Properties

Label 3380.2.a.m.1.2
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
Defining polynomial: \(x^{3} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.339877\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.339877 q^{3} -1.00000 q^{5} +3.88448 q^{7} -2.88448 q^{9} +O(q^{10})\) \(q-0.339877 q^{3} -1.00000 q^{5} +3.88448 q^{7} -2.88448 q^{9} +1.54461 q^{11} +0.339877 q^{15} -2.86485 q^{19} -1.32025 q^{21} -5.42909 q^{23} +1.00000 q^{25} +2.00000 q^{27} -5.20473 q^{29} -6.22436 q^{31} -0.524976 q^{33} -3.88448 q^{35} -8.56424 q^{37} +9.08921 q^{41} -0.980369 q^{43} +2.88448 q^{45} -6.52498 q^{47} +8.08921 q^{49} +6.44872 q^{53} -1.54461 q^{55} +0.973697 q^{57} +4.45539 q^{59} +9.65345 q^{61} -11.2047 q^{63} -6.97370 q^{67} +1.84522 q^{69} +12.6731 q^{71} -3.43576 q^{73} -0.339877 q^{75} +6.00000 q^{77} -13.1285 q^{79} +7.97370 q^{81} -8.56424 q^{83} +1.76897 q^{87} -17.1285 q^{89} +2.11552 q^{93} +2.86485 q^{95} -13.7690 q^{97} -4.45539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{5} + 3 q^{9} - 6 q^{11} - 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} - 6 q^{31} + 6 q^{33} - 12 q^{37} + 6 q^{41} - 6 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} - 30 q^{57} + 24 q^{59} - 6 q^{61} - 24 q^{63} + 12 q^{67} - 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} - 12 q^{83} - 18 q^{87} - 24 q^{89} + 18 q^{93} - 18 q^{97} - 24 q^{99} + O(q^{100}) \)

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
\(3\) −0.339877 −0.196228 −0.0981140 0.995175i \(-0.531281\pi\)
−0.0981140 + 0.995175i \(0.531281\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.88448 1.46820 0.734098 0.679043i \(-0.237605\pi\)
0.734098 + 0.679043i \(0.237605\pi\)
\(8\) 0 0
\(9\) −2.88448 −0.961495
\(10\) 0 0
\(11\) 1.54461 0.465716 0.232858 0.972511i \(-0.425192\pi\)
0.232858 + 0.972511i \(0.425192\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0.339877 0.0877558
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.86485 −0.657242 −0.328621 0.944462i \(-0.606584\pi\)
−0.328621 + 0.944462i \(0.606584\pi\)
\(20\) 0 0
\(21\) −1.32025 −0.288101
\(22\) 0 0
\(23\) −5.42909 −1.13204 −0.566022 0.824390i \(-0.691518\pi\)
−0.566022 + 0.824390i \(0.691518\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.00000 0.384900
\(28\) 0 0
\(29\) −5.20473 −0.966494 −0.483247 0.875484i \(-0.660543\pi\)
−0.483247 + 0.875484i \(0.660543\pi\)
\(30\) 0 0
\(31\) −6.22436 −1.11793 −0.558964 0.829192i \(-0.688801\pi\)
−0.558964 + 0.829192i \(0.688801\pi\)
\(32\) 0 0
\(33\) −0.524976 −0.0913866
\(34\) 0 0
\(35\) −3.88448 −0.656598
\(36\) 0 0
\(37\) −8.56424 −1.40795 −0.703976 0.710224i \(-0.748594\pi\)
−0.703976 + 0.710224i \(0.748594\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.08921 1.41950 0.709748 0.704455i \(-0.248809\pi\)
0.709748 + 0.704455i \(0.248809\pi\)
\(42\) 0 0
\(43\) −0.980369 −0.149505 −0.0747525 0.997202i \(-0.523817\pi\)
−0.0747525 + 0.997202i \(0.523817\pi\)
\(44\) 0 0
\(45\) 2.88448 0.429993
\(46\) 0 0
\(47\) −6.52498 −0.951766 −0.475883 0.879509i \(-0.657871\pi\)
−0.475883 + 0.879509i \(0.657871\pi\)
\(48\) 0 0
\(49\) 8.08921 1.15560
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.44872 0.885800 0.442900 0.896571i \(-0.353950\pi\)
0.442900 + 0.896571i \(0.353950\pi\)
\(54\) 0 0
\(55\) −1.54461 −0.208275
\(56\) 0 0
\(57\) 0.973697 0.128969
\(58\) 0 0
\(59\) 4.45539 0.580043 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(60\) 0 0
\(61\) 9.65345 1.23600 0.617999 0.786179i \(-0.287944\pi\)
0.617999 + 0.786179i \(0.287944\pi\)
\(62\) 0 0
\(63\) −11.2047 −1.41166
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.97370 −0.851973 −0.425986 0.904730i \(-0.640073\pi\)
−0.425986 + 0.904730i \(0.640073\pi\)
\(68\) 0 0
\(69\) 1.84522 0.222139
\(70\) 0 0
\(71\) 12.6731 1.50402 0.752009 0.659153i \(-0.229085\pi\)
0.752009 + 0.659153i \(0.229085\pi\)
\(72\) 0 0
\(73\) −3.43576 −0.402126 −0.201063 0.979578i \(-0.564440\pi\)
−0.201063 + 0.979578i \(0.564440\pi\)
\(74\) 0 0
\(75\) −0.339877 −0.0392456
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −13.1285 −1.47707 −0.738534 0.674216i \(-0.764482\pi\)
