Properties

Label 8400.2.a.dg.1.1
Level $8400$
Weight $2$
Character 8400.1
Self dual yes
Analytic conductor $67.074$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 8400.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.35026 q^{13} +3.35026 q^{17} -5.35026 q^{19} +1.00000 q^{21} +4.96239 q^{23} -1.00000 q^{27} +7.92478 q^{29} -4.57452 q^{31} +2.00000 q^{33} +0.775746 q^{37} +1.35026 q^{39} +3.73813 q^{41} -12.6253 q^{43} +9.92478 q^{47} +1.00000 q^{49} -3.35026 q^{51} -8.57452 q^{53} +5.35026 q^{57} +8.62530 q^{59} -8.70052 q^{61} -1.00000 q^{63} +9.92478 q^{67} -4.96239 q^{69} -2.00000 q^{71} +9.35026 q^{73} +2.00000 q^{77} -10.7005 q^{79} +1.00000 q^{81} +3.22425 q^{83} -7.92478 q^{87} +1.03761 q^{89} +1.35026 q^{91} +4.57452 q^{93} +18.4993 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} + 6 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{63} + 8 q^{67} - 4 q^{69} - 6 q^{71} + 18 q^{73} + 6 q^{77} - 12 q^{79} + 3 q^{81} + 8 q^{83} - 2 q^{87} + 14 q^{89} - 6 q^{91} + 2 q^{93} + 22 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.35026 −0.374495 −0.187248 0.982313i \(-0.559957\pi\)
−0.187248 + 0.982313i \(0.559957\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.35026 0.812558 0.406279 0.913749i \(-0.366826\pi\)
0.406279 + 0.913749i \(0.366826\pi\)
\(18\) 0 0
\(19\) −5.35026 −1.22743 −0.613717 0.789526i \(-0.710326\pi\)
−0.613717 + 0.789526i \(0.710326\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.96239 1.03473 0.517365 0.855765i \(-0.326913\pi\)
0.517365 + 0.855765i \(0.326913\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.92478 1.47159 0.735797 0.677202i \(-0.236808\pi\)
0.735797 + 0.677202i \(0.236808\pi\)
\(30\) 0 0
\(31\) −4.57452 −0.821607 −0.410804 0.911724i \(-0.634752\pi\)
−0.410804 + 0.911724i \(0.634752\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.775746 0.127532 0.0637660 0.997965i \(-0.479689\pi\)
0.0637660 + 0.997965i \(0.479689\pi\)
\(38\) 0 0
\(39\) 1.35026 0.216215
\(40\) 0 0
\(41\) 3.73813 0.583799 0.291899 0.956449i \(-0.405713\pi\)
0.291899 + 0.956449i \(0.405713\pi\)
\(42\) 0 0
\(43\) −12.6253 −1.92534 −0.962670 0.270677i \(-0.912752\pi\)
−0.962670 + 0.270677i \(0.912752\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.92478 1.44768 0.723839 0.689969i \(-0.242376\pi\)
0.723839 + 0.689969i \(0.242376\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.35026 −0.469130
\(52\) 0 0
\(53\) −8.57452 −1.17780 −0.588900 0.808206i \(-0.700439\pi\)
−0.588900 + 0.808206i \(0.700439\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.35026 0.708659
\(58\) 0 0
\(59\) 8.62530 1.12292 0.561459 0.827504i \(-0.310240\pi\)
0.561459 + 0.827504i \(0.310240\pi\)
\(60\) 0 0
\(61\) −8.70052 −1.11399 −0.556994 0.830517i \(-0.688045\pi\)
−0.556994 + 0.830517i \(0.688045\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.92478 1.21250 0.606252 0.795272i \(-0.292672\pi\)
0.606252 + 0.795272i \(0.292672\pi\)
\(68\) 0 0
\(69\) −4.96239 −0.597401
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 9.35026 1.09437 0.547183 0.837013i \(-0.315700\pi\)
0.547183 + 0.837013i \(0.315700\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −10.7005 −1.20390 −0.601951 0.798533i \(-0.705610\pi\)
−0.601951 + 0.798533i \(0.705610\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.22425 0.353908 0.176954 0.984219i \(-0.443376\pi\)
0.176954 + 0.984219i \(0.443376\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.92478 −0.849625
\(88\) 0 0
\(89\) 1.03761 0.109987 0.0549933 0.998487i \(-0.482486\pi\)
0.0549933 + 0.998487i \(0.482486\pi\)
\(90\) 0 0
\(91\) 1.35026 0.141546
\(92\) 0 0
\(93\) 4.57452 0.474355
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.4993 1.87832 0.939159 0.343482i \(-0.111606\pi\)
0.939159 + 0.343482i \(0.111606\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −17.6629 −1.75753 −0.878763 0.477259i \(-0.841630\pi\)
−0.878763 + 0.477259i \(0.841630\pi\)
\(102\) 0 0
