Properties

Label 2352.4.a.v.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -14.0000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -14.0000 q^{5} +9.00000 q^{9} -4.00000 q^{11} -54.0000 q^{13} -42.0000 q^{15} +14.0000 q^{17} +92.0000 q^{19} +152.000 q^{23} +71.0000 q^{25} +27.0000 q^{27} -106.000 q^{29} -144.000 q^{31} -12.0000 q^{33} +158.000 q^{37} -162.000 q^{39} +390.000 q^{41} +508.000 q^{43} -126.000 q^{45} -528.000 q^{47} +42.0000 q^{51} +606.000 q^{53} +56.0000 q^{55} +276.000 q^{57} -364.000 q^{59} -678.000 q^{61} +756.000 q^{65} -844.000 q^{67} +456.000 q^{69} +8.00000 q^{71} +422.000 q^{73} +213.000 q^{75} -384.000 q^{79} +81.0000 q^{81} -548.000 q^{83} -196.000 q^{85} -318.000 q^{87} -1194.00 q^{89} -432.000 q^{93} -1288.00 q^{95} +1502.00 q^{97} -36.0000 q^{99} +O(q^{100})\)

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
\(5\) −14.0000 −1.25220 −0.626099 0.779744i \(-0.715349\pi\)
−0.626099 + 0.779744i \(0.715349\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −0.109640 −0.0548202 0.998496i \(-0.517459\pi\)
−0.0548202 + 0.998496i \(0.517459\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) −42.0000 −0.722957
\(16\) 0 0
\(17\) 14.0000 0.199735 0.0998676 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) 0 0
\(19\) 92.0000 1.11086 0.555428 0.831565i \(-0.312555\pi\)
0.555428 + 0.831565i \(0.312555\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 152.000 1.37801 0.689004 0.724757i \(-0.258048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(24\) 0 0
\(25\) 71.0000 0.568000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −106.000 −0.678748 −0.339374 0.940651i \(-0.610215\pi\)
−0.339374 + 0.940651i \(0.610215\pi\)
\(30\) 0 0
\(31\) −144.000 −0.834296 −0.417148 0.908839i \(-0.636970\pi\)
−0.417148 + 0.908839i \(0.636970\pi\)
\(32\) 0 0
\(33\) −12.0000 −0.0633010
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 158.000 0.702028 0.351014 0.936370i \(-0.385837\pi\)
0.351014 + 0.936370i \(0.385837\pi\)
\(38\) 0 0
\(39\) −162.000 −0.665148
\(40\) 0 0
\(41\) 390.000 1.48556 0.742778 0.669538i \(-0.233508\pi\)
0.742778 + 0.669538i \(0.233508\pi\)
\(42\) 0 0
\(43\) 508.000 1.80161 0.900806 0.434223i \(-0.142977\pi\)
0.900806 + 0.434223i \(0.142977\pi\)
\(44\) 0 0
\(45\) −126.000 −0.417399
\(46\) 0 0
\(47\) −528.000 −1.63865 −0.819327 0.573327i \(-0.805653\pi\)
−0.819327 + 0.573327i \(0.805653\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 42.0000 0.115317
\(52\) 0 0
\(53\) 606.000 1.57058 0.785288 0.619131i \(-0.212515\pi\)
0.785288 + 0.619131i \(0.212515\pi\)
\(54\) 0 0
\(55\) 56.0000 0.137292
\(56\) 0 0
\(57\) 276.000 0.641353
\(58\) 0 0
\(59\) −364.000 −0.803199 −0.401600 0.915815i \(-0.631546\pi\)
−0.401600 + 0.915815i \(0.631546\pi\)
\(60\) 0 0
\(61\) −678.000 −1.42310 −0.711549 0.702636i \(-0.752006\pi\)
−0.711549 + 0.702636i \(0.752006\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 756.000 1.44262
\(66\) 0 0
\(67\) −844.000 −1.53897 −0.769485 0.638665i \(-0.779487\pi\)
−0.769485 + 0.638665i \(0.779487\pi\)
\(68\) 0 0
\(69\) 456.000 0.795593
\(70\) 0 0
\(71\) 8.00000 0.0133722 0.00668609 0.999978i \(-0.497872\pi\)
0.00668609 + 0.999978i \(0.497872\pi\)
\(72\) 0 0
\(73\) 422.000 0.676594 0.338297 0.941039i \(-0.390149\pi\)
0.338297 + 0.941039i \(0.390149\pi\)
\(74\) 0 0
\(75\) 213.000 0.327935
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −384.000 −0.546878 −0.273439 0.961889i \(-0.588161\pi\)
−0.273439 + 0.961889i \(0.588161\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −548.000 −0.724709 −0.362354 0.932040i \(-0.618027\pi\)
−0.362354 + 0.932040i \(0.618027\pi\)
\(84\) 0 0
\(85\) −196.000 −0.250108
\(86\) 0 0
\(87\) −318.000 −0.391876
\(88\) 0 0
\(89\) −1194.00 −1.42206 −0.711032 0.703159i \(-0.751772\pi\)
−0.711032 + 0.703159i \(0.751772\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −432.000 −0.481681
\(94\) 0 0
\(95\) −1288.00 −1.39101
\(96\) 0 0
\(97\) 1502.00 1.57222 0.786108 0.618089i \(-0.212093\pi\)
0.786108 + 0.618089i \(0.212093\pi\)
\(98\) 0 0
\(99\) −36.0000 −0.0365468
