Properties

Label 1024.4.a.n.1.7
Level $1024$
Weight $4$
Character 1024.1
Self dual yes
Analytic conductor $60.418$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.4179558459\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 36 x^{8} + 405 x^{6} - 1380 x^{4} + 420 x^{2} - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 16)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-3.82089\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +O(q^{10})\) \(q+2.80518 q^{3} +0.844070 q^{5} -29.0828 q^{7} -19.1310 q^{9} +17.1532 q^{11} +68.6824 q^{13} +2.36777 q^{15} -86.7193 q^{17} +77.5614 q^{19} -81.5826 q^{21} +70.2145 q^{23} -124.288 q^{25} -129.406 q^{27} +89.6641 q^{29} -8.86868 q^{31} +48.1178 q^{33} -24.5480 q^{35} +30.7089 q^{37} +192.667 q^{39} -153.274 q^{41} -171.050 q^{43} -16.1479 q^{45} +99.9792 q^{47} +502.812 q^{49} -243.263 q^{51} +550.315 q^{53} +14.4785 q^{55} +217.574 q^{57} +459.364 q^{59} -0.479693 q^{61} +556.383 q^{63} +57.9728 q^{65} +799.438 q^{67} +196.964 q^{69} +419.500 q^{71} +374.833 q^{73} -348.649 q^{75} -498.864 q^{77} +705.750 q^{79} +153.530 q^{81} +1339.39 q^{83} -73.1972 q^{85} +251.524 q^{87} +4.72918 q^{89} -1997.48 q^{91} -24.8782 q^{93} +65.4673 q^{95} +379.542 q^{97} -328.157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 28q^{7} + 54q^{9} + O(q^{10}) \) \( 10q + 28q^{7} + 54q^{9} + 124q^{15} + 4q^{17} + 276q^{23} + 50q^{25} + 368q^{31} - 4q^{33} + 732q^{39} + 944q^{47} - 94q^{49} + 1380q^{55} + 108q^{57} + 2628q^{63} - 492q^{65} + 3468q^{71} - 296q^{73} + 4416q^{79} - 482q^{81} + 6036q^{87} + 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.80518 0.539857 0.269929 0.962880i \(-0.413000\pi\)
0.269929 + 0.962880i \(0.413000\pi\)
\(4\) 0 0
\(5\) 0.844070 0.0754959 0.0377480 0.999287i \(-0.487982\pi\)
0.0377480 + 0.999287i \(0.487982\pi\)
\(6\) 0 0
\(7\) −29.0828 −1.57033 −0.785163 0.619289i \(-0.787421\pi\)
−0.785163 + 0.619289i \(0.787421\pi\)
\(8\) 0 0
\(9\) −19.1310 −0.708554
\(10\) 0 0
\(11\) 17.1532 0.470171 0.235086 0.971975i \(-0.424463\pi\)
0.235086 + 0.971975i \(0.424463\pi\)
\(12\) 0 0
\(13\) 68.6824 1.46531 0.732657 0.680598i \(-0.238280\pi\)
0.732657 + 0.680598i \(0.238280\pi\)
\(14\) 0 0
\(15\) 2.36777 0.0407570
\(16\) 0 0
\(17\) −86.7193 −1.23721 −0.618604 0.785703i \(-0.712301\pi\)
−0.618604 + 0.785703i \(0.712301\pi\)
\(18\) 0 0
\(19\) 77.5614 0.936517 0.468258 0.883592i \(-0.344882\pi\)
0.468258 + 0.883592i \(0.344882\pi\)
\(20\) 0 0
\(21\) −81.5826 −0.847752
\(22\) 0 0
\(23\) 70.2145 0.636554 0.318277 0.947998i \(-0.396896\pi\)
0.318277 + 0.947998i \(0.396896\pi\)
\(24\) 0 0
\(25\) −124.288 −0.994300
\(26\) 0 0
\(27\) −129.406 −0.922375
\(28\) 0 0
\(29\) 89.6641 0.574145 0.287072 0.957909i \(-0.407318\pi\)
0.287072 + 0.957909i \(0.407318\pi\)
\(30\) 0 0
\(31\) −8.86868 −0.0513826 −0.0256913 0.999670i \(-0.508179\pi\)
−0.0256913 + 0.999670i \(0.508179\pi\)
\(32\) 0 0
\(33\) 48.1178 0.253825
\(34\) 0 0
\(35\) −24.5480 −0.118553
\(36\) 0 0
\(37\) 30.7089 0.136446 0.0682231 0.997670i \(-0.478267\pi\)
0.0682231 + 0.997670i \(0.478267\pi\)
\(38\) 0 0
\(39\) 192.667 0.791060
\(40\) 0 0
\(41\) −153.274 −0.583840 −0.291920 0.956443i \(-0.594294\pi\)
−0.291920 + 0.956443i \(0.594294\pi\)
\(42\) 0 0
\(43\) −171.050 −0.606625 −0.303312 0.952891i \(-0.598093\pi\)
−0.303312 + 0.952891i \(0.598093\pi\)
\(44\) 0 0
\(45\) −16.1479 −0.0534930
\(46\) 0 0
\(47\) 99.9792 0.310286 0.155143 0.987892i \(-0.450416\pi\)
0.155143 + 0.987892i \(0.450416\pi\)
\(48\) 0 0
\(49\) 502.812 1.46592
\(50\) 0 0
\(51\) −243.263 −0.667915
\(52\) 0 0
\(53\) 550.315 1.42626 0.713128 0.701034i \(-0.247278\pi\)
0.713128 + 0.701034i \(0.247278\pi\)
\(54\) 0 0
\(55\) 14.4785 0.0354960
\(56\) 0 0
\(57\) 217.574 0.505585
\(58\) 0 0
\(59\) 459.364 1.01363 0.506814 0.862055i \(-0.330823\pi\)
0.506814 + 0.862055i \(0.330823\pi\)
\(60\) 0 0
\(61\) −0.479693 −0.00100686 −0.000503430 1.00000i \(-0.500160\pi\)
−0.000503430 1.00000i \(0.500160\pi\)
\(62\) 0 0
\(63\) 556.383 1.11266
\(64\) 0 0
\(65\) 57.9728 0.110625
\(66\) 0 0
\(67\) 799.438 1.45771 0.728857 0.684666i \(-0.240052\pi\)
0.728857 + 0.684666i \(0.240052\pi\)
\(68\) 0 0
\(69\) 196.964 0.343648
\(70\) 0 0
\(71\) 419.500 0.701205 0.350602 0.936524i \(-0.385977\pi\)
0.350602 + 0.936524i \(0.385977\pi\)
