Properties

Label 4006.2.a.g.1.7
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.23215 q^{3} +1.00000 q^{4} -2.75227 q^{5} +2.23215 q^{6} -2.11016 q^{7} -1.00000 q^{8} +1.98248 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.23215 q^{3} +1.00000 q^{4} -2.75227 q^{5} +2.23215 q^{6} -2.11016 q^{7} -1.00000 q^{8} +1.98248 q^{9} +2.75227 q^{10} -1.85331 q^{11} -2.23215 q^{12} -5.20404 q^{13} +2.11016 q^{14} +6.14348 q^{15} +1.00000 q^{16} +3.15695 q^{17} -1.98248 q^{18} -7.18933 q^{19} -2.75227 q^{20} +4.71019 q^{21} +1.85331 q^{22} +5.98802 q^{23} +2.23215 q^{24} +2.57501 q^{25} +5.20404 q^{26} +2.27126 q^{27} -2.11016 q^{28} +0.982740 q^{29} -6.14348 q^{30} +8.19185 q^{31} -1.00000 q^{32} +4.13687 q^{33} -3.15695 q^{34} +5.80774 q^{35} +1.98248 q^{36} +4.83696 q^{37} +7.18933 q^{38} +11.6162 q^{39} +2.75227 q^{40} +1.04417 q^{41} -4.71019 q^{42} -0.753058 q^{43} -1.85331 q^{44} -5.45632 q^{45} -5.98802 q^{46} -3.72764 q^{47} -2.23215 q^{48} -2.54722 q^{49} -2.57501 q^{50} -7.04678 q^{51} -5.20404 q^{52} -5.58254 q^{53} -2.27126 q^{54} +5.10083 q^{55} +2.11016 q^{56} +16.0476 q^{57} -0.982740 q^{58} -8.16846 q^{59} +6.14348 q^{60} +14.4526 q^{61} -8.19185 q^{62} -4.18335 q^{63} +1.00000 q^{64} +14.3230 q^{65} -4.13687 q^{66} +11.4721 q^{67} +3.15695 q^{68} -13.3661 q^{69} -5.80774 q^{70} -0.0465372 q^{71} -1.98248 q^{72} -3.43957 q^{73} -4.83696 q^{74} -5.74780 q^{75} -7.18933 q^{76} +3.91079 q^{77} -11.6162 q^{78} +0.984521 q^{79} -2.75227 q^{80} -11.0172 q^{81} -1.04417 q^{82} -1.05397 q^{83} +4.71019 q^{84} -8.68880 q^{85} +0.753058 q^{86} -2.19362 q^{87} +1.85331 q^{88} -4.09263 q^{89} +5.45632 q^{90} +10.9814 q^{91} +5.98802 q^{92} -18.2854 q^{93} +3.72764 q^{94} +19.7870 q^{95} +2.23215 q^{96} +11.5509 q^{97} +2.54722 q^{98} -3.67416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.23215 −1.28873 −0.644365 0.764718i \(-0.722878\pi\)
−0.644365 + 0.764718i \(0.722878\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.75227 −1.23085 −0.615427 0.788194i \(-0.711016\pi\)
−0.615427 + 0.788194i \(0.711016\pi\)
\(6\) 2.23215 0.911270
\(7\) −2.11016 −0.797566 −0.398783 0.917045i \(-0.630567\pi\)
−0.398783 + 0.917045i \(0.630567\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.98248 0.660826
\(10\) 2.75227 0.870345
\(11\) −1.85331 −0.558795 −0.279398 0.960175i \(-0.590135\pi\)
−0.279398 + 0.960175i \(0.590135\pi\)
\(12\) −2.23215 −0.644365
\(13\) −5.20404 −1.44334 −0.721671 0.692236i \(-0.756626\pi\)
−0.721671 + 0.692236i \(0.756626\pi\)
\(14\) 2.11016 0.563965
\(15\) 6.14348 1.58624
\(16\) 1.00000 0.250000
\(17\) 3.15695 0.765674 0.382837 0.923816i \(-0.374947\pi\)
0.382837 + 0.923816i \(0.374947\pi\)
\(18\) −1.98248 −0.467274
\(19\) −7.18933 −1.64935 −0.824673 0.565610i \(-0.808641\pi\)
−0.824673 + 0.565610i \(0.808641\pi\)
\(20\) −2.75227 −0.615427
\(21\) 4.71019 1.02785
\(22\) 1.85331 0.395128
\(23\) 5.98802 1.24859 0.624294 0.781190i \(-0.285387\pi\)
0.624294 + 0.781190i \(0.285387\pi\)
\(24\) 2.23215 0.455635
\(25\) 2.57501 0.515002
\(26\) 5.20404 1.02060
\(27\) 2.27126 0.437104
\(28\) −2.11016 −0.398783
\(29\) 0.982740 0.182490 0.0912451 0.995828i \(-0.470915\pi\)
0.0912451 + 0.995828i \(0.470915\pi\)
\(30\) −6.14348 −1.12164
\(31\) 8.19185 1.47130 0.735650 0.677362i \(-0.236877\pi\)
0.735650 + 0.677362i \(0.236877\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.13687 0.720137
\(34\) −3.15695 −0.541413
\(35\) 5.80774 0.981688
\(36\) 1.98248 0.330413
\(37\) 4.83696 0.795192 0.397596 0.917561i \(-0.369845\pi\)
0.397596 + 0.917561i \(0.369845\pi\)
\(38\) 7.18933 1.16626
\(39\) 11.6162 1.86008
\(40\) 2.75227 0.435173
\(41\) 1.04417 0.163072 0.0815362 0.996670i \(-0.474017\pi\)
0.0815362 + 0.996670i \(0.474017\pi\)
\(42\) −4.71019 −0.726798
\(43\) −0.753058 −0.114840 −0.0574201 0.998350i \(-0.518287\pi\)
−0.0574201 + 0.998350i \(0.518287\pi\)
\(44\) −1.85331 −0.279398
\(45\) −5.45632 −0.813380
\(46\) −5.98802 −0.882885
\(47\) −3.72764 −0.543733 −0.271866 0.962335i \(-0.587641\pi\)
−0.271866 + 0.962335i \(0.587641\pi\)
\(48\) −2.23215 −0.322183
\(49\) −2.54722 −0.363888
\(50\) −2.57501 −0.364162
\(51\) −7.04678 −0.986747
\(52\) −5.20404 −0.721671
\(53\) −5.58254 −0.766821 −0.383410 0.923578i \(-0.625250\pi\)
−0.383410 + 0.923578i \(0.625250\pi\)
\(54\) −2.27126 −0.309079
\(55\) 5.10083 0.687796
\(56\) 2.11016 0.281982
\(57\) 16.0476 2.12556
\(58\) −0.982740 −0.129040
\(59\) −8.16846 −1.06344 −0.531722 0.846919i \(-0.678455\pi\)
−0.531722 + 0.846919i \(0.678455\pi\)
\(60\) 6.14348 0.793120
\(61\) 14.4526 1.85047 0.925235 0.379395i \(-0.123868\pi\)
0.925235 + 0.379395i \(0.123868\pi\)
\(62\) −8.19185 −1.04037
\(63\) −4.18335 −0.527052
\(64\) 1.00000 0.125000
\(65\) 14.3230 1.77654
\(66\) −4.13687 −0.509213
\(67\) 11.4721 1.40154 0.700771 0.713386i \(-0.252839\pi\)
0.700771 + 0.713386i \(0.252839\pi\)
\(68\) 3.15695 0.382837
\(69\) −13.3661 −1.60909
\(70\) −5.80774 −0.694158
\(71\) −0.0465372 −0.00552295 −0.00276147 0.999996i \(-0.500879\pi\)
−0.00276147 + 0.999996i \(0.500879\pi\)
\(72\) −1.98248 −0.233637
