Properties

Label 1856.4.a.l.1.2
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +O(q^{10})\) \(q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +52.3485 q^{11} +5.40408 q^{13} -33.4495 q^{15} +17.1918 q^{17} -44.2020 q^{19} -95.1918 q^{21} -205.060 q^{23} -30.9694 q^{25} -145.227 q^{27} -29.0000 q^{29} +299.994 q^{31} +180.576 q^{33} +267.596 q^{35} -29.7980 q^{37} +18.6413 q^{39} -43.9592 q^{41} -64.8230 q^{43} +146.434 q^{45} -499.499 q^{47} +418.535 q^{49} +59.3031 q^{51} +351.627 q^{53} -507.620 q^{55} -152.474 q^{57} -522.372 q^{59} -484.606 q^{61} +416.727 q^{63} -52.4031 q^{65} +504.990 q^{67} -707.353 q^{69} +481.283 q^{71} +3.11019 q^{73} -106.829 q^{75} -1444.60 q^{77} -1043.27 q^{79} -93.2316 q^{81} +1007.08 q^{83} -166.708 q^{85} -100.035 q^{87} -295.637 q^{89} -149.131 q^{91} +1034.83 q^{93} +428.624 q^{95} +428.949 q^{97} -790.515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + 90 q^{11} + 50 q^{13} - 62 q^{15} - 44 q^{17} - 108 q^{19} - 112 q^{21} - 28 q^{23} + 232 q^{25} - 70 q^{27} - 58 q^{29} + 66 q^{31} + 126 q^{33} + 496 q^{35} - 40 q^{37} - 46 q^{39} + 304 q^{41} + 130 q^{43} - 344 q^{45} - 514 q^{47} + 210 q^{49} + 148 q^{51} + 958 q^{53} + 234 q^{55} - 60 q^{57} + 180 q^{59} - 1028 q^{61} + 128 q^{63} + 826 q^{65} + 912 q^{67} - 964 q^{69} + 796 q^{71} - 856 q^{73} - 488 q^{75} - 1008 q^{77} - 318 q^{79} + 470 q^{81} + 1828 q^{83} - 1372 q^{85} - 58 q^{87} - 944 q^{89} + 368 q^{91} + 1374 q^{93} - 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.44949 0.663855 0.331927 0.943305i \(-0.392301\pi\)
0.331927 + 0.943305i \(0.392301\pi\)
\(4\) 0 0
\(5\) −9.69694 −0.867321 −0.433660 0.901076i \(-0.642778\pi\)
−0.433660 + 0.901076i \(0.642778\pi\)
\(6\) 0 0
\(7\) −27.5959 −1.49004 −0.745020 0.667042i \(-0.767560\pi\)
−0.745020 + 0.667042i \(0.767560\pi\)
\(8\) 0 0
\(9\) −15.1010 −0.559297
\(10\) 0 0
\(11\) 52.3485 1.43488 0.717439 0.696621i \(-0.245314\pi\)
0.717439 + 0.696621i \(0.245314\pi\)
\(12\) 0 0
\(13\) 5.40408 0.115294 0.0576470 0.998337i \(-0.481640\pi\)
0.0576470 + 0.998337i \(0.481640\pi\)
\(14\) 0 0
\(15\) −33.4495 −0.575775
\(16\) 0 0
\(17\) 17.1918 0.245273 0.122636 0.992452i \(-0.460865\pi\)
0.122636 + 0.992452i \(0.460865\pi\)
\(18\) 0 0
\(19\) −44.2020 −0.533718 −0.266859 0.963736i \(-0.585986\pi\)
−0.266859 + 0.963736i \(0.585986\pi\)
\(20\) 0 0
\(21\) −95.1918 −0.989170
\(22\) 0 0
\(23\) −205.060 −1.85904 −0.929522 0.368767i \(-0.879780\pi\)
−0.929522 + 0.368767i \(0.879780\pi\)
\(24\) 0 0
\(25\) −30.9694 −0.247755
\(26\) 0 0
\(27\) −145.227 −1.03515
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 299.994 1.73808 0.869042 0.494739i \(-0.164736\pi\)
0.869042 + 0.494739i \(0.164736\pi\)
\(32\) 0 0
\(33\) 180.576 0.952550
\(34\) 0 0
\(35\) 267.596 1.29234
\(36\) 0 0
\(37\) −29.7980 −0.132399 −0.0661994 0.997806i \(-0.521087\pi\)
−0.0661994 + 0.997806i \(0.521087\pi\)
\(38\) 0 0
\(39\) 18.6413 0.0765385
\(40\) 0 0
\(41\) −43.9592 −0.167446 −0.0837228 0.996489i \(-0.526681\pi\)
−0.0837228 + 0.996489i \(0.526681\pi\)
\(42\) 0 0
\(43\) −64.8230 −0.229893 −0.114947 0.993372i \(-0.536670\pi\)
−0.114947 + 0.993372i \(0.536670\pi\)
\(44\) 0 0
\(45\) 146.434 0.485090
\(46\) 0 0
\(47\) −499.499 −1.55020 −0.775101 0.631837i \(-0.782301\pi\)
−0.775101 + 0.631837i \(0.782301\pi\)
\(48\) 0 0
\(49\) 418.535 1.22022
\(50\) 0 0
\(51\) 59.3031 0.162825
\(52\) 0 0
\(53\) 351.627 0.911314 0.455657 0.890156i \(-0.349404\pi\)
0.455657 + 0.890156i \(0.349404\pi\)
\(54\) 0 0
\(55\) −507.620 −1.24450
\(56\) 0 0
\(57\) −152.474 −0.354311
\(58\) 0 0
\(59\) −522.372 −1.15266 −0.576331 0.817216i \(-0.695516\pi\)
−0.576331 + 0.817216i \(0.695516\pi\)
\(60\) 0 0
\(61\) −484.606 −1.01717 −0.508586 0.861011i \(-0.669832\pi\)
−0.508586 + 0.861011i \(0.669832\pi\)
\(62\) 0 0
\(63\) 416.727 0.833375
\(64\) 0 0
\(65\) −52.4031 −0.0999969
\(66\) 0 0
\(67\) 504.990 0.920811 0.460405 0.887709i \(-0.347704\pi\)
0.460405 + 0.887709i \(0.347704\pi\)
\(68\) 0 0
\(69\) −707.353 −1.23413
\(70\) 0 0
\(71\) 481.283 0.804475 0.402238 0.915535i \(-0.368233\pi\)
0.402238 + 0.915535i \(0.368233\pi\)
\(72\) 0 0
\(73\) 3.11019 0.00498659 0.00249329 0.999997i \(-0.499206\pi\)
