Properties

Label 8280.2.p.a.1241.19
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 1241.19
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.30

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -1.36101i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -1.36101i q^{7} +2.32855 q^{11} +6.17596 q^{13} +6.91838 q^{17} +0.609544i q^{19} +(4.08664 - 2.50985i) q^{23} +1.00000 q^{25} +0.841999i q^{29} +0.0981227 q^{31} +1.36101i q^{35} +6.85378i q^{37} +6.58549i q^{41} +0.293064i q^{43} -3.82138i q^{47} +5.14766 q^{49} +1.36282 q^{53} -2.32855 q^{55} -14.1519i q^{59} +2.54154i q^{61} -6.17596 q^{65} +13.3006i q^{67} +4.87925i q^{71} +3.41879 q^{73} -3.16917i q^{77} +2.39837i q^{79} +2.10006 q^{83} -6.91838 q^{85} -13.1331 q^{89} -8.40552i q^{91} -0.609544i q^{95} -7.80719i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) 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
\(7\) 1.36101i 0.514412i −0.966357 0.257206i \(-0.917198\pi\)
0.966357 0.257206i \(-0.0828019\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.32855 0.702083 0.351042 0.936360i \(-0.385828\pi\)
0.351042 + 0.936360i \(0.385828\pi\)
\(12\) 0 0
\(13\) 6.17596 1.71290 0.856451 0.516228i \(-0.172664\pi\)
0.856451 + 0.516228i \(0.172664\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.91838 1.67795 0.838977 0.544166i \(-0.183154\pi\)
0.838977 + 0.544166i \(0.183154\pi\)
\(18\) 0 0
\(19\) 0.609544i 0.139839i 0.997553 + 0.0699195i \(0.0222742\pi\)
−0.997553 + 0.0699195i \(0.977726\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.08664 2.50985i 0.852124 0.523340i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.841999i 0.156355i 0.996939 + 0.0781776i \(0.0249101\pi\)
−0.996939 + 0.0781776i \(0.975090\pi\)
\(30\) 0 0
\(31\) 0.0981227 0.0176234 0.00881168 0.999961i \(-0.497195\pi\)
0.00881168 + 0.999961i \(0.497195\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.36101i 0.230052i
\(36\) 0 0
\(37\) 6.85378i 1.12675i 0.826200 + 0.563377i \(0.190498\pi\)
−0.826200 + 0.563377i \(0.809502\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.58549i 1.02848i 0.857646 + 0.514240i \(0.171926\pi\)
−0.857646 + 0.514240i \(0.828074\pi\)
\(42\) 0 0
\(43\) 0.293064i 0.0446919i 0.999750 + 0.0223459i \(0.00711353\pi\)
−0.999750 + 0.0223459i \(0.992886\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.82138i 0.557405i −0.960377 0.278702i \(-0.910096\pi\)
0.960377 0.278702i \(-0.0899043\pi\)
\(48\) 0 0
\(49\) 5.14766 0.735380
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.36282 0.187198 0.0935989 0.995610i \(-0.470163\pi\)
0.0935989 + 0.995610i \(0.470163\pi\)
\(54\) 0 0
\(55\) −2.32855 −0.313981
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.1519i 1.84242i −0.389062 0.921212i \(-0.627201\pi\)
0.389062 0.921212i \(-0.372799\pi\)
\(60\) 0 0
\(61\) 2.54154i 0.325411i 0.986675 + 0.162705i \(0.0520220\pi\)
−0.986675 + 0.162705i \(0.947978\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.17596 −0.766033
\(66\) 0 0
\(67\) 13.3006i 1.62493i 0.583011 + 0.812464i \(0.301875\pi\)
−0.583011 + 0.812464i \(0.698125\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.87925i 0.579060i 0.957169 + 0.289530i \(0.0934991\pi\)
−0.957169 + 0.289530i \(0.906501\pi\)
\(72\) 0 0
\(73\) 3.41879 0.400140 0.200070 0.979782i \(-0.435883\pi\)
0.200070 + 0.979782i \(0.435883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.16917i 0.361160i
\(78\) 0 0
\(79\) 2.39837i 0.269838i 0.990857 + 0.134919i \(0.0430774\pi\)
−0.990857 + 0.134919i \(0.956923\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.10006 0.230511 0.115256 0.993336i \(-0.463231\pi\)
