Properties

Label 384.3.b.c.319.3
Level $384$
Weight $3$
Character 384.319
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.3
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.3.b.c.319.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} +0.898979i q^{5} -2.82843i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +0.898979i q^{5} -2.82843i q^{7} +3.00000 q^{9} -4.38551 q^{11} +13.7980i q^{13} -1.55708i q^{15} +17.5959 q^{17} +4.38551 q^{19} +4.89898i q^{21} -22.0560i q^{23} +24.1918 q^{25} -5.19615 q^{27} +44.4949i q^{29} +53.1687i q^{31} +7.59592 q^{33} +2.54270 q^{35} +35.1918i q^{37} -23.8988i q^{39} +37.5959 q^{41} +49.6403 q^{43} +2.69694i q^{45} +38.4551i q^{47} +41.0000 q^{49} -30.4770 q^{51} -1.70714i q^{53} -3.94248i q^{55} -7.59592 q^{57} -34.6410 q^{59} -24.4041i q^{61} -8.48528i q^{63} -12.4041 q^{65} +93.7523 q^{67} +38.2020i q^{69} -123.879i q^{71} -10.0000 q^{73} -41.9015 q^{75} +12.4041i q^{77} +131.222i q^{79} +9.00000 q^{81} -110.151 q^{83} +15.8184i q^{85} -77.0674i q^{87} -73.1918 q^{89} +39.0265 q^{91} -92.0908i q^{93} +3.94248i q^{95} -105.192 q^{97} -13.1565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} + O(q^{10}) \) \( 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97} + O(q^{100}) \)

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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 0.898979i 0.179796i 0.995951 + 0.0898979i \(0.0286541\pi\)
−0.995951 + 0.0898979i \(0.971346\pi\)
\(6\) 0 0
\(7\) − 2.82843i − 0.404061i −0.979379 0.202031i \(-0.935246\pi\)
0.979379 0.202031i \(-0.0647540\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −4.38551 −0.398682 −0.199341 0.979930i \(-0.563880\pi\)
−0.199341 + 0.979930i \(0.563880\pi\)
\(12\) 0 0
\(13\) 13.7980i 1.06138i 0.847566 + 0.530691i \(0.178067\pi\)
−0.847566 + 0.530691i \(0.821933\pi\)
\(14\) 0 0
\(15\) − 1.55708i − 0.103805i
\(16\) 0 0
\(17\) 17.5959 1.03505 0.517527 0.855667i \(-0.326853\pi\)
0.517527 + 0.855667i \(0.326853\pi\)
\(18\) 0 0
\(19\) 4.38551 0.230816 0.115408 0.993318i \(-0.463182\pi\)
0.115408 + 0.993318i \(0.463182\pi\)
\(20\) 0 0
\(21\) 4.89898i 0.233285i
\(22\) 0 0
\(23\) − 22.0560i − 0.958955i −0.877554 0.479477i \(-0.840826\pi\)
0.877554 0.479477i \(-0.159174\pi\)
\(24\) 0 0
\(25\) 24.1918 0.967673
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 44.4949i 1.53431i 0.641464 + 0.767153i \(0.278328\pi\)
−0.641464 + 0.767153i \(0.721672\pi\)
\(30\) 0 0
\(31\) 53.1687i 1.71512i 0.514386 + 0.857559i \(0.328020\pi\)
−0.514386 + 0.857559i \(0.671980\pi\)
\(32\) 0 0
\(33\) 7.59592 0.230179
\(34\) 0 0
\(35\) 2.54270 0.0726485
\(36\) 0 0
\(37\) 35.1918i 0.951131i 0.879680 + 0.475565i \(0.157756\pi\)
−0.879680 + 0.475565i \(0.842244\pi\)
\(38\) 0 0
\(39\) − 23.8988i − 0.612789i
\(40\) 0 0
\(41\) 37.5959 0.916974 0.458487 0.888701i \(-0.348392\pi\)
0.458487 + 0.888701i \(0.348392\pi\)
\(42\) 0 0
\(43\) 49.6403 1.15443 0.577213 0.816593i \(-0.304140\pi\)
0.577213 + 0.816593i \(0.304140\pi\)
\(44\) 0 0
\(45\) 2.69694i 0.0599320i
\(46\) 0 0
\(47\) 38.4551i 0.818193i 0.912491 + 0.409096i \(0.134156\pi\)
−0.912491 + 0.409096i \(0.865844\pi\)
\(48\) 0 0
\(49\) 41.0000 0.836735
\(50\) 0 0
\(51\) −30.4770 −0.597589
\(52\) 0 0
\(53\) − 1.70714i − 0.0322103i −0.999870 0.0161051i \(-0.994873\pi\)
0.999870 0.0161051i \(-0.00512664\pi\)
\(54\) 0 0
\(55\) − 3.94248i − 0.0716814i
\(56\) 0 0
\(57\) −7.59592 −0.133262
\(58\) 0 0
\(59\) −34.6410 −0.587136 −0.293568 0.955938i \(-0.594843\pi\)
−0.293568 + 0.955938i \(0.594843\pi\)
\(60\) 0 0
\(61\) − 24.4041i − 0.400067i −0.979789 0.200033i \(-0.935895\pi\)
0.979789 0.200033i \(-0.0641051\pi\)
\(62\) 0 0
\(63\) − 8.48528i − 0.134687i
\(64\) 0 0
\(65\) −12.4041 −0.190832
\(66\) 0 0
\(67\) 93.7523 1.39929 0.699644 0.714492i \(-0.253342\pi\)
0.699644 + 0.714492i \(0.253342\pi\)
\(68\) 0 0
\(69\) 38.2020i 0.553653i
\(70\) 0 0
\(71\) − 123.879i − 1.74478i −0.488811 0.872390i \(-0.662569\pi\)
0.488811 0.872390i \(-0.337431\pi\)
\(72\) 0 0
\(73\) −10.0000 −0.136986 −0.0684932 0.997652i \(-0.521819\pi\)
−0.0684932 + 0.997652i \(0.521819\pi\)
\(74\) 0 0
\(75\) −41.9015 −0.558687
\(76\) 0 0
\(77\) 12.4041i 0.161092i
\(78\) 0 0
\(79\) 131.222i 1.66103i 0.556993 + 0.830517i \(0.311955\pi\)
−0.556993 + 0.830517i \(0.688045\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −110.151 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(84\) 0 0
