Properties

Label 2352.4.a.cr.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
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: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.66254\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-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 \(-0.558937\pi\)
−0.184100 + 0.982908i \(0.558937\pi\)
\(6\) 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 \(-0.579048\pi\)
−0.245792 + 0.969323i \(0.579048\pi\)
\(12\) 0 0
\(13\) 23.4236 0.499733 0.249867 0.968280i \(-0.419613\pi\)
0.249867 + 0.968280i \(0.419613\pi\)
\(14\) 0 0
\(15\) −12.3498 −0.212580
\(16\) 0 0
\(17\) 76.2280 1.08753 0.543765 0.839237i \(-0.316998\pi\)
0.543765 + 0.839237i \(0.316998\pi\)
\(18\) 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 \(-0.440802\pi\)
0.184906 + 0.982756i \(0.440802\pi\)
\(24\) 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 \(-0.618216\pi\)
−0.362907 + 0.931825i \(0.618216\pi\)
\(42\) 0 0
\(43\) 131.964 0.468009 0.234004 0.972236i \(-0.424817\pi\)
0.234004 + 0.972236i \(0.424817\pi\)
\(44\) 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 \(-0.410211\pi\)
0.278355 + 0.960478i \(0.410211\pi\)
\(62\) 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 \(-0.581606\pi\)
−0.253572 + 0.967316i \(0.581606\pi\)
\(68\) 0 0
\(69\) 122.375 0.213511
\(70\) 0 0
\(71\) 10.5794 0.0176836 0.00884182 0.999961i \(-0.497186\pi\)
0.00884182 + 0.999961i \(0.497186\pi\)
\(72\) 0 0
\(73\) 584.292 0.936797 0.468398 0.883517i \(-0.344831\pi\)
0.468398 + 0.883517i \(0.344831\pi\)
\(74\) 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 \(-0.458385\pi\)
0.130366 + 0.991466i \(0.458385\pi\)
\(80\) 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 \(-0.491063\pi\)
0.0280743 + 0.999606i \(0.491063\pi\)
\(90\) 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 \(-0.594127\pi\)
−0.291417 + 0.956596i \(0.594127\pi\)
\(98\) 0 0
\(99\) −161.410 −0.163862
\(100\) 0 0
\(101\) 128.804 0.126895 0.0634477 0.997985i \(-0.479790\pi\)
0.0634477 + 0.997985i \(0.479790\pi\)
\(102\) 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 \(-0.461244\pi\)
0.121456 + 0.992597i \(0.461244\pi\)
\(108\) 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 \(-0.692338\pi\)
−0.568144 + 0.822929i \(0.692338\pi\)
\(128\) 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 \(-0.681549\pi\)
−0.539930 + 0.841710i \(0.681549\pi\)
\(152\) 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 \(-0.562997\pi\)
−0.196622 + 0.980479i \(0.562997\pi\)
\(158\) 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 \(-0.505081\pi\)
−0.0159604 + 0.999873i \(0.505081\pi\)
\(164\) 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 \(-0.387832\pi\)
0.345138 + 0.938552i \(0.387832\pi\)
\(174\) 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 \(-0.777944\pi\)
−0.766380 + 0.642387i \(0.777944\pi\)
\(180\) 0 0
\(181\) −802.417 −0.329520 −0.164760 0.986334i \(-0.552685\pi\)
−0.164760 + 0.986334i \(0.552685\pi\)
\(182\) 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 \(-0.168222\pi\)
0.863572 + 0.504225i \(0.168222\pi\)
\(192\) 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 \(-0.676673\pi\)
−0.526974 + 0.849882i \(0.676673\pi\)
\(212\) 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 \(-0.266951\pi\)
0.668467 + 0.743742i \(0.266951\pi\)
\(230\) 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 \(-0.533818\pi\)
−0.106041 + 0.994362i \(0.533818\pi\)
\(240\) 0 0
\(241\) −6338.58 −1.69421 −0.847103 0.531429i \(-0.821655\pi\)
−0.847103 + 0.531429i \(0.821655\pi\)
\(242\) 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 \(-0.670864\pi\)
−0.511376 + 0.859357i \(0.670864\pi\)
