Properties

Label 441.6.a.bb.1.1
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 1.1
Root \(5.59680\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} +O(q^{10})\) \(q-10.0089 q^{2} +68.1779 q^{4} +70.3512 q^{5} -362.101 q^{8} -704.138 q^{10} +731.396 q^{11} -899.645 q^{13} +1442.53 q^{16} -1392.39 q^{17} +190.198 q^{19} +4796.40 q^{20} -7320.46 q^{22} -42.9770 q^{23} +1824.30 q^{25} +9004.45 q^{26} +7746.59 q^{29} -1179.22 q^{31} -2850.94 q^{32} +13936.3 q^{34} +9288.68 q^{37} -1903.67 q^{38} -25474.2 q^{40} +13453.4 q^{41} +6033.68 q^{43} +49865.0 q^{44} +430.152 q^{46} -3244.94 q^{47} -18259.2 q^{50} -61335.9 q^{52} -25675.8 q^{53} +51454.6 q^{55} -77534.8 q^{58} -26450.5 q^{59} +6432.77 q^{61} +11802.7 q^{62} -17626.3 q^{64} -63291.1 q^{65} -23815.7 q^{67} -94930.2 q^{68} +44680.7 q^{71} +39601.3 q^{73} -92969.4 q^{74} +12967.3 q^{76} +23123.0 q^{79} +101484. q^{80} -134654. q^{82} -17267.2 q^{83} -97956.3 q^{85} -60390.4 q^{86} -264839. q^{88} +49929.7 q^{89} -2930.08 q^{92} +32478.3 q^{94} +13380.7 q^{95} +16865.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0089 −1.76934 −0.884669 0.466219i \(-0.845616\pi\)
−0.884669 + 0.466219i \(0.845616\pi\)
\(3\) 0 0
\(4\) 68.1779 2.13056
\(5\) 70.3512 1.25848 0.629241 0.777211i \(-0.283366\pi\)
0.629241 + 0.777211i \(0.283366\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −362.101 −2.00034
\(9\) 0 0
\(10\) −704.138 −2.22668
\(11\) 731.396 1.82251 0.911257 0.411838i \(-0.135113\pi\)
0.911257 + 0.411838i \(0.135113\pi\)
\(12\) 0 0
\(13\) −899.645 −1.47643 −0.738215 0.674566i \(-0.764331\pi\)
−0.738215 + 0.674566i \(0.764331\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1442.53 1.40872
\(17\) −1392.39 −1.16853 −0.584263 0.811564i \(-0.698616\pi\)
−0.584263 + 0.811564i \(0.698616\pi\)
\(18\) 0 0
\(19\) 190.198 0.120871 0.0604354 0.998172i \(-0.480751\pi\)
0.0604354 + 0.998172i \(0.480751\pi\)
\(20\) 4796.40 2.68127
\(21\) 0 0
\(22\) −7320.46 −3.22464
\(23\) −42.9770 −0.0169401 −0.00847006 0.999964i \(-0.502696\pi\)
−0.00847006 + 0.999964i \(0.502696\pi\)
\(24\) 0 0
\(25\) 1824.30 0.583775
\(26\) 9004.45 2.61230
\(27\) 0 0
\(28\) 0 0
\(29\) 7746.59 1.71047 0.855236 0.518239i \(-0.173412\pi\)
0.855236 + 0.518239i \(0.173412\pi\)
\(30\) 0 0
\(31\) −1179.22 −0.220390 −0.110195 0.993910i \(-0.535148\pi\)
−0.110195 + 0.993910i \(0.535148\pi\)
\(32\) −2850.94 −0.492168
\(33\) 0 0
\(34\) 13936.3 2.06752
\(35\) 0 0
\(36\) 0 0
\(37\) 9288.68 1.11545 0.557724 0.830026i \(-0.311675\pi\)
0.557724 + 0.830026i \(0.311675\pi\)
\(38\) −1903.67 −0.213861
\(39\) 0 0
\(40\) −25474.2 −2.51739
\(41\) 13453.4 1.24989 0.624945 0.780668i \(-0.285121\pi\)
0.624945 + 0.780668i \(0.285121\pi\)
\(42\) 0 0
\(43\) 6033.68 0.497635 0.248818 0.968550i \(-0.419958\pi\)
0.248818 + 0.968550i \(0.419958\pi\)
\(44\) 49865.0 3.88297
\(45\) 0 0
\(46\) 430.152 0.0299728
\(47\) −3244.94 −0.214270 −0.107135 0.994244i \(-0.534168\pi\)
−0.107135 + 0.994244i \(0.534168\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −18259.2 −1.03289
\(51\) 0 0
\(52\) −61335.9 −3.14562
\(53\) −25675.8 −1.25555 −0.627775 0.778395i \(-0.716034\pi\)
−0.627775 + 0.778395i \(0.716034\pi\)
\(54\) 0 0
\(55\) 51454.6 2.29360
\(56\) 0 0
\(57\) 0 0
\(58\) −77534.8 −3.02640
\(59\) −26450.5 −0.989244 −0.494622 0.869108i \(-0.664694\pi\)
−0.494622 + 0.869108i \(0.664694\pi\)
\(60\) 0 0
\(61\) 6432.77 0.221347 0.110673 0.993857i \(-0.464699\pi\)
0.110673 + 0.993857i \(0.464699\pi\)
\(62\) 11802.7 0.389944
\(63\) 0 0
\(64\) −17626.3 −0.537913
\(65\) −63291.1 −1.85806
\(66\) 0 0
\(67\) −23815.7 −0.648150 −0.324075 0.946031i \(-0.605053\pi\)
−0.324075 + 0.946031i \(0.605053\pi\)
\(68\) −94930.2 −2.48961
\(69\) 0 0
\(70\) 0 0
\(71\) 44680.7 1.05190 0.525949 0.850516i \(-0.323710\pi\)
0.525949 + 0.850516i \(0.323710\pi\)
\(72\) 0 0
\(73\) 39601.3 0.869765 0.434882 0.900487i \(-0.356790\pi\)
0.434882 + 0.900487i \(0.356790\pi\)
\(74\) −92969.4 −1.97361
\(75\) 0 0
\(76\) 12967.3 0.257523
\(77\) 0 0
\(78\) 0 0
\(79\) 23123.0 0.416846 0.208423 0.978039i \(-0.433167\pi\)
0.208423 + 0.978039i \(0.433167\pi\)
\(80\) 101484. 1.77285
\(81\) 0 0
\(82\) −134654. −2.21148
\(83\) −17267.2 −0.275122 −0.137561 0.990493i \(-0.543926\pi\)
−0.137561 + 0.990493i \(0.543926\pi\)
\(84\) 0 0
\(85\) −97956.3 −1.47057
\(86\) −60390.4 −0.880485
\(87\) 0 0
\(88\) −264839. −3.64565
