Properties

Label 784.6.a.bf.1.3
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(7.22929\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.54802 q^{3} +45.9910 q^{5} -200.123 q^{9} +O(q^{10})\) \(q+6.54802 q^{3} +45.9910 q^{5} -200.123 q^{9} +551.781 q^{11} -1094.10 q^{13} +301.150 q^{15} +1180.71 q^{17} -1166.13 q^{19} -44.3851 q^{23} -1009.82 q^{25} -2901.58 q^{27} +3329.02 q^{29} +8784.01 q^{31} +3613.07 q^{33} -2557.12 q^{37} -7164.17 q^{39} +12761.3 q^{41} +96.7714 q^{43} -9203.89 q^{45} +7679.15 q^{47} +7731.33 q^{51} -11953.3 q^{53} +25377.0 q^{55} -7635.81 q^{57} +9857.24 q^{59} +38517.9 q^{61} -50318.7 q^{65} +67548.9 q^{67} -290.634 q^{69} +61374.6 q^{71} -1850.40 q^{73} -6612.34 q^{75} +8.52913 q^{79} +29630.4 q^{81} -95039.3 q^{83} +54302.3 q^{85} +21798.5 q^{87} -53605.6 q^{89} +57517.9 q^{93} -53631.4 q^{95} -3110.79 q^{97} -110424. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 6.54802 0.420055 0.210028 0.977695i \(-0.432645\pi\)
0.210028 + 0.977695i \(0.432645\pi\)
\(4\) 0 0
\(5\) 45.9910 0.822713 0.411356 0.911475i \(-0.365055\pi\)
0.411356 + 0.911475i \(0.365055\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −200.123 −0.823553
\(10\) 0 0
\(11\) 551.781 1.37494 0.687472 0.726211i \(-0.258720\pi\)
0.687472 + 0.726211i \(0.258720\pi\)
\(12\) 0 0
\(13\) −1094.10 −1.79555 −0.897776 0.440453i \(-0.854818\pi\)
−0.897776 + 0.440453i \(0.854818\pi\)
\(14\) 0 0
\(15\) 301.150 0.345585
\(16\) 0 0
\(17\) 1180.71 0.990883 0.495442 0.868641i \(-0.335006\pi\)
0.495442 + 0.868641i \(0.335006\pi\)
\(18\) 0 0
\(19\) −1166.13 −0.741074 −0.370537 0.928818i \(-0.620826\pi\)
−0.370537 + 0.928818i \(0.620826\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −44.3851 −0.0174951 −0.00874757 0.999962i \(-0.502784\pi\)
−0.00874757 + 0.999962i \(0.502784\pi\)
\(24\) 0 0
\(25\) −1009.82 −0.323143
\(26\) 0 0
\(27\) −2901.58 −0.765993
\(28\) 0 0
\(29\) 3329.02 0.735057 0.367529 0.930012i \(-0.380204\pi\)
0.367529 + 0.930012i \(0.380204\pi\)
\(30\) 0 0
\(31\) 8784.01 1.64168 0.820841 0.571157i \(-0.193505\pi\)
0.820841 + 0.571157i \(0.193505\pi\)
\(32\) 0 0
\(33\) 3613.07 0.577552
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2557.12 −0.307077 −0.153539 0.988143i \(-0.549067\pi\)
−0.153539 + 0.988143i \(0.549067\pi\)
\(38\) 0 0
\(39\) −7164.17 −0.754231
\(40\) 0 0
\(41\) 12761.3 1.18559 0.592794 0.805354i \(-0.298025\pi\)
0.592794 + 0.805354i \(0.298025\pi\)
\(42\) 0 0
\(43\) 96.7714 0.00798135 0.00399067 0.999992i \(-0.498730\pi\)
0.00399067 + 0.999992i \(0.498730\pi\)
\(44\) 0 0
\(45\) −9203.89 −0.677548
\(46\) 0 0
\(47\) 7679.15 0.507071 0.253535 0.967326i \(-0.418407\pi\)
0.253535 + 0.967326i \(0.418407\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7731.33 0.416226
\(52\) 0 0
\(53\) −11953.3 −0.584520 −0.292260 0.956339i \(-0.594407\pi\)
−0.292260 + 0.956339i \(0.594407\pi\)
\(54\) 0 0
\(55\) 25377.0 1.13118
\(56\) 0 0
\(57\) −7635.81 −0.311292
\(58\) 0 0
\(59\) 9857.24 0.368659 0.184330 0.982864i \(-0.440989\pi\)
0.184330 + 0.982864i \(0.440989\pi\)
\(60\) 0 0
\(61\) 38517.9 1.32537 0.662686 0.748897i \(-0.269416\pi\)
0.662686 + 0.748897i \(0.269416\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −50318.7 −1.47722
\(66\) 0 0
\(67\) 67548.9 1.83836 0.919182 0.393833i \(-0.128851\pi\)
0.919182 + 0.393833i \(0.128851\pi\)
\(68\) 0 0
\(69\) −290.634 −0.00734893
\(70\) 0 0
\(71\) 61374.6 1.44492 0.722458 0.691415i \(-0.243012\pi\)
0.722458 + 0.691415i \(0.243012\pi\)
\(72\) 0 0
\(73\) −1850.40 −0.0406404 −0.0203202 0.999794i \(-0.506469\pi\)
−0.0203202 + 0.999794i \(0.506469\pi\)
\(74\) 0 0
\(75\) −6612.34 −0.135738
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.52913 0.000153758 0 7.68788e−5 1.00000i \(-0.499976\pi\)
7.68788e−5 1.00000i \(0.499976\pi\)
\(80\) 0 0
\(81\) 29630.4 0.501794
\(82\) 0 0
\(83\) −95039.3 −1.51429 −0.757143 0.653249i \(-0.773405\pi\)
−0.757143 + 0.653249i \(0.773405\pi\)
\(84\) 0 0
\(85\) 54302.3 0.815212
\(86\) 0 0
\(87\) 21798.5 0.308765
\(88\) 0 0
\(89\) −53605.6 −0.717357 −0.358678 0.933461i \(-0.616773\pi\)
