Properties

Label 240.4.a.g.1.1
Level $240$
Weight $4$
Character 240.1
Self dual yes
Analytic conductor $14.160$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +56.0000 q^{11} -86.0000 q^{13} -15.0000 q^{15} -106.000 q^{17} -4.00000 q^{19} -60.0000 q^{21} -136.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -206.000 q^{29} +152.000 q^{31} +168.000 q^{33} +100.000 q^{35} +282.000 q^{37} -258.000 q^{39} -246.000 q^{41} -412.000 q^{43} -45.0000 q^{45} -40.0000 q^{47} +57.0000 q^{49} -318.000 q^{51} -126.000 q^{53} -280.000 q^{55} -12.0000 q^{57} -56.0000 q^{59} -2.00000 q^{61} -180.000 q^{63} +430.000 q^{65} +388.000 q^{67} -408.000 q^{69} +672.000 q^{71} +1170.00 q^{73} +75.0000 q^{75} -1120.00 q^{77} -408.000 q^{79} +81.0000 q^{81} -668.000 q^{83} +530.000 q^{85} -618.000 q^{87} +66.0000 q^{89} +1720.00 q^{91} +456.000 q^{93} +20.0000 q^{95} -926.000 q^{97} +504.000 q^{99} +O(q^{100})\)

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
\(11\) 56.0000 1.53497 0.767483 0.641069i \(-0.221509\pi\)
0.767483 + 0.641069i \(0.221509\pi\)
\(12\) 0 0
\(13\) −86.0000 −1.83478 −0.917389 0.397992i \(-0.869707\pi\)
−0.917389 + 0.397992i \(0.869707\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −106.000 −1.51228 −0.756140 0.654409i \(-0.772917\pi\)
−0.756140 + 0.654409i \(0.772917\pi\)
\(18\) 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
\(23\) −136.000 −1.23295 −0.616477 0.787373i \(-0.711441\pi\)
−0.616477 + 0.787373i \(0.711441\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −206.000 −1.31908 −0.659539 0.751671i \(-0.729248\pi\)
−0.659539 + 0.751671i \(0.729248\pi\)
\(30\) 0 0
\(31\) 152.000 0.880645 0.440323 0.897840i \(-0.354864\pi\)
0.440323 + 0.897840i \(0.354864\pi\)
\(32\) 0 0
\(33\) 168.000 0.886214
\(34\) 0 0
\(35\) 100.000 0.482945
\(36\) 0 0
\(37\) 282.000 1.25299 0.626493 0.779427i \(-0.284490\pi\)
0.626493 + 0.779427i \(0.284490\pi\)
\(38\) 0 0
\(39\) −258.000 −1.05931
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −412.000 −1.46115 −0.730575 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −40.0000 −0.124140 −0.0620702 0.998072i \(-0.519770\pi\)
−0.0620702 + 0.998072i \(0.519770\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) −318.000 −0.873116
\(52\) 0 0
\(53\) −126.000 −0.326555 −0.163278 0.986580i \(-0.552207\pi\)
−0.163278 + 0.986580i \(0.552207\pi\)
\(54\) 0 0
\(55\) −280.000 −0.686458
\(56\) 0 0
\(57\) −12.0000 −0.0278849
\(58\) 0 0
\(59\) −56.0000 −0.123569 −0.0617846 0.998090i \(-0.519679\pi\)
−0.0617846 + 0.998090i \(0.519679\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.00419793 −0.00209897 0.999998i \(-0.500668\pi\)
−0.00209897 + 0.999998i \(0.500668\pi\)
\(62\) 0 0
\(63\) −180.000 −0.359966
\(64\) 0 0
\(65\) 430.000 0.820537
\(66\) 0 0
\(67\) 388.000 0.707489 0.353744 0.935342i \(-0.384908\pi\)
0.353744 + 0.935342i \(0.384908\pi\)
\(68\) 0 0
\(69\) −408.000 −0.711847
\(70\) 0 0
\(71\) 672.000 1.12326 0.561632 0.827387i \(-0.310174\pi\)
0.561632 + 0.827387i \(0.310174\pi\)
\(72\) 0 0
\(73\) 1170.00 1.87586 0.937932 0.346818i \(-0.112738\pi\)
0.937932 + 0.346818i \(0.112738\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −1120.00 −1.65761
\(78\) 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
\(83\) −668.000 −0.883404 −0.441702 0.897162i \(-0.645625\pi\)
−0.441702 + 0.897162i \(0.645625\pi\)
\(84\) 0 0
\(85\) 530.000 0.676313
\(86\) 0 0
\(87\) −618.000 −0.761570
\(88\) 0 0
\(89\) 66.0000 0.0786066 0.0393033 0.999227i \(-0.487486\pi\)
0.0393033 + 0.999227i \(0.487486\pi\)
\(90\) 0 0
\(91\) 1720.00 1.98137
\(92\) 0 0
\(93\) 456.000 0.508441
\(94\) 0 0
\(95\) 20.0000 0.0215995
\(96\) 0 0
\(97\) −926.000 −0.969289 −0.484645 0.874711i \(-0.661051\pi\)
−0.484645 + 0.874711i \(0.661051\pi\)
\(98\) 0 0
\(99\) 504.000 0.511656
\(100\) 0 0
\(101\) −198.000 −0.195067 −0.0975333 0.995232i \(-0.531095\pi\)
−0.0975333 + 0.995232i \(0.531095\pi\)
\(102\) 0 0
\(103\) 1532.00 1.46556 0.732779 0.680467i \(-0.238223\pi\)
0.732779 + 0.680467i \(0.238223\pi\)
\(104\) 0 0
\(105\) 300.000 0.278829
