Properties

Label 441.6.a.be.1.3
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 1.3
Root \(6.71178\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} +105.150 q^{8} +O(q^{10})\) \(q-1.72287 q^{2} -29.0317 q^{4} -73.1337 q^{5} +105.150 q^{8} +126.000 q^{10} -193.070 q^{11} -151.500 q^{13} +747.857 q^{16} +1760.12 q^{17} -1126.79 q^{19} +2123.20 q^{20} +332.635 q^{22} -2413.33 q^{23} +2223.54 q^{25} +261.015 q^{26} +7729.11 q^{29} +8176.75 q^{31} -4653.24 q^{32} -3032.45 q^{34} +5932.38 q^{37} +1941.31 q^{38} -7689.98 q^{40} +4975.84 q^{41} -5652.51 q^{43} +5605.17 q^{44} +4157.84 q^{46} +4720.93 q^{47} -3830.86 q^{50} +4398.32 q^{52} +32072.1 q^{53} +14120.0 q^{55} -13316.2 q^{58} -28566.9 q^{59} -29003.7 q^{61} -14087.5 q^{62} -15914.5 q^{64} +11079.8 q^{65} -30517.4 q^{67} -51099.3 q^{68} -56944.5 q^{71} -9726.79 q^{73} -10220.7 q^{74} +32712.7 q^{76} -98483.5 q^{79} -54693.5 q^{80} -8572.71 q^{82} +50637.0 q^{83} -128724. q^{85} +9738.52 q^{86} -20301.3 q^{88} +59655.8 q^{89} +70063.0 q^{92} -8133.54 q^{94} +82406.4 q^{95} +134206. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.72287 −0.304563 −0.152281 0.988337i \(-0.548662\pi\)
−0.152281 + 0.988337i \(0.548662\pi\)
\(3\) 0 0
\(4\) −29.0317 −0.907242
\(5\) −73.1337 −1.30826 −0.654128 0.756384i \(-0.726964\pi\)
−0.654128 + 0.756384i \(0.726964\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 105.150 0.580875
\(9\) 0 0
\(10\) 126.000 0.398446
\(11\) −193.070 −0.481099 −0.240549 0.970637i \(-0.577328\pi\)
−0.240549 + 0.970637i \(0.577328\pi\)
\(12\) 0 0
\(13\) −151.500 −0.248631 −0.124315 0.992243i \(-0.539673\pi\)
−0.124315 + 0.992243i \(0.539673\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 747.857 0.730329
\(17\) 1760.12 1.47713 0.738567 0.674180i \(-0.235503\pi\)
0.738567 + 0.674180i \(0.235503\pi\)
\(18\) 0 0
\(19\) −1126.79 −0.716076 −0.358038 0.933707i \(-0.616554\pi\)
−0.358038 + 0.933707i \(0.616554\pi\)
\(20\) 2123.20 1.18690
\(21\) 0 0
\(22\) 332.635 0.146525
\(23\) −2413.33 −0.951254 −0.475627 0.879647i \(-0.657779\pi\)
−0.475627 + 0.879647i \(0.657779\pi\)
\(24\) 0 0
\(25\) 2223.54 0.711532
\(26\) 261.015 0.0757237
\(27\) 0 0
\(28\) 0 0
\(29\) 7729.11 1.70661 0.853305 0.521412i \(-0.174595\pi\)
0.853305 + 0.521412i \(0.174595\pi\)
\(30\) 0 0
\(31\) 8176.75 1.52819 0.764094 0.645105i \(-0.223186\pi\)
0.764094 + 0.645105i \(0.223186\pi\)
\(32\) −4653.24 −0.803306
\(33\) 0 0
\(34\) −3032.45 −0.449880
\(35\) 0 0
\(36\) 0 0
\(37\) 5932.38 0.712401 0.356200 0.934410i \(-0.384072\pi\)
0.356200 + 0.934410i \(0.384072\pi\)
\(38\) 1941.31 0.218090
\(39\) 0 0
\(40\) −7689.98 −0.759932
\(41\) 4975.84 0.462282 0.231141 0.972920i \(-0.425754\pi\)
0.231141 + 0.972920i \(0.425754\pi\)
\(42\) 0 0
\(43\) −5652.51 −0.466198 −0.233099 0.972453i \(-0.574887\pi\)
−0.233099 + 0.972453i \(0.574887\pi\)
\(44\) 5605.17 0.436473
\(45\) 0 0
\(46\) 4157.84 0.289716
\(47\) 4720.93 0.311733 0.155867 0.987778i \(-0.450183\pi\)
0.155867 + 0.987778i \(0.450183\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3830.86 −0.216706
\(51\) 0 0
\(52\) 4398.32 0.225568
\(53\) 32072.1 1.56833 0.784166 0.620551i \(-0.213091\pi\)
0.784166 + 0.620551i \(0.213091\pi\)
\(54\) 0 0
\(55\) 14120.0 0.629400
\(56\) 0 0
\(57\) 0 0
\(58\) −13316.2 −0.519770
\(59\) −28566.9 −1.06840 −0.534200 0.845358i \(-0.679387\pi\)
−0.534200 + 0.845358i \(0.679387\pi\)
\(60\) 0 0
\(61\) −29003.7 −0.997994 −0.498997 0.866604i \(-0.666298\pi\)
−0.498997 + 0.866604i \(0.666298\pi\)
\(62\) −14087.5 −0.465429
\(63\) 0 0
\(64\) −15914.5 −0.485672
\(65\) 11079.8 0.325273
\(66\) 0 0
\(67\) −30517.4 −0.830540 −0.415270 0.909698i \(-0.636313\pi\)
−0.415270 + 0.909698i \(0.636313\pi\)
\(68\) −51099.3 −1.34012
\(69\) 0 0
\(70\) 0 0
\(71\) −56944.5 −1.34062 −0.670310 0.742081i \(-0.733839\pi\)
−0.670310 + 0.742081i \(0.733839\pi\)
\(72\) 0 0
\(73\) −9726.79 −0.213630 −0.106815 0.994279i \(-0.534065\pi\)
−0.106815 + 0.994279i \(0.534065\pi\)
\(74\) −10220.7 −0.216971
\(75\) 0 0
\(76\) 32712.7 0.649654
\(77\) 0 0
\(78\) 0 0
\(79\) −98483.5 −1.77540 −0.887699 0.460424i \(-0.847697\pi\)
−0.887699 + 0.460424i \(0.847697\pi\)
\(80\) −54693.5 −0.955457
\(81\) 0 0
\(82\) −8572.71 −0.140794
\(83\) 50637.0 0.806813 0.403406 0.915021i \(-0.367826\pi\)
0.403406 + 0.915021i \(0.367826\pi\)
\(84\) 0 0
\(85\) −128724. −1.93247
\(86\) 9738.52 0.141986
\(87\) 0 0
