Properties

Label 720.6.f.n.289.1
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 289.1
Root \(0.0965878i\) of defining polynomial
Character \(\chi\) \(=\) 720.289
Dual form 720.6.f.n.289.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-46.7401 - 30.6653i) q^{5} +179.876i q^{7} +O(q^{10})\) \(q+(-46.7401 - 30.6653i) q^{5} +179.876i q^{7} -653.681 q^{11} -284.851i q^{13} +383.672i q^{17} -2563.29 q^{19} +948.124i q^{23} +(1244.27 + 2866.60i) q^{25} -1524.26 q^{29} -3103.88 q^{31} +(5515.97 - 8407.43i) q^{35} -9991.75i q^{37} -15120.7 q^{41} +1754.92i q^{43} +14760.7i q^{47} -15548.5 q^{49} -8704.12i q^{53} +(30553.1 + 20045.4i) q^{55} +12632.1 q^{59} +43098.3 q^{61} +(-8735.07 + 13314.0i) q^{65} +26125.5i q^{67} -46285.8 q^{71} +51303.2i q^{73} -117582. i q^{77} +39314.0 q^{79} +69544.8i q^{83} +(11765.4 - 17932.8i) q^{85} -13092.2 q^{89} +51238.0 q^{91} +(119808. + 78604.0i) q^{95} -26258.1i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100}) \)

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
\(5\) −46.7401 30.6653i −0.836112 0.548558i
\(6\) 0 0
\(7\) 179.876i 1.38749i 0.720223 + 0.693743i \(0.244040\pi\)
−0.720223 + 0.693743i \(0.755960\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −653.681 −1.62886 −0.814431 0.580260i \(-0.802951\pi\)
−0.814431 + 0.580260i \(0.802951\pi\)
\(12\) 0 0
\(13\) 284.851i 0.467477i −0.972300 0.233738i \(-0.924904\pi\)
0.972300 0.233738i \(-0.0750959\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 383.672i 0.321986i 0.986956 + 0.160993i \(0.0514697\pi\)
−0.986956 + 0.160993i \(0.948530\pi\)
\(18\) 0 0
\(19\) −2563.29 −1.62897 −0.814485 0.580185i \(-0.802980\pi\)
−0.814485 + 0.580185i \(0.802980\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 948.124i 0.373719i 0.982387 + 0.186860i \(0.0598309\pi\)
−0.982387 + 0.186860i \(0.940169\pi\)
\(24\) 0 0
\(25\) 1244.27 + 2866.60i 0.398168 + 0.917313i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1524.26 −0.336561 −0.168281 0.985739i \(-0.553822\pi\)
−0.168281 + 0.985739i \(0.553822\pi\)
\(30\) 0 0
\(31\) −3103.88 −0.580098 −0.290049 0.957012i \(-0.593672\pi\)
−0.290049 + 0.957012i \(0.593672\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5515.97 8407.43i 0.761117 1.16009i
\(36\) 0 0
\(37\) 9991.75i 1.19988i −0.800046 0.599939i \(-0.795191\pi\)
0.800046 0.599939i \(-0.204809\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −15120.7 −1.40479 −0.702394 0.711788i \(-0.747886\pi\)
−0.702394 + 0.711788i \(0.747886\pi\)
\(42\) 0 0
\(43\) 1754.92i 0.144739i 0.997378 + 0.0723696i \(0.0230561\pi\)
−0.997378 + 0.0723696i \(0.976944\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 14760.7i 0.974682i 0.873212 + 0.487341i \(0.162033\pi\)
−0.873212 + 0.487341i \(0.837967\pi\)
\(48\) 0 0
\(49\) −15548.5 −0.925118
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8704.12i 0.425633i −0.977092 0.212816i \(-0.931736\pi\)
0.977092 0.212816i \(-0.0682636\pi\)
\(54\) 0 0
\(55\) 30553.1 + 20045.4i 1.36191 + 0.893526i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12632.1 0.472438 0.236219 0.971700i \(-0.424092\pi\)
0.236219 + 0.971700i \(0.424092\pi\)
\(60\) 0 0
\(61\) 43098.3 1.48298 0.741490 0.670964i \(-0.234119\pi\)
0.741490 + 0.670964i \(0.234119\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8735.07 + 13314.0i −0.256438 + 0.390863i
\(66\) 0 0
