Properties

Label 216.4.a.f.1.1
Level $216$
Weight $4$
Character 216.1
Self dual yes
Analytic conductor $12.744$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7444125612\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 216.1

$q$-expansion

\(f(q)\) \(=\) \(q-15.4164 q^{5} +23.8328 q^{7} +O(q^{10})\) \(q-15.4164 q^{5} +23.8328 q^{7} +14.2492 q^{11} -13.8328 q^{13} -80.5836 q^{17} -144.331 q^{19} -141.082 q^{23} +112.666 q^{25} -251.331 q^{29} -16.6687 q^{31} -367.416 q^{35} +305.164 q^{37} +429.325 q^{41} -181.666 q^{43} -79.4164 q^{47} +225.003 q^{49} -663.830 q^{53} -219.672 q^{55} +220.255 q^{59} -473.158 q^{61} +213.252 q^{65} -647.663 q^{67} +14.4922 q^{71} +776.003 q^{73} +339.599 q^{77} -257.827 q^{79} +1285.15 q^{83} +1242.31 q^{85} -156.255 q^{89} -329.675 q^{91} +2225.07 q^{95} +1161.64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} - 6q^{7} + O(q^{10}) \) \( 2q - 4q^{5} - 6q^{7} - 52q^{11} + 26q^{13} - 188q^{17} - 74q^{19} - 148q^{23} + 118q^{25} - 288q^{29} - 248q^{31} - 708q^{35} + 342q^{37} - 256q^{43} - 132q^{47} + 772q^{49} - 952q^{53} - 976q^{55} + 1004q^{59} - 34q^{61} + 668q^{65} - 866q^{67} - 776q^{71} + 1874q^{73} + 2316q^{77} + 182q^{79} + 1336q^{83} + 16q^{85} - 876q^{89} - 1518q^{91} + 3028q^{95} - 38q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −15.4164 −1.37889 −0.689443 0.724340i \(-0.742145\pi\)
−0.689443 + 0.724340i \(0.742145\pi\)
\(6\) 0 0
\(7\) 23.8328 1.28685 0.643426 0.765509i \(-0.277513\pi\)
0.643426 + 0.765509i \(0.277513\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.2492 0.390573 0.195286 0.980746i \(-0.437436\pi\)
0.195286 + 0.980746i \(0.437436\pi\)
\(12\) 0 0
\(13\) −13.8328 −0.295118 −0.147559 0.989053i \(-0.547142\pi\)
−0.147559 + 0.989053i \(0.547142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −80.5836 −1.14967 −0.574835 0.818269i \(-0.694934\pi\)
−0.574835 + 0.818269i \(0.694934\pi\)
\(18\) 0 0
\(19\) −144.331 −1.74273 −0.871365 0.490636i \(-0.836765\pi\)
−0.871365 + 0.490636i \(0.836765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −141.082 −1.27903 −0.639514 0.768780i \(-0.720864\pi\)
−0.639514 + 0.768780i \(0.720864\pi\)
\(24\) 0 0
\(25\) 112.666 0.901325
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −251.331 −1.60935 −0.804673 0.593718i \(-0.797659\pi\)
−0.804673 + 0.593718i \(0.797659\pi\)
\(30\) 0 0
\(31\) −16.6687 −0.0965740 −0.0482870 0.998834i \(-0.515376\pi\)
−0.0482870 + 0.998834i \(0.515376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −367.416 −1.77442
\(36\) 0 0
\(37\) 305.164 1.35591 0.677955 0.735103i \(-0.262866\pi\)
0.677955 + 0.735103i \(0.262866\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 429.325 1.63535 0.817674 0.575681i \(-0.195263\pi\)
0.817674 + 0.575681i \(0.195263\pi\)
\(42\) 0 0
\(43\) −181.666 −0.644273 −0.322137 0.946693i \(-0.604401\pi\)
−0.322137 + 0.946693i \(0.604401\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −79.4164 −0.246470 −0.123235 0.992378i \(-0.539327\pi\)
−0.123235 + 0.992378i \(0.539327\pi\)
\(48\) 0 0
\(49\) 225.003 0.655986
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −663.830 −1.72045 −0.860227 0.509912i \(-0.829678\pi\)
−0.860227 + 0.509912i \(0.829678\pi\)
\(54\) 0 0
\(55\) −219.672 −0.538555
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 220.255 0.486014 0.243007 0.970025i \(-0.421866\pi\)
0.243007 + 0.970025i \(0.421866\pi\)
\(60\) 0 0
\(61\) −473.158 −0.993142 −0.496571 0.867996i \(-0.665408\pi\)
−0.496571 + 0.867996i \(0.665408\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 213.252 0.406934
\(66\) 0 0
\(67\) −647.663 −1.18096 −0.590482 0.807051i \(-0.701062\pi\)
−0.590482 + 0.807051i \(0.701062\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.4922 0.0242241 0.0121121 0.999927i \(-0.496145\pi\)
0.0121121 + 0.999927i \(0.496145\pi\)
\(72\) 0 0
\(73\) 776.003 1.24417 0.622084 0.782950i \(-0.286286\pi\)
0.622084 + 0.782950i \(0.286286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 339.599 0.502609
\(78\) 0 0
