Properties

Label 768.5.g.g.511.2
Level $768$
Weight $5$
Character 768.511
Analytic conductor $79.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.3881316484\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 511.2
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 768.511
Dual form 768.5.g.g.511.4

$q$-expansion

\(f(q)\) \(=\) \(q-5.19615i q^{3} +39.7490 q^{5} -46.0431i q^{7} -27.0000 q^{9} +O(q^{10})\) \(q-5.19615i q^{3} +39.7490 q^{5} -46.0431i q^{7} -27.0000 q^{9} -181.283i q^{11} +183.498 q^{13} -206.542i q^{15} +427.992 q^{17} +668.558i q^{19} -239.247 q^{21} -882.463i q^{23} +954.984 q^{25} +140.296i q^{27} +807.247 q^{29} -391.276i q^{31} -941.976 q^{33} -1830.17i q^{35} +466.980 q^{37} -953.484i q^{39} -2159.93 q^{41} +509.182i q^{43} -1073.22 q^{45} +2056.83i q^{47} +281.031 q^{49} -2223.91i q^{51} -753.725 q^{53} -7205.84i q^{55} +3473.93 q^{57} -1309.43i q^{59} +801.913 q^{61} +1243.16i q^{63} +7293.87 q^{65} -505.813i q^{67} -4585.41 q^{69} +2170.94i q^{71} +2297.97 q^{73} -4962.24i q^{75} -8346.85 q^{77} +10705.3i q^{79} +729.000 q^{81} +2977.81i q^{83} +17012.3 q^{85} -4194.58i q^{87} +6493.92 q^{89} -8448.82i q^{91} -2033.13 q^{93} +26574.5i q^{95} -8189.92 q^{97} +4894.65i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{5} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{5} - 108 q^{9} + 480 q^{13} + 696 q^{17} - 576 q^{21} + 1788 q^{25} + 2848 q^{29} - 720 q^{33} - 672 q^{37} + 504 q^{41} - 864 q^{45} + 5188 q^{49} + 160 q^{53} + 4752 q^{57} - 7968 q^{61} + 11904 q^{65} - 6912 q^{69} + 5128 q^{73} - 14592 q^{77} + 2916 q^{81} + 37824 q^{85} + 15816 q^{89} + 7488 q^{93} - 22600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 5.19615i − 0.577350i
\(4\) 0 0
\(5\) 39.7490 1.58996 0.794980 0.606635i \(-0.207481\pi\)
0.794980 + 0.606635i \(0.207481\pi\)
\(6\) 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
\(11\) − 181.283i − 1.49821i −0.662451 0.749105i \(-0.730484\pi\)
0.662451 0.749105i \(-0.269516\pi\)
\(12\) 0 0
\(13\) 183.498 1.08579 0.542894 0.839801i \(-0.317329\pi\)
0.542894 + 0.839801i \(0.317329\pi\)
\(14\) 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
\(19\) 668.558i 1.85196i 0.377571 + 0.925981i \(0.376759\pi\)
−0.377571 + 0.925981i \(0.623241\pi\)
\(20\) 0 0
\(21\) −239.247 −0.542510
\(22\) 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
\(27\) 140.296i 0.192450i
\(28\) 0 0
\(29\) 807.247 0.959866 0.479933 0.877305i \(-0.340661\pi\)
0.479933 + 0.877305i \(0.340661\pi\)
\(30\) 0 0
\(31\) − 391.276i − 0.407155i −0.979059 0.203577i \(-0.934743\pi\)
0.979059 0.203577i \(-0.0652569\pi\)
\(32\) 0 0
\(33\) −941.976 −0.864992
\(34\) 0 0
\(35\) − 1830.17i − 1.49402i
\(36\) 0 0
\(37\) 466.980 0.341111 0.170555 0.985348i \(-0.445444\pi\)
0.170555 + 0.985348i \(0.445444\pi\)
\(38\) 0 0
\(39\) − 953.484i − 0.626880i
\(40\) 0 0
\(41\) −2159.93 −1.28491 −0.642454 0.766325i \(-0.722083\pi\)
−0.642454 + 0.766325i \(0.722083\pi\)
\(42\) 0 0
\(43\) 509.182i 0.275382i 0.990475 + 0.137691i \(0.0439682\pi\)
−0.990475 + 0.137691i \(0.956032\pi\)
\(44\) 0 0
\(45\) −1073.22 −0.529987
\(46\) 0 0
\(47\) 2056.83i 0.931116i 0.885017 + 0.465558i \(0.154146\pi\)
−0.885017 + 0.465558i \(0.845854\pi\)
\(48\) 0 0
\(49\) 281.031 0.117048
\(50\) 0 0
\(51\) − 2223.91i − 0.855022i
\(52\) 0 0
\(53\) −753.725 −0.268325 −0.134163 0.990959i \(-0.542834\pi\)
−0.134163 + 0.990959i \(0.542834\pi\)
\(54\) 0 0
\(55\) − 7205.84i − 2.38210i
\(56\) 0 0
\(57\) 3473.93 1.06923
\(58\) 0 0
\(59\) − 1309.43i − 0.376165i −0.982153 0.188083i \(-0.939773\pi\)
0.982153 0.188083i \(-0.0602272\pi\)
\(60\) 0 0
\(61\) 801.913 0.215510 0.107755 0.994177i \(-0.465634\pi\)
0.107755 + 0.994177i \(0.465634\pi\)
\(62\) 0 0
\(63\) 1243.16i 0.313218i
\(64\) 0 0
\(65\) 7293.87 1.72636
