Properties

Label 1521.4.a.bd.1.7
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.69212\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.69212 q^{2} +5.63172 q^{4} -18.7574 q^{5} +24.1383 q^{7} -8.74396 q^{8} +O(q^{10})\) \(q+3.69212 q^{2} +5.63172 q^{4} -18.7574 q^{5} +24.1383 q^{7} -8.74396 q^{8} -69.2546 q^{10} -49.2741 q^{11} +89.1214 q^{14} -77.3375 q^{16} +65.3121 q^{17} -109.940 q^{19} -105.637 q^{20} -181.926 q^{22} +83.2974 q^{23} +226.841 q^{25} +135.940 q^{28} -4.99593 q^{29} +255.810 q^{31} -215.587 q^{32} +241.140 q^{34} -452.772 q^{35} -93.6356 q^{37} -405.912 q^{38} +164.014 q^{40} +67.9487 q^{41} +142.543 q^{43} -277.498 q^{44} +307.544 q^{46} -379.275 q^{47} +239.657 q^{49} +837.524 q^{50} +389.560 q^{53} +924.256 q^{55} -211.064 q^{56} -18.4456 q^{58} +133.881 q^{59} +620.792 q^{61} +944.482 q^{62} -177.273 q^{64} -119.002 q^{67} +367.819 q^{68} -1671.69 q^{70} +361.386 q^{71} +748.241 q^{73} -345.714 q^{74} -619.152 q^{76} -1189.39 q^{77} +514.165 q^{79} +1450.65 q^{80} +250.875 q^{82} +260.260 q^{83} -1225.09 q^{85} +526.286 q^{86} +430.851 q^{88} -833.631 q^{89} +469.108 q^{92} -1400.33 q^{94} +2062.19 q^{95} +1481.99 q^{97} +884.841 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.69212 1.30536 0.652680 0.757634i \(-0.273645\pi\)
0.652680 + 0.757634i \(0.273645\pi\)
\(3\) 0 0
\(4\) 5.63172 0.703965
\(5\) −18.7574 −1.67772 −0.838858 0.544351i \(-0.816776\pi\)
−0.838858 + 0.544351i \(0.816776\pi\)
\(6\) 0 0
\(7\) 24.1383 1.30334 0.651672 0.758500i \(-0.274068\pi\)
0.651672 + 0.758500i \(0.274068\pi\)
\(8\) −8.74396 −0.386432
\(9\) 0 0
\(10\) −69.2546 −2.19002
\(11\) −49.2741 −1.35061 −0.675305 0.737539i \(-0.735988\pi\)
−0.675305 + 0.737539i \(0.735988\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 89.1214 1.70133
\(15\) 0 0
\(16\) −77.3375 −1.20840
\(17\) 65.3121 0.931795 0.465897 0.884839i \(-0.345732\pi\)
0.465897 + 0.884839i \(0.345732\pi\)
\(18\) 0 0
\(19\) −109.940 −1.32747 −0.663737 0.747966i \(-0.731030\pi\)
−0.663737 + 0.747966i \(0.731030\pi\)
\(20\) −105.637 −1.18105
\(21\) 0 0
\(22\) −181.926 −1.76303
\(23\) 83.2974 0.755161 0.377581 0.925977i \(-0.376756\pi\)
0.377581 + 0.925977i \(0.376756\pi\)
\(24\) 0 0
\(25\) 226.841 1.81473
\(26\) 0 0
\(27\) 0 0
\(28\) 135.940 0.917509
\(29\) −4.99593 −0.0319904 −0.0159952 0.999872i \(-0.505092\pi\)
−0.0159952 + 0.999872i \(0.505092\pi\)
\(30\) 0 0
\(31\) 255.810 1.48209 0.741047 0.671453i \(-0.234330\pi\)
0.741047 + 0.671453i \(0.234330\pi\)
\(32\) −215.587 −1.19096
\(33\) 0 0
\(34\) 241.140 1.21633
\(35\) −452.772 −2.18664
\(36\) 0 0
\(37\) −93.6356 −0.416043 −0.208022 0.978124i \(-0.566702\pi\)
−0.208022 + 0.978124i \(0.566702\pi\)
\(38\) −405.912 −1.73283
\(39\) 0 0
\(40\) 164.014 0.648323
\(41\) 67.9487 0.258825 0.129412 0.991591i \(-0.458691\pi\)
0.129412 + 0.991591i \(0.458691\pi\)
\(42\) 0 0
\(43\) 142.543 0.505527 0.252763 0.967528i \(-0.418661\pi\)
0.252763 + 0.967528i \(0.418661\pi\)
\(44\) −277.498 −0.950782
\(45\) 0 0
\(46\) 307.544 0.985758
\(47\) −379.275 −1.17708 −0.588541 0.808467i \(-0.700298\pi\)
−0.588541 + 0.808467i \(0.700298\pi\)
\(48\) 0 0
\(49\) 239.657 0.698708
\(50\) 837.524 2.36888
\(51\) 0 0
\(52\) 0 0
\(53\) 389.560 1.00963 0.504813 0.863229i \(-0.331562\pi\)
0.504813 + 0.863229i \(0.331562\pi\)
\(54\) 0 0
\(55\) 924.256 2.26594
\(56\) −211.064 −0.503654
\(57\) 0 0
\(58\) −18.4456 −0.0417590
\(59\) 133.881 0.295420 0.147710 0.989031i \(-0.452810\pi\)
0.147710 + 0.989031i \(0.452810\pi\)
\(60\) 0 0
\(61\) 620.792 1.30302 0.651510 0.758640i \(-0.274136\pi\)
0.651510 + 0.758640i \(0.274136\pi\)
\(62\) 944.482 1.93467
\(63\) 0 0
\(64\) −177.273 −0.346237
\(65\) 0 0
\(66\) 0 0
\(67\) −119.002 −0.216991 −0.108496 0.994097i \(-0.534603\pi\)
−0.108496 + 0.994097i \(0.534603\pi\)
\(68\) 367.819 0.655951
\(69\) 0 0
\(70\) −1671.69 −2.85436
\(71\) 361.386 0.604066 0.302033 0.953298i \(-0.402335\pi\)
0.302033 + 0.953298i \(0.402335\pi\)
\(72\) 0 0
\(73\) 748.241 1.19966 0.599829 0.800128i \(-0.295235\pi\)
0.599829 + 0.800128i \(0.295235\pi\)
\(74\) −345.714 −0.543086
\(75\) 0 0
\(76\) −619.152 −0.934495
\(77\) −1189.39 −1.76031
\(78\) 0 0
\(79\) 514.165 0.732254 0.366127 0.930565i \(-0.380684\pi\)
0.366127 + 0.930565i \(0.380684\pi\)
\(80\) 1450.65 2.02735
\(81\) 0 0
\(82\) 250.875 0.337859
\(83\) 260.260 0.344183 0.172092 0.985081i \(-0.444947\pi\)
