Properties

Label 2352.4.a.b.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} -20.0000 q^{11} +84.0000 q^{13} +36.0000 q^{15} +96.0000 q^{17} +12.0000 q^{19} +176.000 q^{23} +19.0000 q^{25} -27.0000 q^{27} +58.0000 q^{29} -264.000 q^{31} +60.0000 q^{33} +258.000 q^{37} -252.000 q^{39} -156.000 q^{43} -108.000 q^{45} -408.000 q^{47} -288.000 q^{51} -722.000 q^{53} +240.000 q^{55} -36.0000 q^{57} +492.000 q^{59} +492.000 q^{61} -1008.00 q^{65} -412.000 q^{67} -528.000 q^{69} -296.000 q^{71} -240.000 q^{73} -57.0000 q^{75} -776.000 q^{79} +81.0000 q^{81} +924.000 q^{83} -1152.00 q^{85} -174.000 q^{87} +744.000 q^{89} +792.000 q^{93} -144.000 q^{95} +168.000 q^{97} -180.000 q^{99} +O(q^{100})\)

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
\(5\) −12.0000 −1.07331 −0.536656 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) 84.0000 1.79211 0.896054 0.443945i \(-0.146421\pi\)
0.896054 + 0.443945i \(0.146421\pi\)
\(14\) 0 0
\(15\) 36.0000 0.619677
\(16\) 0 0
\(17\) 96.0000 1.36961 0.684806 0.728725i \(-0.259887\pi\)
0.684806 + 0.728725i \(0.259887\pi\)
\(18\) 0 0
\(19\) 12.0000 0.144894 0.0724471 0.997372i \(-0.476919\pi\)
0.0724471 + 0.997372i \(0.476919\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 176.000 1.59559 0.797794 0.602930i \(-0.206000\pi\)
0.797794 + 0.602930i \(0.206000\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −264.000 −1.52954 −0.764771 0.644302i \(-0.777148\pi\)
−0.764771 + 0.644302i \(0.777148\pi\)
\(32\) 0 0
\(33\) 60.0000 0.316505
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 258.000 1.14635 0.573175 0.819433i \(-0.305712\pi\)
0.573175 + 0.819433i \(0.305712\pi\)
\(38\) 0 0
\(39\) −252.000 −1.03467
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −156.000 −0.553251 −0.276625 0.960978i \(-0.589216\pi\)
−0.276625 + 0.960978i \(0.589216\pi\)
\(44\) 0 0
\(45\) −108.000 −0.357771
\(46\) 0 0
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −288.000 −0.790746
\(52\) 0 0
\(53\) −722.000 −1.87121 −0.935607 0.353044i \(-0.885147\pi\)
−0.935607 + 0.353044i \(0.885147\pi\)
\(54\) 0 0
\(55\) 240.000 0.588393
\(56\) 0 0
\(57\) −36.0000 −0.0836547
\(58\) 0 0
\(59\) 492.000 1.08564 0.542822 0.839848i \(-0.317356\pi\)
0.542822 + 0.839848i \(0.317356\pi\)
\(60\) 0 0
\(61\) 492.000 1.03269 0.516345 0.856380i \(-0.327292\pi\)
0.516345 + 0.856380i \(0.327292\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1008.00 −1.92349
\(66\) 0 0
\(67\) −412.000 −0.751251 −0.375625 0.926772i \(-0.622572\pi\)
−0.375625 + 0.926772i \(0.622572\pi\)
\(68\) 0 0
\(69\) −528.000 −0.921213
\(70\) 0 0
\(71\) −296.000 −0.494771 −0.247385 0.968917i \(-0.579571\pi\)
−0.247385 + 0.968917i \(0.579571\pi\)
\(72\) 0 0
\(73\) −240.000 −0.384793 −0.192396 0.981317i \(-0.561626\pi\)
−0.192396 + 0.981317i \(0.561626\pi\)
\(74\) 0 0
\(75\) −57.0000 −0.0877572
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −776.000 −1.10515 −0.552575 0.833463i \(-0.686355\pi\)
−0.552575 + 0.833463i \(0.686355\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 924.000 1.22195 0.610977 0.791648i \(-0.290777\pi\)
0.610977 + 0.791648i \(0.290777\pi\)
\(84\) 0 0
\(85\) −1152.00 −1.47002
\(86\) 0 0
\(87\) −174.000 −0.214423
\(88\) 0 0
\(89\) 744.000 0.886111 0.443055 0.896494i \(-0.353895\pi\)
0.443055 + 0.896494i \(0.353895\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 792.000 0.883081
\(94\) 0 0
\(95\) −144.000 −0.155517
\(96\) 0 0
\(97\) 168.000 0.175854 0.0879269 0.996127i \(-0.471976\pi\)
0.0879269 + 0.996127i \(0.471976\pi\)
\(98\) 0 0
\(99\) −180.000 −0.182734
\(100\) 0 0
\(101\) 1524.00 1.50142 0.750711 0.660630i \(-0.229711\pi\)
0.750711 + 0.660630i \(0.229711\pi\)
\(102\) 0 0
\(103\) −408.000 −0.390305 −0.195153 0.980773i \(-0.562520\pi\)
−0.195153 + 0.980773i \(0.562520\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 820.000 0.740863 0.370432 0.928860i \(-0.379210\pi\)
0.370432 + 0.928860i \(0.379210\pi\)
\(108\) 0 0
