Properties

Label 720.6.f.n.289.8
Level 720
Weight 6
Character 720.289
Analytic conductor 115.476
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.8
Root \(4.73066i\) of defining polynomial
Character \(\chi\) \(=\) 720.289
Dual form 720.6.f.n.289.7

$q$-expansion

\(f(q)\) \(=\) \(q+(53.0051 + 17.7613i) q^{5} +188.968i q^{7} +O(q^{10})\) \(q+(53.0051 + 17.7613i) q^{5} +188.968i q^{7} -501.871 q^{11} -1061.48i q^{13} +29.5861i q^{17} +1578.33 q^{19} +1295.86i q^{23} +(2494.07 + 1882.88i) q^{25} -3586.63 q^{29} +3526.32 q^{31} +(-3356.31 + 10016.2i) q^{35} +8413.79i q^{37} -7015.12 q^{41} +22694.2i q^{43} -3501.99i q^{47} -18901.8 q^{49} +27309.1i q^{53} +(-26601.7 - 8913.88i) q^{55} -7925.39 q^{59} -7020.54 q^{61} +(18853.3 - 56264.0i) q^{65} -17631.2i q^{67} +13432.9 q^{71} -39946.8i q^{73} -94837.3i q^{77} -93321.2 q^{79} +58448.5i q^{83} +(-525.488 + 1568.21i) q^{85} -13989.4 q^{89} +200586. q^{91} +(83659.6 + 28033.2i) q^{95} -110640. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 53.0051 + 17.7613i 0.948183 + 0.317724i
\(6\) 0 0
\(7\) 188.968i 1.45761i 0.684720 + 0.728807i \(0.259925\pi\)
−0.684720 + 0.728807i \(0.740075\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −501.871 −1.25058 −0.625288 0.780394i \(-0.715018\pi\)
−0.625288 + 0.780394i \(0.715018\pi\)
\(12\) 0 0
\(13\) 1061.48i 1.74203i −0.491259 0.871013i \(-0.663463\pi\)
0.491259 0.871013i \(-0.336537\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 29.5861i 0.0248294i 0.999923 + 0.0124147i \(0.00395182\pi\)
−0.999923 + 0.0124147i \(0.996048\pi\)
\(18\) 0 0
\(19\) 1578.33 1.00303 0.501515 0.865149i \(-0.332776\pi\)
0.501515 + 0.865149i \(0.332776\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1295.86i 0.510786i 0.966837 + 0.255393i \(0.0822048\pi\)
−0.966837 + 0.255393i \(0.917795\pi\)
\(24\) 0 0
\(25\) 2494.07 + 1882.88i 0.798103 + 0.602521i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3586.63 −0.791938 −0.395969 0.918264i \(-0.629591\pi\)
−0.395969 + 0.918264i \(0.629591\pi\)
\(30\) 0 0
\(31\) 3526.32 0.659048 0.329524 0.944147i \(-0.393112\pi\)
0.329524 + 0.944147i \(0.393112\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3356.31 + 10016.2i −0.463118 + 1.38208i
\(36\) 0 0
\(37\) 8413.79i 1.01039i 0.863006 + 0.505193i \(0.168579\pi\)
−0.863006 + 0.505193i \(0.831421\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7015.12 −0.651742 −0.325871 0.945414i \(-0.605657\pi\)
−0.325871 + 0.945414i \(0.605657\pi\)
\(42\) 0 0
\(43\) 22694.2i 1.87173i 0.352358 + 0.935865i \(0.385380\pi\)
−0.352358 + 0.935865i \(0.614620\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3501.99i 0.231244i −0.993293 0.115622i \(-0.963114\pi\)
0.993293 0.115622i \(-0.0368861\pi\)
\(48\) 0 0
\(49\) −18901.8 −1.12464
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 27309.1i 1.33542i 0.744422 + 0.667710i \(0.232725\pi\)
−0.744422 + 0.667710i \(0.767275\pi\)
\(54\) 0 0
\(55\) −26601.7 8913.88i −1.18578 0.397338i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7925.39 −0.296409 −0.148204 0.988957i \(-0.547349\pi\)
−0.148204 + 0.988957i \(0.547349\pi\)
\(60\) 0 0
\(61\) −7020.54 −0.241572 −0.120786 0.992679i \(-0.538541\pi\)
−0.120786 + 0.992679i \(0.538541\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 18853.3 56264.0i 0.553483 1.65176i
\(66\) 0 0
