Properties

Label 320.6.d.a.161.5
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.5
Root \(5.36332 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.a.161.4

$q$-expansion

\(f(q)\) \(=\) \(q+9.72664i q^{3} +25.0000i q^{5} +65.1432 q^{7} +148.393 q^{9} +O(q^{10})\) \(q+9.72664i q^{3} +25.0000i q^{5} +65.1432 q^{7} +148.393 q^{9} -71.5642i q^{11} -733.370i q^{13} -243.166 q^{15} -716.077 q^{17} -2333.71i q^{19} +633.624i q^{21} -278.545 q^{23} -625.000 q^{25} +3806.93i q^{27} -2953.16i q^{29} -6047.10 q^{31} +696.079 q^{33} +1628.58i q^{35} -12195.5i q^{37} +7133.22 q^{39} +3629.90 q^{41} +230.070i q^{43} +3709.81i q^{45} +13998.1 q^{47} -12563.4 q^{49} -6965.02i q^{51} -22416.3i q^{53} +1789.10 q^{55} +22699.2 q^{57} +7285.64i q^{59} -19417.5i q^{61} +9666.77 q^{63} +18334.2 q^{65} +1390.88i q^{67} -2709.30i q^{69} +57315.7 q^{71} -35556.0 q^{73} -6079.15i q^{75} -4661.92i q^{77} +93011.5 q^{79} -969.268 q^{81} +1599.04i q^{83} -17901.9i q^{85} +28724.3 q^{87} +2029.74 q^{89} -47774.1i q^{91} -58817.9i q^{93} +58342.9 q^{95} -120706. q^{97} -10619.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 320q^{7} - 1192q^{9} + O(q^{10}) \) \( 8q - 320q^{7} - 1192q^{9} - 2400q^{17} + 1760q^{23} - 5000q^{25} + 31040q^{31} + 3760q^{33} + 4480q^{39} + 21584q^{41} + 47680q^{47} + 82824q^{49} + 18400q^{55} - 106640q^{57} + 322400q^{63} - 44000q^{65} + 246720q^{71} + 46400q^{73} + 325760q^{79} - 82328q^{81} + 636320q^{87} - 78192q^{89} - 40800q^{95} + 24960q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 9.72664i 0.623964i 0.950088 + 0.311982i \(0.100993\pi\)
−0.950088 + 0.311982i \(0.899007\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) 65.1432 0.502486 0.251243 0.967924i \(-0.419161\pi\)
0.251243 + 0.967924i \(0.419161\pi\)
\(8\) 0 0
\(9\) 148.393 0.610669
\(10\) 0 0
\(11\) − 71.5642i − 0.178326i −0.996017 0.0891628i \(-0.971581\pi\)
0.996017 0.0891628i \(-0.0284192\pi\)
\(12\) 0 0
\(13\) − 733.370i − 1.20355i −0.798665 0.601776i \(-0.794460\pi\)
0.798665 0.601776i \(-0.205540\pi\)
\(14\) 0 0
\(15\) −243.166 −0.279045
\(16\) 0 0
\(17\) −716.077 −0.600949 −0.300474 0.953790i \(-0.597145\pi\)
−0.300474 + 0.953790i \(0.597145\pi\)
\(18\) 0 0
\(19\) − 2333.71i − 1.48308i −0.670910 0.741538i \(-0.734096\pi\)
0.670910 0.741538i \(-0.265904\pi\)
\(20\) 0 0
\(21\) 633.624i 0.313533i
\(22\) 0 0
\(23\) −278.545 −0.109793 −0.0548966 0.998492i \(-0.517483\pi\)
−0.0548966 + 0.998492i \(0.517483\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) 3806.93i 1.00500i
\(28\) 0 0
\(29\) − 2953.16i − 0.652066i −0.945358 0.326033i \(-0.894288\pi\)
0.945358 0.326033i \(-0.105712\pi\)
\(30\) 0 0
\(31\) −6047.10 −1.13017 −0.565084 0.825033i \(-0.691156\pi\)
−0.565084 + 0.825033i \(0.691156\pi\)
\(32\) 0 0
\(33\) 696.079 0.111269
\(34\) 0 0
\(35\) 1628.58i 0.224719i
\(36\) 0 0
\(37\) − 12195.5i − 1.46452i −0.681024 0.732261i \(-0.738465\pi\)
0.681024 0.732261i \(-0.261535\pi\)
\(38\) 0 0
\(39\) 7133.22 0.750973
\(40\) 0 0
\(41\) 3629.90 0.337236 0.168618 0.985681i \(-0.446069\pi\)
0.168618 + 0.985681i \(0.446069\pi\)
\(42\) 0 0
\(43\) 230.070i 0.0189753i 0.999955 + 0.00948766i \(0.00302006\pi\)
−0.999955 + 0.00948766i \(0.996980\pi\)
\(44\) 0 0
\(45\) 3709.81i 0.273099i
\(46\) 0 0
\(47\) 13998.1 0.924322 0.462161 0.886796i \(-0.347074\pi\)
0.462161 + 0.886796i \(0.347074\pi\)
\(48\) 0 0
\(49\) −12563.4 −0.747508
\(50\) 0 0
\(51\) − 6965.02i − 0.374971i
\(52\) 0 0
\(53\) − 22416.3i − 1.09616i −0.836426 0.548080i \(-0.815359\pi\)
0.836426 0.548080i \(-0.184641\pi\)
\(54\) 0 0
\(55\) 1789.10 0.0797497
\(56\) 0 0
\(57\) 22699.2 0.925387
\(58\) 0 0
\(59\) 7285.64i 0.272482i 0.990676 + 0.136241i \(0.0435021\pi\)
−0.990676 + 0.136241i \(0.956498\pi\)
\(60\) 0 0
\(61\) − 19417.5i − 0.668141i −0.942548 0.334070i \(-0.891578\pi\)
0.942548 0.334070i \(-0.108422\pi\)
\(62\) 0 0
\(63\) 9666.77 0.306853
\(64\) 0 0
\(65\) 18334.2 0.538245
\(66\) 0 0
\(67\) 1390.88i 0.0378531i 0.999821 + 0.0189265i \(0.00602487\pi\)
−0.999821 + 0.0189265i \(0.993975\pi\)
\(68\) 0 0
\(69\) − 2709.30i − 0.0685070i
\(70\) 0 0
\(71\) 57315.7 1.34936 0.674679 0.738111i \(-0.264282\pi\)
0.674679 + 0.738111i \(0.264282\pi\)
\(72\) 0 0
\(73\) −35556.0 −0.780918 −0.390459 0.920620i \(-0.627684\pi\)
−0.390459 + 0.920620i \(0.627684\pi\)
\(74\) 0 0
\(75\) − 6079.15i − 0.124793i
