Properties

Label 3024.2.k.d.1889.2
Level 3024
Weight 2
Character 3024.1889
Analytic conductor 24.147
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.2
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.d.1889.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} +5.19615i q^{11} -3.46410i q^{13} -6.00000 q^{17} +1.73205i q^{19} -5.19615i q^{23} +4.00000 q^{25} +10.3923i q^{29} +5.19615i q^{31} +(-6.00000 + 5.19615i) q^{35} +1.00000 q^{37} -3.00000 q^{41} -10.0000 q^{43} +6.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +15.5885i q^{55} -6.00000 q^{59} +13.8564i q^{61} -10.3923i q^{65} -2.00000 q^{67} -5.19615i q^{71} +3.46410i q^{73} +(-9.00000 - 10.3923i) q^{77} -14.0000 q^{79} -6.00000 q^{83} -18.0000 q^{85} +9.00000 q^{89} +(6.00000 + 6.92820i) q^{91} +5.19615i q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + 6q^{5} - 4q^{7} - 12q^{17} + 8q^{25} - 12q^{35} + 2q^{37} - 6q^{41} - 20q^{43} + 12q^{47} + 2q^{49} - 12q^{59} - 4q^{67} - 18q^{77} - 28q^{79} - 12q^{83} - 36q^{85} + 18q^{89} + 12q^{91} + O(q^{100}) \)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.19615i 1.56670i 0.621582 + 0.783349i \(0.286490\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615i 1.08347i −0.840548 0.541736i \(-0.817767\pi\)
0.840548 0.541736i \(-0.182233\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.3923i 1.92980i 0.262613 + 0.964901i \(0.415416\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i 0.884454 + 0.466628i \(0.154531\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 + 5.19615i −1.01419 + 0.878310i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.3923i 1.28901i
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.19615i 0.616670i −0.951278 0.308335i \(-0.900228\pi\)
0.951278 0.308335i \(-0.0997717\pi\)
\(72\) 0 0
\(73\) 3.46410i 0.405442i 0.979236 + 0.202721i \(0.0649785\pi\)
−0.979236 + 0.202721i \(0.935021\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.00000 10.3923i −1.02565 1.18431i
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −18.0000 −1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 6.00000 + 6.92820i 0.628971 + 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.19615i 0.533114i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 8.66025i 0.853320i −0.904412 0.426660i \(-0.859690\pi\)
0.904412 0.426660i \(-0.140310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.3923i 1.00466i −0.864675 0.502331i \(-0.832476\pi\)
0.864675 0.502331i \(-0.167524\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 15.5885i 1.45363i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 10.3923i 1.10004 0.952661i
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) −3.00000 3.46410i −0.260133 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.3923i 0.887875i 0.896058 + 0.443937i \(0.146419\pi\)
−0.896058 + 0.443937i \(0.853581\pi\)
\(138\) 0 0
\(139\) 17.3205i 1.46911i 0.678551 + 0.734553i \(0.262608\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.0000 1.50524
\(144\) 0 0
\(145\) 31.1769i 2.58910i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.5885i 1.25210i
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.00000 + 10.3923i 0.709299 + 0.819028i
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) −8.00000 + 6.92820i −0.604743 + 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 31.1769i 2.27988i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.19615i 0.375980i −0.982171 0.187990i \(-0.939803\pi\)
0.982171 0.187990i \(-0.0601973\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7846i 1.48084i −0.672143 0.740421i \(-0.734626\pi\)
0.672143 0.740421i \(-0.265374\pi\)
\(198\) 0 0
\(199\) 1.73205i 0.122782i −0.998114 0.0613909i \(-0.980446\pi\)
0.998114 0.0613909i \(-0.0195536\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −18.0000 20.7846i −1.26335 1.45879i
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −30.0000 −2.04598
\(216\) 0 0
