Properties

Label 1521.4.a.bc.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 \(-0.725932\pi\)
−0.651672 + 0.758500i \(0.725932\pi\)
\(8\) −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 \(-0.654268\pi\)
−0.465897 + 0.884839i \(0.654268\pi\)
\(18\) 0 0
\(19\) 109.940 1.32747 0.663737 0.747966i \(-0.268970\pi\)
0.663737 + 0.747966i \(0.268970\pi\)
\(20\) −105.637 −1.18105
\(21\) 0 0
\(22\) −181.926 −1.76303
\(23\) −83.2974 −0.755161 −0.377581 0.925977i \(-0.623244\pi\)
−0.377581 + 0.925977i \(0.623244\pi\)
\(24\) 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 \(-0.494908\pi\)
0.0159952 + 0.999872i \(0.494908\pi\)
\(30\) 0 0
\(31\) −255.810 −1.48209 −0.741047 0.671453i \(-0.765670\pi\)
−0.741047 + 0.671453i \(0.765670\pi\)
\(32\) −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 \(-0.433298\pi\)
0.208022 + 0.978124i \(0.433298\pi\)
\(38\) 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 \(-0.668438\pi\)
−0.504813 + 0.863229i \(0.668438\pi\)
\(54\) 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 \(-0.465397\pi\)
0.108496 + 0.994097i \(0.465397\pi\)
\(68\) −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 \(-0.704765\pi\)
−0.599829 + 0.800128i \(0.704765\pi\)
\(74\) 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 \(-0.782570\pi\)
−0.775634 + 0.631183i \(0.782570\pi\)
\(98\) 884.841 0.912066
\(99\) 0 0
\(100\) 1277.51 1.27751
\(101\) 1740.21 1.71443 0.857217 0.514956i \(-0.172192\pi\)
0.857217 + 0.514956i \(0.172192\pi\)
\(102\) 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 \(-0.738548\pi\)
−0.681215 + 0.732084i \(0.738548\pi\)
\(108\) 0 0
\(109\) −1074.08 −0.943840 −0.471920 0.881641i \(-0.656439\pi\)
−0.471920 + 0.881641i \(0.656439\pi\)
\(110\) 3412.46 2.95787
\(111\) 0 0
\(112\) 1866.79 1.57496
\(113\) 891.337 0.742035 0.371018 0.928626i \(-0.379009\pi\)
0.371018 + 0.928626i \(0.379009\pi\)
\(114\) 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 \(-0.638903\pi\)
−0.422660 + 0.906288i \(0.638903\pi\)
\(132\) 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 \(-0.521609\pi\)
−0.0678338 + 0.997697i \(0.521609\pi\)
\(152\) −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 \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(164\) 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 \(-0.368928\pi\)
0.400236 + 0.916412i \(0.368928\pi\)
\(174\) 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 \(-0.543045\pi\)
−0.134818 + 0.990870i \(0.543045\pi\)
\(180\) 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 \(-0.568438\pi\)
−0.213351 + 0.976976i \(0.568438\pi\)
\(192\) 0 0
\(193\) −3258.63 −1.21534 −0.607672 0.794188i \(-0.707896\pi\)
−0.607672 + 0.794188i \(0.707896\pi\)
\(194\) −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 \(-0.690609\pi\)
−0.563665 + 0.826004i \(0.690609\pi\)
\(224\) 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 \(-0.436905\pi\)
0.196924 + 0.980419i \(0.436905\pi\)
\(230\) 5768.73 1.65382
\(231\) 0 0
\(232\) −43.6842 −0.0123621
\(233\) −1665.69 −0.468339 −0.234170 0.972196i \(-0.575237\pi\)
−0.234170 + 0.972196i \(0.575237\pi\)
\(234\) 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 \(-0.573261\pi\)
−0.228128 + 0.973631i \(0.573261\pi\)
\(242\) 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 \(-0.512266\pi\)
−0.0385242 + 0.999258i \(0.512266\pi\)
\(252\) 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 \(-0.642040\pi\)
−0.431570 + 0.902079i \(0.642040\pi\)
\(258\) 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 \(-0.185565\pi\)
0.834831 + 0.550506i \(0.185565\pi\)
\(264\) 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 \(-0.759303\pi\)
−0.727467 + 0.686143i \(0.759303\pi\)
\(270\) 0 0
\(271\) 8631.33 1.93474 0.967372 0.253360i \(-0.0815357\pi\)
