Properties

Label 8280.2.p.a.1241.27
Level $8280$
Weight $2$
Character 8280.1241
Analytic conductor $66.116$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.27
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.22

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +0.669780i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +0.669780i q^{7} +6.53199 q^{11} -1.67759 q^{13} +4.86583 q^{17} -7.93620i q^{19} +(-2.87865 - 3.83580i) q^{23} +1.00000 q^{25} -5.55690i q^{29} -2.03194 q^{31} -0.669780i q^{35} +0.940522i q^{37} -2.80249i q^{41} +6.91885i q^{43} -2.23598i q^{47} +6.55139 q^{49} -5.97773 q^{53} -6.53199 q^{55} -2.74853i q^{59} +1.34487i q^{61} +1.67759 q^{65} -2.66113i q^{67} +4.57789i q^{71} -3.52129 q^{73} +4.37500i q^{77} -13.7290i q^{79} -13.0656 q^{83} -4.86583 q^{85} -13.3026 q^{89} -1.12362i q^{91} +7.93620i q^{95} -2.14141i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times\).

\(n\) \(1657\) \(2071\) \(3961\) \(4141\) \(4601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.669780i 0.253153i 0.991957 + 0.126577i \(0.0403989\pi\)
−0.991957 + 0.126577i \(0.959601\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.53199 1.96947 0.984734 0.174064i \(-0.0556898\pi\)
0.984734 + 0.174064i \(0.0556898\pi\)
\(12\) 0 0
\(13\) −1.67759 −0.465280 −0.232640 0.972563i \(-0.574736\pi\)
−0.232640 + 0.972563i \(0.574736\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.86583 1.18014 0.590068 0.807353i \(-0.299101\pi\)
0.590068 + 0.807353i \(0.299101\pi\)
\(18\) 0 0
\(19\) 7.93620i 1.82069i −0.413851 0.910345i \(-0.635817\pi\)
0.413851 0.910345i \(-0.364183\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.87865 3.83580i −0.600241 0.799819i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.55690i 1.03189i −0.856621 0.515946i \(-0.827441\pi\)
0.856621 0.515946i \(-0.172559\pi\)
\(30\) 0 0
\(31\) −2.03194 −0.364948 −0.182474 0.983211i \(-0.558410\pi\)
−0.182474 + 0.983211i \(0.558410\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.669780i 0.113214i
\(36\) 0 0
\(37\) 0.940522i 0.154621i 0.997007 + 0.0773105i \(0.0246333\pi\)
−0.997007 + 0.0773105i \(0.975367\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.80249i 0.437675i −0.975761 0.218838i \(-0.929773\pi\)
0.975761 0.218838i \(-0.0702265\pi\)
\(42\) 0 0
\(43\) 6.91885i 1.05511i 0.849519 + 0.527557i \(0.176892\pi\)
−0.849519 + 0.527557i \(0.823108\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.23598i 0.326151i −0.986614 0.163076i \(-0.947859\pi\)
0.986614 0.163076i \(-0.0521414\pi\)
\(48\) 0 0
\(49\) 6.55139 0.935913
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.97773 −0.821104 −0.410552 0.911837i \(-0.634664\pi\)
−0.410552 + 0.911837i \(0.634664\pi\)
\(54\) 0 0
\(55\) −6.53199 −0.880773
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.74853i 0.357829i −0.983865 0.178914i \(-0.942741\pi\)
0.983865 0.178914i \(-0.0572585\pi\)
\(60\) 0 0
\(61\) 1.34487i 0.172193i 0.996287 + 0.0860967i \(0.0274394\pi\)
−0.996287 + 0.0860967i \(0.972561\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.67759 0.208080
\(66\) 0 0
\(67\) 2.66113i 0.325109i −0.986700 0.162554i \(-0.948027\pi\)
0.986700 0.162554i \(-0.0519733\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.57789i 0.543295i 0.962397 + 0.271648i \(0.0875685\pi\)
−0.962397 + 0.271648i \(0.912432\pi\)
\(72\) 0 0
\(73\) −3.52129 −0.412136 −0.206068 0.978538i \(-0.566067\pi\)
−0.206068 + 0.978538i \(0.566067\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.37500i 0.498577i
\(78\) 0 0
\(79\) 13.7290i 1.54464i −0.635236 0.772318i \(-0.719097\pi\)
