Properties

Label 2015.1.h.c.2014.6
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 2014.6
Root \(-1.77091\) of defining polynomial
Character \(\chi\) \(=\) 2015.2014

$q$-expansion

\(f(q)\) \(=\) \(q+1.77091 q^{2} -0.241073 q^{3} +2.13613 q^{4} +1.00000 q^{5} -0.426920 q^{6} -0.709210 q^{7} +2.01199 q^{8} -0.941884 q^{9} +O(q^{10})\) \(q+1.77091 q^{2} -0.241073 q^{3} +2.13613 q^{4} +1.00000 q^{5} -0.426920 q^{6} -0.709210 q^{7} +2.01199 q^{8} -0.941884 q^{9} +1.77091 q^{10} +1.49702 q^{11} -0.514964 q^{12} -1.00000 q^{13} -1.25595 q^{14} -0.241073 q^{15} +1.42692 q^{16} -1.13613 q^{17} -1.66799 q^{18} +2.13613 q^{20} +0.170972 q^{21} +2.65109 q^{22} +1.94188 q^{23} -0.485036 q^{24} +1.00000 q^{25} -1.77091 q^{26} +0.468136 q^{27} -1.51496 q^{28} -0.426920 q^{30} -1.00000 q^{31} +0.514964 q^{32} -0.360892 q^{33} -2.01199 q^{34} -0.709210 q^{35} -2.01199 q^{36} +0.241073 q^{39} +2.01199 q^{40} +0.302776 q^{42} -1.77091 q^{43} +3.19783 q^{44} -0.941884 q^{45} +3.43891 q^{46} -1.94188 q^{47} -0.343992 q^{48} -0.497021 q^{49} +1.77091 q^{50} +0.273891 q^{51} -2.13613 q^{52} +1.49702 q^{53} +0.829028 q^{54} +1.49702 q^{55} -1.42692 q^{56} -0.514964 q^{60} -1.77091 q^{62} +0.667993 q^{63} -0.514964 q^{64} -1.00000 q^{65} -0.639108 q^{66} -1.49702 q^{67} -2.42692 q^{68} -0.468136 q^{69} -1.25595 q^{70} -1.89506 q^{72} -0.241073 q^{75} -1.06170 q^{77} +0.426920 q^{78} +1.42692 q^{80} +0.829028 q^{81} +0.365217 q^{84} -1.13613 q^{85} -3.13613 q^{86} +3.01199 q^{88} +0.709210 q^{89} -1.66799 q^{90} +0.709210 q^{91} +4.14811 q^{92} +0.241073 q^{93} -3.43891 q^{94} -0.124144 q^{96} +0.241073 q^{97} -0.880181 q^{98} -1.41002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 6q - q^{2} + q^{3} + 5q^{4} + 6q^{5} + 2q^{6} - q^{7} - 2q^{8} + 5q^{9} - q^{10} + q^{11} + 3q^{12} - 6q^{13} - 2q^{14} + q^{15} + 4q^{16} + q^{17} - 3q^{18} + 5q^{20} + 2q^{21} + 2q^{22} + q^{23} - 9q^{24} + 6q^{25} + q^{26} + 2q^{27} - 3q^{28} + 2q^{30} - 6q^{31} - 3q^{32} - 2q^{33} + 2q^{34} - q^{35} + 2q^{36} - q^{39} - 2q^{40} - 9q^{42} + q^{43} + 3q^{44} + 5q^{45} + 2q^{46} - q^{47} + 5q^{48} + 5q^{49} - q^{50} - 2q^{51} - 5q^{52} + q^{53} + 4q^{54} + q^{55} - 4q^{56} + 3q^{60} + q^{62} - 3q^{63} + 3q^{64} - 6q^{65} - 4q^{66} - q^{67} - 10q^{68} - 2q^{69} - 2q^{70} - 6q^{72} + q^{75} + 2q^{77} - 2q^{78} + 4q^{80} + 4q^{81} + 6q^{84} + q^{85} - 11q^{86} + 4q^{88} + q^{89} - 3q^{90} + q^{91} + 3q^{92} - q^{93} - 2q^{94} - 7q^{96} - q^{97} - 3q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(3\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(4\) 2.13613 2.13613
\(5\) 1.00000 1.00000
\(6\) −0.426920 −0.426920
\(7\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(8\) 2.01199 2.01199
\(9\) −0.941884 −0.941884
\(10\) 1.77091 1.77091
\(11\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(12\) −0.514964 −0.514964
\(13\) −1.00000 −1.00000
\(14\) −1.25595 −1.25595
\(15\) −0.241073 −0.241073
\(16\) 1.42692 1.42692
\(17\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(18\) −1.66799 −1.66799
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.13613 2.13613
\(21\) 0.170972 0.170972
\(22\) 2.65109 2.65109
\(23\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(24\) −0.485036 −0.485036
\(25\) 1.00000 1.00000
\(26\) −1.77091 −1.77091
\(27\) 0.468136 0.468136
\(28\) −1.51496 −1.51496
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −0.426920 −0.426920
\(31\) −1.00000 −1.00000
\(32\) 0.514964 0.514964
\(33\) −0.360892 −0.360892
