Properties

Label 1183.1.t.a.699.2
Level $1183$
Weight $1$
Character 1183.699
Analytic conductor $0.590$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1

Embedding invariants

Embedding label 699.2
Root \(-1.56052 + 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 1183.699
Dual form 1183.1.t.a.1161.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.07992 + 0.623490i) q^{2} +(0.277479 - 0.480608i) q^{4} +(-0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.07992 + 0.623490i) q^{2} +(0.277479 - 0.480608i) q^{4} +(-0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.385418 + 0.222521i) q^{11} +1.24698 q^{14} +(0.623490 + 1.07992i) q^{16} -1.24698i q^{18} +(0.277479 - 0.480608i) q^{22} +(-0.900969 - 1.56052i) q^{23} -1.00000 q^{25} +(-0.480608 + 0.277479i) q^{28} +(-0.623490 - 1.07992i) q^{29} +(-0.866025 - 0.500000i) q^{32} +(0.277479 + 0.480608i) q^{36} +(-1.56052 + 0.900969i) q^{37} +(0.623490 - 1.07992i) q^{43} +0.246980i q^{44} +(1.94594 + 1.12349i) q^{46} +(0.500000 + 0.866025i) q^{49} +(1.07992 - 0.623490i) q^{50} -1.80194 q^{53} +(-0.277479 + 0.480608i) q^{56} +(1.34663 + 0.777479i) q^{58} +(0.866025 - 0.500000i) q^{63} +(1.56052 - 0.900969i) q^{67} +(-1.56052 - 0.900969i) q^{71} +(0.480608 + 0.277479i) q^{72} +(1.12349 - 1.94594i) q^{74} +0.445042 q^{77} -0.445042 q^{79} +(-0.500000 - 0.866025i) q^{81} +1.55496i q^{86} +(0.123490 + 0.213891i) q^{88} -1.00000 q^{92} +(-1.07992 - 0.623490i) q^{98} -0.445042i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{4} - 6q^{9} + O(q^{10}) \) \( 12q + 4q^{4} - 6q^{9} - 4q^{14} - 2q^{16} + 4q^{22} - 2q^{23} - 12q^{25} + 2q^{29} + 4q^{36} - 2q^{43} + 6q^{49} - 4q^{53} - 4q^{56} + 4q^{74} + 4q^{77} - 4q^{79} - 6q^{81} - 8q^{88} - 12q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(3\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0.277479 0.480608i 0.277479 0.480608i
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −0.866025 0.500000i −0.866025 0.500000i
\(8\) 0.554958i 0.554958i
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) −0.385418 + 0.222521i −0.385418 + 0.222521i −0.680173 0.733052i \(-0.738095\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.24698 1.24698
\(15\) 0 0
\(16\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 1.24698i 1.24698i
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.277479 0.480608i 0.277479 0.480608i
\(23\) −0.900969 1.56052i −0.900969 1.56052i −0.826239 0.563320i \(-0.809524\pi\)
−0.0747301 0.997204i \(-0.523810\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −0.480608 + 0.277479i −0.480608 + 0.277479i
\(29\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.866025 0.500000i −0.866025 0.500000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(37\) −1.56052 + 0.900969i −1.56052 + 0.900969i −0.563320 + 0.826239i \(0.690476\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 0.623490 1.07992i 0.623490 1.07992i −0.365341 0.930874i \(-0.619048\pi\)
0.988831 0.149042i \(-0.0476190\pi\)
\(44\) 0.246980i 0.246980i
\(45\) 0 0
\(46\) 1.94594 + 1.12349i 1.94594 + 1.12349i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 1.07992 0.623490i 1.07992 0.623490i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.277479 + 0.480608i −0.277479 + 0.480608i
\(57\) 0 0
\(58\) 1.34663 + 0.777479i 1.34663 + 0.777479i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0.866025 0.500000i 0.866025 0.500000i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.56052 0.900969i 1.56052 0.900969i 0.563320 0.826239i \(-0.309524\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.56052 0.900969i −1.56052 0.900969i −0.997204 0.0747301i \(-0.976190\pi\)
−0.563320 0.826239i \(-0.690476\pi\)
\(72\) 0.480608 + 0.277479i 0.480608 + 0.277479i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 1.12349 1.94594i 1.12349 1.94594i
\(75\) 0 0
\(76\) 0 0
\(77\) 0.445042 0.445042
\(78\) 0 0
\(79\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.55496i 1.55496i
\(87\) 0 0
\(88\) 0.123490 + 0.213891i 0.123490 + 0.213891i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(98\) −1.07992 0.623490i −1.07992 0.623490i
\(99\) 0.445042i 0.445042i
\(100\) −0.277479 + 0.480608i −0.277479 + 0.480608i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.94594 1.12349i 1.94594 1.12349i
\(107\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(108\) 0 0
