Properties

Label 6012.2.h.a.3005.5
Level $6012$
Weight $2$
Character 6012.3005
Analytic conductor $48.006$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3005.5
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.47368 q^{5} +3.93299 q^{7} +O(q^{10})\) \(q-3.47368 q^{5} +3.93299 q^{7} -4.34291i q^{11} +0.788280i q^{13} -5.32815 q^{17} +4.26219 q^{19} -6.69278 q^{23} +7.06648 q^{25} +5.99865i q^{29} -6.79664 q^{31} -13.6619 q^{35} +4.66119i q^{37} +4.62083 q^{41} -11.6022i q^{43} +12.6481i q^{47} +8.46837 q^{49} +0.406978 q^{53} +15.0859i q^{55} +6.56137 q^{59} -4.94490 q^{61} -2.73823i q^{65} +3.40874i q^{67} -0.507955 q^{71} -2.74265i q^{73} -17.0806i q^{77} +7.44663i q^{79} +6.51617 q^{83} +18.5083 q^{85} -15.1276i q^{89} +3.10029i q^{91} -14.8055 q^{95} -3.32370 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times\).

\(n\) \(3007\) \(3341\) \(4681\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) −3.47368 −1.55348 −0.776739 0.629822i \(-0.783128\pi\)
−0.776739 + 0.629822i \(0.783128\pi\)
\(6\) 0 0
\(7\) 3.93299 1.48653 0.743264 0.668998i \(-0.233276\pi\)
0.743264 + 0.668998i \(0.233276\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.34291i 1.30944i −0.755873 0.654718i \(-0.772787\pi\)
0.755873 0.654718i \(-0.227213\pi\)
\(12\) 0 0
\(13\) 0.788280i 0.218629i 0.994007 + 0.109315i \(0.0348656\pi\)
−0.994007 + 0.109315i \(0.965134\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.32815 −1.29227 −0.646133 0.763225i \(-0.723615\pi\)
−0.646133 + 0.763225i \(0.723615\pi\)
\(18\) 0 0
\(19\) 4.26219 0.977812 0.488906 0.872336i \(-0.337396\pi\)
0.488906 + 0.872336i \(0.337396\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.69278 −1.39554 −0.697770 0.716321i \(-0.745824\pi\)
−0.697770 + 0.716321i \(0.745824\pi\)
\(24\) 0 0
\(25\) 7.06648 1.41330
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.99865i 1.11392i 0.830539 + 0.556961i \(0.188033\pi\)
−0.830539 + 0.556961i \(0.811967\pi\)
\(30\) 0 0
\(31\) −6.79664 −1.22071 −0.610356 0.792127i \(-0.708974\pi\)
−0.610356 + 0.792127i \(0.708974\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −13.6619 −2.30929
\(36\) 0 0
\(37\) 4.66119i 0.766296i 0.923687 + 0.383148i \(0.125160\pi\)
−0.923687 + 0.383148i \(0.874840\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.62083 0.721652 0.360826 0.932633i \(-0.382495\pi\)
0.360826 + 0.932633i \(0.382495\pi\)
\(42\) 0 0
\(43\) 11.6022i 1.76931i −0.466243 0.884657i \(-0.654393\pi\)
0.466243 0.884657i \(-0.345607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.6481i 1.84491i 0.386106 + 0.922454i \(0.373820\pi\)
−0.386106 + 0.922454i \(0.626180\pi\)
\(48\) 0 0
\(49\) 8.46837 1.20977
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.406978 0.0559028 0.0279514 0.999609i \(-0.491102\pi\)
0.0279514 + 0.999609i \(0.491102\pi\)
\(54\) 0 0
\(55\) 15.0859i 2.03418i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.56137 0.854218 0.427109 0.904200i \(-0.359532\pi\)
0.427109 + 0.904200i \(0.359532\pi\)
\(60\) 0 0
\(61\) −4.94490 −0.633129 −0.316565 0.948571i \(-0.602529\pi\)
−0.316565 + 0.948571i \(0.602529\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.73823i 0.339636i
\(66\) 0 0
\(67\) 3.40874i 0.416444i 0.978082 + 0.208222i \(0.0667677\pi\)
−0.978082 + 0.208222i \(0.933232\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.507955 −0.0602832 −0.0301416 0.999546i \(-0.509596\pi\)
−0.0301416 + 0.999546i \(0.509596\pi\)
\(72\) 0 0
\(73\) 2.74265i 0.321003i −0.987036 0.160501i \(-0.948689\pi\)
0.987036 0.160501i \(-0.0513111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.0806i 1.94652i
\(78\) 0 0
