Properties

Label 405.4.a.k.1.5
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.57457\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.57457 q^{2} +4.77759 q^{4} -5.00000 q^{5} +14.1597 q^{7} -11.5188 q^{8} +O(q^{10})\) \(q+3.57457 q^{2} +4.77759 q^{4} -5.00000 q^{5} +14.1597 q^{7} -11.5188 q^{8} -17.8729 q^{10} -63.6638 q^{11} -10.3959 q^{13} +50.6151 q^{14} -79.3954 q^{16} -108.605 q^{17} +18.6145 q^{19} -23.8879 q^{20} -227.571 q^{22} +128.824 q^{23} +25.0000 q^{25} -37.1609 q^{26} +67.6494 q^{28} -83.6496 q^{29} -10.4728 q^{31} -191.655 q^{32} -388.217 q^{34} -70.7987 q^{35} -81.8885 q^{37} +66.5391 q^{38} +57.5938 q^{40} +307.453 q^{41} -222.439 q^{43} -304.159 q^{44} +460.491 q^{46} -361.931 q^{47} -142.502 q^{49} +89.3644 q^{50} -49.6673 q^{52} -562.215 q^{53} +318.319 q^{55} -163.103 q^{56} -299.012 q^{58} -462.310 q^{59} -649.067 q^{61} -37.4356 q^{62} -49.9208 q^{64} +51.9795 q^{65} +254.180 q^{67} -518.870 q^{68} -253.075 q^{70} +1092.93 q^{71} +1034.52 q^{73} -292.717 q^{74} +88.9326 q^{76} -901.463 q^{77} +1151.85 q^{79} +396.977 q^{80} +1099.01 q^{82} -524.320 q^{83} +543.026 q^{85} -795.126 q^{86} +733.328 q^{88} -656.485 q^{89} -147.203 q^{91} +615.467 q^{92} -1293.75 q^{94} -93.0727 q^{95} +1125.51 q^{97} -509.383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.57457 1.26380 0.631902 0.775049i \(-0.282275\pi\)
0.631902 + 0.775049i \(0.282275\pi\)
\(3\) 0 0
\(4\) 4.77759 0.597198
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 14.1597 0.764554 0.382277 0.924048i \(-0.375140\pi\)
0.382277 + 0.924048i \(0.375140\pi\)
\(8\) −11.5188 −0.509062
\(9\) 0 0
\(10\) −17.8729 −0.565190
\(11\) −63.6638 −1.74503 −0.872516 0.488585i \(-0.837513\pi\)
−0.872516 + 0.488585i \(0.837513\pi\)
\(12\) 0 0
\(13\) −10.3959 −0.221793 −0.110896 0.993832i \(-0.535372\pi\)
−0.110896 + 0.993832i \(0.535372\pi\)
\(14\) 50.6151 0.966246
\(15\) 0 0
\(16\) −79.3954 −1.24055
\(17\) −108.605 −1.54945 −0.774724 0.632300i \(-0.782111\pi\)
−0.774724 + 0.632300i \(0.782111\pi\)
\(18\) 0 0
\(19\) 18.6145 0.224761 0.112381 0.993665i \(-0.464152\pi\)
0.112381 + 0.993665i \(0.464152\pi\)
\(20\) −23.8879 −0.267075
\(21\) 0 0
\(22\) −227.571 −2.20538
\(23\) 128.824 1.16790 0.583949 0.811791i \(-0.301507\pi\)
0.583949 + 0.811791i \(0.301507\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −37.1609 −0.280302
\(27\) 0 0
\(28\) 67.6494 0.456590
\(29\) −83.6496 −0.535632 −0.267816 0.963470i \(-0.586302\pi\)
−0.267816 + 0.963470i \(0.586302\pi\)
\(30\) 0 0
\(31\) −10.4728 −0.0606762 −0.0303381 0.999540i \(-0.509658\pi\)
−0.0303381 + 0.999540i \(0.509658\pi\)
\(32\) −191.655 −1.05875
\(33\) 0 0
\(34\) −388.217 −1.95820
\(35\) −70.7987 −0.341919
\(36\) 0 0
\(37\) −81.8885 −0.363848 −0.181924 0.983313i \(-0.558233\pi\)
−0.181924 + 0.983313i \(0.558233\pi\)
\(38\) 66.5391 0.284054
\(39\) 0 0
\(40\) 57.5938 0.227659
\(41\) 307.453 1.17112 0.585562 0.810628i \(-0.300874\pi\)
0.585562 + 0.810628i \(0.300874\pi\)
\(42\) 0 0
\(43\) −222.439 −0.788876 −0.394438 0.918922i \(-0.629061\pi\)
−0.394438 + 0.918922i \(0.629061\pi\)
\(44\) −304.159 −1.04213
\(45\) 0 0
\(46\) 460.491 1.47599
\(47\) −361.931 −1.12326 −0.561628 0.827390i \(-0.689825\pi\)
−0.561628 + 0.827390i \(0.689825\pi\)
\(48\) 0 0
\(49\) −142.502 −0.415457
\(50\) 89.3644 0.252761
\(51\) 0 0
\(52\) −49.6673 −0.132454
\(53\) −562.215 −1.45710 −0.728548 0.684994i \(-0.759805\pi\)
−0.728548 + 0.684994i \(0.759805\pi\)
\(54\) 0 0
\(55\) 318.319 0.780402
\(56\) −163.103 −0.389206
\(57\) 0 0
\(58\) −299.012 −0.676934
\(59\) −462.310 −1.02013 −0.510065 0.860136i \(-0.670378\pi\)
−0.510065 + 0.860136i \(0.670378\pi\)
\(60\) 0 0
\(61\) −649.067 −1.36237 −0.681184 0.732112i \(-0.738535\pi\)
−0.681184 + 0.732112i \(0.738535\pi\)
\(62\) −37.4356 −0.0766827
\(63\) 0 0
\(64\) −49.9208 −0.0975015
\(65\) 51.9795 0.0991887
\(66\) 0 0
\(67\) 254.180 0.463478 0.231739 0.972778i \(-0.425558\pi\)
0.231739 + 0.972778i \(0.425558\pi\)
\(68\) −518.870 −0.925328
\(69\) 0 0
\(70\) −253.075 −0.432118
\(71\) 1092.93 1.82685 0.913426 0.407006i \(-0.133427\pi\)
0.913426 + 0.407006i \(0.133427\pi\)
\(72\) 0 0
\(73\) 1034.52 1.65865 0.829324 0.558769i \(-0.188726\pi\)
0.829324 + 0.558769i \(0.188726\pi\)
\(74\) −292.717 −0.459833
\(75\) 0 0
\(76\) 88.9326 0.134227
\(77\) −901.463 −1.33417
\(78\) 0 0
\(79\) 1151.85 1.64043 0.820213 0.572059i \(-0.193855\pi\)
0.820213 + 0.572059i \(0.193855\pi\)
\(80\) 396.977 0.554792
