Properties

Label 3525.2.a.bd.1.4
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
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 1.4
Root \(0.936719\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.936719 q^{2} +1.00000 q^{3} -1.12256 q^{4} -0.936719 q^{6} +3.65526 q^{7} +2.92496 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.936719 q^{2} +1.00000 q^{3} -1.12256 q^{4} -0.936719 q^{6} +3.65526 q^{7} +2.92496 q^{8} +1.00000 q^{9} -5.57376 q^{11} -1.12256 q^{12} +2.92869 q^{13} -3.42395 q^{14} -0.494746 q^{16} +2.26461 q^{17} -0.936719 q^{18} -2.64416 q^{19} +3.65526 q^{21} +5.22104 q^{22} -7.72850 q^{23} +2.92496 q^{24} -2.74336 q^{26} +1.00000 q^{27} -4.10324 q^{28} -7.79408 q^{29} -6.03466 q^{31} -5.38648 q^{32} -5.57376 q^{33} -2.12130 q^{34} -1.12256 q^{36} +6.60338 q^{37} +2.47683 q^{38} +2.92869 q^{39} -8.81927 q^{41} -3.42395 q^{42} -9.69632 q^{43} +6.25687 q^{44} +7.23943 q^{46} -1.00000 q^{47} -0.494746 q^{48} +6.36091 q^{49} +2.26461 q^{51} -3.28763 q^{52} +2.97538 q^{53} -0.936719 q^{54} +10.6915 q^{56} -2.64416 q^{57} +7.30086 q^{58} +0.571458 q^{59} +8.30187 q^{61} +5.65278 q^{62} +3.65526 q^{63} +6.03511 q^{64} +5.22104 q^{66} +8.03594 q^{67} -2.54216 q^{68} -7.72850 q^{69} +11.7268 q^{71} +2.92496 q^{72} -5.94373 q^{73} -6.18551 q^{74} +2.96822 q^{76} -20.3735 q^{77} -2.74336 q^{78} -16.6200 q^{79} +1.00000 q^{81} +8.26117 q^{82} -13.8354 q^{83} -4.10324 q^{84} +9.08272 q^{86} -7.79408 q^{87} -16.3030 q^{88} +4.31121 q^{89} +10.7051 q^{91} +8.67569 q^{92} -6.03466 q^{93} +0.936719 q^{94} -5.38648 q^{96} -5.24644 q^{97} -5.95838 q^{98} -5.57376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936719 −0.662360 −0.331180 0.943568i \(-0.607447\pi\)
−0.331180 + 0.943568i \(0.607447\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.12256 −0.561279
\(5\) 0 0
\(6\) −0.936719 −0.382414
\(7\) 3.65526 1.38156 0.690779 0.723066i \(-0.257268\pi\)
0.690779 + 0.723066i \(0.257268\pi\)
\(8\) 2.92496 1.03413
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.57376 −1.68055 −0.840275 0.542160i \(-0.817607\pi\)
−0.840275 + 0.542160i \(0.817607\pi\)
\(12\) −1.12256 −0.324055
\(13\) 2.92869 0.812274 0.406137 0.913812i \(-0.366876\pi\)
0.406137 + 0.913812i \(0.366876\pi\)
\(14\) −3.42395 −0.915089
\(15\) 0 0
\(16\) −0.494746 −0.123686
\(17\) 2.26461 0.549249 0.274624 0.961552i \(-0.411446\pi\)
0.274624 + 0.961552i \(0.411446\pi\)
\(18\) −0.936719 −0.220787
\(19\) −2.64416 −0.606612 −0.303306 0.952893i \(-0.598090\pi\)
−0.303306 + 0.952893i \(0.598090\pi\)
\(20\) 0 0
\(21\) 3.65526 0.797643
\(22\) 5.22104 1.11313
\(23\) −7.72850 −1.61150 −0.805751 0.592254i \(-0.798238\pi\)
−0.805751 + 0.592254i \(0.798238\pi\)
\(24\) 2.92496 0.597055
\(25\) 0 0
\(26\) −2.74336 −0.538018
\(27\) 1.00000 0.192450
\(28\) −4.10324 −0.775440
\(29\) −7.79408 −1.44732 −0.723662 0.690154i \(-0.757543\pi\)
−0.723662 + 0.690154i \(0.757543\pi\)
\(30\) 0 0
\(31\) −6.03466 −1.08386 −0.541928 0.840425i \(-0.682306\pi\)
−0.541928 + 0.840425i \(0.682306\pi\)
\(32\) −5.38648 −0.952204
\(33\) −5.57376 −0.970266
\(34\) −2.12130 −0.363801
\(35\) 0 0
\(36\) −1.12256 −0.187093
\(37\) 6.60338 1.08559 0.542794 0.839866i \(-0.317366\pi\)
0.542794 + 0.839866i \(0.317366\pi\)
\(38\) 2.47683 0.401796
\(39\) 2.92869 0.468966
\(40\) 0 0
\(41\) −8.81927 −1.37734 −0.688669 0.725076i \(-0.741805\pi\)
−0.688669 + 0.725076i \(0.741805\pi\)
\(42\) −3.42395 −0.528327
\(43\) −9.69632 −1.47867 −0.739337 0.673335i \(-0.764861\pi\)
−0.739337 + 0.673335i \(0.764861\pi\)
\(44\) 6.25687 0.943258
\(45\) 0 0
\(46\) 7.23943 1.06740
\(47\) −1.00000 −0.145865
\(48\) −0.494746 −0.0714104
\(49\) 6.36091 0.908702
\(50\) 0 0
\(51\) 2.26461 0.317109
\(52\) −3.28763 −0.455912
\(53\) 2.97538 0.408700 0.204350 0.978898i \(-0.434492\pi\)
0.204350 + 0.978898i \(0.434492\pi\)
\(54\) −0.936719 −0.127471
\(55\) 0 0
\(56\) 10.6915 1.42871
\(57\) −2.64416 −0.350228
\(58\) 7.30086 0.958650
\(59\) 0.571458 0.0743975 0.0371987 0.999308i \(-0.488157\pi\)
0.0371987 + 0.999308i \(0.488157\pi\)
\(60\) 0 0
\(61\) 8.30187 1.06295 0.531473 0.847075i \(-0.321639\pi\)
0.531473 + 0.847075i \(0.321639\pi\)
\(62\) 5.65278 0.717903
\(63\) 3.65526 0.460519
\(64\) 6.03511 0.754388
\(65\) 0 0
\(66\) 5.22104 0.642666
\(67\) 8.03594 0.981746 0.490873 0.871231i \(-0.336678\pi\)
0.490873 + 0.871231i \(0.336678\pi\)
\(68\) −2.54216 −0.308282
\(69\) −7.72850 −0.930402
\(70\) 0 0
\(71\) 11.7268 1.39171 0.695855 0.718183i \(-0.255026\pi\)
0.695855 + 0.718183i \(0.255026\pi\)
\(72\) 2.92496 0.344710
\(73\) −5.94373 −0.695661 −0.347830 0.937558i \(-0.613081\pi\)
−0.347830 + 0.937558i \(0.613081\pi\)
\(74\) −6.18551 −0.719050
\(75\) 0 0
