Properties

Label 1849.4.a.j.1.16
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.12255 q^{2} +9.87196 q^{3} -3.49478 q^{4} -15.5747 q^{5} -20.9537 q^{6} -16.3831 q^{7} +24.3983 q^{8} +70.4556 q^{9} +O(q^{10})\) \(q-2.12255 q^{2} +9.87196 q^{3} -3.49478 q^{4} -15.5747 q^{5} -20.9537 q^{6} -16.3831 q^{7} +24.3983 q^{8} +70.4556 q^{9} +33.0580 q^{10} +42.7049 q^{11} -34.5003 q^{12} -44.4137 q^{13} +34.7739 q^{14} -153.752 q^{15} -23.8283 q^{16} -80.6618 q^{17} -149.546 q^{18} +9.72641 q^{19} +54.4300 q^{20} -161.733 q^{21} -90.6434 q^{22} +81.9683 q^{23} +240.859 q^{24} +117.570 q^{25} +94.2703 q^{26} +428.992 q^{27} +57.2552 q^{28} +170.553 q^{29} +326.348 q^{30} -115.463 q^{31} -144.609 q^{32} +421.581 q^{33} +171.209 q^{34} +255.161 q^{35} -246.227 q^{36} +56.3932 q^{37} -20.6448 q^{38} -438.450 q^{39} -379.995 q^{40} +9.51295 q^{41} +343.287 q^{42} -149.244 q^{44} -1097.32 q^{45} -173.982 q^{46} +9.55328 q^{47} -235.232 q^{48} -74.5945 q^{49} -249.549 q^{50} -796.290 q^{51} +155.216 q^{52} +254.029 q^{53} -910.558 q^{54} -665.115 q^{55} -399.719 q^{56} +96.0187 q^{57} -362.007 q^{58} +288.741 q^{59} +537.331 q^{60} -67.9815 q^{61} +245.075 q^{62} -1154.28 q^{63} +497.567 q^{64} +691.728 q^{65} -894.828 q^{66} +273.745 q^{67} +281.895 q^{68} +809.187 q^{69} -541.592 q^{70} -113.470 q^{71} +1718.99 q^{72} -908.508 q^{73} -119.698 q^{74} +1160.65 q^{75} -33.9916 q^{76} -699.638 q^{77} +930.632 q^{78} -798.537 q^{79} +371.118 q^{80} +2332.69 q^{81} -20.1917 q^{82} -1273.00 q^{83} +565.221 q^{84} +1256.28 q^{85} +1683.69 q^{87} +1041.93 q^{88} +1394.34 q^{89} +2329.12 q^{90} +727.633 q^{91} -286.461 q^{92} -1139.84 q^{93} -20.2773 q^{94} -151.486 q^{95} -1427.58 q^{96} +253.183 q^{97} +158.331 q^{98} +3008.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.12255 −0.750435 −0.375218 0.926937i \(-0.622432\pi\)
−0.375218 + 0.926937i \(0.622432\pi\)
\(3\) 9.87196 1.89986 0.949930 0.312463i \(-0.101154\pi\)
0.949930 + 0.312463i \(0.101154\pi\)
\(4\) −3.49478 −0.436847
\(5\) −15.5747 −1.39304 −0.696520 0.717537i \(-0.745269\pi\)
−0.696520 + 0.717537i \(0.745269\pi\)
\(6\) −20.9537 −1.42572
\(7\) −16.3831 −0.884603 −0.442302 0.896866i \(-0.645838\pi\)
−0.442302 + 0.896866i \(0.645838\pi\)
\(8\) 24.3983 1.07826
\(9\) 70.4556 2.60947
\(10\) 33.0580 1.04539
\(11\) 42.7049 1.17055 0.585274 0.810836i \(-0.300987\pi\)
0.585274 + 0.810836i \(0.300987\pi\)
\(12\) −34.5003 −0.829948
\(13\) −44.4137 −0.947549 −0.473774 0.880646i \(-0.657109\pi\)
−0.473774 + 0.880646i \(0.657109\pi\)
\(14\) 34.7739 0.663837
\(15\) −153.752 −2.64658
\(16\) −23.8283 −0.372318
\(17\) −80.6618 −1.15079 −0.575393 0.817877i \(-0.695151\pi\)
−0.575393 + 0.817877i \(0.695151\pi\)
\(18\) −149.546 −1.95824
\(19\) 9.72641 0.117442 0.0587208 0.998274i \(-0.481298\pi\)
0.0587208 + 0.998274i \(0.481298\pi\)
\(20\) 54.4300 0.608546
\(21\) −161.733 −1.68062
\(22\) −90.6434 −0.878420
\(23\) 81.9683 0.743111 0.371556 0.928411i \(-0.378824\pi\)
0.371556 + 0.928411i \(0.378824\pi\)
\(24\) 240.859 2.04854
\(25\) 117.570 0.940562
\(26\) 94.2703 0.711074
\(27\) 428.992 3.05776
\(28\) 57.2552 0.386436
\(29\) 170.553 1.09210 0.546049 0.837753i \(-0.316131\pi\)
0.546049 + 0.837753i \(0.316131\pi\)
\(30\) 326.348 1.98609
\(31\) −115.463 −0.668958 −0.334479 0.942403i \(-0.608560\pi\)
−0.334479 + 0.942403i \(0.608560\pi\)
\(32\) −144.609 −0.798860
\(33\) 421.581 2.22388
\(34\) 171.209 0.863590
\(35\) 255.161 1.23229
\(36\) −246.227 −1.13994
\(37\) 56.3932 0.250567 0.125284 0.992121i \(-0.460016\pi\)
0.125284 + 0.992121i \(0.460016\pi\)
\(38\) −20.6448 −0.0881323
\(39\) −438.450 −1.80021
\(40\) −379.995 −1.50206
\(41\) 9.51295 0.0362359 0.0181180 0.999836i \(-0.494233\pi\)
0.0181180 + 0.999836i \(0.494233\pi\)
\(42\) 343.287 1.26120
\(43\) 0 0
\(44\) −149.244 −0.511350
\(45\) −1097.32 −3.63509
\(46\) −173.982 −0.557657
\(47\) 9.55328 0.0296487 0.0148243 0.999890i \(-0.495281\pi\)
0.0148243 + 0.999890i \(0.495281\pi\)
\(48\) −235.232 −0.707351
\(49\) −74.5945 −0.217477
\(50\) −249.549 −0.705830
\(51\) −796.290 −2.18633
\(52\) 155.216 0.413934
\(53\) 254.029 0.658370 0.329185 0.944265i \(-0.393226\pi\)
0.329185 + 0.944265i \(0.393226\pi\)
\(54\) −910.558 −2.29465
\(55\) −665.115 −1.63062
\(56\) −399.719 −0.953833
\(57\) 96.0187 0.223123
\(58\) −362.007 −0.819549
\(59\) 288.741 0.637133 0.318566 0.947901i \(-0.396799\pi\)
0.318566 + 0.947901i \(0.396799\pi\)
\(60\) 537.331 1.15615
\(61\) −67.9815 −0.142691 −0.0713454 0.997452i \(-0.522729\pi\)
−0.0713454 + 0.997452i \(0.522729\pi\)
\(62\) 245.075 0.502009
\(63\) −1154.28 −2.30834
\(64\) 497.567 0.971810
\(65\) 691.728 1.31997
\(66\) −894.828 −1.66887
\(67\) 273.745 0.499153 0.249577 0.968355i \(-0.419709\pi\)
0.249577 + 0.968355i \(0.419709\pi\)
\(68\) 281.895 0.502717
\(69\) 809.187 1.41181
\(70\) −541.592 −0.924752
\(71\) −113.470 −0.189668 −0.0948339 0.995493i \(-0.530232\pi\)
−0.0948339 + 0.995493i \(0.530232\pi\)
\(72\) 1718.99 2.81369
