Properties

Label 7.3.b.a.6.1
Level 7
Weight 3
Character 7.6
Self dual Yes
Analytic conductor 0.191
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.190736185052\)
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 6.1
Root \(0\)
Character \(\chi\) = 7.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +5.00000 q^{4} -7.00000 q^{7} -3.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} +5.00000 q^{4} -7.00000 q^{7} -3.00000 q^{8} +9.00000 q^{9} -6.00000 q^{11} +21.0000 q^{14} -11.0000 q^{16} -27.0000 q^{18} +18.0000 q^{22} +18.0000 q^{23} +25.0000 q^{25} -35.0000 q^{28} -54.0000 q^{29} +45.0000 q^{32} +45.0000 q^{36} -38.0000 q^{37} +58.0000 q^{43} -30.0000 q^{44} -54.0000 q^{46} +49.0000 q^{49} -75.0000 q^{50} -6.00000 q^{53} +21.0000 q^{56} +162.000 q^{58} -63.0000 q^{63} -91.0000 q^{64} -118.000 q^{67} +114.000 q^{71} -27.0000 q^{72} +114.000 q^{74} +42.0000 q^{77} -94.0000 q^{79} +81.0000 q^{81} -174.000 q^{86} +18.0000 q^{88} +90.0000 q^{92} -147.000 q^{98} -54.0000 q^{99} +O(q^{100})\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.50000 −0.750000 0.661438i \(-0.769947\pi\)
−0.750000 + 0.661438i \(0.769947\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 5.00000 1.25000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) −3.00000 −0.375000
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −6.00000 −0.545455 −0.272727 0.962091i \(-0.587926\pi\)
−0.272727 + 0.962091i \(0.587926\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 21.0000 1.50000
\(15\) 0 0
\(16\) −11.0000 −0.687500
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −27.0000 −1.50000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 18.0000 0.818182
\(23\) 18.0000 0.782609 0.391304 0.920261i \(-0.372024\pi\)
0.391304 + 0.920261i \(0.372024\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −35.0000 −1.25000
\(29\) −54.0000 −1.86207 −0.931034 0.364931i \(-0.881093\pi\)
−0.931034 + 0.364931i \(0.881093\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 45.0000 1.40625
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 45.0000 1.25000
\(37\) −38.0000 −1.02703 −0.513514 0.858082i \(-0.671656\pi\)
−0.513514 + 0.858082i \(0.671656\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 58.0000 1.34884 0.674419 0.738349i \(-0.264394\pi\)
0.674419 + 0.738349i \(0.264394\pi\)
\(44\) −30.0000 −0.681818
\(45\) 0 0
\(46\) −54.0000 −1.17391
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) −75.0000 −1.50000
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.113208 −0.0566038 0.998397i \(-0.518027\pi\)
−0.0566038 + 0.998397i \(0.518027\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 21.0000 0.375000
\(57\) 0 0
\(58\) 162.000 2.79310
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −63.0000 −1.00000
\(64\) −91.0000 −1.42188
\(65\) 0 0
\(66\) 0 0
\(67\) −118.000 −1.76119 −0.880597 0.473866i \(-0.842858\pi\)
−0.880597 + 0.473866i \(0.842858\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 114.000 1.60563 0.802817 0.596226i \(-0.203334\pi\)
0.802817 + 0.596226i \(0.203334\pi\)
\(72\) −27.0000 −0.375000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 114.000 1.54054
\(75\) 0 0
\(76\) 0 0
\(77\) 42.0000 0.545455
\(78\) 0 0
\(79\) −94.0000 −1.18987 −0.594937 0.803773i \(-0.702823\pi\)
−0.594937 + 0.803773i \(0.702823\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −174.000 −2.02326
\(87\) 0 0
\(88\) 18.0000 0.204545
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 90.0000 0.978261
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −147.000 −1.50000
\(99\) −54.0000 −0.545455
\(100\) 125.000 1.25000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 0.169811
\(107\) 186.000 1.73832 0.869159 0.494533i \(-0.164661\pi\)
0.869159 + 0.494533i \(0.164661\pi\)
