Properties

Label 2312.4.a.k.1.4
Level $2312$
Weight $4$
Character 2312.1
Self dual yes
Analytic conductor $136.412$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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 1.4
Root \(-9.30031\) of defining polynomial
Character \(\chi\) \(=\) 2312.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +O(q^{10})\) \(q-2.62097 q^{3} +11.5318 q^{5} +23.0485 q^{7} -20.1305 q^{9} +1.19452 q^{11} +58.3731 q^{13} -30.2245 q^{15} -138.971 q^{19} -60.4096 q^{21} +180.911 q^{23} +7.98158 q^{25} +123.528 q^{27} -115.027 q^{29} +210.726 q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661 q^{37} -152.994 q^{39} -297.500 q^{41} -174.746 q^{43} -232.140 q^{45} -199.717 q^{47} +188.234 q^{49} +706.217 q^{53} +13.7749 q^{55} +364.238 q^{57} +182.746 q^{59} +489.499 q^{61} -463.978 q^{63} +673.144 q^{65} -167.043 q^{67} -474.163 q^{69} +818.157 q^{71} +1090.15 q^{73} -20.9195 q^{75} +27.5319 q^{77} -110.196 q^{79} +219.760 q^{81} -118.672 q^{83} +301.483 q^{87} +400.332 q^{89} +1345.41 q^{91} -552.306 q^{93} -1602.58 q^{95} -924.543 q^{97} -24.0463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9}+O(q^{10}) \) 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
\(3\) −2.62097 −0.504407 −0.252203 0.967674i \(-0.581155\pi\)
−0.252203 + 0.967674i \(0.581155\pi\)
\(4\) 0 0
\(5\) 11.5318 1.03143 0.515716 0.856759i \(-0.327526\pi\)
0.515716 + 0.856759i \(0.327526\pi\)
\(6\) 0 0
\(7\) 23.0485 1.24450 0.622251 0.782817i \(-0.286218\pi\)
0.622251 + 0.782817i \(0.286218\pi\)
\(8\) 0 0
\(9\) −20.1305 −0.745574
\(10\) 0 0
\(11\) 1.19452 0.0327419 0.0163710 0.999866i \(-0.494789\pi\)
0.0163710 + 0.999866i \(0.494789\pi\)
\(12\) 0 0
\(13\) 58.3731 1.24537 0.622684 0.782474i \(-0.286042\pi\)
0.622684 + 0.782474i \(0.286042\pi\)
\(14\) 0 0
\(15\) −30.2245 −0.520262
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −138.971 −1.67800 −0.839001 0.544130i \(-0.816860\pi\)
−0.839001 + 0.544130i \(0.816860\pi\)
\(20\) 0 0
\(21\) −60.4096 −0.627736
\(22\) 0 0
\(23\) 180.911 1.64011 0.820055 0.572284i \(-0.193943\pi\)
0.820055 + 0.572284i \(0.193943\pi\)
\(24\) 0 0
\(25\) 7.98158 0.0638527
\(26\) 0 0
\(27\) 123.528 0.880479
\(28\) 0 0
\(29\) −115.027 −0.736551 −0.368275 0.929717i \(-0.620052\pi\)
−0.368275 + 0.929717i \(0.620052\pi\)
\(30\) 0 0
\(31\) 210.726 1.22088 0.610442 0.792061i \(-0.290992\pi\)
0.610442 + 0.792061i \(0.290992\pi\)
\(32\) 0 0
\(33\) −3.13080 −0.0165152
\(34\) 0 0
\(35\) 265.790 1.28362
\(36\) 0 0
\(37\) 210.661 0.936010 0.468005 0.883726i \(-0.344973\pi\)
0.468005 + 0.883726i \(0.344973\pi\)
\(38\) 0 0
\(39\) −152.994 −0.628172
\(40\) 0 0
\(41\) −297.500 −1.13321 −0.566605 0.823989i \(-0.691743\pi\)
−0.566605 + 0.823989i \(0.691743\pi\)
\(42\) 0 0
\(43\) −174.746 −0.619734 −0.309867 0.950780i \(-0.600285\pi\)
−0.309867 + 0.950780i \(0.600285\pi\)
\(44\) 0 0
\(45\) −232.140 −0.769009
\(46\) 0 0
\(47\) −199.717 −0.619823 −0.309911 0.950765i \(-0.600299\pi\)
−0.309911 + 0.950765i \(0.600299\pi\)
\(48\) 0 0
\(49\) 188.234 0.548787
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 706.217 1.83031 0.915154 0.403105i \(-0.132069\pi\)
0.915154 + 0.403105i \(0.132069\pi\)
\(54\) 0 0
\(55\) 13.7749 0.0337711
\(56\) 0 0
\(57\) 364.238 0.846396
\(58\) 0 0
\(59\) 182.746 0.403247 0.201623 0.979463i \(-0.435378\pi\)
0.201623 + 0.979463i \(0.435378\pi\)
\(60\) 0 0
\(61\) 489.499 1.02744 0.513720 0.857958i \(-0.328267\pi\)
0.513720 + 0.857958i \(0.328267\pi\)
\(62\) 0 0
\(63\) −463.978 −0.927869
\(64\) 0 0
\(65\) 673.144 1.28451
\(66\) 0 0
\(67\) −167.043 −0.304590 −0.152295 0.988335i \(-0.548666\pi\)
−0.152295 + 0.988335i \(0.548666\pi\)
\(68\) 0 0
\(69\) −474.163 −0.827283
\(70\) 0 0
\(71\) 818.157 1.36757 0.683784 0.729684i \(-0.260333\pi\)
0.683784 + 0.729684i \(0.260333\pi\)
\(72\) 0 0
\(73\) 1090.15 1.74784 0.873919 0.486071i \(-0.161570\pi\)
0.873919 + 0.486071i \(0.161570\pi\)
\(74\) 0 0
\(75\) −20.9195 −0.0322077
\(76\) 0 0
\(77\) 27.5319 0.0407474
\(78\) 0 0
\(79\) −110.196 −0.156937 −0.0784685 0.996917i \(-0.525003\pi\)
−0.0784685 + 0.996917i \(0.525003\pi\)
\(80\) 0 0