\(79\) −257.827 −0.367187 −0.183593 0.983002i \(-0.558773\pi\)
−0.183593 + 0.983002i \(0.558773\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1285.15 1.69957 0.849784 0.527132i \(-0.176733\pi\)
0.849784 + 0.527132i \(0.176733\pi\)
\(84\) 0 0
\(85\) 1242.31 1.58526
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −156.255 −0.186102 −0.0930508 0.995661i \(-0.529662\pi\)
−0.0930508 + 0.995661i \(0.529662\pi\)
\(90\) 0 0
\(91\) −329.675 −0.379773
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2225.07 2.40302
\(96\) 0 0
\(97\) 1161.64 1.21595 0.607975 0.793956i \(-0.291982\pi\)
0.607975 + 0.793956i \(0.291982\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1685.98 1.66100 0.830502 0.557016i \(-0.188054\pi\)
0.830502 + 0.557016i \(0.188054\pi\)
\(102\) 0 0
\(103\) −765.820 −0.732607 −0.366304 0.930495i \(-0.619377\pi\)
−0.366304 + 0.930495i \(0.619377\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1747.89 −1.57920 −0.789602 0.613619i \(-0.789713\pi\)
−0.789602 + 0.613619i \(0.789713\pi\)
\(108\) 0 0
\(109\) −1211.64 −1.06472 −0.532358 0.846519i \(-0.678694\pi\)
−0.532358 + 0.846519i \(0.678694\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 964.426 0.802881 0.401440 0.915885i \(-0.368510\pi\)
0.401440 + 0.915885i \(0.368510\pi\)
\(114\) 0 0
\(115\) 2174.98 1.76363
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1920.53 −1.47945
\(120\) 0 0
\(121\) −1127.96 −0.847453
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 190.152 0.136061
\(126\) 0 0
\(127\) −2238.31 −1.56392 −0.781960 0.623329i \(-0.785780\pi\)
−0.781960 + 0.623329i \(0.785780\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1674.31 1.11668 0.558340 0.829612i \(-0.311438\pi\)
0.558340 + 0.829612i \(0.311438\pi\)
\(132\) 0 0
\(133\) −3439.82 −2.24263
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −95.0758 −0.0592911 −0.0296455 0.999560i \(-0.509438\pi\)
−0.0296455 + 0.999560i \(0.509438\pi\)
\(138\) 0 0
\(139\) 1170.64 0.714334 0.357167 0.934041i \(-0.383743\pi\)
0.357167 + 0.934041i \(0.383743\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −197.107 −0.115265
\(144\) 0 0
\(145\) 3874.63 2.21910
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 859.136 0.472370 0.236185 0.971708i \(-0.424103\pi\)
0.236185 + 0.971708i \(0.424103\pi\)
\(150\) 0 0
\(151\) 2898.83 1.56227 0.781137 0.624359i \(-0.214640\pi\)
0.781137 + 0.624359i \(0.214640\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 256.972 0.133164
\(156\) 0 0
\(157\) −1309.02 −0.665419 −0.332710 0.943029i \(-0.607963\pi\)
−0.332710 + 0.943029i \(0.607963\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3362.38 −1.64592
\(162\) 0 0
\(163\) −190.988 −0.0917749 −0.0458874 0.998947i \(-0.514612\pi\)
−0.0458874 + 0.998947i \(0.514612\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1440.73 −0.667587 −0.333793 0.942646i \(-0.608329\pi\)
−0.333793 + 0.942646i \(0.608329\pi\)
\(168\) 0 0
\(169\) −2005.65 −0.912905
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1581.33 −0.694947 −0.347474 0.937690i \(-0.612960\pi\)
−0.347474 + 0.937690i \(0.612960\pi\)
\(174\) 0 0
\(175\) 2685.14 1.15987
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −741.836 −0.309762 −0.154881 0.987933i \(-0.549499\pi\)
−0.154881 + 0.987933i \(0.549499\pi\)
\(180\) 0 0
\(181\) 626.786 0.257396 0.128698 0.991684i \(-0.458920\pi\)
0.128698 + 0.991684i \(0.458920\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4704.53 −1.86964
\(186\) 0 0
\(187\) −1148.25 −0.449030
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −182.615 −0.0691808 −0.0345904 0.999402i \(-0.511013\pi\)
−0.0345904 + 0.999402i \(0.511013\pi\)
\(192\) 0 0
\(193\) 1718.66 0.640994 0.320497 0.947250i \(-0.396150\pi\)
0.320497 + 0.947250i \(0.396150\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 119.587 0.0432497 0.0216249 0.999766i \(-0.493116\pi\)
0.0216249 + 0.999766i \(0.493116\pi\)
\(198\) 0 0
\(199\) 707.467 0.252015 0.126008 0.992029i \(-0.459784\pi\)
