Properties

Label 975.4.a.l.1.1
Level $975$
Weight $4$
Character 975.1
Self dual yes
Analytic conductor $57.527$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.5268622556\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 975.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +25.8785 q^{11} -29.1483 q^{12} -13.0000 q^{13} -47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -37.8814 q^{18} +154.712 q^{19} -33.7629 q^{21} -108.924 q^{22} +180.418 q^{23} +21.6695 q^{24} +54.7176 q^{26} -27.0000 q^{27} +109.348 q^{28} -20.4522 q^{29} +266.424 q^{31} +256.984 q^{32} -77.6355 q^{33} -85.5599 q^{34} +87.4449 q^{36} -115.984 q^{37} -651.190 q^{38} +39.0000 q^{39} +391.184 q^{41} +142.110 q^{42} -151.407 q^{43} +251.438 q^{44} -759.390 q^{46} +467.365 q^{47} +141.979 q^{48} -216.341 q^{49} -60.9828 q^{51} -126.309 q^{52} -79.9842 q^{53} +113.644 q^{54} -81.2915 q^{56} -464.136 q^{57} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} +101.289 q^{63} -703.047 q^{64} +326.772 q^{66} +609.204 q^{67} +197.505 q^{68} -541.255 q^{69} +248.038 q^{71} -65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} +1503.20 q^{76} +291.244 q^{77} -164.153 q^{78} -331.221 q^{79} +81.0000 q^{81} -1646.51 q^{82} +435.432 q^{83} -328.044 q^{84} +637.281 q^{86} +61.3566 q^{87} -186.924 q^{88} +259.233 q^{89} -146.306 q^{91} +1752.96 q^{92} -799.273 q^{93} -1967.16 q^{94} -770.951 q^{96} -1225.17 q^{97} +910.589 q^{98} +232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 16 q^{11} - 30 q^{12} - 39 q^{13} - 176 q^{14} - 110 q^{16} + 146 q^{17} - 18 q^{18} + 94 q^{19} + 90 q^{21} + 56 q^{22} + 48 q^{23} - 18 q^{24} + 26 q^{26} - 81 q^{27} - 80 q^{28} - 2 q^{29} + 302 q^{31} - 154 q^{32} + 48 q^{33} + 164 q^{34} + 90 q^{36} - 374 q^{37} - 312 q^{38} + 117 q^{39} + 480 q^{41} + 528 q^{42} + 260 q^{43} + 712 q^{44} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 447 q^{49} - 438 q^{51} - 130 q^{52} + 678 q^{53} + 54 q^{54} + 96 q^{56} - 282 q^{57} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} + 460 q^{68} - 144 q^{69} - 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 78 q^{78} - 24 q^{79} + 243 q^{81} - 564 q^{82} + 796 q^{83} + 240 q^{84} + 1800 q^{86} + 6 q^{87} - 1608 q^{88} + 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 906 q^{93} - 1920 q^{94} + 462 q^{96} - 3242 q^{97} + 5070 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.20905 −1.48812 −0.744062 0.668111i \(-0.767103\pi\)
−0.744062 + 0.668111i \(0.767103\pi\)
\(3\) −3.00000 −0.577350
\(4\) 9.71610 1.21451
\(5\) 0 0
\(6\) 12.6271 0.859169
\(7\) 11.2543 0.607675 0.303838 0.952724i \(-0.401732\pi\)
0.303838 + 0.952724i \(0.401732\pi\)
\(8\) −7.22315 −0.319221
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 25.8785 0.709333 0.354666 0.934993i \(-0.384594\pi\)
0.354666 + 0.934993i \(0.384594\pi\)
\(12\) −29.1483 −0.701199
\(13\) −13.0000 −0.277350
\(14\) −47.3699 −0.904296
\(15\) 0 0
\(16\) −47.3262 −0.739472
\(17\) 20.3276 0.290010 0.145005 0.989431i \(-0.453680\pi\)
0.145005 + 0.989431i \(0.453680\pi\)
\(18\) −37.8814 −0.496041
\(19\) 154.712 1.86807 0.934035 0.357181i \(-0.116262\pi\)
0.934035 + 0.357181i \(0.116262\pi\)
\(20\) 0 0
\(21\) −33.7629 −0.350841
\(22\) −108.924 −1.05558
\(23\) 180.418 1.63565 0.817823 0.575471i \(-0.195181\pi\)
0.817823 + 0.575471i \(0.195181\pi\)
\(24\) 21.6695 0.184302
\(25\) 0 0
\(26\) 54.7176 0.412731
\(27\) −27.0000 −0.192450
\(28\) 109.348 0.738029
\(29\) −20.4522 −0.130961 −0.0654806 0.997854i \(-0.520858\pi\)
−0.0654806 + 0.997854i \(0.520858\pi\)
\(30\) 0 0
\(31\) 266.424 1.54359 0.771794 0.635873i \(-0.219360\pi\)
0.771794 + 0.635873i \(0.219360\pi\)
\(32\) 256.984 1.41965
\(33\) −77.6355 −0.409534
\(34\) −85.5599 −0.431571
\(35\) 0 0
\(36\) 87.4449 0.404837
\(37\) −115.984 −0.515340 −0.257670 0.966233i \(-0.582955\pi\)
−0.257670 + 0.966233i \(0.582955\pi\)
\(38\) −651.190 −2.77992
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 391.184 1.49006 0.745032 0.667029i \(-0.232434\pi\)
0.745032 + 0.667029i \(0.232434\pi\)
\(42\) 142.110 0.522095
\(43\) −151.407 −0.536963 −0.268482 0.963285i \(-0.586522\pi\)
−0.268482 + 0.963285i \(0.586522\pi\)
\(44\) 251.438 0.861494
\(45\) 0 0
\(46\) −759.390 −2.43404
\(47\) 467.365 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(48\) 141.979 0.426934
\(49\) −216.341 −0.630731
\(50\) 0 0
\(51\) −60.9828 −0.167437
\(52\) −126.309 −0.336845
\(53\) −79.9842 −0.207296 −0.103648 0.994614i \(-0.533051\pi\)
−0.103648 + 0.994614i \(0.533051\pi\)
\(54\) 113.644 0.286390
\(55\) 0 0
\(56\) −81.2915 −0.193983
\(57\) −464.136 −1.07853
\(58\) 86.0843 0.194887
\(59\) −873.710 −1.92792 −0.963960 0.266045i \(-0.914283\pi\)
−0.963960 + 0.266045i \(0.914283\pi\)
\(60\) 0 0
