Properties

Label 2312.4.a.k.1.1
Level $2312$
Weight $4$
Character 2312.1
Self dual yes
Analytic conductor $136.412$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(136.412415933\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.20783\) of defining polynomial
Character \(\chi\) \(=\) 2312.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.26065 q^{3} +16.4090 q^{5} -34.5010 q^{7} +58.7597 q^{9} +O(q^{10})\) \(q-9.26065 q^{3} +16.4090 q^{5} -34.5010 q^{7} +58.7597 q^{9} -7.42843 q^{11} -42.0242 q^{13} -151.958 q^{15} -59.9095 q^{19} +319.502 q^{21} -49.4728 q^{23} +144.255 q^{25} -294.116 q^{27} -259.369 q^{29} -92.2215 q^{31} +68.7922 q^{33} -566.126 q^{35} +207.564 q^{37} +389.172 q^{39} -176.800 q^{41} +19.0809 q^{43} +964.188 q^{45} +80.1389 q^{47} +847.318 q^{49} -319.869 q^{53} -121.893 q^{55} +554.801 q^{57} -11.0809 q^{59} -712.648 q^{61} -2027.27 q^{63} -689.575 q^{65} +484.980 q^{67} +458.150 q^{69} -443.968 q^{71} +337.468 q^{73} -1335.89 q^{75} +256.288 q^{77} -840.905 q^{79} +1137.19 q^{81} -456.004 q^{83} +2401.93 q^{87} -1205.43 q^{89} +1449.88 q^{91} +854.032 q^{93} -983.055 q^{95} -638.484 q^{97} -436.493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −9.26065 −1.78221 −0.891107 0.453794i \(-0.850070\pi\)
−0.891107 + 0.453794i \(0.850070\pi\)
\(4\) 0 0
\(5\) 16.4090 1.46766 0.733832 0.679331i \(-0.237730\pi\)
0.733832 + 0.679331i \(0.237730\pi\)
\(6\) 0 0
\(7\) −34.5010 −1.86288 −0.931439 0.363898i \(-0.881446\pi\)
−0.931439 + 0.363898i \(0.881446\pi\)
\(8\) 0 0
\(9\) 58.7597 2.17629
\(10\) 0 0
\(11\) −7.42843 −0.203614 −0.101807 0.994804i \(-0.532462\pi\)
−0.101807 + 0.994804i \(0.532462\pi\)
\(12\) 0 0
\(13\) −42.0242 −0.896571 −0.448285 0.893890i \(-0.647965\pi\)
−0.448285 + 0.893890i \(0.647965\pi\)
\(14\) 0 0
\(15\) −151.958 −2.61569
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −59.9095 −0.723378 −0.361689 0.932299i \(-0.617800\pi\)
−0.361689 + 0.932299i \(0.617800\pi\)
\(20\) 0 0
\(21\) 319.502 3.32005
\(22\) 0 0
\(23\) −49.4728 −0.448513 −0.224256 0.974530i \(-0.571995\pi\)
−0.224256 + 0.974530i \(0.571995\pi\)
\(24\) 0 0
\(25\) 144.255 1.15404
\(26\) 0 0
\(27\) −294.116 −2.09639
\(28\) 0 0
\(29\) −259.369 −1.66081 −0.830407 0.557157i \(-0.811892\pi\)
−0.830407 + 0.557157i \(0.811892\pi\)
\(30\) 0 0
\(31\) −92.2215 −0.534306 −0.267153 0.963654i \(-0.586083\pi\)
−0.267153 + 0.963654i \(0.586083\pi\)
\(32\) 0 0
\(33\) 68.7922 0.362884
\(34\) 0 0
\(35\) −566.126 −2.73408
\(36\) 0 0
\(37\) 207.564 0.922252 0.461126 0.887335i \(-0.347446\pi\)
0.461126 + 0.887335i \(0.347446\pi\)
\(38\) 0 0
\(39\) 389.172 1.59788
\(40\) 0 0
\(41\) −176.800 −0.673454 −0.336727 0.941602i \(-0.609320\pi\)
−0.336727 + 0.941602i \(0.609320\pi\)
\(42\) 0 0
\(43\) 19.0809 0.0676700 0.0338350 0.999427i \(-0.489228\pi\)
0.0338350 + 0.999427i \(0.489228\pi\)
\(44\) 0 0
\(45\) 964.188 3.19406
\(46\) 0 0
\(47\) 80.1389 0.248712 0.124356 0.992238i \(-0.460314\pi\)
0.124356 + 0.992238i \(0.460314\pi\)
\(48\) 0 0
\(49\) 847.318 2.47031
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −319.869 −0.829006 −0.414503 0.910048i \(-0.636045\pi\)
−0.414503 + 0.910048i \(0.636045\pi\)
\(54\) 0 0
\(55\) −121.893 −0.298837
\(56\) 0 0
\(57\) 554.801 1.28921
\(58\) 0 0
\(59\) −11.0809 −0.0244510 −0.0122255 0.999925i \(-0.503892\pi\)
−0.0122255 + 0.999925i \(0.503892\pi\)
\(60\) 0 0
\(61\) −712.648 −1.49582 −0.747912 0.663798i \(-0.768943\pi\)
−0.747912 + 0.663798i \(0.768943\pi\)
\(62\) 0 0
\(63\) −2027.27 −4.05415
\(64\) 0 0
\(65\) −689.575 −1.31587
\(66\) 0 0
\(67\) 484.980 0.884325 0.442163 0.896935i \(-0.354211\pi\)
0.442163 + 0.896935i \(0.354211\pi\)
\(68\) 0 0
\(69\) 458.150 0.799345
\(70\) 0 0
\(71\) −443.968 −0.742103 −0.371051 0.928612i \(-0.621003\pi\)
−0.371051 + 0.928612i \(0.621003\pi\)
\(72\) 0 0
\(73\) 337.468 0.541063 0.270531 0.962711i \(-0.412801\pi\)
0.270531 + 0.962711i \(0.412801\pi\)
\(74\) 0 0
\(75\) −1335.89 −2.05674
\(76\) 0 0
\(77\) 256.288 0.379309
\(78\) 0 0
\(79\) −840.905 −1.19759 −0.598793 0.800904i \(-0.704353\pi\)
−0.598793 + 0.800904i \(0.704353\pi\)
