Properties

Label 720.6.f.n.289.3
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 289.3
Root \(-3.98753i\) of defining polynomial
Character \(\chi\) \(=\) 720.289
Dual form 720.6.f.n.289.4

$q$-expansion

\(f(q)\) \(=\) \(q+(-23.4238 - 50.7575i) q^{5} -10.2635i q^{7} +O(q^{10})\) \(q+(-23.4238 - 50.7575i) q^{5} -10.2635i q^{7} +596.423 q^{11} +420.629i q^{13} -974.149i q^{17} -380.528 q^{19} -3543.51i q^{23} +(-2027.65 + 2377.87i) q^{25} +5440.89 q^{29} +3623.54 q^{31} +(-520.948 + 240.409i) q^{35} +1756.01i q^{37} -263.984 q^{41} +14410.5i q^{43} +23464.8i q^{47} +16701.7 q^{49} +33496.0i q^{53} +(-13970.5 - 30273.0i) q^{55} +2906.38 q^{59} +29431.9 q^{61} +(21350.1 - 9852.72i) q^{65} -7163.34i q^{67} -81353.2 q^{71} -55127.9i q^{73} -6121.36i q^{77} +16430.9 q^{79} -116869. i q^{83} +(-49445.4 + 22818.2i) q^{85} +99364.0 q^{89} +4317.11 q^{91} +(8913.39 + 19314.6i) q^{95} +62987.8i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100}) \)

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
\(5\) −23.4238 50.7575i −0.419017 0.907978i
\(6\) 0 0
\(7\) 10.2635i 0.0791678i −0.999216 0.0395839i \(-0.987397\pi\)
0.999216 0.0395839i \(-0.0126032\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 596.423 1.48618 0.743092 0.669189i \(-0.233358\pi\)
0.743092 + 0.669189i \(0.233358\pi\)
\(12\) 0 0
\(13\) 420.629i 0.690305i 0.938547 + 0.345152i \(0.112173\pi\)
−0.938547 + 0.345152i \(0.887827\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 974.149i 0.817529i −0.912640 0.408764i \(-0.865960\pi\)
0.912640 0.408764i \(-0.134040\pi\)
\(18\) 0 0
\(19\) −380.528 −0.241825 −0.120913 0.992663i \(-0.538582\pi\)
−0.120913 + 0.992663i \(0.538582\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3543.51i 1.39673i −0.715740 0.698367i \(-0.753910\pi\)
0.715740 0.698367i \(-0.246090\pi\)
\(24\) 0 0
\(25\) −2027.65 + 2377.87i −0.648849 + 0.760917i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5440.89 1.20136 0.600682 0.799488i \(-0.294896\pi\)
0.600682 + 0.799488i \(0.294896\pi\)
\(30\) 0 0
\(31\) 3623.54 0.677219 0.338609 0.940927i \(-0.390043\pi\)
0.338609 + 0.940927i \(0.390043\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −520.948 + 240.409i −0.0718827 + 0.0331727i
\(36\) 0 0
\(37\) 1756.01i 0.210874i 0.994426 + 0.105437i \(0.0336241\pi\)
−0.994426 + 0.105437i \(0.966376\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −263.984 −0.0245255 −0.0122628 0.999925i \(-0.503903\pi\)
−0.0122628 + 0.999925i \(0.503903\pi\)
\(42\) 0 0
\(43\) 14410.5i 1.18853i 0.804271 + 0.594263i \(0.202556\pi\)
−0.804271 + 0.594263i \(0.797444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 23464.8i 1.54943i 0.632308 + 0.774717i \(0.282108\pi\)
−0.632308 + 0.774717i \(0.717892\pi\)
\(48\) 0 0
\(49\) 16701.7 0.993732
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 33496.0i 1.63796i 0.573823 + 0.818979i \(0.305460\pi\)
−0.573823 + 0.818979i \(0.694540\pi\)
\(54\) 0 0
\(55\) −13970.5 30273.0i −0.622737 1.34942i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2906.38 0.108698 0.0543492 0.998522i \(-0.482692\pi\)
0.0543492 + 0.998522i \(0.482692\pi\)
\(60\) 0 0
\(61\) 29431.9 1.01273 0.506366 0.862319i \(-0.330989\pi\)
0.506366 + 0.862319i \(0.330989\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21350.1 9852.72i 0.626782 0.289250i
\(66\) 0 0
