Properties

Label 441.6.a.bb.1.3
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.10089\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +O(q^{10})\) \(q-3.38033 q^{2} -20.5734 q^{4} -54.5253 q^{5} +177.715 q^{8} +184.313 q^{10} -481.911 q^{11} -512.622 q^{13} +57.6111 q^{16} +590.744 q^{17} -2451.02 q^{19} +1121.77 q^{20} +1629.02 q^{22} -1774.90 q^{23} -151.993 q^{25} +1732.83 q^{26} +4246.44 q^{29} -9767.37 q^{31} -5881.64 q^{32} -1996.91 q^{34} -9969.65 q^{37} +8285.24 q^{38} -9689.98 q^{40} -3377.54 q^{41} -18223.4 q^{43} +9914.53 q^{44} +5999.76 q^{46} +1320.64 q^{47} +513.788 q^{50} +10546.4 q^{52} -34837.0 q^{53} +26276.3 q^{55} -14354.4 q^{58} +11592.5 q^{59} -31406.1 q^{61} +33016.9 q^{62} +18038.3 q^{64} +27950.9 q^{65} +27555.9 q^{67} -12153.6 q^{68} +22868.5 q^{71} -15936.4 q^{73} +33700.7 q^{74} +50425.6 q^{76} +87165.3 q^{79} -3141.26 q^{80} +11417.2 q^{82} +90307.5 q^{83} -32210.5 q^{85} +61601.2 q^{86} -85643.0 q^{88} -126570. q^{89} +36515.7 q^{92} -4464.20 q^{94} +133642. q^{95} +16573.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.38033 −0.597564 −0.298782 0.954321i \(-0.596580\pi\)
−0.298782 + 0.954321i \(0.596580\pi\)
\(3\) 0 0
\(4\) −20.5734 −0.642918
\(5\) −54.5253 −0.975378 −0.487689 0.873017i \(-0.662160\pi\)
−0.487689 + 0.873017i \(0.662160\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 177.715 0.981748
\(9\) 0 0
\(10\) 184.313 0.582850
\(11\) −481.911 −1.20084 −0.600420 0.799685i \(-0.705000\pi\)
−0.600420 + 0.799685i \(0.705000\pi\)
\(12\) 0 0
\(13\) −512.622 −0.841277 −0.420638 0.907228i \(-0.638194\pi\)
−0.420638 + 0.907228i \(0.638194\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 57.6111 0.0562608
\(17\) 590.744 0.495766 0.247883 0.968790i \(-0.420265\pi\)
0.247883 + 0.968790i \(0.420265\pi\)
\(18\) 0 0
\(19\) −2451.02 −1.55762 −0.778811 0.627259i \(-0.784177\pi\)
−0.778811 + 0.627259i \(0.784177\pi\)
\(20\) 1121.77 0.627088
\(21\) 0 0
\(22\) 1629.02 0.717579
\(23\) −1774.90 −0.699609 −0.349804 0.936823i \(-0.613752\pi\)
−0.349804 + 0.936823i \(0.613752\pi\)
\(24\) 0 0
\(25\) −151.993 −0.0486379
\(26\) 1732.83 0.502716
\(27\) 0 0
\(28\) 0 0
\(29\) 4246.44 0.937627 0.468814 0.883297i \(-0.344682\pi\)
0.468814 + 0.883297i \(0.344682\pi\)
\(30\) 0 0
\(31\) −9767.37 −1.82547 −0.912733 0.408558i \(-0.866032\pi\)
−0.912733 + 0.408558i \(0.866032\pi\)
\(32\) −5881.64 −1.01537
\(33\) 0 0
\(34\) −1996.91 −0.296252
\(35\) 0 0
\(36\) 0 0
\(37\) −9969.65 −1.19722 −0.598612 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(38\) 8285.24 0.930778
\(39\) 0 0
\(40\) −9689.98 −0.957575
\(41\) −3377.54 −0.313791 −0.156896 0.987615i \(-0.550149\pi\)
−0.156896 + 0.987615i \(0.550149\pi\)
\(42\) 0 0
\(43\) −18223.4 −1.50300 −0.751500 0.659733i \(-0.770669\pi\)
−0.751500 + 0.659733i \(0.770669\pi\)
\(44\) 9914.53 0.772042
\(45\) 0 0
\(46\) 5999.76 0.418061
\(47\) 1320.64 0.0872046 0.0436023 0.999049i \(-0.486117\pi\)
0.0436023 + 0.999049i \(0.486117\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 513.788 0.0290642
\(51\) 0 0
\(52\) 10546.4 0.540872
\(53\) −34837.0 −1.70353 −0.851767 0.523921i \(-0.824469\pi\)
−0.851767 + 0.523921i \(0.824469\pi\)
\(54\) 0 0
\(55\) 26276.3 1.17127
\(56\) 0 0
\(57\) 0 0
\(58\) −14354.4 −0.560292
\(59\) 11592.5 0.433558 0.216779 0.976221i \(-0.430445\pi\)
0.216779 + 0.976221i \(0.430445\pi\)
\(60\) 0 0
\(61\) −31406.1 −1.08066 −0.540331 0.841452i \(-0.681701\pi\)
−0.540331 + 0.841452i \(0.681701\pi\)
\(62\) 33016.9 1.09083
\(63\) 0 0
\(64\) 18038.3 0.550486
\(65\) 27950.9 0.820563
\(66\) 0 0
\(67\) 27555.9 0.749942 0.374971 0.927036i \(-0.377653\pi\)
0.374971 + 0.927036i \(0.377653\pi\)
\(68\) −12153.6 −0.318737
\(69\) 0 0
\(70\) 0 0
\(71\) 22868.5 0.538384 0.269192 0.963086i \(-0.413243\pi\)
0.269192 + 0.963086i \(0.413243\pi\)
\(72\) 0 0
\(73\) −15936.4 −0.350012 −0.175006 0.984567i \(-0.555995\pi\)
−0.175006 + 0.984567i \(0.555995\pi\)
\(74\) 33700.7 0.715418
\(75\) 0 0
\(76\) 50425.6 1.00142
\(77\) 0 0
\(78\) 0 0
\(79\) 87165.3 1.57136 0.785680 0.618633i \(-0.212313\pi\)
0.785680 + 0.618633i \(0.212313\pi\)
\(80\) −3141.26 −0.0548755
\(81\) 0 0
\(82\) 11417.2 0.187510
\(83\) 90307.5 1.43889 0.719447 0.694548i \(-0.244395\pi\)
0.719447 + 0.694548i \(0.244395\pi\)
\(84\) 0 0
\(85\) −32210.5 −0.483559
\(86\) 61601.2 0.898138
\(87\) 0 0
\(88\) −85643.0 −1.17892
\(89\) −126570. −1.69377 −0.846884 0.531777i \(-0.821524\pi\)
