Properties

Label 448.6.a.c.1.1
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 448.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.0000 q^{3} +56.0000 q^{5} +49.0000 q^{7} -47.0000 q^{9} +O(q^{10})\) \(q-14.0000 q^{3} +56.0000 q^{5} +49.0000 q^{7} -47.0000 q^{9} +232.000 q^{11} +140.000 q^{13} -784.000 q^{15} -1722.00 q^{17} -98.0000 q^{19} -686.000 q^{21} -1824.00 q^{23} +11.0000 q^{25} +4060.00 q^{27} -3418.00 q^{29} +7644.00 q^{31} -3248.00 q^{33} +2744.00 q^{35} +10398.0 q^{37} -1960.00 q^{39} -17962.0 q^{41} +10880.0 q^{43} -2632.00 q^{45} -9324.00 q^{47} +2401.00 q^{49} +24108.0 q^{51} -2262.00 q^{53} +12992.0 q^{55} +1372.00 q^{57} -2730.00 q^{59} -25648.0 q^{61} -2303.00 q^{63} +7840.00 q^{65} -48404.0 q^{67} +25536.0 q^{69} +58560.0 q^{71} +68082.0 q^{73} -154.000 q^{75} +11368.0 q^{77} -31784.0 q^{79} -45419.0 q^{81} -20538.0 q^{83} -96432.0 q^{85} +47852.0 q^{87} -50582.0 q^{89} +6860.00 q^{91} -107016. q^{93} -5488.00 q^{95} -58506.0 q^{97} -10904.0 q^{99} +O(q^{100})\)

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
\(3\) −14.0000 −0.898100 −0.449050 0.893507i \(-0.648238\pi\)
−0.449050 + 0.893507i \(0.648238\pi\)
\(4\) 0 0
\(5\) 56.0000 1.00176 0.500879 0.865517i \(-0.333010\pi\)
0.500879 + 0.865517i \(0.333010\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −47.0000 −0.193416
\(10\) 0 0
\(11\) 232.000 0.578104 0.289052 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(12\) 0 0
\(13\) 140.000 0.229757 0.114879 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 0 0
\(15\) −784.000 −0.899680
\(16\) 0 0
\(17\) −1722.00 −1.44514 −0.722572 0.691296i \(-0.757040\pi\)
−0.722572 + 0.691296i \(0.757040\pi\)
\(18\) 0 0
\(19\) −98.0000 −0.0622791 −0.0311395 0.999515i \(-0.509914\pi\)
−0.0311395 + 0.999515i \(0.509914\pi\)
\(20\) 0 0
\(21\) −686.000 −0.339450
\(22\) 0 0
\(23\) −1824.00 −0.718961 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(24\) 0 0
\(25\) 11.0000 0.00352000
\(26\) 0 0
\(27\) 4060.00 1.07181
\(28\) 0 0
\(29\) −3418.00 −0.754705 −0.377352 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(30\) 0 0
\(31\) 7644.00 1.42862 0.714310 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(32\) 0 0
\(33\) −3248.00 −0.519196
\(34\) 0 0
\(35\) 2744.00 0.378629
\(36\) 0 0
\(37\) 10398.0 1.24866 0.624332 0.781159i \(-0.285371\pi\)
0.624332 + 0.781159i \(0.285371\pi\)
\(38\) 0 0
\(39\) −1960.00 −0.206345
\(40\) 0 0
\(41\) −17962.0 −1.66876 −0.834382 0.551186i \(-0.814175\pi\)
−0.834382 + 0.551186i \(0.814175\pi\)
\(42\) 0 0
\(43\) 10880.0 0.897342 0.448671 0.893697i \(-0.351898\pi\)
0.448671 + 0.893697i \(0.351898\pi\)
\(44\) 0 0
\(45\) −2632.00 −0.193756
\(46\) 0 0
\(47\) −9324.00 −0.615684 −0.307842 0.951438i \(-0.599607\pi\)
−0.307842 + 0.951438i \(0.599607\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 24108.0 1.29788
\(52\) 0 0
\(53\) −2262.00 −0.110612 −0.0553061 0.998469i \(-0.517613\pi\)
−0.0553061 + 0.998469i \(0.517613\pi\)
\(54\) 0 0
\(55\) 12992.0 0.579121
\(56\) 0 0
\(57\) 1372.00 0.0559329
\(58\) 0 0
\(59\) −2730.00 −0.102102 −0.0510508 0.998696i \(-0.516257\pi\)
−0.0510508 + 0.998696i \(0.516257\pi\)
\(60\) 0 0
\(61\) −25648.0 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) 0 0
\(63\) −2303.00 −0.0731042
\(64\) 0 0
\(65\) 7840.00 0.230161
\(66\) 0 0
\(67\) −48404.0 −1.31733 −0.658664 0.752437i \(-0.728878\pi\)
−0.658664 + 0.752437i \(0.728878\pi\)
\(68\) 0 0
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) 58560.0 1.37865 0.689327 0.724450i \(-0.257906\pi\)
0.689327 + 0.724450i \(0.257906\pi\)
\(72\) 0 0
\(73\) 68082.0 1.49529 0.747645 0.664099i \(-0.231185\pi\)
0.747645 + 0.664099i \(0.231185\pi\)
\(74\) 0 0
\(75\) −154.000 −0.00316131
\(76\) 0 0
\(77\) 11368.0 0.218503
\(78\) 0 0
\(79\) −31784.0 −0.572982 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) 0 0
\(83\) −20538.0 −0.327237 −0.163619 0.986524i \(-0.552317\pi\)
−0.163619 + 0.986524i \(0.552317\pi\)
\(84\) 0 0
\(85\) −96432.0 −1.44768
\(86\) 0 0
\(87\) 47852.0 0.677801
\(88\) 0 0
\(89\) −50582.0 −0.676894 −0.338447 0.940985i \(-0.609902\pi\)
−0.338447 + 0.940985i \(0.609902\pi\)
\(90\) 0 0
\(91\) 6860.00 0.0868402
\(92\) 0 0
\(93\) −107016. −1.28304
\(94\) 0 0