\(85\) 15.8184i 0.186098i
\(86\) 0 0
\(87\) − 77.0674i − 0.885832i
\(88\) 0 0
\(89\) −73.1918 −0.822380 −0.411190 0.911550i \(-0.634887\pi\)
−0.411190 + 0.911550i \(0.634887\pi\)
\(90\) 0 0
\(91\) 39.0265 0.428863
\(92\) 0 0
\(93\) − 92.0908i − 0.990224i
\(94\) 0 0
\(95\) 3.94248i 0.0414998i
\(96\) 0 0
\(97\) −105.192 −1.08445 −0.542226 0.840233i \(-0.682418\pi\)
−0.542226 + 0.840233i \(0.682418\pi\)
\(98\) 0 0
\(99\) −13.1565 −0.132894
\(100\) 0 0
\(101\) − 154.879i − 1.53345i −0.641975 0.766726i \(-0.721885\pi\)
0.641975 0.766726i \(-0.278115\pi\)
\(102\) 0 0
\(103\) 37.9125i 0.368082i 0.982919 + 0.184041i \(0.0589180\pi\)
−0.982919 + 0.184041i \(0.941082\pi\)
\(104\) 0 0
\(105\) −4.40408 −0.0419436
\(106\) 0 0
\(107\) −19.3848 −0.181167 −0.0905833 0.995889i \(-0.528873\pi\)
−0.0905833 + 0.995889i \(0.528873\pi\)
\(108\) 0 0
\(109\) − 132.182i − 1.21268i −0.795207 0.606338i \(-0.792638\pi\)
0.795207 0.606338i \(-0.207362\pi\)
\(110\) 0 0
\(111\) − 60.9540i − 0.549136i
\(112\) 0 0
\(113\) −35.5755 −0.314828 −0.157414 0.987533i \(-0.550316\pi\)
−0.157414 + 0.987533i \(0.550316\pi\)
\(114\) 0 0
\(115\) 19.8279 0.172416
\(116\) 0 0
\(117\) 41.3939i 0.353794i
\(118\) 0 0
\(119\) − 49.7688i − 0.418225i
\(120\) 0 0
\(121\) −101.767 −0.841052
\(122\) 0 0
\(123\) −65.1180 −0.529415
\(124\) 0 0
\(125\) 44.2225i 0.353780i
\(126\) 0 0
\(127\) 127.851i 1.00670i 0.864083 + 0.503349i \(0.167899\pi\)
−0.864083 + 0.503349i \(0.832101\pi\)
\(128\) 0 0
\(129\) −85.9796 −0.666508
\(130\) 0 0
\(131\) 86.3810 0.659397 0.329699 0.944086i \(-0.393053\pi\)
0.329699 + 0.944086i \(0.393053\pi\)
\(132\) 0 0
\(133\) − 12.4041i − 0.0932638i
\(134\) 0 0
\(135\) − 4.67123i − 0.0346017i
\(136\) 0 0
\(137\) −47.9796 −0.350216 −0.175108 0.984549i \(-0.556027\pi\)
−0.175108 + 0.984549i \(0.556027\pi\)
\(138\) 0 0
\(139\) 156.549 1.12625 0.563126 0.826371i \(-0.309598\pi\)
0.563126 + 0.826371i \(0.309598\pi\)
\(140\) 0 0
\(141\) − 66.6061i − 0.472384i
\(142\) 0 0
\(143\) − 60.5110i − 0.423154i
\(144\) 0 0
\(145\) −40.0000 −0.275862
\(146\) 0 0
\(147\) −71.0141 −0.483089
\(148\) 0 0
\(149\) − 258.697i − 1.73622i −0.496371 0.868111i \(-0.665334\pi\)
0.496371 0.868111i \(-0.334666\pi\)
\(150\) 0 0
\(151\) 106.880i 0.707814i 0.935281 + 0.353907i \(0.115147\pi\)
−0.935281 + 0.353907i \(0.884853\pi\)
\(152\) 0 0
\(153\) 52.7878 0.345018
\(154\) 0 0
\(155\) −47.7975 −0.308371
\(156\) 0 0
\(157\) − 115.192i − 0.733706i −0.930279 0.366853i \(-0.880435\pi\)
0.930279 0.366853i \(-0.119565\pi\)
\(158\) 0 0
\(159\) 2.95686i 0.0185966i
\(160\) 0 0
\(161\) −62.3837 −0.387476
\(162\) 0 0
\(163\) 248.715 1.52586 0.762931 0.646480i \(-0.223760\pi\)
0.762931 + 0.646480i \(0.223760\pi\)
\(164\) 0 0
\(165\) 6.82857i 0.0413853i
\(166\) 0 0
\(167\) 119.365i 0.714763i 0.933959 + 0.357381i \(0.116330\pi\)
−0.933959 + 0.357381i \(0.883670\pi\)
\(168\) 0 0
\(169\) −21.3837 −0.126531
\(170\) 0 0
\(171\) 13.1565 0.0769387
\(172\) 0 0
\(173\) 265.889i 1.53693i 0.639892 + 0.768465i \(0.278979\pi\)
−0.639892 + 0.768465i \(0.721021\pi\)
\(174\) 0 0
\(175\) − 68.4248i − 0.390999i
\(176\) 0 0
\(177\) 60.0000 0.338983
\(178\) 0 0
\(179\) −225.831 −1.26163 −0.630813 0.775935i \(-0.717279\pi\)
−0.630813 + 0.775935i \(0.717279\pi\)
\(180\) 0 0
\(181\) 48.2225i 0.266422i 0.991088 + 0.133211i \(0.0425289\pi\)
−0.991088 + 0.133211i \(0.957471\pi\)
\(182\) 0 0
\(183\) 42.2691i 0.230979i
\(184\) 0 0
\(185\) −31.6367 −0.171009
\(186\) 0 0
\(187\) −77.1670 −0.412658
\(188\) 0 0
\(189\) 14.6969i 0.0777616i
\(190\) 0 0
\(191\) − 177.591i − 0.929794i −0.885365 0.464897i \(-0.846091\pi\)
0.885365 0.464897i \(-0.153909\pi\)
\(192\) 0 0
\(193\) 130.767 0.677551 0.338776 0.940867i \(-0.389987\pi\)
0.338776 + 0.940867i \(0.389987\pi\)
\(194\) 0 0
\(195\) 21.4845 0.110177
\(196\) 0 0
\(197\) − 149.303i − 0.757884i −0.925421 0.378942i \(-0.876288\pi\)
0.925421 0.378942i \(-0.123712\pi\)
\(198\) 0 0
\(199\) 148.764i 0.747556i 0.927518 + 0.373778i \(0.121938\pi\)
−0.927518 + 0.373778i \(0.878062\pi\)
\(200\) 0 0
\(201\) −162.384 −0.807879
\(202\) 0 0
\(203\) 125.851 0.619954
\(204\) 0 0
\(205\) 33.7980i 0.164868i
\(206\) 0 0
\(207\) − 66.1679i − 0.319652i
\(208\) 0 0
\(209\) −19.2327 −0.0920223
\(210\) 0 0