\(217\) −9.00000 10.3923i −0.610960 0.705476i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.7846i 1.39812i
\(222\) 0 0
\(223\) 25.9808i 1.73980i 0.493228 + 0.869900i \(0.335817\pi\)
−0.493228 + 0.869900i \(0.664183\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i 0.726900 + 0.686743i \(0.240960\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.7846i 1.36165i −0.732448 0.680823i \(-0.761622\pi\)
0.732448 0.680823i \(-0.238378\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) 3.46410i 0.223142i 0.993756 + 0.111571i \(0.0355883\pi\)
−0.993756 + 0.111571i \(0.964412\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.00000 20.7846i 0.191663 1.32788i
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 0.561405 0.280702 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) 0 0
\(259\) −2.00000 + 1.73205i −0.124274 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.19615i 0.320408i −0.987084 0.160204i \(-0.948785\pi\)
0.987084 0.160204i \(-0.0512153\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 3.46410i 0.210429i 0.994450 + 0.105215i \(0.0335529\pi\)
−0.994450 + 0.105215i \(0.966447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.7846i 1.25336i
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.3923i 0.619953i −0.950744 0.309976i \(-0.899679\pi\)
0.950744 0.309976i \(-0.100321\pi\)
\(282\) 0 0
\(283\) 10.3923i 0.617758i −0.951101 0.308879i \(-0.900046\pi\)
0.951101 0.308879i \(-0.0999539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 5.19615i 0.354169 0.306719i
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −18.0000 −1.04800
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) 20.0000 17.3205i 1.15278 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 41.5692i 2.38025i
\(306\) 0 0
\(307\) 15.5885i 0.889680i 0.895610 + 0.444840i \(0.146740\pi\)
−0.895610 + 0.444840i \(0.853260\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) 3.46410i 0.195803i −0.995196 0.0979013i \(-0.968787\pi\)
0.995196 0.0979013i \(-0.0312129\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.1769i 1.75107i 0.483155 + 0.875535i \(0.339491\pi\)
−0.483155 + 0.875535i \(0.660509\pi\)
\(318\) 0 0
\(319\) −54.0000 −3.02342
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.3923i 0.578243i
\(324\) 0 0
\(325\) 13.8564i 0.768615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 + 10.3923i −0.661581 + 0.572946i
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −27.0000 −1.46213
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.19615i 0.278944i 0.990226 + 0.139472i \(0.0445405\pi\)
−0.990226 + 0.139472i \(0.955459\pi\)
\(348\) 0 0
\(349\) 27.7128i 1.48343i −0.670714 0.741716i \(-0.734012\pi\)
0.670714 0.741716i \(-0.265988\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) 15.5885i 0.827349i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923i 0.548485i 0.961661 + 0.274242i \(0.0884271\pi\)
−0.961661 + 0.274242i \(0.911573\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.3923i 0.543958i
\(366\) 0 0
\(367\) 1.73205i 0.0904123i −0.998978 0.0452062i \(-0.985606\pi\)
0.998978 0.0452062i \(-0.0143945\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −27.0000 31.1769i −1.37605 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.7846i 1.05382i −0.849921 0.526911i \(-0.823350\pi\)
0.849921 0.526911i \(-0.176650\pi\)
\(390\) 0 0
\(391\) 31.1769i 1.57668i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −42.0000 −2.11325
\(396\) 0 0
\(397\) 20.7846i 1.04315i 0.853206 + 0.521575i \(0.174655\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.3923i 0.518967i −0.965748 0.259483i \(-0.916448\pi\)
0.965748 0.259483i \(-0.0835523\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.19615i 0.257564i
\(408\) 0 0
\(409\) 17.3205i 0.856444i −0.903674 0.428222i \(-0.859140\pi\)
0.903674 0.428222i \(-0.140860\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000 10.3923i 0.590481 0.511372i
\(414\) 0 0