\(80\) 0 0
\(81\) 1137.19 1.55993
\(82\) 0 0
\(83\) −456.004 −0.603047 −0.301524 0.953459i \(-0.597495\pi\)
−0.301524 + 0.953459i \(0.597495\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2401.93 2.95993
\(88\) 0 0
\(89\) −1205.43 −1.43568 −0.717841 0.696207i \(-0.754870\pi\)
−0.717841 + 0.696207i \(0.754870\pi\)
\(90\) 0 0
\(91\) 1449.88 1.67020
\(92\) 0 0
\(93\) 854.032 0.952247
\(94\) 0 0
\(95\) −983.055 −1.06168
\(96\) 0 0
\(97\) −638.484 −0.668332 −0.334166 0.942514i \(-0.608455\pi\)
−0.334166 + 0.942514i \(0.608455\pi\)
\(98\) 0 0
\(99\) −436.493 −0.443123
\(100\) 0 0
\(101\) 739.395 0.728441 0.364220 0.931313i \(-0.381335\pi\)
0.364220 + 0.931313i \(0.381335\pi\)
\(102\) 0 0
\(103\) 1588.54 1.51965 0.759824 0.650129i \(-0.225285\pi\)
0.759824 + 0.650129i \(0.225285\pi\)
\(104\) 0 0
\(105\) 5242.70 4.87271
\(106\) 0 0
\(107\) 202.298 0.182775 0.0913875 0.995815i \(-0.470870\pi\)
0.0913875 + 0.995815i \(0.470870\pi\)
\(108\) 0 0
\(109\) −1244.71 −1.09378 −0.546890 0.837204i \(-0.684188\pi\)
−0.546890 + 0.837204i \(0.684188\pi\)
\(110\) 0 0
\(111\) −1922.18 −1.64365
\(112\) 0 0
\(113\) 1492.28 1.24232 0.621159 0.783685i \(-0.286662\pi\)
0.621159 + 0.783685i \(0.286662\pi\)
\(114\) 0 0
\(115\) −811.798 −0.658266
\(116\) 0 0
\(117\) −2469.33 −1.95119
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1275.82 −0.958541
\(122\) 0 0
\(123\) 1637.29 1.20024
\(124\) 0 0
\(125\) 315.954 0.226078
\(126\) 0 0
\(127\) −2054.49 −1.43548 −0.717741 0.696311i \(-0.754824\pi\)
−0.717741 + 0.696311i \(0.754824\pi\)
\(128\) 0 0
\(129\) −176.702 −0.120602
\(130\) 0 0
\(131\) −1892.72 −1.26235 −0.631175 0.775640i \(-0.717427\pi\)
−0.631175 + 0.775640i \(0.717427\pi\)
\(132\) 0 0
\(133\) 2066.94 1.34757
\(134\) 0 0
\(135\) −4826.14 −3.07680
\(136\) 0 0
\(137\) −848.618 −0.529214 −0.264607 0.964356i \(-0.585242\pi\)
−0.264607 + 0.964356i \(0.585242\pi\)
\(138\) 0 0
\(139\) −20.6414 −0.0125955 −0.00629777 0.999980i \(-0.502005\pi\)
−0.00629777 + 0.999980i \(0.502005\pi\)
\(140\) 0 0
\(141\) −742.139 −0.443258
\(142\) 0 0
\(143\) 312.174 0.182555
\(144\) 0 0
\(145\) −4255.98 −2.43752
\(146\) 0 0
\(147\) −7846.71 −4.40263
\(148\) 0 0
\(149\) −1202.59 −0.661206 −0.330603 0.943770i \(-0.607252\pi\)
−0.330603 + 0.943770i \(0.607252\pi\)
\(150\) 0 0
\(151\) 2080.77 1.12140 0.560698 0.828020i \(-0.310533\pi\)
0.560698 + 0.828020i \(0.310533\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1513.26 −0.784182
\(156\) 0 0
\(157\) 1153.74 0.586486 0.293243 0.956038i \(-0.405266\pi\)
0.293243 + 0.956038i \(0.405266\pi\)
\(158\) 0 0
\(159\) 2962.19 1.47747
\(160\) 0 0
\(161\) 1706.86 0.835524
\(162\) 0 0
\(163\) −1461.35 −0.702221 −0.351110 0.936334i \(-0.614196\pi\)
−0.351110 + 0.936334i \(0.614196\pi\)
\(164\) 0 0
\(165\) 1128.81 0.532592
\(166\) 0 0
\(167\) −4157.98 −1.92667 −0.963337 0.268296i \(-0.913539\pi\)
−0.963337 + 0.268296i \(0.913539\pi\)
\(168\) 0 0
\(169\) −430.965 −0.196160
\(170\) 0 0
\(171\) −3520.27 −1.57428
\(172\) 0 0
\(173\) 2504.08 1.10047 0.550235 0.835010i \(-0.314538\pi\)
0.550235 + 0.835010i \(0.314538\pi\)
\(174\) 0 0
\(175\) −4976.94 −2.14983
\(176\) 0 0
\(177\) 102.616 0.0435769
\(178\) 0 0
\(179\) −839.685 −0.350620 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(180\) 0 0
\(181\) 500.491 0.205532 0.102766 0.994706i \(-0.467231\pi\)
0.102766 + 0.994706i \(0.467231\pi\)
\(182\) 0 0
\(183\) 6599.59 2.66588
\(184\) 0 0
\(185\) 3405.92 1.35356
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10147.3 3.90532
\(190\) 0 0
\(191\) 199.937 0.0757430 0.0378715 0.999283i \(-0.487942\pi\)
0.0378715 + 0.999283i \(0.487942\pi\)
\(192\) 0 0
\(193\) −2758.52 −1.02882 −0.514411 0.857544i \(-0.671989\pi\)
−0.514411 + 0.857544i \(0.671989\pi\)
\(194\) 0 0
\(195\) 6385.92 2.34515
\(196\) 0 0
\(197\) 3379.87 1.22236 0.611182 0.791490i \(-0.290694\pi\)
0.611182 + 0.791490i \(0.290694\pi\)
\(198\) 0 0
\(199\) 2729.46 0.972293 0.486147 0.873877i \(-0.338402\pi\)
0.486147 + 0.873877i \(0.338402\pi\)
\(200\) 0 0
\(201\) −4491.24 −1.57606
\(202\) 0 0
\(203\) 8948.48 3.09389
\(204\) 0 0
