Properties

Label 441.6.a.be.1.1
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 1.1
Root \(3.13708\) 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 \(-0.549799\pi\)
−0.155811 + 0.987787i \(0.549799\pi\)
\(6\) 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 \(-0.553864\pi\)
−0.168413 + 0.985717i \(0.553864\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −954.857 −0.932477
\(17\) 757.277 0.635524 0.317762 0.948170i \(-0.397069\pi\)
0.317762 + 0.948170i \(0.397069\pi\)
\(18\) 0 0
\(19\) −1013.65 −0.644178 −0.322089 0.946709i \(-0.604385\pi\)
−0.322089 + 0.946709i \(0.604385\pi\)
\(20\) −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 \(-0.220565\pi\)
0.769380 + 0.638791i \(0.220565\pi\)
\(32\) 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 \(-0.836765\pi\)
−0.871365 + 0.490636i \(0.836765\pi\)
\(42\) 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 \(-0.347414\pi\)
0.461214 + 0.887289i \(0.347414\pi\)
\(48\) 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 \(-0.307692\pi\)
0.568067 + 0.822983i \(0.307692\pi\)
\(60\) 0 0
\(61\) 18658.2 0.642014 0.321007 0.947077i \(-0.395979\pi\)
0.321007 + 0.947077i \(0.395979\pi\)
\(62\) −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 \(-0.729879\pi\)
−0.661027 + 0.750362i \(0.729879\pi\)
\(74\) 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 \(-0.742494\pi\)
−0.690239 + 0.723582i \(0.742494\pi\)
\(84\) 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 \(-0.407234\pi\)
0.287326 + 0.957833i \(0.407234\pi\)
\(90\) 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 \(-0.464513\pi\)
0.111254 + 0.993792i \(0.464513\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −96021.8 −0.960218
\(101\) 106588. 1.03970 0.519848 0.854259i \(-0.325988\pi\)
0.519848 + 0.854259i \(0.325988\pi\)
\(102\) 0 0
\(103\) 87392.9 0.811677 0.405839 0.913945i \(-0.366980\pi\)
0.405839 + 0.913945i \(0.366980\pi\)
\(104\) 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 \(-0.659908\pi\)
−0.481501 + 0.876446i \(0.659908\pi\)
\(132\) 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 \(-0.702478\pi\)
−0.594065 + 0.804417i \(0.702478\pi\)
\(140\) 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 \(-0.404865\pi\)
0.294447 + 0.955668i \(0.404865\pi\)
\(158\) −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 \(-0.366620\pi\)
0.406870 + 0.913486i \(0.366620\pi\)
\(168\) 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 \(-0.690144\pi\)
−0.562458 + 0.826826i \(0.690144\pi\)
\(174\) 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 \(-0.449341\pi\)
0.158478 + 0.987363i \(0.449341\pi\)
\(182\) 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 \(-0.677179\pi\)
−0.528324 + 0.849043i \(0.677179\pi\)
\(200\) 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 \(-0.188060\pi\)
0.830491 + 0.557033i \(0.188060\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 525468. 0.684345
\(227\) 1.12039e6 1.44312 0.721562 0.692350i \(-0.243424\pi\)
0.721562 + 0.692350i \(0.243424\pi\)
\(228\) 0 0
\(229\) 412271. 0.519510 0.259755 0.965675i \(-0.416358\pi\)
0.259755 + 0.965675i \(0.416358\pi\)
\(230\) 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 \(-0.679931\pi\)
−0.535644 + 0.844444i \(0.679931\pi\)
\(242\) 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 \(-0.796423\pi\)
−0.802361 + 0.596839i \(0.796423\pi\)
\(252\) 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 \(-0.671140\pi\)
−0.512120 + 0.858914i \(0.671140\pi\)
\(258\) 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 \(-0.296590\pi\)
0.596419 + 0.802673i \(0.296590\pi\)
\(270\) 0 0
\(271\) −884078. −0.731252 −0.365626 0.930762i \(-0.619145\pi\)
−0.365626 + 0.930762i \(0.619145\pi\)
\(272\) −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 \(-0.264453\pi\)
0.674282 + 0.738474i \(0.264453\pi\)
\(284\) 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 \(-0.793373\pi\)
−0.796605 + 0.604500i \(0.793373\pi\)
\(294\) 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 \(-0.416594\pi\)
0.259039 + 0.965867i \(0.416594\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.16548e6 0.688809
\(311\) 2.58071e6 1.51300 0.756498 0.653996i \(-0.226909\pi\)
0.756498 + 0.653996i \(0.226909\pi\)
\(312\) 0 0
\(313\) −3.03363e6 −1.75026 −0.875129 0.483889i \(-0.839224\pi\)
−0.875129 + 0.483889i \(0.839224\pi\)
\(314\) −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 \(-0.208134\pi\)
0.793735 + 0.608263i \(0.208134\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 947088. 0.407412
\(353\) 681893. 0.291259 0.145629 0.989339i \(-0.453479\pi\)
0.145629 + 0.989339i \(0.453479\pi\)
\(354\) 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 \(-0.695276\pi\)
−0.575714 + 0.817651i \(0.695276\pi\)
\(368\) −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 \(-0.720612\pi\)
−0.638905 + 0.769286i \(0.720612\pi\)
\(384\) 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 \(-0.485439\pi\)
0.0457282 + 0.998954i \(0.485439\pi\)
\(398\) 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 \(-0.637325\pi\)
−0.418161 + 0.908373i \(0.637325\pi\)
\(410\) −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 \(-0.0547284\pi\)
0.985256 + 0.171089i \(0.0547284\pi\)
\(420\) 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 \(-0.272563\pi\)
0.655252 + 0.755411i \(0.272563\pi\)
\(434\) 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 \(-0.523302\pi\)
