Properties

Label 2015.1.h.c.2014.1
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 2014.1
Root \(1.94188\) of defining polynomial
Character \(\chi\) \(=\) 2015.2014

$q$-expansion

\(f(q)\) \(=\) \(q-1.94188 q^{2} +1.49702 q^{3} +2.77091 q^{4} +1.00000 q^{5} -2.90704 q^{6} +1.13613 q^{7} -3.43891 q^{8} +1.24107 q^{9} +O(q^{10})\) \(q-1.94188 q^{2} +1.49702 q^{3} +2.77091 q^{4} +1.00000 q^{5} -2.90704 q^{6} +1.13613 q^{7} -3.43891 q^{8} +1.24107 q^{9} -1.94188 q^{10} +0.709210 q^{11} +4.14811 q^{12} -1.00000 q^{13} -2.20623 q^{14} +1.49702 q^{15} +3.90704 q^{16} -1.77091 q^{17} -2.41002 q^{18} +2.77091 q^{20} +1.70081 q^{21} -1.37720 q^{22} -0.241073 q^{23} -5.14811 q^{24} +1.00000 q^{25} +1.94188 q^{26} +0.360892 q^{27} +3.14811 q^{28} -2.90704 q^{30} -1.00000 q^{31} -4.14811 q^{32} +1.06170 q^{33} +3.43891 q^{34} +1.13613 q^{35} +3.43891 q^{36} -1.49702 q^{39} -3.43891 q^{40} -3.30278 q^{42} +1.94188 q^{43} +1.96516 q^{44} +1.24107 q^{45} +0.468136 q^{46} +0.241073 q^{47} +5.84893 q^{48} +0.290790 q^{49} -1.94188 q^{50} -2.65109 q^{51} -2.77091 q^{52} +0.709210 q^{53} -0.700810 q^{54} +0.709210 q^{55} -3.90704 q^{56} +4.14811 q^{60} +1.94188 q^{62} +1.41002 q^{63} +4.14811 q^{64} -1.00000 q^{65} -2.06170 q^{66} -0.709210 q^{67} -4.90704 q^{68} -0.360892 q^{69} -2.20623 q^{70} -4.26793 q^{72} +1.49702 q^{75} +0.805754 q^{77} +2.90704 q^{78} +3.90704 q^{80} -0.700810 q^{81} +4.71280 q^{84} -1.77091 q^{85} -3.77091 q^{86} -2.43891 q^{88} -1.13613 q^{89} -2.41002 q^{90} -1.13613 q^{91} -0.667993 q^{92} -1.49702 q^{93} -0.468136 q^{94} -6.20982 q^{96} -1.49702 q^{97} -0.564681 q^{98} +0.880181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} - q^{10} + q^{11} + 3q^{12} - 6q^{13} - 2q^{14} + q^{15} + 4q^{16} + q^{17} - 3q^{18} + 5q^{20} + 2q^{21} + 2q^{22} + q^{23} - 9q^{24} + 6q^{25} + q^{26} + 2q^{27} - 3q^{28} + 2q^{30} - 6q^{31} - 3q^{32} - 2q^{33} + 2q^{34} - q^{35} + 2q^{36} - q^{39} - 2q^{40} - 9q^{42} + q^{43} + 3q^{44} + 5q^{45} + 2q^{46} - q^{47} + 5q^{48} + 5q^{49} - q^{50} - 2q^{51} - 5q^{52} + q^{53} + 4q^{54} + q^{55} - 4q^{56} + 3q^{60} + q^{62} - 3q^{63} + 3q^{64} - 6q^{65} - 4q^{66} - q^{67} - 10q^{68} - 2q^{69} - 2q^{70} - 6q^{72} + q^{75} + 2q^{77} - 2q^{78} + 4q^{80} + 4q^{81} + 6q^{84} + q^{85} - 11q^{86} + 4q^{88} + q^{89} - 3q^{90} + q^{91} + 3q^{92} - q^{93} - 2q^{94} - 7q^{96} - q^{97} - 3q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(3\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(4\) 2.77091 2.77091
\(5\) 1.00000 1.00000
\(6\) −2.90704 −2.90704
\(7\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(8\) −3.43891 −3.43891
\(9\) 1.24107 1.24107
\(10\) −1.94188 −1.94188
\(11\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(12\) 4.14811 4.14811
\(13\) −1.00000 −1.00000
\(14\) −2.20623 −2.20623
\(15\) 1.49702 1.49702
\(16\) 3.90704 3.90704
\(17\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(18\) −2.41002 −2.41002
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.77091 2.77091
\(21\) 1.70081 1.70081
\(22\) −1.37720 −1.37720
\(23\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(24\) −5.14811 −5.14811
\(25\) 1.00000 1.00000
\(26\) 1.94188 1.94188
\(27\) 0.360892 0.360892
\(28\) 3.14811 3.14811
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −2.90704 −2.90704
\(31\) −1.00000 −1.00000
\(32\) −4.14811 −4.14811
\(33\) 1.06170 1.06170
\(34\) 3.43891 3.43891
\(35\) 1.13613 1.13613
\(36\) 3.43891 3.43891
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.49702 −1.49702
\(40\) −3.43891 −3.43891
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −3.30278 −3.30278
\(43\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(44\) 1.96516 1.96516
\(45\) 1.24107 1.24107
\(46\) 0.468136 0.468136
\(47\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(48\) 5.84893 5.84893
\(49\) 0.290790 0.290790
\(50\) −1.94188 −1.94188
