Properties

Label 1200.4.a.b.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -16.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -16.0000 q^{7} +9.00000 q^{9} -12.0000 q^{11} -38.0000 q^{13} +126.000 q^{17} -20.0000 q^{19} +48.0000 q^{21} +168.000 q^{23} -27.0000 q^{27} +30.0000 q^{29} +88.0000 q^{31} +36.0000 q^{33} -254.000 q^{37} +114.000 q^{39} +42.0000 q^{41} -52.0000 q^{43} -96.0000 q^{47} -87.0000 q^{49} -378.000 q^{51} -198.000 q^{53} +60.0000 q^{57} +660.000 q^{59} -538.000 q^{61} -144.000 q^{63} +884.000 q^{67} -504.000 q^{69} -792.000 q^{71} -218.000 q^{73} +192.000 q^{77} +520.000 q^{79} +81.0000 q^{81} -492.000 q^{83} -90.0000 q^{87} +810.000 q^{89} +608.000 q^{91} -264.000 q^{93} -1154.00 q^{97} -108.000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 126.000 1.79762 0.898808 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) 0 0
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) 88.0000 0.509847 0.254924 0.966961i \(-0.417950\pi\)
0.254924 + 0.966961i \(0.417950\pi\)
\(32\) 0 0
\(33\) 36.0000 0.189903
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) 0 0
\(39\) 114.000 0.468067
\(40\) 0 0
\(41\) 42.0000 0.159983 0.0799914 0.996796i \(-0.474511\pi\)
0.0799914 + 0.996796i \(0.474511\pi\)
\(42\) 0 0
\(43\) −52.0000 −0.184417 −0.0922084 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −96.0000 −0.297937 −0.148969 0.988842i \(-0.547595\pi\)
−0.148969 + 0.988842i \(0.547595\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) −378.000 −1.03785
\(52\) 0 0
\(53\) −198.000 −0.513158 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 60.0000 0.139424
\(58\) 0 0
\(59\) 660.000 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −538.000 −1.12924 −0.564622 0.825350i \(-0.690978\pi\)
−0.564622 + 0.825350i \(0.690978\pi\)
\(62\) 0 0
\(63\) −144.000 −0.287973
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 884.000 1.61191 0.805954 0.591979i \(-0.201653\pi\)
0.805954 + 0.591979i \(0.201653\pi\)
\(68\) 0 0
\(69\) −504.000 −0.879340
\(70\) 0 0
\(71\) −792.000 −1.32385 −0.661923 0.749572i \(-0.730260\pi\)
−0.661923 + 0.749572i \(0.730260\pi\)
\(72\) 0 0
\(73\) −218.000 −0.349520 −0.174760 0.984611i \(-0.555915\pi\)
−0.174760 + 0.984611i \(0.555915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 192.000 0.284161
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −492.000 −0.650651 −0.325325 0.945602i \(-0.605474\pi\)
−0.325325 + 0.945602i \(0.605474\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −90.0000 −0.110908
\(88\) 0 0
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 0 0
\(91\) 608.000 0.700393
\(92\) 0 0
\(93\) −264.000 −0.294360
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1154.00 −1.20795 −0.603974 0.797004i \(-0.706417\pi\)
−0.603974 + 0.797004i \(0.706417\pi\)
\(98\) 0 0
\(99\) −108.000 −0.109640
\(100\) 0 0
\(101\) −618.000 −0.608845 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(102\) 0 0
\(103\) 128.000 0.122449 0.0612243 0.998124i \(-0.480499\pi\)
0.0612243 + 0.998124i \(0.480499\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1476.00 −1.33355 −0.666777 0.745257i \(-0.732327\pi\)
−0.666777 + 0.745257i \(0.732327\pi\)
\(108\) 0 0
\(109\) 1190.00 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(110\) 0 0
\(111\) 762.000 0.651584
\(112\) 0 0
\(113\) 462.000 0.384613 0.192307 0.981335i \(-0.438403\pi\)
0.192307 + 0.981335i \(0.438403\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −342.000 −0.270239
\(118\) 0 0
\(119\) −2016.00 −1.55300
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 0 0
\(123\) −126.000 −0.0923662
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2536.00 −1.77192 −0.885959 0.463763i \(-0.846499\pi\)
−0.885959 + 0.463763i \(0.846499\pi\)
\(128\) 0 0
\(129\) 156.000 0.106473
\(130\) 0 0
\(131\) −2292.00 −1.52865 −0.764324 0.644832i \(-0.776927\pi\)
−0.764324 + 0.644832i \(0.776927\pi\)
\(132\) 0 0
\(133\) 320.000 0.208628
