Properties

Label 441.6.a.be.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 7^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.308653\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} -16.5098 q^{8} +O(q^{10})\) \(q-8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} -16.5098 q^{8} -141.556 q^{10} +114.280 q^{11} +205.240 q^{13} -954.857 q^{16} -757.277 q^{17} +1013.65 q^{19} +592.838 q^{20} -928.635 q^{22} +916.299 q^{23} -2821.54 q^{25} -1667.78 q^{26} +1095.47 q^{29} -8233.32 q^{31} +8287.47 q^{32} +6153.62 q^{34} -10716.4 q^{37} -8236.94 q^{38} -287.604 q^{40} +18758.1 q^{41} -4643.49 q^{43} +3889.13 q^{44} -7445.84 q^{46} -13969.4 q^{47} +22927.8 q^{50} +6984.69 q^{52} +29306.0 q^{53} +1990.77 q^{55} -8901.78 q^{58} -30378.0 q^{59} -18658.2 q^{61} +66903.9 q^{62} -36788.5 q^{64} +3575.32 q^{65} +19933.4 q^{67} -25771.4 q^{68} +57338.5 q^{71} +60194.4 q^{73} +87081.2 q^{74} +34496.4 q^{76} +35715.5 q^{79} -16633.7 q^{80} -152428. q^{82} +86641.2 q^{83} -13191.9 q^{85} +37733.0 q^{86} -1886.73 q^{88} -42941.7 q^{89} +31183.2 q^{92} +113515. q^{94} +17658.0 q^{95} -20619.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.12599 −1.43649 −0.718243 0.695792i \(-0.755053\pi\)
−0.718243 + 0.695792i \(0.755053\pi\)
\(3\) 0 0
\(4\) 34.0317 1.06349
\(5\) 17.4201 0.311621 0.155811 0.987787i \(-0.450201\pi\)
0.155811 + 0.987787i \(0.450201\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −16.5098 −0.0912047
\(9\) 0 0
\(10\) −141.556 −0.447639
\(11\) 114.280 0.284765 0.142383 0.989812i \(-0.454524\pi\)
0.142383 + 0.989812i \(0.454524\pi\)
\(12\) 0 0
\(13\) 205.240 0.336825 0.168413 0.985717i \(-0.446136\pi\)
0.168413 + 0.985717i \(0.446136\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −954.857 −0.932477
\(17\) −757.277 −0.635524 −0.317762 0.948170i \(-0.602931\pi\)
−0.317762 + 0.948170i \(0.602931\pi\)
\(18\) 0 0
\(19\) 1013.65 0.644178 0.322089 0.946709i \(-0.395615\pi\)
0.322089 + 0.946709i \(0.395615\pi\)
\(20\) 592.838 0.331406
\(21\) 0 0
\(22\) −928.635 −0.409061
\(23\) 916.299 0.361175 0.180587 0.983559i \(-0.442200\pi\)
0.180587 + 0.983559i \(0.442200\pi\)
\(24\) 0 0
\(25\) −2821.54 −0.902892
\(26\) −1667.78 −0.483845
\(27\) 0 0
\(28\) 0 0
\(29\) 1095.47 0.241883 0.120942 0.992660i \(-0.461409\pi\)
0.120942 + 0.992660i \(0.461409\pi\)
\(30\) 0 0
\(31\) −8233.32 −1.53876 −0.769380 0.638791i \(-0.779435\pi\)
−0.769380 + 0.638791i \(0.779435\pi\)
\(32\) 8287.47 1.43070
\(33\) 0 0
\(34\) 6153.62 0.912922
\(35\) 0 0
\(36\) 0 0
\(37\) −10716.4 −1.28690 −0.643448 0.765490i \(-0.722497\pi\)
−0.643448 + 0.765490i \(0.722497\pi\)
\(38\) −8236.94 −0.925352
\(39\) 0 0
\(40\) −287.604 −0.0284213
\(41\) 18758.1 1.74273 0.871365 0.490636i \(-0.163235\pi\)
0.871365 + 0.490636i \(0.163235\pi\)
\(42\) 0 0
\(43\) −4643.49 −0.382978 −0.191489 0.981495i \(-0.561332\pi\)
−0.191489 + 0.981495i \(0.561332\pi\)
\(44\) 3889.13 0.302845
\(45\) 0 0
\(46\) −7445.84 −0.518823
\(47\) −13969.4 −0.922428 −0.461214 0.887289i \(-0.652586\pi\)
−0.461214 + 0.887289i \(0.652586\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 22927.8 1.29699
\(51\) 0 0
\(52\) 6984.69 0.358211
\(53\) 29306.0 1.43307 0.716533 0.697553i \(-0.245728\pi\)
0.716533 + 0.697553i \(0.245728\pi\)
\(54\) 0 0
\(55\) 1990.77 0.0887388
\(56\) 0 0
\(57\) 0 0
\(58\) −8901.78 −0.347462
\(59\) −30378.0 −1.13613 −0.568067 0.822983i \(-0.692308\pi\)
−0.568067 + 0.822983i \(0.692308\pi\)
\(60\) 0 0
\(61\) −18658.2 −0.642014 −0.321007 0.947077i \(-0.604021\pi\)
−0.321007 + 0.947077i \(0.604021\pi\)
\(62\) 66903.9 2.21041
\(63\) 0 0
\(64\) −36788.5 −1.12270
\(65\) 3575.32 0.104962
\(66\) 0 0
\(67\) 19933.4 0.542493 0.271246 0.962510i \(-0.412564\pi\)
0.271246 + 0.962510i \(0.412564\pi\)
\(68\) −25771.4 −0.675875
\(69\) 0 0
\(70\) 0 0
\(71\) 57338.5 1.34990 0.674948 0.737866i \(-0.264166\pi\)
0.674948 + 0.737866i \(0.264166\pi\)
\(72\) 0 0
\(73\) 60194.4 1.32205 0.661027 0.750362i \(-0.270121\pi\)
0.661027 + 0.750362i \(0.270121\pi\)
\(74\) 87081.2 1.84861
\(75\) 0 0
\(76\) 34496.4 0.685078
\(77\) 0 0
\(78\) 0 0
\(79\) 35715.5 0.643857 0.321928 0.946764i \(-0.395669\pi\)
0.321928 + 0.946764i \(0.395669\pi\)
\(80\) −16633.7 −0.290580
\(81\) 0 0
\(82\) −152428. −2.50341
\(83\) 86641.2 1.38048 0.690239 0.723582i \(-0.257506\pi\)
0.690239 + 0.723582i \(0.257506\pi\)
\(84\) 0 0
\(85\) −13191.9 −0.198043