\(100\) 0 0
\(101\) −398.000 −0.392104 −0.196052 0.980594i \(-0.562812\pi\)
−0.196052 + 0.980594i \(0.562812\pi\)
\(102\) 0 0
\(103\) 1160.00 1.10969 0.554846 0.831953i \(-0.312777\pi\)
0.554846 + 0.831953i \(0.312777\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −324.000 −0.292731 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(108\) 0 0
\(109\) −938.000 −0.824258 −0.412129 0.911126i \(-0.635215\pi\)
−0.412129 + 0.911126i \(0.635215\pi\)
\(110\) 0 0
\(111\) 474.000 0.405316
\(112\) 0 0
\(113\) −622.000 −0.517813 −0.258906 0.965902i \(-0.583362\pi\)
−0.258906 + 0.965902i \(0.583362\pi\)
\(114\) 0 0
\(115\) −2128.00 −1.72554
\(116\) 0 0
\(117\) −486.000 −0.384023
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1315.00 −0.987979
\(122\) 0 0
\(123\) 1170.00 0.857686
\(124\) 0 0
\(125\) 756.000 0.540950
\(126\) 0 0
\(127\) −1200.00 −0.838447 −0.419224 0.907883i \(-0.637698\pi\)
−0.419224 + 0.907883i \(0.637698\pi\)
\(128\) 0 0
\(129\) 1524.00 1.04016
\(130\) 0 0
\(131\) −1396.00 −0.931062 −0.465531 0.885032i \(-0.654137\pi\)
−0.465531 + 0.885032i \(0.654137\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −378.000 −0.240986
\(136\) 0 0
\(137\) 2810.00 1.75237 0.876184 0.481976i \(-0.160081\pi\)
0.876184 + 0.481976i \(0.160081\pi\)
\(138\) 0 0
\(139\) 4.00000 0.00244083 0.00122042 0.999999i \(-0.499612\pi\)
0.00122042 + 0.999999i \(0.499612\pi\)
\(140\) 0 0
\(141\) −1584.00 −0.946077
\(142\) 0 0
\(143\) 216.000 0.126313
\(144\) 0 0
\(145\) 1484.00 0.849928
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1374.00 0.755453 0.377726 0.925917i \(-0.376706\pi\)
0.377726 + 0.925917i \(0.376706\pi\)
\(150\) 0 0
\(151\) −2104.00 −1.13391 −0.566957 0.823747i \(-0.691880\pi\)
−0.566957 + 0.823747i \(0.691880\pi\)
\(152\) 0 0
\(153\) 126.000 0.0665784
\(154\) 0 0
\(155\) 2016.00 1.04470
\(156\) 0 0
\(157\) −3206.00 −1.62972 −0.814862 0.579655i \(-0.803187\pi\)
−0.814862 + 0.579655i \(0.803187\pi\)
\(158\) 0 0
\(159\) 1818.00 0.906772
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −332.000 −0.159535 −0.0797676 0.996813i \(-0.525418\pi\)
−0.0797676 + 0.996813i \(0.525418\pi\)
\(164\) 0 0
\(165\) 168.000 0.0792653
\(166\) 0 0
\(167\) 1496.00 0.693197 0.346599 0.938014i \(-0.387337\pi\)
0.346599 + 0.938014i \(0.387337\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 828.000 0.370285
\(172\) 0 0
\(173\) 3322.00 1.45992 0.729962 0.683487i \(-0.239538\pi\)
0.729962 + 0.683487i \(0.239538\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1092.00 −0.463727
\(178\) 0 0
\(179\) 900.000 0.375805 0.187903 0.982188i \(-0.439831\pi\)
0.187903 + 0.982188i \(0.439831\pi\)
\(180\) 0 0
\(181\) −1902.00 −0.781075 −0.390537 0.920587i \(-0.627711\pi\)
−0.390537 + 0.920587i \(0.627711\pi\)
\(182\) 0 0
\(183\) −2034.00 −0.821626
\(184\) 0 0
\(185\) −2212.00 −0.879078
\(186\) 0 0
\(187\) −56.0000 −0.0218991
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4128.00 −1.56383 −0.781915 0.623385i \(-0.785757\pi\)
−0.781915 + 0.623385i \(0.785757\pi\)
\(192\) 0 0
\(193\) −1342.00 −0.500514 −0.250257 0.968179i \(-0.580515\pi\)
−0.250257 + 0.968179i \(0.580515\pi\)
\(194\) 0 0
\(195\) 2268.00 0.832897
\(196\) 0 0
\(197\) −3506.00 −1.26798 −0.633990 0.773341i \(-0.718584\pi\)
−0.633990 + 0.773341i \(0.718584\pi\)
\(198\) 0 0
\(199\) 680.000 0.242231 0.121115 0.992638i \(-0.461353\pi\)
0.121115 + 0.992638i \(0.461353\pi\)
\(200\) 0 0
\(201\) −2532.00 −0.888525
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5460.00 −1.86021
\(206\) 0 0
\(207\) 1368.00 0.459336
\(208\) 0 0
\(209\) −368.000 −0.121795
\(210\) 0 0
\(211\) −5372.00 −1.75272 −0.876360 0.481657i \(-0.840035\pi\)
−0.876360 + 0.481657i \(0.840035\pi\)
\(212\) 0 0
\(213\) 24.0000 0.00772044
\(214\) 0 0
\(215\) −7112.00 −2.25597
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1266.00 0.390632
\(220\) 0 0
\(221\) −756.000 −0.230109
\(222\) 0 0
\(223\) −1072.00 −0.321912 −0.160956 0.986962i \(-0.551458\pi\)
−0.160956 + 0.986962i \(0.551458\pi\)
\(224\) 0 0
\(225\) 639.000 0.189333
\(226\) 0 0
