Properties

Label 507.4.b.g.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.158155776.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-0.376763 - 0.376763i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.g.337.6

$q$-expansion

\(f(q)\) \(=\) \(q-4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} +11.4322i q^{5} -12.6271i q^{6} -11.2543i q^{7} +7.22315i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.20905i q^{2} +3.00000 q^{3} -9.71610 q^{4} +11.4322i q^{5} -12.6271i q^{6} -11.2543i q^{7} +7.22315i q^{8} +9.00000 q^{9} +48.1187 q^{10} +25.8785i q^{11} -29.1483 q^{12} -47.3699 q^{14} +34.2966i q^{15} -47.3262 q^{16} +20.3276 q^{17} -37.8814i q^{18} -154.712i q^{19} -111.076i q^{20} -33.7629i q^{21} +108.924 q^{22} +180.418 q^{23} +21.6695i q^{24} -5.69520 q^{25} +27.0000 q^{27} +109.348i q^{28} -20.4522 q^{29} +144.356 q^{30} -266.424i q^{31} +256.984i q^{32} +77.6355i q^{33} -85.5599i q^{34} +128.661 q^{35} -87.4449 q^{36} +115.984i q^{37} -651.190 q^{38} -82.5765 q^{40} -391.184i q^{41} -142.110 q^{42} -151.407 q^{43} -251.438i q^{44} +102.890i q^{45} -759.390i q^{46} -467.365i q^{47} -141.979 q^{48} +216.341 q^{49} +23.9714i q^{50} +60.9828 q^{51} +79.9842 q^{53} -113.644i q^{54} -295.848 q^{55} +81.2915 q^{56} -464.136i q^{57} +86.0843i q^{58} -873.710i q^{59} -333.229i q^{60} -187.068 q^{61} -1121.39 q^{62} -101.289i q^{63} +703.047 q^{64} +326.772 q^{66} +609.204i q^{67} -197.505 q^{68} +541.255 q^{69} -541.542i q^{70} -248.038i q^{71} +65.0084i q^{72} +852.765i q^{73} +488.181 q^{74} -17.0856 q^{75} +1503.20i q^{76} +291.244 q^{77} -331.221 q^{79} -541.043i q^{80} +81.0000 q^{81} -1646.51 q^{82} +435.432i q^{83} +328.044i q^{84} +232.389i q^{85} +637.281i q^{86} -61.3566 q^{87} -186.924 q^{88} +259.233i q^{89} +433.068 q^{90} -1752.96 q^{92} -799.273i q^{93} -1967.16 q^{94} +1768.70 q^{95} +770.951i q^{96} -1225.17i q^{97} -910.589i q^{98} +232.907i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.20905i − 1.48812i −0.668111 0.744062i \(-0.732897\pi\)
0.668111 0.744062i \(-0.267103\pi\)
\(3\) 3.00000 0.577350
\(4\) −9.71610 −1.21451
\(5\) 11.4322i 1.02253i 0.859424 + 0.511264i \(0.170822\pi\)
−0.859424 + 0.511264i \(0.829178\pi\)
\(6\) − 12.6271i − 0.859169i
\(7\) − 11.2543i − 0.607675i −0.952724 0.303838i \(-0.901732\pi\)
0.952724 0.303838i \(-0.0982680\pi\)
\(8\) 7.22315i 0.319221i
\(9\) 9.00000 0.333333
\(10\) 48.1187 1.52165
\(11\) 25.8785i 0.709333i 0.934993 + 0.354666i \(0.115406\pi\)
−0.934993 + 0.354666i \(0.884594\pi\)
\(12\) −29.1483 −0.701199
\(13\) 0 0
\(14\) −47.3699 −0.904296
\(15\) 34.2966i 0.590356i
\(16\) −47.3262 −0.739472
\(17\) 20.3276 0.290010 0.145005 0.989431i \(-0.453680\pi\)
0.145005 + 0.989431i \(0.453680\pi\)
\(18\) − 37.8814i − 0.496041i
\(19\) − 154.712i − 1.86807i −0.357181 0.934035i \(-0.616262\pi\)
0.357181 0.934035i \(-0.383738\pi\)
\(20\) − 111.076i − 1.24187i
\(21\) − 33.7629i − 0.350841i
\(22\) 108.924 1.05558
\(23\) 180.418 1.63565 0.817823 0.575471i \(-0.195181\pi\)
0.817823 + 0.575471i \(0.195181\pi\)
\(24\) 21.6695i 0.184302i
\(25\) −5.69520 −0.0455616
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 109.348i 0.738029i
\(29\) −20.4522 −0.130961 −0.0654806 0.997854i \(-0.520858\pi\)
−0.0654806 + 0.997854i \(0.520858\pi\)
\(30\) 144.356 0.878523
\(31\) − 266.424i − 1.54359i −0.635873 0.771794i \(-0.719360\pi\)
0.635873 0.771794i \(-0.280640\pi\)
\(32\) 256.984i 1.41965i
\(33\) 77.6355i 0.409534i
\(34\) − 85.5599i − 0.431571i
\(35\) 128.661 0.621364
\(36\) −87.4449 −0.404837
\(37\) 115.984i 0.515340i 0.966233 + 0.257670i \(0.0829548\pi\)
−0.966233 + 0.257670i \(0.917045\pi\)
\(38\) −651.190 −2.77992
\(39\) 0 0
\(40\) −82.5765 −0.326412
\(41\) − 391.184i − 1.49006i −0.667029 0.745032i \(-0.732434\pi\)
0.667029 0.745032i \(-0.267566\pi\)
\(42\) −142.110 −0.522095
\(43\) −151.407 −0.536963 −0.268482 0.963285i \(-0.586522\pi\)
−0.268482 + 0.963285i \(0.586522\pi\)
\(44\) − 251.438i − 0.861494i
\(45\) 102.890i 0.340842i
\(46\) − 759.390i − 2.43404i
\(47\) − 467.365i − 1.45047i −0.688500 0.725236i \(-0.741731\pi\)
0.688500 0.725236i \(-0.258269\pi\)
\(48\) −141.979 −0.426934
\(49\) 216.341 0.630731
\(50\) 23.9714i 0.0678012i
\(51\) 60.9828 0.167437
\(52\) 0 0
\(53\) 79.9842 0.207296 0.103648 0.994614i \(-0.466949\pi\)
0.103648 + 0.994614i \(0.466949\pi\)
\(54\) − 113.644i − 0.286390i
\(55\) −295.848 −0.725312
\(56\) 81.2915 0.193983
\(57\) − 464.136i − 1.07853i
\(58\) 86.0843i 0.194887i
\(59\) − 873.710i − 1.92792i −0.266045 0.963960i \(-0.585717\pi\)
0.266045 0.963960i \(-0.414283\pi\)
\(60\) − 333.229i − 0.716995i
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) −1121.39 −2.29705
\(63\) − 101.289i − 0.202558i
\(64\) 703.047 1.37314
\(65\) 0 0
\(66\) 326.772 0.609437
\(67\) 609.204i 1.11084i 0.831571 + 0.555418i \(0.187442\pi\)
−0.831571 + 0.555418i \(0.812558\pi\)
\(68\) −197.505 −0.352221
\(69\) 541.255 0.944340
\(70\) − 541.542i − 0.924667i
\(71\) − 248.038i − 0.414601i −0.978277 0.207301i \(-0.933532\pi\)
0.978277 0.207301i \(-0.0664679\pi\)
\(72\) 65.0084i 0.106407i
