Properties

Label 882.4.a.v.1.1
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1345}) \)
Defining polynomial: \(x^{2} - x - 336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(18.8371\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -20.8371 q^{5} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -20.8371 q^{5} -8.00000 q^{8} +41.6742 q^{10} -15.1629 q^{11} +2.16288 q^{13} +16.0000 q^{16} +119.348 q^{17} -33.5114 q^{19} -83.3485 q^{20} +30.3258 q^{22} -0.651517 q^{23} +309.186 q^{25} -4.32576 q^{26} +163.208 q^{29} -223.326 q^{31} -32.0000 q^{32} -238.697 q^{34} +168.534 q^{37} +67.0227 q^{38} +166.697 q^{40} +323.023 q^{41} +221.557 q^{43} -60.6515 q^{44} +1.30303 q^{46} -508.045 q^{47} -618.371 q^{50} +8.65152 q^{52} +176.511 q^{53} +315.951 q^{55} -326.417 q^{58} -454.928 q^{59} +38.6515 q^{61} +446.652 q^{62} +64.0000 q^{64} -45.0682 q^{65} +141.792 q^{67} +477.394 q^{68} -602.742 q^{71} -1102.30 q^{73} -337.068 q^{74} -134.045 q^{76} -116.303 q^{79} -333.394 q^{80} -646.045 q^{82} +568.928 q^{83} -2486.88 q^{85} -443.114 q^{86} +121.303 q^{88} +383.159 q^{89} -2.60607 q^{92} +1016.09 q^{94} +698.280 q^{95} +334.701 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 5q^{5} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 5q^{5} - 16q^{8} + 10q^{10} - 67q^{11} + 41q^{13} + 32q^{16} + 92q^{17} + 43q^{19} - 20q^{20} + 134q^{22} - 148q^{23} + 435q^{25} - 82q^{26} - 77q^{29} - 520q^{31} - 64q^{32} - 184q^{34} + 7q^{37} - 86q^{38} + 40q^{40} + 426q^{41} - 107q^{43} - 268q^{44} + 296q^{46} - 576q^{47} - 870q^{50} + 164q^{52} + 243q^{53} - 505q^{55} + 154q^{58} + 7q^{59} + 224q^{61} + 1040q^{62} + 128q^{64} + 570q^{65} + 687q^{67} + 368q^{68} - 472q^{71} - 921q^{73} - 14q^{74} + 172q^{76} - 526q^{79} - 80q^{80} - 852q^{82} + 221q^{83} - 2920q^{85} + 214q^{86} + 536q^{88} - 774q^{89} - 592q^{92} + 1152q^{94} + 1910q^{95} + 1953q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −20.8371 −1.86373 −0.931864 0.362807i \(-0.881818\pi\)
−0.931864 + 0.362807i \(0.881818\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 41.6742 1.31786
\(11\) −15.1629 −0.415616 −0.207808 0.978170i \(-0.566633\pi\)
−0.207808 + 0.978170i \(0.566633\pi\)
\(12\) 0 0
\(13\) 2.16288 0.0461442 0.0230721 0.999734i \(-0.492655\pi\)
0.0230721 + 0.999734i \(0.492655\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 119.348 1.70272 0.851361 0.524581i \(-0.175778\pi\)
0.851361 + 0.524581i \(0.175778\pi\)
\(18\) 0 0
\(19\) −33.5114 −0.404633 −0.202317 0.979320i \(-0.564847\pi\)
−0.202317 + 0.979320i \(0.564847\pi\)
\(20\) −83.3485 −0.931864
\(21\) 0 0
\(22\) 30.3258 0.293885
\(23\) −0.651517 −0.00590655 −0.00295327 0.999996i \(-0.500940\pi\)
−0.00295327 + 0.999996i \(0.500940\pi\)
\(24\) 0 0
\(25\) 309.186 2.47348
\(26\) −4.32576 −0.0326289
\(27\) 0 0
\(28\) 0 0
\(29\) 163.208 1.04507 0.522535 0.852618i \(-0.324986\pi\)
0.522535 + 0.852618i \(0.324986\pi\)
\(30\) 0 0
\(31\) −223.326 −1.29389 −0.646943 0.762538i \(-0.723953\pi\)
−0.646943 + 0.762538i \(0.723953\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −238.697 −1.20401
\(35\) 0 0
\(36\) 0 0
\(37\) 168.534 0.748833 0.374417 0.927261i \(-0.377843\pi\)
0.374417 + 0.927261i \(0.377843\pi\)
\(38\) 67.0227 0.286119
\(39\) 0 0
\(40\) 166.697 0.658928
\(41\) 323.023 1.23043 0.615216 0.788359i \(-0.289069\pi\)
0.615216 + 0.788359i \(0.289069\pi\)
\(42\) 0 0
\(43\) 221.557 0.785746 0.392873 0.919593i \(-0.371481\pi\)
0.392873 + 0.919593i \(0.371481\pi\)
\(44\) −60.6515 −0.207808
\(45\) 0 0
\(46\) 1.30303 0.00417656
\(47\) −508.045 −1.57672 −0.788362 0.615211i \(-0.789071\pi\)
−0.788362 + 0.615211i \(0.789071\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −618.371 −1.74902
\(51\) 0 0
\(52\) 8.65152 0.0230721
\(53\) 176.511 0.457466 0.228733 0.973489i \(-0.426542\pi\)
0.228733 + 0.973489i \(0.426542\pi\)
\(54\) 0 0
\(55\) 315.951 0.774596
\(56\) 0 0
\(57\) 0 0
\(58\) −326.417 −0.738976
\(59\) −454.928 −1.00384 −0.501920 0.864914i \(-0.667373\pi\)
−0.501920 + 0.864914i \(0.667373\pi\)
\(60\) 0 0
\(61\) 38.6515 0.0811282 0.0405641 0.999177i \(-0.487085\pi\)
0.0405641 + 0.999177i \(0.487085\pi\)
\(62\) 446.652 0.914916
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −45.0682 −0.0860003
\(66\) 0 0
\(67\) 141.792 0.258546 0.129273 0.991609i \(-0.458736\pi\)
0.129273 + 0.991609i \(0.458736\pi\)
\(68\) 477.394 0.851361
\(69\) 0 0
\(70\) 0 0
\(71\) −602.742 −1.00750 −0.503749 0.863850i \(-0.668046\pi\)