\(76\) 0 0
\(77\) − 4661.92i − 0.0896062i
\(78\) 0 0
\(79\) 93011.5 1.67675 0.838376 0.545092i \(-0.183505\pi\)
0.838376 + 0.545092i \(0.183505\pi\)
\(80\) 0 0
\(81\) −969.268 −0.0164146
\(82\) 0 0
\(83\) 1599.04i 0.0254779i 0.999919 + 0.0127389i \(0.00405504\pi\)
−0.999919 + 0.0127389i \(0.995945\pi\)
\(84\) 0 0
\(85\) − 17901.9i − 0.268753i
\(86\) 0 0
\(87\) 28724.3 0.406866
\(88\) 0 0
\(89\) 2029.74 0.0271622 0.0135811 0.999908i \(-0.495677\pi\)
0.0135811 + 0.999908i \(0.495677\pi\)
\(90\) 0 0
\(91\) − 47774.1i − 0.604768i
\(92\) 0 0
\(93\) − 58817.9i − 0.705184i
\(94\) 0 0
\(95\) 58342.9 0.663252
\(96\) 0 0
\(97\) −120706. −1.30257 −0.651284 0.758834i \(-0.725769\pi\)
−0.651284 + 0.758834i \(0.725769\pi\)
\(98\) 0 0
\(99\) − 10619.6i − 0.108898i
\(100\) 0 0
\(101\) − 173879.i − 1.69607i −0.529942 0.848034i \(-0.677786\pi\)
0.529942 0.848034i \(-0.322214\pi\)
\(102\) 0 0
\(103\) 111795. 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(104\) 0 0
\(105\) −15840.6 −0.140216
\(106\) 0 0
\(107\) 110498.i 0.933030i 0.884513 + 0.466515i \(0.154491\pi\)
−0.884513 + 0.466515i \(0.845509\pi\)
\(108\) 0 0
\(109\) − 45521.0i − 0.366983i −0.983021 0.183491i \(-0.941260\pi\)
0.983021 0.183491i \(-0.0587399\pi\)
\(110\) 0 0
\(111\) 118621. 0.913809
\(112\) 0 0
\(113\) 181355. 1.33608 0.668041 0.744125i \(-0.267133\pi\)
0.668041 + 0.744125i \(0.267133\pi\)
\(114\) 0 0
\(115\) − 6963.62i − 0.0491010i
\(116\) 0 0
\(117\) − 108827.i − 0.734971i
\(118\) 0 0
\(119\) −46647.6 −0.301969
\(120\) 0 0
\(121\) 155930. 0.968200
\(122\) 0 0
\(123\) 35306.7i 0.210423i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) 99975.7 0.550028 0.275014 0.961440i \(-0.411317\pi\)
0.275014 + 0.961440i \(0.411317\pi\)
\(128\) 0 0
\(129\) −2237.81 −0.0118399
\(130\) 0 0
\(131\) − 137164.i − 0.698331i −0.937061 0.349166i \(-0.886465\pi\)
0.937061 0.349166i \(-0.113535\pi\)
\(132\) 0 0
\(133\) − 152026.i − 0.745226i
\(134\) 0 0
\(135\) −95173.3 −0.449449
\(136\) 0 0
\(137\) −181966. −0.828303 −0.414151 0.910208i \(-0.635922\pi\)
−0.414151 + 0.910208i \(0.635922\pi\)
\(138\) 0 0
\(139\) 134495.i 0.590431i 0.955431 + 0.295215i \(0.0953914\pi\)
−0.955431 + 0.295215i \(0.904609\pi\)
\(140\) 0 0
\(141\) 136154.i 0.576744i
\(142\) 0 0
\(143\) −52483.0 −0.214624
\(144\) 0 0
\(145\) 73828.9 0.291613
\(146\) 0 0
\(147\) − 122199.i − 0.466418i
\(148\) 0 0
\(149\) − 348040.i − 1.28429i −0.766583 0.642146i \(-0.778044\pi\)
0.766583 0.642146i \(-0.221956\pi\)
\(150\) 0 0
\(151\) −159935. −0.570821 −0.285411 0.958405i \(-0.592130\pi\)
−0.285411 + 0.958405i \(0.592130\pi\)
\(152\) 0 0
\(153\) −106261. −0.366981
\(154\) 0 0
\(155\) − 151177.i − 0.505426i
\(156\) 0 0
\(157\) − 212838.i − 0.689127i −0.938763 0.344563i \(-0.888027\pi\)
0.938763 0.344563i \(-0.111973\pi\)
\(158\) 0 0
\(159\) 218035. 0.683964
\(160\) 0 0
\(161\) −18145.3 −0.0551696
\(162\) 0 0
\(163\) 410171.i 1.20919i 0.796531 + 0.604597i \(0.206666\pi\)
−0.796531 + 0.604597i \(0.793334\pi\)
\(164\) 0 0
\(165\) 17402.0i 0.0497609i
\(166\) 0 0
\(167\) 35175.3 0.0975994 0.0487997 0.998809i \(-0.484460\pi\)
0.0487997 + 0.998809i \(0.484460\pi\)
\(168\) 0 0
\(169\) −166538. −0.448536
\(170\) 0 0
\(171\) − 346306.i − 0.905669i
\(172\) 0 0
\(173\) 325641.i 0.827224i 0.910453 + 0.413612i \(0.135733\pi\)
−0.910453 + 0.413612i \(0.864267\pi\)
\(174\) 0 0
\(175\) −40714.5 −0.100497
\(176\) 0 0
\(177\) −70864.8 −0.170019
\(178\) 0 0
\(179\) 231118.i 0.539139i 0.962981 + 0.269569i \(0.0868813\pi\)
−0.962981 + 0.269569i \(0.913119\pi\)
\(180\) 0 0
\(181\) 207075.i 0.469819i 0.972017 + 0.234910i \(0.0754794\pi\)
−0.972017 + 0.234910i \(0.924521\pi\)
\(182\) 0 0
\(183\) 188867. 0.416896
\(184\) 0 0
\(185\) 304888. 0.654954
\(186\) 0 0
\(187\) 51245.5i 0.107165i
\(188\) 0 0
\(189\) 247996.i 0.504998i
\(190\) 0 0
\(191\) −190884. −0.378605 −0.189302 0.981919i \(-0.560623\pi\)
−0.189302 + 0.981919i \(0.560623\pi\)
\(192\) 0 0
\(193\) −4008.86 −0.00774690 −0.00387345 0.999992i \(-0.501233\pi\)
−0.00387345 + 0.999992i \(0.501233\pi\)
\(194\) 0 0
\(195\) 178331.i 0.335845i
\(196\) 0 0
\(197\) 517405.i 0.949873i 0.880020 + 0.474936i \(0.157529\pi\)
−0.880020 + 0.474936i \(0.842471\pi\)
\(198\) 0 0
\(199\) −984918. −1.76306 −0.881531 0.472127i \(-0.843487\pi\)
−0.881531 + 0.472127i \(0.843487\pi\)
\(200\) 0 0