\(205\) −2901.12 −0.988404
\(206\) 0 0
\(207\) −2907.01 −0.976092
\(208\) 0 0
\(209\) 445.034 0.147290
\(210\) 0 0
\(211\) −2149.01 −0.701157 −0.350579 0.936533i \(-0.614015\pi\)
−0.350579 + 0.936533i \(0.614015\pi\)
\(212\) 0 0
\(213\) 4111.43 1.32259
\(214\) 0 0
\(215\) 313.098 0.0993168
\(216\) 0 0
\(217\) 3181.73 0.995346
\(218\) 0 0
\(219\) −3125.17 −0.964290
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 115.665 0.0347332 0.0173666 0.999849i \(-0.494472\pi\)
0.0173666 + 0.999849i \(0.494472\pi\)
\(224\) 0 0
\(225\) 8476.38 2.51152
\(226\) 0 0
\(227\) 150.739 0.0440743 0.0220372 0.999757i \(-0.492985\pi\)
0.0220372 + 0.999757i \(0.492985\pi\)
\(228\) 0 0
\(229\) 252.262 0.0727946 0.0363973 0.999337i \(-0.488412\pi\)
0.0363973 + 0.999337i \(0.488412\pi\)
\(230\) 0 0
\(231\) −2373.40 −0.676009
\(232\) 0 0
\(233\) −1025.45 −0.288323 −0.144161 0.989554i \(-0.546048\pi\)
−0.144161 + 0.989554i \(0.546048\pi\)
\(234\) 0 0
\(235\) 1315.00 0.365026
\(236\) 0 0
\(237\) 7787.33 2.13435
\(238\) 0 0
\(239\) 3803.45 1.02939 0.514696 0.857373i \(-0.327905\pi\)
0.514696 + 0.857373i \(0.327905\pi\)
\(240\) 0 0
\(241\) −2073.99 −0.554346 −0.277173 0.960820i \(-0.589397\pi\)
−0.277173 + 0.960820i \(0.589397\pi\)
\(242\) 0 0
\(243\) −2590.02 −0.683743
\(244\) 0 0
\(245\) 13903.6 3.62559
\(246\) 0 0
\(247\) 2517.65 0.648560
\(248\) 0 0
\(249\) 4222.89 1.07476
\(250\) 0 0
\(251\) −1242.05 −0.312341 −0.156170 0.987730i \(-0.549915\pi\)
−0.156170 + 0.987730i \(0.549915\pi\)
\(252\) 0 0
\(253\) 367.505 0.0913236
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1469.82 −0.356751 −0.178375 0.983963i \(-0.557084\pi\)
−0.178375 + 0.983963i \(0.557084\pi\)
\(258\) 0 0
\(259\) −7161.16 −1.71804
\(260\) 0 0
\(261\) −15240.4 −3.61441
\(262\) 0 0
\(263\) 4308.70 1.01021 0.505106 0.863057i \(-0.331453\pi\)
0.505106 + 0.863057i \(0.331453\pi\)
\(264\) 0 0
\(265\) −5248.72 −1.21670
\(266\) 0 0
\(267\) 11163.1 2.55869
\(268\) 0 0
\(269\) 3812.72 0.864184 0.432092 0.901830i \(-0.357776\pi\)
0.432092 + 0.901830i \(0.357776\pi\)
\(270\) 0 0
\(271\) 2859.05 0.640868 0.320434 0.947271i \(-0.396171\pi\)
0.320434 + 0.947271i \(0.396171\pi\)
\(272\) 0 0
\(273\) −13426.8 −2.97666
\(274\) 0 0
\(275\) −1071.59 −0.234979
\(276\) 0 0
\(277\) 5377.92 1.16653 0.583263 0.812283i \(-0.301776\pi\)
0.583263 + 0.812283i \(0.301776\pi\)
\(278\) 0 0
\(279\) −5418.91 −1.16280
\(280\) 0 0
\(281\) −1867.29 −0.396418 −0.198209 0.980160i \(-0.563512\pi\)
−0.198209 + 0.980160i \(0.563512\pi\)
\(282\) 0 0
\(283\) 4296.76 0.902529 0.451265 0.892390i \(-0.350973\pi\)
0.451265 + 0.892390i \(0.350973\pi\)
\(284\) 0 0
\(285\) 9103.73 1.89213
\(286\) 0 0
\(287\) 6099.79 1.25456
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 5912.78 1.19111
\(292\) 0 0
\(293\) 8816.10 1.75782 0.878911 0.476985i \(-0.158270\pi\)
0.878911 + 0.476985i \(0.158270\pi\)
\(294\) 0 0
\(295\) −181.826 −0.0358859
\(296\) 0 0
\(297\) 2184.82 0.426855
\(298\) 0 0
\(299\) 2079.06 0.402123
\(300\) 0 0
\(301\) −658.309 −0.126061
\(302\) 0 0
\(303\) −6847.28 −1.29824
\(304\) 0 0
\(305\) −11693.8 −2.19537
\(306\) 0 0
\(307\) 6143.84 1.14217 0.571087 0.820890i \(-0.306522\pi\)
0.571087 + 0.820890i \(0.306522\pi\)
\(308\) 0 0
\(309\) −14710.9 −2.70834
\(310\) 0 0
\(311\) 7409.80 1.35103 0.675516 0.737345i \(-0.263921\pi\)
0.675516 + 0.737345i \(0.263921\pi\)
\(312\) 0 0
\(313\) −9455.98 −1.70761 −0.853807 0.520589i \(-0.825712\pi\)
−0.853807 + 0.520589i \(0.825712\pi\)
\(314\) 0 0
\(315\) −33265.4 −5.95014
\(316\) 0 0
\(317\) −1045.23 −0.185191 −0.0925957 0.995704i \(-0.529516\pi\)
−0.0925957 + 0.995704i \(0.529516\pi\)
\(318\) 0 0
\(319\) 1926.71 0.338165
\(320\) 0 0
\(321\) −1873.42 −0.325744
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6062.20 −1.03468
\(326\) 0 0
\(327\) 11526.9 1.94935
\(328\) 0 0
\(329\) −2764.87 −0.463320
\(330\) 0 0
\(331\) 6370.04 1.05779 0.528896 0.848687i \(-0.322606\pi\)
0.528896 + 0.848687i \(0.322606\pi\)
\(332\) 0 0
\(333\) 12196.4 2.00708
\(334\) 0 0
\(335\) 7958.04 1.29789
\(336\) 0 0
\(337\) 9216.37 1.48976 0.744878 0.667201i \(-0.232508\pi\)
