Properties

Label 5040.2.t.y.1009.4
Level $5040$
Weight $2$
Character 5040.1009
Analytic conductor $40.245$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.4
Root \(0.432320 - 0.432320i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1009
Dual form 5040.2.t.y.1009.3

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.432320 + 2.19388i) q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+(-0.432320 + 2.19388i) q^{5} -1.00000i q^{7} -0.626198 q^{11} -5.49084i q^{13} -0.896916i q^{17} +6.38776 q^{19} +3.72928i q^{23} +(-4.62620 - 1.89692i) q^{25} -7.87859 q^{29} -7.52311 q^{31} +(2.19388 + 0.432320i) q^{35} +6.00000i q^{37} -7.72928 q^{41} +1.72928i q^{43} +5.87859i q^{47} -1.00000 q^{49} -6.77551i q^{53} +(0.270718 - 1.37380i) q^{55} +0.593923 q^{59} +7.13536 q^{61} +(12.0462 + 2.37380i) q^{65} +5.79383i q^{67} +5.52311 q^{71} +3.72928i q^{73} +0.626198i q^{77} -5.67243 q^{79} +17.4340i q^{83} +(1.96772 + 0.387755i) q^{85} +14.2986 q^{89} -5.49084 q^{91} +(-2.76156 + 14.0140i) q^{95} -10.1493i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q + 14q^{11} + 8q^{19} - 10q^{25} + 6q^{29} - 20q^{31} - 2q^{35} - 36q^{41} - 6q^{49} + 12q^{55} - 12q^{59} + 48q^{61} + 22q^{65} + 8q^{71} + 34q^{79} + 14q^{85} - 10q^{91} - 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.432320 + 2.19388i −0.193340 + 0.981132i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.626198 −0.188806 −0.0944029 0.995534i \(-0.530094\pi\)
−0.0944029 + 0.995534i \(0.530094\pi\)
\(12\) 0 0
\(13\) 5.49084i 1.52288i −0.648233 0.761442i \(-0.724492\pi\)
0.648233 0.761442i \(-0.275508\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.896916i 0.217534i −0.994067 0.108767i \(-0.965310\pi\)
0.994067 0.108767i \(-0.0346903\pi\)
\(18\) 0 0
\(19\) 6.38776 1.46545 0.732726 0.680524i \(-0.238248\pi\)
0.732726 + 0.680524i \(0.238248\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.72928i 0.777609i 0.921320 + 0.388805i \(0.127112\pi\)
−0.921320 + 0.388805i \(0.872888\pi\)
\(24\) 0 0
\(25\) −4.62620 1.89692i −0.925240 0.379383i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.87859 −1.46302 −0.731509 0.681832i \(-0.761184\pi\)
−0.731509 + 0.681832i \(0.761184\pi\)
\(30\) 0 0
\(31\) −7.52311 −1.35119 −0.675596 0.737272i \(-0.736113\pi\)
−0.675596 + 0.737272i \(0.736113\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.19388 + 0.432320i 0.370833 + 0.0730755i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.72928 −1.20711 −0.603556 0.797321i \(-0.706250\pi\)
−0.603556 + 0.797321i \(0.706250\pi\)
\(42\) 0 0
\(43\) 1.72928i 0.263713i 0.991269 + 0.131856i \(0.0420938\pi\)
−0.991269 + 0.131856i \(0.957906\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.87859i 0.857481i 0.903428 + 0.428741i \(0.141043\pi\)
−0.903428 + 0.428741i \(0.858957\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.77551i 0.930688i −0.885130 0.465344i \(-0.845931\pi\)
0.885130 0.465344i \(-0.154069\pi\)
\(54\) 0 0
\(55\) 0.270718 1.37380i 0.0365036 0.185243i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.593923 0.0773221 0.0386611 0.999252i \(-0.487691\pi\)
0.0386611 + 0.999252i \(0.487691\pi\)
\(60\) 0 0
\(61\) 7.13536 0.913589 0.456795 0.889572i \(-0.348997\pi\)
0.456795 + 0.889572i \(0.348997\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0462 + 2.37380i 1.49415 + 0.294434i