\(258\) 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 \(-0.426362\pi\)
0.229283 + 0.973360i \(0.426362\pi\)
\(264\) 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 \(-0.444600\pi\)
0.173166 + 0.984893i \(0.444600\pi\)
\(270\) 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 \(-0.691667\pi\)
−0.566408 + 0.824125i \(0.691667\pi\)
\(294\) 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 \(-0.353726\pi\)
0.443531 + 0.896259i \(0.353726\pi\)
\(314\) 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 \(-0.531547\pi\)
−0.0989470 + 0.995093i \(0.531547\pi\)
\(332\) 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 \(-0.239054\pi\)
0.730999 + 0.682379i \(0.239054\pi\)
\(348\) 0 0
\(349\) −10654.7 −1.63418 −0.817092 0.576507i \(-0.804415\pi\)
−0.817092 + 0.576507i \(0.804415\pi\)
\(350\) 0 0
\(351\) 632.437 0.0961738
\(352\) 0 0
\(353\) −1702.81 −0.256746 −0.128373 0.991726i \(-0.540975\pi\)
−0.128373 + 0.991726i \(0.540975\pi\)
\(354\) 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 \(-0.402342\pi\)
0.302011 + 0.953304i \(0.402342\pi\)
\(360\) 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 \(-0.846640\pi\)
−0.886164 + 0.463372i \(0.846640\pi\)
\(380\) 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 \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(398\) 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 \(-0.657810\pi\)
−0.475713 + 0.879600i \(0.657810\pi\)
\(410\) 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 \(-0.250669\pi\)
0.705618 + 0.708592i \(0.250669\pi\)
\(432\) 0 0
\(433\) −6738.74 −0.747906 −0.373953 0.927448i \(-0.621998\pi\)
−0.373953 + 0.927448i \(0.621998\pi\)
\(434\) 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 \(-0.222180\pi\)
0.766130 + 0.642686i \(0.222180\pi\)
\(444\) 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 \(-0.403842\pi\)
0.297515 + 0.954717i \(0.403842\pi\)
\(462\) 0 0
\(463\) −12716.5 −1.27643 −0.638215 0.769858i \(-0.720327\pi\)
−0.638215 + 0.769858i \(0.720327\pi\)
\(464\) 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 \(-0.334831\pi\)
0.495921 + 0.868368i \(0.334831\pi\)
\(488\) 0 0
\(489\) −199.286 −0.0184295
\(490\) 0 0
\(491\) 277.130 0.0254719 0.0127360 0.999919i \(-0.495946\pi\)
0.0127360 + 0.999919i \(0.495946\pi\)
\(492\) 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 \(-0.801766\pi\)
−0.812266 + 0.583288i \(0.801766\pi\)
\(500\) 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 \(-0.772868\pi\)
−0.756040 + 0.654526i \(0.772868\pi\)
\(510\) 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 \(-0.461490\pi\)
0.120688 + 0.992690i \(0.461490\pi\)
\(522\) 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 \(-0.748470\pi\)
−0.703699 + 0.710498i \(0.748470\pi\)
\(548\) 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 \(-0.619598\pi\)
−0.366951 + 0.930240i \(0.619598\pi\)
\(572\) 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 \(-0.906056\pi\)
−0.956764 + 0.290867i \(0.906056\pi\)
\(578\) 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 \(-0.861066\pi\)
−0.906248 + 0.422746i \(0.861066\pi\)
\(594\) 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 \(-0.435012\pi\)
0.202750 + 0.979231i \(0.435012\pi\)
\(600\) 0 0
\(601\) −1988.96 −0.134994 −0.0674970 0.997719i \(-0.521501\pi\)
−0.0674970 + 0.997719i \(0.521501\pi\)
\(602\) 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 \(-0.744985\pi\)
−0.695880 + 0.718158i \(0.744985\pi\)
\(632\) 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 \(-0.458289\pi\)
0.130665 + 0.991427i \(0.458289\pi\)
\(660\) 0 0
\(661\) 5830.81 0.343105 0.171552 0.985175i \(-0.445122\pi\)
0.171552 + 0.985175i \(0.445122\pi\)
\(662\) 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 \(-0.498816\pi\)
