Properties

Label 675.4.a.q.1.3
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.45938\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.45938 q^{2} +11.8861 q^{4} -5.08123 q^{7} +17.3296 q^{8} +O(q^{10})\) \(q+4.45938 q^{2} +11.8861 q^{4} -5.08123 q^{7} +17.3296 q^{8} -58.3007 q^{11} -21.2119 q^{13} -22.6592 q^{14} -17.8095 q^{16} -68.8451 q^{17} -40.8133 q^{19} -259.985 q^{22} -144.318 q^{23} -94.5921 q^{26} -60.3960 q^{28} +220.058 q^{29} +291.545 q^{31} -218.056 q^{32} -307.006 q^{34} -260.637 q^{37} -182.002 q^{38} +169.766 q^{41} +438.596 q^{43} -692.967 q^{44} -643.571 q^{46} -255.481 q^{47} -317.181 q^{49} -252.127 q^{52} +214.714 q^{53} -88.0557 q^{56} +981.322 q^{58} -331.524 q^{59} +54.9647 q^{61} +1300.11 q^{62} -829.920 q^{64} -758.179 q^{67} -818.299 q^{68} +904.348 q^{71} -866.622 q^{73} -1162.28 q^{74} -485.110 q^{76} +296.239 q^{77} +206.961 q^{79} +757.054 q^{82} -463.397 q^{83} +1955.87 q^{86} -1010.33 q^{88} -601.736 q^{89} +107.783 q^{91} -1715.38 q^{92} -1139.29 q^{94} -229.363 q^{97} -1414.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8} - 38 q^{11} - 28 q^{13} + 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} - 122 q^{22} - 81 q^{23} - 416 q^{26} - 410 q^{28} - 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} - 78 q^{37} - 757 q^{38} + 338 q^{41} - 22 q^{43} - 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} + 1566 q^{52} + 521 q^{53} + 1254 q^{56} + 2096 q^{58} - 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} - 878 q^{67} - 3053 q^{68} + 602 q^{71} - 1294 q^{73} - 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 1682 q^{82} - 1287 q^{83} + 3730 q^{86} + 858 q^{88} - 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} - 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.45938 1.57663 0.788315 0.615272i \(-0.210954\pi\)
0.788315 + 0.615272i \(0.210954\pi\)
\(3\) 0 0
\(4\) 11.8861 1.48576
\(5\) 0 0
\(6\) 0 0
\(7\) −5.08123 −0.274361 −0.137180 0.990546i \(-0.543804\pi\)
−0.137180 + 0.990546i \(0.543804\pi\)
\(8\) 17.3296 0.765867
\(9\) 0 0
\(10\) 0 0
\(11\) −58.3007 −1.59803 −0.799014 0.601312i \(-0.794645\pi\)
−0.799014 + 0.601312i \(0.794645\pi\)
\(12\) 0 0
\(13\) −21.2119 −0.452548 −0.226274 0.974064i \(-0.572655\pi\)
−0.226274 + 0.974064i \(0.572655\pi\)
\(14\) −22.6592 −0.432566
\(15\) 0 0
\(16\) −17.8095 −0.278274
\(17\) −68.8451 −0.982199 −0.491099 0.871104i \(-0.663405\pi\)
−0.491099 + 0.871104i \(0.663405\pi\)
\(18\) 0 0
\(19\) −40.8133 −0.492800 −0.246400 0.969168i \(-0.579248\pi\)
−0.246400 + 0.969168i \(0.579248\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −259.985 −2.51950
\(23\) −144.318 −1.30837 −0.654184 0.756336i \(-0.726988\pi\)
−0.654184 + 0.756336i \(0.726988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −94.5921 −0.713501
\(27\) 0 0
\(28\) −60.3960 −0.407635
\(29\) 220.058 1.40909 0.704547 0.709657i \(-0.251150\pi\)
0.704547 + 0.709657i \(0.251150\pi\)
\(30\) 0 0
\(31\) 291.545 1.68913 0.844566 0.535452i \(-0.179859\pi\)
0.844566 + 0.535452i \(0.179859\pi\)
\(32\) −218.056 −1.20460
\(33\) 0 0
\(34\) −307.006 −1.54856
\(35\) 0 0
\(36\) 0 0
\(37\) −260.637 −1.15807 −0.579033 0.815304i \(-0.696570\pi\)
−0.579033 + 0.815304i \(0.696570\pi\)
\(38\) −182.002 −0.776964
\(39\) 0 0
\(40\) 0 0
\(41\) 169.766 0.646660 0.323330 0.946286i \(-0.395198\pi\)
0.323330 + 0.946286i \(0.395198\pi\)
\(42\) 0 0
\(43\) 438.596 1.55547 0.777735 0.628592i \(-0.216369\pi\)
0.777735 + 0.628592i \(0.216369\pi\)
\(44\) −692.967 −2.37429
\(45\) 0 0
\(46\) −643.571 −2.06281
\(47\) −255.481 −0.792887 −0.396444 0.918059i \(-0.629756\pi\)
−0.396444 + 0.918059i \(0.629756\pi\)
\(48\) 0 0
\(49\) −317.181 −0.924726
\(50\) 0 0
\(51\) 0 0
\(52\) −252.127 −0.672379
\(53\) 214.714 0.556477 0.278239 0.960512i \(-0.410249\pi\)
0.278239 + 0.960512i \(0.410249\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −88.0557 −0.210124
\(57\) 0 0
\(58\) 981.322 2.22162
\(59\) −331.524 −0.731537 −0.365769 0.930706i \(-0.619194\pi\)
−0.365769 + 0.930706i \(0.619194\pi\)
\(60\) 0 0
\(61\) 54.9647 0.115369 0.0576845 0.998335i \(-0.481628\pi\)
0.0576845 + 0.998335i \(0.481628\pi\)
\(62\) 1300.11 2.66314
\(63\) 0 0
\(64\) −829.920 −1.62094
\(65\) 0 0
\(66\) 0 0
\(67\) −758.179 −1.38248 −0.691241 0.722624i \(-0.742936\pi\)
−0.691241 + 0.722624i \(0.742936\pi\)
\(68\) −818.299 −1.45931
\(69\) 0 0
\(70\) 0 0
