Properties

Label 675.4.a.i.1.1
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 675.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -4.00000 q^{4} -24.0000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} -4.00000 q^{4} -24.0000 q^{8} +10.0000 q^{11} +80.0000 q^{13} -16.0000 q^{16} -7.00000 q^{17} -113.000 q^{19} +20.0000 q^{22} +81.0000 q^{23} +160.000 q^{26} -220.000 q^{29} -189.000 q^{31} +160.000 q^{32} -14.0000 q^{34} -170.000 q^{37} -226.000 q^{38} -130.000 q^{41} -10.0000 q^{43} -40.0000 q^{44} +162.000 q^{46} -160.000 q^{47} -343.000 q^{49} -320.000 q^{52} -631.000 q^{53} -440.000 q^{58} -560.000 q^{59} +229.000 q^{61} -378.000 q^{62} +448.000 q^{64} -750.000 q^{67} +28.0000 q^{68} +890.000 q^{71} +890.000 q^{73} -340.000 q^{74} +452.000 q^{76} -27.0000 q^{79} -260.000 q^{82} -429.000 q^{83} -20.0000 q^{86} -240.000 q^{88} -750.000 q^{89} -324.000 q^{92} -320.000 q^{94} +1480.00 q^{97} -686.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −24.0000 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 10.0000 0.274101 0.137051 0.990564i \(-0.456238\pi\)
0.137051 + 0.990564i \(0.456238\pi\)
\(12\) 0 0
\(13\) 80.0000 1.70677 0.853385 0.521281i \(-0.174546\pi\)
0.853385 + 0.521281i \(0.174546\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) −7.00000 −0.0998676 −0.0499338 0.998753i \(-0.515901\pi\)
−0.0499338 + 0.998753i \(0.515901\pi\)
\(18\) 0 0
\(19\) −113.000 −1.36442 −0.682210 0.731156i \(-0.738981\pi\)
−0.682210 + 0.731156i \(0.738981\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 20.0000 0.193819
\(23\) 81.0000 0.734333 0.367167 0.930155i \(-0.380328\pi\)
0.367167 + 0.930155i \(0.380328\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 160.000 1.20687
\(27\) 0 0
\(28\) 0 0
\(29\) −220.000 −1.40872 −0.704362 0.709841i \(-0.748767\pi\)
−0.704362 + 0.709841i \(0.748767\pi\)
\(30\) 0 0
\(31\) −189.000 −1.09501 −0.547506 0.836801i \(-0.684423\pi\)
−0.547506 + 0.836801i \(0.684423\pi\)
\(32\) 160.000 0.883883
\(33\) 0 0
\(34\) −14.0000 −0.0706171
\(35\) 0 0
\(36\) 0 0
\(37\) −170.000 −0.755347 −0.377673 0.925939i \(-0.623276\pi\)
−0.377673 + 0.925939i \(0.623276\pi\)
\(38\) −226.000 −0.964791
\(39\) 0 0
\(40\) 0 0
\(41\) −130.000 −0.495185 −0.247593 0.968864i \(-0.579639\pi\)
−0.247593 + 0.968864i \(0.579639\pi\)
\(42\) 0 0
\(43\) −10.0000 −0.0354648 −0.0177324 0.999843i \(-0.505645\pi\)
−0.0177324 + 0.999843i \(0.505645\pi\)
\(44\) −40.0000 −0.137051
\(45\) 0 0
\(46\) 162.000 0.519252
\(47\) −160.000 −0.496562 −0.248281 0.968688i \(-0.579866\pi\)
−0.248281 + 0.968688i \(0.579866\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −320.000 −0.853385
\(53\) −631.000 −1.63537 −0.817684 0.575667i \(-0.804742\pi\)
−0.817684 + 0.575667i \(0.804742\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −440.000 −0.996118
\(59\) −560.000 −1.23569 −0.617846 0.786299i \(-0.711994\pi\)
−0.617846 + 0.786299i \(0.711994\pi\)
\(60\) 0 0
\(61\) 229.000 0.480663 0.240332 0.970691i \(-0.422744\pi\)
0.240332 + 0.970691i \(0.422744\pi\)
\(62\) −378.000 −0.774291
\(63\) 0 0
\(64\) 448.000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −750.000 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(68\) 28.0000 0.0499338
\(69\) 0 0
\(70\) 0 0
\(71\) 890.000 1.48766 0.743828 0.668371i \(-0.233008\pi\)
0.743828 + 0.668371i \(0.233008\pi\)
\(72\) 0 0
\(73\) 890.000 1.42694 0.713470 0.700686i \(-0.247122\pi\)
0.713470 + 0.700686i \(0.247122\pi\)
\(74\) −340.000 −0.534111
\(75\) 0 0
\(76\) 452.000 0.682210
\(77\) 0 0
\(78\) 0 0
\(79\) −27.0000 −0.0384524 −0.0192262 0.999815i \(-0.506120\pi\)
−0.0192262 + 0.999815i \(0.506120\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −260.000 −0.350149
\(83\) −429.000 −0.567336 −0.283668 0.958923i \(-0.591551\pi\)
−0.283668 + 0.958923i \(0.591551\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −20.0000 −0.0250774
\(87\) 0 0
\(88\) −240.000 −0.290728
\(89\) −750.000 −0.893257 −0.446628 0.894720i \(-0.647375\pi\)
−0.446628 + 0.894720i \(0.647375\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −324.000 −0.367167
\(93\) 0 0
\(94\) −320.000 −0.351122
\(95\) 0 0
\(96\) 0 0