\(67\) 7163.34i 0.194952i −0.995238 0.0974762i \(-0.968923\pi\)
0.995238 0.0974762i \(-0.0310770\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −81353.2 −1.91527 −0.957633 0.287992i \(-0.907012\pi\)
−0.957633 + 0.287992i \(0.907012\pi\)
\(72\) 0 0
\(73\) 55127.9i 1.21078i −0.795930 0.605388i \(-0.793018\pi\)
0.795930 0.605388i \(-0.206982\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6121.36i 0.117658i
\(78\) 0 0
\(79\) 16430.9 0.296205 0.148103 0.988972i \(-0.452683\pi\)
0.148103 + 0.988972i \(0.452683\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 116869.i 1.86210i −0.364891 0.931050i \(-0.618894\pi\)
0.364891 0.931050i \(-0.381106\pi\)
\(84\) 0 0
\(85\) −49445.4 + 22818.2i −0.742299 + 0.342559i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 99364.0 1.32970 0.664851 0.746976i \(-0.268495\pi\)
0.664851 + 0.746976i \(0.268495\pi\)
\(90\) 0 0
\(91\) 4317.11 0.0546499
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8913.39 + 19314.6i 0.101329 + 0.219572i
\(96\) 0 0
\(97\) 62987.8i 0.679715i 0.940477 + 0.339858i \(0.110379\pi\)
−0.940477 + 0.339858i \(0.889621\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −40702.1 −0.397021 −0.198510 0.980099i \(-0.563610\pi\)
−0.198510 + 0.980099i \(0.563610\pi\)
\(102\) 0 0
\(103\) 108113.i 1.00412i −0.864834 0.502058i \(-0.832576\pi\)
0.864834 0.502058i \(-0.167424\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 198380.i 1.67509i −0.546367 0.837546i \(-0.683990\pi\)
0.546367 0.837546i \(-0.316010\pi\)
\(108\) 0 0
\(109\) 89150.3 0.718715 0.359357 0.933200i \(-0.382996\pi\)
0.359357 + 0.933200i \(0.382996\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 165319.i 1.21795i −0.793191 0.608973i \(-0.791582\pi\)
0.793191 0.608973i \(-0.208418\pi\)
\(114\) 0 0
\(115\) −179860. + 83002.3i −1.26820 + 0.585255i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9998.14 −0.0647220
\(120\) 0 0
\(121\) 194669. 1.20874
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 168190. + 47220.2i 0.962775 + 0.270304i
\(126\) 0 0
\(127\) 137238.i 0.755031i −0.926003 0.377516i \(-0.876778\pi\)
0.926003 0.377516i \(-0.123222\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −355564. −1.81026 −0.905128 0.425140i \(-0.860225\pi\)
−0.905128 + 0.425140i \(0.860225\pi\)
\(132\) 0 0
\(133\) 3905.53i 0.0191448i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 232937.i 1.06032i 0.847898 + 0.530160i \(0.177868\pi\)
−0.847898 + 0.530160i \(0.822132\pi\)
\(138\) 0 0
\(139\) −62526.2 −0.274489 −0.137245 0.990537i \(-0.543825\pi\)
−0.137245 + 0.990537i \(0.543825\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 250873.i 1.02592i
\(144\) 0 0
\(145\) −127446. 276166.i −0.503392 1.09081i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 267151. 0.985804 0.492902 0.870085i \(-0.335936\pi\)
0.492902 + 0.870085i \(0.335936\pi\)
\(150\) 0 0
\(151\) 329630. 1.17648 0.588240 0.808686i \(-0.299821\pi\)
0.588240 + 0.808686i \(0.299821\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −84877.0 183922.i −0.283766 0.614900i
\(156\) 0 0
\(157\) 400955.i 1.29822i −0.760696 0.649108i \(-0.775142\pi\)
0.760696 0.649108i \(-0.224858\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −36368.6 −0.110576
\(162\) 0 0
\(163\) 511411.i 1.50765i −0.657074 0.753826i \(-0.728206\pi\)
0.657074 0.753826i \(-0.271794\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 63650.1i 0.176607i −0.996094 0.0883035i \(-0.971855\pi\)