−0.503749 + 0.863850i \(0.668046\pi\)
\(72\) 0 0
\(73\) −1102.30 −1.76732 −0.883660 0.468129i \(-0.844928\pi\)
−0.883660 + 0.468129i \(0.844928\pi\)
\(74\) −337.068 −0.529505
\(75\) 0 0
\(76\) −134.045 −0.202317
\(77\) 0 0
\(78\) 0 0
\(79\) −116.303 −0.165634 −0.0828172 0.996565i \(-0.526392\pi\)
−0.0828172 + 0.996565i \(0.526392\pi\)
\(80\) −333.394 −0.465932
\(81\) 0 0
\(82\) −646.045 −0.870046
\(83\) 568.928 0.752385 0.376193 0.926542i \(-0.377233\pi\)
0.376193 + 0.926542i \(0.377233\pi\)
\(84\) 0 0
\(85\) −2486.88 −3.17341
\(86\) −443.114 −0.555607
\(87\) 0 0
\(88\) 121.303 0.146943
\(89\) 383.159 0.456346 0.228173 0.973621i \(-0.426725\pi\)
0.228173 + 0.973621i \(0.426725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.60607 −0.00295327
\(93\) 0 0
\(94\) 1016.09 1.11491
\(95\) 698.280 0.754127
\(96\) 0 0
\(97\) 334.701 0.350348 0.175174 0.984538i \(-0.443951\pi\)
0.175174 + 0.984538i \(0.443951\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1236.74 1.23674
\(101\) 14.7424 0.0145240 0.00726201 0.999974i \(-0.497688\pi\)
0.00726201 + 0.999974i \(0.497688\pi\)
\(102\) 0 0
\(103\) −841.420 −0.804928 −0.402464 0.915436i \(-0.631846\pi\)
−0.402464 + 0.915436i \(0.631846\pi\)
\(104\) −17.3030 −0.0163144
\(105\) 0 0
\(106\) −353.023 −0.323477
\(107\) 715.670 0.646603 0.323301 0.946296i \(-0.395207\pi\)
0.323301 + 0.946296i \(0.395207\pi\)
\(108\) 0 0
\(109\) 600.019 0.527260 0.263630 0.964624i \(-0.415080\pi\)
0.263630 + 0.964624i \(0.415080\pi\)
\(110\) −631.901 −0.547722
\(111\) 0 0
\(112\) 0 0
\(113\) −622.644 −0.518349 −0.259174 0.965831i \(-0.583450\pi\)
−0.259174 + 0.965831i \(0.583450\pi\)
\(114\) 0 0
\(115\) 13.5757 0.0110082
\(116\) 652.833 0.522535
\(117\) 0 0
\(118\) 909.856 0.709822
\(119\) 0 0
\(120\) 0 0
\(121\) −1101.09 −0.827263
\(122\) −77.3030 −0.0573663
\(123\) 0 0
\(124\) −893.303 −0.646943
\(125\) −3837.90 −2.74618
\(126\) 0 0
\(127\) −180.076 −0.125820 −0.0629100 0.998019i \(-0.520038\pi\)
−0.0629100 + 0.998019i \(0.520038\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 90.1363 0.0608114
\(131\) −217.860 −0.145302 −0.0726508 0.997357i \(-0.523146\pi\)
−0.0726508 + 0.997357i \(0.523146\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −283.583 −0.182820
\(135\) 0 0
\(136\) −954.788 −0.602003
\(137\) 2601.86 1.62257 0.811283 0.584654i \(-0.198770\pi\)
0.811283 + 0.584654i \(0.198770\pi\)
\(138\) 0 0
\(139\) −2651.55 −1.61800 −0.808998 0.587811i \(-0.799990\pi\)
−0.808998 + 0.587811i \(0.799990\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1205.48 0.712409
\(143\) −32.7955 −0.0191783
\(144\) 0 0
\(145\) −3400.79 −1.94773
\(146\) 2204.60 1.24968
\(147\) 0 0
\(148\) 674.136 0.374417
\(149\) −581.023 −0.319458 −0.159729 0.987161i \(-0.551062\pi\)
−0.159729 + 0.987161i \(0.551062\pi\)
\(150\) 0 0
\(151\) −615.390 −0.331654 −0.165827 0.986155i \(-0.553029\pi\)
−0.165827 + 0.986155i \(0.553029\pi\)
\(152\) 268.091 0.143059
\(153\) 0 0
\(154\) 0 0
\(155\) 4653.47 2.41145
\(156\) 0 0
\(157\) −306.932 −0.156024 −0.0780122 0.996952i \(-0.524857\pi\)
−0.0780122 + 0.996952i \(0.524857\pi\)
\(158\) 232.606 0.117121
\(159\) 0 0
\(160\) 666.788 0.329464
\(161\) 0 0
\(162\) 0 0
\(163\) 3514.50 1.68882 0.844408 0.535701i \(-0.179953\pi\)
0.844408 + 0.535701i \(0.179953\pi\)
\(164\) 1292.09 0.615216
\(165\) 0 0
\(166\) −1137.86 −0.532017
\(167\) −1123.30 −0.520502 −0.260251 0.965541i \(-0.583805\pi\)
−0.260251 + 0.965541i \(0.583805\pi\)
\(168\) 0 0
\(169\) −2192.32 −0.997871
\(170\) 4973.76 2.24394
\(171\) 0 0
\(172\) 886.227 0.392873
\(173\) −1530.60 −0.672655 −0.336327 0.941745i \(-0.609185\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −242.606 −0.103904
\(177\) 0 0
\(178\) −766.318 −0.322685
\(179\) −3413.43 −1.42532 −0.712659 0.701511i \(-0.752509\pi\)
−0.712659 + 0.701511i \(0.752509\pi\)
\(180\) 0 0
\(181\) 1286.71 0.528399 0.264200 0.964468i \(-0.414892\pi\)
0.264200 + 0.964468i \(0.414892\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.21213 0.00208828
\(185\) −3511.77 −1.39562
\(186\) 0 0
\(187\) −1809.67 −0.707679
\(188\) −2032.18 −0.788362
\(189\) 0 0
\(190\) −1396.56 −0.533248
\(191\) −1055.30 −0.399783 −0.199891 0.979818i \(-0.564059\pi\)
−0.199891 + 0.979818i \(0.564059\pi\)
\(192\) 0 0
\(193\) −4770.84 −1.77934 −0.889670 0.456604i \(-0.849066\pi\)
−0.889670 + 0.456604i \(0.849066\pi\)
\(194\) −669.402 −0.247733
\(195\) 0 0
\(196\) 0 0
\(197\) −1622.31 −0.586725 −0.293363 0.956001i \(-0.594774\pi\)
