Properties

Label 135.4.a.f.1.3
Level $135$
Weight $4$
Character 135.1
Self dual yes
Analytic conductor $7.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.45938\) of defining polynomial
Character \(\chi\) \(=\) 135.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +O(q^{10})\) \(q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +22.2969 q^{10} +58.3007 q^{11} +21.2119 q^{13} +22.6592 q^{14} -17.8095 q^{16} -68.8451 q^{17} -40.8133 q^{19} +59.4305 q^{20} +259.985 q^{22} -144.318 q^{23} +25.0000 q^{25} +94.5921 q^{26} +60.3960 q^{28} -220.058 q^{29} +291.545 q^{31} -218.056 q^{32} -307.006 q^{34} +25.4062 q^{35} +260.637 q^{37} -182.002 q^{38} +86.6479 q^{40} -169.766 q^{41} -438.596 q^{43} +692.967 q^{44} -643.571 q^{46} -255.481 q^{47} -317.181 q^{49} +111.485 q^{50} +252.127 q^{52} +214.714 q^{53} +291.503 q^{55} +88.0557 q^{56} -981.322 q^{58} +331.524 q^{59} +54.9647 q^{61} +1300.11 q^{62} -829.920 q^{64} +106.060 q^{65} +758.179 q^{67} -818.299 q^{68} +113.296 q^{70} -904.348 q^{71} +866.622 q^{73} +1162.28 q^{74} -485.110 q^{76} +296.239 q^{77} +206.961 q^{79} -89.0475 q^{80} -757.054 q^{82} -463.397 q^{83} -344.225 q^{85} -1955.87 q^{86} +1010.33 q^{88} +601.736 q^{89} +107.783 q^{91} -1715.38 q^{92} -1139.29 q^{94} -204.066 q^{95} +229.363 q^{97} -1414.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) 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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 5.08123 0.274361 0.137180 0.990546i \(-0.456196\pi\)
0.137180 + 0.990546i \(0.456196\pi\)
\(8\) 17.3296 0.765867
\(9\) 0 0
\(10\) 22.2969 0.705090
\(11\) 58.3007 1.59803 0.799014 0.601312i \(-0.205355\pi\)
0.799014 + 0.601312i \(0.205355\pi\)
\(12\) 0 0
\(13\) 21.2119 0.452548 0.226274 0.974064i \(-0.427345\pi\)
0.226274 + 0.974064i \(0.427345\pi\)
\(14\) 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 \(0.579248\pi\)
\(20\) 59.4305 0.664453
\(21\) 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
\(25\) 25.0000 0.200000
\(26\) 94.5921 0.713501
\(27\) 0 0
\(28\) 60.3960 0.407635
\(29\) −220.058 −1.40909 −0.704547 0.709657i \(-0.748850\pi\)
−0.704547 + 0.709657i \(0.748850\pi\)
\(30\) 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
\(35\) 25.4062 0.122698
\(36\) 0 0
\(37\) 260.637 1.15807 0.579033 0.815304i \(-0.303430\pi\)
0.579033 + 0.815304i \(0.303430\pi\)
\(38\) −182.002 −0.776964
\(39\) 0 0
\(40\) 86.6479 0.342506
\(41\) −169.766 −0.646660 −0.323330 0.946286i \(-0.604802\pi\)
−0.323330 + 0.946286i \(0.604802\pi\)
\(42\) 0 0
\(43\) −438.596 −1.55547 −0.777735 0.628592i \(-0.783631\pi\)
−0.777735 + 0.628592i \(0.783631\pi\)
\(44\) 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
\(50\) 111.485 0.315326
\(51\) 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
\(55\) 291.503 0.714660
\(56\) 88.0557 0.210124
\(57\) 0 0
\(58\) −981.322 −2.22162
\(59\) 331.524 0.731537 0.365769 0.930706i \(-0.380806\pi\)
0.365769 + 0.930706i \(0.380806\pi\)
\(60\) 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
\(65\) 106.060 0.202386
\(66\) 0 0
\(67\) 758.179 1.38248 0.691241 0.722624i \(-0.257064\pi\)