\(97\) 1480.00 1.54919 0.774594 0.632459i \(-0.217954\pi\)
0.774594 + 0.632459i \(0.217954\pi\)
\(98\) −686.000 −0.707107
\(99\) 0 0
\(100\) 0 0
\(101\) −1500.00 −1.47778 −0.738889 0.673827i \(-0.764649\pi\)
−0.738889 + 0.673827i \(0.764649\pi\)
\(102\) 0 0
\(103\) 460.000 0.440050 0.220025 0.975494i \(-0.429386\pi\)
0.220025 + 0.975494i \(0.429386\pi\)
\(104\) −1920.00 −1.81030
\(105\) 0 0
\(106\) −1262.00 −1.15638
\(107\) 420.000 0.379467 0.189733 0.981836i \(-0.439238\pi\)
0.189733 + 0.981836i \(0.439238\pi\)
\(108\) 0 0
\(109\) −607.000 −0.533395 −0.266698 0.963780i \(-0.585932\pi\)
−0.266698 + 0.963780i \(0.585932\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2170.00 −1.80652 −0.903259 0.429097i \(-0.858832\pi\)
−0.903259 + 0.429097i \(0.858832\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 880.000 0.704362
\(117\) 0 0
\(118\) −1120.00 −0.873766
\(119\) 0 0
\(120\) 0 0
\(121\) −1231.00 −0.924869
\(122\) 458.000 0.339880
\(123\) 0 0
\(124\) 756.000 0.547506
\(125\) 0 0
\(126\) 0 0
\(127\) 1610.00 1.12492 0.562458 0.826826i \(-0.309856\pi\)
0.562458 + 0.826826i \(0.309856\pi\)
\(128\) −384.000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 2370.00 1.58067 0.790335 0.612674i \(-0.209906\pi\)
0.790335 + 0.612674i \(0.209906\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1500.00 −0.967017
\(135\) 0 0
\(136\) 168.000 0.105926
\(137\) −1797.00 −1.12064 −0.560321 0.828275i \(-0.689322\pi\)
−0.560321 + 0.828275i \(0.689322\pi\)
\(138\) 0 0
\(139\) −124.000 −0.0756658 −0.0378329 0.999284i \(-0.512045\pi\)
−0.0378329 + 0.999284i \(0.512045\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1780.00 1.05193
\(143\) 800.000 0.467828
\(144\) 0 0
\(145\) 0 0
\(146\) 1780.00 1.00900
\(147\) 0 0
\(148\) 680.000 0.377673
\(149\) −70.0000 −0.0384874 −0.0192437 0.999815i \(-0.506126\pi\)
−0.0192437 + 0.999815i \(0.506126\pi\)
\(150\) 0 0
\(151\) 2248.00 1.21152 0.605760 0.795647i \(-0.292869\pi\)
0.605760 + 0.795647i \(0.292869\pi\)
\(152\) 2712.00 1.44719
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1010.00 −0.513419 −0.256709 0.966489i \(-0.582638\pi\)
−0.256709 + 0.966489i \(0.582638\pi\)
\(158\) −54.0000 −0.0271899
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −590.000 −0.283511 −0.141756 0.989902i \(-0.545275\pi\)
−0.141756 + 0.989902i \(0.545275\pi\)
\(164\) 520.000 0.247593
\(165\) 0 0
\(166\) −858.000 −0.401167
\(167\) −2403.00 −1.11347 −0.556736 0.830690i \(-0.687946\pi\)
−0.556736 + 0.830690i \(0.687946\pi\)
\(168\) 0 0
\(169\) 4203.00 1.91306
\(170\) 0 0
\(171\) 0 0
\(172\) 40.0000 0.0177324
\(173\) −801.000 −0.352017 −0.176008 0.984389i \(-0.556319\pi\)
−0.176008 + 0.984389i \(0.556319\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −160.000 −0.0685253
\(177\) 0 0
\(178\) −1500.00 −0.631628
\(179\) −2360.00 −0.985445 −0.492723 0.870186i \(-0.663998\pi\)
−0.492723 + 0.870186i \(0.663998\pi\)
\(180\) 0 0
\(181\) 1241.00 0.509629 0.254814 0.966990i \(-0.417986\pi\)
0.254814 + 0.966990i \(0.417986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1944.00 −0.778878
\(185\) 0 0
\(186\) 0 0
\(187\) −70.0000 −0.0273738
\(188\) 640.000 0.248281
\(189\) 0 0
\(190\) 0 0
\(191\) −4990.00 −1.89039 −0.945193 0.326512i \(-0.894127\pi\)
−0.945193 + 0.326512i \(0.894127\pi\)
\(192\) 0 0
\(193\) 2260.00 0.842893 0.421447 0.906853i \(-0.361523\pi\)
0.421447 + 0.906853i \(0.361523\pi\)
\(194\) 2960.00 1.09544
\(195\) 0 0
\(196\) 1372.00 0.500000
\(197\) 2247.00 0.812650 0.406325 0.913729i \(-0.366810\pi\)
0.406325 + 0.913729i \(0.366810\pi\)
\(198\) 0 0
\(199\) 4564.00 1.62580 0.812898 0.582406i \(-0.197889\pi\)
0.812898 + 0.582406i \(0.197889\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −3000.00 −1.04495
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 920.000 0.311162
\(207\) 0 0
\(208\) −1280.00 −0.426692
\(209\) −1130.00 −0.373989
\(210\) 0 0
\(211\) 4949.00 1.61471 0.807354 0.590068i \(-0.200899\pi\)
0.807354 + 0.590068i \(0.200899\pi\)
\(212\) 2524.00 0.817684
\(213\) 0 0
\(214\) 840.000 0.268323
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1214.00 −0.377167
\(219\) 0 0
\(220\) 0 0
\(221\) −560.000 −0.170451
