Properties

Label 3969.1.ca.a.2782.1
Level $3969$
Weight $1$
Character 3969.2782
Analytic conductor $1.981$
Analytic rank $0$
Dimension $12$
Projective image $D_{42}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.ca (of order \(42\), degree \(12\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \(x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1323)
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

Embedding invariants

Embedding label 2782.1
Root \(0.365341 - 0.930874i\) of defining polynomial
Character \(\chi\) \(=\) 3969.2782
Dual form 3969.1.ca.a.3862.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.826239 - 0.563320i) q^{4} +(0.733052 - 0.680173i) q^{7} +O(q^{10})\) \(q+(-0.826239 - 0.563320i) q^{4} +(0.733052 - 0.680173i) q^{7} +(-0.548760 + 1.77904i) q^{13} +(0.365341 + 0.930874i) q^{16} +(-1.68862 + 0.974928i) q^{19} +(-0.222521 + 0.974928i) q^{25} +(-0.988831 + 0.149042i) q^{28} +(-1.72721 + 0.997204i) q^{31} +(-0.0111692 - 0.149042i) q^{37} +(1.88980 + 0.284841i) q^{43} +(0.0747301 - 0.997204i) q^{49} +(1.45557 - 1.16078i) q^{52} +(-0.634659 - 0.930874i) q^{61} +(0.222521 - 0.974928i) q^{64} +(0.988831 + 1.71271i) q^{67} +(0.590232 - 0.636119i) q^{73} +(1.94440 + 0.145713i) q^{76} +(-0.826239 + 1.43109i) q^{79} +(0.807782 + 1.67738i) q^{91} +(-0.510531 + 0.294755i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{4} - q^{7} + O(q^{10}) \) \( 12 q - q^{4} - q^{7} + 3 q^{13} + q^{16} - 2 q^{25} + q^{28} - 3 q^{31} - 13 q^{37} + q^{43} + q^{49} + 7 q^{52} - 11 q^{61} + 2 q^{64} - q^{67} - q^{79} + 3 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{19}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.294755 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(3\) 0 0
\(4\) −0.826239 0.563320i −0.826239 0.563320i
\(5\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(6\) 0 0
\(7\) 0.733052 0.680173i 0.733052 0.680173i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(12\) 0 0
\(13\) −0.548760 + 1.77904i −0.548760 + 1.77904i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(17\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(18\) 0 0
\(19\) −1.68862 + 0.974928i −1.68862 + 0.974928i −0.733052 + 0.680173i \(0.761905\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(24\) 0 0
\(25\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(29\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(30\) 0 0
\(31\) −1.72721 + 0.997204i −1.72721 + 0.997204i −0.826239 + 0.563320i \(0.809524\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.0111692 0.149042i −0.0111692 0.149042i 0.988831 0.149042i \(-0.0476190\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(42\) 0 0
\(43\) 1.88980 + 0.284841i 1.88980 + 0.284841i 0.988831 0.149042i \(-0.0476190\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(48\) 0 0
\(49\) 0.0747301 0.997204i 0.0747301 0.997204i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.45557 1.16078i 1.45557 1.16078i
\(53\) 0 0 −0.997204 0.0747301i \(-0.976190\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(60\) 0 0
\(61\) −0.634659 0.930874i −0.634659 0.930874i 0.365341 0.930874i \(-0.380952\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.222521 0.974928i 0.222521 0.974928i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.988831 + 1.71271i 0.988831 + 1.71271i 0.623490 + 0.781831i \(0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(72\) 0 0
\(73\) 0.590232 0.636119i 0.590232 0.636119i −0.365341 0.930874i \(-0.619048\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.94440 + 0.145713i 1.94440 + 0.145713i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.826239 + 1.43109i −0.826239 + 1.43109i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(90\) 0 0
\(91\) 0.807782 + 1.67738i 0.807782 + 1.67738i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.510531 + 0.294755i −0.510531 + 0.294755i −0.733052 0.680173i \(-0.761905\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.733052 0.680173i 0.733052 0.680173i
\(101\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(102\) 0 0
\(103\) −1.06356 + 0.848162i −1.06356 + 0.848162i −0.988831 0.149042i \(-0.952381\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.294755 0.955573i \(-0.595238\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(108\) 0 0
\(109\) −0.698220 0.215372i −0.698220 0.215372i −0.0747301 0.997204i \(-0.523810\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(113\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.900969 0.433884i 0.900969 0.433884i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.98883 + 0.149042i 1.98883 + 0.149042i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.32091 + 0.636119i 1.32091 + 0.636119i 0.955573 0.294755i \(-0.0952381\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(132\) 0 0
\(133\) −0.574730 + 1.86323i −0.574730 + 1.86323i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(138\) 0 0
\(139\) 0.0444272 + 0.294755i 0.0444272 + 0.294755i 1.00000 \(0\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.0747301 + 0.129436i −0.0747301 + 0.129436i
\(149\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(150\) 0 0
\(151\) −1.32091 + 0.636119i −1.32091 + 0.636119i −0.955573 0.294755i \(-0.904762\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.233052 1.54620i 0.233052 1.54620i −0.500000 0.866025i \(-0.666667\pi\)
0.733052 0.680173i \(-0.238095\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.722521 0.108903i −0.722521 0.108903i −0.222521 0.974928i \(-0.571429\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(168\) 0 0
\(169\) −2.03759 1.38921i −2.03759 1.38921i
\(170\) 0 0
\(171\) 0 0
\(172\) −1.40097 1.29991i −1.40097 1.29991i
\(173\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(174\) 0 0
\(175\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.563320 0.826239i \(-0.690476\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(180\) 0 0