\(109\) −918.000 −0.806683 −0.403342 0.915050i \(-0.632151\pi\)
−0.403342 + 0.915050i \(0.632151\pi\)
\(110\) 0 0
\(111\) −774.000 −0.661845
\(112\) 0 0
\(113\) −110.000 −0.0915746 −0.0457873 0.998951i \(-0.514580\pi\)
−0.0457873 + 0.998951i \(0.514580\pi\)
\(114\) 0 0
\(115\) −2112.00 −1.71257
\(116\) 0 0
\(117\) 756.000 0.597369
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1272.00 0.910169
\(126\) 0 0
\(127\) −16.0000 −0.0111793 −0.00558965 0.999984i \(-0.501779\pi\)
−0.00558965 + 0.999984i \(0.501779\pi\)
\(128\) 0 0
\(129\) 468.000 0.319419
\(130\) 0 0
\(131\) 1692.00 1.12848 0.564239 0.825611i \(-0.309169\pi\)
0.564239 + 0.825611i \(0.309169\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 324.000 0.206559
\(136\) 0 0
\(137\) 1126.00 0.702195 0.351097 0.936339i \(-0.385809\pi\)
0.351097 + 0.936339i \(0.385809\pi\)
\(138\) 0 0
\(139\) 1092.00 0.666347 0.333173 0.942866i \(-0.391881\pi\)
0.333173 + 0.942866i \(0.391881\pi\)
\(140\) 0 0
\(141\) 1224.00 0.731060
\(142\) 0 0
\(143\) −1680.00 −0.982438
\(144\) 0 0
\(145\) −696.000 −0.398618
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1070.00 0.588307 0.294154 0.955758i \(-0.404962\pi\)
0.294154 + 0.955758i \(0.404962\pi\)
\(150\) 0 0
\(151\) 120.000 0.0646719 0.0323360 0.999477i \(-0.489705\pi\)
0.0323360 + 0.999477i \(0.489705\pi\)
\(152\) 0 0
\(153\) 864.000 0.456538
\(154\) 0 0
\(155\) 3168.00 1.64168
\(156\) 0 0
\(157\) −1836.00 −0.933304 −0.466652 0.884441i \(-0.654540\pi\)
−0.466652 + 0.884441i \(0.654540\pi\)
\(158\) 0 0
\(159\) 2166.00 1.08035
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −916.000 −0.440164 −0.220082 0.975481i \(-0.570632\pi\)
−0.220082 + 0.975481i \(0.570632\pi\)
\(164\) 0 0
\(165\) −720.000 −0.339709
\(166\) 0 0
\(167\) 504.000 0.233537 0.116769 0.993159i \(-0.462746\pi\)
0.116769 + 0.993159i \(0.462746\pi\)
\(168\) 0 0
\(169\) 4859.00 2.21165
\(170\) 0 0
\(171\) 108.000 0.0482980
\(172\) 0 0
\(173\) 1836.00 0.806870 0.403435 0.915008i \(-0.367816\pi\)
0.403435 + 0.915008i \(0.367816\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1476.00 −0.626796
\(178\) 0 0
\(179\) −2372.00 −0.990456 −0.495228 0.868763i \(-0.664915\pi\)
−0.495228 + 0.868763i \(0.664915\pi\)
\(180\) 0 0
\(181\) 1092.00 0.448440 0.224220 0.974539i \(-0.428017\pi\)
0.224220 + 0.974539i \(0.428017\pi\)
\(182\) 0 0
\(183\) −1476.00 −0.596224
\(184\) 0 0
\(185\) −3096.00 −1.23039
\(186\) 0 0
\(187\) −1920.00 −0.750825
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2512.00 −0.951633 −0.475817 0.879545i \(-0.657847\pi\)
−0.475817 + 0.879545i \(0.657847\pi\)
\(192\) 0 0
\(193\) −2430.00 −0.906297 −0.453148 0.891435i \(-0.649699\pi\)
−0.453148 + 0.891435i \(0.649699\pi\)
\(194\) 0 0
\(195\) 3024.00 1.11053
\(196\) 0 0
\(197\) −1762.00 −0.637245 −0.318623 0.947882i \(-0.603220\pi\)
−0.318623 + 0.947882i \(0.603220\pi\)
\(198\) 0 0
\(199\) 3096.00 1.10286 0.551431 0.834220i \(-0.314082\pi\)
0.551431 + 0.834220i \(0.314082\pi\)
\(200\) 0 0
\(201\) 1236.00 0.433735
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1584.00 0.531863
\(208\) 0 0
\(209\) −240.000 −0.0794313
\(210\) 0 0
\(211\) −156.000 −0.0508980 −0.0254490 0.999676i \(-0.508102\pi\)
−0.0254490 + 0.999676i \(0.508102\pi\)
\(212\) 0 0
\(213\) 888.000 0.285656
\(214\) 0 0
\(215\) 1872.00 0.593811
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 720.000 0.222160
\(220\) 0 0
\(221\) 8064.00 2.45449
\(222\) 0 0
\(223\) 5040.00 1.51347 0.756734 0.653723i \(-0.226794\pi\)
0.756734 + 0.653723i \(0.226794\pi\)
\(224\) 0 0
\(225\) 171.000 0.0506667
\(226\) 0 0
\(227\) 2172.00 0.635069 0.317535 0.948247i \(-0.397145\pi\)
0.317535 + 0.948247i \(0.397145\pi\)
\(228\) 0 0
\(229\) −2700.00 −0.779131 −0.389566 0.920999i \(-0.627375\pi\)
−0.389566 + 0.920999i \(0.627375\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3802.00 −1.06900 −0.534501 0.845168i \(-0.679500\pi\)
−0.534501 + 0.845168i \(0.679500\pi\)
