Properties

Label 168.4.a.h.1.1
Level $168$
Weight $4$
Character 168.1
Self dual yes
Analytic conductor $9.912$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.15207\) of defining polynomial
Character \(\chi\) \(=\) 168.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -6.30413 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -6.30413 q^{5} +7.00000 q^{7} +9.00000 q^{9} +48.9124 q^{11} -2.60827 q^{13} -18.9124 q^{15} +136.737 q^{17} +45.2165 q^{19} +21.0000 q^{21} -38.1289 q^{23} -85.2579 q^{25} +27.0000 q^{27} +52.7835 q^{29} -14.7835 q^{31} +146.737 q^{33} -44.1289 q^{35} +333.908 q^{37} -7.82481 q^{39} +227.263 q^{41} -398.433 q^{43} -56.7372 q^{45} -184.608 q^{47} +49.0000 q^{49} +410.212 q^{51} +359.825 q^{53} -308.350 q^{55} +135.650 q^{57} +99.9075 q^{59} -674.516 q^{61} +63.0000 q^{63} +16.4429 q^{65} -376.959 q^{67} -114.387 q^{69} -1187.60 q^{71} -735.825 q^{73} -255.774 q^{75} +342.387 q^{77} -836.774 q^{79} +81.0000 q^{81} +293.732 q^{83} -862.010 q^{85} +158.350 q^{87} +1298.89 q^{89} -18.2579 q^{91} -44.3504 q^{93} -285.051 q^{95} -201.041 q^{97} +440.212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + 18q^{11} + 48q^{13} + 42q^{15} + 34q^{17} - 16q^{19} + 42q^{21} + 110q^{23} + 202q^{25} + 54q^{27} + 212q^{29} - 136q^{31} + 54q^{33} + 98q^{35} - 24q^{37} + 144q^{39} + 694q^{41} - 584q^{43} + 126q^{45} - 316q^{47} + 98q^{49} + 102q^{51} + 560q^{53} - 936q^{55} - 48q^{57} - 492q^{59} - 604q^{61} + 126q^{63} + 1044q^{65} - 1020q^{67} + 330q^{69} - 1710q^{71} - 1312q^{73} + 606q^{75} + 126q^{77} - 556q^{79} + 162q^{81} - 264q^{83} - 2948q^{85} + 636q^{87} + 70q^{89} + 336q^{91} - 408q^{93} - 1528q^{95} - 136q^{97} + 162q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −6.30413 −0.563859 −0.281929 0.959435i \(-0.590974\pi\)
−0.281929 + 0.959435i \(0.590974\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 48.9124 1.34069 0.670347 0.742047i \(-0.266145\pi\)
0.670347 + 0.742047i \(0.266145\pi\)
\(12\) 0 0
\(13\) −2.60827 −0.0556464 −0.0278232 0.999613i \(-0.508858\pi\)
−0.0278232 + 0.999613i \(0.508858\pi\)
\(14\) 0 0
\(15\) −18.9124 −0.325544
\(16\) 0 0
\(17\) 136.737 1.95080 0.975401 0.220436i \(-0.0707482\pi\)
0.975401 + 0.220436i \(0.0707482\pi\)
\(18\) 0 0
\(19\) 45.2165 0.545968 0.272984 0.962019i \(-0.411989\pi\)
0.272984 + 0.962019i \(0.411989\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) −38.1289 −0.345671 −0.172836 0.984951i \(-0.555293\pi\)
−0.172836 + 0.984951i \(0.555293\pi\)
\(24\) 0 0
\(25\) −85.2579 −0.682063
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 52.7835 0.337988 0.168994 0.985617i \(-0.445948\pi\)
0.168994 + 0.985617i \(0.445948\pi\)
\(30\) 0 0
\(31\) −14.7835 −0.0856512 −0.0428256 0.999083i \(-0.513636\pi\)
−0.0428256 + 0.999083i \(0.513636\pi\)
\(32\) 0 0
\(33\) 146.737 0.774051
\(34\) 0 0
\(35\) −44.1289 −0.213119
\(36\) 0 0
\(37\) 333.908 1.48362 0.741812 0.670608i \(-0.233967\pi\)
0.741812 + 0.670608i \(0.233967\pi\)
\(38\) 0 0
\(39\) −7.82481 −0.0321275
\(40\) 0 0
\(41\) 227.263 0.865670 0.432835 0.901473i \(-0.357513\pi\)
0.432835 + 0.901473i \(0.357513\pi\)
\(42\) 0 0
\(43\) −398.433 −1.41303 −0.706517 0.707696i \(-0.749735\pi\)
−0.706517 + 0.707696i \(0.749735\pi\)
\(44\) 0 0
\(45\) −56.7372 −0.187953
\(46\) 0 0
\(47\) −184.608 −0.572934 −0.286467 0.958090i \(-0.592481\pi\)
−0.286467 + 0.958090i \(0.592481\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 410.212 1.12630
\(52\) 0 0
\(53\) 359.825 0.932561 0.466281 0.884637i \(-0.345594\pi\)
0.466281 + 0.884637i \(0.345594\pi\)
\(54\) 0 0
\(55\) −308.350 −0.755963
\(56\) 0 0
\(57\) 135.650 0.315215
\(58\) 0 0
\(59\) 99.9075 0.220455 0.110228 0.993906i \(-0.464842\pi\)
0.110228 + 0.993906i \(0.464842\pi\)
\(60\) 0 0
\(61\) −674.516 −1.41579 −0.707893 0.706320i \(-0.750354\pi\)
−0.707893 + 0.706320i \(0.750354\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 16.4429 0.0313767
\(66\) 0 0
\(67\) −376.959 −0.687356 −0.343678 0.939088i \(-0.611673\pi\)
−0.343678 + 0.939088i \(0.611673\pi\)
\(68\) 0 0
\(69\) −114.387 −0.199573
\(70\) 0 0
\(71\) −1187.60 −1.98511 −0.992553 0.121810i \(-0.961130\pi\)
−0.992553 + 0.121810i \(0.961130\pi\)
