Properties

Label 539.4.a.e.1.1
Level $539$
Weight $4$
Character 539.1
Self dual yes
Analytic conductor $31.802$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.8020294931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 539.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} +O(q^{10})\) \(q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} -9.41154 q^{10} -11.0000 q^{11} +44.2487 q^{12} -74.6410 q^{13} -76.2154 q^{15} +51.4256 q^{16} +82.7846 q^{17} -5.96152 q^{18} +67.9230 q^{19} -95.9615 q^{20} +8.05256 q^{22} +13.3538 q^{23} -67.1103 q^{24} +40.2872 q^{25} +54.6410 q^{26} +111.785 q^{27} +168.995 q^{29} +55.7935 q^{30} +65.4974 q^{31} -128.210 q^{32} +65.2102 q^{33} -60.6025 q^{34} -60.7846 q^{36} +40.8564 q^{37} -49.7231 q^{38} +442.487 q^{39} +145.541 q^{40} -274.928 q^{41} -2.28719 q^{43} +82.1051 q^{44} +104.697 q^{45} -9.77568 q^{46} -71.8461 q^{47} -304.862 q^{48} -29.4923 q^{50} -490.764 q^{51} +557.128 q^{52} -149.005 q^{53} -81.8320 q^{54} -141.420 q^{55} -402.662 q^{57} -123.713 q^{58} -545.631 q^{59} +568.879 q^{60} -101.303 q^{61} -47.9474 q^{62} -317.549 q^{64} -959.615 q^{65} -47.7372 q^{66} +411.641 q^{67} -617.913 q^{68} -79.1642 q^{69} -470.636 q^{71} +92.1896 q^{72} -610.600 q^{73} -29.9090 q^{74} -238.831 q^{75} -506.985 q^{76} -323.923 q^{78} -978.225 q^{79} +661.149 q^{80} -882.559 q^{81} +201.261 q^{82} -26.1539 q^{83} +1064.31 q^{85} +1.67434 q^{86} -1001.84 q^{87} -124.526 q^{88} +352.887 q^{89} -76.6438 q^{90} -99.6743 q^{92} -388.282 q^{93} +52.5950 q^{94} +873.246 q^{95} +760.056 q^{96} -847.585 q^{97} -89.5795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} - 50 q^{10} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 194 q^{15} - 8 q^{16} + 124 q^{17} + 92 q^{18} - 72 q^{19} - 88 q^{20} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} + 40 q^{26} + 182 q^{27} + 144 q^{29} - 266 q^{30} + 34 q^{31} - 104 q^{32} - 22 q^{33} + 52 q^{34} - 80 q^{36} + 54 q^{37} - 432 q^{38} + 400 q^{39} + 492 q^{40} - 536 q^{41} - 60 q^{43} + 88 q^{44} - 428 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 232 q^{50} - 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 22 q^{55} - 1512 q^{57} - 192 q^{58} - 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} + 224 q^{64} - 880 q^{65} - 286 q^{66} + 754 q^{67} - 640 q^{68} - 962 q^{69} - 678 q^{71} - 744 q^{72} + 400 q^{73} + 6 q^{74} + 520 q^{75} - 432 q^{76} - 440 q^{78} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} - 512 q^{82} - 468 q^{83} + 452 q^{85} - 156 q^{86} - 1200 q^{87} + 132 q^{88} + 1842 q^{89} - 1532 q^{90} - 40 q^{92} - 638 q^{93} + 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 484 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.732051 −0.258819 −0.129410 0.991591i \(-0.541308\pi\)
−0.129410 + 0.991591i \(0.541308\pi\)
\(3\) −5.92820 −1.14088 −0.570442 0.821338i \(-0.693228\pi\)
−0.570442 + 0.821338i \(0.693228\pi\)
\(4\) −7.46410 −0.933013
\(5\) 12.8564 1.14991 0.574956 0.818184i \(-0.305019\pi\)
0.574956 + 0.818184i \(0.305019\pi\)
\(6\) 4.33975 0.295282
\(7\) 0 0
\(8\) 11.3205 0.500301
\(9\) 8.14359 0.301615
\(10\) −9.41154 −0.297619
\(11\) −11.0000 −0.301511
\(12\) 44.2487 1.06446
\(13\) −74.6410 −1.59244 −0.796219 0.605009i \(-0.793170\pi\)
−0.796219 + 0.605009i \(0.793170\pi\)
\(14\) 0 0
\(15\) −76.2154 −1.31192
\(16\) 51.4256 0.803525
\(17\) 82.7846 1.18107 0.590536 0.807011i \(-0.298916\pi\)
0.590536 + 0.807011i \(0.298916\pi\)
\(18\) −5.96152 −0.0780636
\(19\) 67.9230 0.820138 0.410069 0.912055i \(-0.365505\pi\)
0.410069 + 0.912055i \(0.365505\pi\)
\(20\) −95.9615 −1.07288
\(21\) 0 0
\(22\) 8.05256 0.0780369
\(23\) 13.3538 0.121064 0.0605319 0.998166i \(-0.480720\pi\)
0.0605319 + 0.998166i \(0.480720\pi\)
\(24\) −67.1103 −0.570784
\(25\) 40.2872 0.322297
\(26\) 54.6410 0.412153
\(27\) 111.785 0.796776
\(28\) 0 0
\(29\) 168.995 1.08212 0.541061 0.840983i \(-0.318023\pi\)
0.541061 + 0.840983i \(0.318023\pi\)
\(30\) 55.7935 0.339549
\(31\) 65.4974 0.379474 0.189737 0.981835i \(-0.439237\pi\)
0.189737 + 0.981835i \(0.439237\pi\)
\(32\) −128.210 −0.708268
\(33\) 65.2102 0.343989
\(34\) −60.6025 −0.305684
\(35\) 0 0
\(36\) −60.7846 −0.281410
\(37\) 40.8564 0.181534 0.0907669 0.995872i \(-0.471068\pi\)
0.0907669 + 0.995872i \(0.471068\pi\)
\(38\) −49.7231 −0.212267
\(39\) 442.487 1.81679
\(40\) 145.541 0.575302
\(41\) −274.928 −1.04723 −0.523617 0.851954i \(-0.675418\pi\)
−0.523617 + 0.851954i \(0.675418\pi\)
\(42\) 0 0
\(43\) −2.28719 −0.00811146 −0.00405573 0.999992i \(-0.501291\pi\)
−0.00405573 + 0.999992i \(0.501291\pi\)
\(44\) 82.1051 0.281314
\(45\) 104.697 0.346830
\(46\) −9.77568 −0.0313336
\(47\) −71.8461 −0.222975 −0.111488 0.993766i \(-0.535562\pi\)
−0.111488 + 0.993766i \(0.535562\pi\)
\(48\) −304.862 −0.916729
\(49\) 0 0
\(50\) −29.4923 −0.0834167
\(51\) −490.764 −1.34746
\(52\) 557.128 1.48576
\(53\) −149.005 −0.386178 −0.193089 0.981181i \(-0.561851\pi\)
−0.193089 + 0.981181i \(0.561851\pi\)
\(54\) −81.8320 −0.206221
\(55\) −141.420 −0.346711
\(56\) 0 0
\(57\) −402.662 −0.935681
\(58\) −123.713 −0.280074