0.744878 + 0.667201i \(0.232508\pi\)
\(338\) 0 0
\(339\) −13819.5 −2.21408
\(340\) 0 0
\(341\) 685.062 0.108792
\(342\) 0 0
\(343\) −17399.4 −2.73901
\(344\) 0 0
\(345\) 7517.79 1.17317
\(346\) 0 0
\(347\) 7810.63 1.20835 0.604174 0.796852i \(-0.293503\pi\)
0.604174 + 0.796852i \(0.293503\pi\)
\(348\) 0 0
\(349\) −8377.96 −1.28499 −0.642495 0.766290i \(-0.722101\pi\)
−0.642495 + 0.766290i \(0.722101\pi\)
\(350\) 0 0
\(351\) 12360.0 1.87956
\(352\) 0 0
\(353\) 500.551 0.0754721 0.0377360 0.999288i \(-0.487985\pi\)
0.0377360 + 0.999288i \(0.487985\pi\)
\(354\) 0 0
\(355\) −7285.06 −1.08916
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12643.0 1.85870 0.929351 0.369198i \(-0.120368\pi\)
0.929351 + 0.369198i \(0.120368\pi\)
\(360\) 0 0
\(361\) −3269.85 −0.476724
\(362\) 0 0
\(363\) 11814.9 1.70833
\(364\) 0 0
\(365\) 5537.50 0.794099
\(366\) 0 0
\(367\) 640.446 0.0910927 0.0455464 0.998962i \(-0.485497\pi\)
0.0455464 + 0.998962i \(0.485497\pi\)
\(368\) 0 0
\(369\) −10388.7 −1.46563
\(370\) 0 0
\(371\) 11035.8 1.54434
\(372\) 0 0
\(373\) −1138.01 −0.157972 −0.0789861 0.996876i \(-0.525168\pi\)
−0.0789861 + 0.996876i \(0.525168\pi\)
\(374\) 0 0
\(375\) −2925.94 −0.402920
\(376\) 0 0
\(377\) 10899.8 1.48904
\(378\) 0 0
\(379\) −12320.8 −1.66987 −0.834933 0.550352i \(-0.814493\pi\)
−0.834933 + 0.550352i \(0.814493\pi\)
\(380\) 0 0
\(381\) 19025.9 2.55833
\(382\) 0 0
\(383\) 9934.77 1.32544 0.662719 0.748868i \(-0.269402\pi\)
0.662719 + 0.748868i \(0.269402\pi\)
\(384\) 0 0
\(385\) 4205.43 0.556698
\(386\) 0 0
\(387\) 1121.19 0.147269
\(388\) 0 0
\(389\) −3323.46 −0.433178 −0.216589 0.976263i \(-0.569493\pi\)
−0.216589 + 0.976263i \(0.569493\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 17527.8 2.24978
\(394\) 0 0
\(395\) −13798.4 −1.75765
\(396\) 0 0
\(397\) 3094.45 0.391199 0.195599 0.980684i \(-0.437335\pi\)
0.195599 + 0.980684i \(0.437335\pi\)
\(398\) 0 0
\(399\) −19141.2 −2.40165
\(400\) 0 0
\(401\) 6261.50 0.779761 0.389881 0.920865i \(-0.372516\pi\)
0.389881 + 0.920865i \(0.372516\pi\)
\(402\) 0 0
\(403\) 3875.54 0.479043
\(404\) 0 0
\(405\) 18660.2 2.28946
\(406\) 0 0
\(407\) −1541.88 −0.187784
\(408\) 0 0
\(409\) 13966.7 1.68853 0.844264 0.535927i \(-0.180038\pi\)
0.844264 + 0.535927i \(0.180038\pi\)
\(410\) 0 0
\(411\) 7858.76 0.943172
\(412\) 0 0
\(413\) 382.302 0.0455492
\(414\) 0 0
\(415\) −7482.56 −0.885071
\(416\) 0 0
\(417\) 191.153 0.0224479
\(418\) 0 0
\(419\) 6308.88 0.735582 0.367791 0.929908i \(-0.380114\pi\)
0.367791 + 0.929908i \(0.380114\pi\)
\(420\) 0 0
\(421\) −3142.63 −0.363806 −0.181903 0.983317i \(-0.558226\pi\)
−0.181903 + 0.983317i \(0.558226\pi\)
\(422\) 0 0
\(423\) 4708.94 0.541268
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24587.1 2.78654
\(428\) 0 0
\(429\) −2890.94 −0.325351
\(430\) 0 0
\(431\) −6900.34 −0.771178 −0.385589 0.922671i \(-0.626002\pi\)
−0.385589 + 0.922671i \(0.626002\pi\)
\(432\) 0 0
\(433\) 15236.0 1.69098 0.845490 0.533991i \(-0.179308\pi\)
0.845490 + 0.533991i \(0.179308\pi\)
\(434\) 0 0
\(435\) 39413.2 4.34418
\(436\) 0 0
\(437\) 2963.89 0.324444
\(438\) 0 0
\(439\) 9594.64 1.04311 0.521557 0.853216i \(-0.325351\pi\)
0.521557 + 0.853216i \(0.325351\pi\)
\(440\) 0 0
\(441\) 49788.1 5.37611
\(442\) 0 0
\(443\) −3375.02 −0.361968 −0.180984 0.983486i \(-0.557928\pi\)
−0.180984 + 0.983486i \(0.557928\pi\)
\(444\) 0 0
\(445\) −19780.0 −2.10710
\(446\) 0 0
\(447\) 11136.7 1.17841
\(448\) 0 0
\(449\) 2918.05 0.306707 0.153354 0.988171i \(-0.450993\pi\)
0.153354 + 0.988171i \(0.450993\pi\)
\(450\) 0 0
\(451\) 1313.35 0.137125
\(452\) 0 0
\(453\) −19269.3 −1.99857
\(454\) 0 0
\(455\) 23791.0 2.45130
\(456\) 0 0
\(457\) 3829.50 0.391983 0.195992 0.980606i \(-0.437207\pi\)
0.195992 + 0.980606i \(0.437207\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6988.84 −0.706080 −0.353040 0.935608i \(-0.614852\pi\)
−0.353040 + 0.935608i \(0.614852\pi\)
\(462\) 0 0
\(463\) 8669.67 0.870224 0.435112 0.900376i \(-0.356709\pi\)
0.435112 + 0.900376i \(0.356709\pi\)
\(464\) 0 0
\(465\) 14013.8 1.39758
\(466\) 0 0
