Properties

Label 3024.2.k.k.1889.5
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.5
Root \(-2.32849 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.k.1889.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.10598 q^{5} +(2.64465 - 0.0763047i) q^{7} +O(q^{10})\) \(q-1.10598 q^{5} +(2.64465 - 0.0763047i) q^{7} +3.42810i q^{11} +4.98427i q^{13} +6.38903 q^{17} +4.65697i q^{19} -8.98172i q^{23} -3.77681 q^{25} -1.51249i q^{29} +6.71632i q^{31} +(-2.92492 + 0.0843913i) q^{35} -2.83122 q^{37} +10.1147 q^{41} -10.8973 q^{43} -12.8935 q^{47} +(6.98836 - 0.403599i) q^{49} -3.02497i q^{53} -3.79140i q^{55} -9.54791 q^{59} +9.16134i q^{61} -5.51249i q^{65} +5.07938 q^{67} +8.94058i q^{71} +7.79378i q^{73} +(0.261580 + 9.06612i) q^{77} +2.05440 q^{79} -5.73444 q^{83} -7.06612 q^{85} -6.47681 q^{89} +(0.380323 + 13.1816i) q^{91} -5.15051i q^{95} -8.04062i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{7} + O(q^{10}) \) \( 16q + 2q^{7} - 12q^{25} - 8q^{37} - 8q^{43} + 2q^{49} + 28q^{67} + 44q^{79} + 16q^{85} - 18q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.10598 −0.494608 −0.247304 0.968938i \(-0.579545\pi\)
−0.247304 + 0.968938i \(0.579545\pi\)
\(6\) 0 0
\(7\) 2.64465 0.0763047i 0.999584 0.0288405i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.42810i 1.03361i 0.856103 + 0.516805i \(0.172879\pi\)
−0.856103 + 0.516805i \(0.827121\pi\)
\(12\) 0 0
\(13\) 4.98427i 1.38239i 0.722670 + 0.691194i \(0.242915\pi\)
−0.722670 + 0.691194i \(0.757085\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.38903 1.54957 0.774783 0.632227i \(-0.217859\pi\)
0.774783 + 0.632227i \(0.217859\pi\)
\(18\) 0 0
\(19\) 4.65697i 1.06838i 0.845363 + 0.534192i \(0.179384\pi\)
−0.845363 + 0.534192i \(0.820616\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.98172i 1.87282i −0.350909 0.936410i \(-0.614127\pi\)
0.350909 0.936410i \(-0.385873\pi\)
\(24\) 0 0
\(25\) −3.77681 −0.755363
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.51249i 0.280862i −0.990090 0.140431i \(-0.955151\pi\)
0.990090 0.140431i \(-0.0448488\pi\)
\(30\) 0 0
\(31\) 6.71632i 1.20629i 0.797633 + 0.603143i \(0.206085\pi\)
−0.797633 + 0.603143i \(0.793915\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.92492 + 0.0843913i −0.494402 + 0.0142647i
\(36\) 0 0
\(37\) −2.83122 −0.465449 −0.232725 0.972543i \(-0.574764\pi\)
−0.232725 + 0.972543i \(0.574764\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.1147 1.57965 0.789826 0.613331i \(-0.210171\pi\)
0.789826 + 0.613331i \(0.210171\pi\)
\(42\) 0 0
\(43\) −10.8973 −1.66183 −0.830914 0.556401i \(-0.812182\pi\)
−0.830914 + 0.556401i \(0.812182\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.8935 −1.88070 −0.940352 0.340202i \(-0.889504\pi\)
−0.940352 + 0.340202i \(0.889504\pi\)
\(48\) 0 0
\(49\) 6.98836 0.403599i 0.998336 0.0576570i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.02497i 0.415512i −0.978181 0.207756i \(-0.933384\pi\)
0.978181 0.207756i \(-0.0666160\pi\)
\(54\) 0 0
\(55\) 3.79140i 0.511232i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.54791 −1.24303 −0.621516 0.783402i \(-0.713483\pi\)
−0.621516 + 0.783402i \(0.713483\pi\)
\(60\) 0 0
\(61\) 9.16134i 1.17299i 0.809953 + 0.586495i \(0.199492\pi\)
