Properties

Label 441.6.a.w.1.4
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 97 x^{2} + 7 x + 294\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(10.1812\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} +O(q^{10})\) \(q+9.18123 q^{2} +52.2950 q^{4} -22.0716 q^{5} +186.333 q^{8} -202.644 q^{10} -416.710 q^{11} +797.918 q^{13} +37.3245 q^{16} +1375.55 q^{17} +2313.03 q^{19} -1154.23 q^{20} -3825.91 q^{22} +955.402 q^{23} -2637.84 q^{25} +7325.87 q^{26} +7035.29 q^{29} +1261.19 q^{31} -5619.96 q^{32} +12629.2 q^{34} +9776.44 q^{37} +21236.4 q^{38} -4112.66 q^{40} +5400.95 q^{41} +19686.6 q^{43} -21791.8 q^{44} +8771.76 q^{46} -2056.56 q^{47} -24218.7 q^{50} +41727.1 q^{52} -18022.7 q^{53} +9197.45 q^{55} +64592.6 q^{58} -7435.68 q^{59} +3495.38 q^{61} +11579.3 q^{62} -52792.5 q^{64} -17611.3 q^{65} +15856.4 q^{67} +71934.4 q^{68} -58133.5 q^{71} +39110.7 q^{73} +89759.8 q^{74} +120960. q^{76} +9760.69 q^{79} -823.812 q^{80} +49587.4 q^{82} +70395.7 q^{83} -30360.6 q^{85} +180747. q^{86} -77646.6 q^{88} -144306. q^{89} +49962.7 q^{92} -18881.8 q^{94} -51052.2 q^{95} -79328.7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + O(q^{10}) \) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.18123 1.62303 0.811514 0.584333i \(-0.198644\pi\)
0.811514 + 0.584333i \(0.198644\pi\)
\(3\) 0 0
\(4\) 52.2950 1.63422
\(5\) −22.0716 −0.394829 −0.197414 0.980320i \(-0.563254\pi\)
−0.197414 + 0.980320i \(0.563254\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 186.333 1.02935
\(9\) 0 0
\(10\) −202.644 −0.640818
\(11\) −416.710 −1.03837 −0.519184 0.854662i \(-0.673764\pi\)
−0.519184 + 0.854662i \(0.673764\pi\)
\(12\) 0 0
\(13\) 797.918 1.30948 0.654742 0.755853i \(-0.272777\pi\)
0.654742 + 0.755853i \(0.272777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 37.3245 0.0364497
\(17\) 1375.55 1.15439 0.577197 0.816605i \(-0.304146\pi\)
0.577197 + 0.816605i \(0.304146\pi\)
\(18\) 0 0
\(19\) 2313.03 1.46993 0.734965 0.678105i \(-0.237199\pi\)
0.734965 + 0.678105i \(0.237199\pi\)
\(20\) −1154.23 −0.645236
\(21\) 0 0
\(22\) −3825.91 −1.68530
\(23\) 955.402 0.376588 0.188294 0.982113i \(-0.439704\pi\)
0.188294 + 0.982113i \(0.439704\pi\)
\(24\) 0 0
\(25\) −2637.84 −0.844110
\(26\) 7325.87 2.12533
\(27\) 0 0
\(28\) 0 0
\(29\) 7035.29 1.55341 0.776707 0.629862i \(-0.216889\pi\)
0.776707 + 0.629862i \(0.216889\pi\)
\(30\) 0 0
\(31\) 1261.19 0.235709 0.117855 0.993031i \(-0.462398\pi\)
0.117855 + 0.993031i \(0.462398\pi\)
\(32\) −5619.96 −0.970194
\(33\) 0 0
\(34\) 12629.2 1.87361
\(35\) 0 0
\(36\) 0 0
\(37\) 9776.44 1.17402 0.587012 0.809579i \(-0.300304\pi\)
0.587012 + 0.809579i \(0.300304\pi\)
\(38\) 21236.4 2.38574
\(39\) 0 0
\(40\) −4112.66 −0.406418
\(41\) 5400.95 0.501777 0.250888 0.968016i \(-0.419277\pi\)
0.250888 + 0.968016i \(0.419277\pi\)
\(42\) 0 0
\(43\) 19686.6 1.62367 0.811837 0.583885i \(-0.198468\pi\)
0.811837 + 0.583885i \(0.198468\pi\)
\(44\) −21791.8 −1.69692
\(45\) 0 0
\(46\) 8771.76 0.611213
\(47\) −2056.56 −0.135799 −0.0678997 0.997692i \(-0.521630\pi\)
−0.0678997 + 0.997692i \(0.521630\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −24218.7 −1.37001
\(51\) 0 0
\(52\) 41727.1 2.13998
\(53\) −18022.7 −0.881315 −0.440658 0.897675i \(-0.645255\pi\)
−0.440658 + 0.897675i \(0.645255\pi\)
\(54\) 0 0
\(55\) 9197.45 0.409978
\(56\) 0 0
\(57\) 0 0
\(58\) 64592.6 2.52123
\(59\) −7435.68 −0.278093 −0.139047 0.990286i \(-0.544404\pi\)
−0.139047 + 0.990286i \(0.544404\pi\)
\(60\) 0 0
\(61\) 3495.38 0.120274 0.0601368 0.998190i \(-0.480846\pi\)
0.0601368 + 0.998190i \(0.480846\pi\)
\(62\) 11579.3 0.382563
\(63\) 0 0
\(64\) −52792.5 −1.61110
\(65\) −17611.3 −0.517022
\(66\) 0 0
\(67\) 15856.4 0.431537 0.215769 0.976445i \(-0.430774\pi\)
0.215769 + 0.976445i \(0.430774\pi\)
\(68\) 71934.4 1.88653
\(69\) 0 0
\(70\) 0 0
\(71\) −58133.5 −1.36861 −0.684306 0.729195i \(-0.739895\pi\)
−0.684306 + 0.729195i \(0.739895\pi\)
\(72\) 0 0
\(73\) 39110.7 0.858990 0.429495 0.903069i \(-0.358692\pi\)
0.429495 + 0.903069i \(0.358692\pi\)
\(74\) 89759.8 1.90547
\(75\) 0 0
\(76\) 120960. 2.40218
\(77\) 0 0
\(78\) 0 0
\(79\) 9760.69 0.175960 0.0879798 0.996122i \(-0.471959\pi\)
0.0879798 + 0.996122i \(0.471959\pi\)
\(80\) −823.812 −0.0143914
\(81\) 0 0
\(82\) 49587.4 0.814397
\(83\) 70395.7 1.12163 0.560816 0.827940i \(-0.310487\pi\)
0.560816 + 0.827940i \(0.310487\pi\)
\(84\) 0 0
\(85\) −30360.6 −0.455788
