Properties

Label 441.6.a.q.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.44622\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.44622 q^{2} -2.33867 q^{4} -36.0000 q^{5} +187.016 q^{8} +O(q^{10})\) \(q-5.44622 q^{2} -2.33867 q^{4} -36.0000 q^{5} +187.016 q^{8} +196.064 q^{10} -184.430 q^{11} +147.872 q^{13} -943.693 q^{16} -1968.38 q^{17} +1892.51 q^{19} +84.1920 q^{20} +1004.45 q^{22} -136.988 q^{23} -1829.00 q^{25} -805.344 q^{26} +1259.58 q^{29} +8969.02 q^{31} -844.949 q^{32} +10720.3 q^{34} +12897.2 q^{37} -10307.0 q^{38} -6732.58 q^{40} -8975.62 q^{41} +13538.9 q^{43} +431.321 q^{44} +746.069 q^{46} +20046.1 q^{47} +9961.14 q^{50} -345.823 q^{52} -9334.33 q^{53} +6639.49 q^{55} -6859.96 q^{58} -8866.46 q^{59} -41148.3 q^{61} -48847.3 q^{62} +34800.0 q^{64} -5323.39 q^{65} -55351.5 q^{67} +4603.39 q^{68} +63866.8 q^{71} +41299.3 q^{73} -70241.1 q^{74} -4425.95 q^{76} +16963.5 q^{79} +33973.0 q^{80} +48883.2 q^{82} +101693. q^{83} +70861.8 q^{85} -73736.0 q^{86} -34491.4 q^{88} -87102.5 q^{89} +320.370 q^{92} -109176. q^{94} -68130.4 q^{95} -118107. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8} - 108 q^{10} - 480 q^{11} + 1296 q^{13} - 1679 q^{16} - 936 q^{17} - 216 q^{19} - 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} + 8892 q^{26} - 6372 q^{29} + 9936 q^{31} - 9039 q^{32} + 19440 q^{34} + 11124 q^{37} - 28116 q^{38} - 8964 q^{40} - 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} - 7920 q^{47} - 5487 q^{50} + 44820 q^{52} - 2220 q^{53} + 17280 q^{55} - 71318 q^{58} - 29736 q^{59} - 17280 q^{61} - 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} + 45216 q^{68} + 92280 q^{71} + 56592 q^{73} - 85218 q^{74} - 87372 q^{76} - 56096 q^{79} + 60444 q^{80} - 52272 q^{82} + 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} - 123192 q^{89} + 25536 q^{92} - 345384 q^{94} + 7776 q^{95} + 35856 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.44622 −0.962765 −0.481383 0.876511i \(-0.659865\pi\)
−0.481383 + 0.876511i \(0.659865\pi\)
\(3\) 0 0
\(4\) −2.33867 −0.0730833
\(5\) −36.0000 −0.643988 −0.321994 0.946742i \(-0.604353\pi\)
−0.321994 + 0.946742i \(0.604353\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 187.016 1.03313
\(9\) 0 0
\(10\) 196.064 0.620009
\(11\) −184.430 −0.459569 −0.229784 0.973242i \(-0.573802\pi\)
−0.229784 + 0.973242i \(0.573802\pi\)
\(12\) 0 0
\(13\) 147.872 0.242676 0.121338 0.992611i \(-0.461281\pi\)
0.121338 + 0.992611i \(0.461281\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −943.693 −0.921576
\(17\) −1968.38 −1.65191 −0.825957 0.563733i \(-0.809365\pi\)
−0.825957 + 0.563733i \(0.809365\pi\)
\(18\) 0 0
\(19\) 1892.51 1.20269 0.601346 0.798989i \(-0.294631\pi\)
0.601346 + 0.798989i \(0.294631\pi\)
\(20\) 84.1920 0.0470647
\(21\) 0 0
\(22\) 1004.45 0.442457
\(23\) −136.988 −0.0539963 −0.0269982 0.999635i \(-0.508595\pi\)
−0.0269982 + 0.999635i \(0.508595\pi\)
\(24\) 0 0
\(25\) −1829.00 −0.585280
\(26\) −805.344 −0.233640
\(27\) 0 0
\(28\) 0 0
\(29\) 1259.58 0.278120 0.139060 0.990284i \(-0.455592\pi\)
0.139060 + 0.990284i \(0.455592\pi\)
\(30\) 0 0
\(31\) 8969.02 1.67626 0.838129 0.545472i \(-0.183650\pi\)
0.838129 + 0.545472i \(0.183650\pi\)
\(32\) −844.949 −0.145866
\(33\) 0 0
\(34\) 10720.3 1.59041
\(35\) 0 0
\(36\) 0 0
\(37\) 12897.2 1.54879 0.774393 0.632705i \(-0.218055\pi\)
0.774393 + 0.632705i \(0.218055\pi\)
\(38\) −10307.0 −1.15791
\(39\) 0 0
\(40\) −6732.58 −0.665321
\(41\) −8975.62 −0.833882 −0.416941 0.908934i \(-0.636898\pi\)
−0.416941 + 0.908934i \(0.636898\pi\)
\(42\) 0 0
\(43\) 13538.9 1.11664 0.558320 0.829626i \(-0.311446\pi\)
0.558320 + 0.829626i \(0.311446\pi\)
\(44\) 431.321 0.0335868
\(45\) 0 0
\(46\) 746.069 0.0519858
\(47\) 20046.1 1.32369 0.661845 0.749641i \(-0.269774\pi\)
0.661845 + 0.749641i \(0.269774\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 9961.14 0.563487
\(51\) 0 0
\(52\) −345.823 −0.0177356
\(53\) −9334.33 −0.456450 −0.228225 0.973608i \(-0.573292\pi\)
−0.228225 + 0.973608i \(0.573292\pi\)
\(54\) 0 0
\(55\) 6639.49 0.295956
\(56\) 0 0
\(57\) 0 0
\(58\) −6859.96 −0.267764
\(59\) −8866.46 −0.331605 −0.165802 0.986159i \(-0.553021\pi\)
−0.165802 + 0.986159i \(0.553021\pi\)
\(60\) 0 0
\(61\) −41148.3 −1.41588 −0.707942 0.706271i \(-0.750376\pi\)
−0.707942 + 0.706271i \(0.750376\pi\)
\(62\) −48847.3 −1.61384
\(63\) 0 0
\(64\) 34800.0 1.06201
\(65\) −5323.39 −0.156281
\(66\) 0 0
\(67\) −55351.5 −1.50641 −0.753204 0.657787i \(-0.771493\pi\)