\(467\) −12728.0 −1.26120 −0.630600 0.776108i \(-0.717191\pi\)
−0.630600 + 0.776108i \(0.717191\pi\)
\(468\) 0 0
\(469\) −16732.3 −1.64739
\(470\) 0 0
\(471\) −10684.4 −1.04524
\(472\) 0 0
\(473\) −141.741 −0.0137786
\(474\) 0 0
\(475\) −8642.24 −0.834807
\(476\) 0 0
\(477\) −18795.4 −1.80415
\(478\) 0 0
\(479\) 13282.8 1.26703 0.633516 0.773730i \(-0.281611\pi\)
0.633516 + 0.773730i \(0.281611\pi\)
\(480\) 0 0
\(481\) −8722.72 −0.826864
\(482\) 0 0
\(483\) −15806.6 −1.48908
\(484\) 0 0
\(485\) −10476.9 −0.980888
\(486\) 0 0
\(487\) −5014.37 −0.466577 −0.233288 0.972408i \(-0.574949\pi\)
−0.233288 + 0.972408i \(0.574949\pi\)
\(488\) 0 0
\(489\) 13533.1 1.25151
\(490\) 0 0
\(491\) −14518.4 −1.33443 −0.667217 0.744863i \(-0.732515\pi\)
−0.667217 + 0.744863i \(0.732515\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −7162.40 −0.650356
\(496\) 0 0
\(497\) 15317.3 1.38245
\(498\) 0 0
\(499\) 18158.2 1.62901 0.814504 0.580158i \(-0.197009\pi\)
0.814504 + 0.580158i \(0.197009\pi\)
\(500\) 0 0
\(501\) 38505.6 3.43374
\(502\) 0 0
\(503\) −12331.2 −1.09309 −0.546543 0.837431i \(-0.684057\pi\)
−0.546543 + 0.837431i \(0.684057\pi\)
\(504\) 0 0
\(505\) 12132.7 1.06911
\(506\) 0 0
\(507\) 3991.01 0.349600
\(508\) 0 0
\(509\) −4051.78 −0.352833 −0.176417 0.984316i \(-0.556451\pi\)
−0.176417 + 0.984316i \(0.556451\pi\)
\(510\) 0 0
\(511\) −11643.0 −1.00793
\(512\) 0 0
\(513\) 17620.3 1.51648
\(514\) 0 0
\(515\) 26066.4 2.23033
\(516\) 0 0
\(517\) −595.307 −0.0506413
\(518\) 0 0
\(519\) −23189.4 −1.96127
\(520\) 0 0
\(521\) −6101.23 −0.513052 −0.256526 0.966537i \(-0.582578\pi\)
−0.256526 + 0.966537i \(0.582578\pi\)
\(522\) 0 0
\(523\) −1233.70 −0.103147 −0.0515734 0.998669i \(-0.516424\pi\)
−0.0515734 + 0.998669i \(0.516424\pi\)
\(524\) 0 0
\(525\) 46089.7 3.83146
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9719.44 −0.798836
\(530\) 0 0
\(531\) −651.110 −0.0532124
\(532\) 0 0
\(533\) 7429.90 0.603799
\(534\) 0 0
\(535\) 3319.51 0.268252
\(536\) 0 0
\(537\) 7776.03 0.624880
\(538\) 0 0
\(539\) −6294.24 −0.502991
\(540\) 0 0
\(541\) 8968.57 0.712734 0.356367 0.934346i \(-0.384015\pi\)
0.356367 + 0.934346i \(0.384015\pi\)
\(542\) 0 0
\(543\) −4634.87 −0.366301
\(544\) 0 0
\(545\) −20424.5 −1.60530
\(546\) 0 0
\(547\) −7187.71 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(548\) 0 0
\(549\) −41875.0 −3.25534
\(550\) 0 0
\(551\) 15538.7 1.20140
\(552\) 0 0
\(553\) 29012.0 2.23095
\(554\) 0 0
\(555\) −31541.0 −2.41233
\(556\) 0 0
\(557\) 12502.2 0.951047 0.475524 0.879703i \(-0.342259\pi\)
0.475524 + 0.879703i \(0.342259\pi\)
\(558\) 0 0
\(559\) −801.860 −0.0606709
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20391.1 −1.52643 −0.763216 0.646143i \(-0.776381\pi\)
−0.763216 + 0.646143i \(0.776381\pi\)
\(564\) 0 0
\(565\) 24486.8 1.82331
\(566\) 0 0
\(567\) −39234.2 −2.90597
\(568\) 0 0
\(569\) −3407.53 −0.251057 −0.125528 0.992090i \(-0.540063\pi\)
−0.125528 + 0.992090i \(0.540063\pi\)
\(570\) 0 0
\(571\) −9682.41 −0.709626 −0.354813 0.934937i \(-0.615455\pi\)
−0.354813 + 0.934937i \(0.615455\pi\)
\(572\) 0 0
\(573\) −1851.54 −0.134990
\(574\) 0 0
\(575\) −7136.69 −0.517601
\(576\) 0 0
\(577\) −10357.7 −0.747309 −0.373654 0.927568i \(-0.621895\pi\)
−0.373654 + 0.927568i \(0.621895\pi\)
\(578\) 0 0
\(579\) 25545.7 1.83358
\(580\) 0 0
\(581\) 15732.6 1.12340
\(582\) 0 0
\(583\) 2376.12 0.168798
\(584\) 0 0
\(585\) −40519.2 −2.86370
\(586\) 0 0
\(587\) 760.786 0.0534940 0.0267470 0.999642i \(-0.491485\pi\)
0.0267470 + 0.999642i \(0.491485\pi\)
\(588\) 0 0
\(589\) 5524.95 0.386505
\(590\) 0 0
\(591\) −31299.8 −2.17851
\(592\) 0 0
\(593\) −23861.5 −1.65241 −0.826203 0.563373i \(-0.809503\pi\)
−0.826203 + 0.563373i \(0.809503\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −25276.6 −1.73283
\(598\) 0 0
\(599\) 40.2985 0.00274884 0.00137442 0.999999i \(-0.499563\pi\)
0.00137442 + 0.999999i \(0.499563\pi\)
\(600\) 0 0
\(601\) 28351.4 1.92426 0.962129 0.272594i \(-0.0878815\pi\)
0.962129 + 0.272594i \(0.0878815\pi\)
\(602\) 0 0
\(603\) 28497.3 1.92454
\(604\) 0 0
\(605\) −20934.9 −1.40682
