Properties

Label 675.4.b.k.649.2
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2033649216.1
Defining polynomial: \( x^{6} + 47x^{4} + 541x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-4.45938i\) of defining polynomial
Character \(\chi\) \(=\) 675.649
Dual form 675.4.b.k.649.5

$q$-expansion

\(f(q)\) \(=\) \(q-4.45938i q^{2} -11.8861 q^{4} +5.08123i q^{7} +17.3296i q^{8} +O(q^{10})\) \(q-4.45938i q^{2} -11.8861 q^{4} +5.08123i q^{7} +17.3296i q^{8} -58.3007 q^{11} -21.2119i q^{13} +22.6592 q^{14} -17.8095 q^{16} +68.8451i q^{17} +40.8133 q^{19} +259.985i q^{22} -144.318i q^{23} -94.5921 q^{26} -60.3960i q^{28} -220.058 q^{29} +291.545 q^{31} +218.056i q^{32} +307.006 q^{34} +260.637i q^{37} -182.002i q^{38} +169.766 q^{41} +438.596i q^{43} +692.967 q^{44} -643.571 q^{46} +255.481i q^{47} +317.181 q^{49} +252.127i q^{52} +214.714i q^{53} -88.0557 q^{56} +981.322i q^{58} +331.524 q^{59} +54.9647 q^{61} -1300.11i q^{62} +829.920 q^{64} +758.179i q^{67} -818.299i q^{68} +904.348 q^{71} -866.622i q^{73} +1162.28 q^{74} -485.110 q^{76} -296.239i q^{77} -206.961 q^{79} -757.054i q^{82} -463.397i q^{83} +1955.87 q^{86} -1010.33i q^{88} +601.736 q^{89} +107.783 q^{91} +1715.38i q^{92} +1139.29 q^{94} +229.363i q^{97} -1414.43i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 46 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.45938i − 1.57663i −0.615272 0.788315i \(-0.710954\pi\)
0.615272 0.788315i \(-0.289046\pi\)
\(3\) 0 0
\(4\) −11.8861 −1.48576
\(5\) 0 0
\(6\) 0 0
\(7\) 5.08123i 0.274361i 0.990546 + 0.137180i \(0.0438040\pi\)
−0.990546 + 0.137180i \(0.956196\pi\)
\(8\) 17.3296i 0.765867i
\(9\) 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
\(13\) − 21.2119i − 0.452548i −0.974064 0.226274i \(-0.927345\pi\)
0.974064 0.226274i \(-0.0726545\pi\)
\(14\) 22.6592 0.432566
\(15\) 0 0
\(16\) −17.8095 −0.278274
\(17\) 68.8451i 0.982199i 0.871104 + 0.491099i \(0.163405\pi\)
−0.871104 + 0.491099i \(0.836595\pi\)
\(18\) 0 0
\(19\) 40.8133 0.492800 0.246400 0.969168i \(-0.420752\pi\)
0.246400 + 0.969168i \(0.420752\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 259.985i 2.51950i
\(23\) − 144.318i − 1.30837i −0.756336 0.654184i \(-0.773012\pi\)
0.756336 0.654184i \(-0.226988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −94.5921 −0.713501
\(27\) 0 0
\(28\) − 60.3960i − 0.407635i
\(29\) −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 \(0.179859\pi\)
\(32\) 218.056i 1.20460i
\(33\) 0 0
\(34\) 307.006 1.54856
\(35\) 0 0
\(36\) 0 0
\(37\) 260.637i 1.15807i 0.815304 + 0.579033i \(0.196570\pi\)
−0.815304 + 0.579033i \(0.803430\pi\)
\(38\) − 182.002i − 0.776964i
\(39\) 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
\(43\) 438.596i 1.55547i 0.628592 + 0.777735i \(0.283631\pi\)
−0.628592 + 0.777735i \(0.716369\pi\)
\(44\) 692.967 2.37429
\(45\) 0 0
\(46\) −643.571 −2.06281
\(47\) 255.481i 0.792887i 0.918059 + 0.396444i \(0.129756\pi\)
−0.918059 + 0.396444i \(0.870244\pi\)
\(48\) 0 0
\(49\) 317.181 0.924726
\(50\) 0 0
\(51\) 0 0
\(52\) 252.127i 0.672379i
\(53\) 214.714i 0.556477i 0.960512 + 0.278239i \(0.0897506\pi\)
−0.960512 + 0.278239i \(0.910249\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −88.0557 −0.210124
\(57\) 0 0
\(58\) 981.322i 2.22162i
\(59\) 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 \(0.481628\pi\)
\(62\) − 1300.11i − 2.66314i
\(63\) 0 0
\(64\) 829.920 1.62094
\(65\) 0 0
\(66\) 0 0
\(67\) 758.179i 1.38248i 0.722624 + 0.691241i \(0.242936\pi\)
−0.722624 + 0.691241i \(0.757064\pi\)
\(68\) − 818.299i − 1.45931i
\(69\) 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
\(73\) − 866.622i − 1.38946i −0.719271 0.694729i \(-0.755524\pi\)
0.719271 0.694729i \(-0.244476\pi\)
\(74\) 1162.28 1.82584
\(75\) 0 0
\(76\) −485.110 −0.732184
\(77\) − 296.239i − 0.438437i
\(78\) 0 0
\(79\) −206.961 −0.294746 −0.147373 0.989081i \(-0.547082\pi\)
−0.147373 + 0.989081i \(0.547082\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 757.054i − 1.01954i