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) −1121.39 −2.29705
\(63\) 101.289 0.202558
\(64\) −703.047 −1.37314
\(65\) 0 0
\(66\) 326.772 0.609437
\(67\) 609.204 1.11084 0.555418 0.831571i \(-0.312558\pi\)
0.555418 + 0.831571i \(0.312558\pi\)
\(68\) 197.505 0.352221
\(69\) −541.255 −0.944340
\(70\) 0 0
\(71\) 248.038 0.414601 0.207301 0.978277i \(-0.433532\pi\)
0.207301 + 0.978277i \(0.433532\pi\)
\(72\) −65.0084 −0.106407
\(73\) −852.765 −1.36724 −0.683621 0.729838i \(-0.739596\pi\)
−0.683621 + 0.729838i \(0.739596\pi\)
\(74\) 488.181 0.766890
\(75\) 0 0
\(76\) 1503.20 2.26880
\(77\) 291.244 0.431044
\(78\) −164.153 −0.238291
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −1646.51 −2.21740
\(83\) 435.432 0.575842 0.287921 0.957654i \(-0.407036\pi\)
0.287921 + 0.957654i \(0.407036\pi\)
\(84\) −328.044 −0.426101
\(85\) 0 0
\(86\) 637.281 0.799067
\(87\) 61.3566 0.0756105
\(88\) −186.924 −0.226434
\(89\) 259.233 0.308749 0.154375 0.988012i \(-0.450664\pi\)
0.154375 + 0.988012i \(0.450664\pi\)
\(90\) 0 0
\(91\) −146.306 −0.168539
\(92\) 1752.96 1.98651
\(93\) −799.273 −0.891191
\(94\) −1967.16 −2.15848
\(95\) 0 0
\(96\) −770.951 −0.819634
\(97\) −1225.17 −1.28245 −0.641223 0.767355i \(-0.721572\pi\)
−0.641223 + 0.767355i \(0.721572\pi\)
\(98\) 910.589 0.938606
\(99\) 232.907 0.236444
\(100\) 0 0
\(101\) 645.416 0.635855 0.317927 0.948115i \(-0.397013\pi\)
0.317927 + 0.948115i \(0.397013\pi\)
\(102\) 256.680 0.249167
\(103\) 511.137 0.488969 0.244484 0.969653i \(-0.421381\pi\)
0.244484 + 0.969653i \(0.421381\pi\)
\(104\) 93.9010 0.0885360
\(105\) 0 0
\(106\) 336.657 0.308482
\(107\) −608.195 −0.549499 −0.274750 0.961516i \(-0.588595\pi\)
−0.274750 + 0.961516i \(0.588595\pi\)
\(108\) −262.335 −0.233733
\(109\) −1300.04 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(110\) 0 0
\(111\) 347.951 0.297532
\(112\) −532.623 −0.449359
\(113\) −42.1953 −0.0351274 −0.0175637 0.999846i \(-0.505591\pi\)
−0.0175637 + 0.999846i \(0.505591\pi\)
\(114\) 1953.57 1.60499
\(115\) 0 0
\(116\) −198.716 −0.159054
\(117\) −117.000 −0.0924500
\(118\) 3677.49 2.86899
\(119\) 228.773 0.176232
\(120\) 0 0
\(121\) −661.303 −0.496847
\(122\) 787.378 0.584311
\(123\) −1173.55 −0.860289
\(124\) 2588.61 1.87471
\(125\) 0 0
\(126\) −426.329 −0.301432
\(127\) 311.018 0.217310 0.108655 0.994080i \(-0.465346\pi\)
0.108655 + 0.994080i \(0.465346\pi\)
\(128\) 903.291 0.623753
\(129\) 454.222 0.310016
\(130\) 0 0
\(131\) 2000.98 1.33456 0.667278 0.744809i \(-0.267459\pi\)
0.667278 + 0.744809i \(0.267459\pi\)
\(132\) −754.314 −0.497384
\(133\) 1741.17 1.13518
\(134\) −2564.17 −1.65306
\(135\) 0 0
\(136\) −146.829 −0.0925773
\(137\) −1038.53 −0.647644 −0.323822 0.946118i \(-0.604968\pi\)
−0.323822 + 0.946118i \(0.604968\pi\)
\(138\) 2278.17 1.40529
\(139\) −2858.46 −1.74426 −0.872128 0.489277i \(-0.837261\pi\)
−0.872128 + 0.489277i \(0.837261\pi\)
\(140\) 0 0
\(141\) −1402.09 −0.837430
\(142\) −1044.00 −0.616978
\(143\) −336.421 −0.196734
\(144\) −425.936 −0.246491
\(145\) 0 0
\(146\) 3589.33 2.03462
\(147\) 649.022 0.364153
\(148\) −1126.91 −0.625887
\(149\) 743.479 0.408780 0.204390 0.978890i \(-0.434479\pi\)
0.204390 + 0.978890i \(0.434479\pi\)
\(150\) 0 0
\(151\) 2277.24 1.22728 0.613640 0.789586i \(-0.289705\pi\)
0.613640 + 0.789586i \(0.289705\pi\)
\(152\) −1117.51 −0.596328
\(153\) 182.948 0.0966700
\(154\) −1225.86 −0.641447
\(155\) 0 0
\(156\) 378.928 0.194478
\(157\) −3173.51 −1.61321 −0.806605 0.591091i \(-0.798697\pi\)
−0.806605 + 0.591091i \(0.798697\pi\)
\(158\) 1394.12 0.701966
\(159\) 239.953 0.119682
\(160\) 0 0
\(161\) 2030.48 0.993941
\(162\) −340.933 −0.165347
\(163\) 2314.65 1.11225 0.556126 0.831098i \(-0.312287\pi\)
0.556126 + 0.831098i \(0.312287\pi\)
\(164\) 3800.78 1.80970
\(165\) 0 0
\(166\) −1832.76 −0.856925
\(167\) 2665.65 1.23517 0.617587 0.786502i \(-0.288110\pi\)
0.617587 + 0.786502i \(0.288110\pi\)
\(168\) 243.874 0.111996
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 1392.41 0.622690
\(172\) −1471.09 −0.652148
\(173\) 165.243 0.0726198 0.0363099 0.999341i \(-0.488440\pi\)
0.0363099 + 0.999341i \(0.488440\pi\)
\(174\) −258.253 −0.112518
\(175\) 0 0
\(176\) −1224.73 −0.524532
\(177\) 2621.13 1.11309
\(178\) −1091.13 −0.459457
\(179\) 712.339 0.297446 0.148723 0.988879i \(-0.452484\pi\)
0.148723 + 0.988879i \(0.452484\pi\)
\(180\) 0 0
\(181\) 2206.53 0.906133 0.453066 0.891477i \(-0.350330\pi\)
0.453066 + 0.891477i \(0.350330\pi\)
\(182\) 615.809 0.250806
\(183\) 561.204 0.226696
\(184\) −1303.19 −0.522132
\(185\) 0 0
\(186\) 3364.18 1.32620
\(187\) 526.048 0.205714
\(188\) 4540.96 1.76162
\(189\) −303.866 −0.116947
\(190\) 0 0
\(191\) 1470.64 0.557129 0.278565 0.960417i \(-0.410141\pi\)
0.278565 + 0.960417i \(0.410141\pi\)