0.126008 + 0.992029i \(0.459784\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5989.93 −2.07099
\(204\) 0 0
\(205\) −6618.65 −2.25496
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2056.61 −0.680663
\(210\) 0 0
\(211\) −2388.62 −0.779332 −0.389666 0.920956i \(-0.627410\pi\)
−0.389666 + 0.920956i \(0.627410\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2800.63 0.888379
\(216\) 0 0
\(217\) −397.263 −0.124276
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1114.70 0.339288
\(222\) 0 0
\(223\) 5622.60 1.68842 0.844209 0.536014i \(-0.180071\pi\)
0.844209 + 0.536014i \(0.180071\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3464.46 1.01297 0.506485 0.862249i \(-0.330944\pi\)
0.506485 + 0.862249i \(0.330944\pi\)
\(228\) 0 0
\(229\) −6082.62 −1.75524 −0.877622 0.479354i \(-0.840871\pi\)
−0.877622 + 0.479354i \(0.840871\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5024.29 1.41267 0.706335 0.707877i \(-0.250347\pi\)
0.706335 + 0.707877i \(0.250347\pi\)
\(234\) 0 0
\(235\) 1224.32 0.339853
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2670.20 −0.722682 −0.361341 0.932434i \(-0.617681\pi\)
−0.361341 + 0.932434i \(0.617681\pi\)
\(240\) 0 0
\(241\) 754.653 0.201707 0.100854 0.994901i \(-0.467843\pi\)
0.100854 + 0.994901i \(0.467843\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3468.74 −0.904529
\(246\) 0 0
\(247\) 1996.51 0.514311
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1588.91 −0.399566 −0.199783 0.979840i \(-0.564024\pi\)
−0.199783 + 0.979840i \(0.564024\pi\)
\(252\) 0 0
\(253\) −2010.31 −0.499554
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4911.76 1.19217 0.596083 0.802923i \(-0.296723\pi\)
0.596083 + 0.802923i \(0.296723\pi\)
\(258\) 0 0
\(259\) 7272.92 1.74485
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5400.41 −1.26617 −0.633087 0.774081i \(-0.718212\pi\)
−0.633087 + 0.774081i \(0.718212\pi\)
\(264\) 0 0
\(265\) 10233.9 2.37231
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3234.09 0.733034 0.366517 0.930411i \(-0.380550\pi\)
0.366517 + 0.930411i \(0.380550\pi\)
\(270\) 0 0
\(271\) −6205.83 −1.39106 −0.695530 0.718497i \(-0.744830\pi\)
−0.695530 + 0.718497i \(0.744830\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1605.40 0.352033
\(276\) 0 0
\(277\) 1030.91 0.223615 0.111808 0.993730i \(-0.464336\pi\)
0.111808 + 0.993730i \(0.464336\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3821.33 0.811250 0.405625 0.914040i \(-0.367054\pi\)
0.405625 + 0.914040i \(0.367054\pi\)
\(282\) 0 0
\(283\) −4402.33 −0.924705 −0.462353 0.886696i \(-0.652995\pi\)
−0.462353 + 0.886696i \(0.652995\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10232.0 2.10445
\(288\) 0 0
\(289\) 1580.72 0.321741
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5973.54 −1.19105 −0.595526 0.803336i \(-0.703056\pi\)
−0.595526 + 0.803336i \(0.703056\pi\)
\(294\) 0 0
\(295\) −3395.55 −0.670157
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1951.56 0.377464
\(300\) 0 0
\(301\) −4329.60 −0.829084
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7294.39 1.36943
\(306\) 0 0
\(307\) −219.622 −0.0408290 −0.0204145 0.999792i \(-0.506499\pi\)
−0.0204145 + 0.999792i \(0.506499\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −957.227 −0.174532 −0.0872659 0.996185i \(-0.527813\pi\)
−0.0872659 + 0.996185i \(0.527813\pi\)
\(312\) 0 0
\(313\) 1874.05 0.338427 0.169214 0.985579i \(-0.445877\pi\)
0.169214 + 0.985579i \(0.445877\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2919.78 0.517322 0.258661 0.965968i \(-0.416719\pi\)
0.258661 + 0.965968i \(0.416719\pi\)
\(318\) 0 0
\(319\) −3581.28 −0.628567
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11630.7 2.00356
\(324\) 0 0
\(325\) −1558.48 −0.265997
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1892.72 −0.317170
\(330\) 0 0
\(331\) −6648.67 −1.10406 −0.552030 0.833824i \(-0.686147\pi\)
−0.552030 + 0.833824i \(0.686147\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9984.63 1.62841
\(336\) 0 0