\(67\) 26125.5i 0.711014i 0.934674 + 0.355507i \(0.115692\pi\)
−0.934674 + 0.355507i \(0.884308\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −46285.8 −1.08969 −0.544844 0.838538i \(-0.683411\pi\)
−0.544844 + 0.838538i \(0.683411\pi\)
\(72\) 0 0
\(73\) 51303.2i 1.12678i 0.826192 + 0.563388i \(0.190502\pi\)
−0.826192 + 0.563388i \(0.809498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 117582.i 2.26002i
\(78\) 0 0
\(79\) 39314.0 0.708729 0.354364 0.935107i \(-0.384697\pi\)
0.354364 + 0.935107i \(0.384697\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 69544.8i 1.10808i 0.832492 + 0.554038i \(0.186914\pi\)
−0.832492 + 0.554038i \(0.813086\pi\)
\(84\) 0 0
\(85\) 11765.4 17932.8i 0.176628 0.269217i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13092.2 −0.175201 −0.0876007 0.996156i \(-0.527920\pi\)
−0.0876007 + 0.996156i \(0.527920\pi\)
\(90\) 0 0
\(91\) 51238.0 0.648618
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 119808. + 78604.0i 1.36200 + 0.893585i
\(96\) 0 0
\(97\) 26258.1i 0.283357i −0.989913 0.141678i \(-0.954750\pi\)
0.989913 0.141678i \(-0.0452499\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7382.54 −0.0720116 −0.0360058 0.999352i \(-0.511463\pi\)
−0.0360058 + 0.999352i \(0.511463\pi\)
\(102\) 0 0
\(103\) 44518.9i 0.413477i 0.978396 + 0.206738i \(0.0662849\pi\)
−0.978396 + 0.206738i \(0.933715\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 194929.i 1.64595i −0.568079 0.822974i \(-0.692313\pi\)
0.568079 0.822974i \(-0.307687\pi\)
\(108\) 0 0
\(109\) −213783. −1.72348 −0.861739 0.507351i \(-0.830625\pi\)
−0.861739 + 0.507351i \(0.830625\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 148946.i 1.09732i −0.836046 0.548660i \(-0.815138\pi\)
0.836046 0.548660i \(-0.184862\pi\)
\(114\) 0 0
\(115\) 29074.5 44315.4i 0.205007 0.312471i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −69013.4 −0.446751
\(120\) 0 0
\(121\) 266248. 1.65319
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 29747.8 172141.i 0.170287 0.985395i
\(126\) 0 0
\(127\) 116757.i 0.642351i −0.947020 0.321175i \(-0.895922\pi\)
0.947020 0.321175i \(-0.104078\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 349111. 1.77740 0.888700 0.458490i \(-0.151610\pi\)
0.888700 + 0.458490i \(0.151610\pi\)
\(132\) 0 0
\(133\) 461074.i 2.26017i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 219776.i 1.00041i −0.865907 0.500205i \(-0.833258\pi\)
0.865907 0.500205i \(-0.166742\pi\)
\(138\) 0 0
\(139\) 126594. 0.555744 0.277872 0.960618i \(-0.410371\pi\)
0.277872 + 0.960618i \(0.410371\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 186202.i 0.761455i
\(144\) 0 0
\(145\) 71244.1 + 46742.0i 0.281403 + 0.184624i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −25809.9 −0.0952403 −0.0476202 0.998866i \(-0.515164\pi\)
−0.0476202 + 0.998866i \(0.515164\pi\)
\(150\) 0 0
\(151\) 311270. 1.11095 0.555475 0.831533i \(-0.312536\pi\)
0.555475 + 0.831533i \(0.312536\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 145076. + 95181.7i 0.485027 + 0.318217i
\(156\) 0 0
\(157\) 429151.i 1.38951i −0.719247 0.694755i \(-0.755513\pi\)
0.719247 0.694755i \(-0.244487\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −170545. −0.518530
\(162\) 0 0
\(163\) 162467.i 0.478957i −0.970902 0.239479i \(-0.923023\pi\)
0.970902 0.239479i \(-0.0769765\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 280057.i 0.777060i 0.921436 + 0.388530i \(0.127017\pi\)