0.115256 + 0.993336i \(0.463231\pi\)
\(84\) 0 0
\(85\) −6.91838 −0.750404
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.1331 −1.39211 −0.696053 0.717991i \(-0.745062\pi\)
−0.696053 + 0.717991i \(0.745062\pi\)
\(90\) 0 0
\(91\) 8.40552i 0.881138i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.609544i 0.0625379i
\(96\) 0 0
\(97\) 7.80719i 0.792700i −0.918100 0.396350i \(-0.870277\pi\)
0.918100 0.396350i \(-0.129723\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.97034i 0.494567i −0.968943 0.247283i \(-0.920462\pi\)
0.968943 0.247283i \(-0.0795379\pi\)
\(102\) 0 0
\(103\) 7.93913i 0.782266i 0.920334 + 0.391133i \(0.127917\pi\)
−0.920334 + 0.391133i \(0.872083\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.38117 −0.616891 −0.308446 0.951242i \(-0.599809\pi\)
−0.308446 + 0.951242i \(0.599809\pi\)
\(108\) 0 0
\(109\) 12.8958i 1.23519i 0.786496 + 0.617596i \(0.211893\pi\)
−0.786496 + 0.617596i \(0.788107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.62398 0.340915 0.170458 0.985365i \(-0.445475\pi\)
0.170458 + 0.985365i \(0.445475\pi\)
\(114\) 0 0
\(115\) −4.08664 + 2.50985i −0.381081 + 0.234045i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.41596i 0.863160i
\(120\) 0 0
\(121\) −5.57787 −0.507079
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 11.6115 1.03035 0.515176 0.857084i \(-0.327727\pi\)
0.515176 + 0.857084i \(0.327727\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.1854i 1.15201i −0.817446 0.576005i \(-0.804611\pi\)
0.817446 0.576005i \(-0.195389\pi\)
\(132\) 0 0
\(133\) 0.829593 0.0719348
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.5459 −1.07187 −0.535934 0.844260i \(-0.680041\pi\)
−0.535934 + 0.844260i \(0.680041\pi\)
\(138\) 0 0
\(139\) −13.0180 −1.10417 −0.552085 0.833788i \(-0.686167\pi\)
−0.552085 + 0.833788i \(0.686167\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.3810 1.20260
\(144\) 0 0
\(145\) 0.841999i 0.0699242i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.638019 0.0522686 0.0261343 0.999658i \(-0.491680\pi\)
0.0261343 + 0.999658i \(0.491680\pi\)
\(150\) 0 0
\(151\) −10.8378 −0.881965 −0.440983 0.897516i \(-0.645370\pi\)
−0.440983 + 0.897516i \(0.645370\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.0981227 −0.00788141
\(156\) 0 0
\(157\) 1.59811i 0.127543i −0.997965 0.0637715i \(-0.979687\pi\)
0.997965 0.0637715i \(-0.0203129\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.41593 5.56195i −0.269213 0.438343i
\(162\) 0 0
\(163\) −14.2578 −1.11675 −0.558377 0.829587i \(-0.688576\pi\)
−0.558377 + 0.829587i \(0.688576\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.30317i 0.178225i −0.996022 0.0891123i \(-0.971597\pi\)
0.996022 0.0891123i \(-0.0284030\pi\)
\(168\) 0 0
\(169\) 25.1425 1.93404
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.9985i 1.06429i 0.846654 + 0.532143i \(0.178613\pi\)
−0.846654 + 0.532143i \(0.821387\pi\)
\(174\) 0 0
\(175\) 1.36101i 0.102882i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.9442i 1.26647i 0.773960 + 0.633235i \(0.218273\pi\)
−0.773960 + 0.633235i \(0.781727\pi\)
\(180\) 0 0
\(181\) 3.00241i 0.223167i −0.993755 0.111584i \(-0.964408\pi\)
0.993755 0.111584i \(-0.0355923\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.85378i 0.503900i
\(186\) 0 0
\(187\) 16.1098 1.17806
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.1388 −0.805974 −0.402987 0.915206i \(-0.632028\pi\)
−0.402987 + 0.915206i \(0.632028\pi\)
\(192\) 0 0
\(193\) 24.7159 1.77909 0.889546 0.456846i \(-0.151021\pi\)
0.889546 + 0.456846i \(0.151021\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.21643i 0.157914i −0.996878 0.0789571i \(-0.974841\pi\)
