Properties

Label 912.6.a.y.1.1
Level $912$
Weight $6$
Character 912.1
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 4725 x^{4} + 92430 x^{3} + 1610577 x^{2} - 16081740 x - 24661341\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-74.1084\) of defining polynomial
Character \(\chi\) \(=\) 912.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.00000 q^{3} -86.8767 q^{5} -2.94760 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} -86.8767 q^{5} -2.94760 q^{7} +81.0000 q^{9} -318.804 q^{11} -638.130 q^{13} -781.890 q^{15} +219.917 q^{17} -361.000 q^{19} -26.5284 q^{21} -3471.66 q^{23} +4422.56 q^{25} +729.000 q^{27} -7206.90 q^{29} -5620.43 q^{31} -2869.24 q^{33} +256.078 q^{35} +13965.2 q^{37} -5743.17 q^{39} +14877.4 q^{41} -1988.69 q^{43} -7037.01 q^{45} -14219.0 q^{47} -16798.3 q^{49} +1979.26 q^{51} -16286.1 q^{53} +27696.7 q^{55} -3249.00 q^{57} +26626.3 q^{59} -11066.0 q^{61} -238.756 q^{63} +55438.6 q^{65} +39426.5 q^{67} -31244.9 q^{69} -52126.0 q^{71} -73711.7 q^{73} +39803.1 q^{75} +939.708 q^{77} +8701.62 q^{79} +6561.00 q^{81} +53001.4 q^{83} -19105.7 q^{85} -64862.1 q^{87} -125964. q^{89} +1880.95 q^{91} -50583.9 q^{93} +31362.5 q^{95} -73604.0 q^{97} -25823.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + O(q^{10}) \) \( 6 q + 54 q^{3} - 65 q^{5} + 149 q^{7} + 486 q^{9} + 203 q^{11} - 298 q^{13} - 585 q^{15} + 1319 q^{17} - 2166 q^{19} + 1341 q^{21} - 1234 q^{23} + 4589 q^{25} + 4374 q^{27} + 7356 q^{29} - 1632 q^{31} + 1827 q^{33} - 4383 q^{35} + 14204 q^{37} - 2682 q^{39} + 14734 q^{41} + 4693 q^{43} - 5265 q^{45} + 10955 q^{47} + 38561 q^{49} + 11871 q^{51} + 47500 q^{53} - 769 q^{55} - 19494 q^{57} + 61744 q^{59} - 81581 q^{61} + 12069 q^{63} - 59686 q^{65} + 45756 q^{67} - 11106 q^{69} + 10416 q^{71} - 54615 q^{73} + 41301 q^{75} - 29515 q^{77} + 145594 q^{79} + 39366 q^{81} + 160548 q^{83} - 53947 q^{85} + 66204 q^{87} - 97728 q^{89} + 418294 q^{91} - 14688 q^{93} + 23465 q^{95} - 760 q^{97} + 16443 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) −86.8767 −1.55410 −0.777049 0.629440i \(-0.783284\pi\)
−0.777049 + 0.629440i \(0.783284\pi\)
\(6\) 0 0
\(7\) −2.94760 −0.0227365 −0.0113682 0.999935i \(-0.503619\pi\)
−0.0113682 + 0.999935i \(0.503619\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −318.804 −0.794406 −0.397203 0.917731i \(-0.630019\pi\)
−0.397203 + 0.917731i \(0.630019\pi\)
\(12\) 0 0
\(13\) −638.130 −1.04725 −0.523625 0.851949i \(-0.675421\pi\)
−0.523625 + 0.851949i \(0.675421\pi\)
\(14\) 0 0
\(15\) −781.890 −0.897259
\(16\) 0 0
\(17\) 219.917 0.184560 0.0922799 0.995733i \(-0.470585\pi\)
0.0922799 + 0.995733i \(0.470585\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) −26.5284 −0.0131269
\(22\) 0 0
\(23\) −3471.66 −1.36841 −0.684207 0.729288i \(-0.739852\pi\)
−0.684207 + 0.729288i \(0.739852\pi\)
\(24\) 0 0
\(25\) 4422.56 1.41522
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −7206.90 −1.59130 −0.795652 0.605753i \(-0.792872\pi\)
−0.795652 + 0.605753i \(0.792872\pi\)
\(30\) 0 0
\(31\) −5620.43 −1.05043 −0.525213 0.850971i \(-0.676014\pi\)
−0.525213 + 0.850971i \(0.676014\pi\)
\(32\) 0 0
\(33\) −2869.24 −0.458651
\(34\) 0 0
\(35\) 256.078 0.0353347
\(36\) 0 0
\(37\) 13965.2 1.67704 0.838518 0.544874i \(-0.183422\pi\)
0.838518 + 0.544874i \(0.183422\pi\)
\(38\) 0 0
\(39\) −5743.17 −0.604630
\(40\) 0 0
\(41\) 14877.4 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(42\) 0 0
\(43\) −1988.69 −0.164020 −0.0820098 0.996632i \(-0.526134\pi\)
−0.0820098 + 0.996632i \(0.526134\pi\)
\(44\) 0 0
\(45\) −7037.01 −0.518033
\(46\) 0 0
\(47\) −14219.0 −0.938909 −0.469455 0.882957i \(-0.655549\pi\)
−0.469455 + 0.882957i \(0.655549\pi\)
\(48\) 0 0
\(49\) −16798.3 −0.999483
\(50\) 0 0
\(51\) 1979.26 0.106556
\(52\) 0 0
\(53\) −16286.1 −0.796395 −0.398197 0.917300i \(-0.630364\pi\)
−0.398197 + 0.917300i \(0.630364\pi\)
\(54\) 0 0
\(55\) 27696.7 1.23459
\(56\) 0 0
\(57\) −3249.00 −0.132453
\(58\) 0 0
\(59\) 26626.3 0.995819 0.497910 0.867229i \(-0.334101\pi\)
0.497910 + 0.867229i \(0.334101\pi\)
\(60\) 0 0
\(61\) −11066.0 −0.380774 −0.190387 0.981709i \(-0.560974\pi\)
−0.190387 + 0.981709i \(0.560974\pi\)
\(62\) 0 0
\(63\) −238.756 −0.00757883
\(64\) 0 0
\(65\) 55438.6 1.62753
\(66\) 0 0