\(73\) −908.508 −1.45661 −0.728307 0.685251i \(-0.759693\pi\)
−0.728307 + 0.685251i \(0.759693\pi\)
\(74\) −119.698 −0.188035
\(75\) 1160.65 1.78694
\(76\) −33.9916 −0.0513040
\(77\) −699.638 −1.03547
\(78\) 930.632 1.35094
\(79\) −798.537 −1.13725 −0.568623 0.822598i \(-0.692524\pi\)
−0.568623 + 0.822598i \(0.692524\pi\)
\(80\) 371.118 0.518653
\(81\) 2332.69 3.19985
\(82\) −20.1917 −0.0271927
\(83\) −1273.00 −1.68349 −0.841744 0.539878i \(-0.818471\pi\)
−0.841744 + 0.539878i \(0.818471\pi\)
\(84\) 565.221 0.734175
\(85\) 1256.28 1.60309
\(86\) 0 0
\(87\) 1683.69 2.07483
\(88\) 1041.93 1.26215
\(89\) 1394.34 1.66067 0.830335 0.557265i \(-0.188149\pi\)
0.830335 + 0.557265i \(0.188149\pi\)
\(90\) 2329.12 2.72790
\(91\) 727.633 0.838205
\(92\) −286.461 −0.324626
\(93\) −1139.84 −1.27093
\(94\) −20.2773 −0.0222494
\(95\) −151.486 −0.163601
\(96\) −1427.58 −1.51772
\(97\) 253.183 0.265018 0.132509 0.991182i \(-0.457697\pi\)
0.132509 + 0.991182i \(0.457697\pi\)
\(98\) 158.331 0.163202
\(99\) 3008.80 3.05450
\(100\) −410.882 −0.410882
\(101\) 623.621 0.614383 0.307191 0.951648i \(-0.400611\pi\)
0.307191 + 0.951648i \(0.400611\pi\)
\(102\) 1690.17 1.64070
\(103\) −1534.71 −1.46815 −0.734076 0.679067i \(-0.762385\pi\)
−0.734076 + 0.679067i \(0.762385\pi\)
\(104\) −1083.62 −1.02170
\(105\) 2518.94 2.34117
\(106\) −539.190 −0.494064
\(107\) 978.902 0.884430 0.442215 0.896909i \(-0.354193\pi\)
0.442215 + 0.896909i \(0.354193\pi\)
\(108\) −1499.23 −1.33577
\(109\) −615.263 −0.540656 −0.270328 0.962768i \(-0.587132\pi\)
−0.270328 + 0.962768i \(0.587132\pi\)
\(110\) 1411.74 1.22367
\(111\) 556.712 0.476043
\(112\) 390.381 0.329353
\(113\) −1889.47 −1.57298 −0.786488 0.617605i \(-0.788103\pi\)
−0.786488 + 0.617605i \(0.788103\pi\)
\(114\) −203.805 −0.167439
\(115\) −1276.63 −1.03518
\(116\) −596.044 −0.477080
\(117\) −3129.19 −2.47260
\(118\) −612.867 −0.478127
\(119\) 1321.49 1.01799
\(120\) −3751.29 −2.85370
\(121\) 492.710 0.370181
\(122\) 144.294 0.107080
\(123\) 93.9115 0.0688432
\(124\) 403.516 0.292232
\(125\) 115.717 0.0828002
\(126\) 2450.02 1.73226
\(127\) −333.272 −0.232859 −0.116430 0.993199i \(-0.537145\pi\)
−0.116430 + 0.993199i \(0.537145\pi\)
\(128\) 100.762 0.0695797
\(129\) 0 0
\(130\) −1468.23 −0.990554
\(131\) −1298.39 −0.865959 −0.432979 0.901404i \(-0.642538\pi\)
−0.432979 + 0.901404i \(0.642538\pi\)
\(132\) −1473.33 −0.971493
\(133\) −159.349 −0.103889
\(134\) −581.038 −0.374582
\(135\) −6681.41 −4.25959
\(136\) −1968.01 −1.24085
\(137\) 1989.18 1.24049 0.620245 0.784408i \(-0.287033\pi\)
0.620245 + 0.784408i \(0.287033\pi\)
\(138\) −1717.54 −1.05947
\(139\) −1554.18 −0.948374 −0.474187 0.880424i \(-0.657258\pi\)
−0.474187 + 0.880424i \(0.657258\pi\)
\(140\) −891.731 −0.538322
\(141\) 94.3096 0.0563284
\(142\) 240.846 0.142333
\(143\) −1896.68 −1.10915
\(144\) −1678.84 −0.971550
\(145\) −2656.30 −1.52134
\(146\) 1928.36 1.09309
\(147\) −736.394 −0.413175
\(148\) −197.082 −0.109460
\(149\) −2018.71 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(150\) −2463.54 −1.34098
\(151\) 219.429 0.118258 0.0591289 0.998250i \(-0.481168\pi\)
0.0591289 + 0.998250i \(0.481168\pi\)
\(152\) 237.307 0.126633
\(153\) −5683.08 −3.00294
\(154\) 1485.02 0.777053
\(155\) 1798.29 0.931885
\(156\) 1532.28 0.786416
\(157\) −2441.15 −1.24092 −0.620462 0.784236i \(-0.713055\pi\)
−0.620462 + 0.784236i \(0.713055\pi\)
\(158\) 1694.94 0.853430
\(159\) 2507.77 1.25081
\(160\) 2252.24 1.11284
\(161\) −1342.89 −0.657359
\(162\) −4951.26 −2.40128
\(163\) 2934.10 1.40992 0.704959 0.709248i \(-0.250965\pi\)
0.704959 + 0.709248i \(0.250965\pi\)
\(164\) −33.2456 −0.0158296
\(165\) −6565.99 −3.09795
\(166\) 2702.00 1.26335
\(167\) −4127.64 −1.91261 −0.956306 0.292368i \(-0.905557\pi\)
−0.956306 + 0.292368i \(0.905557\pi\)
\(168\) −3946.01 −1.81215
\(169\) −224.427 −0.102152
\(170\) −2666.52 −1.20302
\(171\) 685.280 0.306460
\(172\) 0 0
\(173\) −803.747 −0.353224 −0.176612 0.984281i \(-0.556514\pi\)
−0.176612 + 0.984281i \(0.556514\pi\)
\(174\) −3573.72 −1.55703
\(175\) −1926.16 −0.832024
\(176\) −1017.59 −0.435815
\(177\) 2850.44 1.21046
\(178\) −2959.55 −1.24622
\(179\) 3352.74 1.39998 0.699988 0.714155i \(-0.253189\pi\)
0.699988 + 0.714155i \(0.253189\pi\)
\(180\) 3834.90 1.58798
\(181\) −4226.02 −1.73546 −0.867728 0.497040i \(-0.834420\pi\)
−0.867728 + 0.497040i \(0.834420\pi\)
\(182\) −1544.44 −0.629018
\(183\) −671.111 −0.271093
\(184\) 1999.88 0.801268
\(185\) −878.306 −0.349050
\(186\) 2419.37 0.953748
\(187\) −3444.66 −1.34705
\(188\) −33.3866 −0.0129519
\(189\) −7028.21 −2.70491
\(190\) 321.536 0.122772
\(191\) −3645.85 −1.38118 −0.690588 0.723249i \(-0.742648\pi\)
−0.690588 + 0.723249i \(0.742648\pi\)
\(192\) 4911.96 1.84630
\(193\) 3201.57 1.19406 0.597032 0.802218i \(-0.296347\pi\)
0.597032 + 0.802218i \(0.296347\pi\)
\(194\) −537.393 −0.198879
\(195\) 6828.71 2.50776
\(196\) 260.691 0.0950041
\(197\) 92.9451 0.0336145 0.0168073 0.999859i \(-0.494650\pi\)
0.0168073 + 0.999859i \(0.494650\pi\)
