Properties

Label 1176.4.a.bc.1.3
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.145408.2
Defining polynomial: \(x^{4} - 24 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.66254\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +4.11659 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +4.11659 q^{5} +9.00000 q^{9} +17.9344 q^{11} -23.4236 q^{13} +12.3498 q^{15} -76.2280 q^{17} -35.5964 q^{19} -40.7918 q^{23} -108.054 q^{25} +27.0000 q^{27} -178.805 q^{29} -31.6797 q^{31} +53.8032 q^{33} -54.8450 q^{37} -70.2708 q^{39} +190.547 q^{41} -131.964 q^{43} +37.0493 q^{45} -199.573 q^{47} -228.684 q^{51} +321.907 q^{53} +73.8287 q^{55} -106.789 q^{57} -163.392 q^{59} -265.230 q^{61} -96.4254 q^{65} +278.127 q^{67} -122.375 q^{69} -10.5794 q^{71} -584.292 q^{73} -324.161 q^{75} -183.078 q^{79} +81.0000 q^{81} +175.117 q^{83} -313.800 q^{85} -536.416 q^{87} -47.1437 q^{89} -95.0391 q^{93} -146.536 q^{95} +556.805 q^{97} +161.410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} - 40q^{11} - 48q^{13} - 24q^{15} + 152q^{17} - 224q^{19} - 8q^{23} - 28q^{25} + 108q^{27} - 144q^{29} - 400q^{31} - 120q^{33} - 304q^{37} - 144q^{39} + 152q^{41} + 160q^{43} - 72q^{45} - 544q^{47} + 456q^{51} - 1320q^{53} - 16q^{55} - 672q^{57} - 1040q^{59} - 896q^{61} - 648q^{65} - 416q^{67} - 24q^{69} + 248q^{71} + 752q^{73} - 84q^{75} + 864q^{79} + 324q^{81} - 1456q^{83} - 1608q^{85} - 432q^{87} - 2936q^{89} - 1200q^{93} - 80q^{95} - 144q^{97} - 360q^{99} + O(q^{100}) \)

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 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 4.11659 0.368199 0.184100 0.982908i \(-0.441063\pi\)
0.184100 + 0.982908i \(0.441063\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 17.9344 0.491585 0.245792 0.969323i \(-0.420952\pi\)
0.245792 + 0.969323i \(0.420952\pi\)
\(12\) 0 0
\(13\) −23.4236 −0.499733 −0.249867 0.968280i \(-0.580387\pi\)
−0.249867 + 0.968280i \(0.580387\pi\)
\(14\) 0 0
\(15\) 12.3498 0.212580
\(16\) 0 0
\(17\) −76.2280 −1.08753 −0.543765 0.839237i \(-0.683002\pi\)
−0.543765 + 0.839237i \(0.683002\pi\)
\(18\) 0 0
\(19\) −35.5964 −0.429809 −0.214905 0.976635i \(-0.568944\pi\)
−0.214905 + 0.976635i \(0.568944\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −40.7918 −0.369812 −0.184906 0.982756i \(-0.559198\pi\)
−0.184906 + 0.982756i \(0.559198\pi\)
\(24\) 0 0
\(25\) −108.054 −0.864429
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −178.805 −1.14494 −0.572471 0.819925i \(-0.694015\pi\)
−0.572471 + 0.819925i \(0.694015\pi\)
\(30\) 0 0
\(31\) −31.6797 −0.183543 −0.0917717 0.995780i \(-0.529253\pi\)
−0.0917717 + 0.995780i \(0.529253\pi\)
\(32\) 0 0
\(33\) 53.8032 0.283816
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −54.8450 −0.243688 −0.121844 0.992549i \(-0.538881\pi\)
−0.121844 + 0.992549i \(0.538881\pi\)
\(38\) 0 0
\(39\) −70.2708 −0.288521
\(40\) 0 0
\(41\) 190.547 0.725815 0.362907 0.931825i \(-0.381784\pi\)
0.362907 + 0.931825i \(0.381784\pi\)
\(42\) 0 0
\(43\) −131.964 −0.468009 −0.234004 0.972236i \(-0.575183\pi\)
−0.234004 + 0.972236i \(0.575183\pi\)
\(44\) 0 0
\(45\) 37.0493 0.122733
\(46\) 0 0
\(47\) −199.573 −0.619377 −0.309689 0.950838i \(-0.600225\pi\)
−0.309689 + 0.950838i \(0.600225\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −228.684 −0.627886
\(52\) 0 0
\(53\) 321.907 0.834289 0.417145 0.908840i \(-0.363031\pi\)
0.417145 + 0.908840i \(0.363031\pi\)
\(54\) 0 0
\(55\) 73.8287 0.181001
\(56\) 0 0
\(57\) −106.789 −0.248150
\(58\) 0 0
\(59\) −163.392 −0.360540 −0.180270 0.983617i \(-0.557697\pi\)
−0.180270 + 0.983617i \(0.557697\pi\)
\(60\) 0 0
\(61\) −265.230 −0.556710 −0.278355 0.960478i \(-0.589789\pi\)
−0.278355 + 0.960478i \(0.589789\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −96.4254 −0.184001
\(66\) 0 0
\(67\) 278.127 0.507144 0.253572 0.967316i \(-0.418394\pi\)
0.253572 + 0.967316i \(0.418394\pi\)
\(68\) 0 0
\(69\) −122.375 −0.213511
\(70\) 0 0
\(71\) −10.5794 −0.0176836 −0.00884182 0.999961i \(-0.502814\pi\)
−0.00884182 + 0.999961i \(0.502814\pi\)
\(72\) 0 0