\(34\) −2.01199 −2.01199
\(35\) −0.709210 −0.709210
\(36\) −2.01199 −2.01199
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0.241073 0.241073
\(40\) 2.01199 2.01199
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0.302776 0.302776
\(43\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(44\) 3.19783 3.19783
\(45\) −0.941884 −0.941884
\(46\) 3.43891 3.43891
\(47\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(48\) −0.343992 −0.343992
\(49\) −0.497021 −0.497021
\(50\) 1.77091 1.77091
\(51\) 0.273891 0.273891
\(52\) −2.13613 −2.13613
\(53\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(54\) 0.829028 0.829028
\(55\) 1.49702 1.49702
\(56\) −1.42692 −1.42692
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −0.514964 −0.514964
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −1.77091 −1.77091
\(63\) 0.667993 0.667993
\(64\) −0.514964 −0.514964
\(65\) −1.00000 −1.00000
\(66\) −0.639108 −0.639108
\(67\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(68\) −2.42692 −2.42692
\(69\) −0.468136 −0.468136
\(70\) −1.25595 −1.25595
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.89506 −1.89506
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −0.241073 −0.241073
\(76\) 0 0
\(77\) −1.06170 −1.06170
\(78\) 0.426920 0.426920
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.42692 1.42692
\(81\) 0.829028 0.829028
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0.365217 0.365217
\(85\) −1.13613 −1.13613
\(86\) −3.13613 −3.13613
\(87\) 0 0
\(88\) 3.01199 3.01199
\(89\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(90\) −1.66799 −1.66799
\(91\) 0.709210 0.709210
\(92\) 4.14811 4.14811
\(93\) 0.241073 0.241073
\(94\) −3.43891 −3.43891
\(95\) 0 0
\(96\) −0.124144 −0.124144
\(97\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(98\) −0.880181 −0.880181
\(99\) −1.41002 −1.41002
\(100\) 2.13613 2.13613
\(101\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(102\) 0.485036 0.485036
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −2.01199 −2.01199
\(105\) 0.170972 0.170972
\(106\) 2.65109 2.65109
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.00000 1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 2.65109 2.65109
\(111\) 0 0
\(112\) −1.01199 −1.01199
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 1.94188 1.94188
\(116\) 0 0
\(117\) 0.941884 0.941884
\(118\) 0 0
\(119\) 0.805754 0.805754
\(120\) −0.485036 −0.485036
\(121\) 1.24107 1.24107
\(122\) 0 0
\(123\) 0 0
\(124\) −2.13613 −2.13613
\(125\) 1.00000 1.00000
\(126\) 1.18296 1.18296
\(127\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(128\) −1.42692 −1.42692
\(129\) 0.426920 0.426920
\(130\) −1.77091 −1.77091
\(131\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(132\) −0.770912 −0.770912
\(133\) 0 0
\(134\) −2.65109 −2.65109
\(135\) 0.468136 0.468136
\(136\) −2.28588 −2.28588
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −0.829028 −0.829028
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −1.51496 −1.51496
\(141\) 0.468136 0.468136
\(142\) 0 0
\(143\) −1.49702 −1.49702
\(144\) −1.34399 −1.34399
\(145\) 0 0
\(146\) 0 0
\(147\) 0.119819 0.119819
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −0.426920 −0.426920
\(151\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(152\) 0 0
\(153\) 1.07010 1.07010
\(154\) −1.88018 −1.88018
\(155\) −1.00000 −1.00000
\(156\) 0.514964 0.514964
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −0.360892 −0.360892
