Properties

Label 3920.1.fc.a.1999.2
Level $3920$
Weight $1$
Character 3920.1999
Analytic conductor $1.956$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.fc (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
Defining polynomial: \(x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

Embedding invariants

Embedding label 1999.2
Root \(0.149042 + 0.988831i\) of defining polynomial
Character \(\chi\) \(=\) 3920.1999
Dual form 3920.1.fc.a.1759.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.64786 - 1.12349i) q^{3} +(-0.0747301 - 0.997204i) q^{5} +(-0.294755 + 0.955573i) q^{7} +(1.08786 - 2.77183i) q^{9} +O(q^{10})\) \(q+(1.64786 - 1.12349i) q^{3} +(-0.0747301 - 0.997204i) q^{5} +(-0.294755 + 0.955573i) q^{7} +(1.08786 - 2.77183i) q^{9} +(-1.24349 - 1.55929i) q^{15} +(0.587862 + 1.90580i) q^{21} +(-0.825886 + 0.766310i) q^{23} +(-0.988831 + 0.149042i) q^{25} +(-0.877681 - 3.84537i) q^{27} +(0.326239 - 1.42935i) q^{29} +(0.974928 + 0.222521i) q^{35} +(0.658322 + 0.317031i) q^{41} +(1.67738 - 0.807782i) q^{43} +(-2.84537 - 0.877681i) q^{45} +(0.858075 + 0.129334i) q^{47} +(-0.826239 - 0.563320i) q^{49} +(-1.57906 + 0.487076i) q^{61} +(2.32803 + 1.85654i) q^{63} +(-0.866025 + 1.50000i) q^{67} +(-0.500000 + 2.19064i) q^{69} +(-1.46200 + 1.35654i) q^{75} +(-3.58375 - 3.32523i) q^{81} +(0.367554 + 0.460898i) q^{83} +(-1.06826 - 2.72188i) q^{87} +(-0.535628 + 1.36476i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 2q^{5} + 18q^{9} + O(q^{10}) \) \( 24q - 2q^{5} + 18q^{9} + 6q^{21} + 2q^{25} - 10q^{29} - 4q^{41} - 4q^{45} - 2q^{49} + 2q^{61} - 12q^{69} - 12q^{81} - 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{17}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.64786 1.12349i 1.64786 1.12349i 0.781831 0.623490i \(-0.214286\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(4\) 0 0
\(5\) −0.0747301 0.997204i −0.0747301 0.997204i
\(6\) 0 0
\(7\) −0.294755 + 0.955573i −0.294755 + 0.955573i
\(8\) 0 0
\(9\) 1.08786 2.77183i 1.08786 2.77183i
\(10\) 0 0
\(11\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(12\) 0 0
\(13\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(14\) 0 0
\(15\) −1.24349 1.55929i −1.24349 1.55929i
\(16\) 0 0
\(17\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(18\) 0 0
\(19\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0.587862 + 1.90580i 0.587862 + 1.90580i
\(22\) 0 0
\(23\) −0.825886 + 0.766310i −0.825886 + 0.766310i −0.974928 0.222521i \(-0.928571\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(24\) 0 0
\(25\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(26\) 0 0
\(27\) −0.877681 3.84537i −0.877681 3.84537i
\(28\) 0 0
\(29\) 0.326239 1.42935i 0.326239 1.42935i −0.500000 0.866025i \(-0.666667\pi\)
0.826239 0.563320i \(-0.190476\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(36\) 0 0
\(37\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.658322 + 0.317031i 0.658322 + 0.317031i 0.733052 0.680173i \(-0.238095\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(42\) 0 0
\(43\) 1.67738 0.807782i 1.67738 0.807782i 0.680173 0.733052i \(-0.261905\pi\)
0.997204 0.0747301i \(-0.0238095\pi\)
\(44\) 0 0
\(45\) −2.84537 0.877681i −2.84537 0.877681i
\(46\) 0 0
\(47\) 0.858075 + 0.129334i 0.858075 + 0.129334i 0.563320 0.826239i \(-0.309524\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(48\) 0 0
\(49\) −0.826239 0.563320i −0.826239 0.563320i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(60\) 0 0
\(61\) −1.57906 + 0.487076i −1.57906 + 0.487076i −0.955573 0.294755i \(-0.904762\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(62\) 0 0
\(63\) 2.32803 + 1.85654i 2.32803 + 1.85654i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) −0.500000 + 2.19064i −0.500000 + 2.19064i
\(70\) 0 0
\(71\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(72\) 0 0
\(73\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(74\) 0 0
\(75\) −1.46200 + 1.35654i −1.46200 + 1.35654i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −3.58375 3.32523i −3.58375 3.32523i
\(82\) 0 0