\(72\) 0 0
\(73\) −735.825 −1.17975 −0.589875 0.807494i \(-0.700823\pi\)
−0.589875 + 0.807494i \(0.700823\pi\)
\(74\) 0 0
\(75\) −255.774 −0.393789
\(76\) 0 0
\(77\) 342.387 0.506735
\(78\) 0 0
\(79\) −836.774 −1.19170 −0.595851 0.803095i \(-0.703185\pi\)
−0.595851 + 0.803095i \(0.703185\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 293.732 0.388450 0.194225 0.980957i \(-0.437781\pi\)
0.194225 + 0.980957i \(0.437781\pi\)
\(84\) 0 0
\(85\) −862.010 −1.09998
\(86\) 0 0
\(87\) 158.350 0.195137
\(88\) 0 0
\(89\) 1298.89 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(90\) 0 0
\(91\) −18.2579 −0.0210324
\(92\) 0 0
\(93\) −44.3504 −0.0494508
\(94\) 0 0
\(95\) −285.051 −0.307849
\(96\) 0 0
\(97\) −201.041 −0.210440 −0.105220 0.994449i \(-0.533555\pi\)
−0.105220 + 0.994449i \(0.533555\pi\)
\(98\) 0 0
\(99\) 440.212 0.446898
\(100\) 0 0
\(101\) −1053.51 −1.03790 −0.518952 0.854804i \(-0.673678\pi\)
−0.518952 + 0.854804i \(0.673678\pi\)
\(102\) 0 0
\(103\) 1025.73 0.981247 0.490623 0.871372i \(-0.336769\pi\)
0.490623 + 0.871372i \(0.336769\pi\)
\(104\) 0 0
\(105\) −132.387 −0.123044
\(106\) 0 0
\(107\) −103.418 −0.0934377 −0.0467188 0.998908i \(-0.514876\pi\)
−0.0467188 + 0.998908i \(0.514876\pi\)
\(108\) 0 0
\(109\) −677.134 −0.595024 −0.297512 0.954718i \(-0.596157\pi\)
−0.297512 + 0.954718i \(0.596157\pi\)
\(110\) 0 0
\(111\) 1001.72 0.856570
\(112\) 0 0
\(113\) −452.083 −0.376357 −0.188179 0.982135i \(-0.560258\pi\)
−0.188179 + 0.982135i \(0.560258\pi\)
\(114\) 0 0
\(115\) 240.370 0.194910
\(116\) 0 0
\(117\) −23.4744 −0.0185488
\(118\) 0 0
\(119\) 957.160 0.737334
\(120\) 0 0
\(121\) 1061.42 0.797463
\(122\) 0 0
\(123\) 681.788 0.499795
\(124\) 0 0
\(125\) 1325.49 0.948446
\(126\) 0 0
\(127\) −1182.88 −0.826482 −0.413241 0.910622i \(-0.635603\pi\)
−0.413241 + 0.910622i \(0.635603\pi\)
\(128\) 0 0
\(129\) −1195.30 −0.815816
\(130\) 0 0
\(131\) 1257.75 0.838857 0.419429 0.907788i \(-0.362230\pi\)
0.419429 + 0.907788i \(0.362230\pi\)
\(132\) 0 0
\(133\) 316.516 0.206356
\(134\) 0 0
\(135\) −170.212 −0.108515
\(136\) 0 0
\(137\) −56.5256 −0.0352504 −0.0176252 0.999845i \(-0.505611\pi\)
−0.0176252 + 0.999845i \(0.505611\pi\)
\(138\) 0 0
\(139\) 2113.57 1.28972 0.644858 0.764303i \(-0.276917\pi\)
0.644858 + 0.764303i \(0.276917\pi\)
\(140\) 0 0
\(141\) −553.825 −0.330783
\(142\) 0 0
\(143\) −127.577 −0.0746049
\(144\) 0 0
\(145\) −332.754 −0.190577
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 1794.96 0.986904 0.493452 0.869773i \(-0.335735\pi\)
0.493452 + 0.869773i \(0.335735\pi\)
\(150\) 0 0
\(151\) 377.032 0.203195 0.101597 0.994826i \(-0.467605\pi\)
0.101597 + 0.994826i \(0.467605\pi\)
\(152\) 0 0
\(153\) 1230.63 0.650268
\(154\) 0 0
\(155\) 93.1969 0.0482952
\(156\) 0 0
\(157\) −898.701 −0.456842 −0.228421 0.973562i \(-0.573356\pi\)
−0.228421 + 0.973562i \(0.573356\pi\)
\(158\) 0 0
\(159\) 1079.47 0.538414
\(160\) 0 0
\(161\) −266.903 −0.130651
\(162\) 0 0
\(163\) 3863.52 1.85653 0.928264 0.371922i \(-0.121301\pi\)
0.928264 + 0.371922i \(0.121301\pi\)
\(164\) 0 0
\(165\) −925.051 −0.436455
\(166\) 0 0
\(167\) −2861.44 −1.32590 −0.662948 0.748666i \(-0.730695\pi\)
−0.662948 + 0.748666i \(0.730695\pi\)
\(168\) 0 0
\(169\) −2190.20 −0.996903
\(170\) 0 0
\(171\) 406.949 0.181989
\(172\) 0 0
\(173\) 979.005 0.430245 0.215122 0.976587i \(-0.430985\pi\)
0.215122 + 0.976587i \(0.430985\pi\)
\(174\) 0 0
\(175\) −596.805 −0.257796
\(176\) 0 0
\(177\) 299.723 0.127280
\(178\) 0 0
\(179\) −1146.27 −0.478640 −0.239320 0.970941i \(-0.576925\pi\)
−0.239320 + 0.970941i \(0.576925\pi\)
\(180\) 0 0
\(181\) 3929.33 1.61362 0.806809 0.590812i \(-0.201192\pi\)
0.806809 + 0.590812i \(0.201192\pi\)
\(182\) 0 0
\(183\) −2023.55 −0.817404
\(184\) 0 0
\(185\) −2105.00 −0.836554
\(186\) 0 0
\(187\) 6688.15 2.61543
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −2937.36 −1.11277 −0.556386 0.830924i \(-0.687813\pi\)
−0.556386 + 0.830924i \(0.687813\pi\)
\(192\) 0 0
\(193\) −3533.17 −1.31774 −0.658868 0.752259i \(-0.728964\pi\)
−0.658868 + 0.752259i \(0.728964\pi\)
\(194\) 0 0
\(195\) 49.3286 0.0181154
\(196\) 0 0
\(197\) 584.856 0.211519 0.105760 0.994392i \(-0.466273\pi\)