−0.753204 + 0.657787i \(0.771493\pi\)
\(68\) 4603.39 0.120727
\(69\) 0 0
\(70\) 0 0
\(71\) 63866.8 1.50359 0.751794 0.659398i \(-0.229189\pi\)
0.751794 + 0.659398i \(0.229189\pi\)
\(72\) 0 0
\(73\) 41299.3 0.907060 0.453530 0.891241i \(-0.350165\pi\)
0.453530 + 0.891241i \(0.350165\pi\)
\(74\) −70241.1 −1.49112
\(75\) 0 0
\(76\) −4425.95 −0.0878968
\(77\) 0 0
\(78\) 0 0
\(79\) 16963.5 0.305808 0.152904 0.988241i \(-0.451138\pi\)
0.152904 + 0.988241i \(0.451138\pi\)
\(80\) 33973.0 0.593483
\(81\) 0 0
\(82\) 48883.2 0.802833
\(83\) 101693. 1.62030 0.810150 0.586223i \(-0.199386\pi\)
0.810150 + 0.586223i \(0.199386\pi\)
\(84\) 0 0
\(85\) 70861.8 1.06381
\(86\) −73736.0 −1.07506
\(87\) 0 0
\(88\) −34491.4 −0.474793
\(89\) −87102.5 −1.16562 −0.582808 0.812610i \(-0.698046\pi\)
−0.582808 + 0.812610i \(0.698046\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 320.370 0.00394623
\(93\) 0 0
\(94\) −109176. −1.27440
\(95\) −68130.4 −0.774519
\(96\) 0 0
\(97\) −118107. −1.27452 −0.637258 0.770650i \(-0.719932\pi\)
−0.637258 + 0.770650i \(0.719932\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4277.42 0.0427742
\(101\) −22232.7 −0.216865 −0.108432 0.994104i \(-0.534583\pi\)
−0.108432 + 0.994104i \(0.534583\pi\)
\(102\) 0 0
\(103\) 135931. 1.26248 0.631239 0.775588i \(-0.282547\pi\)
0.631239 + 0.775588i \(0.282547\pi\)
\(104\) 27654.4 0.250716
\(105\) 0 0
\(106\) 50836.8 0.439454
\(107\) −117626. −0.993218 −0.496609 0.867974i \(-0.665422\pi\)
−0.496609 + 0.867974i \(0.665422\pi\)
\(108\) 0 0
\(109\) −34664.2 −0.279457 −0.139729 0.990190i \(-0.544623\pi\)
−0.139729 + 0.990190i \(0.544623\pi\)
\(110\) −36160.1 −0.284937
\(111\) 0 0
\(112\) 0 0
\(113\) 26584.5 0.195854 0.0979269 0.995194i \(-0.468779\pi\)
0.0979269 + 0.995194i \(0.468779\pi\)
\(114\) 0 0
\(115\) 4931.58 0.0347730
\(116\) −2945.74 −0.0203259
\(117\) 0 0
\(118\) 48288.7 0.319257
\(119\) 0 0
\(120\) 0 0
\(121\) −127036. −0.788797
\(122\) 224103. 1.36316
\(123\) 0 0
\(124\) −20975.6 −0.122507
\(125\) 178344. 1.02090
\(126\) 0 0
\(127\) −137111. −0.754334 −0.377167 0.926145i \(-0.623102\pi\)
−0.377167 + 0.926145i \(0.623102\pi\)
\(128\) −162490. −0.876600
\(129\) 0 0
\(130\) 28992.4 0.150462
\(131\) 54089.2 0.275380 0.137690 0.990475i \(-0.456032\pi\)
0.137690 + 0.990475i \(0.456032\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 301457. 1.45032
\(135\) 0 0
\(136\) −368119. −1.70664
\(137\) −422849. −1.92479 −0.962395 0.271653i \(-0.912430\pi\)
−0.962395 + 0.271653i \(0.912430\pi\)
\(138\) 0 0
\(139\) −9913.38 −0.0435196 −0.0217598 0.999763i \(-0.506927\pi\)
−0.0217598 + 0.999763i \(0.506927\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −347833. −1.44760
\(143\) −27272.1 −0.111526
\(144\) 0 0
\(145\) −45345.0 −0.179106
\(146\) −224925. −0.873285
\(147\) 0 0
\(148\) −30162.3 −0.113190
\(149\) −505462. −1.86519 −0.932595 0.360925i \(-0.882461\pi\)
−0.932595 + 0.360925i \(0.882461\pi\)
\(150\) 0 0
\(151\) −193103. −0.689201 −0.344601 0.938749i \(-0.611986\pi\)
−0.344601 + 0.938749i \(0.611986\pi\)
\(152\) 353930. 1.24253
\(153\) 0 0
\(154\) 0 0
\(155\) −322885. −1.07949
\(156\) 0 0
\(157\) 264923. 0.857770 0.428885 0.903359i \(-0.358907\pi\)
0.428885 + 0.903359i \(0.358907\pi\)
\(158\) −92387.1 −0.294421
\(159\) 0 0
\(160\) 30418.1 0.0939362
\(161\) 0 0
\(162\) 0 0
\(163\) −539093. −1.58926 −0.794629 0.607095i \(-0.792335\pi\)
−0.794629 + 0.607095i \(0.792335\pi\)
\(164\) 20991.0 0.0609429
\(165\) 0 0
\(166\) −553842. −1.55997
\(167\) −218748. −0.606949 −0.303475 0.952840i \(-0.598147\pi\)
−0.303475 + 0.952840i \(0.598147\pi\)
\(168\) 0 0
\(169\) −349427. −0.941108
\(170\) −385929. −1.02420
\(171\) 0 0
\(172\) −31663.0 −0.0816078
\(173\) −590201. −1.49929 −0.749644 0.661842i \(-0.769775\pi\)
−0.749644 + 0.661842i \(0.769775\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 174046. 0.423527
\(177\) 0 0
\(178\) 474380. 1.12222
\(179\) 217352. 0.507026 0.253513 0.967332i \(-0.418414\pi\)
0.253513 + 0.967332i \(0.418414\pi\)
\(180\) 0 0
\(181\) 188109. 0.426790 0.213395 0.976966i \(-0.431548\pi\)
0.213395 + 0.976966i \(0.431548\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −25619.0 −0.0557851
\(185\) −464300. −0.997399
\(186\) 0 0
\(187\) 363029. 0.759168
\(188\) −46881.2 −0.0967396
\(189\) 0 0
\(190\) 371053. 0.745680
\(191\) −173523. −0.344170 −0.172085 0.985082i \(-0.555050\pi\)
−0.172085 + 0.985082i \(0.555050\pi\)
\(192\) 0 0
\(193\) 117781. 0.227606 0.113803 0.993503i \(-0.463697\pi\)
0.113803 + 0.993503i \(0.463697\pi\)
