Properties

Label 2352.4.a.cm.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.69515\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} -54.0066 q^{11} -50.2350 q^{13} -0.385773 q^{15} -131.487 q^{17} -91.5094 q^{19} -179.487 q^{23} -124.983 q^{25} -27.0000 q^{27} -69.8961 q^{29} -326.847 q^{31} +162.020 q^{33} +301.698 q^{37} +150.705 q^{39} +296.048 q^{41} +144.302 q^{43} +1.15732 q^{45} +360.085 q^{47} +394.462 q^{51} +1.83513 q^{53} -6.94477 q^{55} +274.528 q^{57} +53.2386 q^{59} -108.121 q^{61} -6.45978 q^{65} -842.007 q^{67} +538.462 q^{69} +241.111 q^{71} -206.983 q^{73} +374.950 q^{75} -559.962 q^{79} +81.0000 q^{81} -986.652 q^{83} -16.9081 q^{85} +209.688 q^{87} -443.366 q^{89} +980.541 q^{93} -11.7673 q^{95} -740.815 q^{97} -486.059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} - 14q^{11} + 22q^{13} - 12q^{15} + 96q^{17} + 26q^{19} - 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} + 42q^{33} + 562q^{37} - 66q^{39} + 428q^{41} + 258q^{43} + 36q^{45} + 80q^{47} - 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} - 276q^{61} + 2196q^{65} - 150q^{67} + 288q^{69} + 848q^{71} - 218q^{73} - 330q^{75} - 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} - 344q^{89} + 714q^{93} - 2004q^{95} - 622q^{97} - 126q^{99} + O(q^{100}) \)

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
\(5\) 0.128591 0.0115015 0.00575077 0.999983i \(-0.498169\pi\)
0.00575077 + 0.999983i \(0.498169\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −54.0066 −1.48033 −0.740163 0.672427i \(-0.765252\pi\)
−0.740163 + 0.672427i \(0.765252\pi\)
\(12\) 0 0
\(13\) −50.2350 −1.07175 −0.535873 0.844299i \(-0.680017\pi\)
−0.535873 + 0.844299i \(0.680017\pi\)
\(14\) 0 0
\(15\) −0.385773 −0.00664042
\(16\) 0 0
\(17\) −131.487 −1.87590 −0.937951 0.346767i \(-0.887279\pi\)
−0.937951 + 0.346767i \(0.887279\pi\)
\(18\) 0 0
\(19\) −91.5094 −1.10493 −0.552466 0.833536i \(-0.686313\pi\)
−0.552466 + 0.833536i \(0.686313\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −179.487 −1.62720 −0.813602 0.581423i \(-0.802496\pi\)
−0.813602 + 0.581423i \(0.802496\pi\)
\(24\) 0 0
\(25\) −124.983 −0.999868
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −69.8961 −0.447565 −0.223782 0.974639i \(-0.571841\pi\)
−0.223782 + 0.974639i \(0.571841\pi\)
\(30\) 0 0
\(31\) −326.847 −1.89366 −0.946829 0.321736i \(-0.895734\pi\)
−0.946829 + 0.321736i \(0.895734\pi\)
\(32\) 0 0
\(33\) 162.020 0.854667
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 301.698 1.34051 0.670254 0.742132i \(-0.266185\pi\)
0.670254 + 0.742132i \(0.266185\pi\)
\(38\) 0 0
\(39\) 150.705 0.618773
\(40\) 0 0
\(41\) 296.048 1.12768 0.563840 0.825884i \(-0.309324\pi\)
0.563840 + 0.825884i \(0.309324\pi\)
\(42\) 0 0
\(43\) 144.302 0.511764 0.255882 0.966708i \(-0.417634\pi\)
0.255882 + 0.966708i \(0.417634\pi\)
\(44\) 0 0
\(45\) 1.15732 0.00383385
\(46\) 0 0
\(47\) 360.085 1.11753 0.558764 0.829326i \(-0.311276\pi\)
0.558764 + 0.829326i \(0.311276\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 394.462 1.08305
\(52\) 0 0
\(53\) 1.83513 0.00475613 0.00237807 0.999997i \(-0.499243\pi\)
0.00237807 + 0.999997i \(0.499243\pi\)
\(54\) 0 0
\(55\) −6.94477 −0.0170260
\(56\) 0 0
\(57\) 274.528 0.637932
\(58\) 0 0
\(59\) 53.2386 0.117476 0.0587379 0.998273i \(-0.481292\pi\)
0.0587379 + 0.998273i \(0.481292\pi\)
\(60\) 0 0
\(61\) −108.121 −0.226942 −0.113471 0.993541i \(-0.536197\pi\)
−0.113471 + 0.993541i \(0.536197\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.45978 −0.0123267
\(66\) 0 0
\(67\) −842.007 −1.53534 −0.767668 0.640848i \(-0.778583\pi\)
−0.767668 + 0.640848i \(0.778583\pi\)
\(68\) 0 0
\(69\) 538.462 0.939466
\(70\) 0 0
\(71\) 241.111 0.403023 0.201511 0.979486i \(-0.435415\pi\)
0.201511 + 0.979486i \(0.435415\pi\)
\(72\) 0 0
\(73\) −206.983 −0.331857 −0.165929 0.986138i \(-0.553062\pi\)
−0.165929 + 0.986138i \(0.553062\pi\)
\(74\) 0 0
\(75\) 374.950 0.577274
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −559.962 −0.797477 −0.398738 0.917065i \(-0.630552\pi\)
