Properties

Label 720.6.f.n.289.6
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 289.6
Root \(-1.64654i\) of defining polynomial
Character \(\chi\) \(=\) 720.289
Dual form 720.6.f.n.289.5

$q$-expansion

\(f(q)\) \(=\) \(q+(13.1588 + 54.3309i) q^{5} -146.828i q^{7} +O(q^{10})\) \(q+(13.1588 + 54.3309i) q^{5} -146.828i q^{7} +191.129 q^{11} -83.9971i q^{13} +2000.23i q^{17} +677.481 q^{19} -1296.23i q^{23} +(-2778.69 + 1429.86i) q^{25} -3266.00 q^{29} -6157.98 q^{31} +(7977.29 - 1932.08i) q^{35} -11369.5i q^{37} +10599.8 q^{41} +12926.2i q^{43} +9521.88i q^{47} -4751.45 q^{49} +14786.8i q^{53} +(2515.03 + 10384.2i) q^{55} +38226.9 q^{59} -3581.69 q^{61} +(4563.64 - 1105.30i) q^{65} -21780.0i q^{67} +51390.1 q^{71} +13305.9i q^{73} -28063.1i q^{77} +15944.3 q^{79} +53305.0i q^{83} +(-108675. + 26320.7i) q^{85} -51330.4 q^{89} -12333.1 q^{91} +(8914.84 + 36808.1i) q^{95} +80849.9i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100}) \)

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
\(5\) 13.1588 + 54.3309i 0.235392 + 0.971901i
\(6\) 0 0
\(7\) 146.828i 1.13257i −0.824211 0.566283i \(-0.808381\pi\)
0.824211 0.566283i \(-0.191619\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 191.129 0.476260 0.238130 0.971233i \(-0.423466\pi\)
0.238130 + 0.971233i \(0.423466\pi\)
\(12\) 0 0
\(13\) 83.9971i 0.137850i −0.997622 0.0689249i \(-0.978043\pi\)
0.997622 0.0689249i \(-0.0219569\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2000.23i 1.67864i 0.543635 + 0.839322i \(0.317048\pi\)
−0.543635 + 0.839322i \(0.682952\pi\)
\(18\) 0 0
\(19\) 677.481 0.430540 0.215270 0.976555i \(-0.430937\pi\)
0.215270 + 0.976555i \(0.430937\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1296.23i 0.510931i −0.966818 0.255466i \(-0.917771\pi\)
0.966818 0.255466i \(-0.0822288\pi\)
\(24\) 0 0
\(25\) −2778.69 + 1429.86i −0.889181 + 0.457555i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3266.00 −0.721143 −0.360571 0.932732i \(-0.617418\pi\)
−0.360571 + 0.932732i \(0.617418\pi\)
\(30\) 0 0
\(31\) −6157.98 −1.15089 −0.575445 0.817841i \(-0.695171\pi\)
−0.575445 + 0.817841i \(0.695171\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7977.29 1932.08i 1.10074 0.266597i
\(36\) 0 0
\(37\) 11369.5i 1.36533i −0.730730 0.682667i \(-0.760820\pi\)
0.730730 0.682667i \(-0.239180\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10599.8 0.984774 0.492387 0.870376i \(-0.336124\pi\)
0.492387 + 0.870376i \(0.336124\pi\)
\(42\) 0 0
\(43\) 12926.2i 1.06610i 0.846082 + 0.533052i \(0.178955\pi\)
−0.846082 + 0.533052i \(0.821045\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9521.88i 0.628750i 0.949299 + 0.314375i \(0.101795\pi\)
−0.949299 + 0.314375i \(0.898205\pi\)
\(48\) 0 0
\(49\) −4751.45 −0.282706
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14786.8i 0.723078i 0.932357 + 0.361539i \(0.117749\pi\)
−0.932357 + 0.361539i \(0.882251\pi\)
\(54\) 0 0
\(55\) 2515.03 + 10384.2i 0.112108 + 0.462878i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 38226.9 1.42968 0.714841 0.699287i \(-0.246499\pi\)
0.714841 + 0.699287i \(0.246499\pi\)
\(60\) 0 0
\(61\) −3581.69 −0.123243 −0.0616217 0.998100i \(-0.519627\pi\)
−0.0616217 + 0.998100i \(0.519627\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4563.64 1105.30i 0.133976 0.0324487i
\(66\) 0 0
\(67\) 21780.0i 0.592748i −0.955072 0.296374i \(-0.904223\pi\)
