Properties

Label 8281.2.a.bv.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{23})\)
Defining polynomial: \(x^{4} - 24 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.68406\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -4.09827 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -4.09827 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} -4.09827 q^{10} -3.79583 q^{11} +1.41421 q^{12} +5.79583 q^{15} -1.00000 q^{16} +1.26984 q^{17} -1.00000 q^{18} +2.82843 q^{19} +4.09827 q^{20} -3.79583 q^{22} -7.79583 q^{23} +4.24264 q^{24} +11.7958 q^{25} +5.65685 q^{27} +0.795832 q^{29} +5.79583 q^{30} +1.41421 q^{31} +5.00000 q^{32} +5.36812 q^{33} +1.26984 q^{34} +1.00000 q^{36} +2.79583 q^{37} +2.82843 q^{38} +12.2948 q^{40} -2.97280 q^{41} +7.79583 q^{43} +3.79583 q^{44} +4.09827 q^{45} -7.79583 q^{46} -2.82843 q^{47} +1.41421 q^{48} +11.7958 q^{50} -1.79583 q^{51} -12.5917 q^{53} +5.65685 q^{54} +15.5563 q^{55} -4.00000 q^{57} +0.795832 q^{58} +12.4392 q^{59} -5.79583 q^{60} -8.34091 q^{61} +1.41421 q^{62} +7.00000 q^{64} +5.36812 q^{66} -3.79583 q^{67} -1.26984 q^{68} +11.0250 q^{69} +6.00000 q^{71} +3.00000 q^{72} +12.5836 q^{73} +2.79583 q^{74} -16.6818 q^{75} -2.82843 q^{76} -2.20417 q^{79} +4.09827 q^{80} -5.00000 q^{81} -2.97280 q^{82} +9.89949 q^{83} -5.20417 q^{85} +7.79583 q^{86} -1.12548 q^{87} +11.3875 q^{88} +14.9789 q^{89} +4.09827 q^{90} +7.79583 q^{92} -2.00000 q^{93} -2.82843 q^{94} -11.5917 q^{95} -7.07107 q^{96} +4.24264 q^{97} +3.79583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44} - 12 q^{46} + 28 q^{50} + 12 q^{51} - 12 q^{53} - 16 q^{57} - 16 q^{58} - 4 q^{60} + 28 q^{64} + 4 q^{67} + 24 q^{71} + 12 q^{72} - 8 q^{74} - 28 q^{79} - 20 q^{81} - 40 q^{85} + 12 q^{86} - 12 q^{88} + 12 q^{92} - 8 q^{93} - 8 q^{95} - 4 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) −1.00000 −0.500000
\(5\) −4.09827 −1.83280 −0.916401 0.400260i \(-0.868920\pi\)
−0.916401 + 0.400260i \(0.868920\pi\)
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) −1.00000 −0.333333
\(10\) −4.09827 −1.29599
\(11\) −3.79583 −1.14449 −0.572243 0.820084i \(-0.693927\pi\)
−0.572243 + 0.820084i \(0.693927\pi\)
\(12\) 1.41421 0.408248
\(13\) 0 0
\(14\) 0 0
\(15\) 5.79583 1.49648
\(16\) −1.00000 −0.250000
\(17\) 1.26984 0.307983 0.153991 0.988072i \(-0.450787\pi\)
0.153991 + 0.988072i \(0.450787\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 4.09827 0.916401
\(21\) 0 0
\(22\) −3.79583 −0.809274
\(23\) −7.79583 −1.62554 −0.812772 0.582582i \(-0.802042\pi\)
−0.812772 + 0.582582i \(0.802042\pi\)
\(24\) 4.24264 0.866025
\(25\) 11.7958 2.35917
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0.795832 0.147782 0.0738911 0.997266i \(-0.476458\pi\)
0.0738911 + 0.997266i \(0.476458\pi\)
\(30\) 5.79583 1.05817
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 5.00000 0.883883
\(33\) 5.36812 0.934469
\(34\) 1.26984 0.217777
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.79583 0.459632 0.229816 0.973234i \(-0.426188\pi\)
0.229816 + 0.973234i \(0.426188\pi\)
\(38\) 2.82843 0.458831
\(39\) 0 0
\(40\) 12.2948 1.94398
\(41\) −2.97280 −0.464273 −0.232136 0.972683i \(-0.574572\pi\)
−0.232136 + 0.972683i \(0.574572\pi\)
\(42\) 0 0
\(43\) 7.79583 1.18885 0.594427 0.804150i \(-0.297379\pi\)
0.594427 + 0.804150i \(0.297379\pi\)
\(44\) 3.79583 0.572243
\(45\) 4.09827 0.610934
\(46\) −7.79583 −1.14943
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 1.41421 0.204124
\(49\) 0 0
\(50\) 11.7958 1.66818
\(51\) −1.79583 −0.251467
\(52\) 0 0
\(53\) −12.5917 −1.72960 −0.864799 0.502118i \(-0.832554\pi\)
−0.864799 + 0.502118i \(0.832554\pi\)
\(54\) 5.65685 0.769800
\(55\) 15.5563 2.09762
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0.795832 0.104498
\(59\) 12.4392 1.61944 0.809722 0.586814i \(-0.199618\pi\)
0.809722 + 0.586814i \(0.199618\pi\)
\(60\) −5.79583 −0.748239
\(61\) −8.34091 −1.06794 −0.533972 0.845502i \(-0.679301\pi\)