−0.738534 + 0.674216i \(0.764482\pi\)
\(80\) 0 0
\(81\) 7.97370 0.885966
\(82\) 0 0
\(83\) −8.56424 −0.940047 −0.470024 0.882654i \(-0.655755\pi\)
−0.470024 + 0.882654i \(0.655755\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.76897 0.189653
\(88\) 0 0
\(89\) −17.1285 −1.81561 −0.907807 0.419387i \(-0.862245\pi\)
−0.907807 + 0.419387i \(0.862245\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.11552 0.219369
\(94\) 0 0
\(95\) 2.86485 0.293928
\(96\) 0 0
\(97\) −13.7690 −1.39803 −0.699013 0.715109i \(-0.746377\pi\)
−0.699013 + 0.715109i \(0.746377\pi\)
\(98\) 0 0
\(99\) −4.45539 −0.447784
\(100\) 0 0
\(101\) 6.44872 0.641672 0.320836 0.947135i \(-0.396036\pi\)
0.320836 + 0.947135i \(0.396036\pi\)
\(102\) 0 0
\(103\) −12.1088 −1.19312 −0.596560 0.802569i \(-0.703466\pi\)
−0.596560 + 0.802569i \(0.703466\pi\)
\(104\) 0 0
\(105\) 1.32025 0.128843
\(106\) 0 0
\(107\) 17.4291 1.68493 0.842467 0.538748i \(-0.181103\pi\)
0.842467 + 0.538748i \(0.181103\pi\)
\(108\) 0 0
\(109\) −16.4095 −1.57174 −0.785871 0.618391i \(-0.787785\pi\)
−0.785871 + 0.618391i \(0.787785\pi\)
\(110\) 0 0
\(111\) 2.91079 0.276280
\(112\) 0 0
\(113\) 1.59054 0.149625 0.0748127 0.997198i \(-0.476164\pi\)
0.0748127 + 0.997198i \(0.476164\pi\)
\(114\) 0 0
\(115\) 5.42909 0.506265
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.61419 −0.783108
\(122\) 0 0
\(123\) −3.08921 −0.278545
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.108844 0.00965837 0.00482918 0.999988i \(-0.498463\pi\)
0.00482918 + 0.999988i \(0.498463\pi\)
\(128\) 0 0
\(129\) 0.333205 0.0293371
\(130\) 0 0
\(131\) −10.4095 −0.909479 −0.454739 0.890625i \(-0.650268\pi\)
−0.454739 + 0.890625i \(0.650268\pi\)
\(132\) 0 0
\(133\) −11.1285 −0.964961
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 9.08921 0.776544 0.388272 0.921545i \(-0.373072\pi\)
0.388272 + 0.921545i \(0.373072\pi\)
\(138\) 0 0
\(139\) 9.04995 0.767607 0.383803 0.923415i \(-0.374614\pi\)
0.383803 + 0.923415i \(0.374614\pi\)
\(140\) 0 0
\(141\) 2.21769 0.186763
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.20473 0.432229
\(146\) 0 0
\(147\) −2.74934 −0.226761
\(148\) 0 0
\(149\) −11.1285 −0.911680 −0.455840 0.890062i \(-0.650661\pi\)
−0.455840 + 0.890062i \(0.650661\pi\)
\(150\) 0 0
\(151\) −9.13515 −0.743408 −0.371704 0.928351i \(-0.621226\pi\)
−0.371704 + 0.928351i \(0.621226\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.22436 0.499953
\(156\) 0 0
\(157\) 4.91079 0.391923 0.195962 0.980612i \(-0.437217\pi\)
0.195962 + 0.980612i \(0.437217\pi\)
\(158\) 0 0
\(159\) −2.19177 −0.173819
\(160\) 0 0
\(161\) −21.0892 −1.66206
\(162\) 0 0
\(163\) 16.9344 1.32641 0.663204 0.748439i \(-0.269196\pi\)
0.663204 + 0.748439i \(0.269196\pi\)
\(164\) 0 0
\(165\) 0.524976 0.0408693
\(166\) 0 0
\(167\) −21.6142 −1.67256 −0.836278 0.548306i \(-0.815273\pi\)
−0.836278 + 0.548306i \(0.815273\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 8.26362 0.631935
\(172\) 0 0
\(173\) 8.03926 0.611214 0.305607 0.952158i \(-0.401141\pi\)
0.305607 + 0.952158i \(0.401141\pi\)
\(174\) 0 0
\(175\) 3.88448 0.293639
\(176\) 0 0
\(177\) −1.51429 −0.113821
\(178\) 0 0
\(179\) −10.8582 −0.811579 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(180\) 0 0
\(181\) −10.5642 −0.785234 −0.392617 0.919702i \(-0.628430\pi\)
−0.392617 + 0.919702i \(0.628430\pi\)
\(182\) 0 0
\(183\) −3.28098 −0.242537
\(184\) 0 0
\(185\) 8.56424 0.629655
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 7.76897 0.565109
\(190\) 0 0
\(191\) −8.81892 −0.638115 −0.319057 0.947735i \(-0.603366\pi\)
−0.319057 + 0.947735i \(0.603366\pi\)
\(192\) 0 0
\(193\) −0.270294 −0.0194562 −0.00972809 0.999953i \(-0.503097\pi\)
−0.00972809 + 0.999953i \(0.503097\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.96074 0.282191 0.141095 0.989996i \(-0.454938\pi\)
0.141095 + 0.989996i \(0.454938\pi\)
\(198\) 0 0
\(199\) −6.03926 −0.428112 −0.214056 0.976821i \(-0.568667\pi\)
−0.214056 + 0.976821i \(0.568667\pi\)
\(200\) 0 0
\(201\) 2.37020 0.167181
\(202\) 0 0
\(203\) −20.2177 −1.41900
\(204\) 0 0
