Properties

Label 2015.1.h.c.2014.5
Level 2015
Weight 1
Character 2015.2014
Self dual yes
Analytic conductor 1.006
Analytic rank 0
Dimension 6
Projective image \(D_{13}\)
CM discriminant -2015
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2015.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{26})^+\)
Defining polynomial: \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{13}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

Embedding invariants

Embedding label 2014.5
Root \(-1.13613\) of defining polynomial
Character \(\chi\) \(=\) 2015.2014

$q$-expansion

\(f(q)\) \(=\) \(q+1.13613 q^{2} +1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} +2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +O(q^{10})\) \(q+1.13613 q^{2} +1.94188 q^{3} +0.290790 q^{4} +1.00000 q^{5} +2.20623 q^{6} -1.49702 q^{7} -0.805754 q^{8} +2.77091 q^{9} +1.13613 q^{10} -0.241073 q^{11} +0.564681 q^{12} -1.00000 q^{13} -1.70081 q^{14} +1.94188 q^{15} -1.20623 q^{16} +0.709210 q^{17} +3.14811 q^{18} +0.290790 q^{20} -2.90704 q^{21} -0.273891 q^{22} -1.77091 q^{23} -1.56468 q^{24} +1.00000 q^{25} -1.13613 q^{26} +3.43891 q^{27} -0.435319 q^{28} +2.20623 q^{30} -1.00000 q^{31} -0.564681 q^{32} -0.468136 q^{33} +0.805754 q^{34} -1.49702 q^{35} +0.805754 q^{36} -1.94188 q^{39} -0.805754 q^{40} -3.30278 q^{42} -1.13613 q^{43} -0.0701018 q^{44} +2.77091 q^{45} -2.01199 q^{46} +1.77091 q^{47} -2.34236 q^{48} +1.24107 q^{49} +1.13613 q^{50} +1.37720 q^{51} -0.290790 q^{52} -0.241073 q^{53} +3.90704 q^{54} -0.241073 q^{55} +1.20623 q^{56} +0.564681 q^{60} -1.13613 q^{62} -4.14811 q^{63} +0.564681 q^{64} -1.00000 q^{65} -0.531864 q^{66} +0.241073 q^{67} +0.206231 q^{68} -3.43891 q^{69} -1.70081 q^{70} -2.23267 q^{72} +1.94188 q^{75} +0.360892 q^{77} -2.20623 q^{78} -1.20623 q^{80} +3.90704 q^{81} -0.845339 q^{84} +0.709210 q^{85} -1.29079 q^{86} +0.194246 q^{88} +1.49702 q^{89} +3.14811 q^{90} +1.49702 q^{91} -0.514964 q^{92} -1.94188 q^{93} +2.01199 q^{94} -1.09654 q^{96} -1.94188 q^{97} +1.41002 q^{98} -0.667993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} - q^{10} + q^{11} + 3q^{12} - 6q^{13} - 2q^{14} + q^{15} + 4q^{16} + q^{17} - 3q^{18} + 5q^{20} + 2q^{21} + 2q^{22} + q^{23} - 9q^{24} + 6q^{25} + q^{26} + 2q^{27} - 3q^{28} + 2q^{30} - 6q^{31} - 3q^{32} - 2q^{33} + 2q^{34} - q^{35} + 2q^{36} - q^{39} - 2q^{40} - 9q^{42} + q^{43} + 3q^{44} + 5q^{45} + 2q^{46} - q^{47} + 5q^{48} + 5q^{49} - q^{50} - 2q^{51} - 5q^{52} + q^{53} + 4q^{54} + q^{55} - 4q^{56} + 3q^{60} + q^{62} - 3q^{63} + 3q^{64} - 6q^{65} - 4q^{66} - q^{67} - 10q^{68} - 2q^{69} - 2q^{70} - 6q^{72} + q^{75} + 2q^{77} - 2q^{78} + 4q^{80} + 4q^{81} + 6q^{84} + q^{85} - 11q^{86} + 4q^{88} + q^{89} - 3q^{90} + q^{91} + 3q^{92} - q^{93} - 2q^{94} - 7q^{96} - q^{97} - 3q^{98} + 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times\).