\(160\) 0.514964 0.514964
\(161\) −1.37720 −1.37720
\(162\) 1.46814 1.46814
\(163\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(164\) 0 0
\(165\) −0.360892 −0.360892
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0.343992 0.343992
\(169\) 1.00000 1.00000
\(170\) −2.01199 −2.01199
\(171\) 0 0
\(172\) −3.78290 −3.78290
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −0.709210 −0.709210
\(176\) 2.13613 2.13613
\(177\) 0 0
\(178\) 1.25595 1.25595
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −2.01199 −2.01199
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 1.25595 1.25595
\(183\) 0 0
\(184\) 3.90704 3.90704
\(185\) 0 0
\(186\) 0.426920 0.426920
\(187\) −1.70081 −1.70081
\(188\) −4.14811 −4.14811
\(189\) −0.332007 −0.332007
\(190\) 0 0
\(191\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(192\) 0.124144 0.124144
\(193\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(194\) 0.426920 0.426920
\(195\) 0.241073 0.241073
\(196\) −1.06170 −1.06170
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −2.49702 −2.49702
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 2.01199 2.01199
\(201\) 0.360892 0.360892
\(202\) 3.13613 3.13613
\(203\) 0 0
\(204\) 0.585066 0.585066
\(205\) 0 0
\(206\) 0 0
\(207\) −1.82903 −1.82903
\(208\) −1.42692 −1.42692
\(209\) 0 0
\(210\) 0.302776 0.302776
\(211\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(212\) 3.19783 3.19783
\(213\) 0 0
\(214\) 0 0
\(215\) −1.77091 −1.77091
\(216\) 0.941884 0.941884
\(217\) 0.709210 0.709210
\(218\) 0 0
\(219\) 0 0
\(220\) 3.19783 3.19783
\(221\) 1.13613 1.13613
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −0.365217 −0.365217
\(225\) −0.941884 −0.941884
\(226\) 0 0
\(227\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(228\) 0 0
\(229\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(230\) 3.43891 3.43891
\(231\) 0.255948 0.255948
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 1.66799 1.66799
\(235\) −1.94188 −1.94188
\(236\) 0 0
\(237\) 0 0
\(238\) 1.42692 1.42692
\(239\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(240\) −0.343992 −0.343992
\(241\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(242\) 2.19783 2.19783
\(243\) −0.667993 −0.667993
\(244\) 0 0
\(245\) −0.497021 −0.497021
\(246\) 0 0
\(247\) 0 0
\(248\) −2.01199 −2.01199
\(249\) 0 0
\(250\) 1.77091 1.77091
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.42692 1.42692
\(253\) 2.90704 2.90704
\(254\) −2.01199 −2.01199
\(255\) 0.273891 0.273891
\(256\) −2.01199 −2.01199
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0.756037 0.756037
\(259\) 0 0
\(260\) −2.13613 −2.13613
\(261\) 0 0
\(262\) 2.01199 2.01199
\(263\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(264\) −0.726109 −0.726109
\(265\) 1.49702 1.49702
\(266\) 0 0
\(267\) −0.170972 −0.170972
\(268\) −3.19783 −3.19783
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0.829028 0.829028
\(271\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(272\) −1.62117 −1.62117
\(273\) −0.170972 −0.170972
\(274\) 0 0
\(275\) 1.49702 1.49702
\(276\) −1.00000 −1.00000
\(277\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(278\) 0 0
\(279\) 0.941884 0.941884
\(280\) −1.42692 −1.42692
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0.829028 0.829028
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.65109 −2.65109
\(287\) 0 0
\(288\) −0.485036 −0.485036
\(289\) 0.290790 0.290790
\(290\) 0 0
\(291\) −0.0581164 −0.0581164
\(292\) 0 0