\(205\) −9.08921 −0.634818
\(206\) 0 0
\(207\) 15.6601 1.08845
\(208\) 0 0
\(209\) −4.42507 −0.306089
\(210\) 0 0
\(211\) −16.2177 −1.11647 −0.558236 0.829682i \(-0.688522\pi\)
−0.558236 + 0.829682i \(0.688522\pi\)
\(212\) 0 0
\(213\) −4.30729 −0.295130
\(214\) 0 0
\(215\) 0.980369 0.0668606
\(216\) 0 0
\(217\) −24.1784 −1.64134
\(218\) 0 0
\(219\) 1.16774 0.0789083
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0629 0.673862 0.336931 0.941529i \(-0.390611\pi\)
0.336931 + 0.941529i \(0.390611\pi\)
\(224\) 0 0
\(225\) −2.88448 −0.192299
\(226\) 0 0
\(227\) 3.43576 0.228040 0.114020 0.993478i \(-0.463627\pi\)
0.114020 + 0.993478i \(0.463627\pi\)
\(228\) 0 0
\(229\) −22.8582 −1.51051 −0.755256 0.655430i \(-0.772487\pi\)
−0.755256 + 0.655430i \(0.772487\pi\)
\(230\) 0 0
\(231\) −2.03926 −0.134174
\(232\) 0 0
\(233\) 4.40946 0.288873 0.144437 0.989514i \(-0.453863\pi\)
0.144437 + 0.989514i \(0.453863\pi\)
\(234\) 0 0
\(235\) 6.52498 0.425643
\(236\) 0 0
\(237\) 4.46207 0.289842
\(238\) 0 0
\(239\) 22.8122 1.47560 0.737801 0.675018i \(-0.235864\pi\)
0.737801 + 0.675018i \(0.235864\pi\)
\(240\) 0 0
\(241\) −10.6798 −0.687943 −0.343972 0.938980i \(-0.611772\pi\)
−0.343972 + 0.938980i \(0.611772\pi\)
\(242\) 0 0
\(243\) −8.71008 −0.558752
\(244\) 0 0
\(245\) −8.08921 −0.516801
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 2.91079 0.184464
\(250\) 0 0
\(251\) 21.2676 1.34240 0.671201 0.741276i \(-0.265779\pi\)
0.671201 + 0.741276i \(0.265779\pi\)
\(252\) 0 0
\(253\) −8.38581 −0.527211
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.8974 −1.17879 −0.589395 0.807845i \(-0.700634\pi\)
−0.589395 + 0.807845i \(0.700634\pi\)
\(258\) 0 0
\(259\) −33.2676 −2.06715
\(260\) 0 0
\(261\) 15.0130 0.929279
\(262\) 0 0
\(263\) −13.0196 −0.802825 −0.401412 0.915897i \(-0.631481\pi\)
−0.401412 + 0.915897i \(0.631481\pi\)
\(264\) 0 0
\(265\) −6.44872 −0.396142
\(266\) 0 0
\(267\) 5.82157 0.356274
\(268\) 0 0
\(269\) 14.3702 0.876166 0.438083 0.898934i \(-0.355658\pi\)
0.438083 + 0.898934i \(0.355658\pi\)
\(270\) 0 0
\(271\) −13.7230 −0.833615 −0.416807 0.908995i \(-0.636851\pi\)
−0.416807 + 0.908995i \(0.636851\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.54461 0.0931433
\(276\) 0 0
\(277\) 19.2676 1.15768 0.578840 0.815441i \(-0.303506\pi\)
0.578840 + 0.815441i \(0.303506\pi\)
\(278\) 0 0
\(279\) 17.9541 1.07488
\(280\) 0 0
\(281\) 3.96074 0.236278 0.118139 0.992997i \(-0.462307\pi\)
0.118139 + 0.992997i \(0.462307\pi\)
\(282\) 0 0
\(283\) 21.1981 1.26009 0.630047 0.776557i \(-0.283036\pi\)
0.630047 + 0.776557i \(0.283036\pi\)
\(284\) 0 0
\(285\) −0.973697 −0.0576769
\(286\) 0 0
\(287\) 35.3069 2.08410
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 4.67975 0.274332
\(292\) 0 0
\(293\) −13.6927 −0.799937 −0.399968 0.916529i \(-0.630979\pi\)
−0.399968 + 0.916529i \(0.630979\pi\)
\(294\) 0 0
\(295\) −4.45539 −0.259403
\(296\) 0 0
\(297\) 3.08921 0.179254
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.80823 −0.219503
\(302\) 0 0
\(303\) −2.19177 −0.125914
\(304\) 0 0
\(305\) −9.65345 −0.552755
\(306\) 0 0
\(307\) −12.7953 −0.730265 −0.365132 0.930956i \(-0.618976\pi\)
−0.365132 + 0.930956i \(0.618976\pi\)
\(308\) 0 0
\(309\) 4.11552 0.234124
\(310\) 0 0
\(311\) 22.8582 1.29617 0.648084 0.761569i \(-0.275570\pi\)
0.648084 + 0.761569i \(0.275570\pi\)
\(312\) 0 0
\(313\) −27.2284 −1.53904 −0.769520 0.638623i \(-0.779504\pi\)
−0.769520 + 0.638623i \(0.779504\pi\)
\(314\) 0 0
\(315\) 11.2047 0.631315
\(316\) 0 0
\(317\) −27.7926 −1.56099 −0.780494 0.625163i \(-0.785033\pi\)
−0.780494 + 0.625163i \(0.785033\pi\)
\(318\) 0 0
\(319\) −8.03926 −0.450112
\(320\) 0 0
\(321\) −5.92375 −0.330631
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.57720 0.308420
\(328\) 0 0
\(329\) −25.3462 −1.39738
\(330\) 0 0
\(331\) −19.7230 −1.08408 −0.542038 0.840354i \(-0.682347\pi\)
−0.542038 + 0.840354i \(0.682347\pi\)
\(332\) 0 0
\(333\) 24.7034 1.35374
\(334\) 0 0
\(335\) 6.97370 0.381014
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) −0.540588 −0.0293607
\(340\) 0 0
\(341\) −9.61419 −0.520638
