Properties

Label 6012.2.h.a.3005.1
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 3005.1
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.2

$q$-expansion

\(f(q)\) \(=\) \(q-4.09297 q^{5} +2.29241 q^{7} +O(q^{10})\) \(q-4.09297 q^{5} +2.29241 q^{7} +1.40323i q^{11} +5.26451i q^{13} -0.313267 q^{17} -0.473923 q^{19} +8.04034 q^{23} +11.7524 q^{25} -2.26656i q^{29} +1.40647 q^{31} -9.38276 q^{35} -3.51582i q^{37} -2.02537 q^{41} +7.09211i q^{43} -8.55435i q^{47} -1.74486 q^{49} -8.65647 q^{53} -5.74337i q^{55} +2.04140 q^{59} +11.9929 q^{61} -21.5475i q^{65} +2.40852i q^{67} -10.6701 q^{71} -5.44685i q^{73} +3.21677i q^{77} -3.13005i q^{79} -5.01486 q^{83} +1.28219 q^{85} -3.62776i q^{89} +12.0684i q^{91} +1.93975 q^{95} +13.7994 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) −4.09297 −1.83043 −0.915216 0.402963i \(-0.867980\pi\)
−0.915216 + 0.402963i \(0.867980\pi\)
\(6\) 0 0
\(7\) 2.29241 0.866449 0.433224 0.901286i \(-0.357376\pi\)
0.433224 + 0.901286i \(0.357376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.40323i 0.423089i 0.977368 + 0.211545i \(0.0678494\pi\)
−0.977368 + 0.211545i \(0.932151\pi\)
\(12\) 0 0
\(13\) 5.26451i 1.46011i 0.683387 + 0.730057i \(0.260506\pi\)
−0.683387 + 0.730057i \(0.739494\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.313267 −0.0759783 −0.0379891 0.999278i \(-0.512095\pi\)
−0.0379891 + 0.999278i \(0.512095\pi\)
\(18\) 0 0
\(19\) −0.473923 −0.108725 −0.0543627 0.998521i \(-0.517313\pi\)
−0.0543627 + 0.998521i \(0.517313\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.04034 1.67653 0.838263 0.545265i \(-0.183571\pi\)
0.838263 + 0.545265i \(0.183571\pi\)
\(24\) 0 0
\(25\) 11.7524 2.35048
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.26656i 0.420889i −0.977606 0.210445i \(-0.932509\pi\)
0.977606 0.210445i \(-0.0674912\pi\)
\(30\) 0 0
\(31\) 1.40647 0.252609 0.126304 0.991992i \(-0.459688\pi\)
0.126304 + 0.991992i \(0.459688\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.38276 −1.58598
\(36\) 0 0
\(37\) 3.51582i 0.577997i −0.957330 0.288998i \(-0.906678\pi\)
0.957330 0.288998i \(-0.0933222\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.02537 −0.316310 −0.158155 0.987414i \(-0.550554\pi\)
−0.158155 + 0.987414i \(0.550554\pi\)
\(42\) 0 0
\(43\) 7.09211i 1.08154i 0.841171 + 0.540769i \(0.181867\pi\)
−0.841171 + 0.540769i \(0.818133\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.55435i 1.24778i −0.781512 0.623890i \(-0.785551\pi\)
0.781512 0.623890i \(-0.214449\pi\)
\(48\) 0 0
\(49\) −1.74486 −0.249266
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.65647 −1.18906 −0.594529 0.804074i \(-0.702661\pi\)
−0.594529 + 0.804074i \(0.702661\pi\)
\(54\) 0 0
\(55\) 5.74337i 0.774436i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.04140 0.265767 0.132884 0.991132i \(-0.457576\pi\)
0.132884 + 0.991132i \(0.457576\pi\)
\(60\) 0 0
\(61\) 11.9929 1.53554 0.767770 0.640726i \(-0.221366\pi\)
0.767770 + 0.640726i \(0.221366\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.5475i 2.67264i
\(66\) 0 0
\(67\) 2.40852i 0.294247i 0.989118 + 0.147124i \(0.0470016\pi\)
−0.989118 + 0.147124i \(0.952998\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.6701 −1.26631 −0.633153 0.774027i \(-0.718240\pi\)
−0.633153 + 0.774027i \(0.718240\pi\)
\(72\) 0 0
\(73\) 5.44685i 0.637505i −0.947838 0.318753i \(-0.896736\pi\)
0.947838 0.318753i \(-0.103264\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.21677i 0.366585i
\(78\) 0 0
\(79\) 3.13005i 0.352158i −0.984376 0.176079i \(-0.943659\pi\)