\(89\) 49929.7 0.668166 0.334083 0.942544i \(-0.391573\pi\)
0.334083 + 0.942544i \(0.391573\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2930.08 −0.0360920
\(93\) 0 0
\(94\) 32478.3 0.379117
\(95\) 13380.7 0.152114
\(96\) 0 0
\(97\) 16865.0 0.181994 0.0909970 0.995851i \(-0.470995\pi\)
0.0909970 + 0.995851i \(0.470995\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 124377. 1.24377
\(101\) 105449. 1.02858 0.514290 0.857617i \(-0.328056\pi\)
0.514290 + 0.857617i \(0.328056\pi\)
\(102\) 0 0
\(103\) 20355.8 0.189058 0.0945290 0.995522i \(-0.469865\pi\)
0.0945290 + 0.995522i \(0.469865\pi\)
\(104\) 325762. 2.95337
\(105\) 0 0
\(106\) 256986. 2.22149
\(107\) 3775.43 0.0318792 0.0159396 0.999873i \(-0.494926\pi\)
0.0159396 + 0.999873i \(0.494926\pi\)
\(108\) 0 0
\(109\) 8536.98 0.0688237 0.0344118 0.999408i \(-0.489044\pi\)
0.0344118 + 0.999408i \(0.489044\pi\)
\(110\) −515004. −4.05815
\(111\) 0 0
\(112\) 0 0
\(113\) 100869. 0.743125 0.371562 0.928408i \(-0.378822\pi\)
0.371562 + 0.928408i \(0.378822\pi\)
\(114\) 0 0
\(115\) −3023.49 −0.0213188
\(116\) 528147. 3.64426
\(117\) 0 0
\(118\) 264740. 1.75031
\(119\) 0 0
\(120\) 0 0
\(121\) 373889. 2.32156
\(122\) −64384.9 −0.391638
\(123\) 0 0
\(124\) −80396.9 −0.469554
\(125\) −91506.2 −0.523812
\(126\) 0 0
\(127\) −185451. −1.02028 −0.510142 0.860090i \(-0.670407\pi\)
−0.510142 + 0.860090i \(0.670407\pi\)
\(128\) 267650. 1.44392
\(129\) 0 0
\(130\) 633474. 3.28754
\(131\) 124046. 0.631545 0.315773 0.948835i \(-0.397736\pi\)
0.315773 + 0.948835i \(0.397736\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 238368. 1.14680
\(135\) 0 0
\(136\) 504185. 2.33745
\(137\) −99520.5 −0.453013 −0.226507 0.974010i \(-0.572731\pi\)
−0.226507 + 0.974010i \(0.572731\pi\)
\(138\) 0 0
\(139\) −254481. −1.11717 −0.558583 0.829449i \(-0.688655\pi\)
−0.558583 + 0.829449i \(0.688655\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −447204. −1.86116
\(143\) −657997. −2.69081
\(144\) 0 0
\(145\) 544982. 2.15260
\(146\) −396365. −1.53891
\(147\) 0 0
\(148\) 633283. 2.37653
\(149\) 427260. 1.57662 0.788309 0.615279i \(-0.210957\pi\)
0.788309 + 0.615279i \(0.210957\pi\)
\(150\) 0 0
\(151\) −150472. −0.537047 −0.268523 0.963273i \(-0.586536\pi\)
−0.268523 + 0.963273i \(0.586536\pi\)
\(152\) −68870.8 −0.241783
\(153\) 0 0
\(154\) 0 0
\(155\) −82959.8 −0.277357
\(156\) 0 0
\(157\) −165831. −0.536930 −0.268465 0.963290i \(-0.586516\pi\)
−0.268465 + 0.963290i \(0.586516\pi\)
\(158\) −231435. −0.737542
\(159\) 0 0
\(160\) −200567. −0.619384
\(161\) 0 0
\(162\) 0 0
\(163\) 95718.2 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(164\) 917224. 2.66297
\(165\) 0 0
\(166\) 172825. 0.486784
\(167\) 474209. 1.31577 0.657883 0.753120i \(-0.271452\pi\)
0.657883 + 0.753120i \(0.271452\pi\)
\(168\) 0 0
\(169\) 438068. 1.17984
\(170\) 980434. 2.60193
\(171\) 0 0
\(172\) 411363. 1.06024
\(173\) 258876. 0.657624 0.328812 0.944395i \(-0.393352\pi\)
0.328812 + 0.944395i \(0.393352\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.05506e6 2.56742
\(177\) 0 0
\(178\) −499741. −1.18221
\(179\) 542178. 1.26476 0.632381 0.774657i \(-0.282078\pi\)
0.632381 + 0.774657i \(0.282078\pi\)
\(180\) 0 0
\(181\) −653930. −1.48366 −0.741831 0.670587i \(-0.766042\pi\)
−0.741831 + 0.670587i \(0.766042\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 15562.0 0.0338861
\(185\) 653470. 1.40377
\(186\) 0 0
\(187\) −1.01839e6 −2.12965
\(188\) −221233. −0.456516
\(189\) 0 0
\(190\) −133925. −0.269141
\(191\) −769535. −1.52632 −0.763159 0.646210i \(-0.776353\pi\)
−0.763159 + 0.646210i \(0.776353\pi\)
\(192\) 0 0
\(193\) 814230. 1.57345 0.786727 0.617301i \(-0.211774\pi\)
0.786727 + 0.617301i \(0.211774\pi\)
\(194\) −168800. −0.322009
\(195\) 0 0
\(196\) 0 0
\(197\) 443688. 0.814539 0.407269 0.913308i \(-0.366481\pi\)
0.407269 + 0.913308i \(0.366481\pi\)
\(198\) 0 0
\(199\) −618941. −1.10794 −0.553970 0.832536i \(-0.686888\pi\)
−0.553970 + 0.832536i \(0.686888\pi\)
\(200\) −660579. −1.16775
\(201\) 0 0
\(202\) −1.05542e6 −1.81991
\(203\) 0 0
\(204\) 0 0
\(205\) 946463. 1.57296
\(206\) −203739. −0.334508
\(207\) 0 0
\(208\) −1.29777e6 −2.07988
\(209\) 139110. 0.220289
\(210\) 0 0
\(211\) 769805. 1.19035 0.595175 0.803596i \(-0.297083\pi\)
0.595175 + 0.803596i \(0.297083\pi\)
\(212\) −1.75052e6 −2.67502
\(213\) 0 0
\(214\) −37787.9 −0.0564050
\(215\) 424477. 0.626264
\(216\) 0 0