\(194\) 643236. 1.22706
\(195\) 0 0
\(196\) 0 0
\(197\) −224734. −0.412575 −0.206288 0.978491i \(-0.566138\pi\)
−0.206288 + 0.978491i \(0.566138\pi\)
\(198\) 0 0
\(199\) 740273. 1.32513 0.662567 0.749003i \(-0.269467\pi\)
0.662567 + 0.749003i \(0.269467\pi\)
\(200\) −342052. −0.604669
\(201\) 0 0
\(202\) 121084. 0.208790
\(203\) 0 0
\(204\) 0 0
\(205\) 323122. 0.537010
\(206\) −740308. −1.21547
\(207\) 0 0
\(208\) −139546. −0.223645
\(209\) −349036. −0.552720
\(210\) 0 0
\(211\) −705896. −1.09153 −0.545763 0.837939i \(-0.683760\pi\)
−0.545763 + 0.837939i \(0.683760\pi\)
\(212\) 21829.9 0.0333589
\(213\) 0 0
\(214\) 640618. 0.956236
\(215\) −487402. −0.719102
\(216\) 0 0
\(217\) 0 0
\(218\) 188789. 0.269052
\(219\) 0 0
\(220\) −15527.5 −0.0216295
\(221\) −291069. −0.400881
\(222\) 0 0
\(223\) 42214.3 0.0568456 0.0284228 0.999596i \(-0.490952\pi\)
0.0284228 + 0.999596i \(0.490952\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −144785. −0.188561
\(227\) −742403. −0.956258 −0.478129 0.878290i \(-0.658685\pi\)
−0.478129 + 0.878290i \(0.658685\pi\)
\(228\) 0 0
\(229\) 941236. 1.18607 0.593034 0.805177i \(-0.297930\pi\)
0.593034 + 0.805177i \(0.297930\pi\)
\(230\) −26858.5 −0.0334782
\(231\) 0 0
\(232\) 235562. 0.287333
\(233\) 512331. 0.618246 0.309123 0.951022i \(-0.399965\pi\)
0.309123 + 0.951022i \(0.399965\pi\)
\(234\) 0 0
\(235\) −721661. −0.852440
\(236\) 20735.7 0.0242348
\(237\) 0 0
\(238\) 0 0
\(239\) 115323. 0.130593 0.0652966 0.997866i \(-0.479201\pi\)
0.0652966 + 0.997866i \(0.479201\pi\)
\(240\) 0 0
\(241\) −909864. −1.00910 −0.504549 0.863383i \(-0.668341\pi\)
−0.504549 + 0.863383i \(0.668341\pi\)
\(242\) 691869. 0.759426
\(243\) 0 0
\(244\) 96232.2 0.103477
\(245\) 0 0
\(246\) 0 0
\(247\) 279850. 0.291865
\(248\) 1.67735e6 1.73179
\(249\) 0 0
\(250\) −971301. −0.982888
\(251\) 321264. 0.321868 0.160934 0.986965i \(-0.448549\pi\)
0.160934 + 0.986965i \(0.448549\pi\)
\(252\) 0 0
\(253\) 25264.8 0.0248150
\(254\) 746738. 0.726246
\(255\) 0 0
\(256\) −228642. −0.218050
\(257\) 556492. 0.525565 0.262782 0.964855i \(-0.415360\pi\)
0.262782 + 0.964855i \(0.415360\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 12449.6 0.0114215
\(261\) 0 0
\(262\) −294582. −0.265126
\(263\) 233379. 0.208052 0.104026 0.994575i \(-0.466827\pi\)
0.104026 + 0.994575i \(0.466827\pi\)
\(264\) 0 0
\(265\) 336036. 0.293948
\(266\) 0 0
\(267\) 0 0
\(268\) 129449. 0.110093
\(269\) 1.47857e6 1.24584 0.622919 0.782286i \(-0.285947\pi\)
0.622919 + 0.782286i \(0.285947\pi\)
\(270\) 0 0
\(271\) 177762. 0.147033 0.0735165 0.997294i \(-0.476578\pi\)
0.0735165 + 0.997294i \(0.476578\pi\)
\(272\) 1.85755e6 1.52236
\(273\) 0 0
\(274\) 2.30293e6 1.85312
\(275\) 337323. 0.268976
\(276\) 0 0
\(277\) −1.22252e6 −0.957318 −0.478659 0.878001i \(-0.658877\pi\)
−0.478659 + 0.878001i \(0.658877\pi\)
\(278\) 53990.5 0.0418991
\(279\) 0 0
\(280\) 0 0
\(281\) −177799. −0.134327 −0.0671636 0.997742i \(-0.521395\pi\)
−0.0671636 + 0.997742i \(0.521395\pi\)
\(282\) 0 0
\(283\) −1.09052e6 −0.809406 −0.404703 0.914448i \(-0.632625\pi\)
−0.404703 + 0.914448i \(0.632625\pi\)
\(284\) −149363. −0.109887
\(285\) 0 0
\(286\) 148530. 0.107374
\(287\) 0 0
\(288\) 0 0
\(289\) 2.45468e6 1.72882
\(290\) 246959. 0.172437
\(291\) 0 0
\(292\) −96585.3 −0.0662909
\(293\) −545742. −0.371380 −0.185690 0.982608i \(-0.559452\pi\)
−0.185690 + 0.982608i \(0.559452\pi\)
\(294\) 0 0
\(295\) 319193. 0.213549
\(296\) 2.41198e6 1.60009
\(297\) 0 0
\(298\) 2.75286e6 1.79574
\(299\) −20256.8 −0.0131036
\(300\) 0 0
\(301\) 0 0
\(302\) 1.05168e6 0.663539
\(303\) 0 0
\(304\) −1.78595e6 −1.10837
\(305\) 1.48134e6 0.911811
\(306\) 0 0
\(307\) −3.29000e6 −1.99228 −0.996138 0.0878052i \(-0.972015\pi\)
−0.996138 + 0.0878052i \(0.972015\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.75850e6 1.03929
\(311\) 699712. 0.410221 0.205111 0.978739i \(-0.434245\pi\)
0.205111 + 0.978739i \(0.434245\pi\)
\(312\) 0 0
\(313\) 623219. 0.359567 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(314\) −1.44283e6 −0.825831
\(315\) 0 0
\(316\) −39672.0 −0.0223494
\(317\) 639244. 0.357288 0.178644 0.983914i \(-0.442829\pi\)
0.178644 + 0.983914i \(0.442829\pi\)
\(318\) 0 0
\(319\) −232305. −0.127815
\(320\) −1.25280e6 −0.683922
\(321\) 0 0
\(322\) 0 0
\(323\) −3.72519e6 −1.98675
\(324\) 0 0
\(325\) −270458. −0.142034
\(326\) 2.93602e6 1.53008
\(327\) 0 0
\(328\) −1.67858e6 −0.861506
\(329\) 0 0
\(330\) 0 0
\(331\) 1.40963e6 0.707190 0.353595 0.935399i \(-0.384959\pi\)