0.984376 0.176079i \(-0.0563415\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.01486 −0.550452 −0.275226 0.961380i \(-0.588753\pi\)
−0.275226 + 0.961380i \(0.588753\pi\)
\(84\) 0 0
\(85\) 1.28219 0.139073
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.62776i 0.384542i −0.981342 0.192271i \(-0.938415\pi\)
0.981342 0.192271i \(-0.0615853\pi\)
\(90\) 0 0
\(91\) 12.0684i 1.26511i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.93975 0.199014
\(96\) 0 0
\(97\) 13.7994 1.40111 0.700557 0.713596i \(-0.252935\pi\)
0.700557 + 0.713596i \(0.252935\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.20749 0.418661 0.209330 0.977845i \(-0.432872\pi\)
0.209330 + 0.977845i \(0.432872\pi\)
\(102\) 0 0
\(103\) 17.4177i 1.71621i 0.513470 + 0.858107i \(0.328360\pi\)
−0.513470 + 0.858107i \(0.671640\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.21671i 0.891013i 0.895279 + 0.445507i \(0.146976\pi\)
−0.895279 + 0.445507i \(0.853024\pi\)
\(108\) 0 0
\(109\) 13.7151i 1.31367i 0.754036 + 0.656833i \(0.228105\pi\)
−0.754036 + 0.656833i \(0.771895\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.9552 −1.12465 −0.562326 0.826916i \(-0.690093\pi\)
−0.562326 + 0.826916i \(0.690093\pi\)
\(114\) 0 0
\(115\) −32.9089 −3.06877
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.718135 −0.0658313
\(120\) 0 0
\(121\) 9.03095 0.820996
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −27.6375 −2.47197
\(126\) 0 0
\(127\) 0.269256 0.0238926 0.0119463 0.999929i \(-0.496197\pi\)
0.0119463 + 0.999929i \(0.496197\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.7781 0.941688 0.470844 0.882217i \(-0.343949\pi\)
0.470844 + 0.882217i \(0.343949\pi\)
\(132\) 0 0
\(133\) −1.08642 −0.0942049
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.6657i 1.16754i 0.811920 + 0.583768i \(0.198422\pi\)
−0.811920 + 0.583768i \(0.801578\pi\)
\(138\) 0 0
\(139\) 8.54589i 0.724853i −0.932012 0.362426i \(-0.881948\pi\)
0.932012 0.362426i \(-0.118052\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.38731 −0.617758
\(144\) 0 0
\(145\) 9.27696i 0.770409i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.13771 −0.338974 −0.169487 0.985532i \(-0.554211\pi\)
−0.169487 + 0.985532i \(0.554211\pi\)
\(150\) 0 0
\(151\) 13.7983i 1.12289i 0.827515 + 0.561443i \(0.189754\pi\)
−0.827515 + 0.561443i \(0.810246\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.75662 −0.462383
\(156\) 0 0
\(157\) −15.9049 −1.26935 −0.634675 0.772779i \(-0.718866\pi\)
−0.634675 + 0.772779i \(0.718866\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.4317 1.45263
\(162\) 0 0
\(163\) 9.80969i 0.768354i 0.923259 + 0.384177i \(0.125515\pi\)
−0.923259 + 0.384177i \(0.874485\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.8941 + 5.05270i 0.920395 + 0.390990i
\(168\) 0 0
\(169\) −14.7151 −1.13193
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.08630i 0.310675i 0.987861 + 0.155338i \(0.0496466\pi\)
−0.987861 + 0.155338i \(0.950353\pi\)
\(174\) 0 0
\(175\) 26.9413 2.03657
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.8548i 1.93248i 0.257650 + 0.966238i \(0.417052\pi\)
−0.257650 + 0.966238i \(0.582948\pi\)
\(180\) 0 0
\(181\) 4.62380 0.343685 0.171842 0.985124i \(-0.445028\pi\)
0.171842 + 0.985124i \(0.445028\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.3901i 1.05798i
\(186\) 0 0
\(187\) 0.439584i 0.0321456i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0201i 1.44861i 0.689482 + 0.724303i \(0.257838\pi\)
−0.689482 + 0.724303i \(0.742162\pi\)
\(192\) 0 0
\(193\) 25.5047i 1.83586i −0.396737 0.917932i \(-0.629857\pi\)