\(51\) −2.65109 −2.65109
\(52\) −2.77091 −2.77091
\(53\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(54\) −0.700810 −0.700810
\(55\) 0.709210 0.709210
\(56\) −3.90704 −3.90704
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 4.14811 4.14811
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 1.94188 1.94188
\(63\) 1.41002 1.41002
\(64\) 4.14811 4.14811
\(65\) −1.00000 −1.00000
\(66\) −2.06170 −2.06170
\(67\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(68\) −4.90704 −4.90704
\(69\) −0.360892 −0.360892
\(70\) −2.20623 −2.20623
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −4.26793 −4.26793
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.49702 1.49702
\(76\) 0 0
\(77\) 0.805754 0.805754
\(78\) 2.90704 2.90704
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 3.90704 3.90704
\(81\) −0.700810 −0.700810
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 4.71280 4.71280
\(85\) −1.77091 −1.77091
\(86\) −3.77091 −3.77091
\(87\) 0 0
\(88\) −2.43891 −2.43891
\(89\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(90\) −2.41002 −2.41002
\(91\) −1.13613 −1.13613
\(92\) −0.667993 −0.667993
\(93\) −1.49702 −1.49702
\(94\) −0.468136 −0.468136
\(95\) 0 0
\(96\) −6.20982 −6.20982
\(97\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(98\) −0.564681 −0.564681
\(99\) 0.880181 0.880181
\(100\) 2.77091 2.77091
\(101\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(102\) 5.14811 5.14811
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 3.43891 3.43891
\(105\) 1.70081 1.70081
\(106\) −1.37720 −1.37720
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.00000 1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −1.37720 −1.37720
\(111\) 0 0
\(112\) 4.43891 4.43891
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −0.241073 −0.241073
\(116\) 0 0
\(117\) −1.24107 −1.24107
\(118\) 0 0
\(119\) −2.01199 −2.01199
\(120\) −5.14811 −5.14811
\(121\) −0.497021 −0.497021
\(122\) 0 0
\(123\) 0 0
\(124\) −2.77091 −2.77091
\(125\) 1.00000 1.00000
\(126\) −2.73809 −2.73809
\(127\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(128\) −3.90704 −3.90704
\(129\) 2.90704 2.90704
\(130\) 1.94188 1.94188
\(131\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(132\) 2.94188 2.94188
\(133\) 0 0
\(134\) 1.37720 1.37720
\(135\) 0.360892 0.360892
\(136\) 6.09000 6.09000
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0.700810 0.700810
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 3.14811 3.14811
\(141\) 0.360892 0.360892
\(142\) 0 0
\(143\) −0.709210 −0.709210
\(144\) 4.84893 4.84893
\(145\) 0 0
\(146\) 0 0
\(147\) 0.435319 0.435319
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −2.90704 −2.90704
\(151\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(152\) 0 0
\(153\) −2.19783 −2.19783
\(154\) −1.56468 −1.56468
\(155\) −1.00000 −1.00000
\(156\) −4.14811 −4.14811
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 1.06170 1.06170
\(160\) −4.14811 −4.14811
\(161\) −0.273891 −0.273891
\(162\) 1.36089 1.36089
\(163\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(164\) 0 0
\(165\) 1.06170 1.06170
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −5.84893 −5.84893
\(169\) 1.00000 1.00000
\(170\) 3.43891 3.43891
\(171\) 0 0
\(172\) 5.38079 5.38079
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 1.13613 1.13613
\(176\) 2.77091 2.77091
\(177\) 0 0
\(178\) 2.20623 2.20623
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 3.43891 3.43891
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 2.20623 2.20623
\(183\) 0 0
\(184\) 0.829028 0.829028
\(185\) 0 0
\(186\) 2.90704 2.90704
\(187\) −1.25595 −1.25595
\(188\) 0.667993 0.667993
\(189\) 0.410020 0.410020
\(190\) 0 0
\(191\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(192\) 6.20982 6.20982
\(193\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(194\) 2.90704 2.90704
