Properties

Label 6240.2.a.bw.1.2
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} -3.58774 q^{11} +1.00000 q^{13} +1.00000 q^{15} -7.81756 q^{17} -4.94567 q^{19} +1.35793 q^{21} -0.0737791 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.51396 q^{29} -4.00000 q^{31} +3.58774 q^{33} +1.35793 q^{35} -1.58774 q^{37} -1.00000 q^{39} -5.58774 q^{41} -7.17548 q^{43} -1.00000 q^{45} -2.56829 q^{47} -5.15604 q^{49} +7.81756 q^{51} +12.7632 q^{53} +3.58774 q^{55} +4.94567 q^{57} +8.45963 q^{59} -4.30359 q^{61} -1.35793 q^{63} -1.00000 q^{65} +0.0737791 q^{69} +0.871889 q^{71} +6.79811 q^{73} -1.00000 q^{75} +4.87189 q^{77} -12.8719 q^{79} +1.00000 q^{81} +5.43171 q^{83} +7.81756 q^{85} -5.51396 q^{87} -6.87189 q^{89} -1.35793 q^{91} +4.00000 q^{93} +4.94567 q^{95} +7.35793 q^{97} -3.58774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.35793 −0.513248 −0.256624 0.966511i \(-0.582610\pi\)
−0.256624 + 0.966511i \(0.582610\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.58774 −1.08174 −0.540872 0.841105i \(-0.681906\pi\)
−0.540872 + 0.841105i \(0.681906\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −7.81756 −1.89604 −0.948018 0.318217i \(-0.896916\pi\)
−0.948018 + 0.318217i \(0.896916\pi\)
\(18\) 0 0
\(19\) −4.94567 −1.13461 −0.567307 0.823506i \(-0.692015\pi\)
−0.567307 + 0.823506i \(0.692015\pi\)
\(20\) 0 0
\(21\) 1.35793 0.296324
\(22\) 0 0
\(23\) −0.0737791 −0.0153840 −0.00769200 0.999970i \(-0.502448\pi\)
−0.00769200 + 0.999970i \(0.502448\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.51396 1.02392 0.511959 0.859010i \(-0.328920\pi\)
0.511959 + 0.859010i \(0.328920\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 3.58774 0.624546
\(34\) 0 0
\(35\) 1.35793 0.229531
\(36\) 0 0
\(37\) −1.58774 −0.261023 −0.130512 0.991447i \(-0.541662\pi\)
−0.130512 + 0.991447i \(0.541662\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.58774 −0.872659 −0.436329 0.899787i \(-0.643722\pi\)
−0.436329 + 0.899787i \(0.643722\pi\)
\(42\) 0 0
\(43\) −7.17548 −1.09425 −0.547125 0.837051i \(-0.684278\pi\)
−0.547125 + 0.837051i \(0.684278\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.56829 −0.374624 −0.187312 0.982300i \(-0.559978\pi\)
−0.187312 + 0.982300i \(0.559978\pi\)
\(48\) 0 0
\(49\) −5.15604 −0.736577
\(50\) 0 0
\(51\) 7.81756 1.09468
\(52\) 0 0
\(53\) 12.7632 1.75316 0.876582 0.481253i \(-0.159818\pi\)
0.876582 + 0.481253i \(0.159818\pi\)
\(54\) 0 0
\(55\) 3.58774 0.483771
\(56\) 0 0
\(57\) 4.94567 0.655070
\(58\) 0 0
\(59\) 8.45963 1.10135 0.550675 0.834720i \(-0.314370\pi\)
0.550675 + 0.834720i \(0.314370\pi\)
\(60\) 0 0
\(61\) −4.30359 −0.551019 −0.275509 0.961298i \(-0.588847\pi\)
−0.275509 + 0.961298i \(0.588847\pi\)
\(62\) 0 0
\(63\) −1.35793 −0.171083
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0.0737791 0.00888196
\(70\) 0 0
\(71\) 0.871889 0.103474 0.0517371 0.998661i \(-0.483524\pi\)
0.0517371 + 0.998661i \(0.483524\pi\)
\(72\) 0 0
\(73\) 6.79811 0.795659 0.397829 0.917459i \(-0.369764\pi\)
0.397829 + 0.917459i \(0.369764\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.87189 0.555203
\(78\) 0 0
\(79\) −12.8719 −1.44820 −0.724100 0.689695i \(-0.757745\pi\)
−0.724100 + 0.689695i \(0.757745\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.43171 0.596207 0.298104 0.954534i \(-0.403646\pi\)
0.298104 + 0.954534i \(0.403646\pi\)
\(84\) 0 0
\(85\) 7.81756 0.847933
\(86\) 0 0
\(87\) −5.51396 −0.591159
\(88\) 0 0
\(89\) −6.87189 −0.728419 −0.364209 0.931317i \(-0.618661\pi\)
−0.364209 + 0.931317i \(0.618661\pi\)
\(90\) 0 0
\(91\) −1.35793 −0.142349
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 4.94567 0.507415
\(96\) 0 0
\(97\) 7.35793 0.747084 0.373542 0.927613i \(-0.378143\pi\)
0.373542 + 0.927613i \(0.378143\pi\)
\(98\) 0 0
\(99\) −3.58774 −0.360582
\(100\) 0 0
\(101\) −6.94567 −0.691120 −0.345560 0.938397i \(-0.612311\pi\)
−0.345560 + 0.938397i \(0.612311\pi\)
\(102\) 0 0