\(198\) −6386.33 −2.29221
\(199\) −97.0214 −0.0345611 −0.0172806 0.999851i \(-0.505501\pi\)
−0.0172806 + 0.999851i \(0.505501\pi\)
\(200\) 2868.51 1.01417
\(201\) 2702.40 0.948321
\(202\) −1323.67 −0.461054
\(203\) −2794.18 −0.966074
\(204\) 2782.86 0.955093
\(205\) −148.161 −0.0504781
\(206\) 3257.51 1.10175
\(207\) 5775.12 1.93912
\(208\) 1058.30 0.352789
\(209\) 415.365 0.137471
\(210\) −5346.58 −1.75690
\(211\) 106.820 0.0348522 0.0174261 0.999848i \(-0.494453\pi\)
0.0174261 + 0.999848i \(0.494453\pi\)
\(212\) −887.776 −0.287607
\(213\) −1120.17 −0.360342
\(214\) −2077.77 −0.663708
\(215\) 0 0
\(216\) 10466.7 3.29706
\(217\) 1891.63 0.591762
\(218\) 1305.93 0.405727
\(219\) −8968.76 −2.76736
\(220\) 2324.43 0.712331
\(221\) 3582.48 1.09043
\(222\) −1181.65 −0.357239
\(223\) −3792.84 −1.13896 −0.569478 0.822006i \(-0.692855\pi\)
−0.569478 + 0.822006i \(0.692855\pi\)
\(224\) 2369.14 0.706675
\(225\) 8283.48 2.45436
\(226\) 4010.50 1.18042
\(227\) 2142.14 0.626339 0.313170 0.949697i \(-0.398609\pi\)
0.313170 + 0.949697i \(0.398609\pi\)
\(228\) −335.564 −0.0974705
\(229\) −866.711 −0.250104 −0.125052 0.992150i \(-0.539910\pi\)
−0.125052 + 0.992150i \(0.539910\pi\)
\(230\) 2709.71 0.776839
\(231\) −6906.80 −1.96725
\(232\) 4161.19 1.17757
\(233\) 2698.47 0.758724 0.379362 0.925248i \(-0.376144\pi\)
0.379362 + 0.925248i \(0.376144\pi\)
\(234\) 6641.87 1.85552
\(235\) −148.789 −0.0413018
\(236\) −1009.08 −0.278330
\(237\) −7883.13 −2.16061
\(238\) −2804.93 −0.763935
\(239\) −1856.90 −0.502564 −0.251282 0.967914i \(-0.580852\pi\)
−0.251282 + 0.967914i \(0.580852\pi\)
\(240\) 3663.66 0.985369
\(241\) −4891.12 −1.30732 −0.653661 0.756788i \(-0.726768\pi\)
−0.653661 + 0.756788i \(0.726768\pi\)
\(242\) −1045.80 −0.277796
\(243\) 11445.5 3.02151
\(244\) 237.580 0.0623341
\(245\) 1161.78 0.302954
\(246\) −199.332 −0.0516623
\(247\) −431.985 −0.111282
\(248\) −2817.09 −0.721311
\(249\) −12567.0 −3.19839
\(250\) −245.615 −0.0621362
\(251\) −5679.81 −1.42831 −0.714156 0.699987i \(-0.753189\pi\)
−0.714156 + 0.699987i \(0.753189\pi\)
\(252\) 4033.95 1.00839
\(253\) 3500.45 0.869847
\(254\) 707.387 0.174746
\(255\) 12402.0 3.04565
\(256\) −4194.41 −1.02403
\(257\) −3095.65 −0.751366 −0.375683 0.926748i \(-0.622592\pi\)
−0.375683 + 0.926748i \(0.622592\pi\)
\(258\) 0 0
\(259\) −923.895 −0.221653
\(260\) −2417.43 −0.576627
\(261\) 12016.4 2.84980
\(262\) 2755.89 0.649846
\(263\) 2790.13 0.654170 0.327085 0.944995i \(-0.393934\pi\)
0.327085 + 0.944995i \(0.393934\pi\)
\(264\) 10285.8 2.39792
\(265\) −3956.42 −0.917136
\(266\) 338.225 0.0779622
\(267\) 13764.9 3.15504
\(268\) −956.678 −0.218054
\(269\) −8193.14 −1.85704 −0.928521 0.371279i \(-0.878919\pi\)
−0.928521 + 0.371279i \(0.878919\pi\)
\(270\) 14181.6 3.19654
\(271\) −4239.14 −0.950218 −0.475109 0.879927i \(-0.657591\pi\)
−0.475109 + 0.879927i \(0.657591\pi\)
\(272\) 1922.04 0.428458
\(273\) 7183.16 1.59247
\(274\) −4222.14 −0.930907
\(275\) 5020.83 1.10097
\(276\) −2827.93 −0.616744
\(277\) −8528.69 −1.84996 −0.924981 0.380014i \(-0.875919\pi\)
−0.924981 + 0.380014i \(0.875919\pi\)
\(278\) 3298.83 0.711693
\(279\) −8134.99 −1.74562
\(280\) 6225.48 1.32873
\(281\) −1287.15 −0.273257 −0.136628 0.990622i \(-0.543627\pi\)
−0.136628 + 0.990622i \(0.543627\pi\)
\(282\) −200.177 −0.0422708
\(283\) −8974.14 −1.88501 −0.942504 0.334194i \(-0.891536\pi\)
−0.942504 + 0.334194i \(0.891536\pi\)
\(284\) 396.552 0.0828558
\(285\) −1495.46 −0.310819
\(286\) 4025.80 0.832345
\(287\) −155.851 −0.0320544
\(288\) −10188.5 −2.08460
\(289\) 1593.32 0.324308
\(290\) 5638.14 1.14167
\(291\) 2499.41 0.503498
\(292\) 3175.03 0.636318
\(293\) 1173.60 0.234002 0.117001 0.993132i \(-0.462672\pi\)
0.117001 + 0.993132i \(0.462672\pi\)
\(294\) 1563.03 0.310061
\(295\) −4497.04 −0.887552
\(296\) 1375.90 0.270177
\(297\) 18320.1 3.57925
\(298\) 4284.81 0.832928
\(299\) −3640.51 −0.704134
\(300\) −4056.21 −0.780617
\(301\) 0 0
\(302\) −465.750 −0.0887447
\(303\) 6156.37 1.16724
\(304\) −231.764 −0.0437256
\(305\) 1058.79 0.198774
\(306\) 12062.6 2.25351
\(307\) −8027.77 −1.49241 −0.746204 0.665718i \(-0.768126\pi\)
−0.746204 + 0.665718i \(0.768126\pi\)
\(308\) 2445.08 0.452342
\(309\) −15150.6 −2.78928
\(310\) −3816.97 −0.699320
\(311\) 5900.91 1.07592 0.537958 0.842972i \(-0.319196\pi\)
0.537958 + 0.842972i \(0.319196\pi\)
\(312\) −10697.4 −1.94109
\(313\) 7645.72 1.38071 0.690354 0.723471i \(-0.257455\pi\)
0.690354 + 0.723471i \(0.257455\pi\)
\(314\) 5181.47 0.931234
\(315\) 17977.5 3.21562
\(316\) 2790.71 0.496803
\(317\) 8873.30 1.57216 0.786079 0.618126i \(-0.212108\pi\)
0.786079 + 0.618126i \(0.212108\pi\)
\(318\) −5322.87 −0.938653
\(319\) 7283.45 1.27835
\(320\) −7749.44 −1.35377
\(321\) 9663.68 1.68029
\(322\) 2850.36 0.493305
\(323\) −784.549 −0.135150
\(324\) −8152.24 −1.39785
\(325\) −5221.72 −0.891228
\(326\) −6227.78 −1.05805
\(327\) −6073.85 −1.02717
\(328\) 232.099 0.0390718
\(329\) −156.512 −0.0262273
\(330\) 13936.6 2.32481
