Properties

Label 384.5.b.c.319.1
Level $384$
Weight $5$
Character 384.319
Analytic conductor $39.694$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.1
Root \(1.09445 + 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.5.b.c.319.4

$q$-expansion

\(f(q)\) \(=\) \(q-5.19615 q^{3} -39.7490i q^{5} -46.0431i q^{7} +27.0000 q^{9} +O(q^{10})\) \(q-5.19615 q^{3} -39.7490i q^{5} -46.0431i q^{7} +27.0000 q^{9} +181.283 q^{11} +183.498i q^{13} +206.542i q^{15} +427.992 q^{17} +668.558 q^{19} +239.247i q^{21} -882.463i q^{23} -954.984 q^{25} -140.296 q^{27} +807.247i q^{29} +391.276i q^{31} -941.976 q^{33} -1830.17 q^{35} -466.980i q^{37} -953.484i q^{39} +2159.93 q^{41} -509.182 q^{43} -1073.22i q^{45} -2056.83i q^{47} +281.031 q^{49} -2223.91 q^{51} +753.725i q^{53} -7205.84i q^{55} -3473.93 q^{57} +1309.43 q^{59} +801.913i q^{61} -1243.16i q^{63} +7293.87 q^{65} -505.813 q^{67} +4585.41i q^{69} +2170.94i q^{71} -2297.97 q^{73} +4962.24 q^{75} -8346.85i q^{77} -10705.3i q^{79} +729.000 q^{81} +2977.81 q^{83} -17012.3i q^{85} -4194.58i q^{87} -6493.92 q^{89} +8448.82 q^{91} -2033.13i q^{93} -26574.5i q^{95} -8189.92 q^{97} +4894.65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 216 q^{9} + O(q^{10}) \) \( 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −5.19615 −0.577350
\(4\) 0 0
\(5\) − 39.7490i − 1.58996i −0.606635 0.794980i \(-0.707481\pi\)
0.606635 0.794980i \(-0.292519\pi\)
\(6\) 0 0
\(7\) − 46.0431i − 0.939655i −0.882758 0.469828i \(-0.844316\pi\)
0.882758 0.469828i \(-0.155684\pi\)
\(8\) 0 0
\(9\) 27.0000 0.333333
\(10\) 0 0
\(11\) 181.283 1.49821 0.749105 0.662451i \(-0.230484\pi\)
0.749105 + 0.662451i \(0.230484\pi\)
\(12\) 0 0
\(13\) 183.498i 1.08579i 0.839801 + 0.542894i \(0.182671\pi\)
−0.839801 + 0.542894i \(0.817329\pi\)
\(14\) 0 0
\(15\) 206.542i 0.917964i
\(16\) 0 0
\(17\) 427.992 1.48094 0.740471 0.672089i \(-0.234603\pi\)
0.740471 + 0.672089i \(0.234603\pi\)
\(18\) 0 0
\(19\) 668.558 1.85196 0.925981 0.377571i \(-0.123241\pi\)
0.925981 + 0.377571i \(0.123241\pi\)
\(20\) 0 0
\(21\) 239.247i 0.542510i
\(22\) 0 0
\(23\) − 882.463i − 1.66817i −0.551635 0.834086i \(-0.685996\pi\)
0.551635 0.834086i \(-0.314004\pi\)
\(24\) 0 0
\(25\) −954.984 −1.52797
\(26\) 0 0
\(27\) −140.296 −0.192450
\(28\) 0 0
\(29\) 807.247i 0.959866i 0.877305 + 0.479933i \(0.159339\pi\)
−0.877305 + 0.479933i \(0.840661\pi\)
\(30\) 0 0
\(31\) 391.276i 0.407155i 0.979059 + 0.203577i \(0.0652569\pi\)
−0.979059 + 0.203577i \(0.934743\pi\)
\(32\) 0 0
\(33\) −941.976 −0.864992
\(34\) 0 0
\(35\) −1830.17 −1.49402
\(36\) 0 0
\(37\) − 466.980i − 0.341111i −0.985348 0.170555i \(-0.945444\pi\)
0.985348 0.170555i \(-0.0545561\pi\)
\(38\) 0 0
\(39\) − 953.484i − 0.626880i
\(40\) 0 0
\(41\) 2159.93 1.28491 0.642454 0.766325i \(-0.277917\pi\)
0.642454 + 0.766325i \(0.277917\pi\)
\(42\) 0 0
\(43\) −509.182 −0.275382 −0.137691 0.990475i \(-0.543968\pi\)
−0.137691 + 0.990475i \(0.543968\pi\)
\(44\) 0 0
\(45\) − 1073.22i − 0.529987i
\(46\) 0 0
\(47\) − 2056.83i − 0.931116i −0.885017 0.465558i \(-0.845854\pi\)
0.885017 0.465558i \(-0.154146\pi\)
\(48\) 0 0
\(49\) 281.031 0.117048
\(50\) 0 0
\(51\) −2223.91 −0.855022
\(52\) 0 0
\(53\) 753.725i 0.268325i 0.990959 + 0.134163i \(0.0428344\pi\)
−0.990959 + 0.134163i \(0.957166\pi\)
\(54\) 0 0
\(55\) − 7205.84i − 2.38210i
\(56\) 0 0
\(57\) −3473.93 −1.06923
\(58\) 0 0
\(59\) 1309.43 0.376165 0.188083 0.982153i \(-0.439773\pi\)
0.188083 + 0.982153i \(0.439773\pi\)
\(60\) 0 0
\(61\) 801.913i 0.215510i 0.994177 + 0.107755i \(0.0343662\pi\)
−0.994177 + 0.107755i \(0.965634\pi\)
\(62\) 0 0
\(63\) − 1243.16i − 0.313218i
\(64\) 0 0
\(65\) 7293.87 1.72636
\(66\) 0 0
\(67\) −505.813 −0.112678 −0.0563392 0.998412i \(-0.517943\pi\)
−0.0563392 + 0.998412i \(0.517943\pi\)
\(68\) 0 0
\(69\) 4585.41i 0.963119i
\(70\) 0 0
\(71\) 2170.94i 0.430658i 0.976542 + 0.215329i \(0.0690823\pi\)
−0.976542 + 0.215329i \(0.930918\pi\)
\(72\) 0 0
\(73\) −2297.97 −0.431219 −0.215610 0.976480i \(-0.569174\pi\)
−0.215610 + 0.976480i \(0.569174\pi\)
\(74\) 0 0
\(75\) 4962.24 0.882177
\(76\) 0 0
\(77\) − 8346.85i − 1.40780i
\(78\) 0 0
\(79\) − 10705.3i − 1.71532i −0.514219 0.857659i \(-0.671918\pi\)
0.514219 0.857659i \(-0.328082\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) 2977.81 0.432256 0.216128 0.976365i \(-0.430657\pi\)