\(211\) −381.937 −1.81013 −0.905065 0.425274i \(-0.860178\pi\)
−0.905065 + 0.425274i \(0.860178\pi\)
\(212\) 0 0
\(213\) 214.565i 1.00735i
\(214\) 0 0
\(215\) 44.6256i 0.207561i
\(216\) 0 0
\(217\) 150.384 0.693012
\(218\) 0 0
\(219\) 17.3205 0.0790891
\(220\) 0 0
\(221\) 242.788i 1.09859i
\(222\) 0 0
\(223\) − 268.129i − 1.20237i −0.799109 0.601186i \(-0.794695\pi\)
0.799109 0.601186i \(-0.205305\pi\)
\(224\) 0 0
\(225\) 72.5755 0.322558
\(226\) 0 0
\(227\) 353.082 1.55543 0.777713 0.628620i \(-0.216380\pi\)
0.777713 + 0.628620i \(0.216380\pi\)
\(228\) 0 0
\(229\) 212.545i 0.928144i 0.885798 + 0.464072i \(0.153612\pi\)
−0.885798 + 0.464072i \(0.846388\pi\)
\(230\) 0 0
\(231\) − 21.4845i − 0.0930065i
\(232\) 0 0
\(233\) 355.576 1.52608 0.763038 0.646354i \(-0.223707\pi\)
0.763038 + 0.646354i \(0.223707\pi\)
\(234\) 0 0
\(235\) −34.5703 −0.147108
\(236\) 0 0
\(237\) − 227.283i − 0.958999i
\(238\) 0 0
\(239\) − 42.9690i − 0.179787i −0.995951 0.0898933i \(-0.971347\pi\)
0.995951 0.0898933i \(-0.0286526\pi\)
\(240\) 0 0
\(241\) 125.192 0.519468 0.259734 0.965680i \(-0.416365\pi\)
0.259734 + 0.965680i \(0.416365\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 36.8582i 0.150441i
\(246\) 0 0
\(247\) 60.5110i 0.244984i
\(248\) 0 0
\(249\) 190.788 0.766216
\(250\) 0 0
\(251\) −334.140 −1.33123 −0.665617 0.746294i \(-0.731832\pi\)
−0.665617 + 0.746294i \(0.731832\pi\)
\(252\) 0 0
\(253\) 96.7265i 0.382318i
\(254\) 0 0
\(255\) − 27.3982i − 0.107444i
\(256\) 0 0
\(257\) 290.000 1.12840 0.564202 0.825637i \(-0.309184\pi\)
0.564202 + 0.825637i \(0.309184\pi\)
\(258\) 0 0
\(259\) 99.5375 0.384315
\(260\) 0 0
\(261\) 133.485i 0.511436i
\(262\) 0 0
\(263\) 249.415i 0.948347i 0.880431 + 0.474174i \(0.157253\pi\)
−0.880431 + 0.474174i \(0.842747\pi\)
\(264\) 0 0
\(265\) 1.53469 0.00579127
\(266\) 0 0
\(267\) 126.772 0.474801
\(268\) 0 0
\(269\) 300.858i 1.11843i 0.829022 + 0.559216i \(0.188898\pi\)
−0.829022 + 0.559216i \(0.811102\pi\)
\(270\) 0 0
\(271\) − 386.865i − 1.42755i −0.700376 0.713774i \(-0.746985\pi\)
0.700376 0.713774i \(-0.253015\pi\)
\(272\) 0 0
\(273\) −67.5959 −0.247604
\(274\) 0 0
\(275\) −106.093 −0.385794
\(276\) 0 0
\(277\) 213.798i 0.771834i 0.922534 + 0.385917i \(0.126115\pi\)
−0.922534 + 0.385917i \(0.873885\pi\)
\(278\) 0 0
\(279\) 159.506i 0.571706i
\(280\) 0 0
\(281\) −110.767 −0.394190 −0.197095 0.980384i \(-0.563151\pi\)
−0.197095 + 0.980384i \(0.563151\pi\)
\(282\) 0 0
\(283\) −114.980 −0.406289 −0.203145 0.979149i \(-0.565116\pi\)
−0.203145 + 0.979149i \(0.565116\pi\)
\(284\) 0 0
\(285\) − 6.82857i − 0.0239599i
\(286\) 0 0
\(287\) − 106.337i − 0.370513i
\(288\) 0 0
\(289\) 20.6163 0.0713368
\(290\) 0 0
\(291\) 182.198 0.626109
\(292\) 0 0
\(293\) 108.677i 0.370910i 0.982653 + 0.185455i \(0.0593758\pi\)
−0.982653 + 0.185455i \(0.940624\pi\)
\(294\) 0 0
\(295\) − 31.1416i − 0.105565i
\(296\) 0 0
\(297\) 22.7878 0.0767264
\(298\) 0 0
\(299\) 304.327 1.01782
\(300\) 0 0
\(301\) − 140.404i − 0.466459i
\(302\) 0 0
\(303\) 268.258i 0.885338i
\(304\) 0 0
\(305\) 21.9388 0.0719304
\(306\) 0 0
\(307\) −253.544 −0.825876 −0.412938 0.910759i \(-0.635497\pi\)
−0.412938 + 0.910759i \(0.635497\pi\)
\(308\) 0 0
\(309\) − 65.6663i − 0.212512i
\(310\) 0 0
\(311\) 456.491i 1.46782i 0.679249 + 0.733908i \(0.262306\pi\)
−0.679249 + 0.733908i \(0.737694\pi\)
\(312\) 0 0
\(313\) 371.535 1.18701 0.593506 0.804830i \(-0.297743\pi\)
0.593506 + 0.804830i \(0.297743\pi\)
\(314\) 0 0
\(315\) 7.62809 0.0242162
\(316\) 0 0
\(317\) − 53.4847i − 0.168721i −0.996435 0.0843607i \(-0.973115\pi\)
0.996435 0.0843607i \(-0.0268848\pi\)
\(318\) 0 0
\(319\) − 195.133i − 0.611701i
\(320\) 0 0
\(321\) 33.5755 0.104597
\(322\) 0 0
\(323\) 77.1670 0.238907
\(324\) 0 0
\(325\) 333.798i 1.02707i
\(326\) 0 0
\(327\) 228.945i 0.700139i
\(328\) 0 0
\(329\) 108.767 0.330600
\(330\) 0 0
\(331\) −165.320 −0.499457 −0.249728 0.968316i \(-0.580341\pi\)
−0.249728 + 0.968316i \(0.580341\pi\)
\(332\) 0 0
\(333\) 105.576i 0.317044i
\(334\) 0 0
\(335\) 84.2814i 0.251586i
\(336\) 0 0
\(337\) −210.767 −0.625422 −0.312711 0.949848i \(-0.601237\pi\)
−0.312711 + 0.949848i \(0.601237\pi\)
\(338\) 0 0
\(339\) 61.6186 0.181766
\(340\) 0 0
\(341\) − 233.171i − 0.683787i
\(342\) 0 0
\(343\) − 254.558i − 0.742153i
\(344\) 0 0
\(345\) −34.3429 −0.0995445
\(346\) 0 0