\(195\) −1.49702 −1.49702
\(196\) 0.805754 0.805754
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −1.70921 −1.70921
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −3.43891 −3.43891
\(201\) −1.06170 −1.06170
\(202\) 3.77091 3.77091
\(203\) 0 0
\(204\) −7.34595 −7.34595
\(205\) 0 0
\(206\) 0 0
\(207\) −0.299190 −0.299190
\(208\) −3.90704 −3.90704
\(209\) 0 0
\(210\) −3.30278 −3.30278
\(211\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(212\) 1.96516 1.96516
\(213\) 0 0
\(214\) 0 0
\(215\) 1.94188 1.94188
\(216\) −1.24107 −1.24107
\(217\) −1.13613 −1.13613
\(218\) 0 0
\(219\) 0 0
\(220\) 1.96516 1.96516
\(221\) 1.77091 1.77091
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −4.71280 −4.71280
\(225\) 1.24107 1.24107
\(226\) 0 0
\(227\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(228\) 0 0
\(229\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(230\) 0.468136 0.468136
\(231\) 1.20623 1.20623
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 2.41002 2.41002
\(235\) 0.241073 0.241073
\(236\) 0 0
\(237\) 0 0
\(238\) 3.90704 3.90704
\(239\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(240\) 5.84893 5.84893
\(241\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(242\) 0.965158 0.965158
\(243\) −1.41002 −1.41002
\(244\) 0 0
\(245\) 0.290790 0.290790
\(246\) 0 0
\(247\) 0 0
\(248\) 3.43891 3.43891
\(249\) 0 0
\(250\) −1.94188 −1.94188
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 3.90704 3.90704
\(253\) −0.170972 −0.170972
\(254\) 3.43891 3.43891
\(255\) −2.65109 −2.65109
\(256\) 3.43891 3.43891
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) −5.64514 −5.64514
\(259\) 0 0
\(260\) −2.77091 −2.77091
\(261\) 0 0
\(262\) −3.43891 −3.43891
\(263\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(264\) −3.65109 −3.65109
\(265\) 0.709210 0.709210
\(266\) 0 0
\(267\) −1.70081 −1.70081
\(268\) −1.96516 −1.96516
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) −0.700810 −0.700810
\(271\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(272\) −6.91903 −6.91903
\(273\) −1.70081 −1.70081
\(274\) 0 0
\(275\) 0.709210 0.709210
\(276\) −1.00000 −1.00000
\(277\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(278\) 0 0
\(279\) −1.24107 −1.24107
\(280\) −3.90704 −3.90704
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −0.700810 −0.700810
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.37720 1.37720
\(287\) 0 0
\(288\) −5.14811 −5.14811
\(289\) 2.13613 2.13613
\(290\) 0 0
\(291\) −2.24107 −2.24107
\(292\) 0 0
\(293\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(294\) −0.845339 −0.845339
\(295\) 0 0
\(296\) 0 0
\(297\) 0.255948 0.255948
\(298\) 0 0
\(299\) 0.241073 0.241073
\(300\) 4.14811 4.14811
\(301\) 2.20623 2.20623
\(302\) −3.77091 −3.77091
\(303\) −2.90704 −2.90704
\(304\) 0 0
\(305\) 0 0
\(306\) 4.26793 4.26793
\(307\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(308\) 2.23267 2.23267
\(309\) 0 0
\(310\) 1.94188 1.94188
\(311\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(312\) 5.14811 5.14811
\(313\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(314\) 0 0
\(315\) 1.41002 1.41002
\(316\) 0 0
\(317\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(318\) −2.06170 −2.06170
\(319\) 0 0
\(320\) 4.14811 4.14811
\(321\) 0 0
\(322\) 0.531864 0.531864
\(323\) 0 0
\(324\) −1.94188 −1.94188
\(325\) −1.00000 −1.00000
\(326\) −3.43891 −3.43891
\(327\) 0 0
\(328\) 0 0
\(329\) 0.273891 0.273891
\(330\) −2.06170 −2.06170
\(331\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.709210 −0.709210
\(336\) 6.64514 6.64514
\(337\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(338\) −1.94188 −1.94188
\(339\) 0 0
\(340\) −4.90704 −4.90704
\(341\) −0.709210 −0.709210
\(342\) 0 0
\(343\) −0.805754 −0.805754
\(344\) −6.67795 −6.67795
\(345\) −0.360892 −0.360892
\(346\) 0 0
