Properties

Label 3420.2.a.m.1.1
Level $3420$
Weight $2$
Character 3420.1
Self dual yes
Analytic conductor $27.309$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.3088374913\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 3420.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -1.32088 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -1.32088 q^{7} -3.70739 q^{11} +3.32088 q^{13} -1.61350 q^{17} -1.00000 q^{19} +7.67004 q^{23} +1.00000 q^{25} +1.70739 q^{29} +9.67004 q^{31} +1.32088 q^{35} +5.96265 q^{37} -2.29261 q^{41} -8.73566 q^{43} -11.6700 q^{47} -5.25526 q^{49} -10.4431 q^{53} +3.70739 q^{55} -12.6983 q^{59} +9.02827 q^{61} -3.32088 q^{65} -13.2835 q^{67} +8.69832 q^{71} -12.0565 q^{73} +4.89703 q^{77} -14.0565 q^{79} -5.02827 q^{83} +1.61350 q^{85} +1.70739 q^{89} -4.38650 q^{91} +1.00000 q^{95} -0.679116 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 4 q^{7} - 6 q^{11} + 2 q^{13} - 2 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 4 q^{35} - 6 q^{37} - 12 q^{41} - 8 q^{43} - 6 q^{47} + 3 q^{49} - 8 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} - 2 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{83} + 2 q^{85} - 16 q^{91} + 3 q^{95} - 10 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.32088 −0.499247 −0.249624 0.968343i \(-0.580307\pi\)
−0.249624 + 0.968343i \(0.580307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.70739 −1.11782 −0.558910 0.829228i \(-0.688780\pi\)
−0.558910 + 0.829228i \(0.688780\pi\)
\(12\) 0 0
\(13\) 3.32088 0.921048 0.460524 0.887647i \(-0.347662\pi\)
0.460524 + 0.887647i \(0.347662\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.61350 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.67004 1.59931 0.799657 0.600457i \(-0.205015\pi\)
0.799657 + 0.600457i \(0.205015\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.70739 0.317054 0.158527 0.987355i \(-0.449325\pi\)
0.158527 + 0.987355i \(0.449325\pi\)
\(30\) 0 0
\(31\) 9.67004 1.73679 0.868395 0.495872i \(-0.165152\pi\)
0.868395 + 0.495872i \(0.165152\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.32088 0.223270
\(36\) 0 0
\(37\) 5.96265 0.980254 0.490127 0.871651i \(-0.336950\pi\)
0.490127 + 0.871651i \(0.336950\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.29261 −0.358046 −0.179023 0.983845i \(-0.557294\pi\)
−0.179023 + 0.983845i \(0.557294\pi\)
\(42\) 0 0
\(43\) −8.73566 −1.33218 −0.666088 0.745873i \(-0.732033\pi\)
−0.666088 + 0.745873i \(0.732033\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.6700 −1.70225 −0.851125 0.524963i \(-0.824079\pi\)
−0.851125 + 0.524963i \(0.824079\pi\)
\(48\) 0 0
\(49\) −5.25526 −0.750752
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.4431 −1.43446 −0.717232 0.696835i \(-0.754591\pi\)
−0.717232 + 0.696835i \(0.754591\pi\)
\(54\) 0 0
\(55\) 3.70739 0.499904
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.6983 −1.65318 −0.826590 0.562805i \(-0.809722\pi\)
−0.826590 + 0.562805i \(0.809722\pi\)
\(60\) 0 0
\(61\) 9.02827 1.15595 0.577976 0.816054i \(-0.303843\pi\)
0.577976 + 0.816054i \(0.303843\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.32088 −0.411905
\(66\) 0 0
\(67\) −13.2835 −1.62284 −0.811421 0.584462i \(-0.801306\pi\)
−0.811421 + 0.584462i \(0.801306\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.69832 1.03230 0.516150 0.856498i \(-0.327365\pi\)
0.516150 + 0.856498i \(0.327365\pi\)
\(72\) 0 0
\(73\) −12.0565 −1.41111 −0.705556 0.708654i \(-0.749303\pi\)
−0.705556 + 0.708654i \(0.749303\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.89703 0.558069
\(78\) 0 0
\(79\) −14.0565 −1.58149 −0.790743 0.612149i \(-0.790305\pi\)
−0.790743 + 0.612149i \(0.790305\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.02827 −0.551925 −0.275962 0.961168i \(-0.588997\pi\)
