Properties

Label 1856.4.a.y.1.5
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.03898\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.46343 q^{3} -2.14270 q^{5} +20.3573 q^{7} +14.7760 q^{9} +O(q^{10})\) \(q+6.46343 q^{3} -2.14270 q^{5} +20.3573 q^{7} +14.7760 q^{9} -52.0703 q^{11} -7.04574 q^{13} -13.8492 q^{15} +28.7724 q^{17} -76.4208 q^{19} +131.578 q^{21} +59.7251 q^{23} -120.409 q^{25} -79.0092 q^{27} +29.0000 q^{29} -3.25229 q^{31} -336.553 q^{33} -43.6196 q^{35} -150.673 q^{37} -45.5397 q^{39} -92.3254 q^{41} +100.703 q^{43} -31.6605 q^{45} +324.003 q^{47} +71.4197 q^{49} +185.969 q^{51} -374.774 q^{53} +111.571 q^{55} -493.941 q^{57} -489.567 q^{59} -221.508 q^{61} +300.799 q^{63} +15.0969 q^{65} +427.538 q^{67} +386.029 q^{69} -898.999 q^{71} -1087.35 q^{73} -778.254 q^{75} -1060.01 q^{77} -798.018 q^{79} -909.622 q^{81} +436.713 q^{83} -61.6507 q^{85} +187.440 q^{87} +456.763 q^{89} -143.432 q^{91} -21.0209 q^{93} +163.747 q^{95} +803.714 q^{97} -769.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 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
\(3\) 6.46343 1.24389 0.621944 0.783062i \(-0.286343\pi\)
0.621944 + 0.783062i \(0.286343\pi\)
\(4\) 0 0
\(5\) −2.14270 −0.191649 −0.0958246 0.995398i \(-0.530549\pi\)
−0.0958246 + 0.995398i \(0.530549\pi\)
\(6\) 0 0
\(7\) 20.3573 1.09919 0.549595 0.835431i \(-0.314782\pi\)
0.549595 + 0.835431i \(0.314782\pi\)
\(8\) 0 0
\(9\) 14.7760 0.547258
\(10\) 0 0
\(11\) −52.0703 −1.42725 −0.713627 0.700526i \(-0.752949\pi\)
−0.713627 + 0.700526i \(0.752949\pi\)
\(12\) 0 0
\(13\) −7.04574 −0.150318 −0.0751591 0.997172i \(-0.523946\pi\)
−0.0751591 + 0.997172i \(0.523946\pi\)
\(14\) 0 0
\(15\) −13.8492 −0.238390
\(16\) 0 0
\(17\) 28.7724 0.410490 0.205245 0.978711i \(-0.434201\pi\)
0.205245 + 0.978711i \(0.434201\pi\)
\(18\) 0 0
\(19\) −76.4208 −0.922744 −0.461372 0.887207i \(-0.652643\pi\)
−0.461372 + 0.887207i \(0.652643\pi\)
\(20\) 0 0
\(21\) 131.578 1.36727
\(22\) 0 0
\(23\) 59.7251 0.541458 0.270729 0.962656i \(-0.412735\pi\)
0.270729 + 0.962656i \(0.412735\pi\)
\(24\) 0 0
\(25\) −120.409 −0.963271
\(26\) 0 0
\(27\) −79.0092 −0.563160
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −3.25229 −0.0188428 −0.00942142 0.999956i \(-0.502999\pi\)
−0.00942142 + 0.999956i \(0.502999\pi\)
\(32\) 0 0
\(33\) −336.553 −1.77534
\(34\) 0 0
\(35\) −43.6196 −0.210659
\(36\) 0 0
\(37\) −150.673 −0.669471 −0.334736 0.942312i \(-0.608647\pi\)
−0.334736 + 0.942312i \(0.608647\pi\)
\(38\) 0 0
\(39\) −45.5397 −0.186979
\(40\) 0 0
\(41\) −92.3254 −0.351678 −0.175839 0.984419i \(-0.556264\pi\)
−0.175839 + 0.984419i \(0.556264\pi\)
\(42\) 0 0
\(43\) 100.703 0.357142 0.178571 0.983927i \(-0.442852\pi\)
0.178571 + 0.983927i \(0.442852\pi\)
\(44\) 0 0
\(45\) −31.6605 −0.104882
\(46\) 0 0
\(47\) 324.003 1.00555 0.502774 0.864418i \(-0.332313\pi\)
0.502774 + 0.864418i \(0.332313\pi\)
\(48\) 0 0
\(49\) 71.4197 0.208221
\(50\) 0 0
\(51\) 185.969 0.510604
\(52\) 0 0
\(53\) −374.774 −0.971305 −0.485653 0.874152i \(-0.661418\pi\)
−0.485653 + 0.874152i \(0.661418\pi\)
\(54\) 0 0
\(55\) 111.571 0.273532
\(56\) 0 0
\(57\) −493.941 −1.14779
\(58\) 0 0
\(59\) −489.567 −1.08027 −0.540137 0.841577i \(-0.681628\pi\)
−0.540137 + 0.841577i \(0.681628\pi\)
\(60\) 0 0
\(61\) −221.508 −0.464937 −0.232468 0.972604i \(-0.574680\pi\)
−0.232468 + 0.972604i \(0.574680\pi\)
\(62\) 0 0
\(63\) 300.799 0.601541
\(64\) 0 0
\(65\) 15.0969 0.0288083
\(66\) 0 0
\(67\) 427.538 0.779584 0.389792 0.920903i \(-0.372547\pi\)
0.389792 + 0.920903i \(0.372547\pi\)
\(68\) 0 0
\(69\) 386.029 0.673514
\(70\) 0 0
\(71\) −898.999 −1.50270 −0.751349 0.659905i \(-0.770596\pi\)
−0.751349 + 0.659905i \(0.770596\pi\)
\(72\) 0 0
\(73\) −1087.35 −1.74335 −0.871673 0.490087i \(-0.836965\pi\)
−0.871673 + 0.490087i \(0.836965\pi\)
\(74\) 0 0
\(75\) −778.254 −1.19820
\(76\) 0 0
\(77\) −1060.01 −1.56882
\(78\) 0 0
\(79\) −798.018 −1.13651 −0.568253 0.822854i \(-0.692381\pi\)
−0.568253 + 0.822854i \(0.692381\pi\)
\(80\) 0 0
\(81\) −909.622 −1.24777
\(82\) 0 0
\(83\) 436.713 0.577536 0.288768 0.957399i \(-0.406754\pi\)