0.691241 + 0.722624i \(0.257064\pi\)
\(68\) −818.299 −1.45931
\(69\) 0 0
\(70\) 113.296 0.193449
\(71\) −904.348 −1.51164 −0.755819 0.654780i \(-0.772761\pi\)
−0.755819 + 0.654780i \(0.772761\pi\)
\(72\) 0 0
\(73\) 866.622 1.38946 0.694729 0.719271i \(-0.255524\pi\)
0.694729 + 0.719271i \(0.255524\pi\)
\(74\) 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 \(0.452918\pi\)
\(80\) −89.0475 −0.124448
\(81\) 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
\(85\) −344.225 −0.439253
\(86\) −1955.87 −2.45240
\(87\) 0 0
\(88\) 1010.33 1.22388
\(89\) 601.736 0.716673 0.358337 0.933592i \(-0.383344\pi\)
0.358337 + 0.933592i \(0.383344\pi\)
\(90\) 0 0
\(91\) 107.783 0.124162
\(92\) −1715.38 −1.94392
\(93\) 0 0
\(94\) −1139.29 −1.25009
\(95\) −204.066 −0.220387
\(96\) 0 0
\(97\) 229.363 0.240086 0.120043 0.992769i \(-0.461697\pi\)
0.120043 + 0.992769i \(0.461697\pi\)
\(98\) −1414.43 −1.45795
\(99\) 0 0
\(100\) 297.152 0.297152
\(101\) 1345.66 1.32573 0.662863 0.748740i \(-0.269341\pi\)
0.662863 + 0.748740i \(0.269341\pi\)
\(102\) 0 0
\(103\) −1596.30 −1.52707 −0.763534 0.645768i \(-0.776537\pi\)
−0.763534 + 0.645768i \(0.776537\pi\)
\(104\) 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 \(0.233576\pi\)
\(110\) 1299.93 1.12675
\(111\) 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
\(115\) −721.592 −0.585120
\(116\) −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
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 309.141 0.215999 0.107999 0.994151i \(-0.465556\pi\)
0.107999 + 0.994151i \(0.465556\pi\)
\(128\) −1956.48 −1.35102
\(129\) 0 0
\(130\) 472.960 0.319087
\(131\) 2785.03 1.85747 0.928736 0.370742i \(-0.120897\pi\)
0.928736 + 0.370742i \(0.120897\pi\)
\(132\) 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 \(0.316556\pi\)
\(140\) 301.980 0.182300
\(141\) 0 0
\(142\) −4032.83 −2.38329
\(143\) 1236.67 0.723185
\(144\) 0 0
\(145\) −1100.29 −0.630166
\(146\) 3864.60 2.19066
\(147\) 0 0
\(148\) 3097.95 1.72061
\(149\) 1568.83 0.862575 0.431288 0.902214i \(-0.358059\pi\)
0.431288 + 0.902214i \(0.358059\pi\)
\(150\) 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
\(155\) 1457.73 0.755403
\(156\) 0 0
\(157\) −44.7479 −0.0227469 −0.0113735 0.999935i \(-0.503620\pi\)
−0.0113735 + 0.999935i \(0.503620\pi\)
\(158\) 922.917 0.464705
\(159\) 0 0
\(160\) −1090.28 −0.538714
\(161\) −733.315 −0.358965
\(162\) 0 0
\(163\) −2611.84 −1.25506 −0.627531 0.778591i \(-0.715935\pi\)
−0.627531 + 0.778591i \(0.715935\pi\)
\(164\) −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
\(170\) −1535.03 −0.692539
\(171\) 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
\(175\) 127.031 0.0548722
\(176\) −1038.31 −0.444689
\(177\) 0 0
\(178\) 2683.37 1.12993
\(179\) −3136.62 −1.30973 −0.654865 0.755746i \(-0.727275\pi\)
−0.654865 + 0.755746i \(0.727275\pi\)
\(180\) 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
\(185\) 1303.18 0.517902
\(186\) 0 0
\(187\) −4013.71 −1.56958
\(188\) −3036.67 −1.17804
\(189\) 0 0
\(190\) −910.010 −0.347469
\(191\) 1207.43 0.457418 0.228709 0.973495i \(-0.426550\pi\)
0.228709 + 0.973495i \(0.426550\pi\)