\(234\) 0 0
\(235\) 4896.00 1.35906
\(236\) 0 0
\(237\) 2328.00 0.638058
\(238\) 0 0
\(239\) 4408.00 1.19301 0.596506 0.802609i \(-0.296555\pi\)
0.596506 + 0.802609i \(0.296555\pi\)
\(240\) 0 0
\(241\) −3096.00 −0.827514 −0.413757 0.910387i \(-0.635784\pi\)
−0.413757 + 0.910387i \(0.635784\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1008.00 0.259666
\(248\) 0 0
\(249\) −2772.00 −0.705495
\(250\) 0 0
\(251\) −924.000 −0.232360 −0.116180 0.993228i \(-0.537065\pi\)
−0.116180 + 0.993228i \(0.537065\pi\)
\(252\) 0 0
\(253\) −3520.00 −0.874706
\(254\) 0 0
\(255\) 3456.00 0.848718
\(256\) 0 0
\(257\) 2760.00 0.669899 0.334950 0.942236i \(-0.391281\pi\)
0.334950 + 0.942236i \(0.391281\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 522.000 0.123797
\(262\) 0 0
\(263\) 2360.00 0.553323 0.276661 0.960967i \(-0.410772\pi\)
0.276661 + 0.960967i \(0.410772\pi\)
\(264\) 0 0
\(265\) 8664.00 2.00840
\(266\) 0 0
\(267\) −2232.00 −0.511596
\(268\) 0 0
\(269\) −4020.00 −0.911166 −0.455583 0.890193i \(-0.650569\pi\)
−0.455583 + 0.890193i \(0.650569\pi\)
\(270\) 0 0
\(271\) 4800.00 1.07594 0.537969 0.842965i \(-0.319192\pi\)
0.537969 + 0.842965i \(0.319192\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −380.000 −0.0833268
\(276\) 0 0
\(277\) 6446.00 1.39820 0.699102 0.715022i \(-0.253583\pi\)
0.699102 + 0.715022i \(0.253583\pi\)
\(278\) 0 0
\(279\) −2376.00 −0.509847
\(280\) 0 0
\(281\) −2602.00 −0.552393 −0.276196 0.961101i \(-0.589074\pi\)
−0.276196 + 0.961101i \(0.589074\pi\)
\(282\) 0 0
\(283\) −6900.00 −1.44934 −0.724669 0.689098i \(-0.758007\pi\)
−0.724669 + 0.689098i \(0.758007\pi\)
\(284\) 0 0
\(285\) 432.000 0.0897876
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4303.00 0.875840
\(290\) 0 0
\(291\) −504.000 −0.101529
\(292\) 0 0
\(293\) 4452.00 0.887674 0.443837 0.896107i \(-0.353617\pi\)
0.443837 + 0.896107i \(0.353617\pi\)
\(294\) 0 0
\(295\) −5904.00 −1.16523
\(296\) 0 0
\(297\) 540.000 0.105502
\(298\) 0 0
\(299\) 14784.0 2.85947
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4572.00 −0.866847
\(304\) 0 0
\(305\) −5904.00 −1.10840
\(306\) 0 0
\(307\) −2436.00 −0.452866 −0.226433 0.974027i \(-0.572706\pi\)
−0.226433 + 0.974027i \(0.572706\pi\)
\(308\) 0 0
\(309\) 1224.00 0.225343
\(310\) 0 0
\(311\) −7488.00 −1.36529 −0.682646 0.730750i \(-0.739171\pi\)
−0.682646 + 0.730750i \(0.739171\pi\)
\(312\) 0 0
\(313\) 1752.00 0.316386 0.158193 0.987408i \(-0.449433\pi\)
0.158193 + 0.987408i \(0.449433\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1562.00 −0.276753 −0.138376 0.990380i \(-0.544188\pi\)
−0.138376 + 0.990380i \(0.544188\pi\)
\(318\) 0 0
\(319\) −1160.00 −0.203597
\(320\) 0 0
\(321\) −2460.00 −0.427738
\(322\) 0 0
\(323\) 1152.00 0.198449
\(324\) 0 0
\(325\) 1596.00 0.272400
\(326\) 0 0
\(327\) 2754.00 0.465739
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7092.00 1.17768 0.588839 0.808250i \(-0.299585\pi\)
0.588839 + 0.808250i \(0.299585\pi\)
\(332\) 0 0
\(333\) 2322.00 0.382117
\(334\) 0 0
\(335\) 4944.00 0.806327
\(336\) 0 0
\(337\) 366.000 0.0591611 0.0295805 0.999562i \(-0.490583\pi\)
0.0295805 + 0.999562i \(0.490583\pi\)
\(338\) 0 0
\(339\) 330.000 0.0528706
\(340\) 0 0
\(341\) 5280.00 0.838499
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6336.00 0.988750
\(346\) 0 0
\(347\) 6364.00 0.984546 0.492273 0.870441i \(-0.336166\pi\)
0.492273 + 0.870441i \(0.336166\pi\)
\(348\) 0 0
\(349\) 10500.0 1.61046 0.805232 0.592960i \(-0.202041\pi\)
0.805232 + 0.592960i \(0.202041\pi\)
\(350\) 0 0
\(351\) −2268.00 −0.344891
\(352\) 0 0
\(353\) −408.000 −0.0615174 −0.0307587 0.999527i \(-0.509792\pi\)
−0.0307587 + 0.999527i \(0.509792\pi\)
\(354\) 0 0
\(355\) 3552.00 0.531044
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11936.0 1.75476 0.877379 0.479798i \(-0.159290\pi\)
0.877379 + 0.479798i \(0.159290\pi\)
\(360\) 0 0
\(361\) −6715.00 −0.979006
\(362\) 0 0
\(363\) 2793.00 0.403842
\(364\) 0 0