\(81\) 219.760 0.301454
\(82\) 0 0
\(83\) −118.672 −0.156939 −0.0784696 0.996917i \(-0.525003\pi\)
−0.0784696 + 0.996917i \(0.525003\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 301.483 0.371521
\(88\) 0 0
\(89\) 400.332 0.476799 0.238399 0.971167i \(-0.423377\pi\)
0.238399 + 0.971167i \(0.423377\pi\)
\(90\) 0 0
\(91\) 1345.41 1.54986
\(92\) 0 0
\(93\) −552.306 −0.615823
\(94\) 0 0
\(95\) −1602.58 −1.73075
\(96\) 0 0
\(97\) −924.543 −0.967764 −0.483882 0.875133i \(-0.660774\pi\)
−0.483882 + 0.875133i \(0.660774\pi\)
\(98\) 0 0
\(99\) −24.0463 −0.0244115
\(100\) 0 0
\(101\) 223.816 0.220500 0.110250 0.993904i \(-0.464835\pi\)
0.110250 + 0.993904i \(0.464835\pi\)
\(102\) 0 0
\(103\) −1280.06 −1.22454 −0.612271 0.790648i \(-0.709744\pi\)
−0.612271 + 0.790648i \(0.709744\pi\)
\(104\) 0 0
\(105\) −696.629 −0.647467
\(106\) 0 0
\(107\) −1099.03 −0.992969 −0.496485 0.868046i \(-0.665376\pi\)
−0.496485 + 0.868046i \(0.665376\pi\)
\(108\) 0 0
\(109\) −733.360 −0.644433 −0.322216 0.946666i \(-0.604428\pi\)
−0.322216 + 0.946666i \(0.604428\pi\)
\(110\) 0 0
\(111\) −552.136 −0.472130
\(112\) 0 0
\(113\) 1637.39 1.36312 0.681561 0.731761i \(-0.261301\pi\)
0.681561 + 0.731761i \(0.261301\pi\)
\(114\) 0 0
\(115\) 2086.22 1.69166
\(116\) 0 0
\(117\) −1175.08 −0.928513
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1329.57 −0.998928
\(122\) 0 0
\(123\) 779.739 0.571599
\(124\) 0 0
\(125\) −1349.43 −0.965573
\(126\) 0 0
\(127\) 1502.71 1.04995 0.524977 0.851117i \(-0.324074\pi\)
0.524977 + 0.851117i \(0.324074\pi\)
\(128\) 0 0
\(129\) 458.006 0.312598
\(130\) 0 0
\(131\) −702.381 −0.468453 −0.234226 0.972182i \(-0.575256\pi\)
−0.234226 + 0.972182i \(0.575256\pi\)
\(132\) 0 0
\(133\) −3203.07 −2.08828
\(134\) 0 0
\(135\) 1424.49 0.908155
\(136\) 0 0
\(137\) 18.9889 0.0118418 0.00592092 0.999982i \(-0.498115\pi\)
0.00592092 + 0.999982i \(0.498115\pi\)
\(138\) 0 0
\(139\) −276.969 −0.169009 −0.0845043 0.996423i \(-0.526931\pi\)
−0.0845043 + 0.996423i \(0.526931\pi\)
\(140\) 0 0
\(141\) 523.452 0.312643
\(142\) 0 0
\(143\) 69.7277 0.0407757
\(144\) 0 0
\(145\) −1326.46 −0.759702
\(146\) 0 0
\(147\) −493.357 −0.276812
\(148\) 0 0
\(149\) 128.491 0.0706468 0.0353234 0.999376i \(-0.488754\pi\)
0.0353234 + 0.999376i \(0.488754\pi\)
\(150\) 0 0
\(151\) 2097.44 1.13038 0.565190 0.824961i \(-0.308803\pi\)
0.565190 + 0.824961i \(0.308803\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2430.04 1.25926
\(156\) 0 0
\(157\) 2911.59 1.48007 0.740033 0.672571i \(-0.234810\pi\)
0.740033 + 0.672571i \(0.234810\pi\)
\(158\) 0 0
\(159\) −1850.98 −0.923220
\(160\) 0 0
\(161\) 4169.73 2.04112
\(162\) 0 0
\(163\) −1574.26 −0.756478 −0.378239 0.925708i \(-0.623470\pi\)
−0.378239 + 0.925708i \(0.623470\pi\)
\(164\) 0 0
\(165\) −36.1037 −0.0170344
\(166\) 0 0
\(167\) −1290.10 −0.597788 −0.298894 0.954286i \(-0.596618\pi\)
−0.298894 + 0.954286i \(0.596618\pi\)
\(168\) 0 0
\(169\) 1210.41 0.550939
\(170\) 0 0
\(171\) 2797.55 1.25107
\(172\) 0 0
\(173\) 2370.19 1.04163 0.520816 0.853669i \(-0.325628\pi\)
0.520816 + 0.853669i \(0.325628\pi\)
\(174\) 0 0
\(175\) 183.964 0.0794648
\(176\) 0 0
\(177\) −478.974 −0.203400
\(178\) 0 0
\(179\) −1719.87 −0.718153 −0.359077 0.933308i \(-0.616908\pi\)
−0.359077 + 0.933308i \(0.616908\pi\)
\(180\) 0 0
\(181\) −61.7101 −0.0253418 −0.0126709 0.999920i \(-0.504033\pi\)
−0.0126709 + 0.999920i \(0.504033\pi\)
\(182\) 0 0
\(183\) −1282.96 −0.518248
\(184\) 0 0
\(185\) 2429.29 0.965431
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2847.13 1.09576
\(190\) 0 0
\(191\) −2565.34 −0.971839 −0.485919 0.874004i \(-0.661515\pi\)
−0.485919 + 0.874004i \(0.661515\pi\)
\(192\) 0 0
\(193\) −1991.65 −0.742810 −0.371405 0.928471i \(-0.621124\pi\)
−0.371405 + 0.928471i \(0.621124\pi\)
\(194\) 0 0
\(195\) −1764.29 −0.647917
\(196\) 0 0
\(197\) −3831.76 −1.38579 −0.692897 0.721036i \(-0.743666\pi\)
−0.692897 + 0.721036i \(0.743666\pi\)
\(198\) 0 0
\(199\) 2171.38 0.773492 0.386746 0.922186i \(-0.373599\pi\)
0.386746 + 0.922186i \(0.373599\pi\)
\(200\) 0 0
\(201\) 437.815 0.153637
\(202\) 0 0
\(203\) −2651.20 −0.916639