−0.809953 + 0.586495i \(0.800508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.51249i 0.683740i
\(66\) 0 0
\(67\) 5.07938 0.620545 0.310272 0.950648i \(-0.399580\pi\)
0.310272 + 0.950648i \(0.399580\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.94058i 1.06105i 0.847669 + 0.530526i \(0.178006\pi\)
−0.847669 + 0.530526i \(0.821994\pi\)
\(72\) 0 0
\(73\) 7.79378i 0.912193i 0.889930 + 0.456097i \(0.150753\pi\)
−0.889930 + 0.456097i \(0.849247\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.261580 + 9.06612i 0.0298098 + 1.03318i
\(78\) 0 0
\(79\) 2.05440 0.231138 0.115569 0.993299i \(-0.463131\pi\)
0.115569 + 0.993299i \(0.463131\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.73444 −0.629436 −0.314718 0.949185i \(-0.601910\pi\)
−0.314718 + 0.949185i \(0.601910\pi\)
\(84\) 0 0
\(85\) −7.06612 −0.766428
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.47681 −0.686541 −0.343270 0.939237i \(-0.611535\pi\)
−0.343270 + 0.939237i \(0.611535\pi\)
\(90\) 0 0
\(91\) 0.380323 + 13.1816i 0.0398687 + 1.38181i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.15051i 0.528431i
\(96\) 0 0
\(97\) 8.04062i 0.816401i −0.912892 0.408201i \(-0.866156\pi\)
0.912892 0.408201i \(-0.133844\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3172 1.32511 0.662557 0.749012i \(-0.269471\pi\)
0.662557 + 0.749012i \(0.269471\pi\)
\(102\) 0 0
\(103\) 6.86893i 0.676816i −0.941000 0.338408i \(-0.890112\pi\)
0.941000 0.338408i \(-0.109888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.09109i 0.588848i 0.955675 + 0.294424i \(0.0951278\pi\)
−0.955675 + 0.294424i \(0.904872\pi\)
\(108\) 0 0
\(109\) 14.6580 1.40398 0.701990 0.712187i \(-0.252295\pi\)
0.701990 + 0.712187i \(0.252295\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.1322i 1.70574i 0.522126 + 0.852868i \(0.325139\pi\)
−0.522126 + 0.852868i \(0.674861\pi\)
\(114\) 0 0
\(115\) 9.93358i 0.926311i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.8967 0.487513i 1.54892 0.0446902i
\(120\) 0 0
\(121\) −0.751840 −0.0683491
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.70696 0.868217
\(126\) 0 0
\(127\) 7.23490 0.641993 0.320997 0.947080i \(-0.395982\pi\)
0.320997 + 0.947080i \(0.395982\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.5823 1.27406 0.637029 0.770840i \(-0.280163\pi\)
0.637029 + 0.770840i \(0.280163\pi\)
\(132\) 0 0
\(133\) 0.355349 + 12.3161i 0.0308127 + 1.06794i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.17492i 0.442123i 0.975260 + 0.221062i \(0.0709522\pi\)
−0.975260 + 0.221062i \(0.929048\pi\)
\(138\) 0 0
\(139\) 11.2419i 0.953523i 0.879033 + 0.476761i \(0.158189\pi\)
−0.879033 + 0.476761i \(0.841811\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17.0865 −1.42885
\(144\) 0 0
\(145\) 1.67278i 0.138917i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.66243i 0.627731i −0.949467 0.313866i \(-0.898376\pi\)
0.949467 0.313866i \(-0.101624\pi\)
\(150\) 0 0
\(151\) −9.28930 −0.755953 −0.377976 0.925815i \(-0.623380\pi\)
−0.377976 + 0.925815i \(0.623380\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.42810i 0.596639i
\(156\) 0 0
\(157\) 13.8404i 1.10458i 0.833651 + 0.552292i \(0.186246\pi\)
−0.833651 + 0.552292i \(0.813754\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.685348 23.7535i −0.0540130 1.87204i