\(73\) 852.765i 1.36724i 0.729838 + 0.683621i \(0.239596\pi\)
−0.729838 + 0.683621i \(0.760404\pi\)
\(74\) 488.181 0.766890
\(75\) −17.0856 −0.0263050
\(76\) 1503.20i 2.26880i
\(77\) 291.244 0.431044
\(78\) 0 0
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) − 541.043i − 0.756130i
\(81\) 81.0000 0.111111
\(82\) −1646.51 −2.21740
\(83\) 435.432i 0.575842i 0.957654 + 0.287921i \(0.0929641\pi\)
−0.957654 + 0.287921i \(0.907036\pi\)
\(84\) 328.044i 0.426101i
\(85\) 232.389i 0.296543i
\(86\) 637.281i 0.799067i
\(87\) −61.3566 −0.0756105
\(88\) −186.924 −0.226434
\(89\) 259.233i 0.308749i 0.988012 + 0.154375i \(0.0493362\pi\)
−0.988012 + 0.154375i \(0.950664\pi\)
\(90\) 433.068 0.507216
\(91\) 0 0
\(92\) −1752.96 −1.98651
\(93\) − 799.273i − 0.891191i
\(94\) −1967.16 −2.15848
\(95\) 1768.70 1.91015
\(96\) 770.951i 0.819634i
\(97\) − 1225.17i − 1.28245i −0.767355 0.641223i \(-0.778428\pi\)
0.767355 0.641223i \(-0.221572\pi\)
\(98\) − 910.589i − 0.938606i
\(99\) 232.907i 0.236444i
\(100\) 55.3351 0.0553351
\(101\) −645.416 −0.635855 −0.317927 0.948115i \(-0.602987\pi\)
−0.317927 + 0.948115i \(0.602987\pi\)
\(102\) − 256.680i − 0.249167i
\(103\) 511.137 0.488969 0.244484 0.969653i \(-0.421381\pi\)
0.244484 + 0.969653i \(0.421381\pi\)
\(104\) 0 0
\(105\) 385.984 0.358745
\(106\) − 336.657i − 0.308482i
\(107\) 608.195 0.549499 0.274750 0.961516i \(-0.411405\pi\)
0.274750 + 0.961516i \(0.411405\pi\)
\(108\) −262.335 −0.233733
\(109\) 1300.04i 1.14239i 0.820813 + 0.571197i \(0.193521\pi\)
−0.820813 + 0.571197i \(0.806479\pi\)
\(110\) 1245.24i 1.07935i
\(111\) 347.951i 0.297532i
\(112\) 532.623i 0.449359i
\(113\) 42.1953 0.0351274 0.0175637 0.999846i \(-0.494409\pi\)
0.0175637 + 0.999846i \(0.494409\pi\)
\(114\) −1953.57 −1.60499
\(115\) 2062.58i 1.67249i
\(116\) 198.716 0.159054
\(117\) 0 0
\(118\) −3677.49 −2.86899
\(119\) − 228.773i − 0.176232i
\(120\) −247.729 −0.188454
\(121\) 661.303 0.496847
\(122\) 787.378i 0.584311i
\(123\) − 1173.55i − 0.860289i
\(124\) 2588.61i 1.87471i
\(125\) 1363.92i 0.975939i
\(126\) −426.329 −0.301432
\(127\) 311.018 0.217310 0.108655 0.994080i \(-0.465346\pi\)
0.108655 + 0.994080i \(0.465346\pi\)
\(128\) − 903.291i − 0.623753i
\(129\) −454.222 −0.310016
\(130\) 0 0
\(131\) 2000.98 1.33456 0.667278 0.744809i \(-0.267459\pi\)
0.667278 + 0.744809i \(0.267459\pi\)
\(132\) − 754.314i − 0.497384i
\(133\) −1741.17 −1.13518
\(134\) 2564.17 1.65306
\(135\) 308.669i 0.196785i
\(136\) 146.829i 0.0925773i
\(137\) 1038.53i 0.647644i 0.946118 + 0.323822i \(0.104968\pi\)
−0.946118 + 0.323822i \(0.895032\pi\)
\(138\) − 2278.17i − 1.40529i
\(139\) −2858.46 −1.74426 −0.872128 0.489277i \(-0.837261\pi\)
−0.872128 + 0.489277i \(0.837261\pi\)
\(140\) −1250.09 −0.754655
\(141\) − 1402.09i − 0.837430i
\(142\) −1044.00 −0.616978
\(143\) 0 0
\(144\) −425.936 −0.246491
\(145\) − 233.814i − 0.133911i
\(146\) 3589.33 2.03462
\(147\) 649.022 0.364153
\(148\) − 1126.91i − 0.625887i
\(149\) − 743.479i − 0.408780i −0.978890 0.204390i \(-0.934479\pi\)
0.978890 0.204390i \(-0.0655211\pi\)
\(150\) 71.9141i 0.0391451i
\(151\) 2277.24i 1.22728i 0.789586 + 0.613640i \(0.210295\pi\)
−0.789586 + 0.613640i \(0.789705\pi\)
\(152\) 1117.51 0.596328
\(153\) 182.948 0.0966700
\(154\) − 1225.86i − 0.641447i
\(155\) 3045.82 1.57836
\(156\) 0 0
\(157\) 3173.51 1.61321 0.806605 0.591091i \(-0.201303\pi\)
0.806605 + 0.591091i \(0.201303\pi\)
\(158\) 1394.12i 0.701966i
\(159\) 239.953 0.119682
\(160\) −2937.89 −1.45163
\(161\) − 2030.48i − 0.993941i
\(162\) − 340.933i − 0.165347i
\(163\) − 2314.65i − 1.11225i −0.831098 0.556126i \(-0.812287\pi\)
0.831098 0.556126i \(-0.187713\pi\)
\(164\) 3800.78i 1.80970i
\(165\) −887.545 −0.418759
\(166\) 1832.76 0.856925
\(167\) − 2665.65i − 1.23517i −0.786502 0.617587i \(-0.788110\pi\)
0.786502 0.617587i \(-0.211890\pi\)
\(168\) 243.874 0.111996
\(169\) 0 0
\(170\) 978.138 0.441293
\(171\) − 1392.41i − 0.622690i
\(172\) 1471.09 0.652148
\(173\) 165.243 0.0726198 0.0363099 0.999341i \(-0.488440\pi\)
0.0363099 + 0.999341i \(0.488440\pi\)
\(174\) 258.253i 0.112518i
\(175\) 64.0954i 0.0276866i
\(176\) − 1224.73i − 0.524532i
\(177\) − 2621.13i − 1.11309i
\(178\) 1091.13 0.459457
\(179\) −712.339 −0.297446 −0.148723 0.988879i \(-0.547516\pi\)
−0.148723 + 0.988879i \(0.547516\pi\)
\(180\) − 999.688i − 0.413957i
\(181\) −2206.53 −0.906133 −0.453066 0.891477i \(-0.649670\pi\)
−0.453066 + 0.891477i \(0.649670\pi\)
\(182\) 0 0
\(183\) −561.204 −0.226696
\(184\) 1303.19i 0.522132i
\(185\) −1325.95 −0.526949
\(186\) −3364.18 −1.32620
\(187\) 526.048i 0.205714i
\(188\) 4540.96i 1.76162i
\(189\) − 303.866i − 0.116947i
\(190\) − 7444.54i − 2.84254i
\(191\) 1470.64 0.557129 0.278565 0.960417i \(-0.410141\pi\)
0.278565 + 0.960417i \(0.410141\pi\)
\(192\) 2109.14 0.792782
\(193\) 369.560i 0.137832i 0.997622 + 0.0689158i \(0.0219540\pi\)
−0.997622 + 0.0689158i \(0.978046\pi\)
\(194\) −5156.80 −1.90844
\(195\) 0 0
\(196\) −2101.99 −0.766031
\(197\) 4273.41i 1.54552i 0.634697 + 0.772761i \(0.281125\pi\)
−0.634697 + 0.772761i \(0.718875\pi\)
\(198\) 980.315 0.351858