−0.275962 + 0.961168i \(0.588997\pi\)
\(84\) 0 0
\(85\) 1.61350 0.175008
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.70739 0.180983 0.0904915 0.995897i \(-0.471156\pi\)
0.0904915 + 0.995897i \(0.471156\pi\)
\(90\) 0 0
\(91\) −4.38650 −0.459831
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −0.679116 −0.0689537 −0.0344769 0.999405i \(-0.510977\pi\)
−0.0344769 + 0.999405i \(0.510977\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.6418 −1.25790 −0.628952 0.777445i \(-0.716516\pi\)
−0.628952 + 0.777445i \(0.716516\pi\)
\(102\) 0 0
\(103\) 6.05655 0.596769 0.298385 0.954446i \(-0.403552\pi\)
0.298385 + 0.954446i \(0.403552\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.6983 1.02471 0.512356 0.858773i \(-0.328773\pi\)
0.512356 + 0.858773i \(0.328773\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.3118 1.53449 0.767243 0.641356i \(-0.221628\pi\)
0.767243 + 0.641356i \(0.221628\pi\)
\(114\) 0 0
\(115\) −7.67004 −0.715235
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.13124 0.195371
\(120\) 0 0
\(121\) 2.74474 0.249521
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.29261 −0.375047 −0.187524 0.982260i \(-0.560046\pi\)
−0.187524 + 0.982260i \(0.560046\pi\)
\(132\) 0 0
\(133\) 1.32088 0.114535
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −8.58522 −0.728189 −0.364094 0.931362i \(-0.618621\pi\)
−0.364094 + 0.931362i \(0.618621\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.3118 −1.02957
\(144\) 0 0
\(145\) −1.70739 −0.141791
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.0565 −0.987711 −0.493855 0.869544i \(-0.664413\pi\)
−0.493855 + 0.869544i \(0.664413\pi\)
\(150\) 0 0
\(151\) −22.9536 −1.86794 −0.933968 0.357357i \(-0.883678\pi\)
−0.933968 + 0.357357i \(0.883678\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.67004 −0.776717
\(156\) 0 0
\(157\) −11.8688 −0.947230 −0.473615 0.880732i \(-0.657051\pi\)
−0.473615 + 0.880732i \(0.657051\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.1312 −0.798454
\(162\) 0 0
\(163\) −0.0373465 −0.00292520 −0.00146260 0.999999i \(-0.500466\pi\)
−0.00146260 + 0.999999i \(0.500466\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.0283 0.853393 0.426697 0.904395i \(-0.359677\pi\)
0.426697 + 0.904395i \(0.359677\pi\)
\(168\) 0 0
\(169\) −1.97173 −0.151671
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.8578 1.35771 0.678853 0.734274i \(-0.262477\pi\)
0.678853 + 0.734274i \(0.262477\pi\)
\(174\) 0 0
\(175\) −1.32088 −0.0998495
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.9253 1.48929 0.744644 0.667462i \(-0.232619\pi\)
0.744644 + 0.667462i \(0.232619\pi\)
\(180\) 0 0
\(181\) −15.4713 −1.14997 −0.574987 0.818162i \(-0.694993\pi\)
−0.574987 + 0.818162i \(0.694993\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.96265 −0.438383
\(186\) 0 0
\(187\) 5.98185 0.437437
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.3492 −1.32770 −0.663849 0.747866i \(-0.731078\pi\)
−0.663849 + 0.747866i \(0.731078\pi\)
\(192\) 0 0
\(193\) 12.0192 0.865161 0.432581 0.901595i \(-0.357603\pi\)
0.432581 + 0.901595i \(0.357603\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.84049 −0.344870 −0.172435 0.985021i \(-0.555164\pi\)
−0.172435 + 0.985021i \(0.555164\pi\)
\(198\) 0 0
\(199\) −20.6983 −1.46726 −0.733632 0.679547i \(-0.762177\pi\)
−0.733632 + 0.679547i \(0.762177\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.25526 −0.158289
\(204\) 0 0
\(205\) 2.29261 0.160123
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.70739 0.256445
\(210\) 0 0
\(211\) 2.05655 0.141579 0.0707893 0.997491i \(-0.477448\pi\)
0.0707893 + 0.997491i \(0.477448\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.73566 0.595767