\(103\) 6.70052 0.660222 0.330111 0.943942i \(-0.392914\pi\)
0.330111 + 0.943942i \(0.392914\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.7381 1.32812 0.664058 0.747681i \(-0.268833\pi\)
0.664058 + 0.747681i \(0.268833\pi\)
\(108\) 0 0
\(109\) −2.77575 −0.265868 −0.132934 0.991125i \(-0.542440\pi\)
−0.132934 + 0.991125i \(0.542440\pi\)
\(110\) 0 0
\(111\) −0.775746 −0.0736306
\(112\) 0 0
\(113\) −12.0508 −1.13364 −0.566821 0.823841i \(-0.691827\pi\)
−0.566821 + 0.823841i \(0.691827\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.35026 −0.124832
\(118\) 0 0
\(119\) −3.35026 −0.307118
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −3.73813 −0.337056
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.70052 0.239633 0.119816 0.992796i \(-0.461769\pi\)
0.119816 + 0.992796i \(0.461769\pi\)
\(128\) 0 0
\(129\) 12.6253 1.11160
\(130\) 0 0
\(131\) −20.6253 −1.80204 −0.901020 0.433777i \(-0.857181\pi\)
−0.901020 + 0.433777i \(0.857181\pi\)
\(132\) 0 0
\(133\) 5.35026 0.463927
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −22.4993 −1.92224 −0.961122 0.276124i \(-0.910950\pi\)
−0.961122 + 0.276124i \(0.910950\pi\)
\(138\) 0 0
\(139\) 3.27504 0.277785 0.138893 0.990307i \(-0.455646\pi\)
0.138893 + 0.990307i \(0.455646\pi\)
\(140\) 0 0
\(141\) −9.92478 −0.835817
\(142\) 0 0
\(143\) 2.70052 0.225829
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −4.44851 −0.364436 −0.182218 0.983258i \(-0.558328\pi\)
−0.182218 + 0.983258i \(0.558328\pi\)
\(150\) 0 0
\(151\) −1.29948 −0.105750 −0.0528749 0.998601i \(-0.516838\pi\)
−0.0528749 + 0.998601i \(0.516838\pi\)
\(152\) 0 0
\(153\) 3.35026 0.270853
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.64974 0.211472 0.105736 0.994394i \(-0.466280\pi\)
0.105736 + 0.994394i \(0.466280\pi\)
\(158\) 0 0
\(159\) 8.57452 0.680003
\(160\) 0 0
\(161\) −4.96239 −0.391091
\(162\) 0 0
\(163\) 5.29948 0.415087 0.207544 0.978226i \(-0.433453\pi\)
0.207544 + 0.978226i \(0.433453\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.5501 −1.12592 −0.562959 0.826485i \(-0.690337\pi\)
−0.562959 + 0.826485i \(0.690337\pi\)
\(168\) 0 0
\(169\) −11.1768 −0.859753
\(170\) 0 0
\(171\) −5.35026 −0.409145
\(172\) 0 0
\(173\) 4.49929 0.342075 0.171037 0.985265i \(-0.445288\pi\)
0.171037 + 0.985265i \(0.445288\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.62530 −0.648317
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 10.6253 0.789772 0.394886 0.918730i \(-0.370784\pi\)
0.394886 + 0.918730i \(0.370784\pi\)
\(182\) 0 0
\(183\) 8.70052 0.643161
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.70052 −0.489991
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 13.8496 1.00212 0.501059 0.865413i \(-0.332944\pi\)
0.501059 + 0.865413i \(0.332944\pi\)
\(192\) 0 0
\(193\) −15.3258 −1.10318 −0.551588 0.834116i \(-0.685978\pi\)
−0.551588 + 0.834116i \(0.685978\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.574515 0.0409325 0.0204663 0.999791i \(-0.493485\pi\)
0.0204663 + 0.999791i \(0.493485\pi\)
\(198\) 0 0
\(199\) 0.201231 0.0142649 0.00713244 0.999975i \(-0.497730\pi\)
0.00713244 + 0.999975i \(0.497730\pi\)
\(200\) 0 0
\(201\) −9.92478 −0.700040
\(202\) 0 0
\(203\) −7.92478 −0.556210
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.96239 0.344910
\(208\) 0 0
\(209\) 10.7005 0.740171
\(210\) 0 0
\(211\) −6.44851 −0.443934 −0.221967 0.975054i \(-0.571248\pi\)
−0.221967 + 0.975054i \(0.571248\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.57452 0.310538
\(218\) 0 0
\(219\) −9.35026 −0.631832
\(220\) 0 0
\(221\) −4.52373 −0.304299
\(222\) 0 0
\(223\) 1.55149 0.103896 0.0519478 0.998650i \(-0.483457\pi\)
0.0519478 + 0.998650i \(0.483457\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.1490 −0.872732 −0.436366 0.899769i \(-0.643735\pi\)
−0.436366 + 0.899769i \(0.643735\pi\)
\(228\) 0 0
\(229\) 2.77575 0.183426 0.0917132 0.995785i \(-0.470766\pi\)