−0.293363 + 0.956001i \(0.594774\pi\)
\(198\) 0 0
\(199\) −3550.14 −1.26464 −0.632318 0.774709i \(-0.717896\pi\)
−0.632318 + 0.774709i \(0.717896\pi\)
\(200\) −2473.48 −0.874509
\(201\) 0 0
\(202\) −29.4848 −0.0102700
\(203\) 0 0
\(204\) 0 0
\(205\) −6730.86 −2.29319
\(206\) 1682.84 0.569170
\(207\) 0 0
\(208\) 34.6061 0.0115361
\(209\) 508.129 0.168172
\(210\) 0 0
\(211\) 4653.39 1.51826 0.759129 0.650941i \(-0.225625\pi\)
0.759129 + 0.650941i \(0.225625\pi\)
\(212\) 706.045 0.228733
\(213\) 0 0
\(214\) −1431.34 −0.457217
\(215\) −4616.61 −1.46442
\(216\) 0 0
\(217\) 0 0
\(218\) −1200.04 −0.372829
\(219\) 0 0
\(220\) 1263.80 0.387298
\(221\) 258.136 0.0785707
\(222\) 0 0
\(223\) −4649.53 −1.39621 −0.698107 0.715993i \(-0.745974\pi\)
−0.698107 + 0.715993i \(0.745974\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1245.29 0.366528
\(227\) −4151.72 −1.21392 −0.606958 0.794734i \(-0.707611\pi\)
−0.606958 + 0.794734i \(0.707611\pi\)
\(228\) 0 0
\(229\) 4263.63 1.23034 0.615172 0.788393i \(-0.289087\pi\)
0.615172 + 0.788393i \(0.289087\pi\)
\(230\) −27.1515 −0.00778398
\(231\) 0 0
\(232\) −1305.67 −0.369488
\(233\) −3049.90 −0.857535 −0.428768 0.903415i \(-0.641052\pi\)
−0.428768 + 0.903415i \(0.641052\pi\)
\(234\) 0 0
\(235\) 10586.2 2.93859
\(236\) −1819.71 −0.501920
\(237\) 0 0
\(238\) 0 0
\(239\) −3987.20 −1.07912 −0.539562 0.841946i \(-0.681410\pi\)
−0.539562 + 0.841946i \(0.681410\pi\)
\(240\) 0 0
\(241\) −624.648 −0.166959 −0.0834795 0.996509i \(-0.526603\pi\)
−0.0834795 + 0.996509i \(0.526603\pi\)
\(242\) 2202.17 0.584963
\(243\) 0 0
\(244\) 154.606 0.0405641
\(245\) 0 0
\(246\) 0 0
\(247\) −72.4810 −0.0186715
\(248\) 1786.61 0.457458
\(249\) 0 0
\(250\) 7675.80 1.94184
\(251\) 1328.78 0.334152 0.167076 0.985944i \(-0.446568\pi\)
0.167076 + 0.985944i \(0.446568\pi\)
\(252\) 0 0
\(253\) 9.87887 0.00245486
\(254\) 360.152 0.0889682
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 3226.18 0.783049 0.391525 0.920168i \(-0.371948\pi\)
0.391525 + 0.920168i \(0.371948\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −180.273 −0.0430001
\(261\) 0 0
\(262\) 435.720 0.102744
\(263\) −3250.61 −0.762135 −0.381067 0.924547i \(-0.624443\pi\)
−0.381067 + 0.924547i \(0.624443\pi\)
\(264\) 0 0
\(265\) −3677.99 −0.852593
\(266\) 0 0
\(267\) 0 0
\(268\) 567.167 0.129273
\(269\) −2826.04 −0.640546 −0.320273 0.947325i \(-0.603775\pi\)
−0.320273 + 0.947325i \(0.603775\pi\)
\(270\) 0 0
\(271\) −2396.77 −0.537245 −0.268622 0.963246i \(-0.586568\pi\)
−0.268622 + 0.963246i \(0.586568\pi\)
\(272\) 1909.58 0.425680
\(273\) 0 0
\(274\) −5203.71 −1.14733
\(275\) −4688.14 −1.02802
\(276\) 0 0
\(277\) 1820.47 0.394878 0.197439 0.980315i \(-0.436738\pi\)
0.197439 + 0.980315i \(0.436738\pi\)
\(278\) 5303.10 1.14410
\(279\) 0 0
\(280\) 0 0
\(281\) −3083.81 −0.654679 −0.327339 0.944907i \(-0.606152\pi\)
−0.327339 + 0.944907i \(0.606152\pi\)
\(282\) 0 0
\(283\) 2554.77 0.536626 0.268313 0.963332i \(-0.413534\pi\)
0.268313 + 0.963332i \(0.413534\pi\)
\(284\) −2410.97 −0.503749
\(285\) 0 0
\(286\) 65.5910 0.0135611
\(287\) 0 0
\(288\) 0 0
\(289\) 9331.06 1.89926
\(290\) 6801.58 1.37725
\(291\) 0 0
\(292\) −4409.20 −0.883660
\(293\) 1846.47 0.368163 0.184081 0.982911i \(-0.441069\pi\)
0.184081 + 0.982911i \(0.441069\pi\)
\(294\) 0 0
\(295\) 9479.39 1.87089
\(296\) −1348.27 −0.264753
\(297\) 0 0
\(298\) 1162.05 0.225891
\(299\) −1.40915 −0.000272553 0
\(300\) 0 0
\(301\) 0 0
\(302\) 1230.78 0.234515
\(303\) 0 0
\(304\) −536.182 −0.101158
\(305\) −805.386 −0.151201
\(306\) 0 0
\(307\) 7041.50 1.30905 0.654527 0.756039i \(-0.272868\pi\)
0.654527 + 0.756039i \(0.272868\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9306.93 −1.70516
\(311\) 2685.99 0.489738 0.244869 0.969556i \(-0.421255\pi\)
0.244869 + 0.969556i \(0.421255\pi\)
\(312\) 0 0
\(313\) 2219.19 0.400754 0.200377 0.979719i \(-0.435783\pi\)
0.200377 + 0.979719i \(0.435783\pi\)
\(314\) 613.864 0.110326
\(315\) 0 0
\(316\) −465.212 −0.0828172
\(317\) −2221.26 −0.393560 −0.196780 0.980448i \(-0.563048\pi\)
−0.196780 + 0.980448i \(0.563048\pi\)
\(318\) 0 0
\(319\) −2474.71 −0.434348
\(320\) −1333.58 −0.232966
\(321\) 0 0
\(322\) 0 0
\(323\) −3999.53 −0.688978
\(324\) 0 0
\(325\) 668.731 0.114137
\(326\) −7029.00 −1.19417
\(327\) 0 0
\(328\) −2584.18 −0.435023
\(329\) 0 0
\(330\) 0 0
\(331\) 4154.06 0.689812 0.344906 0.938637i \(-0.387911\pi\)
0.344906 + 0.938637i \(0.387911\pi\)
\(332\) 2275.71 0.376193
