Properties

Label 525.4.a.j.1.2
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.561553 q^{2} +3.00000 q^{3} -7.68466 q^{4} +1.68466 q^{6} +7.00000 q^{7} -8.80776 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.561553 q^{2} +3.00000 q^{3} -7.68466 q^{4} +1.68466 q^{6} +7.00000 q^{7} -8.80776 q^{8} +9.00000 q^{9} -25.8078 q^{11} -23.0540 q^{12} +15.5464 q^{13} +3.93087 q^{14} +56.5312 q^{16} +95.1619 q^{17} +5.05398 q^{18} -143.447 q^{19} +21.0000 q^{21} -14.4924 q^{22} -77.5227 q^{23} -26.4233 q^{24} +8.73012 q^{26} +27.0000 q^{27} -53.7926 q^{28} -204.170 q^{29} -40.6998 q^{31} +102.207 q^{32} -77.4233 q^{33} +53.4384 q^{34} -69.1619 q^{36} -95.6695 q^{37} -80.5530 q^{38} +46.6392 q^{39} +24.7301 q^{41} +11.7926 q^{42} -282.430 q^{43} +198.324 q^{44} -43.5331 q^{46} -257.417 q^{47} +169.594 q^{48} +49.0000 q^{49} +285.486 q^{51} -119.469 q^{52} -257.978 q^{53} +15.1619 q^{54} -61.6543 q^{56} -430.341 q^{57} -114.652 q^{58} -651.365 q^{59} +451.304 q^{61} -22.8551 q^{62} +63.0000 q^{63} -394.855 q^{64} -43.4773 q^{66} -832.071 q^{67} -731.287 q^{68} -232.568 q^{69} -174.965 q^{71} -79.2699 q^{72} +47.4166 q^{73} -53.7235 q^{74} +1102.34 q^{76} -180.654 q^{77} +26.1904 q^{78} -1.16006 q^{79} +81.0000 q^{81} +13.8873 q^{82} -1494.03 q^{83} -161.378 q^{84} -158.599 q^{86} -612.511 q^{87} +227.309 q^{88} +1309.13 q^{89} +108.825 q^{91} +595.736 q^{92} -122.099 q^{93} -144.553 q^{94} +306.622 q^{96} +1365.33 q^{97} +27.5161 q^{98} -232.270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.561553 0.198539 0.0992695 0.995061i \(-0.468349\pi\)
0.0992695 + 0.995061i \(0.468349\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.68466 −0.960582
\(5\) 0 0
\(6\) 1.68466 0.114626
\(7\) 7.00000 0.377964
\(8\) −8.80776 −0.389252
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −25.8078 −0.707394 −0.353697 0.935360i \(-0.615076\pi\)
−0.353697 + 0.935360i \(0.615076\pi\)
\(12\) −23.0540 −0.554592
\(13\) 15.5464 0.331677 0.165838 0.986153i \(-0.446967\pi\)
0.165838 + 0.986153i \(0.446967\pi\)
\(14\) 3.93087 0.0750407
\(15\) 0 0
\(16\) 56.5312 0.883301
\(17\) 95.1619 1.35766 0.678828 0.734297i \(-0.262488\pi\)
0.678828 + 0.734297i \(0.262488\pi\)
\(18\) 5.05398 0.0661796
\(19\) −143.447 −1.73205 −0.866026 0.499999i \(-0.833334\pi\)
−0.866026 + 0.499999i \(0.833334\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) −14.4924 −0.140445
\(23\) −77.5227 −0.702809 −0.351405 0.936224i \(-0.614296\pi\)
−0.351405 + 0.936224i \(0.614296\pi\)
\(24\) −26.4233 −0.224735
\(25\) 0 0
\(26\) 8.73012 0.0658507
\(27\) 27.0000 0.192450
\(28\) −53.7926 −0.363066
\(29\) −204.170 −1.30736 −0.653681 0.756770i \(-0.726776\pi\)
−0.653681 + 0.756770i \(0.726776\pi\)
\(30\) 0 0
\(31\) −40.6998 −0.235803 −0.117902 0.993025i \(-0.537617\pi\)
−0.117902 + 0.993025i \(0.537617\pi\)
\(32\) 102.207 0.564621
\(33\) −77.4233 −0.408414
\(34\) 53.4384 0.269548
\(35\) 0 0
\(36\) −69.1619 −0.320194
\(37\) −95.6695 −0.425080 −0.212540 0.977152i \(-0.568174\pi\)
−0.212540 + 0.977152i \(0.568174\pi\)
\(38\) −80.5530 −0.343880
\(39\) 46.6392 0.191494
\(40\) 0 0
\(41\) 24.7301 0.0941999 0.0471000 0.998890i \(-0.485002\pi\)
0.0471000 + 0.998890i \(0.485002\pi\)
\(42\) 11.7926 0.0433247
\(43\) −282.430 −1.00163 −0.500816 0.865554i \(-0.666967\pi\)
−0.500816 + 0.865554i \(0.666967\pi\)
\(44\) 198.324 0.679510
\(45\) 0 0
\(46\) −43.5331 −0.139535
\(47\) −257.417 −0.798895 −0.399448 0.916756i \(-0.630798\pi\)
−0.399448 + 0.916756i \(0.630798\pi\)
\(48\) 169.594 0.509974
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 285.486 0.783843
\(52\) −119.469 −0.318603
\(53\) −257.978 −0.668604 −0.334302 0.942466i \(-0.608501\pi\)
−0.334302 + 0.942466i \(0.608501\pi\)
\(54\) 15.1619 0.0382088
\(55\) 0 0
\(56\) −61.6543 −0.147123
\(57\) −430.341 −1.00000
\(58\) −114.652 −0.259562
\(59\) −651.365 −1.43730 −0.718648 0.695374i \(-0.755239\pi\)
−0.718648 + 0.695374i \(0.755239\pi\)
\(60\) 0 0
\(61\) 451.304 0.947271 0.473636 0.880721i \(-0.342941\pi\)
0.473636 + 0.880721i \(0.342941\pi\)
\(62\) −22.8551 −0.0468161
\(63\) 63.0000 0.125988
\(64\) −394.855 −0.771201
\(65\) 0 0
\(66\) −43.4773 −0.0810861
\(67\) −832.071 −1.51722 −0.758609 0.651546i \(-0.774121\pi\)
−0.758609 + 0.651546i \(0.774121\pi\)
\(68\) −731.287 −1.30414
\(69\) −232.568 −0.405767
\(70\) 0 0
\(71\) −174.965 −0.292458 −0.146229 0.989251i \(-0.546714\pi\)
−0.146229 + 0.989251i \(0.546714\pi\)
\(72\) −79.2699 −0.129751
\(73\) 47.4166 0.0760233 0.0380116 0.999277i \(-0.487898\pi\)
0.0380116 + 0.999277i \(0.487898\pi\)
\(74\) −53.7235 −0.0843950
\(75\) 0 0
\(76\) 1102.34 1.66378
\(77\) −180.654 −0.267370
\(78\) 26.1904 0.0380189
\(79\) −1.16006 −0.00165211 −0.000826057 1.00000i \(-0.500263\pi\)