\(86\) 37733.0 0.550142
\(87\) 0 0
\(88\) −1886.73 −0.0259719
\(89\) −42941.7 −0.574651 −0.287326 0.957833i \(-0.592766\pi\)
−0.287326 + 0.957833i \(0.592766\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 31183.2 0.384107
\(93\) 0 0
\(94\) 113515. 1.32506
\(95\) 17658.0 0.200739
\(96\) 0 0
\(97\) −20619.4 −0.222508 −0.111254 0.993792i \(-0.535487\pi\)
−0.111254 + 0.993792i \(0.535487\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −96021.8 −0.960218
\(101\) −106588. −1.03970 −0.519848 0.854259i \(-0.674012\pi\)
−0.519848 + 0.854259i \(0.674012\pi\)
\(102\) 0 0
\(103\) −87392.9 −0.811677 −0.405839 0.913945i \(-0.633020\pi\)
−0.405839 + 0.913945i \(0.633020\pi\)
\(104\) −3388.48 −0.0307201
\(105\) 0 0
\(106\) −238140. −2.05858
\(107\) −39306.9 −0.331902 −0.165951 0.986134i \(-0.553069\pi\)
−0.165951 + 0.986134i \(0.553069\pi\)
\(108\) 0 0
\(109\) −44341.3 −0.357472 −0.178736 0.983897i \(-0.557201\pi\)
−0.178736 + 0.983897i \(0.557201\pi\)
\(110\) −16177.0 −0.127472
\(111\) 0 0
\(112\) 0 0
\(113\) −64665.1 −0.476402 −0.238201 0.971216i \(-0.576558\pi\)
−0.238201 + 0.971216i \(0.576558\pi\)
\(114\) 0 0
\(115\) 15962.1 0.112550
\(116\) 37280.7 0.257241
\(117\) 0 0
\(118\) 246851. 1.63204
\(119\) 0 0
\(120\) 0 0
\(121\) −147991. −0.918909
\(122\) 151616. 0.922244
\(123\) 0 0
\(124\) −280194. −1.63646
\(125\) −103590. −0.592981
\(126\) 0 0
\(127\) −155448. −0.855215 −0.427607 0.903965i \(-0.640643\pi\)
−0.427607 + 0.903965i \(0.640643\pi\)
\(128\) 33744.0 0.182042
\(129\) 0 0
\(130\) −29053.0 −0.150776
\(131\) 189150. 0.963002 0.481501 0.876446i \(-0.340092\pi\)
0.481501 + 0.876446i \(0.340092\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −161978. −0.779283
\(135\) 0 0
\(136\) 12502.5 0.0579628
\(137\) −344445. −1.56790 −0.783950 0.620824i \(-0.786798\pi\)
−0.783950 + 0.620824i \(0.786798\pi\)
\(138\) 0 0
\(139\) 270646. 1.18813 0.594065 0.804417i \(-0.297522\pi\)
0.594065 + 0.804417i \(0.297522\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −465932. −1.93911
\(143\) 23454.8 0.0959161
\(144\) 0 0
\(145\) 19083.2 0.0753759
\(146\) −489139. −1.89911
\(147\) 0 0
\(148\) −364697. −1.36860
\(149\) −131911. −0.486759 −0.243379 0.969931i \(-0.578256\pi\)
−0.243379 + 0.969931i \(0.578256\pi\)
\(150\) 0 0
\(151\) −322961. −1.15268 −0.576338 0.817211i \(-0.695519\pi\)
−0.576338 + 0.817211i \(0.695519\pi\)
\(152\) −16735.2 −0.0587521
\(153\) 0 0
\(154\) 0 0
\(155\) −143426. −0.479510
\(156\) 0 0
\(157\) −181881. −0.588894 −0.294447 0.955668i \(-0.595135\pi\)
−0.294447 + 0.955668i \(0.595135\pi\)
\(158\) −290224. −0.924891
\(159\) 0 0
\(160\) 144369. 0.445835
\(161\) 0 0
\(162\) 0 0
\(163\) −196117. −0.578157 −0.289078 0.957305i \(-0.593349\pi\)
−0.289078 + 0.957305i \(0.593349\pi\)
\(164\) 638372. 1.85338
\(165\) 0 0
\(166\) −704046. −1.98304
\(167\) −293276. −0.813741 −0.406870 0.913486i \(-0.633380\pi\)
−0.406870 + 0.913486i \(0.633380\pi\)
\(168\) 0 0
\(169\) −329169. −0.886549
\(170\) 107197. 0.284486
\(171\) 0 0
\(172\) −158026. −0.407294
\(173\) 442828. 1.12492 0.562458 0.826826i \(-0.309856\pi\)
0.562458 + 0.826826i \(0.309856\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −109121. −0.265537
\(177\) 0 0
\(178\) 348944. 0.825479
\(179\) −661446. −1.54298 −0.771492 0.636239i \(-0.780489\pi\)
−0.771492 + 0.636239i \(0.780489\pi\)
\(180\) 0 0
\(181\) −139700. −0.316956 −0.158478 0.987363i \(-0.550659\pi\)
−0.158478 + 0.987363i \(0.550659\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −15127.9 −0.0329409
\(185\) −186681. −0.401024
\(186\) 0 0
\(187\) −86541.2 −0.180975
\(188\) −475402. −0.980995
\(189\) 0 0
\(190\) −143489. −0.288359
\(191\) −751907. −1.49135 −0.745677 0.666308i \(-0.767874\pi\)
−0.745677 + 0.666308i \(0.767874\pi\)
\(192\) 0 0
\(193\) 464929. 0.898449 0.449224 0.893419i \(-0.351700\pi\)
0.449224 + 0.893419i \(0.351700\pi\)
\(194\) 167553. 0.319630
\(195\) 0 0
\(196\) 0 0
\(197\) −518493. −0.951870 −0.475935 0.879480i \(-0.657890\pi\)
−0.475935 + 0.879480i \(0.657890\pi\)
\(198\) 0 0
\(199\) 590287. 1.05665 0.528324 0.849043i \(-0.322821\pi\)
0.528324 + 0.849043i \(0.322821\pi\)
\(200\) 46583.1 0.0823480
\(201\) 0 0
\(202\) 866137. 1.49351
\(203\) 0 0
\(204\) 0 0
\(205\) 326770. 0.543071
\(206\) 710154. 1.16596
\(207\) 0 0
\(208\) −195975. −0.314082
\(209\) 115840. 0.183439
\(210\) 0 0
\(211\) 207037. 0.320142 0.160071 0.987106i \(-0.448828\pi\)