−0.398738 + 0.917065i \(0.630552\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −986.652 −1.30481 −0.652404 0.757871i \(-0.726240\pi\)
−0.652404 + 0.757871i \(0.726240\pi\)
\(84\) 0 0
\(85\) −16.9081 −0.0215758
\(86\) 0 0
\(87\) 209.688 0.258402
\(88\) 0 0
\(89\) −443.366 −0.528053 −0.264026 0.964515i \(-0.585051\pi\)
−0.264026 + 0.964515i \(0.585051\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 980.541 1.09330
\(94\) 0 0
\(95\) −11.7673 −0.0127084
\(96\) 0 0
\(97\) −740.815 −0.775447 −0.387723 0.921776i \(-0.626738\pi\)
−0.387723 + 0.921776i \(0.626738\pi\)
\(98\) 0 0
\(99\) −486.059 −0.493442
\(100\) 0 0
\(101\) 743.777 0.732758 0.366379 0.930466i \(-0.380597\pi\)
0.366379 + 0.930466i \(0.380597\pi\)
\(102\) 0 0
\(103\) −104.651 −0.100112 −0.0500559 0.998746i \(-0.515940\pi\)
−0.0500559 + 0.998746i \(0.515940\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −513.700 −0.464124 −0.232062 0.972701i \(-0.574547\pi\)
−0.232062 + 0.972701i \(0.574547\pi\)
\(108\) 0 0
\(109\) 975.038 0.856805 0.428402 0.903588i \(-0.359077\pi\)
0.428402 + 0.903588i \(0.359077\pi\)
\(110\) 0 0
\(111\) −905.093 −0.773942
\(112\) 0 0
\(113\) 1926.07 1.60345 0.801723 0.597696i \(-0.203917\pi\)
0.801723 + 0.597696i \(0.203917\pi\)
\(114\) 0 0
\(115\) −23.0805 −0.0187153
\(116\) 0 0
\(117\) −452.115 −0.357248
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1585.71 1.19137
\(122\) 0 0
\(123\) −888.143 −0.651066
\(124\) 0 0
\(125\) −32.1457 −0.0230016
\(126\) 0 0
\(127\) 1125.95 0.786709 0.393355 0.919387i \(-0.371314\pi\)
0.393355 + 0.919387i \(0.371314\pi\)
\(128\) 0 0
\(129\) −432.906 −0.295467
\(130\) 0 0
\(131\) −1493.24 −0.995918 −0.497959 0.867201i \(-0.665917\pi\)
−0.497959 + 0.867201i \(0.665917\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.47196 −0.00221347
\(136\) 0 0
\(137\) 1460.77 0.910965 0.455483 0.890245i \(-0.349467\pi\)
0.455483 + 0.890245i \(0.349467\pi\)
\(138\) 0 0
\(139\) −2225.85 −1.35823 −0.679116 0.734031i \(-0.737637\pi\)
−0.679116 + 0.734031i \(0.737637\pi\)
\(140\) 0 0
\(141\) −1080.26 −0.645206
\(142\) 0 0
\(143\) 2713.02 1.58653
\(144\) 0 0
\(145\) −8.98802 −0.00514769
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 394.681 0.217003 0.108502 0.994096i \(-0.465395\pi\)
0.108502 + 0.994096i \(0.465395\pi\)
\(150\) 0 0
\(151\) 3124.20 1.68373 0.841867 0.539685i \(-0.181457\pi\)
0.841867 + 0.539685i \(0.181457\pi\)
\(152\) 0 0
\(153\) −1183.39 −0.625301
\(154\) 0 0
\(155\) −42.0296 −0.0217800
\(156\) 0 0
\(157\) −3600.71 −1.83037 −0.915183 0.403038i \(-0.867954\pi\)
−0.915183 + 0.403038i \(0.867954\pi\)
\(158\) 0 0
\(159\) −5.50540 −0.00274596
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1974.02 −0.948573 −0.474287 0.880370i \(-0.657294\pi\)
−0.474287 + 0.880370i \(0.657294\pi\)
\(164\) 0 0
\(165\) 20.8343 0.00982999
\(166\) 0 0
\(167\) −1067.09 −0.494453 −0.247227 0.968958i \(-0.579519\pi\)
−0.247227 + 0.968958i \(0.579519\pi\)
\(168\) 0 0
\(169\) 326.559 0.148638
\(170\) 0 0
\(171\) −823.584 −0.368310
\(172\) 0 0
\(173\) −274.655 −0.120703 −0.0603515 0.998177i \(-0.519222\pi\)
−0.0603515 + 0.998177i \(0.519222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −159.716 −0.0678247
\(178\) 0 0
\(179\) −554.332 −0.231468 −0.115734 0.993280i \(-0.536922\pi\)
−0.115734 + 0.993280i \(0.536922\pi\)
\(180\) 0 0
\(181\) −685.436 −0.281481 −0.140741 0.990047i \(-0.544948\pi\)
−0.140741 + 0.990047i \(0.544948\pi\)
\(182\) 0 0
\(183\) 324.363 0.131025
\(184\) 0 0
\(185\) 38.7956 0.0154179
\(186\) 0 0
\(187\) 7101.18 2.77695
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4898.87 −1.85586 −0.927932 0.372750i \(-0.878415\pi\)
−0.927932 + 0.372750i \(0.878415\pi\)
\(192\) 0 0
\(193\) 3140.26 1.17120 0.585598 0.810601i \(-0.300860\pi\)
0.585598 + 0.810601i \(0.300860\pi\)
\(194\) 0 0
\(195\) 19.3793 0.00711684
\(196\) 0 0
\(197\) −227.412 −0.0822460 −0.0411230 0.999154i \(-0.513094\pi\)
−0.0411230 + 0.999154i \(0.513094\pi\)
\(198\) 0 0
\(199\) 1214.50 0.432631 0.216316 0.976323i \(-0.430596\pi\)
0.216316 + 0.976323i \(0.430596\pi\)
\(200\) 0 0
\(201\) 2526.02 0.886427
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 38.0691 0.0129700