\(95\) −5488.00 −0.0623886
\(96\) 0 0
\(97\) −58506.0 −0.631351 −0.315676 0.948867i \(-0.602231\pi\)
−0.315676 + 0.948867i \(0.602231\pi\)
\(98\) 0 0
\(99\) −10904.0 −0.111814
\(100\) 0 0
\(101\) −38696.0 −0.377453 −0.188726 0.982030i \(-0.560436\pi\)
−0.188726 + 0.982030i \(0.560436\pi\)
\(102\) 0 0
\(103\) −53060.0 −0.492804 −0.246402 0.969168i \(-0.579248\pi\)
−0.246402 + 0.969168i \(0.579248\pi\)
\(104\) 0 0
\(105\) −38416.0 −0.340047
\(106\) 0 0
\(107\) −146324. −1.23554 −0.617769 0.786360i \(-0.711963\pi\)
−0.617769 + 0.786360i \(0.711963\pi\)
\(108\) 0 0
\(109\) −92898.0 −0.748928 −0.374464 0.927241i \(-0.622173\pi\)
−0.374464 + 0.927241i \(0.622173\pi\)
\(110\) 0 0
\(111\) −145572. −1.12143
\(112\) 0 0
\(113\) −83354.0 −0.614088 −0.307044 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(114\) 0 0
\(115\) −102144. −0.720225
\(116\) 0 0
\(117\) −6580.00 −0.0444387
\(118\) 0 0
\(119\) −84378.0 −0.546213
\(120\) 0 0
\(121\) −107227. −0.665795
\(122\) 0 0
\(123\) 251468. 1.49872
\(124\) 0 0
\(125\) −174384. −0.998232
\(126\) 0 0
\(127\) −60384.0 −0.332210 −0.166105 0.986108i \(-0.553119\pi\)
−0.166105 + 0.986108i \(0.553119\pi\)
\(128\) 0 0
\(129\) −152320. −0.805903
\(130\) 0 0
\(131\) −61586.0 −0.313548 −0.156774 0.987635i \(-0.550109\pi\)
−0.156774 + 0.987635i \(0.550109\pi\)
\(132\) 0 0
\(133\) −4802.00 −0.0235393
\(134\) 0 0
\(135\) 227360. 1.07369
\(136\) 0 0
\(137\) −204462. −0.930703 −0.465352 0.885126i \(-0.654072\pi\)
−0.465352 + 0.885126i \(0.654072\pi\)
\(138\) 0 0
\(139\) −35406.0 −0.155432 −0.0777159 0.996976i \(-0.524763\pi\)
−0.0777159 + 0.996976i \(0.524763\pi\)
\(140\) 0 0
\(141\) 130536. 0.552946
\(142\) 0 0
\(143\) 32480.0 0.132824
\(144\) 0 0
\(145\) −191408. −0.756032
\(146\) 0 0
\(147\) −33614.0 −0.128300
\(148\) 0 0
\(149\) 20226.0 0.0746353 0.0373177 0.999303i \(-0.488119\pi\)
0.0373177 + 0.999303i \(0.488119\pi\)
\(150\) 0 0
\(151\) −70904.0 −0.253063 −0.126531 0.991963i \(-0.540384\pi\)
−0.126531 + 0.991963i \(0.540384\pi\)
\(152\) 0 0
\(153\) 80934.0 0.279513
\(154\) 0 0
\(155\) 428064. 1.43113
\(156\) 0 0
\(157\) −293524. −0.950374 −0.475187 0.879885i \(-0.657620\pi\)
−0.475187 + 0.879885i \(0.657620\pi\)
\(158\) 0 0
\(159\) 31668.0 0.0993408
\(160\) 0 0
\(161\) −89376.0 −0.271742
\(162\) 0 0
\(163\) 13192.0 0.0388903 0.0194452 0.999811i \(-0.493810\pi\)
0.0194452 + 0.999811i \(0.493810\pi\)
\(164\) 0 0
\(165\) −181888. −0.520109
\(166\) 0 0
\(167\) −493612. −1.36960 −0.684801 0.728730i \(-0.740111\pi\)
−0.684801 + 0.728730i \(0.740111\pi\)
\(168\) 0 0
\(169\) −351693. −0.947212
\(170\) 0 0
\(171\) 4606.00 0.0120457
\(172\) 0 0
\(173\) −240716. −0.611490 −0.305745 0.952113i \(-0.598906\pi\)
−0.305745 + 0.952113i \(0.598906\pi\)
\(174\) 0 0
\(175\) 539.000 0.00133043
\(176\) 0 0
\(177\) 38220.0 0.0916975
\(178\) 0 0
\(179\) 294932. 0.688001 0.344001 0.938969i \(-0.388218\pi\)
0.344001 + 0.938969i \(0.388218\pi\)
\(180\) 0 0
\(181\) 336980. 0.764553 0.382277 0.924048i \(-0.375140\pi\)
0.382277 + 0.924048i \(0.375140\pi\)
\(182\) 0 0
\(183\) 359072. 0.792600
\(184\) 0 0
\(185\) 582288. 1.25086
\(186\) 0 0
\(187\) −399504. −0.835444
\(188\) 0 0
\(189\) 198940. 0.405105
\(190\) 0 0
\(191\) −358264. −0.710591 −0.355296 0.934754i \(-0.615620\pi\)
−0.355296 + 0.934754i \(0.615620\pi\)
\(192\) 0 0
\(193\) −989554. −1.91226 −0.956128 0.292948i \(-0.905364\pi\)
−0.956128 + 0.292948i \(0.905364\pi\)
\(194\) 0 0
\(195\) −109760. −0.206708
\(196\) 0 0
\(197\) 990050. 1.81757 0.908786 0.417263i \(-0.137011\pi\)
0.908786 + 0.417263i \(0.137011\pi\)
\(198\) 0 0
\(199\) 840756. 1.50500 0.752501 0.658591i \(-0.228847\pi\)
0.752501 + 0.658591i \(0.228847\pi\)
\(200\) 0 0
\(201\) 677656. 1.18309
\(202\) 0 0
\(203\) −167482. −0.285252
\(204\) 0 0
\(205\) −1.00587e6 −1.67170
\(206\) 0 0
\(207\) 85728.0 0.139058
\(208\) 0 0
\(209\) −22736.0 −0.0360038
\(210\) 0 0
\(211\) 1.15073e6 1.77938 0.889689 0.456568i \(-0.150921\pi\)
0.889689 + 0.456568i \(0.150921\pi\)
\(212\) 0 0
\(213\) −819840. −1.23817
\(214\) 0 0
\(215\) 609280. 0.898919
\(216\) 0 0
\(217\) 374556. 0.539967
\(218\) 0 0
\(219\) −953148. −1.34292
\(220\) 0 0
\(221\) −241080. −0.332032
\(222\) 0 0
\(223\) 824264. 1.10995 0.554976 0.831866i \(-0.312727\pi\)