\(337\) 3886.34 0.628197 0.314098 0.949390i \(-0.398298\pi\)
0.314098 + 0.949390i \(0.398298\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −237.517 −0.0377192
\(342\) 0 0
\(343\) −2812.20 −0.442695
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4207.76 0.650963 0.325481 0.945548i \(-0.394474\pi\)
0.325481 + 0.945548i \(0.394474\pi\)
\(348\) 0 0
\(349\) 3011.42 0.461885 0.230942 0.972967i \(-0.425819\pi\)
0.230942 + 0.972967i \(0.425819\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2499.79 0.376914 0.188457 0.982081i \(-0.439651\pi\)
0.188457 + 0.982081i \(0.439651\pi\)
\(354\) 0 0
\(355\) −223.418 −0.0334023
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8319.17 −1.22303 −0.611517 0.791231i \(-0.709440\pi\)
−0.611517 + 0.791231i \(0.709440\pi\)
\(360\) 0 0
\(361\) 13972.5 2.03711
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11963.2 −1.71557
\(366\) 0 0
\(367\) −4459.85 −0.634339 −0.317169 0.948369i \(-0.602732\pi\)
−0.317169 + 0.948369i \(0.602732\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15820.9 −2.21397
\(372\) 0 0
\(373\) −1179.41 −0.163719 −0.0818596 0.996644i \(-0.526086\pi\)
−0.0818596 + 0.996644i \(0.526086\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3476.62 0.474947
\(378\) 0 0
\(379\) 8413.88 1.14035 0.570174 0.821524i \(-0.306876\pi\)
0.570174 + 0.821524i \(0.306876\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1601.94 −0.213722 −0.106861 0.994274i \(-0.534080\pi\)
−0.106861 + 0.994274i \(0.534080\pi\)
\(384\) 0 0
\(385\) −5235.40 −0.693041
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5202.43 0.678081 0.339041 0.940772i \(-0.389898\pi\)
0.339041 + 0.940772i \(0.389898\pi\)
\(390\) 0 0
\(391\) 11368.9 1.47046
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3974.76 0.506309
\(396\) 0 0
\(397\) 4310.87 0.544979 0.272489 0.962159i \(-0.412153\pi\)
0.272489 + 0.962159i \(0.412153\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6240.59 −0.777157 −0.388579 0.921416i \(-0.627034\pi\)
−0.388579 + 0.921416i \(0.627034\pi\)
\(402\) 0 0
\(403\) 230.576 0.0285007
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4348.35 0.529582
\(408\) 0 0
\(409\) 2047.66 0.247556 0.123778 0.992310i \(-0.460499\pi\)
0.123778 + 0.992310i \(0.460499\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5249.31 0.625427
\(414\) 0 0
\(415\) −19812.5 −2.34351
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5960.12 −0.694919 −0.347459 0.937695i \(-0.612956\pi\)
−0.347459 + 0.937695i \(0.612956\pi\)
\(420\) 0 0
\(421\) 311.567 0.0360685 0.0180342 0.999837i \(-0.494259\pi\)
0.0180342 + 0.999837i \(0.494259\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9079.00 −1.03623
\(426\) 0 0
\(427\) −11276.7 −1.27803
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17105.0 −1.91164 −0.955821 0.293950i \(-0.905030\pi\)
−0.955821 + 0.293950i \(0.905030\pi\)
\(432\) 0 0
\(433\) 2582.96 0.286672 0.143336 0.989674i \(-0.454217\pi\)
0.143336 + 0.989674i \(0.454217\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20362.5 2.22900
\(438\) 0 0
\(439\) −6797.87 −0.739054 −0.369527 0.929220i \(-0.620480\pi\)
−0.369527 + 0.929220i \(0.620480\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10454.1 −1.12119 −0.560596 0.828089i \(-0.689428\pi\)
−0.560596 + 0.828089i \(0.689428\pi\)
\(444\) 0 0
\(445\) 2408.90 0.256613
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17208.6 −1.80873 −0.904367 0.426755i \(-0.859657\pi\)
−0.904367 + 0.426755i \(0.859657\pi\)
\(450\) 0 0
\(451\) 6117.55 0.638723
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5082.40 0.523663
\(456\) 0 0
\(457\) 15788.1 1.61605 0.808025 0.589148i \(-0.200537\pi\)
0.808025 + 0.589148i \(0.200537\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5183.17 −0.523654 −0.261827 0.965115i \(-0.584325\pi\)
−0.261827 + 0.965115i \(0.584325\pi\)
\(462\) 0 0
\(463\) −5833.38 −0.585530 −0.292765 0.956184i \(-0.594575\pi\)
−0.292765 + 0.956184i \(0.594575\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6315.33 −0.625779 −0.312889 0.949790i \(-0.601297\pi\)