−0.846884 + 0.531777i \(0.821524\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 36515.7 0.449791
\(93\) 0 0
\(94\) −4464.20 −0.0521103
\(95\) 133642. 1.51927
\(96\) 0 0
\(97\) 16573.1 0.178844 0.0894219 0.995994i \(-0.471498\pi\)
0.0894219 + 0.995994i \(0.471498\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3127.02 0.0312702
\(101\) −76610.1 −0.747279 −0.373639 0.927574i \(-0.621890\pi\)
−0.373639 + 0.927574i \(0.621890\pi\)
\(102\) 0 0
\(103\) −80435.8 −0.747062 −0.373531 0.927618i \(-0.621853\pi\)
−0.373531 + 0.927618i \(0.621853\pi\)
\(104\) −91100.8 −0.825922
\(105\) 0 0
\(106\) 117760. 1.01797
\(107\) −148042. −1.25005 −0.625023 0.780606i \(-0.714910\pi\)
−0.625023 + 0.780606i \(0.714910\pi\)
\(108\) 0 0
\(109\) −124982. −1.00759 −0.503793 0.863825i \(-0.668062\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(110\) −88822.7 −0.699910
\(111\) 0 0
\(112\) 0 0
\(113\) 194474. 1.43273 0.716366 0.697724i \(-0.245804\pi\)
0.716366 + 0.697724i \(0.245804\pi\)
\(114\) 0 0
\(115\) 96777.1 0.682383
\(116\) −87363.6 −0.602817
\(117\) 0 0
\(118\) −39186.5 −0.259078
\(119\) 0 0
\(120\) 0 0
\(121\) 71187.4 0.442018
\(122\) 106163. 0.645765
\(123\) 0 0
\(124\) 200948. 1.17362
\(125\) 178679. 1.02282
\(126\) 0 0
\(127\) 239766. 1.31910 0.659551 0.751660i \(-0.270746\pi\)
0.659551 + 0.751660i \(0.270746\pi\)
\(128\) 127237. 0.686417
\(129\) 0 0
\(130\) −94483.2 −0.490339
\(131\) −349576. −1.77977 −0.889883 0.456188i \(-0.849214\pi\)
−0.889883 + 0.456188i \(0.849214\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −93148.1 −0.448138
\(135\) 0 0
\(136\) 104984. 0.486717
\(137\) 280966. 1.27894 0.639472 0.768814i \(-0.279153\pi\)
0.639472 + 0.768814i \(0.279153\pi\)
\(138\) 0 0
\(139\) −50042.9 −0.219688 −0.109844 0.993949i \(-0.535035\pi\)
−0.109844 + 0.993949i \(0.535035\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −77303.2 −0.321719
\(143\) 247038. 1.01024
\(144\) 0 0
\(145\) −231538. −0.914541
\(146\) 53870.3 0.209155
\(147\) 0 0
\(148\) 205109. 0.769717
\(149\) 328813. 1.21334 0.606671 0.794953i \(-0.292505\pi\)
0.606671 + 0.794953i \(0.292505\pi\)
\(150\) 0 0
\(151\) 511304. 1.82489 0.912445 0.409200i \(-0.134192\pi\)
0.912445 + 0.409200i \(0.134192\pi\)
\(152\) −435583. −1.52919
\(153\) 0 0
\(154\) 0 0
\(155\) 532569. 1.78052
\(156\) 0 0
\(157\) 75442.1 0.244267 0.122134 0.992514i \(-0.461026\pi\)
0.122134 + 0.992514i \(0.461026\pi\)
\(158\) −294647. −0.938988
\(159\) 0 0
\(160\) 320698. 0.990367
\(161\) 0 0
\(162\) 0 0
\(163\) 340753. 1.00455 0.502274 0.864709i \(-0.332497\pi\)
0.502274 + 0.864709i \(0.332497\pi\)
\(164\) 69487.4 0.201742
\(165\) 0 0
\(166\) −305269. −0.859831
\(167\) −132392. −0.367342 −0.183671 0.982988i \(-0.558798\pi\)
−0.183671 + 0.982988i \(0.558798\pi\)
\(168\) 0 0
\(169\) −108512. −0.292253
\(170\) 108882. 0.288957
\(171\) 0 0
\(172\) 374917. 0.966306
\(173\) 449322. 1.14141 0.570706 0.821154i \(-0.306670\pi\)
0.570706 + 0.821154i \(0.306670\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −27763.4 −0.0675602
\(177\) 0 0
\(178\) 427847. 1.01213
\(179\) 93666.8 0.218501 0.109250 0.994014i \(-0.465155\pi\)
0.109250 + 0.994014i \(0.465155\pi\)
\(180\) 0 0
\(181\) −399795. −0.907071 −0.453535 0.891238i \(-0.649837\pi\)
−0.453535 + 0.891238i \(0.649837\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −315428. −0.686839
\(185\) 543598. 1.16775
\(186\) 0 0
\(187\) −284686. −0.595336
\(188\) −27170.0 −0.0560654
\(189\) 0 0
\(190\) −451755. −0.907861
\(191\) −27822.7 −0.0551844 −0.0275922 0.999619i \(-0.508784\pi\)
−0.0275922 + 0.999619i \(0.508784\pi\)
\(192\) 0 0
\(193\) −66464.3 −0.128438 −0.0642192 0.997936i \(-0.520456\pi\)
−0.0642192 + 0.997936i \(0.520456\pi\)
\(194\) −56022.5 −0.106871
\(195\) 0 0
\(196\) 0 0
\(197\) 492506. 0.904162 0.452081 0.891977i \(-0.350682\pi\)
0.452081 + 0.891977i \(0.350682\pi\)
\(198\) 0 0
\(199\) −349206. −0.625099 −0.312549 0.949902i \(-0.601183\pi\)
−0.312549 + 0.949902i \(0.601183\pi\)
\(200\) −27011.6 −0.0477501
\(201\) 0 0
\(202\) 258967. 0.446547
\(203\) 0 0
\(204\) 0 0
\(205\) 184161. 0.306065
\(206\) 271900. 0.446417
\(207\) 0 0
\(208\) −29532.7 −0.0473309
\(209\) 1.18117e6 1.87046
\(210\) 0 0
\(211\) −218250. −0.337480 −0.168740 0.985661i \(-0.553970\pi\)
−0.168740 + 0.985661i \(0.553970\pi\)
\(212\) 716714. 1.09523
\(213\) 0 0
\(214\) 500431. 0.746982
\(215\) 993638. 1.46599
\(216\) 0 0