\(199\) −4154.31 −1.47985 −0.739927 0.672687i \(-0.765140\pi\)
−0.739927 + 0.672687i \(0.765140\pi\)
\(200\) − 41.1373i − 0.0145442i
\(201\) 1827.61i 0.641342i
\(202\) 2716.59i 0.946230i
\(203\) 230.175i 0.0795819i
\(204\) −592.515 −0.203355
\(205\) 4472.09 1.52363
\(206\) − 2151.40i − 0.727646i
\(207\) 1623.77 0.545215
\(208\) 0 0
\(209\) 4003.71 1.32508
\(210\) − 1624.63i − 0.533857i
\(211\) 1231.59 0.401830 0.200915 0.979609i \(-0.435608\pi\)
0.200915 + 0.979609i \(0.435608\pi\)
\(212\) −777.134 −0.251763
\(213\) − 744.114i − 0.239370i
\(214\) − 2559.92i − 0.817723i
\(215\) − 1730.92i − 0.549059i
\(216\) 195.025i 0.0614341i
\(217\) −2998.42 −0.938000
\(218\) 5471.92 1.70002
\(219\) 2558.30i 0.789377i
\(220\) 2874.49 0.880901
\(221\) 0 0
\(222\) 1464.54 0.442764
\(223\) 2187.24i 0.656809i 0.944537 + 0.328404i \(0.106511\pi\)
−0.944537 + 0.328404i \(0.893489\pi\)
\(224\) 2892.17 0.862684
\(225\) −51.2568 −0.0151872
\(226\) − 177.602i − 0.0522739i
\(227\) − 4138.67i − 1.21010i −0.796187 0.605051i \(-0.793153\pi\)
0.796187 0.605051i \(-0.206847\pi\)
\(228\) 4509.59i 1.30989i
\(229\) − 835.354i − 0.241056i −0.992710 0.120528i \(-0.961541\pi\)
0.992710 0.120528i \(-0.0384587\pi\)
\(230\) 8681.50 2.48887
\(231\) 873.733 0.248863
\(232\) − 147.729i − 0.0418056i
\(233\) −3685.51 −1.03625 −0.518124 0.855305i \(-0.673370\pi\)
−0.518124 + 0.855305i \(0.673370\pi\)
\(234\) 0 0
\(235\) 5343.01 1.48315
\(236\) 8489.05i 2.34148i
\(237\) −993.662 −0.272343
\(238\) −962.917 −0.262255
\(239\) − 3026.21i − 0.819034i −0.912303 0.409517i \(-0.865697\pi\)
0.912303 0.409517i \(-0.134303\pi\)
\(240\) − 1623.13i − 0.436552i
\(241\) 3265.58i 0.872839i 0.899743 + 0.436420i \(0.143754\pi\)
−0.899743 + 0.436420i \(0.856246\pi\)
\(242\) − 2783.46i − 0.739370i
\(243\) 243.000 0.0641500
\(244\) 1817.57 0.476877
\(245\) 2473.25i 0.644940i
\(246\) −4939.53 −1.28022
\(247\) 0 0
\(248\) 1924.42 0.492746
\(249\) 1306.30i 0.332463i
\(250\) 5740.79 1.45232
\(251\) 6363.16 1.60016 0.800078 0.599897i \(-0.204792\pi\)
0.800078 + 0.599897i \(0.204792\pi\)
\(252\) 984.131i 0.246010i
\(253\) 4668.96i 1.16022i
\(254\) − 1309.09i − 0.323385i
\(255\) 697.168i 0.171209i
\(256\) 1822.38 0.444917
\(257\) 6085.36 1.47702 0.738511 0.674242i \(-0.235529\pi\)
0.738511 + 0.674242i \(0.235529\pi\)
\(258\) 1911.84i 0.461342i
\(259\) 1305.31 0.313159
\(260\) 0 0
\(261\) −184.070 −0.0436538
\(262\) − 8422.24i − 1.98598i
\(263\) 123.227 0.0288916 0.0144458 0.999896i \(-0.495402\pi\)
0.0144458 + 0.999896i \(0.495402\pi\)
\(264\) −560.773 −0.130732
\(265\) 914.395i 0.211965i
\(266\) 7328.69i 1.68929i
\(267\) 777.700i 0.178256i
\(268\) − 5919.08i − 1.34913i
\(269\) −1935.79 −0.438763 −0.219381 0.975639i \(-0.570404\pi\)
−0.219381 + 0.975639i \(0.570404\pi\)
\(270\) 1299.20 0.292841
\(271\) − 4612.69i − 1.03395i −0.856000 0.516976i \(-0.827058\pi\)
0.856000 0.516976i \(-0.172942\pi\)
\(272\) −962.028 −0.214454
\(273\) 0 0
\(274\) 4371.20 0.963774
\(275\) − 147.383i − 0.0323183i
\(276\) −5258.89 −1.14691
\(277\) 5834.30 1.26552 0.632761 0.774347i \(-0.281922\pi\)
0.632761 + 0.774347i \(0.281922\pi\)
\(278\) 12031.4i 2.59567i
\(279\) − 2397.82i − 0.514529i
\(280\) 929.341i 0.198353i
\(281\) 4691.91i 0.996071i 0.867157 + 0.498036i \(0.165945\pi\)
−0.867157 + 0.498036i \(0.834055\pi\)
\(282\) −5901.49 −1.24620
\(283\) −3465.60 −0.727945 −0.363973 0.931410i \(-0.618580\pi\)
−0.363973 + 0.931410i \(0.618580\pi\)
\(284\) 2409.96i 0.503539i
\(285\) 5306.09 1.10283
\(286\) 0 0
\(287\) −4402.50 −0.905475
\(288\) 2312.85i 0.473216i
\(289\) −4499.79 −0.915894
\(290\) −984.133 −0.199277
\(291\) − 3675.51i − 0.740420i
\(292\) − 8285.55i − 1.66053i
\(293\) 2677.31i 0.533822i 0.963721 + 0.266911i \(0.0860030\pi\)
−0.963721 + 0.266911i \(0.913997\pi\)
\(294\) − 2731.77i − 0.541904i
\(295\) 9988.43 1.97135
\(296\) −837.767 −0.164507
\(297\) 698.720i 0.136511i
\(298\) −3129.34 −0.608315
\(299\) 0 0
\(300\) 166.005 0.0319477
\(301\) 1703.98i 0.326299i
\(302\) 9585.02 1.82634
\(303\) −1936.25 −0.367111
\(304\) 7321.93i 1.38139i
\(305\) − 2138.60i − 0.401494i
\(306\) − 770.039i − 0.143857i
\(307\) 471.915i 0.0877316i 0.999037 + 0.0438658i \(0.0139674\pi\)
−0.999037 + 0.0438658i \(0.986033\pi\)
\(308\) −2829.76 −0.523508
\(309\) 1533.41 0.282306
\(310\) − 12820.0i − 2.34880i
\(311\) 1518.52 0.276872 0.138436 0.990371i \(-0.455793\pi\)
0.138436 + 0.990371i \(0.455793\pi\)
\(312\) 0 0
\(313\) 4049.86 0.731348 0.365674 0.930743i \(-0.380839\pi\)
0.365674 + 0.930743i \(0.380839\pi\)
\(314\) − 13357.5i − 2.40066i
\(315\) 1157.95 0.207121
\(316\) 3218.17 0.572900
\(317\) − 3253.96i − 0.576532i −0.957550 0.288266i \(-0.906921\pi\)
0.957550 0.288266i \(-0.0930787\pi\)
\(318\) − 1009.97i − 0.178102i
\(319\) − 529.272i − 0.0928951i
\(320\) 8037.37i 1.40407i
\(321\) 1824.59 0.317254
\(322\) −8546.40 −1.47911
\(323\) − 3144.92i − 0.541759i
\(324\) −787.004 −0.134946
\(325\) 0 0
\(326\) −9742.46 −1.65517
\(327\) 3900.11i 0.659561i
\(328\) 2825.58 0.475660
\(329\) −5259.86 −0.881415
\(330\) 3735.72i 0.623165i
\(331\) 3422.45i 0.568322i 0.958777 + 0.284161i \(0.0917150\pi\)