\(76\) 2.96822 0.340479
\(77\) −20.3735 −2.32178
\(78\) −2.74336 −0.310625
\(79\) −16.6200 −1.86990 −0.934950 0.354780i \(-0.884556\pi\)
−0.934950 + 0.354780i \(0.884556\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.26117 0.912294
\(83\) −13.8354 −1.51863 −0.759316 0.650722i \(-0.774466\pi\)
−0.759316 + 0.650722i \(0.774466\pi\)
\(84\) −4.10324 −0.447700
\(85\) 0 0
\(86\) 9.08272 0.979415
\(87\) −7.79408 −0.835613
\(88\) −16.3030 −1.73791
\(89\) 4.31121 0.456988 0.228494 0.973545i \(-0.426620\pi\)
0.228494 + 0.973545i \(0.426620\pi\)
\(90\) 0 0
\(91\) 10.7051 1.12220
\(92\) 8.67569 0.904503
\(93\) −6.03466 −0.625765
\(94\) 0.936719 0.0966151
\(95\) 0 0
\(96\) −5.38648 −0.549755
\(97\) −5.24644 −0.532696 −0.266348 0.963877i \(-0.585817\pi\)
−0.266348 + 0.963877i \(0.585817\pi\)
\(98\) −5.95838 −0.601888
\(99\) −5.57376 −0.560184
\(100\) 0 0
\(101\) −17.9567 −1.78675 −0.893377 0.449308i \(-0.851671\pi\)
−0.893377 + 0.449308i \(0.851671\pi\)
\(102\) −2.12130 −0.210040
\(103\) −13.8334 −1.36305 −0.681525 0.731795i \(-0.738683\pi\)
−0.681525 + 0.731795i \(0.738683\pi\)
\(104\) 8.56631 0.839996
\(105\) 0 0
\(106\) −2.78710 −0.270707
\(107\) −6.17861 −0.597309 −0.298655 0.954361i \(-0.596538\pi\)
−0.298655 + 0.954361i \(0.596538\pi\)
\(108\) −1.12256 −0.108018
\(109\) 14.0831 1.34892 0.674458 0.738314i \(-0.264378\pi\)
0.674458 + 0.738314i \(0.264378\pi\)
\(110\) 0 0
\(111\) 6.60338 0.626765
\(112\) −1.80842 −0.170880
\(113\) 16.6357 1.56496 0.782480 0.622676i \(-0.213955\pi\)
0.782480 + 0.622676i \(0.213955\pi\)
\(114\) 2.47683 0.231977
\(115\) 0 0
\(116\) 8.74931 0.812353
\(117\) 2.92869 0.270758
\(118\) −0.535295 −0.0492779
\(119\) 8.27774 0.758819
\(120\) 0 0
\(121\) 20.0668 1.82425
\(122\) −7.77652 −0.704053
\(123\) −8.81927 −0.795207
\(124\) 6.77426 0.608346
\(125\) 0 0
\(126\) −3.42395 −0.305030
\(127\) 12.1084 1.07445 0.537225 0.843439i \(-0.319473\pi\)
0.537225 + 0.843439i \(0.319473\pi\)
\(128\) 5.11976 0.452527
\(129\) −9.69632 −0.853713
\(130\) 0 0
\(131\) 1.58271 0.138282 0.0691409 0.997607i \(-0.477974\pi\)
0.0691409 + 0.997607i \(0.477974\pi\)
\(132\) 6.25687 0.544590
\(133\) −9.66509 −0.838069
\(134\) −7.52741 −0.650269
\(135\) 0 0
\(136\) 6.62389 0.567994
\(137\) 20.5052 1.75188 0.875938 0.482424i \(-0.160243\pi\)
0.875938 + 0.482424i \(0.160243\pi\)
\(138\) 7.23943 0.616261
\(139\) −16.7180 −1.41800 −0.709002 0.705207i \(-0.750854\pi\)
−0.709002 + 0.705207i \(0.750854\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −10.9847 −0.921813
\(143\) −16.3238 −1.36507
\(144\) −0.494746 −0.0412288
\(145\) 0 0
\(146\) 5.56760 0.460778
\(147\) 6.36091 0.524639
\(148\) −7.41268 −0.609318
\(149\) 10.2172 0.837025 0.418513 0.908211i \(-0.362552\pi\)
0.418513 + 0.908211i \(0.362552\pi\)
\(150\) 0 0
\(151\) −20.9048 −1.70121 −0.850603 0.525808i \(-0.823763\pi\)
−0.850603 + 0.525808i \(0.823763\pi\)
\(152\) −7.73406 −0.627315
\(153\) 2.26461 0.183083
\(154\) 19.0843 1.53785
\(155\) 0 0
\(156\) −3.28763 −0.263221
\(157\) 22.8938 1.82712 0.913561 0.406702i \(-0.133321\pi\)
0.913561 + 0.406702i \(0.133321\pi\)
\(158\) 15.5683 1.23855
\(159\) 2.97538 0.235963
\(160\) 0 0
\(161\) −28.2496 −2.22638
\(162\) −0.936719 −0.0735956
\(163\) −11.2612 −0.882042 −0.441021 0.897497i \(-0.645384\pi\)
−0.441021 + 0.897497i \(0.645384\pi\)
\(164\) 9.90014 0.773071
\(165\) 0 0
\(166\) 12.9599 1.00588
\(167\) −16.8461 −1.30359 −0.651794 0.758396i \(-0.725983\pi\)
−0.651794 + 0.758396i \(0.725983\pi\)
\(168\) 10.6915 0.824865
\(169\) −4.42275 −0.340212
\(170\) 0 0
\(171\) −2.64416 −0.202204
\(172\) 10.8847 0.829949
\(173\) −13.5553 −1.03059 −0.515294 0.857014i \(-0.672317\pi\)
−0.515294 + 0.857014i \(0.672317\pi\)
\(174\) 7.30086 0.553477
\(175\) 0 0
\(176\) 2.75759 0.207861
\(177\) 0.571458 0.0429534
\(178\) −4.03839 −0.302690
\(179\) −5.29177 −0.395525 −0.197763 0.980250i \(-0.563368\pi\)
−0.197763 + 0.980250i \(0.563368\pi\)
\(180\) 0 0
\(181\) −6.93711 −0.515632 −0.257816 0.966194i \(-0.583003\pi\)
−0.257816 + 0.966194i \(0.583003\pi\)
\(182\) −10.0277 −0.743302
\(183\) 8.30187 0.613692
\(184\) −22.6055 −1.66650
\(185\) 0 0
\(186\) 5.65278 0.414482
\(187\) −12.6224 −0.923041
\(188\) 1.12256 0.0818710
\(189\) 3.65526 0.265881
\(190\) 0 0
\(191\) 3.27524 0.236988 0.118494 0.992955i \(-0.462193\pi\)
0.118494 + 0.992955i \(0.462193\pi\)
\(192\) 6.03511 0.435546
\(193\) 6.91822 0.497984 0.248992 0.968506i \(-0.419901\pi\)
0.248992 + 0.968506i \(0.419901\pi\)
\(194\) 4.91444 0.352836
\(195\) 0 0
\(196\) −7.14050 −0.510035
\(197\) −13.2026 −0.940650 −0.470325 0.882493i \(-0.655863\pi\)
−0.470325 + 0.882493i \(0.655863\pi\)
\(198\) 5.22104 0.371043