0.955072 0.296374i \(-0.0957774\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 51390.1 1.20986 0.604928 0.796280i \(-0.293202\pi\)
0.604928 + 0.796280i \(0.293202\pi\)
\(72\) 0 0
\(73\) 13305.9i 0.292238i 0.989267 + 0.146119i \(0.0466782\pi\)
−0.989267 + 0.146119i \(0.953322\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 28063.1i 0.539396i
\(78\) 0 0
\(79\) 15944.3 0.287434 0.143717 0.989619i \(-0.454094\pi\)
0.143717 + 0.989619i \(0.454094\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 53305.0i 0.849323i 0.905352 + 0.424662i \(0.139607\pi\)
−0.905352 + 0.424662i \(0.860393\pi\)
\(84\) 0 0
\(85\) −108675. + 26320.7i −1.63148 + 0.395139i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −51330.4 −0.686910 −0.343455 0.939169i \(-0.611597\pi\)
−0.343455 + 0.939169i \(0.611597\pi\)
\(90\) 0 0
\(91\) −12333.1 −0.156124
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8914.84 + 36808.1i 0.101346 + 0.418442i
\(96\) 0 0
\(97\) 80849.9i 0.872469i 0.899833 + 0.436234i \(0.143688\pi\)
−0.899833 + 0.436234i \(0.856312\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −89471.2 −0.872729 −0.436365 0.899770i \(-0.643734\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(102\) 0 0
\(103\) 44184.6i 0.410372i −0.978723 0.205186i \(-0.934220\pi\)
0.978723 0.205186i \(-0.0657799\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 56446.0i 0.476622i 0.971189 + 0.238311i \(0.0765937\pi\)
−0.971189 + 0.238311i \(0.923406\pi\)
\(108\) 0 0
\(109\) −52651.1 −0.424465 −0.212232 0.977219i \(-0.568073\pi\)
−0.212232 + 0.977219i \(0.568073\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 78399.6i 0.577587i −0.957391 0.288794i \(-0.906746\pi\)
0.957391 0.288794i \(-0.0932542\pi\)
\(114\) 0 0
\(115\) 70425.3 17056.8i 0.496574 0.120269i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 293690. 1.90118
\(120\) 0 0
\(121\) −124521. −0.773176
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −114250. 132154.i −0.654004 0.756491i
\(126\) 0 0
\(127\) 296102.i 1.62904i 0.580133 + 0.814522i \(0.303001\pi\)
−0.580133 + 0.814522i \(0.696999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −199030. −1.01331 −0.506653 0.862150i \(-0.669117\pi\)
−0.506653 + 0.862150i \(0.669117\pi\)
\(132\) 0 0
\(133\) 99473.2i 0.487615i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 361224.i 1.64428i 0.569288 + 0.822138i \(0.307219\pi\)
−0.569288 + 0.822138i \(0.692781\pi\)
\(138\) 0 0
\(139\) −257315. −1.12961 −0.564805 0.825224i \(-0.691049\pi\)
−0.564805 + 0.825224i \(0.691049\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16054.3i 0.0656524i
\(144\) 0 0
\(145\) −42976.7 177445.i −0.169751 0.700879i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −478333. −1.76508 −0.882540 0.470238i \(-0.844168\pi\)
−0.882540 + 0.470238i \(0.844168\pi\)
\(150\) 0 0
\(151\) 193483. 0.690558 0.345279 0.938500i \(-0.387784\pi\)
0.345279 + 0.938500i \(0.387784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −81031.6 334568.i −0.270910 1.11855i
\(156\) 0 0
\(157\) 460294.i 1.49034i 0.666873 + 0.745172i \(0.267633\pi\)
−0.666873 + 0.745172i \(0.732367\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −190323. −0.578663
\(162\) 0 0
\(163\) 372211.i 1.09729i 0.836057 + 0.548643i \(0.184855\pi\)
−0.836057 + 0.548643i \(0.815145\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 423266.i 1.17442i 0.809436 + 0.587208i \(0.199773\pi\)