\(217\) 0 0
\(218\) 422481. 0.602096
\(219\) 0 0
\(220\) −540593. −0.753032
\(221\) −302828. −0.417076
\(222\) 0 0
\(223\) −1.24807e6 −1.68065 −0.840324 0.542084i \(-0.817635\pi\)
−0.840324 + 0.542084i \(0.817635\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −657386. −0.856149
\(227\) −520973. −0.671044 −0.335522 0.942032i \(-0.608913\pi\)
−0.335522 + 0.942032i \(0.608913\pi\)
\(228\) 0 0
\(229\) 525651. 0.662383 0.331191 0.943564i \(-0.392549\pi\)
0.331191 + 0.943564i \(0.392549\pi\)
\(230\) −327139. −0.407767
\(231\) 0 0
\(232\) 754658. 0.920514
\(233\) −210205. −0.253661 −0.126831 0.991924i \(-0.540480\pi\)
−0.126831 + 0.991924i \(0.540480\pi\)
\(234\) 0 0
\(235\) −72008.2 −0.0850575
\(236\) −238497. −0.278742
\(237\) 0 0
\(238\) 0 0
\(239\) −805791. −0.912489 −0.456244 0.889855i \(-0.650806\pi\)
−0.456244 + 0.889855i \(0.650806\pi\)
\(240\) 0 0
\(241\) −1.13795e6 −1.26206 −0.631030 0.775758i \(-0.717368\pi\)
−0.631030 + 0.775758i \(0.717368\pi\)
\(242\) −240637. −0.264134
\(243\) 0 0
\(244\) 646130. 0.694777
\(245\) 0 0
\(246\) 0 0
\(247\) 1.25644e6 1.31039
\(248\) −1.73581e6 −1.79215
\(249\) 0 0
\(250\) −603994. −0.611199
\(251\) −1.16566e6 −1.16785 −0.583927 0.811806i \(-0.698484\pi\)
−0.583927 + 0.811806i \(0.698484\pi\)
\(252\) 0 0
\(253\) 855346. 0.840118
\(254\) −810489. −0.788248
\(255\) 0 0
\(256\) −1.00733e6 −0.960664
\(257\) −1.02881e6 −0.971635 −0.485817 0.874060i \(-0.661478\pi\)
−0.485817 + 0.874060i \(0.661478\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −575043. −0.527554
\(261\) 0 0
\(262\) 1.18168e6 1.06352
\(263\) 657345. 0.586009 0.293004 0.956111i \(-0.405345\pi\)
0.293004 + 0.956111i \(0.405345\pi\)
\(264\) 0 0
\(265\) 1.89950e6 1.66159
\(266\) 0 0
\(267\) 0 0
\(268\) −566918. −0.482151
\(269\) 5475.86 0.00461393 0.00230697 0.999997i \(-0.499266\pi\)
0.00230697 + 0.999997i \(0.499266\pi\)
\(270\) 0 0
\(271\) 769147. 0.636189 0.318094 0.948059i \(-0.396957\pi\)
0.318094 + 0.948059i \(0.396957\pi\)
\(272\) 34033.4 0.0278922
\(273\) 0 0
\(274\) −949757. −0.764251
\(275\) 73247.3 0.0584063
\(276\) 0 0
\(277\) −1.17384e6 −0.919196 −0.459598 0.888127i \(-0.652007\pi\)
−0.459598 + 0.888127i \(0.652007\pi\)
\(278\) 169162. 0.131277
\(279\) 0 0
\(280\) 0 0
\(281\) −837649. −0.632843 −0.316422 0.948619i \(-0.602481\pi\)
−0.316422 + 0.948619i \(0.602481\pi\)
\(282\) 0 0
\(283\) 597221. 0.443271 0.221635 0.975130i \(-0.428861\pi\)
0.221635 + 0.975130i \(0.428861\pi\)
\(284\) −470483. −0.346137
\(285\) 0 0
\(286\) −835071. −0.603682
\(287\) 0 0
\(288\) 0 0
\(289\) −1.07088e6 −0.754216
\(290\) 782677. 0.546496
\(291\) 0 0
\(292\) 327866. 0.225029
\(293\) −851321. −0.579328 −0.289664 0.957128i \(-0.593543\pi\)
−0.289664 + 0.957128i \(0.593543\pi\)
\(294\) 0 0
\(295\) −632084. −0.422883
\(296\) −1.77176e6 −1.17537
\(297\) 0 0
\(298\) −1.11150e6 −0.725049
\(299\) 909855. 0.588565
\(300\) 0 0
\(301\) 0 0
\(302\) −1.72838e6 −1.09049
\(303\) 0 0
\(304\) −141206. −0.0876331
\(305\) 1.71243e6 1.05405
\(306\) 0 0
\(307\) 600906. 0.363882 0.181941 0.983309i \(-0.441762\pi\)
0.181941 + 0.983309i \(0.441762\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.80026e6 −1.06397
\(311\) −465866. −0.273124 −0.136562 0.990632i \(-0.543605\pi\)
−0.136562 + 0.990632i \(0.543605\pi\)
\(312\) 0 0
\(313\) 2.97366e6 1.71565 0.857827 0.513938i \(-0.171814\pi\)
0.857827 + 0.513938i \(0.171814\pi\)
\(314\) −255019. −0.145965
\(315\) 0 0
\(316\) −1.79328e6 −1.01026
\(317\) −2.55521e6 −1.42816 −0.714082 0.700062i \(-0.753156\pi\)
−0.714082 + 0.700062i \(0.753156\pi\)
\(318\) 0 0
\(319\) −2.04641e6 −1.12594
\(320\) −983544. −0.536932
\(321\) 0 0
\(322\) 0 0
\(323\) −1.44792e6 −0.772216
\(324\) 0 0
\(325\) 77915.2 0.0409179
\(326\) −1.15186e6 −0.600281
\(327\) 0 0
\(328\) −600241. −0.308064
\(329\) 0 0
\(330\) 0 0
\(331\) 2.56138e6 1.28500 0.642502 0.766284i \(-0.277896\pi\)
0.642502 + 0.766284i \(0.277896\pi\)
\(332\) −1.85793e6 −0.925090
\(333\) 0 0
\(334\) 447528. 0.219510
\(335\) −1.50249e6 −0.731477
\(336\) 0 0
\(337\) −1.21525e6 −0.582894 −0.291447 0.956587i \(-0.594137\pi\)
−0.291447 + 0.956587i \(0.594137\pi\)
\(338\) 366805. 0.174640
\(339\) 0 0
\(340\) 662678. 0.310889
\(341\) 4.70701e6 2.19209
\(342\) 0 0
\(343\) 0 0
\(344\) −3.23858e6 −1.47557
\(345\) 0 0
\(346\) −1.51886e6 −0.682066
\(347\) −1.25318e6 −0.558714 −0.279357 0.960187i \(-0.590121\pi\)
−0.279357 + 0.960187i \(0.590121\pi\)