\(162\) 0 0
\(163\) 15.6580 1.22643 0.613214 0.789917i \(-0.289876\pi\)
0.613214 + 0.789917i \(0.289876\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.81347 −0.295095 −0.147548 0.989055i \(-0.547138\pi\)
−0.147548 + 0.989055i \(0.547138\pi\)
\(168\) 0 0
\(169\) −11.8429 −0.910995
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.36140 −0.483649 −0.241824 0.970320i \(-0.577746\pi\)
−0.241824 + 0.970320i \(0.577746\pi\)
\(174\) 0 0
\(175\) −9.98836 + 0.288189i −0.755049 + 0.0217850i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.04114i 0.302049i 0.988530 + 0.151025i \(0.0482573\pi\)
−0.988530 + 0.151025i \(0.951743\pi\)
\(180\) 0 0
\(181\) 2.84357i 0.211361i 0.994400 + 0.105681i \(0.0337021\pi\)
−0.994400 + 0.105681i \(0.966298\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.13126 0.230215
\(186\) 0 0
\(187\) 21.9022i 1.60165i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.1505i 0.951537i 0.879570 + 0.475769i \(0.157830\pi\)
−0.879570 + 0.475769i \(0.842170\pi\)
\(192\) 0 0
\(193\) −9.97671 −0.718139 −0.359070 0.933311i \(-0.616906\pi\)
−0.359070 + 0.933311i \(0.616906\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.4448i 1.10040i −0.835034 0.550199i \(-0.814552\pi\)
0.835034 0.550199i \(-0.185448\pi\)
\(198\) 0 0
\(199\) 2.25521i 0.159868i −0.996800 0.0799338i \(-0.974529\pi\)
0.996800 0.0799338i \(-0.0254709\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.115410 4.00000i −0.00810019 0.280745i
\(204\) 0 0
\(205\) −11.1866 −0.781308
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −15.9646 −1.10429
\(210\) 0 0
\(211\) −5.81796 −0.400525 −0.200262 0.979742i \(-0.564179\pi\)
−0.200262 + 0.979742i \(0.564179\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0522 0.821953
\(216\) 0 0
\(217\) 0.512487 + 17.7623i 0.0347899 + 1.20578i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 31.8446i 2.14210i
\(222\) 0 0
\(223\) 1.52241i 0.101948i −0.998700 0.0509741i \(-0.983767\pi\)
0.998700 0.0509741i \(-0.0162326\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.2803 −1.21331 −0.606654 0.794966i \(-0.707488\pi\)
−0.606654 + 0.794966i \(0.707488\pi\)
\(228\) 0 0
\(229\) 10.2738i 0.678909i −0.940622 0.339454i \(-0.889758\pi\)
0.940622 0.339454i \(-0.110242\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 14.2599 0.930212
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.1161i 1.36588i −0.730472 0.682942i \(-0.760700\pi\)
0.730472 0.682942i \(-0.239300\pi\)
\(240\) 0 0
\(241\) 22.9594i 1.47894i 0.673188 + 0.739471i \(0.264924\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.72896 + 0.446371i −0.493785 + 0.0285176i
\(246\) 0 0
\(247\) −23.2116 −1.47692
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.66332 0.609943 0.304972 0.952361i \(-0.401353\pi\)
0.304972 + 0.952361i \(0.401353\pi\)
\(252\) 0 0
\(253\) 30.7902 1.93576
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.34471 −0.582907 −0.291453 0.956585i \(-0.594139\pi\)
−0.291453 + 0.956585i \(0.594139\pi\)
\(258\) 0 0
\(259\) −7.48758 + 0.216035i −0.465256 + 0.0134238i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.8129i 1.03673i 0.855159 + 0.518365i \(0.173459\pi\)
−0.855159 + 0.518365i \(0.826541\pi\)
\(264\) 0 0