0.996878 0.0789571i \(-0.0251590\pi\)
\(198\) 0 0
\(199\) 6.02768i 0.427291i 0.976911 + 0.213646i \(0.0685338\pi\)
−0.976911 + 0.213646i \(0.931466\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.14597 0.0804310
\(204\) 0 0
\(205\) 6.58549i 0.459951i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.41935i 0.0981785i
\(210\) 0 0
\(211\) 7.15202 0.492365 0.246183 0.969223i \(-0.420824\pi\)
0.246183 + 0.969223i \(0.420824\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.293064i 0.0199868i
\(216\) 0 0
\(217\) 0.133546i 0.00906567i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 42.7277 2.87417
\(222\) 0 0
\(223\) 25.2153 1.68854 0.844272 0.535915i \(-0.180033\pi\)
0.844272 + 0.535915i \(0.180033\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.6367 −0.905100 −0.452550 0.891739i \(-0.649486\pi\)
−0.452550 + 0.891739i \(0.649486\pi\)
\(228\) 0 0
\(229\) 14.1505i 0.935090i 0.883969 + 0.467545i \(0.154861\pi\)
−0.883969 + 0.467545i \(0.845139\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.22194i 0.342100i −0.985262 0.171050i \(-0.945284\pi\)
0.985262 0.171050i \(-0.0547160\pi\)
\(234\) 0 0
\(235\) 3.82138i 0.249279i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.86141i 0.637881i −0.947775 0.318941i \(-0.896673\pi\)
0.947775 0.318941i \(-0.103327\pi\)
\(240\) 0 0
\(241\) 4.79014i 0.308560i −0.988027 0.154280i \(-0.950694\pi\)
0.988027 0.154280i \(-0.0493058\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.14766 −0.328872
\(246\) 0 0
\(247\) 3.76452i 0.239530i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.80969 0.240466 0.120233 0.992746i \(-0.461636\pi\)
0.120233 + 0.992746i \(0.461636\pi\)
\(252\) 0 0
\(253\) 9.51594 5.84431i 0.598262 0.367429i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.3778i 0.834484i 0.908795 + 0.417242i \(0.137003\pi\)
−0.908795 + 0.417242i \(0.862997\pi\)
\(258\) 0 0
\(259\) 9.32804 0.579616
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 29.3304 1.80859 0.904295 0.426907i \(-0.140397\pi\)
0.904295 + 0.426907i \(0.140397\pi\)
\(264\) 0 0
\(265\) −1.36282 −0.0837174
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.21872i 0.379162i −0.981865 0.189581i \(-0.939287\pi\)
0.981865 0.189581i \(-0.0607130\pi\)
\(270\) 0 0
\(271\) −0.778174 −0.0472707 −0.0236354 0.999721i \(-0.507524\pi\)
−0.0236354 + 0.999721i \(0.507524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.32855 0.140417
\(276\) 0 0
\(277\) −14.9522 −0.898393 −0.449197 0.893433i \(-0.648290\pi\)
−0.449197 + 0.893433i \(0.648290\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.93159 0.115229 0.0576146 0.998339i \(-0.481651\pi\)
0.0576146 + 0.998339i \(0.481651\pi\)
\(282\) 0 0
\(283\) 6.47867i 0.385117i 0.981285 + 0.192559i \(0.0616785\pi\)
−0.981285 + 0.192559i \(0.938321\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.96289 0.529063
\(288\) 0 0
\(289\) 30.8640 1.81553
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.59861 0.0933920 0.0466960 0.998909i \(-0.485131\pi\)
0.0466960 + 0.998909i \(0.485131\pi\)
\(294\) 0 0
\(295\) 14.1519i 0.823957i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.2389 15.5007i 1.45961 0.896431i
\(300\) 0 0
\(301\) 0.398862 0.0229900
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.54154i 0.145528i
\(306\) 0 0
\(307\) 11.8163 0.674393 0.337196 0.941434i \(-0.390521\pi\)
0.337196 + 0.941434i \(0.390521\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.1530i 0.745836i −0.927864 0.372918i \(-0.878357\pi\)
0.927864 0.372918i \(-0.121643\pi\)
\(312\) 0 0
\(313\) 23.4891i 1.32768i −0.747874 0.663841i \(-0.768925\pi\)