\(192\) 0 0
\(193\) 923.164 0.344305 0.172152 0.985070i \(-0.444928\pi\)
0.172152 + 0.985070i \(0.444928\pi\)
\(194\) 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 \(0.547792\pi\)
\(200\) 433.240 0.153173
\(201\) 0 0
\(202\) 6000.82 2.09018
\(203\) −1118.17 −0.386600
\(204\) 0 0
\(205\) −848.832 −0.289195
\(206\) −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
\(215\) −2192.98 −0.695627
\(216\) 0 0
\(217\) 1481.41 0.463432
\(218\) 7537.38 2.34172
\(219\) 0 0
\(220\) 3464.84 1.06181
\(221\) −1460.34 −0.444492
\(222\) 0 0
\(223\) −4180.76 −1.25544 −0.627722 0.778437i \(-0.716013\pi\)
−0.627722 + 0.778437i \(0.716013\pi\)
\(224\) −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 \(0.585786\pi\)
\(230\) −3217.85 −0.922517
\(231\) 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
\(235\) −1277.40 −0.354590
\(236\) 3940.52 1.08689
\(237\) 0 0
\(238\) −1559.97 −0.424865
\(239\) 2567.64 0.694924 0.347462 0.937694i \(-0.387044\pi\)
0.347462 + 0.937694i \(0.387044\pi\)
\(240\) 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
\(245\) −1585.91 −0.413550
\(246\) 0 0
\(247\) −865.728 −0.223016
\(248\) 5052.36 1.29365
\(249\) 0 0
\(250\) 557.423 0.141018
\(251\) 967.393 0.243272 0.121636 0.992575i \(-0.461186\pi\)
0.121636 + 0.992575i \(0.461186\pi\)
\(252\) 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
\(260\) 1260.63 0.300697
\(261\) 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
\(265\) 1073.57 0.248864
\(266\) −924.795 −0.213168
\(267\) 0 0
\(268\) 9011.79 2.05404
\(269\) −2142.91 −0.485709 −0.242855 0.970063i \(-0.578084\pi\)
−0.242855 + 0.970063i \(0.578084\pi\)
\(270\) 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
\(275\) 1457.52 0.319606
\(276\) 0 0
\(277\) 6777.80 1.47018 0.735088 0.677972i \(-0.237141\pi\)
0.735088 + 0.677972i \(0.237141\pi\)
\(278\) 7964.69 1.71831
\(279\) 0 0
\(280\) 440.278 0.0939702
\(281\) 827.653 0.175707 0.0878535 0.996133i \(-0.471999\pi\)
0.0878535 + 0.996133i \(0.471999\pi\)
\(282\) 0 0
\(283\) −3171.98 −0.666270 −0.333135 0.942879i \(-0.608106\pi\)
−0.333135 + 0.942879i \(0.608106\pi\)
\(284\) −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
\(290\) −4906.61 −0.993539
\(291\) 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
\(295\) 1657.62 0.327153
\(296\) 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
\(305\) 274.823 0.0515946
\(306\) 0 0
\(307\) −119.504 −0.0222165 −0.0111083 0.999938i \(-0.503536\pi\)
−0.0111083 + 0.999938i \(0.503536\pi\)
\(308\) 3521.13 0.651412
\(309\) 0 0
\(310\) 6500.56 1.19099
\(311\) 2139.35 0.390069 0.195035 0.980796i \(-0.437518\pi\)
0.195035 + 0.980796i \(0.437518\pi\)
\(312\) 0 0
\(313\) 5163.50 0.932455 0.466227 0.884665i \(-0.345613\pi\)
0.466227 + 0.884665i \(0.345613\pi\)
\(314\) −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
\(320\) −4149.60 −0.724905
\(321\) 0 0
\(322\) −3270.13 −0.565955
\(323\) 2809.79 0.484028
\(324\) 0 0
\(325\) 530.298 0.0905097
\(326\) −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
\(335\) 3790.90 0.618265
\(336\) 0 0
\(337\) 9897.46 1.59985 0.799924 0.600101i \(-0.204873\pi\)
