Properties

Label 441.6.a.s.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: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7.38987\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.38987 q^{2} +8.83040 q^{4} +38.7291 q^{5} +148.051 q^{8} +O(q^{10})\) \(q-6.38987 q^{2} +8.83040 q^{4} +38.7291 q^{5} +148.051 q^{8} -247.474 q^{10} +576.390 q^{11} -391.491 q^{13} -1228.60 q^{16} -1329.70 q^{17} -942.474 q^{19} +341.993 q^{20} -3683.05 q^{22} +1632.08 q^{23} -1625.06 q^{25} +2501.58 q^{26} +1463.54 q^{29} +3912.42 q^{31} +3112.95 q^{32} +8496.61 q^{34} -16300.3 q^{37} +6022.28 q^{38} +5733.86 q^{40} -13103.8 q^{41} +14733.5 q^{43} +5089.75 q^{44} -10428.8 q^{46} -6814.52 q^{47} +10383.9 q^{50} -3457.02 q^{52} +2011.34 q^{53} +22323.0 q^{55} -9351.85 q^{58} +51453.1 q^{59} -41097.8 q^{61} -24999.8 q^{62} +19423.8 q^{64} -15162.1 q^{65} +50578.2 q^{67} -11741.8 q^{68} -39970.6 q^{71} +55686.6 q^{73} +104157. q^{74} -8322.42 q^{76} -63151.4 q^{79} -47582.4 q^{80} +83731.7 q^{82} +45572.4 q^{83} -51498.1 q^{85} -94145.1 q^{86} +85334.9 q^{88} +15686.7 q^{89} +14411.9 q^{92} +43543.9 q^{94} -36501.1 q^{95} -3128.49 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.38987 −1.12958 −0.564790 0.825235i \(-0.691043\pi\)
−0.564790 + 0.825235i \(0.691043\pi\)
\(3\) 0 0
\(4\) 8.83040 0.275950
\(5\) 38.7291 0.692807 0.346403 0.938086i \(-0.387403\pi\)
0.346403 + 0.938086i \(0.387403\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 148.051 0.817872
\(9\) 0 0
\(10\) −247.474 −0.782580
\(11\) 576.390 1.43627 0.718133 0.695906i \(-0.244997\pi\)
0.718133 + 0.695906i \(0.244997\pi\)
\(12\) 0 0
\(13\) −391.491 −0.642486 −0.321243 0.946997i \(-0.604101\pi\)
−0.321243 + 0.946997i \(0.604101\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1228.60 −1.19980
\(17\) −1329.70 −1.11592 −0.557958 0.829869i \(-0.688415\pi\)
−0.557958 + 0.829869i \(0.688415\pi\)
\(18\) 0 0
\(19\) −942.474 −0.598943 −0.299471 0.954105i \(-0.596810\pi\)
−0.299471 + 0.954105i \(0.596810\pi\)
\(20\) 341.993 0.191180
\(21\) 0 0
\(22\) −3683.05 −1.62238
\(23\) 1632.08 0.643312 0.321656 0.946857i \(-0.395761\pi\)
0.321656 + 0.946857i \(0.395761\pi\)
\(24\) 0 0
\(25\) −1625.06 −0.520019
\(26\) 2501.58 0.725739
\(27\) 0 0
\(28\) 0 0
\(29\) 1463.54 0.323155 0.161577 0.986860i \(-0.448342\pi\)
0.161577 + 0.986860i \(0.448342\pi\)
\(30\) 0 0
\(31\) 3912.42 0.731208 0.365604 0.930770i \(-0.380862\pi\)
0.365604 + 0.930770i \(0.380862\pi\)
\(32\) 3112.95 0.537399
\(33\) 0 0
\(34\) 8496.61 1.26052
\(35\) 0 0
\(36\) 0 0
\(37\) −16300.3 −1.95746 −0.978729 0.205160i \(-0.934229\pi\)
−0.978729 + 0.205160i \(0.934229\pi\)
\(38\) 6022.28 0.676553
\(39\) 0 0
\(40\) 5733.86 0.566627
\(41\) −13103.8 −1.21741 −0.608707 0.793395i \(-0.708312\pi\)
−0.608707 + 0.793395i \(0.708312\pi\)
\(42\) 0 0
\(43\) 14733.5 1.21516 0.607582 0.794257i \(-0.292140\pi\)
0.607582 + 0.794257i \(0.292140\pi\)
\(44\) 5089.75 0.396337
\(45\) 0 0
\(46\) −10428.8 −0.726672
\(47\) −6814.52 −0.449977 −0.224989 0.974361i \(-0.572235\pi\)
−0.224989 + 0.974361i \(0.572235\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 10383.9 0.587403
\(51\) 0 0
\(52\) −3457.02 −0.177294
\(53\) 2011.34 0.0983550 0.0491775 0.998790i \(-0.484340\pi\)
0.0491775 + 0.998790i \(0.484340\pi\)
\(54\) 0 0
\(55\) 22323.0 0.995054
\(56\) 0 0
\(57\) 0 0
\(58\) −9351.85 −0.365029
\(59\) 51453.1 1.92434 0.962170 0.272451i \(-0.0878344\pi\)
0.962170 + 0.272451i \(0.0878344\pi\)
\(60\) 0 0
\(61\) −41097.8 −1.41415 −0.707073 0.707141i \(-0.749985\pi\)
−0.707073 + 0.707141i \(0.749985\pi\)
\(62\) −24999.8 −0.825958
\(63\) 0 0
\(64\) 19423.8 0.592766
\(65\) −15162.1 −0.445119
\(66\) 0 0
\(67\) 50578.2 1.37650 0.688250 0.725473i \(-0.258379\pi\)
0.688250 + 0.725473i \(0.258379\pi\)
\(68\) −11741.8 −0.307937
\(69\) 0 0
\(70\) 0 0
\(71\) −39970.6 −0.941012 −0.470506 0.882397i \(-0.655929\pi\)
−0.470506 + 0.882397i \(0.655929\pi\)
\(72\) 0 0
\(73\) 55686.6 1.22305 0.611524 0.791226i \(-0.290557\pi\)
0.611524 + 0.791226i \(0.290557\pi\)
\(74\) 104157. 2.21110
\(75\) 0 0
\(76\) −8322.42 −0.165278
\(77\) 0 0
\(78\) 0 0
\(79\) −63151.4 −1.13845 −0.569226 0.822181i \(-0.692757\pi\)
−0.569226 + 0.822181i \(0.692757\pi\)
\(80\) −47582.4 −0.831231
\(81\) 0 0
\(82\) 83731.7 1.37517
\(83\) 45572.4 0.726116 0.363058 0.931766i \(-0.381733\pi\)
0.363058 + 0.931766i \(0.381733\pi\)
\(84\) 0 0
\(85\) −51498.1 −0.773114
\(86\) −94145.1 −1.37262
\(87\) 0 0