\(365\) 2880.00 0.413003
\(366\) 0 0
\(367\) −2448.00 −0.348187 −0.174093 0.984729i \(-0.555699\pi\)
−0.174093 + 0.984729i \(0.555699\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11374.0 1.57888 0.789442 0.613826i \(-0.210370\pi\)
0.789442 + 0.613826i \(0.210370\pi\)
\(374\) 0 0
\(375\) −3816.00 −0.525486
\(376\) 0 0
\(377\) 4872.00 0.665572
\(378\) 0 0
\(379\) 5892.00 0.798553 0.399277 0.916830i \(-0.369261\pi\)
0.399277 + 0.916830i \(0.369261\pi\)
\(380\) 0 0
\(381\) 48.0000 0.00645437
\(382\) 0 0
\(383\) −10488.0 −1.39925 −0.699624 0.714511i \(-0.746649\pi\)
−0.699624 + 0.714511i \(0.746649\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1404.00 −0.184417
\(388\) 0 0
\(389\) 4514.00 0.588352 0.294176 0.955751i \(-0.404955\pi\)
0.294176 + 0.955751i \(0.404955\pi\)
\(390\) 0 0
\(391\) 16896.0 2.18534
\(392\) 0 0
\(393\) −5076.00 −0.651528
\(394\) 0 0
\(395\) 9312.00 1.18617
\(396\) 0 0
\(397\) 6036.00 0.763068 0.381534 0.924355i \(-0.375396\pi\)
0.381534 + 0.924355i \(0.375396\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6770.00 −0.843086 −0.421543 0.906808i \(-0.638511\pi\)
−0.421543 + 0.906808i \(0.638511\pi\)
\(402\) 0 0
\(403\) −22176.0 −2.74110
\(404\) 0 0
\(405\) −972.000 −0.119257
\(406\) 0 0
\(407\) −5160.00 −0.628432
\(408\) 0 0
\(409\) −12504.0 −1.51169 −0.755847 0.654748i \(-0.772775\pi\)
−0.755847 + 0.654748i \(0.772775\pi\)
\(410\) 0 0
\(411\) −3378.00 −0.405412
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −11088.0 −1.31154
\(416\) 0 0
\(417\) −3276.00 −0.384716
\(418\) 0 0
\(419\) 9492.00 1.10672 0.553359 0.832943i \(-0.313346\pi\)
0.553359 + 0.832943i \(0.313346\pi\)
\(420\) 0 0
\(421\) 5182.00 0.599894 0.299947 0.953956i \(-0.403031\pi\)
0.299947 + 0.953956i \(0.403031\pi\)
\(422\) 0 0
\(423\) −3672.00 −0.422077
\(424\) 0 0
\(425\) 1824.00 0.208181
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5040.00 0.567211
\(430\) 0 0
\(431\) 5720.00 0.639264 0.319632 0.947542i \(-0.396441\pi\)
0.319632 + 0.947542i \(0.396441\pi\)
\(432\) 0 0
\(433\) −13608.0 −1.51030 −0.755149 0.655554i \(-0.772435\pi\)
−0.755149 + 0.655554i \(0.772435\pi\)
\(434\) 0 0
\(435\) 2088.00 0.230142
\(436\) 0 0
\(437\) 2112.00 0.231191
\(438\) 0 0
\(439\) 12864.0 1.39855 0.699277 0.714851i \(-0.253505\pi\)
0.699277 + 0.714851i \(0.253505\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13252.0 1.42127 0.710634 0.703562i \(-0.248408\pi\)
0.710634 + 0.703562i \(0.248408\pi\)
\(444\) 0 0
\(445\) −8928.00 −0.951074
\(446\) 0 0
\(447\) −3210.00 −0.339659
\(448\) 0 0
\(449\) 226.000 0.0237541 0.0118771 0.999929i \(-0.496219\pi\)
0.0118771 + 0.999929i \(0.496219\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −360.000 −0.0373384
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11334.0 −1.16014 −0.580068 0.814568i \(-0.696974\pi\)
−0.580068 + 0.814568i \(0.696974\pi\)
\(458\) 0 0
\(459\) −2592.00 −0.263582
\(460\) 0 0
\(461\) 1596.00 0.161243 0.0806216 0.996745i \(-0.474309\pi\)
0.0806216 + 0.996745i \(0.474309\pi\)
\(462\) 0 0
\(463\) −12728.0 −1.27758 −0.638791 0.769380i \(-0.720565\pi\)
−0.638791 + 0.769380i \(0.720565\pi\)
\(464\) 0 0
\(465\) −9504.00 −0.947822
\(466\) 0 0
\(467\) −3012.00 −0.298456 −0.149228 0.988803i \(-0.547679\pi\)
−0.149228 + 0.988803i \(0.547679\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5508.00 0.538843
\(472\) 0 0
\(473\) 3120.00 0.303293
\(474\) 0 0
\(475\) 228.000 0.0220239
\(476\) 0 0
\(477\) −6498.00 −0.623738
\(478\) 0 0
\(479\) −4296.00 −0.409790 −0.204895 0.978784i \(-0.565685\pi\)
−0.204895 + 0.978784i \(0.565685\pi\)
\(480\) 0 0
\(481\) 21672.0 2.05438
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2016.00 −0.188746
\(486\) 0 0
\(487\) 8184.00 0.761504 0.380752 0.924677i \(-0.375665\pi\)
0.380752 + 0.924677i \(0.375665\pi\)
\(488\) 0 0
\(489\) 2748.00 0.254129
\(490\) 0 0
\(491\) 12164.0 1.11803 0.559016 0.829157i \(-0.311179\pi\)
0.559016 + 0.829157i \(0.311179\pi\)