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(3\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(4\) 0.290790 0.290790
\(5\) 1.00000 1.00000
\(6\) 2.20623 2.20623
\(7\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(8\) −0.805754 −0.805754
\(9\) 2.77091 2.77091
\(10\) 1.13613 1.13613
\(11\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(12\) 0.564681 0.564681
\(13\) −1.00000 −1.00000
\(14\) −1.70081 −1.70081
\(15\) 1.94188 1.94188
\(16\) −1.20623 −1.20623
\(17\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(18\) 3.14811 3.14811
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.290790 0.290790
\(21\) −2.90704 −2.90704
\(22\) −0.273891 −0.273891
\(23\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(24\) −1.56468 −1.56468
\(25\) 1.00000 1.00000
\(26\) −1.13613 −1.13613
\(27\) 3.43891 3.43891
\(28\) −0.435319 −0.435319
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 2.20623 2.20623
\(31\) −1.00000 −1.00000
\(32\) −0.564681 −0.564681
\(33\) −0.468136 −0.468136
\(34\) 0.805754 0.805754
\(35\) −1.49702 −1.49702
\(36\) 0.805754 0.805754
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.94188 −1.94188
\(40\) −0.805754 −0.805754
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −3.30278 −3.30278
\(43\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(44\) −0.0701018 −0.0701018
\(45\) 2.77091 2.77091
\(46\) −2.01199 −2.01199
\(47\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(48\) −2.34236 −2.34236
\(49\) 1.24107 1.24107
\(50\) 1.13613 1.13613
\(51\) 1.37720 1.37720
\(52\) −0.290790 −0.290790
\(53\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(54\) 3.90704 3.90704
\(55\) −0.241073 −0.241073
\(56\) 1.20623 1.20623
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0.564681 0.564681
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −1.13613 −1.13613
\(63\) −4.14811 −4.14811
\(64\) 0.564681 0.564681
\(65\) −1.00000 −1.00000
\(66\) −0.531864 −0.531864
\(67\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(68\) 0.206231 0.206231
\(69\) −3.43891 −3.43891
\(70\) −1.70081 −1.70081
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −2.23267 −2.23267
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.94188 1.94188
\(76\) 0 0
\(77\) 0.360892 0.360892
\(78\) −2.20623 −2.20623
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.20623 −1.20623
\(81\) 3.90704 3.90704
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −0.845339 −0.845339
\(85\) 0.709210 0.709210
\(86\) −1.29079 −1.29079
\(87\) 0 0
\(88\) 0.194246 0.194246
\(89\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(90\) 3.14811 3.14811
\(91\) 1.49702 1.49702
\(92\) −0.514964 −0.514964
\(93\) −1.94188 −1.94188
\(94\) 2.01199 2.01199
\(95\) 0 0
\(96\) −1.09654 −1.09654
\(97\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(98\) 1.41002 1.41002
\(99\) −0.667993 −0.667993
\(100\) 0.290790 0.290790
\(101\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(102\) 1.56468 1.56468
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0.805754 0.805754
\(105\) −2.90704 −2.90704
\(106\) −0.273891 −0.273891
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.00000 1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −0.273891 −0.273891
\(111\) 0 0
\(112\) 1.80575 1.80575
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −1.77091 −1.77091
\(116\) 0 0
\(117\) −2.77091 −2.77091
\(118\) 0 0
\(119\) −1.06170 −1.06170
\(120\) −1.56468 −1.56468
\(121\) −0.941884 −0.941884
\(122\) 0 0
\(123\) 0 0
\(124\) −0.290790 −0.290790
\(125\) 1.00000 1.00000
\(126\) −4.71280 −4.71280
\(127\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(128\) 1.20623 1.20623
\(129\) −2.20623 −2.20623
\(130\) −1.13613 −1.13613
\(131\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(132\) −0.136129 −0.136129
\(133\) 0 0
\(134\) 0.273891 0.273891
\(135\) 3.43891 3.43891
\(136\) −0.571449 −0.571449
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −3.90704 −3.90704
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −0.435319 −0.435319
\(141\) 3.43891 3.43891
\(142\) 0 0
\(143\) 0.241073 0.241073
\(144\) −3.34236 −3.34236
\(145\) 0 0
\(146\) 0 0