\(66\) 0 0
\(67\) − 505.813i − 0.112678i −0.998412 0.0563392i \(-0.982057\pi\)
0.998412 0.0563392i \(-0.0179428\pi\)
\(68\) 0 0
\(69\) −4585.41 −0.963119
\(70\) 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 \(-0.430826\pi\)
0.215610 + 0.976480i \(0.430826\pi\)
\(74\) 0 0
\(75\) − 4962.24i − 0.882177i
\(76\) 0 0
\(77\) −8346.85 −1.40780
\(78\) 0 0
\(79\) 10705.3i 1.71532i 0.514219 + 0.857659i \(0.328082\pi\)
−0.514219 + 0.857659i \(0.671918\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) 2977.81i 0.432256i 0.976365 + 0.216128i \(0.0693429\pi\)
−0.976365 + 0.216128i \(0.930657\pi\)
\(84\) 0 0
\(85\) 17012.3 2.35464
\(86\) 0 0
\(87\) − 4194.58i − 0.554179i
\(88\) 0 0
\(89\) 6493.92 0.819836 0.409918 0.912122i \(-0.365557\pi\)
0.409918 + 0.912122i \(0.365557\pi\)
\(90\) 0 0
\(91\) − 8448.82i − 1.02027i
\(92\) 0 0
\(93\) −2033.13 −0.235071
\(94\) 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
\(99\) 4894.65i 0.499403i
\(100\) 0 0
\(101\) −15576.5 −1.52696 −0.763479 0.645833i \(-0.776510\pi\)
−0.763479 + 0.645833i \(0.776510\pi\)
\(102\) 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
\(107\) − 11796.4i − 1.03035i −0.857086 0.515173i \(-0.827728\pi\)
0.857086 0.515173i \(-0.172272\pi\)
\(108\) 0 0
\(109\) 13286.1 1.11826 0.559130 0.829080i \(-0.311135\pi\)
0.559130 + 0.829080i \(0.311135\pi\)
\(110\) 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
\(115\) − 35077.0i − 2.65233i
\(116\) 0 0
\(117\) −4954.45 −0.361929
\(118\) 0 0
\(119\) − 19706.1i − 1.39157i
\(120\) 0 0
\(121\) −18222.7 −1.24463
\(122\) 0 0
\(123\) 11223.3i 0.741842i
\(124\) 0 0
\(125\) 13116.5 0.839459
\(126\) 0 0
\(127\) 5640.55i 0.349715i 0.984594 + 0.174858i \(0.0559465\pi\)
−0.984594 + 0.174858i \(0.944054\pi\)
\(128\) 0 0
\(129\) 2645.79 0.158992
\(130\) 0 0
\(131\) − 31922.5i − 1.86018i −0.367335 0.930089i \(-0.619730\pi\)
0.367335 0.930089i \(-0.380270\pi\)
\(132\) 0 0
\(133\) 30782.5 1.74021
\(134\) 0 0
\(135\) 5576.63i 0.305988i
\(136\) 0 0
\(137\) 10515.6 0.560263 0.280132 0.959962i \(-0.409622\pi\)
0.280132 + 0.959962i \(0.409622\pi\)
\(138\) 0 0
\(139\) − 14985.9i − 0.775626i −0.921738 0.387813i \(-0.873231\pi\)
0.921738 0.387813i \(-0.126769\pi\)
\(140\) 0 0
\(141\) 10687.6 0.537580
\(142\) 0 0
\(143\) − 33265.2i − 1.62674i
\(144\) 0 0
\(145\) 32087.3 1.52615
\(146\) 0 0
\(147\) − 1460.28i − 0.0675775i
\(148\) 0 0
\(149\) −11795.4 −0.531298 −0.265649 0.964070i \(-0.585586\pi\)
−0.265649 + 0.964070i \(0.585586\pi\)
\(150\) 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
\(155\) − 15552.8i − 0.647360i
\(156\) 0 0
\(157\) −28788.9 −1.16796 −0.583978 0.811770i \(-0.698504\pi\)
−0.583978 + 0.811770i \(0.698504\pi\)
\(158\) 0 0
\(159\) 3916.47i 0.154918i
\(160\) 0 0
\(161\) −40631.3 −1.56751
\(162\) 0 0
\(163\) 13409.0i 0.504684i 0.967638 + 0.252342i \(0.0812008\pi\)
−0.967638 + 0.252342i \(0.918799\pi\)
\(164\) 0 0
\(165\) −37442.6 −1.37530
\(166\) 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
\(171\) − 18051.1i − 0.617320i
\(172\) 0 0
\(173\) −364.956 −0.0121940 −0.00609702 0.999981i \(-0.501941\pi\)
−0.00609702 + 0.999981i \(0.501941\pi\)
\(174\) 0 0
\(175\) − 43970.5i − 1.43577i
\(176\) 0 0
\(177\) −6804.00 −0.217179
\(178\) 0 0
\(179\) − 14372.0i − 0.448549i −0.974526 0.224275i \(-0.927999\pi\)
0.974526 0.224275i \(-0.0720013\pi\)
\(180\) 0 0
\(181\) −4050.54 −0.123639 −0.0618195 0.998087i \(-0.519690\pi\)
−0.0618195 + 0.998087i \(0.519690\pi\)
\(182\) 0 0
\(183\) − 4166.86i − 0.124425i
\(184\) 0 0
\(185\) 18562.0 0.542352
\(186\) 0 0
\(187\) − 77587.9i − 2.21876i
\(188\) 0 0
\(189\) 6459.67 0.180837
\(190\) 0 0
\(191\) − 49880.9i − 1.36731i −0.729805 0.683656i \(-0.760389\pi\)
0.729805 0.683656i \(-0.239611\pi\)
\(192\) 0 0