\(71\) 904.348 1.51164 0.755819 0.654780i \(-0.227239\pi\)
0.755819 + 0.654780i \(0.227239\pi\)
\(72\) 0 0
\(73\) −866.622 −1.38946 −0.694729 0.719271i \(-0.744476\pi\)
−0.694729 + 0.719271i \(0.744476\pi\)
\(74\) −1162.28 −1.82584
\(75\) 0 0
\(76\) −485.110 −0.732184
\(77\) 296.239 0.438437
\(78\) 0 0
\(79\) 206.961 0.294746 0.147373 0.989081i \(-0.452918\pi\)
0.147373 + 0.989081i \(0.452918\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 757.054 1.01954
\(83\) −463.397 −0.612825 −0.306412 0.951899i \(-0.599129\pi\)
−0.306412 + 0.951899i \(0.599129\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1955.87 2.45240
\(87\) 0 0
\(88\) −1010.33 −1.22388
\(89\) −601.736 −0.716673 −0.358337 0.933592i \(-0.616656\pi\)
−0.358337 + 0.933592i \(0.616656\pi\)
\(90\) 0 0
\(91\) 107.783 0.124162
\(92\) −1715.38 −1.94392
\(93\) 0 0
\(94\) −1139.29 −1.25009
\(95\) 0 0
\(96\) 0 0
\(97\) −229.363 −0.240086 −0.120043 0.992769i \(-0.538303\pi\)
−0.120043 + 0.992769i \(0.538303\pi\)
\(98\) −1414.43 −1.45795
\(99\) 0 0
\(100\) 0 0
\(101\) −1345.66 −1.32573 −0.662863 0.748740i \(-0.730659\pi\)
−0.662863 + 0.748740i \(0.730659\pi\)
\(102\) 0 0
\(103\) 1596.30 1.52707 0.763534 0.645768i \(-0.223463\pi\)
0.763534 + 0.645768i \(0.223463\pi\)
\(104\) −367.594 −0.346592
\(105\) 0 0
\(106\) 957.494 0.877358
\(107\) 958.786 0.866256 0.433128 0.901333i \(-0.357410\pi\)
0.433128 + 0.901333i \(0.357410\pi\)
\(108\) 0 0
\(109\) 1690.23 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 90.4943 0.0763474
\(113\) −11.6211 −0.00967456 −0.00483728 0.999988i \(-0.501540\pi\)
−0.00483728 + 0.999988i \(0.501540\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2615.63 2.09358
\(117\) 0 0
\(118\) −1478.39 −1.15336
\(119\) 349.818 0.269477
\(120\) 0 0
\(121\) 2067.97 1.55370
\(122\) 245.109 0.181894
\(123\) 0 0
\(124\) 3465.34 2.50965
\(125\) 0 0
\(126\) 0 0
\(127\) −309.141 −0.215999 −0.107999 0.994151i \(-0.534444\pi\)
−0.107999 + 0.994151i \(0.534444\pi\)
\(128\) −1956.48 −1.35102
\(129\) 0 0
\(130\) 0 0
\(131\) −2785.03 −1.85747 −0.928736 0.370742i \(-0.879103\pi\)
−0.928736 + 0.370742i \(0.879103\pi\)
\(132\) 0 0
\(133\) 207.382 0.135205
\(134\) −3381.01 −2.17966
\(135\) 0 0
\(136\) −1193.06 −0.752233
\(137\) −2489.41 −1.55244 −0.776222 0.630460i \(-0.782866\pi\)
−0.776222 + 0.630460i \(0.782866\pi\)
\(138\) 0 0
\(139\) 1786.05 1.08986 0.544931 0.838481i \(-0.316556\pi\)
0.544931 + 0.838481i \(0.316556\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4032.83 2.38329
\(143\) 1236.67 0.723185
\(144\) 0 0
\(145\) 0 0
\(146\) −3864.60 −2.19066
\(147\) 0 0
\(148\) −3097.95 −1.72061
\(149\) −1568.83 −0.862575 −0.431288 0.902214i \(-0.641941\pi\)
−0.431288 + 0.902214i \(0.641941\pi\)
\(150\) 0 0
\(151\) −438.327 −0.236229 −0.118114 0.993000i \(-0.537685\pi\)
−0.118114 + 0.993000i \(0.537685\pi\)
\(152\) −707.277 −0.377419
\(153\) 0 0
\(154\) 1321.04 0.691252
\(155\) 0 0
\(156\) 0 0
\(157\) 44.7479 0.0227469 0.0113735 0.999935i \(-0.496380\pi\)
0.0113735 + 0.999935i \(0.496380\pi\)
\(158\) 922.917 0.464705
\(159\) 0 0
\(160\) 0 0
\(161\) 733.315 0.358965
\(162\) 0 0
\(163\) 2611.84 1.25506 0.627531 0.778591i \(-0.284065\pi\)
0.627531 + 0.778591i \(0.284065\pi\)
\(164\) 2017.86 0.960783
\(165\) 0 0
\(166\) −2066.47 −0.966198
\(167\) −188.947 −0.0875516 −0.0437758 0.999041i \(-0.513939\pi\)
−0.0437758 + 0.999041i \(0.513939\pi\)
\(168\) 0 0
\(169\) −1747.05 −0.795200
\(170\) 0 0
\(171\) 0 0
\(172\) 5213.19 2.31106
\(173\) −1505.02 −0.661413 −0.330707 0.943734i \(-0.607287\pi\)
−0.330707 + 0.943734i \(0.607287\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1038.31 0.444689
\(177\) 0 0
\(178\) −2683.37 −1.12993
\(179\) 3136.62 1.30973 0.654865 0.755746i \(-0.272725\pi\)
0.654865 + 0.755746i \(0.272725\pi\)
\(180\) 0 0
\(181\) 4512.67 1.85317 0.926586 0.376084i \(-0.122730\pi\)
0.926586 + 0.376084i \(0.122730\pi\)
\(182\) 480.644 0.195757
\(183\) 0 0
\(184\) −2500.98 −1.00204
\(185\) 0 0
\(186\) 0 0
\(187\) 4013.71 1.56958
\(188\) −3036.67 −1.17804
\(189\) 0 0
\(190\) 0 0
\(191\) −1207.43 −0.457418 −0.228709 0.973495i \(-0.573450\pi\)
−0.228709 + 0.973495i \(0.573450\pi\)
\(192\) 0 0
\(193\) −923.164 −0.344305 −0.172152 0.985070i \(-0.555072\pi\)
−0.172152 + 0.985070i \(0.555072\pi\)
\(194\) −1022.82 −0.378526
\(195\) 0 0
\(196\) −3770.04 −1.37392