0.799924 + 0.600101i \(0.204873\pi\)
\(338\) −7790.79 −1.25374
\(339\) 0 0
\(340\) −4091.49 −0.652625
\(341\) 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 \(0.293002\pi\)
\(350\) 566.479 0.0865131
\(351\) 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
\(355\) −4521.74 −0.676025
\(356\) 7152.30 1.06481
\(357\) 0 0
\(358\) −13987.4 −2.06496
\(359\) −11564.4 −1.70012 −0.850060 0.526685i \(-0.823435\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(360\) 0 0
\(361\) −5193.28 −0.757148
\(362\) 20123.7 2.92177
\(363\) 0 0
\(364\) 1281.12 0.184474
\(365\) 4333.11 0.621385
\(366\) 0 0
\(367\) −1148.57 −0.163365 −0.0816823 0.996658i \(-0.526029\pi\)
−0.0816823 + 0.996658i \(0.526029\pi\)
\(368\) 2570.24 0.364084
\(369\) 0 0
\(370\) 5811.39 0.816540
\(371\) 1091.01 0.152676
\(372\) 0 0
\(373\) −4602.98 −0.638963 −0.319481 0.947593i \(-0.603509\pi\)
−0.319481 + 0.947593i \(0.603509\pi\)
\(374\) −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 \(0.587117\pi\)
\(380\) −2425.55 −0.327443
\(381\) 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
\(385\) 1481.20 0.196075
\(386\) 4116.74 0.542841
\(387\) 0 0
\(388\) 2726.23 0.356710
\(389\) 8524.87 1.11113 0.555563 0.831474i \(-0.312503\pi\)
0.555563 + 0.831474i \(0.312503\pi\)
\(390\) 0 0
\(391\) 9935.60 1.28508
\(392\) −5496.62 −0.708217
\(393\) 0 0
\(394\) −5265.97 −0.673339
\(395\) 1034.80 0.131814
\(396\) 0 0
\(397\) −155.729 −0.0196872 −0.00984361 0.999952i \(-0.503133\pi\)
−0.00984361 + 0.999952i \(0.503133\pi\)
\(398\) −3745.01 −0.471659
\(399\) 0 0
\(400\) −445.238 −0.0556547
\(401\) 5933.96 0.738972 0.369486 0.929236i \(-0.379534\pi\)
0.369486 + 0.929236i \(0.379534\pi\)
\(402\) 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 \(0.172921\pi\)
\(410\) −3785.27 −0.455954
\(411\) 0 0
\(412\) −18973.8 −2.26886
\(413\) 1684.55 0.200705
\(414\) 0 0
\(415\) −2316.99 −0.274064
\(416\) −4625.39 −0.545140
\(417\) 0 0
\(418\) −10610.8 −1.24161
\(419\) 9624.24 1.12214 0.561068 0.827770i \(-0.310391\pi\)
0.561068 + 0.827770i \(0.310391\pi\)
\(420\) 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
\(425\) −1721.13 −0.196440
\(426\) 0 0
\(427\) 279.288 0.0316527
\(428\) 11396.2 1.28705
\(429\) 0 0
\(430\) −9779.33 −1.09675
\(431\) 11582.2 1.29441 0.647207 0.762314i \(-0.275937\pi\)
0.647207 + 0.762314i \(0.275937\pi\)
\(432\) 0 0
\(433\) 14892.6 1.65287 0.826437 0.563029i \(-0.190364\pi\)
0.826437 + 0.563029i \(0.190364\pi\)
\(434\) 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 \(0.528458\pi\)
\(440\) 5051.63 0.547334
\(441\) 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
\(445\) 3008.68 0.320506
\(446\) −18643.6 −1.97937
\(447\) 0 0
\(448\) −4217.02 −0.444722
\(449\) −3985.25 −0.418876 −0.209438 0.977822i \(-0.567163\pi\)
−0.209438 + 0.977822i \(0.567163\pi\)
\(450\) 0 0
\(451\) −9897.50 −1.03338
\(452\) −138.130 −0.0143741
\(453\) 0 0
\(454\) −11606.3 −1.19980
\(455\) 538.914 0.0555267
\(456\) 0 0
\(457\) −14177.8 −1.45122 −0.725611 0.688105i \(-0.758443\pi\)
−0.725611 + 0.688105i \(0.758443\pi\)
\(458\) −8229.13 −0.839567
\(459\) 0 0
\(460\) −8576.91 −0.869349