\(347\) 458.334 1.32085 0.660423 0.750894i \(-0.270377\pi\)
0.660423 + 0.750894i \(0.270377\pi\)
\(348\) 0 0
\(349\) − 33.9388i − 0.0972458i −0.998817 0.0486229i \(-0.984517\pi\)
0.998817 0.0486229i \(-0.0154832\pi\)
\(350\) 0 0
\(351\) − 71.6963i − 0.204263i
\(352\) 0 0
\(353\) 306.727 0.868914 0.434457 0.900693i \(-0.356940\pi\)
0.434457 + 0.900693i \(0.356940\pi\)
\(354\) 0 0
\(355\) 111.365 0.313704
\(356\) 0 0
\(357\) 86.2020i 0.241462i
\(358\) 0 0
\(359\) − 166.848i − 0.464759i −0.972625 0.232379i \(-0.925349\pi\)
0.972625 0.232379i \(-0.0746511\pi\)
\(360\) 0 0
\(361\) −341.767 −0.946724
\(362\) 0 0
\(363\) 176.266 0.485582
\(364\) 0 0
\(365\) − 8.98979i − 0.0246296i
\(366\) 0 0
\(367\) 15.7988i 0.0430485i 0.999768 + 0.0215242i \(0.00685190\pi\)
−0.999768 + 0.0215242i \(0.993148\pi\)
\(368\) 0 0
\(369\) 112.788 0.305658
\(370\) 0 0
\(371\) −4.82853 −0.0130149
\(372\) 0 0
\(373\) − 82.0204i − 0.219894i −0.993937 0.109947i \(-0.964932\pi\)
0.993937 0.109947i \(-0.0350681\pi\)
\(374\) 0 0
\(375\) − 76.5955i − 0.204255i
\(376\) 0 0
\(377\) −613.939 −1.62848
\(378\) 0 0
\(379\) −524.444 −1.38376 −0.691878 0.722014i \(-0.743217\pi\)
−0.691878 + 0.722014i \(0.743217\pi\)
\(380\) 0 0
\(381\) − 221.444i − 0.581218i
\(382\) 0 0
\(383\) 498.946i 1.30273i 0.758764 + 0.651366i \(0.225804\pi\)
−0.758764 + 0.651366i \(0.774196\pi\)
\(384\) 0 0
\(385\) −11.1510 −0.0289637
\(386\) 0 0
\(387\) 148.921 0.384809
\(388\) 0 0
\(389\) − 233.889i − 0.601256i −0.953741 0.300628i \(-0.902804\pi\)
0.953741 0.300628i \(-0.0971963\pi\)
\(390\) 0 0
\(391\) − 388.095i − 0.992570i
\(392\) 0 0
\(393\) −149.616 −0.380703
\(394\) 0 0
\(395\) −117.966 −0.298647
\(396\) 0 0
\(397\) − 348.363i − 0.877489i −0.898612 0.438745i \(-0.855423\pi\)
0.898612 0.438745i \(-0.144577\pi\)
\(398\) 0 0
\(399\) 21.4845i 0.0538459i
\(400\) 0 0
\(401\) −759.049 −1.89289 −0.946445 0.322865i \(-0.895354\pi\)
−0.946445 + 0.322865i \(0.895354\pi\)
\(402\) 0 0
\(403\) −733.619 −1.82039
\(404\) 0 0
\(405\) 8.09082i 0.0199773i
\(406\) 0 0
\(407\) − 154.334i − 0.379199i
\(408\) 0 0
\(409\) 94.0408 0.229929 0.114964 0.993370i \(-0.463325\pi\)
0.114964 + 0.993370i \(0.463325\pi\)
\(410\) 0 0
\(411\) 83.1031 0.202197
\(412\) 0 0
\(413\) 97.9796i 0.237239i
\(414\) 0 0
\(415\) − 99.0238i − 0.238612i
\(416\) 0 0
\(417\) −271.151 −0.650242
\(418\) 0 0
\(419\) 567.413 1.35421 0.677104 0.735888i \(-0.263235\pi\)
0.677104 + 0.735888i \(0.263235\pi\)
\(420\) 0 0
\(421\) − 786.120i − 1.86727i −0.358227 0.933635i \(-0.616618\pi\)
0.358227 0.933635i \(-0.383382\pi\)
\(422\) 0 0
\(423\) 115.365i 0.272731i
\(424\) 0 0
\(425\) 425.678 1.00159
\(426\) 0 0
\(427\) −69.0252 −0.161651
\(428\) 0 0
\(429\) 104.808i 0.244308i
\(430\) 0 0
\(431\) − 402.780i − 0.934523i −0.884119 0.467262i \(-0.845241\pi\)
0.884119 0.467262i \(-0.154759\pi\)
\(432\) 0 0
\(433\) 140.384 0.324212 0.162106 0.986773i \(-0.448171\pi\)
0.162106 + 0.986773i \(0.448171\pi\)
\(434\) 0 0
\(435\) 69.2820 0.159269
\(436\) 0 0
\(437\) − 96.7265i − 0.221342i
\(438\) 0 0
\(439\) − 341.039i − 0.776854i −0.921479 0.388427i \(-0.873019\pi\)
0.921479 0.388427i \(-0.126981\pi\)
\(440\) 0 0
\(441\) 123.000 0.278912
\(442\) 0 0
\(443\) −153.120 −0.345644 −0.172822 0.984953i \(-0.555289\pi\)
−0.172822 + 0.984953i \(0.555289\pi\)
\(444\) 0 0
\(445\) − 65.7980i − 0.147861i
\(446\) 0 0
\(447\) 448.076i 1.00241i
\(448\) 0 0
\(449\) −54.3224 −0.120985 −0.0604927 0.998169i \(-0.519267\pi\)
−0.0604927 + 0.998169i \(0.519267\pi\)
\(450\) 0 0
\(451\) −164.877 −0.365581
\(452\) 0 0
\(453\) − 185.121i − 0.408657i
\(454\) 0 0
\(455\) 35.0840i 0.0771078i
\(456\) 0 0
\(457\) −777.878 −1.70214 −0.851070 0.525053i \(-0.824045\pi\)
−0.851070 + 0.525053i \(0.824045\pi\)
\(458\) 0 0
\(459\) −91.4311 −0.199196
\(460\) 0 0
\(461\) − 635.160i − 1.37779i −0.724862 0.688894i \(-0.758097\pi\)
0.724862 0.688894i \(-0.241903\pi\)
\(462\) 0 0
\(463\) 72.3096i 0.156176i 0.996946 + 0.0780881i \(0.0248815\pi\)
−0.996946 + 0.0780881i \(0.975118\pi\)
\(464\) 0 0
\(465\) 82.7878 0.178038
\(466\) 0 0
\(467\) −671.265 −1.43740 −0.718699 0.695321i \(-0.755262\pi\)
−0.718699 + 0.695321i \(0.755262\pi\)
\(468\) 0 0
\(469\) − 265.171i − 0.565397i
\(470\) 0 0
\(471\) 199.518i 0.423605i
\(472\) 0 0
\(473\) −217.698 −0.460249
\(474\) 0 0
\(475\) 106.093 0.223355
\(476\) 0 0
\(477\) − 5.12143i − 0.0107368i