\(67\) 17631.2i 0.479837i −0.970793 0.239919i \(-0.922879\pi\)
0.970793 0.239919i \(-0.0771208\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13432.9 0.316246 0.158123 0.987419i \(-0.449456\pi\)
0.158123 + 0.987419i \(0.449456\pi\)
\(72\) 0 0
\(73\) 39946.8i 0.877353i −0.898645 0.438677i \(-0.855447\pi\)
0.898645 0.438677i \(-0.144553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 94837.3i 1.82286i
\(78\) 0 0
\(79\) −93321.2 −1.68234 −0.841168 0.540774i \(-0.818131\pi\)
−0.841168 + 0.540774i \(0.818131\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 58448.5i 0.931275i 0.884975 + 0.465638i \(0.154175\pi\)
−0.884975 + 0.465638i \(0.845825\pi\)
\(84\) 0 0
\(85\) −525.488 + 1568.21i −0.00788888 + 0.0235428i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13989.4 −0.187208 −0.0936038 0.995610i \(-0.529839\pi\)
−0.0936038 + 0.995610i \(0.529839\pi\)
\(90\) 0 0
\(91\) 200586. 2.53920
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 83659.6 + 28033.2i 0.951057 + 0.318687i
\(96\) 0 0
\(97\) 110640.i 1.19394i −0.802264 0.596969i \(-0.796371\pi\)
0.802264 0.596969i \(-0.203629\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −198468. −1.93592 −0.967960 0.251103i \(-0.919207\pi\)
−0.967960 + 0.251103i \(0.919207\pi\)
\(102\) 0 0
\(103\) 134780.i 1.25179i 0.779906 + 0.625897i \(0.215267\pi\)
−0.779906 + 0.625897i \(0.784733\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 39785.5i 0.335943i 0.985792 + 0.167971i \(0.0537216\pi\)
−0.985792 + 0.167971i \(0.946278\pi\)
\(108\) 0 0
\(109\) −92692.6 −0.747272 −0.373636 0.927575i \(-0.621889\pi\)
−0.373636 + 0.927575i \(0.621889\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 66923.1i 0.493037i −0.969138 0.246519i \(-0.920713\pi\)
0.969138 0.246519i \(-0.0792866\pi\)
\(114\) 0 0
\(115\) −23016.2 + 68687.2i −0.162289 + 0.484318i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5590.81 −0.0361916
\(120\) 0 0
\(121\) 90823.1 0.563940
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 98756.1 + 144100.i 0.565313 + 0.824877i
\(126\) 0 0
\(127\) 58998.9i 0.324589i 0.986742 + 0.162295i \(0.0518895\pi\)
−0.986742 + 0.162295i \(0.948110\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −209380. −1.06600 −0.533001 0.846115i \(-0.678936\pi\)
−0.533001 + 0.846115i \(0.678936\pi\)
\(132\) 0 0
\(133\) 298254.i 1.46203i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 326432.i 1.48591i −0.669344 0.742953i \(-0.733425\pi\)
0.669344 0.742953i \(-0.266575\pi\)
\(138\) 0 0
\(139\) −233648. −1.02571 −0.512856 0.858475i \(-0.671412\pi\)
−0.512856 + 0.858475i \(0.671412\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 532727.i 2.17854i
\(144\) 0 0
\(145\) −190109. 63703.2i −0.750902 0.251618i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 145336. 0.536299 0.268149 0.963377i \(-0.413588\pi\)
0.268149 + 0.963377i \(0.413588\pi\)
\(150\) 0 0
\(151\) −287343. −1.02555 −0.512777 0.858522i \(-0.671383\pi\)
−0.512777 + 0.858522i \(0.671383\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 186913. + 62632.0i 0.624899 + 0.209395i
\(156\) 0 0
\(157\) 42626.2i 0.138015i −0.997616 0.0690077i \(-0.978017\pi\)
0.997616 0.0690077i \(-0.0219833\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −244876. −0.744528
\(162\) 0 0
\(163\) 221599.i 0.653279i −0.945149 0.326639i \(-0.894084\pi\)