\(83\) − 463.397i − 0.612825i −0.951899 0.306412i \(-0.900871\pi\)
0.951899 0.306412i \(-0.0991287\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1955.87 2.45240
\(87\) 0 0
\(88\) − 1010.33i − 1.22388i
\(89\) 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
\(92\) 1715.38i 1.94392i
\(93\) 0 0
\(94\) 1139.29 1.25009
\(95\) 0 0
\(96\) 0 0
\(97\) 229.363i 0.240086i 0.992769 + 0.120043i \(0.0383032\pi\)
−0.992769 + 0.120043i \(0.961697\pi\)
\(98\) − 1414.43i − 1.45795i
\(99\) 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
\(103\) 1596.30i 1.52707i 0.645768 + 0.763534i \(0.276537\pi\)
−0.645768 + 0.763534i \(0.723463\pi\)
\(104\) 367.594 0.346592
\(105\) 0 0
\(106\) 957.494 0.877358
\(107\) − 958.786i − 0.866256i −0.901333 0.433128i \(-0.857410\pi\)
0.901333 0.433128i \(-0.142590\pi\)
\(108\) 0 0
\(109\) −1690.23 −1.48527 −0.742635 0.669696i \(-0.766424\pi\)
−0.742635 + 0.669696i \(0.766424\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 90.4943i − 0.0763474i
\(113\) − 11.6211i − 0.00967456i −0.999988 0.00483728i \(-0.998460\pi\)
0.999988 0.00483728i \(-0.00153976\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2615.63 2.09358
\(117\) 0 0
\(118\) − 1478.39i − 1.15336i
\(119\) −349.818 −0.269477
\(120\) 0 0
\(121\) 2067.97 1.55370
\(122\) − 245.109i − 0.181894i
\(123\) 0 0
\(124\) −3465.34 −2.50965
\(125\) 0 0
\(126\) 0 0
\(127\) 309.141i 0.215999i 0.994151 + 0.107999i \(0.0344444\pi\)
−0.994151 + 0.107999i \(0.965556\pi\)
\(128\) − 1956.48i − 1.35102i
\(129\) 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
\(133\) 207.382i 0.135205i
\(134\) 3381.01 2.17966
\(135\) 0 0
\(136\) −1193.06 −0.752233
\(137\) 2489.41i 1.55244i 0.630460 + 0.776222i \(0.282866\pi\)
−0.630460 + 0.776222i \(0.717134\pi\)
\(138\) 0 0
\(139\) −1786.05 −1.08986 −0.544931 0.838481i \(-0.683444\pi\)
−0.544931 + 0.838481i \(0.683444\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 4032.83i − 2.38329i
\(143\) 1236.67i 0.723185i
\(144\) 0 0
\(145\) 0 0
\(146\) −3864.60 −2.19066
\(147\) 0 0
\(148\) − 3097.95i − 1.72061i
\(149\) 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 \(0.537685\pi\)
\(152\) 707.277i 0.377419i
\(153\) 0 0
\(154\) −1321.04 −0.691252
\(155\) 0 0
\(156\) 0 0
\(157\) − 44.7479i − 0.0227469i −0.999935 0.0113735i \(-0.996380\pi\)
0.999935 0.0113735i \(-0.00362036\pi\)
\(158\) 922.917i 0.464705i
\(159\) 0 0
\(160\) 0 0
\(161\) 733.315 0.358965
\(162\) 0 0
\(163\) 2611.84i 1.25506i 0.778591 + 0.627531i \(0.215935\pi\)
−0.778591 + 0.627531i \(0.784065\pi\)
\(164\) −2017.86 −0.960783
\(165\) 0 0
\(166\) −2066.47 −0.966198
\(167\) 188.947i 0.0875516i 0.999041 + 0.0437758i \(0.0139387\pi\)
−0.999041 + 0.0437758i \(0.986061\pi\)
\(168\) 0 0
\(169\) 1747.05 0.795200
\(170\) 0 0
\(171\) 0 0
\(172\) − 5213.19i − 2.31106i
\(173\) − 1505.02i − 0.661413i −0.943734 0.330707i \(-0.892713\pi\)
0.943734 0.330707i \(-0.107287\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1038.31 0.444689
\(177\) 0 0
\(178\) − 2683.37i − 1.12993i
\(179\) −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 \(0.122730\pi\)
\(182\) − 480.644i − 0.195757i
\(183\) 0 0
\(184\) 2500.98 1.00204
\(185\) 0 0
\(186\) 0 0
\(187\) − 4013.71i − 1.56958i
\(188\) − 3036.67i − 1.17804i
\(189\) 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
\(193\) − 923.164i − 0.344305i −0.985070 0.172152i \(-0.944928\pi\)
0.985070 0.172152i \(-0.0550721\pi\)
\(194\) 1022.82 0.378526
\(195\) 0 0
\(196\) −3770.04 −1.37392
\(197\) 1180.87i 0.427075i 0.976935 + 0.213537i \(0.0684985\pi\)
−0.976935 + 0.213537i \(0.931501\pi\)
\(198\) 0 0
\(199\) 839.805 0.299157 0.149578 0.988750i \(-0.452208\pi\)
0.149578 + 0.988750i \(0.452208\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6000.82i 2.09018i
\(203\) − 1118.17i − 0.386600i
\(204\) 0 0
\(205\) 0 0
\(206\) 7118.51 2.40762
\(207\) 0 0
\(208\) 377.774i 0.125932i
\(209\) −2379.44 −0.787509
\(210\) 0 0