\(331\) 3998.27 0.663943 0.331971 0.943289i \(-0.392286\pi\)
0.331971 + 0.943289i \(0.392286\pi\)
\(332\) 4448.84 0.735426
\(333\) 3973.22 0.653847
\(334\) 8761.12 1.43529
\(335\) −4263.49 −0.695341
\(336\) 3853.83 0.625725
\(337\) 3151.54 0.509423 0.254711 0.967017i \(-0.418020\pi\)
0.254711 + 0.967017i \(0.418020\pi\)
\(338\) 476.359 0.0766583
\(339\) −18652.8 −2.98844
\(340\) −4390.42 −0.700306
\(341\) −4930.82 −0.783047
\(342\) −1454.54 −0.229978
\(343\) 6841.49 1.07698
\(344\) 0 0
\(345\) −12602.8 −1.96670
\(346\) 1705.99 0.265072
\(347\) −2639.21 −0.408300 −0.204150 0.978940i \(-0.565443\pi\)
−0.204150 + 0.978940i \(0.565443\pi\)
\(348\) −5884.12 −0.906386
\(349\) 8673.47 1.33032 0.665158 0.746702i \(-0.268364\pi\)
0.665158 + 0.746702i \(0.268364\pi\)
\(350\) 4088.38 0.624380
\(351\) −19053.1 −2.89738
\(352\) −6175.52 −0.935104
\(353\) −799.760 −0.120586 −0.0602931 0.998181i \(-0.519204\pi\)
−0.0602931 + 0.998181i \(0.519204\pi\)
\(354\) −6050.20 −0.908374
\(355\) 1767.26 0.264215
\(356\) −4872.90 −0.725459
\(357\) 13045.7 1.93404
\(358\) −7116.36 −1.05059
\(359\) −6892.85 −1.01334 −0.506672 0.862139i \(-0.669125\pi\)
−0.506672 + 0.862139i \(0.669125\pi\)
\(360\) −26772.8 −3.91958
\(361\) −6764.40 −0.986207
\(362\) 8969.94 1.30235
\(363\) 4864.02 0.703291
\(364\) −2542.91 −0.366167
\(365\) 14149.7 2.02912
\(366\) 1424.47 0.203437
\(367\) 10785.9 1.53412 0.767058 0.641578i \(-0.221720\pi\)
0.767058 + 0.641578i \(0.221720\pi\)
\(368\) −1953.17 −0.276673
\(369\) 670.241 0.0945565
\(370\) 1864.25 0.261940
\(371\) −4161.79 −0.582397
\(372\) 3983.49 0.555200
\(373\) −2319.02 −0.321916 −0.160958 0.986961i \(-0.551458\pi\)
−0.160958 + 0.986961i \(0.551458\pi\)
\(374\) 7311.46 1.01087
\(375\) 1142.35 0.157309
\(376\) 233.083 0.0319690
\(377\) −7574.88 −1.03482
\(378\) 14917.7 2.02986
\(379\) −9170.77 −1.24293 −0.621466 0.783441i \(-0.713462\pi\)
−0.621466 + 0.783441i \(0.713462\pi\)
\(380\) 529.408 0.0714686
\(381\) −3290.05 −0.442400
\(382\) 7738.50 1.03648
\(383\) −6403.02 −0.854253 −0.427127 0.904192i \(-0.640474\pi\)
−0.427127 + 0.904192i \(0.640474\pi\)
\(384\) 994.720 0.132192
\(385\) 10896.6 1.44245
\(386\) −6795.50 −0.896067
\(387\) 0 0
\(388\) −884.817 −0.115773
\(389\) −7196.53 −0.937991 −0.468995 0.883201i \(-0.655384\pi\)
−0.468995 + 0.883201i \(0.655384\pi\)
\(390\) −14494.3 −1.88191
\(391\) −6611.71 −0.855162
\(392\) −1819.98 −0.234497
\(393\) −12817.6 −1.64520
\(394\) −197.281 −0.0252255
\(395\) 12436.9 1.58423
\(396\) −10515.1 −1.33435
\(397\) 10186.1 1.28773 0.643863 0.765141i \(-0.277331\pi\)
0.643863 + 0.765141i \(0.277331\pi\)
\(398\) 205.933 0.0259359
\(399\) −1573.08 −0.197375
\(400\) −2801.50 −0.350188
\(401\) 4639.83 0.577810 0.288905 0.957358i \(-0.406709\pi\)
0.288905 + 0.957358i \(0.406709\pi\)
\(402\) −5735.98 −0.711654
\(403\) 5128.12 0.633870
\(404\) −2179.42 −0.268391
\(405\) −36330.9 −4.45752
\(406\) 5930.79 0.724976
\(407\) 2408.27 0.293301
\(408\) −19428.1 −2.35743
\(409\) 9793.28 1.18398 0.591988 0.805947i \(-0.298343\pi\)
0.591988 + 0.805947i \(0.298343\pi\)
\(410\) 314.479 0.0378806
\(411\) 19637.1 2.35676
\(412\) 5363.48 0.641358
\(413\) −4730.46 −0.563610
\(414\) −12258.0 −1.45519
\(415\) 19826.5 2.34517
\(416\) 6422.62 0.756959
\(417\) −15342.8 −1.80178
\(418\) −881.635 −0.103163
\(419\) −10390.7 −1.21150 −0.605748 0.795657i \(-0.707126\pi\)
−0.605748 + 0.795657i \(0.707126\pi\)
\(420\) −8803.13 −1.02274
\(421\) −1278.81 −0.148041 −0.0740206 0.997257i \(-0.523583\pi\)
−0.0740206 + 0.997257i \(0.523583\pi\)
\(422\) −226.732 −0.0261543
\(423\) 673.082 0.0773673
\(424\) 6197.87 0.709895
\(425\) −9483.42 −1.08238
\(426\) 2377.62 0.270414
\(427\) 1113.75 0.126225
\(428\) −3421.04 −0.386361
\(429\) −18724.0 −2.10723
\(430\) 0 0
\(431\) −8345.12 −0.932646 −0.466323 0.884615i \(-0.654422\pi\)
−0.466323 + 0.884615i \(0.654422\pi\)
\(432\) −10222.2 −1.13846
\(433\) −8533.68 −0.947119 −0.473560 0.880762i \(-0.657031\pi\)
−0.473560 + 0.880762i \(0.657031\pi\)
\(434\) −4015.09 −0.444079
\(435\) −26222.9 −2.89033
\(436\) 2150.21 0.236184
\(437\) 797.257 0.0872722
\(438\) 19036.6 2.07673
\(439\) −14123.5 −1.53549 −0.767744 0.640756i \(-0.778621\pi\)
−0.767744 + 0.640756i \(0.778621\pi\)
\(440\) −16227.6 −1.75823
\(441\) −5255.60 −0.567499
\(442\) −7604.01 −0.818293
\(443\) 6033.41 0.647079 0.323539 0.946215i \(-0.395127\pi\)
0.323539 + 0.946215i \(0.395127\pi\)
\(444\) −1945.58 −0.207958
\(445\) −21716.4 −2.31338
\(446\) 8050.49 0.854713
\(447\) −19928.6 −2.10871
\(448\) −8151.68 −0.859667
\(449\) 15345.6 1.61293 0.806464 0.591284i \(-0.201379\pi\)
0.806464 + 0.591284i \(0.201379\pi\)
\(450\) −17582.1 −1.84184
\(451\) 406.250 0.0424159
\(452\) 6603.28 0.687150
\(453\) 2166.20 0.224673
\(454\) −4546.81 −0.470027
\(455\) −11332.6 −1.16765
\(456\) 2342.69 0.240584
\(457\) 16460.5 1.68488 0.842439 0.538792i \(-0.181119\pi\)
0.842439 + 0.538792i \(0.181119\pi\)
\(458\) 1839.64 0.187687
\(459\) −34603.3 −3.51883
\(460\) 4461.53 0.452217