\(333\) 0 0
\(334\) 2246.61 0.368050
\(335\) −2954.53 −0.481860
\(336\) 0 0
\(337\) −254.167 −0.0410841 −0.0205420 0.999789i \(-0.506539\pi\)
−0.0205420 + 0.999789i \(0.506539\pi\)
\(338\) 4384.64 0.705601
\(339\) 0 0
\(340\) −9947.52 −1.58671
\(341\) 3386.26 0.537761
\(342\) 0 0
\(343\) 0 0
\(344\) −1772.45 −0.277803
\(345\) 0 0
\(346\) 3061.20 0.475639
\(347\) 6224.64 0.962986 0.481493 0.876450i \(-0.340095\pi\)
0.481493 + 0.876450i \(0.340095\pi\)
\(348\) 0 0
\(349\) 9732.21 1.49270 0.746352 0.665552i \(-0.231804\pi\)
0.746352 + 0.665552i \(0.231804\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 485.212 0.0734713
\(353\) −1425.61 −0.214951 −0.107476 0.994208i \(-0.534277\pi\)
−0.107476 + 0.994208i \(0.534277\pi\)
\(354\) 0 0
\(355\) 12559.4 1.87770
\(356\) 1532.64 0.228173
\(357\) 0 0
\(358\) 6826.86 1.00785
\(359\) 5766.49 0.847754 0.423877 0.905720i \(-0.360669\pi\)
0.423877 + 0.905720i \(0.360669\pi\)
\(360\) 0 0
\(361\) −5735.99 −0.836272
\(362\) −2573.42 −0.373635
\(363\) 0 0
\(364\) 0 0
\(365\) 22968.7 3.29381
\(366\) 0 0
\(367\) 11545.3 1.64213 0.821065 0.570834i \(-0.193380\pi\)
0.821065 + 0.570834i \(0.193380\pi\)
\(368\) −10.4243 −0.00147664
\(369\) 0 0
\(370\) 7023.53 0.986854
\(371\) 0 0
\(372\) 0 0
\(373\) −6479.57 −0.899463 −0.449731 0.893164i \(-0.648480\pi\)
−0.449731 + 0.893164i \(0.648480\pi\)
\(374\) 3619.33 0.500404
\(375\) 0 0
\(376\) 4064.36 0.557456
\(377\) 353.000 0.0482239
\(378\) 0 0
\(379\) 611.996 0.0829449 0.0414725 0.999140i \(-0.486795\pi\)
0.0414725 + 0.999140i \(0.486795\pi\)
\(380\) 2793.12 0.377063
\(381\) 0 0
\(382\) 2110.59 0.282689
\(383\) −4360.81 −0.581794 −0.290897 0.956754i \(-0.593954\pi\)
−0.290897 + 0.956754i \(0.593954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 9541.68 1.25818
\(387\) 0 0
\(388\) 1338.80 0.175174
\(389\) 13146.9 1.71356 0.856781 0.515681i \(-0.172461\pi\)
0.856781 + 0.515681i \(0.172461\pi\)
\(390\) 0 0
\(391\) −77.7575 −0.0100572
\(392\) 0 0
\(393\) 0 0
\(394\) 3244.62 0.414877
\(395\) 2423.42 0.308697
\(396\) 0 0
\(397\) −8478.04 −1.07179 −0.535895 0.844285i \(-0.680026\pi\)
−0.535895 + 0.844285i \(0.680026\pi\)
\(398\) 7100.27 0.894232
\(399\) 0 0
\(400\) 4946.97 0.618371
\(401\) −2803.00 −0.349065 −0.174533 0.984651i \(-0.555841\pi\)
−0.174533 + 0.984651i \(0.555841\pi\)
\(402\) 0 0
\(403\) −483.027 −0.0597054
\(404\) 58.9697 0.00726201
\(405\) 0 0
\(406\) 0 0
\(407\) −2555.46 −0.311227
\(408\) 0 0
\(409\) −6385.39 −0.771973 −0.385987 0.922504i \(-0.626139\pi\)
−0.385987 + 0.922504i \(0.626139\pi\)
\(410\) 13461.7 1.62153
\(411\) 0 0
\(412\) −3365.68 −0.402464
\(413\) 0 0
\(414\) 0 0
\(415\) −11854.8 −1.40224
\(416\) −69.2121 −0.00815722
\(417\) 0 0
\(418\) −1016.26 −0.118916
\(419\) −4831.66 −0.563346 −0.281673 0.959510i \(-0.590889\pi\)
−0.281673 + 0.959510i \(0.590889\pi\)
\(420\) 0 0
\(421\) 7475.37 0.865385 0.432693 0.901542i \(-0.357564\pi\)
0.432693 + 0.901542i \(0.357564\pi\)
\(422\) −9306.77 −1.07357
\(423\) 0 0
\(424\) −1412.09 −0.161739
\(425\) 36900.8 4.21165
\(426\) 0 0
\(427\) 0 0
\(428\) 2862.68 0.323301
\(429\) 0 0
\(430\) 9233.21 1.03550
\(431\) −6991.93 −0.781414 −0.390707 0.920515i \(-0.627769\pi\)
−0.390707 + 0.920515i \(0.627769\pi\)
\(432\) 0 0
\(433\) −7699.26 −0.854510 −0.427255 0.904131i \(-0.640519\pi\)
−0.427255 + 0.904131i \(0.640519\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2400.08 0.263630
\(437\) 21.8332 0.00238999
\(438\) 0 0
\(439\) 9412.32 1.02329 0.511646 0.859196i \(-0.329036\pi\)
0.511646 + 0.859196i \(0.329036\pi\)
\(440\) −2527.61 −0.273861
\(441\) 0 0
\(442\) −516.273 −0.0555579
\(443\) 6258.18 0.671185 0.335593 0.942007i \(-0.391063\pi\)
0.335593 + 0.942007i \(0.391063\pi\)
\(444\) 0 0
\(445\) −7983.93 −0.850505
\(446\) 9299.07 0.987273
\(447\) 0 0
\(448\) 0 0
\(449\) 11633.8 1.22279 0.611396 0.791325i \(-0.290608\pi\)
0.611396 + 0.791325i \(0.290608\pi\)
\(450\) 0 0
\(451\) −4897.95 −0.511387
\(452\) −2490.58 −0.259174
\(453\) 0 0
\(454\) 8303.43 0.858369
\(455\) 0 0
\(456\) 0 0
\(457\) −13104.6 −1.34138 −0.670688 0.741740i \(-0.734001\pi\)
−0.670688 + 0.741740i \(0.734001\pi\)
\(458\) −8527.26 −0.869985
\(459\) 0 0
\(460\) 54.3029 0.00550410
\(461\) 2594.63 0.262134 0.131067 0.991373i \(-0.458160\pi\)
0.131067 + 0.991373i \(0.458160\pi\)
\(462\) 0 0
\(463\) −14136.2 −1.41893 −0.709465 0.704741i \(-0.751063\pi\)
−0.709465 + 0.704741i \(0.751063\pi\)
\(464\) 2611.33 0.261267
\(465\) 0 0
\(466\) 6099.80 0.606369