\(342\) 0 0
\(343\) 4.23103 0.228454
\(344\) 0 0
\(345\) −1.84522 −0.0993434
\(346\) 0 0
\(347\) −2.61017 −0.140121 −0.0700607 0.997543i \(-0.522319\pi\)
−0.0700607 + 0.997543i \(0.522319\pi\)
\(348\) 0 0
\(349\) −3.08921 −0.165362 −0.0826809 0.996576i \(-0.526348\pi\)
−0.0826809 + 0.996576i \(0.526348\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −20.9211 −1.11352 −0.556759 0.830674i \(-0.687955\pi\)
−0.556759 + 0.830674i \(0.687955\pi\)
\(354\) 0 0
\(355\) −12.6731 −0.672617
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.8515 1.62828 0.814140 0.580668i \(-0.197209\pi\)
0.814140 + 0.580668i \(0.197209\pi\)
\(360\) 0 0
\(361\) −10.7926 −0.568032
\(362\) 0 0
\(363\) 2.92776 0.153668
\(364\) 0 0
\(365\) 3.43576 0.179836
\(366\) 0 0
\(367\) 21.0196 1.09722 0.548608 0.836080i \(-0.315158\pi\)
0.548608 + 0.836080i \(0.315158\pi\)
\(368\) 0 0
\(369\) −26.2177 −1.36484
\(370\) 0 0
\(371\) 25.0500 1.30053
\(372\) 0 0
\(373\) 29.3462 1.51949 0.759743 0.650223i \(-0.225325\pi\)
0.759743 + 0.650223i \(0.225325\pi\)
\(374\) 0 0
\(375\) 0.339877 0.0175512
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.3528 −0.583156 −0.291578 0.956547i \(-0.594180\pi\)
−0.291578 + 0.956547i \(0.594180\pi\)
\(380\) 0 0
\(381\) −0.0369937 −0.00189524
\(382\) 0 0
\(383\) 20.5642 1.05078 0.525392 0.850860i \(-0.323919\pi\)
0.525392 + 0.850860i \(0.323919\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 0 0
\(387\) 2.82786 0.143748
\(388\) 0 0
\(389\) −26.8189 −1.35977 −0.679887 0.733317i \(-0.737971\pi\)
−0.679887 + 0.733317i \(0.737971\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 3.53793 0.178465
\(394\) 0 0
\(395\) 13.1285 0.660565
\(396\) 0 0
\(397\) −36.1022 −1.81192 −0.905958 0.423367i \(-0.860848\pi\)
−0.905958 + 0.423367i \(0.860848\pi\)
\(398\) 0 0
\(399\) 3.78231 0.189352
\(400\) 0 0
\(401\) −4.07852 −0.203672 −0.101836 0.994801i \(-0.532472\pi\)
−0.101836 + 0.994801i \(0.532472\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −7.97370 −0.396216
\(406\) 0 0
\(407\) −13.2284 −0.655706
\(408\) 0 0
\(409\) 0.627148 0.0310105 0.0155052 0.999880i \(-0.495064\pi\)
0.0155052 + 0.999880i \(0.495064\pi\)
\(410\) 0 0
\(411\) −3.08921 −0.152380
\(412\) 0 0
\(413\) 17.3069 0.851617
\(414\) 0 0
\(415\) 8.56424 0.420402
\(416\) 0 0
\(417\) −3.07587 −0.150626
\(418\) 0 0
\(419\) −11.5513 −0.564317 −0.282158 0.959368i \(-0.591050\pi\)
−0.282158 + 0.959368i \(0.591050\pi\)
\(420\) 0 0
\(421\) 35.4853 1.72945 0.864725 0.502246i \(-0.167493\pi\)
0.864725 + 0.502246i \(0.167493\pi\)
\(422\) 0 0
\(423\) 18.8212 0.915117
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 37.4987 1.81469
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.7623 −1.62627 −0.813136 0.582073i \(-0.802242\pi\)
−0.813136 + 0.582073i \(0.802242\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −1.76897 −0.0848155
\(436\) 0 0
\(437\) 15.5535 0.744027
\(438\) 0 0
\(439\) 19.0892 0.911078 0.455539 0.890216i \(-0.349446\pi\)
0.455539 + 0.890216i \(0.349446\pi\)
\(440\) 0 0
\(441\) −23.3332 −1.11110
\(442\) 0 0
\(443\) −10.2873 −0.488763 −0.244382 0.969679i \(-0.578585\pi\)
−0.244382 + 0.969679i \(0.578585\pi\)
\(444\) 0 0
\(445\) 17.1285 0.811968
\(446\) 0 0
\(447\) 3.78231 0.178897
\(448\) 0 0
\(449\) 2.21769 0.104659 0.0523296 0.998630i \(-0.483335\pi\)
0.0523296 + 0.998630i \(0.483335\pi\)
\(450\) 0 0
\(451\) 14.0393 0.661083
\(452\) 0 0
\(453\) 3.10483 0.145877
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.3069 0.809583 0.404791 0.914409i \(-0.367344\pi\)
0.404791 + 0.914409i \(0.367344\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.1392 −0.751676 −0.375838 0.926685i \(-0.622645\pi\)
−0.375838 + 0.926685i \(0.622645\pi\)
\(462\) 0 0
\(463\) −6.52498 −0.303241 −0.151621 0.988439i \(-0.548449\pi\)
−0.151621 + 0.988439i \(0.548449\pi\)
\(464\) 0 0
\(465\) −2.11552 −0.0981047
\(466\) 0 0
\(467\) 35.4291 1.63946 0.819731 0.572748i \(-0.194123\pi\)
0.819731 + 0.572748i \(0.194123\pi\)
\(468\) 0 0
\(469\) −27.0892 −1.25086
\(470\) 0 0
\(471\) −1.66906 −0.0769064
\(472\) 0 0
\(473\) −1.51429 −0.0696269
\(474\) 0 0