0.216128 + 0.976365i \(0.430657\pi\)
\(84\) 0 0
\(85\) − 17012.3i − 2.35464i
\(86\) 0 0
\(87\) − 4194.58i − 0.554179i
\(88\) 0 0
\(89\) −6493.92 −0.819836 −0.409918 0.912122i \(-0.634443\pi\)
−0.409918 + 0.912122i \(0.634443\pi\)
\(90\) 0 0
\(91\) 8448.82 1.02027
\(92\) 0 0
\(93\) − 2033.13i − 0.235071i
\(94\) 0 0
\(95\) − 26574.5i − 2.94455i
\(96\) 0 0
\(97\) −8189.92 −0.870435 −0.435217 0.900325i \(-0.643328\pi\)
−0.435217 + 0.900325i \(0.643328\pi\)
\(98\) 0 0
\(99\) 4894.65 0.499403
\(100\) 0 0
\(101\) 15576.5i 1.52696i 0.645833 + 0.763479i \(0.276510\pi\)
−0.645833 + 0.763479i \(0.723490\pi\)
\(102\) 0 0
\(103\) 6628.58i 0.624807i 0.949950 + 0.312403i \(0.101134\pi\)
−0.949950 + 0.312403i \(0.898866\pi\)
\(104\) 0 0
\(105\) 9509.83 0.862570
\(106\) 0 0
\(107\) 11796.4 1.03035 0.515173 0.857086i \(-0.327728\pi\)
0.515173 + 0.857086i \(0.327728\pi\)
\(108\) 0 0
\(109\) 13286.1i 1.11826i 0.829080 + 0.559130i \(0.188865\pi\)
−0.829080 + 0.559130i \(0.811135\pi\)
\(110\) 0 0
\(111\) 2426.50i 0.196940i
\(112\) 0 0
\(113\) −11713.6 −0.917350 −0.458675 0.888604i \(-0.651676\pi\)
−0.458675 + 0.888604i \(0.651676\pi\)
\(114\) 0 0
\(115\) −35077.0 −2.65233
\(116\) 0 0
\(117\) 4954.45i 0.361929i
\(118\) 0 0
\(119\) − 19706.1i − 1.39157i
\(120\) 0 0
\(121\) 18222.7 1.24463
\(122\) 0 0
\(123\) −11223.3 −0.741842
\(124\) 0 0
\(125\) 13116.5i 0.839459i
\(126\) 0 0
\(127\) − 5640.55i − 0.349715i −0.984594 0.174858i \(-0.944054\pi\)
0.984594 0.174858i \(-0.0559465\pi\)
\(128\) 0 0
\(129\) 2645.79 0.158992
\(130\) 0 0
\(131\) −31922.5 −1.86018 −0.930089 0.367335i \(-0.880270\pi\)
−0.930089 + 0.367335i \(0.880270\pi\)
\(132\) 0 0
\(133\) − 30782.5i − 1.74021i
\(134\) 0 0
\(135\) 5576.63i 0.305988i
\(136\) 0 0
\(137\) −10515.6 −0.560263 −0.280132 0.959962i \(-0.590378\pi\)
−0.280132 + 0.959962i \(0.590378\pi\)
\(138\) 0 0
\(139\) 14985.9 0.775626 0.387813 0.921738i \(-0.373231\pi\)
0.387813 + 0.921738i \(0.373231\pi\)
\(140\) 0 0
\(141\) 10687.6i 0.537580i
\(142\) 0 0
\(143\) 33265.2i 1.62674i
\(144\) 0 0
\(145\) 32087.3 1.52615
\(146\) 0 0
\(147\) −1460.28 −0.0675775
\(148\) 0 0
\(149\) 11795.4i 0.531298i 0.964070 + 0.265649i \(0.0855863\pi\)
−0.964070 + 0.265649i \(0.914414\pi\)
\(150\) 0 0
\(151\) − 33792.9i − 1.48208i −0.671460 0.741040i \(-0.734333\pi\)
0.671460 0.741040i \(-0.265667\pi\)
\(152\) 0 0
\(153\) 11555.8 0.493647
\(154\) 0 0
\(155\) 15552.8 0.647360
\(156\) 0 0
\(157\) − 28788.9i − 1.16796i −0.811770 0.583978i \(-0.801496\pi\)
0.811770 0.583978i \(-0.198504\pi\)
\(158\) 0 0
\(159\) − 3916.47i − 0.154918i
\(160\) 0 0
\(161\) −40631.3 −1.56751
\(162\) 0 0
\(163\) 13409.0 0.504684 0.252342 0.967638i \(-0.418799\pi\)
0.252342 + 0.967638i \(0.418799\pi\)
\(164\) 0 0
\(165\) 37442.6i 1.37530i
\(166\) 0 0
\(167\) 13206.7i 0.473546i 0.971565 + 0.236773i \(0.0760898\pi\)
−0.971565 + 0.236773i \(0.923910\pi\)
\(168\) 0 0
\(169\) −5110.53 −0.178934
\(170\) 0 0
\(171\) 18051.1 0.617320
\(172\) 0 0
\(173\) − 364.956i − 0.0121940i −0.999981 0.00609702i \(-0.998059\pi\)
0.999981 0.00609702i \(-0.00194076\pi\)
\(174\) 0 0
\(175\) 43970.5i 1.43577i
\(176\) 0 0
\(177\) −6804.00 −0.217179
\(178\) 0 0
\(179\) −14372.0 −0.448549 −0.224275 0.974526i \(-0.572001\pi\)
−0.224275 + 0.974526i \(0.572001\pi\)
\(180\) 0 0
\(181\) 4050.54i 0.123639i 0.998087 + 0.0618195i \(0.0196903\pi\)
−0.998087 + 0.0618195i \(0.980310\pi\)
\(182\) 0 0
\(183\) − 4166.86i − 0.124425i
\(184\) 0 0
\(185\) −18562.0 −0.542352
\(186\) 0 0
\(187\) 77587.9 2.21876
\(188\) 0 0
\(189\) 6459.67i 0.180837i
\(190\) 0 0
\(191\) 49880.9i 1.36731i 0.729805 + 0.683656i \(0.239611\pi\)
−0.729805 + 0.683656i \(0.760389\pi\)
\(192\) 0 0
\(193\) −48425.7 −1.30005 −0.650026 0.759912i \(-0.725242\pi\)
−0.650026 + 0.759912i \(0.725242\pi\)
\(194\) 0 0
\(195\) −37900.0 −0.996714
\(196\) 0 0
\(197\) 5556.74i 0.143182i 0.997434 + 0.0715908i \(0.0228076\pi\)
−0.997434 + 0.0715908i \(0.977192\pi\)
\(198\) 0 0
\(199\) 60594.7i 1.53013i 0.643952 + 0.765066i \(0.277294\pi\)
−0.643952 + 0.765066i \(0.722706\pi\)
\(200\) 0 0
\(201\) 2628.28 0.0650549
\(202\) 0 0
\(203\) 37168.2 0.901943
\(204\) 0 0
\(205\) − 85855.1i − 2.04295i
\(206\) 0 0
\(207\) − 23826.5i − 0.556057i
\(208\) 0 0
\(209\) 121198. 2.77463
