Properties

Label 441.6.a.w.1.3
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(1.79080\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.790805 q^{2} -31.3746 q^{4} +104.192 q^{5} -50.1170 q^{8} +O(q^{10})\) \(q+0.790805 q^{2} -31.3746 q^{4} +104.192 q^{5} -50.1170 q^{8} +82.3953 q^{10} +497.660 q^{11} -206.551 q^{13} +964.355 q^{16} -63.1586 q^{17} +1323.95 q^{19} -3268.98 q^{20} +393.552 q^{22} +194.437 q^{23} +7730.91 q^{25} -163.342 q^{26} -4323.14 q^{29} -7525.31 q^{31} +2366.36 q^{32} -49.9461 q^{34} +10355.6 q^{37} +1046.99 q^{38} -5221.77 q^{40} +4180.92 q^{41} +5960.87 q^{43} -15613.9 q^{44} +153.762 q^{46} +4389.74 q^{47} +6113.64 q^{50} +6480.47 q^{52} -17784.8 q^{53} +51852.0 q^{55} -3418.76 q^{58} -3500.47 q^{59} -10632.5 q^{61} -5951.06 q^{62} -28988.0 q^{64} -21520.9 q^{65} -13274.7 q^{67} +1981.58 q^{68} -38811.1 q^{71} +31375.6 q^{73} +8189.30 q^{74} -41538.6 q^{76} +39491.5 q^{79} +100478. q^{80} +3306.29 q^{82} +102372. q^{83} -6580.60 q^{85} +4713.88 q^{86} -24941.2 q^{88} +112821. q^{89} -6100.40 q^{92} +3471.43 q^{94} +137945. q^{95} +30334.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + O(q^{10}) \) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.790805 0.139796 0.0698979 0.997554i \(-0.477733\pi\)
0.0698979 + 0.997554i \(0.477733\pi\)
\(3\) 0 0
\(4\) −31.3746 −0.980457
\(5\) 104.192 1.86384 0.931919 0.362667i \(-0.118134\pi\)
0.931919 + 0.362667i \(0.118134\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −50.1170 −0.276860
\(9\) 0 0
\(10\) 82.3953 0.260557
\(11\) 497.660 1.24008 0.620041 0.784569i \(-0.287116\pi\)
0.620041 + 0.784569i \(0.287116\pi\)
\(12\) 0 0
\(13\) −206.551 −0.338977 −0.169488 0.985532i \(-0.554211\pi\)
−0.169488 + 0.985532i \(0.554211\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 964.355 0.941753
\(17\) −63.1586 −0.0530042 −0.0265021 0.999649i \(-0.508437\pi\)
−0.0265021 + 0.999649i \(0.508437\pi\)
\(18\) 0 0
\(19\) 1323.95 0.841374 0.420687 0.907206i \(-0.361789\pi\)
0.420687 + 0.907206i \(0.361789\pi\)
\(20\) −3268.98 −1.82741
\(21\) 0 0
\(22\) 393.552 0.173358
\(23\) 194.437 0.0766408 0.0383204 0.999266i \(-0.487799\pi\)
0.0383204 + 0.999266i \(0.487799\pi\)
\(24\) 0 0
\(25\) 7730.91 2.47389
\(26\) −163.342 −0.0473875
\(27\) 0 0
\(28\) 0 0
\(29\) −4323.14 −0.954562 −0.477281 0.878751i \(-0.658378\pi\)
−0.477281 + 0.878751i \(0.658378\pi\)
\(30\) 0 0
\(31\) −7525.31 −1.40644 −0.703219 0.710974i \(-0.748255\pi\)
−0.703219 + 0.710974i \(0.748255\pi\)
\(32\) 2366.36 0.408513
\(33\) 0 0
\(34\) −49.9461 −0.00740977
\(35\) 0 0
\(36\) 0 0
\(37\) 10355.6 1.24358 0.621789 0.783185i \(-0.286406\pi\)
0.621789 + 0.783185i \(0.286406\pi\)
\(38\) 1046.99 0.117621
\(39\) 0 0
\(40\) −5221.77 −0.516022
\(41\) 4180.92 0.388429 0.194215 0.980959i \(-0.437784\pi\)
0.194215 + 0.980959i \(0.437784\pi\)
\(42\) 0 0
\(43\) 5960.87 0.491630 0.245815 0.969317i \(-0.420944\pi\)
0.245815 + 0.969317i \(0.420944\pi\)
\(44\) −15613.9 −1.21585
\(45\) 0 0
\(46\) 153.762 0.0107141
\(47\) 4389.74 0.289864 0.144932 0.989442i \(-0.453704\pi\)
0.144932 + 0.989442i \(0.453704\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 6113.64 0.345840
\(51\) 0 0
\(52\) 6480.47 0.332352
\(53\) −17784.8 −0.869679 −0.434839 0.900508i \(-0.643195\pi\)
−0.434839 + 0.900508i \(0.643195\pi\)
\(54\) 0 0
\(55\) 51852.0 2.31131
\(56\) 0 0
\(57\) 0 0
\(58\) −3418.76 −0.133444
\(59\) −3500.47 −0.130917 −0.0654585 0.997855i \(-0.520851\pi\)
−0.0654585 + 0.997855i \(0.520851\pi\)
\(60\) 0 0
\(61\) −10632.5 −0.365855 −0.182928 0.983126i \(-0.558557\pi\)
−0.182928 + 0.983126i \(0.558557\pi\)
\(62\) −5951.06 −0.196614
\(63\) 0 0
\(64\) −28988.0 −0.884645
\(65\) −21520.9 −0.631797
\(66\) 0 0
\(67\) −13274.7 −0.361276 −0.180638 0.983550i \(-0.557816\pi\)
−0.180638 + 0.983550i \(0.557816\pi\)
\(68\) 1981.58 0.0519683
\(69\) 0 0
\(70\) 0 0
\(71\) −38811.1 −0.913713 −0.456857 0.889540i \(-0.651025\pi\)
−0.456857 + 0.889540i \(0.651025\pi\)
\(72\) 0 0
\(73\) 31375.6 0.689105 0.344552 0.938767i \(-0.388031\pi\)
0.344552 + 0.938767i \(0.388031\pi\)
\(74\) 8189.30 0.173847
\(75\) 0 0
\(76\) −41538.6 −0.824931
\(77\) 0 0
\(78\) 0 0
\(79\) 39491.5 0.711927 0.355964 0.934500i \(-0.384153\pi\)
0.355964 + 0.934500i \(0.384153\pi\)
\(80\) 100478. 1.75528
\(81\) 0 0
\(82\) 3306.29 0.0543008
\(83\) 102372. 1.63112 0.815559 0.578675i \(-0.196430\pi\)
0.815559 + 0.578675i \(0.196430\pi\)
\(84\) 0 0
\(85\) −6580.60 −0.0987912
