Properties

Label 507.4.b.k.337.17
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.17
Root \(5.39246i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.2

$q$-expansion

\(f(q)\) \(=\) \(q+4.83750i q^{2} +3.00000 q^{3} -15.4014 q^{4} -21.1983i q^{5} +14.5125i q^{6} +16.2806i q^{7} -35.8043i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.83750i q^{2} +3.00000 q^{3} -15.4014 q^{4} -21.1983i q^{5} +14.5125i q^{6} +16.2806i q^{7} -35.8043i q^{8} +9.00000 q^{9} +102.547 q^{10} +30.7532i q^{11} -46.2042 q^{12} -78.7575 q^{14} -63.5948i q^{15} +49.9922 q^{16} -46.2371 q^{17} +43.5375i q^{18} -144.865i q^{19} +326.483i q^{20} +48.8418i q^{21} -148.769 q^{22} -8.38045 q^{23} -107.413i q^{24} -324.366 q^{25} +27.0000 q^{27} -250.744i q^{28} -242.958 q^{29} +307.640 q^{30} -87.9353i q^{31} -44.5973i q^{32} +92.2597i q^{33} -223.672i q^{34} +345.121 q^{35} -138.613 q^{36} -49.6950i q^{37} +700.783 q^{38} -758.990 q^{40} +107.947i q^{41} -236.272 q^{42} +35.4166 q^{43} -473.643i q^{44} -190.784i q^{45} -40.5404i q^{46} -374.815i q^{47} +149.977 q^{48} +77.9418 q^{49} -1569.12i q^{50} -138.711 q^{51} -348.583 q^{53} +130.613i q^{54} +651.915 q^{55} +582.916 q^{56} -434.594i q^{57} -1175.31i q^{58} -679.430i q^{59} +979.450i q^{60} -230.403 q^{61} +425.387 q^{62} +146.525i q^{63} +615.677 q^{64} -446.306 q^{66} -295.642i q^{67} +712.117 q^{68} -25.1413 q^{69} +1669.52i q^{70} +329.215i q^{71} -322.239i q^{72} -48.9973i q^{73} +240.399 q^{74} -973.099 q^{75} +2231.12i q^{76} -500.681 q^{77} -107.942 q^{79} -1059.75i q^{80} +81.0000 q^{81} -522.192 q^{82} +515.654i q^{83} -752.233i q^{84} +980.147i q^{85} +171.328i q^{86} -728.874 q^{87} +1101.10 q^{88} -984.453i q^{89} +922.919 q^{90} +129.071 q^{92} -263.806i q^{93} +1813.17 q^{94} -3070.88 q^{95} -133.792i q^{96} +487.072i q^{97} +377.043i q^{98} +276.779i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.83750i 1.71031i 0.518368 + 0.855157i \(0.326540\pi\)
−0.518368 + 0.855157i \(0.673460\pi\)
\(3\) 3.00000 0.577350
\(4\) −15.4014 −1.92518
\(5\) − 21.1983i − 1.89603i −0.318225 0.948015i \(-0.603087\pi\)
0.318225 0.948015i \(-0.396913\pi\)
\(6\) 14.5125i 0.987451i
\(7\) 16.2806i 0.879070i 0.898225 + 0.439535i \(0.144857\pi\)
−0.898225 + 0.439535i \(0.855143\pi\)
\(8\) − 35.8043i − 1.58234i
\(9\) 9.00000 0.333333
\(10\) 102.547 3.24281
\(11\) 30.7532i 0.842950i 0.906840 + 0.421475i \(0.138487\pi\)
−0.906840 + 0.421475i \(0.861513\pi\)
\(12\) −46.2042 −1.11150
\(13\) 0 0
\(14\) −78.7575 −1.50349
\(15\) − 63.5948i − 1.09467i
\(16\) 49.9922 0.781128
\(17\) −46.2371 −0.659656 −0.329828 0.944041i \(-0.606991\pi\)
−0.329828 + 0.944041i \(0.606991\pi\)
\(18\) 43.5375i 0.570105i
\(19\) − 144.865i − 1.74917i −0.484873 0.874585i \(-0.661134\pi\)
0.484873 0.874585i \(-0.338866\pi\)
\(20\) 326.483i 3.65019i
\(21\) 48.8418i 0.507531i
\(22\) −148.769 −1.44171
\(23\) −8.38045 −0.0759758 −0.0379879 0.999278i \(-0.512095\pi\)
−0.0379879 + 0.999278i \(0.512095\pi\)
\(24\) − 107.413i − 0.913566i
\(25\) −324.366 −2.59493
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 250.744i − 1.69237i
\(29\) −242.958 −1.55573 −0.777865 0.628431i \(-0.783697\pi\)
−0.777865 + 0.628431i \(0.783697\pi\)
\(30\) 307.640 1.87224
\(31\) − 87.9353i − 0.509472i −0.967011 0.254736i \(-0.918011\pi\)
0.967011 0.254736i \(-0.0819886\pi\)
\(32\) − 44.5973i − 0.246368i
\(33\) 92.2597i 0.486677i
\(34\) − 223.672i − 1.12822i
\(35\) 345.121 1.66674
\(36\) −138.613 −0.641726
\(37\) − 49.6950i − 0.220806i −0.993887 0.110403i \(-0.964786\pi\)
0.993887 0.110403i \(-0.0352141\pi\)
\(38\) 700.783 2.99163
\(39\) 0 0
\(40\) −758.990 −3.00017
\(41\) 107.947i 0.411181i 0.978638 + 0.205591i \(0.0659116\pi\)
−0.978638 + 0.205591i \(0.934088\pi\)
\(42\) −236.272 −0.868039
\(43\) 35.4166 0.125604 0.0628021 0.998026i \(-0.479996\pi\)
0.0628021 + 0.998026i \(0.479996\pi\)
\(44\) − 473.643i − 1.62283i
\(45\) − 190.784i − 0.632010i
\(46\) − 40.5404i − 0.129943i
\(47\) − 374.815i − 1.16324i −0.813460 0.581621i \(-0.802419\pi\)
0.813460 0.581621i \(-0.197581\pi\)
\(48\) 149.977 0.450985
\(49\) 77.9418 0.227236
\(50\) − 1569.12i − 4.43815i
\(51\) −138.711 −0.380853
\(52\) 0 0
\(53\) −348.583 −0.903426 −0.451713 0.892163i \(-0.649187\pi\)
−0.451713 + 0.892163i \(0.649187\pi\)
\(54\) 130.613i 0.329150i
\(55\) 651.915 1.59826
\(56\) 582.916 1.39099
\(57\) − 434.594i − 1.00988i
\(58\) − 1175.31i − 2.66079i
\(59\) − 679.430i − 1.49923i −0.661877 0.749613i \(-0.730240\pi\)
0.661877 0.749613i \(-0.269760\pi\)
\(60\) 979.450i 2.10744i
\(61\) −230.403 −0.483608 −0.241804 0.970325i \(-0.577739\pi\)
−0.241804 + 0.970325i \(0.577739\pi\)
\(62\) 425.387 0.871358
\(63\) 146.525i 0.293023i
\(64\) 615.677 1.20249
\(65\) 0 0
\(66\) −446.306 −0.832372
\(67\) − 295.642i − 0.539082i −0.962989 0.269541i \(-0.913128\pi\)
0.962989 0.269541i \(-0.0868719\pi\)
\(68\) 712.117 1.26995
\(69\) −25.1413 −0.0438647
\(70\) 1669.52i 2.85066i
\(71\) 329.215i 0.550290i 0.961403 + 0.275145i \(0.0887259\pi\)
−0.961403 + 0.275145i \(0.911274\pi\)
\(72\) − 322.239i − 0.527448i
\(73\) − 48.9973i − 0.0785575i −0.999228 0.0392787i \(-0.987494\pi\)