\(222\) 0 0
\(223\) −3890.00 −1.16813 −0.584067 0.811706i \(-0.698539\pi\)
−0.584067 + 0.811706i \(0.698539\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4340.00 −1.27740
\(227\) −2453.00 −0.717231 −0.358615 0.933485i \(-0.616751\pi\)
−0.358615 + 0.933485i \(0.616751\pi\)
\(228\) 0 0
\(229\) −6213.00 −1.79287 −0.896434 0.443178i \(-0.853851\pi\)
−0.896434 + 0.443178i \(0.853851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5280.00 1.49418
\(233\) 3450.00 0.970030 0.485015 0.874506i \(-0.338814\pi\)
0.485015 + 0.874506i \(0.338814\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2240.00 0.617846
\(237\) 0 0
\(238\) 0 0
\(239\) 6490.00 1.75650 0.878249 0.478203i \(-0.158712\pi\)
0.878249 + 0.478203i \(0.158712\pi\)
\(240\) 0 0
\(241\) −3401.00 −0.909036 −0.454518 0.890738i \(-0.650188\pi\)
−0.454518 + 0.890738i \(0.650188\pi\)
\(242\) −2462.00 −0.653981
\(243\) 0 0
\(244\) −916.000 −0.240332
\(245\) 0 0
\(246\) 0 0
\(247\) −9040.00 −2.32875
\(248\) 4536.00 1.16144
\(249\) 0 0
\(250\) 0 0
\(251\) −4980.00 −1.25233 −0.626165 0.779691i \(-0.715376\pi\)
−0.626165 + 0.779691i \(0.715376\pi\)
\(252\) 0 0
\(253\) 810.000 0.201282
\(254\) 3220.00 0.795436
\(255\) 0 0
\(256\) −4352.00 −1.06250
\(257\) −3357.00 −0.814801 −0.407401 0.913250i \(-0.633565\pi\)
−0.407401 + 0.913250i \(0.633565\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 4740.00 1.11770
\(263\) 4540.00 1.06444 0.532221 0.846605i \(-0.321357\pi\)
0.532221 + 0.846605i \(0.321357\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 3000.00 0.683784
\(269\) 8410.00 1.90620 0.953098 0.302662i \(-0.0978752\pi\)
0.953098 + 0.302662i \(0.0978752\pi\)
\(270\) 0 0
\(271\) 259.000 0.0580558 0.0290279 0.999579i \(-0.490759\pi\)
0.0290279 + 0.999579i \(0.490759\pi\)
\(272\) 112.000 0.0249669
\(273\) 0 0
\(274\) −3594.00 −0.792414
\(275\) 0 0
\(276\) 0 0
\(277\) 4170.00 0.904516 0.452258 0.891887i \(-0.350619\pi\)
0.452258 + 0.891887i \(0.350619\pi\)
\(278\) −248.000 −0.0535038
\(279\) 0 0
\(280\) 0 0
\(281\) −1740.00 −0.369394 −0.184697 0.982796i \(-0.559130\pi\)
−0.184697 + 0.982796i \(0.559130\pi\)
\(282\) 0 0
\(283\) 5070.00 1.06495 0.532474 0.846446i \(-0.321262\pi\)
0.532474 + 0.846446i \(0.321262\pi\)
\(284\) −3560.00 −0.743828
\(285\) 0 0
\(286\) 1600.00 0.330804
\(287\) 0 0
\(288\) 0 0
\(289\) −4864.00 −0.990026
\(290\) 0 0
\(291\) 0 0
\(292\) −3560.00 −0.713470
\(293\) 159.000 0.0317027 0.0158513 0.999874i \(-0.494954\pi\)
0.0158513 + 0.999874i \(0.494954\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4080.00 0.801166
\(297\) 0 0
\(298\) −140.000 −0.0272147
\(299\) 6480.00 1.25334
\(300\) 0 0
\(301\) 0 0
\(302\) 4496.00 0.856675
\(303\) 0 0
\(304\) 1808.00 0.341105
\(305\) 0 0
\(306\) 0 0
\(307\) −6490.00 −1.20653 −0.603264 0.797542i \(-0.706133\pi\)
−0.603264 + 0.797542i \(0.706133\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8220.00 −1.49876 −0.749379 0.662142i \(-0.769648\pi\)
−0.749379 + 0.662142i \(0.769648\pi\)
\(312\) 0 0
\(313\) 4660.00 0.841530 0.420765 0.907170i \(-0.361762\pi\)
0.420765 + 0.907170i \(0.361762\pi\)
\(314\) −2020.00 −0.363042
\(315\) 0 0
\(316\) 108.000 0.0192262
\(317\) 6817.00 1.20783 0.603913 0.797050i \(-0.293607\pi\)
0.603913 + 0.797050i \(0.293607\pi\)
\(318\) 0 0
\(319\) −2200.00 −0.386133
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 791.000 0.136261
\(324\) 0 0
\(325\) 0 0
\(326\) −1180.00 −0.200473
\(327\) 0 0
\(328\) 3120.00 0.525223
\(329\) 0 0
\(330\) 0 0
\(331\) 192.000 0.0318830 0.0159415 0.999873i \(-0.494925\pi\)
0.0159415 + 0.999873i \(0.494925\pi\)
\(332\) 1716.00 0.283668
\(333\) 0 0
\(334\) −4806.00 −0.787343
\(335\) 0 0
\(336\) 0 0
\(337\) −4840.00 −0.782349 −0.391174 0.920317i \(-0.627931\pi\)
−0.391174 + 0.920317i \(0.627931\pi\)
\(338\) 8406.00 1.35274
\(339\) 0 0
\(340\) 0 0
\(341\) −1890.00 −0.300144
\(342\) 0 0
\(343\) 0 0
\(344\) 240.000 0.0376161
\(345\) 0 0
\(346\) −1602.00 −0.248913
\(347\) 860.000 0.133047 0.0665234 0.997785i \(-0.478809\pi\)
0.0665234 + 0.997785i \(0.478809\pi\)
\(348\) 0 0
\(349\) 5377.00 0.824711 0.412356 0.911023i \(-0.364706\pi\)
0.412356 + 0.911023i \(0.364706\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1600.00 0.242274