\(88\) 85334.9 1.17468
\(89\) 15686.7 0.209921 0.104961 0.994476i \(-0.466528\pi\)
0.104961 + 0.994476i \(0.466528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 14411.9 0.177522
\(93\) 0 0
\(94\) 43543.9 0.508285
\(95\) −36501.1 −0.414951
\(96\) 0 0
\(97\) −3128.49 −0.0337603 −0.0168801 0.999858i \(-0.505373\pi\)
−0.0168801 + 0.999858i \(0.505373\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −14349.9 −0.143499
\(101\) −169011. −1.64858 −0.824292 0.566164i \(-0.808427\pi\)
−0.824292 + 0.566164i \(0.808427\pi\)
\(102\) 0 0
\(103\) −112820. −1.04784 −0.523918 0.851769i \(-0.675530\pi\)
−0.523918 + 0.851769i \(0.675530\pi\)
\(104\) −57960.5 −0.525471
\(105\) 0 0
\(106\) −12852.2 −0.111100
\(107\) 22309.5 0.188378 0.0941890 0.995554i \(-0.469974\pi\)
0.0941890 + 0.995554i \(0.469974\pi\)
\(108\) 0 0
\(109\) −83819.5 −0.675739 −0.337869 0.941193i \(-0.609706\pi\)
−0.337869 + 0.941193i \(0.609706\pi\)
\(110\) −142641. −1.12399
\(111\) 0 0
\(112\) 0 0
\(113\) 40928.4 0.301529 0.150764 0.988570i \(-0.451826\pi\)
0.150764 + 0.988570i \(0.451826\pi\)
\(114\) 0 0
\(115\) 63208.9 0.445691
\(116\) 12923.7 0.0891746
\(117\) 0 0
\(118\) −328779. −2.17369
\(119\) 0 0
\(120\) 0 0
\(121\) 171174. 1.06286
\(122\) 262610. 1.59739
\(123\) 0 0
\(124\) 34548.2 0.201777
\(125\) −183965. −1.05308
\(126\) 0 0
\(127\) 83270.1 0.458120 0.229060 0.973412i \(-0.426435\pi\)
0.229060 + 0.973412i \(0.426435\pi\)
\(128\) −223730. −1.20698
\(129\) 0 0
\(130\) 96883.7 0.502797
\(131\) −166875. −0.849596 −0.424798 0.905288i \(-0.639655\pi\)
−0.424798 + 0.905288i \(0.639655\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −323188. −1.55487
\(135\) 0 0
\(136\) −196863. −0.912676
\(137\) −38223.9 −0.173994 −0.0869969 0.996209i \(-0.527727\pi\)
−0.0869969 + 0.996209i \(0.527727\pi\)
\(138\) 0 0
\(139\) −106263. −0.466492 −0.233246 0.972418i \(-0.574935\pi\)
−0.233246 + 0.972418i \(0.574935\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 255407. 1.06295
\(143\) −225652. −0.922780
\(144\) 0 0
\(145\) 56681.7 0.223884
\(146\) −355830. −1.38153
\(147\) 0 0
\(148\) −143938. −0.540160
\(149\) −192556. −0.710543 −0.355271 0.934763i \(-0.615612\pi\)
−0.355271 + 0.934763i \(0.615612\pi\)
\(150\) 0 0
\(151\) 141699. 0.505735 0.252868 0.967501i \(-0.418626\pi\)
0.252868 + 0.967501i \(0.418626\pi\)
\(152\) −139534. −0.489858
\(153\) 0 0
\(154\) 0 0
\(155\) 151524. 0.506586
\(156\) 0 0
\(157\) −565771. −1.83186 −0.915928 0.401342i \(-0.868544\pi\)
−0.915928 + 0.401342i \(0.868544\pi\)
\(158\) 403529. 1.28597
\(159\) 0 0
\(160\) 120562. 0.372314
\(161\) 0 0
\(162\) 0 0
\(163\) −430201. −1.26824 −0.634121 0.773233i \(-0.718638\pi\)
−0.634121 + 0.773233i \(0.718638\pi\)
\(164\) −115712. −0.335946
\(165\) 0 0
\(166\) −291201. −0.820206
\(167\) −240265. −0.666653 −0.333327 0.942811i \(-0.608171\pi\)
−0.333327 + 0.942811i \(0.608171\pi\)
\(168\) 0 0
\(169\) −218028. −0.587212
\(170\) 329066. 0.873294
\(171\) 0 0
\(172\) 130103. 0.335324
\(173\) −179300. −0.455476 −0.227738 0.973722i \(-0.573133\pi\)
−0.227738 + 0.973722i \(0.573133\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −708151. −1.72323
\(177\) 0 0
\(178\) −100236. −0.237123
\(179\) 575559. 1.34263 0.671317 0.741170i \(-0.265729\pi\)
0.671317 + 0.741170i \(0.265729\pi\)
\(180\) 0 0
\(181\) −581006. −1.31821 −0.659105 0.752051i \(-0.729065\pi\)
−0.659105 + 0.752051i \(0.729065\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 241630. 0.526147
\(185\) −631297. −1.35614
\(186\) 0 0
\(187\) −766426. −1.60275
\(188\) −60174.9 −0.124171
\(189\) 0 0
\(190\) 233237. 0.468721
\(191\) 660560. 1.31017 0.655087 0.755554i \(-0.272632\pi\)
0.655087 + 0.755554i \(0.272632\pi\)
\(192\) 0 0
\(193\) −557310. −1.07697 −0.538485 0.842635i \(-0.681003\pi\)
−0.538485 + 0.842635i \(0.681003\pi\)
\(194\) 19990.7 0.0381349
\(195\) 0 0
\(196\) 0 0
\(197\) 761400. 1.39781 0.698904 0.715216i \(-0.253672\pi\)
0.698904 + 0.715216i \(0.253672\pi\)
\(198\) 0 0
\(199\) 135860. 0.243197 0.121598 0.992579i \(-0.461198\pi\)
0.121598 + 0.992579i \(0.461198\pi\)
\(200\) −240591. −0.425309
\(201\) 0 0
\(202\) 1.07996e6 1.86221
\(203\) 0 0
\(204\) 0 0
\(205\) −507499. −0.843433
\(206\) 720906. 1.18361
\(207\) 0 0
\(208\) 480985. 0.770856
\(209\) −543232. −0.860240
\(210\) 0 0
\(211\) −991157. −1.53263 −0.766313 0.642467i \(-0.777911\pi\)
−0.766313 + 0.642467i \(0.777911\pi\)
\(212\) 17761.0 0.0271411
\(213\) 0 0
\(214\) −142555. −0.212788
