 0.376108
\(483\) 0 0
\(484\) 3913.70 0.367553
\(485\) −5158.49 −0.482959
\(486\) 0 0
\(487\) −19902.1 −1.85185 −0.925925 0.377708i \(-0.876712\pi\)
−0.925925 + 0.377708i \(0.876712\pi\)
\(488\) 4363.71 0.404787
\(489\) 0 0
\(490\) 0 0
\(491\) 16821.6 1.54613 0.773065 0.634327i \(-0.218723\pi\)
0.773065 + 0.634327i \(0.218723\pi\)
\(492\) 0 0
\(493\) −17436.0 −1.59285
\(494\) −42263.1 −3.84920
\(495\) 0 0
\(496\) −11527.3 −1.04353
\(497\) 0 0
\(498\) 0 0
\(499\) 6031.83 0.541126 0.270563 0.962702i \(-0.412790\pi\)
0.270563 + 0.962702i \(0.412790\pi\)
\(500\) −1243.03 −0.111180
\(501\) 0 0
\(502\) −5947.27 −0.528764
\(503\) −17176.4 −1.52258 −0.761290 0.648412i \(-0.775434\pi\)
−0.761290 + 0.648412i \(0.775434\pi\)
\(504\) 0 0
\(505\) −7235.97 −0.637617
\(506\) 20279.2 1.78166
\(507\) 0 0
\(508\) −25698.5 −2.24446
\(509\) 4706.59 0.409854 0.204927 0.978777i \(-0.434304\pi\)
0.204927 + 0.978777i \(0.434304\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 14336.9 1.23752
\(513\) 0 0
\(514\) −8323.28 −0.714250
\(515\) −817.833 −0.0699768
\(516\) 0 0
\(517\) 23905.9 2.03362
\(518\) 0 0
\(519\) 0 0
\(520\) 3663.84 0.308981
\(521\) −8557.18 −0.719572 −0.359786 0.933035i \(-0.617150\pi\)
−0.359786 + 0.933035i \(0.617150\pi\)
\(522\) 0 0
\(523\) −18248.5 −1.52572 −0.762858 0.646566i \(-0.776204\pi\)
−0.762858 + 0.646566i \(0.776204\pi\)
\(524\) −14131.8 −1.17815
\(525\) 0 0
\(526\) 1668.25 0.138287
\(527\) −31069.3 −2.56813
\(528\) 0 0
\(529\) 1122.16 0.0922302
\(530\) −3903.58 −0.319925
\(531\) 0 0
\(532\) 0 0
\(533\) −17872.9 −1.45246
\(534\) 0 0
\(535\) −648.266 −0.0523869
\(536\) 10.5821 0.000852758 0
\(537\) 0 0
\(538\) −7951.22 −0.637177
\(539\) 0 0
\(540\) 0 0
\(541\) −5734.17 −0.455696 −0.227848 0.973697i \(-0.573169\pi\)
−0.227848 + 0.973697i \(0.573169\pi\)
\(542\) 2922.50 0.231609
\(543\) 0 0
\(544\) 30710.1 2.42038
\(545\) −2833.41 −0.222697
\(546\) 0 0
\(547\) −8002.52 −0.625527 −0.312763 0.949831i \(-0.601255\pi\)
−0.312763 + 0.949831i \(0.601255\pi\)
\(548\) −1042.93 −0.0812991
\(549\) 0 0
\(550\) 4397.87 0.340956
\(551\) 16241.2 1.25572
\(552\) 0 0
\(553\) 0 0
\(554\) 26641.2 2.04310
\(555\) 0 0
\(556\) 9080.71 0.692640
\(557\) −1276.82 −0.0971289 −0.0485644 0.998820i \(-0.515465\pi\)
−0.0485644 + 0.998820i \(0.515465\pi\)
\(558\) 0 0
\(559\) 19414.9 1.46898
\(560\) 0 0
\(561\) 0 0
\(562\) 8280.96 0.621550
\(563\) 11027.7 0.825507 0.412753 0.910843i \(-0.364567\pi\)
0.412753 + 0.910843i \(0.364567\pi\)
\(564\) 0 0
\(565\) 4049.45 0.301525
\(566\) −21324.2 −1.58361
\(567\) 0 0
\(568\) 505.881 0.0373702
\(569\) 4519.03 0.332948 0.166474 0.986046i \(-0.446762\pi\)
0.166474 + 0.986046i \(0.446762\pi\)
\(570\) 0 0
\(571\) 3598.81 0.263758 0.131879 0.991266i \(-0.457899\pi\)
0.131879 + 0.991266i \(0.457899\pi\)
\(572\) −36741.7 −2.68575