\(216\) 0 0
\(217\) −12.7730 −0.867088
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.35823 −0.360434
\(222\) 0 0
\(223\) 6.05655 0.405576 0.202788 0.979223i \(-0.435000\pi\)
0.202788 + 0.979223i \(0.435000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.3118 1.87912 0.939560 0.342383i \(-0.111234\pi\)
0.939560 + 0.342383i \(0.111234\pi\)
\(228\) 0 0
\(229\) −1.80128 −0.119032 −0.0595161 0.998227i \(-0.518956\pi\)
−0.0595161 + 0.998227i \(0.518956\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.94345 0.258344 0.129172 0.991622i \(-0.458768\pi\)
0.129172 + 0.991622i \(0.458768\pi\)
\(234\) 0 0
\(235\) 11.6700 0.761270
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.8031 1.08690 0.543452 0.839440i \(-0.317117\pi\)
0.543452 + 0.839440i \(0.317117\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.25526 0.335747
\(246\) 0 0
\(247\) −3.32088 −0.211303
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.9909 −1.57741 −0.788707 0.614770i \(-0.789249\pi\)
−0.788707 + 0.614770i \(0.789249\pi\)
\(252\) 0 0
\(253\) −28.4358 −1.78775
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.91518 0.181844 0.0909219 0.995858i \(-0.471019\pi\)
0.0909219 + 0.995858i \(0.471019\pi\)
\(258\) 0 0
\(259\) −7.87598 −0.489389
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.3118 1.37581 0.687903 0.725803i \(-0.258532\pi\)
0.687903 + 0.725803i \(0.258532\pi\)
\(264\) 0 0
\(265\) 10.4431 0.641512
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.0656 −0.918567 −0.459284 0.888290i \(-0.651894\pi\)
−0.459284 + 0.888290i \(0.651894\pi\)
\(270\) 0 0
\(271\) 5.47133 0.332359 0.166180 0.986095i \(-0.446857\pi\)
0.166180 + 0.986095i \(0.446857\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.70739 −0.223564
\(276\) 0 0
\(277\) −10.7730 −0.647287 −0.323644 0.946179i \(-0.604908\pi\)
−0.323644 + 0.946179i \(0.604908\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −17.8205 −1.06308 −0.531540 0.847033i \(-0.678387\pi\)
−0.531540 + 0.847033i \(0.678387\pi\)
\(282\) 0 0
\(283\) −30.2070 −1.79562 −0.897810 0.440384i \(-0.854842\pi\)
−0.897810 + 0.440384i \(0.854842\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.02827 0.178753
\(288\) 0 0
\(289\) −14.3966 −0.846861
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.2553 1.06648 0.533242 0.845963i \(-0.320974\pi\)
0.533242 + 0.845963i \(0.320974\pi\)
\(294\) 0 0
\(295\) 12.6983 0.739325
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.4713 1.47304
\(300\) 0 0
\(301\) 11.5388 0.665085
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.02827 −0.516957
\(306\) 0 0
\(307\) −16.5852 −0.946569 −0.473284 0.880910i \(-0.656932\pi\)
−0.473284 + 0.880910i \(0.656932\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.5196 −0.653217 −0.326608 0.945160i \(-0.605906\pi\)
−0.326608 + 0.945160i \(0.605906\pi\)
\(312\) 0 0
\(313\) 32.7549 1.85141 0.925707 0.378241i \(-0.123471\pi\)
0.925707 + 0.378241i \(0.123471\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.2161 −0.854619 −0.427310 0.904105i \(-0.640539\pi\)
−0.427310 + 0.904105i \(0.640539\pi\)
\(318\) 0 0
\(319\) −6.32996 −0.354410
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.61350 0.0897773
\(324\) 0 0
\(325\) 3.32088 0.184210
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.4148 0.849844
\(330\) 0 0
\(331\) −26.9536 −1.48150 −0.740751 0.671779i \(-0.765530\pi\)
−0.740751 + 0.671779i \(0.765530\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.2835 0.725757
\(336\) 0 0
\(337\) −17.5652 −0.956839 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −35.8506 −1.94142
\(342\) 0 0
\(343\) 16.1878 0.874058
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.3118 1.84195 0.920977 0.389616i \(-0.127392\pi\)