\(467\) −15590.2 −1.54482 −0.772409 0.635125i \(-0.780948\pi\)
−0.772409 + 0.635125i \(0.780948\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −21172.4 −2.07789
\(471\) 0 0
\(472\) 3639.42 0.354911
\(473\) −3359.44 −0.326569
\(474\) 0 0
\(475\) −10361.2 −1.00085
\(476\) 0 0
\(477\) 0 0
\(478\) 7974.41 0.763056
\(479\) 8453.51 0.806369 0.403184 0.915119i \(-0.367903\pi\)
0.403184 + 0.915119i \(0.367903\pi\)
\(480\) 0 0
\(481\) 364.519 0.0345543
\(482\) 1249.30 0.118058
\(483\) 0 0
\(484\) −4404.35 −0.413632
\(485\) −6974.20 −0.652953
\(486\) 0 0
\(487\) −4011.07 −0.373221 −0.186611 0.982434i \(-0.559750\pi\)
−0.186611 + 0.982434i \(0.559750\pi\)
\(488\) −309.212 −0.0286831
\(489\) 0 0
\(490\) 0 0
\(491\) −13927.9 −1.28016 −0.640079 0.768309i \(-0.721098\pi\)
−0.640079 + 0.768309i \(0.721098\pi\)
\(492\) 0 0
\(493\) 19478.7 1.77946
\(494\) 144.962 0.0132027
\(495\) 0 0
\(496\) −3573.21 −0.323472
\(497\) 0 0
\(498\) 0 0
\(499\) −3947.55 −0.354141 −0.177071 0.984198i \(-0.556662\pi\)
−0.177071 + 0.984198i \(0.556662\pi\)
\(500\) −15351.6 −1.37309
\(501\) 0 0
\(502\) −2657.57 −0.236281
\(503\) 13725.3 1.21666 0.608331 0.793684i \(-0.291839\pi\)
0.608331 + 0.793684i \(0.291839\pi\)
\(504\) 0 0
\(505\) −307.190 −0.0270688
\(506\) −19.7577 −0.00173585
\(507\) 0 0
\(508\) −720.303 −0.0629100
\(509\) −7830.10 −0.681853 −0.340926 0.940090i \(-0.610741\pi\)
−0.340926 + 0.940090i \(0.610741\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −6452.36 −0.553700
\(515\) 17532.8 1.50017
\(516\) 0 0
\(517\) 7703.43 0.655312
\(518\) 0 0
\(519\) 0 0
\(520\) 360.545 0.0304057
\(521\) 5907.39 0.496751 0.248376 0.968664i \(-0.420103\pi\)
0.248376 + 0.968664i \(0.420103\pi\)
\(522\) 0 0
\(523\) 7908.06 0.661176 0.330588 0.943775i \(-0.392753\pi\)
0.330588 + 0.943775i \(0.392753\pi\)
\(524\) −871.439 −0.0726508
\(525\) 0 0
\(526\) 6501.23 0.538911
\(527\) −26653.6 −2.20313
\(528\) 0 0
\(529\) −12166.6 −0.999965
\(530\) 7355.98 0.602874
\(531\) 0 0
\(532\) 0 0
\(533\) 698.659 0.0567773
\(534\) 0 0
\(535\) −14912.5 −1.20509
\(536\) −1134.33 −0.0914100
\(537\) 0 0
\(538\) 5652.08 0.452934
\(539\) 0 0
\(540\) 0 0
\(541\) −3941.04 −0.313195 −0.156598 0.987662i \(-0.550053\pi\)
−0.156598 + 0.987662i \(0.550053\pi\)
\(542\) 4793.54 0.379889
\(543\) 0 0
\(544\) −3819.15 −0.301001
\(545\) −12502.7 −0.982670
\(546\) 0 0
\(547\) −1828.71 −0.142943 −0.0714717 0.997443i \(-0.522770\pi\)
−0.0714717 + 0.997443i \(0.522770\pi\)
\(548\) 10407.4 0.811283
\(549\) 0 0
\(550\) 9376.29 0.726920
\(551\) −5469.33 −0.422870
\(552\) 0 0
\(553\) 0 0
\(554\) −3640.93 −0.279221
\(555\) 0 0
\(556\) −10606.2 −0.808998
\(557\) −22532.0 −1.71402 −0.857011 0.515298i \(-0.827681\pi\)
−0.857011 + 0.515298i \(0.827681\pi\)
\(558\) 0 0
\(559\) 479.201 0.0362577
\(560\) 0 0
\(561\) 0 0
\(562\) 6167.62 0.462928
\(563\) −23355.7 −1.74836 −0.874179 0.485604i \(-0.838600\pi\)
−0.874179 + 0.485604i \(0.838600\pi\)
\(564\) 0 0
\(565\) 12974.1 0.966062
\(566\) −5109.54 −0.379452
\(567\) 0 0
\(568\) 4821.94 0.356204
\(569\) 20887.6 1.53894 0.769468 0.638686i \(-0.220522\pi\)
0.769468 + 0.638686i \(0.220522\pi\)
\(570\) 0 0
\(571\) 23745.3 1.74029 0.870147 0.492792i \(-0.164024\pi\)
0.870147 + 0.492792i \(0.164024\pi\)
\(572\) −131.182 −0.00958914
\(573\) 0 0
\(574\) 0 0
\(575\) −201.440 −0.0146098
\(576\) 0 0
\(577\) 2454.39 0.177084 0.0885422 0.996072i \(-0.471779\pi\)
0.0885422 + 0.996072i \(0.471779\pi\)
\(578\) −18662.1 −1.34298
\(579\) 0 0
\(580\) −13603.2 −0.973863
\(581\) 0 0
\(582\) 0 0
\(583\) −2676.42 −0.190130
\(584\) 8818.39 0.624842
\(585\) 0 0
\(586\) −3692.93 −0.260330
\(587\) −18567.5 −1.30556 −0.652780 0.757547i \(-0.726397\pi\)
−0.652780 + 0.757547i \(0.726397\pi\)
\(588\) 0 0
\(589\) 7483.95 0.523550
\(590\) −18958.8 −1.32292
\(591\) 0 0
\(592\) 2696.55 0.187208
\(593\) −17112.9 −1.18507 −0.592533 0.805546i \(-0.701872\pi\)
−0.592533 + 0.805546i \(0.701872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2324.09 −0.159729
\(597\) 0 0
\(598\) 2.81830 0.000192724 0
\(599\) −23264.8 −1.58694 −0.793469 0.608611i \(-0.791727\pi\)
−0.793469 + 0.608611i \(0.791727\pi\)
\(600\) 0 0
\(601\) 25322.3 1.71867 0.859334 0.511416i \(-0.170879\pi\)
0.859334 + 0.511416i \(0.170879\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2461.56 −0.165827
\(605\) 22943.5 1.54179
\(606\) 0 0
\(607\) −21734.4 −1.45333 −0.726665 0.686992i \(-0.758931\pi\)
−0.726665 + 0.686992i \(0.758931\pi\)
\(608\) 1072.36 0.0715297