\(265\) 3.34555i 0.205516i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.93981 −0.423127 −0.211564 0.977364i \(-0.567856\pi\)
−0.211564 + 0.977364i \(0.567856\pi\)
\(270\) 0 0
\(271\) 2.35168i 0.142855i −0.997446 0.0714273i \(-0.977245\pi\)
0.997446 0.0714273i \(-0.0227554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.9473i 0.780750i
\(276\) 0 0
\(277\) −1.87948 −0.112927 −0.0564635 0.998405i \(-0.517982\pi\)
−0.0564635 + 0.998405i \(0.517982\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.3072i 0.913148i −0.889685 0.456574i \(-0.849076\pi\)
0.889685 0.456574i \(-0.150924\pi\)
\(282\) 0 0
\(283\) 30.3294i 1.80289i 0.432889 + 0.901447i \(0.357494\pi\)
−0.432889 + 0.901447i \(0.642506\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 26.7499 0.771800i 1.57899 0.0455579i
\(288\) 0 0
\(289\) 23.8196 1.40116
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.1585 −0.827148 −0.413574 0.910471i \(-0.635720\pi\)
−0.413574 + 0.910471i \(0.635720\pi\)
\(294\) 0 0
\(295\) 10.5598 0.614813
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 44.7673 2.58896
\(300\) 0 0
\(301\) −28.8196 + 0.831518i −1.66114 + 0.0479279i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.1322i 0.580170i
\(306\) 0 0
\(307\) 0.267126i 0.0152457i 0.999971 + 0.00762283i \(0.00242645\pi\)
−0.999971 + 0.00762283i \(0.997574\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.962946 −0.0546037 −0.0273018 0.999627i \(-0.508692\pi\)
−0.0273018 + 0.999627i \(0.508692\pi\)
\(312\) 0 0
\(313\) 6.67306i 0.377184i −0.982056 0.188592i \(-0.939608\pi\)
0.982056 0.188592i \(-0.0603923\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.1499i 0.794740i −0.917658 0.397370i \(-0.869923\pi\)
0.917658 0.397370i \(-0.130077\pi\)
\(318\) 0 0
\(319\) 5.18495 0.290301
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 29.7535i 1.65553i
\(324\) 0 0
\(325\) 18.8247i 1.04420i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −34.0987 + 0.983832i −1.87992 + 0.0542404i
\(330\) 0 0
\(331\) −12.5054 −0.687357 −0.343679 0.939087i \(-0.611673\pi\)
−0.343679 + 0.939087i \(0.611673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.61768 −0.306926
\(336\) 0 0
\(337\) 17.1322 0.933252 0.466626 0.884455i \(-0.345469\pi\)
0.466626 + 0.884455i \(0.345469\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −23.0242 −1.24683
\(342\) 0 0
\(343\) 18.4510 1.60062i 0.996258 0.0864255i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.83178i 0.366749i −0.983043 0.183375i \(-0.941298\pi\)
0.983043 0.183375i \(-0.0587021\pi\)
\(348\) 0 0
\(349\) 32.1708i 1.72206i −0.508552 0.861031i \(-0.669819\pi\)
0.508552 0.861031i \(-0.330181\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.7317 0.996988 0.498494 0.866893i \(-0.333887\pi\)
0.498494 + 0.866893i \(0.333887\pi\)
\(354\) 0 0
\(355\) 9.88808i 0.524805i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.9589i 0.842276i −0.906996 0.421138i \(-0.861631\pi\)
0.906996 0.421138i \(-0.138369\pi\)
\(360\) 0 0
\(361\) −2.68741 −0.141443
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.61974i 0.451178i
\(366\) 0 0
\(367\) 25.9175i 1.35288i −0.736498 0.676440i \(-0.763522\pi\)
0.736498 0.676440i \(-0.236478\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.230820 8.00000i −0.0119836 0.415339i