−0.809436 + 0.587208i \(0.800227\pi\)
\(168\) 0 0
\(169\) 364237. 0.980997
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 223915.i 0.568810i 0.958704 + 0.284405i \(0.0917961\pi\)
−0.958704 + 0.284405i \(0.908204\pi\)
\(174\) 0 0
\(175\) 209943. + 407990.i 0.518211 + 1.00706i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 29324.8 0.0684072 0.0342036 0.999415i \(-0.489111\pi\)
0.0342036 + 0.999415i \(0.489111\pi\)
\(180\) 0 0
\(181\) −46594.3 −0.105715 −0.0528574 0.998602i \(-0.516833\pi\)
−0.0528574 + 0.998602i \(0.516833\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 617718. 149610.i 1.32697 0.321388i
\(186\) 0 0
\(187\) 382302.i 0.799472i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 327452. 0.649477 0.324739 0.945804i \(-0.394724\pi\)
0.324739 + 0.945804i \(0.394724\pi\)
\(192\) 0 0
\(193\) 406622.i 0.785773i 0.919587 + 0.392887i \(0.128524\pi\)
−0.919587 + 0.392887i \(0.871476\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 364071.i 0.668376i 0.942506 + 0.334188i \(0.108462\pi\)
−0.942506 + 0.334188i \(0.891538\pi\)
\(198\) 0 0
\(199\) 55181.3 0.0987777 0.0493889 0.998780i \(-0.484273\pi\)
0.0493889 + 0.998780i \(0.484273\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 479540.i 0.816742i
\(204\) 0 0
\(205\) 139480. + 575895.i 0.231808 + 0.957102i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 129486. 0.205049
\(210\) 0 0
\(211\) 831145. 1.28520 0.642600 0.766202i \(-0.277856\pi\)
0.642600 + 0.766202i \(0.277856\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −702292. + 170093.i −1.03615 + 0.250952i
\(216\) 0 0
\(217\) 904163.i 1.30346i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 168014. 0.231401
\(222\) 0 0
\(223\) 458628.i 0.617587i −0.951129 0.308793i \(-0.900075\pi\)
0.951129 0.308793i \(-0.0999252\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22779.7i 0.0293416i −0.999892 0.0146708i \(-0.995330\pi\)
0.999892 0.0146708i \(-0.00467003\pi\)
\(228\) 0 0
\(229\) 1.36560e6 1.72082 0.860408 0.509606i \(-0.170209\pi\)
0.860408 + 0.509606i \(0.170209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 673470.i 0.812696i 0.913718 + 0.406348i \(0.133198\pi\)
−0.913718 + 0.406348i \(0.866802\pi\)
\(234\) 0 0
\(235\) −517332. + 125297.i −0.611082 + 0.148003i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −461339. −0.522426 −0.261213 0.965281i \(-0.584123\pi\)
−0.261213 + 0.965281i \(0.584123\pi\)
\(240\) 0 0
\(241\) −842868. −0.934796 −0.467398 0.884047i \(-0.654809\pi\)
−0.467398 + 0.884047i \(0.654809\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −62523.4 258150.i −0.0665468 0.274763i
\(246\) 0 0
\(247\) 56906.4i 0.0593498i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.34039e6 −1.34291 −0.671457 0.741044i \(-0.734331\pi\)
−0.671457 + 0.741044i \(0.734331\pi\)
\(252\) 0 0
\(253\) 247747.i 0.243336i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 337841.i 0.319066i −0.987193 0.159533i \(-0.949001\pi\)
0.987193 0.159533i \(-0.0509988\pi\)
\(258\) 0 0
\(259\) −1.66937e6 −1.54633
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 836410.i 0.745641i 0.927903 + 0.372821i \(0.121609\pi\)
−0.927903 + 0.372821i \(0.878391\pi\)
\(264\) 0 0
\(265\) −803381. + 194577.i −0.702760 + 0.170207i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −36624.0 −0.0308592 −0.0154296 0.999881i \(-0.504912\pi\)