\(492\) 0 0
\(493\) 5568.00 0.508661
\(494\) 0 0
\(495\) 2160.00 0.196131
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −972.000 −0.0871998 −0.0435999 0.999049i \(-0.513883\pi\)
−0.0435999 + 0.999049i \(0.513883\pi\)
\(500\) 0 0
\(501\) −1512.00 −0.134833
\(502\) 0 0
\(503\) 7728.00 0.685039 0.342519 0.939511i \(-0.388720\pi\)
0.342519 + 0.939511i \(0.388720\pi\)
\(504\) 0 0
\(505\) −18288.0 −1.61150
\(506\) 0 0
\(507\) −14577.0 −1.27690
\(508\) 0 0
\(509\) −11604.0 −1.01049 −0.505244 0.862977i \(-0.668597\pi\)
−0.505244 + 0.862977i \(0.668597\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −324.000 −0.0278849
\(514\) 0 0
\(515\) 4896.00 0.418919
\(516\) 0 0
\(517\) 8160.00 0.694152
\(518\) 0 0
\(519\) −5508.00 −0.465847
\(520\) 0 0
\(521\) 10848.0 0.912206 0.456103 0.889927i \(-0.349245\pi\)
0.456103 + 0.889927i \(0.349245\pi\)
\(522\) 0 0
\(523\) −18132.0 −1.51598 −0.757989 0.652267i \(-0.773818\pi\)
−0.757989 + 0.652267i \(0.773818\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −25344.0 −2.09488
\(528\) 0 0
\(529\) 18809.0 1.54590
\(530\) 0 0
\(531\) 4428.00 0.361881
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −9840.00 −0.795178
\(536\) 0 0
\(537\) 7116.00 0.571840
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6950.00 0.552318 0.276159 0.961112i \(-0.410938\pi\)
0.276159 + 0.961112i \(0.410938\pi\)
\(542\) 0 0
\(543\) −3276.00 −0.258907
\(544\) 0 0
\(545\) 11016.0 0.865823
\(546\) 0 0
\(547\) −17012.0 −1.32976 −0.664882 0.746949i \(-0.731518\pi\)
−0.664882 + 0.746949i \(0.731518\pi\)
\(548\) 0 0
\(549\) 4428.00 0.344230
\(550\) 0 0
\(551\) 696.000 0.0538123
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9288.00 0.710367
\(556\) 0 0
\(557\) 3926.00 0.298653 0.149327 0.988788i \(-0.452289\pi\)
0.149327 + 0.988788i \(0.452289\pi\)
\(558\) 0 0
\(559\) −13104.0 −0.991485
\(560\) 0 0
\(561\) 5760.00 0.433489
\(562\) 0 0
\(563\) −18828.0 −1.40942 −0.704712 0.709494i \(-0.748924\pi\)
−0.704712 + 0.709494i \(0.748924\pi\)
\(564\) 0 0
\(565\) 1320.00 0.0982882
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11990.0 0.883387 0.441693 0.897166i \(-0.354378\pi\)
0.441693 + 0.897166i \(0.354378\pi\)
\(570\) 0 0
\(571\) 15716.0 1.15183 0.575914 0.817510i \(-0.304646\pi\)
0.575914 + 0.817510i \(0.304646\pi\)
\(572\) 0 0
\(573\) 7536.00 0.549426
\(574\) 0 0
\(575\) 3344.00 0.242529
\(576\) 0 0
\(577\) 13872.0 1.00086 0.500432 0.865776i \(-0.333174\pi\)
0.500432 + 0.865776i \(0.333174\pi\)
\(578\) 0 0
\(579\) 7290.00 0.523251
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14440.0 1.02580
\(584\) 0 0
\(585\) −9072.00 −0.641164
\(586\) 0 0
\(587\) 8820.00 0.620171 0.310085 0.950709i \(-0.399642\pi\)
0.310085 + 0.950709i \(0.399642\pi\)
\(588\) 0 0
\(589\) −3168.00 −0.221622
\(590\) 0 0
\(591\) 5286.00 0.367914
\(592\) 0 0
\(593\) 16872.0 1.16838 0.584191 0.811617i \(-0.301412\pi\)
0.584191 + 0.811617i \(0.301412\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9288.00 −0.636738
\(598\) 0 0
\(599\) 6056.00 0.413091 0.206545 0.978437i \(-0.433778\pi\)
0.206545 + 0.978437i \(0.433778\pi\)
\(600\) 0 0
\(601\) −10752.0 −0.729756 −0.364878 0.931055i \(-0.618889\pi\)
−0.364878 + 0.931055i \(0.618889\pi\)
\(602\) 0 0
\(603\) −3708.00 −0.250417
\(604\) 0 0
\(605\) 11172.0 0.750754
\(606\) 0 0
\(607\) 20256.0 1.35447 0.677237 0.735765i \(-0.263177\pi\)
0.677237 + 0.735765i \(0.263177\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34272.0 −2.26923
\(612\) 0 0
\(613\) −28190.0 −1.85740 −0.928698 0.370838i \(-0.879071\pi\)
−0.928698 + 0.370838i \(0.879071\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29318.0 1.91296 0.956482 0.291793i \(-0.0942518\pi\)
0.956482 + 0.291793i \(0.0942518\pi\)
\(618\) 0 0
\(619\) 24348.0 1.58098 0.790492 0.612473i \(-0.209825\pi\)
0.790492 + 0.612473i \(0.209825\pi\)
\(620\) 0 0
\(621\) −4752.00 −0.307071
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 720.000 0.0458597