\(73\) −3.43957 −0.402571 −0.201285 0.979533i \(-0.564512\pi\)
−0.201285 + 0.979533i \(0.564512\pi\)
\(74\) −4.83696 −0.562285
\(75\) −5.74780 −0.663699
\(76\) −7.18933 −0.824673
\(77\) 3.91079 0.445676
\(78\) −11.6162 −1.31527
\(79\) 0.984521 0.110767 0.0553837 0.998465i \(-0.482362\pi\)
0.0553837 + 0.998465i \(0.482362\pi\)
\(80\) −2.75227 −0.307714
\(81\) −11.0172 −1.22414
\(82\) −1.04417 −0.115310
\(83\) −1.05397 −0.115688 −0.0578441 0.998326i \(-0.518423\pi\)
−0.0578441 + 0.998326i \(0.518423\pi\)
\(84\) 4.71019 0.513924
\(85\) −8.68880 −0.942433
\(86\) 0.753058 0.0812043
\(87\) −2.19362 −0.235181
\(88\) 1.85331 0.197564
\(89\) −4.09263 −0.433817 −0.216909 0.976192i \(-0.569597\pi\)
−0.216909 + 0.976192i \(0.569597\pi\)
\(90\) 5.45632 0.575147
\(91\) 10.9814 1.15116
\(92\) 5.98802 0.624294
\(93\) −18.2854 −1.89611
\(94\) 3.72764 0.384477
\(95\) 19.7870 2.03010
\(96\) 2.23215 0.227817
\(97\) 11.5509 1.17281 0.586406 0.810017i \(-0.300542\pi\)
0.586406 + 0.810017i \(0.300542\pi\)
\(98\) 2.54722 0.257308
\(99\) −3.67416 −0.369266
\(100\) 2.57501 0.257501
\(101\) −0.816156 −0.0812106 −0.0406053 0.999175i \(-0.512929\pi\)
−0.0406053 + 0.999175i \(0.512929\pi\)
\(102\) 7.04678 0.697735
\(103\) 11.6081 1.14378 0.571891 0.820330i \(-0.306210\pi\)
0.571891 + 0.820330i \(0.306210\pi\)
\(104\) 5.20404 0.510298
\(105\) −12.9637 −1.26513
\(106\) 5.58254 0.542224
\(107\) 18.4241 1.78113 0.890565 0.454856i \(-0.150309\pi\)
0.890565 + 0.454856i \(0.150309\pi\)
\(108\) 2.27126 0.218552
\(109\) −7.36454 −0.705395 −0.352697 0.935737i \(-0.614735\pi\)
−0.352697 + 0.935737i \(0.614735\pi\)
\(110\) −5.10083 −0.486345
\(111\) −10.7968 −1.02479
\(112\) −2.11016 −0.199392
\(113\) 12.5555 1.18112 0.590560 0.806994i \(-0.298907\pi\)
0.590560 + 0.806994i \(0.298907\pi\)
\(114\) −16.0476 −1.50300
\(115\) −16.4807 −1.53683
\(116\) 0.982740 0.0912451
\(117\) −10.3169 −0.953798
\(118\) 8.16846 0.751968
\(119\) −6.66168 −0.610676
\(120\) −6.14348 −0.560820
\(121\) −7.56522 −0.687748
\(122\) −14.4526 −1.30848
\(123\) −2.33075 −0.210156
\(124\) 8.19185 0.735650
\(125\) 6.67423 0.596962
\(126\) 4.18335 0.372682
\(127\) 11.0872 0.983828 0.491914 0.870644i \(-0.336297\pi\)
0.491914 + 0.870644i \(0.336297\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.68094 0.147998
\(130\) −14.3230 −1.25621
\(131\) 7.46737 0.652427 0.326214 0.945296i \(-0.394227\pi\)
0.326214 + 0.945296i \(0.394227\pi\)
\(132\) 4.13687 0.360068
\(133\) 15.1707 1.31546
\(134\) −11.4721 −0.991040
\(135\) −6.25113 −0.538011
\(136\) −3.15695 −0.270707
\(137\) 2.46456 0.210562 0.105281 0.994443i \(-0.466426\pi\)
0.105281 + 0.994443i \(0.466426\pi\)
\(138\) 13.3661 1.13780
\(139\) −14.9720 −1.26991 −0.634955 0.772549i \(-0.718981\pi\)
−0.634955 + 0.772549i \(0.718981\pi\)
\(140\) 5.80774 0.490844
\(141\) 8.32065 0.700725
\(142\) 0.0465372 0.00390531
\(143\) 9.64473 0.806533
\(144\) 1.98248 0.165206
\(145\) −2.70477 −0.224619
\(146\) 3.43957 0.284661
\(147\) 5.68576 0.468954
\(148\) 4.83696 0.397596
\(149\) −12.9068 −1.05736 −0.528682 0.848820i \(-0.677314\pi\)
−0.528682 + 0.848820i \(0.677314\pi\)
\(150\) 5.74780 0.469306
\(151\) 8.45894 0.688379 0.344190 0.938900i \(-0.388154\pi\)
0.344190 + 0.938900i \(0.388154\pi\)
\(152\) 7.18933 0.583132
\(153\) 6.25859 0.505977
\(154\) −3.91079 −0.315141
\(155\) −22.5462 −1.81096
\(156\) 11.6162 0.930039
\(157\) −9.38080 −0.748670 −0.374335 0.927294i \(-0.622129\pi\)
−0.374335 + 0.927294i \(0.622129\pi\)
\(158\) −0.984521 −0.0783243
\(159\) 12.4610 0.988225
\(160\) 2.75227 0.217586
\(161\) −12.6357 −0.995832
\(162\) 11.0172 0.865594
\(163\) −12.2248 −0.957520 −0.478760 0.877946i \(-0.658914\pi\)
−0.478760 + 0.877946i \(0.658914\pi\)
\(164\) 1.04417 0.0815362
\(165\) −11.3858 −0.886383
\(166\) 1.05397 0.0818039
\(167\) −11.2733 −0.872358 −0.436179 0.899860i \(-0.643668\pi\)
−0.436179 + 0.899860i \(0.643668\pi\)
\(168\) −4.71019 −0.363399
\(169\) 14.0821 1.08324
\(170\) 8.68880 0.666401
\(171\) −14.2527 −1.08993
\(172\) −0.753058 −0.0574201
\(173\) −2.55843 −0.194514 −0.0972569 0.995259i \(-0.531007\pi\)
−0.0972569 + 0.995259i \(0.531007\pi\)
\(174\) 2.19362 0.166298
\(175\) −5.43369 −0.410748
\(176\) −1.85331 −0.139699
\(177\) 18.2332 1.37049
\(178\) 4.09263 0.306755
\(179\) 0.299051 0.0223521 0.0111760 0.999938i \(-0.496442\pi\)
0.0111760 + 0.999938i \(0.496442\pi\)
\(180\) −5.45632 −0.406690
\(181\) 10.4338 0.775538 0.387769 0.921757i \(-0.373246\pi\)
0.387769 + 0.921757i \(0.373246\pi\)
\(182\) −10.9814 −0.813994
\(183\) −32.2604 −2.38476
\(184\) −5.98802 −0.441442
\(185\) −13.3126 −0.978765
\(186\) 18.2854 1.34075
\(187\) −5.85083 −0.427855
\(188\) −3.72764 −0.271866
\(189\) −4.79272 −0.348619
\(190\) −19.7870 −1.43550
\(191\) −13.5586 −0.981066 −0.490533 0.871423i \(-0.663198\pi\)
−0.490533 + 0.871423i \(0.663198\pi\)
\(192\) −2.23215 −0.161091
\(193\) 9.87008 0.710464 0.355232 0.934778i \(-0.384402\pi\)
0.355232 + 0.934778i \(0.384402\pi\)
\(194\) −11.5509 −0.829304
\(195\) −31.9709 −2.28949
\(196\) −2.54722 −0.181944