0.920977 + 0.389616i \(0.127392\pi\)
\(348\) 0 0
\(349\) 35.2835 1.88868 0.944342 0.328965i \(-0.106700\pi\)
0.944342 + 0.328965i \(0.106700\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.71646 0.463930 0.231965 0.972724i \(-0.425484\pi\)
0.231965 + 0.972724i \(0.425484\pi\)
\(354\) 0 0
\(355\) −8.69832 −0.461659
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.4623 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0565 0.631069
\(366\) 0 0
\(367\) −27.5652 −1.43889 −0.719446 0.694548i \(-0.755604\pi\)
−0.719446 + 0.694548i \(0.755604\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.7941 0.716152
\(372\) 0 0
\(373\) 16.0939 0.833310 0.416655 0.909065i \(-0.363202\pi\)
0.416655 + 0.909065i \(0.363202\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.67004 0.292022
\(378\) 0 0
\(379\) 9.01013 0.462819 0.231410 0.972856i \(-0.425666\pi\)
0.231410 + 0.972856i \(0.425666\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.3401 1.19262 0.596311 0.802753i \(-0.296632\pi\)
0.596311 + 0.802753i \(0.296632\pi\)
\(384\) 0 0
\(385\) −4.89703 −0.249576
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.0565 −1.01691 −0.508454 0.861089i \(-0.669783\pi\)
−0.508454 + 0.861089i \(0.669783\pi\)
\(390\) 0 0
\(391\) −12.3756 −0.625860
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.0565 0.707262
\(396\) 0 0
\(397\) −21.7375 −1.09097 −0.545487 0.838119i \(-0.683655\pi\)
−0.545487 + 0.838119i \(0.683655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.17872 0.358488 0.179244 0.983805i \(-0.442635\pi\)
0.179244 + 0.983805i \(0.442635\pi\)
\(402\) 0 0
\(403\) 32.1131 1.59967
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −22.1059 −1.09575
\(408\) 0 0
\(409\) 16.2443 0.803231 0.401615 0.915808i \(-0.368449\pi\)
0.401615 + 0.915808i \(0.368449\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.7730 0.825346
\(414\) 0 0
\(415\) 5.02827 0.246828
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.9909 −0.830061 −0.415031 0.909807i \(-0.636229\pi\)
−0.415031 + 0.909807i \(0.636229\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.61350 −0.0782660
\(426\) 0 0
\(427\) −11.9253 −0.577106
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −22.6418 −1.09062 −0.545308 0.838236i \(-0.683587\pi\)
−0.545308 + 0.838236i \(0.683587\pi\)
\(432\) 0 0
\(433\) −12.2070 −0.586630 −0.293315 0.956016i \(-0.594759\pi\)
−0.293315 + 0.956016i \(0.594759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.67004 −0.366908
\(438\) 0 0
\(439\) −31.8506 −1.52015 −0.760073 0.649837i \(-0.774837\pi\)
−0.760073 + 0.649837i \(0.774837\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.95358 −0.425397 −0.212699 0.977118i \(-0.568225\pi\)
−0.212699 + 0.977118i \(0.568225\pi\)
\(444\) 0 0
\(445\) −1.70739 −0.0809380
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.51960 −0.260486 −0.130243 0.991482i \(-0.541576\pi\)
−0.130243 + 0.991482i \(0.541576\pi\)
\(450\) 0 0
\(451\) 8.49960 0.400231
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.38650 0.205643
\(456\) 0 0
\(457\) 0.131241 0.00613918 0.00306959 0.999995i \(-0.499023\pi\)
0.00306959 + 0.999995i \(0.499023\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1131 0.657312 0.328656 0.944450i \(-0.393404\pi\)
0.328656 + 0.944450i \(0.393404\pi\)
\(462\) 0 0
\(463\) 5.98080 0.277951 0.138976 0.990296i \(-0.455619\pi\)
0.138976 + 0.990296i \(0.455619\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.38650 0.295532 0.147766 0.989022i \(-0.452792\pi\)
0.147766 + 0.989022i \(0.452792\pi\)
\(468\) 0 0
\(469\) 17.5460 0.810200
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 32.3865 1.48913
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.8597 −0.861721 −0.430861 0.902419i \(-0.641790\pi\)