−0.921436 + 0.388530i \(0.872983\pi\)
\(168\) 0 0
\(169\) 290153. 0.781465
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 742577.i 1.88637i 0.332271 + 0.943184i \(0.392185\pi\)
−0.332271 + 0.943184i \(0.607815\pi\)
\(174\) 0 0
\(175\) −515634. + 223815.i −1.27276 + 0.552452i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14556.0 0.0339554 0.0169777 0.999856i \(-0.494596\pi\)
0.0169777 + 0.999856i \(0.494596\pi\)
\(180\) 0 0
\(181\) −243243. −0.551878 −0.275939 0.961175i \(-0.588989\pi\)
−0.275939 + 0.961175i \(0.588989\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −306400. + 467015.i −0.658203 + 1.00323i
\(186\) 0 0
\(187\) 250799.i 0.524471i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 75871.1 0.150485 0.0752425 0.997165i \(-0.476027\pi\)
0.0752425 + 0.997165i \(0.476027\pi\)
\(192\) 0 0
\(193\) 370161.i 0.715314i 0.933853 + 0.357657i \(0.116424\pi\)
−0.933853 + 0.357657i \(0.883576\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 213848.i 0.392590i −0.980545 0.196295i \(-0.937109\pi\)
0.980545 0.196295i \(-0.0628910\pi\)
\(198\) 0 0
\(199\) 466075. 0.834302 0.417151 0.908837i \(-0.363029\pi\)
0.417151 + 0.908837i \(0.363029\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 274178.i 0.466974i
\(204\) 0 0
\(205\) 706741. + 463680.i 1.17456 + 0.770609i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.67557e6 2.65337
\(210\) 0 0
\(211\) 228211. 0.352883 0.176441 0.984311i \(-0.443541\pi\)
0.176441 + 0.984311i \(0.443541\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 53815.2 82025.1i 0.0793979 0.121018i
\(216\) 0 0
\(217\) 558315.i 0.804878i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 109289. 0.150521
\(222\) 0 0
\(223\) 972750.i 1.30990i −0.755671 0.654951i \(-0.772689\pi\)
0.755671 0.654951i \(-0.227311\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 357063.i 0.459917i −0.973200 0.229959i \(-0.926141\pi\)
0.973200 0.229959i \(-0.0738591\pi\)
\(228\) 0 0
\(229\) 295196. 0.371982 0.185991 0.982551i \(-0.440451\pi\)
0.185991 + 0.982551i \(0.440451\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 492373.i 0.594162i −0.954852 0.297081i \(-0.903987\pi\)
0.954852 0.297081i \(-0.0960131\pi\)
\(234\) 0 0
\(235\) 452643. 689918.i 0.534670 0.814944i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 71796.0 0.0813028 0.0406514 0.999173i \(-0.487057\pi\)
0.0406514 + 0.999173i \(0.487057\pi\)
\(240\) 0 0
\(241\) 614304. 0.681304 0.340652 0.940190i \(-0.389352\pi\)
0.340652 + 0.940190i \(0.389352\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 726736. + 476799.i 0.773502 + 0.507481i
\(246\) 0 0
\(247\) 730156.i 0.761505i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −244276. −0.244735 −0.122368 0.992485i \(-0.539049\pi\)
−0.122368 + 0.992485i \(0.539049\pi\)
\(252\) 0 0
\(253\) 619771.i 0.608737i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 951728.i 0.898835i −0.893322 0.449418i \(-0.851632\pi\)
0.893322 0.449418i \(-0.148368\pi\)
\(258\) 0 0
\(259\) 1.79728e6 1.66481
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 258512.i 0.230458i 0.993339 + 0.115229i \(0.0367602\pi\)
−0.993339 + 0.115229i \(0.963240\pi\)
\(264\) 0 0
\(265\) −266915. + 406831.i −0.233484 + 0.355877i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 894716. 0.753884 0.376942 0.926237i \(-0.376976\pi\)