\(353\) 8010.00 1.20773 0.603866 0.797086i \(-0.293626\pi\)
0.603866 + 0.797086i \(0.293626\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3000.00 0.446628
\(357\) 0 0
\(358\) −4720.00 −0.696815
\(359\) 12930.0 1.90089 0.950445 0.310894i \(-0.100628\pi\)
0.950445 + 0.310894i \(0.100628\pi\)
\(360\) 0 0
\(361\) 5910.00 0.861642
\(362\) 2482.00 0.360362
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6000.00 0.853399 0.426700 0.904393i \(-0.359676\pi\)
0.426700 + 0.904393i \(0.359676\pi\)
\(368\) −1296.00 −0.183583
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 140.000 0.0194341 0.00971706 0.999953i \(-0.496907\pi\)
0.00971706 + 0.999953i \(0.496907\pi\)
\(374\) −140.000 −0.0193562
\(375\) 0 0
\(376\) 3840.00 0.526683
\(377\) −17600.0 −2.40437
\(378\) 0 0
\(379\) 6217.00 0.842601 0.421301 0.906921i \(-0.361574\pi\)
0.421301 + 0.906921i \(0.361574\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −9980.00 −1.33670
\(383\) −4551.00 −0.607168 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4520.00 0.596015
\(387\) 0 0
\(388\) −5920.00 −0.774594
\(389\) −2310.00 −0.301084 −0.150542 0.988604i \(-0.548102\pi\)
−0.150542 + 0.988604i \(0.548102\pi\)
\(390\) 0 0
\(391\) −567.000 −0.0733361
\(392\) 8232.00 1.06066
\(393\) 0 0
\(394\) 4494.00 0.574631
\(395\) 0 0
\(396\) 0 0
\(397\) 2900.00 0.366617 0.183308 0.983055i \(-0.441319\pi\)
0.183308 + 0.983055i \(0.441319\pi\)
\(398\) 9128.00 1.14961
\(399\) 0 0
\(400\) 0 0
\(401\) 2250.00 0.280199 0.140099 0.990137i \(-0.455258\pi\)
0.140099 + 0.990137i \(0.455258\pi\)
\(402\) 0 0
\(403\) −15120.0 −1.86894
\(404\) 6000.00 0.738889
\(405\) 0 0
\(406\) 0 0
\(407\) −1700.00 −0.207041
\(408\) 0 0
\(409\) −11263.0 −1.36166 −0.680831 0.732441i \(-0.738381\pi\)
−0.680831 + 0.732441i \(0.738381\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1840.00 −0.220025
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 12800.0 1.50859
\(417\) 0 0
\(418\) −2260.00 −0.264450
\(419\) −6910.00 −0.805670 −0.402835 0.915273i \(-0.631975\pi\)
−0.402835 + 0.915273i \(0.631975\pi\)
\(420\) 0 0
\(421\) −5249.00 −0.607650 −0.303825 0.952728i \(-0.598264\pi\)
−0.303825 + 0.952728i \(0.598264\pi\)
\(422\) 9898.00 1.14177
\(423\) 0 0
\(424\) 15144.0 1.73457
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1680.00 −0.189733
\(429\) 0 0
\(430\) 0 0
\(431\) −11880.0 −1.32770 −0.663851 0.747865i \(-0.731079\pi\)
−0.663851 + 0.747865i \(0.731079\pi\)
\(432\) 0 0
\(433\) 4280.00 0.475020 0.237510 0.971385i \(-0.423669\pi\)
0.237510 + 0.971385i \(0.423669\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2428.00 0.266698
\(437\) −9153.00 −1.00194
\(438\) 0 0
\(439\) 6463.00 0.702647 0.351324 0.936254i \(-0.385732\pi\)
0.351324 + 0.936254i \(0.385732\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1120.00 −0.120527
\(443\) −11721.0 −1.25707 −0.628534 0.777782i \(-0.716345\pi\)
−0.628534 + 0.777782i \(0.716345\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7780.00 −0.825995
\(447\) 0 0
\(448\) 0 0
\(449\) −2180.00 −0.229133 −0.114566 0.993416i \(-0.536548\pi\)
−0.114566 + 0.993416i \(0.536548\pi\)
\(450\) 0 0
\(451\) −1300.00 −0.135731
\(452\) 8680.00 0.903259
\(453\) 0 0
\(454\) −4906.00 −0.507159
\(455\) 0 0
\(456\) 0 0
\(457\) 17840.0 1.82608 0.913042 0.407866i \(-0.133727\pi\)
0.913042 + 0.407866i \(0.133727\pi\)
\(458\) −12426.0 −1.26775
\(459\) 0 0
\(460\) 0 0
\(461\) −2250.00 −0.227317 −0.113658 0.993520i \(-0.536257\pi\)
−0.113658 + 0.993520i \(0.536257\pi\)
\(462\) 0 0
\(463\) −1230.00 −0.123462 −0.0617310 0.998093i \(-0.519662\pi\)
−0.0617310 + 0.998093i \(0.519662\pi\)
\(464\) 3520.00 0.352181
\(465\) 0 0
\(466\) 6900.00 0.685915
\(467\) −5813.00 −0.576003 −0.288002 0.957630i \(-0.592991\pi\)
−0.288002 + 0.957630i \(0.592991\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 13440.0 1.31065
\(473\) −100.000 −0.00972094
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 12980.0 1.24203
\(479\) −6750.00 −0.643873 −0.321937 0.946761i \(-0.604334\pi\)
−0.321937 + 0.946761i \(0.604334\pi\)
\(480\) 0 0
\(481\) −13600.0 −1.28920
\(482\) −6802.00 −0.642785