\(79\) 7.44663i 0.837812i 0.908030 + 0.418906i \(0.137586\pi\)
−0.908030 + 0.418906i \(0.862414\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.51617 0.715242 0.357621 0.933867i \(-0.383588\pi\)
0.357621 + 0.933867i \(0.383588\pi\)
\(84\) 0 0
\(85\) 18.5083 2.00751
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.1276i 1.60352i −0.597648 0.801759i \(-0.703898\pi\)
0.597648 0.801759i \(-0.296102\pi\)
\(90\) 0 0
\(91\) 3.10029i 0.324999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.8055 −1.51901
\(96\) 0 0
\(97\) −3.32370 −0.337471 −0.168735 0.985661i \(-0.553968\pi\)
−0.168735 + 0.985661i \(0.553968\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.28457 −0.923849 −0.461925 0.886919i \(-0.652841\pi\)
−0.461925 + 0.886919i \(0.652841\pi\)
\(102\) 0 0
\(103\) 8.46976i 0.834551i −0.908780 0.417275i \(-0.862985\pi\)
0.908780 0.417275i \(-0.137015\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.4198i 1.49069i 0.666680 + 0.745344i \(0.267715\pi\)
−0.666680 + 0.745344i \(0.732285\pi\)
\(108\) 0 0
\(109\) 9.11253i 0.872822i 0.899747 + 0.436411i \(0.143751\pi\)
−0.899747 + 0.436411i \(0.856249\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.7429 1.76318 0.881591 0.472013i \(-0.156473\pi\)
0.881591 + 0.472013i \(0.156473\pi\)
\(114\) 0 0
\(115\) 23.2486 2.16794
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −20.9555 −1.92099
\(120\) 0 0
\(121\) −7.86087 −0.714625
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.17830 −0.642047
\(126\) 0 0
\(127\) −18.9978 −1.68579 −0.842893 0.538081i \(-0.819149\pi\)
−0.842893 + 0.538081i \(0.819149\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.1680 1.06312 0.531562 0.847019i \(-0.321605\pi\)
0.531562 + 0.847019i \(0.321605\pi\)
\(132\) 0 0
\(133\) 16.7631 1.45355
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.83479i 0.413064i −0.978440 0.206532i \(-0.933782\pi\)
0.978440 0.206532i \(-0.0662178\pi\)
\(138\) 0 0
\(139\) 14.2167i 1.20585i 0.797799 + 0.602923i \(0.205997\pi\)
−0.797799 + 0.602923i \(0.794003\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.42343 0.286281
\(144\) 0 0
\(145\) 20.8374i 1.73045i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.40998 0.197433 0.0987165 0.995116i \(-0.468526\pi\)
0.0987165 + 0.995116i \(0.468526\pi\)
\(150\) 0 0
\(151\) 4.37124i 0.355726i 0.984055 + 0.177863i \(0.0569184\pi\)
−0.984055 + 0.177863i \(0.943082\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.6094 1.89635
\(156\) 0 0
\(157\) 1.86364 0.148735 0.0743673 0.997231i \(-0.476306\pi\)
0.0743673 + 0.997231i \(0.476306\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −26.3226 −2.07451
\(162\) 0 0
\(163\) 22.5643i 1.76737i 0.468079 + 0.883686i \(0.344946\pi\)
−0.468079 + 0.883686i \(0.655054\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.99239 + 8.19464i 0.773234 + 0.634121i
\(168\) 0 0
\(169\) 12.3786 0.952201
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.0210i 0.761885i 0.924599 + 0.380943i \(0.124400\pi\)
−0.924599 + 0.380943i \(0.875600\pi\)
\(174\) 0 0
\(175\) 27.7924 2.10091
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.7422i 1.40086i 0.713723 + 0.700428i \(0.247008\pi\)
−0.713723 + 0.700428i \(0.752992\pi\)
\(180\) 0 0
\(181\) −10.2459 −0.761572 −0.380786 0.924663i \(-0.624347\pi\)
−0.380786 + 0.924663i \(0.624347\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.1915i 1.19042i
\(186\) 0 0
\(187\) 23.1397i 1.69214i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.2270i 1.39122i −0.718421 0.695608i \(-0.755135\pi\)
0.718421 0.695608i \(-0.244865\pi\)
\(192\) 0 0
\(193\) 5.61976i 0.404519i 0.979332 + 0.202259i \(0.0648284\pi\)