\(206\) 0 0
\(207\) −1615.39 −0.542401
\(208\) 0 0
\(209\) 4942.11 1.63566
\(210\) 0 0
\(211\) −5116.07 −1.66922 −0.834608 0.550844i \(-0.814306\pi\)
−0.834608 + 0.550844i \(0.814306\pi\)
\(212\) 0 0
\(213\) −723.333 −0.232685
\(214\) 0 0
\(215\) 18.5560 0.00588608
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 620.950 0.191598
\(220\) 0 0
\(221\) 6605.27 2.01049
\(222\) 0 0
\(223\) −119.384 −0.0358499 −0.0179250 0.999839i \(-0.505706\pi\)
−0.0179250 + 0.999839i \(0.505706\pi\)
\(224\) 0 0
\(225\) −1124.85 −0.333289
\(226\) 0 0
\(227\) −1992.00 −0.582439 −0.291220 0.956656i \(-0.594061\pi\)
−0.291220 + 0.956656i \(0.594061\pi\)
\(228\) 0 0
\(229\) 703.476 0.203000 0.101500 0.994836i \(-0.467636\pi\)
0.101500 + 0.994836i \(0.467636\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 961.783 0.270423 0.135211 0.990817i \(-0.456829\pi\)
0.135211 + 0.990817i \(0.456829\pi\)
\(234\) 0 0
\(235\) 46.3038 0.0128533
\(236\) 0 0
\(237\) 1679.89 0.460423
\(238\) 0 0
\(239\) 4464.71 1.20836 0.604179 0.796848i \(-0.293501\pi\)
0.604179 + 0.796848i \(0.293501\pi\)
\(240\) 0 0
\(241\) −435.311 −0.116352 −0.0581761 0.998306i \(-0.518528\pi\)
−0.0581761 + 0.998306i \(0.518528\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4596.98 1.18421
\(248\) 0 0
\(249\) 2959.95 0.753331
\(250\) 0 0
\(251\) −863.003 −0.217021 −0.108510 0.994095i \(-0.534608\pi\)
−0.108510 + 0.994095i \(0.534608\pi\)
\(252\) 0 0
\(253\) 9693.49 2.40879
\(254\) 0 0
\(255\) 50.7243 0.0124568
\(256\) 0 0
\(257\) −536.693 −0.130264 −0.0651322 0.997877i \(-0.520747\pi\)
−0.0651322 + 0.997877i \(0.520747\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −629.065 −0.149188
\(262\) 0 0
\(263\) −371.908 −0.0871971 −0.0435986 0.999049i \(-0.513882\pi\)
−0.0435986 + 0.999049i \(0.513882\pi\)
\(264\) 0 0
\(265\) 0.235982 5.47029e−5 0
\(266\) 0 0
\(267\) 1330.10 0.304871
\(268\) 0 0
\(269\) −4505.61 −1.02123 −0.510616 0.859809i \(-0.670583\pi\)
−0.510616 + 0.859809i \(0.670583\pi\)
\(270\) 0 0
\(271\) 396.115 0.0887908 0.0443954 0.999014i \(-0.485864\pi\)
0.0443954 + 0.999014i \(0.485864\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6749.93 1.48013
\(276\) 0 0
\(277\) −6508.74 −1.41181 −0.705907 0.708305i \(-0.749460\pi\)
−0.705907 + 0.708305i \(0.749460\pi\)
\(278\) 0 0
\(279\) −2941.62 −0.631220
\(280\) 0 0
\(281\) 2785.80 0.591413 0.295707 0.955279i \(-0.404445\pi\)
0.295707 + 0.955279i \(0.404445\pi\)
\(282\) 0 0
\(283\) −0.611846 −0.000128518 0 −6.42588e−5 1.00000i \(-0.500020\pi\)
−6.42588e−5 1.00000i \(0.500020\pi\)
\(284\) 0 0
\(285\) 35.3019 0.00733720
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12375.9 2.51901
\(290\) 0 0
\(291\) 2222.44 0.447704
\(292\) 0 0
\(293\) −4145.98 −0.826657 −0.413329 0.910582i \(-0.635634\pi\)
−0.413329 + 0.910582i \(0.635634\pi\)
\(294\) 0 0
\(295\) 6.84601 0.00135115
\(296\) 0 0
\(297\) 1458.18 0.284889
\(298\) 0 0
\(299\) 9016.55 1.74395
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2231.33 −0.423058
\(304\) 0 0
\(305\) −13.9034 −0.00261019
\(306\) 0 0
\(307\) −1960.53 −0.364473 −0.182236 0.983255i \(-0.558334\pi\)
−0.182236 + 0.983255i \(0.558334\pi\)
\(308\) 0 0
\(309\) 313.952 0.0577996
\(310\) 0 0
\(311\) −5207.57 −0.949499 −0.474749 0.880121i \(-0.657461\pi\)
−0.474749 + 0.880121i \(0.657461\pi\)
\(312\) 0 0
\(313\) −3991.71 −0.720846 −0.360423 0.932789i \(-0.617368\pi\)
−0.360423 + 0.932789i \(0.617368\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1805.04 0.319814 0.159907 0.987132i \(-0.448881\pi\)
0.159907 + 0.987132i \(0.448881\pi\)
\(318\) 0 0
\(319\) 3774.85 0.662543
\(320\) 0 0
\(321\) 1541.10 0.267962
\(322\) 0 0
\(323\) 12032.3 2.07274
\(324\) 0 0
\(325\) 6278.55 1.07160
\(326\) 0 0
\(327\) −2925.11 −0.494676
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6906.18 −1.14682 −0.573411 0.819268i \(-0.694380\pi\)
−0.573411 + 0.819268i \(0.694380\pi\)
\(332\) 0 0
\(333\) 2715.28 0.446836
\(334\) 0 0
\(335\) −108.275 −0.0176587
\(336\) 0 0
\(337\) −6081.36 −0.983006 −0.491503 0.870876i \(-0.663552\pi\)
−0.491503 + 0.870876i \(0.663552\pi\)