\(106\) 0 0
\(107\) 444.000 0.401150 0.200575 0.979678i \(-0.435719\pi\)
0.200575 + 0.979678i \(0.435719\pi\)
\(108\) 0 0
\(109\) 62.0000 0.0544819 0.0272409 0.999629i \(-0.491328\pi\)
0.0272409 + 0.999629i \(0.491328\pi\)
\(110\) 0 0
\(111\) 846.000 0.723412
\(112\) 0 0
\(113\) 414.000 0.344653 0.172327 0.985040i \(-0.444872\pi\)
0.172327 + 0.985040i \(0.444872\pi\)
\(114\) 0 0
\(115\) 680.000 0.551394
\(116\) 0 0
\(117\) −774.000 −0.611593
\(118\) 0 0
\(119\) 2120.00 1.63311
\(120\) 0 0
\(121\) 1805.00 1.35612
\(122\) 0 0
\(123\) −738.000 −0.541002
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 996.000 0.695911 0.347956 0.937511i \(-0.386876\pi\)
0.347956 + 0.937511i \(0.386876\pi\)
\(128\) 0 0
\(129\) −1236.00 −0.843595
\(130\) 0 0
\(131\) 264.000 0.176075 0.0880374 0.996117i \(-0.471941\pi\)
0.0880374 + 0.996117i \(0.471941\pi\)
\(132\) 0 0
\(133\) 80.0000 0.0521570
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) 2278.00 1.42060 0.710302 0.703897i \(-0.248558\pi\)
0.710302 + 0.703897i \(0.248558\pi\)
\(138\) 0 0
\(139\) −1812.00 −1.10570 −0.552848 0.833282i \(-0.686459\pi\)
−0.552848 + 0.833282i \(0.686459\pi\)
\(140\) 0 0
\(141\) −120.000 −0.0716725
\(142\) 0 0
\(143\) −4816.00 −2.81632
\(144\) 0 0
\(145\) 1030.00 0.589909
\(146\) 0 0
\(147\) 171.000 0.0959445
\(148\) 0 0
\(149\) −1534.00 −0.843424 −0.421712 0.906730i \(-0.638571\pi\)
−0.421712 + 0.906730i \(0.638571\pi\)
\(150\) 0 0
\(151\) 3016.00 1.62542 0.812711 0.582668i \(-0.197991\pi\)
0.812711 + 0.582668i \(0.197991\pi\)
\(152\) 0 0
\(153\) −954.000 −0.504094
\(154\) 0 0
\(155\) −760.000 −0.393837
\(156\) 0 0
\(157\) −1814.00 −0.922121 −0.461060 0.887369i \(-0.652531\pi\)
−0.461060 + 0.887369i \(0.652531\pi\)
\(158\) 0 0
\(159\) −378.000 −0.188537
\(160\) 0 0
\(161\) 2720.00 1.33147
\(162\) 0 0
\(163\) 1844.00 0.886093 0.443047 0.896499i \(-0.353898\pi\)
0.443047 + 0.896499i \(0.353898\pi\)
\(164\) 0 0
\(165\) −840.000 −0.396327
\(166\) 0 0
\(167\) −3768.00 −1.74597 −0.872984 0.487749i \(-0.837818\pi\)
−0.872984 + 0.487749i \(0.837818\pi\)
\(168\) 0 0
\(169\) 5199.00 2.36641
\(170\) 0 0
\(171\) −36.0000 −0.0160993
\(172\) 0 0
\(173\) 938.000 0.412224 0.206112 0.978528i \(-0.433919\pi\)
0.206112 + 0.978528i \(0.433919\pi\)
\(174\) 0 0
\(175\) −500.000 −0.215980
\(176\) 0 0
\(177\) −168.000 −0.0713427
\(178\) 0 0
\(179\) −3968.00 −1.65688 −0.828442 0.560075i \(-0.810772\pi\)
−0.828442 + 0.560075i \(0.810772\pi\)
\(180\) 0 0
\(181\) −3514.00 −1.44306 −0.721529 0.692384i \(-0.756560\pi\)
−0.721529 + 0.692384i \(0.756560\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.00242368
\(184\) 0 0
\(185\) −1410.00 −0.560353
\(186\) 0 0
\(187\) −5936.00 −2.32130
\(188\) 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
\(193\) −2774.00 −1.03460 −0.517298 0.855806i \(-0.673062\pi\)
−0.517298 + 0.855806i \(0.673062\pi\)
\(194\) 0 0
\(195\) 1290.00 0.473738
\(196\) 0 0
\(197\) −3806.00 −1.37648 −0.688239 0.725484i \(-0.741616\pi\)
−0.688239 + 0.725484i \(0.741616\pi\)
\(198\) 0 0
\(199\) 856.000 0.304926 0.152463 0.988309i \(-0.451280\pi\)
0.152463 + 0.988309i \(0.451280\pi\)
\(200\) 0 0
\(201\) 1164.00 0.408469
\(202\) 0 0
\(203\) 4120.00 1.42447
\(204\) 0 0
\(205\) 1230.00 0.419058
\(206\) 0 0
\(207\) −1224.00 −0.410985
\(208\) 0 0
\(209\) −224.000 −0.0741359
\(210\) 0 0
\(211\) −3020.00 −0.985334 −0.492667 0.870218i \(-0.663978\pi\)
−0.492667 + 0.870218i \(0.663978\pi\)
\(212\) 0 0
\(213\) 2016.00 0.648517
\(214\) 0 0
\(215\) 2060.00 0.653446
\(216\) 0 0
\(217\) −3040.00 −0.951008
\(218\) 0 0
\(219\) 3510.00 1.08303
\(220\) 0 0
\(221\) 9116.00 2.77470
\(222\) 0 0
\(223\) 1684.00 0.505690 0.252845 0.967507i \(-0.418634\pi\)
0.252845 + 0.967507i \(0.418634\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −2004.00 −0.585948 −0.292974 0.956120i \(-0.594645\pi\)
−0.292974 + 0.956120i \(0.594645\pi\)
\(228\) 0 0
\(229\) −5042.00 −1.45496 −0.727478 0.686131i \(-0.759307\pi\)
−0.727478 + 0.686131i \(0.759307\pi\)
\(230\) 0 0
\(231\) −3360.00 −0.957021
\(232\) 0 0