\(211\) −2589.65 −0.844923 −0.422461 0.906381i \(-0.638834\pi\)
−0.422461 + 0.906381i \(0.638834\pi\)
\(212\) − 2552.12i − 0.826792i
\(213\) 0 0
\(214\) −4275.59 −1.36576
\(215\) 0 0
\(216\) 0 0
\(217\) 1481.41i 0.463432i
\(218\) 7537.38i 2.34172i
\(219\) 0 0
\(220\) 0 0
\(221\) 1460.34 0.444492
\(222\) 0 0
\(223\) 4180.76i 1.25544i 0.778437 + 0.627722i \(0.216013\pi\)
−0.778437 + 0.627722i \(0.783987\pi\)
\(224\) −1107.99 −0.330495
\(225\) 0 0
\(226\) −51.8232 −0.0152532
\(227\) 2602.67i 0.760993i 0.924782 + 0.380497i \(0.124247\pi\)
−0.924782 + 0.380497i \(0.875753\pi\)
\(228\) 0 0
\(229\) 1845.35 0.532508 0.266254 0.963903i \(-0.414214\pi\)
0.266254 + 0.963903i \(0.414214\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3813.51i − 1.07918i
\(233\) 240.637i 0.0676594i 0.999428 + 0.0338297i \(0.0107704\pi\)
−0.999428 + 0.0338297i \(0.989230\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3940.52 −1.08689
\(237\) 0 0
\(238\) 1559.97i 0.424865i
\(239\) 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 \(0.321078\pi\)
\(242\) − 9221.86i − 2.44960i
\(243\) 0 0
\(244\) −653.315 −0.171411
\(245\) 0 0
\(246\) 0 0
\(247\) − 865.728i − 0.223016i
\(248\) 5052.36i 1.29365i
\(249\) 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
\(253\) 8413.86i 2.09081i
\(254\) 1378.58 0.340550
\(255\) 0 0
\(256\) −2085.34 −0.509116
\(257\) − 3743.03i − 0.908498i −0.890875 0.454249i \(-0.849908\pi\)
0.890875 0.454249i \(-0.150092\pi\)
\(258\) 0 0
\(259\) −1324.36 −0.317728
\(260\) 0 0
\(261\) 0 0
\(262\) 12419.5i 2.92855i
\(263\) − 2497.47i − 0.585554i −0.956181 0.292777i \(-0.905421\pi\)
0.956181 0.292777i \(-0.0945794\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 924.795 0.213168
\(267\) 0 0
\(268\) − 9011.79i − 2.05404i
\(269\) −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 \(0.444760\pi\)
\(272\) − 1226.10i − 0.273320i
\(273\) 0 0
\(274\) 11101.2 2.44763
\(275\) 0 0
\(276\) 0 0
\(277\) 6777.80i 1.47018i 0.677972 + 0.735088i \(0.262859\pi\)
−0.677972 + 0.735088i \(0.737141\pi\)
\(278\) 7964.69i 1.71831i
\(279\) 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
\(283\) 3171.98i 0.666270i 0.942879 + 0.333135i \(0.108106\pi\)
−0.942879 + 0.333135i \(0.891894\pi\)
\(284\) −10749.2 −2.24593
\(285\) 0 0
\(286\) 5514.78 1.14020
\(287\) 862.623i 0.177418i
\(288\) 0 0
\(289\) 173.358 0.0352856
\(290\) 0 0
\(291\) 0 0
\(292\) 10300.8i 2.06440i
\(293\) 1376.02i 0.274362i 0.990546 + 0.137181i \(0.0438043\pi\)
−0.990546 + 0.137181i \(0.956196\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4516.73 −0.886923
\(297\) 0 0
\(298\) − 6996.02i − 1.35996i
\(299\) −3061.27 −0.592099
\(300\) 0 0
\(301\) −2228.61 −0.426760
\(302\) 1954.67i 0.372445i
\(303\) 0 0
\(304\) −726.864 −0.137133
\(305\) 0 0
\(306\) 0 0
\(307\) − 119.504i − 0.0222165i −0.999938 0.0111083i \(-0.996464\pi\)
0.999938 0.0111083i \(-0.00353594\pi\)
\(308\) 3521.13i 0.651412i
\(309\) 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
\(313\) − 5163.50i − 0.932455i −0.884665 0.466227i \(-0.845613\pi\)
0.884665 0.466227i \(-0.154387\pi\)
\(314\) −199.548 −0.0358635
\(315\) 0 0
\(316\) 2459.95 0.437922
\(317\) 8631.69i 1.52935i 0.644416 + 0.764675i \(0.277101\pi\)
−0.644416 + 0.764675i \(0.722899\pi\)
\(318\) 0 0
\(319\) 12829.5 2.25177
\(320\) 0 0
\(321\) 0 0
\(322\) − 3270.13i − 0.565955i
\(323\) 2809.79i 0.484028i
\(324\) 0 0
\(325\) 0 0
\(326\) 11647.2 1.97877
\(327\) 0 0
\(328\) 2941.98i 0.495255i
\(329\) −1298.16 −0.217537
\(330\) 0 0
\(331\) −2942.34 −0.488597 −0.244298 0.969700i \(-0.578558\pi\)
−0.244298 + 0.969700i \(0.578558\pi\)
\(332\) 5507.98i 0.910512i
\(333\) 0 0
\(334\) 842.585 0.138037
\(335\) 0 0
\(336\) 0 0
\(337\) 9897.46i 1.59985i 0.600101 + 0.799924i \(0.295127\pi\)
−0.600101 + 0.799924i \(0.704873\pi\)
\(338\) − 7790.79i − 1.25374i
\(339\) 0 0
\(340\) 0 0
\(341\) −16997.3 −2.69928
\(342\) 0 0
\(343\) 3354.53i 0.528070i
\(344\) −7600.68 −1.19128
\(345\) 0 0
\(346\) −6711.46 −1.04280