\(83\) 0.367554 + 0.460898i 0.367554 + 0.460898i 0.930874 0.365341i \(-0.119048\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.06826 2.72188i −1.06826 2.72188i
\(88\) 0 0
\(89\) −0.535628 + 1.36476i −0.535628 + 1.36476i 0.365341 + 0.930874i \(0.380952\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.57906 1.07659i 1.57906 1.07659i 0.623490 0.781831i \(-0.285714\pi\)
0.955573 0.294755i \(-0.0952381\pi\)
\(102\) 0 0
\(103\) 0.129436 + 1.72721i 0.129436 + 1.72721i 0.563320 + 0.826239i \(0.309524\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(104\) 0 0
\(105\) 1.85654 0.728639i 1.85654 0.728639i
\(106\) 0 0
\(107\) 0.215372 0.548760i 0.215372 0.548760i −0.781831 0.623490i \(-0.785714\pi\)
0.997204 + 0.0747301i \(0.0238095\pi\)
\(108\) 0 0
\(109\) 0.365341 + 0.930874i 0.365341 + 0.930874i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(114\) 0 0
\(115\) 0.825886 + 0.766310i 0.825886 + 0.766310i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(122\) 0 0
\(123\) 1.44100 0.217196i 1.44100 0.217196i
\(124\) 0 0
\(125\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(126\) 0 0
\(127\) −0.347948 + 1.52446i −0.347948 + 1.52446i 0.433884 + 0.900969i \(0.357143\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(128\) 0 0
\(129\) 1.85654 3.21562i 1.85654 3.21562i
\(130\) 0 0
\(131\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.76903 + 1.16259i −3.76903 + 1.16259i
\(136\) 0 0
\(137\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(138\) 0 0
\(139\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(140\) 0 0
\(141\) 1.55929 0.750915i 1.55929 0.750915i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.44973 0.218511i −1.44973 0.218511i
\(146\) 0 0
\(147\) −1.99441 −1.99441
\(148\) 0 0
\(149\) 1.88980 + 0.284841i 1.88980 + 0.284841i 0.988831 0.149042i \(-0.0476190\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(150\) 0 0
\(151\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.488831 1.01507i −0.488831 1.01507i
\(162\) 0 0
\(163\) 1.61105 + 1.09839i 1.61105 + 1.09839i 0.930874 + 0.365341i \(0.119048\pi\)
0.680173 + 0.733052i \(0.261905\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.0663300 + 0.290611i −0.0663300 + 0.290611i −0.997204 0.0747301i \(-0.976190\pi\)
0.930874 + 0.365341i \(0.119048\pi\)
\(168\) 0 0
\(169\) −0.222521 0.974928i −0.222521 0.974928i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(174\) 0 0
\(175\) 0.149042 0.988831i 0.149042 0.988831i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(180\) 0 0
\(181\) −0.623490 0.781831i −0.623490 0.781831i 0.365341 0.930874i \(-0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(182\) 0 0
\(183\) −2.05484 + 2.57669i −2.05484 + 2.57669i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.93323 + 0.294755i 3.93323 + 0.294755i
\(190\) 0 0
\(191\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(192\) 0 0
\(193\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 0.826239 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(200\) 0 0
\(201\) 0.258149 + 3.44476i 0.258149 + 3.44476i
\(202\) 0 0
\(203\) 1.26968 + 0.733052i 1.26968 + 0.733052i
\(204\) 0 0
\(205\) 0.266948 0.680173i 0.266948 0.680173i
\(206\) 0 0
\(207\) 1.22563 + 3.12285i 1.22563 + 3.12285i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.930874 1.61232i −0.930874 1.61232i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.347948 1.52446i −0.347948 1.52446i −0.781831 0.623490i \(-0.785714\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(224\) 0 0
\(225\) −0.662592 + 2.90301i −0.662592 + 2.90301i
\(226\) 0 0
\(227\) −0.781831 + 1.35417i −0.781831 + 1.35417i 0.149042 + 0.988831i \(0.452381\pi\)
−0.930874 + 0.365341i \(0.880952\pi\)
\(228\) 0 0
\(229\) −0.367711 0.250701i −0.367711 0.250701i 0.365341 0.930874i \(-0.380952\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(234\) 0 0
\(235\) 0.0648483 0.865341i 0.0648483 0.865341i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(240\) 0 0
\(241\) −1.19158 0.367554i −1.19158 0.367554i −0.365341 0.930874i \(-0.619048\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(242\) 0 0
\(243\) −5.74116 0.865341i −5.74116 0.865341i