\(204\) 0 0
\(205\) −3430.69 −1.16883
\(206\) 0 0
\(207\) −3641.83 −1.22282
\(208\) 0 0
\(209\) −166.003 −0.0549410
\(210\) 0 0
\(211\) 2069.97 0.675369 0.337685 0.941259i \(-0.390356\pi\)
0.337685 + 0.941259i \(0.390356\pi\)
\(212\) 0 0
\(213\) −2144.37 −0.689811
\(214\) 0 0
\(215\) −2015.13 −0.639214
\(216\) 0 0
\(217\) 4856.91 1.51939
\(218\) 0 0
\(219\) −2857.25 −0.881622
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4297.20 1.29041 0.645206 0.764008i \(-0.276771\pi\)
0.645206 + 0.764008i \(0.276771\pi\)
\(224\) 0 0
\(225\) −160.673 −0.0476069
\(226\) 0 0
\(227\) 336.979 0.0985291 0.0492646 0.998786i \(-0.484312\pi\)
0.0492646 + 0.998786i \(0.484312\pi\)
\(228\) 0 0
\(229\) 3076.57 0.887798 0.443899 0.896077i \(-0.353595\pi\)
0.443899 + 0.896077i \(0.353595\pi\)
\(230\) 0 0
\(231\) −72.1604 −0.0205533
\(232\) 0 0
\(233\) 6537.86 1.83824 0.919119 0.393980i \(-0.128902\pi\)
0.919119 + 0.393980i \(0.128902\pi\)
\(234\) 0 0
\(235\) −2303.09 −0.639305
\(236\) 0 0
\(237\) 288.821 0.0791601
\(238\) 0 0
\(239\) −3432.31 −0.928943 −0.464472 0.885588i \(-0.653756\pi\)
−0.464472 + 0.885588i \(0.653756\pi\)
\(240\) 0 0
\(241\) 1312.48 0.350807 0.175404 0.984497i \(-0.443877\pi\)
0.175404 + 0.984497i \(0.443877\pi\)
\(242\) 0 0
\(243\) −3911.24 −1.03253
\(244\) 0 0
\(245\) 2170.67 0.566037
\(246\) 0 0
\(247\) −8112.14 −2.08973
\(248\) 0 0
\(249\) 311.037 0.0791612
\(250\) 0 0
\(251\) 6498.55 1.63420 0.817101 0.576494i \(-0.195580\pi\)
0.817101 + 0.576494i \(0.195580\pi\)
\(252\) 0 0
\(253\) 216.102 0.0537004
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4874.66 1.18316 0.591582 0.806245i \(-0.298504\pi\)
0.591582 + 0.806245i \(0.298504\pi\)
\(258\) 0 0
\(259\) 4855.41 1.16487
\(260\) 0 0
\(261\) 2315.55 0.549153
\(262\) 0 0
\(263\) −1482.50 −0.347584 −0.173792 0.984782i \(-0.555602\pi\)
−0.173792 + 0.984782i \(0.555602\pi\)
\(264\) 0 0
\(265\) 8143.92 1.88784
\(266\) 0 0
\(267\) −1049.26 −0.240501
\(268\) 0 0
\(269\) 5058.75 1.14661 0.573304 0.819343i \(-0.305662\pi\)
0.573304 + 0.819343i \(0.305662\pi\)
\(270\) 0 0
\(271\) 1133.70 0.254122 0.127061 0.991895i \(-0.459446\pi\)
0.127061 + 0.991895i \(0.459446\pi\)
\(272\) 0 0
\(273\) −3526.29 −0.781761
\(274\) 0 0
\(275\) 9.53415 0.00209066
\(276\) 0 0
\(277\) −3698.34 −0.802208 −0.401104 0.916033i \(-0.631373\pi\)
−0.401104 + 0.916033i \(0.631373\pi\)
\(278\) 0 0
\(279\) −4242.01 −0.910260
\(280\) 0 0
\(281\) −2190.08 −0.464945 −0.232472 0.972603i \(-0.574682\pi\)
−0.232472 + 0.972603i \(0.574682\pi\)
\(282\) 0 0
\(283\) −8159.66 −1.71393 −0.856963 0.515378i \(-0.827652\pi\)
−0.856963 + 0.515378i \(0.827652\pi\)
\(284\) 0 0
\(285\) 4200.31 0.873000
\(286\) 0 0
\(287\) −6856.92 −1.41028
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 2423.20 0.488147
\(292\) 0 0
\(293\) 4777.49 0.952574 0.476287 0.879290i \(-0.341982\pi\)
0.476287 + 0.879290i \(0.341982\pi\)
\(294\) 0 0
\(295\) 2107.39 0.415922
\(296\) 0 0
\(297\) 147.556 0.0288286
\(298\) 0 0
\(299\) 10560.3 2.04254
\(300\) 0 0
\(301\) −4027.65 −0.771261
\(302\) 0 0
\(303\) −586.615 −0.111222
\(304\) 0 0
\(305\) 5644.78 1.05974
\(306\) 0 0
\(307\) 531.103 0.0987351 0.0493675 0.998781i \(-0.484279\pi\)
0.0493675 + 0.998781i \(0.484279\pi\)
\(308\) 0 0
\(309\) 3355.00 0.617667
\(310\) 0 0
\(311\) 4767.26 0.869217 0.434609 0.900619i \(-0.356887\pi\)
0.434609 + 0.900619i \(0.356887\pi\)
\(312\) 0 0
\(313\) 8613.47 1.55547 0.777735 0.628593i \(-0.216369\pi\)
0.777735 + 0.628593i \(0.216369\pi\)
\(314\) 0 0
\(315\) −5350.48 −0.957034
\(316\) 0 0
\(317\) 2453.20 0.434654 0.217327 0.976099i \(-0.430266\pi\)
0.217327 + 0.976099i \(0.430266\pi\)
\(318\) 0 0
\(319\) −137.402 −0.0241161
\(320\) 0 0
\(321\) 2880.54 0.500860
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 465.909 0.0795200
\(326\) 0 0
\(327\) 1922.12 0.325056
\(328\) 0 0
\(329\) −4603.17 −0.771371
\(330\) 0 0
\(331\) 11003.5 1.82721 0.913605 0.406604i \(-0.133287\pi\)
0.913605 + 0.406604i \(0.133287\pi\)
\(332\) 0 0
\(333\) −4240.70 −0.697865
\(334\) 0 0
\(335\) −1926.30 −0.314164
\(336\) 0 0
\(337\) 3664.37 0.592318 0.296159 0.955139i \(-0.404294\pi\)