\(109\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.24698i 1.24698i
\(113\) 0.222521 0.385418i 0.222521 0.385418i −0.733052 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.692021 −0.692021
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.400969 + 0.694498i −0.400969 + 0.694498i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(127\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(128\) 0.866025 0.500000i 0.866025 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.12349 + 1.94594i −1.12349 + 1.94594i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.385418 + 0.222521i 0.385418 + 0.222521i 0.680173 0.733052i \(-0.261905\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.24698 2.24698
\(143\) 0 0
\(144\) −1.24698 −1.24698
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.00000i 1.00000i
\(149\) 1.07992 + 0.623490i 1.07992 + 0.623490i 0.930874 0.365341i \(-0.119048\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(150\) 0 0
\(151\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.480608 + 0.277479i −0.480608 + 0.277479i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0.480608 0.277479i 0.480608 0.277479i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.80194i 1.80194i
\(162\) 1.07992 + 0.623490i 1.07992 + 0.623490i
\(163\) −1.07992 0.623490i −1.07992 0.623490i −0.149042 0.988831i \(-0.547619\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −0.346011 0.599308i −0.346011 0.599308i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(176\) −0.480608 0.277479i −0.480608 0.277479i
\(177\) 0 0
\(178\) 0 0
\(179\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(192\) 0 0
\(193\) 1.07992 0.623490i 1.07992 0.623490i 0.149042 0.988831i \(-0.452381\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.554958 0.554958
\(197\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0.554958i 0.554958i
\(201\) 0 0
\(202\) 0 0
\(203\) 1.24698i 1.24698i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.80194 1.80194
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.900969 + 1.56052i 0.900969 + 1.56052i 0.826239 + 0.563320i \(0.190476\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(212\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(213\) 0 0
\(214\) −0.480608 0.277479i −0.480608 0.277479i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.277479 0.480608i −0.277479 0.480608i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(225\) 0.500000 0.866025i 0.500000 0.866025i
\(226\) 0.554958i 0.554958i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.599308 + 0.346011i −0.599308 + 0.346011i
\(233\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) 1.00000i 1.00000i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) 0.554958i 0.554958i
\(253\) 0.694498 + 0.400969i 0.694498 + 0.400969i
\(254\) −1.34663 0.777479i −1.34663 0.777479i
\(255\) 0 0
\(256\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 1.80194 1.80194
\(260\) 0 0
\(261\) 1.24698 1.24698
\(262\) 0 0
\(263\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.00000i 1.00000i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.554958 −0.554958
\(275\) 0.385418 0.222521i 0.385418 0.222521i
\(276\) 0 0
\(277\) −0.222521 + 0.385418i −0.222521 + 0.385418i −0.955573 0.294755i \(-0.904762\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.866025 0.500000i 0.866025 0.500000i
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(297\) 0 0
\(298\) −1.55496 −1.55496
\(299\) 0 0
\(300\) 0 0
\(301\) −1.07992 + 0.623490i −1.07992 + 0.623490i
\(302\) −0.777479 1.34663i −0.777479 1.34663i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0.123490 0.213891i 0.123490 0.213891i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.123490 + 0.213891i −0.123490 + 0.213891i
\(317\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(318\) 0 0
\(319\) 0.480608 + 0.277479i 0.480608 + 0.277479i
\(320\) 0 0
\(321\) 0 0
\(322\) −1.12349 1.94594i −1.12349 1.94594i
\(323\) 0 0
\(324\) −0.554958 −0.554958
\(325\) 0 0
\(326\) 1.55496 1.55496
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.56052 0.900969i −1.56052 0.900969i −0.997204 0.0747301i \(-0.976190\pi\)
−0.563320 0.826239i \(-0.690476\pi\)
\(332\) 0 0
\(333\) 1.80194i 1.80194i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) −0.599308 0.346011i −0.599308 0.346011i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(348\) 0 0
