Properties

Label 1323.4.a.ba.1.4
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{13})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.684742\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} +O(q^{10})\) \(q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} -87.7802 q^{10} -30.3531 q^{11} +89.5603 q^{13} +132.967 q^{16} -13.4164 q^{17} +5.37355 q^{19} -316.496 q^{20} -156.527 q^{22} +167.449 q^{23} +164.747 q^{25} +461.852 q^{26} -135.085 q^{29} +18.9339 q^{31} +248.664 q^{32} -69.1868 q^{34} +402.560 q^{37} +27.7107 q^{38} -929.889 q^{40} +434.345 q^{41} +53.1868 q^{43} -564.367 q^{44} +863.516 q^{46} +155.218 q^{47} +849.581 q^{50} +1665.23 q^{52} -301.707 q^{53} +516.669 q^{55} -696.615 q^{58} -412.635 q^{59} +571.253 q^{61} +97.6396 q^{62} +218.593 q^{64} -1524.49 q^{65} +820.549 q^{67} -249.456 q^{68} -8.95825 q^{71} -21.9339 q^{73} +2075.95 q^{74} +99.9124 q^{76} +619.428 q^{79} -2263.36 q^{80} +2239.87 q^{82} +259.532 q^{83} +228.374 q^{85} +274.278 q^{86} -1658.15 q^{88} +484.329 q^{89} +3113.45 q^{92} +800.440 q^{94} -91.4683 q^{95} +252.747 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 26q^{4} + O(q^{10}) \) \( 4q + 26q^{4} - 206q^{10} + 68q^{13} + 290q^{16} - 172q^{19} - 94q^{22} + 272q^{25} - 408q^{31} - 180q^{34} + 1320q^{37} - 1446q^{40} + 116q^{43} + 1374q^{46} + 3952q^{52} - 352q^{55} - 1432q^{58} + 2672q^{61} + 826q^{64} + 1444q^{67} + 396q^{73} + 1222q^{76} + 1220q^{79} + 3590q^{82} + 720q^{85} - 6294q^{88} + 3492q^{94} + 624q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.15688 1.82323 0.911616 0.411043i \(-0.134836\pi\)
0.911616 + 0.411043i \(0.134836\pi\)
\(3\) 0 0
\(4\) 18.5934 2.32417
\(5\) −17.0220 −1.52249 −0.761245 0.648464i \(-0.775412\pi\)
−0.761245 + 0.648464i \(0.775412\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 54.6288 2.41427
\(9\) 0 0
\(10\) −87.7802 −2.77585
\(11\) −30.3531 −0.831982 −0.415991 0.909369i \(-0.636565\pi\)
−0.415991 + 0.909369i \(0.636565\pi\)
\(12\) 0 0
\(13\) 89.5603 1.91074 0.955368 0.295419i \(-0.0954592\pi\)
0.955368 + 0.295419i \(0.0954592\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 132.967 2.07761
\(17\) −13.4164 −0.191409 −0.0957046 0.995410i \(-0.530510\pi\)
−0.0957046 + 0.995410i \(0.530510\pi\)
\(18\) 0 0
\(19\) 5.37355 0.0648830 0.0324415 0.999474i \(-0.489672\pi\)
0.0324415 + 0.999474i \(0.489672\pi\)
\(20\) −316.496 −3.53853
\(21\) 0 0
\(22\) −156.527 −1.51690
\(23\) 167.449 1.51807 0.759035 0.651050i \(-0.225671\pi\)
0.759035 + 0.651050i \(0.225671\pi\)
\(24\) 0 0
\(25\) 164.747 1.31798
\(26\) 461.852 3.48371
\(27\) 0 0
\(28\) 0 0
\(29\) −135.085 −0.864986 −0.432493 0.901637i \(-0.642366\pi\)
−0.432493 + 0.901637i \(0.642366\pi\)
\(30\) 0 0
\(31\) 18.9339 0.109698 0.0548488 0.998495i \(-0.482532\pi\)
0.0548488 + 0.998495i \(0.482532\pi\)
\(32\) 248.664 1.37369
\(33\) 0 0
\(34\) −69.1868 −0.348983
\(35\) 0 0
\(36\) 0 0
\(37\) 402.560 1.78866 0.894331 0.447406i \(-0.147652\pi\)
0.894331 + 0.447406i \(0.147652\pi\)
\(38\) 27.7107 0.118297
\(39\) 0 0
\(40\) −929.889 −3.67571
\(41\) 434.345 1.65447 0.827236 0.561855i \(-0.189912\pi\)
0.827236 + 0.561855i \(0.189912\pi\)
\(42\) 0 0
\(43\) 53.1868 0.188626 0.0943129 0.995543i \(-0.469935\pi\)
0.0943129 + 0.995543i \(0.469935\pi\)
\(44\) −564.367 −1.93367
\(45\) 0 0
\(46\) 863.516 2.76779
\(47\) 155.218 0.481720 0.240860 0.970560i \(-0.422570\pi\)
0.240860 + 0.970560i \(0.422570\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 849.581 2.40298
\(51\) 0 0
\(52\) 1665.23 4.44088
\(53\) −301.707 −0.781938 −0.390969 0.920404i \(-0.627860\pi\)
−0.390969 + 0.920404i \(0.627860\pi\)
\(54\) 0 0
\(55\) 516.669 1.26669
\(56\) 0 0
\(57\) 0 0
\(58\) −696.615 −1.57707
\(59\) −412.635 −0.910518 −0.455259 0.890359i \(-0.650453\pi\)
−0.455259 + 0.890359i \(0.650453\pi\)
\(60\) 0 0
\(61\) 571.253 1.19904 0.599520 0.800360i \(-0.295358\pi\)
0.599520 + 0.800360i \(0.295358\pi\)
\(62\) 97.6396 0.200004
\(63\) 0 0
\(64\) 218.593 0.426940
\(65\) −1524.49 −2.90908
\(66\) 0 0
\(67\) 820.549 1.49621 0.748104 0.663581i \(-0.230964\pi\)
0.748104 + 0.663581i \(0.230964\pi\)
\(68\) −249.456 −0.444868
\(69\) 0 0
\(70\) 0 0
\(71\) −8.95825 −0.0149739 −0.00748696 0.999972i \(-0.502383\pi\)
−0.00748696 + 0.999972i \(0.502383\pi\)
\(72\) 0 0
