Properties

Label 240.4.a.a.1.1
Level $240$
Weight $4$
Character 240.1
Self dual yes
Analytic conductor $14.160$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 240.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} -16.0000 q^{11} +58.0000 q^{13} +15.0000 q^{15} +38.0000 q^{17} -4.00000 q^{19} +60.0000 q^{21} +80.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +82.0000 q^{29} +8.00000 q^{31} +48.0000 q^{33} +100.000 q^{35} +426.000 q^{37} -174.000 q^{39} -246.000 q^{41} +524.000 q^{43} -45.0000 q^{45} +464.000 q^{47} +57.0000 q^{49} -114.000 q^{51} -702.000 q^{53} +80.0000 q^{55} +12.0000 q^{57} +592.000 q^{59} +574.000 q^{61} -180.000 q^{63} -290.000 q^{65} +172.000 q^{67} -240.000 q^{69} -768.000 q^{71} -558.000 q^{73} -75.0000 q^{75} +320.000 q^{77} -408.000 q^{79} +81.0000 q^{81} -164.000 q^{83} -190.000 q^{85} -246.000 q^{87} -510.000 q^{89} -1160.00 q^{91} -24.0000 q^{93} +20.0000 q^{95} +514.000 q^{97} -144.000 q^{99} +O(q^{100})\)

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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −16.0000 −0.438562 −0.219281 0.975662i \(-0.570371\pi\)
−0.219281 + 0.975662i \(0.570371\pi\)
\(12\) 0 0
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 38.0000 0.542138 0.271069 0.962560i \(-0.412623\pi\)
0.271069 + 0.962560i \(0.412623\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) 60.0000 0.623480
\(22\) 0 0
\(23\) 80.0000 0.725268 0.362634 0.931932i \(-0.381878\pi\)
0.362634 + 0.931932i \(0.381878\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 82.0000 0.525070 0.262535 0.964923i \(-0.415442\pi\)
0.262535 + 0.964923i \(0.415442\pi\)
\(30\) 0 0
\(31\) 8.00000 0.0463498 0.0231749 0.999731i \(-0.492623\pi\)
0.0231749 + 0.999731i \(0.492623\pi\)
\(32\) 0 0
\(33\) 48.0000 0.253204
\(34\) 0 0
\(35\) 100.000 0.482945
\(36\) 0 0
\(37\) 426.000 1.89281 0.946405 0.322982i \(-0.104685\pi\)
0.946405 + 0.322982i \(0.104685\pi\)
\(38\) 0 0
\(39\) −174.000 −0.714418
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 524.000 1.85835 0.929177 0.369634i \(-0.120517\pi\)
0.929177 + 0.369634i \(0.120517\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 464.000 1.44003 0.720014 0.693959i \(-0.244135\pi\)
0.720014 + 0.693959i \(0.244135\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) −114.000 −0.313004
\(52\) 0 0
\(53\) −702.000 −1.81938 −0.909690 0.415288i \(-0.863681\pi\)
−0.909690 + 0.415288i \(0.863681\pi\)
\(54\) 0 0
\(55\) 80.0000 0.196131
\(56\) 0 0
\(57\) 12.0000 0.0278849
\(58\) 0 0
\(59\) 592.000 1.30630 0.653151 0.757228i \(-0.273447\pi\)
0.653151 + 0.757228i \(0.273447\pi\)
\(60\) 0 0
\(61\) 574.000 1.20481 0.602403 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(62\) 0 0
\(63\) −180.000 −0.359966
\(64\) 0 0
\(65\) −290.000 −0.553386
\(66\) 0 0
\(67\) 172.000 0.313629 0.156815 0.987628i \(-0.449878\pi\)
0.156815 + 0.987628i \(0.449878\pi\)
\(68\) 0 0
\(69\) −240.000 −0.418733
\(70\) 0 0
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) 0 0
\(73\) −558.000 −0.894643 −0.447322 0.894373i \(-0.647622\pi\)
−0.447322 + 0.894373i \(0.647622\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 320.000 0.473602
\(78\) 0 0
\(79\) −408.000 −0.581058 −0.290529 0.956866i \(-0.593831\pi\)
−0.290529 + 0.956866i \(0.593831\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −164.000 −0.216884 −0.108442 0.994103i \(-0.534586\pi\)
−0.108442 + 0.994103i \(0.534586\pi\)
\(84\) 0 0
\(85\) −190.000 −0.242452
\(86\) 0 0
\(87\) −246.000 −0.303149
\(88\) 0 0
\(89\) −510.000 −0.607415 −0.303707 0.952765i \(-0.598224\pi\)
−0.303707 + 0.952765i \(0.598224\pi\)
\(90\) 0 0
\(91\) −1160.00 −1.33628
\(92\) 0 0
\(93\) −24.0000 −0.0267600
\(94\) 0 0
\(95\) 20.0000 0.0215995
\(96\) 0 0
\(97\) 514.000 0.538029 0.269014 0.963136i \(-0.413302\pi\)
0.269014 + 0.963136i \(0.413302\pi\)
\(98\) 0 0
\(99\) −144.000 −0.146187
\(100\) 0 0
\(101\) 666.000 0.656133 0.328067 0.944655i \(-0.393603\pi\)
0.328067 + 0.944655i \(0.393603\pi\)
\(102\) 0 0
\(103\) 1100.00 1.05229 0.526147 0.850394i \(-0.323636\pi\)