\(81\) 0 0
\(82\) 1099.01 1.48007
\(83\) −524.320 −0.693393 −0.346697 0.937977i \(-0.612697\pi\)
−0.346697 + 0.937977i \(0.612697\pi\)
\(84\) 0 0
\(85\) 543.026 0.692934
\(86\) −795.126 −0.996984
\(87\) 0 0
\(88\) 733.328 0.888330
\(89\) −656.485 −0.781879 −0.390940 0.920416i \(-0.627850\pi\)
−0.390940 + 0.920416i \(0.627850\pi\)
\(90\) 0 0
\(91\) −147.203 −0.169573
\(92\) 615.467 0.697466
\(93\) 0 0
\(94\) −1293.75 −1.41957
\(95\) −93.0727 −0.100516
\(96\) 0 0
\(97\) 1125.51 1.17813 0.589063 0.808087i \(-0.299497\pi\)
0.589063 + 0.808087i \(0.299497\pi\)
\(98\) −509.383 −0.525056
\(99\) 0 0
\(100\) 119.440 0.119440
\(101\) 380.324 0.374690 0.187345 0.982294i \(-0.440012\pi\)
0.187345 + 0.982294i \(0.440012\pi\)
\(102\) 0 0
\(103\) 764.277 0.731131 0.365566 0.930786i \(-0.380876\pi\)
0.365566 + 0.930786i \(0.380876\pi\)
\(104\) 119.748 0.112906
\(105\) 0 0
\(106\) −2009.68 −1.84148
\(107\) 816.083 0.737325 0.368662 0.929563i \(-0.379816\pi\)
0.368662 + 0.929563i \(0.379816\pi\)
\(108\) 0 0
\(109\) 606.775 0.533198 0.266599 0.963808i \(-0.414100\pi\)
0.266599 + 0.963808i \(0.414100\pi\)
\(110\) 1137.86 0.986275
\(111\) 0 0
\(112\) −1124.22 −0.948470
\(113\) 27.3226 0.0227460 0.0113730 0.999935i \(-0.496380\pi\)
0.0113730 + 0.999935i \(0.496380\pi\)
\(114\) 0 0
\(115\) −644.120 −0.522300
\(116\) −399.643 −0.319879
\(117\) 0 0
\(118\) −1652.56 −1.28924
\(119\) −1537.82 −1.18464
\(120\) 0 0
\(121\) 2722.08 2.04514
\(122\) −2320.14 −1.72177
\(123\) 0 0
\(124\) −50.0345 −0.0362357
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1206.22 −0.842794 −0.421397 0.906876i \(-0.638460\pi\)
−0.421397 + 0.906876i \(0.638460\pi\)
\(128\) 1354.79 0.935529
\(129\) 0 0
\(130\) 185.805 0.125355
\(131\) 1532.00 1.02177 0.510885 0.859649i \(-0.329318\pi\)
0.510885 + 0.859649i \(0.329318\pi\)
\(132\) 0 0
\(133\) 263.577 0.171842
\(134\) 908.585 0.585745
\(135\) 0 0
\(136\) 1251.00 0.788765
\(137\) −2472.12 −1.54166 −0.770830 0.637041i \(-0.780158\pi\)
−0.770830 + 0.637041i \(0.780158\pi\)
\(138\) 0 0
\(139\) 12.1267 0.00739982 0.00369991 0.999993i \(-0.498822\pi\)
0.00369991 + 0.999993i \(0.498822\pi\)
\(140\) −338.247 −0.204193
\(141\) 0 0
\(142\) 3906.75 2.30878
\(143\) 661.843 0.387036
\(144\) 0 0
\(145\) 418.248 0.239542
\(146\) 3697.96 2.09620
\(147\) 0 0
\(148\) −391.230 −0.217290
\(149\) 3136.51 1.72452 0.862258 0.506469i \(-0.169049\pi\)
0.862258 + 0.506469i \(0.169049\pi\)
\(150\) 0 0
\(151\) −2700.08 −1.45516 −0.727580 0.686023i \(-0.759355\pi\)
−0.727580 + 0.686023i \(0.759355\pi\)
\(152\) −214.416 −0.114418
\(153\) 0 0
\(154\) −3222.35 −1.68613
\(155\) 52.3638 0.0271352
\(156\) 0 0
\(157\) 442.627 0.225003 0.112502 0.993652i \(-0.464114\pi\)
0.112502 + 0.993652i \(0.464114\pi\)
\(158\) 4117.39 2.07317
\(159\) 0 0
\(160\) 958.273 0.473488
\(161\) 1824.11 0.892921
\(162\) 0 0
\(163\) −2116.29 −1.01694 −0.508469 0.861080i \(-0.669788\pi\)
−0.508469 + 0.861080i \(0.669788\pi\)
\(164\) 1468.88 0.699393
\(165\) 0 0
\(166\) −1874.22 −0.876312
\(167\) −1294.62 −0.599886 −0.299943 0.953957i \(-0.596968\pi\)
−0.299943 + 0.953957i \(0.596968\pi\)
\(168\) 0 0
\(169\) −2088.93 −0.950808
\(170\) 1941.09 0.875732
\(171\) 0 0
\(172\) −1062.72 −0.471116
\(173\) −2922.17 −1.28421 −0.642106 0.766616i \(-0.721939\pi\)
−0.642106 + 0.766616i \(0.721939\pi\)
\(174\) 0 0
\(175\) 353.994 0.152911
\(176\) 5054.61 2.16480
\(177\) 0 0
\(178\) −2346.65 −0.988141
\(179\) −3955.51 −1.65167 −0.825834 0.563913i \(-0.809295\pi\)
−0.825834 + 0.563913i \(0.809295\pi\)
\(180\) 0 0
\(181\) −2694.00 −1.10632 −0.553159 0.833076i \(-0.686578\pi\)
−0.553159 + 0.833076i \(0.686578\pi\)
\(182\) −526.189 −0.214306
\(183\) 0 0
\(184\) −1483.89 −0.594532
\(185\) 409.443 0.162718
\(186\) 0 0
\(187\) 6914.22 2.70384
\(188\) −1729.15 −0.670806
\(189\) 0 0
\(190\) −332.695 −0.127033
\(191\) 856.346 0.324414 0.162207 0.986757i \(-0.448139\pi\)
0.162207 + 0.986757i \(0.448139\pi\)
\(192\) 0 0
\(193\) −1735.21 −0.647168 −0.323584 0.946199i \(-0.604888\pi\)
−0.323584 + 0.946199i \(0.604888\pi\)
\(194\) 4023.22 1.48892
\(195\) 0 0
\(196\) −680.814 −0.248110
\(197\) −2943.95 −1.06471 −0.532354 0.846522i \(-0.678692\pi\)
−0.532354 + 0.846522i \(0.678692\pi\)
\(198\) 0 0
\(199\) −4172.06 −1.48618 −0.743088 0.669193i \(-0.766640\pi\)
−0.743088 + 0.669193i \(0.766640\pi\)
\(200\) −287.969 −0.101812
\(201\) 0 0
\(202\) 1359.50 0.473534
\(203\) −1184.46 −0.409520
\(204\) 0 0
\(205\) −1537.26 −0.523742