0.747874 0.663841i \(-0.231075\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.2714i 1.41938i −0.704513 0.709691i \(-0.748835\pi\)
0.704513 0.709691i \(-0.251165\pi\)
\(318\) 0 0
\(319\) 1.96063i 0.109774i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.21706i 0.234643i
\(324\) 0 0
\(325\) 6.17596 0.342581
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.20092 −0.286736
\(330\) 0 0
\(331\) 16.8971 0.928750 0.464375 0.885639i \(-0.346279\pi\)
0.464375 + 0.885639i \(0.346279\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.3006i 0.726690i
\(336\) 0 0
\(337\) 19.2187i 1.04691i 0.852054 + 0.523454i \(0.175357\pi\)
−0.852054 + 0.523454i \(0.824643\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.228483 0.0123731
\(342\) 0 0
\(343\) 16.5330i 0.892700i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.4391i 0.721450i −0.932672 0.360725i \(-0.882529\pi\)
0.932672 0.360725i \(-0.117471\pi\)
\(348\) 0 0
\(349\) 11.6613 0.624213 0.312106 0.950047i \(-0.398965\pi\)
0.312106 + 0.950047i \(0.398965\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.8041i 1.53309i −0.642193 0.766543i \(-0.721975\pi\)
0.642193 0.766543i \(-0.278025\pi\)
\(354\) 0 0
\(355\) 4.87925i 0.258964i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.1562 0.905470 0.452735 0.891645i \(-0.350448\pi\)
0.452735 + 0.891645i \(0.350448\pi\)
\(360\) 0 0
\(361\) 18.6285 0.980445
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.41879 −0.178948
\(366\) 0 0
\(367\) 7.88954i 0.411830i 0.978570 + 0.205915i \(0.0660171\pi\)
−0.978570 + 0.205915i \(0.933983\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.85481i 0.0962968i
\(372\) 0 0
\(373\) 13.8450i 0.716867i 0.933555 + 0.358434i \(0.116689\pi\)
−0.933555 + 0.358434i \(0.883311\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.20015i 0.267821i
\(378\) 0 0
\(379\) 15.7397i 0.808494i −0.914650 0.404247i \(-0.867534\pi\)
0.914650 0.404247i \(-0.132466\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −30.7447 −1.57098 −0.785491 0.618873i \(-0.787589\pi\)
−0.785491 + 0.618873i \(0.787589\pi\)
\(384\) 0 0
\(385\) 3.16917i 0.161516i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.3630 0.981741 0.490870 0.871233i \(-0.336679\pi\)
0.490870 + 0.871233i \(0.336679\pi\)
\(390\) 0 0
\(391\) 28.2730 17.3641i 1.42982 0.878141i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.39837i 0.120675i
\(396\) 0 0
\(397\) 10.5158 0.527774 0.263887 0.964554i \(-0.414995\pi\)
0.263887 + 0.964554i \(0.414995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.7489 −0.586713 −0.293356 0.956003i \(-0.594772\pi\)
−0.293356 + 0.956003i \(0.594772\pi\)
\(402\) 0 0
\(403\) 0.606002 0.0301871
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.9593i 0.791076i
\(408\) 0 0
\(409\) −20.5464 −1.01595 −0.507977 0.861371i \(-0.669606\pi\)
−0.507977 + 0.861371i \(0.669606\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −19.2609 −0.947765
\(414\) 0 0
\(415\) −2.10006 −0.103088
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.3644 0.750600 0.375300 0.926903i \(-0.377540\pi\)
0.375300 + 0.926903i \(0.377540\pi\)
\(420\) 0 0
\(421\) 36.3261i 1.77043i 0.465186 + 0.885213i \(0.345988\pi\)
−0.465186 + 0.885213i \(0.654012\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.91838 0.335591
\(426\) 0 0
\(427\) 3.45905 0.167395
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.1957 1.06913 0.534564 0.845128i \(-0.320476\pi\)
0.534564 + 0.845128i \(0.320476\pi\)
\(432\) 0 0
\(433\) 15.6554i 0.752351i 0.926548 + 0.376176i \(0.122761\pi\)
−0.926548 + 0.376176i \(0.877239\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.52986 + 2.49099i 0.0731834 + 0.119160i