\(348\) 0 0
\(349\) −450403. −0.197942 −0.0989709 0.995090i \(-0.531555\pi\)
−0.0989709 + 0.995090i \(0.531555\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.83443e6 1.21929
\(353\) 545281. 0.232907 0.116454 0.993196i \(-0.462847\pi\)
0.116454 + 0.993196i \(0.462847\pi\)
\(354\) 0 0
\(355\) −1.24691e6 −0.525128
\(356\) 2.60396e6 1.08895
\(357\) 0 0
\(358\) −316625. −0.130568
\(359\) 1.01159e6 0.414254 0.207127 0.978314i \(-0.433589\pi\)
0.207127 + 0.978314i \(0.433589\pi\)
\(360\) 0 0
\(361\) 3.53138e6 1.42619
\(362\) 1.35144e6 0.542033
\(363\) 0 0
\(364\) 0 0
\(365\) 868937. 0.341394
\(366\) 0 0
\(367\) 1.09038e6 0.422584 0.211292 0.977423i \(-0.432233\pi\)
0.211292 + 0.977423i \(0.432233\pi\)
\(368\) −102254. −0.0393605
\(369\) 0 0
\(370\) −1.83754e6 −0.697803
\(371\) 0 0
\(372\) 0 0
\(373\) −3.13003e6 −1.16487 −0.582433 0.812879i \(-0.697899\pi\)
−0.582433 + 0.812879i \(0.697899\pi\)
\(374\) 962333. 0.355751
\(375\) 0 0
\(376\) 234698. 0.0856130
\(377\) −2.17682e6 −0.788804
\(378\) 0 0
\(379\) −1.27933e6 −0.457492 −0.228746 0.973486i \(-0.573463\pi\)
−0.228746 + 0.973486i \(0.573463\pi\)
\(380\) −2.74947e6 −0.976766
\(381\) 0 0
\(382\) 94050.0 0.0329762
\(383\) 1.30666e6 0.455161 0.227580 0.973759i \(-0.426919\pi\)
0.227580 + 0.973759i \(0.426919\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 224671. 0.0767501
\(387\) 0 0
\(388\) −340964. −0.114982
\(389\) 36488.2 0.0122258 0.00611292 0.999981i \(-0.498054\pi\)
0.00611292 + 0.999981i \(0.498054\pi\)
\(390\) 0 0
\(391\) −1.04851e6 −0.346842
\(392\) 0 0
\(393\) 0 0
\(394\) −1.66483e6 −0.540294
\(395\) −4.75271e6 −1.53267
\(396\) 0 0
\(397\) −640204. −0.203865 −0.101932 0.994791i \(-0.532503\pi\)
−0.101932 + 0.994791i \(0.532503\pi\)
\(398\) 1.18043e6 0.373536
\(399\) 0 0
\(400\) −8756.50 −0.00273641
\(401\) 2.69149e6 0.835858 0.417929 0.908480i \(-0.362756\pi\)
0.417929 + 0.908480i \(0.362756\pi\)
\(402\) 0 0
\(403\) 5.00697e6 1.53572
\(404\) 1.57613e6 0.480439
\(405\) 0 0
\(406\) 0 0
\(407\) 4.80449e6 1.43768
\(408\) 0 0
\(409\) −2.92572e6 −0.864817 −0.432409 0.901678i \(-0.642336\pi\)
−0.432409 + 0.901678i \(0.642336\pi\)
\(410\) −622526. −0.182893
\(411\) 0 0
\(412\) 1.65484e6 0.480299
\(413\) 0 0
\(414\) 0 0
\(415\) −4.92404e6 −1.40347
\(416\) 3.01506e6 0.854205
\(417\) 0 0
\(418\) −3.99275e6 −1.11772
\(419\) −5.52810e6 −1.53830 −0.769150 0.639068i \(-0.779320\pi\)
−0.769150 + 0.639068i \(0.779320\pi\)
\(420\) 0 0
\(421\) 5.77477e6 1.58792 0.793961 0.607968i \(-0.208015\pi\)
0.793961 + 0.607968i \(0.208015\pi\)
\(422\) 737756. 0.201666
\(423\) 0 0
\(424\) −6.19106e6 −1.67244
\(425\) −89789.1 −0.0241130
\(426\) 0 0
\(427\) 0 0
\(428\) 3.04573e6 0.803677
\(429\) 0 0
\(430\) −3.35882e6 −0.876024
\(431\) −2.20964e6 −0.572965 −0.286483 0.958085i \(-0.592486\pi\)
−0.286483 + 0.958085i \(0.592486\pi\)
\(432\) 0 0
\(433\) −4.97215e6 −1.27446 −0.637228 0.770676i \(-0.719919\pi\)
−0.637228 + 0.770676i \(0.719919\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.57130e6 0.647794
\(437\) 4.35032e6 1.08973
\(438\) 0 0
\(439\) 2.86405e6 0.709282 0.354641 0.935002i \(-0.384603\pi\)
0.354641 + 0.935002i \(0.384603\pi\)
\(440\) 4.66971e6 1.14990
\(441\) 0 0
\(442\) 1.02366e6 0.249230
\(443\) 5.66234e6 1.37084 0.685419 0.728149i \(-0.259619\pi\)
0.685419 + 0.728149i \(0.259619\pi\)
\(444\) 0 0
\(445\) 6.90124e6 1.65206
\(446\) 4.21889e6 1.00429
\(447\) 0 0
\(448\) 0 0
\(449\) −7.54480e6 −1.76617 −0.883084 0.469215i \(-0.844537\pi\)
−0.883084 + 0.469215i \(0.844537\pi\)
\(450\) 0 0
\(451\) 1.62767e6 0.376813
\(452\) −4.00098e6 −0.921129
\(453\) 0 0
\(454\) 1.76106e6 0.400991
\(455\) 0 0
\(456\) 0 0
\(457\) −589014. −0.131928 −0.0659638 0.997822i \(-0.521012\pi\)
−0.0659638 + 0.997822i \(0.521012\pi\)
\(458\) −1.77687e6 −0.395816
\(459\) 0 0
\(460\) −1.99103e6 −0.438716
\(461\) −5.94951e6 −1.30385 −0.651927 0.758282i \(-0.726039\pi\)
−0.651927 + 0.758282i \(0.726039\pi\)
\(462\) 0 0
\(463\) −2.16083e6 −0.468456 −0.234228 0.972182i \(-0.575256\pi\)
−0.234228 + 0.972182i \(0.575256\pi\)
\(464\) 244642. 0.0527517
\(465\) 0 0
\(466\) 710563. 0.151579
\(467\) 505766. 0.107314 0.0536571 0.998559i \(-0.482912\pi\)
0.0536571 + 0.998559i \(0.482912\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 243412. 0.0508272
\(471\) 0 0
\(472\) 2.06016e6 0.425644
\(473\) 8.78208e6 1.80486
\(474\) 0 0
\(475\) 372538. 0.0757594
\(476\) 0 0