−0.000826057 1.00000i \(0.500263\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 13.8873 0.0187023
\(83\) −1494.03 −1.97580 −0.987898 0.155107i \(-0.950428\pi\)
−0.987898 + 0.155107i \(0.950428\pi\)
\(84\) −161.378 −0.209616
\(85\) 0 0
\(86\) −158.599 −0.198863
\(87\) −612.511 −0.754806
\(88\) 227.309 0.275354
\(89\) 1309.13 1.55919 0.779593 0.626286i \(-0.215426\pi\)
0.779593 + 0.626286i \(0.215426\pi\)
\(90\) 0 0
\(91\) 108.825 0.125362
\(92\) 595.736 0.675106
\(93\) −122.099 −0.136141
\(94\) −144.553 −0.158612
\(95\) 0 0
\(96\) 306.622 0.325984
\(97\) 1365.33 1.42916 0.714580 0.699553i \(-0.246618\pi\)
0.714580 + 0.699553i \(0.246618\pi\)
\(98\) 27.5161 0.0283627
\(99\) −232.270 −0.235798
\(100\) 0 0
\(101\) 1155.99 1.13887 0.569434 0.822037i \(-0.307162\pi\)
0.569434 + 0.822037i \(0.307162\pi\)
\(102\) 160.315 0.155623
\(103\) 78.2074 0.0748156 0.0374078 0.999300i \(-0.488090\pi\)
0.0374078 + 0.999300i \(0.488090\pi\)
\(104\) −136.929 −0.129106
\(105\) 0 0
\(106\) −144.868 −0.132744
\(107\) 94.0814 0.0850018 0.0425009 0.999096i \(-0.486467\pi\)
0.0425009 + 0.999096i \(0.486467\pi\)
\(108\) −207.486 −0.184864
\(109\) −2128.25 −1.87018 −0.935088 0.354417i \(-0.884679\pi\)
−0.935088 + 0.354417i \(0.884679\pi\)
\(110\) 0 0
\(111\) −287.009 −0.245420
\(112\) 395.719 0.333856
\(113\) 596.493 0.496578 0.248289 0.968686i \(-0.420132\pi\)
0.248289 + 0.968686i \(0.420132\pi\)
\(114\) −241.659 −0.198539
\(115\) 0 0
\(116\) 1568.98 1.25583
\(117\) 139.918 0.110559
\(118\) −365.776 −0.285359
\(119\) 666.133 0.513146
\(120\) 0 0
\(121\) −664.959 −0.499594
\(122\) 253.431 0.188070
\(123\) 74.1904 0.0543863
\(124\) 312.764 0.226508
\(125\) 0 0
\(126\) 35.3778 0.0250136
\(127\) −2088.21 −1.45904 −0.729522 0.683957i \(-0.760257\pi\)
−0.729522 + 0.683957i \(0.760257\pi\)
\(128\) −1039.39 −0.717735
\(129\) −847.290 −0.578292
\(130\) 0 0
\(131\) 416.142 0.277546 0.138773 0.990324i \(-0.455684\pi\)
0.138773 + 0.990324i \(0.455684\pi\)
\(132\) 594.972 0.392315
\(133\) −1004.13 −0.654654
\(134\) −467.252 −0.301227
\(135\) 0 0
\(136\) −838.164 −0.528470
\(137\) −619.252 −0.386177 −0.193089 0.981181i \(-0.561850\pi\)
−0.193089 + 0.981181i \(0.561850\pi\)
\(138\) −130.599 −0.0805605
\(139\) −266.994 −0.162922 −0.0814610 0.996677i \(-0.525959\pi\)
−0.0814610 + 0.996677i \(0.525959\pi\)
\(140\) 0 0
\(141\) −772.250 −0.461242
\(142\) −98.2520 −0.0580643
\(143\) −401.218 −0.234626
\(144\) 508.781 0.294434
\(145\) 0 0
\(146\) 26.6270 0.0150936
\(147\) 147.000 0.0824786
\(148\) 735.187 0.408325
\(149\) 2551.80 1.40303 0.701515 0.712655i \(-0.252507\pi\)
0.701515 + 0.712655i \(0.252507\pi\)
\(150\) 0 0
\(151\) 1117.38 0.602195 0.301097 0.953593i \(-0.402647\pi\)
0.301097 + 0.953593i \(0.402647\pi\)
\(152\) 1263.45 0.674204
\(153\) 856.457 0.452552
\(154\) −101.447 −0.0530833
\(155\) 0 0
\(156\) −358.406 −0.183945
\(157\) 456.104 0.231854 0.115927 0.993258i \(-0.463016\pi\)
0.115927 + 0.993258i \(0.463016\pi\)
\(158\) −0.651435 −0.000328009 0
\(159\) −773.935 −0.386019
\(160\) 0 0
\(161\) −542.659 −0.265637
\(162\) 45.4858 0.0220599
\(163\) 105.490 0.0506907 0.0253453 0.999679i \(-0.491931\pi\)
0.0253453 + 0.999679i \(0.491931\pi\)
\(164\) −190.043 −0.0904868
\(165\) 0 0
\(166\) −838.976 −0.392272
\(167\) 2482.54 1.15033 0.575164 0.818038i \(-0.304938\pi\)
0.575164 + 0.818038i \(0.304938\pi\)
\(168\) −184.963 −0.0849417
\(169\) −1955.31 −0.889991
\(170\) 0 0
\(171\) −1291.02 −0.577351
\(172\) 2170.38 0.962150
\(173\) −516.708 −0.227079 −0.113539 0.993534i \(-0.536219\pi\)
−0.113539 + 0.993534i \(0.536219\pi\)
\(174\) −343.957 −0.149858
\(175\) 0 0
\(176\) −1458.94 −0.624842
\(177\) −1954.09 −0.829823
\(178\) 735.146 0.309559
\(179\) 3847.77 1.60668 0.803341 0.595519i \(-0.203054\pi\)
0.803341 + 0.595519i \(0.203054\pi\)
\(180\) 0 0
\(181\) −3026.41 −1.24282 −0.621411 0.783484i \(-0.713440\pi\)
−0.621411 + 0.783484i \(0.713440\pi\)
\(182\) 61.1109 0.0248892
\(183\) 1353.91 0.546907
\(184\) 682.802 0.273570
\(185\) 0 0
\(186\) −68.5653 −0.0270293
\(187\) −2455.92 −0.960398
\(188\) 1978.16 0.767405
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 1271.09 0.481532 0.240766 0.970583i \(-0.422601\pi\)
0.240766 + 0.970583i \(0.422601\pi\)
\(192\) −1184.57 −0.445253
\(193\) 1896.09 0.707170 0.353585 0.935402i \(-0.384963\pi\)
0.353585 + 0.935402i \(0.384963\pi\)
\(194\) 766.707 0.283744
\(195\) 0 0
\(196\) −376.548 −0.137226
\(197\) −4067.89 −1.47119 −0.735597 0.677419i \(-0.763098\pi\)
−0.735597 + 0.677419i \(0.763098\pi\)
\(198\) −130.432 −0.0468151
\(199\) 78.1366 0.0278340 0.0139170 0.999903i \(-0.495570\pi\)
0.0139170 + 0.999903i \(0.495570\pi\)
\(200\) 0 0
\(201\) −2496.21 −0.875967
\(202\) 649.152 0.226110