\(372\) 0 0
\(373\) −18.2848 −0.946753 −0.473377 0.880860i \(-0.656965\pi\)
−0.473377 + 0.880860i \(0.656965\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.53864 0.388260
\(378\) 0 0
\(379\) 22.3454 1.14781 0.573903 0.818923i \(-0.305429\pi\)
0.573903 + 0.818923i \(0.305429\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.7042 1.16013 0.580066 0.814570i \(-0.303027\pi\)
0.580066 + 0.814570i \(0.303027\pi\)
\(384\) 0 0
\(385\) −0.289301 10.0269i −0.0147442 0.511019i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.7947i 0.902225i −0.892467 0.451113i \(-0.851027\pi\)
0.892467 0.451113i \(-0.148973\pi\)
\(390\) 0 0
\(391\) 57.3845i 2.90206i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.27212 −0.114323
\(396\) 0 0
\(397\) 17.8124i 0.893979i 0.894539 + 0.446989i \(0.147504\pi\)
−0.894539 + 0.446989i \(0.852496\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.0195i 1.44917i 0.689187 + 0.724583i \(0.257968\pi\)
−0.689187 + 0.724583i \(0.742032\pi\)
\(402\) 0 0
\(403\) −33.4759 −1.66756
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.70568i 0.481093i
\(408\) 0 0
\(409\) 21.6502i 1.07053i 0.844683 + 0.535266i \(0.179789\pi\)
−0.844683 + 0.535266i \(0.820211\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −25.2509 + 0.728551i −1.24251 + 0.0358496i
\(414\) 0 0
\(415\) 6.34216 0.311324
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.66535 0.325624 0.162812 0.986657i \(-0.447944\pi\)
0.162812 + 0.986657i \(0.447944\pi\)
\(420\) 0 0
\(421\) −3.16878 −0.154437 −0.0772185 0.997014i \(-0.524604\pi\)
−0.0772185 + 0.997014i \(0.524604\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.1302 −1.17048
\(426\) 0 0
\(427\) 0.699054 + 24.2285i 0.0338296 + 1.17250i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.84949i 0.137255i −0.997642 0.0686276i \(-0.978138\pi\)
0.997642 0.0686276i \(-0.0218620\pi\)
\(432\) 0 0
\(433\) 32.2375i 1.54923i −0.632431 0.774617i \(-0.717943\pi\)
0.632431 0.774617i \(-0.282057\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 41.8277 2.00089
\(438\) 0 0
\(439\) 37.6052i 1.79480i −0.441221 0.897398i \(-0.645455\pi\)
0.441221 0.897398i \(-0.354545\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.7407i 0.605328i 0.953097 + 0.302664i \(0.0978760\pi\)
−0.953097 + 0.302664i \(0.902124\pi\)
\(444\) 0 0
\(445\) 7.16321 0.339569
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.18218i 0.386141i 0.981185 + 0.193070i \(0.0618446\pi\)
−0.981185 + 0.193070i \(0.938155\pi\)
\(450\) 0 0
\(451\) 34.6742i 1.63274i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.420629 14.5786i −0.0197194 0.683455i
\(456\) 0 0
\(457\) 9.37313 0.438457 0.219228 0.975674i \(-0.429646\pi\)
0.219228 + 0.975674i \(0.429646\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.7688 0.827574 0.413787 0.910374i \(-0.364206\pi\)
0.413787 + 0.910374i \(0.364206\pi\)
\(462\) 0 0
\(463\) −0.231992 −0.0107816 −0.00539079 0.999985i \(-0.501716\pi\)
−0.00539079 + 0.999985i \(0.501716\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.57569 −0.304287 −0.152143 0.988358i \(-0.548618\pi\)
−0.152143 + 0.988358i \(0.548618\pi\)
\(468\) 0 0
\(469\) 13.4332 0.387581i 0.620287 0.0178968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 37.3571i 1.71768i