0.105760 + 0.994392i \(0.466273\pi\)
\(198\) 0 0
\(199\) 2158.08 0.768756 0.384378 0.923176i \(-0.374416\pi\)
0.384378 + 0.923176i \(0.374416\pi\)
\(200\) 0 0
\(201\) −1130.88 −0.396845
\(202\) 0 0
\(203\) 369.484 0.127747
\(204\) 0 0
\(205\) −1432.70 −0.488116
\(206\) 0 0
\(207\) −343.160 −0.115224
\(208\) 0 0
\(209\) 2211.65 0.731976
\(210\) 0 0
\(211\) −4290.04 −1.39971 −0.699855 0.714285i \(-0.746752\pi\)
−0.699855 + 0.714285i \(0.746752\pi\)
\(212\) 0 0
\(213\) −3562.81 −1.14610
\(214\) 0 0
\(215\) 2511.78 0.796752
\(216\) 0 0
\(217\) −103.484 −0.0323731
\(218\) 0 0
\(219\) −2207.47 −0.681129
\(220\) 0 0
\(221\) −356.647 −0.108555
\(222\) 0 0
\(223\) −2743.78 −0.823932 −0.411966 0.911199i \(-0.635158\pi\)
−0.411966 + 0.911199i \(0.635158\pi\)
\(224\) 0 0
\(225\) −767.321 −0.227354
\(226\) 0 0
\(227\) −1724.79 −0.504311 −0.252155 0.967687i \(-0.581139\pi\)
−0.252155 + 0.967687i \(0.581139\pi\)
\(228\) 0 0
\(229\) 4201.70 1.21247 0.606237 0.795284i \(-0.292678\pi\)
0.606237 + 0.795284i \(0.292678\pi\)
\(230\) 0 0
\(231\) 1027.16 0.292564
\(232\) 0 0
\(233\) 1274.94 0.358472 0.179236 0.983806i \(-0.442637\pi\)
0.179236 + 0.983806i \(0.442637\pi\)
\(234\) 0 0
\(235\) 1163.80 0.323054
\(236\) 0 0
\(237\) −2510.32 −0.688029
\(238\) 0 0
\(239\) −5967.16 −1.61499 −0.807497 0.589872i \(-0.799178\pi\)
−0.807497 + 0.589872i \(0.799178\pi\)
\(240\) 0 0
\(241\) 4881.64 1.30479 0.652395 0.757879i \(-0.273764\pi\)
0.652395 + 0.757879i \(0.273764\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −308.903 −0.0805513
\(246\) 0 0
\(247\) −117.937 −0.0303812
\(248\) 0 0
\(249\) 881.197 0.224271
\(250\) 0 0
\(251\) 2262.63 0.568988 0.284494 0.958678i \(-0.408174\pi\)
0.284494 + 0.958678i \(0.408174\pi\)
\(252\) 0 0
\(253\) −1864.98 −0.463439
\(254\) 0 0
\(255\) −2586.03 −0.635072
\(256\) 0 0
\(257\) −6210.60 −1.50742 −0.753709 0.657209i \(-0.771737\pi\)
−0.753709 + 0.657209i \(0.771737\pi\)
\(258\) 0 0
\(259\) 2337.35 0.560757
\(260\) 0 0
\(261\) 475.051 0.112663
\(262\) 0 0
\(263\) −2972.69 −0.696973 −0.348486 0.937314i \(-0.613304\pi\)
−0.348486 + 0.937314i \(0.613304\pi\)
\(264\) 0 0
\(265\) −2268.38 −0.525833
\(266\) 0 0
\(267\) 3896.68 0.893157
\(268\) 0 0
\(269\) −4443.42 −1.00714 −0.503569 0.863955i \(-0.667980\pi\)
−0.503569 + 0.863955i \(0.667980\pi\)
\(270\) 0 0
\(271\) −6840.25 −1.53327 −0.766634 0.642084i \(-0.778070\pi\)
−0.766634 + 0.642084i \(0.778070\pi\)
\(272\) 0 0
\(273\) −54.7737 −0.0121430
\(274\) 0 0
\(275\) −4170.17 −0.914439
\(276\) 0 0
\(277\) −3228.67 −0.700332 −0.350166 0.936688i \(-0.613875\pi\)
−0.350166 + 0.936688i \(0.613875\pi\)
\(278\) 0 0
\(279\) −133.051 −0.0285504
\(280\) 0 0
\(281\) −6453.83 −1.37012 −0.685059 0.728488i \(-0.740224\pi\)
−0.685059 + 0.728488i \(0.740224\pi\)
\(282\) 0 0
\(283\) 3840.72 0.806739 0.403369 0.915037i \(-0.367839\pi\)
0.403369 + 0.915037i \(0.367839\pi\)
\(284\) 0 0
\(285\) −855.153 −0.177737
\(286\) 0 0
\(287\) 1590.84 0.327193
\(288\) 0 0
\(289\) 13784.1 2.80563
\(290\) 0 0
\(291\) −603.124 −0.121497
\(292\) 0 0
\(293\) 8801.01 1.75481 0.877407 0.479747i \(-0.159272\pi\)
0.877407 + 0.479747i \(0.159272\pi\)
\(294\) 0 0
\(295\) −629.830 −0.124306
\(296\) 0 0
\(297\) 1320.63 0.258017
\(298\) 0 0
\(299\) 99.4506 0.0192354
\(300\) 0 0
\(301\) −2789.03 −0.534077
\(302\) 0 0
\(303\) −3160.53 −0.599234
\(304\) 0 0
\(305\) 4252.24 0.798303
\(306\) 0 0
\(307\) −3926.72 −0.730000 −0.365000 0.931008i \(-0.618931\pi\)
−0.365000 + 0.931008i \(0.618931\pi\)
\(308\) 0 0
\(309\) 3077.20 0.566523
\(310\) 0 0
\(311\) −6143.13 −1.12008 −0.560040 0.828466i \(-0.689214\pi\)
−0.560040 + 0.828466i \(0.689214\pi\)
\(312\) 0 0
\(313\) 3824.19 0.690594 0.345297 0.938493i \(-0.387778\pi\)
0.345297 + 0.938493i \(0.387778\pi\)
\(314\) 0 0
\(315\) −397.160 −0.0710396
\(316\) 0 0
\(317\) 7949.68 1.40851 0.704257 0.709946i \(-0.251280\pi\)
0.704257 + 0.709946i \(0.251280\pi\)
\(318\) 0 0
\(319\) 2581.77 0.453138
\(320\) 0 0
\(321\) −310.255 −0.0539463
\(322\) 0 0
\(323\) 6182.78 1.06508
\(324\) 0 0
\(325\) 222.376 0.0379544
\(326\) 0 0
\(327\) −2031.40 −0.343537
\(328\) 0 0
\(329\) −1292.26 −0.216549
\(330\) 0 0
\(331\) −11236.4 −1.86588 −0.932940 0.360032i \(-0.882766\pi\)