\(203\) −1429.19 −0.494136
\(204\) −2193.86 −0.752946
\(205\) 0 0
\(206\) 43.9176 0.0148538
\(207\) −697.705 −0.234270
\(208\) 878.857 0.292970
\(209\) 3702.05 1.22524
\(210\) 0 0
\(211\) −1293.02 −0.421872 −0.210936 0.977500i \(-0.567651\pi\)
−0.210936 + 0.977500i \(0.567651\pi\)
\(212\) 1982.47 0.642250
\(213\) −524.895 −0.168851
\(214\) 52.8317 0.0168762
\(215\) 0 0
\(216\) −237.810 −0.0749116
\(217\) −284.899 −0.0891253
\(218\) −1195.12 −0.371303
\(219\) 142.250 0.0438921
\(220\) 0 0
\(221\) 1479.43 0.450303
\(222\) −161.170 −0.0487255
\(223\) 1786.65 0.536515 0.268257 0.963347i \(-0.413552\pi\)
0.268257 + 0.963347i \(0.413552\pi\)
\(224\) 715.452 0.213407
\(225\) 0 0
\(226\) 334.962 0.0985901
\(227\) 4242.02 1.24032 0.620161 0.784475i \(-0.287067\pi\)
0.620161 + 0.784475i \(0.287067\pi\)
\(228\) 3307.02 0.960583
\(229\) 3403.36 0.982096 0.491048 0.871132i \(-0.336614\pi\)
0.491048 + 0.871132i \(0.336614\pi\)
\(230\) 0 0
\(231\) −541.963 −0.154366
\(232\) 1798.29 0.508893
\(233\) 3904.09 1.09771 0.548853 0.835919i \(-0.315065\pi\)
0.548853 + 0.835919i \(0.315065\pi\)
\(234\) 78.5711 0.0219502
\(235\) 0 0
\(236\) 5005.51 1.38064
\(237\) −3.48018 −0.000953848 0
\(238\) 374.069 0.101879
\(239\) 6667.83 1.80463 0.902314 0.431079i \(-0.141867\pi\)
0.902314 + 0.431079i \(0.141867\pi\)
\(240\) 0 0
\(241\) −6003.99 −1.60478 −0.802388 0.596803i \(-0.796437\pi\)
−0.802388 + 0.596803i \(0.796437\pi\)
\(242\) −373.410 −0.0991888
\(243\) 243.000 0.0641500
\(244\) −3468.12 −0.909932
\(245\) 0 0
\(246\) 41.6618 0.0107978
\(247\) −2230.08 −0.574481
\(248\) 358.474 0.0917869
\(249\) −4482.09 −1.14073
\(250\) 0 0
\(251\) 728.467 0.183189 0.0915945 0.995796i \(-0.470804\pi\)
0.0915945 + 0.995796i \(0.470804\pi\)
\(252\) −484.133 −0.121022
\(253\) 2000.69 0.497163
\(254\) −1172.64 −0.289677
\(255\) 0 0
\(256\) 2575.17 0.628703
\(257\) 1479.86 0.359187 0.179593 0.983741i \(-0.442522\pi\)
0.179593 + 0.983741i \(0.442522\pi\)
\(258\) −475.798 −0.114814
\(259\) −669.687 −0.160665
\(260\) 0 0
\(261\) −1837.53 −0.435787
\(262\) 233.686 0.0551036
\(263\) 6367.31 1.49287 0.746435 0.665458i \(-0.231764\pi\)
0.746435 + 0.665458i \(0.231764\pi\)
\(264\) 681.926 0.158976
\(265\) 0 0
\(266\) −563.871 −0.129974
\(267\) 3927.39 0.900197
\(268\) 6394.18 1.45741
\(269\) 1787.18 0.405079 0.202539 0.979274i \(-0.435081\pi\)
0.202539 + 0.979274i \(0.435081\pi\)
\(270\) 0 0
\(271\) 3907.65 0.875915 0.437957 0.898996i \(-0.355702\pi\)
0.437957 + 0.898996i \(0.355702\pi\)
\(272\) 5379.62 1.19922
\(273\) 326.474 0.0723778
\(274\) −347.743 −0.0766712
\(275\) 0 0
\(276\) 1787.21 0.389773
\(277\) −3978.33 −0.862942 −0.431471 0.902127i \(-0.642005\pi\)
−0.431471 + 0.902127i \(0.642005\pi\)
\(278\) −149.931 −0.0323464
\(279\) −366.298 −0.0786011
\(280\) 0 0
\(281\) −6488.04 −1.37738 −0.688690 0.725055i \(-0.741814\pi\)
−0.688690 + 0.725055i \(0.741814\pi\)
\(282\) −433.659 −0.0915746
\(283\) 164.336 0.0345185 0.0172592 0.999851i \(-0.494506\pi\)
0.0172592 + 0.999851i \(0.494506\pi\)
\(284\) 1344.55 0.280930
\(285\) 0 0
\(286\) −225.305 −0.0465824
\(287\) 173.111 0.0356042
\(288\) 919.867 0.188207
\(289\) 4142.79 0.843231
\(290\) 0 0
\(291\) 4096.00 0.825126
\(292\) −364.381 −0.0730266
\(293\) −5004.57 −0.997851 −0.498925 0.866645i \(-0.666272\pi\)
−0.498925 + 0.866645i \(0.666272\pi\)
\(294\) 82.5483 0.0163752
\(295\) 0 0
\(296\) 842.634 0.165463
\(297\) −696.810 −0.136138
\(298\) 1432.97 0.278556
\(299\) −1205.20 −0.233105
\(300\) 0 0
\(301\) −1977.01 −0.378581
\(302\) 627.470 0.119559
\(303\) 3467.98 0.657526
\(304\) −8109.23 −1.52992
\(305\) 0 0
\(306\) 480.946 0.0898492
\(307\) 190.167 0.0353531 0.0176765 0.999844i \(-0.494373\pi\)
0.0176765 + 0.999844i \(0.494373\pi\)
\(308\) 1388.27 0.256831
\(309\) 234.622 0.0431948
\(310\) 0 0
\(311\) −1182.57 −0.215619 −0.107810 0.994172i \(-0.534384\pi\)
−0.107810 + 0.994172i \(0.534384\pi\)
\(312\) −410.787 −0.0745392
\(313\) 4659.05 0.841358 0.420679 0.907209i \(-0.361792\pi\)
0.420679 + 0.907209i \(0.361792\pi\)
\(314\) 256.127 0.0460320
\(315\) 0 0
\(316\) 8.91467 0.00158699
\(317\) 3694.55 0.654595 0.327297 0.944921i \(-0.393862\pi\)
0.327297 + 0.944921i \(0.393862\pi\)
\(318\) −434.605 −0.0766398
\(319\) 5269.18 0.924820
\(320\) 0 0
\(321\) 282.244 0.0490758
\(322\) −304.732 −0.0527392
\(323\) −13650.7 −2.35153
\(324\) −622.457 −0.106731
\(325\) 0 0
\(326\) 59.2379 0.0100641
\(327\) −6384.74 −1.07975
\(328\) −217.817 −0.0366675
\(329\) −1801.92 −0.301954
\(330\) 0 0
\(331\) −8632.79 −1.43354 −0.716769 0.697310i \(-0.754380\pi\)
−0.716769 + 0.697310i \(0.754380\pi\)
\(332\) 11481.1 1.89791
\(333\) −861.026 −0.141693
\(334\) 1394.08 0.228385
\(335\) 0 0
\(336\) 1187.16 0.192752