\(199\) −10.3855 −0.736210 −0.368105 0.929784i \(-0.619993\pi\)
−0.368105 + 0.929784i \(0.619993\pi\)
\(200\) 0 0
\(201\) 8.03594 0.566811
\(202\) 16.8203 1.18347
\(203\) −28.4894 −1.99956
\(204\) −2.54216 −0.177987
\(205\) 0 0
\(206\) 12.9580 0.902830
\(207\) −7.72850 −0.537168
\(208\) −1.44896 −0.100467
\(209\) 14.7379 1.01944
\(210\) 0 0
\(211\) 19.5002 1.34245 0.671226 0.741253i \(-0.265768\pi\)
0.671226 + 0.741253i \(0.265768\pi\)
\(212\) −3.34004 −0.229395
\(213\) 11.7268 0.803504
\(214\) 5.78762 0.395634
\(215\) 0 0
\(216\) 2.92496 0.199018
\(217\) −22.0582 −1.49741
\(218\) −13.1919 −0.893467
\(219\) −5.94373 −0.401640
\(220\) 0 0
\(221\) 6.63235 0.446140
\(222\) −6.18551 −0.415144
\(223\) 5.43059 0.363659 0.181830 0.983330i \(-0.441798\pi\)
0.181830 + 0.983330i \(0.441798\pi\)
\(224\) −19.6890 −1.31552
\(225\) 0 0
\(226\) −15.5830 −1.03657
\(227\) −2.68717 −0.178354 −0.0891770 0.996016i \(-0.528424\pi\)
−0.0891770 + 0.996016i \(0.528424\pi\)
\(228\) 2.96822 0.196575
\(229\) 17.6212 1.16444 0.582221 0.813030i \(-0.302184\pi\)
0.582221 + 0.813030i \(0.302184\pi\)
\(230\) 0 0
\(231\) −20.3735 −1.34048
\(232\) −22.7974 −1.49672
\(233\) −14.3094 −0.937437 −0.468719 0.883348i \(-0.655284\pi\)
−0.468719 + 0.883348i \(0.655284\pi\)
\(234\) −2.74336 −0.179339
\(235\) 0 0
\(236\) −0.641495 −0.0417578
\(237\) −16.6200 −1.07959
\(238\) −7.75391 −0.502611
\(239\) −0.608101 −0.0393348 −0.0196674 0.999807i \(-0.506261\pi\)
−0.0196674 + 0.999807i \(0.506261\pi\)
\(240\) 0 0
\(241\) −0.888912 −0.0572598 −0.0286299 0.999590i \(-0.509114\pi\)
−0.0286299 + 0.999590i \(0.509114\pi\)
\(242\) −18.7969 −1.20831
\(243\) 1.00000 0.0641500
\(244\) −9.31934 −0.596609
\(245\) 0 0
\(246\) 8.26117 0.526713
\(247\) −7.74394 −0.492735
\(248\) −17.6511 −1.12085
\(249\) −13.8354 −0.876782
\(250\) 0 0
\(251\) −0.0483724 −0.00305324 −0.00152662 0.999999i \(-0.500486\pi\)
−0.00152662 + 0.999999i \(0.500486\pi\)
\(252\) −4.10324 −0.258480
\(253\) 43.0768 2.70821
\(254\) −11.3422 −0.711672
\(255\) 0 0
\(256\) −16.8660 −1.05412
\(257\) 3.58017 0.223325 0.111662 0.993746i \(-0.464382\pi\)
0.111662 + 0.993746i \(0.464382\pi\)
\(258\) 9.08272 0.565465
\(259\) 24.1370 1.49980
\(260\) 0 0
\(261\) −7.79408 −0.482441
\(262\) −1.48255 −0.0915924
\(263\) 17.6132 1.08608 0.543040 0.839707i \(-0.317273\pi\)
0.543040 + 0.839707i \(0.317273\pi\)
\(264\) −16.3030 −1.00338
\(265\) 0 0
\(266\) 9.05347 0.555104
\(267\) 4.31121 0.263842
\(268\) −9.02081 −0.551034
\(269\) −14.7262 −0.897870 −0.448935 0.893564i \(-0.648196\pi\)
−0.448935 + 0.893564i \(0.648196\pi\)
\(270\) 0 0
\(271\) 7.26580 0.441366 0.220683 0.975346i \(-0.429171\pi\)
0.220683 + 0.975346i \(0.429171\pi\)
\(272\) −1.12041 −0.0679347
\(273\) 10.7051 0.647904
\(274\) −19.2076 −1.16037
\(275\) 0 0
\(276\) 8.67569 0.522215
\(277\) −16.0913 −0.966833 −0.483416 0.875391i \(-0.660604\pi\)
−0.483416 + 0.875391i \(0.660604\pi\)
\(278\) 15.6601 0.939229
\(279\) −6.03466 −0.361286
\(280\) 0 0
\(281\) −2.13975 −0.127647 −0.0638234 0.997961i \(-0.520329\pi\)
−0.0638234 + 0.997961i \(0.520329\pi\)
\(282\) 0.936719 0.0557808
\(283\) −23.8300 −1.41655 −0.708274 0.705937i \(-0.750526\pi\)
−0.708274 + 0.705937i \(0.750526\pi\)
\(284\) −13.1640 −0.781138
\(285\) 0 0
\(286\) 15.2908 0.904166
\(287\) −32.2367 −1.90287
\(288\) −5.38648 −0.317401
\(289\) −11.8715 −0.698326
\(290\) 0 0
\(291\) −5.24644 −0.307552
\(292\) 6.67218 0.390460
\(293\) −17.5931 −1.02780 −0.513901 0.857850i \(-0.671800\pi\)
−0.513901 + 0.857850i \(0.671800\pi\)
\(294\) −5.95838 −0.347500
\(295\) 0 0
\(296\) 19.3146 1.12264
\(297\) −5.57376 −0.323422
\(298\) −9.57064 −0.554412
\(299\) −22.6344 −1.30898
\(300\) 0 0
\(301\) −35.4425 −2.04287
\(302\) 19.5819 1.12681
\(303\) −17.9567 −1.03158
\(304\) 1.30819 0.0750297
\(305\) 0 0
\(306\) −2.12130 −0.121267
\(307\) −20.7229 −1.18272 −0.591360 0.806407i \(-0.701409\pi\)
−0.591360 + 0.806407i \(0.701409\pi\)
\(308\) 22.8705 1.30317
\(309\) −13.8334 −0.786957
\(310\) 0 0
\(311\) −24.3065 −1.37829 −0.689146 0.724622i \(-0.742014\pi\)
−0.689146 + 0.724622i \(0.742014\pi\)
\(312\) 8.56631 0.484972
\(313\) 17.3665 0.981615 0.490808 0.871268i \(-0.336702\pi\)
0.490808 + 0.871268i \(0.336702\pi\)
\(314\) −21.4450 −1.21021
\(315\) 0 0
\(316\) 18.6570 1.04954
\(317\) 2.00730 0.112741 0.0563706 0.998410i \(-0.482047\pi\)
0.0563706 + 0.998410i \(0.482047\pi\)
\(318\) −2.78710 −0.156293
\(319\) 43.4423 2.43230
\(320\) 0 0
\(321\) −6.17861 −0.344857
\(322\) 26.4620 1.47467
\(323\) −5.98799 −0.333181
\(324\) −1.12256 −0.0623644
\(325\) 0 0
\(326\) 10.5485 0.584230
\(327\) 14.0831 0.778797
\(328\) −25.7960 −1.42435
\(329\) −3.65526 −0.201521
\(330\) 0 0