\(206\) 2731.97 0.924006
\(207\) 0 0
\(208\) 825.387 0.275146
\(209\) −1185.07 −0.392216
\(210\) 0 0
\(211\) 3851.04 1.25648 0.628239 0.778021i \(-0.283776\pi\)
0.628239 + 0.778021i \(0.283776\pi\)
\(212\) −2686.03 −0.870176
\(213\) 0 0
\(214\) 2917.15 0.931833
\(215\) 1112.20 0.352796
\(216\) 0 0
\(217\) −148.291 −0.0463902
\(218\) 2168.96 0.673857
\(219\) 0 0
\(220\) 1520.80 0.466055
\(221\) 1129.05 0.343656
\(222\) 0 0
\(223\) 820.714 0.246453 0.123227 0.992379i \(-0.460676\pi\)
0.123227 + 0.992379i \(0.460676\pi\)
\(224\) −2713.78 −0.809473
\(225\) 0 0
\(226\) 97.6667 0.0287464
\(227\) 5261.23 1.53833 0.769163 0.639053i \(-0.220674\pi\)
0.769163 + 0.639053i \(0.220674\pi\)
\(228\) 0 0
\(229\) 19.5989 0.00565560 0.00282780 0.999996i \(-0.499100\pi\)
0.00282780 + 0.999996i \(0.499100\pi\)
\(230\) −2302.45 −0.660084
\(231\) 0 0
\(232\) 963.539 0.272670
\(233\) 3181.79 0.894619 0.447309 0.894379i \(-0.352382\pi\)
0.447309 + 0.894379i \(0.352382\pi\)
\(234\) 0 0
\(235\) 1809.65 0.502335
\(236\) −2208.72 −0.609219
\(237\) 0 0
\(238\) −5497.05 −1.49715
\(239\) −1634.10 −0.442265 −0.221133 0.975244i \(-0.570975\pi\)
−0.221133 + 0.975244i \(0.570975\pi\)
\(240\) 0 0
\(241\) 1414.71 0.378131 0.189065 0.981965i \(-0.439454\pi\)
0.189065 + 0.981965i \(0.439454\pi\)
\(242\) 9730.28 2.58465
\(243\) 0 0
\(244\) −3100.97 −0.813604
\(245\) 712.509 0.185798
\(246\) 0 0
\(247\) −193.515 −0.0498505
\(248\) 120.633 0.0308879
\(249\) 0 0
\(250\) −446.822 −0.113038
\(251\) −5932.53 −1.49186 −0.745932 0.666022i \(-0.767995\pi\)
−0.745932 + 0.666022i \(0.767995\pi\)
\(252\) 0 0
\(253\) −8201.42 −2.03802
\(254\) −4311.73 −1.06513
\(255\) 0 0
\(256\) 5242.17 1.27983
\(257\) −7001.29 −1.69933 −0.849666 0.527321i \(-0.823197\pi\)
−0.849666 + 0.527321i \(0.823197\pi\)
\(258\) 0 0
\(259\) −1159.52 −0.278182
\(260\) 248.337 0.0592353
\(261\) 0 0
\(262\) 5476.26 1.29132
\(263\) −3654.54 −0.856839 −0.428420 0.903580i \(-0.640929\pi\)
−0.428420 + 0.903580i \(0.640929\pi\)
\(264\) 0 0
\(265\) 2811.07 0.651634
\(266\) 942.176 0.217175
\(267\) 0 0
\(268\) 1214.37 0.276788
\(269\) −2589.46 −0.586922 −0.293461 0.955971i \(-0.594807\pi\)
−0.293461 + 0.955971i \(0.594807\pi\)
\(270\) 0 0
\(271\) 4854.10 1.08806 0.544032 0.839064i \(-0.316897\pi\)
0.544032 + 0.839064i \(0.316897\pi\)
\(272\) 8622.74 1.92217
\(273\) 0 0
\(274\) −8836.77 −1.94835
\(275\) −1591.60 −0.349007
\(276\) 0 0
\(277\) −7515.15 −1.63012 −0.815058 0.579380i \(-0.803295\pi\)
−0.815058 + 0.579380i \(0.803295\pi\)
\(278\) 43.3478 0.00935191
\(279\) 0 0
\(280\) 815.513 0.174058
\(281\) 3612.07 0.766825 0.383412 0.923577i \(-0.374749\pi\)
0.383412 + 0.923577i \(0.374749\pi\)
\(282\) 0 0
\(283\) 5166.91 1.08530 0.542652 0.839958i \(-0.317420\pi\)
0.542652 + 0.839958i \(0.317420\pi\)
\(284\) 5221.55 1.09099
\(285\) 0 0
\(286\) 2365.81 0.489137
\(287\) 4353.45 0.895387
\(288\) 0 0
\(289\) 6882.07 1.40079
\(290\) 1495.06 0.302734
\(291\) 0 0
\(292\) 4942.50 0.990541
\(293\) 4226.35 0.842683 0.421342 0.906902i \(-0.361559\pi\)
0.421342 + 0.906902i \(0.361559\pi\)
\(294\) 0 0
\(295\) 2311.55 0.456216
\(296\) 943.255 0.185221
\(297\) 0 0
\(298\) 11211.7 2.17945
\(299\) −1339.24 −0.259031
\(300\) 0 0
\(301\) −3149.68 −0.603139
\(302\) −9651.62 −1.83903
\(303\) 0 0
\(304\) −1477.91 −0.278828
\(305\) 3245.33 0.609270
\(306\) 0 0
\(307\) 5734.25 1.06603 0.533015 0.846106i \(-0.321059\pi\)
0.533015 + 0.846106i \(0.321059\pi\)
\(308\) −4306.82 −0.796765
\(309\) 0 0
\(310\) 187.178 0.0342936
\(311\) −6620.39 −1.20710 −0.603549 0.797326i \(-0.706247\pi\)
−0.603549 + 0.797326i \(0.706247\pi\)
\(312\) 0 0
\(313\) 93.4309 0.0168723 0.00843615 0.999964i \(-0.497315\pi\)
0.00843615 + 0.999964i \(0.497315\pi\)
\(314\) 1582.20 0.284360
\(315\) 0 0
\(316\) 5503.08 0.979659
\(317\) −1993.17 −0.353147 −0.176573 0.984287i \(-0.556501\pi\)
−0.176573 + 0.984287i \(0.556501\pi\)
\(318\) 0 0
\(319\) 5325.45 0.934696
\(320\) 249.604 0.0436040
\(321\) 0 0
\(322\) 6520.43 1.12848
\(323\) −2021.63 −0.348256
\(324\) 0 0
\(325\) −259.898 −0.0443586
\(326\) −7564.85 −1.28521
\(327\) 0 0
\(328\) −3541.47 −0.596174
\(329\) −5124.84 −0.858790
\(330\) 0 0
\(331\) −2389.18 −0.396740 −0.198370 0.980127i \(-0.563565\pi\)
−0.198370 + 0.980127i \(0.563565\pi\)
\(332\) −2504.99 −0.414093
\(333\) 0 0
\(334\) −4627.73 −0.758138
\(335\) −1270.90 −0.207274
\(336\) 0 0
\(337\) 10285.5 1.66257 0.831286 0.555845i \(-0.187605\pi\)