\(73\) −21.9339 −0.0351666 −0.0175833 0.999845i \(-0.505597\pi\)
−0.0175833 + 0.999845i \(0.505597\pi\)
\(74\) 2075.95 3.26115
\(75\) 0 0
\(76\) 99.9124 0.150799
\(77\) 0 0
\(78\) 0 0
\(79\) 619.428 0.882166 0.441083 0.897466i \(-0.354594\pi\)
0.441083 + 0.897466i \(0.354594\pi\)
\(80\) −2263.36 −3.16314
\(81\) 0 0
\(82\) 2239.87 3.01649
\(83\) 259.532 0.343221 0.171610 0.985165i \(-0.445103\pi\)
0.171610 + 0.985165i \(0.445103\pi\)
\(84\) 0 0
\(85\) 228.374 0.291419
\(86\) 274.278 0.343908
\(87\) 0 0
\(88\) −1658.15 −2.00863
\(89\) 484.329 0.576840 0.288420 0.957504i \(-0.406870\pi\)
0.288420 + 0.957504i \(0.406870\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3113.45 3.52826
\(93\) 0 0
\(94\) 800.440 0.878288
\(95\) −91.4683 −0.0987837
\(96\) 0 0
\(97\) 252.747 0.264563 0.132281 0.991212i \(-0.457770\pi\)
0.132281 + 0.991212i \(0.457770\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3063.21 3.06321
\(101\) −629.378 −0.620054 −0.310027 0.950728i \(-0.600338\pi\)
−0.310027 + 0.950728i \(0.600338\pi\)
\(102\) 0 0
\(103\) −1172.54 −1.12168 −0.560842 0.827923i \(-0.689523\pi\)
−0.560842 + 0.827923i \(0.689523\pi\)
\(104\) 4892.57 4.61304
\(105\) 0 0
\(106\) −1555.87 −1.42565
\(107\) 1002.87 0.906087 0.453044 0.891488i \(-0.350338\pi\)
0.453044 + 0.891488i \(0.350338\pi\)
\(108\) 0 0
\(109\) 542.802 0.476981 0.238491 0.971145i \(-0.423347\pi\)
0.238491 + 0.971145i \(0.423347\pi\)
\(110\) 2664.40 2.30946
\(111\) 0 0
\(112\) 0 0
\(113\) 1388.74 1.15612 0.578062 0.815993i \(-0.303809\pi\)
0.578062 + 0.815993i \(0.303809\pi\)
\(114\) 0 0
\(115\) −2850.32 −2.31125
\(116\) −2511.68 −2.01038
\(117\) 0 0
\(118\) −2127.91 −1.66009
\(119\) 0 0
\(120\) 0 0
\(121\) −409.689 −0.307805
\(122\) 2945.88 2.18613
\(123\) 0 0
\(124\) 352.045 0.254956
\(125\) −676.573 −0.484117
\(126\) 0 0
\(127\) −2330.00 −1.62798 −0.813991 0.580877i \(-0.802710\pi\)
−0.813991 + 0.580877i \(0.802710\pi\)
\(128\) −862.051 −0.595276
\(129\) 0 0
\(130\) −7861.62 −5.30392
\(131\) 226.817 0.151276 0.0756379 0.997135i \(-0.475901\pi\)
0.0756379 + 0.997135i \(0.475901\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4231.47 2.72793
\(135\) 0 0
\(136\) −732.922 −0.462114
\(137\) −1900.83 −1.18540 −0.592698 0.805425i \(-0.701937\pi\)
−0.592698 + 0.805425i \(0.701937\pi\)
\(138\) 0 0
\(139\) −1958.99 −1.19539 −0.597695 0.801724i \(-0.703917\pi\)
−0.597695 + 0.801724i \(0.703917\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −46.1966 −0.0273009
\(143\) −2718.43 −1.58970
\(144\) 0 0
\(145\) 2299.40 1.31693
\(146\) −113.110 −0.0641169
\(147\) 0 0
\(148\) 7484.96 4.15716
\(149\) 1518.37 0.834831 0.417416 0.908716i \(-0.362936\pi\)
0.417416 + 0.908716i \(0.362936\pi\)
\(150\) 0 0
\(151\) −1815.18 −0.978262 −0.489131 0.872210i \(-0.662686\pi\)
−0.489131 + 0.872210i \(0.662686\pi\)
\(152\) 293.550 0.156645
\(153\) 0 0
\(154\) 0 0
\(155\) −322.291 −0.167013
\(156\) 0 0
\(157\) 1858.79 0.944889 0.472445 0.881360i \(-0.343372\pi\)
0.472445 + 0.881360i \(0.343372\pi\)
\(158\) 3194.31 1.60839
\(159\) 0 0
\(160\) −4232.75 −2.09142
\(161\) 0 0
\(162\) 0 0
\(163\) 870.089 0.418102 0.209051 0.977905i \(-0.432962\pi\)
0.209051 + 0.977905i \(0.432962\pi\)
\(164\) 8075.95 3.84528
\(165\) 0 0
\(166\) 1338.37 0.625771
\(167\) −1749.13 −0.810488 −0.405244 0.914209i \(-0.632813\pi\)
−0.405244 + 0.914209i \(0.632813\pi\)
\(168\) 0 0
\(169\) 5824.05 2.65091
\(170\) 1177.69 0.531324
\(171\) 0 0
\(172\) 988.922 0.438399
\(173\) −39.3374 −0.0172877 −0.00864383 0.999963i \(-0.502751\pi\)
−0.00864383 + 0.999963i \(0.502751\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4035.96 −1.72853
\(177\) 0 0
\(178\) 2497.62 1.05171
\(179\) −2650.58 −1.10678 −0.553391 0.832922i \(-0.686666\pi\)
−0.553391 + 0.832922i \(0.686666\pi\)
\(180\) 0 0
\(181\) −2719.71 −1.11688 −0.558438 0.829546i \(-0.688599\pi\)
−0.558438 + 0.829546i \(0.688599\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 9147.55 3.66504
\(185\) −6852.37 −2.72322
\(186\) 0 0
\(187\) 407.230 0.159249
\(188\) 2886.03 1.11960
\(189\) 0 0
\(190\) −471.691 −0.180106
\(191\) 4339.00 1.64376 0.821881 0.569659i \(-0.192925\pi\)
0.821881 + 0.569659i \(0.192925\pi\)
\(192\) 0 0
\(193\) 977.689 0.364640 0.182320 0.983239i \(-0.441639\pi\)
0.182320 + 0.983239i \(0.441639\pi\)
\(194\) 1303.39 0.482359
\(195\) 0 0
\(196\) 0 0
