Properties

Label 320.6.d.b.161.2
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 161.2
Root \(-10.3099 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.b.161.7

$q$-expansion

\(f(q)\) \(=\) \(q-21.6198i q^{3} -25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} +O(q^{10})\) \(q-21.6198i q^{3} -25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} +512.594i q^{11} -36.4873i q^{13} -540.496 q^{15} -1260.81 q^{17} -790.251i q^{19} -3663.82i q^{21} -4995.19 q^{23} -625.000 q^{25} -401.768i q^{27} +934.435i q^{29} -6690.60 q^{31} +11082.2 q^{33} -4236.64i q^{35} -4305.71i q^{37} -788.849 q^{39} -10359.3 q^{41} -6371.59i q^{43} +5610.42i q^{45} -22120.7 q^{47} +11911.6 q^{49} +27258.6i q^{51} -23246.6i q^{53} +12814.9 q^{55} -17085.1 q^{57} +27758.9i q^{59} +39950.3i q^{61} -38030.9 q^{63} -912.183 q^{65} -55415.0i q^{67} +107995. i q^{69} -39874.7 q^{71} -435.947 q^{73} +13512.4i q^{75} +86867.1i q^{77} +61654.1 q^{79} -63219.4 q^{81} -2943.22i q^{83} +31520.3i q^{85} +20202.3 q^{87} +64783.7 q^{89} -6183.35i q^{91} +144650. i q^{93} -19756.3 q^{95} +159041. q^{97} -115035. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 320q^{7} - 1192q^{9} + O(q^{10}) \) \( 8q + 320q^{7} - 1192q^{9} - 2400q^{17} - 1760q^{23} - 5000q^{25} - 31040q^{31} + 3760q^{33} - 4480q^{39} + 21584q^{41} - 47680q^{47} + 82824q^{49} - 18400q^{55} - 106640q^{57} - 322400q^{63} - 44000q^{65} - 246720q^{71} + 46400q^{73} - 325760q^{79} - 82328q^{81} - 636320q^{87} - 78192q^{89} + 40800q^{95} + 24960q^{97} + O(q^{100}) \)

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
\(3\) − 21.6198i − 1.38691i −0.720499 0.693456i \(-0.756087\pi\)
0.720499 0.693456i \(-0.243913\pi\)
\(4\) 0 0
\(5\) − 25.0000i − 0.447214i
\(6\) 0 0
\(7\) 169.466 1.30718 0.653592 0.756847i \(-0.273261\pi\)
0.653592 + 0.756847i \(0.273261\pi\)
\(8\) 0 0
\(9\) −224.417 −0.923525
\(10\) 0 0
\(11\) 512.594i 1.27730i 0.769498 + 0.638649i \(0.220506\pi\)
−0.769498 + 0.638649i \(0.779494\pi\)
\(12\) 0 0
\(13\) − 36.4873i − 0.0598802i −0.999552 0.0299401i \(-0.990468\pi\)
0.999552 0.0299401i \(-0.00953166\pi\)
\(14\) 0 0
\(15\) −540.496 −0.620246
\(16\) 0 0
\(17\) −1260.81 −1.05810 −0.529052 0.848589i \(-0.677452\pi\)
−0.529052 + 0.848589i \(0.677452\pi\)
\(18\) 0 0
\(19\) − 790.251i − 0.502205i −0.967960 0.251103i \(-0.919207\pi\)
0.967960 0.251103i \(-0.0807932\pi\)
\(20\) 0 0
\(21\) − 3663.82i − 1.81295i
\(22\) 0 0
\(23\) −4995.19 −1.96894 −0.984470 0.175554i \(-0.943828\pi\)
−0.984470 + 0.175554i \(0.943828\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 401.768i − 0.106064i
\(28\) 0 0
\(29\) 934.435i 0.206326i 0.994664 + 0.103163i \(0.0328963\pi\)
−0.994664 + 0.103163i \(0.967104\pi\)
\(30\) 0 0
\(31\) −6690.60 −1.25043 −0.625217 0.780451i \(-0.714989\pi\)
−0.625217 + 0.780451i \(0.714989\pi\)
\(32\) 0 0
\(33\) 11082.2 1.77150
\(34\) 0 0
\(35\) − 4236.64i − 0.584590i
\(36\) 0 0
\(37\) − 4305.71i − 0.517059i −0.966003 0.258530i \(-0.916762\pi\)
0.966003 0.258530i \(-0.0832379\pi\)
\(38\) 0 0
\(39\) −788.849 −0.0830486
\(40\) 0 0
\(41\) −10359.3 −0.962430 −0.481215 0.876603i \(-0.659804\pi\)
−0.481215 + 0.876603i \(0.659804\pi\)
\(42\) 0 0
\(43\) − 6371.59i − 0.525505i −0.964863 0.262752i \(-0.915370\pi\)
0.964863 0.262752i \(-0.0846303\pi\)
\(44\) 0 0
\(45\) 5610.42i 0.413013i
\(46\) 0 0
\(47\) −22120.7 −1.46068 −0.730338 0.683085i \(-0.760638\pi\)
−0.730338 + 0.683085i \(0.760638\pi\)
\(48\) 0 0
\(49\) 11911.6 0.708730
\(50\) 0 0
\(51\) 27258.6i 1.46750i
\(52\) 0 0
\(53\) − 23246.6i − 1.13676i −0.822766 0.568380i \(-0.807570\pi\)
0.822766 0.568380i \(-0.192430\pi\)
\(54\) 0 0
\(55\) 12814.9 0.571225
\(56\) 0 0
\(57\) −17085.1 −0.696515
\(58\) 0 0
\(59\) 27758.9i 1.03818i 0.854720 + 0.519089i \(0.173729\pi\)
−0.854720 + 0.519089i \(0.826271\pi\)
\(60\) 0 0
\(61\) 39950.3i 1.37466i 0.726345 + 0.687330i \(0.241217\pi\)
−0.726345 + 0.687330i \(0.758783\pi\)
\(62\) 0 0
\(63\) −38030.9 −1.20722
\(64\) 0 0
\(65\) −912.183 −0.0267793
\(66\) 0 0
\(67\) − 55415.0i − 1.50814i −0.656797 0.754068i \(-0.728089\pi\)
0.656797 0.754068i \(-0.271911\pi\)
\(68\) 0 0
\(69\) 107995.i 2.73075i
\(70\) 0 0
\(71\) −39874.7 −0.938752 −0.469376 0.882998i \(-0.655521\pi\)
−0.469376 + 0.882998i \(0.655521\pi\)
\(72\) 0 0
\(73\) −435.947 −0.00957473 −0.00478736 0.999989i \(-0.501524\pi\)