\(197\) −13.9126 −0.991229 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(198\) 3.67416 0.261111
\(199\) 3.99828 0.283431 0.141715 0.989907i \(-0.454738\pi\)
0.141715 + 0.989907i \(0.454738\pi\)
\(200\) −2.57501 −0.182081
\(201\) −25.6074 −1.80621
\(202\) 0.816156 0.0574246
\(203\) −2.07374 −0.145548
\(204\) −7.04678 −0.493373
\(205\) −2.87385 −0.200718
\(206\) −11.6081 −0.808776
\(207\) 11.8711 0.825099
\(208\) −5.20404 −0.360836
\(209\) 13.3241 0.921647
\(210\) 12.9637 0.894583
\(211\) 2.75626 0.189749 0.0948743 0.995489i \(-0.469755\pi\)
0.0948743 + 0.995489i \(0.469755\pi\)
\(212\) −5.58254 −0.383410
\(213\) 0.103878 0.00711759
\(214\) −18.4241 −1.25945
\(215\) 2.07262 0.141352
\(216\) −2.27126 −0.154540
\(217\) −17.2861 −1.17346
\(218\) 7.36454 0.498789
\(219\) 7.67762 0.518805
\(220\) 5.10083 0.343898
\(221\) −16.4289 −1.10513
\(222\) 10.7968 0.724634
\(223\) 1.88498 0.126228 0.0631139 0.998006i \(-0.479897\pi\)
0.0631139 + 0.998006i \(0.479897\pi\)
\(224\) 2.11016 0.140991
\(225\) 5.10490 0.340327
\(226\) −12.5555 −0.835178
\(227\) −1.41936 −0.0942060 −0.0471030 0.998890i \(-0.514999\pi\)
−0.0471030 + 0.998890i \(0.514999\pi\)
\(228\) 16.0476 1.06278
\(229\) −13.1725 −0.870462 −0.435231 0.900319i \(-0.643333\pi\)
−0.435231 + 0.900319i \(0.643333\pi\)
\(230\) 16.4807 1.08670
\(231\) −8.72947 −0.574357
\(232\) −0.982740 −0.0645201
\(233\) −3.47205 −0.227461 −0.113731 0.993512i \(-0.536280\pi\)
−0.113731 + 0.993512i \(0.536280\pi\)
\(234\) 10.3169 0.674437
\(235\) 10.2595 0.669256
\(236\) −8.16846 −0.531722
\(237\) −2.19760 −0.142749
\(238\) 6.66168 0.431813
\(239\) 4.50429 0.291358 0.145679 0.989332i \(-0.453463\pi\)
0.145679 + 0.989332i \(0.453463\pi\)
\(240\) 6.14348 0.396560
\(241\) −4.21948 −0.271801 −0.135900 0.990723i \(-0.543393\pi\)
−0.135900 + 0.990723i \(0.543393\pi\)
\(242\) 7.56522 0.486311
\(243\) 17.7783 1.14048
\(244\) 14.4526 0.925235
\(245\) 7.01064 0.447893
\(246\) 2.33075 0.148603
\(247\) 37.4136 2.38057
\(248\) −8.19185 −0.520183
\(249\) 2.35261 0.149091
\(250\) −6.67423 −0.422116
\(251\) −3.31411 −0.209185 −0.104593 0.994515i \(-0.533354\pi\)
−0.104593 + 0.994515i \(0.533354\pi\)
\(252\) −4.18335 −0.263526
\(253\) −11.0977 −0.697705
\(254\) −11.0872 −0.695672
\(255\) 19.3947 1.21454
\(256\) 1.00000 0.0625000
\(257\) −4.48515 −0.279776 −0.139888 0.990167i \(-0.544674\pi\)
−0.139888 + 0.990167i \(0.544674\pi\)
\(258\) −1.68094 −0.104650
\(259\) −10.2068 −0.634218
\(260\) 14.3230 0.888272
\(261\) 1.94826 0.120594
\(262\) −7.46737 −0.461336
\(263\) 15.9494 0.983484 0.491742 0.870741i \(-0.336360\pi\)
0.491742 + 0.870741i \(0.336360\pi\)
\(264\) −4.13687 −0.254607
\(265\) 15.3647 0.943844
\(266\) −15.1707 −0.930172
\(267\) 9.13534 0.559074
\(268\) 11.4721 0.700771
\(269\) −19.0622 −1.16224 −0.581121 0.813817i \(-0.697386\pi\)
−0.581121 + 0.813817i \(0.697386\pi\)
\(270\) 6.25113 0.380431
\(271\) 21.9042 1.33059 0.665294 0.746582i \(-0.268306\pi\)
0.665294 + 0.746582i \(0.268306\pi\)
\(272\) 3.15695 0.191418
\(273\) −24.5120 −1.48354
\(274\) −2.46456 −0.148890
\(275\) −4.77231 −0.287781
\(276\) −13.3661 −0.804547
\(277\) 22.5490 1.35484 0.677419 0.735597i \(-0.263098\pi\)
0.677419 + 0.735597i \(0.263098\pi\)
\(278\) 14.9720 0.897962
\(279\) 16.2402 0.972273
\(280\) −5.80774 −0.347079
\(281\) −2.43550 −0.145290 −0.0726448 0.997358i \(-0.523144\pi\)
−0.0726448 + 0.997358i \(0.523144\pi\)
\(282\) −8.32065 −0.495487
\(283\) 6.63059 0.394148 0.197074 0.980389i \(-0.436856\pi\)
0.197074 + 0.980389i \(0.436856\pi\)
\(284\) −0.0465372 −0.00276147
\(285\) −44.1675 −2.61626
\(286\) −9.64473 −0.570305
\(287\) −2.20337 −0.130061
\(288\) −1.98248 −0.116819
\(289\) −7.03364 −0.413744
\(290\) 2.70477 0.158830
\(291\) −25.7832 −1.51144
\(292\) −3.43957 −0.201285
\(293\) −22.5352 −1.31652 −0.658259 0.752791i \(-0.728707\pi\)
−0.658259 + 0.752791i \(0.728707\pi\)
\(294\) −5.68576 −0.331600
\(295\) 22.4818 1.30894
\(296\) −4.83696 −0.281143
\(297\) −4.20936 −0.244252
\(298\) 12.9068 0.747669
\(299\) −31.1619 −1.80214
\(300\) −5.74780 −0.331849
\(301\) 1.58907 0.0915927
\(302\) −8.45894 −0.486757
\(303\) 1.82178 0.104659
\(304\) −7.18933 −0.412336
\(305\) −39.7776 −2.27766
\(306\) −6.25859 −0.357780
\(307\) −19.3513 −1.10444 −0.552219 0.833699i \(-0.686219\pi\)
−0.552219 + 0.833699i \(0.686219\pi\)
\(308\) 3.91079 0.222838
\(309\) −25.9110 −1.47403
\(310\) 22.5462 1.28054
\(311\) −30.3012 −1.71822 −0.859112 0.511787i \(-0.828984\pi\)
−0.859112 + 0.511787i \(0.828984\pi\)
\(312\) −11.6162 −0.657637
\(313\) −3.85715 −0.218019 −0.109010 0.994041i \(-0.534768\pi\)
−0.109010 + 0.994041i \(0.534768\pi\)
\(314\) 9.38080 0.529389
\(315\) 11.5137 0.648725
\(316\) 0.984521 0.0553837
\(317\) −8.84967 −0.497047 −0.248523 0.968626i \(-0.579945\pi\)
−0.248523 + 0.968626i \(0.579945\pi\)
\(318\) −12.4610 −0.698781
\(319\) −1.82133 −0.101975
\(320\) −2.75227 −0.153857
\(321\) −41.1254 −2.29540
\(322\) 12.6357 0.704159
\(323\) −22.6964 −1.26286
\(324\) −11.0172 −0.612068
\(325\) −13.4005 −0.743324
\(326\) 12.2248 0.677069
\(327\) 16.4387 0.909063
\(328\) −1.04417 −0.0576548