\(86\) 4713.88 0.0687279
\(87\) 0 0
\(88\) −24941.2 −0.343329
\(89\) 112821. 1.50978 0.754892 0.655849i \(-0.227689\pi\)
0.754892 + 0.655849i \(0.227689\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6100.40 −0.0751430
\(93\) 0 0
\(94\) 3471.43 0.0405218
\(95\) 137945. 1.56818
\(96\) 0 0
\(97\) 30334.3 0.327345 0.163672 0.986515i \(-0.447666\pi\)
0.163672 + 0.986515i \(0.447666\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −242554. −2.42554
\(101\) −106796. −1.04172 −0.520862 0.853641i \(-0.674390\pi\)
−0.520862 + 0.853641i \(0.674390\pi\)
\(102\) 0 0
\(103\) 157536. 1.46314 0.731570 0.681767i \(-0.238788\pi\)
0.731570 + 0.681767i \(0.238788\pi\)
\(104\) 10351.7 0.0938490
\(105\) 0 0
\(106\) −14064.3 −0.121578
\(107\) 89324.3 0.754241 0.377121 0.926164i \(-0.376914\pi\)
0.377121 + 0.926164i \(0.376914\pi\)
\(108\) 0 0
\(109\) 167605. 1.35120 0.675600 0.737268i \(-0.263885\pi\)
0.675600 + 0.737268i \(0.263885\pi\)
\(110\) 41004.8 0.323112
\(111\) 0 0
\(112\) 0 0
\(113\) 115794. 0.853079 0.426539 0.904469i \(-0.359733\pi\)
0.426539 + 0.904469i \(0.359733\pi\)
\(114\) 0 0
\(115\) 20258.8 0.142846
\(116\) 135637. 0.935907
\(117\) 0 0
\(118\) −2768.19 −0.0183017
\(119\) 0 0
\(120\) 0 0
\(121\) 86614.1 0.537805
\(122\) −8408.20 −0.0511451
\(123\) 0 0
\(124\) 236104. 1.37895
\(125\) 479898. 2.74709
\(126\) 0 0
\(127\) 201513. 1.10865 0.554325 0.832300i \(-0.312977\pi\)
0.554325 + 0.832300i \(0.312977\pi\)
\(128\) −98647.4 −0.532183
\(129\) 0 0
\(130\) −17018.9 −0.0883227
\(131\) −38469.6 −0.195857 −0.0979285 0.995193i \(-0.531222\pi\)
−0.0979285 + 0.995193i \(0.531222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10497.7 −0.0505049
\(135\) 0 0
\(136\) 3165.32 0.0146747
\(137\) 241722. 1.10031 0.550155 0.835063i \(-0.314569\pi\)
0.550155 + 0.835063i \(0.314569\pi\)
\(138\) 0 0
\(139\) −53112.2 −0.233162 −0.116581 0.993181i \(-0.537193\pi\)
−0.116581 + 0.993181i \(0.537193\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −30692.0 −0.127733
\(143\) −102792. −0.420359
\(144\) 0 0
\(145\) −450435. −1.77915
\(146\) 24812.0 0.0963340
\(147\) 0 0
\(148\) −324905. −1.21927
\(149\) −129062. −0.476248 −0.238124 0.971235i \(-0.576532\pi\)
−0.238124 + 0.971235i \(0.576532\pi\)
\(150\) 0 0
\(151\) 153206. 0.546808 0.273404 0.961899i \(-0.411850\pi\)
0.273404 + 0.961899i \(0.411850\pi\)
\(152\) −66352.6 −0.232943
\(153\) 0 0
\(154\) 0 0
\(155\) −784075. −2.62137
\(156\) 0 0
\(157\) −151188. −0.489517 −0.244758 0.969584i \(-0.578709\pi\)
−0.244758 + 0.969584i \(0.578709\pi\)
\(158\) 31230.0 0.0995245
\(159\) 0 0
\(160\) 246555. 0.761402
\(161\) 0 0
\(162\) 0 0
\(163\) −32916.7 −0.0970392 −0.0485196 0.998822i \(-0.515450\pi\)
−0.0485196 + 0.998822i \(0.515450\pi\)
\(164\) −131175. −0.380838
\(165\) 0 0
\(166\) 80956.1 0.228023
\(167\) −217586. −0.603725 −0.301862 0.953352i \(-0.597608\pi\)
−0.301862 + 0.953352i \(0.597608\pi\)
\(168\) 0 0
\(169\) −328630. −0.885095
\(170\) −5203.97 −0.0138106
\(171\) 0 0
\(172\) −187020. −0.482022
\(173\) 421242. 1.07008 0.535041 0.844826i \(-0.320296\pi\)
0.535041 + 0.844826i \(0.320296\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 479921. 1.16785
\(177\) 0 0
\(178\) 89219.4 0.211062
\(179\) −3494.06 −0.00815075 −0.00407538 0.999992i \(-0.501297\pi\)
−0.00407538 + 0.999992i \(0.501297\pi\)
\(180\) 0 0
\(181\) 594611. 1.34908 0.674538 0.738240i \(-0.264343\pi\)
0.674538 + 0.738240i \(0.264343\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −9744.61 −0.0212188
\(185\) 1.07897e6 2.31783
\(186\) 0 0
\(187\) −31431.5 −0.0657296
\(188\) −137726. −0.284199
\(189\) 0 0
\(190\) 109088. 0.219226
\(191\) −828644. −1.64356 −0.821778 0.569807i \(-0.807018\pi\)
−0.821778 + 0.569807i \(0.807018\pi\)
\(192\) 0 0
\(193\) 219739. 0.424633 0.212316 0.977201i \(-0.431899\pi\)
0.212316 + 0.977201i \(0.431899\pi\)
\(194\) 23988.5 0.0457614
\(195\) 0 0
\(196\) 0 0
\(197\) −475612. −0.873146 −0.436573 0.899669i \(-0.643808\pi\)
−0.436573 + 0.899669i \(0.643808\pi\)
\(198\) 0 0
\(199\) 627555. 1.12336 0.561681 0.827354i \(-0.310155\pi\)
0.561681 + 0.827354i \(0.310155\pi\)
\(200\) −387450. −0.684921
\(201\) 0 0
\(202\) −84455.1 −0.145629
\(203\) 0 0
\(204\) 0 0
\(205\) 435617. 0.723969
\(206\) 124580. 0.204541
\(207\) 0 0
\(208\) −199189. −0.319232
\(209\) 658879. 1.04337
\(210\) 0 0
\(211\) 570989. 0.882920 0.441460 0.897281i \(-0.354461\pi\)
0.441460 + 0.897281i \(0.354461\pi\)
\(212\) 557991. 0.852683