\(609\) 0 0
\(610\) 1610.77 0.106915
\(611\) −1098.84 −0.0727567
\(612\) 0 0
\(613\) −13572.4 −0.894262 −0.447131 0.894468i \(-0.647554\pi\)
−0.447131 + 0.894468i \(0.647554\pi\)
\(614\) −14083.0 −0.925641
\(615\) 0 0
\(616\) 0 0
\(617\) 8497.12 0.554427 0.277213 0.960808i \(-0.410589\pi\)
0.277213 + 0.960808i \(0.410589\pi\)
\(618\) 0 0
\(619\) −22982.9 −1.49235 −0.746173 0.665752i \(-0.768111\pi\)
−0.746173 + 0.665752i \(0.768111\pi\)
\(620\) 18613.9 1.20573
\(621\) 0 0
\(622\) −5371.98 −0.346297
\(623\) 0 0
\(624\) 0 0
\(625\) 41322.5 2.64464
\(626\) −4438.38 −0.283376
\(627\) 0 0
\(628\) −1227.73 −0.0780122
\(629\) 20114.3 1.27505
\(630\) 0 0
\(631\) −15717.9 −0.991635 −0.495817 0.868427i \(-0.665131\pi\)
−0.495817 + 0.868427i \(0.665131\pi\)
\(632\) 930.424 0.0585606
\(633\) 0 0
\(634\) 4442.52 0.278289
\(635\) 3752.26 0.234494
\(636\) 0 0
\(637\) 0 0
\(638\) 4949.42 0.307131
\(639\) 0 0
\(640\) 2667.15 0.164732
\(641\) −29107.4 −1.79356 −0.896780 0.442478i \(-0.854100\pi\)
−0.896780 + 0.442478i \(0.854100\pi\)
\(642\) 0 0
\(643\) −3112.26 −0.190880 −0.0954398 0.995435i \(-0.530426\pi\)
−0.0954398 + 0.995435i \(0.530426\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 7999.06 0.487181
\(647\) 7857.59 0.477456 0.238728 0.971087i \(-0.423270\pi\)
0.238728 + 0.971087i \(0.423270\pi\)
\(648\) 0 0
\(649\) 6898.02 0.417213
\(650\) −1337.46 −0.0807071
\(651\) 0 0
\(652\) 14058.0 0.844408
\(653\) 19522.0 1.16992 0.584958 0.811063i \(-0.301111\pi\)
0.584958 + 0.811063i \(0.301111\pi\)
\(654\) 0 0
\(655\) 4539.57 0.270803
\(656\) 5168.36 0.307608
\(657\) 0 0
\(658\) 0 0
\(659\) −664.061 −0.0392536 −0.0196268 0.999807i \(-0.506248\pi\)
−0.0196268 + 0.999807i \(0.506248\pi\)
\(660\) 0 0
\(661\) 15921.6 0.936883 0.468442 0.883494i \(-0.344816\pi\)
0.468442 + 0.883494i \(0.344816\pi\)
\(662\) −8308.11 −0.487770
\(663\) 0 0
\(664\) −4551.42 −0.266008
\(665\) 0 0
\(666\) 0 0
\(667\) −106.333 −0.00617276
\(668\) −4493.21 −0.260251
\(669\) 0 0
\(670\) 5909.06 0.340727
\(671\) −586.068 −0.0337182
\(672\) 0 0
\(673\) 24631.0 1.41078 0.705391 0.708819i \(-0.250771\pi\)
0.705391 + 0.708819i \(0.250771\pi\)
\(674\) 508.333 0.0290508
\(675\) 0 0
\(676\) −8769.29 −0.498935
\(677\) −17092.8 −0.970353 −0.485177 0.874416i \(-0.661245\pi\)
−0.485177 + 0.874416i \(0.661245\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 19895.0 1.12197
\(681\) 0 0
\(682\) −6772.52 −0.380254
\(683\) 19163.6 1.07361 0.536804 0.843707i \(-0.319632\pi\)
0.536804 + 0.843707i \(0.319632\pi\)
\(684\) 0 0
\(685\) −54215.2 −3.02402
\(686\) 0 0
\(687\) 0 0
\(688\) 3544.91 0.196437
\(689\) 381.773 0.0211094
\(690\) 0 0
\(691\) 8095.87 0.445704 0.222852 0.974852i \(-0.428463\pi\)
0.222852 + 0.974852i \(0.428463\pi\)
\(692\) −6122.39 −0.336327
\(693\) 0 0
\(694\) −12449.3 −0.680934
\(695\) 55250.7 3.01551
\(696\) 0 0
\(697\) 38552.3 2.09508
\(698\) −19464.4 −1.05550
\(699\) 0 0
\(700\) 0 0
\(701\) −12354.7 −0.665664 −0.332832 0.942986i \(-0.608004\pi\)
−0.332832 + 0.942986i \(0.608004\pi\)
\(702\) 0 0
\(703\) −5647.81 −0.303003
\(704\) −970.424 −0.0519520
\(705\) 0 0
\(706\) 2851.23 0.151993
\(707\) 0 0
\(708\) 0 0
\(709\) 3828.82 0.202813 0.101406 0.994845i \(-0.467666\pi\)
0.101406 + 0.994845i \(0.467666\pi\)
\(710\) −25118.8 −1.32774
\(711\) 0 0
\(712\) −3065.27 −0.161343
\(713\) 145.500 0.00764241
\(714\) 0 0
\(715\) 683.363 0.0357431
\(716\) −13653.7 −0.712659
\(717\) 0 0
\(718\) −11533.0 −0.599453
\(719\) −1223.00 −0.0634356 −0.0317178 0.999497i \(-0.510098\pi\)
−0.0317178 + 0.999497i \(0.510098\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 11472.0 0.591333
\(723\) 0 0
\(724\) 5146.83 0.264200
\(725\) 50461.7 2.58496
\(726\) 0 0
\(727\) 6368.21 0.324875 0.162437 0.986719i \(-0.448064\pi\)
0.162437 + 0.986719i \(0.448064\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −45937.5 −2.32907
\(731\) 26442.5 1.33791
\(732\) 0 0
\(733\) −25154.0 −1.26751 −0.633753 0.773535i \(-0.718487\pi\)
−0.633753 + 0.773535i \(0.718487\pi\)
\(734\) −23090.7 −1.16116
\(735\) 0 0
\(736\) 20.8485 0.00104414
\(737\) −2149.97 −0.107456
\(738\) 0 0
\(739\) −10739.1 −0.534566 −0.267283 0.963618i \(-0.586126\pi\)
−0.267283 + 0.963618i \(0.586126\pi\)
\(740\) −14047.1 −0.697811
\(741\) 0 0
\(742\) 0 0
\(743\) −28166.3 −1.39074 −0.695370 0.718652i \(-0.744760\pi\)
−0.695370 + 0.718652i \(0.744760\pi\)
\(744\) 0 0
\(745\) 12106.8 0.595383
\(746\) 12959.1 0.636016
\(747\) 0 0
\(748\) −7238.67 −0.353839