\(73\) −584.292 −0.936797 −0.468398 0.883517i \(-0.655169\pi\)
−0.468398 + 0.883517i \(0.655169\pi\)
\(74\) 0 0
\(75\) −324.161 −0.499079
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −183.078 −0.260733 −0.130366 0.991466i \(-0.541615\pi\)
−0.130366 + 0.991466i \(0.541615\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 175.117 0.231585 0.115792 0.993273i \(-0.463059\pi\)
0.115792 + 0.993273i \(0.463059\pi\)
\(84\) 0 0
\(85\) −313.800 −0.400428
\(86\) 0 0
\(87\) −536.416 −0.661033
\(88\) 0 0
\(89\) −47.1437 −0.0561485 −0.0280743 0.999606i \(-0.508937\pi\)
−0.0280743 + 0.999606i \(0.508937\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −95.0391 −0.105969
\(94\) 0 0
\(95\) −146.536 −0.158255
\(96\) 0 0
\(97\) 556.805 0.582834 0.291417 0.956596i \(-0.405873\pi\)
0.291417 + 0.956596i \(0.405873\pi\)
\(98\) 0 0
\(99\) 161.410 0.163862
\(100\) 0 0
\(101\) −128.804 −0.126895 −0.0634477 0.997985i \(-0.520210\pi\)
−0.0634477 + 0.997985i \(0.520210\pi\)
\(102\) 0 0
\(103\) −2043.48 −1.95485 −0.977427 0.211271i \(-0.932240\pi\)
−0.977427 + 0.211271i \(0.932240\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −268.860 −0.242913 −0.121456 0.992597i \(-0.538756\pi\)
−0.121456 + 0.992597i \(0.538756\pi\)
\(108\) 0 0
\(109\) 1335.64 1.17368 0.586841 0.809702i \(-0.300371\pi\)
0.586841 + 0.809702i \(0.300371\pi\)
\(110\) 0 0
\(111\) −164.535 −0.140693
\(112\) 0 0
\(113\) −1675.40 −1.39476 −0.697382 0.716700i \(-0.745652\pi\)
−0.697382 + 0.716700i \(0.745652\pi\)
\(114\) 0 0
\(115\) −167.923 −0.136164
\(116\) 0 0
\(117\) −210.812 −0.166578
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1009.36 −0.758345
\(122\) 0 0
\(123\) 571.640 0.419049
\(124\) 0 0
\(125\) −959.387 −0.686481
\(126\) 0 0
\(127\) 1626.28 1.13629 0.568144 0.822929i \(-0.307662\pi\)
0.568144 + 0.822929i \(0.307662\pi\)
\(128\) 0 0
\(129\) −395.893 −0.270205
\(130\) 0 0
\(131\) −1534.98 −1.02375 −0.511877 0.859059i \(-0.671050\pi\)
−0.511877 + 0.859059i \(0.671050\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 111.148 0.0708600
\(136\) 0 0
\(137\) 2466.49 1.53815 0.769076 0.639158i \(-0.220717\pi\)
0.769076 + 0.639158i \(0.220717\pi\)
\(138\) 0 0
\(139\) −1322.68 −0.807107 −0.403554 0.914956i \(-0.632225\pi\)
−0.403554 + 0.914956i \(0.632225\pi\)
\(140\) 0 0
\(141\) −598.719 −0.357598
\(142\) 0 0
\(143\) −420.088 −0.245661
\(144\) 0 0
\(145\) −736.069 −0.421567
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −648.069 −0.356321 −0.178161 0.984001i \(-0.557015\pi\)
−0.178161 + 0.984001i \(0.557015\pi\)
\(150\) 0 0
\(151\) 2003.70 1.07986 0.539930 0.841710i \(-0.318451\pi\)
0.539930 + 0.841710i \(0.318451\pi\)
\(152\) 0 0
\(153\) −686.052 −0.362510
\(154\) 0 0
\(155\) −130.412 −0.0675805
\(156\) 0 0
\(157\) 773.589 0.393243 0.196622 0.980479i \(-0.437003\pi\)
0.196622 + 0.980479i \(0.437003\pi\)
\(158\) 0 0
\(159\) 965.721 0.481677
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 66.4286 0.0319208 0.0159604 0.999873i \(-0.494919\pi\)
0.0159604 + 0.999873i \(0.494919\pi\)
\(164\) 0 0
\(165\) 221.486 0.104501
\(166\) 0 0
\(167\) −842.616 −0.390441 −0.195220 0.980759i \(-0.562542\pi\)
−0.195220 + 0.980759i \(0.562542\pi\)
\(168\) 0 0
\(169\) −1648.34 −0.750266
\(170\) 0 0
\(171\) −320.368 −0.143270
\(172\) 0 0
\(173\) −1570.69 −0.690275 −0.345138 0.938552i \(-0.612168\pi\)
−0.345138 + 0.938552i \(0.612168\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −490.177 −0.208158
\(178\) 0 0
\(179\) 3670.74 1.53276 0.766380 0.642387i \(-0.222056\pi\)
0.766380 + 0.642387i \(0.222056\pi\)
\(180\) 0 0
\(181\) 802.417 0.329520 0.164760 0.986334i \(-0.447315\pi\)
0.164760 + 0.986334i \(0.447315\pi\)
\(182\) 0 0
\(183\) −795.691 −0.321416
\(184\) 0 0
\(185\) −225.775 −0.0897258
\(186\) 0 0
\(187\) −1367.11 −0.534613
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4559.10 −1.72714 −0.863572 0.504225i \(-0.831778\pi\)
−0.863572 + 0.504225i \(0.831778\pi\)
\(192\) 0 0
\(193\) 3133.71 1.16875 0.584377 0.811482i \(-0.301339\pi\)
0.584377 + 0.811482i \(0.301339\pi\)
\(194\) 0 0
\(195\) −289.276 −0.106233
\(196\) 0 0