−0.979332 + 0.202259i \(0.935172\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.1539 −1.07967 −0.539836 0.841770i \(-0.681514\pi\)
−0.539836 + 0.841770i \(0.681514\pi\)
\(198\) 0 0
\(199\) −9.38899 −0.665568 −0.332784 0.943003i \(-0.607988\pi\)
−0.332784 + 0.943003i \(0.607988\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 23.5926i 1.65588i
\(204\) 0 0
\(205\) −16.0513 −1.12107
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 18.5103i 1.28038i
\(210\) 0 0
\(211\) −5.78342 −0.398147 −0.199074 0.979985i \(-0.563793\pi\)
−0.199074 + 0.979985i \(0.563793\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 40.3023i 2.74859i
\(216\) 0 0
\(217\) −26.7311 −1.81462
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.20007i 0.282528i
\(222\) 0 0
\(223\) −19.3088 −1.29301 −0.646506 0.762909i \(-0.723770\pi\)
−0.646506 + 0.762909i \(0.723770\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.98586 0.530040 0.265020 0.964243i \(-0.414621\pi\)
0.265020 + 0.964243i \(0.414621\pi\)
\(228\) 0 0
\(229\) 24.0329 1.58814 0.794070 0.607826i \(-0.207958\pi\)
0.794070 + 0.607826i \(0.207958\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.9229i 0.977630i 0.872388 + 0.488815i \(0.162571\pi\)
−0.872388 + 0.488815i \(0.837429\pi\)
\(234\) 0 0
\(235\) 43.9353i 2.86603i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.14553i 0.0740981i −0.999313 0.0370491i \(-0.988204\pi\)
0.999313 0.0370491i \(-0.0117958\pi\)
\(240\) 0 0
\(241\) 9.14226i 0.588905i −0.955666 0.294452i \(-0.904863\pi\)
0.955666 0.294452i \(-0.0951372\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −29.4164 −1.87935
\(246\) 0 0
\(247\) 3.35979i 0.213779i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.57929i 0.352162i 0.984376 + 0.176081i \(0.0563420\pi\)
−0.984376 + 0.176081i \(0.943658\pi\)
\(252\) 0 0
\(253\) 29.0661i 1.82737i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.5972 1.40957 0.704787 0.709419i \(-0.251043\pi\)
0.704787 + 0.709419i \(0.251043\pi\)
\(258\) 0 0
\(259\) 18.3324i 1.13912i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.8743i 0.855524i 0.903891 + 0.427762i \(0.140698\pi\)
−0.903891 + 0.427762i \(0.859302\pi\)
\(264\) 0 0
\(265\) −1.41371 −0.0868438
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.5090 0.762690 0.381345 0.924433i \(-0.375461\pi\)
0.381345 + 0.924433i \(0.375461\pi\)
\(270\) 0 0
\(271\) 12.7830i 0.776512i −0.921552 0.388256i \(-0.873078\pi\)
0.921552 0.388256i \(-0.126922\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.6891i 1.85062i
\(276\) 0 0
\(277\) 1.74046i 0.104574i 0.998632 + 0.0522869i \(0.0166510\pi\)
−0.998632 + 0.0522869i \(0.983349\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.8524i 1.42292i 0.702728 + 0.711458i \(0.251965\pi\)
−0.702728 + 0.711458i \(0.748035\pi\)
\(282\) 0 0
\(283\) −21.3744 −1.27057 −0.635287 0.772276i \(-0.719118\pi\)
−0.635287 + 0.772276i \(0.719118\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.1736 1.07276
\(288\) 0 0
\(289\) 11.3892 0.669953
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.0998i 0.823719i 0.911247 + 0.411859i \(0.135120\pi\)
−0.911247 + 0.411859i \(0.864880\pi\)
\(294\) 0 0
\(295\) −22.7921 −1.32701
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.27578i 0.305106i
\(300\) 0 0
\(301\) 45.6311i 2.63013i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.1770 0.983553
\(306\) 0 0
\(307\) 9.02299i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.2184i 0.579432i −0.957113 0.289716i \(-0.906439\pi\)
0.957113 0.289716i \(-0.0935608\pi\)
\(312\) 0 0
\(313\) 16.6078i 0.938726i 0.883005 + 0.469363i \(0.155516\pi\)