\(88\) −20301.3 −0.279458
\(89\) 59655.8 0.798321 0.399161 0.916881i \(-0.369302\pi\)
0.399161 + 0.916881i \(0.369302\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 70063.0 0.863017
\(93\) 0 0
\(94\) −8133.54 −0.0949423
\(95\) 82406.4 0.936811
\(96\) 0 0
\(97\) 134206. 1.44825 0.724124 0.689670i \(-0.242244\pi\)
0.724124 + 0.689670i \(0.242244\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −64553.2 −0.645532
\(101\) 135281. 1.31957 0.659786 0.751454i \(-0.270647\pi\)
0.659786 + 0.751454i \(0.270647\pi\)
\(102\) 0 0
\(103\) −199410. −1.85205 −0.926025 0.377461i \(-0.876797\pi\)
−0.926025 + 0.377461i \(0.876797\pi\)
\(104\) −15930.2 −0.144423
\(105\) 0 0
\(106\) −55256.0 −0.477655
\(107\) −93298.1 −0.787795 −0.393898 0.919154i \(-0.628874\pi\)
−0.393898 + 0.919154i \(0.628874\pi\)
\(108\) 0 0
\(109\) −41818.7 −0.337136 −0.168568 0.985690i \(-0.553914\pi\)
−0.168568 + 0.985690i \(0.553914\pi\)
\(110\) −24326.8 −0.191692
\(111\) 0 0
\(112\) 0 0
\(113\) 170560. 1.25655 0.628277 0.777990i \(-0.283760\pi\)
0.628277 + 0.777990i \(0.283760\pi\)
\(114\) 0 0
\(115\) 176495. 1.24448
\(116\) −224389. −1.54831
\(117\) 0 0
\(118\) 49217.0 0.325395
\(119\) 0 0
\(120\) 0 0
\(121\) −123775. −0.768544
\(122\) 49969.4 0.303952
\(123\) 0 0
\(124\) −237385. −1.38644
\(125\) 65927.2 0.377389
\(126\) 0 0
\(127\) −314872. −1.73231 −0.866154 0.499777i \(-0.833415\pi\)
−0.866154 + 0.499777i \(0.833415\pi\)
\(128\) 176322. 0.951223
\(129\) 0 0
\(130\) −19089.0 −0.0990660
\(131\) 91423.9 0.465459 0.232729 0.972542i \(-0.425234\pi\)
0.232729 + 0.972542i \(0.425234\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 52577.4 0.252951
\(135\) 0 0
\(136\) 185076. 0.858030
\(137\) 5959.51 0.0271275 0.0135637 0.999908i \(-0.495682\pi\)
0.0135637 + 0.999908i \(0.495682\pi\)
\(138\) 0 0
\(139\) 13792.3 0.0605479 0.0302740 0.999542i \(-0.490362\pi\)
0.0302740 + 0.999542i \(0.490362\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 98107.8 0.408303
\(143\) 29250.2 0.119616
\(144\) 0 0
\(145\) −565258. −2.23268
\(146\) 16758.0 0.0650637
\(147\) 0 0
\(148\) −172227. −0.646320
\(149\) −400637. −1.47838 −0.739189 0.673498i \(-0.764791\pi\)
−0.739189 + 0.673498i \(0.764791\pi\)
\(150\) 0 0
\(151\) 91744.7 0.327445 0.163723 0.986506i \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(152\) −118482. −0.415951
\(153\) 0 0
\(154\) 0 0
\(155\) −597996. −1.99926
\(156\) 0 0
\(157\) 498424. 1.61380 0.806900 0.590688i \(-0.201143\pi\)
0.806900 + 0.590688i \(0.201143\pi\)
\(158\) 169674. 0.540720
\(159\) 0 0
\(160\) 340309. 1.05093
\(161\) 0 0
\(162\) 0 0
\(163\) −242531. −0.714988 −0.357494 0.933915i \(-0.616369\pi\)
−0.357494 + 0.933915i \(0.616369\pi\)
\(164\) −144457. −0.419401
\(165\) 0 0
\(166\) −87240.8 −0.245725
\(167\) −525601. −1.45836 −0.729181 0.684321i \(-0.760099\pi\)
−0.729181 + 0.684321i \(0.760099\pi\)
\(168\) 0 0
\(169\) −348341. −0.938183
\(170\) 221775. 0.588558
\(171\) 0 0
\(172\) 164102. 0.422954
\(173\) 96423.4 0.244944 0.122472 0.992472i \(-0.460918\pi\)
0.122472 + 0.992472i \(0.460918\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −144389. −0.351360
\(177\) 0 0
\(178\) −102779. −0.243139
\(179\) −178701. −0.416865 −0.208432 0.978037i \(-0.566836\pi\)
−0.208432 + 0.978037i \(0.566836\pi\)
\(180\) 0 0
\(181\) 224533. 0.509428 0.254714 0.967016i \(-0.418019\pi\)
0.254714 + 0.967016i \(0.418019\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −253760. −0.552559
\(185\) −433857. −0.932002
\(186\) 0 0
\(187\) −339827. −0.710647
\(188\) −137057. −0.282817
\(189\) 0 0
\(190\) −141975. −0.285318
\(191\) 905785. 1.79656 0.898280 0.439423i \(-0.144817\pi\)
0.898280 + 0.439423i \(0.144817\pi\)
\(192\) 0 0
\(193\) −717637. −1.38679 −0.693396 0.720556i \(-0.743886\pi\)
−0.693396 + 0.720556i \(0.743886\pi\)
\(194\) −231219. −0.441082
\(195\) 0 0
\(196\) 0 0
\(197\) 57173.0 0.104960 0.0524802 0.998622i \(-0.483287\pi\)
0.0524802 + 0.998622i \(0.483287\pi\)
\(198\) 0 0
\(199\) −263037. −0.470853 −0.235426 0.971892i \(-0.575649\pi\)
−0.235426 + 0.971892i \(0.575649\pi\)
\(200\) 233804. 0.413311
\(201\) 0 0
\(202\) −233071. −0.401892
\(203\) 0 0
\(204\) 0 0
\(205\) −363902. −0.604782
\(206\) 343556. 0.564066
\(207\) 0 0
\(208\) −113301. −0.181582
\(209\) 217550. 0.344503
\(210\) 0 0
\(211\) −495237. −0.765786 −0.382893 0.923793i \(-0.625072\pi\)
−0.382893 + 0.923793i \(0.625072\pi\)