\(347\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −2.20623 −2.20623
\(351\) −0.360892 −0.360892
\(352\) −2.94188 −2.94188
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.14811 −3.14811
\(357\) −3.01199 −3.01199
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −4.26793 −4.26793
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −0.744052 −0.744052
\(364\) −3.14811 −3.14811
\(365\) 0 0
\(366\) 0 0
\(367\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(368\) −0.941884 −0.941884
\(369\) 0 0
\(370\) 0 0
\(371\) 0.805754 0.805754
\(372\) −4.14811 −4.14811
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 2.43891 2.43891
\(375\) 1.49702 1.49702
\(376\) −0.829028 −0.829028
\(377\) 0 0
\(378\) −0.796211 −0.796211
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −2.65109 −2.65109
\(382\) 3.77091 3.77091
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −5.84893 −5.84893
\(385\) 0.805754 0.805754
\(386\) −0.468136 −0.468136
\(387\) 2.41002 2.41002
\(388\) −4.14811 −4.14811
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 2.90704 2.90704
\(391\) 0.426920 0.426920
\(392\) −1.00000 −1.00000
\(393\) 2.65109 2.65109
\(394\) 0 0
\(395\) 0 0
\(396\) 2.43891 2.43891
\(397\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.90704 3.90704
\(401\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(402\) 2.06170 2.06170
\(403\) 1.00000 1.00000
\(404\) −5.38079 −5.38079
\(405\) −0.700810 −0.700810
\(406\) 0 0
\(407\) 0 0
\(408\) 9.11686 9.11686
\(409\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.580992 0.580992
\(415\) 0 0
\(416\) 4.14811 4.14811
\(417\) 0 0
\(418\) 0 0
\(419\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(420\) 4.71280 4.71280
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.90704 2.90704
\(423\) 0.299190 0.299190
\(424\) −2.43891 −2.43891
\(425\) −1.77091 −1.77091
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.06170 −1.06170
\(430\) −3.77091 −3.77091
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.41002 1.41002
\(433\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(434\) 2.20623 2.20623
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(440\) −2.43891 −2.43891
\(441\) 0.360892 0.360892
\(442\) −3.43891 −3.43891
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −1.13613 −1.13613
\(446\) 0 0
\(447\) 0 0
\(448\) 4.71280 4.71280
\(449\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(450\) −2.41002 −2.41002
\(451\) 0 0
\(452\) 0 0
\(453\) 2.90704 2.90704
\(454\) −3.43891 −3.43891
\(455\) −1.13613 −1.13613
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 2.20623 2.20623
\(459\) −0.639108 −0.639108
\(460\) −0.667993 −0.667993
\(461\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(462\) −2.34236 −2.34236
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −1.49702 −1.49702
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −3.43891 −3.43891
\(469\) −0.805754 −0.805754
\(470\) −0.468136 −0.468136
\(471\) 0 0
\(472\) 0 0
\(473\) 1.37720 1.37720
\(474\) 0 0
\(475\) 0 0
\(476\) −5.57503 −5.57503
\(477\) 0.880181 0.880181
\(478\) 2.20623 2.20623
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −6.20982 −6.20982
\(481\) 0 0
\(482\) −2.90704 −2.90704
\(483\) −0.410020 −0.410020
\(484\) −1.37720 −1.37720
\(485\) −1.49702 −1.49702
\(486\) 2.73809 2.73809
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 2.65109 2.65109
\(490\) −0.564681 −0.564681
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0.880181 0.880181
\(496\) −3.90704 −3.90704
\(497\) 0 0
\(498\) 0 0
\(499\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(500\) 2.77091 2.77091
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −4.84893 −4.84893
\(505\) −1.94188 −1.94188
\(506\) 0.332007 0.332007
\(507\) 1.49702 1.49702
\(508\) −4.90704 −4.90704
\(509\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(510\) 5.14811 5.14811
\(511\) 0 0