0.999228 0.0392787i \(-0.0125060\pi\)
\(74\) 240.399 0.377647
\(75\) −973.099 −1.49818
\(76\) 2231.12i 3.36746i
\(77\) −500.681 −0.741012
\(78\) 0 0
\(79\) −107.942 −0.153727 −0.0768636 0.997042i \(-0.524491\pi\)
−0.0768636 + 0.997042i \(0.524491\pi\)
\(80\) − 1059.75i − 1.48104i
\(81\) 81.0000 0.111111
\(82\) −522.192 −0.703250
\(83\) 515.654i 0.681932i 0.940076 + 0.340966i \(0.110754\pi\)
−0.940076 + 0.340966i \(0.889246\pi\)
\(84\) − 752.233i − 0.977088i
\(85\) 980.147i 1.25073i
\(86\) 171.328i 0.214823i
\(87\) −728.874 −0.898201
\(88\) 1101.10 1.33384
\(89\) − 984.453i − 1.17249i −0.810133 0.586246i \(-0.800605\pi\)
0.810133 0.586246i \(-0.199395\pi\)
\(90\) 922.919 1.08094
\(91\) 0 0
\(92\) 129.071 0.146267
\(93\) − 263.806i − 0.294144i
\(94\) 1813.17 1.98951
\(95\) −3070.88 −3.31648
\(96\) − 133.792i − 0.142241i
\(97\) 487.072i 0.509842i 0.966962 + 0.254921i \(0.0820494\pi\)
−0.966962 + 0.254921i \(0.917951\pi\)
\(98\) 377.043i 0.388644i
\(99\) 276.779i 0.280983i
\(100\) 4995.70 4.99570
\(101\) −766.375 −0.755021 −0.377511 0.926005i \(-0.623220\pi\)
−0.377511 + 0.926005i \(0.623220\pi\)
\(102\) − 671.016i − 0.651378i
\(103\) −1229.87 −1.17653 −0.588266 0.808668i \(-0.700189\pi\)
−0.588266 + 0.808668i \(0.700189\pi\)
\(104\) 0 0
\(105\) 1035.36 0.962295
\(106\) − 1686.27i − 1.54514i
\(107\) 76.8261 0.0694117 0.0347059 0.999398i \(-0.488951\pi\)
0.0347059 + 0.999398i \(0.488951\pi\)
\(108\) −415.838 −0.370500
\(109\) − 626.001i − 0.550092i −0.961431 0.275046i \(-0.911307\pi\)
0.961431 0.275046i \(-0.0886930\pi\)
\(110\) 3153.64i 2.73353i
\(111\) − 149.085i − 0.127482i
\(112\) 813.904i 0.686667i
\(113\) −1343.21 −1.11822 −0.559108 0.829095i \(-0.688856\pi\)
−0.559108 + 0.829095i \(0.688856\pi\)
\(114\) 2102.35 1.72722
\(115\) 177.651i 0.144052i
\(116\) 3741.90 2.99506
\(117\) 0 0
\(118\) 3286.74 2.56415
\(119\) − 752.769i − 0.579884i
\(120\) −2276.97 −1.73215
\(121\) 385.238 0.289435
\(122\) − 1114.58i − 0.827122i
\(123\) 323.840i 0.237396i
\(124\) 1354.33i 0.980824i
\(125\) 4226.22i 3.02404i
\(126\) −708.817 −0.501162
\(127\) 2146.69 1.49990 0.749952 0.661492i \(-0.230077\pi\)
0.749952 + 0.661492i \(0.230077\pi\)
\(128\) 2621.56i 1.81028i
\(129\) 106.250 0.0725177
\(130\) 0 0
\(131\) −798.626 −0.532644 −0.266322 0.963884i \(-0.585808\pi\)
−0.266322 + 0.963884i \(0.585808\pi\)
\(132\) − 1420.93i − 0.936940i
\(133\) 2358.48 1.53764
\(134\) 1430.17 0.921999
\(135\) − 572.353i − 0.364891i
\(136\) 1655.49i 1.04380i
\(137\) − 601.153i − 0.374891i −0.982275 0.187445i \(-0.939979\pi\)
0.982275 0.187445i \(-0.0600207\pi\)
\(138\) − 121.621i − 0.0750224i
\(139\) −2134.18 −1.30229 −0.651146 0.758953i \(-0.725711\pi\)
−0.651146 + 0.758953i \(0.725711\pi\)
\(140\) −5315.35 −3.20878
\(141\) − 1124.44i − 0.671598i
\(142\) −1592.58 −0.941169
\(143\) 0 0
\(144\) 449.930 0.260376
\(145\) 5150.29i 2.94971i
\(146\) 237.024 0.134358
\(147\) 233.825 0.131194
\(148\) 765.373i 0.425090i
\(149\) − 3439.93i − 1.89134i −0.325127 0.945670i \(-0.605407\pi\)
0.325127 0.945670i \(-0.394593\pi\)
\(150\) − 4707.37i − 2.56237i
\(151\) 2224.04i 1.19861i 0.800521 + 0.599305i \(0.204556\pi\)
−0.800521 + 0.599305i \(0.795444\pi\)
\(152\) −5186.78 −2.76779
\(153\) −416.134 −0.219885
\(154\) − 2422.05i − 1.26736i
\(155\) −1864.07 −0.965975
\(156\) 0 0
\(157\) 2465.32 1.25321 0.626604 0.779338i \(-0.284444\pi\)
0.626604 + 0.779338i \(0.284444\pi\)
\(158\) − 522.171i − 0.262922i
\(159\) −1045.75 −0.521593
\(160\) −945.386 −0.467121
\(161\) − 136.439i − 0.0667881i
\(162\) 391.838i 0.190035i
\(163\) − 243.565i − 0.117040i −0.998286 0.0585199i \(-0.981362\pi\)
0.998286 0.0585199i \(-0.0186381\pi\)
\(164\) − 1662.53i − 0.791597i
\(165\) 1955.75 0.922755
\(166\) −2494.47 −1.16632
\(167\) 409.099i 0.189563i 0.995498 + 0.0947814i \(0.0302152\pi\)
−0.995498 + 0.0947814i \(0.969785\pi\)
\(168\) 1748.75 0.803089
\(169\) 0 0
\(170\) −4741.46 −2.13914
\(171\) − 1303.78i − 0.583056i
\(172\) −545.466 −0.241810
\(173\) 2618.97 1.15096 0.575481 0.817815i \(-0.304815\pi\)
0.575481 + 0.817815i \(0.304815\pi\)
\(174\) − 3525.93i − 1.53621i
\(175\) − 5280.88i − 2.28113i
\(176\) 1537.42i 0.658452i
\(177\) − 2038.29i − 0.865578i
\(178\) 4762.29 2.00533
\(179\) −163.311 −0.0681925 −0.0340963 0.999419i \(-0.510855\pi\)
−0.0340963 + 0.999419i \(0.510855\pi\)
\(180\) 2938.35i 1.21673i
\(181\) 3313.07 1.36054 0.680272 0.732960i \(-0.261862\pi\)
0.680272 + 0.732960i \(0.261862\pi\)
\(182\) 0 0
\(183\) −691.209 −0.279211
\(184\) 300.056i 0.120220i
\(185\) −1053.45 −0.418654
\(186\) 1276.16 0.503079
\(187\) − 1421.94i − 0.556057i
\(188\) 5772.68i 2.23945i
\(189\) 439.576i 0.169177i
\(190\) − 14855.4i − 5.67222i
\(191\) −4281.90 −1.62213 −0.811066 0.584954i \(-0.801112\pi\)
−0.811066 + 0.584954i \(0.801112\pi\)
\(192\) 1847.03 0.694261
\(193\) 1877.33i 0.700171i 0.936718 + 0.350086i \(0.113848\pi\)
−0.936718 + 0.350086i \(0.886152\pi\)
\(194\) −2356.21 −0.871990
\(195\) 0 0
\(196\) −1200.41 −0.437468
\(197\) 1991.26i 0.720158i 0.932922 + 0.360079i \(0.117250\pi\)
−0.932922 + 0.360079i \(0.882750\pi\)
\(198\) −1338.92 −0.480570