0.526147 + 0.850394i \(0.323636\pi\)
\(104\) 0 0
\(105\) −300.000 −0.278829
\(106\) 0 0
\(107\) −1212.00 −1.09503 −0.547516 0.836795i \(-0.684427\pi\)
−0.547516 + 0.836795i \(0.684427\pi\)
\(108\) 0 0
\(109\) 2078.00 1.82602 0.913011 0.407936i \(-0.133751\pi\)
0.913011 + 0.407936i \(0.133751\pi\)
\(110\) 0 0
\(111\) −1278.00 −1.09281
\(112\) 0 0
\(113\) −1458.00 −1.21378 −0.606890 0.794786i \(-0.707583\pi\)
−0.606890 + 0.794786i \(0.707583\pi\)
\(114\) 0 0
\(115\) −400.000 −0.324349
\(116\) 0 0
\(117\) 522.000 0.412469
\(118\) 0 0
\(119\) −760.000 −0.585455
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) 738.000 0.541002
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 2436.00 1.70205 0.851024 0.525127i \(-0.175982\pi\)
0.851024 + 0.525127i \(0.175982\pi\)
\(128\) 0 0
\(129\) −1572.00 −1.07292
\(130\) 0 0
\(131\) −2544.00 −1.69672 −0.848360 0.529420i \(-0.822410\pi\)
−0.848360 + 0.529420i \(0.822410\pi\)
\(132\) 0 0
\(133\) 80.0000 0.0521570
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 694.000 0.432791 0.216396 0.976306i \(-0.430570\pi\)
0.216396 + 0.976306i \(0.430570\pi\)
\(138\) 0 0
\(139\) −516.000 −0.314867 −0.157434 0.987530i \(-0.550322\pi\)
−0.157434 + 0.987530i \(0.550322\pi\)
\(140\) 0 0
\(141\) −1392.00 −0.831401
\(142\) 0 0
\(143\) −928.000 −0.542680
\(144\) 0 0
\(145\) −410.000 −0.234818
\(146\) 0 0
\(147\) −171.000 −0.0959445
\(148\) 0 0
\(149\) 770.000 0.423361 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(150\) 0 0
\(151\) 424.000 0.228507 0.114254 0.993452i \(-0.463552\pi\)
0.114254 + 0.993452i \(0.463552\pi\)
\(152\) 0 0
\(153\) 342.000 0.180713
\(154\) 0 0
\(155\) −40.0000 −0.0207282
\(156\) 0 0
\(157\) 922.000 0.468685 0.234343 0.972154i \(-0.424706\pi\)
0.234343 + 0.972154i \(0.424706\pi\)
\(158\) 0 0
\(159\) 2106.00 1.05042
\(160\) 0 0
\(161\) −1600.00 −0.783215
\(162\) 0 0
\(163\) 3788.00 1.82024 0.910120 0.414345i \(-0.135989\pi\)
0.910120 + 0.414345i \(0.135989\pi\)
\(164\) 0 0
\(165\) −240.000 −0.113236
\(166\) 0 0
\(167\) 48.0000 0.0222416 0.0111208 0.999938i \(-0.496460\pi\)
0.0111208 + 0.999938i \(0.496460\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) −36.0000 −0.0160993
\(172\) 0 0
\(173\) 3242.00 1.42477 0.712384 0.701790i \(-0.247616\pi\)
0.712384 + 0.701790i \(0.247616\pi\)
\(174\) 0 0
\(175\) −500.000 −0.215980
\(176\) 0 0
\(177\) −1776.00 −0.754194
\(178\) 0 0
\(179\) 2728.00 1.13911 0.569554 0.821954i \(-0.307116\pi\)
0.569554 + 0.821954i \(0.307116\pi\)
\(180\) 0 0
\(181\) −4090.00 −1.67960 −0.839799 0.542897i \(-0.817327\pi\)
−0.839799 + 0.542897i \(0.817327\pi\)
\(182\) 0 0
\(183\) −1722.00 −0.695595
\(184\) 0 0
\(185\) −2130.00 −0.846490
\(186\) 0 0
\(187\) −608.000 −0.237761
\(188\) 0 0
\(189\) 540.000 0.207827
\(190\) 0 0
\(191\) 1480.00 0.560676 0.280338 0.959901i \(-0.409554\pi\)
0.280338 + 0.959901i \(0.409554\pi\)
\(192\) 0 0
\(193\) −1622.00 −0.604944 −0.302472 0.953158i \(-0.597812\pi\)
−0.302472 + 0.953158i \(0.597812\pi\)
\(194\) 0 0
\(195\) 870.000 0.319497
\(196\) 0 0
\(197\) 2530.00 0.915000 0.457500 0.889210i \(-0.348745\pi\)
0.457500 + 0.889210i \(0.348745\pi\)
\(198\) 0 0
\(199\) 2440.00 0.869181 0.434590 0.900628i \(-0.356893\pi\)
0.434590 + 0.900628i \(0.356893\pi\)
\(200\) 0 0
\(201\) −516.000 −0.181074
\(202\) 0 0
\(203\) −1640.00 −0.567022
\(204\) 0 0
\(205\) 1230.00 0.419058
\(206\) 0 0
\(207\) 720.000 0.241756
\(208\) 0 0
\(209\) 64.0000 0.0211817
\(210\) 0 0
\(211\) 148.000 0.0482879 0.0241439 0.999708i \(-0.492314\pi\)
0.0241439 + 0.999708i \(0.492314\pi\)
\(212\) 0 0
\(213\) 2304.00 0.741162
\(214\) 0 0
\(215\) −2620.00 −0.831081
\(216\) 0 0
\(217\) −160.000 −0.0500530
\(218\) 0 0
\(219\) 1674.00 0.516523
\(220\) 0 0
\(221\) 2204.00 0.670847
\(222\) 0 0
\(223\) 676.000 0.202997 0.101498 0.994836i \(-0.467636\pi\)
0.101498 + 0.994836i \(0.467636\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 6276.00 1.83503 0.917517 0.397696i \(-0.130190\pi\)
0.917517 + 0.397696i \(0.130190\pi\)
\(228\) 0 0
\(229\) 6190.00 1.78623 0.893115 0.449828i \(-0.148515\pi\)
0.893115 + 0.449828i \(0.148515\pi\)