\(573\) 0 0
\(574\) 0 0
\(575\) 2881.97 0.209020
\(576\) 0 0
\(577\) 3439.23 0.248140 0.124070 0.992273i \(-0.460405\pi\)
0.124070 + 0.992273i \(0.460405\pi\)
\(578\) 40581.6 2.92037
\(579\) 0 0
\(580\) −7201.28 −0.515547
\(581\) 0 0
\(582\) 0 0
\(583\) 7653.66 0.543709
\(584\) −3992.85 −0.282920
\(585\) 0 0
\(586\) 26980.0 1.90194
\(587\) 21285.2 1.49665 0.748327 0.663330i \(-0.230857\pi\)
0.748327 + 0.663330i \(0.230857\pi\)
\(588\) 0 0
\(589\) 28940.4 2.02456
\(590\) 3208.85 0.223909
\(591\) 0 0
\(592\) −2158.89 −0.149881
\(593\) −14200.8 −0.983404 −0.491702 0.870764i \(-0.663625\pi\)
−0.491702 + 0.870764i \(0.663625\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17715.2 −1.21752
\(597\) 0 0
\(598\) −43446.8 −2.97103
\(599\) 8885.05 0.606065 0.303033 0.952980i \(-0.402001\pi\)
0.303033 + 0.952980i \(0.402001\pi\)
\(600\) 0 0
\(601\) 2052.89 0.139333 0.0696664 0.997570i \(-0.477807\pi\)
0.0696664 + 0.997570i \(0.477807\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 13996.9 0.942920
\(605\) −1967.82 −0.132237
\(606\) 0 0
\(607\) −10280.0 −0.687404 −0.343702 0.939079i \(-0.611681\pi\)
−0.343702 + 0.939079i \(0.611681\pi\)
\(608\) −28605.8 −1.90809
\(609\) 0 0
\(610\) −11222.0 −0.744860
\(611\) −51216.8 −3.39118
\(612\) 0 0
\(613\) 23409.5 1.54242 0.771208 0.636584i \(-0.219653\pi\)
0.771208 + 0.636584i \(0.219653\pi\)
\(614\) 28042.0 1.84314
\(615\) 0 0
\(616\) 0 0
\(617\) 6632.75 0.432779 0.216389 0.976307i \(-0.430572\pi\)
0.216389 + 0.976307i \(0.430572\pi\)
\(618\) 0 0
\(619\) −10734.0 −0.696990 −0.348495 0.937311i \(-0.613307\pi\)
−0.348495 + 0.937311i \(0.613307\pi\)
\(620\) −12832.0 −0.831205
\(621\) 0 0
\(622\) −41976.1 −2.70593
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1789.04 0.114225
\(627\) 0 0
\(628\) 15900.3 1.01034
\(629\) −5818.83 −0.368858
\(630\) 0 0
\(631\) −17071.0 −1.07700 −0.538499 0.842626i \(-0.681008\pi\)
−0.538499 + 0.842626i \(0.681008\pi\)
\(632\) 7233.38 0.455267
\(633\) 0 0
\(634\) 20768.5 1.30098
\(635\) 12921.3 0.807503
\(636\) 0 0
\(637\) 0 0
\(638\) 25478.2 1.58102
\(639\) 0 0
\(640\) 5115.41 0.315945
\(641\) 19389.7 1.19477 0.597386 0.801954i \(-0.296206\pi\)
0.597386 + 0.801954i \(0.296206\pi\)
\(642\) 0 0
\(643\) 25409.3 1.55839 0.779196 0.626780i \(-0.215628\pi\)
0.779196 + 0.626780i \(0.215628\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −57186.6 −3.48294
\(647\) −6039.08 −0.366956 −0.183478 0.983024i \(-0.558736\pi\)
−0.183478 + 0.983024i \(0.558736\pi\)
\(648\) 0 0
\(649\) −6291.54 −0.380531
\(650\) −9422.14 −0.568564
\(651\) 0 0
\(652\) −2037.50 −0.122384
\(653\) 30666.2 1.83776 0.918882 0.394532i \(-0.129093\pi\)
0.918882 + 0.394532i \(0.129093\pi\)
\(654\) 0 0
\(655\) 7105.51 0.423871
\(656\) −8972.67 −0.534031
\(657\) 0 0
\(658\) 0 0