\(478\) 0 0
\(479\) 644.824i 1.34619i 0.739557 + 0.673094i \(0.235035\pi\)
−0.739557 + 0.673094i \(0.764965\pi\)
\(480\) 0 0
\(481\) −485.576 −1.00951
\(482\) 0 0
\(483\) 108.052 0.223710
\(484\) 0 0
\(485\) − 94.5653i − 0.194980i
\(486\) 0 0
\(487\) − 527.715i − 1.08360i −0.840506 0.541802i \(-0.817742\pi\)
0.840506 0.541802i \(-0.182258\pi\)
\(488\) 0 0
\(489\) −430.788 −0.880957
\(490\) 0 0
\(491\) 408.250 0.831467 0.415733 0.909486i \(-0.363525\pi\)
0.415733 + 0.909486i \(0.363525\pi\)
\(492\) 0 0
\(493\) 782.929i 1.58809i
\(494\) 0 0
\(495\) − 11.8274i − 0.0238938i
\(496\) 0 0
\(497\) −350.384 −0.704997
\(498\) 0 0
\(499\) 365.281 0.732027 0.366013 0.930610i \(-0.380722\pi\)
0.366013 + 0.930610i \(0.380722\pi\)
\(500\) 0 0
\(501\) − 206.747i − 0.412669i
\(502\) 0 0
\(503\) − 738.133i − 1.46746i −0.679441 0.733731i \(-0.737777\pi\)
0.679441 0.733731i \(-0.262223\pi\)
\(504\) 0 0
\(505\) 139.233 0.275708
\(506\) 0 0
\(507\) 37.0376 0.0730525
\(508\) 0 0
\(509\) − 96.2724i − 0.189140i −0.995518 0.0945702i \(-0.969852\pi\)
0.995518 0.0945702i \(-0.0301477\pi\)
\(510\) 0 0
\(511\) 28.2843i 0.0553508i
\(512\) 0 0
\(513\) −22.7878 −0.0444206
\(514\) 0 0
\(515\) −34.0825 −0.0661797
\(516\) 0 0
\(517\) − 168.645i − 0.326199i
\(518\) 0 0
\(519\) − 460.533i − 0.887347i
\(520\) 0 0
\(521\) 199.212 0.382365 0.191183 0.981554i \(-0.438768\pi\)
0.191183 + 0.981554i \(0.438768\pi\)
\(522\) 0 0
\(523\) −13.6702 −0.0261381 −0.0130691 0.999915i \(-0.504160\pi\)
−0.0130691 + 0.999915i \(0.504160\pi\)
\(524\) 0 0
\(525\) 118.515i 0.225743i
\(526\) 0 0
\(527\) 935.551i 1.77524i
\(528\) 0 0
\(529\) 42.5347 0.0804058
\(530\) 0 0
\(531\) −103.923 −0.195712
\(532\) 0 0
\(533\) 518.747i 0.973259i
\(534\) 0 0
\(535\) − 17.4266i − 0.0325730i
\(536\) 0 0
\(537\) 391.151 0.728400
\(538\) 0 0
\(539\) −179.806 −0.333591
\(540\) 0 0
\(541\) − 542.443i − 1.00267i −0.865254 0.501333i \(-0.832843\pi\)
0.865254 0.501333i \(-0.167157\pi\)
\(542\) 0 0
\(543\) − 83.5237i − 0.153819i
\(544\) 0 0
\(545\) 118.829 0.218034
\(546\) 0 0
\(547\) 239.058 0.437035 0.218518 0.975833i \(-0.429878\pi\)
0.218518 + 0.975833i \(0.429878\pi\)
\(548\) 0 0
\(549\) − 73.2122i − 0.133356i
\(550\) 0 0
\(551\) 195.133i 0.354143i
\(552\) 0 0
\(553\) 371.151 0.671159
\(554\) 0 0
\(555\) 54.7964 0.0987323
\(556\) 0 0
\(557\) − 521.707i − 0.936638i −0.883560 0.468319i \(-0.844860\pi\)
0.883560 0.468319i \(-0.155140\pi\)
\(558\) 0 0
\(559\) 684.935i 1.22529i
\(560\) 0 0
\(561\) 133.657 0.238248
\(562\) 0 0
\(563\) 1092.42 1.94035 0.970173 0.242413i \(-0.0779388\pi\)
0.970173 + 0.242413i \(0.0779388\pi\)
\(564\) 0 0
\(565\) − 31.9817i − 0.0566047i
\(566\) 0 0
\(567\) − 25.4558i − 0.0448957i
\(568\) 0 0
\(569\) −823.171 −1.44670 −0.723349 0.690482i \(-0.757398\pi\)
−0.723349 + 0.690482i \(0.757398\pi\)
\(570\) 0 0
\(571\) −268.800 −0.470753 −0.235377 0.971904i \(-0.575632\pi\)
−0.235377 + 0.971904i \(0.575632\pi\)
\(572\) 0 0
\(573\) 307.596i 0.536817i
\(574\) 0 0
\(575\) − 533.574i − 0.927955i
\(576\) 0 0
\(577\) −108.465 −0.187981 −0.0939907 0.995573i \(-0.529962\pi\)
−0.0939907 + 0.995573i \(0.529962\pi\)
\(578\) 0 0
\(579\) −226.496 −0.391184
\(580\) 0 0
\(581\) 311.555i 0.536239i
\(582\) 0 0
\(583\) 7.48669i 0.0128417i
\(584\) 0 0
\(585\) −37.2122 −0.0636107
\(586\) 0 0
\(587\) −223.545 −0.380827 −0.190413 0.981704i \(-0.560983\pi\)
−0.190413 + 0.981704i \(0.560983\pi\)
\(588\) 0 0
\(589\) 233.171i 0.395877i
\(590\) 0 0
\(591\) 258.600i 0.437564i
\(592\) 0 0
\(593\) 748.261 1.26182 0.630912 0.775855i \(-0.282681\pi\)
0.630912 + 0.775855i \(0.282681\pi\)
\(594\) 0 0
\(595\) 44.7411 0.0751951
\(596\) 0 0
\(597\) − 257.666i − 0.431602i
\(598\) 0 0
\(599\) 32.2268i 0.0538009i 0.999638 + 0.0269005i \(0.00856372\pi\)
−0.999638 + 0.0269005i \(0.991436\pi\)
\(600\) 0 0
\(601\) −161.918 −0.269415 −0.134707 0.990885i \(-0.543009\pi\)
−0.134707 + 0.990885i \(0.543009\pi\)
\(602\) 0 0
\(603\) 281.257 0.466429
\(604\) 0 0
\(605\) − 91.4868i − 0.151218i
\(606\) 0 0
\(607\) 573.542i 0.944879i 0.881363 + 0.472439i \(0.156626\pi\)
−0.881363 + 0.472439i \(0.843374\pi\)
\(608\) 0 0
\(609\) −217.980 −0.357930
\(610\) 0 0
\(611\) −530.601 −0.868415
\(612\) 0 0
\(613\) 985.857i 1.60825i 0.594460 + 0.804125i \(0.297366\pi\)
−0.594460 + 0.804125i \(0.702634\pi\)
\(614\) 0 0