\(230\) 0 0
\(231\) −960.000 −0.273434
\(232\) 0 0
\(233\) 5406.00 1.52000 0.759998 0.649926i \(-0.225200\pi\)
0.759998 + 0.649926i \(0.225200\pi\)
\(234\) 0 0
\(235\) −2320.00 −0.644000
\(236\) 0 0
\(237\) 1224.00 0.335474
\(238\) 0 0
\(239\) 600.000 0.162388 0.0811941 0.996698i \(-0.474127\pi\)
0.0811941 + 0.996698i \(0.474127\pi\)
\(240\) 0 0
\(241\) −1054.00 −0.281718 −0.140859 0.990030i \(-0.544986\pi\)
−0.140859 + 0.990030i \(0.544986\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −285.000 −0.0743183
\(246\) 0 0
\(247\) −232.000 −0.0597644
\(248\) 0 0
\(249\) 492.000 0.125218
\(250\) 0 0
\(251\) 2232.00 0.561285 0.280643 0.959812i \(-0.409452\pi\)
0.280643 + 0.959812i \(0.409452\pi\)
\(252\) 0 0
\(253\) −1280.00 −0.318075
\(254\) 0 0
\(255\) 570.000 0.139980
\(256\) 0 0
\(257\) 3630.00 0.881063 0.440531 0.897737i \(-0.354790\pi\)
0.440531 + 0.897737i \(0.354790\pi\)
\(258\) 0 0
\(259\) −8520.00 −2.04404
\(260\) 0 0
\(261\) 738.000 0.175023
\(262\) 0 0
\(263\) −6960.00 −1.63183 −0.815916 0.578170i \(-0.803767\pi\)
−0.815916 + 0.578170i \(0.803767\pi\)
\(264\) 0 0
\(265\) 3510.00 0.813651
\(266\) 0 0
\(267\) 1530.00 0.350691
\(268\) 0 0
\(269\) −2062.00 −0.467369 −0.233685 0.972312i \(-0.575078\pi\)
−0.233685 + 0.972312i \(0.575078\pi\)
\(270\) 0 0
\(271\) 2544.00 0.570247 0.285124 0.958491i \(-0.407965\pi\)
0.285124 + 0.958491i \(0.407965\pi\)
\(272\) 0 0
\(273\) 3480.00 0.771499
\(274\) 0 0
\(275\) −400.000 −0.0877124
\(276\) 0 0
\(277\) −694.000 −0.150536 −0.0752679 0.997163i \(-0.523981\pi\)
−0.0752679 + 0.997163i \(0.523981\pi\)
\(278\) 0 0
\(279\) 72.0000 0.0154499
\(280\) 0 0
\(281\) −1982.00 −0.420769 −0.210385 0.977619i \(-0.567472\pi\)
−0.210385 + 0.977619i \(0.567472\pi\)
\(282\) 0 0
\(283\) −5228.00 −1.09814 −0.549068 0.835778i \(-0.685017\pi\)
−0.549068 + 0.835778i \(0.685017\pi\)
\(284\) 0 0
\(285\) −60.0000 −0.0124705
\(286\) 0 0
\(287\) 4920.00 1.01191
\(288\) 0 0
\(289\) −3469.00 −0.706086
\(290\) 0 0
\(291\) −1542.00 −0.310631
\(292\) 0 0
\(293\) −7454.00 −1.48624 −0.743118 0.669160i \(-0.766654\pi\)
−0.743118 + 0.669160i \(0.766654\pi\)
\(294\) 0 0
\(295\) −2960.00 −0.584196
\(296\) 0 0
\(297\) 432.000 0.0844013
\(298\) 0 0
\(299\) 4640.00 0.897452
\(300\) 0 0
\(301\) −10480.0 −2.00683
\(302\) 0 0
\(303\) −1998.00 −0.378819
\(304\) 0 0
\(305\) −2870.00 −0.538806
\(306\) 0 0
\(307\) 1316.00 0.244652 0.122326 0.992490i \(-0.460965\pi\)
0.122326 + 0.992490i \(0.460965\pi\)
\(308\) 0 0
\(309\) −3300.00 −0.607542
\(310\) 0 0
\(311\) 832.000 0.151699 0.0758495 0.997119i \(-0.475833\pi\)
0.0758495 + 0.997119i \(0.475833\pi\)
\(312\) 0 0
\(313\) 6770.00 1.22257 0.611283 0.791412i \(-0.290654\pi\)
0.611283 + 0.791412i \(0.290654\pi\)
\(314\) 0 0
\(315\) 900.000 0.160982
\(316\) 0 0
\(317\) −6582.00 −1.16619 −0.583095 0.812404i \(-0.698158\pi\)
−0.583095 + 0.812404i \(0.698158\pi\)
\(318\) 0 0
\(319\) −1312.00 −0.230276
\(320\) 0 0
\(321\) 3636.00 0.632217
\(322\) 0 0
\(323\) −152.000 −0.0261842
\(324\) 0 0
\(325\) 1450.00 0.247482
\(326\) 0 0
\(327\) −6234.00 −1.05425
\(328\) 0 0
\(329\) −9280.00 −1.55508
\(330\) 0 0
\(331\) −11292.0 −1.87512 −0.937560 0.347825i \(-0.886920\pi\)
−0.937560 + 0.347825i \(0.886920\pi\)
\(332\) 0 0
\(333\) 3834.00 0.630937
\(334\) 0 0
\(335\) −860.000 −0.140259
\(336\) 0 0
\(337\) −8006.00 −1.29411 −0.647054 0.762444i \(-0.723999\pi\)
−0.647054 + 0.762444i \(0.723999\pi\)
\(338\) 0 0
\(339\) 4374.00 0.700776
\(340\) 0 0
\(341\) −128.000 −0.0203272
\(342\) 0 0
\(343\) 5720.00 0.900440
\(344\) 0 0
\(345\) 1200.00 0.187263
\(346\) 0 0
\(347\) 316.000 0.0488869 0.0244435 0.999701i \(-0.492219\pi\)
0.0244435 + 0.999701i \(0.492219\pi\)
\(348\) 0 0
\(349\) 4926.00 0.755538 0.377769 0.925900i \(-0.376691\pi\)
0.377769 + 0.925900i \(0.376691\pi\)
\(350\) 0 0
\(351\) −1566.00 −0.238139
\(352\) 0 0
\(353\) 2438.00 0.367597 0.183798 0.982964i \(-0.441161\pi\)
0.183798 + 0.982964i \(0.441161\pi\)
\(354\) 0 0
\(355\) 3840.00 0.574102
\(356\) 0 0
\(357\) 2280.00 0.338012
\(358\) 0 0
\(359\) 3336.00 0.490438 0.245219 0.969468i \(-0.421140\pi\)