−0.00478736 + 0.999989i \(0.501524\pi\)
\(74\) 0 0
\(75\) 13512.4i 0.277382i
\(76\) 0 0
\(77\) 86867.1i 1.66966i
\(78\) 0 0
\(79\) 61654.1 1.11146 0.555730 0.831363i \(-0.312439\pi\)
0.555730 + 0.831363i \(0.312439\pi\)
\(80\) 0 0
\(81\) −63219.4 −1.07063
\(82\) 0 0
\(83\) − 2943.22i − 0.0468952i −0.999725 0.0234476i \(-0.992536\pi\)
0.999725 0.0234476i \(-0.00746428\pi\)
\(84\) 0 0
\(85\) 31520.3i 0.473199i
\(86\) 0 0
\(87\) 20202.3 0.286156
\(88\) 0 0
\(89\) 64783.7 0.866943 0.433472 0.901167i \(-0.357288\pi\)
0.433472 + 0.901167i \(0.357288\pi\)
\(90\) 0 0
\(91\) − 6183.35i − 0.0782745i
\(92\) 0 0
\(93\) 144650.i 1.73424i
\(94\) 0 0
\(95\) −19756.3 −0.224593
\(96\) 0 0
\(97\) 159041. 1.71625 0.858123 0.513445i \(-0.171631\pi\)
0.858123 + 0.513445i \(0.171631\pi\)
\(98\) 0 0
\(99\) − 115035.i − 1.17962i
\(100\) 0 0
\(101\) − 66973.7i − 0.653282i −0.945148 0.326641i \(-0.894083\pi\)
0.945148 0.326641i \(-0.105917\pi\)
\(102\) 0 0
\(103\) −145452. −1.35091 −0.675454 0.737402i \(-0.736052\pi\)
−0.675454 + 0.737402i \(0.736052\pi\)
\(104\) 0 0
\(105\) −91595.5 −0.810776
\(106\) 0 0
\(107\) − 67124.0i − 0.566785i −0.959004 0.283392i \(-0.908540\pi\)
0.959004 0.283392i \(-0.0914599\pi\)
\(108\) 0 0
\(109\) 102861.i 0.829247i 0.909993 + 0.414624i \(0.136087\pi\)
−0.909993 + 0.414624i \(0.863913\pi\)
\(110\) 0 0
\(111\) −93088.6 −0.717116
\(112\) 0 0
\(113\) −60698.8 −0.447182 −0.223591 0.974683i \(-0.571778\pi\)
−0.223591 + 0.974683i \(0.571778\pi\)
\(114\) 0 0
\(115\) 124880.i 0.880537i
\(116\) 0 0
\(117\) 8188.36i 0.0553009i
\(118\) 0 0
\(119\) −213665. −1.38314
\(120\) 0 0
\(121\) −101702. −0.631489
\(122\) 0 0
\(123\) 223965.i 1.33481i
\(124\) 0 0
\(125\) 15625.0i 0.0894427i
\(126\) 0 0
\(127\) −318984. −1.75493 −0.877464 0.479642i \(-0.840767\pi\)
−0.877464 + 0.479642i \(0.840767\pi\)
\(128\) 0 0
\(129\) −137753. −0.728829
\(130\) 0 0
\(131\) 359449.i 1.83003i 0.403417 + 0.915016i \(0.367822\pi\)
−0.403417 + 0.915016i \(0.632178\pi\)
\(132\) 0 0
\(133\) − 133920.i − 0.656475i
\(134\) 0 0
\(135\) −10044.2 −0.0474331
\(136\) 0 0
\(137\) 289504. 1.31781 0.658906 0.752225i \(-0.271019\pi\)
0.658906 + 0.752225i \(0.271019\pi\)
\(138\) 0 0
\(139\) − 318184.i − 1.39682i −0.715696 0.698412i \(-0.753891\pi\)
0.715696 0.698412i \(-0.246109\pi\)
\(140\) 0 0
\(141\) 478245.i 2.02583i
\(142\) 0 0
\(143\) 18703.2 0.0764849
\(144\) 0 0
\(145\) 23360.9 0.0922718
\(146\) 0 0
\(147\) − 257527.i − 0.982946i
\(148\) 0 0
\(149\) − 464507.i − 1.71406i −0.515266 0.857030i \(-0.672307\pi\)
0.515266 0.857030i \(-0.327693\pi\)
\(150\) 0 0
\(151\) −13229.4 −0.0472171 −0.0236085 0.999721i \(-0.507516\pi\)
−0.0236085 + 0.999721i \(0.507516\pi\)
\(152\) 0 0
\(153\) 282948. 0.977186
\(154\) 0 0
\(155\) 167265.i 0.559211i
\(156\) 0 0
\(157\) − 125487.i − 0.406303i −0.979147 0.203152i \(-0.934882\pi\)
0.979147 0.203152i \(-0.0651184\pi\)
\(158\) 0 0
\(159\) −502587. −1.57659
\(160\) 0 0
\(161\) −846513. −2.57377
\(162\) 0 0
\(163\) − 449649.i − 1.32558i −0.748807 0.662788i \(-0.769373\pi\)
0.748807 0.662788i \(-0.230627\pi\)
\(164\) 0 0
\(165\) − 277055.i − 0.792239i
\(166\) 0 0
\(167\) 157771. 0.437760 0.218880 0.975752i \(-0.429760\pi\)
0.218880 + 0.975752i \(0.429760\pi\)
\(168\) 0 0
\(169\) 369962. 0.996414
\(170\) 0 0
\(171\) 177346.i 0.463799i
\(172\) 0 0
\(173\) 253753.i 0.644608i 0.946636 + 0.322304i \(0.104457\pi\)
−0.946636 + 0.322304i \(0.895543\pi\)
\(174\) 0 0
\(175\) −105916. −0.261437
\(176\) 0 0
\(177\) 600142. 1.43986
\(178\) 0 0
\(179\) 166500.i 0.388403i 0.980962 + 0.194201i \(0.0622115\pi\)
−0.980962 + 0.194201i \(0.937788\pi\)
\(180\) 0 0
\(181\) − 639995.i − 1.45204i −0.687671 0.726022i \(-0.741367\pi\)
0.687671 0.726022i \(-0.258633\pi\)
\(182\) 0 0
\(183\) 863718. 1.90653
\(184\) 0 0
\(185\) −107643. −0.231236
\(186\) 0 0
\(187\) − 646286.i − 1.35151i
\(188\) 0 0
\(189\) − 68086.0i − 0.138645i
\(190\) 0 0
\(191\) 126967. 0.251829 0.125915 0.992041i \(-0.459813\pi\)
0.125915 + 0.992041i \(0.459813\pi\)
\(192\) 0 0
\(193\) 287272. 0.555137 0.277568 0.960706i \(-0.410471\pi\)
0.277568 + 0.960706i \(0.410471\pi\)
\(194\) 0 0
\(195\) 19721.2i 0.0371405i
\(196\) 0 0
\(197\) 500809.i 0.919405i 0.888073 + 0.459702i \(0.152044\pi\)
−0.888073 + 0.459702i \(0.847956\pi\)
\(198\) 0 0
\(199\) 858127. 1.53610 0.768049 0.640392i \(-0.221228\pi\)