0.554976 + 0.831866i \(0.312727\pi\)
\(224\) 0 0
\(225\) −517.000 −0.000680823 0
\(226\) 0 0
\(227\) 74382.0 0.0958083 0.0479042 0.998852i \(-0.484746\pi\)
0.0479042 + 0.998852i \(0.484746\pi\)
\(228\) 0 0
\(229\) −1.13196e6 −1.42640 −0.713199 0.700961i \(-0.752755\pi\)
−0.713199 + 0.700961i \(0.752755\pi\)
\(230\) 0 0
\(231\) −159152. −0.196238
\(232\) 0 0
\(233\) −198726. −0.239809 −0.119904 0.992785i \(-0.538259\pi\)
−0.119904 + 0.992785i \(0.538259\pi\)
\(234\) 0 0
\(235\) −522144. −0.616766
\(236\) 0 0
\(237\) 444976. 0.514595
\(238\) 0 0
\(239\) −482904. −0.546847 −0.273424 0.961894i \(-0.588156\pi\)
−0.273424 + 0.961894i \(0.588156\pi\)
\(240\) 0 0
\(241\) 805910. 0.893807 0.446904 0.894582i \(-0.352527\pi\)
0.446904 + 0.894582i \(0.352527\pi\)
\(242\) 0 0
\(243\) −350714. −0.381011
\(244\) 0 0
\(245\) 134456. 0.143108
\(246\) 0 0
\(247\) −13720.0 −0.0143091
\(248\) 0 0
\(249\) 287532. 0.293892
\(250\) 0 0
\(251\) 430738. 0.431548 0.215774 0.976443i \(-0.430773\pi\)
0.215774 + 0.976443i \(0.430773\pi\)
\(252\) 0 0
\(253\) −423168. −0.415634
\(254\) 0 0
\(255\) 1.35005e6 1.30017
\(256\) 0 0
\(257\) −1.17691e6 −1.11150 −0.555751 0.831349i \(-0.687569\pi\)
−0.555751 + 0.831349i \(0.687569\pi\)
\(258\) 0 0
\(259\) 509502. 0.471951
\(260\) 0 0
\(261\) 160646. 0.145972
\(262\) 0 0
\(263\) −1.29098e6 −1.15088 −0.575438 0.817845i \(-0.695169\pi\)
−0.575438 + 0.817845i \(0.695169\pi\)
\(264\) 0 0
\(265\) −126672. −0.110807
\(266\) 0 0
\(267\) 708148. 0.607919
\(268\) 0 0
\(269\) 1.27756e6 1.07646 0.538232 0.842797i \(-0.319093\pi\)
0.538232 + 0.842797i \(0.319093\pi\)
\(270\) 0 0
\(271\) −1.65054e6 −1.36522 −0.682612 0.730781i \(-0.739156\pi\)
−0.682612 + 0.730781i \(0.739156\pi\)
\(272\) 0 0
\(273\) −96040.0 −0.0779912
\(274\) 0 0
\(275\) 2552.00 0.00203493
\(276\) 0 0
\(277\) 1.06409e6 0.833257 0.416628 0.909077i \(-0.363212\pi\)
0.416628 + 0.909077i \(0.363212\pi\)
\(278\) 0 0
\(279\) −359268. −0.276317
\(280\) 0 0
\(281\) −22342.0 −0.0168794 −0.00843969 0.999964i \(-0.502686\pi\)
−0.00843969 + 0.999964i \(0.502686\pi\)
\(282\) 0 0
\(283\) −2.49574e6 −1.85239 −0.926196 0.377042i \(-0.876941\pi\)
−0.926196 + 0.377042i \(0.876941\pi\)
\(284\) 0 0
\(285\) 76832.0 0.0560312
\(286\) 0 0
\(287\) −880138. −0.630734
\(288\) 0 0
\(289\) 1.54543e6 1.08844
\(290\) 0 0
\(291\) 819084. 0.567017
\(292\) 0 0
\(293\) 1.93178e6 1.31458 0.657291 0.753637i \(-0.271702\pi\)
0.657291 + 0.753637i \(0.271702\pi\)
\(294\) 0 0
\(295\) −152880. −0.102281
\(296\) 0 0
\(297\) 941920. 0.619616
\(298\) 0 0
\(299\) −255360. −0.165187
\(300\) 0 0
\(301\) 533120. 0.339163
\(302\) 0 0
\(303\) 541744. 0.338991
\(304\) 0 0
\(305\) −1.43629e6 −0.884081
\(306\) 0 0
\(307\) −459074. −0.277995 −0.138997 0.990293i \(-0.544388\pi\)
−0.138997 + 0.990293i \(0.544388\pi\)
\(308\) 0 0
\(309\) 742840. 0.442587
\(310\) 0 0
\(311\) −667128. −0.391118 −0.195559 0.980692i \(-0.562652\pi\)
−0.195559 + 0.980692i \(0.562652\pi\)
\(312\) 0 0
\(313\) −111034. −0.0640612 −0.0320306 0.999487i \(-0.510197\pi\)
−0.0320306 + 0.999487i \(0.510197\pi\)
\(314\) 0 0
\(315\) −128968. −0.0732328
\(316\) 0 0
\(317\) 68778.0 0.0384416 0.0192208 0.999815i \(-0.493881\pi\)
0.0192208 + 0.999815i \(0.493881\pi\)
\(318\) 0 0
\(319\) −792976. −0.436298
\(320\) 0 0
\(321\) 2.04854e6 1.10964
\(322\) 0 0
\(323\) 168756. 0.0900022
\(324\) 0 0
\(325\) 1540.00 0.000808746 0
\(326\) 0 0
\(327\) 1.30057e6 0.672613
\(328\) 0 0
\(329\) −456876. −0.232707
\(330\) 0 0
\(331\) −564448. −0.283174 −0.141587 0.989926i \(-0.545221\pi\)
−0.141587 + 0.989926i \(0.545221\pi\)
\(332\) 0 0
\(333\) −488706. −0.241511
\(334\) 0 0
\(335\) −2.71062e6 −1.31965
\(336\) 0 0
\(337\) 2.07729e6 0.996376 0.498188 0.867069i \(-0.333999\pi\)
0.498188 + 0.867069i \(0.333999\pi\)
\(338\) 0 0
\(339\) 1.16696e6 0.551512
\(340\) 0 0
\(341\) 1.77341e6 0.825891
\(342\) 0 0
\(343\) 117649. 0.0539949
\(344\) 0 0
\(345\) 1.43002e6 0.646834
\(346\) 0 0
\(347\) −53248.0 −0.0237399 −0.0118700 0.999930i \(-0.503778\pi\)
−0.0118700 + 0.999930i \(0.503778\pi\)
\(348\) 0 0
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 0 0
\(351\) 568400. 0.246256
\(352\) 0 0
\(353\) 4.00645e6 1.71129 0.855644 0.517565i \(-0.173162\pi\)