\(103\) −10.5683 −1.04133 −0.520663 0.853763i \(-0.674315\pi\)
−0.520663 + 0.853763i \(0.674315\pi\)
\(104\) 0 0
\(105\) −1.35793 −0.132520
\(106\) 0 0
\(107\) 1.69641 0.163998 0.0819989 0.996632i \(-0.473870\pi\)
0.0819989 + 0.996632i \(0.473870\pi\)
\(108\) 0 0
\(109\) −3.40530 −0.326168 −0.163084 0.986612i \(-0.552144\pi\)
−0.163084 + 0.986612i \(0.552144\pi\)
\(110\) 0 0
\(111\) 1.58774 0.150702
\(112\) 0 0
\(113\) 16.6894 1.57001 0.785005 0.619489i \(-0.212660\pi\)
0.785005 + 0.619489i \(0.212660\pi\)
\(114\) 0 0
\(115\) 0.0737791 0.00687994
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 10.6157 0.973137
\(120\) 0 0
\(121\) 1.87189 0.170172
\(122\) 0 0
\(123\) 5.58774 0.503830
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.7827 −1.75543 −0.877714 0.479185i \(-0.840932\pi\)
−0.877714 + 0.479185i \(0.840932\pi\)
\(128\) 0 0
\(129\) 7.17548 0.631766
\(130\) 0 0
\(131\) 0.798110 0.0697312 0.0348656 0.999392i \(-0.488900\pi\)
0.0348656 + 0.999392i \(0.488900\pi\)
\(132\) 0 0
\(133\) 6.71585 0.582338
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 13.0279 1.11305 0.556525 0.830831i \(-0.312134\pi\)
0.556525 + 0.830831i \(0.312134\pi\)
\(138\) 0 0
\(139\) 8.19493 0.695085 0.347542 0.937664i \(-0.387016\pi\)
0.347542 + 0.937664i \(0.387016\pi\)
\(140\) 0 0
\(141\) 2.56829 0.216289
\(142\) 0 0
\(143\) −3.58774 −0.300022
\(144\) 0 0
\(145\) −5.51396 −0.457910
\(146\) 0 0
\(147\) 5.15604 0.425263
\(148\) 0 0
\(149\) 12.1560 0.995861 0.497931 0.867217i \(-0.334093\pi\)
0.497931 + 0.867217i \(0.334093\pi\)
\(150\) 0 0
\(151\) 12.9193 1.05135 0.525677 0.850684i \(-0.323812\pi\)
0.525677 + 0.850684i \(0.323812\pi\)
\(152\) 0 0
\(153\) −7.81756 −0.632012
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 9.32304 0.744060 0.372030 0.928221i \(-0.378662\pi\)
0.372030 + 0.928221i \(0.378662\pi\)
\(158\) 0 0
\(159\) −12.7632 −1.01219
\(160\) 0 0
\(161\) 0.100187 0.00789581
\(162\) 0 0
\(163\) 14.7632 1.15634 0.578172 0.815915i \(-0.303766\pi\)
0.578172 + 0.815915i \(0.303766\pi\)
\(164\) 0 0
\(165\) −3.58774 −0.279305
\(166\) 0 0
\(167\) −4.45963 −0.345097 −0.172548 0.985001i \(-0.555200\pi\)
−0.172548 + 0.985001i \(0.555200\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.94567 −0.378205
\(172\) 0 0
\(173\) −6.45963 −0.491117 −0.245558 0.969382i \(-0.578971\pi\)
−0.245558 + 0.969382i \(0.578971\pi\)
\(174\) 0 0
\(175\) −1.35793 −0.102650
\(176\) 0 0
\(177\) −8.45963 −0.635865
\(178\) 0 0
\(179\) 8.94567 0.668631 0.334315 0.942461i \(-0.391495\pi\)
0.334315 + 0.942461i \(0.391495\pi\)
\(180\) 0 0
\(181\) −19.3315 −1.43690 −0.718450 0.695578i \(-0.755148\pi\)
−0.718450 + 0.695578i \(0.755148\pi\)
\(182\) 0 0
\(183\) 4.30359 0.318131
\(184\) 0 0
\(185\) 1.58774 0.116733
\(186\) 0 0
\(187\) 28.0474 2.05103
\(188\) 0 0
\(189\) 1.35793 0.0987746
\(190\) 0 0
\(191\) −9.74378 −0.705035 −0.352517 0.935805i \(-0.614674\pi\)
−0.352517 + 0.935805i \(0.614674\pi\)
\(192\) 0 0
\(193\) −4.27719 −0.307879 −0.153939 0.988080i \(-0.549196\pi\)
−0.153939 + 0.988080i \(0.549196\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 1.54037 0.109747 0.0548734 0.998493i \(-0.482524\pi\)
0.0548734 + 0.998493i \(0.482524\pi\)
\(198\) 0 0
\(199\) −16.9193 −1.19937 −0.599687 0.800234i \(-0.704708\pi\)
−0.599687 + 0.800234i \(0.704708\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.48755 −0.525523
\(204\) 0 0
\(205\) 5.58774 0.390265
\(206\) 0 0
\(207\) −0.0737791 −0.00512800
\(208\) 0 0
\(209\) 17.7438 1.22736
\(210\) 0 0
\(211\) −6.56829 −0.452180 −0.226090 0.974106i \(-0.572594\pi\)
−0.226090 + 0.974106i \(0.572594\pi\)
\(212\) 0 0
\(213\) −0.871889 −0.0597408
\(214\) 0 0
\(215\) 7.17548 0.489364
\(216\) 0 0
\(217\) 5.43171 0.368728
\(218\) 0 0
\(219\) −6.79811 −0.459374
\(220\) 0 0
\(221\) −7.81756 −0.525866
\(222\) 0 0
\(223\) −4.33848 −0.290526 −0.145263 0.989393i \(-0.546403\pi\)