−0.358678 + 0.933461i \(0.616773\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 57517.9 0.689597
\(94\) 0 0
\(95\) −53631.4 −0.609691
\(96\) 0 0
\(97\) −3110.79 −0.0335693 −0.0167846 0.999859i \(-0.505343\pi\)
−0.0167846 + 0.999859i \(0.505343\pi\)
\(98\) 0 0
\(99\) −110424. −1.13234
\(100\) 0 0
\(101\) −21835.9 −0.212994 −0.106497 0.994313i \(-0.533963\pi\)
−0.106497 + 0.994313i \(0.533963\pi\)
\(102\) 0 0
\(103\) 65341.4 0.606870 0.303435 0.952852i \(-0.401866\pi\)
0.303435 + 0.952852i \(0.401866\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 108957. 0.920020 0.460010 0.887914i \(-0.347846\pi\)
0.460010 + 0.887914i \(0.347846\pi\)
\(108\) 0 0
\(109\) 86728.7 0.699192 0.349596 0.936901i \(-0.386319\pi\)
0.349596 + 0.936901i \(0.386319\pi\)
\(110\) 0 0
\(111\) −16744.1 −0.128989
\(112\) 0 0
\(113\) −101496. −0.747746 −0.373873 0.927480i \(-0.621970\pi\)
−0.373873 + 0.927480i \(0.621970\pi\)
\(114\) 0 0
\(115\) −2041.32 −0.0143935
\(116\) 0 0
\(117\) 218955. 1.47873
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 143411. 0.890470
\(122\) 0 0
\(123\) 83560.9 0.498013
\(124\) 0 0
\(125\) −190165. −1.08857
\(126\) 0 0
\(127\) 3094.61 0.0170253 0.00851267 0.999964i \(-0.497290\pi\)
0.00851267 + 0.999964i \(0.497290\pi\)
\(128\) 0 0
\(129\) 633.661 0.00335261
\(130\) 0 0
\(131\) 253431. 1.29027 0.645136 0.764067i \(-0.276801\pi\)
0.645136 + 0.764067i \(0.276801\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −133447. −0.630193
\(136\) 0 0
\(137\) −97152.9 −0.442236 −0.221118 0.975247i \(-0.570971\pi\)
−0.221118 + 0.975247i \(0.570971\pi\)
\(138\) 0 0
\(139\) 210308. 0.923249 0.461624 0.887076i \(-0.347267\pi\)
0.461624 + 0.887076i \(0.347267\pi\)
\(140\) 0 0
\(141\) 50283.2 0.212998
\(142\) 0 0
\(143\) −603702. −2.46878
\(144\) 0 0
\(145\) 153105. 0.604741
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 140406. 0.518109 0.259055 0.965863i \(-0.416589\pi\)
0.259055 + 0.965863i \(0.416589\pi\)
\(150\) 0 0
\(151\) −119696. −0.427205 −0.213603 0.976921i \(-0.568520\pi\)
−0.213603 + 0.976921i \(0.568520\pi\)
\(152\) 0 0
\(153\) −236289. −0.816045
\(154\) 0 0
\(155\) 403986. 1.35063
\(156\) 0 0
\(157\) −97616.9 −0.316065 −0.158032 0.987434i \(-0.550515\pi\)
−0.158032 + 0.987434i \(0.550515\pi\)
\(158\) 0 0
\(159\) −78270.6 −0.245531
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −182678. −0.538539 −0.269270 0.963065i \(-0.586782\pi\)
−0.269270 + 0.963065i \(0.586782\pi\)
\(164\) 0 0
\(165\) 166169. 0.475160
\(166\) 0 0
\(167\) −451674. −1.25324 −0.626619 0.779326i \(-0.715562\pi\)
−0.626619 + 0.779326i \(0.715562\pi\)
\(168\) 0 0
\(169\) 825757. 2.22400
\(170\) 0 0
\(171\) 233369. 0.610314
\(172\) 0 0
\(173\) −371647. −0.944095 −0.472047 0.881573i \(-0.656485\pi\)
−0.472047 + 0.881573i \(0.656485\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 64545.4 0.154857
\(178\) 0 0
\(179\) −85003.4 −0.198291 −0.0991457 0.995073i \(-0.531611\pi\)
−0.0991457 + 0.995073i \(0.531611\pi\)
\(180\) 0 0
\(181\) 379442. 0.860892 0.430446 0.902616i \(-0.358356\pi\)
0.430446 + 0.902616i \(0.358356\pi\)
\(182\) 0 0
\(183\) 252216. 0.556730
\(184\) 0 0
\(185\) −117605. −0.252636
\(186\) 0 0
\(187\) 651496. 1.36241
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 922196. 1.82911 0.914555 0.404462i \(-0.132541\pi\)
0.914555 + 0.404462i \(0.132541\pi\)
\(192\) 0 0
\(193\) 505107. 0.976090 0.488045 0.872818i \(-0.337710\pi\)
0.488045 + 0.872818i \(0.337710\pi\)
\(194\) 0 0
\(195\) −329488. −0.620516
\(196\) 0 0
\(197\) 251505. 0.461723 0.230861 0.972987i \(-0.425846\pi\)
0.230861 + 0.972987i \(0.425846\pi\)
\(198\) 0 0
\(199\) −208033. −0.372392 −0.186196 0.982513i \(-0.559616\pi\)
−0.186196 + 0.982513i \(0.559616\pi\)
\(200\) 0 0
\(201\) 442311. 0.772215
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 586904. 0.975399
\(206\) 0 0
\(207\) 8882.50 0.0144082
\(208\) 0 0
\(209\) −643446. −1.01893
\(210\) 0 0
\(211\) 640577. 0.990525 0.495262 0.868744i \(-0.335072\pi\)
0.495262 + 0.868744i \(0.335072\pi\)
\(212\) 0 0
\(213\) 401882. 0.606945
\(214\) 0 0
\(215\) 4450.62 0.00656636