0.855644 + 0.517565i \(0.173162\pi\)
\(354\) 0 0
\(355\) 3.27936e6 1.38108
\(356\) 0 0
\(357\) 1.18129e6 0.490554
\(358\) 0 0
\(359\) −73784.0 −0.0302152 −0.0151076 0.999886i \(-0.504809\pi\)
−0.0151076 + 0.999886i \(0.504809\pi\)
\(360\) 0 0
\(361\) −2.46650e6 −0.996121
\(362\) 0 0
\(363\) 1.50118e6 0.597951
\(364\) 0 0
\(365\) 3.81259e6 1.49792
\(366\) 0 0
\(367\) −1.40431e6 −0.544250 −0.272125 0.962262i \(-0.587726\pi\)
−0.272125 + 0.962262i \(0.587726\pi\)
\(368\) 0 0
\(369\) 844214. 0.322765
\(370\) 0 0
\(371\) −110838. −0.0418075
\(372\) 0 0
\(373\) 1.60323e6 0.596657 0.298329 0.954463i \(-0.403571\pi\)
0.298329 + 0.954463i \(0.403571\pi\)
\(374\) 0 0
\(375\) 2.44138e6 0.896513
\(376\) 0 0
\(377\) −478520. −0.173399
\(378\) 0 0
\(379\) −4.77012e6 −1.70581 −0.852906 0.522064i \(-0.825162\pi\)
−0.852906 + 0.522064i \(0.825162\pi\)
\(380\) 0 0
\(381\) 845376. 0.298358
\(382\) 0 0
\(383\) 2.23079e6 0.777072 0.388536 0.921434i \(-0.372981\pi\)
0.388536 + 0.921434i \(0.372981\pi\)
\(384\) 0 0
\(385\) 636608. 0.218887
\(386\) 0 0
\(387\) −511360. −0.173560
\(388\) 0 0
\(389\) −4.84024e6 −1.62178 −0.810892 0.585196i \(-0.801018\pi\)
−0.810892 + 0.585196i \(0.801018\pi\)
\(390\) 0 0
\(391\) 3.14093e6 1.03900
\(392\) 0 0
\(393\) 862204. 0.281597
\(394\) 0 0
\(395\) −1.77990e6 −0.573989
\(396\) 0 0
\(397\) −995820. −0.317106 −0.158553 0.987350i \(-0.550683\pi\)
−0.158553 + 0.987350i \(0.550683\pi\)
\(398\) 0 0
\(399\) 67228.0 0.0211406
\(400\) 0 0
\(401\) −3.31605e6 −1.02982 −0.514909 0.857245i \(-0.672174\pi\)
−0.514909 + 0.857245i \(0.672174\pi\)
\(402\) 0 0
\(403\) 1.07016e6 0.328236
\(404\) 0 0
\(405\) −2.54346e6 −0.770527
\(406\) 0 0
\(407\) 2.41234e6 0.721858
\(408\) 0 0
\(409\) 3.07273e6 0.908274 0.454137 0.890932i \(-0.349948\pi\)
0.454137 + 0.890932i \(0.349948\pi\)
\(410\) 0 0
\(411\) 2.86247e6 0.835865
\(412\) 0 0
\(413\) −133770. −0.0385908
\(414\) 0 0
\(415\) −1.15013e6 −0.327813
\(416\) 0 0
\(417\) 495684. 0.139593
\(418\) 0 0
\(419\) 2.81438e6 0.783154 0.391577 0.920145i \(-0.371930\pi\)
0.391577 + 0.920145i \(0.371930\pi\)
\(420\) 0 0
\(421\) −3.05802e6 −0.840883 −0.420441 0.907320i \(-0.638125\pi\)
−0.420441 + 0.907320i \(0.638125\pi\)
\(422\) 0 0
\(423\) 438228. 0.119083
\(424\) 0 0
\(425\) −18942.0 −0.00508690
\(426\) 0 0
\(427\) −1.25675e6 −0.333565
\(428\) 0 0
\(429\) −454720. −0.119289
\(430\) 0 0
\(431\) −1.93750e6 −0.502398 −0.251199 0.967936i \(-0.580825\pi\)
−0.251199 + 0.967936i \(0.580825\pi\)
\(432\) 0 0
\(433\) 3.94790e6 1.01192 0.505961 0.862557i \(-0.331138\pi\)
0.505961 + 0.862557i \(0.331138\pi\)
\(434\) 0 0
\(435\) 2.67971e6 0.678993
\(436\) 0 0
\(437\) 178752. 0.0447762
\(438\) 0 0
\(439\) 7.41770e6 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(440\) 0 0
\(441\) −112847. −0.0276308
\(442\) 0 0
\(443\) 1.40269e6 0.339589 0.169794 0.985480i \(-0.445690\pi\)
0.169794 + 0.985480i \(0.445690\pi\)
\(444\) 0 0
\(445\) −2.83259e6 −0.678085
\(446\) 0 0
\(447\) −283164. −0.0670300
\(448\) 0 0
\(449\) −590574. −0.138248 −0.0691239 0.997608i \(-0.522020\pi\)
−0.0691239 + 0.997608i \(0.522020\pi\)
\(450\) 0 0
\(451\) −4.16718e6 −0.964720
\(452\) 0 0
\(453\) 992656. 0.227276
\(454\) 0 0
\(455\) 384160. 0.0869929
\(456\) 0 0
\(457\) −2.90484e6 −0.650627 −0.325313 0.945606i \(-0.605470\pi\)
−0.325313 + 0.945606i \(0.605470\pi\)
\(458\) 0 0
\(459\) −6.99132e6 −1.54891
\(460\) 0 0
\(461\) 922684. 0.202209 0.101105 0.994876i \(-0.467762\pi\)
0.101105 + 0.994876i \(0.467762\pi\)
\(462\) 0 0
\(463\) −7.18235e6 −1.55709 −0.778546 0.627588i \(-0.784042\pi\)
−0.778546 + 0.627588i \(0.784042\pi\)
\(464\) 0 0
\(465\) −5.99290e6 −1.28530
\(466\) 0 0
\(467\) −612570. −0.129976 −0.0649881 0.997886i \(-0.520701\pi\)
−0.0649881 + 0.997886i \(0.520701\pi\)
\(468\) 0 0
\(469\) −2.37180e6 −0.497904
\(470\) 0 0
\(471\) 4.10934e6 0.853531
\(472\) 0 0
\(473\) 2.52416e6 0.518757
\(474\) 0 0
\(475\) −1078.00 −0.000219222 0
\(476\) 0 0
\(477\) 106314. 0.0213941
\(478\) 0 0
\(479\) −2.60330e6 −0.518424 −0.259212 0.965820i \(-0.583463\pi\)
−0.259212 + 0.965820i \(0.583463\pi\)
\(480\) 0 0
\(481\) 1.45572e6 0.286890
\(482\) 0 0
\(483\) 1.25126e6 0.244051
\(484\) 0 0
\(485\) −3.27634e6 −0.632461
\(486\) 0 0