\(192\) 2109.14 0.792782
\(193\) −369.560 −0.137832 −0.0689158 0.997622i \(-0.521954\pi\)
−0.0689158 + 0.997622i \(0.521954\pi\)
\(194\) 5156.80 1.90844
\(195\) 0 0
\(196\) −2101.99 −0.766031
\(197\) 4273.41 1.54552 0.772761 0.634697i \(-0.218875\pi\)
0.772761 + 0.634697i \(0.218875\pi\)
\(198\) −980.315 −0.351858
\(199\) 4154.31 1.47985 0.739927 0.672687i \(-0.234860\pi\)
0.739927 + 0.672687i \(0.234860\pi\)
\(200\) 0 0
\(201\) −1827.61 −0.641342
\(202\) −2716.59 −0.946230
\(203\) −230.175 −0.0795819
\(204\) −592.515 −0.203355
\(205\) 0 0
\(206\) −2151.40 −0.727646
\(207\) 1623.77 0.545215
\(208\) 615.241 0.205093
\(209\) 4003.71 1.32508
\(210\) 0 0
\(211\) 1231.59 0.401830 0.200915 0.979609i \(-0.435608\pi\)
0.200915 + 0.979609i \(0.435608\pi\)
\(212\) −777.134 −0.251763
\(213\) −744.114 −0.239370
\(214\) 2559.92 0.817723
\(215\) 0 0
\(216\) 195.025 0.0614341
\(217\) 2998.42 0.938000
\(218\) 5471.92 1.70002
\(219\) 2558.30 0.789377
\(220\) 0 0
\(221\) −264.259 −0.0804343
\(222\) −1464.54 −0.442764
\(223\) 2187.24 0.656809 0.328404 0.944537i \(-0.393489\pi\)
0.328404 + 0.944537i \(0.393489\pi\)
\(224\) 2892.17 0.862684
\(225\) 0 0
\(226\) 177.602 0.0522739
\(227\) −4138.67 −1.21010 −0.605051 0.796187i \(-0.706847\pi\)
−0.605051 + 0.796187i \(0.706847\pi\)
\(228\) −4509.59 −1.30989
\(229\) −835.354 −0.241056 −0.120528 0.992710i \(-0.538459\pi\)
−0.120528 + 0.992710i \(0.538459\pi\)
\(230\) 0 0
\(231\) −873.733 −0.248863
\(232\) 147.729 0.0418056
\(233\) −3685.51 −1.03625 −0.518124 0.855305i \(-0.673370\pi\)
−0.518124 + 0.855305i \(0.673370\pi\)
\(234\) 492.459 0.137577
\(235\) 0 0
\(236\) −8489.05 −2.34148
\(237\) 993.662 0.272343
\(238\) −962.917 −0.262255
\(239\) 3026.21 0.819034 0.409517 0.912303i \(-0.365697\pi\)
0.409517 + 0.912303i \(0.365697\pi\)
\(240\) 0 0
\(241\) 3265.58 0.872839 0.436420 0.899743i \(-0.356246\pi\)
0.436420 + 0.899743i \(0.356246\pi\)
\(242\) 2783.46 0.739370
\(243\) −243.000 −0.0641500
\(244\) −1817.57 −0.476877
\(245\) 0 0
\(246\) 4939.53 1.28022
\(247\) −2011.25 −0.518110
\(248\) −1924.42 −0.492746
\(249\) −1306.30 −0.332463
\(250\) 0 0
\(251\) −6363.16 −1.60016 −0.800078 0.599897i \(-0.795208\pi\)
−0.800078 + 0.599897i \(0.795208\pi\)
\(252\) 984.131 0.246010
\(253\) 4668.96 1.16022
\(254\) −1309.09 −0.323385
\(255\) 0 0
\(256\) 1822.38 0.444917
\(257\) 6085.36 1.47702 0.738511 0.674242i \(-0.235529\pi\)
0.738511 + 0.674242i \(0.235529\pi\)
\(258\) −1911.84 −0.461342
\(259\) −1305.31 −0.313159
\(260\) 0 0
\(261\) −184.070 −0.0436538
\(262\) −8422.24 −1.98598
\(263\) −123.227 −0.0288916 −0.0144458 0.999896i \(-0.504598\pi\)
−0.0144458 + 0.999896i \(0.504598\pi\)
\(264\) 560.773 0.130732
\(265\) 0 0
\(266\) −7328.69 −1.68929
\(267\) −777.700 −0.178256
\(268\) 5919.08 1.34913
\(269\) −1935.79 −0.438763 −0.219381 0.975639i \(-0.570404\pi\)
−0.219381 + 0.975639i \(0.570404\pi\)
\(270\) 0 0
\(271\) −4612.69 −1.03395 −0.516976 0.856000i \(-0.672942\pi\)
−0.516976 + 0.856000i \(0.672942\pi\)
\(272\) −962.028 −0.214454
\(273\) 438.918 0.0973059
\(274\) 4371.20 0.963774
\(275\) 0 0
\(276\) −5258.89 −1.14691
\(277\) 5834.30 1.26552 0.632761 0.774347i \(-0.281922\pi\)
0.632761 + 0.774347i \(0.281922\pi\)
\(278\) 12031.4 2.59567
\(279\) 2397.82 0.514529
\(280\) 0 0
\(281\) 4691.91 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(282\) 5901.49 1.24620
\(283\) −3465.60 −0.727945 −0.363973 0.931410i \(-0.618580\pi\)
−0.363973 + 0.931410i \(0.618580\pi\)
\(284\) 2409.96 0.503539
\(285\) 0 0
\(286\) 1416.01 0.292764
\(287\) 4402.50 0.905475
\(288\) 2312.85 0.473216
\(289\) −4499.79 −0.915894
\(290\) 0 0
\(291\) 3675.51 0.740420
\(292\) −8285.55 −1.66053
\(293\) −2677.31 −0.533822 −0.266911 0.963721i \(-0.586003\pi\)
−0.266911 + 0.963721i \(0.586003\pi\)
\(294\) −2731.77 −0.541904
\(295\) 0 0
\(296\) 837.767 0.164507
\(297\) −698.720 −0.136511
\(298\) −3129.34 −0.608315
\(299\) −2345.44 −0.453646
\(300\) 0 0
\(301\) −1703.98 −0.326299
\(302\) −9585.02 −1.82634
\(303\) −1936.25 −0.367111
\(304\) −7321.93 −1.38139
\(305\) 0 0
\(306\) −770.039 −0.143857
\(307\) −471.915 −0.0877316 −0.0438658 0.999037i \(-0.513967\pi\)
−0.0438658 + 0.999037i \(0.513967\pi\)
\(308\) 2829.76 0.523508
\(309\) −1533.41 −0.282306
\(310\) 0 0
\(311\) −1518.52 −0.276872 −0.138436 0.990371i \(-0.544207\pi\)
−0.138436 + 0.990371i \(0.544207\pi\)
\(312\) −281.703 −0.0511163
\(313\) −4049.86 −0.731348 −0.365674 0.930743i \(-0.619161\pi\)
−0.365674 + 0.930743i \(0.619161\pi\)
\(314\) 13357.5 2.40066
\(315\) 0 0
\(316\) −3218.17 −0.572900
\(317\) −3253.96 −0.576532 −0.288266 0.957550i \(-0.593079\pi\)
−0.288266 + 0.957550i \(0.593079\pi\)
\(318\) −1009.97 −0.178102
\(319\) −529.272 −0.0928951
\(320\) 0 0
\(321\) 1824.59 0.317254