\(217\) 0 0
\(218\) −85445.7 −0.121772
\(219\) 0 0
\(220\) 3.50807e6 4.88665
\(221\) 1.25266e6 1.72525
\(222\) 0 0
\(223\) 1.45664e6 1.96151 0.980755 0.195244i \(-0.0625498\pi\)
0.980755 + 0.195244i \(0.0625498\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.00959e6 −1.31484
\(227\) 949905. 1.22353 0.611767 0.791038i \(-0.290459\pi\)
0.611767 + 0.791038i \(0.290459\pi\)
\(228\) 0 0
\(229\) 312283. 0.393513 0.196757 0.980452i \(-0.436959\pi\)
0.196757 + 0.980452i \(0.436959\pi\)
\(230\) 30261.7 0.0377202
\(231\) 0 0
\(232\) −2.80505e6 −3.42153
\(233\) −396905. −0.478957 −0.239479 0.970902i \(-0.576977\pi\)
−0.239479 + 0.970902i \(0.576977\pi\)
\(234\) 0 0
\(235\) −228286. −0.269655
\(236\) −1.80334e6 −2.10764
\(237\) 0 0
\(238\) 0 0
\(239\) 87033.3 0.0985578 0.0492789 0.998785i \(-0.484308\pi\)
0.0492789 + 0.998785i \(0.484308\pi\)
\(240\) 0 0
\(241\) −1.57173e6 −1.74315 −0.871576 0.490260i \(-0.836902\pi\)
−0.871576 + 0.490260i \(0.836902\pi\)
\(242\) −3.74222e6 −4.10762
\(243\) 0 0
\(244\) 438573. 0.471593
\(245\) 0 0
\(246\) 0 0
\(247\) −171111. −0.178457
\(248\) 426997. 0.440855
\(249\) 0 0
\(250\) 915875. 0.926801
\(251\) 385591. 0.386316 0.193158 0.981168i \(-0.438127\pi\)
0.193158 + 0.981168i \(0.438127\pi\)
\(252\) 0 0
\(253\) −31433.2 −0.0308736
\(254\) 1.85616e6 1.80523
\(255\) 0 0
\(256\) −2.11484e6 −2.01687
\(257\) 441809. 0.417255 0.208627 0.977995i \(-0.433100\pi\)
0.208627 + 0.977995i \(0.433100\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.31506e6 −3.95870
\(261\) 0 0
\(262\) −1.24156e6 −1.11742
\(263\) 600264. 0.535122 0.267561 0.963541i \(-0.413782\pi\)
0.267561 + 0.963541i \(0.413782\pi\)
\(264\) 0 0
\(265\) −1.80632e6 −1.58009
\(266\) 0 0
\(267\) 0 0
\(268\) −1.62370e6 −1.38092
\(269\) −325609. −0.274357 −0.137178 0.990546i \(-0.543803\pi\)
−0.137178 + 0.990546i \(0.543803\pi\)
\(270\) 0 0
\(271\) 1.75592e6 1.45239 0.726193 0.687491i \(-0.241288\pi\)
0.726193 + 0.687491i \(0.241288\pi\)
\(272\) −2.00857e6 −1.64613
\(273\) 0 0
\(274\) 996090. 0.801534
\(275\) 1.33428e6 1.06394
\(276\) 0 0
\(277\) −342363. −0.268094 −0.134047 0.990975i \(-0.542797\pi\)
−0.134047 + 0.990975i \(0.542797\pi\)
\(278\) 2.54707e6 1.97664
\(279\) 0 0
\(280\) 0 0
\(281\) −930671. −0.703122 −0.351561 0.936165i \(-0.614349\pi\)
−0.351561 + 0.936165i \(0.614349\pi\)
\(282\) 0 0
\(283\) −1.64760e6 −1.22288 −0.611442 0.791290i \(-0.709410\pi\)
−0.611442 + 0.791290i \(0.709410\pi\)
\(284\) 3.04623e6 2.24113
\(285\) 0 0
\(286\) 6.58582e6 4.76096
\(287\) 0 0
\(288\) 0 0
\(289\) 518891. 0.365453
\(290\) −5.45467e6 −3.80867
\(291\) 0 0
\(292\) 2.69993e6 1.85309
\(293\) 1.14378e6 0.778347 0.389174 0.921164i \(-0.372761\pi\)
0.389174 + 0.921164i \(0.372761\pi\)
\(294\) 0 0
\(295\) −1.86082e6 −1.24495
\(296\) −3.36344e6 −2.23128
\(297\) 0 0
\(298\) −4.27640e6 −2.78957
\(299\) 38664.1 0.0250109
\(300\) 0 0
\(301\) 0 0
\(302\) 1.50605e6 0.950218
\(303\) 0 0
\(304\) 274367. 0.170274
\(305\) 452553. 0.278561
\(306\) 0 0
\(307\) 2.98831e6 1.80959 0.904795 0.425848i \(-0.140024\pi\)
0.904795 + 0.425848i \(0.140024\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 830335. 0.490738
\(311\) −2.83751e6 −1.66355 −0.831775 0.555113i \(-0.812675\pi\)
−0.831775 + 0.555113i \(0.812675\pi\)
\(312\) 0 0
\(313\) −2.14749e6 −1.23899 −0.619497 0.784999i \(-0.712664\pi\)
−0.619497 + 0.784999i \(0.712664\pi\)
\(314\) 1.65979e6 0.950010
\(315\) 0 0
\(316\) 1.57648e6 0.888116
\(317\) 2.63357e6 1.47196 0.735982 0.677001i \(-0.236721\pi\)
0.735982 + 0.677001i \(0.236721\pi\)
\(318\) 0 0
\(319\) 5.66583e6 3.11736
\(320\) −1.24003e6 −0.676953
\(321\) 0 0
\(322\) 0 0
\(323\) −264829. −0.141241
\(324\) 0 0
\(325\) −1.64122e6 −0.861902
\(326\) −958033. −0.499271
\(327\) 0 0
\(328\) −4.87148e6 −2.50021
\(329\) 0 0
\(330\) 0 0
\(331\) 3.28434e6 1.64770 0.823849 0.566809i \(-0.191822\pi\)
0.823849 + 0.566809i \(0.191822\pi\)
\(332\) −1.17724e6 −0.586164
\(333\) 0 0
\(334\) −4.74631e6 −2.32804
\(335\) −1.67546e6 −0.815685
\(336\) 0 0
\(337\) −238202. −0.114254 −0.0571270 0.998367i \(-0.518194\pi\)
−0.0571270 + 0.998367i \(0.518194\pi\)
\(338\) −4.38457e6 −2.08754
\(339\) 0 0
\(340\) −6.67846e6 −3.13313
\(341\) −862479. −0.401664
\(342\) 0 0
\(343\) 0 0
\(344\) −2.18480e6 −0.995441
\(345\) 0 0
\(346\) −2.59107e6 −1.16356
\(347\) −1.06436e6 −0.474532 −0.237266 0.971445i \(-0.576251\pi\)
−0.237266 + 0.971445i \(0.576251\pi\)
