Properties

Label 4.5.b.a.3.1
Level 4
Weight 5
Character 4.3
Self dual Yes
Analytic conductor 0.413
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.413479852335\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 3.1
Root \(0\)
Character \(\chi\) = 4.3

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -14.0000 q^{5} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} -14.0000 q^{5} -64.0000 q^{8} +81.0000 q^{9} +56.0000 q^{10} -238.000 q^{13} +256.000 q^{16} +322.000 q^{17} -324.000 q^{18} -224.000 q^{20} -429.000 q^{25} +952.000 q^{26} +82.0000 q^{29} -1024.00 q^{32} -1288.00 q^{34} +1296.00 q^{36} +2162.00 q^{37} +896.000 q^{40} -3038.00 q^{41} -1134.00 q^{45} +2401.00 q^{49} +1716.00 q^{50} -3808.00 q^{52} +2482.00 q^{53} -328.000 q^{58} -6958.00 q^{61} +4096.00 q^{64} +3332.00 q^{65} +5152.00 q^{68} -5184.00 q^{72} +1442.00 q^{73} -8648.00 q^{74} -3584.00 q^{80} +6561.00 q^{81} +12152.0 q^{82} -4508.00 q^{85} -9758.00 q^{89} +4536.00 q^{90} -1918.00 q^{97} -9604.00 q^{98} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 16.0000 1.00000
\(5\) −14.0000 −0.560000 −0.280000 0.960000i \(-0.590334\pi\)
−0.280000 + 0.960000i \(0.590334\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −64.0000 −1.00000
\(9\) 81.0000 1.00000
\(10\) 56.0000 0.560000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −238.000 −1.40828 −0.704142 0.710059i \(-0.748668\pi\)
−0.704142 + 0.710059i \(0.748668\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 322.000 1.11419 0.557093 0.830450i \(-0.311917\pi\)
0.557093 + 0.830450i \(0.311917\pi\)
\(18\) −324.000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −224.000 −0.560000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −429.000 −0.686400
\(26\) 952.000 1.40828
\(27\) 0 0
\(28\) 0 0
\(29\) 82.0000 0.0975030 0.0487515 0.998811i \(-0.484476\pi\)
0.0487515 + 0.998811i \(0.484476\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1024.00 −1.00000
\(33\) 0 0
\(34\) −1288.00 −1.11419
\(35\) 0 0
\(36\) 1296.00 1.00000
\(37\) 2162.00 1.57925 0.789627 0.613587i \(-0.210274\pi\)
0.789627 + 0.613587i \(0.210274\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 896.000 0.560000
\(41\) −3038.00 −1.80726 −0.903629 0.428316i \(-0.859107\pi\)
−0.903629 + 0.428316i \(0.859107\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −1134.00 −0.560000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 1716.00 0.686400
\(51\) 0 0
\(52\) −3808.00 −1.40828
\(53\) 2482.00 0.883588 0.441794 0.897116i \(-0.354342\pi\)
0.441794 + 0.897116i \(0.354342\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −328.000 −0.0975030
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −6958.00 −1.86993 −0.934964 0.354743i \(-0.884568\pi\)
−0.934964 + 0.354743i \(0.884568\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 3332.00 0.788639
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 5152.00 1.11419
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −5184.00 −1.00000
\(73\) 1442.00 0.270595 0.135297 0.990805i \(-0.456801\pi\)
0.135297 + 0.990805i \(0.456801\pi\)
\(74\) −8648.00 −1.57925
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −3584.00 −0.560000
\(81\) 6561.00 1.00000
\(82\) 12152.0 1.80726
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −4508.00 −0.623945
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9758.00 −1.23192 −0.615958 0.787779i \(-0.711231\pi\)
−0.615958 + 0.787779i \(0.711231\pi\)
\(90\) 4536.00 0.560000
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1918.00 −0.203847 −0.101924 0.994792i \(-0.532500\pi\)
−0.101924 + 0.994792i \(0.532500\pi\)
\(98\) −9604.00 −1.00000
\(99\) 0 0
\(100\) −6864.00 −0.686400
\(101\) 18802.0 1.84315 0.921576 0.388197i \(-0.126902\pi\)
0.921576 + 0.388197i \(0.126902\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 15232.0 1.40828
\(105\) 0 0
\(106\) −9928.00 −0.883588