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12116.4 −0.0170712
\(220\) 0 0
\(221\) −1.29182e6 −1.77918
\(222\) 0 0
\(223\) 390135. 0.525354 0.262677 0.964884i \(-0.415395\pi\)
0.262677 + 0.964884i \(0.415395\pi\)
\(224\) 0 0
\(225\) 202089. 0.266126
\(226\) 0 0
\(227\) 291353. 0.375279 0.187639 0.982238i \(-0.439916\pi\)
0.187639 + 0.982238i \(0.439916\pi\)
\(228\) 0 0
\(229\) 1.23040e6 1.55045 0.775227 0.631682i \(-0.217635\pi\)
0.775227 + 0.631682i \(0.217635\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 114279. 0.137903 0.0689517 0.997620i \(-0.478035\pi\)
0.0689517 + 0.997620i \(0.478035\pi\)
\(234\) 0 0
\(235\) 353172. 0.417174
\(236\) 0 0
\(237\) 55.8489 6.45867e−5 0
\(238\) 0 0
\(239\) 1.14782e6 1.29981 0.649906 0.760014i \(-0.274808\pi\)
0.649906 + 0.760014i \(0.274808\pi\)
\(240\) 0 0
\(241\) −812708. −0.901346 −0.450673 0.892689i \(-0.648816\pi\)
−0.450673 + 0.892689i \(0.648816\pi\)
\(242\) 0 0
\(243\) 899104. 0.976775
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.27586e6 1.33064
\(248\) 0 0
\(249\) −622318. −0.636084
\(250\) 0 0
\(251\) −406772. −0.407537 −0.203768 0.979019i \(-0.565319\pi\)
−0.203768 + 0.979019i \(0.565319\pi\)
\(252\) 0 0
\(253\) −24490.8 −0.0240548
\(254\) 0 0
\(255\) 355572. 0.342434
\(256\) 0 0
\(257\) 1.69712e6 1.60281 0.801403 0.598125i \(-0.204087\pi\)
0.801403 + 0.598125i \(0.204087\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −666215. −0.605359
\(262\) 0 0
\(263\) −205694. −0.183372 −0.0916859 0.995788i \(-0.529226\pi\)
−0.0916859 + 0.995788i \(0.529226\pi\)
\(264\) 0 0
\(265\) −549746. −0.480892
\(266\) 0 0
\(267\) −351010. −0.301330
\(268\) 0 0
\(269\) −1.73425e6 −1.46127 −0.730635 0.682769i \(-0.760776\pi\)
−0.730635 + 0.682769i \(0.760776\pi\)
\(270\) 0 0
\(271\) 369042. 0.305248 0.152624 0.988284i \(-0.451228\pi\)
0.152624 + 0.988284i \(0.451228\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −557201. −0.444304
\(276\) 0 0
\(277\) 1.22767e6 0.961351 0.480676 0.876899i \(-0.340391\pi\)
0.480676 + 0.876899i \(0.340391\pi\)
\(278\) 0 0
\(279\) −1.75789e6 −1.35201
\(280\) 0 0
\(281\) 2.00671e6 1.51607 0.758035 0.652214i \(-0.226159\pi\)
0.758035 + 0.652214i \(0.226159\pi\)
\(282\) 0 0
\(283\) −1.78581e6 −1.32547 −0.662734 0.748855i \(-0.730604\pi\)
−0.662734 + 0.748855i \(0.730604\pi\)
\(284\) 0 0
\(285\) −351179. −0.256104
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −25770.8 −0.0181503
\(290\) 0 0
\(291\) −20369.5 −0.0141009
\(292\) 0 0
\(293\) −853248. −0.580639 −0.290319 0.956930i \(-0.593762\pi\)
−0.290319 + 0.956930i \(0.593762\pi\)
\(294\) 0 0
\(295\) 453345. 0.303301
\(296\) 0 0
\(297\) −1.60104e6 −1.05320
\(298\) 0 0
\(299\) 48561.6 0.0314134
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −142982. −0.0894693
\(304\) 0 0
\(305\) 1.77148e6 1.09040
\(306\) 0 0
\(307\) −1.96068e6 −1.18730 −0.593652 0.804722i \(-0.702314\pi\)
−0.593652 + 0.804722i \(0.702314\pi\)
\(308\) 0 0
\(309\) 427857. 0.254919
\(310\) 0 0
\(311\) 863604. 0.506307 0.253153 0.967426i \(-0.418532\pi\)
0.253153 + 0.967426i \(0.418532\pi\)
\(312\) 0 0
\(313\) −1.10047e6 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.49591e6 0.836097 0.418048 0.908425i \(-0.362714\pi\)
0.418048 + 0.908425i \(0.362714\pi\)
\(318\) 0 0
\(319\) 1.83689e6 1.01066
\(320\) 0 0
\(321\) 713454. 0.386459
\(322\) 0 0
\(323\) −1.37686e6 −0.734318
\(324\) 0 0
\(325\) 1.10485e6 0.580221
\(326\) 0 0
\(327\) 567901. 0.293699
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.74015e6 1.37469 0.687345 0.726331i \(-0.258776\pi\)
0.687345 + 0.726331i \(0.258776\pi\)
\(332\) 0 0
\(333\) 511741. 0.252894
\(334\) 0 0
\(335\) 3.10665e6 1.51245
\(336\) 0 0
\(337\) −2.31353e6 −1.10968 −0.554842 0.831956i \(-0.687221\pi\)
−0.554842 + 0.831956i \(0.687221\pi\)
\(338\) 0 0
\(339\) −664600. −0.314095
\(340\) 0 0
\(341\) 4.84685e6 2.25722
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −13366.6 −0.00604606
\(346\) 0 0
\(347\) 3.05926e6 1.36393 0.681966 0.731384i \(-0.261125\pi\)
0.681966 + 0.731384i \(0.261125\pi\)
\(348\) 0 0
\(349\) 210232. 0.0923921 0.0461961 0.998932i \(-0.485290\pi\)