0.288768 + 0.957399i \(0.406754\pi\)
\(84\) 0 0
\(85\) −61.6507 −0.0786701
\(86\) 0 0
\(87\) 187.440 0.230984
\(88\) 0 0
\(89\) 456.763 0.544009 0.272004 0.962296i \(-0.412313\pi\)
0.272004 + 0.962296i \(0.412313\pi\)
\(90\) 0 0
\(91\) −143.432 −0.165228
\(92\) 0 0
\(93\) −21.0209 −0.0234384
\(94\) 0 0
\(95\) 163.747 0.176843
\(96\) 0 0
\(97\) 803.714 0.841287 0.420643 0.907226i \(-0.361804\pi\)
0.420643 + 0.907226i \(0.361804\pi\)
\(98\) 0 0
\(99\) −769.390 −0.781077
\(100\) 0 0
\(101\) 738.800 0.727855 0.363928 0.931427i \(-0.381436\pi\)
0.363928 + 0.931427i \(0.381436\pi\)
\(102\) 0 0
\(103\) 2031.60 1.94349 0.971743 0.236042i \(-0.0758501\pi\)
0.971743 + 0.236042i \(0.0758501\pi\)
\(104\) 0 0
\(105\) −281.933 −0.262036
\(106\) 0 0
\(107\) −1594.33 −1.44047 −0.720233 0.693732i \(-0.755965\pi\)
−0.720233 + 0.693732i \(0.755965\pi\)
\(108\) 0 0
\(109\) 229.668 0.201818 0.100909 0.994896i \(-0.467825\pi\)
0.100909 + 0.994896i \(0.467825\pi\)
\(110\) 0 0
\(111\) −973.863 −0.832748
\(112\) 0 0
\(113\) −1584.73 −1.31928 −0.659640 0.751582i \(-0.729291\pi\)
−0.659640 + 0.751582i \(0.729291\pi\)
\(114\) 0 0
\(115\) −127.973 −0.103770
\(116\) 0 0
\(117\) −104.108 −0.0822629
\(118\) 0 0
\(119\) 585.728 0.451207
\(120\) 0 0
\(121\) 1380.32 1.03705
\(122\) 0 0
\(123\) −596.739 −0.437448
\(124\) 0 0
\(125\) 525.838 0.376259
\(126\) 0 0
\(127\) 621.184 0.434025 0.217013 0.976169i \(-0.430369\pi\)
0.217013 + 0.976169i \(0.430369\pi\)
\(128\) 0 0
\(129\) 650.890 0.444245
\(130\) 0 0
\(131\) 1504.34 1.00332 0.501659 0.865066i \(-0.332723\pi\)
0.501659 + 0.865066i \(0.332723\pi\)
\(132\) 0 0
\(133\) −1555.72 −1.01427
\(134\) 0 0
\(135\) 169.293 0.107929
\(136\) 0 0
\(137\) −29.3812 −0.0183227 −0.00916135 0.999958i \(-0.502916\pi\)
−0.00916135 + 0.999958i \(0.502916\pi\)
\(138\) 0 0
\(139\) −1262.60 −0.770450 −0.385225 0.922823i \(-0.625876\pi\)
−0.385225 + 0.922823i \(0.625876\pi\)
\(140\) 0 0
\(141\) 2094.17 1.25079
\(142\) 0 0
\(143\) 366.874 0.214542
\(144\) 0 0
\(145\) −62.1384 −0.0355883
\(146\) 0 0
\(147\) 461.616 0.259003
\(148\) 0 0
\(149\) −826.526 −0.454441 −0.227220 0.973843i \(-0.572964\pi\)
−0.227220 + 0.973843i \(0.572964\pi\)
\(150\) 0 0
\(151\) −2632.03 −1.41849 −0.709243 0.704964i \(-0.750963\pi\)
−0.709243 + 0.704964i \(0.750963\pi\)
\(152\) 0 0
\(153\) 425.140 0.224644
\(154\) 0 0
\(155\) 6.96868 0.00361121
\(156\) 0 0
\(157\) −234.540 −0.119225 −0.0596124 0.998222i \(-0.518986\pi\)
−0.0596124 + 0.998222i \(0.518986\pi\)
\(158\) 0 0
\(159\) −2422.33 −1.20820
\(160\) 0 0
\(161\) 1215.84 0.595166
\(162\) 0 0
\(163\) 1650.78 0.793246 0.396623 0.917982i \(-0.370182\pi\)
0.396623 + 0.917982i \(0.370182\pi\)
\(164\) 0 0
\(165\) 721.133 0.340243
\(166\) 0 0
\(167\) −3684.28 −1.70717 −0.853587 0.520950i \(-0.825578\pi\)
−0.853587 + 0.520950i \(0.825578\pi\)
\(168\) 0 0
\(169\) −2147.36 −0.977404
\(170\) 0 0
\(171\) −1129.19 −0.504979
\(172\) 0 0
\(173\) 3073.58 1.35075 0.675375 0.737474i \(-0.263982\pi\)
0.675375 + 0.737474i \(0.263982\pi\)
\(174\) 0 0
\(175\) −2451.20 −1.05882
\(176\) 0 0
\(177\) −3164.28 −1.34374
\(178\) 0 0
\(179\) −2980.70 −1.24463 −0.622313 0.782769i \(-0.713807\pi\)
−0.622313 + 0.782769i \(0.713807\pi\)
\(180\) 0 0
\(181\) −3145.07 −1.29155 −0.645777 0.763526i \(-0.723467\pi\)
−0.645777 + 0.763526i \(0.723467\pi\)
\(182\) 0 0
\(183\) −1431.70 −0.578329
\(184\) 0 0
\(185\) 322.847 0.128304
\(186\) 0 0
\(187\) −1498.19 −0.585874
\(188\) 0 0
\(189\) −1608.41 −0.619021
\(190\) 0 0
\(191\) −2125.56 −0.805236 −0.402618 0.915368i \(-0.631900\pi\)
−0.402618 + 0.915368i \(0.631900\pi\)
\(192\) 0 0
\(193\) 3085.29 1.15069 0.575347 0.817909i \(-0.304867\pi\)
0.575347 + 0.817909i \(0.304867\pi\)
\(194\) 0 0
\(195\) 97.5779 0.0358344
\(196\) 0 0
\(197\) 2484.45 0.898526 0.449263 0.893400i \(-0.351687\pi\)
0.449263 + 0.893400i \(0.351687\pi\)
\(198\) 0 0
\(199\) 5489.57 1.95550 0.977752 0.209764i \(-0.0672696\pi\)
0.977752 + 0.209764i \(0.0672696\pi\)
\(200\) 0 0
\(201\) 2763.37 0.969715
\(202\) 0 0
\(203\) 590.362 0.204115
\(204\) 0 0
\(205\) 197.826 0.0673988
\(206\) 0 0
\(207\) 882.496 0.296318
\(208\) 0 0