\(331\) 23.0557 1.26725 0.633627 0.773638i \(-0.281565\pi\)
0.633627 + 0.773638i \(0.281565\pi\)
\(332\) 15.5310 0.852376
\(333\) 6.60338 0.361863
\(334\) 15.7800 0.863445
\(335\) 0 0
\(336\) −1.80842 −0.0986576
\(337\) 0.977276 0.0532356 0.0266178 0.999646i \(-0.491526\pi\)
0.0266178 + 0.999646i \(0.491526\pi\)
\(338\) 4.14287 0.225343
\(339\) 16.6357 0.903530
\(340\) 0 0
\(341\) 33.6357 1.82148
\(342\) 2.47683 0.133932
\(343\) −2.33603 −0.126134
\(344\) −28.3613 −1.52914
\(345\) 0 0
\(346\) 12.6975 0.682620
\(347\) 9.11111 0.489110 0.244555 0.969635i \(-0.421358\pi\)
0.244555 + 0.969635i \(0.421358\pi\)
\(348\) 8.74931 0.469012
\(349\) 2.64665 0.141672 0.0708359 0.997488i \(-0.477433\pi\)
0.0708359 + 0.997488i \(0.477433\pi\)
\(350\) 0 0
\(351\) 2.92869 0.156322
\(352\) 30.0229 1.60023
\(353\) 15.8458 0.843386 0.421693 0.906739i \(-0.361436\pi\)
0.421693 + 0.906739i \(0.361436\pi\)
\(354\) −0.535295 −0.0284506
\(355\) 0 0
\(356\) −4.83959 −0.256498
\(357\) 8.27774 0.438104
\(358\) 4.95690 0.261980
\(359\) 11.8670 0.626318 0.313159 0.949701i \(-0.398613\pi\)
0.313159 + 0.949701i \(0.398613\pi\)
\(360\) 0 0
\(361\) −12.0084 −0.632022
\(362\) 6.49812 0.341534
\(363\) 20.0668 1.05323
\(364\) −12.0171 −0.629869
\(365\) 0 0
\(366\) −7.77652 −0.406485
\(367\) −5.06080 −0.264172 −0.132086 0.991238i \(-0.542167\pi\)
−0.132086 + 0.991238i \(0.542167\pi\)
\(368\) 3.82364 0.199321
\(369\) −8.81927 −0.459113
\(370\) 0 0
\(371\) 10.8758 0.564643
\(372\) 6.77426 0.351229
\(373\) 1.19434 0.0618405 0.0309203 0.999522i \(-0.490156\pi\)
0.0309203 + 0.999522i \(0.490156\pi\)
\(374\) 11.8236 0.611385
\(375\) 0 0
\(376\) −2.92496 −0.150843
\(377\) −22.8265 −1.17562
\(378\) −3.42395 −0.176109
\(379\) 16.0321 0.823511 0.411756 0.911294i \(-0.364916\pi\)
0.411756 + 0.911294i \(0.364916\pi\)
\(380\) 0 0
\(381\) 12.1084 0.620334
\(382\) −3.06798 −0.156972
\(383\) 23.4599 1.19875 0.599374 0.800469i \(-0.295416\pi\)
0.599374 + 0.800469i \(0.295416\pi\)
\(384\) 5.11976 0.261267
\(385\) 0 0
\(386\) −6.48042 −0.329845
\(387\) −9.69632 −0.492891
\(388\) 5.88944 0.298991
\(389\) −27.8951 −1.41434 −0.707169 0.707045i \(-0.750028\pi\)
−0.707169 + 0.707045i \(0.750028\pi\)
\(390\) 0 0
\(391\) −17.5020 −0.885116
\(392\) 18.6054 0.939715
\(393\) 1.58271 0.0798371
\(394\) 12.3672 0.623049
\(395\) 0 0
\(396\) 6.25687 0.314419
\(397\) −11.3412 −0.569199 −0.284600 0.958646i \(-0.591861\pi\)
−0.284600 + 0.958646i \(0.591861\pi\)
\(398\) 9.72830 0.487636
\(399\) −9.66509 −0.483860
\(400\) 0 0
\(401\) −12.6214 −0.630283 −0.315142 0.949045i \(-0.602052\pi\)
−0.315142 + 0.949045i \(0.602052\pi\)
\(402\) −7.52741 −0.375433
\(403\) −17.6737 −0.880388
\(404\) 20.1574 1.00287
\(405\) 0 0
\(406\) 26.6865 1.32443
\(407\) −36.8056 −1.82439
\(408\) 6.62389 0.327932
\(409\) −2.98302 −0.147501 −0.0737505 0.997277i \(-0.523497\pi\)
−0.0737505 + 0.997277i \(0.523497\pi\)
\(410\) 0 0
\(411\) 20.5052 1.01145
\(412\) 15.5288 0.765051
\(413\) 2.08883 0.102784
\(414\) 7.23943 0.355798
\(415\) 0 0
\(416\) −15.7754 −0.773450
\(417\) −16.7180 −0.818685
\(418\) −13.8053 −0.675238
\(419\) 15.5076 0.757595 0.378798 0.925480i \(-0.376338\pi\)
0.378798 + 0.925480i \(0.376338\pi\)
\(420\) 0 0
\(421\) −22.2089 −1.08240 −0.541198 0.840895i \(-0.682029\pi\)
−0.541198 + 0.840895i \(0.682029\pi\)
\(422\) −18.2662 −0.889186
\(423\) −1.00000 −0.0486217
\(424\) 8.70287 0.422649
\(425\) 0 0
\(426\) −10.9847 −0.532209
\(427\) 30.3455 1.46852
\(428\) 6.93585 0.335257
\(429\) −16.3238 −0.788122
\(430\) 0 0
\(431\) −20.1349 −0.969865 −0.484932 0.874552i \(-0.661156\pi\)
−0.484932 + 0.874552i \(0.661156\pi\)
\(432\) −0.494746 −0.0238035
\(433\) 17.8238 0.856557 0.428278 0.903647i \(-0.359120\pi\)
0.428278 + 0.903647i \(0.359120\pi\)
\(434\) 20.6624 0.991825
\(435\) 0 0
\(436\) −15.8091 −0.757118
\(437\) 20.4354 0.977557
\(438\) 5.56760 0.266030
\(439\) 11.9909 0.572293 0.286146 0.958186i \(-0.407626\pi\)
0.286146 + 0.958186i \(0.407626\pi\)
\(440\) 0 0
\(441\) 6.36091 0.302901
\(442\) −6.21265 −0.295506
\(443\) −25.2442 −1.19939 −0.599694 0.800230i \(-0.704711\pi\)
−0.599694 + 0.800230i \(0.704711\pi\)
\(444\) −7.41268 −0.351790
\(445\) 0 0
\(446\) −5.08693 −0.240873
\(447\) 10.2172 0.483257
\(448\) 22.0599 1.04223
\(449\) −37.0349 −1.74778 −0.873891 0.486121i \(-0.838411\pi\)
−0.873891 + 0.486121i \(0.838411\pi\)
\(450\) 0 0
\(451\) 49.1564 2.31469
\(452\) −18.6746 −0.878379
\(453\) −20.9048 −0.982192
\(454\) 2.51712 0.118134
\(455\) 0 0
\(456\) −7.73406 −0.362180
\(457\) 13.9630 0.653162 0.326581 0.945169i \(-0.394103\pi\)
0.326581 + 0.945169i \(0.394103\pi\)
\(458\) −16.5061 −0.771280
\(459\) 2.26461 0.105703
\(460\) 0 0