0.0461961 + 0.998932i \(0.485290\pi\)
\(350\) 0 0
\(351\) 3.17461e6 1.37538
\(352\) 0 0
\(353\) 3.76790e6 1.60939 0.804697 0.593686i \(-0.202328\pi\)
0.804697 + 0.593686i \(0.202328\pi\)
\(354\) 0 0
\(355\) 2.82268e6 1.18875
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.00722e6 −0.412465 −0.206232 0.978503i \(-0.566120\pi\)
−0.206232 + 0.978503i \(0.566120\pi\)
\(360\) 0 0
\(361\) −1.11625e6 −0.450809
\(362\) 0 0
\(363\) 939058. 0.374047
\(364\) 0 0
\(365\) −85101.8 −0.0334354
\(366\) 0 0
\(367\) −1.52650e6 −0.591603 −0.295802 0.955249i \(-0.595587\pi\)
−0.295802 + 0.955249i \(0.595587\pi\)
\(368\) 0 0
\(369\) −2.55383e6 −0.976396
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.86297e6 1.80980 0.904898 0.425629i \(-0.139947\pi\)
0.904898 + 0.425629i \(0.139947\pi\)
\(374\) 0 0
\(375\) −1.24520e6 −0.457258
\(376\) 0 0
\(377\) −3.64227e6 −1.31983
\(378\) 0 0
\(379\) −630878. −0.225604 −0.112802 0.993617i \(-0.535983\pi\)
−0.112802 + 0.993617i \(0.535983\pi\)
\(380\) 0 0
\(381\) 20263.5 0.00715159
\(382\) 0 0
\(383\) 565644. 0.197036 0.0985182 0.995135i \(-0.468590\pi\)
0.0985182 + 0.995135i \(0.468590\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −19366.2 −0.00657306
\(388\) 0 0
\(389\) 592212. 0.198428 0.0992140 0.995066i \(-0.468367\pi\)
0.0992140 + 0.995066i \(0.468367\pi\)
\(390\) 0 0
\(391\) −52406.1 −0.0173356
\(392\) 0 0
\(393\) 1.65947e6 0.541986
\(394\) 0 0
\(395\) 392.264 0.000126498 0
\(396\) 0 0
\(397\) −1.34312e6 −0.427698 −0.213849 0.976867i \(-0.568600\pi\)
−0.213849 + 0.976867i \(0.568600\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.68716e6 −1.14507 −0.572534 0.819881i \(-0.694040\pi\)
−0.572534 + 0.819881i \(0.694040\pi\)
\(402\) 0 0
\(403\) −9.61057e6 −2.94772
\(404\) 0 0
\(405\) 1.36273e6 0.412832
\(406\) 0 0
\(407\) −1.41097e6 −0.422214
\(408\) 0 0
\(409\) −1.45630e6 −0.430470 −0.215235 0.976562i \(-0.569052\pi\)
−0.215235 + 0.976562i \(0.569052\pi\)
\(410\) 0 0
\(411\) −636159. −0.185764
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.37096e6 −1.24582
\(416\) 0 0
\(417\) 1.37710e6 0.387816
\(418\) 0 0
\(419\) −2.92192e6 −0.813080 −0.406540 0.913633i \(-0.633265\pi\)
−0.406540 + 0.913633i \(0.633265\pi\)
\(420\) 0 0
\(421\) 2.01999e6 0.555450 0.277725 0.960661i \(-0.410420\pi\)
0.277725 + 0.960661i \(0.410420\pi\)
\(422\) 0 0
\(423\) −1.53678e6 −0.417600
\(424\) 0 0
\(425\) −1.19231e6 −0.320197
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −3.95305e6 −1.03703
\(430\) 0 0
\(431\) 5.43800e6 1.41009 0.705043 0.709164i \(-0.250928\pi\)
0.705043 + 0.709164i \(0.250928\pi\)
\(432\) 0 0
\(433\) 3.77335e6 0.967179 0.483590 0.875295i \(-0.339333\pi\)
0.483590 + 0.875295i \(0.339333\pi\)
\(434\) 0 0
\(435\) 1.00253e6 0.254025
\(436\) 0 0
\(437\) 51758.6 0.0129652
\(438\) 0 0
\(439\) 2.35150e6 0.582350 0.291175 0.956670i \(-0.405954\pi\)
0.291175 + 0.956670i \(0.405954\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.80377e6 −1.16298 −0.581491 0.813553i \(-0.697530\pi\)
−0.581491 + 0.813553i \(0.697530\pi\)
\(444\) 0 0
\(445\) −2.46538e6 −0.590179
\(446\) 0 0
\(447\) 919383. 0.217634
\(448\) 0 0
\(449\) −2.76805e6 −0.647975 −0.323987 0.946061i \(-0.605024\pi\)
−0.323987 + 0.946061i \(0.605024\pi\)
\(450\) 0 0
\(451\) 7.04142e6 1.63012
\(452\) 0 0
\(453\) −783770. −0.179450
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 241566. 0.0541061 0.0270530 0.999634i \(-0.491388\pi\)
0.0270530 + 0.999634i \(0.491388\pi\)
\(458\) 0 0
\(459\) −3.42594e6 −0.759010
\(460\) 0 0
\(461\) −990579. −0.217088 −0.108544 0.994092i \(-0.534619\pi\)
−0.108544 + 0.994092i \(0.534619\pi\)
\(462\) 0 0
\(463\) −6.20488e6 −1.34518 −0.672591 0.740014i \(-0.734819\pi\)
−0.672591 + 0.740014i \(0.734819\pi\)
\(464\) 0 0
\(465\) 2.64531e6 0.567340
\(466\) 0 0
\(467\) −6.88497e6 −1.46086 −0.730432 0.682985i \(-0.760681\pi\)
−0.730432 + 0.682985i \(0.760681\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −639197. −0.132765
\(472\) 0 0
\(473\) 53396.6 0.0109739
\(474\) 0 0
\(475\) 1.17758e6 0.239473
\(476\) 0 0
\(477\) 2.39214e6 0.481383
\(478\) 0 0
\(479\) −5.41288e6 −1.07793 −0.538963 0.842329i \(-0.681184\pi\)