\(72\) 0 0
\(73\) 374.833 0.600971 0.300485 0.953786i \(-0.402851\pi\)
0.300485 + 0.953786i \(0.402851\pi\)
\(74\) 0 0
\(75\) −348.649 −0.536780
\(76\) 0 0
\(77\) −498.864 −0.738322
\(78\) 0 0
\(79\) 705.750 1.00510 0.502551 0.864547i \(-0.332395\pi\)
0.502551 + 0.864547i \(0.332395\pi\)
\(80\) 0 0
\(81\) 153.530 0.210604
\(82\) 0 0
\(83\) 1339.39 1.77129 0.885646 0.464361i \(-0.153716\pi\)
0.885646 + 0.464361i \(0.153716\pi\)
\(84\) 0 0
\(85\) −73.1972 −0.0934041
\(86\) 0 0
\(87\) 251.524 0.309956
\(88\) 0 0
\(89\) 4.72918 0.00563249 0.00281625 0.999996i \(-0.499104\pi\)
0.00281625 + 0.999996i \(0.499104\pi\)
\(90\) 0 0
\(91\) −1997.48 −2.30102
\(92\) 0 0
\(93\) −24.8782 −0.0277393
\(94\) 0 0
\(95\) 65.4673 0.0707032
\(96\) 0 0
\(97\) 379.542 0.397285 0.198643 0.980072i \(-0.436347\pi\)
0.198643 + 0.980072i \(0.436347\pi\)
\(98\) 0 0
\(99\) −328.157 −0.333142
\(100\) 0 0
\(101\) 552.964 0.544772 0.272386 0.962188i \(-0.412187\pi\)
0.272386 + 0.962188i \(0.412187\pi\)
\(102\) 0 0
\(103\) 307.935 0.294580 0.147290 0.989093i \(-0.452945\pi\)
0.147290 + 0.989093i \(0.452945\pi\)
\(104\) 0 0
\(105\) −68.8615 −0.0640018
\(106\) 0 0
\(107\) −850.801 −0.768692 −0.384346 0.923189i \(-0.625573\pi\)
−0.384346 + 0.923189i \(0.625573\pi\)
\(108\) 0 0
\(109\) 1341.93 1.17921 0.589605 0.807692i \(-0.299283\pi\)
0.589605 + 0.807692i \(0.299283\pi\)
\(110\) 0 0
\(111\) 86.1439 0.0736614
\(112\) 0 0
\(113\) −1824.02 −1.51849 −0.759244 0.650807i \(-0.774431\pi\)
−0.759244 + 0.650807i \(0.774431\pi\)
\(114\) 0 0
\(115\) 59.2660 0.0480573
\(116\) 0 0
\(117\) −1313.96 −1.03825
\(118\) 0 0
\(119\) 2522.04 1.94282
\(120\) 0 0
\(121\) −1036.77 −0.778939
\(122\) 0 0
\(123\) −429.962 −0.315190
\(124\) 0 0
\(125\) −210.416 −0.150562
\(126\) 0 0
\(127\) 988.748 0.690844 0.345422 0.938447i \(-0.387736\pi\)
0.345422 + 0.938447i \(0.387736\pi\)
\(128\) 0 0
\(129\) −479.826 −0.327491
\(130\) 0 0
\(131\) 1122.28 0.748505 0.374252 0.927327i \(-0.377899\pi\)
0.374252 + 0.927327i \(0.377899\pi\)
\(132\) 0 0
\(133\) −2255.71 −1.47064
\(134\) 0 0
\(135\) −109.227 −0.0696356
\(136\) 0 0
\(137\) −1595.30 −0.994856 −0.497428 0.867505i \(-0.665722\pi\)
−0.497428 + 0.867505i \(0.665722\pi\)
\(138\) 0 0
\(139\) −392.721 −0.239641 −0.119821 0.992796i \(-0.538232\pi\)
−0.119821 + 0.992796i \(0.538232\pi\)
\(140\) 0 0
\(141\) 280.460 0.167510
\(142\) 0 0
\(143\) 1178.12 0.688948
\(144\) 0 0
\(145\) 75.6828 0.0433456
\(146\) 0 0
\(147\) 1410.48 0.791390
\(148\) 0 0
\(149\) −839.014 −0.461307 −0.230653 0.973036i \(-0.574086\pi\)
−0.230653 + 0.973036i \(0.574086\pi\)
\(150\) 0 0
\(151\) −160.655 −0.0865821 −0.0432911 0.999063i \(-0.513784\pi\)
−0.0432911 + 0.999063i \(0.513784\pi\)
\(152\) 0 0
\(153\) 1659.02 0.876629
\(154\) 0 0
\(155\) −7.48579 −0.00387918
\(156\) 0 0
\(157\) −998.098 −0.507369 −0.253684 0.967287i \(-0.581642\pi\)
−0.253684 + 0.967287i \(0.581642\pi\)
\(158\) 0 0
\(159\) 1543.73 0.769975
\(160\) 0 0
\(161\) −2042.04 −0.999598
\(162\) 0 0
\(163\) 2648.31 1.27259 0.636294 0.771447i \(-0.280467\pi\)
0.636294 + 0.771447i \(0.280467\pi\)
\(164\) 0 0
\(165\) 40.6148 0.0191628
\(166\) 0 0
\(167\) 3852.19 1.78498 0.892490 0.451066i \(-0.148956\pi\)
0.892490 + 0.451066i \(0.148956\pi\)
\(168\) 0 0
\(169\) 2520.28 1.14715
\(170\) 0 0
\(171\) −1483.83 −0.663573
\(172\) 0 0
\(173\) −3713.17 −1.63183 −0.815917 0.578170i \(-0.803767\pi\)
−0.815917 + 0.578170i \(0.803767\pi\)
\(174\) 0 0
\(175\) 3614.64 1.56138
\(176\) 0 0
\(177\) 1288.60 0.547215
\(178\) 0 0
\(179\) 1749.01 0.730318 0.365159 0.930945i \(-0.381015\pi\)
0.365159 + 0.930945i \(0.381015\pi\)
\(180\) 0 0
\(181\) 2227.25 0.914640 0.457320 0.889302i \(-0.348809\pi\)
0.457320 + 0.889302i \(0.348809\pi\)
\(182\) 0 0
\(183\) −1.34563 −0.000543560 0
\(184\) 0 0
\(185\) 25.9204 0.0103011
\(186\) 0 0
\(187\) −1487.51 −0.581699
\(188\) 0 0
\(189\) 3763.48 1.44843
\(190\) 0 0
\(191\) 3585.92 1.35847 0.679236 0.733920i \(-0.262311\pi\)
0.679236 + 0.733920i \(0.262311\pi\)
\(192\) 0 0
\(193\) 523.601 0.195283 0.0976415 0.995222i \(-0.468870\pi\)
0.0976415 + 0.995222i \(0.468870\pi\)
\(194\) 0 0
\(195\) 162.624 0.0597218
\(196\) 0 0
\(197\) 1591.89 0.575724 0.287862 0.957672i \(-0.407056\pi\)
0.287862 + 0.957672i \(0.407056\pi\)
\(198\) 0 0
\(199\) 2312.48 0.823757 0.411878 0.911239i \(-0.364873\pi\)