0.00249329 + 0.999997i \(0.499206\pi\)
\(74\) 0 0
\(75\) −106.829 −0.164473
\(76\) 0 0
\(77\) −1444.60 −2.13802
\(78\) 0 0
\(79\) −1043.27 −1.48578 −0.742890 0.669414i \(-0.766545\pi\)
−0.742890 + 0.669414i \(0.766545\pi\)
\(80\) 0 0
\(81\) −93.2316 −0.127890
\(82\) 0 0
\(83\) 1007.08 1.33182 0.665912 0.746030i \(-0.268042\pi\)
0.665912 + 0.746030i \(0.268042\pi\)
\(84\) 0 0
\(85\) −166.708 −0.212730
\(86\) 0 0
\(87\) −100.035 −0.123275
\(88\) 0 0
\(89\) −295.637 −0.352106 −0.176053 0.984381i \(-0.556333\pi\)
−0.176053 + 0.984381i \(0.556333\pi\)
\(90\) 0 0
\(91\) −149.131 −0.171793
\(92\) 0 0
\(93\) 1034.83 1.15383
\(94\) 0 0
\(95\) 428.624 0.462905
\(96\) 0 0
\(97\) 428.949 0.449002 0.224501 0.974474i \(-0.427925\pi\)
0.224501 + 0.974474i \(0.427925\pi\)
\(98\) 0 0
\(99\) −790.515 −0.802523
\(100\) 0 0
\(101\) 1212.16 1.19420 0.597101 0.802166i \(-0.296319\pi\)
0.597101 + 0.802166i \(0.296319\pi\)
\(102\) 0 0
\(103\) 1032.07 0.987310 0.493655 0.869658i \(-0.335661\pi\)
0.493655 + 0.869658i \(0.335661\pi\)
\(104\) 0 0
\(105\) 923.069 0.857927
\(106\) 0 0
\(107\) 1176.06 1.06256 0.531280 0.847196i \(-0.321711\pi\)
0.531280 + 0.847196i \(0.321711\pi\)
\(108\) 0 0
\(109\) 2167.86 1.90498 0.952491 0.304568i \(-0.0985121\pi\)
0.952491 + 0.304568i \(0.0985121\pi\)
\(110\) 0 0
\(111\) −102.788 −0.0878935
\(112\) 0 0
\(113\) 1623.65 1.35169 0.675843 0.737046i \(-0.263780\pi\)
0.675843 + 0.737046i \(0.263780\pi\)
\(114\) 0 0
\(115\) 1988.46 1.61239
\(116\) 0 0
\(117\) −81.6072 −0.0644836
\(118\) 0 0
\(119\) −474.424 −0.365466
\(120\) 0 0
\(121\) 1409.36 1.05887
\(122\) 0 0
\(123\) −151.637 −0.111160
\(124\) 0 0
\(125\) 1512.43 1.08220
\(126\) 0 0
\(127\) 333.473 0.233000 0.116500 0.993191i \(-0.462833\pi\)
0.116500 + 0.993191i \(0.462833\pi\)
\(128\) 0 0
\(129\) −223.606 −0.152616
\(130\) 0 0
\(131\) 933.492 0.622592 0.311296 0.950313i \(-0.399237\pi\)
0.311296 + 0.950313i \(0.399237\pi\)
\(132\) 0 0
\(133\) 1219.80 0.795261
\(134\) 0 0
\(135\) 1408.26 0.897804
\(136\) 0 0
\(137\) −1090.32 −0.679944 −0.339972 0.940436i \(-0.610418\pi\)
−0.339972 + 0.940436i \(0.610418\pi\)
\(138\) 0 0
\(139\) 1924.60 1.17440 0.587202 0.809440i \(-0.300229\pi\)
0.587202 + 0.809440i \(0.300229\pi\)
\(140\) 0 0
\(141\) −1723.02 −1.02911
\(142\) 0 0
\(143\) 282.895 0.165433
\(144\) 0 0
\(145\) 281.211 0.161057
\(146\) 0 0
\(147\) 1443.73 0.810047
\(148\) 0 0
\(149\) 1703.62 0.936687 0.468343 0.883546i \(-0.344851\pi\)
0.468343 + 0.883546i \(0.344851\pi\)
\(150\) 0 0
\(151\) 1082.81 0.583560 0.291780 0.956486i \(-0.405753\pi\)
0.291780 + 0.956486i \(0.405753\pi\)
\(152\) 0 0
\(153\) −259.614 −0.137180
\(154\) 0 0
\(155\) −2909.03 −1.50748
\(156\) 0 0
\(157\) −2407.90 −1.22402 −0.612010 0.790850i \(-0.709639\pi\)
−0.612010 + 0.790850i \(0.709639\pi\)
\(158\) 0 0
\(159\) 1212.93 0.604980
\(160\) 0 0
\(161\) 5658.82 2.77005
\(162\) 0 0
\(163\) −1061.59 −0.510123 −0.255061 0.966925i \(-0.582096\pi\)
−0.255061 + 0.966925i \(0.582096\pi\)
\(164\) 0 0
\(165\) −1751.03 −0.826166
\(166\) 0 0
\(167\) 1315.63 0.609618 0.304809 0.952414i \(-0.401407\pi\)
0.304809 + 0.952414i \(0.401407\pi\)
\(168\) 0 0
\(169\) −2167.80 −0.986707
\(170\) 0 0
\(171\) 667.496 0.298507
\(172\) 0 0
\(173\) 653.800 0.287327 0.143663 0.989627i \(-0.454112\pi\)
0.143663 + 0.989627i \(0.454112\pi\)
\(174\) 0 0
\(175\) 854.629 0.369165
\(176\) 0 0
\(177\) −1801.92 −0.765200
\(178\) 0 0
\(179\) −2055.28 −0.858208 −0.429104 0.903255i \(-0.641171\pi\)
−0.429104 + 0.903255i \(0.641171\pi\)
\(180\) 0 0
\(181\) −3398.36 −1.39557 −0.697785 0.716307i \(-0.745831\pi\)
−0.697785 + 0.716307i \(0.745831\pi\)
\(182\) 0 0
\(183\) −1671.64 −0.675254
\(184\) 0 0
\(185\) 288.949 0.114832
\(186\) 0 0
\(187\) 899.966 0.351936
\(188\) 0 0
\(189\) 4007.67 1.54241
\(190\) 0 0
\(191\) 4853.92 1.83883 0.919417 0.393285i \(-0.128661\pi\)
0.919417 + 0.393285i \(0.128661\pi\)
\(192\) 0 0
\(193\) −4877.24 −1.81902 −0.909512 0.415677i \(-0.863545\pi\)
−0.909512 + 0.415677i \(0.863545\pi\)
\(194\) 0 0
\(195\) −180.764 −0.0663834
\(196\) 0 0
\(197\) 1031.42 0.373022 0.186511 0.982453i \(-0.440282\pi\)
0.186511 + 0.982453i \(0.440282\pi\)