\(347\) 12015.7i 1.85890i 0.368947 + 0.929451i \(0.379718\pi\)
−0.368947 + 0.929451i \(0.620282\pi\)
\(348\) 0 0
\(349\) −7894.62 −1.21086 −0.605428 0.795900i \(-0.706998\pi\)
−0.605428 + 0.795900i \(0.706998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 12712.8i − 1.92499i
\(353\) 741.154i 0.111750i 0.998438 + 0.0558748i \(0.0177948\pi\)
−0.998438 + 0.0558748i \(0.982205\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7152.30 −1.06481
\(357\) 0 0
\(358\) 13987.4i 2.06496i
\(359\) −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
\(362\) − 20123.7i − 2.92177i
\(363\) 0 0
\(364\) −1281.12 −0.184474
\(365\) 0 0
\(366\) 0 0
\(367\) − 1148.57i − 0.163365i −0.996658 0.0816823i \(-0.973971\pi\)
0.996658 0.0816823i \(-0.0260293\pi\)
\(368\) 2570.24i 0.364084i
\(369\) 0 0
\(370\) 0 0
\(371\) −1091.01 −0.152676
\(372\) 0 0
\(373\) 4602.98i 0.638963i 0.947593 + 0.319481i \(0.103509\pi\)
−0.947593 + 0.319481i \(0.896491\pi\)
\(374\) −17898.7 −2.47465
\(375\) 0 0
\(376\) −4427.38 −0.607246
\(377\) 4667.85i 0.637683i
\(378\) 0 0
\(379\) 3988.46 0.540563 0.270282 0.962781i \(-0.412883\pi\)
0.270282 + 0.962781i \(0.412883\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5384.41i 0.721179i
\(383\) 7788.20i 1.03906i 0.854454 + 0.519528i \(0.173892\pi\)
−0.854454 + 0.519528i \(0.826108\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4116.74 −0.542841
\(387\) 0 0
\(388\) − 2726.23i − 0.356710i
\(389\) 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
\(392\) 5496.62i 0.708217i
\(393\) 0 0
\(394\) 5265.97 0.673339
\(395\) 0 0
\(396\) 0 0
\(397\) − 155.729i − 0.0196872i −0.999952 0.00984361i \(-0.996867\pi\)
0.999952 0.00984361i \(-0.00313337\pi\)
\(398\) − 3745.01i − 0.471659i
\(399\) 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
\(403\) − 6184.23i − 0.764414i
\(404\) 15994.7 1.96971
\(405\) 0 0
\(406\) −4986.33 −0.609526
\(407\) − 15195.3i − 1.85062i
\(408\) 0 0
\(409\) −14161.4 −1.71207 −0.856035 0.516917i \(-0.827079\pi\)
−0.856035 + 0.516917i \(0.827079\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 18973.8i − 2.26886i
\(413\) 1684.55i 0.200705i
\(414\) 0 0
\(415\) 0 0
\(416\) 4625.39 0.545140
\(417\) 0 0
\(418\) 10610.8i 1.24161i
\(419\) 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 \(0.528342\pi\)
\(422\) 11548.2i 1.33213i
\(423\) 0 0
\(424\) −3720.91 −0.426187
\(425\) 0 0
\(426\) 0 0
\(427\) 279.288i 0.0316527i
\(428\) 11396.2i 1.28705i
\(429\) 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
\(433\) − 14892.6i − 1.65287i −0.563029 0.826437i \(-0.690364\pi\)
0.563029 0.826437i \(-0.309636\pi\)
\(434\) 6606.17 0.730660
\(435\) 0 0
\(436\) 20090.2 2.20676
\(437\) − 5890.10i − 0.644764i
\(438\) 0 0
\(439\) 1642.51 0.178571 0.0892853 0.996006i \(-0.471542\pi\)
0.0892853 + 0.996006i \(0.471542\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6512.20i − 0.700800i
\(443\) 3916.31i 0.420021i 0.977699 + 0.210011i \(0.0673499\pi\)
−0.977699 + 0.210011i \(0.932650\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 18643.6 1.97937
\(447\) 0 0
\(448\) 4217.02i 0.444722i
\(449\) −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
\(452\) 138.130i 0.0143741i
\(453\) 0 0
\(454\) 11606.3 1.19980
\(455\) 0 0
\(456\) 0 0
\(457\) − 14177.8i − 1.45122i −0.688105 0.725611i \(-0.741557\pi\)
0.688105 0.725611i \(-0.258443\pi\)
\(458\) − 8229.13i − 0.839567i
\(459\) 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
\(463\) − 3319.60i − 0.333207i −0.986024 0.166603i \(-0.946720\pi\)
0.986024 0.166603i \(-0.0532800\pi\)
\(464\) 3919.12 0.392114
\(465\) 0 0
\(466\) 1073.09 0.106674
\(467\) 2529.79i 0.250674i 0.992114 + 0.125337i \(0.0400012\pi\)
−0.992114 + 0.125337i \(0.959999\pi\)
\(468\) 0 0
\(469\) −3852.49 −0.379299
\(470\) 0 0
\(471\) 0 0
\(472\) 5745.17i 0.560260i
\(473\) − 25570.4i − 2.48569i
\(474\) 0 0
\(475\) 0 0
\(476\) 4157.97 0.400379
\(477\) 0 0
\(478\) − 11450.1i − 1.09564i
\(479\) −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