\(461\) −1574.00 −0.159021 −0.0795103 0.996834i \(-0.525336\pi\)
−0.0795103 + 0.996834i \(0.525336\pi\)
\(462\) 14660.0 1.47629
\(463\) 10676.6 1.07167 0.535834 0.844323i \(-0.319997\pi\)
0.535834 + 0.844323i \(0.319997\pi\)
\(464\) −4063.99 −0.406608
\(465\) 17752.7 1.77045
\(466\) −5727.64 −0.569373
\(467\) 6834.42 0.677215 0.338607 0.940928i \(-0.390044\pi\)
0.338607 + 0.940928i \(0.390044\pi\)
\(468\) 10935.8 1.08015
\(469\) −4484.79 −0.441553
\(470\) 315.812 0.0309943
\(471\) −24099.0 −2.35758
\(472\) 7044.77 0.686995
\(473\) 0 0
\(474\) 16732.3 1.62140
\(475\) 1143.54 0.110461
\(476\) −4618.31 −0.444706
\(477\) 17897.8 1.71800
\(478\) 3941.36 0.377142
\(479\) −3790.83 −0.361602 −0.180801 0.983520i \(-0.557869\pi\)
−0.180801 + 0.983520i \(0.557869\pi\)
\(480\) 22234.0 2.11425
\(481\) −2504.63 −0.237425
\(482\) 10381.6 0.981060
\(483\) −13257.0 −1.24889
\(484\) −1721.91 −0.161712
\(485\) −3943.23 −0.369181
\(486\) −24293.6 −2.26745
\(487\) 6937.25 0.645497 0.322748 0.946485i \(-0.395393\pi\)
0.322748 + 0.946485i \(0.395393\pi\)
\(488\) −1658.63 −0.153858
\(489\) 28965.4 2.67865
\(490\) −2465.95 −0.227347
\(491\) 3418.47 0.314203 0.157101 0.987582i \(-0.449785\pi\)
0.157101 + 0.987582i \(0.449785\pi\)
\(492\) −328.200 −0.0300739
\(493\) −13757.1 −1.25677
\(494\) 916.911 0.0835097
\(495\) −46861.1 −4.25505
\(496\) 2751.28 0.249065
\(497\) 1858.99 0.167781
\(498\) 26674.0 2.40018
\(499\) −8330.03 −0.747301 −0.373651 0.927570i \(-0.621894\pi\)
−0.373651 + 0.927570i \(0.621894\pi\)
\(500\) −404.404 −0.0361710
\(501\) −40747.9 −3.63369
\(502\) 12055.7 1.07186
\(503\) 1435.22 0.127223 0.0636116 0.997975i \(-0.479738\pi\)
0.0636116 + 0.997975i \(0.479738\pi\)
\(504\) −28162.4 −2.48900
\(505\) −9712.69 −0.855860
\(506\) −7429.88 −0.652764
\(507\) −2215.54 −0.194074
\(508\) 1164.71 0.101724
\(509\) −1358.45 −0.118295 −0.0591477 0.998249i \(-0.518838\pi\)
−0.0591477 + 0.998249i \(0.518838\pi\)
\(510\) −26323.8 −2.28556
\(511\) 14884.2 1.28853
\(512\) 8096.75 0.698885
\(513\) 4172.55 0.359109
\(514\) 6570.67 0.563851
\(515\) 23902.6 2.04520
\(516\) 0 0
\(517\) 407.972 0.0347052
\(518\) 1961.02 0.166336
\(519\) −7934.56 −0.671076
\(520\) 16876.9 1.42328
\(521\) −852.637 −0.0716981 −0.0358490 0.999357i \(-0.511414\pi\)
−0.0358490 + 0.999357i \(0.511414\pi\)
\(522\) −25505.4 −2.13859
\(523\) 9749.01 0.815094 0.407547 0.913184i \(-0.366384\pi\)
0.407547 + 0.913184i \(0.366384\pi\)
\(524\) 4537.57 0.378292
\(525\) −19015.0 −1.58073
\(526\) −5922.19 −0.490912
\(527\) 9313.42 0.769827
\(528\) −10045.6 −0.827988
\(529\) −5448.21 −0.447785
\(530\) 8397.71 0.688251
\(531\) 20343.4 1.66258
\(532\) 556.888 0.0453837
\(533\) −422.505 −0.0343353
\(534\) −29216.6 −2.36765
\(535\) −15246.1 −1.23205
\(536\) 6678.90 0.538217
\(537\) 33098.1 2.65976
\(538\) 17390.4 1.39359
\(539\) −3185.55 −0.254567
\(540\) 23350.0 1.86079
\(541\) 4019.47 0.319428 0.159714 0.987163i \(-0.448943\pi\)
0.159714 + 0.987163i \(0.448943\pi\)
\(542\) 8997.78 0.713077
\(543\) −41719.1 −3.29712
\(544\) 11664.4 0.919317
\(545\) 9582.52 0.753156
\(546\) −15246.6 −1.19505
\(547\) 4687.88 0.366434 0.183217 0.983073i \(-0.441349\pi\)
0.183217 + 0.983073i \(0.441349\pi\)
\(548\) −6951.74 −0.541904
\(549\) −4789.68 −0.372347
\(550\) −10657.0 −0.826208
\(551\) 1658.87 0.128258
\(552\) 19742.8 1.52230
\(553\) 13082.5 1.00601
\(554\) 18102.6 1.38828
\(555\) −8670.60 −0.663147
\(556\) 5431.52 0.414294
\(557\) 8917.98 0.678397 0.339198 0.940715i \(-0.389844\pi\)
0.339198 + 0.940715i \(0.389844\pi\)
\(558\) 17266.9 1.30998
\(559\) 0 0
\(560\) −6080.06 −0.458803
\(561\) −34005.5 −2.55920
\(562\) 2732.05 0.205061
\(563\) 4096.21 0.306634 0.153317 0.988177i \(-0.451004\pi\)
0.153317 + 0.988177i \(0.451004\pi\)
\(564\) −329.591 −0.0246069
\(565\) 29427.9 2.19122
\(566\) 19048.1 1.41458
\(567\) −38216.7 −2.83060
\(568\) −2768.47 −0.204511
\(569\) 2683.32 0.197699 0.0988495 0.995102i \(-0.468484\pi\)
0.0988495 + 0.995102i \(0.468484\pi\)
\(570\) 3174.19 0.233249
\(571\) −10667.4 −0.781815 −0.390908 0.920430i \(-0.627839\pi\)
−0.390908 + 0.920430i \(0.627839\pi\)
\(572\) 6628.48 0.484529
\(573\) −35991.7 −2.62404
\(574\) 330.803 0.0240548
\(575\) 9637.02 0.698942
\(576\) 35056.4 2.53591
\(577\) 21766.6 1.57046 0.785230 0.619205i \(-0.212545\pi\)
0.785230 + 0.619205i \(0.212545\pi\)
\(578\) −3381.91 −0.243372
\(579\) 31605.8 2.26855
\(580\) 9283.19 0.664592
\(581\) 20855.6 1.48922
\(582\) −5305.12 −0.377843
\(583\) 10848.3 0.770653
\(584\) −22166.0 −1.57061
\(585\) 48736.1 3.44443
\(586\) −2491.04 −0.175604
\(587\) −24126.0 −1.69640 −0.848201 0.529675i \(-0.822314\pi\)
−0.848201 + 0.529675i \(0.822314\pi\)
\(588\) 2573.53 0.180494
\(589\) −1123.04 −0.0785635
\(590\) 9545.20 0.666050
\(591\) 917.551 0.0638629
\(592\) −1343.76 −0.0932906
\(593\) −23547.7 −1.63067 −0.815337 0.578987i \(-0.803448\pi\)
−0.815337 + 0.578987i \(0.803448\pi\)
\(594\) −38885.3 −2.68600
\(595\) −20581.7 −1.41810
\(596\) 7054.94 0.484868
\(597\) −957.791 −0.0656613