\(483\) 0 0
\(484\) 4924.00 0.462434
\(485\) 0 0
\(486\) 0 0
\(487\) 6610.00 0.615047 0.307523 0.951541i \(-0.400500\pi\)
0.307523 + 0.951541i \(0.400500\pi\)
\(488\) −5496.00 −0.509820
\(489\) 0 0
\(490\) 0 0
\(491\) −4990.00 −0.458647 −0.229323 0.973350i \(-0.573651\pi\)
−0.229323 + 0.973350i \(0.573651\pi\)
\(492\) 0 0
\(493\) 1540.00 0.140686
\(494\) −18080.0 −1.64668
\(495\) 0 0
\(496\) 3024.00 0.273753
\(497\) 0 0
\(498\) 0 0
\(499\) 1483.00 0.133042 0.0665212 0.997785i \(-0.478810\pi\)
0.0665212 + 0.997785i \(0.478810\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −9960.00 −0.885531
\(503\) 11641.0 1.03190 0.515951 0.856618i \(-0.327439\pi\)
0.515951 + 0.856618i \(0.327439\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1620.00 0.142328
\(507\) 0 0
\(508\) −6440.00 −0.562458
\(509\) 2620.00 0.228152 0.114076 0.993472i \(-0.463609\pi\)
0.114076 + 0.993472i \(0.463609\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5632.00 −0.486136
\(513\) 0 0
\(514\) −6714.00 −0.576151
\(515\) 0 0
\(516\) 0 0
\(517\) −1600.00 −0.136108
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13690.0 −1.15119 −0.575595 0.817735i \(-0.695229\pi\)
−0.575595 + 0.817735i \(0.695229\pi\)
\(522\) 0 0
\(523\) 10220.0 0.854473 0.427237 0.904140i \(-0.359487\pi\)
0.427237 + 0.904140i \(0.359487\pi\)
\(524\) −9480.00 −0.790335
\(525\) 0 0
\(526\) 9080.00 0.752675
\(527\) 1323.00 0.109356
\(528\) 0 0
\(529\) −5606.00 −0.460754
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10400.0 −0.845167
\(534\) 0 0
\(535\) 0 0
\(536\) 18000.0 1.45053
\(537\) 0 0
\(538\) 16820.0 1.34788
\(539\) −3430.00 −0.274101
\(540\) 0 0
\(541\) −2778.00 −0.220768 −0.110384 0.993889i \(-0.535208\pi\)
−0.110384 + 0.993889i \(0.535208\pi\)
\(542\) 518.000 0.0410517
\(543\) 0 0
\(544\) −1120.00 −0.0882713
\(545\) 0 0
\(546\) 0 0
\(547\) 12830.0 1.00287 0.501436 0.865195i \(-0.332805\pi\)
0.501436 + 0.865195i \(0.332805\pi\)
\(548\) 7188.00 0.560321
\(549\) 0 0
\(550\) 0 0
\(551\) 24860.0 1.92209
\(552\) 0 0
\(553\) 0 0
\(554\) 8340.00 0.639590
\(555\) 0 0
\(556\) 496.000 0.0378329
\(557\) 4950.00 0.376550 0.188275 0.982116i \(-0.439710\pi\)
0.188275 + 0.982116i \(0.439710\pi\)
\(558\) 0 0
\(559\) −800.000 −0.0605302
\(560\) 0 0
\(561\) 0 0
\(562\) −3480.00 −0.261201
\(563\) −6540.00 −0.489570 −0.244785 0.969577i \(-0.578717\pi\)
−0.244785 + 0.969577i \(0.578717\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10140.0 0.753032
\(567\) 0 0
\(568\) −21360.0 −1.57790
\(569\) −15240.0 −1.12284 −0.561418 0.827532i \(-0.689744\pi\)
−0.561418 + 0.827532i \(0.689744\pi\)
\(570\) 0 0
\(571\) −5281.00 −0.387045 −0.193523 0.981096i \(-0.561991\pi\)
−0.193523 + 0.981096i \(0.561991\pi\)
\(572\) −3200.00 −0.233914
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10510.0 0.758296 0.379148 0.925336i \(-0.376217\pi\)
0.379148 + 0.925336i \(0.376217\pi\)
\(578\) −9728.00 −0.700054
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6310.00 −0.448256
\(584\) −21360.0 −1.51350
\(585\) 0 0
\(586\) 318.000 0.0224172
\(587\) −4107.00 −0.288780 −0.144390 0.989521i \(-0.546122\pi\)
−0.144390 + 0.989521i \(0.546122\pi\)
\(588\) 0 0
\(589\) 21357.0 1.49406
\(590\) 0 0
\(591\) 0 0
\(592\) 2720.00 0.188837
\(593\) −26129.0 −1.80943 −0.904713 0.426022i \(-0.859915\pi\)
−0.904713 + 0.426022i \(0.859915\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 280.000 0.0192437
\(597\) 0 0
\(598\) 12960.0 0.886244
\(599\) −4360.00 −0.297404 −0.148702 0.988882i \(-0.547509\pi\)
−0.148702 + 0.988882i \(0.547509\pi\)
\(600\) 0 0
\(601\) −16639.0 −1.12932 −0.564658 0.825325i \(-0.690992\pi\)
−0.564658 + 0.825325i \(0.690992\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8992.00 −0.605760
\(605\) 0 0
\(606\) 0 0
\(607\) −490.000 −0.0327652 −0.0163826 0.999866i \(-0.505215\pi\)
−0.0163826 + 0.999866i \(0.505215\pi\)
\(608\) −18080.0 −1.20599
\(609\) 0 0
\(610\) 0 0
\(611\) −12800.0 −0.847516
\(612\) 0 0
\(613\) −18400.0 −1.21235 −0.606174 0.795332i \(-0.707296\pi\)
−0.606174 + 0.795332i \(0.707296\pi\)
\(614\) −12980.0 −0.853144
\(615\) 0 0
\(616\) 0 0
\(617\) −7827.00 −0.510702 −0.255351 0.966848i \(-0.582191\pi\)