0.376942 + 0.926237i \(0.376976\pi\)
\(270\) 0 0
\(271\) −474149. −0.392186 −0.196093 0.980585i \(-0.562825\pi\)
−0.196093 + 0.980585i \(0.562825\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −813358. 1.87384e6i −0.648560 1.49418i
\(276\) 0 0
\(277\) 142752.i 0.111785i 0.998437 + 0.0558923i \(0.0178003\pi\)
−0.998437 + 0.0558923i \(0.982200\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 427838. 0.323231 0.161616 0.986854i \(-0.448330\pi\)
0.161616 + 0.986854i \(0.448330\pi\)
\(282\) 0 0
\(283\) 2.30233e6i 1.70884i −0.519585 0.854419i \(-0.673913\pi\)
0.519585 0.854419i \(-0.326087\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.71985e6i 1.94913i
\(288\) 0 0
\(289\) 1.27265e6 0.896325
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.59390e6i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(294\) 0 0
\(295\) −590424. 387367.i −0.395011 0.259160i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 270074. 0.174705
\(300\) 0 0
\(301\) −315668. −0.200824
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.01442e6 1.32162e6i −1.23994 0.813501i
\(306\) 0 0
\(307\) 322680.i 0.195401i 0.995216 + 0.0977005i \(0.0311487\pi\)
−0.995216 + 0.0977005i \(0.968851\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −660204. −0.387059 −0.193529 0.981094i \(-0.561994\pi\)
−0.193529 + 0.981094i \(0.561994\pi\)
\(312\) 0 0
\(313\) 1.15378e6i 0.665677i −0.942984 0.332838i \(-0.891994\pi\)
0.942984 0.332838i \(-0.108006\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.20296e6i 0.672364i 0.941797 + 0.336182i \(0.109136\pi\)
−0.941797 + 0.336182i \(0.890864\pi\)
\(318\) 0 0
\(319\) 996381. 0.548212
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 983460.i 0.524506i
\(324\) 0 0
\(325\) 816556. 354433.i 0.428822 0.186134i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.65510e6 −1.35236
\(330\) 0 0
\(331\) −1.82424e6 −0.915193 −0.457597 0.889160i \(-0.651290\pi\)
−0.457597 + 0.889160i \(0.651290\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 801149. 1.22111e6i 0.390033 0.594488i
\(336\) 0 0
\(337\) 485867.i 0.233047i 0.993188 + 0.116523i \(0.0371750\pi\)
−0.993188 + 0.116523i \(0.962825\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.02895e6 0.944899
\(342\) 0 0
\(343\) 226383.i 0.103898i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.35066e6i 1.04801i 0.851715 + 0.524006i \(0.175563\pi\)
−0.851715 + 0.524006i \(0.824437\pi\)
\(348\) 0 0
\(349\) 2.62295e6 1.15273 0.576364 0.817193i \(-0.304471\pi\)
0.576364 + 0.817193i \(0.304471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 583638.i 0.249291i 0.992201 + 0.124646i \(0.0397794\pi\)
−0.992201 + 0.124646i \(0.960221\pi\)
\(354\) 0 0
\(355\) 2.16340e6 + 1.41937e6i 0.911101 + 0.597757i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.64503e6 −0.673654 −0.336827 0.941567i \(-0.609354\pi\)
−0.336827 + 0.941567i \(0.609354\pi\)
\(360\) 0 0
\(361\) 4.09433e6 1.65354
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.57323e6 2.39792e6i 0.618102 0.942111i
\(366\) 0 0
\(367\) 2.62251e6i 1.01637i 0.861247 + 0.508186i \(0.169684\pi\)
−0.861247 + 0.508186i \(0.830316\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.56566e6 0.590559
\(372\) 0 0
\(373\) 3.73068e6i 1.38840i 0.719780 + 0.694202i \(0.244243\pi\)
−0.719780 + 0.694202i \(0.755757\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 434188.i 0.157335i