\(199\) −1345.05 −0.479137 −0.239568 0.970879i \(-0.577006\pi\)
−0.239568 + 0.970879i \(0.577006\pi\)
\(200\) 11613.7i 4.10607i
\(201\) − 886.927i − 0.311239i
\(202\) − 3707.34i − 1.29132i
\(203\) − 3955.51i − 1.36760i
\(204\) 2136.35 0.733208
\(205\) 2288.28 0.779612
\(206\) − 5949.50i − 2.01224i
\(207\) −75.4240 −0.0253253
\(208\) 0 0
\(209\) 4455.05 1.47446
\(210\) 5008.56i 1.64583i
\(211\) −288.763 −0.0942147 −0.0471073 0.998890i \(-0.515000\pi\)
−0.0471073 + 0.998890i \(0.515000\pi\)
\(212\) 5368.67 1.73925
\(213\) 987.644i 0.317710i
\(214\) 371.646i 0.118716i
\(215\) − 750.771i − 0.238150i
\(216\) − 966.717i − 0.304522i
\(217\) 1431.64 0.447862
\(218\) 3028.28 0.940830
\(219\) − 146.992i − 0.0453552i
\(220\) −10040.4 −3.07693
\(221\) 0 0
\(222\) 721.198 0.218035
\(223\) 1798.41i 0.540046i 0.962854 + 0.270023i \(0.0870314\pi\)
−0.962854 + 0.270023i \(0.912969\pi\)
\(224\) 726.072 0.216575
\(225\) −2919.30 −0.864977
\(226\) − 6497.77i − 1.91250i
\(227\) − 2486.10i − 0.726909i −0.931612 0.363454i \(-0.881597\pi\)
0.931612 0.363454i \(-0.118403\pi\)
\(228\) 6693.36i 1.94420i
\(229\) 6074.71i 1.75296i 0.481438 + 0.876480i \(0.340115\pi\)
−0.481438 + 0.876480i \(0.659885\pi\)
\(230\) −859.386 −0.246375
\(231\) −1502.04 −0.427824
\(232\) 8698.95i 2.46170i
\(233\) −6367.16 −1.79024 −0.895121 0.445822i \(-0.852911\pi\)
−0.895121 + 0.445822i \(0.852911\pi\)
\(234\) 0 0
\(235\) −7945.42 −2.20554
\(236\) 10464.2i 2.88627i
\(237\) −323.827 −0.0887544
\(238\) 3641.52 0.991784
\(239\) 1886.43i 0.510556i 0.966868 + 0.255278i \(0.0821670\pi\)
−0.966868 + 0.255278i \(0.917833\pi\)
\(240\) − 3179.24i − 0.855081i
\(241\) 5847.91i 1.56306i 0.623869 + 0.781529i \(0.285560\pi\)
−0.623869 + 0.781529i \(0.714440\pi\)
\(242\) 1863.59i 0.495025i
\(243\) 243.000 0.0641500
\(244\) 3548.53 0.931031
\(245\) − 1652.23i − 0.430845i
\(246\) −1566.58 −0.406021
\(247\) 0 0
\(248\) −3148.46 −0.806160
\(249\) 1546.96i 0.393713i
\(250\) −20444.3 −5.17205
\(251\) 2388.65 0.600678 0.300339 0.953832i \(-0.402900\pi\)
0.300339 + 0.953832i \(0.402900\pi\)
\(252\) − 2256.70i − 0.564122i
\(253\) − 257.726i − 0.0640438i
\(254\) 10384.6i 2.56531i
\(255\) 2940.44i 0.722108i
\(256\) −7756.38 −1.89365
\(257\) −4447.21 −1.07941 −0.539707 0.841853i \(-0.681465\pi\)
−0.539707 + 0.841853i \(0.681465\pi\)
\(258\) 513.984i 0.124028i
\(259\) 809.064 0.194104
\(260\) 0 0
\(261\) −2186.62 −0.518577
\(262\) − 3863.36i − 0.910988i
\(263\) 7590.87 1.77975 0.889873 0.456208i \(-0.150793\pi\)
0.889873 + 0.456208i \(0.150793\pi\)
\(264\) 3303.30 0.770091
\(265\) 7389.36i 1.71292i
\(266\) 11409.2i 2.62985i
\(267\) − 2953.36i − 0.676939i
\(268\) 4553.31i 1.03783i
\(269\) −557.911 −0.126455 −0.0632275 0.997999i \(-0.520139\pi\)
−0.0632275 + 0.997999i \(0.520139\pi\)
\(270\) 2768.76 0.624079
\(271\) − 2707.41i − 0.606877i −0.952851 0.303439i \(-0.901865\pi\)
0.952851 0.303439i \(-0.0981347\pi\)
\(272\) −2311.50 −0.515276
\(273\) 0 0
\(274\) 2908.08 0.641181
\(275\) − 9975.31i − 2.18740i
\(276\) 387.212 0.0844472
\(277\) 1231.35 0.267092 0.133546 0.991043i \(-0.457364\pi\)
0.133546 + 0.991043i \(0.457364\pi\)
\(278\) − 10324.1i − 2.22733i
\(279\) − 791.417i − 0.169824i
\(280\) − 12356.8i − 2.63736i
\(281\) − 4601.44i − 0.976864i −0.872602 0.488432i \(-0.837569\pi\)
0.872602 0.488432i \(-0.162431\pi\)
\(282\) 5439.50 1.14864
\(283\) 54.5677 0.0114619 0.00573094 0.999984i \(-0.498176\pi\)
0.00573094 + 0.999984i \(0.498176\pi\)
\(284\) − 5070.37i − 1.05941i
\(285\) −9212.63 −1.91477
\(286\) 0 0
\(287\) −1757.44 −0.361457
\(288\) − 401.376i − 0.0821226i
\(289\) −2775.13 −0.564854
\(290\) −24914.5 −5.04494
\(291\) 1461.22i 0.294357i
\(292\) 754.627i 0.151237i
\(293\) 4745.24i 0.946144i 0.881024 + 0.473072i \(0.156855\pi\)
−0.881024 + 0.473072i \(0.843145\pi\)
\(294\) 1131.13i 0.224384i
\(295\) −14402.7 −2.84258
\(296\) −1779.30 −0.349390
\(297\) 830.337i 0.162226i
\(298\) 16640.6 3.23479
\(299\) 0 0
\(300\) 14987.1 2.88427
\(301\) 576.604i 0.110415i
\(302\) −10758.8 −2.05000
\(303\) −2299.13 −0.435912
\(304\) − 7242.10i − 1.36633i
\(305\) 4884.15i 0.916936i
\(306\) − 2013.05i − 0.376073i
\(307\) 1028.82i 0.191264i 0.995417 + 0.0956320i \(0.0304872\pi\)
−0.995417 + 0.0956320i \(0.969513\pi\)
\(308\) 7711.20 1.42658
\(309\) −3689.61 −0.679271
\(310\) − 9017.46i − 1.65212i
\(311\) 2437.84 0.444492 0.222246 0.974991i \(-0.428661\pi\)
0.222246 + 0.974991i \(0.428661\pi\)
\(312\) 0 0
\(313\) 7934.20 1.43280 0.716402 0.697688i \(-0.245788\pi\)
0.716402 + 0.697688i \(0.245788\pi\)
\(314\) 11926.0i 2.14338i
\(315\) 3106.09 0.555581
\(316\) 1662.46 0.295952
\(317\) − 6942.72i − 1.23010i −0.788488 0.615050i \(-0.789136\pi\)
0.788488 0.615050i \(-0.210864\pi\)
\(318\) − 5058.81i − 0.892089i
\(319\) − 7471.75i − 1.31140i
\(320\) − 13051.3i − 2.27997i
\(321\) 230.478 0.0400749
\(322\) 660.023 0.114229
\(323\) 6698.12i 1.15385i
\(324\) −1247.51 −0.213909
\(325\) 0 0
\(326\) 1178.25 0.200175
\(327\) − 1878.00i − 0.317596i
\(328\) 3864.96 0.650630
\(329\) 6102.21 1.02257
\(330\) 9460.92i 1.57820i
\(331\) 5439.27i 0.903230i 0.892213 + 0.451615i \(0.149152\pi\)