0.768049 + 0.640392i \(0.221228\pi\)
\(200\) 0 0
\(201\) −1.19806e6 −2.09165
\(202\) 0 0
\(203\) 158355.i 0.269706i
\(204\) 0 0
\(205\) 258982.i 0.430412i
\(206\) 0 0
\(207\) 1.12100e6 1.81837
\(208\) 0 0
\(209\) 405078. 0.641465
\(210\) 0 0
\(211\) 982183.i 1.51875i 0.650653 + 0.759375i \(0.274495\pi\)
−0.650653 + 0.759375i \(0.725505\pi\)
\(212\) 0 0
\(213\) 862083.i 1.30197i
\(214\) 0 0
\(215\) −159290. −0.235013
\(216\) 0 0
\(217\) −1.13383e6 −1.63455
\(218\) 0 0
\(219\) 9425.09i 0.0132793i
\(220\) 0 0
\(221\) 46003.7i 0.0633595i
\(222\) 0 0
\(223\) −188728. −0.254141 −0.127071 0.991894i \(-0.540557\pi\)
−0.127071 + 0.991894i \(0.540557\pi\)
\(224\) 0 0
\(225\) 140260. 0.184705
\(226\) 0 0
\(227\) − 517079.i − 0.666027i −0.942922 0.333014i \(-0.891934\pi\)
0.942922 0.333014i \(-0.108066\pi\)
\(228\) 0 0
\(229\) − 160000.i − 0.201619i −0.994906 0.100809i \(-0.967857\pi\)
0.994906 0.100809i \(-0.0321432\pi\)
\(230\) 0 0
\(231\) 1.87805e6 2.31568
\(232\) 0 0
\(233\) 1.49124e6 1.79953 0.899765 0.436375i \(-0.143738\pi\)
0.899765 + 0.436375i \(0.143738\pi\)
\(234\) 0 0
\(235\) 553017.i 0.653235i
\(236\) 0 0
\(237\) − 1.33295e6i − 1.54150i
\(238\) 0 0
\(239\) 142742. 0.161643 0.0808215 0.996729i \(-0.474246\pi\)
0.0808215 + 0.996729i \(0.474246\pi\)
\(240\) 0 0
\(241\) −12428.9 −0.0137844 −0.00689222 0.999976i \(-0.502194\pi\)
−0.00689222 + 0.999976i \(0.502194\pi\)
\(242\) 0 0
\(243\) 1.26916e6i 1.37880i
\(244\) 0 0
\(245\) − 297791.i − 0.316954i
\(246\) 0 0
\(247\) −28834.2 −0.0300722
\(248\) 0 0
\(249\) −63632.0 −0.0650395
\(250\) 0 0
\(251\) − 759309.i − 0.760736i −0.924835 0.380368i \(-0.875797\pi\)
0.924835 0.380368i \(-0.124203\pi\)
\(252\) 0 0
\(253\) − 2.56051e6i − 2.51492i
\(254\) 0 0
\(255\) 681464. 0.656285
\(256\) 0 0
\(257\) 500270. 0.472468 0.236234 0.971696i \(-0.424087\pi\)
0.236234 + 0.971696i \(0.424087\pi\)
\(258\) 0 0
\(259\) − 729670.i − 0.675891i
\(260\) 0 0
\(261\) − 209703.i − 0.190547i
\(262\) 0 0
\(263\) 549782. 0.490119 0.245059 0.969508i \(-0.421193\pi\)
0.245059 + 0.969508i \(0.421193\pi\)
\(264\) 0 0
\(265\) −581164. −0.508375
\(266\) 0 0
\(267\) − 1.40061e6i − 1.20237i
\(268\) 0 0
\(269\) − 1.02271e6i − 0.861731i −0.902416 0.430865i \(-0.858208\pi\)
0.902416 0.430865i \(-0.141792\pi\)
\(270\) 0 0
\(271\) −1.12050e6 −0.926807 −0.463404 0.886147i \(-0.653372\pi\)
−0.463404 + 0.886147i \(0.653372\pi\)
\(272\) 0 0
\(273\) −133683. −0.108560
\(274\) 0 0
\(275\) − 320371.i − 0.255459i
\(276\) 0 0
\(277\) − 323208.i − 0.253095i −0.991961 0.126547i \(-0.959610\pi\)
0.991961 0.126547i \(-0.0403896\pi\)
\(278\) 0 0
\(279\) 1.50148e6 1.15481
\(280\) 0 0
\(281\) 325436. 0.245867 0.122933 0.992415i \(-0.460770\pi\)
0.122933 + 0.992415i \(0.460770\pi\)
\(282\) 0 0
\(283\) 2.41107e6i 1.78955i 0.446518 + 0.894775i \(0.352664\pi\)
−0.446518 + 0.894775i \(0.647336\pi\)
\(284\) 0 0
\(285\) 427127.i 0.311491i
\(286\) 0 0
\(287\) −1.75554e6 −1.25807
\(288\) 0 0
\(289\) 169793. 0.119585
\(290\) 0 0
\(291\) − 3.43843e6i − 2.38028i
\(292\) 0 0
\(293\) − 1.89437e6i − 1.28913i −0.764550 0.644565i \(-0.777039\pi\)
0.764550 0.644565i \(-0.222961\pi\)
\(294\) 0 0
\(295\) 693972. 0.464287
\(296\) 0 0
\(297\) 205944. 0.135475
\(298\) 0 0
\(299\) 182261.i 0.117901i
\(300\) 0 0
\(301\) − 1.07977e6i − 0.686932i
\(302\) 0 0
\(303\) −1.44796e6 −0.906045
\(304\) 0 0
\(305\) 998757. 0.614767
\(306\) 0 0
\(307\) 1.37530e6i 0.832823i 0.909176 + 0.416411i \(0.136712\pi\)
−0.909176 + 0.416411i \(0.863288\pi\)
\(308\) 0 0
\(309\) 3.14464e6i 1.87359i
\(310\) 0 0
\(311\) −2.60469e6 −1.52705 −0.763527 0.645776i \(-0.776534\pi\)
−0.763527 + 0.645776i \(0.776534\pi\)
\(312\) 0 0
\(313\) 780226. 0.450152 0.225076 0.974341i \(-0.427737\pi\)
0.225076 + 0.974341i \(0.427737\pi\)
\(314\) 0 0
\(315\) 950773.i 0.539884i
\(316\) 0 0
\(317\) − 1.28045e6i − 0.715673i −0.933784 0.357836i \(-0.883515\pi\)
0.933784 0.357836i \(-0.116485\pi\)
\(318\) 0 0
\(319\) −478986. −0.263540
\(320\) 0 0
\(321\) −1.45121e6 −0.786081
\(322\) 0 0
\(323\) 996359.i 0.531386i
\(324\) 0 0
\(325\) 22804.6i 0.0119760i
\(326\) 0 0
\(327\) 2.22383e6 1.15009
\(328\) 0 0
\(329\) −3.74870e6 −1.90937
\(330\) 0 0
\(331\) − 2.35495e6i − 1.18144i −0.806876 0.590721i \(-0.798843\pi\)
0.806876 0.590721i \(-0.201157\pi\)
\(332\) 0 0
\(333\) 966273.i 0.477517i