\(233\) −3090.00 −0.868810 −0.434405 0.900718i \(-0.643041\pi\)
−0.434405 + 0.900718i \(0.643041\pi\)
\(234\) 0 0
\(235\) 200.000 0.0555173
\(236\) 0 0
\(237\) −1224.00 −0.335474
\(238\) 0 0
\(239\) −2136.00 −0.578102 −0.289051 0.957314i \(-0.593340\pi\)
−0.289051 + 0.957314i \(0.593340\pi\)
\(240\) 0 0
\(241\) 98.0000 0.0261939 0.0130970 0.999914i \(-0.495831\pi\)
0.0130970 + 0.999914i \(0.495831\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −285.000 −0.0743183
\(246\) 0 0
\(247\) 344.000 0.0886162
\(248\) 0 0
\(249\) −2004.00 −0.510033
\(250\) 0 0
\(251\) 5040.00 1.26742 0.633709 0.773571i \(-0.281532\pi\)
0.633709 + 0.773571i \(0.281532\pi\)
\(252\) 0 0
\(253\) −7616.00 −1.89254
\(254\) 0 0
\(255\) 1590.00 0.390469
\(256\) 0 0
\(257\) −1986.00 −0.482036 −0.241018 0.970521i \(-0.577481\pi\)
−0.241018 + 0.970521i \(0.577481\pi\)
\(258\) 0 0
\(259\) −5640.00 −1.35310
\(260\) 0 0
\(261\) −1854.00 −0.439692
\(262\) 0 0
\(263\) −1416.00 −0.331994 −0.165997 0.986126i \(-0.553084\pi\)
−0.165997 + 0.986126i \(0.553084\pi\)
\(264\) 0 0
\(265\) 630.000 0.146040
\(266\) 0 0
\(267\) 198.000 0.0453835
\(268\) 0 0
\(269\) −6670.00 −1.51181 −0.755905 0.654681i \(-0.772803\pi\)
−0.755905 + 0.654681i \(0.772803\pi\)
\(270\) 0 0
\(271\) −48.0000 −0.0107594 −0.00537969 0.999986i \(-0.501712\pi\)
−0.00537969 + 0.999986i \(0.501712\pi\)
\(272\) 0 0
\(273\) 5160.00 1.14395
\(274\) 0 0
\(275\) 1400.00 0.306993
\(276\) 0 0
\(277\) 6938.00 1.50492 0.752462 0.658636i \(-0.228866\pi\)
0.752462 + 0.658636i \(0.228866\pi\)
\(278\) 0 0
\(279\) 1368.00 0.293548
\(280\) 0 0
\(281\) −1694.00 −0.359628 −0.179814 0.983701i \(-0.557550\pi\)
−0.179814 + 0.983701i \(0.557550\pi\)
\(282\) 0 0
\(283\) 6364.00 1.33675 0.668376 0.743824i \(-0.266990\pi\)
0.668376 + 0.743824i \(0.266990\pi\)
\(284\) 0 0
\(285\) 60.0000 0.0124705
\(286\) 0 0
\(287\) 4920.00 1.01191
\(288\) 0 0
\(289\) 6323.00 1.28699
\(290\) 0 0
\(291\) −2778.00 −0.559619
\(292\) 0 0
\(293\) −3134.00 −0.624881 −0.312441 0.949937i \(-0.601147\pi\)
−0.312441 + 0.949937i \(0.601147\pi\)
\(294\) 0 0
\(295\) 280.000 0.0552618
\(296\) 0 0
\(297\) 1512.00 0.295405
\(298\) 0 0
\(299\) 11696.0 2.26220
\(300\) 0 0
\(301\) 8240.00 1.57789
\(302\) 0 0
\(303\) −594.000 −0.112622
\(304\) 0 0
\(305\) 10.0000 0.00187737
\(306\) 0 0
\(307\) 236.000 0.0438737 0.0219369 0.999759i \(-0.493017\pi\)
0.0219369 + 0.999759i \(0.493017\pi\)
\(308\) 0 0
\(309\) 4596.00 0.846140
\(310\) 0 0
\(311\) −3776.00 −0.688480 −0.344240 0.938882i \(-0.611863\pi\)
−0.344240 + 0.938882i \(0.611863\pi\)
\(312\) 0 0
\(313\) −7918.00 −1.42988 −0.714939 0.699187i \(-0.753546\pi\)
−0.714939 + 0.699187i \(0.753546\pi\)
\(314\) 0 0
\(315\) 900.000 0.160982
\(316\) 0 0
\(317\) 4362.00 0.772853 0.386426 0.922320i \(-0.373709\pi\)
0.386426 + 0.922320i \(0.373709\pi\)
\(318\) 0 0
\(319\) −11536.0 −2.02474
\(320\) 0 0
\(321\) 1332.00 0.231604
\(322\) 0 0
\(323\) 424.000 0.0730402
\(324\) 0 0
\(325\) −2150.00 −0.366956
\(326\) 0 0
\(327\) 186.000 0.0314551
\(328\) 0 0
\(329\) 800.000 0.134059
\(330\) 0 0
\(331\) −7980.00 −1.32514 −0.662569 0.749001i \(-0.730534\pi\)
−0.662569 + 0.749001i \(0.730534\pi\)
\(332\) 0 0
\(333\) 2538.00 0.417662
\(334\) 0 0
\(335\) −1940.00 −0.316399
\(336\) 0 0
\(337\) −8294.00 −1.34066 −0.670331 0.742062i \(-0.733848\pi\)
−0.670331 + 0.742062i \(0.733848\pi\)
\(338\) 0 0
\(339\) 1242.00 0.198986
\(340\) 0 0
\(341\) 8512.00 1.35176
\(342\) 0 0
\(343\) 5720.00 0.900440
\(344\) 0 0
\(345\) 2040.00 0.318348
\(346\) 0 0
\(347\) 964.000 0.149136 0.0745681 0.997216i \(-0.476242\pi\)
0.0745681 + 0.997216i \(0.476242\pi\)
\(348\) 0 0
\(349\) 8670.00 1.32978 0.664892 0.746940i \(-0.268478\pi\)
0.664892 + 0.746940i \(0.268478\pi\)
\(350\) 0 0
\(351\) −2322.00 −0.353103
\(352\) 0 0
\(353\) −2314.00 −0.348900 −0.174450 0.984666i \(-0.555815\pi\)
−0.174450 + 0.984666i \(0.555815\pi\)
\(354\) 0 0
\(355\) −3360.00 −0.502339
\(356\) 0 0
\(357\) 6360.00 0.942876
\(358\) 0 0
\(359\) 1896.00 0.278738 0.139369 0.990240i \(-0.455493\pi\)