\(197\) −2805.70 −1.01471 −0.507354 0.861738i \(-0.669376\pi\)
−0.507354 + 0.861738i \(0.669376\pi\)
\(198\) 0 0
\(199\) −5257.19 −1.87272 −0.936362 0.351036i \(-0.885829\pi\)
−0.936362 + 0.351036i \(0.885829\pi\)
\(200\) 0 0
\(201\) 834.382 0.292800
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 784.403 0.267244
\(206\) 0 0
\(207\) −367.126 −0.123271
\(208\) 0 0
\(209\) −638.400 −0.211287
\(210\) 0 0
\(211\) 3230.30 1.05395 0.526974 0.849882i \(-0.323327\pi\)
0.526974 + 0.849882i \(0.323327\pi\)
\(212\) 0 0
\(213\) −31.7381 −0.0102097
\(214\) 0 0
\(215\) −543.243 −0.172320
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1752.87 −0.540860
\(220\) 0 0
\(221\) 1785.53 0.543475
\(222\) 0 0
\(223\) −433.169 −0.130077 −0.0650385 0.997883i \(-0.520717\pi\)
−0.0650385 + 0.997883i \(0.520717\pi\)
\(224\) 0 0
\(225\) −972.483 −0.288143
\(226\) 0 0
\(227\) −4108.60 −1.20131 −0.600655 0.799509i \(-0.705093\pi\)
−0.600655 + 0.799509i \(0.705093\pi\)
\(228\) 0 0
\(229\) −4633.01 −1.33693 −0.668467 0.743742i \(-0.733049\pi\)
−0.668467 + 0.743742i \(0.733049\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 643.222 0.180853 0.0904267 0.995903i \(-0.471177\pi\)
0.0904267 + 0.995903i \(0.471177\pi\)
\(234\) 0 0
\(235\) −821.561 −0.228054
\(236\) 0 0
\(237\) −549.235 −0.150534
\(238\) 0 0
\(239\) 783.612 0.212082 0.106041 0.994362i \(-0.466182\pi\)
0.106041 + 0.994362i \(0.466182\pi\)
\(240\) 0 0
\(241\) 6338.58 1.69421 0.847103 0.531429i \(-0.178345\pi\)
0.847103 + 0.531429i \(0.178345\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 833.795 0.214790
\(248\) 0 0
\(249\) 525.350 0.133706
\(250\) 0 0
\(251\) −5863.01 −1.47438 −0.737191 0.675685i \(-0.763848\pi\)
−0.737191 + 0.675685i \(0.763848\pi\)
\(252\) 0 0
\(253\) −731.577 −0.181794
\(254\) 0 0
\(255\) −941.399 −0.231187
\(256\) 0 0
\(257\) 4213.76 1.02275 0.511376 0.859357i \(-0.329136\pi\)
0.511376 + 0.859357i \(0.329136\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1609.25 −0.381648
\(262\) 0 0
\(263\) −1955.85 −0.458566 −0.229283 0.973360i \(-0.573638\pi\)
−0.229283 + 0.973360i \(0.573638\pi\)
\(264\) 0 0
\(265\) 1325.16 0.307185
\(266\) 0 0
\(267\) −141.431 −0.0324174
\(268\) 0 0
\(269\) −1527.99 −0.346332 −0.173166 0.984893i \(-0.555400\pi\)
−0.173166 + 0.984893i \(0.555400\pi\)
\(270\) 0 0
\(271\) −7673.88 −1.72013 −0.860064 0.510185i \(-0.829577\pi\)
−0.860064 + 0.510185i \(0.829577\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1937.88 −0.424940
\(276\) 0 0
\(277\) 6312.20 1.36918 0.684591 0.728927i \(-0.259981\pi\)
0.684591 + 0.728927i \(0.259981\pi\)
\(278\) 0 0
\(279\) −285.117 −0.0611811
\(280\) 0 0
\(281\) −3334.66 −0.707932 −0.353966 0.935258i \(-0.615167\pi\)
−0.353966 + 0.935258i \(0.615167\pi\)
\(282\) 0 0
\(283\) 725.900 0.152474 0.0762372 0.997090i \(-0.475709\pi\)
0.0762372 + 0.997090i \(0.475709\pi\)
\(284\) 0 0
\(285\) −439.607 −0.0913688
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 897.713 0.182722
\(290\) 0 0
\(291\) 1670.41 0.336500
\(292\) 0 0
\(293\) 5681.47 1.13282 0.566408 0.824125i \(-0.308333\pi\)
0.566408 + 0.824125i \(0.308333\pi\)
\(294\) 0 0
\(295\) −672.620 −0.132751
\(296\) 0 0
\(297\) 484.229 0.0946055
\(298\) 0 0
\(299\) 955.490 0.184807
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −386.411 −0.0732631
\(304\) 0 0
\(305\) −1091.85 −0.204980
\(306\) 0 0
\(307\) 2469.04 0.459008 0.229504 0.973308i \(-0.426290\pi\)
0.229504 + 0.973308i \(0.426290\pi\)
\(308\) 0 0
\(309\) −6130.44 −1.12864
\(310\) 0 0
\(311\) −4068.00 −0.741721 −0.370860 0.928689i \(-0.620937\pi\)
−0.370860 + 0.928689i \(0.620937\pi\)
\(312\) 0 0
\(313\) −4912.13 −0.887062 −0.443531 0.896259i \(-0.646274\pi\)
−0.443531 + 0.896259i \(0.646274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2761.17 0.489219 0.244610 0.969622i \(-0.421340\pi\)
0.244610 + 0.969622i \(0.421340\pi\)
\(318\) 0 0
\(319\) −3206.77 −0.562836
\(320\) 0 0
\(321\) −806.579 −0.140246
\(322\) 0 0
\(323\) 2713.44 0.467430
\(324\) 0 0
\(325\) 2531.00 0.431984
\(326\) 0 0
\(327\) 4006.93 0.677626
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1191.72 0.197894 0.0989470 0.995093i \(-0.468453\pi\)