\(197\) −1556.32 −0.562859 −0.281430 0.959582i \(-0.590809\pi\)
−0.281430 + 0.959582i \(0.590809\pi\)
\(198\) 0 0
\(199\) −1723.21 −0.613846 −0.306923 0.951734i \(-0.599299\pi\)
−0.306923 + 0.951734i \(0.599299\pi\)
\(200\) 8999.94 3.18196
\(201\) 0 0
\(202\) −3245.62 −1.13050
\(203\) 0 0
\(204\) 0 0
\(205\) −7393.41 −2.51892
\(206\) −6046.63 −2.04509
\(207\) 0 0
\(208\) 11908.6 3.96976
\(209\) −163.104 −0.0539815
\(210\) 0 0
\(211\) 1295.19 0.422581 0.211290 0.977423i \(-0.432233\pi\)
0.211290 + 0.977423i \(0.432233\pi\)
\(212\) −5609.76 −1.81736
\(213\) 0 0
\(214\) 5171.69 1.65201
\(215\) −905.343 −0.287181
\(216\) 0 0
\(217\) 0 0
\(218\) 2799.16 0.869647
\(219\) 0 0
\(220\) 9606.63 2.94400
\(221\) −1201.58 −0.365732
\(222\) 0 0
\(223\) 483.611 0.145224 0.0726121 0.997360i \(-0.476866\pi\)
0.0726121 + 0.997360i \(0.476866\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7161.58 2.10788
\(227\) 4019.86 1.17536 0.587681 0.809092i \(-0.300041\pi\)
0.587681 + 0.809092i \(0.300041\pi\)
\(228\) 0 0
\(229\) −5070.37 −1.46314 −0.731570 0.681766i \(-0.761212\pi\)
−0.731570 + 0.681766i \(0.761212\pi\)
\(230\) −14698.7 −4.21394
\(231\) 0 0
\(232\) −7379.51 −2.08831
\(233\) −4418.26 −1.24227 −0.621137 0.783702i \(-0.713329\pi\)
−0.621137 + 0.783702i \(0.713329\pi\)
\(234\) 0 0
\(235\) −2642.11 −0.733415
\(236\) −7672.29 −2.11620
\(237\) 0 0
\(238\) 0 0
\(239\) −3228.15 −0.873690 −0.436845 0.899537i \(-0.643904\pi\)
−0.436845 + 0.899537i \(0.643904\pi\)
\(240\) 0 0
\(241\) 1537.14 0.410854 0.205427 0.978672i \(-0.434142\pi\)
0.205427 + 0.978672i \(0.434142\pi\)
\(242\) −2112.72 −0.561200
\(243\) 0 0
\(244\) 10621.5 2.78678
\(245\) 0 0
\(246\) 0 0
\(247\) 481.257 0.123974
\(248\) 1034.33 0.264840
\(249\) 0 0
\(250\) −3489.01 −0.882657
\(251\) −178.384 −0.0448587 −0.0224293 0.999748i \(-0.507140\pi\)
−0.0224293 + 0.999748i \(0.507140\pi\)
\(252\) 0 0
\(253\) −5082.61 −1.26301
\(254\) −12015.5 −2.96819
\(255\) 0 0
\(256\) −6194.24 −1.51227
\(257\) −3271.92 −0.794152 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −28345.5 −6.76120
\(261\) 0 0
\(262\) 1169.67 0.275811
\(263\) −4738.45 −1.11097 −0.555486 0.831526i \(-0.687468\pi\)
−0.555486 + 0.831526i \(0.687468\pi\)
\(264\) 0 0
\(265\) 5135.65 1.19049
\(266\) 0 0
\(267\) 0 0
\(268\) 15256.8 3.47745
\(269\) −418.456 −0.0948466 −0.0474233 0.998875i \(-0.515101\pi\)
−0.0474233 + 0.998875i \(0.515101\pi\)
\(270\) 0 0
\(271\) −5263.95 −1.17993 −0.589967 0.807428i \(-0.700859\pi\)
−0.589967 + 0.807428i \(0.700859\pi\)
\(272\) −1783.94 −0.397673
\(273\) 0 0
\(274\) −9802.37 −2.16125
\(275\) −5000.59 −1.09653
\(276\) 0 0
\(277\) 2855.61 0.619412 0.309706 0.950832i \(-0.399769\pi\)
0.309706 + 0.950832i \(0.399769\pi\)
\(278\) −10102.3 −2.17947
\(279\) 0 0
\(280\) 0 0
\(281\) 8589.72 1.82356 0.911779 0.410682i \(-0.134709\pi\)
0.911779 + 0.410682i \(0.134709\pi\)
\(282\) 0 0
\(283\) −3681.18 −0.773228 −0.386614 0.922242i \(-0.626355\pi\)
−0.386614 + 0.922242i \(0.626355\pi\)
\(284\) −166.564 −0.0348020
\(285\) 0 0
\(286\) −14018.6 −2.89839
\(287\) 0 0
\(288\) 0 0
\(289\) −4733.00 −0.963363
\(290\) 11857.7 2.40107
\(291\) 0 0
\(292\) −407.825 −0.0817334
\(293\) −1958.68 −0.390536 −0.195268 0.980750i \(-0.562558\pi\)
−0.195268 + 0.980750i \(0.562558\pi\)
\(294\) 0 0
\(295\) 7023.86 1.38625
\(296\) 21991.4 4.31832
\(297\) 0 0
\(298\) 7830.06 1.52209
\(299\) 14996.8 2.90063
\(300\) 0 0
\(301\) 0 0
\(302\) −9360.68 −1.78360
\(303\) 0 0
\(304\) 714.504 0.134801
\(305\) −9723.84 −1.82553
\(306\) 0 0
\(307\) 5672.32 1.05452 0.527258 0.849706i \(-0.323220\pi\)
0.527258 + 0.849706i \(0.323220\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1662.02 −0.304504
\(311\) −1271.71 −0.231872 −0.115936 0.993257i \(-0.536987\pi\)
−0.115936 + 0.993257i \(0.536987\pi\)
\(312\) 0 0
\(313\) 4868.38 0.879160 0.439580 0.898204i \(-0.355127\pi\)
0.439580 + 0.898204i \(0.355127\pi\)
\(314\) 9585.55 1.72275
\(315\) 0 0
\(316\) 11517.3 2.05031
\(317\) 3224.18 0.571256 0.285628 0.958341i \(-0.407798\pi\)
0.285628 + 0.958341i \(0.407798\pi\)
\(318\) 0 0
\(319\) 4100.24 0.719653
\(320\) −3720.89 −0.650012
\(321\) 0 0
\(322\) 0 0
\(323\) −72.0937 −0.0124192
\(324\) 0 0
\(325\) 14754.8 2.51831
\(326\) 4486.94 0.762297
\(327\) 0 0
\(328\) 23727.8 3.99435
\(329\) 0 0
\(330\) 0 0