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 9362.00 0.787981 0.393990 0.919115i \(-0.371094\pi\)
0.393990 + 0.919115i \(0.371094\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −24638.0 −1.92952 −0.964758 0.263137i \(-0.915243\pi\)
−0.964758 + 0.263137i \(0.915243\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1312.00 0.0975030
\(117\) −19278.0 −1.40828
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 27832.0 1.86993
\(123\) 0 0
\(124\) 0 0
\(125\) 14756.0 0.944384
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −16384.0 −1.00000
\(129\) 0 0
\(130\) −13328.0 −0.788639
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −20608.0 −1.11419
\(137\) 6562.00 0.349619 0.174810 0.984602i \(-0.444069\pi\)
0.174810 + 0.984602i \(0.444069\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 20736.0 1.00000
\(145\) −1148.00 −0.0546017
\(146\) −5768.00 −0.270595
\(147\) 0 0
\(148\) 34592.0 1.57925
\(149\) −33998.0 −1.53137 −0.765686 0.643214i \(-0.777600\pi\)
−0.765686 + 0.643214i \(0.777600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 26082.0 1.11419
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20398.0 −0.827539 −0.413769 0.910382i \(-0.635788\pi\)
−0.413769 + 0.910382i \(0.635788\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 14336.0 0.560000
\(161\) 0 0
\(162\) −26244.0 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −48608.0 −1.80726
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28083.0 0.983264
\(170\) 18032.0 0.623945
\(171\) 0 0
\(172\) 0 0
\(173\) 49042.0 1.63861 0.819306 0.573357i \(-0.194359\pi\)
0.819306 + 0.573357i \(0.194359\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 39032.0 1.23192
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −18144.0 −0.560000
\(181\) −64078.0 −1.95592 −0.977962 0.208785i \(-0.933049\pi\)
−0.977962 + 0.208785i \(0.933049\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −30268.0 −0.884383
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −38398.0 −1.03085 −0.515423 0.856936i \(-0.672365\pi\)
−0.515423 + 0.856936i \(0.672365\pi\)
\(194\) 7672.00 0.203847
\(195\) 0 0
\(196\) 38416.0 1.00000
\(197\) 74482.0 1.91919 0.959597 0.281378i \(-0.0907915\pi\)
0.959597 + 0.281378i \(0.0907915\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 27456.0 0.686400
\(201\) 0 0
\(202\) −75208.0 −1.84315
\(203\) 0 0
\(204\) 0 0
\(205\) 42532.0 1.01206
\(206\) 0 0
\(207\) 0 0
\(208\) −60928.0 −1.40828
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 39712.0 0.883588
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −37448.0 −0.787981
\(219\) 0 0
\(220\) 0 0
\(221\) −76636.0 −1.56909
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −34749.0 −0.686400
\(226\) 98552.0 1.92952
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 90482.0 1.72541 0.862703 0.505711i \(-0.168770\pi\)
0.862703 + 0.505711i \(0.168770\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5248.00 −0.0975030
\(233\) −64478.0 −1.18768 −0.593840 0.804583i \(-0.702389\pi\)
−0.593840 + 0.804583i \(0.702389\pi\)
\(234\) 77112.0 1.40828
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 58562.0 1.00828 0.504141 0.863621i \(-0.331809\pi\)
0.504141 + 0.863621i \(0.331809\pi\)
\(242\) −58564.0 −1.00000
\(243\) 0 0
\(244\) −111328. −1.86993
\(245\) −33614.0 −0.560000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −59024.0 −0.944384
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 128002. 1.93799 0.968993 0.247089i \(-0.0794741\pi\)
0.968993 + 0.247089i \(0.0794741\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 53312.0 0.788639
\(261\) 6642.00 0.0975030
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −34748.0 −0.494810
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −125678. −1.73682 −0.868410 0.495847i \(-0.834858\pi\)
−0.868410 + 0.495847i \(0.834858\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 82432.0 1.11419
\(273\) 0 0