\(329\) 7.86593 0.433663
\(330\) 11.3858 0.626768
\(331\) −3.49270 −0.191976 −0.0959881 0.995382i \(-0.530601\pi\)
−0.0959881 + 0.995382i \(0.530601\pi\)
\(332\) −1.05397 −0.0578441
\(333\) 9.58917 0.525483
\(334\) 11.2733 0.616850
\(335\) −31.5744 −1.72509
\(336\) 4.71019 0.256962
\(337\) −5.32473 −0.290057 −0.145028 0.989428i \(-0.546327\pi\)
−0.145028 + 0.989428i \(0.546327\pi\)
\(338\) −14.0821 −0.765964
\(339\) −28.0257 −1.52215
\(340\) −8.68880 −0.471216
\(341\) −15.1821 −0.822155
\(342\) 14.2527 0.770697
\(343\) 20.1462 1.08779
\(344\) 0.753058 0.0406022
\(345\) 36.7873 1.98056
\(346\) 2.55843 0.137542
\(347\) 19.5881 1.05154 0.525771 0.850626i \(-0.323777\pi\)
0.525771 + 0.850626i \(0.323777\pi\)
\(348\) −2.19362 −0.117590
\(349\) −6.21124 −0.332480 −0.166240 0.986085i \(-0.553163\pi\)
−0.166240 + 0.986085i \(0.553163\pi\)
\(350\) 5.43369 0.290443
\(351\) −11.8197 −0.630891
\(352\) 1.85331 0.0987820
\(353\) −34.8009 −1.85226 −0.926132 0.377200i \(-0.876887\pi\)
−0.926132 + 0.377200i \(0.876887\pi\)
\(354\) −18.2332 −0.969084
\(355\) 0.128083 0.00679794
\(356\) −4.09263 −0.216909
\(357\) 14.8699 0.786996
\(358\) −0.299051 −0.0158053
\(359\) −3.75205 −0.198025 −0.0990127 0.995086i \(-0.531568\pi\)
−0.0990127 + 0.995086i \(0.531568\pi\)
\(360\) 5.45632 0.287573
\(361\) 32.6865 1.72034
\(362\) −10.4338 −0.548388
\(363\) 16.8867 0.886321
\(364\) 10.9814 0.575580
\(365\) 9.46663 0.495506
\(366\) 32.2604 1.68628
\(367\) 9.86089 0.514734 0.257367 0.966314i \(-0.417145\pi\)
0.257367 + 0.966314i \(0.417145\pi\)
\(368\) 5.98802 0.312147
\(369\) 2.07005 0.107762
\(370\) 13.3126 0.692091
\(371\) 11.7801 0.611590
\(372\) −18.2854 −0.948054
\(373\) 30.1102 1.55905 0.779523 0.626374i \(-0.215462\pi\)
0.779523 + 0.626374i \(0.215462\pi\)
\(374\) 5.85083 0.302539
\(375\) −14.8979 −0.769322
\(376\) 3.72764 0.192239
\(377\) −5.11422 −0.263396
\(378\) 4.79272 0.246511
\(379\) 26.1492 1.34319 0.671597 0.740917i \(-0.265609\pi\)
0.671597 + 0.740917i \(0.265609\pi\)
\(380\) 19.7870 1.01505
\(381\) −24.7482 −1.26789
\(382\) 13.5586 0.693719
\(383\) −2.23479 −0.114192 −0.0570962 0.998369i \(-0.518184\pi\)
−0.0570962 + 0.998369i \(0.518184\pi\)
\(384\) 2.23215 0.113909
\(385\) −10.7636 −0.548563
\(386\) −9.87008 −0.502374
\(387\) −1.49292 −0.0758894
\(388\) 11.5509 0.586406
\(389\) −19.4143 −0.984346 −0.492173 0.870497i \(-0.663797\pi\)
−0.492173 + 0.870497i \(0.663797\pi\)
\(390\) 31.9709 1.61891
\(391\) 18.9039 0.956011
\(392\) 2.54722 0.128654
\(393\) −16.6683 −0.840803
\(394\) 13.9126 0.700904
\(395\) −2.70967 −0.136338
\(396\) −3.67416 −0.184633
\(397\) −7.46893 −0.374855 −0.187427 0.982278i \(-0.560015\pi\)
−0.187427 + 0.982278i \(0.560015\pi\)
\(398\) −3.99828 −0.200416
\(399\) −33.8631 −1.69528
\(400\) 2.57501 0.128751
\(401\) 24.0473 1.20086 0.600432 0.799676i \(-0.294995\pi\)
0.600432 + 0.799676i \(0.294995\pi\)
\(402\) 25.6074 1.27718
\(403\) −42.6307 −2.12359
\(404\) −0.816156 −0.0406053
\(405\) 30.3224 1.50673
\(406\) 2.07374 0.102918
\(407\) −8.96441 −0.444349
\(408\) 7.04678 0.348868
\(409\) −31.3735 −1.55132 −0.775660 0.631151i \(-0.782583\pi\)
−0.775660 + 0.631151i \(0.782583\pi\)
\(410\) 2.87385 0.141929
\(411\) −5.50126 −0.271357
\(412\) 11.6081 0.571891
\(413\) 17.2368 0.848166
\(414\) −11.8711 −0.583433
\(415\) 2.90081 0.142395
\(416\) 5.20404 0.255149
\(417\) 33.4198 1.63657
\(418\) −13.3241 −0.651703
\(419\) −0.844969 −0.0412794 −0.0206397 0.999787i \(-0.506570\pi\)
−0.0206397 + 0.999787i \(0.506570\pi\)
\(420\) −12.9637 −0.632565
\(421\) −0.0109361 −0.000532995 0 −0.000266497 1.00000i \(-0.500085\pi\)
−0.000266497 1.00000i \(0.500085\pi\)
\(422\) −2.75626 −0.134172
\(423\) −7.38997 −0.359313
\(424\) 5.58254 0.271112
\(425\) 8.12919 0.394324
\(426\) −0.103878 −0.00503290
\(427\) −30.4974 −1.47587
\(428\) 18.4241 0.890565
\(429\) −21.5285 −1.03940
\(430\) −2.07262 −0.0999507
\(431\) 11.7557 0.566250 0.283125 0.959083i \(-0.408629\pi\)
0.283125 + 0.959083i \(0.408629\pi\)
\(432\) 2.27126 0.109276
\(433\) 30.5511 1.46819 0.734095 0.679046i \(-0.237607\pi\)
0.734095 + 0.679046i \(0.237607\pi\)
\(434\) 17.2861 0.829761
\(435\) 6.03744 0.289473
\(436\) −7.36454 −0.352697
\(437\) −43.0498 −2.05935
\(438\) −7.67762 −0.366851
\(439\) 13.4317 0.641062 0.320531 0.947238i \(-0.396139\pi\)
0.320531 + 0.947238i \(0.396139\pi\)
\(440\) −5.10083 −0.243173
\(441\) −5.04980 −0.240467
\(442\) 16.4289 0.781444
\(443\) −21.7955 −1.03553 −0.517767 0.855521i \(-0.673237\pi\)
−0.517767 + 0.855521i \(0.673237\pi\)
\(444\) −10.7968 −0.512394
\(445\) 11.2640 0.533966
\(446\) −1.88498 −0.0892566
\(447\) 28.8098 1.36266
\(448\) −2.11016 −0.0996958
\(449\) 8.83987 0.417179 0.208590 0.978003i \(-0.433113\pi\)
0.208590 + 0.978003i \(0.433113\pi\)
\(450\) −5.10490 −0.240647
\(451\) −1.93518 −0.0911241
\(452\) 12.5555 0.590560
\(453\) −18.8816 −0.887135
\(454\) 1.41936 0.0666137
\(455\) −30.2238 −1.41691
\(456\) −16.0476 −0.751500
\(457\) −7.96665 −0.372664 −0.186332 0.982487i \(-0.559660\pi\)
−0.186332 + 0.982487i \(0.559660\pi\)
\(458\) 13.1725 0.615509