\(438\) 0 0
\(439\) −13.6179 −0.649946 −0.324973 0.945723i \(-0.605355\pi\)
−0.324973 + 0.945723i \(0.605355\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.1520i 0.909940i −0.890507 0.454970i \(-0.849650\pi\)
0.890507 0.454970i \(-0.150350\pi\)
\(444\) 0 0
\(445\) 13.1331 0.622569
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.0825i 1.27810i 0.769164 + 0.639051i \(0.220673\pi\)
−0.769164 + 0.639051i \(0.779327\pi\)
\(450\) 0 0
\(451\) 15.3346i 0.722079i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.40552i 0.394057i
\(456\) 0 0
\(457\) 29.6948i 1.38906i 0.719463 + 0.694531i \(0.244388\pi\)
−0.719463 + 0.694531i \(0.755612\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.77211i 0.455133i 0.973763 + 0.227566i \(0.0730768\pi\)
−0.973763 + 0.227566i \(0.926923\pi\)
\(462\) 0 0
\(463\) −10.2661 −0.477106 −0.238553 0.971129i \(-0.576673\pi\)
−0.238553 + 0.971129i \(0.576673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −21.4893 −0.994406 −0.497203 0.867634i \(-0.665640\pi\)
−0.497203 + 0.867634i \(0.665640\pi\)
\(468\) 0 0
\(469\) 18.1022 0.835883
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.682414i 0.0313774i
\(474\) 0 0
\(475\) 0.609544i 0.0279678i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.13018 0.0973302 0.0486651 0.998815i \(-0.484503\pi\)
0.0486651 + 0.998815i \(0.484503\pi\)
\(480\) 0 0
\(481\) 42.3287i 1.93002i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.80719i 0.354506i
\(486\) 0 0
\(487\) −23.8695 −1.08163 −0.540816 0.841141i \(-0.681885\pi\)
−0.540816 + 0.841141i \(0.681885\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 40.6477i 1.83441i −0.398420 0.917203i \(-0.630441\pi\)
0.398420 0.917203i \(-0.369559\pi\)
\(492\) 0 0
\(493\) 5.82527i 0.262357i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.64069 0.297876
\(498\) 0 0
\(499\) 21.3107 0.953997 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.2994 1.35098 0.675491 0.737368i \(-0.263932\pi\)
0.675491 + 0.737368i \(0.263932\pi\)
\(504\) 0 0
\(505\) 4.97034i 0.221177i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.4954i 0.731146i 0.930783 + 0.365573i \(0.119127\pi\)
−0.930783 + 0.365573i \(0.880873\pi\)
\(510\) 0 0
\(511\) 4.65300i 0.205837i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.93913i 0.349840i
\(516\) 0 0
\(517\) 8.89825i 0.391345i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.9661 0.699489 0.349745 0.936845i \(-0.386268\pi\)
0.349745 + 0.936845i \(0.386268\pi\)
\(522\) 0 0
\(523\) 30.6409i 1.33983i −0.742437 0.669916i \(-0.766330\pi\)
0.742437 0.669916i \(-0.233670\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.678851 0.0295712
\(528\) 0 0
\(529\) 10.4013 20.5137i 0.452230 0.891902i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 40.6717i 1.76169i
\(534\) 0 0
\(535\) 6.38117 0.275882
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.9866 0.516298
\(540\) 0 0
\(541\) 4.65641 0.200195 0.100097 0.994978i \(-0.468085\pi\)
0.100097 + 0.994978i \(0.468085\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.8958i 0.552394i
\(546\) 0 0
\(547\) −21.7546 −0.930158 −0.465079 0.885269i \(-0.653974\pi\)
−0.465079 + 0.885269i \(0.653974\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.513235 −0.0218645
\(552\) 0 0
\(553\) 3.26420 0.138808
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −40.5716 −1.71907 −0.859536 0.511075i \(-0.829247\pi\)
−0.859536 + 0.511075i \(0.829247\pi\)
\(558\) 0 0
\(559\) 1.80995i 0.0765529i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.4731 −0.567822 −0.283911 0.958851i \(-0.591632\pi\)