\(337\) 8136.61 1.31522 0.657610 0.753358i \(-0.271567\pi\)
0.657610 + 0.753358i \(0.271567\pi\)
\(338\) −1098.01 −0.176698
\(339\) 1789.48 0.286700
\(340\) 0 0
\(341\) 1050.37 0.166806
\(342\) −724.977 −0.114627
\(343\) 343.000 0.0539949
\(344\) 2487.58 0.389887
\(345\) 0 0
\(346\) −290.159 −0.0450839
\(347\) −8646.33 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(348\) 4706.94 0.725053
\(349\) −7455.43 −1.14350 −0.571748 0.820429i \(-0.693734\pi\)
−0.571748 + 0.820429i \(0.693734\pi\)
\(350\) 0 0
\(351\) 419.753 0.0638312
\(352\) −2637.74 −0.399410
\(353\) −5450.21 −0.821771 −0.410886 0.911687i \(-0.634780\pi\)
−0.410886 + 0.911687i \(0.634780\pi\)
\(354\) −1097.33 −0.164752
\(355\) 0 0
\(356\) −10060.2 −1.49773
\(357\) 1998.40 0.296265
\(358\) 2160.73 0.318989
\(359\) 4775.79 0.702107 0.351053 0.936355i \(-0.385824\pi\)
0.351053 + 0.936355i \(0.385824\pi\)
\(360\) 0 0
\(361\) 13718.0 2.00000
\(362\) −1699.49 −0.246749
\(363\) −1994.88 −0.288441
\(364\) −836.281 −0.120420
\(365\) 0 0
\(366\) 760.293 0.108582
\(367\) −9636.30 −1.37060 −0.685301 0.728260i \(-0.740329\pi\)
−0.685301 + 0.728260i \(0.740329\pi\)
\(368\) −4382.46 −0.620792
\(369\) 222.571 0.0314000
\(370\) 0 0
\(371\) −1805.85 −0.252709
\(372\) 938.292 0.130775
\(373\) −12180.4 −1.69082 −0.845412 0.534115i \(-0.820645\pi\)
−0.845412 + 0.534115i \(0.820645\pi\)
\(374\) −1379.13 −0.190676
\(375\) 0 0
\(376\) 2267.27 0.310971
\(377\) −3174.11 −0.433621
\(378\) 106.133 0.0144416
\(379\) 1689.39 0.228966 0.114483 0.993425i \(-0.463479\pi\)
0.114483 + 0.993425i \(0.463479\pi\)
\(380\) 0 0
\(381\) −6264.63 −0.842380
\(382\) 713.783 0.0956029
\(383\) −1513.13 −0.201872 −0.100936 0.994893i \(-0.532184\pi\)
−0.100936 + 0.994893i \(0.532184\pi\)
\(384\) −3118.17 −0.414384
\(385\) 0 0
\(386\) 1064.76 0.140401
\(387\) −2541.87 −0.333877
\(388\) −10492.1 −1.37283
\(389\) 12165.5 1.58564 0.792822 0.609453i \(-0.208611\pi\)
0.792822 + 0.609453i \(0.208611\pi\)
\(390\) 0 0
\(391\) −7377.21 −0.954173
\(392\) −431.580 −0.0556074
\(393\) 1248.43 0.160241
\(394\) −2284.34 −0.292089
\(395\) 0 0
\(396\) 1784.91 0.226503
\(397\) −7353.40 −0.929613 −0.464807 0.885412i \(-0.653876\pi\)
−0.464807 + 0.885412i \(0.653876\pi\)
\(398\) 43.8778 0.00552612
\(399\) −3012.39 −0.377965
\(400\) 0 0
\(401\) 7481.62 0.931707 0.465853 0.884862i \(-0.345747\pi\)
0.465853 + 0.884862i \(0.345747\pi\)
\(402\) −1401.76 −0.173913
\(403\) −632.735 −0.0782104
\(404\) −8883.42 −1.09398
\(405\) 0 0
\(406\) −802.567 −0.0981053
\(407\) 2469.02 0.300699
\(408\) −2514.49 −0.305112
\(409\) 9248.94 1.11817 0.559084 0.829111i \(-0.311153\pi\)
0.559084 + 0.829111i \(0.311153\pi\)
\(410\) 0 0
\(411\) −1857.76 −0.222959
\(412\) −600.997 −0.0718665
\(413\) −4559.55 −0.543247
\(414\) −391.798 −0.0465116
\(415\) 0 0
\(416\) 1588.96 0.187272
\(417\) −800.983 −0.0940631
\(418\) 2078.89 0.243258
\(419\) −3363.39 −0.392154 −0.196077 0.980589i \(-0.562820\pi\)
−0.196077 + 0.980589i \(0.562820\pi\)
\(420\) 0 0
\(421\) −3638.86 −0.421252 −0.210626 0.977567i \(-0.567550\pi\)
−0.210626 + 0.977567i \(0.567550\pi\)
\(422\) −726.097 −0.0837579
\(423\) −2316.75 −0.266298
\(424\) 2272.21 0.260255
\(425\) 0 0
\(426\) −294.756 −0.0335234
\(427\) 3159.13 0.358035
\(428\) −722.983 −0.0816512
\(429\) −1203.65 −0.135461
\(430\) 0 0
\(431\) 11243.2 1.25653 0.628264 0.778000i \(-0.283766\pi\)
0.628264 + 0.778000i \(0.283766\pi\)
\(432\) 1526.34 0.169991
\(433\) −187.332 −0.0207912 −0.0103956 0.999946i \(-0.503309\pi\)
−0.0103956 + 0.999946i \(0.503309\pi\)
\(434\) −159.986 −0.0176948
\(435\) 0 0
\(436\) 16354.9 1.79646
\(437\) 11120.4 1.21730
\(438\) 79.8809 0.00871428
\(439\) −11479.4 −1.24802 −0.624011 0.781415i \(-0.714498\pi\)
−0.624011 + 0.781415i \(0.714498\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 830.775 0.0894026
\(443\) 9490.56 1.01786 0.508928 0.860809i \(-0.330042\pi\)
0.508928 + 0.860809i \(0.330042\pi\)
\(444\) 2205.56 0.235746
\(445\) 0 0
\(446\) 1003.30 0.106519
\(447\) 7655.39 0.810039
\(448\) −2763.99 −0.291487
\(449\) −5546.58 −0.582982 −0.291491 0.956574i \(-0.594151\pi\)
−0.291491 + 0.956574i \(0.594151\pi\)
\(450\) 0 0
\(451\) −638.229 −0.0666364
\(452\) −4583.85 −0.477004
\(453\) 3352.15 0.347677
\(454\) 2382.12 0.246252
\(455\) 0 0
\(456\) 3790.34 0.389252
\(457\) −1776.94 −0.181885 −0.0909426 0.995856i \(-0.528988\pi\)
−0.0909426 + 0.995856i \(0.528988\pi\)
\(458\) 1911.16 0.194984
\(459\) 2569.37 0.261281
\(460\) 0 0
\(461\) −9059.81 −0.915309 −0.457655 0.889130i \(-0.651310\pi\)
−0.457655 + 0.889130i \(0.651310\pi\)
\(462\) −304.341 −0.0306477
\(463\) −14878.5 −1.49344 −0.746719 0.665139i \(-0.768372\pi\)
−0.746719 + 0.665139i \(0.768372\pi\)
\(464\) −11542.0 −1.15479