−0.255351 + 0.966848i \(0.582191\pi\)
\(618\) 0 0
\(619\) −19756.0 −1.28281 −0.641406 0.767202i \(-0.721649\pi\)
−0.641406 + 0.767202i \(0.721649\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −16440.0 −1.05978
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 9320.00 0.595051
\(627\) 0 0
\(628\) 4040.00 0.256709
\(629\) 1190.00 0.0754347
\(630\) 0 0
\(631\) 9829.00 0.620105 0.310053 0.950719i \(-0.399653\pi\)
0.310053 + 0.950719i \(0.399653\pi\)
\(632\) 648.000 0.0407849
\(633\) 0 0
\(634\) 13634.0 0.854062
\(635\) 0 0
\(636\) 0 0
\(637\) −27440.0 −1.70677
\(638\) −4400.00 −0.273037
\(639\) 0 0
\(640\) 0 0
\(641\) −6000.00 −0.369713 −0.184856 0.982766i \(-0.559182\pi\)
−0.184856 + 0.982766i \(0.559182\pi\)
\(642\) 0 0
\(643\) −8280.00 −0.507825 −0.253912 0.967227i \(-0.581717\pi\)
−0.253912 + 0.967227i \(0.581717\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1582.00 0.0963513
\(647\) 16637.0 1.01092 0.505462 0.862849i \(-0.331322\pi\)
0.505462 + 0.862849i \(0.331322\pi\)
\(648\) 0 0
\(649\) −5600.00 −0.338705
\(650\) 0 0
\(651\) 0 0
\(652\) 2360.00 0.141756
\(653\) −19751.0 −1.18364 −0.591820 0.806070i \(-0.701590\pi\)
−0.591820 + 0.806070i \(0.701590\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2080.00 0.123796
\(657\) 0 0
\(658\) 0 0
\(659\) 14260.0 0.842930 0.421465 0.906845i \(-0.361516\pi\)
0.421465 + 0.906845i \(0.361516\pi\)
\(660\) 0 0
\(661\) 22318.0 1.31327 0.656634 0.754210i \(-0.271980\pi\)
0.656634 + 0.754210i \(0.271980\pi\)
\(662\) 384.000 0.0225447
\(663\) 0 0
\(664\) 10296.0 0.601750
\(665\) 0 0
\(666\) 0 0
\(667\) −17820.0 −1.03447
\(668\) 9612.00 0.556736
\(669\) 0 0
\(670\) 0 0
\(671\) 2290.00 0.131750
\(672\) 0 0
\(673\) −20040.0 −1.14782 −0.573912 0.818917i \(-0.694575\pi\)
−0.573912 + 0.818917i \(0.694575\pi\)
\(674\) −9680.00 −0.553204
\(675\) 0 0
\(676\) −16812.0 −0.956532
\(677\) 2310.00 0.131138 0.0655691 0.997848i \(-0.479114\pi\)
0.0655691 + 0.997848i \(0.479114\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −3780.00 −0.212234
\(683\) 26739.0 1.49801 0.749004 0.662566i \(-0.230532\pi\)
0.749004 + 0.662566i \(0.230532\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 160.000 0.00886620
\(689\) −50480.0 −2.79120
\(690\) 0 0
\(691\) 5101.00 0.280827 0.140413 0.990093i \(-0.455157\pi\)
0.140413 + 0.990093i \(0.455157\pi\)
\(692\) 3204.00 0.176008
\(693\) 0 0
\(694\) 1720.00 0.0940783
\(695\) 0 0
\(696\) 0 0
\(697\) 910.000 0.0494530
\(698\) 10754.0 0.583159
\(699\) 0 0
\(700\) 0 0
\(701\) 26030.0 1.40248 0.701241 0.712925i \(-0.252630\pi\)
0.701241 + 0.712925i \(0.252630\pi\)
\(702\) 0 0
\(703\) 19210.0 1.03061
\(704\) 4480.00 0.239839
\(705\) 0 0
\(706\) 16020.0 0.853995
\(707\) 0 0
\(708\) 0 0
\(709\) −3854.00 −0.204147 −0.102073 0.994777i \(-0.532548\pi\)
−0.102073 + 0.994777i \(0.532548\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18000.0 0.947442
\(713\) −15309.0 −0.804105
\(714\) 0 0
\(715\) 0 0
\(716\) 9440.00 0.492723
\(717\) 0 0
\(718\) 25860.0 1.34413
\(719\) −870.000 −0.0451259 −0.0225630 0.999745i \(-0.507183\pi\)
−0.0225630 + 0.999745i \(0.507183\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 11820.0 0.609273
\(723\) 0 0
\(724\) −4964.00 −0.254814
\(725\) 0 0
\(726\) 0 0
\(727\) −35780.0 −1.82532 −0.912659 0.408721i \(-0.865975\pi\)
−0.912659 + 0.408721i \(0.865975\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 70.0000 0.00354178
\(732\) 0 0
\(733\) 3400.00 0.171326 0.0856629 0.996324i \(-0.472699\pi\)
0.0856629 + 0.996324i \(0.472699\pi\)
\(734\) 12000.0 0.603444
\(735\) 0 0
\(736\) 12960.0 0.649065
\(737\) −7500.00 −0.374852
\(738\) 0 0
\(739\) −683.000 −0.0339981 −0.0169990 0.999856i \(-0.505411\pi\)
−0.0169990 + 0.999856i \(0.505411\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13400.0 0.661640 0.330820 0.943694i \(-0.392675\pi\)
0.330820 + 0.943694i \(0.392675\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 280.000 0.0137420
\(747\) 0 0
\(748\) 280.000 0.0136869
\(749\) 0 0
\(750\) 0 0
\(751\) −23219.0 −1.12819 −0.564097 0.825709i \(-0.690776\pi\)
−0.564097 + 0.825709i \(0.690776\pi\)
\(752\) 2560.00 0.124140
\(753\) 0 0