\(331\) 5309.27 0.881643 0.440821 0.897595i \(-0.354687\pi\)
0.440821 + 0.897595i \(0.354687\pi\)
\(332\) 4825.57 0.797704
\(333\) 0 0
\(334\) −9020.04 −1.47771
\(335\) −13967.3 −2.27796
\(336\) 0 0
\(337\) −2304.33 −0.372478 −0.186239 0.982505i \(-0.559630\pi\)
−0.186239 + 0.982505i \(0.559630\pi\)
\(338\) 30033.9 4.83322
\(339\) 0 0
\(340\) 4246.24 0.677308
\(341\) −574.702 −0.0912664
\(342\) 0 0
\(343\) 0 0
\(344\) 2905.53 0.455394
\(345\) 0 0
\(346\) −202.858 −0.0315194
\(347\) −5196.20 −0.803881 −0.401940 0.915666i \(-0.631664\pi\)
−0.401940 + 0.915666i \(0.631664\pi\)
\(348\) 0 0
\(349\) 9030.60 1.38509 0.692546 0.721374i \(-0.256489\pi\)
0.692546 + 0.721374i \(0.256489\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7547.72 −1.14288
\(353\) 8885.51 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(354\) 0 0
\(355\) 152.487 0.0227976
\(356\) 9005.31 1.34068
\(357\) 0 0
\(358\) −13668.7 −2.01792
\(359\) −37.2983 −0.00548337 −0.00274169 0.999996i \(-0.500873\pi\)
−0.00274169 + 0.999996i \(0.500873\pi\)
\(360\) 0 0
\(361\) −6830.12 −0.995790
\(362\) −14025.2 −2.03632
\(363\) 0 0
\(364\) 0 0
\(365\) 373.357 0.0535409
\(366\) 0 0
\(367\) −3335.79 −0.474460 −0.237230 0.971454i \(-0.576240\pi\)
−0.237230 + 0.971454i \(0.576240\pi\)
\(368\) 22265.2 3.15395
\(369\) 0 0
\(370\) −35336.8 −4.96506
\(371\) 0 0
\(372\) 0 0
\(373\) 3935.81 0.546349 0.273175 0.961964i \(-0.411926\pi\)
0.273175 + 0.961964i \(0.411926\pi\)
\(374\) 2100.03 0.290348
\(375\) 0 0
\(376\) 8479.37 1.16301
\(377\) −12098.2 −1.65276
\(378\) 0 0
\(379\) 2308.00 0.312808 0.156404 0.987693i \(-0.450010\pi\)
0.156404 + 0.987693i \(0.450010\pi\)
\(380\) −1700.71 −0.229590
\(381\) 0 0
\(382\) 22375.7 2.99696
\(383\) 2931.62 0.391119 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5041.82 0.664824
\(387\) 0 0
\(388\) 4699.42 0.614890
\(389\) −7610.12 −0.991898 −0.495949 0.868352i \(-0.665180\pi\)
−0.495949 + 0.868352i \(0.665180\pi\)
\(390\) 0 0
\(391\) −2246.57 −0.290572
\(392\) 0 0
\(393\) 0 0
\(394\) −8025.76 −1.02622
\(395\) −10543.9 −1.34309
\(396\) 0 0
\(397\) 4269.74 0.539778 0.269889 0.962891i \(-0.413013\pi\)
0.269889 + 0.962891i \(0.413013\pi\)
\(398\) −8886.41 −1.11918
\(399\) 0 0
\(400\) 21905.9 2.73824
\(401\) −11128.8 −1.38590 −0.692948 0.720988i \(-0.743688\pi\)
−0.692948 + 0.720988i \(0.743688\pi\)
\(402\) 0 0
\(403\) 1695.72 0.209603
\(404\) −11702.3 −1.44111
\(405\) 0 0
\(406\) 0 0
\(407\) −12219.0 −1.48814
\(408\) 0 0
\(409\) −7220.39 −0.872922 −0.436461 0.899723i \(-0.643768\pi\)
−0.436461 + 0.899723i \(0.643768\pi\)
\(410\) −38126.9 −4.59257
\(411\) 0 0
\(412\) −21801.4 −2.60699
\(413\) 0 0
\(414\) 0 0
\(415\) −4417.74 −0.522550
\(416\) 22270.4 2.62475
\(417\) 0 0
\(418\) −841.106 −0.0984207
\(419\) 16631.2 1.93911 0.969556 0.244870i \(-0.0787454\pi\)
0.969556 + 0.244870i \(0.0787454\pi\)
\(420\) 0 0
\(421\) −654.498 −0.0757679 −0.0378839 0.999282i \(-0.512062\pi\)
−0.0378839 + 0.999282i \(0.512062\pi\)
\(422\) 6679.14 0.770463
\(423\) 0 0
\(424\) −16481.9 −1.88781
\(425\) −2210.31 −0.252273
\(426\) 0 0
\(427\) 0 0
\(428\) 18646.8 2.10590
\(429\) 0 0
\(430\) −4668.74 −0.523597
\(431\) −3049.91 −0.340856 −0.170428 0.985370i \(-0.554515\pi\)
−0.170428 + 0.985370i \(0.554515\pi\)
\(432\) 0 0
\(433\) 1599.41 0.177512 0.0887562 0.996053i \(-0.471711\pi\)
0.0887562 + 0.996053i \(0.471711\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10092.5 1.10859
\(437\) 899.797 0.0984968
\(438\) 0 0
\(439\) 15893.9 1.72796 0.863979 0.503528i \(-0.167965\pi\)
0.863979 + 0.503528i \(0.167965\pi\)
\(440\) 28225.0 3.05813
\(441\) 0 0
\(442\) −6196.39 −0.666815
\(443\) −6160.22 −0.660679 −0.330339 0.943862i \(-0.607163\pi\)
−0.330339 + 0.943862i \(0.607163\pi\)
\(444\) 0 0
\(445\) −8244.23 −0.878233
\(446\) 2493.92 0.264777
\(447\) 0 0
\(448\) 0 0
\(449\) 8742.47 0.918892 0.459446 0.888206i \(-0.348048\pi\)
0.459446 + 0.888206i \(0.348048\pi\)
\(450\) 0 0
\(451\) −13183.7 −1.37649
\(452\) 25821.4 2.68703
\(453\) 0 0
\(454\) 20729.9 2.14296
\(455\) 0 0
\(456\) 0 0
\(457\) −580.112 −0.0593796 −0.0296898 0.999559i \(-0.509452\pi\)
−0.0296898 + 0.999559i \(0.509452\pi\)
\(458\) −26147.3 −2.66764
\(459\) 0 0
\(460\) −52997.0 −5.37174
\(461\) −1957.05 −0.197720 −0.0988599 0.995101i \(-0.531520\pi\)
−0.0988599 + 0.995101i \(0.531520\pi\)
\(462\) 0 0