\(487\) −5.46309e6 −1.04380 −0.521898 0.853008i \(-0.674776\pi\)
−0.521898 + 0.853008i \(0.674776\pi\)
\(488\) 0 0
\(489\) −184688. −0.0349274
\(490\) 0 0
\(491\) 1.64090e6 0.307170 0.153585 0.988135i \(-0.450918\pi\)
0.153585 + 0.988135i \(0.450918\pi\)
\(492\) 0 0
\(493\) 5.88580e6 1.09066
\(494\) 0 0
\(495\) −610624. −0.112011
\(496\) 0 0
\(497\) 2.86944e6 0.521082
\(498\) 0 0
\(499\) 2.99796e6 0.538983 0.269491 0.963003i \(-0.413144\pi\)
0.269491 + 0.963003i \(0.413144\pi\)
\(500\) 0 0
\(501\) 6.91057e6 1.23004
\(502\) 0 0
\(503\) 6.89405e6 1.21494 0.607469 0.794343i \(-0.292185\pi\)
0.607469 + 0.794343i \(0.292185\pi\)
\(504\) 0 0
\(505\) −2.16698e6 −0.378117
\(506\) 0 0
\(507\) 4.92370e6 0.850691
\(508\) 0 0
\(509\) −2.30476e6 −0.394305 −0.197152 0.980373i \(-0.563169\pi\)
−0.197152 + 0.980373i \(0.563169\pi\)
\(510\) 0 0
\(511\) 3.33602e6 0.565166
\(512\) 0 0
\(513\) −397880. −0.0667511
\(514\) 0 0
\(515\) −2.97136e6 −0.493671
\(516\) 0 0
\(517\) −2.16317e6 −0.355929
\(518\) 0 0
\(519\) 3.37002e6 0.549180
\(520\) 0 0
\(521\) −1.20960e7 −1.95231 −0.976155 0.217073i \(-0.930349\pi\)
−0.976155 + 0.217073i \(0.930349\pi\)
\(522\) 0 0
\(523\) 5.48443e6 0.876753 0.438377 0.898791i \(-0.355554\pi\)
0.438377 + 0.898791i \(0.355554\pi\)
\(524\) 0 0
\(525\) −7546.00 −0.00119486
\(526\) 0 0
\(527\) −1.31630e7 −2.06456
\(528\) 0 0
\(529\) −3.10937e6 −0.483095
\(530\) 0 0
\(531\) 128310. 0.0197480
\(532\) 0 0
\(533\) −2.51468e6 −0.383411
\(534\) 0 0
\(535\) −8.19414e6 −1.23771
\(536\) 0 0
\(537\) −4.12905e6 −0.617894
\(538\) 0 0
\(539\) 557032. 0.0825863
\(540\) 0 0
\(541\) 6.71799e6 0.986839 0.493420 0.869791i \(-0.335747\pi\)
0.493420 + 0.869791i \(0.335747\pi\)
\(542\) 0 0
\(543\) −4.71772e6 −0.686646
\(544\) 0 0
\(545\) −5.20229e6 −0.750245
\(546\) 0 0
\(547\) −5.00235e6 −0.714835 −0.357418 0.933945i \(-0.616343\pi\)
−0.357418 + 0.933945i \(0.616343\pi\)
\(548\) 0 0
\(549\) 1.20546e6 0.170695
\(550\) 0 0
\(551\) 334964. 0.0470023
\(552\) 0 0
\(553\) −1.55742e6 −0.216567
\(554\) 0 0
\(555\) −8.15203e6 −1.12340
\(556\) 0 0
\(557\) −9.01961e6 −1.23183 −0.615913 0.787814i \(-0.711213\pi\)
−0.615913 + 0.787814i \(0.711213\pi\)
\(558\) 0 0
\(559\) 1.52320e6 0.206171
\(560\) 0 0
\(561\) 5.59306e6 0.750312
\(562\) 0 0
\(563\) 1.24051e7 1.64941 0.824707 0.565561i \(-0.191340\pi\)
0.824707 + 0.565561i \(0.191340\pi\)
\(564\) 0 0
\(565\) −4.66782e6 −0.615167
\(566\) 0 0
\(567\) −2.22553e6 −0.290721
\(568\) 0 0
\(569\) 6.48804e6 0.840103 0.420052 0.907500i \(-0.362012\pi\)
0.420052 + 0.907500i \(0.362012\pi\)
\(570\) 0 0
\(571\) −1.02285e7 −1.31287 −0.656435 0.754382i \(-0.727936\pi\)
−0.656435 + 0.754382i \(0.727936\pi\)
\(572\) 0 0
\(573\) 5.01570e6 0.638182
\(574\) 0 0
\(575\) −20064.0 −0.00253074
\(576\) 0 0
\(577\) 2.65338e6 0.331787 0.165894 0.986144i \(-0.446949\pi\)
0.165894 + 0.986144i \(0.446949\pi\)
\(578\) 0 0
\(579\) 1.38538e7 1.71740
\(580\) 0 0
\(581\) −1.00636e6 −0.123684
\(582\) 0 0
\(583\) −524784. −0.0639454
\(584\) 0 0
\(585\) −368480. −0.0445168
\(586\) 0 0
\(587\) −1.43044e7 −1.71346 −0.856729 0.515766i \(-0.827507\pi\)
−0.856729 + 0.515766i \(0.827507\pi\)
\(588\) 0 0
\(589\) −749112. −0.0889731
\(590\) 0 0
\(591\) −1.38607e7 −1.63236
\(592\) 0 0
\(593\) −1.00265e7 −1.17088 −0.585442 0.810714i \(-0.699079\pi\)
−0.585442 + 0.810714i \(0.699079\pi\)
\(594\) 0 0
\(595\) −4.72517e6 −0.547173
\(596\) 0 0
\(597\) −1.17706e7 −1.35164
\(598\) 0 0
\(599\) 7.52292e6 0.856681 0.428341 0.903617i \(-0.359098\pi\)
0.428341 + 0.903617i \(0.359098\pi\)
\(600\) 0 0
\(601\) 3.38625e6 0.382413 0.191207 0.981550i \(-0.438760\pi\)
0.191207 + 0.981550i \(0.438760\pi\)
\(602\) 0 0
\(603\) 2.27499e6 0.254792
\(604\) 0 0
\(605\) −6.00471e6 −0.666966
\(606\) 0 0
\(607\) 6.90861e6 0.761060 0.380530 0.924769i \(-0.375742\pi\)
0.380530 + 0.924769i \(0.375742\pi\)
\(608\) 0 0
\(609\) 2.34475e6 0.256185
\(610\) 0 0
\(611\) −1.30536e6 −0.141458
\(612\) 0 0
\(613\) 9.68896e6 1.04142 0.520710 0.853734i \(-0.325667\pi\)
0.520710 + 0.853734i \(0.325667\pi\)
\(614\) 0 0
\(615\) 1.40822e7 1.50135
\(616\) 0 0
\(617\) −7.84742e6 −0.829877 −0.414939 0.909849i \(-0.636197\pi\)
−0.414939 + 0.909849i \(0.636197\pi\)
\(618\) 0 0
\(619\) −1.01972e7 −1.06968 −0.534840 0.844953i \(-0.679628\pi\)