\(198\) 0 0
\(199\) −1167.70 −0.415960 −0.207980 0.978133i \(-0.566689\pi\)
−0.207980 + 0.978133i \(0.566689\pi\)
\(200\) 0 0
\(201\) 1741.96 0.611284
\(202\) 0 0
\(203\) 800.282 0.276693
\(204\) 0 0
\(205\) 426.269 0.145229
\(206\) 0 0
\(207\) 3096.62 1.03976
\(208\) 0 0
\(209\) −2313.91 −0.765820
\(210\) 0 0
\(211\) −794.341 −0.259169 −0.129585 0.991568i \(-0.541364\pi\)
−0.129585 + 0.991568i \(0.541364\pi\)
\(212\) 0 0
\(213\) 1660.18 0.534055
\(214\) 0 0
\(215\) 628.584 0.199391
\(216\) 0 0
\(217\) −8278.62 −2.58981
\(218\) 0 0
\(219\) 10.7286 0.00331037
\(220\) 0 0
\(221\) 92.9061 0.0282785
\(222\) 0 0
\(223\) 5136.08 1.54232 0.771161 0.636641i \(-0.219676\pi\)
0.771161 + 0.636641i \(0.219676\pi\)
\(224\) 0 0
\(225\) 467.669 0.138569
\(226\) 0 0
\(227\) 5032.84 1.47155 0.735774 0.677228i \(-0.236819\pi\)
0.735774 + 0.677228i \(0.236819\pi\)
\(228\) 0 0
\(229\) 6213.12 1.79290 0.896451 0.443142i \(-0.146136\pi\)
0.896451 + 0.443142i \(0.146136\pi\)
\(230\) 0 0
\(231\) −4983.15 −1.41934
\(232\) 0 0
\(233\) −3117.59 −0.876566 −0.438283 0.898837i \(-0.644413\pi\)
−0.438283 + 0.898837i \(0.644413\pi\)
\(234\) 0 0
\(235\) 4843.62 1.34452
\(236\) 0 0
\(237\) −3598.73 −0.986342
\(238\) 0 0
\(239\) 7193.09 1.94679 0.973394 0.229136i \(-0.0735902\pi\)
0.973394 + 0.229136i \(0.0735902\pi\)
\(240\) 0 0
\(241\) −837.347 −0.223810 −0.111905 0.993719i \(-0.535695\pi\)
−0.111905 + 0.993719i \(0.535695\pi\)
\(242\) 0 0
\(243\) 3599.53 0.950246
\(244\) 0 0
\(245\) −4058.51 −1.05832
\(246\) 0 0
\(247\) −238.871 −0.0615345
\(248\) 0 0
\(249\) 3473.91 0.884138
\(250\) 0 0
\(251\) −4306.56 −1.08298 −0.541489 0.840708i \(-0.682139\pi\)
−0.541489 + 0.840708i \(0.682139\pi\)
\(252\) 0 0
\(253\) −10734.6 −2.66750
\(254\) 0 0
\(255\) −575.058 −0.141222
\(256\) 0 0
\(257\) 401.238 0.0973873 0.0486936 0.998814i \(-0.484494\pi\)
0.0486936 + 0.998814i \(0.484494\pi\)
\(258\) 0 0
\(259\) 822.302 0.197279
\(260\) 0 0
\(261\) 437.930 0.103859
\(262\) 0 0
\(263\) −2682.72 −0.628988 −0.314494 0.949260i \(-0.601835\pi\)
−0.314494 + 0.949260i \(0.601835\pi\)
\(264\) 0 0
\(265\) −3409.70 −0.790401
\(266\) 0 0
\(267\) −1019.80 −0.233747
\(268\) 0 0
\(269\) 4732.38 1.07263 0.536316 0.844017i \(-0.319816\pi\)
0.536316 + 0.844017i \(0.319816\pi\)
\(270\) 0 0
\(271\) 4529.66 1.01534 0.507670 0.861552i \(-0.330507\pi\)
0.507670 + 0.861552i \(0.330507\pi\)
\(272\) 0 0
\(273\) −514.424 −0.114045
\(274\) 0 0
\(275\) −1621.20 −0.355498
\(276\) 0 0
\(277\) −7158.69 −1.55279 −0.776397 0.630245i \(-0.782955\pi\)
−0.776397 + 0.630245i \(0.782955\pi\)
\(278\) 0 0
\(279\) −4530.22 −0.972105
\(280\) 0 0
\(281\) −5769.75 −1.22489 −0.612446 0.790512i \(-0.709814\pi\)
−0.612446 + 0.790512i \(0.709814\pi\)
\(282\) 0 0
\(283\) 3815.25 0.801389 0.400694 0.916212i \(-0.368769\pi\)
0.400694 + 0.916212i \(0.368769\pi\)
\(284\) 0 0
\(285\) 1478.54 0.307301
\(286\) 0 0
\(287\) 1213.09 0.249501
\(288\) 0 0
\(289\) −4617.44 −0.939841
\(290\) 0 0
\(291\) 1479.66 0.298072
\(292\) 0 0
\(293\) −5416.66 −1.08002 −0.540008 0.841660i \(-0.681579\pi\)
−0.540008 + 0.841660i \(0.681579\pi\)
\(294\) 0 0
\(295\) 5065.41 0.999728
\(296\) 0 0
\(297\) −7602.41 −1.48531
\(298\) 0 0
\(299\) −1108.16 −0.214337
\(300\) 0 0
\(301\) 1788.85 0.342550
\(302\) 0 0
\(303\) 4181.33 0.792776
\(304\) 0 0
\(305\) 4699.20 0.882214
\(306\) 0 0
\(307\) −4929.06 −0.916339 −0.458169 0.888865i \(-0.651495\pi\)
−0.458169 + 0.888865i \(0.651495\pi\)
\(308\) 0 0
\(309\) 3560.12 0.655430
\(310\) 0 0
\(311\) 1117.93 0.203832 0.101916 0.994793i \(-0.467503\pi\)
0.101916 + 0.994793i \(0.467503\pi\)
\(312\) 0 0
\(313\) 2032.21 0.366988 0.183494 0.983021i \(-0.441259\pi\)
0.183494 + 0.983021i \(0.441259\pi\)
\(314\) 0 0
\(315\) −4040.97 −0.722803
\(316\) 0 0
\(317\) −2051.75 −0.363526 −0.181763 0.983342i \(-0.558180\pi\)
−0.181763 + 0.983342i \(0.558180\pi\)
\(318\) 0 0
\(319\) −1518.11 −0.266450
\(320\) 0 0
\(321\) 4056.80 0.705385
\(322\) 0 0
\(323\) −759.914 −0.130906
\(324\) 0 0
\(325\) −167.361 −0.0285647
\(326\) 0 0
\(327\) 7478.00 1.26463
\(328\) 0 0
\(329\) 13784.1 2.30986
\(330\) 0 0
\(331\) 9652.92 1.60294 0.801469 0.598036i \(-0.204052\pi\)