−0.538963 + 0.842329i \(0.681184\pi\)
\(480\) 0 0
\(481\) 2.79774e6 0.551373
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −143069. −0.0276179
\(486\) 0 0
\(487\) 3.01043e6 0.575182 0.287591 0.957753i \(-0.407146\pi\)
0.287591 + 0.957753i \(0.407146\pi\)
\(488\) 0 0
\(489\) −1.19618e6 −0.226216
\(490\) 0 0
\(491\) 7.24498e6 1.35623 0.678115 0.734956i \(-0.262797\pi\)
0.678115 + 0.734956i \(0.262797\pi\)
\(492\) 0 0
\(493\) 3.93062e6 0.728356
\(494\) 0 0
\(495\) −5.07853e6 −0.931591
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.12788e6 0.921906 0.460953 0.887425i \(-0.347508\pi\)
0.460953 + 0.887425i \(0.347508\pi\)
\(500\) 0 0
\(501\) −2.95757e6 −0.526429
\(502\) 0 0
\(503\) 1.05978e7 1.86766 0.933830 0.357718i \(-0.116445\pi\)
0.933830 + 0.357718i \(0.116445\pi\)
\(504\) 0 0
\(505\) −1.00426e6 −0.175233
\(506\) 0 0
\(507\) 5.40707e6 0.934205
\(508\) 0 0
\(509\) −8.78840e6 −1.50354 −0.751770 0.659425i \(-0.770800\pi\)
−0.751770 + 0.659425i \(0.770800\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.38361e6 0.567658
\(514\) 0 0
\(515\) 3.00512e6 0.499280
\(516\) 0 0
\(517\) 4.23721e6 0.697194
\(518\) 0 0
\(519\) −2.43355e6 −0.396572
\(520\) 0 0
\(521\) −162133. −0.0261684 −0.0130842 0.999914i \(-0.504165\pi\)
−0.0130842 + 0.999914i \(0.504165\pi\)
\(522\) 0 0
\(523\) −7.14844e6 −1.14277 −0.571383 0.820684i \(-0.693593\pi\)
−0.571383 + 0.820684i \(0.693593\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.03714e7 1.62671
\(528\) 0 0
\(529\) −6.43437e6 −0.999694
\(530\) 0 0
\(531\) −1.97267e6 −0.303611
\(532\) 0 0
\(533\) −1.39621e7 −2.12879
\(534\) 0 0
\(535\) 5.01106e6 0.756912
\(536\) 0 0
\(537\) −556604. −0.0832934
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4.83604e6 −0.710390 −0.355195 0.934792i \(-0.615586\pi\)
−0.355195 + 0.934792i \(0.615586\pi\)
\(542\) 0 0
\(543\) 2.48459e6 0.361622
\(544\) 0 0
\(545\) 3.98874e6 0.575234
\(546\) 0 0
\(547\) −9.98777e6 −1.42725 −0.713626 0.700527i \(-0.752948\pi\)
−0.713626 + 0.700527i \(0.752948\pi\)
\(548\) 0 0
\(549\) −7.70833e6 −1.09151
\(550\) 0 0
\(551\) −3.88205e6 −0.544732
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −770078. −0.106121
\(556\) 0 0
\(557\) 1.74619e6 0.238481 0.119241 0.992865i \(-0.461954\pi\)
0.119241 + 0.992865i \(0.461954\pi\)
\(558\) 0 0
\(559\) −105877. −0.0143309
\(560\) 0 0
\(561\) 4.26600e6 0.572287
\(562\) 0 0
\(563\) −755218. −0.100416 −0.0502078 0.998739i \(-0.515988\pi\)
−0.0502078 + 0.998739i \(0.515988\pi\)
\(564\) 0 0
\(565\) −4.66792e6 −0.615181
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.39534e6 −0.569131 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(570\) 0 0
\(571\) −1.16104e7 −1.49024 −0.745121 0.666930i \(-0.767608\pi\)
−0.745121 + 0.666930i \(0.767608\pi\)
\(572\) 0 0
\(573\) 6.03855e6 0.768327
\(574\) 0 0
\(575\) 44821.1 0.00565344
\(576\) 0 0
\(577\) −1.06643e7 −1.33350 −0.666748 0.745283i \(-0.732314\pi\)
−0.666748 + 0.745283i \(0.732314\pi\)
\(578\) 0 0
\(579\) 3.30745e6 0.410012
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.59562e6 −0.803682
\(584\) 0 0
\(585\) 1.00700e7 1.21657
\(586\) 0 0
\(587\) −1.39482e7 −1.67079 −0.835396 0.549648i \(-0.814762\pi\)
−0.835396 + 0.549648i \(0.814762\pi\)
\(588\) 0 0
\(589\) −1.02433e7 −1.21661
\(590\) 0 0
\(591\) 1.64686e6 0.193949
\(592\) 0 0
\(593\) −1.17933e7 −1.37720 −0.688600 0.725142i \(-0.741774\pi\)
−0.688600 + 0.725142i \(0.741774\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.36220e6 −0.156425
\(598\) 0 0
\(599\) 4.38057e6 0.498843 0.249421 0.968395i \(-0.419760\pi\)
0.249421 + 0.968395i \(0.419760\pi\)
\(600\) 0 0
\(601\) −688570. −0.0777610 −0.0388805 0.999244i \(-0.512379\pi\)
−0.0388805 + 0.999244i \(0.512379\pi\)
\(602\) 0 0
\(603\) −1.35181e7 −1.51399
\(604\) 0 0
\(605\) 6.59563e6 0.732601
\(606\) 0 0
\(607\) −9.37319e6 −1.03256 −0.516281 0.856420i \(-0.672684\pi\)
−0.516281 + 0.856420i \(0.672684\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.40174e6 −0.910472
\(612\) 0 0
\(613\) −2.16685e6 −0.232904 −0.116452 0.993196i \(-0.537152\pi\)