0.245219 + 0.969468i \(0.421140\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 3225.00 0.466305
\(364\) 0 0
\(365\) 2790.00 0.400097
\(366\) 0 0
\(367\) −44.0000 −0.00625826 −0.00312913 0.999995i \(-0.500996\pi\)
−0.00312913 + 0.999995i \(0.500996\pi\)
\(368\) 0 0
\(369\) −2214.00 −0.312348
\(370\) 0 0
\(371\) 14040.0 1.96475
\(372\) 0 0
\(373\) −11966.0 −1.66106 −0.830531 0.556973i \(-0.811963\pi\)
−0.830531 + 0.556973i \(0.811963\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 4756.00 0.649725
\(378\) 0 0
\(379\) −12676.0 −1.71800 −0.859001 0.511975i \(-0.828914\pi\)
−0.859001 + 0.511975i \(0.828914\pi\)
\(380\) 0 0
\(381\) −7308.00 −0.982678
\(382\) 0 0
\(383\) −6672.00 −0.890139 −0.445070 0.895496i \(-0.646821\pi\)
−0.445070 + 0.895496i \(0.646821\pi\)
\(384\) 0 0
\(385\) −1600.00 −0.211801
\(386\) 0 0
\(387\) 4716.00 0.619452
\(388\) 0 0
\(389\) 354.000 0.0461401 0.0230701 0.999734i \(-0.492656\pi\)
0.0230701 + 0.999734i \(0.492656\pi\)
\(390\) 0 0
\(391\) 3040.00 0.393195
\(392\) 0 0
\(393\) 7632.00 0.979602
\(394\) 0 0
\(395\) 2040.00 0.259857
\(396\) 0 0
\(397\) −5054.00 −0.638924 −0.319462 0.947599i \(-0.603502\pi\)
−0.319462 + 0.947599i \(0.603502\pi\)
\(398\) 0 0
\(399\) −240.000 −0.0301129
\(400\) 0 0
\(401\) 10266.0 1.27845 0.639226 0.769019i \(-0.279255\pi\)
0.639226 + 0.769019i \(0.279255\pi\)
\(402\) 0 0
\(403\) 464.000 0.0573536
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) −6816.00 −0.830114
\(408\) 0 0
\(409\) −1526.00 −0.184489 −0.0922443 0.995736i \(-0.529404\pi\)
−0.0922443 + 0.995736i \(0.529404\pi\)
\(410\) 0 0
\(411\) −2082.00 −0.249872
\(412\) 0 0
\(413\) −11840.0 −1.41067
\(414\) 0 0
\(415\) 820.000 0.0969933
\(416\) 0 0
\(417\) 1548.00 0.181789
\(418\) 0 0
\(419\) −2064.00 −0.240652 −0.120326 0.992734i \(-0.538394\pi\)
−0.120326 + 0.992734i \(0.538394\pi\)
\(420\) 0 0
\(421\) 4590.00 0.531361 0.265680 0.964061i \(-0.414403\pi\)
0.265680 + 0.964061i \(0.414403\pi\)
\(422\) 0 0
\(423\) 4176.00 0.480010
\(424\) 0 0
\(425\) 950.000 0.108428
\(426\) 0 0
\(427\) −11480.0 −1.30107
\(428\) 0 0
\(429\) 2784.00 0.313317
\(430\) 0 0
\(431\) 5536.00 0.618700 0.309350 0.950948i \(-0.399889\pi\)
0.309350 + 0.950948i \(0.399889\pi\)
\(432\) 0 0
\(433\) 1850.00 0.205324 0.102662 0.994716i \(-0.467264\pi\)
0.102662 + 0.994716i \(0.467264\pi\)
\(434\) 0 0
\(435\) 1230.00 0.135572
\(436\) 0 0
\(437\) −320.000 −0.0350290
\(438\) 0 0
\(439\) −11704.0 −1.27244 −0.636220 0.771507i \(-0.719503\pi\)
−0.636220 + 0.771507i \(0.719503\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 0 0
\(443\) −6948.00 −0.745168 −0.372584 0.927998i \(-0.621528\pi\)
−0.372584 + 0.927998i \(0.621528\pi\)
\(444\) 0 0
\(445\) 2550.00 0.271644
\(446\) 0 0
\(447\) −2310.00 −0.244428
\(448\) 0 0
\(449\) 12090.0 1.27074 0.635370 0.772208i \(-0.280848\pi\)
0.635370 + 0.772208i \(0.280848\pi\)
\(450\) 0 0
\(451\) 3936.00 0.410951
\(452\) 0 0
\(453\) −1272.00 −0.131929
\(454\) 0 0
\(455\) 5800.00 0.597600
\(456\) 0 0
\(457\) 11626.0 1.19002 0.595012 0.803717i \(-0.297147\pi\)
0.595012 + 0.803717i \(0.297147\pi\)
\(458\) 0 0
\(459\) −1026.00 −0.104335
\(460\) 0 0
\(461\) 16314.0 1.64820 0.824098 0.566447i \(-0.191682\pi\)
0.824098 + 0.566447i \(0.191682\pi\)
\(462\) 0 0
\(463\) 15756.0 1.58152 0.790760 0.612127i \(-0.209686\pi\)
0.790760 + 0.612127i \(0.209686\pi\)
\(464\) 0 0
\(465\) 120.000 0.0119675
\(466\) 0 0
\(467\) −5684.00 −0.563221 −0.281610 0.959529i \(-0.590869\pi\)
−0.281610 + 0.959529i \(0.590869\pi\)
\(468\) 0 0
\(469\) −3440.00 −0.338688
\(470\) 0 0
\(471\) −2766.00 −0.270596
\(472\) 0 0
\(473\) −8384.00 −0.815004
\(474\) 0 0
\(475\) −100.000 −0.00965961
\(476\) 0 0
\(477\) −6318.00 −0.606460
\(478\) 0 0
\(479\) 3368.00 0.321269 0.160634 0.987014i \(-0.448646\pi\)
0.160634 + 0.987014i \(0.448646\pi\)
\(480\) 0 0
\(481\) 24708.0 2.34218
\(482\) 0 0
\(483\) 4800.00 0.452190
\(484\) 0 0
\(485\) −2570.00 −0.240614
\(486\) 0 0
\(487\) 5588.00 0.519952 0.259976 0.965615i \(-0.416285\pi\)
0.259976 + 0.965615i \(0.416285\pi\)
\(488\) 0 0
\(489\) −11364.0 −1.05092