−0.0154296 + 0.999881i \(0.504912\pi\)
\(270\) 0 0
\(271\) −1.92834e6 −1.59500 −0.797500 0.603319i \(-0.793845\pi\)
−0.797500 + 0.603319i \(0.793845\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −531088. + 273287.i −0.423482 + 0.217915i
\(276\) 0 0
\(277\) 1.42694e6i 1.11739i 0.829373 + 0.558696i \(0.188698\pi\)
−0.829373 + 0.558696i \(0.811302\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.36724e6 1.78845 0.894223 0.447621i \(-0.147729\pi\)
0.894223 + 0.447621i \(0.147729\pi\)
\(282\) 0 0
\(283\) 1.35021e6i 1.00215i 0.865403 + 0.501077i \(0.167063\pi\)
−0.865403 + 0.501077i \(0.832937\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.55634e6i 1.11532i
\(288\) 0 0
\(289\) −2.58108e6 −1.81785
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 506336.i 0.344564i 0.985048 + 0.172282i \(0.0551140\pi\)
−0.985048 + 0.172282i \(0.944886\pi\)
\(294\) 0 0
\(295\) 503021. + 2.07690e6i 0.336536 + 1.38951i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −108880. −0.0704317
\(300\) 0 0
\(301\) 1.89793e6 1.20743
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −47130.8 194596.i −0.0290105 0.119780i
\(306\) 0 0
\(307\) 1.15053e6i 0.696712i −0.937362 0.348356i \(-0.886740\pi\)
0.937362 0.348356i \(-0.113260\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −742231. −0.435149 −0.217574 0.976044i \(-0.569815\pi\)
−0.217574 + 0.976044i \(0.569815\pi\)
\(312\) 0 0
\(313\) 1.67101e6i 0.964090i −0.876146 0.482045i \(-0.839894\pi\)
0.876146 0.482045i \(-0.160106\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 646531.i 0.361361i 0.983542 + 0.180681i \(0.0578300\pi\)
−0.983542 + 0.180681i \(0.942170\pi\)
\(318\) 0 0
\(319\) −624227. −0.343452
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.35512e6i 0.722723i
\(324\) 0 0
\(325\) 120104. + 233402.i 0.0630738 + 0.122573i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.39808e6 0.712101
\(330\) 0 0
\(331\) −2.73052e6 −1.36986 −0.684928 0.728610i \(-0.740166\pi\)
−0.684928 + 0.728610i \(0.740166\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.18332e6 286598.i 0.576092 0.139528i
\(336\) 0 0
\(337\) 390252.i 0.187185i 0.995611 + 0.0935923i \(0.0298350\pi\)
−0.995611 + 0.0935923i \(0.970165\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.17697e6 −0.548123
\(342\) 0 0
\(343\) 1.77009e6i 0.812382i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.23128e6i 0.994788i −0.867525 0.497394i \(-0.834290\pi\)
0.867525 0.497394i \(-0.165710\pi\)
\(348\) 0 0
\(349\) 502679. 0.220916 0.110458 0.993881i \(-0.464768\pi\)
0.110458 + 0.993881i \(0.464768\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.57148e6i 1.95263i 0.216350 + 0.976316i \(0.430585\pi\)
−0.216350 + 0.976316i \(0.569415\pi\)
\(354\) 0 0
\(355\) 676232. + 2.79207e6i 0.284790 + 1.17586i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 448308. 0.183586 0.0917931 0.995778i \(-0.470740\pi\)
0.0917931 + 0.995778i \(0.470740\pi\)
\(360\) 0 0
\(361\) −2.01712e6 −0.814636
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −722920. + 175089.i −0.284026 + 0.0687903i
\(366\) 0 0
\(367\) 2.56484e6i 0.994018i −0.867745 0.497009i \(-0.834432\pi\)
0.867745 0.497009i \(-0.165568\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.17112e6 0.818934
\(372\) 0 0
\(373\) 484728.i 0.180396i −0.995924 0.0901978i \(-0.971250\pi\)