0.831286 + 0.555845i \(0.187605\pi\)
\(338\) −7467.02 −1.20163
\(339\) 0 0
\(340\) 2594.35 0.413819
\(341\) 666.735 0.105882
\(342\) 0 0
\(343\) −6874.58 −1.08219
\(344\) 2562.23 0.401587
\(345\) 0 0
\(346\) −10445.5 −1.62299
\(347\) −4230.21 −0.654436 −0.327218 0.944949i \(-0.606111\pi\)
−0.327218 + 0.944949i \(0.606111\pi\)
\(348\) 0 0
\(349\) −11210.9 −1.71950 −0.859752 0.510712i \(-0.829382\pi\)
−0.859752 + 0.510712i \(0.829382\pi\)
\(350\) 1265.38 0.193249
\(351\) 0 0
\(352\) 12201.5 1.84756
\(353\) 11383.0 1.71631 0.858154 0.513393i \(-0.171612\pi\)
0.858154 + 0.513393i \(0.171612\pi\)
\(354\) 0 0
\(355\) −5464.63 −0.816993
\(356\) −3136.41 −0.466937
\(357\) 0 0
\(358\) −14139.3 −2.08738
\(359\) −8942.57 −1.31468 −0.657341 0.753593i \(-0.728319\pi\)
−0.657341 + 0.753593i \(0.728319\pi\)
\(360\) 0 0
\(361\) −6512.50 −0.949482
\(362\) −9629.90 −1.39817
\(363\) 0 0
\(364\) −703.277 −0.101268
\(365\) −5172.59 −0.741770
\(366\) 0 0
\(367\) −444.170 −0.0631757 −0.0315878 0.999501i \(-0.510056\pi\)
−0.0315878 + 0.999501i \(0.510056\pi\)
\(368\) −10228.0 −1.44884
\(369\) 0 0
\(370\) 1463.58 0.205643
\(371\) −7960.82 −1.11403
\(372\) 0 0
\(373\) 3416.09 0.474205 0.237103 0.971485i \(-0.423802\pi\)
0.237103 + 0.971485i \(0.423802\pi\)
\(374\) 24715.4 3.41712
\(375\) 0 0
\(376\) 4168.99 0.571807
\(377\) 869.613 0.118799
\(378\) 0 0
\(379\) 598.277 0.0810855 0.0405428 0.999178i \(-0.487091\pi\)
0.0405428 + 0.999178i \(0.487091\pi\)
\(380\) −444.663 −0.0600282
\(381\) 0 0
\(382\) 3061.07 0.409995
\(383\) 9057.17 1.20836 0.604178 0.796850i \(-0.293502\pi\)
0.604178 + 0.796850i \(0.293502\pi\)
\(384\) 0 0
\(385\) 4507.31 0.596660
\(386\) −6202.65 −0.817893
\(387\) 0 0
\(388\) 5377.22 0.703575
\(389\) −631.406 −0.0822971 −0.0411485 0.999153i \(-0.513102\pi\)
−0.0411485 + 0.999153i \(0.513102\pi\)
\(390\) 0 0
\(391\) −13990.9 −1.80960
\(392\) 1641.44 0.211493
\(393\) 0 0
\(394\) −10523.4 −1.34558
\(395\) −5759.27 −0.733620
\(396\) 0 0
\(397\) −4615.39 −0.583475 −0.291738 0.956498i \(-0.594233\pi\)
−0.291738 + 0.956498i \(0.594233\pi\)
\(398\) −14913.3 −1.87824
\(399\) 0 0
\(400\) −1984.88 −0.248111
\(401\) −4691.20 −0.584207 −0.292104 0.956387i \(-0.594355\pi\)
−0.292104 + 0.956387i \(0.594355\pi\)
\(402\) 0 0
\(403\) 108.874 0.0134575
\(404\) 1817.03 0.223764
\(405\) 0 0
\(406\) −4233.93 −0.517552
\(407\) 5213.34 0.634927
\(408\) 0 0
\(409\) −15563.3 −1.88155 −0.940776 0.339028i \(-0.889902\pi\)
−0.940776 + 0.339028i \(0.889902\pi\)
\(410\) −5495.06 −0.661907
\(411\) 0 0
\(412\) 3651.40 0.436630
\(413\) −6546.19 −0.779944
\(414\) 0 0
\(415\) 2621.60 0.310095
\(416\) 1992.42 0.234824
\(417\) 0 0
\(418\) −4236.13 −0.495684
\(419\) −4278.44 −0.498843 −0.249422 0.968395i \(-0.580240\pi\)
−0.249422 + 0.968395i \(0.580240\pi\)
\(420\) 0 0
\(421\) −97.7063 −0.0113110 −0.00565548 0.999984i \(-0.501800\pi\)
−0.00565548 + 0.999984i \(0.501800\pi\)
\(422\) 13765.8 1.58794
\(423\) 0 0
\(424\) 6476.02 0.741753
\(425\) −2715.13 −0.309890
\(426\) 0 0
\(427\) −9190.62 −1.04160
\(428\) 3898.91 0.440329
\(429\) 0 0
\(430\) 3975.63 0.445865
\(431\) −11932.9 −1.33361 −0.666806 0.745231i \(-0.732339\pi\)
−0.666806 + 0.745231i \(0.732339\pi\)
\(432\) 0 0
\(433\) 6304.32 0.699691 0.349845 0.936807i \(-0.386234\pi\)
0.349845 + 0.936807i \(0.386234\pi\)
\(434\) −530.079 −0.0586281
\(435\) 0 0
\(436\) 2898.92 0.318425
\(437\) 2398.00 0.262498
\(438\) 0 0
\(439\) −12922.7 −1.40493 −0.702466 0.711717i \(-0.747918\pi\)
−0.702466 + 0.711717i \(0.747918\pi\)
\(440\) −3666.64 −0.397273
\(441\) 0 0
\(442\) 4035.87 0.434314
\(443\) 3824.41 0.410165 0.205083 0.978745i \(-0.434254\pi\)
0.205083 + 0.978745i \(0.434254\pi\)
\(444\) 0 0
\(445\) 3282.42 0.349667
\(446\) 2933.70 0.311468
\(447\) 0 0
\(448\) −706.865 −0.0745452
\(449\) 530.656 0.0557756 0.0278878 0.999611i \(-0.491122\pi\)
0.0278878 + 0.999611i \(0.491122\pi\)
\(450\) 0 0
\(451\) −19573.6 −2.04365
\(452\) 130.536 0.0135839
\(453\) 0 0
\(454\) 18806.6 1.94414
\(455\) 736.017 0.0758352
\(456\) 0 0
\(457\) −15593.6 −1.59614 −0.798072 0.602562i \(-0.794147\pi\)
−0.798072 + 0.602562i \(0.794147\pi\)
\(458\) 70.0577 0.00714756
\(459\) 0 0
\(460\) −3077.34 −0.311916
\(461\) 7274.61 0.734951 0.367475 0.930033i \(-0.380222\pi\)
0.367475 + 0.930033i \(0.380222\pi\)
\(462\) 0 0
\(463\) 2685.14 0.269523 0.134761 0.990878i \(-0.456973\pi\)
0.134761 + 0.990878i \(0.456973\pi\)
\(464\) 6641.39 0.664480
\(465\) 0 0
\(466\) 11373.6 1.13062