\(213\) 0 0
\(214\) 70638.1 0.105440
\(215\) 621073. 0.916319
\(216\) 0 0
\(217\) 0 0
\(218\) 132543. 0.188892
\(219\) 0 0
\(220\) −1.62684e6 −2.26614
\(221\) 13045.5 0.0179672
\(222\) 0 0
\(223\) 4233.11 0.00570029 0.00285015 0.999996i \(-0.499093\pi\)
0.00285015 + 0.999996i \(0.499093\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 91570.3 0.119257
\(227\) −1.12986e6 −1.45533 −0.727664 0.685934i \(-0.759394\pi\)
−0.727664 + 0.685934i \(0.759394\pi\)
\(228\) 0 0
\(229\) 804643. 1.01395 0.506973 0.861962i \(-0.330764\pi\)
0.506973 + 0.861962i \(0.330764\pi\)
\(230\) 16020.7 0.0199693
\(231\) 0 0
\(232\) 216663. 0.264280
\(233\) −1.16927e6 −1.41100 −0.705498 0.708712i \(-0.749277\pi\)
−0.705498 + 0.708712i \(0.749277\pi\)
\(234\) 0 0
\(235\) 457374. 0.540259
\(236\) 109826. 0.128358
\(237\) 0 0
\(238\) 0 0
\(239\) 1.70554e6 1.93138 0.965689 0.259700i \(-0.0836238\pi\)
0.965689 + 0.259700i \(0.0836238\pi\)
\(240\) 0 0
\(241\) 951196. 1.05494 0.527470 0.849574i \(-0.323141\pi\)
0.527470 + 0.849574i \(0.323141\pi\)
\(242\) 68494.9 0.0751830
\(243\) 0 0
\(244\) 333590. 0.358705
\(245\) 0 0
\(246\) 0 0
\(247\) −273465. −0.285206
\(248\) 377146. 0.389386
\(249\) 0 0
\(250\) 379505. 0.384032
\(251\) −1.14498e6 −1.14713 −0.573566 0.819159i \(-0.694440\pi\)
−0.573566 + 0.819159i \(0.694440\pi\)
\(252\) 0 0
\(253\) 96763.6 0.0950409
\(254\) 159358. 0.154985
\(255\) 0 0
\(256\) 849606. 0.810248
\(257\) −1.18374e6 −1.11795 −0.558976 0.829184i \(-0.688806\pi\)
−0.558976 + 0.829184i \(0.688806\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 675211. 0.619450
\(261\) 0 0
\(262\) −30421.9 −0.0273800
\(263\) −448316. −0.399664 −0.199832 0.979830i \(-0.564040\pi\)
−0.199832 + 0.979830i \(0.564040\pi\)
\(264\) 0 0
\(265\) −1.85303e6 −1.62094
\(266\) 0 0
\(267\) 0 0
\(268\) 416490. 0.354216
\(269\) −747474. −0.629819 −0.314909 0.949122i \(-0.601974\pi\)
−0.314909 + 0.949122i \(0.601974\pi\)
\(270\) 0 0
\(271\) −232637. −0.192423 −0.0962114 0.995361i \(-0.530672\pi\)
−0.0962114 + 0.995361i \(0.530672\pi\)
\(272\) −60907.3 −0.0499169
\(273\) 0 0
\(274\) 191155. 0.153819
\(275\) 3.84736e6 3.06783
\(276\) 0 0
\(277\) −2.42924e6 −1.90227 −0.951134 0.308778i \(-0.900080\pi\)
−0.951134 + 0.308778i \(0.900080\pi\)
\(278\) −42001.4 −0.0325951
\(279\) 0 0
\(280\) 0 0
\(281\) −2.51704e6 −1.90163 −0.950813 0.309766i \(-0.899749\pi\)
−0.950813 + 0.309766i \(0.899749\pi\)
\(282\) 0 0
\(283\) 260994. 0.193716 0.0968579 0.995298i \(-0.469121\pi\)
0.0968579 + 0.995298i \(0.469121\pi\)
\(284\) 1.21768e6 0.895857
\(285\) 0 0
\(286\) −81288.6 −0.0587645
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41587e6 −0.997191
\(290\) −356206. −0.248718
\(291\) 0 0
\(292\) −984399. −0.675638
\(293\) −65011.7 −0.0442408 −0.0221204 0.999755i \(-0.507042\pi\)
−0.0221204 + 0.999755i \(0.507042\pi\)
\(294\) 0 0
\(295\) −364720. −0.244008
\(296\) −518994. −0.344297
\(297\) 0 0
\(298\) −102063. −0.0665775
\(299\) −40161.3 −0.0259794
\(300\) 0 0
\(301\) 0 0
\(302\) 121156. 0.0764414
\(303\) 0 0
\(304\) 1.27676e6 0.792367
\(305\) −1.10781e6 −0.681895
\(306\) 0 0
\(307\) −2.35599e6 −1.42668 −0.713342 0.700816i \(-0.752819\pi\)
−0.713342 + 0.700816i \(0.752819\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −620051. −0.366457
\(311\) 2.11805e6 1.24176 0.620878 0.783907i \(-0.286776\pi\)
0.620878 + 0.783907i \(0.286776\pi\)
\(312\) 0 0
\(313\) −187514. −0.108186 −0.0540931 0.998536i \(-0.517227\pi\)
−0.0540931 + 0.998536i \(0.517227\pi\)
\(314\) −119560. −0.0684324
\(315\) 0 0
\(316\) −1.23903e6 −0.698014
\(317\) 1.00541e6 0.561947 0.280974 0.959716i \(-0.409343\pi\)
0.280974 + 0.959716i \(0.409343\pi\)
\(318\) 0 0
\(319\) −2.15145e6 −1.18374
\(320\) −3.02031e6 −1.64883
\(321\) 0 0
\(322\) 0 0
\(323\) −83619.1 −0.0445964
\(324\) 0 0
\(325\) −1.59683e6 −0.838591
\(326\) −26030.7 −0.0135657
\(327\) 0 0
\(328\) −209535. −0.107540
\(329\) 0 0
\(330\) 0 0
\(331\) 1.67984e6 0.842748 0.421374 0.906887i \(-0.361548\pi\)
0.421374 + 0.906887i \(0.361548\pi\)
\(332\) −3.21188e6 −1.59924
\(333\) 0 0
\(334\) −172068. −0.0843982
\(335\) −1.38312e6 −0.673360
\(336\) 0 0
\(337\) −995036. −0.477270 −0.238635 0.971109i \(-0.576700\pi\)
−0.238635 + 0.971109i \(0.576700\pi\)
\(338\) −259882. −0.123733
\(339\) 0 0
\(340\) 206464. 0.0968605
\(341\) −3.74505e6 −1.74410
\(342\) 0 0
\(343\) 0 0
\(344\) −298741. −0.136113
\(345\) 0 0
\(346\) 333121. 0.149593
\(347\) −4.05665e6 −1.80861 −0.904304 0.426890i \(-0.859609\pi\)