\(86\) 180747. 2.63527
\(87\) 0 0
\(88\) −77646.6 −1.06885
\(89\) −144306. −1.93112 −0.965562 0.260173i \(-0.916220\pi\)
−0.965562 + 0.260173i \(0.916220\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 49962.7 0.615427
\(93\) 0 0
\(94\) −18881.8 −0.220406
\(95\) −51052.2 −0.580371
\(96\) 0 0
\(97\) −79328.7 −0.856053 −0.428027 0.903766i \(-0.640791\pi\)
−0.428027 + 0.903766i \(0.640791\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −137946. −1.37946
\(101\) 84833.7 0.827495 0.413747 0.910392i \(-0.364220\pi\)
0.413747 + 0.910392i \(0.364220\pi\)
\(102\) 0 0
\(103\) 20332.3 0.188839 0.0944197 0.995532i \(-0.469900\pi\)
0.0944197 + 0.995532i \(0.469900\pi\)
\(104\) 148678. 1.34792
\(105\) 0 0
\(106\) −165471. −1.43040
\(107\) −6962.19 −0.0587877 −0.0293938 0.999568i \(-0.509358\pi\)
−0.0293938 + 0.999568i \(0.509358\pi\)
\(108\) 0 0
\(109\) 112651. 0.908177 0.454088 0.890957i \(-0.349965\pi\)
0.454088 + 0.890957i \(0.349965\pi\)
\(110\) 84443.9 0.665406
\(111\) 0 0
\(112\) 0 0
\(113\) −112005. −0.825167 −0.412583 0.910920i \(-0.635373\pi\)
−0.412583 + 0.910920i \(0.635373\pi\)
\(114\) 0 0
\(115\) −21087.3 −0.148688
\(116\) 367910. 2.53862
\(117\) 0 0
\(118\) −68268.7 −0.451353
\(119\) 0 0
\(120\) 0 0
\(121\) 12595.8 0.0782102
\(122\) 32091.9 0.195207
\(123\) 0 0
\(124\) 65954.0 0.385200
\(125\) 127195. 0.728108
\(126\) 0 0
\(127\) 82224.5 0.452368 0.226184 0.974085i \(-0.427375\pi\)
0.226184 + 0.974085i \(0.427375\pi\)
\(128\) −304862. −1.64467
\(129\) 0 0
\(130\) −161694. −0.839140
\(131\) 175812. 0.895097 0.447548 0.894260i \(-0.352297\pi\)
0.447548 + 0.894260i \(0.352297\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 145581. 0.700397
\(135\) 0 0
\(136\) 256310. 1.18828
\(137\) −31330.3 −0.142614 −0.0713072 0.997454i \(-0.522717\pi\)
−0.0713072 + 0.997454i \(0.522717\pi\)
\(138\) 0 0
\(139\) 152234. 0.668305 0.334152 0.942519i \(-0.391550\pi\)
0.334152 + 0.942519i \(0.391550\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −533737. −2.22129
\(143\) −332500. −1.35973
\(144\) 0 0
\(145\) −155280. −0.613332
\(146\) 359084. 1.39416
\(147\) 0 0
\(148\) 511259. 1.91861
\(149\) −362860. −1.33898 −0.669489 0.742822i \(-0.733487\pi\)
−0.669489 + 0.742822i \(0.733487\pi\)
\(150\) 0 0
\(151\) 205126. 0.732112 0.366056 0.930593i \(-0.380708\pi\)
0.366056 + 0.930593i \(0.380708\pi\)
\(152\) 430992. 1.51308
\(153\) 0 0
\(154\) 0 0
\(155\) −27836.5 −0.0930648
\(156\) 0 0
\(157\) −77272.0 −0.250192 −0.125096 0.992145i \(-0.539924\pi\)
−0.125096 + 0.992145i \(0.539924\pi\)
\(158\) 89615.1 0.285587
\(159\) 0 0
\(160\) 124042. 0.383060
\(161\) 0 0
\(162\) 0 0
\(163\) 184931. 0.545182 0.272591 0.962130i \(-0.412119\pi\)
0.272591 + 0.962130i \(0.412119\pi\)
\(164\) 282442. 0.820012
\(165\) 0 0
\(166\) 646319. 1.82044
\(167\) −129262. −0.358657 −0.179329 0.983789i \(-0.557393\pi\)
−0.179329 + 0.983789i \(0.557393\pi\)
\(168\) 0 0
\(169\) 265380. 0.714746
\(170\) −278748. −0.739757
\(171\) 0 0
\(172\) 1.02951e6 2.65344
\(173\) 507867. 1.29013 0.645067 0.764126i \(-0.276829\pi\)
0.645067 + 0.764126i \(0.276829\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −15553.5 −0.0378483
\(177\) 0 0
\(178\) −1.32491e6 −3.13427
\(179\) 132589. 0.309296 0.154648 0.987970i \(-0.450576\pi\)
0.154648 + 0.987970i \(0.450576\pi\)
\(180\) 0 0
\(181\) −740060. −1.67908 −0.839538 0.543301i \(-0.817174\pi\)
−0.839538 + 0.543301i \(0.817174\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 178023. 0.387642
\(185\) −215782. −0.463538
\(186\) 0 0
\(187\) −573205. −1.19869
\(188\) −107548. −0.221926
\(189\) 0 0
\(190\) −468722. −0.941957
\(191\) 582732. 1.15581 0.577904 0.816105i \(-0.303871\pi\)
0.577904 + 0.816105i \(0.303871\pi\)
\(192\) 0 0
\(193\) −400904. −0.774725 −0.387362 0.921928i \(-0.626614\pi\)
−0.387362 + 0.921928i \(0.626614\pi\)
\(194\) −728335. −1.38940
\(195\) 0 0
\(196\) 0 0
\(197\) −671589. −1.23293 −0.616464 0.787383i \(-0.711436\pi\)
−0.616464 + 0.787383i \(0.711436\pi\)
\(198\) 0 0
\(199\) −455022. −0.814515 −0.407258 0.913313i \(-0.633515\pi\)
−0.407258 + 0.913313i \(0.633515\pi\)
\(200\) −491517. −0.868887
\(201\) 0 0
\(202\) 778878. 1.34305
\(203\) 0 0
\(204\) 0 0
\(205\) −119208. −0.198116
\(206\) 186675. 0.306492
\(207\) 0 0
\(208\) 29781.9 0.0477303
\(209\) −963860. −1.52633
\(210\) 0 0
\(211\) −1.19545e6 −1.84852 −0.924260 0.381764i \(-0.875317\pi\)
−0.924260 + 0.381764i \(0.875317\pi\)