−0.883005 + 0.469363i \(0.844484\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.5050i 1.43250i 0.697842 + 0.716252i \(0.254144\pi\)
−0.697842 + 0.716252i \(0.745856\pi\)
\(318\) 0 0
\(319\) 26.0516 1.45861
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −22.7096 −1.26359
\(324\) 0 0
\(325\) 5.57036i 0.308988i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 49.7446i 2.74251i
\(330\) 0 0
\(331\) 27.8955i 1.53328i 0.642080 + 0.766638i \(0.278072\pi\)
−0.642080 + 0.766638i \(0.721928\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.8409i 0.646937i
\(336\) 0 0
\(337\) 25.8957 1.41063 0.705314 0.708895i \(-0.250806\pi\)
0.705314 + 0.708895i \(0.250806\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 29.5172i 1.59845i
\(342\) 0 0
\(343\) 5.77508 0.311825
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −27.8704 −1.49616 −0.748081 0.663608i \(-0.769024\pi\)
−0.748081 + 0.663608i \(0.769024\pi\)
\(348\) 0 0
\(349\) 28.6541i 1.53382i 0.641757 + 0.766908i \(0.278206\pi\)
−0.641757 + 0.766908i \(0.721794\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.19480i 0.223267i −0.993749 0.111633i \(-0.964392\pi\)
0.993749 0.111633i \(-0.0356083\pi\)
\(354\) 0 0
\(355\) 1.76447 0.0936486
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.52346i 0.449851i −0.974376 0.224925i \(-0.927786\pi\)
0.974376 0.224925i \(-0.0722138\pi\)
\(360\) 0 0
\(361\) −0.833772 −0.0438827
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.52710i 0.498671i
\(366\) 0 0
\(367\) −0.593513 −0.0309812 −0.0154906 0.999880i \(-0.504931\pi\)
−0.0154906 + 0.999880i \(0.504931\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.60064 0.0831011
\(372\) 0 0
\(373\) 24.8382i 1.28607i 0.765836 + 0.643036i \(0.222325\pi\)
−0.765836 + 0.643036i \(0.777675\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.72861 −0.243536
\(378\) 0 0
\(379\) 30.1994i 1.55124i 0.631200 + 0.775620i \(0.282563\pi\)
−0.631200 + 0.775620i \(0.717437\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 33.7964i 1.72692i 0.504420 + 0.863458i \(0.331706\pi\)
−0.504420 + 0.863458i \(0.668294\pi\)
\(384\) 0 0
\(385\) 59.3326i 3.02387i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.1882 −1.02358 −0.511791 0.859110i \(-0.671018\pi\)
−0.511791 + 0.859110i \(0.671018\pi\)
\(390\) 0 0
\(391\) 35.6601 1.80341
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.8673i 1.30152i
\(396\) 0 0
\(397\) −18.5320 −0.930093 −0.465046 0.885286i \(-0.653962\pi\)
−0.465046 + 0.885286i \(0.653962\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.0402 −0.801007 −0.400504 0.916295i \(-0.631165\pi\)
−0.400504 + 0.916295i \(0.631165\pi\)
\(402\) 0 0
\(403\) 5.35765i 0.266884i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.2432 1.00342
\(408\) 0 0
\(409\) 28.4443 1.40648 0.703239 0.710953i \(-0.251736\pi\)
0.703239 + 0.710953i \(0.251736\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 25.8058 1.26982
\(414\) 0 0
\(415\) −22.6351 −1.11111
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.24873i 0.451830i −0.974147 0.225915i \(-0.927463\pi\)
0.974147 0.225915i \(-0.0725372\pi\)
\(420\) 0 0
\(421\) 35.8984 1.74958 0.874791 0.484501i \(-0.160999\pi\)
0.874791 + 0.484501i \(0.160999\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −37.6513 −1.82636
\(426\) 0 0
\(427\) −19.4482 −0.941165
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0918i 0.678779i 0.940646 + 0.339389i \(0.110220\pi\)
−0.940646 + 0.339389i \(0.889780\pi\)
\(432\) 0 0
\(433\) −0.284485 −0.0136715 −0.00683574 0.999977i \(-0.502176\pi\)
−0.00683574 + 0.999977i \(0.502176\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −28.5259 −1.36458
\(438\) 0 0