0.139369 + 0.990240i \(0.455493\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 5415.00 0.782958
\(364\) 0 0
\(365\) −5850.00 −0.838912
\(366\) 0 0
\(367\) −1484.00 −0.211074 −0.105537 0.994415i \(-0.533656\pi\)
−0.105537 + 0.994415i \(0.533656\pi\)
\(368\) 0 0
\(369\) −2214.00 −0.312348
\(370\) 0 0
\(371\) 2520.00 0.352647
\(372\) 0 0
\(373\) 12370.0 1.71714 0.858571 0.512694i \(-0.171352\pi\)
0.858571 + 0.512694i \(0.171352\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) 17716.0 2.42021
\(378\) 0 0
\(379\) −5620.00 −0.761689 −0.380844 0.924639i \(-0.624367\pi\)
−0.380844 + 0.924639i \(0.624367\pi\)
\(380\) 0 0
\(381\) 2988.00 0.401784
\(382\) 0 0
\(383\) −5880.00 −0.784475 −0.392238 0.919864i \(-0.628299\pi\)
−0.392238 + 0.919864i \(0.628299\pi\)
\(384\) 0 0
\(385\) 5600.00 0.741305
\(386\) 0 0
\(387\) −3708.00 −0.487050
\(388\) 0 0
\(389\) 2082.00 0.271367 0.135683 0.990752i \(-0.456677\pi\)
0.135683 + 0.990752i \(0.456677\pi\)
\(390\) 0 0
\(391\) 14416.0 1.86457
\(392\) 0 0
\(393\) 792.000 0.101657
\(394\) 0 0
\(395\) 2040.00 0.259857
\(396\) 0 0
\(397\) −1742.00 −0.220223 −0.110111 0.993919i \(-0.535121\pi\)
−0.110111 + 0.993919i \(0.535121\pi\)
\(398\) 0 0
\(399\) 240.000 0.0301129
\(400\) 0 0
\(401\) −3270.00 −0.407222 −0.203611 0.979052i \(-0.565268\pi\)
−0.203611 + 0.979052i \(0.565268\pi\)
\(402\) 0 0
\(403\) −13072.0 −1.61579
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 15792.0 1.92329
\(408\) 0 0
\(409\) −6134.00 −0.741581 −0.370791 0.928716i \(-0.620913\pi\)
−0.370791 + 0.928716i \(0.620913\pi\)
\(410\) 0 0
\(411\) 6834.00 0.820186
\(412\) 0 0
\(413\) 1120.00 0.133442
\(414\) 0 0
\(415\) 3340.00 0.395070
\(416\) 0 0
\(417\) −5436.00 −0.638374
\(418\) 0 0
\(419\) 10392.0 1.21165 0.605826 0.795597i \(-0.292843\pi\)
0.605826 + 0.795597i \(0.292843\pi\)
\(420\) 0 0
\(421\) −12690.0 −1.46906 −0.734528 0.678578i \(-0.762596\pi\)
−0.734528 + 0.678578i \(0.762596\pi\)
\(422\) 0 0
\(423\) −360.000 −0.0413801
\(424\) 0 0
\(425\) −2650.00 −0.302456
\(426\) 0 0
\(427\) 40.0000 0.00453334
\(428\) 0 0
\(429\) −14448.0 −1.62600
\(430\) 0 0
\(431\) 7408.00 0.827914 0.413957 0.910297i \(-0.364146\pi\)
0.413957 + 0.910297i \(0.364146\pi\)
\(432\) 0 0
\(433\) −5062.00 −0.561811 −0.280906 0.959735i \(-0.590635\pi\)
−0.280906 + 0.959735i \(0.590635\pi\)
\(434\) 0 0
\(435\) 3090.00 0.340584
\(436\) 0 0
\(437\) 544.000 0.0595493
\(438\) 0 0
\(439\) 7160.00 0.778424 0.389212 0.921148i \(-0.372747\pi\)
0.389212 + 0.921148i \(0.372747\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 0 0
\(443\) −17100.0 −1.83396 −0.916981 0.398930i \(-0.869382\pi\)
−0.916981 + 0.398930i \(0.869382\pi\)
\(444\) 0 0
\(445\) −330.000 −0.0351539
\(446\) 0 0
\(447\) −4602.00 −0.486951
\(448\) 0 0
\(449\) 8634.00 0.907491 0.453746 0.891131i \(-0.350087\pi\)
0.453746 + 0.891131i \(0.350087\pi\)
\(450\) 0 0
\(451\) −13776.0 −1.43833
\(452\) 0 0
\(453\) 9048.00 0.938437
\(454\) 0 0
\(455\) −8600.00 −0.886097
\(456\) 0 0
\(457\) 2986.00 0.305644 0.152822 0.988254i \(-0.451164\pi\)
0.152822 + 0.988254i \(0.451164\pi\)
\(458\) 0 0
\(459\) −2862.00 −0.291039
\(460\) 0 0
\(461\) −2406.00 −0.243077 −0.121539 0.992587i \(-0.538783\pi\)
−0.121539 + 0.992587i \(0.538783\pi\)
\(462\) 0 0
\(463\) 14316.0 1.43698 0.718489 0.695538i \(-0.244834\pi\)
0.718489 + 0.695538i \(0.244834\pi\)
\(464\) 0 0
\(465\) −2280.00 −0.227382
\(466\) 0 0
\(467\) 292.000 0.0289339 0.0144670 0.999895i \(-0.495395\pi\)
0.0144670 + 0.999895i \(0.495395\pi\)
\(468\) 0 0
\(469\) −7760.00 −0.764016
\(470\) 0 0
\(471\) −5442.00 −0.532387
\(472\) 0 0
\(473\) −23072.0 −2.24282
\(474\) 0 0
\(475\) −100.000 −0.00965961
\(476\) 0 0
\(477\) −1134.00 −0.108852
\(478\) 0 0
\(479\) −14056.0 −1.34078 −0.670391 0.742008i \(-0.733874\pi\)
−0.670391 + 0.742008i \(0.733874\pi\)
\(480\) 0 0
\(481\) −24252.0 −2.29895
\(482\) 0 0
\(483\) 8160.00 0.768722
\(484\) 0 0
\(485\) 4630.00 0.433479
\(486\) 0 0
\(487\) 11204.0 1.04251 0.521254 0.853401i \(-0.325464\pi\)
0.521254 + 0.853401i \(0.325464\pi\)
\(488\) 0 0
\(489\) 5532.00 0.511586
\(490\) 0 0