\(274\) −26248.0 −0.349619
\(275\) 0 0
\(276\) 0 0
\(277\) −100558. −1.31056 −0.655280 0.755386i \(-0.727449\pi\)
−0.655280 + 0.755386i \(0.727449\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 55522.0 0.703157 0.351579 0.936158i \(-0.385645\pi\)
0.351579 + 0.936158i \(0.385645\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −82944.0 −1.00000
\(289\) 20163.0 0.241412
\(290\) 4592.00 0.0546017
\(291\) 0 0
\(292\) 23072.0 0.270595
\(293\) 153202. 1.78455 0.892276 0.451490i \(-0.149107\pi\)
0.892276 + 0.451490i \(0.149107\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −138368. −1.57925
\(297\) 0 0
\(298\) 135992. 1.53137
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 97412.0 1.04716
\(306\) −104328. −1.11419
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −193438. −1.97448 −0.987241 0.159234i \(-0.949098\pi\)
−0.987241 + 0.159234i \(0.949098\pi\)
\(314\) 81592.0 0.827539
\(315\) 0 0
\(316\) 0 0
\(317\) −178478. −1.77609 −0.888047 0.459752i \(-0.847938\pi\)
−0.888047 + 0.459752i \(0.847938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −57344.0 −0.560000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 104976. 1.00000
\(325\) 102102. 0.966646
\(326\) 0 0
\(327\) 0 0
\(328\) 194432. 1.80726
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 175122. 1.57925
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −104638. −0.921361 −0.460680 0.887566i \(-0.652395\pi\)
−0.460680 + 0.887566i \(0.652395\pi\)
\(338\) −112332. −0.983264
\(339\) 0 0
\(340\) −72128.0 −0.623945
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −196168. −1.63861
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 114002. 0.935969 0.467985 0.883737i \(-0.344980\pi\)
0.467985 + 0.883737i \(0.344980\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −46718.0 −0.374917 −0.187458 0.982273i \(-0.560025\pi\)
−0.187458 + 0.982273i \(0.560025\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −156128. −1.23192
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 72576.0 0.560000
\(361\) 130321. 1.00000
\(362\) 256312. 1.95592
\(363\) 0 0
\(364\) 0 0
\(365\) −20188.0 −0.151533
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −246078. −1.80726
\(370\) 121072. 0.884383
\(371\) 0 0
\(372\) 0 0
\(373\) 24242.0 0.174241 0.0871206 0.996198i \(-0.472233\pi\)
0.0871206 + 0.996198i \(0.472233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19516.0 −0.137312
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 153592. 1.03085
\(387\) 0 0
\(388\) −30688.0 −0.203847
\(389\) −159758. −1.05576 −0.527878 0.849320i \(-0.677012\pi\)
−0.527878 + 0.849320i \(0.677012\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −153664. −1.00000
\(393\) 0 0
\(394\) −297928. −1.91919
\(395\) 0 0
\(396\) 0 0
\(397\) 107282. 0.680684 0.340342 0.940302i \(-0.389457\pi\)
0.340342 + 0.940302i \(0.389457\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −109824. −0.686400
\(401\) 315202. 1.96020 0.980100 0.198506i \(-0.0636090\pi\)
0.980100 + 0.198506i \(0.0636090\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 300832. 1.84315
\(405\) −91854.0 −0.560000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 276962. 1.65567 0.827835 0.560972i \(-0.189573\pi\)
0.827835 + 0.560972i \(0.189573\pi\)
\(410\) −170128. −1.01206
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 243712. 1.40828
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −351118. −1.98102 −0.990510 0.137440i \(-0.956113\pi\)
−0.990510 + 0.137440i \(0.956113\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −158848. −0.883588
\(425\) −138138. −0.764778
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −290878. −1.55144 −0.775720 0.631077i \(-0.782613\pi\)
−0.775720 + 0.631077i \(0.782613\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 149792. 0.787981
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 194481. 1.00000
\(442\) 306544. 1.56909