\(212\) −942499. −1.44026
\(213\) 0 0
\(214\) −63921.4 −0.0954140
\(215\) −434514. −0.641073
\(216\) 0 0
\(217\) 0 0
\(218\) 1.03428e6 1.47400
\(219\) 0 0
\(220\) 480980. 0.669993
\(221\) 1.09758e6 1.51166
\(222\) 0 0
\(223\) 296529. 0.399305 0.199653 0.979867i \(-0.436019\pi\)
0.199653 + 0.979867i \(0.436019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.02834e6 −1.33927
\(227\) −218146. −0.280985 −0.140492 0.990082i \(-0.544869\pi\)
−0.140492 + 0.990082i \(0.544869\pi\)
\(228\) 0 0
\(229\) −1.22920e6 −1.54894 −0.774471 0.632609i \(-0.781984\pi\)
−0.774471 + 0.632609i \(0.781984\pi\)
\(230\) −193607. −0.241324
\(231\) 0 0
\(232\) 1.31090e6 1.59901
\(233\) 62944.2 0.0759567 0.0379784 0.999279i \(-0.487908\pi\)
0.0379784 + 0.999279i \(0.487908\pi\)
\(234\) 0 0
\(235\) 45391.7 0.0536175
\(236\) −388849. −0.454465
\(237\) 0 0
\(238\) 0 0
\(239\) −219330. −0.248372 −0.124186 0.992259i \(-0.539632\pi\)
−0.124186 + 0.992259i \(0.539632\pi\)
\(240\) 0 0
\(241\) 433864. 0.481183 0.240592 0.970626i \(-0.422659\pi\)
0.240592 + 0.970626i \(0.422659\pi\)
\(242\) 115645. 0.126937
\(243\) 0 0
\(244\) 182791. 0.196553
\(245\) 0 0
\(246\) 0 0
\(247\) 1.84561e6 1.92485
\(248\) 235001. 0.242628
\(249\) 0 0
\(250\) 1.16781e6 1.18174
\(251\) 1.71109e6 1.71431 0.857155 0.515059i \(-0.172230\pi\)
0.857155 + 0.515059i \(0.172230\pi\)
\(252\) 0 0
\(253\) −398125. −0.391037
\(254\) 754922. 0.734206
\(255\) 0 0
\(256\) −1.10964e6 −1.05824
\(257\) −984057. −0.929368 −0.464684 0.885477i \(-0.653832\pi\)
−0.464684 + 0.885477i \(0.653832\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −920984. −0.844926
\(261\) 0 0
\(262\) 1.61417e6 1.45277
\(263\) −547095. −0.487724 −0.243862 0.969810i \(-0.578414\pi\)
−0.243862 + 0.969810i \(0.578414\pi\)
\(264\) 0 0
\(265\) 397791. 0.347969
\(266\) 0 0
\(267\) 0 0
\(268\) 829211. 0.705226
\(269\) 641112. 0.540198 0.270099 0.962833i \(-0.412944\pi\)
0.270099 + 0.962833i \(0.412944\pi\)
\(270\) 0 0
\(271\) −548012. −0.453280 −0.226640 0.973979i \(-0.572774\pi\)
−0.226640 + 0.973979i \(0.572774\pi\)
\(272\) 51341.8 0.0420774
\(273\) 0 0
\(274\) −287651. −0.231467
\(275\) 1.09921e6 0.876498
\(276\) 0 0
\(277\) 1.67414e6 1.31097 0.655485 0.755208i \(-0.272464\pi\)
0.655485 + 0.755208i \(0.272464\pi\)
\(278\) 1.39769e6 1.08468
\(279\) 0 0
\(280\) 0 0
\(281\) −1.81078e6 −1.36804 −0.684021 0.729462i \(-0.739770\pi\)
−0.684021 + 0.729462i \(0.739770\pi\)
\(282\) 0 0
\(283\) 2.51315e6 1.86531 0.932657 0.360764i \(-0.117484\pi\)
0.932657 + 0.360764i \(0.117484\pi\)
\(284\) −3.04009e6 −2.23661
\(285\) 0 0
\(286\) −3.05276e6 −2.20687
\(287\) 0 0
\(288\) 0 0
\(289\) 472283. 0.332627
\(290\) −1.42566e6 −0.995455
\(291\) 0 0
\(292\) 2.04529e6 1.40378
\(293\) −107228. −0.0729691 −0.0364845 0.999334i \(-0.511616\pi\)
−0.0364845 + 0.999334i \(0.511616\pi\)
\(294\) 0 0
\(295\) 164117. 0.109799
\(296\) 1.82167e6 1.20848
\(297\) 0 0
\(298\) −3.33150e6 −2.17320
\(299\) 762332. 0.493136
\(300\) 0 0
\(301\) 0 0
\(302\) 1.88330e6 1.18824
\(303\) 0 0
\(304\) 86332.6 0.0535785
\(305\) −77148.7 −0.0474875
\(306\) 0 0
\(307\) 1.49622e6 0.906042 0.453021 0.891500i \(-0.350346\pi\)
0.453021 + 0.891500i \(0.350346\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −255573. −0.151047
\(311\) −861202. −0.504899 −0.252449 0.967610i \(-0.581236\pi\)
−0.252449 + 0.967610i \(0.581236\pi\)
\(312\) 0 0
\(313\) −503937. −0.290747 −0.145374 0.989377i \(-0.546438\pi\)
−0.145374 + 0.989377i \(0.546438\pi\)
\(314\) −709452. −0.406068
\(315\) 0 0
\(316\) 510435. 0.287556
\(317\) 480009. 0.268288 0.134144 0.990962i \(-0.457172\pi\)
0.134144 + 0.990962i \(0.457172\pi\)
\(318\) 0 0
\(319\) −2.93167e6 −1.61302
\(320\) 1.16522e6 0.636109
\(321\) 0 0
\(322\) 0 0
\(323\) 3.18168e6 1.69688
\(324\) 0 0
\(325\) −2.10478e6 −1.10535
\(326\) 1.69790e6 0.884845
\(327\) 0 0
\(328\) 1.00637e6 0.516505
\(329\) 0 0
\(330\) 0 0
\(331\) −2.19922e6 −1.10331 −0.551656 0.834072i \(-0.686004\pi\)
−0.551656 + 0.834072i \(0.686004\pi\)
\(332\) 3.68134e6 1.83299
\(333\) 0 0
\(334\) −1.18678e6 −0.582110
\(335\) −349977. −0.170383
\(336\) 0 0
\(337\) −1.35725e6 −0.651008 −0.325504 0.945541i \(-0.605534\pi\)
−0.325504 + 0.945541i \(0.605534\pi\)
\(338\) 2.43652e6 1.16005
\(339\) 0 0
\(340\) −1.58771e6 −0.744857
\(341\) −525550. −0.244753
\(342\) 0 0
\(343\) 0 0
\(344\) 3.66825e6 1.67133
\(345\) 0 0
\(346\) 4.66285e6 2.09392
\(347\) −3.79941e6 −1.69392 −0.846959 0.531659i \(-0.821569\pi\)