0.996094 0.0883035i \(-0.0281445\pi\)
\(168\) 0 0
\(169\) 194364. 0.523479
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 33351.0i 0.0847214i −0.999102 0.0423607i \(-0.986512\pi\)
0.999102 0.0423607i \(-0.0134879\pi\)
\(174\) 0 0
\(175\) 24405.1 + 20810.7i 0.0602401 + 0.0513680i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 314739. 0.734207 0.367103 0.930180i \(-0.380350\pi\)
0.367103 + 0.930180i \(0.380350\pi\)
\(180\) 0 0
\(181\) 415818. 0.943425 0.471712 0.881753i \(-0.343636\pi\)
0.471712 + 0.881753i \(0.343636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 89130.7 41132.3i 0.191469 0.0883596i
\(186\) 0 0
\(187\) 581005.i 1.21500i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 494250. 0.980310 0.490155 0.871635i \(-0.336940\pi\)
0.490155 + 0.871635i \(0.336940\pi\)
\(192\) 0 0
\(193\) 62426.1i 0.120635i 0.998179 + 0.0603175i \(0.0192113\pi\)
−0.998179 + 0.0603175i \(0.980789\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 513844.i 0.943334i −0.881777 0.471667i \(-0.843652\pi\)
0.881777 0.471667i \(-0.156348\pi\)
\(198\) 0 0
\(199\) −29132.1 −0.0521481 −0.0260741 0.999660i \(-0.508301\pi\)
−0.0260741 + 0.999660i \(0.508301\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 55842.3i 0.0951094i
\(204\) 0 0
\(205\) 6183.51 + 13399.2i 0.0102766 + 0.0222687i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −226955. −0.359397
\(210\) 0 0
\(211\) 330044. 0.510347 0.255173 0.966895i \(-0.417867\pi\)
0.255173 + 0.966895i \(0.417867\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 731443. 337549.i 1.07916 0.498013i
\(216\) 0 0
\(217\) 37190.1i 0.0536139i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 409756. 0.564344
\(222\) 0 0
\(223\) 930113.i 1.25249i 0.779627 + 0.626244i \(0.215409\pi\)
−0.779627 + 0.626244i \(0.784591\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 37009.2i 0.0476699i −0.999716 0.0238350i \(-0.992412\pi\)
0.999716 0.0238350i \(-0.00758762\pi\)
\(228\) 0 0
\(229\) −506969. −0.638841 −0.319421 0.947613i \(-0.603488\pi\)
−0.319421 + 0.947613i \(0.603488\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 378560.i 0.456819i 0.973565 + 0.228410i \(0.0733526\pi\)
−0.973565 + 0.228410i \(0.926647\pi\)
\(234\) 0 0
\(235\) 1.19102e6 549635.i 1.40685 0.649239i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −105854. −0.119870 −0.0599351 0.998202i \(-0.519089\pi\)
−0.0599351 + 0.998202i \(0.519089\pi\)
\(240\) 0 0
\(241\) −1.14897e6 −1.27429 −0.637144 0.770745i \(-0.719884\pi\)
−0.637144 + 0.770745i \(0.719884\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −391216. 847735.i −0.416391 0.902288i
\(246\) 0 0
\(247\) 160061.i 0.166933i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 655956. 0.657189 0.328595 0.944471i \(-0.393425\pi\)
0.328595 + 0.944471i \(0.393425\pi\)
\(252\) 0 0
\(253\) 2.11343e6i 2.07580i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 986720.i 0.931883i −0.884816 0.465941i \(-0.845716\pi\)
0.884816 0.465941i \(-0.154284\pi\)
\(258\) 0 0
\(259\) 18022.7 0.0166944
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 865331.i 0.771424i −0.922619 0.385712i \(-0.873956\pi\)
0.922619 0.385712i \(-0.126044\pi\)
\(264\) 0 0
\(265\) 1.70017e6 784602.i 1.48723 0.686333i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.75595e6 −1.47956 −0.739780 0.672849i \(-0.765070\pi\)