\(474\) 0 0
\(475\) 17.5885i 0.807017i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.3544 0.747253 0.373626 0.927579i \(-0.378114\pi\)
0.373626 + 0.927579i \(0.378114\pi\)
\(480\) 0 0
\(481\) 14.1115i 0.643431i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.89274i 0.403799i
\(486\) 0 0
\(487\) 13.1617 0.596412 0.298206 0.954502i \(-0.403612\pi\)
0.298206 + 0.954502i \(0.403612\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.4692i 0.698117i −0.937101 0.349058i \(-0.886501\pi\)
0.937101 0.349058i \(-0.113499\pi\)
\(492\) 0 0
\(493\) 9.66332i 0.435214i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.682209 + 23.6447i 0.0306013 + 1.06061i
\(498\) 0 0
\(499\) 8.94728 0.400535 0.200268 0.979741i \(-0.435819\pi\)
0.200268 + 0.979741i \(0.435819\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.5654 −1.22908 −0.614539 0.788886i \(-0.710658\pi\)
−0.614539 + 0.788886i \(0.710658\pi\)
\(504\) 0 0
\(505\) −14.7286 −0.655412
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.68961 0.119215 0.0596075 0.998222i \(-0.481015\pi\)
0.0596075 + 0.998222i \(0.481015\pi\)
\(510\) 0 0
\(511\) 0.594703 + 20.6118i 0.0263081 + 0.911814i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.59688i 0.334758i
\(516\) 0 0
\(517\) 44.2000i 1.94391i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.5721 1.42701 0.713505 0.700651i \(-0.247107\pi\)
0.713505 + 0.700651i \(0.247107\pi\)
\(522\) 0 0
\(523\) 9.00962i 0.393963i −0.980407 0.196982i \(-0.936886\pi\)
0.980407 0.196982i \(-0.0631139\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.9107i 1.86922i
\(528\) 0 0
\(529\) −57.6714 −2.50745
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 50.4144i 2.18369i
\(534\) 0 0
\(535\) 6.73661i 0.291249i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.38358 + 23.9568i 0.0595948 + 1.03189i
\(540\) 0 0
\(541\) 21.9884 0.945356 0.472678 0.881235i \(-0.343287\pi\)
0.472678 + 0.881235i \(0.343287\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.2114 −0.694420
\(546\) 0 0
\(547\) −1.28930 −0.0551266 −0.0275633 0.999620i \(-0.508775\pi\)
−0.0275633 + 0.999620i \(0.508775\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.04361 0.300068
\(552\) 0 0
\(553\) 5.43318 0.156761i 0.231042 0.00666614i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.44482i 0.315447i 0.987483 + 0.157724i \(0.0504155\pi\)
−0.987483 + 0.157724i \(0.949584\pi\)
\(558\) 0 0
\(559\) 54.3152i 2.29729i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −29.2542 −1.23292 −0.616458 0.787388i \(-0.711433\pi\)
−0.616458 + 0.787388i \(0.711433\pi\)
\(564\) 0 0
\(565\) 20.0538i 0.843671i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.4698i 1.10967i −0.831960 0.554836i \(-0.812781\pi\)
0.831960 0.554836i \(-0.187219\pi\)
\(570\) 0 0
\(571\) −29.6626 −1.24134 −0.620670 0.784072i \(-0.713139\pi\)
−0.620670 + 0.784072i \(0.713139\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 33.9223i 1.41466i
\(576\) 0 0
\(577\) 1.65473i 0.0688875i 0.999407 + 0.0344437i \(0.0109660\pi\)
−0.999407 + 0.0344437i \(0.989034\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.1656 + 0.437565i −0.629174 + 0.0181532i
\(582\) 0 0
\(583\) 10.3699 0.429477
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.45826 0.390384 0.195192 0.980765i \(-0.437467\pi\)