\(210\) 0 0
\(211\) −27539.9 −0.618583 −0.309292 0.950967i \(-0.600092\pi\)
−0.309292 + 0.950967i \(0.600092\pi\)
\(212\) 0 0
\(213\) − 11280.6i − 0.248640i
\(214\) 0 0
\(215\) 20239.5i 0.437847i
\(216\) 0 0
\(217\) 18015.6 0.382585
\(218\) 0 0
\(219\) 11940.6 0.248965
\(220\) 0 0
\(221\) 78535.7i 1.60799i
\(222\) 0 0
\(223\) 3021.35i 0.0607564i 0.999538 + 0.0303782i \(0.00967116\pi\)
−0.999538 + 0.0303782i \(0.990329\pi\)
\(224\) 0 0
\(225\) −25784.6 −0.509325
\(226\) 0 0
\(227\) 7077.28 0.137346 0.0686728 0.997639i \(-0.478124\pi\)
0.0686728 + 0.997639i \(0.478124\pi\)
\(228\) 0 0
\(229\) − 102761.i − 1.95956i −0.200088 0.979778i \(-0.564123\pi\)
0.200088 0.979778i \(-0.435877\pi\)
\(230\) 0 0
\(231\) 43371.5i 0.812795i
\(232\) 0 0
\(233\) 22976.6 0.423227 0.211613 0.977353i \(-0.432128\pi\)
0.211613 + 0.977353i \(0.432128\pi\)
\(234\) 0 0
\(235\) −81757.1 −1.48044
\(236\) 0 0
\(237\) 55626.4i 0.990339i
\(238\) 0 0
\(239\) 35566.7i 0.622656i 0.950303 + 0.311328i \(0.100774\pi\)
−0.950303 + 0.311328i \(0.899226\pi\)
\(240\) 0 0
\(241\) −92683.4 −1.59576 −0.797881 0.602816i \(-0.794045\pi\)
−0.797881 + 0.602816i \(0.794045\pi\)
\(242\) 0 0
\(243\) −3788.00 −0.0641500
\(244\) 0 0
\(245\) − 11170.7i − 0.186101i
\(246\) 0 0
\(247\) 122679.i 2.01084i
\(248\) 0 0
\(249\) −15473.2 −0.249563
\(250\) 0 0
\(251\) −27590.5 −0.437937 −0.218968 0.975732i \(-0.570269\pi\)
−0.218968 + 0.975732i \(0.570269\pi\)
\(252\) 0 0
\(253\) − 159976.i − 2.49927i
\(254\) 0 0
\(255\) 88398.3i 1.35945i
\(256\) 0 0
\(257\) 92304.6 1.39752 0.698758 0.715358i \(-0.253736\pi\)
0.698758 + 0.715358i \(0.253736\pi\)
\(258\) 0 0
\(259\) −21501.2 −0.320526
\(260\) 0 0
\(261\) 21795.7i 0.319955i
\(262\) 0 0
\(263\) 13884.4i 0.200732i 0.994951 + 0.100366i \(0.0320014\pi\)
−0.994951 + 0.100366i \(0.967999\pi\)
\(264\) 0 0
\(265\) 29959.8 0.426626
\(266\) 0 0
\(267\) 33743.4 0.473333
\(268\) 0 0
\(269\) 52698.1i 0.728266i 0.931347 + 0.364133i \(0.118635\pi\)
−0.931347 + 0.364133i \(0.881365\pi\)
\(270\) 0 0
\(271\) − 47028.8i − 0.640361i −0.947356 0.320181i \(-0.896256\pi\)
0.947356 0.320181i \(-0.103744\pi\)
\(272\) 0 0
\(273\) −43901.4 −0.589051
\(274\) 0 0
\(275\) −173123. −2.28923
\(276\) 0 0
\(277\) 108393.i 1.41268i 0.707873 + 0.706340i \(0.249655\pi\)
−0.707873 + 0.706340i \(0.750345\pi\)
\(278\) 0 0
\(279\) 10564.4i 0.135718i
\(280\) 0 0
\(281\) −73090.7 −0.925656 −0.462828 0.886448i \(-0.653165\pi\)
−0.462828 + 0.886448i \(0.653165\pi\)
\(282\) 0 0
\(283\) −44763.1 −0.558917 −0.279458 0.960158i \(-0.590155\pi\)
−0.279458 + 0.960158i \(0.590155\pi\)
\(284\) 0 0
\(285\) 138085.i 1.70003i
\(286\) 0 0
\(287\) − 99449.9i − 1.20737i
\(288\) 0 0
\(289\) 99656.3 1.19319
\(290\) 0 0
\(291\) 42556.1 0.502546
\(292\) 0 0
\(293\) 42030.2i 0.489583i 0.969576 + 0.244792i \(0.0787195\pi\)
−0.969576 + 0.244792i \(0.921280\pi\)
\(294\) 0 0
\(295\) − 52048.6i − 0.598088i
\(296\) 0 0
\(297\) −25433.4 −0.288331
\(298\) 0 0
\(299\) 161930. 1.81128
\(300\) 0 0
\(301\) 23444.3i 0.258765i
\(302\) 0 0
\(303\) − 80937.9i − 0.881590i
\(304\) 0 0
\(305\) 31875.3 0.342653
\(306\) 0 0
\(307\) −8820.31 −0.0935852 −0.0467926 0.998905i \(-0.514900\pi\)
−0.0467926 + 0.998905i \(0.514900\pi\)
\(308\) 0 0
\(309\) − 34443.1i − 0.360732i
\(310\) 0 0
\(311\) 155059.i 1.60315i 0.597893 + 0.801576i \(0.296005\pi\)
−0.597893 + 0.801576i \(0.703995\pi\)
\(312\) 0 0
\(313\) −179666. −1.83390 −0.916951 0.398999i \(-0.869358\pi\)
−0.916951 + 0.398999i \(0.869358\pi\)
\(314\) 0 0
\(315\) −49414.6 −0.498005
\(316\) 0 0
\(317\) 6188.77i 0.0615865i 0.999526 + 0.0307933i \(0.00980335\pi\)
−0.999526 + 0.0307933i \(0.990197\pi\)
\(318\) 0 0
\(319\) 146341.i 1.43808i
\(320\) 0 0
\(321\) −61296.0 −0.594870
\(322\) 0 0
\(323\) 286138. 2.74265
\(324\) 0 0
\(325\) − 175238.i − 1.65906i
\(326\) 0 0
\(327\) − 69036.4i − 0.645628i
\(328\) 0 0
\(329\) −94703.1 −0.874928
\(330\) 0 0
\(331\) 132159. 1.20626 0.603130 0.797643i \(-0.293920\pi\)
0.603130 + 0.797643i \(0.293920\pi\)
\(332\) 0 0
\(333\) − 12608.5i − 0.113704i
\(334\) 0 0
\(335\) 20105.6i 0.179154i
\(336\) 0 0
\(337\) 70040.0 0.616718 0.308359 0.951270i \(-0.400220\pi\)
0.308359 + 0.951270i \(0.400220\pi\)
\(338\) 0 0
\(339\) 60865.8 0.529632
\(340\) 0 0
\(341\) 70931.8i 0.610004i
\(342\) 0 0