\(628\) 0 0
\(629\) 24768.0 1.57006
\(630\) 0 0
\(631\) 25184.0 1.58884 0.794421 0.607368i \(-0.207774\pi\)
0.794421 + 0.607368i \(0.207774\pi\)
\(632\) 0 0
\(633\) 468.000 0.0293860
\(634\) 0 0
\(635\) 192.000 0.0119989
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2664.00 −0.164924
\(640\) 0 0
\(641\) 32318.0 1.99140 0.995698 0.0926628i \(-0.0295379\pi\)
0.995698 + 0.0926628i \(0.0295379\pi\)
\(642\) 0 0
\(643\) −3948.00 −0.242137 −0.121068 0.992644i \(-0.538632\pi\)
−0.121068 + 0.992644i \(0.538632\pi\)
\(644\) 0 0
\(645\) −5616.00 −0.342837
\(646\) 0 0
\(647\) 13848.0 0.841454 0.420727 0.907187i \(-0.361775\pi\)
0.420727 + 0.907187i \(0.361775\pi\)
\(648\) 0 0
\(649\) −9840.00 −0.595152
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3158.00 −0.189253 −0.0946264 0.995513i \(-0.530166\pi\)
−0.0946264 + 0.995513i \(0.530166\pi\)
\(654\) 0 0
\(655\) −20304.0 −1.21121
\(656\) 0 0
\(657\) −2160.00 −0.128264
\(658\) 0 0
\(659\) 24596.0 1.45391 0.726953 0.686687i \(-0.240936\pi\)
0.726953 + 0.686687i \(0.240936\pi\)
\(660\) 0 0
\(661\) 15468.0 0.910190 0.455095 0.890443i \(-0.349605\pi\)
0.455095 + 0.890443i \(0.349605\pi\)
\(662\) 0 0
\(663\) −24192.0 −1.41710
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10208.0 0.592587
\(668\) 0 0
\(669\) −15120.0 −0.873801
\(670\) 0 0
\(671\) −9840.00 −0.566124
\(672\) 0 0
\(673\) 13470.0 0.771516 0.385758 0.922600i \(-0.373940\pi\)
0.385758 + 0.922600i \(0.373940\pi\)
\(674\) 0 0
\(675\) −513.000 −0.0292524
\(676\) 0 0
\(677\) 9564.00 0.542946 0.271473 0.962446i \(-0.412489\pi\)
0.271473 + 0.962446i \(0.412489\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6516.00 −0.366657
\(682\) 0 0
\(683\) −13852.0 −0.776035 −0.388018 0.921652i \(-0.626840\pi\)
−0.388018 + 0.921652i \(0.626840\pi\)
\(684\) 0 0
\(685\) −13512.0 −0.753674
\(686\) 0 0
\(687\) 8100.00 0.449832
\(688\) 0 0
\(689\) −60648.0 −3.35342
\(690\) 0 0
\(691\) −324.000 −0.0178373 −0.00891863 0.999960i \(-0.502839\pi\)
−0.00891863 + 0.999960i \(0.502839\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −13104.0 −0.715199
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 11406.0 0.617188
\(700\) 0 0
\(701\) 24922.0 1.34278 0.671392 0.741103i \(-0.265697\pi\)
0.671392 + 0.741103i \(0.265697\pi\)
\(702\) 0 0
\(703\) 3096.00 0.166099
\(704\) 0 0
\(705\) −14688.0 −0.784655
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17886.0 −0.947423 −0.473711 0.880680i \(-0.657086\pi\)
−0.473711 + 0.880680i \(0.657086\pi\)
\(710\) 0 0
\(711\) −6984.00 −0.368383
\(712\) 0 0
\(713\) −46464.0 −2.44052
\(714\) 0 0
\(715\) 20160.0 1.05446
\(716\) 0 0
\(717\) −13224.0 −0.688786
\(718\) 0 0
\(719\) −6792.00 −0.352293 −0.176147 0.984364i \(-0.556363\pi\)
−0.176147 + 0.984364i \(0.556363\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9288.00 0.477765
\(724\) 0 0
\(725\) 1102.00 0.0564514
\(726\) 0 0
\(727\) 1512.00 0.0771348 0.0385674 0.999256i \(-0.487721\pi\)
0.0385674 + 0.999256i \(0.487721\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −14976.0 −0.757739
\(732\) 0 0
\(733\) 11244.0 0.566585 0.283292 0.959034i \(-0.408573\pi\)
0.283292 + 0.959034i \(0.408573\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8240.00 0.411838
\(738\) 0 0
\(739\) 1996.00 0.0993559 0.0496780 0.998765i \(-0.484180\pi\)
0.0496780 + 0.998765i \(0.484180\pi\)
\(740\) 0 0
\(741\) −3024.00 −0.149918
\(742\) 0 0
\(743\) 656.000 0.0323907 0.0161954 0.999869i \(-0.494845\pi\)
0.0161954 + 0.999869i \(0.494845\pi\)
\(744\) 0 0
\(745\) −12840.0 −0.631438
\(746\) 0 0
\(747\) 8316.00 0.407318
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1056.00 −0.0513102 −0.0256551 0.999671i \(-0.508167\pi\)
−0.0256551 + 0.999671i \(0.508167\pi\)
\(752\) 0 0
\(753\) 2772.00 0.134153
\(754\) 0 0
\(755\) −1440.00 −0.0694132
\(756\) 0 0
\(757\) −18702.0 −0.897934 −0.448967 0.893548i \(-0.648208\pi\)
−0.448967 + 0.893548i \(0.648208\pi\)
\(758\) 0 0