0.296159 + 0.955139i \(0.404294\pi\)
\(338\) 0 0
\(339\) −4291.56 −0.687568
\(340\) 0 0
\(341\) 251.716 0.0399741
\(342\) 0 0
\(343\) −3567.12 −0.561535
\(344\) 0 0
\(345\) −5467.94 −0.853286
\(346\) 0 0
\(347\) −3173.37 −0.490938 −0.245469 0.969404i \(-0.578942\pi\)
−0.245469 + 0.969404i \(0.578942\pi\)
\(348\) 0 0
\(349\) 2185.47 0.335202 0.167601 0.985855i \(-0.446398\pi\)
0.167601 + 0.985855i \(0.446398\pi\)
\(350\) 0 0
\(351\) 7210.70 1.09652
\(352\) 0 0
\(353\) 1216.36 0.183400 0.0917002 0.995787i \(-0.470770\pi\)
0.0917002 + 0.995787i \(0.470770\pi\)
\(354\) 0 0
\(355\) 9434.80 1.41055
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7448.93 1.09510 0.547548 0.836774i \(-0.315561\pi\)
0.547548 + 0.836774i \(0.315561\pi\)
\(360\) 0 0
\(361\) 12453.8 1.81569
\(362\) 0 0
\(363\) 3484.78 0.503866
\(364\) 0 0
\(365\) 12571.3 1.80278
\(366\) 0 0
\(367\) −2374.59 −0.337746 −0.168873 0.985638i \(-0.554013\pi\)
−0.168873 + 0.985638i \(0.554013\pi\)
\(368\) 0 0
\(369\) 5988.81 0.844892
\(370\) 0 0
\(371\) 16277.2 2.27782
\(372\) 0 0
\(373\) 12231.7 1.69795 0.848974 0.528435i \(-0.177221\pi\)
0.848974 + 0.528435i \(0.177221\pi\)
\(374\) 0 0
\(375\) 3536.82 0.487041
\(376\) 0 0
\(377\) −6714.48 −0.917276
\(378\) 0 0
\(379\) 14129.9 1.91505 0.957523 0.288358i \(-0.0931094\pi\)
0.957523 + 0.288358i \(0.0931094\pi\)
\(380\) 0 0
\(381\) −3938.57 −0.529604
\(382\) 0 0
\(383\) −13306.4 −1.77526 −0.887631 0.460556i \(-0.847650\pi\)
−0.887631 + 0.460556i \(0.847650\pi\)
\(384\) 0 0
\(385\) 317.491 0.0420282
\(386\) 0 0
\(387\) 3517.73 0.462058
\(388\) 0 0
\(389\) 8007.84 1.04374 0.521868 0.853026i \(-0.325235\pi\)
0.521868 + 0.853026i \(0.325235\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1840.92 0.236291
\(394\) 0 0
\(395\) −1270.76 −0.161870
\(396\) 0 0
\(397\) 13185.2 1.66687 0.833434 0.552619i \(-0.186372\pi\)
0.833434 + 0.552619i \(0.186372\pi\)
\(398\) 0 0
\(399\) 8395.15 1.05334
\(400\) 0 0
\(401\) −3205.81 −0.399228 −0.199614 0.979875i \(-0.563969\pi\)
−0.199614 + 0.979875i \(0.563969\pi\)
\(402\) 0 0
\(403\) 12300.7 1.52045
\(404\) 0 0
\(405\) 2534.22 0.310929
\(406\) 0 0
\(407\) 251.638 0.0306468
\(408\) 0 0
\(409\) −15693.3 −1.89727 −0.948636 0.316369i \(-0.897536\pi\)
−0.948636 + 0.316369i \(0.897536\pi\)
\(410\) 0 0
\(411\) −49.7695 −0.00597311
\(412\) 0 0
\(413\) 4212.03 0.501842
\(414\) 0 0
\(415\) −1368.50 −0.161872
\(416\) 0 0
\(417\) 725.929 0.0852491
\(418\) 0 0
\(419\) 6910.65 0.805745 0.402873 0.915256i \(-0.368012\pi\)
0.402873 + 0.915256i \(0.368012\pi\)
\(420\) 0 0
\(421\) 9225.05 1.06794 0.533968 0.845505i \(-0.320700\pi\)
0.533968 + 0.845505i \(0.320700\pi\)
\(422\) 0 0
\(423\) 4020.40 0.462124
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11282.2 1.27865
\(428\) 0 0
\(429\) −182.755 −0.0205675
\(430\) 0 0
\(431\) −8822.74 −0.986024 −0.493012 0.870022i \(-0.664104\pi\)
−0.493012 + 0.870022i \(0.664104\pi\)
\(432\) 0 0
\(433\) 3190.95 0.354151 0.177075 0.984197i \(-0.443336\pi\)
0.177075 + 0.984197i \(0.443336\pi\)
\(434\) 0 0
\(435\) 3476.63 0.383199
\(436\) 0 0
\(437\) −25141.3 −2.75211
\(438\) 0 0
\(439\) −15069.5 −1.63834 −0.819168 0.573553i \(-0.805565\pi\)
−0.819168 + 0.573553i \(0.805565\pi\)
\(440\) 0 0
\(441\) −3789.24 −0.409161
\(442\) 0 0
\(443\) −3026.24 −0.324562 −0.162281 0.986745i \(-0.551885\pi\)
−0.162281 + 0.986745i \(0.551885\pi\)
\(444\) 0 0
\(445\) 4616.53 0.491786
\(446\) 0 0
\(447\) −336.771 −0.0356347
\(448\) 0 0
\(449\) 4641.02 0.487802 0.243901 0.969800i \(-0.421573\pi\)
0.243901 + 0.969800i \(0.421573\pi\)
\(450\) 0 0
\(451\) −355.369 −0.0371035
\(452\) 0 0
\(453\) −5497.34 −0.570171
\(454\) 0 0
\(455\) 15515.0 1.59858
\(456\) 0 0
\(457\) −17393.2 −1.78035 −0.890174 0.455620i \(-0.849418\pi\)
−0.890174 + 0.455620i \(0.849418\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2451.93 −0.247718 −0.123859 0.992300i \(-0.539527\pi\)
−0.123859 + 0.992300i \(0.539527\pi\)
\(462\) 0 0
\(463\) −14586.3 −1.46411 −0.732054 0.681246i \(-0.761438\pi\)
−0.732054 + 0.681246i \(0.761438\pi\)
\(464\) 0 0
\(465\) −6369.07 −0.635179
\(466\) 0 0