\(439\) 0.454366i 0.0216857i 0.999941 + 0.0108428i \(0.00345145\pi\)
−0.999941 + 0.0108428i \(0.996549\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −33.4825 −1.59080 −0.795400 0.606085i \(-0.792739\pi\)
−0.795400 + 0.606085i \(0.792739\pi\)
\(444\) 0 0
\(445\) 52.5483i 2.49103i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.3532i 0.866143i −0.901360 0.433072i \(-0.857430\pi\)
0.901360 0.433072i \(-0.142570\pi\)
\(450\) 0 0
\(451\) 20.0678i 0.944957i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.7694i 0.504879i
\(456\) 0 0
\(457\) 12.3285i 0.576704i −0.957525 0.288352i \(-0.906893\pi\)
0.957525 0.288352i \(-0.0931073\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.7458i 0.547058i −0.961864 0.273529i \(-0.911809\pi\)
0.961864 0.273529i \(-0.0881910\pi\)
\(462\) 0 0
\(463\) 19.8903i 0.924380i −0.886781 0.462190i \(-0.847064\pi\)
0.886781 0.462190i \(-0.152936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.2627i 1.30784i −0.756563 0.653921i \(-0.773123\pi\)
0.756563 0.653921i \(-0.226877\pi\)
\(468\) 0 0
\(469\) 13.4065i 0.619056i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −50.3872 −2.31680
\(474\) 0 0
\(475\) 30.1187 1.38194
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.7430 1.54176 0.770879 0.636981i \(-0.219817\pi\)
0.770879 + 0.636981i \(0.219817\pi\)
\(480\) 0 0
\(481\) −3.67432 −0.167535
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.5455 0.524254
\(486\) 0 0
\(487\) 23.3213i 1.05679i 0.848999 + 0.528395i \(0.177206\pi\)
−0.848999 + 0.528395i \(0.822794\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.22659i 0.145614i −0.997346 0.0728070i \(-0.976804\pi\)
0.997346 0.0728070i \(-0.0231957\pi\)
\(492\) 0 0
\(493\) 31.9617i 1.43948i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.99778 −0.0896126
\(498\) 0 0
\(499\) 35.1020i 1.57138i −0.618621 0.785690i \(-0.712308\pi\)
0.618621 0.785690i \(-0.287692\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.35604i 0.105051i 0.998620 + 0.0525253i \(0.0167270\pi\)
−0.998620 + 0.0525253i \(0.983273\pi\)
\(504\) 0 0
\(505\) 32.2517 1.43518
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.04422i 0.179257i −0.995975 0.0896284i \(-0.971432\pi\)
0.995975 0.0896284i \(-0.0285679\pi\)
\(510\) 0 0
\(511\) 10.7868i 0.477180i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 29.4213i 1.29646i
\(516\) 0 0
\(517\) 54.9294 2.41579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −34.7536 −1.52258 −0.761292 0.648410i \(-0.775434\pi\)
−0.761292 + 0.648410i \(0.775434\pi\)
\(522\) 0 0
\(523\) 6.98394 0.305386 0.152693 0.988274i \(-0.451205\pi\)
0.152693 + 0.988274i \(0.451205\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 36.2135 1.57749
\(528\) 0 0
\(529\) 21.7933 0.947534
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.64250i 0.157774i
\(534\) 0 0
\(535\) 53.5635i 2.31575i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.7774i 1.58411i
\(540\) 0 0
\(541\) 13.1092i 0.563606i 0.959472 + 0.281803i \(0.0909325\pi\)
−0.959472 + 0.281803i \(0.909067\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 31.6541i 1.35591i
\(546\) 0 0
\(547\) 23.7068i 1.01363i −0.862055 0.506815i \(-0.830823\pi\)
0.862055 0.506815i \(-0.169177\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 25.5674i 1.08921i
\(552\) 0 0
\(553\) 29.2875i 1.24543i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.9063i 0.589230i 0.955616 + 0.294615i \(0.0951914\pi\)
−0.955616 + 0.294615i \(0.904809\pi\)
\(558\) 0 0
\(559\) 9.14575 0.386824
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.83747i 0.203875i −0.994791 0.101938i \(-0.967496\pi\)
0.994791 0.101938i \(-0.0325042\pi\)