0.945149 0.326639i \(-0.105916\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 334835.i 0.929051i −0.885560 0.464525i \(-0.846225\pi\)
0.885560 0.464525i \(-0.153775\pi\)
\(168\) 0 0
\(169\) −755454. −2.03466
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 268475.i 0.682006i −0.940062 0.341003i \(-0.889233\pi\)
0.940062 0.341003i \(-0.110767\pi\)
\(174\) 0 0
\(175\) −355803. + 471299.i −0.878242 + 1.16333i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 268052. 0.625297 0.312649 0.949869i \(-0.398784\pi\)
0.312649 + 0.949869i \(0.398784\pi\)
\(180\) 0 0
\(181\) −90565.5 −0.205479 −0.102739 0.994708i \(-0.532761\pi\)
−0.102739 + 0.994708i \(0.532761\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −149440. + 445973.i −0.321024 + 0.958031i
\(186\) 0 0
\(187\) 14848.4i 0.0310510i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −706725. −1.40174 −0.700870 0.713290i \(-0.747204\pi\)
−0.700870 + 0.713290i \(0.747204\pi\)
\(192\) 0 0
\(193\) 236999.i 0.457988i −0.973428 0.228994i \(-0.926456\pi\)
0.973428 0.228994i \(-0.0735435\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 607901.i 1.11601i 0.829838 + 0.558004i \(0.188433\pi\)
−0.829838 + 0.558004i \(0.811567\pi\)
\(198\) 0 0
\(199\) 621060. 1.11173 0.555867 0.831272i \(-0.312387\pi\)
0.555867 + 0.831272i \(0.312387\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 677756.i 1.15434i
\(204\) 0 0
\(205\) −371837. 124598.i −0.617971 0.207074i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −792118. −1.25437
\(210\) 0 0
\(211\) −1.06177e6 −1.64181 −0.820906 0.571063i \(-0.806531\pi\)
−0.820906 + 0.571063i \(0.806531\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −403078. + 1.20291e6i −0.594693 + 1.77474i
\(216\) 0 0
\(217\) 666360.i 0.960638i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 31405.2 0.0432534
\(222\) 0 0
\(223\) 240721.i 0.324154i 0.986778 + 0.162077i \(0.0518193\pi\)
−0.986778 + 0.162077i \(0.948181\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 202494.i 0.260824i −0.991460 0.130412i \(-0.958370\pi\)
0.991460 0.130412i \(-0.0416299\pi\)
\(228\) 0 0
\(229\) 284279. 0.358225 0.179112 0.983829i \(-0.442677\pi\)
0.179112 + 0.983829i \(0.442677\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 345405.i 0.416811i 0.978042 + 0.208406i \(0.0668274\pi\)
−0.978042 + 0.208406i \(0.933173\pi\)
\(234\) 0 0
\(235\) 62199.9 185623.i 0.0734717 0.219262i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.59920e6 −1.81095 −0.905476 0.424398i \(-0.860486\pi\)
−0.905476 + 0.424398i \(0.860486\pi\)
\(240\) 0 0
\(241\) −42246.4 −0.0468541 −0.0234270 0.999726i \(-0.507458\pi\)
−0.0234270 + 0.999726i \(0.507458\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00189e6 335720.i −1.06636 0.357324i
\(246\) 0 0
\(247\) 1.67537e6i 1.74731i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 802841. 0.804351 0.402175 0.915563i \(-0.368254\pi\)
0.402175 + 0.915563i \(0.368254\pi\)
\(252\) 0 0
\(253\) 650354.i 0.638776i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 941153.i 0.888848i 0.895817 + 0.444424i \(0.146592\pi\)
−0.895817 + 0.444424i \(0.853408\pi\)
\(258\) 0 0
\(259\) −1.58993e6 −1.47275
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.53130e6i 1.36512i −0.730830 0.682560i \(-0.760867\pi\)
0.730830 0.682560i \(-0.239133\pi\)
\(264\) 0 0