\(193\) −48425.7 −1.30005 −0.650026 0.759912i \(-0.725242\pi\)
−0.650026 + 0.759912i \(0.725242\pi\)
\(194\) 0 0
\(195\) − 37900.0i − 0.996714i
\(196\) 0 0
\(197\) −5556.74 −0.143182 −0.0715908 0.997434i \(-0.522808\pi\)
−0.0715908 + 0.997434i \(0.522808\pi\)
\(198\) 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
\(203\) − 37168.2i − 0.901943i
\(204\) 0 0
\(205\) −85855.1 −2.04295
\(206\) 0 0
\(207\) 23826.5i 0.556057i
\(208\) 0 0
\(209\) 121198. 2.77463
\(210\) 0 0
\(211\) − 27539.9i − 0.618583i −0.950967 0.309292i \(-0.899908\pi\)
0.950967 0.309292i \(-0.100092\pi\)
\(212\) 0 0
\(213\) 11280.6 0.248640
\(214\) 0 0
\(215\) 20239.5i 0.437847i
\(216\) 0 0
\(217\) −18015.6 −0.382585
\(218\) 0 0
\(219\) − 11940.6i − 0.248965i
\(220\) 0 0
\(221\) 78535.7 1.60799
\(222\) 0 0
\(223\) − 3021.35i − 0.0607564i −0.999538 0.0303782i \(-0.990329\pi\)
0.999538 0.0303782i \(-0.00967116\pi\)
\(224\) 0 0
\(225\) −25784.6 −0.509325
\(226\) 0 0
\(227\) 7077.28i 0.137346i 0.997639 + 0.0686728i \(0.0218765\pi\)
−0.997639 + 0.0686728i \(0.978124\pi\)
\(228\) 0 0
\(229\) 102761. 1.95956 0.979778 0.200088i \(-0.0641229\pi\)
0.979778 + 0.200088i \(0.0641229\pi\)
\(230\) 0 0
\(231\) 43371.5i 0.812795i
\(232\) 0 0
\(233\) −22976.6 −0.423227 −0.211613 0.977353i \(-0.567872\pi\)
−0.211613 + 0.977353i \(0.567872\pi\)
\(234\) 0 0
\(235\) 81757.1i 1.48044i
\(236\) 0 0
\(237\) 55626.4 0.990339
\(238\) 0 0
\(239\) − 35566.7i − 0.622656i −0.950303 0.311328i \(-0.899226\pi\)
0.950303 0.311328i \(-0.100774\pi\)
\(240\) 0 0
\(241\) −92683.4 −1.59576 −0.797881 0.602816i \(-0.794045\pi\)
−0.797881 + 0.602816i \(0.794045\pi\)
\(242\) 0 0
\(243\) − 3788.00i − 0.0641500i
\(244\) 0 0
\(245\) 11170.7 0.186101
\(246\) 0 0
\(247\) 122679.i 2.01084i
\(248\) 0 0
\(249\) 15473.2 0.249563
\(250\) 0 0
\(251\) 27590.5i 0.437937i 0.975732 + 0.218968i \(0.0702692\pi\)
−0.975732 + 0.218968i \(0.929731\pi\)
\(252\) 0 0
\(253\) −159976. −2.49927
\(254\) 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
\(259\) − 21501.2i − 0.320526i
\(260\) 0 0
\(261\) −21795.7 −0.319955
\(262\) 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
\(267\) − 33743.4i − 0.473333i
\(268\) 0 0
\(269\) 52698.1 0.728266 0.364133 0.931347i \(-0.381365\pi\)
0.364133 + 0.931347i \(0.381365\pi\)
\(270\) 0 0
\(271\) 47028.8i 0.640361i 0.947356 + 0.320181i \(0.103744\pi\)
−0.947356 + 0.320181i \(0.896256\pi\)
\(272\) 0 0
\(273\) −43901.4 −0.589051
\(274\) 0 0
\(275\) − 173123.i − 2.28923i
\(276\) 0 0
\(277\) −108393. −1.41268 −0.706340 0.707873i \(-0.749655\pi\)
−0.706340 + 0.707873i \(0.749655\pi\)
\(278\) 0 0
\(279\) 10564.4i 0.135718i
\(280\) 0 0
\(281\) 73090.7 0.925656 0.462828 0.886448i \(-0.346835\pi\)
0.462828 + 0.886448i \(0.346835\pi\)
\(282\) 0 0
\(283\) 44763.1i 0.558917i 0.960158 + 0.279458i \(0.0901549\pi\)
−0.960158 + 0.279458i \(0.909845\pi\)
\(284\) 0 0
\(285\) 138085. 1.70003
\(286\) 0 0
\(287\) 99449.9i 1.20737i
\(288\) 0 0
\(289\) 99656.3 1.19319
\(290\) 0 0
\(291\) 42556.1i 0.502546i
\(292\) 0 0
\(293\) −42030.2 −0.489583 −0.244792 0.969576i \(-0.578720\pi\)
−0.244792 + 0.969576i \(0.578720\pi\)
\(294\) 0 0
\(295\) − 52048.6i − 0.598088i
\(296\) 0 0
\(297\) 25433.4 0.288331
\(298\) 0 0
\(299\) − 161930.i − 1.81128i
\(300\) 0 0
\(301\) 23444.3 0.258765
\(302\) 0 0
\(303\) 80937.9i 0.881590i
\(304\) 0 0
\(305\) 31875.3 0.342653
\(306\) 0 0
\(307\) − 8820.31i − 0.0935852i −0.998905 0.0467926i \(-0.985100\pi\)
0.998905 0.0467926i \(-0.0149000\pi\)
\(308\) 0 0
\(309\) 34443.1 0.360732
\(310\) 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 \(-0.130642\pi\)
0.916951 + 0.398999i \(0.130642\pi\)
\(314\) 0 0
\(315\) 49414.6i 0.498005i
\(316\) 0 0
\(317\) 6188.77 0.0615865 0.0307933 0.999526i \(-0.490197\pi\)
0.0307933 + 0.999526i \(0.490197\pi\)
\(318\) 0 0