0.635236 0.772318i \(-0.280903\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −13.0656 −1.43414 −0.717068 0.697003i \(-0.754516\pi\)
−0.717068 + 0.697003i \(0.754516\pi\)
\(84\) 0 0
\(85\) −4.86583 −0.527773
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.3026 −1.41007 −0.705034 0.709173i \(-0.749068\pi\)
−0.705034 + 0.709173i \(0.749068\pi\)
\(90\) 0 0
\(91\) 1.12362i 0.117787i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.93620i 0.814237i
\(96\) 0 0
\(97\) 2.14141i 0.217428i −0.994073 0.108714i \(-0.965327\pi\)
0.994073 0.108714i \(-0.0346732\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.83776i 0.481375i 0.970603 + 0.240688i \(0.0773729\pi\)
−0.970603 + 0.240688i \(0.922627\pi\)
\(102\) 0 0
\(103\) 12.2379i 1.20584i −0.797802 0.602919i \(-0.794004\pi\)
0.797802 0.602919i \(-0.205996\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.8635 −1.24356 −0.621782 0.783190i \(-0.713591\pi\)
−0.621782 + 0.783190i \(0.713591\pi\)
\(108\) 0 0
\(109\) 7.24388i 0.693838i −0.937895 0.346919i \(-0.887228\pi\)
0.937895 0.346919i \(-0.112772\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24581 0.399412 0.199706 0.979856i \(-0.436001\pi\)
0.199706 + 0.979856i \(0.436001\pi\)
\(114\) 0 0
\(115\) 2.87865 + 3.83580i 0.268436 + 0.357690i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.25903i 0.298755i
\(120\) 0 0
\(121\) 31.6669 2.87881
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.97859 −0.530514 −0.265257 0.964178i \(-0.585457\pi\)
−0.265257 + 0.964178i \(0.585457\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.7326i 0.937710i 0.883275 + 0.468855i \(0.155333\pi\)
−0.883275 + 0.468855i \(0.844667\pi\)
\(132\) 0 0
\(133\) 5.31551 0.460913
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.24036 0.191407 0.0957033 0.995410i \(-0.469490\pi\)
0.0957033 + 0.995410i \(0.469490\pi\)
\(138\) 0 0
\(139\) 20.0068 1.69695 0.848476 0.529234i \(-0.177520\pi\)
0.848476 + 0.529234i \(0.177520\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.9580 −0.916355
\(144\) 0 0
\(145\) 5.55690i 0.461476i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.8878 −1.13773 −0.568866 0.822430i \(-0.692618\pi\)
−0.568866 + 0.822430i \(0.692618\pi\)
\(150\) 0 0
\(151\) −20.1907 −1.64309 −0.821547 0.570141i \(-0.806889\pi\)
−0.821547 + 0.570141i \(0.806889\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.03194 0.163210
\(156\) 0 0
\(157\) 18.9416i 1.51170i 0.654743 + 0.755852i \(0.272777\pi\)
−0.654743 + 0.755852i \(0.727223\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.56914 1.92807i 0.202477 0.151953i
\(162\) 0 0
\(163\) 18.1105 1.41852 0.709261 0.704946i \(-0.249029\pi\)
0.709261 + 0.704946i \(0.249029\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.2004i 1.64054i 0.571978 + 0.820269i \(0.306176\pi\)
−0.571978 + 0.820269i \(0.693824\pi\)
\(168\) 0 0
\(169\) −10.1857 −0.783514
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.5587i 1.63908i −0.573021 0.819540i \(-0.694229\pi\)
0.573021 0.819540i \(-0.305771\pi\)
\(174\) 0 0
\(175\) 0.669780i 0.0506306i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.3629i 0.998788i 0.866375 + 0.499394i \(0.166444\pi\)
−0.866375 + 0.499394i \(0.833556\pi\)
\(180\) 0 0
\(181\) 17.3173i 1.28719i −0.765368 0.643593i \(-0.777443\pi\)
0.765368 0.643593i \(-0.222557\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.940522i 0.0691486i
\(186\) 0 0
\(187\) 31.7835 2.32424
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.07752 0.222682 0.111341 0.993782i \(-0.464485\pi\)
0.111341 + 0.993782i \(0.464485\pi\)
\(192\) 0 0
\(193\) 16.7424 1.20514 0.602572 0.798065i \(-0.294143\pi\)