0.967372 + 0.253360i \(0.0815357\pi\)
\(272\) 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 \(-0.481394\pi\)
0.0584187 + 0.998292i \(0.481394\pi\)
\(308\) 6698.33 1.23920
\(309\) 0 0
\(310\) 17716.1 3.24582
\(311\) −85.8693 −0.0156566 −0.00782830 0.999969i \(-0.502492\pi\)
−0.00782830 + 0.999969i \(0.502492\pi\)
\(312\) 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 \(-0.433624\pi\)
0.207020 + 0.978337i \(0.433624\pi\)
\(332\) 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 \(-0.666796\pi\)
−0.500351 + 0.865823i \(0.666796\pi\)
\(348\) 0 0
\(349\) 5246.94 0.804763 0.402382 0.915472i \(-0.368183\pi\)
0.402382 + 0.915472i \(0.368183\pi\)
\(350\) −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 \(-0.343775\pi\)
0.471328 + 0.881958i \(0.343775\pi\)
\(380\) −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 \(-0.635093\pi\)
−0.411780 + 0.911283i \(0.635093\pi\)
\(390\) 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 \(-0.437513\pi\)
0.195049 + 0.980793i \(0.437513\pi\)
\(398\) 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 \(-0.347341\pi\)
0.461418 + 0.887183i \(0.347341\pi\)
\(410\) −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 \(-0.434428\pi\)
0.204547 + 0.978857i \(0.434428\pi\)
\(420\) 0 0
\(421\) 12477.7 1.44448 0.722241 0.691642i \(-0.243112\pi\)
0.722241 + 0.691642i \(0.243112\pi\)
\(422\) 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 \(-0.239095\pi\)
0.730913 + 0.682471i \(0.239095\pi\)
\(444\) 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 \(-0.638961\pi\)
−0.422824 + 0.906212i \(0.638961\pi\)
\(458\) 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 \(-0.428028\pi\)
0.224186 + 0.974546i \(0.428028\pi\)
\(464\) −386.373 −0.0386571
\(465\) 0 0
\(466\) −6149.92 −0.611351
\(467\) −2731.37 −0.270649 −0.135324 0.990801i \(-0.543208\pi\)
−0.135324 + 0.990801i \(0.543208\pi\)
\(468\) 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 \(-0.705090\pi\)
−0.600647 + 0.799514i \(0.705090\pi\)
\(488\) −5428.18 −0.503529
\(489\) 0 0
\(490\) −16597.3 −1.53019
\(491\) 3114.67 0.286279 0.143139 0.989703i \(-0.454280\pi\)
0.143139 + 0.989703i \(0.454280\pi\)
\(492\) 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 \(-0.674821\pi\)
−0.522018 + 0.852934i \(0.674821\pi\)
\(500\) −10758.2 −0.962239
\(501\) 0 0
\(502\) −1131.23 −0.100576
\(503\) 2832.94 0.251122 0.125561 0.992086i \(-0.459927\pi\)
0.125561 + 0.992086i \(0.459927\pi\)
\(504\) 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 \(-0.711881\pi\)
−0.617565 + 0.786520i \(0.711881\pi\)
\(522\) 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 \(-0.193287\pi\)
0.821232 + 0.570594i \(0.193287\pi\)
\(542\) 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 \(-0.821566\pi\)
−0.846953 + 0.531667i \(0.821566\pi\)
\(564\) 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 \(-0.895023\pi\)
−0.946109 + 0.323850i \(0.895023\pi\)
\(570\) 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 \(-0.719598\pi\)
−0.636449 + 0.771319i \(0.719598\pi\)
\(578\) −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 \(-0.256925\pi\)
0.691556 + 0.722323i \(0.256925\pi\)
\(600\) 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 \(-0.440844\pi\)
0.184777 + 0.982780i \(0.440844\pi\)
\(614\) 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 \(-0.377803\pi\)
0.374533 + 0.927214i \(0.377803\pi\)
\(620\) 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 \(-0.231486\pi\)
0.747016 + 0.664806i \(0.231486\pi\)
\(632\) −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 \(-0.355632\pi\)
0.438156 + 0.898899i \(0.355632\pi\)
\(642\) 0 0
\(643\) 16022.4 0.982679 0.491340 0.870968i \(-0.336507\pi\)