\(348\) 0 0
\(349\) −1.85940e6 −0.817163 −0.408582 0.912722i \(-0.633977\pi\)
−0.408582 + 0.912722i \(0.633977\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.08517e6 −0.896983
\(353\) −1.66487e6 −0.711123 −0.355561 0.934653i \(-0.615710\pi\)
−0.355561 + 0.934653i \(0.615710\pi\)
\(354\) 0 0
\(355\) 3.14334e6 1.32379
\(356\) 3.40411e6 1.42357
\(357\) 0 0
\(358\) −5.42660e6 −2.23779
\(359\) −2.67457e6 −1.09526 −0.547632 0.836720i \(-0.684470\pi\)
−0.547632 + 0.836720i \(0.684470\pi\)
\(360\) 0 0
\(361\) −2.43992e6 −0.985390
\(362\) 6.54511e6 2.62510
\(363\) 0 0
\(364\) 0 0
\(365\) 2.78600e6 1.09458
\(366\) 0 0
\(367\) 2.14384e6 0.830860 0.415430 0.909625i \(-0.363631\pi\)
0.415430 + 0.909625i \(0.363631\pi\)
\(368\) −61995.8 −0.0238640
\(369\) 0 0
\(370\) −6.54051e6 −2.48375
\(371\) 0 0
\(372\) 0 0
\(373\) −759867. −0.282791 −0.141395 0.989953i \(-0.545159\pi\)
−0.141395 + 0.989953i \(0.545159\pi\)
\(374\) 1.01929e7 3.76808
\(375\) 0 0
\(376\) 1.17500e6 0.428614
\(377\) −6.96918e6 −2.52539
\(378\) 0 0
\(379\) 3.03679e6 1.08597 0.542984 0.839743i \(-0.317294\pi\)
0.542984 + 0.839743i \(0.317294\pi\)
\(380\) 912265. 0.324087
\(381\) 0 0
\(382\) 7.70220e6 2.70057
\(383\) 2.50929e6 0.874086 0.437043 0.899441i \(-0.356026\pi\)
0.437043 + 0.899441i \(0.356026\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.14954e6 −2.78397
\(387\) 0 0
\(388\) 1.14982e6 0.387749
\(389\) 985137. 0.330082 0.165041 0.986287i \(-0.447224\pi\)
0.165041 + 0.986287i \(0.447224\pi\)
\(390\) 0 0
\(391\) 59840.7 0.0197950
\(392\) 0 0
\(393\) 0 0
\(394\) −4.44082e6 −1.44119
\(395\) 1.62673e6 0.524593
\(396\) 0 0
\(397\) 2.90510e6 0.925092 0.462546 0.886595i \(-0.346936\pi\)
0.462546 + 0.886595i \(0.346936\pi\)
\(398\) 6.19491e6 1.96032
\(399\) 0 0
\(400\) 2.63161e6 0.822377
\(401\) 4.23124e6 1.31404 0.657018 0.753875i \(-0.271818\pi\)
0.657018 + 0.753875i \(0.271818\pi\)
\(402\) 0 0
\(403\) 1.06088e6 0.325390
\(404\) 7.18927e6 2.19145
\(405\) 0 0
\(406\) 0 0
\(407\) 6.79370e6 2.03292
\(408\) 0 0
\(409\) 4.13890e6 1.22342 0.611711 0.791081i \(-0.290482\pi\)
0.611711 + 0.791081i \(0.290482\pi\)
\(410\) −9.47304e6 −2.78311
\(411\) 0 0
\(412\) 1.38782e6 0.402799
\(413\) 0 0
\(414\) 0 0
\(415\) −1.21477e6 −0.346236
\(416\) 2.56483e6 0.726651
\(417\) 0 0
\(418\) −1.39234e6 −0.389766
\(419\) −5.99852e6 −1.66920 −0.834601 0.550855i \(-0.814302\pi\)
−0.834601 + 0.550855i \(0.814302\pi\)
\(420\) 0 0
\(421\) −1.00759e6 −0.277062 −0.138531 0.990358i \(-0.544238\pi\)
−0.138531 + 0.990358i \(0.544238\pi\)
\(422\) −7.70489e6 −2.10613
\(423\) 0 0
\(424\) 9.29722e6 2.51153
\(425\) −2.54013e6 −0.682156
\(426\) 0 0
\(427\) 0 0
\(428\) 257401. 0.0679205
\(429\) 0 0
\(430\) −4.24854e6 −1.10807
\(431\) −303149. −0.0786073 −0.0393037 0.999227i \(-0.512514\pi\)
−0.0393037 + 0.999227i \(0.512514\pi\)
\(432\) 0 0
\(433\) 3.97130e6 1.01792 0.508960 0.860790i \(-0.330030\pi\)
0.508960 + 0.860790i \(0.330030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 582033. 0.146633
\(437\) −8174.14 −0.00204757
\(438\) 0 0
\(439\) 3.74799e6 0.928192 0.464096 0.885785i \(-0.346379\pi\)
0.464096 + 0.885785i \(0.346379\pi\)
\(440\) −1.86318e7 −4.58798
\(441\) 0 0
\(442\) −1.25377e7 −3.05255
\(443\) 1.31077e6 0.317334 0.158667 0.987332i \(-0.449280\pi\)
0.158667 + 0.987332i \(0.449280\pi\)
\(444\) 0 0
\(445\) 3.51262e6 0.840874
\(446\) −1.45794e7 −3.47057
\(447\) 0 0
\(448\) 0 0
\(449\) −6.31311e6 −1.47784 −0.738920 0.673793i \(-0.764664\pi\)
−0.738920 + 0.673793i \(0.764664\pi\)
\(450\) 0 0
\(451\) 9.83976e6 2.27794
\(452\) 6.87704e6 1.58327
\(453\) 0 0
\(454\) −9.50750e6 −2.16484
\(455\) 0 0
\(456\) 0 0
\(457\) −583476. −0.130687 −0.0653435 0.997863i \(-0.520814\pi\)
−0.0653435 + 0.997863i \(0.520814\pi\)
\(458\) −3.12561e6 −0.696259
\(459\) 0 0
\(460\) −206135. −0.0454210
\(461\) 2.93381e6 0.642953 0.321476 0.946918i \(-0.395821\pi\)
0.321476 + 0.946918i \(0.395821\pi\)
\(462\) 0 0
\(463\) 3.76716e6 0.816699 0.408349 0.912826i \(-0.366105\pi\)
0.408349 + 0.912826i \(0.366105\pi\)
\(464\) 1.11747e7 2.40958
\(465\) 0 0
\(466\) 3.97258e6 0.847438
\(467\) 2.90596e6 0.616592 0.308296 0.951290i \(-0.400241\pi\)
0.308296 + 0.951290i \(0.400241\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 2.28489e6 0.477111
\(471\) 0 0
\(472\) 9.57774e6 1.97883
\(473\) 4.41301e6 0.906947
\(474\) 0 0
\(475\) 346977. 0.0705613