\(147\) 2.41002 2.41002
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 2.20623 2.20623
\(151\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(152\) 0 0
\(153\) 1.96516 1.96516
\(154\) 0.410020 0.410020
\(155\) −1.00000 −1.00000
\(156\) −0.564681 −0.564681
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −0.468136 −0.468136
\(160\) −0.564681 −0.564681
\(161\) 2.65109 2.65109
\(162\) 4.43891 4.43891
\(163\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(164\) 0 0
\(165\) −0.468136 −0.468136
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.34236 2.34236
\(169\) 1.00000 1.00000
\(170\) 0.805754 0.805754
\(171\) 0 0
\(172\) −0.330375 −0.330375
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.49702 −1.49702
\(176\) 0.290790 0.290790
\(177\) 0 0
\(178\) 1.70081 1.70081
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0.805754 0.805754
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 1.70081 1.70081
\(183\) 0 0
\(184\) 1.42692 1.42692
\(185\) 0 0
\(186\) −2.20623 −2.20623
\(187\) −0.170972 −0.170972
\(188\) 0.514964 0.514964
\(189\) −5.14811 −5.14811
\(190\) 0 0
\(191\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(192\) 1.09654 1.09654
\(193\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(194\) −2.20623 −2.20623
\(195\) −1.94188 −1.94188
\(196\) 0.360892 0.360892
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −0.758927 −0.758927
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −0.805754 −0.805754
\(201\) 0.468136 0.468136
\(202\) 1.29079 1.29079
\(203\) 0 0
\(204\) 0.400477 0.400477
\(205\) 0 0
\(206\) 0 0
\(207\) −4.90704 −4.90704
\(208\) 1.20623 1.20623
\(209\) 0 0
\(210\) −3.30278 −3.30278
\(211\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(212\) −0.0701018 −0.0701018
\(213\) 0 0
\(214\) 0 0
\(215\) −1.13613 −1.13613
\(216\) −2.77091 −2.77091
\(217\) 1.49702 1.49702
\(218\) 0 0
\(219\) 0 0
\(220\) −0.0701018 −0.0701018
\(221\) −0.709210 −0.709210
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0.845339 0.845339
\(225\) 2.77091 2.77091
\(226\) 0 0
\(227\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(228\) 0 0
\(229\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(230\) −2.01199 −2.01199
\(231\) 0.700810 0.700810
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −3.14811 −3.14811
\(235\) 1.77091 1.77091
\(236\) 0 0
\(237\) 0 0
\(238\) −1.20623 −1.20623
\(239\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(240\) −2.34236 −2.34236
\(241\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(242\) −1.07010 −1.07010
\(243\) 4.14811 4.14811
\(244\) 0 0
\(245\) 1.24107 1.24107
\(246\) 0 0
\(247\) 0 0
\(248\) 0.805754 0.805754
\(249\) 0 0
\(250\) 1.13613 1.13613
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.20623 −1.20623
\(253\) 0.426920 0.426920
\(254\) 0.805754 0.805754
\(255\) 1.37720 1.37720
\(256\) 0.805754 0.805754
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) −2.50656 −2.50656
\(259\) 0 0
\(260\) −0.290790 −0.290790
\(261\) 0 0
\(262\) −0.805754 −0.805754
\(263\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(264\) 0.377203 0.377203
\(265\) −0.241073 −0.241073
\(266\) 0 0
\(267\) 2.90704 2.90704
\(268\) 0.0701018 0.0701018
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 3.90704 3.90704
\(271\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(272\) −0.855471 −0.855471
\(273\) 2.90704 2.90704
\(274\) 0 0
\(275\) −0.241073 −0.241073
\(276\) −1.00000 −1.00000
\(277\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(278\) 0 0
\(279\) −2.77091 −2.77091
\(280\) 1.20623 1.20623
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 3.90704 3.90704
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0.273891 0.273891
\(287\) 0 0
\(288\) −1.56468 −1.56468
\(289\) −0.497021 −0.497021
\(290\) 0 0
\(291\) −3.77091 −3.77091
\(292\) 0 0
\(293\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(294\) 2.73809 2.73809
\(295\) 0 0
\(296\) 0 0
\(297\) −0.829028 −0.829028
\(298\) 0 0
\(299\) 1.77091 1.77091
\(300\) 0.564681 0.564681
\(301\) 1.70081 1.70081
\(302\) −1.29079 −1.29079
\(303\) 2.20623 2.20623
\(304\) 0 0
\(305\) 0 0
\(306\) 2.23267 2.23267