−0.534840 + 0.844953i \(0.679628\pi\)
\(620\) 0 0
\(621\) −7.40544e6 −0.770587
\(622\) 0 0
\(623\) −2.47852e6 −0.255842
\(624\) 0 0
\(625\) −9.79988e6 −1.00351
\(626\) 0 0
\(627\) 318304. 0.0323350
\(628\) 0 0
\(629\) −1.79054e7 −1.80450
\(630\) 0 0
\(631\) 8.36258e6 0.836116 0.418058 0.908420i \(-0.362711\pi\)
0.418058 + 0.908420i \(0.362711\pi\)
\(632\) 0 0
\(633\) −1.61102e7 −1.59806
\(634\) 0 0
\(635\) −3.38150e6 −0.332794
\(636\) 0 0
\(637\) 336140. 0.0328225
\(638\) 0 0
\(639\) −2.75232e6 −0.266653
\(640\) 0 0
\(641\) 1.10283e6 0.106014 0.0530070 0.998594i \(-0.483119\pi\)
0.0530070 + 0.998594i \(0.483119\pi\)
\(642\) 0 0
\(643\) 1.71354e7 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(644\) 0 0
\(645\) −8.52992e6 −0.807320
\(646\) 0 0
\(647\) 54964.0 0.00516200 0.00258100 0.999997i \(-0.499178\pi\)
0.00258100 + 0.999997i \(0.499178\pi\)
\(648\) 0 0
\(649\) −633360. −0.0590254
\(650\) 0 0
\(651\) −5.24378e6 −0.484945
\(652\) 0 0
\(653\) 485166. 0.0445254 0.0222627 0.999752i \(-0.492913\pi\)
0.0222627 + 0.999752i \(0.492913\pi\)
\(654\) 0 0
\(655\) −3.44882e6 −0.314099
\(656\) 0 0
\(657\) −3.19985e6 −0.289212
\(658\) 0 0
\(659\) −2.72136e6 −0.244103 −0.122051 0.992524i \(-0.538947\pi\)
−0.122051 + 0.992524i \(0.538947\pi\)
\(660\) 0 0
\(661\) 2.14525e6 0.190974 0.0954869 0.995431i \(-0.469559\pi\)
0.0954869 + 0.995431i \(0.469559\pi\)
\(662\) 0 0
\(663\) 3.37512e6 0.298198
\(664\) 0 0
\(665\) −268912. −0.0235807
\(666\) 0 0
\(667\) 6.23443e6 0.542603
\(668\) 0 0
\(669\) −1.15397e7 −0.996848
\(670\) 0 0
\(671\) −5.95034e6 −0.510194
\(672\) 0 0
\(673\) 2.92796e6 0.249188 0.124594 0.992208i \(-0.460237\pi\)
0.124594 + 0.992208i \(0.460237\pi\)
\(674\) 0 0
\(675\) 44660.0 0.00377276
\(676\) 0 0
\(677\) 1.34992e7 1.13198 0.565988 0.824414i \(-0.308495\pi\)
0.565988 + 0.824414i \(0.308495\pi\)
\(678\) 0 0
\(679\) −2.86679e6 −0.238628
\(680\) 0 0
\(681\) −1.04135e6 −0.0860455
\(682\) 0 0
\(683\) −5.42972e6 −0.445375 −0.222688 0.974890i \(-0.571483\pi\)
−0.222688 + 0.974890i \(0.571483\pi\)
\(684\) 0 0
\(685\) −1.14499e7 −0.932340
\(686\) 0 0
\(687\) 1.58474e7 1.28105
\(688\) 0 0
\(689\) −316680. −0.0254140
\(690\) 0 0
\(691\) 2.08280e7 1.65940 0.829702 0.558207i \(-0.188510\pi\)
0.829702 + 0.558207i \(0.188510\pi\)
\(692\) 0 0
\(693\) −534296. −0.0422619
\(694\) 0 0
\(695\) −1.98274e6 −0.155705
\(696\) 0 0
\(697\) 3.09306e7 2.41160
\(698\) 0 0
\(699\) 2.78216e6 0.215372
\(700\) 0 0
\(701\) −2.35141e7 −1.80731 −0.903655 0.428261i \(-0.859126\pi\)
−0.903655 + 0.428261i \(0.859126\pi\)
\(702\) 0 0
\(703\) −1.01900e6 −0.0777656
\(704\) 0 0
\(705\) 7.31002e6 0.553918
\(706\) 0 0
\(707\) −1.89610e6 −0.142664
\(708\) 0 0
\(709\) 1.95747e7 1.46244 0.731221 0.682140i \(-0.238951\pi\)
0.731221 + 0.682140i \(0.238951\pi\)
\(710\) 0 0
\(711\) 1.49385e6 0.110824
\(712\) 0 0
\(713\) −1.39427e7 −1.02712
\(714\) 0 0
\(715\) 1.81888e6 0.133057
\(716\) 0 0
\(717\) 6.76066e6 0.491124
\(718\) 0 0
\(719\) 2.61152e7 1.88396 0.941978 0.335674i \(-0.108964\pi\)
0.941978 + 0.335674i \(0.108964\pi\)
\(720\) 0 0
\(721\) −2.59994e6 −0.186262
\(722\) 0 0
\(723\) −1.12827e7 −0.802729
\(724\) 0 0
\(725\) −37598.0 −0.00265656
\(726\) 0 0
\(727\) −1.54126e7 −1.08154 −0.540768 0.841172i \(-0.681866\pi\)
−0.540768 + 0.841172i \(0.681866\pi\)
\(728\) 0 0
\(729\) 1.59468e7 1.11136
\(730\) 0 0
\(731\) −1.87354e7 −1.29679
\(732\) 0 0
\(733\) 1.69868e7 1.16776 0.583878 0.811841i \(-0.301535\pi\)
0.583878 + 0.811841i \(0.301535\pi\)
\(734\) 0 0
\(735\) −1.88238e6 −0.128526
\(736\) 0 0
\(737\) −1.12297e7 −0.761554
\(738\) 0 0
\(739\) 2.01511e6 0.135734 0.0678669 0.997694i \(-0.478381\pi\)
0.0678669 + 0.997694i \(0.478381\pi\)
\(740\) 0 0
\(741\) 192080. 0.0128510
\(742\) 0 0
\(743\) 1.51381e7 1.00600 0.503001 0.864286i \(-0.332229\pi\)
0.503001 + 0.864286i \(0.332229\pi\)
\(744\) 0 0
\(745\) 1.13266e6 0.0747666
\(746\) 0 0
\(747\) 965286. 0.0632928
\(748\) 0 0
\(749\) −7.16988e6 −0.466989
\(750\) 0 0
\(751\) −7.21401e6 −0.466742 −0.233371 0.972388i \(-0.574976\pi\)
−0.233371 + 0.972388i \(0.574976\pi\)
\(752\) 0 0
\(753\) −6.03033e6 −0.387573
\(754\) 0 0
\(755\) −3.97062e6 −0.253508
\(756\) 0 0
\(757\) 1.09697e7 0.695755 0.347877 0.937540i \(-0.386903\pi\)