0.0989470 + 0.995093i \(0.468453\pi\)
\(332\) 0 0
\(333\) −493.605 −0.0812294
\(334\) 0 0
\(335\) 1144.94 0.186730
\(336\) 0 0
\(337\) 4336.34 0.700937 0.350468 0.936575i \(-0.386022\pi\)
0.350468 + 0.936575i \(0.386022\pi\)
\(338\) 0 0
\(339\) −5026.20 −0.805267
\(340\) 0 0
\(341\) −568.157 −0.0902271
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −503.769 −0.0786146
\(346\) 0 0
\(347\) −9450.20 −1.46200 −0.730999 0.682379i \(-0.760946\pi\)
−0.730999 + 0.682379i \(0.760946\pi\)
\(348\) 0 0
\(349\) 10654.7 1.63418 0.817092 0.576507i \(-0.195585\pi\)
0.817092 + 0.576507i \(0.195585\pi\)
\(350\) 0 0
\(351\) −632.437 −0.0961738
\(352\) 0 0
\(353\) 1702.81 0.256746 0.128373 0.991726i \(-0.459025\pi\)
0.128373 + 0.991726i \(0.459025\pi\)
\(354\) 0 0
\(355\) −43.5509 −0.00651110
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4108.61 −0.604023 −0.302011 0.953304i \(-0.597658\pi\)
−0.302011 + 0.953304i \(0.597658\pi\)
\(360\) 0 0
\(361\) −5591.90 −0.815264
\(362\) 0 0
\(363\) −3028.07 −0.437830
\(364\) 0 0
\(365\) −2405.29 −0.344928
\(366\) 0 0
\(367\) −1179.35 −0.167743 −0.0838715 0.996477i \(-0.526729\pi\)
−0.0838715 + 0.996477i \(0.526729\pi\)
\(368\) 0 0
\(369\) 1714.92 0.241938
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7865.22 1.09181 0.545906 0.837846i \(-0.316186\pi\)
0.545906 + 0.837846i \(0.316186\pi\)
\(374\) 0 0
\(375\) −2878.16 −0.396340
\(376\) 0 0
\(377\) 4188.27 0.572166
\(378\) 0 0
\(379\) 13076.8 1.77233 0.886164 0.463372i \(-0.153360\pi\)
0.886164 + 0.463372i \(0.153360\pi\)
\(380\) 0 0
\(381\) 4878.83 0.656036
\(382\) 0 0
\(383\) 2410.38 0.321578 0.160789 0.986989i \(-0.448596\pi\)
0.160789 + 0.986989i \(0.448596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1187.68 −0.156003
\(388\) 0 0
\(389\) 6178.78 0.805339 0.402669 0.915346i \(-0.368082\pi\)
0.402669 + 0.915346i \(0.368082\pi\)
\(390\) 0 0
\(391\) 3109.48 0.402182
\(392\) 0 0
\(393\) −4604.94 −0.591064
\(394\) 0 0
\(395\) −753.658 −0.0960017
\(396\) 0 0
\(397\) −4809.52 −0.608017 −0.304009 0.952669i \(-0.598325\pi\)
−0.304009 + 0.952669i \(0.598325\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6088.29 0.758191 0.379096 0.925357i \(-0.376235\pi\)
0.379096 + 0.925357i \(0.376235\pi\)
\(402\) 0 0
\(403\) 742.053 0.0917228
\(404\) 0 0
\(405\) 333.444 0.0409110
\(406\) 0 0
\(407\) −983.613 −0.119793
\(408\) 0 0
\(409\) 7869.74 0.951426 0.475713 0.879600i \(-0.342190\pi\)
0.475713 + 0.879600i \(0.342190\pi\)
\(410\) 0 0
\(411\) 7399.48 0.888052
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 720.884 0.0852694
\(416\) 0 0
\(417\) −3968.03 −0.465984
\(418\) 0 0
\(419\) −15279.0 −1.78145 −0.890724 0.454544i \(-0.849802\pi\)
−0.890724 + 0.454544i \(0.849802\pi\)
\(420\) 0 0
\(421\) 6909.70 0.799900 0.399950 0.916537i \(-0.369027\pi\)
0.399950 + 0.916537i \(0.369027\pi\)
\(422\) 0 0
\(423\) −1796.16 −0.206459
\(424\) 0 0
\(425\) 8236.72 0.940093
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1260.27 −0.141833
\(430\) 0 0
\(431\) −12627.4 −1.41124 −0.705618 0.708592i \(-0.749331\pi\)
−0.705618 + 0.708592i \(0.749331\pi\)
\(432\) 0 0
\(433\) 6738.74 0.747906 0.373953 0.927448i \(-0.378002\pi\)
0.373953 + 0.927448i \(0.378002\pi\)
\(434\) 0 0
\(435\) −2208.21 −0.243392
\(436\) 0 0
\(437\) 1452.04 0.158948
\(438\) 0 0
\(439\) 1752.54 0.190533 0.0952667 0.995452i \(-0.469630\pi\)
0.0952667 + 0.995452i \(0.469630\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14286.9 −1.53226 −0.766130 0.642686i \(-0.777820\pi\)
−0.766130 + 0.642686i \(0.777820\pi\)
\(444\) 0 0
\(445\) −194.071 −0.0206738
\(446\) 0 0
\(447\) −1944.21 −0.205722
\(448\) 0 0
\(449\) −7524.11 −0.790835 −0.395417 0.918501i \(-0.629400\pi\)
−0.395417 + 0.918501i \(0.629400\pi\)
\(450\) 0 0
\(451\) 3417.35 0.356799
\(452\) 0 0
\(453\) 6011.10 0.623457
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10579.4 1.08290 0.541448 0.840734i \(-0.317876\pi\)
0.541448 + 0.840734i \(0.317876\pi\)
\(458\) 0 0
\(459\) −2058.16 −0.209295
\(460\) 0 0
\(461\) −5889.66 −0.595030 −0.297515 0.954717i \(-0.596158\pi\)