\(343\) − 123489.i − 1.04964i
\(344\) 0 0
\(345\) 182266. 1.53132
\(346\) 0 0
\(347\) −157722. −1.30988 −0.654942 0.755680i \(-0.727307\pi\)
−0.654942 + 0.755680i \(0.727307\pi\)
\(348\) 0 0
\(349\) − 198747.i − 1.63174i −0.578238 0.815868i \(-0.696259\pi\)
0.578238 0.815868i \(-0.303741\pi\)
\(350\) 0 0
\(351\) − 25744.1i − 0.208960i
\(352\) 0 0
\(353\) 13376.1 0.107345 0.0536724 0.998559i \(-0.482907\pi\)
0.0536724 + 0.998559i \(0.482907\pi\)
\(354\) 0 0
\(355\) 86292.9 0.684729
\(356\) 0 0
\(357\) 102396.i 0.803426i
\(358\) 0 0
\(359\) − 190011.i − 1.47432i −0.675721 0.737158i \(-0.736168\pi\)
0.675721 0.737158i \(-0.263832\pi\)
\(360\) 0 0
\(361\) 316649. 2.42976
\(362\) 0 0
\(363\) −94687.8 −0.718590
\(364\) 0 0
\(365\) 91342.0i 0.685622i
\(366\) 0 0
\(367\) − 94140.5i − 0.698947i −0.936946 0.349474i \(-0.886360\pi\)
0.936946 0.349474i \(-0.113640\pi\)
\(368\) 0 0
\(369\) 58318.1 0.428302
\(370\) 0 0
\(371\) 34703.9 0.252133
\(372\) 0 0
\(373\) 66152.4i 0.475475i 0.971329 + 0.237738i \(0.0764058\pi\)
−0.971329 + 0.237738i \(0.923594\pi\)
\(374\) 0 0
\(375\) − 68155.6i − 0.484662i
\(376\) 0 0
\(377\) −148128. −1.04221
\(378\) 0 0
\(379\) −64395.2 −0.448306 −0.224153 0.974554i \(-0.571962\pi\)
−0.224153 + 0.974554i \(0.571962\pi\)
\(380\) 0 0
\(381\) 29309.2i 0.201908i
\(382\) 0 0
\(383\) − 30714.5i − 0.209385i −0.994505 0.104693i \(-0.966614\pi\)
0.994505 0.104693i \(-0.0333859\pi\)
\(384\) 0 0
\(385\) −331779. −2.23835
\(386\) 0 0
\(387\) −13747.9 −0.0917941
\(388\) 0 0
\(389\) 199122.i 1.31589i 0.753066 + 0.657945i \(0.228574\pi\)
−0.753066 + 0.657945i \(0.771426\pi\)
\(390\) 0 0
\(391\) − 377687.i − 2.47046i
\(392\) 0 0
\(393\) 165874. 1.07397
\(394\) 0 0
\(395\) −425525. −2.72729
\(396\) 0 0
\(397\) − 68565.8i − 0.435037i −0.976056 0.217519i \(-0.930204\pi\)
0.976056 0.217519i \(-0.0697963\pi\)
\(398\) 0 0
\(399\) 159951.i 1.00471i
\(400\) 0 0
\(401\) −21797.1 −0.135553 −0.0677765 0.997701i \(-0.521590\pi\)
−0.0677765 + 0.997701i \(0.521590\pi\)
\(402\) 0 0
\(403\) −71798.3 −0.442084
\(404\) 0 0
\(405\) − 28977.0i − 0.176662i
\(406\) 0 0
\(407\) − 84655.8i − 0.511055i
\(408\) 0 0
\(409\) 230211. 1.37620 0.688098 0.725618i \(-0.258446\pi\)
0.688098 + 0.725618i \(0.258446\pi\)
\(410\) 0 0
\(411\) 54640.6 0.323468
\(412\) 0 0
\(413\) − 60290.3i − 0.353465i
\(414\) 0 0
\(415\) − 118365.i − 0.687270i
\(416\) 0 0
\(417\) −77868.9 −0.447808
\(418\) 0 0
\(419\) −17360.5 −0.0988859 −0.0494430 0.998777i \(-0.515745\pi\)
−0.0494430 + 0.998777i \(0.515745\pi\)
\(420\) 0 0
\(421\) − 186829.i − 1.05410i −0.849836 0.527048i \(-0.823299\pi\)
0.849836 0.527048i \(-0.176701\pi\)
\(422\) 0 0
\(423\) − 55534.5i − 0.310372i
\(424\) 0 0
\(425\) −408726. −2.26284
\(426\) 0 0
\(427\) 36922.6 0.202505
\(428\) 0 0
\(429\) − 172851.i − 0.939197i
\(430\) 0 0
\(431\) − 153753.i − 0.827693i −0.910347 0.413846i \(-0.864185\pi\)
0.910347 0.413846i \(-0.135815\pi\)
\(432\) 0 0
\(433\) 9168.87 0.0489035 0.0244517 0.999701i \(-0.492216\pi\)
0.0244517 + 0.999701i \(0.492216\pi\)
\(434\) 0 0
\(435\) −166730. −0.881122
\(436\) 0 0
\(437\) − 589978.i − 3.08939i
\(438\) 0 0
\(439\) 178760.i 0.927561i 0.885950 + 0.463780i \(0.153507\pi\)
−0.885950 + 0.463780i \(0.846493\pi\)
\(440\) 0 0
\(441\) 7587.85 0.0390159
\(442\) 0 0
\(443\) 17798.2 0.0906920 0.0453460 0.998971i \(-0.485561\pi\)
0.0453460 + 0.998971i \(0.485561\pi\)
\(444\) 0 0
\(445\) 258127.i 1.30351i
\(446\) 0 0
\(447\) − 61290.5i − 0.306745i
\(448\) 0 0
\(449\) 242915. 1.20493 0.602464 0.798146i \(-0.294186\pi\)
0.602464 + 0.798146i \(0.294186\pi\)
\(450\) 0 0
\(451\) 391559. 1.92506
\(452\) 0 0
\(453\) 175593.i 0.855680i
\(454\) 0 0
\(455\) − 335832.i − 1.62218i
\(456\) 0 0
\(457\) −190762. −0.913396 −0.456698 0.889622i \(-0.650968\pi\)
−0.456698 + 0.889622i \(0.650968\pi\)
\(458\) 0 0
\(459\) −60045.6 −0.285007
\(460\) 0 0
\(461\) − 155121.i − 0.729909i −0.931025 0.364955i \(-0.881084\pi\)
0.931025 0.364955i \(-0.118916\pi\)
\(462\) 0 0
\(463\) − 230925.i − 1.07723i −0.842551 0.538616i \(-0.818947\pi\)
0.842551 0.538616i \(-0.181053\pi\)
\(464\) 0 0
\(465\) −80814.9 −0.373754
\(466\) 0 0
\(467\) 154112. 0.706647 0.353323 0.935501i \(-0.385052\pi\)
0.353323 + 0.935501i \(0.385052\pi\)
\(468\) 0 0
\(469\) 23289.2i 0.105879i
\(470\) 0 0
