Properties

Label 8001.2.a.z.1.28
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.28
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23843 q^{2} +3.01059 q^{4} +0.715002 q^{5} -1.00000 q^{7} +2.26213 q^{8} +O(q^{10})\) \(q+2.23843 q^{2} +3.01059 q^{4} +0.715002 q^{5} -1.00000 q^{7} +2.26213 q^{8} +1.60049 q^{10} +3.48743 q^{11} -3.88025 q^{13} -2.23843 q^{14} -0.957541 q^{16} -2.02085 q^{17} +1.32595 q^{19} +2.15258 q^{20} +7.80638 q^{22} -4.77444 q^{23} -4.48877 q^{25} -8.68568 q^{26} -3.01059 q^{28} -8.26004 q^{29} -4.73204 q^{31} -6.66766 q^{32} -4.52353 q^{34} -0.715002 q^{35} -10.4983 q^{37} +2.96805 q^{38} +1.61743 q^{40} +0.827289 q^{41} +2.69972 q^{43} +10.4992 q^{44} -10.6873 q^{46} -9.44282 q^{47} +1.00000 q^{49} -10.0478 q^{50} -11.6818 q^{52} +7.12745 q^{53} +2.49352 q^{55} -2.26213 q^{56} -18.4895 q^{58} -6.64479 q^{59} +9.79645 q^{61} -10.5924 q^{62} -13.0100 q^{64} -2.77439 q^{65} +10.4839 q^{67} -6.08393 q^{68} -1.60049 q^{70} +8.80029 q^{71} +6.52998 q^{73} -23.4997 q^{74} +3.99189 q^{76} -3.48743 q^{77} -1.16978 q^{79} -0.684644 q^{80} +1.85183 q^{82} -4.68620 q^{83} -1.44491 q^{85} +6.04313 q^{86} +7.88902 q^{88} -3.95955 q^{89} +3.88025 q^{91} -14.3739 q^{92} -21.1371 q^{94} +0.948058 q^{95} +6.83283 q^{97} +2.23843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23843 1.58281 0.791406 0.611291i \(-0.209349\pi\)
0.791406 + 0.611291i \(0.209349\pi\)
\(3\) 0 0
\(4\) 3.01059 1.50529
\(5\) 0.715002 0.319759 0.159879 0.987137i \(-0.448889\pi\)
0.159879 + 0.987137i \(0.448889\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.26213 0.799784
\(9\) 0 0
\(10\) 1.60049 0.506118
\(11\) 3.48743 1.05150 0.525750 0.850639i \(-0.323785\pi\)
0.525750 + 0.850639i \(0.323785\pi\)
\(12\) 0 0
\(13\) −3.88025 −1.07619 −0.538093 0.842885i \(-0.680855\pi\)
−0.538093 + 0.842885i \(0.680855\pi\)
\(14\) −2.23843 −0.598247
\(15\) 0 0
\(16\) −0.957541 −0.239385
\(17\) −2.02085 −0.490127 −0.245064 0.969507i \(-0.578809\pi\)
−0.245064 + 0.969507i \(0.578809\pi\)
\(18\) 0 0
\(19\) 1.32595 0.304194 0.152097 0.988366i \(-0.451397\pi\)
0.152097 + 0.988366i \(0.451397\pi\)
\(20\) 2.15258 0.481331
\(21\) 0 0
\(22\) 7.80638 1.66433
\(23\) −4.77444 −0.995540 −0.497770 0.867309i \(-0.665848\pi\)
−0.497770 + 0.867309i \(0.665848\pi\)
\(24\) 0 0
\(25\) −4.48877 −0.897754
\(26\) −8.68568 −1.70340
\(27\) 0 0
\(28\) −3.01059 −0.568947
\(29\) −8.26004 −1.53385 −0.766925 0.641737i \(-0.778214\pi\)
−0.766925 + 0.641737i \(0.778214\pi\)
\(30\) 0 0
\(31\) −4.73204 −0.849900 −0.424950 0.905217i \(-0.639708\pi\)
−0.424950 + 0.905217i \(0.639708\pi\)
\(32\) −6.66766 −1.17869
\(33\) 0 0
\(34\) −4.52353 −0.775779
\(35\) −0.715002 −0.120857
\(36\) 0 0
\(37\) −10.4983 −1.72591 −0.862954 0.505283i \(-0.831388\pi\)
−0.862954 + 0.505283i \(0.831388\pi\)
\(38\) 2.96805 0.481482
\(39\) 0 0
\(40\) 1.61743 0.255738
\(41\) 0.827289 0.129201 0.0646004 0.997911i \(-0.479423\pi\)
0.0646004 + 0.997911i \(0.479423\pi\)
\(42\) 0 0
\(43\) 2.69972 0.411703 0.205851 0.978583i \(-0.434004\pi\)
0.205851 + 0.978583i \(0.434004\pi\)
\(44\) 10.4992 1.58281
\(45\) 0 0
\(46\) −10.6873 −1.57575
\(47\) −9.44282 −1.37738 −0.688688 0.725058i \(-0.741813\pi\)
−0.688688 + 0.725058i \(0.741813\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.0478 −1.42098
\(51\) 0 0
\(52\) −11.6818 −1.61998
\(53\) 7.12745 0.979030 0.489515 0.871995i \(-0.337174\pi\)
0.489515 + 0.871995i \(0.337174\pi\)
\(54\) 0 0
\(55\) 2.49352 0.336226
\(56\) −2.26213 −0.302290
\(57\) 0 0
\(58\) −18.4895 −2.42780
\(59\) −6.64479 −0.865078 −0.432539 0.901615i \(-0.642382\pi\)
−0.432539 + 0.901615i \(0.642382\pi\)
\(60\) 0 0
\(61\) 9.79645 1.25431 0.627153 0.778896i \(-0.284220\pi\)
0.627153 + 0.778896i \(0.284220\pi\)
\(62\) −10.5924 −1.34523
\(63\) 0 0
\(64\) −13.0100 −1.62625
\(65\) −2.77439 −0.344120
\(66\) 0 0
\(67\) 10.4839 1.28081 0.640407 0.768036i \(-0.278766\pi\)
0.640407 + 0.768036i \(0.278766\pi\)
\(68\) −6.08393 −0.737785
\(69\) 0 0
\(70\) −1.60049 −0.191295
\(71\) 8.80029 1.04440 0.522201 0.852823i \(-0.325111\pi\)
0.522201 + 0.852823i \(0.325111\pi\)
\(72\) 0 0
\(73\) 6.52998 0.764276 0.382138 0.924105i \(-0.375188\pi\)
0.382138 + 0.924105i \(0.375188\pi\)
\(74\) −23.4997 −2.73179
\(75\) 0 0
\(76\) 3.99189 0.457901
\(77\) −3.48743 −0.397429
\(78\) 0 0
\(79\) −1.16978 −0.131611 −0.0658053 0.997832i \(-0.520962\pi\)
−0.0658053 + 0.997832i \(0.520962\pi\)