\(66\) 0 0
\(67\) 5.79383i 0.707829i 0.935278 + 0.353915i \(0.115150\pi\)
−0.935278 + 0.353915i \(0.884850\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.52311 0.655473 0.327737 0.944769i \(-0.393714\pi\)
0.327737 + 0.944769i \(0.393714\pi\)
\(72\) 0 0
\(73\) 3.72928i 0.436479i 0.975895 + 0.218240i \(0.0700315\pi\)
−0.975895 + 0.218240i \(0.929969\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.626198i 0.0713619i
\(78\) 0 0
\(79\) −5.67243 −0.638198 −0.319099 0.947721i \(-0.603380\pi\)
−0.319099 + 0.947721i \(0.603380\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.4340i 1.91363i 0.290700 + 0.956814i \(0.406112\pi\)
−0.290700 + 0.956814i \(0.593888\pi\)
\(84\) 0 0
\(85\) 1.96772 + 0.387755i 0.213430 + 0.0420580i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.2986 1.51565 0.757826 0.652457i \(-0.226262\pi\)
0.757826 + 0.652457i \(0.226262\pi\)
\(90\) 0 0
\(91\) −5.49084 −0.575596
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.76156 + 14.0140i −0.283330 + 1.43780i
\(96\) 0 0
\(97\) 10.1493i 1.03051i −0.857038 0.515253i \(-0.827698\pi\)
0.857038 0.515253i \(-0.172302\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.64015 −0.959231 −0.479615 0.877479i \(-0.659224\pi\)
−0.479615 + 0.877479i \(0.659224\pi\)
\(102\) 0 0
\(103\) 0.626198i 0.0617011i −0.999524 0.0308506i \(-0.990178\pi\)
0.999524 0.0308506i \(-0.00982160\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −18.9248 −1.81267 −0.906335 0.422561i \(-0.861131\pi\)
−0.906335 + 0.422561i \(0.861131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.04623i 0.0984209i −0.998788 0.0492105i \(-0.984329\pi\)
0.998788 0.0492105i \(-0.0156705\pi\)
\(114\) 0 0
\(115\) −8.18159 1.61224i −0.762937 0.150343i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.896916 −0.0822202
\(120\) 0 0
\(121\) −10.6079 −0.964352
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.16160 9.32924i 0.551110 0.834432i
\(126\) 0 0
\(127\) 21.2803i 1.88832i 0.329485 + 0.944161i \(0.393125\pi\)
−0.329485 + 0.944161i \(0.606875\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.91087 −0.865917 −0.432958 0.901414i \(-0.642530\pi\)
−0.432958 + 0.901414i \(0.642530\pi\)
\(132\) 0 0
\(133\) 6.38776i 0.553889i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.45856i 0.295485i −0.989026 0.147743i \(-0.952799\pi\)
0.989026 0.147743i \(-0.0472007\pi\)
\(138\) 0 0
\(139\) −20.4157 −1.73163 −0.865817 0.500361i \(-0.833201\pi\)
−0.865817 + 0.500361i \(0.833201\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.43835i 0.287530i
\(144\) 0 0
\(145\) 3.40608 17.2847i 0.282859 1.43541i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.0462 −1.39648 −0.698241 0.715863i \(-0.746033\pi\)
−0.698241 + 0.715863i \(0.746033\pi\)
\(150\) 0 0
\(151\) −2.89692 −0.235748 −0.117874 0.993029i \(-0.537608\pi\)
−0.117874 + 0.993029i \(0.537608\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.25240 16.5048i 0.261239 1.32570i
\(156\) 0 0
\(157\) 10.1170i 0.807427i 0.914885 + 0.403714i \(0.132281\pi\)
−0.914885 + 0.403714i \(0.867719\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.72928 0.293909
\(162\) 0 0
\(163\) 0.476886i 0.0373526i 0.999826 + 0.0186763i \(0.00594519\pi\)