\(67\) 39426.5 1.07300 0.536502 0.843899i \(-0.319746\pi\)
0.536502 + 0.843899i \(0.319746\pi\)
\(68\) 0 0
\(69\) −31244.9 −0.790054
\(70\) 0 0
\(71\) −52126.0 −1.22718 −0.613590 0.789625i \(-0.710275\pi\)
−0.613590 + 0.789625i \(0.710275\pi\)
\(72\) 0 0
\(73\) −73711.7 −1.61893 −0.809467 0.587165i \(-0.800244\pi\)
−0.809467 + 0.587165i \(0.800244\pi\)
\(74\) 0 0
\(75\) 39803.1 0.817078
\(76\) 0 0
\(77\) 939.708 0.0180620
\(78\) 0 0
\(79\) 8701.62 0.156867 0.0784336 0.996919i \(-0.475008\pi\)
0.0784336 + 0.996919i \(0.475008\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 53001.4 0.844485 0.422242 0.906483i \(-0.361243\pi\)
0.422242 + 0.906483i \(0.361243\pi\)
\(84\) 0 0
\(85\) −19105.7 −0.286824
\(86\) 0 0
\(87\) −64862.1 −0.918740
\(88\) 0 0
\(89\) −125964. −1.68567 −0.842833 0.538175i \(-0.819114\pi\)
−0.842833 + 0.538175i \(0.819114\pi\)
\(90\) 0 0
\(91\) 1880.95 0.0238108
\(92\) 0 0
\(93\) −50583.9 −0.606464
\(94\) 0 0
\(95\) 31362.5 0.356535
\(96\) 0 0
\(97\) −73604.0 −0.794277 −0.397139 0.917759i \(-0.629997\pi\)
−0.397139 + 0.917759i \(0.629997\pi\)
\(98\) 0 0
\(99\) −25823.2 −0.264802
\(100\) 0 0
\(101\) 127732. 1.24594 0.622968 0.782247i \(-0.285927\pi\)
0.622968 + 0.782247i \(0.285927\pi\)
\(102\) 0 0
\(103\) 127852. 1.18745 0.593723 0.804669i \(-0.297657\pi\)
0.593723 + 0.804669i \(0.297657\pi\)
\(104\) 0 0
\(105\) 2304.70 0.0204005
\(106\) 0 0
\(107\) −95172.5 −0.803623 −0.401811 0.915722i \(-0.631619\pi\)
−0.401811 + 0.915722i \(0.631619\pi\)
\(108\) 0 0
\(109\) 235632. 1.89962 0.949812 0.312820i \(-0.101274\pi\)
0.949812 + 0.312820i \(0.101274\pi\)
\(110\) 0 0
\(111\) 125687. 0.968237
\(112\) 0 0
\(113\) 256972. 1.89317 0.946584 0.322457i \(-0.104509\pi\)
0.946584 + 0.322457i \(0.104509\pi\)
\(114\) 0 0
\(115\) 301606. 2.12665
\(116\) 0 0
\(117\) −51688.5 −0.349083
\(118\) 0 0
\(119\) −648.229 −0.00419624
\(120\) 0 0
\(121\) −59414.7 −0.368919
\(122\) 0 0
\(123\) 133897. 0.798007
\(124\) 0 0
\(125\) −112728. −0.645293
\(126\) 0 0
\(127\) 57230.7 0.314862 0.157431 0.987530i \(-0.449679\pi\)
0.157431 + 0.987530i \(0.449679\pi\)
\(128\) 0 0
\(129\) −17898.2 −0.0946967
\(130\) 0 0
\(131\) −5007.18 −0.0254927 −0.0127463 0.999919i \(-0.504057\pi\)
−0.0127463 + 0.999919i \(0.504057\pi\)
\(132\) 0 0
\(133\) 1064.08 0.00521611
\(134\) 0 0
\(135\) −63333.1 −0.299086
\(136\) 0 0
\(137\) −402860. −1.83380 −0.916901 0.399115i \(-0.869317\pi\)
−0.916901 + 0.399115i \(0.869317\pi\)
\(138\) 0 0
\(139\) 395813. 1.73761 0.868807 0.495150i \(-0.164887\pi\)
0.868807 + 0.495150i \(0.164887\pi\)
\(140\) 0 0
\(141\) −127971. −0.542080
\(142\) 0 0
\(143\) 203439. 0.831942
\(144\) 0 0
\(145\) 626112. 2.47304
\(146\) 0 0
\(147\) −151185. −0.577052
\(148\) 0 0
\(149\) −86000.9 −0.317349 −0.158675 0.987331i \(-0.550722\pi\)
−0.158675 + 0.987331i \(0.550722\pi\)
\(150\) 0 0
\(151\) 257590. 0.919362 0.459681 0.888084i \(-0.347964\pi\)
0.459681 + 0.888084i \(0.347964\pi\)
\(152\) 0 0
\(153\) 17813.3 0.0615199
\(154\) 0 0
\(155\) 488285. 1.63246
\(156\) 0 0
\(157\) 80428.3 0.260411 0.130206 0.991487i \(-0.458436\pi\)
0.130206 + 0.991487i \(0.458436\pi\)
\(158\) 0 0
\(159\) −146575. −0.459799
\(160\) 0 0
\(161\) 10233.1 0.0311129
\(162\) 0 0
\(163\) 482818. 1.42336 0.711680 0.702504i \(-0.247935\pi\)
0.711680 + 0.702504i \(0.247935\pi\)
\(164\) 0 0
\(165\) 249270. 0.712788
\(166\) 0 0
\(167\) −219499. −0.609035 −0.304517 0.952507i \(-0.598495\pi\)
−0.304517 + 0.952507i \(0.598495\pi\)
\(168\) 0 0
\(169\) 35916.4 0.0967332
\(170\) 0 0
\(171\) −29241.0 −0.0764719
\(172\) 0 0
\(173\) 361299. 0.917807 0.458904 0.888486i \(-0.348242\pi\)
0.458904 + 0.888486i \(0.348242\pi\)
\(174\) 0 0
\(175\) −13036.0 −0.0321772
\(176\) 0 0
\(177\) 239637. 0.574937
\(178\) 0 0
\(179\) 622262. 1.45158 0.725790 0.687916i \(-0.241474\pi\)
0.725790 + 0.687916i \(0.241474\pi\)
\(180\) 0 0
\(181\) −104136. −0.236267 −0.118134 0.992998i \(-0.537691\pi\)
−0.118134 + 0.992998i \(0.537691\pi\)
\(182\) 0 0
\(183\) −99594.3 −0.219840
\(184\) 0 0
\(185\) −1.21325e6 −2.60628
\(186\) 0 0
\(187\) −70110.6 −0.146615
\(188\) 0 0
\(189\) −2148.80 −0.00437564
\(190\) 0 0
\(191\) −183238. −0.363439 −0.181719 0.983350i \(-0.558166\pi\)
−0.181719 + 0.983350i \(0.558166\pi\)
\(192\) 0 0