−0.430861 + 0.902419i \(0.641790\pi\)
\(480\) 0 0
\(481\) 19.8013 0.902861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.679116 0.0308370
\(486\) 0 0
\(487\) 10.1312 0.459090 0.229545 0.973298i \(-0.426276\pi\)
0.229545 + 0.973298i \(0.426276\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.06562 −0.0480908 −0.0240454 0.999711i \(-0.507655\pi\)
−0.0240454 + 0.999711i \(0.507655\pi\)
\(492\) 0 0
\(493\) −2.75486 −0.124073
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.4895 −0.515373
\(498\) 0 0
\(499\) −14.2443 −0.637664 −0.318832 0.947811i \(-0.603291\pi\)
−0.318832 + 0.947811i \(0.603291\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −30.9717 −1.38096 −0.690481 0.723351i \(-0.742601\pi\)
−0.690481 + 0.723351i \(0.742601\pi\)
\(504\) 0 0
\(505\) 12.6418 0.562551
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 30.4057 1.34771 0.673855 0.738864i \(-0.264637\pi\)
0.673855 + 0.738864i \(0.264637\pi\)
\(510\) 0 0
\(511\) 15.9253 0.704494
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.05655 −0.266883
\(516\) 0 0
\(517\) 43.2654 1.90281
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.34916 0.190540 0.0952700 0.995451i \(-0.469629\pi\)
0.0952700 + 0.995451i \(0.469629\pi\)
\(522\) 0 0
\(523\) −8.69832 −0.380351 −0.190175 0.981750i \(-0.560906\pi\)
−0.190175 + 0.981750i \(0.560906\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.6026 −0.679659
\(528\) 0 0
\(529\) 35.8296 1.55781
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.61350 −0.329777
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 19.4833 0.839206
\(540\) 0 0
\(541\) 8.45398 0.363465 0.181733 0.983348i \(-0.441830\pi\)
0.181733 + 0.983348i \(0.441830\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.6983 −0.458266
\(546\) 0 0
\(547\) 11.5279 0.492896 0.246448 0.969156i \(-0.420736\pi\)
0.246448 + 0.969156i \(0.420736\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.70739 −0.0727372
\(552\) 0 0
\(553\) 18.5671 0.789552
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.22699 −0.0519892 −0.0259946 0.999662i \(-0.508275\pi\)
−0.0259946 + 0.999662i \(0.508275\pi\)
\(558\) 0 0
\(559\) −29.0101 −1.22700
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.9144 −0.839291 −0.419646 0.907688i \(-0.637846\pi\)
−0.419646 + 0.907688i \(0.637846\pi\)
\(564\) 0 0
\(565\) −16.3118 −0.686243
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.5015 1.15292 0.576460 0.817125i \(-0.304433\pi\)
0.576460 + 0.817125i \(0.304433\pi\)
\(570\) 0 0
\(571\) −9.47133 −0.396363 −0.198181 0.980165i \(-0.563503\pi\)
−0.198181 + 0.980165i \(0.563503\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.67004 0.319863
\(576\) 0 0
\(577\) 36.6418 1.52542 0.762708 0.646743i \(-0.223869\pi\)
0.762708 + 0.646743i \(0.223869\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.64177 0.275547
\(582\) 0 0
\(583\) 38.7165 1.60347
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.67004 0.151479 0.0757394 0.997128i \(-0.475868\pi\)
0.0757394 + 0.997128i \(0.475868\pi\)
\(588\) 0 0
\(589\) −9.67004 −0.398447
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.2270 0.707428 0.353714 0.935354i \(-0.384919\pi\)
0.353714 + 0.935354i \(0.384919\pi\)
\(594\) 0 0
\(595\) −2.13124 −0.0873724
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.11310 0.331492 0.165746 0.986168i \(-0.446997\pi\)
0.165746 + 0.986168i \(0.446997\pi\)
\(600\) 0 0
\(601\) 7.35823 0.300149 0.150074 0.988675i \(-0.452049\pi\)
0.150074 + 0.988675i \(0.452049\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.74474 −0.111589
\(606\) 0 0
\(607\) 19.9253 0.808743 0.404372 0.914595i \(-0.367490\pi\)
0.404372 + 0.914595i \(0.367490\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −38.7549 −1.56785