0.602572 + 0.798065i \(0.294143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.57646i 0.254812i 0.991851 + 0.127406i \(0.0406651\pi\)
−0.991851 + 0.127406i \(0.959335\pi\)
\(198\) 0 0
\(199\) 25.5351i 1.81014i −0.425264 0.905069i \(-0.639819\pi\)
0.425264 0.905069i \(-0.360181\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.72190 0.261226
\(204\) 0 0
\(205\) 2.80249i 0.195734i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 51.8392i 3.58579i
\(210\) 0 0
\(211\) 13.6623 0.940550 0.470275 0.882520i \(-0.344155\pi\)
0.470275 + 0.882520i \(0.344155\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.91885i 0.471862i
\(216\) 0 0
\(217\) 1.36096i 0.0923877i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.16287 −0.549094
\(222\) 0 0
\(223\) 1.76892 0.118456 0.0592278 0.998244i \(-0.481136\pi\)
0.0592278 + 0.998244i \(0.481136\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.40585 −0.159682 −0.0798409 0.996808i \(-0.525441\pi\)
−0.0798409 + 0.996808i \(0.525441\pi\)
\(228\) 0 0
\(229\) 11.5202i 0.761279i 0.924723 + 0.380640i \(0.124296\pi\)
−0.924723 + 0.380640i \(0.875704\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.11652i 0.531731i 0.964010 + 0.265866i \(0.0856577\pi\)
−0.964010 + 0.265866i \(0.914342\pi\)
\(234\) 0 0
\(235\) 2.23598i 0.145859i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.1582i 0.786451i −0.919442 0.393226i \(-0.871359\pi\)
0.919442 0.393226i \(-0.128641\pi\)
\(240\) 0 0
\(241\) 3.09653i 0.199465i −0.995014 0.0997327i \(-0.968201\pi\)
0.995014 0.0997327i \(-0.0317988\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.55139 −0.418553
\(246\) 0 0
\(247\) 13.3137i 0.847130i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.07435 0.509649 0.254824 0.966987i \(-0.417982\pi\)
0.254824 + 0.966987i \(0.417982\pi\)
\(252\) 0 0
\(253\) −18.8033 25.0554i −1.18216 1.57522i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.0766i 0.690942i 0.938429 + 0.345471i \(0.112281\pi\)
−0.938429 + 0.345471i \(0.887719\pi\)
\(258\) 0 0
\(259\) −0.629943 −0.0391428
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.352558 −0.0217397 −0.0108698 0.999941i \(-0.503460\pi\)
−0.0108698 + 0.999941i \(0.503460\pi\)
\(264\) 0 0
\(265\) 5.97773 0.367209
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.6847i 0.712432i −0.934404 0.356216i \(-0.884067\pi\)
0.934404 0.356216i \(-0.115933\pi\)
\(270\) 0 0
\(271\) 6.62513 0.402448 0.201224 0.979545i \(-0.435508\pi\)
0.201224 + 0.979545i \(0.435508\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.53199 0.393894
\(276\) 0 0
\(277\) 2.25382 0.135419 0.0677093 0.997705i \(-0.478431\pi\)
0.0677093 + 0.997705i \(0.478431\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.2793 −0.672870 −0.336435 0.941707i \(-0.609221\pi\)
−0.336435 + 0.941707i \(0.609221\pi\)
\(282\) 0 0
\(283\) 1.03403i 0.0614668i 0.999528 + 0.0307334i \(0.00978428\pi\)
−0.999528 + 0.0307334i \(0.990216\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.87705 0.110799
\(288\) 0 0
\(289\) 6.67627 0.392722
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.84176 −0.458121 −0.229060 0.973412i \(-0.573565\pi\)
−0.229060 + 0.973412i \(0.573565\pi\)
\(294\) 0 0
\(295\) 2.74853i 0.160026i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.82920 + 6.43490i 0.279280 + 0.372140i
\(300\) 0 0
\(301\) −4.63411 −0.267106
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.34487i 0.0770072i
\(306\) 0 0
\(307\) −27.0617 −1.54449 −0.772247 0.635323i \(-0.780867\pi\)
−0.772247 + 0.635323i \(0.780867\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.8664i 1.41005i −0.709184 0.705023i \(-0.750937\pi\)
0.709184 0.705023i \(-0.249063\pi\)
\(312\) 0 0