\(467\) 8844.77 0.876418 0.438209 0.898873i \(-0.355613\pi\)
0.438209 + 0.898873i \(0.355613\pi\)
\(468\) 0 0
\(469\) 3599.12 0.354354
\(470\) 6468.74 0.634853
\(471\) 0 0
\(472\) 5325.24 0.519309
\(473\) 14161.3 1.37662
\(474\) 0 0
\(475\) 465.363 0.0449523
\(476\) −7347.07 −0.707463
\(477\) 0 0
\(478\) −5841.23 −0.558936
\(479\) −12334.3 −1.17656 −0.588278 0.808659i \(-0.700194\pi\)
−0.588278 + 0.808659i \(0.700194\pi\)
\(480\) 0 0
\(481\) 851.306 0.0806990
\(482\) 5056.99 0.477883
\(483\) 0 0
\(484\) 13005.0 1.22135
\(485\) −5627.55 −0.526874
\(486\) 0 0
\(487\) 3540.31 0.329419 0.164709 0.986342i \(-0.447331\pi\)
0.164709 + 0.986342i \(0.447331\pi\)
\(488\) 7476.45 0.693530
\(489\) 0 0
\(490\) 2546.92 0.234812
\(491\) −7711.25 −0.708765 −0.354383 0.935100i \(-0.615309\pi\)
−0.354383 + 0.935100i \(0.615309\pi\)
\(492\) 0 0
\(493\) 9084.77 0.829934
\(494\) −691.734 −0.0630012
\(495\) 0 0
\(496\) 831.488 0.0752720
\(497\) 15475.5 1.39673
\(498\) 0 0
\(499\) −12162.6 −1.09113 −0.545563 0.838070i \(-0.683684\pi\)
−0.545563 + 0.838070i \(0.683684\pi\)
\(500\) −597.198 −0.0534150
\(501\) 0 0
\(502\) −21206.3 −1.88542
\(503\) 8796.85 0.779786 0.389893 0.920860i \(-0.372512\pi\)
0.389893 + 0.920860i \(0.372512\pi\)
\(504\) 0 0
\(505\) −1901.62 −0.167566
\(506\) −29316.6 −2.57566
\(507\) 0 0
\(508\) −5762.83 −0.503315
\(509\) 21948.2 1.91127 0.955634 0.294558i \(-0.0951723\pi\)
0.955634 + 0.294558i \(0.0951723\pi\)
\(510\) 0 0
\(511\) 14648.5 1.26813
\(512\) 7900.20 0.681919
\(513\) 0 0
\(514\) −25026.6 −2.14762
\(515\) −3821.39 −0.326972
\(516\) 0 0
\(517\) 23041.9 1.96012
\(518\) −4144.79 −0.351567
\(519\) 0 0
\(520\) −598.740 −0.0504932
\(521\) 11232.4 0.944531 0.472265 0.881456i \(-0.343436\pi\)
0.472265 + 0.881456i \(0.343436\pi\)
\(522\) 0 0
\(523\) −1962.88 −0.164112 −0.0820560 0.996628i \(-0.526149\pi\)
−0.0820560 + 0.996628i \(0.526149\pi\)
\(524\) 7319.28 0.610199
\(525\) 0 0
\(526\) −13063.4 −1.08288
\(527\) 1137.39 0.0940146
\(528\) 0 0
\(529\) 4428.60 0.363985
\(530\) 10048.4 0.823537
\(531\) 0 0
\(532\) 1259.26 0.102624
\(533\) −3196.25 −0.259747
\(534\) 0 0
\(535\) −4080.42 −0.329742
\(536\) −2927.84 −0.235939
\(537\) 0 0
\(538\) −9256.21 −0.741754
\(539\) 9072.20 0.724986
\(540\) 0 0
\(541\) 2816.86 0.223856 0.111928 0.993716i \(-0.464297\pi\)
0.111928 + 0.993716i \(0.464297\pi\)
\(542\) 17351.3 1.37510
\(543\) 0 0
\(544\) 20814.7 1.64048
\(545\) −3033.88 −0.238453
\(546\) 0 0
\(547\) 4528.91 0.354007 0.177004 0.984210i \(-0.443360\pi\)
0.177004 + 0.984210i \(0.443360\pi\)
\(548\) −11810.8 −0.920676
\(549\) 0 0
\(550\) −5689.28 −0.441076
\(551\) −1557.10 −0.120389
\(552\) 0 0
\(553\) 16309.9 1.25419
\(554\) −26863.5 −2.06014
\(555\) 0 0
\(556\) 57.9364 0.00441916
\(557\) 6267.47 0.476771 0.238385 0.971171i \(-0.423382\pi\)
0.238385 + 0.971171i \(0.423382\pi\)
\(558\) 0 0
\(559\) 2312.46 0.174967
\(560\) 5621.09 0.424168
\(561\) 0 0
\(562\) 12911.6 0.969116
\(563\) −9794.00 −0.733158 −0.366579 0.930387i \(-0.619471\pi\)
−0.366579 + 0.930387i \(0.619471\pi\)
\(564\) 0 0
\(565\) −136.613 −0.0101723
\(566\) 18469.5 1.37161
\(567\) 0 0
\(568\) −12589.2 −0.929981
\(569\) 7599.88 0.559936 0.279968 0.960009i \(-0.409676\pi\)
0.279968 + 0.960009i \(0.409676\pi\)
\(570\) 0 0
\(571\) −12094.1 −0.886382 −0.443191 0.896427i \(-0.646154\pi\)
−0.443191 + 0.896427i \(0.646154\pi\)
\(572\) 3162.01 0.231137
\(573\) 0 0
\(574\) 15561.7 1.13159
\(575\) 3220.60 0.233580
\(576\) 0 0
\(577\) 3113.85 0.224665 0.112332 0.993671i \(-0.464168\pi\)
0.112332 + 0.993671i \(0.464168\pi\)
\(578\) 24600.5 1.77032
\(579\) 0 0
\(580\) 1998.21 0.143054
\(581\) −7424.24 −0.530137
\(582\) 0 0
\(583\) 35792.7 2.54268
\(584\) −11916.4 −0.844354
\(585\) 0 0
\(586\) 15107.4 1.06499
\(587\) 5947.91 0.418222 0.209111 0.977892i \(-0.432943\pi\)
0.209111 + 0.977892i \(0.432943\pi\)
\(588\) 0 0
\(589\) −194.945 −0.0136377
\(590\) 8262.81 0.576567
\(591\) 0 0
\(592\) 6501.57 0.451373
\(593\) 22153.0 1.53409 0.767043 0.641596i \(-0.221727\pi\)
0.767043 + 0.641596i \(0.221727\pi\)
\(594\) 0 0
\(595\) 7689.10 0.529786
\(596\) 14984.9 1.02988
\(597\) 0 0
\(598\) −4787.22 −0.327364
\(599\) −26876.4 −1.83329 −0.916643 0.399708i \(-0.869112\pi\)
−0.916643 + 0.399708i \(0.869112\pi\)
\(600\) 0 0
\(601\) 25840.2 1.75381 0.876907 0.480659i \(-0.159603\pi\)
0.876907 + 0.480659i \(0.159603\pi\)
\(602\) −11258.8 −0.762249
\(603\) 0 0
\(604\) −12899.8 −0.869019
\(605\) −13610.4 −0.914614