0.411878 + 0.911239i \(0.364873\pi\)
\(200\) 0 0
\(201\) 2242.57 0.786957
\(202\) 0 0
\(203\) −2607.69 −0.901595
\(204\) 0 0
\(205\) −129.374 −0.0440775
\(206\) 0 0
\(207\) −1343.27 −0.451033
\(208\) 0 0
\(209\) 1330.43 0.440323
\(210\) 0 0
\(211\) −2006.19 −0.654558 −0.327279 0.944928i \(-0.606132\pi\)
−0.327279 + 0.944928i \(0.606132\pi\)
\(212\) 0 0
\(213\) 1176.77 0.378550
\(214\) 0 0
\(215\) −144.378 −0.0457977
\(216\) 0 0
\(217\) 257.926 0.0806875
\(218\) 0 0
\(219\) 1051.47 0.324438
\(220\) 0 0
\(221\) −5956.10 −1.81290
\(222\) 0 0
\(223\) −4315.08 −1.29578 −0.647890 0.761734i \(-0.724349\pi\)
−0.647890 + 0.761734i \(0.724349\pi\)
\(224\) 0 0
\(225\) 2377.74 0.704516
\(226\) 0 0
\(227\) 991.651 0.289948 0.144974 0.989435i \(-0.453690\pi\)
0.144974 + 0.989435i \(0.453690\pi\)
\(228\) 0 0
\(229\) 938.121 0.270711 0.135355 0.990797i \(-0.456782\pi\)
0.135355 + 0.990797i \(0.456782\pi\)
\(230\) 0 0
\(231\) −1399.40 −0.398588
\(232\) 0 0
\(233\) −3490.15 −0.981318 −0.490659 0.871352i \(-0.663244\pi\)
−0.490659 + 0.871352i \(0.663244\pi\)
\(234\) 0 0
\(235\) 84.3895 0.0234254
\(236\) 0 0
\(237\) 1979.76 0.542612
\(238\) 0 0
\(239\) −2950.43 −0.798525 −0.399263 0.916837i \(-0.630734\pi\)
−0.399263 + 0.916837i \(0.630734\pi\)
\(240\) 0 0
\(241\) 1128.96 0.301755 0.150877 0.988552i \(-0.451790\pi\)
0.150877 + 0.988552i \(0.451790\pi\)
\(242\) 0 0
\(243\) 3924.63 1.03607
\(244\) 0 0
\(245\) 424.409 0.110671
\(246\) 0 0
\(247\) 5327.11 1.37229
\(248\) 0 0
\(249\) 3757.23 0.956244
\(250\) 0 0
\(251\) −6536.30 −1.64370 −0.821848 0.569706i \(-0.807057\pi\)
−0.821848 + 0.569706i \(0.807057\pi\)
\(252\) 0 0
\(253\) 1204.40 0.299289
\(254\) 0 0
\(255\) −205.331 −0.0504249
\(256\) 0 0
\(257\) 610.977 0.148295 0.0741473 0.997247i \(-0.476377\pi\)
0.0741473 + 0.997247i \(0.476377\pi\)
\(258\) 0 0
\(259\) −893.101 −0.214265
\(260\) 0 0
\(261\) −1715.36 −0.406813
\(262\) 0 0
\(263\) −4973.57 −1.16610 −0.583048 0.812438i \(-0.698140\pi\)
−0.583048 + 0.812438i \(0.698140\pi\)
\(264\) 0 0
\(265\) 464.505 0.107677
\(266\) 0 0
\(267\) 13.2662 0.00304074
\(268\) 0 0
\(269\) 1327.12 0.300803 0.150401 0.988625i \(-0.451943\pi\)
0.150401 + 0.988625i \(0.451943\pi\)
\(270\) 0 0
\(271\) 4010.64 0.898999 0.449500 0.893280i \(-0.351602\pi\)
0.449500 + 0.893280i \(0.351602\pi\)
\(272\) 0 0
\(273\) −5603.29 −1.24222
\(274\) 0 0
\(275\) −2131.93 −0.467491
\(276\) 0 0
\(277\) −4999.23 −1.08439 −0.542193 0.840254i \(-0.682406\pi\)
−0.542193 + 0.840254i \(0.682406\pi\)
\(278\) 0 0
\(279\) 169.666 0.0364074
\(280\) 0 0
\(281\) −7468.35 −1.58550 −0.792748 0.609550i \(-0.791350\pi\)
−0.792748 + 0.609550i \(0.791350\pi\)
\(282\) 0 0
\(283\) 3180.88 0.668140 0.334070 0.942548i \(-0.391578\pi\)
0.334070 + 0.942548i \(0.391578\pi\)
\(284\) 0 0
\(285\) 183.648 0.0381696
\(286\) 0 0
\(287\) 4457.66 0.916819
\(288\) 0 0
\(289\) 2607.24 0.530682
\(290\) 0 0
\(291\) 1064.68 0.214477
\(292\) 0 0
\(293\) 5590.09 1.11460 0.557298 0.830312i \(-0.311838\pi\)
0.557298 + 0.830312i \(0.311838\pi\)
\(294\) 0 0
\(295\) 387.736 0.0765249
\(296\) 0 0
\(297\) −2219.72 −0.433674
\(298\) 0 0
\(299\) 4822.51 0.932752
\(300\) 0 0
\(301\) 4974.62 0.952599
\(302\) 0 0
\(303\) 1551.16 0.294099
\(304\) 0 0
\(305\) −0.404895 −7.60138e−5 0
\(306\) 0 0
\(307\) 5452.13 1.01358 0.506791 0.862069i \(-0.330832\pi\)
0.506791 + 0.862069i \(0.330832\pi\)
\(308\) 0 0
\(309\) 863.814 0.159031
\(310\) 0 0
\(311\) −5194.39 −0.947096 −0.473548 0.880768i \(-0.657027\pi\)
−0.473548 + 0.880768i \(0.657027\pi\)
\(312\) 0 0
\(313\) −4710.01 −0.850561 −0.425281 0.905062i \(-0.639825\pi\)
−0.425281 + 0.905062i \(0.639825\pi\)
\(314\) 0 0
\(315\) 469.626 0.0840014
\(316\) 0 0
\(317\) 8033.04 1.42328 0.711641 0.702544i \(-0.247952\pi\)
0.711641 + 0.702544i \(0.247952\pi\)
\(318\) 0 0
\(319\) 1538.03 0.269946
\(320\) 0 0
\(321\) −2386.65 −0.414984
\(322\) 0 0
\(323\) −6726.08 −1.15867
\(324\) 0 0
\(325\) −8536.37 −1.45696
\(326\) 0 0
\(327\) 3764.36 0.636605
\(328\) 0 0
\(329\) −2907.68 −0.487251
\(330\) 0 0
\(331\) 2567.92 0.426423 0.213211 0.977006i \(-0.431608\pi\)
0.213211 + 0.977006i \(0.431608\pi\)
\(332\) 0 0
\(333\) −587.490 −0.0966795
\(334\) 0 0
\(335\) 674.782 0.110052
\(336\) 0 0
\(337\) 2683.29 0.433733 0.216867 0.976201i \(-0.430416\pi\)