−0.932940 + 0.360032i \(0.882766\pi\)
\(332\) 0 0
\(333\) 3005.17 0.494541
\(334\) 0 0
\(335\) 2376.40 0.387572
\(336\) 0 0
\(337\) 8425.41 1.36190 0.680951 0.732329i \(-0.261567\pi\)
0.680951 + 0.732329i \(0.261567\pi\)
\(338\) 0 0
\(339\) −1356.25 −0.217290
\(340\) 0 0
\(341\) −723.095 −0.114832
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 721.110 0.112531
\(346\) 0 0
\(347\) 7177.15 1.11034 0.555172 0.831735i \(-0.312652\pi\)
0.555172 + 0.831735i \(0.312652\pi\)
\(348\) 0 0
\(349\) 1549.41 0.237644 0.118822 0.992916i \(-0.462088\pi\)
0.118822 + 0.992916i \(0.462088\pi\)
\(350\) 0 0
\(351\) −70.4233 −0.0107092
\(352\) 0 0
\(353\) −566.231 −0.0853752 −0.0426876 0.999088i \(-0.513592\pi\)
−0.0426876 + 0.999088i \(0.513592\pi\)
\(354\) 0 0
\(355\) 7486.81 1.11932
\(356\) 0 0
\(357\) 2871.48 0.425700
\(358\) 0 0
\(359\) 3848.19 0.565738 0.282869 0.959159i \(-0.408714\pi\)
0.282869 + 0.959159i \(0.408714\pi\)
\(360\) 0 0
\(361\) −4814.46 −0.701919
\(362\) 0 0
\(363\) 3184.27 0.460415
\(364\) 0 0
\(365\) 4638.74 0.665213
\(366\) 0 0
\(367\) 12542.2 1.78392 0.891960 0.452113i \(-0.149330\pi\)
0.891960 + 0.452113i \(0.149330\pi\)
\(368\) 0 0
\(369\) 2045.37 0.288557
\(370\) 0 0
\(371\) 2518.77 0.352475
\(372\) 0 0
\(373\) 6345.87 0.880903 0.440452 0.897776i \(-0.354818\pi\)
0.440452 + 0.897776i \(0.354818\pi\)
\(374\) 0 0
\(375\) 3976.48 0.547586
\(376\) 0 0
\(377\) −137.673 −0.0188078
\(378\) 0 0
\(379\) −12200.2 −1.65351 −0.826757 0.562559i \(-0.809817\pi\)
−0.826757 + 0.562559i \(0.809817\pi\)
\(380\) 0 0
\(381\) −3548.63 −0.477170
\(382\) 0 0
\(383\) −2770.91 −0.369679 −0.184839 0.982769i \(-0.559176\pi\)
−0.184839 + 0.982769i \(0.559176\pi\)
\(384\) 0 0
\(385\) −2158.45 −0.285727
\(386\) 0 0
\(387\) −3585.90 −0.471011
\(388\) 0 0
\(389\) 1581.89 0.206183 0.103091 0.994672i \(-0.467127\pi\)
0.103091 + 0.994672i \(0.467127\pi\)
\(390\) 0 0
\(391\) −5213.65 −0.674336
\(392\) 0 0
\(393\) 3773.26 0.484314
\(394\) 0 0
\(395\) 5275.13 0.671951
\(396\) 0 0
\(397\) −14235.9 −1.79970 −0.899848 0.436203i \(-0.856323\pi\)
−0.899848 + 0.436203i \(0.856323\pi\)
\(398\) 0 0
\(399\) 949.547 0.119140
\(400\) 0 0
\(401\) −9556.18 −1.19006 −0.595028 0.803705i \(-0.702859\pi\)
−0.595028 + 0.803705i \(0.702859\pi\)
\(402\) 0 0
\(403\) 38.5592 0.00476619
\(404\) 0 0
\(405\) −510.635 −0.0626510
\(406\) 0 0
\(407\) 16332.2 1.98909
\(408\) 0 0
\(409\) 2858.17 0.345544 0.172772 0.984962i \(-0.444728\pi\)
0.172772 + 0.984962i \(0.444728\pi\)
\(410\) 0 0
\(411\) −169.577 −0.0203518
\(412\) 0 0
\(413\) 699.353 0.0833242
\(414\) 0 0
\(415\) −1851.73 −0.219031
\(416\) 0 0
\(417\) 6340.70 0.744617
\(418\) 0 0
\(419\) −13333.3 −1.55460 −0.777299 0.629132i \(-0.783411\pi\)
−0.777299 + 0.629132i \(0.783411\pi\)
\(420\) 0 0
\(421\) −13567.4 −1.57063 −0.785314 0.619098i \(-0.787499\pi\)
−0.785314 + 0.619098i \(0.787499\pi\)
\(422\) 0 0
\(423\) −1661.47 −0.190978
\(424\) 0 0
\(425\) −11657.9 −1.33057
\(426\) 0 0
\(427\) −4721.61 −0.535116
\(428\) 0 0
\(429\) −382.730 −0.0430732
\(430\) 0 0
\(431\) 14207.3 1.58780 0.793898 0.608051i \(-0.208048\pi\)
0.793898 + 0.608051i \(0.208048\pi\)
\(432\) 0 0
\(433\) −10530.6 −1.16875 −0.584375 0.811484i \(-0.698660\pi\)
−0.584375 + 0.811484i \(0.698660\pi\)
\(434\) 0 0
\(435\) −998.262 −0.110030
\(436\) 0 0
\(437\) −1724.06 −0.188725
\(438\) 0 0
\(439\) −4038.23 −0.439030 −0.219515 0.975609i \(-0.570447\pi\)
−0.219515 + 0.975609i \(0.570447\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −4574.15 −0.490574 −0.245287 0.969450i \(-0.578882\pi\)
−0.245287 + 0.969450i \(0.578882\pi\)
\(444\) 0 0
\(445\) −8188.40 −0.872286
\(446\) 0 0
\(447\) 5384.88 0.569789
\(448\) 0 0
\(449\) 14957.1 1.57209 0.786044 0.618171i \(-0.212126\pi\)
0.786044 + 0.618171i \(0.212126\pi\)
\(450\) 0 0
\(451\) 11116.0 1.16060
\(452\) 0 0
\(453\) 1131.09 0.117314
\(454\) 0 0
\(455\) 115.100 0.0118593
\(456\) 0 0
\(457\) 3027.65 0.309907 0.154954 0.987922i \(-0.450477\pi\)
0.154954 + 0.987922i \(0.450477\pi\)
\(458\) 0 0
\(459\) 3691.90 0.375432
\(460\) 0 0
\(461\) −10877.7 −1.09897 −0.549484 0.835504i \(-0.685176\pi\)
−0.549484 + 0.835504i \(0.685176\pi\)
\(462\) 0 0