0.160071 + 0.987106i \(0.448828\pi\)
\(212\) 997333. 1.52405
\(213\) 0 0
\(214\) 319408. 0.476772
\(215\) −80890.3 −0.119344
\(216\) 0 0
\(217\) 0 0
\(218\) 360317. 0.513503
\(219\) 0 0
\(220\) 67749.2 0.0943730
\(221\) −155424. −0.214061
\(222\) 0 0
\(223\) −1.23347e6 −1.66098 −0.830491 0.557033i \(-0.811940\pi\)
−0.830491 + 0.557033i \(0.811940\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 525468. 0.684345
\(227\) −1.12039e6 −1.44312 −0.721562 0.692350i \(-0.756576\pi\)
−0.721562 + 0.692350i \(0.756576\pi\)
\(228\) 0 0
\(229\) −412271. −0.519510 −0.259755 0.965675i \(-0.583642\pi\)
−0.259755 + 0.965675i \(0.583642\pi\)
\(230\) −129708. −0.161676
\(231\) 0 0
\(232\) −18086.0 −0.0220609
\(233\) −836448. −1.00937 −0.504684 0.863304i \(-0.668391\pi\)
−0.504684 + 0.863304i \(0.668391\pi\)
\(234\) 0 0
\(235\) −243349. −0.287448
\(236\) −1.03382e6 −1.20827
\(237\) 0 0
\(238\) 0 0
\(239\) 319694. 0.362026 0.181013 0.983481i \(-0.442062\pi\)
0.181013 + 0.983481i \(0.442062\pi\)
\(240\) 0 0
\(241\) 965937. 1.07129 0.535644 0.844444i \(-0.320069\pi\)
0.535644 + 0.844444i \(0.320069\pi\)
\(242\) 1.20258e6 1.32000
\(243\) 0 0
\(244\) −634970. −0.682776
\(245\) 0 0
\(246\) 0 0
\(247\) 208043. 0.216975
\(248\) 135931. 0.140342
\(249\) 0 0
\(250\) 841768. 0.851809
\(251\) 1.60171e6 1.60472 0.802361 0.596839i \(-0.203577\pi\)
0.802361 + 0.596839i \(0.203577\pi\)
\(252\) 0 0
\(253\) 104714. 0.102850
\(254\) 1.26317e6 1.22850
\(255\) 0 0
\(256\) 903029. 0.861196
\(257\) 1.08451e6 1.02424 0.512120 0.858914i \(-0.328860\pi\)
0.512120 + 0.858914i \(0.328860\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 121674. 0.111626
\(261\) 0 0
\(262\) −1.53703e6 −1.38334
\(263\) 1.04604e6 0.932517 0.466259 0.884648i \(-0.345602\pi\)
0.466259 + 0.884648i \(0.345602\pi\)
\(264\) 0 0
\(265\) 510514. 0.446574
\(266\) 0 0
\(267\) 0 0
\(268\) 678368. 0.576937
\(269\) −1.41567e6 −1.19284 −0.596419 0.802673i \(-0.703410\pi\)
−0.596419 + 0.802673i \(0.703410\pi\)
\(270\) 0 0
\(271\) 884078. 0.731252 0.365626 0.930762i \(-0.380855\pi\)
0.365626 + 0.930762i \(0.380855\pi\)
\(272\) 723091. 0.592612
\(273\) 0 0
\(274\) 2.79896e6 2.25227
\(275\) −322444. −0.257112
\(276\) 0 0
\(277\) −1.77062e6 −1.38652 −0.693258 0.720690i \(-0.743825\pi\)
−0.693258 + 0.720690i \(0.743825\pi\)
\(278\) −2.19926e6 −1.70673
\(279\) 0 0
\(280\) 0 0
\(281\) 802195. 0.606058 0.303029 0.952981i \(-0.402002\pi\)
0.303029 + 0.952981i \(0.402002\pi\)
\(282\) 0 0
\(283\) −1.81693e6 −1.34856 −0.674282 0.738474i \(-0.735547\pi\)
−0.674282 + 0.738474i \(0.735547\pi\)
\(284\) 1.95133e6 1.43560
\(285\) 0 0
\(286\) −190593. −0.137782
\(287\) 0 0
\(288\) 0 0
\(289\) −846389. −0.596109
\(290\) −155070. −0.108276
\(291\) 0 0
\(292\) 2.04852e6 1.40599
\(293\) 2.34122e6 1.59321 0.796605 0.604500i \(-0.206627\pi\)
0.796605 + 0.604500i \(0.206627\pi\)
\(294\) 0 0
\(295\) −529189. −0.354043
\(296\) 176925. 0.117371
\(297\) 0 0
\(298\) 1.07190e6 0.699222
\(299\) 188062. 0.121653
\(300\) 0 0
\(301\) 0 0
\(302\) 2.62438e6 1.65580
\(303\) 0 0
\(304\) −967894. −0.600681
\(305\) −325028. −0.200065
\(306\) 0 0
\(307\) −855540. −0.518077 −0.259039 0.965867i \(-0.583406\pi\)
−0.259039 + 0.965867i \(0.583406\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.16548e6 0.688809
\(311\) −2.58071e6 −1.51300 −0.756498 0.653996i \(-0.773091\pi\)
−0.756498 + 0.653996i \(0.773091\pi\)
\(312\) 0 0
\(313\) 3.03363e6 1.75026 0.875129 0.483889i \(-0.160776\pi\)
0.875129 + 0.483889i \(0.160776\pi\)
\(314\) 1.47796e6 0.845938
\(315\) 0 0
\(316\) 1.21546e6 0.684736
\(317\) −751440. −0.419997 −0.209999 0.977702i \(-0.567346\pi\)
−0.209999 + 0.977702i \(0.567346\pi\)
\(318\) 0 0
\(319\) 125190. 0.0688799
\(320\) −640861. −0.349856
\(321\) 0 0
\(322\) 0 0
\(323\) −767616. −0.409391
\(324\) 0 0
\(325\) −579094. −0.304117
\(326\) 1.59364e6 0.830514
\(327\) 0 0
\(328\) −309693. −0.158945
\(329\) 0 0
\(330\) 0 0
\(331\) −726862. −0.364655 −0.182327 0.983238i \(-0.558363\pi\)
−0.182327 + 0.983238i \(0.558363\pi\)
\(332\) 2.94855e6 1.46813
\(333\) 0 0
\(334\) 2.38316e6 1.16893
\(335\) 347243. 0.169052
\(336\) 0 0
\(337\) 650213. 0.311875 0.155938 0.987767i \(-0.450160\pi\)
0.155938 + 0.987767i \(0.450160\pi\)
\(338\) 2.67483e6 1.27351
\(339\) 0 0
\(340\) −448942. −0.210617
\(341\) −940900. −0.438185
\(342\) 0 0
\(343\) 0 0
\(344\) 76663.2 0.0349294