\(459\) 7.17026 0.334679
\(460\) −16.4807 −0.768415
\(461\) −4.27698 −0.199199 −0.0995993 0.995028i \(-0.531756\pi\)
−0.0995993 + 0.995028i \(0.531756\pi\)
\(462\) 8.72947 0.406132
\(463\) 2.49886 0.116132 0.0580660 0.998313i \(-0.481507\pi\)
0.0580660 + 0.998313i \(0.481507\pi\)
\(464\) 0.982740 0.0456226
\(465\) 50.3264 2.33383
\(466\) 3.47205 0.160840
\(467\) 39.3888 1.82270 0.911349 0.411634i \(-0.135042\pi\)
0.911349 + 0.411634i \(0.135042\pi\)
\(468\) −10.3169 −0.476899
\(469\) −24.2080 −1.11782
\(470\) −10.2595 −0.473235
\(471\) 20.9393 0.964833
\(472\) 8.16846 0.375984
\(473\) 1.39565 0.0641722
\(474\) 2.19760 0.100939
\(475\) −18.5126 −0.849417
\(476\) −6.66168 −0.305338
\(477\) −11.0673 −0.506735
\(478\) −4.50429 −0.206022
\(479\) −7.14596 −0.326507 −0.163254 0.986584i \(-0.552199\pi\)
−0.163254 + 0.986584i \(0.552199\pi\)
\(480\) −6.14348 −0.280410
\(481\) −25.1718 −1.14773
\(482\) 4.21948 0.192192
\(483\) 28.2047 1.28336
\(484\) −7.56522 −0.343874
\(485\) −31.7911 −1.44356
\(486\) −17.7783 −0.806438
\(487\) −23.0636 −1.04511 −0.522555 0.852605i \(-0.675021\pi\)
−0.522555 + 0.852605i \(0.675021\pi\)
\(488\) −14.4526 −0.654240
\(489\) 27.2875 1.23399
\(490\) −7.01064 −0.316708
\(491\) −25.1874 −1.13669 −0.568346 0.822790i \(-0.692417\pi\)
−0.568346 + 0.822790i \(0.692417\pi\)
\(492\) −2.33075 −0.105078
\(493\) 3.10247 0.139728
\(494\) −37.4136 −1.68332
\(495\) 10.1123 0.454513
\(496\) 8.19185 0.367825
\(497\) 0.0982010 0.00440492
\(498\) −2.35261 −0.105423
\(499\) 26.4449 1.18384 0.591918 0.805998i \(-0.298371\pi\)
0.591918 + 0.805998i \(0.298371\pi\)
\(500\) 6.67423 0.298481
\(501\) 25.1638 1.12423
\(502\) 3.31411 0.147916
\(503\) −12.8377 −0.572403 −0.286202 0.958169i \(-0.592393\pi\)
−0.286202 + 0.958169i \(0.592393\pi\)
\(504\) 4.18335 0.186341
\(505\) 2.24629 0.0999584
\(506\) 11.0977 0.493352
\(507\) −31.4332 −1.39600
\(508\) 11.0872 0.491914
\(509\) 4.34117 0.192419 0.0962095 0.995361i \(-0.469328\pi\)
0.0962095 + 0.995361i \(0.469328\pi\)
\(510\) −19.3947 −0.858811
\(511\) 7.25805 0.321077
\(512\) −1.00000 −0.0441942
\(513\) −16.3288 −0.720936
\(514\) 4.48515 0.197831
\(515\) −31.9487 −1.40783
\(516\) 1.68094 0.0739990
\(517\) 6.90850 0.303835
\(518\) 10.2068 0.448460
\(519\) 5.71079 0.250676
\(520\) −14.3230 −0.628103
\(521\) −22.7965 −0.998735 −0.499367 0.866390i \(-0.666434\pi\)
−0.499367 + 0.866390i \(0.666434\pi\)
\(522\) −1.94826 −0.0852731
\(523\) 8.37167 0.366068 0.183034 0.983107i \(-0.441408\pi\)
0.183034 + 0.983107i \(0.441408\pi\)
\(524\) 7.46737 0.326214
\(525\) 12.1288 0.529344
\(526\) −15.9494 −0.695428
\(527\) 25.8613 1.12654
\(528\) 4.13687 0.180034
\(529\) 12.8563 0.558972
\(530\) −15.3647 −0.667399
\(531\) −16.1938 −0.702751
\(532\) 15.1707 0.657731
\(533\) −5.43392 −0.235369
\(534\) −9.13534 −0.395325
\(535\) −50.7083 −2.19231
\(536\) −11.4721 −0.495520
\(537\) −0.667525 −0.0288058
\(538\) 19.0622 0.821829
\(539\) 4.72079 0.203339
\(540\) −6.25113 −0.269006
\(541\) 42.1599 1.81259 0.906297 0.422641i \(-0.138897\pi\)
0.906297 + 0.422641i \(0.138897\pi\)
\(542\) −21.9042 −0.940867
\(543\) −23.2898 −0.999459
\(544\) −3.15695 −0.135353
\(545\) 20.2692 0.868238
\(546\) 24.5120 1.04902
\(547\) −20.2458 −0.865647 −0.432824 0.901479i \(-0.642483\pi\)
−0.432824 + 0.901479i \(0.642483\pi\)
\(548\) 2.46456 0.105281
\(549\) 28.6520 1.22284
\(550\) 4.77231 0.203492
\(551\) −7.06525 −0.300990
\(552\) 13.3661 0.568900
\(553\) −2.07750 −0.0883443
\(554\) −22.5490 −0.958015
\(555\) 29.7158 1.26136
\(556\) −14.9720 −0.634955
\(557\) −29.0344 −1.23023 −0.615113 0.788439i \(-0.710890\pi\)
−0.615113 + 0.788439i \(0.710890\pi\)
\(558\) −16.2402 −0.687501
\(559\) 3.91895 0.165754
\(560\) 5.80774 0.245422
\(561\) 13.0599 0.551390
\(562\) 2.43550 0.102735
\(563\) 5.37912 0.226703 0.113352 0.993555i \(-0.463841\pi\)
0.113352 + 0.993555i \(0.463841\pi\)
\(564\) 8.32065 0.350362
\(565\) −34.5561 −1.45379
\(566\) −6.63059 −0.278705
\(567\) 23.2481 0.976329
\(568\) 0.0465372 0.00195266
\(569\) −0.890630 −0.0373371 −0.0186686 0.999826i \(-0.505943\pi\)
−0.0186686 + 0.999826i \(0.505943\pi\)
\(570\) 44.1675 1.84997
\(571\) 25.3434 1.06059 0.530294 0.847814i \(-0.322081\pi\)
0.530294 + 0.847814i \(0.322081\pi\)
\(572\) 9.64473 0.403266
\(573\) 30.2648 1.26433
\(574\) 2.20337 0.0919670
\(575\) 15.4192 0.643026
\(576\) 1.98248 0.0826032
\(577\) −28.0868 −1.16927 −0.584635 0.811297i \(-0.698762\pi\)
−0.584635 + 0.811297i \(0.698762\pi\)
\(578\) 7.03364 0.292561
\(579\) −22.0315 −0.915596
\(580\) −2.70477 −0.112309
\(581\) 2.22405 0.0922690
\(582\) 25.7832 1.06875
\(583\) 10.3462 0.428496
\(584\) 3.43957 0.142330
\(585\) 28.3949 1.17399
\(586\) 22.5352 0.930919
\(587\) −11.0790 −0.457281 −0.228640 0.973511i \(-0.573428\pi\)
−0.228640 + 0.973511i \(0.573428\pi\)
\(588\) 5.68576 0.234477
\(589\) −58.8939 −2.42668
\(590\) −22.4818 −0.925563
\(591\) 31.0549 1.27743
\(592\) 4.83696 0.198798
\(593\) 29.3332 1.20457 0.602284 0.798282i \(-0.294257\pi\)
0.602284 + 0.798282i \(0.294257\pi\)
\(594\) 4.20936 0.172712
\(595\) 18.3348 0.751653