\(108\) 0 0
\(109\) 106.000 0.972477 0.486239 0.873826i \(-0.338369\pi\)
0.486239 + 0.873826i \(0.338369\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 77.0000 0.687500
\(113\) −222.000 −1.96460 −0.982301 0.187310i \(-0.940023\pi\)
−0.982301 + 0.187310i \(0.940023\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −270.000 −2.32759
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 189.000 1.50000
\(127\) 2.00000 0.0157480 0.00787402 0.999969i \(-0.497494\pi\)
0.00787402 + 0.999969i \(0.497494\pi\)
\(128\) 93.0000 0.726562
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 354.000 2.64179
\(135\) 0 0
\(136\) 0 0
\(137\) −174.000 −1.27007 −0.635036 0.772482i \(-0.719015\pi\)
−0.635036 + 0.772482i \(0.719015\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −342.000 −2.40845
\(143\) 0 0
\(144\) −99.0000 −0.687500
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −190.000 −1.28378
\(149\) 186.000 1.24832 0.624161 0.781296i \(-0.285441\pi\)
0.624161 + 0.781296i \(0.285441\pi\)
\(150\) 0 0
\(151\) 274.000 1.81457 0.907285 0.420517i \(-0.138151\pi\)
0.907285 + 0.420517i \(0.138151\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −126.000 −0.818182
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 282.000 1.78481
\(159\) 0 0
\(160\) 0 0
\(161\) −126.000 −0.782609
\(162\) −243.000 −1.50000
\(163\) 74.0000 0.453988 0.226994 0.973896i \(-0.427110\pi\)
0.226994 + 0.973896i \(0.427110\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 290.000 1.68605
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 66.0000 0.375000
\(177\) 0 0
\(178\) 0 0
\(179\) −342.000 −1.91061 −0.955307 0.295615i \(-0.904476\pi\)
−0.955307 + 0.295615i \(0.904476\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −54.0000 −0.293478
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −318.000 −1.66492 −0.832461 0.554084i \(-0.813069\pi\)
−0.832461 + 0.554084i \(0.813069\pi\)
\(192\) 0 0
\(193\) −62.0000 −0.321244 −0.160622 0.987016i \(-0.551350\pi\)
−0.160622 + 0.987016i \(0.551350\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 245.000 1.25000
\(197\) 282.000 1.43147 0.715736 0.698371i \(-0.246091\pi\)
0.715736 + 0.698371i \(0.246091\pi\)
\(198\) 162.000 0.818182
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −75.0000 −0.375000
\(201\) 0 0
\(202\) 0 0
\(203\) 378.000 1.86207
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 162.000 0.782609
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −278.000 −1.31754 −0.658768 0.752346i \(-0.728922\pi\)
−0.658768 + 0.752346i \(0.728922\pi\)
\(212\) −30.0000 −0.141509
\(213\) 0 0
\(214\) −558.000 −2.60748
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −318.000 −1.45872
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −315.000 −1.40625
\(225\) 225.000 1.00000
\(226\) 666.000 2.94690
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 162.000 0.698276
\(233\) 18.0000 0.0772532 0.0386266 0.999254i \(-0.487702\pi\)
0.0386266 + 0.999254i \(0.487702\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −222.000 −0.928870 −0.464435 0.885607i \(-0.653743\pi\)
−0.464435 + 0.885607i \(0.653743\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 255.000 1.05372
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −315.000 −1.25000
\(253\) −108.000 −0.426877
\(254\) −6.00000 −0.0236220
\(255\) 0 0
\(256\) 85.0000 0.332031
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 266.000 1.02703
\(260\) 0 0
\(261\) −486.000 −1.86207
\(262\) 0 0
\(263\) 498.000 1.89354 0.946768 0.321917i \(-0.104327\pi\)
0.946768 + 0.321917i \(0.104327\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −590.000 −2.20149
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 522.000 1.90511
\(275\) −150.000 −0.545455
\(276\) 0 0