\(465\) 0 0
\(466\) 2192.35 0.217937
\(467\) −14275.8 −1.41458 −0.707288 0.706926i \(-0.750082\pi\)
−0.707288 + 0.706926i \(0.750082\pi\)
\(468\) −1075.22 −0.106201
\(469\) −5824.50 −0.573455
\(470\) 0 0
\(471\) 1368.31 0.133861
\(472\) 5737.07 0.559470
\(473\) 7288.89 0.708548
\(474\) −1.95431 −0.000189376 0
\(475\) 0 0
\(476\) −5119.01 −0.492919
\(477\) −2321.80 −0.222868
\(478\) 3744.34 0.358289
\(479\) −15377.8 −1.46687 −0.733433 0.679761i \(-0.762083\pi\)
−0.733433 + 0.679761i \(0.762083\pi\)
\(480\) 0 0
\(481\) −1487.32 −0.140989
\(482\) −3371.56 −0.318610
\(483\) −1627.98 −0.153365
\(484\) 5109.99 0.479901
\(485\) 0 0
\(486\) 136.457 0.0127363
\(487\) −19683.3 −1.83149 −0.915747 0.401756i \(-0.868400\pi\)
−0.915747 + 0.401756i \(0.868400\pi\)
\(488\) −3974.98 −0.368727
\(489\) 316.469 0.0292663
\(490\) 0 0
\(491\) 19585.1 1.80013 0.900065 0.435755i \(-0.143519\pi\)
0.900065 + 0.435755i \(0.143519\pi\)
\(492\) −570.128 −0.0522426
\(493\) −19429.3 −1.77495
\(494\) −1252.31 −0.114057
\(495\) 0 0
\(496\) −2300.81 −0.208285
\(497\) −1224.75 −0.110539
\(498\) −2516.93 −0.226478
\(499\) −6888.47 −0.617977 −0.308988 0.951066i \(-0.599990\pi\)
−0.308988 + 0.951066i \(0.599990\pi\)
\(500\) 0 0
\(501\) 7447.62 0.664142
\(502\) 409.073 0.0363701
\(503\) −14878.0 −1.31885 −0.659423 0.751772i \(-0.729199\pi\)
−0.659423 + 0.751772i \(0.729199\pi\)
\(504\) −554.889 −0.0490411
\(505\) 0 0
\(506\) 1123.49 0.0987062
\(507\) −5865.93 −0.513836
\(508\) 16047.2 1.40153
\(509\) 1935.28 0.168526 0.0842629 0.996444i \(-0.473146\pi\)
0.0842629 + 0.996444i \(0.473146\pi\)
\(510\) 0 0
\(511\) 331.917 0.0287341
\(512\) 9761.22 0.842557
\(513\) −3873.07 −0.333334
\(514\) 831.019 0.0713126
\(515\) 0 0
\(516\) 6511.13 0.555497
\(517\) 6643.35 0.565134
\(518\) −376.064 −0.0318983
\(519\) −1550.12 −0.131104
\(520\) 0 0
\(521\) −6892.28 −0.579570 −0.289785 0.957092i \(-0.593584\pi\)
−0.289785 + 0.957092i \(0.593584\pi\)
\(522\) −1031.87 −0.0865207
\(523\) 1074.60 0.0898447 0.0449223 0.998990i \(-0.485696\pi\)
0.0449223 + 0.998990i \(0.485696\pi\)
\(524\) −3197.91 −0.266606
\(525\) 0 0
\(526\) 3575.58 0.296393
\(527\) −3873.07 −0.320140
\(528\) −4376.83 −0.360752
\(529\) −6157.23 −0.506060
\(530\) 0 0
\(531\) −5862.28 −0.479099
\(532\) 7716.39 0.628849
\(533\) 384.464 0.0312439
\(534\) 2205.44 0.178724
\(535\) 0 0
\(536\) 7328.69 0.590580
\(537\) 11543.3 0.927619
\(538\) 1003.60 0.0804239
\(539\) −1264.58 −0.101056
\(540\) 0 0
\(541\) 8660.23 0.688230 0.344115 0.938928i \(-0.388179\pi\)
0.344115 + 0.938928i \(0.388179\pi\)
\(542\) 2194.35 0.173903
\(543\) −9079.22 −0.717544
\(544\) 9726.25 0.766562
\(545\) 0 0
\(546\) 183.333 0.0143698
\(547\) 15346.1 1.19954 0.599771 0.800171i \(-0.295258\pi\)
0.599771 + 0.800171i \(0.295258\pi\)
\(548\) 4758.74 0.370955
\(549\) 4061.74 0.315757
\(550\) 0 0
\(551\) 29287.6 2.26442
\(552\) 2048.41 0.157946
\(553\) −8.12042 −0.000624440 0
\(554\) −2234.04 −0.171328
\(555\) 0 0
\(556\) 2051.76 0.156500
\(557\) −15544.9 −1.18251 −0.591257 0.806483i \(-0.701368\pi\)
−0.591257 + 0.806483i \(0.701368\pi\)
\(558\) −205.696 −0.0156054
\(559\) −4390.77 −0.332218
\(560\) 0 0
\(561\) −7367.75 −0.554486
\(562\) −3643.38 −0.273464
\(563\) 18511.9 1.38576 0.692881 0.721052i \(-0.256341\pi\)
0.692881 + 0.721052i \(0.256341\pi\)
\(564\) 5934.48 0.443061
\(565\) 0 0
\(566\) 92.2831 0.00685326
\(567\) 567.000 0.0419961
\(568\) 1541.05 0.113840
\(569\) 2157.36 0.158948 0.0794738 0.996837i \(-0.474676\pi\)
0.0794738 + 0.996837i \(0.474676\pi\)
\(570\) 0 0
\(571\) 16010.3 1.17340 0.586700 0.809805i \(-0.300427\pi\)
0.586700 + 0.809805i \(0.300427\pi\)
\(572\) 3083.22 0.225378
\(573\) 3813.26 0.278013
\(574\) 97.2109 0.00706882
\(575\) 0 0
\(576\) −3553.70 −0.257067
\(577\) 2164.01 0.156134 0.0780668 0.996948i \(-0.475125\pi\)
0.0780668 + 0.996948i \(0.475125\pi\)
\(578\) 2326.40 0.167414
\(579\) 5688.28 0.408285
\(580\) 0 0
\(581\) −10458.2 −0.746780
\(582\) 2300.12 0.163820
\(583\) 6657.84 0.472967
\(584\) −417.635 −0.0295922
\(585\) 0 0
\(586\) −2810.33 −0.198112
\(587\) −27616.6 −1.94184 −0.970918 0.239412i \(-0.923045\pi\)
−0.970918 + 0.239412i \(0.923045\pi\)
\(588\) −1129.64 −0.0792275
\(589\) 5838.26 0.408424
\(590\) 0 0
\(591\) −12203.7 −0.849395
\(592\) −5408.32 −0.375474
\(593\) 10205.4 0.706718 0.353359 0.935488i \(-0.385039\pi\)
0.353359 + 0.935488i \(0.385039\pi\)
\(594\) −391.295 −0.0270287
\(595\) 0 0
\(596\) −19609.7 −1.34773
\(597\) 234.410 0.0160699
\(598\) −676.783 −0.0462805
\(599\) 11090.5 0.756501 0.378251 0.925703i \(-0.376526\pi\)
0.378251 + 0.925703i \(0.376526\pi\)
\(600\) 0 0
\(601\) −26537.0 −1.80111 −0.900555 0.434742i \(-0.856839\pi\)
−0.900555 + 0.434742i \(0.856839\pi\)