\(349\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(350\) −1.24698 −1.24698
\(351\) 0 0
\(352\) 0.445042 0.445042
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.480608 + 0.277479i 0.480608 + 0.277479i
\(359\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(360\) 0 0
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 1.12349 1.94594i 1.12349 1.94594i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.56052 + 0.900969i 1.56052 + 0.900969i
\(372\) 0 0
\(373\) 0.900969 1.56052i 0.900969 1.56052i 0.0747301 0.997204i \(-0.476190\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.55496i 1.55496i
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.777479 + 1.34663i −0.777479 + 1.34663i
\(387\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(388\) 0 0
\(389\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.480608 0.277479i 0.480608 0.277479i
\(393\) 0 0
\(394\) 1.24698 2.15983i 1.24698 2.15983i
\(395\) 0 0
\(396\) −0.213891 0.123490i −0.213891 0.123490i
\(397\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.623490 1.07992i −0.623490 1.07992i
\(401\) 1.07992 0.623490i 1.07992 0.623490i 0.149042 0.988831i \(-0.452381\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −0.777479 1.34663i −0.777479 1.34663i
\(407\) 0.400969 0.694498i 0.400969 0.694498i
\(408\) 0 0
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.94594 + 1.12349i −1.94594 + 1.12349i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(422\) −1.94594 1.12349i −1.94594 1.12349i
\(423\) 0 0
\(424\) 1.00000i 1.00000i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.246980 0.246980
\(429\) 0 0
\(430\) 0 0
\(431\) 0.385418 0.222521i 0.385418 0.222521i −0.294755 0.955573i \(-0.595238\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(432\) 0 0
\(433\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.213891 + 0.123490i 0.213891 + 0.123490i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.56052 + 0.900969i 1.56052 + 0.900969i 0.997204 + 0.0747301i \(0.0238095\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(450\) 1.24698i 1.24698i
\(451\) 0 0
\(452\) −0.123490 0.213891i −0.123490 0.213891i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.385418 0.222521i 0.385418 0.222521i −0.294755 0.955573i \(-0.595238\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(462\) 0 0
\(463\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(464\) 0.777479 1.34663i 0.777479 1.34663i
\(465\) 0 0
\(466\) 1.34663 0.777479i 1.34663 0.777479i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.80194 −1.80194
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.554958i 0.554958i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.900969 1.56052i 0.900969 1.56052i
\(478\) 0.777479 + 1.34663i 0.777479 + 1.34663i
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.385418 0.222521i −0.385418 0.222521i 0.294755 0.955573i \(-0.404762\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.900969 1.56052i −0.900969 1.56052i −0.826239 0.563320i \(-0.809524\pi\)
−0.0747301 0.997204i \(-0.523810\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(498\) 0 0
\(499\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) −0.277479 0.480608i −0.277479 0.480608i
\(505\) 0 0
\(506\) −1.00000 −1.00000
\(507\) 0 0
\(508\) 0.692021 0.692021
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.554958i 0.554958i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −1.94594 + 1.12349i −1.94594 + 1.12349i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.34663 + 0.777479i −1.34663 + 0.777479i
\(523\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.480608 0.277479i −0.480608 0.277479i
\(527\) 0 0
\(528\) 0 0
\(529\) −1.12349 + 1.94594i −1.12349 + 1.94594i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.500000 0.866025i −0.500000 0.866025i
\(537\) 0 0
\(538\) 0 0
\(539\) −0.385418 0.222521i −0.385418 0.222521i
\(540\) 0 0
\(541\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(548\) 0.213891 0.123490i 0.213891 0.123490i
\(549\) 0 0
\(550\) −0.277479 + 0.480608i −0.277479 + 0.480608i
\(551\) 0 0
\(552\) 0 0
\(553\) 0.385418 + 0.222521i 0.385418 + 0.222521i
\(554\) 0.554958i 0.554958i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.56052 + 0.900969i −1.56052 + 0.900969i −0.563320 + 0.826239i \(0.690476\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.24698 2.15983i −1.24698 2.15983i
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 1.00000i