\(476\) 0 0
\(477\) 0 0
\(478\) −871107. −0.174382
\(479\) 7.06622e6 1.40718 0.703588 0.710608i \(-0.251580\pi\)
0.703588 + 0.710608i \(0.251580\pi\)
\(480\) 0 0
\(481\) −8.35651e6 −1.64688
\(482\) 1.57313e7 3.08423
\(483\) 0 0
\(484\) 2.54910e7 4.94622
\(485\) 1.18647e6 0.229036
\(486\) 0 0
\(487\) 2.72981e6 0.521566 0.260783 0.965397i \(-0.416019\pi\)
0.260783 + 0.965397i \(0.416019\pi\)
\(488\) −2.32931e6 −0.442770
\(489\) 0 0
\(490\) 0 0
\(491\) 1.02859e7 1.92548 0.962740 0.270428i \(-0.0871651\pi\)
0.962740 + 0.270428i \(0.0871651\pi\)
\(492\) 0 0
\(493\) −1.07863e7 −1.99873
\(494\) 1.71263e6 0.315751
\(495\) 0 0
\(496\) −1.70107e6 −0.310469
\(497\) 0 0
\(498\) 0 0
\(499\) −3.63591e6 −0.653674 −0.326837 0.945081i \(-0.605983\pi\)
−0.326837 + 0.945081i \(0.605983\pi\)
\(500\) −6.23870e6 −1.11601
\(501\) 0 0
\(502\) −3.85934e6 −0.683524
\(503\) 1.65499e6 0.291658 0.145829 0.989310i \(-0.453415\pi\)
0.145829 + 0.989310i \(0.453415\pi\)
\(504\) 0 0
\(505\) 7.41845e6 1.29445
\(506\) 314612. 0.0546259
\(507\) 0 0
\(508\) −1.26437e7 −2.17377
\(509\) −731110. −0.125080 −0.0625401 0.998042i \(-0.519920\pi\)
−0.0625401 + 0.998042i \(0.519920\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.26024e7 2.12460
\(513\) 0 0
\(514\) −4.42202e6 −0.738265
\(515\) 1.43206e6 0.237926
\(516\) 0 0
\(517\) −2.37334e6 −0.390511
\(518\) 0 0
\(519\) 0 0
\(520\) 2.29178e7 3.71675
\(521\) 4.73095e6 0.763579 0.381790 0.924249i \(-0.375308\pi\)
0.381790 + 0.924249i \(0.375308\pi\)
\(522\) 0 0
\(523\) 9.72563e6 1.55476 0.777380 0.629031i \(-0.216548\pi\)
0.777380 + 0.629031i \(0.216548\pi\)
\(524\) 8.45719e6 1.34554
\(525\) 0 0
\(526\) −6.00798e6 −0.946812
\(527\) 1.64194e6 0.257531
\(528\) 0 0
\(529\) −6.43450e6 −0.999713
\(530\) 1.80793e7 2.79571
\(531\) 0 0
\(532\) 0 0
\(533\) −1.21033e7 −1.84538
\(534\) 0 0
\(535\) 265606. 0.0401193
\(536\) 8.62367e6 1.29652
\(537\) 0 0
\(538\) 3.25898e6 0.485430
\(539\) 0 0
\(540\) 0 0
\(541\) 8.19002e6 1.20307 0.601536 0.798845i \(-0.294556\pi\)
0.601536 + 0.798845i \(0.294556\pi\)
\(542\) −1.75748e7 −2.56976
\(543\) 0 0
\(544\) 3.96962e6 0.575111
\(545\) 600587. 0.0866133
\(546\) 0 0
\(547\) −4.17136e6 −0.596087 −0.298043 0.954552i \(-0.596334\pi\)
−0.298043 + 0.954552i \(0.596334\pi\)
\(548\) −6.78510e6 −0.965172
\(549\) 0 0
\(550\) −1.33547e7 −1.88247
\(551\) 1.47339e6 0.206746
\(552\) 0 0
\(553\) 0 0
\(554\) 3.42668e6 0.474350
\(555\) 0 0
\(556\) −1.73500e7 −2.38019
\(557\) 1.26658e6 0.172980 0.0864899 0.996253i \(-0.472435\pi\)
0.0864899 + 0.996253i \(0.472435\pi\)
\(558\) 0 0
\(559\) −5.42817e6 −0.734723
\(560\) 0 0
\(561\) 0 0
\(562\) 9.31499e6 1.24406
\(563\) −6.37431e6 −0.847544 −0.423772 0.905769i \(-0.639294\pi\)
−0.423772 + 0.905769i \(0.639294\pi\)
\(564\) 0 0
\(565\) 7.09626e6 0.935208
\(566\) 1.64906e7 2.16369
\(567\) 0 0
\(568\) −1.61789e7 −2.10416
\(569\) −7.64040e6 −0.989317 −0.494659 0.869087i \(-0.664707\pi\)
−0.494659 + 0.869087i \(0.664707\pi\)
\(570\) 0 0
\(571\) 790914. 0.101517 0.0507585 0.998711i \(-0.483836\pi\)
0.0507585 + 0.998711i \(0.483836\pi\)
\(572\) −4.48608e7 −5.73294
\(573\) 0 0
\(574\) 0 0
\(575\) −78402.8 −0.00988922
\(576\) 0 0
\(577\) 8.55670e6 1.06996 0.534979 0.844865i \(-0.320320\pi\)
0.534979 + 0.844865i \(0.320320\pi\)
\(578\) −5.19352e6 −0.646610
\(579\) 0 0
\(580\) 3.71558e7 4.58623
\(581\) 0 0
\(582\) 0 0
\(583\) −1.87792e7 −2.28826
\(584\) −1.43396e7 −1.73983
\(585\) 0 0
\(586\) −1.14480e7 −1.37716
\(587\) −1.17153e7 −1.40333 −0.701663 0.712509i \(-0.747559\pi\)
−0.701663 + 0.712509i \(0.747559\pi\)
\(588\) 0 0
\(589\) −224286. −0.0266387
\(590\) 1.86248e7 2.20273
\(591\) 0 0
\(592\) 1.33992e7 1.57136
\(593\) 1.55617e7 1.81728 0.908639 0.417583i \(-0.137123\pi\)
0.908639 + 0.417583i \(0.137123\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.91297e7 3.35908
\(597\) 0 0
\(598\) −386984. −0.0442528
\(599\) 9.18930e6 1.04644 0.523221 0.852197i \(-0.324730\pi\)
0.523221 + 0.852197i \(0.324730\pi\)
\(600\) 0 0
\(601\) −1.61226e7 −1.82075 −0.910373 0.413789i \(-0.864205\pi\)
−0.910373 + 0.413789i \(0.864205\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.02588e7 −1.14421
\(605\) 2.63036e7 2.92164
\(606\) 0 0
\(607\) −1.04056e7 −1.14629 −0.573144 0.819455i \(-0.694276\pi\)
−0.573144 + 0.819455i \(0.694276\pi\)
\(608\) −542243. −0.0594887
\(609\) 0 0
\(610\) −4.52956e6 −0.492868
\(611\) 2.91929e6 0.316355