\(197\) −1180.87 −0.427075 −0.213537 0.976935i \(-0.568499\pi\)
−0.213537 + 0.976935i \(0.568499\pi\)
\(198\) 0 0
\(199\) −839.805 −0.299157 −0.149578 0.988750i \(-0.547792\pi\)
−0.149578 + 0.988750i \(0.547792\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6000.82 −2.09018
\(203\) −1118.17 −0.386600
\(204\) 0 0
\(205\) 0 0
\(206\) 7118.51 2.40762
\(207\) 0 0
\(208\) 377.774 0.125932
\(209\) 2379.44 0.787509
\(210\) 0 0
\(211\) −2589.65 −0.844923 −0.422461 0.906381i \(-0.638834\pi\)
−0.422461 + 0.906381i \(0.638834\pi\)
\(212\) 2552.12 0.826792
\(213\) 0 0
\(214\) 4275.59 1.36576
\(215\) 0 0
\(216\) 0 0
\(217\) −1481.41 −0.463432
\(218\) 7537.38 2.34172
\(219\) 0 0
\(220\) 0 0
\(221\) 1460.34 0.444492
\(222\) 0 0
\(223\) 4180.76 1.25544 0.627722 0.778437i \(-0.283987\pi\)
0.627722 + 0.778437i \(0.283987\pi\)
\(224\) 1107.99 0.330495
\(225\) 0 0
\(226\) −51.8232 −0.0152532
\(227\) −2602.67 −0.760993 −0.380497 0.924782i \(-0.624247\pi\)
−0.380497 + 0.924782i \(0.624247\pi\)
\(228\) 0 0
\(229\) −1845.35 −0.532508 −0.266254 0.963903i \(-0.585786\pi\)
−0.266254 + 0.963903i \(0.585786\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3813.51 1.07918
\(233\) 240.637 0.0676594 0.0338297 0.999428i \(-0.489230\pi\)
0.0338297 + 0.999428i \(0.489230\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3940.52 −1.08689
\(237\) 0 0
\(238\) 1559.97 0.424865
\(239\) −2567.64 −0.694924 −0.347462 0.937694i \(-0.612956\pi\)
−0.347462 + 0.937694i \(0.612956\pi\)
\(240\) 0 0
\(241\) 3987.99 1.06593 0.532965 0.846137i \(-0.321078\pi\)
0.532965 + 0.846137i \(0.321078\pi\)
\(242\) 9221.86 2.44960
\(243\) 0 0
\(244\) 653.315 0.171411
\(245\) 0 0
\(246\) 0 0
\(247\) 865.728 0.223016
\(248\) 5052.36 1.29365
\(249\) 0 0
\(250\) 0 0
\(251\) −967.393 −0.243272 −0.121636 0.992575i \(-0.538814\pi\)
−0.121636 + 0.992575i \(0.538814\pi\)
\(252\) 0 0
\(253\) 8413.86 2.09081
\(254\) −1378.58 −0.340550
\(255\) 0 0
\(256\) −2085.34 −0.509116
\(257\) 3743.03 0.908498 0.454249 0.890875i \(-0.349908\pi\)
0.454249 + 0.890875i \(0.349908\pi\)
\(258\) 0 0
\(259\) 1324.36 0.317728
\(260\) 0 0
\(261\) 0 0
\(262\) −12419.5 −2.92855
\(263\) −2497.47 −0.585554 −0.292777 0.956181i \(-0.594579\pi\)
−0.292777 + 0.956181i \(0.594579\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 924.795 0.213168
\(267\) 0 0
\(268\) −9011.79 −2.05404
\(269\) 2142.91 0.485709 0.242855 0.970063i \(-0.421916\pi\)
0.242855 + 0.970063i \(0.421916\pi\)
\(270\) 0 0
\(271\) 1540.64 0.345341 0.172671 0.984980i \(-0.444760\pi\)
0.172671 + 0.984980i \(0.444760\pi\)
\(272\) 1226.10 0.273320
\(273\) 0 0
\(274\) −11101.2 −2.44763
\(275\) 0 0
\(276\) 0 0
\(277\) −6777.80 −1.47018 −0.735088 0.677972i \(-0.762859\pi\)
−0.735088 + 0.677972i \(0.762859\pi\)
\(278\) 7964.69 1.71831
\(279\) 0 0
\(280\) 0 0
\(281\) −827.653 −0.175707 −0.0878535 0.996133i \(-0.528001\pi\)
−0.0878535 + 0.996133i \(0.528001\pi\)
\(282\) 0 0
\(283\) 3171.98 0.666270 0.333135 0.942879i \(-0.391894\pi\)
0.333135 + 0.942879i \(0.391894\pi\)
\(284\) 10749.2 2.24593
\(285\) 0 0
\(286\) 5514.78 1.14020
\(287\) −862.623 −0.177418
\(288\) 0 0
\(289\) −173.358 −0.0352856
\(290\) 0 0
\(291\) 0 0
\(292\) −10300.8 −2.06440
\(293\) 1376.02 0.274362 0.137181 0.990546i \(-0.456196\pi\)
0.137181 + 0.990546i \(0.456196\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4516.73 −0.886923
\(297\) 0 0
\(298\) −6996.02 −1.35996
\(299\) 3061.27 0.592099
\(300\) 0 0
\(301\) −2228.61 −0.426760
\(302\) −1954.67 −0.372445
\(303\) 0 0
\(304\) 726.864 0.137133
\(305\) 0 0
\(306\) 0 0
\(307\) 119.504 0.0222165 0.0111083 0.999938i \(-0.496464\pi\)
0.0111083 + 0.999938i \(0.496464\pi\)
\(308\) 3521.13 0.651412
\(309\) 0 0
\(310\) 0 0
\(311\) −2139.35 −0.390069 −0.195035 0.980796i \(-0.562482\pi\)
−0.195035 + 0.980796i \(0.562482\pi\)
\(312\) 0 0
\(313\) −5163.50 −0.932455 −0.466227 0.884665i \(-0.654387\pi\)
−0.466227 + 0.884665i \(0.654387\pi\)
\(314\) 199.548 0.0358635
\(315\) 0 0
\(316\) 2459.95 0.437922
\(317\) −8631.69 −1.52935 −0.764675 0.644416i \(-0.777101\pi\)
−0.764675 + 0.644416i \(0.777101\pi\)
\(318\) 0 0
\(319\) −12829.5 −2.25177
\(320\) 0 0
\(321\) 0 0
\(322\) 3270.13 0.565955
\(323\) 2809.79 0.484028
\(324\) 0 0
\(325\) 0 0
\(326\) 11647.2 1.97877
\(327\) 0 0
\(328\) 2941.98 0.495255
\(329\) 1298.16 0.217537
\(330\) 0 0
\(331\) −2942.34 −0.488597 −0.244298 0.969700i \(-0.578558\pi\)
