Properties

Label 1176.4.a.z.1.4
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.391168.1
Defining polynomial: \(x^{4} - 40 x^{2} + 382\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.92368\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +13.3405 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +13.3405 q^{5} +9.00000 q^{9} -1.42531 q^{11} -38.7658 q^{13} -40.0215 q^{15} -27.3562 q^{17} +65.4495 q^{19} +2.54332 q^{23} +52.9686 q^{25} -27.0000 q^{27} -63.7544 q^{29} -51.9548 q^{31} +4.27592 q^{33} -335.494 q^{37} +116.297 q^{39} -447.541 q^{41} -170.339 q^{43} +120.064 q^{45} +116.813 q^{47} +82.0685 q^{51} +86.3042 q^{53} -19.0143 q^{55} -196.349 q^{57} -380.892 q^{59} +199.624 q^{61} -517.155 q^{65} +951.939 q^{67} -7.62996 q^{69} +830.527 q^{71} +332.753 q^{73} -158.906 q^{75} +755.996 q^{79} +81.0000 q^{81} +15.2261 q^{83} -364.945 q^{85} +191.263 q^{87} -1554.11 q^{89} +155.864 q^{93} +873.128 q^{95} -101.205 q^{97} -12.8278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} - 8q^{5} + 36q^{9} + 40q^{11} - 48q^{13} + 24q^{15} - 72q^{17} - 32q^{19} + 8q^{23} + 164q^{25} - 108q^{27} + 144q^{29} - 48q^{31} - 120q^{33} + 48q^{37} + 144q^{39} - 72q^{41} + 512q^{43} - 72q^{45} - 160q^{47} + 216q^{51} + 536q^{53} + 336q^{55} + 96q^{57} - 240q^{59} - 896q^{61} - 136q^{65} + 1088q^{67} - 24q^{69} + 1288q^{71} - 1488q^{73} - 492q^{75} + 416q^{79} + 324q^{81} + 112q^{83} - 1512q^{85} - 432q^{87} - 3160q^{89} + 144q^{93} - 240q^{95} - 2384q^{97} + 360q^{99} + O(q^{100}) \)

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
\(5\) 13.3405 1.19321 0.596605 0.802535i \(-0.296516\pi\)
0.596605 + 0.802535i \(0.296516\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −1.42531 −0.0390678 −0.0195339 0.999809i \(-0.506218\pi\)
−0.0195339 + 0.999809i \(0.506218\pi\)
\(12\) 0 0
\(13\) −38.7658 −0.827054 −0.413527 0.910492i \(-0.635703\pi\)
−0.413527 + 0.910492i \(0.635703\pi\)
\(14\) 0 0
\(15\) −40.0215 −0.688900
\(16\) 0 0
\(17\) −27.3562 −0.390285 −0.195143 0.980775i \(-0.562517\pi\)
−0.195143 + 0.980775i \(0.562517\pi\)
\(18\) 0 0
\(19\) 65.4495 0.790271 0.395135 0.918623i \(-0.370698\pi\)
0.395135 + 0.918623i \(0.370698\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.54332 0.0230573 0.0115287 0.999934i \(-0.496330\pi\)
0.0115287 + 0.999934i \(0.496330\pi\)
\(24\) 0 0
\(25\) 52.9686 0.423749
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −63.7544 −0.408238 −0.204119 0.978946i \(-0.565433\pi\)
−0.204119 + 0.978946i \(0.565433\pi\)
\(30\) 0 0
\(31\) −51.9548 −0.301011 −0.150506 0.988609i \(-0.548090\pi\)
−0.150506 + 0.988609i \(0.548090\pi\)
\(32\) 0 0
\(33\) 4.27592 0.0225558
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −335.494 −1.49067 −0.745337 0.666688i \(-0.767711\pi\)
−0.745337 + 0.666688i \(0.767711\pi\)
\(38\) 0 0
\(39\) 116.297 0.477500
\(40\) 0 0
\(41\) −447.541 −1.70473 −0.852367 0.522944i \(-0.824834\pi\)
−0.852367 + 0.522944i \(0.824834\pi\)
\(42\) 0 0
\(43\) −170.339 −0.604103 −0.302052 0.953292i \(-0.597672\pi\)
−0.302052 + 0.953292i \(0.597672\pi\)
\(44\) 0 0
\(45\) 120.064 0.397737
\(46\) 0 0
\(47\) 116.813 0.362531 0.181266 0.983434i \(-0.441981\pi\)
0.181266 + 0.983434i \(0.441981\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 82.0685 0.225331
\(52\) 0 0
\(53\) 86.3042 0.223675 0.111838 0.993726i \(-0.464326\pi\)
0.111838 + 0.993726i \(0.464326\pi\)
\(54\) 0 0
\(55\) −19.0143 −0.0466161
\(56\) 0 0
\(57\) −196.349 −0.456263
\(58\) 0 0
\(59\) −380.892 −0.840473 −0.420236 0.907415i \(-0.638053\pi\)
−0.420236 + 0.907415i \(0.638053\pi\)
\(60\) 0 0
\(61\) 199.624 0.419004 0.209502 0.977808i \(-0.432816\pi\)
0.209502 + 0.977808i \(0.432816\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −517.155 −0.986848
\(66\) 0 0
\(67\) 951.939 1.73579 0.867894 0.496749i \(-0.165473\pi\)
0.867894 + 0.496749i \(0.165473\pi\)
\(68\) 0 0
\(69\) −7.62996 −0.0133122
\(70\) 0 0
\(71\) 830.527 1.38825 0.694123 0.719857i \(-0.255793\pi\)
0.694123 + 0.719857i \(0.255793\pi\)
\(72\) 0 0
\(73\) 332.753 0.533504 0.266752 0.963765i \(-0.414050\pi\)