\(659\) 2765.96 0.163500 0.0817500 0.996653i \(-0.473949\pi\)
0.0817500 + 0.996653i \(0.473949\pi\)
\(660\) 0 0
\(661\) −27261.8 −1.60418 −0.802089 0.597204i \(-0.796278\pi\)
−0.802089 + 0.597204i \(0.796278\pi\)
\(662\) −22373.8 −1.31357
\(663\) 0 0
\(664\) 4050.74 0.236746
\(665\) 0 0
\(666\) 0 0
\(667\) 16696.1 0.969230
\(668\) −11594.4 −0.671560
\(669\) 0 0
\(670\) −27.2136 −0.00156918
\(671\) 22002.7 1.26588
\(672\) 0 0
\(673\) −1048.17 −0.0600356 −0.0300178 0.999549i \(-0.509556\pi\)
−0.0300178 + 0.999549i \(0.509556\pi\)
\(674\) 18891.2 1.07961
\(675\) 0 0
\(676\) 56869.0 3.23560
\(677\) −34554.7 −1.96166 −0.980831 0.194860i \(-0.937575\pi\)
−0.980831 + 0.194860i \(0.937575\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4957.58 0.279580
\(681\) 0 0
\(682\) 45399.9 2.54905
\(683\) −14711.6 −0.824192 −0.412096 0.911140i \(-0.635203\pi\)
−0.412096 + 0.911140i \(0.635203\pi\)
\(684\) 0 0
\(685\) 524.389 0.0292494
\(686\) 0 0
\(687\) 0 0
\(688\) 9746.79 0.540106
\(689\) −16397.4 −0.906666
\(690\) 0 0
\(691\) 24522.6 1.35005 0.675024 0.737796i \(-0.264133\pi\)
0.675024 + 0.737796i \(0.264133\pi\)
\(692\) −25238.6 −1.38646
\(693\) 0 0
\(694\) −22290.1 −1.21919
\(695\) −4565.80 −0.249195
\(696\) 0 0
\(697\) −24184.0 −1.31425
\(698\) −4068.00 −0.220596
\(699\) 0 0
\(700\) 0 0
\(701\) −19912.2 −1.07286 −0.536429 0.843946i \(-0.680227\pi\)
−0.536429 + 0.843946i \(0.680227\pi\)
\(702\) 0 0
\(703\) 5420.11 0.290787
\(704\) −30036.1 −1.60799
\(705\) 0 0
\(706\) −37810.3 −2.01559
\(707\) 0 0
\(708\) 0 0
\(709\) 6208.79 0.328880 0.164440 0.986387i \(-0.447418\pi\)
0.164440 + 0.986387i \(0.447418\pi\)
\(710\) −1300.95 −0.0687660
\(711\) 0 0
\(712\) −3422.58 −0.180150
\(713\) 29751.0 1.56267
\(714\) 0 0
\(715\) 18473.8 0.966267
\(716\) −3900.69 −0.203597
\(717\) 0 0
\(718\) 12946.6 0.672927
\(719\) 13063.6 0.677593 0.338797 0.940860i \(-0.389980\pi\)
0.338797 + 0.940860i \(0.389980\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24212.9 1.24807
\(723\) 0 0
\(724\) 29614.8 1.52020
\(725\) 3620.82 0.185481
\(726\) 0 0
\(727\) 12897.0 0.657940 0.328970 0.944340i \(-0.393298\pi\)
0.328970 + 0.944340i \(0.393298\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10268.2 0.520609
\(731\) 26270.5 1.32920
\(732\) 0 0
\(733\) −11699.6 −0.589540 −0.294770 0.955568i \(-0.595243\pi\)
−0.294770 + 0.955568i \(0.595243\pi\)
\(734\) 7682.19 0.386315
\(735\) 0 0
\(736\) −29407.0 −1.47277
\(737\) 53.3571 0.00266681
\(738\) 0 0
\(739\) 14974.0 0.745368 0.372684 0.927958i \(-0.378438\pi\)
0.372684 + 0.927958i \(0.378438\pi\)
\(740\) −2403.25 −0.119385
\(741\) 0 0
\(742\) 0 0
\(743\) −18500.7 −0.913492 −0.456746 0.889597i \(-0.650985\pi\)
−0.456746 + 0.889597i \(0.650985\pi\)
\(744\) 0 0
\(745\) 8907.24 0.438035
\(746\) −19136.3 −0.939184
\(747\) 0 0
\(748\) −49715.5 −2.43019
\(749\) 0 0