−0.999826 + 0.0186763i \(0.994055\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.1676i 1.01894i −0.860488 0.509471i \(-0.829841\pi\)
0.860488 0.509471i \(-0.170159\pi\)
\(168\) 0 0
\(169\) −17.1493 −1.31918
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.9677i 1.06195i 0.847389 + 0.530973i \(0.178174\pi\)
−0.847389 + 0.530973i \(0.821826\pi\)
\(174\) 0 0
\(175\) −1.89692 + 4.62620i −0.143393 + 0.349708i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.45856 −0.557479 −0.278740 0.960367i \(-0.589917\pi\)
−0.278740 + 0.960367i \(0.589917\pi\)
\(180\) 0 0
\(181\) −14.1170 −1.04931 −0.524656 0.851315i \(-0.675806\pi\)
−0.524656 + 0.851315i \(0.675806\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.1633 2.59392i −0.967783 0.190709i
\(186\) 0 0
\(187\) 0.561647i 0.0410717i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.42003 −0.609252 −0.304626 0.952472i \(-0.598531\pi\)
−0.304626 + 0.952472i \(0.598531\pi\)
\(192\) 0 0
\(193\) 22.2986i 1.60509i 0.596591 + 0.802545i \(0.296521\pi\)
−0.596591 + 0.802545i \(0.703479\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.3632i 1.87830i −0.343509 0.939149i \(-0.611616\pi\)
0.343509 0.939149i \(-0.388384\pi\)
\(198\) 0 0
\(199\) 22.4402 1.59075 0.795373 0.606120i \(-0.207275\pi\)
0.795373 + 0.606120i \(0.207275\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.87859i 0.552969i
\(204\) 0 0
\(205\) 3.34153 16.9571i 0.233382 1.18434i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 15.1955 1.04610 0.523052 0.852301i \(-0.324793\pi\)
0.523052 + 0.852301i \(0.324793\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.79383 0.747604i −0.258737 0.0509862i
\(216\) 0 0
\(217\) 7.52311i 0.510702i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.92482 −0.331279
\(222\) 0 0
\(223\) 6.89692i 0.461852i −0.972971 0.230926i \(-0.925825\pi\)
0.972971 0.230926i \(-0.0741755\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.77988i 0.184507i −0.995736 0.0922535i \(-0.970593\pi\)
0.995736 0.0922535i \(-0.0294070\pi\)
\(228\) 0 0
\(229\) −18.6585 −1.23299 −0.616493 0.787360i \(-0.711447\pi\)
−0.616493 + 0.787360i \(0.711447\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.0462i 0.723663i −0.932244 0.361831i \(-0.882152\pi\)
0.932244 0.361831i \(-0.117848\pi\)
\(234\) 0 0
\(235\) −12.8969 2.54144i −0.841302 0.165785i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.1493 1.30335 0.651675 0.758498i \(-0.274066\pi\)
0.651675 + 0.758498i \(0.274066\pi\)
\(240\) 0 0
\(241\) 3.72928 0.240224 0.120112 0.992760i \(-0.461675\pi\)
0.120112 + 0.992760i \(0.461675\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.432320 2.19388i 0.0276199 0.140162i
\(246\) 0 0
\(247\) 35.0741i 2.23171i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.4985 −1.35698 −0.678488 0.734612i \(-0.737364\pi\)
−0.678488 + 0.734612i \(0.737364\pi\)
\(252\) 0 0
\(253\) 2.33527i 0.146817i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.27072i 0.266400i −0.991089 0.133200i \(-0.957475\pi\)
0.991089 0.133200i \(-0.0425253\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.4586i 0.706565i 0.935517 + 0.353283i \(0.114935\pi\)
−0.935517 + 0.353283i \(0.885065\pi\)
\(264\) 0 0
\(265\) 14.8646 + 2.92919i 0.913128 + 0.179939i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.07081 −0.431115 −0.215557 0.976491i \(-0.569157\pi\)