\(80\) −0.684644 −0.0765455
\(81\) 0 0
\(82\) 1.85183 0.204501
\(83\) −4.68620 −0.514377 −0.257189 0.966361i \(-0.582796\pi\)
−0.257189 + 0.966361i \(0.582796\pi\)
\(84\) 0 0
\(85\) −1.44491 −0.156722
\(86\) 6.04313 0.651648
\(87\) 0 0
\(88\) 7.88902 0.840973
\(89\) −3.95955 −0.419712 −0.209856 0.977732i \(-0.567299\pi\)
−0.209856 + 0.977732i \(0.567299\pi\)
\(90\) 0 0
\(91\) 3.88025 0.406760
\(92\) −14.3739 −1.49858
\(93\) 0 0
\(94\) −21.1371 −2.18013
\(95\) 0.948058 0.0972687
\(96\) 0 0
\(97\) 6.83283 0.693768 0.346884 0.937908i \(-0.387240\pi\)
0.346884 + 0.937908i \(0.387240\pi\)
\(98\) 2.23843 0.226116
\(99\) 0 0
\(100\) −13.5138 −1.35138
\(101\) −14.4018 −1.43303 −0.716517 0.697569i \(-0.754265\pi\)
−0.716517 + 0.697569i \(0.754265\pi\)
\(102\) 0 0
\(103\) 4.78657 0.471635 0.235817 0.971797i \(-0.424223\pi\)
0.235817 + 0.971797i \(0.424223\pi\)
\(104\) −8.77763 −0.860717
\(105\) 0 0
\(106\) 15.9543 1.54962
\(107\) 8.11171 0.784189 0.392094 0.919925i \(-0.371751\pi\)
0.392094 + 0.919925i \(0.371751\pi\)
\(108\) 0 0
\(109\) 0.982270 0.0940844 0.0470422 0.998893i \(-0.485020\pi\)
0.0470422 + 0.998893i \(0.485020\pi\)
\(110\) 5.58158 0.532183
\(111\) 0 0
\(112\) 0.957541 0.0904791
\(113\) 5.77508 0.543273 0.271637 0.962400i \(-0.412435\pi\)
0.271637 + 0.962400i \(0.412435\pi\)
\(114\) 0 0
\(115\) −3.41374 −0.318333
\(116\) −24.8676 −2.30889
\(117\) 0 0
\(118\) −14.8739 −1.36926
\(119\) 2.02085 0.185251
\(120\) 0 0
\(121\) 1.16215 0.105650
\(122\) 21.9287 1.98533
\(123\) 0 0
\(124\) −14.2462 −1.27935
\(125\) −6.78450 −0.606824
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −15.7868 −1.39537
\(129\) 0 0
\(130\) −6.21028 −0.544678
\(131\) −1.06804 −0.0933152 −0.0466576 0.998911i \(-0.514857\pi\)
−0.0466576 + 0.998911i \(0.514857\pi\)
\(132\) 0 0
\(133\) −1.32595 −0.114975
\(134\) 23.4675 2.02729
\(135\) 0 0
\(136\) −4.57142 −0.391996
\(137\) 12.8058 1.09408 0.547038 0.837108i \(-0.315755\pi\)
0.547038 + 0.837108i \(0.315755\pi\)
\(138\) 0 0
\(139\) −4.13929 −0.351090 −0.175545 0.984471i \(-0.556169\pi\)
−0.175545 + 0.984471i \(0.556169\pi\)
\(140\) −2.15258 −0.181926
\(141\) 0 0
\(142\) 19.6989 1.65309
\(143\) −13.5321 −1.13161
\(144\) 0 0
\(145\) −5.90595 −0.490462
\(146\) 14.6169 1.20971
\(147\) 0 0
\(148\) −31.6060 −2.59800
\(149\) 13.5650 1.11129 0.555646 0.831419i \(-0.312471\pi\)
0.555646 + 0.831419i \(0.312471\pi\)
\(150\) 0 0
\(151\) 3.40613 0.277187 0.138594 0.990349i \(-0.455742\pi\)
0.138594 + 0.990349i \(0.455742\pi\)
\(152\) 2.99948 0.243290
\(153\) 0 0
\(154\) −7.80638 −0.629056
\(155\) −3.38342 −0.271763
\(156\) 0 0
\(157\) −20.5876 −1.64307 −0.821536 0.570157i \(-0.806882\pi\)
−0.821536 + 0.570157i \(0.806882\pi\)
\(158\) −2.61848 −0.208315
\(159\) 0 0
\(160\) −4.76739 −0.376895
\(161\) 4.77444 0.376279
\(162\) 0 0
\(163\) 14.3948 1.12749 0.563744 0.825950i \(-0.309360\pi\)
0.563744 + 0.825950i \(0.309360\pi\)
\(164\) 2.49063 0.194485
\(165\) 0 0
\(166\) −10.4898 −0.814163
\(167\) 14.7266 1.13958 0.569788 0.821792i \(-0.307025\pi\)
0.569788 + 0.821792i \(0.307025\pi\)
\(168\) 0 0
\(169\) 2.05631 0.158178
\(170\) −3.23433 −0.248062
\(171\) 0 0
\(172\) 8.12773 0.619733
\(173\) −12.1096 −0.920678 −0.460339 0.887743i \(-0.652272\pi\)
−0.460339 + 0.887743i \(0.652272\pi\)
\(174\) 0 0
\(175\) 4.48877 0.339319
\(176\) −3.33935 −0.251713
\(177\) 0 0
\(178\) −8.86320 −0.664325
\(179\) 10.2816 0.768484 0.384242 0.923232i \(-0.374463\pi\)
0.384242 + 0.923232i \(0.374463\pi\)
\(180\) 0 0
\(181\) −6.97609 −0.518529 −0.259264 0.965806i \(-0.583480\pi\)
−0.259264 + 0.965806i \(0.583480\pi\)
\(182\) 8.68568 0.643825
\(183\) 0 0
\(184\) −10.8004 −0.796217
\(185\) −7.50630 −0.551874
\(186\) 0 0
\(187\) −7.04755 −0.515368
\(188\) −28.4284 −2.07336
\(189\) 0 0
\(190\) 2.12217 0.153958
\(191\) −16.8925 −1.22230 −0.611150 0.791515i \(-0.709293\pi\)
−0.611150 + 0.791515i \(0.709293\pi\)
\(192\) 0 0
\(193\) 5.01576 0.361043 0.180521 0.983571i \(-0.442222\pi\)
0.180521 + 0.983571i \(0.442222\pi\)
\(194\) 15.2948 1.09810
\(195\) 0 0
\(196\) 3.01059 0.215042
\(197\) 8.79497 0.626616 0.313308 0.949652i \(-0.398563\pi\)
0.313308 + 0.949652i \(0.398563\pi\)
\(198\) 0 0
\(199\) −21.9618 −1.55683 −0.778417 0.627747i \(-0.783977\pi\)
−0.778417 + 0.627747i \(0.783977\pi\)
\(200\) −10.1542 −0.718010
\(201\) 0 0
\(202\) −32.2375 −2.26822
\(203\) 8.26004 0.579741
\(204\) 0 0