−0.904304 + 0.426890i \(0.859609\pi\)
\(348\) 0 0
\(349\) 2.86202e6 1.25779 0.628897 0.777488i \(-0.283507\pi\)
0.628897 + 0.777488i \(0.283507\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.17764e6 0.506590
\(353\) −832685. −0.355667 −0.177834 0.984061i \(-0.556909\pi\)
−0.177834 + 0.984061i \(0.556909\pi\)
\(354\) 0 0
\(355\) −4.04379e6 −1.70301
\(356\) −3.53972e6 −1.48028
\(357\) 0 0
\(358\) −2763.12 −0.00113944
\(359\) 2.69751e6 1.10465 0.552327 0.833627i \(-0.313740\pi\)
0.552327 + 0.833627i \(0.313740\pi\)
\(360\) 0 0
\(361\) −723243. −0.292090
\(362\) 470221. 0.188595
\(363\) 0 0
\(364\) 0 0
\(365\) 3.26908e6 1.28438
\(366\) 0 0
\(367\) 1.45482e6 0.563826 0.281913 0.959440i \(-0.409031\pi\)
0.281913 + 0.959440i \(0.409031\pi\)
\(368\) 187507. 0.0721767
\(369\) 0 0
\(370\) 853257. 0.324023
\(371\) 0 0
\(372\) 0 0
\(373\) −2.04689e6 −0.761766 −0.380883 0.924623i \(-0.624380\pi\)
−0.380883 + 0.924623i \(0.624380\pi\)
\(374\) −24856.2 −0.00918872
\(375\) 0 0
\(376\) −220000. −0.0802516
\(377\) 892950. 0.323574
\(378\) 0 0
\(379\) −416898. −0.149084 −0.0745421 0.997218i \(-0.523750\pi\)
−0.0745421 + 0.997218i \(0.523750\pi\)
\(380\) −4.32798e6 −1.53754
\(381\) 0 0
\(382\) −655296. −0.229762
\(383\) 3.87626e6 1.35025 0.675127 0.737701i \(-0.264089\pi\)
0.675127 + 0.737701i \(0.264089\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 173771. 0.0593619
\(387\) 0 0
\(388\) −951729. −0.320947
\(389\) −2.83802e6 −0.950915 −0.475458 0.879739i \(-0.657717\pi\)
−0.475458 + 0.879739i \(0.657717\pi\)
\(390\) 0 0
\(391\) −12280.4 −0.00406228
\(392\) 0 0
\(393\) 0 0
\(394\) −376116. −0.122062
\(395\) 4.11468e6 1.32692
\(396\) 0 0
\(397\) 4.34266e6 1.38286 0.691432 0.722442i \(-0.256980\pi\)
0.691432 + 0.722442i \(0.256980\pi\)
\(398\) 496274. 0.157041
\(399\) 0 0
\(400\) 7.45534e6 2.32979
\(401\) −3.40304e6 −1.05683 −0.528417 0.848985i \(-0.677214\pi\)
−0.528417 + 0.848985i \(0.677214\pi\)
\(402\) 0 0
\(403\) 1.55436e6 0.476749
\(404\) 3.35070e6 1.02137
\(405\) 0 0
\(406\) 0 0
\(407\) 5.15359e6 1.54214
\(408\) 0 0
\(409\) −5.29431e6 −1.56495 −0.782477 0.622680i \(-0.786044\pi\)
−0.782477 + 0.622680i \(0.786044\pi\)
\(410\) 344488. 0.101208
\(411\) 0 0
\(412\) −4.94262e6 −1.43455
\(413\) 0 0
\(414\) 0 0
\(415\) 1.06663e7 3.04014
\(416\) −488775. −0.138476
\(417\) 0 0
\(418\) 521045. 0.145859
\(419\) −2.87267e6 −0.799376 −0.399688 0.916651i \(-0.630881\pi\)
−0.399688 + 0.916651i \(0.630881\pi\)
\(420\) 0 0
\(421\) 2.08688e6 0.573843 0.286921 0.957954i \(-0.407368\pi\)
0.286921 + 0.957954i \(0.407368\pi\)
\(422\) 451541. 0.123429
\(423\) 0 0
\(424\) 891319. 0.240779
\(425\) −488273. −0.131127
\(426\) 0 0
\(427\) 0 0
\(428\) −2.80252e6 −0.739501
\(429\) 0 0
\(430\) 491148. 0.128098
\(431\) 2.42863e6 0.629751 0.314876 0.949133i \(-0.398037\pi\)
0.314876 + 0.949133i \(0.398037\pi\)
\(432\) 0 0
\(433\) 956219. 0.245097 0.122548 0.992463i \(-0.460893\pi\)
0.122548 + 0.992463i \(0.460893\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.25853e6 −1.32479
\(437\) 257426. 0.0644836
\(438\) 0 0
\(439\) 2.64205e6 0.654304 0.327152 0.944972i \(-0.393911\pi\)
0.327152 + 0.944972i \(0.393911\pi\)
\(440\) −2.59867e6 −0.639910
\(441\) 0 0
\(442\) 10316.4 0.00251174
\(443\) 3.43966e6 0.832733 0.416366 0.909197i \(-0.363303\pi\)
0.416366 + 0.909197i \(0.363303\pi\)
\(444\) 0 0
\(445\) 1.17550e7 2.81399
\(446\) 3347.56 0.000796878 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.39903e6 −1.02977 −0.514886 0.857259i \(-0.672166\pi\)
−0.514886 + 0.857259i \(0.672166\pi\)
\(450\) 0 0
\(451\) 2.08067e6 0.481684
\(452\) −3.63299e6 −0.836407
\(453\) 0 0
\(454\) −893501. −0.203449
\(455\) 0 0
\(456\) 0 0
\(457\) −2.25110e6 −0.504202 −0.252101 0.967701i \(-0.581122\pi\)
−0.252101 + 0.967701i \(0.581122\pi\)
\(458\) 636316. 0.141745
\(459\) 0 0
\(460\) −635611. −0.140054
\(461\) −1.85307e6 −0.406107 −0.203053 0.979168i \(-0.565087\pi\)
−0.203053 + 0.979168i \(0.565087\pi\)
\(462\) 0 0
\(463\) −3.01089e6 −0.652744 −0.326372 0.945241i \(-0.605826\pi\)
−0.326372 + 0.945241i \(0.605826\pi\)
\(464\) −4.16904e6 −0.898962
\(465\) 0 0
\(466\) −924667. −0.197252
\(467\) 1.06725e6 0.226452 0.113226 0.993569i \(-0.463882\pi\)
0.113226 + 0.993569i \(0.463882\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 361694. 0.0755260
\(471\) 0 0
\(472\) 175433. 0.0362456
\(473\) 2.96648e6 0.609662
\(474\) 0 0
\(475\) 1.02354e7 2.08147
\(476\) 0 0