−0.739780 + 0.672849i \(0.765070\pi\)
\(270\) 0 0
\(271\) −1.07635e6 −0.890287 −0.445143 0.895459i \(-0.646847\pi\)
−0.445143 + 0.895459i \(0.646847\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.20934e6 + 1.41821e6i −0.964310 + 1.13086i
\(276\) 0 0
\(277\) 620129.i 0.485605i 0.970076 + 0.242802i \(0.0780666\pi\)
−0.970076 + 0.242802i \(0.921933\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −884924. −0.668560 −0.334280 0.942474i \(-0.608493\pi\)
−0.334280 + 0.942474i \(0.608493\pi\)
\(282\) 0 0
\(283\) 1.06088e6i 0.787408i −0.919237 0.393704i \(-0.871194\pi\)
0.919237 0.393704i \(-0.128806\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2709.39i 0.00194163i
\(288\) 0 0
\(289\) 470891. 0.331646
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 585847.i 0.398672i −0.979931 0.199336i \(-0.936122\pi\)
0.979931 0.199336i \(-0.0638784\pi\)
\(294\) 0 0
\(295\) −68078.5 147521.i −0.0455465 0.0986958i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.49050e6 0.964172
\(300\) 0 0
\(301\) 147902. 0.0940931
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −689407. 1.49389e6i −0.424352 0.919539i
\(306\) 0 0
\(307\) 2.33467e6i 1.41377i −0.707328 0.706886i \(-0.750099\pi\)
0.707328 0.706886i \(-0.249901\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.18712e6 1.28225 0.641123 0.767438i \(-0.278469\pi\)
0.641123 + 0.767438i \(0.278469\pi\)
\(312\) 0 0
\(313\) 2.76800e6i 1.59700i −0.601993 0.798501i \(-0.705626\pi\)
0.601993 0.798501i \(-0.294374\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.71952e6i 0.961077i 0.876974 + 0.480538i \(0.159559\pi\)
−0.876974 + 0.480538i \(0.840441\pi\)
\(318\) 0 0
\(319\) 3.24507e6 1.78545
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 370691.i 0.197699i
\(324\) 0 0
\(325\) −1.00020e6 852891.i −0.525265 0.447904i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 240830. 0.122665
\(330\) 0 0
\(331\) 1.24759e6 0.625895 0.312948 0.949770i \(-0.398684\pi\)
0.312948 + 0.949770i \(0.398684\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −363593. + 167792.i −0.177013 + 0.0816884i
\(336\) 0 0
\(337\) 1.96509e6i 0.942556i 0.881985 + 0.471278i \(0.156207\pi\)
−0.881985 + 0.471278i \(0.843793\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.16116e6 1.00647
\(342\) 0 0
\(343\) 343915.i 0.157839i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.27623e6i 1.46067i −0.683091 0.730333i \(-0.739365\pi\)
0.683091 0.730333i \(-0.260635\pi\)
\(348\) 0 0
\(349\) −101966. −0.0448117 −0.0224058 0.999749i \(-0.507133\pi\)
−0.0224058 + 0.999749i \(0.507133\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.03342e6i 1.29567i −0.761779 0.647837i \(-0.775674\pi\)
0.761779 0.647837i \(-0.224326\pi\)
\(354\) 0 0
\(355\) 1.90560e6 + 4.12929e6i 0.802529 + 1.73902i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.68388e6 −1.09908 −0.549538 0.835469i \(-0.685196\pi\)
−0.549538 + 0.835469i \(0.685196\pi\)
\(360\) 0 0
\(361\) −2.33130e6 −0.941520
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.79815e6 + 1.29130e6i −1.09936 + 0.507336i
\(366\) 0 0
\(367\) 1.17487e6i 0.455329i −0.973740 0.227664i \(-0.926891\pi\)
0.973740 0.227664i \(-0.0731089\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 343784. 0.129674
\(372\) 0 0
\(373\) 4.13661e6i 1.53947i 0.638361 + 0.769737i \(0.279613\pi\)