0.801469 + 0.598036i \(0.204052\pi\)
\(332\) 0 0
\(333\) 449.980 0.0740502
\(334\) 0 0
\(335\) −4896.85 −0.798638
\(336\) 0 0
\(337\) 4277.57 0.691437 0.345719 0.938338i \(-0.387635\pi\)
0.345719 + 0.938338i \(0.387635\pi\)
\(338\) 0 0
\(339\) 5600.77 0.897322
\(340\) 0 0
\(341\) 15704.2 2.49394
\(342\) 0 0
\(343\) −2084.45 −0.328133
\(344\) 0 0
\(345\) 6859.16 1.07039
\(346\) 0 0
\(347\) 9475.83 1.46596 0.732982 0.680248i \(-0.238128\pi\)
0.732982 + 0.680248i \(0.238128\pi\)
\(348\) 0 0
\(349\) −2968.57 −0.455313 −0.227656 0.973742i \(-0.573106\pi\)
−0.227656 + 0.973742i \(0.573106\pi\)
\(350\) 0 0
\(351\) −784.819 −0.119346
\(352\) 0 0
\(353\) 3969.16 0.598461 0.299231 0.954181i \(-0.403270\pi\)
0.299231 + 0.954181i \(0.403270\pi\)
\(354\) 0 0
\(355\) −4666.97 −0.697738
\(356\) 0 0
\(357\) −1636.52 −0.242616
\(358\) 0 0
\(359\) −3503.03 −0.514994 −0.257497 0.966279i \(-0.582898\pi\)
−0.257497 + 0.966279i \(0.582898\pi\)
\(360\) 0 0
\(361\) −4905.18 −0.715145
\(362\) 0 0
\(363\) 4861.58 0.702939
\(364\) 0 0
\(365\) −30.1594 −0.00432497
\(366\) 0 0
\(367\) −8253.53 −1.17393 −0.586963 0.809614i \(-0.699677\pi\)
−0.586963 + 0.809614i \(0.699677\pi\)
\(368\) 0 0
\(369\) 663.828 0.0936518
\(370\) 0 0
\(371\) −9703.46 −1.35789
\(372\) 0 0
\(373\) −1253.95 −0.174067 −0.0870335 0.996205i \(-0.527739\pi\)
−0.0870335 + 0.996205i \(0.527739\pi\)
\(374\) 0 0
\(375\) 5217.10 0.718426
\(376\) 0 0
\(377\) −156.718 −0.0214096
\(378\) 0 0
\(379\) 6966.52 0.944185 0.472092 0.881549i \(-0.343499\pi\)
0.472092 + 0.881549i \(0.343499\pi\)
\(380\) 0 0
\(381\) 1150.31 0.154678
\(382\) 0 0
\(383\) −9821.84 −1.31037 −0.655186 0.755467i \(-0.727410\pi\)
−0.655186 + 0.755467i \(0.727410\pi\)
\(384\) 0 0
\(385\) 14008.2 1.85435
\(386\) 0 0
\(387\) 978.893 0.128579
\(388\) 0 0
\(389\) −3942.96 −0.513923 −0.256962 0.966422i \(-0.582721\pi\)
−0.256962 + 0.966422i \(0.582721\pi\)
\(390\) 0 0
\(391\) −3525.36 −0.455972
\(392\) 0 0
\(393\) 3220.07 0.413311
\(394\) 0 0
\(395\) 10116.5 1.28865
\(396\) 0 0
\(397\) −8725.50 −1.10307 −0.551537 0.834150i \(-0.685959\pi\)
−0.551537 + 0.834150i \(0.685959\pi\)
\(398\) 0 0
\(399\) 4207.67 0.527938
\(400\) 0 0
\(401\) 3986.86 0.496494 0.248247 0.968697i \(-0.420146\pi\)
0.248247 + 0.968697i \(0.420146\pi\)
\(402\) 0 0
\(403\) 1621.19 0.200391
\(404\) 0 0
\(405\) 904.061 0.110921
\(406\) 0 0
\(407\) −1559.88 −0.189976
\(408\) 0 0
\(409\) −704.847 −0.0852138 −0.0426069 0.999092i \(-0.513566\pi\)
−0.0426069 + 0.999092i \(0.513566\pi\)
\(410\) 0 0
\(411\) −3761.05 −0.451384
\(412\) 0 0
\(413\) 14415.3 1.71751
\(414\) 0 0
\(415\) −9765.60 −1.15512
\(416\) 0 0
\(417\) 6638.88 0.779634
\(418\) 0 0
\(419\) −6705.33 −0.781807 −0.390903 0.920432i \(-0.627837\pi\)
−0.390903 + 0.920432i \(0.627837\pi\)
\(420\) 0 0
\(421\) −8996.55 −1.04148 −0.520742 0.853714i \(-0.674345\pi\)
−0.520742 + 0.853714i \(0.674345\pi\)
\(422\) 0 0
\(423\) 7542.95 0.867023
\(424\) 0 0
\(425\) −532.421 −0.0607675
\(426\) 0 0
\(427\) 13373.2 1.51563
\(428\) 0 0
\(429\) 975.845 0.109823
\(430\) 0 0
\(431\) 5383.16 0.601619 0.300810 0.953684i \(-0.402743\pi\)
0.300810 + 0.953684i \(0.402743\pi\)
\(432\) 0 0
\(433\) −6901.64 −0.765985 −0.382993 0.923751i \(-0.625107\pi\)
−0.382993 + 0.923751i \(0.625107\pi\)
\(434\) 0 0
\(435\) 970.035 0.106919
\(436\) 0 0
\(437\) 9064.08 0.992205
\(438\) 0 0
\(439\) 12119.4 1.31760 0.658802 0.752316i \(-0.271063\pi\)
0.658802 + 0.752316i \(0.271063\pi\)
\(440\) 0 0
\(441\) −6320.30 −0.682464
\(442\) 0 0
\(443\) −5229.40 −0.560849 −0.280425 0.959876i \(-0.590475\pi\)
−0.280425 + 0.959876i \(0.590475\pi\)
\(444\) 0 0
\(445\) 2866.77 0.305389
\(446\) 0 0
\(447\) 5876.64 0.621824
\(448\) 0 0
\(449\) −10220.3 −1.07422 −0.537112 0.843511i \(-0.680485\pi\)
−0.537112 + 0.843511i \(0.680485\pi\)
\(450\) 0 0
\(451\) −2301.20 −0.240264
\(452\) 0 0
\(453\) 3735.13 0.387399
\(454\) 0 0
\(455\) 1446.11 0.148999
\(456\) 0 0
\(457\) 12030.6 1.23143 0.615717 0.787967i \(-0.288866\pi\)
0.615717 + 0.787967i \(0.288866\pi\)
\(458\) 0 0
\(459\) −2496.72 −0.253893
\(460\) 0 0
\(461\) −1063.24 −0.107418 −0.0537092 0.998557i \(-0.517104\pi\)
−0.0537092 + 0.998557i \(0.517104\pi\)