−0.145263 + 0.989393i \(0.546403\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.0947 −1.33373 −0.666867 0.745176i \(-0.732365\pi\)
−0.666867 + 0.745176i \(0.732365\pi\)
\(228\) 0 0
\(229\) −0.837003 −0.0553107 −0.0276554 0.999618i \(-0.508804\pi\)
−0.0276554 + 0.999618i \(0.508804\pi\)
\(230\) 0 0
\(231\) −4.87189 −0.320547
\(232\) 0 0
\(233\) −3.21037 −0.210318 −0.105159 0.994455i \(-0.533535\pi\)
−0.105159 + 0.994455i \(0.533535\pi\)
\(234\) 0 0
\(235\) 2.56829 0.167537
\(236\) 0 0
\(237\) 12.8719 0.836119
\(238\) 0 0
\(239\) 10.3036 0.666484 0.333242 0.942841i \(-0.391857\pi\)
0.333242 + 0.942841i \(0.391857\pi\)
\(240\) 0 0
\(241\) 23.5264 1.51547 0.757736 0.652561i \(-0.226306\pi\)
0.757736 + 0.652561i \(0.226306\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.15604 0.329407
\(246\) 0 0
\(247\) −4.94567 −0.314685
\(248\) 0 0
\(249\) −5.43171 −0.344220
\(250\) 0 0
\(251\) 17.7702 1.12164 0.560822 0.827936i \(-0.310485\pi\)
0.560822 + 0.827936i \(0.310485\pi\)
\(252\) 0 0
\(253\) 0.264700 0.0166416
\(254\) 0 0
\(255\) −7.81756 −0.489554
\(256\) 0 0
\(257\) −17.4442 −1.08814 −0.544069 0.839040i \(-0.683117\pi\)
−0.544069 + 0.839040i \(0.683117\pi\)
\(258\) 0 0
\(259\) 2.15604 0.133970
\(260\) 0 0
\(261\) 5.51396 0.341306
\(262\) 0 0
\(263\) −11.1491 −0.687481 −0.343741 0.939065i \(-0.611694\pi\)
−0.343741 + 0.939065i \(0.611694\pi\)
\(264\) 0 0
\(265\) −12.7632 −0.784039
\(266\) 0 0
\(267\) 6.87189 0.420553
\(268\) 0 0
\(269\) −13.2966 −0.810710 −0.405355 0.914159i \(-0.632852\pi\)
−0.405355 + 0.914159i \(0.632852\pi\)
\(270\) 0 0
\(271\) 12.0947 0.734703 0.367352 0.930082i \(-0.380265\pi\)
0.367352 + 0.930082i \(0.380265\pi\)
\(272\) 0 0
\(273\) 1.35793 0.0821854
\(274\) 0 0
\(275\) −3.58774 −0.216349
\(276\) 0 0
\(277\) 27.0668 1.62629 0.813144 0.582063i \(-0.197754\pi\)
0.813144 + 0.582063i \(0.197754\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 17.7827 1.06083 0.530413 0.847740i \(-0.322037\pi\)
0.530413 + 0.847740i \(0.322037\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) −4.94567 −0.292956
\(286\) 0 0
\(287\) 7.58774 0.447890
\(288\) 0 0
\(289\) 44.1142 2.59495
\(290\) 0 0
\(291\) −7.35793 −0.431329
\(292\) 0 0
\(293\) 16.7159 0.976551 0.488275 0.872690i \(-0.337626\pi\)
0.488275 + 0.872690i \(0.337626\pi\)
\(294\) 0 0
\(295\) −8.45963 −0.492539
\(296\) 0 0
\(297\) 3.58774 0.208182
\(298\) 0 0
\(299\) −0.0737791 −0.00426676
\(300\) 0 0
\(301\) 9.74378 0.561622
\(302\) 0 0
\(303\) 6.94567 0.399018
\(304\) 0 0
\(305\) 4.30359 0.246423
\(306\) 0 0
\(307\) 22.0863 1.26053 0.630265 0.776380i \(-0.282946\pi\)
0.630265 + 0.776380i \(0.282946\pi\)
\(308\) 0 0
\(309\) 10.5683 0.601209
\(310\) 0 0
\(311\) 4.60719 0.261250 0.130625 0.991432i \(-0.458302\pi\)
0.130625 + 0.991432i \(0.458302\pi\)
\(312\) 0 0
\(313\) −16.8106 −0.950191 −0.475096 0.879934i \(-0.657587\pi\)
−0.475096 + 0.879934i \(0.657587\pi\)
\(314\) 0 0
\(315\) 1.35793 0.0765105
\(316\) 0 0
\(317\) 32.3510 1.81701 0.908506 0.417873i \(-0.137224\pi\)
0.908506 + 0.417873i \(0.137224\pi\)
\(318\) 0 0
\(319\) −19.7827 −1.10762
\(320\) 0 0
\(321\) −1.69641 −0.0946841
\(322\) 0 0
\(323\) 38.6630 2.15127
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 3.40530 0.188313
\(328\) 0 0
\(329\) 3.48755 0.192275
\(330\) 0 0
\(331\) −28.2159 −1.55089 −0.775443 0.631418i \(-0.782473\pi\)
−0.775443 + 0.631418i \(0.782473\pi\)
\(332\) 0 0
\(333\) −1.58774 −0.0870077
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.5962 1.06747 0.533737 0.845651i \(-0.320787\pi\)
0.533737 + 0.845651i \(0.320787\pi\)
\(338\) 0 0
\(339\) −16.6894 −0.906446
\(340\) 0 0
\(341\) 14.3510 0.777148
\(342\) 0 0
\(343\) 16.5070 0.891294
\(344\) 0 0
\(345\) −0.0737791 −0.00397213
\(346\) 0 0
\(347\) 33.6436 1.80608 0.903041 0.429554i \(-0.141329\pi\)
0.903041 + 0.429554i \(0.141329\pi\)
\(348\) 0 0
\(349\) 23.5529 1.26076 0.630378 0.776289i \(-0.282900\pi\)
0.630378 + 0.776289i \(0.282900\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 20.2034 1.07532 0.537659 0.843162i \(-0.319309\pi\)