\(313\) 27.0768i 1.53047i −0.643750 0.765236i \(-0.722622\pi\)
0.643750 0.765236i \(-0.277378\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.3848i 0.920261i −0.887851 0.460131i \(-0.847803\pi\)
0.887851 0.460131i \(-0.152197\pi\)
\(318\) 0 0
\(319\) 36.2976i 2.03228i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 38.6162i 2.14866i
\(324\) 0 0
\(325\) −1.67759 −0.0930560
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.49761 0.0825662
\(330\) 0 0
\(331\) 23.8287 1.30974 0.654872 0.755740i \(-0.272722\pi\)
0.654872 + 0.755740i \(0.272722\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.66113i 0.145393i
\(336\) 0 0
\(337\) 1.72528i 0.0939817i −0.998895 0.0469909i \(-0.985037\pi\)
0.998895 0.0469909i \(-0.0149632\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.2726 −0.718753
\(342\) 0 0
\(343\) 9.07646i 0.490083i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.1433i 1.72554i −0.505594 0.862772i \(-0.668726\pi\)
0.505594 0.862772i \(-0.331274\pi\)
\(348\) 0 0
\(349\) −10.1946 −0.545704 −0.272852 0.962056i \(-0.587967\pi\)
−0.272852 + 0.962056i \(0.587967\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.09406i 0.0582310i 0.999576 + 0.0291155i \(0.00926906\pi\)
−0.999576 + 0.0291155i \(0.990731\pi\)
\(354\) 0 0
\(355\) 4.57789i 0.242969i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.0047 1.21414 0.607070 0.794649i \(-0.292345\pi\)
0.607070 + 0.794649i \(0.292345\pi\)
\(360\) 0 0
\(361\) −43.9833 −2.31491
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.52129 0.184313
\(366\) 0 0
\(367\) 30.5207i 1.59317i −0.604529 0.796583i \(-0.706639\pi\)
0.604529 0.796583i \(-0.293361\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00377i 0.207865i
\(372\) 0 0
\(373\) 16.8273i 0.871282i 0.900120 + 0.435641i \(0.143478\pi\)
−0.900120 + 0.435641i \(0.856522\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.32221i 0.480118i
\(378\) 0 0
\(379\) 6.56133i 0.337033i 0.985699 + 0.168516i \(0.0538976\pi\)
−0.985699 + 0.168516i \(0.946102\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 37.2581 1.90380 0.951899 0.306411i \(-0.0991282\pi\)
0.951899 + 0.306411i \(0.0991282\pi\)
\(384\) 0 0
\(385\) 4.37500i 0.222970i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −32.3098 −1.63817 −0.819087 0.573670i \(-0.805519\pi\)
−0.819087 + 0.573670i \(0.805519\pi\)
\(390\) 0 0
\(391\) −14.0070 18.6643i −0.708366 0.943896i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.7290i 0.690783i
\(396\) 0 0
\(397\) −11.5346 −0.578904 −0.289452 0.957193i \(-0.593473\pi\)
−0.289452 + 0.957193i \(0.593473\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.03309 −0.401154 −0.200577 0.979678i \(-0.564282\pi\)
−0.200577 + 0.979678i \(0.564282\pi\)
\(402\) 0 0
\(403\) 3.40877 0.169803
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.14348i 0.304521i
\(408\) 0 0
\(409\) −10.7487 −0.531490 −0.265745 0.964043i \(-0.585618\pi\)
−0.265745 + 0.964043i \(0.585618\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.84091 0.0905855
\(414\) 0 0
\(415\) 13.0656 0.641365
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.2851 0.942140 0.471070 0.882096i \(-0.343868\pi\)
0.471070 + 0.882096i \(0.343868\pi\)
\(420\) 0 0
\(421\) 16.4742i 0.802903i −0.915880 0.401451i \(-0.868506\pi\)
0.915880 0.401451i \(-0.131494\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.86583 0.236027
\(426\) 0 0
\(427\) −0.900770 −0.0435913
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.94237 0.141729 0.0708645 0.997486i \(-0.477424\pi\)
0.0708645 + 0.997486i \(0.477424\pi\)
\(432\) 0 0
\(433\) 21.8972i 1.05231i −0.850389 0.526155i \(-0.823633\pi\)