0.347877 + 0.937540i \(0.386903\pi\)
\(758\) 0 0
\(759\) 5.92435e6 0.373281
\(760\) 0 0
\(761\) 1.92442e7 1.20459 0.602293 0.798275i \(-0.294254\pi\)
0.602293 + 0.798275i \(0.294254\pi\)
\(762\) 0 0
\(763\) −4.55200e6 −0.283068
\(764\) 0 0
\(765\) 4.53230e6 0.280005
\(766\) 0 0
\(767\) −382200. −0.0234586
\(768\) 0 0
\(769\) 8.21185e6 0.500755 0.250378 0.968148i \(-0.419445\pi\)
0.250378 + 0.968148i \(0.419445\pi\)
\(770\) 0 0
\(771\) 1.64767e7 0.998241
\(772\) 0 0
\(773\) −1.86187e7 −1.12073 −0.560363 0.828247i \(-0.689338\pi\)
−0.560363 + 0.828247i \(0.689338\pi\)
\(774\) 0 0
\(775\) 84084.0 0.00502874
\(776\) 0 0
\(777\) −7.13303e6 −0.423859
\(778\) 0 0
\(779\) 1.76028e6 0.103929
\(780\) 0 0
\(781\) 1.35859e7 0.797006
\(782\) 0 0
\(783\) −1.38771e7 −0.808898
\(784\) 0 0
\(785\) −1.64373e7 −0.952045
\(786\) 0 0
\(787\) 2.62501e7 1.51075 0.755377 0.655291i \(-0.227454\pi\)
0.755377 + 0.655291i \(0.227454\pi\)
\(788\) 0 0
\(789\) 1.80737e7 1.03360
\(790\) 0 0
\(791\) −4.08435e6 −0.232103
\(792\) 0 0
\(793\) −3.59072e6 −0.202768
\(794\) 0 0
\(795\) 1.77341e6 0.0995155
\(796\) 0 0
\(797\) 1.00373e7 0.559720 0.279860 0.960041i \(-0.409712\pi\)
0.279860 + 0.960041i \(0.409712\pi\)
\(798\) 0 0
\(799\) 1.60559e7 0.889751
\(800\) 0 0
\(801\) 2.37735e6 0.130922
\(802\) 0 0
\(803\) 1.57950e7 0.864433
\(804\) 0 0
\(805\) −5.00506e6 −0.272220
\(806\) 0 0
\(807\) −1.78858e7 −0.966772
\(808\) 0 0
\(809\) 1.40884e7 0.756816 0.378408 0.925639i \(-0.376472\pi\)
0.378408 + 0.925639i \(0.376472\pi\)
\(810\) 0 0
\(811\) 1.81433e7 0.968646 0.484323 0.874889i \(-0.339066\pi\)
0.484323 + 0.874889i \(0.339066\pi\)
\(812\) 0 0
\(813\) 2.31076e7 1.22611
\(814\) 0 0
\(815\) 738752. 0.0389587
\(816\) 0 0
\(817\) −1.06624e6 −0.0558856
\(818\) 0 0
\(819\) −322420. −0.0167962
\(820\) 0 0
\(821\) 2.13669e7 1.10633 0.553164 0.833072i \(-0.313420\pi\)
0.553164 + 0.833072i \(0.313420\pi\)
\(822\) 0 0
\(823\) −1.78017e7 −0.916142 −0.458071 0.888916i \(-0.651459\pi\)
−0.458071 + 0.888916i \(0.651459\pi\)
\(824\) 0 0
\(825\) −35728.0 −0.00182757
\(826\) 0 0
\(827\) 1.62921e7 0.828350 0.414175 0.910197i \(-0.364070\pi\)
0.414175 + 0.910197i \(0.364070\pi\)
\(828\) 0 0
\(829\) 2.08499e6 0.105370 0.0526851 0.998611i \(-0.483222\pi\)
0.0526851 + 0.998611i \(0.483222\pi\)
\(830\) 0 0
\(831\) −1.48973e7 −0.748348
\(832\) 0 0
\(833\) −4.13452e6 −0.206449
\(834\) 0 0
\(835\) −2.76423e7 −1.37201
\(836\) 0 0
\(837\) 3.10346e7 1.53120
\(838\) 0 0
\(839\) 2.27850e7 1.11749 0.558745 0.829340i \(-0.311283\pi\)
0.558745 + 0.829340i \(0.311283\pi\)
\(840\) 0 0
\(841\) −8.82842e6 −0.430421
\(842\) 0 0
\(843\) 312788. 0.0151594
\(844\) 0 0
\(845\) −1.96948e7 −0.948877
\(846\) 0 0
\(847\) −5.25412e6 −0.251647
\(848\) 0 0
\(849\) 3.49403e7 1.66363
\(850\) 0 0
\(851\) −1.89660e7 −0.897740
\(852\) 0 0
\(853\) 2.26975e7 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(854\) 0 0
\(855\) 257936. 0.0120669
\(856\) 0 0
\(857\) 2.52900e7 1.17624 0.588120 0.808774i \(-0.299868\pi\)
0.588120 + 0.808774i \(0.299868\pi\)
\(858\) 0 0
\(859\) −1.03947e7 −0.480652 −0.240326 0.970692i \(-0.577254\pi\)
−0.240326 + 0.970692i \(0.577254\pi\)
\(860\) 0 0
\(861\) 1.23219e7 0.566462
\(862\) 0 0
\(863\) −4.33399e7 −1.98089 −0.990447 0.137892i \(-0.955967\pi\)
−0.990447 + 0.137892i \(0.955967\pi\)
\(864\) 0 0
\(865\) −1.34801e7 −0.612566
\(866\) 0 0
\(867\) −2.16360e7 −0.977527
\(868\) 0 0
\(869\) −7.37389e6 −0.331243
\(870\) 0 0
\(871\) −6.77656e6 −0.302666
\(872\) 0 0
\(873\) 2.74978e6 0.122113
\(874\) 0 0
\(875\) −8.54482e6 −0.377296
\(876\) 0 0
\(877\) −3.71659e7 −1.63172 −0.815861 0.578248i \(-0.803736\pi\)
−0.815861 + 0.578248i \(0.803736\pi\)
\(878\) 0 0
\(879\) −2.70449e7 −1.18063
\(880\) 0 0
\(881\) 9.04785e6 0.392740 0.196370 0.980530i \(-0.437085\pi\)
0.196370 + 0.980530i \(0.437085\pi\)
\(882\) 0 0
\(883\) 3.29679e7 1.42295 0.711474 0.702712i \(-0.248028\pi\)
0.711474 + 0.702712i \(0.248028\pi\)
\(884\) 0 0
\(885\) 2.14032e6 0.0918588
\(886\) 0 0
\(887\) 1.61099e7 0.687517 0.343758 0.939058i \(-0.388300\pi\)
0.343758 + 0.939058i \(0.388300\pi\)
\(888\) 0 0
\(889\) −2.95882e6 −0.125564
\(890\) 0 0
\(891\) −1.05372e7 −0.444663
\(892\) 0 0
\(893\) 913752. 0.0383442