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 726.000 0.452747 0.226374 0.974041i \(-0.427313\pi\)
0.226374 + 0.974041i \(0.427313\pi\)
\(138\) 0 0
\(139\) −380.000 −0.231879 −0.115939 0.993256i \(-0.536988\pi\)
−0.115939 + 0.993256i \(0.536988\pi\)
\(140\) 0 0
\(141\) 288.000 0.172014
\(142\) 0 0
\(143\) 456.000 0.266662
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 261.000 0.146442
\(148\) 0 0
\(149\) 1590.00 0.874214 0.437107 0.899410i \(-0.356003\pi\)
0.437107 + 0.899410i \(0.356003\pi\)
\(150\) 0 0
\(151\) −2432.00 −1.31068 −0.655342 0.755332i \(-0.727476\pi\)
−0.655342 + 0.755332i \(0.727476\pi\)
\(152\) 0 0
\(153\) 1134.00 0.599206
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −614.000 −0.312118 −0.156059 0.987748i \(-0.549879\pi\)
−0.156059 + 0.987748i \(0.549879\pi\)
\(158\) 0 0
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) −2688.00 −1.31580
\(162\) 0 0
\(163\) −1852.00 −0.889938 −0.444969 0.895546i \(-0.646785\pi\)
−0.444969 + 0.895546i \(0.646785\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2136.00 −0.989752 −0.494876 0.868964i \(-0.664787\pi\)
−0.494876 + 0.868964i \(0.664787\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) 0 0
\(173\) −1758.00 −0.772591 −0.386296 0.922375i \(-0.626246\pi\)
−0.386296 + 0.922375i \(0.626246\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1980.00 −0.840824
\(178\) 0 0
\(179\) 540.000 0.225483 0.112742 0.993624i \(-0.464037\pi\)
0.112742 + 0.993624i \(0.464037\pi\)
\(180\) 0 0
\(181\) 1982.00 0.813928 0.406964 0.913444i \(-0.366588\pi\)
0.406964 + 0.913444i \(0.366588\pi\)
\(182\) 0 0
\(183\) 1614.00 0.651969
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1512.00 −0.591275
\(188\) 0 0
\(189\) 432.000 0.166261
\(190\) 0 0
\(191\) 2688.00 1.01831 0.509154 0.860675i \(-0.329958\pi\)
0.509154 + 0.860675i \(0.329958\pi\)
\(192\) 0 0
\(193\) 2302.00 0.858557 0.429279 0.903172i \(-0.358768\pi\)
0.429279 + 0.903172i \(0.358768\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4374.00 −1.58190 −0.790951 0.611880i \(-0.790414\pi\)
−0.790951 + 0.611880i \(0.790414\pi\)
\(198\) 0 0
\(199\) 1600.00 0.569955 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(200\) 0 0
\(201\) −2652.00 −0.930635
\(202\) 0 0
\(203\) −480.000 −0.165958
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1512.00 0.507687
\(208\) 0 0
\(209\) 240.000 0.0794313
\(210\) 0 0
\(211\) −3332.00 −1.08713 −0.543565 0.839367i \(-0.682926\pi\)
−0.543565 + 0.839367i \(0.682926\pi\)
\(212\) 0 0
\(213\) 2376.00 0.764323
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1408.00 −0.440467
\(218\) 0 0
\(219\) 654.000 0.201796
\(220\) 0 0
\(221\) −4788.00 −1.45736
\(222\) 0 0
\(223\) 2648.00 0.795171 0.397586 0.917565i \(-0.369848\pi\)
0.397586 + 0.917565i \(0.369848\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2244.00 0.656121 0.328061 0.944657i \(-0.393605\pi\)
0.328061 + 0.944657i \(0.393605\pi\)
\(228\) 0 0
\(229\) −5650.00 −1.63040 −0.815202 0.579177i \(-0.803374\pi\)
−0.815202 + 0.579177i \(0.803374\pi\)
\(230\) 0 0
\(231\) −576.000 −0.164061
\(232\) 0 0
\(233\) −4698.00 −1.32093 −0.660464 0.750858i \(-0.729640\pi\)
−0.660464 + 0.750858i \(0.729640\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1560.00 −0.427565
\(238\) 0 0
\(239\) 1200.00 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(240\) 0 0
\(241\) −718.000 −0.191911 −0.0959553 0.995386i \(-0.530591\pi\)
−0.0959553 + 0.995386i \(0.530591\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 760.000 0.195780
\(248\) 0 0
\(249\) 1476.00 0.375653
\(250\) 0 0
\(251\) −6012.00 −1.51185 −0.755924 0.654659i \(-0.772812\pi\)
−0.755924 + 0.654659i \(0.772812\pi\)
\(252\) 0 0
\(253\) −2016.00 −0.500968
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2046.00 0.496599 0.248300 0.968683i \(-0.420128\pi\)
0.248300 + 0.968683i \(0.420128\pi\)
\(258\) 0 0
\(259\) 4064.00 0.974999
\(260\) 0 0
\(261\) 270.000 0.0640329
\(262\) 0 0
\(263\) −6072.00 −1.42363 −0.711817 0.702365i \(-0.752127\pi\)