\(59\) −545.631 −1.20398 −0.601992 0.798502i \(-0.705626\pi\)
−0.601992 + 0.798502i \(0.705626\pi\)
\(60\) 568.879 1.22403
\(61\) −101.303 −0.212631 −0.106315 0.994332i \(-0.533905\pi\)
−0.106315 + 0.994332i \(0.533905\pi\)
\(62\) −47.9474 −0.0982150
\(63\) 0 0
\(64\) −317.549 −0.620212
\(65\) −959.615 −1.83116
\(66\) −47.7372 −0.0890310
\(67\) 411.641 0.750596 0.375298 0.926904i \(-0.377540\pi\)
0.375298 + 0.926904i \(0.377540\pi\)
\(68\) −617.913 −1.10195
\(69\) −79.1642 −0.138120
\(70\) 0 0
\(71\) −470.636 −0.786679 −0.393339 0.919393i \(-0.628680\pi\)
−0.393339 + 0.919393i \(0.628680\pi\)
\(72\) 92.1896 0.150898
\(73\) −610.600 −0.978977 −0.489488 0.872010i \(-0.662816\pi\)
−0.489488 + 0.872010i \(0.662816\pi\)
\(74\) −29.9090 −0.0469844
\(75\) −238.831 −0.367704
\(76\) −506.985 −0.765199
\(77\) 0 0
\(78\) −323.923 −0.470219
\(79\) −978.225 −1.39315 −0.696576 0.717483i \(-0.745294\pi\)
−0.696576 + 0.717483i \(0.745294\pi\)
\(80\) 661.149 0.923983
\(81\) −882.559 −1.21064
\(82\) 201.261 0.271044
\(83\) −26.1539 −0.0345875 −0.0172938 0.999850i \(-0.505505\pi\)
−0.0172938 + 0.999850i \(0.505505\pi\)
\(84\) 0 0
\(85\) 1064.31 1.35813
\(86\) 1.67434 0.00209940
\(87\) −1001.84 −1.23458
\(88\) −124.526 −0.150846
\(89\) 352.887 0.420292 0.210146 0.977670i \(-0.432606\pi\)
0.210146 + 0.977670i \(0.432606\pi\)
\(90\) −76.6438 −0.0897663
\(91\) 0 0
\(92\) −99.6743 −0.112954
\(93\) −388.282 −0.432935
\(94\) 52.5950 0.0577102
\(95\) 873.246 0.943086
\(96\) 760.056 0.808051
\(97\) −847.585 −0.887208 −0.443604 0.896223i \(-0.646300\pi\)
−0.443604 + 0.896223i \(0.646300\pi\)
\(98\) 0 0
\(99\) −89.5795 −0.0909402
\(100\) −300.708 −0.300708
\(101\) −1293.46 −1.27430 −0.637150 0.770740i \(-0.719887\pi\)
−0.637150 + 0.770740i \(0.719887\pi\)
\(102\) 359.264 0.348750
\(103\) 1725.24 1.65042 0.825209 0.564828i \(-0.191057\pi\)
0.825209 + 0.564828i \(0.191057\pi\)
\(104\) −844.974 −0.796697
\(105\) 0 0
\(106\) 109.079 0.0999502
\(107\) −484.179 −0.437452 −0.218726 0.975786i \(-0.570190\pi\)
−0.218726 + 0.975786i \(0.570190\pi\)
\(108\) −834.372 −0.743402
\(109\) −64.2563 −0.0564645 −0.0282323 0.999601i \(-0.508988\pi\)
−0.0282323 + 0.999601i \(0.508988\pi\)
\(110\) 103.527 0.0897355
\(111\) −242.205 −0.207109
\(112\) 0 0
\(113\) −2005.08 −1.66922 −0.834612 0.550839i \(-0.814308\pi\)
−0.834612 + 0.550839i \(0.814308\pi\)
\(114\) 294.769 0.242172
\(115\) 171.682 0.139213
\(116\) −1261.39 −1.00963
\(117\) −607.846 −0.480302
\(118\) 399.429 0.311614
\(119\) 0 0
\(120\) −862.797 −0.656352
\(121\) 121.000 0.0909091
\(122\) 74.1587 0.0550329
\(123\) 1629.83 1.19477
\(124\) −488.879 −0.354054
\(125\) −1089.10 −0.779298
\(126\) 0 0
\(127\) 109.605 0.0765816 0.0382908 0.999267i \(-0.487809\pi\)
0.0382908 + 0.999267i \(0.487809\pi\)
\(128\) 1258.14 0.868791
\(129\) 13.5589 0.00925423
\(130\) 702.487 0.473940
\(131\) −1156.71 −0.771469 −0.385734 0.922610i \(-0.626052\pi\)
−0.385734 + 0.922610i \(0.626052\pi\)
\(132\) −486.736 −0.320946
\(133\) 0 0
\(134\) −301.342 −0.194269
\(135\) 1437.15 0.916223
\(136\) 937.164 0.590891
\(137\) 198.323 0.123678 0.0618391 0.998086i \(-0.480303\pi\)
0.0618391 + 0.998086i \(0.480303\pi\)
\(138\) 57.9522 0.0357480
\(139\) 2900.14 1.76969 0.884844 0.465888i \(-0.154265\pi\)
0.884844 + 0.465888i \(0.154265\pi\)
\(140\) 0 0
\(141\) 425.918 0.254389
\(142\) 344.529 0.203607
\(143\) 821.051 0.480138
\(144\) 418.789 0.242355
\(145\) 2172.67 1.24435
\(146\) 446.990 0.253378
\(147\) 0 0
\(148\) −304.956 −0.169373
\(149\) 3488.34 1.91796 0.958980 0.283472i \(-0.0914864\pi\)
0.958980 + 0.283472i \(0.0914864\pi\)
\(150\) 174.836 0.0951687
\(151\) −1163.32 −0.626953 −0.313477 0.949596i \(-0.601494\pi\)
−0.313477 + 0.949596i \(0.601494\pi\)
\(152\) 768.923 0.410315
\(153\) 674.164 0.356228
\(154\) 0 0
\(155\) 842.061 0.436361
\(156\) −3302.77 −1.69508
\(157\) −342.057 −0.173880 −0.0869398 0.996214i \(-0.527709\pi\)
−0.0869398 + 0.996214i \(0.527709\pi\)
\(158\) 716.111 0.360574
\(159\) 883.333 0.440584
\(160\) −1648.32 −0.814446
\(161\) 0 0
\(162\) 646.078 0.313338
\(163\) −1394.89 −0.670285 −0.335142 0.942167i \(-0.608784\pi\)
−0.335142 + 0.942167i \(0.608784\pi\)
\(164\) 2052.09 0.977082
\(165\) 838.369 0.395557
\(166\) 19.1460 0.00895191
\(167\) −478.703 −0.221815 −0.110908 0.993831i \(-0.535376\pi\)
−0.110908 + 0.993831i \(0.535376\pi\)
\(168\) 0 0
\(169\) 3374.28 1.53586
\(170\) −779.131 −0.351509
\(171\) 553.138 0.247365
\(172\) 17.0718 0.00756809
\(173\) −1808.58 −0.794822 −0.397411 0.917641i \(-0.630091\pi\)
−0.397411 + 0.917641i \(0.630091\pi\)
\(174\) 733.395 0.319532
\(175\) 0 0
\(176\) −565.682 −0.242272
\(177\) 3234.61 1.37361
\(178\) −258.331 −0.108780
\(179\) −4429.85 −1.84973 −0.924867 0.380292i \(-0.875824\pi\)
−0.924867 + 0.380292i \(0.875824\pi\)
\(180\) −781.472 −0.323597
\(181\) −3409.17 −1.40001 −0.700005 0.714138i \(-0.746819\pi\)
−0.700005 + 0.714138i \(0.746819\pi\)
\(182\) 0 0
\(183\) 600.543 0.242587
\(184\) 151.172 0.0605682
\(185\) 525.267 0.208748
\(186\) 284.242 0.112052
\(187\) −910.631 −0.356106
\(188\) 536.267 0.208039