0.850389 0.526155i \(-0.176367\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −30.4417 + 22.8456i −1.45622 + 1.09285i
\(438\) 0 0
\(439\) 23.5219 1.12264 0.561320 0.827599i \(-0.310294\pi\)
0.561320 + 0.827599i \(0.310294\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.84084i 0.420041i 0.977697 + 0.210020i \(0.0673530\pi\)
−0.977697 + 0.210020i \(0.932647\pi\)
\(444\) 0 0
\(445\) 13.3026 0.630602
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.3766i 1.29198i −0.763346 0.645990i \(-0.776445\pi\)
0.763346 0.645990i \(-0.223555\pi\)
\(450\) 0 0
\(451\) 18.3058i 0.861988i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.12362i 0.0526760i
\(456\) 0 0
\(457\) 36.5250i 1.70857i 0.519808 + 0.854283i \(0.326004\pi\)
−0.519808 + 0.854283i \(0.673996\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 35.1897i 1.63895i −0.573115 0.819475i \(-0.694265\pi\)
0.573115 0.819475i \(-0.305735\pi\)
\(462\) 0 0
\(463\) 18.7369 0.870778 0.435389 0.900243i \(-0.356611\pi\)
0.435389 + 0.900243i \(0.356611\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.8152 −1.19459 −0.597293 0.802023i \(-0.703757\pi\)
−0.597293 + 0.802023i \(0.703757\pi\)
\(468\) 0 0
\(469\) 1.78237 0.0823023
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 45.1938i 2.07802i
\(474\) 0 0
\(475\) 7.93620i 0.364138i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.91066 0.224374 0.112187 0.993687i \(-0.464214\pi\)
0.112187 + 0.993687i \(0.464214\pi\)
\(480\) 0 0
\(481\) 1.57781i 0.0719420i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.14141i 0.0972366i
\(486\) 0 0
\(487\) 20.7414 0.939881 0.469941 0.882698i \(-0.344275\pi\)
0.469941 + 0.882698i \(0.344275\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.7369i 0.935844i −0.883770 0.467922i \(-0.845003\pi\)
0.883770 0.467922i \(-0.154997\pi\)
\(492\) 0 0
\(493\) 27.0389i 1.21777i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.06618 −0.137537
\(498\) 0 0
\(499\) −23.0473 −1.03174 −0.515870 0.856667i \(-0.672531\pi\)
−0.515870 + 0.856667i \(0.672531\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −22.6799 −1.01125 −0.505624 0.862754i \(-0.668738\pi\)
−0.505624 + 0.862754i \(0.668738\pi\)
\(504\) 0 0
\(505\) 4.83776i 0.215278i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.03904i 0.267676i 0.991003 + 0.133838i \(0.0427302\pi\)
−0.991003 + 0.133838i \(0.957270\pi\)
\(510\) 0 0
\(511\) 2.35849i 0.104334i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.2379i 0.539267i
\(516\) 0 0
\(517\) 14.6054i 0.642344i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.0984 −1.36244 −0.681222 0.732077i \(-0.738551\pi\)
−0.681222 + 0.732077i \(0.738551\pi\)
\(522\) 0 0
\(523\) 26.6962i 1.16734i −0.811989 0.583672i \(-0.801615\pi\)
0.811989 0.583672i \(-0.198385\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.88708 −0.430688
\(528\) 0 0
\(529\) −6.42670 + 22.0839i −0.279422 + 0.960168i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.70143i 0.203642i
\(534\) 0 0
\(535\) 12.8635 0.556139
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 42.7936 1.84325
\(540\) 0 0
\(541\) 13.9894 0.601452 0.300726 0.953711i \(-0.402771\pi\)
0.300726 + 0.953711i \(0.402771\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.24388i 0.310294i
\(546\) 0 0
\(547\) 37.5455 1.60533 0.802665 0.596429i \(-0.203414\pi\)
0.802665 + 0.596429i \(0.203414\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −44.1007 −1.87875
\(552\) 0 0
\(553\) 9.19543 0.391030
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.71665 0.115108 0.0575541 0.998342i \(-0.481670\pi\)
0.0575541 + 0.998342i \(0.481670\pi\)
\(558\) 0 0