−0.711817 + 0.702365i \(0.752127\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2430.00 −0.556980
\(268\) 0 0
\(269\) −6930.00 −1.57074 −0.785371 0.619025i \(-0.787528\pi\)
−0.785371 + 0.619025i \(0.787528\pi\)
\(270\) 0 0
\(271\) −1352.00 −0.303056 −0.151528 0.988453i \(-0.548419\pi\)
−0.151528 + 0.988453i \(0.548419\pi\)
\(272\) 0 0
\(273\) −1824.00 −0.404372
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1186.00 0.257256 0.128628 0.991693i \(-0.458943\pi\)
0.128628 + 0.991693i \(0.458943\pi\)
\(278\) 0 0
\(279\) 792.000 0.169949
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) 0 0
\(283\) 2828.00 0.594018 0.297009 0.954875i \(-0.404011\pi\)
0.297009 + 0.954875i \(0.404011\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −672.000 −0.138212
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) 3462.00 0.697409
\(292\) 0 0
\(293\) −4758.00 −0.948687 −0.474344 0.880340i \(-0.657315\pi\)
−0.474344 + 0.880340i \(0.657315\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 324.000 0.0633010
\(298\) 0 0
\(299\) −6384.00 −1.23477
\(300\) 0 0
\(301\) 832.000 0.159321
\(302\) 0 0
\(303\) 1854.00 0.351517
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8476.00 −1.57574 −0.787868 0.615844i \(-0.788815\pi\)
−0.787868 + 0.615844i \(0.788815\pi\)
\(308\) 0 0
\(309\) −384.000 −0.0706958
\(310\) 0 0
\(311\) −4632.00 −0.844555 −0.422278 0.906467i \(-0.638769\pi\)
−0.422278 + 0.906467i \(0.638769\pi\)
\(312\) 0 0
\(313\) 4822.00 0.870785 0.435392 0.900241i \(-0.356610\pi\)
0.435392 + 0.900241i \(0.356610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3426.00 0.607014 0.303507 0.952829i \(-0.401842\pi\)
0.303507 + 0.952829i \(0.401842\pi\)
\(318\) 0 0
\(319\) −360.000 −0.0631854
\(320\) 0 0
\(321\) 4428.00 0.769928
\(322\) 0 0
\(323\) −2520.00 −0.434107
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3570.00 −0.603735
\(328\) 0 0
\(329\) 1536.00 0.257393
\(330\) 0 0
\(331\) 2788.00 0.462968 0.231484 0.972839i \(-0.425642\pi\)
0.231484 + 0.972839i \(0.425642\pi\)
\(332\) 0 0
\(333\) −2286.00 −0.376192
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −434.000 −0.0701528 −0.0350764 0.999385i \(-0.511167\pi\)
−0.0350764 + 0.999385i \(0.511167\pi\)
\(338\) 0 0
\(339\) −1386.00 −0.222057
\(340\) 0 0
\(341\) −1056.00 −0.167700
\(342\) 0 0
\(343\) 6880.00 1.08305
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6684.00 1.03405 0.517026 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(348\) 0 0
\(349\) 2630.00 0.403383 0.201692 0.979449i \(-0.435356\pi\)
0.201692 + 0.979449i \(0.435356\pi\)
\(350\) 0 0
\(351\) 1026.00 0.156022
\(352\) 0 0
\(353\) 7422.00 1.11907 0.559537 0.828805i \(-0.310979\pi\)
0.559537 + 0.828805i \(0.310979\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6048.00 0.896622
\(358\) 0 0
\(359\) 10440.0 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) 3561.00 0.514887
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10424.0 1.48264 0.741319 0.671153i \(-0.234200\pi\)
0.741319 + 0.671153i \(0.234200\pi\)
\(368\) 0 0
\(369\) 378.000 0.0533276
\(370\) 0 0
\(371\) 3168.00 0.443327
\(372\) 0 0
\(373\) −3278.00 −0.455036 −0.227518 0.973774i \(-0.573061\pi\)
−0.227518 + 0.973774i \(0.573061\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1140.00 −0.155737
\(378\) 0 0
\(379\) −6140.00 −0.832165 −0.416083 0.909327i \(-0.636597\pi\)
−0.416083 + 0.909327i \(0.636597\pi\)
\(380\) 0 0
\(381\) 7608.00 1.02302
\(382\) 0 0
\(383\) −3072.00 −0.409848 −0.204924 0.978778i \(-0.565695\pi\)
−0.204924 + 0.978778i \(0.565695\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −468.000 −0.0614723
\(388\) 0 0
\(389\) 6150.00 0.801587 0.400794 0.916168i \(-0.368734\pi\)
0.400794 + 0.916168i \(0.368734\pi\)
\(390\) 0 0
\(391\) 21168.0 2.73788
\(392\) 0 0
\(393\) 6876.00 0.882566
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 106.000 0.0134005 0.00670024 0.999978i \(-0.497867\pi\)
0.00670024 + 0.999978i \(0.497867\pi\)
\(398\) 0 0