\(606\) 0 0
\(607\) −2768.74 −0.185139 −0.0925697 0.995706i \(-0.529508\pi\)
−0.0925697 + 0.995706i \(0.529508\pi\)
\(608\) −3567.56 −0.237967
\(609\) 0 0
\(610\) 11600.7 0.769997
\(611\) 3762.60 0.249130
\(612\) 0 0
\(613\) 10177.3 0.670563 0.335282 0.942118i \(-0.391169\pi\)
0.335282 + 0.942118i \(0.391169\pi\)
\(614\) 20497.5 1.34725
\(615\) 0 0
\(616\) 10383.7 0.679176
\(617\) 8417.23 0.549213 0.274607 0.961557i \(-0.411452\pi\)
0.274607 + 0.961557i \(0.411452\pi\)
\(618\) 0 0
\(619\) −13280.3 −0.862324 −0.431162 0.902274i \(-0.641896\pi\)
−0.431162 + 0.902274i \(0.641896\pi\)
\(620\) 250.172 0.0162051
\(621\) 0 0
\(622\) −23665.1 −1.52553
\(623\) −9295.66 −0.597789
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 333.976 0.0213233
\(627\) 0 0
\(628\) 2114.69 0.134371
\(629\) 8893.52 0.563764
\(630\) 0 0
\(631\) −14454.1 −0.911903 −0.455951 0.890005i \(-0.650701\pi\)
−0.455951 + 0.890005i \(0.650701\pi\)
\(632\) −13267.9 −0.835078
\(633\) 0 0
\(634\) −7124.73 −0.446308
\(635\) 6031.11 0.376909
\(636\) 0 0
\(637\) 1481.43 0.0921453
\(638\) 19036.2 1.18127
\(639\) 0 0
\(640\) −6773.96 −0.418381
\(641\) 21064.9 1.29799 0.648997 0.760791i \(-0.275189\pi\)
0.648997 + 0.760791i \(0.275189\pi\)
\(642\) 0 0
\(643\) −7128.81 −0.437221 −0.218610 0.975812i \(-0.570152\pi\)
−0.218610 + 0.975812i \(0.570152\pi\)
\(644\) 8714.86 0.533251
\(645\) 0 0
\(646\) −7226.48 −0.440127
\(647\) 1364.92 0.0829376 0.0414688 0.999140i \(-0.486796\pi\)
0.0414688 + 0.999140i \(0.486796\pi\)
\(648\) 0 0
\(649\) 29432.4 1.78016
\(650\) −929.024 −0.0560605
\(651\) 0 0
\(652\) −10110.8 −0.607313
\(653\) 12160.6 0.728763 0.364381 0.931250i \(-0.381280\pi\)
0.364381 + 0.931250i \(0.381280\pi\)
\(654\) 0 0
\(655\) −7660.02 −0.456949
\(656\) −24410.3 −1.45284
\(657\) 0 0
\(658\) −18319.1 −1.08534
\(659\) −6168.57 −0.364633 −0.182317 0.983240i \(-0.558360\pi\)
−0.182317 + 0.983240i \(0.558360\pi\)
\(660\) 0 0
\(661\) 18137.7 1.06728 0.533642 0.845711i \(-0.320823\pi\)
0.533642 + 0.845711i \(0.320823\pi\)
\(662\) −8540.29 −0.501402
\(663\) 0 0
\(664\) 6039.52 0.352980
\(665\) −1317.89 −0.0768502
\(666\) 0 0
\(667\) −10776.1 −0.625564
\(668\) −6185.18 −0.358251
\(669\) 0 0
\(670\) −4542.93 −0.261953
\(671\) 41322.1 2.37738
\(672\) 0 0
\(673\) −21702.1 −1.24302 −0.621512 0.783404i \(-0.713481\pi\)
−0.621512 + 0.783404i \(0.713481\pi\)
\(674\) 36766.3 2.10116
\(675\) 0 0
\(676\) −9980.02 −0.567821
\(677\) −10470.6 −0.594412 −0.297206 0.954813i \(-0.596055\pi\)
−0.297206 + 0.954813i \(0.596055\pi\)
\(678\) 0 0
\(679\) 15936.9 0.900742
\(680\) −6254.98 −0.352747
\(681\) 0 0
\(682\) 2383.29 0.133814
\(683\) −15386.0 −0.861974 −0.430987 0.902358i \(-0.641834\pi\)
−0.430987 + 0.902358i \(0.641834\pi\)
\(684\) 0 0
\(685\) 12360.6 0.689451
\(686\) −24573.7 −1.36768
\(687\) 0 0
\(688\) 17660.7 0.978643
\(689\) 5844.73 0.323174
\(690\) 0 0
\(691\) 15191.1 0.836322 0.418161 0.908373i \(-0.362675\pi\)
0.418161 + 0.908373i \(0.362675\pi\)
\(692\) −13960.9 −0.766929
\(693\) 0 0
\(694\) −15121.2 −0.827079
\(695\) −60.6336 −0.00330930
\(696\) 0 0
\(697\) −33390.9 −1.81459
\(698\) −40074.3 −2.17311
\(699\) 0 0
\(700\) 1691.23 0.0913181
\(701\) −5181.93 −0.279199 −0.139600 0.990208i \(-0.544582\pi\)
−0.139600 + 0.990208i \(0.544582\pi\)
\(702\) 0 0
\(703\) −1524.32 −0.0817791
\(704\) 3178.15 0.170143
\(705\) 0 0
\(706\) 40689.4 2.16907
\(707\) 5385.30 0.286471
\(708\) 0 0
\(709\) −8786.53 −0.465423 −0.232712 0.972546i \(-0.574760\pi\)
−0.232712 + 0.972546i \(0.574760\pi\)
\(710\) −19533.7 −1.03252
\(711\) 0 0
\(712\) 7561.89 0.398025
\(713\) −1349.14 −0.0708636
\(714\) 0 0
\(715\) −3309.21 −0.173088
\(716\) −18897.8 −0.986373
\(717\) 0 0
\(718\) −31965.9 −1.66150
\(719\) −14433.0 −0.748625 −0.374312 0.927303i \(-0.622121\pi\)
−0.374312 + 0.927303i \(0.622121\pi\)
\(720\) 0 0
\(721\) 10822.0 0.558989
\(722\) −23279.4 −1.19996
\(723\) 0 0
\(724\) −12870.8 −0.660691
\(725\) −2091.24 −0.107126
\(726\) 0 0
\(727\) 26698.9 1.36205 0.681023 0.732262i \(-0.261535\pi\)
0.681023 + 0.732262i \(0.261535\pi\)
\(728\) 1695.60 0.0863230
\(729\) 0 0
\(730\) −18489.8 −0.937451
\(731\) 24158.1 1.22232
\(732\) 0 0
\(733\) −7733.84 −0.389708 −0.194854 0.980832i \(-0.562423\pi\)
−0.194854 + 0.980832i \(0.562423\pi\)
\(734\) −1587.72 −0.0798416
\(735\) 0 0
\(736\) −24689.7 −1.23651
\(737\) −16182.1 −0.808784
\(738\) 0 0
\(739\) 28142.2 1.40085 0.700426 0.713725i \(-0.252994\pi\)