\(477\) 0 0
\(478\) 1.34875e6 0.269999
\(479\) 5.68788e6 1.13269 0.566346 0.824168i \(-0.308357\pi\)
0.566346 + 0.824168i \(0.308357\pi\)
\(480\) 0 0
\(481\) −2.13897e6 −0.421544
\(482\) 752211. 0.147476
\(483\) 0 0
\(484\) −2.71749e6 −0.527295
\(485\) 3.16059e6 0.610117
\(486\) 0 0
\(487\) −6.84477e6 −1.30779 −0.653893 0.756587i \(-0.726865\pi\)
−0.653893 + 0.756587i \(0.726865\pi\)
\(488\) 532867. 0.101291
\(489\) 0 0
\(490\) 0 0
\(491\) −5.60132e6 −1.04854 −0.524272 0.851551i \(-0.675662\pi\)
−0.524272 + 0.851551i \(0.675662\pi\)
\(492\) 0 0
\(493\) 273043. 0.0505958
\(494\) −216257. −0.0398706
\(495\) 0 0
\(496\) −7.25708e6 −1.32452
\(497\) 0 0
\(498\) 0 0
\(499\) −2.33529e6 −0.419845 −0.209923 0.977718i \(-0.567321\pi\)
−0.209923 + 0.977718i \(0.567321\pi\)
\(500\) −1.50566e7 −2.69341
\(501\) 0 0
\(502\) −905455. −0.160364
\(503\) 1.08278e7 1.90819 0.954093 0.299510i \(-0.0968233\pi\)
0.954093 + 0.299510i \(0.0968233\pi\)
\(504\) 0 0
\(505\) −1.11273e7 −1.94161
\(506\) 76521.1 0.0132863
\(507\) 0 0
\(508\) −6.32240e6 −1.08698
\(509\) −3.17847e6 −0.543781 −0.271890 0.962328i \(-0.587649\pi\)
−0.271890 + 0.962328i \(0.587649\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3.82859e6 0.645452
\(513\) 0 0
\(514\) −936107. −0.156285
\(515\) 1.64139e7 2.72705
\(516\) 0 0
\(517\) 2.18460e6 0.359455
\(518\) 0 0
\(519\) 0 0
\(520\) 1.07856e6 0.174919
\(521\) 6.35837e6 1.02625 0.513123 0.858315i \(-0.328489\pi\)
0.513123 + 0.858315i \(0.328489\pi\)
\(522\) 0 0
\(523\) −7.22187e6 −1.15450 −0.577252 0.816566i \(-0.695875\pi\)
−0.577252 + 0.816566i \(0.695875\pi\)
\(524\) 1.20697e6 0.192029
\(525\) 0 0
\(526\) −354531. −0.0558714
\(527\) 475288. 0.0745471
\(528\) 0 0
\(529\) −6.39854e6 −0.994126
\(530\) −1.46538e6 −0.226601
\(531\) 0 0
\(532\) 0 0
\(533\) −863574. −0.131668
\(534\) 0 0
\(535\) 9.30685e6 1.40578
\(536\) 665290. 0.100023
\(537\) 0 0
\(538\) −591106. −0.0880461
\(539\) 0 0
\(540\) 0 0
\(541\) −9.43165e6 −1.38546 −0.692731 0.721196i \(-0.743593\pi\)
−0.692731 + 0.721196i \(0.743593\pi\)
\(542\) −183971. −0.0268999
\(543\) 0 0
\(544\) −149456. −0.0216529
\(545\) 1.74630e7 2.51842
\(546\) 0 0
\(547\) 9.91568e6 1.41695 0.708474 0.705737i \(-0.249384\pi\)
0.708474 + 0.705737i \(0.249384\pi\)
\(548\) −7.58394e6 −1.07881
\(549\) 0 0
\(550\) 3.04251e6 0.428870
\(551\) −5.72364e6 −0.803144
\(552\) 0 0
\(553\) 0 0
\(554\) −1.92106e6 −0.265929
\(555\) 0 0
\(556\) 1.66638e6 0.228605
\(557\) −9.64613e6 −1.31739 −0.658696 0.752409i \(-0.728892\pi\)
−0.658696 + 0.752409i \(0.728892\pi\)
\(558\) 0 0
\(559\) −1.23123e6 −0.166651
\(560\) 0 0
\(561\) 0 0
\(562\) −1.99049e6 −0.265839
\(563\) −3.09504e6 −0.411524 −0.205762 0.978602i \(-0.565967\pi\)
−0.205762 + 0.978602i \(0.565967\pi\)
\(564\) 0 0
\(565\) 1.20648e7 1.59000
\(566\) 206396. 0.0270807
\(567\) 0 0
\(568\) 1.94509e6 0.252970
\(569\) 1.23185e7 1.59507 0.797533 0.603275i \(-0.206138\pi\)
0.797533 + 0.603275i \(0.206138\pi\)
\(570\) 0 0
\(571\) −3.23510e6 −0.415238 −0.207619 0.978210i \(-0.566571\pi\)
−0.207619 + 0.978210i \(0.566571\pi\)
\(572\) 3.22507e6 0.412144
\(573\) 0 0
\(574\) 0 0
\(575\) 1.50318e6 0.189601
\(576\) 0 0
\(577\) −9.05049e6 −1.13170 −0.565852 0.824507i \(-0.691453\pi\)
−0.565852 + 0.824507i \(0.691453\pi\)
\(578\) −1.11968e6 −0.139403
\(579\) 0 0
\(580\) 1.41322e7 1.74438
\(581\) 0 0
\(582\) 0 0
\(583\) −8.85077e6 −1.07847
\(584\) −1.57245e6 −0.190785
\(585\) 0 0
\(586\) −51411.6 −0.00618468
\(587\) 2.13170e6 0.255347 0.127673 0.991816i \(-0.459249\pi\)
0.127673 + 0.991816i \(0.459249\pi\)
\(588\) 0 0
\(589\) −9.96318e6 −1.18334
\(590\) −288422. −0.0341113
\(591\) 0 0
\(592\) 9.98652e6 1.17114
\(593\) 7.26933e6 0.848903 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.04927e6 0.466941
\(597\) 0 0
\(598\) −31759.8 −0.00363182
\(599\) 2.20565e6 0.251171 0.125585 0.992083i \(-0.459919\pi\)
0.125585 + 0.992083i \(0.459919\pi\)
\(600\) 0 0
\(601\) −1.00121e7 −1.13067 −0.565336 0.824860i \(-0.691254\pi\)
−0.565336 + 0.824860i \(0.691254\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.80679e6 −0.536121
\(605\) 9.02447e6 1.00238
\(606\) 0 0
\(607\) 6.32139e6 0.696371 0.348185 0.937426i \(-0.386798\pi\)
0.348185 + 0.937426i \(0.386798\pi\)
\(608\) 3.13295e6 0.343712
\(609\) 0 0
\(610\) −876065. −0.0953261
\(611\) −906706. −0.0982570
\(612\) 0 0
\(613\) 1.43078e7 1.53788 0.768941 0.639319i \(-0.220784\pi\)
0.768941 + 0.639319i \(0.220784\pi\)