\(612\) 0 0
\(613\) −1.13921e7 −1.22449 −0.612243 0.790670i \(-0.709732\pi\)
−0.612243 + 0.790670i \(0.709732\pi\)
\(614\) −2.99097e7 −3.20178
\(615\) 0 0
\(616\) 0 0
\(617\) −6.20517e6 −0.656206 −0.328103 0.944642i \(-0.606409\pi\)
−0.328103 + 0.944642i \(0.606409\pi\)
\(618\) 0 0
\(619\) 9.85608e6 1.03390 0.516949 0.856016i \(-0.327068\pi\)
0.516949 + 0.856016i \(0.327068\pi\)
\(620\) −5.65602e6 −0.590925
\(621\) 0 0
\(622\) 2.84003e7 2.94338
\(623\) 0 0
\(624\) 0 0
\(625\) −1.21385e7 −1.24298
\(626\) 2.14940e7 2.19220
\(627\) 0 0
\(628\) −1.13060e7 −1.14396
\(629\) −1.29335e7 −1.30343
\(630\) 0 0
\(631\) 7.26873e6 0.726750 0.363375 0.931643i \(-0.381624\pi\)
0.363375 + 0.931643i \(0.381624\pi\)
\(632\) −8.37284e6 −0.833835
\(633\) 0 0
\(634\) −2.63591e7 −2.60440
\(635\) −1.30467e7 −1.28401
\(636\) 0 0
\(637\) 0 0
\(638\) −5.67087e7 −5.51566
\(639\) 0 0
\(640\) 1.88295e7 1.81714
\(641\) −8.30119e6 −0.797986 −0.398993 0.916954i \(-0.630640\pi\)
−0.398993 + 0.916954i \(0.630640\pi\)
\(642\) 0 0
\(643\) 4.14322e6 0.395194 0.197597 0.980283i \(-0.436686\pi\)
0.197597 + 0.980283i \(0.436686\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.65065e6 0.249903
\(647\) −1.47906e7 −1.38907 −0.694535 0.719459i \(-0.744390\pi\)
−0.694535 + 0.719459i \(0.744390\pi\)
\(648\) 0 0
\(649\) −1.93458e7 −1.80291
\(650\) 1.64268e7 1.52500
\(651\) 0 0
\(652\) 6.52587e6 0.601200
\(653\) 6.08931e6 0.558837 0.279419 0.960169i \(-0.409858\pi\)
0.279419 + 0.960169i \(0.409858\pi\)
\(654\) 0 0
\(655\) 8.72679e6 0.794788
\(656\) 1.94070e7 1.76075
\(657\) 0 0
\(658\) 0 0
\(659\) −4.14447e6 −0.371754 −0.185877 0.982573i \(-0.559513\pi\)
−0.185877 + 0.982573i \(0.559513\pi\)
\(660\) 0 0
\(661\) −1.80378e7 −1.60576 −0.802881 0.596140i \(-0.796700\pi\)
−0.802881 + 0.596140i \(0.796700\pi\)
\(662\) −3.28726e7 −2.91534
\(663\) 0 0
\(664\) 6.25245e6 0.550339
\(665\) 0 0
\(666\) 0 0
\(667\) −332926. −0.0289756
\(668\) 3.23306e7 2.80332
\(669\) 0 0
\(670\) 1.67695e7 1.44322
\(671\) 4.70490e6 0.403408
\(672\) 0 0
\(673\) 5.62885e6 0.479051 0.239526 0.970890i \(-0.423008\pi\)
0.239526 + 0.970890i \(0.423008\pi\)
\(674\) 2.38414e6 0.202154
\(675\) 0 0
\(676\) 2.98666e7 2.51373
\(677\) −1.50877e7 −1.26518 −0.632588 0.774488i \(-0.718007\pi\)
−0.632588 + 0.774488i \(0.718007\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.54700e7 2.94164
\(681\) 0 0
\(682\) 8.63246e6 0.710679
\(683\) −7.20414e6 −0.590922 −0.295461 0.955355i \(-0.595473\pi\)
−0.295461 + 0.955355i \(0.595473\pi\)
\(684\) 0 0
\(685\) −7.00139e6 −0.570109
\(686\) 0 0
\(687\) 0 0
\(688\) 8.70378e6 0.701031
\(689\) 2.30991e7 1.85373
\(690\) 0 0
\(691\) −9.08625e6 −0.723918 −0.361959 0.932194i \(-0.617892\pi\)
−0.361959 + 0.932194i \(0.617892\pi\)
\(692\) 1.76497e7 1.40111
\(693\) 0 0
\(694\) 1.06531e7 0.839607
\(695\) −1.79030e7 −1.40593
\(696\) 0 0
\(697\) −1.87324e7 −1.46053
\(698\) 1.86105e7 1.44584
\(699\) 0 0
\(700\) 0 0
\(701\) 1.41913e7 1.09075 0.545376 0.838192i \(-0.316387\pi\)
0.545376 + 0.838192i \(0.316387\pi\)
\(702\) 0 0
\(703\) 1.76669e6 0.134825
\(704\) −1.28918e7 −0.980354
\(705\) 0 0
\(706\) 1.66635e7 1.25822
\(707\) 0 0
\(708\) 0 0
\(709\) −2.21070e6 −0.165164 −0.0825819 0.996584i \(-0.526317\pi\)
−0.0825819 + 0.996584i \(0.526317\pi\)
\(710\) −3.14613e7 −2.34224
\(711\) 0 0
\(712\) −1.80796e7 −1.33656
\(713\) 50679.5 0.00373343
\(714\) 0 0
\(715\) −4.62909e7 −3.38634
\(716\) 3.69645e7 2.69465
\(717\) 0 0
\(718\) 2.67695e7 1.93789
\(719\) −2.99082e6 −0.215758 −0.107879 0.994164i \(-0.534406\pi\)
−0.107879 + 0.994164i \(0.534406\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.44209e7 1.74349
\(723\) 0 0
\(724\) −4.45836e7 −3.16103
\(725\) 1.41321e7 0.998530
\(726\) 0 0
\(727\) −8.38066e6 −0.588088 −0.294044 0.955792i \(-0.595001\pi\)
−0.294044 + 0.955792i \(0.595001\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.78847e7 −1.93669
\(731\) −8.40123e6 −0.581499
\(732\) 0 0
\(733\) −7.70245e6 −0.529504 −0.264752 0.964317i \(-0.585290\pi\)
−0.264752 + 0.964317i \(0.585290\pi\)
\(734\) −2.14575e7 −1.47007
\(735\) 0 0
\(736\) 122525. 0.00833739
\(737\) −1.74187e7 −1.18126
\(738\) 0 0
\(739\) −7.59899e6 −0.511852 −0.255926 0.966696i \(-0.582380\pi\)
−0.255926 + 0.966696i \(0.582380\pi\)
\(740\) 4.45522e7 2.99082
\(741\) 0 0
\(742\) 0 0
\(743\) 2.40124e7 1.59574 0.797871 0.602828i \(-0.205959\pi\)
0.797871 + 0.602828i \(0.205959\pi\)