\(307\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(308\) 0.104944 0.104944
\(309\) 0 0
\(310\) −1.13613 −1.13613
\(311\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(312\) 1.56468 1.56468
\(313\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(314\) 0 0
\(315\) −4.14811 −4.14811
\(316\) 0 0
\(317\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(318\) −0.531864 −0.531864
\(319\) 0 0
\(320\) 0.564681 0.564681
\(321\) 0 0
\(322\) 3.01199 3.01199
\(323\) 0 0
\(324\) 1.13613 1.13613
\(325\) −1.00000 −1.00000
\(326\) −0.805754 −0.805754
\(327\) 0 0
\(328\) 0 0
\(329\) −2.65109 −2.65109
\(330\) −0.531864 −0.531864
\(331\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.241073 0.241073
\(336\) 3.50656 3.50656
\(337\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(338\) 1.13613 1.13613
\(339\) 0 0
\(340\) 0.206231 0.206231
\(341\) 0.241073 0.241073
\(342\) 0 0
\(343\) −0.360892 −0.360892
\(344\) 0.915441 0.915441
\(345\) −3.43891 −3.43891
\(346\) 0 0
\(347\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −1.70081 −1.70081
\(351\) −3.43891 −3.43891
\(352\) 0.136129 0.136129
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.435319 0.435319
\(357\) −2.06170 −2.06170
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −2.23267 −2.23267
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −1.82903 −1.82903
\(364\) 0.435319 0.435319
\(365\) 0 0
\(366\) 0 0
\(367\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(368\) 2.13613 2.13613
\(369\) 0 0
\(370\) 0 0
\(371\) 0.360892 0.360892
\(372\) −0.564681 −0.564681
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −0.194246 −0.194246
\(375\) 1.94188 1.94188
\(376\) −1.42692 −1.42692
\(377\) 0 0
\(378\) −5.84893 −5.84893
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 1.37720 1.37720
\(382\) 1.29079 1.29079
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 2.34236 2.34236
\(385\) 0.360892 0.360892
\(386\) 2.01199 2.01199
\(387\) −3.14811 −3.14811
\(388\) −0.564681 −0.564681
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −2.20623 −2.20623
\(391\) −1.25595 −1.25595
\(392\) −1.00000 −1.00000
\(393\) −1.37720 −1.37720
\(394\) 0 0
\(395\) 0 0
\(396\) −0.194246 −0.194246
\(397\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.20623 −1.20623
\(401\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(402\) 0.531864 0.531864
\(403\) 1.00000 1.00000
\(404\) 0.330375 0.330375
\(405\) 3.90704 3.90704
\(406\) 0 0
\(407\) 0 0
\(408\) −1.10969 −1.10969
\(409\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −5.57503 −5.57503
\(415\) 0 0
\(416\) 0.564681 0.564681
\(417\) 0 0
\(418\) 0 0
\(419\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(420\) −0.845339 −0.845339
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −2.20623 −2.20623
\(423\) 4.90704 4.90704
\(424\) 0.194246 0.194246
\(425\) 0.709210 0.709210
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.468136 0.468136
\(430\) −1.29079 −1.29079
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −4.14811 −4.14811
\(433\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(434\) 1.70081 1.70081
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(440\) 0.194246 0.194246
\(441\) 3.43891 3.43891
\(442\) −0.805754 −0.805754
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1.49702 1.49702
\(446\) 0 0
\(447\) 0 0
\(448\) −0.845339 −0.845339
\(449\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(450\) 3.14811 3.14811
\(451\) 0 0
\(452\) 0 0
\(453\) −2.20623 −2.20623
\(454\) −0.805754 −0.805754
\(455\) 1.49702 1.49702
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 1.70081 1.70081
\(459\) 2.43891 2.43891
\(460\) −0.514964 −0.514964
\(461\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(462\) 0.796211 0.796211
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −1.94188 −1.94188
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −0.805754 −0.805754
\(469\) −0.360892 −0.360892
\(470\) 2.01199 2.01199
\(471\) 0 0
\(472\) 0 0
\(473\) 0.273891 0.273891
\(474\) 0 0
\(475\) 0 0
\(476\) −0.308733 −0.308733
\(477\) −0.667993 −0.667993
\(478\) 1.70081 1.70081