0.0917132 + 0.995785i \(0.470766\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) 0.0507852 0.00332705 0.00166353 0.999999i \(-0.499470\pi\)
0.00166353 + 0.999999i \(0.499470\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.7005 0.695074
\(238\) 0 0
\(239\) −5.84955 −0.378376 −0.189188 0.981941i \(-0.560586\pi\)
−0.189188 + 0.981941i \(0.560586\pi\)
\(240\) 0 0
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.22425 0.459668
\(248\) 0 0
\(249\) −3.22425 −0.204329
\(250\) 0 0
\(251\) −19.2243 −1.21342 −0.606712 0.794922i \(-0.707512\pi\)
−0.606712 + 0.794922i \(0.707512\pi\)
\(252\) 0 0
\(253\) −9.92478 −0.623965
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.35026 −0.458497 −0.229248 0.973368i \(-0.573627\pi\)
−0.229248 + 0.973368i \(0.573627\pi\)
\(258\) 0 0
\(259\) −0.775746 −0.0482025
\(260\) 0 0
\(261\) 7.92478 0.490531
\(262\) 0 0
\(263\) −12.9624 −0.799295 −0.399648 0.916669i \(-0.630867\pi\)
−0.399648 + 0.916669i \(0.630867\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.03761 −0.0635008
\(268\) 0 0
\(269\) 4.11142 0.250678 0.125339 0.992114i \(-0.459998\pi\)
0.125339 + 0.992114i \(0.459998\pi\)
\(270\) 0 0
\(271\) 16.4241 0.997691 0.498846 0.866691i \(-0.333757\pi\)
0.498846 + 0.866691i \(0.333757\pi\)
\(272\) 0 0
\(273\) −1.35026 −0.0817216
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.0738 0.665361 0.332680 0.943040i \(-0.392047\pi\)
0.332680 + 0.943040i \(0.392047\pi\)
\(278\) 0 0
\(279\) −4.57452 −0.273869
\(280\) 0 0
\(281\) 14.3733 0.857438 0.428719 0.903438i \(-0.358965\pi\)
0.428719 + 0.903438i \(0.358965\pi\)
\(282\) 0 0
\(283\) 1.14903 0.0683028 0.0341514 0.999417i \(-0.489127\pi\)
0.0341514 + 0.999417i \(0.489127\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.73813 −0.220655
\(288\) 0 0
\(289\) −5.77575 −0.339750
\(290\) 0 0
\(291\) −18.4993 −1.08445
\(292\) 0 0
\(293\) −0.649738 −0.0379581 −0.0189791 0.999820i \(-0.506042\pi\)
−0.0189791 + 0.999820i \(0.506042\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −6.70052 −0.387501
\(300\) 0 0
\(301\) 12.6253 0.727710
\(302\) 0 0
\(303\) 17.6629 1.01471
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 24.1016 1.37555 0.687775 0.725924i \(-0.258588\pi\)
0.687775 + 0.725924i \(0.258588\pi\)
\(308\) 0 0
\(309\) −6.70052 −0.381179
\(310\) 0 0
\(311\) −8.25202 −0.467929 −0.233964 0.972245i \(-0.575170\pi\)
−0.233964 + 0.972245i \(0.575170\pi\)
\(312\) 0 0
\(313\) 14.9018 0.842297 0.421148 0.906992i \(-0.361627\pi\)
0.421148 + 0.906992i \(0.361627\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.1260 −0.568733 −0.284367 0.958716i \(-0.591783\pi\)
−0.284367 + 0.958716i \(0.591783\pi\)
\(318\) 0 0
\(319\) −15.8496 −0.887405
\(320\) 0 0
\(321\) −13.7381 −0.766788
\(322\) 0 0
\(323\) −17.9248 −0.997361
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.77575 0.153499
\(328\) 0 0
\(329\) −9.92478 −0.547171
\(330\) 0 0
\(331\) −27.8496 −1.53075 −0.765375 0.643585i \(-0.777446\pi\)
−0.765375 + 0.643585i \(0.777446\pi\)
\(332\) 0 0
\(333\) 0.775746 0.0425106
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.84955 0.209699 0.104849 0.994488i \(-0.466564\pi\)
0.104849 + 0.994488i \(0.466564\pi\)
\(338\) 0 0
\(339\) 12.0508 0.654509
\(340\) 0 0
\(341\) 9.14903 0.495448
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.58769 0.514694 0.257347 0.966319i \(-0.417152\pi\)
0.257347 + 0.966319i \(0.417152\pi\)
\(348\) 0 0
\(349\) −15.1490 −0.810909 −0.405455 0.914115i \(-0.632887\pi\)
−0.405455 + 0.914115i \(0.632887\pi\)
\(350\) 0 0
\(351\) 1.35026 0.0720716
\(352\) 0 0
\(353\) 20.3488 1.08306 0.541530 0.840681i \(-0.317845\pi\)
0.541530 + 0.840681i \(0.317845\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.35026 0.177315
\(358\) 0 0
\(359\) 31.4010 1.65728 0.828642 0.559779i \(-0.189114\pi\)
0.828642 + 0.559779i \(0.189114\pi\)
\(360\) 0 0
\(361\) 9.62530 0.506595
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −29.4010 −1.53472 −0.767361 0.641215i \(-0.778431\pi\)