−0.892213 + 0.451615i \(0.850848\pi\)
\(332\) − 7941.79i − 1.31284i
\(333\) − 447.255i − 0.0736018i
\(334\) −1979.01 −0.324212
\(335\) −6267.10 −1.02211
\(336\) 2441.71i 0.396447i
\(337\) 1663.87 0.268952 0.134476 0.990917i \(-0.457065\pi\)
0.134476 + 0.990917i \(0.457065\pi\)
\(338\) 0 0
\(339\) −4029.63 −0.645602
\(340\) − 15095.6i − 2.40787i
\(341\) 2704.29 0.429460
\(342\) 6307.04 0.997210
\(343\) 6853.19i 1.07883i
\(344\) − 1268.07i − 0.198749i
\(345\) 532.953i 0.0831687i
\(346\) 12669.3i 1.96851i
\(347\) 9268.10 1.43383 0.716913 0.697163i \(-0.245554\pi\)
0.716913 + 0.697163i \(0.245554\pi\)
\(348\) 11225.7 1.72920
\(349\) 5239.45i 0.803614i 0.915724 + 0.401807i \(0.131618\pi\)
−0.915724 + 0.401807i \(0.868382\pi\)
\(350\) 25546.3 3.90144
\(351\) 0 0
\(352\) 1371.51 0.207676
\(353\) − 1218.01i − 0.183649i −0.995775 0.0918244i \(-0.970730\pi\)
0.995775 0.0918244i \(-0.0292698\pi\)
\(354\) 9860.23 1.48041
\(355\) 6978.78 1.04337
\(356\) 15162.0i 2.25726i
\(357\) − 2258.31i − 0.334796i
\(358\) − 790.018i − 0.116631i
\(359\) 5316.06i 0.781534i 0.920490 + 0.390767i \(0.127790\pi\)
−0.920490 + 0.390767i \(0.872210\pi\)
\(360\) −6830.91 −1.00006
\(361\) −14126.7 −2.05959
\(362\) 16027.0i 2.32696i
\(363\) 1155.72 0.167106
\(364\) 0 0
\(365\) −1038.66 −0.148947
\(366\) − 3343.73i − 0.477539i
\(367\) −289.101 −0.0411197 −0.0205599 0.999789i \(-0.506545\pi\)
−0.0205599 + 0.999789i \(0.506545\pi\)
\(368\) −418.957 −0.0593469
\(369\) 971.520i 0.137060i
\(370\) − 5096.05i − 0.716030i
\(371\) − 5675.15i − 0.794175i
\(372\) 4062.98i 0.566279i
\(373\) 6163.98 0.855654 0.427827 0.903861i \(-0.359279\pi\)
0.427827 + 0.903861i \(0.359279\pi\)
\(374\) 6878.64 0.951032
\(375\) 12678.7i 1.74593i
\(376\) −13420.0 −1.84065
\(377\) 0 0
\(378\) −2126.45 −0.289346
\(379\) 2440.03i 0.330702i 0.986235 + 0.165351i \(0.0528756\pi\)
−0.986235 + 0.165351i \(0.947124\pi\)
\(380\) 47295.9 6.38481
\(381\) 6440.06 0.865970
\(382\) − 20713.7i − 2.77436i
\(383\) 6577.28i 0.877502i 0.898609 + 0.438751i \(0.144579\pi\)
−0.898609 + 0.438751i \(0.855421\pi\)
\(384\) 7864.68i 1.04516i
\(385\) 10613.6i 1.40498i
\(386\) −9081.58 −1.19751
\(387\) 318.750 0.0418681
\(388\) − 7501.59i − 0.981535i
\(389\) −4964.94 −0.647127 −0.323563 0.946206i \(-0.604881\pi\)
−0.323563 + 0.946206i \(0.604881\pi\)
\(390\) 0 0
\(391\) 387.488 0.0501179
\(392\) − 2790.65i − 0.359565i
\(393\) −2395.88 −0.307522
\(394\) −9632.70 −1.23170
\(395\) 2288.19i 0.291471i
\(396\) − 4262.79i − 0.540943i
\(397\) − 12073.9i − 1.52637i −0.646178 0.763187i \(-0.723634\pi\)
0.646178 0.763187i \(-0.276366\pi\)
\(398\) − 6506.69i − 0.819475i
\(399\) 7075.45 0.887758
\(400\) −16215.8 −2.02697
\(401\) − 4916.90i − 0.612315i −0.951981 0.306158i \(-0.900957\pi\)
0.951981 0.306158i \(-0.0990434\pi\)
\(402\) 4290.51 0.532316
\(403\) 0 0
\(404\) 11803.3 1.45355
\(405\) − 1717.06i − 0.210670i
\(406\) 19134.8 2.33902
\(407\) 1528.28 0.186128
\(408\) 4966.47i 0.602639i
\(409\) − 15350.9i − 1.85588i −0.372733 0.927939i \(-0.621579\pi\)
0.372733 0.927939i \(-0.378421\pi\)
\(410\) 11069.6i 1.33338i
\(411\) − 1803.46i − 0.216443i
\(412\) 18941.7 2.26503
\(413\) 11061.5 1.31792
\(414\) − 364.864i − 0.0433142i
\(415\) 10931.0 1.29296
\(416\) 0 0
\(417\) −6402.53 −0.751878
\(418\) 21551.3i 2.52179i
\(419\) −5488.58 −0.639939 −0.319970 0.947428i \(-0.603673\pi\)
−0.319970 + 0.947428i \(0.603673\pi\)
\(420\) −15946.0 −1.85259
\(421\) 927.681i 0.107393i 0.998557 + 0.0536964i \(0.0171003\pi\)
−0.998557 + 0.0536964i \(0.982900\pi\)
\(422\) − 1396.89i − 0.161137i
\(423\) − 3373.33i − 0.387747i
\(424\) 12480.8i 1.42953i
\(425\) 14997.8 1.71176
\(426\) −4777.73 −0.543384
\(427\) − 3751.10i − 0.425126i
\(428\) −1183.23 −0.133630
\(429\) 0 0
\(430\) 3631.85 0.407311
\(431\) − 11002.3i − 1.22961i −0.788678 0.614806i \(-0.789234\pi\)
0.788678 0.614806i \(-0.210766\pi\)
\(432\) 1349.79 0.150328
\(433\) 9596.76 1.06511 0.532553 0.846397i \(-0.321233\pi\)
0.532553 + 0.846397i \(0.321233\pi\)
\(434\) 6925.56i 0.765985i
\(435\) 15450.9i 1.70302i
\(436\) 9641.30i 1.05902i
\(437\) 1214.03i 0.132895i
\(438\) 711.073 0.0775716
\(439\) 10346.3 1.12483 0.562416 0.826854i \(-0.309872\pi\)
0.562416 + 0.826854i \(0.309872\pi\)
\(440\) − 23341.4i − 2.52899i
\(441\) 701.476 0.0757452
\(442\) 0 0
\(443\) 5650.38 0.606000 0.303000 0.952991i \(-0.402012\pi\)
0.303000 + 0.952991i \(0.402012\pi\)
\(444\) 2296.12i 0.245426i
\(445\) −20868.7 −2.22308
\(446\) −8699.80 −0.923649
\(447\) − 10319.8i − 1.09197i
\(448\) 10023.6i 1.05708i
\(449\) − 11987.8i − 1.25999i −0.776597 0.629997i \(-0.783056\pi\)
0.776597 0.629997i \(-0.216944\pi\)
\(450\) − 14122.1i − 1.47938i
\(451\) −3319.71 −0.346605
\(452\) 20687.3 2.15276
\(453\) 6672.13i 0.692018i
\(454\) 12026.5 1.24324
\(455\) 0 0
\(456\) −15560.3 −1.59798
\(457\) − 16437.6i − 1.68253i −0.540623 0.841265i \(-0.681811\pi\)
0.540623 0.841265i \(-0.318189\pi\)
\(458\) −29386.4 −2.99811
\(459\) −1248.40 −0.126951
\(460\) − 2736.08i − 0.277326i
\(461\) 8847.70i 0.893880i 0.894564 + 0.446940i \(0.147486\pi\)
−0.894564 + 0.446940i \(0.852514\pi\)
\(462\) − 7266.14i − 0.731713i