\(490\) 0 0
\(491\) −10584.0 −0.972809 −0.486405 0.873734i \(-0.661692\pi\)
−0.486405 + 0.873734i \(0.661692\pi\)
\(492\) 0 0
\(493\) 3116.00 0.284660
\(494\) 0 0
\(495\) 720.000 0.0653770
\(496\) 0 0
\(497\) 15360.0 1.38630
\(498\) 0 0
\(499\) 12220.0 1.09628 0.548139 0.836388i \(-0.315337\pi\)
0.548139 + 0.836388i \(0.315337\pi\)
\(500\) 0 0
\(501\) −144.000 −0.0128412
\(502\) 0 0
\(503\) −16152.0 −1.43177 −0.715887 0.698216i \(-0.753977\pi\)
−0.715887 + 0.698216i \(0.753977\pi\)
\(504\) 0 0
\(505\) −3330.00 −0.293432
\(506\) 0 0
\(507\) −3501.00 −0.306676
\(508\) 0 0
\(509\) 10642.0 0.926716 0.463358 0.886171i \(-0.346644\pi\)
0.463358 + 0.886171i \(0.346644\pi\)
\(510\) 0 0
\(511\) 11160.0 0.966124
\(512\) 0 0
\(513\) 108.000 0.00929496
\(514\) 0 0
\(515\) −5500.00 −0.470600
\(516\) 0 0
\(517\) −7424.00 −0.631542
\(518\) 0 0
\(519\) −9726.00 −0.822590
\(520\) 0 0
\(521\) 22882.0 1.92414 0.962072 0.272797i \(-0.0879487\pi\)
0.962072 + 0.272797i \(0.0879487\pi\)
\(522\) 0 0
\(523\) 10052.0 0.840427 0.420213 0.907425i \(-0.361955\pi\)
0.420213 + 0.907425i \(0.361955\pi\)
\(524\) 0 0
\(525\) 1500.00 0.124696
\(526\) 0 0
\(527\) 304.000 0.0251280
\(528\) 0 0
\(529\) −5767.00 −0.473987
\(530\) 0 0
\(531\) 5328.00 0.435434
\(532\) 0 0
\(533\) −14268.0 −1.15950
\(534\) 0 0
\(535\) 6060.00 0.489713
\(536\) 0 0
\(537\) −8184.00 −0.657664
\(538\) 0 0
\(539\) −912.000 −0.0728806
\(540\) 0 0
\(541\) −6530.00 −0.518940 −0.259470 0.965751i \(-0.583548\pi\)
−0.259470 + 0.965751i \(0.583548\pi\)
\(542\) 0 0
\(543\) 12270.0 0.969717
\(544\) 0 0
\(545\) −10390.0 −0.816621
\(546\) 0 0
\(547\) −16652.0 −1.30162 −0.650812 0.759239i \(-0.725571\pi\)
−0.650812 + 0.759239i \(0.725571\pi\)
\(548\) 0 0
\(549\) 5166.00 0.401602
\(550\) 0 0
\(551\) −328.000 −0.0253598
\(552\) 0 0
\(553\) 8160.00 0.627484
\(554\) 0 0
\(555\) 6390.00 0.488721
\(556\) 0 0
\(557\) −12886.0 −0.980247 −0.490123 0.871653i \(-0.663048\pi\)
−0.490123 + 0.871653i \(0.663048\pi\)
\(558\) 0 0
\(559\) 30392.0 2.29954
\(560\) 0 0
\(561\) 1824.00 0.137272
\(562\) 0 0
\(563\) 11108.0 0.831521 0.415761 0.909474i \(-0.363515\pi\)
0.415761 + 0.909474i \(0.363515\pi\)
\(564\) 0 0
\(565\) 7290.00 0.542819
\(566\) 0 0
\(567\) −1620.00 −0.119989
\(568\) 0 0
\(569\) −9214.00 −0.678859 −0.339430 0.940631i \(-0.610234\pi\)
−0.339430 + 0.940631i \(0.610234\pi\)
\(570\) 0 0
\(571\) 4052.00 0.296972 0.148486 0.988915i \(-0.452560\pi\)
0.148486 + 0.988915i \(0.452560\pi\)
\(572\) 0 0
\(573\) −4440.00 −0.323706
\(574\) 0 0
\(575\) 2000.00 0.145054
\(576\) 0 0
\(577\) −8446.00 −0.609379 −0.304689 0.952452i \(-0.598553\pi\)
−0.304689 + 0.952452i \(0.598553\pi\)
\(578\) 0 0
\(579\) 4866.00 0.349264
\(580\) 0 0
\(581\) 3280.00 0.234212
\(582\) 0 0
\(583\) 11232.0 0.797911
\(584\) 0 0
\(585\) −2610.00 −0.184462
\(586\) 0 0
\(587\) −2172.00 −0.152722 −0.0763612 0.997080i \(-0.524330\pi\)
−0.0763612 + 0.997080i \(0.524330\pi\)
\(588\) 0 0
\(589\) −32.0000 −0.00223860
\(590\) 0 0
\(591\) −7590.00 −0.528276
\(592\) 0 0
\(593\) −1218.00 −0.0843461 −0.0421731 0.999110i \(-0.513428\pi\)
−0.0421731 + 0.999110i \(0.513428\pi\)
\(594\) 0 0
\(595\) 3800.00 0.261823
\(596\) 0 0
\(597\) −7320.00 −0.501822
\(598\) 0 0
\(599\) −21240.0 −1.44882 −0.724410 0.689370i \(-0.757888\pi\)
−0.724410 + 0.689370i \(0.757888\pi\)
\(600\) 0 0
\(601\) 17626.0 1.19631 0.598153 0.801382i \(-0.295902\pi\)
0.598153 + 0.801382i \(0.295902\pi\)
\(602\) 0 0
\(603\) 1548.00 0.104543
\(604\) 0 0
\(605\) 5375.00 0.361198
\(606\) 0 0
\(607\) −2580.00 −0.172519 −0.0862594 0.996273i \(-0.527491\pi\)
−0.0862594 + 0.996273i \(0.527491\pi\)
\(608\) 0 0
\(609\) 4920.00 0.327370
\(610\) 0 0
\(611\) 26912.0 1.78190
\(612\) 0 0
\(613\) −14166.0 −0.933376 −0.466688 0.884422i \(-0.654553\pi\)
−0.466688 + 0.884422i \(0.654553\pi\)
\(614\) 0 0
\(615\) −3690.00 −0.241943
\(616\) 0 0
\(617\) −21426.0 −1.39802 −0.699010 0.715112i \(-0.746376\pi\)
−0.699010 + 0.715112i \(0.746376\pi\)
\(618\) 0 0
\(619\) −3668.00 −0.238173 −0.119087 0.992884i \(-0.537997\pi\)
−0.119087 + 0.992884i \(0.537997\pi\)