0.266752 + 0.963765i \(0.414050\pi\)
\(74\) 0 0
\(75\) −158.906 −0.244652
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 755.996 1.07666 0.538330 0.842734i \(-0.319055\pi\)
0.538330 + 0.842734i \(0.319055\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 15.2261 0.0201360 0.0100680 0.999949i \(-0.496795\pi\)
0.0100680 + 0.999949i \(0.496795\pi\)
\(84\) 0 0
\(85\) −364.945 −0.465692
\(86\) 0 0
\(87\) 191.263 0.235696
\(88\) 0 0
\(89\) −1554.11 −1.85096 −0.925481 0.378795i \(-0.876339\pi\)
−0.925481 + 0.378795i \(0.876339\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 155.864 0.173789
\(94\) 0 0
\(95\) 873.128 0.942959
\(96\) 0 0
\(97\) −101.205 −0.105937 −0.0529683 0.998596i \(-0.516868\pi\)
−0.0529683 + 0.998596i \(0.516868\pi\)
\(98\) 0 0
\(99\) −12.8278 −0.0130226
\(100\) 0 0
\(101\) −1439.24 −1.41792 −0.708961 0.705248i \(-0.750836\pi\)
−0.708961 + 0.705248i \(0.750836\pi\)
\(102\) 0 0
\(103\) −1605.32 −1.53569 −0.767847 0.640633i \(-0.778672\pi\)
−0.767847 + 0.640633i \(0.778672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −308.070 −0.278339 −0.139169 0.990269i \(-0.544443\pi\)
−0.139169 + 0.990269i \(0.544443\pi\)
\(108\) 0 0
\(109\) 708.959 0.622991 0.311495 0.950248i \(-0.399170\pi\)
0.311495 + 0.950248i \(0.399170\pi\)
\(110\) 0 0
\(111\) 1006.48 0.860641
\(112\) 0 0
\(113\) 1107.48 0.921975 0.460988 0.887407i \(-0.347495\pi\)
0.460988 + 0.887407i \(0.347495\pi\)
\(114\) 0 0
\(115\) 33.9291 0.0275122
\(116\) 0 0
\(117\) −348.892 −0.275685
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1328.97 −0.998474
\(122\) 0 0
\(123\) 1342.62 0.984229
\(124\) 0 0
\(125\) −960.934 −0.687588
\(126\) 0 0
\(127\) −2330.27 −1.62817 −0.814087 0.580743i \(-0.802762\pi\)
−0.814087 + 0.580743i \(0.802762\pi\)
\(128\) 0 0
\(129\) 511.017 0.348779
\(130\) 0 0
\(131\) −2494.68 −1.66383 −0.831913 0.554907i \(-0.812754\pi\)
−0.831913 + 0.554907i \(0.812754\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −360.193 −0.229633
\(136\) 0 0
\(137\) 260.269 0.162309 0.0811543 0.996702i \(-0.474139\pi\)
0.0811543 + 0.996702i \(0.474139\pi\)
\(138\) 0 0
\(139\) −244.245 −0.149040 −0.0745202 0.997220i \(-0.523743\pi\)
−0.0745202 + 0.997220i \(0.523743\pi\)
\(140\) 0 0
\(141\) −350.440 −0.209308
\(142\) 0 0
\(143\) 55.2531 0.0323112
\(144\) 0 0
\(145\) −850.514 −0.487113
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1326.35 −0.729255 −0.364628 0.931153i \(-0.618804\pi\)
−0.364628 + 0.931153i \(0.618804\pi\)
\(150\) 0 0
\(151\) 406.949 0.219318 0.109659 0.993969i \(-0.465024\pi\)
0.109659 + 0.993969i \(0.465024\pi\)
\(152\) 0 0
\(153\) −246.206 −0.130095
\(154\) 0 0
\(155\) −693.102 −0.359170
\(156\) 0 0
\(157\) −3135.93 −1.59410 −0.797052 0.603911i \(-0.793608\pi\)
−0.797052 + 0.603911i \(0.793608\pi\)
\(158\) 0 0
\(159\) −258.913 −0.129139
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −130.980 −0.0629394 −0.0314697 0.999505i \(-0.510019\pi\)
−0.0314697 + 0.999505i \(0.510019\pi\)
\(164\) 0 0
\(165\) 57.0428 0.0269138
\(166\) 0 0
\(167\) 1787.80 0.828407 0.414204 0.910184i \(-0.364060\pi\)
0.414204 + 0.910184i \(0.364060\pi\)
\(168\) 0 0
\(169\) −694.213 −0.315982
\(170\) 0 0
\(171\) 589.046 0.263424
\(172\) 0 0
\(173\) 609.165 0.267711 0.133855 0.991001i \(-0.457264\pi\)
0.133855 + 0.991001i \(0.457264\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1142.68 0.485247
\(178\) 0 0
\(179\) 1488.76 0.621649 0.310825 0.950467i \(-0.399395\pi\)
0.310825 + 0.950467i \(0.399395\pi\)
\(180\) 0 0
\(181\) −452.765 −0.185932 −0.0929661 0.995669i \(-0.529635\pi\)
−0.0929661 + 0.995669i \(0.529635\pi\)
\(182\) 0 0
\(183\) −598.872 −0.241912
\(184\) 0 0
\(185\) −4475.66 −1.77869
\(186\) 0 0
\(187\) 38.9909 0.0152476
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2723.03 −1.03158 −0.515789 0.856716i \(-0.672501\pi\)
−0.515789 + 0.856716i \(0.672501\pi\)
\(192\) 0 0
\(193\) 2577.69 0.961380 0.480690 0.876891i \(-0.340386\pi\)
0.480690 + 0.876891i \(0.340386\pi\)
\(194\) 0 0
\(195\) 1551.46 0.569757
\(196\) 0 0
\(197\) −1433.02 −0.518266 −0.259133 0.965842i \(-0.583437\pi\)