\(205\) 0.591514 0.0413131
\(206\) 10.7144 0.746509
\(207\) 0 0
\(208\) 3.71549 0.257623
\(209\) 4.62416 0.319860
\(210\) 0 0
\(211\) 10.3601 0.713221 0.356610 0.934253i \(-0.383932\pi\)
0.356610 + 0.934253i \(0.383932\pi\)
\(212\) 21.4578 1.47373
\(213\) 0 0
\(214\) 18.1575 1.24122
\(215\) 1.93030 0.131646
\(216\) 0 0
\(217\) 4.73204 0.321232
\(218\) 2.19875 0.148918
\(219\) 0 0
\(220\) 7.50696 0.506119
\(221\) 7.84138 0.527468
\(222\) 0 0
\(223\) −6.35678 −0.425681 −0.212841 0.977087i \(-0.568272\pi\)
−0.212841 + 0.977087i \(0.568272\pi\)
\(224\) 6.66766 0.445501
\(225\) 0 0
\(226\) 12.9271 0.859900
\(227\) −7.68768 −0.510249 −0.255124 0.966908i \(-0.582116\pi\)
−0.255124 + 0.966908i \(0.582116\pi\)
\(228\) 0 0
\(229\) −10.5663 −0.698244 −0.349122 0.937077i \(-0.613520\pi\)
−0.349122 + 0.937077i \(0.613520\pi\)
\(230\) −7.64142 −0.503861
\(231\) 0 0
\(232\) −18.6853 −1.22675
\(233\) −25.1211 −1.64574 −0.822869 0.568231i \(-0.807628\pi\)
−0.822869 + 0.568231i \(0.807628\pi\)
\(234\) 0 0
\(235\) −6.75164 −0.440428
\(236\) −20.0047 −1.30220
\(237\) 0 0
\(238\) 4.52353 0.293217
\(239\) 10.8412 0.701257 0.350629 0.936515i \(-0.385968\pi\)
0.350629 + 0.936515i \(0.385968\pi\)
\(240\) 0 0
\(241\) −17.8749 −1.15142 −0.575712 0.817652i \(-0.695275\pi\)
−0.575712 + 0.817652i \(0.695275\pi\)
\(242\) 2.60141 0.167225
\(243\) 0 0
\(244\) 29.4931 1.88810
\(245\) 0.715002 0.0456798
\(246\) 0 0
\(247\) −5.14502 −0.327369
\(248\) −10.7045 −0.679737
\(249\) 0 0
\(250\) −15.1866 −0.960488
\(251\) −4.79178 −0.302455 −0.151227 0.988499i \(-0.548323\pi\)
−0.151227 + 0.988499i \(0.548323\pi\)
\(252\) 0 0
\(253\) −16.6505 −1.04681
\(254\) −2.23843 −0.140452
\(255\) 0 0
\(256\) −9.31760 −0.582350
\(257\) −6.08661 −0.379672 −0.189836 0.981816i \(-0.560796\pi\)
−0.189836 + 0.981816i \(0.560796\pi\)
\(258\) 0 0
\(259\) 10.4983 0.652332
\(260\) −8.35253 −0.518002
\(261\) 0 0
\(262\) −2.39074 −0.147700
\(263\) −25.1166 −1.54876 −0.774378 0.632723i \(-0.781937\pi\)
−0.774378 + 0.632723i \(0.781937\pi\)
\(264\) 0 0
\(265\) 5.09614 0.313053
\(266\) −2.96805 −0.181983
\(267\) 0 0
\(268\) 31.5627 1.92800
\(269\) −12.2844 −0.748995 −0.374498 0.927228i \(-0.622185\pi\)
−0.374498 + 0.927228i \(0.622185\pi\)
\(270\) 0 0
\(271\) −0.354378 −0.0215269 −0.0107635 0.999942i \(-0.503426\pi\)
−0.0107635 + 0.999942i \(0.503426\pi\)
\(272\) 1.93504 0.117329
\(273\) 0 0
\(274\) 28.6650 1.73172
\(275\) −15.6543 −0.943988
\(276\) 0 0
\(277\) −18.8044 −1.12984 −0.564922 0.825144i \(-0.691094\pi\)
−0.564922 + 0.825144i \(0.691094\pi\)
\(278\) −9.26552 −0.555709
\(279\) 0 0
\(280\) −1.61743 −0.0966599
\(281\) −22.7302 −1.35597 −0.677985 0.735076i \(-0.737146\pi\)
−0.677985 + 0.735076i \(0.737146\pi\)
\(282\) 0 0
\(283\) −9.78820 −0.581848 −0.290924 0.956746i \(-0.593963\pi\)
−0.290924 + 0.956746i \(0.593963\pi\)
\(284\) 26.4940 1.57213
\(285\) 0 0
\(286\) −30.2907 −1.79112
\(287\) −0.827289 −0.0488333
\(288\) 0 0
\(289\) −12.9162 −0.759775
\(290\) −13.2201 −0.776309
\(291\) 0 0
\(292\) 19.6591 1.15046
\(293\) 6.16187 0.359981 0.179990 0.983668i \(-0.442393\pi\)
0.179990 + 0.983668i \(0.442393\pi\)
\(294\) 0 0
\(295\) −4.75104 −0.276616
\(296\) −23.7485 −1.38035
\(297\) 0 0
\(298\) 30.3645 1.75897
\(299\) 18.5260 1.07139
\(300\) 0 0
\(301\) −2.69972 −0.155609
\(302\) 7.62440 0.438735
\(303\) 0 0
\(304\) −1.26965 −0.0728195
\(305\) 7.00449 0.401076
\(306\) 0 0
\(307\) −13.7893 −0.786999 −0.393500 0.919325i \(-0.628736\pi\)
−0.393500 + 0.919325i \(0.628736\pi\)
\(308\) −10.4992 −0.598248
\(309\) 0 0
\(310\) −7.57357 −0.430150
\(311\) 7.99152 0.453158 0.226579 0.973993i \(-0.427246\pi\)
0.226579 + 0.973993i \(0.427246\pi\)
\(312\) 0 0
\(313\) 8.19550 0.463237 0.231618 0.972807i \(-0.425598\pi\)
0.231618 + 0.972807i \(0.425598\pi\)
\(314\) −46.0840 −2.60067
\(315\) 0 0
\(316\) −3.52173 −0.198113
\(317\) −9.71596 −0.545703 −0.272851 0.962056i \(-0.587967\pi\)
−0.272851 + 0.962056i \(0.587967\pi\)
\(318\) 0 0
\(319\) −28.8063 −1.61284
\(320\) −9.30220 −0.520009
\(321\) 0 0
\(322\) 10.6873 0.595578
\(323\) −2.67954 −0.149094
\(324\) 0 0
\(325\) 17.4175 0.966151
\(326\) 32.2218 1.78460
\(327\) 0 0
\(328\) 1.87144 0.103333
\(329\) 9.44282 0.520599
\(330\) 0 0
\(331\) 26.7460 1.47009 0.735047 0.678017i \(-0.237160\pi\)
0.735047 + 0.678017i \(0.237160\pi\)
\(332\) −14.1082 −0.774289
\(333\) 0 0
\(334\) 32.9644 1.80373
\(335\) 7.49602 0.409551