\(265\) −485045. + 1.44752e6i −0.424294 + 1.26622i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.25712e6 1.05924 0.529621 0.848234i \(-0.322334\pi\)
0.529621 + 0.848234i \(0.322334\pi\)
\(270\) 0 0
\(271\) −843976. −0.698083 −0.349041 0.937107i \(-0.613493\pi\)
−0.349041 + 0.937107i \(0.613493\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.25170e6 944961.i −0.998089 0.753498i
\(276\) 0 0
\(277\) 1.52070e6i 1.19082i 0.803423 + 0.595408i \(0.203010\pi\)
−0.803423 + 0.595408i \(0.796990\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.97975e6 1.49570 0.747848 0.663870i \(-0.231087\pi\)
0.747848 + 0.663870i \(0.231087\pi\)
\(282\) 0 0
\(283\) 2.02901e6i 1.50597i 0.658036 + 0.752986i \(0.271387\pi\)
−0.658036 + 0.752986i \(0.728613\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.32563e6i 0.949987i
\(288\) 0 0
\(289\) 1.41898e6 0.999384
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.23140e6i 1.51848i 0.650813 + 0.759238i \(0.274428\pi\)
−0.650813 + 0.759238i \(0.725572\pi\)
\(294\) 0 0
\(295\) −420086. 140765.i −0.281050 0.0941761i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.37553e6 0.889802
\(300\) 0 0
\(301\) −4.28847e6 −2.72826
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −372124. 124694.i −0.229054 0.0767531i
\(306\) 0 0
\(307\) 334318.i 0.202448i 0.994864 + 0.101224i \(0.0322759\pi\)
−0.994864 + 0.101224i \(0.967724\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.39090e6 1.40172 0.700858 0.713300i \(-0.252800\pi\)
0.700858 + 0.713300i \(0.252800\pi\)
\(312\) 0 0
\(313\) 1.88563e6i 1.08791i 0.839113 + 0.543957i \(0.183075\pi\)
−0.839113 + 0.543957i \(0.816925\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 179636.i 0.100403i −0.998739 0.0502014i \(-0.984014\pi\)
0.998739 0.0502014i \(-0.0159863\pi\)
\(318\) 0 0
\(319\) 1.80002e6 0.990378
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 46696.7i 0.0249046i
\(324\) 0 0
\(325\) 1.99864e6 2.64742e6i 1.04961 1.39032i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 661763. 0.337064
\(330\) 0 0
\(331\) −168620. −0.0845937 −0.0422968 0.999105i \(-0.513468\pi\)
−0.0422968 + 0.999105i \(0.513468\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 313152. 934541.i 0.152456 0.454974i
\(336\) 0 0
\(337\) 1.74240e6i 0.835744i 0.908506 + 0.417872i \(0.137224\pi\)
−0.908506 + 0.417872i \(0.862776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.76976e6 −0.824190
\(342\) 0 0
\(343\) 395842.i 0.181672i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.89447e6i 1.29046i 0.763987 + 0.645232i \(0.223239\pi\)
−0.763987 + 0.645232i \(0.776761\pi\)
\(348\) 0 0
\(349\) −63803.8 −0.0280403 −0.0140202 0.999902i \(-0.504463\pi\)
−0.0140202 + 0.999902i \(0.504463\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.79206e6i 1.19258i −0.802769 0.596290i \(-0.796641\pi\)
0.802769 0.596290i \(-0.203359\pi\)
\(354\) 0 0
\(355\) 712014. + 238586.i 0.299859 + 0.100479i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.48391e6 −0.607674 −0.303837 0.952724i \(-0.598268\pi\)
−0.303837 + 0.952724i \(0.598268\pi\)
\(360\) 0 0
\(361\) 15032.0 0.00607083
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 709507. 2.11738e6i 0.278756 0.831892i
\(366\) 0 0
\(367\) 1.08188e6i 0.419289i −0.977778 0.209644i \(-0.932769\pi\)
0.977778 0.209644i \(-0.0672306\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.16053e6 −1.94652
\(372\) 0 0