\(399\) −960.000 −0.120451
\(400\) 0 0
\(401\) −1758.00 −0.218929 −0.109464 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(402\) 0 0
\(403\) −3344.00 −0.413341
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3048.00 0.371213
\(408\) 0 0
\(409\) −3670.00 −0.443691 −0.221846 0.975082i \(-0.571208\pi\)
−0.221846 + 0.975082i \(0.571208\pi\)
\(410\) 0 0
\(411\) −2178.00 −0.261394
\(412\) 0 0
\(413\) −10560.0 −1.25817
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1140.00 0.133875
\(418\) 0 0
\(419\) 9660.00 1.12631 0.563153 0.826353i \(-0.309588\pi\)
0.563153 + 0.826353i \(0.309588\pi\)
\(420\) 0 0
\(421\) 8462.00 0.979602 0.489801 0.871834i \(-0.337069\pi\)
0.489801 + 0.871834i \(0.337069\pi\)
\(422\) 0 0
\(423\) −864.000 −0.0993123
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8608.00 0.975575
\(428\) 0 0
\(429\) −1368.00 −0.153957
\(430\) 0 0
\(431\) −9792.00 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(432\) 0 0
\(433\) 7342.00 0.814859 0.407430 0.913237i \(-0.366425\pi\)
0.407430 + 0.913237i \(0.366425\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3360.00 −0.367805
\(438\) 0 0
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) −17412.0 −1.86742 −0.933712 0.358024i \(-0.883451\pi\)
−0.933712 + 0.358024i \(0.883451\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4770.00 −0.504728
\(448\) 0 0
\(449\) −1710.00 −0.179732 −0.0898662 0.995954i \(-0.528644\pi\)
−0.0898662 + 0.995954i \(0.528644\pi\)
\(450\) 0 0
\(451\) −504.000 −0.0526218
\(452\) 0 0
\(453\) 7296.00 0.756724
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 646.000 0.0661239 0.0330619 0.999453i \(-0.489474\pi\)
0.0330619 + 0.999453i \(0.489474\pi\)
\(458\) 0 0
\(459\) −3402.00 −0.345952
\(460\) 0 0
\(461\) −6018.00 −0.607996 −0.303998 0.952673i \(-0.598322\pi\)
−0.303998 + 0.952673i \(0.598322\pi\)
\(462\) 0 0
\(463\) −6712.00 −0.673722 −0.336861 0.941554i \(-0.609365\pi\)
−0.336861 + 0.941554i \(0.609365\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5364.00 0.531512 0.265756 0.964040i \(-0.414378\pi\)
0.265756 + 0.964040i \(0.414378\pi\)
\(468\) 0 0
\(469\) −14144.0 −1.39256
\(470\) 0 0
\(471\) 1842.00 0.180201
\(472\) 0 0
\(473\) 624.000 0.0606587
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1782.00 −0.171053
\(478\) 0 0
\(479\) −9840.00 −0.938624 −0.469312 0.883032i \(-0.655498\pi\)
−0.469312 + 0.883032i \(0.655498\pi\)
\(480\) 0 0
\(481\) 9652.00 0.914955
\(482\) 0 0
\(483\) 8064.00 0.759678
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1424.00 0.132500 0.0662501 0.997803i \(-0.478896\pi\)
0.0662501 + 0.997803i \(0.478896\pi\)
\(488\) 0 0
\(489\) 5556.00 0.513806
\(490\) 0 0
\(491\) 4548.00 0.418021 0.209011 0.977913i \(-0.432976\pi\)
0.209011 + 0.977913i \(0.432976\pi\)
\(492\) 0 0
\(493\) 3780.00 0.345320
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12672.0 1.14370
\(498\) 0 0
\(499\) −6500.00 −0.583126 −0.291563 0.956552i \(-0.594175\pi\)
−0.291563 + 0.956552i \(0.594175\pi\)
\(500\) 0 0
\(501\) 6408.00 0.571434
\(502\) 0 0
\(503\) 12168.0 1.07862 0.539308 0.842108i \(-0.318686\pi\)
0.539308 + 0.842108i \(0.318686\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2259.00 0.197881
\(508\) 0 0
\(509\) −21090.0 −1.83654 −0.918269 0.395957i \(-0.870413\pi\)
−0.918269 + 0.395957i \(0.870413\pi\)
\(510\) 0 0
\(511\) 3488.00 0.301957
\(512\) 0 0
\(513\) 540.000 0.0464748
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1152.00 0.0979979
\(518\) 0 0
\(519\) 5274.00 0.446056
\(520\) 0 0
\(521\) −5238.00 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(522\) 0 0
\(523\) 8588.00 0.718025 0.359012 0.933333i \(-0.383114\pi\)
0.359012 + 0.933333i \(0.383114\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11088.0 0.916510
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) 5940.00 0.485450
\(532\) 0 0
\(533\) −1596.00 −0.129701
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1620.00 −0.130183
\(538\) 0 0
\(539\) 1044.00 0.0834291