\(461\) −16394.3 −1.65631 −0.828154 0.560500i \(-0.810609\pi\)
−0.828154 + 0.560500i \(0.810609\pi\)
\(462\) 0 0
\(463\) 3319.60 0.333207 0.166603 0.986024i \(-0.446720\pi\)
0.166603 + 0.986024i \(0.446720\pi\)
\(464\) 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
\(470\) −5696.43 −0.559057
\(471\) 0 0
\(472\) 5745.17 0.560260
\(473\) −25570.4 −2.48569
\(474\) 0 0
\(475\) −1020.33 −0.0985601
\(476\) −4157.97 −0.400379
\(477\) 0 0
\(478\) 11450.1 1.09564
\(479\) −8646.75 −0.824802 −0.412401 0.911002i \(-0.635310\pi\)
−0.412401 + 0.911002i \(0.635310\pi\)
\(480\) 0 0
\(481\) 5528.60 0.524080
\(482\) 17784.0 1.68058
\(483\) 0 0
\(484\) 24580.1 2.30842
\(485\) 1146.82 0.107370
\(486\) 0 0
\(487\) −15251.7 −1.41914 −0.709569 0.704636i \(-0.751110\pi\)
−0.709569 + 0.704636i \(0.751110\pi\)
\(488\) 952.515 0.0883572
\(489\) 0 0
\(490\) −7072.16 −0.652015
\(491\) −1766.87 −0.162398 −0.0811992 0.996698i \(-0.525875\pi\)
−0.0811992 + 0.996698i \(0.525875\pi\)
\(492\) 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 \(0.323567\pi\)
\(500\) 1485.76 0.132891
\(501\) 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
\(505\) 6728.31 0.592883
\(506\) −37520.6 −3.29643
\(507\) 0 0
\(508\) 3674.48 0.320923
\(509\) 9674.72 0.842484 0.421242 0.906948i \(-0.361594\pi\)
0.421242 + 0.906948i \(0.361594\pi\)
\(510\) 0 0
\(511\) 4403.51 0.381213
\(512\) 6352.52 0.548329
\(513\) 0 0
\(514\) 16691.6 1.43237
\(515\) −7981.49 −0.682926
\(516\) 0 0
\(517\) −14894.7 −1.26706
\(518\) 5905.81 0.500939
\(519\) 0 0
\(520\) 1837.97 0.155000
\(521\) 5178.95 0.435497 0.217748 0.976005i \(-0.430129\pi\)
0.217748 + 0.976005i \(0.430129\pi\)
\(522\) 0 0
\(523\) 14280.7 1.19398 0.596992 0.802248i \(-0.296363\pi\)
0.596992 + 0.802248i \(0.296363\pi\)
\(524\) 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
\(530\) 4787.47 0.392367
\(531\) 0 0
\(532\) −2464.96 −0.200883
\(533\) −3601.07 −0.292645
\(534\) 0 0
\(535\) 4793.93 0.387401
\(536\) 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
\(545\) 8451.14 0.664233
\(546\) 0 0
\(547\) 13653.3 1.06723 0.533614 0.845728i \(-0.320834\pi\)
0.533614 + 0.845728i \(0.320834\pi\)
\(548\) −29589.4 −2.30656
\(549\) 0 0
\(550\) 6499.63 0.503900
\(551\) 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
\(560\) −452.471 −0.0341436
\(561\) 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
\(565\) −58.1057 −0.00432660
\(566\) −14145.1 −1.05046
\(567\) 0 0
\(568\) −15672.0 −1.15771
\(569\) 9060.50 0.667550 0.333775 0.942653i \(-0.391677\pi\)
0.333775 + 0.942653i \(0.391677\pi\)
\(570\) 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
\(575\) −3607.96 −0.261674
\(576\) 0 0
\(577\) 6347.76 0.457991 0.228996 0.973427i \(-0.426456\pi\)
0.228996 + 0.973427i \(0.426456\pi\)
\(578\) −773.071 −0.0556324
\(579\) 0 0
\(580\) −13078.1 −0.936277
\(581\) −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
\(590\) 7391.95 0.515800
\(591\) 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
\(595\) −1749.09 −0.120514
\(596\) 18647.3 1.28158
\(597\) 0 0
\(598\) −13651.4 −0.933522
\(599\) −1749.83 −0.119359 −0.0596794 0.998218i \(-0.519008\pi\)