\(322\) −8546.40 −1.47911
\(323\) 3144.92 0.541759
\(324\) 787.004 0.134946
\(325\) 0 0
\(326\) −9742.46 −1.65517
\(327\) 3900.11 0.659561
\(328\) −2825.58 −0.475660
\(329\) 5259.86 0.881415
\(330\) 0 0
\(331\) −3422.45 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(332\) 4230.71 0.699368
\(333\) −1043.85 −0.171780
\(334\) −11219.8 −1.83809
\(335\) 0 0
\(336\) 1597.87 0.259437
\(337\) 9301.67 1.50354 0.751772 0.659423i \(-0.229199\pi\)
0.751772 + 0.659423i \(0.229199\pi\)
\(338\) −711.329 −0.114471
\(339\) 126.586 0.0202808
\(340\) 0 0
\(341\) 6894.66 1.09492
\(342\) −5860.71 −0.926640
\(343\) −6294.99 −0.990955
\(344\) 1093.64 0.171410
\(345\) 0 0
\(346\) −695.518 −0.108067
\(347\) −216.898 −0.0335554 −0.0167777 0.999859i \(-0.505341\pi\)
−0.0167777 + 0.999859i \(0.505341\pi\)
\(348\) 596.147 0.0918299
\(349\) −4809.84 −0.737721 −0.368861 0.929485i \(-0.620252\pi\)
−0.368861 + 0.929485i \(0.620252\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 6650.35 1.00700
\(353\) 2859.64 0.431170 0.215585 0.976485i \(-0.430834\pi\)
0.215585 + 0.976485i \(0.430834\pi\)
\(354\) −11032.5 −1.65641
\(355\) 0 0
\(356\) 2518.74 0.374980
\(357\) −686.319 −0.101747
\(358\) −2998.27 −0.442636
\(359\) 3686.04 0.541899 0.270949 0.962594i \(-0.412662\pi\)
0.270949 + 0.962594i \(0.412662\pi\)
\(360\) 0 0
\(361\) 17076.8 2.48969
\(362\) −9287.39 −1.34844
\(363\) 1983.91 0.286855
\(364\) −1421.52 −0.204692
\(365\) 0 0
\(366\) −2362.14 −0.337352
\(367\) 3470.59 0.493633 0.246816 0.969062i \(-0.420616\pi\)
0.246816 + 0.969062i \(0.420616\pi\)
\(368\) −8538.52 −1.20951
\(369\) 3520.65 0.496688
\(370\) 0 0
\(371\) −900.166 −0.125968
\(372\) −7765.82 −1.08236
\(373\) 11963.4 1.66070 0.830352 0.557240i \(-0.188140\pi\)
0.830352 + 0.557240i \(0.188140\pi\)
\(374\) −2214.16 −0.306127
\(375\) 0 0
\(376\) −3375.85 −0.463021
\(377\) 265.879 0.0363221
\(378\) 1278.99 0.174032
\(379\) 345.604 0.0468403 0.0234202 0.999726i \(-0.492544\pi\)
0.0234202 + 0.999726i \(0.492544\pi\)
\(380\) 0 0
\(381\) −933.055 −0.125464
\(382\) −6189.99 −0.829078
\(383\) 3386.40 0.451793 0.225897 0.974151i \(-0.427469\pi\)
0.225897 + 0.974151i \(0.427469\pi\)
\(384\) −2709.87 −0.360124
\(385\) 0 0
\(386\) 1555.49 0.205110
\(387\) −1362.67 −0.178988
\(388\) −11903.9 −1.55755
\(389\) −1629.88 −0.212438 −0.106219 0.994343i \(-0.533874\pi\)
−0.106219 + 0.994343i \(0.533874\pi\)
\(390\) 0 0
\(391\) 3667.47 0.474353
\(392\) 1562.66 0.201343
\(393\) −6002.95 −0.770506
\(394\) −17987.0 −2.29993
\(395\) 0 0
\(396\) 2262.94 0.287165
\(397\) −7938.94 −1.00364 −0.501819 0.864973i \(-0.667336\pi\)
−0.501819 + 0.864973i \(0.667336\pi\)
\(398\) −17485.7 −2.20221
\(399\) −5223.52 −0.655396
\(400\) 0 0
\(401\) 214.402 0.0267001 0.0133500 0.999911i \(-0.495750\pi\)
0.0133500 + 0.999911i \(0.495750\pi\)
\(402\) 7692.51 0.954396
\(403\) −3463.52 −0.428114
\(404\) 6270.93 0.772253
\(405\) 0 0
\(406\) 968.819 0.118428
\(407\) −3001.48 −0.365548
\(408\) 440.488 0.0534495
\(409\) −4783.73 −0.578338 −0.289169 0.957278i \(-0.593379\pi\)
−0.289169 + 0.957278i \(0.593379\pi\)
\(410\) 0 0
\(411\) 3115.58 0.373917
\(412\) 4966.25 0.593859
\(413\) −9832.99 −1.17155
\(414\) −6834.51 −0.811347
\(415\) 0 0
\(416\) −3340.79 −0.393739
\(417\) 8575.39 1.00705
\(418\) −16851.8 −1.97189
\(419\) −9903.67 −1.15472 −0.577358 0.816491i \(-0.695916\pi\)
−0.577358 + 0.816491i \(0.695916\pi\)
\(420\) 0 0
\(421\) −12120.6 −1.40314 −0.701572 0.712598i \(-0.747518\pi\)
−0.701572 + 0.712598i \(0.747518\pi\)
\(422\) −5183.82 −0.597973
\(423\) 4206.28 0.483491
\(424\) 577.738 0.0661732
\(425\) 0 0
\(426\) 3132.01 0.356213
\(427\) −2105.32 −0.238603
\(428\) −5909.28 −0.667374
\(429\) 1009.26 0.113584
\(430\) 0 0
\(431\) −13672.6 −1.52805 −0.764023 0.645189i \(-0.776779\pi\)
−0.764023 + 0.645189i \(0.776779\pi\)
\(432\) 1277.81 0.142311
\(433\) −7113.10 −0.789455 −0.394727 0.918798i \(-0.629161\pi\)
−0.394727 + 0.918798i \(0.629161\pi\)
\(434\) −12620.5 −1.39586
\(435\) 0 0
\(436\) −12631.3 −1.38745
\(437\) 27912.9 3.05550
\(438\) −10768.0 −1.17469
\(439\) −6022.04 −0.654707 −0.327353 0.944902i \(-0.606157\pi\)
−0.327353 + 0.944902i \(0.606157\pi\)
\(440\) 0 0
\(441\) −1947.07 −0.210244
\(442\) 1112.28 0.119696
\(443\) 12994.4 1.39364 0.696821 0.717245i \(-0.254597\pi\)
0.696821 + 0.717245i \(0.254597\pi\)
\(444\) 3380.72 0.361356
\(445\) 0 0
\(446\) −9206.20 −0.977413
\(447\) −2230.44 −0.236009
\(448\) −7912.30 −0.834422
\(449\) 10984.3 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(450\) 0 0
\(451\) 10123.2 1.05695
\(452\) −409.973 −0.0426627
\(453\) −6831.72 −0.708570
\(454\) 17419.9 1.80078
\(455\) 0 0
\(456\) 3352.52 0.344290
\(457\) −9834.10 −1.00661 −0.503304 0.864109i \(-0.667882\pi\)
−0.503304 + 0.864109i \(0.667882\pi\)