0.216867 + 0.976201i \(0.430416\pi\)
\(338\) 0 0
\(339\) −5116.69 −0.819766
\(340\) 0 0
\(341\) −152.126 −0.0241586
\(342\) 0 0
\(343\) −4647.79 −0.731653
\(344\) 0 0
\(345\) 166.252 0.0259440
\(346\) 0 0
\(347\) 7483.74 1.15778 0.578888 0.815407i \(-0.303487\pi\)
0.578888 + 0.815407i \(0.303487\pi\)
\(348\) 0 0
\(349\) 104.239 0.0159880 0.00799400 0.999968i \(-0.497455\pi\)
0.00799400 + 0.999968i \(0.497455\pi\)
\(350\) 0 0
\(351\) −8887.90 −1.35157
\(352\) 0 0
\(353\) −5067.25 −0.764030 −0.382015 0.924156i \(-0.624770\pi\)
−0.382015 + 0.924156i \(0.624770\pi\)
\(354\) 0 0
\(355\) 354.088 0.0529381
\(356\) 0 0
\(357\) 7074.79 1.04884
\(358\) 0 0
\(359\) −970.230 −0.142637 −0.0713186 0.997454i \(-0.522721\pi\)
−0.0713186 + 0.997454i \(0.522721\pi\)
\(360\) 0 0
\(361\) −843.224 −0.122937
\(362\) 0 0
\(363\) −2908.32 −0.420516
\(364\) 0 0
\(365\) 316.385 0.0453709
\(366\) 0 0
\(367\) −13451.4 −1.91323 −0.956617 0.291347i \(-0.905896\pi\)
−0.956617 + 0.291347i \(0.905896\pi\)
\(368\) 0 0
\(369\) 2932.29 0.413682
\(370\) 0 0
\(371\) −16004.7 −2.23969
\(372\) 0 0
\(373\) 8341.34 1.15790 0.578952 0.815362i \(-0.303462\pi\)
0.578952 + 0.815362i \(0.303462\pi\)
\(374\) 0 0
\(375\) −590.255 −0.0812817
\(376\) 0 0
\(377\) 6158.35 0.841302
\(378\) 0 0
\(379\) 6288.62 0.852308 0.426154 0.904651i \(-0.359868\pi\)
0.426154 + 0.904651i \(0.359868\pi\)
\(380\) 0 0
\(381\) 2773.62 0.372957
\(382\) 0 0
\(383\) −6417.68 −0.856209 −0.428105 0.903729i \(-0.640819\pi\)
−0.428105 + 0.903729i \(0.640819\pi\)
\(384\) 0 0
\(385\) −421.076 −0.0557403
\(386\) 0 0
\(387\) 3272.35 0.429827
\(388\) 0 0
\(389\) −9271.04 −1.20838 −0.604191 0.796840i \(-0.706503\pi\)
−0.604191 + 0.796840i \(0.706503\pi\)
\(390\) 0 0
\(391\) −6088.96 −0.787549
\(392\) 0 0
\(393\) 3148.20 0.404086
\(394\) 0 0
\(395\) 595.703 0.0758812
\(396\) 0 0
\(397\) −12590.0 −1.59163 −0.795814 0.605541i \(-0.792957\pi\)
−0.795814 + 0.605541i \(0.792957\pi\)
\(398\) 0 0
\(399\) −6327.66 −0.793933
\(400\) 0 0
\(401\) 6425.77 0.800218 0.400109 0.916468i \(-0.368972\pi\)
0.400109 + 0.916468i \(0.368972\pi\)
\(402\) 0 0
\(403\) −609.122 −0.0752917
\(404\) 0 0
\(405\) 129.590 0.0158997
\(406\) 0 0
\(407\) 526.755 0.0641530
\(408\) 0 0
\(409\) 12796.0 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(410\) 0 0
\(411\) −4475.09 −0.537080
\(412\) 0 0
\(413\) −13359.6 −1.59173
\(414\) 0 0
\(415\) 1130.54 0.133725
\(416\) 0 0
\(417\) −1101.65 −0.129372
\(418\) 0 0
\(419\) −9256.33 −1.07924 −0.539620 0.841909i \(-0.681432\pi\)
−0.539620 + 0.841909i \(0.681432\pi\)
\(420\) 0 0
\(421\) 9036.83 1.04615 0.523074 0.852287i \(-0.324785\pi\)
0.523074 + 0.852287i \(0.324785\pi\)
\(422\) 0 0
\(423\) −1912.70 −0.219855
\(424\) 0 0
\(425\) 10778.1 1.23016
\(426\) 0 0
\(427\) 13.9508 0.00158110
\(428\) 0 0
\(429\) 3304.85 0.371934
\(430\) 0 0
\(431\) 10639.3 1.18904 0.594519 0.804081i \(-0.297342\pi\)
0.594519 + 0.804081i \(0.297342\pi\)
\(432\) 0 0
\(433\) −3806.14 −0.422428 −0.211214 0.977440i \(-0.567742\pi\)
−0.211214 + 0.977440i \(0.567742\pi\)
\(434\) 0 0
\(435\) 212.304 0.0234004
\(436\) 0 0
\(437\) 5445.94 0.596143
\(438\) 0 0
\(439\) −14102.8 −1.53323 −0.766616 0.642106i \(-0.778061\pi\)
−0.766616 + 0.642106i \(0.778061\pi\)
\(440\) 0 0
\(441\) −9619.28 −1.03869
\(442\) 0 0
\(443\) 10836.3 1.16219 0.581095 0.813836i \(-0.302625\pi\)
0.581095 + 0.813836i \(0.302625\pi\)
\(444\) 0 0
\(445\) 3.99176 0.000425230 0
\(446\) 0 0
\(447\) −2353.58 −0.249040
\(448\) 0 0
\(449\) 13679.4 1.43779 0.718897 0.695117i \(-0.244647\pi\)
0.718897 + 0.695117i \(0.244647\pi\)
\(450\) 0 0
\(451\) −2629.14 −0.274505
\(452\) 0 0
\(453\) −450.666 −0.0467420
\(454\) 0 0
\(455\) −1686.01 −0.173718
\(456\) 0 0
\(457\) −7913.48 −0.810016 −0.405008 0.914313i \(-0.632731\pi\)
−0.405008 + 0.914313i \(0.632731\pi\)
\(458\) 0 0
\(459\) 11222.0 1.14117
\(460\) 0 0
\(461\) 820.548 0.0828996 0.0414498 0.999141i \(-0.486802\pi\)
0.0414498 + 0.999141i \(0.486802\pi\)
\(462\) 0 0
\(463\) −14236.5 −1.42899 −0.714497 0.699638i \(-0.753344\pi\)
−0.714497 + 0.699638i \(0.753344\pi\)
\(464\) 0 0
\(465\) −20.9990 −0.00209420
\(466\) 0 0
\(467\) 11801.0 1.16935 0.584674 0.811269i \(-0.301223\pi\)
0.584674 + 0.811269i \(0.301223\pi\)
\(468\) 0 0
\(469\) −23249.9 −2.28909
\(470\) 0 0
\(471\) −2799.85 −0.273907
\(472\) 0 0