−0.244298 + 0.969700i \(0.578558\pi\)
\(332\) −5507.98 −0.910512
\(333\) 0 0
\(334\) −842.585 −0.138037
\(335\) 0 0
\(336\) 0 0
\(337\) −9897.46 −1.59985 −0.799924 0.600101i \(-0.795127\pi\)
−0.799924 + 0.600101i \(0.795127\pi\)
\(338\) −7790.79 −1.25374
\(339\) 0 0
\(340\) 0 0
\(341\) −16997.3 −2.69928
\(342\) 0 0
\(343\) 3354.53 0.528070
\(344\) 7600.68 1.19128
\(345\) 0 0
\(346\) −6711.46 −1.04280
\(347\) −12015.7 −1.85890 −0.929451 0.368947i \(-0.879718\pi\)
−0.929451 + 0.368947i \(0.879718\pi\)
\(348\) 0 0
\(349\) 7894.62 1.21086 0.605428 0.795900i \(-0.293002\pi\)
0.605428 + 0.795900i \(0.293002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12712.8 1.92499
\(353\) 741.154 0.111750 0.0558748 0.998438i \(-0.482205\pi\)
0.0558748 + 0.998438i \(0.482205\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7152.30 −1.06481
\(357\) 0 0
\(358\) 13987.4 2.06496
\(359\) 11564.4 1.70012 0.850060 0.526685i \(-0.176565\pi\)
0.850060 + 0.526685i \(0.176565\pi\)
\(360\) 0 0
\(361\) −5193.28 −0.757148
\(362\) 20123.7 2.92177
\(363\) 0 0
\(364\) 1281.12 0.184474
\(365\) 0 0
\(366\) 0 0
\(367\) 1148.57 0.163365 0.0816823 0.996658i \(-0.473971\pi\)
0.0816823 + 0.996658i \(0.473971\pi\)
\(368\) 2570.24 0.364084
\(369\) 0 0
\(370\) 0 0
\(371\) −1091.01 −0.152676
\(372\) 0 0
\(373\) 4602.98 0.638963 0.319481 0.947593i \(-0.396491\pi\)
0.319481 + 0.947593i \(0.396491\pi\)
\(374\) 17898.7 2.47465
\(375\) 0 0
\(376\) −4427.38 −0.607246
\(377\) −4667.85 −0.637683
\(378\) 0 0
\(379\) −3988.46 −0.540563 −0.270282 0.962781i \(-0.587117\pi\)
−0.270282 + 0.962781i \(0.587117\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5384.41 −0.721179
\(383\) 7788.20 1.03906 0.519528 0.854454i \(-0.326108\pi\)
0.519528 + 0.854454i \(0.326108\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4116.74 −0.542841
\(387\) 0 0
\(388\) −2726.23 −0.356710
\(389\) −8524.87 −1.11113 −0.555563 0.831474i \(-0.687497\pi\)
−0.555563 + 0.831474i \(0.687497\pi\)
\(390\) 0 0
\(391\) 9935.60 1.28508
\(392\) −5496.62 −0.708217
\(393\) 0 0
\(394\) −5265.97 −0.673339
\(395\) 0 0
\(396\) 0 0
\(397\) 155.729 0.0196872 0.00984361 0.999952i \(-0.496867\pi\)
0.00984361 + 0.999952i \(0.496867\pi\)
\(398\) −3745.01 −0.471659
\(399\) 0 0
\(400\) 0 0
\(401\) −5933.96 −0.738972 −0.369486 0.929236i \(-0.620466\pi\)
−0.369486 + 0.929236i \(0.620466\pi\)
\(402\) 0 0
\(403\) −6184.23 −0.764414
\(404\) −15994.7 −1.96971
\(405\) 0 0
\(406\) −4986.33 −0.609526
\(407\) 15195.3 1.85062
\(408\) 0 0
\(409\) 14161.4 1.71207 0.856035 0.516917i \(-0.172921\pi\)
0.856035 + 0.516917i \(0.172921\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 18973.8 2.26886
\(413\) 1684.55 0.200705
\(414\) 0 0
\(415\) 0 0
\(416\) 4625.39 0.545140
\(417\) 0 0
\(418\) 10610.8 1.24161
\(419\) −9624.24 −1.12214 −0.561068 0.827770i \(-0.689609\pi\)
−0.561068 + 0.827770i \(0.689609\pi\)
\(420\) 0 0
\(421\) −1536.26 −0.177845 −0.0889223 0.996039i \(-0.528342\pi\)
−0.0889223 + 0.996039i \(0.528342\pi\)
\(422\) −11548.2 −1.33213
\(423\) 0 0
\(424\) 3720.91 0.426187
\(425\) 0 0
\(426\) 0 0
\(427\) −279.288 −0.0316527
\(428\) 11396.2 1.28705
\(429\) 0 0
\(430\) 0 0
\(431\) −11582.2 −1.29441 −0.647207 0.762314i \(-0.724063\pi\)
−0.647207 + 0.762314i \(0.724063\pi\)
\(432\) 0 0
\(433\) −14892.6 −1.65287 −0.826437 0.563029i \(-0.809636\pi\)
−0.826437 + 0.563029i \(0.809636\pi\)
\(434\) −6606.17 −0.730660
\(435\) 0 0
\(436\) 20090.2 2.20676
\(437\) 5890.10 0.644764
\(438\) 0 0
\(439\) −1642.51 −0.178571 −0.0892853 0.996006i \(-0.528458\pi\)
−0.0892853 + 0.996006i \(0.528458\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6512.20 0.700800
\(443\) 3916.31 0.420021 0.210011 0.977699i \(-0.432650\pi\)
0.210011 + 0.977699i \(0.432650\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 18643.6 1.97937
\(447\) 0 0
\(448\) 4217.02 0.444722
\(449\) 3985.25 0.418876 0.209438 0.977822i \(-0.432837\pi\)
0.209438 + 0.977822i \(0.432837\pi\)
\(450\) 0 0
\(451\) −9897.50 −1.03338
\(452\) −138.130 −0.0143741
\(453\) 0 0
\(454\) −11606.3 −1.19980
\(455\) 0 0
\(456\) 0 0
\(457\) 14177.8 1.45122 0.725611 0.688105i \(-0.241557\pi\)
0.725611 + 0.688105i \(0.241557\pi\)
\(458\) −8229.13 −0.839567
\(459\) 0 0
\(460\) 0 0
\(461\) 16394.3 1.65631 0.828154 0.560500i \(-0.189391\pi\)
0.828154 + 0.560500i \(0.189391\pi\)
\(462\) 0 0
\(463\) −3319.60 −0.333207 −0.166603 0.986024i \(-0.553280\pi\)