−0.259133 + 0.965842i \(0.583437\pi\)
\(198\) 0 0
\(199\) −1771.08 −0.630897 −0.315448 0.948943i \(-0.602155\pi\)
−0.315448 + 0.948943i \(0.602155\pi\)
\(200\) 0 0
\(201\) −2855.82 −1.00216
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5970.41 −2.03411
\(206\) 0 0
\(207\) 22.8899 0.00768578
\(208\) 0 0
\(209\) −93.2856 −0.0308742
\(210\) 0 0
\(211\) −5366.25 −1.75084 −0.875421 0.483362i \(-0.839416\pi\)
−0.875421 + 0.483362i \(0.839416\pi\)
\(212\) 0 0
\(213\) −2491.58 −0.801504
\(214\) 0 0
\(215\) −2272.40 −0.720822
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −998.259 −0.308019
\(220\) 0 0
\(221\) 1060.48 0.322787
\(222\) 0 0
\(223\) −2827.33 −0.849023 −0.424511 0.905423i \(-0.639554\pi\)
−0.424511 + 0.905423i \(0.639554\pi\)
\(224\) 0 0
\(225\) 476.718 0.141250
\(226\) 0 0
\(227\) 3565.64 1.04255 0.521277 0.853387i \(-0.325456\pi\)
0.521277 + 0.853387i \(0.325456\pi\)
\(228\) 0 0
\(229\) 4743.28 1.36876 0.684378 0.729128i \(-0.260074\pi\)
0.684378 + 0.729128i \(0.260074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3471.07 0.975955 0.487978 0.872856i \(-0.337735\pi\)
0.487978 + 0.872856i \(0.337735\pi\)
\(234\) 0 0
\(235\) 1558.35 0.432576
\(236\) 0 0
\(237\) −2267.99 −0.621610
\(238\) 0 0
\(239\) −6089.40 −1.64808 −0.824039 0.566533i \(-0.808284\pi\)
−0.824039 + 0.566533i \(0.808284\pi\)
\(240\) 0 0
\(241\) −4936.25 −1.31938 −0.659692 0.751536i \(-0.729313\pi\)
−0.659692 + 0.751536i \(0.729313\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2537.20 −0.653596
\(248\) 0 0
\(249\) −45.6784 −0.0116255
\(250\) 0 0
\(251\) −1263.24 −0.317668 −0.158834 0.987305i \(-0.550774\pi\)
−0.158834 + 0.987305i \(0.550774\pi\)
\(252\) 0 0
\(253\) −3.62501 −0.000900800 0
\(254\) 0 0
\(255\) 1094.83 0.268867
\(256\) 0 0
\(257\) −6152.93 −1.49342 −0.746711 0.665149i \(-0.768368\pi\)
−0.746711 + 0.665149i \(0.768368\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −573.789 −0.136079
\(262\) 0 0
\(263\) 4184.54 0.981102 0.490551 0.871412i \(-0.336796\pi\)
0.490551 + 0.871412i \(0.336796\pi\)
\(264\) 0 0
\(265\) 1151.34 0.266892
\(266\) 0 0
\(267\) 4662.34 1.06865
\(268\) 0 0
\(269\) −718.978 −0.162962 −0.0814812 0.996675i \(-0.525965\pi\)
−0.0814812 + 0.996675i \(0.525965\pi\)
\(270\) 0 0
\(271\) 1294.42 0.290150 0.145075 0.989421i \(-0.453658\pi\)
0.145075 + 0.989421i \(0.453658\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −75.4965 −0.0165549
\(276\) 0 0
\(277\) −2320.92 −0.503431 −0.251716 0.967801i \(-0.580995\pi\)
−0.251716 + 0.967801i \(0.580995\pi\)
\(278\) 0 0
\(279\) −467.593 −0.100337
\(280\) 0 0
\(281\) 6565.23 1.39377 0.696884 0.717184i \(-0.254569\pi\)
0.696884 + 0.717184i \(0.254569\pi\)
\(282\) 0 0
\(283\) 3728.49 0.783165 0.391583 0.920143i \(-0.371928\pi\)
0.391583 + 0.920143i \(0.371928\pi\)
\(284\) 0 0
\(285\) −2619.39 −0.544418
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4164.64 −0.847678
\(290\) 0 0
\(291\) 303.616 0.0611626
\(292\) 0 0
\(293\) −269.734 −0.0537817 −0.0268909 0.999638i \(-0.508561\pi\)
−0.0268909 + 0.999638i \(0.508561\pi\)
\(294\) 0 0
\(295\) −5081.28 −1.00286
\(296\) 0 0
\(297\) 38.4833 0.00751860
\(298\) 0 0
\(299\) −98.5938 −0.0190697
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4317.73 0.818638
\(304\) 0 0
\(305\) 2663.08 0.499959
\(306\) 0 0
\(307\) −3151.93 −0.585961 −0.292980 0.956118i \(-0.594647\pi\)
−0.292980 + 0.956118i \(0.594647\pi\)
\(308\) 0 0
\(309\) 4815.95 0.886633
\(310\) 0 0
\(311\) 8137.23 1.48366 0.741832 0.670585i \(-0.233957\pi\)
0.741832 + 0.670585i \(0.233957\pi\)
\(312\) 0 0
\(313\) −3975.26 −0.717875 −0.358938 0.933362i \(-0.616861\pi\)
−0.358938 + 0.933362i \(0.616861\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6079.93 1.07723 0.538617 0.842551i \(-0.318947\pi\)
0.538617 + 0.842551i \(0.318947\pi\)
\(318\) 0 0
\(319\) 90.8695 0.0159489
\(320\) 0 0
\(321\) 924.210 0.160699
\(322\) 0 0
\(323\) −1790.45 −0.308431
\(324\) 0 0
\(325\) −2053.37 −0.350463
\(326\) 0 0
\(327\) −2126.88 −0.359684
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7579.85 1.25869 0.629345 0.777126i \(-0.283323\pi\)