\(559\) 11.6070i 0.490924i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.4941 1.32732 0.663659 0.748035i \(-0.269002\pi\)
0.663659 + 0.748035i \(0.269002\pi\)
\(564\) 0 0
\(565\) −4.24581 −0.178622
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.4946 0.607646 0.303823 0.952728i \(-0.401737\pi\)
0.303823 + 0.952728i \(0.401737\pi\)
\(570\) 0 0
\(571\) 11.0190i 0.461130i −0.973057 0.230565i \(-0.925943\pi\)
0.973057 0.230565i \(-0.0740574\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.87865 3.83580i −0.120048 0.159964i
\(576\) 0 0
\(577\) 28.4467 1.18425 0.592125 0.805846i \(-0.298289\pi\)
0.592125 + 0.805846i \(0.298289\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.75108i 0.363056i
\(582\) 0 0
\(583\) −39.0465 −1.61714
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.28618i 0.342007i 0.985270 + 0.171004i \(0.0547010\pi\)
−0.985270 + 0.171004i \(0.945299\pi\)
\(588\) 0 0
\(589\) 16.1259i 0.664456i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.9404i 1.27057i −0.772278 0.635285i \(-0.780883\pi\)
0.772278 0.635285i \(-0.219117\pi\)
\(594\) 0 0
\(595\) 3.25903i 0.133607i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.8074i 0.441576i −0.975322 0.220788i \(-0.929137\pi\)
0.975322 0.220788i \(-0.0708630\pi\)
\(600\) 0 0
\(601\) 18.7779 0.765967 0.382983 0.923755i \(-0.374897\pi\)
0.382983 + 0.923755i \(0.374897\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −31.6669 −1.28744
\(606\) 0 0
\(607\) 22.6596 0.919725 0.459862 0.887990i \(-0.347899\pi\)
0.459862 + 0.887990i \(0.347899\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.75106i 0.151752i
\(612\) 0 0
\(613\) 15.0213i 0.606705i 0.952878 + 0.303353i \(0.0981060\pi\)
−0.952878 + 0.303353i \(0.901894\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.9666 −1.24667 −0.623333 0.781956i \(-0.714222\pi\)
−0.623333 + 0.781956i \(0.714222\pi\)
\(618\) 0 0
\(619\) 28.2693i 1.13624i 0.822946 + 0.568119i \(0.192329\pi\)
−0.822946 + 0.568119i \(0.807671\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.90979i 0.356963i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.57642i 0.182474i
\(630\) 0 0
\(631\) 9.29055i 0.369851i 0.982753 + 0.184925i \(0.0592043\pi\)
−0.982753 + 0.184925i \(0.940796\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.97859 0.237253
\(636\) 0 0
\(637\) −10.9906 −0.435462
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.1330 0.597717 0.298859 0.954297i \(-0.403394\pi\)
0.298859 + 0.954297i \(0.403394\pi\)
\(642\) 0 0
\(643\) 6.49585i 0.256171i −0.991763 0.128086i \(-0.959117\pi\)
0.991763 0.128086i \(-0.0408832\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.9943i 0.668116i 0.942552 + 0.334058i \(0.108418\pi\)
−0.942552 + 0.334058i \(0.891582\pi\)
\(648\) 0 0
\(649\) 17.9534i 0.704733i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.3036i 0.872807i −0.899751 0.436403i \(-0.856252\pi\)
0.899751 0.436403i \(-0.143748\pi\)
\(654\) 0 0
\(655\) 10.7326i 0.419357i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −29.4672 −1.14788 −0.573940 0.818897i \(-0.694586\pi\)
−0.573940 + 0.818897i \(0.694586\pi\)
\(660\) 0 0
\(661\) 5.62468i 0.218775i −0.993999 0.109387i \(-0.965111\pi\)
0.993999 0.109387i \(-0.0348889\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.31551 −0.206127
\(666\) 0 0
\(667\) −21.3152 + 15.9964i −0.825326 + 0.619383i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.78470i 0.339130i
\(672\) 0 0
\(673\) 4.85261 0.187055 0.0935273 0.995617i \(-0.470186\pi\)
0.0935273 + 0.995617i \(0.470186\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.5803 −0.521932 −0.260966 0.965348i \(-0.584041\pi\)