−0.283911 + 0.958851i \(0.591632\pi\)
\(564\) 0 0
\(565\) −3.62398 −0.152462
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.4298 0.604928 0.302464 0.953161i \(-0.402191\pi\)
0.302464 + 0.953161i \(0.402191\pi\)
\(570\) 0 0
\(571\) 16.9229i 0.708201i −0.935207 0.354100i \(-0.884787\pi\)
0.935207 0.354100i \(-0.115213\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.08664 2.50985i 0.170425 0.104668i
\(576\) 0 0
\(577\) −28.0757 −1.16881 −0.584403 0.811463i \(-0.698671\pi\)
−0.584403 + 0.811463i \(0.698671\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.85819i 0.118578i
\(582\) 0 0
\(583\) 3.17339 0.131428
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.6563i 0.481105i −0.970636 0.240553i \(-0.922671\pi\)
0.970636 0.240553i \(-0.0773287\pi\)
\(588\) 0 0
\(589\) 0.0598101i 0.00246443i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.5861i 0.968565i −0.874912 0.484283i \(-0.839081\pi\)
0.874912 0.484283i \(-0.160919\pi\)
\(594\) 0 0
\(595\) 9.41596i 0.386017i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.5437i 0.512522i −0.966608 0.256261i \(-0.917509\pi\)
0.966608 0.256261i \(-0.0824907\pi\)
\(600\) 0 0
\(601\) 8.82023 0.359785 0.179892 0.983686i \(-0.442425\pi\)
0.179892 + 0.983686i \(0.442425\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.57787 0.226773
\(606\) 0 0
\(607\) −13.9016 −0.564248 −0.282124 0.959378i \(-0.591039\pi\)
−0.282124 + 0.959378i \(0.591039\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 23.6007i 0.954781i
\(612\) 0 0
\(613\) 21.8968i 0.884405i 0.896915 + 0.442203i \(0.145803\pi\)
−0.896915 + 0.442203i \(0.854197\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.33977 0.174712 0.0873562 0.996177i \(-0.472158\pi\)
0.0873562 + 0.996177i \(0.472158\pi\)
\(618\) 0 0
\(619\) 22.0721i 0.887152i −0.896237 0.443576i \(-0.853710\pi\)
0.896237 0.443576i \(-0.146290\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.8742i 0.716116i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 47.4171i 1.89064i
\(630\) 0 0
\(631\) 32.4052i 1.29003i −0.764170 0.645015i \(-0.776851\pi\)
0.764170 0.645015i \(-0.223149\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.6115 −0.460787
\(636\) 0 0
\(637\) 31.7918 1.25964
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −39.4591 −1.55854 −0.779270 0.626688i \(-0.784410\pi\)
−0.779270 + 0.626688i \(0.784410\pi\)
\(642\) 0 0
\(643\) 39.2510i 1.54791i 0.633241 + 0.773954i \(0.281724\pi\)
−0.633241 + 0.773954i \(0.718276\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.6651i 1.08763i −0.839206 0.543814i \(-0.816980\pi\)
0.839206 0.543814i \(-0.183020\pi\)
\(648\) 0 0
\(649\) 32.9534i 1.29353i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.9791i 0.507910i 0.967216 + 0.253955i \(0.0817314\pi\)
−0.967216 + 0.253955i \(0.918269\pi\)
\(654\) 0 0
\(655\) 13.1854i 0.515195i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 25.1712 0.980532 0.490266 0.871573i \(-0.336900\pi\)
0.490266 + 0.871573i \(0.336900\pi\)
\(660\) 0 0
\(661\) 3.49951i 0.136115i −0.997681 0.0680577i \(-0.978320\pi\)
0.997681 0.0680577i \(-0.0216802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.829593 −0.0321702
\(666\) 0 0
\(667\) 2.11329 + 3.44095i 0.0818270 + 0.133234i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.91810i 0.228466i
\(672\) 0 0
\(673\) −46.5139 −1.79298 −0.896489 0.443066i \(-0.853891\pi\)
−0.896489 + 0.443066i \(0.853891\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.2333 −0.508598 −0.254299 0.967126i \(-0.581845\pi\)
−0.254299 + 0.967126i \(0.581845\pi\)
\(678\) 0 0
\(679\) −10.6256 −0.407774