0.995924 0.0901978i \(-0.0287499\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 274335.i 0.0994094i
\(378\) 0 0
\(379\) 4.26934e6 1.52673 0.763366 0.645966i \(-0.223545\pi\)
0.763366 + 0.645966i \(0.223545\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.27791e6i 0.445147i 0.974916 + 0.222573i \(0.0714457\pi\)
−0.974916 + 0.222573i \(0.928554\pi\)
\(384\) 0 0
\(385\) 1.52469e6 369276.i 0.524240 0.126970i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.37341e6 1.13030 0.565151 0.824987i \(-0.308818\pi\)
0.565151 + 0.824987i \(0.308818\pi\)
\(390\) 0 0
\(391\) 2.59276e6 0.857672
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 209808. + 866270.i 0.0676597 + 0.279357i
\(396\) 0 0
\(397\) 1.82804e6i 0.582117i 0.956705 + 0.291058i \(0.0940074\pi\)
−0.956705 + 0.291058i \(0.905993\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.34118e6 −0.416511 −0.208256 0.978074i \(-0.566779\pi\)
−0.208256 + 0.978074i \(0.566779\pi\)
\(402\) 0 0
\(403\) 517252.i 0.158650i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.17305e6i 0.650254i
\(408\) 0 0
\(409\) 803937. 0.237637 0.118818 0.992916i \(-0.462089\pi\)
0.118818 + 0.992916i \(0.462089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.61278e6i 1.61921i
\(414\) 0 0
\(415\) −2.89611e6 + 701430.i −0.825458 + 0.199924i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.90295e6 1.08607 0.543035 0.839710i \(-0.317275\pi\)
0.543035 + 0.839710i \(0.317275\pi\)
\(420\) 0 0
\(421\) 3.04306e6 0.836769 0.418384 0.908270i \(-0.362597\pi\)
0.418384 + 0.908270i \(0.362597\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.86005e6 5.55804e6i −0.768072 1.49262i
\(426\) 0 0
\(427\) 525892.i 0.139581i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 772580. 0.200332 0.100166 0.994971i \(-0.468063\pi\)
0.100166 + 0.994971i \(0.468063\pi\)
\(432\) 0 0
\(433\) 4.83430e6i 1.23912i −0.784948 0.619561i \(-0.787311\pi\)
0.784948 0.619561i \(-0.212689\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 878171.i 0.219976i
\(438\) 0 0
\(439\) 4.98655e6 1.23492 0.617460 0.786602i \(-0.288162\pi\)
0.617460 + 0.786602i \(0.288162\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.63238e6i 1.84778i 0.382655 + 0.923891i \(0.375010\pi\)
−0.382655 + 0.923891i \(0.624990\pi\)
\(444\) 0 0
\(445\) −675447. 2.78883e6i −0.161693 0.667608i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.92156e6 0.449819 0.224910 0.974380i \(-0.427791\pi\)
0.224910 + 0.974380i \(0.427791\pi\)
\(450\) 0 0
\(451\) 2.02592e6 0.469009
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −162289. 670070.i −0.0367503 0.151737i
\(456\) 0 0
\(457\) 6.64057e6i 1.48735i 0.668539 + 0.743677i \(0.266920\pi\)
−0.668539 + 0.743677i \(0.733080\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.78658e6 −1.26815 −0.634074 0.773272i \(-0.718619\pi\)
−0.634074 + 0.773272i \(0.718619\pi\)
\(462\) 0 0
\(463\) 4.55738e6i 0.988013i −0.869458 0.494007i \(-0.835532\pi\)
0.869458 0.494007i \(-0.164468\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.02351e6i 0.217171i 0.994087 + 0.108585i \(0.0346321\pi\)
−0.994087 + 0.108585i \(0.965368\pi\)
\(468\) 0 0
\(469\) −3.19791e6 −0.671326
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.47057e6i 0.507743i
\(474\) 0 0
\(475\) −1.88251e6 + 968703.i −0.382828 + 0.196996i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.42421e6 −1.08018 −0.540092 0.841606i \(-0.681610\pi\)