\(606\) 0 0
\(607\) 380.879 0.0254686 0.0127343 0.999919i \(-0.495946\pi\)
0.0127343 + 0.999919i \(0.495946\pi\)
\(608\) 0 0
\(609\) −82868.8 −5.51398
\(610\) 0 0
\(611\) −3367.78 −0.222988
\(612\) 0 0
\(613\) −18449.5 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(614\) 0 0
\(615\) 26866.2 1.76155
\(616\) 0 0
\(617\) 13692.2 0.893397 0.446698 0.894685i \(-0.352600\pi\)
0.446698 + 0.894685i \(0.352600\pi\)
\(618\) 0 0
\(619\) 15000.2 0.974004 0.487002 0.873401i \(-0.338090\pi\)
0.487002 + 0.873401i \(0.338090\pi\)
\(620\) 0 0
\(621\) 14550.7 0.940259
\(622\) 0 0
\(623\) 41588.7 2.67450
\(624\) 0 0
\(625\) −12847.4 −0.822232
\(626\) 0 0
\(627\) −4121.31 −0.262503
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 25659.3 1.61883 0.809413 0.587240i \(-0.199785\pi\)
0.809413 + 0.587240i \(0.199785\pi\)
\(632\) 0 0
\(633\) 19901.3 1.24961
\(634\) 0 0
\(635\) −33712.0 −2.10680
\(636\) 0 0
\(637\) −35607.9 −2.21481
\(638\) 0 0
\(639\) −26087.4 −1.61503
\(640\) 0 0
\(641\) 6709.60 0.413437 0.206719 0.978400i \(-0.433722\pi\)
0.206719 + 0.978400i \(0.433722\pi\)
\(642\) 0 0
\(643\) −11976.4 −0.734528 −0.367264 0.930117i \(-0.619705\pi\)
−0.367264 + 0.930117i \(0.619705\pi\)
\(644\) 0 0
\(645\) −2899.49 −0.177004
\(646\) 0 0
\(647\) 21265.3 1.29215 0.646077 0.763272i \(-0.276408\pi\)
0.646077 + 0.763272i \(0.276408\pi\)
\(648\) 0 0
\(649\) 82.3137 0.00497857
\(650\) 0 0
\(651\) −29464.9 −1.77392
\(652\) 0 0
\(653\) 19671.9 1.17890 0.589449 0.807806i \(-0.299345\pi\)
0.589449 + 0.807806i \(0.299345\pi\)
\(654\) 0 0
\(655\) −31057.6 −1.85271
\(656\) 0 0
\(657\) 19829.5 1.17751
\(658\) 0 0
\(659\) −14782.6 −0.873824 −0.436912 0.899504i \(-0.643928\pi\)
−0.436912 + 0.899504i \(0.643928\pi\)
\(660\) 0 0
\(661\) −27770.8 −1.63413 −0.817063 0.576548i \(-0.804399\pi\)
−0.817063 + 0.576548i \(0.804399\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33916.4 1.97777
\(666\) 0 0
\(667\) 12831.7 0.744896
\(668\) 0 0
\(669\) −1071.14 −0.0619021
\(670\) 0 0
\(671\) 5293.86 0.304571
\(672\) 0 0
\(673\) −21891.3 −1.25386 −0.626929 0.779077i \(-0.715688\pi\)
−0.626929 + 0.779077i \(0.715688\pi\)
\(674\) 0 0
\(675\) −42427.6 −2.41932
\(676\) 0 0
\(677\) −1946.42 −0.110498 −0.0552488 0.998473i \(-0.517595\pi\)
−0.0552488 + 0.998473i \(0.517595\pi\)
\(678\) 0 0
\(679\) 22028.3 1.24502
\(680\) 0 0
\(681\) −1395.94 −0.0785498
\(682\) 0 0
\(683\) 10516.5 0.589169 0.294584 0.955625i \(-0.404819\pi\)
0.294584 + 0.955625i \(0.404819\pi\)
\(684\) 0 0
\(685\) −13925.0 −0.776709
\(686\) 0 0
\(687\) −2336.12 −0.129736
\(688\) 0 0
\(689\) 13442.2 0.743263
\(690\) 0 0
\(691\) −26695.9 −1.46970 −0.734848 0.678232i \(-0.762747\pi\)
−0.734848 + 0.678232i \(0.762747\pi\)
\(692\) 0 0
\(693\) 15059.4 0.825484
\(694\) 0 0
\(695\) −338.704 −0.0184860
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 9496.30 0.513853
\(700\) 0 0
\(701\) −22317.0 −1.20243 −0.601214 0.799088i \(-0.705316\pi\)
−0.601214 + 0.799088i \(0.705316\pi\)
\(702\) 0 0
\(703\) −12435.1 −0.667137
\(704\) 0 0
\(705\) −12177.7 −0.650554
\(706\) 0 0
\(707\) −25509.8 −1.35700
\(708\) 0 0
\(709\) −28249.7 −1.49639 −0.748196 0.663478i \(-0.769080\pi\)
−0.748196 + 0.663478i \(0.769080\pi\)
\(710\) 0 0
\(711\) −49411.3 −2.60629
\(712\) 0 0
\(713\) 4562.46 0.239643
\(714\) 0 0
\(715\) 5122.46 0.267929
\(716\) 0 0
\(717\) −35222.4 −1.83460
\(718\) 0 0
\(719\) 30991.2 1.60748 0.803739 0.594982i \(-0.202841\pi\)
0.803739 + 0.594982i \(0.202841\pi\)
\(720\) 0 0
\(721\) −54806.3 −2.83092
\(722\) 0 0
\(723\) 19206.5 0.987963
\(724\) 0 0
\(725\) −37415.2 −1.91664
\(726\) 0 0
\(727\) 13328.0 0.679929 0.339965 0.940438i \(-0.389585\pi\)
0.339965 + 0.940438i \(0.389585\pi\)
\(728\) 0 0
\(729\) −6718.94 −0.341358
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 27976.7 1.40974 0.704872 0.709334i \(-0.251004\pi\)
0.704872 + 0.709334i \(0.251004\pi\)
\(734\) 0 0
\(735\) −128757. −6.46158
\(736\) 0 0
\(737\) −3602.64 −0.180061
\(738\) 0 0
\(739\) −33971.2 −1.69100 −0.845501 0.533974i \(-0.820698\pi\)
−0.845501 + 0.533974i \(0.820698\pi\)
\(740\) 0 0
\(741\) −23315.1 −1.15587