\(463\) −4038.28 −0.405345 −0.202673 0.979247i \(-0.564963\pi\)
−0.202673 + 0.979247i \(0.564963\pi\)
\(464\) 0 0
\(465\) 279.591 0.0278833
\(466\) 0 0
\(467\) 8411.80 0.833515 0.416758 0.909018i \(-0.363166\pi\)
0.416758 + 0.909018i \(0.363166\pi\)
\(468\) 0 0
\(469\) −2638.71 −0.259796
\(470\) 0 0
\(471\) −2696.10 −0.263758
\(472\) 0 0
\(473\) −19488.3 −1.89445
\(474\) 0 0
\(475\) −3855.07 −0.372384
\(476\) 0 0
\(477\) 3238.42 0.310854
\(478\) 0 0
\(479\) 7172.70 0.684194 0.342097 0.939665i \(-0.388863\pi\)
0.342097 + 0.939665i \(0.388863\pi\)
\(480\) 0 0
\(481\) −870.921 −0.0825584
\(482\) 0 0
\(483\) −800.708 −0.0754316
\(484\) 0 0
\(485\) 1267.39 0.118658
\(486\) 0 0
\(487\) −5580.52 −0.519256 −0.259628 0.965709i \(-0.583600\pi\)
−0.259628 + 0.965709i \(0.583600\pi\)
\(488\) 0 0
\(489\) 11590.6 1.07187
\(490\) 0 0
\(491\) 12489.4 1.14794 0.573972 0.818875i \(-0.305402\pi\)
0.573972 + 0.818875i \(0.305402\pi\)
\(492\) 0 0
\(493\) 7217.46 0.659347
\(494\) 0 0
\(495\) −2775.15 −0.251988
\(496\) 0 0
\(497\) −8313.22 −0.750300
\(498\) 0 0
\(499\) −15216.6 −1.36511 −0.682556 0.730834i \(-0.739131\pi\)
−0.682556 + 0.730834i \(0.739131\pi\)
\(500\) 0 0
\(501\) −8584.31 −0.765506
\(502\) 0 0
\(503\) −1814.89 −0.160879 −0.0804393 0.996760i \(-0.525632\pi\)
−0.0804393 + 0.996760i \(0.525632\pi\)
\(504\) 0 0
\(505\) 6641.47 0.585231
\(506\) 0 0
\(507\) −6570.59 −0.575562
\(508\) 0 0
\(509\) −4853.68 −0.422663 −0.211332 0.977414i \(-0.567780\pi\)
−0.211332 + 0.977414i \(0.567780\pi\)
\(510\) 0 0
\(511\) −5150.77 −0.445904
\(512\) 0 0
\(513\) 1220.85 0.105072
\(514\) 0 0
\(515\) −6466.35 −0.553285
\(516\) 0 0
\(517\) −9029.63 −0.768129
\(518\) 0 0
\(519\) 2937.01 0.248402
\(520\) 0 0
\(521\) −9913.18 −0.833598 −0.416799 0.908999i \(-0.636848\pi\)
−0.416799 + 0.908999i \(0.636848\pi\)
\(522\) 0 0
\(523\) −4524.29 −0.378267 −0.189133 0.981951i \(-0.560568\pi\)
−0.189133 + 0.981951i \(0.560568\pi\)
\(524\) 0 0
\(525\) −1790.42 −0.148838
\(526\) 0 0
\(527\) −2021.45 −0.167089
\(528\) 0 0
\(529\) −10713.2 −0.880512
\(530\) 0 0
\(531\) 899.168 0.0734850
\(532\) 0 0
\(533\) −592.763 −0.0481715
\(534\) 0 0
\(535\) 651.963 0.0526857
\(536\) 0 0
\(537\) −3438.82 −0.276343
\(538\) 0 0
\(539\) 2396.71 0.191528
\(540\) 0 0
\(541\) 3724.94 0.296022 0.148011 0.988986i \(-0.452713\pi\)
0.148011 + 0.988986i \(0.452713\pi\)
\(542\) 0 0
\(543\) 11788.0 0.931623
\(544\) 0 0
\(545\) 4268.74 0.335510
\(546\) 0 0
\(547\) 16121.6 1.26016 0.630082 0.776528i \(-0.283021\pi\)
0.630082 + 0.776528i \(0.283021\pi\)
\(548\) 0 0
\(549\) −6070.64 −0.471928
\(550\) 0 0
\(551\) 2386.69 0.184530
\(552\) 0 0
\(553\) −5857.42 −0.450421
\(554\) 0 0
\(555\) −6314.99 −0.482985
\(556\) 0 0
\(557\) 20451.7 1.55577 0.777887 0.628405i \(-0.216292\pi\)
0.777887 + 0.628405i \(0.216292\pi\)
\(558\) 0 0
\(559\) 1039.22 0.0786303
\(560\) 0 0
\(561\) 20064.4 1.51002
\(562\) 0 0
\(563\) −10046.7 −0.752078 −0.376039 0.926604i \(-0.622714\pi\)
−0.376039 + 0.926604i \(0.622714\pi\)
\(564\) 0 0
\(565\) 2849.99 0.212212
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −15356.4 −1.13141 −0.565705 0.824608i \(-0.691396\pi\)
−0.565705 + 0.824608i \(0.691396\pi\)
\(570\) 0 0
\(571\) −19333.6 −1.41696 −0.708480 0.705731i \(-0.750619\pi\)
−0.708480 + 0.705731i \(0.750619\pi\)
\(572\) 0 0
\(573\) −8812.07 −0.642460
\(574\) 0 0
\(575\) 3250.79 0.235769
\(576\) 0 0
\(577\) 26258.8 1.89458 0.947288 0.320384i \(-0.103812\pi\)
0.947288 + 0.320384i \(0.103812\pi\)
\(578\) 0 0
\(579\) −10599.5 −0.760795
\(580\) 0 0
\(581\) 2056.13 0.146820
\(582\) 0 0
\(583\) 17599.9 1.25028
\(584\) 0 0
\(585\) 147.986 0.0104589
\(586\) 0 0
\(587\) −4868.98 −0.342358 −0.171179 0.985240i \(-0.554758\pi\)
−0.171179 + 0.985240i \(0.554758\pi\)
\(588\) 0 0
\(589\) −668.457 −0.0467628
\(590\) 0 0
\(591\) 1754.57 0.122121
\(592\) 0 0
\(593\) −13647.1 −0.945055 −0.472528 0.881316i \(-0.656658\pi\)
−0.472528 + 0.881316i \(0.656658\pi\)
\(594\) 0 0
\(595\) −6034.07 −0.415752
\(596\) 0 0
\(597\) 6474.25 0.443841
\(598\) 0 0
\(599\) 7543.11 0.514529 0.257265 0.966341i \(-0.417179\pi\)
0.257265 + 0.966341i \(0.417179\pi\)
\(600\) 0 0
\(601\) −19522.4 −1.32501 −0.662507 0.749056i \(-0.730507\pi\)