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −1.09654 −1.09654
\(481\) 0 0
\(482\) 2.20623 2.20623
\(483\) 5.14811 5.14811
\(484\) −0.273891 −0.273891
\(485\) −1.94188 −1.94188
\(486\) 4.71280 4.71280
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −1.37720 −1.37720
\(490\) 1.41002 1.41002
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −0.667993 −0.667993
\(496\) 1.20623 1.20623
\(497\) 0 0
\(498\) 0 0
\(499\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(500\) 0.290790 0.290790
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 3.34236 3.34236
\(505\) 1.13613 1.13613
\(506\) 0.485036 0.485036
\(507\) 1.94188 1.94188
\(508\) 0.206231 0.206231
\(509\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(510\) 1.56468 1.56468
\(511\) 0 0
\(512\) −0.290790 −0.290790
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −0.641550 −0.641550
\(517\) −0.426920 −0.426920
\(518\) 0 0
\(519\) 0 0
\(520\) 0.805754 0.805754
\(521\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(522\) 0 0
\(523\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(524\) −0.206231 −0.206231
\(525\) −2.90704 −2.90704
\(526\) −1.29079 −1.29079
\(527\) −0.709210 −0.709210
\(528\) 0.564681 0.564681
\(529\) 2.13613 2.13613
\(530\) −0.273891 −0.273891
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 3.30278 3.30278
\(535\) 0 0
\(536\) −0.194246 −0.194246
\(537\) 0 0
\(538\) 0 0
\(539\) −0.299190 −0.299190
\(540\) 1.00000 1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0.805754 0.805754
\(543\) 0 0
\(544\) −0.400477 −0.400477
\(545\) 0 0
\(546\) 3.30278 3.30278
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −0.273891 −0.273891
\(551\) 0 0
\(552\) 2.77091 2.77091
\(553\) 0 0
\(554\) −2.01199 −2.01199
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −3.14811 −3.14811
\(559\) 1.13613 1.13613
\(560\) 1.80575 1.80575
\(561\) −0.332007 −0.332007
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 1.00000 1.00000
\(565\) 0 0
\(566\) 0 0
\(567\) −5.84893 −5.84893
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0.0701018 0.0701018
\(573\) 2.20623 2.20623
\(574\) 0 0
\(575\) −1.77091 −1.77091
\(576\) 1.56468 1.56468
\(577\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(578\) −0.564681 −0.564681
\(579\) 3.43891 3.43891
\(580\) 0 0
\(581\) 0 0
\(582\) −4.28424 −4.28424
\(583\) 0.0581164 0.0581164
\(584\) 0 0
\(585\) −2.77091 −2.77091
\(586\) −1.70081 −1.70081
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0.700810 0.700810
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(594\) −0.941884 −0.941884
\(595\) −1.06170 −1.06170
\(596\) 0 0
\(597\) 0 0
\(598\) 2.01199 2.01199
\(599\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(600\) −1.56468 −1.56468
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 1.93234 1.93234
\(603\) 0.667993 0.667993
\(604\) −0.330375 −0.330375
\(605\) −0.941884 −0.941884
\(606\) 2.50656 2.50656
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.77091 −1.77091
\(612\) 0.571449 0.571449
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.29079 1.29079
\(615\) 0 0
\(616\) −0.290790 −0.290790
\(617\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(618\) 0 0
\(619\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(620\) −0.290790 −0.290790
\(621\) −6.09000 −6.09000
\(622\) 2.01199 2.01199
\(623\) −2.24107 −2.24107
\(624\) 2.34236 2.34236
\(625\) 1.00000 1.00000
\(626\) −2.01199 −2.01199
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −4.71280 −4.71280
\(631\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(632\) 0 0
\(633\) −3.77091 −3.77091
\(634\) −0.805754 −0.805754
\(635\) 0.709210 0.709210
\(636\) −0.136129 −0.136129
\(637\) −1.24107 −1.24107
\(638\) 0 0
\(639\) 0 0
\(640\) 1.20623 1.20623
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0.770912 0.770912
\(645\) −2.20623 −2.20623
\(646\) 0 0
\(647\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(648\) −3.14811 −3.14811
\(649\) 0 0
\(650\) −1.13613 −1.13613
\(651\) 2.90704 2.90704
\(652\) −0.206231 −0.206231
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −0.709210 −0.709210
\(656\) 0 0
\(657\) 0 0
\(658\) −3.01199 −3.01199