\(338\) 0 0
\(339\) −5778.21 −0.925750
\(340\) 0 0
\(341\) 17651.9 2.80323
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 69.2414 0.0108053
\(346\) 0 0
\(347\) 7787.02 1.20469 0.602347 0.798234i \(-0.294232\pi\)
0.602347 + 0.798234i \(0.294232\pi\)
\(348\) 0 0
\(349\) −1928.25 −0.295751 −0.147875 0.989006i \(-0.547243\pi\)
−0.147875 + 0.989006i \(0.547243\pi\)
\(350\) 0 0
\(351\) 1356.35 0.206258
\(352\) 0 0
\(353\) −3404.27 −0.513289 −0.256645 0.966506i \(-0.582617\pi\)
−0.256645 + 0.966506i \(0.582617\pi\)
\(354\) 0 0
\(355\) 31.0047 0.00463538
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4417.05 −0.649368 −0.324684 0.945823i \(-0.605258\pi\)
−0.324684 + 0.945823i \(0.605258\pi\)
\(360\) 0 0
\(361\) 1514.97 0.220873
\(362\) 0 0
\(363\) −4757.13 −0.687837
\(364\) 0 0
\(365\) −26.6162 −0.00381687
\(366\) 0 0
\(367\) 5076.65 0.722068 0.361034 0.932553i \(-0.382424\pi\)
0.361034 + 0.932553i \(0.382424\pi\)
\(368\) 0 0
\(369\) 2664.43 0.375893
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6876.64 −0.954582 −0.477291 0.878745i \(-0.658381\pi\)
−0.477291 + 0.878745i \(0.658381\pi\)
\(374\) 0 0
\(375\) 96.4370 0.0132800
\(376\) 0 0
\(377\) 3511.23 0.479676
\(378\) 0 0
\(379\) 9285.61 1.25850 0.629248 0.777205i \(-0.283363\pi\)
0.629248 + 0.777205i \(0.283363\pi\)
\(380\) 0 0
\(381\) −3377.85 −0.454207
\(382\) 0 0
\(383\) 7681.65 1.02484 0.512420 0.858735i \(-0.328749\pi\)
0.512420 + 0.858735i \(0.328749\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1298.72 0.170588
\(388\) 0 0
\(389\) 8627.72 1.12453 0.562266 0.826956i \(-0.309930\pi\)
0.562266 + 0.826956i \(0.309930\pi\)
\(390\) 0 0
\(391\) 23600.3 3.05247
\(392\) 0 0
\(393\) 4479.73 0.574993
\(394\) 0 0
\(395\) −72.0062 −0.00917221
\(396\) 0 0
\(397\) 5735.61 0.725094 0.362547 0.931966i \(-0.381907\pi\)
0.362547 + 0.931966i \(0.381907\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7217.79 −0.898850 −0.449425 0.893318i \(-0.648371\pi\)
−0.449425 + 0.893318i \(0.648371\pi\)
\(402\) 0 0
\(403\) 16419.2 2.02952
\(404\) 0 0
\(405\) 10.4159 0.00127795
\(406\) 0 0
\(407\) −16293.7 −1.98439
\(408\) 0 0
\(409\) −10802.9 −1.30604 −0.653019 0.757341i \(-0.726498\pi\)
−0.653019 + 0.757341i \(0.726498\pi\)
\(410\) 0 0
\(411\) −4382.32 −0.525946
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −126.875 −0.0150073
\(416\) 0 0
\(417\) 6677.55 0.784175
\(418\) 0 0
\(419\) −13257.1 −1.54571 −0.772856 0.634582i \(-0.781172\pi\)
−0.772856 + 0.634582i \(0.781172\pi\)
\(420\) 0 0
\(421\) −6252.11 −0.723774 −0.361887 0.932222i \(-0.617867\pi\)
−0.361887 + 0.932222i \(0.617867\pi\)
\(422\) 0 0
\(423\) 3240.77 0.372510
\(424\) 0 0
\(425\) 16433.7 1.87565
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8139.07 −0.915986
\(430\) 0 0
\(431\) −4949.08 −0.553106 −0.276553 0.960999i \(-0.589192\pi\)
−0.276553 + 0.960999i \(0.589192\pi\)
\(432\) 0 0
\(433\) −16602.8 −1.84267 −0.921337 0.388764i \(-0.872902\pi\)
−0.921337 + 0.388764i \(0.872902\pi\)
\(434\) 0 0
\(435\) 26.9641 0.00297202
\(436\) 0 0
\(437\) 16424.8 1.79795
\(438\) 0 0
\(439\) 4708.38 0.511887 0.255944 0.966692i \(-0.417614\pi\)
0.255944 + 0.966692i \(0.417614\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1700.44 0.182370 0.0911852 0.995834i \(-0.470934\pi\)
0.0911852 + 0.995834i \(0.470934\pi\)
\(444\) 0 0
\(445\) −57.0129 −0.00607342
\(446\) 0 0
\(447\) −1184.04 −0.125287
\(448\) 0 0
\(449\) 10050.2 1.05635 0.528173 0.849137i \(-0.322877\pi\)
0.528173 + 0.849137i \(0.322877\pi\)
\(450\) 0 0
\(451\) −15988.5 −1.66933
\(452\) 0 0
\(453\) −9372.60 −0.972104
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10991.9 −1.12512 −0.562560 0.826757i \(-0.690183\pi\)
−0.562560 + 0.826757i \(0.690183\pi\)
\(458\) 0 0
\(459\) 3550.16 0.361018
\(460\) 0 0
\(461\) −548.440 −0.0554086 −0.0277043 0.999616i \(-0.508820\pi\)
−0.0277043 + 0.999616i \(0.508820\pi\)
\(462\) 0 0
\(463\) −4028.04 −0.404317 −0.202159 0.979353i \(-0.564796\pi\)
−0.202159 + 0.979353i \(0.564796\pi\)
\(464\) 0 0
\(465\) 126.089 0.0125747
\(466\) 0 0
\(467\) −2070.69 −0.205182 −0.102591 0.994724i \(-0.532713\pi\)