\(750\) 0 0
\(751\) −26348.4 −1.28025 −0.640125 0.768271i \(-0.721117\pi\)
−0.640125 + 0.768271i \(0.721117\pi\)
\(752\) −25712.2 −1.24684
\(753\) 0 0
\(754\) −54585.4 −2.63645
\(755\) −7037.65 −0.339240
\(756\) 0 0
\(757\) −28061.7 −1.34732 −0.673659 0.739042i \(-0.735278\pi\)
−0.673659 + 0.739042i \(0.735278\pi\)
\(758\) −20889.1 −1.00096
\(759\) 0 0
\(760\) −4617.87 −0.220405
\(761\) −3579.22 −0.170495 −0.0852476 0.996360i \(-0.527168\pi\)
−0.0852476 + 0.996360i \(0.527168\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10912.5 0.516757
\(765\) 0 0
\(766\) −6279.44 −0.296195
\(767\) 13479.2 0.634557
\(768\) 0 0
\(769\) −4339.61 −0.203499 −0.101749 0.994810i \(-0.532444\pi\)
−0.101749 + 0.994810i \(0.532444\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 34808.0 1.62276
\(773\) 10005.1 0.465537 0.232769 0.972532i \(-0.425222\pi\)
0.232769 + 0.972532i \(0.425222\pi\)
\(774\) 0 0
\(775\) 6451.97 0.299047
\(776\) 8497.14 0.393079
\(777\) 0 0
\(778\) 23082.3 1.06368
\(779\) 22526.8 1.03608
\(780\) 0 0
\(781\) 2550.75 0.116867
\(782\) −58788.4 −2.68832
\(783\) 0 0
\(784\) 0 0
\(785\) −7994.70 −0.363494
\(786\) 0 0
\(787\) −17826.8 −0.807443 −0.403721 0.914882i \(-0.632284\pi\)
−0.403721 + 0.914882i \(0.632284\pi\)
\(788\) −15651.8 −0.707581
\(789\) 0 0
\(790\) −18601.8 −0.837749
\(791\) 0 0
\(792\) 0 0
\(793\) −47139.3 −2.11093
\(794\) 57932.2 2.58934
\(795\) 0 0
\(796\) 33777.0 1.50401
\(797\) 36723.0 1.63211 0.816057 0.577971i \(-0.196155\pi\)
0.816057 + 0.577971i \(0.196155\pi\)
\(798\) 0 0
\(799\) −69301.9 −3.06849
\(800\) −6377.38 −0.281843
\(801\) 0 0
\(802\) −59770.4 −2.63163
\(803\) −20132.7 −0.884767
\(804\) 0 0
\(805\) 0 0
\(806\) −97266.2 −4.25069
\(807\) 0 0
\(808\) 11919.2 0.518955
\(809\) −5657.55 −0.245870 −0.122935 0.992415i \(-0.539231\pi\)
−0.122935 + 0.992415i \(0.539231\pi\)
\(810\) 0 0
\(811\) −7532.41 −0.326139 −0.163070 0.986615i \(-0.552139\pi\)
−0.163070 + 0.986615i \(0.552139\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8502.73 0.366119
\(815\) 1024.46 0.0440309
\(816\) 0 0
\(817\) −24470.3 −1.04787
\(818\) 59286.5 2.53411
\(819\) 0 0
\(820\) −9988.30 −0.425374
\(821\) 6489.25 0.275854 0.137927 0.990442i \(-0.455956\pi\)
0.137927 + 0.990442i \(0.455956\pi\)
\(822\) 0 0
\(823\) 7901.57 0.334668 0.167334 0.985900i \(-0.446484\pi\)
0.167334 + 0.985900i \(0.446484\pi\)
\(824\) 1347.15 0.0569539
\(825\) 0 0
\(826\) 0 0
\(827\) 37815.8 1.59007 0.795033 0.606566i \(-0.207453\pi\)
0.795033 + 0.606566i \(0.207453\pi\)
\(828\) 0 0
\(829\) −26073.5 −1.09236 −0.546182 0.837667i \(-0.683919\pi\)
−0.546182 + 0.837667i \(0.683919\pi\)
\(830\) −10417.1 −0.435643
\(831\) 0 0
\(832\) 64350.2 2.68142
\(833\) 0 0
\(834\) 0 0
\(835\) 5829.71 0.241611
\(836\) 46308.9 1.91582
\(837\) 0 0
\(838\) −41685.5 −1.71838