\(759\) 10560.0 0.505011
\(760\) 0 0
\(761\) −17904.0 −0.852851 −0.426425 0.904523i \(-0.640227\pi\)
−0.426425 + 0.904523i \(0.640227\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −10368.0 −0.490008
\(766\) 0 0
\(767\) 41328.0 1.94559
\(768\) 0 0
\(769\) 7560.00 0.354513 0.177257 0.984165i \(-0.443278\pi\)
0.177257 + 0.984165i \(0.443278\pi\)
\(770\) 0 0
\(771\) −8280.00 −0.386766
\(772\) 0 0
\(773\) 14292.0 0.665003 0.332502 0.943103i \(-0.392107\pi\)
0.332502 + 0.943103i \(0.392107\pi\)
\(774\) 0 0
\(775\) −5016.00 −0.232490
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 5920.00 0.271235
\(782\) 0 0
\(783\) −1566.00 −0.0714742
\(784\) 0 0
\(785\) 22032.0 1.00173
\(786\) 0 0
\(787\) 26364.0 1.19412 0.597062 0.802195i \(-0.296335\pi\)
0.597062 + 0.802195i \(0.296335\pi\)
\(788\) 0 0
\(789\) −7080.00 −0.319461
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 41328.0 1.85069
\(794\) 0 0
\(795\) −25992.0 −1.15955
\(796\) 0 0
\(797\) −17220.0 −0.765325 −0.382662 0.923888i \(-0.624993\pi\)
−0.382662 + 0.923888i \(0.624993\pi\)
\(798\) 0 0
\(799\) −39168.0 −1.73425
\(800\) 0 0
\(801\) 6696.00 0.295370
\(802\) 0 0
\(803\) 4800.00 0.210944
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12060.0 0.526062
\(808\) 0 0
\(809\) 16442.0 0.714549 0.357274 0.933999i \(-0.383706\pi\)
0.357274 + 0.933999i \(0.383706\pi\)
\(810\) 0 0
\(811\) −31332.0 −1.35662 −0.678308 0.734778i \(-0.737286\pi\)
−0.678308 + 0.734778i \(0.737286\pi\)
\(812\) 0 0
\(813\) −14400.0 −0.621193
\(814\) 0 0
\(815\) 10992.0 0.472433
\(816\) 0 0
\(817\) −1872.00 −0.0801628
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25810.0 −1.09717 −0.548584 0.836095i \(-0.684833\pi\)
−0.548584 + 0.836095i \(0.684833\pi\)
\(822\) 0 0
\(823\) −12368.0 −0.523841 −0.261921 0.965089i \(-0.584356\pi\)
−0.261921 + 0.965089i \(0.584356\pi\)
\(824\) 0 0
\(825\) 1140.00 0.0481087
\(826\) 0 0
\(827\) −6316.00 −0.265573 −0.132786 0.991145i \(-0.542392\pi\)
−0.132786 + 0.991145i \(0.542392\pi\)
\(828\) 0 0
\(829\) 23868.0 0.999964 0.499982 0.866036i \(-0.333340\pi\)
0.499982 + 0.866036i \(0.333340\pi\)
\(830\) 0 0
\(831\) −19338.0 −0.807254
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −6048.00 −0.250658
\(836\) 0 0
\(837\) 7128.00 0.294360
\(838\) 0 0
\(839\) −48216.0 −1.98403 −0.992015 0.126120i \(-0.959748\pi\)
−0.992015 + 0.126120i \(0.959748\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 0 0
\(843\) 7806.00 0.318924
\(844\) 0 0
\(845\) −58308.0 −2.37379
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20700.0 0.836775
\(850\) 0 0
\(851\) 45408.0 1.82910
\(852\) 0 0
\(853\) 27300.0 1.09582 0.547910 0.836537i \(-0.315424\pi\)
0.547910 + 0.836537i \(0.315424\pi\)
\(854\) 0 0
\(855\) −1296.00 −0.0518389
\(856\) 0 0
\(857\) −8640.00 −0.344384 −0.172192 0.985063i \(-0.555085\pi\)
−0.172192 + 0.985063i \(0.555085\pi\)
\(858\) 0 0
\(859\) 24372.0 0.968058 0.484029 0.875052i \(-0.339173\pi\)
0.484029 + 0.875052i \(0.339173\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2176.00 −0.0858307 −0.0429154 0.999079i \(-0.513665\pi\)
−0.0429154 + 0.999079i \(0.513665\pi\)
\(864\) 0 0
\(865\) −22032.0 −0.866024
\(866\) 0 0
\(867\) −12909.0 −0.505666
\(868\) 0 0
\(869\) 15520.0 0.605846
\(870\) 0 0
\(871\) −34608.0 −1.34632
\(872\) 0 0
\(873\) 1512.00 0.0586179
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −27574.0 −1.06170 −0.530848 0.847467i \(-0.678127\pi\)
−0.530848 + 0.847467i \(0.678127\pi\)
\(878\) 0 0
\(879\) −13356.0 −0.512499
\(880\) 0 0
\(881\) −16968.0 −0.648884 −0.324442 0.945906i \(-0.605176\pi\)
−0.324442 + 0.945906i \(0.605176\pi\)
\(882\) 0 0
\(883\) 1860.00 0.0708879 0.0354439 0.999372i \(-0.488715\pi\)
0.0354439 + 0.999372i \(0.488715\pi\)
\(884\) 0 0
\(885\) 17712.0 0.672748
\(886\) 0 0
\(887\) 2280.00 0.0863077 0.0431538 0.999068i \(-0.486259\pi\)
0.0431538 + 0.999068i \(0.486259\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1620.00 −0.0609114