\(596\) −12.9068 −0.528682
\(597\) −8.92475 −0.365266
\(598\) 31.1619 1.27430
\(599\) −10.5150 −0.429631 −0.214816 0.976655i \(-0.568915\pi\)
−0.214816 + 0.976655i \(0.568915\pi\)
\(600\) 5.74780 0.234653
\(601\) −7.21670 −0.294375 −0.147188 0.989109i \(-0.547022\pi\)
−0.147188 + 0.989109i \(0.547022\pi\)
\(602\) −1.58907 −0.0647658
\(603\) 22.7432 0.926175
\(604\) 8.45894 0.344190
\(605\) 20.8216 0.846517
\(606\) −1.82178 −0.0740048
\(607\) −14.5597 −0.590962 −0.295481 0.955349i \(-0.595480\pi\)
−0.295481 + 0.955349i \(0.595480\pi\)
\(608\) 7.18933 0.291566
\(609\) 4.62889 0.187572
\(610\) 39.7776 1.61055
\(611\) 19.3988 0.784792
\(612\) 6.25859 0.252989
\(613\) 34.4849 1.39283 0.696416 0.717639i \(-0.254777\pi\)
0.696416 + 0.717639i \(0.254777\pi\)
\(614\) 19.3513 0.780956
\(615\) 6.41485 0.258672
\(616\) −3.91079 −0.157570
\(617\) −24.7820 −0.997686 −0.498843 0.866692i \(-0.666242\pi\)
−0.498843 + 0.866692i \(0.666242\pi\)
\(618\) 25.9110 1.04229
\(619\) −20.9678 −0.842768 −0.421384 0.906882i \(-0.638456\pi\)
−0.421384 + 0.906882i \(0.638456\pi\)
\(620\) −22.5462 −0.905478
\(621\) 13.6003 0.545763
\(622\) 30.3012 1.21497
\(623\) 8.63610 0.345998
\(624\) 11.6162 0.465020
\(625\) −31.2444 −1.24977
\(626\) 3.85715 0.154163
\(627\) −29.7413 −1.18775
\(628\) −9.38080 −0.374335
\(629\) 15.2701 0.608857
\(630\) −11.5137 −0.458718
\(631\) 42.6444 1.69765 0.848823 0.528677i \(-0.177312\pi\)
0.848823 + 0.528677i \(0.177312\pi\)
\(632\) −0.984521 −0.0391622
\(633\) −6.15237 −0.244535
\(634\) 8.84967 0.351465
\(635\) −30.5150 −1.21095
\(636\) 12.4610 0.494112
\(637\) 13.2558 0.525215
\(638\) 1.82133 0.0721070
\(639\) −0.0922589 −0.00364971
\(640\) 2.75227 0.108793
\(641\) −27.7579 −1.09637 −0.548186 0.836357i \(-0.684681\pi\)
−0.548186 + 0.836357i \(0.684681\pi\)
\(642\) 41.1254 1.62309
\(643\) 10.5592 0.416415 0.208208 0.978085i \(-0.433237\pi\)
0.208208 + 0.978085i \(0.433237\pi\)
\(644\) −12.6357 −0.497916
\(645\) −4.62639 −0.182164
\(646\) 22.6964 0.892977
\(647\) 22.1479 0.870726 0.435363 0.900255i \(-0.356620\pi\)
0.435363 + 0.900255i \(0.356620\pi\)
\(648\) 11.0172 0.432797
\(649\) 15.1387 0.594247
\(650\) 13.4005 0.525610
\(651\) 38.5852 1.51227
\(652\) −12.2248 −0.478760
\(653\) 12.1877 0.476943 0.238472 0.971149i \(-0.423354\pi\)
0.238472 + 0.971149i \(0.423354\pi\)
\(654\) −16.4387 −0.642805
\(655\) −20.5523 −0.803043
\(656\) 1.04417 0.0407681
\(657\) −6.81887 −0.266029
\(658\) −7.86593 −0.306646
\(659\) 11.0872 0.431898 0.215949 0.976405i \(-0.430716\pi\)
0.215949 + 0.976405i \(0.430716\pi\)
\(660\) −11.3858 −0.443192
\(661\) −46.6454 −1.81429 −0.907147 0.420814i \(-0.861745\pi\)
−0.907147 + 0.420814i \(0.861745\pi\)
\(662\) 3.49270 0.135748
\(663\) 36.6718 1.42421
\(664\) 1.05397 0.0409019
\(665\) −41.7538 −1.61914
\(666\) −9.58917 −0.371573
\(667\) 5.88467 0.227855
\(668\) −11.2733 −0.436179
\(669\) −4.20756 −0.162674
\(670\) 31.5744 1.21983
\(671\) −26.7853 −1.03403
\(672\) −4.71019 −0.181700
\(673\) 40.8156 1.57333 0.786664 0.617382i \(-0.211807\pi\)
0.786664 + 0.617382i \(0.211807\pi\)
\(674\) 5.32473 0.205101
\(675\) 5.84852 0.225110
\(676\) 14.0821 0.541618
\(677\) 8.61164 0.330972 0.165486 0.986212i \(-0.447081\pi\)
0.165486 + 0.986212i \(0.447081\pi\)
\(678\) 28.0257 1.07632
\(679\) −24.3742 −0.935396
\(680\) 8.68880 0.333200
\(681\) 3.16821 0.121406
\(682\) 15.1821 0.581352
\(683\) 6.70095 0.256405 0.128202 0.991748i \(-0.459079\pi\)
0.128202 + 0.991748i \(0.459079\pi\)
\(684\) −14.2527 −0.544965
\(685\) −6.78315 −0.259171
\(686\) −20.1462 −0.769184
\(687\) 29.4029 1.12179
\(688\) −0.753058 −0.0287101
\(689\) 29.0518 1.10678
\(690\) −36.7873 −1.40047
\(691\) 17.3752 0.660984 0.330492 0.943809i \(-0.392785\pi\)
0.330492 + 0.943809i \(0.392785\pi\)
\(692\) −2.55843 −0.0972569
\(693\) 7.75306 0.294514
\(694\) −19.5881 −0.743553
\(695\) 41.2071 1.56307
\(696\) 2.19362 0.0831490
\(697\) 3.29640 0.124860
\(698\) 6.21124 0.235099
\(699\) 7.75012 0.293137
\(700\) −5.43369 −0.205374
\(701\) 19.8369 0.749230 0.374615 0.927181i \(-0.377775\pi\)
0.374615 + 0.927181i \(0.377775\pi\)
\(702\) 11.8197 0.446107
\(703\) −34.7745 −1.31155
\(704\) −1.85331 −0.0698494
\(705\) −22.9007 −0.862490
\(706\) 34.8009 1.30975
\(707\) 1.72222 0.0647708
\(708\) 18.2332 0.685246
\(709\) −24.5811 −0.923161 −0.461581 0.887098i \(-0.652717\pi\)
−0.461581 + 0.887098i \(0.652717\pi\)
\(710\) −0.128083 −0.00480687
\(711\) 1.95179 0.0731979
\(712\) 4.09263 0.153378
\(713\) 49.0529 1.83705
\(714\) −14.8699 −0.556490
\(715\) −26.5449 −0.992724
\(716\) 0.299051 0.0111760
\(717\) −10.0542 −0.375483
\(718\) 3.75205 0.140025
\(719\) −21.0684 −0.785718 −0.392859 0.919599i \(-0.628514\pi\)
−0.392859 + 0.919599i \(0.628514\pi\)
\(720\) −5.45632 −0.203345
\(721\) −24.4950 −0.912242
\(722\) −32.6865 −1.21647
\(723\) 9.41850 0.350278
\(724\) 10.4338 0.387769
\(725\) 2.53057 0.0939829
\(726\) −16.8867 −0.626724
\(727\) −21.9040 −0.812373 −0.406186 0.913790i \(-0.633142\pi\)
−0.406186 + 0.913790i \(0.633142\pi\)
\(728\) −10.9814 −0.406997
\(729\) −6.63204 −0.245631