\(744\) 0 0
\(745\) 3.00583e7 1.98414
\(746\) 7.60543e6 0.500353
\(747\) 0 0
\(748\) −6.94316e7 −4.53736
\(749\) 0 0
\(750\) 0 0
\(751\) 1.69479e7 1.09652 0.548259 0.836308i \(-0.315291\pi\)
0.548259 + 0.836308i \(0.315291\pi\)
\(752\) −4.68094e6 −0.301848
\(753\) 0 0
\(754\) 6.97538e7 4.46827
\(755\) −1.05859e7 −0.675863
\(756\) 0 0
\(757\) 4.56402e6 0.289473 0.144736 0.989470i \(-0.453767\pi\)
0.144736 + 0.989470i \(0.453767\pi\)
\(758\) −3.03949e7 −1.92145
\(759\) 0 0
\(760\) −4.84514e6 −0.304279
\(761\) 4.74018e6 0.296711 0.148355 0.988934i \(-0.452602\pi\)
0.148355 + 0.988934i \(0.452602\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.24653e7 −3.25191
\(765\) 0 0
\(766\) −2.51152e7 −1.54655
\(767\) 2.37960e7 1.46055
\(768\) 0 0
\(769\) −1.39469e7 −0.850476 −0.425238 0.905082i \(-0.639810\pi\)
−0.425238 + 0.905082i \(0.639810\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.55125e7 3.35234
\(773\) −1.49561e7 −0.900264 −0.450132 0.892962i \(-0.648623\pi\)
−0.450132 + 0.892962i \(0.648623\pi\)
\(774\) 0 0
\(775\) −2.15125e6 −0.128658
\(776\) −6.10683e6 −0.364050
\(777\) 0 0
\(778\) −9.86012e6 −0.584028
\(779\) 2.55881e6 0.151075
\(780\) 0 0
\(781\) 3.26793e7 1.91710
\(782\) −598940. −0.0350240
\(783\) 0 0
\(784\) 0 0
\(785\) −1.16664e7 −0.675716
\(786\) 0 0
\(787\) 2.18273e7 1.25621 0.628105 0.778128i \(-0.283831\pi\)
0.628105 + 0.778128i \(0.283831\pi\)
\(788\) 3.02497e7 1.73542
\(789\) 0 0
\(790\) −1.62818e7 −0.928183
\(791\) 0 0
\(792\) 0 0
\(793\) −5.78721e6 −0.326803
\(794\) −2.90768e7 −1.63680
\(795\) 0 0
\(796\) −4.21981e7 −2.36053
\(797\) 8.79229e6 0.490294 0.245147 0.969486i \(-0.421164\pi\)
0.245147 + 0.969486i \(0.421164\pi\)
\(798\) 0 0
\(799\) 4.51822e6 0.250381
\(800\) −5.20096e6 −0.287315
\(801\) 0 0
\(802\) −4.23501e7 −2.32497
\(803\) 2.89642e7 1.58516
\(804\) 0 0
\(805\) 0 0
\(806\) −1.06183e7 −0.575725
\(807\) 0 0
\(808\) −3.81831e7 −2.05751
\(809\) 2.87451e6 0.154416 0.0772081 0.997015i \(-0.475399\pi\)
0.0772081 + 0.997015i \(0.475399\pi\)
\(810\) 0 0
\(811\) −660356. −0.0352554 −0.0176277 0.999845i \(-0.505611\pi\)
−0.0176277 + 0.999845i \(0.505611\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.79974e7 −3.59693
\(815\) 6.73389e6 0.355118
\(816\) 0 0
\(817\) 1.14759e6 0.0601496
\(818\) −4.14258e7 −2.16465
\(819\) 0 0
\(820\) 6.45278e7 3.35129
\(821\) 3.05738e7 1.58304 0.791519 0.611144i \(-0.209290\pi\)
0.791519 + 0.611144i \(0.209290\pi\)
\(822\) 0 0
\(823\) 1.11281e7 0.572693 0.286346 0.958126i \(-0.407559\pi\)
0.286346 + 0.958126i \(0.407559\pi\)
\(824\) −7.37085e6 −0.378181
\(825\) 0 0
\(826\) 0 0
\(827\) 2.78304e7 1.41500 0.707500 0.706714i \(-0.249823\pi\)
0.707500 + 0.706714i \(0.249823\pi\)
\(828\) 0 0
\(829\) 7.06133e6 0.356862 0.178431 0.983952i \(-0.442898\pi\)
0.178431 + 0.983952i \(0.442898\pi\)
\(830\) 1.21585e7 0.612609
\(831\) 0 0
\(832\) 1.58574e7 0.794190
\(833\) 0 0
\(834\) 0 0
\(835\) 3.33612e7 1.65587
\(836\) 9.48422e6 0.469338
\(837\) 0 0
\(838\) 6.00385e7 2.95338
\(839\) 2.27299e7 1.11479 0.557395 0.830247i \(-0.311801\pi\)
0.557395 + 0.830247i \(0.311801\pi\)
\(840\) 0 0
\(841\) 3.94986e7 1.92571
\(842\) 1.00848e7 0.490217
\(843\) 0 0
\(844\) 5.24837e7 2.53611
\(845\) 3.08186e7 1.48481
\(846\) 0 0
\(847\) 0 0
\(848\) −3.70382e7 −1.76872
\(849\) 0 0
\(850\) 2.54239e7 1.20696
\(851\) −399200. −0.0188958
\(852\) 0 0
\(853\) −7.94309e6 −0.373781 −0.186890 0.982381i \(-0.559841\pi\)
−0.186890 + 0.982381i \(0.559841\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.36709e6 −0.0637693
\(857\) 9.91766e6 0.461272 0.230636 0.973040i \(-0.425919\pi\)
0.230636 + 0.973040i \(0.425919\pi\)
\(858\) 0 0
\(859\) −61754.4 −0.00285552 −0.00142776 0.999999i \(-0.500454\pi\)
−0.00142776 + 0.999999i \(0.500454\pi\)
\(860\) 2.89399e7 1.33429
\(861\) 0 0
\(862\) 3.03419e6 0.139083
\(863\) −3.99007e7 −1.82370 −0.911851 0.410522i \(-0.865347\pi\)
−0.911851 + 0.410522i \(0.865347\pi\)
\(864\) 0 0
\(865\) 1.82123e7 0.827607
\(866\) −3.97484e7 −1.80104
\(867\) 0 0
\(868\) 0 0
\(869\) 1.69120e7 0.759708
\(870\) 0 0
\(871\) 2.14256e7 0.956949
\(872\) −3.09125e6 −0.137671
\(873\) 0 0
\(874\) 81814.0 0.00362284
\(875\) 0 0
\(876\) 0 0
\(877\) −1.89184e7 −0.830586 −0.415293 0.909688i \(-0.636321\pi\)
−0.415293 + 0.909688i \(0.636321\pi\)
\(878\) −3.75133e7 −1.64229
\(879\) 0 0
\(880\) 7.42250e7 3.23105