\(336\) 0 0
\(337\) 7.80788 0.425322 0.212661 0.977126i \(-0.431787\pi\)
0.212661 + 0.977126i \(0.431787\pi\)
\(338\) 4.60292 0.250366
\(339\) 0 0
\(340\) −4.35003 −0.235913
\(341\) −16.5027 −0.893669
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.10711 0.329273
\(345\) 0 0
\(346\) −27.1066 −1.45726
\(347\) 16.8929 0.906859 0.453429 0.891292i \(-0.350200\pi\)
0.453429 + 0.891292i \(0.350200\pi\)
\(348\) 0 0
\(349\) −14.8133 −0.792937 −0.396468 0.918048i \(-0.629764\pi\)
−0.396468 + 0.918048i \(0.629764\pi\)
\(350\) 10.0478 0.537078
\(351\) 0 0
\(352\) −23.2530 −1.23939
\(353\) 32.0204 1.70427 0.852137 0.523319i \(-0.175306\pi\)
0.852137 + 0.523319i \(0.175306\pi\)
\(354\) 0 0
\(355\) 6.29223 0.333957
\(356\) −11.9206 −0.631790
\(357\) 0 0
\(358\) 23.0147 1.21637
\(359\) 7.14129 0.376903 0.188451 0.982083i \(-0.439653\pi\)
0.188451 + 0.982083i \(0.439653\pi\)
\(360\) 0 0
\(361\) −17.2419 −0.907466
\(362\) −15.6155 −0.820733
\(363\) 0 0
\(364\) 11.6818 0.612294
\(365\) 4.66895 0.244384
\(366\) 0 0
\(367\) 19.0108 0.992356 0.496178 0.868221i \(-0.334736\pi\)
0.496178 + 0.868221i \(0.334736\pi\)
\(368\) 4.57172 0.238317
\(369\) 0 0
\(370\) −16.8024 −0.873513
\(371\) −7.12745 −0.370039
\(372\) 0 0
\(373\) 32.6219 1.68910 0.844550 0.535476i \(-0.179868\pi\)
0.844550 + 0.535476i \(0.179868\pi\)
\(374\) −15.7755 −0.815731
\(375\) 0 0
\(376\) −21.3609 −1.10160
\(377\) 32.0510 1.65071
\(378\) 0 0
\(379\) 9.55764 0.490943 0.245471 0.969404i \(-0.421057\pi\)
0.245471 + 0.969404i \(0.421057\pi\)
\(380\) 2.85421 0.146418
\(381\) 0 0
\(382\) −37.8128 −1.93467
\(383\) 19.9990 1.02190 0.510950 0.859610i \(-0.329294\pi\)
0.510950 + 0.859610i \(0.329294\pi\)
\(384\) 0 0
\(385\) −2.49352 −0.127082
\(386\) 11.2275 0.571462
\(387\) 0 0
\(388\) 20.5708 1.04432
\(389\) 4.07409 0.206565 0.103282 0.994652i \(-0.467066\pi\)
0.103282 + 0.994652i \(0.467066\pi\)
\(390\) 0 0
\(391\) 9.64841 0.487941
\(392\) 2.26213 0.114255
\(393\) 0 0
\(394\) 19.6870 0.991814
\(395\) −0.836396 −0.0420837
\(396\) 0 0
\(397\) 21.7833 1.09327 0.546636 0.837370i \(-0.315908\pi\)
0.546636 + 0.837370i \(0.315908\pi\)
\(398\) −49.1602 −2.46418
\(399\) 0 0
\(400\) 4.29818 0.214909
\(401\) −17.3330 −0.865568 −0.432784 0.901498i \(-0.642469\pi\)
−0.432784 + 0.901498i \(0.642469\pi\)
\(402\) 0 0
\(403\) 18.3615 0.914651
\(404\) −43.3579 −2.15714
\(405\) 0 0
\(406\) 18.4895 0.917621
\(407\) −36.6120 −1.81479
\(408\) 0 0
\(409\) −12.4744 −0.616821 −0.308410 0.951253i \(-0.599797\pi\)
−0.308410 + 0.951253i \(0.599797\pi\)
\(410\) 1.32406 0.0653909
\(411\) 0 0
\(412\) 14.4104 0.709948
\(413\) 6.64479 0.326969
\(414\) 0 0
\(415\) −3.35065 −0.164477
\(416\) 25.8721 1.26849
\(417\) 0 0
\(418\) 10.3509 0.506278
\(419\) 10.7981 0.527524 0.263762 0.964588i \(-0.415037\pi\)
0.263762 + 0.964588i \(0.415037\pi\)
\(420\) 0 0
\(421\) −7.59997 −0.370400 −0.185200 0.982701i \(-0.559293\pi\)
−0.185200 + 0.982701i \(0.559293\pi\)
\(422\) 23.1905 1.12889
\(423\) 0 0
\(424\) 16.1232 0.783013
\(425\) 9.07111 0.440014
\(426\) 0 0
\(427\) −9.79645 −0.474083
\(428\) 24.4210 1.18043
\(429\) 0 0
\(430\) 4.32086 0.208370
\(431\) 34.8208 1.67726 0.838630 0.544701i \(-0.183357\pi\)
0.838630 + 0.544701i \(0.183357\pi\)
\(432\) 0 0
\(433\) −33.9006 −1.62916 −0.814578 0.580054i \(-0.803032\pi\)
−0.814578 + 0.580054i \(0.803032\pi\)
\(434\) 10.5924 0.508450
\(435\) 0 0
\(436\) 2.95721 0.141625
\(437\) −6.33067 −0.302837
\(438\) 0 0
\(439\) 6.04032 0.288289 0.144144 0.989557i \(-0.453957\pi\)
0.144144 + 0.989557i \(0.453957\pi\)
\(440\) 5.64067 0.268908
\(441\) 0 0
\(442\) 17.5524 0.834883
\(443\) 32.0832 1.52432 0.762159 0.647390i \(-0.224140\pi\)
0.762159 + 0.647390i \(0.224140\pi\)
\(444\) 0 0
\(445\) −2.83109 −0.134207
\(446\) −14.2292 −0.673774
\(447\) 0 0
\(448\) 13.0100 0.614666
\(449\) 7.01644 0.331126 0.165563 0.986199i \(-0.447056\pi\)
0.165563 + 0.986199i \(0.447056\pi\)
\(450\) 0 0
\(451\) 2.88511 0.135855
\(452\) 17.3864 0.817786
\(453\) 0 0
\(454\) −17.2084 −0.807628
\(455\) 2.77439 0.130065
\(456\) 0 0
\(457\) 16.8482 0.788126 0.394063 0.919083i \(-0.371069\pi\)
0.394063 + 0.919083i \(0.371069\pi\)
\(458\) −23.6521 −1.10519
\(459\) 0 0
\(460\) −10.2774 −0.479184
\(461\) 17.7603 0.827181 0.413591 0.910463i \(-0.364274\pi\)
0.413591 + 0.910463i \(0.364274\pi\)
\(462\) 0 0
\(463\) 10.1828 0.473236 0.236618 0.971603i \(-0.423961\pi\)