\(615\) − 58.5398i − 0.0951866i
\(616\) 0 0
\(617\) 761.151 1.23363 0.616816 0.787107i \(-0.288422\pi\)
0.616816 + 0.787107i \(0.288422\pi\)
\(618\) 0 0
\(619\) 538.929 0.870645 0.435323 0.900275i \(-0.356634\pi\)
0.435323 + 0.900275i \(0.356634\pi\)
\(620\) 0 0
\(621\) 114.606i 0.184551i
\(622\) 0 0
\(623\) 207.018i 0.332292i
\(624\) 0 0
\(625\) 565.041 0.904065
\(626\) 0 0
\(627\) 33.3119 0.0531291
\(628\) 0 0
\(629\) 619.233i 0.984472i
\(630\) 0 0
\(631\) 278.756i 0.441768i 0.975300 + 0.220884i \(0.0708943\pi\)
−0.975300 + 0.220884i \(0.929106\pi\)
\(632\) 0 0
\(633\) 661.535 1.04508
\(634\) 0 0
\(635\) −114.935 −0.181000
\(636\) 0 0
\(637\) 565.716i 0.888095i
\(638\) 0 0
\(639\) − 371.638i − 0.581593i
\(640\) 0 0
\(641\) 318.363 0.496667 0.248333 0.968675i \(-0.420117\pi\)
0.248333 + 0.968675i \(0.420117\pi\)
\(642\) 0 0
\(643\) −647.752 −1.00739 −0.503695 0.863882i \(-0.668026\pi\)
−0.503695 + 0.863882i \(0.668026\pi\)
\(644\) 0 0
\(645\) − 77.2939i − 0.119835i
\(646\) 0 0
\(647\) − 1084.29i − 1.67587i −0.545772 0.837934i \(-0.683764\pi\)
0.545772 0.837934i \(-0.316236\pi\)
\(648\) 0 0
\(649\) 151.918 0.234081
\(650\) 0 0
\(651\) −260.472 −0.400111
\(652\) 0 0
\(653\) 23.3826i 0.0358080i 0.999840 + 0.0179040i \(0.00569933\pi\)
−0.999840 + 0.0179040i \(0.994301\pi\)
\(654\) 0 0
\(655\) 77.6548i 0.118557i
\(656\) 0 0
\(657\) −30.0000 −0.0456621
\(658\) 0 0
\(659\) −496.846 −0.753940 −0.376970 0.926225i \(-0.623034\pi\)
−0.376970 + 0.926225i \(0.623034\pi\)
\(660\) 0 0
\(661\) 113.576i 0.171824i 0.996303 + 0.0859119i \(0.0273804\pi\)
−0.996303 + 0.0859119i \(0.972620\pi\)
\(662\) 0 0
\(663\) − 420.521i − 0.634270i
\(664\) 0 0
\(665\) 11.1510 0.0167684
\(666\) 0 0
\(667\) 981.378 1.47133
\(668\) 0 0
\(669\) 464.413i 0.694190i
\(670\) 0 0
\(671\) 107.024i 0.159500i
\(672\) 0 0
\(673\) 554.604 0.824077 0.412039 0.911166i \(-0.364817\pi\)
0.412039 + 0.911166i \(0.364817\pi\)
\(674\) 0 0
\(675\) −125.704 −0.186229
\(676\) 0 0
\(677\) 902.474i 1.33305i 0.745483 + 0.666525i \(0.232219\pi\)
−0.745483 + 0.666525i \(0.767781\pi\)
\(678\) 0 0
\(679\) 297.527i 0.438185i
\(680\) 0 0
\(681\) −611.555 −0.898025
\(682\) 0 0
\(683\) −924.738 −1.35394 −0.676968 0.736012i \(-0.736706\pi\)
−0.676968 + 0.736012i \(0.736706\pi\)
\(684\) 0 0
\(685\) − 43.1327i − 0.0629674i
\(686\) 0 0
\(687\) − 368.139i − 0.535864i
\(688\) 0 0
\(689\) 23.5551 0.0341874
\(690\) 0 0
\(691\) 1155.87 1.67275 0.836373 0.548161i \(-0.184672\pi\)
0.836373 + 0.548161i \(0.184672\pi\)
\(692\) 0 0
\(693\) 37.2122i 0.0536973i
\(694\) 0 0
\(695\) 140.734i 0.202496i
\(696\) 0 0
\(697\) 661.535 0.949117
\(698\) 0 0
\(699\) −615.875 −0.881080
\(700\) 0 0
\(701\) − 591.383i − 0.843627i −0.906683 0.421814i \(-0.861394\pi\)
0.906683 0.421814i \(-0.138606\pi\)
\(702\) 0 0
\(703\) 154.334i 0.219536i
\(704\) 0 0
\(705\) 59.8775 0.0849327
\(706\) 0 0
\(707\) −438.063 −0.619608
\(708\) 0 0
\(709\) − 1216.87i − 1.71631i −0.513387 0.858157i \(-0.671609\pi\)
0.513387 0.858157i \(-0.328391\pi\)
\(710\) 0 0
\(711\) 393.665i 0.553678i
\(712\) 0 0
\(713\) 1172.69 1.64472
\(714\) 0 0
\(715\) 54.3982 0.0760814
\(716\) 0 0
\(717\) 74.4245i 0.103800i
\(718\) 0 0
\(719\) − 1202.62i − 1.67263i −0.548246 0.836317i \(-0.684704\pi\)
0.548246 0.836317i \(-0.315296\pi\)
\(720\) 0 0
\(721\) 107.233 0.148728
\(722\) 0 0
\(723\) −216.839 −0.299915
\(724\) 0 0
\(725\) 1076.41i 1.48471i
\(726\) 0 0
\(727\) − 194.019i − 0.266876i −0.991057 0.133438i \(-0.957398\pi\)
0.991057 0.133438i \(-0.0426016\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 873.467 1.19489
\(732\) 0 0
\(733\) 1070.52i 1.46047i 0.683196 + 0.730235i \(0.260589\pi\)
−0.683196 + 0.730235i \(0.739411\pi\)
\(734\) 0 0
\(735\) − 63.8402i − 0.0868574i
\(736\) 0 0
\(737\) −411.151 −0.557871
\(738\) 0 0
\(739\) 376.710 0.509757 0.254879 0.966973i \(-0.417965\pi\)
0.254879 + 0.966973i \(0.417965\pi\)
\(740\) 0 0
\(741\) − 104.808i − 0.141442i
\(742\) 0 0
\(743\) 608.141i 0.818494i 0.912424 + 0.409247i \(0.134208\pi\)
−0.912424 + 0.409247i \(0.865792\pi\)
\(744\) 0 0
\(745\) 232.563 0.312165
\(746\) 0 0
\(747\) −330.454 −0.442375
\(748\) 0 0
\(749\) 54.8286i 0.0732024i
\(750\) 0 0
\(751\) − 382.236i − 0.508969i −0.967077 0.254485i \(-0.918094\pi\)
0.967077 0.254485i \(-0.0819058\pi\)
\(752\) 0 0
\(753\) 578.747 0.768588
\(754\) 0 0