\(181\) 1.52446 + 0.347948i 1.52446 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.149042 0.988831i \(-0.547619\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(192\) 0 0
\(193\) −0.266948 + 0.680173i −0.266948 + 0.680173i 0.733052 + 0.680173i \(0.238095\pi\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −0.202749 1.34515i −0.202749 1.34515i −0.826239 0.563320i \(-0.809524\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.85654 + 0.139129i −1.85654 + 0.139129i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.21135 + 1.12397i 1.21135 + 1.12397i 0.988831 + 0.149042i \(0.0476190\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.587862 + 1.90580i −0.587862 + 1.90580i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.488831 + 0.716983i −0.488831 + 0.716983i −0.988831 0.149042i \(-0.952381\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0.460898 + 0.367554i 0.460898 + 0.367554i 0.826239 0.563320i \(-0.190476\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.997204 0.0747301i \(-0.0238095\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(240\) 0 0
\(241\) −0.129334 0.268565i −0.129334 0.268565i 0.826239 0.563320i \(-0.190476\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.12664i 1.12664i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.807782 3.53912i −0.807782 3.53912i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(257\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(258\) 0 0
\(259\) −0.109562 0.101659i −0.109562 0.101659i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.147791 1.97213i 0.147791 1.97213i
\(269\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(270\) 0 0
\(271\) 0.167917 0.246289i 0.167917 0.246289i −0.733052 0.680173i \(-0.761905\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.78181 + 0.858075i 1.78181 + 0.858075i 0.955573 + 0.294755i \(0.0952381\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.680173 0.733052i \(-0.738095\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(282\) 0 0
\(283\) −0.766310 0.825886i −0.766310 0.825886i 0.222521 0.974928i \(-0.428571\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.365341 0.930874i 0.365341 0.930874i
\(290\) 0 0
\(291\) 0 0
\(292\) −0.846011 + 0.193096i −0.846011 + 0.193096i
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.57906 1.07659i 1.57906 1.07659i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.52446 1.21572i −1.52446 1.21572i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.574730 0.131178i 0.574730 0.131178i 0.0747301 0.997204i \(-0.476190\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(312\) 0 0
\(313\) −0.751509 0.433884i −0.751509 0.433884i 0.0747301 0.997204i \(-0.476190\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.48883 0.716983i 1.48883 0.716983i
\(317\) 0 0 −0.997204 0.0747301i \(-0.976190\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.61232 0.930874i −1.61232 0.930874i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.367711 0.250701i 0.367711 0.250701i −0.365341 0.930874i \(-0.619048\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.455573 1.16078i −0.455573 1.16078i −0.955573 0.294755i \(-0.904762\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.623490 0.781831i −0.623490 0.781831i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.563320 0.826239i \(-0.309524\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(348\) 0 0
\(349\) −0.202749 + 1.34515i −0.202749 + 1.34515i 0.623490 + 0.781831i \(0.285714\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.149042 0.988831i \(-0.452381\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(360\) 0 0
\(361\) 1.40097 2.42655i 1.40097 2.42655i
\(362\) 0 0
\(363\) 0 0
\(364\) 0.277479 1.84095i 0.277479 1.84095i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.52446 + 0.347948i 1.52446 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.425270 + 1.86323i 0.425270 + 1.86323i 0.500000 + 0.866025i \(0.333333\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0.587862 + 0.0440542i 0.587862 + 0.0440542i
\(389\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.04876 + 0.411608i 1.04876 + 0.411608i 0.826239 0.563320i \(-0.190476\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(401\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(402\) 0 0
\(403\) −0.826239 3.61999i −0.826239 3.61999i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.12349 + 0.0841939i −1.12349 + 0.0841939i −0.623490 0.781831i \(-0.714286\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.35654 0.101659i 1.35654 0.101659i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(420\) 0 0
\(421\) 0.0931869 + 1.24349i 0.0931869 + 1.24349i 0.826239 + 0.563320i \(0.190476\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.09839 0.250701i −1.09839 0.250701i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.930874 0.365341i \(-0.119048\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(432\) 0 0
\(433\) −1.06356 + 0.848162i −1.06356 + 0.848162i −0.988831 0.149042i \(-0.952381\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.455573 + 0.571270i 0.455573 + 0.571270i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.460898 1.49419i 0.460898 1.49419i −0.365341 0.930874i \(-0.619048\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.294755 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.500000 0.866025i −0.500000 0.866025i
\(449\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.0546039 0.139129i −0.0546039 0.139129i 0.900969 0.433884i \(-0.142857\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(462\) 0 0
\(463\) 0.0931869 1.24349i 0.0931869 1.24349i −0.733052 0.680173i \(-0.761905\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(468\) 0 0