−0.662507 + 0.749056i \(0.730507\pi\)
\(602\) 0 0
\(603\) −3392.63 −0.229119
\(604\) 0 0
\(605\) −6691.36 −0.449657
\(606\) 0 0
\(607\) 13804.5 0.923079 0.461539 0.887120i \(-0.347297\pi\)
0.461539 + 0.887120i \(0.347297\pi\)
\(608\) 0 0
\(609\) 1108.45 0.0737550
\(610\) 0 0
\(611\) 481.508 0.0318817
\(612\) 0 0
\(613\) 21718.8 1.43102 0.715508 0.698605i \(-0.246195\pi\)
0.715508 + 0.698605i \(0.246195\pi\)
\(614\) 0 0
\(615\) −4298.09 −0.281814
\(616\) 0 0
\(617\) −5183.85 −0.338240 −0.169120 0.985595i \(-0.554093\pi\)
−0.169120 + 0.985595i \(0.554093\pi\)
\(618\) 0 0
\(619\) 22003.7 1.42876 0.714382 0.699756i \(-0.246708\pi\)
0.714382 + 0.699756i \(0.246708\pi\)
\(620\) 0 0
\(621\) −1029.48 −0.0665244
\(622\) 0 0
\(623\) 9092.25 0.584708
\(624\) 0 0
\(625\) 2301.14 0.147273
\(626\) 0 0
\(627\) 6634.95 0.422607
\(628\) 0 0
\(629\) 45657.6 2.89426
\(630\) 0 0
\(631\) −985.836 −0.0621957 −0.0310979 0.999516i \(-0.509900\pi\)
−0.0310979 + 0.999516i \(0.509900\pi\)
\(632\) 0 0
\(633\) −12870.1 −0.808123
\(634\) 0 0
\(635\) 7457.01 0.466020
\(636\) 0 0
\(637\) −127.805 −0.00794949
\(638\) 0 0
\(639\) −10688.4 −0.661702
\(640\) 0 0
\(641\) 24282.6 1.49626 0.748131 0.663551i \(-0.230951\pi\)
0.748131 + 0.663551i \(0.230951\pi\)
\(642\) 0 0
\(643\) −4743.12 −0.290903 −0.145451 0.989365i \(-0.546463\pi\)
−0.145451 + 0.989365i \(0.546463\pi\)
\(644\) 0 0
\(645\) 7535.33 0.460005
\(646\) 0 0
\(647\) 29641.1 1.80110 0.900549 0.434754i \(-0.143165\pi\)
0.900549 + 0.434754i \(0.143165\pi\)
\(648\) 0 0
\(649\) 4886.72 0.295563
\(650\) 0 0
\(651\) −310.453 −0.0186906
\(652\) 0 0
\(653\) −23046.9 −1.38116 −0.690578 0.723258i \(-0.742644\pi\)
−0.690578 + 0.723258i \(0.742644\pi\)
\(654\) 0 0
\(655\) −7929.04 −0.472997
\(656\) 0 0
\(657\) −6622.42 −0.393250
\(658\) 0 0
\(659\) 5795.12 0.342558 0.171279 0.985223i \(-0.445210\pi\)
0.171279 + 0.985223i \(0.445210\pi\)
\(660\) 0 0
\(661\) 2592.59 0.152557 0.0762784 0.997087i \(-0.475696\pi\)
0.0762784 + 0.997087i \(0.475696\pi\)
\(662\) 0 0
\(663\) −1069.94 −0.0626744
\(664\) 0 0
\(665\) −1995.36 −0.116356
\(666\) 0 0
\(667\) −2012.58 −0.116833
\(668\) 0 0
\(669\) −8231.33 −0.475697
\(670\) 0 0
\(671\) −32992.2 −1.89814
\(672\) 0 0
\(673\) 28156.0 1.61268 0.806341 0.591451i \(-0.201445\pi\)
0.806341 + 0.591451i \(0.201445\pi\)
\(674\) 0 0
\(675\) −2301.96 −0.131263
\(676\) 0 0
\(677\) 20271.4 1.15080 0.575402 0.817871i \(-0.304846\pi\)
0.575402 + 0.817871i \(0.304846\pi\)
\(678\) 0 0
\(679\) −1407.29 −0.0795388
\(680\) 0 0
\(681\) −5174.38 −0.291164
\(682\) 0 0
\(683\) −13267.0 −0.743264 −0.371632 0.928380i \(-0.621202\pi\)
−0.371632 + 0.928380i \(0.621202\pi\)
\(684\) 0 0
\(685\) 356.345 0.0198763
\(686\) 0 0
\(687\) 12605.1 0.700022
\(688\) 0 0
\(689\) −938.520 −0.0518937
\(690\) 0 0
\(691\) −15966.3 −0.878995 −0.439497 0.898244i \(-0.644843\pi\)
−0.439497 + 0.898244i \(0.644843\pi\)
\(692\) 0 0
\(693\) 3081.48 0.168912
\(694\) 0 0
\(695\) −13324.2 −0.727217
\(696\) 0 0
\(697\) 31075.3 1.68875
\(698\) 0 0
\(699\) 3824.82 0.206964
\(700\) 0 0
\(701\) 7525.29 0.405458 0.202729 0.979235i \(-0.435019\pi\)
0.202729 + 0.979235i \(0.435019\pi\)
\(702\) 0 0
\(703\) 15098.1 0.810010
\(704\) 0 0
\(705\) 3491.39 0.186515
\(706\) 0 0
\(707\) −7374.58 −0.392291
\(708\) 0 0
\(709\) 20033.6 1.06118 0.530591 0.847628i \(-0.321970\pi\)
0.530591 + 0.847628i \(0.321970\pi\)
\(710\) 0 0
\(711\) −7530.96 −0.397234
\(712\) 0 0
\(713\) 563.678 0.0296071
\(714\) 0 0
\(715\) 804.261 0.0420666
\(716\) 0 0
\(717\) −17901.5 −0.932417
\(718\) 0 0
\(719\) −8081.69 −0.419188 −0.209594 0.977788i \(-0.567214\pi\)
−0.209594 + 0.977788i \(0.567214\pi\)
\(720\) 0 0
\(721\) 7180.13 0.370876
\(722\) 0 0
\(723\) 14644.9 0.753321
\(724\) 0 0
\(725\) −4500.21 −0.230529
\(726\) 0 0
\(727\) 34117.8 1.74052 0.870262 0.492590i \(-0.163950\pi\)
0.870262 + 0.492590i \(0.163950\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −54480.6 −2.75655
\(732\) 0 0
\(733\) 20048.0 1.01022 0.505110 0.863055i \(-0.331452\pi\)
0.505110 + 0.863055i \(0.331452\pi\)
\(734\) 0 0
\(735\) −926.708 −0.0465063
\(736\) 0 0
\(737\) −18438.0 −0.921534
\(738\) 0 0
\(739\) −407.607 −0.0202897 −0.0101448 0.999949i \(-0.503229\pi\)