\(463\) 8111.80 0.814227 0.407114 0.913378i \(-0.366535\pi\)
0.407114 + 0.913378i \(0.366535\pi\)
\(464\) −17961.8 −1.79710
\(465\) 0 0
\(466\) −22784.4 −2.26495
\(467\) −1240.29 −0.122899 −0.0614493 0.998110i \(-0.519572\pi\)
−0.0614493 + 0.998110i \(0.519572\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −13625.1 −1.33718
\(471\) 0 0
\(472\) −22541.8 −2.19824
\(473\) −1614.38 −0.156933
\(474\) 0 0
\(475\) 885.276 0.0855142
\(476\) 0 0
\(477\) 0 0
\(478\) −16647.2 −1.59294
\(479\) −18605.0 −1.77470 −0.887352 0.461092i \(-0.847458\pi\)
−0.887352 + 0.461092i \(0.847458\pi\)
\(480\) 0 0
\(481\) 36053.4 3.41766
\(482\) 7926.84 0.749083
\(483\) 0 0
\(484\) −7617.51 −0.715393
\(485\) −4302.25 −0.402794
\(486\) 0 0
\(487\) 874.431 0.0813640 0.0406820 0.999172i \(-0.487047\pi\)
0.0406820 + 0.999172i \(0.487047\pi\)
\(488\) 31206.9 2.89481
\(489\) 0 0
\(490\) 0 0
\(491\) 5849.95 0.537688 0.268844 0.963184i \(-0.413358\pi\)
0.268844 + 0.963184i \(0.413358\pi\)
\(492\) 0 0
\(493\) 1812.35 0.165566
\(494\) 2481.78 0.226034
\(495\) 0 0
\(496\) 2517.58 0.227908
\(497\) 0 0
\(498\) 0 0
\(499\) −1095.76 −0.0983029 −0.0491514 0.998791i \(-0.515652\pi\)
−0.0491514 + 0.998791i \(0.515652\pi\)
\(500\) −12579.8 −1.12517
\(501\) 0 0
\(502\) −919.907 −0.0817878
\(503\) −4988.25 −0.442177 −0.221088 0.975254i \(-0.570961\pi\)
−0.221088 + 0.975254i \(0.570961\pi\)
\(504\) 0 0
\(505\) 10713.2 0.944025
\(506\) −26210.4 −2.30275
\(507\) 0 0
\(508\) −43322.5 −3.78371
\(509\) 16831.7 1.46572 0.732859 0.680381i \(-0.238186\pi\)
0.732859 + 0.680381i \(0.238186\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −25046.5 −2.16193
\(513\) 0 0
\(514\) −16872.9 −1.44792
\(515\) 19958.9 1.70775
\(516\) 0 0
\(517\) −4711.35 −0.400783
\(518\) 0 0
\(519\) 0 0
\(520\) −83281.2 −7.02331
\(521\) 12826.2 1.07855 0.539277 0.842129i \(-0.318698\pi\)
0.539277 + 0.842129i \(0.318698\pi\)
\(522\) 0 0
\(523\) 3377.82 0.282412 0.141206 0.989980i \(-0.454902\pi\)
0.141206 + 0.989980i \(0.454902\pi\)
\(524\) 4217.30 0.351591
\(525\) 0 0
\(526\) −24435.6 −2.02556
\(527\) −254.024 −0.0209971
\(528\) 0 0
\(529\) 15872.3 1.30453
\(530\) 26483.9 2.17054
\(531\) 0 0
\(532\) 0 0
\(533\) 38900.1 3.16126
\(534\) 0 0
\(535\) −17070.8 −1.37951
\(536\) 44825.6 3.61226
\(537\) 0 0
\(538\) −2157.93 −0.172927
\(539\) 0 0
\(540\) 0 0
\(541\) 4866.90 0.386774 0.193387 0.981123i \(-0.438053\pi\)
0.193387 + 0.981123i \(0.438053\pi\)
\(542\) −27145.5 −2.15129
\(543\) 0 0
\(544\) −3336.18 −0.262936
\(545\) −9239.55 −0.726199
\(546\) 0 0
\(547\) 6870.41 0.537034 0.268517 0.963275i \(-0.413467\pi\)
0.268517 + 0.963275i \(0.413467\pi\)
\(548\) −35342.9 −2.75506
\(549\) 0 0
\(550\) −25787.4 −1.99923
\(551\) −725.883 −0.0561228
\(552\) 0 0
\(553\) 0 0
\(554\) 14726.0 1.12933
\(555\) 0 0
\(556\) −36424.2 −2.77829
\(557\) −19795.9 −1.50589 −0.752943 0.658086i \(-0.771366\pi\)
−0.752943 + 0.658086i \(0.771366\pi\)
\(558\) 0 0
\(559\) 4763.42 0.360414
\(560\) 0 0
\(561\) 0 0
\(562\) 44296.1 3.32477
\(563\) −6981.58 −0.522626 −0.261313 0.965254i \(-0.584156\pi\)
−0.261313 + 0.965254i \(0.584156\pi\)
\(564\) 0 0
\(565\) −23639.1 −1.76019
\(566\) −18983.4 −1.40977
\(567\) 0 0
\(568\) −489.378 −0.0361512
\(569\) −3412.25 −0.251404 −0.125702 0.992068i \(-0.540118\pi\)
−0.125702 + 0.992068i \(0.540118\pi\)
\(570\) 0 0
\(571\) 20647.3 1.51325 0.756624 0.653850i \(-0.226847\pi\)
0.756624 + 0.653850i \(0.226847\pi\)
\(572\) −50544.9 −3.69473
\(573\) 0 0
\(574\) 0 0
\(575\) 27586.8 2.00078
\(576\) 0 0
\(577\) −12938.7 −0.933530 −0.466765 0.884381i \(-0.654581\pi\)
−0.466765 + 0.884381i \(0.654581\pi\)
\(578\) −24407.5 −1.75643
\(579\) 0 0
\(580\) 42753.7 3.06078
\(581\) 0 0
\(582\) 0 0
\(583\) 9157.75 0.650558
\(584\) −1198.22 −0.0849019
\(585\) 0 0
\(586\) −10100.7 −0.712038
\(587\) −14032.4 −0.986676 −0.493338 0.869838i \(-0.664223\pi\)
−0.493338 + 0.869838i \(0.664223\pi\)
\(588\) 0 0
\(589\) 101.742 0.00711750
\(590\) 36221.2 2.52746
\(591\) 0 0
\(592\) 53527.2 3.71614
\(593\) −23344.9 −1.61663 −0.808313 0.588752i \(-0.799619\pi\)
−0.808313 + 0.588752i \(0.799619\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 28231.7 1.94029
\(597\) 0 0
\(598\) 77336.7 5.28852
\(599\) 21813.4 1.48793 0.743966 0.668217i \(-0.232942\pi\)
0.743966 + 0.668217i \(0.232942\pi\)
\(600\) 0 0