\(463\) − 10269.6i − 1.03081i −0.856945 0.515407i \(-0.827641\pi\)
0.856945 0.515407i \(-0.172359\pi\)
\(464\) −12146.0 −1.21523
\(465\) −5592.22 −0.557706
\(466\) − 30801.1i − 3.06188i
\(467\) −14730.4 −1.45962 −0.729811 0.683649i \(-0.760392\pi\)
−0.729811 + 0.683649i \(0.760392\pi\)
\(468\) 0 0
\(469\) 4813.24 0.473891
\(470\) − 38436.0i − 3.77217i
\(471\) 7395.95 0.723540
\(472\) −24326.6 −2.37229
\(473\) 1089.18i 0.105878i
\(474\) − 1566.51i − 0.151798i
\(475\) 46989.2i 4.53897i
\(476\) 11593.7i 1.11638i
\(477\) −3137.25 −0.301142
\(478\) −9125.59 −0.873211
\(479\) − 635.077i − 0.0605792i −0.999541 0.0302896i \(-0.990357\pi\)
0.999541 0.0302896i \(-0.00964295\pi\)
\(480\) −2836.16 −0.269692
\(481\) 0 0
\(482\) −28289.3 −2.67332
\(483\) − 409.316i − 0.0385601i
\(484\) −5933.22 −0.557214
\(485\) 10325.1 0.966675
\(486\) 1175.51i 0.109717i
\(487\) − 16611.1i − 1.54563i −0.634632 0.772815i \(-0.718848\pi\)
0.634632 0.772815i \(-0.281152\pi\)
\(488\) 8249.43i 0.765234i
\(489\) − 730.696i − 0.0675730i
\(490\) 7992.66 0.736881
\(491\) 2584.14 0.237516 0.118758 0.992923i \(-0.462109\pi\)
0.118758 + 0.992923i \(0.462109\pi\)
\(492\) − 4987.59i − 0.457029i
\(493\) 11233.7 1.02625
\(494\) 0 0
\(495\) 5867.24 0.532753
\(496\) − 4396.08i − 0.397963i
\(497\) −5359.82 −0.483744
\(498\) −7483.42 −0.673374
\(499\) 2183.88i 0.195919i 0.995190 + 0.0979597i \(0.0312316\pi\)
−0.995190 + 0.0979597i \(0.968768\pi\)
\(500\) − 65089.8i − 5.82180i
\(501\) 1227.30i 0.109444i
\(502\) 11555.1i 1.02735i
\(503\) −17214.9 −1.52600 −0.762998 0.646401i \(-0.776273\pi\)
−0.762998 + 0.646401i \(0.776273\pi\)
\(504\) 5246.25 0.463664
\(505\) 16245.8i 1.43154i
\(506\) 1246.75 0.109535
\(507\) 0 0
\(508\) −33062.0 −2.88758
\(509\) 20260.6i 1.76431i 0.470955 + 0.882157i \(0.343909\pi\)
−0.470955 + 0.882157i \(0.656091\pi\)
\(510\) −14224.4 −1.23503
\(511\) 797.705 0.0690575
\(512\) − 16549.0i − 1.42846i
\(513\) − 3911.34i − 0.336628i
\(514\) − 21513.4i − 1.84614i
\(515\) 26071.1i 2.23074i
\(516\) −1636.40 −0.139609
\(517\) 11526.8 0.980554
\(518\) 3913.85i 0.331978i
\(519\) 7856.90 0.664508
\(520\) 0 0
\(521\) −20731.5 −1.74331 −0.871653 0.490124i \(-0.836952\pi\)
−0.871653 + 0.490124i \(0.836952\pi\)
\(522\) − 10577.8i − 0.886930i
\(523\) −944.353 −0.0789554 −0.0394777 0.999220i \(-0.512569\pi\)
−0.0394777 + 0.999220i \(0.512569\pi\)
\(524\) 12300.0 1.02543
\(525\) − 15842.6i − 1.31701i
\(526\) 36720.9i 3.04393i
\(527\) 4065.87i 0.336076i
\(528\) 4612.27i 0.380158i
\(529\) −12096.8 −0.994228
\(530\) −35746.0 −2.92964
\(531\) − 6114.87i − 0.499742i
\(532\) −36324.0 −2.96023
\(533\) 0 0
\(534\) 14286.9 1.15778
\(535\) − 1628.58i − 0.131607i
\(536\) −10585.3 −0.853012
\(537\) −489.934 −0.0393710
\(538\) − 2698.89i − 0.216278i
\(539\) 2396.96i 0.191548i
\(540\) 8815.05i 0.702480i
\(541\) − 4883.06i − 0.388058i −0.980996 0.194029i \(-0.937844\pi\)
0.980996 0.194029i \(-0.0621556\pi\)
\(542\) 13097.1 1.03795
\(543\) 9939.21 0.785511
\(544\) 2062.05i 0.162518i
\(545\) −13270.1 −1.04299
\(546\) 0 0
\(547\) −16269.1 −1.27169 −0.635847 0.771815i \(-0.719349\pi\)
−0.635847 + 0.771815i \(0.719349\pi\)
\(548\) 9258.61i 0.721730i
\(549\) −2073.63 −0.161203
\(550\) 48255.6 3.74114
\(551\) 35196.0i 2.72124i
\(552\) 900.169i 0.0694090i
\(553\) − 1757.37i − 0.135137i
\(554\) 5956.63i 0.456811i
\(555\) −3160.34 −0.241710
\(556\) 32869.3 2.50714
\(557\) 15661.0i 1.19134i 0.803229 + 0.595670i \(0.203113\pi\)
−0.803229 + 0.595670i \(0.796887\pi\)
\(558\) 3828.48 0.290453
\(559\) 0 0
\(560\) 17253.3 1.30194
\(561\) − 4265.82i − 0.321040i
\(562\) 22259.5 1.67074
\(563\) 9915.18 0.742230 0.371115 0.928587i \(-0.378976\pi\)
0.371115 + 0.928587i \(0.378976\pi\)
\(564\) 17318.0i 1.29294i
\(565\) 28473.7i 2.12017i
\(566\) 263.971i 0.0196034i
\(567\) 1318.73i 0.0976745i
\(568\) 11787.3 0.870748
\(569\) 11299.1 0.832482 0.416241 0.909254i \(-0.363347\pi\)
0.416241 + 0.909254i \(0.363347\pi\)
\(570\) − 44566.1i − 3.27486i
\(571\) 17619.6 1.29134 0.645672 0.763615i \(-0.276577\pi\)
0.645672 + 0.763615i \(0.276577\pi\)
\(572\) 0 0
\(573\) −12845.7 −0.936539
\(574\) − 8501.61i − 0.618206i
\(575\) 2718.33 0.197152
\(576\) 5541.10 0.400832
\(577\) 22153.5i 1.59837i 0.601083 + 0.799186i \(0.294736\pi\)
−0.601083 + 0.799186i \(0.705264\pi\)
\(578\) − 13424.7i − 0.966078i
\(579\) 5631.99i 0.404244i
\(580\) − 79321.7i − 5.67872i
\(581\) −8395.15 −0.599466
\(582\) −7068.63 −0.503444
\(583\) − 10720.1i − 0.761543i
\(584\) −1754.31 −0.124305
\(585\) 0 0
\(586\) −22955.1 −1.61820
\(587\) 9891.28i 0.695497i 0.937588 + 0.347748i \(0.113054\pi\)
−0.937588 + 0.347748i \(0.886946\pi\)
\(588\) −3601.24 −0.252573
\(589\) −12738.7 −0.891153
\(590\) − 69673.3i − 4.86170i
\(591\) 5973.77i 0.415783i
\(592\) − 2484.36i − 0.172477i
\(593\) − 9746.23i − 0.674924i −0.941339 0.337462i \(-0.890432\pi\)
0.941339 0.337462i \(-0.109568\pi\)
\(594\) −4016.76 −0.277457
\(595\) −15957.4 −1.09948
\(596\) 52979.7i 3.64116i
\(597\) −4035.16 −0.276630
\(598\) 0 0
\(599\) −8460.53 −0.577109 −0.288554 0.957464i \(-0.593175\pi\)
−0.288554 + 0.957464i \(0.593175\pi\)
\(600\) 34841.2i 2.37064i