\(378\) 0 0
\(379\) −3.00539e6 −1.07474 −0.537368 0.843348i \(-0.680582\pi\)
−0.537368 + 0.843348i \(0.680582\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 343090.i 0.119512i 0.998213 + 0.0597560i \(0.0190323\pi\)
−0.998213 + 0.0597560i \(0.980968\pi\)
\(384\) 0 0
\(385\) −3.60568e6 + 5.49578e6i −1.23975 + 1.88963i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 190583. 0.0638572 0.0319286 0.999490i \(-0.489835\pi\)
0.0319286 + 0.999490i \(0.489835\pi\)
\(390\) 0 0
\(391\) −363768. −0.120332
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.83754e6 1.20558e6i −0.592577 0.388779i
\(396\) 0 0
\(397\) 5.22551e6i 1.66400i −0.554778 0.831999i \(-0.687197\pi\)
0.554778 0.831999i \(-0.312803\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.60312e6 0.497859 0.248929 0.968522i \(-0.419921\pi\)
0.248929 + 0.968522i \(0.419921\pi\)
\(402\) 0 0
\(403\) 884146.i 0.271182i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.53142e6i 1.95444i
\(408\) 0 0
\(409\) −4.58732e6 −1.35597 −0.677986 0.735075i \(-0.737147\pi\)
−0.677986 + 0.735075i \(0.737147\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.27221e6i 0.655501i
\(414\) 0 0
\(415\) 2.13261e6 3.25053e6i 0.607844 0.926476i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.16854e6 1.71651 0.858256 0.513221i \(-0.171548\pi\)
0.858256 + 0.513221i \(0.171548\pi\)
\(420\) 0 0
\(421\) −6.15589e6 −1.69272 −0.846361 0.532610i \(-0.821211\pi\)
−0.846361 + 0.532610i \(0.821211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.09983e6 + 477393.i −0.295362 + 0.128205i
\(426\) 0 0
\(427\) 7.75236e6i 2.05761i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.15900e6 0.300532 0.150266 0.988646i \(-0.451987\pi\)
0.150266 + 0.988646i \(0.451987\pi\)
\(432\) 0 0
\(433\) 6.73906e6i 1.72735i −0.504052 0.863673i \(-0.668158\pi\)
0.504052 0.863673i \(-0.331842\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.43031e6i 0.608777i
\(438\) 0 0
\(439\) −3.69068e6 −0.913998 −0.456999 0.889467i \(-0.651076\pi\)
−0.456999 + 0.889467i \(0.651076\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.55124e6i 1.10185i 0.834556 + 0.550923i \(0.185724\pi\)
−0.834556 + 0.550923i \(0.814276\pi\)
\(444\) 0 0
\(445\) 611931. + 401477.i 0.146488 + 0.0961082i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.31419e6 −0.775822 −0.387911 0.921697i \(-0.626803\pi\)
−0.387911 + 0.921697i \(0.626803\pi\)
\(450\) 0 0
\(451\) 9.88409e6 2.28821
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.39487e6 1.57123e6i −0.542317 0.355805i
\(456\) 0 0
\(457\) 4.10489e6i 0.919414i 0.888071 + 0.459707i \(0.152046\pi\)
−0.888071 + 0.459707i \(0.847954\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.16525e6 −1.78944 −0.894720 0.446628i \(-0.852625\pi\)
−0.894720 + 0.446628i \(0.852625\pi\)
\(462\) 0 0
\(463\) 5.65614e6i 1.22622i −0.789998 0.613109i \(-0.789919\pi\)
0.789998 0.613109i \(-0.210081\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.77216e6i 1.86129i 0.365921 + 0.930646i \(0.380754\pi\)
−0.365921 + 0.930646i \(0.619246\pi\)
\(468\) 0 0
\(469\) −4.69936e6 −0.986522
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.14716e6i 0.235760i
\(474\) 0 0
\(475\) −3.18943e6 7.34792e6i −0.648603 1.49427i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −417978. −0.0832367 −0.0416184 0.999134i \(-0.513251\pi\)