−0.166603 + 0.986024i \(0.553280\pi\)
\(464\) −3919.12 −0.392114
\(465\) 0 0
\(466\) 1073.09 0.106674
\(467\) −2529.79 −0.250674 −0.125337 0.992114i \(-0.540001\pi\)
−0.125337 + 0.992114i \(0.540001\pi\)
\(468\) 0 0
\(469\) 3852.49 0.379299
\(470\) 0 0
\(471\) 0 0
\(472\) −5745.17 −0.560260
\(473\) −25570.4 −2.48569
\(474\) 0 0
\(475\) 0 0
\(476\) 4157.97 0.400379
\(477\) 0 0
\(478\) −11450.1 −1.09564
\(479\) 8646.75 0.824802 0.412401 0.911002i \(-0.364690\pi\)
0.412401 + 0.911002i \(0.364690\pi\)
\(480\) 0 0
\(481\) 5528.60 0.524080
\(482\) 17784.0 1.68058
\(483\) 0 0
\(484\) 24580.1 2.30842
\(485\) 0 0
\(486\) 0 0
\(487\) 15251.7 1.41914 0.709569 0.704636i \(-0.248890\pi\)
0.709569 + 0.704636i \(0.248890\pi\)
\(488\) 952.515 0.0883572
\(489\) 0 0
\(490\) 0 0
\(491\) 1766.87 0.162398 0.0811992 0.996698i \(-0.474125\pi\)
0.0811992 + 0.996698i \(0.474125\pi\)
\(492\) 0 0
\(493\) −15149.9 −1.38401
\(494\) 3860.61 0.351614
\(495\) 0 0
\(496\) −5192.28 −0.470041
\(497\) −4595.20 −0.414734
\(498\) 0 0
\(499\) 11733.8 1.05266 0.526332 0.850279i \(-0.323567\pi\)
0.526332 + 0.850279i \(0.323567\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4313.98 −0.383550
\(503\) −977.608 −0.0866589 −0.0433294 0.999061i \(-0.513797\pi\)
−0.0433294 + 0.999061i \(0.513797\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 37520.6 3.29643
\(507\) 0 0
\(508\) −3674.48 −0.320923
\(509\) −9674.72 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(510\) 0 0
\(511\) 4403.51 0.381213
\(512\) 6352.52 0.548329
\(513\) 0 0
\(514\) 16691.6 1.43237
\(515\) 0 0
\(516\) 0 0
\(517\) 14894.7 1.26706
\(518\) 5905.81 0.500939
\(519\) 0 0
\(520\) 0 0
\(521\) −5178.95 −0.435497 −0.217748 0.976005i \(-0.569871\pi\)
−0.217748 + 0.976005i \(0.569871\pi\)
\(522\) 0 0
\(523\) −14280.7 −1.19398 −0.596992 0.802248i \(-0.703637\pi\)
−0.596992 + 0.802248i \(0.703637\pi\)
\(524\) −33103.1 −2.75976
\(525\) 0 0
\(526\) −11137.2 −0.923203
\(527\) −20071.5 −1.65906
\(528\) 0 0
\(529\) 8660.78 0.711826
\(530\) 0 0
\(531\) 0 0
\(532\) 2464.96 0.200883
\(533\) −3601.07 −0.292645
\(534\) 0 0
\(535\) 0 0
\(536\) −13138.9 −1.05880
\(537\) 0 0
\(538\) 9556.08 0.765784
\(539\) 18491.9 1.47774
\(540\) 0 0
\(541\) 12923.8 1.02706 0.513529 0.858072i \(-0.328338\pi\)
0.513529 + 0.858072i \(0.328338\pi\)
\(542\) 6870.32 0.544475
\(543\) 0 0
\(544\) 15012.1 1.18316
\(545\) 0 0
\(546\) 0 0
\(547\) −13653.3 −1.06723 −0.533614 0.845728i \(-0.679166\pi\)
−0.533614 + 0.845728i \(0.679166\pi\)
\(548\) −29589.4 −2.30656
\(549\) 0 0
\(550\) 0 0
\(551\) −8981.28 −0.694402
\(552\) 0 0
\(553\) −1051.62 −0.0808666
\(554\) −30224.8 −2.31792
\(555\) 0 0
\(556\) 21229.2 1.61928
\(557\) −9313.63 −0.708494 −0.354247 0.935152i \(-0.615263\pi\)
−0.354247 + 0.935152i \(0.615263\pi\)
\(558\) 0 0
\(559\) −9303.46 −0.703925
\(560\) 0 0
\(561\) 0 0
\(562\) −3690.82 −0.277025
\(563\) −10625.4 −0.795394 −0.397697 0.917517i \(-0.630190\pi\)
−0.397697 + 0.917517i \(0.630190\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14145.1 1.05046
\(567\) 0 0
\(568\) 15672.0 1.15771
\(569\) −9060.50 −0.667550 −0.333775 0.942653i \(-0.608323\pi\)
−0.333775 + 0.942653i \(0.608323\pi\)
\(570\) 0 0
\(571\) −21379.1 −1.56688 −0.783440 0.621467i \(-0.786537\pi\)
−0.783440 + 0.621467i \(0.786537\pi\)
\(572\) 14699.2 1.07448
\(573\) 0 0
\(574\) −3846.77 −0.279723
\(575\) 0 0
\(576\) 0 0
\(577\) −6347.76 −0.457991 −0.228996 0.973427i \(-0.573544\pi\)
−0.228996 + 0.973427i \(0.573544\pi\)
\(578\) −773.071 −0.0556324
\(579\) 0 0
\(580\) 0 0
\(581\) 2354.63 0.168135
\(582\) 0 0
\(583\) −12518.0 −0.889266
\(584\) −15018.2 −1.06414
\(585\) 0 0
\(586\) 6136.22 0.432568
\(587\) 14773.7 1.03880 0.519401 0.854531i \(-0.326155\pi\)
0.519401 + 0.854531i \(0.326155\pi\)
\(588\) 0 0
\(589\) −11898.9 −0.832405
\(590\) 0 0
\(591\) 0 0
\(592\) 4641.81 0.322259
\(593\) 26868.1 1.86061 0.930304 0.366790i \(-0.119543\pi\)
0.930304 + 0.366790i \(0.119543\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18647.3 −1.28158
\(597\) 0 0
\(598\) 13651.4 0.933522
\(599\) 1749.83 0.119359 0.0596794 0.998218i \(-0.480992\pi\)
0.0596794 + 0.998218i \(0.480992\pi\)
\(600\) 0 0
\(601\) −17964.0 −1.21924 −0.609622 0.792692i \(-0.708679\pi\)
−0.609622 + 0.792692i \(0.708679\pi\)
\(602\) −9938.21 −0.672843
\(603\) 0 0
\(604\) −5209.99 −0.350979
\(605\) 0 0
\(606\) 0 0