\(473\) −2934.05 −0.285217
\(474\) 0 0
\(475\) −9639.92 −0.931179
\(476\) 0 0
\(477\) −10528.1 −1.01058
\(478\) 0 0
\(479\) −5563.77 −0.530720 −0.265360 0.964149i \(-0.585491\pi\)
−0.265360 + 0.964149i \(0.585491\pi\)
\(480\) 0 0
\(481\) 2109.16 0.199936
\(482\) 0 0
\(483\) −5728.29 −0.539640
\(484\) 0 0
\(485\) 320.360 0.0299934
\(486\) 0 0
\(487\) −18150.5 −1.68886 −0.844432 0.535662i \(-0.820062\pi\)
−0.844432 + 0.535662i \(0.820062\pi\)
\(488\) 0 0
\(489\) 7428.99 0.687015
\(490\) 0 0
\(491\) 16395.0 1.50692 0.753459 0.657495i \(-0.228384\pi\)
0.753459 + 0.657495i \(0.228384\pi\)
\(492\) 0 0
\(493\) −7775.61 −0.710336
\(494\) 0 0
\(495\) −276.988 −0.0251509
\(496\) 0 0
\(497\) −12200.3 −1.10112
\(498\) 0 0
\(499\) −4397.61 −0.394517 −0.197259 0.980351i \(-0.563204\pi\)
−0.197259 + 0.980351i \(0.563204\pi\)
\(500\) 0 0
\(501\) 10806.1 0.963635
\(502\) 0 0
\(503\) 6221.21 0.551471 0.275736 0.961233i \(-0.411079\pi\)
0.275736 + 0.961233i \(0.411079\pi\)
\(504\) 0 0
\(505\) 466.740 0.0411281
\(506\) 0 0
\(507\) 7069.83 0.619295
\(508\) 0 0
\(509\) 19516.8 1.69954 0.849770 0.527154i \(-0.176741\pi\)
0.849770 + 0.527154i \(0.176741\pi\)
\(510\) 0 0
\(511\) −10901.2 −0.943720
\(512\) 0 0
\(513\) −10036.9 −0.863820
\(514\) 0 0
\(515\) 259.919 0.0222396
\(516\) 0 0
\(517\) 1714.96 0.145888
\(518\) 0 0
\(519\) −10416.1 −0.880957
\(520\) 0 0
\(521\) 6874.63 0.578086 0.289043 0.957316i \(-0.406663\pi\)
0.289043 + 0.957316i \(0.406663\pi\)
\(522\) 0 0
\(523\) 3261.91 0.272722 0.136361 0.990659i \(-0.456459\pi\)
0.136361 + 0.990659i \(0.456459\pi\)
\(524\) 0 0
\(525\) 10139.7 0.842920
\(526\) 0 0
\(527\) 769.086 0.0635710
\(528\) 0 0
\(529\) −7236.92 −0.594799
\(530\) 0 0
\(531\) −8788.08 −0.718211
\(532\) 0 0
\(533\) −10527.3 −0.855509
\(534\) 0 0
\(535\) −718.136 −0.0580332
\(536\) 0 0
\(537\) 4906.28 0.394267
\(538\) 0 0
\(539\) 8624.83 0.689235
\(540\) 0 0
\(541\) −18724.1 −1.48801 −0.744005 0.668174i \(-0.767076\pi\)
−0.744005 + 0.668174i \(0.767076\pi\)
\(542\) 0 0
\(543\) 6247.82 0.493775
\(544\) 0 0
\(545\) 1132.69 0.0890256
\(546\) 0 0
\(547\) −18768.5 −1.46706 −0.733530 0.679657i \(-0.762129\pi\)
−0.733530 + 0.679657i \(0.762129\pi\)
\(548\) 0 0
\(549\) 9.17700 0.000713415 0
\(550\) 0 0
\(551\) 6954.47 0.537696
\(552\) 0 0
\(553\) −20525.2 −1.57834
\(554\) 0 0
\(555\) 72.7115 0.00556114
\(556\) 0 0
\(557\) 12021.7 0.914497 0.457249 0.889339i \(-0.348835\pi\)
0.457249 + 0.889339i \(0.348835\pi\)
\(558\) 0 0
\(559\) −11748.1 −0.888896
\(560\) 0 0
\(561\) −4172.74 −0.314034
\(562\) 0 0
\(563\) −24504.4 −1.83434 −0.917172 0.398491i \(-0.869534\pi\)
−0.917172 + 0.398491i \(0.869534\pi\)
\(564\) 0 0
\(565\) −1539.60 −0.114640
\(566\) 0 0
\(567\) −4465.09 −0.330716
\(568\) 0 0
\(569\) 8998.54 0.662985 0.331492 0.943458i \(-0.392448\pi\)
0.331492 + 0.943458i \(0.392448\pi\)
\(570\) 0 0
\(571\) 13928.9 1.02085 0.510427 0.859921i \(-0.329487\pi\)
0.510427 + 0.859921i \(0.329487\pi\)
\(572\) 0 0
\(573\) 10059.1 0.733380
\(574\) 0 0
\(575\) −8726.79 −0.632926
\(576\) 0 0
\(577\) −20584.4 −1.48516 −0.742580 0.669757i \(-0.766398\pi\)
−0.742580 + 0.669757i \(0.766398\pi\)
\(578\) 0 0
\(579\) 1468.79 0.105425
\(580\) 0 0
\(581\) −38953.3 −2.78151
\(582\) 0 0
\(583\) 9439.66 0.670585
\(584\) 0 0
\(585\) −1109.08 −0.0783840
\(586\) 0 0
\(587\) −3658.25 −0.257227 −0.128613 0.991695i \(-0.541053\pi\)
−0.128613 + 0.991695i \(0.541053\pi\)
\(588\) 0 0
\(589\) −687.867 −0.0481207
\(590\) 0 0
\(591\) 4465.54 0.310809
\(592\) 0 0
\(593\) 6035.89 0.417984 0.208992 0.977917i \(-0.432982\pi\)
0.208992 + 0.977917i \(0.432982\pi\)
\(594\) 0 0
\(595\) 2128.78 0.146675
\(596\) 0 0
\(597\) 6486.93 0.444711
\(598\) 0 0
\(599\) 5427.20 0.370199 0.185100 0.982720i \(-0.440739\pi\)
0.185100 + 0.982720i \(0.440739\pi\)
\(600\) 0 0
\(601\) −17725.7 −1.20307 −0.601535 0.798847i \(-0.705444\pi\)
−0.601535 + 0.798847i \(0.705444\pi\)
\(602\) 0 0
\(603\) −15294.0 −1.03287
\(604\) 0 0
\(605\) −875.105 −0.0588067
\(606\) 0 0
\(607\) 13487.6 0.901884 0.450942 0.892553i \(-0.351088\pi\)
0.450942 + 0.892553i \(0.351088\pi\)
\(608\) 0 0
\(609\) −7315.03 −0.486732
\(610\) 0 0
\(611\) 6866.81 0.454667
\(612\) 0 0
\(613\) 23830.0 1.57012 0.785062 0.619417i \(-0.212631\pi\)
0.785062 + 0.619417i \(0.212631\pi\)
\(614\) 0 0
\(615\) −362.918 −0.0237956