\(189\) 0 0
\(190\) −639.261 −0.244089
\(191\) 2923.75 1.10762 0.553810 0.832643i \(-0.313173\pi\)
0.553810 + 0.832643i \(0.313173\pi\)
\(192\) 1882.49 0.707590
\(193\) −2484.18 −0.926505 −0.463253 0.886226i \(-0.653318\pi\)
−0.463253 + 0.886226i \(0.653318\pi\)
\(194\) 620.475 0.229626
\(195\) 5688.79 2.08914
\(196\) 0 0
\(197\) −5125.67 −1.85375 −0.926876 0.375369i \(-0.877516\pi\)
−0.926876 + 0.375369i \(0.877516\pi\)
\(198\) 65.5768 0.0235371
\(199\) 7.69219 0.00274013 0.00137006 0.999999i \(-0.499564\pi\)
0.00137006 + 0.999999i \(0.499564\pi\)
\(200\) 456.071 0.161246
\(201\) −2440.29 −0.856343
\(202\) 946.879 0.329813
\(203\) 0 0
\(204\) 3663.11 1.25720
\(205\) −3534.59 −1.20423
\(206\) −1262.96 −0.427160
\(207\) 108.748 0.0365146
\(208\) −3838.46 −1.27956
\(209\) −747.154 −0.247281
\(210\) 0 0
\(211\) 3107.34 1.01383 0.506915 0.861996i \(-0.330786\pi\)
0.506915 + 0.861996i \(0.330786\pi\)
\(212\) 1112.19 0.360309
\(213\) 2790.03 0.897509
\(214\) 354.444 0.113221
\(215\) −29.4050 −0.00932746
\(216\) 1265.46 0.398628
\(217\) 0 0
\(218\) 47.0388 0.0146141
\(219\) 3619.76 1.11690
\(220\) 1055.58 0.323486
\(221\) −6179.13 −1.88078
\(222\) 177.306 0.0536037
\(223\) 12.3185 0.00369913 0.00184957 0.999998i \(-0.499411\pi\)
0.00184957 + 0.999998i \(0.499411\pi\)
\(224\) 0 0
\(225\) 328.082 0.0972096
\(226\) 1467.82 0.432027
\(227\) −4615.90 −1.34964 −0.674820 0.737983i \(-0.735779\pi\)
−0.674820 + 0.737983i \(0.735779\pi\)
\(228\) 3005.51 0.873003
\(229\) −5074.63 −1.46437 −0.732186 0.681105i \(-0.761500\pi\)
−0.732186 + 0.681105i \(0.761500\pi\)
\(230\) −125.680 −0.0360309
\(231\) 0 0
\(232\) 1913.11 0.541386
\(233\) 211.683 0.0595184 0.0297592 0.999557i \(-0.490526\pi\)
0.0297592 + 0.999557i \(0.490526\pi\)
\(234\) 444.974 0.124311
\(235\) −923.683 −0.256402
\(236\) 4072.64 1.12333
\(237\) 5799.12 1.58942
\(238\) 0 0
\(239\) 4312.49 1.16716 0.583581 0.812055i \(-0.301651\pi\)
0.583581 + 0.812055i \(0.301651\pi\)
\(240\) −3919.42 −1.05416
\(241\) 996.584 0.266372 0.133186 0.991091i \(-0.457479\pi\)
0.133186 + 0.991091i \(0.457479\pi\)
\(242\) −88.5781 −0.0235290
\(243\) 2213.80 0.584426
\(244\) 756.133 0.198387
\(245\) 0 0
\(246\) −1193.12 −0.309230
\(247\) −5069.85 −1.30602
\(248\) 741.464 0.189851
\(249\) 155.046 0.0394603
\(250\) 797.278 0.201697
\(251\) 276.892 0.0696306 0.0348153 0.999394i \(-0.488916\pi\)
0.0348153 + 0.999394i \(0.488916\pi\)
\(252\) 0 0
\(253\) −146.892 −0.0365021
\(254\) −80.2364 −0.0198208
\(255\) −6309.46 −1.54947
\(256\) 1619.36 0.395352
\(257\) 3235.18 0.785233 0.392617 0.919702i \(-0.371570\pi\)
0.392617 + 0.919702i \(0.371570\pi\)
\(258\) −9.92581 −0.00239517
\(259\) 0 0
\(260\) 7162.67 1.70850
\(261\) 1376.23 0.326384
\(262\) 846.772 0.199671
\(263\) 207.944 0.0487544 0.0243772 0.999703i \(-0.492240\pi\)
0.0243772 + 0.999703i \(0.492240\pi\)
\(264\) 738.213 0.172098
\(265\) −1915.67 −0.444071
\(266\) 0 0
\(267\) −2091.99 −0.479504
\(268\) −3072.53 −0.700316
\(269\) −5033.04 −1.14078 −0.570390 0.821374i \(-0.693208\pi\)
−0.570390 + 0.821374i \(0.693208\pi\)
\(270\) −1052.07 −0.237136
\(271\) −1487.01 −0.333319 −0.166660 0.986015i \(-0.553298\pi\)
−0.166660 + 0.986015i \(0.553298\pi\)
\(272\) 4257.25 0.949021
\(273\) 0 0
\(274\) −145.183 −0.0320102
\(275\) −443.159 −0.0971764
\(276\) 590.890 0.128867
\(277\) −235.836 −0.0511552 −0.0255776 0.999673i \(-0.508142\pi\)
−0.0255776 + 0.999673i \(0.508142\pi\)
\(278\) −2123.05 −0.458029
\(279\) 533.384 0.114455
\(280\) 0 0
\(281\) −4915.01 −1.04343 −0.521717 0.853118i \(-0.674708\pi\)
−0.521717 + 0.853118i \(0.674708\pi\)
\(282\) −311.794 −0.0658406
\(283\) 5199.56 1.09216 0.546081 0.837733i \(-0.316119\pi\)
0.546081 + 0.837733i \(0.316119\pi\)
\(284\) 3512.87 0.733981
\(285\) −5176.78 −1.07595
\(286\) −601.051 −0.124269
\(287\) 0 0
\(288\) −1044.09 −0.213624
\(289\) 1940.29 0.394930
\(290\) −1590.50 −0.322060
\(291\) 5024.65 1.01220
\(292\) 4557.58 0.913398
\(293\) 8880.92 1.77075 0.885373 0.464881i \(-0.153903\pi\)
0.885373 + 0.464881i \(0.153903\pi\)
\(294\) 0 0
\(295\) −7014.85 −1.38448
\(296\) 462.515 0.0908215
\(297\) −1229.63 −0.240237
\(298\) −2553.64 −0.496405
\(299\) −996.743 −0.192786
\(300\) 1782.66 0.343072
\(301\) 0 0
\(302\) 851.612 0.162267
\(303\) 7667.90 1.45383
\(304\) 3492.99 0.659001
\(305\) −1302.39 −0.244507
\(306\) −493.522 −0.0921987
\(307\) 1497.93 0.278474 0.139237 0.990259i \(-0.455535\pi\)
0.139237 + 0.990259i \(0.455535\pi\)
\(308\) 0 0
\(309\) −10227.6 −1.88293
\(310\) −616.432 −0.112939
\(311\) 7484.71 1.36469 0.682345 0.731030i \(-0.260960\pi\)
0.682345 + 0.731030i \(0.260960\pi\)
\(312\) 5009.18 0.908939
\(313\) 658.363 0.118891 0.0594455 0.998232i \(-0.481067\pi\)
0.0594455 + 0.998232i \(0.481067\pi\)
\(314\) 250.403 0.0450034
\(315\) 0 0
\(316\) 7301.57 1.29983
\(317\) 233.708 0.0414080 0.0207040 0.999786i \(-0.493409\pi\)
0.0207040 + 0.999786i \(0.493409\pi\)
\(318\) −646.645 −0.114032
\(319\) −1858.94 −0.326272
\(320\) −4082.53 −0.713189
\(321\) 2870.31 0.499082
\(322\) 0 0
\(323\) 5622.98 0.968641
\(324\) 6587.51 1.12955
\(325\) −3007.08 −0.513239