\(193\) −96854.3 −0.187165 −0.0935827 0.995612i \(-0.529832\pi\)
−0.0935827 + 0.995612i \(0.529832\pi\)
\(194\) 0 0
\(195\) 498947. 0.939655
\(196\) 0 0
\(197\) −275049. −0.504945 −0.252472 0.967604i \(-0.581244\pi\)
−0.252472 + 0.967604i \(0.581244\pi\)
\(198\) 0 0
\(199\) 256443. 0.459047 0.229524 0.973303i \(-0.426283\pi\)
0.229524 + 0.973303i \(0.426283\pi\)
\(200\) 0 0
\(201\) 354838. 0.619499
\(202\) 0 0
\(203\) 21243.1 0.0361807
\(204\) 0 0
\(205\) −1.29250e6 −2.14806
\(206\) 0 0
\(207\) −281204. −0.456138
\(208\) 0 0
\(209\) 115088. 0.182249
\(210\) 0 0
\(211\) −150209. −0.232268 −0.116134 0.993234i \(-0.537050\pi\)
−0.116134 + 0.993234i \(0.537050\pi\)
\(212\) 0 0
\(213\) −469134. −0.708513
\(214\) 0 0
\(215\) 172771. 0.254902
\(216\) 0 0
\(217\) 16566.8 0.0238830
\(218\) 0 0
\(219\) −663405. −0.934692
\(220\) 0 0
\(221\) −140336. −0.193280
\(222\) 0 0
\(223\) −1.00599e6 −1.35466 −0.677330 0.735679i \(-0.736863\pi\)
−0.677330 + 0.735679i \(0.736863\pi\)
\(224\) 0 0
\(225\) 358228. 0.471740
\(226\) 0 0
\(227\) −666788. −0.858862 −0.429431 0.903100i \(-0.641286\pi\)
−0.429431 + 0.903100i \(0.641286\pi\)
\(228\) 0 0
\(229\) 161657. 0.203708 0.101854 0.994799i \(-0.467523\pi\)
0.101854 + 0.994799i \(0.467523\pi\)
\(230\) 0 0
\(231\) 8457.37 0.0104281
\(232\) 0 0
\(233\) 932958. 1.12583 0.562914 0.826515i \(-0.309680\pi\)
0.562914 + 0.826515i \(0.309680\pi\)
\(234\) 0 0
\(235\) 1.23530e6 1.45916
\(236\) 0 0
\(237\) 78314.5 0.0905673
\(238\) 0 0
\(239\) −1.01934e6 −1.15431 −0.577156 0.816634i \(-0.695838\pi\)
−0.577156 + 0.816634i \(0.695838\pi\)
\(240\) 0 0
\(241\) −21598.0 −0.0239537 −0.0119768 0.999928i \(-0.503812\pi\)
−0.0119768 + 0.999928i \(0.503812\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) 1.45938e6 1.55329
\(246\) 0 0
\(247\) 230365. 0.240256
\(248\) 0 0
\(249\) 477012. 0.487564
\(250\) 0 0
\(251\) 1.00782e6 1.00972 0.504858 0.863202i \(-0.331545\pi\)
0.504858 + 0.863202i \(0.331545\pi\)
\(252\) 0 0
\(253\) 1.10678e6 1.08708
\(254\) 0 0
\(255\) −171951. −0.165598
\(256\) 0 0
\(257\) 112989. 0.106709 0.0533547 0.998576i \(-0.483009\pi\)
0.0533547 + 0.998576i \(0.483009\pi\)
\(258\) 0 0
\(259\) −41163.8 −0.0381299
\(260\) 0 0
\(261\) −583759. −0.530435
\(262\) 0 0
\(263\) 289482. 0.258067 0.129033 0.991640i \(-0.458813\pi\)
0.129033 + 0.991640i \(0.458813\pi\)
\(264\) 0 0
\(265\) 1.41489e6 1.23768
\(266\) 0 0
\(267\) −1.13368e6 −0.973220
\(268\) 0 0
\(269\) −885103. −0.745784 −0.372892 0.927875i \(-0.621634\pi\)
−0.372892 + 0.927875i \(0.621634\pi\)
\(270\) 0 0
\(271\) 1.75726e6 1.45349 0.726746 0.686906i \(-0.241032\pi\)
0.726746 + 0.686906i \(0.241032\pi\)
\(272\) 0 0
\(273\) 16928.6 0.0137472
\(274\) 0 0
\(275\) −1.40993e6 −1.12426
\(276\) 0 0
\(277\) 365208. 0.285984 0.142992 0.989724i \(-0.454328\pi\)
0.142992 + 0.989724i \(0.454328\pi\)
\(278\) 0 0
\(279\) −455255. −0.350142
\(280\) 0 0
\(281\) 1.54749e6 1.16913 0.584566 0.811346i \(-0.301265\pi\)
0.584566 + 0.811346i \(0.301265\pi\)
\(282\) 0 0
\(283\) −389877. −0.289375 −0.144688 0.989477i \(-0.546218\pi\)
−0.144688 + 0.989477i \(0.546218\pi\)
\(284\) 0 0
\(285\) 282262. 0.205845
\(286\) 0 0
\(287\) −43852.7 −0.0314262
\(288\) 0 0
\(289\) −1.37149e6 −0.965938
\(290\) 0 0
\(291\) −662436. −0.458576
\(292\) 0 0
\(293\) 289148. 0.196766 0.0983831 0.995149i \(-0.468633\pi\)
0.0983831 + 0.995149i \(0.468633\pi\)
\(294\) 0 0
\(295\) −2.31320e6 −1.54760
\(296\) 0 0
\(297\) −232408. −0.152884
\(298\) 0 0
\(299\) 2.21537e6 1.43307
\(300\) 0 0
\(301\) 5861.86 0.00372923
\(302\) 0 0
\(303\) 1.14959e6 0.719342
\(304\) 0 0
\(305\) 961380. 0.591760
\(306\) 0 0
\(307\) 3.12009e6 1.88939 0.944695 0.327950i \(-0.106358\pi\)
0.944695 + 0.327950i \(0.106358\pi\)
\(308\) 0 0
\(309\) 1.15067e6 0.685573
\(310\) 0 0
\(311\) −555696. −0.325789 −0.162895 0.986643i \(-0.552083\pi\)
−0.162895 + 0.986643i \(0.552083\pi\)
\(312\) 0 0
\(313\) 1.68130e6 0.970028 0.485014 0.874506i \(-0.338814\pi\)
0.485014 + 0.874506i \(0.338814\pi\)
\(314\) 0 0
\(315\) 20742.3 0.0117782
\(316\) 0 0
\(317\) −1.70075e6 −0.950586 −0.475293 0.879828i \(-0.657658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(318\) 0 0
\(319\) 2.29759e6 1.26414
\(320\) 0 0
\(321\) −856553. −0.463972
\(322\) 0 0
\(323\) −79390.2 −0.0423409