0.172092 + 0.985081i \(0.444947\pi\)
\(84\) 0 0
\(85\) −1225.09 −1.56329
\(86\) 526.286 0.659894
\(87\) 0 0
\(88\) 430.851 0.521919
\(89\) −833.631 −0.992861 −0.496431 0.868076i \(-0.665356\pi\)
−0.496431 + 0.868076i \(0.665356\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 469.108 0.531607
\(93\) 0 0
\(94\) −1400.33 −1.53652
\(95\) 2062.19 2.22712
\(96\) 0 0
\(97\) 1481.99 1.55127 0.775634 0.631183i \(-0.217430\pi\)
0.775634 + 0.631183i \(0.217430\pi\)
\(98\) 884.841 0.912066
\(99\) 0 0
\(100\) 1277.51 1.27751
\(101\) −1740.21 −1.71443 −0.857217 0.514956i \(-0.827808\pi\)
−0.857217 + 0.514956i \(0.827808\pi\)
\(102\) 0 0
\(103\) 1020.16 0.975917 0.487959 0.872867i \(-0.337742\pi\)
0.487959 + 0.872867i \(0.337742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1438.30 1.31792
\(107\) 1507.96 1.36243 0.681215 0.732084i \(-0.261452\pi\)
0.681215 + 0.732084i \(0.261452\pi\)
\(108\) 0 0
\(109\) 1074.08 0.943840 0.471920 0.881641i \(-0.343561\pi\)
0.471920 + 0.881641i \(0.343561\pi\)
\(110\) 3412.46 2.95787
\(111\) 0 0
\(112\) −1866.79 −1.57496
\(113\) −891.337 −0.742035 −0.371018 0.928626i \(-0.620991\pi\)
−0.371018 + 0.928626i \(0.620991\pi\)
\(114\) 0 0
\(115\) −1562.45 −1.26695
\(116\) −28.1357 −0.0225201
\(117\) 0 0
\(118\) 494.304 0.385630
\(119\) 1576.52 1.21445
\(120\) 0 0
\(121\) 1096.94 0.824146
\(122\) 2292.03 1.70091
\(123\) 0 0
\(124\) 1440.65 1.04334
\(125\) −1910.28 −1.36688
\(126\) 0 0
\(127\) −6.55994 −0.00458347 −0.00229173 0.999997i \(-0.500729\pi\)
−0.00229173 + 0.999997i \(0.500729\pi\)
\(128\) 1070.18 0.738999
\(129\) 0 0
\(130\) 0 0
\(131\) 1267.44 0.845320 0.422660 0.906288i \(-0.361097\pi\)
0.422660 + 0.906288i \(0.361097\pi\)
\(132\) 0 0
\(133\) −2653.77 −1.73016
\(134\) −439.369 −0.283252
\(135\) 0 0
\(136\) −571.086 −0.360075
\(137\) −1350.85 −0.842416 −0.421208 0.906964i \(-0.638394\pi\)
−0.421208 + 0.906964i \(0.638394\pi\)
\(138\) 0 0
\(139\) −990.606 −0.604476 −0.302238 0.953233i \(-0.597734\pi\)
−0.302238 + 0.953233i \(0.597734\pi\)
\(140\) −2549.89 −1.53932
\(141\) 0 0
\(142\) 1334.28 0.788523
\(143\) 0 0
\(144\) 0 0
\(145\) 93.7108 0.0536708
\(146\) 2762.59 1.56599
\(147\) 0 0
\(148\) −527.330 −0.292880
\(149\) 2132.38 1.17243 0.586214 0.810156i \(-0.300618\pi\)
0.586214 + 0.810156i \(0.300618\pi\)
\(150\) 0 0
\(151\) 251.734 0.135668 0.0678338 0.997697i \(-0.478391\pi\)
0.0678338 + 0.997697i \(0.478391\pi\)
\(152\) 961.312 0.512978
\(153\) 0 0
\(154\) −4391.38 −2.29784
\(155\) −4798.35 −2.48653
\(156\) 0 0
\(157\) −3686.27 −1.87386 −0.936931 0.349515i \(-0.886346\pi\)
−0.936931 + 0.349515i \(0.886346\pi\)
\(158\) 1898.36 0.955855
\(159\) 0 0
\(160\) 4043.86 1.99810
\(161\) 2010.66 0.984236
\(162\) 0 0
\(163\) 1267.79 0.609207 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(164\) 382.668 0.182204
\(165\) 0 0
\(166\) 960.910 0.449283
\(167\) −938.997 −0.435100 −0.217550 0.976049i \(-0.569807\pi\)
−0.217550 + 0.976049i \(0.569807\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −4523.16 −2.04065
\(171\) 0 0
\(172\) 802.764 0.355873
\(173\) −1821.44 −0.800472 −0.400236 0.916412i \(-0.631072\pi\)
−0.400236 + 0.916412i \(0.631072\pi\)
\(174\) 0 0
\(175\) 5475.56 2.36522
\(176\) 3810.74 1.63207
\(177\) 0 0
\(178\) −3077.86 −1.29604
\(179\) 645.740 0.269636 0.134818 0.990870i \(-0.456955\pi\)
0.134818 + 0.990870i \(0.456955\pi\)
\(180\) 0 0
\(181\) 2387.31 0.980370 0.490185 0.871618i \(-0.336929\pi\)
0.490185 + 0.871618i \(0.336929\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −728.349 −0.291819
\(185\) 1756.36 0.698002
\(186\) 0 0
\(187\) −3218.19 −1.25849
\(188\) −2135.97 −0.828625
\(189\) 0 0
\(190\) 7613.86 2.90720
\(191\) 1126.36 0.426703 0.213351 0.976976i \(-0.431562\pi\)
0.213351 + 0.976976i \(0.431562\pi\)
\(192\) 0 0
\(193\) 3258.63 1.21534 0.607672 0.794188i \(-0.292104\pi\)
0.607672 + 0.794188i \(0.292104\pi\)
\(194\) 5471.67 2.02496
\(195\) 0 0
\(196\) 1349.68 0.491866
\(197\) −179.494 −0.0649159 −0.0324580 0.999473i \(-0.510334\pi\)
−0.0324580 + 0.999473i \(0.510334\pi\)
\(198\) 0 0
\(199\) 2953.42 1.05207 0.526036 0.850462i \(-0.323678\pi\)
0.526036 + 0.850462i \(0.323678\pi\)
\(200\) −1983.49 −0.701270
\(201\) 0 0
\(202\) −6425.07 −2.23795
\(203\) −120.593 −0.0416945
\(204\) 0 0
\(205\) −1274.54 −0.434234
\(206\) 3766.55 1.27392
\(207\) 0 0