\(471\) 149592.i 0.674319i
\(472\) 0 0
\(473\) −92306.3 −0.412581
\(474\) 0 0
\(475\) −638462. −2.82975
\(476\) 0 0
\(477\) 20350.6i 0.0894417i
\(478\) 0 0
\(479\) − 405668.i − 1.76807i −0.467420 0.884035i \(-0.654816\pi\)
0.467420 0.884035i \(-0.345184\pi\)
\(480\) 0 0
\(481\) 85690.0 0.370373
\(482\) 0 0
\(483\) 211127. 0.905000
\(484\) 0 0
\(485\) 325541.i 1.38396i
\(486\) 0 0
\(487\) 426114.i 1.79667i 0.439314 + 0.898334i \(0.355222\pi\)
−0.439314 + 0.898334i \(0.644778\pi\)
\(488\) 0 0
\(489\) −69675.0 −0.291379
\(490\) 0 0
\(491\) 273125. 1.13292 0.566459 0.824090i \(-0.308313\pi\)
0.566459 + 0.824090i \(0.308313\pi\)
\(492\) 0 0
\(493\) 345495.i 1.42151i
\(494\) 0 0
\(495\) − 194558.i − 0.794032i
\(496\) 0 0
\(497\) 99957.1 0.404670
\(498\) 0 0
\(499\) 103548. 0.415855 0.207928 0.978144i \(-0.433328\pi\)
0.207928 + 0.978144i \(0.433328\pi\)
\(500\) 0 0
\(501\) − 68624.2i − 0.273402i
\(502\) 0 0
\(503\) 217063.i 0.857925i 0.903322 + 0.428963i \(0.141121\pi\)
−0.903322 + 0.428963i \(0.858879\pi\)
\(504\) 0 0
\(505\) 619151. 2.42780
\(506\) 0 0
\(507\) 26555.1 0.103307
\(508\) 0 0
\(509\) 115098.i 0.444256i 0.975018 + 0.222128i \(0.0713003\pi\)
−0.975018 + 0.222128i \(0.928700\pi\)
\(510\) 0 0
\(511\) 105806.i 0.405198i
\(512\) 0 0
\(513\) −93796.1 −0.356410
\(514\) 0 0
\(515\) 263479. 0.993418
\(516\) 0 0
\(517\) − 372870.i − 1.39501i
\(518\) 0 0
\(519\) 1896.37i 0.00704024i
\(520\) 0 0
\(521\) 288543. 1.06300 0.531502 0.847057i \(-0.321628\pi\)
0.531502 + 0.847057i \(0.321628\pi\)
\(522\) 0 0
\(523\) 176826. 0.646461 0.323231 0.946320i \(-0.395231\pi\)
0.323231 + 0.946320i \(0.395231\pi\)
\(524\) 0 0
\(525\) − 228477.i − 0.828942i
\(526\) 0 0
\(527\) 167463.i 0.602973i
\(528\) 0 0
\(529\) −498900. −1.78280
\(530\) 0 0
\(531\) 35354.6 0.125388
\(532\) 0 0
\(533\) 396343.i 1.39514i
\(534\) 0 0
\(535\) − 468896.i − 1.63821i
\(536\) 0 0
\(537\) 74678.9 0.258970
\(538\) 0 0
\(539\) 50946.4 0.175362
\(540\) 0 0
\(541\) 425220.i 1.45284i 0.687249 + 0.726422i \(0.258818\pi\)
−0.687249 + 0.726422i \(0.741182\pi\)
\(542\) 0 0
\(543\) − 21047.2i − 0.0713830i
\(544\) 0 0
\(545\) 528108. 1.77799
\(546\) 0 0
\(547\) 495505. 1.65605 0.828024 0.560692i \(-0.189465\pi\)
0.828024 + 0.560692i \(0.189465\pi\)
\(548\) 0 0
\(549\) 21651.7i 0.0718367i
\(550\) 0 0
\(551\) 539691.i 1.77763i
\(552\) 0 0
\(553\) −492905. −1.61181
\(554\) 0 0
\(555\) 96451.0 0.313127
\(556\) 0 0
\(557\) − 257895.i − 0.831251i −0.909536 0.415626i \(-0.863563\pi\)
0.909536 0.415626i \(-0.136437\pi\)
\(558\) 0 0
\(559\) − 93433.9i − 0.299007i
\(560\) 0 0
\(561\) −403158. −1.28100
\(562\) 0 0
\(563\) −417799. −1.31811 −0.659054 0.752096i \(-0.729043\pi\)
−0.659054 + 0.752096i \(0.729043\pi\)
\(564\) 0 0
\(565\) 465606.i 1.45855i
\(566\) 0 0
\(567\) − 33565.4i − 0.104406i
\(568\) 0 0
\(569\) −100935. −0.311756 −0.155878 0.987776i \(-0.549821\pi\)
−0.155878 + 0.987776i \(0.549821\pi\)
\(570\) 0 0
\(571\) −453320. −1.39038 −0.695188 0.718828i \(-0.744679\pi\)
−0.695188 + 0.718828i \(0.744679\pi\)
\(572\) 0 0
\(573\) − 259189.i − 0.789418i
\(574\) 0 0
\(575\) 842738.i 2.54892i
\(576\) 0 0
\(577\) 12119.6 0.0364031 0.0182015 0.999834i \(-0.494206\pi\)
0.0182015 + 0.999834i \(0.494206\pi\)
\(578\) 0 0
\(579\) 251627. 0.750586
\(580\) 0 0
\(581\) − 137108.i − 0.406172i
\(582\) 0 0
\(583\) 136638.i 0.402008i
\(584\) 0 0
\(585\) 196934. 0.575453
\(586\) 0 0
\(587\) −327468. −0.950370 −0.475185 0.879886i \(-0.657619\pi\)
−0.475185 + 0.879886i \(0.657619\pi\)
\(588\) 0 0
\(589\) 261591.i 0.754035i
\(590\) 0 0
\(591\) − 28873.7i − 0.0826660i
\(592\) 0 0
\(593\) 610182. 1.73520 0.867601 0.497260i \(-0.165661\pi\)
0.867601 + 0.497260i \(0.165661\pi\)
\(594\) 0 0
\(595\) −783298. −2.21255
\(596\) 0 0
\(597\) − 314859.i − 0.883422i
\(598\) 0 0
\(599\) − 499941.i − 1.39337i −0.717379 0.696683i \(-0.754658\pi\)
0.717379 0.696683i \(-0.245342\pi\)
\(600\) 0 0
\(601\) −486965. −1.34818 −0.674091 0.738649i \(-0.735464\pi\)
−0.674091 + 0.738649i \(0.735464\pi\)
\(602\) 0 0
\(603\) −13657.0 −0.0375595
\(604\) 0 0
\(605\) − 724334.i − 1.97892i
\(606\) 0 0
\(607\) − 41693.4i − 0.113159i −0.998398 0.0565796i \(-0.981981\pi\)
0.998398 0.0565796i \(-0.0180195\pi\)
\(608\) 0 0
\(609\) −193131. −0.520737
\(610\) 0 0
\(611\) 377425. 1.01099
\(612\) 0 0