\(742\) 0 0
\(743\) −25922.4 −1.27995 −0.639974 0.768397i \(-0.721055\pi\)
−0.639974 + 0.768397i \(0.721055\pi\)
\(744\) 0 0
\(745\) −19733.2 −0.970428
\(746\) 0 0
\(747\) −26794.6 −1.31240
\(748\) 0 0
\(749\) −6979.49 −0.340488
\(750\) 0 0
\(751\) −18950.5 −0.920792 −0.460396 0.887714i \(-0.652293\pi\)
−0.460396 + 0.887714i \(0.652293\pi\)
\(752\) 0 0
\(753\) 11502.2 0.556658
\(754\) 0 0
\(755\) 34143.4 1.64583
\(756\) 0 0
\(757\) 8184.07 0.392940 0.196470 0.980510i \(-0.437052\pi\)
0.196470 + 0.980510i \(0.437052\pi\)
\(758\) 0 0
\(759\) −3403.34 −0.162758
\(760\) 0 0
\(761\) −25803.9 −1.22916 −0.614579 0.788855i \(-0.710674\pi\)
−0.614579 + 0.788855i \(0.710674\pi\)
\(762\) 0 0
\(763\) 42943.9 2.03758
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 465.666 0.0219221
\(768\) 0 0
\(769\) 5222.53 0.244901 0.122451 0.992475i \(-0.460925\pi\)
0.122451 + 0.992475i \(0.460925\pi\)
\(770\) 0 0
\(771\) 13611.5 0.635806
\(772\) 0 0
\(773\) −32636.6 −1.51857 −0.759286 0.650757i \(-0.774452\pi\)
−0.759286 + 0.650757i \(0.774452\pi\)
\(774\) 0 0
\(775\) −13303.4 −0.616610
\(776\) 0 0
\(777\) 66317.0 3.06192
\(778\) 0 0
\(779\) 10592.0 0.487162
\(780\) 0 0
\(781\) 3297.99 0.151103
\(782\) 0 0
\(783\) 76284.5 3.48172
\(784\) 0 0
\(785\) 18931.7 0.860764
\(786\) 0 0
\(787\) −17014.8 −0.770664 −0.385332 0.922778i \(-0.625913\pi\)
−0.385332 + 0.922778i \(0.625913\pi\)
\(788\) 0 0
\(789\) −39901.4 −1.80042
\(790\) 0 0
\(791\) −51485.1 −2.31429
\(792\) 0 0
\(793\) 29948.5 1.34111
\(794\) 0 0
\(795\) 48606.6 2.16842
\(796\) 0 0
\(797\) 20826.6 0.925616 0.462808 0.886458i \(-0.346842\pi\)
0.462808 + 0.886458i \(0.346842\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −70831.0 −3.12446
\(802\) 0 0
\(803\) −2506.86 −0.110168
\(804\) 0 0
\(805\) 28007.8 1.22627
\(806\) 0 0
\(807\) −35308.2 −1.54016
\(808\) 0 0
\(809\) 8639.25 0.375451 0.187726 0.982222i \(-0.439888\pi\)
0.187726 + 0.982222i \(0.439888\pi\)
\(810\) 0 0
\(811\) 22319.6 0.966397 0.483199 0.875511i \(-0.339475\pi\)
0.483199 + 0.875511i \(0.339475\pi\)
\(812\) 0 0
\(813\) −26476.7 −1.14216
\(814\) 0 0
\(815\) −23979.3 −1.03062
\(816\) 0 0
\(817\) −1143.13 −0.0489510
\(818\) 0 0
\(819\) 85194.4 3.63484
\(820\) 0 0
\(821\) 9361.94 0.397971 0.198985 0.980002i \(-0.436235\pi\)
0.198985 + 0.980002i \(0.436235\pi\)
\(822\) 0 0
\(823\) 19128.8 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(824\) 0 0
\(825\) 9923.61 0.418783
\(826\) 0 0
\(827\) 40640.9 1.70886 0.854428 0.519570i \(-0.173908\pi\)
0.854428 + 0.519570i \(0.173908\pi\)
\(828\) 0 0
\(829\) −25220.3 −1.05662 −0.528310 0.849052i \(-0.677174\pi\)
−0.528310 + 0.849052i \(0.677174\pi\)
\(830\) 0 0
\(831\) −49803.0 −2.07900
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −68228.3 −2.82771
\(836\) 0 0
\(837\) 27123.8 1.12011
\(838\) 0 0
\(839\) 4544.50 0.187001 0.0935004 0.995619i \(-0.470194\pi\)
0.0935004 + 0.995619i \(0.470194\pi\)
\(840\) 0 0
\(841\) 42883.3 1.75830
\(842\) 0 0
\(843\) 17292.4 0.706501
\(844\) 0 0
\(845\) −7071.69 −0.287898
\(846\) 0 0
\(847\) 44017.0 1.78565
\(848\) 0 0
\(849\) −39790.8 −1.60850
\(850\) 0 0
\(851\) −10268.8 −0.413641
\(852\) 0 0
\(853\) 22395.8 0.898964 0.449482 0.893289i \(-0.351609\pi\)
0.449482 + 0.893289i \(0.351609\pi\)
\(854\) 0 0
\(855\) −57764.0 −2.31051
\(856\) 0 0
\(857\) −5735.42 −0.228610 −0.114305 0.993446i \(-0.536464\pi\)
−0.114305 + 0.993446i \(0.536464\pi\)
\(858\) 0 0
\(859\) 34718.6 1.37903 0.689513 0.724273i \(-0.257825\pi\)
0.689513 + 0.724273i \(0.257825\pi\)
\(860\) 0 0
\(861\) −56488.0 −2.23590
\(862\) 0 0
\(863\) −22239.9 −0.877237 −0.438619 0.898673i \(-0.644532\pi\)
−0.438619 + 0.898673i \(0.644532\pi\)
\(864\) 0 0
\(865\) 41089.4 1.61512
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6246.61 0.243845
\(870\) 0 0
\(871\) −20380.9 −0.792860
\(872\) 0 0
\(873\) −37517.1 −1.45448
\(874\) 0 0
\(875\) −10900.7 −0.421156
\(876\) 0 0
\(877\) −10379.6 −0.399650 −0.199825 0.979832i \(-0.564037\pi\)
−0.199825 + 0.979832i \(0.564037\pi\)
\(878\) 0 0
\(879\) −81642.9 −3.13282
\(880\) 0 0