\(212\) −931109. −1.42286
\(213\) 0 0
\(214\) 160740. 0.239933
\(215\) 413389. 0.609906
\(216\) 0 0
\(217\) 0 0
\(218\) 72048.1 0.102679
\(219\) 0 0
\(220\) −409927. −0.571018
\(221\) −266659. −0.367261
\(222\) 0 0
\(223\) 373295. 0.502679 0.251339 0.967899i \(-0.419129\pi\)
0.251339 + 0.967899i \(0.419129\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −293852. −0.382700
\(227\) 284573. 0.366547 0.183273 0.983062i \(-0.441331\pi\)
0.183273 + 0.983062i \(0.441331\pi\)
\(228\) 0 0
\(229\) 989364. 1.24672 0.623358 0.781937i \(-0.285768\pi\)
0.623358 + 0.781937i \(0.285768\pi\)
\(230\) −304078. −0.379023
\(231\) 0 0
\(232\) 812712. 0.991327
\(233\) −602273. −0.726781 −0.363391 0.931637i \(-0.618381\pi\)
−0.363391 + 0.931637i \(0.618381\pi\)
\(234\) 0 0
\(235\) −345259. −0.407827
\(236\) 829347. 0.969296
\(237\) 0 0
\(238\) 0 0
\(239\) 1.44372e6 1.63489 0.817447 0.576004i \(-0.195389\pi\)
0.817447 + 0.576004i \(0.195389\pi\)
\(240\) 0 0
\(241\) −538796. −0.597560 −0.298780 0.954322i \(-0.596580\pi\)
−0.298780 + 0.954322i \(0.596580\pi\)
\(242\) 213248. 0.234070
\(243\) 0 0
\(244\) 842026. 0.905422
\(245\) 0 0
\(246\) 0 0
\(247\) 170709. 0.178039
\(248\) 859782. 0.887685
\(249\) 0 0
\(250\) −113584. −0.114939
\(251\) 1.63982e6 1.64290 0.821449 0.570282i \(-0.193166\pi\)
0.821449 + 0.570282i \(0.193166\pi\)
\(252\) 0 0
\(253\) 465942. 0.457647
\(254\) 542483. 0.527596
\(255\) 0 0
\(256\) 205484. 0.195965
\(257\) −1.82127e6 −1.72005 −0.860026 0.510251i \(-0.829553\pi\)
−0.860026 + 0.510251i \(0.829553\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −321665. −0.295101
\(261\) 0 0
\(262\) −157511. −0.141761
\(263\) −1.77477e6 −1.58217 −0.791085 0.611706i \(-0.790483\pi\)
−0.791085 + 0.611706i \(0.790483\pi\)
\(264\) 0 0
\(265\) −2.34555e6 −2.05178
\(266\) 0 0
\(267\) 0 0
\(268\) 885972. 0.753500
\(269\) −769025. −0.647977 −0.323989 0.946061i \(-0.605024\pi\)
−0.323989 + 0.946061i \(0.605024\pi\)
\(270\) 0 0
\(271\) 1.81374e6 1.50021 0.750106 0.661317i \(-0.230002\pi\)
0.750106 + 0.661317i \(0.230002\pi\)
\(272\) 1.31632e6 1.07879
\(273\) 0 0
\(274\) −10267.4 −0.00826202
\(275\) −429299. −0.342317
\(276\) 0 0
\(277\) 1.12526e6 0.881157 0.440578 0.897714i \(-0.354773\pi\)
0.440578 + 0.897714i \(0.354773\pi\)
\(278\) −23762.3 −0.0184406
\(279\) 0 0
\(280\) 0 0
\(281\) 243202. 0.183739 0.0918696 0.995771i \(-0.470716\pi\)
0.0918696 + 0.995771i \(0.470716\pi\)
\(282\) 0 0
\(283\) 1.06625e6 0.791396 0.395698 0.918381i \(-0.370503\pi\)
0.395698 + 0.918381i \(0.370503\pi\)
\(284\) 1.65320e6 1.21627
\(285\) 0 0
\(286\) −50394.3 −0.0364306
\(287\) 0 0
\(288\) 0 0
\(289\) 1.67817e6 1.18193
\(290\) 973865. 0.679992
\(291\) 0 0
\(292\) 282385. 0.193814
\(293\) −2.32478e6 −1.58203 −0.791013 0.611799i \(-0.790446\pi\)
−0.791013 + 0.611799i \(0.790446\pi\)
\(294\) 0 0
\(295\) 2.08921e6 1.39774
\(296\) 623787. 0.413816
\(297\) 0 0
\(298\) 690244. 0.450259
\(299\) 365620. 0.236511
\(300\) 0 0
\(301\) 0 0
\(302\) −158064. −0.0997275
\(303\) 0 0
\(304\) −842678. −0.522971
\(305\) 2.12114e6 1.30563
\(306\) 0 0
\(307\) −2.37882e6 −1.44051 −0.720255 0.693710i \(-0.755975\pi\)
−0.720255 + 0.693710i \(0.755975\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.03027e6 0.608900
\(311\) −2.22414e6 −1.30395 −0.651976 0.758240i \(-0.726060\pi\)
−0.651976 + 0.758240i \(0.726060\pi\)
\(312\) 0 0
\(313\) 1.03945e6 0.599713 0.299857 0.953984i \(-0.403061\pi\)
0.299857 + 0.953984i \(0.403061\pi\)
\(314\) −858718. −0.491503
\(315\) 0 0
\(316\) 2.85915e6 1.61072
\(317\) 428476. 0.239485 0.119743 0.992805i \(-0.461793\pi\)
0.119743 + 0.992805i \(0.461793\pi\)
\(318\) 0 0
\(319\) −1.49226e6 −0.821048
\(320\) 1.16389e6 0.635383
\(321\) 0 0
\(322\) 0 0
\(323\) −1.98329e6 −1.05774
\(324\) 0 0
\(325\) −336867. −0.176909
\(326\) 417849. 0.217759
\(327\) 0 0
\(328\) 523207. 0.268528
\(329\) 0 0
\(330\) 0 0
\(331\) 2.79057e6 1.39998 0.699991 0.714152i \(-0.253187\pi\)
0.699991 + 0.714152i \(0.253187\pi\)
\(332\) −1.47008e6 −0.731974
\(333\) 0 0
\(334\) 905541. 0.444163
\(335\) 2.23185e6 1.08656
\(336\) 0 0
\(337\) 1.30809e6 0.627427 0.313714 0.949518i \(-0.398427\pi\)
0.313714 + 0.949518i \(0.398427\pi\)
\(338\) 600145. 0.285735
\(339\) 0 0
\(340\) 3.73708e6 1.75322
\(341\) −1.57869e6 −0.735209
\(342\) 0 0
\(343\) 0 0
\(344\) −594359. −0.270802
\(345\) 0 0
\(346\) −166125. −0.0746009