\(215\) 570615. 0.841873
\(216\) 0 0
\(217\) 0 0
\(218\) 535596. 0.763301
\(219\) 0 0
\(220\) 197121. 0.274585
\(221\) 520566. 0.716960
\(222\) 0 0
\(223\) −543344. −0.731666 −0.365833 0.930681i \(-0.619216\pi\)
−0.365833 + 0.930681i \(0.619216\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −261527. −0.340601
\(227\) 16.3341 2.10393e−5 0 1.05196e−5 1.00000i \(-0.499997\pi\)
1.05196e−5 1.00000i \(0.499997\pi\)
\(228\) 0 0
\(229\) 77379.0 0.0975067 0.0487534 0.998811i \(-0.484475\pi\)
0.0487534 + 0.998811i \(0.484475\pi\)
\(230\) −403896. −0.503443
\(231\) 0 0
\(232\) 216679. 0.264299
\(233\) 103657. 0.125086 0.0625432 0.998042i \(-0.480079\pi\)
0.0625432 + 0.998042i \(0.480079\pi\)
\(234\) 0 0
\(235\) −263920. −0.311747
\(236\) 454351. 0.531021
\(237\) 0 0
\(238\) 0 0
\(239\) −689109. −0.780356 −0.390178 0.920739i \(-0.627587\pi\)
−0.390178 + 0.920739i \(0.627587\pi\)
\(240\) 0 0
\(241\) −220296. −0.244323 −0.122161 0.992510i \(-0.538983\pi\)
−0.122161 + 0.992510i \(0.538983\pi\)
\(242\) −1.09378e6 −1.20058
\(243\) 0 0
\(244\) −362910. −0.390234
\(245\) 0 0
\(246\) 0 0
\(247\) 368970. 0.384812
\(248\) 579236. 0.598035
\(249\) 0 0
\(250\) 1.17551e6 1.18954
\(251\) 1.43641e6 1.43912 0.719558 0.694433i \(-0.244345\pi\)
0.719558 + 0.694433i \(0.244345\pi\)
\(252\) 0 0
\(253\) 940714. 0.923966
\(254\) −532085. −0.517483
\(255\) 0 0
\(256\) 808042. 0.770609
\(257\) −909197. −0.858668 −0.429334 0.903146i \(-0.641252\pi\)
−0.429334 + 0.903146i \(0.641252\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −133887. −0.122830
\(261\) 0 0
\(262\) 1.06631e6 0.959687
\(263\) −749057. −0.667768 −0.333884 0.942614i \(-0.608359\pi\)
−0.333884 + 0.942614i \(0.608359\pi\)
\(264\) 0 0
\(265\) 77897.4 0.0681410
\(266\) 0 0
\(267\) 0 0
\(268\) 446626. 0.379845
\(269\) 669945. 0.564493 0.282246 0.959342i \(-0.408920\pi\)
0.282246 + 0.959342i \(0.408920\pi\)
\(270\) 0 0
\(271\) −540659. −0.447199 −0.223599 0.974681i \(-0.571781\pi\)
−0.223599 + 0.974681i \(0.571781\pi\)
\(272\) 1.63367e6 1.33888
\(273\) 0 0
\(274\) 244246. 0.196540
\(275\) −936668. −0.746885
\(276\) 0 0
\(277\) −401910. −0.314723 −0.157362 0.987541i \(-0.550299\pi\)
−0.157362 + 0.987541i \(0.550299\pi\)
\(278\) 679005. 0.526940
\(279\) 0 0
\(280\) 0 0
\(281\) 429139. 0.324214 0.162107 0.986773i \(-0.448171\pi\)
0.162107 + 0.986773i \(0.448171\pi\)
\(282\) 0 0
\(283\) 340927. 0.253044 0.126522 0.991964i \(-0.459619\pi\)
0.126522 + 0.991964i \(0.459619\pi\)
\(284\) −352957. −0.259672
\(285\) 0 0
\(286\) 1.44188e6 1.04235
\(287\) 0 0
\(288\) 0 0
\(289\) 348246. 0.245268
\(290\) −362188. −0.252895
\(291\) 0 0
\(292\) 491735. 0.337500
\(293\) −388847. −0.264612 −0.132306 0.991209i \(-0.542238\pi\)
−0.132306 + 0.991209i \(0.542238\pi\)
\(294\) 0 0
\(295\) 1.99273e6 1.33319
\(296\) −2.41328e6 −1.60095
\(297\) 0 0
\(298\) 1.23040e6 0.802615
\(299\) −638945. −0.413319
\(300\) 0 0
\(301\) 0 0
\(302\) −905435. −0.571268
\(303\) 0 0
\(304\) 1.15792e6 0.718612
\(305\) −1.59168e6 −0.979730
\(306\) 0 0
\(307\) 2.35747e6 1.42758 0.713789 0.700361i \(-0.246978\pi\)
0.713789 + 0.700361i \(0.246978\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −968220. −0.572229
\(311\) −1.43663e6 −0.842254 −0.421127 0.907002i \(-0.638365\pi\)
−0.421127 + 0.907002i \(0.638365\pi\)
\(312\) 0 0
\(313\) 822800. 0.474715 0.237358 0.971422i \(-0.423719\pi\)
0.237358 + 0.971422i \(0.423719\pi\)
\(314\) 3.61520e6 2.06923
\(315\) 0 0
\(316\) −557652. −0.314156
\(317\) 1.76693e6 0.987580 0.493790 0.869581i \(-0.335611\pi\)
0.493790 + 0.869581i \(0.335611\pi\)
\(318\) 0 0
\(319\) 843572. 0.464136
\(320\) 752264. 0.410672
\(321\) 0 0
\(322\) 0 0
\(323\) 1.25321e6 0.668370
\(324\) 0 0
\(325\) 636196. 0.334105
\(326\) 2.74893e6 1.43258
\(327\) 0 0
\(328\) −1.94003e6 −0.995689
\(329\) 0 0
\(330\) 0 0
\(331\) −3052.27 −0.00153128 −0.000765638 1.00000i \(-0.500244\pi\)
−0.000765638 1.00000i \(0.500244\pi\)
\(332\) 402422. 0.200372
\(333\) 0 0
\(334\) 1.53526e6 0.753038
\(335\) 1.95885e6 0.953649
\(336\) 0 0
\(337\) 2.02939e6 0.973398 0.486699 0.873570i \(-0.338201\pi\)
0.486699 + 0.873570i \(0.338201\pi\)
\(338\) 1.39317e6 0.663302
\(339\) 0 0
\(340\) −454748. −0.213341
\(341\) 2.25508e6 1.05021
\(342\) 0 0
\(343\) 0 0
\(344\) 2.18130e6 0.993848
\(345\) 0 0
\(346\) 1.14570e6 0.514496
\(347\) −3.78218e6 −1.68624 −0.843119 0.537727i \(-0.819283\pi\)
−0.843119 + 0.537727i \(0.819283\pi\)
\(348\) 0 0