0.396737 0.917932i \(-0.370143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.7184 1.33363 0.666817 0.745222i \(-0.267656\pi\)
0.666817 + 0.745222i \(0.267656\pi\)
\(198\) 0 0
\(199\) 1.99785 0.141624 0.0708120 0.997490i \(-0.477441\pi\)
0.0708120 + 0.997490i \(0.477441\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.19588i 0.364679i
\(204\) 0 0
\(205\) 8.28978 0.578983
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.665022i 0.0460005i
\(210\) 0 0
\(211\) −7.85976 −0.541088 −0.270544 0.962708i \(-0.587204\pi\)
−0.270544 + 0.962708i \(0.587204\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 29.0278i 1.97968i
\(216\) 0 0
\(217\) 3.22419 0.218873
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.64920i 0.110937i
\(222\) 0 0
\(223\) −9.96950 −0.667607 −0.333803 0.942643i \(-0.608332\pi\)
−0.333803 + 0.942643i \(0.608332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.58319 −0.171452 −0.0857262 0.996319i \(-0.527321\pi\)
−0.0857262 + 0.996319i \(0.527321\pi\)
\(228\) 0 0
\(229\) −0.0922633 −0.00609693 −0.00304846 0.999995i \(-0.500970\pi\)
−0.00304846 + 0.999995i \(0.500970\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.0534i 1.44477i −0.691492 0.722384i \(-0.743046\pi\)
0.691492 0.722384i \(-0.256954\pi\)
\(234\) 0 0
\(235\) 35.0127i 2.28398i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.9512i 1.41991i 0.704249 + 0.709953i \(0.251284\pi\)
−0.704249 + 0.709953i \(0.748716\pi\)
\(240\) 0 0
\(241\) 20.8816i 1.34510i −0.740052 0.672550i \(-0.765199\pi\)
0.740052 0.672550i \(-0.234801\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.14168 0.456265
\(246\) 0 0
\(247\) 2.49497i 0.158751i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.8849i 0.939526i 0.882793 + 0.469763i \(0.155661\pi\)
−0.882793 + 0.469763i \(0.844339\pi\)
\(252\) 0 0
\(253\) 11.2824i 0.709320i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.502162 0.0313240 0.0156620 0.999877i \(-0.495014\pi\)
0.0156620 + 0.999877i \(0.495014\pi\)
\(258\) 0 0
\(259\) 8.05969i 0.500804i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.6073i 1.08571i −0.839826 0.542855i \(-0.817343\pi\)
0.839826 0.542855i \(-0.182657\pi\)
\(264\) 0 0
\(265\) 35.4307 2.17649
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.56273 0.217223 0.108612 0.994084i \(-0.465359\pi\)
0.108612 + 0.994084i \(0.465359\pi\)
\(270\) 0 0
\(271\) 16.9568i 1.03005i −0.857174 0.515027i \(-0.827782\pi\)
0.857174 0.515027i \(-0.172218\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.4913i 0.994464i
\(276\) 0 0
\(277\) 16.7583i 1.00691i 0.864022 + 0.503454i \(0.167938\pi\)
−0.864022 + 0.503454i \(0.832062\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.5849i 0.750752i 0.926873 + 0.375376i \(0.122486\pi\)
−0.926873 + 0.375376i \(0.877514\pi\)
\(282\) 0 0
\(283\) 5.95647 0.354075 0.177038 0.984204i \(-0.443349\pi\)
0.177038 + 0.984204i \(0.443349\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.64297 −0.274066
\(288\) 0 0
\(289\) −16.9019 −0.994227
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.1176i 0.649494i −0.945801 0.324747i \(-0.894721\pi\)
0.945801 0.324747i \(-0.105279\pi\)
\(294\) 0 0
\(295\) −8.35539 −0.486469
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 42.3285i 2.44792i
\(300\) 0 0
\(301\) 16.2580i 0.937097i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −49.0868 −2.81070
\(306\) 0 0
\(307\) 2.18955i 0.124964i −0.998046 0.0624821i \(-0.980098\pi\)
0.998046 0.0624821i \(-0.0199016\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.8065i 1.34994i 0.737844 + 0.674971i \(0.235844\pi\)