\(319\) − 146341.i − 1.43808i
\(320\) 0 0
\(321\) −61296.0 −0.594870
\(322\) 0 0
\(323\) 286138.i 2.74265i
\(324\) 0 0
\(325\) 175238. 1.65906
\(326\) 0 0
\(327\) − 69036.4i − 0.645628i
\(328\) 0 0
\(329\) 94703.1 0.874928
\(330\) 0 0
\(331\) − 132159.i − 1.20626i −0.797643 0.603130i \(-0.793920\pi\)
0.797643 0.603130i \(-0.206080\pi\)
\(332\) 0 0
\(333\) −12608.5 −0.113704
\(334\) 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
\(339\) 60865.8i 0.529632i
\(340\) 0 0
\(341\) −70931.8 −0.610004
\(342\) 0 0
\(343\) − 123489.i − 1.04964i
\(344\) 0 0
\(345\) −182266. −1.53132
\(346\) 0 0
\(347\) 157722.i 1.30988i 0.755680 + 0.654942i \(0.227307\pi\)
−0.755680 + 0.654942i \(0.772693\pi\)
\(348\) 0 0
\(349\) −198747. −1.63174 −0.815868 0.578238i \(-0.803741\pi\)
−0.815868 + 0.578238i \(0.803741\pi\)
\(350\) 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
\(355\) 86292.9i 0.684729i
\(356\) 0 0
\(357\) −102396. −0.803426
\(358\) 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
\(363\) 94687.8i 0.718590i
\(364\) 0 0
\(365\) 91342.0 0.685622
\(366\) 0 0
\(367\) 94140.5i 0.698947i 0.936946 + 0.349474i \(0.113640\pi\)
−0.936946 + 0.349474i \(0.886360\pi\)
\(368\) 0 0
\(369\) 58318.1 0.428302
\(370\) 0 0
\(371\) 34703.9i 0.252133i
\(372\) 0 0
\(373\) −66152.4 −0.475475 −0.237738 0.971329i \(-0.576406\pi\)
−0.237738 + 0.971329i \(0.576406\pi\)
\(374\) 0 0
\(375\) − 68155.6i − 0.484662i
\(376\) 0 0
\(377\) 148128. 1.04221
\(378\) 0 0
\(379\) 64395.2i 0.448306i 0.974554 + 0.224153i \(0.0719616\pi\)
−0.974554 + 0.224153i \(0.928038\pi\)
\(380\) 0 0
\(381\) 29309.2 0.201908
\(382\) 0 0
\(383\) 30714.5i 0.209385i 0.994505 + 0.104693i \(0.0333859\pi\)
−0.994505 + 0.104693i \(0.966614\pi\)
\(384\) 0 0
\(385\) −331779. −2.23835
\(386\) 0 0
\(387\) − 13747.9i − 0.0917941i
\(388\) 0 0
\(389\) −199122. −1.31589 −0.657945 0.753066i \(-0.728574\pi\)
−0.657945 + 0.753066i \(0.728574\pi\)
\(390\) 0 0
\(391\) − 377687.i − 2.47046i
\(392\) 0 0
\(393\) −165874. −1.07397
\(394\) 0 0
\(395\) 425525.i 2.72729i
\(396\) 0 0
\(397\) −68565.8 −0.435037 −0.217519 0.976056i \(-0.569796\pi\)
−0.217519 + 0.976056i \(0.569796\pi\)
\(398\) 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
\(403\) − 71798.3i − 0.442084i
\(404\) 0 0
\(405\) 28977.0 0.176662
\(406\) 0 0
\(407\) − 84655.8i − 0.511055i
\(408\) 0 0
\(409\) −230211. −1.37620 −0.688098 0.725618i \(-0.741554\pi\)
−0.688098 + 0.725618i \(0.741554\pi\)
\(410\) 0 0
\(411\) − 54640.6i − 0.323468i
\(412\) 0 0
\(413\) −60290.3 −0.353465
\(414\) 0 0
\(415\) 118365.i 0.687270i
\(416\) 0 0
\(417\) −77868.9 −0.447808
\(418\) 0 0
\(419\) − 17360.5i − 0.0988859i −0.998777 0.0494430i \(-0.984255\pi\)
0.998777 0.0494430i \(-0.0157446\pi\)
\(420\) 0 0
\(421\) 186829. 1.05410 0.527048 0.849836i \(-0.323299\pi\)
0.527048 + 0.849836i \(0.323299\pi\)
\(422\) 0 0
\(423\) − 55534.5i − 0.310372i
\(424\) 0 0
\(425\) 408726. 2.26284
\(426\) 0 0
\(427\) − 36922.6i − 0.202505i
\(428\) 0 0
\(429\) −172851. −0.939197
\(430\) 0 0
\(431\) 153753.i 0.827693i 0.910347 + 0.413846i \(0.135815\pi\)
−0.910347 + 0.413846i \(0.864185\pi\)
\(432\) 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.i − 0.881122i
\(436\) 0 0
\(437\) 589978. 3.08939
\(438\) 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
\(443\) − 17798.2i − 0.0906920i −0.998971 0.0453460i \(-0.985561\pi\)
0.998971 0.0453460i \(-0.0144390\pi\)
\(444\) 0 0
\(445\) 258127. 1.30351
\(446\) 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.i 1.92506i
\(452\) 0 0
\(453\) −175593. −0.855680
\(454\) 0 0
\(455\) − 335832.i − 1.62218i
\(456\) 0 0
\(457\) 190762. 0.913396 0.456698 0.889622i \(-0.349032\pi\)
0.456698 + 0.889622i \(0.349032\pi\)
\(458\) 0 0
\(459\) 60045.6i 0.285007i
\(460\) 0 0