\(475\) −2.86485 −0.131448
\(476\) 0 0
\(477\) −18.6012 −0.851692
\(478\) 0 0
\(479\) 26.7730 1.22329 0.611644 0.791133i \(-0.290508\pi\)
0.611644 + 0.791133i \(0.290508\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 7.16774 0.326143
\(484\) 0 0
\(485\) 13.7690 0.625217
\(486\) 0 0
\(487\) 18.3725 0.832536 0.416268 0.909242i \(-0.363338\pi\)
0.416268 + 0.909242i \(0.363338\pi\)
\(488\) 0 0
\(489\) −5.75562 −0.260278
\(490\) 0 0
\(491\) 8.81892 0.397992 0.198996 0.980000i \(-0.436232\pi\)
0.198996 + 0.980000i \(0.436232\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 4.45539 0.200255
\(496\) 0 0
\(497\) 49.2284 2.20819
\(498\) 0 0
\(499\) 23.7756 1.06434 0.532172 0.846636i \(-0.321376\pi\)
0.532172 + 0.846636i \(0.321376\pi\)
\(500\) 0 0
\(501\) 7.34616 0.328202
\(502\) 0 0
\(503\) 39.1455 1.74541 0.872705 0.488248i \(-0.162364\pi\)
0.872705 + 0.488248i \(0.162364\pi\)
\(504\) 0 0
\(505\) −6.44872 −0.286964
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 44.5745 1.97573 0.987866 0.155309i \(-0.0496373\pi\)
0.987866 + 0.155309i \(0.0496373\pi\)
\(510\) 0 0
\(511\) −13.3462 −0.590400
\(512\) 0 0
\(513\) −5.72971 −0.252973
\(514\) 0 0
\(515\) 12.1088 0.533579
\(516\) 0 0
\(517\) −10.0785 −0.443253
\(518\) 0 0
\(519\) −2.73236 −0.119937
\(520\) 0 0
\(521\) −17.2047 −0.753753 −0.376876 0.926264i \(-0.623002\pi\)
−0.376876 + 0.926264i \(0.623002\pi\)
\(522\) 0 0
\(523\) 3.19806 0.139841 0.0699207 0.997553i \(-0.477725\pi\)
0.0699207 + 0.997553i \(0.477725\pi\)
\(524\) 0 0
\(525\) −1.32025 −0.0576203
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.47502 0.281523
\(530\) 0 0
\(531\) −12.8515 −0.557708
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −17.4291 −0.753525
\(536\) 0 0
\(537\) 3.69044 0.159254
\(538\) 0 0
\(539\) 12.4947 0.538183
\(540\) 0 0
\(541\) −1.65118 −0.0709899 −0.0354950 0.999370i \(-0.511301\pi\)
−0.0354950 + 0.999370i \(0.511301\pi\)
\(542\) 0 0
\(543\) 3.59054 0.154085
\(544\) 0 0
\(545\) 16.4095 0.702904
\(546\) 0 0
\(547\) 38.3265 1.63872 0.819362 0.573276i \(-0.194328\pi\)
0.819362 + 0.573276i \(0.194328\pi\)
\(548\) 0 0
\(549\) −27.8452 −1.18841
\(550\) 0 0
\(551\) 14.9108 0.635221
\(552\) 0 0
\(553\) −50.9973 −2.16863
\(554\) 0 0
\(555\) −2.91079 −0.123556
\(556\) 0 0
\(557\) 31.8711 1.35042 0.675212 0.737624i \(-0.264052\pi\)
0.675212 + 0.737624i \(0.264052\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.87781 −0.247720 −0.123860 0.992300i \(-0.539527\pi\)
−0.123860 + 0.992300i \(0.539527\pi\)
\(564\) 0 0
\(565\) −1.59054 −0.0669145
\(566\) 0 0
\(567\) 30.9737 1.30077
\(568\) 0 0
\(569\) 16.0629 0.673392 0.336696 0.941613i \(-0.390690\pi\)
0.336696 + 0.941613i \(0.390690\pi\)
\(570\) 0 0
\(571\) −5.04995 −0.211334 −0.105667 0.994402i \(-0.533698\pi\)
−0.105667 + 0.994402i \(0.533698\pi\)
\(572\) 0 0
\(573\) 2.99735 0.125216
\(574\) 0 0
\(575\) −5.42909 −0.226409
\(576\) 0 0
\(577\) 17.3832 0.723670 0.361835 0.932242i \(-0.382150\pi\)
0.361835 + 0.932242i \(0.382150\pi\)
\(578\) 0 0
\(579\) 0.0918667 0.00381785
\(580\) 0 0
\(581\) −33.2676 −1.38017
\(582\) 0 0
\(583\) 9.96074 0.412532
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.57493 0.312651 0.156325 0.987706i \(-0.450035\pi\)
0.156325 + 0.987706i \(0.450035\pi\)
\(588\) 0 0
\(589\) 17.8319 0.734750
\(590\) 0 0
\(591\) −1.34616 −0.0553738
\(592\) 0 0
\(593\) 16.9500 0.696055 0.348028 0.937484i \(-0.386852\pi\)
0.348028 + 0.937484i \(0.386852\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.05261 0.0840075
\(598\) 0 0
\(599\) −43.6771 −1.78460 −0.892299 0.451445i \(-0.850909\pi\)
−0.892299 + 0.451445i \(0.850909\pi\)
\(600\) 0 0
\(601\) 2.07852 0.0847847 0.0423924 0.999101i \(-0.486502\pi\)
0.0423924 + 0.999101i \(0.486502\pi\)
\(602\) 0 0
\(603\) 20.1155 0.819167
\(604\) 0 0
\(605\) 8.61419 0.350217
\(606\) 0 0
\(607\) −34.3265 −1.39327 −0.696635 0.717425i \(-0.745320\pi\)
−0.696635 + 0.717425i \(0.745320\pi\)
\(608\) 0 0
\(609\) 6.87153 0.278448
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 18.3569 0.741426 0.370713 0.928747i \(-0.379113\pi\)
0.370713 + 0.928747i \(0.379113\pi\)