\(477\) 0 0
\(478\) 2.72384e6 0.545270
\(479\) −547067. −0.108944 −0.0544718 0.998515i \(-0.517348\pi\)
−0.0544718 + 0.998515i \(0.517348\pi\)
\(480\) 0 0
\(481\) 5.11066e6 1.00720
\(482\) 3.84665e6 0.754162
\(483\) 0 0
\(484\) −1.46456e6 −0.284181
\(485\) −903652. −0.174440
\(486\) 0 0
\(487\) 2.37889e6 0.454519 0.227259 0.973834i \(-0.427024\pi\)
0.227259 + 0.973834i \(0.427024\pi\)
\(488\) −5.58135e6 −1.06094
\(489\) 0 0
\(490\) 0 0
\(491\) 7.19719e6 1.34728 0.673642 0.739058i \(-0.264729\pi\)
0.673642 + 0.739058i \(0.264729\pi\)
\(492\) 0 0
\(493\) 2.50856e6 0.464844
\(494\) −4.24720e6 −0.783042
\(495\) 0 0
\(496\) −562709. −0.102702
\(497\) 0 0
\(498\) 0 0
\(499\) 2.22236e6 0.399543 0.199771 0.979843i \(-0.435980\pi\)
0.199771 + 0.979843i \(0.435980\pi\)
\(500\) −3.67603e6 −0.657588
\(501\) 0 0
\(502\) 3.94032e6 0.697867
\(503\) −305857. −0.0539013 −0.0269506 0.999637i \(-0.508580\pi\)
−0.0269506 + 0.999637i \(0.508580\pi\)
\(504\) 0 0
\(505\) 4.17719e6 0.728879
\(506\) −2.89135e6 −0.502024
\(507\) 0 0
\(508\) −4.93280e6 −0.848074
\(509\) −9.06420e6 −1.55073 −0.775363 0.631516i \(-0.782433\pi\)
−0.775363 + 0.631516i \(0.782433\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −666475. −0.112359
\(513\) 0 0
\(514\) 3.47772e6 0.580614
\(515\) 4.38579e6 0.728668
\(516\) 0 0
\(517\) −636431. −0.104719
\(518\) 0 0
\(519\) 0 0
\(520\) 4.96730e6 0.805586
\(521\) 8.44649e6 1.36327 0.681635 0.731692i \(-0.261269\pi\)
0.681635 + 0.731692i \(0.261269\pi\)
\(522\) 0 0
\(523\) −3.91585e6 −0.625997 −0.312998 0.949754i \(-0.601333\pi\)
−0.312998 + 0.949754i \(0.601333\pi\)
\(524\) 7.19195e6 1.14424
\(525\) 0 0
\(526\) −2.22204e6 −0.350178
\(527\) −5.77001e6 −0.905003
\(528\) 0 0
\(529\) −3.28606e6 −0.510548
\(530\) −6.42092e6 −0.992905
\(531\) 0 0
\(532\) 0 0
\(533\) 1.73140e6 0.263985
\(534\) 0 0
\(535\) 8.07204e6 1.21927
\(536\) 4.89711e6 0.736254
\(537\) 0 0
\(538\) −18510.2 −0.00275712
\(539\) 0 0
\(540\) 0 0
\(541\) −2.94802e6 −0.433049 −0.216524 0.976277i \(-0.569472\pi\)
−0.216524 + 0.976277i \(0.569472\pi\)
\(542\) −2.59997e6 −0.380163
\(543\) 0 0
\(544\) −3.47454e6 −0.503385
\(545\) 6.81469e6 0.982776
\(546\) 0 0
\(547\) −5.50670e6 −0.786906 −0.393453 0.919345i \(-0.628720\pi\)
−0.393453 + 0.919345i \(0.628720\pi\)
\(548\) −5.78041e6 −0.822256
\(549\) 0 0
\(550\) −247600. −0.0349015
\(551\) −1.04081e7 −1.46047
\(552\) 0 0
\(553\) 0 0
\(554\) 3.96796e6 0.549278
\(555\) 0 0
\(556\) 1.02955e6 0.141241
\(557\) 4.97235e6 0.679085 0.339542 0.940591i \(-0.389728\pi\)
0.339542 + 0.940591i \(0.389728\pi\)
\(558\) 0 0
\(559\) 9.34174e6 1.26444
\(560\) 0 0
\(561\) 0 0
\(562\) 2.83153e6 0.378164
\(563\) 1.31397e7 1.74709 0.873544 0.486745i \(-0.161816\pi\)
0.873544 + 0.486745i \(0.161816\pi\)
\(564\) 0 0
\(565\) −1.06037e7 −1.39746
\(566\) −2.01880e6 −0.264882
\(567\) 0 0
\(568\) 4.06409e6 0.528558
\(569\) 1.38589e7 1.79452 0.897261 0.441500i \(-0.145553\pi\)
0.897261 + 0.441500i \(0.145553\pi\)
\(570\) 0 0
\(571\) −1.06977e7 −1.37309 −0.686547 0.727085i \(-0.740874\pi\)
−0.686547 + 0.727085i \(0.740874\pi\)
\(572\) −5.08241e6 −0.649501
\(573\) 0 0
\(574\) 0 0
\(575\) 269774. 0.0340275
\(576\) 0 0
\(577\) 5.61898e6 0.702616 0.351308 0.936260i \(-0.385737\pi\)
0.351308 + 0.936260i \(0.385737\pi\)
\(578\) 3.61993e6 0.450692
\(579\) 0 0
\(580\) 4.76353e6 0.587975
\(581\) 0 0
\(582\) 0 0
\(583\) 1.67883e7 2.04567
\(584\) −2.83214e6 −0.343624
\(585\) 0 0
\(586\) 2.87775e6 0.346185
\(587\) −6.86832e6 −0.822726 −0.411363 0.911471i \(-0.634947\pi\)
−0.411363 + 0.911471i \(0.634947\pi\)
\(588\) 0 0
\(589\) 2.39400e7 2.84338
\(590\) 2.13665e6 0.252699
\(591\) 0 0
\(592\) −574362. −0.0673568
\(593\) 4.07737e6 0.476149 0.238075 0.971247i \(-0.423484\pi\)
0.238075 + 0.971247i \(0.423484\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.76479e6 −0.780079
\(597\) 0 0
\(598\) −3.07561e6 −0.351705
\(599\) −582993. −0.0663890 −0.0331945 0.999449i \(-0.510568\pi\)
−0.0331945 + 0.999449i \(0.510568\pi\)
\(600\) 0 0
\(601\) 6.87250e6 0.776119 0.388060 0.921634i \(-0.373145\pi\)
0.388060 + 0.921634i \(0.373145\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.05192e7 −1.17325
\(605\) −3.88151e6 −0.431134
\(606\) 0 0
\(607\) 2.40563e6 0.265006 0.132503 0.991183i \(-0.457699\pi\)
0.132503 + 0.991183i \(0.457699\pi\)
\(608\) 1.44160e7 1.58156
\(609\) 0 0
\(610\) −5.78858e6 −0.629865
\(611\) −676989. −0.0733632
\(612\) 0 0