\(349\) −291147. −0.127953 −0.0639763 0.997951i \(-0.520378\pi\)
−0.0639763 + 0.997951i \(0.520378\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.79427e6 0.771848
\(353\) 385076. 0.164479 0.0822394 0.996613i \(-0.473793\pi\)
0.0822394 + 0.996613i \(0.473793\pi\)
\(354\) 0 0
\(355\) −1.54803e6 −0.651939
\(356\) 138520. 0.0579277
\(357\) 0 0
\(358\) −3.67775e6 −1.51661
\(359\) 3.23014e6 1.32277 0.661385 0.750046i \(-0.269969\pi\)
0.661385 + 0.750046i \(0.269969\pi\)
\(360\) 0 0
\(361\) −1.58784e6 −0.641268
\(362\) 3.71255e6 1.48902
\(363\) 0 0
\(364\) 0 0
\(365\) 2.15669e6 0.847336
\(366\) 0 0
\(367\) 479559. 0.185856 0.0929280 0.995673i \(-0.470377\pi\)
0.0929280 + 0.995673i \(0.470377\pi\)
\(368\) −2.00517e6 −0.771847
\(369\) 0 0
\(370\) 4.03390e6 1.53187
\(371\) 0 0
\(372\) 0 0
\(373\) 872666. 0.324770 0.162385 0.986727i \(-0.448081\pi\)
0.162385 + 0.986727i \(0.448081\pi\)
\(374\) 4.89736e6 1.81043
\(375\) 0 0
\(376\) −1.00889e6 −0.368024
\(377\) −572965. −0.207622
\(378\) 0 0
\(379\) −2.43493e6 −0.870742 −0.435371 0.900251i \(-0.643383\pi\)
−0.435371 + 0.900251i \(0.643383\pi\)
\(380\) −322320. −0.114506
\(381\) 0 0
\(382\) −4.22089e6 −1.47995
\(383\) −3.61169e6 −1.25809 −0.629047 0.777367i \(-0.716555\pi\)
−0.629047 + 0.777367i \(0.716555\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.56114e6 1.21652
\(387\) 0 0
\(388\) −27625.9 −0.00931615
\(389\) 175232. 0.0587138 0.0293569 0.999569i \(-0.490654\pi\)
0.0293569 + 0.999569i \(0.490654\pi\)
\(390\) 0 0
\(391\) −2.17018e6 −0.717882
\(392\) 0 0
\(393\) 0 0
\(394\) −4.86525e6 −1.57893
\(395\) −2.44579e6 −0.788727
\(396\) 0 0
\(397\) −1.88515e6 −0.600303 −0.300152 0.953892i \(-0.597037\pi\)
−0.300152 + 0.953892i \(0.597037\pi\)
\(398\) −868126. −0.274710
\(399\) 0 0
\(400\) 1.99654e6 0.623920
\(401\) −1.33983e6 −0.416091 −0.208046 0.978119i \(-0.566710\pi\)
−0.208046 + 0.978119i \(0.566710\pi\)
\(402\) 0 0
\(403\) −1.53168e6 −0.469791
\(404\) −1.49243e6 −0.454927
\(405\) 0 0
\(406\) 0 0
\(407\) −9.39535e6 −2.81143
\(408\) 0 0
\(409\) −6.58628e6 −1.94685 −0.973423 0.229013i \(-0.926450\pi\)
−0.973423 + 0.229013i \(0.926450\pi\)
\(410\) 3.24285e6 0.952725
\(411\) 0 0
\(412\) −996247. −0.289150
\(413\) 0 0
\(414\) 0 0
\(415\) 1.76497e6 0.503058
\(416\) −1.21869e6 −0.345271
\(417\) 0 0
\(418\) 3.47118e6 0.971710
\(419\) −6.96869e6 −1.93917 −0.969585 0.244754i \(-0.921293\pi\)
−0.969585 + 0.244754i \(0.921293\pi\)
\(420\) 0 0
\(421\) 3.84041e6 1.05602 0.528010 0.849238i \(-0.322938\pi\)
0.528010 + 0.849238i \(0.322938\pi\)
\(422\) 6.33336e6 1.73122
\(423\) 0 0
\(424\) 297781. 0.0804418
\(425\) 2.16084e6 0.580297
\(426\) 0 0
\(427\) 0 0
\(428\) 197002. 0.0519829
\(429\) 0 0
\(430\) −3.64615e6 −0.950963
\(431\) −3.03636e6 −0.787337 −0.393668 0.919253i \(-0.628794\pi\)
−0.393668 + 0.919253i \(0.628794\pi\)
\(432\) 0 0
\(433\) 941529. 0.241332 0.120666 0.992693i \(-0.461497\pi\)
0.120666 + 0.992693i \(0.461497\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −740160. −0.186470
\(437\) −1.53819e6 −0.385307
\(438\) 0 0
\(439\) −1.34109e6 −0.332122 −0.166061 0.986116i \(-0.553105\pi\)
−0.166061 + 0.986116i \(0.553105\pi\)
\(440\) 3.30494e6 0.813827
\(441\) 0 0
\(442\) −3.32635e6 −0.809864
\(443\) 772341. 0.186982 0.0934910 0.995620i \(-0.470197\pi\)
0.0934910 + 0.995620i \(0.470197\pi\)
\(444\) 0 0
\(445\) 607531. 0.145435
\(446\) 3.47190e6 0.826475
\(447\) 0 0
\(448\) 0 0
\(449\) −2.25684e6 −0.528304 −0.264152 0.964481i \(-0.585092\pi\)
−0.264152 + 0.964481i \(0.585092\pi\)
\(450\) 0 0
\(451\) −7.55291e6 −1.74853
\(452\) 361414. 0.0832069
\(453\) 0 0
\(454\) −104.373 −2.37655e−5 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.28470e6 0.959688 0.479844 0.877354i \(-0.340693\pi\)
0.479844 + 0.877354i \(0.340693\pi\)
\(458\) −494442. −0.110142
\(459\) 0 0
\(460\) 558160. 0.122988
\(461\) −3.10462e6 −0.680387 −0.340193 0.940355i \(-0.610493\pi\)
−0.340193 + 0.940355i \(0.610493\pi\)
\(462\) 0 0
\(463\) −3.53386e6 −0.766121 −0.383060 0.923723i \(-0.625130\pi\)
−0.383060 + 0.923723i \(0.625130\pi\)
\(464\) −1.79811e6 −0.387722
\(465\) 0 0
\(466\) −662356. −0.141295
\(467\) 2.72459e6 0.578109 0.289054 0.957313i \(-0.406659\pi\)
0.289054 + 0.957313i \(0.406659\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.68641e6 0.352143
\(471\) 0 0
\(472\) 7.61767e6 1.57386
\(473\) 8.49224e6 1.74530
\(474\) 0 0
\(475\) 1.53158e6 0.311461
\(476\) 0 0
\(477\) 0 0
\(478\) 4.40331e6 0.881475