−0.102591 + 0.994724i \(0.532713\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 10802.1 1.05676
\(472\) 0 0
\(473\) −7793.26 −0.757579
\(474\) 0 0
\(475\) 11437.2 1.10479
\(476\) 0 0
\(477\) 16.5162 0.00158538
\(478\) 0 0
\(479\) 178.785 0.0170541 0.00852704 0.999964i \(-0.497286\pi\)
0.00852704 + 0.999964i \(0.497286\pi\)
\(480\) 0 0
\(481\) −15155.8 −1.43668
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −95.2622 −0.00891883
\(486\) 0 0
\(487\) 16597.0 1.54431 0.772156 0.635433i \(-0.219178\pi\)
0.772156 + 0.635433i \(0.219178\pi\)
\(488\) 0 0
\(489\) 5922.07 0.547659
\(490\) 0 0
\(491\) 4547.22 0.417949 0.208975 0.977921i \(-0.432987\pi\)
0.208975 + 0.977921i \(0.432987\pi\)
\(492\) 0 0
\(493\) 9190.45 0.839588
\(494\) 0 0
\(495\) −62.5029 −0.00567535
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1025.97 −0.0920414 −0.0460207 0.998940i \(-0.514654\pi\)
−0.0460207 + 0.998940i \(0.514654\pi\)
\(500\) 0 0
\(501\) 3201.26 0.285473
\(502\) 0 0
\(503\) −6745.26 −0.597925 −0.298962 0.954265i \(-0.596641\pi\)
−0.298962 + 0.954265i \(0.596641\pi\)
\(504\) 0 0
\(505\) 95.6431 0.00842785
\(506\) 0 0
\(507\) −979.676 −0.0858164
\(508\) 0 0
\(509\) −6001.02 −0.522575 −0.261287 0.965261i \(-0.584147\pi\)
−0.261287 + 0.965261i \(0.584147\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2470.75 0.212644
\(514\) 0 0
\(515\) −13.4571 −0.00115144
\(516\) 0 0
\(517\) −19447.0 −1.65431
\(518\) 0 0
\(519\) 823.964 0.0696879
\(520\) 0 0
\(521\) 4695.21 0.394820 0.197410 0.980321i \(-0.436747\pi\)
0.197410 + 0.980321i \(0.436747\pi\)
\(522\) 0 0
\(523\) −771.767 −0.0645258 −0.0322629 0.999479i \(-0.510271\pi\)
−0.0322629 + 0.999479i \(0.510271\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42976.2 3.55232
\(528\) 0 0
\(529\) 20048.7 1.64779
\(530\) 0 0
\(531\) 479.147 0.0391586
\(532\) 0 0
\(533\) −14872.0 −1.20859
\(534\) 0 0
\(535\) −66.0573 −0.00533814
\(536\) 0 0
\(537\) 1663.00 0.133638
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3385.66 0.269059 0.134529 0.990910i \(-0.457048\pi\)
0.134529 + 0.990910i \(0.457048\pi\)
\(542\) 0 0
\(543\) 2056.31 0.162513
\(544\) 0 0
\(545\) 125.381 0.00985457
\(546\) 0 0
\(547\) −4988.75 −0.389952 −0.194976 0.980808i \(-0.562463\pi\)
−0.194976 + 0.980808i \(0.562463\pi\)
\(548\) 0 0
\(549\) −973.090 −0.0756475
\(550\) 0 0
\(551\) 6396.15 0.494529
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −116.387 −0.00890153
\(556\) 0 0
\(557\) −20207.3 −1.53718 −0.768591 0.639740i \(-0.779042\pi\)
−0.768591 + 0.639740i \(0.779042\pi\)
\(558\) 0 0
\(559\) −7249.02 −0.548481
\(560\) 0 0
\(561\) −21303.5 −1.60327
\(562\) 0 0
\(563\) 11690.4 0.875118 0.437559 0.899190i \(-0.355843\pi\)
0.437559 + 0.899190i \(0.355843\pi\)
\(564\) 0 0
\(565\) 247.675 0.0184421
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17928.9 −1.32095 −0.660473 0.750849i \(-0.729644\pi\)
−0.660473 + 0.750849i \(0.729644\pi\)
\(570\) 0 0
\(571\) 15673.7 1.14873 0.574364 0.818600i \(-0.305249\pi\)
0.574364 + 0.818600i \(0.305249\pi\)
\(572\) 0 0
\(573\) 14696.6 1.07148
\(574\) 0 0
\(575\) 22432.9 1.62699
\(576\) 0 0
\(577\) −13653.0 −0.985064 −0.492532 0.870294i \(-0.663929\pi\)
−0.492532 + 0.870294i \(0.663929\pi\)
\(578\) 0 0
\(579\) −9420.78 −0.676191
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −99.1093 −0.00704063
\(584\) 0 0
\(585\) −58.1380 −0.00410891
\(586\) 0 0
\(587\) 18021.5 1.26717 0.633583 0.773675i \(-0.281583\pi\)
0.633583 + 0.773675i \(0.281583\pi\)
\(588\) 0 0
\(589\) 29909.6 2.09236
\(590\) 0 0
\(591\) 682.237 0.0474847
\(592\) 0 0
\(593\) 21137.3 1.46375 0.731877 0.681437i \(-0.238645\pi\)
0.731877 + 0.681437i \(0.238645\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3643.50 −0.249780
\(598\) 0 0
\(599\) −9369.59 −0.639117 −0.319559 0.947567i \(-0.603535\pi\)
−0.319559 + 0.947567i \(0.603535\pi\)
\(600\) 0 0
\(601\) −22750.0 −1.54408 −0.772040 0.635573i \(-0.780764\pi\)
−0.772040 + 0.635573i \(0.780764\pi\)
\(602\) 0 0
\(603\) −7578.06 −0.511779
\(604\) 0 0
\(605\) 203.908 0.0137026
\(606\) 0 0
\(607\) 5973.65 0.399445 0.199722 0.979853i \(-0.435996\pi\)