\(227\) −2868.00 −0.838572 −0.419286 0.907854i \(-0.637720\pi\)
−0.419286 + 0.907854i \(0.637720\pi\)
\(228\) 0 0
\(229\) −4798.00 −1.38454 −0.692272 0.721636i \(-0.743390\pi\)
−0.692272 + 0.721636i \(0.743390\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5126.00 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 7392.00 2.05192
\(236\) 0 0
\(237\) −1152.00 −0.315740
\(238\) 0 0
\(239\) 528.000 0.142902 0.0714508 0.997444i \(-0.477237\pi\)
0.0714508 + 0.997444i \(0.477237\pi\)
\(240\) 0 0
\(241\) 814.000 0.217570 0.108785 0.994065i \(-0.465304\pi\)
0.108785 + 0.994065i \(0.465304\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4968.00 −1.27978
\(248\) 0 0
\(249\) −1644.00 −0.418411
\(250\) 0 0
\(251\) −1932.00 −0.485844 −0.242922 0.970046i \(-0.578106\pi\)
−0.242922 + 0.970046i \(0.578106\pi\)
\(252\) 0 0
\(253\) −608.000 −0.151086
\(254\) 0 0
\(255\) −588.000 −0.144400
\(256\) 0 0
\(257\) 3294.00 0.799510 0.399755 0.916622i \(-0.369095\pi\)
0.399755 + 0.916622i \(0.369095\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −954.000 −0.226249
\(262\) 0 0
\(263\) 7080.00 1.65997 0.829984 0.557787i \(-0.188350\pi\)
0.829984 + 0.557787i \(0.188350\pi\)
\(264\) 0 0
\(265\) −8484.00 −1.96667
\(266\) 0 0
\(267\) −3582.00 −0.821029
\(268\) 0 0
\(269\) −7814.00 −1.77111 −0.885554 0.464537i \(-0.846221\pi\)
−0.885554 + 0.464537i \(0.846221\pi\)
\(270\) 0 0
\(271\) 3168.00 0.710119 0.355060 0.934844i \(-0.384461\pi\)
0.355060 + 0.934844i \(0.384461\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −284.000 −0.0622758
\(276\) 0 0
\(277\) −7858.00 −1.70448 −0.852241 0.523150i \(-0.824757\pi\)
−0.852241 + 0.523150i \(0.824757\pi\)
\(278\) 0 0
\(279\) −1296.00 −0.278099
\(280\) 0 0
\(281\) 6730.00 1.42875 0.714374 0.699764i \(-0.246712\pi\)
0.714374 + 0.699764i \(0.246712\pi\)
\(282\) 0 0
\(283\) −3020.00 −0.634348 −0.317174 0.948367i \(-0.602734\pi\)
−0.317174 + 0.948367i \(0.602734\pi\)
\(284\) 0 0
\(285\) −3864.00 −0.803100
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 4506.00 0.907720
\(292\) 0 0
\(293\) 6834.00 1.36262 0.681308 0.731997i \(-0.261411\pi\)
0.681308 + 0.731997i \(0.261411\pi\)
\(294\) 0 0
\(295\) 5096.00 1.00576
\(296\) 0 0
\(297\) −108.000 −0.0211003
\(298\) 0 0
\(299\) −8208.00 −1.58756
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1194.00 −0.226381
\(304\) 0 0
\(305\) 9492.00 1.78200
\(306\) 0 0
\(307\) 2332.00 0.433532 0.216766 0.976224i \(-0.430449\pi\)
0.216766 + 0.976224i \(0.430449\pi\)
\(308\) 0 0
\(309\) 3480.00 0.640681
\(310\) 0 0
\(311\) 8840.00 1.61180 0.805901 0.592050i \(-0.201681\pi\)
0.805901 + 0.592050i \(0.201681\pi\)
\(312\) 0 0
\(313\) 1046.00 0.188893 0.0944464 0.995530i \(-0.469892\pi\)
0.0944464 + 0.995530i \(0.469892\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7542.00 1.33628 0.668140 0.744035i \(-0.267091\pi\)
0.668140 + 0.744035i \(0.267091\pi\)
\(318\) 0 0
\(319\) 424.000 0.0744183
\(320\) 0 0
\(321\) −972.000 −0.169009
\(322\) 0 0
\(323\) 1288.00 0.221877
\(324\) 0 0
\(325\) −3834.00 −0.654376
\(326\) 0 0
\(327\) −2814.00 −0.475885
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2756.00 −0.457654 −0.228827 0.973467i \(-0.573489\pi\)
−0.228827 + 0.973467i \(0.573489\pi\)
\(332\) 0 0
\(333\) 1422.00 0.234009
\(334\) 0 0
\(335\) 11816.0 1.92710
\(336\) 0 0
\(337\) 3954.00 0.639134 0.319567 0.947564i \(-0.396463\pi\)
0.319567 + 0.947564i \(0.396463\pi\)
\(338\) 0 0
\(339\) −1866.00 −0.298959
\(340\) 0 0
\(341\) 576.000 0.0914726
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −6384.00 −0.996241
\(346\) 0 0
\(347\) −6900.00 −1.06747 −0.533734 0.845652i \(-0.679212\pi\)
−0.533734 + 0.845652i \(0.679212\pi\)
\(348\) 0 0
\(349\) 2426.00 0.372094 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(350\) 0 0
\(351\) −1458.00 −0.221716
\(352\) 0 0
\(353\) 1470.00 0.221644 0.110822 0.993840i \(-0.464652\pi\)
0.110822 + 0.993840i \(0.464652\pi\)
\(354\) 0 0
\(355\) −112.000 −0.0167446
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6872.00 −1.01028 −0.505140 0.863038i \(-0.668559\pi\)