0.700426 + 0.713725i \(0.252994\pi\)
\(740\) 1956.15 0.0971749
\(741\) 0 0
\(742\) −28456.5 −1.40791
\(743\) 22769.1 1.12425 0.562125 0.827052i \(-0.309984\pi\)
0.562125 + 0.827052i \(0.309984\pi\)
\(744\) 0 0
\(745\) −15682.6 −0.771227
\(746\) 12211.1 0.599302
\(747\) 0 0
\(748\) 33033.3 1.61473
\(749\) 11555.5 0.563725
\(750\) 0 0
\(751\) 509.837 0.0247726 0.0123863 0.999923i \(-0.496057\pi\)
0.0123863 + 0.999923i \(0.496057\pi\)
\(752\) 28735.6 1.39346
\(753\) 0 0
\(754\) 3108.50 0.150139
\(755\) 13500.4 0.650767
\(756\) 0 0
\(757\) −28105.6 −1.34943 −0.674713 0.738080i \(-0.735733\pi\)
−0.674713 + 0.738080i \(0.735733\pi\)
\(758\) 2138.58 0.102476
\(759\) 0 0
\(760\) 1072.08 0.0511691
\(761\) 8695.64 0.414214 0.207107 0.978318i \(-0.433595\pi\)
0.207107 + 0.978318i \(0.433595\pi\)
\(762\) 0 0
\(763\) 8591.78 0.407659
\(764\) 4091.27 0.193739
\(765\) 0 0
\(766\) 32375.5 1.52712
\(767\) 4806.13 0.226257
\(768\) 0 0
\(769\) 5461.87 0.256125 0.128062 0.991766i \(-0.459124\pi\)
0.128062 + 0.991766i \(0.459124\pi\)
\(770\) 16111.7 0.754061
\(771\) 0 0
\(772\) −8290.13 −0.386487
\(773\) 37952.9 1.76594 0.882970 0.469430i \(-0.155541\pi\)
0.882970 + 0.469430i \(0.155541\pi\)
\(774\) 0 0
\(775\) −261.819 −0.0121352
\(776\) −12964.5 −0.599740
\(777\) 0 0
\(778\) −2257.01 −0.104007
\(779\) 5723.09 0.263223
\(780\) 0 0
\(781\) −69579.8 −3.18791
\(782\) −50011.7 −2.28697
\(783\) 0 0
\(784\) 11314.0 0.515396
\(785\) −2213.14 −0.100624
\(786\) 0 0
\(787\) −18215.1 −0.825032 −0.412516 0.910950i \(-0.635350\pi\)
−0.412516 + 0.910950i \(0.635350\pi\)
\(788\) −14065.0 −0.635842
\(789\) 0 0
\(790\) −20586.9 −0.927152
\(791\) 386.881 0.0173905
\(792\) 0 0
\(793\) 6747.64 0.302164
\(794\) −16498.1 −0.737398
\(795\) 0 0
\(796\) −19932.4 −0.887542
\(797\) 28549.5 1.26885 0.634427 0.772983i \(-0.281236\pi\)
0.634427 + 0.772983i \(0.281236\pi\)
\(798\) 0 0
\(799\) 39307.5 1.74043
\(800\) −4791.36 −0.211750
\(801\) 0 0
\(802\) −16769.0 −0.738323
\(803\) −65861.4 −2.89439
\(804\) 0 0
\(805\) −9120.57 −0.399326
\(806\) 389.177 0.0170077
\(807\) 0 0
\(808\) −4380.87 −0.190741
\(809\) 2685.20 0.116695 0.0583476 0.998296i \(-0.481417\pi\)
0.0583476 + 0.998296i \(0.481417\pi\)
\(810\) 0 0
\(811\) −12491.4 −0.540854 −0.270427 0.962740i \(-0.587165\pi\)
−0.270427 + 0.962740i \(0.587165\pi\)
\(812\) −5658.84 −0.244565
\(813\) 0 0
\(814\) 18635.5 0.802423
\(815\) 10581.5 0.454788
\(816\) 0 0
\(817\) −4140.61 −0.177309
\(818\) −55632.1 −2.37791
\(819\) 0 0
\(820\) −7344.41 −0.312778
\(821\) 23517.5 0.999715 0.499858 0.866108i \(-0.333386\pi\)
0.499858 + 0.866108i \(0.333386\pi\)
\(822\) 0 0
\(823\) −18126.2 −0.767727 −0.383863 0.923390i \(-0.625407\pi\)
−0.383863 + 0.923390i \(0.625407\pi\)
\(824\) −8803.53 −0.372191
\(825\) 0 0
\(826\) −23399.8 −0.985696
\(827\) −45677.1 −1.92062 −0.960308 0.278940i \(-0.910017\pi\)
−0.960308 + 0.278940i \(0.910017\pi\)
\(828\) 0 0
\(829\) 9130.02 0.382507 0.191254 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(830\) 9371.11 0.391899
\(831\) 0 0
\(832\) 518.972 0.0216251
\(833\) 15476.4 0.643729
\(834\) 0 0
\(835\) 6473.12 0.268277
\(836\) −5661.78 −0.234231
\(837\) 0 0
\(838\) −15293.6 −0.630440
\(839\) −18206.4 −0.749172 −0.374586 0.927192i \(-0.622215\pi\)
−0.374586 + 0.927192i \(0.622215\pi\)
\(840\) 0 0
\(841\) −17391.8 −0.713098
\(842\) −349.258 −0.0142948
\(843\) 0 0
\(844\) 18398.7 0.750366
\(845\) 10444.6 0.425214
\(846\) 0 0
\(847\) 38543.9 1.56362
\(848\) 44637.2 1.80761
\(849\) 0 0
\(850\) −9705.43 −0.391639
\(851\) −10549.2 −0.424938
\(852\) 0 0
\(853\) 29792.9 1.19589 0.597943 0.801539i \(-0.295985\pi\)
0.597943 + 0.801539i \(0.295985\pi\)
\(854\) −32852.6 −1.31638
\(855\) 0 0
\(856\) −9400.27 −0.375344
\(857\) −21499.5 −0.856954 −0.428477 0.903553i \(-0.640950\pi\)
−0.428477 + 0.903553i \(0.640950\pi\)
\(858\) 0 0
\(859\) −30413.5 −1.20803 −0.604013 0.796974i \(-0.706433\pi\)
−0.604013 + 0.796974i \(0.706433\pi\)
\(860\) 5313.62 0.210689
\(861\) 0 0
\(862\) −42655.0 −1.68542
\(863\) 7021.71 0.276966 0.138483 0.990365i \(-0.455777\pi\)
0.138483 + 0.990365i \(0.455777\pi\)
\(864\) 0 0
\(865\) 14610.9 0.574317
\(866\) 22535.3 0.884272
\(867\) 0 0
\(868\) −708.475 −0.0277042
\(869\) −73331.3 −2.86260
\(870\) 0 0
\(871\) −2642.43 −0.102796
\(872\) −6989.30 −0.271431
\(873\) 0 0
\(874\) 8571.82 0.331746
\(875\) −1769.97 −0.0683838
\(876\) 0 0
\(877\) −12018.7 −0.462764 −0.231382 0.972863i \(-0.574325\pi\)