−0.0416184 + 0.999134i \(0.513251\pi\)
\(480\) 0 0
\(481\) −2.84616e6 −0.560915
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −805213. + 1.22731e6i −0.155438 + 0.236918i
\(486\) 0 0
\(487\) 7.07076e6i 1.35096i 0.737376 + 0.675482i \(0.236064\pi\)
−0.737376 + 0.675482i \(0.763936\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.00127e7 −1.87433 −0.937166 0.348883i \(-0.886561\pi\)
−0.937166 + 0.348883i \(0.886561\pi\)
\(492\) 0 0
\(493\) 584816.i 0.108368i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.32571e6i 1.51193i
\(498\) 0 0
\(499\) −7.74452e6 −1.39233 −0.696166 0.717881i \(-0.745112\pi\)
−0.696166 + 0.717881i \(0.745112\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.20376e6i 0.388370i 0.980965 + 0.194185i \(0.0622061\pi\)
−0.980965 + 0.194185i \(0.937794\pi\)
\(504\) 0 0
\(505\) 345061. + 226388.i 0.0602098 + 0.0395025i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.40053e6 0.239607 0.119803 0.992798i \(-0.461774\pi\)
0.119803 + 0.992798i \(0.461774\pi\)
\(510\) 0 0
\(511\) −9.22823e6 −1.56339
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.36519e6 2.08082e6i 0.226816 0.345713i
\(516\) 0 0
\(517\) 9.64881e6i 1.58762i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.60571e6 −0.259162 −0.129581 0.991569i \(-0.541363\pi\)
−0.129581 + 0.991569i \(0.541363\pi\)
\(522\) 0 0
\(523\) 5.80485e6i 0.927977i −0.885841 0.463988i \(-0.846418\pi\)
0.885841 0.463988i \(-0.153582\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.19087e6i 0.186784i
\(528\) 0 0
\(529\) 5.53740e6 0.860334
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.30714e6i 0.656706i
\(534\) 0 0
\(535\) −5.97755e6 + 9.11098e6i −0.902898 + 1.37620i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.01637e7 1.50689
\(540\) 0 0
\(541\) −9.37930e6 −1.37777 −0.688886 0.724870i \(-0.741900\pi\)
−0.688886 + 0.724870i \(0.741900\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.99222e6 + 6.55571e6i 1.44102 + 0.945429i
\(546\) 0 0
\(547\) 2.33685e6i 0.333936i 0.985962 + 0.166968i \(0.0533976\pi\)
−0.985962 + 0.166968i \(0.946602\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.90712e6 0.548248
\(552\) 0 0
\(553\) 7.07166e6i 0.983351i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.02238e6i 0.959061i −0.877525 0.479530i \(-0.840807\pi\)
0.877525 0.479530i \(-0.159193\pi\)
\(558\) 0 0
\(559\) 499891. 0.0676622
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.81010e6i 0.240675i 0.992733 + 0.120338i \(0.0383977\pi\)
−0.992733 + 0.120338i \(0.961602\pi\)
\(564\) 0 0
\(565\) −4.56748e6 + 6.96175e6i −0.601943 + 0.917482i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.62751e6 0.728678 0.364339 0.931266i \(-0.381295\pi\)
0.364339 + 0.931266i \(0.381295\pi\)
\(570\) 0 0
\(571\) 2.16702e6 0.278145 0.139073 0.990282i \(-0.455588\pi\)
0.139073 + 0.990282i \(0.455588\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.71789e6 + 1.17973e6i −0.342817 + 0.148803i
\(576\) 0 0
\(577\) 1.04992e7i 1.31286i −0.754387 0.656430i \(-0.772066\pi\)
0.754387 0.656430i \(-0.227934\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.25095e7 −1.53744
\(582\) 0 0
\(583\) 5.68972e6i 0.693297i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.47459e7i 1.76634i −0.469050 0.883172i \(-0.655404\pi\)
0.469050 0.883172i \(-0.344596\pi\)
\(588\) 0 0
\(589\) 7.95614e6 0.944962