−0.505140 + 0.863038i \(0.668559\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 0 0
\(363\) −3945.00 −0.570410
\(364\) 0 0
\(365\) −5908.00 −0.847230
\(366\) 0 0
\(367\) −7072.00 −1.00587 −0.502937 0.864323i \(-0.667747\pi\)
−0.502937 + 0.864323i \(0.667747\pi\)
\(368\) 0 0
\(369\) 3510.00 0.495185
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −818.000 −0.113551 −0.0567754 0.998387i \(-0.518082\pi\)
−0.0567754 + 0.998387i \(0.518082\pi\)
\(374\) 0 0
\(375\) 2268.00 0.312317
\(376\) 0 0
\(377\) 5724.00 0.781966
\(378\) 0 0
\(379\) 5132.00 0.695549 0.347775 0.937578i \(-0.386937\pi\)
0.347775 + 0.937578i \(0.386937\pi\)
\(380\) 0 0
\(381\) −3600.00 −0.484078
\(382\) 0 0
\(383\) −8576.00 −1.14416 −0.572080 0.820198i \(-0.693863\pi\)
−0.572080 + 0.820198i \(0.693863\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4572.00 0.600537
\(388\) 0 0
\(389\) −3730.00 −0.486166 −0.243083 0.970006i \(-0.578159\pi\)
−0.243083 + 0.970006i \(0.578159\pi\)
\(390\) 0 0
\(391\) 2128.00 0.275237
\(392\) 0 0
\(393\) −4188.00 −0.537549
\(394\) 0 0
\(395\) 5376.00 0.684800
\(396\) 0 0
\(397\) −6678.00 −0.844230 −0.422115 0.906542i \(-0.638712\pi\)
−0.422115 + 0.906542i \(0.638712\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3054.00 −0.380323 −0.190161 0.981753i \(-0.560901\pi\)
−0.190161 + 0.981753i \(0.560901\pi\)
\(402\) 0 0
\(403\) 7776.00 0.961167
\(404\) 0 0
\(405\) −1134.00 −0.139133
\(406\) 0 0
\(407\) −632.000 −0.0769707
\(408\) 0 0
\(409\) −266.000 −0.0321586 −0.0160793 0.999871i \(-0.505118\pi\)
−0.0160793 + 0.999871i \(0.505118\pi\)
\(410\) 0 0
\(411\) 8430.00 1.01173
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7672.00 0.907479
\(416\) 0 0
\(417\) 12.0000 0.00140921
\(418\) 0 0
\(419\) 8844.00 1.03116 0.515582 0.856840i \(-0.327576\pi\)
0.515582 + 0.856840i \(0.327576\pi\)
\(420\) 0 0
\(421\) −4482.00 −0.518858 −0.259429 0.965762i \(-0.583534\pi\)
−0.259429 + 0.965762i \(0.583534\pi\)
\(422\) 0 0
\(423\) −4752.00 −0.546218
\(424\) 0 0
\(425\) 994.000 0.113450
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 648.000 0.0729271
\(430\) 0 0
\(431\) −9936.00 −1.11044 −0.555221 0.831703i \(-0.687366\pi\)
−0.555221 + 0.831703i \(0.687366\pi\)
\(432\) 0 0
\(433\) 11758.0 1.30497 0.652487 0.757800i \(-0.273726\pi\)
0.652487 + 0.757800i \(0.273726\pi\)
\(434\) 0 0
\(435\) 4452.00 0.490706
\(436\) 0 0
\(437\) 13984.0 1.53077
\(438\) 0 0
\(439\) −4104.00 −0.446180 −0.223090 0.974798i \(-0.571614\pi\)
−0.223090 + 0.974798i \(0.571614\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9748.00 −1.04547 −0.522733 0.852496i \(-0.675088\pi\)
−0.522733 + 0.852496i \(0.675088\pi\)
\(444\) 0 0
\(445\) 16716.0 1.78071
\(446\) 0 0
\(447\) 4122.00 0.436161
\(448\) 0 0
\(449\) −478.000 −0.0502410 −0.0251205 0.999684i \(-0.507997\pi\)
−0.0251205 + 0.999684i \(0.507997\pi\)
\(450\) 0 0
\(451\) −1560.00 −0.162877
\(452\) 0 0
\(453\) −6312.00 −0.654666
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11174.0 −1.14376 −0.571879 0.820338i \(-0.693785\pi\)
−0.571879 + 0.820338i \(0.693785\pi\)
\(458\) 0 0
\(459\) 378.000 0.0384391
\(460\) 0 0
\(461\) 11674.0 1.17942 0.589710 0.807615i \(-0.299242\pi\)
0.589710 + 0.807615i \(0.299242\pi\)
\(462\) 0 0
\(463\) −10528.0 −1.05676 −0.528378 0.849009i \(-0.677199\pi\)
−0.528378 + 0.849009i \(0.677199\pi\)
\(464\) 0 0
\(465\) 6048.00 0.603160
\(466\) 0 0
\(467\) 16604.0 1.64527 0.822635 0.568569i \(-0.192503\pi\)
0.822635 + 0.568569i \(0.192503\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9618.00 −0.940922
\(472\) 0 0
\(473\) −2032.00 −0.197530
\(474\) 0 0
\(475\) 6532.00 0.630966
\(476\) 0 0
\(477\) 5454.00 0.523525
\(478\) 0 0
\(479\) −8576.00 −0.818053 −0.409027 0.912522i \(-0.634132\pi\)
−0.409027 + 0.912522i \(0.634132\pi\)
\(480\) 0 0
\(481\) −8532.00 −0.808785
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21028.0 −1.96873
\(486\) 0 0
\(487\) −9704.00 −0.902937 −0.451468 0.892287i \(-0.649100\pi\)
−0.451468 + 0.892287i \(0.649100\pi\)
\(488\) 0 0
\(489\) −996.000 −0.0921077
\(490\) 0 0