−0.638361 + 0.769737i \(0.720387\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.28860e6i 0.829308i
\(378\) 0 0
\(379\) −4.02996e6 −1.44113 −0.720565 0.693387i \(-0.756117\pi\)
−0.720565 + 0.693387i \(0.756117\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.29187e6i 1.14669i 0.819314 + 0.573345i \(0.194355\pi\)
−0.819314 + 0.573345i \(0.805645\pi\)
\(384\) 0 0
\(385\) −310705. + 143385.i −0.106831 + 0.0493007i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.23387e6 1.08355 0.541774 0.840524i \(-0.317753\pi\)
0.541774 + 0.840524i \(0.317753\pi\)
\(390\) 0 0
\(391\) −3.45190e6 −1.14187
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −384872. 833990.i −0.124115 0.268948i
\(396\) 0 0
\(397\) 1.50766e6i 0.480095i 0.970761 + 0.240047i \(0.0771630\pi\)
−0.970761 + 0.240047i \(0.922837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 132613. 0.0411837 0.0205918 0.999788i \(-0.493445\pi\)
0.0205918 + 0.999788i \(0.493445\pi\)
\(402\) 0 0
\(403\) 1.52417e6i 0.467487i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.04732e6i 0.313397i
\(408\) 0 0
\(409\) 4.30354e6 1.27209 0.636044 0.771653i \(-0.280570\pi\)
0.636044 + 0.771653i \(0.280570\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29829.6i 0.00860541i
\(414\) 0 0
\(415\) −5.93197e6 + 2.73751e6i −1.69075 + 0.780252i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.35654e6 1.49056 0.745280 0.666752i \(-0.232316\pi\)
0.745280 + 0.666752i \(0.232316\pi\)
\(420\) 0 0
\(421\) 2.95420e6 0.812335 0.406167 0.913799i \(-0.366865\pi\)
0.406167 + 0.913799i \(0.366865\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.31640e6 + 1.97524e6i 0.622071 + 0.530453i
\(426\) 0 0
\(427\) 302074.i 0.0801758i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.93581e6 −1.53917 −0.769586 0.638543i \(-0.779537\pi\)
−0.769586 + 0.638543i \(0.779537\pi\)
\(432\) 0 0
\(433\) 2.72633e6i 0.698810i 0.936972 + 0.349405i \(0.113616\pi\)
−0.936972 + 0.349405i \(0.886384\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.34840e6i 0.337766i
\(438\) 0 0
\(439\) 6.39220e6 1.58303 0.791515 0.611150i \(-0.209293\pi\)
0.791515 + 0.611150i \(0.209293\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 990808.i 0.239872i 0.992782 + 0.119936i \(0.0382690\pi\)
−0.992782 + 0.119936i \(0.961731\pi\)
\(444\) 0 0
\(445\) −2.32748e6 5.04347e6i −0.557168 1.20734i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.79185e6 1.12173 0.560864 0.827908i \(-0.310469\pi\)
0.560864 + 0.827908i \(0.310469\pi\)
\(450\) 0 0
\(451\) −157446. −0.0364495
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −101123. 219126.i −0.0228993 0.0496210i
\(456\) 0 0
\(457\) 1.50517e6i 0.337128i 0.985691 + 0.168564i \(0.0539130\pi\)
−0.985691 + 0.168564i \(0.946087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.28168e6 −0.280883 −0.140442 0.990089i \(-0.544852\pi\)
−0.140442 + 0.990089i \(0.544852\pi\)
\(462\) 0 0
\(463\) 700105.i 0.151779i 0.997116 + 0.0758893i \(0.0241796\pi\)
−0.997116 + 0.0758893i \(0.975820\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.89776e6i 0.402669i −0.979523 0.201334i \(-0.935472\pi\)
0.979523 0.201334i \(-0.0645278\pi\)
\(468\) 0 0
\(469\) −73520.6 −0.0154340
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.59478e6i 1.76637i
\(474\) 0 0
\(475\) 771578. 904843.i 0.156908 0.184009i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.88878e6 −1.37184 −0.685920 0.727677i \(-0.740600\pi\)