\(461\) −155121. −0.729909 −0.364955 0.931025i \(-0.618916\pi\)
−0.364955 + 0.931025i \(0.618916\pi\)
\(462\) 0 0
\(463\) 230925.i 1.07723i 0.842551 + 0.538616i \(0.181053\pi\)
−0.842551 + 0.538616i \(0.818947\pi\)
\(464\) 0 0
\(465\) −80814.9 −0.373754
\(466\) 0 0
\(467\) 154112.i 0.706647i 0.935501 + 0.353323i \(0.114948\pi\)
−0.935501 + 0.353323i \(0.885052\pi\)
\(468\) 0 0
\(469\) −23289.2 −0.105879
\(470\) 0 0
\(471\) 149592.i 0.674319i
\(472\) 0 0
\(473\) 92306.3 0.412581
\(474\) 0 0
\(475\) 638462.i 2.82975i
\(476\) 0 0
\(477\) 20350.6 0.0894417
\(478\) 0 0
\(479\) 405668.i 1.76807i 0.467420 + 0.884035i \(0.345184\pi\)
−0.467420 + 0.884035i \(0.654816\pi\)
\(480\) 0 0
\(481\) 85690.0 0.370373
\(482\) 0 0
\(483\) 211127.i 0.905000i
\(484\) 0 0
\(485\) −325541. −1.38396
\(486\) 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.i − 1.13292i −0.824090 0.566459i \(-0.808313\pi\)
0.824090 0.566459i \(-0.191687\pi\)
\(492\) 0 0
\(493\) 345495. 1.42151
\(494\) 0 0
\(495\) 194558.i 0.794032i
\(496\) 0 0
\(497\) 99957.1 0.404670
\(498\) 0 0
\(499\) 103548.i 0.415855i 0.978144 + 0.207928i \(0.0666719\pi\)
−0.978144 + 0.207928i \(0.933328\pi\)
\(500\) 0 0
\(501\) 68624.2 0.273402
\(502\) 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
\(507\) − 26555.1i − 0.103307i
\(508\) 0 0
\(509\) 115098. 0.444256 0.222128 0.975018i \(-0.428700\pi\)
0.222128 + 0.975018i \(0.428700\pi\)
\(510\) 0 0
\(511\) − 105806.i − 0.405198i
\(512\) 0 0
\(513\) −93796.1 −0.356410
\(514\) 0 0
\(515\) 263479.i 0.993418i
\(516\) 0 0
\(517\) 372870. 1.39501
\(518\) 0 0
\(519\) 1896.37i 0.00704024i
\(520\) 0 0
\(521\) −288543. −1.06300 −0.531502 0.847057i \(-0.678372\pi\)
−0.531502 + 0.847057i \(0.678372\pi\)
\(522\) 0 0
\(523\) − 176826.i − 0.646461i −0.946320 0.323231i \(-0.895231\pi\)
0.946320 0.323231i \(-0.104769\pi\)
\(524\) 0 0
\(525\) −228477. −0.828942
\(526\) 0 0
\(527\) − 167463.i − 0.602973i
\(528\) 0 0
\(529\) −498900. −1.78280
\(530\) 0 0
\(531\) 35354.6i 0.125388i
\(532\) 0 0
\(533\) −396343. −1.39514
\(534\) 0 0
\(535\) − 468896.i − 1.63821i
\(536\) 0 0
\(537\) −74678.9 −0.258970
\(538\) 0 0
\(539\) − 50946.4i − 0.175362i
\(540\) 0 0
\(541\) 425220. 1.45284 0.726422 0.687249i \(-0.241182\pi\)
0.726422 + 0.687249i \(0.241182\pi\)
\(542\) 0 0
\(543\) 21047.2i 0.0713830i
\(544\) 0 0
\(545\) 528108. 1.77799
\(546\) 0 0
\(547\) 495505.i 1.65605i 0.560692 + 0.828024i \(0.310535\pi\)
−0.560692 + 0.828024i \(0.689465\pi\)
\(548\) 0 0
\(549\) −21651.7 −0.0718367
\(550\) 0 0
\(551\) 539691.i 1.77763i
\(552\) 0 0
\(553\) 492905. 1.61181
\(554\) 0 0
\(555\) − 96451.0i − 0.313127i
\(556\) 0 0
\(557\) −257895. −0.831251 −0.415626 0.909536i \(-0.636437\pi\)
−0.415626 + 0.909536i \(0.636437\pi\)
\(558\) 0 0
\(559\) 93433.9i 0.299007i
\(560\) 0 0
\(561\) −403158. −1.28100
\(562\) 0 0
\(563\) − 417799.i − 1.31811i −0.752096 0.659054i \(-0.770957\pi\)
0.752096 0.659054i \(-0.229043\pi\)
\(564\) 0 0
\(565\) −465606. −1.45855
\(566\) 0 0
\(567\) − 33565.4i − 0.104406i
\(568\) 0 0
\(569\) 100935. 0.311756 0.155878 0.987776i \(-0.450179\pi\)
0.155878 + 0.987776i \(0.450179\pi\)
\(570\) 0 0
\(571\) 453320.i 1.39038i 0.718828 + 0.695188i \(0.244679\pi\)
−0.718828 + 0.695188i \(0.755321\pi\)
\(572\) 0 0
\(573\) −259189. −0.789418
\(574\) 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.i 0.750586i
\(580\) 0 0
\(581\) 137108. 0.406172
\(582\) 0 0
\(583\) 136638.i 0.402008i
\(584\) 0 0
\(585\) −196934. −0.575453
\(586\) 0 0
\(587\) 327468.i 0.950370i 0.879886 + 0.475185i \(0.157619\pi\)
−0.879886 + 0.475185i \(0.842381\pi\)
\(588\) 0 0
\(589\) 261591. 0.754035
\(590\) 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.i − 2.21255i
\(596\) 0 0
\(597\) 314859. 0.883422
\(598\) 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 \(-0.264536\pi\)