−0.737844 + 0.674971i \(0.764156\pi\)
\(312\) 0 0
\(313\) 21.6598i 1.22428i 0.790748 + 0.612142i \(0.209692\pi\)
−0.790748 + 0.612142i \(0.790308\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.9496i 1.62597i 0.582282 + 0.812987i \(0.302160\pi\)
−0.582282 + 0.812987i \(0.697840\pi\)
\(318\) 0 0
\(319\) 3.18050 0.178074
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.148464 0.00826076
\(324\) 0 0
\(325\) 61.8708i 3.43197i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.6101i 1.08114i
\(330\) 0 0
\(331\) 30.2737i 1.66400i 0.554779 + 0.831998i \(0.312803\pi\)
−0.554779 + 0.831998i \(0.687197\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.85800i 0.538600i
\(336\) 0 0
\(337\) 1.69185 0.0921611 0.0460805 0.998938i \(-0.485327\pi\)
0.0460805 + 0.998938i \(0.485327\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.97359i 0.106876i
\(342\) 0 0
\(343\) −20.0468 −1.08243
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.9707 1.01840 0.509201 0.860648i \(-0.329941\pi\)
0.509201 + 0.860648i \(0.329941\pi\)
\(348\) 0 0
\(349\) 10.5336i 0.563849i 0.959437 + 0.281925i \(0.0909728\pi\)
−0.959437 + 0.281925i \(0.909027\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.52080i 0.506741i −0.967369 0.253370i \(-0.918461\pi\)
0.967369 0.253370i \(-0.0815392\pi\)
\(354\) 0 0
\(355\) 43.6724 2.31789
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.5816i 0.822366i −0.911553 0.411183i \(-0.865116\pi\)
0.911553 0.411183i \(-0.134884\pi\)
\(360\) 0 0
\(361\) −18.7754 −0.988179
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.2938i 1.16691i
\(366\) 0 0
\(367\) −31.7003 −1.65474 −0.827371 0.561655i \(-0.810165\pi\)
−0.827371 + 0.561655i \(0.810165\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.8442 −1.03026
\(372\) 0 0
\(373\) 18.7719i 0.971973i 0.873966 + 0.485986i \(0.161540\pi\)
−0.873966 + 0.485986i \(0.838460\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.9323 0.614546
\(378\) 0 0
\(379\) 24.3145i 1.24895i −0.781045 0.624475i \(-0.785313\pi\)
0.781045 0.624475i \(-0.214687\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.0531i 1.33125i 0.746285 + 0.665627i \(0.231836\pi\)
−0.746285 + 0.665627i \(0.768164\pi\)
\(384\) 0 0
\(385\) 13.1662i 0.671009i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.3750 −1.54007 −0.770036 0.638001i \(-0.779762\pi\)
−0.770036 + 0.638001i \(0.779762\pi\)
\(390\) 0 0
\(391\) −2.51877 −0.127380
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.8112i 0.644602i
\(396\) 0 0
\(397\) −19.8263 −0.995053 −0.497527 0.867449i \(-0.665758\pi\)
−0.497527 + 0.867449i \(0.665758\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.0482552 −0.00240975 −0.00120487 0.999999i \(-0.500384\pi\)
−0.00120487 + 0.999999i \(0.500384\pi\)
\(402\) 0 0
\(403\) 7.40436i 0.368837i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.93349 0.244544
\(408\) 0 0
\(409\) 13.5257 0.668804 0.334402 0.942431i \(-0.391466\pi\)
0.334402 + 0.942431i \(0.391466\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.67972 0.230274
\(414\) 0 0
\(415\) 20.5257 1.00757
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.3643i 0.604036i −0.953302 0.302018i \(-0.902340\pi\)
0.953302 0.302018i \(-0.0976603\pi\)
\(420\) 0 0
\(421\) −33.8891 −1.65166 −0.825828 0.563923i \(-0.809292\pi\)
−0.825828 + 0.563923i \(0.809292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.68164 −0.178586
\(426\) 0 0
\(427\) 27.4927 1.33047
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.6335i 0.608534i 0.952587 + 0.304267i \(0.0984115\pi\)
−0.952587 + 0.304267i \(0.901589\pi\)