−0.846959 + 0.531659i \(0.821569\pi\)
\(348\) 0 0
\(349\) −1.31753e6 −0.579024 −0.289512 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.34189e6 1.00742
\(353\) 3.42505e6 1.46295 0.731477 0.681866i \(-0.238832\pi\)
0.731477 + 0.681866i \(0.238832\pi\)
\(354\) 0 0
\(355\) 1.28310e6 0.540367
\(356\) −7.54649e6 −3.15588
\(357\) 0 0
\(358\) 1.21733e6 0.501997
\(359\) −1.47084e6 −0.602322 −0.301161 0.953573i \(-0.597374\pi\)
−0.301161 + 0.953573i \(0.597374\pi\)
\(360\) 0 0
\(361\) 2.87399e6 1.16069
\(362\) −6.79466e6 −2.72519
\(363\) 0 0
\(364\) 0 0
\(365\) −863235. −0.339154
\(366\) 0 0
\(367\) −4.98079e6 −1.93034 −0.965168 0.261630i \(-0.915740\pi\)
−0.965168 + 0.261630i \(0.915740\pi\)
\(368\) 35659.9 0.0137265
\(369\) 0 0
\(370\) −1.98114e6 −0.752335
\(371\) 0 0
\(372\) 0 0
\(373\) 3.96047e6 1.47392 0.736962 0.675934i \(-0.236260\pi\)
0.736962 + 0.675934i \(0.236260\pi\)
\(374\) −5.26273e6 −1.94550
\(375\) 0 0
\(376\) −383205. −0.139785
\(377\) 5.61359e6 2.03417
\(378\) 0 0
\(379\) −1.75155e6 −0.626359 −0.313179 0.949694i \(-0.601394\pi\)
−0.313179 + 0.949694i \(0.601394\pi\)
\(380\) −2.66977e6 −0.948452
\(381\) 0 0
\(382\) 5.35020e6 1.87591
\(383\) −3.13834e6 −1.09321 −0.546604 0.837391i \(-0.684080\pi\)
−0.546604 + 0.837391i \(0.684080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.68079e6 −1.25740
\(387\) 0 0
\(388\) −4.14849e6 −1.39898
\(389\) 1.05252e6 0.352661 0.176331 0.984331i \(-0.443577\pi\)
0.176331 + 0.984331i \(0.443577\pi\)
\(390\) 0 0
\(391\) 1.31420e6 0.434731
\(392\) 0 0
\(393\) 0 0
\(394\) −6.16601e6 −2.00108
\(395\) −215434. −0.0694739
\(396\) 0 0
\(397\) 454724. 0.144801 0.0724005 0.997376i \(-0.476934\pi\)
0.0724005 + 0.997376i \(0.476934\pi\)
\(398\) −4.17766e6 −1.32198
\(399\) 0 0
\(400\) −98456.3 −0.0307676
\(401\) 2.88431e6 0.895739 0.447870 0.894099i \(-0.352183\pi\)
0.447870 + 0.894099i \(0.352183\pi\)
\(402\) 0 0
\(403\) 1.00633e6 0.308657
\(404\) 4.43638e6 1.35231
\(405\) 0 0
\(406\) 0 0
\(407\) −4.07394e6 −1.21907
\(408\) 0 0
\(409\) −225914. −0.0667782 −0.0333891 0.999442i \(-0.510630\pi\)
−0.0333891 + 0.999442i \(0.510630\pi\)
\(410\) −1.09447e6 −0.321548
\(411\) 0 0
\(412\) 1.06328e6 0.308605
\(413\) 0 0
\(414\) 0 0
\(415\) −1.55375e6 −0.442853
\(416\) −4.48427e6 −1.27045
\(417\) 0 0
\(418\) −8.84942e6 −2.47727
\(419\) −4.31027e6 −1.19941 −0.599707 0.800220i \(-0.704716\pi\)
−0.599707 + 0.800220i \(0.704716\pi\)
\(420\) 0 0
\(421\) 1.25088e6 0.343962 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(422\) −1.09757e7 −3.00020
\(423\) 0 0
\(424\) −3.35823e6 −0.907184
\(425\) −3.62849e6 −0.974436
\(426\) 0 0
\(427\) 0 0
\(428\) −364087. −0.0960719
\(429\) 0 0
\(430\) −3.98937e6 −1.04048
\(431\) 4.40793e6 1.14299 0.571494 0.820606i \(-0.306364\pi\)
0.571494 + 0.820606i \(0.306364\pi\)
\(432\) 0 0
\(433\) −1.60951e6 −0.412549 −0.206274 0.978494i \(-0.566134\pi\)
−0.206274 + 0.978494i \(0.566134\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.89110e6 1.48416
\(437\) 2.20987e6 0.553558
\(438\) 0 0
\(439\) −4.42452e6 −1.09573 −0.547867 0.836566i \(-0.684560\pi\)
−0.547867 + 0.836566i \(0.684560\pi\)
\(440\) 1.71379e6 0.422012
\(441\) 0 0
\(442\) 1.00771e7 2.45347
\(443\) 3.61438e6 0.875034 0.437517 0.899210i \(-0.355858\pi\)
0.437517 + 0.899210i \(0.355858\pi\)
\(444\) 0 0
\(445\) 3.18507e6 0.762464
\(446\) 2.72250e6 0.648084
\(447\) 0 0
\(448\) 0 0
\(449\) 467024. 0.109326 0.0546630 0.998505i \(-0.482592\pi\)
0.0546630 + 0.998505i \(0.482592\pi\)
\(450\) 0 0
\(451\) −2.25063e6 −0.521029
\(452\) −5.85730e6 −1.34850
\(453\) 0 0
\(454\) −2.00285e6 −0.456046
\(455\) 0 0
\(456\) 0 0
\(457\) −601252. −0.134668 −0.0673342 0.997730i \(-0.521449\pi\)
−0.0673342 + 0.997730i \(0.521449\pi\)
\(458\) −1.12856e7 −2.51398
\(459\) 0 0
\(460\) −1.10276e6 −0.242988
\(461\) −2.87193e6 −0.629392 −0.314696 0.949192i \(-0.601903\pi\)
−0.314696 + 0.949192i \(0.601903\pi\)
\(462\) 0 0
\(463\) −2.91502e6 −0.631959 −0.315979 0.948766i \(-0.602333\pi\)
−0.315979 + 0.948766i \(0.602333\pi\)
\(464\) 262589. 0.0566215
\(465\) 0 0
\(466\) 577906. 0.123280
\(467\) 7.19376e6 1.52638 0.763192 0.646172i \(-0.223631\pi\)
0.763192 + 0.646172i \(0.223631\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 416751. 0.0870227
\(471\) 0 0
\(472\) −1.38551e6 −0.286256
\(473\) −8.20358e6 −1.68597
\(474\) 0 0
\(475\) −6.10140e6 −1.24078
\(476\) 0 0
\(477\) 0 0
\(478\) −2.01372e6 −0.403115