\(881\) −9.12061e6 −0.395899 −0.197949 0.980212i \(-0.563428\pi\)
−0.197949 + 0.980212i \(0.563428\pi\)
\(882\) 0 0
\(883\) −4.49665e7 −1.94083 −0.970414 0.241449i \(-0.922377\pi\)
−0.970414 + 0.241449i \(0.922377\pi\)
\(884\) 8.54035e7 3.67574
\(885\) 0 0
\(886\) −1.31193e7 −0.561472
\(887\) −2.82985e7 −1.20769 −0.603845 0.797102i \(-0.706365\pi\)
−0.603845 + 0.797102i \(0.706365\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −3.51574e7 −1.48779
\(891\) 0 0
\(892\) 9.93107e7 4.17911
\(893\) −617181. −0.0258990
\(894\) 0 0
\(895\) 3.81429e7 1.59168
\(896\) 0 0
\(897\) 0 0
\(898\) 6.31873e7 2.61480
\(899\) −9.13496e6 −0.376971
\(900\) 0 0
\(901\) 3.57507e7 1.46714
\(902\) −9.84850e7 −4.03045
\(903\) 0 0
\(904\) −3.65247e7 −1.48650
\(905\) −4.60048e7 −1.86716
\(906\) 0 0
\(907\) −768537. −0.0310204 −0.0155102 0.999880i \(-0.504937\pi\)
−0.0155102 + 0.999880i \(0.504937\pi\)
\(908\) 6.47626e7 2.60681
\(909\) 0 0
\(910\) 0 0
\(911\) −3.45594e6 −0.137965 −0.0689827 0.997618i \(-0.521975\pi\)
−0.0689827 + 0.997618i \(0.521975\pi\)
\(912\) 0 0
\(913\) −1.26291e7 −0.501414
\(914\) 5.83994e6 0.231229
\(915\) 0 0
\(916\) 2.12908e7 0.838404
\(917\) 0 0
\(918\) 0 0
\(919\) 5.65327e6 0.220806 0.110403 0.993887i \(-0.464786\pi\)
0.110403 + 0.993887i \(0.464786\pi\)
\(920\) 1.09481e6 0.0426450
\(921\) 0 0
\(922\) −2.93641e7 −1.13760
\(923\) −4.01967e7 −1.55305
\(924\) 0 0
\(925\) 1.69453e7 0.651171
\(926\) −3.77051e7 −1.44502
\(927\) 0 0
\(928\) −2.20851e7 −0.841839
\(929\) −2.42253e7 −0.920937 −0.460469 0.887676i \(-0.652319\pi\)
−0.460469 + 0.887676i \(0.652319\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.70602e7 −1.02045
\(933\) 0 0
\(934\) −2.90855e7 −1.09096
\(935\) −7.16448e7 −2.68013
\(936\) 0 0
\(937\) −3.02477e7 −1.12549 −0.562746 0.826630i \(-0.690255\pi\)
−0.562746 + 0.826630i \(0.690255\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.55640e7 −0.574517
\(941\) 1.38949e7 0.511541 0.255770 0.966738i \(-0.417671\pi\)
0.255770 + 0.966738i \(0.417671\pi\)
\(942\) 0 0
\(943\) −578187. −0.0211733
\(944\) −3.81557e7 −1.39357
\(945\) 0 0
\(946\) −4.41693e7 −1.60470
\(947\) 1.78680e7 0.647441 0.323721 0.946153i \(-0.395066\pi\)
0.323721 + 0.946153i \(0.395066\pi\)
\(948\) 0 0
\(949\) −3.56271e7 −1.28415
\(950\) −3.47286e6 −0.124847
\(951\) 0 0
\(952\) 0 0
\(953\) −1.37348e7 −0.489880 −0.244940 0.969538i \(-0.578768\pi\)
−0.244940 + 0.969538i \(0.578768\pi\)
\(954\) 0 0
\(955\) −5.41378e7 −1.92084
\(956\) 5.93375e6 0.209983
\(957\) 0 0
\(958\) −7.07250e7 −2.48977
\(959\) 0 0
\(960\) 0 0
\(961\) −2.72386e7 −0.951428
\(962\) 8.36394e7 2.91389
\(963\) 0 0
\(964\) −1.07157e8 −3.71389
\(965\) 5.72821e7 1.98016
\(966\) 0 0
\(967\) −7.63195e6 −0.262464 −0.131232 0.991352i \(-0.541893\pi\)
−0.131232 + 0.991352i \(0.541893\pi\)
\(968\) −1.35386e8 −4.64391
\(969\) 0 0
\(970\) −1.18753e7 −0.405242
\(971\) −2.22480e7 −0.757257 −0.378629 0.925549i \(-0.623604\pi\)
−0.378629 + 0.925549i \(0.623604\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.73223e7 −0.922827
\(975\) 0 0
\(976\) 9.27949e6 0.311817
\(977\) 1.82385e7 0.611297 0.305648 0.952144i \(-0.401127\pi\)
0.305648 + 0.952144i \(0.401127\pi\)
\(978\) 0 0
\(979\) 3.65184e7 1.21774
\(980\) 0 0
\(981\) 0 0
\(982\) −1.02951e8 −3.40683
\(983\) −3.84459e7 −1.26902 −0.634508 0.772917i \(-0.718797\pi\)
−0.634508 + 0.772917i \(0.718797\pi\)
\(984\) 0 0
\(985\) 3.12140e7 1.02508
\(986\) 1.07959e8 3.53643
\(987\) 0 0
\(988\) −1.16660e7 −0.380214
\(989\) −259309. −0.00843000
\(990\) 0 0
\(991\) −2.49658e7 −0.807533 −0.403767 0.914862i \(-0.632299\pi\)
−0.403767 + 0.914862i \(0.632299\pi\)
\(992\) 3.36189e6 0.108469
\(993\) 0 0
\(994\) 0 0
\(995\) −4.35433e7 −1.39432
\(996\) 0 0
\(997\) −3.22763e7 −1.02836 −0.514181 0.857682i \(-0.671904\pi\)
−0.514181 + 0.857682i \(0.671904\pi\)
\(998\) 3.63914e7 1.15657
\(999\) 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.1.1 6
3.2 odd 2 147.6.a.n.1.6 6
7.6 odd 2 441.6.a.ba.1.1 6
21.2 odd 6 147.6.e.q.67.1 12
21.5 even 6 147.6.e.p.67.1 12
21.11 odd 6 147.6.e.q.79.1 12
21.17 even 6 147.6.e.p.79.1 12
21.20 even 2 147.6.a.o.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.6 6 3.2 odd 2
147.6.a.o.1.6 yes 6 21.20 even 2
147.6.e.p.67.1 12 21.5 even 6
147.6.e.p.79.1 12 21.17 even 6
147.6.e.q.67.1 12 21.2 odd 6
147.6.e.q.79.1 12 21.11 odd 6
441.6.a.ba.1.1 6 7.6 odd 2
441.6.a.bb.1.1 6 1.1 even 1 trivial