\(491\) −4608.00 −0.423536 −0.211768 0.977320i \(-0.567922\pi\)
−0.211768 + 0.977320i \(0.567922\pi\)
\(492\) 0 0
\(493\) 21836.0 1.99482
\(494\) 0 0
\(495\) −2520.00 −0.228819
\(496\) 0 0
\(497\) −13440.0 −1.21301
\(498\) 0 0
\(499\) −2468.00 −0.221409 −0.110704 0.993853i \(-0.535311\pi\)
−0.110704 + 0.993853i \(0.535311\pi\)
\(500\) 0 0
\(501\) −11304.0 −1.00803
\(502\) 0 0
\(503\) −12192.0 −1.08074 −0.540372 0.841426i \(-0.681717\pi\)
−0.540372 + 0.841426i \(0.681717\pi\)
\(504\) 0 0
\(505\) 990.000 0.0872365
\(506\) 0 0
\(507\) 15597.0 1.36625
\(508\) 0 0
\(509\) 1714.00 0.149257 0.0746284 0.997211i \(-0.476223\pi\)
0.0746284 + 0.997211i \(0.476223\pi\)
\(510\) 0 0
\(511\) −23400.0 −2.02574
\(512\) 0 0
\(513\) −108.000 −0.00929496
\(514\) 0 0
\(515\) −7660.00 −0.655417
\(516\) 0 0
\(517\) −2240.00 −0.190551
\(518\) 0 0
\(519\) 2814.00 0.237998
\(520\) 0 0
\(521\) −18014.0 −1.51479 −0.757397 0.652955i \(-0.773529\pi\)
−0.757397 + 0.652955i \(0.773529\pi\)
\(522\) 0 0
\(523\) 16748.0 1.40027 0.700133 0.714013i \(-0.253124\pi\)
0.700133 + 0.714013i \(0.253124\pi\)
\(524\) 0 0
\(525\) −1500.00 −0.124696
\(526\) 0 0
\(527\) −16112.0 −1.33178
\(528\) 0 0
\(529\) 6329.00 0.520178
\(530\) 0 0
\(531\) −504.000 −0.0411897
\(532\) 0 0
\(533\) 21156.0 1.71926
\(534\) 0 0
\(535\) −2220.00 −0.179400
\(536\) 0 0
\(537\) −11904.0 −0.956602
\(538\) 0 0
\(539\) 3192.00 0.255082
\(540\) 0 0
\(541\) −14018.0 −1.11401 −0.557006 0.830508i \(-0.688050\pi\)
−0.557006 + 0.830508i \(0.688050\pi\)
\(542\) 0 0
\(543\) −10542.0 −0.833150
\(544\) 0 0
\(545\) −310.000 −0.0243650
\(546\) 0 0
\(547\) 412.000 0.0322045 0.0161022 0.999870i \(-0.494874\pi\)
0.0161022 + 0.999870i \(0.494874\pi\)
\(548\) 0 0
\(549\) −18.0000 −0.00139931
\(550\) 0 0
\(551\) 824.000 0.0637089
\(552\) 0 0
\(553\) 8160.00 0.627484
\(554\) 0 0
\(555\) −4230.00 −0.323520
\(556\) 0 0
\(557\) 18218.0 1.38586 0.692928 0.721007i \(-0.256321\pi\)
0.692928 + 0.721007i \(0.256321\pi\)
\(558\) 0 0
\(559\) 35432.0 2.68088
\(560\) 0 0
\(561\) −17808.0 −1.34020
\(562\) 0 0
\(563\) −23524.0 −1.76096 −0.880478 0.474087i \(-0.842778\pi\)
−0.880478 + 0.474087i \(0.842778\pi\)
\(564\) 0 0
\(565\) −2070.00 −0.154134
\(566\) 0 0
\(567\) −1620.00 −0.119989
\(568\) 0 0
\(569\) 23330.0 1.71888 0.859442 0.511234i \(-0.170811\pi\)
0.859442 + 0.511234i \(0.170811\pi\)
\(570\) 0 0
\(571\) 13124.0 0.961860 0.480930 0.876759i \(-0.340299\pi\)
0.480930 + 0.876759i \(0.340299\pi\)
\(572\) 0 0
\(573\) 4440.00 0.323706
\(574\) 0 0
\(575\) −3400.00 −0.246591
\(576\) 0 0
\(577\) 11714.0 0.845165 0.422582 0.906324i \(-0.361124\pi\)
0.422582 + 0.906324i \(0.361124\pi\)
\(578\) 0 0
\(579\) −8322.00 −0.597324
\(580\) 0 0
\(581\) 13360.0 0.953987
\(582\) 0 0
\(583\) −7056.00 −0.501252
\(584\) 0 0
\(585\) 3870.00 0.273512
\(586\) 0 0
\(587\) 17628.0 1.23950 0.619749 0.784800i \(-0.287234\pi\)
0.619749 + 0.784800i \(0.287234\pi\)
\(588\) 0 0
\(589\) −608.000 −0.0425335
\(590\) 0 0
\(591\) −11418.0 −0.794710
\(592\) 0 0
\(593\) −2802.00 −0.194038 −0.0970188 0.995283i \(-0.530931\pi\)
−0.0970188 + 0.995283i \(0.530931\pi\)
\(594\) 0 0
\(595\) −10600.0 −0.730349
\(596\) 0 0
\(597\) 2568.00 0.176049
\(598\) 0 0
\(599\) 2664.00 0.181716 0.0908582 0.995864i \(-0.471039\pi\)
0.0908582 + 0.995864i \(0.471039\pi\)
\(600\) 0 0
\(601\) 23962.0 1.62634 0.813170 0.582026i \(-0.197740\pi\)
0.813170 + 0.582026i \(0.197740\pi\)
\(602\) 0 0
\(603\) 3492.00 0.235830
\(604\) 0 0
\(605\) −9025.00 −0.606477
\(606\) 0 0
\(607\) −11940.0 −0.798401 −0.399201 0.916864i \(-0.630712\pi\)
−0.399201 + 0.916864i \(0.630712\pi\)
\(608\) 0 0
\(609\) 12360.0 0.822418
\(610\) 0 0
\(611\) 3440.00 0.227770
\(612\) 0 0
\(613\) 16794.0 1.10653 0.553265 0.833005i \(-0.313382\pi\)
0.553265 + 0.833005i \(0.313382\pi\)
\(614\) 0 0
\(615\) 3690.00 0.241943
\(616\) 0 0
\(617\) −20706.0 −1.35104 −0.675520 0.737341i \(-0.736081\pi\)
−0.675520 + 0.737341i \(0.736081\pi\)
\(618\) 0 0
\(619\) −10724.0 −0.696339 −0.348170 0.937432i \(-0.613197\pi\)
−0.348170 + 0.937432i \(0.613197\pi\)
\(620\) 0 0
\(621\) −3672.00 −0.237282