\(659\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(660\) −0.136129 −0.136129
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0.805754 0.805754
\(663\) −1.37720 −1.37720
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0.273891 0.273891
\(671\) 0 0
\(672\) 1.64155 1.64155
\(673\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(674\) −0.273891 −0.273891
\(675\) 3.43891 3.43891
\(676\) 0.290790 0.290790
\(677\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(678\) 0 0
\(679\) 2.90704 2.90704
\(680\) −0.571449 −0.571449
\(681\) −1.37720 −1.37720
\(682\) 0.273891 0.273891
\(683\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.410020 −0.410020
\(687\) 2.90704 2.90704
\(688\) 1.37043 1.37043
\(689\) 0.241073 0.241073
\(690\) −3.90704 −3.90704
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.00000 1.00000
\(694\) −0.273891 −0.273891
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.435319 −0.435319
\(701\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(702\) −3.90704 −3.90704
\(703\) 0 0
\(704\) −0.136129 −0.136129
\(705\) 3.43891 3.43891
\(706\) 0 0
\(707\) −1.70081 −1.70081
\(708\) 0 0
\(709\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.20623 −1.20623
\(713\) 1.77091 1.77091
\(714\) −2.34236 −2.34236
\(715\) 0.241073 0.241073
\(716\) 0 0
\(717\) 2.90704 2.90704
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −3.34236 −3.34236
\(721\) 0 0
\(722\) 1.13613 1.13613
\(723\) 3.77091 3.77091
\(724\) 0 0
\(725\) 0 0
\(726\) −2.07801 −2.07801
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −1.20623 −1.20623
\(729\) 4.14811 4.14811
\(730\) 0 0
\(731\) −0.805754 −0.805754
\(732\) 0 0
\(733\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(734\) 1.70081 1.70081
\(735\) 2.41002 2.41002
\(736\) 1.00000 1.00000
\(737\) −0.0581164 −0.0581164
\(738\) 0 0
\(739\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.410020 0.410020
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 1.56468 1.56468
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) −0.0497169 −0.0497169
\(749\) 0 0
\(750\) 2.20623 2.20623
\(751\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(752\) −2.13613 −2.13613
\(753\) 0 0
\(754\) 0 0
\(755\) −1.13613 −1.13613
\(756\) −1.49702 −1.49702
\(757\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(758\) 0 0
\(759\) 0.829028 0.829028
\(760\) 0 0
\(761\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(762\) 1.56468 1.56468
\(763\) 0 0
\(764\) 0.330375 0.330375
\(765\) 1.96516 1.96516
\(766\) 0 0
\(767\) 0 0
\(768\) 1.56468 1.56468
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0.410020 0.410020
\(771\) 0 0
\(772\) 0.514964 0.514964
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −3.57667 −3.57667
\(775\) −1.00000 −1.00000
\(776\) 1.56468 1.56468
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −0.564681 −0.564681
\(781\) 0 0
\(782\) −1.42692 −1.42692
\(783\) 0 0
\(784\) −1.49702 −1.49702
\(785\) 0 0
\(786\) −1.56468 −1.56468
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) −2.20623 −2.20623
\(790\) 0 0
\(791\) 0 0
\(792\) 0.538238 0.538238
\(793\) 0 0
\(794\) 0.273891 0.273891
\(795\) −0.468136 −0.468136
\(796\) 0 0
\(797\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(798\) 0 0
\(799\) 1.25595 1.25595
\(800\) −0.564681 −0.564681
\(801\) 4.14811 4.14811
\(802\) −2.01199 −2.01199
\(803\) 0 0
\(804\) 0.136129 0.136129
\(805\) 2.65109 2.65109
\(806\) 1.13613 1.13613
\(807\) 0 0
\(808\) −0.915441 −0.915441
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 4.43891 4.43891
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 1.37720 1.37720
\(814\) 0 0
\(815\) −0.709210 −0.709210
\(816\) −1.66123 −1.66123
\(817\) 0 0
\(818\) −2.01199 −2.01199
\(819\) 4.14811 4.14811
\(820\) 0 0
\(821\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(822\) 0 0
\(823\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(824\) 0 0
\(825\) −0.468136 −0.468136
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −1.42692 −1.42692
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −3.43891 −3.43891
\(832\) −0.564681 −0.564681
\(833\) 0.880181 0.880181