\(755\) −96.0828 −0.127262
\(756\) 0 0
\(757\) 206.969i 0.273407i 0.990612 + 0.136704i \(0.0436508\pi\)
−0.990612 + 0.136704i \(0.956349\pi\)
\(758\) 0 0
\(759\) − 167.535i − 0.220732i
\(760\) 0 0
\(761\) −959.980 −1.26147 −0.630736 0.775998i \(-0.717247\pi\)
−0.630736 + 0.775998i \(0.717247\pi\)
\(762\) 0 0
\(763\) −373.866 −0.489995
\(764\) 0 0
\(765\) 47.4551i 0.0620328i
\(766\) 0 0
\(767\) − 477.975i − 0.623175i
\(768\) 0 0
\(769\) −513.918 −0.668294 −0.334147 0.942521i \(-0.608448\pi\)
−0.334147 + 0.942521i \(0.608448\pi\)
\(770\) 0 0
\(771\) −502.295 −0.651485
\(772\) 0 0
\(773\) 1389.24i 1.79721i 0.438763 + 0.898603i \(0.355417\pi\)
−0.438763 + 0.898603i \(0.644583\pi\)
\(774\) 0 0
\(775\) 1286.25i 1.65967i
\(776\) 0 0
\(777\) −172.404 −0.221884
\(778\) 0 0
\(779\) 164.877 0.211652
\(780\) 0 0
\(781\) 543.273i 0.695613i
\(782\) 0 0
\(783\) − 231.202i − 0.295277i
\(784\) 0 0
\(785\) 103.555 0.131917
\(786\) 0 0
\(787\) 526.588 0.669108 0.334554 0.942377i \(-0.391414\pi\)
0.334554 + 0.942377i \(0.391414\pi\)
\(788\) 0 0
\(789\) − 432.000i − 0.547529i
\(790\) 0 0
\(791\) 100.623i 0.127210i
\(792\) 0 0
\(793\) 336.727 0.424624
\(794\) 0 0
\(795\) −2.65816 −0.00334359
\(796\) 0 0
\(797\) − 1141.22i − 1.43190i −0.698153 0.715948i \(-0.745995\pi\)
0.698153 0.715948i \(-0.254005\pi\)
\(798\) 0 0
\(799\) 676.652i 0.846874i
\(800\) 0 0
\(801\) −219.576 −0.274127
\(802\) 0 0
\(803\) 43.8551 0.0546140
\(804\) 0 0
\(805\) − 56.0816i − 0.0696666i
\(806\) 0 0
\(807\) − 521.102i − 0.645727i
\(808\) 0 0
\(809\) 944.020 1.16690 0.583449 0.812150i \(-0.301703\pi\)
0.583449 + 0.812150i \(0.301703\pi\)
\(810\) 0 0
\(811\) 58.7837 0.0724830 0.0362415 0.999343i \(-0.488461\pi\)
0.0362415 + 0.999343i \(0.488461\pi\)
\(812\) 0 0
\(813\) 670.070i 0.824195i
\(814\) 0 0
\(815\) 223.590i 0.274344i
\(816\) 0 0
\(817\) 217.698 0.266460
\(818\) 0 0
\(819\) 117.080 0.142954
\(820\) 0 0
\(821\) − 1246.67i − 1.51848i −0.650809 0.759241i \(-0.725570\pi\)
0.650809 0.759241i \(-0.274430\pi\)
\(822\) 0 0
\(823\) 762.960i 0.927047i 0.886085 + 0.463523i \(0.153415\pi\)
−0.886085 + 0.463523i \(0.846585\pi\)
\(824\) 0 0
\(825\) 183.759 0.222738
\(826\) 0 0
\(827\) −1251.44 −1.51322 −0.756612 0.653864i \(-0.773147\pi\)
−0.756612 + 0.653864i \(0.773147\pi\)
\(828\) 0 0
\(829\) − 13.8796i − 0.0167426i −0.999965 0.00837129i \(-0.997335\pi\)
0.999965 0.00837129i \(-0.00266470\pi\)
\(830\) 0 0
\(831\) − 370.309i − 0.445618i
\(832\) 0 0
\(833\) 721.433 0.866066
\(834\) 0 0
\(835\) −107.307 −0.128511
\(836\) 0 0
\(837\) − 276.272i − 0.330075i
\(838\) 0 0
\(839\) − 153.249i − 0.182656i −0.995821 0.0913282i \(-0.970889\pi\)
0.995821 0.0913282i \(-0.0291112\pi\)
\(840\) 0 0
\(841\) −1138.80 −1.35410
\(842\) 0 0
\(843\) 191.855 0.227586
\(844\) 0 0
\(845\) − 19.2235i − 0.0227497i
\(846\) 0 0
\(847\) 287.842i 0.339836i
\(848\) 0 0
\(849\) 199.151 0.234571
\(850\) 0 0
\(851\) 776.190 0.912091
\(852\) 0 0
\(853\) − 869.616i − 1.01948i −0.860329 0.509740i \(-0.829742\pi\)
0.860329 0.509740i \(-0.170258\pi\)
\(854\) 0 0
\(855\) 11.8274i 0.0138333i
\(856\) 0 0
\(857\) −883.939 −1.03143 −0.515717 0.856759i \(-0.672474\pi\)
−0.515717 + 0.856759i \(0.672474\pi\)
\(858\) 0 0
\(859\) 1592.65 1.85407 0.927035 0.374976i \(-0.122349\pi\)
0.927035 + 0.374976i \(0.122349\pi\)
\(860\) 0 0
\(861\) 184.182i 0.213916i
\(862\) 0 0
\(863\) − 1209.31i − 1.40128i −0.713513 0.700642i \(-0.752897\pi\)
0.713513 0.700642i \(-0.247103\pi\)
\(864\) 0 0
\(865\) −239.029 −0.276334
\(866\) 0 0
\(867\) −35.7085 −0.0411863
\(868\) 0 0
\(869\) − 575.473i − 0.662225i
\(870\) 0 0
\(871\) 1293.59i 1.48518i
\(872\) 0 0
\(873\) −315.576 −0.361484
\(874\) 0 0
\(875\) 125.080 0.142949
\(876\) 0 0
\(877\) − 922.584i − 1.05198i −0.850492 0.525988i \(-0.823696\pi\)
0.850492 0.525988i \(-0.176304\pi\)
\(878\) 0 0
\(879\) − 188.233i − 0.214145i
\(880\) 0 0
\(881\) −1564.22 −1.77551 −0.887753 0.460321i \(-0.847734\pi\)
−0.887753 + 0.460321i \(0.847734\pi\)
\(882\) 0 0
\(883\) 836.284 0.947094 0.473547 0.880769i \(-0.342973\pi\)
0.473547 + 0.880769i \(0.342973\pi\)
\(884\) 0 0
\(885\) 53.9388i 0.0609478i
\(886\) 0 0
\(887\) 596.712i 0.672730i 0.941732 + 0.336365i \(0.109198\pi\)
−0.941732 + 0.336365i \(0.890802\pi\)
\(888\) 0 0
\(889\) 361.616 0.406768
\(890\) 0 0
\(891\) −39.4695 −0.0442980
\(892\) 0 0
