Properties

Label 3024.2.b.h.1567.1
Level $3024$
Weight $2$
Character 3024.1567
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.b (of order \(2\), degree \(1\), 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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.h.1567.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{5} +(-0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q-1.73205i q^{5} +(-0.500000 - 2.59808i) q^{7} -5.19615i q^{11} -3.46410i q^{13} -3.46410i q^{17} +8.00000 q^{19} +6.92820i q^{23} +2.00000 q^{25} +6.00000 q^{29} -5.00000 q^{31} +(-4.50000 + 0.866025i) q^{35} +4.00000 q^{37} -6.92820i q^{41} -3.46410i q^{43} -6.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +3.00000 q^{53} -9.00000 q^{55} -12.0000 q^{59} +13.8564i q^{61} -6.00000 q^{65} +3.46410i q^{67} -3.46410i q^{71} +1.73205i q^{73} +(-13.5000 + 2.59808i) q^{77} -3.46410i q^{79} -15.0000 q^{83} -6.00000 q^{85} -10.3923i q^{89} +(-9.00000 + 1.73205i) q^{91} -13.8564i q^{95} -8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} + 16q^{19} + 4q^{25} + 12q^{29} - 10q^{31} - 9q^{35} + 8q^{37} - 12q^{47} - 13q^{49} + 6q^{53} - 18q^{55} - 24q^{59} - 12q^{65} - 27q^{77} - 30q^{83} - 12q^{85} - 18q^{91} + O(q^{100}) \)

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
\(5\) 1.73205i 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.19615i 1.56670i −0.621582 0.783349i \(-0.713510\pi\)
0.621582 0.783349i \(-0.286490\pi\)
\(12\) 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
\(17\) 3.46410i 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820i 1.44463i 0.691564 + 0.722315i \(0.256922\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.50000 + 0.866025i −0.760639 + 0.146385i
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 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
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.5000 + 2.59808i −1.53847 + 0.296078i
\(78\) 0 0
\(79\) 3.46410i 0.389742i −0.980829 0.194871i \(-0.937571\pi\)
0.980829 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3923i 1.10158i −0.834643 0.550791i \(-0.814326\pi\)
0.834643 0.550791i \(-0.185674\pi\)
\(90\) 0 0
\(91\) −9.00000 + 1.73205i −0.943456 + 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.8564i 1.42164i
\(96\) 0 0
\(97\) 8.66025i 0.879316i −0.898165 0.439658i \(-0.855100\pi\)
0.898165 0.439658i \(-0.144900\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.66025i 0.861727i 0.902417 + 0.430864i \(0.141791\pi\)
−0.902417 + 0.430864i \(0.858209\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.66025i 0.837218i −0.908166 0.418609i \(-0.862518\pi\)
0.908166 0.418609i \(-0.137482\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.00000 + 1.73205i −0.825029 + 0.158777i
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 12.1244i 1.07586i 0.842989 + 0.537931i \(0.180794\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) −4.00000 20.7846i −0.346844 1.80225i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −18.0000 −1.50524
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 12.1244i 0.986666i 0.869841 + 0.493333i \(0.164222\pi\)
−0.869841 + 0.493333i \(0.835778\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.66025i 0.695608i
\(156\) 0 0
\(157\) 6.92820i 0.552931i −0.961024 0.276465i \(-0.910837\pi\)
0.961024 0.276465i \(-0.0891631\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 3.46410i 1.41860 0.273009i
\(162\) 0 0
\(163\) 10.3923i 0.813988i 0.913431 + 0.406994i \(0.133423\pi\)
−0.913431 + 0.406994i \(0.866577\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.73205i 0.131685i −0.997830 0.0658427i \(-0.979026\pi\)
0.997830 0.0658427i \(-0.0209736\pi\)
\(174\) 0 0
\(175\) −1.00000 5.19615i −0.0755929 0.392792i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.0526i 1.42406i 0.702152 + 0.712028i \(0.252223\pi\)
−0.702152 + 0.712028i \(0.747777\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.92820i 0.509372i
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3923i 0.751961i −0.926628 0.375980i \(-0.877306\pi\)
0.926628 0.375980i \(-0.122694\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 15.5885i −0.210559 1.09410i
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 41.5692i 2.87540i
\(210\) 0 0
\(211\) 13.8564i 0.953914i −0.878927 0.476957i \(-0.841740\pi\)
0.878927 0.476957i \(-0.158260\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 2.50000 + 12.9904i 0.169711 + 0.881845i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 3.46410i 0.228914i 0.993428 + 0.114457i \(0.0365129\pi\)
−0.993428 + 0.114457i \(0.963487\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 10.3923i 0.677919i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820i 0.448148i 0.974572 + 0.224074i \(0.0719358\pi\)
−0.974572 + 0.224074i \(0.928064\pi\)
\(240\) 0 0
\(241\) 20.7846i 1.33885i 0.742878 + 0.669427i \(0.233460\pi\)
−0.742878 + 0.669427i \(0.766540\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.50000 + 11.2583i 0.287494 + 0.719268i
\(246\) 0 0
\(247\) 27.7128i 1.76332i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.2487i 1.51259i −0.654229 0.756297i \(-0.727007\pi\)