0.353595 + 0.935399i \(0.384959\pi\)
\(332\) −237826. −0.118417
\(333\) 0 0
\(334\) 1.19135e6 0.584350
\(335\) 1.99265e6 0.970108
\(336\) 0 0
\(337\) −1.55677e6 −0.746704 −0.373352 0.927690i \(-0.621792\pi\)
−0.373352 + 0.927690i \(0.621792\pi\)
\(338\) 1.90306e6 0.906066
\(339\) 0 0
\(340\) −165722. −0.0777469
\(341\) −1.65416e6 −0.770356
\(342\) 0 0
\(343\) 0 0
\(344\) 2.53200e6 1.15363
\(345\) 0 0
\(346\) 3.21437e6 1.44346
\(347\) −248297. −0.110700 −0.0553500 0.998467i \(-0.517627\pi\)
−0.0553500 + 0.998467i \(0.517627\pi\)
\(348\) 0 0
\(349\) 1.86169e6 0.818171 0.409086 0.912496i \(-0.365848\pi\)
0.409086 + 0.912496i \(0.365848\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 155834. 0.0670356
\(353\) −1.77128e6 −0.756574 −0.378287 0.925688i \(-0.623487\pi\)
−0.378287 + 0.925688i \(0.623487\pi\)
\(354\) 0 0
\(355\) −2.29920e6 −0.968292
\(356\) 203704. 0.0851871
\(357\) 0 0
\(358\) −1.18375e6 −0.488147
\(359\) 4.47560e6 1.83280 0.916401 0.400262i \(-0.131081\pi\)
0.916401 + 0.400262i \(0.131081\pi\)
\(360\) 0 0
\(361\) 1.10550e6 0.446469
\(362\) −1.02449e6 −0.410898
\(363\) 0 0
\(364\) 0 0
\(365\) −1.48678e6 −0.584135
\(366\) 0 0
\(367\) 3.17097e6 1.22893 0.614465 0.788944i \(-0.289372\pi\)
0.614465 + 0.788944i \(0.289372\pi\)
\(368\) 129275. 0.0497617
\(369\) 0 0
\(370\) 2.52868e6 0.960261
\(371\) 0 0
\(372\) 0 0
\(373\) −2.31179e6 −0.860353 −0.430177 0.902745i \(-0.641549\pi\)
−0.430177 + 0.902745i \(0.641549\pi\)
\(374\) −1.97714e6 −0.730900
\(375\) 0 0
\(376\) 3.74895e6 1.36754
\(377\) 186257. 0.0674931
\(378\) 0 0
\(379\) −591840. −0.211644 −0.105822 0.994385i \(-0.533747\pi\)
−0.105822 + 0.994385i \(0.533747\pi\)
\(380\) 159334. 0.0566044
\(381\) 0 0
\(382\) 945042. 0.331354
\(383\) 3.59766e6 1.25321 0.626605 0.779337i \(-0.284444\pi\)
0.626605 + 0.779337i \(0.284444\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −641464. −0.219131
\(387\) 0 0
\(388\) 276212. 0.0931459
\(389\) 2.24746e6 0.753040 0.376520 0.926409i \(-0.377121\pi\)
0.376520 + 0.926409i \(0.377121\pi\)
\(390\) 0 0
\(391\) 269646. 0.0891973
\(392\) 0 0
\(393\) 0 0
\(394\) 1.22395e6 0.397213
\(395\) −610687. −0.196936
\(396\) 0 0
\(397\) −4.58670e6 −1.46058 −0.730288 0.683139i \(-0.760614\pi\)
−0.730288 + 0.683139i \(0.760614\pi\)
\(398\) −4.03169e6 −1.27579
\(399\) 0 0
\(400\) 1.72602e6 0.539380
\(401\) −6.00432e6 −1.86468 −0.932338 0.361589i \(-0.882234\pi\)
−0.932338 + 0.361589i \(0.882234\pi\)
\(402\) 0 0
\(403\) 1.32627e6 0.406788
\(404\) 51994.9 0.0158492
\(405\) 0 0
\(406\) 0 0
\(407\) −2.37864e6 −0.711774
\(408\) 0 0
\(409\) −3.55340e6 −1.05035 −0.525177 0.850993i \(-0.676001\pi\)
−0.525177 + 0.850993i \(0.676001\pi\)
\(410\) −1.75980e6 −0.517014
\(411\) 0 0
\(412\) −317896. −0.0922661
\(413\) 0 0
\(414\) 0 0
\(415\) −3.66094e6 −1.04345
\(416\) −124944. −0.0353983
\(417\) 0 0
\(418\) 1.90093e6 0.532139
\(419\) −2.01375e6 −0.560365 −0.280182 0.959947i \(-0.590395\pi\)
−0.280182 + 0.959947i \(0.590395\pi\)
\(420\) 0 0
\(421\) −5.89987e6 −1.62232 −0.811161 0.584823i \(-0.801164\pi\)
−0.811161 + 0.584823i \(0.801164\pi\)
\(422\) 3.84446e6 1.05088
\(423\) 0 0
\(424\) −1.74567e6 −0.471571
\(425\) 3.60017e6 0.966832
\(426\) 0 0
\(427\) 0 0
\(428\) 275088. 0.0725877
\(429\) 0 0
\(430\) 2.65450e6 0.692327
\(431\) 3.81048e6 0.988066 0.494033 0.869443i \(-0.335522\pi\)
0.494033 + 0.869443i \(0.335522\pi\)
\(432\) 0 0
\(433\) 6.59449e6 1.69029 0.845146 0.534536i \(-0.179513\pi\)
0.845146 + 0.534536i \(0.179513\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 81068.1 0.0204237
\(437\) −259252. −0.0649410
\(438\) 0 0
\(439\) −4.55028e6 −1.12688 −0.563438 0.826158i \(-0.690522\pi\)
−0.563438 + 0.826158i \(0.690522\pi\)
\(440\) 1.24169e6 0.305761
\(441\) 0 0
\(442\) 1.58523e6 0.385954
\(443\) −5.11537e6 −1.23842 −0.619210 0.785225i \(-0.712547\pi\)
−0.619210 + 0.785225i \(0.712547\pi\)
\(444\) 0 0
\(445\) 3.13569e6 0.750643
\(446\) −229908. −0.0547290
\(447\) 0 0
\(448\) 0 0
\(449\) −5.95600e6 −1.39424 −0.697122 0.716953i \(-0.745536\pi\)
−0.697122 + 0.716953i \(0.745536\pi\)
\(450\) 0 0
\(451\) 1.65537e6 0.383226
\(452\) −62172.2 −0.0143136
\(453\) 0 0
\(454\) 4.04329e6 0.920652
\(455\) 0 0
\(456\) 0 0
\(457\) −2.59834e6 −0.581976 −0.290988 0.956727i \(-0.593984\pi\)
−0.290988 + 0.956727i \(0.593984\pi\)
\(458\) −5.12618e6 −1.14191
\(459\) 0 0
\(460\) −11533.3 −0.00254132
\(461\) −4.51513e6 −0.989505 −0.494752 0.869034i \(-0.664741\pi\)
−0.494752 + 0.869034i \(0.664741\pi\)