0.537659 + 0.843162i \(0.319309\pi\)
\(354\) 0 0
\(355\) −0.871889 −0.0462750
\(356\) 0 0
\(357\) −10.6157 −0.561841
\(358\) 0 0
\(359\) 1.43171 0.0755625 0.0377813 0.999286i \(-0.487971\pi\)
0.0377813 + 0.999286i \(0.487971\pi\)
\(360\) 0 0
\(361\) 5.45963 0.287349
\(362\) 0 0
\(363\) −1.87189 −0.0982487
\(364\) 0 0
\(365\) −6.79811 −0.355829
\(366\) 0 0
\(367\) 9.74378 0.508621 0.254311 0.967123i \(-0.418151\pi\)
0.254311 + 0.967123i \(0.418151\pi\)
\(368\) 0 0
\(369\) −5.58774 −0.290886
\(370\) 0 0
\(371\) −17.3315 −0.899808
\(372\) 0 0
\(373\) 15.6740 0.811569 0.405785 0.913969i \(-0.366998\pi\)
0.405785 + 0.913969i \(0.366998\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 5.51396 0.283984
\(378\) 0 0
\(379\) −27.1491 −1.39455 −0.697277 0.716802i \(-0.745605\pi\)
−0.697277 + 0.716802i \(0.745605\pi\)
\(380\) 0 0
\(381\) 19.7827 1.01350
\(382\) 0 0
\(383\) 9.06682 0.463293 0.231646 0.972800i \(-0.425589\pi\)
0.231646 + 0.972800i \(0.425589\pi\)
\(384\) 0 0
\(385\) −4.87189 −0.248294
\(386\) 0 0
\(387\) −7.17548 −0.364750
\(388\) 0 0
\(389\) 15.7004 0.796043 0.398021 0.917376i \(-0.369697\pi\)
0.398021 + 0.917376i \(0.369697\pi\)
\(390\) 0 0
\(391\) 0.576772 0.0291686
\(392\) 0 0
\(393\) −0.798110 −0.0402593
\(394\) 0 0
\(395\) 12.8719 0.647655
\(396\) 0 0
\(397\) 11.5488 0.579620 0.289810 0.957084i \(-0.406408\pi\)
0.289810 + 0.957084i \(0.406408\pi\)
\(398\) 0 0
\(399\) −6.71585 −0.336213
\(400\) 0 0
\(401\) −16.8106 −0.839481 −0.419741 0.907644i \(-0.637879\pi\)
−0.419741 + 0.907644i \(0.637879\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 5.69641 0.282360
\(408\) 0 0
\(409\) 16.2034 0.801207 0.400603 0.916252i \(-0.368801\pi\)
0.400603 + 0.916252i \(0.368801\pi\)
\(410\) 0 0
\(411\) −13.0279 −0.642620
\(412\) 0 0
\(413\) −11.4876 −0.565266
\(414\) 0 0
\(415\) −5.43171 −0.266632
\(416\) 0 0
\(417\) −8.19493 −0.401307
\(418\) 0 0
\(419\) −34.1600 −1.66883 −0.834414 0.551139i \(-0.814194\pi\)
−0.834414 + 0.551139i \(0.814194\pi\)
\(420\) 0 0
\(421\) −1.80908 −0.0881691 −0.0440846 0.999028i \(-0.514037\pi\)
−0.0440846 + 0.999028i \(0.514037\pi\)
\(422\) 0 0
\(423\) −2.56829 −0.124875
\(424\) 0 0
\(425\) −7.81756 −0.379207
\(426\) 0 0
\(427\) 5.84396 0.282809
\(428\) 0 0
\(429\) 3.58774 0.173218
\(430\) 0 0
\(431\) 13.1366 0.632767 0.316384 0.948631i \(-0.397531\pi\)
0.316384 + 0.948631i \(0.397531\pi\)
\(432\) 0 0
\(433\) −18.7547 −0.901296 −0.450648 0.892702i \(-0.648807\pi\)
−0.450648 + 0.892702i \(0.648807\pi\)
\(434\) 0 0
\(435\) 5.51396 0.264374
\(436\) 0 0
\(437\) 0.364887 0.0174549
\(438\) 0 0
\(439\) 15.2229 0.726547 0.363274 0.931683i \(-0.381659\pi\)
0.363274 + 0.931683i \(0.381659\pi\)
\(440\) 0 0
\(441\) −5.15604 −0.245526
\(442\) 0 0
\(443\) −17.1142 −0.813120 −0.406560 0.913624i \(-0.633272\pi\)
−0.406560 + 0.913624i \(0.633272\pi\)
\(444\) 0 0
\(445\) 6.87189 0.325759
\(446\) 0 0
\(447\) −12.1560 −0.574961
\(448\) 0 0
\(449\) −22.9666 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(450\) 0 0
\(451\) 20.0474 0.943994
\(452\) 0 0
\(453\) −12.9193 −0.607000
\(454\) 0 0
\(455\) 1.35793 0.0636606
\(456\) 0 0
\(457\) −11.2104 −0.524399 −0.262199 0.965014i \(-0.584448\pi\)
−0.262199 + 0.965014i \(0.584448\pi\)
\(458\) 0 0
\(459\) 7.81756 0.364892
\(460\) 0 0
\(461\) −20.6157 −0.960167 −0.480084 0.877223i \(-0.659394\pi\)
−0.480084 + 0.877223i \(0.659394\pi\)
\(462\) 0 0
\(463\) −5.81756 −0.270365 −0.135182 0.990821i \(-0.543162\pi\)
−0.135182 + 0.990821i \(0.543162\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 12.2647 0.567543 0.283771 0.958892i \(-0.408414\pi\)
0.283771 + 0.958892i \(0.408414\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9.32304 −0.429583
\(472\) 0 0
\(473\) 25.7438 1.18370
\(474\) 0 0
\(475\) −4.94567 −0.226923
\(476\) 0 0
\(477\) 12.7632 0.584388
\(478\) 0 0
\(479\) 14.9108 0.681291 0.340646 0.940192i \(-0.389354\pi\)