\(482\) − 17784.0i − 1.68058i
\(483\) 0 0
\(484\) −24580.1 −2.30842
\(485\) 0 0
\(486\) 0 0
\(487\) − 15251.7i − 1.41914i −0.704636 0.709569i \(-0.748890\pi\)
0.704636 0.709569i \(-0.251110\pi\)
\(488\) 952.515i 0.0883572i
\(489\) 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
\(493\) − 15149.9i − 1.38401i
\(494\) −3860.61 −0.351614
\(495\) 0 0
\(496\) −5192.28 −0.470041
\(497\) 4595.20i 0.414734i
\(498\) 0 0
\(499\) −11733.8 −1.05266 −0.526332 0.850279i \(-0.676433\pi\)
−0.526332 + 0.850279i \(0.676433\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4313.98i 0.383550i
\(503\) − 977.608i − 0.0866589i −0.999061 0.0433294i \(-0.986203\pi\)
0.999061 0.0433294i \(-0.0137965\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 37520.6 3.29643
\(507\) 0 0
\(508\) − 3674.48i − 0.320923i
\(509\) 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
\(512\) − 6352.52i − 0.548329i
\(513\) 0 0
\(514\) −16691.6 −1.43237
\(515\) 0 0
\(516\) 0 0
\(517\) − 14894.7i − 1.26706i
\(518\) 5905.81i 0.500939i
\(519\) 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
\(523\) − 14280.7i − 1.19398i −0.802248 0.596992i \(-0.796363\pi\)
0.802248 0.596992i \(-0.203637\pi\)
\(524\) 33103.1 2.75976
\(525\) 0 0
\(526\) −11137.2 −0.923203
\(527\) 20071.5i 1.65906i
\(528\) 0 0
\(529\) −8660.78 −0.711826
\(530\) 0 0
\(531\) 0 0
\(532\) − 2464.96i − 0.200883i
\(533\) − 3601.07i − 0.292645i
\(534\) 0 0
\(535\) 0 0
\(536\) −13138.9 −1.05880
\(537\) 0 0
\(538\) 9556.08i 0.765784i
\(539\) −18491.9 −1.47774
\(540\) 0 0
\(541\) 12923.8 1.02706 0.513529 0.858072i \(-0.328338\pi\)
0.513529 + 0.858072i \(0.328338\pi\)
\(542\) − 6870.32i − 0.544475i
\(543\) 0 0
\(544\) −15012.1 −1.18316
\(545\) 0 0
\(546\) 0 0
\(547\) 13653.3i 1.06723i 0.845728 + 0.533614i \(0.179166\pi\)
−0.845728 + 0.533614i \(0.820834\pi\)
\(548\) − 29589.4i − 2.30656i
\(549\) 0 0
\(550\) 0 0
\(551\) −8981.28 −0.694402
\(552\) 0 0
\(553\) − 1051.62i − 0.0808666i
\(554\) 30224.8 2.31792
\(555\) 0 0
\(556\) 21229.2 1.61928
\(557\) 9313.63i 0.708494i 0.935152 + 0.354247i \(0.115263\pi\)
−0.935152 + 0.354247i \(0.884737\pi\)
\(558\) 0 0
\(559\) 9303.46 0.703925
\(560\) 0 0
\(561\) 0 0
\(562\) 3690.82i 0.277025i
\(563\) − 10625.4i − 0.795394i −0.917517 0.397697i \(-0.869810\pi\)
0.917517 0.397697i \(-0.130190\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14145.1 1.05046
\(567\) 0 0
\(568\) 15672.0i 1.15771i
\(569\) 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 \(0.786537\pi\)
\(572\) − 14699.2i − 1.07448i
\(573\) 0 0
\(574\) 3846.77 0.279723
\(575\) 0 0
\(576\) 0 0
\(577\) 6347.76i 0.457991i 0.973427 + 0.228996i \(0.0735441\pi\)
−0.973427 + 0.228996i \(0.926456\pi\)
\(578\) − 773.071i − 0.0556324i
\(579\) 0 0
\(580\) 0 0
\(581\) 2354.63 0.168135
\(582\) 0 0
\(583\) − 12518.0i − 0.889266i
\(584\) 15018.2 1.06414
\(585\) 0 0
\(586\) 6136.22 0.432568
\(587\) − 14773.7i − 1.03880i −0.854531 0.519401i \(-0.826155\pi\)
0.854531 0.519401i \(-0.173845\pi\)
\(588\) 0 0
\(589\) 11898.9 0.832405
\(590\) 0 0
\(591\) 0 0
\(592\) − 4641.81i − 0.322259i
\(593\) 26868.1i 1.86061i 0.366790 + 0.930304i \(0.380457\pi\)
−0.366790 + 0.930304i \(0.619543\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18647.3 −1.28158
\(597\) 0 0
\(598\) 13651.4i 0.933522i
\(599\) −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 \(0.708679\pi\)
\(602\) 9938.21i 0.672843i
\(603\) 0 0
\(604\) 5209.99 0.350979
\(605\) 0 0
\(606\) 0 0
\(607\) − 4418.22i − 0.295437i −0.989029 0.147718i \(-0.952807\pi\)
0.989029 0.147718i \(-0.0471929\pi\)
\(608\) 8899.58i 0.593628i
\(609\) 0 0
\(610\) 0 0
\(611\) 5419.24 0.358820
\(612\) 0 0
\(613\) 20179.7i 1.32961i 0.747018 + 0.664804i \(0.231485\pi\)
−0.747018 + 0.664804i \(0.768515\pi\)
\(614\) −532.915 −0.0350272
\(615\) 0 0
\(616\) 5133.71 0.335784
\(617\) 4673.01i 0.304908i 0.988311 + 0.152454i \(0.0487176\pi\)
−0.988311 + 0.152454i \(0.951282\pi\)
\(618\) 0 0