\(461\) −41.9963 −1.95596 −0.977982 0.208689i \(-0.933080\pi\)
−0.977982 + 0.208689i \(0.933080\pi\)
\(462\) 19.0843 0.887880
\(463\) 23.1267 1.07479 0.537394 0.843331i \(-0.319409\pi\)
0.537394 + 0.843331i \(0.319409\pi\)
\(464\) 3.85609 0.179014
\(465\) 0 0
\(466\) 13.4038 0.620921
\(467\) 13.4945 0.624451 0.312225 0.950008i \(-0.398926\pi\)
0.312225 + 0.950008i \(0.398926\pi\)
\(468\) −3.28763 −0.151971
\(469\) 29.3734 1.35634
\(470\) 0 0
\(471\) 22.8938 1.05489
\(472\) 1.67149 0.0769366
\(473\) 54.0449 2.48499
\(474\) 15.5683 0.715075
\(475\) 0 0
\(476\) −9.29224 −0.425909
\(477\) 2.97538 0.136233
\(478\) 0.569620 0.0260538
\(479\) 6.69141 0.305739 0.152869 0.988246i \(-0.451149\pi\)
0.152869 + 0.988246i \(0.451149\pi\)
\(480\) 0 0
\(481\) 19.3393 0.881795
\(482\) 0.832660 0.0379266
\(483\) −28.2496 −1.28540
\(484\) −22.5261 −1.02391
\(485\) 0 0
\(486\) −0.936719 −0.0424904
\(487\) −28.6972 −1.30039 −0.650197 0.759765i \(-0.725314\pi\)
−0.650197 + 0.759765i \(0.725314\pi\)
\(488\) 24.2826 1.09922
\(489\) −11.2612 −0.509247
\(490\) 0 0
\(491\) −6.00762 −0.271120 −0.135560 0.990769i \(-0.543283\pi\)
−0.135560 + 0.990769i \(0.543283\pi\)
\(492\) 9.90014 0.446333
\(493\) −17.6506 −0.794941
\(494\) 7.25389 0.326368
\(495\) 0 0
\(496\) 2.98562 0.134058
\(497\) 42.8643 1.92273
\(498\) 12.9599 0.580746
\(499\) 40.6329 1.81898 0.909489 0.415727i \(-0.136473\pi\)
0.909489 + 0.415727i \(0.136473\pi\)
\(500\) 0 0
\(501\) −16.8461 −0.752627
\(502\) 0.0453114 0.00202235
\(503\) 15.2536 0.680124 0.340062 0.940403i \(-0.389552\pi\)
0.340062 + 0.940403i \(0.389552\pi\)
\(504\) 10.6915 0.476236
\(505\) 0 0
\(506\) −40.3508 −1.79381
\(507\) −4.42275 −0.196421
\(508\) −13.5924 −0.603066
\(509\) −12.9027 −0.571904 −0.285952 0.958244i \(-0.592310\pi\)
−0.285952 + 0.958244i \(0.592310\pi\)
\(510\) 0 0
\(511\) −21.7259 −0.961095
\(512\) 5.55916 0.245683
\(513\) −2.64416 −0.116743
\(514\) −3.35361 −0.147921
\(515\) 0 0
\(516\) 10.8847 0.479171
\(517\) 5.57376 0.245134
\(518\) −22.6096 −0.993410
\(519\) −13.5553 −0.595010
\(520\) 0 0
\(521\) 8.30560 0.363875 0.181937 0.983310i \(-0.441763\pi\)
0.181937 + 0.983310i \(0.441763\pi\)
\(522\) 7.30086 0.319550
\(523\) 36.5557 1.59847 0.799235 0.601019i \(-0.205238\pi\)
0.799235 + 0.601019i \(0.205238\pi\)
\(524\) −1.77668 −0.0776147
\(525\) 0 0
\(526\) −16.4987 −0.719375
\(527\) −13.6662 −0.595307
\(528\) 2.75759 0.120009
\(529\) 36.7297 1.59694
\(530\) 0 0
\(531\) 0.571458 0.0247992
\(532\) 10.8496 0.470391
\(533\) −25.8289 −1.11878
\(534\) −4.03839 −0.174758
\(535\) 0 0
\(536\) 23.5048 1.01525
\(537\) −5.29177 −0.228357
\(538\) 13.7943 0.594713
\(539\) −35.4542 −1.52712
\(540\) 0 0
\(541\) 17.6937 0.760713 0.380356 0.924840i \(-0.375801\pi\)
0.380356 + 0.924840i \(0.375801\pi\)
\(542\) −6.80601 −0.292343
\(543\) −6.93711 −0.297700
\(544\) −12.1983 −0.522997
\(545\) 0 0
\(546\) −10.0277 −0.429146
\(547\) −15.8840 −0.679150 −0.339575 0.940579i \(-0.610283\pi\)
−0.339575 + 0.940579i \(0.610283\pi\)
\(548\) −23.0183 −0.983291
\(549\) 8.30187 0.354315
\(550\) 0 0
\(551\) 20.6088 0.877964
\(552\) −22.6055 −0.962155
\(553\) −60.7505 −2.58337
\(554\) 15.0730 0.640391
\(555\) 0 0
\(556\) 18.7669 0.795896
\(557\) −8.53325 −0.361566 −0.180783 0.983523i \(-0.557863\pi\)
−0.180783 + 0.983523i \(0.557863\pi\)
\(558\) 5.65278 0.239301
\(559\) −28.3975 −1.20109
\(560\) 0 0
\(561\) −12.6224 −0.532918
\(562\) 2.00434 0.0845482
\(563\) 20.6854 0.871785 0.435893 0.899999i \(-0.356433\pi\)
0.435893 + 0.899999i \(0.356433\pi\)
\(564\) 1.12256 0.0472682
\(565\) 0 0
\(566\) 22.3220 0.938265
\(567\) 3.65526 0.153506
\(568\) 34.3003 1.43921
\(569\) 10.7355 0.450057 0.225029 0.974352i \(-0.427752\pi\)
0.225029 + 0.974352i \(0.427752\pi\)
\(570\) 0 0
\(571\) −11.4293 −0.478303 −0.239152 0.970982i \(-0.576869\pi\)
−0.239152 + 0.970982i \(0.576869\pi\)
\(572\) 18.3245 0.766184
\(573\) 3.27524 0.136825
\(574\) 30.1967 1.26039
\(575\) 0 0
\(576\) 6.03511 0.251463
\(577\) −18.3590 −0.764296 −0.382148 0.924101i \(-0.624816\pi\)
−0.382148 + 0.924101i \(0.624816\pi\)
\(578\) 11.1203 0.462543
\(579\) 6.91822 0.287511
\(580\) 0 0
\(581\) −50.5719 −2.09808
\(582\) 4.91444 0.203710
\(583\) −16.5841 −0.686841
\(584\) −17.3852 −0.719403
\(585\) 0 0
\(586\) 16.4798 0.680774
\(587\) 15.6571 0.646237 0.323119 0.946358i \(-0.395269\pi\)
0.323119 + 0.946358i \(0.395269\pi\)
\(588\) −7.14050 −0.294469
\(589\) 15.9566 0.657480
\(590\) 0 0
\(591\) −13.2026 −0.543084
\(592\) −3.26699 −0.134273
\(593\) 15.5623 0.639067 0.319533 0.947575i \(-0.396474\pi\)
0.319533 + 0.947575i \(0.396474\pi\)
\(594\) 5.22104 0.214222
\(595\) 0 0
\(596\) −11.4694 −0.469805
\(597\) −10.3855 −0.425051
\(598\) 21.2021 0.867017