\(607\) 4418.22 0.295437 0.147718 0.989029i \(-0.452807\pi\)
0.147718 + 0.989029i \(0.452807\pi\)
\(608\) 8899.58 0.593628
\(609\) 0 0
\(610\) 0 0
\(611\) 5419.24 0.358820
\(612\) 0 0
\(613\) 20179.7 1.32961 0.664804 0.747018i \(-0.268515\pi\)
0.664804 + 0.747018i \(0.268515\pi\)
\(614\) 532.915 0.0350272
\(615\) 0 0
\(616\) 5133.71 0.335784
\(617\) −4673.01 −0.304908 −0.152454 0.988311i \(-0.548718\pi\)
−0.152454 + 0.988311i \(0.548718\pi\)
\(618\) 0 0
\(619\) −19976.8 −1.29715 −0.648574 0.761151i \(-0.724634\pi\)
−0.648574 + 0.761151i \(0.724634\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −9540.19 −0.614994
\(623\) 3057.56 0.196627
\(624\) 0 0
\(625\) 0 0
\(626\) −23026.0 −1.47014
\(627\) 0 0
\(628\) 531.877 0.0337965
\(629\) 17943.5 1.13745
\(630\) 0 0
\(631\) 10457.5 0.659757 0.329879 0.944023i \(-0.392992\pi\)
0.329879 + 0.944023i \(0.392992\pi\)
\(632\) 3586.54 0.225736
\(633\) 0 0
\(634\) −38492.0 −2.41122
\(635\) 0 0
\(636\) 0 0
\(637\) 6728.02 0.418483
\(638\) −57211.8 −3.55021
\(639\) 0 0
\(640\) 0 0
\(641\) −4395.86 −0.270867 −0.135434 0.990786i \(-0.543243\pi\)
−0.135434 + 0.990786i \(0.543243\pi\)
\(642\) 0 0
\(643\) −5786.36 −0.354886 −0.177443 0.984131i \(-0.556783\pi\)
−0.177443 + 0.984131i \(0.556783\pi\)
\(644\) 8716.26 0.533336
\(645\) 0 0
\(646\) 12529.9 0.763133
\(647\) −25367.7 −1.54143 −0.770717 0.637178i \(-0.780102\pi\)
−0.770717 + 0.637178i \(0.780102\pi\)
\(648\) 0 0
\(649\) 19328.1 1.16902
\(650\) 0 0
\(651\) 0 0
\(652\) 31044.6 1.86472
\(653\) −21633.2 −1.29644 −0.648218 0.761454i \(-0.724486\pi\)
−0.648218 + 0.761454i \(0.724486\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3023.46 −0.179948
\(657\) 0 0
\(658\) 5788.98 0.342976
\(659\) −18312.4 −1.08248 −0.541238 0.840870i \(-0.682044\pi\)
−0.541238 + 0.840870i \(0.682044\pi\)
\(660\) 0 0
\(661\) −5526.08 −0.325174 −0.162587 0.986694i \(-0.551984\pi\)
−0.162587 + 0.986694i \(0.551984\pi\)
\(662\) −13121.0 −0.770337
\(663\) 0 0
\(664\) −8030.48 −0.469342
\(665\) 0 0
\(666\) 0 0
\(667\) −31758.4 −1.84361
\(668\) −2245.84 −0.130081
\(669\) 0 0
\(670\) 0 0
\(671\) −3204.48 −0.184363
\(672\) 0 0
\(673\) −1437.24 −0.0823204 −0.0411602 0.999153i \(-0.513105\pi\)
−0.0411602 + 0.999153i \(0.513105\pi\)
\(674\) −44136.6 −2.52237
\(675\) 0 0
\(676\) −20765.7 −1.18148
\(677\) −23405.9 −1.32875 −0.664373 0.747401i \(-0.731301\pi\)
−0.664373 + 0.747401i \(0.731301\pi\)
\(678\) 0 0
\(679\) 1165.45 0.0658701
\(680\) 0 0
\(681\) 0 0
\(682\) −75797.4 −4.25577
\(683\) 19227.4 1.07719 0.538593 0.842566i \(-0.318956\pi\)
0.538593 + 0.842566i \(0.318956\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 14959.2 0.832570
\(687\) 0 0
\(688\) −7811.17 −0.432846
\(689\) −4554.50 −0.251833
\(690\) 0 0
\(691\) −35284.8 −1.94254 −0.971271 0.237975i \(-0.923516\pi\)
−0.971271 + 0.237975i \(0.923516\pi\)
\(692\) −17888.8 −0.982703
\(693\) 0 0
\(694\) −53582.8 −2.93080
\(695\) 0 0
\(696\) 0 0
\(697\) −11687.6 −0.635149
\(698\) 35205.1 1.90907
\(699\) 0 0
\(700\) 0 0
\(701\) −9173.00 −0.494236 −0.247118 0.968985i \(-0.579484\pi\)
−0.247118 + 0.968985i \(0.579484\pi\)
\(702\) 0 0
\(703\) 10637.4 0.570695
\(704\) 48384.9 2.59030
\(705\) 0 0
\(706\) 3305.09 0.176188
\(707\) 6837.63 0.363728
\(708\) 0 0
\(709\) −33951.6 −1.79842 −0.899210 0.437517i \(-0.855858\pi\)
−0.899210 + 0.437517i \(0.855858\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10427.8 −0.548876
\(713\) −42075.3 −2.21001
\(714\) 0 0
\(715\) 0 0
\(716\) 37282.1 1.94595
\(717\) 0 0
\(718\) 51569.9 2.68046
\(719\) 6727.90 0.348968 0.174484 0.984660i \(-0.444174\pi\)
0.174484 + 0.984660i \(0.444174\pi\)
\(720\) 0 0
\(721\) −8111.17 −0.418968
\(722\) −23158.8 −1.19374
\(723\) 0 0
\(724\) 53638.0 2.75337
\(725\) 0 0
\(726\) 0 0
\(727\) 36726.1 1.87359 0.936793 0.349885i \(-0.113779\pi\)
0.936793 + 0.349885i \(0.113779\pi\)
\(728\) 1867.83 0.0950912
\(729\) 0 0
\(730\) 0 0
\(731\) −30195.1 −1.52778
\(732\) 0 0
\(733\) −26691.4 −1.34498 −0.672489 0.740108i \(-0.734775\pi\)
−0.672489 + 0.740108i \(0.734775\pi\)
\(734\) 5121.91 0.257566
\(735\) 0 0
\(736\) 31469.5 1.57606
\(737\) 44202.4 2.20925
\(738\) 0 0
\(739\) 12207.0 0.607634 0.303817 0.952730i \(-0.401739\pi\)
0.303817 + 0.952730i \(0.401739\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4865.25 −0.240713
\(743\) −12473.3 −0.615882 −0.307941 0.951405i \(-0.599640\pi\)