0.340646 + 0.940192i \(0.389354\pi\)
\(480\) 0 0
\(481\) −1.58774 −0.0723948
\(482\) 0 0
\(483\) −0.100187 −0.00455865
\(484\) 0 0
\(485\) −7.35793 −0.334106
\(486\) 0 0
\(487\) −24.5334 −1.11171 −0.555857 0.831278i \(-0.687610\pi\)
−0.555857 + 0.831278i \(0.687610\pi\)
\(488\) 0 0
\(489\) −14.7632 −0.667616
\(490\) 0 0
\(491\) 30.3246 1.36853 0.684264 0.729234i \(-0.260124\pi\)
0.684264 + 0.729234i \(0.260124\pi\)
\(492\) 0 0
\(493\) −43.1057 −1.94138
\(494\) 0 0
\(495\) 3.58774 0.161257
\(496\) 0 0
\(497\) −1.18396 −0.0531079
\(498\) 0 0
\(499\) 10.0823 0.451344 0.225672 0.974203i \(-0.427542\pi\)
0.225672 + 0.974203i \(0.427542\pi\)
\(500\) 0 0
\(501\) 4.45963 0.199242
\(502\) 0 0
\(503\) −38.9317 −1.73588 −0.867940 0.496668i \(-0.834557\pi\)
−0.867940 + 0.496668i \(0.834557\pi\)
\(504\) 0 0
\(505\) 6.94567 0.309078
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −14.3594 −0.636471 −0.318236 0.948012i \(-0.603090\pi\)
−0.318236 + 0.948012i \(0.603090\pi\)
\(510\) 0 0
\(511\) −9.23133 −0.408370
\(512\) 0 0
\(513\) 4.94567 0.218357
\(514\) 0 0
\(515\) 10.5683 0.465695
\(516\) 0 0
\(517\) 9.21438 0.405248
\(518\) 0 0
\(519\) 6.45963 0.283546
\(520\) 0 0
\(521\) 3.59622 0.157553 0.0787766 0.996892i \(-0.474899\pi\)
0.0787766 + 0.996892i \(0.474899\pi\)
\(522\) 0 0
\(523\) −7.68793 −0.336170 −0.168085 0.985773i \(-0.553758\pi\)
−0.168085 + 0.985773i \(0.553758\pi\)
\(524\) 0 0
\(525\) 1.35793 0.0592648
\(526\) 0 0
\(527\) 31.2702 1.36215
\(528\) 0 0
\(529\) −22.9946 −0.999763
\(530\) 0 0
\(531\) 8.45963 0.367117
\(532\) 0 0
\(533\) −5.58774 −0.242032
\(534\) 0 0
\(535\) −1.69641 −0.0733420
\(536\) 0 0
\(537\) −8.94567 −0.386034
\(538\) 0 0
\(539\) 18.4985 0.796788
\(540\) 0 0
\(541\) −1.41922 −0.0610170 −0.0305085 0.999535i \(-0.509713\pi\)
−0.0305085 + 0.999535i \(0.509713\pi\)
\(542\) 0 0
\(543\) 19.3315 0.829595
\(544\) 0 0
\(545\) 3.40530 0.145867
\(546\) 0 0
\(547\) 6.86341 0.293458 0.146729 0.989177i \(-0.453125\pi\)
0.146729 + 0.989177i \(0.453125\pi\)
\(548\) 0 0
\(549\) −4.30359 −0.183673
\(550\) 0 0
\(551\) −27.2702 −1.16175
\(552\) 0 0
\(553\) 17.4791 0.743286
\(554\) 0 0
\(555\) −1.58774 −0.0673959
\(556\) 0 0
\(557\) 35.7438 1.51451 0.757256 0.653118i \(-0.226539\pi\)
0.757256 + 0.653118i \(0.226539\pi\)
\(558\) 0 0
\(559\) −7.17548 −0.303491
\(560\) 0 0
\(561\) −28.0474 −1.18416
\(562\) 0 0
\(563\) 36.8021 1.55102 0.775512 0.631333i \(-0.217492\pi\)
0.775512 + 0.631333i \(0.217492\pi\)
\(564\) 0 0
\(565\) −16.6894 −0.702130
\(566\) 0 0
\(567\) −1.35793 −0.0570275
\(568\) 0 0
\(569\) 6.45963 0.270802 0.135401 0.990791i \(-0.456768\pi\)
0.135401 + 0.990791i \(0.456768\pi\)
\(570\) 0 0
\(571\) 31.3704 1.31281 0.656405 0.754408i \(-0.272076\pi\)
0.656405 + 0.754408i \(0.272076\pi\)
\(572\) 0 0
\(573\) 9.74378 0.407052
\(574\) 0 0
\(575\) −0.0737791 −0.00307680
\(576\) 0 0
\(577\) 40.9541 1.70494 0.852472 0.522773i \(-0.175103\pi\)
0.852472 + 0.522773i \(0.175103\pi\)
\(578\) 0 0
\(579\) 4.27719 0.177754
\(580\) 0 0
\(581\) −7.37586 −0.306002
\(582\) 0 0
\(583\) −45.7911 −1.89648
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 36.3899 1.50197 0.750985 0.660319i \(-0.229579\pi\)
0.750985 + 0.660319i \(0.229579\pi\)
\(588\) 0 0
\(589\) 19.7827 0.815131
\(590\) 0 0
\(591\) −1.54037 −0.0633623
\(592\) 0 0
\(593\) 8.03889 0.330118 0.165059 0.986284i \(-0.447219\pi\)
0.165059 + 0.986284i \(0.447219\pi\)
\(594\) 0 0
\(595\) −10.6157 −0.435200
\(596\) 0 0
\(597\) 16.9193 0.692459
\(598\) 0 0
\(599\) 44.7019 1.82647 0.913236 0.407432i \(-0.133576\pi\)
0.913236 + 0.407432i \(0.133576\pi\)
\(600\) 0 0
\(601\) −23.9387 −0.976480 −0.488240 0.872709i \(-0.662361\pi\)
−0.488240 + 0.872709i \(0.662361\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.87189 −0.0761031
\(606\) 0 0
\(607\) −6.25622 −0.253932 −0.126966 0.991907i \(-0.540524\pi\)
−0.126966 + 0.991907i \(0.540524\pi\)
\(608\) 0 0
\(609\) 7.48755 0.303411
\(610\) 0 0