\(616\) 0 0
\(617\) −535.243 −0.0349239 −0.0174620 0.999848i \(-0.505559\pi\)
−0.0174620 + 0.999848i \(0.505559\pi\)
\(618\) 0 0
\(619\) −27847.2 −1.80820 −0.904098 0.427324i \(-0.859456\pi\)
−0.904098 + 0.427324i \(0.859456\pi\)
\(620\) 0 0
\(621\) −9086.16 −0.587142
\(622\) 0 0
\(623\) −137.538 −0.00884485
\(624\) 0 0
\(625\) 15358.3 0.982934
\(626\) 0 0
\(627\) 3732.08 0.237711
\(628\) 0 0
\(629\) −2663.05 −0.168812
\(630\) 0 0
\(631\) −11880.2 −0.749511 −0.374755 0.927124i \(-0.622273\pi\)
−0.374755 + 0.927124i \(0.622273\pi\)
\(632\) 0 0
\(633\) −5627.72 −0.353368
\(634\) 0 0
\(635\) 834.573 0.0521560
\(636\) 0 0
\(637\) 34534.4 2.14804
\(638\) 0 0
\(639\) −8025.45 −0.496842
\(640\) 0 0
\(641\) 18341.0 1.13015 0.565074 0.825040i \(-0.308848\pi\)
0.565074 + 0.825040i \(0.308848\pi\)
\(642\) 0 0
\(643\) 9936.56 0.609424 0.304712 0.952445i \(-0.401440\pi\)
0.304712 + 0.952445i \(0.401440\pi\)
\(644\) 0 0
\(645\) −405.007 −0.0247242
\(646\) 0 0
\(647\) 21429.7 1.30215 0.651073 0.759015i \(-0.274319\pi\)
0.651073 + 0.759015i \(0.274319\pi\)
\(648\) 0 0
\(649\) 7879.56 0.476579
\(650\) 0 0
\(651\) 723.530 0.0435597
\(652\) 0 0
\(653\) −12277.1 −0.735742 −0.367871 0.929877i \(-0.619913\pi\)
−0.367871 + 0.929877i \(0.619913\pi\)
\(654\) 0 0
\(655\) 947.284 0.0565091
\(656\) 0 0
\(657\) −7170.92 −0.425821
\(658\) 0 0
\(659\) 5871.25 0.347058 0.173529 0.984829i \(-0.444483\pi\)
0.173529 + 0.984829i \(0.444483\pi\)
\(660\) 0 0
\(661\) 17308.5 1.01849 0.509246 0.860621i \(-0.329924\pi\)
0.509246 + 0.860621i \(0.329924\pi\)
\(662\) 0 0
\(663\) −16707.9 −0.978706
\(664\) 0 0
\(665\) −1903.98 −0.111027
\(666\) 0 0
\(667\) 6295.72 0.365474
\(668\) 0 0
\(669\) −12104.6 −0.699536
\(670\) 0 0
\(671\) −8.22827 −0.000473396 0
\(672\) 0 0
\(673\) 6528.62 0.373937 0.186969 0.982366i \(-0.440134\pi\)
0.186969 + 0.982366i \(0.440134\pi\)
\(674\) 0 0
\(675\) 16083.5 0.917118
\(676\) 0 0
\(677\) 20110.9 1.14169 0.570847 0.821057i \(-0.306615\pi\)
0.570847 + 0.821057i \(0.306615\pi\)
\(678\) 0 0
\(679\) −11038.2 −0.623867
\(680\) 0 0
\(681\) 2781.76 0.156530
\(682\) 0 0
\(683\) 30291.8 1.69705 0.848524 0.529157i \(-0.177492\pi\)
0.848524 + 0.529157i \(0.177492\pi\)
\(684\) 0 0
\(685\) −1346.54 −0.0751076
\(686\) 0 0
\(687\) 2631.60 0.146145
\(688\) 0 0
\(689\) 37797.0 2.08991
\(690\) 0 0
\(691\) 23387.1 1.28753 0.643767 0.765222i \(-0.277371\pi\)
0.643767 + 0.765222i \(0.277371\pi\)
\(692\) 0 0
\(693\) 9543.74 0.523141
\(694\) 0 0
\(695\) −331.484 −0.0180920
\(696\) 0 0
\(697\) 13291.8 0.722331
\(698\) 0 0
\(699\) −9790.49 −0.529772
\(700\) 0 0
\(701\) −4280.14 −0.230612 −0.115306 0.993330i \(-0.536785\pi\)
−0.115306 + 0.993330i \(0.536785\pi\)
\(702\) 0 0
\(703\) 2381.82 0.127784
\(704\) 0 0
\(705\) 236.728 0.0126464
\(706\) 0 0
\(707\) −16081.8 −0.855470
\(708\) 0 0
\(709\) −5630.18 −0.298231 −0.149115 0.988820i \(-0.547643\pi\)
−0.149115 + 0.988820i \(0.547643\pi\)
\(710\) 0 0
\(711\) −13501.7 −0.712170
\(712\) 0 0
\(713\) −622.710 −0.0327078
\(714\) 0 0
\(715\) 994.419 0.0520128
\(716\) 0 0
\(717\) −8276.49 −0.431090
\(718\) 0 0
\(719\) 5682.25 0.294732 0.147366 0.989082i \(-0.452921\pi\)
0.147366 + 0.989082i \(0.452921\pi\)
\(720\) 0 0
\(721\) −8955.64 −0.462587
\(722\) 0 0
\(723\) 3166.94 0.162904
\(724\) 0 0
\(725\) −11144.1 −0.570872
\(726\) 0 0
\(727\) 18883.0 0.963317 0.481658 0.876359i \(-0.340035\pi\)
0.481658 + 0.876359i \(0.340035\pi\)
\(728\) 0 0
\(729\) 6863.99 0.348727
\(730\) 0 0
\(731\) 14833.3 0.750521
\(732\) 0 0
\(733\) −35302.3 −1.77888 −0.889441 0.457049i \(-0.848906\pi\)
−0.889441 + 0.457049i \(0.848906\pi\)
\(734\) 0 0
\(735\) 1190.54 0.0597467
\(736\) 0 0
\(737\) 13712.9 0.685375
\(738\) 0 0
\(739\) 8771.48 0.436623 0.218311 0.975879i \(-0.429945\pi\)
0.218311 + 0.975879i \(0.429945\pi\)
\(740\) 0 0
\(741\) 14943.5 0.740841
\(742\) 0 0
\(743\) 30.9140 0.00152641 0.000763205 1.00000i \(-0.499757\pi\)
0.000763205 1.00000i \(0.499757\pi\)
\(744\) 0 0
\(745\) −708.187 −0.0348268
\(746\) 0 0
\(747\) −25623.8 −1.25506
\(748\) 0 0
\(749\) 24743.7 1.20710
\(750\) 0 0
\(751\) 16318.5 0.792905 0.396453 0.918055i \(-0.370241\pi\)
0.396453 + 0.918055i \(0.370241\pi\)
\(752\) 0 0
\(753\) −18335.5 −0.887361
\(754\) 0 0
\(755\) −135.604 −0.00653660
\(756\) 0 0