\(293\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(294\) 0.212188 0.212188
\(295\) 0 0
\(296\) 0 0
\(297\) 0.700810 0.700810
\(298\) 0 0
\(299\) −1.94188 −1.94188
\(300\) −0.514964 −0.514964
\(301\) 1.25595 1.25595
\(302\) −3.13613 −3.13613
\(303\) −0.426920 −0.426920
\(304\) 0 0
\(305\) 0 0
\(306\) 1.89506 1.89506
\(307\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(308\) −2.26793 −2.26793
\(309\) 0 0
\(310\) −1.77091 −1.77091
\(311\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(312\) 0.485036 0.485036
\(313\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(314\) 0 0
\(315\) 0.667993 0.667993
\(316\) 0 0
\(317\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(318\) −0.639108 −0.639108
\(319\) 0 0
\(320\) −0.514964 −0.514964
\(321\) 0 0
\(322\) −2.43891 −2.43891
\(323\) 0 0
\(324\) 1.77091 1.77091
\(325\) −1.00000 −1.00000
\(326\) 2.01199 2.01199
\(327\) 0 0
\(328\) 0 0
\(329\) 1.37720 1.37720
\(330\) −0.639108 −0.639108
\(331\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.49702 −1.49702
\(336\) 0.243963 0.243963
\(337\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(338\) 1.77091 1.77091
\(339\) 0 0
\(340\) −2.42692 −2.42692
\(341\) −1.49702 −1.49702
\(342\) 0 0
\(343\) 1.06170 1.06170
\(344\) −3.56305 −3.56305
\(345\) −0.468136 −0.468136
\(346\) 0 0
\(347\) 1.49702 1.49702 0.748511 0.663123i \(-0.230769\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −1.25595 −1.25595
\(351\) −0.468136 −0.468136
\(352\) 0.770912 0.770912
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.51496 1.51496
\(357\) −0.194246 −0.194246
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −1.89506 −1.89506
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −0.299190 −0.299190
\(364\) 1.51496 1.51496
\(365\) 0 0
\(366\) 0 0
\(367\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(368\) 2.77091 2.77091
\(369\) 0 0
\(370\) 0 0
\(371\) −1.06170 −1.06170
\(372\) 0.514964 0.514964
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −3.01199 −3.01199
\(375\) −0.241073 −0.241073
\(376\) −3.90704 −3.90704
\(377\) 0 0
\(378\) −0.587955 −0.587955
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0.273891 0.273891
\(382\) 3.13613 3.13613
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0.343992 0.343992
\(385\) −1.06170 −1.06170
\(386\) −3.43891 −3.43891
\(387\) 1.66799 1.66799
\(388\) 0.514964 0.514964
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0.426920 0.426920
\(391\) −2.20623 −2.20623
\(392\) −1.00000 −1.00000
\(393\) −0.273891 −0.273891
\(394\) 0 0
\(395\) 0 0
\(396\) −3.01199 −3.01199
\(397\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.42692 1.42692
\(401\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(402\) 0.639108 0.639108
\(403\) 1.00000 1.00000
\(404\) 3.78290 3.78290
\(405\) 0.829028 0.829028
\(406\) 0 0
\(407\) 0 0
\(408\) 0.551064 0.551064
\(409\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −3.23905 −3.23905
\(415\) 0 0
\(416\) −0.514964 −0.514964
\(417\) 0 0
\(418\) 0 0
\(419\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(420\) 0.365217 0.365217
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0.426920 0.426920
\(423\) 1.82903 1.82903
\(424\) 3.01199 3.01199
\(425\) −1.13613 −1.13613
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.360892 0.360892
\(430\) −3.13613 −3.13613
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0.667993 0.667993
\(433\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(434\) 1.25595 1.25595