\(208\) 0 0
\(209\) 5417.20 1.79290
\(210\) 0 0
\(211\) 3128.41 1.02070 0.510352 0.859965i \(-0.329515\pi\)
0.510352 + 0.859965i \(0.329515\pi\)
\(212\) 2193.89 0.710741
\(213\) 0 0
\(214\) 5567.56 1.77846
\(215\) −2673.75 −0.848130
\(216\) 0 0
\(217\) 6174.83 1.93168
\(218\) 3965.64 1.23205
\(219\) 0 0
\(220\) 5205.15 1.59514
\(221\) 0 0
\(222\) 0 0
\(223\) 3754.12 1.12733 0.563665 0.826004i \(-0.309391\pi\)
0.563665 + 0.826004i \(0.309391\pi\)
\(224\) −5203.91 −1.55224
\(225\) 0 0
\(226\) −3290.92 −0.968623
\(227\) 1549.52 0.453062 0.226531 0.974004i \(-0.427262\pi\)
0.226531 + 0.974004i \(0.427262\pi\)
\(228\) 0 0
\(229\) −1364.84 −0.393847 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(230\) −5768.73 −1.65382
\(231\) 0 0
\(232\) 43.6842 0.0123621
\(233\) 1665.69 0.468339 0.234170 0.972196i \(-0.424763\pi\)
0.234170 + 0.972196i \(0.424763\pi\)
\(234\) 0 0
\(235\) 7114.22 1.97481
\(236\) 753.980 0.207966
\(237\) 0 0
\(238\) 5820.70 1.58529
\(239\) −5950.06 −1.61037 −0.805183 0.593026i \(-0.797933\pi\)
−0.805183 + 0.593026i \(0.797933\pi\)
\(240\) 0 0
\(241\) 1707.01 0.456257 0.228128 0.973631i \(-0.426739\pi\)
0.228128 + 0.973631i \(0.426739\pi\)
\(242\) 4050.02 1.07581
\(243\) 0 0
\(244\) 3496.12 0.917281
\(245\) −4495.35 −1.17223
\(246\) 0 0
\(247\) 0 0
\(248\) −2236.80 −0.572729
\(249\) 0 0
\(250\) −7052.97 −1.78428
\(251\) 306.390 0.0770484 0.0385242 0.999258i \(-0.487734\pi\)
0.0385242 + 0.999258i \(0.487734\pi\)
\(252\) 0 0
\(253\) −4104.41 −1.01993
\(254\) −24.2200 −0.00598307
\(255\) 0 0
\(256\) 5369.43 1.31090
\(257\) 3556.16 0.863140 0.431570 0.902079i \(-0.357960\pi\)
0.431570 + 0.902079i \(0.357960\pi\)
\(258\) 0 0
\(259\) −2260.20 −0.542248
\(260\) 0 0
\(261\) 0 0
\(262\) 4679.54 1.10345
\(263\) −7121.35 −1.66966 −0.834831 0.550506i \(-0.814435\pi\)
−0.834831 + 0.550506i \(0.814435\pi\)
\(264\) 0 0
\(265\) −7307.14 −1.69386
\(266\) −9798.01 −2.25848
\(267\) 0 0
\(268\) −670.186 −0.152754
\(269\) 6419.06 1.45493 0.727467 0.686143i \(-0.240697\pi\)
0.727467 + 0.686143i \(0.240697\pi\)
\(270\) 0 0
\(271\) −8631.33 −1.93474 −0.967372 0.253360i \(-0.918464\pi\)
−0.967372 + 0.253360i \(0.918464\pi\)
\(272\) −5051.07 −1.12598
\(273\) 0 0
\(274\) −4987.50 −1.09966
\(275\) −11177.4 −2.45099
\(276\) 0 0
\(277\) 727.474 0.157797 0.0788983 0.996883i \(-0.474860\pi\)
0.0788983 + 0.996883i \(0.474860\pi\)
\(278\) −3657.43 −0.789059
\(279\) 0 0
\(280\) 3959.02 0.844989
\(281\) −5588.39 −1.18639 −0.593194 0.805059i \(-0.702133\pi\)
−0.593194 + 0.805059i \(0.702133\pi\)
\(282\) 0 0
\(283\) −1432.76 −0.300949 −0.150474 0.988614i \(-0.548080\pi\)
−0.150474 + 0.988614i \(0.548080\pi\)
\(284\) 2035.23 0.425241
\(285\) 0 0
\(286\) 0 0
\(287\) 1640.17 0.337338
\(288\) 0 0
\(289\) −647.332 −0.131759
\(290\) 345.991 0.0700597
\(291\) 0 0
\(292\) 4213.89 0.844517
\(293\) −8274.51 −1.64984 −0.824918 0.565253i \(-0.808779\pi\)
−0.824918 + 0.565253i \(0.808779\pi\)
\(294\) 0 0
\(295\) −2511.26 −0.495631
\(296\) 818.746 0.160772
\(297\) 0 0
\(298\) 7873.01 1.53044
\(299\) 0 0
\(300\) 0 0
\(301\) 3440.75 0.658876
\(302\) 929.430 0.177095
\(303\) 0 0
\(304\) 8502.49 1.60412
\(305\) −11644.5 −2.18610
\(306\) 0 0
\(307\) −628.477 −0.116837 −0.0584187 0.998292i \(-0.518606\pi\)
−0.0584187 + 0.998292i \(0.518606\pi\)
\(308\) −6698.33 −1.23920
\(309\) 0 0
\(310\) −17716.1 −3.24582
\(311\) 85.8693 0.0156566 0.00782830 0.999969i \(-0.497508\pi\)
0.00782830 + 0.999969i \(0.497508\pi\)
\(312\) 0 0
\(313\) −2279.49 −0.411643 −0.205821 0.978590i \(-0.565987\pi\)
−0.205821 + 0.978590i \(0.565987\pi\)
\(314\) −13610.1 −2.44606
\(315\) 0 0
\(316\) 2895.63 0.515481
\(317\) 6576.19 1.16516 0.582580 0.812774i \(-0.302043\pi\)
0.582580 + 0.812774i \(0.302043\pi\)
\(318\) 0 0
\(319\) 246.170 0.0432065
\(320\) 3325.19 0.580887
\(321\) 0 0
\(322\) 7423.58 1.28478
\(323\) −7180.42 −1.23693
\(324\) 0 0
\(325\) 0 0
\(326\) 4680.82 0.795235
\(327\) 0 0
\(328\) −594.141 −0.100018
\(329\) −9155.04 −1.53414
\(330\) 0 0
\(331\) −2493.35 −0.414039 −0.207020 0.978337i \(-0.566376\pi\)
−0.207020 + 0.978337i \(0.566376\pi\)
\(332\) 1465.71 0.242293
\(333\) 0 0
\(334\) −3466.89 −0.567963
\(335\) 2232.17 0.364049
\(336\) 0 0
\(337\) −2089.60 −0.337767 −0.168884 0.985636i \(-0.554016\pi\)
−0.168884 + 0.985636i \(0.554016\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6899.35 −1.10050