\(334\) 0 0
\(335\) −1.38537e6 −0.674459
\(336\) 0 0
\(337\) −4.10943e6 −1.97109 −0.985546 0.169411i \(-0.945814\pi\)
−0.985546 + 0.169411i \(0.945814\pi\)
\(338\) 0 0
\(339\) 1.31230e6i 0.620202i
\(340\) 0 0
\(341\) − 3.42956e6i − 1.59718i
\(342\) 0 0
\(343\) −829599. −0.380744
\(344\) 0 0
\(345\) 2.69988e6 1.22123
\(346\) 0 0
\(347\) 1.41845e6i 0.632398i 0.948693 + 0.316199i \(0.102407\pi\)
−0.948693 + 0.316199i \(0.897593\pi\)
\(348\) 0 0
\(349\) − 2.03977e6i − 0.896434i −0.893925 0.448217i \(-0.852059\pi\)
0.893925 0.448217i \(-0.147941\pi\)
\(350\) 0 0
\(351\) −14659.5 −0.00635112
\(352\) 0 0
\(353\) 129749. 0.0554201 0.0277101 0.999616i \(-0.491178\pi\)
0.0277101 + 0.999616i \(0.491178\pi\)
\(354\) 0 0
\(355\) 996866.i 0.419823i
\(356\) 0 0
\(357\) 4.61939e6i 1.91829i
\(358\) 0 0
\(359\) 1.54034e6 0.630782 0.315391 0.948962i \(-0.397864\pi\)
0.315391 + 0.948962i \(0.397864\pi\)
\(360\) 0 0
\(361\) 1.85160e6 0.747790
\(362\) 0 0
\(363\) 2.19878e6i 0.875819i
\(364\) 0 0
\(365\) 10898.7i 0.00428195i
\(366\) 0 0
\(367\) −1.89303e6 −0.733655 −0.366828 0.930289i \(-0.619556\pi\)
−0.366828 + 0.930289i \(0.619556\pi\)
\(368\) 0 0
\(369\) 2.32479e6 0.888829
\(370\) 0 0
\(371\) − 3.93950e6i − 1.48596i
\(372\) 0 0
\(373\) 3.29601e6i 1.22664i 0.789836 + 0.613319i \(0.210166\pi\)
−0.789836 + 0.613319i \(0.789834\pi\)
\(374\) 0 0
\(375\) 337810. 0.124049
\(376\) 0 0
\(377\) 34095.0 0.0123549
\(378\) 0 0
\(379\) − 1.38042e6i − 0.493643i −0.969061 0.246821i \(-0.920614\pi\)
0.969061 0.246821i \(-0.0793861\pi\)
\(380\) 0 0
\(381\) 6.89637e6i 2.43393i
\(382\) 0 0
\(383\) −3.13515e6 −1.09210 −0.546049 0.837753i \(-0.683869\pi\)
−0.546049 + 0.837753i \(0.683869\pi\)
\(384\) 0 0
\(385\) 2.17168e6 0.746696
\(386\) 0 0
\(387\) 1.42989e6i 0.485317i
\(388\) 0 0
\(389\) 4.57462e6i 1.53279i 0.642372 + 0.766393i \(0.277950\pi\)
−0.642372 + 0.766393i \(0.722050\pi\)
\(390\) 0 0
\(391\) 6.29800e6 2.08334
\(392\) 0 0
\(393\) 7.77122e6 2.53809
\(394\) 0 0
\(395\) − 1.54135e6i − 0.497060i
\(396\) 0 0
\(397\) − 2.13071e6i − 0.678496i −0.940697 0.339248i \(-0.889827\pi\)
0.940697 0.339248i \(-0.110173\pi\)
\(398\) 0 0
\(399\) −2.89534e6 −0.910473
\(400\) 0 0
\(401\) −2.03442e6 −0.631799 −0.315900 0.948793i \(-0.602306\pi\)
−0.315900 + 0.948793i \(0.602306\pi\)
\(402\) 0 0
\(403\) 244122.i 0.0748763i
\(404\) 0 0
\(405\) 1.58049e6i 0.478799i
\(406\) 0 0
\(407\) 2.20708e6 0.660438
\(408\) 0 0
\(409\) −4.79858e6 −1.41842 −0.709210 0.704997i \(-0.750948\pi\)
−0.709210 + 0.704997i \(0.750948\pi\)
\(410\) 0 0
\(411\) − 6.25903e6i − 1.82769i
\(412\) 0 0
\(413\) 4.70418e6i 1.35709i
\(414\) 0 0
\(415\) −73580.6 −0.0209722
\(416\) 0 0
\(417\) −6.87908e6 −1.93727
\(418\) 0 0
\(419\) 1.63789e6i 0.455773i 0.973688 + 0.227887i \(0.0731816\pi\)
−0.973688 + 0.227887i \(0.926818\pi\)
\(420\) 0 0
\(421\) 6.15276e6i 1.69186i 0.533293 + 0.845931i \(0.320954\pi\)
−0.533293 + 0.845931i \(0.679046\pi\)
\(422\) 0 0
\(423\) 4.96425e6 1.34897
\(424\) 0 0
\(425\) 788008. 0.211621
\(426\) 0 0
\(427\) 6.77020e6i 1.79693i
\(428\) 0 0
\(429\) − 404360.i − 0.106078i
\(430\) 0 0
\(431\) −6.74591e6 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(432\) 0 0
\(433\) 6.05459e6 1.55190 0.775952 0.630792i \(-0.217270\pi\)
0.775952 + 0.630792i \(0.217270\pi\)
\(434\) 0 0
\(435\) − 505058.i − 0.127973i
\(436\) 0 0
\(437\) 3.94746e6i 0.988812i
\(438\) 0 0
\(439\) −6.84810e6 −1.69593 −0.847967 0.530049i \(-0.822174\pi\)
−0.847967 + 0.530049i \(0.822174\pi\)
\(440\) 0 0
\(441\) −2.67317e6 −0.654530
\(442\) 0 0
\(443\) 4.22271e6i 1.02231i 0.859489 + 0.511154i \(0.170782\pi\)
−0.859489 + 0.511154i \(0.829218\pi\)
\(444\) 0 0
\(445\) − 1.61959e6i − 0.387709i
\(446\) 0 0
\(447\) −1.00425e7 −2.37725
\(448\) 0 0
\(449\) 733420. 0.171687 0.0858434 0.996309i \(-0.472642\pi\)
0.0858434 + 0.996309i \(0.472642\pi\)
\(450\) 0 0
\(451\) − 5.31010e6i − 1.22931i
\(452\) 0 0
\(453\) 286018.i 0.0654859i
\(454\) 0 0
\(455\) −154584. −0.0350054
\(456\) 0 0
\(457\) −5.36667e6 −1.20203 −0.601014 0.799239i \(-0.705236\pi\)
−0.601014 + 0.799239i \(0.705236\pi\)
\(458\) 0 0
\(459\) 506555.i 0.112226i
\(460\) 0 0
\(461\) − 1.80941e6i − 0.396538i −0.980148 0.198269i \(-0.936468\pi\)
0.980148 0.198269i \(-0.0635320\pi\)
\(462\) 0 0
\(463\) 4.86270e6 1.05421 0.527103 0.849802i \(-0.323278\pi\)