\(892\) 0 0
\(893\) −4896.00 −0.183470
\(894\) 0 0
\(895\) 28464.0 1.06307
\(896\) 0 0
\(897\) −44352.0 −1.65091
\(898\) 0 0
\(899\) −15312.0 −0.568058
\(900\) 0 0
\(901\) −69312.0 −2.56284
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13104.0 −0.481317
\(906\) 0 0
\(907\) −36084.0 −1.32100 −0.660501 0.750825i \(-0.729656\pi\)
−0.660501 + 0.750825i \(0.729656\pi\)
\(908\) 0 0
\(909\) 13716.0 0.500474
\(910\) 0 0
\(911\) −24152.0 −0.878366 −0.439183 0.898398i \(-0.644732\pi\)
−0.439183 + 0.898398i \(0.644732\pi\)
\(912\) 0 0
\(913\) −18480.0 −0.669878
\(914\) 0 0
\(915\) 17712.0 0.639935
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −36336.0 −1.30426 −0.652130 0.758108i \(-0.726124\pi\)
−0.652130 + 0.758108i \(0.726124\pi\)
\(920\) 0 0
\(921\) 7308.00 0.261462
\(922\) 0 0
\(923\) −24864.0 −0.886683
\(924\) 0 0
\(925\) 4902.00 0.174245
\(926\) 0 0
\(927\) −3672.00 −0.130102
\(928\) 0 0
\(929\) −432.000 −0.0152567 −0.00762834 0.999971i \(-0.502428\pi\)
−0.00762834 + 0.999971i \(0.502428\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 22464.0 0.788251
\(934\) 0 0
\(935\) 23040.0 0.805870
\(936\) 0 0
\(937\) −22176.0 −0.773168 −0.386584 0.922254i \(-0.626345\pi\)
−0.386584 + 0.922254i \(0.626345\pi\)
\(938\) 0 0
\(939\) −5256.00 −0.182666
\(940\) 0 0
\(941\) 43524.0 1.50780 0.753901 0.656988i \(-0.228170\pi\)
0.753901 + 0.656988i \(0.228170\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1868.00 −0.0640991 −0.0320495 0.999486i \(-0.510203\pi\)
−0.0320495 + 0.999486i \(0.510203\pi\)
\(948\) 0 0
\(949\) −20160.0 −0.689590
\(950\) 0 0
\(951\) 4686.00 0.159783
\(952\) 0 0
\(953\) −9238.00 −0.314006 −0.157003 0.987598i \(-0.550183\pi\)
−0.157003 + 0.987598i \(0.550183\pi\)
\(954\) 0 0
\(955\) 30144.0 1.02140
\(956\) 0 0
\(957\) 3480.00 0.117547
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 39905.0 1.33950
\(962\) 0 0
\(963\) 7380.00 0.246954
\(964\) 0 0
\(965\) 29160.0 0.972739
\(966\) 0 0
\(967\) 30616.0 1.01814 0.509071 0.860724i \(-0.329989\pi\)
0.509071 + 0.860724i \(0.329989\pi\)
\(968\) 0 0
\(969\) −3456.00 −0.114575
\(970\) 0 0
\(971\) −27540.0 −0.910196 −0.455098 0.890441i \(-0.650396\pi\)
−0.455098 + 0.890441i \(0.650396\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4788.00 −0.157270
\(976\) 0 0
\(977\) −16402.0 −0.537100 −0.268550 0.963266i \(-0.586544\pi\)
−0.268550 + 0.963266i \(0.586544\pi\)
\(978\) 0 0
\(979\) −14880.0 −0.485768
\(980\) 0 0
\(981\) −8262.00 −0.268894
\(982\) 0 0
\(983\) 55176.0 1.79028 0.895138 0.445789i \(-0.147077\pi\)
0.895138 + 0.445789i \(0.147077\pi\)
\(984\) 0 0
\(985\) 21144.0 0.683963
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −27456.0 −0.882760
\(990\) 0 0
\(991\) −27096.0 −0.868550 −0.434275 0.900780i \(-0.642995\pi\)
−0.434275 + 0.900780i \(0.642995\pi\)
\(992\) 0 0
\(993\) −21276.0 −0.679933
\(994\) 0 0
\(995\) −37152.0 −1.18372
\(996\) 0 0
\(997\) 16812.0 0.534044 0.267022 0.963691i \(-0.413960\pi\)
0.267022 + 0.963691i \(0.413960\pi\)
\(998\) 0 0
\(999\) −6966.00 −0.220615
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.b.1.1 1
4.3 odd 2 147.4.a.e.1.1 yes 1
7.6 odd 2 2352.4.a.bi.1.1 1
12.11 even 2 441.4.a.h.1.1 1
28.3 even 6 147.4.e.f.79.1 2
28.11 odd 6 147.4.e.e.79.1 2
28.19 even 6 147.4.e.f.67.1 2
28.23 odd 6 147.4.e.e.67.1 2
28.27 even 2 147.4.a.d.1.1 1
84.11 even 6 441.4.e.f.226.1 2
84.23 even 6 441.4.e.f.361.1 2
84.47 odd 6 441.4.e.g.361.1 2
84.59 odd 6 441.4.e.g.226.1 2
84.83 odd 2 441.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.d.1.1 1 28.27 even 2
147.4.a.e.1.1 yes 1 4.3 odd 2
147.4.e.e.67.1 2 28.23 odd 6
147.4.e.e.79.1 2 28.11 odd 6
147.4.e.f.67.1 2 28.19 even 6
147.4.e.f.79.1 2 28.3 even 6
441.4.a.g.1.1 1 84.83 odd 2
441.4.a.h.1.1 1 12.11 even 2
441.4.e.f.226.1 2 84.11 even 6
441.4.e.f.361.1 2 84.23 even 6
441.4.e.g.226.1 2 84.59 odd 6
441.4.e.g.361.1 2 84.47 odd 6
2352.4.a.b.1.1 1 1.1 even 1 trivial
2352.4.a.bi.1.1 1 7.6 odd 2