\(341\) −12604.8 −2.00173
\(342\) 0 0
\(343\) −2494.53 −0.392687
\(344\) −1246.39 −0.195352
\(345\) 0 0
\(346\) −6724.97 −1.04490
\(347\) 6468.43 1.00070 0.500351 0.865823i \(-0.333204\pi\)
0.500351 + 0.865823i \(0.333204\pi\)
\(348\) 0 0
\(349\) −5246.94 −0.804763 −0.402382 0.915472i \(-0.631817\pi\)
−0.402382 + 0.915472i \(0.631817\pi\)
\(350\) 20216.4 3.08746
\(351\) 0 0
\(352\) 10622.9 1.60853
\(353\) 10269.9 1.54847 0.774235 0.632899i \(-0.218135\pi\)
0.774235 + 0.632899i \(0.218135\pi\)
\(354\) 0 0
\(355\) −6778.68 −1.01345
\(356\) −4694.78 −0.698940
\(357\) 0 0
\(358\) 2384.15 0.351972
\(359\) 3200.22 0.470477 0.235238 0.971938i \(-0.424413\pi\)
0.235238 + 0.971938i \(0.424413\pi\)
\(360\) 0 0
\(361\) 5227.83 0.762185
\(362\) 8814.21 1.27974
\(363\) 0 0
\(364\) 0 0
\(365\) −14035.1 −2.01268
\(366\) 0 0
\(367\) 22.9581 0.00326540 0.00163270 0.999999i \(-0.499480\pi\)
0.00163270 + 0.999999i \(0.499480\pi\)
\(368\) −6442.01 −0.912536
\(369\) 0 0
\(370\) 6484.70 0.911144
\(371\) 9403.30 1.31589
\(372\) 0 0
\(373\) 6194.91 0.859947 0.429973 0.902842i \(-0.358523\pi\)
0.429973 + 0.902842i \(0.358523\pi\)
\(374\) −11881.9 −1.64278
\(375\) 0 0
\(376\) 3316.36 0.454863
\(377\) 0 0
\(378\) 0 0
\(379\) −6955.24 −0.942656 −0.471328 0.881958i \(-0.656225\pi\)
−0.471328 + 0.881958i \(0.656225\pi\)
\(380\) 11613.7 1.56782
\(381\) 0 0
\(382\) 4158.64 0.557001
\(383\) 1221.06 0.162907 0.0814534 0.996677i \(-0.474044\pi\)
0.0814534 + 0.996677i \(0.474044\pi\)
\(384\) 0 0
\(385\) 22309.9 2.95330
\(386\) 12031.2 1.58646
\(387\) 0 0
\(388\) 8346.13 1.09204
\(389\) 6318.58 0.823560 0.411780 0.911283i \(-0.364907\pi\)
0.411780 + 0.911283i \(0.364907\pi\)
\(390\) 0 0
\(391\) 5440.33 0.703655
\(392\) −2095.55 −0.270003
\(393\) 0 0
\(394\) −662.714 −0.0847387
\(395\) −9644.41 −1.22851
\(396\) 0 0
\(397\) −3085.74 −0.390098 −0.195049 0.980793i \(-0.562487\pi\)
−0.195049 + 0.980793i \(0.562487\pi\)
\(398\) 10904.4 1.37333
\(399\) 0 0
\(400\) −17543.3 −2.19292
\(401\) −10780.0 −1.34246 −0.671231 0.741248i \(-0.734234\pi\)
−0.671231 + 0.741248i \(0.734234\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −9800.40 −1.20690
\(405\) 0 0
\(406\) −445.244 −0.0544264
\(407\) 4613.81 0.561912
\(408\) 0 0
\(409\) −7633.26 −0.922837 −0.461418 0.887183i \(-0.652659\pi\)
−0.461418 + 0.887183i \(0.652659\pi\)
\(410\) −4705.76 −0.566832
\(411\) 0 0
\(412\) 5745.26 0.687012
\(413\) 3231.65 0.385035
\(414\) 0 0
\(415\) −4881.81 −0.577442
\(416\) 0 0
\(417\) 0 0
\(418\) 20000.9 2.34038
\(419\) −3508.69 −0.409095 −0.204547 0.978857i \(-0.565572\pi\)
−0.204547 + 0.978857i \(0.565572\pi\)
\(420\) 0 0
\(421\) −12477.7 −1.44448 −0.722241 0.691642i \(-0.756888\pi\)
−0.722241 + 0.691642i \(0.756888\pi\)
\(422\) 11550.5 1.33239
\(423\) 0 0
\(424\) −3406.29 −0.390152
\(425\) 14815.5 1.69096
\(426\) 0 0
\(427\) 14984.8 1.69828
\(428\) 8492.41 0.959103
\(429\) 0 0
\(430\) −9871.78 −1.10711
\(431\) −17745.5 −1.98322 −0.991612 0.129249i \(-0.958743\pi\)
−0.991612 + 0.129249i \(0.958743\pi\)
\(432\) 0 0
\(433\) 9696.17 1.07614 0.538069 0.842901i \(-0.319154\pi\)
0.538069 + 0.842901i \(0.319154\pi\)
\(434\) 22798.2 2.52154
\(435\) 0 0
\(436\) 6048.94 0.664430
\(437\) −9157.73 −1.00246
\(438\) 0 0
\(439\) 13398.1 1.45662 0.728308 0.685250i \(-0.240307\pi\)
0.728308 + 0.685250i \(0.240307\pi\)
\(440\) −8081.66 −0.875631
\(441\) 0 0
\(442\) 0 0
\(443\) −13630.2 −1.46183 −0.730913 0.682471i \(-0.760905\pi\)
−0.730913 + 0.682471i \(0.760905\pi\)
\(444\) 0 0
\(445\) 15636.8 1.66574
\(446\) 13860.6 1.47157
\(447\) 0 0
\(448\) −4279.08 −0.451266
\(449\) 1381.79 0.145235 0.0726176 0.997360i \(-0.476865\pi\)
0.0726176 + 0.997360i \(0.476865\pi\)
\(450\) 0 0
\(451\) −3348.11 −0.349571
\(452\) −5019.76 −0.522367
\(453\) 0 0
\(454\) 5720.99 0.591409
\(455\) 0 0
\(456\) 0 0
\(457\) 8261.60 0.845648 0.422824 0.906212i \(-0.361039\pi\)
0.422824 + 0.906212i \(0.361039\pi\)
\(458\) −5039.14 −0.514113
\(459\) 0 0
\(460\) −8799.26 −0.891886
\(461\) 16874.2 1.70480 0.852399 0.522892i \(-0.175147\pi\)
0.852399 + 0.522892i \(0.175147\pi\)
\(462\) 0 0
\(463\) −4466.93 −0.448371 −0.224186 0.974546i \(-0.571972\pi\)
−0.224186 + 0.974546i \(0.571972\pi\)
\(464\) 386.373 0.0386571
\(465\) 0 0
\(466\) 6149.92 0.611351
\(467\) 2731.37 0.270649 0.135324 0.990801i \(-0.456792\pi\)
0.135324 + 0.990801i \(0.456792\pi\)