\(754\) −35200.0 −1.70014
\(755\) 0 0
\(756\) 0 0
\(757\) −19630.0 −0.942489 −0.471245 0.882003i \(-0.656195\pi\)
−0.471245 + 0.882003i \(0.656195\pi\)
\(758\) 12434.0 0.595809
\(759\) 0 0
\(760\) 0 0
\(761\) 2940.00 0.140046 0.0700229 0.997545i \(-0.477693\pi\)
0.0700229 + 0.997545i \(0.477693\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 19960.0 0.945193
\(765\) 0 0
\(766\) −9102.00 −0.429332
\(767\) −44800.0 −2.10904
\(768\) 0 0
\(769\) −13987.0 −0.655896 −0.327948 0.944696i \(-0.606357\pi\)
−0.327948 + 0.944696i \(0.606357\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9040.00 −0.421447
\(773\) −19839.0 −0.923104 −0.461552 0.887113i \(-0.652707\pi\)
−0.461552 + 0.887113i \(0.652707\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −35520.0 −1.64316
\(777\) 0 0
\(778\) −4620.00 −0.212898
\(779\) 14690.0 0.675640
\(780\) 0 0
\(781\) 8900.00 0.407768
\(782\) −1134.00 −0.0518565
\(783\) 0 0
\(784\) 5488.00 0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 38390.0 1.73883 0.869413 0.494086i \(-0.164497\pi\)
0.869413 + 0.494086i \(0.164497\pi\)
\(788\) −8988.00 −0.406325
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18320.0 0.820381
\(794\) 5800.00 0.259237
\(795\) 0 0
\(796\) −18256.0 −0.812898
\(797\) 28027.0 1.24563 0.622815 0.782369i \(-0.285989\pi\)
0.622815 + 0.782369i \(0.285989\pi\)
\(798\) 0 0
\(799\) 1120.00 0.0495904
\(800\) 0 0
\(801\) 0 0
\(802\) 4500.00 0.198130
\(803\) 8900.00 0.391126
\(804\) 0 0
\(805\) 0 0
\(806\) −30240.0 −1.32154
\(807\) 0 0
\(808\) 36000.0 1.56742
\(809\) 8630.00 0.375049 0.187525 0.982260i \(-0.439954\pi\)
0.187525 + 0.982260i \(0.439954\pi\)
\(810\) 0 0
\(811\) −1932.00 −0.0836519 −0.0418260 0.999125i \(-0.513317\pi\)
−0.0418260 + 0.999125i \(0.513317\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3400.00 −0.146400
\(815\) 0 0
\(816\) 0 0
\(817\) 1130.00 0.0483889
\(818\) −22526.0 −0.962840
\(819\) 0 0
\(820\) 0 0
\(821\) −18090.0 −0.768996 −0.384498 0.923126i \(-0.625625\pi\)
−0.384498 + 0.923126i \(0.625625\pi\)
\(822\) 0 0
\(823\) −12890.0 −0.545950 −0.272975 0.962021i \(-0.588008\pi\)
−0.272975 + 0.962021i \(0.588008\pi\)
\(824\) −11040.0 −0.466743
\(825\) 0 0
\(826\) 0 0
\(827\) −14887.0 −0.625963 −0.312982 0.949759i \(-0.601328\pi\)
−0.312982 + 0.949759i \(0.601328\pi\)
\(828\) 0 0
\(829\) 12666.0 0.530649 0.265325 0.964159i \(-0.414521\pi\)
0.265325 + 0.964159i \(0.414521\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 35840.0 1.49342
\(833\) 2401.00 0.0998676
\(834\) 0 0
\(835\) 0 0
\(836\) 4520.00 0.186995
\(837\) 0 0
\(838\) −13820.0 −0.569694
\(839\) 43820.0 1.80314 0.901570 0.432633i \(-0.142416\pi\)
0.901570 + 0.432633i \(0.142416\pi\)
\(840\) 0 0
\(841\) 24011.0 0.984501
\(842\) −10498.0 −0.429673
\(843\) 0 0
\(844\) −19796.0 −0.807354
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 10096.0 0.408842
\(849\) 0 0
\(850\) 0 0
\(851\) −13770.0 −0.554676
\(852\) 0 0
\(853\) −19320.0 −0.775503 −0.387752 0.921764i \(-0.626748\pi\)
−0.387752 + 0.921764i \(0.626748\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10080.0 −0.402485
\(857\) 3653.00 0.145606 0.0728029 0.997346i \(-0.476806\pi\)
0.0728029 + 0.997346i \(0.476806\pi\)
\(858\) 0 0
\(859\) 24373.0 0.968098 0.484049 0.875041i \(-0.339166\pi\)
0.484049 + 0.875041i \(0.339166\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −23760.0 −0.938827
\(863\) 17629.0 0.695363 0.347681 0.937613i \(-0.386969\pi\)
0.347681 + 0.937613i \(0.386969\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8560.00 0.335890
\(867\) 0 0
\(868\) 0 0
\(869\) −270.000 −0.0105398
\(870\) 0 0
\(871\) −60000.0 −2.33412
\(872\) 14568.0 0.565751
\(873\) 0 0
\(874\) −18306.0 −0.708478
\(875\) 0 0
\(876\) 0 0
\(877\) 21210.0 0.816660 0.408330 0.912834i \(-0.366111\pi\)
0.408330 + 0.912834i \(0.366111\pi\)
\(878\) 12926.0 0.496847
\(879\) 0 0
\(880\) 0 0
\(881\) 39340.0 1.50442 0.752212 0.658921i \(-0.228987\pi\)
0.752212 + 0.658921i \(0.228987\pi\)
\(882\) 0 0
\(883\) 4240.00 0.161594 0.0807969 0.996731i \(-0.474253\pi\)
0.0807969 + 0.996731i \(0.474253\pi\)
\(884\) 2240.00 0.0852255
\(885\) 0 0
\(886\) −23442.0 −0.888882