\(345\) 0 0
\(346\) −3.59842e6 −1.61593
\(347\) −60293.3 −0.0268810 −0.0134405 0.999910i \(-0.504278\pi\)
−0.0134405 + 0.999910i \(0.504278\pi\)
\(348\) 0 0
\(349\) −3.61218e6 −1.58747 −0.793735 0.608263i \(-0.791866\pi\)
−0.793735 + 0.608263i \(0.791866\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 947088. 0.407412
\(353\) −681893. −0.291259 −0.145629 0.989339i \(-0.546521\pi\)
−0.145629 + 0.989339i \(0.546521\pi\)
\(354\) 0 0
\(355\) 998844. 0.420656
\(356\) −1.46138e6 −0.611137
\(357\) 0 0
\(358\) 5.37490e6 2.21648
\(359\) −3.14620e6 −1.28840 −0.644200 0.764857i \(-0.722810\pi\)
−0.644200 + 0.764857i \(0.722810\pi\)
\(360\) 0 0
\(361\) −1.44860e6 −0.585035
\(362\) 1.13520e6 0.455303
\(363\) 0 0
\(364\) 0 0
\(365\) 1.04860e6 0.411980
\(366\) 0 0
\(367\) 2.97099e6 1.15143 0.575714 0.817651i \(-0.304724\pi\)
0.575714 + 0.817651i \(0.304724\pi\)
\(368\) −874934. −0.336787
\(369\) 0 0
\(370\) 1.51697e6 0.576066
\(371\) 0 0
\(372\) 0 0
\(373\) 3.00412e6 1.11801 0.559005 0.829164i \(-0.311183\pi\)
0.559005 + 0.829164i \(0.311183\pi\)
\(374\) 703233. 0.259968
\(375\) 0 0
\(376\) 230632. 0.0841298
\(377\) 224835. 0.0814723
\(378\) 0 0
\(379\) −754946. −0.269972 −0.134986 0.990848i \(-0.543099\pi\)
−0.134986 + 0.990848i \(0.543099\pi\)
\(380\) 600932. 0.213485
\(381\) 0 0
\(382\) 6.10999e6 2.14231
\(383\) 3.66829e6 1.27781 0.638905 0.769286i \(-0.279388\pi\)
0.638905 + 0.769286i \(0.279388\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.77801e6 −1.29061
\(387\) 0 0
\(388\) −701712. −0.236635
\(389\) −3.00960e6 −1.00841 −0.504203 0.863585i \(-0.668214\pi\)
−0.504203 + 0.863585i \(0.668214\pi\)
\(390\) 0 0
\(391\) −693892. −0.229536
\(392\) 0 0
\(393\) 0 0
\(394\) 4.21327e6 1.36735
\(395\) 622170. 0.200639
\(396\) 0 0
\(397\) −287204. −0.0914564 −0.0457282 0.998954i \(-0.514561\pi\)
−0.0457282 + 0.998954i \(0.514561\pi\)
\(398\) −4.79666e6 −1.51786
\(399\) 0 0
\(400\) 2.69416e6 0.841927
\(401\) 863171. 0.268063 0.134031 0.990977i \(-0.457208\pi\)
0.134031 + 0.990977i \(0.457208\pi\)
\(402\) 0 0
\(403\) −1.68981e6 −0.518293
\(404\) −3.62739e6 −1.10571
\(405\) 0 0
\(406\) 0 0
\(407\) −1.22466e6 −0.366463
\(408\) 0 0
\(409\) 2.82932e6 0.836321 0.418161 0.908373i \(-0.362675\pi\)
0.418161 + 0.908373i \(0.362675\pi\)
\(410\) −2.65533e6 −0.780114
\(411\) 0 0
\(412\) −2.97413e6 −0.863212
\(413\) 0 0
\(414\) 0 0
\(415\) 1.50930e6 0.430186
\(416\) 1.70092e6 0.481894
\(417\) 0 0
\(418\) −941314. −0.263508
\(419\) −7.08132e6 −1.97051 −0.985256 0.171089i \(-0.945272\pi\)
−0.985256 + 0.171089i \(0.945272\pi\)
\(420\) 0 0
\(421\) −2.95296e6 −0.811994 −0.405997 0.913874i \(-0.633076\pi\)
−0.405997 + 0.913874i \(0.633076\pi\)
\(422\) −1.68238e6 −0.459879
\(423\) 0 0
\(424\) −483836. −0.130702
\(425\) 2.13668e6 0.573810
\(426\) 0 0
\(427\) 0 0
\(428\) −1.33768e6 −0.352975
\(429\) 0 0
\(430\) 657314. 0.171436
\(431\) −4.47340e6 −1.15996 −0.579981 0.814630i \(-0.696940\pi\)
−0.579981 + 0.814630i \(0.696940\pi\)
\(432\) 0 0
\(433\) −5.11279e6 −1.31050 −0.655252 0.755411i \(-0.727437\pi\)
−0.655252 + 0.755411i \(0.727437\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.50901e6 −0.380168
\(437\) 928810. 0.232661
\(438\) 0 0
\(439\) 590666. 0.146279 0.0731393 0.997322i \(-0.476698\pi\)
0.0731393 + 0.997322i \(0.476698\pi\)
\(440\) −32867.2 −0.00809340
\(441\) 0 0
\(442\) 1.26297e6 0.307495
\(443\) −160765. −0.0389209 −0.0194605 0.999811i \(-0.506195\pi\)
−0.0194605 + 0.999811i \(0.506195\pi\)
\(444\) 0 0
\(445\) −748051. −0.179073
\(446\) 1.00231e7 2.38598
\(447\) 0 0
\(448\) 0 0
\(449\) −7.98274e6 −1.86869 −0.934343 0.356376i \(-0.884012\pi\)
−0.934343 + 0.356376i \(0.884012\pi\)
\(450\) 0 0
\(451\) 2.14367e6 0.496269
\(452\) −2.20067e6 −0.506650
\(453\) 0 0
\(454\) 9.10426e6 2.07303
\(455\) 0 0
\(456\) 0 0
\(457\) −1.22463e6 −0.274292 −0.137146 0.990551i \(-0.543793\pi\)
−0.137146 + 0.990551i \(0.543793\pi\)
\(458\) 3.35011e6 0.746269
\(459\) 0 0
\(460\) 543217. 0.119696
\(461\) −1.99446e6 −0.437091 −0.218546 0.975827i \(-0.570131\pi\)
−0.218546 + 0.975827i \(0.570131\pi\)
\(462\) 0 0
\(463\) −125144. −0.0271304 −0.0135652 0.999908i \(-0.504318\pi\)
−0.0135652 + 0.999908i \(0.504318\pi\)
\(464\) −1.04602e6 −0.225550
\(465\) 0 0
\(466\) 6.79697e6 1.44994
\(467\) −7.06372e6 −1.49879 −0.749396 0.662122i \(-0.769656\pi\)
−0.749396 + 0.662122i \(0.769656\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.97745e6 0.412915