\(468\) 0 0
\(469\) −2872.50 −0.282814
\(470\) 26266.5 2.57784
\(471\) 0 0
\(472\) −1170.65 −0.114160
\(473\) −7023.69 −0.682769
\(474\) 0 0
\(475\) −24938.9 −2.40900
\(476\) 8878.53 0.854930
\(477\) 0 0
\(478\) −21968.3 −2.10211
\(479\) 5630.91 0.537124 0.268562 0.963262i \(-0.413452\pi\)
0.268562 + 0.963262i \(0.413452\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 6302.46 0.595580
\(483\) 0 0
\(484\) 6177.65 0.580170
\(485\) −27798.3 −2.60259
\(486\) 0 0
\(487\) 12910.5 1.20129 0.600647 0.799514i \(-0.294910\pi\)
0.600647 + 0.799514i \(0.294910\pi\)
\(488\) −5428.18 −0.503529
\(489\) 0 0
\(490\) −16597.3 −1.53019
\(491\) −3114.67 −0.286279 −0.143139 0.989703i \(-0.545720\pi\)
−0.143139 + 0.989703i \(0.545720\pi\)
\(492\) 0 0
\(493\) −326.295 −0.0298085
\(494\) 0 0
\(495\) 0 0
\(496\) −19783.7 −1.79096
\(497\) 8723.24 0.787306
\(498\) 0 0
\(499\) 11637.7 1.04404 0.522018 0.852934i \(-0.325179\pi\)
0.522018 + 0.852934i \(0.325179\pi\)
\(500\) −10758.2 −0.962239
\(501\) 0 0
\(502\) 1131.23 0.100576
\(503\) −2832.94 −0.251122 −0.125561 0.992086i \(-0.540073\pi\)
−0.125561 + 0.992086i \(0.540073\pi\)
\(504\) 0 0
\(505\) 32641.9 2.87633
\(506\) −15153.9 −1.33137
\(507\) 0 0
\(508\) −36.9437 −0.00322660
\(509\) −7699.95 −0.670519 −0.335260 0.942126i \(-0.608824\pi\)
−0.335260 + 0.942126i \(0.608824\pi\)
\(510\) 0 0
\(511\) 18061.3 1.56357
\(512\) 11263.1 0.972193
\(513\) 0 0
\(514\) 13129.8 1.12671
\(515\) −19135.6 −1.63731
\(516\) 0 0
\(517\) 18688.4 1.58978
\(518\) −8344.93 −0.707829
\(519\) 0 0
\(520\) 0 0
\(521\) 14688.2 1.23513 0.617565 0.786520i \(-0.288119\pi\)
0.617565 + 0.786520i \(0.288119\pi\)
\(522\) 0 0
\(523\) −8895.47 −0.743732 −0.371866 0.928286i \(-0.621282\pi\)
−0.371866 + 0.928286i \(0.621282\pi\)
\(524\) 7137.88 0.595076
\(525\) 0 0
\(526\) −26292.9 −2.17951
\(527\) 16707.5 1.38101
\(528\) 0 0
\(529\) −5228.54 −0.429731
\(530\) −26978.8 −2.21110
\(531\) 0 0
\(532\) −14945.3 −1.21797
\(533\) 0 0
\(534\) 0 0
\(535\) −28285.4 −2.28577
\(536\) 1040.55 0.0838523
\(537\) 0 0
\(538\) 23699.9 1.89921
\(539\) −11808.9 −0.943682
\(540\) 0 0
\(541\) −20667.7 −1.64246 −0.821232 0.570594i \(-0.806713\pi\)
−0.821232 + 0.570594i \(0.806713\pi\)
\(542\) −31867.9 −2.52554
\(543\) 0 0
\(544\) −14080.5 −1.10973
\(545\) −20147.1 −1.58350
\(546\) 0 0
\(547\) −903.226 −0.0706018 −0.0353009 0.999377i \(-0.511239\pi\)
−0.0353009 + 0.999377i \(0.511239\pi\)
\(548\) −7607.61 −0.593031
\(549\) 0 0
\(550\) −41268.3 −3.19943
\(551\) 549.253 0.0424664
\(552\) 0 0
\(553\) 12411.1 0.954380
\(554\) 2685.92 0.205981
\(555\) 0 0
\(556\) −5578.82 −0.425530
\(557\) 19269.0 1.46580 0.732902 0.680334i \(-0.238165\pi\)
0.732902 + 0.680334i \(0.238165\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 35016.3 2.64233
\(561\) 0 0
\(562\) −20633.0 −1.54866
\(563\) 22628.3 1.69391 0.846953 0.531667i \(-0.178434\pi\)
0.846953 + 0.531667i \(0.178434\pi\)
\(564\) 0 0
\(565\) 16719.2 1.24492
\(566\) −5289.90 −0.392847
\(567\) 0 0
\(568\) −3159.95 −0.233430
\(569\) 25682.6 1.89222 0.946109 0.323850i \(-0.104977\pi\)
0.946109 + 0.323850i \(0.104977\pi\)
\(570\) 0 0
\(571\) 301.979 0.0221321 0.0110660 0.999939i \(-0.496477\pi\)
0.0110660 + 0.999939i \(0.496477\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6055.68 0.440347
\(575\) 18895.3 1.37041
\(576\) 0 0
\(577\) 17642.4 1.27290 0.636449 0.771319i \(-0.280402\pi\)
0.636449 + 0.771319i \(0.280402\pi\)
\(578\) −2390.02 −0.171993
\(579\) 0 0
\(580\) 527.753 0.0377824
\(581\) 6282.23 0.448590
\(582\) 0 0
\(583\) −19195.2 −1.36361
\(584\) −6542.59 −0.463586
\(585\) 0 0
\(586\) −30550.4 −2.15363
\(587\) 22820.1 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(588\) 0 0
\(589\) −28123.8 −1.96744
\(590\) −9271.87 −0.646977
\(591\) 0 0
\(592\) 7241.54 0.502746
\(593\) 13688.7 0.947940 0.473970 0.880541i \(-0.342821\pi\)
0.473970 + 0.880541i \(0.342821\pi\)
\(594\) 0 0
\(595\) −29571.5 −2.03750
\(596\) 12009.0 0.825348
\(597\) 0 0
\(598\) 0 0
\(599\) −20276.7 −1.38311 −0.691556 0.722323i \(-0.743075\pi\)
−0.691556 + 0.722323i \(0.743075\pi\)
\(600\) 0 0
\(601\) 3596.78 0.244119 0.122060 0.992523i \(-0.461050\pi\)
0.122060 + 0.992523i \(0.461050\pi\)
\(602\) 12703.6 0.860070
\(603\) 0 0
\(604\) 1417.69 0.0955052
\(605\) −20575.7 −1.38268
\(606\) 0 0
\(607\) 5556.39 0.371543 0.185772 0.982593i \(-0.440522\pi\)