\(326\) 1021.13 0.173482
\(327\) 380.924 0.0644194
\(328\) −3112.33 −0.523931
\(329\) 0 0
\(330\) −613.729 −0.102378
\(331\) 8532.95 1.41696 0.708480 0.705731i \(-0.249381\pi\)
0.708480 + 0.705731i \(0.249381\pi\)
\(332\) 195.215 0.0322706
\(333\) 332.718 0.0547533
\(334\) 350.435 0.0574100
\(335\) 5292.22 0.863120
\(336\) 0 0
\(337\) 11691.2 1.88979 0.944895 0.327373i \(-0.106163\pi\)
0.944895 + 0.327373i \(0.106163\pi\)
\(338\) −2470.15 −0.397509
\(339\) 11886.5 1.90439
\(340\) −7944.14 −1.26715
\(341\) −720.472 −0.114416
\(342\) −404.925 −0.0640229
\(343\) 0 0
\(344\) −25.8921 −0.00405817
\(345\) −1017.77 −0.158825
\(346\) 1323.98 0.205715
\(347\) 4598.79 0.711459 0.355729 0.934589i \(-0.384232\pi\)
0.355729 + 0.934589i \(0.384232\pi\)
\(348\) 7477.80 1.15187
\(349\) −6720.27 −1.03074 −0.515369 0.856968i \(-0.672345\pi\)
−0.515369 + 0.856968i \(0.672345\pi\)
\(350\) 0 0
\(351\) −8343.72 −1.26882
\(352\) 1410.31 0.213551
\(353\) −5738.70 −0.865270 −0.432635 0.901569i \(-0.642416\pi\)
−0.432635 + 0.901569i \(0.642416\pi\)
\(354\) −2367.90 −0.355515
\(355\) −6050.69 −0.904611
\(356\) −2633.99 −0.392138
\(357\) 0 0
\(358\) 3242.87 0.478746
\(359\) −4115.27 −0.605001 −0.302501 0.953149i \(-0.597821\pi\)
−0.302501 + 0.953149i \(0.597821\pi\)
\(360\) 1185.23 0.173519
\(361\) −2245.46 −0.327374
\(362\) 2495.69 0.362349
\(363\) −717.313 −0.103717
\(364\) 0 0
\(365\) −7850.12 −1.12574
\(366\) −439.628 −0.0627861
\(367\) −9662.99 −1.37440 −0.687199 0.726469i \(-0.741160\pi\)
−0.687199 + 0.726469i \(0.741160\pi\)
\(368\) 686.729 0.0972778
\(369\) −2238.90 −0.315861
\(370\) −384.522 −0.0540279
\(371\) 0 0
\(372\) 2898.18 0.403934
\(373\) −141.780 −0.0196812 −0.00984062 0.999952i \(-0.503132\pi\)
−0.00984062 + 0.999952i \(0.503132\pi\)
\(374\) 666.628 0.0921671
\(375\) 6456.42 0.889088
\(376\) −813.334 −0.111555
\(377\) −12613.9 −1.72321
\(378\) 0 0
\(379\) −2819.73 −0.382163 −0.191082 0.981574i \(-0.561200\pi\)
−0.191082 + 0.981574i \(0.561200\pi\)
\(380\) −6518.00 −0.879911
\(381\) −649.760 −0.0873707
\(382\) −2140.34 −0.286673
\(383\) 6337.84 0.845557 0.422778 0.906233i \(-0.361055\pi\)
0.422778 + 0.906233i \(0.361055\pi\)
\(384\) −7458.53 −0.991189
\(385\) 0 0
\(386\) 1818.55 0.239797
\(387\) −18.6259 −0.00244653
\(388\) 6326.46 0.827776
\(389\) −8805.25 −1.14767 −0.573836 0.818970i \(-0.694545\pi\)
−0.573836 + 0.818970i \(0.694545\pi\)
\(390\) −4164.49 −0.540710
\(391\) 1105.49 0.142985
\(392\) 0 0
\(393\) 6857.23 0.880156
\(394\) 3752.25 0.479786
\(395\) −12576.5 −1.60200
\(396\) 668.631 0.0848484
\(397\) −4315.26 −0.545534 −0.272767 0.962080i \(-0.587939\pi\)
−0.272767 + 0.962080i \(0.587939\pi\)
\(398\) −5.63108 −0.000709197 0
\(399\) 0 0
\(400\) 2071.79 0.258974
\(401\) 361.681 0.0450411 0.0225206 0.999746i \(-0.492831\pi\)
0.0225206 + 0.999746i \(0.492831\pi\)
\(402\) 1786.42 0.221638
\(403\) −4888.79 −0.604288
\(404\) 9654.53 1.18894
\(405\) −11346.5 −1.39213
\(406\) 0 0
\(407\) −449.420 −0.0547345
\(408\) −5555.70 −0.674137
\(409\) −9220.50 −1.11473 −0.557365 0.830268i \(-0.688188\pi\)
−0.557365 + 0.830268i \(0.688188\pi\)
\(410\) 2587.50 0.311677
\(411\) −1175.70 −0.141102
\(412\) −12877.4 −1.53986
\(413\) 0 0
\(414\) −79.6092 −0.00945067
\(415\) −336.245 −0.0397726
\(416\) 9569.74 1.12787
\(417\) −17192.6 −2.01901
\(418\) 546.954 0.0640010
\(419\) 14912.9 1.73876 0.869380 0.494144i \(-0.164519\pi\)
0.869380 + 0.494144i \(0.164519\pi\)
\(420\) 0 0
\(421\) −13486.0 −1.56121 −0.780603 0.625027i \(-0.785088\pi\)
−0.780603 + 0.625027i \(0.785088\pi\)
\(422\) −2274.73 −0.262399
\(423\) −585.085 −0.0672525
\(424\) −1686.81 −0.193205
\(425\) 3335.16 0.380656
\(426\) −2042.44 −0.232292
\(427\) 0 0
\(428\) 3613.96 0.408148
\(429\) −4867.36 −0.547782
\(430\) 21.5260 0.00241413
\(431\) 406.334 0.0454116 0.0227058 0.999742i \(-0.492772\pi\)
0.0227058 + 0.999742i \(0.492772\pi\)
\(432\) 5748.59 0.640230
\(433\) 1766.69 0.196078 0.0980391 0.995183i \(-0.468743\pi\)
0.0980391 + 0.995183i \(0.468743\pi\)
\(434\) 0 0
\(435\) −12880.0 −1.41965
\(436\) 479.615 0.0526821
\(437\) 907.033 0.0992889
\(438\) −2649.85 −0.289075
\(439\) −7824.19 −0.850634 −0.425317 0.905044i \(-0.639837\pi\)
−0.425317 + 0.905044i \(0.639837\pi\)
\(440\) −1600.95 −0.173460
\(441\) 0 0
\(442\) 4523.44 0.486783
\(443\) 11667.9 1.25137 0.625686 0.780075i \(-0.284819\pi\)
0.625686 + 0.780075i \(0.284819\pi\)
\(444\) 1807.84 0.193235
\(445\) 4536.86 0.483299
\(446\) −9.01776 −0.000957406 0
\(447\) −20679.6 −2.18817
\(448\) 0 0
\(449\) 16975.3 1.78421 0.892107 0.451825i \(-0.149227\pi\)
0.892107 + 0.451825i \(0.149227\pi\)
\(450\) −240.173 −0.0251597
\(451\) 3024.21 0.315753
\(452\) 14966.1 1.55741
\(453\) 6896.42 0.715280
\(454\) 3379.07 0.349312
\(455\) 0 0
\(456\) −4558.33 −0.468122
\(457\) −16192.9 −1.65748 −0.828741 0.559632i \(-0.810943\pi\)
−0.828741 + 0.559632i \(0.810943\pi\)
\(458\) 3714.89 0.379007
\(459\) 9254.05 0.941050
\(460\) −1281.45 −0.129887
\(461\) −8586.04 −0.867444 −0.433722 0.901047i \(-0.642800\pi\)
−0.433722 + 0.901047i \(0.642800\pi\)
\(462\) 0 0