−0.540092 + 0.841606i \(0.681610\pi\)
\(480\) 0 0
\(481\) −955009. −0.188211
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.39265e6 + 1.06389e6i −0.847953 + 0.205372i
\(486\) 0 0
\(487\) 8.76364e6i 1.67441i −0.546888 0.837206i \(-0.684188\pi\)
0.546888 0.837206i \(-0.315812\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.44420e6 0.457545 0.228772 0.973480i \(-0.426529\pi\)
0.228772 + 0.973480i \(0.426529\pi\)
\(492\) 0 0
\(493\) 6.53277e6i 1.21054i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.54550e6i 1.37024i
\(498\) 0 0
\(499\) 5.77888e6 1.03894 0.519472 0.854487i \(-0.326129\pi\)
0.519472 + 0.854487i \(0.326129\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.58777e6i 0.984733i −0.870388 0.492367i \(-0.836132\pi\)
0.870388 0.492367i \(-0.163868\pi\)
\(504\) 0 0
\(505\) −1.17733e6 4.86105e6i −0.205433 0.848206i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.32211e6 −1.59485 −0.797425 0.603418i \(-0.793805\pi\)
−0.797425 + 0.603418i \(0.793805\pi\)
\(510\) 0 0
\(511\) 1.95367e6 0.330978
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.40059e6 581416.i 0.398841 0.0965983i
\(516\) 0 0
\(517\) 1.81990e6i 0.299449i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.51696e6 −1.21324 −0.606622 0.794991i \(-0.707476\pi\)
−0.606622 + 0.794991i \(0.707476\pi\)
\(522\) 0 0
\(523\) 618460.i 0.0988683i −0.998777 0.0494342i \(-0.984258\pi\)
0.998777 0.0494342i \(-0.0157418\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.23174e7i 1.93193i
\(528\) 0 0
\(529\) 4.75613e6 0.738949
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 890350.i 0.135751i
\(534\) 0 0
\(535\) −3.06676e6 + 742762.i −0.463229 + 0.112193i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −908138. −0.134642
\(540\) 0 0
\(541\) −7.78044e6 −1.14291 −0.571454 0.820634i \(-0.693620\pi\)
−0.571454 + 0.820634i \(0.693620\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −692826. 2.86058e6i −0.0999155 0.412537i
\(546\) 0 0
\(547\) 6.38809e6i 0.912857i −0.889760 0.456428i \(-0.849128\pi\)
0.889760 0.456428i \(-0.150872\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.21265e6 −0.310481
\(552\) 0 0
\(553\) 2.34107e6i 0.325538i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.07422e6i 1.10271i 0.834270 + 0.551356i \(0.185890\pi\)
−0.834270 + 0.551356i \(0.814110\pi\)
\(558\) 0 0
\(559\) 1.08576e6 0.146962
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.02445e6i 0.668064i −0.942562 0.334032i \(-0.891591\pi\)
0.942562 0.334032i \(-0.108409\pi\)
\(564\) 0 0
\(565\) 4.25952e6 1.03165e6i 0.561357 0.135959i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.36647e6 1.08333 0.541666 0.840594i \(-0.317794\pi\)
0.541666 + 0.840594i \(0.317794\pi\)
\(570\) 0 0
\(571\) 831558. 0.106734 0.0533670 0.998575i \(-0.483005\pi\)
0.0533670 + 0.998575i \(0.483005\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.85343e6 + 3.60182e6i 0.233779 + 0.454310i
\(576\) 0 0
\(577\) 5.63868e6i 0.705080i −0.935797 0.352540i \(-0.885318\pi\)
0.935797 0.352540i \(-0.114682\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.82667e6 0.961915
\(582\) 0 0
\(583\) 2.82619e6i 0.344373i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.13074e6i 0.494803i −0.968913 0.247402i \(-0.920423\pi\)
0.968913 0.247402i \(-0.0795767\pi\)
\(588\) 0 0
\(589\) −4.17191e6 −0.495504