\(598\) 7727.17 0.528407
\(599\) −10000.3 −0.682139 −0.341070 0.940038i \(-0.610789\pi\)
−0.341070 + 0.940038i \(0.610789\pi\)
\(600\) 28317.8 1.92678
\(601\) 4062.49 0.275728 0.137864 0.990451i \(-0.455976\pi\)
0.137864 + 0.990451i \(0.455976\pi\)
\(602\) 0 0
\(603\) 19286.9 1.30252
\(604\) −766.857 −0.0516605
\(605\) −7673.80 −0.515676
\(606\) −13067.2 −0.875939
\(607\) 12695.2 0.848902 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(608\) −1406.53 −0.0938195
\(609\) −27584.1 −1.83541
\(610\) −2247.34 −0.149167
\(611\) −424.296 −0.0280936
\(612\) 19861.1 1.31182
\(613\) −4025.43 −0.265230 −0.132615 0.991168i \(-0.542337\pi\)
−0.132615 + 0.991168i \(0.542337\pi\)
\(614\) 17039.4 1.11996
\(615\) −1462.64 −0.0959013
\(616\) −17070.0 −1.11651
\(617\) −21979.7 −1.43415 −0.717074 0.696997i \(-0.754519\pi\)
−0.717074 + 0.696997i \(0.754519\pi\)
\(618\) 32158.0 2.09318
\(619\) 3508.41 0.227811 0.113906 0.993492i \(-0.463664\pi\)
0.113906 + 0.993492i \(0.463664\pi\)
\(620\) −6284.63 −0.407091
\(621\) 35163.7 2.27226
\(622\) −12525.0 −0.807405
\(623\) −22843.6 −1.46903
\(624\) 10447.5 0.670250
\(625\) −16498.5 −1.05591
\(626\) −16228.4 −1.03613
\(627\) 4100.47 0.261176
\(628\) 8531.28 0.542094
\(629\) −4548.78 −0.288349
\(630\) −38158.2 −2.41311
\(631\) 12163.9 0.767412 0.383706 0.923455i \(-0.374648\pi\)
0.383706 + 0.923455i \(0.374648\pi\)
\(632\) −19482.9 −1.22625
\(633\) 1054.53 0.0662143
\(634\) −18834.0 −1.17980
\(635\) 5190.60 0.324382
\(636\) −8764.09 −0.546413
\(637\) 3313.02 0.206070
\(638\) −15459.5 −0.959321
\(639\) −7994.60 −0.494932
\(640\) −1569.34 −0.0969273
\(641\) −19508.5 −1.20209 −0.601045 0.799216i \(-0.705249\pi\)
−0.601045 + 0.799216i \(0.705249\pi\)
\(642\) −20511.7 −1.26095
\(643\) 8349.25 0.512072 0.256036 0.966667i \(-0.417583\pi\)
0.256036 + 0.966667i \(0.417583\pi\)
\(644\) 4693.11 0.287165
\(645\) 0 0
\(646\) 1665.25 0.101421
\(647\) 24420.9 1.48390 0.741952 0.670453i \(-0.233900\pi\)
0.741952 + 0.670453i \(0.233900\pi\)
\(648\) 56913.6 3.45027
\(649\) 12330.6 0.745794
\(650\) 11083.4 0.668809
\(651\) 18674.1 1.12427
\(652\) −10254.0 −0.615919
\(653\) −17813.5 −1.06753 −0.533763 0.845634i \(-0.679223\pi\)
−0.533763 + 0.845634i \(0.679223\pi\)
\(654\) 12892.1 0.770825
\(655\) 20221.9 1.20632
\(656\) −226.678 −0.0134913
\(657\) −64009.5 −3.80099
\(658\) 332.205 0.0196819
\(659\) −10172.2 −0.601294 −0.300647 0.953735i \(-0.597203\pi\)
−0.300647 + 0.953735i \(0.597203\pi\)
\(660\) 22946.7 1.35333
\(661\) −10378.7 −0.610716 −0.305358 0.952238i \(-0.598776\pi\)
−0.305358 + 0.952238i \(0.598776\pi\)
\(662\) −8486.54 −0.498246
\(663\) 35366.1 2.07166
\(664\) −31058.9 −1.81524
\(665\) 2481.80 0.144722
\(666\) −8433.36 −0.490670
\(667\) 13979.9 0.811551
\(668\) 14425.2 0.835519
\(669\) −37442.8 −2.16386
\(670\) 9049.47 0.521808
\(671\) −2903.15 −0.167026
\(672\) 23388.1 1.34258
\(673\) 20924.5 1.19849 0.599243 0.800567i \(-0.295468\pi\)
0.599243 + 0.800567i \(0.295468\pi\)
\(674\) −6689.31 −0.382289
\(675\) 50436.7 2.87601
\(676\) 784.324 0.0446247
\(677\) 9393.92 0.533290 0.266645 0.963795i \(-0.414085\pi\)
0.266645 + 0.963795i \(0.414085\pi\)
\(678\) 39591.5 2.24263
\(679\) −4147.91 −0.234436
\(680\) 30651.0 1.72855
\(681\) 21147.1 1.18996
\(682\) 10465.9 0.587626
\(683\) −4288.63 −0.240263 −0.120132 0.992758i \(-0.538332\pi\)
−0.120132 + 0.992758i \(0.538332\pi\)
\(684\) −2394.90 −0.133876
\(685\) −30980.8 −1.72805
\(686\) −14521.4 −0.808207
\(687\) −8556.14 −0.475163
\(688\) 0 0
\(689\) −11282.4 −0.623838
\(690\) 26750.1 1.47588
\(691\) 6084.89 0.334993 0.167496 0.985873i \(-0.446432\pi\)
0.167496 + 0.985873i \(0.446432\pi\)
\(692\) 2808.92 0.154305
\(693\) −49293.4 −2.70202
\(694\) 5601.86 0.306403
\(695\) 24205.9 1.32112
\(696\) 41079.1 2.23721
\(697\) −767.332 −0.0416998
\(698\) −18409.9 −0.998316
\(699\) 26639.2 1.44147
\(700\) 6731.51 0.363467
\(701\) 19240.3 1.03666 0.518329 0.855182i \(-0.326554\pi\)
0.518329 + 0.855182i \(0.326554\pi\)
\(702\) 40441.2 2.17429
\(703\) 548.504 0.0294270
\(704\) 21248.6 1.13755
\(705\) −1468.84 −0.0784677
\(706\) 1697.53 0.0904922
\(707\) −10216.8 −0.543485
\(708\) −9961.64 −0.528787
\(709\) −22023.7 −1.16660 −0.583299 0.812258i \(-0.698238\pi\)
−0.583299 + 0.812258i \(0.698238\pi\)
\(710\) −3751.10 −0.198276
\(711\) −56261.4 −2.96761
\(712\) 34019.4 1.79063
\(713\) −9464.27 −0.497110
\(714\) −27690.1 −1.45137
\(715\) 29540.2 1.54509
\(716\) −11717.1 −0.611575
\(717\) −18331.2 −0.954801
\(718\) 14630.4 0.760449
\(719\) −9873.52 −0.512128 −0.256064 0.966660i \(-0.582426\pi\)
−0.256064 + 0.966660i \(0.582426\pi\)
\(720\) 26147.4 1.35341
\(721\) 25143.3 1.29873
\(722\) 14357.8 0.740085
\(723\) −48284.9 −2.48373
\(724\) 14769.0 0.758129
\(725\) 20051.9 1.02719
\(726\) −10324.1 −0.527774
\(727\) 22548.2 1.15030 0.575150 0.818048i \(-0.304944\pi\)
0.575150 + 0.818048i \(0.304944\pi\)
\(728\) 17753.0 0.903803
\(729\) 50006.4 2.54059
\(730\) −30033.5 −1.52272
\(731\) 0 0
\(732\) 2345.38 0.118426