\(619\) 19976.8 1.29715 0.648574 0.761151i \(-0.275366\pi\)
0.648574 + 0.761151i \(0.275366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9540.19i 0.614994i
\(623\) 3057.56i 0.196627i
\(624\) 0 0
\(625\) 0 0
\(626\) −23026.0 −1.47014
\(627\) 0 0
\(628\) 531.877i 0.0337965i
\(629\) −17943.5 −1.13745
\(630\) 0 0
\(631\) 10457.5 0.659757 0.329879 0.944023i \(-0.392992\pi\)
0.329879 + 0.944023i \(0.392992\pi\)
\(632\) − 3586.54i − 0.225736i
\(633\) 0 0
\(634\) 38492.0 2.41122
\(635\) 0 0
\(636\) 0 0
\(637\) − 6728.02i − 0.418483i
\(638\) − 57211.8i − 3.55021i
\(639\) 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
\(643\) − 5786.36i − 0.354886i −0.984131 0.177443i \(-0.943217\pi\)
0.984131 0.177443i \(-0.0567825\pi\)
\(644\) −8716.26 −0.533336
\(645\) 0 0
\(646\) 12529.9 0.763133
\(647\) 25367.7i 1.54143i 0.637178 + 0.770717i \(0.280102\pi\)
−0.637178 + 0.770717i \(0.719898\pi\)
\(648\) 0 0
\(649\) −19328.1 −1.16902
\(650\) 0 0
\(651\) 0 0
\(652\) − 31044.6i − 1.86472i
\(653\) − 21633.2i − 1.29644i −0.761454 0.648218i \(-0.775514\pi\)
0.761454 0.648218i \(-0.224486\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3023.46 −0.179948
\(657\) 0 0
\(658\) 5788.98i 0.342976i
\(659\) 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 \(0.551984\pi\)
\(662\) 13121.0i 0.770337i
\(663\) 0 0
\(664\) 8030.48 0.469342
\(665\) 0 0
\(666\) 0 0
\(667\) 31758.4i 1.84361i
\(668\) − 2245.84i − 0.130081i
\(669\) 0 0
\(670\) 0 0
\(671\) −3204.48 −0.184363
\(672\) 0 0
\(673\) − 1437.24i − 0.0823204i −0.999153 0.0411602i \(-0.986895\pi\)
0.999153 0.0411602i \(-0.0131054\pi\)
\(674\) 44136.6 2.52237
\(675\) 0 0
\(676\) −20765.7 −1.18148
\(677\) 23405.9i 1.32875i 0.747401 + 0.664373i \(0.231301\pi\)
−0.747401 + 0.664373i \(0.768699\pi\)
\(678\) 0 0
\(679\) −1165.45 −0.0658701
\(680\) 0 0
\(681\) 0 0
\(682\) 75797.4i 4.25577i
\(683\) 19227.4i 1.07719i 0.842566 + 0.538593i \(0.181044\pi\)
−0.842566 + 0.538593i \(0.818956\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 14959.2 0.832570
\(687\) 0 0
\(688\) − 7811.17i − 0.432846i
\(689\) 4554.50 0.251833
\(690\) 0 0
\(691\) −35284.8 −1.94254 −0.971271 0.237975i \(-0.923516\pi\)
−0.971271 + 0.237975i \(0.923516\pi\)
\(692\) 17888.8i 0.982703i
\(693\) 0 0
\(694\) 53582.8 2.93080
\(695\) 0 0
\(696\) 0 0
\(697\) 11687.6i 0.635149i
\(698\) 35205.1i 1.90907i
\(699\) 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
\(703\) 10637.4i 0.570695i
\(704\) −48384.9 −2.59030
\(705\) 0 0
\(706\) 3305.09 0.176188
\(707\) − 6837.63i − 0.363728i
\(708\) 0 0
\(709\) 33951.6 1.79842 0.899210 0.437517i \(-0.144142\pi\)
0.899210 + 0.437517i \(0.144142\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10427.8i 0.548876i
\(713\) − 42075.3i − 2.21001i
\(714\) 0 0
\(715\) 0 0
\(716\) 37282.1 1.94595
\(717\) 0 0
\(718\) 51569.9i 2.68046i
\(719\) −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
\(722\) 23158.8i 1.19374i
\(723\) 0 0
\(724\) −53638.0 −2.75337
\(725\) 0 0
\(726\) 0 0
\(727\) − 36726.1i − 1.87359i −0.349885 0.936793i \(-0.613779\pi\)
0.349885 0.936793i \(-0.386221\pi\)
\(728\) 1867.83i 0.0950912i
\(729\) 0 0
\(730\) 0 0
\(731\) −30195.1 −1.52778
\(732\) 0 0
\(733\) − 26691.4i − 1.34498i −0.740108 0.672489i \(-0.765225\pi\)
0.740108 0.672489i \(-0.234775\pi\)
\(734\) −5121.91 −0.257566
\(735\) 0 0
\(736\) 31469.5 1.57606
\(737\) − 44202.4i − 2.20925i
\(738\) 0 0
\(739\) −12207.0 −0.607634 −0.303817 0.952730i \(-0.598261\pi\)
−0.303817 + 0.952730i \(0.598261\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4865.25i 0.240713i
\(743\) − 12473.3i − 0.615882i −0.951405 0.307941i \(-0.900360\pi\)
0.951405 0.307941i \(-0.0996400\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20526.4 1.00741
\(747\) 0 0
\(748\) 47707.4i 2.33202i
\(749\) 4871.82 0.237667
\(750\) 0 0
\(751\) −15102.6 −0.733825 −0.366913 0.930255i \(-0.619585\pi\)
−0.366913 + 0.930255i \(0.619585\pi\)
\(752\) − 4549.99i − 0.220640i
\(753\) 0 0
\(754\) 20815.7 1.00539