\(244\) 0 0
\(245\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.12349 + 0.346551i 1.12349 + 0.346551i
\(250\) 0 0
\(251\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.60700 2.45921i −3.60700 2.45921i
\(262\) 0 0
\(263\) 0.680173 1.17809i 0.680173 1.17809i −0.294755 0.955573i \(-0.595238\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.650653 + 2.85070i 0.650653 + 2.85070i
\(268\) 0 0
\(269\) −1.95557 + 0.294755i −1.95557 + 0.294755i −0.955573 + 0.294755i \(0.904762\pi\)
−1.00000 \(1.00000\pi\)
\(270\) 0 0
\(271\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.12349 1.40881i 1.12349 1.40881i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(282\) 0 0
\(283\) 0.317031 + 0.807782i 0.317031 + 0.807782i 0.997204 + 0.0747301i \(0.0238095\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.496990 + 0.535628i −0.496990 + 0.535628i
\(288\) 0 0
\(289\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.277479 + 1.84095i 0.277479 + 1.84095i
\(302\) 0 0
\(303\) 1.39254 3.54812i 1.39254 3.54812i
\(304\) 0 0
\(305\) 0.603718 + 1.53825i 0.603718 + 1.53825i
\(306\) 0 0
\(307\) 0.185853 0.233052i 0.185853 0.233052i −0.680173 0.733052i \(-0.738095\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(308\) 0 0
\(309\) 2.15379 + 2.70077i 2.15379 + 2.70077i
\(310\) 0 0
\(311\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(312\) 0 0
\(313\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 0 0
\(315\) 1.67738 2.46026i 1.67738 2.46026i
\(316\) 0 0
\(317\) 0 0 0.733052 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.261623 1.14625i −0.261623 1.14625i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.64786 + 1.12349i 1.64786 + 1.12349i
\(328\) 0 0
\(329\) −0.376510 + 0.781831i −0.376510 + 0.781831i
\(330\) 0 0
\(331\) 0 0 0.955573 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.56052 + 0.751509i 1.56052 + 0.751509i
\(336\) 0 0
\(337\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.781831 0.623490i 0.781831 0.623490i
\(344\) 0 0
\(345\) 2.22188 + 0.334895i 2.22188 + 0.334895i
\(346\) 0 0
\(347\) 1.77904 + 0.548760i 1.77904 + 0.548760i 0.997204 0.0747301i \(-0.0238095\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(348\) 0 0
\(349\) −0.658322 + 0.317031i −0.658322 + 0.317031i −0.733052 0.680173i \(-0.761905\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(360\) 0 0
\(361\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(362\) 0 0
\(363\) −0.443797 + 1.94440i −0.443797 + 1.94440i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.294755 0.0444272i 0.294755 0.0444272i 1.00000i \(-0.5\pi\)
0.294755 + 0.955573i \(0.404762\pi\)
\(368\) 0 0
\(369\) 1.59492 1.47987i 1.59492 1.47987i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 1.46200 + 1.35654i 1.46200 + 1.35654i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(380\) 0 0
\(381\) 1.13935 + 2.90301i 1.13935 + 2.90301i
\(382\) 0 0
\(383\) 0.496990 1.26631i 0.496990 1.26631i −0.433884 0.900969i \(-0.642857\pi\)
0.930874 0.365341i \(-0.119048\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.414278 5.52815i −0.414278 5.52815i
\(388\) 0 0
\(389\) 0.367711 0.250701i 0.367711 0.250701i −0.365341 0.930874i \(-0.619048\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.0546039 0.139129i 0.0546039 0.139129i −0.900969 0.433884i \(-0.857143\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −3.04812 + 3.82222i −3.04812 + 3.82222i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.44973 1.34515i −1.44973 1.34515i −0.826239 0.563320i \(-0.809524\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.432142 0.400969i 0.432142 0.400969i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(420\) 0 0
\(421\) 0.0332580 0.145713i 0.0332580 0.145713i −0.955573 0.294755i \(-0.904762\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(422\) 0 0
\(423\) 1.29196 2.23774i 1.29196 2.23774i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.65248i 1.65248i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.0747301 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(432\) 0 0
\(433\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(434\) 0 0