\(612\) 0 0
\(613\) 39.3219 1.58820 0.794099 0.607788i \(-0.207943\pi\)
0.794099 + 0.607788i \(0.207943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.0674757 −0.00271647 −0.00135823 0.999999i \(-0.500432\pi\)
−0.00135823 + 0.999999i \(0.500432\pi\)
\(618\) 0 0
\(619\) 38.8680 1.56224 0.781118 0.624384i \(-0.214650\pi\)
0.781118 + 0.624384i \(0.214650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.25526 −0.0903552
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.62071 −0.383603
\(630\) 0 0
\(631\) 29.2835 1.16576 0.582880 0.812559i \(-0.301926\pi\)
0.582880 + 0.812559i \(0.301926\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −17.4521 −0.691478
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.4057 −0.411001 −0.205500 0.978657i \(-0.565882\pi\)
−0.205500 + 0.978657i \(0.565882\pi\)
\(642\) 0 0
\(643\) −11.3774 −0.448682 −0.224341 0.974511i \(-0.572023\pi\)
−0.224341 + 0.974511i \(0.572023\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −45.5388 −1.79032 −0.895158 0.445750i \(-0.852937\pi\)
−0.895158 + 0.445750i \(0.852937\pi\)
\(648\) 0 0
\(649\) 47.0776 1.84796
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.27341 0.0889654 0.0444827 0.999010i \(-0.485836\pi\)
0.0444827 + 0.999010i \(0.485836\pi\)
\(654\) 0 0
\(655\) 4.29261 0.167726
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.77301 0.341748 0.170874 0.985293i \(-0.445341\pi\)
0.170874 + 0.985293i \(0.445341\pi\)
\(660\) 0 0
\(661\) −44.4924 −1.73055 −0.865277 0.501295i \(-0.832857\pi\)
−0.865277 + 0.501295i \(0.832857\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.32088 −0.0512217
\(666\) 0 0
\(667\) 13.0957 0.507069
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33.4713 −1.29215
\(672\) 0 0
\(673\) −5.96265 −0.229843 −0.114922 0.993375i \(-0.536662\pi\)
−0.114922 + 0.993375i \(0.536662\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.32996 −0.243280 −0.121640 0.992574i \(-0.538815\pi\)
−0.121640 + 0.992574i \(0.538815\pi\)
\(678\) 0 0
\(679\) 0.897033 0.0344250
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −39.8506 −1.52484 −0.762421 0.647082i \(-0.775989\pi\)
−0.762421 + 0.647082i \(0.775989\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −34.6802 −1.32121
\(690\) 0 0
\(691\) 8.07469 0.307176 0.153588 0.988135i \(-0.450917\pi\)
0.153588 + 0.988135i \(0.450917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.58522 0.325656
\(696\) 0 0
\(697\) 3.69912 0.140114
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.77301 0.104735 0.0523676 0.998628i \(-0.483323\pi\)
0.0523676 + 0.998628i \(0.483323\pi\)
\(702\) 0 0
\(703\) −5.96265 −0.224886
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.6983 0.628005
\(708\) 0 0
\(709\) 10.1987 0.383021 0.191510 0.981491i \(-0.438661\pi\)
0.191510 + 0.981491i \(0.438661\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 74.1696 2.77767
\(714\) 0 0
\(715\) 12.3118 0.460436
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.2745 −1.27822 −0.639111 0.769115i \(-0.720698\pi\)
−0.639111 + 0.769115i \(0.720698\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.70739 0.0634108
\(726\) 0 0
\(727\) 22.0939 0.819417 0.409709 0.912216i \(-0.365630\pi\)
0.409709 + 0.912216i \(0.365630\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14.0950 0.521321
\(732\) 0 0
\(733\) −13.3401 −0.492727 −0.246364 0.969177i \(-0.579236\pi\)
−0.246364 + 0.969177i \(0.579236\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 49.2472 1.81405
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.8861 −1.20647 −0.603237 0.797562i \(-0.706123\pi\)
−0.603237 + 0.797562i \(0.706123\pi\)
\(744\) 0 0
\(745\) 12.0565 0.441718
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.4996 0.602079 0.301039 0.953612i \(-0.402666\pi\)