\(462\) 0 0
\(463\) −5.55129e6 −1.20349 −0.601744 0.798689i \(-0.705527\pi\)
−0.601744 + 0.798689i \(0.705527\pi\)
\(464\) −1.18866e6 −0.256308
\(465\) 0 0
\(466\) −2.79027e6 −0.595225
\(467\) 7.95350e6 1.68759 0.843794 0.536668i \(-0.180317\pi\)
0.843794 + 0.536668i \(0.180317\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.93033e6 0.820699
\(471\) 0 0
\(472\) −1.65817e6 −0.342590
\(473\) −2.49699e6 −0.513173
\(474\) 0 0
\(475\) −3.46140e6 −0.703912
\(476\) 0 0
\(477\) 0 0
\(478\) −628074. −0.125731
\(479\) 6.56370e6 1.30710 0.653552 0.756882i \(-0.273278\pi\)
0.653552 + 0.756882i \(0.273278\pi\)
\(480\) 0 0
\(481\) 1.90714e6 0.375854
\(482\) 4.95532e6 0.971525
\(483\) 0 0
\(484\) 297096. 0.0576479
\(485\) 4.25185e6 0.820773
\(486\) 0 0
\(487\) 4.66370e6 0.891062 0.445531 0.895266i \(-0.353015\pi\)
0.445531 + 0.895266i \(0.353015\pi\)
\(488\) −7.69539e6 −1.46279
\(489\) 0 0
\(490\) 0 0
\(491\) 917227. 0.171701 0.0858506 0.996308i \(-0.472639\pi\)
0.0858506 + 0.996308i \(0.472639\pi\)
\(492\) 0 0
\(493\) −2.47934e6 −0.459430
\(494\) −1.52412e6 −0.280998
\(495\) 0 0
\(496\) −8.46401e6 −1.54480
\(497\) 0 0
\(498\) 0 0
\(499\) 313436. 0.0563504 0.0281752 0.999603i \(-0.491030\pi\)
0.0281752 + 0.999603i \(0.491030\pi\)
\(500\) −417087. −0.0746108
\(501\) 0 0
\(502\) −1.74967e6 −0.309883
\(503\) 3.74110e6 0.659294 0.329647 0.944104i \(-0.393070\pi\)
0.329647 + 0.944104i \(0.393070\pi\)
\(504\) 0 0
\(505\) 800378. 0.139658
\(506\) −137598. −0.0238910
\(507\) 0 0
\(508\) 320657. 0.0551292
\(509\) −1.03017e7 −1.76244 −0.881220 0.472707i \(-0.843277\pi\)
−0.881220 + 0.472707i \(0.843277\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6.44492e6 1.08653
\(513\) 0 0
\(514\) −3.03078e6 −0.505995
\(515\) −4.89350e6 −0.813021
\(516\) 0 0
\(517\) −3.69711e6 −0.608326
\(518\) 0 0
\(519\) 0 0
\(520\) −995560. −0.161458
\(521\) 6.57325e6 1.06093 0.530464 0.847707i \(-0.322018\pi\)
0.530464 + 0.847707i \(0.322018\pi\)
\(522\) 0 0
\(523\) 8.36532e6 1.33730 0.668650 0.743578i \(-0.266873\pi\)
0.668650 + 0.743578i \(0.266873\pi\)
\(524\) −126497. −0.0201257
\(525\) 0 0
\(526\) −1.27103e6 −0.200306
\(527\) −1.76545e7 −2.76904
\(528\) 0 0
\(529\) −6.41758e6 −0.997084
\(530\) −1.83013e6 −0.283003
\(531\) 0 0
\(532\) 0 0
\(533\) −1.32724e6 −0.202364
\(534\) 0 0
\(535\) 4.23454e6 0.639620
\(536\) −1.03516e7 −1.55631
\(537\) 0 0
\(538\) −8.05263e6 −1.19945
\(539\) 0 0
\(540\) 0 0
\(541\) −8.06623e6 −1.18489 −0.592444 0.805612i \(-0.701837\pi\)
−0.592444 + 0.805612i \(0.701837\pi\)
\(542\) −968129. −0.141558
\(543\) 0 0
\(544\) 1.66318e6 0.240959
\(545\) 1.24791e6 0.179967
\(546\) 0 0
\(547\) 3.90775e6 0.558416 0.279208 0.960231i \(-0.409928\pi\)
0.279208 + 0.960231i \(0.409928\pi\)
\(548\) 988902. 0.140670
\(549\) 0 0
\(550\) −1.83714e6 −0.258961
\(551\) 2.38377e6 0.334492
\(552\) 0 0
\(553\) 0 0
\(554\) 6.65811e6 0.921672
\(555\) 0 0
\(556\) 23184.1 0.00318056
\(557\) 1.15862e7 1.58235 0.791177 0.611588i \(-0.209469\pi\)
0.791177 + 0.611588i \(0.209469\pi\)
\(558\) 0 0
\(559\) 2.00203e6 0.270982
\(560\) 0 0
\(561\) 0 0
\(562\) 968334. 0.129326
\(563\) −1.91176e6 −0.254192 −0.127096 0.991890i \(-0.540566\pi\)
−0.127096 + 0.991890i \(0.540566\pi\)
\(564\) 0 0
\(565\) −957041. −0.126127
\(566\) 5.93920e6 0.779268
\(567\) 0 0
\(568\) 1.19441e7 1.55340
\(569\) −8.16817e6 −1.05766 −0.528828 0.848729i \(-0.677368\pi\)
−0.528828 + 0.848729i \(0.677368\pi\)
\(570\) 0 0
\(571\) −7.46593e6 −0.958283 −0.479141 0.877738i \(-0.659052\pi\)
−0.479141 + 0.877738i \(0.659052\pi\)
\(572\) 63780.3 0.00815072
\(573\) 0 0
\(574\) 0 0
\(575\) 250552. 0.0316030
\(576\) 0 0
\(577\) −6.88438e6 −0.860845 −0.430423 0.902627i \(-0.641635\pi\)
−0.430423 + 0.902627i \(0.641635\pi\)
\(578\) −1.33687e7 −1.66445
\(579\) 0 0
\(580\) 106047. 0.0130896
\(581\) 0 0
\(582\) 0 0
\(583\) 1.72153e6 0.209770
\(584\) 7.72363e6 0.937108
\(585\) 0 0
\(586\) 2.97223e6 0.357551
\(587\) 8.91086e6 1.06739 0.533696 0.845676i \(-0.320803\pi\)
0.533696 + 0.845676i \(0.320803\pi\)
\(588\) 0 0
\(589\) 1.69740e7 2.01602
\(590\) −1.73839e6 −0.205598
\(591\) 0 0
\(592\) −1.21710e7 −1.42732
\(593\) −1.23430e7 −1.44140 −0.720700 0.693247i \(-0.756180\pi\)
−0.720700 + 0.693247i \(0.756180\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.18211e6 0.136314
\(597\) 0 0
\(598\) 110323. 0.0126157
\(599\) −9.05732e6 −1.03141 −0.515707 0.856765i \(-0.672471\pi\)
−0.515707 + 0.856765i \(0.672471\pi\)
\(600\) 0 0
\(601\) 7.41700e6 0.837611 0.418805 0.908076i \(-0.362449\pi\)