\(894\) 0 0
\(895\) 1.65162e7 0.689211
\(896\) 0 0
\(897\) 3.57504e6 0.148354
\(898\) 0 0
\(899\) −2.61272e7 −1.07819
\(900\) 0 0
\(901\) 3.89516e6 0.159850
\(902\) 0 0
\(903\) −7.46368e6 −0.304603
\(904\) 0 0
\(905\) 1.88709e7 0.765898
\(906\) 0 0
\(907\) −4.47286e7 −1.80537 −0.902686 0.430300i \(-0.858408\pi\)
−0.902686 + 0.430300i \(0.858408\pi\)
\(908\) 0 0
\(909\) 1.81871e6 0.0730053
\(910\) 0 0
\(911\) 6.60518e6 0.263687 0.131844 0.991271i \(-0.457910\pi\)
0.131844 + 0.991271i \(0.457910\pi\)
\(912\) 0 0
\(913\) −4.76482e6 −0.189177
\(914\) 0 0
\(915\) 2.01080e7 0.793993
\(916\) 0 0
\(917\) −3.01771e6 −0.118510
\(918\) 0 0
\(919\) 3.08930e7 1.20662 0.603311 0.797506i \(-0.293848\pi\)
0.603311 + 0.797506i \(0.293848\pi\)
\(920\) 0 0
\(921\) 6.42704e6 0.249667
\(922\) 0 0
\(923\) 8.19840e6 0.316756
\(924\) 0 0
\(925\) 114378. 0.00439530
\(926\) 0 0
\(927\) 2.49382e6 0.0953160
\(928\) 0 0
\(929\) −4.87215e6 −0.185217 −0.0926087 0.995703i \(-0.529521\pi\)
−0.0926087 + 0.995703i \(0.529521\pi\)
\(930\) 0 0
\(931\) −235298. −0.00889701
\(932\) 0 0
\(933\) 9.33979e6 0.351264
\(934\) 0 0
\(935\) −2.23722e7 −0.836913
\(936\) 0 0
\(937\) −3.25004e7 −1.20932 −0.604658 0.796485i \(-0.706690\pi\)
−0.604658 + 0.796485i \(0.706690\pi\)
\(938\) 0 0
\(939\) 1.55448e6 0.0575334
\(940\) 0 0
\(941\) 2.64040e6 0.0972066 0.0486033 0.998818i \(-0.484523\pi\)
0.0486033 + 0.998818i \(0.484523\pi\)
\(942\) 0 0
\(943\) 3.27627e7 1.19978
\(944\) 0 0
\(945\) 1.11406e7 0.405817
\(946\) 0 0
\(947\) −4.08179e7 −1.47903 −0.739513 0.673142i \(-0.764944\pi\)
−0.739513 + 0.673142i \(0.764944\pi\)
\(948\) 0 0
\(949\) 9.53148e6 0.343554
\(950\) 0 0
\(951\) −962892. −0.0345244
\(952\) 0 0
\(953\) −6.71983e6 −0.239677 −0.119838 0.992793i \(-0.538238\pi\)
−0.119838 + 0.992793i \(0.538238\pi\)
\(954\) 0 0
\(955\) −2.00628e7 −0.711841
\(956\) 0 0
\(957\) 1.11017e7 0.391840
\(958\) 0 0
\(959\) −1.00186e7 −0.351773
\(960\) 0 0
\(961\) 2.98016e7 1.04095
\(962\) 0 0
\(963\) 6.87723e6 0.238972
\(964\) 0 0
\(965\) −5.54150e7 −1.91562
\(966\) 0 0
\(967\) 2.78979e6 0.0959413 0.0479707 0.998849i \(-0.484725\pi\)
0.0479707 + 0.998849i \(0.484725\pi\)
\(968\) 0 0
\(969\) −2.36258e6 −0.0808310
\(970\) 0 0
\(971\) 3.33594e7 1.13545 0.567727 0.823217i \(-0.307823\pi\)
0.567727 + 0.823217i \(0.307823\pi\)
\(972\) 0 0
\(973\) −1.73489e6 −0.0587477
\(974\) 0 0
\(975\) −21560.0 −0.000726335 0
\(976\) 0 0
\(977\) −7.60033e6 −0.254739 −0.127370 0.991855i \(-0.540653\pi\)
−0.127370 + 0.991855i \(0.540653\pi\)
\(978\) 0 0
\(979\) −1.17350e7 −0.391316
\(980\) 0 0
\(981\) 4.36621e6 0.144854
\(982\) 0 0
\(983\) 5.79760e6 0.191366 0.0956829 0.995412i \(-0.469497\pi\)
0.0956829 + 0.995412i \(0.469497\pi\)
\(984\) 0 0
\(985\) 5.54428e7 1.82077
\(986\) 0 0
\(987\) 6.39626e6 0.208994
\(988\) 0 0
\(989\) −1.98451e7 −0.645153
\(990\) 0 0
\(991\) −1.26825e7 −0.410224 −0.205112 0.978739i \(-0.565756\pi\)
−0.205112 + 0.978739i \(0.565756\pi\)
\(992\) 0 0
\(993\) 7.90227e6 0.254319
\(994\) 0 0
\(995\) 4.70823e7 1.50765
\(996\) 0 0
\(997\) −1.44400e7 −0.460077 −0.230039 0.973182i \(-0.573885\pi\)
−0.230039 + 0.973182i \(0.573885\pi\)
\(998\) 0 0
\(999\) 4.22159e7 1.33833
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 448.6.a.c.1.1 1
4.3 odd 2 448.6.a.m.1.1 1
8.3 odd 2 7.6.a.a.1.1 1
8.5 even 2 112.6.a.g.1.1 1
24.5 odd 2 1008.6.a.y.1.1 1
24.11 even 2 63.6.a.e.1.1 1
40.3 even 4 175.6.b.a.99.2 2
40.19 odd 2 175.6.a.b.1.1 1
40.27 even 4 175.6.b.a.99.1 2
56.3 even 6 49.6.c.b.30.1 2
56.11 odd 6 49.6.c.c.30.1 2
56.13 odd 2 784.6.a.c.1.1 1
56.19 even 6 49.6.c.b.18.1 2
56.27 even 2 49.6.a.a.1.1 1
56.51 odd 6 49.6.c.c.18.1 2
88.43 even 2 847.6.a.b.1.1 1
168.83 odd 2 441.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 8.3 odd 2
49.6.a.a.1.1 1 56.27 even 2
49.6.c.b.18.1 2 56.19 even 6
49.6.c.b.30.1 2 56.3 even 6
49.6.c.c.18.1 2 56.51 odd 6
49.6.c.c.30.1 2 56.11 odd 6
63.6.a.e.1.1 1 24.11 even 2
112.6.a.g.1.1 1 8.5 even 2
175.6.a.b.1.1 1 40.19 odd 2
175.6.b.a.99.1 2 40.27 even 4
175.6.b.a.99.2 2 40.3 even 4
441.6.a.k.1.1 1 168.83 odd 2
448.6.a.c.1.1 1 1.1 even 1 trivial
448.6.a.m.1.1 1 4.3 odd 2
784.6.a.c.1.1 1 56.13 odd 2
847.6.a.b.1.1 1 88.43 even 2
1008.6.a.y.1.1 1 24.5 odd 2