\(601\) 6792.99 0.461052 0.230526 0.973066i \(-0.425955\pi\)
0.230526 + 0.973066i \(0.425955\pi\)
\(602\) −2789.32 −0.188844
\(603\) − 2660.78i − 0.179694i
\(604\) − 34253.4i − 2.30754i
\(605\) − 8166.38i − 0.548778i
\(606\) − 11122.0i − 0.745546i
\(607\) 19073.4 1.27539 0.637697 0.770287i \(-0.279887\pi\)
0.637697 + 0.770287i \(0.279887\pi\)
\(608\) −6460.58 −0.430939
\(609\) − 11866.5i − 0.789582i
\(610\) −23627.1 −1.56825
\(611\) 0 0
\(612\) 6409.05 0.423318
\(613\) 14465.2i 0.953091i 0.879150 + 0.476545i \(0.158111\pi\)
−0.879150 + 0.476545i \(0.841889\pi\)
\(614\) −4976.93 −0.327122
\(615\) 6864.85 0.450109
\(616\) 17926.6i 1.17254i
\(617\) − 12897.3i − 0.841536i −0.907168 0.420768i \(-0.861761\pi\)
0.907168 0.420768i \(-0.138239\pi\)
\(618\) − 17848.5i − 1.16177i
\(619\) − 27735.8i − 1.80096i −0.434895 0.900481i \(-0.643214\pi\)
0.434895 0.900481i \(-0.356786\pi\)
\(620\) 28709.4 1.85967
\(621\) −226.272 −0.0146216
\(622\) 11793.0i 0.760222i
\(623\) 16027.5 1.03070
\(624\) 0 0
\(625\) 49042.7 3.13873
\(626\) 38381.7i 2.45055i
\(627\) 13365.2 0.851281
\(628\) −37969.3 −2.41265
\(629\) 2297.75i 0.145656i
\(630\) 15025.7i 0.950219i
\(631\) 28560.1i 1.80184i 0.433990 + 0.900918i \(0.357105\pi\)
−0.433990 + 0.900918i \(0.642895\pi\)
\(632\) 3864.80i 0.243249i
\(633\) −866.290 −0.0543949
\(634\) 33585.4 2.10386
\(635\) − 45506.1i − 2.84386i
\(636\) 16106.0 1.00416
\(637\) 0 0
\(638\) 36144.6 2.24291
\(639\) 2962.93i 0.183430i
\(640\) 55572.5 3.43234
\(641\) −1862.53 −0.114767 −0.0573833 0.998352i \(-0.518276\pi\)
−0.0573833 + 0.998352i \(0.518276\pi\)
\(642\) 1114.94i 0.0685407i
\(643\) − 29495.2i − 1.80899i −0.426487 0.904494i \(-0.640249\pi\)
0.426487 0.904494i \(-0.359751\pi\)
\(644\) 2101.35i 0.128579i
\(645\) − 2252.31i − 0.137496i
\(646\) −32402.2 −1.97345
\(647\) −27780.7 −1.68805 −0.844027 0.536301i \(-0.819821\pi\)
−0.844027 + 0.536301i \(0.819821\pi\)
\(648\) − 2900.15i − 0.175816i
\(649\) 20894.7 1.26377
\(650\) 0 0
\(651\) 4294.92 0.258573
\(652\) 3751.25i 0.225322i
\(653\) 276.678 0.0165808 0.00829039 0.999966i \(-0.497361\pi\)
0.00829039 + 0.999966i \(0.497361\pi\)
\(654\) 9084.84 0.543188
\(655\) 16929.5i 1.00991i
\(656\) 5396.49i 0.321185i
\(657\) − 440.975i − 0.0261858i
\(658\) 29519.5i 1.74892i
\(659\) 18114.9 1.07080 0.535398 0.844600i \(-0.320162\pi\)
0.535398 + 0.844600i \(0.320162\pi\)
\(660\) −30121.2 −1.77647
\(661\) 27094.4i 1.59433i 0.603764 + 0.797163i \(0.293667\pi\)
−0.603764 + 0.797163i \(0.706333\pi\)
\(662\) −26312.5 −1.54481
\(663\) 0 0
\(664\) 18462.6 1.07905
\(665\) − 49995.8i − 2.91542i
\(666\) 2163.60 0.125882
\(667\) 2036.10 0.118198
\(668\) − 6300.70i − 0.364942i
\(669\) 5395.23i 0.311796i
\(670\) − 30317.1i − 1.74814i
\(671\) − 7085.64i − 0.407657i
\(672\) 2178.22 0.125039
\(673\) −2410.31 −0.138054 −0.0690272 0.997615i \(-0.521990\pi\)
−0.0690272 + 0.997615i \(0.521990\pi\)
\(674\) 8048.99i 0.459993i
\(675\) −8757.89 −0.499395
\(676\) 0 0
\(677\) 12508.3 0.710091 0.355046 0.934849i \(-0.384465\pi\)
0.355046 + 0.934849i \(0.384465\pi\)
\(678\) − 19493.3i − 1.10418i
\(679\) −7929.83 −0.448187
\(680\) 35093.5 1.97908
\(681\) − 7458.30i − 0.419681i
\(682\) 13082.0i 0.734511i
\(683\) − 16793.8i − 0.940844i −0.882442 0.470422i \(-0.844102\pi\)
0.882442 0.470422i \(-0.155898\pi\)
\(684\) 20080.1i 1.12249i
\(685\) −12743.4 −0.710804
\(686\) −33152.3 −1.84513
\(687\) 18224.1i 1.01207i
\(688\) 1770.56 0.0981131
\(689\) 0 0
\(690\) −2578.16 −0.142245
\(691\) 26026.9i 1.43286i 0.697657 + 0.716432i \(0.254226\pi\)
−0.697657 + 0.716432i \(0.745774\pi\)
\(692\) −40335.8 −2.21580
\(693\) −4506.13 −0.247004
\(694\) 44834.4i 2.45229i
\(695\) 45240.8i 2.46918i
\(696\) 26096.9i 1.42126i
\(697\) − 4991.14i − 0.271238i
\(698\) −25345.8 −1.37443
\(699\) −19101.5 −1.03360
\(700\) 81333.0i 4.39157i
\(701\) −5079.08 −0.273658 −0.136829 0.990595i \(-0.543691\pi\)
−0.136829 + 0.990595i \(0.543691\pi\)
\(702\) 0 0
\(703\) −7199.04 −0.386226
\(704\) 18934.1i 1.01364i
\(705\) −23836.3 −1.27337
\(706\) 5892.11 0.314097
\(707\) − 12477.1i − 0.663717i
\(708\) 31392.6i 1.66639i
\(709\) − 2530.72i − 0.134052i −0.997751 0.0670262i \(-0.978649\pi\)
0.997751 0.0670262i \(-0.0213511\pi\)
\(710\) 33759.9i 1.78449i
\(711\) −971.480 −0.0512424
\(712\) −35247.7 −1.85529
\(713\) 736.937i 0.0387076i
\(714\) 10924.6 0.572607
\(715\) 0 0
\(716\) 2515.22 0.131283
\(717\) 5659.28i 0.294770i
\(718\) −25716.4 −1.33667
\(719\) −9503.67 −0.492944 −0.246472 0.969150i \(-0.579271\pi\)
−0.246472 + 0.969150i \(0.579271\pi\)
\(720\) − 9537.73i − 0.493681i
\(721\) − 20023.0i − 1.03425i
\(722\) − 68338.2i − 3.52255i
\(723\) 17543.7i 0.902432i
\(724\) −51026.0 −2.61929
\(725\) 78807.4 4.03701
\(726\) 5590.77i 0.285803i
\(727\) −33343.1 −1.70100 −0.850500 0.525975i \(-0.823700\pi\)
−0.850500 + 0.525975i \(0.823700\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 5024.50i − 0.254747i
\(731\) −1637.56 −0.0828556
\(732\) 10645.6 0.537531
\(733\) − 32288.1i − 1.62699i −0.581569 0.813497i \(-0.697561\pi\)
0.581569 0.813497i \(-0.302439\pi\)
\(734\) − 1398.53i − 0.0703277i
\(735\) − 4956.69i − 0.248749i