−0.958777 + 0.284161i \(0.908285\pi\)
\(332\) − 4230.71i − 0.699368i
\(333\) 1043.85i 0.171780i
\(334\) −11219.8 −1.83809
\(335\) −6964.54 −1.13586
\(336\) 1597.87i 0.259437i
\(337\) 9301.67 1.50354 0.751772 0.659423i \(-0.229199\pi\)
0.751772 + 0.659423i \(0.229199\pi\)
\(338\) 0 0
\(339\) 126.586 0.0202808
\(340\) − 2257.92i − 0.360155i
\(341\) 6894.66 1.09492
\(342\) −5860.71 −0.926640
\(343\) − 6294.99i − 0.990955i
\(344\) − 1093.64i − 0.171410i
\(345\) 6187.74i 0.965613i
\(346\) − 695.518i − 0.108067i
\(347\) 216.898 0.0335554 0.0167777 0.999859i \(-0.494659\pi\)
0.0167777 + 0.999859i \(0.494659\pi\)
\(348\) 596.147 0.0918299
\(349\) − 4809.84i − 0.737721i −0.929485 0.368861i \(-0.879748\pi\)
0.929485 0.368861i \(-0.120252\pi\)
\(350\) 269.781 0.0412011
\(351\) 0 0
\(352\) −6650.35 −1.00700
\(353\) 2859.64i 0.431170i 0.976485 + 0.215585i \(0.0691659\pi\)
−0.976485 + 0.215585i \(0.930834\pi\)
\(354\) −11032.5 −1.65641
\(355\) 2835.62 0.423941
\(356\) − 2518.74i − 0.374980i
\(357\) − 686.319i − 0.101747i
\(358\) 2998.27i 0.442636i
\(359\) 3686.04i 0.541899i 0.962594 + 0.270949i \(0.0873376\pi\)
−0.962594 + 0.270949i \(0.912662\pi\)
\(360\) −743.188 −0.108804
\(361\) −17076.8 −2.48969
\(362\) 9287.39i 1.34844i
\(363\) 1983.91 0.286855
\(364\) 0 0
\(365\) −9748.98 −1.39804
\(366\) 2362.14i 0.337352i
\(367\) −3470.59 −0.493633 −0.246816 0.969062i \(-0.579384\pi\)
−0.246816 + 0.969062i \(0.579384\pi\)
\(368\) −8538.52 −1.20951
\(369\) − 3520.65i − 0.496688i
\(370\) 5580.98i 0.784166i
\(371\) − 900.166i − 0.125968i
\(372\) 7765.82i 1.08236i
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) 2214.16 0.306127
\(375\) 4091.75i 0.563459i
\(376\) 3375.85 0.463021
\(377\) 0 0
\(378\) −1278.99 −0.174032
\(379\) − 345.604i − 0.0468403i −0.999726 0.0234202i \(-0.992544\pi\)
0.999726 0.0234202i \(-0.00745555\pi\)
\(380\) −17184.8 −2.31990
\(381\) 933.055 0.125464
\(382\) − 6189.99i − 0.829078i
\(383\) 3386.40i 0.451793i 0.974151 + 0.225897i \(0.0725312\pi\)
−0.974151 + 0.225897i \(0.927469\pi\)
\(384\) − 2709.87i − 0.360124i
\(385\) 3329.56i 0.440754i
\(386\) 1555.49 0.205110
\(387\) −1362.67 −0.178988
\(388\) 11903.9i 1.55755i
\(389\) 1629.88 0.212438 0.106219 0.994343i \(-0.466126\pi\)
0.106219 + 0.994343i \(0.466126\pi\)
\(390\) 0 0
\(391\) 3667.47 0.474353
\(392\) 1562.66i 0.201343i
\(393\) 6002.95 0.770506
\(394\) 17987.0 2.29993
\(395\) − 3786.58i − 0.482338i
\(396\) − 2262.94i − 0.287165i
\(397\) 7938.94i 1.00364i 0.864973 + 0.501819i \(0.167336\pi\)
−0.864973 + 0.501819i \(0.832664\pi\)
\(398\) 17485.7i 2.20221i
\(399\) −5223.52 −0.655396
\(400\) 269.532 0.0336915
\(401\) 214.402i 0.0267001i 0.999911 + 0.0133500i \(0.00424958\pi\)
−0.999911 + 0.0133500i \(0.995750\pi\)
\(402\) 7692.51 0.954396
\(403\) 0 0
\(404\) 6270.93 0.772253
\(405\) 926.008i 0.113614i
\(406\) 968.819 0.118428
\(407\) −3001.48 −0.365548
\(408\) 440.488i 0.0534495i
\(409\) 4783.73i 0.578338i 0.957278 + 0.289169i \(0.0933789\pi\)
−0.957278 + 0.289169i \(0.906621\pi\)
\(410\) − 18823.2i − 2.26735i
\(411\) 3115.58i 0.373917i
\(412\) −4966.25 −0.593859
\(413\) −9832.99 −1.17155
\(414\) − 6834.51i − 0.811347i
\(415\) −4977.95 −0.588815
\(416\) 0 0
\(417\) −8575.39 −1.00705
\(418\) − 16851.8i − 1.97189i
\(419\) −9903.67 −1.15472 −0.577358 0.816491i \(-0.695916\pi\)
−0.577358 + 0.816491i \(0.695916\pi\)
\(420\) −3750.26 −0.435700
\(421\) 12120.6i 1.40314i 0.712598 + 0.701572i \(0.247518\pi\)
−0.712598 + 0.701572i \(0.752482\pi\)
\(422\) − 5183.82i − 0.597973i
\(423\) − 4206.28i − 0.483491i
\(424\) 577.738i 0.0661732i
\(425\) −115.770 −0.0132133
\(426\) −3132.01 −0.356213
\(427\) 2105.32i 0.238603i
\(428\) −5909.28 −0.667374
\(429\) 0 0
\(430\) −7285.53 −0.817068
\(431\) 13672.6i 1.52805i 0.645189 + 0.764023i \(0.276779\pi\)
−0.645189 + 0.764023i \(0.723221\pi\)
\(432\) −1277.81 −0.142311
\(433\) −7113.10 −0.789455 −0.394727 0.918798i \(-0.629161\pi\)
−0.394727 + 0.918798i \(0.629161\pi\)
\(434\) 12620.5i 1.39586i
\(435\) − 701.441i − 0.0773138i
\(436\) − 12631.3i − 1.38745i
\(437\) − 27912.9i − 3.05550i
\(438\) 10768.0 1.17469
\(439\) 6022.04 0.654707 0.327353 0.944902i \(-0.393843\pi\)
0.327353 + 0.944902i \(0.393843\pi\)
\(440\) − 2136.96i − 0.231535i
\(441\) 1947.07 0.210244
\(442\) 0 0
\(443\) −12994.4 −1.39364 −0.696821 0.717245i \(-0.745403\pi\)
−0.696821 + 0.717245i \(0.745403\pi\)
\(444\) − 3380.72i − 0.361356i
\(445\) −2963.61 −0.315704
\(446\) 9206.20 0.977413
\(447\) − 2230.44i − 0.236009i
\(448\) − 7912.30i − 0.834422i
\(449\) 10984.3i 1.15452i 0.816560 + 0.577260i \(0.195878\pi\)
−0.816560 + 0.577260i \(0.804122\pi\)
\(450\) 215.742i 0.0226004i
\(451\) 10123.2 1.05695
\(452\) −409.973 −0.0426627
\(453\) 6831.72i 0.708570i
\(454\) −17419.9 −1.80078
\(455\) 0 0
\(456\) 3352.52 0.344290
\(457\) − 9834.10i − 1.00661i −0.864109 0.503304i \(-0.832118\pi\)
0.864109 0.503304i \(-0.167882\pi\)
\(458\) −3516.05 −0.358721
\(459\) 548.845 0.0558124
\(460\) − 20040.2i − 2.03126i
\(461\) − 3401.42i − 0.343644i −0.985128 0.171822i \(-0.945035\pi\)
0.985128 0.171822i \(-0.0549653\pi\)
\(462\) − 3677.59i − 0.370339i