0.00371810 + 0.999993i \(0.498816\pi\)
\(678\) 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 \(-0.456056\pi\)
0.137617 + 0.990485i \(0.456056\pi\)
\(684\) 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 \(-0.214423\pi\)
0.781563 + 0.623826i \(0.214423\pi\)
\(734\) 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 \(-0.604814\pi\)
−0.323365 + 0.946274i \(0.604814\pi\)
\(740\) 0 0
\(741\) −2501.39 −0.124009
\(742\) 0 0
\(743\) 23950.7 1.18259 0.591297 0.806454i \(-0.298616\pi\)
0.591297 + 0.806454i \(0.298616\pi\)
\(744\) 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 \(-0.607821\pi\)
−0.332290 + 0.943177i \(0.607821\pi\)
\(752\) 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 \(-0.654480\pi\)
−0.466485 + 0.884529i \(0.654480\pi\)
\(762\) 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 \(-0.455431\pi\)
0.139561 + 0.990213i \(0.455431\pi\)
\(770\) 0 0
\(771\) −12641.3 −0.590486
\(772\) 0 0
\(773\) 34379.6 1.59968 0.799838 0.600216i \(-0.204919\pi\)
0.799838 + 0.600216i \(0.204919\pi\)
\(774\) 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 \(-0.481037\pi\)
0.0595380 + 0.998226i \(0.481037\pi\)
\(798\) 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 \(-0.493490\pi\)
0.0204517 + 0.999791i \(0.493490\pi\)
\(824\) 0 0
\(825\) 5813.64 0.245339
\(826\) 0 0
\(827\) 7538.11 0.316960 0.158480 0.987362i \(-0.449341\pi\)
0.158480 + 0.987362i \(0.449341\pi\)
\(828\) 0 0
\(829\) 12841.2 0.537991 0.268995 0.963141i \(-0.413308\pi\)
0.268995 + 0.963141i \(0.413308\pi\)
\(830\) 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 \(-0.269270\pi\)
0.663030 + 0.748593i \(0.269270\pi\)
\(854\) 0 0
\(855\) 1318.82 0.0527518
\(856\) 0 0
\(857\) 15827.5 0.630873 0.315437 0.948947i \(-0.397849\pi\)
0.315437 + 0.948947i \(0.397849\pi\)
\(858\) 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 \(-0.331568\pi\)
0.504795 + 0.863239i \(0.331568\pi\)
\(864\) 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 \(-0.464786\pi\)
0.110403 + 0.993887i \(0.464786\pi\)
\(882\) 0 0
\(883\) 48171.4 1.83590 0.917948 0.396701i \(-0.129845\pi\)
0.917948 + 0.396701i \(0.129845\pi\)
\(884\) 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 \(-0.699144\pi\)
−0.585608 + 0.810595i \(0.699144\pi\)
\(908\) 0 0
\(909\) 1159.23 0.0422985
\(910\) 0 0
\(911\) 21184.7 0.770450 0.385225 0.922823i \(-0.374124\pi\)
0.385225 + 0.922823i \(0.374124\pi\)
\(912\) 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 \(-0.627014\pi\)
−0.388520 + 0.921440i \(0.627014\pi\)
\(920\) 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 \(-0.446457\pi\)
0.167417 + 0.985886i \(0.446457\pi\)
\(930\) 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 \(-0.516331\pi\)
−0.0512841 + 0.998684i \(0.516331\pi\)
\(938\) 0 0
\(939\) 14736.4 0.512145
\(940\) 0 0
\(941\) 52683.8 1.82513 0.912563 0.408937i \(-0.134100\pi\)
0.912563 + 0.408937i \(0.134100\pi\)
\(942\) 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 \(-0.219194\pi\)
0.772125 + 0.635471i \(0.219194\pi\)
\(948\) 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 \(-0.253506\pi\)
0.699276 + 0.714852i \(0.253506\pi\)
\(968\) 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 \(-0.782837\pi\)
−0.776163 + 0.630533i \(0.782837\pi\)
\(992\) 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 \(-0.346102\pi\)
0.464867 + 0.885381i \(0.346102\pi\)
\(998\) 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 2352.4.a.cr.1.2 4
4.3 odd 2 1176.4.a.bb.1.2 4
7.6 odd 2 2352.4.a.ck.1.3 4
28.27 even 2 1176.4.a.bc.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.2 4 4.3 odd 2
1176.4.a.bc.1.3 yes 4 28.27 even 2
2352.4.a.ck.1.3 4 7.6 odd 2
2352.4.a.cr.1.2 4 1.1 even 1 trivial