\(209\) 3979.26 1.31699
\(210\) 0 0
\(211\) 1884.64 0.614902 0.307451 0.951564i \(-0.400524\pi\)
0.307451 + 0.951564i \(0.400524\pi\)
\(212\) 0 0
\(213\) −5810.62 −1.86919
\(214\) 0 0
\(215\) −215.777 −0.0684460
\(216\) 0 0
\(217\) −66.2078 −0.0207119
\(218\) 0 0
\(219\) −7027.99 −2.16853
\(220\) 0 0
\(221\) −202.723 −0.0617041
\(222\) 0 0
\(223\) −5208.50 −1.56407 −0.782033 0.623237i \(-0.785817\pi\)
−0.782033 + 0.623237i \(0.785817\pi\)
\(224\) 0 0
\(225\) −1779.16 −0.527158
\(226\) 0 0
\(227\) 5243.29 1.53308 0.766540 0.642196i \(-0.221977\pi\)
0.766540 + 0.642196i \(0.221977\pi\)
\(228\) 0 0
\(229\) 846.348 0.244228 0.122114 0.992516i \(-0.461033\pi\)
0.122114 + 0.992516i \(0.461033\pi\)
\(230\) 0 0
\(231\) −6851.31 −1.95144
\(232\) 0 0
\(233\) 823.620 0.231576 0.115788 0.993274i \(-0.463061\pi\)
0.115788 + 0.993274i \(0.463061\pi\)
\(234\) 0 0
\(235\) −694.243 −0.192712
\(236\) 0 0
\(237\) −5157.93 −1.41369
\(238\) 0 0
\(239\) −2471.50 −0.668903 −0.334452 0.942413i \(-0.608551\pi\)
−0.334452 + 0.942413i \(0.608551\pi\)
\(240\) 0 0
\(241\) −1148.79 −0.307055 −0.153527 0.988144i \(-0.549063\pi\)
−0.153527 + 0.988144i \(0.549063\pi\)
\(242\) 0 0
\(243\) −3746.03 −0.988922
\(244\) 0 0
\(245\) −153.031 −0.0399053
\(246\) 0 0
\(247\) 538.441 0.138705
\(248\) 0 0
\(249\) 2822.67 0.718391
\(250\) 0 0
\(251\) 1686.20 0.424032 0.212016 0.977266i \(-0.431997\pi\)
0.212016 + 0.977266i \(0.431997\pi\)
\(252\) 0 0
\(253\) −3109.91 −0.772799
\(254\) 0 0
\(255\) −398.475 −0.0978568
\(256\) 0 0
\(257\) 3593.97 0.872318 0.436159 0.899870i \(-0.356338\pi\)
0.436159 + 0.899870i \(0.356338\pi\)
\(258\) 0 0
\(259\) −3067.29 −0.735877
\(260\) 0 0
\(261\) 428.503 0.101623
\(262\) 0 0
\(263\) −2321.87 −0.544382 −0.272191 0.962243i \(-0.587748\pi\)
−0.272191 + 0.962243i \(0.587748\pi\)
\(264\) 0 0
\(265\) 803.029 0.186150
\(266\) 0 0
\(267\) 2952.26 0.676686
\(268\) 0 0
\(269\) −2365.41 −0.536140 −0.268070 0.963399i \(-0.586386\pi\)
−0.268070 + 0.963399i \(0.586386\pi\)
\(270\) 0 0
\(271\) 1732.51 0.388348 0.194174 0.980967i \(-0.437797\pi\)
0.194174 + 0.980967i \(0.437797\pi\)
\(272\) 0 0
\(273\) −927.065 −0.205526
\(274\) 0 0
\(275\) 6269.73 1.37483
\(276\) 0 0
\(277\) 688.616 0.149368 0.0746839 0.997207i \(-0.476205\pi\)
0.0746839 + 0.997207i \(0.476205\pi\)
\(278\) 0 0
\(279\) −48.0557 −0.0103119
\(280\) 0 0
\(281\) 2224.44 0.472238 0.236119 0.971724i \(-0.424125\pi\)
0.236119 + 0.971724i \(0.424125\pi\)
\(282\) 0 0
\(283\) −5037.52 −1.05813 −0.529063 0.848582i \(-0.677457\pi\)
−0.529063 + 0.848582i \(0.677457\pi\)
\(284\) 0 0
\(285\) 1058.37 0.219973
\(286\) 0 0
\(287\) −1879.50 −0.386561
\(288\) 0 0
\(289\) −4085.15 −0.831498
\(290\) 0 0
\(291\) 5194.75 1.04647
\(292\) 0 0
\(293\) 6761.01 1.34806 0.674032 0.738702i \(-0.264561\pi\)
0.674032 + 0.738702i \(0.264561\pi\)
\(294\) 0 0
\(295\) 1049.00 0.207034
\(296\) 0 0
\(297\) 4114.03 0.803773
\(298\) 0 0
\(299\) −420.807 −0.0813910
\(300\) 0 0
\(301\) 2050.05 0.392568
\(302\) 0 0
\(303\) 4775.19 0.905371
\(304\) 0 0
\(305\) 474.625 0.0891047
\(306\) 0 0
\(307\) −8860.99 −1.64731 −0.823653 0.567093i \(-0.808068\pi\)
−0.823653 + 0.567093i \(0.808068\pi\)
\(308\) 0 0
\(309\) 13131.1 2.41748
\(310\) 0 0
\(311\) 4127.77 0.752619 0.376309 0.926494i \(-0.377193\pi\)
0.376309 + 0.926494i \(0.377193\pi\)
\(312\) 0 0
\(313\) −5384.53 −0.972369 −0.486184 0.873856i \(-0.661612\pi\)
−0.486184 + 0.873856i \(0.661612\pi\)
\(314\) 0 0
\(315\) −644.522 −0.115285
\(316\) 0 0
\(317\) 5379.49 0.953129 0.476565 0.879139i \(-0.341882\pi\)
0.476565 + 0.879139i \(0.341882\pi\)
\(318\) 0 0
\(319\) −1510.04 −0.265034
\(320\) 0 0
\(321\) −10304.9 −1.79178
\(322\) 0 0
\(323\) −2198.81 −0.378778
\(324\) 0 0
\(325\) 848.369 0.144797
\(326\) 0 0
\(327\) 1484.44 0.251039
\(328\) 0 0
\(329\) 6595.83 1.10529
\(330\) 0 0
\(331\) −5825.09 −0.967298 −0.483649 0.875262i \(-0.660689\pi\)
−0.483649 + 0.875262i \(0.660689\pi\)
\(332\) 0 0
\(333\) −2226.34 −0.366374
\(334\) 0 0
\(335\) −916.087 −0.149407
\(336\) 0 0
\(337\) −3627.26 −0.586318 −0.293159 0.956064i \(-0.594707\pi\)
−0.293159 + 0.956064i \(0.594707\pi\)
\(338\) 0 0