0.654229 0.756297i \(-0.272993\pi\)
\(258\) 0 0
\(259\) −2.00000 10.3923i −0.124274 0.645746i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.8564i 0.854423i 0.904152 + 0.427211i \(0.140504\pi\)
−0.904152 + 0.427211i \(0.859496\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.8564i 0.844840i 0.906400 + 0.422420i \(0.138819\pi\)
−0.906400 + 0.422420i \(0.861181\pi\)
\(270\) 0 0
\(271\) −17.0000 −1.03268 −0.516338 0.856385i \(-0.672705\pi\)
−0.516338 + 0.856385i \(0.672705\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3923i 0.626680i
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.0000 + 3.46410i −1.06251 + 0.204479i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 20.7846i 1.21013i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −9.00000 + 1.73205i −0.518751 + 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 15.5885i 0.881112i −0.897725 0.440556i \(-0.854781\pi\)
0.897725 0.440556i \(-0.145219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 31.1769i 1.74557i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.7128i 1.54198i
\(324\) 0 0
\(325\) 6.92820i 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.00000 + 15.5885i 0.165395 + 0.859419i
\(330\) 0 0
\(331\) 24.2487i 1.33283i −0.745581 0.666415i \(-0.767828\pi\)
0.745581 0.666415i \(-0.232172\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 25.9808i 1.40694i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.9808i 1.39472i 0.716721 + 0.697360i \(0.245642\pi\)
−0.716721 + 0.697360i \(0.754358\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.92820i 0.368751i 0.982856 + 0.184376i \(0.0590263\pi\)
−0.982856 + 0.184376i \(0.940974\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.46410i 0.182828i 0.995813 + 0.0914141i \(0.0291387\pi\)
−0.995813 + 0.0914141i \(0.970861\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.50000 7.79423i −0.0778761 0.404656i
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 4.50000 + 23.3827i 0.229341 + 1.19169i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 3.46410i 0.173858i −0.996214 0.0869291i \(-0.972295\pi\)
0.996214 0.0869291i \(-0.0277054\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 17.3205i 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.7846i 1.03025i
\(408\) 0 0
\(409\) 15.5885i 0.770800i −0.922750 0.385400i \(-0.874064\pi\)
0.922750 0.385400i \(-0.125936\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.00000 + 31.1769i 0.295241 + 1.53412i
\(414\) 0 0
\(415\) 25.9808i 1.27535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.92820i 0.336067i
\(426\) 0 0
\(427\) 36.0000 6.92820i 1.74216 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.3205i 0.834300i −0.908838 0.417150i \(-0.863029\pi\)
0.908838 0.417150i \(-0.136971\pi\)
\(432\) 0 0
\(433\) 22.5167i 1.08208i −0.840996 0.541041i \(-0.818030\pi\)
0.840996 0.541041i \(-0.181970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 55.4256i 2.65137i
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.3205i 0.822922i 0.911427 + 0.411461i \(0.134981\pi\)
−0.911427 + 0.411461i \(0.865019\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 + 15.5885i 0.140642 + 0.730798i
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.19615i 0.242009i −0.992652 0.121004i \(-0.961388\pi\)
0.992652 0.121004i \(-0.0386115\pi\)
\(462\) 0 0
\(463\) 12.1244i 0.563467i −0.959493 0.281733i \(-0.909091\pi\)
0.959493 0.281733i \(-0.0909093\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 9.00000 1.73205i 0.415581 0.0799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −18.0000 −0.827641
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 13.8564i 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.0000 −0.681115
\(486\) 0 0
\(487\) 3.46410i 0.156973i −0.996915 0.0784867i \(-0.974991\pi\)
0.996915 0.0784867i \(-0.0250088\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.5167i 1.01616i 0.861309 + 0.508081i \(0.169645\pi\)
−0.861309 + 0.508081i \(0.830355\pi\)
\(492\) 0 0
\(493\) 20.7846i 0.936092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 + 1.73205i −0.403705 + 0.0776931i
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 15.0000 0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.66025i 0.383859i −0.981409 0.191930i \(-0.938526\pi\)
0.981409 0.191930i \(-0.0614745\pi\)
\(510\) 0 0
\(511\) 4.50000 0.866025i 0.199068 0.0383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 27.7128i 1.22117i
\(516\) 0 0
\(517\) 31.1769i 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 31.1769i 1.36589i 0.730472 + 0.682943i \(0.239300\pi\)
−0.730472 + 0.682943i \(0.760700\pi\)
\(522\) 0 0
\(523\) −2.00000 −0.0874539 −0.0437269 0.999044i \(-0.513923\pi\)
−0.0437269 + 0.999044i \(0.513923\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.3205i 0.754493i
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) −15.0000 −0.648507
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.5000 + 33.7750i 0.581486 + 1.45479i
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 34.6410i 1.48386i
\(546\) 0 0