\(749\) 0 0
\(750\) 0 0
\(751\) 28657.0 1.39242 0.696211 0.717837i \(-0.254868\pi\)
0.696211 + 0.717837i \(0.254868\pi\)
\(752\) −8128.73 −0.394181
\(753\) 0 0
\(754\) −706.000 −0.0340995
\(755\) 12823.0 0.618113
\(756\) 0 0
\(757\) −23604.1 −1.13330 −0.566648 0.823960i \(-0.691760\pi\)
−0.566648 + 0.823960i \(0.691760\pi\)
\(758\) −1223.99 −0.0586509
\(759\) 0 0
\(760\) −5586.24 −0.266624
\(761\) −4630.97 −0.220595 −0.110297 0.993899i \(-0.535180\pi\)
−0.110297 + 0.993899i \(0.535180\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4221.18 −0.199891
\(765\) 0 0
\(766\) 8721.62 0.411390
\(767\) −983.954 −0.0463214
\(768\) 0 0
\(769\) 33276.8 1.56046 0.780228 0.625495i \(-0.215103\pi\)
0.780228 + 0.625495i \(0.215103\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19083.4 −0.889670
\(773\) 22938.8 1.06734 0.533668 0.845694i \(-0.320813\pi\)
0.533668 + 0.845694i \(0.320813\pi\)
\(774\) 0 0
\(775\) −69049.1 −3.20041
\(776\) −2677.61 −0.123867
\(777\) 0 0
\(778\) −26293.8 −1.21167
\(779\) −10824.9 −0.497873
\(780\) 0 0
\(781\) 9139.31 0.418733
\(782\) 155.515 0.00711152
\(783\) 0 0
\(784\) 0 0
\(785\) 6395.58 0.290787
\(786\) 0 0
\(787\) −13514.5 −0.612120 −0.306060 0.952012i \(-0.599011\pi\)
−0.306060 + 0.952012i \(0.599011\pi\)
\(788\) −6489.24 −0.293363
\(789\) 0 0
\(790\) −4846.84 −0.218282
\(791\) 0 0
\(792\) 0 0
\(793\) 83.5986 0.00374360
\(794\) 16956.1 0.757870
\(795\) 0 0
\(796\) −14200.5 −0.632318
\(797\) −10473.4 −0.465480 −0.232740 0.972539i \(-0.574769\pi\)
−0.232740 + 0.972539i \(0.574769\pi\)
\(798\) 0 0
\(799\) −60634.5 −2.68472
\(800\) −9893.94 −0.437254
\(801\) 0 0
\(802\) 5606.00 0.246826
\(803\) 16714.0 0.734527
\(804\) 0 0
\(805\) 0 0
\(806\) 966.053 0.0422181
\(807\) 0 0
\(808\) −117.939 −0.00513501
\(809\) −23568.0 −1.02423 −0.512117 0.858916i \(-0.671139\pi\)
−0.512117 + 0.858916i \(0.671139\pi\)
\(810\) 0 0
\(811\) −6704.22 −0.290280 −0.145140 0.989411i \(-0.546363\pi\)
−0.145140 + 0.989411i \(0.546363\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 5110.92 0.220071
\(815\) −73232.1 −3.14749
\(816\) 0 0
\(817\) −7424.67 −0.317939
\(818\) 12770.8 0.545867
\(819\) 0 0
\(820\) −26923.5 −1.14659
\(821\) −24539.4 −1.04316 −0.521579 0.853203i \(-0.674657\pi\)
−0.521579 + 0.853203i \(0.674657\pi\)
\(822\) 0 0
\(823\) −31117.0 −1.31795 −0.658973 0.752167i \(-0.729009\pi\)
−0.658973 + 0.752167i \(0.729009\pi\)
\(824\) 6731.36 0.284585
\(825\) 0 0
\(826\) 0 0
\(827\) −31244.9 −1.31377 −0.656887 0.753989i \(-0.728127\pi\)
−0.656887 + 0.753989i \(0.728127\pi\)
\(828\) 0 0
\(829\) 4231.50 0.177281 0.0886405 0.996064i \(-0.471748\pi\)
0.0886405 + 0.996064i \(0.471748\pi\)
\(830\) 23709.6 0.991535
\(831\) 0 0
\(832\) 138.424 0.00576803
\(833\) 0 0
\(834\) 0 0
\(835\) 23406.4 0.970074
\(836\) 2032.51 0.0840861
\(837\) 0 0
\(838\) 9663.32 0.398346
\(839\) 38670.4 1.59124 0.795621 0.605795i \(-0.207145\pi\)
0.795621 + 0.605795i \(0.207145\pi\)
\(840\) 0 0
\(841\) 2247.96 0.0921710
\(842\) −14950.7 −0.611920
\(843\) 0 0
\(844\) 18613.5 0.759129
\(845\) 45681.7 1.85976
\(846\) 0 0
\(847\) 0 0
\(848\) 2824.18 0.114367
\(849\) 0 0
\(850\) −73801.7 −2.97809
\(851\) −109.803 −0.00442302
\(852\) 0 0
\(853\) −19944.4 −0.800565 −0.400282 0.916392i \(-0.631088\pi\)
−0.400282 + 0.916392i \(0.631088\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5725.36 −0.228609
\(857\) −13882.2 −0.553334 −0.276667 0.960966i \(-0.589230\pi\)
−0.276667 + 0.960966i \(0.589230\pi\)
\(858\) 0 0
\(859\) −4157.16 −0.165123 −0.0825614 0.996586i \(-0.526310\pi\)
−0.0825614 + 0.996586i \(0.526310\pi\)
\(860\) −18466.4 −0.732209
\(861\) 0 0
\(862\) 13983.9 0.552543
\(863\) 16237.9 0.640493 0.320246 0.947334i \(-0.396234\pi\)
0.320246 + 0.947334i \(0.396234\pi\)
\(864\) 0 0
\(865\) 31893.3 1.25365
\(866\) 15398.5 0.604230
\(867\) 0 0
\(868\) 0 0
\(869\) 1763.49 0.0688403
\(870\) 0 0
\(871\) 306.678 0.0119304
\(872\) −4800.15 −0.186415
\(873\) 0 0
\(874\) −43.6664 −0.00168998
\(875\) 0 0
\(876\) 0 0
\(877\) −16489.4 −0.634900 −0.317450 0.948275i \(-0.602827\pi\)
−0.317450 + 0.948275i \(0.602827\pi\)
\(878\) −18824.6 −0.723577
\(879\) 0 0
\(880\) 5055.21 0.193649
\(881\) −45411.7 −1.73662 −0.868309 0.496023i \(-0.834793\pi\)
−0.868309 + 0.496023i \(0.834793\pi\)
\(882\) 0 0
\(883\) −2206.85 −0.0841070 −0.0420535 0.999115i \(-0.513390\pi\)
−0.0420535 + 0.999115i \(0.513390\pi\)
\(884\) 1032.55 0.0392854
\(885\) 0 0
\(886\) −12516.4 −0.474600