0.236618 + 0.971603i \(0.423961\pi\)
\(464\) 7.90932 0.367181
\(465\) 0 0
\(466\) −56.2319 −2.60489
\(467\) 1.65415 0.0765449 0.0382724 0.999267i \(-0.487815\pi\)
0.0382724 + 0.999267i \(0.487815\pi\)
\(468\) 0 0
\(469\) −10.4839 −0.484102
\(470\) −15.1131 −0.697115
\(471\) 0 0
\(472\) −15.0314 −0.691876
\(473\) 9.41506 0.432905
\(474\) 0 0
\(475\) −5.95189 −0.273091
\(476\) 6.08393 0.278857
\(477\) 0 0
\(478\) 24.2672 1.10996
\(479\) 12.7961 0.584670 0.292335 0.956316i \(-0.405568\pi\)
0.292335 + 0.956316i \(0.405568\pi\)
\(480\) 0 0
\(481\) 40.7359 1.85740
\(482\) −40.0118 −1.82249
\(483\) 0 0
\(484\) 3.49877 0.159035
\(485\) 4.88549 0.221839
\(486\) 0 0
\(487\) 5.81280 0.263403 0.131702 0.991289i \(-0.457956\pi\)
0.131702 + 0.991289i \(0.457956\pi\)
\(488\) 22.1609 1.00317
\(489\) 0 0
\(490\) 1.60049 0.0723026
\(491\) −5.28049 −0.238305 −0.119153 0.992876i \(-0.538018\pi\)
−0.119153 + 0.992876i \(0.538018\pi\)
\(492\) 0 0
\(493\) 16.6923 0.751782
\(494\) −11.5168 −0.518164
\(495\) 0 0
\(496\) 4.53113 0.203454
\(497\) −8.80029 −0.394747
\(498\) 0 0
\(499\) −20.0549 −0.897778 −0.448889 0.893587i \(-0.648180\pi\)
−0.448889 + 0.893587i \(0.648180\pi\)
\(500\) −20.4253 −0.913448
\(501\) 0 0
\(502\) −10.7261 −0.478729
\(503\) −14.1949 −0.632920 −0.316460 0.948606i \(-0.602494\pi\)
−0.316460 + 0.948606i \(0.602494\pi\)
\(504\) 0 0
\(505\) −10.2973 −0.458225
\(506\) −37.2711 −1.65690
\(507\) 0 0
\(508\) −3.01059 −0.133573
\(509\) −34.0499 −1.50923 −0.754617 0.656166i \(-0.772177\pi\)
−0.754617 + 0.656166i \(0.772177\pi\)
\(510\) 0 0
\(511\) −6.52998 −0.288869
\(512\) 10.7167 0.473617
\(513\) 0 0
\(514\) −13.6245 −0.600950
\(515\) 3.42241 0.150809
\(516\) 0 0
\(517\) −32.9311 −1.44831
\(518\) 23.4997 1.03252
\(519\) 0 0
\(520\) −6.27603 −0.275222
\(521\) −4.82296 −0.211298 −0.105649 0.994403i \(-0.533692\pi\)
−0.105649 + 0.994403i \(0.533692\pi\)
\(522\) 0 0
\(523\) −2.29591 −0.100393 −0.0501965 0.998739i \(-0.515985\pi\)
−0.0501965 + 0.998739i \(0.515985\pi\)
\(524\) −3.21543 −0.140467
\(525\) 0 0
\(526\) −56.2219 −2.45139
\(527\) 9.56273 0.416559
\(528\) 0 0
\(529\) −0.204717 −0.00890074
\(530\) 11.4074 0.495505
\(531\) 0 0
\(532\) −3.99189 −0.173070
\(533\) −3.21009 −0.139044
\(534\) 0 0
\(535\) 5.79989 0.250751
\(536\) 23.7160 1.02437
\(537\) 0 0
\(538\) −27.4979 −1.18552
\(539\) 3.48743 0.150214
\(540\) 0 0
\(541\) −13.1848 −0.566857 −0.283429 0.958993i \(-0.591472\pi\)
−0.283429 + 0.958993i \(0.591472\pi\)
\(542\) −0.793252 −0.0340731
\(543\) 0 0
\(544\) 13.4743 0.577706
\(545\) 0.702325 0.0300843
\(546\) 0 0
\(547\) −0.144810 −0.00619164 −0.00309582 0.999995i \(-0.500985\pi\)
−0.00309582 + 0.999995i \(0.500985\pi\)
\(548\) 38.5531 1.64691
\(549\) 0 0
\(550\) −35.0410 −1.49416
\(551\) −10.9524 −0.466588
\(552\) 0 0
\(553\) 1.16978 0.0497441
\(554\) −42.0923 −1.78833
\(555\) 0 0
\(556\) −12.4617 −0.528493
\(557\) 6.35824 0.269407 0.134704 0.990886i \(-0.456992\pi\)
0.134704 + 0.990886i \(0.456992\pi\)
\(558\) 0 0
\(559\) −10.4756 −0.443069
\(560\) 0.684644 0.0289315
\(561\) 0 0
\(562\) −50.8800 −2.14624
\(563\) 27.7532 1.16966 0.584829 0.811157i \(-0.301162\pi\)
0.584829 + 0.811157i \(0.301162\pi\)
\(564\) 0 0
\(565\) 4.12919 0.173716
\(566\) −21.9102 −0.920956
\(567\) 0 0
\(568\) 19.9074 0.835296
\(569\) −6.89351 −0.288991 −0.144495 0.989505i \(-0.546156\pi\)
−0.144495 + 0.989505i \(0.546156\pi\)
\(570\) 0 0
\(571\) 15.4924 0.648337 0.324168 0.945999i \(-0.394916\pi\)
0.324168 + 0.945999i \(0.394916\pi\)
\(572\) −40.7395 −1.70340
\(573\) 0 0
\(574\) −1.85183 −0.0772940
\(575\) 21.4314 0.893750
\(576\) 0 0
\(577\) 26.1702 1.08948 0.544741 0.838605i \(-0.316628\pi\)
0.544741 + 0.838605i \(0.316628\pi\)
\(578\) −28.9120 −1.20258
\(579\) 0 0
\(580\) −17.7804 −0.738289
\(581\) 4.68620 0.194416
\(582\) 0 0
\(583\) 24.8565 1.02945
\(584\) 14.7717 0.611256
\(585\) 0 0
\(586\) 13.7929 0.569781
\(587\) −34.8064 −1.43662 −0.718308 0.695725i \(-0.755083\pi\)
−0.718308 + 0.695725i \(0.755083\pi\)
\(588\) 0 0
\(589\) −6.27446 −0.258534
\(590\) −10.6349 −0.437831
\(591\) 0 0
\(592\) 10.0525 0.413157
\(593\) 24.0643 0.988203 0.494101 0.869404i \(-0.335497\pi\)
0.494101 + 0.869404i \(0.335497\pi\)
\(594\) 0 0
\(595\) 1.44491 0.0592355
\(596\) 40.8387 1.67282
\(597\) 0 0
\(598\) 41.4692 1.69580
\(599\) 0.508407 0.0207729 0.0103865 0.999946i \(-0.496694\pi\)
0.0103865 + 0.999946i \(0.496694\pi\)
\(600\) 0 0