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(440\) 3.01199 3.01199
\(441\) 0.468136 0.468136
\(442\) 2.01199 2.01199
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0.709210 0.709210
\(446\) 0 0
\(447\) 0 0
\(448\) 0.365217 0.365217
\(449\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(450\) −1.66799 −1.66799
\(451\) 0 0
\(452\) 0 0
\(453\) 0.426920 0.426920
\(454\) 2.01199 2.01199
\(455\) 0.709210 0.709210
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 1.25595 1.25595
\(459\) −0.531864 −0.531864
\(460\) 4.14811 4.14811
\(461\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(462\) 0.453262 0.453262
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0.241073 0.241073
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 2.01199 2.01199
\(469\) 1.06170 1.06170
\(470\) −3.43891 −3.43891
\(471\) 0 0
\(472\) 0 0
\(473\) −2.65109 −2.65109
\(474\) 0 0
\(475\) 0 0
\(476\) 1.72120 1.72120
\(477\) −1.41002 −1.41002
\(478\) 1.25595 1.25595
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −0.124144 −0.124144
\(481\) 0 0
\(482\) −0.426920 −0.426920
\(483\) 0.332007 0.332007
\(484\) 2.65109 2.65109
\(485\) 0.241073 0.241073
\(486\) −1.18296 −1.18296
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −0.273891 −0.273891
\(490\) −0.880181 −0.880181
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.41002 −1.41002
\(496\) −1.42692 −1.42692
\(497\) 0 0
\(498\) 0 0
\(499\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(500\) 2.13613 2.13613
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 1.34399 1.34399
\(505\) 1.77091 1.77091
\(506\) 5.14811 5.14811
\(507\) −0.241073 −0.241073
\(508\) −2.42692 −2.42692
\(509\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(510\) 0.485036 0.485036
\(511\) 0 0
\(512\) −2.13613 −2.13613
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0.911956 0.911956
\(517\) −2.90704 −2.90704
\(518\) 0 0
\(519\) 0 0
\(520\) −2.01199 −2.01199
\(521\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(522\) 0 0
\(523\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(524\) 2.42692 2.42692
\(525\) 0.170972 0.170972
\(526\) −3.13613 −3.13613
\(527\) 1.13613 1.13613
\(528\) −0.514964 −0.514964
\(529\) 2.77091 2.77091
\(530\) 2.65109 2.65109
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −0.302776 −0.302776
\(535\) 0 0
\(536\) −3.01199 −3.01199
\(537\) 0 0
\(538\) 0 0
\(539\) −0.744052 −0.744052
\(540\) 1.00000 1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −2.01199 −2.01199
\(543\) 0 0
\(544\) −0.585066 −0.585066
\(545\) 0 0
\(546\) −0.302776 −0.302776
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 2.65109 2.65109
\(551\) 0 0
\(552\) −0.941884 −0.941884
\(553\) 0 0
\(554\) 3.43891 3.43891
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 1.66799 1.66799
\(559\) 1.77091 1.77091
\(560\) −1.01199 −1.01199
\(561\) 0.410020 0.410020
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 1.00000 1.00000
\(565\) 0 0
\(566\) 0 0
\(567\) −0.587955 −0.587955
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −3.19783 −3.19783
\(573\) −0.426920 −0.426920
\(574\) 0 0
\(575\) 1.94188 1.94188
\(576\) 0.485036 0.485036
\(577\) 1.77091 1.77091 0.885456 0.464723i \(-0.153846\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(578\) 0.514964 0.514964
\(579\) 0.468136 0.468136
\(580\) 0 0
\(581\) 0 0
\(582\) −0.102919 −0.102919
\(583\) 2.24107 2.24107
\(584\) 0 0
\(585\) 0.941884 0.941884
\(586\) −1.25595 −1.25595