\(733\) −23342.2 −1.17621 −0.588106 0.808784i \(-0.700126\pi\)
−0.588106 + 0.808784i \(0.700126\pi\)
\(734\) −22893.7 −1.15125
\(735\) 11469.1 0.575570
\(736\) −11853.4 −0.593642
\(737\) 11690.3 0.584282
\(738\) −1422.62 −0.0709585
\(739\) −2588.54 −0.128851 −0.0644255 0.997923i \(-0.520521\pi\)
−0.0644255 + 0.997923i \(0.520521\pi\)
\(740\) 3069.48 0.152482
\(741\) −4264.54 −0.211420
\(742\) 8833.60 0.437051
\(743\) 8452.43 0.417348 0.208674 0.977985i \(-0.433085\pi\)
0.208674 + 0.977985i \(0.433085\pi\)
\(744\) −27810.2 −1.37039
\(745\) 31440.7 1.54617
\(746\) 4922.25 0.241577
\(747\) −89689.7 −4.39300
\(748\) 12038.3 0.588454
\(749\) −16037.4 −0.782370
\(750\) −2424.70 −0.118050
\(751\) 3358.11 0.163168 0.0815841 0.996666i \(-0.474002\pi\)
0.0815841 + 0.996666i \(0.474002\pi\)
\(752\) −227.639 −0.0110387
\(753\) −56070.8 −2.71359
\(754\) 16078.1 0.776563
\(755\) −3417.54 −0.164738
\(756\) 24562.0 1.18163
\(757\) 14867.1 0.713810 0.356905 0.934141i \(-0.383832\pi\)
0.356905 + 0.934141i \(0.383832\pi\)
\(758\) 19465.4 0.932739
\(759\) 34556.3 1.65259
\(760\) −3695.98 −0.176404
\(761\) −1721.42 −0.0819991 −0.0409995 0.999159i \(-0.513054\pi\)
−0.0409995 + 0.999159i \(0.513054\pi\)
\(762\) 6983.30 0.331992
\(763\) 10079.9 0.478266
\(764\) 12741.4 0.603362
\(765\) 88512.0 4.18321
\(766\) 13590.7 0.641062
\(767\) −12824.0 −0.603714
\(768\) −41407.0 −1.94550
\(769\) 13213.3 0.619616 0.309808 0.950799i \(-0.399735\pi\)
0.309808 + 0.950799i \(0.399735\pi\)
\(770\) −23128.7 −1.08247
\(771\) −30560.1 −1.42749
\(772\) −11188.8 −0.521623
\(773\) −26452.9 −1.23085 −0.615424 0.788196i \(-0.711015\pi\)
−0.615424 + 0.788196i \(0.711015\pi\)
\(774\) 0 0
\(775\) −13575.0 −0.629196
\(776\) 6177.21 0.285759
\(777\) −9120.66 −0.421109
\(778\) 15275.0 0.703901
\(779\) 92.5268 0.00425561
\(780\) −23864.8 −1.09551
\(781\) −4845.73 −0.222015
\(782\) 14033.7 0.641744
\(783\) 73165.8 3.33938
\(784\) 1777.46 0.0809704
\(785\) 38020.1 1.72866
\(786\) 27206.1 1.23462
\(787\) 22609.2 1.02405 0.512027 0.858970i \(-0.328895\pi\)
0.512027 + 0.858970i \(0.328895\pi\)
\(788\) −324.822 −0.0146844
\(789\) 27544.0 1.24283
\(790\) −26398.1 −1.18886
\(791\) 30955.3 1.39146
\(792\) 73409.5 3.29355
\(793\) 3019.31 0.135207
\(794\) −21620.6 −0.966355
\(795\) −39057.7 −1.74243
\(796\) 339.068 0.0150979
\(797\) 16151.3 0.717829 0.358915 0.933370i \(-0.383147\pi\)
0.358915 + 0.933370i \(0.383147\pi\)
\(798\) 3338.95 0.148117
\(799\) −770.584 −0.0341193
\(800\) −17001.7 −0.751377
\(801\) 98239.0 4.33346
\(802\) −9848.27 −0.433609
\(803\) −38797.8 −1.70504
\(804\) −9444.28 −0.414271
\(805\) 20915.1 0.915727
\(806\) −10884.7 −0.475678
\(807\) −80882.4 −3.52812
\(808\) 15215.3 0.662465
\(809\) 8389.19 0.364584 0.182292 0.983244i \(-0.441648\pi\)
0.182292 + 0.983244i \(0.441648\pi\)
\(810\) 77114.2 3.34508
\(811\) 687.716 0.0297768 0.0148884 0.999889i \(-0.495261\pi\)
0.0148884 + 0.999889i \(0.495261\pi\)
\(812\) 9765.04 0.422027
\(813\) −41848.6 −1.80528
\(814\) −5111.67 −0.220103
\(815\) −45697.7 −1.96407
\(816\) 18974.3 0.814010
\(817\) 0 0
\(818\) −20786.7 −0.888498
\(819\) 51265.8 2.18727
\(820\) 517.790 0.0220512
\(821\) −36524.3 −1.55263 −0.776313 0.630348i \(-0.782912\pi\)
−0.776313 + 0.630348i \(0.782912\pi\)
\(822\) −41680.8 −1.76859
\(823\) −2950.87 −0.124983 −0.0624915 0.998045i \(-0.519905\pi\)
−0.0624915 + 0.998045i \(0.519905\pi\)
\(824\) −37444.3 −1.58305
\(825\) 49565.4 2.09169
\(826\) 10040.6 0.422953
\(827\) −7907.73 −0.332501 −0.166251 0.986084i \(-0.553166\pi\)
−0.166251 + 0.986084i \(0.553166\pi\)
\(828\) −20182.8 −0.847101
\(829\) 36919.5 1.54677 0.773383 0.633939i \(-0.218563\pi\)
0.773383 + 0.633939i \(0.218563\pi\)
\(830\) −42082.7 −1.75989
\(831\) −84194.9 −3.51467
\(832\) −22098.8 −0.920838
\(833\) 6016.93 0.250269
\(834\) 32565.9 1.35212
\(835\) 64286.6 2.66435
\(836\) −1451.61 −0.0600538
\(837\) −49532.5 −2.04551
\(838\) 22054.7 0.909149
\(839\) 21841.4 0.898747 0.449373 0.893344i \(-0.351647\pi\)
0.449373 + 0.893344i \(0.351647\pi\)
\(840\) 61457.7 2.52440
\(841\) 4699.28 0.192680
\(842\) 2714.34 0.111095
\(843\) −12706.7 −0.519149
\(844\) −373.313 −0.0152251
\(845\) 3495.38 0.142302
\(846\) −1428.65 −0.0580591
\(847\) −8072.11 −0.327463
\(848\) −6053.10 −0.245123
\(849\) −88592.4 −3.58125
\(850\) 20129.0 0.812260
\(851\) 4622.46 0.186199
\(852\) 3914.75 0.157414
\(853\) −43215.2 −1.73466 −0.867328 0.497737i \(-0.834165\pi\)
−0.867328 + 0.497737i \(0.834165\pi\)
\(854\) −2363.99 −0.0947236
\(855\) −10673.0 −0.426911
\(856\) 23883.5 0.953646
\(857\) 14793.5 0.589657 0.294828 0.955550i \(-0.404738\pi\)
0.294828 + 0.955550i \(0.404738\pi\)
\(858\) 39742.6 1.58134
\(859\) −16459.1 −0.653758 −0.326879 0.945066i \(-0.605997\pi\)
−0.326879 + 0.945066i \(0.605997\pi\)
\(860\) 0 0
\(861\) −1538.56 −0.0608989
\(862\) 17712.9 0.699890
\(863\) 32877.1 1.29681 0.648406 0.761295i \(-0.275436\pi\)
0.648406 + 0.761295i \(0.275436\pi\)
\(864\) −62036.2 −2.44272
\(865\) 12518.1 0.492055