−0.116452 + 0.993196i \(0.537152\pi\)
\(614\) 0 0
\(615\) 3.84306e6 0.409722
\(616\) 0 0
\(617\) −5.07951e6 −0.537166 −0.268583 0.963256i \(-0.586555\pi\)
−0.268583 + 0.963256i \(0.586555\pi\)
\(618\) 0 0
\(619\) −2.19034e6 −0.229766 −0.114883 0.993379i \(-0.536649\pi\)
−0.114883 + 0.993379i \(0.536649\pi\)
\(620\) 0 0
\(621\) 128787. 0.0134012
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.59018e6 −0.572435
\(626\) 0 0
\(627\) −4.21329e6 −0.428009
\(628\) 0 0
\(629\) −3.01923e6 −0.304278
\(630\) 0 0
\(631\) 7.18693e6 0.718572 0.359286 0.933228i \(-0.383020\pi\)
0.359286 + 0.933228i \(0.383020\pi\)
\(632\) 0 0
\(633\) 4.19451e6 0.416075
\(634\) 0 0
\(635\) 142324. 0.0140070
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.22825e7 −1.18997
\(640\) 0 0
\(641\) −1.76500e7 −1.69668 −0.848340 0.529452i \(-0.822398\pi\)
−0.848340 + 0.529452i \(0.822398\pi\)
\(642\) 0 0
\(643\) 898309. 0.0856837 0.0428419 0.999082i \(-0.486359\pi\)
0.0428419 + 0.999082i \(0.486359\pi\)
\(644\) 0 0
\(645\) 29142.7 0.00275823
\(646\) 0 0
\(647\) −1.38642e6 −0.130207 −0.0651035 0.997879i \(-0.520738\pi\)
−0.0651035 + 0.997879i \(0.520738\pi\)
\(648\) 0 0
\(649\) 5.43904e6 0.506886
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.75425e7 −1.60994 −0.804968 0.593318i \(-0.797818\pi\)
−0.804968 + 0.593318i \(0.797818\pi\)
\(654\) 0 0
\(655\) 1.16556e7 1.06152
\(656\) 0 0
\(657\) 370308. 0.0334696
\(658\) 0 0
\(659\) −9.87522e6 −0.885795 −0.442898 0.896572i \(-0.646050\pi\)
−0.442898 + 0.896572i \(0.646050\pi\)
\(660\) 0 0
\(661\) 8.06792e6 0.718221 0.359110 0.933295i \(-0.383080\pi\)
0.359110 + 0.933295i \(0.383080\pi\)
\(662\) 0 0
\(663\) −8.45884e6 −0.747355
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −147759. −0.0128599
\(668\) 0 0
\(669\) 2.55461e6 0.220678
\(670\) 0 0
\(671\) 2.12534e7 1.82231
\(672\) 0 0
\(673\) −1.12772e7 −0.959762 −0.479881 0.877334i \(-0.659320\pi\)
−0.479881 + 0.877334i \(0.659320\pi\)
\(674\) 0 0
\(675\) 2.93008e6 0.247526
\(676\) 0 0
\(677\) −5.20372e6 −0.436357 −0.218179 0.975909i \(-0.570012\pi\)
−0.218179 + 0.975909i \(0.570012\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.90778e6 0.157638
\(682\) 0 0
\(683\) −6.05915e6 −0.497004 −0.248502 0.968631i \(-0.579938\pi\)
−0.248502 + 0.968631i \(0.579938\pi\)
\(684\) 0 0
\(685\) −4.46816e6 −0.363833
\(686\) 0 0
\(687\) 8.05671e6 0.651277
\(688\) 0 0
\(689\) 1.30781e7 1.04954
\(690\) 0 0
\(691\) 7.36498e6 0.586781 0.293391 0.955993i \(-0.405216\pi\)
0.293391 + 0.955993i \(0.405216\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.67228e6 0.759569
\(696\) 0 0
\(697\) 1.50674e7 1.17478
\(698\) 0 0
\(699\) 748298. 0.0579271
\(700\) 0 0
\(701\) 7.80919e6 0.600221 0.300110 0.953904i \(-0.402977\pi\)
0.300110 + 0.953904i \(0.402977\pi\)
\(702\) 0 0
\(703\) 2.98193e6 0.227567
\(704\) 0 0
\(705\) 2.31258e6 0.175236
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.75650e7 1.31230 0.656150 0.754631i \(-0.272184\pi\)
0.656150 + 0.754631i \(0.272184\pi\)
\(710\) 0 0
\(711\) −1706.88 −0.000126628 0
\(712\) 0 0
\(713\) −389879. −0.0287214
\(714\) 0 0
\(715\) −2.77649e7 −2.03110
\(716\) 0 0
\(717\) 7.51597e6 0.545993
\(718\) 0 0
\(719\) −8.09220e6 −0.583773 −0.291887 0.956453i \(-0.594283\pi\)
−0.291887 + 0.956453i \(0.594283\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −5.32162e6 −0.378615
\(724\) 0 0
\(725\) −3.36172e6 −0.237529
\(726\) 0 0
\(727\) 1.51986e7 1.06652 0.533258 0.845952i \(-0.320967\pi\)
0.533258 + 0.845952i \(0.320967\pi\)
\(728\) 0 0
\(729\) −1.31285e6 −0.0914944
\(730\) 0 0
\(731\) 114259. 0.00790858
\(732\) 0 0
\(733\) −5.83402e6 −0.401059 −0.200530 0.979688i \(-0.564266\pi\)
−0.200530 + 0.979688i \(0.564266\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.72722e7 2.52765
\(738\) 0 0
\(739\) −6.47719e6 −0.436290 −0.218145 0.975916i \(-0.570001\pi\)
−0.218145 + 0.975916i \(0.570001\pi\)
\(740\) 0 0
\(741\) 8.35433e6 0.558941
\(742\) 0 0
\(743\) 1.50899e7 1.00280 0.501401 0.865215i \(-0.332818\pi\)
0.501401 + 0.865215i \(0.332818\pi\)
\(744\) 0 0
\(745\) 6.45744e6 0.426255
\(746\) 0 0
\(747\) 1.90196e7 1.24710