\(602\) −1110.20 −0.0751631
\(603\) −7488.64 −0.505740
\(604\) −8586.71 −0.578457
\(605\) 0 0
\(606\) 1947.46 0.130544
\(607\) 11234.5 0.751226 0.375613 0.926777i \(-0.377432\pi\)
0.375613 + 0.926777i \(0.377432\pi\)
\(608\) −14661.3 −0.977954
\(609\) −4287.58 −0.285290
\(610\) 0 0
\(611\) −4001.90 −0.264975
\(612\) −6581.58 −0.434714
\(613\) 10622.5 0.699900 0.349950 0.936768i \(-0.386199\pi\)
0.349950 + 0.936768i \(0.386199\pi\)
\(614\) 106.789 0.00701896
\(615\) 0 0
\(616\) 1591.16 0.104074
\(617\) −20433.3 −1.33325 −0.666623 0.745395i \(-0.732261\pi\)
−0.666623 + 0.745395i \(0.732261\pi\)
\(618\) 131.753 0.00857585
\(619\) −7963.57 −0.517097 −0.258549 0.965998i \(-0.583244\pi\)
−0.258549 + 0.965998i \(0.583244\pi\)
\(620\) 0 0
\(621\) −2093.11 −0.135256
\(622\) −664.077 −0.0428088
\(623\) 9163.91 0.589317
\(624\) 2636.57 0.169146
\(625\) 0 0
\(626\) 2616.30 0.167042
\(627\) 11106.1 0.707394
\(628\) −3505.01 −0.222715
\(629\) −9104.09 −0.577113
\(630\) 0 0
\(631\) −14703.1 −0.927608 −0.463804 0.885938i \(-0.653516\pi\)
−0.463804 + 0.885938i \(0.653516\pi\)
\(632\) 10.2175 0.000643088 0
\(633\) −3879.05 −0.243568
\(634\) 2074.68 0.129963
\(635\) 0 0
\(636\) 5947.42 0.370803
\(637\) 761.773 0.0473824
\(638\) 2958.92 0.183613
\(639\) −1574.68 −0.0974860
\(640\) 0 0
\(641\) 3353.41 0.206633 0.103317 0.994649i \(-0.467055\pi\)
0.103317 + 0.994649i \(0.467055\pi\)
\(642\) 158.495 0.00974345
\(643\) −31862.0 −1.95415 −0.977073 0.212906i \(-0.931707\pi\)
−0.977073 + 0.212906i \(0.931707\pi\)
\(644\) 4170.15 0.255166
\(645\) 0 0
\(646\) −7665.58 −0.466870
\(647\) 8518.46 0.517612 0.258806 0.965929i \(-0.416671\pi\)
0.258806 + 0.965929i \(0.416671\pi\)
\(648\) −713.429 −0.0432502
\(649\) 16810.3 1.01673
\(650\) 0 0
\(651\) −854.696 −0.0514565
\(652\) −810.651 −0.0486925
\(653\) 28509.3 1.70851 0.854254 0.519856i \(-0.174014\pi\)
0.854254 + 0.519856i \(0.174014\pi\)
\(654\) −3585.37 −0.214372
\(655\) 0 0
\(656\) 1398.02 0.0832068
\(657\) 426.750 0.0253411
\(658\) −1011.87 −0.0599496
\(659\) −27632.4 −1.63339 −0.816697 0.577066i \(-0.804197\pi\)
−0.816697 + 0.577066i \(0.804197\pi\)
\(660\) 0 0
\(661\) −27052.8 −1.59188 −0.795941 0.605374i \(-0.793023\pi\)
−0.795941 + 0.605374i \(0.793023\pi\)
\(662\) −4847.77 −0.284613
\(663\) 4438.28 0.259982
\(664\) 13159.1 0.769082
\(665\) 0 0
\(666\) −483.511 −0.0281317
\(667\) 15827.9 0.918826
\(668\) −19077.5 −1.10498
\(669\) 5359.94 0.309757
\(670\) 0 0
\(671\) −11647.1 −0.670094
\(672\) 2146.36 0.123210
\(673\) 1569.99 0.0899235 0.0449618 0.998989i \(-0.485683\pi\)
0.0449618 + 0.998989i \(0.485683\pi\)
\(674\) 4569.13 0.261122
\(675\) 0 0
\(676\) 15025.9 0.854909
\(677\) −13853.9 −0.786482 −0.393241 0.919435i \(-0.628646\pi\)
−0.393241 + 0.919435i \(0.628646\pi\)
\(678\) 1004.89 0.0569210
\(679\) 9557.33 0.540172
\(680\) 0 0
\(681\) 12726.1 0.716100
\(682\) 589.839 0.0331174
\(683\) 28337.1 1.58754 0.793769 0.608219i \(-0.208116\pi\)
0.793769 + 0.608219i \(0.208116\pi\)
\(684\) 9921.07 0.554593
\(685\) 0 0
\(686\) 192.613 0.0107201
\(687\) 10210.1 0.567014
\(688\) −15966.1 −0.884742
\(689\) −4010.63 −0.221760
\(690\) 0 0
\(691\) 16936.5 0.932410 0.466205 0.884677i \(-0.345621\pi\)
0.466205 + 0.884677i \(0.345621\pi\)
\(692\) 3970.73 0.218128
\(693\) −1625.89 −0.0891233
\(694\) −4855.37 −0.265572
\(695\) 0 0
\(696\) 5394.86 0.293810
\(697\) 2353.37 0.127891
\(698\) −4186.62 −0.227028
\(699\) 11712.3 0.633760
\(700\) 0 0
\(701\) −29744.0 −1.60259 −0.801294 0.598271i \(-0.795855\pi\)
−0.801294 + 0.598271i \(0.795855\pi\)
\(702\) 235.713 0.0126730
\(703\) 13723.5 0.736261
\(704\) 10190.3 0.545543
\(705\) 0 0
\(706\) −3060.58 −0.163154
\(707\) 8091.96 0.430452
\(708\) 15016.5 0.797113
\(709\) 34264.7 1.81500 0.907502 0.420047i \(-0.137986\pi\)
0.907502 + 0.420047i \(0.137986\pi\)
\(710\) 0 0
\(711\) −10.4405 −0.000550705 0
\(712\) −11530.5 −0.606916
\(713\) 3155.16 0.165725
\(714\) 1122.21 0.0588201
\(715\) 0 0
\(716\) −29568.8 −1.54335
\(717\) 20003.5 1.04190
\(718\) 2681.86 0.139396
\(719\) −25469.2 −1.32106 −0.660529 0.750801i \(-0.729668\pi\)
−0.660529 + 0.750801i \(0.729668\pi\)
\(720\) 0 0
\(721\) 547.452 0.0282776
\(722\) 7703.40 0.397079
\(723\) −18012.0 −0.926518
\(724\) 23256.9 1.19383
\(725\) 0 0
\(726\) −1120.23 −0.0572667
\(727\) 24294.6 1.23939 0.619695 0.784843i \(-0.287256\pi\)
0.619695 + 0.784843i \(0.287256\pi\)
\(728\) −958.503 −0.0487974
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −26876.6 −1.35987
\(732\) −10404.3 −0.525349
\(733\) 34870.8 1.75714 0.878569 0.477615i \(-0.158499\pi\)
0.878569 + 0.477615i \(0.158499\pi\)
\(734\) −5411.29 −0.272118
\(735\) 0 0
\(736\) −7923.40 −0.396821
\(737\) 21473.9 1.07327
\(738\) 124.985 0.00623412