\(491\) 4092.00 0.376109 0.188054 0.982159i \(-0.439782\pi\)
0.188054 + 0.982159i \(0.439782\pi\)
\(492\) 0 0
\(493\) −1484.00 −0.135570
\(494\) 0 0
\(495\) 504.000 0.0457639
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17884.0 −1.60440 −0.802202 0.597052i \(-0.796338\pi\)
−0.802202 + 0.597052i \(0.796338\pi\)
\(500\) 0 0
\(501\) 4488.00 0.400218
\(502\) 0 0
\(503\) −7704.00 −0.682911 −0.341456 0.939898i \(-0.610920\pi\)
−0.341456 + 0.939898i \(0.610920\pi\)
\(504\) 0 0
\(505\) 5572.00 0.490992
\(506\) 0 0
\(507\) 2157.00 0.188946
\(508\) 0 0
\(509\) −14358.0 −1.25031 −0.625154 0.780501i \(-0.714964\pi\)
−0.625154 + 0.780501i \(0.714964\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2484.00 0.213784
\(514\) 0 0
\(515\) −16240.0 −1.38955
\(516\) 0 0
\(517\) 2112.00 0.179663
\(518\) 0 0
\(519\) 9966.00 0.842888
\(520\) 0 0
\(521\) −5082.00 −0.427344 −0.213672 0.976905i \(-0.568542\pi\)
−0.213672 + 0.976905i \(0.568542\pi\)
\(522\) 0 0
\(523\) −1756.00 −0.146816 −0.0734078 0.997302i \(-0.523387\pi\)
−0.0734078 + 0.997302i \(0.523387\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2016.00 −0.166638
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) −3276.00 −0.267733
\(532\) 0 0
\(533\) −21060.0 −1.71146
\(534\) 0 0
\(535\) 4536.00 0.366558
\(536\) 0 0
\(537\) 2700.00 0.216971
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 16230.0 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) −5706.00 −0.450954
\(544\) 0 0
\(545\) 13132.0 1.03213
\(546\) 0 0
\(547\) −17676.0 −1.38167 −0.690833 0.723014i \(-0.742756\pi\)
−0.690833 + 0.723014i \(0.742756\pi\)
\(548\) 0 0
\(549\) −6102.00 −0.474366
\(550\) 0 0
\(551\) −9752.00 −0.753991
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6636.00 −0.507536
\(556\) 0 0
\(557\) −12250.0 −0.931866 −0.465933 0.884820i \(-0.654281\pi\)
−0.465933 + 0.884820i \(0.654281\pi\)
\(558\) 0 0
\(559\) −27432.0 −2.07558
\(560\) 0 0
\(561\) −168.000 −0.0126434
\(562\) 0 0
\(563\) −10052.0 −0.752471 −0.376236 0.926524i \(-0.622782\pi\)
−0.376236 + 0.926524i \(0.622782\pi\)
\(564\) 0 0
\(565\) 8708.00 0.648404
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25674.0 1.89158 0.945791 0.324776i \(-0.105289\pi\)
0.945791 + 0.324776i \(0.105289\pi\)
\(570\) 0 0
\(571\) −3732.00 −0.273519 −0.136759 0.990604i \(-0.543669\pi\)
−0.136759 + 0.990604i \(0.543669\pi\)
\(572\) 0 0
\(573\) −12384.0 −0.902878
\(574\) 0 0
\(575\) 10792.0 0.782709
\(576\) 0 0
\(577\) 1214.00 0.0875901 0.0437950 0.999041i \(-0.486055\pi\)
0.0437950 + 0.999041i \(0.486055\pi\)
\(578\) 0 0
\(579\) −4026.00 −0.288972
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2424.00 −0.172199
\(584\) 0 0
\(585\) 6804.00 0.480873
\(586\) 0 0
\(587\) 7108.00 0.499793 0.249897 0.968273i \(-0.419603\pi\)
0.249897 + 0.968273i \(0.419603\pi\)
\(588\) 0 0
\(589\) −13248.0 −0.926782
\(590\) 0 0
\(591\) −10518.0 −0.732069
\(592\) 0 0
\(593\) −6162.00 −0.426717 −0.213358 0.976974i \(-0.568440\pi\)
−0.213358 + 0.976974i \(0.568440\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2040.00 0.139852
\(598\) 0 0
\(599\) −2472.00 −0.168620 −0.0843098 0.996440i \(-0.526869\pi\)
−0.0843098 + 0.996440i \(0.526869\pi\)
\(600\) 0 0
\(601\) 13750.0 0.933235 0.466617 0.884459i \(-0.345472\pi\)
0.466617 + 0.884459i \(0.345472\pi\)
\(602\) 0 0
\(603\) −7596.00 −0.512990
\(604\) 0 0
\(605\) 18410.0 1.23715
\(606\) 0 0
\(607\) −11376.0 −0.760688 −0.380344 0.924845i \(-0.624194\pi\)
−0.380344 + 0.924845i \(0.624194\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28512.0 1.88784
\(612\) 0 0
\(613\) 20382.0 1.34294 0.671469 0.741032i \(-0.265664\pi\)
0.671469 + 0.741032i \(0.265664\pi\)
\(614\) 0 0
\(615\) −16380.0 −1.07399
\(616\) 0 0
\(617\) 21178.0 1.38184 0.690919 0.722932i \(-0.257206\pi\)
0.690919 + 0.722932i \(0.257206\pi\)
\(618\) 0 0
\(619\) −4700.00 −0.305184 −0.152592 0.988289i \(-0.548762\pi\)
−0.152592 + 0.988289i \(0.548762\pi\)
\(620\) 0 0
\(621\) 4104.00 0.265198
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19459.0 −1.24538
\(626\) 0 0
\(627\) −1104.00 −0.0703182
\(628\) 0 0