\(479\) 2.79649e6 0.556896 0.278448 0.960451i \(-0.410180\pi\)
0.278448 + 0.960451i \(0.410180\pi\)
\(480\) 0 0
\(481\) 7.80080e6 1.53736
\(482\) 3.98340e6 0.780974
\(483\) 0 0
\(484\) 658698. 0.127812
\(485\) 1.75091e6 0.337995
\(486\) 0 0
\(487\) −2.93247e6 −0.560289 −0.280144 0.959958i \(-0.590382\pi\)
−0.280144 + 0.959958i \(0.590382\pi\)
\(488\) 651305. 0.123804
\(489\) 0 0
\(490\) 0 0
\(491\) 3.06121e6 0.573046 0.286523 0.958073i \(-0.407501\pi\)
0.286523 + 0.958073i \(0.407501\pi\)
\(492\) 0 0
\(493\) 9.67740e6 1.79325
\(494\) 1.69449e7 3.12408
\(495\) 0 0
\(496\) 47073.4 0.00859154
\(497\) 0 0
\(498\) 0 0
\(499\) −6.55154e6 −1.17786 −0.588928 0.808186i \(-0.700450\pi\)
−0.588928 + 0.808186i \(0.700450\pi\)
\(500\) 6.65167e6 1.18989
\(501\) 0 0
\(502\) 1.57099e7 2.78237
\(503\) 1.58524e6 0.279367 0.139684 0.990196i \(-0.455391\pi\)
0.139684 + 0.990196i \(0.455391\pi\)
\(504\) 0 0
\(505\) −1.87242e6 −0.326719
\(506\) −3.65528e6 −0.634664
\(507\) 0 0
\(508\) 4.29993e6 0.739268
\(509\) −6.77981e6 −1.15991 −0.579953 0.814650i \(-0.696929\pi\)
−0.579953 + 0.814650i \(0.696929\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −432315. −0.0728829
\(513\) 0 0
\(514\) −9.03486e6 −1.50839
\(515\) −448766. −0.0745593
\(516\) 0 0
\(517\) 856990. 0.141010
\(518\) 0 0
\(519\) 0 0
\(520\) −3.28157e6 −0.532198
\(521\) 1.01384e7 1.63634 0.818170 0.574977i \(-0.194989\pi\)
0.818170 + 0.574977i \(0.194989\pi\)
\(522\) 0 0
\(523\) 99997.3 0.0159858 0.00799289 0.999968i \(-0.497456\pi\)
0.00799289 + 0.999968i \(0.497456\pi\)
\(524\) 9.19408e6 1.46278
\(525\) 0 0
\(526\) −5.02301e6 −0.791589
\(527\) 1.73483e6 0.272102
\(528\) 0 0
\(529\) −5.52355e6 −0.858181
\(530\) 3.65221e6 0.564763
\(531\) 0 0
\(532\) 0 0
\(533\) 4.30952e6 0.657068
\(534\) 0 0
\(535\) 153667. 0.0232111
\(536\) 2.95457e6 0.444204
\(537\) 0 0
\(538\) 5.88620e6 0.876756
\(539\) 0 0
\(540\) 0 0
\(541\) −119743. −0.0175896 −0.00879481 0.999961i \(-0.502800\pi\)
−0.00879481 + 0.999961i \(0.502800\pi\)
\(542\) −5.03143e6 −0.735687
\(543\) 0 0
\(544\) −7.73054e6 −1.11999
\(545\) −2.48640e6 −0.358574
\(546\) 0 0
\(547\) 236568. 0.0338056 0.0169028 0.999857i \(-0.494619\pi\)
0.0169028 + 0.999857i \(0.494619\pi\)
\(548\) −1.63842e6 −0.233063
\(549\) 0 0
\(550\) 1.00921e7 1.42258
\(551\) 1.62728e7 2.28341
\(552\) 0 0
\(553\) 0 0
\(554\) 1.53707e7 2.12774
\(555\) 0 0
\(556\) 7.96107e6 1.09216
\(557\) 4.83666e6 0.660553 0.330277 0.943884i \(-0.392858\pi\)
0.330277 + 0.943884i \(0.392858\pi\)
\(558\) 0 0
\(559\) 1.57083e7 2.12617
\(560\) 0 0
\(561\) 0 0
\(562\) −1.66252e7 −2.22037
\(563\) 5.48295e6 0.729026 0.364513 0.931198i \(-0.381235\pi\)
0.364513 + 0.931198i \(0.381235\pi\)
\(564\) 0 0
\(565\) 2.47213e6 0.325800
\(566\) 2.30738e7 3.02746
\(567\) 0 0
\(568\) −1.08322e7 −1.40878
\(569\) 6.97660e6 0.903364 0.451682 0.892179i \(-0.350824\pi\)
0.451682 + 0.892179i \(0.350824\pi\)
\(570\) 0 0
\(571\) −1.17715e7 −1.51092 −0.755458 0.655197i \(-0.772585\pi\)
−0.755458 + 0.655197i \(0.772585\pi\)
\(572\) −1.73881e7 −2.22209
\(573\) 0 0
\(574\) 0 0
\(575\) −2.52020e6 −0.317882
\(576\) 0 0
\(577\) 6.77292e6 0.846909 0.423454 0.905917i \(-0.360817\pi\)
0.423454 + 0.905917i \(0.360817\pi\)
\(578\) 4.33614e6 0.539863
\(579\) 0 0
\(580\) −8.12037e6 −1.00232
\(581\) 0 0
\(582\) 0 0
\(583\) 7.51025e6 0.915130
\(584\) 7.28760e6 0.884203
\(585\) 0 0
\(586\) −984484. −0.118431
\(587\) 1.05020e7 1.25799 0.628996 0.777408i \(-0.283466\pi\)
0.628996 + 0.777408i \(0.283466\pi\)
\(588\) 0 0
\(589\) 2.91717e6 0.346476
\(590\) 1.50680e6 0.178207
\(591\) 0 0
\(592\) 364901. 0.0427928
\(593\) 7.59074e6 0.886436 0.443218 0.896414i \(-0.353837\pi\)
0.443218 + 0.896414i \(0.353837\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.89758e7 −2.18818
\(597\) 0 0
\(598\) 6.99915e6 0.800373
\(599\) 1.30198e7 1.48265 0.741325 0.671146i \(-0.234198\pi\)
0.741325 + 0.671146i \(0.234198\pi\)
\(600\) 0 0
\(601\) 1.41821e7 1.60160 0.800801 0.598931i \(-0.204408\pi\)
0.800801 + 0.598931i \(0.204408\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.07270e7 1.19643
\(605\) −278010. −0.0308796
\(606\) 0 0
\(607\) 1.20772e7 1.33044 0.665221 0.746646i \(-0.268337\pi\)
0.665221 + 0.746646i \(0.268337\pi\)
\(608\) −1.29991e7 −1.42612
\(609\) 0 0
\(610\) −708320. −0.0770735
\(611\) −1.64097e6 −0.177827
\(612\) 0 0
\(613\) −2.72530e6 −0.292929 −0.146465 0.989216i \(-0.546789\pi\)
−0.146465 + 0.989216i \(0.546789\pi\)