\(757\) 13936.5 0.669129 0.334564 0.942373i \(-0.391411\pi\)
0.334564 + 0.942373i \(0.391411\pi\)
\(758\) 0 0
\(759\) 3378.57 0.161573
\(760\) 0 0
\(761\) −3823.42 −0.182127 −0.0910637 0.995845i \(-0.529027\pi\)
−0.0910637 + 0.995845i \(0.529027\pi\)
\(762\) 0 0
\(763\) −39027.2 −1.85174
\(764\) 0 0
\(765\) 1400.33 0.0661819
\(766\) 0 0
\(767\) 31550.2 1.48528
\(768\) 0 0
\(769\) 31689.1 1.48601 0.743003 0.669288i \(-0.233401\pi\)
0.743003 + 0.669288i \(0.233401\pi\)
\(770\) 0 0
\(771\) 1713.90 0.0800578
\(772\) 0 0
\(773\) 1846.74 0.0859282 0.0429641 0.999077i \(-0.486320\pi\)
0.0429641 + 0.999077i \(0.486320\pi\)
\(774\) 0 0
\(775\) 1102.27 0.0510898
\(776\) 0 0
\(777\) −2505.31 −0.115672
\(778\) 0 0
\(779\) −11888.2 −0.546776
\(780\) 0 0
\(781\) 7195.77 0.329686
\(782\) 0 0
\(783\) −11603.0 −0.529577
\(784\) 0 0
\(785\) −842.465 −0.0383043
\(786\) 0 0
\(787\) −20364.0 −0.922363 −0.461181 0.887306i \(-0.652574\pi\)
−0.461181 + 0.887306i \(0.652574\pi\)
\(788\) 0 0
\(789\) −13951.7 −0.629525
\(790\) 0 0
\(791\) 53047.6 2.38452
\(792\) 0 0
\(793\) −32.9465 −0.00147537
\(794\) 0 0
\(795\) 1303.02 0.0581300
\(796\) 0 0
\(797\) −7880.27 −0.350230 −0.175115 0.984548i \(-0.556030\pi\)
−0.175115 + 0.984548i \(0.556030\pi\)
\(798\) 0 0
\(799\) −8670.13 −0.383889
\(800\) 0 0
\(801\) −90.4737 −0.00399093
\(802\) 0 0
\(803\) 6429.58 0.282559
\(804\) 0 0
\(805\) −1723.62 −0.0754656
\(806\) 0 0
\(807\) 3722.81 0.162390
\(808\) 0 0
\(809\) 5081.28 0.220826 0.110413 0.993886i \(-0.464783\pi\)
0.110413 + 0.993886i \(0.464783\pi\)
\(810\) 0 0
\(811\) 6354.33 0.275130 0.137565 0.990493i \(-0.456072\pi\)
0.137565 + 0.990493i \(0.456072\pi\)
\(812\) 0 0
\(813\) 11250.6 0.485331
\(814\) 0 0
\(815\) 2235.36 0.0960752
\(816\) 0 0
\(817\) −13266.9 −0.568114
\(818\) 0 0
\(819\) 38213.7 1.63040
\(820\) 0 0
\(821\) 2991.83 0.127181 0.0635904 0.997976i \(-0.479745\pi\)
0.0635904 + 0.997976i \(0.479745\pi\)
\(822\) 0 0
\(823\) 24432.6 1.03483 0.517416 0.855734i \(-0.326894\pi\)
0.517416 + 0.855734i \(0.326894\pi\)
\(824\) 0 0
\(825\) −5980.44 −0.252378
\(826\) 0 0
\(827\) −29771.2 −1.25181 −0.625904 0.779900i \(-0.715270\pi\)
−0.625904 + 0.779900i \(0.715270\pi\)
\(828\) 0 0
\(829\) 35980.5 1.50742 0.753711 0.657206i \(-0.228262\pi\)
0.753711 + 0.657206i \(0.228262\pi\)
\(830\) 0 0
\(831\) −14023.7 −0.585413
\(832\) 0 0
\(833\) −43603.5 −1.81365
\(834\) 0 0
\(835\) 3251.52 0.134759
\(836\) 0 0
\(837\) 1147.66 0.0473941
\(838\) 0 0
\(839\) 18757.2 0.771837 0.385919 0.922533i \(-0.373885\pi\)
0.385919 + 0.922533i \(0.373885\pi\)
\(840\) 0 0
\(841\) −16349.4 −0.670358
\(842\) 0 0
\(843\) −20950.1 −0.855941
\(844\) 0 0
\(845\) 2127.29 0.0866048
\(846\) 0 0
\(847\) 30152.2 1.22319
\(848\) 0 0
\(849\) 8922.94 0.360700
\(850\) 0 0
\(851\) 2156.21 0.0868553
\(852\) 0 0
\(853\) −8626.94 −0.346285 −0.173142 0.984897i \(-0.555392\pi\)
−0.173142 + 0.984897i \(0.555392\pi\)
\(854\) 0 0
\(855\) −1252.45 −0.0500971
\(856\) 0 0
\(857\) −2079.04 −0.0828687 −0.0414344 0.999141i \(-0.513193\pi\)
−0.0414344 + 0.999141i \(0.513193\pi\)
\(858\) 0 0
\(859\) −11740.7 −0.466343 −0.233171 0.972436i \(-0.574910\pi\)
−0.233171 + 0.972436i \(0.574910\pi\)
\(860\) 0 0
\(861\) 12504.5 0.494951
\(862\) 0 0
\(863\) 43830.8 1.72887 0.864436 0.502743i \(-0.167676\pi\)
0.864436 + 0.502743i \(0.167676\pi\)
\(864\) 0 0
\(865\) −3134.18 −0.123197
\(866\) 0 0
\(867\) 7313.78 0.286493
\(868\) 0 0
\(869\) 12105.9 0.472570
\(870\) 0 0
\(871\) 54907.3 2.13601
\(872\) 0 0
\(873\) −7261.01 −0.281498
\(874\) 0 0
\(875\) 6119.50 0.236431
\(876\) 0 0
\(877\) 3628.71 0.139718 0.0698591 0.997557i \(-0.477745\pi\)
0.0698591 + 0.997557i \(0.477745\pi\)
\(878\) 0 0
\(879\) 15681.2 0.601723
\(880\) 0 0
\(881\) −26429.8 −1.01072 −0.505360 0.862909i \(-0.668640\pi\)
−0.505360 + 0.862909i \(0.668640\pi\)
\(882\) 0 0
\(883\) 29286.1 1.11614 0.558071 0.829793i \(-0.311542\pi\)
0.558071 + 0.829793i \(0.311542\pi\)
\(884\) 0 0
\(885\) 1087.67 0.0413125
\(886\) 0 0
\(887\) −22350.0 −0.846042 −0.423021 0.906120i \(-0.639030\pi\)
−0.423021 + 0.906120i \(0.639030\pi\)
\(888\) 0 0
\(889\) −28755.6 −1.08485
\(890\) 0 0
\(891\) 2633.53 0.0990197
\(892\) 0 0
\(893\) 7754.53 0.290588
\(894\) 0 0
\(895\) 1476.28 0.0551360
\(896\) 0 0
\(897\) 13528.0 0.503553
\(898\) 0 0
\(899\) −795.202 −0.0295011
\(900\) 0 0