\(463\) 1739.42i 0.174596i 0.996182 + 0.0872979i \(0.0278232\pi\)
−0.996182 + 0.0872979i \(0.972177\pi\)
\(464\) 967.925 0.0968422
\(465\) 9137.45 0.911267
\(466\) 15512.5i 1.54207i
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) 0 0
\(469\) 6856.16 0.675028
\(470\) − 22489.0i − 2.20711i
\(471\) 9520.54 0.931387
\(472\) 6310.94 0.615433
\(473\) − 3918.20i − 0.380886i
\(474\) 4182.37i 0.405280i
\(475\) 881.114i 0.0851122i
\(476\) 2222.78i 0.214036i
\(477\) 719.858 0.0690986
\(478\) −12737.5 −1.21882
\(479\) 8431.98i 0.804315i 0.915570 + 0.402158i \(0.131740\pi\)
−0.915570 + 0.402158i \(0.868260\pi\)
\(480\) −8813.66 −0.838097
\(481\) 0 0
\(482\) 13745.0 1.29889
\(483\) − 6091.45i − 0.573852i
\(484\) −6425.29 −0.603427
\(485\) 14006.4 1.31133
\(486\) − 1022.80i − 0.0954632i
\(487\) 11684.7i 1.08723i 0.839334 + 0.543617i \(0.182945\pi\)
−0.839334 + 0.543617i \(0.817055\pi\)
\(488\) − 1351.22i − 0.125342i
\(489\) − 6943.94i − 0.642159i
\(490\) 10410.0 0.959750
\(491\) 3954.70 0.363489 0.181745 0.983346i \(-0.441826\pi\)
0.181745 + 0.983346i \(0.441826\pi\)
\(492\) 11402.3i 1.04483i
\(493\) −415.744 −0.0379801
\(494\) 0 0
\(495\) −2662.63 −0.241771
\(496\) 12608.8i 1.14144i
\(497\) −2791.49 −0.251943
\(498\) 5498.27 0.494746
\(499\) 5690.37i 0.510493i 0.966876 + 0.255246i \(0.0821566\pi\)
−0.966876 + 0.255246i \(0.917843\pi\)
\(500\) − 13251.9i − 1.18529i
\(501\) − 7996.95i − 0.713128i
\(502\) − 26782.8i − 2.38123i
\(503\) 10859.1 0.962595 0.481298 0.876557i \(-0.340166\pi\)
0.481298 + 0.876557i \(0.340166\pi\)
\(504\) 731.623 0.0646609
\(505\) − 7378.53i − 0.650178i
\(506\) 19651.9 1.72655
\(507\) 0 0
\(508\) −3021.88 −0.263926
\(509\) 18558.6i 1.61610i 0.589115 + 0.808049i \(0.299476\pi\)
−0.589115 + 0.808049i \(0.700524\pi\)
\(510\) 2934.41 0.254780
\(511\) 9597.27 0.830838
\(512\) − 14896.8i − 1.28584i
\(513\) − 4177.22i − 0.359510i
\(514\) − 25613.6i − 2.19799i
\(515\) 5843.42i 0.499984i
\(516\) 4413.27 0.376518
\(517\) 12094.7 1.02887
\(518\) − 5494.13i − 0.466020i
\(519\) 495.730 0.0419271
\(520\) 0 0
\(521\) 17297.5 1.45454 0.727271 0.686350i \(-0.240788\pi\)
0.727271 + 0.686350i \(0.240788\pi\)
\(522\) 774.759i 0.0649622i
\(523\) −5016.11 −0.419386 −0.209693 0.977767i \(-0.567247\pi\)
−0.209693 + 0.977767i \(0.567247\pi\)
\(524\) −19441.8 −1.62084
\(525\) 192.286i 0.0159849i
\(526\) − 518.667i − 0.0429942i
\(527\) − 5415.77i − 0.447656i
\(528\) − 3674.19i − 0.302839i
\(529\) 20383.8 1.67533
\(530\) 3848.73 0.315431
\(531\) − 7863.39i − 0.642640i
\(532\) 16917.4 1.37869
\(533\) 0 0
\(534\) 3273.38 0.265268
\(535\) 6953.01i 0.561878i
\(536\) −4400.37 −0.354603
\(537\) −2137.02 −0.171730
\(538\) 8147.83i 0.652933i
\(539\) 5598.57i 0.447398i
\(540\) − 2999.06i − 0.238998i
\(541\) 17642.3i 1.40204i 0.713144 + 0.701018i \(0.247271\pi\)
−0.713144 + 0.701018i \(0.752729\pi\)
\(542\) −19415.0 −1.53865
\(543\) −6619.59 −0.523156
\(544\) 5223.86i 0.411712i
\(545\) −14862.3 −1.16813
\(546\) 0 0
\(547\) −18414.9 −1.43943 −0.719713 0.694271i \(-0.755727\pi\)
−0.719713 + 0.694271i \(0.755727\pi\)
\(548\) − 10090.4i − 0.786571i
\(549\) −1683.61 −0.130883
\(550\) −620.343 −0.0480937
\(551\) 3164.20i 0.244645i
\(552\) 3909.57i 0.301453i
\(553\) 3727.66i 0.286648i
\(554\) − 24556.9i − 1.88325i
\(555\) −3977.84 −0.304234
\(556\) 27773.1 2.11842
\(557\) 8179.15i 0.622193i 0.950378 + 0.311096i \(0.100696\pi\)
−0.950378 + 0.311096i \(0.899304\pi\)
\(558\) −10092.5 −0.765683
\(559\) 0 0
\(560\) −6089.05 −0.459481
\(561\) 1578.14i 0.118769i
\(562\) 19748.5 1.48228
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) 13622.9i 1.01707i
\(565\) 482.385i 0.0359187i
\(566\) 14586.9i 1.08327i
\(567\) − 911.598i − 0.0675194i
\(568\) 1791.62 0.132350
\(569\) −10118.3 −0.745485 −0.372743 0.927935i \(-0.621583\pi\)
−0.372743 + 0.927935i \(0.621583\pi\)
\(570\) − 22333.6i − 1.64114i
\(571\) −23428.9 −1.71711 −0.858555 0.512721i \(-0.828638\pi\)
−0.858555 + 0.512721i \(0.828638\pi\)
\(572\) 0 0
\(573\) 4411.92 0.321659
\(574\) 18530.3i 1.34746i
\(575\) −1027.52 −0.0745225
\(576\) 6327.42 0.457713
\(577\) − 20508.1i − 1.47966i −0.672793 0.739831i \(-0.734906\pi\)
0.672793 0.739831i \(-0.265094\pi\)
\(578\) 18939.8i 1.36296i
\(579\) 1108.68i 0.0795771i
\(580\) 2271.76i 0.162637i
\(581\) 4900.49 0.349925
\(582\) −15470.4 −1.10184
\(583\) 2069.87i 0.147042i
\(584\) −6159.65 −0.436452
\(585\) 0 0
\(586\) 11268.9 0.794394
\(587\) 5968.43i 0.419665i 0.977737 + 0.209833i \(0.0672919\pi\)
−0.977737 + 0.209833i \(0.932708\pi\)
\(588\) −6305.97 −0.442268
\(589\) −41219.0 −2.88353
\(590\) − 42041.8i − 2.93361i
\(591\) 12820.2i 0.892308i
\(592\) − 5489.06i − 0.381080i
\(593\) − 14659.5i − 1.01517i −0.861602 0.507584i \(-0.830539\pi\)
0.861602 0.507584i \(-0.169461\pi\)
\(594\) 2940.95 0.203146
\(595\) 2615.38 0.180202
\(596\) 7223.72i 0.496468i
\(597\) −12462.9 −0.854394
\(598\) 0 0
\(599\) 23635.9 1.61225 0.806125 0.591746i \(-0.201561\pi\)
0.806125 + 0.591746i \(0.201561\pi\)
\(600\) − 123.412i − 0.00839711i
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) 7172.15 0.485573
\(603\) 5482.83i 0.370279i