−0.685920 + 0.727677i \(0.740600\pi\)
\(480\) 0 0
\(481\) −738628. −0.145567
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.19710e6 1.47541e6i 0.617167 0.284812i
\(486\) 0 0
\(487\) 3.31559e6i 0.633488i −0.948511 0.316744i \(-0.897410\pi\)
0.948511 0.316744i \(-0.102590\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.55075e6 −1.22627 −0.613137 0.789977i \(-0.710092\pi\)
−0.613137 + 0.789977i \(0.710092\pi\)
\(492\) 0 0
\(493\) 5.30024e6i 0.982150i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 834966.i 0.151627i
\(498\) 0 0
\(499\) −4.14654e6 −0.745477 −0.372738 0.927936i \(-0.621581\pi\)
−0.372738 + 0.927936i \(0.621581\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 386446.i 0.0681035i −0.999420 0.0340517i \(-0.989159\pi\)
0.999420 0.0340517i \(-0.0108411\pi\)
\(504\) 0 0
\(505\) 953396. + 2.06594e6i 0.166358 + 0.360486i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.04725e6 0.521331 0.260665 0.965429i \(-0.416058\pi\)
0.260665 + 0.965429i \(0.416058\pi\)
\(510\) 0 0
\(511\) −565802. −0.0958545
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.48753e6 + 2.53241e6i −0.911715 + 0.420742i
\(516\) 0 0
\(517\) 1.39950e7i 2.30274i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8.88552e6 −1.43413 −0.717065 0.697007i \(-0.754515\pi\)
−0.717065 + 0.697007i \(0.754515\pi\)
\(522\) 0 0
\(523\) 4.92458e6i 0.787254i 0.919270 + 0.393627i \(0.128780\pi\)
−0.919270 + 0.393627i \(0.871220\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.52987e6i 0.553646i
\(528\) 0 0
\(529\) −6.12010e6 −0.950866
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 111040.i 0.0169301i
\(534\) 0 0
\(535\) −1.00693e7 + 4.64681e6i −1.52095 + 0.701892i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.96126e6 1.47687
\(540\) 0 0
\(541\) 5.45948e6 0.801970 0.400985 0.916085i \(-0.368668\pi\)
0.400985 + 0.916085i \(0.368668\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.08824e6 4.52505e6i −0.301154 0.652577i
\(546\) 0 0
\(547\) 9.70824e6i 1.38731i −0.720309 0.693653i \(-0.756000\pi\)
0.720309 0.693653i \(-0.244000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.07041e6 −0.290521
\(552\) 0 0
\(553\) 168637.i 0.0234499i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.77260e6i 0.651804i 0.945404 + 0.325902i \(0.105668\pi\)
−0.945404 + 0.325902i \(0.894332\pi\)
\(558\) 0 0
\(559\) −6.06149e6 −0.820446
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.16768e6i 0.288220i −0.989562 0.144110i \(-0.953968\pi\)
0.989562 0.144110i \(-0.0460319\pi\)
\(564\) 0 0
\(565\) −8.39121e6 + 3.87240e6i −1.10587 + 0.510340i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.17864e6 0.541070 0.270535 0.962710i \(-0.412799\pi\)
0.270535 + 0.962710i \(0.412799\pi\)
\(570\) 0 0
\(571\) 7.89574e6 1.01345 0.506725 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.42598e6 + 7.18501e6i 1.06280 + 0.906270i
\(576\) 0 0
\(577\) 1.34605e7i 1.68315i 0.540141 + 0.841575i \(0.318371\pi\)
−0.540141 + 0.841575i \(0.681629\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.19948e6 −0.147418
\(582\) 0 0
\(583\) 1.99778e7i 2.43431i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.43350e6i 0.171713i −0.996308 0.0858564i \(-0.972637\pi\)
0.996308 0.0858564i \(-0.0273626\pi\)
\(588\) 0 0
\(589\) −1.37886e6 −0.163769