\(458\) 3516.05 0.358721
\(459\) −548.845 −0.0558124
\(460\) 0 0
\(461\) 3401.42 0.343644 0.171822 0.985128i \(-0.445035\pi\)
0.171822 + 0.985128i \(0.445035\pi\)
\(462\) 3677.59 0.370339
\(463\) −1739.42 −0.174596 −0.0872979 0.996182i \(-0.527823\pi\)
−0.0872979 + 0.996182i \(0.527823\pi\)
\(464\) 967.925 0.0968422
\(465\) 0 0
\(466\) 15512.5 1.54207
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) −1136.78 −0.112282
\(469\) 6856.16 0.675028
\(470\) 0 0
\(471\) 9520.54 0.931387
\(472\) 6310.94 0.615433
\(473\) −3918.20 −0.380886
\(474\) −4182.37 −0.405280
\(475\) 0 0
\(476\) 2222.78 0.214036
\(477\) −719.858 −0.0690986
\(478\) −12737.5 −1.21882
\(479\) 8431.98 0.804315 0.402158 0.915570i \(-0.368260\pi\)
0.402158 + 0.915570i \(0.368260\pi\)
\(480\) 0 0
\(481\) 1507.79 0.142930
\(482\) −13745.0 −1.29889
\(483\) −6091.45 −0.573852
\(484\) −6425.29 −0.603427
\(485\) 0 0
\(486\) 1022.80 0.0954632
\(487\) 11684.7 1.08723 0.543617 0.839334i \(-0.317055\pi\)
0.543617 + 0.839334i \(0.317055\pi\)
\(488\) 1351.22 0.125342
\(489\) −6943.94 −0.642159
\(490\) 0 0
\(491\) −3954.70 −0.363489 −0.181745 0.983346i \(-0.558174\pi\)
−0.181745 + 0.983346i \(0.558174\pi\)
\(492\) −11402.3 −1.04483
\(493\) −415.744 −0.0379801
\(494\) 8465.47 0.771011
\(495\) 0 0
\(496\) −12608.8 −1.14144
\(497\) 2791.49 0.251943
\(498\) 5498.27 0.494746
\(499\) −5690.37 −0.510493 −0.255246 0.966876i \(-0.582157\pi\)
−0.255246 + 0.966876i \(0.582157\pi\)
\(500\) 0 0
\(501\) −7996.95 −0.713128
\(502\) 26782.8 2.38123
\(503\) −10859.1 −0.962595 −0.481298 0.876557i \(-0.659834\pi\)
−0.481298 + 0.876557i \(0.659834\pi\)
\(504\) −731.623 −0.0646609
\(505\) 0 0
\(506\) −19651.9 −1.72655
\(507\) −507.000 −0.0444116
\(508\) 3021.88 0.263926
\(509\) −18558.6 −1.61610 −0.808049 0.589115i \(-0.799476\pi\)
−0.808049 + 0.589115i \(0.799476\pi\)
\(510\) 0 0
\(511\) −9597.27 −0.830838
\(512\) −14896.8 −1.28584
\(513\) −4177.22 −0.359510
\(514\) −25613.6 −2.19799
\(515\) 0 0
\(516\) 4413.27 0.376518
\(517\) 12094.7 1.02887
\(518\) 5494.13 0.466020
\(519\) −495.730 −0.0419271
\(520\) 0 0
\(521\) 17297.5 1.45454 0.727271 0.686350i \(-0.240788\pi\)
0.727271 + 0.686350i \(0.240788\pi\)
\(522\) 774.759 0.0649622
\(523\) 5016.11 0.419386 0.209693 0.977767i \(-0.432753\pi\)
0.209693 + 0.977767i \(0.432753\pi\)
\(524\) 19441.8 1.62084
\(525\) 0 0
\(526\) 518.667 0.0429942
\(527\) 5415.77 0.447656
\(528\) 3674.19 0.302839
\(529\) 20383.8 1.67533
\(530\) 0 0
\(531\) −7863.39 −0.642640
\(532\) 16917.4 1.37869
\(533\) −5085.39 −0.413269
\(534\) 3273.38 0.265268
\(535\) 0 0
\(536\) −4400.37 −0.354603
\(537\) −2137.02 −0.171730
\(538\) 8147.83 0.652933
\(539\) −5598.57 −0.447398
\(540\) 0 0
\(541\) 17642.3 1.40204 0.701018 0.713144i \(-0.252729\pi\)
0.701018 + 0.713144i \(0.252729\pi\)
\(542\) 19415.0 1.53865
\(543\) −6619.59 −0.523156
\(544\) 5223.86 0.411712
\(545\) 0 0
\(546\) −1847.43 −0.144803
\(547\) 18414.9 1.43943 0.719713 0.694271i \(-0.244273\pi\)
0.719713 + 0.694271i \(0.244273\pi\)
\(548\) −10090.4 −0.786571
\(549\) −1683.61 −0.130883
\(550\) 0 0
\(551\) −3164.20 −0.244645
\(552\) 3909.57 0.301453
\(553\) −3727.66 −0.286648
\(554\) −24556.9 −1.88325
\(555\) 0 0
\(556\) −27773.1 −2.11842
\(557\) −8179.15 −0.622193 −0.311096 0.950378i \(-0.600696\pi\)
−0.311096 + 0.950378i \(0.600696\pi\)
\(558\) −10092.5 −0.765683
\(559\) 1968.30 0.148927
\(560\) 0 0
\(561\) −1578.14 −0.118769
\(562\) −19748.5 −1.48228
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) −13622.9 −1.01707
\(565\) 0 0
\(566\) 14586.9 1.08327
\(567\) 911.598 0.0675194
\(568\) −1791.62 −0.132350
\(569\) 10118.3 0.745485 0.372743 0.927935i \(-0.378417\pi\)
0.372743 + 0.927935i \(0.378417\pi\)
\(570\) 0 0
\(571\) 23428.9 1.71711 0.858555 0.512721i \(-0.171362\pi\)
0.858555 + 0.512721i \(0.171362\pi\)
\(572\) −3268.70 −0.238935
\(573\) −4411.92 −0.321659
\(574\) −18530.3 −1.34746
\(575\) 0 0
\(576\) −6327.42 −0.457713
\(577\) −20508.1 −1.47966 −0.739831 0.672793i \(-0.765094\pi\)
−0.739831 + 0.672793i \(0.765094\pi\)
\(578\) 18939.8 1.36296
\(579\) 1108.68 0.0795771
\(580\) 0 0
\(581\) 4900.49 0.349925
\(582\) −15470.4 −1.10184
\(583\) −2069.87 −0.147042
\(584\) 6159.65 0.436452
\(585\) 0 0
\(586\) 11268.9 0.794394
\(587\) 5968.43 0.419665 0.209833 0.977737i \(-0.432708\pi\)
0.209833 + 0.977737i \(0.432708\pi\)
\(588\) 6305.97 0.442268
\(589\) 41219.0 2.88353
\(590\) 0 0
\(591\) −12820.2 −0.892308
\(592\) 5489.06 0.381080
\(593\) 14659.5 1.01517 0.507584 0.861602i \(-0.330539\pi\)
0.507584 + 0.861602i \(0.330539\pi\)
\(594\) 2940.95 0.203146
\(595\) 0 0
\(596\) 7223.72 0.496468
\(597\) −12462.9 −0.854394
\(598\) 9872.07 0.675082
\(599\) 23635.9 1.61225 0.806125 0.591746i \(-0.201561\pi\)