\(614\) −1.86313e6 −0.199445
\(615\) 0 0
\(616\) 0 0
\(617\) −1.73991e7 −1.83999 −0.919993 0.391936i \(-0.871806\pi\)
−0.919993 + 0.391936i \(0.871806\pi\)
\(618\) 0 0
\(619\) 8.26736e6 0.867242 0.433621 0.901095i \(-0.357236\pi\)
0.433621 + 0.901095i \(0.357236\pi\)
\(620\) 2.46001e7 2.57014
\(621\) 0 0
\(622\) 1.67497e6 0.173592
\(623\) 0 0
\(624\) 0 0
\(625\) 2.58422e7 2.64625
\(626\) −148287. −0.0151240
\(627\) 0 0
\(628\) 4.74346e6 0.479950
\(629\) −654048. −0.0659148
\(630\) 0 0
\(631\) 2.83238e6 0.283190 0.141595 0.989925i \(-0.454777\pi\)
0.141595 + 0.989925i \(0.454777\pi\)
\(632\) −1.97919e6 −0.197104
\(633\) 0 0
\(634\) 795084. 0.0785579
\(635\) 2.09960e7 2.06634
\(636\) 0 0
\(637\) 0 0
\(638\) −1.70138e6 −0.165481
\(639\) 0 0
\(640\) −1.02782e7 −0.991902
\(641\) 3.57176e6 0.343350 0.171675 0.985154i \(-0.445082\pi\)
0.171675 + 0.985154i \(0.445082\pi\)
\(642\) 0 0
\(643\) −5.96911e6 −0.569354 −0.284677 0.958624i \(-0.591886\pi\)
−0.284677 + 0.958624i \(0.591886\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −66126.4 −0.00623439
\(647\) 1.68276e6 0.158038 0.0790189 0.996873i \(-0.474821\pi\)
0.0790189 + 0.996873i \(0.474821\pi\)
\(648\) 0 0
\(649\) −1.74204e6 −0.162348
\(650\) −1.26278e6 −0.117232
\(651\) 0 0
\(652\) 1.03275e6 0.0951427
\(653\) −1.70305e7 −1.56295 −0.781475 0.623937i \(-0.785532\pi\)
−0.781475 + 0.623937i \(0.785532\pi\)
\(654\) 0 0
\(655\) −4.00821e6 −0.365046
\(656\) 4.03189e6 0.365804
\(657\) 0 0
\(658\) 0 0
\(659\) 1.99303e7 1.78772 0.893862 0.448342i \(-0.147985\pi\)
0.893862 + 0.448342i \(0.147985\pi\)
\(660\) 0 0
\(661\) 9.55735e6 0.850812 0.425406 0.905003i \(-0.360131\pi\)
0.425406 + 0.905003i \(0.360131\pi\)
\(662\) 1.32843e6 0.117813
\(663\) 0 0
\(664\) −5.13056e6 −0.451591
\(665\) 0 0
\(666\) 0 0
\(667\) −840579. −0.0731584
\(668\) 6.82666e6 0.591926
\(669\) 0 0
\(670\) −1.09378e6 −0.0941330
\(671\) −5.29135e6 −0.453691
\(672\) 0 0
\(673\) 1.80397e7 1.53530 0.767648 0.640872i \(-0.221427\pi\)
0.767648 + 0.640872i \(0.221427\pi\)
\(674\) −786879. −0.0667204
\(675\) 0 0
\(676\) 1.03106e7 0.867798
\(677\) 1.44741e7 1.21372 0.606861 0.794808i \(-0.292429\pi\)
0.606861 + 0.794808i \(0.292429\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 329800. 0.0273513
\(681\) 0 0
\(682\) −2.96160e6 −0.243818
\(683\) −2.68785e6 −0.220472 −0.110236 0.993905i \(-0.535161\pi\)
−0.110236 + 0.993905i \(0.535161\pi\)
\(684\) 0 0
\(685\) 2.51854e7 2.05080
\(686\) 0 0
\(687\) 0 0
\(688\) 5.74839e6 0.462994
\(689\) 3.67347e6 0.294801
\(690\) 0 0
\(691\) 4.67627e6 0.372567 0.186284 0.982496i \(-0.440356\pi\)
0.186284 + 0.982496i \(0.440356\pi\)
\(692\) −1.32163e7 −1.04917
\(693\) 0 0
\(694\) −3.20802e6 −0.252836
\(695\) −5.53385e6 −0.434576
\(696\) 0 0
\(697\) −264061. −0.0205884
\(698\) 2.26330e6 0.175834
\(699\) 0 0
\(700\) 0 0
\(701\) −6.34801e6 −0.487913 −0.243957 0.969786i \(-0.578445\pi\)
−0.243957 + 0.969786i \(0.578445\pi\)
\(702\) 0 0
\(703\) 1.37104e7 1.04631
\(704\) −1.44262e7 −1.09703
\(705\) 0 0
\(706\) −658492. −0.0497208
\(707\) 0 0
\(708\) 0 0
\(709\) −6.41052e6 −0.478936 −0.239468 0.970904i \(-0.576973\pi\)
−0.239468 + 0.970904i \(0.576973\pi\)
\(710\) −3.19785e6 −0.238074
\(711\) 0 0
\(712\) −5.65425e6 −0.417998
\(713\) −1.46320e6 −0.107790
\(714\) 0 0
\(715\) −1.07101e7 −0.783481
\(716\) 109625. 0.00799146
\(717\) 0 0
\(718\) 2.13320e6 0.154426
\(719\) −7.34505e6 −0.529874 −0.264937 0.964266i \(-0.585351\pi\)
−0.264937 + 0.964266i \(0.585351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −571944. −0.0408329
\(723\) 0 0
\(724\) −1.86557e7 −1.32271
\(725\) −3.34218e7 −2.36148
\(726\) 0 0
\(727\) 1.57839e7 1.10759 0.553793 0.832655i \(-0.313180\pi\)
0.553793 + 0.832655i \(0.313180\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.58520e6 0.179551
\(731\) −376480. −0.0260584
\(732\) 0 0
\(733\) 1.35367e7 0.930579 0.465289 0.885159i \(-0.345950\pi\)
0.465289 + 0.885159i \(0.345950\pi\)
\(734\) 1.15048e6 0.0788205
\(735\) 0 0
\(736\) 460109. 0.0313088
\(737\) −6.60631e6 −0.448012
\(738\) 0 0
\(739\) −7.33702e6 −0.494207 −0.247103 0.968989i \(-0.579479\pi\)
−0.247103 + 0.968989i \(0.579479\pi\)
\(740\) −3.38524e7 −2.27253
\(741\) 0 0
\(742\) 0 0
\(743\) −1.20844e7 −0.803068 −0.401534 0.915844i \(-0.631523\pi\)
−0.401534 + 0.915844i \(0.631523\pi\)
\(744\) 0 0
\(745\) −1.34472e7 −0.887649
\(746\) −1.61869e6 −0.106492
\(747\) 0 0
\(748\) 986151. 0.0644450
\(749\) 0 0
\(750\) 0 0