−0.215557 + 0.976491i \(0.569157\pi\)
\(270\) 0 0
\(271\) −17.0096 −1.03326 −0.516629 0.856209i \(-0.672813\pi\)
−0.516629 + 0.856209i \(0.672813\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.89692 + 1.18785i 0.174691 + 0.0716298i
\(276\) 0 0
\(277\) 18.7755i 1.12811i −0.825737 0.564056i \(-0.809240\pi\)
0.825737 0.564056i \(-0.190760\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.59829 0.274311 0.137156 0.990550i \(-0.456204\pi\)
0.137156 + 0.990550i \(0.456204\pi\)
\(282\) 0 0
\(283\) 13.5833i 0.807443i −0.914882 0.403722i \(-0.867716\pi\)
0.914882 0.403722i \(-0.132284\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.72928i 0.456245i
\(288\) 0 0
\(289\) 16.1955 0.952679
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.8261i 0.690889i 0.938439 + 0.345444i \(0.112272\pi\)
−0.938439 + 0.345444i \(0.887728\pi\)
\(294\) 0 0
\(295\) −0.256765 + 1.30299i −0.0149494 + 0.0758632i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 20.4769 1.18421
\(300\) 0 0
\(301\) 1.72928 0.0996741
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.08476 + 15.6541i −0.176633 + 0.896351i
\(306\) 0 0
\(307\) 20.9860i 1.19774i 0.800847 + 0.598868i \(0.204383\pi\)
−0.800847 + 0.598868i \(0.795617\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.5048 −1.27613 −0.638065 0.769983i \(-0.720265\pi\)
−0.638065 + 0.769983i \(0.720265\pi\)
\(312\) 0 0
\(313\) 12.4846i 0.705670i 0.935686 + 0.352835i \(0.114782\pi\)
−0.935686 + 0.352835i \(0.885218\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.5231i 1.43352i −0.697319 0.716760i \(-0.745624\pi\)
0.697319 0.716760i \(-0.254376\pi\)
\(318\) 0 0
\(319\) 4.93356 0.276226
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.72928i 0.318786i
\(324\) 0 0
\(325\) −10.4157 + 25.4017i −0.577757 + 1.40903i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.87859 0.324097
\(330\) 0 0
\(331\) 2.68305 0.147474 0.0737370 0.997278i \(-0.476507\pi\)
0.0737370 + 0.997278i \(0.476507\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.7110 2.50479i −0.694474 0.136851i
\(336\) 0 0
\(337\) 12.5048i 0.681179i 0.940212 + 0.340590i \(0.110627\pi\)
−0.940212 + 0.340590i \(0.889373\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.71096 0.255113
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.9450i 1.55385i −0.629593 0.776925i \(-0.716778\pi\)
0.629593 0.776925i \(-0.283222\pi\)
\(348\) 0 0
\(349\) −32.1449 −1.72068 −0.860340 0.509721i \(-0.829749\pi\)
−0.860340 + 0.509721i \(0.829749\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.17722i 0.435229i 0.976035 + 0.217615i \(0.0698276\pi\)
−0.976035 + 0.217615i \(0.930172\pi\)
\(354\) 0 0
\(355\) −2.38776 + 12.1170i −0.126729 + 0.643106i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.5048 −0.976646 −0.488323 0.872663i \(-0.662391\pi\)
−0.488323 + 0.872663i \(0.662391\pi\)
\(360\) 0 0
\(361\) 21.8034 1.14755
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.18159 1.61224i −0.428244 0.0843887i
\(366\) 0 0
\(367\) 27.4942i 1.43518i 0.696464 + 0.717592i \(0.254756\pi\)
−0.696464 + 0.717592i \(0.745244\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.77551 −0.351767
\(372\) 0 0
\(373\) 6.06455i 0.314011i −0.987598 0.157005i \(-0.949816\pi\)