−0.0731393 + 0.997322i \(0.523302\pi\)
\(440\) 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 \(-0.429869\pi\)
0.218546 + 0.975827i \(0.429869\pi\)
\(462\) 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 \(-0.230344\pi\)
0.749396 + 0.662122i \(0.230344\pi\)
\(468\) 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 \(-0.653160\pi\)
−0.462814 + 0.886455i \(0.653160\pi\)
\(480\) 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 \(-0.585225\pi\)
−0.264555 + 0.964371i \(0.585225\pi\)
\(504\) 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 \(-0.597466\pi\)
−0.301435 + 0.953487i \(0.597466\pi\)
\(510\) 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 \(-0.583263\pi\)
−0.258605 + 0.965983i \(0.583263\pi\)
\(522\) 0 0
\(523\) −8.14584e6 −1.30221 −0.651106 0.758987i \(-0.725695\pi\)
−0.651106 + 0.758987i \(0.725695\pi\)
\(524\) −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 \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(564\) 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 \(-0.613141\pi\)
−0.348004 + 0.937493i \(0.613141\pi\)
\(578\) 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 \(-0.530035\pi\)
−0.0942173 + 0.995552i \(0.530035\pi\)
\(588\) 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 \(-0.102501\pi\)
0.948599 + 0.316481i \(0.102501\pi\)
\(594\) 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 \(-0.512588\pi\)
−0.0395347 + 0.999218i \(0.512588\pi\)
\(602\) 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 \(-0.444793\pi\)
0.172570 + 0.984997i \(0.444793\pi\)
\(608\) −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 \(-0.644000\pi\)
−0.437115 + 0.899406i \(0.644000\pi\)
\(620\) −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 \(-0.147281\pi\)
0.894852 + 0.446364i \(0.147281\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.23764e6 0.588084
\(647\) −1.59559e7 −1.49851 −0.749255 0.662282i \(-0.769588\pi\)
−0.749255 + 0.662282i \(0.769588\pi\)
\(648\) 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 \(-0.841305\pi\)
−0.878275 + 0.478157i \(0.841305\pi\)
\(662\) 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 \(-0.470302\pi\)
0.0931647 + 0.995651i \(0.470302\pi\)
\(678\) 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 \(-0.306293\pi\)
0.571677 + 0.820478i \(0.306293\pi\)
\(692\) −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 \(-0.368464\pi\)
0.401572 + 0.915827i \(0.368464\pi\)
\(720\) 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 \(-0.601714\pi\)
−0.314135 + 0.949378i \(0.601714\pi\)
\(728\) 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 \(-0.853199\pi\)
−0.895524 + 0.445014i \(0.853199\pi\)
\(734\) 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 \(-0.680710\pi\)
−0.537710 + 0.843130i \(0.680710\pi\)
\(762\) 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 \(-0.359655\pi\)
0.426758 + 0.904366i \(0.359655\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.58223e7 0.955493
\(773\) −2.54449e7 −1.53162 −0.765812 0.643065i \(-0.777663\pi\)
−0.765812 + 0.643065i \(0.777663\pi\)
\(774\) 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 \(-0.407507\pi\)
0.286505 + 0.958079i \(0.407507\pi\)
\(788\) −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 \(-0.290008\pi\)
0.612888 + 0.790170i \(0.290008\pi\)
\(798\) 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 \(-0.591981\pi\)
−0.284961 + 0.958539i \(0.591981\pi\)
\(812\) 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 \(-0.445162\pi\)
0.171427 + 0.985197i \(0.445162\pi\)
\(830\) −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 \(-0.346624\pi\)
0.463415 + 0.886141i \(0.346624\pi\)
\(840\) 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 \(-0.369664\pi\)
0.398118 + 0.917334i \(0.369664\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 648950. 0.0302710
\(857\) −2.87564e7 −1.33747 −0.668733 0.743502i \(-0.733163\pi\)
−0.668733 + 0.743502i \(0.733163\pi\)
\(858\) 0 0
\(859\) −8.92517e6 −0.412699 −0.206350 0.978478i \(-0.566158\pi\)
−0.206350 + 0.978478i \(0.566158\pi\)
\(860\) 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 \(-0.478221\pi\)
0.0683665 + 0.997660i \(0.478221\pi\)
\(882\) 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 \(-0.486102\pi\)
0.0436473 + 0.999047i \(0.486102\pi\)
\(888\) 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 \(-0.448721\pi\)
0.160401 + 0.987052i \(0.448721\pi\)
\(930\) 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 \(-0.504049\pi\)
−0.0127210 + 0.999919i \(0.504049\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.28158e6 −0.305699
\(941\) 4.97436e7 1.83132 0.915658 0.401958i \(-0.131670\pi\)
0.915658 + 0.401958i \(0.131670\pi\)
\(942\) 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 \(-0.612814\pi\)
−0.347043 + 0.937849i \(0.612814\pi\)
\(972\) 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 \(-0.457793\pi\)
0.132209 + 0.991222i \(0.457793\pi\)
\(984\) 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 \(-0.656899\pi\)
−0.473193 + 0.880959i \(0.656899\pi\)
\(998\) 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.1.1 8
3.2 odd 2 inner 441.6.a.be.1.8 yes 8
7.6 odd 2 inner 441.6.a.be.1.2 yes 8
21.20 even 2 inner 441.6.a.be.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.be.1.1 8 1.1 even 1 trivial
441.6.a.be.1.2 yes 8 7.6 odd 2 inner
441.6.a.be.1.7 yes 8 21.20 even 2 inner
441.6.a.be.1.8 yes 8 3.2 odd 2 inner