\(432\) 0 0
\(433\) 8.31009 0.399357 0.199679 0.979861i \(-0.436010\pi\)
0.199679 + 0.979861i \(0.436010\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.81050 −0.182281
\(438\) 0 0
\(439\) 32.4531i 1.54890i 0.632634 + 0.774451i \(0.281974\pi\)
−0.632634 + 0.774451i \(0.718026\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.84588 −0.325258 −0.162629 0.986687i \(-0.551997\pi\)
−0.162629 + 0.986687i \(0.551997\pi\)
\(444\) 0 0
\(445\) 14.8483i 0.703879i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17.2495i 0.814056i 0.913416 + 0.407028i \(0.133435\pi\)
−0.913416 + 0.407028i \(0.866565\pi\)
\(450\) 0 0
\(451\) 2.84205i 0.133827i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 49.3957i 2.31571i
\(456\) 0 0
\(457\) 23.2886i 1.08939i 0.838633 + 0.544697i \(0.183355\pi\)
−0.838633 + 0.544697i \(0.816645\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.93021i 0.0898990i 0.998989 + 0.0449495i \(0.0143127\pi\)
−0.998989 + 0.0449495i \(0.985687\pi\)
\(462\) 0 0
\(463\) 7.27902i 0.338285i −0.985592 0.169142i \(-0.945900\pi\)
0.985592 0.169142i \(-0.0540998\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.77061i 0.220757i −0.993890 0.110379i \(-0.964794\pi\)
0.993890 0.110379i \(-0.0352064\pi\)
\(468\) 0 0
\(469\) 5.52131i 0.254950i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9.95185 −0.457587
\(474\) 0 0
\(475\) −5.56974 −0.255557
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.73270 −0.0791691 −0.0395846 0.999216i \(-0.512603\pi\)
−0.0395846 + 0.999216i \(0.512603\pi\)
\(480\) 0 0
\(481\) 18.5091 0.843940
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −56.4805 −2.56465
\(486\) 0 0
\(487\) 11.9288i 0.540544i −0.962784 0.270272i \(-0.912886\pi\)
0.962784 0.270272i \(-0.0871136\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.7052i 0.753895i 0.926235 + 0.376948i \(0.123026\pi\)
−0.926235 + 0.376948i \(0.876974\pi\)
\(492\) 0 0
\(493\) 0.710037i 0.0319784i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.4602 −1.09719
\(498\) 0 0
\(499\) 13.7439i 0.615263i −0.951506 0.307631i \(-0.900464\pi\)
0.951506 0.307631i \(-0.0995364\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.44347i 0.331888i 0.986135 + 0.165944i \(0.0530671\pi\)
−0.986135 + 0.165944i \(0.946933\pi\)
\(504\) 0 0
\(505\) −17.2211 −0.766330
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.1790i 0.894418i −0.894429 0.447209i \(-0.852418\pi\)
0.894429 0.447209i \(-0.147582\pi\)
\(510\) 0 0
\(511\) 12.4864i 0.552366i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 71.2901i 3.14142i
\(516\) 0 0
\(517\) 12.0037 0.527922
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.8880 1.17798 0.588991 0.808139i \(-0.299525\pi\)
0.588991 + 0.808139i \(0.299525\pi\)
\(522\) 0 0
\(523\) −20.0539 −0.876895 −0.438448 0.898757i \(-0.644472\pi\)
−0.438448 + 0.898757i \(0.644472\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.440599 −0.0191928
\(528\) 0 0
\(529\) 41.6471 1.81074
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.6626i 0.461848i
\(534\) 0 0
\(535\) 37.7238i 1.63094i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.44844i 0.105462i
\(540\) 0 0
\(541\) 21.8696i 0.940248i 0.882600 + 0.470124i \(0.155791\pi\)
−0.882600 + 0.470124i \(0.844209\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 56.1355i 2.40458i
\(546\) 0 0
\(547\) 29.8845i 1.27777i −0.769303 0.638884i \(-0.779396\pi\)
0.769303 0.638884i \(-0.220604\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.07417i 0.0457613i
\(552\) 0 0