−0.312889 + 0.949790i \(0.601297\pi\)
\(468\) 0 0
\(469\) −15435.6 −1.51972
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2588.59 −0.251636
\(474\) 0 0
\(475\) −16261.2 −1.57077
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9735.88 −0.928692 −0.464346 0.885654i \(-0.653711\pi\)
−0.464346 + 0.885654i \(0.653711\pi\)
\(480\) 0 0
\(481\) −4221.28 −0.400153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17908.4 −1.67665
\(486\) 0 0
\(487\) 13447.9 1.25130 0.625649 0.780104i \(-0.284834\pi\)
0.625649 + 0.780104i \(0.284834\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7028.09 0.645974 0.322987 0.946403i \(-0.395313\pi\)
0.322987 + 0.946403i \(0.395313\pi\)
\(492\) 0 0
\(493\) 20253.2 1.85022
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 345.391 0.0311728
\(498\) 0 0
\(499\) 1655.44 0.148513 0.0742563 0.997239i \(-0.476342\pi\)
0.0742563 + 0.997239i \(0.476342\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1909.16 −0.169235 −0.0846174 0.996414i \(-0.526967\pi\)
−0.0846174 + 0.996414i \(0.526967\pi\)
\(504\) 0 0
\(505\) −25991.8 −2.29033
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8865.70 −0.772034 −0.386017 0.922492i \(-0.626149\pi\)
−0.386017 + 0.922492i \(0.626149\pi\)
\(510\) 0 0
\(511\) 18494.3 1.60106
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11806.2 1.01018
\(516\) 0 0
\(517\) −1131.62 −0.0962644
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9256.80 −0.778403 −0.389201 0.921153i \(-0.627249\pi\)
−0.389201 + 0.921153i \(0.627249\pi\)
\(522\) 0 0
\(523\) 13607.5 1.13770 0.568849 0.822442i \(-0.307389\pi\)
0.568849 + 0.822442i \(0.307389\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1343.23 0.111028
\(528\) 0 0
\(529\) 7737.14 0.635912
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5938.77 −0.482621
\(534\) 0 0
\(535\) 26946.2 2.17754
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3206.12 0.256210
\(540\) 0 0
\(541\) −483.548 −0.0384276 −0.0192138 0.999815i \(-0.506116\pi\)
−0.0192138 + 0.999815i \(0.506116\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18679.1 1.46812
\(546\) 0 0
\(547\) −4711.28 −0.368263 −0.184131 0.982902i \(-0.558947\pi\)
−0.184131 + 0.982902i \(0.558947\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 36275.0 2.80466
\(552\) 0 0
\(553\) −6144.73 −0.472515
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21650.7 1.64698 0.823492 0.567327i \(-0.192023\pi\)
0.823492 + 0.567327i \(0.192023\pi\)
\(558\) 0 0
\(559\) 2512.95 0.190137
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14023.0 −1.04973 −0.524866 0.851185i \(-0.675884\pi\)
−0.524866 + 0.851185i \(0.675884\pi\)
\(564\) 0 0
\(565\) −14868.0 −1.10708
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4875.33 −0.359200 −0.179600 0.983740i \(-0.557480\pi\)
−0.179600 + 0.983740i \(0.557480\pi\)
\(570\) 0 0
\(571\) −25969.1 −1.90328 −0.951640 0.307215i \(-0.900603\pi\)
−0.951640 + 0.307215i \(0.900603\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15895.1 −1.15282
\(576\) 0 0
\(577\) −23113.1 −1.66761 −0.833806 0.552058i \(-0.813843\pi\)
−0.833806 + 0.552058i \(0.813843\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 30628.9 2.18709
\(582\) 0 0
\(583\) −9459.06 −0.671963
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5471.66 0.384735 0.192368 0.981323i \(-0.438383\pi\)
0.192368 + 0.981323i \(0.438383\pi\)
\(588\) 0 0
\(589\) 2405.82 0.168302
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22609.2 −1.56568 −0.782841 0.622222i \(-0.786230\pi\)
−0.782841 + 0.622222i \(0.786230\pi\)
\(594\) 0 0
\(595\) 29607.7 2.04000
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3708.89 0.252990 0.126495 0.991967i \(-0.459627\pi\)
0.126495 + 0.991967i \(0.459627\pi\)
\(600\) 0 0
\(601\) −10478.2 −0.711170 −0.355585 0.934644i \(-0.615718\pi\)
−0.355585 + 0.934644i \(0.615718\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17389.1 1.16854
\(606\) 0 0
\(607\) 10713.6 0.716392 0.358196 0.933646i \(-0.383392\pi\)