0.987598 0.157005i \(-0.0501840\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.2601i 2.22801i
\(378\) 0 0
\(379\) −5.72928 −0.294293 −0.147147 0.989115i \(-0.547009\pi\)
−0.147147 + 0.989115i \(0.547009\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.72928i 0.0883622i 0.999024 + 0.0441811i \(0.0140679\pi\)
−0.999024 + 0.0441811i \(0.985932\pi\)
\(384\) 0 0
\(385\) −1.37380 0.270718i −0.0700154 0.0137971i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −36.7187 −1.86171 −0.930855 0.365389i \(-0.880936\pi\)
−0.930855 + 0.365389i \(0.880936\pi\)
\(390\) 0 0
\(391\) 3.34485 0.169157
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.45231 12.4446i 0.123389 0.626156i
\(396\) 0 0
\(397\) 2.03228i 0.101997i −0.998699 0.0509985i \(-0.983760\pi\)
0.998699 0.0509985i \(-0.0162404\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.64452 0.0821234 0.0410617 0.999157i \(-0.486926\pi\)
0.0410617 + 0.999157i \(0.486926\pi\)
\(402\) 0 0
\(403\) 41.3082i 2.05771i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.75719i 0.186237i
\(408\) 0 0
\(409\) 4.98168 0.246328 0.123164 0.992386i \(-0.460696\pi\)
0.123164 + 0.992386i \(0.460696\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.593923i 0.0292250i
\(414\) 0 0
\(415\) −38.2480 7.53707i −1.87752 0.369980i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −22.9205 −1.11974 −0.559869 0.828581i \(-0.689148\pi\)
−0.559869 + 0.828581i \(0.689148\pi\)
\(420\) 0 0
\(421\) −20.9527 −1.02117 −0.510587 0.859826i \(-0.670572\pi\)
−0.510587 + 0.859826i \(0.670572\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.70138 + 4.14931i −0.0825288 + 0.201271i
\(426\) 0 0
\(427\) 7.13536i 0.345304i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.1589 1.59721 0.798604 0.601857i \(-0.205572\pi\)
0.798604 + 0.601857i \(0.205572\pi\)
\(432\) 0 0
\(433\) 18.5414i 0.891045i 0.895271 + 0.445522i \(0.146982\pi\)
−0.895271 + 0.445522i \(0.853018\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.8217i 1.13955i
\(438\) 0 0
\(439\) −20.8401 −0.994642 −0.497321 0.867567i \(-0.665683\pi\)
−0.497321 + 0.867567i \(0.665683\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.5510i 0.833874i −0.908935 0.416937i \(-0.863104\pi\)
0.908935 0.416937i \(-0.136896\pi\)
\(444\) 0 0
\(445\) −6.18159 + 31.3694i −0.293035 + 1.48705i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13.1955 −0.622736 −0.311368 0.950289i \(-0.600787\pi\)
−0.311368 + 0.950289i \(0.600787\pi\)
\(450\) 0 0
\(451\) 4.84006 0.227910
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.37380 12.0462i 0.111286 0.564736i
\(456\) 0 0
\(457\) 29.1020i 1.36134i −0.732592 0.680668i \(-0.761690\pi\)
0.732592 0.680668i \(-0.238310\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.8280 0.876907 0.438454 0.898754i \(-0.355526\pi\)
0.438454 + 0.898754i \(0.355526\pi\)
\(462\) 0 0
\(463\) 14.0925i 0.654932i −0.944863 0.327466i \(-0.893805\pi\)
0.944863 0.327466i \(-0.106195\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.2663i 0.567619i −0.958881 0.283809i \(-0.908402\pi\)
0.958881 0.283809i \(-0.0915983\pi\)
\(468\) 0 0
\(469\) 5.79383 0.267534
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.08287i 0.0497905i
\(474\) 0 0