0.527103 + 0.849802i \(0.323278\pi\)
\(464\) 0 0
\(465\) 3.61624e6 0.775576
\(466\) 0 0
\(467\) − 8.18481e6i − 1.73667i −0.495981 0.868334i \(-0.665191\pi\)
0.495981 0.868334i \(-0.334809\pi\)
\(468\) 0 0
\(469\) − 9.39094e6i − 1.97141i
\(470\) 0 0
\(471\) −2.71301e6 −0.563507
\(472\) 0 0
\(473\) 3.26604e6 0.671226
\(474\) 0 0
\(475\) 493907.i 0.100441i
\(476\) 0 0
\(477\) 5.21692e6i 1.04983i
\(478\) 0 0
\(479\) −5.20370e6 −1.03627 −0.518136 0.855298i \(-0.673374\pi\)
−0.518136 + 0.855298i \(0.673374\pi\)
\(480\) 0 0
\(481\) −157104. −0.0309616
\(482\) 0 0
\(483\) 1.83015e7i 3.56959i
\(484\) 0 0
\(485\) − 3.97602e6i − 0.767528i
\(486\) 0 0
\(487\) 3.82182e6 0.730211 0.365105 0.930966i \(-0.381033\pi\)
0.365105 + 0.930966i \(0.381033\pi\)
\(488\) 0 0
\(489\) −9.72133e6 −1.83846
\(490\) 0 0
\(491\) − 4.68377e6i − 0.876782i −0.898784 0.438391i \(-0.855549\pi\)
0.898784 0.438391i \(-0.144451\pi\)
\(492\) 0 0
\(493\) − 1.17815e6i − 0.218314i
\(494\) 0 0
\(495\) −2.87587e6 −0.527541
\(496\) 0 0
\(497\) −6.75739e6 −1.22712
\(498\) 0 0
\(499\) − 129527.i − 0.0232867i −0.999932 0.0116433i \(-0.996294\pi\)
0.999932 0.0116433i \(-0.00370628\pi\)
\(500\) 0 0
\(501\) − 3.41098e6i − 0.607135i
\(502\) 0 0
\(503\) −4.59305e6 −0.809434 −0.404717 0.914442i \(-0.632630\pi\)
−0.404717 + 0.914442i \(0.632630\pi\)
\(504\) 0 0
\(505\) −1.67434e6 −0.292157
\(506\) 0 0
\(507\) − 7.99851e6i − 1.38194i
\(508\) 0 0
\(509\) 7.25700e6i 1.24155i 0.783991 + 0.620773i \(0.213181\pi\)
−0.783991 + 0.620773i \(0.786819\pi\)
\(510\) 0 0
\(511\) −73878.0 −0.0125159
\(512\) 0 0
\(513\) −317498. −0.0532657
\(514\) 0 0
\(515\) 3.63629e6i 0.604145i
\(516\) 0 0
\(517\) − 1.13389e7i − 1.86572i
\(518\) 0 0
\(519\) 5.48609e6 0.894015
\(520\) 0 0
\(521\) 3.82096e6 0.616706 0.308353 0.951272i \(-0.400222\pi\)
0.308353 + 0.951272i \(0.400222\pi\)
\(522\) 0 0
\(523\) − 3.14789e6i − 0.503228i −0.967828 0.251614i \(-0.919039\pi\)
0.967828 0.251614i \(-0.0809613\pi\)
\(524\) 0 0
\(525\) 2.28989e6i 0.362590i
\(526\) 0 0
\(527\) 8.43559e6 1.32309
\(528\) 0 0
\(529\) 1.85156e7 2.87672
\(530\) 0 0
\(531\) − 6.22956e6i − 0.958784i
\(532\) 0 0
\(533\) 377982.i 0.0576306i
\(534\) 0 0
\(535\) −1.67810e6 −0.253474
\(536\) 0 0
\(537\) 3.59971e6 0.538681
\(538\) 0 0
\(539\) 6.10583e6i 0.905259i
\(540\) 0 0
\(541\) − 6.65423e6i − 0.977472i −0.872432 0.488736i \(-0.837458\pi\)
0.872432 0.488736i \(-0.162542\pi\)
\(542\) 0 0
\(543\) −1.38366e7 −2.01386
\(544\) 0 0
\(545\) 2.57152e6 0.370851
\(546\) 0 0
\(547\) − 2.73564e6i − 0.390923i −0.980711 0.195461i \(-0.937380\pi\)
0.980711 0.195461i \(-0.0626204\pi\)
\(548\) 0 0
\(549\) − 8.96551e6i − 1.26953i
\(550\) 0 0
\(551\) 738438. 0.103618
\(552\) 0 0
\(553\) 1.04483e7 1.45288
\(554\) 0 0
\(555\) 2.32722e6i 0.320704i
\(556\) 0 0
\(557\) 8.08022e6i 1.10353i 0.833999 + 0.551766i \(0.186046\pi\)
−0.833999 + 0.551766i \(0.813954\pi\)
\(558\) 0 0
\(559\) −232482. −0.0314674
\(560\) 0 0
\(561\) −1.39726e7 −1.87443
\(562\) 0 0
\(563\) 8.04656e6i 1.06989i 0.844887 + 0.534945i \(0.179668\pi\)
−0.844887 + 0.534945i \(0.820332\pi\)
\(564\) 0 0
\(565\) 1.51747e6i 0.199986i
\(566\) 0 0
\(567\) −1.07135e7 −1.39951
\(568\) 0 0
\(569\) −5.03321e6 −0.651725 −0.325862 0.945417i \(-0.605655\pi\)
−0.325862 + 0.945417i \(0.605655\pi\)
\(570\) 0 0
\(571\) − 1.04770e7i − 1.34476i −0.740205 0.672381i \(-0.765272\pi\)
0.740205 0.672381i \(-0.234728\pi\)
\(572\) 0 0
\(573\) − 2.74499e6i − 0.349265i
\(574\) 0 0
\(575\) 3.12199e6 0.393788
\(576\) 0 0
\(577\) 6.04045e6 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(578\) 0 0
\(579\) − 6.21077e6i − 0.769926i
\(580\) 0 0
\(581\) − 498776.i − 0.0613006i
\(582\) 0 0
\(583\) 1.19161e7 1.45198
\(584\) 0 0
\(585\) 204709. 0.0247313
\(586\) 0 0
\(587\) 1.24894e6i 0.149605i 0.997198 + 0.0748026i \(0.0238327\pi\)
−0.997198 + 0.0748026i \(0.976167\pi\)
\(588\) 0 0
\(589\) 5.28725e6i 0.627974i
\(590\) 0 0
\(591\) 1.08274e7 1.27513
\(592\) 0 0
\(593\) −5.62448e6 −0.656819 −0.328410 0.944535i \(-0.606513\pi\)
−0.328410 + 0.944535i \(0.606513\pi\)
\(594\) 0 0
\(595\) 5.34162e6i 0.618558i
\(596\) 0 0
\(597\) − 1.85525e7i − 2.13043i
\(598\) 0 0
\(599\) 1.21046e7 1.37842 0.689211 0.724560i \(-0.257957\pi\)
0.689211 + 0.724560i \(0.257957\pi\)
\(600\) 0 0
\(601\) −1.35567e7 −1.53098 −0.765488 0.643450i \(-0.777502\pi\)