\(736\) 373.746i 0.0187180i
\(737\) 9091.96 0.454419
\(738\) −4699.73 −0.234417
\(739\) − 14168.8i − 0.705288i −0.935758 0.352644i \(-0.885283\pi\)
0.935758 0.352644i \(-0.114717\pi\)
\(740\) 16224.6 0.805983
\(741\) 0 0
\(742\) 27453.5 1.35829
\(743\) − 13248.6i − 0.654165i −0.944996 0.327083i \(-0.893934\pi\)
0.944996 0.327083i \(-0.106066\pi\)
\(744\) −9445.39 −0.465437
\(745\) −72920.5 −3.58604
\(746\) 29818.3i 1.46344i
\(747\) 4640.88i 0.227311i
\(748\) 21899.9i 1.07051i
\(749\) 1250.78i 0.0610178i
\(750\) −61333.0 −2.98609
\(751\) −5113.98 −0.248484 −0.124242 0.992252i \(-0.539650\pi\)
−0.124242 + 0.992252i \(0.539650\pi\)
\(752\) − 18737.8i − 0.908641i
\(753\) 7165.95 0.346802
\(754\) 0 0
\(755\) 47145.9 2.27260
\(756\) − 6770.10i − 0.325696i
\(757\) 27380.4 1.31461 0.657303 0.753626i \(-0.271697\pi\)
0.657303 + 0.753626i \(0.271697\pi\)
\(758\) −11803.6 −0.565604
\(759\) − 773.178i − 0.0369757i
\(760\) 109951.i 5.24781i
\(761\) − 10613.8i − 0.505586i −0.967520 0.252793i \(-0.918651\pi\)
0.967520 0.252793i \(-0.0813492\pi\)
\(762\) 31153.8i 1.48108i
\(763\) 10191.7 0.483569
\(764\) 65947.3 3.12289
\(765\) 8821.32i 0.416909i
\(766\) −31817.6 −1.50080
\(767\) 0 0
\(768\) −23269.2 −1.09330
\(769\) − 9172.22i − 0.430115i −0.976601 0.215058i \(-0.931006\pi\)
0.976601 0.215058i \(-0.0689939\pi\)
\(770\) −51343.2 −2.40296
\(771\) −13341.6 −0.623200
\(772\) − 28913.5i − 1.34795i
\(773\) − 14952.1i − 0.695718i −0.937547 0.347859i \(-0.886909\pi\)
0.937547 0.347859i \(-0.113091\pi\)
\(774\) 1541.95i 0.0716076i
\(775\) 28523.2i 1.32205i
\(776\) 17439.3 0.806745
\(777\) 2427.19 0.112066
\(778\) − 24017.9i − 1.10679i
\(779\) 15637.7 0.719226
\(780\) 0 0
\(781\) −10124.4 −0.463867
\(782\) 1874.47i 0.0857174i
\(783\) −6559.87 −0.299400
\(784\) 3896.48 0.177500
\(785\) − 52260.4i − 2.37612i
\(786\) − 11590.1i − 0.525959i
\(787\) − 2671.52i − 0.121003i −0.998168 0.0605015i \(-0.980730\pi\)
0.998168 0.0605015i \(-0.0192700\pi\)
\(788\) − 30668.1i − 1.38643i
\(789\) 22772.6 1.02754
\(790\) −11069.1 −0.498508
\(791\) − 21868.3i − 0.982991i
\(792\) 9909.89 0.444612
\(793\) 0 0
\(794\) 58407.4 2.61058
\(795\) 22168.1i 0.988957i
\(796\) 20715.7 0.922423
\(797\) 12949.7 0.575536 0.287768 0.957700i \(-0.407087\pi\)
0.287768 + 0.957700i \(0.407087\pi\)
\(798\) 34227.5i 1.51835i
\(799\) 17330.4i 0.767339i
\(800\) 14465.9i 0.639307i
\(801\) − 8860.08i − 0.390831i
\(802\) 23785.5 1.04725
\(803\) 1506.82 0.0662200
\(804\) 13659.9i 0.599190i
\(805\) −2892.27 −0.126632
\(806\) 0 0
\(807\) −1673.73 −0.0730089
\(808\) 27439.6i 1.19470i
\(809\) −11640.4 −0.505876 −0.252938 0.967482i \(-0.581397\pi\)
−0.252938 + 0.967482i \(0.581397\pi\)
\(810\) 8306.28 0.360312
\(811\) − 18135.3i − 0.785225i −0.919704 0.392613i \(-0.871571\pi\)
0.919704 0.392613i \(-0.128429\pi\)
\(812\) 60920.4i 2.63286i
\(813\) − 8122.24i − 0.350381i
\(814\) 7393.06i 0.318337i
\(815\) −5163.16 −0.221911
\(816\) −6934.49 −0.297495
\(817\) − 5130.61i − 0.219703i
\(818\) 74260.0 3.17413
\(819\) 0 0
\(820\) −35242.8 −1.50089
\(821\) 10789.1i 0.458640i 0.973351 + 0.229320i \(0.0736503\pi\)
−0.973351 + 0.229320i \(0.926350\pi\)
\(822\) 8724.24 0.370186
\(823\) 12284.7 0.520312 0.260156 0.965567i \(-0.416226\pi\)
0.260156 + 0.965567i \(0.416226\pi\)
\(824\) 44034.7i 1.86168i
\(825\) − 29925.9i − 1.26289i
\(826\) 53510.2i 2.25407i
\(827\) − 36077.2i − 1.51696i −0.651696 0.758480i \(-0.725942\pi\)
0.651696 0.758480i \(-0.274058\pi\)
\(828\) 1161.64 0.0487556
\(829\) 8861.83 0.371271 0.185636 0.982619i \(-0.440566\pi\)
0.185636 + 0.982619i \(0.440566\pi\)
\(830\) 52878.5i 2.21137i
\(831\) 3694.04 0.154205
\(832\) 0 0
\(833\) −3603.80 −0.149897
\(834\) − 30972.2i − 1.28595i
\(835\) 8672.18 0.359417
\(836\) −68614.1 −2.83860
\(837\) − 2374.25i − 0.0980480i
\(838\) − 26551.0i − 1.09450i
\(839\) − 5833.37i − 0.240036i −0.992772 0.120018i \(-0.961705\pi\)
0.992772 0.120018i \(-0.0382952\pi\)
\(840\) − 37070.4i − 1.52268i
\(841\) 34639.6 1.42030
\(842\) −4487.66 −0.183676
\(843\) − 13804.3i − 0.563993i
\(844\) 4447.37 0.181380
\(845\) 0 0
\(846\) 16318.5 0.663170
\(847\) 6271.92i 0.254434i
\(848\) −17426.4 −0.705692
\(849\) 163.703 0.00661752
\(850\) 72551.7i 2.92765i
\(851\) 416.466i 0.0167759i
\(852\) − 15211.1i − 0.611648i
\(853\) 28649.5i 1.14999i 0.818157 + 0.574995i \(0.194996\pi\)
−0.818157 + 0.574995i \(0.805004\pi\)
\(854\) 18146.0 0.727098
\(855\) −27637.9 −1.10549
\(856\) − 2750.71i − 0.109833i
\(857\) −32336.1 −1.28889 −0.644445 0.764651i \(-0.722911\pi\)
−0.644445 + 0.764651i \(0.722911\pi\)
\(858\) 0 0
\(859\) −13878.5 −0.551254 −0.275627 0.961265i \(-0.588885\pi\)
−0.275627 + 0.961265i \(0.588885\pi\)
\(860\) 11562.9i 0.458480i
\(861\) −5272.31 −0.208688
\(862\) 53223.7 2.10302
\(863\) − 32934.7i − 1.29908i −0.760326 0.649542i \(-0.774961\pi\)
0.760326 0.649542i \(-0.225039\pi\)
\(864\) − 1204.13i − 0.0474135i
\(865\) − 55517.5i − 2.18226i
\(866\) 46424.3i 1.82167i
\(867\) −8325.38 −0.326119
\(868\) −22049.3 −0.862213
\(869\) − 3319.57i − 0.129584i
\(870\) −74743.6 −2.91270
\(871\) 0 0
\(872\) −22413.5 −0.870434
\(873\) 4383.65i 0.169947i