\(479\) 978685. 0.194896 0.0974482 0.995241i \(-0.468932\pi\)
0.0974482 + 0.995241i \(0.468932\pi\)
\(480\) 0 0
\(481\) 6.38144e6 1.25764
\(482\) 1.40766e6 0.275982
\(483\) 0 0
\(484\) 1.51154e6 0.293296
\(485\) −121164. −0.0233893
\(486\) 0 0
\(487\) 3.92744e6 0.750390 0.375195 0.926946i \(-0.377576\pi\)
0.375195 + 0.926946i \(0.377576\pi\)
\(488\) −6.08456e6 −1.15659
\(489\) 0 0
\(490\) 0 0
\(491\) −2.63241e6 −0.492777 −0.246388 0.969171i \(-0.579244\pi\)
−0.246388 + 0.969171i \(0.579244\pi\)
\(492\) 0 0
\(493\) −1.94607e6 −0.360614
\(494\) −2.35767e6 −0.434676
\(495\) 0 0
\(496\) −4.80678e6 −0.877305
\(497\) 0 0
\(498\) 0 0
\(499\) 2.12544e6 0.382118 0.191059 0.981579i \(-0.438808\pi\)
0.191059 + 0.981579i \(0.438808\pi\)
\(500\) −1.62449e6 −0.290597
\(501\) 0 0
\(502\) −9.17850e6 −1.62560
\(503\) 2.60929e6 0.459835 0.229917 0.973210i \(-0.426154\pi\)
0.229917 + 0.973210i \(0.426154\pi\)
\(504\) 0 0
\(505\) −6.54564e6 −1.14215
\(506\) −6.01104e6 −1.04369
\(507\) 0 0
\(508\) 735308. 0.126418
\(509\) −1.00182e7 −1.71394 −0.856970 0.515366i \(-0.827656\pi\)
−0.856970 + 0.515366i \(0.827656\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.99607e6 0.336512
\(513\) 0 0
\(514\) 5.80965e6 0.969933
\(515\) −4.36942e6 −0.725948
\(516\) 0 0
\(517\) −3.92782e6 −0.646287
\(518\) 0 0
\(519\) 0 0
\(520\) −2.24476e6 −0.364050
\(521\) −4.17941e6 −0.674560 −0.337280 0.941404i \(-0.609507\pi\)
−0.337280 + 0.941404i \(0.609507\pi\)
\(522\) 0 0
\(523\) −3.60525e6 −0.576343 −0.288172 0.957579i \(-0.593047\pi\)
−0.288172 + 0.957579i \(0.593047\pi\)
\(524\) −1.47357e6 −0.234446
\(525\) 0 0
\(526\) 4.78637e6 0.754297
\(527\) −5.20234e6 −0.815967
\(528\) 0 0
\(529\) −3.77266e6 −0.586150
\(530\) −497754. −0.0769707
\(531\) 0 0
\(532\) 0 0
\(533\) 5.13003e6 0.782172
\(534\) 0 0
\(535\) 864025. 0.130509
\(536\) 7.48814e6 1.12580
\(537\) 0 0
\(538\) −4.28086e6 −0.637640
\(539\) 0 0
\(540\) 0 0
\(541\) −6.68166e6 −0.981501 −0.490751 0.871300i \(-0.663277\pi\)
−0.490751 + 0.871300i \(0.663277\pi\)
\(542\) 3.45474e6 0.505146
\(543\) 0 0
\(544\) −4.13929e6 −0.599692
\(545\) −3.24625e6 −0.468156
\(546\) 0 0
\(547\) 8.69076e6 1.24191 0.620954 0.783847i \(-0.286745\pi\)
0.620954 + 0.783847i \(0.286745\pi\)
\(548\) −337532. −0.0480136
\(549\) 0 0
\(550\) 5.98518e6 0.843666
\(551\) −1.37935e6 −0.193551
\(552\) 0 0
\(553\) 0 0
\(554\) 2.56815e6 0.355505
\(555\) 0 0
\(556\) −938342. −0.128728
\(557\) −6.24742e6 −0.853223 −0.426612 0.904435i \(-0.640293\pi\)
−0.426612 + 0.904435i \(0.640293\pi\)
\(558\) 0 0
\(559\) −5.76803e6 −0.780725
\(560\) 0 0
\(561\) 0 0
\(562\) −2.74214e6 −0.366226
\(563\) −1.19045e7 −1.58285 −0.791423 0.611269i \(-0.790659\pi\)
−0.791423 + 0.611269i \(0.790659\pi\)
\(564\) 0 0
\(565\) 1.58512e6 0.208901
\(566\) −2.17848e6 −0.285833
\(567\) 0 0
\(568\) −5.91768e6 −0.769627
\(569\) −21414.4 −0.00277284 −0.00138642 0.999999i \(-0.500441\pi\)
−0.00138642 + 0.999999i \(0.500441\pi\)
\(570\) 0 0
\(571\) 7.11647e6 0.913428 0.456714 0.889614i \(-0.349026\pi\)
0.456714 + 0.889614i \(0.349026\pi\)
\(572\) −1.99259e6 −0.254641
\(573\) 0 0
\(574\) 0 0
\(575\) −2.65223e6 −0.334534
\(576\) 0 0
\(577\) 1.06652e7 1.33361 0.666805 0.745232i \(-0.267661\pi\)
0.666805 + 0.745232i \(0.267661\pi\)
\(578\) −2.22524e6 −0.277050
\(579\) 0 0
\(580\) 500522. 0.0617808
\(581\) 0 0
\(582\) 0 0
\(583\) 1.15932e6 0.141264
\(584\) 8.24444e6 1.00030
\(585\) 0 0
\(586\) 2.48468e6 0.298901
\(587\) −1.30101e7 −1.55843 −0.779213 0.626759i \(-0.784381\pi\)
−0.779213 + 0.626759i \(0.784381\pi\)
\(588\) 0 0
\(589\) −3.68735e6 −0.437952
\(590\) −1.27333e7 −1.50595
\(591\) 0 0
\(592\) 2.00265e7 2.34856
\(593\) 4.26086e6 0.497578 0.248789 0.968558i \(-0.419967\pi\)
0.248789 + 0.968558i \(0.419967\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.70034e6 −0.196074
\(597\) 0 0
\(598\) 4.08277e6 0.466877
\(599\) 1.37958e7 1.57101 0.785507 0.618853i \(-0.212402\pi\)
0.785507 + 0.618853i \(0.212402\pi\)
\(600\) 0 0
\(601\) −4.99695e6 −0.564311 −0.282155 0.959369i \(-0.591049\pi\)
−0.282155 + 0.959369i \(0.591049\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.25126e6 0.139558
\(605\) 6.62942e6 0.736355
\(606\) 0 0
\(607\) 3.04946e6 0.335932 0.167966 0.985793i \(-0.446280\pi\)
0.167966 + 0.985793i \(0.446280\pi\)
\(608\) −2.93387e6 −0.321871
\(609\) 0 0
\(610\) 1.01706e7 1.10668
\(611\) 2.66782e6 0.289104
\(612\) 0 0
\(613\) −7.03625e6 −0.756293 −0.378147 0.925746i \(-0.623438\pi\)