\(568\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(569\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(570\) 0 0
\(571\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.900969 + 1.56052i 0.900969 + 1.56052i
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 1.07992 + 0.623490i 1.07992 + 0.623490i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.694498 0.400969i 0.694498 0.400969i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.94594 1.12349i −1.94594 1.12349i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.599308 0.346011i 0.599308 0.346011i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(600\) 0 0
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0.777479 1.34663i 0.777479 1.34663i
\(603\) 1.80194i 1.80194i
\(604\) 0.599308 + 0.346011i 0.599308 + 0.346011i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.246980i 0.246980i
\(617\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.385418 + 0.222521i 0.385418 + 0.222521i 0.680173 0.733052i \(-0.261905\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(632\) 0.246980i 0.246980i
\(633\) 0 0
\(634\) −0.277479 0.480608i −0.277479 0.480608i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −0.692021 −0.692021
\(639\) 1.56052 0.900969i 1.56052 0.900969i
\(640\) 0 0
\(641\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(642\) 0 0
\(643\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(644\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) −0.480608 + 0.277479i −0.480608 + 0.277479i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.599308 + 0.346011i −0.599308 + 0.346011i
\(653\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 2.24698 2.24698
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(667\) −1.12349 + 1.94594i −1.12349 + 1.94594i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(674\) 1.34663 0.777479i 1.34663 0.777479i
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(687\) 0 0
\(688\) 1.55496 1.55496
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) −0.222521 + 0.385418i −0.222521 + 0.385418i
\(694\) 2.24698i 2.24698i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.480608 0.277479i 0.480608 0.277479i
\(701\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.07992 0.623490i −1.07992 0.623490i −0.149042 0.988831i \(-0.547619\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(710\) 0 0
\(711\) 0.222521 0.385418i 0.222521 0.385418i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.246980 −0.246980
\(717\) 0 0
\(718\) 1.12349 + 1.94594i 1.12349 + 1.94594i
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.07992 0.623490i −1.07992 0.623490i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.623490 + 1.07992i 0.623490 + 1.07992i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.80194i 1.80194i
\(737\) −0.400969 + 0.694498i −0.400969 + 0.694498i
\(738\) 0 0
\(739\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.24698 −2.24698
\(743\) −1.07992 + 0.623490i −1.07992 + 0.623490i −0.930874 0.365341i \(-0.880952\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.24698i 2.24698i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.445042i 0.445042i
\(750\) 0 0
\(751\) −0.222521 0.385418i −0.222521 0.385418i 0.733052 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.623490 1.07992i −0.623490 1.07992i −0.988831 0.149042i \(-0.952381\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(758\) 0.777479 1.34663i 0.777479 1.34663i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0.222521 0.385418i 0.222521 0.385418i
\(764\) 0.346011 + 0.599308i 0.346011 + 0.599308i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.692021i 0.692021i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) −1.34663 0.777479i −1.34663 0.777479i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.480608 + 0.277479i −0.480608 + 0.277479i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.801938 0.801938
\(782\) 0 0
\(783\) 0 0
\(784\) −0.623490 + 1.07992i −0.623490 + 1.07992i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 1.10992i 1.10992i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.385418 + 0.222521i −0.385418 + 0.222521i
\(792\) −0.246980 −0.246980
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(801\) 0 0
\(802\) −0.777479 + 1.34663i −0.777479 + 1.34663i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0