\(540\) 0 0
\(541\) 3062.00 0.243338 0.121669 0.992571i \(-0.461175\pi\)
0.121669 + 0.992571i \(0.461175\pi\)
\(542\) 0 0
\(543\) −5946.00 −0.469921
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8476.00 −0.662537 −0.331268 0.943537i \(-0.607477\pi\)
−0.331268 + 0.943537i \(0.607477\pi\)
\(548\) 0 0
\(549\) −4842.00 −0.376414
\(550\) 0 0
\(551\) −600.000 −0.0463899
\(552\) 0 0
\(553\) −8320.00 −0.639787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12546.0 0.954383 0.477191 0.878799i \(-0.341655\pi\)
0.477191 + 0.878799i \(0.341655\pi\)
\(558\) 0 0
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) 0 0
\(563\) −12.0000 −0.000898294 0 −0.000449147 1.00000i \(-0.500143\pi\)
−0.000449147 1.00000i \(0.500143\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1296.00 −0.0959910
\(568\) 0 0
\(569\) 19290.0 1.42123 0.710614 0.703582i \(-0.248417\pi\)
0.710614 + 0.703582i \(0.248417\pi\)
\(570\) 0 0
\(571\) 12148.0 0.890329 0.445165 0.895449i \(-0.353145\pi\)
0.445165 + 0.895449i \(0.353145\pi\)
\(572\) 0 0
\(573\) −8064.00 −0.587920
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10366.0 0.747907 0.373953 0.927447i \(-0.378002\pi\)
0.373953 + 0.927447i \(0.378002\pi\)
\(578\) 0 0
\(579\) −6906.00 −0.495688
\(580\) 0 0
\(581\) 7872.00 0.562109
\(582\) 0 0
\(583\) 2376.00 0.168789
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7644.00 0.537482 0.268741 0.963213i \(-0.413393\pi\)
0.268741 + 0.963213i \(0.413393\pi\)
\(588\) 0 0
\(589\) −1760.00 −0.123123
\(590\) 0 0
\(591\) 13122.0 0.913311
\(592\) 0 0
\(593\) −8658.00 −0.599564 −0.299782 0.954008i \(-0.596914\pi\)
−0.299782 + 0.954008i \(0.596914\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4800.00 −0.329064
\(598\) 0 0
\(599\) −25800.0 −1.75987 −0.879933 0.475098i \(-0.842413\pi\)
−0.879933 + 0.475098i \(0.842413\pi\)
\(600\) 0 0
\(601\) 16202.0 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(602\) 0 0
\(603\) 7956.00 0.537302
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −24136.0 −1.61392 −0.806960 0.590605i \(-0.798889\pi\)
−0.806960 + 0.590605i \(0.798889\pi\)
\(608\) 0 0
\(609\) 1440.00 0.0958157
\(610\) 0 0
\(611\) 3648.00 0.241542
\(612\) 0 0
\(613\) 4642.00 0.305854 0.152927 0.988237i \(-0.451130\pi\)
0.152927 + 0.988237i \(0.451130\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6726.00 0.438863 0.219432 0.975628i \(-0.429580\pi\)
0.219432 + 0.975628i \(0.429580\pi\)
\(618\) 0 0
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 0 0
\(623\) −12960.0 −0.833437
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −720.000 −0.0458597
\(628\) 0 0
\(629\) −32004.0 −2.02875
\(630\) 0 0
\(631\) −29792.0 −1.87956 −0.939779 0.341783i \(-0.888969\pi\)
−0.939779 + 0.341783i \(0.888969\pi\)
\(632\) 0 0
\(633\) 9996.00 0.627655
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3306.00 0.205633
\(638\) 0 0
\(639\) −7128.00 −0.441282
\(640\) 0 0
\(641\) −10158.0 −0.625923 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(642\) 0 0
\(643\) 29828.0 1.82940 0.914698 0.404138i \(-0.132429\pi\)
0.914698 + 0.404138i \(0.132429\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1944.00 0.118124 0.0590622 0.998254i \(-0.481189\pi\)
0.0590622 + 0.998254i \(0.481189\pi\)
\(648\) 0 0
\(649\) −7920.00 −0.479025
\(650\) 0 0
\(651\) 4224.00 0.254304
\(652\) 0 0
\(653\) −26718.0 −1.60116 −0.800579 0.599227i \(-0.795475\pi\)
−0.800579 + 0.599227i \(0.795475\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1962.00 −0.116507
\(658\) 0 0
\(659\) −4260.00 −0.251815 −0.125907 0.992042i \(-0.540184\pi\)
−0.125907 + 0.992042i \(0.540184\pi\)
\(660\) 0 0
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) 0 0
\(663\) 14364.0 0.841405
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5040.00 0.292578
\(668\) 0 0
\(669\) −7944.00 −0.459092
\(670\) 0 0
\(671\) 6456.00 0.371432
\(672\) 0 0
\(673\) 32542.0 1.86390 0.931948 0.362592i \(-0.118108\pi\)