\(475\) −29.5510 12.1170i −1.35589 0.555968i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.1570 0.646850 0.323425 0.946254i \(-0.395166\pi\)
0.323425 + 0.946254i \(0.395166\pi\)
\(480\) 0 0
\(481\) 32.9450 1.50216
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 22.2663 + 4.38776i 1.01106 + 0.199238i
\(486\) 0 0
\(487\) 30.2341i 1.37004i −0.728526 0.685018i \(-0.759794\pi\)
0.728526 0.685018i \(-0.240206\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.9065 1.80096 0.900478 0.434902i \(-0.143217\pi\)
0.900478 + 0.434902i \(0.143217\pi\)
\(492\) 0 0
\(493\) 7.06644i 0.318256i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.52311i 0.247746i
\(498\) 0 0
\(499\) 27.3246 1.22322 0.611610 0.791160i \(-0.290522\pi\)
0.611610 + 0.791160i \(0.290522\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.40171i 0.0624991i −0.999512 0.0312495i \(-0.990051\pi\)
0.999512 0.0312495i \(-0.00994866\pi\)
\(504\) 0 0
\(505\) 4.16763 21.1493i 0.185457 0.941132i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.6585 0.827022 0.413511 0.910499i \(-0.364302\pi\)
0.413511 + 0.910499i \(0.364302\pi\)
\(510\) 0 0
\(511\) 3.72928 0.164974
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.37380 + 0.270718i 0.0605369 + 0.0119293i
\(516\) 0 0
\(517\) 3.68116i 0.161897i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.02791 0.176466 0.0882329 0.996100i \(-0.471878\pi\)
0.0882329 + 0.996100i \(0.471878\pi\)
\(522\) 0 0
\(523\) 38.4436i 1.68102i 0.541796 + 0.840510i \(0.317744\pi\)
−0.541796 + 0.840510i \(0.682256\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.74760i 0.293930i
\(528\) 0 0
\(529\) 9.09246 0.395324
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 42.4402i 1.83829i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.626198 0.0269723
\(540\) 0 0
\(541\) 12.4846 0.536754 0.268377 0.963314i \(-0.413513\pi\)
0.268377 + 0.963314i \(0.413513\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.18159 41.5187i 0.350461 1.77847i
\(546\) 0 0
\(547\) 37.7851i 1.61557i −0.589474 0.807787i \(-0.700665\pi\)
0.589474 0.807787i \(-0.299335\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −50.3265 −2.14398
\(552\) 0 0
\(553\) 5.67243i 0.241216i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.93545i 0.0820076i −0.999159 0.0410038i \(-0.986944\pi\)
0.999159 0.0410038i \(-0.0130556\pi\)
\(558\) 0 0
\(559\) 9.49521 0.401604
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 29.8463i 1.25787i 0.777457 + 0.628936i \(0.216509\pi\)
−0.777457 + 0.628936i \(0.783491\pi\)
\(564\) 0 0
\(565\) 2.29530 + 0.452306i 0.0965639 + 0.0190287i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.3449 0.643290 0.321645 0.946860i \(-0.395764\pi\)
0.321645 + 0.946860i \(0.395764\pi\)
\(570\) 0 0
\(571\) −35.8217 −1.49909 −0.749547 0.661952i \(-0.769728\pi\)
−0.749547 + 0.661952i \(0.769728\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.07414 17.2524i 0.295012 0.719475i
\(576\) 0 0
\(577\) 2.92482i 0.121762i 0.998145 + 0.0608810i \(0.0193910\pi\)
−0.998145 + 0.0608810i \(0.980609\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.4340 0.723284
\(582\) 0 0
\(583\) 4.24281i 0.175719i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.5756i 0.807972i 0.914765 + 0.403986i \(0.132375\pi\)
−0.914765 + 0.403986i \(0.867625\pi\)
\(588\) 0 0