\(415\) −18.0000 −0.883585
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 5.00000 0.243685 0.121843 0.992549i \(-0.461120\pi\)
0.121843 + 0.992549i \(0.461120\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) −24.0000 27.7128i −1.16144 1.34112i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.3731i 1.75203i 0.482285 + 0.876014i \(0.339807\pi\)
−0.482285 + 0.876014i \(0.660193\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.00000 0.430528
\(438\) 0 0
\(439\) 31.1769i 1.48799i 0.668184 + 0.743996i \(0.267072\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.5885i 0.740630i −0.928906 0.370315i \(-0.879250\pi\)
0.928906 0.370315i \(-0.120750\pi\)
\(444\) 0 0
\(445\) 27.0000 1.27992
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.7846i 0.980886i 0.871473 + 0.490443i \(0.163165\pi\)
−0.871473 + 0.490443i \(0.836835\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 18.0000 + 20.7846i 0.843853 + 0.974398i
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) 4.00000 3.46410i 0.184703 0.159957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 51.9615i 2.38919i
\(474\) 0 0
\(475\) 6.92820i 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.7846i 0.943781i
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5885i 0.703497i −0.936094 0.351749i \(-0.885587\pi\)
0.936094 0.351749i \(-0.114413\pi\)
\(492\) 0 0
\(493\) 62.3538i 2.80828i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.00000 + 10.3923i 0.403705 + 0.466159i
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) −6.00000 6.92820i −0.265424 0.306486i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 25.9808i 1.14485i
\(516\) 0 0
\(517\) 31.1769i 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 0 0
\(523\) 12.1244i 0.530161i −0.964226 0.265081i \(-0.914601\pi\)
0.964226 0.265081i \(-0.0853985\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.1769i 1.35809i
\(528\) 0 0
\(529\) −4.00000 −0.173913
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.3923i 0.450141i
\(534\) 0 0
\(535\) 31.1769i 1.34790i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 + 5.19615i 1.55063 + 0.223814i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 28.0000 24.2487i 1.19068 1.03116i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.7846i 0.880672i −0.897833 0.440336i \(-0.854859\pi\)
0.897833 0.440336i \(-0.145141\pi\)
\(558\) 0 0
\(559\) 34.6410i 1.46516i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.5692i 1.74267i −0.490687 0.871336i \(-0.663254\pi\)
0.490687 0.871336i \(-0.336746\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.7846i 0.866778i
\(576\) 0 0
\(577\) 34.6410i 1.44212i 0.692870 + 0.721062i \(0.256346\pi\)
−0.692870 + 0.721062i \(0.743654\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 10.3923i 0.497844 0.431145i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 36.0000 31.1769i 1.47586 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25.9808i 1.06155i −0.847514 0.530773i \(-0.821902\pi\)
0.847514 0.530773i \(-0.178098\pi\)
\(600\) 0 0
\(601\) 10.3923i 0.423911i 0.977279 + 0.211955i \(0.0679832\pi\)
−0.977279 + 0.211955i \(0.932017\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −48.0000 −1.95148
\(606\) 0 0
\(607\) 17.3205i 0.703018i −0.936185 0.351509i \(-0.885669\pi\)
0.936185 0.351509i \(-0.114331\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.7846i 0.840855i
\(612\) 0 0
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.1769i 1.25514i 0.778562 + 0.627568i \(0.215949\pi\)
−0.778562 + 0.627568i \(0.784051\pi\)
\(618\) 0 0
\(619\) 32.9090i 1.32272i −0.750067 0.661361i \(-0.769979\pi\)
0.750067 0.661361i \(-0.230021\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18.0000 + 15.5885i −0.721155 + 0.624538i
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −26.0000 −1.03504 −0.517522 0.855670i \(-0.673145\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −24.0000 3.46410i −0.950915 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.7846i 0.820943i −0.911873 0.410471i \(-0.865364\pi\)