\(622\) 0 0
\(623\) −1320.00 −0.0848871
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −672.000 −0.0428024
\(628\) 0 0
\(629\) −29892.0 −1.89487
\(630\) 0 0
\(631\) 5744.00 0.362385 0.181193 0.983448i \(-0.442004\pi\)
0.181193 + 0.983448i \(0.442004\pi\)
\(632\) 0 0
\(633\) −9060.00 −0.568883
\(634\) 0 0
\(635\) −4980.00 −0.311221
\(636\) 0 0
\(637\) −4902.00 −0.304905
\(638\) 0 0
\(639\) 6048.00 0.374421
\(640\) 0 0
\(641\) 27906.0 1.71953 0.859767 0.510687i \(-0.170609\pi\)
0.859767 + 0.510687i \(0.170609\pi\)
\(642\) 0 0
\(643\) −20556.0 −1.26073 −0.630365 0.776299i \(-0.717095\pi\)
−0.630365 + 0.776299i \(0.717095\pi\)
\(644\) 0 0
\(645\) 6180.00 0.377267
\(646\) 0 0
\(647\) 10224.0 0.621247 0.310624 0.950533i \(-0.399462\pi\)
0.310624 + 0.950533i \(0.399462\pi\)
\(648\) 0 0
\(649\) −3136.00 −0.189675
\(650\) 0 0
\(651\) −9120.00 −0.549064
\(652\) 0 0
\(653\) −12982.0 −0.777986 −0.388993 0.921241i \(-0.627177\pi\)
−0.388993 + 0.921241i \(0.627177\pi\)
\(654\) 0 0
\(655\) −1320.00 −0.0787430
\(656\) 0 0
\(657\) 10530.0 0.625288
\(658\) 0 0
\(659\) 1512.00 0.0893766 0.0446883 0.999001i \(-0.485771\pi\)
0.0446883 + 0.999001i \(0.485771\pi\)
\(660\) 0 0
\(661\) 16710.0 0.983273 0.491637 0.870800i \(-0.336399\pi\)
0.491637 + 0.870800i \(0.336399\pi\)
\(662\) 0 0
\(663\) 27348.0 1.60197
\(664\) 0 0
\(665\) −400.000 −0.0233253
\(666\) 0 0
\(667\) 28016.0 1.62636
\(668\) 0 0
\(669\) 5052.00 0.291961
\(670\) 0 0
\(671\) −112.000 −0.00644368
\(672\) 0 0
\(673\) 7962.00 0.456036 0.228018 0.973657i \(-0.426775\pi\)
0.228018 + 0.973657i \(0.426775\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 12226.0 0.694067 0.347033 0.937853i \(-0.387189\pi\)
0.347033 + 0.937853i \(0.387189\pi\)
\(678\) 0 0
\(679\) 18520.0 1.04673
\(680\) 0 0
\(681\) −6012.00 −0.338297
\(682\) 0 0
\(683\) 8748.00 0.490092 0.245046 0.969511i \(-0.421197\pi\)
0.245046 + 0.969511i \(0.421197\pi\)
\(684\) 0 0
\(685\) −11390.0 −0.635313
\(686\) 0 0
\(687\) −15126.0 −0.840019
\(688\) 0 0
\(689\) 10836.0 0.599156
\(690\) 0 0
\(691\) 7324.00 0.403210 0.201605 0.979467i \(-0.435384\pi\)
0.201605 + 0.979467i \(0.435384\pi\)
\(692\) 0 0
\(693\) −10080.0 −0.552536
\(694\) 0 0
\(695\) 9060.00 0.494483
\(696\) 0 0
\(697\) 26076.0 1.41707
\(698\) 0 0
\(699\) −9270.00 −0.501607
\(700\) 0 0
\(701\) −21934.0 −1.18179 −0.590896 0.806748i \(-0.701226\pi\)
−0.590896 + 0.806748i \(0.701226\pi\)
\(702\) 0 0
\(703\) −1128.00 −0.0605168
\(704\) 0 0
\(705\) 600.000 0.0320529
\(706\) 0 0
\(707\) 3960.00 0.210652
\(708\) 0 0
\(709\) −10690.0 −0.566250 −0.283125 0.959083i \(-0.591371\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(710\) 0 0
\(711\) −3672.00 −0.193686
\(712\) 0 0
\(713\) −20672.0 −1.08580
\(714\) 0 0
\(715\) 24080.0 1.25950
\(716\) 0 0
\(717\) −6408.00 −0.333767
\(718\) 0 0
\(719\) −13792.0 −0.715375 −0.357688 0.933841i \(-0.616435\pi\)
−0.357688 + 0.933841i \(0.616435\pi\)
\(720\) 0 0
\(721\) −30640.0 −1.58265
\(722\) 0 0
\(723\) 294.000 0.0151231
\(724\) 0 0
\(725\) −5150.00 −0.263815
\(726\) 0 0
\(727\) 24004.0 1.22457 0.612283 0.790639i \(-0.290251\pi\)
0.612283 + 0.790639i \(0.290251\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 43672.0 2.20967
\(732\) 0 0
\(733\) 8562.00 0.431439 0.215719 0.976455i \(-0.430790\pi\)
0.215719 + 0.976455i \(0.430790\pi\)
\(734\) 0 0
\(735\) −855.000 −0.0429077
\(736\) 0 0
\(737\) 21728.0 1.08597
\(738\) 0 0
\(739\) 13836.0 0.688722 0.344361 0.938837i \(-0.388096\pi\)
0.344361 + 0.938837i \(0.388096\pi\)
\(740\) 0 0
\(741\) 1032.00 0.0511626
\(742\) 0 0
\(743\) 22224.0 1.09733 0.548667 0.836041i \(-0.315135\pi\)
0.548667 + 0.836041i \(0.315135\pi\)
\(744\) 0 0
\(745\) 7670.00 0.377191
\(746\) 0 0
\(747\) −6012.00 −0.294468
\(748\) 0 0
\(749\) −8880.00 −0.433202
\(750\) 0 0
\(751\) −11544.0 −0.560914 −0.280457 0.959867i \(-0.590486\pi\)
−0.280457 + 0.959867i \(0.590486\pi\)
\(752\) 0 0
\(753\) 15120.0 0.731744
\(754\) 0 0
\(755\) −15080.0 −0.726910
\(756\) 0 0
\(757\) −3814.00 −0.183120 −0.0915602 0.995800i \(-0.529185\pi\)