\(613\) −1.16245e7 −1.24946 −0.624731 0.780840i \(-0.714791\pi\)
−0.624731 + 0.780840i \(0.714791\pi\)
\(614\) −2.03126e6 −0.217443
\(615\) 0 0
\(616\) 0 0
\(617\) 1.37370e7 1.45271 0.726353 0.687322i \(-0.241214\pi\)
0.726353 + 0.687322i \(0.241214\pi\)
\(618\) 0 0
\(619\) −1.64008e7 −1.72044 −0.860220 0.509923i \(-0.829674\pi\)
−0.860220 + 0.509923i \(0.829674\pi\)
\(620\) −1.09567e7 −1.14473
\(621\) 0 0
\(622\) 1.57478e6 0.163209
\(623\) 0 0
\(624\) 0 0
\(625\) −9.26754e6 −0.948996
\(626\) −1.00519e7 −1.02521
\(627\) 0 0
\(628\) −1.55210e6 −0.157044
\(629\) −5.88951e6 −0.593543
\(630\) 0 0
\(631\) −7.28252e6 −0.728129 −0.364065 0.931374i \(-0.618611\pi\)
−0.364065 + 0.931374i \(0.618611\pi\)
\(632\) 1.54906e7 1.54268
\(633\) 0 0
\(634\) 8.63745e6 0.853419
\(635\) −1.30733e7 −1.28662
\(636\) 0 0
\(637\) 0 0
\(638\) 6.91754e6 0.672821
\(639\) 0 0
\(640\) −6.93763e6 −0.669516
\(641\) 107514. 0.0103352 0.00516762 0.999987i \(-0.498355\pi\)
0.00516762 + 0.999987i \(0.498355\pi\)
\(642\) 0 0
\(643\) −9.75913e6 −0.930858 −0.465429 0.885085i \(-0.654100\pi\)
−0.465429 + 0.885085i \(0.654100\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.89445e6 0.461448
\(647\) −6.75588e6 −0.634485 −0.317243 0.948344i \(-0.602757\pi\)
−0.317243 + 0.948344i \(0.602757\pi\)
\(648\) 0 0
\(649\) −5.58655e6 −0.520633
\(650\) −263379. −0.0244511
\(651\) 0 0
\(652\) −7.01043e6 −0.645841
\(653\) −2.20346e6 −0.202219 −0.101110 0.994875i \(-0.532239\pi\)
−0.101110 + 0.994875i \(0.532239\pi\)
\(654\) 0 0
\(655\) 1.90607e7 1.73595
\(656\) −194584. −0.0176542
\(657\) 0 0
\(658\) 0 0
\(659\) 1.39630e7 1.25247 0.626233 0.779636i \(-0.284596\pi\)
0.626233 + 0.779636i \(0.284596\pi\)
\(660\) 0 0
\(661\) 51052.8 0.00454482 0.00227241 0.999997i \(-0.499277\pi\)
0.00227241 + 0.999997i \(0.499277\pi\)
\(662\) −8.65832e6 −0.767872
\(663\) 0 0
\(664\) 1.60490e7 1.41263
\(665\) 0 0
\(666\) 0 0
\(667\) −7.53703e6 −0.655972
\(668\) 2.72375e6 0.236170
\(669\) 0 0
\(670\) 5.07893e6 0.437104
\(671\) 1.51350e7 1.29770
\(672\) 0 0
\(673\) 6.71688e6 0.571650 0.285825 0.958282i \(-0.407732\pi\)
0.285825 + 0.958282i \(0.407732\pi\)
\(674\) 4.10793e6 0.348316
\(675\) 0 0
\(676\) 2.23245e6 0.187895
\(677\) 1.45655e7 1.22139 0.610693 0.791867i \(-0.290891\pi\)
0.610693 + 0.791867i \(0.290891\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.72429e6 −0.474733
\(681\) 0 0
\(682\) −1.59112e7 −1.30991
\(683\) 1.62565e6 0.133344 0.0666722 0.997775i \(-0.478762\pi\)
0.0666722 + 0.997775i \(0.478762\pi\)
\(684\) 0 0
\(685\) −1.53197e7 −1.24745
\(686\) 0 0
\(687\) 0 0
\(688\) −1.04987e6 −0.0845600
\(689\) 1.78582e7 1.43314
\(690\) 0 0
\(691\) 2.81556e6 0.224321 0.112160 0.993690i \(-0.464223\pi\)
0.112160 + 0.993690i \(0.464223\pi\)
\(692\) −9.24407e6 −0.733834
\(693\) 0 0
\(694\) 4.23616e6 0.333867
\(695\) 2.72860e6 0.214278
\(696\) 0 0
\(697\) −1.99526e6 −0.155567
\(698\) 1.52251e6 0.118283
\(699\) 0 0
\(700\) 0 0
\(701\) −1.40996e7 −1.08371 −0.541855 0.840472i \(-0.682278\pi\)
−0.541855 + 0.840472i \(0.682278\pi\)
\(702\) 0 0
\(703\) 2.44358e7 1.86482
\(704\) −8.69287e6 −0.661046
\(705\) 0 0
\(706\) −1.84323e6 −0.139177
\(707\) 0 0
\(708\) 0 0
\(709\) −8.44943e6 −0.631265 −0.315633 0.948881i \(-0.602217\pi\)
−0.315633 + 0.948881i \(0.602217\pi\)
\(710\) 4.21498e6 0.313797
\(711\) 0 0
\(712\) −2.24933e7 −1.66285
\(713\) 1.73361e7 1.27711
\(714\) 0 0
\(715\) −1.34698e7 −0.985365
\(716\) −1.92704e6 −0.140478
\(717\) 0 0
\(718\) −3.41950e6 −0.247543
\(719\) 1.82657e7 1.31769 0.658846 0.752278i \(-0.271045\pi\)
0.658846 + 0.752278i \(0.271045\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.19372e7 −0.852237
\(723\) 0 0
\(724\) 8.22513e6 0.583172
\(725\) −645431. −0.0456042
\(726\) 0 0
\(727\) −1.37476e6 −0.0964700 −0.0482350 0.998836i \(-0.515360\pi\)
−0.0482350 + 0.998836i \(0.515360\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.93729e6 −0.204005
\(731\) −1.07654e7 −0.745136
\(732\) 0 0
\(733\) −1.64625e7 −1.13171 −0.565857 0.824503i \(-0.691455\pi\)
−0.565857 + 0.824503i \(0.691455\pi\)
\(734\) −3.68585e6 −0.252521
\(735\) 0 0
\(736\) 1.04393e7 0.710360
\(737\) −1.32795e7 −0.900561
\(738\) 0 0
\(739\) 2.72997e6 0.183885 0.0919425 0.995764i \(-0.470692\pi\)
0.0919425 + 0.995764i \(0.470692\pi\)
\(740\) −1.11836e7 −0.750765
\(741\) 0 0
\(742\) 0 0
\(743\) 2.34396e7 1.55768 0.778839 0.627224i \(-0.215809\pi\)
0.778839 + 0.627224i \(0.215809\pi\)