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.13997e6 0.138455 0.0692273 0.997601i \(-0.477947\pi\)
0.0692273 + 0.997601i \(0.477947\pi\)
\(752\) 0 0
\(753\) −2.66355e6 −0.171188
\(754\) 0 0
\(755\) −5.50494e6 −0.351467
\(756\) 0 0
\(757\) 2.10943e7 1.33791 0.668954 0.743304i \(-0.266742\pi\)
0.668954 + 0.743304i \(0.266742\pi\)
\(758\) 0 0
\(759\) −160366. −0.0101044
\(760\) 0 0
\(761\) −9.79958e6 −0.613403 −0.306701 0.951806i \(-0.599225\pi\)
−0.306701 + 0.951806i \(0.599225\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.08672e7 −0.671371
\(766\) 0 0
\(767\) −1.07848e7 −0.661947
\(768\) 0 0
\(769\) 3.23493e7 1.97265 0.986323 0.164825i \(-0.0527060\pi\)
0.986323 + 0.164825i \(0.0527060\pi\)
\(770\) 0 0
\(771\) 1.11128e7 0.673267
\(772\) 0 0
\(773\) −9.19713e6 −0.553609 −0.276805 0.960926i \(-0.589276\pi\)
−0.276805 + 0.960926i \(0.589276\pi\)
\(774\) 0 0
\(775\) −8.87030e6 −0.530499
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.48812e7 −0.878609
\(780\) 0 0
\(781\) 3.38653e7 1.98668
\(782\) 0 0
\(783\) −9.65941e6 −0.563049
\(784\) 0 0
\(785\) −4.48950e6 −0.260030
\(786\) 0 0
\(787\) 1.46950e7 0.845731 0.422866 0.906192i \(-0.361024\pi\)
0.422866 + 0.906192i \(0.361024\pi\)
\(788\) 0 0
\(789\) −1.34689e6 −0.0770263
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.21423e7 −2.37977
\(794\) 0 0
\(795\) −3.59975e6 −0.202001
\(796\) 0 0
\(797\) −2.33344e7 −1.30122 −0.650610 0.759412i \(-0.725487\pi\)
−0.650610 + 0.759412i \(0.725487\pi\)
\(798\) 0 0
\(799\) 9.06688e6 0.502448
\(800\) 0 0
\(801\) 1.07277e7 0.590782
\(802\) 0 0
\(803\) −1.02101e6 −0.0558783
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.13559e7 −0.613814
\(808\) 0 0
\(809\) 1.69301e6 0.0909468 0.0454734 0.998966i \(-0.485520\pi\)
0.0454734 + 0.998966i \(0.485520\pi\)
\(810\) 0 0
\(811\) −2.12400e7 −1.13397 −0.566987 0.823727i \(-0.691891\pi\)
−0.566987 + 0.823727i \(0.691891\pi\)
\(812\) 0 0
\(813\) 2.41649e6 0.128221
\(814\) 0 0
\(815\) −8.40155e6 −0.443063
\(816\) 0 0
\(817\) −112848. −0.00591477
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.73550e6 0.452304 0.226152 0.974092i \(-0.427385\pi\)
0.226152 + 0.974092i \(0.427385\pi\)
\(822\) 0 0
\(823\) 3.27964e7 1.68782 0.843910 0.536485i \(-0.180248\pi\)
0.843910 + 0.536485i \(0.180248\pi\)
\(824\) 0 0
\(825\) −3.64856e6 −0.186632
\(826\) 0 0
\(827\) −1.31248e7 −0.667311 −0.333656 0.942695i \(-0.608282\pi\)
−0.333656 + 0.942695i \(0.608282\pi\)
\(828\) 0 0
\(829\) −2.05402e7 −1.03805 −0.519026 0.854759i \(-0.673705\pi\)
−0.519026 + 0.854759i \(0.673705\pi\)
\(830\) 0 0
\(831\) 8.03880e6 0.403821
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −2.07729e7 −1.03106
\(836\) 0 0
\(837\) −2.54875e7 −1.25752
\(838\) 0 0
\(839\) 2.83736e7 1.39159 0.695793 0.718243i \(-0.255053\pi\)
0.695793 + 0.718243i \(0.255053\pi\)
\(840\) 0 0
\(841\) −9.42879e6 −0.459691
\(842\) 0 0
\(843\) 1.31400e7 0.636834
\(844\) 0 0
\(845\) 3.79774e7 1.82972
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.16935e7 −0.556770
\(850\) 0 0
\(851\) 113498. 0.00537236
\(852\) 0 0
\(853\) −1.02093e7 −0.480424 −0.240212 0.970720i \(-0.577217\pi\)
−0.240212 + 0.970720i \(0.577217\pi\)
\(854\) 0 0
\(855\) 1.07329e7 0.502113
\(856\) 0 0
\(857\) 8.33206e6 0.387525 0.193763 0.981048i \(-0.437931\pi\)
0.193763 + 0.981048i \(0.437931\pi\)
\(858\) 0 0
\(859\) 3.12766e7 1.44623 0.723113 0.690729i \(-0.242710\pi\)
0.723113 + 0.690729i \(0.242710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.73573e7 1.70745 0.853726 0.520722i \(-0.174337\pi\)
0.853726 + 0.520722i \(0.174337\pi\)
\(864\) 0 0
\(865\) −1.70924e7 −0.776719
\(866\) 0 0
\(867\) −168747. −0.00762411
\(868\) 0 0
\(869\) 4706.21 0.000211408 0
\(870\) 0 0
\(871\) −7.39051e7 −3.30088
\(872\) 0 0
\(873\) 622543. 0.0276461
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38996.7 0.00171210 0.000856049 1.00000i \(-0.499728\pi\)
0.000856049 1.00000i \(0.499728\pi\)
\(878\) 0 0
\(879\) −5.58708e6 −0.243900
\(880\) 0 0
\(881\) −3.15554e7 −1.36973 −0.684864 0.728671i \(-0.740138\pi\)
−0.684864 + 0.728671i \(0.740138\pi\)