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.85519e6i 0.566982i −0.958975 0.283491i \(-0.908507\pi\)
0.958975 0.283491i \(-0.0914926\pi\)
\(594\) 0 0
\(595\) 3.22569e6 + 2.11632e6i 0.373534 + 0.245069i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.48204e6 0.510398 0.255199 0.966889i \(-0.417859\pi\)
0.255199 + 0.966889i \(0.417859\pi\)
\(600\) 0 0
\(601\) 4.29562e6 0.485110 0.242555 0.970138i \(-0.422015\pi\)
0.242555 + 0.970138i \(0.422015\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.24445e7 8.16459e6i −1.38225 0.906872i
\(606\) 0 0
\(607\) 1.05706e7i 1.16447i 0.813020 + 0.582235i \(0.197822\pi\)
−0.813020 + 0.582235i \(0.802178\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.20461e6 0.455641
\(612\) 0 0
\(613\) 1.30960e7i 1.40763i 0.710383 + 0.703816i \(0.248522\pi\)
−0.710383 + 0.703816i \(0.751478\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.69519e7i 1.79270i 0.443351 + 0.896348i \(0.353789\pi\)
−0.443351 + 0.896348i \(0.646211\pi\)
\(618\) 0 0
\(619\) 1.27214e7 1.33447 0.667236 0.744846i \(-0.267477\pi\)
0.667236 + 0.744846i \(0.267477\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.35498e6i 0.243090i
\(624\) 0 0
\(625\) −6.66919e6 + 7.13368e6i −0.682925 + 0.730488i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.83355e6 0.386344
\(630\) 0 0
\(631\) −8.32823e6 −0.832682 −0.416341 0.909209i \(-0.636688\pi\)
−0.416341 + 0.909209i \(0.636688\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.58038e6 + 5.45722e6i −0.352367 + 0.537078i
\(636\) 0 0
\(637\) 4.42900e6i 0.432471i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.18862e6 0.402648 0.201324 0.979525i \(-0.435476\pi\)
0.201324 + 0.979525i \(0.435476\pi\)
\(642\) 0 0
\(643\) 271198.i 0.0258677i −0.999916 0.0129339i \(-0.995883\pi\)
0.999916 0.0129339i \(-0.00411709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.01692e7i 0.955052i 0.878618 + 0.477526i \(0.158466\pi\)
−0.878618 + 0.477526i \(0.841534\pi\)
\(648\) 0 0
\(649\) −8.25735e6 −0.769536
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.95062e6i 0.821429i 0.911764 + 0.410714i \(0.134721\pi\)
−0.911764 + 0.410714i \(0.865279\pi\)
\(654\) 0 0
\(655\) −1.63175e7 1.07056e7i −1.48611 0.975007i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.07182e7 −0.961409 −0.480705 0.876883i \(-0.659619\pi\)
−0.480705 + 0.876883i \(0.659619\pi\)
\(660\) 0 0
\(661\) 4.65131e6 0.414068 0.207034 0.978334i \(-0.433619\pi\)
0.207034 + 0.978334i \(0.433619\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.41390e7 + 2.15506e7i −1.23984 + 1.88976i
\(666\) 0 0
\(667\) 1.44519e6i 0.125779i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.81725e7 −2.41557
\(672\) 0 0
\(673\) 9.92139e6i 0.844374i 0.906509 + 0.422187i \(0.138737\pi\)
−0.906509 + 0.422187i \(0.861263\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.45936e6i 0.541649i −0.962629 0.270824i \(-0.912704\pi\)
0.962629 0.270824i \(-0.0872963\pi\)
\(678\) 0 0
\(679\) 4.72321e6 0.393154
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.26684e7i 1.03913i −0.854432 0.519564i \(-0.826095\pi\)
0.854432 0.519564i \(-0.173905\pi\)
\(684\) 0 0
\(685\) −6.73950e6 + 1.02723e7i −0.548783 + 0.836455i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.47938e6 −0.198973
\(690\) 0 0
\(691\) −4.07160e6 −0.324392 −0.162196 0.986759i \(-0.551858\pi\)