\(620\) 0 0
\(621\) −2160.00 −0.139578
\(622\) 0 0
\(623\) 10200.0 0.655946
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −192.000 −0.0122293
\(628\) 0 0
\(629\) 16188.0 1.02617
\(630\) 0 0
\(631\) −20032.0 −1.26381 −0.631903 0.775048i \(-0.717726\pi\)
−0.631903 + 0.775048i \(0.717726\pi\)
\(632\) 0 0
\(633\) −444.000 −0.0278790
\(634\) 0 0
\(635\) −12180.0 −0.761179
\(636\) 0 0
\(637\) 3306.00 0.205633
\(638\) 0 0
\(639\) −6912.00 −0.427910
\(640\) 0 0
\(641\) 7458.00 0.459553 0.229776 0.973243i \(-0.426201\pi\)
0.229776 + 0.973243i \(0.426201\pi\)
\(642\) 0 0
\(643\) −7092.00 −0.434963 −0.217481 0.976064i \(-0.569784\pi\)
−0.217481 + 0.976064i \(0.569784\pi\)
\(644\) 0 0
\(645\) 7860.00 0.479825
\(646\) 0 0
\(647\) −3384.00 −0.205624 −0.102812 0.994701i \(-0.532784\pi\)
−0.102812 + 0.994701i \(0.532784\pi\)
\(648\) 0 0
\(649\) −9472.00 −0.572894
\(650\) 0 0
\(651\) 480.000 0.0288981
\(652\) 0 0
\(653\) −29398.0 −1.76177 −0.880883 0.473335i \(-0.843050\pi\)
−0.880883 + 0.473335i \(0.843050\pi\)
\(654\) 0 0
\(655\) 12720.0 0.758796
\(656\) 0 0
\(657\) −5022.00 −0.298214
\(658\) 0 0
\(659\) 6624.00 0.391554 0.195777 0.980648i \(-0.437277\pi\)
0.195777 + 0.980648i \(0.437277\pi\)
\(660\) 0 0
\(661\) 8646.00 0.508760 0.254380 0.967104i \(-0.418129\pi\)
0.254380 + 0.967104i \(0.418129\pi\)
\(662\) 0 0
\(663\) −6612.00 −0.387313
\(664\) 0 0
\(665\) −400.000 −0.0233253
\(666\) 0 0
\(667\) 6560.00 0.380816
\(668\) 0 0
\(669\) −2028.00 −0.117200
\(670\) 0 0
\(671\) −9184.00 −0.528382
\(672\) 0 0
\(673\) 28698.0 1.64372 0.821862 0.569686i \(-0.192935\pi\)
0.821862 + 0.569686i \(0.192935\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 19426.0 1.10281 0.551405 0.834238i \(-0.314092\pi\)
0.551405 + 0.834238i \(0.314092\pi\)
\(678\) 0 0
\(679\) −10280.0 −0.581016
\(680\) 0 0
\(681\) −18828.0 −1.05946
\(682\) 0 0
\(683\) −8604.00 −0.482025 −0.241012 0.970522i \(-0.577479\pi\)
−0.241012 + 0.970522i \(0.577479\pi\)
\(684\) 0 0
\(685\) −3470.00 −0.193550
\(686\) 0 0
\(687\) −18570.0 −1.03128
\(688\) 0 0
\(689\) −40716.0 −2.25132
\(690\) 0 0
\(691\) −12980.0 −0.714591 −0.357296 0.933991i \(-0.616301\pi\)
−0.357296 + 0.933991i \(0.616301\pi\)
\(692\) 0 0
\(693\) 2880.00 0.157867
\(694\) 0 0
\(695\) 2580.00 0.140813
\(696\) 0 0
\(697\) −9348.00 −0.508007
\(698\) 0 0
\(699\) −16218.0 −0.877570
\(700\) 0 0
\(701\) −19630.0 −1.05765 −0.528827 0.848730i \(-0.677368\pi\)
−0.528827 + 0.848730i \(0.677368\pi\)
\(702\) 0 0
\(703\) −1704.00 −0.0914190
\(704\) 0 0
\(705\) 6960.00 0.371814
\(706\) 0 0
\(707\) −13320.0 −0.708558
\(708\) 0 0
\(709\) 8030.00 0.425350 0.212675 0.977123i \(-0.431782\pi\)
0.212675 + 0.977123i \(0.431782\pi\)
\(710\) 0 0
\(711\) −3672.00 −0.193686
\(712\) 0 0
\(713\) 640.000 0.0336160
\(714\) 0 0
\(715\) 4640.00 0.242694
\(716\) 0 0
\(717\) −1800.00 −0.0937549
\(718\) 0 0
\(719\) −22720.0 −1.17846 −0.589230 0.807965i \(-0.700569\pi\)
−0.589230 + 0.807965i \(0.700569\pi\)
\(720\) 0 0
\(721\) −22000.0 −1.13637
\(722\) 0 0
\(723\) 3162.00 0.162650
\(724\) 0 0
\(725\) 2050.00 0.105014
\(726\) 0 0
\(727\) −27116.0 −1.38332 −0.691662 0.722221i \(-0.743121\pi\)
−0.691662 + 0.722221i \(0.743121\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19912.0 1.00749
\(732\) 0 0
\(733\) 30882.0 1.55614 0.778071 0.628176i \(-0.216198\pi\)
0.778071 + 0.628176i \(0.216198\pi\)
\(734\) 0 0
\(735\) 855.000 0.0429077
\(736\) 0 0
\(737\) −2752.00 −0.137546
\(738\) 0 0
\(739\) 13836.0 0.688722 0.344361 0.938837i \(-0.388096\pi\)
0.344361 + 0.938837i \(0.388096\pi\)
\(740\) 0 0
\(741\) 696.000 0.0345050
\(742\) 0 0
\(743\) −32712.0 −1.61519 −0.807595 0.589737i \(-0.799231\pi\)
−0.807595 + 0.589737i \(0.799231\pi\)
\(744\) 0 0
\(745\) −3850.00 −0.189333
\(746\) 0 0
\(747\) −1476.00 −0.0722945
\(748\) 0 0
\(749\) 24240.0 1.18252
\(750\) 0 0
\(751\) 8472.00 0.411648 0.205824 0.978589i \(-0.434013\pi\)
0.205824 + 0.978589i \(0.434013\pi\)
\(752\) 0 0
\(753\) −6696.00 −0.324058
\(754\) 0 0
\(755\) −2120.00 −0.102192
\(756\) 0 0
\(757\) 9866.00 0.473693 0.236847 0.971547i \(-0.423886\pi\)