\(463\) −7917.20 −0.794694 −0.397347 0.917668i \(-0.630069\pi\)
−0.397347 + 0.917668i \(0.630069\pi\)
\(464\) 8690.67 0.869513
\(465\) −4991.91 −0.497837
\(466\) −154.962 −0.0154045
\(467\) 15155.0 1.50169 0.750844 0.660480i \(-0.229647\pi\)
0.750844 + 0.660480i \(0.229647\pi\)
\(468\) 4537.03 0.448128
\(469\) 0 0
\(470\) 676.183 0.0663617
\(471\) 2027.78 0.198376
\(472\) −6176.82 −0.602354
\(473\) 25.1591 0.00244570
\(474\) −4245.25 −0.411373
\(475\) 2736.43 0.264328
\(476\) 0 0
\(477\) −1213.44 −0.116477
\(478\) −3156.96 −0.302084
\(479\) −10001.1 −0.953993 −0.476996 0.878905i \(-0.658275\pi\)
−0.476996 + 0.878905i \(0.658275\pi\)
\(480\) 9771.59 0.929188
\(481\) −3049.56 −0.289081
\(482\) −729.550 −0.0689421
\(483\) 0 0
\(484\) −903.156 −0.0848193
\(485\) −10896.9 −1.02021
\(486\) −1620.62 −0.151261
\(487\) 7044.54 0.655480 0.327740 0.944768i \(-0.393713\pi\)
0.327740 + 0.944768i \(0.393713\pi\)
\(488\) −1146.80 −0.106379
\(489\) 8269.21 0.764717
\(490\) 0 0
\(491\) −13326.4 −1.22487 −0.612437 0.790520i \(-0.709811\pi\)
−0.612437 + 0.790520i \(0.709811\pi\)
\(492\) −12165.2 −1.11474
\(493\) 13990.2 1.27806
\(494\) 3711.38 0.338022
\(495\) −1151.67 −0.104573
\(496\) 3368.25 0.304917
\(497\) 0 0
\(498\) −113.501 −0.0102131
\(499\) −20069.1 −1.80044 −0.900218 0.435440i \(-0.856593\pi\)
−0.900218 + 0.435440i \(0.856593\pi\)
\(500\) 8129.17 0.727095
\(501\) 2837.85 0.253065
\(502\) −202.699 −0.0180217
\(503\) −7782.35 −0.689856 −0.344928 0.938629i \(-0.612097\pi\)
−0.344928 + 0.938629i \(0.612097\pi\)
\(504\) 0 0
\(505\) −16629.3 −1.46533
\(506\) 107.532 0.00944744
\(507\) −20003.4 −1.75224
\(508\) −818.102 −0.0714516
\(509\) 1475.93 0.128526 0.0642628 0.997933i \(-0.479530\pi\)
0.0642628 + 0.997933i \(0.479530\pi\)
\(510\) 4618.85 0.401031
\(511\) 0 0
\(512\) −11250.6 −0.971116
\(513\) 7592.75 0.653466
\(514\) −2368.32 −0.203233
\(515\) 22180.4 1.89784
\(516\) −101.205 −0.00863431
\(517\) 790.307 0.0672295
\(518\) 0 0
\(519\) 10721.7 0.906799
\(520\) −10863.3 −0.916132
\(521\) −7609.43 −0.639875 −0.319938 0.947439i \(-0.603662\pi\)
−0.319938 + 0.947439i \(0.603662\pi\)
\(522\) −1007.47 −0.0844744
\(523\) −12452.9 −1.04116 −0.520581 0.853812i \(-0.674285\pi\)
−0.520581 + 0.853812i \(0.674285\pi\)
\(524\) 8633.82 0.719790
\(525\) 0 0
\(526\) −152.226 −0.0126186
\(527\) 5422.18 0.448186
\(528\) 3353.48 0.276404
\(529\) −11988.7 −0.985344
\(530\) 1402.37 0.114934
\(531\) −4443.39 −0.363139
\(532\) 0 0
\(533\) 20520.9 1.66765
\(534\) 1531.44 0.124105
\(535\) −6224.81 −0.503031
\(536\) 4659.99 0.375524
\(537\) 26261.0 2.11033
\(538\) 3684.44 0.295255
\(539\) 0 0
\(540\) −10727.0 −0.854847
\(541\) 9312.17 0.740039 0.370020 0.929024i \(-0.379351\pi\)
0.370020 + 0.929024i \(0.379351\pi\)
\(542\) 1088.57 0.0862693
\(543\) 20210.3 1.59725
\(544\) −10613.8 −0.836515
\(545\) −826.105 −0.0649292
\(546\) 0 0
\(547\) −11018.6 −0.861278 −0.430639 0.902524i \(-0.641712\pi\)
−0.430639 + 0.902524i \(0.641712\pi\)
\(548\) −1480.31 −0.115393
\(549\) −824.968 −0.0641325
\(550\) 324.415 0.0251511
\(551\) 11478.6 0.887490
\(552\) −896.179 −0.0691013
\(553\) 0 0
\(554\) 172.644 0.0132399
\(555\) −3113.89 −0.238157
\(556\) −21646.9 −1.65114
\(557\) −12018.4 −0.914250 −0.457125 0.889403i \(-0.651121\pi\)
−0.457125 + 0.889403i \(0.651121\pi\)
\(558\) −390.464 −0.0296231
\(559\) 170.718 0.0129170
\(560\) 0 0
\(561\) 5398.40 0.406276
\(562\) 3598.04 0.270061
\(563\) 8763.89 0.656046 0.328023 0.944670i \(-0.393618\pi\)
0.328023 + 0.944670i \(0.393618\pi\)
\(564\) −3179.10 −0.237348
\(565\) −25778.1 −1.91946
\(566\) −3806.34 −0.282672
\(567\) 0 0
\(568\) −5327.84 −0.393576
\(569\) −10273.2 −0.756895 −0.378447 0.925623i \(-0.623542\pi\)
−0.378447 + 0.925623i \(0.623542\pi\)
\(570\) 3789.67 0.278477
\(571\) 2602.62 0.190747 0.0953734 0.995442i \(-0.469596\pi\)
0.0953734 + 0.995442i \(0.469596\pi\)
\(572\) −6128.41 −0.447975
\(573\) −17332.6 −1.26366
\(574\) 0 0
\(575\) 537.988 0.0390185
\(576\) −2585.99 −0.187065
\(577\) 19727.0 1.42331 0.711653 0.702532i \(-0.247947\pi\)
0.711653 + 0.702532i \(0.247947\pi\)
\(578\) −1420.39 −0.102215
\(579\) 14726.7 1.05703
\(580\) −16217.0 −1.16099
\(581\) 0 0
\(582\) −3678.30 −0.261977
\(583\) 1639.06 0.116437
\(584\) −6912.30 −0.489783
\(585\) −7814.72 −0.552306
\(586\) −6501.28 −0.458303
\(587\) 10116.2 0.711309 0.355654 0.934618i \(-0.384258\pi\)
0.355654 + 0.934618i \(0.384258\pi\)
\(588\) 0 0
\(589\) 4448.78 0.311221
\(590\) 5135.23 0.358329
\(591\) 30386.0 2.11491
\(592\) 2101.07 0.145867
\(593\) −3130.32 −0.216774 −0.108387 0.994109i \(-0.534569\pi\)
−0.108387 + 0.994109i \(0.534569\pi\)
\(594\) 900.152 0.0621779
\(595\) 0 0
\(596\) −26037.3 −1.78948
\(597\) −45.6009 −0.00312616
\(598\) 729.667 0.0498968
\(599\) 10080.1 0.687581 0.343790 0.939046i \(-0.388289\pi\)
0.343790 + 0.939046i \(0.388289\pi\)
\(600\) −2703.68 −0.183962
\(601\) −4777.02 −0.324224 −0.162112 0.986772i \(-0.551831\pi\)
−0.162112 + 0.986772i \(0.551831\pi\)
\(602\) 0 0
\(603\) 3352.24 0.226391
\(604\) 8683.16 0.584955
\(605\) 1555.63 0.104537
\(606\) −5613.29 −0.376278
\(607\) 2571.35 0.171941 0.0859703 0.996298i \(-0.472601\pi\)