\(471\) 0 0
\(472\) 501535. 0.103621
\(473\) −530656. −0.109059
\(474\) 0 0
\(475\) −2.86006e6 −0.581623
\(476\) 0 0
\(477\) 0 0
\(478\) −2.59783e6 −0.520045
\(479\) 4.64810e6 0.925629 0.462814 0.886455i \(-0.346840\pi\)
0.462814 + 0.886455i \(0.346840\pi\)
\(480\) 0 0
\(481\) −2.19943e6 −0.433459
\(482\) −7.84919e6 −1.53889
\(483\) 0 0
\(484\) −5.03640e6 −0.977252
\(485\) −359192. −0.0693382
\(486\) 0 0
\(487\) −5.57906e6 −1.06595 −0.532977 0.846130i \(-0.678927\pi\)
−0.532977 + 0.846130i \(0.678927\pi\)
\(488\) 308043. 0.0585547
\(489\) 0 0
\(490\) 0 0
\(491\) 5.81454e6 1.08846 0.544229 0.838937i \(-0.316822\pi\)
0.544229 + 0.838937i \(0.316822\pi\)
\(492\) 0 0
\(493\) −829574. −0.153723
\(494\) −1.69055e6 −0.311682
\(495\) 0 0
\(496\) 7.86164e6 1.43486
\(497\) 0 0
\(498\) 0 0
\(499\) −7.17797e6 −1.29048 −0.645238 0.763982i \(-0.723242\pi\)
−0.645238 + 0.763982i \(0.723242\pi\)
\(500\) −3.52533e6 −0.630631
\(501\) 0 0
\(502\) −1.30155e7 −2.30516
\(503\) 3.00238e6 0.529109 0.264555 0.964371i \(-0.414775\pi\)
0.264555 + 0.964371i \(0.414775\pi\)
\(504\) 0 0
\(505\) −1.85679e6 −0.323991
\(506\) −850907. −0.147743
\(507\) 0 0
\(508\) −5.29016e6 −0.909514
\(509\) 3.52386e6 0.602870 0.301435 0.953487i \(-0.402534\pi\)
0.301435 + 0.953487i \(0.402534\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8.41781e6 −1.41914
\(513\) 0 0
\(514\) −8.81274e6 −1.47131
\(515\) −1.52240e6 −0.252936
\(516\) 0 0
\(517\) −1.59641e6 −0.262675
\(518\) 0 0
\(519\) 0 0
\(520\) −59027.9 −0.00957302
\(521\) 3.20451e6 0.517210 0.258605 0.965983i \(-0.416737\pi\)
0.258605 + 0.965983i \(0.416737\pi\)
\(522\) 0 0
\(523\) 8.14584e6 1.30221 0.651106 0.758987i \(-0.274305\pi\)
0.651106 + 0.758987i \(0.274305\pi\)
\(524\) 6.43709e6 1.02414
\(525\) 0 0
\(526\) −8.50007e6 −1.33955
\(527\) 6.23490e6 0.977920
\(528\) 0 0
\(529\) −5.59674e6 −0.869553
\(530\) −4.14843e6 −0.641497
\(531\) 0 0
\(532\) 0 0
\(533\) 3.84993e6 0.586995
\(534\) 0 0
\(535\) −684733. −0.103428
\(536\) −329097. −0.0494779
\(537\) 0 0
\(538\) 1.15037e7 1.71349
\(539\) 0 0
\(540\) 0 0
\(541\) −9.33721e6 −1.37159 −0.685794 0.727795i \(-0.740545\pi\)
−0.685794 + 0.727795i \(0.740545\pi\)
\(542\) −7.18401e6 −1.05043
\(543\) 0 0
\(544\) −6.27591e6 −0.909242
\(545\) −772432. −0.111396
\(546\) 0 0
\(547\) −7.27605e6 −1.03975 −0.519873 0.854244i \(-0.674021\pi\)
−0.519873 + 0.854244i \(0.674021\pi\)
\(548\) −1.17221e7 −1.66745
\(549\) 0 0
\(550\) 2.62018e6 0.369338
\(551\) 1.11043e6 0.155816
\(552\) 0 0
\(553\) 0 0
\(554\) 1.43880e7 1.99171
\(555\) 0 0
\(556\) 9.21054e6 1.26357
\(557\) 9.10577e6 1.24359 0.621797 0.783178i \(-0.286403\pi\)
0.621797 + 0.783178i \(0.286403\pi\)
\(558\) 0 0
\(559\) −953032. −0.128997
\(560\) 0 0
\(561\) 0 0
\(562\) −6.51863e6 −0.870594
\(563\) −1.93485e6 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(564\) 0 0
\(565\) −1.12648e6 −0.148457
\(566\) 1.47643e7 1.93719
\(567\) 0 0
\(568\) −946648. −0.123117
\(569\) 8.73948e6 1.13163 0.565816 0.824532i \(-0.308561\pi\)
0.565816 + 0.824532i \(0.308561\pi\)
\(570\) 0 0
\(571\) −1.03137e7 −1.32380 −0.661902 0.749591i \(-0.730250\pi\)
−0.661902 + 0.749591i \(0.730250\pi\)
\(572\) 798207. 0.102006
\(573\) 0 0
\(574\) 0 0
\(575\) −2.58537e6 −0.326102
\(576\) 0 0
\(577\) 5.56614e6 0.696009 0.348004 0.937493i \(-0.386859\pi\)
0.348004 + 0.937493i \(0.386859\pi\)
\(578\) 6.87775e6 0.856302
\(579\) 0 0
\(580\) 649436. 0.0801616
\(581\) 0 0
\(582\) 0 0
\(583\) 3.34907e6 0.408087
\(584\) −993799. −0.120578
\(585\) 0 0
\(586\) −1.90247e7 −2.28862
\(587\) 1.57310e6 0.188435 0.0942173 0.995552i \(-0.469965\pi\)
0.0942173 + 0.995552i \(0.469965\pi\)
\(588\) 0 0
\(589\) −8.34574e6 −0.991235
\(590\) 4.30019e6 0.508578
\(591\) 0 0
\(592\) 1.02326e7 1.20000
\(593\) −1.62461e7 −1.89720 −0.948599 0.316481i \(-0.897499\pi\)
−0.948599 + 0.316481i \(0.897499\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.48914e6 −0.517664
\(597\) 0 0
\(598\) −1.52819e6 −0.174753
\(599\) 1.69388e7 1.92893 0.964465 0.264210i \(-0.0851113\pi\)
0.964465 + 0.264210i \(0.0851113\pi\)
\(600\) 0 0
\(601\) 700155. 0.0790693 0.0395347 0.999218i \(-0.487412\pi\)
0.0395347 + 0.999218i \(0.487412\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.09909e7 −1.22586
\(605\) −2.57803e6 −0.286351
\(606\) 0 0
\(607\) −3.13306e6 −0.345141 −0.172570 0.984997i \(-0.555207\pi\)