\(373\) 127750.i 0.0475434i 0.999717 + 0.0237717i \(0.00756748\pi\)
−0.999717 + 0.0237717i \(0.992433\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.80714e6i 1.37958i
\(378\) 0 0
\(379\) −1.61295e6 −0.576798 −0.288399 0.957510i \(-0.593123\pi\)
−0.288399 + 0.957510i \(0.593123\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.96656e6i 1.38171i 0.722993 + 0.690855i \(0.242766\pi\)
−0.722993 + 0.690855i \(0.757234\pi\)
\(384\) 0 0
\(385\) 1.68443e6 5.02686e6i 0.579165 1.72840i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.54852e6 −1.18898 −0.594488 0.804104i \(-0.702645\pi\)
−0.594488 + 0.804104i \(0.702645\pi\)
\(390\) 0 0
\(391\) −38339.5 −0.0126825
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.94650e6 1.65751e6i −1.59516 0.534518i
\(396\) 0 0
\(397\) 580854.i 0.184965i −0.995714 0.0924827i \(-0.970520\pi\)
0.995714 0.0924827i \(-0.0294803\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.10413e6 −0.653449 −0.326725 0.945120i \(-0.605945\pi\)
−0.326725 + 0.945120i \(0.605945\pi\)
\(402\) 0 0
\(403\) 3.74313e6i 1.14808i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.22263e6i 1.26356i
\(408\) 0 0
\(409\) −3.23687e6 −0.956790 −0.478395 0.878145i \(-0.658781\pi\)
−0.478395 + 0.878145i \(0.658781\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.49764e6i 0.432049i
\(414\) 0 0
\(415\) −1.03812e6 + 3.09807e6i −0.295888 + 0.883020i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.03059e6 0.565049 0.282525 0.959260i \(-0.408828\pi\)
0.282525 + 0.959260i \(0.408828\pi\)
\(420\) 0 0
\(421\) 2.15883e6 0.593625 0.296813 0.954936i \(-0.404076\pi\)
0.296813 + 0.954936i \(0.404076\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −55707.0 + 73789.9i −0.0149602 + 0.0198164i
\(426\) 0 0
\(427\) 1.32665e6i 0.352118i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.49812e6 1.42568 0.712838 0.701328i \(-0.247409\pi\)
0.712838 + 0.701328i \(0.247409\pi\)
\(432\) 0 0
\(433\) 3.87201e6i 0.992470i −0.868188 0.496235i \(-0.834716\pi\)
0.868188 0.496235i \(-0.165284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.04530e6i 0.512334i
\(438\) 0 0
\(439\) 3.15465e6 0.781250 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.07021e6i 0.985389i 0.870202 + 0.492694i \(0.163988\pi\)
−0.870202 + 0.492694i \(0.836012\pi\)
\(444\) 0 0
\(445\) −741508. 248470.i −0.177507 0.0594803i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 957718. 0.224193 0.112096 0.993697i \(-0.464243\pi\)
0.112096 + 0.993697i \(0.464243\pi\)
\(450\) 0 0
\(451\) 3.52068e6 0.815052
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.06321e7 + 3.56267e6i 2.40763 + 0.806765i
\(456\) 0 0
\(457\) 7.38167e6i 1.65335i 0.562682 + 0.826673i \(0.309769\pi\)
−0.562682 + 0.826673i \(0.690231\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.96862e6 0.650582 0.325291 0.945614i \(-0.394538\pi\)
0.325291 + 0.945614i \(0.394538\pi\)
\(462\) 0 0
\(463\) 6.84593e6i 1.48416i 0.670313 + 0.742079i \(0.266160\pi\)
−0.670313 + 0.742079i \(0.733840\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.20816e6i 0.892895i 0.894810 + 0.446447i \(0.147311\pi\)
−0.894810 + 0.446447i \(0.852689\pi\)
\(468\) 0 0
\(469\) 3.33172e6 0.699417
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.13895e7i 2.34074i
\(474\) 0 0
\(475\) 3.93647e6 + 2.97181e6i 0.800522 + 0.604347i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.73077e6 1.14123 0.570617 0.821217i \(-0.306704\pi\)