\(739\) 7028.81 0.349877 0.174938 0.984579i \(-0.444027\pi\)
0.174938 + 0.984579i \(0.444027\pi\)
\(740\) 0 0
\(741\) −6690.25 −0.331677
\(742\) −1014.08 −0.0501725
\(743\) 1368.74 0.0675831 0.0337915 0.999429i \(-0.489242\pi\)
0.0337915 + 0.999429i \(0.489242\pi\)
\(744\) 1075.42 0.0529932
\(745\) 0 0
\(746\) −6839.94 −0.335694
\(747\) −13446.3 −0.658598
\(748\) 18872.9 0.922541
\(749\) 658.570 0.0321276
\(750\) 0 0
\(751\) 37144.0 1.80480 0.902400 0.430900i \(-0.141804\pi\)
0.902400 + 0.430900i \(0.141804\pi\)
\(752\) −14552.1 −0.705665
\(753\) 2185.40 0.105764
\(754\) −1782.43 −0.0860907
\(755\) 0 0
\(756\) −1452.40 −0.0698721
\(757\) 12042.4 0.578189 0.289095 0.957301i \(-0.406646\pi\)
0.289095 + 0.957301i \(0.406646\pi\)
\(758\) 948.683 0.0454588
\(759\) 6002.07 0.287037
\(760\) 0 0
\(761\) 6112.92 0.291187 0.145593 0.989345i \(-0.453491\pi\)
0.145593 + 0.989345i \(0.453491\pi\)
\(762\) −3517.92 −0.167245
\(763\) −14897.7 −0.706860
\(764\) −9767.88 −0.462552
\(765\) 0 0
\(766\) −849.700 −0.0400795
\(767\) −10126.4 −0.476717
\(768\) 7725.50 0.362982
\(769\) −957.146 −0.0448837 −0.0224418 0.999748i \(-0.507144\pi\)
−0.0224418 + 0.999748i \(0.507144\pi\)
\(770\) 0 0
\(771\) 4439.58 0.207377
\(772\) −14570.8 −0.679295
\(773\) 32867.9 1.52934 0.764668 0.644424i \(-0.222903\pi\)
0.764668 + 0.644424i \(0.222903\pi\)
\(774\) −1427.39 −0.0662876
\(775\) 0 0
\(776\) −12025.5 −0.556303
\(777\) −2009.06 −0.0927601
\(778\) 6831.58 0.314812
\(779\) −3547.46 −0.163159
\(780\) 0 0
\(781\) 4515.45 0.206883
\(782\) −4142.69 −0.189440
\(783\) −5512.60 −0.251602
\(784\) 2770.03 0.126186
\(785\) 0 0
\(786\) 701.057 0.0318141
\(787\) 15320.0 0.693900 0.346950 0.937884i \(-0.387217\pi\)
0.346950 + 0.937884i \(0.387217\pi\)
\(788\) 31260.4 1.41320
\(789\) 19101.9 0.861909
\(790\) 0 0
\(791\) 4175.45 0.187689
\(792\) 2045.78 0.0917848
\(793\) 7016.15 0.314188
\(794\) −4129.32 −0.184564
\(795\) 0 0
\(796\) −600.453 −0.0267368
\(797\) 43629.0 1.93904 0.969522 0.245004i \(-0.0787893\pi\)
0.969522 + 0.245004i \(0.0787893\pi\)
\(798\) −1691.61 −0.0750407
\(799\) −24496.3 −1.08463
\(800\) 0 0
\(801\) 11782.2 0.519729
\(802\) 4201.33 0.184980
\(803\) −1223.72 −0.0537784
\(804\) 19182.5 0.841438
\(805\) 0 0
\(806\) −355.314 −0.0155278
\(807\) 5361.54 0.233872
\(808\) −10181.7 −0.443307
\(809\) 127.735 0.00555119 0.00277560 0.999996i \(-0.499116\pi\)
0.00277560 + 0.999996i \(0.499116\pi\)
\(810\) 0 0
\(811\) 16227.5 0.702618 0.351309 0.936260i \(-0.385737\pi\)
0.351309 + 0.936260i \(0.385737\pi\)
\(812\) 10982.9 0.474659
\(813\) 11723.0 0.505710
\(814\) 1386.48 0.0597005
\(815\) 0 0
\(816\) 16138.9 0.692369
\(817\) 40513.7 1.73488
\(818\) 5193.77 0.222000
\(819\) 979.423 0.0417873
\(820\) 0 0
\(821\) 34249.2 1.45591 0.727957 0.685623i \(-0.240470\pi\)
0.727957 + 0.685623i \(0.240470\pi\)
\(822\) −1043.23 −0.0442661
\(823\) −6624.51 −0.280578 −0.140289 0.990111i \(-0.544803\pi\)
−0.140289 + 0.990111i \(0.544803\pi\)
\(824\) −688.832 −0.0291221
\(825\) 0 0
\(826\) −2560.43 −0.107856
\(827\) −33786.8 −1.42065 −0.710327 0.703872i \(-0.751453\pi\)
−0.710327 + 0.703872i \(0.751453\pi\)
\(828\) 5361.62 0.225035
\(829\) −30283.8 −1.26876 −0.634378 0.773023i \(-0.718744\pi\)
−0.634378 + 0.773023i \(0.718744\pi\)
\(830\) 0 0
\(831\) −11935.0 −0.498220
\(832\) −6138.57 −0.255789
\(833\) 4662.93 0.193951
\(834\) −449.794 −0.0186752
\(835\) 0 0
\(836\) −28449.0 −1.17695
\(837\) −1098.89 −0.0453804
\(838\) −1888.72 −0.0778578
\(839\) 16810.9 0.691750 0.345875 0.938281i \(-0.387582\pi\)
0.345875 + 0.938281i \(0.387582\pi\)
\(840\) 0 0
\(841\) 17296.6 0.709195
\(842\) −2043.41 −0.0836350
\(843\) −19464.1 −0.795231
\(844\) 9936.39 0.405242
\(845\) 0 0
\(846\) −1300.98 −0.0528706
\(847\) −4654.72 −0.188829
\(848\) −14583.8 −0.590579
\(849\) 493.007 0.0199293
\(850\) 0 0
\(851\) 7416.56 0.298750
\(852\) 4033.64 0.162195
\(853\) 14875.8 0.597112 0.298556 0.954392i \(-0.403495\pi\)
0.298556 + 0.954392i \(0.403495\pi\)
\(854\) 1774.02 0.0710839
\(855\) 0 0
\(856\) −828.647 −0.0330871
\(857\) −3987.55 −0.158941 −0.0794703 0.996837i \(-0.525323\pi\)
−0.0794703 + 0.996837i \(0.525323\pi\)
\(858\) −675.915 −0.0268944
\(859\) 39344.7 1.56278 0.781388 0.624046i \(-0.214512\pi\)
0.781388 + 0.624046i \(0.214512\pi\)
\(860\) 0 0
\(861\) 519.333 0.0205561
\(862\) 6313.63 0.249470
\(863\) −15627.0 −0.616397 −0.308198 0.951322i \(-0.599726\pi\)
−0.308198 + 0.951322i \(0.599726\pi\)
\(864\) 2759.60 0.108661
\(865\) 0 0
\(866\) −105.197 −0.00412787
\(867\) 12428.4 0.486839
\(868\) 2189.35 0.0856122
\(869\) 29.9386 0.00116870
\(870\) 0 0
\(871\) −12935.7 −0.503226
\(872\) 18745.1 0.727969
\(873\) 12288.0 0.476387
\(874\) 6244.69 0.241682
\(875\) 0 0
\(876\) −1093.14 −0.0421619