0.0859703 + 0.996298i \(0.472601\pi\)
\(608\) −8708.43 −0.580877
\(609\) 0 0
\(610\) 953.414 0.0632830
\(611\) 5362.67 0.355074
\(612\) −5032.03 −0.332366
\(613\) 12711.9 0.837564 0.418782 0.908087i \(-0.362457\pi\)
0.418782 + 0.908087i \(0.362457\pi\)
\(614\) −1096.56 −0.0720744
\(615\) 20953.8 1.37388
\(616\) 0 0
\(617\) 16236.1 1.05939 0.529693 0.848189i \(-0.322307\pi\)
0.529693 + 0.848189i \(0.322307\pi\)
\(618\) 7487.11 0.487339
\(619\) −12657.3 −0.821874 −0.410937 0.911664i \(-0.634798\pi\)
−0.410937 + 0.911664i \(0.634798\pi\)
\(620\) −6285.23 −0.407131
\(621\) 1492.75 0.0964607
\(622\) −5479.19 −0.353208
\(623\) 0 0
\(624\) 22755.2 1.45983
\(625\) −19037.8 −1.21842
\(626\) −481.955 −0.0307713
\(627\) 4429.28 0.282119
\(628\) 2553.15 0.162232
\(629\) 3382.28 0.214404
\(630\) 0 0
\(631\) −3949.97 −0.249201 −0.124600 0.992207i \(-0.539765\pi\)
−0.124600 + 0.992207i \(0.539765\pi\)
\(632\) −11074.0 −0.696994
\(633\) −18421.0 −1.15666
\(634\) −171.086 −0.0107172
\(635\) 1409.13 0.0880621
\(636\) −6593.29 −0.411070
\(637\) 0 0
\(638\) 1360.84 0.0844455
\(639\) −3832.67 −0.237274
\(640\) 16175.2 0.999033
\(641\) −7398.27 −0.455872 −0.227936 0.973676i \(-0.573198\pi\)
−0.227936 + 0.973676i \(0.573198\pi\)
\(642\) −2101.22 −0.129172
\(643\) 12491.7 0.766134 0.383067 0.923721i \(-0.374868\pi\)
0.383067 + 0.923721i \(0.374868\pi\)
\(644\) 0 0
\(645\) 174.319 0.0106415
\(646\) −4116.31 −0.250703
\(647\) 10472.0 0.636315 0.318158 0.948038i \(-0.396936\pi\)
0.318158 + 0.948038i \(0.396936\pi\)
\(648\) −9991.02 −0.605685
\(649\) 6001.94 0.363015
\(650\) 2201.33 0.132836
\(651\) 0 0
\(652\) 10411.6 0.625384
\(653\) 6337.94 0.379820 0.189910 0.981801i \(-0.439180\pi\)
0.189910 + 0.981801i \(0.439180\pi\)
\(654\) −278.856 −0.0166730
\(655\) −14871.2 −0.887121
\(656\) −14138.4 −0.841479
\(657\) −4972.48 −0.295274
\(658\) 0 0
\(659\) 15196.7 0.898302 0.449151 0.893456i \(-0.351726\pi\)
0.449151 + 0.893456i \(0.351726\pi\)
\(660\) −6257.67 −0.369060
\(661\) −2298.17 −0.135232 −0.0676161 0.997711i \(-0.521539\pi\)
−0.0676161 + 0.997711i \(0.521539\pi\)
\(662\) −6246.55 −0.366736
\(663\) 36631.1 2.14575
\(664\) −296.075 −0.0173042
\(665\) 0 0
\(666\) −243.566 −0.0141712
\(667\) 2256.73 0.131006
\(668\) 3573.09 0.206956
\(669\) −73.0265 −0.00422028
\(670\) −3874.18 −0.223392
\(671\) 1114.33 0.0641106
\(672\) 0 0
\(673\) 23199.6 1.32880 0.664398 0.747379i \(-0.268688\pi\)
0.664398 + 0.747379i \(0.268688\pi\)
\(674\) −8558.54 −0.489114
\(675\) 4503.49 0.256799
\(676\) −25186.0 −1.43298
\(677\) 2145.38 0.121793 0.0608963 0.998144i \(-0.480604\pi\)
0.0608963 + 0.998144i \(0.480604\pi\)
\(678\) −8701.55 −0.492892
\(679\) 0 0
\(680\) 12048.6 0.679472
\(681\) 27364.0 1.53978
\(682\) 527.422 0.0296129
\(683\) −29544.6 −1.65519 −0.827593 0.561329i \(-0.810290\pi\)
−0.827593 + 0.561329i \(0.810290\pi\)
\(684\) −4128.68 −0.230795
\(685\) 2549.72 0.142219
\(686\) 0 0
\(687\) 30083.4 1.67068
\(688\) −117.620 −0.00651776
\(689\) 11121.9 0.614964
\(690\) 745.057 0.0411070
\(691\) −27803.1 −1.53065 −0.765325 0.643644i \(-0.777422\pi\)
−0.765325 + 0.643644i \(0.777422\pi\)
\(692\) 13499.5 0.741579
\(693\) 0 0
\(694\) −3366.55 −0.184139
\(695\) 37285.4 2.03498
\(696\) −11341.3 −0.617659
\(697\) −22759.8 −1.23686
\(698\) 4919.58 0.266775
\(699\) −1254.90 −0.0679036
\(700\) 0 0
\(701\) −19697.8 −1.06130 −0.530652 0.847590i \(-0.678053\pi\)
−0.530652 + 0.847590i \(0.678053\pi\)
\(702\) 6108.02 0.328394
\(703\) 2775.09 0.148883
\(704\) 3493.03 0.187001
\(705\) 5475.78 0.292524
\(706\) 4201.02 0.223948
\(707\) 0 0
\(708\) −24143.5 −1.28159
\(709\) 19122.5 1.01292 0.506460 0.862263i \(-0.330954\pi\)
0.506460 + 0.862263i \(0.330954\pi\)
\(710\) 4429.41 0.234131
\(711\) −7966.27 −0.420195
\(712\) 3994.86 0.210272
\(713\) 874.641 0.0459405
\(714\) 0 0
\(715\) 10555.8 0.552117
\(716\) 33064.8 1.72582
\(717\) −25565.3 −1.33160
\(718\) 3012.59 0.156586
\(719\) −1837.44 −0.0953060 −0.0476530 0.998864i \(-0.515174\pi\)
−0.0476530 + 0.998864i \(0.515174\pi\)
\(720\) 5384.13 0.278687
\(721\) 0 0
\(722\) 1643.79 0.0847307
\(723\) −5907.95 −0.303899
\(724\) 25446.4 1.30623
\(725\) 6808.33 0.348765
\(726\) 525.109 0.0268438
\(727\) 7555.46 0.385442 0.192721 0.981254i \(-0.438269\pi\)
0.192721 + 0.981254i \(0.438269\pi\)
\(728\) 0 0
\(729\) 10705.2 0.543881
\(730\) 5746.69 0.291362
\(731\) −189.344 −0.00958021
\(732\) −4482.51 −0.226337
\(733\) −11984.6 −0.603905 −0.301952 0.953323i \(-0.597638\pi\)
−0.301952 + 0.953323i \(0.597638\pi\)
\(734\) 7073.80 0.355720
\(735\) 0 0
\(736\) −1712.10 −0.0857456
\(737\) −4528.05 −0.226313
\(738\) 1638.99 0.0817508
\(739\) −27142.5 −1.35109 −0.675543 0.737321i \(-0.736091\pi\)
−0.675543 + 0.737321i \(0.736091\pi\)
\(740\) −3920.64 −0.194764
\(741\) 30055.1 1.49001
\(742\) 0 0
\(743\) −29222.6 −1.44290 −0.721450 0.692467i \(-0.756524\pi\)
−0.721450 + 0.692467i \(0.756524\pi\)
\(744\) −4395.55 −0.216598
\(745\) 44847.6 2.20549
\(746\) 103.790 0.00509388
\(747\) −212.987 −0.0104321
\(748\) 6797.04 0.332252
\(749\) 0 0
\(750\) −4726.43 −0.230113