0.570617 + 0.821217i \(0.306704\pi\)
\(480\) 0 0
\(481\) 8.93110e6 1.76012
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.96511e6 5.86447e6i 0.379343 1.13207i
\(486\) 0 0
\(487\) 6.63249e6i 1.26723i −0.773650 0.633613i \(-0.781571\pi\)
0.773650 0.633613i \(-0.218429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.26463e6 −0.611125 −0.305563 0.952172i \(-0.598844\pi\)
−0.305563 + 0.952172i \(0.598844\pi\)
\(492\) 0 0
\(493\) 106114.i 0.0196633i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.53839e6i 0.460965i
\(498\) 0 0
\(499\) −8.51527e6 −1.53090 −0.765450 0.643495i \(-0.777484\pi\)
−0.765450 + 0.643495i \(0.777484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.24174e6i 0.571292i −0.958335 0.285646i \(-0.907792\pi\)
0.958335 0.285646i \(-0.0922082\pi\)
\(504\) 0 0
\(505\) −1.05198e7 3.52505e6i −1.83561 0.615088i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.04569e7 1.78900 0.894499 0.447070i \(-0.147533\pi\)
0.894499 + 0.447070i \(0.147533\pi\)
\(510\) 0 0
\(511\) 7.54865e6 1.27884
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.39387e6 + 7.14403e6i −0.397725 + 1.18693i
\(516\) 0 0
\(517\) 1.75755e6i 0.289188i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 328142. 0.0529625 0.0264812 0.999649i \(-0.491570\pi\)
0.0264812 + 0.999649i \(0.491570\pi\)
\(522\) 0 0
\(523\) 9.42102e6i 1.50606i 0.657984 + 0.753032i \(0.271410\pi\)
−0.657984 + 0.753032i \(0.728590\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 104330.i 0.0163637i
\(528\) 0 0
\(529\) 4.75709e6 0.739098
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.44643e6i 1.13535i
\(534\) 0 0
\(535\) −706642. + 2.10883e6i −0.106737 + 0.318535i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.48624e6 1.40644
\(540\) 0 0
\(541\) 6.08041e6 0.893182 0.446591 0.894738i \(-0.352638\pi\)
0.446591 + 0.894738i \(0.352638\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.91318e6 1.64634e6i −0.708551 0.237426i
\(546\) 0 0
\(547\) 4.32182e6i 0.617587i −0.951129 0.308794i \(-0.900075\pi\)
0.951129 0.308794i \(-0.0999253\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.66089e6 −0.794338
\(552\) 0 0
\(553\) 1.76347e7i 2.45220i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.41019e6i 1.01203i −0.862526 0.506013i \(-0.831119\pi\)
0.862526 0.506013i \(-0.168881\pi\)
\(558\) 0 0
\(559\) 2.40895e7 3.26061
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.08039e7i 1.43652i −0.695775 0.718260i \(-0.744939\pi\)
0.695775 0.718260i \(-0.255061\pi\)
\(564\) 0 0
\(565\) 1.18864e6 3.54726e6i 0.156650 0.467490i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.77404e6 0.229712 0.114856 0.993382i \(-0.463359\pi\)
0.114856 + 0.993382i \(0.463359\pi\)
\(570\) 0 0
\(571\) 70981.0 0.00911070 0.00455535 0.999990i \(-0.498550\pi\)
0.00455535 + 0.999990i \(0.498550\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.43995e6 + 3.23197e6i −0.307759 + 0.407660i
\(576\) 0 0
\(577\) 4.04410e6i 0.505688i −0.967507 0.252844i \(-0.918634\pi\)
0.967507 0.252844i \(-0.0813659\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.10449e7 −1.35744
\(582\) 0 0
\(583\) 1.37056e7i 1.67004i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.86862e6i 0.343620i −0.985130 0.171810i \(-0.945038\pi\)
0.985130 0.171810i \(-0.0549615\pi\)