\(201\) −13528.5 −0.0236190
\(202\) 0 0
\(203\) − 192378.i − 0.327654i
\(204\) 0 0
\(205\) 90747.4i 0.150817i
\(206\) 0 0
\(207\) −41334.0 −0.0670473
\(208\) 0 0
\(209\) −167010. −0.264471
\(210\) 0 0
\(211\) − 920941.i − 1.42405i −0.702154 0.712025i \(-0.747778\pi\)
0.702154 0.712025i \(-0.252222\pi\)
\(212\) 0 0
\(213\) 557489.i 0.841951i
\(214\) 0 0
\(215\) −5751.75 −0.00848602
\(216\) 0 0
\(217\) −393927. −0.567894
\(218\) 0 0
\(219\) − 345840.i − 0.487265i
\(220\) 0 0
\(221\) 525149.i 0.723273i
\(222\) 0 0
\(223\) −186992. −0.251803 −0.125902 0.992043i \(-0.540182\pi\)
−0.125902 + 0.992043i \(0.540182\pi\)
\(224\) 0 0
\(225\) −92745.3 −0.122134
\(226\) 0 0
\(227\) 796832.i 1.02637i 0.858279 + 0.513183i \(0.171534\pi\)
−0.858279 + 0.513183i \(0.828466\pi\)
\(228\) 0 0
\(229\) 1.14511e6i 1.44298i 0.692426 + 0.721489i \(0.256542\pi\)
−0.692426 + 0.721489i \(0.743458\pi\)
\(230\) 0 0
\(231\) 45344.8 0.0559110
\(232\) 0 0
\(233\) 1.25471e6 1.51410 0.757048 0.653359i \(-0.226641\pi\)
0.757048 + 0.653359i \(0.226641\pi\)
\(234\) 0 0
\(235\) 349952.i 0.413369i
\(236\) 0 0
\(237\) 904689.i 1.04623i
\(238\) 0 0
\(239\) 454297. 0.514452 0.257226 0.966351i \(-0.417191\pi\)
0.257226 + 0.966351i \(0.417191\pi\)
\(240\) 0 0
\(241\) 241513. 0.267853 0.133927 0.990991i \(-0.457241\pi\)
0.133927 + 0.990991i \(0.457241\pi\)
\(242\) 0 0
\(243\) 915657.i 0.994757i
\(244\) 0 0
\(245\) − 314084.i − 0.334296i
\(246\) 0 0
\(247\) −1.71148e6 −1.78496
\(248\) 0 0
\(249\) −15553.2 −0.0158973
\(250\) 0 0
\(251\) 22467.7i 0.0225099i 0.999937 + 0.0112550i \(0.00358264\pi\)
−0.999937 + 0.0112550i \(0.996417\pi\)
\(252\) 0 0
\(253\) 19933.8i 0.0195790i
\(254\) 0 0
\(255\) 174126. 0.167692
\(256\) 0 0
\(257\) −391165. −0.369426 −0.184713 0.982793i \(-0.559135\pi\)
−0.184713 + 0.982793i \(0.559135\pi\)
\(258\) 0 0
\(259\) − 794455.i − 0.735902i
\(260\) 0 0
\(261\) − 438227.i − 0.398196i
\(262\) 0 0
\(263\) −2.20817e6 −1.96854 −0.984269 0.176677i \(-0.943465\pi\)
−0.984269 + 0.176677i \(0.943465\pi\)
\(264\) 0 0
\(265\) 560407. 0.490217
\(266\) 0 0
\(267\) 19742.6i 0.0169483i
\(268\) 0 0
\(269\) − 290107.i − 0.244443i −0.992503 0.122221i \(-0.960998\pi\)
0.992503 0.122221i \(-0.0390018\pi\)
\(270\) 0 0
\(271\) 252397. 0.208766 0.104383 0.994537i \(-0.466713\pi\)
0.104383 + 0.994537i \(0.466713\pi\)
\(272\) 0 0
\(273\) 464681. 0.377353
\(274\) 0 0
\(275\) 44727.6i 0.0356651i
\(276\) 0 0
\(277\) 698922.i 0.547305i 0.961829 + 0.273653i \(0.0882318\pi\)
−0.961829 + 0.273653i \(0.911768\pi\)
\(278\) 0 0
\(279\) −897344. −0.690158
\(280\) 0 0
\(281\) −2.49919e6 −1.88813 −0.944067 0.329754i \(-0.893034\pi\)
−0.944067 + 0.329754i \(0.893034\pi\)
\(282\) 0 0
\(283\) 1.39956e6i 1.03878i 0.854536 + 0.519392i \(0.173842\pi\)
−0.854536 + 0.519392i \(0.826158\pi\)
\(284\) 0 0
\(285\) 567480.i 0.413845i
\(286\) 0 0
\(287\) 236463. 0.169457
\(288\) 0 0
\(289\) −907090. −0.638860
\(290\) 0 0
\(291\) − 1.17407e6i − 0.812755i
\(292\) 0 0
\(293\) 407532.i 0.277327i 0.990340 + 0.138664i \(0.0442807\pi\)
−0.990340 + 0.138664i \(0.955719\pi\)
\(294\) 0 0
\(295\) −182141. −0.121858
\(296\) 0 0
\(297\) 272440. 0.179217
\(298\) 0 0
\(299\) 204276.i 0.132142i
\(300\) 0 0
\(301\) 14987.5i 0.00953483i
\(302\) 0 0
\(303\) 1.69126e6 1.05828
\(304\) 0 0
\(305\) 485437. 0.298802
\(306\) 0 0
\(307\) − 3.06612e6i − 1.85671i −0.371699 0.928353i \(-0.621225\pi\)
0.371699 0.928353i \(-0.378775\pi\)
\(308\) 0 0
\(309\) 1.08739e6i 0.647874i
\(310\) 0 0
\(311\) −3.37340e6 −1.97773 −0.988864 0.148822i \(-0.952452\pi\)
−0.988864 + 0.148822i \(0.952452\pi\)
\(312\) 0 0
\(313\) 1.20582e6 0.695702 0.347851 0.937550i \(-0.386911\pi\)
0.347851 + 0.937550i \(0.386911\pi\)
\(314\) 0 0
\(315\) 241669.i 0.137229i
\(316\) 0 0
\(317\) − 2.43849e6i − 1.36293i −0.731851 0.681464i \(-0.761343\pi\)
0.731851 0.681464i \(-0.238657\pi\)
\(318\) 0 0
\(319\) −211340. −0.116280
\(320\) 0 0
\(321\) −1.07478e6 −0.582177
\(322\) 0 0
\(323\) 1.67112e6i 0.891254i
\(324\) 0 0
\(325\) 458356.i 0.240710i
\(326\) 0 0
\(327\) 442766. 0.228984
\(328\) 0 0
\(329\) 911879. 0.464459
\(330\) 0 0
\(331\) − 465766.i − 0.233667i −0.993152 0.116834i \(-0.962726\pi\)
0.993152 0.116834i \(-0.0372744\pi\)
\(332\) 0 0
\(333\) − 1.80972e6i − 0.894338i
\(334\) 0 0
\(335\) −34771.9 −0.0169284
\(336\) 0 0