0.629345 + 0.777126i \(0.283323\pi\)
\(332\) 0 0
\(333\) −3019.45 −0.496891
\(334\) 0 0
\(335\) 12699.3 2.07116
\(336\) 0 0
\(337\) 10921.5 1.76537 0.882685 0.469965i \(-0.155733\pi\)
0.882685 + 0.469965i \(0.155733\pi\)
\(338\) 0 0
\(339\) −3322.45 −0.532303
\(340\) 0 0
\(341\) 74.0515 0.0117599
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −101.787 −0.0158842
\(346\) 0 0
\(347\) 8410.73 1.30119 0.650593 0.759426i \(-0.274520\pi\)
0.650593 + 0.759426i \(0.274520\pi\)
\(348\) 0 0
\(349\) −12309.3 −1.88797 −0.943985 0.329988i \(-0.892955\pi\)
−0.943985 + 0.329988i \(0.892955\pi\)
\(350\) 0 0
\(351\) 1046.68 0.159167
\(352\) 0 0
\(353\) 4136.83 0.623742 0.311871 0.950124i \(-0.399044\pi\)
0.311871 + 0.950124i \(0.399044\pi\)
\(354\) 0 0
\(355\) 11079.6 1.65647
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3539.56 0.520365 0.260182 0.965559i \(-0.416217\pi\)
0.260182 + 0.965559i \(0.416217\pi\)
\(360\) 0 0
\(361\) −2575.36 −0.375472
\(362\) 0 0
\(363\) 3986.91 0.576469
\(364\) 0 0
\(365\) 4439.09 0.636582
\(366\) 0 0
\(367\) −7873.94 −1.11994 −0.559968 0.828514i \(-0.689186\pi\)
−0.559968 + 0.828514i \(0.689186\pi\)
\(368\) 0 0
\(369\) −4027.87 −0.568245
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7059.21 −0.979925 −0.489963 0.871743i \(-0.662990\pi\)
−0.489963 + 0.871743i \(0.662990\pi\)
\(374\) 0 0
\(375\) 2882.80 0.396979
\(376\) 0 0
\(377\) 2471.49 0.337634
\(378\) 0 0
\(379\) −7629.57 −1.03405 −0.517025 0.855971i \(-0.672960\pi\)
−0.517025 + 0.855971i \(0.672960\pi\)
\(380\) 0 0
\(381\) 6990.81 0.940027
\(382\) 0 0
\(383\) −533.655 −0.0711971 −0.0355985 0.999366i \(-0.511334\pi\)
−0.0355985 + 0.999366i \(0.511334\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1533.05 −0.201368
\(388\) 0 0
\(389\) −5716.76 −0.745119 −0.372559 0.928008i \(-0.621520\pi\)
−0.372559 + 0.928008i \(0.621520\pi\)
\(390\) 0 0
\(391\) −69.5755 −0.00899894
\(392\) 0 0
\(393\) 7484.04 0.960610
\(394\) 0 0
\(395\) 10085.4 1.28468
\(396\) 0 0
\(397\) 7873.77 0.995398 0.497699 0.867350i \(-0.334178\pi\)
0.497699 + 0.867350i \(0.334178\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 619.386 0.0771338 0.0385669 0.999256i \(-0.487721\pi\)
0.0385669 + 0.999256i \(0.487721\pi\)
\(402\) 0 0
\(403\) 2014.07 0.248953
\(404\) 0 0
\(405\) 1080.58 0.132579
\(406\) 0 0
\(407\) 478.182 0.0582374
\(408\) 0 0
\(409\) 135.639 0.0163983 0.00819916 0.999966i \(-0.497390\pi\)
0.00819916 + 0.999966i \(0.497390\pi\)
\(410\) 0 0
\(411\) −780.807 −0.0937089
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 203.124 0.0240264
\(416\) 0 0
\(417\) 732.736 0.0860486
\(418\) 0 0
\(419\) −4903.63 −0.571738 −0.285869 0.958269i \(-0.592282\pi\)
−0.285869 + 0.958269i \(0.592282\pi\)
\(420\) 0 0
\(421\) −3294.67 −0.381407 −0.190704 0.981648i \(-0.561077\pi\)
−0.190704 + 0.981648i \(0.561077\pi\)
\(422\) 0 0
\(423\) 1051.32 0.120844
\(424\) 0 0
\(425\) −1449.02 −0.165383
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −165.759 −0.0186549
\(430\) 0 0
\(431\) −9838.69 −1.09957 −0.549783 0.835308i \(-0.685290\pi\)
−0.549783 + 0.835308i \(0.685290\pi\)
\(432\) 0 0
\(433\) −8862.06 −0.983564 −0.491782 0.870718i \(-0.663654\pi\)
−0.491782 + 0.870718i \(0.663654\pi\)
\(434\) 0 0
\(435\) 2551.54 0.281235
\(436\) 0 0
\(437\) 166.459 0.0182215
\(438\) 0 0
\(439\) 14282.7 1.55280 0.776399 0.630242i \(-0.217044\pi\)
0.776399 + 0.630242i \(0.217044\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9833.36 1.05462 0.527310 0.849673i \(-0.323201\pi\)
0.527310 + 0.849673i \(0.323201\pi\)
\(444\) 0 0
\(445\) −20732.6 −2.20858
\(446\) 0 0
\(447\) 3979.06 0.421036
\(448\) 0 0
\(449\) 7326.49 0.770063 0.385032 0.922903i \(-0.374190\pi\)
0.385032 + 0.922903i \(0.374190\pi\)
\(450\) 0 0
\(451\) 637.882 0.0666002
\(452\) 0 0
\(453\) −1220.85 −0.126624
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1533.54 −0.156971 −0.0784856 0.996915i \(-0.525008\pi\)
−0.0784856 + 0.996915i \(0.525008\pi\)
\(458\) 0 0
\(459\) 738.617 0.0751104
\(460\) 0 0