0.185772 + 0.982593i \(0.440522\pi\)
\(608\) 23701.7 1.58097
\(609\) 0 0
\(610\) −42992.7 −2.85364
\(611\) 0 0
\(612\) 0 0
\(613\) −5608.79 −0.369555 −0.184777 0.982780i \(-0.559156\pi\)
−0.184777 + 0.982780i \(0.559156\pi\)
\(614\) −2320.41 −0.152515
\(615\) 0 0
\(616\) 10400.0 0.680240
\(617\) 23125.5 1.50891 0.754454 0.656353i \(-0.227902\pi\)
0.754454 + 0.656353i \(0.227902\pi\)
\(618\) 0 0
\(619\) −11536.0 −0.749065 −0.374533 0.927214i \(-0.622197\pi\)
−0.374533 + 0.927214i \(0.622197\pi\)
\(620\) −27022.9 −1.75043
\(621\) 0 0
\(622\) 317.039 0.0204375
\(623\) −20122.4 −1.29404
\(624\) 0 0
\(625\) 7476.78 0.478514
\(626\) −8416.13 −0.537342
\(627\) 0 0
\(628\) −20760.0 −1.31913
\(629\) −6115.54 −0.387667
\(630\) 0 0
\(631\) −23681.2 −1.49403 −0.747016 0.664806i \(-0.768514\pi\)
−0.747016 + 0.664806i \(0.768514\pi\)
\(632\) −4495.84 −0.282967
\(633\) 0 0
\(634\) 24280.0 1.52095
\(635\) 123.048 0.00768975
\(636\) 0 0
\(637\) 0 0
\(638\) 908.888 0.0564001
\(639\) 0 0
\(640\) −20073.9 −1.23983
\(641\) −14221.5 −0.876312 −0.438156 0.898899i \(-0.644368\pi\)
−0.438156 + 0.898899i \(0.644368\pi\)
\(642\) 0 0
\(643\) −16022.4 −0.982679 −0.491340 0.870968i \(-0.663493\pi\)
−0.491340 + 0.870968i \(0.663493\pi\)
\(644\) 11323.5 0.692868
\(645\) 0 0
\(646\) −26510.9 −1.61464
\(647\) 25539.8 1.55189 0.775945 0.630801i \(-0.217274\pi\)
0.775945 + 0.630801i \(0.217274\pi\)
\(648\) 0 0
\(649\) −6596.86 −0.398998
\(650\) 0 0
\(651\) 0 0
\(652\) 7139.83 0.428861
\(653\) −11206.7 −0.671597 −0.335798 0.941934i \(-0.609006\pi\)
−0.335798 + 0.941934i \(0.609006\pi\)
\(654\) 0 0
\(655\) −23773.9 −1.41821
\(656\) −5254.99 −0.312763
\(657\) 0 0
\(658\) −33801.5 −2.00261
\(659\) 2760.89 0.163200 0.0816002 0.996665i \(-0.473997\pi\)
0.0816002 + 0.996665i \(0.473997\pi\)
\(660\) 0 0
\(661\) 29868.4 1.75756 0.878781 0.477226i \(-0.158358\pi\)
0.878781 + 0.477226i \(0.158358\pi\)
\(662\) −9205.74 −0.540470
\(663\) 0 0
\(664\) −2275.70 −0.133004
\(665\) 49777.8 2.90271
\(666\) 0 0
\(667\) −416.148 −0.0241579
\(668\) −5288.17 −0.306296
\(669\) 0 0
\(670\) 8241.44 0.475216
\(671\) −30589.0 −1.75987
\(672\) 0 0
\(673\) −2238.65 −0.128222 −0.0641110 0.997943i \(-0.520421\pi\)
−0.0641110 + 0.997943i \(0.520421\pi\)
\(674\) −7715.03 −0.440908
\(675\) 0 0
\(676\) 0 0
\(677\) 27574.0 1.56537 0.782685 0.622418i \(-0.213849\pi\)
0.782685 + 0.622418i \(0.213849\pi\)
\(678\) 0 0
\(679\) 35772.6 2.02184
\(680\) 10712.1 0.604104
\(681\) 0 0
\(682\) −46538.5 −2.61298
\(683\) 1862.60 0.104349 0.0521745 0.998638i \(-0.483385\pi\)
0.0521745 + 0.998638i \(0.483385\pi\)
\(684\) 0 0
\(685\) 25338.5 1.41333
\(686\) −9210.08 −0.512598
\(687\) 0 0
\(688\) −11023.9 −0.610877
\(689\) 0 0
\(690\) 0 0
\(691\) 5999.62 0.330299 0.165149 0.986269i \(-0.447189\pi\)
0.165149 + 0.986269i \(0.447189\pi\)
\(692\) −10257.8 −0.563504
\(693\) 0 0
\(694\) 23882.2 1.30628
\(695\) 18581.2 1.01414
\(696\) 0 0
\(697\) 4437.87 0.241171
\(698\) −19372.3 −1.05051
\(699\) 0 0
\(700\) 30836.8 1.66503
\(701\) −13737.7 −0.740177 −0.370088 0.928997i \(-0.620673\pi\)
−0.370088 + 0.928997i \(0.620673\pi\)
\(702\) 0 0
\(703\) 10294.3 0.552286
\(704\) 8734.99 0.467631
\(705\) 0 0
\(706\) 37917.5 2.02131
\(707\) −42005.8 −2.23450
\(708\) 0 0
\(709\) 1556.55 0.0824507 0.0412254 0.999150i \(-0.486874\pi\)
0.0412254 + 0.999150i \(0.486874\pi\)
\(710\) −25027.7 −1.32292
\(711\) 0 0
\(712\) 7289.23 0.383674
\(713\) 21308.3 1.11922
\(714\) 0 0
\(715\) 0 0
\(716\) 3636.63 0.189814
\(717\) 0 0
\(718\) 11815.6 0.614142
\(719\) 15740.2 0.816428 0.408214 0.912886i \(-0.366152\pi\)
0.408214 + 0.912886i \(0.366152\pi\)
\(720\) 0 0
\(721\) 24624.9 1.27196
\(722\) 19301.7 0.994926
\(723\) 0 0
\(724\) 13444.6 0.690146
\(725\) −1133.28 −0.0580539
\(726\) 0 0
\(727\) 17533.2 0.894459 0.447230 0.894419i \(-0.352411\pi\)
0.447230 + 0.894419i \(0.352411\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −51819.2 −2.62728
\(731\) 9309.80 0.471047
\(732\) 0 0
\(733\) −32766.4 −1.65110 −0.825549 0.564330i \(-0.809135\pi\)
−0.825549 + 0.564330i \(0.809135\pi\)
\(734\) 84.7639 0.00426252
\(735\) 0 0
\(736\) −17957.9 −0.899369
\(737\) 5863.72 0.293070
\(738\) 0 0
\(739\) 31683.2 1.57711 0.788557 0.614962i \(-0.210829\pi\)
0.788557 + 0.614962i \(0.210829\pi\)
\(740\) 9891.35 0.491369
\(741\) 0 0
\(742\) 34718.1 1.71771