\(839\) −15590.3 −0.641523 −0.320762 0.947160i \(-0.603939\pi\)
−0.320762 + 0.947160i \(0.603939\pi\)
\(840\) 0 0
\(841\) −3412.46 −0.139918
\(842\) −53630.8 −2.19506
\(843\) 0 0
\(844\) 33192.1 1.35370
\(845\) −28593.8 −1.16409
\(846\) 0 0
\(847\) 0 0
\(848\) −8231.96 −0.333357
\(849\) 0 0
\(850\) −12749.2 −0.514463
\(851\) 5571.92 0.224445
\(852\) 0 0
\(853\) 17476.1 0.701488 0.350744 0.936471i \(-0.385929\pi\)
0.350744 + 0.936471i \(0.385929\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1067.83 0.0426376
\(857\) −5694.54 −0.226980 −0.113490 0.993539i \(-0.536203\pi\)
−0.113490 + 0.993539i \(0.536203\pi\)
\(858\) 0 0
\(859\) −27313.6 −1.08490 −0.542448 0.840089i \(-0.682503\pi\)
−0.542448 + 0.840089i \(0.682503\pi\)
\(860\) 10850.0 0.430213
\(861\) 0 0
\(862\) 19396.6 0.766415
\(863\) 9046.07 0.356815 0.178408 0.983957i \(-0.442905\pi\)
0.178408 + 0.983957i \(0.442905\pi\)
\(864\) 0 0
\(865\) 12690.0 0.498813
\(866\) 13893.8 0.545185
\(867\) 0 0
\(868\) 0 0
\(869\) 36472.1 1.42374
\(870\) 0 0
\(871\) −114.314 −0.00444705
\(872\) 4667.22 0.181252
\(873\) 0 0
\(874\) 54760.0 2.11932
\(875\) 0 0
\(876\) 0 0
\(877\) −2104.29 −0.0810224 −0.0405112 0.999179i \(-0.512899\pi\)
−0.0405112 + 0.999179i \(0.512899\pi\)
\(878\) 1812.22 0.0696576
\(879\) 0 0
\(880\) 9274.34 0.355270
\(881\) −22589.6 −0.863861 −0.431931 0.901907i \(-0.642167\pi\)
−0.431931 + 0.901907i \(0.642167\pi\)
\(882\) 0 0
\(883\) −2419.71 −0.0922193 −0.0461096 0.998936i \(-0.514682\pi\)
−0.0461096 + 0.998936i \(0.514682\pi\)
\(884\) 106512. 4.05248
\(885\) 0 0
\(886\) −65409.8 −2.48023
\(887\) 13177.0 0.498806 0.249403 0.968400i \(-0.419766\pi\)
0.249403 + 0.968400i \(0.419766\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8801.71 0.331499
\(891\) 0 0
\(892\) −1266.19 −0.0475281
\(893\) 64553.2 2.41902
\(894\) 0 0
\(895\) 1961.28 0.0732495
\(896\) 0 0
\(897\) 0 0
\(898\) −39742.9 −1.47688
\(899\) 37378.3 1.38669
\(900\) 0 0
\(901\) −22187.5 −0.820393
\(902\) 35338.7 1.30449
\(903\) 0 0
\(904\) −6670.31 −0.245411
\(905\) −14890.4 −0.546932
\(906\) 0 0
\(907\) 9189.14 0.336406 0.168203 0.985752i \(-0.446204\pi\)
0.168203 + 0.985752i \(0.446204\pi\)
\(908\) 38227.0 1.39714
\(909\) 0 0
\(910\) 0 0
\(911\) 17045.8 0.619928 0.309964 0.950748i \(-0.399683\pi\)
0.309964 + 0.950748i \(0.399683\pi\)
\(912\) 0 0
\(913\) 20424.6 0.740369
\(914\) 57485.0 2.08034
\(915\) 0 0
\(916\) −25228.1 −0.910001
\(917\) 0 0
\(918\) 0 0
\(919\) −30825.0 −1.10645 −0.553223 0.833033i \(-0.686602\pi\)
−0.553223 + 0.833033i \(0.686602\pi\)
\(920\) −4747.21 −0.170121
\(921\) 0 0
\(922\) 5303.06 0.189422
\(923\) −5464.81 −0.194882
\(924\) 0 0
\(925\) 1208.36 0.0429520
\(926\) 33610.5 1.19277
\(927\) 0 0
\(928\) −36946.2 −1.30691
\(929\) −5785.88 −0.204336 −0.102168 0.994767i \(-0.532578\pi\)