\(874\) −5872.87 −0.227292
\(875\) −68805.4 −2.65834
\(876\) 2263.88i 0.0873167i
\(877\) − 18375.7i − 0.707531i −0.935334 0.353765i \(-0.884901\pi\)
0.935334 0.353765i \(-0.115099\pi\)
\(878\) 50050.2i 1.92382i
\(879\) 14235.7i 0.546256i
\(880\) 32590.7 1.24845
\(881\) −46883.4 −1.79290 −0.896448 0.443150i \(-0.853861\pi\)
−0.896448 + 0.443150i \(0.853861\pi\)
\(882\) 3393.39i 0.129548i
\(883\) −1050.05 −0.0400194 −0.0200097 0.999800i \(-0.506370\pi\)
−0.0200097 + 0.999800i \(0.506370\pi\)
\(884\) 0 0
\(885\) −43208.2 −1.64116
\(886\) 27333.7i 1.03645i
\(887\) 27916.8 1.05677 0.528385 0.849005i \(-0.322798\pi\)
0.528385 + 0.849005i \(0.322798\pi\)
\(888\) −5337.89 −0.201720
\(889\) 34949.4i 1.31852i
\(890\) − 100952.i − 3.80217i
\(891\) 2491.01i 0.0936611i
\(892\) − 27698.0i − 1.03968i
\(893\) −54297.4 −2.03471
\(894\) 49921.9 1.86761
\(895\) 3461.92i 0.129295i
\(896\) −42680.6 −1.59136
\(897\) 0 0
\(898\) 57990.8 2.15499
\(899\) 21364.6i 0.792601i
\(900\) 44961.3 1.66523
\(901\) 16117.5 0.595950
\(902\) − 16059.1i − 0.592804i
\(903\) 1729.81i 0.0637481i
\(904\) 48092.7i 1.76940i
\(905\) − 70231.3i − 2.57963i
\(906\) −32276.4 −1.18357
\(907\) −23220.1 −0.850066 −0.425033 0.905178i \(-0.639737\pi\)
−0.425033 + 0.905178i \(0.639737\pi\)
\(908\) 38289.5i 1.39943i
\(909\) −6897.38 −0.251674
\(910\) 0 0
\(911\) 16344.9 0.594435 0.297217 0.954810i \(-0.403941\pi\)
0.297217 + 0.954810i \(0.403941\pi\)
\(912\) − 21726.3i − 0.788848i
\(913\) −15858.0 −0.574834
\(914\) 79516.7 2.87766
\(915\) 14652.4i 0.529393i
\(916\) − 93559.0i − 3.37476i
\(917\) − 13002.1i − 0.468231i
\(918\) − 6039.15i − 0.217126i
\(919\) −32931.7 −1.18206 −0.591032 0.806648i \(-0.701279\pi\)
−0.591032 + 0.806648i \(0.701279\pi\)
\(920\) 6360.67 0.227940
\(921\) 3086.47i 0.110426i
\(922\) −42800.8 −1.52882
\(923\) 0 0
\(924\) 23133.6 0.823636
\(925\) 16119.4i 0.572975i
\(926\) 49679.0 1.76302
\(927\) −11068.8 −0.392177
\(928\) 10835.3i 0.383282i
\(929\) − 56305.5i − 1.98851i −0.107056 0.994253i \(-0.534142\pi\)
0.107056 0.994253i \(-0.465858\pi\)
\(930\) − 27052.4i − 0.953852i
\(931\) − 11291.0i − 0.397473i
\(932\) 98063.3 3.44653
\(933\) 7313.51 0.256628
\(934\) − 71258.5i − 2.49641i
\(935\) −30142.7 −1.05430
\(936\) 0 0
\(937\) 7095.61 0.247389 0.123695 0.992320i \(-0.460526\pi\)
0.123695 + 0.992320i \(0.460526\pi\)
\(938\) 23284.0i 0.810502i
\(939\) 23802.6 0.827230
\(940\) 122371. 4.24606
\(941\) − 5185.17i − 0.179630i −0.995958 0.0898150i \(-0.971372\pi\)
0.995958 0.0898150i \(-0.0286276\pi\)
\(942\) 35777.9i 1.23748i
\(943\) − 904.641i − 0.0312398i
\(944\) − 33966.2i − 1.17109i
\(945\) 9318.26 0.320765
\(946\) −5268.89 −0.181085
\(947\) − 38451.7i − 1.31944i −0.751510 0.659722i \(-0.770674\pi\)
0.751510 0.659722i \(-0.229326\pi\)
\(948\) 4987.39 0.170868
\(949\) 0 0
\(950\) −227310. −7.76307
\(951\) − 20828.2i − 0.710199i
\(952\) −26952.4 −0.917575
\(953\) 20930.5 0.711445 0.355722 0.934592i \(-0.384235\pi\)
0.355722 + 0.934592i \(0.384235\pi\)
\(954\) − 15176.4i − 0.515048i
\(955\) 90768.8i 3.07561i
\(956\) − 29053.7i − 0.982910i
\(957\) − 22415.2i − 0.757139i
\(958\) 3072.19 0.103609
\(959\) 9787.14 0.329555
\(960\) − 39153.9i − 1.31634i
\(961\) 22058.4 0.740438
\(962\) 0 0
\(963\) 691.435 0.0231372
\(964\) − 90066.1i − 3.00916i
\(965\) 39796.1 1.32755
\(966\) 1980.07 0.0659499
\(967\) − 50788.1i − 1.68897i −0.535578 0.844486i \(-0.679906\pi\)
0.535578 0.844486i \(-0.320094\pi\)
\(968\) − 13793.2i − 0.457986i
\(969\) 20094.4i 0.666175i
\(970\) 49947.6i 1.65332i
\(971\) 15277.5 0.504922 0.252461 0.967607i \(-0.418760\pi\)
0.252461 + 0.967607i \(0.418760\pi\)
\(972\) −3742.54 −0.123500
\(973\) − 34745.7i − 1.14481i
\(974\) 80356.3 2.64351
\(975\) 0 0
\(976\) −11518.4 −0.377760
\(977\) 1293.19i 0.0423469i 0.999776 + 0.0211734i \(0.00674022\pi\)
−0.999776 + 0.0211734i \(0.993260\pi\)
\(978\) 3534.74 0.115571
\(979\) 30275.1 0.988353
\(980\) 25446.7i 0.829453i
\(981\) − 5634.01i − 0.183364i
\(982\) 12500.8i 0.406227i
\(983\) − 8474.63i − 0.274973i −0.990504 0.137487i \(-0.956098\pi\)
0.990504 0.137487i \(-0.0439024\pi\)
\(984\) 11594.9 0.375641
\(985\) 42211.1 1.36544
\(986\) 54343.0i 1.75521i
\(987\) 18306.6 0.590382
\(988\) 0 0
\(989\) −296.807 −0.00954289
\(990\) 28382.8i 0.911175i
\(991\) −7080.71 −0.226969 −0.113484 0.993540i \(-0.536201\pi\)
−0.113484 + 0.993540i \(0.536201\pi\)
\(992\) −3921.68 −0.125518
\(993\) 16317.8i 0.521480i
\(994\) − 25928.1i − 0.827354i
\(995\) 28512.8i 0.908458i
\(996\) − 23825.4i − 0.757968i
\(997\) 19423.1 0.616985 0.308493 0.951227i \(-0.400175\pi\)
0.308493 + 0.951227i \(0.400175\pi\)
\(998\) −10564.5 −0.335084
\(999\) − 1341.76i − 0.0424940i
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 507.4.b.k.337.17 18
13.5 odd 4 507.4.a.o.1.9 9
13.8 odd 4 507.4.a.p.1.1 yes 9
13.12 even 2 inner 507.4.b.k.337.2 18
39.5 even 4 1521.4.a.bi.1.1 9
39.8 even 4 1521.4.a.bf.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.9 9 13.5 odd 4
507.4.a.p.1.1 yes 9 13.8 odd 4
507.4.b.k.337.2 18 13.12 even 2 inner
507.4.b.k.337.17 18 1.1 even 1 trivial
1521.4.a.bf.1.9 9 39.8 even 4
1521.4.a.bi.1.1 9 39.5 even 4