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.07710e6i 0.242561i −0.992618 0.121280i \(-0.961300\pi\)
0.992618 0.121280i \(-0.0387000\pi\)
\(594\) 0 0
\(595\) 3.86461e6 + 1.59565e7i 0.447521 + 1.84775i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.44039e7 1.64026 0.820128 0.572180i \(-0.193902\pi\)
0.820128 + 0.572180i \(0.193902\pi\)
\(600\) 0 0
\(601\) 1.16092e7 1.31104 0.655522 0.755176i \(-0.272449\pi\)
0.655522 + 0.755176i \(0.272449\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.63854e6 6.76533e6i −0.181999 0.751450i
\(606\) 0 0
\(607\) 1.75662e7i 1.93511i −0.252654 0.967557i \(-0.581304\pi\)
0.252654 0.967557i \(-0.418696\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 799810. 0.0866730
\(612\) 0 0
\(613\) 1.76294e7i 1.89490i −0.319906 0.947449i \(-0.603651\pi\)
0.319906 0.947449i \(-0.396349\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.44774e6i 0.681859i 0.940089 + 0.340930i \(0.110742\pi\)
−0.940089 + 0.340930i \(0.889258\pi\)
\(618\) 0 0
\(619\) −1.48717e7 −1.56003 −0.780017 0.625758i \(-0.784790\pi\)
−0.780017 + 0.625758i \(0.784790\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.53674e6i 0.777971i
\(624\) 0 0
\(625\) 5.67663e6 7.94628e6i 0.581287 0.813699i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.27418e7 2.29191
\(630\) 0 0
\(631\) 5.05549e6 0.505464 0.252732 0.967536i \(-0.418671\pi\)
0.252732 + 0.967536i \(0.418671\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.60875e7 + 3.89635e6i −1.58327 + 0.383464i
\(636\) 0 0
\(637\) 399108.i 0.0389710i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.66243e6 −0.640453 −0.320226 0.947341i \(-0.603759\pi\)
−0.320226 + 0.947341i \(0.603759\pi\)
\(642\) 0 0
\(643\) 3.29414e6i 0.314206i 0.987582 + 0.157103i \(0.0502155\pi\)
−0.987582 + 0.157103i \(0.949784\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.79219e6i 0.450063i 0.974351 + 0.225032i \(0.0722486\pi\)
−0.974351 + 0.225032i \(0.927751\pi\)
\(648\) 0 0
\(649\) 7.30627e6 0.680901
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.27962e7i 1.17435i −0.809459 0.587177i \(-0.800239\pi\)
0.809459 0.587177i \(-0.199761\pi\)
\(654\) 0 0
\(655\) −2.61900e6 1.08135e7i −0.238524 0.984832i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.06887e6 0.275274 0.137637 0.990483i \(-0.456049\pi\)
0.137637 + 0.990483i \(0.456049\pi\)
\(660\) 0 0
\(661\) −1.79240e6 −0.159563 −0.0797814 0.996812i \(-0.525422\pi\)
−0.0797814 + 0.996812i \(0.525422\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.40447e6 1.30895e6i 0.473913 0.114781i
\(666\) 0 0
\(667\) 4.23349e6i 0.368454i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −684564. −0.0586959
\(672\) 0 0
\(673\) 1.37614e7i 1.17119i 0.810605 + 0.585593i \(0.199138\pi\)
−0.810605 + 0.585593i \(0.800862\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.01097e7i 1.68629i 0.537684 + 0.843147i \(0.319300\pi\)
−0.537684 + 0.843147i \(0.680700\pi\)
\(678\) 0 0
\(679\) 1.18710e7 0.988129
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.08331e6i 0.498986i −0.968377 0.249493i \(-0.919736\pi\)
0.968377 0.249493i \(-0.0802639\pi\)
\(684\) 0 0
\(685\) −1.96256e7 + 4.75327e6i −1.59807 + 0.387049i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.24205e6 0.0996761
\(690\) 0 0
\(691\) −2.23063e7 −1.77719 −0.888593 0.458697i \(-0.848316\pi\)