\(339\) −10242.8 −1.64104
\(340\) 0 0
\(341\) 169.348 0.0268935
\(342\) 0 0
\(343\) −5528.64 −0.870317
\(344\) 0 0
\(345\) −827.145 −0.129078
\(346\) 0 0
\(347\) −1172.68 −0.181421 −0.0907104 0.995877i \(-0.528914\pi\)
−0.0907104 + 0.995877i \(0.528914\pi\)
\(348\) 0 0
\(349\) 173.078 0.0265463 0.0132731 0.999912i \(-0.495775\pi\)
0.0132731 + 0.999912i \(0.495775\pi\)
\(350\) 0 0
\(351\) 556.678 0.0846532
\(352\) 0 0
\(353\) 13008.4 1.96137 0.980687 0.195582i \(-0.0626596\pi\)
0.980687 + 0.195582i \(0.0626596\pi\)
\(354\) 0 0
\(355\) 1926.29 0.287991
\(356\) 0 0
\(357\) 3785.82 0.561251
\(358\) 0 0
\(359\) −4487.40 −0.659710 −0.329855 0.944032i \(-0.607000\pi\)
−0.329855 + 0.944032i \(0.607000\pi\)
\(360\) 0 0
\(361\) −1018.85 −0.148543
\(362\) 0 0
\(363\) 8921.60 1.28998
\(364\) 0 0
\(365\) 2329.86 0.334111
\(366\) 0 0
\(367\) 1674.11 0.238114 0.119057 0.992887i \(-0.462013\pi\)
0.119057 + 0.992887i \(0.462013\pi\)
\(368\) 0 0
\(369\) −1364.20 −0.192459
\(370\) 0 0
\(371\) −7629.39 −1.06765
\(372\) 0 0
\(373\) 6800.89 0.944066 0.472033 0.881581i \(-0.343520\pi\)
0.472033 + 0.881581i \(0.343520\pi\)
\(374\) 0 0
\(375\) 3398.72 0.468024
\(376\) 0 0
\(377\) −204.326 −0.0279134
\(378\) 0 0
\(379\) −10216.7 −1.38468 −0.692341 0.721571i \(-0.743421\pi\)
−0.692341 + 0.721571i \(0.743421\pi\)
\(380\) 0 0
\(381\) 4014.98 0.539879
\(382\) 0 0
\(383\) −7184.47 −0.958509 −0.479255 0.877676i \(-0.659093\pi\)
−0.479255 + 0.877676i \(0.659093\pi\)
\(384\) 0 0
\(385\) 2271.29 0.300664
\(386\) 0 0
\(387\) 1487.99 0.195449
\(388\) 0 0
\(389\) 3848.67 0.501633 0.250817 0.968035i \(-0.419301\pi\)
0.250817 + 0.968035i \(0.419301\pi\)
\(390\) 0 0
\(391\) 1718.43 0.222263
\(392\) 0 0
\(393\) 9723.18 1.24801
\(394\) 0 0
\(395\) 1709.91 0.217810
\(396\) 0 0
\(397\) 1622.51 0.205117 0.102559 0.994727i \(-0.467297\pi\)
0.102559 + 0.994727i \(0.467297\pi\)
\(398\) 0 0
\(399\) −10055.3 −1.26164
\(400\) 0 0
\(401\) 13842.9 1.72389 0.861947 0.506999i \(-0.169245\pi\)
0.861947 + 0.506999i \(0.169245\pi\)
\(402\) 0 0
\(403\) 22.9148 0.00283242
\(404\) 0 0
\(405\) 1949.05 0.239133
\(406\) 0 0
\(407\) 7845.58 0.955506
\(408\) 0 0
\(409\) 3357.24 0.405880 0.202940 0.979191i \(-0.434950\pi\)
0.202940 + 0.979191i \(0.434950\pi\)
\(410\) 0 0
\(411\) −189.904 −0.0227914
\(412\) 0 0
\(413\) −9966.26 −1.18743
\(414\) 0 0
\(415\) −935.746 −0.110684
\(416\) 0 0
\(417\) −8160.74 −0.958353
\(418\) 0 0
\(419\) −1652.11 −0.192627 −0.0963134 0.995351i \(-0.530705\pi\)
−0.0963134 + 0.995351i \(0.530705\pi\)
\(420\) 0 0
\(421\) 6405.13 0.741490 0.370745 0.928735i \(-0.379102\pi\)
0.370745 + 0.928735i \(0.379102\pi\)
\(422\) 0 0
\(423\) 4787.46 0.550294
\(424\) 0 0
\(425\) −3464.45 −0.395413
\(426\) 0 0
\(427\) −4509.30 −0.511054
\(428\) 0 0
\(429\) 2371.27 0.266867
\(430\) 0 0
\(431\) 252.536 0.0282233 0.0141116 0.999900i \(-0.495508\pi\)
0.0141116 + 0.999900i \(0.495508\pi\)
\(432\) 0 0
\(433\) −12779.1 −1.41831 −0.709153 0.705055i \(-0.750922\pi\)
−0.709153 + 0.705055i \(0.750922\pi\)
\(434\) 0 0
\(435\) −401.627 −0.0442679
\(436\) 0 0
\(437\) −4564.24 −0.499628
\(438\) 0 0
\(439\) −3760.84 −0.408873 −0.204437 0.978880i \(-0.565536\pi\)
−0.204437 + 0.978880i \(0.565536\pi\)
\(440\) 0 0
\(441\) 1055.29 0.113950
\(442\) 0 0
\(443\) −3713.77 −0.398299 −0.199149 0.979969i \(-0.563818\pi\)
−0.199149 + 0.979969i \(0.563818\pi\)
\(444\) 0 0
\(445\) −978.707 −0.104259
\(446\) 0 0
\(447\) −5342.20 −0.565274
\(448\) 0 0
\(449\) −12043.9 −1.26589 −0.632947 0.774195i \(-0.718155\pi\)
−0.632947 + 0.774195i \(0.718155\pi\)
\(450\) 0 0
\(451\) 4807.41 0.501934
\(452\) 0 0
\(453\) −17011.9 −1.76444
\(454\) 0 0
\(455\) 307.333 0.0316659
\(456\) 0 0
\(457\) −12995.7 −1.33023 −0.665113 0.746743i \(-0.731617\pi\)
−0.665113 + 0.746743i \(0.731617\pi\)
\(458\) 0 0
\(459\) −2273.28 −0.231172
\(460\) 0 0
\(461\) −7874.30 −0.795538 −0.397769 0.917486i \(-0.630215\pi\)
−0.397769 + 0.917486i \(0.630215\pi\)
\(462\) 0 0
\(463\) −3466.08 −0.347910 −0.173955 0.984754i \(-0.555655\pi\)
−0.173955 + 0.984754i \(0.555655\pi\)
\(464\) 0 0
\(465\) 45.0416 0.00449195
\(466\) 0 0