0.418805 + 0.908076i \(0.362449\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 451603. 0.0503691
\(605\) 4.57331e6 0.507975
\(606\) 0 0
\(607\) −5.77813e6 −0.636525 −0.318263 0.948003i \(-0.603099\pi\)
−0.318263 + 0.948003i \(0.603099\pi\)
\(608\) −1.59908e6 −0.175432
\(609\) 0 0
\(610\) −8.06770e6 −0.877860
\(611\) 2.96426e6 0.321228
\(612\) 0 0
\(613\) 6.29264e6 0.676366 0.338183 0.941080i \(-0.390188\pi\)
0.338183 + 0.941080i \(0.390188\pi\)
\(614\) 1.79180e7 1.91809
\(615\) 0 0
\(616\) 0 0
\(617\) 9.79133e6 1.03545 0.517725 0.855547i \(-0.326779\pi\)
0.517725 + 0.855547i \(0.326779\pi\)
\(618\) 0 0
\(619\) −1.32677e7 −1.39178 −0.695889 0.718150i \(-0.744989\pi\)
−0.695889 + 0.718150i \(0.744989\pi\)
\(620\) 755120. 0.0788927
\(621\) 0 0
\(622\) −3.81079e6 −0.394947
\(623\) 0 0
\(624\) 0 0
\(625\) −704759. −0.0721673
\(626\) −3.39419e6 −0.346178
\(627\) 0 0
\(628\) −619567. −0.0626886
\(629\) −2.53867e7 −2.55846
\(630\) 0 0
\(631\) 4.41233e6 0.441158 0.220579 0.975369i \(-0.429205\pi\)
0.220579 + 0.975369i \(0.429205\pi\)
\(632\) 3.17245e6 0.315938
\(633\) 0 0
\(634\) −3.48146e6 −0.343984
\(635\) 4.93600e6 0.485782
\(636\) 0 0
\(637\) 0 0
\(638\) 1.26518e6 0.123056
\(639\) 0 0
\(640\) 5.84964e6 0.564520
\(641\) −8.13035e6 −0.781564 −0.390782 0.920483i \(-0.627795\pi\)
−0.390782 + 0.920483i \(0.627795\pi\)
\(642\) 0 0
\(643\) −3.12961e6 −0.298513 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.02882e7 1.91277
\(647\) −1.34660e7 −1.26467 −0.632336 0.774695i \(-0.717904\pi\)
−0.632336 + 0.774695i \(0.717904\pi\)
\(648\) 0 0
\(649\) 1.63524e6 0.152395
\(650\) 1.47297e6 0.136745
\(651\) 0 0
\(652\) 1.26076e6 0.116148
\(653\) −1.44772e7 −1.32862 −0.664312 0.747455i \(-0.731275\pi\)
−0.664312 + 0.747455i \(0.731275\pi\)
\(654\) 0 0
\(655\) −1.94721e6 −0.177341
\(656\) 8.47023e6 0.768485
\(657\) 0 0
\(658\) 0 0
\(659\) −432708. −0.0388134 −0.0194067 0.999812i \(-0.506178\pi\)
−0.0194067 + 0.999812i \(0.506178\pi\)
\(660\) 0 0
\(661\) −29965.2 −0.00266756 −0.00133378 0.999999i \(-0.500425\pi\)
−0.00133378 + 0.999999i \(0.500425\pi\)
\(662\) −7.67718e6 −0.680858
\(663\) 0 0
\(664\) 1.90182e7 1.67398
\(665\) 0 0
\(666\) 0 0
\(667\) −172548. −0.0150174
\(668\) 511578. 0.0443579
\(669\) 0 0
\(670\) −1.08524e7 −0.933986
\(671\) 7.58899e6 0.650696
\(672\) 0 0
\(673\) −6.71329e6 −0.571344 −0.285672 0.958327i \(-0.592217\pi\)
−0.285672 + 0.958327i \(0.592217\pi\)
\(674\) 8.47849e6 0.718901
\(675\) 0 0
\(676\) 817193. 0.0687793
\(677\) 4.45929e6 0.373933 0.186967 0.982366i \(-0.440134\pi\)
0.186967 + 0.982366i \(0.440134\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.32523e7 1.09905
\(681\) 0 0
\(682\) 9.00892e6 0.741672
\(683\) 1.79839e7 1.47514 0.737569 0.675272i \(-0.235974\pi\)
0.737569 + 0.675272i \(0.235974\pi\)
\(684\) 0 0
\(685\) 1.52225e7 1.23954
\(686\) 0 0
\(687\) 0 0
\(688\) −1.27766e7 −1.02907
\(689\) −1.38029e6 −0.110770
\(690\) 0 0
\(691\) 2.50935e6 0.199925 0.0999624 0.994991i \(-0.468128\pi\)
0.0999624 + 0.994991i \(0.468128\pi\)
\(692\) 1.38028e6 0.109573
\(693\) 0 0
\(694\) 1.35228e6 0.106578
\(695\) 356882. 0.0280261
\(696\) 0 0
\(697\) 1.76675e7 1.37750
\(698\) −1.01392e7 −0.787707
\(699\) 0 0
\(700\) 0 0
\(701\) −9.68649e6 −0.744512 −0.372256 0.928130i \(-0.621416\pi\)
−0.372256 + 0.928130i \(0.621416\pi\)
\(702\) 0 0
\(703\) 2.44081e7 1.86271
\(704\) −6.41817e6 −0.488067
\(705\) 0 0
\(706\) 9.64681e6 0.728403
\(707\) 0 0
\(708\) 0 0
\(709\) −9.52720e6 −0.711786 −0.355893 0.934527i \(-0.615823\pi\)
−0.355893 + 0.934527i \(0.615823\pi\)
\(710\) 1.25220e7 0.932238
\(711\) 0 0
\(712\) −1.62896e7 −1.20423
\(713\) −1.22865e6 −0.0905118
\(714\) 0 0
\(715\) 981794. 0.0718217
\(716\) −508313. −0.0370552
\(717\) 0 0
\(718\) −2.43751e7 −1.76456
\(719\) −757176. −0.0546229 −0.0273114 0.999627i \(-0.508695\pi\)
−0.0273114 + 0.999627i \(0.508695\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −6.02081e6 −0.429845
\(723\) 0 0
\(724\) −439925. −0.0311912
\(725\) −2.30378e6 −0.162778
\(726\) 0 0
\(727\) −2.66570e7 −1.87058 −0.935288 0.353888i \(-0.884859\pi\)
−0.935288 + 0.353888i \(0.884859\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 8.09731e6 0.562385
\(731\) −2.66498e7 −1.84459
\(732\) 0 0
\(733\) 2.70857e7 1.86200 0.931001 0.365017i \(-0.118937\pi\)
0.931001 + 0.365017i \(0.118937\pi\)
\(734\) −1.72698e7 −1.18317
\(735\) 0 0
\(736\) 115748. 0.00787625
\(737\) 1.02085e7 0.692298
\(738\) 0 0
\(739\) 2.19507e7 1.47855 0.739276 0.673402i \(-0.235168\pi\)