−0.767361 + 0.641215i \(0.778431\pi\)
\(368\) 0 0
\(369\) 3.73813 0.194600
\(370\) 0 0
\(371\) 8.57452 0.445167
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.7005 −0.551105
\(378\) 0 0
\(379\) −10.7005 −0.549649 −0.274824 0.961494i \(-0.588620\pi\)
−0.274824 + 0.961494i \(0.588620\pi\)
\(380\) 0 0
\(381\) −2.70052 −0.138352
\(382\) 0 0
\(383\) −16.7757 −0.857201 −0.428600 0.903494i \(-0.640993\pi\)
−0.428600 + 0.903494i \(0.640993\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.6253 −0.641780
\(388\) 0 0
\(389\) −29.3258 −1.48688 −0.743439 0.668804i \(-0.766807\pi\)
−0.743439 + 0.668804i \(0.766807\pi\)
\(390\) 0 0
\(391\) 16.6253 0.840778
\(392\) 0 0
\(393\) 20.6253 1.04041
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −18.3488 −0.920902 −0.460451 0.887685i \(-0.652312\pi\)
−0.460451 + 0.887685i \(0.652312\pi\)
\(398\) 0 0
\(399\) −5.35026 −0.267848
\(400\) 0 0
\(401\) −37.3258 −1.86396 −0.931981 0.362506i \(-0.881921\pi\)
−0.931981 + 0.362506i \(0.881921\pi\)
\(402\) 0 0
\(403\) 6.17679 0.307688
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.55149 −0.0769046
\(408\) 0 0
\(409\) −22.3733 −1.10629 −0.553144 0.833086i \(-0.686572\pi\)
−0.553144 + 0.833086i \(0.686572\pi\)
\(410\) 0 0
\(411\) 22.4993 1.10981
\(412\) 0 0
\(413\) −8.62530 −0.424423
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.27504 −0.160379
\(418\) 0 0
\(419\) −23.4763 −1.14689 −0.573445 0.819244i \(-0.694394\pi\)
−0.573445 + 0.819244i \(0.694394\pi\)
\(420\) 0 0
\(421\) −25.2243 −1.22935 −0.614677 0.788779i \(-0.710714\pi\)
−0.614677 + 0.788779i \(0.710714\pi\)
\(422\) 0 0
\(423\) 9.92478 0.482559
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.70052 0.421048
\(428\) 0 0
\(429\) −2.70052 −0.130383
\(430\) 0 0
\(431\) 19.4010 0.934516 0.467258 0.884121i \(-0.345242\pi\)
0.467258 + 0.884121i \(0.345242\pi\)
\(432\) 0 0
\(433\) −6.49929 −0.312336 −0.156168 0.987731i \(-0.549914\pi\)
−0.156168 + 0.987731i \(0.549914\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.5501 −1.27006
\(438\) 0 0
\(439\) 14.6497 0.699194 0.349597 0.936900i \(-0.386319\pi\)
0.349597 + 0.936900i \(0.386319\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 19.1392 0.909330 0.454665 0.890663i \(-0.349759\pi\)
0.454665 + 0.890663i \(0.349759\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.44851 0.210407
\(448\) 0 0
\(449\) −32.8021 −1.54803 −0.774013 0.633169i \(-0.781754\pi\)
−0.774013 + 0.633169i \(0.781754\pi\)
\(450\) 0 0
\(451\) −7.47627 −0.352044
\(452\) 0 0
\(453\) 1.29948 0.0610547
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.7005 0.874774 0.437387 0.899273i \(-0.355904\pi\)
0.437387 + 0.899273i \(0.355904\pi\)
\(458\) 0 0
\(459\) −3.35026 −0.156377
\(460\) 0 0
\(461\) −6.96239 −0.324271 −0.162135 0.986769i \(-0.551838\pi\)
−0.162135 + 0.986769i \(0.551838\pi\)
\(462\) 0 0
\(463\) 5.29948 0.246288 0.123144 0.992389i \(-0.460702\pi\)
0.123144 + 0.992389i \(0.460702\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13.1490 −0.608465 −0.304232 0.952598i \(-0.598400\pi\)
−0.304232 + 0.952598i \(0.598400\pi\)
\(468\) 0 0
\(469\) −9.92478 −0.458284
\(470\) 0 0
\(471\) −2.64974 −0.122093
\(472\) 0 0
\(473\) 25.2506 1.16102
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.57452 −0.392600
\(478\) 0 0
\(479\) −5.14903 −0.235265 −0.117633 0.993057i \(-0.537531\pi\)
−0.117633 + 0.993057i \(0.537531\pi\)
\(480\) 0 0
\(481\) −1.04746 −0.0477601
\(482\) 0 0
\(483\) 4.96239 0.225797
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −22.1768 −1.00493 −0.502463 0.864599i \(-0.667573\pi\)
−0.502463 + 0.864599i \(0.667573\pi\)
\(488\) 0 0
\(489\) −5.29948 −0.239651
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 26.5501 1.19576
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) 6.55008 0.293222 0.146611 0.989194i \(-0.453163\pi\)
0.146611 + 0.989194i \(0.453163\pi\)
\(500\) 0 0
\(501\) 14.5501 0.650050
\(502\) 0 0
\(503\) 8.77575 0.391291 0.195646 0.980675i \(-0.437320\pi\)