−0.378147 + 0.925746i \(0.623438\pi\)
\(614\) −1.50639e7 −1.61256
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00066e7 −1.05822 −0.529108 0.848554i \(-0.677473\pi\)
−0.529108 + 0.848554i \(0.677473\pi\)
\(618\) 0 0
\(619\) 6.55067e6 0.687161 0.343581 0.939123i \(-0.388360\pi\)
0.343581 + 0.939123i \(0.388360\pi\)
\(620\) 1.33802e6 0.139792
\(621\) 0 0
\(622\) 9.17986e6 0.951393
\(623\) 0 0
\(624\) 0 0
\(625\) −2.04650e6 −0.209561
\(626\) −5.25758e6 −0.536229
\(627\) 0 0
\(628\) −4.99598e6 −0.505501
\(629\) 2.16746e7 2.18436
\(630\) 0 0
\(631\) 2.22672e6 0.222635 0.111317 0.993785i \(-0.464493\pi\)
0.111317 + 0.993785i \(0.464493\pi\)
\(632\) −9.34960e6 −0.931109
\(633\) 0 0
\(634\) −1.12905e7 −1.11555
\(635\) 3.22497e6 0.317389
\(636\) 0 0
\(637\) 0 0
\(638\) −5.39031e6 −0.524279
\(639\) 0 0
\(640\) −8.66484e6 −0.836201
\(641\) 1.59340e7 1.53172 0.765859 0.643008i \(-0.222314\pi\)
0.765859 + 0.643008i \(0.222314\pi\)
\(642\) 0 0
\(643\) 1.49933e7 1.43011 0.715056 0.699067i \(-0.246401\pi\)
0.715056 + 0.699067i \(0.246401\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00783e6 −0.754977
\(647\) 1.29805e7 1.21908 0.609540 0.792755i \(-0.291354\pi\)
0.609540 + 0.792755i \(0.291354\pi\)
\(648\) 0 0
\(649\) 2.96571e7 2.76386
\(650\) −4.06521e6 −0.377398
\(651\) 0 0
\(652\) −3.79885e6 −0.349972
\(653\) 1.65492e7 1.51878 0.759389 0.650637i \(-0.225498\pi\)
0.759389 + 0.650637i \(0.225498\pi\)
\(654\) 0 0
\(655\) −6.46291e6 −0.588606
\(656\) 1.60993e7 1.46066
\(657\) 0 0
\(658\) 0 0
\(659\) −5.86879e6 −0.526423 −0.263212 0.964738i \(-0.584782\pi\)
−0.263212 + 0.964738i \(0.584782\pi\)
\(660\) 0 0
\(661\) −7.27687e6 −0.647800 −0.323900 0.946091i \(-0.604994\pi\)
−0.323900 + 0.946091i \(0.604994\pi\)
\(662\) 19503.6 0.00172970
\(663\) 0 0
\(664\) 6.74702e6 0.593870
\(665\) 0 0
\(666\) 0 0
\(667\) 2.38862e6 0.207889
\(668\) −2.12164e6 −0.183963
\(669\) 0 0
\(670\) −1.25168e7 −1.07722
\(671\) −2.36884e7 −2.03109
\(672\) 0 0
\(673\) −1.82417e7 −1.55248 −0.776241 0.630437i \(-0.782876\pi\)
−0.776241 + 0.630437i \(0.782876\pi\)
\(674\) −1.29675e7 −1.09953
\(675\) 0 0
\(676\) −1.92527e6 −0.162041
\(677\) −7.76406e6 −0.651054 −0.325527 0.945533i \(-0.605542\pi\)
−0.325527 + 0.945533i \(0.605542\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −7.62432e6 −0.632308
\(681\) 0 0
\(682\) −1.44096e7 −1.18629
\(683\) −8.56324e6 −0.702403 −0.351201 0.936300i \(-0.614227\pi\)
−0.351201 + 0.936300i \(0.614227\pi\)
\(684\) 0 0
\(685\) −1.48038e6 −0.120544
\(686\) 0 0
\(687\) 0 0
\(688\) −1.81015e7 −1.45796
\(689\) −787423. −0.0631917
\(690\) 0 0
\(691\) −1.64509e7 −1.31068 −0.655338 0.755336i \(-0.727474\pi\)
−0.655338 + 0.755336i \(0.727474\pi\)
\(692\) −1.58329e6 −0.125689
\(693\) 0 0
\(694\) 2.41677e7 1.90474
\(695\) −4.11546e6 −0.323189
\(696\) 0 0
\(697\) 1.74242e7 1.35853
\(698\) 1.86039e6 0.144533
\(699\) 0 0
\(700\) 0 0
\(701\) 1.66928e7 1.28302 0.641512 0.767113i \(-0.278307\pi\)
0.641512 + 0.767113i \(0.278307\pi\)
\(702\) 0 0
\(703\) 1.53626e7 1.17240
\(704\) 1.11957e7 0.851370
\(705\) 0 0
\(706\) −2.46059e6 −0.185792
\(707\) 0 0
\(708\) 0 0
\(709\) 5.61779e6 0.419711 0.209855 0.977732i \(-0.432701\pi\)
0.209855 + 0.977732i \(0.432701\pi\)
\(710\) 9.89167e6 0.736417
\(711\) 0 0
\(712\) 2.32242e6 0.171689
\(713\) 6.38537e6 0.470395
\(714\) 0 0
\(715\) −8.73927e6 −0.639308
\(716\) 5.08242e6 0.370500
\(717\) 0 0
\(718\) −2.06401e7 −1.49417
\(719\) 1.03718e7 0.748227 0.374113 0.927383i \(-0.377947\pi\)
0.374113 + 0.927383i \(0.377947\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.01461e7 0.724363
\(723\) 0 0
\(724\) −5.13052e6 −0.363760
\(725\) −2.37835e6 −0.168047
\(726\) 0 0
\(727\) −1.15369e7 −0.809565 −0.404783 0.914413i \(-0.632653\pi\)
−0.404783 + 0.914413i \(0.632653\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.37810e7 −0.957134
\(731\) −1.95911e7 −1.35602
\(732\) 0 0
\(733\) −1.49470e7 −1.02753 −0.513764 0.857932i \(-0.671749\pi\)
−0.513764 + 0.857932i \(0.671749\pi\)
\(734\) −3.06432e6 −0.209939
\(735\) 0 0
\(736\) 5.08058e6 0.345715
\(737\) 2.91528e7 1.97702
\(738\) 0 0
\(739\) 9.01364e6 0.607140 0.303570 0.952809i \(-0.401821\pi\)
0.303570 + 0.952809i \(0.401821\pi\)
\(740\) −5.57460e6 −0.374227
\(741\) 0 0
\(742\) 0 0
\(743\) 2.10239e7 1.39714 0.698571 0.715541i \(-0.253820\pi\)
0.698571 + 0.715541i \(0.253820\pi\)
\(744\) 0 0
\(745\) −7.45750e6 −0.492269
\(746\) −5.57622e6 −0.366854
\(747\) 0 0