−0.0915602 + 0.995800i \(0.529185\pi\)
\(758\) 0 0
\(759\) −22848.0 −1.09266
\(760\) 0 0
\(761\) −25662.0 −1.22240 −0.611200 0.791476i \(-0.709313\pi\)
−0.611200 + 0.791476i \(0.709313\pi\)
\(762\) 0 0
\(763\) −1240.00 −0.0588349
\(764\) 0 0
\(765\) 4770.00 0.225438
\(766\) 0 0
\(767\) 4816.00 0.226722
\(768\) 0 0
\(769\) 30658.0 1.43765 0.718827 0.695189i \(-0.244679\pi\)
0.718827 + 0.695189i \(0.244679\pi\)
\(770\) 0 0
\(771\) −5958.00 −0.278304
\(772\) 0 0
\(773\) −30894.0 −1.43749 −0.718745 0.695274i \(-0.755283\pi\)
−0.718745 + 0.695274i \(0.755283\pi\)
\(774\) 0 0
\(775\) 3800.00 0.176129
\(776\) 0 0
\(777\) −16920.0 −0.781212
\(778\) 0 0
\(779\) 984.000 0.0452573
\(780\) 0 0
\(781\) 37632.0 1.72417
\(782\) 0 0
\(783\) −5562.00 −0.253857
\(784\) 0 0
\(785\) 9070.00 0.412385
\(786\) 0 0
\(787\) −21596.0 −0.978163 −0.489081 0.872238i \(-0.662668\pi\)
−0.489081 + 0.872238i \(0.662668\pi\)
\(788\) 0 0
\(789\) −4248.00 −0.191677
\(790\) 0 0
\(791\) −8280.00 −0.372191
\(792\) 0 0
\(793\) 172.000 0.00770227
\(794\) 0 0
\(795\) 1890.00 0.0843162
\(796\) 0 0
\(797\) −8646.00 −0.384262 −0.192131 0.981369i \(-0.561540\pi\)
−0.192131 + 0.981369i \(0.561540\pi\)
\(798\) 0 0
\(799\) 4240.00 0.187735
\(800\) 0 0
\(801\) 594.000 0.0262022
\(802\) 0 0
\(803\) 65520.0 2.87939
\(804\) 0 0
\(805\) −13600.0 −0.595450
\(806\) 0 0
\(807\) −20010.0 −0.872844
\(808\) 0 0
\(809\) 24954.0 1.08447 0.542235 0.840227i \(-0.317578\pi\)
0.542235 + 0.840227i \(0.317578\pi\)
\(810\) 0 0
\(811\) −40004.0 −1.73210 −0.866048 0.499960i \(-0.833348\pi\)
−0.866048 + 0.499960i \(0.833348\pi\)
\(812\) 0 0
\(813\) −144.000 −0.00621193
\(814\) 0 0
\(815\) −9220.00 −0.396273
\(816\) 0 0
\(817\) 1648.00 0.0705707
\(818\) 0 0
\(819\) 15480.0 0.660458
\(820\) 0 0
\(821\) 16570.0 0.704381 0.352191 0.935928i \(-0.385437\pi\)
0.352191 + 0.935928i \(0.385437\pi\)
\(822\) 0 0
\(823\) 4388.00 0.185852 0.0929259 0.995673i \(-0.470378\pi\)
0.0929259 + 0.995673i \(0.470378\pi\)
\(824\) 0 0
\(825\) 4200.00 0.177243
\(826\) 0 0
\(827\) −14364.0 −0.603972 −0.301986 0.953312i \(-0.597650\pi\)
−0.301986 + 0.953312i \(0.597650\pi\)
\(828\) 0 0
\(829\) −21170.0 −0.886929 −0.443465 0.896292i \(-0.646251\pi\)
−0.443465 + 0.896292i \(0.646251\pi\)
\(830\) 0 0
\(831\) 20814.0 0.868868
\(832\) 0 0
\(833\) −6042.00 −0.251312
\(834\) 0 0
\(835\) 18840.0 0.780820
\(836\) 0 0
\(837\) 4104.00 0.169480
\(838\) 0 0
\(839\) −10664.0 −0.438811 −0.219405 0.975634i \(-0.570412\pi\)
−0.219405 + 0.975634i \(0.570412\pi\)
\(840\) 0 0
\(841\) 18047.0 0.739965
\(842\) 0 0
\(843\) −5082.00 −0.207632
\(844\) 0 0
\(845\) −25995.0 −1.05829
\(846\) 0 0
\(847\) −36100.0 −1.46448
\(848\) 0 0
\(849\) 19092.0 0.771774
\(850\) 0 0
\(851\) −38352.0 −1.54488
\(852\) 0 0
\(853\) −3190.00 −0.128046 −0.0640232 0.997948i \(-0.520393\pi\)
−0.0640232 + 0.997948i \(0.520393\pi\)
\(854\) 0 0
\(855\) 180.000 0.00719985
\(856\) 0 0
\(857\) 20814.0 0.829630 0.414815 0.909906i \(-0.363846\pi\)
0.414815 + 0.909906i \(0.363846\pi\)
\(858\) 0 0
\(859\) 18988.0 0.754205 0.377103 0.926172i \(-0.376920\pi\)
0.377103 + 0.926172i \(0.376920\pi\)
\(860\) 0 0
\(861\) 14760.0 0.584227
\(862\) 0 0
\(863\) −11664.0 −0.460078 −0.230039 0.973181i \(-0.573885\pi\)
−0.230039 + 0.973181i \(0.573885\pi\)
\(864\) 0 0
\(865\) −4690.00 −0.184352
\(866\) 0 0
\(867\) 18969.0 0.743046
\(868\) 0 0
\(869\) −22848.0 −0.891905
\(870\) 0 0
\(871\) −33368.0 −1.29808
\(872\) 0 0
\(873\) −8334.00 −0.323096
\(874\) 0 0
\(875\) 2500.00 0.0965891
\(876\) 0 0
\(877\) −8246.00 −0.317500 −0.158750 0.987319i \(-0.550746\pi\)
−0.158750 + 0.987319i \(0.550746\pi\)
\(878\) 0 0
\(879\) −9402.00 −0.360775
\(880\) 0 0
\(881\) 22890.0 0.875350 0.437675 0.899133i \(-0.355802\pi\)
0.437675 + 0.899133i \(0.355802\pi\)
\(882\) 0 0
\(883\) −33548.0 −1.27857 −0.639287 0.768969i \(-0.720770\pi\)
−0.639287 + 0.768969i \(0.720770\pi\)
\(884\) 0 0
\(885\) 840.000 0.0319054
\(886\) 0 0
\(887\) 32264.0 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(888\) 0 0
\(889\) −19920.0 −0.751513
\(890\) 0 0
\(891\) 4536.00 0.170552