\(893\) 168.645i 0.188852i
\(894\) 0 0
\(895\) − 203.018i − 0.226835i
\(896\) 0 0
\(897\) −527.110 −0.587637
\(898\) 0 0
\(899\) −2365.73 −2.63152
\(900\) 0 0
\(901\) − 30.0388i − 0.0333394i
\(902\) 0 0
\(903\) 243.187i 0.269310i
\(904\) 0 0
\(905\) −43.3510 −0.0479017
\(906\) 0 0
\(907\) −399.595 −0.440567 −0.220284 0.975436i \(-0.570698\pi\)
−0.220284 + 0.975436i \(0.570698\pi\)
\(908\) 0 0
\(909\) − 464.636i − 0.511150i
\(910\) 0 0
\(911\) − 612.995i − 0.672882i −0.941705 0.336441i \(-0.890777\pi\)
0.941705 0.336441i \(-0.109223\pi\)
\(912\) 0 0
\(913\) 483.069 0.529101
\(914\) 0 0
\(915\) −37.9991 −0.0415290
\(916\) 0 0
\(917\) − 244.322i − 0.266437i
\(918\) 0 0
\(919\) 1095.74i 1.19232i 0.802865 + 0.596161i \(0.203308\pi\)
−0.802865 + 0.596161i \(0.796692\pi\)
\(920\) 0 0
\(921\) 439.151 0.476820
\(922\) 0 0
\(923\) 1709.28 1.85188
\(924\) 0 0
\(925\) 851.355i 0.920384i
\(926\) 0 0
\(927\) 113.737i 0.122694i
\(928\) 0 0
\(929\) 180.102 0.193867 0.0969333 0.995291i \(-0.469097\pi\)
0.0969333 + 0.995291i \(0.469097\pi\)
\(930\) 0 0
\(931\) 179.806 0.193132
\(932\) 0 0
\(933\) − 790.665i − 0.847444i
\(934\) 0 0
\(935\) − 69.3715i − 0.0741942i
\(936\) 0 0
\(937\) −1393.84 −1.48755 −0.743776 0.668429i \(-0.766967\pi\)
−0.743776 + 0.668429i \(0.766967\pi\)
\(938\) 0 0
\(939\) −643.517 −0.685322
\(940\) 0 0
\(941\) − 659.968i − 0.701348i −0.936498 0.350674i \(-0.885953\pi\)
0.936498 0.350674i \(-0.114047\pi\)
\(942\) 0 0
\(943\) − 829.214i − 0.879336i
\(944\) 0 0
\(945\) −13.2122 −0.0139812
\(946\) 0 0
\(947\) 1584.65 1.67333 0.836666 0.547714i \(-0.184502\pi\)
0.836666 + 0.547714i \(0.184502\pi\)
\(948\) 0 0
\(949\) − 137.980i − 0.145395i
\(950\) 0 0
\(951\) 92.6382i 0.0974114i
\(952\) 0 0
\(953\) −1212.02 −1.27179 −0.635897 0.771774i \(-0.719370\pi\)
−0.635897 + 0.771774i \(0.719370\pi\)
\(954\) 0 0
\(955\) 159.650 0.167173
\(956\) 0 0
\(957\) 337.980i 0.353166i
\(958\) 0 0
\(959\) 135.707i 0.141509i
\(960\) 0 0
\(961\) −1865.91 −1.94163
\(962\) 0 0
\(963\) −58.1545 −0.0603889
\(964\) 0 0
\(965\) 117.557i 0.121821i
\(966\) 0 0
\(967\) − 529.943i − 0.548028i −0.961726 0.274014i \(-0.911648\pi\)
0.961726 0.274014i \(-0.0883515\pi\)
\(968\) 0 0
\(969\) −133.657 −0.137933
\(970\) 0 0
\(971\) −1364.60 −1.40535 −0.702677 0.711509i \(-0.748012\pi\)
−0.702677 + 0.711509i \(0.748012\pi\)
\(972\) 0 0
\(973\) − 442.788i − 0.455075i
\(974\) 0 0
\(975\) − 578.155i − 0.592980i
\(976\) 0 0
\(977\) 141.637 0.144971 0.0724855 0.997369i \(-0.476907\pi\)
0.0724855 + 0.997369i \(0.476907\pi\)
\(978\) 0 0
\(979\) 320.983 0.327868
\(980\) 0 0
\(981\) − 396.545i − 0.404225i
\(982\) 0 0
\(983\) 354.667i 0.360801i 0.983593 + 0.180401i \(0.0577394\pi\)
−0.983593 + 0.180401i \(0.942261\pi\)
\(984\) 0 0
\(985\) 134.220 0.136264
\(986\) 0 0
\(987\) −188.391 −0.190872
\(988\) 0 0
\(989\) − 1094.87i − 1.10704i
\(990\) 0 0
\(991\) − 1596.00i − 1.61050i −0.592939 0.805248i \(-0.702032\pi\)
0.592939 0.805248i \(-0.297968\pi\)
\(992\) 0 0
\(993\) 286.343 0.288361
\(994\) 0 0
\(995\) −133.736 −0.134408
\(996\) 0 0
\(997\) − 1122.02i − 1.12540i −0.826662 0.562698i \(-0.809763\pi\)
0.826662 0.562698i \(-0.190237\pi\)
\(998\) 0 0
\(999\) − 182.862i − 0.183045i
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 384.3.b.c.319.3 yes 8
3.2 odd 2 1152.3.b.j.703.3 8
4.3 odd 2 inner 384.3.b.c.319.7 yes 8
8.3 odd 2 inner 384.3.b.c.319.2 8
8.5 even 2 inner 384.3.b.c.319.6 yes 8
12.11 even 2 1152.3.b.j.703.4 8
16.3 odd 4 768.3.g.c.511.4 4
16.5 even 4 768.3.g.g.511.3 4
16.11 odd 4 768.3.g.g.511.1 4
16.13 even 4 768.3.g.c.511.2 4
24.5 odd 2 1152.3.b.j.703.5 8
24.11 even 2 1152.3.b.j.703.6 8
48.5 odd 4 2304.3.g.o.1279.4 4
48.11 even 4 2304.3.g.o.1279.3 4
48.29 odd 4 2304.3.g.x.1279.2 4
48.35 even 4 2304.3.g.x.1279.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.c.319.2 8 8.3 odd 2 inner
384.3.b.c.319.3 yes 8 1.1 even 1 trivial
384.3.b.c.319.6 yes 8 8.5 even 2 inner
384.3.b.c.319.7 yes 8 4.3 odd 2 inner
768.3.g.c.511.2 4 16.13 even 4
768.3.g.c.511.4 4 16.3 odd 4
768.3.g.g.511.1 4 16.11 odd 4
768.3.g.g.511.3 4 16.5 even 4
1152.3.b.j.703.3 8 3.2 odd 2
1152.3.b.j.703.4 8 12.11 even 2
1152.3.b.j.703.5 8 24.5 odd 2
1152.3.b.j.703.6 8 24.11 even 2
2304.3.g.o.1279.3 4 48.11 even 4
2304.3.g.o.1279.4 4 48.5 odd 4
2304.3.g.x.1279.1 4 48.35 even 4
2304.3.g.x.1279.2 4 48.29 odd 4