\(467\) −15195.6 −1.50572 −0.752859 0.658182i \(-0.771326\pi\)
−0.752859 + 0.658182i \(0.771326\pi\)
\(468\) 0 0
\(469\) −3850.09 −0.379063
\(470\) 0 0
\(471\) −7631.21 −0.746555
\(472\) 0 0
\(473\) −208.738 −0.0202913
\(474\) 0 0
\(475\) −1109.21 −0.107145
\(476\) 0 0
\(477\) −14216.5 −1.36463
\(478\) 0 0
\(479\) −4422.10 −0.421819 −0.210909 0.977506i \(-0.567642\pi\)
−0.210909 + 0.977506i \(0.567642\pi\)
\(480\) 0 0
\(481\) 12296.9 1.16568
\(482\) 0 0
\(483\) −10928.8 −1.02956
\(484\) 0 0
\(485\) −10661.6 −0.998183
\(486\) 0 0
\(487\) 17772.7 1.65371 0.826856 0.562414i \(-0.190127\pi\)
0.826856 + 0.562414i \(0.190127\pi\)
\(488\) 0 0
\(489\) 4126.11 0.381573
\(490\) 0 0
\(491\) −4845.34 −0.445350 −0.222675 0.974893i \(-0.571479\pi\)
−0.222675 + 0.974893i \(0.571479\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −277.296 −0.0251788
\(496\) 0 0
\(497\) 18857.3 1.70194
\(498\) 0 0
\(499\) 1060.23 0.0951153 0.0475576 0.998868i \(-0.484856\pi\)
0.0475576 + 0.998868i \(0.484856\pi\)
\(500\) 0 0
\(501\) 3381.31 0.301528
\(502\) 0 0
\(503\) −2803.19 −0.248485 −0.124243 0.992252i \(-0.539650\pi\)
−0.124243 + 0.992252i \(0.539650\pi\)
\(504\) 0 0
\(505\) 2580.99 0.227431
\(506\) 0 0
\(507\) −3172.46 −0.277898
\(508\) 0 0
\(509\) 1658.97 0.144465 0.0722323 0.997388i \(-0.476988\pi\)
0.0722323 + 0.997388i \(0.476988\pi\)
\(510\) 0 0
\(511\) 25126.3 2.17519
\(512\) 0 0
\(513\) −17166.7 −1.47745
\(514\) 0 0
\(515\) −14761.3 −1.26303
\(516\) 0 0
\(517\) −238.565 −0.0202942
\(518\) 0 0
\(519\) −6212.21 −0.525406
\(520\) 0 0
\(521\) 12553.6 1.05563 0.527815 0.849359i \(-0.323011\pi\)
0.527815 + 0.849359i \(0.323011\pi\)
\(522\) 0 0
\(523\) −12031.6 −1.00593 −0.502967 0.864306i \(-0.667758\pi\)
−0.502967 + 0.864306i \(0.667758\pi\)
\(524\) 0 0
\(525\) −482.164 −0.0400826
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 20561.8 1.68996
\(530\) 0 0
\(531\) −3678.78 −0.300650
\(532\) 0 0
\(533\) −17366.0 −1.41126
\(534\) 0 0
\(535\) −12673.8 −1.02418
\(536\) 0 0
\(537\) 4507.75 0.362241
\(538\) 0 0
\(539\) 224.849 0.0179683
\(540\) 0 0
\(541\) −10632.7 −0.844984 −0.422492 0.906367i \(-0.638845\pi\)
−0.422492 + 0.906367i \(0.638845\pi\)
\(542\) 0 0
\(543\) 161.741 0.0127826
\(544\) 0 0
\(545\) −8456.94 −0.664689
\(546\) 0 0
\(547\) 17370.0 1.35774 0.678872 0.734257i \(-0.262469\pi\)
0.678872 + 0.734257i \(0.262469\pi\)
\(548\) 0 0
\(549\) −9853.85 −0.766033
\(550\) 0 0
\(551\) 15985.4 1.23593
\(552\) 0 0
\(553\) −2539.86 −0.195309
\(554\) 0 0
\(555\) −6367.10 −0.486970
\(556\) 0 0
\(557\) 1084.77 0.0825190 0.0412595 0.999148i \(-0.486863\pi\)
0.0412595 + 0.999148i \(0.486863\pi\)
\(558\) 0 0
\(559\) −10200.5 −0.771797
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5494.27 0.411289 0.205645 0.978627i \(-0.434071\pi\)
0.205645 + 0.978627i \(0.434071\pi\)
\(564\) 0 0
\(565\) 18882.0 1.40597
\(566\) 0 0
\(567\) 5065.14 0.375160
\(568\) 0 0
\(569\) 11612.4 0.855569 0.427784 0.903881i \(-0.359294\pi\)
0.427784 + 0.903881i \(0.359294\pi\)
\(570\) 0 0
\(571\) −3413.97 −0.250210 −0.125105 0.992143i \(-0.539927\pi\)
−0.125105 + 0.992143i \(0.539927\pi\)
\(572\) 0 0
\(573\) 6723.68 0.490202
\(574\) 0 0
\(575\) 1443.96 0.104725
\(576\) 0 0
\(577\) 1355.70 0.0978141 0.0489070 0.998803i \(-0.484426\pi\)
0.0489070 + 0.998803i \(0.484426\pi\)
\(578\) 0 0
\(579\) 5220.07 0.374678
\(580\) 0 0
\(581\) −2735.22 −0.195311
\(582\) 0 0
\(583\) 843.589 0.0599278
\(584\) 0 0
\(585\) −13550.7 −0.957698
\(586\) 0 0
\(587\) −25232.2 −1.77418 −0.887092 0.461593i \(-0.847278\pi\)
−0.887092 + 0.461593i \(0.847278\pi\)
\(588\) 0 0
\(589\) −29284.7 −2.04865
\(590\) 0 0
\(591\) 10042.9 0.699004
\(592\) 0 0
\(593\) 14327.4 0.992171 0.496086 0.868274i \(-0.334770\pi\)
0.496086 + 0.868274i \(0.334770\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5691.13 −0.390155
\(598\) 0 0
\(599\) −2523.52 −0.172134 −0.0860671 0.996289i \(-0.527430\pi\)
−0.0860671 + 0.996289i \(0.527430\pi\)
\(600\) 0 0
\(601\) 17286.6 1.17327 0.586634 0.809853i \(-0.300453\pi\)
0.586634 + 0.809853i \(0.300453\pi\)
\(602\) 0 0
\(603\) 3362.66 0.227095
\(604\) 0 0
\(605\) −15332.3 −1.03033
\(606\) 0 0