\(601\) −1010.28 −0.0685690 −0.0342845 0.999412i \(-0.510915\pi\)
−0.0342845 + 0.999412i \(0.510915\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −33750.4 −2.27365
\(605\) 6973.71 0.468631
\(606\) 0 0
\(607\) 1763.19 0.117901 0.0589504 0.998261i \(-0.481225\pi\)
0.0589504 + 0.998261i \(0.481225\pi\)
\(608\) 1336.21 0.0891288
\(609\) 0 0
\(610\) −50144.7 −3.32836
\(611\) 13901.4 0.920440
\(612\) 0 0
\(613\) 264.802 0.0174474 0.00872368 0.999962i \(-0.497223\pi\)
0.00872368 + 0.999962i \(0.497223\pi\)
\(614\) 29251.4 1.92263
\(615\) 0 0
\(616\) 0 0
\(617\) −17930.8 −1.16996 −0.584980 0.811048i \(-0.698898\pi\)
−0.584980 + 0.811048i \(0.698898\pi\)
\(618\) 0 0
\(619\) 19581.1 1.27146 0.635729 0.771912i \(-0.280700\pi\)
0.635729 + 0.771912i \(0.280700\pi\)
\(620\) −5992.49 −0.388168
\(621\) 0 0
\(622\) −6558.07 −0.422757
\(623\) 0 0
\(624\) 0 0
\(625\) −9076.78 −0.580914
\(626\) 25105.6 1.60291
\(627\) 0 0
\(628\) 34561.2 2.19609
\(629\) −5400.91 −0.342366
\(630\) 0 0
\(631\) 13595.9 0.857754 0.428877 0.903363i \(-0.358909\pi\)
0.428877 + 0.903363i \(0.358909\pi\)
\(632\) 33838.6 2.12979
\(633\) 0 0
\(634\) 16626.7 1.04153
\(635\) 39661.1 2.47859
\(636\) 0 0
\(637\) 0 0
\(638\) 21144.4 1.31209
\(639\) 0 0
\(640\) 14673.8 0.906301
\(641\) −9626.58 −0.593178 −0.296589 0.955005i \(-0.595849\pi\)
−0.296589 + 0.955005i \(0.595849\pi\)
\(642\) 0 0
\(643\) −26571.5 −1.62967 −0.814834 0.579695i \(-0.803172\pi\)
−0.814834 + 0.579695i \(0.803172\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −371.778 −0.0226431
\(647\) 29729.0 1.80644 0.903220 0.429179i \(-0.141197\pi\)
0.903220 + 0.429179i \(0.141197\pi\)
\(648\) 0 0
\(649\) 12524.8 0.757535
\(650\) 76088.7 4.59145
\(651\) 0 0
\(652\) 16177.9 0.971742
\(653\) −16796.1 −1.00656 −0.503278 0.864125i \(-0.667873\pi\)
−0.503278 + 0.864125i \(0.667873\pi\)
\(654\) 0 0
\(655\) −3860.88 −0.230316
\(656\) 57753.6 3.43734
\(657\) 0 0
\(658\) 0 0
\(659\) −2078.96 −0.122891 −0.0614453 0.998110i \(-0.519571\pi\)
−0.0614453 + 0.998110i \(0.519571\pi\)
\(660\) 0 0
\(661\) 6600.40 0.388390 0.194195 0.980963i \(-0.437791\pi\)
0.194195 + 0.980963i \(0.437791\pi\)
\(662\) 27379.2 1.60744
\(663\) 0 0
\(664\) 14177.9 0.828629
\(665\) 0 0
\(666\) 0 0
\(667\) −22619.8 −1.31311
\(668\) −32522.2 −1.88372
\(669\) 0 0
\(670\) −72027.9 −4.15325
\(671\) −17339.3 −0.997580
\(672\) 0 0
\(673\) −21644.8 −1.23974 −0.619871 0.784704i \(-0.712815\pi\)
−0.619871 + 0.784704i \(0.712815\pi\)
\(674\) −11883.2 −0.679113
\(675\) 0 0
\(676\) 108289. 6.16118
\(677\) −349.464 −0.0198390 −0.00991949 0.999951i \(-0.503158\pi\)
−0.00991949 + 0.999951i \(0.503158\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 12475.8 0.703565
\(681\) 0 0
\(682\) −2963.67 −0.166400
\(683\) −14807.9 −0.829589 −0.414795 0.909915i \(-0.636147\pi\)
−0.414795 + 0.909915i \(0.636147\pi\)
\(684\) 0 0
\(685\) 32355.9 1.80475
\(686\) 0 0
\(687\) 0 0
\(688\) 7072.08 0.391890
\(689\) −27021.0 −1.49408
\(690\) 0 0
\(691\) −21679.4 −1.19352 −0.596760 0.802420i \(-0.703546\pi\)
−0.596760 + 0.802420i \(0.703546\pi\)
\(692\) −731.415 −0.0401795
\(693\) 0 0
\(694\) −26796.2 −1.46566
\(695\) 33345.8 1.81997
\(696\) 0 0
\(697\) −5827.35 −0.316681
\(698\) 46569.7 2.52534
\(699\) 0 0
\(700\) 0 0
\(701\) 5065.92 0.272949 0.136474 0.990644i \(-0.456423\pi\)
0.136474 + 0.990644i \(0.456423\pi\)
\(702\) 0 0
\(703\) 2163.18 0.116054
\(704\) −6634.99 −0.355207
\(705\) 0 0
\(706\) 45821.5 2.44265
\(707\) 0 0
\(708\) 0 0
\(709\) −16981.8 −0.899527 −0.449764 0.893148i \(-0.648492\pi\)
−0.449764 + 0.893148i \(0.648492\pi\)
\(710\) 786.356 0.0415654
\(711\) 0 0
\(712\) 26458.3 1.39265
\(713\) 3170.46 0.166528
\(714\) 0 0
\(715\) 46273.1 2.42030
\(716\) −49283.3 −2.57235
\(717\) 0 0
\(718\) −192.343 −0.00999746
\(719\) 31450.6 1.63130 0.815652 0.578543i \(-0.196378\pi\)
0.815652 + 0.578543i \(0.196378\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −35222.1 −1.81556
\(723\) 0 0
\(724\) −50568.7 −2.59581
\(725\) −22254.8 −1.14003
\(726\) 0 0
\(727\) −14930.2 −0.761665 −0.380833 0.924644i \(-0.624363\pi\)
−0.380833 + 0.924644i \(0.624363\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1925.36 0.0976174
\(731\) −713.575 −0.0361047
\(732\) 0 0
\(733\) −26799.5 −1.35043 −0.675213 0.737623i \(-0.735948\pi\)
−0.675213 + 0.737623i \(0.735948\pi\)