\(347\) −664031. −0.296050 −0.148025 0.988984i \(-0.547292\pi\)
−0.148025 + 0.988984i \(0.547292\pi\)
\(348\) 0 0
\(349\) −2.27119e6 −0.998136 −0.499068 0.866563i \(-0.666324\pi\)
−0.499068 + 0.866563i \(0.666324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 898404. 0.386469
\(353\) −1.96361e6 −0.838723 −0.419361 0.907819i \(-0.637746\pi\)
−0.419361 + 0.907819i \(0.637746\pi\)
\(354\) 0 0
\(355\) 4.16456e6 1.75387
\(356\) −1.73191e6 −0.724270
\(357\) 0 0
\(358\) 307879. 0.126961
\(359\) 2.09222e6 0.856783 0.428391 0.903593i \(-0.359080\pi\)
0.428391 + 0.903593i \(0.359080\pi\)
\(360\) 0 0
\(361\) −1.20644e6 −0.487235
\(362\) −386840. −0.155153
\(363\) 0 0
\(364\) 0 0
\(365\) 711356. 0.279483
\(366\) 0 0
\(367\) −1.33844e6 −0.518719 −0.259360 0.965781i \(-0.583512\pi\)
−0.259360 + 0.965781i \(0.583512\pi\)
\(368\) −1.80482e6 −0.694728
\(369\) 0 0
\(370\) 747477. 0.283853
\(371\) 0 0
\(372\) 0 0
\(373\) −1.32657e6 −0.493694 −0.246847 0.969054i \(-0.579394\pi\)
−0.246847 + 0.969054i \(0.579394\pi\)
\(374\) 585477. 0.216437
\(375\) 0 0
\(376\) 496404. 0.181078
\(377\) −1.17096e6 −0.424316
\(378\) 0 0
\(379\) 863514. 0.308796 0.154398 0.988009i \(-0.450656\pi\)
0.154398 + 0.988009i \(0.450656\pi\)
\(380\) −2.39240e6 −0.849914
\(381\) 0 0
\(382\) −1.56055e6 −0.547165
\(383\) 2.53709e6 0.883769 0.441884 0.897072i \(-0.354310\pi\)
0.441884 + 0.897072i \(0.354310\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.23639e6 0.422365
\(387\) 0 0
\(388\) −3.89624e6 −1.31391
\(389\) 253814. 0.0850434 0.0425217 0.999096i \(-0.486461\pi\)
0.0425217 + 0.999096i \(0.486461\pi\)
\(390\) 0 0
\(391\) −4.24774e6 −1.40513
\(392\) 0 0
\(393\) 0 0
\(394\) −98501.4 −0.0319670
\(395\) 7.20246e6 2.32267
\(396\) 0 0
\(397\) 4.57767e6 1.45770 0.728850 0.684673i \(-0.240055\pi\)
0.728850 + 0.684673i \(0.240055\pi\)
\(398\) 453179. 0.143404
\(399\) 0 0
\(400\) 1.66289e6 0.519653
\(401\) −4.38670e6 −1.36231 −0.681156 0.732138i \(-0.738523\pi\)
−0.681156 + 0.732138i \(0.738523\pi\)
\(402\) 0 0
\(403\) −1.23878e6 −0.379955
\(404\) −3.92744e6 −1.19717
\(405\) 0 0
\(406\) 0 0
\(407\) −1.14537e6 −0.342735
\(408\) 0 0
\(409\) −222245. −0.0656936 −0.0328468 0.999460i \(-0.510457\pi\)
−0.0328468 + 0.999460i \(0.510457\pi\)
\(410\) 626954. 0.184194
\(411\) 0 0
\(412\) 5.78920e6 1.68026
\(413\) 0 0
\(414\) 0 0
\(415\) −3.70327e6 −1.05552
\(416\) 704968. 0.199727
\(417\) 0 0
\(418\) −374810. −0.104923
\(419\) 4.33564e6 1.20647 0.603237 0.797562i \(-0.293877\pi\)
0.603237 + 0.797562i \(0.293877\pi\)
\(420\) 0 0
\(421\) 572536. 0.157434 0.0787168 0.996897i \(-0.474918\pi\)
0.0787168 + 0.996897i \(0.474918\pi\)
\(422\) 853228. 0.233230
\(423\) 0 0
\(424\) 3.37237e6 0.911004
\(425\) 3.91370e6 1.05103
\(426\) 0 0
\(427\) 0 0
\(428\) 2.70860e6 0.714720
\(429\) 0 0
\(430\) −712214. −0.185754
\(431\) −3.30224e6 −0.856279 −0.428139 0.903713i \(-0.640831\pi\)
−0.428139 + 0.903713i \(0.640831\pi\)
\(432\) 0 0
\(433\) 19285.5 0.00494322 0.00247161 0.999997i \(-0.499213\pi\)
0.00247161 + 0.999997i \(0.499213\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.21407e6 0.305863
\(437\) 2.71931e6 0.681170
\(438\) 0 0
\(439\) −376815. −0.0933182 −0.0466591 0.998911i \(-0.514857\pi\)
−0.0466591 + 0.998911i \(0.514857\pi\)
\(440\) 1.48471e6 0.365602
\(441\) 0 0
\(442\) 459418. 0.111854
\(443\) 1.24690e6 0.301871 0.150936 0.988544i \(-0.451771\pi\)
0.150936 + 0.988544i \(0.451771\pi\)
\(444\) 0 0
\(445\) −4.36285e6 −1.04441
\(446\) −643138. −0.153097
\(447\) 0 0
\(448\) 0 0
\(449\) 2.85298e6 0.667857 0.333928 0.942598i \(-0.391626\pi\)
0.333928 + 0.942598i \(0.391626\pi\)
\(450\) 0 0
\(451\) −960687. −0.222403
\(452\) −4.95165e6 −1.14000
\(453\) 0 0
\(454\) −490282. −0.111636
\(455\) 0 0
\(456\) 0 0
\(457\) 2.10513e6 0.471506 0.235753 0.971813i \(-0.424244\pi\)
0.235753 + 0.971813i \(0.424244\pi\)
\(458\) −1.70454e6 −0.379703
\(459\) 0 0
\(460\) −5.12397e6 −1.12905
\(461\) −8.96535e6 −1.96478 −0.982392 0.186832i \(-0.940178\pi\)
−0.982392 + 0.186832i \(0.940178\pi\)
\(462\) 0 0
\(463\) −7.01066e6 −1.51987 −0.759935 0.649999i \(-0.774769\pi\)
−0.759935 + 0.649999i \(0.774769\pi\)
\(464\) 5.78026e6 1.24639
\(465\) 0 0
\(466\) 1.03764e6 0.221350
\(467\) 3.26840e6 0.693495 0.346747 0.937959i \(-0.387286\pi\)
0.346747 + 0.937959i \(0.387286\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 594836. 0.124209
\(471\) 0 0
\(472\) −3.00380e6 −0.620606