\(882\) 0 0
\(883\) 3.42253e7 1.47722 0.738611 0.674132i \(-0.235482\pi\)
0.738611 + 0.674132i \(0.235482\pi\)
\(884\) 0 0
\(885\) 2.96851e6 0.127403
\(886\) 0 0
\(887\) −2.69886e7 −1.15178 −0.575892 0.817526i \(-0.695345\pi\)
−0.575892 + 0.817526i \(0.695345\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.63495e7 0.689938
\(892\) 0 0
\(893\) −8.95486e6 −0.375777
\(894\) 0 0
\(895\) −3.90940e6 −0.163137
\(896\) 0 0
\(897\) 317982. 0.0131954
\(898\) 0 0
\(899\) 2.92421e7 1.20673
\(900\) 0 0
\(901\) −1.41135e7 −0.579191
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.74509e7 0.708267
\(906\) 0 0
\(907\) −1.92103e7 −0.775381 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(908\) 0 0
\(909\) 4.36988e6 0.175412
\(910\) 0 0
\(911\) −2.86013e7 −1.14180 −0.570899 0.821020i \(-0.693405\pi\)
−0.570899 + 0.821020i \(0.693405\pi\)
\(912\) 0 0
\(913\) −5.24408e7 −2.08206
\(914\) 0 0
\(915\) 1.15997e7 0.458029
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4.21754e7 1.64729 0.823645 0.567106i \(-0.191937\pi\)
0.823645 + 0.567106i \(0.191937\pi\)
\(920\) 0 0
\(921\) −1.28386e7 −0.498733
\(922\) 0 0
\(923\) −6.71498e7 −2.59442
\(924\) 0 0
\(925\) 2.58224e6 0.0992299
\(926\) 0 0
\(927\) −1.30764e7 −0.499790
\(928\) 0 0
\(929\) 3.01886e7 1.14763 0.573817 0.818983i \(-0.305462\pi\)
0.573817 + 0.818983i \(0.305462\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 5.65489e6 0.212677
\(934\) 0 0
\(935\) 2.99630e7 1.12087
\(936\) 0 0
\(937\) 3.64068e6 0.135467 0.0677335 0.997703i \(-0.478423\pi\)
0.0677335 + 0.997703i \(0.478423\pi\)
\(938\) 0 0
\(939\) −7.20589e6 −0.266701
\(940\) 0 0
\(941\) −1.88601e7 −0.694336 −0.347168 0.937803i \(-0.612857\pi\)
−0.347168 + 0.937803i \(0.612857\pi\)
\(942\) 0 0
\(943\) −566410. −0.0207420
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.82172e7 0.660094 0.330047 0.943965i \(-0.392935\pi\)
0.330047 + 0.943965i \(0.392935\pi\)
\(948\) 0 0
\(949\) 2.02452e6 0.0729720
\(950\) 0 0
\(951\) 9.79522e6 0.351207
\(952\) 0 0
\(953\) 2.76898e7 0.987616 0.493808 0.869571i \(-0.335604\pi\)
0.493808 + 0.869571i \(0.335604\pi\)
\(954\) 0 0
\(955\) 4.24127e7 1.50483
\(956\) 0 0
\(957\) 1.20280e7 0.424534
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.85298e7 1.69512
\(962\) 0 0
\(963\) −2.18049e7 −0.757685
\(964\) 0 0
\(965\) 2.32304e7 0.803042
\(966\) 0 0
\(967\) 2.44768e7 0.841761 0.420881 0.907116i \(-0.361721\pi\)
0.420881 + 0.907116i \(0.361721\pi\)
\(968\) 0 0
\(969\) −9.01571e6 −0.308454
\(970\) 0 0
\(971\) 9.50151e6 0.323403 0.161702 0.986840i \(-0.448302\pi\)
0.161702 + 0.986840i \(0.448302\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7.23455e6 0.243725
\(976\) 0 0
\(977\) 4.69012e7 1.57198 0.785991 0.618238i \(-0.212153\pi\)
0.785991 + 0.618238i \(0.212153\pi\)
\(978\) 0 0
\(979\) −2.95785e7 −0.986325
\(980\) 0 0
\(981\) −1.73564e7 −0.575822
\(982\) 0 0
\(983\) 2.35382e7 0.776945 0.388473 0.921460i \(-0.373003\pi\)
0.388473 + 0.921460i \(0.373003\pi\)
\(984\) 0 0
\(985\) 1.15670e7 0.379865
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4295.21 −0.000139635 0
\(990\) 0 0
\(991\) 2.64104e7 0.854261 0.427130 0.904190i \(-0.359525\pi\)
0.427130 + 0.904190i \(0.359525\pi\)
\(992\) 0 0
\(993\) 1.79426e7 0.577446
\(994\) 0 0
\(995\) −9.56766e6 −0.306371
\(996\) 0 0
\(997\) 1.95164e7 0.621815 0.310907 0.950440i \(-0.399367\pi\)
0.310907 + 0.950440i \(0.399367\pi\)
\(998\) 0 0
\(999\) 7.41970e6 0.235219
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 784.6.a.bf.1.3 4
4.3 odd 2 49.6.a.g.1.3 4
7.6 odd 2 inner 784.6.a.bf.1.2 4
12.11 even 2 441.6.a.z.1.1 4
28.3 even 6 49.6.c.h.30.1 8
28.11 odd 6 49.6.c.h.30.2 8
28.19 even 6 49.6.c.h.18.1 8
28.23 odd 6 49.6.c.h.18.2 8
28.27 even 2 49.6.a.g.1.4 yes 4
84.83 odd 2 441.6.a.z.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 4.3 odd 2
49.6.a.g.1.4 yes 4 28.27 even 2
49.6.c.h.18.1 8 28.19 even 6
49.6.c.h.18.2 8 28.23 odd 6
49.6.c.h.30.1 8 28.3 even 6
49.6.c.h.30.2 8 28.11 odd 6
441.6.a.z.1.1 4 12.11 even 2
441.6.a.z.1.2 4 84.83 odd 2
784.6.a.bf.1.2 4 7.6 odd 2 inner
784.6.a.bf.1.3 4 1.1 even 1 trivial