\(614\) 0 0
\(615\) 3.08921 0.124569
\(616\) 0 0
\(617\) 37.5246 1.51068 0.755342 0.655331i \(-0.227471\pi\)
0.755342 + 0.655331i \(0.227471\pi\)
\(618\) 0 0
\(619\) −12.8256 −0.515504 −0.257752 0.966211i \(-0.582982\pi\)
−0.257752 + 0.966211i \(0.582982\pi\)
\(620\) 0 0
\(621\) −10.8582 −0.435724
\(622\) 0 0
\(623\) −66.5353 −2.66568
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.50398 0.0600632
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −12.6472 −0.503476 −0.251738 0.967795i \(-0.581002\pi\)
−0.251738 + 0.967795i \(0.581002\pi\)
\(632\) 0 0
\(633\) 5.51202 0.219083
\(634\) 0 0
\(635\) −0.108844 −0.00431935
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −36.5553 −1.44611
\(640\) 0 0
\(641\) −5.10256 −0.201539 −0.100769 0.994910i \(-0.532130\pi\)
−0.100769 + 0.994910i \(0.532130\pi\)
\(642\) 0 0
\(643\) −0.346549 −0.0136666 −0.00683328 0.999977i \(-0.502175\pi\)
−0.00683328 + 0.999977i \(0.502175\pi\)
\(644\) 0 0
\(645\) −0.333205 −0.0131199
\(646\) 0 0
\(647\) −44.2480 −1.73957 −0.869784 0.493432i \(-0.835742\pi\)
−0.869784 + 0.493432i \(0.835742\pi\)
\(648\) 0 0
\(649\) 6.88183 0.270135
\(650\) 0 0
\(651\) 8.21769 0.322077
\(652\) 0 0
\(653\) −14.0393 −0.549399 −0.274699 0.961530i \(-0.588578\pi\)
−0.274699 + 0.961530i \(0.588578\pi\)
\(654\) 0 0
\(655\) 10.4095 0.406731
\(656\) 0 0
\(657\) 9.91040 0.386642
\(658\) 0 0
\(659\) 12.4487 0.484933 0.242467 0.970160i \(-0.422043\pi\)
0.242467 + 0.970160i \(0.422043\pi\)
\(660\) 0 0
\(661\) 11.8216 0.459806 0.229903 0.973214i \(-0.426159\pi\)
0.229903 + 0.973214i \(0.426159\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.1285 0.431544
\(666\) 0 0
\(667\) 28.2569 1.09411
\(668\) 0 0
\(669\) −3.42015 −0.132231
\(670\) 0 0
\(671\) 14.9108 0.575625
\(672\) 0 0
\(673\) −31.3069 −1.20679 −0.603396 0.797442i \(-0.706186\pi\)
−0.603396 + 0.797442i \(0.706186\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 28.8582 1.10911 0.554555 0.832147i \(-0.312889\pi\)
0.554555 + 0.832147i \(0.312889\pi\)
\(678\) 0 0
\(679\) −53.4853 −2.05258
\(680\) 0 0
\(681\) −1.16774 −0.0447478
\(682\) 0 0
\(683\) 20.5642 0.786869 0.393434 0.919353i \(-0.371287\pi\)
0.393434 + 0.919353i \(0.371287\pi\)
\(684\) 0 0
\(685\) −9.08921 −0.347281
\(686\) 0 0
\(687\) 7.76897 0.296405
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.67308 −0.253856 −0.126928 0.991912i \(-0.540512\pi\)
−0.126928 + 0.991912i \(0.540512\pi\)
\(692\) 0 0
\(693\) −17.3069 −0.657435
\(694\) 0 0
\(695\) −9.04995 −0.343284
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.49867 −0.0566850
\(700\) 0 0
\(701\) −9.62980 −0.363713 −0.181856 0.983325i \(-0.558211\pi\)
−0.181856 + 0.983325i \(0.558211\pi\)
\(702\) 0 0
\(703\) 24.5353 0.925366
\(704\) 0 0
\(705\) −2.21769 −0.0835230
\(706\) 0 0
\(707\) 25.0500 0.942100
\(708\) 0 0
\(709\) −33.8082 −1.26969 −0.634847 0.772638i \(-0.718937\pi\)
−0.634847 + 0.772638i \(0.718937\pi\)
\(710\) 0 0
\(711\) 37.8689 1.42019
\(712\) 0 0
\(713\) 33.7926 1.26554
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.75336 −0.289554
\(718\) 0 0
\(719\) 36.2043 1.35019 0.675097 0.737729i \(-0.264102\pi\)
0.675097 + 0.737729i \(0.264102\pi\)
\(720\) 0 0
\(721\) −47.0366 −1.75173
\(722\) 0 0
\(723\) 3.62980 0.134994
\(724\) 0 0
\(725\) −5.20473 −0.193299
\(726\) 0 0
\(727\) 46.2480 1.71524 0.857622 0.514281i \(-0.171941\pi\)
0.857622 + 0.514281i \(0.171941\pi\)
\(728\) 0 0
\(729\) −20.9607 −0.776324
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 20.0393 0.740167 0.370084 0.928998i \(-0.379329\pi\)
0.370084 + 0.928998i \(0.379329\pi\)
\(734\) 0 0
\(735\) 2.74934 0.101411
\(736\) 0 0
\(737\) −10.7716 −0.396778
\(738\) 0 0
\(739\) 52.6338 1.93617 0.968083 0.250629i \(-0.0806374\pi\)
0.968083 + 0.250629i \(0.0806374\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −47.9497 −1.75910 −0.879551 0.475804i \(-0.842157\pi\)
−0.879551 + 0.475804i \(0.842157\pi\)
\(744\) 0 0
\(745\) 11.1285 0.407716
\(746\) 0 0
\(747\) 24.7034 0.903850
\(748\) 0 0
\(749\) 67.7030 2.47381
\(750\) 0 0
\(751\) −23.0892 −0.842537 −0.421269 0.906936i \(-0.638415\pi\)