\(467\) −14835.9 −1.47007 −0.735034 0.678030i \(-0.762834\pi\)
−0.735034 + 0.678030i \(0.762834\pi\)
\(468\) 0 0
\(469\) 8703.53 0.856912
\(470\) 0 0
\(471\) −1515.93 −0.148302
\(472\) 0 0
\(473\) −5243.66 −0.509733
\(474\) 0 0
\(475\) 9201.74 0.888853
\(476\) 0 0
\(477\) −5537.65 −0.531555
\(478\) 0 0
\(479\) 8119.86 0.774542 0.387271 0.921966i \(-0.373418\pi\)
0.387271 + 0.921966i \(0.373418\pi\)
\(480\) 0 0
\(481\) 1061.60 0.100634
\(482\) 0 0
\(483\) 7858.51 0.740320
\(484\) 0 0
\(485\) −1722.12 −0.161232
\(486\) 0 0
\(487\) 9698.92 0.902464 0.451232 0.892407i \(-0.350985\pi\)
0.451232 + 0.892407i \(0.350985\pi\)
\(488\) 0 0
\(489\) 10669.7 0.986709
\(490\) 0 0
\(491\) 6844.25 0.629076 0.314538 0.949245i \(-0.398150\pi\)
0.314538 + 0.949245i \(0.398150\pi\)
\(492\) 0 0
\(493\) 834.400 0.0762261
\(494\) 0 0
\(495\) 1648.57 0.149693
\(496\) 0 0
\(497\) −18301.2 −1.65175
\(498\) 0 0
\(499\) 18603.4 1.66895 0.834473 0.551049i \(-0.185772\pi\)
0.834473 + 0.551049i \(0.185772\pi\)
\(500\) 0 0
\(501\) −23813.1 −2.12353
\(502\) 0 0
\(503\) 10624.8 0.941822 0.470911 0.882181i \(-0.343925\pi\)
0.470911 + 0.882181i \(0.343925\pi\)
\(504\) 0 0
\(505\) −1583.03 −0.139493
\(506\) 0 0
\(507\) −13879.3 −1.21578
\(508\) 0 0
\(509\) 7931.22 0.690658 0.345329 0.938482i \(-0.387767\pi\)
0.345329 + 0.938482i \(0.387767\pi\)
\(510\) 0 0
\(511\) −22135.4 −1.91627
\(512\) 0 0
\(513\) 6037.95 0.519653
\(514\) 0 0
\(515\) −4353.10 −0.372467
\(516\) 0 0
\(517\) −16871.0 −1.43517
\(518\) 0 0
\(519\) 19865.9 1.68018
\(520\) 0 0
\(521\) 13771.1 1.15801 0.579005 0.815324i \(-0.303441\pi\)
0.579005 + 0.815324i \(0.303441\pi\)
\(522\) 0 0
\(523\) −20397.1 −1.70536 −0.852680 0.522433i \(-0.825024\pi\)
−0.852680 + 0.522433i \(0.825024\pi\)
\(524\) 0 0
\(525\) −15843.2 −1.31705
\(526\) 0 0
\(527\) −93.5761 −0.00773480
\(528\) 0 0
\(529\) −8599.91 −0.706823
\(530\) 0 0
\(531\) −7233.83 −0.591189
\(532\) 0 0
\(533\) 650.501 0.0528636
\(534\) 0 0
\(535\) 3416.18 0.276064
\(536\) 0 0
\(537\) −19265.6 −1.54817
\(538\) 0 0
\(539\) −3718.85 −0.297184
\(540\) 0 0
\(541\) −1883.72 −0.149699 −0.0748496 0.997195i \(-0.523848\pi\)
−0.0748496 + 0.997195i \(0.523848\pi\)
\(542\) 0 0
\(543\) −20328.0 −1.60655
\(544\) 0 0
\(545\) −492.110 −0.0386783
\(546\) 0 0
\(547\) 9079.06 0.709676 0.354838 0.934928i \(-0.384536\pi\)
0.354838 + 0.934928i \(0.384536\pi\)
\(548\) 0 0
\(549\) −3272.99 −0.254440
\(550\) 0 0
\(551\) −2216.20 −0.171349
\(552\) 0 0
\(553\) −16245.5 −1.24924
\(554\) 0 0
\(555\) 2086.70 0.159595
\(556\) 0 0
\(557\) 5982.38 0.455084 0.227542 0.973768i \(-0.426931\pi\)
0.227542 + 0.973768i \(0.426931\pi\)
\(558\) 0 0
\(559\) −709.530 −0.0536850
\(560\) 0 0
\(561\) −9683.44 −0.728762
\(562\) 0 0
\(563\) 7837.88 0.586727 0.293363 0.956001i \(-0.405225\pi\)
0.293363 + 0.956001i \(0.405225\pi\)
\(564\) 0 0
\(565\) 3395.60 0.252839
\(566\) 0 0
\(567\) −18517.4 −1.37153
\(568\) 0 0
\(569\) 10019.2 0.738182 0.369091 0.929393i \(-0.379669\pi\)
0.369091 + 0.929393i \(0.379669\pi\)
\(570\) 0 0
\(571\) −8832.38 −0.647327 −0.323663 0.946172i \(-0.604915\pi\)
−0.323663 + 0.946172i \(0.604915\pi\)
\(572\) 0 0
\(573\) −13738.4 −1.00162
\(574\) 0 0
\(575\) −7191.43 −0.521571
\(576\) 0 0
\(577\) −17583.3 −1.26863 −0.634317 0.773073i \(-0.718719\pi\)
−0.634317 + 0.773073i \(0.718719\pi\)
\(578\) 0 0
\(579\) 19941.6 1.43133
\(580\) 0 0
\(581\) 8890.30 0.634823
\(582\) 0 0
\(583\) 19514.6 1.38630
\(584\) 0 0
\(585\) 223.072 0.0157656
\(586\) 0 0
\(587\) 10120.7 0.711630 0.355815 0.934556i \(-0.384203\pi\)
0.355815 + 0.934556i \(0.384203\pi\)
\(588\) 0 0
\(589\) 248.543 0.0173871
\(590\) 0 0
\(591\) 16058.1 1.11767
\(592\) 0 0
\(593\) 3597.06 0.249095 0.124548 0.992214i \(-0.460252\pi\)
0.124548 + 0.992214i \(0.460252\pi\)
\(594\) 0 0
\(595\) −1255.04 −0.0864734
\(596\) 0 0
\(597\) 35481.5 2.43243
\(598\) 0 0
\(599\) −7987.90 −0.544870 −0.272435 0.962174i \(-0.587829\pi\)
−0.272435 + 0.962174i \(0.587829\pi\)
\(600\) 0 0
\(601\) 2646.60 0.179629 0.0898147 0.995958i \(-0.471373\pi\)
0.0898147 + 0.995958i \(0.471373\pi\)
\(602\) 0 0
\(603\) 6317.29 0.426634
\(604\) 0 0
\(605\) −2957.61 −0.198751