\(734\) −17202.3 −0.865050
\(735\) 0 0
\(736\) 41638.6 2.08535
\(737\) −24906.2 −1.24482
\(738\) 0 0
\(739\) 22511.2 1.12055 0.560275 0.828307i \(-0.310695\pi\)
0.560275 + 0.828307i \(0.310695\pi\)
\(740\) −127409. −6.32924
\(741\) 0 0
\(742\) 0 0
\(743\) −23783.0 −1.17431 −0.587155 0.809474i \(-0.699752\pi\)
−0.587155 + 0.809474i \(0.699752\pi\)
\(744\) 0 0
\(745\) −25845.7 −1.27102
\(746\) 20296.5 0.996121
\(747\) 0 0
\(748\) 7571.78 0.370123
\(749\) 0 0
\(750\) 0 0
\(751\) −6729.39 −0.326976 −0.163488 0.986545i \(-0.552274\pi\)
−0.163488 + 0.986545i \(0.552274\pi\)
\(752\) 20638.8 1.00083
\(753\) 0 0
\(754\) −62389.0 −3.01336
\(755\) 30898.0 1.48939
\(756\) 0 0
\(757\) 34646.1 1.66345 0.831726 0.555187i \(-0.187353\pi\)
0.831726 + 0.555187i \(0.187353\pi\)
\(758\) 11902.1 0.570321
\(759\) 0 0
\(760\) −4996.80 −0.238491
\(761\) −24096.3 −1.14782 −0.573908 0.818919i \(-0.694573\pi\)
−0.573908 + 0.818919i \(0.694573\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 80676.6 3.82039
\(765\) 0 0
\(766\) 15118.0 0.713100
\(767\) −36955.8 −1.73976
\(768\) 0 0
\(769\) 28099.8 1.31769 0.658846 0.752278i \(-0.271045\pi\)
0.658846 + 0.752278i \(0.271045\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18178.5 0.847487
\(773\) −9455.98 −0.439984 −0.219992 0.975502i \(-0.570603\pi\)
−0.219992 + 0.975502i \(0.570603\pi\)
\(774\) 0 0
\(775\) 3119.30 0.144579
\(776\) 13807.3 0.638727
\(777\) 0 0
\(778\) −39244.5 −1.80846
\(779\) 2333.98 0.107347
\(780\) 0 0
\(781\) 271.911 0.0124580
\(782\) −11585.3 −0.529781
\(783\) 0 0
\(784\) 0 0
\(785\) −31640.2 −1.43858
\(786\) 0 0
\(787\) 16846.0 0.763017 0.381509 0.924365i \(-0.375405\pi\)
0.381509 + 0.924365i \(0.375405\pi\)
\(788\) −28937.3 −1.30818
\(789\) 0 0
\(790\) −54373.5 −2.44876
\(791\) 0 0
\(792\) 0 0
\(793\) 51161.6 2.29105
\(794\) 22018.5 0.984140
\(795\) 0 0
\(796\) −32040.4 −1.42669
\(797\) 35955.5 1.59801 0.799003 0.601327i \(-0.205361\pi\)
0.799003 + 0.601327i \(0.205361\pi\)
\(798\) 0 0
\(799\) −2082.47 −0.0922057
\(800\) 40966.6 1.81049
\(801\) 0 0
\(802\) −57389.7 −2.52681
\(803\) 665.761 0.0292580
\(804\) 0 0
\(805\) 0 0
\(806\) 8744.64 0.382155
\(807\) 0 0
\(808\) −34382.1 −1.49698
\(809\) 13680.6 0.594543 0.297272 0.954793i \(-0.403923\pi\)
0.297272 + 0.954793i \(0.403923\pi\)
\(810\) 0 0
\(811\) −10215.1 −0.442294 −0.221147 0.975241i \(-0.570980\pi\)
−0.221147 + 0.975241i \(0.570980\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −63011.7 −2.71322
\(815\) −14810.6 −0.636557
\(816\) 0 0
\(817\) 285.802 0.0122386
\(818\) −37234.6 −1.59154
\(819\) 0 0
\(820\) −137469. −5.85440
\(821\) 14255.6 0.605996 0.302998 0.952991i \(-0.402012\pi\)
0.302998 + 0.952991i \(0.402012\pi\)
\(822\) 0 0
\(823\) −31349.9 −1.32781 −0.663905 0.747817i \(-0.731102\pi\)
−0.663905 + 0.747817i \(0.731102\pi\)
\(824\) −64054.3 −2.70805
\(825\) 0 0
\(826\) 0 0
\(827\) −6114.87 −0.257116 −0.128558 0.991702i \(-0.541035\pi\)
−0.128558 + 0.991702i \(0.541035\pi\)
\(828\) 0 0
\(829\) −32418.4 −1.35819 −0.679093 0.734052i \(-0.737627\pi\)
−0.679093 + 0.734052i \(0.737627\pi\)
\(830\) −22781.7 −0.952730
\(831\) 0 0
\(832\) 19577.3 0.815770
\(833\) 0 0
\(834\) 0 0
\(835\) 29773.6 1.23396
\(836\) −3032.65 −0.125462
\(837\) 0 0
\(838\) 85765.1 3.53545
\(839\) 44683.5 1.83867 0.919336 0.393474i \(-0.128727\pi\)
0.919336 + 0.393474i \(0.128727\pi\)
\(840\) 0 0
\(841\) −6141.15 −0.251800
\(842\) −3375.17 −0.138142
\(843\) 0 0
\(844\) 24082.0 0.982151
\(845\) −99136.8 −4.03599
\(846\) 0 0
\(847\) 0 0
\(848\) −40117.1 −1.62456
\(849\) 0 0
\(850\) −11398.3 −0.459952
\(851\) 67408.4 2.71531
\(852\) 0 0
\(853\) 9965.17 0.400001 0.200001 0.979796i \(-0.435906\pi\)
0.200001 + 0.979796i \(0.435906\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 54785.7 2.18754
\(857\) −8639.89 −0.344379 −0.172190 0.985064i \(-0.555084\pi\)
−0.172190 + 0.985064i \(0.555084\pi\)
\(858\) 0 0
\(859\) −15551.5 −0.617707 −0.308853 0.951110i \(-0.599945\pi\)
−0.308853 + 0.951110i \(0.599945\pi\)
\(860\) −16833.4 −0.667458
\(861\) 0 0
\(862\) −15728.0 −0.621460
\(863\) −47746.4 −1.88332 −0.941661 0.336563i \(-0.890735\pi\)
−0.941661 + 0.336563i \(0.890735\pi\)
\(864\) 0 0
\(865\) 669.599 0.0263203
\(866\) 8247.97 0.323646
\(867\) 0 0
\(868\) 0 0
\(869\) −18801.6 −0.733946
\(870\) 0 0
\(871\) 73488.6 2.85886