\(324\) 0 0
\(325\) −2.82217e6 −1.48209
\(326\) 0 0
\(327\) 2.12069e6 1.09675
\(328\) 0 0
\(329\) 41911.9 0.0213475
\(330\) 0 0
\(331\) −268046. −0.134475 −0.0672373 0.997737i \(-0.521418\pi\)
−0.0672373 + 0.997737i \(0.521418\pi\)
\(332\) 0 0
\(333\) 1.13118e6 0.559012
\(334\) 0 0
\(335\) −3.42524e6 −1.66755
\(336\) 0 0
\(337\) 52952.3 0.0253986 0.0126993 0.999919i \(-0.495958\pi\)
0.0126993 + 0.999919i \(0.495958\pi\)
\(338\) 0 0
\(339\) 2.31275e6 1.09302
\(340\) 0 0
\(341\) 1.79182e6 0.834465
\(342\) 0 0
\(343\) 99055.1 0.0454612
\(344\) 0 0
\(345\) 2.71446e6 1.22782
\(346\) 0 0
\(347\) −2.74946e6 −1.22581 −0.612905 0.790157i \(-0.709999\pi\)
−0.612905 + 0.790157i \(0.709999\pi\)
\(348\) 0 0
\(349\) 1.23589e6 0.543147 0.271573 0.962418i \(-0.412456\pi\)
0.271573 + 0.962418i \(0.412456\pi\)
\(350\) 0 0
\(351\) −465196. −0.201543
\(352\) 0 0
\(353\) 1.89335e6 0.808712 0.404356 0.914602i \(-0.367496\pi\)
0.404356 + 0.914602i \(0.367496\pi\)
\(354\) 0 0
\(355\) 4.52854e6 1.90716
\(356\) 0 0
\(357\) −5834.06 −0.00242270
\(358\) 0 0
\(359\) 2.28422e6 0.935411 0.467706 0.883884i \(-0.345081\pi\)
0.467706 + 0.883884i \(0.345081\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) −534733. −0.212995
\(364\) 0 0
\(365\) 6.40383e6 2.51598
\(366\) 0 0
\(367\) −858143. −0.332579 −0.166289 0.986077i \(-0.553179\pi\)
−0.166289 + 0.986077i \(0.553179\pi\)
\(368\) 0 0
\(369\) 1.20507e6 0.460730
\(370\) 0 0
\(371\) 48005.0 0.0181072
\(372\) 0 0
\(373\) 3.42136e6 1.27329 0.636644 0.771158i \(-0.280322\pi\)
0.636644 + 0.771158i \(0.280322\pi\)
\(374\) 0 0
\(375\) −1.01455e6 −0.372560
\(376\) 0 0
\(377\) 4.59893e6 1.66649
\(378\) 0 0
\(379\) 4.11049e6 1.46993 0.734963 0.678107i \(-0.237199\pi\)
0.734963 + 0.678107i \(0.237199\pi\)
\(380\) 0 0
\(381\) 515076. 0.181785
\(382\) 0 0
\(383\) −287612. −0.100187 −0.0500933 0.998745i \(-0.515952\pi\)
−0.0500933 + 0.998745i \(0.515952\pi\)
\(384\) 0 0
\(385\) −81638.8 −0.0280701
\(386\) 0 0
\(387\) −161084. −0.0546732
\(388\) 0 0
\(389\) −4.31602e6 −1.44614 −0.723068 0.690777i \(-0.757269\pi\)
−0.723068 + 0.690777i \(0.757269\pi\)
\(390\) 0 0
\(391\) −763478. −0.252554
\(392\) 0 0
\(393\) −45064.6 −0.0147182
\(394\) 0 0
\(395\) −755968. −0.243787
\(396\) 0 0
\(397\) −3.42142e6 −1.08951 −0.544754 0.838596i \(-0.683377\pi\)
−0.544754 + 0.838596i \(0.683377\pi\)
\(398\) 0 0
\(399\) 9576.76 0.00301152
\(400\) 0 0
\(401\) 4.54215e6 1.41059 0.705294 0.708914i \(-0.250815\pi\)
0.705294 + 0.708914i \(0.250815\pi\)
\(402\) 0 0
\(403\) 3.58656e6 1.10006
\(404\) 0 0
\(405\) −569998. −0.172678
\(406\) 0 0
\(407\) −4.45216e6 −1.33225
\(408\) 0 0
\(409\) −2.71102e6 −0.801354 −0.400677 0.916219i \(-0.631225\pi\)
−0.400677 + 0.916219i \(0.631225\pi\)
\(410\) 0 0
\(411\) −3.62574e6 −1.05875
\(412\) 0 0
\(413\) −78483.7 −0.0226414
\(414\) 0 0
\(415\) −4.60459e6 −1.31241
\(416\) 0 0
\(417\) 3.56232e6 1.00321
\(418\) 0 0
\(419\) 2.42371e6 0.674442 0.337221 0.941425i \(-0.390513\pi\)
0.337221 + 0.941425i \(0.390513\pi\)
\(420\) 0 0
\(421\) −3.06050e6 −0.841564 −0.420782 0.907162i \(-0.638244\pi\)
−0.420782 + 0.907162i \(0.638244\pi\)
\(422\) 0 0
\(423\) −1.15174e6 −0.312970
\(424\) 0 0
\(425\) 972598. 0.261193
\(426\) 0 0
\(427\) 32618.2 0.00865747
\(428\) 0 0
\(429\) 1.83095e6 0.480322
\(430\) 0 0
\(431\) 2.08114e6 0.539645 0.269823 0.962910i \(-0.413035\pi\)
0.269823 + 0.962910i \(0.413035\pi\)
\(432\) 0 0
\(433\) −4.21567e6 −1.08055 −0.540277 0.841487i \(-0.681681\pi\)
−0.540277 + 0.841487i \(0.681681\pi\)
\(434\) 0 0
\(435\) 5.63500e6 1.42781
\(436\) 0 0
\(437\) 1.25327e6 0.313936
\(438\) 0 0
\(439\) −2.24036e6 −0.554826 −0.277413 0.960751i \(-0.589477\pi\)
−0.277413 + 0.960751i \(0.589477\pi\)
\(440\) 0 0
\(441\) −1.36066e6 −0.333161
\(442\) 0 0
\(443\) −5.08323e6 −1.23064 −0.615319 0.788278i \(-0.710973\pi\)
−0.615319 + 0.788278i \(0.710973\pi\)
\(444\) 0 0
\(445\) 1.09433e7 2.61969
\(446\) 0 0
\(447\) −774008. −0.183222
\(448\) 0 0
\(449\) 3.67478e6 0.860232 0.430116 0.902774i \(-0.358473\pi\)
0.430116 + 0.902774i \(0.358473\pi\)
\(450\) 0 0
\(451\) −4.74298e6 −1.09802
\(452\) 0 0
\(453\) 2.31831e6 0.530794
\(454\) 0 0
\(455\) −163411. −0.0370043
\(456\) 0 0
\(457\) −2.30561e6 −0.516412 −0.258206 0.966090i \(-0.583131\pi\)