\(553\) 7.17536i 0.305127i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.7789i 1.30414i −0.758158 0.652071i \(-0.773900\pi\)
0.758158 0.652071i \(-0.226100\pi\)
\(558\) 0 0
\(559\) −37.3365 −1.57917
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.0579i 1.05607i −0.849224 0.528033i \(-0.822930\pi\)
0.849224 0.528033i \(-0.177070\pi\)
\(564\) 0 0
\(565\) 48.9323 2.05860
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.04649 −0.211560 −0.105780 0.994390i \(-0.533734\pi\)
−0.105780 + 0.994390i \(0.533734\pi\)
\(570\) 0 0
\(571\) 5.72341i 0.239517i 0.992803 + 0.119759i \(0.0382120\pi\)
−0.992803 + 0.119759i \(0.961788\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 94.4935 3.94065
\(576\) 0 0
\(577\) 18.9725 0.789835 0.394917 0.918717i \(-0.370773\pi\)
0.394917 + 0.918717i \(0.370773\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11.4961 −0.476939
\(582\) 0 0
\(583\) 12.1470i 0.503077i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.7081 −1.26746 −0.633730 0.773554i \(-0.718477\pi\)
−0.633730 + 0.773554i \(0.718477\pi\)
\(588\) 0 0
\(589\) −0.666556 −0.0274650
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −42.8134 −1.75814 −0.879068 0.476696i \(-0.841834\pi\)
−0.879068 + 0.476696i \(0.841834\pi\)
\(594\) 0 0
\(595\) 2.93931 0.120500
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.22309i 0.213410i −0.994291 0.106705i \(-0.965970\pi\)
0.994291 0.106705i \(-0.0340300\pi\)
\(600\) 0 0
\(601\) 36.4071 1.48508 0.742538 0.669804i \(-0.233622\pi\)
0.742538 + 0.669804i \(0.233622\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −36.9634 −1.50278
\(606\) 0 0
\(607\) 16.7018i 0.677904i −0.940804 0.338952i \(-0.889928\pi\)
0.940804 0.338952i \(-0.110072\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 45.0345 1.82190
\(612\) 0 0
\(613\) −17.1592 −0.693055 −0.346527 0.938040i \(-0.612639\pi\)
−0.346527 + 0.938040i \(0.612639\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0578i 1.37112i 0.728017 + 0.685559i \(0.240442\pi\)
−0.728017 + 0.685559i \(0.759558\pi\)
\(618\) 0 0
\(619\) 21.0690i 0.846833i 0.905935 + 0.423416i \(0.139169\pi\)
−0.905935 + 0.423416i \(0.860831\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.31632i 0.333186i
\(624\) 0 0
\(625\) 54.3573 2.17429
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.10139i 0.0439152i
\(630\) 0 0
\(631\) 16.2976 0.648797 0.324398 0.945921i \(-0.394838\pi\)
0.324398 + 0.945921i \(0.394838\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.10206 −0.0437337
\(636\) 0 0
\(637\) 9.18586i 0.363957i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.8706 −0.547858 −0.273929 0.961750i \(-0.588323\pi\)
−0.273929 + 0.961750i \(0.588323\pi\)
\(642\) 0 0
\(643\) 18.1271i 0.714865i −0.933939 0.357432i \(-0.883652\pi\)
0.933939 0.357432i \(-0.116348\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −34.0926 −1.34032 −0.670159 0.742218i \(-0.733774\pi\)
−0.670159 + 0.742218i \(0.733774\pi\)
\(648\) 0 0
\(649\) 2.86455i 0.112443i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.3491i 1.10939i 0.832055 + 0.554693i \(0.187164\pi\)
−0.832055 + 0.554693i \(0.812836\pi\)
\(654\) 0 0
\(655\) −44.1145 −1.72370
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.2879 −0.829260 −0.414630 0.909990i \(-0.636089\pi\)
−0.414630 + 0.909990i \(0.636089\pi\)
\(660\) 0 0
\(661\) 15.5357i 0.604267i 0.953266 + 0.302134i \(0.0976989\pi\)
−0.953266 + 0.302134i \(0.902301\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.44671 0.172436
\(666\) 0 0
\(667\) 18.2239i 0.705632i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16.8288i 0.649670i
\(672\) 0 0