0.195192 + 0.980765i \(0.437467\pi\)
\(588\) 0 0
\(589\) −31.2777 −1.28878
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.17049 0.130196 0.0650981 0.997879i \(-0.479264\pi\)
0.0650981 + 0.997879i \(0.479264\pi\)
\(594\) 0 0
\(595\) −18.6874 + 0.539178i −0.766109 + 0.0221042i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.2827i 0.951307i 0.879633 + 0.475653i \(0.157788\pi\)
−0.879633 + 0.475653i \(0.842212\pi\)
\(600\) 0 0
\(601\) 23.0358i 0.939649i 0.882760 + 0.469825i \(0.155683\pi\)
−0.882760 + 0.469825i \(0.844317\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.831518 0.0338060
\(606\) 0 0
\(607\) 0.737290i 0.0299257i −0.999888 0.0149628i \(-0.995237\pi\)
0.999888 0.0149628i \(-0.00476300\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 64.2645i 2.59986i
\(612\) 0 0
\(613\) 36.4494 1.47218 0.736089 0.676885i \(-0.236670\pi\)
0.736089 + 0.676885i \(0.236670\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.19990i 0.330115i −0.986284 0.165058i \(-0.947219\pi\)
0.986284 0.165058i \(-0.0527810\pi\)
\(618\) 0 0
\(619\) 18.3236i 0.736487i 0.929729 + 0.368243i \(0.120041\pi\)
−0.929729 + 0.368243i \(0.879959\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.1289 + 0.494212i −0.686255 + 0.0198002i
\(624\) 0 0
\(625\) 8.14840 0.325936
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.0887 −0.721244
\(630\) 0 0
\(631\) 6.47425 0.257736 0.128868 0.991662i \(-0.458866\pi\)
0.128868 + 0.991662i \(0.458866\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00163 −0.317535
\(636\) 0 0
\(637\) 2.01164 + 34.8318i 0.0797043 + 1.38009i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.9519i 1.22253i 0.791428 + 0.611263i \(0.209338\pi\)
−0.791428 + 0.611263i \(0.790662\pi\)
\(642\) 0 0
\(643\) 11.2017i 0.441754i −0.975302 0.220877i \(-0.929108\pi\)
0.975302 0.220877i \(-0.0708919\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.3689 −1.27255 −0.636276 0.771461i \(-0.719526\pi\)
−0.636276 + 0.771461i \(0.719526\pi\)
\(648\) 0 0
\(649\) 32.7311i 1.28481i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 44.9373i 1.75853i −0.476332 0.879265i \(-0.658034\pi\)
0.476332 0.879265i \(-0.341966\pi\)
\(654\) 0 0
\(655\) −16.1276 −0.630159
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.5969i 0.451750i −0.974156 0.225875i \(-0.927476\pi\)
0.974156 0.225875i \(-0.0725241\pi\)
\(660\) 0 0
\(661\) 13.9488i 0.542547i −0.962502 0.271274i \(-0.912555\pi\)
0.962502 0.271274i \(-0.0874448\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.393008 13.6213i −0.0152402 0.528211i
\(666\) 0 0
\(667\) −13.5847 −0.526003
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −31.4059 −1.21241
\(672\) 0 0
\(673\) −16.9194 −0.652195 −0.326097 0.945336i \(-0.605734\pi\)
−0.326097 + 0.945336i \(0.605734\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.6171 −0.753947 −0.376973 0.926224i \(-0.623035\pi\)
−0.376973 + 0.926224i \(0.623035\pi\)
\(678\) 0 0
\(679\) −0.613537 21.2646i −0.0235454 0.816062i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 41.4738i 1.58695i −0.608602 0.793476i \(-0.708269\pi\)
0.608602 0.793476i \(-0.291731\pi\)
\(684\) 0 0
\(685\) 5.72335i 0.218678i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.0773 0.574399