−0.258206 + 0.966090i \(0.583131\pi\)
\(458\) 0 0
\(459\) 160320. 0.0355186
\(460\) 0 0
\(461\) −4.00465e6 −0.877631 −0.438816 0.898577i \(-0.644602\pi\)
−0.438816 + 0.898577i \(0.644602\pi\)
\(462\) 0 0
\(463\) 573291. 0.124286 0.0621430 0.998067i \(-0.480207\pi\)
0.0621430 + 0.998067i \(0.480207\pi\)
\(464\) 0 0
\(465\) 4.39456e6 0.942504
\(466\) 0 0
\(467\) −4.13715e6 −0.877827 −0.438914 0.898529i \(-0.644637\pi\)
−0.438914 + 0.898529i \(0.644637\pi\)
\(468\) 0 0
\(469\) −116214. −0.0243963
\(470\) 0 0
\(471\) 723855. 0.150348
\(472\) 0 0
\(473\) 634003. 0.130298
\(474\) 0 0
\(475\) −1.59655e6 −0.324674
\(476\) 0 0
\(477\) −1.31918e6 −0.265465
\(478\) 0 0
\(479\) 9.05846e6 1.80391 0.901956 0.431827i \(-0.142131\pi\)
0.901956 + 0.431827i \(0.142131\pi\)
\(480\) 0 0
\(481\) −8.91160e6 −1.75628
\(482\) 0 0
\(483\) 92097.6 0.0179631
\(484\) 0 0
\(485\) 6.39448e6 1.23439
\(486\) 0 0
\(487\) 8.05132e6 1.53831 0.769157 0.639060i \(-0.220677\pi\)
0.769157 + 0.639060i \(0.220677\pi\)
\(488\) 0 0
\(489\) 4.34536e6 0.821777
\(490\) 0 0
\(491\) −5.43572e6 −1.01754 −0.508772 0.860901i \(-0.669900\pi\)
−0.508772 + 0.860901i \(0.669900\pi\)
\(492\) 0 0
\(493\) −1.58492e6 −0.293691
\(494\) 0 0
\(495\) 2.24343e6 0.411528
\(496\) 0 0
\(497\) 153647. 0.0279018
\(498\) 0 0
\(499\) −9.75490e6 −1.75377 −0.876883 0.480704i \(-0.840381\pi\)
−0.876883 + 0.480704i \(0.840381\pi\)
\(500\) 0 0
\(501\) −1.97549e6 −0.351626
\(502\) 0 0
\(503\) −61342.4 −0.0108104 −0.00540519 0.999985i \(-0.501721\pi\)
−0.00540519 + 0.999985i \(0.501721\pi\)
\(504\) 0 0
\(505\) −1.10969e7 −1.93631
\(506\) 0 0
\(507\) 323247. 0.0558490
\(508\) 0 0
\(509\) −9.96677e6 −1.70514 −0.852570 0.522613i \(-0.824957\pi\)
−0.852570 + 0.522613i \(0.824957\pi\)
\(510\) 0 0
\(511\) 217273. 0.0368089
\(512\) 0 0
\(513\) −263169. −0.0441511
\(514\) 0 0
\(515\) −1.11074e7 −1.84541
\(516\) 0 0
\(517\) 4.53307e6 0.745876
\(518\) 0 0
\(519\) 3.25169e6 0.529896
\(520\) 0 0
\(521\) −9.39975e6 −1.51713 −0.758563 0.651599i \(-0.774098\pi\)
−0.758563 + 0.651599i \(0.774098\pi\)
\(522\) 0 0
\(523\) 6.85510e6 1.09587 0.547936 0.836520i \(-0.315414\pi\)
0.547936 + 0.836520i \(0.315414\pi\)
\(524\) 0 0
\(525\) −117324. −0.0185775
\(526\) 0 0
\(527\) −1.23603e6 −0.193866
\(528\) 0 0
\(529\) 5.61608e6 0.872557
\(530\) 0 0
\(531\) 2.15673e6 0.331940
\(532\) 0 0
\(533\) −9.49371e6 −1.44750
\(534\) 0 0
\(535\) 8.26828e6 1.24891
\(536\) 0 0
\(537\) 5.60036e6 0.838070
\(538\) 0 0
\(539\) 5.35538e6 0.793996
\(540\) 0 0
\(541\) −5.56585e6 −0.817595 −0.408797 0.912625i \(-0.634052\pi\)
−0.408797 + 0.912625i \(0.634052\pi\)
\(542\) 0 0
\(543\) −937221. −0.136409
\(544\) 0 0
\(545\) −2.04709e7 −2.95220
\(546\) 0 0
\(547\) 1.05049e7 1.50115 0.750573 0.660787i \(-0.229777\pi\)
0.750573 + 0.660787i \(0.229777\pi\)
\(548\) 0 0
\(549\) −896349. −0.126925
\(550\) 0 0
\(551\) 2.60169e6 0.365070
\(552\) 0 0
\(553\) −25648.9 −0.00356661
\(554\) 0 0
\(555\) −1.09192e7 −1.50474
\(556\) 0 0
\(557\) −8.51679e6 −1.16316 −0.581578 0.813490i \(-0.697565\pi\)
−0.581578 + 0.813490i \(0.697565\pi\)
\(558\) 0 0
\(559\) 1.26904e6 0.171769
\(560\) 0 0
\(561\) −630996. −0.0846485
\(562\) 0 0
\(563\) 8.65897e6 1.15132 0.575659 0.817690i \(-0.304746\pi\)
0.575659 + 0.817690i \(0.304746\pi\)
\(564\) 0 0
\(565\) −2.23249e7 −2.94217
\(566\) 0 0
\(567\) −19339.2 −0.00252628
\(568\) 0 0
\(569\) −5.60596e6 −0.725887 −0.362944 0.931811i \(-0.618228\pi\)
−0.362944 + 0.931811i \(0.618228\pi\)
\(570\) 0 0
\(571\) −7.13232e6 −0.915463 −0.457731 0.889090i \(-0.651338\pi\)
−0.457731 + 0.889090i \(0.651338\pi\)
\(572\) 0 0
\(573\) −1.64914e6 −0.209831
\(574\) 0 0
\(575\) −1.53536e7 −1.93661
\(576\) 0 0
\(577\) −3.42233e6 −0.427940 −0.213970 0.976840i \(-0.568639\pi\)
−0.213970 + 0.976840i \(0.568639\pi\)
\(578\) 0 0
\(579\) −871689. −0.108060
\(580\) 0 0
\(581\) −156227. −0.0192006
\(582\) 0 0
\(583\) 5.19209e6 0.632661
\(584\) 0 0
\(585\) 4.49053e6 0.542510
\(586\) 0 0
\(587\) 1.09057e7 1.30634 0.653171 0.757210i \(-0.273438\pi\)
0.653171 + 0.757210i \(0.273438\pi\)
\(588\) 0 0
\(589\) 2.02898e6 0.240984
\(590\) 0 0
\(591\) −2.47544e6 −0.291530
\(592\) 0 0
\(593\) −8.51651e6 −0.994546 −0.497273 0.867594i \(-0.665665\pi\)
−0.497273 + 0.867594i \(0.665665\pi\)
\(594\) 0 0