\(614\) 1.37371e7 1.47053
\(615\) 0 0
\(616\) 0 0
\(617\) −8.47094e6 −0.895816 −0.447908 0.894080i \(-0.647831\pi\)
−0.447908 + 0.894080i \(0.647831\pi\)
\(618\) 0 0
\(619\) 1.73835e7 1.82352 0.911759 0.410726i \(-0.134725\pi\)
0.911759 + 0.410726i \(0.134725\pi\)
\(620\) −1.45571e6 −0.152088
\(621\) 0 0
\(622\) −7.90690e6 −0.819464
\(623\) 0 0
\(624\) 0 0
\(625\) 5.43586e6 0.556632
\(626\) −4.62676e6 −0.471891
\(627\) 0 0
\(628\) −4.04093e6 −0.408868
\(629\) 1.34480e7 1.35529
\(630\) 0 0
\(631\) −6.45149e6 −0.645040 −0.322520 0.946563i \(-0.604530\pi\)
−0.322520 + 0.946563i \(0.604530\pi\)
\(632\) 1.81874e6 0.181124
\(633\) 0 0
\(634\) 4.40707e6 0.435438
\(635\) −1.81483e6 −0.178608
\(636\) 0 0
\(637\) 0 0
\(638\) −2.69164e7 −2.61797
\(639\) 0 0
\(640\) 6.72879e6 0.649362
\(641\) −1.77716e7 −1.70837 −0.854185 0.519969i \(-0.825943\pi\)
−0.854185 + 0.519969i \(0.825943\pi\)
\(642\) 0 0
\(643\) −9.34806e6 −0.891649 −0.445825 0.895120i \(-0.647090\pi\)
−0.445825 + 0.895120i \(0.647090\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.92118e7 2.75408
\(647\) −5.34386e6 −0.501874 −0.250937 0.968003i \(-0.580739\pi\)
−0.250937 + 0.968003i \(0.580739\pi\)
\(648\) 0 0
\(649\) 3.09852e6 0.288764
\(650\) −1.93245e7 −1.79401
\(651\) 0 0
\(652\) 9.67098e6 0.890946
\(653\) 1.19701e6 0.109854 0.0549270 0.998490i \(-0.482507\pi\)
0.0549270 + 0.998490i \(0.482507\pi\)
\(654\) 0 0
\(655\) −3.88045e6 −0.353410
\(656\) 201588. 0.0182896
\(657\) 0 0
\(658\) 0 0
\(659\) 1.17541e7 1.05433 0.527163 0.849764i \(-0.323256\pi\)
0.527163 + 0.849764i \(0.323256\pi\)
\(660\) 0 0
\(661\) −1.80115e7 −1.60341 −0.801706 0.597719i \(-0.796074\pi\)
−0.801706 + 0.597719i \(0.796074\pi\)
\(662\) −2.01915e7 −1.79071
\(663\) 0 0
\(664\) 1.31170e7 1.15456
\(665\) 0 0
\(666\) 0 0
\(667\) 6.72153e6 0.584997
\(668\) −6.75975e6 −0.586124
\(669\) 0 0
\(670\) −3.21322e6 −0.276537
\(671\) −1.45656e6 −0.124888
\(672\) 0 0
\(673\) 1.40977e7 1.19981 0.599904 0.800072i \(-0.295206\pi\)
0.599904 + 0.800072i \(0.295206\pi\)
\(674\) −1.24613e7 −1.05660
\(675\) 0 0
\(676\) 1.38780e7 1.16805
\(677\) −7.65587e6 −0.641982 −0.320991 0.947082i \(-0.604016\pi\)
−0.320991 + 0.947082i \(0.604016\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.65718e6 −0.469167
\(681\) 0 0
\(682\) −4.82520e6 −0.397241
\(683\) 1.08676e7 0.891415 0.445708 0.895179i \(-0.352952\pi\)
0.445708 + 0.895179i \(0.352952\pi\)
\(684\) 0 0
\(685\) 691510. 0.0563083
\(686\) 0 0
\(687\) 0 0
\(688\) 734791. 0.0591825
\(689\) −1.43807e7 −1.15407
\(690\) 0 0
\(691\) −1.46118e7 −1.16415 −0.582073 0.813136i \(-0.697758\pi\)
−0.582073 + 0.813136i \(0.697758\pi\)
\(692\) 2.65589e7 2.10836
\(693\) 0 0
\(694\) −3.48832e7 −2.74927
\(695\) −3.36005e6 −0.263866
\(696\) 0 0
\(697\) 7.42928e6 0.579248
\(698\) −1.20965e7 −0.939772
\(699\) 0 0
\(700\) 0 0
\(701\) −1.90104e7 −1.46115 −0.730577 0.682830i \(-0.760749\pi\)
−0.730577 + 0.682830i \(0.760749\pi\)
\(702\) 0 0
\(703\) 2.26132e7 1.72573
\(704\) 2.19992e7 1.67292
\(705\) 0 0
\(706\) 3.14462e7 2.37441
\(707\) 0 0
\(708\) 0 0
\(709\) −1.15929e7 −0.866116 −0.433058 0.901366i \(-0.642565\pi\)
−0.433058 + 0.901366i \(0.642565\pi\)
\(710\) 1.17804e7 0.877031
\(711\) 0 0
\(712\) −2.68890e7 −1.98781
\(713\) 1.20494e6 0.0887653
\(714\) 0 0
\(715\) 7.33881e6 0.536859
\(716\) 6.93374e6 0.505458
\(717\) 0 0
\(718\) −1.35041e7 −0.977585
\(719\) 1.02861e7 0.742040 0.371020 0.928625i \(-0.379008\pi\)
0.371020 + 0.928625i \(0.379008\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.63868e7 1.88384
\(723\) 0 0
\(724\) −3.87014e7 −2.74398
\(725\) −1.85580e7 −1.31125
\(726\) 0 0
\(727\) 1.00970e7 0.708526 0.354263 0.935146i \(-0.384732\pi\)
0.354263 + 0.935146i \(0.384732\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7.92556e6 −0.550456
\(731\) 2.70799e7 1.87436
\(732\) 0 0
\(733\) 1.92402e6 0.132266 0.0661332 0.997811i \(-0.478934\pi\)
0.0661332 + 0.997811i \(0.478934\pi\)
\(734\) −4.57298e7 −3.13299
\(735\) 0 0
\(736\) −5.36932e6 −0.365363
\(737\) −6.60752e6 −0.448095
\(738\) 0 0
\(739\) 4.20273e6 0.283087 0.141543 0.989932i \(-0.454793\pi\)
0.141543 + 0.989932i \(0.454793\pi\)
\(740\) −1.12843e7 −0.757522
\(741\) 0 0
\(742\) 0 0
\(743\) 1.99659e7 1.32684 0.663418 0.748249i \(-0.269105\pi\)
0.663418 + 0.748249i \(0.269105\pi\)
\(744\) 0 0
\(745\) 8.00891e6 0.528667
\(746\) 3.63620e7 2.39222
\(747\) 0 0
\(748\) −2.99757e7 −1.95892
\(749\) 0 0