0.674091 + 0.738649i \(0.264536\pi\)
\(602\) 0 0
\(603\) 13657.0i 0.0375595i
\(604\) 0 0
\(605\) −724334. −1.97892
\(606\) 0 0
\(607\) 41693.4i 0.113159i 0.998398 + 0.0565796i \(0.0180195\pi\)
−0.998398 + 0.0565796i \(0.981981\pi\)
\(608\) 0 0
\(609\) −193131. −0.520737
\(610\) 0 0
\(611\) 377425.i 1.01099i
\(612\) 0 0
\(613\) −542042. −1.44249 −0.721244 0.692681i \(-0.756429\pi\)
−0.721244 + 0.692681i \(0.756429\pi\)
\(614\) 0 0
\(615\) 446116.i 1.17950i
\(616\) 0 0
\(617\) 146814. 0.385653 0.192827 0.981233i \(-0.438235\pi\)
0.192827 + 0.981233i \(0.438235\pi\)
\(618\) 0 0
\(619\) 72720.6i 0.189791i 0.995487 + 0.0948956i \(0.0302517\pi\)
−0.995487 + 0.0948956i \(0.969748\pi\)
\(620\) 0 0
\(621\) 123806. 0.321040
\(622\) 0 0
\(623\) − 299000.i − 0.770363i
\(624\) 0 0
\(625\) −75495.2 −0.193268
\(626\) 0 0
\(627\) − 629766.i − 1.60193i
\(628\) 0 0
\(629\) 199864. 0.505165
\(630\) 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.i 0.556033i
\(636\) 0 0
\(637\) 51568.7 0.127089
\(638\) 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.i − 0.249514i −0.992187 0.124757i \(-0.960185\pi\)
0.992187 0.124757i \(-0.0398151\pi\)
\(644\) 0 0
\(645\) 105167. 0.252791
\(646\) 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
\(651\) 93611.6i 0.220886i
\(652\) 0 0
\(653\) 85619.0 0.200791 0.100395 0.994948i \(-0.467989\pi\)
0.100395 + 0.994948i \(0.467989\pi\)
\(654\) 0 0
\(655\) − 1.26889e6i − 2.95761i
\(656\) 0 0
\(657\) −62045.1 −0.143740
\(658\) 0 0
\(659\) − 258812.i − 0.595954i −0.954573 0.297977i \(-0.903688\pi\)
0.954573 0.297977i \(-0.0963119\pi\)
\(660\) 0 0
\(661\) −327660. −0.749928 −0.374964 0.927039i \(-0.622345\pi\)
−0.374964 + 0.927039i \(0.622345\pi\)
\(662\) 0 0
\(663\) − 408084.i − 0.928372i
\(664\) 0 0
\(665\) 1.22357e6 2.76686
\(666\) 0 0
\(667\) − 712366.i − 1.60122i
\(668\) 0 0
\(669\) −15699.4 −0.0350777
\(670\) 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.i 0.294059i
\(676\) 0 0
\(677\) 573451. 1.25118 0.625589 0.780153i \(-0.284859\pi\)
0.625589 + 0.780153i \(0.284859\pi\)
\(678\) 0 0
\(679\) 377089.i 0.817909i
\(680\) 0 0
\(681\) 36774.6 0.0792965
\(682\) 0 0
\(683\) − 175872.i − 0.377012i −0.982072 0.188506i \(-0.939636\pi\)
0.982072 0.188506i \(-0.0603644\pi\)
\(684\) 0 0
\(685\) 417984. 0.890797
\(686\) 0 0
\(687\) − 533962.i − 1.13135i
\(688\) 0 0
\(689\) −138307. −0.291344
\(690\) 0 0
\(691\) 7216.66i 0.0151140i 0.999971 + 0.00755702i \(0.00240550\pi\)
−0.999971 + 0.00755702i \(0.997595\pi\)
\(692\) 0 0
\(693\) 225365. 0.469267
\(694\) 0 0
\(695\) − 595673.i − 1.23321i
\(696\) 0 0
\(697\) −924433. −1.90287
\(698\) 0 0
\(699\) 119390.i 0.244350i
\(700\) 0 0
\(701\) 502543. 1.02267 0.511337 0.859380i \(-0.329150\pi\)
0.511337 + 0.859380i \(0.329150\pi\)
\(702\) 0 0
\(703\) 312203.i 0.631723i
\(704\) 0 0
\(705\) 424823. 0.854731
\(706\) 0 0
\(707\) 717191.i 1.43481i
\(708\) 0 0
\(709\) −95591.1 −0.190163 −0.0950813 0.995470i \(-0.530311\pi\)
−0.0950813 + 0.995470i \(0.530311\pi\)
\(710\) 0 0
\(711\) − 289043.i − 0.571772i
\(712\) 0 0
\(713\) −345286. −0.679204
\(714\) 0 0
\(715\) − 1.32226e6i − 2.58645i
\(716\) 0 0
\(717\) −184810. −0.359491
\(718\) 0 0
\(719\) 475789.i 0.920357i 0.887826 + 0.460178i \(0.152215\pi\)
−0.887826 + 0.460178i \(0.847785\pi\)
\(720\) 0 0
\(721\) 305200. 0.587103
\(722\) 0 0
\(723\) 481597.i 0.921313i
\(724\) 0 0
\(725\) 770908. 1.46665
\(726\) 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.i 0.407825i
\(732\) 0 0
\(733\) −650799. −1.21126 −0.605632 0.795745i \(-0.707080\pi\)
−0.605632 + 0.795745i \(0.707080\pi\)
\(734\) 0 0
\(735\) − 58044.8i − 0.107446i
\(736\) 0 0
\(737\) −91695.6 −0.168816
\(738\) 0 0
\(739\) − 99472.6i − 0.182144i −0.995844 0.0910719i \(-0.970971\pi\)