\(866\) 18113.2 0.710752
\(867\) 15729.2 0.616139
\(868\) −6610.84 −0.258510
\(869\) −34101.5 −1.33120
\(870\) 55659.5 2.16900
\(871\) −12158.0 −0.472972
\(872\) −15011.3 −0.582968
\(873\) 17838.1 0.691557
\(874\) −1692.22 −0.0654921
\(875\) −1895.80 −0.0732453
\(876\) 31343.8 1.20891
\(877\) −13873.2 −0.534168 −0.267084 0.963673i \(-0.586060\pi\)
−0.267084 + 0.963673i \(0.586060\pi\)
\(878\) 29977.9 1.15228
\(879\) 11585.8 0.444572
\(880\) 15848.6 0.607108
\(881\) 36781.0 1.40657 0.703283 0.710910i \(-0.251717\pi\)
0.703283 + 0.710910i \(0.251717\pi\)
\(882\) 11155.3 0.425871
\(883\) 14471.7 0.551542 0.275771 0.961223i \(-0.411067\pi\)
0.275771 + 0.961223i \(0.411067\pi\)
\(884\) −12520.0 −0.476349
\(885\) −44394.6 −1.68622
\(886\) −12806.2 −0.485591
\(887\) −29063.0 −1.10016 −0.550080 0.835112i \(-0.685403\pi\)
−0.550080 + 0.835112i \(0.685403\pi\)
\(888\) 13582.8 0.513298
\(889\) 5460.03 0.205988
\(890\) 46094.1 1.73604
\(891\) 99617.4 3.74558
\(892\) 13255.1 0.497550
\(893\) 92.9191 0.00348199
\(894\) 42299.5 1.58245
\(895\) −52217.8 −1.95022
\(896\) −1650.79 −0.0615504
\(897\) −35939.0 −1.33776
\(898\) −32571.8 −1.21040
\(899\) −19692.5 −0.730568
\(900\) −28948.9 −1.07218
\(901\) −20490.5 −0.757643
\(902\) −862.286 −0.0318304
\(903\) 0 0
\(904\) −46099.8 −1.69608
\(905\) 65818.8 2.41756
\(906\) −4597.87 −0.168603
\(907\) −5383.63 −0.197090 −0.0985450 0.995133i \(-0.531419\pi\)
−0.0985450 + 0.995133i \(0.531419\pi\)
\(908\) −7486.31 −0.273614
\(909\) 43937.6 1.60321
\(910\) 24054.1 0.876248
\(911\) 11716.3 0.426100 0.213050 0.977041i \(-0.431660\pi\)
0.213050 + 0.977041i \(0.431660\pi\)
\(912\) −2287.97 −0.0830725
\(913\) −54363.2 −1.97060
\(914\) −34938.2 −1.26439
\(915\) 10452.3 0.377643
\(916\) 3028.96 0.109257
\(917\) 21271.6 0.766030
\(918\) 73447.2 2.64065
\(919\) 18077.4 0.648879 0.324439 0.945906i \(-0.394824\pi\)
0.324439 + 0.945906i \(0.394824\pi\)
\(920\) −31147.5 −1.11620
\(921\) −79249.8 −2.83537
\(922\) 3340.90 0.119335
\(923\) 5039.62 0.179719
\(924\) 24137.7 0.859386
\(925\) 6630.16 0.235674
\(926\) −22661.6 −0.804217
\(927\) −108129. −3.83110
\(928\) −24663.5 −0.872435
\(929\) −2703.22 −0.0954680 −0.0477340 0.998860i \(-0.515200\pi\)
−0.0477340 + 0.998860i \(0.515200\pi\)
\(930\) −37680.9 −1.32861
\(931\) −725.537 −0.0255408
\(932\) −9430.55 −0.331446
\(933\) 58253.6 2.04409
\(934\) −14506.4 −0.508206
\(935\) 53649.4 1.87649
\(936\) −76346.8 −2.66610
\(937\) −5610.28 −0.195603 −0.0978015 0.995206i \(-0.531181\pi\)
−0.0978015 + 0.995206i \(0.531181\pi\)
\(938\) 9519.19 0.331357
\(939\) 75478.3 2.62315
\(940\) 519.985 0.0180426
\(941\) −1782.55 −0.0617528 −0.0308764 0.999523i \(-0.509830\pi\)
−0.0308764 + 0.999523i \(0.509830\pi\)
\(942\) 51151.3 1.76921
\(943\) 779.760 0.0269273
\(944\) −6880.21 −0.237216
\(945\) 109462. 3.76804
\(946\) 0 0
\(947\) −17721.3 −0.608094 −0.304047 0.952657i \(-0.598338\pi\)
−0.304047 + 0.952657i \(0.598338\pi\)
\(948\) 27549.8 0.943855
\(949\) 40350.2 1.38021
\(950\) −2427.21 −0.0828939
\(951\) 87596.9 2.98688
\(952\) 32242.0 1.09766
\(953\) 12699.3 0.431658 0.215829 0.976431i \(-0.430755\pi\)
0.215829 + 0.976431i \(0.430755\pi\)
\(954\) −37989.0 −1.28924
\(955\) 56782.9 1.92403
\(956\) 6489.45 0.219544
\(957\) 71901.9 2.42869
\(958\) 8046.22 0.271359
\(959\) −32588.9 −1.09734
\(960\) −76502.2 −2.57198
\(961\) −16459.4 −0.552495
\(962\) 5316.21 0.178172
\(963\) 68969.1 2.30789
\(964\) 17093.4 0.571100
\(965\) −49863.4 −1.66338
\(966\) 28138.6 0.937211
\(967\) 30667.6 1.01986 0.509929 0.860216i \(-0.329671\pi\)
0.509929 + 0.860216i \(0.329671\pi\)
\(968\) 12021.3 0.399151
\(969\) −7745.04 −0.256766
\(970\) 8369.71 0.277047
\(971\) 38789.5 1.28199 0.640996 0.767544i \(-0.278521\pi\)
0.640996 + 0.767544i \(0.278521\pi\)
\(972\) −39999.3 −1.31994
\(973\) 25462.3 0.838935
\(974\) −14724.7 −0.484404
\(975\) −51548.6 −1.69321
\(976\) 1619.89 0.0531263
\(977\) 34788.6 1.13919 0.569594 0.821926i \(-0.307101\pi\)
0.569594 + 0.821926i \(0.307101\pi\)
\(978\) −61480.4 −2.01015
\(979\) 59545.1 1.94389
\(980\) −4060.18 −0.132345
\(981\) −43348.7 −1.41082
\(982\) −7255.89 −0.235789
\(983\) 34284.6 1.11242 0.556210 0.831042i \(-0.312255\pi\)
0.556210 + 0.831042i \(0.312255\pi\)
\(984\) 2291.28 0.0742309
\(985\) −1447.59 −0.0468264
\(986\) 29200.1 0.943126
\(987\) −1545.08 −0.0498283
\(988\) 1509.69 0.0486131
\(989\) 0 0
\(990\) 99465.0 3.19314
\(991\) 6213.41 0.199168 0.0995840 0.995029i \(-0.468249\pi\)
0.0995840 + 0.995029i \(0.468249\pi\)
\(992\) 16697.0 0.534404
\(993\) 39470.8 1.26140
\(994\) −3945.80 −0.125909
\(995\) 1511.08 0.0481450
\(996\) 43918.7 1.39721
\(997\) 38148.6 1.21181 0.605907 0.795535i \(-0.292810\pi\)
0.605907 + 0.795535i \(0.292810\pi\)
\(998\) 17680.9 0.560801
\(999\) 24192.3 0.766175
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 1849.4.a.j.1.16 yes 50
43.42 odd 2 1849.4.a.i.1.35 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.35 50 43.42 odd 2
1849.4.a.j.1.16 yes 50 1.1 even 1 trivial