\(588\) 0 0
\(589\) 5.56570e6 0.661046
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.23131e6i 0.377347i −0.982040 0.188674i \(-0.939581\pi\)
0.982040 0.188674i \(-0.0604188\pi\)
\(594\) 0 0
\(595\) −296341. 99300.1i −0.0343163 0.0114989i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.80078e6 −0.432819 −0.216409 0.976303i \(-0.569435\pi\)
−0.216409 + 0.976303i \(0.569435\pi\)
\(600\) 0 0
\(601\) 5.45289e6 0.615801 0.307900 0.951419i \(-0.400374\pi\)
0.307900 + 0.951419i \(0.400374\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.81408e6 + 1.61314e6i 0.534719 + 0.179177i
\(606\) 0 0
\(607\) 1.50483e7i 1.65774i −0.559441 0.828870i \(-0.688984\pi\)
0.559441 0.828870i \(-0.311016\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.71731e6 −0.402833
\(612\) 0 0
\(613\) 2.30782e6i 0.248056i 0.992279 + 0.124028i \(0.0395813\pi\)
−0.992279 + 0.124028i \(0.960419\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 381761.i 0.0403718i −0.999796 0.0201859i \(-0.993574\pi\)
0.999796 0.0201859i \(-0.00642581\pi\)
\(618\) 0 0
\(619\) −1.07208e7 −1.12460 −0.562302 0.826932i \(-0.690084\pi\)
−0.562302 + 0.826932i \(0.690084\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.64354e6i 0.272876i
\(624\) 0 0
\(625\) 2.67517e6 + 9.39207e6i 0.273937 + 0.961748i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −248931. −0.0250872
\(630\) 0 0
\(631\) 4.56189e6 0.456112 0.228056 0.973648i \(-0.426763\pi\)
0.228056 + 0.973648i \(0.426763\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.04790e6 + 3.12724e6i −0.103130 + 0.307770i
\(636\) 0 0
\(637\) 2.00639e7i 1.95915i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.70983e6 0.260494 0.130247 0.991482i \(-0.458423\pi\)
0.130247 + 0.991482i \(0.458423\pi\)
\(642\) 0 0
\(643\) 6.10481e6i 0.582297i 0.956678 + 0.291149i \(0.0940374\pi\)
−0.956678 + 0.291149i \(0.905963\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.22081e7i 1.14654i −0.819367 0.573269i \(-0.805675\pi\)
0.819367 0.573269i \(-0.194325\pi\)
\(648\) 0 0
\(649\) 3.97752e6 0.370681
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.26514e6i 0.574973i 0.957785 + 0.287487i \(0.0928197\pi\)
−0.957785 + 0.287487i \(0.907180\pi\)
\(654\) 0 0
\(655\) −1.10982e7 3.71887e6i −1.01076 0.338694i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.85107e7 1.66039 0.830194 0.557474i \(-0.188229\pi\)
0.830194 + 0.557474i \(0.188229\pi\)
\(660\) 0 0
\(661\) −8.46041e6 −0.753161 −0.376581 0.926384i \(-0.622900\pi\)
−0.376581 + 0.926384i \(0.622900\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.29737e6 + 1.58089e7i −0.464522 + 1.38627i
\(666\) 0 0
\(667\) 4.64777e6i 0.404511i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.52340e6 0.302104
\(672\) 0 0
\(673\) 7.25182e6i 0.617177i 0.951196 + 0.308588i \(0.0998566\pi\)
−0.951196 + 0.308588i \(0.900143\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.74893e6i 0.398221i 0.979977 + 0.199110i \(0.0638052\pi\)
−0.979977 + 0.199110i \(0.936195\pi\)
\(678\) 0 0
\(679\) 2.09073e7 1.74030
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.37712e7i 1.12959i 0.825233 + 0.564793i \(0.191044\pi\)
−0.825233 + 0.564793i \(0.808956\pi\)
\(684\) 0 0
\(685\) 5.79786e6 1.73025e7i 0.472108 1.40891i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.89882e7 2.32634
\(690\) 0 0
\(691\) −1.64305e7 −1.30905 −0.654526 0.756040i \(-0.727132\pi\)