\(462\) 0 0
\(463\) 15498.8 1.55570 0.777851 0.628449i \(-0.216310\pi\)
0.777851 + 0.628449i \(0.216310\pi\)
\(464\) 0 0
\(465\) −10034.7 −1.00074
\(466\) 0 0
\(467\) −13215.6 −1.30952 −0.654759 0.755838i \(-0.727230\pi\)
−0.654759 + 0.755838i \(0.727230\pi\)
\(468\) 0 0
\(469\) −13935.7 −1.37204
\(470\) 0 0
\(471\) −8306.01 −0.812571
\(472\) 0 0
\(473\) −3393.38 −0.329869
\(474\) 0 0
\(475\) 1368.91 0.132231
\(476\) 0 0
\(477\) −5309.92 −0.509695
\(478\) 0 0
\(479\) −19058.5 −1.81796 −0.908981 0.416837i \(-0.863139\pi\)
−0.908981 + 0.416837i \(0.863139\pi\)
\(480\) 0 0
\(481\) −161.031 −0.0152648
\(482\) 0 0
\(483\) 19520.1 1.83891
\(484\) 0 0
\(485\) −4159.49 −0.389428
\(486\) 0 0
\(487\) −487.214 −0.0453343 −0.0226671 0.999743i \(-0.507216\pi\)
−0.0226671 + 0.999743i \(0.507216\pi\)
\(488\) 0 0
\(489\) −3661.94 −0.338647
\(490\) 0 0
\(491\) 6462.89 0.594025 0.297013 0.954874i \(-0.404010\pi\)
0.297013 + 0.954874i \(0.404010\pi\)
\(492\) 0 0
\(493\) −498.563 −0.0455460
\(494\) 0 0
\(495\) 7665.58 0.696045
\(496\) 0 0
\(497\) −13281.4 −1.19870
\(498\) 0 0
\(499\) 7947.79 0.713010 0.356505 0.934293i \(-0.383968\pi\)
0.356505 + 0.934293i \(0.383968\pi\)
\(500\) 0 0
\(501\) 4538.24 0.404698
\(502\) 0 0
\(503\) −2307.81 −0.204573 −0.102287 0.994755i \(-0.532616\pi\)
−0.102287 + 0.994755i \(0.532616\pi\)
\(504\) 0 0
\(505\) −11754.2 −1.03576
\(506\) 0 0
\(507\) −7477.79 −0.655030
\(508\) 0 0
\(509\) 16496.2 1.43651 0.718253 0.695782i \(-0.244942\pi\)
0.718253 + 0.695782i \(0.244942\pi\)
\(510\) 0 0
\(511\) −85.8287 −0.00743021
\(512\) 0 0
\(513\) 6419.33 0.552476
\(514\) 0 0
\(515\) −10007.9 −0.856314
\(516\) 0 0
\(517\) −26148.0 −2.22435
\(518\) 0 0
\(519\) 2255.28 0.190743
\(520\) 0 0
\(521\) −10386.5 −0.873398 −0.436699 0.899608i \(-0.643853\pi\)
−0.436699 + 0.899608i \(0.643853\pi\)
\(522\) 0 0
\(523\) 918.174 0.0767666 0.0383833 0.999263i \(-0.487779\pi\)
0.0383833 + 0.999263i \(0.487779\pi\)
\(524\) 0 0
\(525\) 2948.03 0.245072
\(526\) 0 0
\(527\) 5157.45 0.426304
\(528\) 0 0
\(529\) 29882.7 2.45604
\(530\) 0 0
\(531\) 7888.36 0.644681
\(532\) 0 0
\(533\) −237.559 −0.0193055
\(534\) 0 0
\(535\) −11404.2 −0.921580
\(536\) 0 0
\(537\) −7089.68 −0.569725
\(538\) 0 0
\(539\) 21909.7 1.75086
\(540\) 0 0
\(541\) 12860.2 1.02200 0.511001 0.859580i \(-0.329275\pi\)
0.511001 + 0.859580i \(0.329275\pi\)
\(542\) 0 0
\(543\) −11722.6 −0.926456
\(544\) 0 0
\(545\) −21021.6 −1.65223
\(546\) 0 0
\(547\) −4471.84 −0.349547 −0.174774 0.984609i \(-0.555919\pi\)
−0.174774 + 0.984609i \(0.555919\pi\)
\(548\) 0 0
\(549\) 7318.05 0.568901
\(550\) 0 0
\(551\) 1281.86 0.0991090
\(552\) 0 0
\(553\) 28789.9 2.21387
\(554\) 0 0
\(555\) 996.727 0.0762319
\(556\) 0 0
\(557\) −2649.40 −0.201542 −0.100771 0.994910i \(-0.532131\pi\)
−0.100771 + 0.994910i \(0.532131\pi\)
\(558\) 0 0
\(559\) −350.309 −0.0265053
\(560\) 0 0
\(561\) 3104.42 0.233634
\(562\) 0 0
\(563\) −12615.8 −0.944395 −0.472198 0.881493i \(-0.656539\pi\)
−0.472198 + 0.881493i \(0.656539\pi\)
\(564\) 0 0
\(565\) −15744.5 −1.17234
\(566\) 0 0
\(567\) 2572.81 0.190561
\(568\) 0 0
\(569\) 617.443 0.0454913 0.0227456 0.999741i \(-0.492759\pi\)
0.0227456 + 0.999741i \(0.492759\pi\)
\(570\) 0 0
\(571\) −8005.50 −0.586725 −0.293362 0.956001i \(-0.594774\pi\)
−0.293362 + 0.956001i \(0.594774\pi\)
\(572\) 0 0
\(573\) 16743.5 1.22072
\(574\) 0 0
\(575\) 6350.59 0.460588
\(576\) 0 0
\(577\) 11761.1 0.848565 0.424282 0.905530i \(-0.360526\pi\)
0.424282 + 0.905530i \(0.360526\pi\)
\(578\) 0 0
\(579\) −16824.0 −1.20757
\(580\) 0 0
\(581\) −27791.3 −1.98447
\(582\) 0 0
\(583\) 18407.1 1.30762
\(584\) 0 0
\(585\) 791.340 0.0559280
\(586\) 0 0
\(587\) 1211.03 0.0851528 0.0425764 0.999093i \(-0.486443\pi\)
0.0425764 + 0.999093i \(0.486443\pi\)
\(588\) 0 0
\(589\) −13260.4 −0.927646
\(590\) 0 0
\(591\) 3557.86 0.247633
\(592\) 0 0
\(593\) 459.376 0.0318117 0.0159058 0.999873i \(-0.494937\pi\)
0.0159058 + 0.999873i \(0.494937\pi\)
\(594\) 0 0
\(595\) 4600.47 0.316976
\(596\) 0 0
\(597\) −4027.97 −0.276137
\(598\) 0 0
\(599\) 16686.2 1.13820 0.569099 0.822269i \(-0.307292\pi\)
0.569099 + 0.822269i \(0.307292\pi\)
\(600\) 0 0