\(277\) −454.000 −1.63899 −0.819495 0.573087i \(-0.805746\pi\)
−0.819495 + 0.573087i \(0.805746\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 114.000 0.405694 0.202847 0.979210i \(-0.434981\pi\)
0.202847 + 0.979210i \(0.434981\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 570.000 2.00704
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 405.000 1.40625
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 114.000 0.385135
\(297\) 0 0
\(298\) −558.000 −1.87248
\(299\) 0 0
\(300\) 0 0
\(301\) −406.000 −1.34884
\(302\) −822.000 −2.72185
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 210.000 0.681818
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −470.000 −1.48734
\(317\) 522.000 1.64669 0.823344 0.567543i \(-0.192106\pi\)
0.823344 + 0.567543i \(0.192106\pi\)
\(318\) 0 0
\(319\) 324.000 1.01567
\(320\) 0 0
\(321\) 0 0
\(322\) 378.000 1.17391
\(323\) 0 0
\(324\) 405.000 1.25000
\(325\) 0 0
\(326\) −222.000 −0.680982
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 634.000 1.91541 0.957704 0.287755i \(-0.0929090\pi\)
0.957704 + 0.287755i \(0.0929090\pi\)
\(332\) 0 0
\(333\) −342.000 −1.02703
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 226.000 0.670623 0.335312 0.942107i \(-0.391158\pi\)
0.335312 + 0.942107i \(0.391158\pi\)
\(338\) −507.000 −1.50000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) −174.000 −0.505814
\(345\) 0 0
\(346\) 0 0
\(347\) −678.000 −1.95389 −0.976945 0.213490i \(-0.931517\pi\)
−0.976945 + 0.213490i \(0.931517\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 525.000 1.50000
\(351\) 0 0
\(352\) −270.000 −0.767045
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1026.00 2.86592
\(359\) −654.000 −1.82173 −0.910864 0.412708i \(-0.864583\pi\)
−0.910864 + 0.412708i \(0.864583\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −198.000 −0.538043
\(369\) 0 0
\(370\) 0 0
\(371\) 42.0000 0.113208
\(372\) 0 0
\(373\) −262.000 −0.702413 −0.351206 0.936298i \(-0.614228\pi\)
−0.351206 + 0.936298i \(0.614228\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −614.000 −1.62005 −0.810026 0.586393i \(-0.800547\pi\)
−0.810026 + 0.586393i \(0.800547\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 954.000 2.49738
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 186.000 0.481865
\(387\) 522.000 1.34884
\(388\) 0 0
\(389\) 666.000 1.71208 0.856041 0.516908i \(-0.172917\pi\)
0.856041 + 0.516908i \(0.172917\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −147.000 −0.375000
\(393\) 0 0
\(394\) −846.000 −2.14721
\(395\) 0 0
\(396\) −270.000 −0.681818
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −275.000 −0.687500
\(401\) 354.000 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1134.00 −2.79310
\(407\) 228.000 0.560197
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −486.000 −1.17391
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −166.000 −0.394299 −0.197150 0.980373i \(-0.563168\pi\)
−0.197150 + 0.980373i \(0.563168\pi\)
\(422\) 834.000 1.97630
\(423\) 0 0
\(424\) 18.0000 0.0424528
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 930.000 2.17290
\(429\) 0 0
\(430\) 0 0
\(431\) 162.000 0.375870 0.187935 0.982181i \(-0.439821\pi\)
0.187935 + 0.982181i \(0.439821\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 530.000 1.21560
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) −486.000 −1.09707 −0.548533 0.836129i \(-0.684813\pi\)
−0.548533 + 0.836129i \(0.684813\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 637.000 1.42188
\(449\) −894.000 −1.99109 −0.995546 0.0942807i \(-0.969945\pi\)
−0.995546 + 0.0942807i \(0.969945\pi\)
\(450\) −675.000 −1.50000
\(451\) 0 0
\(452\) −1110.00 −2.45575
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −878.000 −1.92123 −0.960613 0.277891i \(-0.910365\pi\)