\(606\) 0 0
\(607\) −15181.8 −1.01517 −0.507587 0.861600i \(-0.669463\pi\)
−0.507587 + 0.861600i \(0.669463\pi\)
\(608\) 0 0
\(609\) 3815.76 0.253896
\(610\) 0 0
\(611\) −2282.84 −0.151152
\(612\) 0 0
\(613\) 9721.86 0.640558 0.320279 0.947323i \(-0.396223\pi\)
0.320279 + 0.947323i \(0.396223\pi\)
\(614\) 0 0
\(615\) 1278.63 0.0838366
\(616\) 0 0
\(617\) −14150.2 −0.923282 −0.461641 0.887067i \(-0.652739\pi\)
−0.461641 + 0.887067i \(0.652739\pi\)
\(618\) 0 0
\(619\) 10804.4 0.701560 0.350780 0.936458i \(-0.385917\pi\)
0.350780 + 0.936458i \(0.385917\pi\)
\(620\) 0 0
\(621\) −4718.83 −0.304928
\(622\) 0 0
\(623\) 9298.46 0.597970
\(624\) 0 0
\(625\) 13924.4 0.891161
\(626\) 0 0
\(627\) 25719.7 1.63819
\(628\) 0 0
\(629\) −4335.22 −0.274811
\(630\) 0 0
\(631\) 20692.1 1.30545 0.652725 0.757595i \(-0.273626\pi\)
0.652725 + 0.757595i \(0.273626\pi\)
\(632\) 0 0
\(633\) 12181.3 0.764869
\(634\) 0 0
\(635\) −1331.01 −0.0831805
\(636\) 0 0
\(637\) −503.204 −0.0312993
\(638\) 0 0
\(639\) −13283.6 −0.822364
\(640\) 0 0
\(641\) −1995.59 −0.122966 −0.0614828 0.998108i \(-0.519583\pi\)
−0.0614828 + 0.998108i \(0.519583\pi\)
\(642\) 0 0
\(643\) −10911.2 −0.669203 −0.334602 0.942360i \(-0.608602\pi\)
−0.334602 + 0.942360i \(0.608602\pi\)
\(644\) 0 0
\(645\) −1394.66 −0.0851392
\(646\) 0 0
\(647\) 20046.3 1.21808 0.609042 0.793138i \(-0.291554\pi\)
0.609042 + 0.793138i \(0.291554\pi\)
\(648\) 0 0
\(649\) 25491.9 1.54183
\(650\) 0 0
\(651\) −427.930 −0.0257633
\(652\) 0 0
\(653\) −20205.6 −1.21088 −0.605441 0.795890i \(-0.707003\pi\)
−0.605441 + 0.795890i \(0.707003\pi\)
\(654\) 0 0
\(655\) −3223.35 −0.192285
\(656\) 0 0
\(657\) −16066.6 −0.954061
\(658\) 0 0
\(659\) 9268.73 0.547888 0.273944 0.961746i \(-0.411672\pi\)
0.273944 + 0.961746i \(0.411672\pi\)
\(660\) 0 0
\(661\) 18209.9 1.07153 0.535765 0.844367i \(-0.320023\pi\)
0.535765 + 0.844367i \(0.320023\pi\)
\(662\) 0 0
\(663\) −1310.29 −0.0767531
\(664\) 0 0
\(665\) 3333.45 0.194384
\(666\) 0 0
\(667\) 1732.03 0.100546
\(668\) 0 0
\(669\) −33664.8 −1.94552
\(670\) 0 0
\(671\) 11534.0 0.663583
\(672\) 0 0
\(673\) −63.9154 −0.00366086 −0.00183043 0.999998i \(-0.500583\pi\)
−0.00183043 + 0.999998i \(0.500583\pi\)
\(674\) 0 0
\(675\) 9513.40 0.542476
\(676\) 0 0
\(677\) −11436.2 −0.649231 −0.324616 0.945846i \(-0.605235\pi\)
−0.324616 + 0.945846i \(0.605235\pi\)
\(678\) 0 0
\(679\) 16361.5 0.924735
\(680\) 0 0
\(681\) 33889.6 1.90698
\(682\) 0 0
\(683\) 22719.5 1.27282 0.636410 0.771351i \(-0.280419\pi\)
0.636410 + 0.771351i \(0.280419\pi\)
\(684\) 0 0
\(685\) 62.9553 0.00351153
\(686\) 0 0
\(687\) 5470.31 0.303793
\(688\) 0 0
\(689\) 2640.56 0.146005
\(690\) 0 0
\(691\) −10495.0 −0.577783 −0.288891 0.957362i \(-0.593287\pi\)
−0.288891 + 0.957362i \(0.593287\pi\)
\(692\) 0 0
\(693\) −15662.7 −0.858552
\(694\) 0 0
\(695\) 2705.38 0.147656
\(696\) 0 0
\(697\) −2656.42 −0.144360
\(698\) 0 0
\(699\) 5323.41 0.288054
\(700\) 0 0
\(701\) 6530.49 0.351859 0.175930 0.984403i \(-0.443707\pi\)
0.175930 + 0.984403i \(0.443707\pi\)
\(702\) 0 0
\(703\) 11514.5 0.617751
\(704\) 0 0
\(705\) −4487.19 −0.239713
\(706\) 0 0
\(707\) 15040.0 0.800052
\(708\) 0 0
\(709\) 16597.7 0.879183 0.439591 0.898198i \(-0.355123\pi\)
0.439591 + 0.898198i \(0.355123\pi\)
\(710\) 0 0
\(711\) −11791.5 −0.621962
\(712\) 0 0
\(713\) −194.243 −0.0102026
\(714\) 0 0
\(715\) −786.102 −0.0411168
\(716\) 0 0
\(717\) −15974.4 −0.832041
\(718\) 0 0
\(719\) 6525.25 0.338457 0.169229 0.985577i \(-0.445872\pi\)
0.169229 + 0.985577i \(0.445872\pi\)
\(720\) 0 0
\(721\) 41357.8 2.13626
\(722\) 0 0
\(723\) −7425.14 −0.381942
\(724\) 0 0
\(725\) −3491.86 −0.178875
\(726\) 0 0
\(727\) 1066.84 0.0544249 0.0272125 0.999630i \(-0.491337\pi\)
0.0272125 + 0.999630i \(0.491337\pi\)
\(728\) 0 0
\(729\) 347.561 0.0176579
\(730\) 0 0
\(731\) 2897.48 0.146603
\(732\) 0 0
\(733\) 6226.75 0.313765 0.156883 0.987617i \(-0.449856\pi\)
0.156883 + 0.987617i \(0.449856\pi\)
\(734\) 0 0
\(735\) −989.106 −0.0496377
\(736\) 0 0
\(737\) −22262.1 −1.11266
\(738\) 0 0
\(739\) −31226.3 −1.55437 −0.777183 0.629274i \(-0.783352\pi\)