\(473\) 1.09133e6 0.224287
\(474\) 0 0
\(475\) −2.50546e6 −0.509511
\(476\) 0 0
\(477\) 0 0
\(478\) −2.48734e6 −0.497928
\(479\) −4.89425e6 −0.974646 −0.487323 0.873222i \(-0.662027\pi\)
−0.487323 + 0.873222i \(0.662027\pi\)
\(480\) 0 0
\(481\) −898757. −0.177125
\(482\) 928274. 0.181995
\(483\) 0 0
\(484\) 3.59340e6 0.697255
\(485\) −9.81499e6 −1.89468
\(486\) 0 0
\(487\) −4.60536e6 −0.879916 −0.439958 0.898018i \(-0.645007\pi\)
−0.439958 + 0.898018i \(0.645007\pi\)
\(488\) −3.04972e6 −0.579710
\(489\) 0 0
\(490\) 0 0
\(491\) −5.58108e6 −1.04475 −0.522377 0.852714i \(-0.674955\pi\)
−0.522377 + 0.852714i \(0.674955\pi\)
\(492\) 0 0
\(493\) 1.36042e7 2.52089
\(494\) −294109. −0.0542240
\(495\) 0 0
\(496\) 6.11504e6 1.11608
\(497\) 0 0
\(498\) 0 0
\(499\) −1.49822e6 −0.269354 −0.134677 0.990890i \(-0.543000\pi\)
−0.134677 + 0.990890i \(0.543000\pi\)
\(500\) −1.91398e6 −0.342383
\(501\) 0 0
\(502\) −2.82518e6 −0.500366
\(503\) −479890. −0.0845710 −0.0422855 0.999106i \(-0.513464\pi\)
−0.0422855 + 0.999106i \(0.513464\pi\)
\(504\) 0 0
\(505\) −9.89359e6 −1.72634
\(506\) −802756. −0.139382
\(507\) 0 0
\(508\) 9.14129e6 1.57162
\(509\) 1.21062e6 0.207117 0.103558 0.994623i \(-0.466977\pi\)
0.103558 + 0.994623i \(0.466977\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.99634e6 −1.01091
\(513\) 0 0
\(514\) 3.13780e6 0.523863
\(515\) 1.45836e7 2.42296
\(516\) 0 0
\(517\) −911472. −0.149974
\(518\) 0 0
\(519\) 0 0
\(520\) 1.16503e6 0.188943
\(521\) 69333.1 0.0111904 0.00559521 0.999984i \(-0.498219\pi\)
0.00559521 + 0.999984i \(0.498219\pi\)
\(522\) 0 0
\(523\) −8.95346e6 −1.43132 −0.715660 0.698449i \(-0.753874\pi\)
−0.715660 + 0.698449i \(0.753874\pi\)
\(524\) −2.65419e6 −0.422284
\(525\) 0 0
\(526\) 3.05770e6 0.481870
\(527\) 1.43921e7 2.25734
\(528\) 0 0
\(529\) −612203. −0.0951166
\(530\) 4.04107e6 0.624895
\(531\) 0 0
\(532\) 0 0
\(533\) −753841. −0.114938
\(534\) 0 0
\(535\) 6.82323e6 1.03064
\(536\) −3.20889e6 −0.482439
\(537\) 0 0
\(538\) 1.32493e6 0.197350
\(539\) 0 0
\(540\) 0 0
\(541\) −2.38913e6 −0.350951 −0.175476 0.984484i \(-0.556146\pi\)
−0.175476 + 0.984484i \(0.556146\pi\)
\(542\) −3.12484e6 −0.456909
\(543\) 0 0
\(544\) −8.19027e6 −1.18659
\(545\) 3.05836e6 0.441059
\(546\) 0 0
\(547\) −7.19431e6 −1.02807 −0.514033 0.857770i \(-0.671849\pi\)
−0.514033 + 0.857770i \(0.671849\pi\)
\(548\) −173015. −0.0246112
\(549\) 0 0
\(550\) 739626. 0.104257
\(551\) −8.70909e6 −1.22206
\(552\) 0 0
\(553\) 0 0
\(554\) −1.93867e6 −0.268367
\(555\) 0 0
\(556\) −400414. −0.0549316
\(557\) −753706. −0.102935 −0.0514676 0.998675i \(-0.516390\pi\)
−0.0514676 + 0.998675i \(0.516390\pi\)
\(558\) 0 0
\(559\) 856357. 0.115911
\(560\) 0 0
\(561\) 0 0
\(562\) −419005. −0.0559601
\(563\) −1.22890e6 −0.163397 −0.0816985 0.996657i \(-0.526034\pi\)
−0.0816985 + 0.996657i \(0.526034\pi\)
\(564\) 0 0
\(565\) −1.24737e7 −1.64389
\(566\) −1.83701e6 −0.241030
\(567\) 0 0
\(568\) −5.98769e6 −0.778733
\(569\) −4.30545e6 −0.557491 −0.278745 0.960365i \(-0.589919\pi\)
−0.278745 + 0.960365i \(0.589919\pi\)
\(570\) 0 0
\(571\) −3.96799e6 −0.509307 −0.254654 0.967032i \(-0.581961\pi\)
−0.254654 + 0.967032i \(0.581961\pi\)
\(572\) −849185. −0.108521
\(573\) 0 0
\(574\) 0 0
\(575\) −5.36612e6 −0.676848
\(576\) 0 0
\(577\) 596027. 0.0745292 0.0372646 0.999305i \(-0.488136\pi\)
0.0372646 + 0.999305i \(0.488136\pi\)
\(578\) −2.89126e6 −0.359971
\(579\) 0 0
\(580\) 1.64104e7 2.02558
\(581\) 0 0
\(582\) 0 0
\(583\) −6.19218e6 −0.754522
\(584\) −1.02277e6 −0.124092
\(585\) 0 0
\(586\) 4.00529e6 0.481826
\(587\) 2.47516e6 0.296489 0.148244 0.988951i \(-0.452638\pi\)
0.148244 + 0.988951i \(0.452638\pi\)
\(588\) 0 0
\(589\) −9.21349e6 −1.09430
\(590\) −3.59942e6 −0.425699
\(591\) 0 0
\(592\) 4.43657e6 0.520287
\(593\) −1.14108e7 −1.33253 −0.666267 0.745713i \(-0.732109\pi\)
−0.666267 + 0.745713i \(0.732109\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.16312e7 1.34125
\(597\) 0 0
\(598\) −629914. −0.0720325
\(599\) −8.60394e6 −0.979784 −0.489892 0.871783i \(-0.662964\pi\)
−0.489892 + 0.871783i \(0.662964\pi\)
\(600\) 0 0
\(601\) −1.00463e7 −1.13454 −0.567270 0.823532i \(-0.692001\pi\)
−0.567270 + 0.823532i \(0.692001\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.66351e6 −0.297072
\(605\) 9.05211e6 1.00545
\(606\) 0 0
\(607\) −2.22979e6 −0.245636 −0.122818 0.992429i \(-0.539193\pi\)