\(337\) −119437. −0.0572881 −0.0286441 0.999590i \(-0.509119\pi\)
−0.0286441 + 0.999590i \(0.509119\pi\)
\(338\) 0 0
\(339\) 1.76397e6i 0.833667i
\(340\) 0 0
\(341\) 432756.i 0.201538i
\(342\) 0 0
\(343\) −1.91328e6 −0.878098
\(344\) 0 0
\(345\) 67732.6 0.0306373
\(346\) 0 0
\(347\) − 2.44229e6i − 1.08886i −0.838805 0.544431i \(-0.816746\pi\)
0.838805 0.544431i \(-0.183254\pi\)
\(348\) 0 0
\(349\) − 908422.i − 0.399231i −0.979874 0.199615i \(-0.936031\pi\)
0.979874 0.199615i \(-0.0639693\pi\)
\(350\) 0 0
\(351\) 2.79189e6 1.20957
\(352\) 0 0
\(353\) 4.33295e6 1.85075 0.925374 0.379055i \(-0.123751\pi\)
0.925374 + 0.379055i \(0.123751\pi\)
\(354\) 0 0
\(355\) 1.43289e6i 0.603452i
\(356\) 0 0
\(357\) − 453724.i − 0.188417i
\(358\) 0 0
\(359\) 3.84357e6 1.57398 0.786988 0.616968i \(-0.211639\pi\)
0.786988 + 0.616968i \(0.211639\pi\)
\(360\) 0 0
\(361\) −2.97012e6 −1.19952
\(362\) 0 0
\(363\) 1.51667e6i 0.604122i
\(364\) 0 0
\(365\) − 888899.i − 0.349237i
\(366\) 0 0
\(367\) 3.40048e6 1.31788 0.658938 0.752197i \(-0.271006\pi\)
0.658938 + 0.752197i \(0.271006\pi\)
\(368\) 0 0
\(369\) 538649. 0.205940
\(370\) 0 0
\(371\) − 1.46027e6i − 0.550805i
\(372\) 0 0
\(373\) − 728064.i − 0.270955i −0.990780 0.135478i \(-0.956743\pi\)
0.990780 0.135478i \(-0.0432569\pi\)
\(374\) 0 0
\(375\) 151979. 0.0558090
\(376\) 0 0
\(377\) −2.16576e6 −0.784795
\(378\) 0 0
\(379\) − 1.03272e6i − 0.369306i −0.982804 0.184653i \(-0.940884\pi\)
0.982804 0.184653i \(-0.0591161\pi\)
\(380\) 0 0
\(381\) 972427.i 0.343198i
\(382\) 0 0
\(383\) 4.21824e6 1.46938 0.734690 0.678403i \(-0.237328\pi\)
0.734690 + 0.678403i \(0.237328\pi\)
\(384\) 0 0
\(385\) 116548. 0.0400731
\(386\) 0 0
\(387\) 34140.7i 0.0115876i
\(388\) 0 0
\(389\) − 3.25817e6i − 1.09169i −0.837886 0.545845i \(-0.816209\pi\)
0.837886 0.545845i \(-0.183791\pi\)
\(390\) 0 0
\(391\) 199460. 0.0659801
\(392\) 0 0
\(393\) 1.33414e6 0.435734
\(394\) 0 0
\(395\) 2.32529e6i 0.749866i
\(396\) 0 0
\(397\) − 174369.i − 0.0555255i −0.999615 0.0277628i \(-0.991162\pi\)
0.999615 0.0277628i \(-0.00883830\pi\)
\(398\) 0 0
\(399\) 1.47870e6 0.464994
\(400\) 0 0
\(401\) −5.38282e6 −1.67166 −0.835831 0.548986i \(-0.815014\pi\)
−0.835831 + 0.548986i \(0.815014\pi\)
\(402\) 0 0
\(403\) 4.43476e6i 1.36021i
\(404\) 0 0
\(405\) − 24231.7i − 0.00734085i
\(406\) 0 0
\(407\) −872762. −0.261162
\(408\) 0 0
\(409\) 3.44650e6 1.01876 0.509378 0.860543i \(-0.329875\pi\)
0.509378 + 0.860543i \(0.329875\pi\)
\(410\) 0 0
\(411\) − 1.76992e6i − 0.516831i
\(412\) 0 0
\(413\) 474610.i 0.136918i
\(414\) 0 0
\(415\) −39975.9 −0.0113941
\(416\) 0 0
\(417\) −1.30818e6 −0.368407
\(418\) 0 0
\(419\) − 4.73672e6i − 1.31808i −0.752107 0.659041i \(-0.770962\pi\)
0.752107 0.659041i \(-0.229038\pi\)
\(420\) 0 0
\(421\) − 2.96214e6i − 0.814516i −0.913313 0.407258i \(-0.866485\pi\)
0.913313 0.407258i \(-0.133515\pi\)
\(422\) 0 0
\(423\) 2.07721e6 0.564455
\(424\) 0 0
\(425\) 447548. 0.120190
\(426\) 0 0
\(427\) − 1.26492e6i − 0.335731i
\(428\) 0 0
\(429\) − 510483.i − 0.133918i
\(430\) 0 0
\(431\) 6.11636e6 1.58599 0.792995 0.609229i \(-0.208521\pi\)
0.792995 + 0.609229i \(0.208521\pi\)
\(432\) 0 0
\(433\) 7.55385e6 1.93619 0.968097 0.250577i \(-0.0806204\pi\)
0.968097 + 0.250577i \(0.0806204\pi\)
\(434\) 0 0
\(435\) 718107.i 0.181956i
\(436\) 0 0
\(437\) 650044.i 0.162832i
\(438\) 0 0
\(439\) −4.84092e6 −1.19886 −0.599428 0.800429i \(-0.704605\pi\)
−0.599428 + 0.800429i \(0.704605\pi\)
\(440\) 0 0
\(441\) −1.86431e6 −0.456480
\(442\) 0 0
\(443\) 6.96723e6i 1.68675i 0.537325 + 0.843375i \(0.319435\pi\)
−0.537325 + 0.843375i \(0.680565\pi\)
\(444\) 0 0
\(445\) 50743.5i 0.0121473i
\(446\) 0 0
\(447\) 3.38526e6 0.801352
\(448\) 0 0
\(449\) −3.85998e6 −0.903586 −0.451793 0.892123i \(-0.649215\pi\)
−0.451793 + 0.892123i \(0.649215\pi\)
\(450\) 0 0
\(451\) − 259770.i − 0.0601379i
\(452\) 0 0
\(453\) − 1.55563e6i − 0.356172i
\(454\) 0 0
\(455\) 1.19435e6 0.270460
\(456\) 0 0
\(457\) 1.52936e6 0.342547 0.171273 0.985224i \(-0.445212\pi\)
0.171273 + 0.985224i \(0.445212\pi\)
\(458\) 0 0
\(459\) − 2.72606e6i − 0.603953i
\(460\) 0 0
\(461\) − 7.36871e6i − 1.61488i −0.589952 0.807438i \(-0.700854\pi\)
0.589952 0.807438i \(-0.299146\pi\)
\(462\) 0 0
\(463\) 2.95746e6 0.641161 0.320580 0.947221i \(-0.396122\pi\)
0.320580 + 0.947221i \(0.396122\pi\)
\(464\) 0 0
\(465\) 1.47045e6 0.315368