\(750\) 0 0
\(751\) −3.63975e6 −0.235490 −0.117745 0.993044i \(-0.537567\pi\)
−0.117745 + 0.993044i \(0.537567\pi\)
\(752\) −76760.3 −0.00494985
\(753\) 0 0
\(754\) 5.15396e7 3.30151
\(755\) −4.52745e6 −0.289059
\(756\) 0 0
\(757\) 1.73429e7 1.09997 0.549986 0.835174i \(-0.314633\pi\)
0.549986 + 0.835174i \(0.314633\pi\)
\(758\) −1.60813e7 −1.01660
\(759\) 0 0
\(760\) −9.51270e6 −0.597406
\(761\) 1.03568e7 0.648284 0.324142 0.946009i \(-0.394924\pi\)
0.324142 + 0.946009i \(0.394924\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3.04740e7 1.88884
\(765\) 0 0
\(766\) −2.88138e7 −1.77431
\(767\) −5.93306e6 −0.364159
\(768\) 0 0
\(769\) −1.81548e7 −1.10707 −0.553534 0.832826i \(-0.686721\pi\)
−0.553534 + 0.832826i \(0.686721\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.09653e7 −1.26607
\(773\) 1.63566e7 0.984565 0.492282 0.870435i \(-0.336163\pi\)
0.492282 + 0.870435i \(0.336163\pi\)
\(774\) 0 0
\(775\) −3.32683e6 −0.198965
\(776\) −1.47815e7 −0.881181
\(777\) 0 0
\(778\) 9.66346e6 0.572379
\(779\) 1.24925e7 0.737576
\(780\) 0 0
\(781\) 2.42248e7 1.42112
\(782\) 1.20660e7 0.705581
\(783\) 0 0
\(784\) 0 0
\(785\) 1.70552e6 0.0987829
\(786\) 0 0
\(787\) −4.44728e6 −0.255952 −0.127976 0.991777i \(-0.540848\pi\)
−0.127976 + 0.991777i \(0.540848\pi\)
\(788\) −3.51207e7 −2.01487
\(789\) 0 0
\(790\) −1.97795e6 −0.112758
\(791\) 0 0
\(792\) 0 0
\(793\) 2.78903e6 0.157496
\(794\) 4.17493e6 0.235016
\(795\) 0 0
\(796\) −2.37953e7 −1.33110
\(797\) 9.15303e6 0.510410 0.255205 0.966887i \(-0.417857\pi\)
0.255205 + 0.966887i \(0.417857\pi\)
\(798\) 0 0
\(799\) −2.82891e6 −0.156766
\(800\) 1.48246e7 0.818950
\(801\) 0 0
\(802\) 2.64815e7 1.45381
\(803\) −1.62978e7 −0.891948
\(804\) 0 0
\(805\) 0 0
\(806\) 9.23932e6 0.500959
\(807\) 0 0
\(808\) 1.58073e7 0.851784
\(809\) −2.51923e7 −1.35331 −0.676654 0.736301i \(-0.736571\pi\)
−0.676654 + 0.736301i \(0.736571\pi\)
\(810\) 0 0
\(811\) −8.90585e6 −0.475470 −0.237735 0.971330i \(-0.576405\pi\)
−0.237735 + 0.971330i \(0.576405\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.74038e7 −1.97858
\(815\) −4.08173e6 −0.215254
\(816\) 0 0
\(817\) 4.55355e7 2.38669
\(818\) −2.07417e6 −0.108383
\(819\) 0 0
\(820\) −6.23396e6 −0.323765
\(821\) −3.35490e7 −1.73709 −0.868543 0.495613i \(-0.834943\pi\)
−0.868543 + 0.495613i \(0.834943\pi\)
\(822\) 0 0
\(823\) −1.46001e7 −0.751376 −0.375688 0.926746i \(-0.622593\pi\)
−0.375688 + 0.926746i \(0.622593\pi\)
\(824\) 3.78857e6 0.194382
\(825\) 0 0
\(826\) 0 0
\(827\) −1.97530e7 −1.00431 −0.502156 0.864777i \(-0.667460\pi\)
−0.502156 + 0.864777i \(0.667460\pi\)
\(828\) 0 0
\(829\) −4.06255e6 −0.205311 −0.102655 0.994717i \(-0.532734\pi\)
−0.102655 + 0.994717i \(0.532734\pi\)
\(830\) −1.42653e7 −0.718762
\(831\) 0 0
\(832\) −4.21241e7 −2.10971
\(833\) 0 0
\(834\) 0 0
\(835\) 2.85302e6 0.141608
\(836\) −5.04050e7 −2.49435
\(837\) 0 0
\(838\) −3.95735e7 −1.94668
\(839\) 1.60509e7 0.787217 0.393609 0.919278i \(-0.371227\pi\)
0.393609 + 0.919278i \(0.371227\pi\)
\(840\) 0 0
\(841\) 2.89842e7 1.41309
\(842\) 1.14846e7 0.558260
\(843\) 0 0
\(844\) −6.25159e7 −3.02088
\(845\) −5.85737e6 −0.282202
\(846\) 0 0
\(847\) 0 0
\(848\) −672690. −0.0321237
\(849\) 0 0
\(850\) −3.33140e7 −1.58154
\(851\) 9.34043e6 0.442123
\(852\) 0 0
\(853\) −1.23887e7 −0.582979 −0.291489 0.956574i \(-0.594151\pi\)
−0.291489 + 0.956574i \(0.594151\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.29728e6 −0.0605133
\(857\) −2.63519e7 −1.22563 −0.612816 0.790226i \(-0.709963\pi\)
−0.612816 + 0.790226i \(0.709963\pi\)
\(858\) 0 0
\(859\) −1.21249e7 −0.560654 −0.280327 0.959905i \(-0.590443\pi\)
−0.280327 + 0.959905i \(0.590443\pi\)
\(860\) −2.27229e7 −1.04765
\(861\) 0 0
\(862\) 4.04702e7 1.85510
\(863\) −2.55952e6 −0.116985 −0.0584927 0.998288i \(-0.518629\pi\)
−0.0584927 + 0.998288i \(0.518629\pi\)
\(864\) 0 0
\(865\) −1.12094e7 −0.509382
\(866\) −1.47773e7 −0.669578
\(867\) 0 0
\(868\) 0 0
\(869\) −4.06737e6 −0.182711
\(870\) 0 0
\(871\) 1.26521e7 0.565091
\(872\) 2.09906e7 0.934834
\(873\) 0 0
\(874\) 2.02893e7 0.898439
\(875\) 0 0
\(876\) 0 0
\(877\) 2.65820e7 1.16705 0.583523 0.812096i \(-0.301674\pi\)
0.583523 + 0.812096i \(0.301674\pi\)
\(878\) −4.06225e7 −1.77841
\(879\) 0 0
\(880\) 343290. 0.0149436
\(881\) 8.32262e6 0.361260 0.180630 0.983551i \(-0.442186\pi\)
0.180630 + 0.983551i \(0.442186\pi\)
\(882\) 0 0
\(883\) 1.81133e7 0.781798 0.390899 0.920434i \(-0.372164\pi\)