\(751\) −8859.39 −0.430471 −0.215236 0.976562i \(-0.569052\pi\)
−0.215236 + 0.976562i \(0.569052\pi\)
\(752\) −3694.73 −0.179166
\(753\) −1641.47 −0.0794404
\(754\) 9234.05 0.446000
\(755\) −14956.2 −0.720941
\(756\) 0 0
\(757\) 35734.4 1.71571 0.857853 0.513896i \(-0.171798\pi\)
0.857853 + 0.513896i \(0.171798\pi\)
\(758\) 2064.19 0.0989112
\(759\) 870.806 0.0416446
\(760\) 9885.59 0.471826
\(761\) −34394.7 −1.63838 −0.819189 0.573524i \(-0.805576\pi\)
−0.819189 + 0.573524i \(0.805576\pi\)
\(762\) 475.658 0.0226132
\(763\) 0 0
\(764\) −21823.2 −1.03342
\(765\) 8667.33 0.409631
\(766\) −4639.62 −0.218846
\(767\) 40726.4 1.91727
\(768\) −9599.92 −0.451051
\(769\) 11602.7 0.544091 0.272045 0.962284i \(-0.412300\pi\)
0.272045 + 0.962284i \(0.412300\pi\)
\(770\) 0 0
\(771\) −19178.8 −0.895859
\(772\) 18542.2 0.864441
\(773\) 12680.6 0.590026 0.295013 0.955493i \(-0.404676\pi\)
0.295013 + 0.955493i \(0.404676\pi\)
\(774\) 13.6351 0.000633210 0
\(775\) 2638.71 0.122303
\(776\) −9595.09 −0.443871
\(777\) 0 0
\(778\) 6445.89 0.297039
\(779\) −18674.0 −0.858876
\(780\) −42461.7 −1.94920
\(781\) 5176.99 0.237193
\(782\) −809.276 −0.0370072
\(783\) 18891.0 0.862210
\(784\) 0 0
\(785\) −4397.62 −0.199946
\(786\) −5019.84 −0.227801
\(787\) 4417.61 0.200090 0.100045 0.994983i \(-0.468101\pi\)
0.100045 + 0.994983i \(0.468101\pi\)
\(788\) 38258.5 1.72957
\(789\) −1232.74 −0.0556231
\(790\) 9206.61 0.414628
\(791\) 0 0
\(792\) −1014.09 −0.0454974
\(793\) 7561.33 0.338601
\(794\) 3158.99 0.141194
\(795\) 11356.5 0.506633
\(796\) −57.4153 −0.00255657
\(797\) 27030.1 1.20132 0.600661 0.799504i \(-0.294904\pi\)
0.600661 + 0.799504i \(0.294904\pi\)
\(798\) 0 0
\(799\) −5947.75 −0.263350
\(800\) −5165.23 −0.228273
\(801\) 2873.77 0.126766
\(802\) −264.769 −0.0116575
\(803\) 6716.60 0.295173
\(804\) 18214.6 0.798979
\(805\) 0 0
\(806\) 3578.85 0.156401
\(807\) 29836.9 1.30150
\(808\) −14642.6 −0.637533
\(809\) 23647.0 1.02767 0.513835 0.857889i \(-0.328224\pi\)
0.513835 + 0.857889i \(0.328224\pi\)
\(810\) 8306.24 0.360311
\(811\) −33486.1 −1.44988 −0.724941 0.688811i \(-0.758133\pi\)
−0.724941 + 0.688811i \(0.758133\pi\)
\(812\) 0 0
\(813\) 8815.30 0.380278
\(814\) 328.999 0.0141663
\(815\) −17933.3 −0.770768
\(816\) −25237.8 −1.08272
\(817\) −155.353 −0.00665251
\(818\) 6749.88 0.288513
\(819\) 0 0
\(820\) 26382.5 1.12356
\(821\) 2605.69 0.110766 0.0553832 0.998465i \(-0.482362\pi\)
0.0553832 + 0.998465i \(0.482362\pi\)
\(822\) 860.673 0.0365200
\(823\) 31976.2 1.35434 0.677169 0.735828i \(-0.263207\pi\)
0.677169 + 0.735828i \(0.263207\pi\)
\(824\) 19530.6 0.825705
\(825\) 2627.14 0.110867
\(826\) 0 0
\(827\) −37759.0 −1.58768 −0.793839 0.608128i \(-0.791921\pi\)
−0.793839 + 0.608128i \(0.791921\pi\)
\(828\) −811.707 −0.0340686
\(829\) 1137.55 0.0476584 0.0238292 0.999716i \(-0.492414\pi\)
0.0238292 + 0.999716i \(0.492414\pi\)
\(830\) 246.149 0.0102939
\(831\) 1398.08 0.0583621
\(832\) 23702.2 0.987649
\(833\) 0 0
\(834\) 12585.9 0.522557
\(835\) −6154.40 −0.255068
\(836\) 5576.83 0.230716
\(837\) 7321.60 0.302356
\(838\) −10917.0 −0.450024
\(839\) 37372.2 1.53782 0.768911 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(840\) 0 0
\(841\) 4170.26 0.170989
\(842\) 9872.44 0.404070
\(843\) 29137.2 1.19044
\(844\) −23193.5 −0.945917
\(845\) 43381.1 1.76610
\(846\) 428.312 0.0174062
\(847\) 0 0
\(848\) −7662.68 −0.310304
\(849\) −30824.0 −1.24603
\(850\) −2441.51 −0.0985211
\(851\) 545.589 0.0219772
\(852\) −20825.0 −0.837387
\(853\) 22490.8 0.902780 0.451390 0.892327i \(-0.350928\pi\)
0.451390 + 0.892327i \(0.350928\pi\)
\(854\) 0 0
\(855\) 7111.36 0.284449
\(856\) −5481.16 −0.218858
\(857\) −43409.5 −1.73027 −0.865135 0.501539i \(-0.832767\pi\)
−0.865135 + 0.501539i \(0.832767\pi\)
\(858\) 3563.15 0.141776
\(859\) −29533.2 −1.17306 −0.586532 0.809926i \(-0.699507\pi\)
−0.586532 + 0.809926i \(0.699507\pi\)
\(860\) 219.482 0.00870264
\(861\) 0 0
\(862\) −297.457 −0.0117534
\(863\) 14351.6 0.566090 0.283045 0.959107i \(-0.408655\pi\)
0.283045 + 0.959107i \(0.408655\pi\)
\(864\) −14331.9 −0.564331
\(865\) −23251.9 −0.913975
\(866\) −1293.31 −0.0507488
\(867\) −11502.4 −0.450569
\(868\) 0 0
\(869\) 10760.5 0.420051
\(870\) 9428.82 0.367433
\(871\) −30725.3 −1.19528
\(872\) −727.413 −0.0282492
\(873\) −6902.39 −0.267595
\(874\) −663.994 −0.0256979
\(875\) 0 0
\(876\) −27018.3 −1.04208
\(877\) −43248.7 −1.66523 −0.832614 0.553854i \(-0.813156\pi\)
−0.832614 + 0.553854i \(0.813156\pi\)
\(878\) 5727.71 0.220160
\(879\) −52647.9 −2.02021
\(880\) −7272.64 −0.278591
\(881\) −3816.13 −0.145935 −0.0729675 0.997334i \(-0.523247\pi\)
−0.0729675 + 0.997334i \(0.523247\pi\)
\(882\) 0 0
\(883\) 48787.6 1.85938 0.929690 0.368343i \(-0.120075\pi\)
0.929690 + 0.368343i \(0.120075\pi\)
\(884\) 46121.6 1.75479
\(885\) 41585.5 1.57953
\(886\) −8541.49 −0.323879
\(887\) −41495.1 −1.57077 −0.785384 0.619009i \(-0.787534\pi\)
−0.785384 + 0.619009i \(0.787534\pi\)
\(888\) −2741.88 −0.103617
\(889\) 0 0
\(890\) −3321.21 −0.125087
\(891\) 9708.15 0.365023
\(892\) −91.9464 −0.00345134
\(893\) −4880.01 −0.182870