\(466\) 0 0
\(467\) − 6.04555e6i − 1.28275i −0.767226 0.641377i \(-0.778363\pi\)
0.767226 0.641377i \(-0.221637\pi\)
\(468\) 0 0
\(469\) 90606.1i 0.0190207i
\(470\) 0 0
\(471\) 2.07019e6 0.429990
\(472\) 0 0
\(473\) 16464.8 0.00338379
\(474\) 0 0
\(475\) 1.45857e6i 0.296615i
\(476\) 0 0
\(477\) − 3.32641e6i − 0.669391i
\(478\) 0 0
\(479\) 3.59287e6 0.715489 0.357745 0.933819i \(-0.383546\pi\)
0.357745 + 0.933819i \(0.383546\pi\)
\(480\) 0 0
\(481\) −8.94383e6 −1.76263
\(482\) 0 0
\(483\) − 176493.i − 0.0344238i
\(484\) 0 0
\(485\) − 3.01765e6i − 0.582526i
\(486\) 0 0
\(487\) −6.30036e6 −1.20377 −0.601884 0.798584i \(-0.705583\pi\)
−0.601884 + 0.798584i \(0.705583\pi\)
\(488\) 0 0
\(489\) −3.98959e6 −0.754494
\(490\) 0 0
\(491\) − 8.99067e6i − 1.68302i −0.540245 0.841508i \(-0.681668\pi\)
0.540245 0.841508i \(-0.318332\pi\)
\(492\) 0 0
\(493\) 2.11469e6i 0.391858i
\(494\) 0 0
\(495\) 265490. 0.0487006
\(496\) 0 0
\(497\) 3.73373e6 0.678034
\(498\) 0 0
\(499\) 3.96986e6i 0.713713i 0.934159 + 0.356856i \(0.116151\pi\)
−0.934159 + 0.356856i \(0.883849\pi\)
\(500\) 0 0
\(501\) 342138.i 0.0608985i
\(502\) 0 0
\(503\) −4.44728e6 −0.783745 −0.391873 0.920020i \(-0.628173\pi\)
−0.391873 + 0.920020i \(0.628173\pi\)
\(504\) 0 0
\(505\) 4.34697e6 0.758504
\(506\) 0 0
\(507\) − 1.61986e6i − 0.279870i
\(508\) 0 0
\(509\) − 3.69162e6i − 0.631572i −0.948830 0.315786i \(-0.897732\pi\)
0.948830 0.315786i \(-0.102268\pi\)
\(510\) 0 0
\(511\) −2.31623e6 −0.392400
\(512\) 0 0
\(513\) 8.88429e6 1.49049
\(514\) 0 0
\(515\) 2.79489e6i 0.464350i
\(516\) 0 0
\(517\) − 1.00176e6i − 0.164830i
\(518\) 0 0
\(519\) −3.16739e6 −0.516158
\(520\) 0 0
\(521\) −3.62317e6 −0.584783 −0.292391 0.956299i \(-0.594451\pi\)
−0.292391 + 0.956299i \(0.594451\pi\)
\(522\) 0 0
\(523\) 6.82738e6i 1.09144i 0.837968 + 0.545720i \(0.183744\pi\)
−0.837968 + 0.545720i \(0.816256\pi\)
\(524\) 0 0
\(525\) − 396015.i − 0.0627067i
\(526\) 0 0
\(527\) 4.33019e6 0.679173
\(528\) 0 0
\(529\) −6.35876e6 −0.987945
\(530\) 0 0
\(531\) 1.08113e6i 0.166396i
\(532\) 0 0
\(533\) − 2.66206e6i − 0.405881i
\(534\) 0 0
\(535\) −2.76245e6 −0.417264
\(536\) 0 0
\(537\) −2.24800e6 −0.336403
\(538\) 0 0
\(539\) 899087.i 0.133300i
\(540\) 0 0
\(541\) 1.20284e7i 1.76692i 0.468509 + 0.883459i \(0.344791\pi\)
−0.468509 + 0.883459i \(0.655209\pi\)
\(542\) 0 0
\(543\) −2.01414e6 −0.293150
\(544\) 0 0
\(545\) 1.13803e6 0.164120
\(546\) 0 0
\(547\) 742984.i 0.106172i 0.998590 + 0.0530861i \(0.0169058\pi\)
−0.998590 + 0.0530861i \(0.983094\pi\)
\(548\) 0 0
\(549\) − 2.88141e6i − 0.408013i
\(550\) 0 0
\(551\) −6.89182e6 −0.967064
\(552\) 0 0
\(553\) 6.05907e6 0.842545
\(554\) 0 0
\(555\) 2.96554e6i 0.408668i
\(556\) 0 0
\(557\) 9.35350e6i 1.27743i 0.769444 + 0.638714i \(0.220533\pi\)
−0.769444 + 0.638714i \(0.779467\pi\)
\(558\) 0 0
\(559\) 168726. 0.0228378
\(560\) 0 0
\(561\) −498446. −0.0668669
\(562\) 0 0
\(563\) 9.70970e6i 1.29103i 0.763749 + 0.645513i \(0.223356\pi\)
−0.763749 + 0.645513i \(0.776644\pi\)
\(564\) 0 0
\(565\) 4.53387e6i 0.597514i
\(566\) 0 0
\(567\) −63141.3 −0.00824813
\(568\) 0 0
\(569\) 1.09705e7 1.42052 0.710258 0.703941i \(-0.248578\pi\)
0.710258 + 0.703941i \(0.248578\pi\)
\(570\) 0 0
\(571\) − 9.87288e6i − 1.26722i −0.773651 0.633612i \(-0.781572\pi\)
0.773651 0.633612i \(-0.218428\pi\)
\(572\) 0 0
\(573\) − 1.85666e6i − 0.236236i
\(574\) 0 0
\(575\) 174091. 0.0219586
\(576\) 0 0
\(577\) −9.27844e6 −1.16021 −0.580103 0.814543i \(-0.696988\pi\)
−0.580103 + 0.814543i \(0.696988\pi\)
\(578\) 0 0
\(579\) − 38992.8i − 0.00483379i
\(580\) 0 0
\(581\) 104166.i 0.0128023i
\(582\) 0 0
\(583\) −1.60420e6 −0.195473
\(584\) 0 0
\(585\) 2.72067e6 0.328689
\(586\) 0 0
\(587\) 2.34170e6i 0.280502i 0.990116 + 0.140251i \(0.0447910\pi\)
−0.990116 + 0.140251i \(0.955209\pi\)
\(588\) 0 0
\(589\) 1.41122e7i 1.67613i
\(590\) 0 0
\(591\) −5.03261e6 −0.592686
\(592\) 0 0
\(593\) 3.14474e6 0.367238 0.183619 0.982997i \(-0.441219\pi\)
0.183619 + 0.982997i \(0.441219\pi\)
\(594\) 0 0
\(595\) − 1.16619e6i − 0.135044i
\(596\) 0 0
\(597\) − 9.57994e6i − 1.10009i
\(598\) 0 0
\(599\) −3.65098e6 −0.415759 −0.207880 0.978154i \(-0.566656\pi\)
−0.207880 + 0.978154i \(0.566656\pi\)
\(600\) 0 0
\(601\) −1.23821e7 −1.39832 −0.699161 0.714965i \(-0.746443\pi\)
−0.699161 + 0.714965i \(0.746443\pi\)