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.38150e6i 0.161330i −0.996741 0.0806649i \(-0.974296\pi\)
0.996741 0.0806649i \(-0.0257044\pi\)
\(594\) 0 0
\(595\) 234194. + 507481.i 0.0271196 + 0.0587661i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.53001e6 −0.743612 −0.371806 0.928310i \(-0.621261\pi\)
−0.371806 + 0.928310i \(0.621261\pi\)
\(600\) 0 0
\(601\) −3.81467e6 −0.430795 −0.215398 0.976526i \(-0.569105\pi\)
−0.215398 + 0.976526i \(0.569105\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.55989e6 9.88094e6i −0.506484 1.09751i
\(606\) 0 0
\(607\) 1.06018e7i 1.16791i 0.811787 + 0.583954i \(0.198495\pi\)
−0.811787 + 0.583954i \(0.801505\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.87000e6 −1.06958
\(612\) 0 0
\(613\) 5.80630e6i 0.624091i −0.950067 0.312046i \(-0.898986\pi\)
0.950067 0.312046i \(-0.101014\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.80508e6i 0.719648i 0.933020 + 0.359824i \(0.117163\pi\)
−0.933020 + 0.359824i \(0.882837\pi\)
\(618\) 0 0
\(619\) 852530. 0.0894299 0.0447150 0.999000i \(-0.485762\pi\)
0.0447150 + 0.999000i \(0.485762\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.01982e6i 0.105270i
\(624\) 0 0
\(625\) −1.54286e6 9.64298e6i −0.157989 0.987441i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.71061e6 0.172395
\(630\) 0 0
\(631\) −8.01341e6 −0.801205 −0.400603 0.916252i \(-0.631199\pi\)
−0.400603 + 0.916252i \(0.631199\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.96586e6 + 3.21463e6i −0.685552 + 0.316371i
\(636\) 0 0
\(637\) 7.02521e6i 0.685978i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.68737e6 0.354464 0.177232 0.984169i \(-0.443286\pi\)
0.177232 + 0.984169i \(0.443286\pi\)
\(642\) 0 0
\(643\) 9.97031e6i 0.951001i −0.879715 0.475501i \(-0.842267\pi\)
0.879715 0.475501i \(-0.157733\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.73097e6i 0.162566i 0.996691 + 0.0812830i \(0.0259018\pi\)
−0.996691 + 0.0812830i \(0.974098\pi\)
\(648\) 0 0
\(649\) 1.73343e6 0.161546
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.15248e6i 0.105767i −0.998601 0.0528836i \(-0.983159\pi\)
0.998601 0.0528836i \(-0.0168412\pi\)
\(654\) 0 0
\(655\) 8.32865e6 + 1.80476e7i 0.758528 + 1.64367i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.53161e7 1.37384 0.686919 0.726734i \(-0.258963\pi\)
0.686919 + 0.726734i \(0.258963\pi\)
\(660\) 0 0
\(661\) 1.69450e6 0.150848 0.0754238 0.997152i \(-0.475969\pi\)
0.0754238 + 0.997152i \(0.475969\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 198235. 91482.2i 0.0173831 0.00802199i
\(666\) 0 0
\(667\) 1.92798e7i 1.67799i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.75539e7 1.50511
\(672\) 0 0
\(673\) 1.53955e6i 0.131026i −0.997852 0.0655129i \(-0.979132\pi\)
0.997852 0.0655129i \(-0.0208683\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.16833e7i 0.979702i 0.871806 + 0.489851i \(0.162949\pi\)
−0.871806 + 0.489851i \(0.837051\pi\)
\(678\) 0 0
\(679\) 646473. 0.0538116
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.49762e7i 1.22843i 0.789139 + 0.614215i \(0.210527\pi\)
−0.789139 + 0.614215i \(0.789473\pi\)
\(684\) 0 0
\(685\) 1.18233e7 5.45626e6i 0.962747 0.444292i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.40894e7 −1.13069
\(690\) 0 0
\(691\) 9.90115e6 0.788843 0.394422 0.918930i \(-0.370945\pi\)