−0.960613 + 0.277891i \(0.910365\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 674.000 1.45572 0.727862 0.685724i \(-0.240514\pi\)
0.727862 + 0.685724i \(0.240514\pi\)
\(464\) 594.000 1.28017
\(465\) 0 0
\(466\) −54.0000 −0.115880
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 826.000 1.76119
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −348.000 −0.735729
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −54.0000 −0.113208
\(478\) 666.000 1.39331
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −425.000 −0.878099
\(485\) 0 0
\(486\) 0 0
\(487\) −398.000 −0.817248 −0.408624 0.912703i \(-0.633991\pi\)
−0.408624 + 0.912703i \(0.633991\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 954.000 1.94297 0.971487 0.237094i \(-0.0761949\pi\)
0.971487 + 0.237094i \(0.0761949\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −798.000 −1.60563
\(498\) 0 0
\(499\) 298.000 0.597194 0.298597 0.954379i \(-0.403481\pi\)
0.298597 + 0.954379i \(0.403481\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 189.000 0.375000
\(505\) 0 0
\(506\) 324.000 0.640316
\(507\) 0 0
\(508\) 10.0000 0.0196850
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −627.000 −1.22461
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −798.000 −1.54054
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 1458.00 2.79310
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1494.00 −2.84030
\(527\) 0 0
\(528\) 0 0
\(529\) −205.000 −0.387524
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 354.000 0.660448
\(537\) 0 0
\(538\) 0 0
\(539\) −294.000 −0.545455
\(540\) 0 0
\(541\) 74.0000 0.136784 0.0683919 0.997659i \(-0.478213\pi\)
0.0683919 + 0.997659i \(0.478213\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 842.000 1.53931 0.769653 0.638463i \(-0.220429\pi\)
0.769653 + 0.638463i \(0.220429\pi\)
\(548\) −870.000 −1.58759
\(549\) 0 0
\(550\) 450.000 0.818182
\(551\) 0 0
\(552\) 0 0
\(553\) 658.000 1.18987
\(554\) 1362.00 2.45848
\(555\) 0 0
\(556\) 0 0
\(557\) 1002.00 1.79892 0.899461 0.437000i \(-0.143959\pi\)
0.899461 + 0.437000i \(0.143959\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −342.000 −0.608541
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −567.000 −1.00000
\(568\) −342.000 −0.602113
\(569\) −654.000 −1.14938 −0.574692 0.818369i \(-0.694878\pi\)
−0.574692 + 0.818369i \(0.694878\pi\)
\(570\) 0 0
\(571\) −1126.00 −1.97198 −0.985989 0.166807i \(-0.946654\pi\)
−0.985989 + 0.166807i \(0.946654\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 450.000 0.782609
\(576\) −819.000 −1.42188
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −867.000 −1.50000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 36.0000 0.0617496
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 418.000 0.706081
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 930.000 1.56040
\(597\) 0 0
\(598\) 0 0
\(599\) −174.000 −0.290484 −0.145242 0.989396i \(-0.546396\pi\)
−0.145242 + 0.989396i \(0.546396\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 1218.00 2.02326
\(603\) −1062.00 −1.76119
\(604\) 1370.00 2.26821
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 218.000 0.355628 0.177814 0.984064i \(-0.443097\pi\)
0.177814 + 0.984064i \(0.443097\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −126.000 −0.204545
\(617\) −558.000 −0.904376 −0.452188 0.891923i \(-0.649356\pi\)
−0.452188 + 0.891923i \(0.649356\pi\)
\(618\) 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
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1006.00 −1.59429 −0.797147 0.603785i \(-0.793659\pi\)
−0.797147 + 0.603785i \(0.793659\pi\)
\(632\) 282.000 0.446203
\(633\) 0 0
\(634\) −1566.00 −2.47003
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −972.000 −1.52351