\(611\) −2.56829 −0.103902
\(612\) 0 0
\(613\) −18.4123 −0.743664 −0.371832 0.928300i \(-0.621270\pi\)
−0.371832 + 0.928300i \(0.621270\pi\)
\(614\) 0 0
\(615\) −5.58774 −0.225319
\(616\) 0 0
\(617\) −11.5962 −0.466846 −0.233423 0.972375i \(-0.574993\pi\)
−0.233423 + 0.972375i \(0.574993\pi\)
\(618\) 0 0
\(619\) 19.8260 0.796876 0.398438 0.917195i \(-0.369552\pi\)
0.398438 + 0.917195i \(0.369552\pi\)
\(620\) 0 0
\(621\) 0.0737791 0.00296065
\(622\) 0 0
\(623\) 9.33152 0.373859
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.7438 −0.708618
\(628\) 0 0
\(629\) 12.4123 0.494909
\(630\) 0 0
\(631\) −33.0838 −1.31704 −0.658522 0.752561i \(-0.728818\pi\)
−0.658522 + 0.752561i \(0.728818\pi\)
\(632\) 0 0
\(633\) 6.56829 0.261066
\(634\) 0 0
\(635\) 19.7827 0.785051
\(636\) 0 0
\(637\) −5.15604 −0.204290
\(638\) 0 0
\(639\) 0.871889 0.0344914
\(640\) 0 0
\(641\) −11.4317 −0.451525 −0.225763 0.974182i \(-0.572487\pi\)
−0.225763 + 0.974182i \(0.572487\pi\)
\(642\) 0 0
\(643\) −8.26470 −0.325928 −0.162964 0.986632i \(-0.552105\pi\)
−0.162964 + 0.986632i \(0.552105\pi\)
\(644\) 0 0
\(645\) −7.17548 −0.282534
\(646\) 0 0
\(647\) −19.7089 −0.774837 −0.387418 0.921904i \(-0.626633\pi\)
−0.387418 + 0.921904i \(0.626633\pi\)
\(648\) 0 0
\(649\) −30.3510 −1.19138
\(650\) 0 0
\(651\) −5.43171 −0.212885
\(652\) 0 0
\(653\) −22.0778 −0.863971 −0.431985 0.901881i \(-0.642187\pi\)
−0.431985 + 0.901881i \(0.642187\pi\)
\(654\) 0 0
\(655\) −0.798110 −0.0311847
\(656\) 0 0
\(657\) 6.79811 0.265220
\(658\) 0 0
\(659\) −35.6087 −1.38712 −0.693559 0.720400i \(-0.743958\pi\)
−0.693559 + 0.720400i \(0.743958\pi\)
\(660\) 0 0
\(661\) −51.2577 −1.99370 −0.996848 0.0793414i \(-0.974718\pi\)
−0.996848 + 0.0793414i \(0.974718\pi\)
\(662\) 0 0
\(663\) 7.81756 0.303609
\(664\) 0 0
\(665\) −6.71585 −0.260430
\(666\) 0 0
\(667\) −0.406815 −0.0157519
\(668\) 0 0
\(669\) 4.33848 0.167735
\(670\) 0 0
\(671\) 15.4402 0.596062
\(672\) 0 0
\(673\) 25.1755 0.970444 0.485222 0.874391i \(-0.338739\pi\)
0.485222 + 0.874391i \(0.338739\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 13.7353 0.527890 0.263945 0.964538i \(-0.414976\pi\)
0.263945 + 0.964538i \(0.414976\pi\)
\(678\) 0 0
\(679\) −9.99152 −0.383439
\(680\) 0 0
\(681\) 20.0947 0.770032
\(682\) 0 0
\(683\) 40.3899 1.54548 0.772738 0.634726i \(-0.218887\pi\)
0.772738 + 0.634726i \(0.218887\pi\)
\(684\) 0 0
\(685\) −13.0279 −0.497771
\(686\) 0 0
\(687\) 0.837003 0.0319337
\(688\) 0 0
\(689\) 12.7632 0.486240
\(690\) 0 0
\(691\) −41.9427 −1.59558 −0.797788 0.602938i \(-0.793997\pi\)
−0.797788 + 0.602938i \(0.793997\pi\)
\(692\) 0 0
\(693\) 4.87189 0.185068
\(694\) 0 0
\(695\) −8.19493 −0.310851
\(696\) 0 0
\(697\) 43.6825 1.65459
\(698\) 0 0
\(699\) 3.21037 0.121427
\(700\) 0 0
\(701\) −42.7283 −1.61383 −0.806914 0.590670i \(-0.798864\pi\)
−0.806914 + 0.590670i \(0.798864\pi\)
\(702\) 0 0
\(703\) 7.85244 0.296160
\(704\) 0 0
\(705\) −2.56829 −0.0967276
\(706\) 0 0
\(707\) 9.43171 0.354716
\(708\) 0 0
\(709\) −11.1102 −0.417252 −0.208626 0.977996i \(-0.566899\pi\)
−0.208626 + 0.977996i \(0.566899\pi\)
\(710\) 0 0
\(711\) −12.8719 −0.482734
\(712\) 0 0
\(713\) 0.295116 0.0110522
\(714\) 0 0
\(715\) 3.58774 0.134174
\(716\) 0 0
\(717\) −10.3036 −0.384795
\(718\) 0 0
\(719\) 8.67696 0.323596 0.161798 0.986824i \(-0.448271\pi\)
0.161798 + 0.986824i \(0.448271\pi\)
\(720\) 0 0
\(721\) 14.3510 0.534458
\(722\) 0 0
\(723\) −23.5264 −0.874958
\(724\) 0 0
\(725\) 5.51396 0.204783
\(726\) 0 0
\(727\) 14.4985 0.537720 0.268860 0.963179i \(-0.413353\pi\)
0.268860 + 0.963179i \(0.413353\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 56.0947 2.07474
\(732\) 0 0
\(733\) −0.668481 −0.0246909 −0.0123455 0.999924i \(-0.503930\pi\)
−0.0123455 + 0.999924i \(0.503930\pi\)
\(734\) 0 0
\(735\) −5.15604 −0.190183
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −17.1102 −0.629408 −0.314704 0.949190i \(-0.601905\pi\)
−0.314704 + 0.949190i \(0.601905\pi\)