−0.0596794 + 0.998218i \(0.519008\pi\)
\(600\) 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
\(605\) 10339.8 0.694834
\(606\) 0 0
\(607\) −4418.22 −0.295437 −0.147718 0.989029i \(-0.547193\pi\)
−0.147718 + 0.989029i \(0.547193\pi\)
\(608\) 8899.58 0.593628
\(609\) 0 0
\(610\) 1225.54 0.0813455
\(611\) −5419.24 −0.358820
\(612\) 0 0
\(613\) −20179.7 −1.32961 −0.664804 0.747018i \(-0.731485\pi\)
−0.664804 + 0.747018i \(0.731485\pi\)
\(614\) −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 \(0.724634\pi\)
\(620\) 17326.7 1.12235
\(621\) 0 0
\(622\) 9540.19 0.614994
\(623\) 3057.56 0.196627
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 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
\(635\) 1545.70 0.0965975
\(636\) 0 0
\(637\) −6728.02 −0.418483
\(638\) −57211.8 −3.55021
\(639\) 0 0
\(640\) −9782.40 −0.604193
\(641\) 4395.86 0.270867 0.135434 0.990786i \(-0.456757\pi\)
0.135434 + 0.990786i \(0.456757\pi\)
\(642\) 0 0
\(643\) 5786.36 0.354886 0.177443 0.984131i \(-0.443217\pi\)
0.177443 + 0.984131i \(0.443217\pi\)
\(644\) −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
\(650\) 2364.80 0.142700
\(651\) 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
\(655\) 13925.1 0.830687
\(656\) 3023.46 0.179948
\(657\) 0 0
\(658\) −5788.98 −0.342976
\(659\) 18312.4 1.08248 0.541238 0.840870i \(-0.317956\pi\)
0.541238 + 0.840870i \(0.317956\pi\)
\(660\) 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
\(665\) −1036.91 −0.0604656
\(666\) 0 0
\(667\) 31758.4 1.84361
\(668\) −2245.84 −0.130081
\(669\) 0 0
\(670\) 16905.1 0.974775
\(671\) 3204.48 0.184363
\(672\) 0 0
\(673\) 1437.24 0.0823204 0.0411602 0.999153i \(-0.486895\pi\)
0.0411602 + 0.999153i \(0.486895\pi\)
\(674\) 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
\(680\) −5965.28 −0.336409
\(681\) 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
\(685\) −12447.1 −0.694274
\(686\) −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
\(695\) 8930.26 0.487401
\(696\) 0 0
\(697\) 11687.6 0.635149
\(698\) 35205.1 1.90907
\(699\) 0 0
\(700\) 1509.90 0.0815270
\(701\) 9173.00 0.494236 0.247118 0.968985i \(-0.420516\pi\)
0.247118 + 0.968985i \(0.420516\pi\)
\(702\) 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 \(0.855858\pi\)
\(710\) −20164.2 −1.06584
\(711\) 0 0
\(712\) 10427.8 0.548876
\(713\) −42075.3 −2.21001
\(714\) 0 0
\(715\) 6183.35 0.323418
\(716\) −37282.1 −1.94595
\(717\) 0 0
\(718\) −51569.9 −2.68046
\(719\) −6727.90 −0.348968 −0.174484 0.984660i \(-0.555826\pi\)
−0.174484 + 0.984660i \(0.555826\pi\)
\(720\) 0 0
\(721\) −8111.17 −0.418968
\(722\) −23158.8 −1.19374
\(723\) 0 0
\(724\) 53638.0 2.75337
\(725\) −5501.45 −0.281819
\(726\) 0 0
\(727\) −36726.1 −1.87359 −0.936793 0.349885i \(-0.886221\pi\)
−0.936793 + 0.349885i \(0.886221\pi\)
\(728\) 1867.83 0.0950912
\(729\) 0 0
\(730\) 19323.0 0.979694
\(731\) 30195.1 1.52778
\(732\) 0 0
\(733\) 26691.4 1.34498 0.672489 0.740108i \(-0.265225\pi\)
0.672489 + 0.740108i \(0.265225\pi\)
\(734\) −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 \(0.401739\pi\)
\(740\) 15489.8 0.769480