−0.888593 + 0.458697i \(0.848316\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.38596e6 1.39802e7i −0.265901 1.09787i
\(696\) 0 0
\(697\) 2.12020e7i 1.65308i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.01349e7 1.54758 0.773792 0.633439i \(-0.218357\pi\)
0.773792 + 0.633439i \(0.218357\pi\)
\(702\) 0 0
\(703\) 7.70265e6i 0.587830i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.31369e7i 0.988424i
\(708\) 0 0
\(709\) 7.09436e6 0.530027 0.265013 0.964245i \(-0.414624\pi\)
0.265013 + 0.964245i \(0.414624\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.98215e6i 0.588026i
\(714\) 0 0
\(715\) 872243. 211255.i 0.0638076 0.0154540i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.57591e7 1.13686 0.568432 0.822730i \(-0.307550\pi\)
0.568432 + 0.822730i \(0.307550\pi\)
\(720\) 0 0
\(721\) −6.48753e6 −0.464774
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.07521e6 4.66992e6i 0.641227 0.329963i
\(726\) 0 0
\(727\) 1.93222e7i 1.35588i −0.735118 0.677939i \(-0.762873\pi\)
0.735118 0.677939i \(-0.237127\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.58554e7 −1.78961
\(732\) 0 0
\(733\) 7.53614e6i 0.518071i 0.965868 + 0.259035i \(0.0834046\pi\)
−0.965868 + 0.259035i \(0.916595\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.16278e6i 0.282302i
\(738\) 0 0
\(739\) −6.08996e6 −0.410207 −0.205104 0.978740i \(-0.565753\pi\)
−0.205104 + 0.978740i \(0.565753\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 5.29284e6i 0.351736i −0.984414 0.175868i \(-0.943727\pi\)
0.984414 0.175868i \(-0.0562731\pi\)
\(744\) 0 0
\(745\) −6.29429e6 2.59882e7i −0.415485 1.71548i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.28785e6 0.539806
\(750\) 0 0
\(751\) −1.58109e7 −1.02296 −0.511478 0.859296i \(-0.670902\pi\)
−0.511478 + 0.859296i \(0.670902\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.54600e6 + 1.05121e7i 0.162552 + 0.671154i
\(756\) 0 0
\(757\) 2.01450e7i 1.27770i 0.769332 + 0.638849i \(0.220589\pi\)
−0.769332 + 0.638849i \(0.779411\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.10320e7 1.31650 0.658248 0.752801i \(-0.271298\pi\)
0.658248 + 0.752801i \(0.271298\pi\)
\(762\) 0 0
\(763\) 7.73066e6i 0.480734i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.21095e6i 0.197081i
\(768\) 0 0
\(769\) 4.09332e6 0.249609 0.124804 0.992181i \(-0.460170\pi\)
0.124804 + 0.992181i \(0.460170\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.09247e7i 0.657599i −0.944400 0.328799i \(-0.893356\pi\)
0.944400 0.328799i \(-0.106644\pi\)
\(774\) 0 0
\(775\) 1.71111e7 8.80504e6i 1.02335 0.526595i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.18114e6 0.423984
\(780\) 0 0
\(781\) 9.82213e6 0.576206
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.50082e7 + 6.05692e6i −1.44847 + 0.350815i
\(786\) 0 0
\(787\) 1.38257e7i 0.795700i 0.917450 + 0.397850i \(0.130244\pi\)
−0.917450 + 0.397850i \(0.869756\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.15113e7 −0.654156
\(792\) 0 0
\(793\) 300852.i 0.0169891i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.38429e6i 0.0771937i 0.999255 + 0.0385969i \(0.0122888\pi\)
−0.999255 + 0.0385969i \(0.987711\pi\)
\(798\) 0 0
\(799\) −1.90460e7 −1.05545
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.54313e6i 0.139181i
\(804\) 0 0
\(805\) −2.50442e6 1.03404e7i −0.136213 0.562403i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0