−0.777183 + 0.629274i \(0.783352\pi\)
\(740\) 0 0
\(741\) 3480.18 0.172534
\(742\) 0 0
\(743\) −30790.4 −1.52031 −0.760156 0.649741i \(-0.774877\pi\)
−0.760156 + 0.649741i \(0.774877\pi\)
\(744\) 0 0
\(745\) 1771.00 0.0870932
\(746\) 0 0
\(747\) 6452.86 0.316061
\(748\) 0 0
\(749\) −32456.3 −1.58335
\(750\) 0 0
\(751\) 11908.0 0.578598 0.289299 0.957239i \(-0.406578\pi\)
0.289299 + 0.957239i \(0.406578\pi\)
\(752\) 0 0
\(753\) 10898.6 0.527448
\(754\) 0 0
\(755\) 5639.65 0.271852
\(756\) 0 0
\(757\) 7197.74 0.345583 0.172792 0.984958i \(-0.444721\pi\)
0.172792 + 0.984958i \(0.444721\pi\)
\(758\) 0 0
\(759\) −20100.7 −0.961275
\(760\) 0 0
\(761\) 9185.20 0.437534 0.218767 0.975777i \(-0.429797\pi\)
0.218767 + 0.975777i \(0.429797\pi\)
\(762\) 0 0
\(763\) 4675.42 0.221837
\(764\) 0 0
\(765\) −910.949 −0.0430528
\(766\) 0 0
\(767\) 3449.36 0.162385
\(768\) 0 0
\(769\) 32791.1 1.53768 0.768840 0.639441i \(-0.220834\pi\)
0.768840 + 0.639441i \(0.220834\pi\)
\(770\) 0 0
\(771\) 23229.4 1.08507
\(772\) 0 0
\(773\) 28237.1 1.31386 0.656932 0.753950i \(-0.271854\pi\)
0.656932 + 0.753950i \(0.271854\pi\)
\(774\) 0 0
\(775\) 391.604 0.0181508
\(776\) 0 0
\(777\) −19825.2 −0.915349
\(778\) 0 0
\(779\) 7055.58 0.324509
\(780\) 0 0
\(781\) 46811.2 2.14473
\(782\) 0 0
\(783\) −2291.27 −0.104576
\(784\) 0 0
\(785\) 502.548 0.0228493
\(786\) 0 0
\(787\) −12082.3 −0.547254 −0.273627 0.961836i \(-0.588223\pi\)
−0.273627 + 0.961836i \(0.588223\pi\)
\(788\) 0 0
\(789\) −15007.2 −0.677151
\(790\) 0 0
\(791\) −32260.8 −1.45014
\(792\) 0 0
\(793\) 1560.68 0.0698884
\(794\) 0 0
\(795\) 5190.33 0.231550
\(796\) 0 0
\(797\) 1248.33 0.0554807 0.0277404 0.999615i \(-0.491169\pi\)
0.0277404 + 0.999615i \(0.491169\pi\)
\(798\) 0 0
\(799\) 9322.35 0.412767
\(800\) 0 0
\(801\) 6749.12 0.297713
\(802\) 0 0
\(803\) 56618.5 2.48820
\(804\) 0 0
\(805\) −2605.19 −0.114063
\(806\) 0 0
\(807\) −15288.7 −0.666898
\(808\) 0 0
\(809\) −23512.6 −1.02183 −0.510914 0.859632i \(-0.670693\pi\)
−0.510914 + 0.859632i \(0.670693\pi\)
\(810\) 0 0
\(811\) −23248.4 −1.00661 −0.503306 0.864108i \(-0.667883\pi\)
−0.503306 + 0.864108i \(0.667883\pi\)
\(812\) 0 0
\(813\) 11197.9 0.483062
\(814\) 0 0
\(815\) −3537.13 −0.152025
\(816\) 0 0
\(817\) −7695.84 −0.329551
\(818\) 0 0
\(819\) −2119.35 −0.0904226
\(820\) 0 0
\(821\) −26898.0 −1.14342 −0.571708 0.820457i \(-0.693719\pi\)
−0.571708 + 0.820457i \(0.693719\pi\)
\(822\) 0 0
\(823\) 12422.7 0.526158 0.263079 0.964774i \(-0.415262\pi\)
0.263079 + 0.964774i \(0.415262\pi\)
\(824\) 0 0
\(825\) 40524.0 1.71014
\(826\) 0 0
\(827\) 4915.14 0.206670 0.103335 0.994647i \(-0.467049\pi\)
0.103335 + 0.994647i \(0.467049\pi\)
\(828\) 0 0
\(829\) 45226.4 1.89478 0.947392 0.320074i \(-0.103708\pi\)
0.947392 + 0.320074i \(0.103708\pi\)
\(830\) 0 0
\(831\) 4450.82 0.185797
\(832\) 0 0
\(833\) 2054.92 0.0854725
\(834\) 0 0
\(835\) 7894.31 0.327178
\(836\) 0 0
\(837\) 256.961 0.0106115
\(838\) 0 0
\(839\) 3982.56 0.163877 0.0819387 0.996637i \(-0.473889\pi\)
0.0819387 + 0.996637i \(0.473889\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 14377.5 0.587411
\(844\) 0 0
\(845\) 4601.15 0.187319
\(846\) 0 0
\(847\) 28099.6 1.13992
\(848\) 0 0
\(849\) −32559.7 −1.31619
\(850\) 0 0
\(851\) −8998.94 −0.362491
\(852\) 0 0
\(853\) 37142.6 1.49090 0.745450 0.666562i \(-0.232235\pi\)
0.745450 + 0.666562i \(0.232235\pi\)
\(854\) 0 0
\(855\) 2419.52 0.0967789
\(856\) 0 0
\(857\) 18690.5 0.744990 0.372495 0.928034i \(-0.378502\pi\)
0.372495 + 0.928034i \(0.378502\pi\)
\(858\) 0 0
\(859\) 22852.1 0.907687 0.453843 0.891081i \(-0.350053\pi\)
0.453843 + 0.891081i \(0.350053\pi\)
\(860\) 0 0
\(861\) −12148.0 −0.480839
\(862\) 0 0
\(863\) 41976.4 1.65573 0.827865 0.560928i \(-0.189555\pi\)
0.827865 + 0.560928i \(0.189555\pi\)
\(864\) 0 0
\(865\) −6585.77 −0.258870
\(866\) 0 0
\(867\) −26404.1 −1.03429
\(868\) 0 0
\(869\) 41553.0 1.62208
\(870\) 0 0
\(871\) −3012.32 −0.117186
\(872\) 0 0
\(873\) 11875.7 0.460401
\(874\) 0 0
\(875\) 10704.6 0.413581
\(876\) 0 0
\(877\) −44394.0 −1.70932 −0.854662 0.519184i \(-0.826236\pi\)
−0.854662 + 0.519184i \(0.826236\pi\)
\(878\) 0 0