−0.172570 + 0.984997i \(0.555207\pi\)
\(608\) 8.40063e6 0.921622
\(609\) 0 0
\(610\) 2.64118e6 0.287391
\(611\) −2.86708e6 −0.310697
\(612\) 0 0
\(613\) 2.65140e6 0.284986 0.142493 0.989796i \(-0.454488\pi\)
0.142493 + 0.989796i \(0.454488\pi\)
\(614\) 6.95211e6 0.744210
\(615\) 0 0
\(616\) 0 0
\(617\) 1.47657e7 1.56149 0.780747 0.624848i \(-0.214839\pi\)
0.780747 + 0.624848i \(0.214839\pi\)
\(618\) 0 0
\(619\) 8.33398e6 0.874230 0.437115 0.899406i \(-0.356000\pi\)
0.437115 + 0.899406i \(0.356000\pi\)
\(620\) −4.88103e6 −0.509955
\(621\) 0 0
\(622\) 2.09708e7 2.17340
\(623\) 0 0
\(624\) 0 0
\(625\) 7.01276e6 0.718107
\(626\) −2.46513e7 −2.51422
\(627\) 0 0
\(628\) −6.18971e6 −0.626284
\(629\) 8.11526e6 0.817854
\(630\) 0 0
\(631\) −1.17243e7 −1.17223 −0.586116 0.810227i \(-0.699344\pi\)
−0.586116 + 0.810227i \(0.699344\pi\)
\(632\) −589657. −0.0587228
\(633\) 0 0
\(634\) 6.10620e6 0.603320
\(635\) −2.70792e6 −0.266503
\(636\) 0 0
\(637\) 0 0
\(638\) −1.01729e6 −0.0989449
\(639\) 0 0
\(640\) 587825. 0.0567281
\(641\) 8.58605e6 0.825369 0.412685 0.910874i \(-0.364591\pi\)
0.412685 + 0.910874i \(0.364591\pi\)
\(642\) 0 0
\(643\) −1.87633e7 −1.78970 −0.894852 0.446364i \(-0.852719\pi\)
−0.894852 + 0.446364i \(0.852719\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.23764e6 0.588084
\(647\) 1.59559e7 1.49851 0.749255 0.662282i \(-0.230412\pi\)
0.749255 + 0.662282i \(0.230412\pi\)
\(648\) 0 0
\(649\) −3.47159e6 −0.323531
\(650\) 4.70571e6 0.436860
\(651\) 0 0
\(652\) −6.67419e6 −0.614865
\(653\) 1.36524e7 1.25293 0.626464 0.779451i \(-0.284502\pi\)
0.626464 + 0.779451i \(0.284502\pi\)
\(654\) 0 0
\(655\) 3.29501e6 0.300092
\(656\) −1.79113e7 −1.62506
\(657\) 0 0
\(658\) 0 0
\(659\) 4.32613e6 0.388049 0.194024 0.980997i \(-0.437846\pi\)
0.194024 + 0.980997i \(0.437846\pi\)
\(660\) 0 0
\(661\) 1.97317e7 1.75655 0.878275 0.478157i \(-0.158695\pi\)
0.878275 + 0.478157i \(0.158695\pi\)
\(662\) 5.90647e6 0.523821
\(663\) 0 0
\(664\) −1.43043e6 −0.125906
\(665\) 0 0
\(666\) 0 0
\(667\) 1.00378e6 0.0873621
\(668\) −9.98070e6 −0.865406
\(669\) 0 0
\(670\) −2.82169e6 −0.242841
\(671\) −2.13225e6 −0.182823
\(672\) 0 0
\(673\) 9.88089e6 0.840927 0.420464 0.907309i \(-0.361867\pi\)
0.420464 + 0.907309i \(0.361867\pi\)
\(674\) −5.28362e6 −0.448004
\(675\) 0 0
\(676\) −1.12022e7 −0.942837
\(677\) −2.22205e6 −0.186329 −0.0931647 0.995651i \(-0.529698\pi\)
−0.0931647 + 0.995651i \(0.529698\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 217795. 0.0180624
\(681\) 0 0
\(682\) 7.64575e6 0.629447
\(683\) 5.54335e6 0.454695 0.227347 0.973814i \(-0.426995\pi\)
0.227347 + 0.973814i \(0.426995\pi\)
\(684\) 0 0
\(685\) −6.00028e6 −0.488591
\(686\) 0 0
\(687\) 0 0
\(688\) 4.43387e6 0.357118
\(689\) 6.01477e6 0.482693
\(690\) 0 0
\(691\) −1.43508e7 −1.14335 −0.571677 0.820478i \(-0.693707\pi\)
−0.571677 + 0.820478i \(0.693707\pi\)
\(692\) 1.50702e7 1.19634
\(693\) 0 0
\(694\) 489943. 0.0386142
\(695\) 4.71469e6 0.370246
\(696\) 0 0
\(697\) −1.42051e7 −1.10755
\(698\) 2.93525e7 2.28038
\(699\) 0 0
\(700\) 0 0
\(701\) −1.50894e7 −1.15978 −0.579891 0.814694i \(-0.696905\pi\)
−0.579891 + 0.814694i \(0.696905\pi\)
\(702\) 0 0
\(703\) −1.08627e7 −0.828990
\(704\) −4.20417e6 −0.319705
\(705\) 0 0
\(706\) 5.54105e6 0.418389
\(707\) 0 0
\(708\) 0 0
\(709\) 1.04016e7 0.777115 0.388557 0.921425i \(-0.372974\pi\)
0.388557 + 0.921425i \(0.372974\pi\)
\(710\) −8.11660e6 −0.604266
\(711\) 0 0
\(712\) 708960. 0.0524109
\(713\) −7.54419e6 −0.555762
\(714\) 0 0
\(715\) 408586. 0.0298895
\(716\) −2.25101e7 −1.64095
\(717\) 0 0
\(718\) 2.55660e7 1.85077
\(719\) −1.11331e7 −0.803145 −0.401572 0.915827i \(-0.631536\pi\)
−0.401572 + 0.915827i \(0.631536\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.17713e7 0.840395
\(723\) 0 0
\(724\) −4.75422e6 −0.337080
\(725\) −3.09091e6 −0.218394
\(726\) 0 0
\(727\) 8.95327e6 0.628269 0.314135 0.949378i \(-0.398286\pi\)
0.314135 + 0.949378i \(0.398286\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −8.52088e6 −0.591803
\(731\) 3.51641e6 0.243392
\(732\) 0 0
\(733\) 2.60536e7 1.79105 0.895524 0.445014i \(-0.146801\pi\)
0.895524 + 0.445014i \(0.146801\pi\)
\(734\) −2.41423e7 −1.65401
\(735\) 0 0
\(736\) 7.59380e6 0.516731
\(737\) 2.27798e6 0.154483
\(738\) 0 0
\(739\) −5.97111e6 −0.402202 −0.201101 0.979571i \(-0.564452\pi\)
−0.201101 + 0.979571i \(0.564452\pi\)