\(755\) 0 0
\(756\) 0 0
\(757\) − 3418.34i − 0.164124i −0.996627 0.0820618i \(-0.973850\pi\)
0.996627 0.0820618i \(-0.0261505\pi\)
\(758\) − 17786.1i − 0.852268i
\(759\) 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
\(763\) − 8588.45i − 0.407500i
\(764\) 14351.7 0.679614
\(765\) 0 0
\(766\) 34730.6 1.63821
\(767\) − 7032.25i − 0.331056i
\(768\) 0 0
\(769\) 27580.2 1.29333 0.646663 0.762776i \(-0.276164\pi\)
0.646663 + 0.762776i \(0.276164\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10972.8i 0.511555i
\(773\) 17386.0i 0.808966i 0.914546 + 0.404483i \(0.132548\pi\)
−0.914546 + 0.404483i \(0.867452\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3974.77 −0.183874
\(777\) 0 0
\(778\) − 38015.7i − 1.75183i
\(779\) 6928.72 0.318674
\(780\) 0 0
\(781\) −52724.1 −2.41564
\(782\) − 44306.7i − 2.02609i
\(783\) 0 0
\(784\) −5648.84 −0.257327
\(785\) 0 0
\(786\) 0 0
\(787\) 4680.29i 0.211988i 0.994367 + 0.105994i \(0.0338024\pi\)
−0.994367 + 0.105994i \(0.966198\pi\)
\(788\) − 14036.0i − 0.634531i
\(789\) 0 0
\(790\) 0 0
\(791\) 59.0498 0.00265432
\(792\) 0 0
\(793\) − 1165.91i − 0.0522100i
\(794\) −694.456 −0.0310395
\(795\) 0 0
\(796\) −9982.00 −0.444476
\(797\) − 7278.62i − 0.323490i −0.986833 0.161745i \(-0.948288\pi\)
0.986833 0.161745i \(-0.0517123\pi\)
\(798\) 0 0
\(799\) −17588.6 −0.778773
\(800\) 0 0
\(801\) 0 0
\(802\) 26461.8i 1.16508i
\(803\) 50524.7i 2.22039i
\(804\) 0 0
\(805\) 0 0
\(806\) −27577.9 −1.20520
\(807\) 0 0
\(808\) − 23319.8i − 1.01533i
\(809\) 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 \(0.136775\pi\)
\(812\) 13290.6i 0.574396i
\(813\) 0 0
\(814\) −67761.6 −2.91774
\(815\) 0 0
\(816\) 0 0
\(817\) 17900.5i 0.766536i
\(818\) 63151.2i 2.69930i
\(819\) 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
\(823\) − 13584.8i − 0.575379i −0.957724 0.287690i \(-0.907113\pi\)
0.957724 0.287690i \(-0.0928871\pi\)
\(824\) −27663.2 −1.16953
\(825\) 0 0
\(826\) 7512.05 0.316438
\(827\) − 26573.6i − 1.11736i −0.829384 0.558679i \(-0.811308\pi\)
0.829384 0.558679i \(-0.188692\pi\)
\(828\) 0 0
\(829\) 43238.4 1.81150 0.905748 0.423816i \(-0.139310\pi\)
0.905748 + 0.423816i \(0.139310\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 17604.2i − 0.733552i
\(833\) 21836.3i 0.908265i
\(834\) 0 0
\(835\) 0 0
\(836\) 28282.3 1.17005
\(837\) 0 0
\(838\) − 42918.2i − 1.76919i
\(839\) −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
\(842\) 6850.75i 0.280395i
\(843\) 0 0
\(844\) 30780.8 1.25535
\(845\) 0 0
\(846\) 0 0
\(847\) 10507.8i 0.426273i
\(848\) − 3823.96i − 0.154853i
\(849\) 0 0
\(850\) 0 0
\(851\) 37614.7 1.51517
\(852\) 0 0
\(853\) 20201.5i 0.810886i 0.914120 + 0.405443i \(0.132883\pi\)
−0.914120 + 0.405443i \(0.867117\pi\)
\(854\) 1245.45 0.0499046
\(855\) 0 0
\(856\) 16615.4 0.663436
\(857\) − 4551.65i − 0.181425i −0.995877 0.0907126i \(-0.971086\pi\)
0.995877 0.0907126i \(-0.0289145\pi\)
\(858\) 0 0
\(859\) −11962.6 −0.475154 −0.237577 0.971369i \(-0.576353\pi\)
−0.237577 + 0.971369i \(0.576353\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 51649.3i 2.04081i
\(863\) − 7164.61i − 0.282603i −0.989967 0.141301i \(-0.954871\pi\)
0.989967 0.141301i \(-0.0451287\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −66412.0 −2.60597
\(867\) 0 0
\(868\) − 17608.2i − 0.688549i
\(869\) 12065.9 0.471012
\(870\) 0 0
\(871\) 16082.4 0.625640
\(872\) − 29291.0i − 1.13752i
\(873\) 0 0
\(874\) −26266.2 −1.01655
\(875\) 0 0
\(876\) 0 0
\(877\) − 19218.7i − 0.739987i −0.929034 0.369994i \(-0.879360\pi\)
0.929034 0.369994i \(-0.120640\pi\)
\(878\) − 7324.56i − 0.281540i
\(879\) 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
\(883\) 44656.7i 1.70194i 0.525211 + 0.850972i \(0.323986\pi\)
−0.525211 + 0.850972i \(0.676014\pi\)
\(884\) −17357.7 −0.660410
\(885\) 0 0
\(886\) 17464.3 0.662218
\(887\) − 14975.2i − 0.566873i −0.958991 0.283437i \(-0.908525\pi\)
0.958991 0.283437i \(-0.0914745\pi\)
\(888\) 0 0
\(889\) −1570.82 −0.0592616