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 136612. 0.689872
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 89602.0 0.444452 0.222226 0.974995i \(-0.428668\pi\)
0.222226 + 0.974995i \(0.428668\pi\)
\(450\) 138996. 0.686400
\(451\) 0 0
\(452\) −394208. −1.92952
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 304802. 1.45944 0.729719 0.683748i \(-0.239651\pi\)
0.729719 + 0.683748i \(0.239651\pi\)
\(458\) −361928. −1.72541
\(459\) 0 0
\(460\) 0 0
\(461\) −152558. −0.717849 −0.358925 0.933367i \(-0.616856\pi\)
−0.358925 + 0.933367i \(0.616856\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 20992.0 0.0975030
\(465\) 0 0
\(466\) 257912. 1.18768
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −308448. −1.40828
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 201042. 0.883588
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −514556. −2.22404
\(482\) −234248. −1.00828
\(483\) 0 0
\(484\) 234256. 1.00000
\(485\) 26852.0 0.114155
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 445312. 1.86993
\(489\) 0 0
\(490\) 134456. 0.560000
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 26404.0 0.108637
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 236096. 0.944384
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −263228. −1.03217
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 324562. 1.25274 0.626372 0.779525i \(-0.284539\pi\)
0.626372 + 0.779525i \(0.284539\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −262144. −1.00000
\(513\) 0 0
\(514\) −512008. −1.93799
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −213248. −0.788639
\(521\) −231518. −0.852922 −0.426461 0.904506i \(-0.640240\pi\)
−0.426461 + 0.904506i \(0.640240\pi\)
\(522\) −26568.0 −0.0975030
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 138992. 0.494810
\(531\) 0 0
\(532\) 0 0
\(533\) 723044. 2.54513
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 502712. 1.73682
\(539\) 0 0
\(540\) 0 0
\(541\) −120238. −0.410816 −0.205408 0.978676i \(-0.565852\pi\)
−0.205408 + 0.978676i \(0.565852\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −329728. −1.11419
\(545\) −131068. −0.441269
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 104992. 0.349619
\(549\) −563598. −1.86993
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 402232. 1.31056
\(555\) 0 0
\(556\) 0 0
\(557\) −511598. −1.64899 −0.824496 0.565868i \(-0.808541\pi\)
−0.824496 + 0.565868i \(0.808541\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −222088. −0.703157
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 344932. 1.08053
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −434078. −1.34074 −0.670368 0.742029i \(-0.733864\pi\)
−0.670368 + 0.742029i \(0.733864\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 331776. 1.00000
\(577\) 656642. 1.97232 0.986159 0.165801i \(-0.0530210\pi\)
0.986159 + 0.165801i \(0.0530210\pi\)
\(578\) −80652.0 −0.241412
\(579\) 0 0
\(580\) −18368.0 −0.0546017
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −92288.0 −0.270595
\(585\) 269892. 0.788639
\(586\) −612808. −1.78455
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 553472. 1.57925
\(593\) 161602. 0.459555 0.229777 0.973243i \(-0.426200\pi\)
0.229777 + 0.973243i \(0.426200\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −543968. −1.53137
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 492002. 1.36213 0.681064 0.732224i \(-0.261518\pi\)
0.681064 + 0.732224i \(0.261518\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −204974. −0.560000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −389648. −1.04716
\(611\) 0 0
\(612\) 417312. 1.11419
\(613\) −746638. −1.98696 −0.993480 0.114006i \(-0.963632\pi\)
−0.993480 + 0.114006i \(0.963632\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −717278. −1.88416 −0.942079 0.335392i \(-0.891131\pi\)