\(629\) 2212.00 0.140220
\(630\) 0 0
\(631\) 21736.0 1.37131 0.685655 0.727927i \(-0.259516\pi\)
0.685655 + 0.727927i \(0.259516\pi\)
\(632\) 0 0
\(633\) −16116.0 −1.01193
\(634\) 0 0
\(635\) 16800.0 1.04990
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 72.0000 0.00445740
\(640\) 0 0
\(641\) −13022.0 −0.802399 −0.401200 0.915991i \(-0.631407\pi\)
−0.401200 + 0.915991i \(0.631407\pi\)
\(642\) 0 0
\(643\) 3308.00 0.202885 0.101442 0.994841i \(-0.467654\pi\)
0.101442 + 0.994841i \(0.467654\pi\)
\(644\) 0 0
\(645\) −21336.0 −1.30249
\(646\) 0 0
\(647\) −13800.0 −0.838538 −0.419269 0.907862i \(-0.637714\pi\)
−0.419269 + 0.907862i \(0.637714\pi\)
\(648\) 0 0
\(649\) 1456.00 0.0880632
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2682.00 −0.160727 −0.0803635 0.996766i \(-0.525608\pi\)
−0.0803635 + 0.996766i \(0.525608\pi\)
\(654\) 0 0
\(655\) 19544.0 1.16587
\(656\) 0 0
\(657\) 3798.00 0.225531
\(658\) 0 0
\(659\) −23836.0 −1.40898 −0.704491 0.709713i \(-0.748825\pi\)
−0.704491 + 0.709713i \(0.748825\pi\)
\(660\) 0 0
\(661\) 11282.0 0.663871 0.331936 0.943302i \(-0.392298\pi\)
0.331936 + 0.943302i \(0.392298\pi\)
\(662\) 0 0
\(663\) −2268.00 −0.132853
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16112.0 −0.935321
\(668\) 0 0
\(669\) −3216.00 −0.185856
\(670\) 0 0
\(671\) 2712.00 0.156029
\(672\) 0 0
\(673\) −13726.0 −0.786179 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(674\) 0 0
\(675\) 1917.00 0.109312
\(676\) 0 0
\(677\) −4974.00 −0.282373 −0.141186 0.989983i \(-0.545092\pi\)
−0.141186 + 0.989983i \(0.545092\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8604.00 −0.484150
\(682\) 0 0
\(683\) 8988.00 0.503538 0.251769 0.967787i \(-0.418988\pi\)
0.251769 + 0.967787i \(0.418988\pi\)
\(684\) 0 0
\(685\) −39340.0 −2.19431
\(686\) 0 0
\(687\) −14394.0 −0.799367
\(688\) 0 0
\(689\) −32724.0 −1.80941
\(690\) 0 0
\(691\) 10172.0 0.560002 0.280001 0.960000i \(-0.409665\pi\)
0.280001 + 0.960000i \(0.409665\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −56.0000 −0.00305640
\(696\) 0 0
\(697\) 5460.00 0.296718
\(698\) 0 0
\(699\) −15378.0 −0.832116
\(700\) 0 0
\(701\) 27446.0 1.47877 0.739387 0.673280i \(-0.235115\pi\)
0.739387 + 0.673280i \(0.235115\pi\)
\(702\) 0 0
\(703\) 14536.0 0.779852
\(704\) 0 0
\(705\) 22176.0 1.18468
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3998.00 0.211774 0.105887 0.994378i \(-0.466232\pi\)
0.105887 + 0.994378i \(0.466232\pi\)
\(710\) 0 0
\(711\) −3456.00 −0.182293
\(712\) 0 0
\(713\) −21888.0 −1.14967
\(714\) 0 0
\(715\) −3024.00 −0.158169
\(716\) 0 0
\(717\) 1584.00 0.0825043
\(718\) 0 0
\(719\) 25872.0 1.34195 0.670976 0.741480i \(-0.265876\pi\)
0.670976 + 0.741480i \(0.265876\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2442.00 0.125614
\(724\) 0 0
\(725\) −7526.00 −0.385529
\(726\) 0 0
\(727\) 12088.0 0.616670 0.308335 0.951278i \(-0.400228\pi\)
0.308335 + 0.951278i \(0.400228\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 7112.00 0.359845
\(732\) 0 0
\(733\) −7974.00 −0.401810 −0.200905 0.979611i \(-0.564388\pi\)
−0.200905 + 0.979611i \(0.564388\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3376.00 0.168733
\(738\) 0 0
\(739\) 31764.0 1.58113 0.790567 0.612376i \(-0.209786\pi\)
0.790567 + 0.612376i \(0.209786\pi\)
\(740\) 0 0
\(741\) −14904.0 −0.738883
\(742\) 0 0
\(743\) −888.000 −0.0438460 −0.0219230 0.999760i \(-0.506979\pi\)
−0.0219230 + 0.999760i \(0.506979\pi\)
\(744\) 0 0
\(745\) −19236.0 −0.945977
\(746\) 0 0
\(747\) −4932.00 −0.241570
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 34656.0 1.68391 0.841954 0.539549i \(-0.181405\pi\)
0.841954 + 0.539549i \(0.181405\pi\)
\(752\) 0 0
\(753\) −5796.00 −0.280502
\(754\) 0 0
\(755\) 29456.0 1.41989
\(756\) 0 0
\(757\) −22866.0 −1.09786 −0.548929 0.835869i \(-0.684964\pi\)
−0.548929 + 0.835869i \(0.684964\pi\)
\(758\) 0 0
\(759\) −1824.00 −0.0872293
\(760\) 0 0
\(761\) −22570.0 −1.07511 −0.537557 0.843227i \(-0.680653\pi\)
−0.537557 + 0.843227i \(0.680653\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1764.00 −0.0833694