\(604\) − 22125.9i − 1.49055i
\(605\) 7560.15i 0.508039i
\(606\) 8149.77i 0.546306i
\(607\) −5098.56 −0.340930 −0.170465 0.985364i \(-0.554527\pi\)
−0.170465 + 0.985364i \(0.554527\pi\)
\(608\) 39758.4 2.65200
\(609\) 690.525i 0.0459466i
\(610\) −9001.47 −0.597473
\(611\) 0 0
\(612\) −1777.55 −0.117407
\(613\) − 1516.39i − 0.0999128i −0.998751 0.0499564i \(-0.984092\pi\)
0.998751 0.0499564i \(-0.0159082\pi\)
\(614\) 1986.31 0.130556
\(615\) 13416.3 0.879668
\(616\) 2103.70i 0.137598i
\(617\) − 18539.3i − 1.20966i −0.796353 0.604832i \(-0.793240\pi\)
0.796353 0.604832i \(-0.206760\pi\)
\(618\) − 6454.20i − 0.420107i
\(619\) 25684.9i 1.66779i 0.551920 + 0.833897i \(0.313895\pi\)
−0.551920 + 0.833897i \(0.686105\pi\)
\(620\) −29593.5 −1.91694
\(621\) 4871.30 0.314780
\(622\) − 6391.51i − 0.412020i
\(623\) 2917.49 0.187619
\(624\) 0 0
\(625\) −16304.5 −1.04349
\(626\) − 17046.1i − 1.08834i
\(627\) 12011.1 0.765038
\(628\) −30834.2 −1.95926
\(629\) 2357.67i 0.149454i
\(630\) − 4873.88i − 0.308222i
\(631\) − 22410.9i − 1.41389i −0.707269 0.706945i \(-0.750073\pi\)
0.707269 0.706945i \(-0.249927\pi\)
\(632\) − 2392.46i − 0.150580i
\(633\) 3694.77 0.231997
\(634\) −13696.1 −0.857950
\(635\) 3555.62i 0.222206i
\(636\) −2331.40 −0.145356
\(637\) 0 0
\(638\) −2227.73 −0.138239
\(639\) − 2232.34i − 0.138200i
\(640\) 10326.6 0.637804
\(641\) −6827.81 −0.420721 −0.210361 0.977624i \(-0.567464\pi\)
−0.210361 + 0.977624i \(0.567464\pi\)
\(642\) − 7679.77i − 0.472113i
\(643\) 23264.3i 1.42684i 0.700738 + 0.713418i \(0.252854\pi\)
−0.700738 + 0.713418i \(0.747146\pi\)
\(644\) 19728.4i 1.20715i
\(645\) − 5192.76i − 0.316999i
\(646\) −13237.1 −0.806204
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) 585.075i 0.0354690i
\(649\) 22610.3 1.36754
\(650\) 0 0
\(651\) −8995.26 −0.541554
\(652\) 22489.3i 1.35084i
\(653\) 10909.0 0.653755 0.326878 0.945067i \(-0.394003\pi\)
0.326878 + 0.945067i \(0.394003\pi\)
\(654\) 16415.8 0.981509
\(655\) 22875.7i 1.36462i
\(656\) 18513.2i 1.10186i
\(657\) 7674.89i 0.455747i
\(658\) 22139.0i 1.31166i
\(659\) −4182.99 −0.247263 −0.123631 0.992328i \(-0.539454\pi\)
−0.123631 + 0.992328i \(0.539454\pi\)
\(660\) 8623.47 0.508588
\(661\) 2224.23i 0.130881i 0.997856 + 0.0654406i \(0.0208453\pi\)
−0.997856 + 0.0654406i \(0.979155\pi\)
\(662\) 14405.2 0.845734
\(663\) 0 0
\(664\) −3145.19 −0.183821
\(665\) − 19905.4i − 1.16075i
\(666\) 4393.63 0.255630
\(667\) −3689.95 −0.214206
\(668\) 25899.7i 1.50013i
\(669\) 6561.72i 0.379209i
\(670\) 29314.1i 1.69030i
\(671\) − 4841.04i − 0.278519i
\(672\) 8676.51 0.498071
\(673\) 24152.5 1.38337 0.691687 0.722197i \(-0.256868\pi\)
0.691687 + 0.722197i \(0.256868\pi\)
\(674\) − 39151.2i − 2.23746i
\(675\) −153.770 −0.00876833
\(676\) 0 0
\(677\) −15310.7 −0.869187 −0.434593 0.900627i \(-0.643108\pi\)
−0.434593 + 0.900627i \(0.643108\pi\)
\(678\) − 532.806i − 0.0301804i
\(679\) −13788.4 −0.779310
\(680\) −1678.58 −0.0946628
\(681\) − 12416.0i − 0.698652i
\(682\) − 29020.0i − 1.62937i
\(683\) 11399.6i 0.638646i 0.947646 + 0.319323i \(0.103455\pi\)
−0.947646 + 0.319323i \(0.896545\pi\)
\(684\) 13528.8i 0.756265i
\(685\) −11872.6 −0.662233
\(686\) −26495.9 −1.47466
\(687\) − 2506.06i − 0.139174i
\(688\) 7165.54 0.397069
\(689\) 0 0
\(690\) 26044.5 1.43695
\(691\) 3323.23i 0.182955i 0.995807 + 0.0914773i \(0.0291589\pi\)
−0.995807 + 0.0914773i \(0.970841\pi\)
\(692\) −1605.52 −0.0881976
\(693\) 2621.20 0.143681
\(694\) − 912.936i − 0.0499345i
\(695\) − 32678.5i − 1.78355i
\(696\) − 443.188i − 0.0241365i
\(697\) − 7951.82i − 0.432133i
\(698\) −20244.8 −1.09782
\(699\) −11056.5 −0.598279
\(700\) − 622.758i − 0.0336257i
\(701\) 12670.4 0.682673 0.341336 0.939941i \(-0.389120\pi\)
0.341336 + 0.939941i \(0.389120\pi\)
\(702\) 0 0
\(703\) 17944.0 0.962692
\(704\) 18193.8i 0.974012i
\(705\) 16029.0 0.856295
\(706\) 12036.4 0.641635
\(707\) 7263.71i 0.386393i
\(708\) 25467.2i 1.35186i
\(709\) 13075.2i 0.692594i 0.938125 + 0.346297i \(0.112561\pi\)
−0.938125 + 0.346297i \(0.887439\pi\)
\(710\) − 11935.3i − 0.630877i
\(711\) −2980.99 −0.157237
\(712\) −1872.48 −0.0985592
\(713\) − 48067.8i − 2.52476i
\(714\) −2888.75 −0.151413
\(715\) 0 0
\(716\) 6921.16 0.361251
\(717\) − 9078.62i − 0.472869i
\(718\) 15514.7 0.806412
\(719\) 2988.41 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(720\) − 4869.38i − 0.252043i
\(721\) − 5752.48i − 0.297134i
\(722\) 71877.0i 3.70496i
\(723\) 9796.73i 0.503934i
\(724\) 21438.9 1.10051
\(725\) 116.479 0.00596680
\(726\) − 8350.37i − 0.426875i
\(727\) 5507.46 0.280963 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 41033.9i 2.08046i
\(731\) −3077.75 −0.155725
\(732\) 5452.71 0.275325
\(733\) 36585.2i 1.84353i 0.387751 + 0.921764i \(0.373252\pi\)
−0.387751 + 0.921764i \(0.626748\pi\)
\(734\) 14607.9i 0.734587i
\(735\) 7419.75i 0.372356i
\(736\) 46364.6i 2.32204i
\(737\) −15765.3 −0.787953
\(738\) −14818.6 −0.739133
\(739\) 6425.89i 0.319865i 0.987128 + 0.159933i \(0.0511277\pi\)
−0.987128 + 0.159933i \(0.948872\pi\)
\(740\) 12883.0 0.639986
\(741\) 0 0
\(742\) −3788.84 −0.187457