0.806125 + 0.591746i \(0.201561\pi\)
\(600\) 0 0
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) 7172.15 0.485573
\(603\) 5482.83 0.370279
\(604\) 22125.9 1.49055
\(605\) 0 0
\(606\) 8149.77 0.546306
\(607\) 5098.56 0.340930 0.170465 0.985364i \(-0.445473\pi\)
0.170465 + 0.985364i \(0.445473\pi\)
\(608\) 39758.4 2.65200
\(609\) 690.525 0.0459466
\(610\) 0 0
\(611\) −6075.74 −0.402288
\(612\) 1777.55 0.117407
\(613\) −1516.39 −0.0999128 −0.0499564 0.998751i \(-0.515908\pi\)
−0.0499564 + 0.998751i \(0.515908\pi\)
\(614\) 1986.31 0.130556
\(615\) 0 0
\(616\) −2103.70 −0.137598
\(617\) −18539.3 −1.20966 −0.604832 0.796353i \(-0.706760\pi\)
−0.604832 + 0.796353i \(0.706760\pi\)
\(618\) 6454.20 0.420107
\(619\) 25684.9 1.66779 0.833897 0.551920i \(-0.186105\pi\)
0.833897 + 0.551920i \(0.186105\pi\)
\(620\) 0 0
\(621\) −4871.30 −0.314780
\(622\) 6391.51 0.412020
\(623\) 2917.49 0.187619
\(624\) −1845.72 −0.118410
\(625\) 0 0
\(626\) 17046.1 1.08834
\(627\) −12011.1 −0.765038
\(628\) −30834.2 −1.95926
\(629\) −2357.67 −0.149454
\(630\) 0 0
\(631\) −22410.9 −1.41389 −0.706945 0.707269i \(-0.749927\pi\)
−0.706945 + 0.707269i \(0.749927\pi\)
\(632\) 2392.46 0.150580
\(633\) −3694.77 −0.231997
\(634\) 13696.1 0.857950
\(635\) 0 0
\(636\) 2331.40 0.145356
\(637\) 2812.43 0.174933
\(638\) 2227.73 0.138239
\(639\) 2232.34 0.138200
\(640\) 0 0
\(641\) 6827.81 0.420721 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(642\) −7679.77 −0.472113
\(643\) 23264.3 1.42684 0.713418 0.700738i \(-0.247146\pi\)
0.713418 + 0.700738i \(0.247146\pi\)
\(644\) 19728.4 1.20715
\(645\) 0 0
\(646\) −13237.1 −0.806204
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) −585.075 −0.0354690
\(649\) −22610.3 −1.36754
\(650\) 0 0
\(651\) −8995.26 −0.541554
\(652\) 22489.3 1.35084
\(653\) −10909.0 −0.653755 −0.326878 0.945067i \(-0.605997\pi\)
−0.326878 + 0.945067i \(0.605997\pi\)
\(654\) −16415.8 −0.981509
\(655\) 0 0
\(656\) −18513.2 −1.10186
\(657\) −7674.89 −0.455747
\(658\) −22139.0 −1.31166
\(659\) −4182.99 −0.247263 −0.123631 0.992328i \(-0.539454\pi\)
−0.123631 + 0.992328i \(0.539454\pi\)
\(660\) 0 0
\(661\) 2224.23 0.130881 0.0654406 0.997856i \(-0.479155\pi\)
0.0654406 + 0.997856i \(0.479155\pi\)
\(662\) 14405.2 0.845734
\(663\) 792.776 0.0464387
\(664\) −3145.19 −0.183821
\(665\) 0 0
\(666\) 4393.63 0.255630
\(667\) −3689.95 −0.214206
\(668\) 25899.7 1.50013
\(669\) −6561.72 −0.379209
\(670\) 0 0
\(671\) −4841.04 −0.278519
\(672\) −8676.51 −0.498071
\(673\) 24152.5 1.38337 0.691687 0.722197i \(-0.256868\pi\)
0.691687 + 0.722197i \(0.256868\pi\)
\(674\) −39151.2 −2.23746
\(675\) 0 0
\(676\) 1642.02 0.0934240
\(677\) 15310.7 0.869187 0.434593 0.900627i \(-0.356892\pi\)
0.434593 + 0.900627i \(0.356892\pi\)
\(678\) −532.806 −0.0301804
\(679\) −13788.4 −0.779310
\(680\) 0 0
\(681\) 12416.0 0.698652
\(682\) −29020.0 −1.62937
\(683\) −11399.6 −0.638646 −0.319323 0.947646i \(-0.603455\pi\)
−0.319323 + 0.947646i \(0.603455\pi\)
\(684\) 13528.8 0.756265
\(685\) 0 0
\(686\) 26495.9 1.47466
\(687\) 2506.06 0.139174
\(688\) 7165.54 0.397069
\(689\) 1039.79 0.0574935
\(690\) 0 0
\(691\) −3323.23 −0.182955 −0.0914773 0.995807i \(-0.529159\pi\)
−0.0914773 + 0.995807i \(0.529159\pi\)
\(692\) 1605.52 0.0881976
\(693\) 2621.20 0.143681
\(694\) 912.936 0.0499345
\(695\) 0 0
\(696\) −443.188 −0.0241365
\(697\) 7951.82 0.432133
\(698\) 20244.8 1.09782
\(699\) 11056.5 0.598279
\(700\) 0 0
\(701\) −12670.4 −0.682673 −0.341336 0.939941i \(-0.610880\pi\)
−0.341336 + 0.939941i \(0.610880\pi\)
\(702\) −1477.38 −0.0794302
\(703\) −17944.0 −0.962692
\(704\) −18193.8 −0.974012
\(705\) 0 0
\(706\) −12036.4 −0.641635
\(707\) 7263.71 0.386393
\(708\) 25467.2 1.35186
\(709\) 13075.2 0.692594 0.346297 0.938125i \(-0.387439\pi\)
0.346297 + 0.938125i \(0.387439\pi\)
\(710\) 0 0
\(711\) −2980.99 −0.157237
\(712\) −1872.48 −0.0985592
\(713\) 48067.8 2.52476
\(714\) 2888.75 0.151413
\(715\) 0 0
\(716\) 6921.16 0.361251
\(717\) −9078.62 −0.472869
\(718\) −15514.7 −0.806412
\(719\) −2988.41 −0.155005 −0.0775026 0.996992i \(-0.524695\pi\)
−0.0775026 + 0.996992i \(0.524695\pi\)
\(720\) 0 0
\(721\) 5752.48 0.297134
\(722\) −71877.0 −3.70496
\(723\) −9796.73 −0.503934
\(724\) 21438.9 1.10051
\(725\) 0 0
\(726\) −8350.37 −0.426875
\(727\) 5507.46 0.280963 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(728\) 1056.79 0.0538011
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −3077.75 −0.155725
\(732\) 5452.71 0.275325
\(733\) 36585.2 1.84353 0.921764 0.387751i \(-0.126748\pi\)
0.921764 + 0.387751i \(0.126748\pi\)
\(734\) −14607.9 −0.734587
\(735\) 0 0
\(736\) 46364.6 2.32204
\(737\) 15765.3 0.787953
\(738\) −14818.6 −0.739133
\(739\) 6425.89 0.319865 0.159933 0.987128i \(-0.448872\pi\)