\(602\) 0 0
\(603\) 206396.i 0.0231157i
\(604\) 0 0
\(605\) 3.89824e6i 0.432992i
\(606\) 0 0
\(607\) −1.06103e7 −1.16884 −0.584421 0.811451i \(-0.698678\pi\)
−0.584421 + 0.811451i \(0.698678\pi\)
\(608\) 0 0
\(609\) 1.87119e6 0.204444
\(610\) 0 0
\(611\) − 1.02658e7i − 1.11247i
\(612\) 0 0
\(613\) 1.01958e7i 1.09590i 0.836511 + 0.547950i \(0.184592\pi\)
−0.836511 + 0.547950i \(0.815408\pi\)
\(614\) 0 0
\(615\) −882667. −0.0941042
\(616\) 0 0
\(617\) −1.30889e7 −1.38417 −0.692086 0.721815i \(-0.743308\pi\)
−0.692086 + 0.721815i \(0.743308\pi\)
\(618\) 0 0
\(619\) − 8.92428e6i − 0.936152i −0.883688 0.468076i \(-0.844947\pi\)
0.883688 0.468076i \(-0.155053\pi\)
\(620\) 0 0
\(621\) − 1.06040e6i − 0.110342i
\(622\) 0 0
\(623\) 132224. 0.0136486
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 1.62445e6i − 0.165020i
\(628\) 0 0
\(629\) 8.73294e6i 0.880103i
\(630\) 0 0
\(631\) −6.87718e6 −0.687602 −0.343801 0.939043i \(-0.611715\pi\)
−0.343801 + 0.939043i \(0.611715\pi\)
\(632\) 0 0
\(633\) 8.95765e6 0.888556
\(634\) 0 0
\(635\) 2.49939e6i 0.245980i
\(636\) 0 0
\(637\) 9.21359e6i 0.899664i
\(638\) 0 0
\(639\) 8.50522e6 0.824011
\(640\) 0 0
\(641\) 1.39914e7 1.34498 0.672489 0.740107i \(-0.265225\pi\)
0.672489 + 0.740107i \(0.265225\pi\)
\(642\) 0 0
\(643\) 6.77228e6i 0.645962i 0.946405 + 0.322981i \(0.104685\pi\)
−0.946405 + 0.322981i \(0.895315\pi\)
\(644\) 0 0
\(645\) − 55945.2i − 0.00529497i
\(646\) 0 0
\(647\) 1.16544e7 1.09454 0.547268 0.836958i \(-0.315668\pi\)
0.547268 + 0.836958i \(0.315668\pi\)
\(648\) 0 0
\(649\) 521391. 0.0485905
\(650\) 0 0
\(651\) − 3.83159e6i − 0.354345i
\(652\) 0 0
\(653\) 6.39818e6i 0.587183i 0.955931 + 0.293591i \(0.0948505\pi\)
−0.955931 + 0.293591i \(0.905149\pi\)
\(654\) 0 0
\(655\) 3.42910e6 0.312303
\(656\) 0 0
\(657\) −5.27624e6 −0.476882
\(658\) 0 0
\(659\) 2.45934e6i 0.220600i 0.993898 + 0.110300i \(0.0351812\pi\)
−0.993898 + 0.110300i \(0.964819\pi\)
\(660\) 0 0
\(661\) 1.99968e7i 1.78015i 0.455816 + 0.890074i \(0.349348\pi\)
−0.455816 + 0.890074i \(0.650652\pi\)
\(662\) 0 0
\(663\) −5.10794e6 −0.451296
\(664\) 0 0
\(665\) 3.80064e6 0.333275
\(666\) 0 0
\(667\) 822587.i 0.0715924i
\(668\) 0 0
\(669\) − 1.81881e6i − 0.157116i
\(670\) 0 0
\(671\) −1.38959e6 −0.119147
\(672\) 0 0
\(673\) 8.57475e6 0.729766 0.364883 0.931053i \(-0.381109\pi\)
0.364883 + 0.931053i \(0.381109\pi\)
\(674\) 0 0
\(675\) − 2.37933e6i − 0.201000i
\(676\) 0 0
\(677\) 5.45187e6i 0.457166i 0.973524 + 0.228583i \(0.0734092\pi\)
−0.973524 + 0.228583i \(0.926591\pi\)
\(678\) 0 0
\(679\) −7.86319e6 −0.654522
\(680\) 0 0
\(681\) −7.75050e6 −0.640416
\(682\) 0 0
\(683\) 1.77200e7i 1.45349i 0.686909 + 0.726743i \(0.258967\pi\)
−0.686909 + 0.726743i \(0.741033\pi\)
\(684\) 0 0
\(685\) − 4.54915e6i − 0.370428i
\(686\) 0 0
\(687\) −1.11381e7 −0.900366
\(688\) 0 0
\(689\) −1.64394e7 −1.31928
\(690\) 0 0
\(691\) − 1.58363e7i − 1.26171i −0.775902 0.630853i \(-0.782705\pi\)
0.775902 0.630853i \(-0.217295\pi\)
\(692\) 0 0
\(693\) − 691794.i − 0.0547197i
\(694\) 0 0
\(695\) −3.36237e6 −0.264049
\(696\) 0 0
\(697\) −2.59929e6 −0.202662
\(698\) 0 0
\(699\) 1.22041e7i 0.944742i
\(700\) 0 0
\(701\) − 1.17168e7i − 0.900565i −0.892886 0.450283i \(-0.851323\pi\)
0.892886 0.450283i \(-0.148677\pi\)
\(702\) 0 0
\(703\) −2.84609e7 −2.17200
\(704\) 0 0
\(705\) −3.40385e6 −0.257928
\(706\) 0 0
\(707\) − 1.13270e7i − 0.852250i
\(708\) 0 0
\(709\) 5.36977e6i 0.401181i 0.979675 + 0.200591i \(0.0642861\pi\)
−0.979675 + 0.200591i \(0.935714\pi\)
\(710\) 0 0
\(711\) 1.38022e7 1.02394
\(712\) 0 0
\(713\) 1.68439e6 0.124085
\(714\) 0 0
\(715\) − 1.31207e6i − 0.0959828i
\(716\) 0 0
\(717\) 4.41878e6i 0.321000i
\(718\) 0 0
\(719\) −7.02495e6 −0.506782 −0.253391 0.967364i \(-0.581546\pi\)
−0.253391 + 0.967364i \(0.581546\pi\)
\(720\) 0 0
\(721\) 7.28271e6 0.521741
\(722\) 0 0
\(723\) 2.34911e6i 0.167131i
\(724\) 0 0
\(725\) 1.84572e6i 0.130413i
\(726\) 0 0
\(727\) −1.57023e7 −1.10187 −0.550933 0.834550i \(-0.685728\pi\)
−0.550933 + 0.834550i \(0.685728\pi\)
\(728\) 0 0
\(729\) −9.14180e6 −0.637107
\(730\) 0 0
\(731\) − 164748.i − 0.0114032i
\(732\) 0 0
\(733\) 1.01796e6i 0.0699794i 0.999388 + 0.0349897i \(0.0111398\pi\)
−0.999388 + 0.0349897i \(0.988860\pi\)
\(734\) 0 0
\(735\) 3.05498e6 0.208588
\(736\) 0 0
\(737\) 99536.9 0.00675018
\(738\) 0 0