0.236847 + 0.971547i \(0.423886\pi\)
\(758\) 0 0
\(759\) 3840.00 0.183641
\(760\) 0 0
\(761\) −3774.00 −0.179773 −0.0898866 0.995952i \(-0.528650\pi\)
−0.0898866 + 0.995952i \(0.528650\pi\)
\(762\) 0 0
\(763\) −41560.0 −1.97192
\(764\) 0 0
\(765\) −1710.00 −0.0808172
\(766\) 0 0
\(767\) 34336.0 1.61643
\(768\) 0 0
\(769\) −28670.0 −1.34443 −0.672215 0.740356i \(-0.734657\pi\)
−0.672215 + 0.740356i \(0.734657\pi\)
\(770\) 0 0
\(771\) −10890.0 −0.508682
\(772\) 0 0
\(773\) −3246.00 −0.151036 −0.0755178 0.997144i \(-0.524061\pi\)
−0.0755178 + 0.997144i \(0.524061\pi\)
\(774\) 0 0
\(775\) 200.000 0.00926995
\(776\) 0 0
\(777\) 25560.0 1.18013
\(778\) 0 0
\(779\) 984.000 0.0452573
\(780\) 0 0
\(781\) 12288.0 0.562995
\(782\) 0 0
\(783\) −2214.00 −0.101050
\(784\) 0 0
\(785\) −4610.00 −0.209602
\(786\) 0 0
\(787\) 19372.0 0.877430 0.438715 0.898626i \(-0.355434\pi\)
0.438715 + 0.898626i \(0.355434\pi\)
\(788\) 0 0
\(789\) 20880.0 0.942139
\(790\) 0 0
\(791\) 29160.0 1.31076
\(792\) 0 0
\(793\) 33292.0 1.49084
\(794\) 0 0
\(795\) −10530.0 −0.469762
\(796\) 0 0
\(797\) −11814.0 −0.525061 −0.262530 0.964924i \(-0.584557\pi\)
−0.262530 + 0.964924i \(0.584557\pi\)
\(798\) 0 0
\(799\) 17632.0 0.780695
\(800\) 0 0
\(801\) −4590.00 −0.202472
\(802\) 0 0
\(803\) 8928.00 0.392357
\(804\) 0 0
\(805\) 8000.00 0.350265
\(806\) 0 0
\(807\) 6186.00 0.269836
\(808\) 0 0
\(809\) −30054.0 −1.30611 −0.653055 0.757311i \(-0.726513\pi\)
−0.653055 + 0.757311i \(0.726513\pi\)
\(810\) 0 0
\(811\) −2852.00 −0.123486 −0.0617431 0.998092i \(-0.519666\pi\)
−0.0617431 + 0.998092i \(0.519666\pi\)
\(812\) 0 0
\(813\) −7632.00 −0.329232
\(814\) 0 0
\(815\) −18940.0 −0.814036
\(816\) 0 0
\(817\) −2096.00 −0.0897549
\(818\) 0 0
\(819\) −10440.0 −0.445425
\(820\) 0 0
\(821\) 2170.00 0.0922455 0.0461227 0.998936i \(-0.485313\pi\)
0.0461227 + 0.998936i \(0.485313\pi\)
\(822\) 0 0
\(823\) −19804.0 −0.838790 −0.419395 0.907804i \(-0.637758\pi\)
−0.419395 + 0.907804i \(0.637758\pi\)
\(824\) 0 0
\(825\) 1200.00 0.0506408
\(826\) 0 0
\(827\) −5508.00 −0.231598 −0.115799 0.993273i \(-0.536943\pi\)
−0.115799 + 0.993273i \(0.536943\pi\)
\(828\) 0 0
\(829\) 33262.0 1.39353 0.696765 0.717299i \(-0.254622\pi\)
0.696765 + 0.717299i \(0.254622\pi\)
\(830\) 0 0
\(831\) 2082.00 0.0869119
\(832\) 0 0
\(833\) 2166.00 0.0900930
\(834\) 0 0
\(835\) −240.000 −0.00994676
\(836\) 0 0
\(837\) −216.000 −0.00892001
\(838\) 0 0
\(839\) 4600.00 0.189284 0.0946422 0.995511i \(-0.469829\pi\)
0.0946422 + 0.995511i \(0.469829\pi\)
\(840\) 0 0
\(841\) −17665.0 −0.724302
\(842\) 0 0
\(843\) 5946.00 0.242931
\(844\) 0 0
\(845\) −5835.00 −0.237550
\(846\) 0 0
\(847\) 21500.0 0.872195
\(848\) 0 0
\(849\) 15684.0 0.634009
\(850\) 0 0
\(851\) 34080.0 1.37279
\(852\) 0 0
\(853\) −4198.00 −0.168507 −0.0842537 0.996444i \(-0.526851\pi\)
−0.0842537 + 0.996444i \(0.526851\pi\)
\(854\) 0 0
\(855\) 180.000 0.00719985
\(856\) 0 0
\(857\) −5826.00 −0.232220 −0.116110 0.993236i \(-0.537042\pi\)
−0.116110 + 0.993236i \(0.537042\pi\)
\(858\) 0 0
\(859\) 3004.00 0.119319 0.0596596 0.998219i \(-0.480998\pi\)
0.0596596 + 0.998219i \(0.480998\pi\)
\(860\) 0 0
\(861\) −14760.0 −0.584227
\(862\) 0 0
\(863\) 36936.0 1.45691 0.728457 0.685092i \(-0.240238\pi\)
0.728457 + 0.685092i \(0.240238\pi\)
\(864\) 0 0
\(865\) −16210.0 −0.637175
\(866\) 0 0
\(867\) 10407.0 0.407659
\(868\) 0 0
\(869\) 6528.00 0.254830
\(870\) 0 0
\(871\) 9976.00 0.388087
\(872\) 0 0
\(873\) 4626.00 0.179343
\(874\) 0 0
\(875\) 2500.00 0.0965891
\(876\) 0 0
\(877\) 5434.00 0.209228 0.104614 0.994513i \(-0.466639\pi\)
0.104614 + 0.994513i \(0.466639\pi\)
\(878\) 0 0
\(879\) 22362.0 0.858079
\(880\) 0 0
\(881\) −4758.00 −0.181954 −0.0909768 0.995853i \(-0.528999\pi\)
−0.0909768 + 0.995853i \(0.528999\pi\)
\(882\) 0 0
\(883\) −15476.0 −0.589818 −0.294909 0.955525i \(-0.595289\pi\)
−0.294909 + 0.955525i \(0.595289\pi\)
\(884\) 0 0
\(885\) 8880.00 0.337286
\(886\) 0 0
\(887\) 27440.0 1.03872 0.519360 0.854555i \(-0.326170\pi\)
0.519360 + 0.854555i \(0.326170\pi\)
\(888\) 0 0
\(889\) −48720.0 −1.83804
\(890\) 0 0