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0.255948 0.255948
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(594\) 1.24107 1.24107
\(595\) 0.805754 0.805754
\(596\) 0 0
\(597\) 0 0
\(598\) −3.43891 −3.43891
\(599\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(600\) −0.485036 −0.485036
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 2.22417 2.22417
\(603\) 1.41002 1.41002
\(604\) −3.78290 −3.78290
\(605\) 1.24107 1.24107
\(606\) −0.756037 −0.756037
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.94188 1.94188
\(612\) 2.28588 2.28588
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 3.13613 3.13613
\(615\) 0 0
\(616\) −2.13613 −2.13613
\(617\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(618\) 0 0
\(619\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(620\) −2.13613 −2.13613
\(621\) 0.909066 0.909066
\(622\) −3.43891 −3.43891
\(623\) −0.502979 −0.502979
\(624\) 0.343992 0.343992
\(625\) 1.00000 1.00000
\(626\) 3.43891 3.43891
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 1.18296 1.18296
\(631\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(632\) 0 0
\(633\) −0.0581164 −0.0581164
\(634\) 2.01199 2.01199
\(635\) −1.13613 −1.13613
\(636\) −0.770912 −0.770912
\(637\) 0.497021 0.497021
\(638\) 0 0
\(639\) 0 0
\(640\) −1.42692 −1.42692
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −2.94188 −2.94188
\(645\) 0.426920 0.426920
\(646\) 0 0
\(647\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(648\) 1.66799 1.66799
\(649\) 0 0
\(650\) −1.77091 −1.77091
\(651\) −0.170972 −0.170972
\(652\) 2.42692 2.42692
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 1.13613 1.13613
\(656\) 0 0
\(657\) 0 0
\(658\) 2.43891 2.43891
\(659\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(660\) −0.770912 −0.770912
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −2.01199 −2.01199
\(663\) −0.273891 −0.273891
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) −2.65109 −2.65109
\(671\) 0 0
\(672\) 0.0880442 0.0880442
\(673\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(674\) 2.65109 2.65109
\(675\) 0.468136 0.468136
\(676\) 2.13613 2.13613
\(677\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(678\) 0 0
\(679\) −0.170972 −0.170972
\(680\) −2.28588 −2.28588
\(681\) −0.273891 −0.273891
\(682\) −2.65109 −2.65109
\(683\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.88018 1.88018
\(687\) −0.170972 −0.170972
\(688\) −2.52695 −2.52695
\(689\) −1.49702 −1.49702
\(690\) −0.829028 −0.829028
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.00000 1.00000
\(694\) 2.65109 2.65109
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.51496 −1.51496
\(701\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(702\) −0.829028 −0.829028
\(703\) 0 0
\(704\) −0.770912 −0.770912
\(705\) 0.468136 0.468136
\(706\) 0 0
\(707\) −1.25595 −1.25595
\(708\) 0 0
\(709\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.42692 1.42692
\(713\) −1.94188 −1.94188
\(714\) −0.343992 −0.343992
\(715\) −1.49702 −1.49702
\(716\) 0 0
\(717\) −0.170972 −0.170972
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.34399 −1.34399
\(721\) 0 0
\(722\) 1.77091 1.77091
\(723\) 0.0581164 0.0581164
\(724\) 0 0
\(725\) 0 0
\(726\) −0.529839 −0.529839
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 1.42692 1.42692
\(729\) −0.667993 −0.667993
\(730\) 0 0
\(731\) 2.01199 2.01199
\(732\) 0 0
\(733\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(734\) 1.25595 1.25595