\(887\) 15933.0 0.603132 0.301566 0.953445i \(-0.402491\pi\)
0.301566 + 0.953445i \(0.402491\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 15560.0 0.584067
\(893\) 18080.0 0.677519
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −4360.00 −0.162021
\(899\) 41580.0 1.54257
\(900\) 0 0
\(901\) 4417.00 0.163320
\(902\) −2600.00 −0.0959762
\(903\) 0 0
\(904\) 52080.0 1.91610
\(905\) 0 0
\(906\) 0 0
\(907\) 6780.00 0.248210 0.124105 0.992269i \(-0.460394\pi\)
0.124105 + 0.992269i \(0.460394\pi\)
\(908\) 9812.00 0.358615
\(909\) 0 0
\(910\) 0 0
\(911\) −24740.0 −0.899751 −0.449875 0.893091i \(-0.648532\pi\)
−0.449875 + 0.893091i \(0.648532\pi\)
\(912\) 0 0
\(913\) −4290.00 −0.155507
\(914\) 35680.0 1.29124
\(915\) 0 0
\(916\) 24852.0 0.896434
\(917\) 0 0
\(918\) 0 0
\(919\) −48344.0 −1.73528 −0.867640 0.497194i \(-0.834364\pi\)
−0.867640 + 0.497194i \(0.834364\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −4500.00 −0.160737
\(923\) 71200.0 2.53909
\(924\) 0 0
\(925\) 0 0
\(926\) −2460.00 −0.0873009
\(927\) 0 0
\(928\) −35200.0 −1.24515
\(929\) −29650.0 −1.04713 −0.523566 0.851985i \(-0.675399\pi\)
−0.523566 + 0.851985i \(0.675399\pi\)
\(930\) 0 0
\(931\) 38759.0 1.36442
\(932\) −13800.0 −0.485015
\(933\) 0 0
\(934\) −11626.0 −0.407296
\(935\) 0 0
\(936\) 0 0
\(937\) −10260.0 −0.357716 −0.178858 0.983875i \(-0.557240\pi\)
−0.178858 + 0.983875i \(0.557240\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −16270.0 −0.563642 −0.281821 0.959467i \(-0.590938\pi\)
−0.281821 + 0.959467i \(0.590938\pi\)
\(942\) 0 0
\(943\) −10530.0 −0.363631
\(944\) 8960.00 0.308923
\(945\) 0 0
\(946\) −200.000 −0.00687374
\(947\) 23103.0 0.792763 0.396382 0.918086i \(-0.370266\pi\)
0.396382 + 0.918086i \(0.370266\pi\)
\(948\) 0 0
\(949\) 71200.0 2.43546
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 32090.0 1.09076 0.545381 0.838188i \(-0.316385\pi\)
0.545381 + 0.838188i \(0.316385\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −25960.0 −0.878249
\(957\) 0 0
\(958\) −13500.0 −0.455287
\(959\) 0 0
\(960\) 0 0
\(961\) 5930.00 0.199053
\(962\) −27200.0 −0.911604
\(963\) 0 0
\(964\) 13604.0 0.454518
\(965\) 0 0
\(966\) 0 0
\(967\) 42010.0 1.39705 0.698527 0.715584i \(-0.253839\pi\)
0.698527 + 0.715584i \(0.253839\pi\)
\(968\) 29544.0 0.980971
\(969\) 0 0
\(970\) 0 0
\(971\) 17490.0 0.578044 0.289022 0.957322i \(-0.406670\pi\)
0.289022 + 0.957322i \(0.406670\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 13220.0 0.434904
\(975\) 0 0
\(976\) −3664.00 −0.120166
\(977\) 22130.0 0.724669 0.362334 0.932048i \(-0.381980\pi\)
0.362334 + 0.932048i \(0.381980\pi\)
\(978\) 0 0
\(979\) −7500.00 −0.244843
\(980\) 0 0
\(981\) 0 0
\(982\) −9980.00 −0.324312
\(983\) −40959.0 −1.32898 −0.664491 0.747296i \(-0.731352\pi\)
−0.664491 + 0.747296i \(0.731352\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3080.00 0.0994799
\(987\) 0 0
\(988\) 36160.0 1.16438
\(989\) −810.000 −0.0260430
\(990\) 0 0
\(991\) 61169.0 1.96074 0.980372 0.197157i \(-0.0631707\pi\)
0.980372 + 0.197157i \(0.0631707\pi\)
\(992\) −30240.0 −0.967864
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −26190.0 −0.831941 −0.415971 0.909378i \(-0.636558\pi\)
−0.415971 + 0.909378i \(0.636558\pi\)
\(998\) 2966.00 0.0940752
\(999\) 0 0
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 675.4.a.i.1.1 1
3.2 odd 2 675.4.a.b.1.1 1
5.2 odd 4 675.4.b.d.649.2 2
5.3 odd 4 675.4.b.d.649.1 2
5.4 even 2 135.4.a.a.1.1 1
15.2 even 4 675.4.b.c.649.1 2
15.8 even 4 675.4.b.c.649.2 2
15.14 odd 2 135.4.a.d.1.1 yes 1
20.19 odd 2 2160.4.a.n.1.1 1
45.4 even 6 405.4.e.j.136.1 2
45.14 odd 6 405.4.e.e.136.1 2
45.29 odd 6 405.4.e.e.271.1 2
45.34 even 6 405.4.e.j.271.1 2
60.59 even 2 2160.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.a.1.1 1 5.4 even 2
135.4.a.d.1.1 yes 1 15.14 odd 2
405.4.e.e.136.1 2 45.14 odd 6
405.4.e.e.271.1 2 45.29 odd 6
405.4.e.j.136.1 2 45.4 even 6
405.4.e.j.271.1 2 45.34 even 6
675.4.a.b.1.1 1 3.2 odd 2
675.4.a.i.1.1 1 1.1 even 1 trivial
675.4.b.c.649.1 2 15.2 even 4
675.4.b.c.649.2 2 15.8 even 4
675.4.b.d.649.1 2 5.3 odd 4
675.4.b.d.649.2 2 5.2 odd 4
2160.4.a.d.1.1 1 60.59 even 2
2160.4.a.n.1.1 1 20.19 odd 2