\(613\) 542042.i 1.44249i 0.692681 + 0.721244i \(0.256429\pi\)
−0.692681 + 0.721244i \(0.743571\pi\)
\(614\) 0 0
\(615\) 446116.i 1.17950i
\(616\) 0 0
\(617\) −146814. −0.385653 −0.192827 0.981233i \(-0.561765\pi\)
−0.192827 + 0.981233i \(0.561765\pi\)
\(618\) 0 0
\(619\) −72720.6 −0.189791 −0.0948956 0.995487i \(-0.530252\pi\)
−0.0948956 + 0.995487i \(0.530252\pi\)
\(620\) 0 0
\(621\) 123806.i 0.321040i
\(622\) 0 0
\(623\) 299000.i 0.770363i
\(624\) 0 0
\(625\) −75495.2 −0.193268
\(626\) 0 0
\(627\) −629766. −1.60193
\(628\) 0 0
\(629\) − 199864.i − 0.505165i
\(630\) 0 0
\(631\) 568570.i 1.42799i 0.700151 + 0.713995i \(0.253116\pi\)
−0.700151 + 0.713995i \(0.746884\pi\)
\(632\) 0 0
\(633\) 143102. 0.357139
\(634\) 0 0
\(635\) −224206. −0.556033
\(636\) 0 0
\(637\) 51568.7i 0.127089i
\(638\) 0 0
\(639\) 58615.5i 0.143553i
\(640\) 0 0
\(641\) 292494. 0.711870 0.355935 0.934511i \(-0.384162\pi\)
0.355935 + 0.934511i \(0.384162\pi\)
\(642\) 0 0
\(643\) −103161. −0.249514 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(644\) 0 0
\(645\) − 105167.i − 0.252791i
\(646\) 0 0
\(647\) 372569.i 0.890017i 0.895526 + 0.445009i \(0.146799\pi\)
−0.895526 + 0.445009i \(0.853201\pi\)
\(648\) 0 0
\(649\) 237378. 0.563574
\(650\) 0 0
\(651\) −93611.6 −0.220886
\(652\) 0 0
\(653\) 85619.0i 0.200791i 0.994948 + 0.100395i \(0.0320108\pi\)
−0.994948 + 0.100395i \(0.967989\pi\)
\(654\) 0 0
\(655\) 1.26889e6i 2.95761i
\(656\) 0 0
\(657\) −62045.1 −0.143740
\(658\) 0 0
\(659\) −258812. −0.595954 −0.297977 0.954573i \(-0.596312\pi\)
−0.297977 + 0.954573i \(0.596312\pi\)
\(660\) 0 0
\(661\) 327660.i 0.749928i 0.927039 + 0.374964i \(0.122345\pi\)
−0.927039 + 0.374964i \(0.877655\pi\)
\(662\) 0 0
\(663\) − 408084.i − 0.928372i
\(664\) 0 0
\(665\) −1.22357e6 −2.76686
\(666\) 0 0
\(667\) 712366. 1.60122
\(668\) 0 0
\(669\) − 15699.4i − 0.0350777i
\(670\) 0 0
\(671\) 145374.i 0.322880i
\(672\) 0 0
\(673\) 137121. 0.302742 0.151371 0.988477i \(-0.451631\pi\)
0.151371 + 0.988477i \(0.451631\pi\)
\(674\) 0 0
\(675\) 133981. 0.294059
\(676\) 0 0
\(677\) − 573451.i − 1.25118i −0.780153 0.625589i \(-0.784859\pi\)
0.780153 0.625589i \(-0.215141\pi\)
\(678\) 0 0
\(679\) 377089.i 0.817909i
\(680\) 0 0
\(681\) −36774.6 −0.0792965
\(682\) 0 0
\(683\) 175872. 0.377012 0.188506 0.982072i \(-0.439636\pi\)
0.188506 + 0.982072i \(0.439636\pi\)
\(684\) 0 0
\(685\) 417984.i 0.890797i
\(686\) 0 0
\(687\) 533962.i 1.13135i
\(688\) 0 0
\(689\) −138307. −0.291344
\(690\) 0 0
\(691\) 7216.66 0.0151140 0.00755702 0.999971i \(-0.497595\pi\)
0.00755702 + 0.999971i \(0.497595\pi\)
\(692\) 0 0
\(693\) − 225365.i − 0.469267i
\(694\) 0 0
\(695\) − 595673.i − 1.23321i
\(696\) 0 0
\(697\) 924433. 1.90287
\(698\) 0 0
\(699\) −119390. −0.244350
\(700\) 0 0
\(701\) 502543.i 1.02267i 0.859380 + 0.511337i \(0.170850\pi\)
−0.859380 + 0.511337i \(0.829150\pi\)
\(702\) 0 0
\(703\) − 312203.i − 0.631723i
\(704\) 0 0
\(705\) 424823. 0.854731
\(706\) 0 0
\(707\) 717191. 1.43481
\(708\) 0 0
\(709\) 95591.1i 0.190163i 0.995470 + 0.0950813i \(0.0303111\pi\)
−0.995470 + 0.0950813i \(0.969689\pi\)
\(710\) 0 0
\(711\) − 289043.i − 0.571772i
\(712\) 0 0
\(713\) 345286. 0.679204
\(714\) 0 0
\(715\) 1.32226e6 2.58645
\(716\) 0 0
\(717\) − 184810.i − 0.359491i
\(718\) 0 0
\(719\) − 475789.i − 0.920357i −0.887826 0.460178i \(-0.847785\pi\)
0.887826 0.460178i \(-0.152215\pi\)
\(720\) 0 0
\(721\) 305200. 0.587103
\(722\) 0 0
\(723\) 481597. 0.921313
\(724\) 0 0
\(725\) − 770908.i − 1.46665i
\(726\) 0 0
\(727\) − 193262.i − 0.365661i −0.983144 0.182830i \(-0.941474\pi\)
0.983144 0.182830i \(-0.0585259\pi\)
\(728\) 0 0
\(729\) 19683.0 0.0370370
\(730\) 0 0
\(731\) −217926. −0.407825
\(732\) 0 0
\(733\) − 650799.i − 1.21126i −0.795745 0.605632i \(-0.792920\pi\)
0.795745 0.605632i \(-0.207080\pi\)
\(734\) 0 0
\(735\) 58044.8i 0.107446i
\(736\) 0 0
\(737\) −91695.6 −0.168816
\(738\) 0 0
\(739\) −99472.6 −0.182144 −0.0910719 0.995844i \(-0.529029\pi\)
−0.0910719 + 0.995844i \(0.529029\pi\)
\(740\) 0 0
\(741\) − 637459.i − 1.16096i
\(742\) 0 0
\(743\) 715681.i 1.29641i 0.761466 + 0.648205i \(0.224480\pi\)
−0.761466 + 0.648205i \(0.775520\pi\)
\(744\) 0 0
\(745\) 468854. 0.844744
\(746\) 0 0
\(747\) 80401.0 0.144085
\(748\) 0 0
\(749\) − 543144.i − 0.968170i