\(601\) 23.1016 0.942333 0.471166 0.882044i \(-0.343833\pi\)
0.471166 + 0.882044i \(0.343833\pi\)
\(602\) −6.04313 −0.246300
\(603\) 0 0
\(604\) 10.2545 0.417248
\(605\) 0.830943 0.0337826
\(606\) 0 0
\(607\) −35.7095 −1.44940 −0.724701 0.689063i \(-0.758022\pi\)
−0.724701 + 0.689063i \(0.758022\pi\)
\(608\) −8.84098 −0.358549
\(609\) 0 0
\(610\) 15.6791 0.634827
\(611\) 36.6405 1.48231
\(612\) 0 0
\(613\) −9.75584 −0.394035 −0.197017 0.980400i \(-0.563126\pi\)
−0.197017 + 0.980400i \(0.563126\pi\)
\(614\) −30.8665 −1.24567
\(615\) 0 0
\(616\) −7.88902 −0.317858
\(617\) −4.62163 −0.186060 −0.0930299 0.995663i \(-0.529655\pi\)
−0.0930299 + 0.995663i \(0.529655\pi\)
\(618\) 0 0
\(619\) −46.6480 −1.87494 −0.937471 0.348062i \(-0.886840\pi\)
−0.937471 + 0.348062i \(0.886840\pi\)
\(620\) −10.1861 −0.409083
\(621\) 0 0
\(622\) 17.8885 0.717263
\(623\) 3.95955 0.158636
\(624\) 0 0
\(625\) 17.5929 0.703717
\(626\) 18.3451 0.733217
\(627\) 0 0
\(628\) −61.9808 −2.47330
\(629\) 21.2154 0.845914
\(630\) 0 0
\(631\) −40.7101 −1.62064 −0.810322 0.585984i \(-0.800708\pi\)
−0.810322 + 0.585984i \(0.800708\pi\)
\(632\) −2.64620 −0.105260
\(633\) 0 0
\(634\) −21.7485 −0.863745
\(635\) −0.715002 −0.0283740
\(636\) 0 0
\(637\) −3.88025 −0.153741
\(638\) −64.4810 −2.55283
\(639\) 0 0
\(640\) −11.2876 −0.446181
\(641\) −16.0725 −0.634826 −0.317413 0.948287i \(-0.602814\pi\)
−0.317413 + 0.948287i \(0.602814\pi\)
\(642\) 0 0
\(643\) 6.75968 0.266576 0.133288 0.991077i \(-0.457446\pi\)
0.133288 + 0.991077i \(0.457446\pi\)
\(644\) 14.3739 0.566410
\(645\) 0 0
\(646\) −5.99798 −0.235987
\(647\) −26.3074 −1.03425 −0.517125 0.855910i \(-0.672998\pi\)
−0.517125 + 0.855910i \(0.672998\pi\)
\(648\) 0 0
\(649\) −23.1732 −0.909628
\(650\) 38.9880 1.52924
\(651\) 0 0
\(652\) 43.3368 1.69720
\(653\) −15.4078 −0.602954 −0.301477 0.953473i \(-0.597480\pi\)
−0.301477 + 0.953473i \(0.597480\pi\)
\(654\) 0 0
\(655\) −0.763652 −0.0298384
\(656\) −0.792163 −0.0309288
\(657\) 0 0
\(658\) 21.1371 0.824011
\(659\) 11.1639 0.434885 0.217442 0.976073i \(-0.430229\pi\)
0.217442 + 0.976073i \(0.430229\pi\)
\(660\) 0 0
\(661\) 16.7973 0.653338 0.326669 0.945139i \(-0.394074\pi\)
0.326669 + 0.945139i \(0.394074\pi\)
\(662\) 59.8692 2.32688
\(663\) 0 0
\(664\) −10.6008 −0.411391
\(665\) −0.948058 −0.0367641
\(666\) 0 0
\(667\) 39.4371 1.52701
\(668\) 44.3356 1.71539
\(669\) 0 0
\(670\) 16.7794 0.648243
\(671\) 34.1644 1.31890
\(672\) 0 0
\(673\) 36.1418 1.39317 0.696583 0.717476i \(-0.254703\pi\)
0.696583 + 0.717476i \(0.254703\pi\)
\(674\) 17.4774 0.673205
\(675\) 0 0
\(676\) 6.19071 0.238104
\(677\) −36.2462 −1.39306 −0.696528 0.717530i \(-0.745273\pi\)
−0.696528 + 0.717530i \(0.745273\pi\)
\(678\) 0 0
\(679\) −6.83283 −0.262220
\(680\) −3.26858 −0.125344
\(681\) 0 0
\(682\) −36.9401 −1.41451
\(683\) 21.5032 0.822796 0.411398 0.911456i \(-0.365041\pi\)
0.411398 + 0.911456i \(0.365041\pi\)
\(684\) 0 0
\(685\) 9.15620 0.349840
\(686\) −2.23843 −0.0854638
\(687\) 0 0
\(688\) −2.58509 −0.0985555
\(689\) −27.6562 −1.05362
\(690\) 0 0
\(691\) −16.8476 −0.640913 −0.320456 0.947263i \(-0.603836\pi\)
−0.320456 + 0.947263i \(0.603836\pi\)
\(692\) −36.4571 −1.38589
\(693\) 0 0
\(694\) 37.8137 1.43539
\(695\) −2.95960 −0.112264
\(696\) 0 0
\(697\) −1.67182 −0.0633248
\(698\) −33.1585 −1.25507
\(699\) 0 0
\(700\) 13.5138 0.510775
\(701\) −8.39278 −0.316991 −0.158495 0.987360i \(-0.550664\pi\)
−0.158495 + 0.987360i \(0.550664\pi\)
\(702\) 0 0
\(703\) −13.9202 −0.525011
\(704\) −45.3715 −1.71000
\(705\) 0 0
\(706\) 71.6756 2.69755
\(707\) 14.4018 0.541636
\(708\) 0 0
\(709\) −22.8508 −0.858181 −0.429091 0.903261i \(-0.641166\pi\)
−0.429091 + 0.903261i \(0.641166\pi\)
\(710\) 14.0847 0.528591
\(711\) 0 0
\(712\) −8.95703 −0.335679
\(713\) 22.5929 0.846109
\(714\) 0 0
\(715\) −9.67547 −0.361842
\(716\) 30.9537 1.15679
\(717\) 0 0
\(718\) 15.9853 0.596566
\(719\) 29.0142 1.08205 0.541023 0.841008i \(-0.318037\pi\)
0.541023 + 0.841008i \(0.318037\pi\)
\(720\) 0 0
\(721\) −4.78657 −0.178261
\(722\) −38.5948 −1.43635
\(723\) 0 0
\(724\) −21.0021 −0.780538
\(725\) 37.0774 1.37702
\(726\) 0 0
\(727\) 1.71536 0.0636190 0.0318095 0.999494i \(-0.489873\pi\)
0.0318095 + 0.999494i \(0.489873\pi\)
\(728\) 8.77763 0.325321
\(729\) 0 0
\(730\) 10.4511 0.386814
\(731\) −5.45571 −0.201787
\(732\) 0 0
\(733\) 32.0548 1.18397 0.591986 0.805948i \(-0.298344\pi\)