0.394422 + 0.918930i \(0.370945\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.46460e6 + 3.17368e6i 0.115016 + 0.249230i
\(696\) 0 0
\(697\) 257160.i 0.0200503i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.48933e7 −1.14471 −0.572357 0.820005i \(-0.693971\pi\)
−0.572357 + 0.820005i \(0.693971\pi\)
\(702\) 0 0
\(703\) 668209.i 0.0509946i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 417744.i 0.0314313i
\(708\) 0 0
\(709\) −3.63694e6 −0.271719 −0.135860 0.990728i \(-0.543380\pi\)
−0.135860 + 0.990728i \(0.543380\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.28400e7i 0.945894i
\(714\) 0 0
\(715\) 1.27337e7 5.87639e6i 0.931514 0.429878i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.17971e7 0.851044 0.425522 0.904948i \(-0.360091\pi\)
0.425522 + 0.904948i \(0.360091\pi\)
\(720\) 0 0
\(721\) −1.10961e6 −0.0794936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.10322e7 + 1.29377e7i −0.779505 + 0.914139i
\(726\) 0 0
\(727\) 7.53381e6i 0.528663i 0.964432 + 0.264331i \(0.0851513\pi\)
−0.964432 + 0.264331i \(0.914849\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.40380e7 0.971655
\(732\) 0 0
\(733\) 1.36302e7i 0.937003i 0.883463 + 0.468501i \(0.155206\pi\)
−0.883463 + 0.468501i \(0.844794\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.27238e6i 0.289735i
\(738\) 0 0
\(739\) 1.07655e7 0.725141 0.362571 0.931956i \(-0.381899\pi\)
0.362571 + 0.931956i \(0.381899\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.53921e7i 1.68743i 0.536788 + 0.843717i \(0.319637\pi\)
−0.536788 + 0.843717i \(0.680363\pi\)
\(744\) 0 0
\(745\) −6.25768e6 1.35599e7i −0.413069 0.895089i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.03607e6 −0.132613
\(750\) 0 0
\(751\) −8.06289e6 −0.521664 −0.260832 0.965384i \(-0.583997\pi\)
−0.260832 + 0.965384i \(0.583997\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.72118e6 1.67312e7i −0.492965 1.06822i
\(756\) 0 0
\(757\) 1.08891e7i 0.690639i −0.938485 0.345319i \(-0.887771\pi\)
0.938485 0.345319i \(-0.112229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.07800e7 0.674773 0.337387 0.941366i \(-0.390457\pi\)
0.337387 + 0.941366i \(0.390457\pi\)
\(762\) 0 0
\(763\) 914990.i 0.0568991i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.22251e6i 0.0750350i
\(768\) 0 0
\(769\) 8.24398e6 0.502714 0.251357 0.967894i \(-0.419123\pi\)
0.251357 + 0.967894i \(0.419123\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.76264e6i 0.226487i −0.993567 0.113244i \(-0.963876\pi\)
0.993567 0.113244i \(-0.0361240\pi\)
\(774\) 0 0
\(775\) −7.34729e6 + 8.61629e6i −0.439413 + 0.515307i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 100453. 0.00593090
\(780\) 0 0
\(781\) −4.85209e7 −2.84644
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.03515e7 + 9.39189e6i −1.17875 + 0.543975i
\(786\) 0 0
\(787\) 1.14314e7i 0.657906i −0.944346 0.328953i \(-0.893304\pi\)
0.944346 0.328953i \(-0.106696\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.69675e6 −0.0964221
\(792\) 0 0
\(793\) 1.23799e7i 0.699094i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.82556e7i 1.01801i −0.860765 0.509003i \(-0.830014\pi\)
0.860765 0.509003i \(-0.169986\pi\)
\(798\) 0 0
\(799\) 2.28583e7 1.26671
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.28795e7i 1.79944i
\(804\) 0 0
\(805\) 851890. + 1.84598e6i 0.0463334 + 0.100401i
\(806\) 0 0
\(807\) 0 0
\(808\) 0