\(750\) 0 0
\(751\) 603532.i 1.07009i 0.844824 + 0.535045i \(0.179705\pi\)
−0.844824 + 0.535045i \(0.820295\pi\)
\(752\) 0 0
\(753\) 143364. 0.252843
\(754\) 0 0
\(755\) −1.34324e6 −2.35645
\(756\) 0 0
\(757\) 784192.i 1.36846i 0.729268 + 0.684228i \(0.239861\pi\)
−0.729268 + 0.684228i \(0.760139\pi\)
\(758\) 0 0
\(759\) 831259.i 1.44296i
\(760\) 0 0
\(761\) −568076. −0.980927 −0.490464 0.871462i \(-0.663173\pi\)
−0.490464 + 0.871462i \(0.663173\pi\)
\(762\) 0 0
\(763\) 611731. 1.05078
\(764\) 0 0
\(765\) − 459331.i − 0.784880i
\(766\) 0 0
\(767\) 240278.i 0.408435i
\(768\) 0 0
\(769\) −531757. −0.899209 −0.449605 0.893228i \(-0.648435\pi\)
−0.449605 + 0.893228i \(0.648435\pi\)
\(770\) 0 0
\(771\) −479629. −0.806856
\(772\) 0 0
\(773\) − 260161.i − 0.435394i −0.976016 0.217697i \(-0.930145\pi\)
0.976016 0.217697i \(-0.0698545\pi\)
\(774\) 0 0
\(775\) − 373662.i − 0.622122i
\(776\) 0 0
\(777\) 111724. 0.185056
\(778\) 0 0
\(779\) 1.44404e6 2.37960
\(780\) 0 0
\(781\) 393556.i 0.645216i
\(782\) 0 0
\(783\) − 113254.i − 0.184726i
\(784\) 0 0
\(785\) −1.14433e6 −1.85700
\(786\) 0 0
\(787\) −724342. −1.16948 −0.584742 0.811219i \(-0.698804\pi\)
−0.584742 + 0.811219i \(0.698804\pi\)
\(788\) 0 0
\(789\) − 72145.7i − 0.115893i
\(790\) 0 0
\(791\) 539332.i 0.861993i
\(792\) 0 0
\(793\) −147150. −0.233998
\(794\) 0 0
\(795\) −155676. −0.246313
\(796\) 0 0
\(797\) 1.04249e6i 1.64117i 0.571524 + 0.820586i \(0.306353\pi\)
−0.571524 + 0.820586i \(0.693647\pi\)
\(798\) 0 0
\(799\) − 880309.i − 1.37893i
\(800\) 0 0
\(801\) −175336. −0.273279
\(802\) 0 0
\(803\) −416584. −0.646057
\(804\) 0 0
\(805\) 1.61506e6i 2.49227i
\(806\) 0 0
\(807\) − 273827.i − 0.420465i
\(808\) 0 0
\(809\) −647222. −0.988909 −0.494455 0.869204i \(-0.664632\pi\)
−0.494455 + 0.869204i \(0.664632\pi\)
\(810\) 0 0
\(811\) −1.24087e6 −1.88662 −0.943309 0.331916i \(-0.892305\pi\)
−0.943309 + 0.331916i \(0.892305\pi\)
\(812\) 0 0
\(813\) 244369.i 0.369713i
\(814\) 0 0
\(815\) − 532993.i − 0.802428i
\(816\) 0 0
\(817\) −340418. −0.509997
\(818\) 0 0
\(819\) 228118. 0.340089
\(820\) 0 0
\(821\) 819276.i 1.21547i 0.794141 + 0.607734i \(0.207921\pi\)
−0.794141 + 0.607734i \(0.792079\pi\)
\(822\) 0 0
\(823\) − 515929.i − 0.761710i −0.924635 0.380855i \(-0.875630\pi\)
0.924635 0.380855i \(-0.124370\pi\)
\(824\) 0 0
\(825\) 899573. 1.32169
\(826\) 0 0
\(827\) 140674. 0.205685 0.102842 0.994698i \(-0.467206\pi\)
0.102842 + 0.994698i \(0.467206\pi\)
\(828\) 0 0
\(829\) 137443.i 0.199993i 0.994988 + 0.0999964i \(0.0318831\pi\)
−0.994988 + 0.0999964i \(0.968117\pi\)
\(830\) 0 0
\(831\) − 563229.i − 0.815611i
\(832\) 0 0
\(833\) 120279. 0.173341
\(834\) 0 0
\(835\) 524955. 0.752920
\(836\) 0 0
\(837\) − 54894.5i − 0.0783570i
\(838\) 0 0
\(839\) 591422.i 0.840182i 0.907482 + 0.420091i \(0.138002\pi\)
−0.907482 + 0.420091i \(0.861998\pi\)
\(840\) 0 0
\(841\) 55633.2 0.0786579
\(842\) 0 0
\(843\) 379790. 0.534428
\(844\) 0 0
\(845\) 203138.i 0.284498i
\(846\) 0 0
\(847\) − 839029.i − 1.16953i
\(848\) 0 0
\(849\) 232596. 0.322691
\(850\) 0 0
\(851\) −412093. −0.569031
\(852\) 0 0
\(853\) − 169773.i − 0.233331i −0.993171 0.116665i \(-0.962780\pi\)
0.993171 0.116665i \(-0.0372205\pi\)
\(854\) 0 0
\(855\) − 717512.i − 0.981515i
\(856\) 0 0
\(857\) 1.05083e6 1.43077 0.715384 0.698732i \(-0.246252\pi\)
0.715384 + 0.698732i \(0.246252\pi\)
\(858\) 0 0
\(859\) −1.05896e6 −1.43514 −0.717569 0.696487i \(-0.754745\pi\)
−0.717569 + 0.696487i \(0.754745\pi\)
\(860\) 0 0
\(861\) 516757.i 0.697075i
\(862\) 0 0
\(863\) − 1.22011e6i − 1.63824i −0.573621 0.819121i \(-0.694462\pi\)
0.573621 0.819121i \(-0.305538\pi\)
\(864\) 0 0
\(865\) −14506.6 −0.0193881
\(866\) 0 0
\(867\) −517829. −0.688887
\(868\) 0 0
\(869\) − 1.94069e6i − 2.56991i
\(870\) 0 0
\(871\) − 92815.8i − 0.122345i
\(872\) 0 0
\(873\) −221128. −0.290145
\(874\) 0 0
\(875\) 603927. 0.788802
\(876\) 0 0
\(877\) − 707060.i − 0.919299i −0.888100 0.459650i \(-0.847975\pi\)
0.888100 0.459650i \(-0.152025\pi\)
\(878\) 0 0
\(879\) − 218395.i − 0.282661i
\(880\) 0 0
\(881\) 430965. 0.555252 0.277626 0.960689i \(-0.410452\pi\)
0.277626 + 0.960689i \(0.410452\pi\)
\(882\) 0 0
\(883\) −1.08382e6 −1.39007 −0.695037 0.718974i \(-0.744612\pi\)
−0.695037 + 0.718974i \(0.744612\pi\)
\(884\) 0 0