\(461\) 1528.92 0.154466 0.0772330 0.997013i \(-0.475391\pi\)
0.0772330 + 0.997013i \(0.475391\pi\)
\(462\) 0 0
\(463\) −8497.20 −0.852912 −0.426456 0.904508i \(-0.640238\pi\)
−0.426456 + 0.904508i \(0.640238\pi\)
\(464\) 0 0
\(465\) 2079.31 0.207367
\(466\) 0 0
\(467\) 8349.59 0.827351 0.413676 0.910424i \(-0.364245\pi\)
0.413676 + 0.910424i \(0.364245\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 9407.78 0.920356
\(472\) 0 0
\(473\) 242.785 0.0236010
\(474\) 0 0
\(475\) 3466.77 0.334877
\(476\) 0 0
\(477\) 776.738 0.0745585
\(478\) 0 0
\(479\) 9408.01 0.897417 0.448709 0.893678i \(-0.351884\pi\)
0.448709 + 0.893678i \(0.351884\pi\)
\(480\) 0 0
\(481\) 13005.7 1.23287
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1350.13 −0.126405
\(486\) 0 0
\(487\) −9034.53 −0.840644 −0.420322 0.907375i \(-0.638083\pi\)
−0.420322 + 0.907375i \(0.638083\pi\)
\(488\) 0 0
\(489\) 392.939 0.0363381
\(490\) 0 0
\(491\) 3232.81 0.297138 0.148569 0.988902i \(-0.452533\pi\)
0.148569 + 0.988902i \(0.452533\pi\)
\(492\) 0 0
\(493\) 1744.08 0.159329
\(494\) 0 0
\(495\) −171.129 −0.0155387
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9718.05 −0.871823 −0.435912 0.899989i \(-0.643574\pi\)
−0.435912 + 0.899989i \(0.643574\pi\)
\(500\) 0 0
\(501\) −5363.40 −0.478281
\(502\) 0 0
\(503\) 3056.84 0.270970 0.135485 0.990779i \(-0.456741\pi\)
0.135485 + 0.990779i \(0.456741\pi\)
\(504\) 0 0
\(505\) −19200.2 −1.69188
\(506\) 0 0
\(507\) 2082.64 0.182432
\(508\) 0 0
\(509\) 8692.26 0.756931 0.378465 0.925615i \(-0.376452\pi\)
0.378465 + 0.925615i \(0.376452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1767.14 −0.152088
\(514\) 0 0
\(515\) −21415.7 −1.83240
\(516\) 0 0
\(517\) −166.495 −0.0141633
\(518\) 0 0
\(519\) −1827.49 −0.154563
\(520\) 0 0
\(521\) −15864.8 −1.33407 −0.667036 0.745025i \(-0.732437\pi\)
−0.667036 + 0.745025i \(0.732437\pi\)
\(522\) 0 0
\(523\) 23437.5 1.95956 0.979779 0.200085i \(-0.0641218\pi\)
0.979779 + 0.200085i \(0.0641218\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1421.28 0.117480
\(528\) 0 0
\(529\) −12160.5 −0.999468
\(530\) 0 0
\(531\) −3428.03 −0.280158
\(532\) 0 0
\(533\) 17349.3 1.40991
\(534\) 0 0
\(535\) −4109.80 −0.332116
\(536\) 0 0
\(537\) −4466.28 −0.358909
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7285.87 0.579009 0.289505 0.957177i \(-0.406509\pi\)
0.289505 + 0.957177i \(0.406509\pi\)
\(542\) 0 0
\(543\) 1358.29 0.107348
\(544\) 0 0
\(545\) 9457.87 0.743359
\(546\) 0 0
\(547\) −13089.8 −1.02318 −0.511589 0.859230i \(-0.670943\pi\)
−0.511589 + 0.859230i \(0.670943\pi\)
\(548\) 0 0
\(549\) 1796.62 0.139668
\(550\) 0 0
\(551\) −4172.69 −0.322618
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 13427.0 1.02693
\(556\) 0 0
\(557\) −11167.2 −0.849495 −0.424747 0.905312i \(-0.639637\pi\)
−0.424747 + 0.905312i \(0.639637\pi\)
\(558\) 0 0
\(559\) 6603.32 0.499626
\(560\) 0 0
\(561\) −116.973 −0.00880320
\(562\) 0 0
\(563\) −839.230 −0.0628230 −0.0314115 0.999507i \(-0.510000\pi\)
−0.0314115 + 0.999507i \(0.510000\pi\)
\(564\) 0 0
\(565\) 14774.4 1.10011
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5207.13 0.383645 0.191823 0.981430i \(-0.438560\pi\)
0.191823 + 0.981430i \(0.438560\pi\)
\(570\) 0 0
\(571\) 19877.8 1.45685 0.728425 0.685125i \(-0.240253\pi\)
0.728425 + 0.685125i \(0.240253\pi\)
\(572\) 0 0
\(573\) 8169.09 0.595582
\(574\) 0 0
\(575\) 134.716 0.00977052
\(576\) 0 0
\(577\) −2014.01 −0.145311 −0.0726553 0.997357i \(-0.523147\pi\)
−0.0726553 + 0.997357i \(0.523147\pi\)
\(578\) 0 0
\(579\) −7733.08 −0.555053
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −123.010 −0.00873851
\(584\) 0 0
\(585\) −4654.39 −0.328949
\(586\) 0 0
\(587\) −12502.2 −0.879085 −0.439542 0.898222i \(-0.644859\pi\)
−0.439542 + 0.898222i \(0.644859\pi\)
\(588\) 0 0
\(589\) −3400.41 −0.237881
\(590\) 0 0
\(591\) 4299.06 0.299221
\(592\) 0 0
\(593\) −3322.15 −0.230058 −0.115029 0.993362i \(-0.536696\pi\)
−0.115029 + 0.993362i \(0.536696\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5313.24 0.364248
\(598\) 0 0