0.301039 + 0.953612i \(0.402666\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.9536 0.835366
\(756\) 0 0
\(757\) −42.5489 −1.54647 −0.773234 0.634121i \(-0.781362\pi\)
−0.773234 + 0.634121i \(0.781362\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5.34009 −0.193578 −0.0967890 0.995305i \(-0.530857\pi\)
−0.0967890 + 0.995305i \(0.530857\pi\)
\(762\) 0 0
\(763\) −14.1312 −0.511585
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −42.1696 −1.52266
\(768\) 0 0
\(769\) −7.28354 −0.262651 −0.131326 0.991339i \(-0.541923\pi\)
−0.131326 + 0.991339i \(0.541923\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.1806 0.653910 0.326955 0.945040i \(-0.393977\pi\)
0.326955 + 0.945040i \(0.393977\pi\)
\(774\) 0 0
\(775\) 9.67004 0.347358
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.29261 0.0821413
\(780\) 0 0
\(781\) −32.2480 −1.15393
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.8688 0.423614
\(786\) 0 0
\(787\) 20.8114 0.741847 0.370923 0.928663i \(-0.379041\pi\)
0.370923 + 0.928663i \(0.379041\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −21.5460 −0.766088
\(792\) 0 0
\(793\) 29.9819 1.06469
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −40.9354 −1.45001 −0.725004 0.688745i \(-0.758162\pi\)
−0.725004 + 0.688745i \(0.758162\pi\)
\(798\) 0 0
\(799\) 18.8296 0.666142
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 44.6983 1.57737
\(804\) 0 0
\(805\) 10.1312 0.357079
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.48947 0.0523670 0.0261835 0.999657i \(-0.491665\pi\)
0.0261835 + 0.999657i \(0.491665\pi\)
\(810\) 0 0
\(811\) 21.5460 0.756583 0.378292 0.925687i \(-0.376512\pi\)
0.378292 + 0.925687i \(0.376512\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.0373465 0.00130819
\(816\) 0 0
\(817\) 8.73566 0.305622
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35.9819 1.25578 0.627888 0.778304i \(-0.283920\pi\)
0.627888 + 0.778304i \(0.283920\pi\)
\(822\) 0 0
\(823\) 15.8880 0.553819 0.276910 0.960896i \(-0.410690\pi\)
0.276910 + 0.960896i \(0.410690\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −26.8789 −0.934671 −0.467335 0.884080i \(-0.654786\pi\)
−0.467335 + 0.884080i \(0.654786\pi\)
\(828\) 0 0
\(829\) 17.9253 0.622572 0.311286 0.950316i \(-0.399240\pi\)
0.311286 + 0.950316i \(0.399240\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.47934 0.293792
\(834\) 0 0
\(835\) −11.0283 −0.381649
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.30168 −0.252082 −0.126041 0.992025i \(-0.540227\pi\)
−0.126041 + 0.992025i \(0.540227\pi\)
\(840\) 0 0
\(841\) −26.0848 −0.899477
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.97173 0.0678294
\(846\) 0 0
\(847\) −3.62548 −0.124573
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 45.7338 1.56773
\(852\) 0 0
\(853\) −37.4148 −1.28106 −0.640529 0.767934i \(-0.721285\pi\)
−0.640529 + 0.767934i \(0.721285\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.1806 0.347762 0.173881 0.984767i \(-0.444369\pi\)
0.173881 + 0.984767i \(0.444369\pi\)
\(858\) 0 0
\(859\) −40.7367 −1.38992 −0.694959 0.719049i \(-0.744578\pi\)
−0.694959 + 0.719049i \(0.744578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.11310 0.276173 0.138086 0.990420i \(-0.455905\pi\)
0.138086 + 0.990420i \(0.455905\pi\)
\(864\) 0 0
\(865\) −17.8578 −0.607184
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 52.1131 1.76782
\(870\) 0 0
\(871\) −44.1131 −1.49472
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.32088 0.0446540
\(876\) 0 0
\(877\) 29.8880 1.00924 0.504622 0.863340i \(-0.331632\pi\)
0.504622 + 0.863340i \(0.331632\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −34.8114 −1.17283 −0.586413 0.810012i \(-0.699460\pi\)