\(743\) − 20411.0i − 1.00782i −0.863757 0.503908i \(-0.831895\pi\)
0.863757 0.503908i \(-0.168105\pi\)
\(744\) 5773.27 0.284487
\(745\) 8499.60 0.417988
\(746\) 50354.6i 2.47133i
\(747\) 3918.89i 0.191947i
\(748\) − 5111.13i − 0.249842i
\(749\) − 6844.81i − 0.333917i
\(750\) 17222.4 0.838496
\(751\) 24259.5 1.17875 0.589375 0.807860i \(-0.299374\pi\)
0.589375 + 0.807860i \(0.299374\pi\)
\(752\) 22118.6i 1.07258i
\(753\) 19089.5 0.923850
\(754\) 0 0
\(755\) −26033.9 −1.25493
\(756\) 2952.39i 0.142034i
\(757\) 9295.39 0.446297 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(758\) −1454.66 −0.0697042
\(759\) 14006.9i 0.669851i
\(760\) 12775.6i 0.609761i
\(761\) − 21974.7i − 1.04676i −0.852101 0.523378i \(-0.824672\pi\)
0.852101 0.523378i \(-0.175328\pi\)
\(762\) − 3927.27i − 0.186706i
\(763\) 14631.0 0.694204
\(764\) −14288.9 −0.676641
\(765\) 2091.50i 0.0988476i
\(766\) 14253.5 0.672324
\(767\) 0 0
\(768\) 5467.13 0.256873
\(769\) − 22987.4i − 1.07795i −0.842320 0.538977i \(-0.818811\pi\)
0.842320 0.538977i \(-0.181189\pi\)
\(770\) 14014.3 0.655897
\(771\) 18256.1 0.852759
\(772\) − 3590.68i − 0.167398i
\(773\) − 31970.9i − 1.48760i −0.668404 0.743799i \(-0.733022\pi\)
0.668404 0.743799i \(-0.266978\pi\)
\(774\) 5735.53i 0.266356i
\(775\) 1517.34i 0.0703283i
\(776\) 8849.59 0.409384
\(777\) 3915.94 0.180803
\(778\) − 6860.25i − 0.316133i
\(779\) −60520.8 −2.78354
\(780\) 0 0
\(781\) 6418.85 0.294090
\(782\) − 15436.6i − 0.705896i
\(783\) −552.209 −0.0252035
\(784\) −10238.6 −0.466408
\(785\) 36280.2i 1.64955i
\(786\) − 25266.7i − 1.14661i
\(787\) 6087.26i 0.275715i 0.990452 + 0.137857i \(0.0440215\pi\)
−0.990452 + 0.137857i \(0.955978\pi\)
\(788\) − 41520.9i − 1.87706i
\(789\) 369.680 0.0166805
\(790\) −15937.9 −0.717779
\(791\) − 474.878i − 0.0213460i
\(792\) −1682.32 −0.0754780
\(793\) 0 0
\(794\) 33415.4 1.49354
\(795\) 2743.19i 0.122378i
\(796\) 40363.6 1.79730
\(797\) −23080.0 −1.02577 −0.512883 0.858458i \(-0.671423\pi\)
−0.512883 + 0.858458i \(0.671423\pi\)
\(798\) 21986.1i 0.975311i
\(799\) − 9500.41i − 0.420651i
\(800\) − 1463.57i − 0.0646813i
\(801\) 2333.10i 0.102916i
\(802\) 902.429 0.0397330
\(803\) −22068.3 −0.969829
\(804\) − 17757.3i − 0.778918i
\(805\) 23212.9 1.01633
\(806\) 0 0
\(807\) −5807.37 −0.253320
\(808\) − 4661.94i − 0.202978i
\(809\) −32377.8 −1.40710 −0.703550 0.710646i \(-0.748403\pi\)
−0.703550 + 0.710646i \(0.748403\pi\)
\(810\) 3897.61 0.169072
\(811\) − 26352.8i − 1.14103i −0.821288 0.570513i \(-0.806744\pi\)
0.821288 0.570513i \(-0.193256\pi\)
\(812\) − 2236.40i − 0.0966532i
\(813\) − 13838.1i − 0.596952i
\(814\) 12633.4i 0.543980i
\(815\) 26461.5 1.13731
\(816\) −2886.08 −0.123815
\(817\) 23424.5i 1.00308i
\(818\) 20135.0 0.860638
\(819\) 0 0
\(820\) −43451.3 −1.85047
\(821\) − 35355.3i − 1.50294i −0.659770 0.751468i \(-0.729346\pi\)
0.659770 0.751468i \(-0.270654\pi\)
\(822\) 13113.6 0.556435
\(823\) 12663.3 0.536347 0.268173 0.963371i \(-0.413580\pi\)
0.268173 + 0.963371i \(0.413580\pi\)
\(824\) 3692.02i 0.156089i
\(825\) − 442.149i − 0.0186590i
\(826\) 41387.6i 1.74341i
\(827\) − 16295.2i − 0.685176i −0.939486 0.342588i \(-0.888697\pi\)
0.939486 0.342588i \(-0.111303\pi\)
\(828\) −15776.7 −0.662170
\(829\) −13638.9 −0.571411 −0.285705 0.958318i \(-0.592228\pi\)
−0.285705 + 0.958318i \(0.592228\pi\)
\(830\) 20952.4i 0.876229i
\(831\) 17502.9 0.730649
\(832\) 0 0
\(833\) 4397.69 0.182918
\(834\) 36094.2i 1.49861i
\(835\) 30474.2 1.26300
\(836\) −38900.5 −1.60933
\(837\) − 7193.46i − 0.297064i
\(838\) 41685.0i 1.71836i
\(839\) − 1890.31i − 0.0777838i −0.999243 0.0388919i \(-0.987617\pi\)
0.999243 0.0388919i \(-0.0123828\pi\)
\(840\) 2788.02i 0.114519i
\(841\) −23970.7 −0.982849
\(842\) 51016.4 2.08805
\(843\) 14075.7i 0.575082i
\(844\) −11966.3 −0.488028
\(845\) 0 0
\(846\) −17704.5 −0.719494
\(847\) − 7442.50i − 0.301921i
\(848\) −3785.35 −0.153289
\(849\) −10396.8 −0.420279
\(850\) 487.280i 0.0196630i
\(851\) 20925.6i 0.842914i
\(852\) 7229.89i 0.290718i
\(853\) 1620.21i 0.0650351i 0.999471 + 0.0325175i \(0.0103525\pi\)
−0.999471 + 0.0325175i \(0.989648\pi\)
\(854\) 8861.39 0.355071
\(855\) 15918.3 0.636718
\(856\) 4393.08i 0.175412i
\(857\) 14508.4 0.578292 0.289146 0.957285i \(-0.406629\pi\)
0.289146 + 0.957285i \(0.406629\pi\)
\(858\) 0 0
\(859\) 29639.8 1.17730 0.588648 0.808389i \(-0.299660\pi\)
0.588648 + 0.808389i \(0.299660\pi\)
\(860\) 16817.8i 0.666839i
\(861\) −13207.5 −0.522776
\(862\) 57548.8 2.27392
\(863\) − 21528.8i − 0.849186i −0.905384 0.424593i \(-0.860417\pi\)
0.905384 0.424593i \(-0.139583\pi\)
\(864\) 6938.56i 0.273211i
\(865\) 1889.10i 0.0742557i
\(866\) 29939.4i 1.17481i
\(867\) −13499.4 −0.528792
\(868\) 29132.9 1.13921
\(869\) − 8571.50i − 0.334601i
\(870\) −2952.40 −0.115053
\(871\) 0 0
\(872\) −9390.36 −0.364676
\(873\) − 11026.5i − 0.427482i
\(874\) −117487. −4.54696
\(875\) 15349.9 0.593054
\(876\) − 24856.7i − 0.958708i
\(877\) − 14865.3i − 0.572366i −0.958175 0.286183i \(-0.907613\pi\)
0.958175 0.286183i \(-0.0923866\pi\)
\(878\) − 25347.1i − 0.974285i
\(879\) 8031.92i 0.308202i
\(880\) 14001.4 0.536348