\(891\) −1296.00 −0.0487291
\(892\) 0 0
\(893\) −1856.00 −0.0695506
\(894\) 0 0
\(895\) −13640.0 −0.509424
\(896\) 0 0
\(897\) −13920.0 −0.518144
\(898\) 0 0
\(899\) 656.000 0.0243368
\(900\) 0 0
\(901\) −26676.0 −0.986356
\(902\) 0 0
\(903\) 31440.0 1.15865
\(904\) 0 0
\(905\) 20450.0 0.751139
\(906\) 0 0
\(907\) 48924.0 1.79106 0.895532 0.444997i \(-0.146795\pi\)
0.895532 + 0.444997i \(0.146795\pi\)
\(908\) 0 0
\(909\) 5994.00 0.218711
\(910\) 0 0
\(911\) 3440.00 0.125107 0.0625534 0.998042i \(-0.480076\pi\)
0.0625534 + 0.998042i \(0.480076\pi\)
\(912\) 0 0
\(913\) 2624.00 0.0951169
\(914\) 0 0
\(915\) 8610.00 0.311080
\(916\) 0 0
\(917\) 50880.0 1.83229
\(918\) 0 0
\(919\) 27184.0 0.975753 0.487877 0.872913i \(-0.337772\pi\)
0.487877 + 0.872913i \(0.337772\pi\)
\(920\) 0 0
\(921\) −3948.00 −0.141250
\(922\) 0 0
\(923\) −44544.0 −1.58850
\(924\) 0 0
\(925\) 10650.0 0.378562
\(926\) 0 0
\(927\) 9900.00 0.350764
\(928\) 0 0
\(929\) 42490.0 1.50059 0.750297 0.661101i \(-0.229911\pi\)
0.750297 + 0.661101i \(0.229911\pi\)
\(930\) 0 0
\(931\) −228.000 −0.00802621
\(932\) 0 0
\(933\) −2496.00 −0.0875835
\(934\) 0 0
\(935\) 3040.00 0.106330
\(936\) 0 0
\(937\) 37354.0 1.30235 0.651175 0.758928i \(-0.274276\pi\)
0.651175 + 0.758928i \(0.274276\pi\)
\(938\) 0 0
\(939\) −20310.0 −0.705849
\(940\) 0 0
\(941\) −24470.0 −0.847714 −0.423857 0.905729i \(-0.639324\pi\)
−0.423857 + 0.905729i \(0.639324\pi\)
\(942\) 0 0
\(943\) −19680.0 −0.679607
\(944\) 0 0
\(945\) −2700.00 −0.0929429
\(946\) 0 0
\(947\) 34100.0 1.17012 0.585059 0.810991i \(-0.301071\pi\)
0.585059 + 0.810991i \(0.301071\pi\)
\(948\) 0 0
\(949\) −32364.0 −1.10704
\(950\) 0 0
\(951\) 19746.0 0.673300
\(952\) 0 0
\(953\) 1878.00 0.0638346 0.0319173 0.999491i \(-0.489839\pi\)
0.0319173 + 0.999491i \(0.489839\pi\)
\(954\) 0 0
\(955\) −7400.00 −0.250742
\(956\) 0 0
\(957\) 3936.00 0.132950
\(958\) 0 0
\(959\) −13880.0 −0.467371
\(960\) 0 0
\(961\) −29727.0 −0.997852
\(962\) 0 0
\(963\) −10908.0 −0.365011
\(964\) 0 0
\(965\) 8110.00 0.270539
\(966\) 0 0
\(967\) −38484.0 −1.27980 −0.639898 0.768460i \(-0.721023\pi\)
−0.639898 + 0.768460i \(0.721023\pi\)
\(968\) 0 0
\(969\) 456.000 0.0151175
\(970\) 0 0
\(971\) −45272.0 −1.49624 −0.748119 0.663564i \(-0.769043\pi\)
−0.748119 + 0.663564i \(0.769043\pi\)
\(972\) 0 0
\(973\) 10320.0 0.340025
\(974\) 0 0
\(975\) −4350.00 −0.142884
\(976\) 0 0
\(977\) −25354.0 −0.830242 −0.415121 0.909766i \(-0.636261\pi\)
−0.415121 + 0.909766i \(0.636261\pi\)
\(978\) 0 0
\(979\) 8160.00 0.266389
\(980\) 0 0
\(981\) 18702.0 0.608674
\(982\) 0 0
\(983\) 18744.0 0.608180 0.304090 0.952643i \(-0.401648\pi\)
0.304090 + 0.952643i \(0.401648\pi\)
\(984\) 0 0
\(985\) −12650.0 −0.409201
\(986\) 0 0
\(987\) 27840.0 0.897829
\(988\) 0 0
\(989\) 41920.0 1.34780
\(990\) 0 0
\(991\) −59600.0 −1.91045 −0.955225 0.295880i \(-0.904387\pi\)
−0.955225 + 0.295880i \(0.904387\pi\)
\(992\) 0 0
\(993\) 33876.0 1.08260
\(994\) 0 0
\(995\) −12200.0 −0.388710
\(996\) 0 0
\(997\) −17886.0 −0.568160 −0.284080 0.958801i \(-0.591688\pi\)
−0.284080 + 0.958801i \(0.591688\pi\)
\(998\) 0 0
\(999\) −11502.0 −0.364271
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 240.4.a.a.1.1 1
3.2 odd 2 720.4.a.s.1.1 1
4.3 odd 2 120.4.a.e.1.1 1
5.2 odd 4 1200.4.f.h.49.2 2
5.3 odd 4 1200.4.f.h.49.1 2
5.4 even 2 1200.4.a.bj.1.1 1
8.3 odd 2 960.4.a.q.1.1 1
8.5 even 2 960.4.a.bd.1.1 1
12.11 even 2 360.4.a.m.1.1 1
20.3 even 4 600.4.f.f.49.2 2
20.7 even 4 600.4.f.f.49.1 2
20.19 odd 2 600.4.a.a.1.1 1
60.23 odd 4 1800.4.f.k.649.1 2
60.47 odd 4 1800.4.f.k.649.2 2
60.59 even 2 1800.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.e.1.1 1 4.3 odd 2
240.4.a.a.1.1 1 1.1 even 1 trivial
360.4.a.m.1.1 1 12.11 even 2
600.4.a.a.1.1 1 20.19 odd 2
600.4.f.f.49.1 2 20.7 even 4
600.4.f.f.49.2 2 20.3 even 4
720.4.a.s.1.1 1 3.2 odd 2
960.4.a.q.1.1 1 8.3 odd 2
960.4.a.bd.1.1 1 8.5 even 2
1200.4.a.bj.1.1 1 5.4 even 2
1200.4.f.h.49.1 2 5.3 odd 4
1200.4.f.h.49.2 2 5.2 odd 4
1800.4.a.e.1.1 1 60.59 even 2
1800.4.f.k.649.1 2 60.23 odd 4
1800.4.f.k.649.2 2 60.47 odd 4