0.159933 + 0.987128i \(0.448872\pi\)
\(740\) 0 0
\(741\) 6033.76 0.299131
\(742\) 3788.84 0.187457
\(743\) −20411.0 −1.00782 −0.503908 0.863757i \(-0.668105\pi\)
−0.503908 + 0.863757i \(0.668105\pi\)
\(744\) 5773.27 0.284487
\(745\) 0 0
\(746\) −50354.6 −2.47133
\(747\) 3918.89 0.191947
\(748\) 5111.13 0.249842
\(749\) −6844.81 −0.333917
\(750\) 0 0
\(751\) −24259.5 −1.17875 −0.589375 0.807860i \(-0.700626\pi\)
−0.589375 + 0.807860i \(0.700626\pi\)
\(752\) −22118.6 −1.07258
\(753\) 19089.5 0.923850
\(754\) −1119.10 −0.0540518
\(755\) 0 0
\(756\) −2952.39 −0.142034
\(757\) −9295.39 −0.446297 −0.223148 0.974785i \(-0.571633\pi\)
−0.223148 + 0.974785i \(0.571633\pi\)
\(758\) −1454.66 −0.0697042
\(759\) −14006.9 −0.669851
\(760\) 0 0
\(761\) −21974.7 −1.04676 −0.523378 0.852101i \(-0.675328\pi\)
−0.523378 + 0.852101i \(0.675328\pi\)
\(762\) 3927.27 0.186706
\(763\) −14631.0 −0.694204
\(764\) 14288.9 0.676641
\(765\) 0 0
\(766\) −14253.5 −0.672324
\(767\) 11358.2 0.534709
\(768\) −5467.13 −0.256873
\(769\) 22987.4 1.07795 0.538977 0.842320i \(-0.318811\pi\)
0.538977 + 0.842320i \(0.318811\pi\)
\(770\) 0 0
\(771\) −18256.1 −0.852759
\(772\) −3590.68 −0.167398
\(773\) −31970.9 −1.48760 −0.743799 0.668404i \(-0.766978\pi\)
−0.743799 + 0.668404i \(0.766978\pi\)
\(774\) 5735.53 0.266356
\(775\) 0 0
\(776\) 8849.59 0.409384
\(777\) 3915.94 0.180803
\(778\) 6860.25 0.316133
\(779\) 60520.8 2.78354
\(780\) 0 0
\(781\) 6418.85 0.294090
\(782\) −15436.6 −0.705896
\(783\) 552.209 0.0252035
\(784\) 10238.6 0.466408
\(785\) 0 0
\(786\) 25266.7 1.14661
\(787\) −6087.26 −0.275715 −0.137857 0.990452i \(-0.544022\pi\)
−0.137857 + 0.990452i \(0.544022\pi\)
\(788\) 41520.9 1.87706
\(789\) 369.680 0.0166805
\(790\) 0 0
\(791\) −474.878 −0.0213460
\(792\) −1682.32 −0.0754780
\(793\) 2431.88 0.108901
\(794\) 33415.4 1.49354
\(795\) 0 0
\(796\) 40363.6 1.79730
\(797\) −23080.0 −1.02577 −0.512883 0.858458i \(-0.671423\pi\)
−0.512883 + 0.858458i \(0.671423\pi\)
\(798\) 21986.1 0.975311
\(799\) 9500.41 0.420651
\(800\) 0 0
\(801\) 2333.10 0.102916
\(802\) −902.429 −0.0397330
\(803\) −22068.3 −0.969829
\(804\) −17757.3 −0.778918
\(805\) 0 0
\(806\) 14578.1 0.637087
\(807\) 5807.37 0.253320
\(808\) −4661.94 −0.202978
\(809\) −32377.8 −1.40710 −0.703550 0.710646i \(-0.748403\pi\)
−0.703550 + 0.710646i \(0.748403\pi\)
\(810\) 0 0
\(811\) 26352.8 1.14103 0.570513 0.821288i \(-0.306744\pi\)
0.570513 + 0.821288i \(0.306744\pi\)
\(812\) −2236.40 −0.0966532
\(813\) 13838.1 0.596952
\(814\) 12633.4 0.543980
\(815\) 0 0
\(816\) 2886.08 0.123815
\(817\) −23424.5 −1.00308
\(818\) 20135.0 0.860638
\(819\) −1316.75 −0.0561796
\(820\) 0 0
\(821\) 35355.3 1.50294 0.751468 0.659770i \(-0.229346\pi\)
0.751468 + 0.659770i \(0.229346\pi\)
\(822\) −13113.6 −0.556435
\(823\) 12663.3 0.536347 0.268173 0.963371i \(-0.413580\pi\)
0.268173 + 0.963371i \(0.413580\pi\)
\(824\) −3692.02 −0.156089
\(825\) 0 0
\(826\) 41387.6 1.74341
\(827\) 16295.2 0.685176 0.342588 0.939486i \(-0.388697\pi\)
0.342588 + 0.939486i \(0.388697\pi\)
\(828\) 15776.7 0.662170
\(829\) 13638.9 0.571411 0.285705 0.958318i \(-0.407772\pi\)
0.285705 + 0.958318i \(0.407772\pi\)
\(830\) 0 0
\(831\) −17502.9 −0.730649
\(832\) 9139.61 0.380840
\(833\) −4397.69 −0.182918
\(834\) −36094.2 −1.49861
\(835\) 0 0
\(836\) 38900.5 1.60933
\(837\) −7193.46 −0.297064
\(838\) 41685.0 1.71836
\(839\) −1890.31 −0.0777838 −0.0388919 0.999243i \(-0.512383\pi\)
−0.0388919 + 0.999243i \(0.512383\pi\)
\(840\) 0 0
\(841\) −23970.7 −0.982849
\(842\) 51016.4 2.08805
\(843\) −14075.7 −0.575082
\(844\) 11966.3 0.488028
\(845\) 0 0
\(846\) −17704.5 −0.719494
\(847\) −7442.50 −0.301921
\(848\) 3785.35 0.153289
\(849\) 10396.8 0.420279
\(850\) 0 0
\(851\) −20925.6 −0.842914
\(852\) −7229.89 −0.290718
\(853\) −1620.21 −0.0650351 −0.0325175 0.999471i \(-0.510352\pi\)
−0.0325175 + 0.999471i \(0.510352\pi\)
\(854\) 8861.39 0.355071
\(855\) 0 0
\(856\) 4393.08 0.175412
\(857\) 14508.4 0.578292 0.289146 0.957285i \(-0.406629\pi\)
0.289146 + 0.957285i \(0.406629\pi\)
\(858\) −4248.03 −0.169027
\(859\) 29639.8 1.17730 0.588648 0.808389i \(-0.299660\pi\)
0.588648 + 0.808389i \(0.299660\pi\)
\(860\) 0 0
\(861\) −13207.5 −0.522776
\(862\) 57548.8 2.27392
\(863\) −21528.8 −0.849186 −0.424593 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(864\) −6938.56 −0.273211
\(865\) 0 0
\(866\) 29939.4 1.17481
\(867\) 13499.4 0.528792
\(868\) 29132.9 1.13921
\(869\) −8571.50 −0.334601
\(870\) 0 0
\(871\) −7919.65 −0.308091
\(872\) 9390.36 0.364676
\(873\) −11026.5 −0.427482
\(874\) −117487. −4.54696
\(875\) 0 0
\(876\) 24856.7 0.958708
\(877\) −14865.3 −0.572366 −0.286183 0.958175i \(-0.592387\pi\)
−0.286183 + 0.958175i \(0.592387\pi\)
\(878\) 25347.1 0.974285