\(607\) 6413.65 0.428867 0.214433 0.976739i \(-0.431210\pi\)
0.214433 + 0.976739i \(0.431210\pi\)
\(608\) 0 0
\(609\) 6948.73 0.462359
\(610\) 0 0
\(611\) −11658.1 −0.771907
\(612\) 0 0
\(613\) 485.788 0.0320078 0.0160039 0.999872i \(-0.494906\pi\)
0.0160039 + 0.999872i \(0.494906\pi\)
\(614\) 0 0
\(615\) 8991.76 0.589566
\(616\) 0 0
\(617\) 6830.32 0.445670 0.222835 0.974856i \(-0.428469\pi\)
0.222835 + 0.974856i \(0.428469\pi\)
\(618\) 0 0
\(619\) 10667.8 0.692689 0.346344 0.938107i \(-0.387423\pi\)
0.346344 + 0.938107i \(0.387423\pi\)
\(620\) 0 0
\(621\) 22347.5 1.44408
\(622\) 0 0
\(623\) 9227.05 0.593377
\(624\) 0 0
\(625\) −16559.0 −1.05978
\(626\) 0 0
\(627\) 435.090 0.0277126
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 15251.2 0.962188 0.481094 0.876669i \(-0.340240\pi\)
0.481094 + 0.876669i \(0.340240\pi\)
\(632\) 0 0
\(633\) −5425.35 −0.340661
\(634\) 0 0
\(635\) 17328.9 1.08296
\(636\) 0 0
\(637\) 10987.8 0.683442
\(638\) 0 0
\(639\) −16469.9 −1.01962
\(640\) 0 0
\(641\) 12000.9 0.739480 0.369740 0.929135i \(-0.379447\pi\)
0.369740 + 0.929135i \(0.379447\pi\)
\(642\) 0 0
\(643\) −5642.30 −0.346051 −0.173025 0.984917i \(-0.555354\pi\)
−0.173025 + 0.984917i \(0.555354\pi\)
\(644\) 0 0
\(645\) 5281.62 0.322424
\(646\) 0 0
\(647\) 7790.69 0.473391 0.236695 0.971584i \(-0.423936\pi\)
0.236695 + 0.971584i \(0.423936\pi\)
\(648\) 0 0
\(649\) 218.294 0.0132031
\(650\) 0 0
\(651\) −12729.8 −0.766393
\(652\) 0 0
\(653\) 6755.02 0.404815 0.202408 0.979301i \(-0.435123\pi\)
0.202408 + 0.979301i \(0.435123\pi\)
\(654\) 0 0
\(655\) −8099.69 −0.483177
\(656\) 0 0
\(657\) −21945.2 −1.30314
\(658\) 0 0
\(659\) −14303.3 −0.845492 −0.422746 0.906248i \(-0.638934\pi\)
−0.422746 + 0.906248i \(0.638934\pi\)
\(660\) 0 0
\(661\) 26136.0 1.53793 0.768966 0.639289i \(-0.220771\pi\)
0.768966 + 0.639289i \(0.220771\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −36937.0 −2.15392
\(666\) 0 0
\(667\) −20809.6 −1.20802
\(668\) 0 0
\(669\) −11262.9 −0.650893
\(670\) 0 0
\(671\) 584.715 0.0336404
\(672\) 0 0
\(673\) 19026.5 1.08977 0.544887 0.838509i \(-0.316572\pi\)
0.544887 + 0.838509i \(0.316572\pi\)
\(674\) 0 0
\(675\) 985.948 0.0562210
\(676\) 0 0
\(677\) −25683.6 −1.45805 −0.729026 0.684486i \(-0.760026\pi\)
−0.729026 + 0.684486i \(0.760026\pi\)
\(678\) 0 0
\(679\) −21309.3 −1.20439
\(680\) 0 0
\(681\) −883.214 −0.0496988
\(682\) 0 0
\(683\) 28196.4 1.57966 0.789829 0.613327i \(-0.210169\pi\)
0.789829 + 0.613327i \(0.210169\pi\)
\(684\) 0 0
\(685\) 218.976 0.0122141
\(686\) 0 0
\(687\) −8063.62 −0.447811
\(688\) 0 0
\(689\) 41224.0 2.27941
\(690\) 0 0
\(691\) −35037.4 −1.92892 −0.964462 0.264221i \(-0.914885\pi\)
−0.964462 + 0.264221i \(0.914885\pi\)
\(692\) 0 0
\(693\) −554.230 −0.0303802
\(694\) 0 0
\(695\) −3193.94 −0.174321
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −17135.6 −0.927220
\(700\) 0 0
\(701\) −3539.51 −0.190707 −0.0953535 0.995443i \(-0.530398\pi\)
−0.0953535 + 0.995443i \(0.530398\pi\)
\(702\) 0 0
\(703\) −29275.6 −1.57063
\(704\) 0 0
\(705\) 6036.33 0.322470
\(706\) 0 0
\(707\) 5158.62 0.274413
\(708\) 0 0
\(709\) 7264.39 0.384795 0.192398 0.981317i \(-0.438374\pi\)
0.192398 + 0.981317i \(0.438374\pi\)
\(710\) 0 0
\(711\) 2218.30 0.117008
\(712\) 0 0
\(713\) 38122.6 2.00239
\(714\) 0 0
\(715\) 804.084 0.0420574
\(716\) 0 0
\(717\) 8995.99 0.468565
\(718\) 0 0
\(719\) −9704.92 −0.503383 −0.251692 0.967807i \(-0.580987\pi\)
−0.251692 + 0.967807i \(0.580987\pi\)
\(720\) 0 0
\(721\) −29503.4 −1.52395
\(722\) 0 0
\(723\) −3439.99 −0.176949
\(724\) 0 0
\(725\) −918.097 −0.0470307
\(726\) 0 0
\(727\) 10937.3 0.557967 0.278983 0.960296i \(-0.410003\pi\)
0.278983 + 0.960296i \(0.410003\pi\)
\(728\) 0 0
\(729\) 4317.73 0.219364
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −14196.3 −0.715351 −0.357675 0.933846i \(-0.616431\pi\)
−0.357675 + 0.933846i \(0.616431\pi\)
\(734\) 0 0
\(735\) −5689.27 −0.285513
\(736\) 0 0
\(737\) −199.536 −0.00997287
\(738\) 0 0
\(739\) −8477.05 −0.421966 −0.210983 0.977490i \(-0.567667\pi\)
−0.210983 + 0.977490i \(0.567667\pi\)
\(740\) 0 0
\(741\) 21261.7 1.05407