\(881\) 21336.0 0.815921 0.407961 0.913000i \(-0.366240\pi\)
0.407961 + 0.913000i \(0.366240\pi\)
\(882\) − 8195.30i − 0.312869i
\(883\) −37538.2 −1.43065 −0.715323 0.698794i \(-0.753720\pi\)
−0.715323 + 0.698794i \(0.753720\pi\)
\(884\) 0 0
\(885\) 29965.3 1.13816
\(886\) 54694.1i 2.07391i
\(887\) 34575.0 1.30881 0.654406 0.756144i \(-0.272919\pi\)
0.654406 + 0.756144i \(0.272919\pi\)
\(888\) −2513.30 −0.0949784
\(889\) − 3500.29i − 0.132054i
\(890\) 12474.0i 0.469807i
\(891\) 2096.16i 0.0788148i
\(892\) − 21251.4i − 0.797703i
\(893\) −72306.9 −2.70958
\(894\) −9388.02 −0.351211
\(895\) − 8143.61i − 0.304146i
\(896\) −10165.9 −0.379039
\(897\) 0 0
\(898\) 46233.3 1.71807
\(899\) 5448.96i 0.202150i
\(900\) 498.016 0.0184450
\(901\) 1625.89 0.0601178
\(902\) − 42609.2i − 1.57287i
\(903\) 5111.95i 0.188389i
\(904\) 304.783i 0.0112134i
\(905\) − 25225.5i − 0.926545i
\(906\) 28755.0 1.05444
\(907\) 10424.8 0.381641 0.190820 0.981625i \(-0.438885\pi\)
0.190820 + 0.981625i \(0.438885\pi\)
\(908\) 40211.7i 1.46968i
\(909\) −5808.75 −0.211952
\(910\) 0 0
\(911\) 10961.8 0.398661 0.199331 0.979932i \(-0.436123\pi\)
0.199331 + 0.979932i \(0.436123\pi\)
\(912\) 21965.8i 0.797543i
\(913\) −11268.3 −0.408464
\(914\) −41392.2 −1.49796
\(915\) − 6415.80i − 0.231803i
\(916\) 8116.38i 0.292765i
\(917\) − 22519.7i − 0.810976i
\(918\) − 2310.12i − 0.0830558i
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) −14898.3 −0.533895
\(921\) 1415.74i 0.0506519i
\(922\) −14316.7 −0.511384
\(923\) 0 0
\(924\) −8489.28 −0.302248
\(925\) − 660.549i − 0.0234797i
\(926\) 7321.32 0.259820
\(927\) 4600.23 0.162990
\(928\) − 5255.88i − 0.185919i
\(929\) − 5429.07i − 0.191735i −0.995394 0.0958675i \(-0.969437\pi\)
0.995394 0.0958675i \(-0.0305625\pi\)
\(930\) − 38460.0i − 1.35608i
\(931\) − 33470.5i − 1.17825i
\(932\) 35808.8 1.25854
\(933\) 4555.55 0.159852
\(934\) − 33499.1i − 1.17358i
\(935\) −6013.88 −0.210348
\(936\) 0 0
\(937\) −21300.1 −0.742631 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(938\) − 28857.9i − 1.00453i
\(939\) 12149.6 0.422244
\(940\) −51913.2 −1.80130
\(941\) − 26851.2i − 0.930207i −0.885256 0.465103i \(-0.846017\pi\)
0.885256 0.465103i \(-0.153983\pi\)
\(942\) − 40072.4i − 1.38602i
\(943\) − 70576.7i − 2.43722i
\(944\) 41349.4i 1.42564i
\(945\) 3473.86 0.119582
\(946\) −16491.9 −0.566805
\(947\) − 8021.68i − 0.275258i −0.990484 0.137629i \(-0.956052\pi\)
0.990484 0.137629i \(-0.0439482\pi\)
\(948\) 9654.52 0.330764
\(949\) 0 0
\(950\) 3708.65 0.126658
\(951\) − 9761.88i − 0.332861i
\(952\) 1652.46 0.0562569
\(953\) −35715.0 −1.21398 −0.606990 0.794709i \(-0.707623\pi\)
−0.606990 + 0.794709i \(0.707623\pi\)
\(954\) − 3029.92i − 0.102827i
\(955\) 16812.6i 0.569680i
\(956\) 29402.9i 0.994726i
\(957\) − 1587.82i − 0.0536330i
\(958\) 35490.6 1.19692
\(959\) 11687.9 0.393557
\(960\) 24112.1i 0.810641i
\(961\) −41190.9 −1.38266
\(962\) 0 0
\(963\) 5473.76 0.183166
\(964\) − 31728.7i − 1.06007i
\(965\) −4224.88 −0.140936
\(966\) −25639.2 −0.853963
\(967\) − 53338.8i − 1.77380i −0.461965 0.886898i \(-0.652855\pi\)
0.461965 0.886898i \(-0.347145\pi\)
\(968\) 4776.69i 0.158604i
\(969\) − 9434.77i − 0.312785i
\(970\) − 58953.6i − 1.95143i
\(971\) 23112.9 0.763882 0.381941 0.924187i \(-0.375256\pi\)
0.381941 + 0.924187i \(0.375256\pi\)
\(972\) −2361.01 −0.0779110
\(973\) 32170.0i 1.05994i
\(974\) 49181.3 1.61794
\(975\) 0 0
\(976\) 8853.22 0.290353
\(977\) 52874.6i 1.73143i 0.500538 + 0.865715i \(0.333136\pi\)
−0.500538 + 0.865715i \(0.666864\pi\)
\(978\) −29227.4 −0.955612
\(979\) −6708.57 −0.219006
\(980\) − 24030.4i − 0.783287i
\(981\) 11700.3i 0.380798i
\(982\) − 16645.5i − 0.540917i
\(983\) 45173.1i 1.46572i 0.680381 + 0.732858i \(0.261814\pi\)
−0.680381 + 0.732858i \(0.738186\pi\)
\(984\) 8476.73 0.274622
\(985\) −48854.5 −1.58034
\(986\) 1749.89i 0.0565190i
\(987\) −15779.6 −0.508885
\(988\) 0 0
\(989\) −27316.7 −0.878281
\(990\) 11207.2i 0.359785i
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) 68466.7 2.19135
\(993\) 10267.3i 0.328121i
\(994\) 11749.5i 0.374922i
\(995\) − 47492.8i − 1.51319i
\(996\) − 12692.1i − 0.403780i
\(997\) −18108.1 −0.575214 −0.287607 0.957749i \(-0.592860\pi\)
−0.287607 + 0.957749i \(0.592860\pi\)
\(998\) 23951.1 0.759677
\(999\) 3131.56i 0.0991773i
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.g.337.1 6
13.5 odd 4 507.4.a.h.1.1 3
13.8 odd 4 39.4.a.c.1.3 3
13.12 even 2 inner 507.4.b.g.337.6 6
39.5 even 4 1521.4.a.u.1.3 3
39.8 even 4 117.4.a.f.1.1 3
52.47 even 4 624.4.a.t.1.1 3
65.34 odd 4 975.4.a.l.1.1 3
91.34 even 4 1911.4.a.k.1.3 3
104.21 odd 4 2496.4.a.bl.1.3 3
104.99 even 4 2496.4.a.bp.1.3 3
156.47 odd 4 1872.4.a.bk.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 13.8 odd 4
117.4.a.f.1.1 3 39.8 even 4
507.4.a.h.1.1 3 13.5 odd 4
507.4.b.g.337.1 6 1.1 even 1 trivial
507.4.b.g.337.6 6 13.12 even 2 inner
624.4.a.t.1.1 3 52.47 even 4
975.4.a.l.1.1 3 65.34 odd 4
1521.4.a.u.1.3 3 39.5 even 4
1872.4.a.bk.1.3 3 156.47 odd 4
1911.4.a.k.1.3 3 91.34 even 4
2496.4.a.bl.1.3 3 104.21 odd 4
2496.4.a.bp.1.3 3 104.99 even 4