−0.0101448 + 0.999949i \(0.503229\pi\)
\(740\) 0 0
\(741\) −353.811 −0.0175406
\(742\) 0 0
\(743\) −128.374 −0.00633861 −0.00316930 0.999995i \(-0.501009\pi\)
−0.00316930 + 0.999995i \(0.501009\pi\)
\(744\) 0 0
\(745\) −11315.7 −0.556475
\(746\) 0 0
\(747\) 2643.59 0.129483
\(748\) 0 0
\(749\) −723.929 −0.0353161
\(750\) 0 0
\(751\) 25076.0 1.21842 0.609212 0.793007i \(-0.291486\pi\)
0.609212 + 0.793007i \(0.291486\pi\)
\(752\) 0 0
\(753\) 6787.90 0.328506
\(754\) 0 0
\(755\) −2376.86 −0.114573
\(756\) 0 0
\(757\) −21824.1 −1.04783 −0.523916 0.851770i \(-0.675530\pi\)
−0.523916 + 0.851770i \(0.675530\pi\)
\(758\) 0 0
\(759\) −5594.93 −0.267567
\(760\) 0 0
\(761\) −38939.2 −1.85485 −0.927427 0.374004i \(-0.877985\pi\)
−0.927427 + 0.374004i \(0.877985\pi\)
\(762\) 0 0
\(763\) −4739.94 −0.224898
\(764\) 0 0
\(765\) −7758.09 −0.366659
\(766\) 0 0
\(767\) −260.586 −0.0122675
\(768\) 0 0
\(769\) 3079.42 0.144404 0.0722021 0.997390i \(-0.476997\pi\)
0.0722021 + 0.997390i \(0.476997\pi\)
\(770\) 0 0
\(771\) −18631.8 −0.870308
\(772\) 0 0
\(773\) 31770.9 1.47829 0.739146 0.673545i \(-0.235229\pi\)
0.739146 + 0.673545i \(0.235229\pi\)
\(774\) 0 0
\(775\) 1260.41 0.0584195
\(776\) 0 0
\(777\) 7012.06 0.323753
\(778\) 0 0
\(779\) 10276.0 0.472628
\(780\) 0 0
\(781\) −58088.5 −2.66142
\(782\) 0 0
\(783\) 1425.15 0.0650458
\(784\) 0 0
\(785\) 5665.53 0.257594
\(786\) 0 0
\(787\) 6736.02 0.305099 0.152550 0.988296i \(-0.451252\pi\)
0.152550 + 0.988296i \(0.451252\pi\)
\(788\) 0 0
\(789\) −8918.06 −0.402397
\(790\) 0 0
\(791\) −3164.58 −0.142250
\(792\) 0 0
\(793\) 1759.32 0.0787834
\(794\) 0 0
\(795\) −6805.15 −0.303590
\(796\) 0 0
\(797\) 13445.1 0.597552 0.298776 0.954323i \(-0.403422\pi\)
0.298776 + 0.954323i \(0.403422\pi\)
\(798\) 0 0
\(799\) −25242.8 −1.11768
\(800\) 0 0
\(801\) 11690.0 0.515664
\(802\) 0 0
\(803\) −35991.0 −1.58169
\(804\) 0 0
\(805\) 1682.59 0.0736689
\(806\) 0 0
\(807\) −13330.3 −0.581472
\(808\) 0 0
\(809\) 20337.7 0.883853 0.441927 0.897051i \(-0.354295\pi\)
0.441927 + 0.897051i \(0.354295\pi\)
\(810\) 0 0
\(811\) −23069.9 −0.998884 −0.499442 0.866347i \(-0.666462\pi\)
−0.499442 + 0.866347i \(0.666462\pi\)
\(812\) 0 0
\(813\) −20520.8 −0.885233
\(814\) 0 0
\(815\) −24356.1 −1.04682
\(816\) 0 0
\(817\) −18015.8 −0.771471
\(818\) 0 0
\(819\) −164.321 −0.00701079
\(820\) 0 0
\(821\) 8230.74 0.349884 0.174942 0.984579i \(-0.444026\pi\)
0.174942 + 0.984579i \(0.444026\pi\)
\(822\) 0 0
\(823\) 17577.9 0.744506 0.372253 0.928131i \(-0.378585\pi\)
0.372253 + 0.928131i \(0.378585\pi\)
\(824\) 0 0
\(825\) −12510.5 −0.527951
\(826\) 0 0
\(827\) −6440.14 −0.270793 −0.135396 0.990792i \(-0.543231\pi\)
−0.135396 + 0.990792i \(0.543231\pi\)
\(828\) 0 0
\(829\) 5084.23 0.213007 0.106503 0.994312i \(-0.466034\pi\)
0.106503 + 0.994312i \(0.466034\pi\)
\(830\) 0 0
\(831\) −9686.01 −0.404337
\(832\) 0 0
\(833\) 6700.12 0.278686
\(834\) 0 0
\(835\) 18038.9 0.747618
\(836\) 0 0
\(837\) −399.153 −0.0164836
\(838\) 0 0
\(839\) −25150.5 −1.03491 −0.517456 0.855710i \(-0.673121\pi\)
−0.517456 + 0.855710i \(0.673121\pi\)
\(840\) 0 0
\(841\) −21602.9 −0.885764
\(842\) 0 0
\(843\) −19361.5 −0.791038
\(844\) 0 0
\(845\) 13807.3 0.562113
\(846\) 0 0
\(847\) 7429.96 0.301413
\(848\) 0 0
\(849\) 11522.2 0.465771
\(850\) 0 0
\(851\) −12731.5 −0.512846
\(852\) 0 0
\(853\) −6408.37 −0.257232 −0.128616 0.991694i \(-0.541053\pi\)
−0.128616 + 0.991694i \(0.541053\pi\)
\(854\) 0 0
\(855\) −2565.46 −0.102616
\(856\) 0 0
\(857\) 17248.6 0.687515 0.343758 0.939058i \(-0.388300\pi\)
0.343758 + 0.939058i \(0.388300\pi\)
\(858\) 0 0
\(859\) 3159.07 0.125479 0.0627393 0.998030i \(-0.480016\pi\)
0.0627393 + 0.998030i \(0.480016\pi\)
\(860\) 0 0
\(861\) 4772.52 0.188905
\(862\) 0 0
\(863\) 41071.9 1.62005 0.810025 0.586395i \(-0.199453\pi\)
0.810025 + 0.586395i \(0.199453\pi\)
\(864\) 0 0
\(865\) −6171.78 −0.242597
\(866\) 0 0
\(867\) 41352.2 1.61983
\(868\) 0 0
\(869\) −40928.6 −1.59771
\(870\) 0 0
\(871\) 983.210 0.0382489
\(872\) 0 0
\(873\) −1809.37 −0.0701466
\(874\) 0 0
\(875\) 9278.46 0.358479
\(876\) 0 0
\(877\) −1034.90 −0.0398472 −0.0199236 0.999802i \(-0.506342\pi\)