−0.297515 + 0.954717i \(0.596158\pi\)
\(462\) 0 0
\(463\) 12716.5 1.27643 0.638215 0.769858i \(-0.279673\pi\)
0.638215 + 0.769858i \(0.279673\pi\)
\(464\) 0 0
\(465\) −391.237 −0.0390176
\(466\) 0 0
\(467\) 8362.46 0.828626 0.414313 0.910134i \(-0.364022\pi\)
0.414313 + 0.910134i \(0.364022\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2320.77 0.227039
\(472\) 0 0
\(473\) −2366.70 −0.230066
\(474\) 0 0
\(475\) 3846.32 0.371540
\(476\) 0 0
\(477\) 2897.16 0.278096
\(478\) 0 0
\(479\) 5404.57 0.515535 0.257768 0.966207i \(-0.417013\pi\)
0.257768 + 0.966207i \(0.417013\pi\)
\(480\) 0 0
\(481\) 1284.67 0.121779
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2292.14 0.214599
\(486\) 0 0
\(487\) −10659.5 −0.991842 −0.495921 0.868368i \(-0.665169\pi\)
−0.495921 + 0.868368i \(0.665169\pi\)
\(488\) 0 0
\(489\) 199.286 0.0184295
\(490\) 0 0
\(491\) −277.130 −0.0254719 −0.0127360 0.999919i \(-0.504054\pi\)
−0.0127360 + 0.999919i \(0.504054\pi\)
\(492\) 0 0
\(493\) 13630.0 1.24516
\(494\) 0 0
\(495\) 664.458 0.0603337
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18108.3 1.62453 0.812266 0.583288i \(-0.198234\pi\)
0.812266 + 0.583288i \(0.198234\pi\)
\(500\) 0 0
\(501\) −2527.85 −0.225421
\(502\) 0 0
\(503\) 3671.00 0.325411 0.162706 0.986675i \(-0.447978\pi\)
0.162706 + 0.986675i \(0.447978\pi\)
\(504\) 0 0
\(505\) −530.232 −0.0467228
\(506\) 0 0
\(507\) −4945.01 −0.433167
\(508\) 0 0
\(509\) 17364.1 1.51208 0.756040 0.654526i \(-0.227132\pi\)
0.756040 + 0.654526i \(0.227132\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −961.103 −0.0827168
\(514\) 0 0
\(515\) −8412.17 −0.719776
\(516\) 0 0
\(517\) −3579.23 −0.304476
\(518\) 0 0
\(519\) −4712.08 −0.398531
\(520\) 0 0
\(521\) −2870.45 −0.241376 −0.120688 0.992690i \(-0.538510\pi\)
−0.120688 + 0.992690i \(0.538510\pi\)
\(522\) 0 0
\(523\) −5735.39 −0.479524 −0.239762 0.970832i \(-0.577069\pi\)
−0.239762 + 0.970832i \(0.577069\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2414.88 0.199609
\(528\) 0 0
\(529\) −10503.0 −0.863239
\(530\) 0 0
\(531\) −1470.53 −0.120180
\(532\) 0 0
\(533\) −4463.29 −0.362714
\(534\) 0 0
\(535\) −1106.79 −0.0894402
\(536\) 0 0
\(537\) 11012.2 0.884940
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2014.27 0.160074 0.0800372 0.996792i \(-0.474496\pi\)
0.0800372 + 0.996792i \(0.474496\pi\)
\(542\) 0 0
\(543\) 2407.25 0.190249
\(544\) 0 0
\(545\) 5498.30 0.432149
\(546\) 0 0
\(547\) 18005.2 1.40740 0.703699 0.710498i \(-0.251530\pi\)
0.703699 + 0.710498i \(0.251530\pi\)
\(548\) 0 0
\(549\) −2387.07 −0.185570
\(550\) 0 0
\(551\) 6364.83 0.492107
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −677.324 −0.0518032
\(556\) 0 0
\(557\) 3093.70 0.235340 0.117670 0.993053i \(-0.462458\pi\)
0.117670 + 0.993053i \(0.462458\pi\)
\(558\) 0 0
\(559\) 3091.08 0.233880
\(560\) 0 0
\(561\) −4101.32 −0.308659
\(562\) 0 0
\(563\) 18245.4 1.36581 0.682907 0.730505i \(-0.260715\pi\)
0.682907 + 0.730505i \(0.260715\pi\)
\(564\) 0 0
\(565\) −6896.94 −0.513551
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17456.7 1.28616 0.643079 0.765800i \(-0.277657\pi\)
0.643079 + 0.765800i \(0.277657\pi\)
\(570\) 0 0
\(571\) 10013.7 0.733902 0.366951 0.930240i \(-0.380402\pi\)
0.366951 + 0.930240i \(0.380402\pi\)
\(572\) 0 0
\(573\) −13677.3 −0.997167
\(574\) 0 0
\(575\) 4407.70 0.319676
\(576\) 0 0
\(577\) 26521.5 1.91353 0.956764 0.290867i \(-0.0939436\pi\)
0.956764 + 0.290867i \(0.0939436\pi\)
\(578\) 0 0
\(579\) 9401.14 0.674781
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5773.21 0.410124
\(584\) 0 0
\(585\) −867.828 −0.0613338
\(586\) 0 0
\(587\) −6734.74 −0.473547 −0.236774 0.971565i \(-0.576090\pi\)
−0.236774 + 0.971565i \(0.576090\pi\)
\(588\) 0 0
\(589\) 1127.68 0.0788886
\(590\) 0 0
\(591\) −8417.09 −0.585842
\(592\) 0 0
\(593\) 26173.3 1.81250 0.906248 0.422746i \(-0.138934\pi\)
0.906248 + 0.422746i \(0.138934\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15771.6 −1.08122
\(598\) 0 0
\(599\) −5944.71 −0.405499 −0.202750 0.979231i \(-0.564988\pi\)