\(512\) −2.77091 −2.77091
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 8.05516 8.05516
\(517\) 0.170972 0.170972
\(518\) 0 0
\(519\) 0 0
\(520\) 3.43891 3.43891
\(521\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(522\) 0 0
\(523\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(524\) 4.90704 4.90704
\(525\) 1.70081 1.70081
\(526\) −3.77091 −3.77091
\(527\) 1.77091 1.77091
\(528\) 4.14811 4.14811
\(529\) −0.941884 −0.941884
\(530\) −1.37720 −1.37720
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 3.30278 3.30278
\(535\) 0 0
\(536\) 2.43891 2.43891
\(537\) 0 0
\(538\) 0 0
\(539\) 0.206231 0.206231
\(540\) 1.00000 1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 3.43891 3.43891
\(543\) 0 0
\(544\) 7.34595 7.34595
\(545\) 0 0
\(546\) 3.30278 3.30278
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.37720 −1.37720
\(551\) 0 0
\(552\) 1.24107 1.24107
\(553\) 0 0
\(554\) 0.468136 0.468136
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 2.41002 2.41002
\(559\) −1.94188 −1.94188
\(560\) 4.43891 4.43891
\(561\) −1.88018 −1.88018
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 1.00000 1.00000
\(565\) 0 0
\(566\) 0 0
\(567\) −0.796211 −0.796211
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −1.96516 −1.96516
\(573\) −2.90704 −2.90704
\(574\) 0 0
\(575\) −0.241073 −0.241073
\(576\) 5.14811 5.14811
\(577\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(578\) −4.14811 −4.14811
\(579\) 0.360892 0.360892
\(580\) 0 0
\(581\) 0 0
\(582\) 4.35190 4.35190
\(583\) 0.502979 0.502979
\(584\) 0 0
\(585\) −1.24107 −1.24107
\(586\) −2.20623 −2.20623
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.20623 1.20623
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(594\) −0.497021 −0.497021
\(595\) −2.01199 −2.01199
\(596\) 0 0
\(597\) 0 0
\(598\) −0.468136 −0.468136
\(599\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(600\) −5.14811 −5.14811
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −4.28424 −4.28424
\(603\) −0.880181 −0.880181
\(604\) 5.38079 5.38079
\(605\) −0.497021 −0.497021
\(606\) 5.64514 5.64514
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.241073 −0.241073
\(612\) −6.09000 −6.09000
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 3.77091 3.77091
\(615\) 0 0
\(616\) −2.77091 −2.77091
\(617\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(618\) 0 0
\(619\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(620\) −2.77091 −2.77091
\(621\) −0.0870014 −0.0870014
\(622\) −0.468136 −0.468136
\(623\) −1.29079 −1.29079
\(624\) −5.84893 −5.84893
\(625\) 1.00000 1.00000
\(626\) 0.468136 0.468136
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −2.73809 −2.73809
\(631\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(632\) 0 0
\(633\) −2.24107 −2.24107
\(634\) −3.43891 −3.43891
\(635\) −1.77091 −1.77091
\(636\) 2.94188 2.94188
\(637\) −0.290790 −0.290790
\(638\) 0 0
\(639\) 0 0
\(640\) −3.90704 −3.90704
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −0.758927 −0.758927
\(645\) 2.90704 2.90704
\(646\) 0 0
\(647\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(648\) 2.41002 2.41002
\(649\) 0 0
\(650\) 1.94188 1.94188
\(651\) −1.70081 −1.70081
\(652\) 4.90704 4.90704
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 1.77091 1.77091
\(656\) 0 0
\(657\) 0 0
\(658\) −0.531864 −0.531864
\(659\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(660\) 2.94188 2.94188
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 3.43891 3.43891
\(663\) 2.65109 2.65109
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 1.37720 1.37720
\(671\) 0 0
\(672\) −7.05516 −7.05516
\(673\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(674\) −1.37720 −1.37720
\(675\) 0.360892 0.360892
\(676\) 2.77091 2.77091
\(677\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(678\) 0 0
\(679\) −1.70081 −1.70081