0.591986 + 0.805948i \(0.298344\pi\)
\(734\) 42.5544 1.57071
\(735\) 0 0
\(736\) 31.8343 1.17343
\(737\) 36.5619 1.34677
\(738\) 0 0
\(739\) 15.0497 0.553613 0.276807 0.960926i \(-0.410724\pi\)
0.276807 + 0.960926i \(0.410724\pi\)
\(740\) −22.5984 −0.830732
\(741\) 0 0
\(742\) −15.9543 −0.585701
\(743\) −1.95125 −0.0715845 −0.0357922 0.999359i \(-0.511395\pi\)
−0.0357922 + 0.999359i \(0.511395\pi\)
\(744\) 0 0
\(745\) 9.69904 0.355345
\(746\) 73.0221 2.67353
\(747\) 0 0
\(748\) −21.2173 −0.775780
\(749\) −8.11171 −0.296395
\(750\) 0 0
\(751\) −13.9594 −0.509385 −0.254692 0.967022i \(-0.581974\pi\)
−0.254692 + 0.967022i \(0.581974\pi\)
\(752\) 9.04188 0.329724
\(753\) 0 0
\(754\) 71.7440 2.61276
\(755\) 2.43539 0.0886330
\(756\) 0 0
\(757\) −23.9434 −0.870238 −0.435119 0.900373i \(-0.643294\pi\)
−0.435119 + 0.900373i \(0.643294\pi\)
\(758\) 21.3941 0.777070
\(759\) 0 0
\(760\) 2.14463 0.0777940
\(761\) 17.6709 0.640569 0.320285 0.947321i \(-0.396221\pi\)
0.320285 + 0.947321i \(0.396221\pi\)
\(762\) 0 0
\(763\) −0.982270 −0.0355606
\(764\) −50.8564 −1.83992
\(765\) 0 0
\(766\) 44.7664 1.61748
\(767\) 25.7834 0.930985
\(768\) 0 0
\(769\) −3.34345 −0.120568 −0.0602839 0.998181i \(-0.519201\pi\)
−0.0602839 + 0.998181i \(0.519201\pi\)
\(770\) −5.58158 −0.201146
\(771\) 0 0
\(772\) 15.1004 0.543475
\(773\) 35.6247 1.28133 0.640665 0.767821i \(-0.278659\pi\)
0.640665 + 0.767821i \(0.278659\pi\)
\(774\) 0 0
\(775\) 21.2411 0.763002
\(776\) 15.4568 0.554865
\(777\) 0 0
\(778\) 9.11958 0.326953
\(779\) 1.09694 0.0393021
\(780\) 0 0
\(781\) 30.6904 1.09819
\(782\) 21.5973 0.772319
\(783\) 0 0
\(784\) −0.957541 −0.0341979
\(785\) −14.7202 −0.525386
\(786\) 0 0
\(787\) −49.3147 −1.75788 −0.878939 0.476935i \(-0.841748\pi\)
−0.878939 + 0.476935i \(0.841748\pi\)
\(788\) 26.4780 0.943240
\(789\) 0 0
\(790\) −1.87222 −0.0666105
\(791\) −5.77508 −0.205338
\(792\) 0 0
\(793\) −38.0126 −1.34987
\(794\) 48.7605 1.73045
\(795\) 0 0
\(796\) −66.1181 −2.34349
\(797\) −18.8764 −0.668637 −0.334318 0.942460i \(-0.608506\pi\)
−0.334318 + 0.942460i \(0.608506\pi\)
\(798\) 0 0
\(799\) 19.0825 0.675089
\(800\) 29.9296 1.05817
\(801\) 0 0
\(802\) −38.7988 −1.37003
\(803\) 22.7728 0.803636
\(804\) 0 0
\(805\) 3.41374 0.120318
\(806\) 41.1010 1.44772
\(807\) 0 0
\(808\) −32.5788 −1.14612
\(809\) −27.6464 −0.971994 −0.485997 0.873960i \(-0.661543\pi\)
−0.485997 + 0.873960i \(0.661543\pi\)
\(810\) 0 0
\(811\) −40.0155 −1.40514 −0.702568 0.711617i \(-0.747963\pi\)
−0.702568 + 0.711617i \(0.747963\pi\)
\(812\) 24.8676 0.872680
\(813\) 0 0
\(814\) −81.9536 −2.87247
\(815\) 10.2923 0.360524
\(816\) 0 0
\(817\) 3.57969 0.125237
\(818\) −27.9232 −0.976311
\(819\) 0 0
\(820\) 1.78080 0.0621884
\(821\) 7.00349 0.244423 0.122212 0.992504i \(-0.461001\pi\)
0.122212 + 0.992504i \(0.461001\pi\)
\(822\) 0 0
\(823\) 7.48955 0.261069 0.130535 0.991444i \(-0.458331\pi\)
0.130535 + 0.991444i \(0.458331\pi\)
\(824\) 10.8278 0.377206
\(825\) 0 0
\(826\) 14.8739 0.517530
\(827\) −5.69107 −0.197898 −0.0989490 0.995093i \(-0.531548\pi\)
−0.0989490 + 0.995093i \(0.531548\pi\)
\(828\) 0 0
\(829\) −46.9867 −1.63192 −0.815958 0.578111i \(-0.803790\pi\)
−0.815958 + 0.578111i \(0.803790\pi\)
\(830\) −7.50020 −0.260336
\(831\) 0 0
\(832\) 50.4821 1.75015
\(833\) −2.02085 −0.0700182
\(834\) 0 0
\(835\) 10.5295 0.364389
\(836\) 13.9214 0.481483
\(837\) 0 0
\(838\) 24.1709 0.834971
\(839\) 23.2141 0.801441 0.400721 0.916200i \(-0.368760\pi\)
0.400721 + 0.916200i \(0.368760\pi\)
\(840\) 0 0
\(841\) 39.2282 1.35270
\(842\) −17.0120 −0.586273
\(843\) 0 0
\(844\) 31.1901 1.07361
\(845\) 1.47027 0.0505788
\(846\) 0 0
\(847\) −1.16215 −0.0399321
\(848\) −6.82482 −0.234365
\(849\) 0 0
\(850\) 20.3051 0.696459
\(851\) 50.1234 1.71821
\(852\) 0 0
\(853\) −30.3267 −1.03837 −0.519183 0.854663i \(-0.673764\pi\)
−0.519183 + 0.854663i \(0.673764\pi\)
\(854\) −21.9287 −0.750385
\(855\) 0 0
\(856\) 18.3498 0.627182
\(857\) 22.4832 0.768011 0.384005 0.923331i \(-0.374544\pi\)
0.384005 + 0.923331i \(0.374544\pi\)
\(858\) 0 0
\(859\) −36.3023 −1.23862 −0.619309 0.785147i \(-0.712587\pi\)
−0.619309 + 0.785147i \(0.712587\pi\)
\(860\) 5.81134 0.198165
\(861\) 0 0
\(862\) 77.9441 2.65479
\(863\) 32.5087 1.10661 0.553305 0.832979i \(-0.313367\pi\)
0.553305 + 0.832979i \(0.313367\pi\)
\(864\) 0 0
\(865\) −8.65842 −0.294395
\(866\) −75.8841 −2.57865
\(867\) 0 0