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 27.9323i 1.06880i −0.845232 0.534400i \(-0.820538\pi\)
0.845232 0.534400i \(-0.179462\pi\)
\(684\) 0 0
\(685\) 12.5459 0.479354
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.41672 0.320652
\(690\) 0 0
\(691\) −12.9018 −0.490808 −0.245404 0.969421i \(-0.578921\pi\)
−0.245404 + 0.969421i \(0.578921\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.0180 0.493799
\(696\) 0 0
\(697\) 45.5609i 1.72574i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.822196 −0.0310539 −0.0155270 0.999879i \(-0.504943\pi\)
−0.0155270 + 0.999879i \(0.504943\pi\)
\(702\) 0 0
\(703\) −4.17768 −0.157564
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.76466 −0.254411
\(708\) 0 0
\(709\) 3.99096i 0.149883i 0.997188 + 0.0749417i \(0.0238771\pi\)
−0.997188 + 0.0749417i \(0.976123\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.400992 0.246274i 0.0150173 0.00922302i
\(714\) 0 0
\(715\) −14.3810 −0.537819
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 52.1805i 1.94600i 0.230800 + 0.973001i \(0.425866\pi\)
−0.230800 + 0.973001i \(0.574134\pi\)
\(720\) 0 0
\(721\) 10.8052 0.402407
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.841999i 0.0312710i
\(726\) 0 0
\(727\) 32.5482i 1.20715i −0.797308 0.603573i \(-0.793743\pi\)
0.797308 0.603573i \(-0.206257\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.02753i 0.0749909i
\(732\) 0 0
\(733\) 3.25906i 0.120376i 0.998187 + 0.0601881i \(0.0191701\pi\)
−0.998187 + 0.0601881i \(0.980830\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.9711i 1.14084i
\(738\) 0 0
\(739\) 10.9443 0.402593 0.201296 0.979530i \(-0.435485\pi\)
0.201296 + 0.979530i \(0.435485\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −17.7242 −0.650236 −0.325118 0.945673i \(-0.605404\pi\)
−0.325118 + 0.945673i \(0.605404\pi\)
\(744\) 0 0
\(745\) −0.638019 −0.0233752
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.68482i 0.317336i
\(750\) 0 0
\(751\) 1.88536i 0.0687978i 0.999408 + 0.0343989i \(0.0109517\pi\)
−0.999408 + 0.0343989i \(0.989048\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.8378 0.394427
\(756\) 0 0
\(757\) 13.7509i 0.499783i 0.968274 + 0.249892i \(0.0803950\pi\)
−0.968274 + 0.249892i \(0.919605\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.2184i 1.49417i 0.664731 + 0.747083i \(0.268546\pi\)
−0.664731 + 0.747083i \(0.731454\pi\)
\(762\) 0 0
\(763\) 17.5512 0.635397
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 87.4017i 3.15589i
\(768\) 0 0
\(769\) 34.0574i 1.22814i −0.789252 0.614070i \(-0.789531\pi\)
0.789252 0.614070i \(-0.210469\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.13234 −0.184597 −0.0922987 0.995731i \(-0.529421\pi\)
−0.0922987 + 0.995731i \(0.529421\pi\)
\(774\) 0 0
\(775\) 0.0981227 0.00352467
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.01414 −0.143822
\(780\) 0 0
\(781\) 11.3616i 0.406549i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.59811i 0.0570390i
\(786\) 0 0
\(787\) 15.3569i 0.547415i 0.961813 + 0.273707i \(0.0882500\pi\)
−0.961813 + 0.273707i \(0.911750\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.93225i 0.175371i
\(792\) 0 0
\(793\) 15.6965i 0.557397i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.7015 −1.19377 −0.596884 0.802328i \(-0.703595\pi\)
−0.596884 + 0.802328i \(0.703595\pi\)
\(798\) 0 0
\(799\) 26.4377i 0.935300i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.96082 0.280931
\(804\) 0 0
\(805\) 3.41593 + 5.56195i 0.120396 + 0.196033i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13.2530i 0.465951i −0.972483 0.232976i \(-0.925154\pi\)
0.972483