\(680\) 6.09000 6.09000
\(681\) 2.65109 2.65109
\(682\) 1.37720 1.37720
\(683\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.56468 1.56468
\(687\) −1.70081 −1.70081
\(688\) 7.58702 7.58702
\(689\) −0.709210 −0.709210
\(690\) 0.700810 0.700810
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.00000 1.00000
\(694\) −1.37720 −1.37720
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.14811 3.14811
\(701\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(702\) 0.700810 0.700810
\(703\) 0 0
\(704\) 2.94188 2.94188
\(705\) 0.360892 0.360892
\(706\) 0 0
\(707\) −2.20623 −2.20623
\(708\) 0 0
\(709\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.90704 3.90704
\(713\) 0.241073 0.241073
\(714\) 5.84893 5.84893
\(715\) −0.709210 −0.709210
\(716\) 0 0
\(717\) −1.70081 −1.70081
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 4.84893 4.84893
\(721\) 0 0
\(722\) −1.94188 −1.94188
\(723\) 2.24107 2.24107
\(724\) 0 0
\(725\) 0 0
\(726\) 1.44486 1.44486
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 3.90704 3.90704
\(729\) −1.41002 −1.41002
\(730\) 0 0
\(731\) −3.43891 −3.43891
\(732\) 0 0
\(733\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(734\) 2.20623 2.20623
\(735\) 0.435319 0.435319
\(736\) 1.00000 1.00000
\(737\) −0.502979 −0.502979
\(738\) 0 0
\(739\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.56468 −1.56468
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 5.14811 5.14811
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) −3.48012 −3.48012
\(749\) 0 0
\(750\) −2.90704 −2.90704
\(751\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(752\) 0.941884 0.941884
\(753\) 0 0
\(754\) 0 0
\(755\) 1.94188 1.94188
\(756\) 1.13613 1.13613
\(757\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(758\) 0 0
\(759\) −0.255948 −0.255948
\(760\) 0 0
\(761\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(762\) 5.14811 5.14811
\(763\) 0 0
\(764\) −5.38079 −5.38079
\(765\) −2.19783 −2.19783
\(766\) 0 0
\(767\) 0 0
\(768\) 5.14811 5.14811
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) −1.56468 −1.56468
\(771\) 0 0
\(772\) 0.667993 0.667993
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −4.67998 −4.67998
\(775\) −1.00000 −1.00000
\(776\) 5.14811 5.14811
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −4.14811 −4.14811
\(781\) 0 0
\(782\) −0.829028 −0.829028
\(783\) 0 0
\(784\) 1.13613 1.13613
\(785\) 0 0
\(786\) −5.14811 −5.14811
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 2.90704 2.90704
\(790\) 0 0
\(791\) 0 0
\(792\) −3.02686 −3.02686
\(793\) 0 0
\(794\) 1.37720 1.37720
\(795\) 1.06170 1.06170
\(796\) 0 0
\(797\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(798\) 0 0
\(799\) −0.426920 −0.426920
\(800\) −4.14811 −4.14811
\(801\) −1.41002 −1.41002
\(802\) 0.468136 0.468136
\(803\) 0 0
\(804\) −2.94188 −2.94188
\(805\) −0.273891 −0.273891
\(806\) −1.94188 −1.94188
\(807\) 0 0
\(808\) 6.67795 6.67795
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 1.36089 1.36089
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −2.65109 −2.65109
\(814\) 0 0
\(815\) 1.77091 1.77091
\(816\) −10.3579 −10.3579
\(817\) 0 0
\(818\) 0.468136 0.468136
\(819\) −1.41002 −1.41002
\(820\) 0 0
\(821\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(822\) 0 0
\(823\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(824\) 0 0
\(825\) 1.06170 1.06170
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −0.829028 −0.829028
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −0.360892 −0.360892
\(832\) −4.14811 −4.14811
\(833\) −0.514964 −0.514964
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.360892 −0.360892
\(838\) 2.90704 2.90704
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) −5.84893 −5.84893
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −4.14811 −4.14811
\(845\) 1.00000 1.00000
\(846\) −0.580992 −0.580992