\(639\) 1026.00 1.60563
\(640\) 0 0
\(641\) 834.000 1.30109 0.650546 0.759467i \(-0.274540\pi\)
0.650546 + 0.759467i \(0.274540\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −630.000 −0.978261
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −243.000 −0.375000
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 370.000 0.567485
\(653\) 1194.00 1.82848 0.914242 0.405169i \(-0.132787\pi\)
0.914242 + 0.405169i \(0.132787\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 618.000 0.937785 0.468892 0.883255i \(-0.344653\pi\)
0.468892 + 0.883255i \(0.344653\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −1902.00 −2.87311
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1026.00 1.54054
\(667\) −972.000 −1.45727
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −446.000 −0.662704 −0.331352 0.943507i \(-0.607505\pi\)
−0.331352 + 0.943507i \(0.607505\pi\)
\(674\) −678.000 −1.00593
\(675\) 0 0
\(676\) 845.000 1.25000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1338.00 1.95900 0.979502 0.201433i \(-0.0645600\pi\)
0.979502 + 0.201433i \(0.0645600\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1029.00 1.50000
\(687\) 0 0
\(688\) −638.000 −0.927326
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 378.000 0.545455
\(694\) 2034.00 2.93084
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −875.000 −1.25000
\(701\) −1398.00 −1.99429 −0.997147 0.0754851i \(-0.975949\pi\)
−0.997147 + 0.0754851i \(0.975949\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 546.000 0.775568
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1382.00 −1.94922 −0.974612 0.223900i \(-0.928121\pi\)
−0.974612 + 0.223900i \(0.928121\pi\)
\(710\) 0 0
\(711\) −846.000 −1.18987
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1710.00 −2.38827
\(717\) 0 0
\(718\) 1962.00 2.73259
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1083.00 −1.50000
\(723\) 0 0
\(724\) 0 0
\(725\) −1350.00 −1.86207
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 810.000 1.10054
\(737\) 708.000 0.960651
\(738\) 0 0
\(739\) 1226.00 1.65900 0.829499 0.558508i \(-0.188626\pi\)
0.829499 + 0.558508i \(0.188626\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −126.000 −0.169811
\(743\) 114.000 0.153432 0.0767160 0.997053i \(-0.475557\pi\)
0.0767160 + 0.997053i \(0.475557\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 786.000 1.05362
\(747\) 0 0
\(748\) 0 0
\(749\) −1302.00 −1.73832
\(750\) 0 0
\(751\) 802.000 1.06791 0.533955 0.845513i \(-0.320705\pi\)
0.533955 + 0.845513i \(0.320705\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1402.00 1.85205 0.926024 0.377465i \(-0.123204\pi\)
0.926024 + 0.377465i \(0.123204\pi\)
\(758\) 1842.00 2.43008
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −742.000 −0.972477
\(764\) −1590.00 −2.08115
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −310.000 −0.401554
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −1566.00 −2.02326
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1998.00 −2.56812
\(779\) 0 0
\(780\) 0 0
\(781\) −684.000 −0.875800
\(782\) 0 0
\(783\) 0 0
\(784\) −539.000 −0.687500
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1410.00 1.78934
\(789\) 0 0
\(790\) 0 0
\(791\) 1554.00 1.96460
\(792\) 162.000 0.204545
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1125.00 1.40625
\(801\) 0 0
\(802\) −1062.00 −1.32419
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −174.000 −0.215080 −0.107540 0.994201i \(-0.534297\pi\)
−0.107540 + 0.994201i \(0.534297\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1890.00 2.32759
\(813\) 0 0
\(814\) −684.000 −0.840295
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1158.00 −1.41048 −0.705238 0.708971i \(-0.749160\pi\)
−0.705238 + 0.708971i \(0.749160\pi\)