0.995844 0.0910719i \(-0.0290293\pi\)
\(740\) 0 0
\(741\) 637459. 1.16096
\(742\) 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
\(747\) − 80401.0i − 0.144085i
\(748\) 0 0
\(749\) −543144. −0.968170
\(750\) 0 0
\(751\) − 603532.i − 1.07009i −0.844824 0.535045i \(-0.820295\pi\)
0.844824 0.535045i \(-0.179705\pi\)
\(752\) 0 0
\(753\) 143364. 0.252843
\(754\) 0 0
\(755\) − 1.34324e6i − 2.35645i
\(756\) 0 0
\(757\) −784192. −1.36846 −0.684228 0.729268i \(-0.739861\pi\)
−0.684228 + 0.729268i \(0.739861\pi\)
\(758\) 0 0
\(759\) 831259.i 1.44296i
\(760\) 0 0
\(761\) 568076. 0.980927 0.490464 0.871462i \(-0.336827\pi\)
0.490464 + 0.871462i \(0.336827\pi\)
\(762\) 0 0
\(763\) − 611731.i − 1.05078i
\(764\) 0 0
\(765\) −459331. −0.784880
\(766\) 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.i − 0.806856i
\(772\) 0 0
\(773\) 260161. 0.435394 0.217697 0.976016i \(-0.430145\pi\)
0.217697 + 0.976016i \(0.430145\pi\)
\(774\) 0 0
\(775\) − 373662.i − 0.622122i
\(776\) 0 0
\(777\) −111724. −0.185056
\(778\) 0 0
\(779\) − 1.44404e6i − 2.37960i
\(780\) 0 0
\(781\) 393556. 0.645216
\(782\) 0 0
\(783\) 113254.i 0.184726i
\(784\) 0 0
\(785\) −1.14433e6 −1.85700
\(786\) 0 0
\(787\) − 724342.i − 1.16948i −0.811219 0.584742i \(-0.801196\pi\)
0.811219 0.584742i \(-0.198804\pi\)
\(788\) 0 0
\(789\) 72145.7 0.115893
\(790\) 0 0
\(791\) 539332.i 0.861993i
\(792\) 0 0
\(793\) 147150. 0.233998
\(794\) 0 0
\(795\) 155676.i 0.246313i
\(796\) 0 0
\(797\) 1.04249e6 1.64117 0.820586 0.571524i \(-0.193647\pi\)
0.820586 + 0.571524i \(0.193647\pi\)
\(798\) 0 0
\(799\) 880309.i 1.37893i
\(800\) 0 0
\(801\) −175336. −0.273279
\(802\) 0 0
\(803\) − 416584.i − 0.646057i
\(804\) 0 0
\(805\) −1.61506e6 −2.49227
\(806\) 0 0
\(807\) − 273827.i − 0.420465i
\(808\) 0 0
\(809\) 647222. 0.988909 0.494455 0.869204i \(-0.335368\pi\)
0.494455 + 0.869204i \(0.335368\pi\)
\(810\) 0 0
\(811\) 1.24087e6i 1.88662i 0.331916 + 0.943309i \(0.392305\pi\)
−0.331916 + 0.943309i \(0.607695\pi\)
\(812\) 0 0
\(813\) 244369. 0.369713
\(814\) 0 0
\(815\) 532993.i 0.802428i
\(816\) 0 0
\(817\) −340418. −0.509997
\(818\) 0 0
\(819\) 228118.i 0.340089i
\(820\) 0 0
\(821\) −819276. −1.21547 −0.607734 0.794141i \(-0.707921\pi\)
−0.607734 + 0.794141i \(0.707921\pi\)
\(822\) 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.i − 0.205685i −0.994698 0.102842i \(-0.967206\pi\)
0.994698 0.102842i \(-0.0327938\pi\)
\(828\) 0 0
\(829\) 137443. 0.199993 0.0999964 0.994988i \(-0.468117\pi\)
0.0999964 + 0.994988i \(0.468117\pi\)
\(830\) 0 0
\(831\) 563229.i 0.815611i
\(832\) 0 0
\(833\) 120279. 0.173341
\(834\) 0 0
\(835\) 524955.i 0.752920i
\(836\) 0 0
\(837\) 54894.5 0.0783570
\(838\) 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.i − 0.534428i
\(844\) 0 0
\(845\) 203138. 0.284498
\(846\) 0 0
\(847\) 839029.i 1.16953i
\(848\) 0 0
\(849\) 232596. 0.322691
\(850\) 0 0
\(851\) − 412093.i − 0.569031i
\(852\) 0 0
\(853\) 169773. 0.233331 0.116665 0.993171i \(-0.462780\pi\)
0.116665 + 0.993171i \(0.462780\pi\)
\(854\) 0 0
\(855\) − 717512.i − 0.981515i
\(856\) 0 0
\(857\) −1.05083e6 −1.43077 −0.715384 0.698732i \(-0.753748\pi\)
−0.715384 + 0.698732i \(0.753748\pi\)
\(858\) 0 0
\(859\) 1.05896e6i 1.43514i 0.696487 + 0.717569i \(0.254745\pi\)
−0.696487 + 0.717569i \(0.745255\pi\)
\(860\) 0 0
\(861\) 516757. 0.697075
\(862\) 0 0
\(863\) 1.22011e6i 1.63824i 0.573621 + 0.819121i \(0.305538\pi\)
−0.573621 + 0.819121i \(0.694462\pi\)
\(864\) 0 0
\(865\) −14506.6 −0.0193881
\(866\) 0 0
\(867\) − 517829.i − 0.688887i
\(868\) 0 0
\(869\) 1.94069e6 2.56991
\(870\) 0 0
\(871\) − 92815.8i − 0.122345i
\(872\) 0 0
\(873\) 221128. 0.290145
\(874\) 0 0
\(875\) − 603927.i − 0.788802i
\(876\) 0 0
\(877\) −707060. −0.919299 −0.459650 0.888100i \(-0.652025\pi\)