\(740\) 0 0
\(741\) 4.94567 0.181684
\(742\) 0 0
\(743\) −5.57926 −0.204683 −0.102342 0.994749i \(-0.532634\pi\)
−0.102342 + 0.994749i \(0.532634\pi\)
\(744\) 0 0
\(745\) −12.1560 −0.445363
\(746\) 0 0
\(747\) 5.43171 0.198736
\(748\) 0 0
\(749\) −2.30359 −0.0841715
\(750\) 0 0
\(751\) −5.47908 −0.199934 −0.0999672 0.994991i \(-0.531874\pi\)
−0.0999672 + 0.994991i \(0.531874\pi\)
\(752\) 0 0
\(753\) −17.7702 −0.647582
\(754\) 0 0
\(755\) −12.9193 −0.470180
\(756\) 0 0
\(757\) −48.7408 −1.77152 −0.885758 0.464148i \(-0.846361\pi\)
−0.885758 + 0.464148i \(0.846361\pi\)
\(758\) 0 0
\(759\) −0.264700 −0.00960801
\(760\) 0 0
\(761\) −29.9472 −1.08558 −0.542792 0.839867i \(-0.682633\pi\)
−0.542792 + 0.839867i \(0.682633\pi\)
\(762\) 0 0
\(763\) 4.62414 0.167405
\(764\) 0 0
\(765\) 7.81756 0.282644
\(766\) 0 0
\(767\) 8.45963 0.305460
\(768\) 0 0
\(769\) −24.1336 −0.870281 −0.435141 0.900363i \(-0.643301\pi\)
−0.435141 + 0.900363i \(0.643301\pi\)
\(770\) 0 0
\(771\) 17.4442 0.628237
\(772\) 0 0
\(773\) 42.5544 1.53057 0.765287 0.643689i \(-0.222597\pi\)
0.765287 + 0.643689i \(0.222597\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) −2.15604 −0.0773474
\(778\) 0 0
\(779\) 27.6351 0.990131
\(780\) 0 0
\(781\) −3.12811 −0.111933
\(782\) 0 0
\(783\) −5.51396 −0.197053
\(784\) 0 0
\(785\) −9.32304 −0.332754
\(786\) 0 0
\(787\) 0.676959 0.0241310 0.0120655 0.999927i \(-0.496159\pi\)
0.0120655 + 0.999927i \(0.496159\pi\)
\(788\) 0 0
\(789\) 11.1491 0.396918
\(790\) 0 0
\(791\) −22.6630 −0.805805
\(792\) 0 0
\(793\) −4.30359 −0.152825
\(794\) 0 0
\(795\) 12.7632 0.452665
\(796\) 0 0
\(797\) −7.84396 −0.277847 −0.138924 0.990303i \(-0.544364\pi\)
−0.138924 + 0.990303i \(0.544364\pi\)
\(798\) 0 0
\(799\) 20.0778 0.710301
\(800\) 0 0
\(801\) −6.87189 −0.242806
\(802\) 0 0
\(803\) −24.3899 −0.860699
\(804\) 0 0
\(805\) −0.100187 −0.00353111
\(806\) 0 0
\(807\) 13.2966 0.468064
\(808\) 0 0
\(809\) 7.13659 0.250909 0.125455 0.992099i \(-0.459961\pi\)
0.125455 + 0.992099i \(0.459961\pi\)
\(810\) 0 0
\(811\) 16.4860 0.578903 0.289452 0.957193i \(-0.406527\pi\)
0.289452 + 0.957193i \(0.406527\pi\)
\(812\) 0 0
\(813\) −12.0947 −0.424181
\(814\) 0 0
\(815\) −14.7632 −0.517133
\(816\) 0 0
\(817\) 35.4876 1.24155
\(818\) 0 0
\(819\) −1.35793 −0.0474498
\(820\) 0 0
\(821\) −9.80507 −0.342199 −0.171100 0.985254i \(-0.554732\pi\)
−0.171100 + 0.985254i \(0.554732\pi\)
\(822\) 0 0
\(823\) −26.5683 −0.926113 −0.463056 0.886329i \(-0.653247\pi\)
−0.463056 + 0.886329i \(0.653247\pi\)
\(824\) 0 0
\(825\) 3.58774 0.124909
\(826\) 0 0
\(827\) 17.6490 0.613717 0.306859 0.951755i \(-0.400722\pi\)
0.306859 + 0.951755i \(0.400722\pi\)
\(828\) 0 0
\(829\) 54.1895 1.88208 0.941039 0.338297i \(-0.109851\pi\)
0.941039 + 0.338297i \(0.109851\pi\)
\(830\) 0 0
\(831\) −27.0668 −0.938938
\(832\) 0 0
\(833\) 40.3076 1.39658
\(834\) 0 0
\(835\) 4.45963 0.154332
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) −47.8300 −1.65128 −0.825638 0.564200i \(-0.809185\pi\)
−0.825638 + 0.564200i \(0.809185\pi\)
\(840\) 0 0
\(841\) 1.40378 0.0484062
\(842\) 0 0
\(843\) −17.7827 −0.612468
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −2.54189 −0.0873403
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.117142 0.00401558
\(852\) 0 0
\(853\) 9.07530 0.310732 0.155366 0.987857i \(-0.450344\pi\)
0.155366 + 0.987857i \(0.450344\pi\)
\(854\) 0 0
\(855\) 4.94567 0.169138
\(856\) 0 0
\(857\) 1.54438 0.0527550 0.0263775 0.999652i \(-0.491603\pi\)
0.0263775 + 0.999652i \(0.491603\pi\)
\(858\) 0 0
\(859\) −9.69641 −0.330837 −0.165419 0.986223i \(-0.552897\pi\)
−0.165419 + 0.986223i \(0.552897\pi\)
\(860\) 0 0
\(861\) −7.58774 −0.258590
\(862\) 0 0
\(863\) 37.9083 1.29041 0.645207 0.764008i \(-0.276771\pi\)
0.645207 + 0.764008i \(0.276771\pi\)
\(864\) 0 0
\(865\) 6.45963 0.219634
\(866\) 0 0
\(867\) −44.1142 −1.49820
\(868\) 0 0
\(869\) 46.1810 1.56658