0.195646 + 0.980675i \(0.437320\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 11.1768 0.496379
\(508\) 0 0
\(509\) 13.1392 0.582384 0.291192 0.956665i \(-0.405948\pi\)
0.291192 + 0.956665i \(0.405948\pi\)
\(510\) 0 0
\(511\) −9.35026 −0.413631
\(512\) 0 0
\(513\) 5.35026 0.236220
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −19.8496 −0.872982
\(518\) 0 0
\(519\) −4.49929 −0.197497
\(520\) 0 0
\(521\) −37.6629 −1.65004 −0.825021 0.565102i \(-0.808837\pi\)
−0.825021 + 0.565102i \(0.808837\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.3258 −0.667603
\(528\) 0 0
\(529\) 1.62530 0.0706652
\(530\) 0 0
\(531\) 8.62530 0.374306
\(532\) 0 0
\(533\) −5.04746 −0.218630
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −22.4749 −0.966269 −0.483135 0.875546i \(-0.660502\pi\)
−0.483135 + 0.875546i \(0.660502\pi\)
\(542\) 0 0
\(543\) −10.6253 −0.455975
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −25.9248 −1.10846 −0.554232 0.832362i \(-0.686988\pi\)
−0.554232 + 0.832362i \(0.686988\pi\)
\(548\) 0 0
\(549\) −8.70052 −0.371329
\(550\) 0 0
\(551\) −42.3996 −1.80629
\(552\) 0 0
\(553\) 10.7005 0.455033
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.5256 1.20867 0.604335 0.796730i \(-0.293439\pi\)
0.604335 + 0.796730i \(0.293439\pi\)
\(558\) 0 0
\(559\) 17.0475 0.721031
\(560\) 0 0
\(561\) 6.70052 0.282896
\(562\) 0 0
\(563\) −11.6267 −0.490008 −0.245004 0.969522i \(-0.578789\pi\)
−0.245004 + 0.969522i \(0.578789\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 9.32582 0.390959 0.195479 0.980708i \(-0.437374\pi\)
0.195479 + 0.980708i \(0.437374\pi\)
\(570\) 0 0
\(571\) 19.6991 0.824382 0.412191 0.911097i \(-0.364764\pi\)
0.412191 + 0.911097i \(0.364764\pi\)
\(572\) 0 0
\(573\) −13.8496 −0.578573
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −32.7974 −1.36537 −0.682686 0.730712i \(-0.739188\pi\)
−0.682686 + 0.730712i \(0.739188\pi\)
\(578\) 0 0
\(579\) 15.3258 0.636920
\(580\) 0 0
\(581\) −3.22425 −0.133765
\(582\) 0 0
\(583\) 17.1490 0.710240
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.8218 0.776859 0.388429 0.921479i \(-0.373018\pi\)
0.388429 + 0.921479i \(0.373018\pi\)
\(588\) 0 0
\(589\) 24.4749 1.00847
\(590\) 0 0
\(591\) −0.574515 −0.0236324
\(592\) 0 0
\(593\) −33.7499 −1.38594 −0.692971 0.720965i \(-0.743699\pi\)
−0.692971 + 0.720965i \(0.743699\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.201231 −0.00823583
\(598\) 0 0
\(599\) 20.2981 0.829356 0.414678 0.909968i \(-0.363894\pi\)
0.414678 + 0.909968i \(0.363894\pi\)
\(600\) 0 0
\(601\) −13.8496 −0.564935 −0.282468 0.959277i \(-0.591153\pi\)
−0.282468 + 0.959277i \(0.591153\pi\)
\(602\) 0 0
\(603\) 9.92478 0.404168
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −25.2506 −1.02489 −0.512445 0.858720i \(-0.671260\pi\)
−0.512445 + 0.858720i \(0.671260\pi\)
\(608\) 0 0
\(609\) 7.92478 0.321128
\(610\) 0 0
\(611\) −13.4010 −0.542148
\(612\) 0 0
\(613\) 9.14903 0.369526 0.184763 0.982783i \(-0.440848\pi\)
0.184763 + 0.982783i \(0.440848\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.9492 −0.642091 −0.321046 0.947064i \(-0.604034\pi\)
−0.321046 + 0.947064i \(0.604034\pi\)
\(618\) 0 0
\(619\) −11.1735 −0.449100 −0.224550 0.974463i \(-0.572091\pi\)
−0.224550 + 0.974463i \(0.572091\pi\)
\(620\) 0 0
\(621\) −4.96239 −0.199134
\(622\) 0 0
\(623\) −1.03761 −0.0415710
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.7005 −0.427338
\(628\) 0 0
\(629\) 2.59895 0.103627
\(630\) 0 0
\(631\) 14.5501 0.579229 0.289615 0.957143i \(-0.406473\pi\)
0.289615 + 0.957143i \(0.406473\pi\)
\(632\) 0 0
\(633\) 6.44851 0.256305
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.35026 −0.0534993
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −38.7269 −1.52962 −0.764810 0.644256i \(-0.777167\pi\)
−0.764810 + 0.644256i \(0.777167\pi\)
\(642\) 0 0
\(643\) −11.9511 −0.471306 −0.235653 0.971837i \(-0.575723\pi\)