0.199722 + 0.979853i \(0.435996\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18088.9 −1.19771
\(612\) 0 0
\(613\) −23347.4 −1.53832 −0.769162 0.639054i \(-0.779326\pi\)
−0.769162 + 0.639054i \(0.779326\pi\)
\(614\) 0 0
\(615\) −114.207 −0.00748826
\(616\) 0 0
\(617\) 28199.1 1.83996 0.919979 0.391968i \(-0.128206\pi\)
0.919979 + 0.391968i \(0.128206\pi\)
\(618\) 0 0
\(619\) 2975.56 0.193211 0.0966057 0.995323i \(-0.469201\pi\)
0.0966057 + 0.995323i \(0.469201\pi\)
\(620\) 0 0
\(621\) 4846.16 0.313155
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15618.8 0.999603
\(626\) 0 0
\(627\) −14826.3 −0.944348
\(628\) 0 0
\(629\) −39669.4 −2.51466
\(630\) 0 0
\(631\) 2631.33 0.166009 0.0830044 0.996549i \(-0.473548\pi\)
0.0830044 + 0.996549i \(0.473548\pi\)
\(632\) 0 0
\(633\) 15348.2 0.963723
\(634\) 0 0
\(635\) 144.787 0.00904837
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2170.00 0.134341
\(640\) 0 0
\(641\) 14494.2 0.893117 0.446559 0.894754i \(-0.352649\pi\)
0.446559 + 0.894754i \(0.352649\pi\)
\(642\) 0 0
\(643\) 15176.0 0.930767 0.465383 0.885109i \(-0.345916\pi\)
0.465383 + 0.885109i \(0.345916\pi\)
\(644\) 0 0
\(645\) −55.6679 −0.00339833
\(646\) 0 0
\(647\) 7569.18 0.459931 0.229965 0.973199i \(-0.426139\pi\)
0.229965 + 0.973199i \(0.426139\pi\)
\(648\) 0 0
\(649\) −2875.23 −0.173903
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13933.5 −0.835007 −0.417504 0.908675i \(-0.637095\pi\)
−0.417504 + 0.908675i \(0.637095\pi\)
\(654\) 0 0
\(655\) −192.018 −0.0114546
\(656\) 0 0
\(657\) −1862.85 −0.110619
\(658\) 0 0
\(659\) 17015.5 1.00581 0.502904 0.864342i \(-0.332265\pi\)
0.502904 + 0.864342i \(0.332265\pi\)
\(660\) 0 0
\(661\) −16535.4 −0.972999 −0.486500 0.873681i \(-0.661726\pi\)
−0.486500 + 0.873681i \(0.661726\pi\)
\(662\) 0 0
\(663\) −19815.8 −1.16076
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12545.5 0.728279
\(668\) 0 0
\(669\) 358.152 0.0206980
\(670\) 0 0
\(671\) 5839.25 0.335949
\(672\) 0 0
\(673\) 4571.05 0.261814 0.130907 0.991395i \(-0.458211\pi\)
0.130907 + 0.991395i \(0.458211\pi\)
\(674\) 0 0
\(675\) 3374.55 0.192425
\(676\) 0 0
\(677\) 26284.5 1.49217 0.746083 0.665853i \(-0.231932\pi\)
0.746083 + 0.665853i \(0.231932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5976.00 0.336271
\(682\) 0 0
\(683\) −6797.22 −0.380803 −0.190402 0.981706i \(-0.560979\pi\)
−0.190402 + 0.981706i \(0.560979\pi\)
\(684\) 0 0
\(685\) 187.842 0.0104775
\(686\) 0 0
\(687\) −2110.43 −0.117202
\(688\) 0 0
\(689\) −92.1880 −0.00509737
\(690\) 0 0
\(691\) −27758.7 −1.52821 −0.764103 0.645094i \(-0.776818\pi\)
−0.764103 + 0.645094i \(0.776818\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −286.225 −0.0156218
\(696\) 0 0
\(697\) −38926.5 −2.11542
\(698\) 0 0
\(699\) −2885.35 −0.156129
\(700\) 0 0
\(701\) −21638.3 −1.16586 −0.582929 0.812523i \(-0.698093\pi\)
−0.582929 + 0.812523i \(0.698093\pi\)
\(702\) 0 0
\(703\) −27608.2 −1.48117
\(704\) 0 0
\(705\) −138.911 −0.00742086
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −16288.7 −0.862815 −0.431408 0.902157i \(-0.641983\pi\)
−0.431408 + 0.902157i \(0.641983\pi\)
\(710\) 0 0
\(711\) −5039.66 −0.265826
\(712\) 0 0
\(713\) 58664.8 3.08137
\(714\) 0 0
\(715\) 348.871 0.0182476
\(716\) 0 0
\(717\) −13394.1 −0.697646
\(718\) 0 0
\(719\) 3391.74 0.175926 0.0879628 0.996124i \(-0.471964\pi\)
0.0879628 + 0.996124i \(0.471964\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1305.93 0.0671760
\(724\) 0 0
\(725\) 8735.86 0.447506
\(726\) 0 0
\(727\) −23158.7 −1.18144 −0.590722 0.806875i \(-0.701157\pi\)
−0.590722 + 0.806875i \(0.701157\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −18973.9 −0.960020
\(732\) 0 0
\(733\) −27629.2 −1.39223 −0.696116 0.717929i \(-0.745090\pi\)
−0.696116 + 0.717929i \(0.745090\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45473.9 2.27280
\(738\) 0 0
\(739\) 9461.65 0.470978 0.235489 0.971877i \(-0.424331\pi\)
0.235489 + 0.971877i \(0.424331\pi\)
\(740\) 0 0
\(741\) −13790.9 −0.683701
\(742\) 0 0
\(743\) 14316.9 0.706913 0.353457 0.935451i \(-0.385006\pi\)