\(595\) 56316.0 0.00652137
\(596\) 0 0
\(597\) 2.30798e6 0.265031
\(598\) 0 0
\(599\) 30851.7 0.00351327 0.00175664 0.999998i \(-0.499441\pi\)
0.00175664 + 0.999998i \(0.499441\pi\)
\(600\) 0 0
\(601\) 8.18341e6 0.924162 0.462081 0.886838i \(-0.347103\pi\)
0.462081 + 0.886838i \(0.347103\pi\)
\(602\) 0 0
\(603\) 3.19355e6 0.357668
\(604\) 0 0
\(605\) 5.16176e6 0.573336
\(606\) 0 0
\(607\) −4.22459e6 −0.465385 −0.232693 0.972550i \(-0.574754\pi\)
−0.232693 + 0.972550i \(0.574754\pi\)
\(608\) 0 0
\(609\) 191188. 0.0208889
\(610\) 0 0
\(611\) 9.07355e6 0.983273
\(612\) 0 0
\(613\) 1.66199e6 0.178640 0.0893199 0.996003i \(-0.471531\pi\)
0.0893199 + 0.996003i \(0.471531\pi\)
\(614\) 0 0
\(615\) −1.16325e7 −1.24018
\(616\) 0 0
\(617\) 6.20540e6 0.656231 0.328115 0.944638i \(-0.393587\pi\)
0.328115 + 0.944638i \(0.393587\pi\)
\(618\) 0 0
\(619\) −1.59871e7 −1.67704 −0.838519 0.544872i \(-0.816578\pi\)
−0.838519 + 0.544872i \(0.816578\pi\)
\(620\) 0 0
\(621\) −2.53084e6 −0.263351
\(622\) 0 0
\(623\) 371292. 0.0383261
\(624\) 0 0
\(625\) −4.02706e6 −0.412371
\(626\) 0 0
\(627\) 1.03580e6 0.105222
\(628\) 0 0
\(629\) 3.07119e6 0.309513
\(630\) 0 0
\(631\) 1.12004e7 1.11985 0.559923 0.828544i \(-0.310831\pi\)
0.559923 + 0.828544i \(0.310831\pi\)
\(632\) 0 0
\(633\) −1.35188e6 −0.134100
\(634\) 0 0
\(635\) −4.97201e6 −0.489326
\(636\) 0 0
\(637\) 1.07195e7 1.04671
\(638\) 0 0
\(639\) −4.22221e6 −0.409060
\(640\) 0 0
\(641\) 9.74956e6 0.937217 0.468608 0.883406i \(-0.344756\pi\)
0.468608 + 0.883406i \(0.344756\pi\)
\(642\) 0 0
\(643\) 6.53602e6 0.623427 0.311714 0.950176i \(-0.399097\pi\)
0.311714 + 0.950176i \(0.399097\pi\)
\(644\) 0 0
\(645\) 1.55494e6 0.147168
\(646\) 0 0
\(647\) −8.79430e6 −0.825925 −0.412962 0.910748i \(-0.635506\pi\)
−0.412962 + 0.910748i \(0.635506\pi\)
\(648\) 0 0
\(649\) −8.48858e6 −0.791085
\(650\) 0 0
\(651\) 149101. 0.0137889
\(652\) 0 0
\(653\) −1.15593e7 −1.06084 −0.530418 0.847736i \(-0.677965\pi\)
−0.530418 + 0.847736i \(0.677965\pi\)
\(654\) 0 0
\(655\) 435008. 0.0396181
\(656\) 0 0
\(657\) −5.97065e6 −0.539645
\(658\) 0 0
\(659\) −1.52916e7 −1.37163 −0.685817 0.727774i \(-0.740555\pi\)
−0.685817 + 0.727774i \(0.740555\pi\)
\(660\) 0 0
\(661\) −5.89824e6 −0.525072 −0.262536 0.964922i \(-0.584559\pi\)
−0.262536 + 0.964922i \(0.584559\pi\)
\(662\) 0 0
\(663\) −1.26302e6 −0.111590
\(664\) 0 0
\(665\) −92444.1 −0.00810635
\(666\) 0 0
\(667\) 2.50199e7 2.17756
\(668\) 0 0
\(669\) −9.05389e6 −0.782114
\(670\) 0 0
\(671\) 3.52790e6 0.302489
\(672\) 0 0
\(673\) −1.20024e7 −1.02148 −0.510740 0.859735i \(-0.670628\pi\)
−0.510740 + 0.859735i \(0.670628\pi\)
\(674\) 0 0
\(675\) 3.22405e6 0.272359
\(676\) 0 0
\(677\) −1.65605e7 −1.38868 −0.694338 0.719649i \(-0.744303\pi\)
−0.694338 + 0.719649i \(0.744303\pi\)
\(678\) 0 0
\(679\) 216955. 0.0180591
\(680\) 0 0
\(681\) −6.00109e6 −0.495864
\(682\) 0 0
\(683\) −401168. −0.0329060 −0.0164530 0.999865i \(-0.505237\pi\)
−0.0164530 + 0.999865i \(0.505237\pi\)
\(684\) 0 0
\(685\) 3.49991e7 2.84991
\(686\) 0 0
\(687\) 1.45492e6 0.117611
\(688\) 0 0
\(689\) 1.03927e7 0.834025
\(690\) 0 0
\(691\) −1.65388e6 −0.131767 −0.0658837 0.997827i \(-0.520987\pi\)
−0.0658837 + 0.997827i \(0.520987\pi\)
\(692\) 0 0
\(693\) 76116.4 0.00602067
\(694\) 0 0
\(695\) −3.43870e7 −2.70042
\(696\) 0 0
\(697\) 3.27180e6 0.255097
\(698\) 0 0
\(699\) 8.39662e6 0.649998
\(700\) 0 0
\(701\) −2.31470e6 −0.177910 −0.0889549 0.996036i \(-0.528353\pi\)
−0.0889549 + 0.996036i \(0.528353\pi\)
\(702\) 0 0
\(703\) −5.04143e6 −0.384739
\(704\) 0 0
\(705\) 1.11177e7 0.842445
\(706\) 0 0
\(707\) −376503. −0.0283282
\(708\) 0 0
\(709\) 167819. 0.0125379 0.00626897 0.999980i \(-0.498005\pi\)
0.00626897 + 0.999980i \(0.498005\pi\)
\(710\) 0 0
\(711\) 704831. 0.0522891
\(712\) 0 0
\(713\) 1.95122e7 1.43742
\(714\) 0 0
\(715\) −1.76741e7 −1.29292
\(716\) 0 0
\(717\) −9.17404e6 −0.666443
\(718\) 0 0
\(719\) −5.38526e6 −0.388494 −0.194247 0.980953i \(-0.562226\pi\)
−0.194247 + 0.980953i \(0.562226\pi\)
\(720\) 0 0
\(721\) −376856. −0.0269984
\(722\) 0 0
\(723\) −194382. −0.0138297
\(724\) 0 0
\(725\) −3.18730e7 −2.25205
\(726\) 0 0
\(727\) 2.38223e7 1.67166 0.835828 0.548991i \(-0.184988\pi\)
0.835828 + 0.548991i \(0.184988\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −437347. −0.0302714