\(741\) 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
\(745\) 7844.16 0.385755
\(746\) −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
\(755\) −2191.63 −0.105645
\(756\) 0 0
\(757\) −3418.34 −0.164124 −0.0820618 0.996627i \(-0.526150\pi\)
−0.0820618 + 0.996627i \(0.526150\pi\)
\(758\) −17786.1 −0.852268
\(759\) 0 0
\(760\) −3536.39 −0.168787
\(761\) −12684.5 −0.604224 −0.302112 0.953272i \(-0.597692\pi\)
−0.302112 + 0.953272i \(0.597692\pi\)
\(762\) 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 \(0.723836\pi\)
\(770\) 6605.22 0.309137
\(771\) 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
\(775\) 7288.63 0.337826
\(776\) 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
\(785\) −223.739 −0.0101727
\(786\) 0 0
\(787\) 4680.29 0.211988 0.105994 0.994367i \(-0.466198\pi\)
0.105994 + 0.994367i \(0.466198\pi\)
\(788\) −14036.0 −0.634531
\(789\) 0 0
\(790\) 4614.58 0.207822
\(791\) −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
\(800\) −5451.40 −0.240920
\(801\) 0 0
\(802\) 26461.8 1.16508
\(803\) 50524.7 2.22039
\(804\) 0 0
\(805\) −3666.58 −0.160534
\(806\) 27577.9 1.20520
\(807\) 0 0
\(808\) 23319.8 1.01533
\(809\) 29212.5 1.26954 0.634770 0.772701i \(-0.281095\pi\)
0.634770 + 0.772701i \(0.281095\pi\)
\(810\) 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
\(815\) −13059.2 −0.561281
\(816\) 0 0
\(817\) 17900.5 0.766536
\(818\) 63151.2 2.69930
\(819\) 0 0
\(820\) −10089.3 −0.429675
\(821\) −8722.99 −0.370809 −0.185405 0.982662i \(-0.559360\pi\)
−0.185405 + 0.982662i \(0.559360\pi\)
\(822\) 0 0
\(823\) 13584.8 0.575379 0.287690 0.957724i \(-0.407113\pi\)
0.287690 + 0.957724i \(0.407113\pi\)
\(824\) −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 \(0.860690\pi\)
\(830\) −10332.3 −0.432097
\(831\) 0 0
\(832\) −17604.2 −0.733552
\(833\) 21836.3 0.908265
\(834\) 0 0
\(835\) −944.733 −0.0391543
\(836\) −28282.3 −1.17005
\(837\) 0 0
\(838\) 42918.2 1.76919
\(839\) −4093.21 −0.168431 −0.0842154 0.996448i \(-0.526838\pi\)
−0.0842154 + 0.996448i \(0.526838\pi\)
\(840\) 0 0
\(841\) 24036.5 0.985546
\(842\) −6850.75 −0.280395
\(843\) 0 0
\(844\) −30780.8 −1.25535
\(845\) −8735.27 −0.355624
\(846\) 0 0
\(847\) 10507.8 0.426273
\(848\) −3823.96 −0.154853
\(849\) 0 0
\(850\) −7675.16 −0.309713
\(851\) −37614.7 −1.51517
\(852\) 0 0
\(853\) −20201.5 −0.810886 −0.405443 0.914120i \(-0.632883\pi\)
−0.405443 + 0.914120i \(0.632883\pi\)
\(854\) 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 \(0.423647\pi\)
\(860\) −26065.9 −1.03354
\(861\) 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
\(865\) −7525.10 −0.295793
\(866\) 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
\(875\) 635.154 0.0245396
\(876\) 0 0
\(877\) −19218.7 −0.739987 −0.369994 0.929034i \(-0.620640\pi\)
−0.369994 + 0.929034i \(0.620640\pi\)
\(878\) −7324.56 −0.281540
\(879\) 0 0
\(880\) −5191.53 −0.198871
\(881\) −45914.5 −1.75585 −0.877923 0.478802i \(-0.841071\pi\)
−0.877923 + 0.478802i \(0.841071\pi\)
\(882\) 0 0
\(883\) −44656.7 −1.70194 −0.850972 0.525211i \(-0.823986\pi\)
−0.850972 + 0.525211i \(0.823986\pi\)
\(884\) −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