\(735\) 0.119819 0.119819
\(736\) 1.00000 1.00000
\(737\) −2.24107 −2.24107
\(738\) 0 0
\(739\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.88018 −1.88018
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0.485036 0.485036
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) −3.63315 −3.63315
\(749\) 0 0
\(750\) −0.426920 −0.426920
\(751\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(752\) −2.77091 −2.77091
\(753\) 0 0
\(754\) 0 0
\(755\) −1.77091 −1.77091
\(756\) −0.709210 −0.709210
\(757\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(758\) 0 0
\(759\) −0.700810 −0.700810
\(760\) 0 0
\(761\) −1.77091 −1.77091 −0.885456 0.464723i \(-0.846154\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(762\) 0.485036 0.485036
\(763\) 0 0
\(764\) 3.78290 3.78290
\(765\) 1.07010 1.07010
\(766\) 0 0
\(767\) 0 0
\(768\) 0.485036 0.485036
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) −1.88018 −1.88018
\(771\) 0 0
\(772\) −4.14811 −4.14811
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.95387 2.95387
\(775\) −1.00000 −1.00000
\(776\) 0.485036 0.485036
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0.514964 0.514964
\(781\) 0 0
\(782\) −3.90704 −3.90704
\(783\) 0 0
\(784\) −0.709210 −0.709210
\(785\) 0 0
\(786\) −0.485036 −0.485036
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0.426920 0.426920
\(790\) 0 0
\(791\) 0 0
\(792\) −2.83694 −2.83694
\(793\) 0 0
\(794\) −2.65109 −2.65109
\(795\) −0.360892 −0.360892
\(796\) 0 0
\(797\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(798\) 0 0
\(799\) 2.20623 2.20623
\(800\) 0.514964 0.514964
\(801\) −0.667993 −0.667993
\(802\) 3.43891 3.43891
\(803\) 0 0
\(804\) 0.770912 0.770912
\(805\) −1.37720 −1.37720
\(806\) 1.77091 1.77091
\(807\) 0 0
\(808\) 3.56305 3.56305
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 1.46814 1.46814
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0.273891 0.273891
\(814\) 0 0
\(815\) 1.13613 1.13613
\(816\) 0.390820 0.390820
\(817\) 0 0
\(818\) 3.43891 3.43891
\(819\) −0.667993 −0.667993
\(820\) 0 0
\(821\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(822\) 0 0
\(823\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(824\) 0 0
\(825\) −0.360892 −0.360892
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −3.90704 −3.90704
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −0.468136 −0.468136
\(832\) 0.514964 0.514964
\(833\) 0.564681 0.564681
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.468136 −0.468136
\(838\) 0.426920 0.426920
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0.343992 0.343992
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0.514964 0.514964
\(845\) 1.00000 1.00000
\(846\) 3.23905 3.23905
\(847\) −0.880181 −0.880181
\(848\) 2.13613 2.13613
\(849\) 0 0
\(850\) −2.01199 −2.01199
\(851\) 0 0
\(852\) 0 0
\(853\) 0.241073 0.241073 0.120537 0.992709i \(-0.461538\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0.639108 0.639108
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −3.78290 −3.78290
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0.241073 0.241073
\(865\) 0 0
\(866\) −3.13613 −3.13613
\(867\) −0.0701018 −0.0701018
\(868\) 1.51496 1.51496
\(869\) 0 0
\(870\) 0 0
\(871\) 1.49702 1.49702
\(872\) 0 0
\(873\) −0.227063 −0.227063
\(874\) 0 0
\(875\) −0.709210 −0.709210
\(876\) 0 0
\(877\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(878\) −3.43891 −3.43891
\(879\) 0.170972 0.170972
\(880\) 2.13613 2.13613