0.491340 + 0.870968i \(0.336507\pi\)
\(644\) 11323.5 0.692868
\(645\) 0 0
\(646\) −26510.9 −1.61464
\(647\) −25539.8 −1.55189 −0.775945 0.630801i \(-0.782726\pi\)
−0.775945 + 0.630801i \(0.782726\pi\)
\(648\) 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 \(-0.390994\pi\)
0.335798 + 0.941934i \(0.390994\pi\)
\(654\) 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 \(-0.526003\pi\)
−0.0816002 + 0.996665i \(0.526003\pi\)
\(660\) 0 0
\(661\) −29868.4 −1.75756 −0.878781 0.477226i \(-0.841642\pi\)
−0.878781 + 0.477226i \(0.841642\pi\)
\(662\) 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 \(-0.786151\pi\)
−0.782685 + 0.622418i \(0.786151\pi\)
\(678\) 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 \(-0.552811\pi\)
−0.165149 + 0.986269i \(0.552811\pi\)
\(692\) 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 \(-0.379327\pi\)
0.370088 + 0.928997i \(0.379327\pi\)
\(702\) 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 \(-0.513126\pi\)
−0.0412254 + 0.999150i \(0.513126\pi\)
\(710\) −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 \(-0.633848\pi\)
−0.408214 + 0.912886i \(0.633848\pi\)
\(720\) 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 \(-0.190865\pi\)
0.825549 + 0.564330i \(0.190865\pi\)
\(734\) 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 \(-0.789171\pi\)
−0.788557 + 0.614962i \(0.789171\pi\)
\(740\) −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 \(-0.665709\pi\)
−0.497391 + 0.867526i \(0.665709\pi\)
\(770\) −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 \(-0.531135\pi\)
−0.0976562 + 0.995220i \(0.531135\pi\)
\(788\) −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 \(-0.507142\pi\)
−0.0224357 + 0.999748i \(0.507142\pi\)
\(798\) 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 \(-0.448806\pi\)
0.160138 + 0.987095i \(0.448806\pi\)
\(810\) 0 0
\(811\) 6713.40 0.290678 0.145339 0.989382i \(-0.453573\pi\)
0.145339 + 0.989382i \(0.453573\pi\)
\(812\) −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 \(-0.565626\pi\)
−0.204713 + 0.978822i \(0.565626\pi\)
\(854\) −55325.8 −2.21687
\(855\) 0 0
\(856\) 13185.5 0.526486
\(857\) 2379.48 0.0948443 0.0474222 0.998875i \(-0.484899\pi\)
0.0474222 + 0.998875i \(0.484899\pi\)
\(858\) 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 \(-0.487872\pi\)
0.0380928 + 0.999274i \(0.487872\pi\)
\(878\) 49467.2 1.90141
\(879\) 0 0
\(880\) −71479.6 −2.73816
\(881\) 4802.77 0.183665 0.0918327 0.995774i \(-0.470727\pi\)
0.0918327 + 0.995774i \(0.470727\pi\)
\(882\) 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 \(-0.546670\pi\)
−0.146093 + 0.989271i \(0.546670\pi\)
\(888\) 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 \(-0.547872\pi\)
−0.149828 + 0.988712i \(0.547872\pi\)
\(912\) 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 \(-0.621660\pi\)
−0.372968 + 0.927844i \(0.621660\pi\)
\(954\) 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 \(-0.936883\pi\)
−0.980405 + 0.196991i \(0.936883\pi\)
\(968\) −9591.58 −0.318476
\(969\) 0 0
\(970\) 102634. 3.39731
\(971\) 51664.3 1.70751 0.853753 0.520679i \(-0.174321\pi\)
0.853753 + 0.520679i \(0.174321\pi\)
\(972\) 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.bc.1.7 8
3.2 odd 2 inner 1521.4.a.bc.1.2 8
13.3 even 3 117.4.g.f.100.2 yes 16
13.9 even 3 117.4.g.f.55.2 16
13.12 even 2 1521.4.a.bd.1.2 8
39.29 odd 6 117.4.g.f.100.7 yes 16
39.35 odd 6 117.4.g.f.55.7 yes 16
39.38 odd 2 1521.4.a.bd.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.2 16 13.9 even 3
117.4.g.f.55.7 yes 16 39.35 odd 6
117.4.g.f.100.2 yes 16 13.3 even 3
117.4.g.f.100.7 yes 16 39.29 odd 6
1521.4.a.bc.1.2 8 3.2 odd 2 inner
1521.4.a.bc.1.7 8 1.1 even 1 trivial
1521.4.a.bd.1.2 8 13.12 even 2
1521.4.a.bd.1.7 8 39.38 odd 2