0.911873 0.410471i \(-0.134636\pi\)
\(642\) 0 0
\(643\) 46.7654i 1.84425i −0.386897 0.922123i \(-0.626453\pi\)
0.386897 0.922123i \(-0.373547\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 31.1769i 1.22380i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.3923i 0.406682i 0.979108 + 0.203341i \(0.0651801\pi\)
−0.979108 + 0.203341i \(0.934820\pi\)
\(654\) 0 0
\(655\) 54.0000 2.10995
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.3731i 1.41689i −0.705764 0.708447i \(-0.749396\pi\)
0.705764 0.708447i \(-0.250604\pi\)
\(660\) 0 0
\(661\) 24.2487i 0.943166i −0.881822 0.471583i \(-0.843683\pi\)
0.881822 0.471583i \(-0.156317\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9.00000 10.3923i −0.349005 0.402996i
\(666\) 0 0
\(667\) 54.0000 2.09089
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −72.0000 −2.77953
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) −12.0000 13.8564i −0.460518 0.531760i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.5885i 0.596476i −0.954492 0.298238i \(-0.903601\pi\)
0.954492 0.298238i \(-0.0963989\pi\)
\(684\) 0 0
\(685\) 31.1769i 1.19121i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.3923i 0.395342i −0.980268 0.197671i \(-0.936662\pi\)
0.980268 0.197671i \(-0.0633378\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 51.9615i 1.97101i
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.1769i 1.17754i −0.808302 0.588768i \(-0.799613\pi\)
0.808302 0.588768i \(-0.200387\pi\)
\(702\) 0 0
\(703\) 1.73205i 0.0653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −36.0000 + 31.1769i −1.35392 + 1.17253i
\(708\) 0 0
\(709\) −23.0000 −0.863783 −0.431892 0.901926i \(-0.642154\pi\)
−0.431892 + 0.901926i \(0.642154\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.0000 1.01116
\(714\) 0 0
\(715\) 54.0000 2.01949
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 15.0000 + 17.3205i 0.558629 + 0.645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 41.5692i 1.54384i
\(726\) 0 0
\(727\) 31.1769i 1.15629i −0.815935 0.578144i \(-0.803777\pi\)
0.815935 0.578144i \(-0.196223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 60.0000 2.21918
\(732\) 0 0
\(733\) 10.3923i 0.383849i −0.981410 0.191924i \(-0.938527\pi\)
0.981410 0.191924i \(-0.0614728\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.3923i 0.382805i
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.3731i 1.33440i 0.744879 + 0.667199i \(0.232507\pi\)
−0.744879 + 0.667199i \(0.767493\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.0000 + 20.7846i 0.657706 + 0.759453i
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 22.0000 19.0526i 0.796453 0.689749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.7846i 0.750489i
\(768\) 0 0
\(769\) 3.46410i 0.124919i 0.998048 + 0.0624593i \(0.0198944\pi\)
−0.998048 + 0.0624593i \(0.980106\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) 20.7846i 0.746605i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.19615i 0.186171i
\(780\) 0 0
\(781\) 27.0000 0.966136
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 31.1769i 1.11275i
\(786\) 0 0
\(787\) 38.1051i 1.35830i −0.733999 0.679150i \(-0.762348\pi\)
0.733999 0.679150i \(-0.237652\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −45.0000 −1.59398 −0.796991 0.603991i \(-0.793576\pi\)
−0.796991 + 0.603991i \(0.793576\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 27.0000 + 31.1769i 0.951625 + 1.09884i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.1769i 1.09612i −0.836438 0.548061i \(-0.815366\pi\)
0.836438 0.548061i \(-0.184634\pi\)
\(810\) 0 0
\(811\) 5.19615i 0.182462i −0.995830 0.0912308i \(-0.970920\pi\)
0.995830 0.0912308i \(-0.0290801\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 30.0000 1.05085
\(816\) 0 0
\(817\) 17.3205i 0.605968i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0