0.931948 + 0.362592i \(0.118108\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14214.0 −0.806925 −0.403463 0.914996i \(-0.632193\pi\)
−0.403463 + 0.914996i \(0.632193\pi\)
\(678\) 0 0
\(679\) 18464.0 1.04357
\(680\) 0 0
\(681\) −6732.00 −0.378812
\(682\) 0 0
\(683\) −7092.00 −0.397317 −0.198659 0.980069i \(-0.563659\pi\)
−0.198659 + 0.980069i \(0.563659\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16950.0 0.941314
\(688\) 0 0
\(689\) 7524.00 0.416026
\(690\) 0 0
\(691\) 13228.0 0.728244 0.364122 0.931351i \(-0.381369\pi\)
0.364122 + 0.931351i \(0.381369\pi\)
\(692\) 0 0
\(693\) 1728.00 0.0947205
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5292.00 0.287588
\(698\) 0 0
\(699\) 14094.0 0.762638
\(700\) 0 0
\(701\) 28062.0 1.51196 0.755982 0.654592i \(-0.227160\pi\)
0.755982 + 0.654592i \(0.227160\pi\)
\(702\) 0 0
\(703\) 5080.00 0.272540
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9888.00 0.525992
\(708\) 0 0
\(709\) −27250.0 −1.44343 −0.721717 0.692188i \(-0.756647\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(710\) 0 0
\(711\) 4680.00 0.246855
\(712\) 0 0
\(713\) 14784.0 0.776529
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3600.00 −0.187510
\(718\) 0 0
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) 0 0
\(721\) −2048.00 −0.105786
\(722\) 0 0
\(723\) 2154.00 0.110800
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17984.0 0.917455 0.458727 0.888577i \(-0.348305\pi\)
0.458727 + 0.888577i \(0.348305\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6552.00 −0.331511
\(732\) 0 0
\(733\) −16598.0 −0.836373 −0.418186 0.908361i \(-0.637334\pi\)
−0.418186 + 0.908361i \(0.637334\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10608.0 −0.530191
\(738\) 0 0
\(739\) −1460.00 −0.0726752 −0.0363376 0.999340i \(-0.511569\pi\)
−0.0363376 + 0.999340i \(0.511569\pi\)
\(740\) 0 0
\(741\) −2280.00 −0.113034
\(742\) 0 0
\(743\) −30072.0 −1.48484 −0.742419 0.669936i \(-0.766322\pi\)
−0.742419 + 0.669936i \(0.766322\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4428.00 −0.216884
\(748\) 0 0
\(749\) 23616.0 1.15208
\(750\) 0 0
\(751\) 18088.0 0.878882 0.439441 0.898271i \(-0.355177\pi\)
0.439441 + 0.898271i \(0.355177\pi\)
\(752\) 0 0
\(753\) 18036.0 0.872866
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −24734.0 −1.18755 −0.593773 0.804633i \(-0.702362\pi\)
−0.593773 + 0.804633i \(0.702362\pi\)
\(758\) 0 0
\(759\) 6048.00 0.289234
\(760\) 0 0
\(761\) −22278.0 −1.06120 −0.530602 0.847621i \(-0.678034\pi\)
−0.530602 + 0.847621i \(0.678034\pi\)
\(762\) 0 0
\(763\) −19040.0 −0.903400
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25080.0 −1.18069
\(768\) 0 0
\(769\) 16130.0 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(770\) 0 0
\(771\) −6138.00 −0.286712
\(772\) 0 0
\(773\) −29718.0 −1.38277 −0.691386 0.722486i \(-0.742999\pi\)
−0.691386 + 0.722486i \(0.742999\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −12192.0 −0.562916
\(778\) 0 0
\(779\) −840.000 −0.0386343
\(780\) 0 0
\(781\) 9504.00 0.435442
\(782\) 0 0
\(783\) −810.000 −0.0369694
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9524.00 0.431377 0.215689 0.976462i \(-0.430800\pi\)
0.215689 + 0.976462i \(0.430800\pi\)
\(788\) 0 0
\(789\) 18216.0 0.821935
\(790\) 0 0
\(791\) −7392.00 −0.332275
\(792\) 0 0
\(793\) 20444.0 0.915495
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33906.0 1.50692 0.753458 0.657496i \(-0.228384\pi\)
0.753458 + 0.657496i \(0.228384\pi\)
\(798\) 0 0
\(799\) −12096.0 −0.535577
\(800\) 0 0
\(801\) 7290.00 0.321572
\(802\) 0 0
\(803\) 2616.00 0.114965
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20790.0 0.906868
\(808\) 0 0
\(809\) −630.000 −0.0273790 −0.0136895 0.999906i \(-0.504358\pi\)
−0.0136895 + 0.999906i \(0.504358\pi\)
\(810\) 0 0
\(811\) 20788.0 0.900081 0.450040 0.893008i \(-0.351410\pi\)
0.450040 + 0.893008i \(0.351410\pi\)
\(812\) 0 0
\(813\) 4056.00 0.174969
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1040.00 0.0445349