\(744\) 0 0
\(745\) −1.79286e7 −1.18347
\(746\) 1.05805e7 0.696082
\(747\) 0 0
\(748\) 5.85695e6 0.382752
\(749\) 0 0
\(750\) 0 0
\(751\) 1.79407e7 1.16075 0.580376 0.814349i \(-0.302906\pi\)
0.580376 + 0.814349i \(0.302906\pi\)
\(752\) 76083.4 0.00490620
\(753\) 0 0
\(754\) 7.35837e6 0.471361
\(755\) −2.78790e7 −1.77996
\(756\) 0 0
\(757\) −2.30823e7 −1.46399 −0.731997 0.681308i \(-0.761412\pi\)
−0.731997 + 0.681308i \(0.761412\pi\)
\(758\) 4.32455e6 0.273381
\(759\) 0 0
\(760\) 2.37503e7 1.49154
\(761\) 1.91587e7 1.19924 0.599618 0.800286i \(-0.295319\pi\)
0.599618 + 0.800286i \(0.295319\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 572407. 0.0354790
\(765\) 0 0
\(766\) −4.41693e6 −0.271988
\(767\) −5.94257e6 −0.364742
\(768\) 0 0
\(769\) −1.73017e7 −1.05505 −0.527526 0.849539i \(-0.676880\pi\)
−0.527526 + 0.849539i \(0.676880\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.36739e6 0.0825753
\(773\) 3.12287e7 1.87977 0.939885 0.341491i \(-0.110932\pi\)
0.939885 + 0.341491i \(0.110932\pi\)
\(774\) 0 0
\(775\) 1.48458e6 0.0887868
\(776\) 2.94529e6 0.175580
\(777\) 0 0
\(778\) −123342. −0.00730572
\(779\) 8.27840e6 0.488768
\(780\) 0 0
\(781\) −1.10206e7 −0.646514
\(782\) 3.54432e6 0.207260
\(783\) 0 0
\(784\) 0 0
\(785\) −4.11350e6 −0.238253
\(786\) 0 0
\(787\) 2.48691e7 1.43128 0.715638 0.698471i \(-0.246136\pi\)
0.715638 + 0.698471i \(0.246136\pi\)
\(788\) −1.01325e7 −0.581302
\(789\) 0 0
\(790\) 1.60657e7 0.915868
\(791\) 0 0
\(792\) 0 0
\(793\) 1.60995e7 0.909136
\(794\) 2.16410e6 0.121822
\(795\) 0 0
\(796\) 7.18433e6 0.401887
\(797\) 584574. 0.0325982 0.0162991 0.999867i \(-0.494812\pi\)
0.0162991 + 0.999867i \(0.494812\pi\)
\(798\) 0 0
\(799\) 780159. 0.0432331
\(800\) 893970. 0.0493853
\(801\) 0 0
\(802\) −9.09814e6 −0.499478
\(803\) 7.67993e6 0.420309
\(804\) 0 0
\(805\) 0 0
\(806\) −1.69252e7 −0.917691
\(807\) 0 0
\(808\) −1.36148e7 −0.733639
\(809\) 1.10552e7 0.593878 0.296939 0.954897i \(-0.404034\pi\)
0.296939 + 0.954897i \(0.404034\pi\)
\(810\) 0 0
\(811\) −1.62780e7 −0.869060 −0.434530 0.900657i \(-0.643086\pi\)
−0.434530 + 0.900657i \(0.643086\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.62408e7 −0.859103
\(815\) −1.85796e7 −0.979813
\(816\) 0 0
\(817\) 4.46659e7 2.34111
\(818\) 9.88989e6 0.516783
\(819\) 0 0
\(820\) −3.78882e6 −0.196775
\(821\) −3.28574e7 −1.70128 −0.850638 0.525752i \(-0.823784\pi\)
−0.850638 + 0.525752i \(0.823784\pi\)
\(822\) 0 0
\(823\) 1.06046e7 0.545752 0.272876 0.962049i \(-0.412025\pi\)
0.272876 + 0.962049i \(0.412025\pi\)
\(824\) −1.42947e7 −0.733426
\(825\) 0 0
\(826\) 0 0
\(827\) −1.57582e7 −0.801203 −0.400602 0.916252i \(-0.631199\pi\)
−0.400602 + 0.916252i \(0.631199\pi\)
\(828\) 0 0
\(829\) 2.70135e7 1.36520 0.682598 0.730794i \(-0.260850\pi\)
0.682598 + 0.730794i \(0.260850\pi\)
\(830\) 1.66449e7 0.838660
\(831\) 0 0
\(832\) −9.24684e6 −0.463111
\(833\) 0 0
\(834\) 0 0
\(835\) 7.21871e6 0.358297
\(836\) −2.43007e7 −1.20255
\(837\) 0 0
\(838\) 1.86868e7 0.919232
\(839\) −3.50490e7 −1.71898 −0.859489 0.511154i \(-0.829218\pi\)
−0.859489 + 0.511154i \(0.829218\pi\)
\(840\) 0 0
\(841\) −2.47888e6 −0.120855
\(842\) −1.95206e7 −0.948885
\(843\) 0 0
\(844\) 4.49013e6 0.216972
\(845\) 5.91663e6 0.285057
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00699e6 −0.0958422
\(849\) 0 0
\(850\) 303517. 0.0144091
\(851\) 1.76952e7 0.837589
\(852\) 0 0
\(853\) 538313. 0.0253316 0.0126658 0.999920i \(-0.495968\pi\)
0.0126658 + 0.999920i \(0.495968\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.63094e7 −1.22723
\(857\) −3.68275e6 −0.171285 −0.0856426 0.996326i \(-0.527294\pi\)
−0.0856426 + 0.996326i \(0.527294\pi\)
\(858\) 0 0
\(859\) 1.88883e7 0.873394 0.436697 0.899609i \(-0.356148\pi\)
0.436697 + 0.899609i \(0.356148\pi\)
\(860\) −2.04425e7 −0.942513
\(861\) 0 0
\(862\) 7.46931e6 0.342383
\(863\) −2.78544e7 −1.27311 −0.636557 0.771230i \(-0.719642\pi\)
−0.636557 + 0.771230i \(0.719642\pi\)
\(864\) 0 0
\(865\) −2.44994e7 −1.11331
\(866\) 1.68075e7 0.761568
\(867\) 0 0
\(868\) 0 0
\(869\) −4.20059e7 −1.88695
\(870\) 0 0
\(871\) −1.41258e7 −0.630909
\(872\) −2.22113e7 −0.989195
\(873\) 0 0
\(874\) −1.47055e7 −0.651180
\(875\) 0 0
\(876\) 0 0
\(877\) −1.75052e7 −0.768544 −0.384272 0.923220i \(-0.625548\pi\)
−0.384272 + 0.923220i \(0.625548\pi\)
\(878\) −9.68143e6 −0.423841
\(879\) 0 0
\(880\) 1.51381e6 0.0658968