\(748\) −6.76785e6 −0.442279
\(749\) 0 0
\(750\) 0 0
\(751\) 4.04219e6 0.261527 0.130764 0.991414i \(-0.458257\pi\)
0.130764 + 0.991414i \(0.458257\pi\)
\(752\) 8.37230e6 0.539884
\(753\) 0 0
\(754\) 3.66117e6 0.234526
\(755\) 5.48786e6 0.350377
\(756\) 0 0
\(757\) −1.82059e7 −1.15471 −0.577353 0.816495i \(-0.695914\pi\)
−0.577353 + 0.816495i \(0.695914\pi\)
\(758\) 1.55589e7 0.983572
\(759\) 0 0
\(760\) −5.40402e6 −0.339377
\(761\) −1.89200e7 −1.18429 −0.592146 0.805831i \(-0.701719\pi\)
−0.592146 + 0.805831i \(0.701719\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 5.83301e6 0.361542
\(765\) 0 0
\(766\) 2.30782e7 1.42112
\(767\) −2.01434e7 −1.23636
\(768\) 0 0
\(769\) 1.12831e7 0.688037 0.344019 0.938963i \(-0.388212\pi\)
0.344019 + 0.938963i \(0.388212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.92127e6 −0.297190
\(773\) −3.68565e6 −0.221853 −0.110926 0.993829i \(-0.535382\pi\)
−0.110926 + 0.993829i \(0.535382\pi\)
\(774\) 0 0
\(775\) −6.35791e6 −0.380242
\(776\) −463176. −0.0276116
\(777\) 0 0
\(778\) −1.11971e6 −0.0663219
\(779\) 1.23500e7 0.729161
\(780\) 0 0
\(781\) −2.30387e7 −1.35154
\(782\) 1.38671e7 0.810905
\(783\) 0 0
\(784\) 0 0
\(785\) −2.19118e7 −1.26912
\(786\) 0 0
\(787\) 1.40748e7 0.810039 0.405019 0.914308i \(-0.367265\pi\)
0.405019 + 0.914308i \(0.367265\pi\)
\(788\) 6.72347e6 0.385725
\(789\) 0 0
\(790\) 1.56283e7 0.890930
\(791\) 0 0
\(792\) 0 0
\(793\) 1.60894e7 0.908569
\(794\) 1.20459e7 0.678090
\(795\) 0 0
\(796\) 1.19970e6 0.0671102
\(797\) −1.75191e7 −0.976937 −0.488469 0.872582i \(-0.662444\pi\)
−0.488469 + 0.872582i \(0.662444\pi\)
\(798\) 0 0
\(799\) 9.06127e6 0.502137
\(800\) −5.05873e6 −0.279458
\(801\) 0 0
\(802\) 8.56134e6 0.470008
\(803\) 3.20972e7 1.75662
\(804\) 0 0
\(805\) 0 0
\(806\) 9.78721e6 0.530666
\(807\) 0 0
\(808\) −2.50222e7 −1.34833
\(809\) −814521. −0.0437553 −0.0218777 0.999761i \(-0.506964\pi\)
−0.0218777 + 0.999761i \(0.506964\pi\)
\(810\) 0 0
\(811\) −1.26533e7 −0.675540 −0.337770 0.941229i \(-0.609673\pi\)
−0.337770 + 0.941229i \(0.609673\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00350e7 3.17573
\(815\) −1.66613e7 −0.878647
\(816\) 0 0
\(817\) −1.38859e7 −0.727813
\(818\) 4.20854e7 2.19912
\(819\) 0 0
\(820\) −4.48142e6 −0.232745
\(821\) −4.27393e6 −0.221294 −0.110647 0.993860i \(-0.535292\pi\)
−0.110647 + 0.993860i \(0.535292\pi\)
\(822\) 0 0
\(823\) 1.70463e7 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(824\) −1.67031e7 −0.856996
\(825\) 0 0
\(826\) 0 0
\(827\) −2.60828e6 −0.132614 −0.0663071 0.997799i \(-0.521122\pi\)
−0.0663071 + 0.997799i \(0.521122\pi\)
\(828\) 0 0
\(829\) −2.13865e7 −1.08082 −0.540409 0.841403i \(-0.681730\pi\)
−0.540409 + 0.841403i \(0.681730\pi\)
\(830\) −1.12780e7 −0.568244
\(831\) 0 0
\(832\) −7.60423e6 −0.380844
\(833\) 0 0
\(834\) 0 0
\(835\) −9.30525e6 −0.461862
\(836\) −4.79696e6 −0.237383
\(837\) 0 0
\(838\) 4.45290e7 2.19045
\(839\) 771393. 0.0378330 0.0189165 0.999821i \(-0.493978\pi\)
0.0189165 + 0.999821i \(0.493978\pi\)
\(840\) 0 0
\(841\) −1.83692e7 −0.895571
\(842\) −2.45397e7 −1.19286
\(843\) 0 0
\(844\) −8.75231e6 −0.422928
\(845\) −8.44401e6 −0.406824
\(846\) 0 0
\(847\) 0 0
\(848\) −2.47113e6 −0.118006
\(849\) 0 0
\(850\) −1.38075e7 −0.655492
\(851\) −2.66034e7 −1.25926
\(852\) 0 0
\(853\) −2.94032e7 −1.38364 −0.691818 0.722072i \(-0.743190\pi\)
−0.691818 + 0.722072i \(0.743190\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.30293e6 0.154069
\(857\) −2.65010e7 −1.23257 −0.616283 0.787525i \(-0.711362\pi\)
−0.616283 + 0.787525i \(0.711362\pi\)
\(858\) 0 0
\(859\) 3.67519e7 1.69940 0.849702 0.527264i \(-0.176782\pi\)
0.849702 + 0.527264i \(0.176782\pi\)
\(860\) 5.03876e6 0.232315
\(861\) 0 0
\(862\) 1.94020e7 0.889360
\(863\) −1.18189e7 −0.540196 −0.270098 0.962833i \(-0.587056\pi\)
−0.270098 + 0.962833i \(0.587056\pi\)
\(864\) 0 0
\(865\) −6.94413e6 −0.315557
\(866\) −6.01625e6 −0.272603
\(867\) 0 0
\(868\) 0 0
\(869\) −3.63998e7 −1.63512
\(870\) 0 0
\(871\) −1.98009e7 −0.884382
\(872\) −1.24095e7 −0.552668
\(873\) 0 0
\(874\) 9.82884e6 0.435235
\(875\) 0 0
\(876\) 0 0
\(877\) 7.93509e6 0.348380 0.174190 0.984712i \(-0.444269\pi\)
0.174190 + 0.984712i \(0.444269\pi\)
\(878\) 8.56939e6 0.375158
\(879\) 0 0
\(880\) −2.74260e7 −1.19387
\(881\) 4.20152e7 1.82375 0.911877 0.410464i \(-0.134633\pi\)
0.911877 + 0.410464i \(0.134633\pi\)
\(882\) 0 0
\(883\) 2.12461e7 0.917016 0.458508 0.888690i \(-0.348384\pi\)