\(881\) 44934.3 1.71836 0.859181 0.511672i \(-0.170974\pi\)
0.859181 + 0.511672i \(0.170974\pi\)
\(882\) 0 0
\(883\) −15846.0 −0.603918 −0.301959 0.953321i \(-0.597640\pi\)
−0.301959 + 0.953321i \(0.597640\pi\)
\(884\) 0 0
\(885\) 1683.83 0.0639563
\(886\) 0 0
\(887\) −30415.0 −1.15134 −0.575669 0.817683i \(-0.695258\pi\)
−0.575669 + 0.817683i \(0.695258\pi\)
\(888\) 0 0
\(889\) 70881.8 2.67413
\(890\) 0 0
\(891\) −8447.56 −0.317625
\(892\) 0 0
\(893\) −4801.08 −0.179913
\(894\) 0 0
\(895\) −13778.4 −0.514593
\(896\) 0 0
\(897\) −19253.4 −0.716670
\(898\) 0 0
\(899\) 23919.4 0.887382
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 6096.38 0.224667
\(904\) 0 0
\(905\) 8212.55 0.301651
\(906\) 0 0
\(907\) −7288.75 −0.266835 −0.133417 0.991060i \(-0.542595\pi\)
−0.133417 + 0.991060i \(0.542595\pi\)
\(908\) 0 0
\(909\) 43446.6 1.58530
\(910\) 0 0
\(911\) −10232.6 −0.372140 −0.186070 0.982536i \(-0.559575\pi\)
−0.186070 + 0.982536i \(0.559575\pi\)
\(912\) 0 0
\(913\) 3387.39 0.122789
\(914\) 0 0
\(915\) 108293. 3.91261
\(916\) 0 0
\(917\) 65300.8 2.35160
\(918\) 0 0
\(919\) −26533.8 −0.952415 −0.476207 0.879333i \(-0.657989\pi\)
−0.476207 + 0.879333i \(0.657989\pi\)
\(920\) 0 0
\(921\) −56895.9 −2.03560
\(922\) 0 0
\(923\) 18657.4 0.665348
\(924\) 0 0
\(925\) 29942.1 1.06431
\(926\) 0 0
\(927\) 93342.3 3.30719
\(928\) 0 0
\(929\) 7037.69 0.248546 0.124273 0.992248i \(-0.460340\pi\)
0.124273 + 0.992248i \(0.460340\pi\)
\(930\) 0 0
\(931\) −50762.4 −1.78697
\(932\) 0 0
\(933\) −68619.6 −2.40783
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −16895.0 −0.589047 −0.294523 0.955644i \(-0.595161\pi\)
−0.294523 + 0.955644i \(0.595161\pi\)
\(938\) 0 0
\(939\) 87568.5 3.04333
\(940\) 0 0
\(941\) 51103.6 1.77038 0.885190 0.465229i \(-0.154028\pi\)
0.885190 + 0.465229i \(0.154028\pi\)
\(942\) 0 0
\(943\) 8746.81 0.302052
\(944\) 0 0
\(945\) 166507. 5.73170
\(946\) 0 0
\(947\) 22510.4 0.772429 0.386215 0.922409i \(-0.373782\pi\)
0.386215 + 0.922409i \(0.373782\pi\)
\(948\) 0 0
\(949\) −14181.8 −0.485101
\(950\) 0 0
\(951\) 9679.47 0.330051
\(952\) 0 0
\(953\) −4260.98 −0.144834 −0.0724169 0.997374i \(-0.523071\pi\)
−0.0724169 + 0.997374i \(0.523071\pi\)
\(954\) 0 0
\(955\) 3280.76 0.111165
\(956\) 0 0
\(957\) −17842.5 −0.602683
\(958\) 0 0
\(959\) 29278.1 0.985861
\(960\) 0 0
\(961\) −21286.2 −0.714517
\(962\) 0 0
\(963\) 11887.0 0.397771
\(964\) 0 0
\(965\) −45264.5 −1.50996
\(966\) 0 0
\(967\) −5646.95 −0.187791 −0.0938953 0.995582i \(-0.529932\pi\)
−0.0938953 + 0.995582i \(0.529932\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3320.02 −0.109726 −0.0548632 0.998494i \(-0.517472\pi\)
−0.0548632 + 0.998494i \(0.517472\pi\)
\(972\) 0 0
\(973\) 712.148 0.0234639
\(974\) 0 0
\(975\) 56140.0 1.84402
\(976\) 0 0
\(977\) 5821.02 0.190615 0.0953076 0.995448i \(-0.469617\pi\)
0.0953076 + 0.995448i \(0.469617\pi\)
\(978\) 0 0
\(979\) 8954.49 0.292326
\(980\) 0 0
\(981\) −73139.1 −2.38038
\(982\) 0 0
\(983\) 234.506 0.00760894 0.00380447 0.999993i \(-0.498789\pi\)
0.00380447 + 0.999993i \(0.498789\pi\)
\(984\) 0 0
\(985\) 55460.3 1.79402
\(986\) 0 0
\(987\) 25604.5 0.825735
\(988\) 0 0
\(989\) −943.985 −0.0303508
\(990\) 0 0
\(991\) 16708.2 0.535573 0.267787 0.963478i \(-0.413708\pi\)
0.267787 + 0.963478i \(0.413708\pi\)
\(992\) 0 0
\(993\) −58990.8 −1.88521
\(994\) 0 0
\(995\) 44787.7 1.42700
\(996\) 0 0
\(997\) −10281.9 −0.326611 −0.163305 0.986576i \(-0.552216\pi\)
−0.163305 + 0.986576i \(0.552216\pi\)
\(998\) 0 0
\(999\) −61047.8 −1.93340
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 2312.4.a.k.1.1 8
17.4 even 4 136.4.b.b.33.8 yes 8
17.13 even 4 136.4.b.b.33.1 8
17.16 even 2 inner 2312.4.a.k.1.8 8
51.38 odd 4 1224.4.c.e.577.7 8
51.47 odd 4 1224.4.c.e.577.2 8
68.47 odd 4 272.4.b.f.33.8 8
68.55 odd 4 272.4.b.f.33.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.1 8 17.13 even 4
136.4.b.b.33.8 yes 8 17.4 even 4
272.4.b.f.33.1 8 68.55 odd 4
272.4.b.f.33.8 8 68.47 odd 4
1224.4.c.e.577.2 8 51.47 odd 4
1224.4.c.e.577.7 8 51.38 odd 4
2312.4.a.k.1.1 8 1.1 even 1 trivial
2312.4.a.k.1.8 8 17.16 even 2 inner