\(599\) −13057.6 −0.890683 −0.445342 0.895361i \(-0.646918\pi\)
−0.445342 + 0.895361i \(0.646918\pi\)
\(600\) 0 0
\(601\) −15364.1 −1.04278 −0.521392 0.853317i \(-0.674587\pi\)
−0.521392 + 0.853317i \(0.674587\pi\)
\(602\) 0 0
\(603\) 8567.45 0.578596
\(604\) 0 0
\(605\) −17729.1 −1.19139
\(606\) 0 0
\(607\) 17502.3 1.17034 0.585170 0.810911i \(-0.301028\pi\)
0.585170 + 0.810911i \(0.301028\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4528.36 −0.299833
\(612\) 0 0
\(613\) −8668.35 −0.571144 −0.285572 0.958357i \(-0.592184\pi\)
−0.285572 + 0.958357i \(0.592184\pi\)
\(614\) 0 0
\(615\) 17911.2 1.17439
\(616\) 0 0
\(617\) 13110.8 0.855462 0.427731 0.903906i \(-0.359313\pi\)
0.427731 + 0.903906i \(0.359313\pi\)
\(618\) 0 0
\(619\) 24444.2 1.58723 0.793615 0.608421i \(-0.208197\pi\)
0.793615 + 0.608421i \(0.208197\pi\)
\(620\) 0 0
\(621\) −68.6696 −0.00443739
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19440.4 −1.24419
\(626\) 0 0
\(627\) 279.857 0.0178252
\(628\) 0 0
\(629\) 9177.84 0.581788
\(630\) 0 0
\(631\) 18458.3 1.16452 0.582262 0.813002i \(-0.302168\pi\)
0.582262 + 0.813002i \(0.302168\pi\)
\(632\) 0 0
\(633\) 16098.7 1.01085
\(634\) 0 0
\(635\) −31087.0 −1.94275
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7474.74 0.462748
\(640\) 0 0
\(641\) 8862.99 0.546126 0.273063 0.961996i \(-0.411963\pi\)
0.273063 + 0.961996i \(0.411963\pi\)
\(642\) 0 0
\(643\) 27011.4 1.65665 0.828324 0.560249i \(-0.189294\pi\)
0.828324 + 0.560249i \(0.189294\pi\)
\(644\) 0 0
\(645\) 6817.21 0.416167
\(646\) 0 0
\(647\) 28930.3 1.75791 0.878954 0.476906i \(-0.158242\pi\)
0.878954 + 0.476906i \(0.158242\pi\)
\(648\) 0 0
\(649\) 542.888 0.0328354
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23824.8 1.42778 0.713888 0.700260i \(-0.246933\pi\)
0.713888 + 0.700260i \(0.246933\pi\)
\(654\) 0 0
\(655\) −33280.2 −1.98529
\(656\) 0 0
\(657\) 2994.78 0.177835
\(658\) 0 0
\(659\) 19306.5 1.14124 0.570618 0.821216i \(-0.306704\pi\)
0.570618 + 0.821216i \(0.306704\pi\)
\(660\) 0 0
\(661\) 6414.99 0.377480 0.188740 0.982027i \(-0.439560\pi\)
0.188740 + 0.982027i \(0.439560\pi\)
\(662\) 0 0
\(663\) −3181.45 −0.186361
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −162.148 −0.00941287
\(668\) 0 0
\(669\) 8481.99 0.490184
\(670\) 0 0
\(671\) −284.525 −0.0163696
\(672\) 0 0
\(673\) −24090.1 −1.37980 −0.689899 0.723906i \(-0.742345\pi\)
−0.689899 + 0.723906i \(0.742345\pi\)
\(674\) 0 0
\(675\) −1430.15 −0.0815505
\(676\) 0 0
\(677\) 18477.5 1.04897 0.524483 0.851421i \(-0.324259\pi\)
0.524483 + 0.851421i \(0.324259\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −10696.9 −0.601919
\(682\) 0 0
\(683\) −12404.8 −0.694961 −0.347480 0.937687i \(-0.612963\pi\)
−0.347480 + 0.937687i \(0.612963\pi\)
\(684\) 0 0
\(685\) 3472.11 0.193668
\(686\) 0 0
\(687\) −14229.9 −0.790251
\(688\) 0 0
\(689\) −3345.65 −0.184992
\(690\) 0 0
\(691\) −3690.71 −0.203186 −0.101593 0.994826i \(-0.532394\pi\)
−0.101593 + 0.994826i \(0.532394\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3258.35 −0.177837
\(696\) 0 0
\(697\) 12243.0 0.665332
\(698\) 0 0
\(699\) −10413.2 −0.563468
\(700\) 0 0
\(701\) 30121.5 1.62293 0.811465 0.584401i \(-0.198671\pi\)
0.811465 + 0.584401i \(0.198671\pi\)
\(702\) 0 0
\(703\) −21957.9 −1.17804
\(704\) 0 0
\(705\) −4675.04 −0.249748
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 36821.1 1.95042 0.975209 0.221287i \(-0.0710259\pi\)
0.975209 + 0.221287i \(0.0710259\pi\)
\(710\) 0 0
\(711\) 6803.96 0.358887
\(712\) 0 0
\(713\) −132.138 −0.00694052
\(714\) 0 0
\(715\) 737.104 0.0385540
\(716\) 0 0
\(717\) 18268.2 0.951518
\(718\) 0 0
\(719\) −29581.9 −1.53438 −0.767188 0.641422i \(-0.778345\pi\)
−0.767188 + 0.641422i \(0.778345\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 14808.7 0.761747
\(724\) 0 0
\(725\) −3376.98 −0.172990
\(726\) 0 0
\(727\) −22956.0 −1.17110 −0.585551 0.810636i \(-0.699122\pi\)
−0.585551 + 0.810636i \(0.699122\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 4659.82 0.235773
\(732\) 0 0
\(733\) 10773.6 0.542880 0.271440 0.962455i \(-0.412500\pi\)