\(589\) −48.0558 −1.98011
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.00770i 0.328837i 0.986391 + 0.164418i \(0.0525747\pi\)
−0.986391 + 0.164418i \(0.947425\pi\)
\(594\) 0 0
\(595\) 0.387755 1.96772i 0.0158964 0.0806688i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.3005 1.19719 0.598593 0.801053i \(-0.295727\pi\)
0.598593 + 0.801053i \(0.295727\pi\)
\(600\) 0 0
\(601\) 21.3449 0.870675 0.435337 0.900267i \(-0.356629\pi\)
0.435337 + 0.900267i \(0.356629\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.58600 23.2724i 0.186447 0.946157i
\(606\) 0 0
\(607\) 9.53081i 0.386844i −0.981116 0.193422i \(-0.938041\pi\)
0.981116 0.193422i \(-0.0619586\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.2784 1.30584
\(612\) 0 0
\(613\) 9.75719i 0.394089i 0.980395 + 0.197045i \(0.0631344\pi\)
−0.980395 + 0.197045i \(0.936866\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 37.9267i 1.52687i 0.645884 + 0.763436i \(0.276489\pi\)
−0.645884 + 0.763436i \(0.723511\pi\)
\(618\) 0 0
\(619\) 41.9109 1.68454 0.842270 0.539056i \(-0.181219\pi\)
0.842270 + 0.539056i \(0.181219\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.2986i 0.572862i
\(624\) 0 0
\(625\) 17.8034 + 17.5510i 0.712137 + 0.702041i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.38150 0.214574
\(630\) 0 0
\(631\) 8.42003 0.335196 0.167598 0.985855i \(-0.446399\pi\)
0.167598 + 0.985855i \(0.446399\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −46.6864 9.19991i −1.85269 0.365087i
\(636\) 0 0
\(637\) 5.49084i 0.217555i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.07707 0.0820392 0.0410196 0.999158i \(-0.486939\pi\)
0.0410196 + 0.999158i \(0.486939\pi\)
\(642\) 0 0
\(643\) 27.2480i 1.07456i −0.843405 0.537279i \(-0.819452\pi\)
0.843405 0.537279i \(-0.180548\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.3632i 0.486047i 0.970020 + 0.243023i \(0.0781391\pi\)
−0.970020 + 0.243023i \(0.921861\pi\)
\(648\) 0 0
\(649\) −0.371913 −0.0145989
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 37.3449i 1.46142i 0.682690 + 0.730709i \(0.260810\pi\)
−0.682690 + 0.730709i \(0.739190\pi\)
\(654\) 0 0
\(655\) 4.28467 21.7432i 0.167416 0.849578i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.1772 0.552266 0.276133 0.961119i \(-0.410947\pi\)
0.276133 + 0.961119i \(0.410947\pi\)
\(660\) 0 0
\(661\) −1.76925 −0.0688160 −0.0344080 0.999408i \(-0.510955\pi\)
−0.0344080 + 0.999408i \(0.510955\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.0140 + 2.76156i 0.543438 + 0.107089i
\(666\) 0 0
\(667\) 29.3815i 1.13766i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.46815 −0.172491
\(672\) 0 0
\(673\) 4.05581i 0.156340i −0.996940 0.0781701i \(-0.975092\pi\)
0.996940 0.0781701i \(-0.0249077\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.37713i 0.360392i 0.983631 + 0.180196i \(0.0576733\pi\)
−0.983631 + 0.180196i \(0.942327\pi\)
\(678\) 0 0
\(679\) −10.1493 −0.389495
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.49521i 0.210268i 0.994458 + 0.105134i \(0.0335272\pi\)
−0.994458 + 0.105134i \(0.966473\pi\)
\(684\) 0 0
\(685\) 7.58767 + 1.49521i 0.289910 + 0.0571290i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −37.2032 −1.41733
\(690\) 0 0
\(691\) 3.03416 0.115425 0.0577125 0.998333i \(-0.481619\pi\)