\(469\) 1.88980 + 0.582926i 1.88980 + 0.582926i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.574730 1.86323i −0.574730 1.86323i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(480\) 0 0
\(481\) 0.271281 + 0.0619180i 0.271281 + 0.0619180i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.988831 0.149042i −0.988831 0.149042i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.78181 + 0.268565i −1.78181 + 0.268565i −0.955573 0.294755i \(-0.904762\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.55929 1.24349i −1.55929 1.24349i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.0332580 0.145713i 0.0332580 0.145713i −0.955573 0.294755i \(-0.904762\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.733052 1.26968i −0.733052 1.26968i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0.867767i 0.867767i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −0.297251 + 1.97213i −0.297251 + 1.97213i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.52446 1.21572i 1.52446 1.21572i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.326239 0.302705i 0.326239 0.302705i −0.500000 0.866025i \(-0.666667\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.0546039 0.139129i 0.0546039 0.139129i −0.900969 0.433884i \(-0.857143\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.367711 + 1.61105i 0.367711 + 1.61105i
\(554\) 0 0
\(555\) 0 0
\(556\) 0.129334 0.268565i 0.129334 0.268565i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) −1.54379 + 3.20571i −1.54379 + 3.20571i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −0.134659 1.79690i −0.134659 1.79690i −0.500000 0.866025i \(-0.666667\pi\)
0.365341 0.930874i \(-0.380952\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.0878620 0.284841i 0.0878620 0.284841i −0.900969 0.433884i \(-0.857143\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 1.94440 3.36781i 1.94440 3.36781i
\(590\) 0 0
\(591\) 0 0
\(592\) 0.134659 0.0648483i 0.134659 0.0648483i
\(593\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.680173 0.733052i \(-0.738095\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(600\) 0 0
\(601\) −1.17809 1.26968i −1.17809 1.26968i −0.955573 0.294755i \(-0.904762\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.44973 + 0.218511i 1.44973 + 0.218511i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.21135 1.12397i −1.21135 1.12397i −0.988831 0.149042i \(-0.952381\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.563320 0.826239i \(-0.690476\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(618\) 0 0
\(619\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.900969 0.433884i −0.900969 0.433884i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.06356 + 1.14625i −1.06356 + 1.14625i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.0931869 0.116853i 0.0931869 0.116853i −0.733052 0.680173i \(-0.761905\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.73305 + 0.680173i 1.73305 + 0.680173i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(642\) 0 0
\(643\) 0.277479 0.108903i 0.277479 0.108903i −0.222521 0.974928i \(-0.571429\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.535628 + 0.496990i 0.535628 + 0.496990i
\(653\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.563320 0.826239i \(-0.309524\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(660\) 0 0
\(661\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.914101 0.848162i 0.914101 0.848162i −0.0747301 0.997204i \(-0.523810\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.900969 + 2.29563i 0.900969 + 2.29563i
\(677\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(678\) 0 0
\(679\) −0.173761 + 0.563320i −0.173761 + 0.563320i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.149042 0.988831i \(-0.547619\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0.425270 + 1.86323i 0.425270 + 1.86323i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.258149 + 1.71271i 0.258149 + 1.71271i 0.623490 + 0.781831i \(0.285714\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.0747301 0.997204i 0.0747301 0.997204i
\(701\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(702\) 0 0
\(703\) 0.164166 + 0.240787i 0.164166 + 0.240787i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.142820 + 1.90580i −0.142820 + 1.90580i 0.222521 + 0.974928i \(0.428571\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(720\) 0 0
\(721\) −0.202749 + 1.34515i −0.202749 + 1.34515i
\(722\) 0 0
\(723\) 0 0
\(724\) −1.06356 1.14625i −1.06356 1.14625i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.61232 0.632789i −1.61232 0.632789i −0.623490 0.781831i \(-0.714286\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.32624 0.302705i 1.32624 0.302705i 0.500000 0.866025i \(-0.333333\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.147791 + 1.97213i −0.147791 + 1.97213i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.149042 0.988831i \(-0.547619\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.658322 0.317031i 0.658322 0.317031i −0.0747301 0.997204i \(-0.523810\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(762\) 0 0
\(763\) −0.658322 + 0.317031i −0.658322 + 0.317031i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.603718 0.411608i 0.603718 0.411608i
\(773\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(774\) 0 0
\(775\) −0.587862 1.90580i −0.587862 1.90580i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.955573 0.294755i 0.955573 0.294755i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.85654 + 0.728639i 1.85654 + 0.728639i 0.955573 + 0.294755i \(0.0952381\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00433 0.618255i 2.00433 0.618255i
\(794\) 0 0
\(795\) 0 0
\(796\) −0.590232 + 1.22563i −0.590232 + 1.22563i
\(797\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.680173 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)