−0.0199236 + 0.999802i \(0.506342\pi\)
\(878\) 0 0
\(879\) 26403.0 1.01314
\(880\) 0 0
\(881\) −3109.73 −0.118921 −0.0594606 0.998231i \(-0.518938\pi\)
−0.0594606 + 0.998231i \(0.518938\pi\)
\(882\) 0 0
\(883\) −19782.8 −0.753955 −0.376978 0.926222i \(-0.623037\pi\)
−0.376978 + 0.926222i \(0.623037\pi\)
\(884\) 0 0
\(885\) −1889.49 −0.0717678
\(886\) 0 0
\(887\) 9355.56 0.354148 0.177074 0.984198i \(-0.443337\pi\)
0.177074 + 0.984198i \(0.443337\pi\)
\(888\) 0 0
\(889\) −8280.13 −0.312381
\(890\) 0 0
\(891\) 3961.90 0.148966
\(892\) 0 0
\(893\) −8347.35 −0.312803
\(894\) 0 0
\(895\) 7226.27 0.269886
\(896\) 0 0
\(897\) 298.352 0.0111055
\(898\) 0 0
\(899\) −780.322 −0.0289491
\(900\) 0 0
\(901\) 49201.4 1.81924
\(902\) 0 0
\(903\) −8367.09 −0.308349
\(904\) 0 0
\(905\) −24771.0 −0.909853
\(906\) 0 0
\(907\) 16211.9 0.593505 0.296752 0.954954i \(-0.404096\pi\)
0.296752 + 0.954954i \(0.404096\pi\)
\(908\) 0 0
\(909\) −9481.60 −0.345968
\(910\) 0 0
\(911\) 14540.6 0.528816 0.264408 0.964411i \(-0.414823\pi\)
0.264408 + 0.964411i \(0.414823\pi\)
\(912\) 0 0
\(913\) 14367.2 0.520792
\(914\) 0 0
\(915\) 12756.7 0.460901
\(916\) 0 0
\(917\) 8804.26 0.317058
\(918\) 0 0
\(919\) −37239.2 −1.33668 −0.668339 0.743857i \(-0.732995\pi\)
−0.668339 + 0.743857i \(0.732995\pi\)
\(920\) 0 0
\(921\) −11780.2 −0.421466
\(922\) 0 0
\(923\) 3097.59 0.110464
\(924\) 0 0
\(925\) −28468.2 −1.01192
\(926\) 0 0
\(927\) 9231.59 0.327082
\(928\) 0 0
\(929\) 22062.5 0.779167 0.389583 0.920991i \(-0.372619\pi\)
0.389583 + 0.920991i \(0.372619\pi\)
\(930\) 0 0
\(931\) 2215.61 0.0779954
\(932\) 0 0
\(933\) −18429.4 −0.646678
\(934\) 0 0
\(935\) −42163.0 −1.47473
\(936\) 0 0
\(937\) −15597.3 −0.543800 −0.271900 0.962326i \(-0.587652\pi\)
−0.271900 + 0.962326i \(0.587652\pi\)
\(938\) 0 0
\(939\) 11472.6 0.398715
\(940\) 0 0
\(941\) −15887.1 −0.550377 −0.275188 0.961390i \(-0.588740\pi\)
−0.275188 + 0.961390i \(0.588740\pi\)
\(942\) 0 0
\(943\) −8665.29 −0.299237
\(944\) 0 0
\(945\) −1191.48 −0.0410147
\(946\) 0 0
\(947\) 54754.6 1.87887 0.939433 0.342733i \(-0.111352\pi\)
0.939433 + 0.342733i \(0.111352\pi\)
\(948\) 0 0
\(949\) 1919.23 0.0656489
\(950\) 0 0
\(951\) 23849.0 0.813205
\(952\) 0 0
\(953\) 12091.3 0.410992 0.205496 0.978658i \(-0.434119\pi\)
0.205496 + 0.978658i \(0.434119\pi\)
\(954\) 0 0
\(955\) 18517.5 0.627447
\(956\) 0 0
\(957\) 7745.30 0.261620
\(958\) 0 0
\(959\) −395.679 −0.0133234
\(960\) 0 0
\(961\) −29572.4 −0.992664
\(962\) 0 0
\(963\) −930.765 −0.0311459
\(964\) 0 0
\(965\) 22273.6 0.743017
\(966\) 0 0
\(967\) −27415.1 −0.911695 −0.455848 0.890058i \(-0.650664\pi\)
−0.455848 + 0.890058i \(0.650664\pi\)
\(968\) 0 0
\(969\) 18548.4 0.614921
\(970\) 0 0
\(971\) 55397.0 1.83087 0.915435 0.402466i \(-0.131847\pi\)
0.915435 + 0.402466i \(0.131847\pi\)
\(972\) 0 0
\(973\) 14795.0 0.487467
\(974\) 0 0
\(975\) 667.127 0.0219130
\(976\) 0 0
\(977\) 18294.3 0.599065 0.299532 0.954086i \(-0.403169\pi\)
0.299532 + 0.954086i \(0.403169\pi\)
\(978\) 0 0
\(979\) 63532.0 2.07405
\(980\) 0 0
\(981\) −6094.20 −0.198341
\(982\) 0 0
\(983\) −23803.2 −0.772334 −0.386167 0.922429i \(-0.626201\pi\)
−0.386167 + 0.922429i \(0.626201\pi\)
\(984\) 0 0
\(985\) −3687.01 −0.119267
\(986\) 0 0
\(987\) −3876.77 −0.125024
\(988\) 0 0
\(989\) 15191.8 0.488445
\(990\) 0 0
\(991\) −13624.5 −0.436726 −0.218363 0.975868i \(-0.570072\pi\)
−0.218363 + 0.975868i \(0.570072\pi\)
\(992\) 0 0
\(993\) −33709.1 −1.07727
\(994\) 0 0
\(995\) −13604.8 −0.433470
\(996\) 0 0
\(997\) −46834.4 −1.48772 −0.743861 0.668334i \(-0.767008\pi\)
−0.743861 + 0.668334i \(0.767008\pi\)
\(998\) 0 0
\(999\) 9015.50 0.285523
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 168.4.a.h.1.1 2
3.2 odd 2 504.4.a.j.1.2 2
4.3 odd 2 336.4.a.n.1.1 2
7.6 odd 2 1176.4.a.p.1.2 2
8.3 odd 2 1344.4.a.bl.1.2 2
8.5 even 2 1344.4.a.bd.1.2 2
12.11 even 2 1008.4.a.y.1.2 2
28.27 even 2 2352.4.a.bv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.h.1.1 2 1.1 even 1 trivial
336.4.a.n.1.1 2 4.3 odd 2
504.4.a.j.1.2 2 3.2 odd 2
1008.4.a.y.1.2 2 12.11 even 2
1176.4.a.p.1.2 2 7.6 odd 2
1344.4.a.bd.1.2 2 8.5 even 2
1344.4.a.bl.1.2 2 8.3 odd 2
2352.4.a.bv.1.2 2 28.27 even 2