\(743\) −19757.8 −0.975563 −0.487781 0.872966i \(-0.662194\pi\)
−0.487781 + 0.872966i \(0.662194\pi\)
\(744\) 0 0
\(745\) −39998.0 −1.96700
\(746\) 22872.3 1.12254
\(747\) 0 0
\(748\) −18124.0 −0.885933
\(749\) 36399.6 1.77572
\(750\) 0 0
\(751\) 170.877 0.00830279 0.00415140 0.999991i \(-0.498679\pi\)
0.00415140 + 0.999991i \(0.498679\pi\)
\(752\) 29332.1 1.42238
\(753\) 0 0
\(754\) 0 0
\(755\) −4721.88 −0.227612
\(756\) 0 0
\(757\) 32442.0 1.55763 0.778815 0.627253i \(-0.215821\pi\)
0.778815 + 0.627253i \(0.215821\pi\)
\(758\) −25679.6 −1.23051
\(759\) 0 0
\(760\) −18031.7 −0.860632
\(761\) 4594.95 0.218879 0.109439 0.993993i \(-0.465094\pi\)
0.109439 + 0.993993i \(0.465094\pi\)
\(762\) 0 0
\(763\) 25926.5 1.23015
\(764\) 6343.32 0.300384
\(765\) 0 0
\(766\) 4508.30 0.212652
\(767\) 0 0
\(768\) 0 0
\(769\) 21213.7 0.994782 0.497391 0.867526i \(-0.334291\pi\)
0.497391 + 0.867526i \(0.334291\pi\)
\(770\) 82370.9 3.85512
\(771\) 0 0
\(772\) 18351.7 0.855560
\(773\) −2816.60 −0.131056 −0.0655278 0.997851i \(-0.520873\pi\)
−0.0655278 + 0.997851i \(0.520873\pi\)
\(774\) 0 0
\(775\) 58028.3 2.68960
\(776\) −12958.4 −0.599459
\(777\) 0 0
\(778\) 23328.9 1.07504
\(779\) −7470.29 −0.343583
\(780\) 0 0
\(781\) −17807.0 −0.815857
\(782\) 20086.3 0.918524
\(783\) 0 0
\(784\) −18534.5 −0.844318
\(785\) 69144.9 3.14381
\(786\) 0 0
\(787\) 4312.13 0.195312 0.0976562 0.995220i \(-0.468865\pi\)
0.0976562 + 0.995220i \(0.468865\pi\)
\(788\) −1010.86 −0.0456985
\(789\) 0 0
\(790\) −35608.3 −1.60365
\(791\) −21515.4 −0.967128
\(792\) 0 0
\(793\) 0 0
\(794\) −11392.9 −0.509219
\(795\) 0 0
\(796\) 16632.8 0.740622
\(797\) 1009.62 0.0448714 0.0224357 0.999748i \(-0.492858\pi\)
0.0224357 + 0.999748i \(0.492858\pi\)
\(798\) 0 0
\(799\) −24771.2 −1.09680
\(800\) −48904.1 −2.16128
\(801\) 0 0
\(802\) −39801.0 −1.75240
\(803\) −36868.9 −1.62027
\(804\) 0 0
\(805\) −37714.8 −1.65127
\(806\) 0 0
\(807\) 0 0
\(808\) 15216.4 0.662512
\(809\) −7369.64 −0.320275 −0.160138 0.987095i \(-0.551194\pi\)
−0.160138 + 0.987095i \(0.551194\pi\)
\(810\) 0 0
\(811\) −6713.40 −0.290678 −0.145339 0.989382i \(-0.546427\pi\)
−0.145339 + 0.989382i \(0.546427\pi\)
\(812\) −679.147 −0.0293515
\(813\) 0 0
\(814\) 17034.7 0.733497
\(815\) −23780.4 −1.02208
\(816\) 0 0
\(817\) −15671.2 −0.671073
\(818\) −28182.9 −1.20463
\(819\) 0 0
\(820\) −7177.88 −0.305686
\(821\) 2409.11 0.102410 0.0512049 0.998688i \(-0.483694\pi\)
0.0512049 + 0.998688i \(0.483694\pi\)
\(822\) 0 0
\(823\) −2370.79 −0.100414 −0.0502069 0.998739i \(-0.515988\pi\)
−0.0502069 + 0.998739i \(0.515988\pi\)
\(824\) −8920.25 −0.377126
\(825\) 0 0
\(826\) 11931.6 0.502609
\(827\) −10168.6 −0.427567 −0.213783 0.976881i \(-0.568579\pi\)
−0.213783 + 0.976881i \(0.568579\pi\)
\(828\) 0 0
\(829\) −19837.3 −0.831095 −0.415547 0.909572i \(-0.636410\pi\)
−0.415547 + 0.909572i \(0.636410\pi\)
\(830\) −18024.2 −0.753770
\(831\) 0 0
\(832\) 0 0
\(833\) 15652.5 0.651052
\(834\) 0 0
\(835\) 17613.2 0.729975
\(836\) 30508.2 1.26214
\(837\) 0 0
\(838\) −12954.5 −0.534016
\(839\) 1470.08 0.0604919 0.0302459 0.999542i \(-0.490371\pi\)
0.0302459 + 0.999542i \(0.490371\pi\)
\(840\) 0 0
\(841\) −24364.0 −0.998977
\(842\) −46069.2 −1.88557
\(843\) 0 0
\(844\) 17618.3 0.718540
\(845\) 0 0
\(846\) 0 0
\(847\) 26478.2 1.07415
\(848\) −30127.6 −1.22003
\(849\) 0 0
\(850\) 54700.4 2.20731
\(851\) −7799.61 −0.314180
\(852\) 0 0
\(853\) 10200.0 0.409426 0.204713 0.978822i \(-0.434374\pi\)
0.204713 + 0.978822i \(0.434374\pi\)
\(854\) 55325.8 2.21687
\(855\) 0 0
\(856\) −13185.5 −0.526486
\(857\) −2379.48 −0.0948443 −0.0474222 0.998875i \(-0.515101\pi\)
−0.0474222 + 0.998875i \(0.515101\pi\)
\(858\) 0 0
\(859\) 9651.19 0.383346 0.191673 0.981459i \(-0.438609\pi\)
0.191673 + 0.981459i \(0.438609\pi\)
\(860\) −15057.8 −0.597054
\(861\) 0 0
\(862\) −65518.4 −2.58882
\(863\) 19568.7 0.771872 0.385936 0.922526i \(-0.373879\pi\)
0.385936 + 0.922526i \(0.373879\pi\)
\(864\) 0 0
\(865\) 34165.5 1.34296
\(866\) 35799.4 1.40475
\(867\) 0 0
\(868\) 34774.9 1.35984
\(869\) −25335.0 −0.988989
\(870\) 0 0
\(871\) 0 0
\(872\) −9391.75 −0.364730
\(873\) 0 0
\(874\) −33811.4 −1.30857
\(875\) −46110.9 −1.78152
\(876\) 0 0
\(877\) −1978.67 −0.0761856 −0.0380928 0.999274i \(-0.512128\pi\)
−0.0380928 + 0.999274i \(0.512128\pi\)
\(878\) 49467.2 1.90141
\(879\) 0 0
\(880\) −71479.6 −2.73816