−0.307941 + 0.951405i \(0.599640\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20526.4 1.00741
\(747\) 0 0
\(748\) 47707.4 2.33202
\(749\) −4871.82 −0.237667
\(750\) 0 0
\(751\) −15102.6 −0.733825 −0.366913 0.930255i \(-0.619585\pi\)
−0.366913 + 0.930255i \(0.619585\pi\)
\(752\) 4549.99 0.220640
\(753\) 0 0
\(754\) −20815.7 −1.00539
\(755\) 0 0
\(756\) 0 0
\(757\) 3418.34 0.164124 0.0820618 0.996627i \(-0.473850\pi\)
0.0820618 + 0.996627i \(0.473850\pi\)
\(758\) −17786.1 −0.852268
\(759\) 0 0
\(760\) 0 0
\(761\) 12684.5 0.604224 0.302112 0.953272i \(-0.402308\pi\)
0.302112 + 0.953272i \(0.402308\pi\)
\(762\) 0 0
\(763\) −8588.45 −0.407500
\(764\) −14351.7 −0.679614
\(765\) 0 0
\(766\) 34730.6 1.63821
\(767\) 7032.25 0.331056
\(768\) 0 0
\(769\) −27580.2 −1.29333 −0.646663 0.762776i \(-0.723836\pi\)
−0.646663 + 0.762776i \(0.723836\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10972.8 −0.511555
\(773\) 17386.0 0.808966 0.404483 0.914546i \(-0.367452\pi\)
0.404483 + 0.914546i \(0.367452\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3974.77 −0.183874
\(777\) 0 0
\(778\) −38015.7 −1.75183
\(779\) −6928.72 −0.318674
\(780\) 0 0
\(781\) −52724.1 −2.41564
\(782\) 44306.7 2.02609
\(783\) 0 0
\(784\) 5648.84 0.257327
\(785\) 0 0
\(786\) 0 0
\(787\) −4680.29 −0.211988 −0.105994 0.994367i \(-0.533802\pi\)
−0.105994 + 0.994367i \(0.533802\pi\)
\(788\) −14036.0 −0.634531
\(789\) 0 0
\(790\) 0 0
\(791\) 59.0498 0.00265432
\(792\) 0 0
\(793\) −1165.91 −0.0522100
\(794\) 694.456 0.0310395
\(795\) 0 0
\(796\) −9982.00 −0.444476
\(797\) 7278.62 0.323490 0.161745 0.986833i \(-0.448288\pi\)
0.161745 + 0.986833i \(0.448288\pi\)
\(798\) 0 0
\(799\) 17588.6 0.778773
\(800\) 0 0
\(801\) 0 0
\(802\) −26461.8 −1.16508
\(803\) 50524.7 2.22039
\(804\) 0 0
\(805\) 0 0
\(806\) −27577.9 −1.20520
\(807\) 0 0
\(808\) −23319.8 −1.01533
\(809\) −29212.5 −1.26954 −0.634770 0.772701i \(-0.718905\pi\)
−0.634770 + 0.772701i \(0.718905\pi\)
\(810\) 0 0
\(811\) 41992.4 1.81819 0.909094 0.416590i \(-0.136775\pi\)
0.909094 + 0.416590i \(0.136775\pi\)
\(812\) −13290.6 −0.574396
\(813\) 0 0
\(814\) 67761.6 2.91774
\(815\) 0 0
\(816\) 0 0
\(817\) −17900.5 −0.766536
\(818\) 63151.2 2.69930
\(819\) 0 0
\(820\) 0 0
\(821\) 8722.99 0.370809 0.185405 0.982662i \(-0.440640\pi\)
0.185405 + 0.982662i \(0.440640\pi\)
\(822\) 0 0
\(823\) −13584.8 −0.575379 −0.287690 0.957724i \(-0.592887\pi\)
−0.287690 + 0.957724i \(0.592887\pi\)
\(824\) 27663.2 1.16953
\(825\) 0 0
\(826\) 7512.05 0.316438
\(827\) 26573.6 1.11736 0.558679 0.829384i \(-0.311308\pi\)
0.558679 + 0.829384i \(0.311308\pi\)
\(828\) 0 0
\(829\) −43238.4 −1.81150 −0.905748 0.423816i \(-0.860690\pi\)
−0.905748 + 0.423816i \(0.860690\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 17604.2 0.733552
\(833\) 21836.3 0.908265
\(834\) 0 0
\(835\) 0 0
\(836\) 28282.3 1.17005
\(837\) 0 0
\(838\) −42918.2 −1.76919
\(839\) 4093.21 0.168431 0.0842154 0.996448i \(-0.473162\pi\)
0.0842154 + 0.996448i \(0.473162\pi\)
\(840\) 0 0
\(841\) 24036.5 0.985546
\(842\) −6850.75 −0.280395
\(843\) 0 0
\(844\) −30780.8 −1.25535
\(845\) 0 0
\(846\) 0 0
\(847\) −10507.8 −0.426273
\(848\) −3823.96 −0.154853
\(849\) 0 0
\(850\) 0 0
\(851\) 37614.7 1.51517
\(852\) 0 0
\(853\) 20201.5 0.810886 0.405443 0.914120i \(-0.367117\pi\)
0.405443 + 0.914120i \(0.367117\pi\)
\(854\) −1245.45 −0.0499046
\(855\) 0 0
\(856\) 16615.4 0.663436
\(857\) 4551.65 0.181425 0.0907126 0.995877i \(-0.471086\pi\)
0.0907126 + 0.995877i \(0.471086\pi\)
\(858\) 0 0
\(859\) 11962.6 0.475154 0.237577 0.971369i \(-0.423647\pi\)
0.237577 + 0.971369i \(0.423647\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −51649.3 −2.04081
\(863\) −7164.61 −0.282603 −0.141301 0.989967i \(-0.545129\pi\)
−0.141301 + 0.989967i \(0.545129\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −66412.0 −2.60597
\(867\) 0 0
\(868\) −17608.2 −0.688549
\(869\) −12065.9 −0.471012
\(870\) 0 0
\(871\) 16082.4 0.625640
\(872\) 29291.0 1.13752
\(873\) 0 0
\(874\) 26266.2 1.01655
\(875\) 0 0
\(876\) 0 0
\(877\) 19218.7 0.739987 0.369994 0.929034i \(-0.379360\pi\)
0.369994 + 0.929034i \(0.379360\pi\)
\(878\) −7324.56 −0.281540
\(879\) 0 0
\(880\) 0 0
\(881\) 45914.5 1.75585 0.877923 0.478802i \(-0.158929\pi\)
0.877923 + 0.478802i \(0.158929\pi\)
\(882\) 0 0
\(883\) 44656.7 1.70194 0.850972 0.525211i \(-0.176014\pi\)