0.739276 + 0.673402i \(0.235168\pi\)
\(740\) 1.08584e6 0.0728932
\(741\) 0 0
\(742\) 0 0
\(743\) −5.77370e6 −0.383691 −0.191846 0.981425i \(-0.561447\pi\)
−0.191846 + 0.981425i \(0.561447\pi\)
\(744\) 0 0
\(745\) 1.81966e7 1.20116
\(746\) 1.25905e7 0.828318
\(747\) 0 0
\(748\) −849005. −0.0554825
\(749\) 0 0
\(750\) 0 0
\(751\) −8.37749e6 −0.542018 −0.271009 0.962577i \(-0.587357\pi\)
−0.271009 + 0.962577i \(0.587357\pi\)
\(752\) −1.89174e7 −1.21988
\(753\) 0 0
\(754\) −1.01440e6 −0.0649800
\(755\) 6.95170e6 0.443837
\(756\) 0 0
\(757\) 1.18828e7 0.753665 0.376833 0.926281i \(-0.377013\pi\)
0.376833 + 0.926281i \(0.377013\pi\)
\(758\) 3.22329e6 0.203764
\(759\) 0 0
\(760\) −1.27415e7 −0.800177
\(761\) −1.76940e7 −1.10755 −0.553777 0.832665i \(-0.686814\pi\)
−0.553777 + 0.832665i \(0.686814\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 405811. 0.0251531
\(765\) 0 0
\(766\) −1.95937e7 −1.20655
\(767\) −1.31110e6 −0.0804726
\(768\) 0 0
\(769\) −4.12006e6 −0.251239 −0.125620 0.992078i \(-0.540092\pi\)
−0.125620 + 0.992078i \(0.540092\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −275452. −0.0166342
\(773\) −1.26864e7 −0.763642 −0.381821 0.924236i \(-0.624703\pi\)
−0.381821 + 0.924236i \(0.624703\pi\)
\(774\) 0 0
\(775\) −1.64043e7 −0.981080
\(776\) −2.20879e7 −1.31674
\(777\) 0 0
\(778\) −1.22402e7 −0.725001
\(779\) −1.69865e7 −1.00290
\(780\) 0 0
\(781\) −1.17790e7 −0.691002
\(782\) −1.46855e6 −0.0858761
\(783\) 0 0
\(784\) 0 0
\(785\) −9.53723e6 −0.552393
\(786\) 0 0
\(787\) 9.14809e6 0.526494 0.263247 0.964728i \(-0.415207\pi\)
0.263247 + 0.964728i \(0.415207\pi\)
\(788\) 525578. 0.0301524
\(789\) 0 0
\(790\) 3.32594e6 0.189603
\(791\) 0 0
\(792\) 0 0
\(793\) −6.08468e6 −0.343602
\(794\) 2.49802e7 1.40619
\(795\) 0 0
\(796\) −1.73125e6 −0.0968451
\(797\) 1.10180e7 0.614408 0.307204 0.951644i \(-0.400607\pi\)
0.307204 + 0.951644i \(0.400607\pi\)
\(798\) 0 0
\(799\) −3.94585e7 −2.18662
\(800\) 1.54541e6 0.0853727
\(801\) 0 0
\(802\) 3.27009e7 1.79524
\(803\) −7.61684e6 −0.416856
\(804\) 0 0
\(805\) 0 0
\(806\) −7.22315e6 −0.391642
\(807\) 0 0
\(808\) −4.15788e6 −0.224049
\(809\) 3.10273e7 1.66676 0.833378 0.552703i \(-0.186404\pi\)
0.833378 + 0.552703i \(0.186404\pi\)
\(810\) 0 0
\(811\) −2.94456e7 −1.57206 −0.786028 0.618191i \(-0.787866\pi\)
−0.786028 + 0.618191i \(0.787866\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.29546e7 0.685271
\(815\) 1.94073e7 1.02346
\(816\) 0 0
\(817\) 2.56226e7 1.34297
\(818\) 1.93526e7 1.01124
\(819\) 0 0
\(820\) −755675. −0.0392464
\(821\) 5.23311e6 0.270958 0.135479 0.990780i \(-0.456743\pi\)
0.135479 + 0.990780i \(0.456743\pi\)
\(822\) 0 0
\(823\) 1.41631e7 0.728884 0.364442 0.931226i \(-0.381260\pi\)
0.364442 + 0.931226i \(0.381260\pi\)
\(824\) 2.54212e7 1.30430
\(825\) 0 0
\(826\) 0 0
\(827\) −2.98006e7 −1.51517 −0.757585 0.652737i \(-0.773621\pi\)
−0.757585 + 0.652737i \(0.773621\pi\)
\(828\) 0 0
\(829\) −1.99304e7 −1.00723 −0.503617 0.863927i \(-0.667998\pi\)
−0.503617 + 0.863927i \(0.667998\pi\)
\(830\) 1.99383e7 1.00460
\(831\) 0 0
\(832\) 5.14594e6 0.257725
\(833\) 0 0
\(834\) 0 0
\(835\) 7.87492e6 0.390868
\(836\) 816280. 0.0403946
\(837\) 0 0
\(838\) 1.09673e7 0.539500
\(839\) 1.88103e7 0.922552 0.461276 0.887257i \(-0.347392\pi\)
0.461276 + 0.887257i \(0.347392\pi\)
\(840\) 0 0
\(841\) −1.89246e7 −0.922650
\(842\) 3.21320e7 1.56191
\(843\) 0 0
\(844\) 1.65085e6 0.0797724
\(845\) 1.25794e7 0.606062
\(846\) 0 0
\(847\) 0 0
\(848\) 8.80874e6 0.420653
\(849\) 0 0
\(850\) −1.96073e7 −0.930833
\(851\) −1.76677e6 −0.0836288
\(852\) 0 0
\(853\) 2.05980e7 0.969285 0.484643 0.874712i \(-0.338950\pi\)
0.484643 + 0.874712i \(0.338950\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.19980e7 −1.02612
\(857\) 2.47572e7 1.15146 0.575731 0.817639i \(-0.304718\pi\)
0.575731 + 0.817639i \(0.304718\pi\)
\(858\) 0 0
\(859\) −3.91065e7 −1.80828 −0.904141 0.427234i \(-0.859488\pi\)
−0.904141 + 0.427234i \(0.859488\pi\)
\(860\) 1.13987e6 0.0525544
\(861\) 0 0
\(862\) −2.07527e7 −0.951275
\(863\) 3.93363e7 1.79790 0.898952 0.438048i \(-0.144330\pi\)
0.898952 + 0.438048i \(0.144330\pi\)
\(864\) 0 0
\(865\) 2.12472e7 0.965522
\(866\) −3.59151e7 −1.62735
\(867\) 0 0
\(868\) 0 0
\(869\) −3.12859e6 −0.140540
\(870\) 0 0
\(871\) −8.18494e6 −0.365570
\(872\) −6.48277e6 −0.288715
\(873\) 0 0
\(874\) 1.41195e6 0.0625229
\(875\) 0 0
\(876\) 0 0
\(877\) 8.09253e6 0.355292 0.177646 0.984094i \(-0.443152\pi\)