0.390899 + 0.920434i \(0.372164\pi\)
\(884\) 5.73977e7 2.47038
\(885\) 0 0
\(886\) 3.31845e7 1.42020
\(887\) 2.04255e7 0.871694 0.435847 0.900021i \(-0.356449\pi\)
0.435847 + 0.900021i \(0.356449\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.92429e7 1.23750
\(891\) 0 0
\(892\) 1.55070e7 0.652552
\(893\) −4.75689e6 −0.199616
\(894\) 0 0
\(895\) −2.92645e6 −0.122119
\(896\) 0 0
\(897\) 0 0
\(898\) 4.28785e6 0.177439
\(899\) 8.87285e6 0.366154
\(900\) 0 0
\(901\) −2.47912e7 −1.01739
\(902\) −2.06635e7 −0.845645
\(903\) 0 0
\(904\) −2.08702e7 −0.849388
\(905\) 1.63343e7 0.662948
\(906\) 0 0
\(907\) −2.01197e7 −0.812089 −0.406044 0.913853i \(-0.633092\pi\)
−0.406044 + 0.913853i \(0.633092\pi\)
\(908\) −1.14079e7 −0.459190
\(909\) 0 0
\(910\) 0 0
\(911\) −3.17075e7 −1.26580 −0.632902 0.774232i \(-0.718137\pi\)
−0.632902 + 0.774232i \(0.718137\pi\)
\(912\) 0 0
\(913\) −2.93345e7 −1.16467
\(914\) −5.52023e6 −0.218571
\(915\) 0 0
\(916\) −6.42812e7 −2.53131
\(917\) 0 0
\(918\) 0 0
\(919\) −8.70727e6 −0.340090 −0.170045 0.985436i \(-0.554391\pi\)
−0.170045 + 0.985436i \(0.554391\pi\)
\(920\) −3.92924e6 −0.153052
\(921\) 0 0
\(922\) −2.63678e7 −1.02152
\(923\) −4.63857e7 −1.79217
\(924\) 0 0
\(925\) −2.57887e7 −0.991005
\(926\) −2.67634e7 −1.02569
\(927\) 0 0
\(928\) −3.95381e7 −1.50711
\(929\) −3.24042e7 −1.23186 −0.615931 0.787800i \(-0.711220\pi\)
−0.615931 + 0.787800i \(0.711220\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.29167e6 0.124130
\(933\) 0 0
\(934\) 6.60475e7 2.47736
\(935\) 1.26516e7 0.473277
\(936\) 0 0
\(937\) −2.19155e7 −0.815458 −0.407729 0.913103i \(-0.633679\pi\)
−0.407729 + 0.913103i \(0.633679\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.37376e6 0.0876227
\(941\) −1.46019e7 −0.537571 −0.268785 0.963200i \(-0.586622\pi\)
−0.268785 + 0.963200i \(0.586622\pi\)
\(942\) 0 0
\(943\) 5.16008e6 0.188963
\(944\) −277533. −0.0101364
\(945\) 0 0
\(946\) −7.53189e7 −2.73638
\(947\) 2.08378e7 0.755053 0.377527 0.925999i \(-0.376775\pi\)
0.377527 + 0.925999i \(0.376775\pi\)
\(948\) 0 0
\(949\) 3.12071e7 1.12483
\(950\) −5.60184e7 −2.01382
\(951\) 0 0
\(952\) 0 0
\(953\) 942012. 0.0335988 0.0167994 0.999859i \(-0.494652\pi\)
0.0167994 + 0.999859i \(0.494652\pi\)
\(954\) 0 0
\(955\) −1.28618e7 −0.456346
\(956\) −1.14698e7 −0.405894
\(957\) 0 0
\(958\) 2.56752e7 0.903857
\(959\) 0 0
\(960\) 0 0
\(961\) −2.70385e7 −0.944441
\(962\) 7.16209e7 2.49518
\(963\) 0 0
\(964\) 2.26889e7 0.786359
\(965\) 8.84860e6 0.305884
\(966\) 0 0
\(967\) −2.56570e7 −0.882346 −0.441173 0.897422i \(-0.645438\pi\)
−0.441173 + 0.897422i \(0.645438\pi\)
\(968\) 2.34701e6 0.0805059
\(969\) 0 0
\(970\) 1.60755e7 0.548575
\(971\) 2.14274e7 0.729324 0.364662 0.931140i \(-0.381184\pi\)
0.364662 + 0.931140i \(0.381184\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.69237e7 −0.909364
\(975\) 0 0
\(976\) 130464. 0.00438394
\(977\) 2.04841e7 0.686562 0.343281 0.939233i \(-0.388462\pi\)
0.343281 + 0.939233i \(0.388462\pi\)
\(978\) 0 0
\(979\) 6.01338e7 2.00522
\(980\) 0 0
\(981\) 0 0
\(982\) 2.81057e7 0.930069
\(983\) −2.77650e7 −0.916459 −0.458230 0.888834i \(-0.651516\pi\)
−0.458230 + 0.888834i \(0.651516\pi\)
\(984\) 0 0
\(985\) 1.48230e7 0.486796
\(986\) 8.88504e7 2.91050
\(987\) 0 0
\(988\) 9.65159e7 3.14562
\(989\) 1.88086e7 0.611456
\(990\) 0 0
\(991\) 3.46832e7 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(992\) −7.08785e6 −0.228684
\(993\) 0 0
\(994\) 0 0
\(995\) 1.00431e7 0.321594
\(996\) 0 0
\(997\) −1.48572e7 −0.473369 −0.236685 0.971587i \(-0.576061\pi\)
−0.236685 + 0.971587i \(0.576061\pi\)
\(998\) −6.01512e7 −1.91169
\(999\) 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.w.1.4 4
3.2 odd 2 147.6.a.m.1.1 4
7.2 even 3 63.6.e.e.46.1 8
7.4 even 3 63.6.e.e.37.1 8
7.6 odd 2 441.6.a.v.1.4 4
21.2 odd 6 21.6.e.c.4.4 8
21.5 even 6 147.6.e.o.67.4 8
21.11 odd 6 21.6.e.c.16.4 yes 8
21.17 even 6 147.6.e.o.79.4 8
21.20 even 2 147.6.a.l.1.1 4
84.11 even 6 336.6.q.j.289.3 8
84.23 even 6 336.6.q.j.193.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.4 8 21.2 odd 6
21.6.e.c.16.4 yes 8 21.11 odd 6
63.6.e.e.37.1 8 7.4 even 3
63.6.e.e.46.1 8 7.2 even 3
147.6.a.l.1.1 4 21.20 even 2
147.6.a.m.1.1 4 3.2 odd 2
147.6.e.o.67.4 8 21.5 even 6
147.6.e.o.79.4 8 21.17 even 6
336.6.q.j.193.3 8 84.23 even 6
336.6.q.j.289.3 8 84.11 even 6
441.6.a.v.1.4 4 7.6 odd 2
441.6.a.w.1.4 4 1.1 even 1 trivial