\(751\) 8.78176e6 0.568175 0.284087 0.958798i \(-0.408309\pi\)
0.284087 + 0.958798i \(0.408309\pi\)
\(752\) 4.23327e6 0.272980
\(753\) 0 0
\(754\) 706149. 0.0452343
\(755\) 1.59628e7 1.01916
\(756\) 0 0
\(757\) 465874. 0.0295480 0.0147740 0.999891i \(-0.495297\pi\)
0.0147740 + 0.999891i \(0.495297\pi\)
\(758\) −329685. −0.0208414
\(759\) 0 0
\(760\) −6.91339e6 −0.434167
\(761\) −1.49978e7 −0.938786 −0.469393 0.882989i \(-0.655527\pi\)
−0.469393 + 0.882989i \(0.655527\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.59984e7 1.61144
\(765\) 0 0
\(766\) 3.06536e6 0.188760
\(767\) 723026. 0.0443778
\(768\) 0 0
\(769\) −1.85606e7 −1.13181 −0.565907 0.824469i \(-0.691474\pi\)
−0.565907 + 0.824469i \(0.691474\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.89422e6 −0.416334
\(773\) −1.22760e7 −0.738941 −0.369470 0.929243i \(-0.620461\pi\)
−0.369470 + 0.929243i \(0.620461\pi\)
\(774\) 0 0
\(775\) −5.81775e7 −3.47937
\(776\) −1.52027e6 −0.0906286
\(777\) 0 0
\(778\) −2.24432e6 −0.132934
\(779\) 5.53534e6 0.326814
\(780\) 0 0
\(781\) −1.93147e7 −1.13308
\(782\) −9711.39 −0.000567890 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.57525e7 −0.912380
\(786\) 0 0
\(787\) 5.26991e6 0.303296 0.151648 0.988435i \(-0.451542\pi\)
0.151648 + 0.988435i \(0.451542\pi\)
\(788\) 1.49221e7 0.856082
\(789\) 0 0
\(790\) 3.25391e6 0.185497
\(791\) 0 0
\(792\) 0 0
\(793\) 2.19615e6 0.124016
\(794\) 3.43420e6 0.193319
\(795\) 0 0
\(796\) −1.96893e7 −1.10141
\(797\) −1.11889e7 −0.623940 −0.311970 0.950092i \(-0.600989\pi\)
−0.311970 + 0.950092i \(0.600989\pi\)
\(798\) 0 0
\(799\) −277250. −0.0153640
\(800\) 1.82941e7 1.01062
\(801\) 0 0
\(802\) −2.69114e6 −0.147741
\(803\) 1.56144e7 0.854547
\(804\) 0 0
\(805\) 0 0
\(806\) 1.22920e6 0.0666476
\(807\) 0 0
\(808\) 5.35231e6 0.288412
\(809\) −7.05051e6 −0.378747 −0.189374 0.981905i \(-0.560646\pi\)
−0.189374 + 0.981905i \(0.560646\pi\)
\(810\) 0 0
\(811\) −2.54873e7 −1.36073 −0.680364 0.732875i \(-0.738178\pi\)
−0.680364 + 0.732875i \(0.738178\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.07548e6 0.215585
\(815\) −3.42964e6 −0.180865
\(816\) 0 0
\(817\) 7.89192e6 0.413645
\(818\) −4.18677e6 −0.218774
\(819\) 0 0
\(820\) −1.36673e7 −0.709820
\(821\) 2.34879e7 1.21615 0.608073 0.793881i \(-0.291943\pi\)
0.608073 + 0.793881i \(0.291943\pi\)
\(822\) 0 0
\(823\) −1.20903e7 −0.622213 −0.311107 0.950375i \(-0.600700\pi\)
−0.311107 + 0.950375i \(0.600700\pi\)
\(824\) −7.89521e6 −0.405084
\(825\) 0 0
\(826\) 0 0
\(827\) −3.33335e6 −0.169480 −0.0847398 0.996403i \(-0.527006\pi\)
−0.0847398 + 0.996403i \(0.527006\pi\)
\(828\) 0 0
\(829\) −7.70534e6 −0.389408 −0.194704 0.980862i \(-0.562375\pi\)
−0.194704 + 0.980862i \(0.562375\pi\)
\(830\) 8.43496e6 0.424999
\(831\) 0 0
\(832\) 5.98752e6 0.299874
\(833\) 0 0
\(834\) 0 0
\(835\) −2.26706e7 −1.12524
\(836\) −2.06721e7 −1.02298
\(837\) 0 0
\(838\) −2.27172e6 −0.111749
\(839\) −2.40843e7 −1.18121 −0.590607 0.806959i \(-0.701112\pi\)
−0.590607 + 0.806959i \(0.701112\pi\)
\(840\) 0 0
\(841\) −1.82163e6 −0.0888116
\(842\) 1.65032e6 0.0802208
\(843\) 0 0
\(844\) −1.79146e7 −0.865665
\(845\) −3.42405e7 −1.64967
\(846\) 0 0
\(847\) 0 0
\(848\) −1.71509e7 −0.819023
\(849\) 0 0
\(850\) −386129. −0.0183310
\(851\) 2.01352e6 0.0953088
\(852\) 0 0
\(853\) −2.82938e7 −1.33143 −0.665716 0.746206i \(-0.731874\pi\)
−0.665716 + 0.746206i \(0.731874\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.47666e6 −0.208819
\(857\) −2.16176e7 −1.00544 −0.502719 0.864450i \(-0.667667\pi\)
−0.502719 + 0.864450i \(0.667667\pi\)
\(858\) 0 0
\(859\) 3.25382e7 1.50456 0.752282 0.658841i \(-0.228953\pi\)
0.752282 + 0.658841i \(0.228953\pi\)
\(860\) −1.94859e7 −0.898411
\(861\) 0 0
\(862\) 1.92058e6 0.0880366
\(863\) −8.06384e6 −0.368566 −0.184283 0.982873i \(-0.558996\pi\)
−0.184283 + 0.982873i \(0.558996\pi\)
\(864\) 0 0
\(865\) 4.38900e7 1.99446
\(866\) 756183. 0.0342635
\(867\) 0 0
\(868\) 0 0
\(869\) 1.96533e7 0.882849
\(870\) 0 0
\(871\) 2.74192e6 0.122464
\(872\) −8.39983e6 −0.374093
\(873\) 0 0
\(874\) 203574. 0.00901454
\(875\) 0 0
\(876\) 0 0
\(877\) 1.70312e7 0.747732 0.373866 0.927483i \(-0.378032\pi\)
0.373866 + 0.927483i \(0.378032\pi\)
\(878\) 2.08934e6 0.0914690
\(879\) 0 0
\(880\) 5.00038e7 2.17669
\(881\) −1.24740e7 −0.541459 −0.270729 0.962655i \(-0.587265\pi\)
−0.270729 + 0.962655i \(0.587265\pi\)
\(882\) 0 0
\(883\) 7.28955e6 0.314629 0.157315 0.987549i \(-0.449716\pi\)