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.43891 −3.43891
\(838\) −2.20623 −2.20623
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 2.34236 2.34236
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −0.564681 −0.564681
\(845\) 1.00000 1.00000
\(846\) 5.57503 5.57503
\(847\) 1.41002 1.41002
\(848\) 0.290790 0.290790
\(849\) 0 0
\(850\) 0.805754 0.805754
\(851\) 0 0
\(852\) 0 0
\(853\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0.531864 0.531864
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −0.330375 −0.330375
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.94188 −1.94188
\(865\) 0 0
\(866\) −1.29079 −1.29079
\(867\) −0.965158 −0.965158
\(868\) 0.435319 0.435319
\(869\) 0 0
\(870\) 0 0
\(871\) −0.241073 −0.241073
\(872\) 0 0
\(873\) −5.38079 −5.38079
\(874\) 0 0
\(875\) −1.49702 −1.49702
\(876\) 0 0
\(877\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(878\) 2.01199 2.01199
\(879\) −2.90704 −2.90704
\(880\) 0.290790 0.290790
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 3.90704 3.90704
\(883\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(884\) −0.206231 −0.206231
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.06170 −1.06170
\(890\) 1.70081 1.70081
\(891\) −0.941884 −0.941884
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.80575 −1.80575
\(897\) 3.43891 3.43891
\(898\) −1.29079 −1.29079
\(899\) 0 0
\(900\) 0.805754 0.805754
\(901\) −0.170972 −0.170972
\(902\) 0 0
\(903\) 3.30278 3.30278
\(904\) 0 0
\(905\) 0 0
\(906\) −2.50656 −2.50656
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −0.206231 −0.206231
\(909\) 3.14811 3.14811
\(910\) 1.70081 1.70081
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.435319 0.435319
\(917\) 1.06170 1.06170
\(918\) 2.77091 2.77091
\(919\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(920\) 1.42692 1.42692
\(921\) 2.20623 2.20623
\(922\) 2.20623 2.20623
\(923\) 0 0
\(924\) 0.203789 0.203789
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(930\) −2.20623 −2.20623
\(931\) 0 0
\(932\) 0 0
\(933\) 3.43891 3.43891
\(934\) 0 0
\(935\) −0.170972 −0.170972
\(936\) 2.23267 2.23267
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −0.410020 −0.410020
\(939\) −3.43891 −3.43891
\(940\) 0.514964 0.514964
\(941\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −5.14811 −5.14811
\(946\) 0.311175 0.311175
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.37720 −1.37720
\(952\) 0.855471 0.855471
\(953\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(954\) −0.758927 −0.758927
\(955\) 1.13613 1.13613
\(956\) 0.435319 0.435319
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.09654 1.09654
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0.564681 0.564681
\(965\) 1.77091 1.77091
\(966\) 5.84893 5.84893
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0.758927 0.758927
\(969\) 0 0
\(970\) −2.20623 −2.20623
\(971\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(972\) 1.20623 1.20623
\(973\) 0 0
\(974\) 0 0
\(975\) −1.94188 −1.94188
\(976\) 0 0
\(977\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(978\) −1.56468 −1.56468
\(979\) −0.360892 −0.360892
\(980\) 0.360892 0.360892
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.14811 −5.14811
\(988\) 0 0
\(989\) 2.01199 2.01199
\(990\) −0.758927 −0.758927
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0.564681 0.564681
\(993\) 1.37720 1.37720
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −1.29079 −1.29079
\(999\) 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 2015.1.h.c.2014.5 yes 6
5.4 even 2 2015.1.h.d.2014.2 yes 6
13.12 even 2 2015.1.h.e.2014.2 yes 6
31.30 odd 2 2015.1.h.b.2014.5 6
65.64 even 2 2015.1.h.b.2014.5 6
155.154 odd 2 2015.1.h.e.2014.2 yes 6
403.402 odd 2 2015.1.h.d.2014.2 yes 6
2015.2014 odd 2 CM 2015.1.h.c.2014.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2015.1.h.b.2014.5 6 31.30 odd 2
2015.1.h.b.2014.5 6 65.64 even 2
2015.1.h.c.2014.5 yes 6 1.1 even 1 trivial
2015.1.h.c.2014.5 yes 6 2015.2014 odd 2 CM
2015.1.h.d.2014.2 yes 6 5.4 even 2
2015.1.h.d.2014.2 yes 6 403.402 odd 2
2015.1.h.e.2014.2 yes 6 13.12 even 2
2015.1.h.e.2014.2 yes 6 155.154 odd 2