−0.102168 + 0.994767i \(0.532578\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −39652.2 −1.39362
\(933\) 0 0
\(934\) 32125.3 1.12545
\(935\) 24997.1 0.874323
\(936\) 0 0
\(937\) 13680.9 0.476986 0.238493 0.971144i \(-0.423347\pi\)
0.238493 + 0.971144i \(0.423347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −28622.6 −0.993155
\(941\) −45448.8 −1.57448 −0.787242 0.616644i \(-0.788492\pi\)
−0.787242 + 0.616644i \(0.788492\pi\)
\(942\) 0 0
\(943\) 23157.8 0.799705
\(944\) 6766.91 0.233310
\(945\) 0 0
\(946\) −38387.6 −1.31933
\(947\) 7788.45 0.267255 0.133628 0.991032i \(-0.457337\pi\)
0.133628 + 0.991032i \(0.457337\pi\)
\(948\) 0 0
\(949\) 43133.0 1.47540
\(950\) 11875.6 0.405573
\(951\) 0 0
\(952\) 0 0
\(953\) −6149.43 −0.209024 −0.104512 0.994524i \(-0.533328\pi\)
−0.104512 + 0.994524i \(0.533328\pi\)
\(954\) 0 0
\(955\) −5486.85 −0.185917
\(956\) 33484.2 1.13280
\(957\) 0 0
\(958\) −24693.5 −0.832788
\(959\) 0 0
\(960\) 0 0
\(961\) 36813.7 1.23573
\(962\) −18216.5 −0.610524
\(963\) 0 0
\(964\) 9343.15 0.312160
\(965\) −17501.5 −0.583829
\(966\) 0 0
\(967\) 23902.9 0.794896 0.397448 0.917625i \(-0.369896\pi\)
0.397448 + 0.917625i \(0.369896\pi\)
\(968\) 3241.42 0.107627
\(969\) 0 0
\(970\) −21851.7 −0.723316
\(971\) −8015.06 −0.264898 −0.132449 0.991190i \(-0.542284\pi\)
−0.132449 + 0.991190i \(0.542284\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −84306.7 −2.77347
\(975\) 0 0
\(976\) −23665.2 −0.776131
\(977\) 34861.1 1.14156 0.570780 0.821103i \(-0.306641\pi\)
0.570780 + 0.821103i \(0.306641\pi\)
\(978\) 0 0
\(979\) −17257.3 −0.563378
\(980\) 0 0
\(981\) 0 0
\(982\) 71257.6 2.31560
\(983\) 6620.83 0.214824 0.107412 0.994215i \(-0.465744\pi\)
0.107412 + 0.994215i \(0.465744\pi\)
\(984\) 0 0
\(985\) 7869.78 0.254570
\(986\) −73860.0 −2.38558
\(987\) 0 0
\(988\) −99213.6 −3.19474
\(989\) −25155.7 −0.808802
\(990\) 0 0
\(991\) 10360.1 0.332089 0.166045 0.986118i \(-0.446900\pi\)
0.166045 + 0.986118i \(0.446900\pi\)
\(992\) −65834.7 −2.10711
\(993\) 0 0
\(994\) 0 0
\(995\) −16983.1 −0.541107
\(996\) 0 0
\(997\) 40309.3 1.28045 0.640225 0.768188i \(-0.278841\pi\)
0.640225 + 0.768188i \(0.278841\pi\)
\(998\) 25551.3 0.810432
\(999\) 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 2205.4.a.be.1.2 2
3.2 odd 2 735.4.a.m.1.1 2
7.6 odd 2 315.4.a.l.1.2 2
21.20 even 2 105.4.a.d.1.1 2
35.34 odd 2 1575.4.a.n.1.1 2
84.83 odd 2 1680.4.a.bd.1.1 2
105.62 odd 4 525.4.d.k.274.1 4
105.83 odd 4 525.4.d.k.274.4 4
105.104 even 2 525.4.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.d.1.1 2 21.20 even 2
315.4.a.l.1.2 2 7.6 odd 2
525.4.a.o.1.2 2 105.104 even 2
525.4.d.k.274.1 4 105.62 odd 4
525.4.d.k.274.4 4 105.83 odd 4
735.4.a.m.1.1 2 3.2 odd 2
1575.4.a.n.1.1 2 35.34 odd 2
1680.4.a.bd.1.1 2 84.83 odd 2
2205.4.a.be.1.2 2 1.1 even 1 trivial