\(887\) −28146.2 −1.06545 −0.532727 0.846287i \(-0.678833\pi\)
−0.532727 + 0.846287i \(0.678833\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 15967.9 0.601398
\(891\) 0 0
\(892\) −18598.1 −0.698107
\(893\) 17025.3 0.637995
\(894\) 0 0
\(895\) 71126.1 2.65641
\(896\) 0 0
\(897\) 0 0
\(898\) −23267.6 −0.864644
\(899\) −36448.6 −1.35220
\(900\) 0 0
\(901\) 21066.4 0.778937
\(902\) 9795.91 0.361605
\(903\) 0 0
\(904\) 4981.15 0.183264
\(905\) −26811.3 −0.984793
\(906\) 0 0
\(907\) −5042.25 −0.184592 −0.0922960 0.995732i \(-0.529421\pi\)
−0.0922960 + 0.995732i \(0.529421\pi\)
\(908\) −16606.9 −0.606958
\(909\) 0 0
\(910\) 0 0
\(911\) −29647.3 −1.07822 −0.539110 0.842235i \(-0.681239\pi\)
−0.539110 + 0.842235i \(0.681239\pi\)
\(912\) 0 0
\(913\) −8626.59 −0.312704
\(914\) 26209.3 0.948496
\(915\) 0 0
\(916\) 17054.5 0.615172
\(917\) 0 0
\(918\) 0 0
\(919\) −11891.3 −0.426830 −0.213415 0.976962i \(-0.568459\pi\)
−0.213415 + 0.976962i \(0.568459\pi\)
\(920\) −108.606 −0.00389199
\(921\) 0 0
\(922\) −5189.26 −0.185357
\(923\) −1303.66 −0.0464902
\(924\) 0 0
\(925\) 52108.3 1.85223
\(926\) 28272.4 1.00333
\(927\) 0 0
\(928\) −5222.67 −0.184744
\(929\) 39188.5 1.38400 0.691999 0.721898i \(-0.256730\pi\)
0.691999 + 0.721898i \(0.256730\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12199.6 −0.428768
\(933\) 0 0
\(934\) 31180.5 1.09235
\(935\) 37708.2 1.31892
\(936\) 0 0
\(937\) 9716.23 0.338757 0.169379 0.985551i \(-0.445824\pi\)
0.169379 + 0.985551i \(0.445824\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 42344.8 1.46929
\(941\) 6995.87 0.242358 0.121179 0.992631i \(-0.461333\pi\)
0.121179 + 0.992631i \(0.461333\pi\)
\(942\) 0 0
\(943\) −210.455 −0.00726760
\(944\) −7278.85 −0.250960
\(945\) 0 0
\(946\) 6718.88 0.230919
\(947\) −14979.2 −0.514002 −0.257001 0.966411i \(-0.582734\pi\)
−0.257001 + 0.966411i \(0.582734\pi\)
\(948\) 0 0
\(949\) −2384.14 −0.0815516
\(950\) 20722.5 0.707711
\(951\) 0 0
\(952\) 0 0
\(953\) −29393.3 −0.999100 −0.499550 0.866285i \(-0.666501\pi\)
−0.499550 + 0.866285i \(0.666501\pi\)
\(954\) 0 0
\(955\) 21989.3 0.745087
\(956\) −15948.8 −0.539562
\(957\) 0 0
\(958\) −16907.0 −0.570189
\(959\) 0 0
\(960\) 0 0
\(961\) 20083.4 0.674143
\(962\) −729.038 −0.0244336
\(963\) 0 0
\(964\) −2498.59 −0.0834795
\(965\) 99410.6 3.31621
\(966\) 0 0
\(967\) −7133.95 −0.237241 −0.118621 0.992940i \(-0.537847\pi\)
−0.118621 + 0.992940i \(0.537847\pi\)
\(968\) 8808.70 0.292482
\(969\) 0 0
\(970\) 13948.4 0.461707
\(971\) 9688.13 0.320192 0.160096 0.987101i \(-0.448820\pi\)
0.160096 + 0.987101i \(0.448820\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8022.14 0.263907
\(975\) 0 0
\(976\) 618.424 0.0202820
\(977\) −21305.7 −0.697676 −0.348838 0.937183i \(-0.613424\pi\)
−0.348838 + 0.937183i \(0.613424\pi\)
\(978\) 0 0
\(979\) −5809.79 −0.189665
\(980\) 0 0
\(981\) 0 0
\(982\) 27855.8 0.905208
\(983\) 37280.8 1.20964 0.604818 0.796364i \(-0.293246\pi\)
0.604818 + 0.796364i \(0.293246\pi\)
\(984\) 0 0
\(985\) 33804.3 1.09350
\(986\) −38957.3 −1.25827
\(987\) 0 0
\(988\) −289.924 −0.00933574
\(989\) −144.348 −0.00464105
\(990\) 0 0
\(991\) −51397.1 −1.64751 −0.823755 0.566946i \(-0.808125\pi\)
−0.823755 + 0.566946i \(0.808125\pi\)
\(992\) 7146.42 0.228729
\(993\) 0 0
\(994\) 0 0
\(995\) 73974.6 2.35694
\(996\) 0 0
\(997\) −34373.8 −1.09191 −0.545953 0.837816i \(-0.683832\pi\)
−0.545953 + 0.837816i \(0.683832\pi\)
\(998\) 7895.10 0.250416
\(999\) 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 882.4.a.v.1.1 2
3.2 odd 2 294.4.a.n.1.2 2
7.2 even 3 126.4.g.g.109.2 4
7.3 odd 6 882.4.g.bf.667.1 4
7.4 even 3 126.4.g.g.37.2 4
7.5 odd 6 882.4.g.bf.361.1 4
7.6 odd 2 882.4.a.z.1.2 2
12.11 even 2 2352.4.a.bq.1.2 2
21.2 odd 6 42.4.e.c.25.1 4
21.5 even 6 294.4.e.l.67.2 4
21.11 odd 6 42.4.e.c.37.1 yes 4
21.17 even 6 294.4.e.l.79.2 4
21.20 even 2 294.4.a.m.1.1 2
84.11 even 6 336.4.q.j.289.1 4
84.23 even 6 336.4.q.j.193.1 4
84.83 odd 2 2352.4.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.c.25.1 4 21.2 odd 6
42.4.e.c.37.1 yes 4 21.11 odd 6
126.4.g.g.37.2 4 7.4 even 3
126.4.g.g.109.2 4 7.2 even 3
294.4.a.m.1.1 2 21.20 even 2
294.4.a.n.1.2 2 3.2 odd 2
294.4.e.l.67.2 4 21.5 even 6
294.4.e.l.79.2 4 21.17 even 6
336.4.q.j.193.1 4 84.23 even 6
336.4.q.j.289.1 4 84.11 even 6
882.4.a.v.1.1 2 1.1 even 1 trivial
882.4.a.z.1.2 2 7.6 odd 2
882.4.g.bf.361.1 4 7.5 odd 6
882.4.g.bf.667.1 4 7.3 odd 6
2352.4.a.bq.1.2 2 12.11 even 2
2352.4.a.ca.1.1 2 84.83 odd 2