\(601\) 11625.9 0.789072 0.394536 0.918881i \(-0.370905\pi\)
0.394536 + 0.918881i \(0.370905\pi\)
\(602\) 0 0
\(603\) −7625.86 −0.515007
\(604\) 0 0
\(605\) −13666.5 −0.918384
\(606\) 0 0
\(607\) −21556.9 −1.44146 −0.720731 0.693215i \(-0.756194\pi\)
−0.720731 + 0.693215i \(0.756194\pi\)
\(608\) 0 0
\(609\) 2760.56 0.183684
\(610\) 0 0
\(611\) −2699.34 −0.178729
\(612\) 0 0
\(613\) 19273.2 1.26988 0.634940 0.772561i \(-0.281025\pi\)
0.634940 + 0.772561i \(0.281025\pi\)
\(614\) 0 0
\(615\) 1470.41 0.0964110
\(616\) 0 0
\(617\) −3270.70 −0.213409 −0.106704 0.994291i \(-0.534030\pi\)
−0.106704 + 0.994291i \(0.534030\pi\)
\(618\) 0 0
\(619\) −75.8068 −0.00492234 −0.00246117 0.999997i \(-0.500783\pi\)
−0.00246117 + 0.999997i \(0.500783\pi\)
\(620\) 0 0
\(621\) 29780.3 1.92438
\(622\) 0 0
\(623\) 8158.37 0.524652
\(624\) 0 0
\(625\) −10794.7 −0.690862
\(626\) 0 0
\(627\) −7981.81 −0.508393
\(628\) 0 0
\(629\) −512.282 −0.0324738
\(630\) 0 0
\(631\) −4915.71 −0.310129 −0.155064 0.987904i \(-0.549559\pi\)
−0.155064 + 0.987904i \(0.549559\pi\)
\(632\) 0 0
\(633\) −2740.07 −0.172051
\(634\) 0 0
\(635\) −3233.67 −0.202086
\(636\) 0 0
\(637\) 2261.80 0.140684
\(638\) 0 0
\(639\) −7267.86 −0.449941
\(640\) 0 0
\(641\) 3317.63 0.204428 0.102214 0.994762i \(-0.467407\pi\)
0.102214 + 0.994762i \(0.467407\pi\)
\(642\) 0 0
\(643\) 27222.7 1.66961 0.834805 0.550546i \(-0.185580\pi\)
0.834805 + 0.550546i \(0.185580\pi\)
\(644\) 0 0
\(645\) 2168.29 0.132367
\(646\) 0 0
\(647\) 24433.6 1.48468 0.742338 0.670025i \(-0.233717\pi\)
0.742338 + 0.670025i \(0.233717\pi\)
\(648\) 0 0
\(649\) −27345.4 −1.65393
\(650\) 0 0
\(651\) −28557.0 −1.71926
\(652\) 0 0
\(653\) −5755.36 −0.344907 −0.172454 0.985018i \(-0.555170\pi\)
−0.172454 + 0.985018i \(0.555170\pi\)
\(654\) 0 0
\(655\) −9052.01 −0.539987
\(656\) 0 0
\(657\) −46.9671 −0.00278898
\(658\) 0 0
\(659\) −965.300 −0.0570603 −0.0285301 0.999593i \(-0.509083\pi\)
−0.0285301 + 0.999593i \(0.509083\pi\)
\(660\) 0 0
\(661\) 22791.0 1.34110 0.670549 0.741866i \(-0.266059\pi\)
0.670549 + 0.741866i \(0.266059\pi\)
\(662\) 0 0
\(663\) 320.479 0.0187728
\(664\) 0 0
\(665\) −11828.3 −0.689746
\(666\) 0 0
\(667\) 5946.75 0.345216
\(668\) 0 0
\(669\) 17716.9 1.02388
\(670\) 0 0
\(671\) −25368.4 −1.45952
\(672\) 0 0
\(673\) 12412.1 0.710925 0.355463 0.934690i \(-0.384323\pi\)
0.355463 + 0.934690i \(0.384323\pi\)
\(674\) 0 0
\(675\) 4497.59 0.256463
\(676\) 0 0
\(677\) −8998.03 −0.510816 −0.255408 0.966833i \(-0.582210\pi\)
−0.255408 + 0.966833i \(0.582210\pi\)
\(678\) 0 0
\(679\) −11837.2 −0.669030
\(680\) 0 0
\(681\) 17360.7 0.976893
\(682\) 0 0
\(683\) −13439.1 −0.752904 −0.376452 0.926436i \(-0.622856\pi\)
−0.376452 + 0.926436i \(0.622856\pi\)
\(684\) 0 0
\(685\) 10572.8 0.589730
\(686\) 0 0
\(687\) 21432.1 1.19023
\(688\) 0 0
\(689\) 1900.22 0.105069
\(690\) 0 0
\(691\) 16220.4 0.892983 0.446492 0.894788i \(-0.352673\pi\)
0.446492 + 0.894788i \(0.352673\pi\)
\(692\) 0 0
\(693\) 21815.0 1.19579
\(694\) 0 0
\(695\) −18662.7 −1.01858
\(696\) 0 0
\(697\) −755.739 −0.0410698
\(698\) 0 0
\(699\) −10754.1 −0.581912
\(700\) 0 0
\(701\) 18919.6 1.01938 0.509689 0.860359i \(-0.329761\pi\)
0.509689 + 0.860359i \(0.329761\pi\)
\(702\) 0 0
\(703\) 1317.13 0.0706636
\(704\) 0 0
\(705\) 16708.0 0.892567
\(706\) 0 0
\(707\) −33450.6 −1.77941
\(708\) 0 0
\(709\) 4781.99 0.253302 0.126651 0.991947i \(-0.459577\pi\)
0.126651 + 0.991947i \(0.459577\pi\)
\(710\) 0 0
\(711\) 15754.4 0.830992
\(712\) 0 0
\(713\) −61516.9 −3.23117
\(714\) 0 0
\(715\) −2743.22 −0.143483
\(716\) 0 0
\(717\) 24812.5 1.29238
\(718\) 0 0
\(719\) 13020.1 0.675336 0.337668 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(720\) 0 0
\(721\) −28480.9 −1.47113
\(722\) 0 0
\(723\) −2888.42 −0.148577
\(724\) 0 0
\(725\) 898.112 0.0460070
\(726\) 0 0
\(727\) −8738.68 −0.445804 −0.222902 0.974841i \(-0.571553\pi\)
−0.222902 + 0.974841i \(0.571553\pi\)
\(728\) 0 0
\(729\) 14933.8 0.758715
\(730\) 0 0
\(731\) −1114.43 −0.0563865
\(732\) 0 0
\(733\) 12775.2 0.643744 0.321872 0.946783i \(-0.395688\pi\)
0.321872 + 0.946783i \(0.395688\pi\)
\(734\) 0 0
\(735\) −13999.8 −0.702571
\(736\) 0 0
\(737\) 26435.4 1.32125