−0.765488 + 0.643450i \(0.777502\pi\)
\(602\) 0 0
\(603\) 1.24360e7i 1.39280i
\(604\) 0 0
\(605\) 2.54255e6i 0.282410i
\(606\) 0 0
\(607\) 9.54199e6 1.05116 0.525578 0.850745i \(-0.323849\pi\)
0.525578 + 0.850745i \(0.323849\pi\)
\(608\) 0 0
\(609\) 3.42360e6 0.374059
\(610\) 0 0
\(611\) 807125.i 0.0874657i
\(612\) 0 0
\(613\) − 8.07109e6i − 0.867523i −0.901028 0.433762i \(-0.857186\pi\)
0.901028 0.433762i \(-0.142814\pi\)
\(614\) 0 0
\(615\) 5.59914e6 0.596943
\(616\) 0 0
\(617\) 277031. 0.0292964 0.0146482 0.999893i \(-0.495337\pi\)
0.0146482 + 0.999893i \(0.495337\pi\)
\(618\) 0 0
\(619\) − 6.49138e6i − 0.680943i −0.940255 0.340471i \(-0.889413\pi\)
0.940255 0.340471i \(-0.110587\pi\)
\(620\) 0 0
\(621\) 2.00691e6i 0.208833i
\(622\) 0 0
\(623\) 1.09786e7 1.13325
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 8.75772e6i − 0.889656i
\(628\) 0 0
\(629\) 5.42869e6i 0.547103i
\(630\) 0 0
\(631\) 1.43239e7 1.43215 0.716074 0.698025i \(-0.245937\pi\)
0.716074 + 0.698025i \(0.245937\pi\)
\(632\) 0 0
\(633\) 2.12346e7 2.10637
\(634\) 0 0
\(635\) 7.97460e6i 0.784828i
\(636\) 0 0
\(637\) − 434623.i − 0.0424389i
\(638\) 0 0
\(639\) 8.94854e6 0.866962
\(640\) 0 0
\(641\) −1.52078e7 −1.46191 −0.730956 0.682425i \(-0.760925\pi\)
−0.730956 + 0.682425i \(0.760925\pi\)
\(642\) 0 0
\(643\) − 2.09843e6i − 0.200155i −0.994980 0.100078i \(-0.968091\pi\)
0.994980 0.100078i \(-0.0319091\pi\)
\(644\) 0 0
\(645\) 3.44382e6i 0.325942i
\(646\) 0 0
\(647\) 1.04496e7 0.981384 0.490692 0.871333i \(-0.336744\pi\)
0.490692 + 0.871333i \(0.336744\pi\)
\(648\) 0 0
\(649\) −1.42290e7 −1.32606
\(650\) 0 0
\(651\) 2.45131e7i 2.26697i
\(652\) 0 0
\(653\) − 5.78680e6i − 0.531075i −0.964101 0.265537i \(-0.914451\pi\)
0.964101 0.265537i \(-0.0855494\pi\)
\(654\) 0 0
\(655\) 8.98622e6 0.818415
\(656\) 0 0
\(657\) 97833.7 0.00884250
\(658\) 0 0
\(659\) − 6.05285e6i − 0.542933i −0.962448 0.271467i \(-0.912491\pi\)
0.962448 0.271467i \(-0.0875087\pi\)
\(660\) 0 0
\(661\) 5.13781e6i 0.457377i 0.973500 + 0.228689i \(0.0734438\pi\)
−0.973500 + 0.228689i \(0.926556\pi\)
\(662\) 0 0
\(663\) 994592. 0.0878741
\(664\) 0 0
\(665\) −3.34801e6 −0.293584
\(666\) 0 0
\(667\) − 4.66768e6i − 0.406243i
\(668\) 0 0
\(669\) 4.08027e6i 0.352471i
\(670\) 0 0
\(671\) −2.04783e7 −1.75585
\(672\) 0 0
\(673\) −1.06833e6 −0.0909214 −0.0454607 0.998966i \(-0.514476\pi\)
−0.0454607 + 0.998966i \(0.514476\pi\)
\(674\) 0 0
\(675\) 251105.i 0.0212127i
\(676\) 0 0
\(677\) 1.78231e7i 1.49456i 0.664510 + 0.747279i \(0.268640\pi\)
−0.664510 + 0.747279i \(0.731360\pi\)
\(678\) 0 0
\(679\) 2.69520e7 2.24345
\(680\) 0 0
\(681\) −1.11792e7 −0.923721
\(682\) 0 0
\(683\) − 1.21938e7i − 1.00020i −0.865968 0.500099i \(-0.833297\pi\)
0.865968 0.500099i \(-0.166703\pi\)
\(684\) 0 0
\(685\) − 7.23761e6i − 0.589344i
\(686\) 0 0
\(687\) −3.45917e6 −0.279628
\(688\) 0 0
\(689\) −848205. −0.0680695
\(690\) 0 0
\(691\) 7.75412e6i 0.617785i 0.951097 + 0.308893i \(0.0999583\pi\)
−0.951097 + 0.308893i \(0.900042\pi\)
\(692\) 0 0
\(693\) − 1.94944e7i − 1.54198i
\(694\) 0 0
\(695\) −7.95460e6 −0.624678
\(696\) 0 0
\(697\) 1.30611e7 1.01835
\(698\) 0 0
\(699\) − 3.22404e7i − 2.49579i
\(700\) 0 0
\(701\) − 2.02808e7i − 1.55880i −0.626528 0.779399i \(-0.715524\pi\)
0.626528 0.779399i \(-0.284476\pi\)
\(702\) 0 0
\(703\) −3.40259e6 −0.259670
\(704\) 0 0
\(705\) 1.19561e7 0.905979
\(706\) 0 0
\(707\) − 1.13497e7i − 0.853960i
\(708\) 0 0
\(709\) 2.36091e6i 0.176386i 0.996103 + 0.0881928i \(0.0281092\pi\)
−0.996103 + 0.0881928i \(0.971891\pi\)
\(710\) 0 0
\(711\) −1.38362e7 −1.02646
\(712\) 0 0
\(713\) 3.34208e7 2.46203
\(714\) 0 0
\(715\) − 467580.i − 0.0342051i
\(716\) 0 0
\(717\) − 3.08606e6i − 0.224185i
\(718\) 0 0
\(719\) −4.33349e6 −0.312619 −0.156310 0.987708i \(-0.549960\pi\)
−0.156310 + 0.987708i \(0.549960\pi\)
\(720\) 0 0
\(721\) −2.46491e7 −1.76589
\(722\) 0 0
\(723\) 268710.i 0.0191178i
\(724\) 0 0
\(725\) − 584022.i − 0.0412652i
\(726\) 0 0
\(727\) 4.45026e6 0.312284 0.156142 0.987735i \(-0.450094\pi\)
0.156142 + 0.987735i \(0.450094\pi\)
\(728\) 0 0
\(729\) 1.20768e7 0.841650
\(730\) 0 0
\(731\) 8.03339e6i 0.556039i
\(732\) 0 0
\(733\) − 838509.i − 0.0576432i −0.999585 0.0288216i \(-0.990825\pi\)
0.999585 0.0288216i \(-0.00917547\pi\)
\(734\) 0 0
\(735\) −6.43818e6 −0.439587
\(736\) 0 0
\(737\) 2.84054e7 1.92634