\(740\) −6.35307e6 −0.426486
\(741\) 0 0
\(742\) 0 0
\(743\) 6.80330e6 0.452114 0.226057 0.974114i \(-0.427417\pi\)
0.226057 + 0.974114i \(0.427417\pi\)
\(744\) 0 0
\(745\) −2.29790e6 −0.151684
\(746\) −2.44115e7 −1.60601
\(747\) 0 0
\(748\) −2.94515e6 −0.192466
\(749\) 0 0
\(750\) 0 0
\(751\) −2.12648e7 −1.37582 −0.687911 0.725795i \(-0.741472\pi\)
−0.687911 + 0.725795i \(0.741472\pi\)
\(752\) 1.33388e7 0.860143
\(753\) 0 0
\(754\) −1.82701e6 −0.117034
\(755\) −5.62602e6 −0.359198
\(756\) 0 0
\(757\) −9.49238e6 −0.602054 −0.301027 0.953616i \(-0.597329\pi\)
−0.301027 + 0.953616i \(0.597329\pi\)
\(758\) 6.13469e6 0.387810
\(759\) 0 0
\(760\) −291530. −0.0183084
\(761\) 1.71806e7 1.07542 0.537710 0.843130i \(-0.319290\pi\)
0.537710 + 0.843130i \(0.319290\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.55887e7 −1.58604
\(765\) 0 0
\(766\) −2.98085e7 −1.83556
\(767\) −6.23480e6 −0.382678
\(768\) 0 0
\(769\) −1.39968e7 −0.853517 −0.426758 0.904366i \(-0.640345\pi\)
−0.426758 + 0.904366i \(0.640345\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.58223e7 0.955493
\(773\) 2.54449e7 1.53162 0.765812 0.643065i \(-0.222337\pi\)
0.765812 + 0.643065i \(0.222337\pi\)
\(774\) 0 0
\(775\) 2.32306e7 1.38933
\(776\) 340422. 0.0202938
\(777\) 0 0
\(778\) 2.44560e7 1.44856
\(779\) 1.90143e7 1.12263
\(780\) 0 0
\(781\) 6.55261e6 0.384403
\(782\) 5.63856e6 0.329725
\(783\) 0 0
\(784\) 0 0
\(785\) −3.16839e6 −0.183512
\(786\) 0 0
\(787\) −9.95632e6 −0.573010 −0.286505 0.958079i \(-0.592493\pi\)
−0.286505 + 0.958079i \(0.592493\pi\)
\(788\) −1.76452e7 −1.01231
\(789\) 0 0
\(790\) −5.05575e6 −0.288216
\(791\) 0 0
\(792\) 0 0
\(793\) −3.82941e6 −0.216246
\(794\) 2.33382e6 0.131376
\(795\) 0 0
\(796\) 2.00885e7 1.12374
\(797\) −2.19815e7 −1.22578 −0.612888 0.790170i \(-0.709992\pi\)
−0.612888 + 0.790170i \(0.709992\pi\)
\(798\) 0 0
\(799\) 1.05787e7 0.586226
\(800\) −2.33834e7 −1.29176
\(801\) 0 0
\(802\) −7.01412e6 −0.385068
\(803\) 6.87899e6 0.376475
\(804\) 0 0
\(805\) 0 0
\(806\) 1.37314e7 0.744521
\(807\) 0 0
\(808\) 1.75976e6 0.0948253
\(809\) −1.01820e7 −0.546969 −0.273485 0.961876i \(-0.588176\pi\)
−0.273485 + 0.961876i \(0.588176\pi\)
\(810\) 0 0
\(811\) 1.06750e7 0.569923 0.284961 0.958539i \(-0.408019\pi\)
0.284961 + 0.958539i \(0.408019\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9.95160e6 0.526419
\(815\) −3.41638e6 −0.180166
\(816\) 0 0
\(817\) −4.70689e6 −0.246706
\(818\) −2.29910e7 −1.20136
\(819\) 0 0
\(820\) 1.11205e7 0.577552
\(821\) −2.22260e7 −1.15081 −0.575404 0.817869i \(-0.695155\pi\)
−0.575404 + 0.817869i \(0.695155\pi\)
\(822\) 0 0
\(823\) −2.42908e7 −1.25009 −0.625046 0.780588i \(-0.714920\pi\)
−0.625046 + 0.780588i \(0.714920\pi\)
\(824\) 1.44284e6 0.0740288
\(825\) 0 0
\(826\) 0 0
\(827\) −2.66722e7 −1.35611 −0.678055 0.735011i \(-0.737177\pi\)
−0.678055 + 0.735011i \(0.737177\pi\)
\(828\) 0 0
\(829\) −6.78413e6 −0.342853 −0.171427 0.985197i \(-0.554838\pi\)
−0.171427 + 0.985197i \(0.554838\pi\)
\(830\) −1.22646e7 −0.617956
\(831\) 0 0
\(832\) −7.55049e6 −0.378152
\(833\) 0 0
\(834\) 0 0
\(835\) −5.10892e6 −0.253579
\(836\) 3.94223e6 0.195086
\(837\) 0 0
\(838\) 5.75427e7 2.83061
\(839\) −1.88975e7 −0.926830 −0.463415 0.886141i \(-0.653376\pi\)
−0.463415 + 0.886141i \(0.653376\pi\)
\(840\) 0 0
\(841\) −1.93111e7 −0.941493
\(842\) 2.39958e7 1.16642
\(843\) 0 0
\(844\) 7.04584e6 0.340468
\(845\) −5.73418e6 −0.276267
\(846\) 0 0
\(847\) 0 0
\(848\) −2.79830e7 −1.33630
\(849\) 0 0
\(850\) −1.73627e7 −0.824270
\(851\) −9.81941e6 −0.464795
\(852\) 0 0
\(853\) −1.69205e7 −0.796235 −0.398118 0.917334i \(-0.630336\pi\)
−0.398118 + 0.917334i \(0.630336\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 648950. 0.0302710
\(857\) 2.87564e7 1.33747 0.668733 0.743502i \(-0.266837\pi\)
0.668733 + 0.743502i \(0.266837\pi\)
\(858\) 0 0
\(859\) 8.92517e6 0.412699 0.206350 0.978478i \(-0.433842\pi\)
0.206350 + 0.978478i \(0.433842\pi\)
\(860\) −2.75284e6 −0.126921
\(861\) 0 0
\(862\) 3.63508e7 1.66627
\(863\) −7.18205e6 −0.328262 −0.164131 0.986439i \(-0.552482\pi\)
−0.164131 + 0.986439i \(0.552482\pi\)
\(864\) 0 0
\(865\) 7.71413e6 0.350547
\(866\) 4.15465e7 1.88252
\(867\) 0 0
\(868\) 0 0
\(869\) 4.08155e6 0.183348
\(870\) 0 0
\(871\) 4.09114e6 0.182725
\(872\) 732066. 0.0326031
\(873\) 0 0
\(874\) −7.54750e6 −0.334214
\(875\) 0 0
\(876\) 0 0