0.358196 + 0.933646i \(0.383392\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1098.55 0.0727376
\(612\) 0 0
\(613\) 7860.62 0.517924 0.258962 0.965887i \(-0.416619\pi\)
0.258962 + 0.965887i \(0.416619\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8528.00 0.556442 0.278221 0.960517i \(-0.410255\pi\)
0.278221 + 0.960517i \(0.410255\pi\)
\(618\) 0 0
\(619\) −8022.89 −0.520949 −0.260474 0.965481i \(-0.583879\pi\)
−0.260474 + 0.965481i \(0.583879\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3724.01 −0.239485
\(624\) 0 0
\(625\) −17014.7 −1.08894
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24591.2 −1.55885
\(630\) 0 0
\(631\) −3984.40 −0.251373 −0.125687 0.992070i \(-0.540113\pi\)
−0.125687 + 0.992070i \(0.540113\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34506.7 2.15647
\(636\) 0 0
\(637\) −3112.43 −0.193593
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14750.2 0.908892 0.454446 0.890774i \(-0.349837\pi\)
0.454446 + 0.890774i \(0.349837\pi\)
\(642\) 0 0
\(643\) 9810.74 0.601707 0.300854 0.953670i \(-0.402728\pi\)
0.300854 + 0.953670i \(0.402728\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5683.55 0.345353 0.172677 0.984979i \(-0.444758\pi\)
0.172677 + 0.984979i \(0.444758\pi\)
\(648\) 0 0
\(649\) 3138.47 0.189824
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3405.52 0.204086 0.102043 0.994780i \(-0.467462\pi\)
0.102043 + 0.994780i \(0.467462\pi\)
\(654\) 0 0
\(655\) −25811.8 −1.53977
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19049.1 −1.12602 −0.563011 0.826449i \(-0.690357\pi\)
−0.563011 + 0.826449i \(0.690357\pi\)
\(660\) 0 0
\(661\) 20422.5 1.20173 0.600864 0.799351i \(-0.294823\pi\)
0.600864 + 0.799351i \(0.294823\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 53029.7 3.09233
\(666\) 0 0
\(667\) 35458.3 2.05840
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6742.13 −0.387894
\(672\) 0 0
\(673\) 8613.45 0.493349 0.246675 0.969098i \(-0.420662\pi\)
0.246675 + 0.969098i \(0.420662\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10469.1 0.594329 0.297164 0.954826i \(-0.403959\pi\)
0.297164 + 0.954826i \(0.403959\pi\)
\(678\) 0 0
\(679\) 27685.2 1.56475
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9490.28 −0.531677 −0.265839 0.964018i \(-0.585649\pi\)
−0.265839 + 0.964018i \(0.585649\pi\)
\(684\) 0 0
\(685\) 1465.73 0.0817556
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9182.63 0.507737
\(690\) 0 0
\(691\) 15814.7 0.870652 0.435326 0.900273i \(-0.356633\pi\)
0.435326 + 0.900273i \(0.356633\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18047.1 −0.984985
\(696\) 0 0
\(697\) −34596.6 −1.88011
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15790.6 −0.850791 −0.425395 0.905008i \(-0.639865\pi\)
−0.425395 + 0.905008i \(0.639865\pi\)
\(702\) 0 0
\(703\) −44044.7 −2.36298
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 40181.7 2.13746
\(708\) 0 0
\(709\) 27542.9 1.45895 0.729474 0.684009i \(-0.239765\pi\)
0.729474 + 0.684009i \(0.239765\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2351.66 0.123521
\(714\) 0 0
\(715\) 3038.68 0.158937
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19578.6 1.01552 0.507761 0.861498i \(-0.330473\pi\)
0.507761 + 0.861498i \(0.330473\pi\)
\(720\) 0 0
\(721\) −18251.7 −0.942756
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −28316.4 −1.45054
\(726\) 0 0
\(727\) 20948.0 1.06866 0.534330 0.845276i \(-0.320564\pi\)
0.534330 + 0.845276i \(0.320564\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14639.3 0.740702
\(732\) 0 0
\(733\) −825.349 −0.0415893 −0.0207946 0.999784i \(-0.506620\pi\)
−0.0207946 + 0.999784i \(0.506620\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9228.69 −0.461253
\(738\) 0 0
\(739\) 22821.1 1.13598 0.567988 0.823037i \(-0.307722\pi\)
0.567988 + 0.823037i \(0.307722\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12056.5 0.595301 0.297650 0.954675i \(-0.403797\pi\)
0.297650 + 0.954675i \(0.403797\pi\)
\(744\) 0 0