0.0577125 + 0.998333i \(0.481619\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.82611 44.7895i 0.334793 1.69896i
\(696\) 0 0
\(697\) 6.93252i 0.262588i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −35.2234 −1.33037 −0.665186 0.746678i \(-0.731648\pi\)
−0.665186 + 0.746678i \(0.731648\pi\)
\(702\) 0 0
\(703\) 38.3265i 1.44551i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.64015i 0.362555i
\(708\) 0 0
\(709\) −28.7187 −1.07855 −0.539276 0.842129i \(-0.681302\pi\)
−0.539276 + 0.842129i \(0.681302\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.0558i 1.05070i
\(714\) 0 0
\(715\) −7.54333 1.48647i −0.282104 0.0555908i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.60599 0.320949 0.160475 0.987040i \(-0.448698\pi\)
0.160475 + 0.987040i \(0.448698\pi\)
\(720\) 0 0
\(721\) −0.626198 −0.0233208
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 36.4479 + 14.9450i 1.35364 + 0.555045i
\(726\) 0 0
\(727\) 15.2803i 0.566715i 0.959014 + 0.283358i \(0.0914483\pi\)
−0.959014 + 0.283358i \(0.908552\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.55102 0.0573666
\(732\) 0 0
\(733\) 46.0235i 1.69992i 0.526849 + 0.849959i \(0.323373\pi\)
−0.526849 + 0.849959i \(0.676627\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.62809i 0.133642i
\(738\) 0 0
\(739\) 46.3467 1.70489 0.852446 0.522815i \(-0.175118\pi\)
0.852446 + 0.522815i \(0.175118\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.7755i 0.395315i −0.980271 0.197658i \(-0.936667\pi\)
0.980271 0.197658i \(-0.0633334\pi\)
\(744\) 0 0
\(745\) 7.36943 37.3973i 0.269995 1.37013i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.76781 −0.100999 −0.0504995 0.998724i \(-0.516081\pi\)
−0.0504995 + 0.998724i \(0.516081\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.25240 6.35548i 0.0455794 0.231300i
\(756\) 0 0
\(757\) 14.8401i 0.539371i 0.962948 + 0.269686i \(0.0869198\pi\)
−0.962948 + 0.269686i \(0.913080\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.3169 −1.13524 −0.567619 0.823291i \(-0.692135\pi\)
−0.567619 + 0.823291i \(0.692135\pi\)
\(762\) 0 0
\(763\) 18.9248i 0.685125i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.26113i 0.117753i
\(768\) 0 0
\(769\) −8.92586 −0.321875 −0.160937 0.986965i \(-0.551452\pi\)
−0.160937 + 0.986965i \(0.551452\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.7711i 1.32257i 0.750136 + 0.661283i \(0.229988\pi\)
−0.750136 + 0.661283i \(0.770012\pi\)
\(774\) 0 0
\(775\) 34.8034 + 14.2707i 1.25018 + 0.512619i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −49.3728 −1.76896
\(780\) 0 0
\(781\) −3.45856 −0.123757
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −22.1955 4.37380i −0.792193 0.156108i
\(786\) 0 0
\(787\) 6.53707i 0.233021i −0.993189 0.116511i \(-0.962829\pi\)
0.993189 0.116511i \(-0.0371709\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.04623 −0.0371996
\(792\) 0 0
\(793\) 39.1791i 1.39129i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.82611i 0.277215i −0.990347 0.138607i \(-0.955737\pi\)
0.990347 0.138607i \(-0.0442626\pi\)
\(798\) 0 0
\(799\) 5.27261 0.186531
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.33527i 0.0824099i
\(804\) 0 0
\(805\) −1.61224 + 8.18159i −0.0568242 + 0.288363i
\(806\) 0 0
\(807\) 0 0
\(808\)