\(877\) 5.05341e6 0.221863 0.110932 0.993828i \(-0.464617\pi\)
0.110932 + 0.993828i \(0.464617\pi\)
\(878\) −4.79975e6 −0.210127
\(879\) 0 0
\(880\) −1.90090e6 −0.0827469
\(881\) −3.15002e6 −0.136733 −0.0683665 0.997660i \(-0.521779\pi\)
−0.0683665 + 0.997660i \(0.521779\pi\)
\(882\) 0 0
\(883\) 3.33141e7 1.43789 0.718946 0.695066i \(-0.244625\pi\)
0.718946 + 0.695066i \(0.244625\pi\)
\(884\) −5.28934e6 −0.227652
\(885\) 0 0
\(886\) 1.30638e6 0.0559094
\(887\) −2.04549e6 −0.0872946 −0.0436473 0.999047i \(-0.513898\pi\)
−0.0436473 + 0.999047i \(0.513898\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.07866e6 0.257237
\(891\) 0 0
\(892\) −4.19770e7 −1.76644
\(893\) −1.41601e7 −0.594208
\(894\) 0 0
\(895\) −1.15225e7 −0.480827
\(896\) 0 0
\(897\) 0 0
\(898\) 6.48677e7 2.68434
\(899\) −9.01936e6 −0.372200
\(900\) 0 0
\(901\) −2.21927e7 −0.910749
\(902\) −1.74195e7 −0.712883
\(903\) 0 0
\(904\) 1.06761e6 0.0434502
\(905\) −2.43359e6 −0.0987702
\(906\) 0 0
\(907\) 2.69774e7 1.08888 0.544442 0.838798i \(-0.316741\pi\)
0.544442 + 0.838798i \(0.316741\pi\)
\(908\) −3.81287e7 −1.53475
\(909\) 0 0
\(910\) 0 0
\(911\) 5.42821e6 0.216701 0.108350 0.994113i \(-0.465443\pi\)
0.108350 + 0.994113i \(0.465443\pi\)
\(912\) 0 0
\(913\) 9.90132e6 0.393112
\(914\) 9.95129e6 0.394016
\(915\) 0 0
\(916\) −1.40303e7 −0.552495
\(917\) 0 0
\(918\) 0 0
\(919\) 9.41696e6 0.367809 0.183904 0.982944i \(-0.441126\pi\)
0.183904 + 0.982944i \(0.441126\pi\)
\(920\) −263531. −0.0102651
\(921\) 0 0
\(922\) 1.62069e7 0.627876
\(923\) 1.17682e7 0.454679
\(924\) 0 0
\(925\) 3.02367e7 1.16193
\(926\) 1.01692e6 0.0389724
\(927\) 0 0
\(928\) 9.07868e6 0.346061
\(929\) −8.43871e6 −0.320802 −0.160401 0.987052i \(-0.551279\pi\)
−0.160401 + 0.987052i \(0.551279\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.84658e7 −1.07345
\(933\) 0 0
\(934\) 5.73997e7 2.15299
\(935\) −1.50756e6 −0.0563957
\(936\) 0 0
\(937\) 683755. 0.0254420 0.0127210 0.999919i \(-0.495951\pi\)
0.0127210 + 0.999919i \(0.495951\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.28158e6 −0.305699
\(941\) −4.97436e7 −1.83132 −0.915658 0.401958i \(-0.868330\pi\)
−0.915658 + 0.401958i \(0.868330\pi\)
\(942\) 0 0
\(943\) 1.71881e7 0.629430
\(944\) 2.90066e7 1.05942
\(945\) 0 0
\(946\) 4.31211e6 0.156661
\(947\) 2.21967e7 0.804292 0.402146 0.915575i \(-0.368264\pi\)
0.402146 + 0.915575i \(0.368264\pi\)
\(948\) 0 0
\(949\) 1.23543e7 0.445301
\(950\) 2.32408e7 0.835493
\(951\) 0 0
\(952\) 0 0
\(953\) 1.34934e7 0.481270 0.240635 0.970616i \(-0.422644\pi\)
0.240635 + 0.970616i \(0.422644\pi\)
\(954\) 0 0
\(955\) −1.30983e7 −0.464737
\(956\) 1.08797e7 0.385012
\(957\) 0 0
\(958\) −3.77704e7 −1.32965
\(959\) 0 0
\(960\) 0 0
\(961\) 3.91585e7 1.36778
\(962\) 1.78726e7 0.622658
\(963\) 0 0
\(964\) 3.28725e7 1.13931
\(965\) 8.09913e6 0.279976
\(966\) 0 0
\(967\) −1.59641e7 −0.549008 −0.274504 0.961586i \(-0.588514\pi\)
−0.274504 + 0.961586i \(0.588514\pi\)
\(968\) 2.44331e6 0.0838088
\(969\) 0 0
\(970\) 2.91879e6 0.0996033
\(971\) 2.03921e7 0.694087 0.347043 0.937849i \(-0.387186\pi\)
0.347043 + 0.937849i \(0.387186\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4.53354e7 1.53123
\(975\) 0 0
\(976\) 1.78159e7 0.598663
\(977\) 2.62595e7 0.880136 0.440068 0.897964i \(-0.354954\pi\)
0.440068 + 0.897964i \(0.354954\pi\)
\(978\) 0 0
\(979\) −4.90736e6 −0.163641
\(980\) 0 0
\(981\) 0 0
\(982\) −4.72489e7 −1.56355
\(983\) −8.01076e6 −0.264417 −0.132209 0.991222i \(-0.542207\pi\)
−0.132209 + 0.991222i \(0.542207\pi\)
\(984\) 0 0
\(985\) −9.03223e6 −0.296623
\(986\) 6.74111e6 0.220820
\(987\) 0 0
\(988\) 7.08006e6 0.230751
\(989\) −4.25483e6 −0.138322
\(990\) 0 0
\(991\) −1.60083e7 −0.517798 −0.258899 0.965904i \(-0.583360\pi\)
−0.258899 + 0.965904i \(0.583360\pi\)
\(992\) −6.82334e7 −2.20150
\(993\) 0 0
\(994\) 0 0
\(995\) 1.02829e7 0.329274
\(996\) 0 0
\(997\) 2.97034e7 0.946386 0.473193 0.880959i \(-0.343101\pi\)
0.473193 + 0.880959i \(0.343101\pi\)
\(998\) 5.83281e7 1.85375
\(999\) 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.be.1.2 yes 8
3.2 odd 2 inner 441.6.a.be.1.7 yes 8
7.6 odd 2 inner 441.6.a.be.1.1 8
21.20 even 2 inner 441.6.a.be.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.be.1.1 8 7.6 odd 2 inner
441.6.a.be.1.2 yes 8 1.1 even 1 trivial
441.6.a.be.1.7 yes 8 3.2 odd 2 inner
441.6.a.be.1.8 yes 8 21.20 even 2 inner