\(892\) 0 0
\(893\) 160.000 0.00599574
\(894\) 0 0
\(895\) 19840.0 0.740981
\(896\) 0 0
\(897\) 35088.0 1.30608
\(898\) 0 0
\(899\) −31312.0 −1.16164
\(900\) 0 0
\(901\) 13356.0 0.493843
\(902\) 0 0
\(903\) 24720.0 0.910997
\(904\) 0 0
\(905\) 17570.0 0.645355
\(906\) 0 0
\(907\) −51228.0 −1.87541 −0.937706 0.347431i \(-0.887054\pi\)
−0.937706 + 0.347431i \(0.887054\pi\)
\(908\) 0 0
\(909\) −1782.00 −0.0650222
\(910\) 0 0
\(911\) 2144.00 0.0779735 0.0389868 0.999240i \(-0.487587\pi\)
0.0389868 + 0.999240i \(0.487587\pi\)
\(912\) 0 0
\(913\) −37408.0 −1.35600
\(914\) 0 0
\(915\) 30.0000 0.00108390
\(916\) 0 0
\(917\) −5280.00 −0.190143
\(918\) 0 0
\(919\) −33584.0 −1.20548 −0.602739 0.797939i \(-0.705924\pi\)
−0.602739 + 0.797939i \(0.705924\pi\)
\(920\) 0 0
\(921\) 708.000 0.0253305
\(922\) 0 0
\(923\) −57792.0 −2.06094
\(924\) 0 0
\(925\) 7050.00 0.250597
\(926\) 0 0
\(927\) 13788.0 0.488519
\(928\) 0 0
\(929\) −3590.00 −0.126786 −0.0633929 0.997989i \(-0.520192\pi\)
−0.0633929 + 0.997989i \(0.520192\pi\)
\(930\) 0 0
\(931\) −228.000 −0.00802621
\(932\) 0 0
\(933\) −11328.0 −0.397494
\(934\) 0 0
\(935\) 29680.0 1.03812
\(936\) 0 0
\(937\) −21686.0 −0.756084 −0.378042 0.925788i \(-0.623403\pi\)
−0.378042 + 0.925788i \(0.623403\pi\)
\(938\) 0 0
\(939\) −23754.0 −0.825540
\(940\) 0 0
\(941\) −5174.00 −0.179243 −0.0896215 0.995976i \(-0.528566\pi\)
−0.0896215 + 0.995976i \(0.528566\pi\)
\(942\) 0 0
\(943\) 33456.0 1.15533
\(944\) 0 0
\(945\) 2700.00 0.0929429
\(946\) 0 0
\(947\) −35524.0 −1.21898 −0.609490 0.792793i \(-0.708626\pi\)
−0.609490 + 0.792793i \(0.708626\pi\)
\(948\) 0 0
\(949\) −100620. −3.44179
\(950\) 0 0
\(951\) 13086.0 0.446207
\(952\) 0 0
\(953\) −16122.0 −0.547999 −0.273999 0.961730i \(-0.588347\pi\)
−0.273999 + 0.961730i \(0.588347\pi\)
\(954\) 0 0
\(955\) −7400.00 −0.250742
\(956\) 0 0
\(957\) −34608.0 −1.16898
\(958\) 0 0
\(959\) −45560.0 −1.53411
\(960\) 0 0
\(961\) −6687.00 −0.224464
\(962\) 0 0
\(963\) 3996.00 0.133717
\(964\) 0 0
\(965\) 13870.0 0.462685
\(966\) 0 0
\(967\) −19188.0 −0.638102 −0.319051 0.947738i \(-0.603364\pi\)
−0.319051 + 0.947738i \(0.603364\pi\)
\(968\) 0 0
\(969\) 1272.00 0.0421698
\(970\) 0 0
\(971\) 38464.0 1.27123 0.635617 0.772004i \(-0.280746\pi\)
0.635617 + 0.772004i \(0.280746\pi\)
\(972\) 0 0
\(973\) 36240.0 1.19404
\(974\) 0 0
\(975\) −6450.00 −0.211862
\(976\) 0 0
\(977\) −43930.0 −1.43853 −0.719266 0.694735i \(-0.755522\pi\)
−0.719266 + 0.694735i \(0.755522\pi\)
\(978\) 0 0
\(979\) 3696.00 0.120659
\(980\) 0 0
\(981\) 558.000 0.0181606
\(982\) 0 0
\(983\) −17328.0 −0.562235 −0.281118 0.959673i \(-0.590705\pi\)
−0.281118 + 0.959673i \(0.590705\pi\)
\(984\) 0 0
\(985\) 19030.0 0.615580
\(986\) 0 0
\(987\) 2400.00 0.0773990
\(988\) 0 0
\(989\) 56032.0 1.80153
\(990\) 0 0
\(991\) 18160.0 0.582110 0.291055 0.956706i \(-0.405994\pi\)
0.291055 + 0.956706i \(0.405994\pi\)
\(992\) 0 0
\(993\) −23940.0 −0.765068
\(994\) 0 0
\(995\) −4280.00 −0.136367
\(996\) 0 0
\(997\) −9102.00 −0.289131 −0.144565 0.989495i \(-0.546178\pi\)
−0.144565 + 0.989495i \(0.546178\pi\)
\(998\) 0 0
\(999\) 7614.00 0.241137
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.g.1.1 1
3.2 odd 2 720.4.a.q.1.1 1
4.3 odd 2 120.4.a.b.1.1 1
5.2 odd 4 1200.4.f.t.49.1 2
5.3 odd 4 1200.4.f.t.49.2 2
5.4 even 2 1200.4.a.p.1.1 1
8.3 odd 2 960.4.a.bj.1.1 1
8.5 even 2 960.4.a.k.1.1 1
12.11 even 2 360.4.a.n.1.1 1
20.3 even 4 600.4.f.a.49.1 2
20.7 even 4 600.4.f.a.49.2 2
20.19 odd 2 600.4.a.i.1.1 1
60.23 odd 4 1800.4.f.v.649.1 2
60.47 odd 4 1800.4.f.v.649.2 2
60.59 even 2 1800.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 1 4.3 odd 2
240.4.a.g.1.1 1 1.1 even 1 trivial
360.4.a.n.1.1 1 12.11 even 2
600.4.a.i.1.1 1 20.19 odd 2
600.4.f.a.49.1 2 20.3 even 4
600.4.f.a.49.2 2 20.7 even 4
720.4.a.q.1.1 1 3.2 odd 2
960.4.a.k.1.1 1 8.5 even 2
960.4.a.bj.1.1 1 8.3 odd 2
1200.4.a.p.1.1 1 5.4 even 2
1200.4.f.t.49.1 2 5.2 odd 4
1200.4.f.t.49.2 2 5.3 odd 4
1800.4.a.f.1.1 1 60.59 even 2
1800.4.f.v.649.1 2 60.23 odd 4
1800.4.f.v.649.2 2 60.47 odd 4