−0.122818 + 0.992429i \(0.539193\pi\)
\(608\) 5.24323e6 0.575228
\(609\) 0 0
\(610\) −3.65445e6 −0.397647
\(611\) −715223. −0.0775065
\(612\) 0 0
\(613\) −1.63670e7 −1.75921 −0.879607 0.475701i \(-0.842194\pi\)
−0.879607 + 0.475701i \(0.842194\pi\)
\(614\) 4.09840e6 0.438726
\(615\) 0 0
\(616\) 0 0
\(617\) 1.21424e7 1.28407 0.642037 0.766674i \(-0.278090\pi\)
0.642037 + 0.766674i \(0.278090\pi\)
\(618\) 0 0
\(619\) 1.12279e7 1.17780 0.588898 0.808207i \(-0.299562\pi\)
0.588898 + 0.808207i \(0.299562\pi\)
\(620\) 1.73609e7 1.81381
\(621\) 0 0
\(622\) 3.83190e6 0.397135
\(623\) 0 0
\(624\) 0 0
\(625\) −1.17701e7 −1.20525
\(626\) −1.79084e6 −0.182650
\(627\) 0 0
\(628\) −1.44701e7 −1.46411
\(629\) 1.04417e7 1.05231
\(630\) 0 0
\(631\) −5.69644e6 −0.569548 −0.284774 0.958595i \(-0.591918\pi\)
−0.284774 + 0.958595i \(0.591918\pi\)
\(632\) −1.03555e7 −1.03128
\(633\) 0 0
\(634\) −738208. −0.0729383
\(635\) 2.30278e7 2.26630
\(636\) 0 0
\(637\) 0 0
\(638\) 2.57097e6 0.250060
\(639\) 0 0
\(640\) −1.28951e7 −1.24444
\(641\) −2.96921e6 −0.285427 −0.142714 0.989764i \(-0.545583\pi\)
−0.142714 + 0.989764i \(0.545583\pi\)
\(642\) 0 0
\(643\) 1.22682e7 1.17018 0.585091 0.810968i \(-0.301059\pi\)
0.585091 + 0.810968i \(0.301059\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.41694e6 0.322149
\(647\) −1.21684e6 −0.114281 −0.0571404 0.998366i \(-0.518198\pi\)
−0.0571404 + 0.998366i \(0.518198\pi\)
\(648\) 0 0
\(649\) 5.51543e6 0.514005
\(650\) 580377. 0.0538799
\(651\) 0 0
\(652\) 7.04110e6 0.648667
\(653\) 1.10265e7 1.01194 0.505969 0.862552i \(-0.331135\pi\)
0.505969 + 0.862552i \(0.331135\pi\)
\(654\) 0 0
\(655\) −6.68617e6 −0.608939
\(656\) 3.72121e6 0.337618
\(657\) 0 0
\(658\) 0 0
\(659\) 1.32297e7 1.18668 0.593342 0.804951i \(-0.297808\pi\)
0.593342 + 0.804951i \(0.297808\pi\)
\(660\) 0 0
\(661\) −6.60257e6 −0.587773 −0.293886 0.955840i \(-0.594949\pi\)
−0.293886 + 0.955840i \(0.594949\pi\)
\(662\) −4.80777e6 −0.426382
\(663\) 0 0
\(664\) 5.32446e6 0.468657
\(665\) 0 0
\(666\) 0 0
\(667\) −1.86529e7 −1.62342
\(668\) 1.52591e7 1.32309
\(669\) 0 0
\(670\) −3.84518e6 −0.330925
\(671\) 5.59975e6 0.480134
\(672\) 0 0
\(673\) −1.42013e7 −1.20862 −0.604310 0.796749i \(-0.706551\pi\)
−0.604310 + 0.796749i \(0.706551\pi\)
\(674\) −2.25367e6 −0.191091
\(675\) 0 0
\(676\) 1.01129e7 0.851158
\(677\) −1.55878e7 −1.30712 −0.653559 0.756876i \(-0.726725\pi\)
−0.653559 + 0.756876i \(0.726725\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.35353e7 −1.12252
\(681\) 0 0
\(682\) 2.71987e6 0.223917
\(683\) −1.37838e7 −1.13063 −0.565313 0.824877i \(-0.691244\pi\)
−0.565313 + 0.824877i \(0.691244\pi\)
\(684\) 0 0
\(685\) −435841. −0.0354897
\(686\) 0 0
\(687\) 0 0
\(688\) −4.22727e6 −0.340478
\(689\) −4.85894e6 −0.389936
\(690\) 0 0
\(691\) 1.13345e7 0.903043 0.451522 0.892260i \(-0.350881\pi\)
0.451522 + 0.892260i \(0.350881\pi\)
\(692\) −2.79934e6 −0.222224
\(693\) 0 0
\(694\) 1.14404e6 0.0901657
\(695\) −1.00868e6 −0.0792121
\(696\) 0 0
\(697\) 8.75808e6 0.682852
\(698\) 3.91296e6 0.303995
\(699\) 0 0
\(700\) 0 0
\(701\) 4.21970e6 0.324330 0.162165 0.986764i \(-0.448152\pi\)
0.162165 + 0.986764i \(0.448152\pi\)
\(702\) 0 0
\(703\) −6.68455e6 −0.510133
\(704\) 3.07262e6 0.233656
\(705\) 0 0
\(706\) 3.38304e6 0.255444
\(707\) 0 0
\(708\) 0 0
\(709\) 1.25786e7 0.939757 0.469878 0.882731i \(-0.344298\pi\)
0.469878 + 0.882731i \(0.344298\pi\)
\(710\) −7.17499e6 −0.534165
\(711\) 0 0
\(712\) 6.27278e6 0.463724
\(713\) −1.97332e7 −1.45369
\(714\) 0 0
\(715\) −2.13918e6 −0.156488
\(716\) 5.18801e6 0.378197
\(717\) 0 0
\(718\) −3.60461e6 −0.260944
\(719\) −2.55041e7 −1.83988 −0.919938 0.392065i \(-0.871761\pi\)
−0.919938 + 0.392065i \(0.871761\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.07854e6 0.148393
\(723\) 0 0
\(724\) −6.51857e6 −0.462175
\(725\) 1.71860e7 1.21431
\(726\) 0 0
\(727\) −2.25405e7 −1.58171 −0.790857 0.612001i \(-0.790365\pi\)
−0.790857 + 0.612001i \(0.790365\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.22557e6 −0.0851200
\(731\) −9.94909e6 −0.688637
\(732\) 0 0
\(733\) −1.07168e7 −0.736724 −0.368362 0.929682i \(-0.620081\pi\)
−0.368362 + 0.929682i \(0.620081\pi\)
\(734\) 2.30595e6 0.157983
\(735\) 0 0
\(736\) 1.12298e7 0.764147
\(737\) 5.89200e6 0.399571
\(738\) 0 0
\(739\) −1.67756e7 −1.12997 −0.564987 0.825100i \(-0.691119\pi\)
−0.564987 + 0.825100i \(0.691119\pi\)