−0.202750 + 0.979231i \(0.564988\pi\)
\(600\) 0 0
\(601\) 1988.96 0.134994 0.0674970 0.997719i \(-0.478499\pi\)
0.0674970 + 0.997719i \(0.478499\pi\)
\(602\) 0 0
\(603\) 2503.15 0.169048
\(604\) 0 0
\(605\) −4155.11 −0.279222
\(606\) 0 0
\(607\) −1383.89 −0.0925376 −0.0462688 0.998929i \(-0.514733\pi\)
−0.0462688 + 0.998929i \(0.514733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4674.72 0.309524
\(612\) 0 0
\(613\) 1500.35 0.0988558 0.0494279 0.998778i \(-0.484260\pi\)
0.0494279 + 0.998778i \(0.484260\pi\)
\(614\) 0 0
\(615\) 2353.21 0.154294
\(616\) 0 0
\(617\) 15497.7 1.01121 0.505604 0.862766i \(-0.331270\pi\)
0.505604 + 0.862766i \(0.331270\pi\)
\(618\) 0 0
\(619\) −515.002 −0.0334405 −0.0167203 0.999860i \(-0.505322\pi\)
−0.0167203 + 0.999860i \(0.505322\pi\)
\(620\) 0 0
\(621\) −1101.38 −0.0711703
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9557.30 0.611667
\(626\) 0 0
\(627\) −1915.20 −0.121987
\(628\) 0 0
\(629\) 4180.73 0.265018
\(630\) 0 0
\(631\) 22060.1 1.39176 0.695880 0.718158i \(-0.255015\pi\)
0.695880 + 0.718158i \(0.255015\pi\)
\(632\) 0 0
\(633\) 9690.89 0.608497
\(634\) 0 0
\(635\) 6694.71 0.418380
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −95.2142 −0.00589455
\(640\) 0 0
\(641\) −17094.1 −1.05332 −0.526660 0.850076i \(-0.676556\pi\)
−0.526660 + 0.850076i \(0.676556\pi\)
\(642\) 0 0
\(643\) −2999.98 −0.183993 −0.0919965 0.995759i \(-0.529325\pi\)
−0.0919965 + 0.995759i \(0.529325\pi\)
\(644\) 0 0
\(645\) −1629.73 −0.0994892
\(646\) 0 0
\(647\) 8255.72 0.501648 0.250824 0.968033i \(-0.419299\pi\)
0.250824 + 0.968033i \(0.419299\pi\)
\(648\) 0 0
\(649\) −2930.35 −0.177236
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7953.02 −0.476609 −0.238304 0.971191i \(-0.576592\pi\)
−0.238304 + 0.971191i \(0.576592\pi\)
\(654\) 0 0
\(655\) −6318.88 −0.376945
\(656\) 0 0
\(657\) −5258.62 −0.312266
\(658\) 0 0
\(659\) −4420.98 −0.261331 −0.130665 0.991427i \(-0.541711\pi\)
−0.130665 + 0.991427i \(0.541711\pi\)
\(660\) 0 0
\(661\) −5830.81 −0.343105 −0.171552 0.985175i \(-0.554878\pi\)
−0.171552 + 0.985175i \(0.554878\pi\)
\(662\) 0 0
\(663\) 5356.60 0.313776
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7293.79 0.423413
\(668\) 0 0
\(669\) −1299.51 −0.0751000
\(670\) 0 0
\(671\) −4756.75 −0.273670
\(672\) 0 0
\(673\) 24614.2 1.40982 0.704910 0.709297i \(-0.250987\pi\)
0.704910 + 0.709297i \(0.250987\pi\)
\(674\) 0 0
\(675\) −2917.45 −0.166360
\(676\) 0 0
\(677\) −130.989 −0.00743619 −0.00371810 0.999993i \(-0.501184\pi\)
−0.00371810 + 0.999993i \(0.501184\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12325.8 −0.693576
\(682\) 0 0
\(683\) −4912.86 −0.275235 −0.137617 0.990485i \(-0.543944\pi\)
−0.137617 + 0.990485i \(0.543944\pi\)
\(684\) 0 0
\(685\) 10153.5 0.566346
\(686\) 0 0
\(687\) −13899.0 −0.771879
\(688\) 0 0
\(689\) −7540.22 −0.416922
\(690\) 0 0
\(691\) −9639.37 −0.530678 −0.265339 0.964155i \(-0.585484\pi\)
−0.265339 + 0.964155i \(0.585484\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5444.92 −0.297176
\(696\) 0 0
\(697\) −14525.0 −0.789346
\(698\) 0 0
\(699\) 1929.66 0.104416
\(700\) 0 0
\(701\) −18377.9 −0.990188 −0.495094 0.868839i \(-0.664866\pi\)
−0.495094 + 0.868839i \(0.664866\pi\)
\(702\) 0 0
\(703\) 1952.28 0.104739
\(704\) 0 0
\(705\) −2464.68 −0.131667
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8667.23 −0.459104 −0.229552 0.973296i \(-0.573726\pi\)
−0.229552 + 0.973296i \(0.573726\pi\)
\(710\) 0 0
\(711\) −1647.70 −0.0869110
\(712\) 0 0
\(713\) 1292.27 0.0678765
\(714\) 0 0
\(715\) −1729.33 −0.0904523
\(716\) 0 0
\(717\) 2350.84 0.122446
\(718\) 0 0
\(719\) −13439.1 −0.697070 −0.348535 0.937296i \(-0.613321\pi\)
−0.348535 + 0.937296i \(0.613321\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 19015.7 0.978150
\(724\) 0 0
\(725\) 19320.6 0.989722
\(726\) 0 0
\(727\) 25085.9 1.27976 0.639879 0.768476i \(-0.278985\pi\)
0.639879 + 0.768476i \(0.278985\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 10059.4 0.508973
\(732\) 0 0
\(733\) −31020.6 −1.56313 −0.781563 0.623826i \(-0.785577\pi\)