\(901\) −47723.0 −1.76457
\(902\) 0 0
\(903\) 13954.7 0.514267
\(904\) 0 0
\(905\) 1879.95 0.0690516
\(906\) 0 0
\(907\) −5779.79 −0.211593 −0.105796 0.994388i \(-0.533739\pi\)
−0.105796 + 0.994388i \(0.533739\pi\)
\(908\) 0 0
\(909\) −10578.7 −0.386000
\(910\) 0 0
\(911\) 20024.6 0.728259 0.364130 0.931348i \(-0.381367\pi\)
0.364130 + 0.931348i \(0.381367\pi\)
\(912\) 0 0
\(913\) 22974.8 0.832810
\(914\) 0 0
\(915\) −1.13580 −4.10366e−5 0
\(916\) 0 0
\(917\) −32639.1 −1.17540
\(918\) 0 0
\(919\) −26977.7 −0.968349 −0.484175 0.874971i \(-0.660880\pi\)
−0.484175 + 0.874971i \(0.660880\pi\)
\(920\) 0 0
\(921\) 15294.2 0.547189
\(922\) 0 0
\(923\) 28812.3 1.02748
\(924\) 0 0
\(925\) −3816.73 −0.135668
\(926\) 0 0
\(927\) −5891.10 −0.208726
\(928\) 0 0
\(929\) −34371.4 −1.21387 −0.606937 0.794750i \(-0.707602\pi\)
−0.606937 + 0.794750i \(0.707602\pi\)
\(930\) 0 0
\(931\) 38998.8 1.37286
\(932\) 0 0
\(933\) −14571.2 −0.511297
\(934\) 0 0
\(935\) −1255.57 −0.0439159
\(936\) 0 0
\(937\) 32901.8 1.14712 0.573561 0.819163i \(-0.305561\pi\)
0.573561 + 0.819163i \(0.305561\pi\)
\(938\) 0 0
\(939\) −13212.4 −0.459181
\(940\) 0 0
\(941\) −47208.7 −1.63545 −0.817726 0.575607i \(-0.804766\pi\)
−0.817726 + 0.575607i \(0.804766\pi\)
\(942\) 0 0
\(943\) −10762.1 −0.371646
\(944\) 0 0
\(945\) 3176.65 0.109351
\(946\) 0 0
\(947\) −19250.2 −0.660557 −0.330279 0.943884i \(-0.607143\pi\)
−0.330279 + 0.943884i \(0.607143\pi\)
\(948\) 0 0
\(949\) 25744.4 0.880611
\(950\) 0 0
\(951\) 22534.1 0.768369
\(952\) 0 0
\(953\) −4572.49 −0.155422 −0.0777112 0.996976i \(-0.524761\pi\)
−0.0777112 + 0.996976i \(0.524761\pi\)
\(954\) 0 0
\(955\) 3026.77 0.102559
\(956\) 0 0
\(957\) 4314.44 0.145732
\(958\) 0 0
\(959\) 46395.7 1.56225
\(960\) 0 0
\(961\) −29712.3 −0.997360
\(962\) 0 0
\(963\) 16276.7 0.544660
\(964\) 0 0
\(965\) 441.956 0.0147431
\(966\) 0 0
\(967\) −33060.9 −1.09945 −0.549725 0.835346i \(-0.685268\pi\)
−0.549725 + 0.835346i \(0.685268\pi\)
\(968\) 0 0
\(969\) −18867.9 −0.625514
\(970\) 0 0
\(971\) −21995.9 −0.726964 −0.363482 0.931601i \(-0.618412\pi\)
−0.363482 + 0.931601i \(0.618412\pi\)
\(972\) 0 0
\(973\) 11421.4 0.376315
\(974\) 0 0
\(975\) −23946.1 −0.786551
\(976\) 0 0
\(977\) −1241.54 −0.0406555 −0.0203277 0.999793i \(-0.506471\pi\)
−0.0203277 + 0.999793i \(0.506471\pi\)
\(978\) 0 0
\(979\) 81.1205 0.00264823
\(980\) 0 0
\(981\) −25672.5 −0.835534
\(982\) 0 0
\(983\) −10797.7 −0.350349 −0.175175 0.984537i \(-0.556049\pi\)
−0.175175 + 0.984537i \(0.556049\pi\)
\(984\) 0 0
\(985\) 1343.67 0.0434648
\(986\) 0 0
\(987\) −8156.56 −0.263046
\(988\) 0 0
\(989\) −12010.2 −0.386149
\(990\) 0 0
\(991\) −10204.8 −0.327112 −0.163556 0.986534i \(-0.552296\pi\)
−0.163556 + 0.986534i \(0.552296\pi\)
\(992\) 0 0
\(993\) 7203.49 0.230207
\(994\) 0 0
\(995\) 1951.90 0.0621903
\(996\) 0 0
\(997\) −21279.2 −0.675947 −0.337973 0.941156i \(-0.609741\pi\)
−0.337973 + 0.941156i \(0.609741\pi\)
\(998\) 0 0
\(999\) −3973.90 −0.125855
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 1024.4.a.n.1.7 10
4.3 odd 2 1024.4.a.m.1.4 10
8.3 odd 2 1024.4.a.m.1.7 10
8.5 even 2 inner 1024.4.a.n.1.4 10
16.3 odd 4 1024.4.b.k.513.4 10
16.5 even 4 1024.4.b.j.513.4 10
16.11 odd 4 1024.4.b.k.513.7 10
16.13 even 4 1024.4.b.j.513.7 10
32.3 odd 8 64.4.e.a.17.2 10
32.5 even 8 128.4.e.b.97.2 10
32.11 odd 8 64.4.e.a.49.2 10
32.13 even 8 128.4.e.b.33.2 10
32.19 odd 8 128.4.e.a.33.4 10
32.21 even 8 16.4.e.a.5.4 10
32.27 odd 8 128.4.e.a.97.4 10
32.29 even 8 16.4.e.a.13.4 yes 10
96.11 even 8 576.4.k.a.433.3 10
96.29 odd 8 144.4.k.a.109.2 10
96.35 even 8 576.4.k.a.145.3 10
96.53 odd 8 144.4.k.a.37.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.4 10 32.21 even 8
16.4.e.a.13.4 yes 10 32.29 even 8
64.4.e.a.17.2 10 32.3 odd 8
64.4.e.a.49.2 10 32.11 odd 8
128.4.e.a.33.4 10 32.19 odd 8
128.4.e.a.97.4 10 32.27 odd 8
128.4.e.b.33.2 10 32.13 even 8
128.4.e.b.97.2 10 32.5 even 8
144.4.k.a.37.2 10 96.53 odd 8
144.4.k.a.109.2 10 96.29 odd 8
576.4.k.a.145.3 10 96.35 even 8
576.4.k.a.433.3 10 96.11 even 8
1024.4.a.m.1.4 10 4.3 odd 2
1024.4.a.m.1.7 10 8.3 odd 2
1024.4.a.n.1.4 10 8.5 even 2 inner
1024.4.a.n.1.7 10 1.1 even 1 trivial
1024.4.b.j.513.4 10 16.5 even 4
1024.4.b.j.513.7 10 16.13 even 4
1024.4.b.k.513.4 10 16.3 odd 4
1024.4.b.k.513.7 10 16.11 odd 4