\(742\) 0 0
\(743\) −5251.86 −0.259316 −0.129658 0.991559i \(-0.541388\pi\)
−0.129658 + 0.991559i \(0.541388\pi\)
\(744\) 0 0
\(745\) 1481.72 0.0728674
\(746\) 0 0
\(747\) 2388.93 0.117010
\(748\) 0 0
\(749\) −25331.1 −1.23575
\(750\) 0 0
\(751\) −16018.0 −0.778302 −0.389151 0.921174i \(-0.627232\pi\)
−0.389151 + 0.921174i \(0.627232\pi\)
\(752\) 0 0
\(753\) −17032.5 −0.824303
\(754\) 0 0
\(755\) 24187.2 1.16591
\(756\) 0 0
\(757\) 9344.23 0.448642 0.224321 0.974515i \(-0.427984\pi\)
0.224321 + 0.974515i \(0.427984\pi\)
\(758\) 0 0
\(759\) −566.397 −0.0270868
\(760\) 0 0
\(761\) 35043.1 1.66927 0.834633 0.550807i \(-0.185680\pi\)
0.834633 + 0.550807i \(0.185680\pi\)
\(762\) 0 0
\(763\) −16902.9 −0.801998
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10667.5 0.502190
\(768\) 0 0
\(769\) −6719.33 −0.315091 −0.157546 0.987512i \(-0.550358\pi\)
−0.157546 + 0.987512i \(0.550358\pi\)
\(770\) 0 0
\(771\) −12776.4 −0.596796
\(772\) 0 0
\(773\) −18716.9 −0.870892 −0.435446 0.900215i \(-0.643409\pi\)
−0.435446 + 0.900215i \(0.643409\pi\)
\(774\) 0 0
\(775\) 1681.92 0.0779568
\(776\) 0 0
\(777\) −12725.9 −0.587567
\(778\) 0 0
\(779\) 41343.7 1.90153
\(780\) 0 0
\(781\) 977.304 0.0447768
\(782\) 0 0
\(783\) −14209.0 −0.648518
\(784\) 0 0
\(785\) 33575.8 1.52659
\(786\) 0 0
\(787\) −15916.1 −0.720898 −0.360449 0.932779i \(-0.617377\pi\)
−0.360449 + 0.932779i \(0.617377\pi\)
\(788\) 0 0
\(789\) 3885.59 0.175324
\(790\) 0 0
\(791\) 37739.4 1.69641
\(792\) 0 0
\(793\) 28573.5 1.27954
\(794\) 0 0
\(795\) −21345.0 −0.952239
\(796\) 0 0
\(797\) −39802.4 −1.76898 −0.884489 0.466562i \(-0.845492\pi\)
−0.884489 + 0.466562i \(0.845492\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −8058.88 −0.355489
\(802\) 0 0
\(803\) 1302.20 0.0572276
\(804\) 0 0
\(805\) 48084.3 2.10528
\(806\) 0 0
\(807\) −13258.9 −0.578357
\(808\) 0 0
\(809\) −12250.3 −0.532384 −0.266192 0.963920i \(-0.585766\pi\)
−0.266192 + 0.963920i \(0.585766\pi\)
\(810\) 0 0
\(811\) −15936.1 −0.690002 −0.345001 0.938602i \(-0.612121\pi\)
−0.345001 + 0.938602i \(0.612121\pi\)
\(812\) 0 0
\(813\) −2971.39 −0.128181
\(814\) 0 0
\(815\) −18154.0 −0.780256
\(816\) 0 0
\(817\) 24284.6 1.03992
\(818\) 0 0
\(819\) −27083.8 −1.15554
\(820\) 0 0
\(821\) −10389.6 −0.441656 −0.220828 0.975313i \(-0.570876\pi\)
−0.220828 + 0.975313i \(0.570876\pi\)
\(822\) 0 0
\(823\) 1915.45 0.0811281 0.0405640 0.999177i \(-0.487085\pi\)
0.0405640 + 0.999177i \(0.487085\pi\)
\(824\) 0 0
\(825\) −24.9888 −0.00105454
\(826\) 0 0
\(827\) 1406.72 0.0591492 0.0295746 0.999563i \(-0.490585\pi\)
0.0295746 + 0.999563i \(0.490585\pi\)
\(828\) 0 0
\(829\) −22131.8 −0.927226 −0.463613 0.886038i \(-0.653447\pi\)
−0.463613 + 0.886038i \(0.653447\pi\)
\(830\) 0 0
\(831\) 9693.25 0.404639
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −14877.1 −0.616578
\(836\) 0 0
\(837\) 26030.5 1.07496
\(838\) 0 0
\(839\) 5716.82 0.235240 0.117620 0.993059i \(-0.462473\pi\)
0.117620 + 0.993059i \(0.462473\pi\)
\(840\) 0 0
\(841\) −11157.8 −0.457493
\(842\) 0 0
\(843\) 5740.16 0.234521
\(844\) 0 0
\(845\) 13958.2 0.568257
\(846\) 0 0
\(847\) −30644.7 −1.24317
\(848\) 0 0
\(849\) 21386.3 0.864516
\(850\) 0 0
\(851\) 38110.8 1.53516
\(852\) 0 0
\(853\) −26801.3 −1.07580 −0.537900 0.843008i \(-0.680782\pi\)
−0.537900 + 0.843008i \(0.680782\pi\)
\(854\) 0 0
\(855\) 32260.6 1.29040
\(856\) 0 0
\(857\) −6701.95 −0.267135 −0.133567 0.991040i \(-0.542643\pi\)
−0.133567 + 0.991040i \(0.542643\pi\)
\(858\) 0 0
\(859\) −33950.5 −1.34852 −0.674259 0.738495i \(-0.735537\pi\)
−0.674259 + 0.738495i \(0.735537\pi\)
\(860\) 0 0
\(861\) 17971.8 0.711357
\(862\) 0 0
\(863\) −43090.0 −1.69965 −0.849827 0.527062i \(-0.823294\pi\)
−0.849827 + 0.527062i \(0.823294\pi\)
\(864\) 0 0
\(865\) 27332.5 1.07437
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −131.631 −0.00513842
\(870\) 0 0
\(871\) −9750.81 −0.379327
\(872\) 0 0
\(873\) 18611.5 0.721540
\(874\) 0 0
\(875\) −31102.3 −1.20166
\(876\) 0 0
\(877\) −38515.6 −1.48299 −0.741493 0.670960i \(-0.765882\pi\)
−0.741493 + 0.670960i \(0.765882\pi\)
\(878\) 0 0
\(879\) −12521.7 −0.480485