\(730\) −9.46663 −0.350376
\(731\) −2.37737 −0.0879301
\(732\) −32.2604 −1.19238
\(733\) −16.9824 −0.627259 −0.313629 0.949545i \(-0.601545\pi\)
−0.313629 + 0.949545i \(0.601545\pi\)
\(734\) −9.86089 −0.363972
\(735\) −15.6488 −0.577213
\(736\) −5.98802 −0.220721
\(737\) −21.2614 −0.783175
\(738\) −2.07005 −0.0761995
\(739\) 43.7947 1.61101 0.805507 0.592586i \(-0.201893\pi\)
0.805507 + 0.592586i \(0.201893\pi\)
\(740\) −13.3126 −0.489382
\(741\) −83.5126 −3.06791
\(742\) −11.7801 −0.432460
\(743\) 48.9409 1.79547 0.897734 0.440538i \(-0.145212\pi\)
0.897734 + 0.440538i \(0.145212\pi\)
\(744\) 18.2854 0.670375
\(745\) 35.5230 1.30146
\(746\) −30.1102 −1.10241
\(747\) −2.08947 −0.0764497
\(748\) −5.85083 −0.213927
\(749\) −38.8779 −1.42057
\(750\) 14.8979 0.543993
\(751\) 22.3322 0.814915 0.407457 0.913224i \(-0.366415\pi\)
0.407457 + 0.913224i \(0.366415\pi\)
\(752\) −3.72764 −0.135933
\(753\) 7.39759 0.269583
\(754\) 5.11422 0.186249
\(755\) −23.2813 −0.847294
\(756\) −4.79272 −0.174310
\(757\) −30.2187 −1.09832 −0.549158 0.835719i \(-0.685051\pi\)
−0.549158 + 0.835719i \(0.685051\pi\)
\(758\) −26.1492 −0.949782
\(759\) 24.7716 0.899154
\(760\) −19.7870 −0.717750
\(761\) 25.8559 0.937277 0.468638 0.883390i \(-0.344745\pi\)
0.468638 + 0.883390i \(0.344745\pi\)
\(762\) 24.7482 0.896533
\(763\) 15.5404 0.562599
\(764\) −13.5586 −0.490533
\(765\) −17.2254 −0.622784
\(766\) 2.23479 0.0807462
\(767\) 42.5090 1.53491
\(768\) −2.23215 −0.0805456
\(769\) 6.47879 0.233631 0.116816 0.993154i \(-0.462731\pi\)
0.116816 + 0.993154i \(0.462731\pi\)
\(770\) 10.7636 0.387892
\(771\) 10.0115 0.360556
\(772\) 9.87008 0.355232
\(773\) −16.2589 −0.584793 −0.292397 0.956297i \(-0.594453\pi\)
−0.292397 + 0.956297i \(0.594453\pi\)
\(774\) 1.49292 0.0536619
\(775\) 21.0941 0.757722
\(776\) −11.5509 −0.414652
\(777\) 22.7830 0.817336
\(778\) 19.4143 0.696038
\(779\) −7.50690 −0.268963
\(780\) −31.9709 −1.14474
\(781\) 0.0862481 0.00308620
\(782\) −18.9039 −0.676002
\(783\) 2.23206 0.0797672
\(784\) −2.54722 −0.0909720
\(785\) 25.8185 0.921503
\(786\) 16.6683 0.594538
\(787\) 36.1243 1.28769 0.643847 0.765154i \(-0.277337\pi\)
0.643847 + 0.765154i \(0.277337\pi\)
\(788\) −13.9126 −0.495614
\(789\) −35.6015 −1.26745
\(790\) 2.70967 0.0964058
\(791\) −26.4941 −0.942022
\(792\) 3.67416 0.130555
\(793\) −75.2121 −2.67086
\(794\) 7.46893 0.265062
\(795\) −34.2962 −1.21636
\(796\) 3.99828 0.141715
\(797\) 35.2385 1.24821 0.624105 0.781340i \(-0.285464\pi\)
0.624105 + 0.781340i \(0.285464\pi\)
\(798\) 33.8631 1.19874
\(799\) −11.7680 −0.416322
\(800\) −2.57501 −0.0910404
\(801\) −8.11354 −0.286678
\(802\) −24.0473 −0.849139
\(803\) 6.37460 0.224955
\(804\) −25.6074 −0.903105
\(805\) 34.7769 1.22572
\(806\) 42.6307 1.50160
\(807\) 42.5496 1.49782
\(808\) 0.816156 0.0287123
\(809\) 9.45066 0.332268 0.166134 0.986103i \(-0.446872\pi\)
0.166134 + 0.986103i \(0.446872\pi\)
\(810\) −30.3224 −1.06542
\(811\) 54.2957 1.90658 0.953290 0.302057i \(-0.0976732\pi\)
0.953290 + 0.302057i \(0.0976732\pi\)
\(812\) −2.07374 −0.0727741
\(813\) −48.8934 −1.71477
\(814\) 8.96441 0.314202
\(815\) 33.6460 1.17857
\(816\) −7.04678 −0.246687
\(817\) 5.41398 0.189411
\(818\) 31.3735 1.09695
\(819\) 21.7703 0.760717
\(820\) −2.87385 −0.100359
\(821\) −24.8819 −0.868385 −0.434192 0.900820i \(-0.642966\pi\)
−0.434192 + 0.900820i \(0.642966\pi\)
\(822\) 5.50126 0.191879
\(823\) 31.5703 1.10047 0.550237 0.835009i \(-0.314538\pi\)
0.550237 + 0.835009i \(0.314538\pi\)
\(824\) −11.6081 −0.404388
\(825\) 10.6525 0.370872
\(826\) −17.2368 −0.599744
\(827\) 13.1657 0.457816 0.228908 0.973448i \(-0.426484\pi\)
0.228908 + 0.973448i \(0.426484\pi\)
\(828\) 11.8711 0.412550
\(829\) −45.8047 −1.59086 −0.795431 0.606044i \(-0.792756\pi\)
−0.795431 + 0.606044i \(0.792756\pi\)
\(830\) −2.90081 −0.100689
\(831\) −50.3327 −1.74602
\(832\) −5.20404 −0.180418
\(833\) −8.04144 −0.278620
\(834\) −33.4198 −1.15723
\(835\) 31.0273 1.07375
\(836\) 13.3241 0.460823
\(837\) 18.6058 0.643111
\(838\) 0.844969 0.0291890
\(839\) −45.8675 −1.58352 −0.791762 0.610830i \(-0.790836\pi\)
−0.791762 + 0.610830i \(0.790836\pi\)
\(840\) 12.9637 0.447291
\(841\) −28.0342 −0.966697
\(842\) 0.0109361 0.000376884 0
\(843\) 5.43639 0.187239
\(844\) 2.75626 0.0948743
\(845\) −38.7577 −1.33331
\(846\) 7.38997 0.254072
\(847\) 15.9638 0.548524
\(848\) −5.58254 −0.191705
\(849\) −14.8005 −0.507950
\(850\) −8.12919 −0.278829
\(851\) 28.9638 0.992867
\(852\) 0.103878 0.00355880
\(853\) 32.1839 1.10196 0.550978 0.834520i \(-0.314255\pi\)
0.550978 + 0.834520i \(0.314255\pi\)
\(854\) 30.4974 1.04360
\(855\) 39.2273 1.34155
\(856\) −18.4241 −0.629724
\(857\) −14.6138 −0.499199 −0.249600 0.968349i \(-0.580299\pi\)
−0.249600 + 0.968349i \(0.580299\pi\)
\(858\) 21.5285 0.734969
\(859\) −18.1537 −0.619397 −0.309699 0.950835i \(-0.600228\pi\)
−0.309699 + 0.950835i \(0.600228\pi\)
\(860\) 2.07262 0.0706758
\(861\) 4.91825 0.167614
\(862\) −11.7557 −0.400399
\(863\) −6.79309 −0.231239 −0.115620 0.993294i \(-0.536885\pi\)