−0.459650 + 0.888100i \(0.652025\pi\)
\(878\) 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
\(883\) − 1.08382e6i − 1.39007i −0.718974 0.695037i \(-0.755388\pi\)
0.718974 0.695037i \(-0.244612\pi\)
\(884\) 0 0
\(885\) −270452. −0.345306
\(886\) 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.i − 0.166468i
\(892\) 0 0
\(893\) −1.37511e6 −1.72439
\(894\) 0 0
\(895\) − 571272.i − 0.713176i
\(896\) 0 0
\(897\) −841414. −1.04574
\(898\) 0 0
\(899\) − 315856.i − 0.390814i
\(900\) 0 0
\(901\) −322589. −0.397374
\(902\) 0 0
\(903\) − 121820.i − 0.149398i
\(904\) 0 0
\(905\) −161005. −0.196581
\(906\) 0 0
\(907\) − 155091.i − 0.188526i −0.995547 0.0942629i \(-0.969951\pi\)
0.995547 0.0942629i \(-0.0300494\pi\)
\(908\) 0 0
\(909\) 420566. 0.508986
\(910\) 0 0
\(911\) 670134.i 0.807467i 0.914877 + 0.403734i \(0.132288\pi\)
−0.914877 + 0.403734i \(0.867712\pi\)
\(912\) 0 0
\(913\) 539828. 0.647611
\(914\) 0 0
\(915\) − 165629.i − 0.197831i
\(916\) 0 0
\(917\) −1.46981e6 −1.74793
\(918\) 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.i 0.467602i
\(924\) 0 0
\(925\) 445959. 0.521208
\(926\) 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.i 0.216768i
\(932\) 0 0
\(933\) 805708. 0.925580
\(934\) 0 0
\(935\) − 3.08404e6i − 3.52774i
\(936\) 0 0
\(937\) 1.18305e6 1.34748 0.673741 0.738967i \(-0.264686\pi\)
0.673741 + 0.738967i \(0.264686\pi\)
\(938\) 0 0
\(939\) − 933570.i − 1.05880i
\(940\) 0 0
\(941\) −159241. −0.179836 −0.0899179 0.995949i \(-0.528660\pi\)
−0.0899179 + 0.995949i \(0.528660\pi\)
\(942\) 0 0
\(943\) 1.90606e6i 2.14345i
\(944\) 0 0
\(945\) 256766. 0.287523
\(946\) 0 0
\(947\) − 940316.i − 1.04851i −0.851560 0.524257i \(-0.824343\pi\)
0.851560 0.524257i \(-0.175657\pi\)
\(948\) 0 0
\(949\) 421673. 0.468213
\(950\) 0 0
\(951\) − 32157.8i − 0.0355570i
\(952\) 0 0
\(953\) −577950. −0.636362 −0.318181 0.948030i \(-0.603072\pi\)
−0.318181 + 0.948030i \(0.603072\pi\)
\(954\) 0 0
\(955\) − 1.98272e6i − 2.17397i
\(956\) 0 0
\(957\) −760408. −0.830276
\(958\) 0 0
\(959\) − 484170.i − 0.526454i
\(960\) 0 0
\(961\) 770424. 0.834225
\(962\) 0 0
\(963\) 318504.i 0.343449i
\(964\) 0 0
\(965\) −1.92487e6 −2.06703
\(966\) 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
\(971\) − 1.05236e6i − 1.11616i −0.829786 0.558082i \(-0.811538\pi\)
0.829786 0.558082i \(-0.188462\pi\)
\(972\) 0 0
\(973\) −689996. −0.728821
\(974\) 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
\(979\) − 1.17724e6i − 1.22829i
\(980\) 0 0
\(981\) −358723. −0.372754
\(982\) 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.i − 0.505140i
\(988\) 0 0
\(989\) 449334. 0.459385
\(990\) 0 0
\(991\) − 787866.i − 0.802241i −0.916025 0.401120i \(-0.868621\pi\)
0.916025 0.401120i \(-0.131379\pi\)
\(992\) 0 0
\(993\) −686718. −0.696434
\(994\) 0 0
\(995\) 2.40858e6i 2.43285i
\(996\) 0 0
\(997\) 453709. 0.456444 0.228222 0.973609i \(-0.426709\pi\)
0.228222 + 0.973609i \(0.426709\pi\)
\(998\) 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 768.5.g.g.511.2 4
4.3 odd 2 inner 768.5.g.g.511.4 4
8.3 odd 2 768.5.g.c.511.1 4
8.5 even 2 768.5.g.c.511.3 4
16.3 odd 4 384.5.b.c.319.1 8
16.5 even 4 384.5.b.c.319.4 yes 8
16.11 odd 4 384.5.b.c.319.8 yes 8
16.13 even 4 384.5.b.c.319.5 yes 8
48.5 odd 4 1152.5.b.k.703.2 8
48.11 even 4 1152.5.b.k.703.1 8
48.29 odd 4 1152.5.b.k.703.8 8
48.35 even 4 1152.5.b.k.703.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.1 8 16.3 odd 4
384.5.b.c.319.4 yes 8 16.5 even 4
384.5.b.c.319.5 yes 8 16.13 even 4
384.5.b.c.319.8 yes 8 16.11 odd 4
768.5.g.c.511.1 4 8.3 odd 2
768.5.g.c.511.3 4 8.5 even 2
768.5.g.g.511.2 4 1.1 even 1 trivial
768.5.g.g.511.4 4 4.3 odd 2 inner
1152.5.b.k.703.1 8 48.11 even 4
1152.5.b.k.703.2 8 48.5 odd 4
1152.5.b.k.703.7 8 48.35 even 4
1152.5.b.k.703.8 8 48.29 odd 4