−0.231382 + 0.972863i \(0.574325\pi\)
\(878\) −46193.1 −1.77556
\(879\) 0 0
\(880\) −25273.1 −0.968130
\(881\) 21990.9 0.840968 0.420484 0.907300i \(-0.361860\pi\)
0.420484 + 0.907300i \(0.361860\pi\)
\(882\) 0 0
\(883\) −6023.11 −0.229551 −0.114776 0.993391i \(-0.536615\pi\)
−0.114776 + 0.993391i \(0.536615\pi\)
\(884\) 5394.13 0.205231
\(885\) 0 0
\(886\) 13670.6 0.518368
\(887\) −8578.09 −0.324717 −0.162359 0.986732i \(-0.551910\pi\)
−0.162359 + 0.986732i \(0.551910\pi\)
\(888\) 0 0
\(889\) −17079.8 −0.644362
\(890\) 11733.3 0.441910
\(891\) 0 0
\(892\) 3921.03 0.147181
\(893\) −6737.17 −0.252465
\(894\) 0 0
\(895\) 19777.5 0.738648
\(896\) 19183.5 0.715263
\(897\) 0 0
\(898\) 1896.87 0.0704893
\(899\) 876.041 0.0325001
\(900\) 0 0
\(901\) 61059.4 2.25770
\(902\) −69967.3 −2.58277
\(903\) 0 0
\(904\) −314.723 −0.0115791
\(905\) 13470.0 0.494760
\(906\) 0 0
\(907\) 9340.62 0.341952 0.170976 0.985275i \(-0.445308\pi\)
0.170976 + 0.985275i \(0.445308\pi\)
\(908\) 25136.0 0.918685
\(909\) 0 0
\(910\) 2630.95 0.0958407
\(911\) 16352.9 0.594727 0.297363 0.954764i \(-0.403893\pi\)
0.297363 + 0.954764i \(0.403893\pi\)
\(912\) 0 0
\(913\) 33380.2 1.20999
\(914\) −55740.5 −2.01721
\(915\) 0 0
\(916\) 93.6354 0.00337751
\(917\) 21692.8 0.781198
\(918\) 0 0
\(919\) 1321.13 0.0474211 0.0237105 0.999719i \(-0.492452\pi\)
0.0237105 + 0.999719i \(0.492452\pi\)
\(920\) 7419.46 0.265883
\(921\) 0 0
\(922\) 26003.6 0.928833
\(923\) −11362.0 −0.405182
\(924\) 0 0
\(925\) −2047.21 −0.0727697
\(926\) 9598.23 0.340623
\(927\) 0 0
\(928\) 16031.8 0.567102
\(929\) −38318.9 −1.35329 −0.676643 0.736311i \(-0.736566\pi\)
−0.676643 + 0.736311i \(0.736566\pi\)
\(930\) 0 0
\(931\) −2652.60 −0.0933787
\(932\) 15201.3 0.534265
\(933\) 0 0
\(934\) 31616.3 1.10762
\(935\) −34571.1 −1.20919
\(936\) 0 0
\(937\) −24307.8 −0.847494 −0.423747 0.905781i \(-0.639285\pi\)
−0.423747 + 0.905781i \(0.639285\pi\)
\(938\) 12865.3 0.447834
\(939\) 0 0
\(940\) 8645.77 0.299994
\(941\) −23022.5 −0.797568 −0.398784 0.917045i \(-0.630568\pi\)
−0.398784 + 0.917045i \(0.630568\pi\)
\(942\) 0 0
\(943\) 39607.3 1.36775
\(944\) 36705.3 1.26552
\(945\) 0 0
\(946\) 50620.8 1.73977
\(947\) −28324.8 −0.971945 −0.485972 0.873974i \(-0.661534\pi\)
−0.485972 + 0.873974i \(0.661534\pi\)
\(948\) 0 0
\(949\) −10754.8 −0.367876
\(950\) 1663.48 0.0568109
\(951\) 0 0
\(952\) 17713.8 0.603054
\(953\) 12986.4 0.441417 0.220709 0.975340i \(-0.429163\pi\)
0.220709 + 0.975340i \(0.429163\pi\)
\(954\) 0 0
\(955\) −4281.73 −0.145082
\(956\) −7807.08 −0.264120
\(957\) 0 0
\(958\) −44090.0 −1.48694
\(959\) −35004.6 −1.17868
\(960\) 0 0
\(961\) −29681.3 −0.996318
\(962\) 3043.06 0.101988
\(963\) 0 0
\(964\) 6758.90 0.225819
\(965\) 8676.07 0.289422
\(966\) 0 0
\(967\) −39251.9 −1.30533 −0.652666 0.757645i \(-0.726350\pi\)
−0.652666 + 0.757645i \(0.726350\pi\)
\(968\) −31355.0 −1.04110
\(969\) 0 0
\(970\) −20116.1 −0.665865
\(971\) 2404.97 0.0794843 0.0397422 0.999210i \(-0.487346\pi\)
0.0397422 + 0.999210i \(0.487346\pi\)
\(972\) 0 0
\(973\) 171.711 0.00565756
\(974\) 12655.1 0.416321
\(975\) 0 0
\(976\) 51532.9 1.69009
\(977\) 17640.0 0.577641 0.288820 0.957383i \(-0.406737\pi\)
0.288820 + 0.957383i \(0.406737\pi\)
\(978\) 0 0
\(979\) 41794.3 1.36440
\(980\) 3404.07 0.110958
\(981\) 0 0
\(982\) −27564.4 −0.895740
\(983\) 53222.8 1.72690 0.863451 0.504433i \(-0.168298\pi\)
0.863451 + 0.504433i \(0.168298\pi\)
\(984\) 0 0
\(985\) 14719.7 0.476152
\(986\) 32474.2 1.04887
\(987\) 0 0
\(988\) −924.535 −0.0297706
\(989\) −28655.5 −0.921327
\(990\) 0 0
\(991\) −33124.1 −1.06178 −0.530889 0.847441i \(-0.678142\pi\)
−0.530889 + 0.847441i \(0.678142\pi\)
\(992\) 2007.15 0.0642410
\(993\) 0 0
\(994\) 55318.5 1.76519
\(995\) 20860.3 0.664639
\(996\) 0 0
\(997\) −11736.9 −0.372830 −0.186415 0.982471i \(-0.559687\pi\)
−0.186415 + 0.982471i \(0.559687\pi\)
\(998\) −43476.1 −1.37897
\(999\) 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 405.4.a.k.1.5 6
3.2 odd 2 405.4.a.l.1.2 yes 6
5.4 even 2 2025.4.a.z.1.2 6
9.2 odd 6 405.4.e.w.271.5 12
9.4 even 3 405.4.e.x.136.2 12
9.5 odd 6 405.4.e.w.136.5 12
9.7 even 3 405.4.e.x.271.2 12
15.14 odd 2 2025.4.a.y.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.k.1.5 6 1.1 even 1 trivial
405.4.a.l.1.2 yes 6 3.2 odd 2
405.4.e.w.136.5 12 9.5 odd 6
405.4.e.w.271.5 12 9.2 odd 6
405.4.e.x.136.2 12 9.4 even 3
405.4.e.x.271.2 12 9.7 even 3
2025.4.a.y.1.5 6 15.14 odd 2
2025.4.a.z.1.2 6 5.4 even 2