\(877\) −14519.0 −0.559034 −0.279517 0.960141i \(-0.590174\pi\)
−0.279517 + 0.960141i \(0.590174\pi\)
\(878\) −6446.29 −0.247781
\(879\) −15013.7 −0.576109
\(880\) 0 0
\(881\) 24177.7 0.924592 0.462296 0.886726i \(-0.347026\pi\)
0.462296 + 0.886726i \(0.347026\pi\)
\(882\) 247.645 0.00945423
\(883\) 10340.3 0.394089 0.197044 0.980395i \(-0.436866\pi\)
0.197044 + 0.980395i \(0.436866\pi\)
\(884\) −11368.9 −0.432553
\(885\) 0 0
\(886\) 5329.45 0.202084
\(887\) −3222.62 −0.121990 −0.0609949 0.998138i \(-0.519427\pi\)
−0.0609949 + 0.998138i \(0.519427\pi\)
\(888\) 2527.90 0.0955303
\(889\) −14617.5 −0.551467
\(890\) 0 0
\(891\) −2090.43 −0.0785993
\(892\) −13729.8 −0.515367
\(893\) 36925.6 1.38373
\(894\) 4298.91 0.160824
\(895\) 0 0
\(896\) −7275.74 −0.271278
\(897\) −3615.60 −0.134583
\(898\) −3114.70 −0.115745
\(899\) 8309.70 0.308280
\(900\) 0 0
\(901\) −24549.7 −0.907735
\(902\) −358.399 −0.0132299
\(903\) −5931.03 −0.218574
\(904\) −5253.77 −0.193294
\(905\) 0 0
\(906\) 1882.41 0.0690274
\(907\) −31692.8 −1.16025 −0.580123 0.814529i \(-0.696995\pi\)
−0.580123 + 0.814529i \(0.696995\pi\)
\(908\) −32598.5 −1.19143
\(909\) 10403.9 0.379623
\(910\) 0 0
\(911\) 2403.15 0.0873985 0.0436993 0.999045i \(-0.486086\pi\)
0.0436993 + 0.999045i \(0.486086\pi\)
\(912\) −24327.7 −0.883301
\(913\) 38557.6 1.39767
\(914\) −997.843 −0.0361113
\(915\) 0 0
\(916\) −26153.6 −0.943385
\(917\) 2912.99 0.104902
\(918\) 1442.84 0.0518745
\(919\) 19622.5 0.704339 0.352170 0.935936i \(-0.385444\pi\)
0.352170 + 0.935936i \(0.385444\pi\)
\(920\) 0 0
\(921\) 570.501 0.0204111
\(922\) −5087.56 −0.181724
\(923\) −2720.07 −0.0970014
\(924\) 4164.80 0.148281
\(925\) 0 0
\(926\) −8355.06 −0.296506
\(927\) 703.867 0.0249385
\(928\) −20867.7 −0.738165
\(929\) −12930.5 −0.456660 −0.228330 0.973584i \(-0.573327\pi\)
−0.228330 + 0.973584i \(0.573327\pi\)
\(930\) 0 0
\(931\) −7028.90 −0.247436
\(932\) −30001.6 −1.05444
\(933\) −3547.72 −0.124488
\(934\) −8016.64 −0.280848
\(935\) 0 0
\(936\) −1232.36 −0.0430352
\(937\) −18717.1 −0.652573 −0.326287 0.945271i \(-0.605797\pi\)
−0.326287 + 0.945271i \(0.605797\pi\)
\(938\) −3270.76 −0.113853
\(939\) 13977.2 0.485758
\(940\) 0 0
\(941\) −10152.9 −0.351727 −0.175863 0.984415i \(-0.556272\pi\)
−0.175863 + 0.984415i \(0.556272\pi\)
\(942\) 768.380 0.0265766
\(943\) −1917.15 −0.0662045
\(944\) −36822.4 −1.26956
\(945\) 0 0
\(946\) 4093.09 0.140674
\(947\) −15836.2 −0.543408 −0.271704 0.962381i \(-0.587587\pi\)
−0.271704 + 0.962381i \(0.587587\pi\)
\(948\) 26.7440 0.000916250 0
\(949\) 737.158 0.0252151
\(950\) 0 0
\(951\) 11083.6 0.377930
\(952\) −5867.15 −0.199743
\(953\) −4847.86 −0.164782 −0.0823911 0.996600i \(-0.526256\pi\)
−0.0823911 + 0.996600i \(0.526256\pi\)
\(954\) −1303.82 −0.0442480
\(955\) 0 0
\(956\) −51240.0 −1.73349
\(957\) 15807.5 0.533945
\(958\) −8635.44 −0.291230
\(959\) −4334.76 −0.145961
\(960\) 0 0
\(961\) −28134.5 −0.944397
\(962\) −835.207 −0.0279918
\(963\) 846.732 0.0283339
\(964\) 46138.6 1.54152
\(965\) 0 0
\(966\) −914.195 −0.0304490
\(967\) −17153.8 −0.570454 −0.285227 0.958460i \(-0.592069\pi\)
−0.285227 + 0.958460i \(0.592069\pi\)
\(968\) 5856.80 0.194468
\(969\) −40952.1 −1.35766
\(970\) 0 0
\(971\) 50352.2 1.66414 0.832070 0.554670i \(-0.187156\pi\)
0.832070 + 0.554670i \(0.187156\pi\)
\(972\) −1867.37 −0.0616214
\(973\) −1868.96 −0.0615788
\(974\) −11053.2 −0.363623
\(975\) 0 0
\(976\) 25512.8 0.836725
\(977\) −1510.03 −0.0494474 −0.0247237 0.999694i \(-0.507871\pi\)
−0.0247237 + 0.999694i \(0.507871\pi\)
\(978\) 177.714 0.00581049
\(979\) −33785.7 −1.10296
\(980\) 0 0
\(981\) −19154.2 −0.623392
\(982\) 10998.1 0.357396
\(983\) 5310.89 0.172321 0.0861603 0.996281i \(-0.472540\pi\)
0.0861603 + 0.996281i \(0.472540\pi\)
\(984\) −653.451 −0.0211700
\(985\) 0 0
\(986\) −10910.6 −0.352396
\(987\) −5405.75 −0.174333
\(988\) 17137.4 0.551836
\(989\) 21894.7 0.703956
\(990\) 0 0
\(991\) 35845.4 1.14901 0.574505 0.818501i \(-0.305195\pi\)
0.574505 + 0.818501i \(0.305195\pi\)
\(992\) −4159.82 −0.133140
\(993\) −25898.4 −0.827654
\(994\) −687.764 −0.0219462
\(995\) 0 0
\(996\) 34443.3 1.09576
\(997\) 17857.5 0.567253 0.283627 0.958935i \(-0.408462\pi\)
0.283627 + 0.958935i \(0.408462\pi\)
\(998\) −3868.24 −0.122692
\(999\) −2583.08 −0.0818067
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 525.4.a.j.1.2 2
3.2 odd 2 1575.4.a.x.1.1 2
5.2 odd 4 525.4.d.m.274.3 4
5.3 odd 4 525.4.d.m.274.2 4
5.4 even 2 525.4.a.m.1.1 yes 2
15.14 odd 2 1575.4.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.2 2 1.1 even 1 trivial
525.4.a.m.1.1 yes 2 5.4 even 2
525.4.d.m.274.2 4 5.3 odd 4
525.4.d.m.274.3 4 5.2 odd 4
1575.4.a.o.1.2 2 15.14 odd 2
1575.4.a.x.1.1 2 3.2 odd 2