\(599\) −30.3904 −1.24172 −0.620859 0.783922i \(-0.713216\pi\)
−0.620859 + 0.783922i \(0.713216\pi\)
\(600\) 0 0
\(601\) −30.1251 −1.22883 −0.614414 0.788984i \(-0.710608\pi\)
−0.614414 + 0.788984i \(0.710608\pi\)
\(602\) 33.1997 1.35312
\(603\) 8.03594 0.327249
\(604\) 23.4668 0.954852
\(605\) 0 0
\(606\) 16.8203 0.683279
\(607\) 44.1714 1.79286 0.896430 0.443184i \(-0.146151\pi\)
0.896430 + 0.443184i \(0.146151\pi\)
\(608\) 14.2427 0.577618
\(609\) −28.4894 −1.15445
\(610\) 0 0
\(611\) −2.92869 −0.118482
\(612\) −2.54216 −0.102761
\(613\) 24.8671 1.00437 0.502186 0.864759i \(-0.332529\pi\)
0.502186 + 0.864759i \(0.332529\pi\)
\(614\) 19.4116 0.783387
\(615\) 0 0
\(616\) −59.5917 −2.40102
\(617\) 24.8433 1.00015 0.500077 0.865981i \(-0.333305\pi\)
0.500077 + 0.865981i \(0.333305\pi\)
\(618\) 12.9580 0.521249
\(619\) 18.9581 0.761988 0.380994 0.924577i \(-0.375582\pi\)
0.380994 + 0.924577i \(0.375582\pi\)
\(620\) 0 0
\(621\) −7.72850 −0.310134
\(622\) 22.7683 0.912926
\(623\) 15.7586 0.631355
\(624\) −1.44896 −0.0580048
\(625\) 0 0
\(626\) −16.2676 −0.650183
\(627\) 14.7379 0.588575
\(628\) −25.6996 −1.02553
\(629\) 14.9541 0.596258
\(630\) 0 0
\(631\) 39.5889 1.57601 0.788004 0.615670i \(-0.211115\pi\)
0.788004 + 0.615670i \(0.211115\pi\)
\(632\) −48.6129 −1.93372
\(633\) 19.5002 0.775065
\(634\) −1.88027 −0.0746752
\(635\) 0 0
\(636\) −3.34004 −0.132441
\(637\) 18.6292 0.738114
\(638\) −40.6932 −1.61106
\(639\) 11.7268 0.463903
\(640\) 0 0
\(641\) −7.69114 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(642\) 5.78762 0.228419
\(643\) −32.5033 −1.28180 −0.640902 0.767623i \(-0.721439\pi\)
−0.640902 + 0.767623i \(0.721439\pi\)
\(644\) 31.7119 1.24962
\(645\) 0 0
\(646\) 5.60907 0.220686
\(647\) −33.1357 −1.30270 −0.651349 0.758778i \(-0.725797\pi\)
−0.651349 + 0.758778i \(0.725797\pi\)
\(648\) 2.92496 0.114903
\(649\) −3.18517 −0.125029
\(650\) 0 0
\(651\) −22.0582 −0.864530
\(652\) 12.6413 0.495072
\(653\) 1.78903 0.0700102 0.0350051 0.999387i \(-0.488855\pi\)
0.0350051 + 0.999387i \(0.488855\pi\)
\(654\) −13.1919 −0.515844
\(655\) 0 0
\(656\) 4.36330 0.170358
\(657\) −5.94373 −0.231887
\(658\) 3.42395 0.133479
\(659\) −13.1926 −0.513912 −0.256956 0.966423i \(-0.582720\pi\)
−0.256956 + 0.966423i \(0.582720\pi\)
\(660\) 0 0
\(661\) 48.8401 1.89966 0.949830 0.312766i \(-0.101256\pi\)
0.949830 + 0.312766i \(0.101256\pi\)
\(662\) −21.5967 −0.839379
\(663\) 6.63235 0.257579
\(664\) −40.4679 −1.57046
\(665\) 0 0
\(666\) −6.18551 −0.239683
\(667\) 60.2365 2.33237
\(668\) 18.9107 0.731677
\(669\) 5.43059 0.209959
\(670\) 0 0
\(671\) −46.2726 −1.78633
\(672\) −19.6890 −0.759519
\(673\) −7.24521 −0.279282 −0.139641 0.990202i \(-0.544595\pi\)
−0.139641 + 0.990202i \(0.544595\pi\)
\(674\) −0.915433 −0.0352611
\(675\) 0 0
\(676\) 4.96479 0.190954
\(677\) 17.0194 0.654109 0.327054 0.945006i \(-0.393944\pi\)
0.327054 + 0.945006i \(0.393944\pi\)
\(678\) −15.5830 −0.598462
\(679\) −19.1771 −0.735950
\(680\) 0 0
\(681\) −2.68717 −0.102973
\(682\) −31.5072 −1.20647
\(683\) 1.41734 0.0542331 0.0271166 0.999632i \(-0.491367\pi\)
0.0271166 + 0.999632i \(0.491367\pi\)
\(684\) 2.96822 0.113493
\(685\) 0 0
\(686\) 2.18820 0.0835460
\(687\) 17.6212 0.672291
\(688\) 4.79721 0.182892
\(689\) 8.71398 0.331976
\(690\) 0 0
\(691\) 8.82922 0.335879 0.167940 0.985797i \(-0.446289\pi\)
0.167940 + 0.985797i \(0.446289\pi\)
\(692\) 15.2166 0.578447
\(693\) −20.3735 −0.773926
\(694\) −8.53455 −0.323967
\(695\) 0 0
\(696\) −22.7974 −0.864132
\(697\) −19.9722 −0.756501
\(698\) −2.47916 −0.0938377
\(699\) −14.3094 −0.541230
\(700\) 0 0
\(701\) −29.1496 −1.10096 −0.550482 0.834847i \(-0.685556\pi\)
−0.550482 + 0.834847i \(0.685556\pi\)
\(702\) −2.74336 −0.103542
\(703\) −17.4604 −0.658531
\(704\) −33.6382 −1.26779
\(705\) 0 0
\(706\) −14.8430 −0.558625
\(707\) −65.6362 −2.46850
\(708\) −0.641495 −0.0241088
\(709\) −19.7491 −0.741694 −0.370847 0.928694i \(-0.620933\pi\)
−0.370847 + 0.928694i \(0.620933\pi\)
\(710\) 0 0
\(711\) −16.6200 −0.623300
\(712\) 12.6101 0.472584
\(713\) 46.6388 1.74664
\(714\) −7.75391 −0.290183
\(715\) 0 0
\(716\) 5.94032 0.222000
\(717\) −0.608101 −0.0227100
\(718\) −11.1161 −0.414848
\(719\) 16.8239 0.627426 0.313713 0.949518i \(-0.398427\pi\)
0.313713 + 0.949518i \(0.398427\pi\)
\(720\) 0 0
\(721\) −50.5648 −1.88313
\(722\) 11.2485 0.418626
\(723\) −0.888912 −0.0330590
\(724\) 7.78732 0.289413
\(725\) 0 0
\(726\) −18.7969 −0.697619
\(727\) 21.9005 0.812244 0.406122 0.913819i \(-0.366881\pi\)
0.406122 + 0.913819i \(0.366881\pi\)
\(728\) 31.3121 1.16050
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.9584 −0.812160
\(732\) −9.31934 −0.344453