\(872\) 29652.6 1.15156
\(873\) 0 0
\(874\) 4640.14 0.179583
\(875\) 0 0
\(876\) 0 0
\(877\) −37958.8 −1.46155 −0.730773 0.682620i \(-0.760840\pi\)
−0.730773 + 0.682620i \(0.760840\pi\)
\(878\) 81962.8 3.15047
\(879\) 0 0
\(880\) 68699.9 2.63168
\(881\) −32323.6 −1.23611 −0.618053 0.786136i \(-0.712078\pi\)
−0.618053 + 0.786136i \(0.712078\pi\)
\(882\) 0 0
\(883\) 27760.0 1.05798 0.528991 0.848628i \(-0.322571\pi\)
0.528991 + 0.848628i \(0.322571\pi\)
\(884\) −22341.4 −0.850026
\(885\) 0 0
\(886\) −31767.5 −1.20457
\(887\) 32759.6 1.24009 0.620044 0.784567i \(-0.287115\pi\)
0.620044 + 0.784567i \(0.287115\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −42514.5 −1.60122
\(891\) 0 0
\(892\) 8991.97 0.337526
\(893\) 834.071 0.0312554
\(894\) 0 0
\(895\) 45118.1 1.68506
\(896\) 0 0
\(897\) 0 0
\(898\) 45083.8 1.67535
\(899\) −2557.67 −0.0948868
\(900\) 0 0
\(901\) 4047.83 0.149670
\(902\) −67986.9 −2.50966
\(903\) 0 0
\(904\) 75865.3 2.79120
\(905\) 46294.8 1.70043
\(906\) 0 0
\(907\) −42682.6 −1.56257 −0.781286 0.624174i \(-0.785436\pi\)
−0.781286 + 0.624174i \(0.785436\pi\)
\(908\) 74742.8 2.73175
\(909\) 0 0
\(910\) 0 0
\(911\) −50130.0 −1.82314 −0.911570 0.411144i \(-0.865129\pi\)
−0.911570 + 0.411144i \(0.865129\pi\)
\(912\) 0 0
\(913\) −7877.60 −0.285554
\(914\) −2991.57 −0.108263
\(915\) 0 0
\(916\) −94275.3 −3.40059
\(917\) 0 0
\(918\) 0 0
\(919\) −33784.7 −1.21268 −0.606341 0.795205i \(-0.707363\pi\)
−0.606341 + 0.795205i \(0.707363\pi\)
\(920\) −155709. −5.57998
\(921\) 0 0
\(922\) −10092.3 −0.360489
\(923\) −802.303 −0.0286112
\(924\) 0 0
\(925\) 66320.6 2.35742
\(926\) 41831.5 1.48452
\(927\) 0 0
\(928\) −33590.7 −1.18822
\(929\) 50854.8 1.79601 0.898004 0.439987i \(-0.145017\pi\)
0.898004 + 0.439987i \(0.145017\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −82150.4 −2.88726
\(933\) 0 0
\(934\) −6396.01 −0.224073
\(935\) −6931.85 −0.242455
\(936\) 0 0
\(937\) 3684.00 0.128443 0.0642215 0.997936i \(-0.479544\pi\)
0.0642215 + 0.997936i \(0.479544\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −49125.8 −1.70458
\(941\) 28660.5 0.992886 0.496443 0.868069i \(-0.334639\pi\)
0.496443 + 0.868069i \(0.334639\pi\)
\(942\) 0 0
\(943\) 72730.8 2.51160
\(944\) −54866.9 −1.89170
\(945\) 0 0
\(946\) −8325.18 −0.286126
\(947\) 31873.0 1.09370 0.546849 0.837231i \(-0.315827\pi\)
0.546849 + 0.837231i \(0.315827\pi\)
\(948\) 0 0
\(949\) −1964.40 −0.0671942
\(950\) 4565.26 0.155912
\(951\) 0 0
\(952\) 0 0
\(953\) −3310.08 −0.112512 −0.0562561 0.998416i \(-0.517916\pi\)
−0.0562561 + 0.998416i \(0.517916\pi\)
\(954\) 0 0
\(955\) −73858.2 −2.50261
\(956\) −60022.3 −2.03061
\(957\) 0 0
\(958\) −95943.6 −3.23570
\(959\) 0 0
\(960\) 0 0
\(961\) −29432.5 −0.987966
\(962\) 185923. 6.23119
\(963\) 0 0
\(964\) 28580.6 0.954897
\(965\) −16642.2 −0.555161
\(966\) 0 0
\(967\) −168.413 −0.00560061 −0.00280031 0.999996i \(-0.500891\pi\)
−0.00280031 + 0.999996i \(0.500891\pi\)
\(968\) −22380.8 −0.743127
\(969\) 0 0
\(970\) −22186.2 −0.734387
\(971\) −37669.9 −1.24499 −0.622495 0.782624i \(-0.713881\pi\)
−0.622495 + 0.782624i \(0.713881\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4509.33 0.148345
\(975\) 0 0
\(976\) 75957.7 2.49114
\(977\) 26688.7 0.873948 0.436974 0.899474i \(-0.356050\pi\)
0.436974 + 0.899474i \(0.356050\pi\)
\(978\) 0 0
\(979\) −14700.9 −0.479921
\(980\) 0 0
\(981\) 0 0
\(982\) 30167.5 0.980329
\(983\) 1102.29 0.0357655 0.0178828 0.999840i \(-0.494307\pi\)
0.0178828 + 0.999840i \(0.494307\pi\)
\(984\) 0 0
\(985\) 26491.6 0.856948
\(986\) 9346.07 0.301866
\(987\) 0 0
\(988\) 8948.19 0.288137
\(989\) 8906.09 0.286347
\(990\) 0 0
\(991\) 16523.6 0.529656 0.264828 0.964296i \(-0.414685\pi\)
0.264828 + 0.964296i \(0.414685\pi\)
\(992\) 4708.17 0.150690
\(993\) 0 0
\(994\) 0 0
\(995\) 29332.5 0.934575
\(996\) 0 0
\(997\) −17994.5 −0.571605 −0.285802 0.958289i \(-0.592260\pi\)
−0.285802 + 0.958289i \(0.592260\pi\)
\(998\) −5650.72 −0.179229
\(999\) 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 1323.4.a.ba.1.4 4
3.2 odd 2 inner 1323.4.a.ba.1.1 4
7.6 odd 2 189.4.a.l.1.4 yes 4
21.20 even 2 189.4.a.l.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.l.1.1 4 21.20 even 2
189.4.a.l.1.4 yes 4 7.6 odd 2
1323.4.a.ba.1.1 4 3.2 odd 2 inner
1323.4.a.ba.1.4 4 1.1 even 1 trivial