0.458508 + 0.888690i \(0.348384\pi\)
\(884\) 4.59681e6 0.197845
\(885\) 0 0
\(886\) −4.93516e6 −0.211211
\(887\) 1.88490e7 0.804415 0.402208 0.915548i \(-0.368243\pi\)
0.402208 + 0.915548i \(0.368243\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −3.88204e6 −0.164280
\(891\) 0 0
\(892\) −4.79795e6 −0.201903
\(893\) 6.42251e6 0.269511
\(894\) 0 0
\(895\) 2.22909e7 0.930186
\(896\) 0 0
\(897\) 0 0
\(898\) 1.44209e7 0.596762
\(899\) 5.72600e6 0.236294
\(900\) 0 0
\(901\) −2.67448e6 −0.109756
\(902\) 4.82621e7 1.97510
\(903\) 0 0
\(904\) 6.05948e6 0.246612
\(905\) −2.25018e7 −0.913264
\(906\) 0 0
\(907\) −6.19446e6 −0.250026 −0.125013 0.992155i \(-0.539897\pi\)
−0.125013 + 0.992155i \(0.539897\pi\)
\(908\) 144.237 5.80578e−6 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.50171e7 0.998712 0.499356 0.866397i \(-0.333570\pi\)
0.499356 + 0.866397i \(0.333570\pi\)
\(912\) 0 0
\(913\) 2.62674e7 1.04290
\(914\) −2.73787e7 −1.08404
\(915\) 0 0
\(916\) 683288. 0.0269070
\(917\) 0 0
\(918\) 0 0
\(919\) 1.54992e7 0.605370 0.302685 0.953091i \(-0.402117\pi\)
0.302685 + 0.953091i \(0.402117\pi\)
\(920\) 9.35812e6 0.364518
\(921\) 0 0
\(922\) 1.98381e7 0.768551
\(923\) 1.56481e7 0.604587
\(924\) 0 0
\(925\) 2.64890e7 1.01791
\(926\) 2.25809e7 0.865394
\(927\) 0 0
\(928\) 4.55594e6 0.173663
\(929\) 3.75285e7 1.42667 0.713333 0.700826i \(-0.247185\pi\)
0.713333 + 0.700826i \(0.247185\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 915335. 0.0345176
\(933\) 0 0
\(934\) −1.74098e7 −0.653020
\(935\) −2.96830e7 −1.11040
\(936\) 0 0
\(937\) 1.08298e7 0.402969 0.201485 0.979492i \(-0.435423\pi\)
0.201485 + 0.979492i \(0.435423\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.33052e6 −0.0860267
\(941\) 3.24591e7 1.19498 0.597492 0.801875i \(-0.296164\pi\)
0.597492 + 0.801875i \(0.296164\pi\)
\(942\) 0 0
\(943\) −2.13865e7 −0.783177
\(944\) −6.32151e7 −2.30883
\(945\) 0 0
\(946\) −5.42643e7 −1.97145
\(947\) −7.53877e6 −0.273165 −0.136583 0.990629i \(-0.543612\pi\)
−0.136583 + 0.990629i \(0.543612\pi\)
\(948\) 0 0
\(949\) −2.18008e7 −0.785792
\(950\) −9.78656e6 −0.351821
\(951\) 0 0
\(952\) 0 0
\(953\) 3.01356e7 1.07485 0.537424 0.843312i \(-0.319398\pi\)
0.537424 + 0.843312i \(0.319398\pi\)
\(954\) 0 0
\(955\) 2.55829e7 0.907697
\(956\) −6.08510e6 −0.215339
\(957\) 0 0
\(958\) −6.25366e6 −0.220151
\(959\) 0 0
\(960\) 0 0
\(961\) −1.33221e7 −0.465335
\(962\) −4.07765e7 −1.42060
\(963\) 0 0
\(964\) −1.94530e6 −0.0674208
\(965\) −2.15841e7 −0.746132
\(966\) 0 0
\(967\) 2.88021e6 0.0990509 0.0495255 0.998773i \(-0.484229\pi\)
0.0495255 + 0.998773i \(0.484229\pi\)
\(968\) 2.53425e7 0.869282
\(969\) 0 0
\(970\) 774220. 0.0264201
\(971\) −2.74490e7 −0.934283 −0.467142 0.884183i \(-0.654716\pi\)
−0.467142 + 0.884183i \(0.654716\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.50958e7 −0.847626
\(975\) 0 0
\(976\) 5.04927e7 1.69669
\(977\) 5.88524e7 1.97255 0.986274 0.165118i \(-0.0528004\pi\)
0.986274 + 0.165118i \(0.0528004\pi\)
\(978\) 0 0
\(979\) 9.04164e6 0.301502
\(980\) 0 0
\(981\) 0 0
\(982\) 1.68208e7 0.556631
\(983\) 1.86077e7 0.614198 0.307099 0.951678i \(-0.400642\pi\)
0.307099 + 0.951678i \(0.400642\pi\)
\(984\) 0 0
\(985\) 2.94883e7 0.968410
\(986\) 1.24352e7 0.407342
\(987\) 0 0
\(988\) 3.25815e6 0.106189
\(989\) 2.40462e7 0.781729
\(990\) 0 0
\(991\) 2.47154e7 0.799435 0.399718 0.916638i \(-0.369108\pi\)
0.399718 + 0.916638i \(0.369108\pi\)
\(992\) 1.21792e7 0.392951
\(993\) 0 0
\(994\) 0 0
\(995\) 5.26172e6 0.168488
\(996\) 0 0
\(997\) 1.79581e7 0.572167 0.286084 0.958205i \(-0.407646\pi\)
0.286084 + 0.958205i \(0.407646\pi\)
\(998\) −1.35813e7 −0.431633
\(999\) 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.s.1.1 2
3.2 odd 2 147.6.a.k.1.2 2
7.3 odd 6 63.6.e.c.37.2 4
7.5 odd 6 63.6.e.c.46.2 4
7.6 odd 2 441.6.a.t.1.1 2
21.2 odd 6 147.6.e.l.67.1 4
21.5 even 6 21.6.e.b.4.1 4
21.11 odd 6 147.6.e.l.79.1 4
21.17 even 6 21.6.e.b.16.1 yes 4
21.20 even 2 147.6.a.i.1.2 2
84.47 odd 6 336.6.q.e.193.1 4
84.59 odd 6 336.6.q.e.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 4 21.5 even 6
21.6.e.b.16.1 yes 4 21.17 even 6
63.6.e.c.37.2 4 7.3 odd 6
63.6.e.c.46.2 4 7.5 odd 6
147.6.a.i.1.2 2 21.20 even 2
147.6.a.k.1.2 2 3.2 odd 2
147.6.e.l.67.1 4 21.2 odd 6
147.6.e.l.79.1 4 21.11 odd 6
336.6.q.e.193.1 4 84.47 odd 6
336.6.q.e.289.1 4 84.59 odd 6
441.6.a.s.1.1 2 1.1 even 1 trivial
441.6.a.t.1.1 2 7.6 odd 2