0.271440 + 0.962455i \(0.412500\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1356.80 −0.0678135
\(738\) 0 0
\(739\) 2440.70 0.121492 0.0607459 0.998153i \(-0.480652\pi\)
0.0607459 + 0.998153i \(0.480652\pi\)
\(740\) 0 0
\(741\) 7611.61 0.377354
\(742\) 0 0
\(743\) 13502.7 0.666712 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(744\) 0 0
\(745\) −17694.2 −0.870154
\(746\) 0 0
\(747\) 137.035 0.00671199
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22874.0 −1.11143 −0.555716 0.831373i \(-0.687556\pi\)
−0.555716 + 0.831373i \(0.687556\pi\)
\(752\) 0 0
\(753\) 3789.71 0.183406
\(754\) 0 0
\(755\) 5428.90 0.261693
\(756\) 0 0
\(757\) −3770.12 −0.181014 −0.0905069 0.995896i \(-0.528849\pi\)
−0.0905069 + 0.995896i \(0.528849\pi\)
\(758\) 0 0
\(759\) 10.8750 0.000520077 0
\(760\) 0 0
\(761\) 23825.4 1.13491 0.567456 0.823404i \(-0.307928\pi\)
0.567456 + 0.823404i \(0.307928\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3284.50 −0.155231
\(766\) 0 0
\(767\) 14765.6 0.695116
\(768\) 0 0
\(769\) −15661.3 −0.734408 −0.367204 0.930140i \(-0.619685\pi\)
−0.367204 + 0.930140i \(0.619685\pi\)
\(770\) 0 0
\(771\) 18458.8 0.862227
\(772\) 0 0
\(773\) 5265.53 0.245004 0.122502 0.992468i \(-0.460908\pi\)
0.122502 + 0.992468i \(0.460908\pi\)
\(774\) 0 0
\(775\) −2751.97 −0.127553
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −29291.3 −1.34720
\(780\) 0 0
\(781\) −1183.75 −0.0542357
\(782\) 0 0
\(783\) 1721.37 0.0785653
\(784\) 0 0
\(785\) −41834.8 −1.90210
\(786\) 0 0
\(787\) −5288.51 −0.239536 −0.119768 0.992802i \(-0.538215\pi\)
−0.119768 + 0.992802i \(0.538215\pi\)
\(788\) 0 0
\(789\) −12553.6 −0.566440
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7738.58 −0.346539
\(794\) 0 0
\(795\) −3454.02 −0.154090
\(796\) 0 0
\(797\) 19536.6 0.868285 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(798\) 0 0
\(799\) −3195.57 −0.141491
\(800\) 0 0
\(801\) −13987.0 −0.616987
\(802\) 0 0
\(803\) −474.275 −0.0208428
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2156.93 0.0940863
\(808\) 0 0
\(809\) 37594.9 1.63383 0.816913 0.576761i \(-0.195684\pi\)
0.816913 + 0.576761i \(0.195684\pi\)
\(810\) 0 0
\(811\) −32059.6 −1.38812 −0.694060 0.719917i \(-0.744180\pi\)
−0.694060 + 0.719917i \(0.744180\pi\)
\(812\) 0 0
\(813\) −3883.27 −0.167518
\(814\) 0 0
\(815\) −1747.33 −0.0750998
\(816\) 0 0
\(817\) −11148.6 −0.477405
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37886.9 1.61055 0.805276 0.592900i \(-0.202017\pi\)
0.805276 + 0.592900i \(0.202017\pi\)
\(822\) 0 0
\(823\) 33341.7 1.41217 0.706087 0.708125i \(-0.250459\pi\)
0.706087 + 0.708125i \(0.250459\pi\)
\(824\) 0 0
\(825\) 226.490 0.00955800
\(826\) 0 0
\(827\) 15589.5 0.655501 0.327750 0.944764i \(-0.393710\pi\)
0.327750 + 0.944764i \(0.393710\pi\)
\(828\) 0 0
\(829\) 34075.0 1.42759 0.713796 0.700353i \(-0.246974\pi\)
0.713796 + 0.700353i \(0.246974\pi\)
\(830\) 0 0
\(831\) 6962.75 0.290656
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 23850.1 0.988464
\(836\) 0 0
\(837\) 1402.78 0.0579297
\(838\) 0 0
\(839\) −29579.3 −1.21715 −0.608577 0.793495i \(-0.708259\pi\)
−0.608577 + 0.793495i \(0.708259\pi\)
\(840\) 0 0
\(841\) −20324.4 −0.833342
\(842\) 0 0
\(843\) −19695.7 −0.804692
\(844\) 0 0
\(845\) −9261.14 −0.377033
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11185.5 −0.452161
\(850\) 0 0
\(851\) −853.269 −0.0343710
\(852\) 0 0
\(853\) −22610.0 −0.907563 −0.453782 0.891113i \(-0.649925\pi\)
−0.453782 + 0.891113i \(0.649925\pi\)
\(854\) 0 0
\(855\) 7858.16 0.314320
\(856\) 0 0
\(857\) −23598.3 −0.940610 −0.470305 0.882504i \(-0.655856\pi\)
−0.470305 + 0.882504i \(0.655856\pi\)
\(858\) 0 0
\(859\) 21212.4 0.842558 0.421279 0.906931i \(-0.361581\pi\)
0.421279 + 0.906931i \(0.361581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7004.76 −0.276298 −0.138149 0.990411i \(-0.544115\pi\)
−0.138149 + 0.990411i \(0.544115\pi\)
\(864\) 0 0
\(865\) 8126.55 0.319435
\(866\) 0 0
\(867\) 12493.9 0.489407
\(868\) 0 0
\(869\) −1077.53 −0.0420628
\(870\) 0 0
\(871\) −36902.7 −1.43559
\(872\) 0 0