\(880\) 0 0
\(881\) −32178.8 −1.23057 −0.615284 0.788305i \(-0.710959\pi\)
−0.615284 + 0.788305i \(0.710959\pi\)
\(882\) 0 0
\(883\) −33578.7 −1.27974 −0.639871 0.768482i \(-0.721012\pi\)
−0.639871 + 0.768482i \(0.721012\pi\)
\(884\) 0 0
\(885\) −5523.41 −0.209794
\(886\) 0 0
\(887\) 17336.8 0.656273 0.328137 0.944630i \(-0.393579\pi\)
0.328137 + 0.944630i \(0.393579\pi\)
\(888\) 0 0
\(889\) 34635.3 1.30667
\(890\) 0 0
\(891\) 262.507 0.00987018
\(892\) 0 0
\(893\) 27754.7 1.04006
\(894\) 0 0
\(895\) −19833.2 −0.740726
\(896\) 0 0
\(897\) −27678.4 −1.03027
\(898\) 0 0
\(899\) −24239.1 −0.899244
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 10556.4 0.389029
\(904\) 0 0
\(905\) −711.626 −0.0261384
\(906\) 0 0
\(907\) −2865.79 −0.104914 −0.0524570 0.998623i \(-0.516705\pi\)
−0.0524570 + 0.998623i \(0.516705\pi\)
\(908\) 0 0
\(909\) −4505.52 −0.164399
\(910\) 0 0
\(911\) −23972.0 −0.871819 −0.435910 0.899990i \(-0.643573\pi\)
−0.435910 + 0.899990i \(0.643573\pi\)
\(912\) 0 0
\(913\) −141.756 −0.00513849
\(914\) 0 0
\(915\) −14794.8 −0.534538
\(916\) 0 0
\(917\) −16188.8 −0.582991
\(918\) 0 0
\(919\) 49371.5 1.77216 0.886081 0.463530i \(-0.153418\pi\)
0.886081 + 0.463530i \(0.153418\pi\)
\(920\) 0 0
\(921\) −1392.01 −0.0498026
\(922\) 0 0
\(923\) 47758.3 1.70313
\(924\) 0 0
\(925\) 1681.40 0.0597668
\(926\) 0 0
\(927\) 25768.2 0.912986
\(928\) 0 0
\(929\) 47434.7 1.67522 0.837612 0.546266i \(-0.183951\pi\)
0.837612 + 0.546266i \(0.183951\pi\)
\(930\) 0 0
\(931\) −26159.0 −0.920866
\(932\) 0 0
\(933\) −12494.9 −0.438439
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37330.2 −1.30152 −0.650760 0.759283i \(-0.725550\pi\)
−0.650760 + 0.759283i \(0.725550\pi\)
\(938\) 0 0
\(939\) −22575.7 −0.784589
\(940\) 0 0
\(941\) −32070.6 −1.11102 −0.555511 0.831509i \(-0.687477\pi\)
−0.555511 + 0.831509i \(0.687477\pi\)
\(942\) 0 0
\(943\) −53820.9 −1.85859
\(944\) 0 0
\(945\) 32832.5 1.13020
\(946\) 0 0
\(947\) −23171.6 −0.795118 −0.397559 0.917577i \(-0.630143\pi\)
−0.397559 + 0.917577i \(0.630143\pi\)
\(948\) 0 0
\(949\) 63635.3 2.17670
\(950\) 0 0
\(951\) −6429.77 −0.219242
\(952\) 0 0
\(953\) −39914.2 −1.35671 −0.678357 0.734733i \(-0.737307\pi\)
−0.678357 + 0.734733i \(0.737307\pi\)
\(954\) 0 0
\(955\) −29582.8 −1.00239
\(956\) 0 0
\(957\) 360.127 0.0121643
\(958\) 0 0
\(959\) 437.666 0.0147372
\(960\) 0 0
\(961\) 14614.3 0.490560
\(962\) 0 0
\(963\) 22124.1 0.740332
\(964\) 0 0
\(965\) −22967.3 −0.766158
\(966\) 0 0
\(967\) 40591.6 1.34988 0.674941 0.737871i \(-0.264169\pi\)
0.674941 + 0.737871i \(0.264169\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29860.5 −0.986890 −0.493445 0.869777i \(-0.664263\pi\)
−0.493445 + 0.869777i \(0.664263\pi\)
\(972\) 0 0
\(973\) −6383.73 −0.210332
\(974\) 0 0
\(975\) −1221.14 −0.0401104
\(976\) 0 0
\(977\) 35041.0 1.14745 0.573726 0.819048i \(-0.305498\pi\)
0.573726 + 0.819048i \(0.305498\pi\)
\(978\) 0 0
\(979\) 478.204 0.0156113
\(980\) 0 0
\(981\) 14762.9 0.480472
\(982\) 0 0
\(983\) 54091.0 1.75507 0.877535 0.479513i \(-0.159187\pi\)
0.877535 + 0.479513i \(0.159187\pi\)
\(984\) 0 0
\(985\) −44187.0 −1.42935
\(986\) 0 0
\(987\) 12064.8 0.389085
\(988\) 0 0
\(989\) −31613.5 −1.01643
\(990\) 0 0
\(991\) −15918.2 −0.510251 −0.255126 0.966908i \(-0.582117\pi\)
−0.255126 + 0.966908i \(0.582117\pi\)
\(992\) 0 0
\(993\) −28839.9 −0.921657
\(994\) 0 0
\(995\) 25039.8 0.797805
\(996\) 0 0
\(997\) 15294.7 0.485845 0.242923 0.970046i \(-0.421894\pi\)
0.242923 + 0.970046i \(0.421894\pi\)
\(998\) 0 0
\(999\) 26022.4 0.824138
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.1.4 8
17.4 even 4 136.4.b.b.33.5 yes 8
17.13 even 4 136.4.b.b.33.4 8
17.16 even 2 inner 2312.4.a.k.1.5 8
51.38 odd 4 1224.4.c.e.577.6 8
51.47 odd 4 1224.4.c.e.577.3 8
68.47 odd 4 272.4.b.f.33.5 8
68.55 odd 4 272.4.b.f.33.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.4 8 17.13 even 4
136.4.b.b.33.5 yes 8 17.4 even 4
272.4.b.f.33.4 8 68.55 odd 4
272.4.b.f.33.5 8 68.47 odd 4
1224.4.c.e.577.3 8 51.47 odd 4
1224.4.c.e.577.6 8 51.38 odd 4
2312.4.a.k.1.4 8 1.1 even 1 trivial
2312.4.a.k.1.5 8 17.16 even 2 inner