\(733\) −36.6289 −1.35292 −0.676460 0.736479i \(-0.736487\pi\)
−0.676460 + 0.736479i \(0.736487\pi\)
\(734\) 4.74055 0.174977
\(735\) 0 0
\(736\) 41.6294 1.53448
\(737\) −44.7904 −1.64987
\(738\) 8.26117 0.304098
\(739\) −9.51491 −0.350011 −0.175006 0.984567i \(-0.555994\pi\)
−0.175006 + 0.984567i \(0.555994\pi\)
\(740\) 0 0
\(741\) −7.74394 −0.284481
\(742\) −10.1876 −0.373997
\(743\) −4.47470 −0.164161 −0.0820805 0.996626i \(-0.526156\pi\)
−0.0820805 + 0.996626i \(0.526156\pi\)
\(744\) −17.6511 −0.647122
\(745\) 0 0
\(746\) −1.11876 −0.0409607
\(747\) −13.8354 −0.506211
\(748\) 14.1694 0.518084
\(749\) −22.5844 −0.825217
\(750\) 0 0
\(751\) 33.1075 1.20811 0.604055 0.796943i \(-0.293551\pi\)
0.604055 + 0.796943i \(0.293551\pi\)
\(752\) 0.494746 0.0180415
\(753\) −0.0483724 −0.00176279
\(754\) 21.3820 0.778686
\(755\) 0 0
\(756\) −4.10324 −0.149233
\(757\) 12.0208 0.436902 0.218451 0.975848i \(-0.429900\pi\)
0.218451 + 0.975848i \(0.429900\pi\)
\(758\) −15.0175 −0.545461
\(759\) 43.0768 1.56359
\(760\) 0 0
\(761\) 26.4998 0.960616 0.480308 0.877100i \(-0.340525\pi\)
0.480308 + 0.877100i \(0.340525\pi\)
\(762\) −11.3422 −0.410884
\(763\) 51.4773 1.86360
\(764\) −3.67665 −0.133017
\(765\) 0 0
\(766\) −21.9754 −0.794002
\(767\) 1.67363 0.0604311
\(768\) −16.8660 −0.608599
\(769\) −7.70999 −0.278029 −0.139015 0.990290i \(-0.544394\pi\)
−0.139015 + 0.990290i \(0.544394\pi\)
\(770\) 0 0
\(771\) 3.58017 0.128937
\(772\) −7.76610 −0.279508
\(773\) −20.8018 −0.748190 −0.374095 0.927390i \(-0.622047\pi\)
−0.374095 + 0.927390i \(0.622047\pi\)
\(774\) 9.08272 0.326472
\(775\) 0 0
\(776\) −15.3456 −0.550876
\(777\) 24.1370 0.865912
\(778\) 26.1299 0.936801
\(779\) 23.3196 0.835510
\(780\) 0 0
\(781\) −65.3621 −2.33884
\(782\) 16.3945 0.586266
\(783\) −7.79408 −0.278538
\(784\) −3.14704 −0.112394
\(785\) 0 0
\(786\) −1.48255 −0.0528809
\(787\) −46.7576 −1.66673 −0.833364 0.552725i \(-0.813588\pi\)
−0.833364 + 0.552725i \(0.813588\pi\)
\(788\) 14.8207 0.527967
\(789\) 17.6132 0.627048
\(790\) 0 0
\(791\) 60.8079 2.16208
\(792\) −16.3030 −0.579302
\(793\) 24.3136 0.863403
\(794\) 10.6235 0.377015
\(795\) 0 0
\(796\) 11.6583 0.413219
\(797\) 24.6265 0.872314 0.436157 0.899870i \(-0.356339\pi\)
0.436157 + 0.899870i \(0.356339\pi\)
\(798\) 9.05347 0.320489
\(799\) −2.26461 −0.0801162
\(800\) 0 0
\(801\) 4.31121 0.152329
\(802\) 11.8227 0.417474
\(803\) 33.1289 1.16909
\(804\) −9.02081 −0.318139
\(805\) 0 0
\(806\) 16.5553 0.583134
\(807\) −14.7262 −0.518385
\(808\) −52.5225 −1.84773
\(809\) 32.1274 1.12954 0.564770 0.825248i \(-0.308965\pi\)
0.564770 + 0.825248i \(0.308965\pi\)
\(810\) 0 0
\(811\) −0.000411906 0 −1.44640e−5 0 −7.23199e−6 1.00000i \(-0.500002\pi\)
−7.23199e−6 1.00000i \(0.500002\pi\)
\(812\) 31.9810 1.12231
\(813\) 7.26580 0.254823
\(814\) 34.4765 1.20840
\(815\) 0 0
\(816\) −1.12041 −0.0392221
\(817\) 25.6386 0.896981
\(818\) 2.79425 0.0976988
\(819\) 10.7051 0.374068
\(820\) 0 0
\(821\) 53.7005 1.87416 0.937081 0.349113i \(-0.113517\pi\)
0.937081 + 0.349113i \(0.113517\pi\)
\(822\) −19.2076 −0.669941
\(823\) −30.1058 −1.04942 −0.524711 0.851280i \(-0.675827\pi\)
−0.524711 + 0.851280i \(0.675827\pi\)
\(824\) −40.4622 −1.40957
\(825\) 0 0
\(826\) −1.95664 −0.0680803
\(827\) 4.74104 0.164862 0.0824310 0.996597i \(-0.473732\pi\)
0.0824310 + 0.996597i \(0.473732\pi\)
\(828\) 8.67569 0.301501
\(829\) 24.2241 0.841337 0.420669 0.907214i \(-0.361796\pi\)
0.420669 + 0.907214i \(0.361796\pi\)
\(830\) 0 0
\(831\) −16.0913 −0.558201
\(832\) 17.6750 0.612770
\(833\) 14.4050 0.499103
\(834\) 15.6601 0.542264
\(835\) 0 0
\(836\) −16.5442 −0.572192
\(837\) −6.03466 −0.208588
\(838\) −14.5262 −0.501801
\(839\) 13.1024 0.452345 0.226172 0.974087i \(-0.427379\pi\)
0.226172 + 0.974087i \(0.427379\pi\)
\(840\) 0 0
\(841\) 31.7477 1.09475
\(842\) 20.8035 0.716936
\(843\) −2.13975 −0.0736969
\(844\) −21.8902 −0.753490
\(845\) 0 0
\(846\) 0.936719 0.0322050
\(847\) 73.3492 2.52031
\(848\) −1.47206 −0.0505507
\(849\) −23.8300 −0.817845
\(850\) 0 0
\(851\) −51.0342 −1.74943
\(852\) −13.1640 −0.450990
\(853\) −50.2328 −1.71994 −0.859969 0.510347i \(-0.829517\pi\)
−0.859969 + 0.510347i \(0.829517\pi\)
\(854\) −28.4252 −0.972690
\(855\) 0 0
\(856\) −18.0722 −0.617695
\(857\) 11.5763 0.395438 0.197719 0.980259i \(-0.436647\pi\)
0.197719 + 0.980259i \(0.436647\pi\)
\(858\) 15.2908 0.522020
\(859\) 5.35315 0.182647 0.0913236 0.995821i \(-0.470890\pi\)
0.0913236 + 0.995821i \(0.470890\pi\)
\(860\) 0 0
\(861\) −32.2367 −1.09862
\(862\) 18.8608 0.642400
\(863\) 27.8072 0.946569 0.473285 0.880910i \(-0.343068\pi\)
0.473285 + 0.880910i \(0.343068\pi\)
\(864\) −5.38648 −0.183252
\(865\) 0 0