\(673\) 28.1672i 1.08577i 0.839808 + 0.542883i \(0.182667\pi\)
−0.839808 + 0.542883i \(0.817333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.4868i 1.70977i 0.518820 + 0.854883i \(0.326371\pi\)
−0.518820 + 0.854883i \(0.673629\pi\)
\(678\) 0 0
\(679\) 31.6338 1.21399
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.976098 −0.0373493 −0.0186747 0.999826i \(-0.505945\pi\)
−0.0186747 + 0.999826i \(0.505945\pi\)
\(684\) 0 0
\(685\) 55.9332i 2.13710i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 45.5721i 1.73616i
\(690\) 0 0
\(691\) 48.3214i 1.83823i −0.393985 0.919117i \(-0.628904\pi\)
0.393985 0.919117i \(-0.371096\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 34.9781i 1.32679i
\(696\) 0 0
\(697\) 0.634480 0.0240327
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.1624i 1.02591i 0.858416 + 0.512955i \(0.171449\pi\)
−0.858416 + 0.512955i \(0.828551\pi\)
\(702\) 0 0
\(703\) 1.66623i 0.0628429i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.64528 0.362748
\(708\) 0 0
\(709\) 0.178679i 0.00671042i 0.999994 + 0.00335521i \(0.00106800\pi\)
−0.999994 + 0.00335521i \(0.998932\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.3085 0.423505
\(714\) 0 0
\(715\) 30.2361 1.13076
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 21.2941 0.794137 0.397068 0.917789i \(-0.370028\pi\)
0.397068 + 0.917789i \(0.370028\pi\)
\(720\) 0 0
\(721\) 39.9284i 1.48701i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 26.6375i 0.989293i
\(726\) 0 0
\(727\) 18.3771i 0.681570i −0.940141 0.340785i \(-0.889307\pi\)
0.940141 0.340785i \(-0.110693\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.22172i 0.0821733i
\(732\) 0 0
\(733\) 32.1446 1.18729 0.593644 0.804727i \(-0.297689\pi\)
0.593644 + 0.804727i \(0.297689\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.37970 −0.124493
\(738\) 0 0
\(739\) 16.9885i 0.624931i 0.949929 + 0.312465i \(0.101155\pi\)
−0.949929 + 0.312465i \(0.898845\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.7879i 0.689262i 0.938738 + 0.344631i \(0.111996\pi\)
−0.938738 + 0.344631i \(0.888004\pi\)
\(744\) 0 0
\(745\) 16.9355 0.620469
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 21.1285i 0.772018i
\(750\) 0 0
\(751\) 3.66376i 0.133692i 0.997763 + 0.0668462i \(0.0212937\pi\)
−0.997763 + 0.0668462i \(0.978706\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 56.4759i 2.05537i
\(756\) 0 0
\(757\) −54.8790 −1.99461 −0.997306 0.0733524i \(-0.976630\pi\)
−0.997306 + 0.0733524i \(0.976630\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.2256i 0.805678i −0.915271 0.402839i \(-0.868024\pi\)
0.915271 0.402839i \(-0.131976\pi\)
\(762\) 0 0
\(763\) 31.4406i 1.13823i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.7470i 0.388050i
\(768\) 0 0
\(769\) 12.6337i 0.455583i −0.973710 0.227792i \(-0.926849\pi\)
0.973710 0.227792i \(-0.0731505\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.6153 1.02922 0.514611 0.857424i \(-0.327936\pi\)
0.514611 + 0.857424i \(0.327936\pi\)
\(774\) 0 0
\(775\) 16.5294 0.593753
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.959869 0.0343909
\(780\) 0 0
\(781\) 14.9726i 0.535760i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 65.0984 2.32346
\(786\) 0 0
\(787\) 21.4638i 0.765103i −0.923934 0.382552i \(-0.875045\pi\)
0.923934 0.382552i \(-0.124955\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −27.4062 −0.974453
\(792\) 0 0
\(793\) 63.1370i 2.24206i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.7969 1.05546 0.527730 0.849412i \(-0.323043\pi\)
0.527730 + 0.849412i \(0.323043\pi\)
\(798\) 0 0