−0.654526 + 0.756040i \(0.727132\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.23845e7 4.14990e6i −0.972563 0.325893i
\(696\) 0 0
\(697\) 207550.i 0.0161823i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.05170e7 0.808342 0.404171 0.914683i \(-0.367560\pi\)
0.404171 + 0.914683i \(0.367560\pi\)
\(702\) 0 0
\(703\) 1.32797e7i 1.01345i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.75041e7i 2.82182i
\(708\) 0 0
\(709\) 1.32691e6 0.0991349 0.0495674 0.998771i \(-0.484216\pi\)
0.0495674 + 0.998771i \(0.484216\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.56962e6i 0.336633i
\(714\) 0 0
\(715\) −9.46193e6 + 2.82372e7i −0.692173 + 2.06565i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.65753e6 −0.335995 −0.167998 0.985787i \(-0.553730\pi\)
−0.167998 + 0.985787i \(0.553730\pi\)
\(720\) 0 0
\(721\) −2.54691e7 −1.82463
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.94531e6 6.75318e6i −0.632048 0.477159i
\(726\) 0 0
\(727\) 1.47884e6i 0.103773i −0.998653 0.0518867i \(-0.983477\pi\)
0.998653 0.0518867i \(-0.0165235\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −671432. −0.0464739
\(732\) 0 0
\(733\) 1.41118e7i 0.970115i 0.874482 + 0.485057i \(0.161201\pi\)
−0.874482 + 0.485057i \(0.838799\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.84856e6i 0.600073i
\(738\) 0 0
\(739\) −5.94045e6 −0.400137 −0.200068 0.979782i \(-0.564116\pi\)
−0.200068 + 0.979782i \(0.564116\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.46589e7i 0.974156i 0.873359 + 0.487078i \(0.161937\pi\)
−0.873359 + 0.487078i \(0.838063\pi\)
\(744\) 0 0
\(745\) 7.70353e6 + 2.58135e6i 0.508510 + 0.170395i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.51817e6 −0.489674
\(750\) 0 0
\(751\) 6.08263e6 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.52306e7 5.10359e6i −0.972413 0.325843i
\(756\) 0 0
\(757\) 8.11757e6i 0.514856i −0.966297 0.257428i \(-0.917125\pi\)
0.966297 0.257428i \(-0.0828751\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.63567e7 −1.02384 −0.511921 0.859033i \(-0.671066\pi\)
−0.511921 + 0.859033i \(0.671066\pi\)
\(762\) 0 0
\(763\) 1.75159e7i 1.08923i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.41267e6i 0.516352i
\(768\) 0 0
\(769\) 2.09586e7 1.27805 0.639024 0.769187i \(-0.279339\pi\)
0.639024 + 0.769187i \(0.279339\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.95651e6i 0.539126i 0.962983 + 0.269563i \(0.0868793\pi\)
−0.962983 + 0.269563i \(0.913121\pi\)
\(774\) 0 0
\(775\) 8.79489e6 + 6.63963e6i 0.525989 + 0.397090i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.10722e7 −0.653717
\(780\) 0 0
\(781\) −6.74160e6 −0.395490
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 757097. 2.25940e6i 0.0438508 0.130864i
\(786\) 0 0
\(787\) 2.42009e7i 1.39282i −0.717643 0.696411i \(-0.754779\pi\)
0.717643 0.696411i \(-0.245221\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.26463e7 0.718657
\(792\) 0 0
\(793\) 7.45219e6i 0.420824i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.91659e6i 0.552989i 0.961016 + 0.276494i \(0.0891728\pi\)
−0.961016 + 0.276494i \(0.910827\pi\)
\(798\) 0 0
\(799\) 103610. 0.00574164
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.00481e7i 1.09720i
\(804\) 0 0
\(805\) −1.29797e7 4.34931e6i −0.705949 0.236554i
\(806\) 0 0
\(807\) 0 0
\(808\) 0