−0.942079 + 0.335392i \(0.891131\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 61541.0 0.157545
\(626\) 773752. 1.97448
\(627\) 0 0
\(628\) −326368. −0.827539
\(629\) 696164. 1.75959
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 713912. 1.77609
\(635\) 0 0
\(636\) 0 0
\(637\) −571438. −1.40828
\(638\) 0 0
\(639\) 0 0
\(640\) 229376. 0.560000
\(641\) 661762. 1.61059 0.805296 0.592872i \(-0.202006\pi\)
0.805296 + 0.592872i \(0.202006\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −419904. −1.00000
\(649\) 0 0
\(650\) −408408. −0.966646
\(651\) 0 0
\(652\) 0 0
\(653\) −455918. −1.06920 −0.534602 0.845104i \(-0.679538\pi\)
−0.534602 + 0.845104i \(0.679538\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −777728. −1.80726
\(657\) 116802. 0.270595
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 513842. 1.17605 0.588026 0.808842i \(-0.299905\pi\)
0.588026 + 0.808842i \(0.299905\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −700488. −1.57925
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −312958. −0.690965 −0.345482 0.938425i \(-0.612285\pi\)
−0.345482 + 0.938425i \(0.612285\pi\)
\(674\) 418552. 0.921361
\(675\) 0 0
\(676\) 449328. 0.983264
\(677\) 905842. 1.97640 0.988201 0.153165i \(-0.0489466\pi\)
0.988201 + 0.153165i \(0.0489466\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 288512. 0.623945
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −91868.0 −0.195787
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −590716. −1.24434
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 784672. 1.63861
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −978236. −2.01362
\(698\) −456008. −0.935969
\(699\) 0 0
\(700\) 0 0
\(701\) 712402. 1.44974 0.724868 0.688887i \(-0.241901\pi\)
0.724868 + 0.688887i \(0.241901\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 186872. 0.374917
\(707\) 0 0
\(708\) 0 0
\(709\) −737038. −1.46621 −0.733107 0.680113i \(-0.761931\pi\)
−0.733107 + 0.680113i \(0.761931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 624512. 1.23192
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −290304. −0.560000
\(721\) 0 0
\(722\) −521284. −1.00000
\(723\) 0 0
\(724\) −1.02525e6 −1.95592
\(725\) −35178.0 −0.0669260
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 80752.0 0.151533
\(731\) 0 0
\(732\) 0 0
\(733\) 1.02792e6 1.91316 0.956582 0.291463i \(-0.0941421\pi\)
0.956582 + 0.291463i \(0.0941421\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 984312. 1.80726
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −484288. −0.884383
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 475972. 0.857569
\(746\) −96968.0 −0.174241
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 78064.0 0.137312
\(755\) 0 0
\(756\) 0 0
\(757\) 270002. 0.471167 0.235584 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.15216e6 −1.98949 −0.994747 0.102362i \(-0.967360\pi\)
−0.994747 + 0.102362i \(0.967360\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −365148. −0.623945
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −257278. −0.435061 −0.217530 0.976054i \(-0.569800\pi\)
−0.217530 + 0.976054i \(0.569800\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −614368. −1.03085
\(773\) −1.04296e6 −1.74545 −0.872726 0.488211i \(-0.837650\pi\)
−0.872726 + 0.488211i \(0.837650\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 122752. 0.203847
\(777\) 0 0
\(778\) 639032. 1.05576
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 614656. 1.00000
\(785\) 285572. 0.463422
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.19171e6 1.91919
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.65600e6 2.63339
\(794\) −429128. −0.680684
\(795\) 0 0
\(796\) 0 0
\(797\) −38318.0 −0.0603235 −0.0301617 0.999545i \(-0.509602\pi\)
−0.0301617 + 0.999545i \(0.509602\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 439296. 0.686400
\(801\) −790398. −1.23192
\(802\) −1.26081e6 −1.96020