\(847\) −0.564681 −0.564681
\(848\) 2.77091 2.77091
\(849\) 0 0
\(850\) 3.43891 3.43891
\(851\) 0 0
\(852\) 0 0
\(853\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 2.06170 2.06170
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 5.38079 5.38079
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.49702 −1.49702
\(865\) 0 0
\(866\) −3.77091 −3.77091
\(867\) 3.19783 3.19783
\(868\) −3.14811 −3.14811
\(869\) 0 0
\(870\) 0 0
\(871\) 0.709210 0.709210
\(872\) 0 0
\(873\) −1.85791 −1.85791
\(874\) 0 0
\(875\) 1.13613 1.13613
\(876\) 0 0
\(877\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(878\) −0.468136 −0.468136
\(879\) 1.70081 1.70081
\(880\) 2.77091 2.77091
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −0.700810 −0.700810
\(883\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(884\) 4.90704 4.90704
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2.01199 −2.01199
\(890\) 2.20623 2.20623
\(891\) −0.497021 −0.497021
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −4.43891 −4.43891
\(897\) 0.360892 0.360892
\(898\) −3.77091 −3.77091
\(899\) 0 0
\(900\) 3.43891 3.43891
\(901\) −1.25595 −1.25595
\(902\) 0 0
\(903\) 3.30278 3.30278
\(904\) 0 0
\(905\) 0 0
\(906\) −5.64514 −5.64514
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 4.90704 4.90704
\(909\) −2.41002 −2.41002
\(910\) 2.20623 2.20623
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −3.14811 −3.14811
\(917\) 2.01199 2.01199
\(918\) 1.24107 1.24107
\(919\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(920\) 0.829028 0.829028
\(921\) −2.90704 −2.90704
\(922\) −2.90704 −2.90704
\(923\) 0 0
\(924\) 3.34236 3.34236
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(930\) 2.90704 2.90704
\(931\) 0 0
\(932\) 0 0
\(933\) 0.360892 0.360892
\(934\) 0 0
\(935\) −1.25595 −1.25595
\(936\) 4.26793 4.26793
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 1.56468 1.56468
\(939\) −0.360892 −0.360892
\(940\) 0.667993 0.667993
\(941\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0.410020 0.410020
\(946\) −2.67437 −2.67437
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.65109 2.65109
\(952\) 6.91903 6.91903
\(953\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(954\) −1.70921 −1.70921
\(955\) −1.94188 −1.94188
\(956\) −3.14811 −3.14811
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 6.20982 6.20982
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 4.14811 4.14811
\(965\) 0.241073 0.241073
\(966\) 0.796211 0.796211
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 1.70921 1.70921
\(969\) 0 0
\(970\) 2.90704 2.90704
\(971\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(972\) −3.90704 −3.90704
\(973\) 0 0
\(974\) 0 0
\(975\) −1.49702 −1.49702
\(976\) 0 0
\(977\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(978\) −5.14811 −5.14811
\(979\) −0.805754 −0.805754
\(980\) 0.805754 0.805754
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.410020 0.410020
\(988\) 0 0
\(989\) −0.468136 −0.468136
\(990\) −1.70921 −1.70921
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 4.14811 4.14811
\(993\) −2.65109 −2.65109
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −3.77091 −3.77091
\(999\) 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.2014.1 yes 6
5.4 even 2 2015.1.h.d.2014.6 yes 6
13.12 even 2 2015.1.h.e.2014.6 yes 6
31.30 odd 2 2015.1.h.b.2014.1 6
65.64 even 2 2015.1.h.b.2014.1 6
155.154 odd 2 2015.1.h.e.2014.6 yes 6
403.402 odd 2 2015.1.h.d.2014.6 yes 6
2015.2014 odd 2 CM 2015.1.h.c.2014.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2015.1.h.b.2014.1 6 31.30 odd 2
2015.1.h.b.2014.1 6 65.64 even 2
2015.1.h.c.2014.1 yes 6 1.1 even 1 trivial
2015.1.h.c.2014.1 yes 6 2015.2014 odd 2 CM
2015.1.h.d.2014.6 yes 6 5.4 even 2
2015.1.h.d.2014.6 yes 6 403.402 odd 2
2015.1.h.e.2014.6 yes 6 13.12 even 2
2015.1.h.e.2014.6 yes 6 155.154 odd 2