\(738\) 0 0
\(739\) 2.44547e7i 1.64722i 0.567158 + 0.823609i \(0.308043\pi\)
−0.567158 + 0.823609i \(0.691957\pi\)
\(740\) 0 0
\(741\) 623389.i 0.0417075i
\(742\) 0 0
\(743\) −1.34515e7 −0.893920 −0.446960 0.894554i \(-0.647493\pi\)
−0.446960 + 0.894554i \(0.647493\pi\)
\(744\) 0 0
\(745\) −1.16127e7 −0.766551
\(746\) 0 0
\(747\) 660509.i 0.0433089i
\(748\) 0 0
\(749\) − 1.13752e7i − 0.740892i
\(750\) 0 0
\(751\) 3.02598e7 1.95779 0.978896 0.204359i \(-0.0655111\pi\)
0.978896 + 0.204359i \(0.0655111\pi\)
\(752\) 0 0
\(753\) −1.64161e7 −1.05507
\(754\) 0 0
\(755\) 330736.i 0.0211161i
\(756\) 0 0
\(757\) − 2.18417e7i − 1.38531i −0.721269 0.692655i \(-0.756441\pi\)
0.721269 0.692655i \(-0.243559\pi\)
\(758\) 0 0
\(759\) −5.53577e7 −3.48798
\(760\) 0 0
\(761\) −2.66377e6 −0.166738 −0.0833692 0.996519i \(-0.526568\pi\)
−0.0833692 + 0.996519i \(0.526568\pi\)
\(762\) 0 0
\(763\) 1.74314e7i 1.08398i
\(764\) 0 0
\(765\) − 7.07369e6i − 0.437011i
\(766\) 0 0
\(767\) 1.01285e6 0.0621664
\(768\) 0 0
\(769\) 1.79300e6 0.109336 0.0546681 0.998505i \(-0.482590\pi\)
0.0546681 + 0.998505i \(0.482590\pi\)
\(770\) 0 0
\(771\) − 1.08158e7i − 0.655271i
\(772\) 0 0
\(773\) 3.03672e7i 1.82792i 0.405810 + 0.913958i \(0.366990\pi\)
−0.405810 + 0.913958i \(0.633010\pi\)
\(774\) 0 0
\(775\) 4.18162e6 0.250087
\(776\) 0 0
\(777\) −1.57753e7 −0.937402
\(778\) 0 0
\(779\) 8.18642e6i 0.483337i
\(780\) 0 0
\(781\) − 2.04395e7i − 1.19907i
\(782\) 0 0
\(783\) 375426. 0.0218837
\(784\) 0 0
\(785\) −3.13718e6 −0.181704
\(786\) 0 0
\(787\) − 1.37601e7i − 0.791925i −0.918267 0.395962i \(-0.870411\pi\)
0.918267 0.395962i \(-0.129589\pi\)
\(788\) 0 0
\(789\) − 1.18862e7i − 0.679752i
\(790\) 0 0
\(791\) −1.02864e7 −0.584549
\(792\) 0 0
\(793\) 1.45768e6 0.0823150
\(794\) 0 0
\(795\) 1.25647e7i 0.705071i
\(796\) 0 0
\(797\) 1.94451e6i 0.108434i 0.998529 + 0.0542169i \(0.0172662\pi\)
−0.998529 + 0.0542169i \(0.982734\pi\)
\(798\) 0 0
\(799\) 2.78901e7 1.54555
\(800\) 0 0
\(801\) −1.45385e7 −0.800644
\(802\) 0 0
\(803\) − 223464.i − 0.0122298i
\(804\) 0 0
\(805\) 2.11628e7i 1.15102i
\(806\) 0 0
\(807\) −2.21108e7 −1.19514
\(808\) 0 0
\(809\) 4.71536e6 0.253305 0.126652 0.991947i \(-0.459577\pi\)
0.126652 + 0.991947i \(0.459577\pi\)
\(810\) 0 0
\(811\) 7.50002e6i 0.400415i 0.979754 + 0.200207i \(0.0641616\pi\)
−0.979754 + 0.200207i \(0.935838\pi\)
\(812\) 0 0
\(813\) 2.42251e7i 1.28540i
\(814\) 0 0
\(815\) −1.12412e7 −0.592816
\(816\) 0 0
\(817\) −5.03516e6 −0.263911
\(818\) 0 0
\(819\) 1.38765e6i 0.0722885i
\(820\) 0 0
\(821\) 2.84972e7i 1.47552i 0.675063 + 0.737760i \(0.264116\pi\)
−0.675063 + 0.737760i \(0.735884\pi\)
\(822\) 0 0
\(823\) −6.20948e6 −0.319562 −0.159781 0.987152i \(-0.551079\pi\)
−0.159781 + 0.987152i \(0.551079\pi\)
\(824\) 0 0
\(825\) −6.92637e6 −0.354300
\(826\) 0 0
\(827\) − 1.27490e7i − 0.648207i −0.946022 0.324104i \(-0.894937\pi\)
0.946022 0.324104i \(-0.105063\pi\)
\(828\) 0 0
\(829\) − 2.60957e7i − 1.31881i −0.751787 0.659406i \(-0.770808\pi\)
0.751787 0.659406i \(-0.229192\pi\)
\(830\) 0 0
\(831\) −6.98771e6 −0.351020
\(832\) 0 0
\(833\) −1.50183e7 −0.749910
\(834\) 0 0
\(835\) − 3.94428e6i − 0.195772i
\(836\) 0 0
\(837\) 2.68807e6i 0.132626i
\(838\) 0 0
\(839\) −2.88675e7 −1.41581 −0.707903 0.706310i \(-0.750359\pi\)
−0.707903 + 0.706310i \(0.750359\pi\)
\(840\) 0 0
\(841\) 1.96380e7 0.957430
\(842\) 0 0
\(843\) − 7.03587e6i − 0.340996i
\(844\) 0 0
\(845\) − 9.24904e6i − 0.445610i
\(846\) 0 0
\(847\) −1.72350e7 −0.825472
\(848\) 0 0
\(849\) 5.21269e7 2.48195
\(850\) 0 0
\(851\) 2.15078e7i 1.01806i
\(852\) 0 0
\(853\) − 1.83501e7i − 0.863505i −0.901992 0.431753i \(-0.857895\pi\)
0.901992 0.431753i \(-0.142105\pi\)
\(854\) 0 0
\(855\) 4.43364e6 0.207417
\(856\) 0 0
\(857\) −2.16867e7 −1.00865 −0.504327 0.863513i \(-0.668259\pi\)
−0.504327 + 0.863513i \(0.668259\pi\)
\(858\) 0 0
\(859\) 7.71806e6i 0.356883i 0.983951 + 0.178441i \(0.0571055\pi\)
−0.983951 + 0.178441i \(0.942895\pi\)
\(860\) 0 0
\(861\) 3.79545e7i 1.74484i
\(862\) 0 0
\(863\) 2.51914e7 1.15140 0.575700 0.817661i \(-0.304730\pi\)
0.575700 + 0.817661i \(0.304730\pi\)
\(864\) 0 0
\(865\) 6.34382e6 0.288277
\(866\) 0 0
\(867\) − 3.67090e6i − 0.165854i
\(868\) 0 0
\(869\) 3.16035e7i 1.41967i
\(870\) 0 0
\(871\) −2.02194e6 −0.0903075
\(872\) 0 0
\(873\) −3.56914e7 −1.58500