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0.829028 0.829028
\(883\) −1.13613 −1.13613 −0.568065 0.822984i \(-0.692308\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(884\) 2.42692 2.42692
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0.805754 0.805754
\(890\) 1.25595 1.25595
\(891\) 1.24107 1.24107
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.01199 1.01199
\(897\) 0.468136 0.468136
\(898\) −3.13613 −3.13613
\(899\) 0 0
\(900\) −2.01199 −2.01199
\(901\) −1.70081 −1.70081
\(902\) 0 0
\(903\) −0.302776 −0.302776
\(904\) 0 0
\(905\) 0 0
\(906\) 0.756037 0.756037
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 2.42692 2.42692
\(909\) −1.66799 −1.66799
\(910\) 1.25595 1.25595
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.51496 1.51496
\(917\) −0.805754 −0.805754
\(918\) −0.941884 −0.941884
\(919\) −1.49702 −1.49702 −0.748511 0.663123i \(-0.769231\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(920\) 3.90704 3.90704
\(921\) −0.426920 −0.426920
\(922\) −0.426920 −0.426920
\(923\) 0 0
\(924\) 0.546738 0.546738
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.94188 1.94188 0.970942 0.239316i \(-0.0769231\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(930\) 0.426920 0.426920
\(931\) 0 0
\(932\) 0 0
\(933\) 0.468136 0.468136
\(934\) 0 0
\(935\) −1.70081 −1.70081
\(936\) 1.89506 1.89506
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 1.88018 1.88018
\(939\) −0.468136 −0.468136
\(940\) −4.14811 −4.14811
\(941\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −0.332007 −0.332007
\(946\) −4.69485 −4.69485
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −0.273891 −0.273891
\(952\) 1.62117 1.62117
\(953\) 0.709210 0.709210 0.354605 0.935016i \(-0.384615\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(954\) −2.49702 −2.49702
\(955\) 1.77091 1.77091
\(956\) 1.51496 1.51496
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0.124144 0.124144
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −0.514964 −0.514964
\(965\) −1.94188 −1.94188
\(966\) 0.587955 0.587955
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 2.49702 2.49702
\(969\) 0 0
\(970\) 0.426920 0.426920
\(971\) −1.94188 −1.94188 −0.970942 0.239316i \(-0.923077\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(972\) −1.42692 −1.42692
\(973\) 0 0
\(974\) 0 0
\(975\) 0.241073 0.241073
\(976\) 0 0
\(977\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(978\) −0.485036 −0.485036
\(979\) 1.06170 1.06170
\(980\) −1.06170 −1.06170
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.332007 −0.332007
\(988\) 0 0
\(989\) −3.43891 −3.43891
\(990\) −2.49702 −2.49702
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −0.514964 −0.514964
\(993\) 0.273891 0.273891
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −3.13613 −3.13613
\(999\) 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.2014.6 yes 6
5.4 even 2 2015.1.h.d.2014.1 yes 6
13.12 even 2 2015.1.h.e.2014.1 yes 6
31.30 odd 2 2015.1.h.b.2014.6 6
65.64 even 2 2015.1.h.b.2014.6 6
155.154 odd 2 2015.1.h.e.2014.1 yes 6
403.402 odd 2 2015.1.h.d.2014.1 yes 6
2015.2014 odd 2 CM 2015.1.h.c.2014.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2015.1.h.b.2014.6 6 31.30 odd 2
2015.1.h.b.2014.6 6 65.64 even 2
2015.1.h.c.2014.6 yes 6 1.1 even 1 trivial
2015.1.h.c.2014.6 yes 6 2015.2014 odd 2 CM
2015.1.h.d.2014.1 yes 6 5.4 even 2
2015.1.h.d.2014.1 yes 6 403.402 odd 2
2015.1.h.e.2014.1 yes 6 13.12 even 2
2015.1.h.e.2014.1 yes 6 155.154 odd 2