\(739\) 1.30624e7i 0.879857i 0.898033 + 0.439929i \(0.144996\pi\)
−0.898033 + 0.439929i \(0.855004\pi\)
\(740\) 0 0
\(741\) − 1.66469e7i − 1.11375i
\(742\) 0 0
\(743\) 2.27890e7 1.51444 0.757222 0.653157i \(-0.226556\pi\)
0.757222 + 0.653157i \(0.226556\pi\)
\(744\) 0 0
\(745\) 8.70100e6 0.574352
\(746\) 0 0
\(747\) 237285.i 0.0155585i
\(748\) 0 0
\(749\) 7.19821e6i 0.468835i
\(750\) 0 0
\(751\) −6.67237e6 −0.431699 −0.215849 0.976427i \(-0.569252\pi\)
−0.215849 + 0.976427i \(0.569252\pi\)
\(752\) 0 0
\(753\) −218535. −0.0140454
\(754\) 0 0
\(755\) − 3.99837e6i − 0.255279i
\(756\) 0 0
\(757\) 1.93030e7i 1.22429i 0.790745 + 0.612146i \(0.209693\pi\)
−0.790745 + 0.612146i \(0.790307\pi\)
\(758\) 0 0
\(759\) −193889. −0.0122166
\(760\) 0 0
\(761\) 171195. 0.0107159 0.00535795 0.999986i \(-0.498295\pi\)
0.00535795 + 0.999986i \(0.498295\pi\)
\(762\) 0 0
\(763\) − 2.96538e6i − 0.184404i
\(764\) 0 0
\(765\) − 2.65651e6i − 0.164119i
\(766\) 0 0
\(767\) 5.34307e6 0.327946
\(768\) 0 0
\(769\) 5.91465e6 0.360673 0.180336 0.983605i \(-0.442281\pi\)
0.180336 + 0.983605i \(0.442281\pi\)
\(770\) 0 0
\(771\) − 3.80472e6i − 0.230508i
\(772\) 0 0
\(773\) − 2.30974e7i − 1.39032i −0.718856 0.695160i \(-0.755334\pi\)
0.718856 0.695160i \(-0.244666\pi\)
\(774\) 0 0
\(775\) 3.77944e6 0.226034
\(776\) 0 0
\(777\) 7.72738e6 0.459177
\(778\) 0 0
\(779\) − 8.47114e6i − 0.500148i
\(780\) 0 0
\(781\) − 4.10175e6i − 0.240625i
\(782\) 0 0
\(783\) 1.12425e7 0.655326
\(784\) 0 0
\(785\) 5.32094e6 0.308187
\(786\) 0 0
\(787\) 1.06371e7i 0.612192i 0.952001 + 0.306096i \(0.0990229\pi\)
−0.952001 + 0.306096i \(0.900977\pi\)
\(788\) 0 0
\(789\) − 2.14781e7i − 1.22830i
\(790\) 0 0
\(791\) 1.18140e7 0.671362
\(792\) 0 0
\(793\) −1.42402e7 −0.804142
\(794\) 0 0
\(795\) 5.45087e6i 0.305878i
\(796\) 0 0
\(797\) − 3.28380e7i − 1.83118i −0.402115 0.915589i \(-0.631725\pi\)
0.402115 0.915589i \(-0.368275\pi\)
\(798\) 0 0
\(799\) −1.00237e7 −0.555470
\(800\) 0 0
\(801\) 301198. 0.0165871
\(802\) 0 0
\(803\) 2.54453e6i 0.139258i
\(804\) 0 0
\(805\) − 453633.i − 0.0246726i
\(806\) 0 0
\(807\) 2.82177e6 0.152524
\(808\) 0 0
\(809\) 1.93286e7 1.03831 0.519157 0.854679i \(-0.326246\pi\)
0.519157 + 0.854679i \(0.326246\pi\)
\(810\) 0 0
\(811\) − 95836.0i − 0.00511654i −0.999997 0.00255827i \(-0.999186\pi\)
0.999997 0.00255827i \(-0.000814324\pi\)
\(812\) 0 0
\(813\) 2.45497e6i 0.130263i
\(814\) 0 0
\(815\) −1.02543e7 −0.540768
\(816\) 0 0
\(817\) 536918. 0.0281419
\(818\) 0 0
\(819\) − 7.08931e6i − 0.369313i
\(820\) 0 0
\(821\) 3.19867e6i 0.165620i 0.996565 + 0.0828098i \(0.0263894\pi\)
−0.996565 + 0.0828098i \(0.973611\pi\)
\(822\) 0 0
\(823\) 4.78006e6 0.245999 0.123000 0.992407i \(-0.460749\pi\)
0.123000 + 0.992407i \(0.460749\pi\)
\(824\) 0 0
\(825\) −435049. −0.0222538
\(826\) 0 0
\(827\) 2.64456e7i 1.34459i 0.740283 + 0.672296i \(0.234692\pi\)
−0.740283 + 0.672296i \(0.765308\pi\)
\(828\) 0 0
\(829\) − 1.72636e7i − 0.872460i −0.899835 0.436230i \(-0.856313\pi\)
0.899835 0.436230i \(-0.143687\pi\)
\(830\) 0 0
\(831\) −6.79816e6 −0.341499
\(832\) 0 0
\(833\) 8.99634e6 0.449214
\(834\) 0 0
\(835\) 879384.i 0.0436478i
\(836\) 0 0
\(837\) − 2.30209e7i − 1.13582i
\(838\) 0 0
\(839\) 2.59843e7 1.27440 0.637200 0.770699i \(-0.280093\pi\)
0.637200 + 0.770699i \(0.280093\pi\)
\(840\) 0 0
\(841\) 1.17900e7 0.574810
\(842\) 0 0
\(843\) − 2.43087e7i − 1.17813i
\(844\) 0 0
\(845\) − 4.16346e6i − 0.200591i
\(846\) 0 0
\(847\) 1.01578e7 0.486507
\(848\) 0 0
\(849\) −1.36130e7 −0.648164
\(850\) 0 0
\(851\) 3.39700e6i 0.160795i
\(852\) 0 0
\(853\) 3.60072e7i 1.69440i 0.531271 + 0.847202i \(0.321715\pi\)
−0.531271 + 0.847202i \(0.678285\pi\)
\(854\) 0 0
\(855\) 8.65764e6 0.405027
\(856\) 0 0
\(857\) −1.44304e7 −0.671158 −0.335579 0.942012i \(-0.608932\pi\)
−0.335579 + 0.942012i \(0.608932\pi\)
\(858\) 0 0
\(859\) − 2.88104e7i − 1.33219i −0.745867 0.666095i \(-0.767965\pi\)
0.745867 0.666095i \(-0.232035\pi\)
\(860\) 0 0
\(861\) 2.29999e6i 0.105735i
\(862\) 0 0
\(863\) 1.48508e7 0.678772 0.339386 0.940647i \(-0.389781\pi\)
0.339386 + 0.940647i \(0.389781\pi\)
\(864\) 0 0
\(865\) −8.14101e6 −0.369946
\(866\) 0 0
\(867\) − 8.82294e6i − 0.398626i
\(868\) 0 0
\(869\) − 6.65629e6i − 0.299008i
\(870\) 0 0
\(871\) 1.02003e6 0.0455581
\(872\) 0 0
\(873\) −1.79119e7 −0.795437
\(874\) 0 0