0.177646 + 0.984094i \(0.443152\pi\)
\(878\) 2.47818e7 1.08492
\(879\) 0 0
\(880\) −6.26564e6 −0.272746
\(881\) 4.05755e6 0.176126 0.0880631 0.996115i \(-0.471932\pi\)
0.0880631 + 0.996115i \(0.471932\pi\)
\(882\) 0 0
\(883\) 1.79813e7 0.776102 0.388051 0.921638i \(-0.373148\pi\)
0.388051 + 0.921638i \(0.373148\pi\)
\(884\) 680713. 0.0292977
\(885\) 0 0
\(886\) 2.78595e7 1.19231
\(887\) 1.53139e7 0.653547 0.326773 0.945103i \(-0.394039\pi\)
0.326773 + 0.945103i \(0.394039\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.70777e7 −0.722693
\(891\) 0 0
\(892\) −98725.0 −0.00415447
\(893\) 3.79376e7 1.59199
\(894\) 0 0
\(895\) −7.82466e6 −0.326519
\(896\) 0 0
\(897\) 0 0
\(898\) 3.24377e7 1.34233
\(899\) 1.12972e7 0.466200
\(900\) 0 0
\(901\) 1.83735e7 0.754016
\(902\) −9.01554e6 −0.368957
\(903\) 0 0
\(904\) 4.97172e6 0.202342
\(905\) −6.77194e6 −0.274847
\(906\) 0 0
\(907\) −2.17757e7 −0.878930 −0.439465 0.898260i \(-0.644832\pi\)
−0.439465 + 0.898260i \(0.644832\pi\)
\(908\) 1.73623e6 0.0698865
\(909\) 0 0
\(910\) 0 0
\(911\) 5.35112e6 0.213623 0.106812 0.994279i \(-0.465936\pi\)
0.106812 + 0.994279i \(0.465936\pi\)
\(912\) 0 0
\(913\) −1.87552e7 −0.744639
\(914\) 1.41511e7 0.560306
\(915\) 0 0
\(916\) −2.20124e6 −0.0866818
\(917\) 0 0
\(918\) 0 0
\(919\) −2.67858e7 −1.04620 −0.523102 0.852270i \(-0.675225\pi\)
−0.523102 + 0.852270i \(0.675225\pi\)
\(920\) 922285. 0.0359249
\(921\) 0 0
\(922\) 2.45904e7 0.952661
\(923\) 9.44411e6 0.364886
\(924\) 0 0
\(925\) −2.35890e7 −0.906474
\(926\) 3.02336e7 1.15868
\(927\) 0 0
\(928\) −1.06428e6 −0.0405683
\(929\) −739974. −0.0281305 −0.0140652 0.999901i \(-0.504477\pi\)
−0.0140652 + 0.999901i \(0.504477\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.19817e6 −0.0451834
\(933\) 0 0
\(934\) −4.33166e7 −1.62475
\(935\) −1.30691e7 −0.488895
\(936\) 0 0
\(937\) −122654. −0.00456385 −0.00228193 0.999997i \(-0.500726\pi\)
−0.00228193 + 0.999997i \(0.500726\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.68772e6 0.0622991
\(941\) −1.22466e7 −0.450861 −0.225431 0.974259i \(-0.572379\pi\)
−0.225431 + 0.974259i \(0.572379\pi\)
\(942\) 0 0
\(943\) 1.22956e6 0.0450266
\(944\) 8.36722e6 0.305599
\(945\) 0 0
\(946\) 1.35992e7 0.494065
\(947\) 3.71174e6 0.134494 0.0672470 0.997736i \(-0.478578\pi\)
0.0672470 + 0.997736i \(0.478578\pi\)
\(948\) 0 0
\(949\) 6.10701e6 0.220122
\(950\) 1.88516e7 0.677702
\(951\) 0 0
\(952\) 0 0
\(953\) 2.32857e7 0.830533 0.415266 0.909700i \(-0.363688\pi\)
0.415266 + 0.909700i \(0.363688\pi\)
\(954\) 0 0
\(955\) 6.24681e6 0.221641
\(956\) −269702. −0.00954418
\(957\) 0 0
\(958\) −3.57474e7 −1.25843
\(959\) 0 0
\(960\) 0 0
\(961\) 5.18142e7 1.80984
\(962\) −1.03867e7 −0.361859
\(963\) 0 0
\(964\) 2.12787e6 0.0737483
\(965\) −4.24013e6 −0.146575
\(966\) 0 0
\(967\) 2.25257e7 0.774661 0.387330 0.921941i \(-0.373397\pi\)
0.387330 + 0.921941i \(0.373397\pi\)
\(968\) −2.37579e7 −0.814927
\(969\) 0 0
\(970\) −2.31565e7 −0.790212
\(971\) −1.99456e7 −0.678890 −0.339445 0.940626i \(-0.610239\pi\)
−0.339445 + 0.940626i \(0.610239\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.53995e7 −0.857884
\(975\) 0 0
\(976\) 3.88314e7 1.30484
\(977\) 959342. 0.0321542 0.0160771 0.999871i \(-0.494882\pi\)
0.0160771 + 0.999871i \(0.494882\pi\)
\(978\) 0 0
\(979\) 1.60643e7 0.535681
\(980\) 0 0
\(981\) 0 0
\(982\) −4.99542e6 −0.165308
\(983\) −5.22097e7 −1.72333 −0.861663 0.507481i \(-0.830577\pi\)
−0.861663 + 0.507481i \(0.830577\pi\)
\(984\) 0 0
\(985\) 8.09042e6 0.265693
\(986\) 1.35030e7 0.442323
\(987\) 0 0
\(988\) −654475. −0.0213305
\(989\) −1.85468e6 −0.0602945
\(990\) 0 0
\(991\) −1.76305e7 −0.570269 −0.285134 0.958488i \(-0.592038\pi\)
−0.285134 + 0.958488i \(0.592038\pi\)
\(992\) −7.57836e6 −0.244510
\(993\) 0 0
\(994\) 0 0
\(995\) −2.66498e7 −0.853369
\(996\) 0 0
\(997\) 4.04875e7 1.28998 0.644990 0.764191i \(-0.276862\pi\)
0.644990 + 0.764191i \(0.276862\pi\)
\(998\) −1.70704e6 −0.0542522
\(999\) 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.q.1.1 2
3.2 odd 2 147.6.a.h.1.2 2
7.6 odd 2 441.6.a.r.1.1 2
21.2 odd 6 147.6.e.n.67.1 4
21.5 even 6 147.6.e.m.67.1 4
21.11 odd 6 147.6.e.n.79.1 4
21.17 even 6 147.6.e.m.79.1 4
21.20 even 2 147.6.a.j.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.2 2 3.2 odd 2
147.6.a.j.1.2 yes 2 21.20 even 2
147.6.e.m.67.1 4 21.5 even 6
147.6.e.m.79.1 4 21.17 even 6
147.6.e.n.67.1 4 21.2 odd 6
147.6.e.n.79.1 4 21.11 odd 6
441.6.a.q.1.1 2 1.1 even 1 trivial
441.6.a.r.1.1 2 7.6 odd 2