\(766\) 0 0
\(767\) 19656.0 0.925342
\(768\) 0 0
\(769\) 1790.00 0.0839389 0.0419695 0.999119i \(-0.486637\pi\)
0.0419695 + 0.999119i \(0.486637\pi\)
\(770\) 0 0
\(771\) 9882.00 0.461597
\(772\) 0 0
\(773\) −2990.00 −0.139124 −0.0695620 0.997578i \(-0.522160\pi\)
−0.0695620 + 0.997578i \(0.522160\pi\)
\(774\) 0 0
\(775\) −10224.0 −0.473880
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 35880.0 1.65024
\(780\) 0 0
\(781\) −32.0000 −0.00146613
\(782\) 0 0
\(783\) −2862.00 −0.130625
\(784\) 0 0
\(785\) 44884.0 2.04074
\(786\) 0 0
\(787\) −30756.0 −1.39305 −0.696527 0.717531i \(-0.745272\pi\)
−0.696527 + 0.717531i \(0.745272\pi\)
\(788\) 0 0
\(789\) 21240.0 0.958383
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36612.0 1.63951
\(794\) 0 0
\(795\) −25452.0 −1.13546
\(796\) 0 0
\(797\) −15126.0 −0.672259 −0.336129 0.941816i \(-0.609118\pi\)
−0.336129 + 0.941816i \(0.609118\pi\)
\(798\) 0 0
\(799\) −7392.00 −0.327297
\(800\) 0 0
\(801\) −10746.0 −0.474022
\(802\) 0 0
\(803\) −1688.00 −0.0741821
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −23442.0 −1.02255
\(808\) 0 0
\(809\) −6502.00 −0.282569 −0.141284 0.989969i \(-0.545123\pi\)
−0.141284 + 0.989969i \(0.545123\pi\)
\(810\) 0 0
\(811\) −8252.00 −0.357296 −0.178648 0.983913i \(-0.557172\pi\)
−0.178648 + 0.983913i \(0.557172\pi\)
\(812\) 0 0
\(813\) 9504.00 0.409987
\(814\) 0 0
\(815\) 4648.00 0.199770
\(816\) 0 0
\(817\) 46736.0 2.00133
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21986.0 −0.934612 −0.467306 0.884096i \(-0.654775\pi\)
−0.467306 + 0.884096i \(0.654775\pi\)
\(822\) 0 0
\(823\) −3736.00 −0.158237 −0.0791183 0.996865i \(-0.525210\pi\)
−0.0791183 + 0.996865i \(0.525210\pi\)
\(824\) 0 0
\(825\) −852.000 −0.0359549
\(826\) 0 0
\(827\) 23820.0 1.00158 0.500788 0.865570i \(-0.333044\pi\)
0.500788 + 0.865570i \(0.333044\pi\)
\(828\) 0 0
\(829\) −7942.00 −0.332735 −0.166367 0.986064i \(-0.553204\pi\)
−0.166367 + 0.986064i \(0.553204\pi\)
\(830\) 0 0
\(831\) −23574.0 −0.984083
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −20944.0 −0.868020
\(836\) 0 0
\(837\) −3888.00 −0.160560
\(838\) 0 0
\(839\) 21016.0 0.864783 0.432391 0.901686i \(-0.357670\pi\)
0.432391 + 0.901686i \(0.357670\pi\)
\(840\) 0 0
\(841\) −13153.0 −0.539301
\(842\) 0 0
\(843\) 20190.0 0.824888
\(844\) 0 0
\(845\) −10066.0 −0.409800
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9060.00 −0.366241
\(850\) 0 0
\(851\) 24016.0 0.967401
\(852\) 0 0
\(853\) −24878.0 −0.998601 −0.499300 0.866429i \(-0.666410\pi\)
−0.499300 + 0.866429i \(0.666410\pi\)
\(854\) 0 0
\(855\) −11592.0 −0.463670
\(856\) 0 0
\(857\) 6390.00 0.254700 0.127350 0.991858i \(-0.459353\pi\)
0.127350 + 0.991858i \(0.459353\pi\)
\(858\) 0 0
\(859\) −46444.0 −1.84476 −0.922380 0.386284i \(-0.873759\pi\)
−0.922380 + 0.386284i \(0.873759\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25408.0 1.00220 0.501100 0.865389i \(-0.332929\pi\)
0.501100 + 0.865389i \(0.332929\pi\)
\(864\) 0 0
\(865\) −46508.0 −1.82811
\(866\) 0 0
\(867\) −14151.0 −0.554317
\(868\) 0 0
\(869\) 1536.00 0.0599600
\(870\) 0 0
\(871\) 45576.0 1.77300
\(872\) 0 0
\(873\) 13518.0 0.524072
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1078.00 0.0415068 0.0207534 0.999785i \(-0.493394\pi\)
0.0207534 + 0.999785i \(0.493394\pi\)
\(878\) 0 0
\(879\) 20502.0 0.786707
\(880\) 0 0
\(881\) 45006.0 1.72110 0.860551 0.509364i \(-0.170119\pi\)
0.860551 + 0.509364i \(0.170119\pi\)
\(882\) 0 0
\(883\) −4028.00 −0.153514 −0.0767571 0.997050i \(-0.524457\pi\)
−0.0767571 + 0.997050i \(0.524457\pi\)
\(884\) 0 0
\(885\) 15288.0 0.580679
\(886\) 0 0
\(887\) −29304.0 −1.10928 −0.554640 0.832090i \(-0.687144\pi\)
−0.554640 + 0.832090i \(0.687144\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −324.000 −0.0121823
\(892\) 0 0
\(893\) −48576.0 −1.82031
\(894\) 0 0
\(895\) −12600.0 −0.470583
\(896\) 0 0
\(897\) −24624.0 −0.916579
\(898\) 0 0
\(899\) 15264.0 0.566277
\(900\) 0 0
\(901\) 8484.00 0.313699
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26628.0 0.978060
\(906\) 0 0