−0.235653 + 0.971837i \(0.575723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.5501 −0.572023 −0.286011 0.958226i \(-0.592329\pi\)
−0.286011 + 0.958226i \(0.592329\pi\)
\(648\) 0 0
\(649\) −17.2506 −0.677145
\(650\) 0 0
\(651\) −4.57452 −0.179289
\(652\) 0 0
\(653\) −49.9756 −1.95569 −0.977847 0.209319i \(-0.932875\pi\)
−0.977847 + 0.209319i \(0.932875\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.35026 0.364788
\(658\) 0 0
\(659\) 16.9525 0.660377 0.330189 0.943915i \(-0.392888\pi\)
0.330189 + 0.943915i \(0.392888\pi\)
\(660\) 0 0
\(661\) −15.6531 −0.608834 −0.304417 0.952539i \(-0.598462\pi\)
−0.304417 + 0.952539i \(0.598462\pi\)
\(662\) 0 0
\(663\) 4.52373 0.175687
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 39.3258 1.52270
\(668\) 0 0
\(669\) −1.55149 −0.0599842
\(670\) 0 0
\(671\) 17.4010 0.671760
\(672\) 0 0
\(673\) −26.0263 −1.00324 −0.501621 0.865088i \(-0.667263\pi\)
−0.501621 + 0.865088i \(0.667263\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.4518 1.36252 0.681262 0.732039i \(-0.261431\pi\)
0.681262 + 0.732039i \(0.261431\pi\)
\(678\) 0 0
\(679\) −18.4993 −0.709938
\(680\) 0 0
\(681\) 13.1490 0.503872
\(682\) 0 0
\(683\) −23.6629 −0.905436 −0.452718 0.891654i \(-0.649546\pi\)
−0.452718 + 0.891654i \(0.649546\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.77575 −0.105901
\(688\) 0 0
\(689\) 11.5778 0.441081
\(690\) 0 0
\(691\) 0.574515 0.0218556 0.0109278 0.999940i \(-0.496522\pi\)
0.0109278 + 0.999940i \(0.496522\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.5237 0.474370
\(698\) 0 0
\(699\) −0.0507852 −0.00192087
\(700\) 0 0
\(701\) 42.7269 1.61377 0.806886 0.590707i \(-0.201151\pi\)
0.806886 + 0.590707i \(0.201151\pi\)
\(702\) 0 0
\(703\) −4.15045 −0.156537
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.6629 0.664282
\(708\) 0 0
\(709\) 27.2506 1.02342 0.511709 0.859159i \(-0.329013\pi\)
0.511709 + 0.859159i \(0.329013\pi\)
\(710\) 0 0
\(711\) −10.7005 −0.401301
\(712\) 0 0
\(713\) −22.7005 −0.850141
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.84955 0.218456
\(718\) 0 0
\(719\) 10.7005 0.399062 0.199531 0.979891i \(-0.436058\pi\)
0.199531 + 0.979891i \(0.436058\pi\)
\(720\) 0 0
\(721\) −6.70052 −0.249541
\(722\) 0 0
\(723\) 0.0752228 0.00279757
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 39.9511 1.48171 0.740853 0.671668i \(-0.234422\pi\)
0.740853 + 0.671668i \(0.234422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −42.2981 −1.56445
\(732\) 0 0
\(733\) 30.3488 1.12096 0.560480 0.828168i \(-0.310617\pi\)
0.560480 + 0.828168i \(0.310617\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.8496 −0.731168
\(738\) 0 0
\(739\) −37.2506 −1.37029 −0.685143 0.728409i \(-0.740260\pi\)
−0.685143 + 0.728409i \(0.740260\pi\)
\(740\) 0 0
\(741\) −7.22425 −0.265390
\(742\) 0 0
\(743\) 26.3634 0.967181 0.483590 0.875294i \(-0.339332\pi\)
0.483590 + 0.875294i \(0.339332\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.22425 0.117969
\(748\) 0 0
\(749\) −13.7381 −0.501981
\(750\) 0 0
\(751\) −50.6516 −1.84830 −0.924152 0.382024i \(-0.875227\pi\)
−0.924152 + 0.382024i \(0.875227\pi\)
\(752\) 0 0
\(753\) 19.2243 0.700571
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.9525 −1.41575 −0.707877 0.706336i \(-0.750347\pi\)
−0.707877 + 0.706336i \(0.750347\pi\)
\(758\) 0 0
\(759\) 9.92478 0.360247
\(760\) 0 0
\(761\) 48.2130 1.74772 0.873860 0.486178i \(-0.161609\pi\)
0.873860 + 0.486178i \(0.161609\pi\)
\(762\) 0 0
\(763\) 2.77575 0.100489
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.6464 −0.420528
\(768\) 0 0
\(769\) 4.44851 0.160417 0.0802086 0.996778i \(-0.474441\pi\)
0.0802086 + 0.996778i \(0.474441\pi\)
\(770\) 0 0
\(771\) 7.35026 0.264713
\(772\) 0 0
\(773\) −39.3014 −1.41357 −0.706786 0.707427i \(-0.749856\pi\)
−0.706786 + 0.707427i \(0.749856\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.775746 0.0278297
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −7.92478 −0.283208