\(740\) 1.25956e7 0.845551
\(741\) 0 0
\(742\) 0 0
\(743\) −8.93625e6 −0.593859 −0.296929 0.954899i \(-0.595963\pi\)
−0.296929 + 0.954899i \(0.595963\pi\)
\(744\) 0 0
\(745\) 2.93001e7 1.93409
\(746\) 2.28550e6 0.150361
\(747\) 0 0
\(748\) 9.86577e6 0.644729
\(749\) 0 0
\(750\) 0 0
\(751\) 1.36875e7 0.885571 0.442785 0.896628i \(-0.353990\pi\)
0.442785 + 0.896628i \(0.353990\pi\)
\(752\) 3.53058e6 0.227668
\(753\) 0 0
\(754\) 2.01741e6 0.129231
\(755\) −6.70963e6 −0.428382
\(756\) 0 0
\(757\) 1.97877e7 1.25504 0.627518 0.778602i \(-0.284071\pi\)
0.627518 + 0.778602i \(0.284071\pi\)
\(758\) −1.48772e6 −0.0940477
\(759\) 0 0
\(760\) 8.66499e6 0.544170
\(761\) 5.23103e6 0.327435 0.163718 0.986507i \(-0.447651\pi\)
0.163718 + 0.986507i \(0.447651\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.62965e7 −1.62991
\(765\) 0 0
\(766\) −4.37107e6 −0.269163
\(767\) 4.32790e6 0.265637
\(768\) 0 0
\(769\) −1.04244e7 −0.635673 −0.317837 0.948145i \(-0.602956\pi\)
−0.317837 + 0.948145i \(0.602956\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.08342e7 1.25816
\(773\) −8.04863e6 −0.484477 −0.242238 0.970217i \(-0.577882\pi\)
−0.242238 + 0.970217i \(0.577882\pi\)
\(774\) 0 0
\(775\) 1.81813e7 1.08735
\(776\) 1.41117e7 0.841251
\(777\) 0 0
\(778\) −437287. −0.0259011
\(779\) −5.60673e6 −0.331029
\(780\) 0 0
\(781\) 1.09943e7 0.644971
\(782\) 7.31830e6 0.427950
\(783\) 0 0
\(784\) 0 0
\(785\) −3.64516e7 −2.11126
\(786\) 0 0
\(787\) −1.51619e7 −0.872602 −0.436301 0.899801i \(-0.643712\pi\)
−0.436301 + 0.899801i \(0.643712\pi\)
\(788\) −1.65983e6 −0.0952244
\(789\) 0 0
\(790\) −1.24089e7 −0.707400
\(791\) 0 0
\(792\) 0 0
\(793\) 4.39406e6 0.248132
\(794\) −7.88672e6 −0.443961
\(795\) 0 0
\(796\) 7.63643e6 0.427177
\(797\) −1.70525e7 −0.950915 −0.475458 0.879739i \(-0.657717\pi\)
−0.475458 + 0.879739i \(0.657717\pi\)
\(798\) 0 0
\(799\) 8.30941e6 0.460472
\(800\) −1.03467e7 −0.571578
\(801\) 0 0
\(802\) 7.55770e6 0.414910
\(803\) 1.87795e6 0.102777
\(804\) 0 0
\(805\) 0 0
\(806\) 2.13425e6 0.115720
\(807\) 0 0
\(808\) 1.42247e7 0.766506
\(809\) −1.81012e7 −0.972381 −0.486190 0.873853i \(-0.661614\pi\)
−0.486190 + 0.873853i \(0.661614\pi\)
\(810\) 0 0
\(811\) −1.37346e7 −0.733271 −0.366636 0.930365i \(-0.619490\pi\)
−0.366636 + 0.930365i \(0.619490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.97331e6 0.104384
\(815\) 1.77372e7 0.935387
\(816\) 0 0
\(817\) 6.36919e6 0.333833
\(818\) 382898. 0.0200078
\(819\) 0 0
\(820\) 1.05647e7 0.548684
\(821\) 2.27546e7 1.17818 0.589089 0.808068i \(-0.299487\pi\)
0.589089 + 0.808068i \(0.299487\pi\)
\(822\) 0 0
\(823\) 8.91287e6 0.458689 0.229344 0.973345i \(-0.426342\pi\)
0.229344 + 0.973345i \(0.426342\pi\)
\(824\) −2.09678e7 −1.07581
\(825\) 0 0
\(826\) 0 0
\(827\) 2.85147e7 1.44979 0.724896 0.688859i \(-0.241888\pi\)
0.724896 + 0.688859i \(0.241888\pi\)
\(828\) 0 0
\(829\) −1.53299e7 −0.774733 −0.387366 0.921926i \(-0.626615\pi\)
−0.387366 + 0.921926i \(0.626615\pi\)
\(830\) 6.38024e6 0.321471
\(831\) 0 0
\(832\) 2.41105e6 0.120753
\(833\) 0 0
\(834\) 0 0
\(835\) 3.84392e7 1.90791
\(836\) −6.31585e6 −0.312548
\(837\) 0 0
\(838\) −7.46973e6 −0.367447
\(839\) −2.17291e7 −1.06571 −0.532853 0.846208i \(-0.678880\pi\)
−0.532853 + 0.846208i \(0.678880\pi\)
\(840\) 0 0
\(841\) 3.92279e7 1.91252
\(842\) −986403. −0.0479484
\(843\) 0 0
\(844\) 1.43776e7 0.694753
\(845\) 2.54754e7 1.22738
\(846\) 0 0
\(847\) 0 0
\(848\) 2.39853e7 1.14540
\(849\) 0 0
\(850\) −6.74278e6 −0.320104
\(851\) −1.43168e7 −0.677674
\(852\) 0 0
\(853\) −2.60934e7 −1.22789 −0.613943 0.789350i \(-0.710418\pi\)
−0.613943 + 0.789350i \(0.710418\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.81025e6 −0.457610
\(857\) −1.07218e7 −0.498675 −0.249337 0.968417i \(-0.580213\pi\)
−0.249337 + 0.968417i \(0.580213\pi\)
\(858\) 0 0
\(859\) 3.54531e7 1.63935 0.819675 0.572829i \(-0.194154\pi\)
0.819675 + 0.572829i \(0.194154\pi\)
\(860\) −1.20014e7 −0.553332
\(861\) 0 0
\(862\) 5.68932e6 0.260791
\(863\) −1.63278e7 −0.746278 −0.373139 0.927775i \(-0.621719\pi\)
−0.373139 + 0.927775i \(0.621719\pi\)
\(864\) 0 0
\(865\) −7.05180e6 −0.320450
\(866\) −33226.3 −0.00150552
\(867\) 0 0
\(868\) 0 0
\(869\) 1.90143e7 0.854142
\(870\) 0 0
\(871\) 4.62339e6 0.206498
\(872\) −4.39722e6 −0.195834
\(873\) 0 0
\(874\) −4.68501e6 −0.207459
\(875\) 0 0
\(876\) 0 0