−0.781563 + 0.623826i \(0.785577\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4988.05 0.249304
\(738\) 0 0
\(739\) 12992.4 0.646730 0.323365 0.946274i \(-0.395186\pi\)
0.323365 + 0.946274i \(0.395186\pi\)
\(740\) 0 0
\(741\) 2501.39 0.124009
\(742\) 0 0
\(743\) −23950.7 −1.18259 −0.591297 0.806454i \(-0.701384\pi\)
−0.591297 + 0.806454i \(0.701384\pi\)
\(744\) 0 0
\(745\) −2667.83 −0.131197
\(746\) 0 0
\(747\) 1576.05 0.0771950
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13677.5 0.664580 0.332290 0.943177i \(-0.392179\pi\)
0.332290 + 0.943177i \(0.392179\pi\)
\(752\) 0 0
\(753\) −17589.0 −0.851235
\(754\) 0 0
\(755\) 8248.41 0.397603
\(756\) 0 0
\(757\) −9295.64 −0.446309 −0.223154 0.974783i \(-0.571635\pi\)
−0.223154 + 0.974783i \(0.571635\pi\)
\(758\) 0 0
\(759\) −2194.73 −0.104959
\(760\) 0 0
\(761\) 19586.0 0.932970 0.466485 0.884529i \(-0.345520\pi\)
0.466485 + 0.884529i \(0.345520\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2824.20 −0.133476
\(766\) 0 0
\(767\) 3827.24 0.180174
\(768\) 0 0
\(769\) −5952.29 −0.279122 −0.139561 0.990213i \(-0.544569\pi\)
−0.139561 + 0.990213i \(0.544569\pi\)
\(770\) 0 0
\(771\) 12641.3 0.590486
\(772\) 0 0
\(773\) −34379.6 −1.59968 −0.799838 0.600216i \(-0.795081\pi\)
−0.799838 + 0.600216i \(0.795081\pi\)
\(774\) 0 0
\(775\) 3423.11 0.158660
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6782.78 −0.311962
\(780\) 0 0
\(781\) −189.735 −0.00869300
\(782\) 0 0
\(783\) −4827.75 −0.220344
\(784\) 0 0
\(785\) 3184.55 0.144792
\(786\) 0 0
\(787\) 19167.6 0.868172 0.434086 0.900872i \(-0.357071\pi\)
0.434086 + 0.900872i \(0.357071\pi\)
\(788\) 0 0
\(789\) −5867.55 −0.264753
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6212.65 0.278206
\(794\) 0 0
\(795\) 3975.48 0.177353
\(796\) 0 0
\(797\) −2679.24 −0.119076 −0.0595380 0.998226i \(-0.518963\pi\)
−0.0595380 + 0.998226i \(0.518963\pi\)
\(798\) 0 0
\(799\) 15213.1 0.673592
\(800\) 0 0
\(801\) −424.293 −0.0187162
\(802\) 0 0
\(803\) −10478.9 −0.460515
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4583.98 −0.199955
\(808\) 0 0
\(809\) 8075.98 0.350972 0.175486 0.984482i \(-0.443850\pi\)
0.175486 + 0.984482i \(0.443850\pi\)
\(810\) 0 0
\(811\) 31378.3 1.35862 0.679311 0.733851i \(-0.262279\pi\)
0.679311 + 0.733851i \(0.262279\pi\)
\(812\) 0 0
\(813\) −23021.6 −0.993117
\(814\) 0 0
\(815\) 273.459 0.0117532
\(816\) 0 0
\(817\) 4697.45 0.201154
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3099.20 0.131745 0.0658726 0.997828i \(-0.479017\pi\)
0.0658726 + 0.997828i \(0.479017\pi\)
\(822\) 0 0
\(823\) −965.740 −0.0409035 −0.0204517 0.999791i \(-0.506510\pi\)
−0.0204517 + 0.999791i \(0.506510\pi\)
\(824\) 0 0
\(825\) −5813.64 −0.245339
\(826\) 0 0
\(827\) −7538.11 −0.316960 −0.158480 0.987362i \(-0.550659\pi\)
−0.158480 + 0.987362i \(0.550659\pi\)
\(828\) 0 0
\(829\) −12841.2 −0.537991 −0.268995 0.963141i \(-0.586692\pi\)
−0.268995 + 0.963141i \(0.586692\pi\)
\(830\) 0 0
\(831\) 18936.6 0.790498
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3468.71 −0.143760
\(836\) 0 0
\(837\) −855.352 −0.0353229
\(838\) 0 0
\(839\) −13909.3 −0.572351 −0.286176 0.958177i \(-0.592384\pi\)
−0.286176 + 0.958177i \(0.592384\pi\)
\(840\) 0 0
\(841\) 7582.39 0.310894
\(842\) 0 0
\(843\) −10004.0 −0.408725
\(844\) 0 0
\(845\) −6785.52 −0.276248
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2177.70 0.0880311
\(850\) 0 0
\(851\) 2237.22 0.0901188
\(852\) 0 0
\(853\) −33036.0 −1.32606 −0.663030 0.748593i \(-0.730730\pi\)
−0.663030 + 0.748593i \(0.730730\pi\)
\(854\) 0 0
\(855\) −1318.82 −0.0527518
\(856\) 0 0
\(857\) −15827.5 −0.630873 −0.315437 0.948947i \(-0.602151\pi\)
−0.315437 + 0.948947i \(0.602151\pi\)
\(858\) 0 0
\(859\) −6933.95 −0.275417 −0.137709 0.990473i \(-0.543974\pi\)
−0.137709 + 0.990473i \(0.543974\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25595.4 −1.00959 −0.504795 0.863239i \(-0.668432\pi\)
−0.504795 + 0.863239i \(0.668432\pi\)
\(864\) 0 0
\(865\) −6465.91 −0.254159
\(866\) 0 0
\(867\) 2693.14 0.105495
\(868\) 0 0
\(869\) −3283.40 −0.128172
\(870\) 0 0