−0.115620 + 0.993294i \(0.536885\pi\)
\(864\) −2.27126 −0.0772698
\(865\) 7.04150 0.239418
\(866\) −30.5511 −1.03817
\(867\) 15.7001 0.533204
\(868\) −17.2861 −0.586729
\(869\) −1.82463 −0.0618963
\(870\) −6.03744 −0.204689
\(871\) −59.7014 −2.02290
\(872\) 7.36454 0.249395
\(873\) 22.8993 0.775025
\(874\) 43.0498 1.45618
\(875\) −14.0837 −0.476116
\(876\) 7.67762 0.259403
\(877\) −28.5280 −0.963322 −0.481661 0.876358i \(-0.659966\pi\)
−0.481661 + 0.876358i \(0.659966\pi\)
\(878\) −13.4317 −0.453299
\(879\) 50.3018 1.69664
\(880\) 5.10083 0.171949
\(881\) 33.6221 1.13276 0.566379 0.824145i \(-0.308344\pi\)
0.566379 + 0.824145i \(0.308344\pi\)
\(882\) 5.04980 0.170036
\(883\) 37.7700 1.27106 0.635531 0.772075i \(-0.280781\pi\)
0.635531 + 0.772075i \(0.280781\pi\)
\(884\) −16.4289 −0.552565
\(885\) −50.1828 −1.68688
\(886\) 21.7955 0.732234
\(887\) 25.3961 0.852717 0.426359 0.904554i \(-0.359796\pi\)
0.426359 + 0.904554i \(0.359796\pi\)
\(888\) 10.7968 0.362317
\(889\) −23.3957 −0.784668
\(890\) −11.2640 −0.377571
\(891\) 20.4184 0.684041
\(892\) 1.88498 0.0631139
\(893\) 26.7993 0.896803
\(894\) −28.8098 −0.963544
\(895\) −0.823069 −0.0275122
\(896\) 2.11016 0.0704956
\(897\) 69.5579 2.32247
\(898\) −8.83987 −0.294990
\(899\) 8.05046 0.268498
\(900\) 5.10490 0.170163
\(901\) −17.6238 −0.587134
\(902\) 1.93518 0.0644344
\(903\) −3.54705 −0.118038
\(904\) −12.5555 −0.417589
\(905\) −28.7167 −0.954574
\(906\) 18.8816 0.627299
\(907\) 2.39378 0.0794843 0.0397421 0.999210i \(-0.487346\pi\)
0.0397421 + 0.999210i \(0.487346\pi\)
\(908\) −1.41936 −0.0471030
\(909\) −1.61801 −0.0536661
\(910\) 30.2238 1.00191
\(911\) −46.3233 −1.53476 −0.767380 0.641193i \(-0.778440\pi\)
−0.767380 + 0.641193i \(0.778440\pi\)
\(912\) 16.0476 0.531390
\(913\) 1.95334 0.0646460
\(914\) 7.96665 0.263513
\(915\) 88.7894 2.93529
\(916\) −13.1725 −0.435231
\(917\) −15.7574 −0.520354
\(918\) −7.17026 −0.236654
\(919\) −7.69608 −0.253870 −0.126935 0.991911i \(-0.540514\pi\)
−0.126935 + 0.991911i \(0.540514\pi\)
\(920\) 16.4807 0.543351
\(921\) 43.1950 1.42332
\(922\) 4.27698 0.140855
\(923\) 0.242182 0.00797150
\(924\) −8.72947 −0.287178
\(925\) 12.4552 0.409525
\(926\) −2.49886 −0.0821177
\(927\) 23.0128 0.755841
\(928\) −0.982740 −0.0322600
\(929\) 40.2897 1.32186 0.660931 0.750447i \(-0.270162\pi\)
0.660931 + 0.750447i \(0.270162\pi\)
\(930\) −50.3264 −1.65027
\(931\) 18.3128 0.600177
\(932\) −3.47205 −0.113731
\(933\) 67.6368 2.21433
\(934\) −39.3888 −1.28884
\(935\) 16.1031 0.526627
\(936\) 10.3169 0.337218
\(937\) 14.2365 0.465085 0.232543 0.972586i \(-0.425295\pi\)
0.232543 + 0.972586i \(0.425295\pi\)
\(938\) 24.2080 0.790420
\(939\) 8.60973 0.280968
\(940\) 10.2595 0.334628
\(941\) 52.9745 1.72692 0.863459 0.504418i \(-0.168293\pi\)
0.863459 + 0.504418i \(0.168293\pi\)
\(942\) −20.9393 −0.682240
\(943\) 6.25252 0.203610
\(944\) −8.16846 −0.265861
\(945\) 13.1909 0.429100
\(946\) −1.39565 −0.0453766
\(947\) 0.634653 0.0206234 0.0103117 0.999947i \(-0.496718\pi\)
0.0103117 + 0.999947i \(0.496718\pi\)
\(948\) −2.19760 −0.0713746
\(949\) 17.8997 0.581048
\(950\) 18.5126 0.600628
\(951\) 19.7538 0.640559
\(952\) 6.66168 0.215906
\(953\) 37.5100 1.21507 0.607535 0.794293i \(-0.292159\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(954\) 11.0673 0.358316
\(955\) 37.3170 1.20755
\(956\) 4.50429 0.145679
\(957\) 4.06547 0.131418
\(958\) 7.14596 0.230876
\(959\) −5.20062 −0.167937
\(960\) 6.14348 0.198280
\(961\) 36.1064 1.16472
\(962\) 25.1718 0.811570
\(963\) 36.5255 1.17702
\(964\) −4.21948 −0.135900
\(965\) −27.1652 −0.874477
\(966\) −28.2047 −0.907471
\(967\) 15.3024 0.492091 0.246046 0.969258i \(-0.420869\pi\)
0.246046 + 0.969258i \(0.420869\pi\)
\(968\) 7.56522 0.243156
\(969\) 50.6617 1.62749
\(970\) 31.7911 1.02075
\(971\) −29.6941 −0.952929 −0.476465 0.879194i \(-0.658082\pi\)
−0.476465 + 0.879194i \(0.658082\pi\)
\(972\) 17.7783 0.570238
\(973\) 31.5934 1.01284
\(974\) 23.0636 0.739005
\(975\) 29.9118 0.957945
\(976\) 14.4526 0.462617
\(977\) −48.4235 −1.54920 −0.774602 0.632449i \(-0.782050\pi\)
−0.774602 + 0.632449i \(0.782050\pi\)
\(978\) −27.2875 −0.872559
\(979\) 7.58492 0.242415
\(980\) 7.01064 0.223947
\(981\) −14.6000 −0.466143
\(982\) 25.1874 0.803763
\(983\) −52.5449 −1.67592 −0.837960 0.545731i \(-0.816252\pi\)
−0.837960 + 0.545731i \(0.816252\pi\)
\(984\) 2.33075 0.0743015
\(985\) 38.2912 1.22006
\(986\) −3.10247 −0.0988026
\(987\) −17.5579 −0.558875
\(988\) 37.4136 1.19029
\(989\) −4.50932 −0.143388
\(990\) −10.1123 −0.321389
\(991\) −32.7726 −1.04105 −0.520527 0.853845i \(-0.674265\pi\)
−0.520527 + 0.853845i \(0.674265\pi\)
\(992\) −8.19185 −0.260091
\(993\) 7.79622 0.247406
\(994\) −0.0982010 −0.00311475
\(995\) −11.0044 −0.348862
\(996\) 2.35261 0.0745454
\(997\) −38.4466 −1.21762 −0.608808 0.793318i \(-0.708352\pi\)
−0.608808 + 0.793318i \(0.708352\pi\)
\(998\) −26.4449 −0.837099
\(999\) 10.9860 0.347581
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 4006.2.a.g.1.7 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.7 40 1.1 even 1 trivial