\(435\) −2.63444 + 1.26868i −2.63444 + 1.26868i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(440\) 0 0
\(441\) −2.46026 + 1.67738i −2.46026 + 1.67738i
\(442\) 0 0
\(443\) −1.34515 0.202749i −1.34515 0.202749i −0.563320 0.826239i \(-0.690476\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(444\) 0 0
\(445\) 1.40097 + 0.432142i 1.40097 + 0.432142i
\(446\) 0 0
\(447\) 3.43414 1.65379i 3.43414 1.65379i
\(448\) 0 0
\(449\) 1.48883 + 0.716983i 1.48883 + 0.716983i 0.988831 0.149042i \(-0.0476190\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(462\) 0 0
\(463\) −0.131178 0.574730i −0.131178 0.574730i −0.997204 0.0747301i \(-0.976190\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.26968 + 1.17809i −1.26968 + 1.17809i −0.294755 + 0.955573i \(0.595238\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(468\) 0 0
\(469\) −1.17809 1.26968i −1.17809 1.26968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.94594 1.12349i −1.94594 1.12349i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.716983 + 0.488831i −0.716983 + 0.488831i −0.866025 0.500000i \(-0.833333\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(488\) 0 0
\(489\) 3.88881 3.88881
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(500\) 0 0
\(501\) 0.217196 + 0.553406i 0.217196 + 0.553406i
\(502\) 0 0
\(503\) −1.16078 + 1.45557i −1.16078 + 1.45557i −0.294755 + 0.955573i \(0.595238\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) −1.19158 1.49419i −1.19158 1.49419i
\(506\) 0 0
\(507\) −1.46200 1.35654i −1.46200 1.35654i
\(508\) 0 0
\(509\) −0.826239 1.43109i −0.826239 1.43109i −0.900969 0.433884i \(-0.857143\pi\)
0.0747301 0.997204i \(-0.476190\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.71271 0.258149i 1.71271 0.258149i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.222521 0.385418i 0.222521 0.385418i −0.733052 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(522\) 0 0
\(523\) −1.61105 1.09839i −1.61105 1.09839i −0.930874 0.365341i \(-0.880952\pi\)
−0.680173 0.733052i \(-0.738095\pi\)
\(524\) 0 0
\(525\) −0.865341 1.79690i −0.865341 1.79690i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.0201262 0.268565i 0.0201262 0.268565i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.563320 0.173761i −0.563320 0.173761i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.147791 0.0222759i −0.147791 0.0222759i 0.0747301 0.997204i \(-0.476190\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(542\) 0 0
\(543\) −1.90580 0.587862i −1.90580 0.587862i
\(544\) 0 0
\(545\) 0.900969 0.433884i 0.900969 0.433884i
\(546\) 0 0
\(547\) 1.01507 + 0.488831i 1.01507 + 0.488831i 0.866025 0.500000i \(-0.166667\pi\)
0.149042 + 0.988831i \(0.452381\pi\)
\(548\) 0 0
\(549\) −0.367711 + 4.90676i −0.367711 + 4.90676i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.34515 0.202749i 1.34515 0.202749i 0.563320 0.826239i \(-0.309524\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.23383 2.44440i 4.23383 2.44440i
\(568\) 0 0
\(569\) 0.623490 + 1.07992i 0.623490 + 1.07992i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(570\) 0 0
\(571\) 0 0 −0.733052 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.702449 0.880843i 0.702449 0.880843i
\(576\) 0 0
\(577\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.548760 + 0.215372i −0.548760 + 0.215372i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.94986 −1.94986 −0.974928 0.222521i \(-0.928571\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(600\) 0 0
\(601\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(602\) 0 0
\(603\) 3.21562 + 4.03227i 3.21562 + 4.03227i
\(604\) 0 0
\(605\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(606\) 0 0
\(607\) −0.680173 1.17809i −0.680173 1.17809i −0.974928 0.222521i \(-0.928571\pi\)
0.294755 0.955573i \(-0.404762\pi\)
\(608\) 0 0
\(609\) 2.91583 0.218511i 2.91583 0.218511i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.988831 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(614\) 0 0
\(615\) −0.324275 1.42074i −0.324275 1.42074i
\(616\) 0 0