−0.421269 + 0.906936i \(0.638415\pi\)
\(752\) 0 0
\(753\) −7.22838 −0.263417
\(754\) 0 0
\(755\) 9.13515 0.332462
\(756\) 0 0
\(757\) 44.3176 1.61075 0.805375 0.592765i \(-0.201964\pi\)
0.805375 + 0.592765i \(0.201964\pi\)
\(758\) 0 0
\(759\) 2.85014 0.103454
\(760\) 0 0
\(761\) −41.4853 −1.50384 −0.751921 0.659253i \(-0.770873\pi\)
−0.751921 + 0.659253i \(0.770873\pi\)
\(762\) 0 0
\(763\) −63.7423 −2.30763
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −28.5879 −1.03091 −0.515453 0.856918i \(-0.672376\pi\)
−0.515453 + 0.856918i \(0.672376\pi\)
\(770\) 0 0
\(771\) 6.42280 0.231312
\(772\) 0 0
\(773\) 15.4358 0.555186 0.277593 0.960699i \(-0.410463\pi\)
0.277593 + 0.960699i \(0.410463\pi\)
\(774\) 0 0
\(775\) −6.22436 −0.223586
\(776\) 0 0
\(777\) 11.3069 0.405633
\(778\) 0 0
\(779\) −26.0393 −0.932953
\(780\) 0 0
\(781\) 19.5749 0.700446
\(782\) 0 0
\(783\) −10.4095 −0.372004
\(784\) 0 0
\(785\) −4.91079 −0.175273
\(786\) 0 0
\(787\) 5.38316 0.191889 0.0959444 0.995387i \(-0.469413\pi\)
0.0959444 + 0.995387i \(0.469413\pi\)
\(788\) 0 0
\(789\) 4.42507 0.157537
\(790\) 0 0
\(791\) 6.17843 0.219680
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.19177 0.0777341
\(796\) 0 0
\(797\) −35.7556 −1.26653 −0.633265 0.773935i \(-0.718286\pi\)
−0.633265 + 0.773935i \(0.718286\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 49.4068 1.74570
\(802\) 0 0
\(803\) −5.30690 −0.187277
\(804\) 0 0
\(805\) 21.0892 0.743297
\(806\) 0 0
\(807\) −4.88410 −0.171928
\(808\) 0 0
\(809\) −35.7712 −1.25765 −0.628825 0.777547i \(-0.716464\pi\)
−0.628825 + 0.777547i \(0.716464\pi\)
\(810\) 0 0
\(811\) 30.2244 1.06132 0.530660 0.847585i \(-0.321944\pi\)
0.530660 + 0.847585i \(0.321944\pi\)
\(812\) 0 0
\(813\) 4.66414 0.163579
\(814\) 0 0
\(815\) −16.9344 −0.593187
\(816\) 0 0
\(817\) 2.80861 0.0982610
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.2284 −0.880477 −0.440238 0.897881i \(-0.645106\pi\)
−0.440238 + 0.897881i \(0.645106\pi\)
\(822\) 0 0
\(823\) −30.9804 −1.07991 −0.539954 0.841695i \(-0.681558\pi\)
−0.539954 + 0.841695i \(0.681558\pi\)
\(824\) 0 0
\(825\) −0.524976 −0.0182773
\(826\) 0 0
\(827\) −6.82119 −0.237196 −0.118598 0.992942i \(-0.537840\pi\)
−0.118598 + 0.992942i \(0.537840\pi\)
\(828\) 0 0
\(829\) 3.39650 0.117965 0.0589827 0.998259i \(-0.481214\pi\)
0.0589827 + 0.998259i \(0.481214\pi\)
\(830\) 0 0
\(831\) −6.54863 −0.227169
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 21.6142 0.747990
\(836\) 0 0
\(837\) −12.4487 −0.430291
\(838\) 0 0
\(839\) −9.76230 −0.337032 −0.168516 0.985699i \(-0.553897\pi\)
−0.168516 + 0.985699i \(0.553897\pi\)
\(840\) 0 0
\(841\) −1.91079 −0.0658892
\(842\) 0 0
\(843\) −1.34616 −0.0463643
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −33.4617 −1.14976
\(848\) 0 0
\(849\) −7.20473 −0.247266
\(850\) 0 0
\(851\) 46.4960 1.59386
\(852\) 0 0
\(853\) 18.2806 0.625916 0.312958 0.949767i \(-0.398680\pi\)
0.312958 + 0.949767i \(0.398680\pi\)
\(854\) 0 0
\(855\) −8.26362 −0.282610
\(856\) 0 0
\(857\) 11.6691 0.398608 0.199304 0.979938i \(-0.436132\pi\)
0.199304 + 0.979938i \(0.436132\pi\)
\(858\) 0 0
\(859\) −3.92148 −0.133799 −0.0668995 0.997760i \(-0.521311\pi\)
−0.0668995 + 0.997760i \(0.521311\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −5.83188 −0.198519 −0.0992597 0.995062i \(-0.531647\pi\)
−0.0992597 + 0.995062i \(0.531647\pi\)
\(864\) 0 0
\(865\) −8.03926 −0.273343
\(866\) 0 0
\(867\) 5.77791 0.196228
\(868\) 0 0
\(869\) −20.2783 −0.687895
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 39.7164 1.34420
\(874\) 0 0
\(875\) −3.88448 −0.131320
\(876\) 0 0
\(877\) −57.5379 −1.94292 −0.971459 0.237208i \(-0.923768\pi\)
−0.971459 + 0.237208i \(0.923768\pi\)
\(878\) 0 0
\(879\) 4.65384 0.156970
\(880\) 0 0
\(881\) 55.4483 1.86810 0.934051 0.357140i \(-0.116248\pi\)
0.934051 + 0.357140i \(0.116248\pi\)
\(882\) 0 0
\(883\) −1.75199 −0.0589591 −0.0294796 0.999565i \(-0.509385\pi\)
−0.0294796 + 0.999565i \(0.509385\pi\)
\(884\) 0 0
\(885\) 1.51429 0.0509021
\(886\) 0 0
\(887\) 36.3265 1.21973 0.609863 0.792507i \(-0.291225\pi\)