\(874\) 0 0
\(875\) 2.64790e6i 0.116918i
\(876\) 0 0
\(877\) 1.45084e7i 0.636971i 0.947928 + 0.318485i \(0.103174\pi\)
−0.947928 + 0.318485i \(0.896826\pi\)
\(878\) 0 0
\(879\) −4.09560e7 −1.78791
\(880\) 0 0
\(881\) 4.13732e7 1.79589 0.897943 0.440111i \(-0.145061\pi\)
0.897943 + 0.440111i \(0.145061\pi\)
\(882\) 0 0
\(883\) − 5.77268e6i − 0.249158i −0.992210 0.124579i \(-0.960242\pi\)
0.992210 0.124579i \(-0.0397581\pi\)
\(884\) 0 0
\(885\) − 1.50036e7i − 0.643926i
\(886\) 0 0
\(887\) −9.78721e6 −0.417686 −0.208843 0.977949i \(-0.566970\pi\)
−0.208843 + 0.977949i \(0.566970\pi\)
\(888\) 0 0
\(889\) −5.40568e7 −2.29401
\(890\) 0 0
\(891\) − 3.24059e7i − 1.36751i
\(892\) 0 0
\(893\) 1.74809e7i 0.733560i
\(894\) 0 0
\(895\) 4.16251e6 0.173699
\(896\) 0 0
\(897\) 3.94045e6 0.163518
\(898\) 0 0
\(899\) − 6.25193e6i − 0.257997i
\(900\) 0 0
\(901\) 2.93096e7i 1.20281i
\(902\) 0 0
\(903\) −2.33444e7 −0.952714
\(904\) 0 0
\(905\) −1.59999e7 −0.649374
\(906\) 0 0
\(907\) − 1.90645e6i − 0.0769497i −0.999260 0.0384749i \(-0.987750\pi\)
0.999260 0.0384749i \(-0.0122500\pi\)
\(908\) 0 0
\(909\) 1.50300e7i 0.603323i
\(910\) 0 0
\(911\) 6.24825e6 0.249438 0.124719 0.992192i \(-0.460197\pi\)
0.124719 + 0.992192i \(0.460197\pi\)
\(912\) 0 0
\(913\) 1.50868e6 0.0598991
\(914\) 0 0
\(915\) − 2.15929e7i − 0.852627i
\(916\) 0 0
\(917\) 6.09142e7i 2.39219i
\(918\) 0 0
\(919\) 3.90166e6 0.152391 0.0761957 0.997093i \(-0.475723\pi\)
0.0761957 + 0.997093i \(0.475723\pi\)
\(920\) 0 0
\(921\) 2.97338e7 1.15505
\(922\) 0 0
\(923\) 1.45492e6i 0.0562127i
\(924\) 0 0
\(925\) 2.69107e6i 0.103412i
\(926\) 0 0
\(927\) 3.26418e7 1.24760
\(928\) 0 0
\(929\) 2.94138e6 0.111818 0.0559090 0.998436i \(-0.482194\pi\)
0.0559090 + 0.998436i \(0.482194\pi\)
\(930\) 0 0
\(931\) − 9.41318e6i − 0.355928i
\(932\) 0 0
\(933\) 5.63129e7i 2.11789i
\(934\) 0 0
\(935\) −1.61571e7 −0.604415
\(936\) 0 0
\(937\) 2.73011e7 1.01585 0.507926 0.861401i \(-0.330412\pi\)
0.507926 + 0.861401i \(0.330412\pi\)
\(938\) 0 0
\(939\) − 1.68683e7i − 0.624322i
\(940\) 0 0
\(941\) 2.99537e6i 0.110275i 0.998479 + 0.0551374i \(0.0175597\pi\)
−0.998479 + 0.0551374i \(0.982440\pi\)
\(942\) 0 0
\(943\) 5.17465e7 1.89497
\(944\) 0 0
\(945\) −1.70215e6 −0.0620038
\(946\) 0 0
\(947\) 2.96401e7i 1.07400i 0.843582 + 0.537000i \(0.180442\pi\)
−0.843582 + 0.537000i \(0.819558\pi\)
\(948\) 0 0
\(949\) 15906.5i 0 0.000573337i
\(950\) 0 0
\(951\) −2.76831e7 −0.992575
\(952\) 0 0
\(953\) 2.57409e7 0.918101 0.459051 0.888410i \(-0.348190\pi\)
0.459051 + 0.888410i \(0.348190\pi\)
\(954\) 0 0
\(955\) − 3.17416e6i − 0.112621i
\(956\) 0 0
\(957\) 1.03556e7i 0.365506i
\(958\) 0 0
\(959\) 4.90611e7 1.72262
\(960\) 0 0
\(961\) 1.61349e7 0.563584
\(962\) 0 0
\(963\) 1.50637e7i 0.523440i
\(964\) 0 0
\(965\) − 7.18180e6i − 0.248265i
\(966\) 0 0
\(967\) 2.11539e7 0.727487 0.363743 0.931499i \(-0.381499\pi\)
0.363743 + 0.931499i \(0.381499\pi\)
\(968\) 0 0
\(969\) 2.15411e7 0.736985
\(970\) 0 0
\(971\) − 4.18119e7i − 1.42315i −0.702608 0.711577i \(-0.747981\pi\)
0.702608 0.711577i \(-0.252019\pi\)
\(972\) 0 0
\(973\) − 5.39213e7i − 1.82590i
\(974\) 0 0
\(975\) 493031. 0.0166097
\(976\) 0 0
\(977\) −3.93721e7 −1.31963 −0.659815 0.751428i \(-0.729365\pi\)
−0.659815 + 0.751428i \(0.729365\pi\)
\(978\) 0 0
\(979\) 3.32078e7i 1.10734i
\(980\) 0 0
\(981\) − 2.30837e7i − 0.765831i
\(982\) 0 0
\(983\) −7.10636e6 −0.234565 −0.117283 0.993099i \(-0.537418\pi\)
−0.117283 + 0.993099i \(0.537418\pi\)
\(984\) 0 0
\(985\) 1.25202e7 0.411170
\(986\) 0 0
\(987\) 8.10462e7i 2.64813i
\(988\) 0 0
\(989\) 3.18273e7i 1.03469i
\(990\) 0 0
\(991\) −2.70666e6 −0.0875487 −0.0437744 0.999041i \(-0.513938\pi\)
−0.0437744 + 0.999041i \(0.513938\pi\)
\(992\) 0 0
\(993\) −5.09137e7 −1.63856
\(994\) 0 0
\(995\) − 2.14532e7i − 0.686964i
\(996\) 0 0
\(997\) − 1.58535e7i − 0.505112i −0.967582 0.252556i \(-0.918729\pi\)
0.967582 0.252556i \(-0.0812713\pi\)
\(998\) 0 0
\(999\) −1.72990e6 −0.0548412
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.b.161.2 yes 8
4.3 odd 2 320.6.d.a.161.7 yes 8
8.3 odd 2 320.6.d.a.161.2 8
8.5 even 2 inner 320.6.d.b.161.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.a.161.2 8 8.3 odd 2
320.6.d.a.161.7 yes 8 4.3 odd 2
320.6.d.b.161.2 yes 8 1.1 even 1 trivial
320.6.d.b.161.7 yes 8 8.5 even 2 inner