\(617\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(618\) 0 0
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 3.67161 + 2.50326i 3.67161 + 2.50326i
\(622\) 0 0
\(623\) −1.14625 0.914101i −1.14625 0.914101i
\(624\) 0 0
\(625\) 0.955573 0.294755i 0.955573 0.294755i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.54620 + 0.233052i 1.54620 + 0.233052i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.142820 0.0440542i −0.142820 0.0440542i 0.222521 0.974928i \(-0.428571\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(642\) 0 0
\(643\) 1.40881 0.678448i 1.40881 0.678448i 0.433884 0.900969i \(-0.357143\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(644\) 0 0
\(645\) −3.34537 1.61105i −3.34537 1.61105i
\(646\) 0 0
\(647\) −0.0222759 + 0.297251i −0.0222759 + 0.297251i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.997204 + 0.0747301i \(0.976190\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(660\) 0 0
\(661\) 1.88980 0.284841i 1.88980 0.284841i 0.900969 0.433884i \(-0.142857\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.825886 + 1.43048i 0.825886 + 1.43048i
\(668\) 0 0
\(669\) −2.28608 2.12117i −2.28608 2.12117i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(674\) 0 0
\(675\) 1.44100 + 3.67161i 1.44100 + 3.67161i
\(676\) 0 0
\(677\) 0 0 0.365341 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.233052 + 3.10986i 0.233052 + 3.10986i
\(682\) 0 0
\(683\) −0.246289 + 0.167917i −0.246289 + 0.167917i −0.680173 0.733052i \(-0.738095\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.887595 −0.887595
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.0747301 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(692\) 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 0
\(701\) −0.0931869 0.116853i −0.0931869 0.116853i 0.733052 0.680173i \(-0.238095\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.865341 1.49881i −0.865341 1.49881i
\(706\) 0 0
\(707\) 0.563320 + 1.82624i 0.563320 + 1.82624i
\(708\) 0 0
\(709\) −1.40097 + 1.29991i −1.40097 + 1.29991i −0.500000 + 0.866025i \(0.666667\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(710\) 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 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(720\) 0 0
\(721\) −1.68862 0.385418i −1.68862 0.385418i
\(722\) 0 0
\(723\) −2.37650 + 0.733052i −2.37650 + 0.733052i
\(724\) 0 0
\(725\) −0.109562 + 1.46200i −0.109562 + 1.46200i
\(726\) 0 0
\(727\) −0.531130 0.255779i −0.531130 0.255779i 0.149042 0.988831i \(-0.452381\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(728\) 0 0
\(729\) −6.02815 + 2.90301i −6.02815 + 2.90301i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.988831 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(734\) 0 0
\(735\) 0.149042 + 1.98883i 0.149042 + 1.98883i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.955573 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.268565 0.129334i −0.268565 0.129334i 0.294755 0.955573i \(-0.404762\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(744\) 0 0
\(745\) 0.142820 1.90580i 0.142820 1.90580i
\(746\) 0 0
\(747\) 1.67738 0.517402i 1.67738 0.517402i
\(748\) 0 0
\(749\) 0.460898 + 0.367554i 0.460898 + 0.367554i
\(750\) 0 0
\(751\) 0 0 −0.826239 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.914101 + 0.848162i −0.914101 + 0.848162i −0.988831 0.149042i \(-0.952381\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(762\) 0 0
\(763\) −0.997204 + 0.0747301i −0.997204 + 0.0747301i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.777479 + 0.974928i 0.777479 + 0.974928i 1.00000 \(0\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.365341 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.78270 −5.78270
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.12397 + 0.766310i −1.12397 + 0.766310i −0.974928 0.222521i \(-0.928571\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(788\) 0 0
\(789\) −0.202749 2.70550i −0.202749 2.70550i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 3.20018 + 2.96934i 3.20018 + 2.96934i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0