\(818\) 0 0
\(819\) 5472.00 0.233464
\(820\) 0 0
\(821\) −43098.0 −1.83207 −0.916036 0.401097i \(-0.868629\pi\)
−0.916036 + 0.401097i \(0.868629\pi\)
\(822\) 0 0
\(823\) −14272.0 −0.604484 −0.302242 0.953231i \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13644.0 0.573698 0.286849 0.957976i \(-0.407392\pi\)
0.286849 + 0.957976i \(0.407392\pi\)
\(828\) 0 0
\(829\) −2410.00 −0.100968 −0.0504842 0.998725i \(-0.516076\pi\)
−0.0504842 + 0.998725i \(0.516076\pi\)
\(830\) 0 0
\(831\) −3558.00 −0.148527
\(832\) 0 0
\(833\) −10962.0 −0.455955
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2376.00 −0.0981202
\(838\) 0 0
\(839\) −23160.0 −0.953006 −0.476503 0.879173i \(-0.658096\pi\)
−0.476503 + 0.879173i \(0.658096\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 0 0
\(843\) −7326.00 −0.299313
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 18992.0 0.770452
\(848\) 0 0
\(849\) −8484.00 −0.342957
\(850\) 0 0
\(851\) −42672.0 −1.71889
\(852\) 0 0
\(853\) −32078.0 −1.28761 −0.643804 0.765190i \(-0.722645\pi\)
−0.643804 + 0.765190i \(0.722645\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14406.0 0.574212 0.287106 0.957899i \(-0.407307\pi\)
0.287106 + 0.957899i \(0.407307\pi\)
\(858\) 0 0
\(859\) −30620.0 −1.21623 −0.608115 0.793849i \(-0.708074\pi\)
−0.608115 + 0.793849i \(0.708074\pi\)
\(860\) 0 0
\(861\) 2016.00 0.0797969
\(862\) 0 0
\(863\) 17568.0 0.692957 0.346478 0.938058i \(-0.387377\pi\)
0.346478 + 0.938058i \(0.387377\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −32889.0 −1.28831
\(868\) 0 0
\(869\) −6240.00 −0.243587
\(870\) 0 0
\(871\) −33592.0 −1.30680
\(872\) 0 0
\(873\) −10386.0 −0.402649
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21706.0 0.835758 0.417879 0.908503i \(-0.362774\pi\)
0.417879 + 0.908503i \(0.362774\pi\)
\(878\) 0 0
\(879\) 14274.0 0.547725
\(880\) 0 0
\(881\) −14958.0 −0.572018 −0.286009 0.958227i \(-0.592329\pi\)
−0.286009 + 0.958227i \(0.592329\pi\)
\(882\) 0 0
\(883\) −32812.0 −1.25052 −0.625261 0.780415i \(-0.715008\pi\)
−0.625261 + 0.780415i \(0.715008\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −38856.0 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(888\) 0 0
\(889\) 40576.0 1.53079
\(890\) 0 0
\(891\) −972.000 −0.0365468
\(892\) 0 0
\(893\) 1920.00 0.0719489
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 19152.0 0.712895
\(898\) 0 0
\(899\) 2640.00 0.0979410
\(900\) 0 0
\(901\) −24948.0 −0.922462
\(902\) 0 0
\(903\) −2496.00 −0.0919841
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −28276.0 −1.03516 −0.517579 0.855635i \(-0.673167\pi\)
−0.517579 + 0.855635i \(0.673167\pi\)
\(908\) 0 0
\(909\) −5562.00 −0.202948
\(910\) 0 0
\(911\) −8112.00 −0.295019 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(912\) 0 0
\(913\) 5904.00 0.214013
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36672.0 1.32063
\(918\) 0 0
\(919\) 26080.0 0.936126 0.468063 0.883695i \(-0.344952\pi\)
0.468063 + 0.883695i \(0.344952\pi\)
\(920\) 0 0
\(921\) 25428.0 0.909751
\(922\) 0 0
\(923\) 30096.0 1.07326
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1152.00 0.0408162
\(928\) 0 0
\(929\) 49170.0 1.73651 0.868254 0.496120i \(-0.165243\pi\)
0.868254 + 0.496120i \(0.165243\pi\)
\(930\) 0 0
\(931\) 1740.00 0.0612526
\(932\) 0 0
\(933\) 13896.0 0.487604
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −48314.0 −1.68447 −0.842236 0.539110i \(-0.818761\pi\)
−0.842236 + 0.539110i \(0.818761\pi\)
\(938\) 0 0
\(939\) −14466.0 −0.502748
\(940\) 0 0
\(941\) 34782.0 1.20495 0.602477 0.798137i \(-0.294181\pi\)
0.602477 + 0.798137i \(0.294181\pi\)
\(942\) 0 0
\(943\) 7056.00 0.243664
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25116.0 −0.861838 −0.430919 0.902391i \(-0.641810\pi\)
−0.430919 + 0.902391i \(0.641810\pi\)
\(948\) 0 0
\(949\) 8284.00 0.283361
\(950\) 0 0
\(951\) −10278.0 −0.350460
\(952\) 0 0
\(953\) 15462.0 0.525565 0.262782 0.964855i \(-0.415360\pi\)