\(879\) 43699.4 1.67684
\(880\) 0 0
\(881\) −6337.13 −0.242342 −0.121171 0.992632i \(-0.538665\pi\)
−0.121171 + 0.992632i \(0.538665\pi\)
\(882\) 0 0
\(883\) 2834.24 0.108018 0.0540090 0.998540i \(-0.482800\pi\)
0.0540090 + 0.998540i \(0.482800\pi\)
\(884\) 0 0
\(885\) 6780.12 0.257527
\(886\) 0 0
\(887\) 76.3532 0.00289029 0.00144515 0.999999i \(-0.499540\pi\)
0.00144515 + 0.999999i \(0.499540\pi\)
\(888\) 0 0
\(889\) 12645.6 0.477077
\(890\) 0 0
\(891\) 47364.3 1.78088
\(892\) 0 0
\(893\) −24760.6 −0.927863
\(894\) 0 0
\(895\) 6386.75 0.238531
\(896\) 0 0
\(897\) −2719.86 −0.101241
\(898\) 0 0
\(899\) −94.3163 −0.00349903
\(900\) 0 0
\(901\) −10783.2 −0.398711
\(902\) 0 0
\(903\) 13250.4 0.488310
\(904\) 0 0
\(905\) 6738.95 0.247525
\(906\) 0 0
\(907\) −18209.7 −0.666641 −0.333320 0.942814i \(-0.608169\pi\)
−0.333320 + 0.942814i \(0.608169\pi\)
\(908\) 0 0
\(909\) 10916.5 0.398325
\(910\) 0 0
\(911\) 33348.8 1.21284 0.606419 0.795145i \(-0.292605\pi\)
0.606419 + 0.795145i \(0.292605\pi\)
\(912\) 0 0
\(913\) −22739.8 −0.824291
\(914\) 0 0
\(915\) 3067.71 0.110836
\(916\) 0 0
\(917\) 30624.2 1.10284
\(918\) 0 0
\(919\) 20190.6 0.724730 0.362365 0.932036i \(-0.381969\pi\)
0.362365 + 0.932036i \(0.381969\pi\)
\(920\) 0 0
\(921\) −57272.4 −2.04907
\(922\) 0 0
\(923\) 6334.11 0.225883
\(924\) 0 0
\(925\) 18142.3 0.644882
\(926\) 0 0
\(927\) 30018.8 1.06359
\(928\) 0 0
\(929\) 9089.58 0.321011 0.160506 0.987035i \(-0.448688\pi\)
0.160506 + 0.987035i \(0.448688\pi\)
\(930\) 0 0
\(931\) −5457.95 −0.192134
\(932\) 0 0
\(933\) 26679.6 0.936173
\(934\) 0 0
\(935\) 3210.17 0.112282
\(936\) 0 0
\(937\) 39646.5 1.38228 0.691140 0.722721i \(-0.257109\pi\)
0.691140 + 0.722721i \(0.257109\pi\)
\(938\) 0 0
\(939\) −34802.5 −1.20952
\(940\) 0 0
\(941\) 51013.4 1.76726 0.883629 0.468188i \(-0.155093\pi\)
0.883629 + 0.468188i \(0.155093\pi\)
\(942\) 0 0
\(943\) −5514.14 −0.190419
\(944\) 0 0
\(945\) 3446.35 0.118635
\(946\) 0 0
\(947\) −43867.1 −1.50527 −0.752634 0.658439i \(-0.771217\pi\)
−0.752634 + 0.658439i \(0.771217\pi\)
\(948\) 0 0
\(949\) 7661.16 0.262057
\(950\) 0 0
\(951\) 34769.9 1.18559
\(952\) 0 0
\(953\) −4167.10 −0.141643 −0.0708214 0.997489i \(-0.522562\pi\)
−0.0708214 + 0.997489i \(0.522562\pi\)
\(954\) 0 0
\(955\) 4554.44 0.154323
\(956\) 0 0
\(957\) −9760.04 −0.329673
\(958\) 0 0
\(959\) −598.123 −0.0201401
\(960\) 0 0
\(961\) −29780.4 −0.999645
\(962\) 0 0
\(963\) −23557.8 −0.788307
\(964\) 0 0
\(965\) −6610.85 −0.220529
\(966\) 0 0
\(967\) −21935.0 −0.729454 −0.364727 0.931115i \(-0.618838\pi\)
−0.364727 + 0.931115i \(0.618838\pi\)
\(968\) 0 0
\(969\) −14211.9 −0.471157
\(970\) 0 0
\(971\) 24169.6 0.798805 0.399403 0.916776i \(-0.369218\pi\)
0.399403 + 0.916776i \(0.369218\pi\)
\(972\) 0 0
\(973\) −25703.2 −0.846871
\(974\) 0 0
\(975\) 5483.38 0.180111
\(976\) 0 0
\(977\) 16106.2 0.527415 0.263707 0.964603i \(-0.415055\pi\)
0.263707 + 0.964603i \(0.415055\pi\)
\(978\) 0 0
\(979\) −23783.8 −0.776439
\(980\) 0 0
\(981\) 3393.57 0.110447
\(982\) 0 0
\(983\) −22527.2 −0.730933 −0.365466 0.930825i \(-0.619090\pi\)
−0.365466 + 0.930825i \(0.619090\pi\)
\(984\) 0 0
\(985\) −5323.43 −0.172202
\(986\) 0 0
\(987\) 42631.7 1.37486
\(988\) 0 0
\(989\) 6014.52 0.193378
\(990\) 0 0
\(991\) 8337.94 0.267269 0.133634 0.991031i \(-0.457335\pi\)
0.133634 + 0.991031i \(0.457335\pi\)
\(992\) 0 0
\(993\) −37650.1 −1.20321
\(994\) 0 0
\(995\) −11762.5 −0.374771
\(996\) 0 0
\(997\) 18827.2 0.598058 0.299029 0.954244i \(-0.403337\pi\)
0.299029 + 0.954244i \(0.403337\pi\)
\(998\) 0 0
\(999\) 11904.5 0.377020
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 1856.4.a.y.1.5 5
4.3 odd 2 1856.4.a.bb.1.1 5
8.3 odd 2 464.4.a.l.1.5 5
8.5 even 2 29.4.a.b.1.1 5
24.5 odd 2 261.4.a.f.1.5 5
40.29 even 2 725.4.a.c.1.5 5
56.13 odd 2 1421.4.a.e.1.1 5
232.173 even 2 841.4.a.b.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.1 5 8.5 even 2
261.4.a.f.1.5 5 24.5 odd 2
464.4.a.l.1.5 5 8.3 odd 2
725.4.a.c.1.5 5 40.29 even 2
841.4.a.b.1.5 5 232.173 even 2
1421.4.a.e.1.1 5 56.13 odd 2
1856.4.a.y.1.5 5 1.1 even 1 trivial
1856.4.a.bb.1.1 5 4.3 odd 2