\(890\) 0 0
\(891\) 0 0
\(892\) − 49692.9i − 1.86529i
\(893\) 10427.0i 0.390735i
\(894\) 0 0
\(895\) 0 0
\(896\) 9941.33 0.370666
\(897\) 0 0
\(898\) 17771.7i 0.660413i
\(899\) −64156.8 −2.38015
\(900\) 0 0
\(901\) −14782.0 −0.546571
\(902\) 44136.7i 1.62926i
\(903\) 0 0
\(904\) 201.390 0.00740943
\(905\) 0 0
\(906\) 0 0
\(907\) − 14818.1i − 0.542479i −0.962512 0.271240i \(-0.912566\pi\)
0.962512 0.271240i \(-0.0874336\pi\)
\(908\) − 30935.6i − 1.13065i
\(909\) 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
\(913\) 27016.4i 0.979312i
\(914\) −63224.2 −2.28804
\(915\) 0 0
\(916\) −21934.0 −0.791179
\(917\) − 14151.4i − 0.509618i
\(918\) 0 0
\(919\) −28878.9 −1.03659 −0.518296 0.855201i \(-0.673434\pi\)
−0.518296 + 0.855201i \(0.673434\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 73108.4i − 2.61139i
\(923\) − 19182.9i − 0.684089i
\(924\) 0 0
\(925\) 0 0
\(926\) −14803.4 −0.525344
\(927\) 0 0
\(928\) − 47985.0i − 1.69740i
\(929\) 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
\(932\) − 2860.23i − 0.100526i
\(933\) 0 0
\(934\) 11281.3 0.395220
\(935\) 0 0
\(936\) 0 0
\(937\) − 21063.8i − 0.734391i −0.930144 0.367195i \(-0.880318\pi\)
0.930144 0.367195i \(-0.119682\pi\)
\(938\) 17179.7i 0.598014i
\(939\) 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
\(943\) − 24500.4i − 0.846069i
\(944\) −5904.27 −0.203567
\(945\) 0 0
\(946\) −114028. −3.91901
\(947\) 29978.3i 1.02868i 0.857585 + 0.514342i \(0.171964\pi\)
−0.857585 + 0.514342i \(0.828036\pi\)
\(948\) 0 0
\(949\) −18382.7 −0.628797
\(950\) 0 0
\(951\) 0 0
\(952\) − 6062.20i − 0.206383i
\(953\) 5782.65i 0.196556i 0.995159 + 0.0982782i \(0.0313335\pi\)
−0.995159 + 0.0982782i \(0.968666\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −30519.2 −1.03249
\(957\) 0 0
\(958\) 38559.2i 1.30041i
\(959\) −12649.3 −0.425930
\(960\) 0 0
\(961\) 55207.7 1.85317
\(962\) − 24654.2i − 0.826281i
\(963\) 0 0
\(964\) −47401.6 −1.58372
\(965\) 0 0
\(966\) 0 0
\(967\) − 26119.2i − 0.868600i −0.900768 0.434300i \(-0.856996\pi\)
0.900768 0.434300i \(-0.143004\pi\)
\(968\) 35837.0i 1.18992i
\(969\) 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
\(973\) − 9075.34i − 0.299016i
\(974\) −68013.1 −2.23746
\(975\) 0 0
\(976\) −978.894 −0.0321041
\(977\) 45902.4i 1.50312i 0.659666 + 0.751559i \(0.270698\pi\)
−0.659666 + 0.751559i \(0.729302\pi\)
\(978\) 0 0
\(979\) −35081.6 −1.14526
\(980\) 0 0
\(981\) 0 0
\(982\) − 7879.14i − 0.256042i
\(983\) 13416.5i 0.435321i 0.976025 + 0.217661i \(0.0698426\pi\)
−0.976025 + 0.217661i \(0.930157\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −67559.2 −2.18207
\(987\) 0 0
\(988\) 10290.1i 0.331349i
\(989\) 63297.4 2.03513
\(990\) 0 0
\(991\) 3806.66 0.122021 0.0610104 0.998137i \(-0.480568\pi\)
0.0610104 + 0.998137i \(0.480568\pi\)
\(992\) 63573.2i 2.03473i
\(993\) 0 0
\(994\) 20491.8 0.653883
\(995\) 0 0
\(996\) 0 0
\(997\) − 22523.8i − 0.715483i −0.933821 0.357742i \(-0.883547\pi\)
0.933821 0.357742i \(-0.116453\pi\)
\(998\) 52325.7i 1.65966i
\(999\) 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.b.k.649.2 6
3.2 odd 2 675.4.b.l.649.5 6
5.2 odd 4 675.4.a.q.1.3 3
5.3 odd 4 135.4.a.g.1.1 yes 3
5.4 even 2 inner 675.4.b.k.649.5 6
15.2 even 4 675.4.a.r.1.1 3
15.8 even 4 135.4.a.f.1.3 3
15.14 odd 2 675.4.b.l.649.2 6
20.3 even 4 2160.4.a.be.1.3 3
45.13 odd 12 405.4.e.r.136.3 6
45.23 even 12 405.4.e.t.136.1 6
45.38 even 12 405.4.e.t.271.1 6
45.43 odd 12 405.4.e.r.271.3 6
60.23 odd 4 2160.4.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.3 3 15.8 even 4
135.4.a.g.1.1 yes 3 5.3 odd 4
405.4.e.r.136.3 6 45.13 odd 12
405.4.e.r.271.3 6 45.43 odd 12
405.4.e.t.136.1 6 45.23 even 12
405.4.e.t.271.1 6 45.38 even 12
675.4.a.q.1.3 3 5.2 odd 4
675.4.a.r.1.1 3 15.2 even 4
675.4.b.k.649.2 6 1.1 even 1 trivial
675.4.b.k.649.5 6 5.4 even 2 inner
675.4.b.l.649.2 6 15.14 odd 2
675.4.b.l.649.5 6 3.2 odd 2
2160.4.a.be.1.3 3 20.3 even 4
2160.4.a.bm.1.3 3 60.23 odd 4