\(890\) 13416.9 0.505319
\(891\) 0 0
\(892\) −49692.9 −1.86529
\(893\) 10427.0 0.390735
\(894\) 0 0
\(895\) −15683.1 −0.585729
\(896\) −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
\(905\) 22563.3 0.828763
\(906\) 0 0
\(907\) −14818.1 −0.542479 −0.271240 0.962512i \(-0.587434\pi\)
−0.271240 + 0.962512i \(0.587434\pi\)
\(908\) −30935.6 −1.13065
\(909\) 0 0
\(910\) 2403.22 0.0875451
\(911\) 21846.5 0.794518 0.397259 0.917707i \(-0.369961\pi\)
0.397259 + 0.917707i \(0.369961\pi\)
\(912\) 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 \(0.326566\pi\)
\(920\) −12504.9 −0.448124
\(921\) 0 0
\(922\) −73108.4 −2.61139
\(923\) −19182.9 −0.684089
\(924\) 0 0
\(925\) 6515.92 0.231613
\(926\) 14803.4 0.525344
\(927\) 0 0
\(928\) 47985.0 1.69740
\(929\) 4608.79 0.162766 0.0813829 0.996683i \(-0.474066\pi\)
0.0813829 + 0.996683i \(0.474066\pi\)
\(930\) 0 0
\(931\) 12945.2 0.455705
\(932\) 2860.23 0.100526
\(933\) 0 0
\(934\) −11281.3 −0.395220
\(935\) −20068.6 −0.701938
\(936\) 0 0
\(937\) −21063.8 −0.734391 −0.367195 0.930144i \(-0.619682\pi\)
−0.367195 + 0.930144i \(0.619682\pi\)
\(938\) 17179.7 0.598014
\(939\) 0 0
\(940\) −15183.3 −0.526836
\(941\) −20244.8 −0.701339 −0.350670 0.936499i \(-0.614046\pi\)
−0.350670 + 0.936499i \(0.614046\pi\)
\(942\) 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
\(950\) −4550.05 −0.155393
\(951\) 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
\(955\) 6037.17 0.204564
\(956\) 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
\(965\) 4615.82 0.153978
\(966\) 0 0
\(967\) −26119.2 −0.868600 −0.434300 0.900768i \(-0.643004\pi\)
−0.434300 + 0.900768i \(0.643004\pi\)
\(968\) 35837.0 1.18992
\(969\) 0 0
\(970\) 5114.09 0.169282
\(971\) 5101.93 0.168619 0.0843093 0.996440i \(-0.473132\pi\)
0.0843093 + 0.996440i \(0.473132\pi\)
\(972\) 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
\(980\) −18850.2 −0.614437
\(981\) 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
\(985\) −5904.37 −0.190994
\(986\) 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
\(995\) −4199.02 −0.133787
\(996\) 0 0
\(997\) −22523.8 −0.715483 −0.357742 0.933821i \(-0.616453\pi\)
−0.357742 + 0.933821i \(0.616453\pi\)
\(998\) 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 135.4.a.f.1.3 3
3.2 odd 2 135.4.a.g.1.1 yes 3
4.3 odd 2 2160.4.a.bm.1.3 3
5.2 odd 4 675.4.b.l.649.5 6
5.3 odd 4 675.4.b.l.649.2 6
5.4 even 2 675.4.a.r.1.1 3
9.2 odd 6 405.4.e.r.271.3 6
9.4 even 3 405.4.e.t.136.1 6
9.5 odd 6 405.4.e.r.136.3 6
9.7 even 3 405.4.e.t.271.1 6
12.11 even 2 2160.4.a.be.1.3 3
15.2 even 4 675.4.b.k.649.2 6
15.8 even 4 675.4.b.k.649.5 6
15.14 odd 2 675.4.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.3 3 1.1 even 1 trivial
135.4.a.g.1.1 yes 3 3.2 odd 2
405.4.e.r.136.3 6 9.5 odd 6
405.4.e.r.271.3 6 9.2 odd 6
405.4.e.t.136.1 6 9.4 even 3
405.4.e.t.271.1 6 9.7 even 3
675.4.a.q.1.3 3 15.14 odd 2
675.4.a.r.1.1 3 5.4 even 2
675.4.b.k.649.2 6 15.2 even 4
675.4.b.k.649.5 6 15.8 even 4
675.4.b.l.649.2 6 5.3 odd 4
675.4.b.l.649.5 6 5.2 odd 4
2160.4.a.be.1.3 3 12.11 even 2
2160.4.a.bm.1.3 3 4.3 odd 2