\(738\) 0 0
\(739\) −5397.68 −0.268683 −0.134342 0.990935i \(-0.542892\pi\)
−0.134342 + 0.990935i \(0.542892\pi\)
\(740\) 0 0
\(741\) −823.985 −0.0408500
\(742\) 0 0
\(743\) −37984.2 −1.87551 −0.937757 0.347293i \(-0.887101\pi\)
−0.937757 + 0.347293i \(0.887101\pi\)
\(744\) 0 0
\(745\) −16519.9 −0.812408
\(746\) 0 0
\(747\) −15207.9 −0.744886
\(748\) 0 0
\(749\) −32454.4 −1.58326
\(750\) 0 0
\(751\) 26384.8 1.28202 0.641009 0.767534i \(-0.278516\pi\)
0.641009 + 0.767534i \(0.278516\pi\)
\(752\) 0 0
\(753\) −14855.4 −0.718940
\(754\) 0 0
\(755\) −10499.9 −0.506133
\(756\) 0 0
\(757\) −6889.95 −0.330805 −0.165403 0.986226i \(-0.552892\pi\)
−0.165403 + 0.986226i \(0.552892\pi\)
\(758\) 0 0
\(759\) −37028.8 −1.77083
\(760\) 0 0
\(761\) 31579.3 1.50427 0.752134 0.659010i \(-0.229025\pi\)
0.752134 + 0.659010i \(0.229025\pi\)
\(762\) 0 0
\(763\) −59824.0 −2.83850
\(764\) 0 0
\(765\) 2517.46 0.118979
\(766\) 0 0
\(767\) −2822.94 −0.132895
\(768\) 0 0
\(769\) −12232.7 −0.573632 −0.286816 0.957986i \(-0.592597\pi\)
−0.286816 + 0.957986i \(0.592597\pi\)
\(770\) 0 0
\(771\) 1384.07 0.0646510
\(772\) 0 0
\(773\) −9958.46 −0.463365 −0.231682 0.972792i \(-0.574423\pi\)
−0.231682 + 0.972792i \(0.574423\pi\)
\(774\) 0 0
\(775\) −9290.64 −0.430619
\(776\) 0 0
\(777\) 2836.52 0.130965
\(778\) 0 0
\(779\) 1943.09 0.0893688
\(780\) 0 0
\(781\) 25194.4 1.15432
\(782\) 0 0
\(783\) 4211.58 0.192222
\(784\) 0 0
\(785\) 23349.2 1.06162
\(786\) 0 0
\(787\) −103.911 −0.00470653 −0.00235326 0.999997i \(-0.500749\pi\)
−0.00235326 + 0.999997i \(0.500749\pi\)
\(788\) 0 0
\(789\) −9254.02 −0.417556
\(790\) 0 0
\(791\) −44806.2 −2.01406
\(792\) 0 0
\(793\) −2618.85 −0.117274
\(794\) 0 0
\(795\) −11761.7 −0.524711
\(796\) 0 0
\(797\) −30624.4 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(798\) 0 0
\(799\) −8587.31 −0.380222
\(800\) 0 0
\(801\) 4464.42 0.196932
\(802\) 0 0
\(803\) 162.814 0.00715514
\(804\) 0 0
\(805\) −54873.3 −2.40252
\(806\) 0 0
\(807\) 16324.3 0.712072
\(808\) 0 0
\(809\) −7737.50 −0.336262 −0.168131 0.985765i \(-0.553773\pi\)
−0.168131 + 0.985765i \(0.553773\pi\)
\(810\) 0 0
\(811\) 12974.1 0.561756 0.280878 0.959744i \(-0.409374\pi\)
0.280878 + 0.959744i \(0.409374\pi\)
\(812\) 0 0
\(813\) 15625.0 0.674038
\(814\) 0 0
\(815\) 10294.2 0.442440
\(816\) 0 0
\(817\) 2865.31 0.122698
\(818\) 0 0
\(819\) 2252.02 0.0960831
\(820\) 0 0
\(821\) −3264.26 −0.138762 −0.0693809 0.997590i \(-0.522102\pi\)
−0.0693809 + 0.997590i \(0.522102\pi\)
\(822\) 0 0
\(823\) 23509.7 0.995745 0.497872 0.867250i \(-0.334115\pi\)
0.497872 + 0.867250i \(0.334115\pi\)
\(824\) 0 0
\(825\) −5592.31 −0.235999
\(826\) 0 0
\(827\) −27116.5 −1.14019 −0.570093 0.821580i \(-0.693093\pi\)
−0.570093 + 0.821580i \(0.693093\pi\)
\(828\) 0 0
\(829\) −15393.6 −0.644923 −0.322462 0.946583i \(-0.604510\pi\)
−0.322462 + 0.946583i \(0.604510\pi\)
\(830\) 0 0
\(831\) −24693.8 −1.03083
\(832\) 0 0
\(833\) 7195.38 0.299286
\(834\) 0 0
\(835\) −12757.5 −0.528734
\(836\) 0 0
\(837\) −43567.3 −1.79917
\(838\) 0 0
\(839\) −7906.51 −0.325343 −0.162672 0.986680i \(-0.552011\pi\)
−0.162672 + 0.986680i \(0.552011\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −19902.7 −0.813150
\(844\) 0 0
\(845\) 21021.0 0.855791
\(846\) 0 0
\(847\) −38892.6 −1.57777
\(848\) 0 0
\(849\) 13160.7 0.532005
\(850\) 0 0
\(851\) 6110.38 0.246135
\(852\) 0 0
\(853\) 11887.1 0.477146 0.238573 0.971125i \(-0.423320\pi\)
0.238573 + 0.971125i \(0.423320\pi\)
\(854\) 0 0
\(855\) −6472.67 −0.258901
\(856\) 0 0
\(857\) −655.328 −0.0261208 −0.0130604 0.999915i \(-0.504157\pi\)
−0.0130604 + 0.999915i \(0.504157\pi\)
\(858\) 0 0
\(859\) 48758.4 1.93669 0.968345 0.249617i \(-0.0803048\pi\)
0.968345 + 0.249617i \(0.0803048\pi\)
\(860\) 0 0
\(861\) 4184.55 0.165632
\(862\) 0 0
\(863\) −10398.2 −0.410150 −0.205075 0.978746i \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(864\) 0 0
\(865\) −6339.86 −0.249204
\(866\) 0 0
\(867\) −15927.8 −0.623918
\(868\) 0 0
\(869\) −54613.4 −2.13191
\(870\) 0 0
\(871\) 2729.01 0.106164
\(872\) 0 0
\(873\) −6477.57 −0.251125
\(874\) 0 0
\(875\) −41736.8 −1.61253
\(876\) 0 0