0.609863 + 0.792507i \(0.291225\pi\)
\(888\) 0 0
\(889\) 0.422804 0.0141804
\(890\) 0 0
\(891\) 12.3162 0.412609
\(892\) 0 0
\(893\) 18.6931 0.625541
\(894\) 0 0
\(895\) 10.8582 0.362949
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.3961 1.08047
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.29433 0.0430726
\(904\) 0 0
\(905\) 10.5642 0.351167
\(906\) 0 0
\(907\) 10.0696 0.334355 0.167178 0.985927i \(-0.446535\pi\)
0.167178 + 0.985927i \(0.446535\pi\)
\(908\) 0 0
\(909\) −18.6012 −0.616964
\(910\) 0 0
\(911\) −39.8341 −1.31976 −0.659882 0.751369i \(-0.729394\pi\)
−0.659882 + 0.751369i \(0.729394\pi\)
\(912\) 0 0
\(913\) −13.2284 −0.437795
\(914\) 0 0
\(915\) 3.28098 0.108466
\(916\) 0 0
\(917\) −40.4354 −1.33529
\(918\) 0 0
\(919\) 11.8608 0.391253 0.195626 0.980678i \(-0.437326\pi\)
0.195626 + 0.980678i \(0.437326\pi\)
\(920\) 0 0
\(921\) 4.34882 0.143298
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.56424 −0.281590
\(926\) 0 0
\(927\) 34.9278 1.14718
\(928\) 0 0
\(929\) −7.34616 −0.241020 −0.120510 0.992712i \(-0.538453\pi\)
−0.120510 + 0.992712i \(0.538453\pi\)
\(930\) 0 0
\(931\) −23.1744 −0.759511
\(932\) 0 0
\(933\) −7.76897 −0.254345
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 45.3069 1.48011 0.740056 0.672545i \(-0.234799\pi\)
0.740056 + 0.672545i \(0.234799\pi\)
\(938\) 0 0
\(939\) 9.25430 0.302003
\(940\) 0 0
\(941\) −5.88222 −0.191755 −0.0958774 0.995393i \(-0.530566\pi\)
−0.0958774 + 0.995393i \(0.530566\pi\)
\(942\) 0 0
\(943\) −49.3462 −1.60693
\(944\) 0 0
\(945\) −7.76897 −0.252725
\(946\) 0 0
\(947\) −14.7427 −0.479072 −0.239536 0.970887i \(-0.576995\pi\)
−0.239536 + 0.970887i \(0.576995\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 9.44607 0.306310
\(952\) 0 0
\(953\) 37.2284 1.20595 0.602973 0.797762i \(-0.293983\pi\)
0.602973 + 0.797762i \(0.293983\pi\)
\(954\) 0 0
\(955\) 8.81892 0.285374
\(956\) 0 0
\(957\) 2.73236 0.0883246
\(958\) 0 0
\(959\) 35.3069 1.14012
\(960\) 0 0
\(961\) 7.74266 0.249763
\(962\) 0 0
\(963\) −50.2739 −1.62005
\(964\) 0 0
\(965\) 0.270294 0.00870107
\(966\) 0 0
\(967\) 31.8711 1.02491 0.512453 0.858715i \(-0.328737\pi\)
0.512453 + 0.858715i \(0.328737\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.6531 1.17625 0.588126 0.808769i \(-0.299866\pi\)
0.588126 + 0.808769i \(0.299866\pi\)
\(972\) 0 0
\(973\) 35.1544 1.12700
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.61419 −0.307585 −0.153793 0.988103i \(-0.549149\pi\)
−0.153793 + 0.988103i \(0.549149\pi\)
\(978\) 0 0
\(979\) −26.4568 −0.845562
\(980\) 0 0
\(981\) 47.3328 1.51122
\(982\) 0 0
\(983\) −4.78193 −0.152520 −0.0762599 0.997088i \(-0.524298\pi\)
−0.0762599 + 0.997088i \(0.524298\pi\)
\(984\) 0 0
\(985\) −3.96074 −0.126200
\(986\) 0 0
\(987\) 8.61458 0.274205
\(988\) 0 0
\(989\) 5.32251 0.169246
\(990\) 0 0
\(991\) 51.8814 1.64807 0.824034 0.566540i \(-0.191718\pi\)
0.824034 + 0.566540i \(0.191718\pi\)
\(992\) 0 0
\(993\) 6.70340 0.212726
\(994\) 0 0
\(995\) 6.03926 0.191457
\(996\) 0 0
\(997\) 12.3176 0.390102 0.195051 0.980793i \(-0.437513\pi\)
0.195051 + 0.980793i \(0.437513\pi\)
\(998\) 0 0
\(999\) −17.1285 −0.541921
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 3380.2.a.m.1.2 3
13.5 odd 4 260.2.f.a.181.4 yes 6
13.8 odd 4 260.2.f.a.181.3 6
13.12 even 2 3380.2.a.n.1.2 3
39.5 even 4 2340.2.c.d.181.3 6
39.8 even 4 2340.2.c.d.181.4 6
52.31 even 4 1040.2.k.c.961.4 6
52.47 even 4 1040.2.k.c.961.3 6
65.8 even 4 1300.2.d.d.649.3 6
65.18 even 4 1300.2.d.c.649.3 6
65.34 odd 4 1300.2.f.e.701.4 6
65.44 odd 4 1300.2.f.e.701.3 6
65.47 even 4 1300.2.d.c.649.4 6
65.57 even 4 1300.2.d.d.649.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.3 6 13.8 odd 4
260.2.f.a.181.4 yes 6 13.5 odd 4
1040.2.k.c.961.3 6 52.47 even 4
1040.2.k.c.961.4 6 52.31 even 4
1300.2.d.c.649.3 6 65.18 even 4
1300.2.d.c.649.4 6 65.47 even 4
1300.2.d.d.649.3 6 65.8 even 4
1300.2.d.d.649.4 6 65.57 even 4
1300.2.f.e.701.3 6 65.44 odd 4
1300.2.f.e.701.4 6 65.34 odd 4
2340.2.c.d.181.3 6 39.5 even 4
2340.2.c.d.181.4 6 39.8 even 4
3380.2.a.m.1.2 3 1.1 even 1 trivial
3380.2.a.n.1.2 3 13.12 even 2