−0.162196 + 0.986759i \(0.551858\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.91700e6 3.88204e6i −0.464665 0.304858i
\(696\) 0 0
\(697\) 5.80137e6i 0.452323i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.11804e7 1.62795 0.813973 0.580903i \(-0.197300\pi\)
0.813973 + 0.580903i \(0.197300\pi\)
\(702\) 0 0
\(703\) 2.56117e7i 1.95457i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.32794e6i 0.0999150i
\(708\) 0 0
\(709\) 1.87573e7 1.40138 0.700689 0.713467i \(-0.252876\pi\)
0.700689 + 0.713467i \(0.252876\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.94287e6i 0.216794i
\(714\) 0 0
\(715\) 5.70995e6 8.70310e6i 0.417703 0.636662i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.82800e7 −1.31872 −0.659362 0.751826i \(-0.729174\pi\)
−0.659362 + 0.751826i \(0.729174\pi\)
\(720\) 0 0
\(721\) −8.00788e6 −0.573693
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.89660e6 4.36945e6i −0.134008 0.308732i
\(726\) 0 0
\(727\) 6.45855e6i 0.453210i −0.973987 0.226605i \(-0.927237\pi\)
0.973987 0.226605i \(-0.0727626\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −673313. −0.0466040
\(732\) 0 0
\(733\) 1.88365e6i 0.129491i −0.997902 0.0647454i \(-0.979376\pi\)
0.997902 0.0647454i \(-0.0206235\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.70778e7i 1.15814i
\(738\) 0 0
\(739\) 2.24002e7 1.50883 0.754417 0.656395i \(-0.227920\pi\)
0.754417 + 0.656395i \(0.227920\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.22827e6i 0.148080i 0.997255 + 0.0740400i \(0.0235893\pi\)
−0.997255 + 0.0740400i \(0.976411\pi\)
\(744\) 0 0
\(745\) 1.20636e6 + 791470.i 0.0796316 + 0.0522449i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.50630e7 2.28373
\(750\) 0 0
\(751\) −1.92986e7 −1.24861 −0.624303 0.781183i \(-0.714617\pi\)
−0.624303 + 0.781183i \(0.714617\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.45488e7 9.54520e6i −0.928879 0.609421i
\(756\) 0 0
\(757\) 7.22211e6i 0.458062i −0.973419 0.229031i \(-0.926444\pi\)
0.973419 0.229031i \(-0.0735557\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.08001e7 −0.676028 −0.338014 0.941141i \(-0.609755\pi\)
−0.338014 + 0.941141i \(0.609755\pi\)
\(762\) 0 0
\(763\) 3.84544e7i 2.39130i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.59827e6i 0.220854i
\(768\) 0 0
\(769\) −5.50354e6 −0.335603 −0.167802 0.985821i \(-0.553667\pi\)
−0.167802 + 0.985821i \(0.553667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.91304e7i 1.75347i 0.480977 + 0.876733i \(0.340282\pi\)
−0.480977 + 0.876733i \(0.659718\pi\)
\(774\) 0 0
\(775\) −3.86208e6 8.89760e6i −0.230976 0.532131i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.87586e7 2.28836
\(780\) 0 0
\(781\) 3.02562e7 1.77495
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.31601e7 + 2.00586e7i −0.762227 + 1.16179i
\(786\) 0 0
\(787\) 5.40251e6i 0.310927i −0.987842 0.155464i \(-0.950313\pi\)
0.987842 0.155464i \(-0.0496871\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.67919e7 1.52251
\(792\) 0 0
\(793\) 1.22766e7i 0.693259i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.19820e7i 1.22580i −0.790159 0.612902i \(-0.790002\pi\)
0.790159 0.612902i \(-0.209998\pi\)
\(798\) 0 0
\(799\) −5.66327e6 −0.313834
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.35360e7i 1.83536i
\(804\) 0 0
\(805\) 7.97128e6 + 5.22982e6i 0.433549 + 0.284444i
\(806\) 0 0
\(807\) 0 0
\(808\) 0