−0.260966 + 0.965348i \(0.584041\pi\)
\(678\) 0 0
\(679\) 1.43428 0.0550425
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.30907i 0.0500901i −0.999686 0.0250451i \(-0.992027\pi\)
0.999686 0.0250451i \(-0.00797293\pi\)
\(684\) 0 0
\(685\) −2.24036 −0.0855996
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0282 0.382043
\(690\) 0 0
\(691\) 47.9593 1.82446 0.912229 0.409681i \(-0.134360\pi\)
0.912229 + 0.409681i \(0.134360\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.0068 −0.758900
\(696\) 0 0
\(697\) 13.6364i 0.516517i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.6365 0.930509 0.465254 0.885177i \(-0.345963\pi\)
0.465254 + 0.885177i \(0.345963\pi\)
\(702\) 0 0
\(703\) 7.46417 0.281517
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.24024 −0.121862
\(708\) 0 0
\(709\) 5.58949i 0.209918i 0.994477 + 0.104959i \(0.0334711\pi\)
−0.994477 + 0.104959i \(0.966529\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.84926 + 7.79412i 0.219057 + 0.291892i
\(714\) 0 0
\(715\) 10.9580 0.409806
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.4071i 1.54422i 0.635487 + 0.772112i \(0.280799\pi\)
−0.635487 + 0.772112i \(0.719201\pi\)
\(720\) 0 0
\(721\) 8.19672 0.305262
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.55690i 0.206378i
\(726\) 0 0
\(727\) 3.25287i 0.120642i −0.998179 0.0603211i \(-0.980788\pi\)
0.998179 0.0603211i \(-0.0192125\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 33.6659i 1.24518i
\(732\) 0 0
\(733\) 32.9050i 1.21537i 0.794177 + 0.607687i \(0.207902\pi\)
−0.794177 + 0.607687i \(0.792098\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.3825i 0.640291i
\(738\) 0 0
\(739\) −28.1180 −1.03434 −0.517168 0.855884i \(-0.673014\pi\)
−0.517168 + 0.855884i \(0.673014\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.2581 1.33018 0.665090 0.746763i \(-0.268393\pi\)
0.665090 + 0.746763i \(0.268393\pi\)
\(744\) 0 0
\(745\) 13.8878 0.508809
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.61574i 0.314812i
\(750\) 0 0
\(751\) 46.0578i 1.68067i 0.542066 + 0.840336i \(0.317642\pi\)
−0.542066 + 0.840336i \(0.682358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.1907 0.734814
\(756\) 0 0
\(757\) 5.16778i 0.187826i −0.995580 0.0939132i \(-0.970062\pi\)
0.995580 0.0939132i \(-0.0299376\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 14.8894i 0.539741i −0.962897 0.269871i \(-0.913019\pi\)
0.962897 0.269871i \(-0.0869809\pi\)
\(762\) 0 0
\(763\) 4.85181 0.175647
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.61092i 0.166491i
\(768\) 0 0
\(769\) 20.3215i 0.732812i 0.930455 + 0.366406i \(0.119412\pi\)
−0.930455 + 0.366406i \(0.880588\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.0773 −1.00987 −0.504935 0.863158i \(-0.668483\pi\)
−0.504935 + 0.863158i \(0.668483\pi\)
\(774\) 0 0
\(775\) −2.03194 −0.0729895
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −22.2411 −0.796871
\(780\) 0 0
\(781\) 29.9027i 1.07000i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.9416i 0.676054i
\(786\) 0 0
\(787\) 32.6563i 1.16407i −0.813163 0.582036i \(-0.802256\pi\)
0.813163 0.582036i \(-0.197744\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.84376i 0.101112i
\(792\) 0 0
\(793\) 2.25615i 0.0801182i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −50.7925 −1.79916 −0.899582 0.436753i \(-0.856128\pi\)
−0.899582 + 0.436753i \(0.856128\pi\)
\(798\) 0 0
\(799\) 10.8799i 0.384903i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −23.0011 −0.811690
\(804\) 0 0
\(805\) −2.56914 + 1.92807i −0.0905504 + 0.0679554i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 51.8554i 1.82314i 0.411145 + 0.911570i