\(885\) 270452.i 0.345306i
\(886\) 0 0
\(887\) 620229.i 0.788324i 0.919041 + 0.394162i \(0.128965\pi\)
−0.919041 + 0.394162i \(0.871035\pi\)
\(888\) 0 0
\(889\) −259709. −0.328612
\(890\) 0 0
\(891\) 132156. 0.166468
\(892\) 0 0
\(893\) − 1.37511e6i − 1.72439i
\(894\) 0 0
\(895\) 571272.i 0.713176i
\(896\) 0 0
\(897\) −841414. −1.04574
\(898\) 0 0
\(899\) −315856. −0.390814
\(900\) 0 0
\(901\) 322589.i 0.397374i
\(902\) 0 0
\(903\) − 121820.i − 0.149398i
\(904\) 0 0
\(905\) 161005. 0.196581
\(906\) 0 0
\(907\) 155091. 0.188526 0.0942629 0.995547i \(-0.469951\pi\)
0.0942629 + 0.995547i \(0.469951\pi\)
\(908\) 0 0
\(909\) 420566.i 0.508986i
\(910\) 0 0
\(911\) − 670134.i − 0.807467i −0.914877 0.403734i \(-0.867712\pi\)
0.914877 0.403734i \(-0.132288\pi\)
\(912\) 0 0
\(913\) 539828. 0.647611
\(914\) 0 0
\(915\) −165629. −0.197831
\(916\) 0 0
\(917\) 1.46981e6i 1.74793i
\(918\) 0 0
\(919\) 631844.i 0.748134i 0.927402 + 0.374067i \(0.122037\pi\)
−0.927402 + 0.374067i \(0.877963\pi\)
\(920\) 0 0
\(921\) 45831.7 0.0540314
\(922\) 0 0
\(923\) −398364. −0.467602
\(924\) 0 0
\(925\) 445959.i 0.521208i
\(926\) 0 0
\(927\) 178972.i 0.208269i
\(928\) 0 0
\(929\) 1.16204e6 1.34645 0.673225 0.739437i \(-0.264908\pi\)
0.673225 + 0.739437i \(0.264908\pi\)
\(930\) 0 0
\(931\) 187886. 0.216768
\(932\) 0 0
\(933\) − 805708.i − 0.925580i
\(934\) 0 0
\(935\) − 3.08404e6i − 3.52774i
\(936\) 0 0
\(937\) −1.18305e6 −1.34748 −0.673741 0.738967i \(-0.735314\pi\)
−0.673741 + 0.738967i \(0.735314\pi\)
\(938\) 0 0
\(939\) 933570. 1.05880
\(940\) 0 0
\(941\) − 159241.i − 0.179836i −0.995949 0.0899179i \(-0.971340\pi\)
0.995949 0.0899179i \(-0.0286605\pi\)
\(942\) 0 0
\(943\) − 1.90606e6i − 2.14345i
\(944\) 0 0
\(945\) 256766. 0.287523
\(946\) 0 0
\(947\) −940316. −1.04851 −0.524257 0.851560i \(-0.675657\pi\)
−0.524257 + 0.851560i \(0.675657\pi\)
\(948\) 0 0
\(949\) − 421673.i − 0.468213i
\(950\) 0 0
\(951\) − 32157.8i − 0.0355570i
\(952\) 0 0
\(953\) 577950. 0.636362 0.318181 0.948030i \(-0.396928\pi\)
0.318181 + 0.948030i \(0.396928\pi\)
\(954\) 0 0
\(955\) 1.98272e6 2.17397
\(956\) 0 0
\(957\) − 760408.i − 0.830276i
\(958\) 0 0
\(959\) 484170.i 0.526454i
\(960\) 0 0
\(961\) 770424. 0.834225
\(962\) 0 0
\(963\) 318504. 0.343449
\(964\) 0 0
\(965\) 1.92487e6i 2.06703i
\(966\) 0 0
\(967\) − 785047.i − 0.839542i −0.907630 0.419771i \(-0.862110\pi\)
0.907630 0.419771i \(-0.137890\pi\)
\(968\) 0 0
\(969\) −1.48681e6 −1.58347
\(970\) 0 0
\(971\) 1.05236e6 1.11616 0.558082 0.829786i \(-0.311538\pi\)
0.558082 + 0.829786i \(0.311538\pi\)
\(972\) 0 0
\(973\) − 689996.i − 0.728821i
\(974\) 0 0
\(975\) 910562.i 0.957856i
\(976\) 0 0
\(977\) 20197.7 0.0211598 0.0105799 0.999944i \(-0.496632\pi\)
0.0105799 + 0.999944i \(0.496632\pi\)
\(978\) 0 0
\(979\) −1.17724e6 −1.22829
\(980\) 0 0
\(981\) 358723.i 0.372754i
\(982\) 0 0
\(983\) − 1.43076e6i − 1.48068i −0.672234 0.740338i \(-0.734665\pi\)
0.672234 0.740338i \(-0.265335\pi\)
\(984\) 0 0
\(985\) 220875. 0.227653
\(986\) 0 0
\(987\) 492092. 0.505140
\(988\) 0 0
\(989\) 449334.i 0.459385i
\(990\) 0 0
\(991\) 787866.i 0.802241i 0.916025 + 0.401120i \(0.131379\pi\)
−0.916025 + 0.401120i \(0.868621\pi\)
\(992\) 0 0
\(993\) −686718. −0.696434
\(994\) 0 0
\(995\) 2.40858e6 2.43285
\(996\) 0 0
\(997\) − 453709.i − 0.456444i −0.973609 0.228222i \(-0.926709\pi\)
0.973609 0.228222i \(-0.0732912\pi\)
\(998\) 0 0
\(999\) 65515.5i 0.0656468i
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 384.5.b.c.319.1 8
3.2 odd 2 1152.5.b.k.703.7 8
4.3 odd 2 inner 384.5.b.c.319.5 yes 8
8.3 odd 2 inner 384.5.b.c.319.4 yes 8
8.5 even 2 inner 384.5.b.c.319.8 yes 8
12.11 even 2 1152.5.b.k.703.8 8
16.3 odd 4 768.5.g.c.511.3 4
16.5 even 4 768.5.g.g.511.4 4
16.11 odd 4 768.5.g.g.511.2 4
16.13 even 4 768.5.g.c.511.1 4
24.5 odd 2 1152.5.b.k.703.1 8
24.11 even 2 1152.5.b.k.703.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.1 8 1.1 even 1 trivial
384.5.b.c.319.4 yes 8 8.3 odd 2 inner
384.5.b.c.319.5 yes 8 4.3 odd 2 inner
384.5.b.c.319.8 yes 8 8.5 even 2 inner
768.5.g.c.511.1 4 16.13 even 4
768.5.g.c.511.3 4 16.3 odd 4
768.5.g.g.511.2 4 16.11 odd 4
768.5.g.g.511.4 4 16.5 even 4
1152.5.b.k.703.1 8 24.5 odd 2
1152.5.b.k.703.2 8 24.11 even 2
1152.5.b.k.703.7 8 3.2 odd 2
1152.5.b.k.703.8 8 12.11 even 2