\(871\) −6514.74 −0.253437
\(872\) 0 0
\(873\) 5011.24 0.194278
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −44274.9 −1.70474 −0.852369 0.522940i \(-0.824835\pi\)
−0.852369 + 0.522940i \(0.824835\pi\)
\(878\) 0 0
\(879\) 17044.4 0.654032
\(880\) 0 0
\(881\) −5773.97 −0.220806 −0.110403 0.993887i \(-0.535214\pi\)
−0.110403 + 0.993887i \(0.535214\pi\)
\(882\) 0 0
\(883\) −48171.4 −1.83590 −0.917948 0.396701i \(-0.870155\pi\)
−0.917948 + 0.396701i \(0.870155\pi\)
\(884\) 0 0
\(885\) −2017.86 −0.0766437
\(886\) 0 0
\(887\) −48168.5 −1.82338 −0.911692 0.410875i \(-0.865223\pi\)
−0.911692 + 0.410875i \(0.865223\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1452.69 0.0546205
\(892\) 0 0
\(893\) 7104.08 0.266214
\(894\) 0 0
\(895\) 15110.9 0.564361
\(896\) 0 0
\(897\) 2866.47 0.106699
\(898\) 0 0
\(899\) 5664.51 0.210147
\(900\) 0 0
\(901\) −24538.3 −0.907315
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3303.22 0.121329
\(906\) 0 0
\(907\) 31992.5 1.17122 0.585608 0.810595i \(-0.300856\pi\)
0.585608 + 0.810595i \(0.300856\pi\)
\(908\) 0 0
\(909\) −1159.23 −0.0422985
\(910\) 0 0
\(911\) −21184.7 −0.770450 −0.385225 0.922823i \(-0.625876\pi\)
−0.385225 + 0.922823i \(0.625876\pi\)
\(912\) 0 0
\(913\) 3140.61 0.113844
\(914\) 0 0
\(915\) −3275.54 −0.118345
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 21647.9 0.777040 0.388520 0.921440i \(-0.372986\pi\)
0.388520 + 0.921440i \(0.372986\pi\)
\(920\) 0 0
\(921\) 7407.11 0.265008
\(922\) 0 0
\(923\) 247.807 0.00883711
\(924\) 0 0
\(925\) 5926.20 0.210651
\(926\) 0 0
\(927\) −18391.3 −0.651618
\(928\) 0 0
\(929\) −9480.99 −0.334834 −0.167417 0.985886i \(-0.553543\pi\)
−0.167417 + 0.985886i \(0.553543\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −12204.0 −0.428233
\(934\) 0 0
\(935\) −5627.82 −0.196844
\(936\) 0 0
\(937\) 2941.86 0.102568 0.0512841 0.998684i \(-0.483669\pi\)
0.0512841 + 0.998684i \(0.483669\pi\)
\(938\) 0 0
\(939\) −14736.4 −0.512145
\(940\) 0 0
\(941\) −52683.8 −1.82513 −0.912563 0.408937i \(-0.865900\pi\)
−0.912563 + 0.408937i \(0.865900\pi\)
\(942\) 0 0
\(943\) −7772.74 −0.268415
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45003.1 −1.54425 −0.772125 0.635471i \(-0.780806\pi\)
−0.772125 + 0.635471i \(0.780806\pi\)
\(948\) 0 0
\(949\) 13686.2 0.468149
\(950\) 0 0
\(951\) 8283.50 0.282451
\(952\) 0 0
\(953\) 40137.2 1.36429 0.682146 0.731216i \(-0.261047\pi\)
0.682146 + 0.731216i \(0.261047\pi\)
\(954\) 0 0
\(955\) −18767.9 −0.635933
\(956\) 0 0
\(957\) −9620.31 −0.324954
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28787.4 −0.966312
\(962\) 0 0
\(963\) −2419.74 −0.0809709
\(964\) 0 0
\(965\) 12900.2 0.430335
\(966\) 0 0
\(967\) −42055.0 −1.39855 −0.699276 0.714852i \(-0.746494\pi\)
−0.699276 + 0.714852i \(0.746494\pi\)
\(968\) 0 0
\(969\) 8140.33 0.269871
\(970\) 0 0
\(971\) 9772.69 0.322987 0.161494 0.986874i \(-0.448369\pi\)
0.161494 + 0.986874i \(0.448369\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7593.01 0.249406
\(976\) 0 0
\(977\) −11978.1 −0.392236 −0.196118 0.980580i \(-0.562834\pi\)
−0.196118 + 0.980580i \(0.562834\pi\)
\(978\) 0 0
\(979\) −845.494 −0.0276018
\(980\) 0 0
\(981\) 12020.8 0.391228
\(982\) 0 0
\(983\) 56474.0 1.83239 0.916197 0.400729i \(-0.131243\pi\)
0.916197 + 0.400729i \(0.131243\pi\)
\(984\) 0 0
\(985\) −11549.9 −0.373615
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5383.06 0.173075
\(990\) 0 0
\(991\) 48427.6 1.55233 0.776163 0.630533i \(-0.217163\pi\)
0.776163 + 0.630533i \(0.217163\pi\)
\(992\) 0 0
\(993\) 3575.16 0.114254
\(994\) 0 0
\(995\) −21641.7 −0.689535
\(996\) 0 0
\(997\) −29268.6 −0.929734 −0.464867 0.885381i \(-0.653898\pi\)
−0.464867 + 0.885381i \(0.653898\pi\)
\(998\) 0 0
\(999\) −1480.81 −0.0468978
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 1176.4.a.bc.1.3 yes 4
4.3 odd 2 2352.4.a.ck.1.3 4
7.6 odd 2 1176.4.a.bb.1.2 4
28.27 even 2 2352.4.a.cr.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.2 4 7.6 odd 2
1176.4.a.bc.1.3 yes 4 1.1 even 1 trivial
2352.4.a.ck.1.3 4 4.3 odd 2
2352.4.a.cr.1.2 4 28.27 even 2