\(881\) −5.66292e6 −0.245811 −0.122905 0.992418i \(-0.539221\pi\)
−0.122905 + 0.992418i \(0.539221\pi\)
\(882\) 0 0
\(883\) 1.43933e7 0.621239 0.310620 0.950534i \(-0.399463\pi\)
0.310620 + 0.950534i \(0.399463\pi\)
\(884\) 6.23020e6 0.268146
\(885\) 0 0
\(886\) −1.91406e7 −0.819163
\(887\) 5.50193e6 0.234804 0.117402 0.993084i \(-0.462543\pi\)
0.117402 + 0.993084i \(0.462543\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.33285e7 −0.987214
\(891\) 0 0
\(892\) 2.56770e7 1.08052
\(893\) −3.23691e6 −0.135832
\(894\) 0 0
\(895\) −5.10721e6 −0.213121
\(896\) 0 0
\(897\) 0 0
\(898\) 2.55039e7 1.05540
\(899\) −4.14766e7 −1.71161
\(900\) 0 0
\(901\) −2.05797e7 −0.844554
\(902\) −5.50208e6 −0.225170
\(903\) 0 0
\(904\) 3.45610e7 1.40658
\(905\) 2.17990e7 0.884737
\(906\) 0 0
\(907\) 6.01555e6 0.242805 0.121402 0.992603i \(-0.461261\pi\)
0.121402 + 0.992603i \(0.461261\pi\)
\(908\) 1.07182e7 0.431426
\(909\) 0 0
\(910\) 0 0
\(911\) 1.47513e7 0.588892 0.294446 0.955668i \(-0.404865\pi\)
0.294446 + 0.955668i \(0.404865\pi\)
\(912\) 0 0
\(913\) −4.35202e7 −1.72788
\(914\) 1.99106e6 0.0788351
\(915\) 0 0
\(916\) −1.08144e7 −0.425858
\(917\) 0 0
\(918\) 0 0
\(919\) −3.54001e7 −1.38266 −0.691331 0.722538i \(-0.742976\pi\)
−0.691331 + 0.722538i \(0.742976\pi\)
\(920\) 1.71988e7 0.669928
\(921\) 0 0
\(922\) 2.01113e7 0.779135
\(923\) −1.17229e7 −0.452930
\(924\) 0 0
\(925\) 1.51532e6 0.0582305
\(926\) 7.30434e6 0.279932
\(927\) 0 0
\(928\) −2.49760e7 −0.952036
\(929\) −3.83266e7 −1.45700 −0.728502 0.685044i \(-0.759783\pi\)
−0.728502 + 0.685044i \(0.759783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.32463e6 0.163083
\(933\) 0 0
\(934\) −1.70966e6 −0.0641271
\(935\) 1.55226e7 0.580677
\(936\) 0 0
\(937\) −1.48693e7 −0.553274 −0.276637 0.960974i \(-0.589220\pi\)
−0.276637 + 0.960974i \(0.589220\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.48145e6 0.0546849
\(941\) −3.50894e6 −0.129182 −0.0645909 0.997912i \(-0.520574\pi\)
−0.0645909 + 0.997912i \(0.520574\pi\)
\(942\) 0 0
\(943\) 5.99481e6 0.219531
\(944\) 667856. 0.0243923
\(945\) 0 0
\(946\) −2.96863e7 −1.07852
\(947\) −3.89774e7 −1.41234 −0.706168 0.708044i \(-0.749578\pi\)
−0.706168 + 0.708044i \(0.749578\pi\)
\(948\) 0 0
\(949\) 8.16935e6 0.294457
\(950\) −1.25930e6 −0.0452711
\(951\) 0 0
\(952\) 0 0
\(953\) −7.11438e6 −0.253749 −0.126875 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(954\) 0 0
\(955\) 1.51704e6 0.0538256
\(956\) 1.65778e7 0.586655
\(957\) 0 0
\(958\) 1.84927e6 0.0651008
\(959\) 0 0
\(960\) 0 0
\(961\) 6.67724e7 2.33232
\(962\) −1.72757e7 −0.601865
\(963\) 0 0
\(964\) 2.34114e7 0.811401
\(965\) 3.62398e6 0.125276
\(966\) 0 0
\(967\) −1.99357e7 −0.685592 −0.342796 0.939410i \(-0.611374\pi\)
−0.342796 + 0.939410i \(0.611374\pi\)
\(968\) 1.26511e7 0.433950
\(969\) 0 0
\(970\) 3.05464e6 0.104239
\(971\) 4.95888e7 1.68786 0.843928 0.536456i \(-0.180237\pi\)
0.843928 + 0.536456i \(0.180237\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8.04143e6 −0.271604
\(975\) 0 0
\(976\) −1.80934e6 −0.0607989
\(977\) 355348. 0.0119102 0.00595508 0.999982i \(-0.498104\pi\)
0.00595508 + 0.999982i \(0.498104\pi\)
\(978\) 0 0
\(979\) 6.09953e7 2.03395
\(980\) 0 0
\(981\) 0 0
\(982\) −2.43289e7 −0.805088
\(983\) −991754. −0.0327356 −0.0163678 0.999866i \(-0.505210\pi\)
−0.0163678 + 0.999866i \(0.505210\pi\)
\(984\) 0 0
\(985\) −2.68541e7 −0.881900
\(986\) −8.47976e6 −0.277774
\(987\) 0 0
\(988\) −2.58493e7 −0.842474
\(989\) 3.23448e7 1.05151
\(990\) 0 0
\(991\) 1.06674e7 0.345045 0.172523 0.985006i \(-0.444808\pi\)
0.172523 + 0.985006i \(0.444808\pi\)
\(992\) 5.74481e7 1.85352
\(993\) 0 0
\(994\) 0 0
\(995\) 1.90405e7 0.609707
\(996\) 0 0
\(997\) −5.58283e7 −1.77876 −0.889379 0.457171i \(-0.848863\pi\)
−0.889379 + 0.457171i \(0.848863\pi\)
\(998\) −7.51232e6 −0.238752
\(999\) 0 0
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 441.6.a.bb.1.3 6
3.2 odd 2 147.6.a.n.1.4 6
7.6 odd 2 441.6.a.ba.1.3 6
21.2 odd 6 147.6.e.q.67.3 12
21.5 even 6 147.6.e.p.67.3 12
21.11 odd 6 147.6.e.q.79.3 12
21.17 even 6 147.6.e.p.79.3 12
21.20 even 2 147.6.a.o.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.4 6 3.2 odd 2
147.6.a.o.1.4 yes 6 21.20 even 2
147.6.e.p.67.3 12 21.5 even 6
147.6.e.p.79.3 12 21.17 even 6
147.6.e.q.67.3 12 21.2 odd 6
147.6.e.q.79.3 12 21.11 odd 6
441.6.a.ba.1.3 6 7.6 odd 2
441.6.a.bb.1.3 6 1.1 even 1 trivial