\(868\) 14.2462 0.483549
\(869\) −4.07953 −0.138388
\(870\) 0 0
\(871\) −40.6802 −1.37839
\(872\) 2.22202 0.0752472
\(873\) 0 0
\(874\) −14.1708 −0.479334
\(875\) 6.78450 0.229358
\(876\) 0 0
\(877\) −6.43887 −0.217425 −0.108713 0.994073i \(-0.534673\pi\)
−0.108713 + 0.994073i \(0.534673\pi\)
\(878\) 13.5209 0.456307
\(879\) 0 0
\(880\) −2.38765 −0.0804876
\(881\) 50.9542 1.71669 0.858346 0.513072i \(-0.171493\pi\)
0.858346 + 0.513072i \(0.171493\pi\)
\(882\) 0 0
\(883\) −1.41232 −0.0475284 −0.0237642 0.999718i \(-0.507565\pi\)
−0.0237642 + 0.999718i \(0.507565\pi\)
\(884\) 23.6072 0.793994
\(885\) 0 0
\(886\) 71.8161 2.41271
\(887\) 9.49697 0.318877 0.159438 0.987208i \(-0.449032\pi\)
0.159438 + 0.987208i \(0.449032\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −6.33721 −0.212424
\(891\) 0 0
\(892\) −19.1376 −0.640775
\(893\) −12.5207 −0.418990
\(894\) 0 0
\(895\) 7.35138 0.245730
\(896\) 15.7868 0.527399
\(897\) 0 0
\(898\) 15.7058 0.524110
\(899\) 39.0869 1.30362
\(900\) 0 0
\(901\) −14.4035 −0.479849
\(902\) 6.45813 0.215032
\(903\) 0 0
\(904\) 13.0640 0.434502
\(905\) −4.98792 −0.165804
\(906\) 0 0
\(907\) −48.9510 −1.62539 −0.812695 0.582689i \(-0.802000\pi\)
−0.812695 + 0.582689i \(0.802000\pi\)
\(908\) −23.1444 −0.768074
\(909\) 0 0
\(910\) 6.21028 0.205869
\(911\) 5.99796 0.198721 0.0993607 0.995051i \(-0.468320\pi\)
0.0993607 + 0.995051i \(0.468320\pi\)
\(912\) 0 0
\(913\) −16.3428 −0.540867
\(914\) 37.7136 1.24746
\(915\) 0 0
\(916\) −31.8109 −1.05106
\(917\) 1.06804 0.0352698
\(918\) 0 0
\(919\) −0.999658 −0.0329757 −0.0164878 0.999864i \(-0.505248\pi\)
−0.0164878 + 0.999864i \(0.505248\pi\)
\(920\) −7.72232 −0.254597
\(921\) 0 0
\(922\) 39.7553 1.30927
\(923\) −34.1473 −1.12397
\(924\) 0 0
\(925\) 47.1244 1.54944
\(926\) 22.7936 0.749044
\(927\) 0 0
\(928\) 55.0751 1.80793
\(929\) 55.3468 1.81587 0.907935 0.419110i \(-0.137658\pi\)
0.907935 + 0.419110i \(0.137658\pi\)
\(930\) 0 0
\(931\) 1.32595 0.0434563
\(932\) −75.6293 −2.47732
\(933\) 0 0
\(934\) 3.70270 0.121156
\(935\) −5.03902 −0.164794
\(936\) 0 0
\(937\) −1.27880 −0.0417766 −0.0208883 0.999782i \(-0.506649\pi\)
−0.0208883 + 0.999782i \(0.506649\pi\)
\(938\) −23.4675 −0.766242
\(939\) 0 0
\(940\) −20.3264 −0.662974
\(941\) 22.7871 0.742837 0.371419 0.928466i \(-0.378872\pi\)
0.371419 + 0.928466i \(0.378872\pi\)
\(942\) 0 0
\(943\) −3.94984 −0.128625
\(944\) 6.36265 0.207087
\(945\) 0 0
\(946\) 21.0750 0.685207
\(947\) 8.91484 0.289693 0.144847 0.989454i \(-0.453731\pi\)
0.144847 + 0.989454i \(0.453731\pi\)
\(948\) 0 0
\(949\) −25.3379 −0.822504
\(950\) −13.3229 −0.432252
\(951\) 0 0
\(952\) 4.57142 0.148161
\(953\) 8.13381 0.263480 0.131740 0.991284i \(-0.457944\pi\)
0.131740 + 0.991284i \(0.457944\pi\)
\(954\) 0 0
\(955\) −12.0782 −0.390841
\(956\) 32.6383 1.05560
\(957\) 0 0
\(958\) 28.6433 0.925423
\(959\) −12.8058 −0.413522
\(960\) 0 0
\(961\) −8.60776 −0.277670
\(962\) 91.1847 2.93991
\(963\) 0 0
\(964\) −53.8140 −1.73323
\(965\) 3.58628 0.115447
\(966\) 0 0
\(967\) −15.9426 −0.512679 −0.256339 0.966587i \(-0.582516\pi\)
−0.256339 + 0.966587i \(0.582516\pi\)
\(968\) 2.62895 0.0844975
\(969\) 0 0
\(970\) 10.9358 0.351129
\(971\) −43.2242 −1.38713 −0.693566 0.720393i \(-0.743961\pi\)
−0.693566 + 0.720393i \(0.743961\pi\)
\(972\) 0 0
\(973\) 4.13929 0.132699
\(974\) 13.0116 0.416918
\(975\) 0 0
\(976\) −9.38050 −0.300262
\(977\) −11.4772 −0.367190 −0.183595 0.983002i \(-0.558773\pi\)
−0.183595 + 0.983002i \(0.558773\pi\)
\(978\) 0 0
\(979\) −13.8087 −0.441327
\(980\) 2.15258 0.0687616
\(981\) 0 0
\(982\) −11.8200 −0.377192
\(983\) −32.2991 −1.03018 −0.515090 0.857136i \(-0.672242\pi\)
−0.515090 + 0.857136i \(0.672242\pi\)
\(984\) 0 0
\(985\) 6.28842 0.200366
\(986\) 37.3645 1.18993
\(987\) 0 0
\(988\) −15.4895 −0.492787
\(989\) −12.8896 −0.409866
\(990\) 0 0
\(991\) −46.4069 −1.47416 −0.737081 0.675804i \(-0.763796\pi\)
−0.737081 + 0.675804i \(0.763796\pi\)
\(992\) 31.5516 1.00177
\(993\) 0 0
\(994\) −19.6989 −0.624810
\(995\) −15.7028 −0.497812
\(996\) 0 0
\(997\) 36.8281 1.16636 0.583179 0.812344i \(-0.301809\pi\)
0.583179 + 0.812344i \(0.301809\pi\)
\(998\) −44.8915 −1.42101
\(999\) 0 0
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 8001.2.a.z.1.28 yes 32
3.2 odd 2 inner 8001.2.a.z.1.5 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.5 32 3.2 odd 2 inner
8001.2.a.z.1.28 yes 32 1.1 even 1 trivial