\(879\) 8031.92 0.308202
\(880\) 0 0
\(881\) −21336.0 −0.815921 −0.407961 0.913000i \(-0.633760\pi\)
−0.407961 + 0.913000i \(0.633760\pi\)
\(882\) 8195.30 0.312869
\(883\) −37538.2 −1.43065 −0.715323 0.698794i \(-0.753720\pi\)
−0.715323 + 0.698794i \(0.753720\pi\)
\(884\) −2567.57 −0.0976884
\(885\) 0 0
\(886\) −54694.1 −2.07391
\(887\) −34575.0 −1.30881 −0.654406 0.756144i \(-0.727081\pi\)
−0.654406 + 0.756144i \(0.727081\pi\)
\(888\) −2513.30 −0.0949784
\(889\) 3500.29 0.132054
\(890\) 0 0
\(891\) 2096.16 0.0788148
\(892\) 21251.4 0.797703
\(893\) 72306.9 2.70958
\(894\) 9388.02 0.351211
\(895\) 0 0
\(896\) 10165.9 0.379039
\(897\) 7036.32 0.261913
\(898\) −46233.3 −1.71807
\(899\) −5448.96 −0.202150
\(900\) 0 0
\(901\) −1625.89 −0.0601178
\(902\) −42609.2 −1.57287
\(903\) 5111.95 0.188389
\(904\) 304.783 0.0112134
\(905\) 0 0
\(906\) 28755.0 1.05444
\(907\) 10424.8 0.381641 0.190820 0.981625i \(-0.438885\pi\)
0.190820 + 0.981625i \(0.438885\pi\)
\(908\) −40211.7 −1.46968
\(909\) 5808.75 0.211952
\(910\) 0 0
\(911\) 10961.8 0.398661 0.199331 0.979932i \(-0.436123\pi\)
0.199331 + 0.979932i \(0.436123\pi\)
\(912\) 21965.8 0.797543
\(913\) 11268.3 0.408464
\(914\) 41392.2 1.49796
\(915\) 0 0
\(916\) −8116.38 −0.292765
\(917\) 22519.7 0.810976
\(918\) 2310.12 0.0830558
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) 0 0
\(921\) 1415.74 0.0506519
\(922\) −14316.7 −0.511384
\(923\) −3224.49 −0.114990
\(924\) −8489.28 −0.302248
\(925\) 0 0
\(926\) 7321.32 0.259820
\(927\) 4600.23 0.162990
\(928\) −5255.88 −0.185919
\(929\) 5429.07 0.191735 0.0958675 0.995394i \(-0.469437\pi\)
0.0958675 + 0.995394i \(0.469437\pi\)
\(930\) 0 0
\(931\) −33470.5 −1.17825
\(932\) −35808.8 −1.25854
\(933\) 4555.55 0.159852
\(934\) −33499.1 −1.17358
\(935\) 0 0
\(936\) 845.109 0.0295120
\(937\) 21300.1 0.742631 0.371315 0.928507i \(-0.378907\pi\)
0.371315 + 0.928507i \(0.378907\pi\)
\(938\) −28857.9 −1.00453
\(939\) 12149.6 0.422244
\(940\) 0 0
\(941\) 26851.2 0.930207 0.465103 0.885256i \(-0.346017\pi\)
0.465103 + 0.885256i \(0.346017\pi\)
\(942\) −40072.4 −1.38602
\(943\) 70576.7 2.43722
\(944\) 41349.4 1.42564
\(945\) 0 0
\(946\) 16491.9 0.566805
\(947\) 8021.68 0.275258 0.137629 0.990484i \(-0.456052\pi\)
0.137629 + 0.990484i \(0.456052\pi\)
\(948\) 9654.52 0.330764
\(949\) 11085.9 0.379204
\(950\) 0 0
\(951\) 9761.88 0.332861
\(952\) −1652.46 −0.0562569
\(953\) −35715.0 −1.21398 −0.606990 0.794709i \(-0.707623\pi\)
−0.606990 + 0.794709i \(0.707623\pi\)
\(954\) 3029.92 0.102827
\(955\) 0 0
\(956\) 29402.9 0.994726
\(957\) 1587.82 0.0536330
\(958\) −35490.6 −1.19692
\(959\) −11687.9 −0.393557
\(960\) 0 0
\(961\) 41190.9 1.38266
\(962\) −6346.35 −0.212697
\(963\) −5473.76 −0.183166
\(964\) 31728.7 1.06007
\(965\) 0 0
\(966\) 25639.2 0.853963
\(967\) −53338.8 −1.77380 −0.886898 0.461965i \(-0.847145\pi\)
−0.886898 + 0.461965i \(0.847145\pi\)
\(968\) 4776.69 0.158604
\(969\) −9434.77 −0.312785
\(970\) 0 0
\(971\) 23112.9 0.763882 0.381941 0.924187i \(-0.375256\pi\)
0.381941 + 0.924187i \(0.375256\pi\)
\(972\) −2361.01 −0.0779110
\(973\) −32170.0 −1.05994
\(974\) −49181.3 −1.61794
\(975\) 0 0
\(976\) 8853.22 0.290353
\(977\) 52874.6 1.73143 0.865715 0.500538i \(-0.166864\pi\)
0.865715 + 0.500538i \(0.166864\pi\)
\(978\) 29227.4 0.955612
\(979\) 6708.57 0.219006
\(980\) 0 0
\(981\) −11700.3 −0.380798
\(982\) 16645.5 0.540917
\(983\) −45173.1 −1.46572 −0.732858 0.680381i \(-0.761814\pi\)
−0.732858 + 0.680381i \(0.761814\pi\)
\(984\) 8476.73 0.274622
\(985\) 0 0
\(986\) 1749.89 0.0565190
\(987\) −15779.6 −0.508885
\(988\) −19541.6 −0.629251
\(989\) −27316.7 −0.878281
\(990\) 0 0
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) 68466.7 2.19135
\(993\) 10267.3 0.328121
\(994\) −11749.5 −0.374922
\(995\) 0 0
\(996\) −12692.1 −0.403780
\(997\) 18108.1 0.575214 0.287607 0.957749i \(-0.407140\pi\)
0.287607 + 0.957749i \(0.407140\pi\)
\(998\) 23951.1 0.759677
\(999\) 3131.56 0.0991773
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 975.4.a.l.1.1 3
5.4 even 2 39.4.a.c.1.3 3
15.14 odd 2 117.4.a.f.1.1 3
20.19 odd 2 624.4.a.t.1.1 3
35.34 odd 2 1911.4.a.k.1.3 3
40.19 odd 2 2496.4.a.bp.1.3 3
40.29 even 2 2496.4.a.bl.1.3 3
60.59 even 2 1872.4.a.bk.1.3 3
65.34 odd 4 507.4.b.g.337.6 6
65.44 odd 4 507.4.b.g.337.1 6
65.64 even 2 507.4.a.h.1.1 3
195.194 odd 2 1521.4.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 5.4 even 2
117.4.a.f.1.1 3 15.14 odd 2
507.4.a.h.1.1 3 65.64 even 2
507.4.b.g.337.1 6 65.44 odd 4
507.4.b.g.337.6 6 65.34 odd 4
624.4.a.t.1.1 3 20.19 odd 2
975.4.a.l.1.1 3 1.1 even 1 trivial
1521.4.a.u.1.3 3 195.194 odd 2
1872.4.a.bk.1.3 3 60.59 even 2
1911.4.a.k.1.3 3 35.34 odd 2
2496.4.a.bl.1.3 3 40.29 even 2
2496.4.a.bp.1.3 3 40.19 odd 2