0.850972 + 0.525211i \(0.176014\pi\)
\(884\) 17357.7 0.660410
\(885\) 0 0
\(886\) 17464.3 0.662218
\(887\) 14975.2 0.566873 0.283437 0.958991i \(-0.408525\pi\)
0.283437 + 0.958991i \(0.408525\pi\)
\(888\) 0 0
\(889\) 1570.82 0.0592616
\(890\) 0 0
\(891\) 0 0
\(892\) 49692.9 1.86529
\(893\) 10427.0 0.390735
\(894\) 0 0
\(895\) 0 0
\(896\) 9941.33 0.370666
\(897\) 0 0
\(898\) 17771.7 0.660413
\(899\) 64156.8 2.38015
\(900\) 0 0
\(901\) −14782.0 −0.546571
\(902\) −44136.7 −1.62926
\(903\) 0 0
\(904\) −201.390 −0.00740943
\(905\) 0 0
\(906\) 0 0
\(907\) 14818.1 0.542479 0.271240 0.962512i \(-0.412566\pi\)
0.271240 + 0.962512i \(0.412566\pi\)
\(908\) −30935.6 −1.13065
\(909\) 0 0
\(910\) 0 0
\(911\) −21846.5 −0.794518 −0.397259 0.917707i \(-0.630039\pi\)
−0.397259 + 0.917707i \(0.630039\pi\)
\(912\) 0 0
\(913\) 27016.4 0.979312
\(914\) 63224.2 2.28804
\(915\) 0 0
\(916\) −21934.0 −0.791179
\(917\) 14151.4 0.509618
\(918\) 0 0
\(919\) 28878.9 1.03659 0.518296 0.855201i \(-0.326566\pi\)
0.518296 + 0.855201i \(0.326566\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 73108.4 2.61139
\(923\) −19182.9 −0.684089
\(924\) 0 0
\(925\) 0 0
\(926\) −14803.4 −0.525344
\(927\) 0 0
\(928\) −47985.0 −1.69740
\(929\) −4608.79 −0.162766 −0.0813829 0.996683i \(-0.525934\pi\)
−0.0813829 + 0.996683i \(0.525934\pi\)
\(930\) 0 0
\(931\) 12945.2 0.455705
\(932\) 2860.23 0.100526
\(933\) 0 0
\(934\) −11281.3 −0.395220
\(935\) 0 0
\(936\) 0 0
\(937\) 21063.8 0.734391 0.367195 0.930144i \(-0.380318\pi\)
0.367195 + 0.930144i \(0.380318\pi\)
\(938\) 17179.7 0.598014
\(939\) 0 0
\(940\) 0 0
\(941\) 20244.8 0.701339 0.350670 0.936499i \(-0.385954\pi\)
0.350670 + 0.936499i \(0.385954\pi\)
\(942\) 0 0
\(943\) −24500.4 −0.846069
\(944\) 5904.27 0.203567
\(945\) 0 0
\(946\) −114028. −3.91901
\(947\) −29978.3 −1.02868 −0.514342 0.857585i \(-0.671964\pi\)
−0.514342 + 0.857585i \(0.671964\pi\)
\(948\) 0 0
\(949\) 18382.7 0.628797
\(950\) 0 0
\(951\) 0 0
\(952\) 6062.20 0.206383
\(953\) 5782.65 0.196556 0.0982782 0.995159i \(-0.468666\pi\)
0.0982782 + 0.995159i \(0.468666\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −30519.2 −1.03249
\(957\) 0 0
\(958\) 38559.2 1.30041
\(959\) 12649.3 0.425930
\(960\) 0 0
\(961\) 55207.7 1.85317
\(962\) 24654.2 0.826281
\(963\) 0 0
\(964\) 47401.6 1.58372
\(965\) 0 0
\(966\) 0 0
\(967\) 26119.2 0.868600 0.434300 0.900768i \(-0.356996\pi\)
0.434300 + 0.900768i \(0.356996\pi\)
\(968\) 35837.0 1.18992
\(969\) 0 0
\(970\) 0 0
\(971\) −5101.93 −0.168619 −0.0843093 0.996440i \(-0.526868\pi\)
−0.0843093 + 0.996440i \(0.526868\pi\)
\(972\) 0 0
\(973\) −9075.34 −0.299016
\(974\) 68013.1 2.23746
\(975\) 0 0
\(976\) −978.894 −0.0321041
\(977\) −45902.4 −1.50312 −0.751559 0.659666i \(-0.770698\pi\)
−0.751559 + 0.659666i \(0.770698\pi\)
\(978\) 0 0
\(979\) 35081.6 1.14526
\(980\) 0 0
\(981\) 0 0
\(982\) 7879.14 0.256042
\(983\) 13416.5 0.435321 0.217661 0.976025i \(-0.430157\pi\)
0.217661 + 0.976025i \(0.430157\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −67559.2 −2.18207
\(987\) 0 0
\(988\) 10290.1 0.331349
\(989\) −63297.4 −2.03513
\(990\) 0 0
\(991\) 3806.66 0.122021 0.0610104 0.998137i \(-0.480568\pi\)
0.0610104 + 0.998137i \(0.480568\pi\)
\(992\) −63573.2 −2.03473
\(993\) 0 0
\(994\) −20491.8 −0.653883
\(995\) 0 0
\(996\) 0 0
\(997\) 22523.8 0.715483 0.357742 0.933821i \(-0.383547\pi\)
0.357742 + 0.933821i \(0.383547\pi\)
\(998\) 52325.7 1.65966
\(999\) 0 0
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 675.4.a.q.1.3 3
3.2 odd 2 675.4.a.r.1.1 3
5.2 odd 4 675.4.b.k.649.5 6
5.3 odd 4 675.4.b.k.649.2 6
5.4 even 2 135.4.a.g.1.1 yes 3
15.2 even 4 675.4.b.l.649.2 6
15.8 even 4 675.4.b.l.649.5 6
15.14 odd 2 135.4.a.f.1.3 3
20.19 odd 2 2160.4.a.be.1.3 3
45.4 even 6 405.4.e.r.136.3 6
45.14 odd 6 405.4.e.t.136.1 6
45.29 odd 6 405.4.e.t.271.1 6
45.34 even 6 405.4.e.r.271.3 6
60.59 even 2 2160.4.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.3 3 15.14 odd 2
135.4.a.g.1.1 yes 3 5.4 even 2
405.4.e.r.136.3 6 45.4 even 6
405.4.e.r.271.3 6 45.34 even 6
405.4.e.t.136.1 6 45.14 odd 6
405.4.e.t.271.1 6 45.29 odd 6
675.4.a.q.1.3 3 1.1 even 1 trivial
675.4.a.r.1.1 3 3.2 odd 2
675.4.b.k.649.2 6 5.3 odd 4
675.4.b.k.649.5 6 5.2 odd 4
675.4.b.l.649.2 6 15.2 even 4
675.4.b.l.649.5 6 15.8 even 4
2160.4.a.be.1.3 3 20.19 odd 2
2160.4.a.bm.1.3 3 60.59 even 2