\(866\) −16.6959 −0.567349
\(867\) −11.8715 −0.403178
\(868\) 24.7617 0.840465
\(869\) 92.6360 3.14246
\(870\) 0 0
\(871\) 23.5348 0.797447
\(872\) 41.1924 1.39495
\(873\) −5.24644 −0.177565
\(874\) −19.1422 −0.647495
\(875\) 0 0
\(876\) 6.67218 0.225432
\(877\) −24.8181 −0.838048 −0.419024 0.907975i \(-0.637628\pi\)
−0.419024 + 0.907975i \(0.637628\pi\)
\(878\) −11.2321 −0.379064
\(879\) −17.5931 −0.593401
\(880\) 0 0
\(881\) 35.1845 1.18540 0.592698 0.805424i \(-0.298063\pi\)
0.592698 + 0.805424i \(0.298063\pi\)
\(882\) −5.95838 −0.200629
\(883\) 18.2761 0.615039 0.307520 0.951542i \(-0.400501\pi\)
0.307520 + 0.951542i \(0.400501\pi\)
\(884\) −7.44520 −0.250409
\(885\) 0 0
\(886\) 23.6467 0.794426
\(887\) 21.5403 0.723254 0.361627 0.932323i \(-0.382221\pi\)
0.361627 + 0.932323i \(0.382221\pi\)
\(888\) 19.3146 0.648156
\(889\) 44.2594 1.48441
\(890\) 0 0
\(891\) −5.57376 −0.186728
\(892\) −6.09615 −0.204114
\(893\) 2.64416 0.0884834
\(894\) −9.57064 −0.320090
\(895\) 0 0
\(896\) 18.7140 0.625193
\(897\) −22.6344 −0.755741
\(898\) 34.6912 1.15766
\(899\) 47.0346 1.56869
\(900\) 0 0
\(901\) 6.73808 0.224478
\(902\) −46.0458 −1.53316
\(903\) −35.4425 −1.17945
\(904\) 48.6589 1.61837
\(905\) 0 0
\(906\) 19.5819 0.650565
\(907\) −17.7267 −0.588605 −0.294302 0.955712i \(-0.595087\pi\)
−0.294302 + 0.955712i \(0.595087\pi\)
\(908\) 3.01651 0.100106
\(909\) −17.9567 −0.595585
\(910\) 0 0
\(911\) −5.58606 −0.185074 −0.0925372 0.995709i \(-0.529498\pi\)
−0.0925372 + 0.995709i \(0.529498\pi\)
\(912\) 1.30819 0.0433184
\(913\) 77.1151 2.55214
\(914\) −13.0794 −0.432628
\(915\) 0 0
\(916\) −19.7808 −0.653577
\(917\) 5.78521 0.191044
\(918\) −2.12130 −0.0700134
\(919\) 9.14763 0.301752 0.150876 0.988553i \(-0.451791\pi\)
0.150876 + 0.988553i \(0.451791\pi\)
\(920\) 0 0
\(921\) −20.7229 −0.682844
\(922\) 39.3387 1.29555
\(923\) 34.3441 1.13045
\(924\) 22.8705 0.752383
\(925\) 0 0
\(926\) −21.6632 −0.711896
\(927\) −13.8334 −0.454350
\(928\) 41.9826 1.37815
\(929\) 19.9482 0.654480 0.327240 0.944941i \(-0.393881\pi\)
0.327240 + 0.944941i \(0.393881\pi\)
\(930\) 0 0
\(931\) −16.8193 −0.551229
\(932\) 16.0631 0.526164
\(933\) −24.3065 −0.795758
\(934\) −12.6405 −0.413611
\(935\) 0 0
\(936\) 8.56631 0.279999
\(937\) −28.6210 −0.935007 −0.467504 0.883991i \(-0.654847\pi\)
−0.467504 + 0.883991i \(0.654847\pi\)
\(938\) −27.5146 −0.898385
\(939\) 17.3665 0.566736
\(940\) 0 0
\(941\) 17.6221 0.574464 0.287232 0.957861i \(-0.407265\pi\)
0.287232 + 0.957861i \(0.407265\pi\)
\(942\) −21.4450 −0.698716
\(943\) 68.1597 2.21958
\(944\) −0.282726 −0.00920196
\(945\) 0 0
\(946\) −50.6249 −1.64596
\(947\) 41.9598 1.36351 0.681754 0.731581i \(-0.261217\pi\)
0.681754 + 0.731581i \(0.261217\pi\)
\(948\) 18.6570 0.605950
\(949\) −17.4074 −0.565067
\(950\) 0 0
\(951\) 2.00730 0.0650911
\(952\) 24.2120 0.784717
\(953\) −26.1334 −0.846545 −0.423273 0.906002i \(-0.639119\pi\)
−0.423273 + 0.906002i \(0.639119\pi\)
\(954\) −2.78710 −0.0902355
\(955\) 0 0
\(956\) 0.682629 0.0220778
\(957\) 43.4423 1.40429
\(958\) −6.26797 −0.202509
\(959\) 74.9517 2.42032
\(960\) 0 0
\(961\) 5.41711 0.174745
\(962\) −18.1155 −0.584066
\(963\) −6.17861 −0.199103
\(964\) 0.997855 0.0321388
\(965\) 0 0
\(966\) 26.4620 0.851400
\(967\) 9.86542 0.317251 0.158625 0.987339i \(-0.449294\pi\)
0.158625 + 0.987339i \(0.449294\pi\)
\(968\) 58.6944 1.88651
\(969\) −5.98799 −0.192362
\(970\) 0 0
\(971\) 3.91113 0.125514 0.0627571 0.998029i \(-0.480011\pi\)
0.0627571 + 0.998029i \(0.480011\pi\)
\(972\) −1.12256 −0.0360061
\(973\) −61.1087 −1.95905
\(974\) 26.8812 0.861329
\(975\) 0 0
\(976\) −4.10732 −0.131472
\(977\) 8.80390 0.281662 0.140831 0.990034i \(-0.455023\pi\)
0.140831 + 0.990034i \(0.455023\pi\)
\(978\) 10.5485 0.337305
\(979\) −24.0297 −0.767991
\(980\) 0 0
\(981\) 14.0831 0.449638
\(982\) 5.62745 0.179579
\(983\) −2.42901 −0.0774735 −0.0387368 0.999249i \(-0.512333\pi\)
−0.0387368 + 0.999249i \(0.512333\pi\)
\(984\) −25.7960 −0.822346
\(985\) 0 0
\(986\) 16.5336 0.526537
\(987\) −3.65526 −0.116348
\(988\) 8.69302 0.276562
\(989\) 74.9379 2.38289
\(990\) 0 0
\(991\) −12.4176 −0.394459 −0.197230 0.980357i \(-0.563194\pi\)
−0.197230 + 0.980357i \(0.563194\pi\)
\(992\) 32.5056 1.03205
\(993\) 23.0557 0.731650
\(994\) −40.1518 −1.27354
\(995\) 0 0
\(996\) 15.5310 0.492120
\(997\) −14.3515 −0.454518 −0.227259 0.973834i \(-0.572976\pi\)
−0.227259 + 0.973834i \(0.572976\pi\)
\(998\) −38.0616 −1.20482
\(999\) 6.60338 0.208922
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 3525.2.a.bd.1.4 8
5.4 even 2 3525.2.a.be.1.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.4 8 1.1 even 1 trivial
3525.2.a.be.1.5 yes 8 5.4 even 2