\(881\) −4802.77 −0.183665 −0.0918327 0.995774i \(-0.529273\pi\)
−0.0918327 + 0.995774i \(0.529273\pi\)
\(882\) 0 0
\(883\) −49531.7 −1.88774 −0.943871 0.330315i \(-0.892845\pi\)
−0.943871 + 0.330315i \(0.892845\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −50324.1 −1.90821
\(887\) 7718.74 0.292187 0.146093 0.989271i \(-0.453330\pi\)
0.146093 + 0.989271i \(0.453330\pi\)
\(888\) 0 0
\(889\) −158.346 −0.00597384
\(890\) 57732.8 2.17439
\(891\) 0 0
\(892\) 21142.2 0.793600
\(893\) 41697.5 1.56255
\(894\) 0 0
\(895\) −12112.4 −0.452373
\(896\) 25832.4 0.963170
\(897\) 0 0
\(898\) 5101.72 0.189584
\(899\) −1278.01 −0.0474128
\(900\) 0 0
\(901\) 25442.9 0.940763
\(902\) −12361.6 −0.456316
\(903\) 0 0
\(904\) 7793.82 0.286746
\(905\) −44779.7 −1.64478
\(906\) 0 0
\(907\) 1788.58 0.0654783 0.0327392 0.999464i \(-0.489577\pi\)
0.0327392 + 0.999464i \(0.489577\pi\)
\(908\) 8726.44 0.318940
\(909\) 0 0
\(910\) 0 0
\(911\) 8239.52 0.299657 0.149828 0.988712i \(-0.452128\pi\)
0.149828 + 0.988712i \(0.452128\pi\)
\(912\) 0 0
\(913\) −12824.1 −0.464857
\(914\) 30502.8 1.10388
\(915\) 0 0
\(916\) −7686.39 −0.277255
\(917\) 30593.9 1.10174
\(918\) 0 0
\(919\) 21053.8 0.755713 0.377857 0.925864i \(-0.376661\pi\)
0.377857 + 0.925864i \(0.376661\pi\)
\(920\) 13662.0 0.489589
\(921\) 0 0
\(922\) 62301.7 2.22538
\(923\) 0 0
\(924\) 0 0
\(925\) −21240.4 −0.755006
\(926\) −16492.4 −0.585286
\(927\) 0 0
\(928\) 1077.06 0.0380994
\(929\) 34539.0 1.21979 0.609896 0.792481i \(-0.291211\pi\)
0.609896 + 0.792481i \(0.291211\pi\)
\(930\) 0 0
\(931\) −26347.9 −0.927516
\(932\) 9380.70 0.329694
\(933\) 0 0
\(934\) 10084.6 0.353294
\(935\) 60365.1 2.11139
\(936\) 0 0
\(937\) −6368.78 −0.222048 −0.111024 0.993818i \(-0.535413\pi\)
−0.111024 + 0.993818i \(0.535413\pi\)
\(938\) −10605.6 −0.369175
\(939\) 0 0
\(940\) 40065.3 1.39020
\(941\) 4504.64 0.156054 0.0780272 0.996951i \(-0.475138\pi\)
0.0780272 + 0.996951i \(0.475138\pi\)
\(942\) 0 0
\(943\) 5659.96 0.195454
\(944\) −10354.0 −0.356985
\(945\) 0 0
\(946\) −25932.3 −0.891259
\(947\) −38765.3 −1.33020 −0.665101 0.746753i \(-0.731612\pi\)
−0.665101 + 0.746753i \(0.731612\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −92077.5 −3.14462
\(951\) 0 0
\(952\) −13785.0 −0.469302
\(953\) 21945.3 0.745936 0.372968 0.927844i \(-0.378340\pi\)
0.372968 + 0.927844i \(0.378340\pi\)
\(954\) 0 0
\(955\) −21127.5 −0.715886
\(956\) −33509.1 −1.13364
\(957\) 0 0
\(958\) 20790.0 0.701141
\(959\) −32607.2 −1.09796
\(960\) 0 0
\(961\) 35648.0 1.19660
\(962\) 0 0
\(963\) 0 0
\(964\) 9613.38 0.321189
\(965\) −61123.5 −2.03900
\(966\) 0 0
\(967\) 58962.4 1.96081 0.980405 0.196991i \(-0.0631169\pi\)
0.980405 + 0.196991i \(0.0631169\pi\)
\(968\) −9591.58 −0.318476
\(969\) 0 0
\(970\) −102634. −3.39731
\(971\) −51664.3 −1.70751 −0.853753 0.520679i \(-0.825679\pi\)
−0.853753 + 0.520679i \(0.825679\pi\)
\(972\) 0 0
\(973\) −23911.5 −0.787841
\(974\) 47667.0 1.56812
\(975\) 0 0
\(976\) −48010.5 −1.57457
\(977\) −17888.8 −0.585788 −0.292894 0.956145i \(-0.594618\pi\)
−0.292894 + 0.956145i \(0.594618\pi\)
\(978\) 0 0
\(979\) 41076.4 1.34097
\(980\) −25316.5 −0.825212
\(981\) 0 0
\(982\) −11499.7 −0.373697
\(983\) 40916.4 1.32760 0.663800 0.747910i \(-0.268942\pi\)
0.663800 + 0.747910i \(0.268942\pi\)
\(984\) 0 0
\(985\) 3366.85 0.108910
\(986\) −1204.72 −0.0389108
\(987\) 0 0
\(988\) 0 0
\(989\) 11873.5 0.381754
\(990\) 0 0
\(991\) 27804.1 0.891249 0.445624 0.895220i \(-0.352982\pi\)
0.445624 + 0.895220i \(0.352982\pi\)
\(992\) −55149.5 −1.76512
\(993\) 0 0
\(994\) 32207.2 1.02772
\(995\) −55398.5 −1.76508
\(996\) 0 0
\(997\) 41605.5 1.32162 0.660812 0.750552i \(-0.270212\pi\)
0.660812 + 0.750552i \(0.270212\pi\)
\(998\) 42967.7 1.36284
\(999\) 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 1521.4.a.bd.1.7 8
3.2 odd 2 inner 1521.4.a.bd.1.2 8
13.4 even 6 117.4.g.f.55.7 yes 16
13.10 even 6 117.4.g.f.100.7 yes 16
13.12 even 2 1521.4.a.bc.1.2 8
39.17 odd 6 117.4.g.f.55.2 16
39.23 odd 6 117.4.g.f.100.2 yes 16
39.38 odd 2 1521.4.a.bc.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.2 16 39.17 odd 6
117.4.g.f.55.7 yes 16 13.4 even 6
117.4.g.f.100.2 yes 16 39.23 odd 6
117.4.g.f.100.7 yes 16 13.10 even 6
1521.4.a.bc.1.2 8 13.12 even 2
1521.4.a.bc.1.7 8 39.38 odd 2
1521.4.a.bd.1.2 8 3.2 odd 2 inner
1521.4.a.bd.1.7 8 1.1 even 1 trivial