\(690\) 0 0
\(691\) 20.2864i 0.771733i 0.922555 + 0.385866i \(0.126097\pi\)
−0.922555 + 0.385866i \(0.873903\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.4332i 0.471620i
\(696\) 0 0
\(697\) 64.6231 2.44777
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.4302i 0.507252i −0.967302 0.253626i \(-0.918377\pi\)
0.967302 0.253626i \(-0.0816232\pi\)
\(702\) 0 0
\(703\) 13.1849i 0.497278i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 35.2194 1.01617i 1.32456 0.0382169i
\(708\) 0 0
\(709\) 37.0046 1.38974 0.694868 0.719137i \(-0.255463\pi\)
0.694868 + 0.719137i \(0.255463\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 60.3241 2.25916
\(714\) 0 0
\(715\) 18.8973 0.706720
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.3967 1.09631 0.548156 0.836376i \(-0.315330\pi\)
0.548156 + 0.836376i \(0.315330\pi\)
\(720\) 0 0
\(721\) −0.524132 18.1659i −0.0195197 0.676534i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.71238i 0.212153i
\(726\) 0 0
\(727\) 19.8248i 0.735261i −0.929972 0.367631i \(-0.880169\pi\)
0.929972 0.367631i \(-0.119831\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −69.6233 −2.57511
\(732\) 0 0
\(733\) 11.8040i 0.435991i 0.975950 + 0.217996i \(0.0699518\pi\)
−0.975950 + 0.217996i \(0.930048\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.4126i 0.641401i
\(738\) 0 0
\(739\) 43.8446 1.61285 0.806425 0.591336i \(-0.201399\pi\)
0.806425 + 0.591336i \(0.201399\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.4603i 0.934049i 0.884244 + 0.467024i \(0.154674\pi\)
−0.884244 + 0.467024i \(0.845326\pi\)
\(744\) 0 0
\(745\) 8.47448i 0.310481i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.464779 + 16.1088i 0.0169827 + 0.588603i
\(750\) 0 0
\(751\) 22.4277 0.818397 0.409199 0.912445i \(-0.365808\pi\)
0.409199 + 0.912445i \(0.365808\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.2738 0.373900
\(756\) 0 0
\(757\) −9.44806 −0.343395 −0.171698 0.985150i \(-0.554925\pi\)
−0.171698 + 0.985150i \(0.554925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.6274 0.928991 0.464496 0.885575i \(-0.346236\pi\)
0.464496 + 0.885575i \(0.346236\pi\)
\(762\) 0 0
\(763\) 38.7652 1.11847i 1.40340 0.0404915i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 47.5893i 1.71835i
\(768\) 0 0
\(769\) 33.4382i 1.20581i −0.797812 0.602906i \(-0.794009\pi\)
0.797812 0.602906i \(-0.205991\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 51.5205 1.85306 0.926531 0.376218i \(-0.122776\pi\)
0.926531 + 0.376218i \(0.122776\pi\)
\(774\) 0 0
\(775\) 25.3663i 0.911184i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.1039i 1.68767i
\(780\) 0 0
\(781\) −30.6492 −1.09671
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.3072i 0.546336i
\(786\) 0 0
\(787\) 14.7759i 0.526703i −0.964700 0.263352i \(-0.915172\pi\)
0.964700 0.263352i \(-0.0848279\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.38358 + 47.9534i 0.0491943 + 1.70503i
\(792\) 0 0
\(793\) −45.6626 −1.62153
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.9822 0.849493 0.424747 0.905312i \(-0.360363\pi\)
0.424747 + 0.905312i \(0.360363\pi\)
\(798\) 0 0
\(799\) −82.3766 −2.91428
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −26.7178 −0.942852
\(804\) 0