−0.586413 + 0.810012i \(0.699460\pi\)
\(882\) 0 0
\(883\) 27.8496 0.937212 0.468606 0.883407i \(-0.344756\pi\)
0.468606 + 0.883407i \(0.344756\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 50.7933 1.70547 0.852735 0.522343i \(-0.174942\pi\)
0.852735 + 0.522343i \(0.174942\pi\)
\(888\) 0 0
\(889\) 5.28354 0.177204
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.6700 0.390523
\(894\) 0 0
\(895\) −19.9253 −0.666030
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.5105 0.550657
\(900\) 0 0
\(901\) 16.8498 0.561349
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.4713 0.514284
\(906\) 0 0
\(907\) −32.3009 −1.07253 −0.536267 0.844049i \(-0.680166\pi\)
−0.536267 + 0.844049i \(0.680166\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 52.7367 1.74725 0.873623 0.486604i \(-0.161764\pi\)
0.873623 + 0.486604i \(0.161764\pi\)
\(912\) 0 0
\(913\) 18.6418 0.616953
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.67004 0.187241
\(918\) 0 0
\(919\) 52.5105 1.73216 0.866081 0.499903i \(-0.166631\pi\)
0.866081 + 0.499903i \(0.166631\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 28.8861 0.950798
\(924\) 0 0
\(925\) 5.96265 0.196051
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.8688 1.17682 0.588408 0.808564i \(-0.299755\pi\)
0.588408 + 0.808564i \(0.299755\pi\)
\(930\) 0 0
\(931\) 5.25526 0.172234
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.98185 −0.195628
\(936\) 0 0
\(937\) 10.6983 0.349499 0.174749 0.984613i \(-0.444088\pi\)
0.174749 + 0.984613i \(0.444088\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 22.3673 0.729153 0.364577 0.931173i \(-0.381214\pi\)
0.364577 + 0.931173i \(0.381214\pi\)
\(942\) 0 0
\(943\) −17.5844 −0.572628
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.8223 −1.13157 −0.565787 0.824551i \(-0.691428\pi\)
−0.565787 + 0.824551i \(0.691428\pi\)
\(948\) 0 0
\(949\) −40.0384 −1.29970
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.4823 0.955024 0.477512 0.878625i \(-0.341539\pi\)
0.477512 + 0.878625i \(0.341539\pi\)
\(954\) 0 0
\(955\) 18.3492 0.593765
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.64177 0.0853072
\(960\) 0 0
\(961\) 62.5097 2.01644
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.0192 −0.386912
\(966\) 0 0
\(967\) 18.8669 0.606719 0.303359 0.952876i \(-0.401892\pi\)
0.303359 + 0.952876i \(0.401892\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31.2270 −1.00212 −0.501061 0.865412i \(-0.667057\pi\)
−0.501061 + 0.865412i \(0.667057\pi\)
\(972\) 0 0
\(973\) 11.3401 0.363546
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.44305 −0.206132 −0.103066 0.994675i \(-0.532865\pi\)
−0.103066 + 0.994675i \(0.532865\pi\)
\(978\) 0 0
\(979\) −6.32996 −0.202306
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.57429 −0.145897 −0.0729486 0.997336i \(-0.523241\pi\)
−0.0729486 + 0.997336i \(0.523241\pi\)
\(984\) 0 0
\(985\) 4.84049 0.154231
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −67.0029 −2.13057
\(990\) 0 0
\(991\) −38.8296 −1.23346 −0.616731 0.787174i \(-0.711543\pi\)
−0.616731 + 0.787174i \(0.711543\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.6983 0.656181
\(996\) 0 0
\(997\) 54.9245 1.73948 0.869738 0.493513i \(-0.164288\pi\)
0.869738 + 0.493513i \(0.164288\pi\)
\(998\) 0 0
\(999\) 0 0
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 3420.2.a.m.1.1 3
3.2 odd 2 1140.2.a.g.1.1 3
12.11 even 2 4560.2.a.br.1.3 3
15.2 even 4 5700.2.f.q.3649.1 6
15.8 even 4 5700.2.f.q.3649.6 6
15.14 odd 2 5700.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.g.1.1 3 3.2 odd 2
3420.2.a.m.1.1 3 1.1 even 1 trivial
4560.2.a.br.1.3 3 12.11 even 2
5700.2.a.w.1.3 3 15.14 odd 2
5700.2.f.q.3649.1 6 15.2 even 4
5700.2.f.q.3649.6 6 15.8 even 4