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.897015 0.0319751 0.0159876 0.999872i \(-0.494911\pi\)
0.0159876 + 0.999872i \(0.494911\pi\)
\(788\) 0 0
\(789\) 12.9624 0.461473
\(790\) 0 0
\(791\) 12.0508 0.428477
\(792\) 0 0
\(793\) 11.7480 0.417183
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.19982 −0.113343 −0.0566717 0.998393i \(-0.518049\pi\)
−0.0566717 + 0.998393i \(0.518049\pi\)
\(798\) 0 0
\(799\) 33.2506 1.17632
\(800\) 0 0
\(801\) 1.03761 0.0366622
\(802\) 0 0
\(803\) −18.7005 −0.659927
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.11142 −0.144729
\(808\) 0 0
\(809\) 4.44851 0.156401 0.0782006 0.996938i \(-0.475083\pi\)
0.0782006 + 0.996938i \(0.475083\pi\)
\(810\) 0 0
\(811\) −37.6747 −1.32294 −0.661468 0.749973i \(-0.730066\pi\)
−0.661468 + 0.749973i \(0.730066\pi\)
\(812\) 0 0
\(813\) −16.4241 −0.576017
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 67.5487 2.36323
\(818\) 0 0
\(819\) 1.35026 0.0471820
\(820\) 0 0
\(821\) −0.749399 −0.0261542 −0.0130771 0.999914i \(-0.504163\pi\)
−0.0130771 + 0.999914i \(0.504163\pi\)
\(822\) 0 0
\(823\) −26.3996 −0.920233 −0.460117 0.887858i \(-0.652192\pi\)
−0.460117 + 0.887858i \(0.652192\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.43724 0.189071 0.0945357 0.995521i \(-0.469863\pi\)
0.0945357 + 0.995521i \(0.469863\pi\)
\(828\) 0 0
\(829\) 22.7757 0.791034 0.395517 0.918459i \(-0.370565\pi\)
0.395517 + 0.918459i \(0.370565\pi\)
\(830\) 0 0
\(831\) −11.0738 −0.384146
\(832\) 0 0
\(833\) 3.35026 0.116080
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.57452 0.158118
\(838\) 0 0
\(839\) −15.8496 −0.547187 −0.273594 0.961845i \(-0.588212\pi\)
−0.273594 + 0.961845i \(0.588212\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) 0 0
\(843\) −14.3733 −0.495042
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) −1.14903 −0.0394346
\(850\) 0 0
\(851\) 3.84955 0.131961
\(852\) 0 0
\(853\) 21.0494 0.720717 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −50.1524 −1.71317 −0.856586 0.516004i \(-0.827419\pi\)
−0.856586 + 0.516004i \(0.827419\pi\)
\(858\) 0 0
\(859\) 5.35026 0.182549 0.0912743 0.995826i \(-0.470906\pi\)
0.0912743 + 0.995826i \(0.470906\pi\)
\(860\) 0 0
\(861\) 3.73813 0.127395
\(862\) 0 0
\(863\) −33.6893 −1.14680 −0.573398 0.819277i \(-0.694375\pi\)
−0.573398 + 0.819277i \(0.694375\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5.77575 0.196155
\(868\) 0 0
\(869\) 21.4010 0.725981
\(870\) 0 0
\(871\) −13.4010 −0.454077
\(872\) 0 0
\(873\) 18.4993 0.626106
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21.5026 0.726092 0.363046 0.931771i \(-0.381737\pi\)
0.363046 + 0.931771i \(0.381737\pi\)
\(878\) 0 0
\(879\) 0.649738 0.0219151
\(880\) 0 0
\(881\) 32.3634 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(882\) 0 0
\(883\) −2.59895 −0.0874617 −0.0437309 0.999043i \(-0.513924\pi\)
−0.0437309 + 0.999043i \(0.513924\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.2784 1.28526 0.642631 0.766176i \(-0.277843\pi\)
0.642631 + 0.766176i \(0.277843\pi\)
\(888\) 0 0
\(889\) −2.70052 −0.0905727
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) −53.1002 −1.77693
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.70052 0.223724
\(898\) 0 0
\(899\) −36.2520 −1.20907
\(900\) 0 0
\(901\) −28.7269 −0.957031
\(902\) 0 0
\(903\) −12.6253 −0.420144
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −49.9972 −1.66013 −0.830064 0.557668i \(-0.811696\pi\)
−0.830064 + 0.557668i \(0.811696\pi\)
\(908\) 0 0
\(909\) −17.6629 −0.585842
\(910\) 0 0
\(911\) 24.9525 0.826715 0.413357 0.910569i \(-0.364356\pi\)
0.413357 + 0.910569i \(0.364356\pi\)
\(912\) 0 0
\(913\) −6.44851 −0.213414
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.6253 0.681107
\(918\) 0 0
\(919\) 11.6991 0.385918 0.192959 0.981207i \(-0.438192\pi\)
0.192959 + 0.981207i \(0.438192\pi\)
\(920\) 0 0
\(921\) −24.1016 −0.794174
\(922\) 0 0
\(923\) 2.70052 0.0888888