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −1.20333e6 −1.84315
\(809\) 995362. 1.52084 0.760421 0.649431i \(-0.224993\pi\)
0.760421 + 0.649431i \(0.224993\pi\)
\(810\) 367416. 0.560000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.10785e6 −1.65567
\(819\) 0 0
\(820\) 680512. 1.01206
\(821\) −611918. −0.907835 −0.453917 0.891044i \(-0.649974\pi\)
−0.453917 + 0.891044i \(0.649974\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 208082. 0.302779 0.151389 0.988474i \(-0.451625\pi\)
0.151389 + 0.988474i \(0.451625\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −974848. −1.40828
\(833\) 773122. 1.11419
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −700557. −0.990493
\(842\) 1.40447e6 1.98102
\(843\) 0 0
\(844\) 0 0
\(845\) −393162. −0.550628
\(846\) 0 0
\(847\) 0 0
\(848\) 635392. 0.883588
\(849\) 0 0
\(850\) 552552. 0.764778
\(851\) 0 0
\(852\) 0 0
\(853\) −1.28712e6 −1.76897 −0.884485 0.466569i \(-0.845490\pi\)
−0.884485 + 0.466569i \(0.845490\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.25360e6 1.70686 0.853430 0.521207i \(-0.174518\pi\)
0.853430 + 0.521207i \(0.174518\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −686588. −0.917622
\(866\) 1.16351e6 1.55144
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −599168. −0.787981
\(873\) −155358. −0.203847
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.05384e6 1.37018 0.685088 0.728460i \(-0.259764\pi\)
0.685088 + 0.728460i \(0.259764\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.00768e6 −1.29828 −0.649142 0.760667i \(-0.724872\pi\)
−0.649142 + 0.760667i \(0.724872\pi\)
\(882\) −777924. −1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −1.22618e6 −1.56909
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −546448. −0.689872
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −358408. −0.444452
\(899\) 0 0
\(900\) −555984. −0.686400
\(901\) 799204. 0.984483
\(902\) 0 0
\(903\) 0 0
\(904\) 1.57683e6 1.92952
\(905\) 897092. 1.09532
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1.52296e6 1.84315
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.21921e6 −1.45944
\(915\) 0 0
\(916\) 1.44771e6 1.72541
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 610232. 0.717849
\(923\) 0 0
\(924\) 0 0
\(925\) −927498. −1.08400
\(926\) 0 0
\(927\) 0 0
\(928\) −83968.0 −0.0975030
\(929\) −1.65952e6 −1.92287 −0.961436 0.275027i \(-0.911313\pi\)
−0.961436 + 0.275027i \(0.911313\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.03165e6 −1.18768
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.23379e6 1.40828
\(937\) −1.57104e6 −1.78940 −0.894700 0.446667i \(-0.852611\pi\)
−0.894700 + 0.446667i \(0.852611\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 425362. 0.480374 0.240187 0.970727i \(-0.422791\pi\)
0.240187 + 0.970727i \(0.422791\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −343196. −0.381074
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −303518. −0.334194 −0.167097 0.985940i \(-0.553439\pi\)
−0.167097 + 0.985940i \(0.553439\pi\)
\(954\) −804168. −0.883588
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 2.05822e6 2.22404
\(963\) 0 0
\(964\) 936992. 1.00828
\(965\) 537572. 0.577274
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −937024. −1.00000
\(969\) 0 0
\(970\) −107408. −0.114155
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.78125e6 −1.86993
\(977\) 1.66304e6 1.74226 0.871132 0.491048i \(-0.163386\pi\)
0.871132 + 0.491048i \(0.163386\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −537824. −0.560000
\(981\) 758322. 0.787981
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1.04275e6 −1.07475
\(986\) −105616. −0.108637
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.43448e6 1.44313 0.721564 0.692348i \(-0.243424\pi\)
0.721564 + 0.692348i \(0.243424\pi\)
\(998\) 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 \(a_p\))