\(564\) 0 0
\(565\) −65.1069 −2.73907
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.14692 −0.0900037 −0.0450019 0.998987i \(-0.514329\pi\)
−0.0450019 + 0.998987i \(0.514329\pi\)
\(570\) 0 0
\(571\) 37.4846i 1.56868i 0.620329 + 0.784342i \(0.286999\pi\)
−0.620329 + 0.784342i \(0.713001\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −47.2944 −1.97231
\(576\) 0 0
\(577\) 12.9723 0.540042 0.270021 0.962854i \(-0.412969\pi\)
0.270021 + 0.962854i \(0.412969\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 25.6280 1.06323
\(582\) 0 0
\(583\) 1.76747i 0.0732012i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.5888 1.17999 0.589994 0.807408i \(-0.299130\pi\)
0.589994 + 0.807408i \(0.299130\pi\)
\(588\) 0 0
\(589\) −28.9685 −1.19363
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.67910 −0.192148 −0.0960739 0.995374i \(-0.530629\pi\)
−0.0960739 + 0.995374i \(0.530629\pi\)
\(594\) 0 0
\(595\) 72.7929 2.98422
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.9092i 1.09948i −0.835336 0.549740i \(-0.814727\pi\)
0.835336 0.549740i \(-0.185273\pi\)
\(600\) 0 0
\(601\) 15.5403 0.633902 0.316951 0.948442i \(-0.397341\pi\)
0.316951 + 0.948442i \(0.397341\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 27.3062 1.11015
\(606\) 0 0
\(607\) 4.06302i 0.164913i −0.996595 0.0824564i \(-0.973723\pi\)
0.996595 0.0824564i \(-0.0262765\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.97020 −0.403351
\(612\) 0 0
\(613\) 23.8370 0.962769 0.481384 0.876510i \(-0.340134\pi\)
0.481384 + 0.876510i \(0.340134\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.21006i 0.0487152i −0.999703 0.0243576i \(-0.992246\pi\)
0.999703 0.0243576i \(-0.00775403\pi\)
\(618\) 0 0
\(619\) 18.5597i 0.745977i 0.927836 + 0.372988i \(0.121667\pi\)
−0.927836 + 0.372988i \(0.878333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 59.4965i 2.38367i
\(624\) 0 0
\(625\) −10.3973 −0.415890
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.8356i 0.990258i
\(630\) 0 0
\(631\) −41.6100 −1.65647 −0.828234 0.560382i \(-0.810654\pi\)
−0.828234 + 0.560382i \(0.810654\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 65.9925 2.61883
\(636\) 0 0
\(637\) 6.67544i 0.264491i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.20133 −0.363431 −0.181715 0.983351i \(-0.558165\pi\)
−0.181715 + 0.983351i \(0.558165\pi\)
\(642\) 0 0
\(643\) 5.18930i 0.204646i −0.994751 0.102323i \(-0.967372\pi\)
0.994751 0.102323i \(-0.0326275\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.7966 1.25005 0.625027 0.780603i \(-0.285088\pi\)
0.625027 + 0.780603i \(0.285088\pi\)
\(648\) 0 0
\(649\) 28.4954i 1.11854i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 41.2069i 1.61255i −0.591542 0.806275i \(-0.701480\pi\)
0.591542 0.806275i \(-0.298520\pi\)
\(654\) 0 0
\(655\) −42.2678 −1.65154
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −34.0275 −1.32552 −0.662761 0.748831i \(-0.730616\pi\)
−0.662761 + 0.748831i \(0.730616\pi\)
\(660\) 0 0
\(661\) 8.80722i 0.342561i −0.985222 0.171281i \(-0.945210\pi\)
0.985222 0.171281i \(-0.0547905\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −58.2298 −2.25805
\(666\) 0 0
\(667\) 40.1476i 1.55452i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 21.4752i 0.829043i
\(672\) 0 0
\(673\) 8.37773i 0.322938i −0.986878 0.161469i \(-0.948377\pi\)
0.986878 0.161469i \(-0.0516231\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.16558i 0.198529i 0.995061 + 0.0992647i \(0.0316490\pi\)
−0.995061 + 0.0992647i \(0.968351\pi\)
\(678\) 0 0
\(679\) −13.0721 −0.501660
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17.6447 −0.675157 −0.337579 0.941297i \(-0.609608\pi\)