\(745\) −13244.8 −0.651345
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −41657.1 −2.03220
\(750\) 0 0
\(751\) −7722.32 −0.375221 −0.187611 0.982243i \(-0.560074\pi\)
−0.187611 + 0.982243i \(0.560074\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −44689.5 −2.15420
\(756\) 0 0
\(757\) −1440.60 −0.0691671 −0.0345835 0.999402i \(-0.511010\pi\)
−0.0345835 + 0.999402i \(0.511010\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30722.6 −1.46346 −0.731729 0.681595i \(-0.761286\pi\)
−0.731729 + 0.681595i \(0.761286\pi\)
\(762\) 0 0
\(763\) −28876.8 −1.37013
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3046.75 −0.143431
\(768\) 0 0
\(769\) −40280.9 −1.88890 −0.944451 0.328652i \(-0.893406\pi\)
−0.944451 + 0.328652i \(0.893406\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4282.45 −0.199261 −0.0996306 0.995024i \(-0.531766\pi\)
−0.0996306 + 0.995024i \(0.531766\pi\)
\(774\) 0 0
\(775\) −1877.99 −0.0870446
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −61965.0 −2.84997
\(780\) 0 0
\(781\) 206.503 0.00946128
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20180.3 0.917537
\(786\) 0 0
\(787\) −12571.6 −0.569416 −0.284708 0.958614i \(-0.591897\pi\)
−0.284708 + 0.958614i \(0.591897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 22985.0 1.03319
\(792\) 0 0
\(793\) 6545.11 0.293094
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28074.0 −1.24772 −0.623860 0.781536i \(-0.714436\pi\)
−0.623860 + 0.781536i \(0.714436\pi\)
\(798\) 0 0
\(799\) 6399.66 0.283359
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11057.4 0.485939
\(804\) 0 0
\(805\) 51835.9 2.26953
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −293.254 −0.0127445 −0.00637223 0.999980i \(-0.502028\pi\)
−0.00637223 + 0.999980i \(0.502028\pi\)
\(810\) 0 0
\(811\) −45634.2 −1.97588 −0.987938 0.154851i \(-0.950510\pi\)
−0.987938 + 0.154851i \(0.950510\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2944.34 0.126547
\(816\) 0 0
\(817\) 26220.0 1.12279
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5406.68 0.229835 0.114917 0.993375i \(-0.463340\pi\)
0.114917 + 0.993375i \(0.463340\pi\)
\(822\) 0 0
\(823\) 1228.88 0.0520487 0.0260243 0.999661i \(-0.491715\pi\)
0.0260243 + 0.999661i \(0.491715\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45769.8 1.92451 0.962256 0.272147i \(-0.0877337\pi\)
0.962256 + 0.272147i \(0.0877337\pi\)
\(828\) 0 0
\(829\) −13630.9 −0.571076 −0.285538 0.958367i \(-0.592172\pi\)
−0.285538 + 0.958367i \(0.592172\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18131.6 −0.754167
\(834\) 0 0
\(835\) 22210.9 0.920525
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −44987.6 −1.85119 −0.925593 0.378520i \(-0.876433\pi\)
−0.925593 + 0.378520i \(0.876433\pi\)
\(840\) 0 0
\(841\) 38778.4 1.59000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30920.0 1.25879
\(846\) 0 0
\(847\) −26882.5 −1.09055
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −43053.2 −1.73425
\(852\) 0 0
\(853\) 26043.2 1.04537 0.522685 0.852526i \(-0.324930\pi\)
0.522685 + 0.852526i \(0.324930\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13706.4 −0.546327 −0.273163 0.961968i \(-0.588070\pi\)
−0.273163 + 0.961968i \(0.588070\pi\)
\(858\) 0 0
\(859\) −45319.1 −1.80008 −0.900039 0.435810i \(-0.856462\pi\)
−0.900039 + 0.435810i \(0.856462\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28057.3 −1.10670 −0.553350 0.832949i \(-0.686651\pi\)
−0.553350 + 0.832949i \(0.686651\pi\)
\(864\) 0 0
\(865\) 24378.4 0.958253
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3673.83 −0.143413
\(870\) 0 0
\(871\) 8959.00 0.348524
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4531.85 0.175091
\(876\) 0 0
\(877\) 3923.18 0.151056 0.0755282 0.997144i \(-0.475936\pi\)
0.0755282 + 0.997144i \(0.475936\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2441.97 0.0933849 0.0466924 0.998909i \(-0.485132\pi\)
0.0466924 + 0.998909i \(0.485132\pi\)
\(882\) 0 0
\(883\) 44576.9 1.69890 0.849452 0.527667i \(-0.176933\pi\)