\(732\) 0 0
\(733\) 3.43335e6 0.236025 0.118013 0.993012i \(-0.462348\pi\)
0.118013 + 0.993012i \(0.462348\pi\)
\(734\) 0 0
\(735\) 1.31344e7 0.896795
\(736\) 0 0
\(737\) −1.25693e7 −0.852401
\(738\) 0 0
\(739\) 1.72898e7 1.16460 0.582302 0.812973i \(-0.302152\pi\)
0.582302 + 0.812973i \(0.302152\pi\)
\(740\) 0 0
\(741\) 2.07328e6 0.138712
\(742\) 0 0
\(743\) −6.69285e6 −0.444773 −0.222387 0.974959i \(-0.571385\pi\)
−0.222387 + 0.974959i \(0.571385\pi\)
\(744\) 0 0
\(745\) 7.47147e6 0.493192
\(746\) 0 0
\(747\) 4.29311e6 0.281495
\(748\) 0 0
\(749\) 280531. 0.0182716
\(750\) 0 0
\(751\) 1.81435e7 1.17387 0.586937 0.809633i \(-0.300334\pi\)
0.586937 + 0.809633i \(0.300334\pi\)
\(752\) 0 0
\(753\) 9.07039e6 0.582960
\(754\) 0 0
\(755\) −2.23786e7 −1.42878
\(756\) 0 0
\(757\) 2.79893e6 0.177522 0.0887610 0.996053i \(-0.471709\pi\)
0.0887610 + 0.996053i \(0.471709\pi\)
\(758\) 0 0
\(759\) 9.96102e6 0.627624
\(760\) 0 0
\(761\) −6.44239e6 −0.403261 −0.201630 0.979462i \(-0.564624\pi\)
−0.201630 + 0.979462i \(0.564624\pi\)
\(762\) 0 0
\(763\) −694549. −0.0431908
\(764\) 0 0
\(765\) −1.54756e6 −0.0956080
\(766\) 0 0
\(767\) −1.69910e7 −1.04287
\(768\) 0 0
\(769\) −1.03180e7 −0.629188 −0.314594 0.949226i \(-0.601868\pi\)
−0.314594 + 0.949226i \(0.601868\pi\)
\(770\) 0 0
\(771\) 1.01690e6 0.0616087
\(772\) 0 0
\(773\) 2.30416e6 0.138696 0.0693480 0.997593i \(-0.477908\pi\)
0.0693480 + 0.997593i \(0.477908\pi\)
\(774\) 0 0
\(775\) −2.48567e7 −1.48658
\(776\) 0 0
\(777\) −370474. −0.0220143
\(778\) 0 0
\(779\) −5.37074e6 −0.317096
\(780\) 0 0
\(781\) 1.66180e7 0.974880
\(782\) 0 0
\(783\) −5.25383e6 −0.306247
\(784\) 0 0
\(785\) −6.98735e6 −0.404705
\(786\) 0 0
\(787\) −1.05381e7 −0.606495 −0.303247 0.952912i \(-0.598071\pi\)
−0.303247 + 0.952912i \(0.598071\pi\)
\(788\) 0 0
\(789\) 2.60534e6 0.148995
\(790\) 0 0
\(791\) −757450. −0.0430440
\(792\) 0 0
\(793\) 7.06156e6 0.398766
\(794\) 0 0
\(795\) 1.27340e7 0.714572
\(796\) 0 0
\(797\) 1.28930e6 0.0718966 0.0359483 0.999354i \(-0.488555\pi\)
0.0359483 + 0.999354i \(0.488555\pi\)
\(798\) 0 0
\(799\) −3.12700e6 −0.173285
\(800\) 0 0
\(801\) −1.02031e7 −0.561889
\(802\) 0 0
\(803\) 2.34996e7 1.28609
\(804\) 0 0
\(805\) −889015. −0.0483526
\(806\) 0 0
\(807\) −7.96593e6 −0.430579
\(808\) 0 0
\(809\) 8.06076e6 0.433017 0.216508 0.976281i \(-0.430533\pi\)
0.216508 + 0.976281i \(0.430533\pi\)
\(810\) 0 0
\(811\) 3.63148e7 1.93879 0.969397 0.245499i \(-0.0789519\pi\)
0.969397 + 0.245499i \(0.0789519\pi\)
\(812\) 0 0
\(813\) 1.58153e7 0.839174
\(814\) 0 0
\(815\) −4.19457e7 −2.21204
\(816\) 0 0
\(817\) 717916. 0.0376287
\(818\) 0 0
\(819\) 152357. 0.00793694
\(820\) 0 0
\(821\) −2.27674e7 −1.17884 −0.589420 0.807827i \(-0.700644\pi\)
−0.589420 + 0.807827i \(0.700644\pi\)
\(822\) 0 0
\(823\) −2.46863e7 −1.27045 −0.635224 0.772328i \(-0.719092\pi\)
−0.635224 + 0.772328i \(0.719092\pi\)
\(824\) 0 0
\(825\) −1.26894e7 −0.649092
\(826\) 0 0
\(827\) 2.14922e6 0.109274 0.0546369 0.998506i \(-0.482600\pi\)
0.0546369 + 0.998506i \(0.482600\pi\)
\(828\) 0 0
\(829\) 2.31358e7 1.16922 0.584612 0.811313i \(-0.301247\pi\)
0.584612 + 0.811313i \(0.301247\pi\)
\(830\) 0 0
\(831\) 3.28687e6 0.165113
\(832\) 0 0
\(833\) −3.69424e6 −0.184464
\(834\) 0 0
\(835\) 1.90694e7 0.946500
\(836\) 0 0
\(837\) −4.09729e6 −0.202155
\(838\) 0 0
\(839\) 2.41020e7 1.18208 0.591042 0.806641i \(-0.298717\pi\)
0.591042 + 0.806641i \(0.298717\pi\)
\(840\) 0 0
\(841\) 3.14282e7 1.53225
\(842\) 0 0
\(843\) 1.39275e7 0.674998
\(844\) 0 0
\(845\) −3.12030e6 −0.150333
\(846\) 0 0
\(847\) 175131. 0.00838792
\(848\) 0 0
\(849\) −3.50889e6 −0.167071
\(850\) 0 0
\(851\) −4.84824e7 −2.29488
\(852\) 0 0
\(853\) −2.00928e7 −0.945515 −0.472758 0.881192i \(-0.656741\pi\)
−0.472758 + 0.881192i \(0.656741\pi\)
\(854\) 0 0
\(855\) 2.54036e6 0.118845
\(856\) 0 0
\(857\) −2.38204e7 −1.10789 −0.553944 0.832554i \(-0.686878\pi\)
−0.553944 + 0.832554i \(0.686878\pi\)
\(858\) 0 0
\(859\) −5.90779e6 −0.273176 −0.136588 0.990628i \(-0.543614\pi\)
−0.136588 + 0.990628i \(0.543614\pi\)
\(860\) 0 0
\(861\) −394674. −0.0181439
\(862\) 0 0
\(863\) −3.98849e7 −1.82298 −0.911490 0.411322i \(-0.865067\pi\)
−0.911490 + 0.411322i \(0.865067\pi\)