\(822\) 0 0
\(823\) −622.000 −0.755772 −0.377886 0.925852i \(-0.623349\pi\)
−0.377886 + 0.925852i \(0.623349\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 282.000 0.340992 0.170496 0.985358i \(-0.445463\pi\)
0.170496 + 0.985358i \(0.445463\pi\)
\(828\) 810.000 0.978261
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 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
\(841\) 2075.00 2.46730
\(842\) 498.000 0.591449
\(843\) 0 0
\(844\) −1390.00 −1.64692
\(845\) 0 0
\(846\) 0 0
\(847\) 595.000 0.702479
\(848\) 66.0000 0.0778302
\(849\) 0 0
\(850\) 0 0
\(851\) −684.000 −0.803760
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −558.000 −0.651869
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −486.000 −0.563805
\(863\) −1662.00 −1.92584 −0.962920 0.269787i \(-0.913047\pi\)
−0.962920 + 0.269787i \(0.913047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 564.000 0.649022
\(870\) 0 0
\(871\) 0 0
\(872\) −318.000 −0.364679
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 746.000 0.850627 0.425314 0.905046i \(-0.360164\pi\)
0.425314 + 0.905046i \(0.360164\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −1323.00 −1.50000
\(883\) −1622.00 −1.83692 −0.918460 0.395514i \(-0.870566\pi\)
−0.918460 + 0.395514i \(0.870566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1458.00 1.64560
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.0157480
\(890\) 0 0
\(891\) −486.000 −0.545455
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −651.000 −0.726562
\(897\) 0 0
\(898\) 2682.00 2.98664
\(899\) 0 0
\(900\) 1125.00 1.25000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 666.000 0.736726
\(905\) 0 0
\(906\) 0 0
\(907\) 1786.00 1.96913 0.984564 0.175022i \(-0.0559997\pi\)
0.984564 + 0.175022i \(0.0559997\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1566.00 −1.71899 −0.859495 0.511144i \(-0.829222\pi\)
−0.859495 + 0.511144i \(0.829222\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2634.00 2.88184
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 466.000 0.507073 0.253536 0.967326i \(-0.418406\pi\)
0.253536 + 0.967326i \(0.418406\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −950.000 −1.02703
\(926\) −2022.00 −2.18359
\(927\) 0 0
\(928\) −2430.00 −2.61853
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 90.0000 0.0965665
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −2478.00 −2.64179
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 1044.00 1.10359
\(947\) −1494.00 −1.57761 −0.788807 0.614641i \(-0.789301\pi\)
−0.788807 + 0.614641i \(0.789301\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1458.00 1.52991 0.764953 0.644086i \(-0.222762\pi\)
0.764953 + 0.644086i \(0.222762\pi\)
\(954\) 162.000 0.169811
\(955\) 0 0
\(956\) −1110.00 −1.16109
\(957\) 0 0
\(958\) 0 0
\(959\) 1218.00 1.27007
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 1674.00 1.73832
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −334.000 −0.345398 −0.172699 0.984975i \(-0.555249\pi\)
−0.172699 + 0.984975i \(0.555249\pi\)
\(968\) 255.000 0.263430
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1194.00 1.22587
\(975\) 0 0
\(976\) 0 0
\(977\) 162.000 0.165814 0.0829069 0.996557i \(-0.473580\pi\)
0.0829069 + 0.996557i \(0.473580\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 954.000 0.972477
\(982\) −2862.00 −2.91446
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1044.00 1.05561
\(990\) 0 0
\(991\) −1406.00 −1.41877 −0.709384 0.704822i \(-0.751027\pi\)
−0.709384 + 0.704822i \(0.751027\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 2394.00 2.40845
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −894.000 −0.895792
\(999\) 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 \(a_p\))