0.262782 + 0.964855i \(0.415360\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1080.00 0.0364801
\(958\) 0 0
\(959\) −11616.0 −0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 0 0
\(963\) −13284.0 −0.444518
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −736.000 −0.0244759 −0.0122379 0.999925i \(-0.503896\pi\)
−0.0122379 + 0.999925i \(0.503896\pi\)
\(968\) 0 0
\(969\) 7560.00 0.250632
\(970\) 0 0
\(971\) 29268.0 0.967307 0.483653 0.875260i \(-0.339310\pi\)
0.483653 + 0.875260i \(0.339310\pi\)
\(972\) 0 0
\(973\) 6080.00 0.200325
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16674.0 −0.546007 −0.273003 0.962013i \(-0.588017\pi\)
−0.273003 + 0.962013i \(0.588017\pi\)
\(978\) 0 0
\(979\) −9720.00 −0.317316
\(980\) 0 0
\(981\) 10710.0 0.348567
\(982\) 0 0
\(983\) −31272.0 −1.01467 −0.507336 0.861749i \(-0.669370\pi\)
−0.507336 + 0.861749i \(0.669370\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4608.00 −0.148606
\(988\) 0 0
\(989\) −8736.00 −0.280878
\(990\) 0 0
\(991\) 15928.0 0.510565 0.255282 0.966867i \(-0.417832\pi\)
0.255282 + 0.966867i \(0.417832\pi\)
\(992\) 0 0
\(993\) −8364.00 −0.267295
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42014.0 −1.33460 −0.667300 0.744789i \(-0.732550\pi\)
−0.667300 + 0.744789i \(0.732550\pi\)
\(998\) 0 0
\(999\) 6858.00 0.217195
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 1200.4.a.b.1.1 1
4.3 odd 2 150.4.a.i.1.1 1
5.2 odd 4 1200.4.f.j.49.2 2
5.3 odd 4 1200.4.f.j.49.1 2
5.4 even 2 48.4.a.c.1.1 1
12.11 even 2 450.4.a.h.1.1 1
15.14 odd 2 144.4.a.c.1.1 1
20.3 even 4 150.4.c.d.49.1 2
20.7 even 4 150.4.c.d.49.2 2
20.19 odd 2 6.4.a.a.1.1 1
35.34 odd 2 2352.4.a.e.1.1 1
40.19 odd 2 192.4.a.i.1.1 1
40.29 even 2 192.4.a.c.1.1 1
60.23 odd 4 450.4.c.e.199.2 2
60.47 odd 4 450.4.c.e.199.1 2
60.59 even 2 18.4.a.a.1.1 1
80.19 odd 4 768.4.d.n.385.1 2
80.29 even 4 768.4.d.c.385.2 2
80.59 odd 4 768.4.d.n.385.2 2
80.69 even 4 768.4.d.c.385.1 2
120.29 odd 2 576.4.a.r.1.1 1
120.59 even 2 576.4.a.q.1.1 1
140.19 even 6 294.4.e.g.67.1 2
140.39 odd 6 294.4.e.h.79.1 2
140.59 even 6 294.4.e.g.79.1 2
140.79 odd 6 294.4.e.h.67.1 2
140.139 even 2 294.4.a.e.1.1 1
180.59 even 6 162.4.c.c.55.1 2
180.79 odd 6 162.4.c.f.109.1 2
180.119 even 6 162.4.c.c.109.1 2
180.139 odd 6 162.4.c.f.55.1 2
220.219 even 2 726.4.a.f.1.1 1
260.99 even 4 1014.4.b.d.337.1 2
260.239 even 4 1014.4.b.d.337.2 2
260.259 odd 2 1014.4.a.g.1.1 1
340.339 odd 2 1734.4.a.d.1.1 1
380.379 even 2 2166.4.a.i.1.1 1
420.59 odd 6 882.4.g.f.667.1 2
420.179 even 6 882.4.g.i.667.1 2
420.299 odd 6 882.4.g.f.361.1 2
420.359 even 6 882.4.g.i.361.1 2
420.419 odd 2 882.4.a.n.1.1 1
660.659 odd 2 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 20.19 odd 2
18.4.a.a.1.1 1 60.59 even 2
48.4.a.c.1.1 1 5.4 even 2
144.4.a.c.1.1 1 15.14 odd 2
150.4.a.i.1.1 1 4.3 odd 2
150.4.c.d.49.1 2 20.3 even 4
150.4.c.d.49.2 2 20.7 even 4
162.4.c.c.55.1 2 180.59 even 6
162.4.c.c.109.1 2 180.119 even 6
162.4.c.f.55.1 2 180.139 odd 6
162.4.c.f.109.1 2 180.79 odd 6
192.4.a.c.1.1 1 40.29 even 2
192.4.a.i.1.1 1 40.19 odd 2
294.4.a.e.1.1 1 140.139 even 2
294.4.e.g.67.1 2 140.19 even 6
294.4.e.g.79.1 2 140.59 even 6
294.4.e.h.67.1 2 140.79 odd 6
294.4.e.h.79.1 2 140.39 odd 6
450.4.a.h.1.1 1 12.11 even 2
450.4.c.e.199.1 2 60.47 odd 4
450.4.c.e.199.2 2 60.23 odd 4
576.4.a.q.1.1 1 120.59 even 2
576.4.a.r.1.1 1 120.29 odd 2
726.4.a.f.1.1 1 220.219 even 2
768.4.d.c.385.1 2 80.69 even 4
768.4.d.c.385.2 2 80.29 even 4
768.4.d.n.385.1 2 80.19 odd 4
768.4.d.n.385.2 2 80.59 odd 4
882.4.a.n.1.1 1 420.419 odd 2
882.4.g.f.361.1 2 420.299 odd 6
882.4.g.f.667.1 2 420.59 odd 6
882.4.g.i.361.1 2 420.359 even 6
882.4.g.i.667.1 2 420.179 even 6
1014.4.a.g.1.1 1 260.259 odd 2
1014.4.b.d.337.1 2 260.99 even 4
1014.4.b.d.337.2 2 260.239 even 4
1200.4.a.b.1.1 1 1.1 even 1 trivial
1200.4.f.j.49.1 2 5.3 odd 4
1200.4.f.j.49.2 2 5.2 odd 4
1734.4.a.d.1.1 1 340.339 odd 2
2166.4.a.i.1.1 1 380.379 even 2
2178.4.a.e.1.1 1 660.659 odd 2
2352.4.a.e.1.1 1 35.34 odd 2