\(894\) 15138.5 0.566340
\(895\) −56951.9 −2.12703
\(896\) 0 0
\(897\) 5908.90 0.219947
\(898\) −12426.7 −0.461788
\(899\) 11068.7 0.410637
\(900\) −2448.84 −0.0906978
\(901\) −12335.3 −0.456104
\(902\) −2213.88 −0.0817228
\(903\) 0 0
\(904\) −22698.5 −0.835113
\(905\) −43829.7 −1.60989
\(906\) −5048.53 −0.185128
\(907\) 21615.3 0.791316 0.395658 0.918398i \(-0.370517\pi\)
0.395658 + 0.918398i \(0.370517\pi\)
\(908\) 34453.6 1.25923
\(909\) −10533.4 −0.384347
\(910\) 0 0
\(911\) 3646.35 0.132611 0.0663057 0.997799i \(-0.478879\pi\)
0.0663057 + 0.997799i \(0.478879\pi\)
\(912\) −20707.1 −0.751844
\(913\) 287.693 0.0104285
\(914\) 11854.0 0.428988
\(915\) 7720.82 0.278954
\(916\) 37877.6 1.36628
\(917\) 0 0
\(918\) −6774.43 −0.243562
\(919\) 31280.0 1.12278 0.561388 0.827553i \(-0.310267\pi\)
0.561388 + 0.827553i \(0.310267\pi\)
\(920\) 1943.53 0.0696482
\(921\) −8880.05 −0.317707
\(922\) 6285.42 0.224511
\(923\) 35128.7 1.25274
\(924\) 0 0
\(925\) 1645.99 0.0585079
\(926\) 5795.79 0.205682
\(927\) 14049.7 0.497790
\(928\) −21666.9 −0.766433
\(929\) −6557.92 −0.231602 −0.115801 0.993272i \(-0.536944\pi\)
−0.115801 + 0.993272i \(0.536944\pi\)
\(930\) 3654.33 0.128850
\(931\) 0 0
\(932\) −1580.02 −0.0555314
\(933\) −44370.9 −1.55695
\(934\) −11094.2 −0.388665
\(935\) −11707.4 −0.409491
\(936\) −6881.13 −0.240296
\(937\) 24473.3 0.853265 0.426632 0.904425i \(-0.359700\pi\)
0.426632 + 0.904425i \(0.359700\pi\)
\(938\) 0 0
\(939\) −3902.91 −0.135641
\(940\) 6894.46 0.239226
\(941\) −15420.8 −0.534224 −0.267112 0.963665i \(-0.586069\pi\)
−0.267112 + 0.963665i \(0.586069\pi\)
\(942\) −1484.44 −0.0513436
\(943\) −3671.34 −0.126782
\(944\) −28059.4 −0.967432
\(945\) 0 0
\(946\) −18.4177 −0.000632993 0
\(947\) −33141.2 −1.13722 −0.568608 0.822609i \(-0.692518\pi\)
−0.568608 + 0.822609i \(0.692518\pi\)
\(948\) −43285.2 −1.48295
\(949\) 45575.8 1.55896
\(950\) −2003.20 −0.0684132
\(951\) −1385.47 −0.0472417
\(952\) 0 0
\(953\) −20735.4 −0.704813 −0.352406 0.935847i \(-0.614637\pi\)
−0.352406 + 0.935847i \(0.614637\pi\)
\(954\) 888.298 0.0301464
\(955\) 37589.0 1.27367
\(956\) −32188.9 −1.08898
\(957\) 11020.2 0.372239
\(958\) 7321.32 0.246911
\(959\) 0 0
\(960\) 24202.1 0.813666
\(961\) −25501.1 −0.856000
\(962\) 2232.44 0.0748198
\(963\) −3942.96 −0.131942
\(964\) −7438.60 −0.248528
\(965\) −31937.7 −1.06540
\(966\) 0 0
\(967\) −8178.87 −0.271990 −0.135995 0.990710i \(-0.543423\pi\)
−0.135995 + 0.990710i \(0.543423\pi\)
\(968\) 1369.78 0.0454819
\(969\) −33334.2 −1.10511
\(970\) 7977.08 0.264050
\(971\) 20576.1 0.680039 0.340020 0.940418i \(-0.389566\pi\)
0.340020 + 0.940418i \(0.389566\pi\)
\(972\) −16524.1 −0.545277
\(973\) 0 0
\(974\) −5156.96 −0.169651
\(975\) 17826.6 0.585546
\(976\) −5209.55 −0.170854
\(977\) 14541.9 0.476188 0.238094 0.971242i \(-0.423477\pi\)
0.238094 + 0.971242i \(0.423477\pi\)
\(978\) −6053.48 −0.197923
\(979\) −3881.76 −0.126723
\(980\) 0 0
\(981\) −523.277 −0.0170305
\(982\) 9755.62 0.317021
\(983\) 29285.7 0.950223 0.475111 0.879926i \(-0.342408\pi\)
0.475111 + 0.879926i \(0.342408\pi\)
\(984\) 18450.5 0.597745
\(985\) −65897.7 −2.13165
\(986\) −10241.5 −0.330787
\(987\) 0 0
\(988\) 37841.8 1.21853
\(989\) −30.5427 −0.000982004 0
\(990\) 843.082 0.0270655
\(991\) −38085.9 −1.22083 −0.610413 0.792083i \(-0.708997\pi\)
−0.610413 + 0.792083i \(0.708997\pi\)
\(992\) −8397.44 −0.268769
\(993\) −50585.1 −1.61658
\(994\) 0 0
\(995\) 98.8940 0.00315090
\(996\) −1157.28 −0.0368170
\(997\) 26803.6 0.851434 0.425717 0.904856i \(-0.360022\pi\)
0.425717 + 0.904856i \(0.360022\pi\)
\(998\) 14691.6 0.465987
\(999\) 4567.12 0.144642
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 539.4.a.e.1.1 2
7.6 odd 2 11.4.a.a.1.1 2
21.20 even 2 99.4.a.c.1.2 2
28.27 even 2 176.4.a.i.1.1 2
35.13 even 4 275.4.b.c.199.3 4
35.27 even 4 275.4.b.c.199.2 4
35.34 odd 2 275.4.a.b.1.2 2
56.13 odd 2 704.4.a.p.1.1 2
56.27 even 2 704.4.a.n.1.2 2
77.6 even 10 121.4.c.f.3.1 8
77.13 even 10 121.4.c.f.81.1 8
77.20 odd 10 121.4.c.c.81.2 8
77.27 odd 10 121.4.c.c.3.2 8
77.41 even 10 121.4.c.f.9.2 8
77.48 odd 10 121.4.c.c.27.1 8
77.62 even 10 121.4.c.f.27.2 8
77.69 odd 10 121.4.c.c.9.1 8
77.76 even 2 121.4.a.c.1.2 2
84.83 odd 2 1584.4.a.bc.1.2 2
91.90 odd 2 1859.4.a.a.1.2 2
105.104 even 2 2475.4.a.q.1.1 2
231.230 odd 2 1089.4.a.v.1.1 2
308.307 odd 2 1936.4.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.1 2 7.6 odd 2
99.4.a.c.1.2 2 21.20 even 2
121.4.a.c.1.2 2 77.76 even 2
121.4.c.c.3.2 8 77.27 odd 10
121.4.c.c.9.1 8 77.69 odd 10
121.4.c.c.27.1 8 77.48 odd 10
121.4.c.c.81.2 8 77.20 odd 10
121.4.c.f.3.1 8 77.6 even 10
121.4.c.f.9.2 8 77.41 even 10
121.4.c.f.27.2 8 77.62 even 10
121.4.c.f.81.1 8 77.13 even 10
176.4.a.i.1.1 2 28.27 even 2
275.4.a.b.1.2 2 35.34 odd 2
275.4.b.c.199.2 4 35.27 even 4
275.4.b.c.199.3 4 35.13 even 4
539.4.a.e.1.1 2 1.1 even 1 trivial
704.4.a.n.1.2 2 56.27 even 2
704.4.a.p.1.1 2 56.13 odd 2
1089.4.a.v.1.1 2 231.230 odd 2
1584.4.a.bc.1.2 2 84.83 odd 2
1859.4.a.a.1.2 2 91.90 odd 2
1936.4.a.w.1.1 2 308.307 odd 2
2475.4.a.q.1.1 2 105.104 even 2