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.70052 0.220074
\(928\) 0 0
\(929\) 23.7090 0.777866 0.388933 0.921266i \(-0.372844\pi\)
0.388933 + 0.921266i \(0.372844\pi\)
\(930\) 0 0
\(931\) −5.35026 −0.175348
\(932\) 0 0
\(933\) 8.25202 0.270159
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.9003 0.650116 0.325058 0.945694i \(-0.394616\pi\)
0.325058 + 0.945694i \(0.394616\pi\)
\(938\) 0 0
\(939\) −14.9018 −0.486300
\(940\) 0 0
\(941\) −6.28821 −0.204990 −0.102495 0.994734i \(-0.532683\pi\)
−0.102495 + 0.994734i \(0.532683\pi\)
\(942\) 0 0
\(943\) 18.5501 0.604074
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40.0362 −1.30100 −0.650501 0.759506i \(-0.725441\pi\)
−0.650501 + 0.759506i \(0.725441\pi\)
\(948\) 0 0
\(949\) −12.6253 −0.409835
\(950\) 0 0
\(951\) 10.1260 0.328358
\(952\) 0 0
\(953\) −40.9478 −1.32643 −0.663215 0.748429i \(-0.730808\pi\)
−0.663215 + 0.748429i \(0.730808\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 15.8496 0.512343
\(958\) 0 0
\(959\) 22.4993 0.726540
\(960\) 0 0
\(961\) −10.0738 −0.324962
\(962\) 0 0
\(963\) 13.7381 0.442705
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 38.2784 1.23095 0.615475 0.788157i \(-0.288964\pi\)
0.615475 + 0.788157i \(0.288964\pi\)
\(968\) 0 0
\(969\) 17.9248 0.575827
\(970\) 0 0
\(971\) 28.7269 0.921889 0.460945 0.887429i \(-0.347511\pi\)
0.460945 + 0.887429i \(0.347511\pi\)
\(972\) 0 0
\(973\) −3.27504 −0.104993
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41.3014 1.32135 0.660674 0.750673i \(-0.270270\pi\)
0.660674 + 0.750673i \(0.270270\pi\)
\(978\) 0 0
\(979\) −2.07522 −0.0663244
\(980\) 0 0
\(981\) −2.77575 −0.0886228
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.92478 0.315909
\(988\) 0 0
\(989\) −62.6516 −1.99221
\(990\) 0 0
\(991\) −25.1002 −0.797333 −0.398666 0.917096i \(-0.630527\pi\)
−0.398666 + 0.917096i \(0.630527\pi\)
\(992\) 0 0
\(993\) 27.8496 0.883779
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −32.7974 −1.03870 −0.519351 0.854561i \(-0.673826\pi\)
−0.519351 + 0.854561i \(0.673826\pi\)
\(998\) 0 0
\(999\) −0.775746 −0.0245435
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8400.2.a.dg.1.1 3
4.3 odd 2 525.2.a.j.1.2 3
5.2 odd 4 1680.2.t.k.1009.4 6
5.3 odd 4 1680.2.t.k.1009.1 6
5.4 even 2 8400.2.a.dj.1.3 3
12.11 even 2 1575.2.a.x.1.2 3
15.2 even 4 5040.2.t.v.1009.6 6
15.8 even 4 5040.2.t.v.1009.5 6
20.3 even 4 105.2.d.b.64.4 yes 6
20.7 even 4 105.2.d.b.64.3 6
20.19 odd 2 525.2.a.k.1.2 3
28.27 even 2 3675.2.a.bi.1.2 3
60.23 odd 4 315.2.d.e.64.3 6
60.47 odd 4 315.2.d.e.64.4 6
60.59 even 2 1575.2.a.w.1.2 3
140.3 odd 12 735.2.q.f.79.3 12
140.23 even 12 735.2.q.e.214.4 12
140.27 odd 4 735.2.d.b.589.3 6
140.47 odd 12 735.2.q.f.214.3 12
140.67 even 12 735.2.q.e.79.4 12
140.83 odd 4 735.2.d.b.589.4 6
140.87 odd 12 735.2.q.f.79.4 12
140.103 odd 12 735.2.q.f.214.4 12
140.107 even 12 735.2.q.e.214.3 12
140.123 even 12 735.2.q.e.79.3 12
140.139 even 2 3675.2.a.bj.1.2 3
420.83 even 4 2205.2.d.l.1324.3 6
420.167 even 4 2205.2.d.l.1324.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.3 6 20.7 even 4
105.2.d.b.64.4 yes 6 20.3 even 4
315.2.d.e.64.3 6 60.23 odd 4
315.2.d.e.64.4 6 60.47 odd 4
525.2.a.j.1.2 3 4.3 odd 2
525.2.a.k.1.2 3 20.19 odd 2
735.2.d.b.589.3 6 140.27 odd 4
735.2.d.b.589.4 6 140.83 odd 4
735.2.q.e.79.3 12 140.123 even 12
735.2.q.e.79.4 12 140.67 even 12
735.2.q.e.214.3 12 140.107 even 12
735.2.q.e.214.4 12 140.23 even 12
735.2.q.f.79.3 12 140.3 odd 12
735.2.q.f.79.4 12 140.87 odd 12
735.2.q.f.214.3 12 140.47 odd 12
735.2.q.f.214.4 12 140.103 odd 12
1575.2.a.w.1.2 3 60.59 even 2
1575.2.a.x.1.2 3 12.11 even 2
1680.2.t.k.1009.1 6 5.3 odd 4
1680.2.t.k.1009.4 6 5.2 odd 4
2205.2.d.l.1324.3 6 420.83 even 4
2205.2.d.l.1324.4 6 420.167 even 4
3675.2.a.bi.1.2 3 28.27 even 2
3675.2.a.bj.1.2 3 140.139 even 2
5040.2.t.v.1009.5 6 15.8 even 4
5040.2.t.v.1009.6 6 15.2 even 4
8400.2.a.dg.1.1 3 1.1 even 1 trivial
8400.2.a.dj.1.3 3 5.4 even 2