0.157315 + 0.987549i \(0.449716\pi\)
\(884\) −409298. −0.0176160
\(885\) 0 0
\(886\) 2.72010e6 0.116413
\(887\) −1.87675e7 −0.800937 −0.400469 0.916310i \(-0.631153\pi\)
−0.400469 + 0.916310i \(0.631153\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 9.29592e6 0.393385
\(891\) 0 0
\(892\) −132812. −0.00558889
\(893\) 5.81181e6 0.243884
\(894\) 0 0
\(895\) −364052. −0.0151917
\(896\) 0 0
\(897\) 0 0
\(898\) −3.47878e6 −0.143958
\(899\) 3.25330e7 1.34253
\(900\) 0 0
\(901\) 1.12326e6 0.0460966
\(902\) 1.64541e6 0.0673375
\(903\) 0 0
\(904\) −5.80323e6 −0.236183
\(905\) 6.19535e7 2.51446
\(906\) 0 0
\(907\) 1.26367e6 0.0510052 0.0255026 0.999675i \(-0.491881\pi\)
0.0255026 + 0.999675i \(0.491881\pi\)
\(908\) 3.54490e7 1.42689
\(909\) 0 0
\(910\) 0 0
\(911\) 2.47718e7 0.988921 0.494461 0.869200i \(-0.335366\pi\)
0.494461 + 0.869200i \(0.335366\pi\)
\(912\) 0 0
\(913\) 5.09463e7 2.02272
\(914\) −1.78018e6 −0.0704854
\(915\) 0 0
\(916\) −2.52454e7 −0.994130
\(917\) 0 0
\(918\) 0 0
\(919\) −9.36676e6 −0.365848 −0.182924 0.983127i \(-0.558556\pi\)
−0.182924 + 0.983127i \(0.558556\pi\)
\(920\) −1.01531e6 −0.0395483
\(921\) 0 0
\(922\) −1.46542e6 −0.0567721
\(923\) 8.01648e6 0.309727
\(924\) 0 0
\(925\) 8.00586e7 3.07648
\(926\) −2.38103e6 −0.0912509
\(927\) 0 0
\(928\) −1.02301e7 −0.389951
\(929\) 6.87350e6 0.261300 0.130650 0.991429i \(-0.458294\pi\)
0.130650 + 0.991429i \(0.458294\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.66855e7 1.38342
\(933\) 0 0
\(934\) 843990. 0.0316570
\(935\) −3.27490e6 −0.122509
\(936\) 0 0
\(937\) −4.26197e7 −1.58585 −0.792923 0.609322i \(-0.791442\pi\)
−0.792923 + 0.609322i \(0.791442\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.43499e7 −0.529701
\(941\) −4.50226e6 −0.165751 −0.0828755 0.996560i \(-0.526410\pi\)
−0.0828755 + 0.996560i \(0.526410\pi\)
\(942\) 0 0
\(943\) 812926. 0.0297695
\(944\) −3.37569e6 −0.123291
\(945\) 0 0
\(946\) 2.34591e6 0.0852282
\(947\) −3.37725e7 −1.22374 −0.611870 0.790959i \(-0.709582\pi\)
−0.611870 + 0.790959i \(0.709582\pi\)
\(948\) 0 0
\(949\) −6.48068e6 −0.233590
\(950\) 8.09419e6 0.290981
\(951\) 0 0
\(952\) 0 0
\(953\) −4.81813e7 −1.71849 −0.859244 0.511566i \(-0.829066\pi\)
−0.859244 + 0.511566i \(0.829066\pi\)
\(954\) 0 0
\(955\) −8.63379e7 −3.06332
\(956\) −5.35107e7 −1.89363
\(957\) 0 0
\(958\) 4.49800e6 0.158346
\(959\) 0 0
\(960\) 0 0
\(961\) 2.80012e7 0.978066
\(962\) −1.69151e6 −0.0589301
\(963\) 0 0
\(964\) −2.98434e7 −1.03432
\(965\) 2.28950e7 0.791446
\(966\) 0 0
\(967\) −3.54640e7 −1.21961 −0.609805 0.792551i \(-0.708752\pi\)
−0.609805 + 0.792551i \(0.708752\pi\)
\(968\) −4.34084e6 −0.148897
\(969\) 0 0
\(970\) 2.49941e6 0.0852919
\(971\) 7.84296e6 0.266951 0.133476 0.991052i \(-0.457386\pi\)
0.133476 + 0.991052i \(0.457386\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −5.41288e6 −0.182823
\(975\) 0 0
\(976\) −1.02535e7 −0.344545
\(977\) 2.47534e7 0.829658 0.414829 0.909899i \(-0.363841\pi\)
0.414829 + 0.909899i \(0.363841\pi\)
\(978\) 0 0
\(979\) 5.61464e7 1.87226
\(980\) 0 0
\(981\) 0 0
\(982\) −4.42955e6 −0.146582
\(983\) −4.53129e7 −1.49568 −0.747838 0.663881i \(-0.768908\pi\)
−0.747838 + 0.663881i \(0.768908\pi\)
\(984\) 0 0
\(985\) −4.95548e7 −1.62740
\(986\) 215924. 0.00707308
\(987\) 0 0
\(988\) 8.57985e6 0.279632
\(989\) 1.15902e6 0.0376789
\(990\) 0 0
\(991\) −3.74143e7 −1.21019 −0.605095 0.796154i \(-0.706865\pi\)
−0.605095 + 0.796154i \(0.706865\pi\)
\(992\) −1.78076e7 −0.574548
\(993\) 0 0
\(994\) 0 0
\(995\) 6.53861e7 2.09376
\(996\) 0 0
\(997\) 4.65278e7 1.48243 0.741216 0.671267i \(-0.234249\pi\)
0.741216 + 0.671267i \(0.234249\pi\)
\(998\) −1.84676e6 −0.0586926
\(999\) 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.1.3 4
3.2 odd 2 147.6.a.m.1.2 4
7.2 even 3 63.6.e.e.46.2 8
7.4 even 3 63.6.e.e.37.2 8
7.6 odd 2 441.6.a.v.1.3 4
21.2 odd 6 21.6.e.c.4.3 8
21.5 even 6 147.6.e.o.67.3 8
21.11 odd 6 21.6.e.c.16.3 yes 8
21.17 even 6 147.6.e.o.79.3 8
21.20 even 2 147.6.a.l.1.2 4
84.11 even 6 336.6.q.j.289.4 8
84.23 even 6 336.6.q.j.193.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.3 8 21.2 odd 6
21.6.e.c.16.3 yes 8 21.11 odd 6
63.6.e.e.37.2 8 7.4 even 3
63.6.e.e.46.2 8 7.2 even 3
147.6.a.l.1.2 4 21.20 even 2
147.6.a.m.1.2 4 3.2 odd 2
147.6.e.o.67.3 8 21.5 even 6
147.6.e.o.79.3 8 21.17 even 6
336.6.q.j.193.4 8 84.23 even 6
336.6.q.j.289.4 8 84.11 even 6
441.6.a.v.1.3 4 7.6 odd 2
441.6.a.w.1.3 4 1.1 even 1 trivial