\(873\) −910.849 −0.0353122
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −47506.2 −1.82916 −0.914578 0.404411i \(-0.867477\pi\)
−0.914578 + 0.404411i \(0.867477\pi\)
\(878\) 0 0
\(879\) 809.203 0.0310509
\(880\) 0 0
\(881\) −3901.92 −0.149216 −0.0746079 0.997213i \(-0.523771\pi\)
−0.0746079 + 0.997213i \(0.523771\pi\)
\(882\) 0 0
\(883\) −18715.1 −0.713266 −0.356633 0.934245i \(-0.616075\pi\)
−0.356633 + 0.934245i \(0.616075\pi\)
\(884\) 0 0
\(885\) 15243.9 0.579002
\(886\) 0 0
\(887\) −2727.69 −0.103255 −0.0516274 0.998666i \(-0.516441\pi\)
−0.0516274 + 0.998666i \(0.516441\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −115.450 −0.00434087
\(892\) 0 0
\(893\) 7645.37 0.286498
\(894\) 0 0
\(895\) 19860.8 0.741758
\(896\) 0 0
\(897\) 295.781 0.0110099
\(898\) 0 0
\(899\) 3312.34 0.122884
\(900\) 0 0
\(901\) −2360.95 −0.0872972
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6040.10 −0.221856
\(906\) 0 0
\(907\) 17941.6 0.656825 0.328412 0.944534i \(-0.393486\pi\)
0.328412 + 0.944534i \(0.393486\pi\)
\(908\) 0 0
\(909\) −12953.2 −0.472641
\(910\) 0 0
\(911\) −35439.6 −1.28887 −0.644437 0.764657i \(-0.722908\pi\)
−0.644437 + 0.764657i \(0.722908\pi\)
\(912\) 0 0
\(913\) −21.7019 −0.000786669 0
\(914\) 0 0
\(915\) −7989.24 −0.288652
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17329.3 −0.622023 −0.311012 0.950406i \(-0.600668\pi\)
−0.311012 + 0.950406i \(0.600668\pi\)
\(920\) 0 0
\(921\) 9455.78 0.338305
\(922\) 0 0
\(923\) −32196.0 −1.14815
\(924\) 0 0
\(925\) −17770.7 −0.631672
\(926\) 0 0
\(927\) −14447.8 −0.511898
\(928\) 0 0
\(929\) −18347.2 −0.647959 −0.323979 0.946064i \(-0.605021\pi\)
−0.323979 + 0.946064i \(0.605021\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24411.7 −0.856594
\(934\) 0 0
\(935\) 520.158 0.0181936
\(936\) 0 0
\(937\) −11020.9 −0.384245 −0.192123 0.981371i \(-0.561537\pi\)
−0.192123 + 0.981371i \(0.561537\pi\)
\(938\) 0 0
\(939\) 11925.8 0.414465
\(940\) 0 0
\(941\) 35859.7 1.24229 0.621144 0.783697i \(-0.286668\pi\)
0.621144 + 0.783697i \(0.286668\pi\)
\(942\) 0 0
\(943\) −1138.24 −0.0393066
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41681.1 −1.43026 −0.715129 0.698992i \(-0.753632\pi\)
−0.715129 + 0.698992i \(0.753632\pi\)
\(948\) 0 0
\(949\) −12899.4 −0.441237
\(950\) 0 0
\(951\) −18239.8 −0.621941
\(952\) 0 0
\(953\) −51773.4 −1.75982 −0.879908 0.475143i \(-0.842396\pi\)
−0.879908 + 0.475143i \(0.842396\pi\)
\(954\) 0 0
\(955\) −36326.5 −1.23089
\(956\) 0 0
\(957\) −272.608 −0.00920813
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27091.7 −0.909392
\(962\) 0 0
\(963\) −2772.63 −0.0927796
\(964\) 0 0
\(965\) 34387.7 1.14713
\(966\) 0 0
\(967\) 55592.6 1.84874 0.924372 0.381491i \(-0.124589\pi\)
0.924372 + 0.381491i \(0.124589\pi\)
\(968\) 0 0
\(969\) 5371.34 0.178073
\(970\) 0 0
\(971\) 22136.6 0.731613 0.365807 0.930691i \(-0.380793\pi\)
0.365807 + 0.930691i \(0.380793\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 6160.11 0.202340
\(976\) 0 0
\(977\) 40501.1 1.32625 0.663125 0.748509i \(-0.269230\pi\)
0.663125 + 0.748509i \(0.269230\pi\)
\(978\) 0 0
\(979\) 2215.09 0.0723130
\(980\) 0 0
\(981\) 6380.64 0.207664
\(982\) 0 0
\(983\) −11785.9 −0.382414 −0.191207 0.981550i \(-0.561240\pi\)
−0.191207 + 0.981550i \(0.561240\pi\)
\(984\) 0 0
\(985\) −19117.2 −0.618400
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −433.226 −0.0139290
\(990\) 0 0
\(991\) −8009.47 −0.256740 −0.128370 0.991726i \(-0.540974\pi\)
−0.128370 + 0.991726i \(0.540974\pi\)
\(992\) 0 0
\(993\) −22739.6 −0.726705
\(994\) 0 0
\(995\) −23627.1 −0.752792
\(996\) 0 0
\(997\) 22894.7 0.727265 0.363633 0.931542i \(-0.381536\pi\)
0.363633 + 0.931542i \(0.381536\pi\)
\(998\) 0 0
\(999\) 9058.35 0.286880
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 1176.4.a.z.1.4 4
4.3 odd 2 2352.4.a.co.1.4 4
7.6 odd 2 1176.4.a.be.1.1 yes 4
28.27 even 2 2352.4.a.cn.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.z.1.4 4 1.1 even 1 trivial
1176.4.a.be.1.1 yes 4 7.6 odd 2
2352.4.a.cn.1.1 4 28.27 even 2
2352.4.a.co.1.4 4 4.3 odd 2