0.353457 + 0.935451i \(0.385006\pi\)
\(744\) 0 0
\(745\) 50.7524 0.00249587
\(746\) 0 0
\(747\) −8879.86 −0.434936
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7486.07 0.363742 0.181871 0.983322i \(-0.441785\pi\)
0.181871 + 0.983322i \(0.441785\pi\)
\(752\) 0 0
\(753\) 2589.01 0.125297
\(754\) 0 0
\(755\) 401.744 0.0193655
\(756\) 0 0
\(757\) −17416.8 −0.836227 −0.418114 0.908395i \(-0.637309\pi\)
−0.418114 + 0.908395i \(0.637309\pi\)
\(758\) 0 0
\(759\) −29080.5 −1.39072
\(760\) 0 0
\(761\) −33231.0 −1.58295 −0.791474 0.611203i \(-0.790686\pi\)
−0.791474 + 0.611203i \(0.790686\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −152.173 −0.00719192
\(766\) 0 0
\(767\) −2674.44 −0.125904
\(768\) 0 0
\(769\) 13714.2 0.643103 0.321552 0.946892i \(-0.395796\pi\)
0.321552 + 0.946892i \(0.395796\pi\)
\(770\) 0 0
\(771\) 1610.08 0.0752082
\(772\) 0 0
\(773\) 15209.3 0.707685 0.353842 0.935305i \(-0.384875\pi\)
0.353842 + 0.935305i \(0.384875\pi\)
\(774\) 0 0
\(775\) 40850.5 1.89341
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −27091.1 −1.24601
\(780\) 0 0
\(781\) −13021.6 −0.596605
\(782\) 0 0
\(783\) 1887.20 0.0861339
\(784\) 0 0
\(785\) −463.019 −0.0210520
\(786\) 0 0
\(787\) 13221.5 0.598850 0.299425 0.954120i \(-0.403205\pi\)
0.299425 + 0.954120i \(0.403205\pi\)
\(788\) 0 0
\(789\) 1115.72 0.0503433
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5431.47 0.243224
\(794\) 0 0
\(795\) −0.707946 −3.15827e−5 0
\(796\) 0 0
\(797\) 13373.9 0.594387 0.297194 0.954817i \(-0.403949\pi\)
0.297194 + 0.954817i \(0.403949\pi\)
\(798\) 0 0
\(799\) −47346.6 −2.09637
\(800\) 0 0
\(801\) −3990.29 −0.176018
\(802\) 0 0
\(803\) 11178.5 0.491257
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13516.8 0.589609
\(808\) 0 0
\(809\) −11229.3 −0.488011 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(810\) 0 0
\(811\) 4301.42 0.186243 0.0931215 0.995655i \(-0.470315\pi\)
0.0931215 + 0.995655i \(0.470315\pi\)
\(812\) 0 0
\(813\) −1188.35 −0.0512634
\(814\) 0 0
\(815\) −253.842 −0.0109101
\(816\) 0 0
\(817\) −13205.0 −0.565464
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −40910.6 −1.73909 −0.869543 0.493857i \(-0.835587\pi\)
−0.869543 + 0.493857i \(0.835587\pi\)
\(822\) 0 0
\(823\) −28043.2 −1.18776 −0.593879 0.804554i \(-0.702404\pi\)
−0.593879 + 0.804554i \(0.702404\pi\)
\(824\) 0 0
\(825\) −20249.8 −0.854554
\(826\) 0 0
\(827\) −17598.9 −0.739993 −0.369996 0.929033i \(-0.620641\pi\)
−0.369996 + 0.929033i \(0.620641\pi\)
\(828\) 0 0
\(829\) −14565.5 −0.610231 −0.305116 0.952315i \(-0.598695\pi\)
−0.305116 + 0.952315i \(0.598695\pi\)
\(830\) 0 0
\(831\) 19526.2 0.815111
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −137.218 −0.00568697
\(836\) 0 0
\(837\) 8824.86 0.364435
\(838\) 0 0
\(839\) −711.056 −0.0292591 −0.0146295 0.999893i \(-0.504657\pi\)
−0.0146295 + 0.999893i \(0.504657\pi\)
\(840\) 0 0
\(841\) −19503.5 −0.799686
\(842\) 0 0
\(843\) −8357.41 −0.341453
\(844\) 0 0
\(845\) 41.9925 0.00170957
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.83554 7.41996e−5 0
\(850\) 0 0
\(851\) −54150.9 −2.18128
\(852\) 0 0
\(853\) −7459.48 −0.299423 −0.149712 0.988730i \(-0.547834\pi\)
−0.149712 + 0.988730i \(0.547834\pi\)
\(854\) 0 0
\(855\) −105.906 −0.00423614
\(856\) 0 0
\(857\) 3560.92 0.141936 0.0709678 0.997479i \(-0.477391\pi\)
0.0709678 + 0.997479i \(0.477391\pi\)
\(858\) 0 0
\(859\) −13519.1 −0.536981 −0.268491 0.963282i \(-0.586525\pi\)
−0.268491 + 0.963282i \(0.586525\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20395.9 −0.804502 −0.402251 0.915530i \(-0.631772\pi\)
−0.402251 + 0.915530i \(0.631772\pi\)
\(864\) 0 0
\(865\) −35.3182 −0.00138827
\(866\) 0 0
\(867\) −37127.7 −1.45435
\(868\) 0 0
\(869\) 30241.6 1.18053
\(870\) 0 0
\(871\) 42298.3 1.64549
\(872\) 0 0
\(873\) −6667.33 −0.258482
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7197.26 0.277120 0.138560 0.990354i \(-0.455753\pi\)
0.138560 + 0.990354i \(0.455753\pi\)
\(878\) 0 0
\(879\) 12437.9 0.477271
\(880\) 0 0
\(881\) 11588.3 0.443156 0.221578 0.975143i \(-0.428879\pi\)