\(907\) −50916.0 −1.86399 −0.931995 0.362472i \(-0.881933\pi\)
−0.931995 + 0.362472i \(0.881933\pi\)
\(908\) 0 0
\(909\) −3582.00 −0.130701
\(910\) 0 0
\(911\) 24432.0 0.888549 0.444275 0.895891i \(-0.353461\pi\)
0.444275 + 0.895891i \(0.353461\pi\)
\(912\) 0 0
\(913\) 2192.00 0.0794574
\(914\) 0 0
\(915\) 28476.0 1.02884
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 20360.0 0.730810 0.365405 0.930849i \(-0.380930\pi\)
0.365405 + 0.930849i \(0.380930\pi\)
\(920\) 0 0
\(921\) 6996.00 0.250300
\(922\) 0 0
\(923\) −432.000 −0.0154057
\(924\) 0 0
\(925\) 11218.0 0.398752
\(926\) 0 0
\(927\) 10440.0 0.369897
\(928\) 0 0
\(929\) −23202.0 −0.819411 −0.409706 0.912218i \(-0.634369\pi\)
−0.409706 + 0.912218i \(0.634369\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26520.0 0.930574
\(934\) 0 0
\(935\) 784.000 0.0274220
\(936\) 0 0
\(937\) 1990.00 0.0693815 0.0346908 0.999398i \(-0.488955\pi\)
0.0346908 + 0.999398i \(0.488955\pi\)
\(938\) 0 0
\(939\) 3138.00 0.109057
\(940\) 0 0
\(941\) 51130.0 1.77130 0.885648 0.464356i \(-0.153714\pi\)
0.885648 + 0.464356i \(0.153714\pi\)
\(942\) 0 0
\(943\) 59280.0 2.04711
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47044.0 1.61428 0.807141 0.590359i \(-0.201014\pi\)
0.807141 + 0.590359i \(0.201014\pi\)
\(948\) 0 0
\(949\) −22788.0 −0.779483
\(950\) 0 0
\(951\) 22626.0 0.771502
\(952\) 0 0
\(953\) 46858.0 1.59274 0.796369 0.604811i \(-0.206751\pi\)
0.796369 + 0.604811i \(0.206751\pi\)
\(954\) 0 0
\(955\) 57792.0 1.95823
\(956\) 0 0
\(957\) 1272.00 0.0429654
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9055.00 −0.303951
\(962\) 0 0
\(963\) −2916.00 −0.0975771
\(964\) 0 0
\(965\) 18788.0 0.626743
\(966\) 0 0
\(967\) −30632.0 −1.01867 −0.509337 0.860567i \(-0.670110\pi\)
−0.509337 + 0.860567i \(0.670110\pi\)
\(968\) 0 0
\(969\) 3864.00 0.128101
\(970\) 0 0
\(971\) −3804.00 −0.125722 −0.0628611 0.998022i \(-0.520022\pi\)
−0.0628611 + 0.998022i \(0.520022\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −11502.0 −0.377804
\(976\) 0 0
\(977\) −49326.0 −1.61523 −0.807614 0.589711i \(-0.799242\pi\)
−0.807614 + 0.589711i \(0.799242\pi\)
\(978\) 0 0
\(979\) 4776.00 0.155916
\(980\) 0 0
\(981\) −8442.00 −0.274753
\(982\) 0 0
\(983\) 11112.0 0.360547 0.180274 0.983617i \(-0.442302\pi\)
0.180274 + 0.983617i \(0.442302\pi\)
\(984\) 0 0
\(985\) 49084.0 1.58776
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 77216.0 2.48263
\(990\) 0 0
\(991\) −13616.0 −0.436455 −0.218227 0.975898i \(-0.570027\pi\)
−0.218227 + 0.975898i \(0.570027\pi\)
\(992\) 0 0
\(993\) −8268.00 −0.264227
\(994\) 0 0
\(995\) −9520.00 −0.303321
\(996\) 0 0
\(997\) 56674.0 1.80028 0.900142 0.435596i \(-0.143462\pi\)
0.900142 + 0.435596i \(0.143462\pi\)
\(998\) 0 0
\(999\) 4266.00 0.135105
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.v.1.1 1
4.3 odd 2 588.4.a.a.1.1 1
7.6 odd 2 336.4.a.e.1.1 1
12.11 even 2 1764.4.a.l.1.1 1
21.20 even 2 1008.4.a.d.1.1 1
28.3 even 6 588.4.i.a.373.1 2
28.11 odd 6 588.4.i.h.373.1 2
28.19 even 6 588.4.i.a.361.1 2
28.23 odd 6 588.4.i.h.361.1 2
28.27 even 2 84.4.a.b.1.1 1
56.13 odd 2 1344.4.a.p.1.1 1
56.27 even 2 1344.4.a.b.1.1 1
84.11 even 6 1764.4.k.c.1549.1 2
84.23 even 6 1764.4.k.c.361.1 2
84.47 odd 6 1764.4.k.n.361.1 2
84.59 odd 6 1764.4.k.n.1549.1 2
84.83 odd 2 252.4.a.a.1.1 1
140.27 odd 4 2100.4.k.g.1849.1 2
140.83 odd 4 2100.4.k.g.1849.2 2
140.139 even 2 2100.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.b.1.1 1 28.27 even 2
252.4.a.a.1.1 1 84.83 odd 2
336.4.a.e.1.1 1 7.6 odd 2
588.4.a.a.1.1 1 4.3 odd 2
588.4.i.a.361.1 2 28.19 even 6
588.4.i.a.373.1 2 28.3 even 6
588.4.i.h.361.1 2 28.23 odd 6
588.4.i.h.373.1 2 28.11 odd 6
1008.4.a.d.1.1 1 21.20 even 2
1344.4.a.b.1.1 1 56.27 even 2
1344.4.a.p.1.1 1 56.13 odd 2
1764.4.a.l.1.1 1 12.11 even 2
1764.4.k.c.361.1 2 84.23 even 6
1764.4.k.c.1549.1 2 84.11 even 6
1764.4.k.n.361.1 2 84.47 odd 6
1764.4.k.n.1549.1 2 84.59 odd 6
2100.4.a.g.1.1 1 140.139 even 2
2100.4.k.g.1849.1 2 140.27 odd 4
2100.4.k.g.1849.2 2 140.83 odd 4
2352.4.a.v.1.1 1 1.1 even 1 trivial