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 7.35793 0.249028
\(874\) 0 0
\(875\) 1.35793 0.0459063
\(876\) 0 0
\(877\) −22.4068 −0.756624 −0.378312 0.925678i \(-0.623495\pi\)
−0.378312 + 0.925678i \(0.623495\pi\)
\(878\) 0 0
\(879\) −16.7159 −0.563812
\(880\) 0 0
\(881\) −32.7159 −1.10223 −0.551113 0.834431i \(-0.685797\pi\)
−0.551113 + 0.834431i \(0.685797\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 8.45963 0.284367
\(886\) 0 0
\(887\) 49.5305 1.66307 0.831535 0.555472i \(-0.187463\pi\)
0.831535 + 0.555472i \(0.187463\pi\)
\(888\) 0 0
\(889\) 26.8634 0.900970
\(890\) 0 0
\(891\) −3.58774 −0.120194
\(892\) 0 0
\(893\) 12.7019 0.425054
\(894\) 0 0
\(895\) −8.94567 −0.299021
\(896\) 0 0
\(897\) 0.0737791 0.00246341
\(898\) 0 0
\(899\) −22.0558 −0.735604
\(900\) 0 0
\(901\) −99.7772 −3.32406
\(902\) 0 0
\(903\) −9.74378 −0.324253
\(904\) 0 0
\(905\) 19.3315 0.642601
\(906\) 0 0
\(907\) 50.3679 1.67244 0.836220 0.548395i \(-0.184761\pi\)
0.836220 + 0.548395i \(0.184761\pi\)
\(908\) 0 0
\(909\) −6.94567 −0.230373
\(910\) 0 0
\(911\) −16.7717 −0.555671 −0.277836 0.960629i \(-0.589617\pi\)
−0.277836 + 0.960629i \(0.589617\pi\)
\(912\) 0 0
\(913\) −19.4876 −0.644944
\(914\) 0 0
\(915\) −4.30359 −0.142272
\(916\) 0 0
\(917\) −1.08377 −0.0357894
\(918\) 0 0
\(919\) 30.1032 0.993014 0.496507 0.868033i \(-0.334616\pi\)
0.496507 + 0.868033i \(0.334616\pi\)
\(920\) 0 0
\(921\) −22.0863 −0.727767
\(922\) 0 0
\(923\) 0.871889 0.0286986
\(924\) 0 0
\(925\) −1.58774 −0.0522046
\(926\) 0 0
\(927\) −10.5683 −0.347108
\(928\) 0 0
\(929\) 16.3983 0.538012 0.269006 0.963139i \(-0.413305\pi\)
0.269006 + 0.963139i \(0.413305\pi\)
\(930\) 0 0
\(931\) 25.5000 0.835730
\(932\) 0 0
\(933\) −4.60719 −0.150833
\(934\) 0 0
\(935\) −28.0474 −0.917247
\(936\) 0 0
\(937\) −23.3091 −0.761476 −0.380738 0.924683i \(-0.624330\pi\)
−0.380738 + 0.924683i \(0.624330\pi\)
\(938\) 0 0
\(939\) 16.8106 0.548593
\(940\) 0 0
\(941\) 33.1531 1.08076 0.540380 0.841421i \(-0.318281\pi\)
0.540380 + 0.841421i \(0.318281\pi\)
\(942\) 0 0
\(943\) 0.412259 0.0134250
\(944\) 0 0
\(945\) −1.35793 −0.0441733
\(946\) 0 0
\(947\) 43.7827 1.42275 0.711373 0.702815i \(-0.248074\pi\)
0.711373 + 0.702815i \(0.248074\pi\)
\(948\) 0 0
\(949\) 6.79811 0.220676
\(950\) 0 0
\(951\) −32.3510 −1.04905
\(952\) 0 0
\(953\) −29.2493 −0.947477 −0.473738 0.880666i \(-0.657096\pi\)
−0.473738 + 0.880666i \(0.657096\pi\)
\(954\) 0 0
\(955\) 9.74378 0.315301
\(956\) 0 0
\(957\) 19.7827 0.639483
\(958\) 0 0
\(959\) −17.6910 −0.571271
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 1.69641 0.0546659
\(964\) 0 0
\(965\) 4.27719 0.137688
\(966\) 0 0
\(967\) −28.5808 −0.919096 −0.459548 0.888153i \(-0.651989\pi\)
−0.459548 + 0.888153i \(0.651989\pi\)
\(968\) 0 0
\(969\) −38.6630 −1.24204
\(970\) 0 0
\(971\) −31.6785 −1.01661 −0.508305 0.861177i \(-0.669728\pi\)
−0.508305 + 0.861177i \(0.669728\pi\)
\(972\) 0 0
\(973\) −11.1281 −0.356751
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 13.2702 0.424552 0.212276 0.977210i \(-0.431912\pi\)
0.212276 + 0.977210i \(0.431912\pi\)
\(978\) 0 0
\(979\) 24.6546 0.787963
\(980\) 0 0
\(981\) −3.40530 −0.108723
\(982\) 0 0
\(983\) −2.18645 −0.0697370 −0.0348685 0.999392i \(-0.511101\pi\)
−0.0348685 + 0.999392i \(0.511101\pi\)
\(984\) 0 0
\(985\) −1.54037 −0.0490803
\(986\) 0 0
\(987\) −3.48755 −0.111010
\(988\) 0 0
\(989\) 0.529401 0.0168340
\(990\) 0 0
\(991\) 16.9497 0.538424 0.269212 0.963081i \(-0.413237\pi\)
0.269212 + 0.963081i \(0.413237\pi\)
\(992\) 0 0
\(993\) 28.2159 0.895404
\(994\) 0 0
\(995\) 16.9193 0.536377
\(996\) 0 0
\(997\) −20.7936 −0.658541 −0.329271 0.944236i \(-0.606803\pi\)
−0.329271 + 0.944236i \(0.606803\pi\)
\(998\) 0 0
\(999\) 1.58774 0.0502339
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 6240.2.a.bw.1.2 3
4.3 odd 2 6240.2.a.cb.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bw.1.2 3 1.1 even 1 trivial
6240.2.a.cb.1.2 yes 3 4.3 odd 2