\(877\) 1.94566e7 0.854216 0.427108 0.904201i \(-0.359532\pi\)
0.427108 + 0.904201i \(0.359532\pi\)
\(878\) 649201. 0.0284213
\(879\) 0 0
\(880\) 1.05597e7 0.459669
\(881\) 4.31688e7 1.87383 0.936916 0.349555i \(-0.113667\pi\)
0.936916 + 0.349555i \(0.113667\pi\)
\(882\) 0 0
\(883\) −7.63478e6 −0.329530 −0.164765 0.986333i \(-0.552687\pi\)
−0.164765 + 0.986333i \(0.552687\pi\)
\(884\) 7.74157e6 0.333195
\(885\) 0 0
\(886\) −2.14824e6 −0.0919387
\(887\) −4.08512e6 −0.174340 −0.0871698 0.996193i \(-0.527782\pi\)
−0.0871698 + 0.996193i \(0.527782\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.51661e6 0.318088
\(891\) 0 0
\(892\) −1.08374e7 −0.456051
\(893\) −5.31950e6 −0.223225
\(894\) 0 0
\(895\) 1.30691e7 0.545366
\(896\) 0 0
\(897\) 0 0
\(898\) −4.91531e6 −0.203404
\(899\) 6.31990e7 2.60802
\(900\) 0 0
\(901\) 5.64508e7 2.31664
\(902\) 1.65514e6 0.0677357
\(903\) 0 0
\(904\) 1.79343e7 0.729900
\(905\) −1.64209e7 −0.666462
\(906\) 0 0
\(907\) 9.00583e6 0.363501 0.181751 0.983345i \(-0.441824\pi\)
0.181751 + 0.983345i \(0.441824\pi\)
\(908\) −8.26165e6 −0.332546
\(909\) 0 0
\(910\) 0 0
\(911\) −2.17438e7 −0.868041 −0.434020 0.900903i \(-0.642905\pi\)
−0.434020 + 0.900903i \(0.642905\pi\)
\(912\) 0 0
\(913\) −9.77650e6 −0.388156
\(914\) −3.62685e6 −0.143603
\(915\) 0 0
\(916\) −2.87229e7 −1.13107
\(917\) 0 0
\(918\) 0 0
\(919\) −3.84235e7 −1.50075 −0.750374 0.661013i \(-0.770127\pi\)
−0.750374 + 0.661013i \(0.770127\pi\)
\(920\) 1.85584e7 0.722888
\(921\) 0 0
\(922\) 1.54461e7 0.598400
\(923\) 8.62711e6 0.333320
\(924\) 0 0
\(925\) 1.31909e7 0.506896
\(926\) 1.20784e7 0.462896
\(927\) 0 0
\(928\) −3.59654e7 −1.37093
\(929\) 3.67005e7 1.39519 0.697594 0.716493i \(-0.254254\pi\)
0.697594 + 0.716493i \(0.254254\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.74850e7 0.659366
\(933\) 0 0
\(934\) −5.63102e6 −0.211213
\(935\) 2.48528e7 0.929708
\(936\) 0 0
\(937\) 1.37847e7 0.512919 0.256459 0.966555i \(-0.417444\pi\)
0.256459 + 0.966555i \(0.417444\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.00235e7 0.369997
\(941\) −6.91322e6 −0.254511 −0.127255 0.991870i \(-0.540617\pi\)
−0.127255 + 0.991870i \(0.540617\pi\)
\(942\) 0 0
\(943\) −1.20083e7 −0.439747
\(944\) −2.13640e7 −0.780283
\(945\) 0 0
\(946\) −1.88022e6 −0.0683094
\(947\) −1.67481e7 −0.606863 −0.303431 0.952853i \(-0.598132\pi\)
−0.303431 + 0.952853i \(0.598132\pi\)
\(948\) 0 0
\(949\) 1.47361e6 0.0531150
\(950\) 4.31658e6 0.155178
\(951\) 0 0
\(952\) 0 0
\(953\) −1.09625e7 −0.390999 −0.195500 0.980704i \(-0.562633\pi\)
−0.195500 + 0.980704i \(0.562633\pi\)
\(954\) 0 0
\(955\) −6.62434e7 −2.35036
\(956\) −4.19138e7 −1.48324
\(957\) 0 0
\(958\) 8.43213e6 0.296841
\(959\) 0 0
\(960\) 0 0
\(961\) 3.82302e7 1.33536
\(962\) 1.54844e6 0.0539456
\(963\) 0 0
\(964\) 1.56422e7 0.542132
\(965\) 5.24835e7 1.81428
\(966\) 0 0
\(967\) 2.98311e7 1.02589 0.512947 0.858420i \(-0.328554\pi\)
0.512947 + 0.858420i \(0.328554\pi\)
\(968\) −1.30149e7 −0.446428
\(969\) 0 0
\(970\) 1.69099e7 0.577049
\(971\) −1.74542e7 −0.594090 −0.297045 0.954864i \(-0.596001\pi\)
−0.297045 + 0.954864i \(0.596001\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 7.93442e6 0.267989
\(975\) 0 0
\(976\) −2.16906e7 −0.728864
\(977\) 2.21922e7 0.743812 0.371906 0.928270i \(-0.378704\pi\)
0.371906 + 0.928270i \(0.378704\pi\)
\(978\) 0 0
\(979\) −1.15178e7 −0.384071
\(980\) 0 0
\(981\) 0 0
\(982\) 9.61545e6 0.318193
\(983\) −5.05328e7 −1.66798 −0.833988 0.551783i \(-0.813948\pi\)
−0.833988 + 0.551783i \(0.813948\pi\)
\(984\) 0 0
\(985\) −4.18127e6 −0.137315
\(986\) −2.34382e7 −0.767770
\(987\) 0 0
\(988\) −4.95598e6 −0.161524
\(989\) 1.36413e7 0.443472
\(990\) 0 0
\(991\) −7.91699e6 −0.256080 −0.128040 0.991769i \(-0.540869\pi\)
−0.128040 + 0.991769i \(0.540869\pi\)
\(992\) −3.80484e7 −1.22760
\(993\) 0 0
\(994\) 0 0
\(995\) 1.92369e7 0.615995
\(996\) 0 0
\(997\) 3.28160e7 1.04556 0.522778 0.852469i \(-0.324896\pi\)
0.522778 + 0.852469i \(0.324896\pi\)
\(998\) 2.58123e6 0.0820352
\(999\) 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.1.3 8
3.2 odd 2 inner 441.6.a.be.1.6 yes 8
7.6 odd 2 inner 441.6.a.be.1.4 yes 8
21.20 even 2 inner 441.6.a.be.1.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.be.1.3 8 1.1 even 1 trivial
441.6.a.be.1.4 yes 8 7.6 odd 2 inner
441.6.a.be.1.5 yes 8 21.20 even 2 inner
441.6.a.be.1.6 yes 8 3.2 odd 2 inner