\(864\) 0 0
\(865\) −3.13885e7 −1.42636
\(866\) 0 0
\(867\) −1.23434e7 −0.557684
\(868\) 0 0
\(869\) −2.77411e6 −0.124616
\(870\) 0 0
\(871\) −2.51592e7 −1.12370
\(872\) 0 0
\(873\) −5.96193e6 −0.264759
\(874\) 0 0
\(875\) 332278. 0.0146717
\(876\) 0 0
\(877\) 2.37640e7 1.04333 0.521664 0.853151i \(-0.325311\pi\)
0.521664 + 0.853151i \(0.325311\pi\)
\(878\) 0 0
\(879\) 2.60233e6 0.113603
\(880\) 0 0
\(881\) −3.96169e7 −1.71965 −0.859827 0.510586i \(-0.829429\pi\)
−0.859827 + 0.510586i \(0.829429\pi\)
\(882\) 0 0
\(883\) −8.43589e6 −0.364107 −0.182054 0.983289i \(-0.558274\pi\)
−0.182054 + 0.983289i \(0.558274\pi\)
\(884\) 0 0
\(885\) −2.08188e7 −0.893508
\(886\) 0 0
\(887\) −6.39703e6 −0.273004 −0.136502 0.990640i \(-0.543586\pi\)
−0.136502 + 0.990640i \(0.543586\pi\)
\(888\) 0 0
\(889\) −168693. −0.00715885
\(890\) 0 0
\(891\) −2.09168e6 −0.0882674
\(892\) 0 0
\(893\) 5.13305e6 0.215401
\(894\) 0 0
\(895\) −5.40601e7 −2.25590
\(896\) 0 0
\(897\) 1.99383e7 0.827385
\(898\) 0 0
\(899\) 4.05059e7 1.67155
\(900\) 0 0
\(901\) −3.58160e6 −0.146982
\(902\) 0 0
\(903\) 52756.7 0.00215307
\(904\) 0 0
\(905\) 9.04697e6 0.367182
\(906\) 0 0
\(907\) −2.81799e7 −1.13742 −0.568711 0.822538i \(-0.692558\pi\)
−0.568711 + 0.822538i \(0.692558\pi\)
\(908\) 0 0
\(909\) 1.03463e7 0.415312
\(910\) 0 0
\(911\) −2.13420e7 −0.852000 −0.426000 0.904723i \(-0.640078\pi\)
−0.426000 + 0.904723i \(0.640078\pi\)
\(912\) 0 0
\(913\) −1.68971e7 −0.670864
\(914\) 0 0
\(915\) 8.65242e6 0.341653
\(916\) 0 0
\(917\) 14759.2 0.000579614 0
\(918\) 0 0
\(919\) −2.45601e6 −0.0959272 −0.0479636 0.998849i \(-0.515273\pi\)
−0.0479636 + 0.998849i \(0.515273\pi\)
\(920\) 0 0
\(921\) 2.80808e7 1.09084
\(922\) 0 0
\(923\) 3.32631e7 1.28517
\(924\) 0 0
\(925\) 6.17619e7 2.37338
\(926\) 0 0
\(927\) 1.03560e7 0.395816
\(928\) 0 0
\(929\) −1.58818e7 −0.603755 −0.301877 0.953347i \(-0.597613\pi\)
−0.301877 + 0.953347i \(0.597613\pi\)
\(930\) 0 0
\(931\) 6.06419e6 0.229297
\(932\) 0 0
\(933\) −5.00127e6 −0.188094
\(934\) 0 0
\(935\) 6.09098e6 0.227855
\(936\) 0 0
\(937\) −2.86186e7 −1.06488 −0.532439 0.846469i \(-0.678724\pi\)
−0.532439 + 0.846469i \(0.678724\pi\)
\(938\) 0 0
\(939\) 1.51317e7 0.560046
\(940\) 0 0
\(941\) 5.22102e7 1.92212 0.961061 0.276337i \(-0.0891205\pi\)
0.961061 + 0.276337i \(0.0891205\pi\)
\(942\) 0 0
\(943\) −5.16493e7 −1.89141
\(944\) 0 0
\(945\) 186681. 0.00680018
\(946\) 0 0
\(947\) 3.01070e7 1.09092 0.545459 0.838138i \(-0.316355\pi\)
0.545459 + 0.838138i \(0.316355\pi\)
\(948\) 0 0
\(949\) 4.70376e7 1.69543
\(950\) 0 0
\(951\) −1.53067e7 −0.548821
\(952\) 0 0
\(953\) 4.11241e7 1.46678 0.733388 0.679810i \(-0.237938\pi\)
0.733388 + 0.679810i \(0.237938\pi\)
\(954\) 0 0
\(955\) 1.59191e7 0.564819
\(956\) 0 0
\(957\) 2.06783e7 0.729853
\(958\) 0 0
\(959\) 1.18747e6 0.0416942
\(960\) 0 0
\(961\) 2.96010e6 0.103395
\(962\) 0 0
\(963\) −7.70898e6 −0.267874
\(964\) 0 0
\(965\) 8.41438e6 0.290873
\(966\) 0 0
\(967\) −8.91988e6 −0.306756 −0.153378 0.988168i \(-0.549015\pi\)
−0.153378 + 0.988168i \(0.549015\pi\)
\(968\) 0 0
\(969\) −714511. −0.0244455
\(970\) 0 0
\(971\) 7.68387e6 0.261536 0.130768 0.991413i \(-0.458256\pi\)
0.130768 + 0.991413i \(0.458256\pi\)
\(972\) 0 0
\(973\) −1.16670e6 −0.0395073
\(974\) 0 0
\(975\) −2.53995e7 −0.855685
\(976\) 0 0
\(977\) −4.77434e7 −1.60021 −0.800105 0.599860i \(-0.795223\pi\)
−0.800105 + 0.599860i \(0.795223\pi\)
\(978\) 0 0
\(979\) 4.01579e7 1.33910
\(980\) 0 0
\(981\) 1.90862e7 0.633208
\(982\) 0 0
\(983\) −2.36245e7 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(984\) 0 0
\(985\) 2.38953e7 0.784734
\(986\) 0 0
\(987\) 377207. 0.0123250
\(988\) 0 0
\(989\) 6.90405e6 0.224447
\(990\) 0 0
\(991\) −1.16867e6 −0.0378014 −0.0189007 0.999821i \(-0.506017\pi\)
−0.0189007 + 0.999821i \(0.506017\pi\)
\(992\) 0 0
\(993\) −2.41242e6 −0.0776389
\(994\) 0 0
\(995\) −2.22789e7 −0.713405
\(996\) 0 0
\(997\) 7.06390e6 0.225064 0.112532 0.993648i \(-0.464104\pi\)
0.112532 + 0.993648i \(0.464104\pi\)
\(998\) 0 0
\(999\) 1.01806e7 0.322746
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 912.6.a.y.1.1 6
4.3 odd 2 456.6.a.f.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.6.a.f.1.1 6 4.3 odd 2
912.6.a.y.1.1 6 1.1 even 1 trivial