\(875\) − 1.01786e6i − 0.0449437i
\(876\) 0 0
\(877\) 8.37357e6i 0.367630i 0.982961 + 0.183815i \(0.0588448\pi\)
−0.982961 + 0.183815i \(0.941155\pi\)
\(878\) 0 0
\(879\) −3.96391e6 −0.173042
\(880\) 0 0
\(881\) −2.64394e7 −1.14766 −0.573829 0.818975i \(-0.694543\pi\)
−0.573829 + 0.818975i \(0.694543\pi\)
\(882\) 0 0
\(883\) 595075.i 0.0256844i 0.999918 + 0.0128422i \(0.00408792\pi\)
−0.999918 + 0.0128422i \(0.995912\pi\)
\(884\) 0 0
\(885\) − 1.77162e6i − 0.0760348i
\(886\) 0 0
\(887\) 4.09777e7 1.74879 0.874396 0.485213i \(-0.161258\pi\)
0.874396 + 0.485213i \(0.161258\pi\)
\(888\) 0 0
\(889\) 6.51274e6 0.276382
\(890\) 0 0
\(891\) 69364.9i 0.00292715i
\(892\) 0 0
\(893\) − 3.26675e7i − 1.37084i
\(894\) 0 0
\(895\) −5.77794e6 −0.241110
\(896\) 0 0
\(897\) −1.98692e6 −0.0824517
\(898\) 0 0
\(899\) 1.78580e7i 0.736944i
\(900\) 0 0
\(901\) 1.60518e7i 0.658736i
\(902\) 0 0
\(903\) −145778. −0.00594939
\(904\) 0 0
\(905\) −5.17687e6 −0.210110
\(906\) 0 0
\(907\) 1.30125e7i 0.525222i 0.964902 + 0.262611i \(0.0845836\pi\)
−0.964902 + 0.262611i \(0.915416\pi\)
\(908\) 0 0
\(909\) − 2.58023e7i − 1.03574i
\(910\) 0 0
\(911\) 2.09280e7 0.835472 0.417736 0.908568i \(-0.362824\pi\)
0.417736 + 0.908568i \(0.362824\pi\)
\(912\) 0 0
\(913\) 114434. 0.00454336
\(914\) 0 0
\(915\) 4.72167e6i 0.186441i
\(916\) 0 0
\(917\) − 8.93530e6i − 0.350902i
\(918\) 0 0
\(919\) 2.58899e7 1.01121 0.505605 0.862765i \(-0.331269\pi\)
0.505605 + 0.862765i \(0.331269\pi\)
\(920\) 0 0
\(921\) 2.98230e7 1.15852
\(922\) 0 0
\(923\) − 4.20336e7i − 1.62402i
\(924\) 0 0
\(925\) 7.62220e6i 0.292905i
\(926\) 0 0
\(927\) 1.65896e7 0.634069
\(928\) 0 0
\(929\) −6.83846e6 −0.259967 −0.129984 0.991516i \(-0.541493\pi\)
−0.129984 + 0.991516i \(0.541493\pi\)
\(930\) 0 0
\(931\) 2.93193e7i 1.10861i
\(932\) 0 0
\(933\) − 3.28118e7i − 1.23403i
\(934\) 0 0
\(935\) −1.28114e6 −0.0479255
\(936\) 0 0
\(937\) 2.70839e7 1.00777 0.503886 0.863770i \(-0.331903\pi\)
0.503886 + 0.863770i \(0.331903\pi\)
\(938\) 0 0
\(939\) 1.17286e7i 0.434093i
\(940\) 0 0
\(941\) − 1.67445e7i − 0.616452i −0.951313 0.308226i \(-0.900265\pi\)
0.951313 0.308226i \(-0.0997353\pi\)
\(942\) 0 0
\(943\) −1.01109e6 −0.0370263
\(944\) 0 0
\(945\) −6.19990e6 −0.225842
\(946\) 0 0
\(947\) − 1.30092e7i − 0.471386i −0.971828 0.235693i \(-0.924264\pi\)
0.971828 0.235693i \(-0.0757360\pi\)
\(948\) 0 0
\(949\) 2.60757e7i 0.939875i
\(950\) 0 0
\(951\) 2.37183e7 0.850419
\(952\) 0 0
\(953\) −1.37164e7 −0.489224 −0.244612 0.969621i \(-0.578661\pi\)
−0.244612 + 0.969621i \(0.578661\pi\)
\(954\) 0 0
\(955\) − 4.77210e6i − 0.169317i
\(956\) 0 0
\(957\) − 2.05563e6i − 0.0725546i
\(958\) 0 0
\(959\) −1.18539e7 −0.416211
\(960\) 0 0
\(961\) 7.93826e6 0.277279
\(962\) 0 0
\(963\) 1.63971e7i 0.569772i
\(964\) 0 0
\(965\) − 100222.i − 0.00346452i
\(966\) 0 0
\(967\) 7.73160e6 0.265891 0.132945 0.991123i \(-0.457557\pi\)
0.132945 + 0.991123i \(0.457557\pi\)
\(968\) 0 0
\(969\) −1.62544e7 −0.556110
\(970\) 0 0
\(971\) 4.66660e7i 1.58837i 0.607674 + 0.794187i \(0.292103\pi\)
−0.607674 + 0.794187i \(0.707897\pi\)
\(972\) 0 0
\(973\) 8.76143e6i 0.296683i
\(974\) 0 0
\(975\) −4.45826e6 −0.150195
\(976\) 0 0
\(977\) 3.88117e7 1.30085 0.650424 0.759572i \(-0.274591\pi\)
0.650424 + 0.759572i \(0.274591\pi\)
\(978\) 0 0
\(979\) − 145257.i − 0.00484373i
\(980\) 0 0
\(981\) − 6.75498e6i − 0.224105i
\(982\) 0 0
\(983\) 4.79223e7 1.58181 0.790904 0.611940i \(-0.209611\pi\)
0.790904 + 0.611940i \(0.209611\pi\)
\(984\) 0 0
\(985\) −1.29351e7 −0.424796
\(986\) 0 0
\(987\) 8.86951e6i 0.289806i
\(988\) 0 0
\(989\) − 64084.8i − 0.00208336i
\(990\) 0 0
\(991\) −2.44268e7 −0.790101 −0.395050 0.918659i \(-0.629273\pi\)
−0.395050 + 0.918659i \(0.629273\pi\)
\(992\) 0 0
\(993\) 4.53034e6 0.145800
\(994\) 0 0
\(995\) − 2.46230e7i − 0.788465i
\(996\) 0 0
\(997\) − 1.25248e7i − 0.399057i −0.979892 0.199528i \(-0.936059\pi\)
0.979892 0.199528i \(-0.0639410\pi\)
\(998\) 0 0
\(999\) 4.64275e7 1.47184
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 320.6.d.a.161.5 yes 8
4.3 odd 2 320.6.d.b.161.4 yes 8
8.3 odd 2 320.6.d.b.161.5 yes 8
8.5 even 2 inner 320.6.d.a.161.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.a.161.4 8 8.5 even 2 inner
320.6.d.a.161.5 yes 8 1.1 even 1 trivial
320.6.d.b.161.4 yes 8 4.3 odd 2
320.6.d.b.161.5 yes 8 8.3 odd 2