\(877\) 31024.2 1.19454 0.597272 0.802039i \(-0.296251\pi\)
0.597272 + 0.802039i \(0.296251\pi\)
\(878\) 0 0
\(879\) −18684.7 −0.716973
\(880\) 0 0
\(881\) −1510.50 −0.0577641 −0.0288821 0.999583i \(-0.509195\pi\)
−0.0288821 + 0.999583i \(0.509195\pi\)
\(882\) 0 0
\(883\) 32376.4 1.23392 0.616961 0.786993i \(-0.288363\pi\)
0.616961 + 0.786993i \(0.288363\pi\)
\(884\) 0 0
\(885\) 17473.1 0.663674
\(886\) 0 0
\(887\) 10474.6 0.396510 0.198255 0.980150i \(-0.436473\pi\)
0.198255 + 0.980150i \(0.436473\pi\)
\(888\) 0 0
\(889\) −9202.51 −0.347179
\(890\) 0 0
\(891\) −4880.53 −0.183506
\(892\) 0 0
\(893\) 22078.9 0.827371
\(894\) 0 0
\(895\) 19930.0 0.744341
\(896\) 0 0
\(897\) −3822.59 −0.142288
\(898\) 0 0
\(899\) −8699.84 −0.322754
\(900\) 0 0
\(901\) 6045.11 0.223520
\(902\) 0 0
\(903\) 6170.62 0.227403
\(904\) 0 0
\(905\) 32953.7 1.21041
\(906\) 0 0
\(907\) 34634.3 1.26793 0.633965 0.773362i \(-0.281426\pi\)
0.633965 + 0.773362i \(0.281426\pi\)
\(908\) 0 0
\(909\) −18304.8 −0.667913
\(910\) 0 0
\(911\) 1186.14 0.0431377 0.0215688 0.999767i \(-0.493134\pi\)
0.0215688 + 0.999767i \(0.493134\pi\)
\(912\) 0 0
\(913\) 52719.1 1.91101
\(914\) 0 0
\(915\) 16209.8 0.585662
\(916\) 0 0
\(917\) −25760.6 −0.927687
\(918\) 0 0
\(919\) −5685.58 −0.204080 −0.102040 0.994780i \(-0.532537\pi\)
−0.102040 + 0.994780i \(0.532537\pi\)
\(920\) 0 0
\(921\) −17002.7 −0.608316
\(922\) 0 0
\(923\) 2600.89 0.0927512
\(924\) 0 0
\(925\) 922.824 0.0328025
\(926\) 0 0
\(927\) −15585.3 −0.552200
\(928\) 0 0
\(929\) −19143.7 −0.676087 −0.338044 0.941130i \(-0.609765\pi\)
−0.338044 + 0.941130i \(0.609765\pi\)
\(930\) 0 0
\(931\) −18500.1 −0.651252
\(932\) 0 0
\(933\) 3856.28 0.135315
\(934\) 0 0
\(935\) −8726.92 −0.305241
\(936\) 0 0
\(937\) −53623.5 −1.86959 −0.934794 0.355191i \(-0.884416\pi\)
−0.934794 + 0.355191i \(0.884416\pi\)
\(938\) 0 0
\(939\) 7010.08 0.243626
\(940\) 0 0
\(941\) 7947.57 0.275328 0.137664 0.990479i \(-0.456041\pi\)
0.137664 + 0.990479i \(0.456041\pi\)
\(942\) 0 0
\(943\) 9014.28 0.311289
\(944\) 0 0
\(945\) −38862.2 −1.33776
\(946\) 0 0
\(947\) 41047.8 1.40853 0.704263 0.709939i \(-0.251277\pi\)
0.704263 + 0.709939i \(0.251277\pi\)
\(948\) 0 0
\(949\) 16.8077 0.000574924 0
\(950\) 0 0
\(951\) −7077.49 −0.241328
\(952\) 0 0
\(953\) −12000.8 −0.407915 −0.203957 0.978980i \(-0.565380\pi\)
−0.203957 + 0.978980i \(0.565380\pi\)
\(954\) 0 0
\(955\) −47068.1 −1.59486
\(956\) 0 0
\(957\) −5236.69 −0.176884
\(958\) 0 0
\(959\) 30088.4 1.01314
\(960\) 0 0
\(961\) 60205.6 2.02093
\(962\) 0 0
\(963\) −17759.7 −0.594287
\(964\) 0 0
\(965\) 47294.3 1.57768
\(966\) 0 0
\(967\) −55391.0 −1.84204 −0.921021 0.389514i \(-0.872643\pi\)
−0.921021 + 0.389514i \(0.872643\pi\)
\(968\) 0 0
\(969\) −2621.32 −0.0869028
\(970\) 0 0
\(971\) 23937.0 0.791117 0.395558 0.918441i \(-0.370551\pi\)
0.395558 + 0.918441i \(0.370551\pi\)
\(972\) 0 0
\(973\) −53111.0 −1.74991
\(974\) 0 0
\(975\) −577.310 −0.0189628
\(976\) 0 0
\(977\) −43982.3 −1.44024 −0.720122 0.693847i \(-0.755914\pi\)
−0.720122 + 0.693847i \(0.755914\pi\)
\(978\) 0 0
\(979\) −15476.1 −0.505229
\(980\) 0 0
\(981\) −32736.8 −1.06545
\(982\) 0 0
\(983\) 2724.78 0.0884100 0.0442050 0.999022i \(-0.485925\pi\)
0.0442050 + 0.999022i \(0.485925\pi\)
\(984\) 0 0
\(985\) −10001.6 −0.323530
\(986\) 0 0
\(987\) 47548.3 1.53341
\(988\) 0 0
\(989\) 13292.6 0.427382
\(990\) 0 0
\(991\) −23748.6 −0.761250 −0.380625 0.924729i \(-0.624291\pi\)
−0.380625 + 0.924729i \(0.624291\pi\)
\(992\) 0 0
\(993\) 33297.7 1.06412
\(994\) 0 0
\(995\) 11323.1 0.360771
\(996\) 0 0
\(997\) −24718.0 −0.785182 −0.392591 0.919713i \(-0.628421\pi\)
−0.392591 + 0.919713i \(0.628421\pi\)
\(998\) 0 0
\(999\) 4327.47 0.137052
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 1856.4.a.l.1.2 2
4.3 odd 2 1856.4.a.i.1.1 2
8.3 odd 2 464.4.a.e.1.2 2
8.5 even 2 58.4.a.c.1.1 2
24.5 odd 2 522.4.a.j.1.1 2
40.29 even 2 1450.4.a.g.1.2 2
232.173 even 2 1682.4.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.4.a.c.1.1 2 8.5 even 2
464.4.a.e.1.2 2 8.3 odd 2
522.4.a.j.1.1 2 24.5 odd 2
1450.4.a.g.1.2 2 40.29 even 2
1682.4.a.c.1.2 2 232.173 even 2
1856.4.a.i.1.1 2 4.3 odd 2
1856.4.a.l.1.2 2 1.1 even 1 trivial