\(547\) 24.2487i 1.03680i −0.855138 0.518400i \(-0.826528\pi\)
0.855138 0.518400i \(-0.173472\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) −9.00000 + 1.73205i −0.382719 + 0.0736543i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.0000 0.889799 0.444899 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 10.3923i 0.434904i 0.976071 + 0.217452i \(0.0697746\pi\)
−0.976071 + 0.217452i \(0.930225\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.8564i 0.577852i
\(576\) 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
\(581\) 7.50000 + 38.9711i 0.311152 + 1.61680i
\(582\) 0 0
\(583\) 15.5885i 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 39.0000 1.60970 0.804851 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.1769i 1.28028i −0.768257 0.640141i \(-0.778876\pi\)
0.768257 0.640141i \(-0.221124\pi\)
\(594\) 0 0
\(595\) 3.00000 + 15.5885i 0.122988 + 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 38.1051i 1.55693i −0.627686 0.778466i \(-0.715998\pi\)
0.627686 0.778466i \(-0.284002\pi\)
\(600\) 0 0
\(601\) 19.0526i 0.777170i 0.921413 + 0.388585i \(0.127036\pi\)
−0.921413 + 0.388585i \(0.872964\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 27.7128i 1.12669i
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.7846i 0.840855i
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −27.0000 + 5.19615i −1.08173 + 0.208179i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.8564i 0.552491i
\(630\) 0 0
\(631\) 5.19615i 0.206856i 0.994637 + 0.103428i \(0.0329811\pi\)
−0.994637 + 0.103428i \(0.967019\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.0000 0.833360
\(636\) 0 0
\(637\) 9.00000 + 22.5167i 0.356593 + 0.892143i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 0 0
\(649\) 62.3538i 2.44760i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.0000 −1.05659 −0.528296 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) 25.9808i 1.01515i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19.0526i 0.742182i 0.928596 + 0.371091i \(0.121016\pi\)
−0.928596 + 0.371091i \(0.878984\pi\)
\(660\) 0 0
\(661\) 13.8564i 0.538952i −0.963007 0.269476i \(-0.913150\pi\)
0.963007 0.269476i \(-0.0868504\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −36.0000 + 6.92820i −1.39602 + 0.268664i
\(666\) 0 0
\(667\) 41.5692i 1.60957i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 72.0000 2.77953
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 41.5692i 1.59763i 0.601574 + 0.798817i \(0.294541\pi\)
−0.601574 + 0.798817i \(0.705459\pi\)
\(678\) 0 0
\(679\) −22.5000 + 4.33013i −0.863471 + 0.166175i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3205i 0.662751i −0.943499 0.331375i \(-0.892487\pi\)
0.943499 0.331375i \(-0.107513\pi\)
\(684\) 0 0
\(685\) 10.3923i 0.397070i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.3923i 0.395915i
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.8564i 0.525603i
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.5000 4.33013i 0.846200 0.162851i
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 34.6410i 1.29732i
\(714\) 0 0
\(715\) 31.1769i 1.16595i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −8.00000 41.5692i −0.297936 1.54812i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 6.92820i 0.255899i −0.991781 0.127950i \(-0.959160\pi\)
0.991781 0.127950i \(-0.0408395\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.0000 0.663039
\(738\) 0 0
\(739\) 45.0333i 1.65658i −0.560301 0.828289i \(-0.689315\pi\)
0.560301 0.828289i \(-0.310685\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.3923i 0.381257i −0.981662 0.190628i \(-0.938947\pi\)
0.981662 0.190628i \(-0.0610525\pi\)
\(744\) 0 0
\(745\) 36.3731i 1.33261i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −22.5000 + 4.33013i −0.822132 + 0.158219i
\(750\) 0 0
\(751\) 25.9808i 0.948051i −0.880511 0.474026i \(-0.842800\pi\)
0.880511 0.474026i \(-0.157200\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.0000 0.764268
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.8564i 0.502294i 0.967949 + 0.251147i \(0.0808078\pi\)
−0.967949 + 0.251147i \(0.919192\pi\)
\(762\) 0 0
\(763\) −10.0000 51.9615i −0.362024 1.88113i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 41.5692i 1.50098i
\(768\) 0 0
\(769\) 39.8372i 1.43657i −0.695752 0.718283i \(-0.744929\pi\)
0.695752 0.718283i \(-0.255071\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.4974i 1.74433i 0.489211 + 0.872166i \(0.337285\pi\)
−0.489211 + 0.872166i \(0.662715\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 55.4256i 1.98583i
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 31.1769i −0.213335 1.10852i
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.0526i 0.674876i −0.941348 0.337438i \(-0.890440\pi\)
0.941348 0.337438i \(-0.109560\pi\)
\(798\) 0 0
\(799\) 20.7846i 0.735307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.00000 0.317603
\(804\) 0 0
\(805\) −6.00000 31.1769i −0.211472 1.09884i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.0000 0.630512
\(816\) 0 0
\(817\) 27.7128i 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)