0.849452 + 0.527667i \(0.176933\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17095.5 −0.647137 −0.323568 0.946205i \(-0.604883\pi\)
−0.323568 + 0.946205i \(0.604883\pi\)
\(888\) 0 0
\(889\) −53345.2 −2.01253
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11462.3 0.429530
\(894\) 0 0
\(895\) 11436.4 0.427126
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4189.37 0.155421
\(900\) 0 0
\(901\) 53493.8 1.97795
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9662.79 −0.354919
\(906\) 0 0
\(907\) −33142.7 −1.21332 −0.606662 0.794960i \(-0.707492\pi\)
−0.606662 + 0.794960i \(0.707492\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −32750.5 −1.19108 −0.595539 0.803326i \(-0.703061\pi\)
−0.595539 + 0.803326i \(0.703061\pi\)
\(912\) 0 0
\(913\) 18312.5 0.663805
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 39903.5 1.43700
\(918\) 0 0
\(919\) 6095.76 0.218804 0.109402 0.993998i \(-0.465106\pi\)
0.109402 + 0.993998i \(0.465106\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −200.468 −0.00714897
\(924\) 0 0
\(925\) 34381.5 1.22212
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24708.0 −0.872597 −0.436298 0.899802i \(-0.643711\pi\)
−0.436298 + 0.899802i \(0.643711\pi\)
\(930\) 0 0
\(931\) −32475.0 −1.14321
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 17701.9 0.619161
\(936\) 0 0
\(937\) −44713.2 −1.55893 −0.779465 0.626446i \(-0.784509\pi\)
−0.779465 + 0.626446i \(0.784509\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10172.7 0.352414 0.176207 0.984353i \(-0.443617\pi\)
0.176207 + 0.984353i \(0.443617\pi\)
\(942\) 0 0
\(943\) −60570.1 −2.09166
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13346.6 0.457978 0.228989 0.973429i \(-0.426458\pi\)
0.228989 + 0.973429i \(0.426458\pi\)
\(948\) 0 0
\(949\) −10734.3 −0.367176
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36232.4 −1.23157 −0.615783 0.787916i \(-0.711160\pi\)
−0.615783 + 0.787916i \(0.711160\pi\)
\(954\) 0 0
\(955\) 2815.26 0.0953924
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2265.92 −0.0762988
\(960\) 0 0
\(961\) −29513.2 −0.990673
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −26495.6 −0.883857
\(966\) 0 0
\(967\) 5429.03 0.180544 0.0902718 0.995917i \(-0.471226\pi\)
0.0902718 + 0.995917i \(0.471226\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5645.37 0.186579 0.0932897 0.995639i \(-0.470262\pi\)
0.0932897 + 0.995639i \(0.470262\pi\)
\(972\) 0 0
\(973\) 27899.7 0.919242
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51103.6 1.67344 0.836719 0.547632i \(-0.184471\pi\)
0.836719 + 0.547632i \(0.184471\pi\)
\(978\) 0 0
\(979\) −2226.52 −0.0726863
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32082.5 1.04097 0.520485 0.853871i \(-0.325751\pi\)
0.520485 + 0.853871i \(0.325751\pi\)
\(984\) 0 0
\(985\) −1843.60 −0.0596364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25629.8 0.824043
\(990\) 0 0
\(991\) −27696.1 −0.887785 −0.443893 0.896080i \(-0.646403\pi\)
−0.443893 + 0.896080i \(0.646403\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −10906.6 −0.347500
\(996\) 0 0
\(997\) −41284.9 −1.31144 −0.655720 0.755004i \(-0.727635\pi\)
−0.655720 + 0.755004i \(0.727635\pi\)
\(998\) 0 0
\(999\) 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 216.4.a.f.1.1 2
3.2 odd 2 216.4.a.g.1.2 yes 2
4.3 odd 2 432.4.a.p.1.1 2
8.3 odd 2 1728.4.a.br.1.2 2
8.5 even 2 1728.4.a.bq.1.2 2
9.2 odd 6 648.4.i.o.433.1 4
9.4 even 3 648.4.i.r.217.2 4
9.5 odd 6 648.4.i.o.217.1 4
9.7 even 3 648.4.i.r.433.2 4
12.11 even 2 432.4.a.r.1.2 2
24.5 odd 2 1728.4.a.bi.1.1 2
24.11 even 2 1728.4.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.f.1.1 2 1.1 even 1 trivial
216.4.a.g.1.2 yes 2 3.2 odd 2
432.4.a.p.1.1 2 4.3 odd 2
432.4.a.r.1.2 2 12.11 even 2
648.4.i.o.217.1 4 9.5 odd 6
648.4.i.o.433.1 4 9.2 odd 6
648.4.i.r.217.2 4 9.4 even 3
648.4.i.r.433.2 4 9.7 even 3
1728.4.a.bi.1.1 2 24.5 odd 2
1728.4.a.bj.1.1 2 24.11 even 2
1728.4.a.bq.1.2 2 8.5 even 2
1728.4.a.br.1.2 2 8.3 odd 2