−0.337579 + 0.941297i \(0.609608\pi\)
\(684\) 0 0
\(685\) 16.7945i 0.641686i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.320813i 0.0122220i
\(690\) 0 0
\(691\) 44.5617i 1.69521i 0.530629 + 0.847604i \(0.321956\pi\)
−0.530629 + 0.847604i \(0.678044\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 49.3844i 1.87326i
\(696\) 0 0
\(697\) −24.6205 −0.932567
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.36642i 0.240456i −0.992746 0.120228i \(-0.961637\pi\)
0.992746 0.120228i \(-0.0383626\pi\)
\(702\) 0 0
\(703\) 19.8669i 0.749293i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −36.5161 −1.37333
\(708\) 0 0
\(709\) 12.1344i 0.455718i −0.973694 0.227859i \(-0.926828\pi\)
0.973694 0.227859i \(-0.0731725\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 45.4884 1.70355
\(714\) 0 0
\(715\) −11.8919 −0.444732
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.35686 0.162483 0.0812417 0.996694i \(-0.474111\pi\)
0.0812417 + 0.996694i \(0.474111\pi\)
\(720\) 0 0
\(721\) 33.3115i 1.24058i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 42.3893i 1.57430i
\(726\) 0 0
\(727\) 47.6676i 1.76789i −0.467587 0.883947i \(-0.654876\pi\)
0.467587 0.883947i \(-0.345124\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 61.8181i 2.28642i
\(732\) 0 0
\(733\) −17.4188 −0.643377 −0.321688 0.946846i \(-0.604250\pi\)
−0.321688 + 0.946846i \(0.604250\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.8039 0.545307
\(738\) 0 0
\(739\) 10.8508i 0.399153i 0.979882 + 0.199576i \(0.0639566\pi\)
−0.979882 + 0.199576i \(0.936043\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.2937i 0.487699i 0.969813 + 0.243850i \(0.0784103\pi\)
−0.969813 + 0.243850i \(0.921590\pi\)
\(744\) 0 0
\(745\) −8.37150 −0.306708
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 60.6458i 2.21595i
\(750\) 0 0
\(751\) 7.66618i 0.279743i −0.990170 0.139871i \(-0.955331\pi\)
0.990170 0.139871i \(-0.0446689\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.1843i 0.552613i
\(756\) 0 0
\(757\) 1.28794 0.0468109 0.0234055 0.999726i \(-0.492549\pi\)
0.0234055 + 0.999726i \(0.492549\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.1314i 0.693511i −0.937956 0.346756i \(-0.887283\pi\)
0.937956 0.346756i \(-0.112717\pi\)
\(762\) 0 0
\(763\) 35.8395i 1.29748i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.17219i 0.186757i
\(768\) 0 0
\(769\) 25.6056i 0.923360i 0.887047 + 0.461680i \(0.152753\pi\)
−0.887047 + 0.461680i \(0.847247\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.6184 −1.53288 −0.766439 0.642317i \(-0.777973\pi\)
−0.766439 + 0.642317i \(0.777973\pi\)
\(774\) 0 0
\(775\) −48.0283 −1.72523
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.6948 0.705640
\(780\) 0 0
\(781\) 2.20600i 0.0789370i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.47369 −0.231056
\(786\) 0 0
\(787\) 54.9021i 1.95705i −0.206132 0.978524i \(-0.566088\pi\)
0.206132 0.978524i \(-0.433912\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 73.7155 2.62102
\(792\) 0 0
\(793\) 3.89796i 0.138421i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.3457 −0.933211 −0.466606 0.884465i \(-0.654523\pi\)
−0.466606 + 0.884465i \(0.654523\pi\)
\(798\) 0 0
\(799\) 67.3908i 2.38411i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.9111 −0.420333
\(804\) 0 0
\(805\) 91.4364 3.22271
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 44.8080i 1.57537i 0.616081 + 0.787683i \(0.288719\pi\)
−0.616081 + 0.787683i \(0.711281\pi\)
\(810\) 0 0
\(811\) 8.29490i 0.291273i 0.989338 + 0.145637i \(0.0465230\pi\)
−0.989338 + 0.145637i \(0.953477\pi\)