0.221578 + 0.975143i \(0.428879\pi\)
\(882\) 0 0
\(883\) −21959.0 −0.836898 −0.418449 0.908240i \(-0.637426\pi\)
−0.418449 + 0.908240i \(0.637426\pi\)
\(884\) 0 0
\(885\) −20.5380 −0.000780088 0
\(886\) 0 0
\(887\) −12518.9 −0.473894 −0.236947 0.971523i \(-0.576147\pi\)
−0.236947 + 0.971523i \(0.576147\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4374.53 −0.164481
\(892\) 0 0
\(893\) −32951.2 −1.23479
\(894\) 0 0
\(895\) −71.2822 −0.00266224
\(896\) 0 0
\(897\) −27049.6 −1.00687
\(898\) 0 0
\(899\) 22845.3 0.847535
\(900\) 0 0
\(901\) −241.297 −0.00892204
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −88.1410 −0.00323747
\(906\) 0 0
\(907\) −28270.0 −1.03494 −0.517470 0.855701i \(-0.673126\pi\)
−0.517470 + 0.855701i \(0.673126\pi\)
\(908\) 0 0
\(909\) 6693.99 0.244253
\(910\) 0 0
\(911\) 7908.58 0.287621 0.143811 0.989605i \(-0.454064\pi\)
0.143811 + 0.989605i \(0.454064\pi\)
\(912\) 0 0
\(913\) 53285.7 1.93154
\(914\) 0 0
\(915\) 41.7102 0.00150699
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −31450.6 −1.12890 −0.564451 0.825467i \(-0.690912\pi\)
−0.564451 + 0.825467i \(0.690912\pi\)
\(920\) 0 0
\(921\) 5881.58 0.210428
\(922\) 0 0
\(923\) −12112.2 −0.431938
\(924\) 0 0
\(925\) −37707.2 −1.34033
\(926\) 0 0
\(927\) −941.855 −0.0333706
\(928\) 0 0
\(929\) −3469.19 −0.122519 −0.0612596 0.998122i \(-0.519512\pi\)
−0.0612596 + 0.998122i \(0.519512\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 15622.7 0.548193
\(934\) 0 0
\(935\) 913.148 0.0319392
\(936\) 0 0
\(937\) 50935.7 1.77588 0.887939 0.459962i \(-0.152137\pi\)
0.887939 + 0.459962i \(0.152137\pi\)
\(938\) 0 0
\(939\) 11975.1 0.416181
\(940\) 0 0
\(941\) 47894.9 1.65922 0.829612 0.558341i \(-0.188562\pi\)
0.829612 + 0.558341i \(0.188562\pi\)
\(942\) 0 0
\(943\) −53136.7 −1.83496
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10538.9 −0.361634 −0.180817 0.983517i \(-0.557874\pi\)
−0.180817 + 0.983517i \(0.557874\pi\)
\(948\) 0 0
\(949\) 10397.8 0.355667
\(950\) 0 0
\(951\) −5415.11 −0.184645
\(952\) 0 0
\(953\) 31123.5 1.05791 0.528955 0.848650i \(-0.322584\pi\)
0.528955 + 0.848650i \(0.322584\pi\)
\(954\) 0 0
\(955\) −629.951 −0.0213453
\(956\) 0 0
\(957\) −11324.6 −0.382519
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 77037.9 2.58594
\(962\) 0 0
\(963\) −4623.30 −0.154708
\(964\) 0 0
\(965\) 403.810 0.0134706
\(966\) 0 0
\(967\) 1626.14 0.0540776 0.0270388 0.999634i \(-0.491392\pi\)
0.0270388 + 0.999634i \(0.491392\pi\)
\(968\) 0 0
\(969\) −36096.9 −1.19670
\(970\) 0 0
\(971\) −35500.1 −1.17328 −0.586638 0.809849i \(-0.699549\pi\)
−0.586638 + 0.809849i \(0.699549\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −18835.6 −0.618691
\(976\) 0 0
\(977\) 26570.2 0.870068 0.435034 0.900414i \(-0.356736\pi\)
0.435034 + 0.900414i \(0.356736\pi\)
\(978\) 0 0
\(979\) 23944.7 0.781691
\(980\) 0 0
\(981\) 8775.34 0.285602
\(982\) 0 0
\(983\) 53851.6 1.74730 0.873652 0.486552i \(-0.161746\pi\)
0.873652 + 0.486552i \(0.161746\pi\)
\(984\) 0 0
\(985\) −29.2432 −0.000945955 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −25900.4 −0.832745
\(990\) 0 0
\(991\) −2829.75 −0.0907063 −0.0453531 0.998971i \(-0.514441\pi\)
−0.0453531 + 0.998971i \(0.514441\pi\)
\(992\) 0 0
\(993\) 20718.6 0.662118
\(994\) 0 0
\(995\) 156.174 0.00497593
\(996\) 0 0
\(997\) 12904.3 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(998\) 0 0
\(999\) −8145.83 −0.257981
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 2352.4.a.cm.1.2 4
4.3 odd 2 1176.4.a.bd.1.2 4
7.2 even 3 336.4.q.m.193.3 8
7.4 even 3 336.4.q.m.289.3 8
7.6 odd 2 2352.4.a.cp.1.3 4
28.11 odd 6 168.4.q.f.121.3 yes 8
28.23 odd 6 168.4.q.f.25.3 8
28.27 even 2 1176.4.a.ba.1.3 4
84.11 even 6 504.4.s.j.289.2 8
84.23 even 6 504.4.s.j.361.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.3 8 28.23 odd 6
168.4.q.f.121.3 yes 8 28.11 odd 6
336.4.q.m.193.3 8 7.2 even 3
336.4.q.m.289.3 8 7.4 even 3
504.4.s.j.289.2 8 84.11 even 6
504.4.s.j.361.2 8 84.23 even 6
1176.4.a.ba.1.3 4 28.27 even 2
1176.4.a.bd.1.2 4 4.3 odd 2
2352.4.a.cm.1.2 4 1.1 even 1 trivial
2352.4.a.cp.1.3 4 7.6 odd 2