−0.533972 + 0.845502i \(0.679301\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 5.36812 0.660769
\(67\) −3.79583 −0.463735 −0.231867 0.972747i \(-0.574484\pi\)
−0.231867 + 0.972747i \(0.574484\pi\)
\(68\) −1.26984 −0.153991
\(69\) 11.0250 1.32725
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) 12.5836 1.47279 0.736397 0.676550i \(-0.236526\pi\)
0.736397 + 0.676550i \(0.236526\pi\)
\(74\) 2.79583 0.325009
\(75\) −16.6818 −1.92625
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) 0 0
\(79\) −2.20417 −0.247988 −0.123994 0.992283i \(-0.539570\pi\)
−0.123994 + 0.992283i \(0.539570\pi\)
\(80\) 4.09827 0.458201
\(81\) −5.00000 −0.555556
\(82\) −2.97280 −0.328290
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) −5.20417 −0.564471
\(86\) 7.79583 0.840646
\(87\) −1.12548 −0.120664
\(88\) 11.3875 1.21391
\(89\) 14.9789 1.58776 0.793879 0.608076i \(-0.208058\pi\)
0.793879 + 0.608076i \(0.208058\pi\)
\(90\) 4.09827 0.431996
\(91\) 0 0
\(92\) 7.79583 0.812772
\(93\) −2.00000 −0.207390
\(94\) −2.82843 −0.291730
\(95\) −11.5917 −1.18928
\(96\) −7.07107 −0.721688
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 0 0
\(99\) 3.79583 0.381495
\(100\) −11.7958 −1.17958
\(101\) 9.75513 0.970671 0.485336 0.874328i \(-0.338697\pi\)
0.485336 + 0.874328i \(0.338697\pi\)
\(102\) −1.79583 −0.177814
\(103\) −5.36812 −0.528936 −0.264468 0.964394i \(-0.585196\pi\)
−0.264468 + 0.964394i \(0.585196\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.5917 −1.22301
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −5.65685 −0.544331
\(109\) 1.59166 0.152454 0.0762268 0.997091i \(-0.475713\pi\)
0.0762268 + 0.997091i \(0.475713\pi\)
\(110\) 15.5563 1.48324
\(111\) −3.95390 −0.375288
\(112\) 0 0
\(113\) 2.59166 0.243803 0.121902 0.992542i \(-0.461101\pi\)
0.121902 + 0.992542i \(0.461101\pi\)
\(114\) −4.00000 −0.374634
\(115\) 31.9494 2.97930
\(116\) −0.795832 −0.0738911
\(117\) 0 0
\(118\) 12.4392 1.14512
\(119\) 0 0
\(120\) −17.3875 −1.58725
\(121\) 3.40834 0.309849
\(122\) −8.34091 −0.755151
\(123\) 4.20417 0.379077
\(124\) −1.41421 −0.127000
\(125\) −27.8512 −2.49108
\(126\) 0 0
\(127\) −11.5917 −1.02859 −0.514297 0.857612i \(-0.671947\pi\)
−0.514297 + 0.857612i \(0.671947\pi\)
\(128\) −3.00000 −0.265165
\(129\) −11.0250 −0.970695
\(130\) 0 0
\(131\) −5.36812 −0.469015 −0.234507 0.972114i \(-0.575348\pi\)
−0.234507 + 0.972114i \(0.575348\pi\)
\(132\) −5.36812 −0.467235
\(133\) 0 0
\(134\) −3.79583 −0.327910
\(135\) −23.1833 −1.99530
\(136\) −3.80953 −0.326665
\(137\) −18.7958 −1.60584 −0.802918 0.596089i \(-0.796720\pi\)
−0.802918 + 0.596089i \(0.796720\pi\)
\(138\) 11.0250 0.938508
\(139\) 1.70295 0.144442 0.0722212 0.997389i \(-0.476991\pi\)
0.0722212 + 0.997389i \(0.476991\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.26153 −0.270856
\(146\) 12.5836 1.04142
\(147\) 0 0
\(148\) −2.79583 −0.229816
\(149\) −14.5917 −1.19540 −0.597698 0.801721i \(-0.703918\pi\)
−0.597698 + 0.801721i \(0.703918\pi\)
\(150\) −16.6818 −1.36207
\(151\) 1.59166 0.129528 0.0647639 0.997901i \(-0.479371\pi\)
0.0647639 + 0.997901i \(0.479371\pi\)
\(152\) −8.48528 −0.688247
\(153\) −1.26984 −0.102661
\(154\) 0 0
\(155\) −5.79583 −0.465532
\(156\) 0 0
\(157\) 0.144369 0.0115219 0.00576095 0.999983i \(-0.498166\pi\)
0.00576095 + 0.999983i \(0.498166\pi\)
\(158\) −2.20417 −0.175354
\(159\) 17.8073 1.41221
\(160\) −20.4914 −1.61998
\(161\) 0 0
\(162\) −5.00000 −0.392837
\(163\) 15.3875 1.20524 0.602621 0.798028i \(-0.294123\pi\)
0.602621 + 0.798028i \(0.294123\pi\)
\(164\) 2.97280 0.232136
\(165\) −22.0000 −1.71270
\(166\) 9.89949 0.768350
\(167\) −1.70295 −0.131778 −0.0658892 0.997827i \(-0.520988\pi\)
−0.0658892 + 0.997827i \(0.520988\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −5.20417 −0.399142
\(171\) −2.82843 −0.216295
\(172\) −7.79583 −0.594427
\(173\) 17.8073 1.35386 0.676932 0.736046i \(-0.263309\pi\)
0.676932 + 0.736046i \(0.263309\pi\)
\(174\) −1.12548 −0.0853221
\(175\) 0 0
\(176\) 3.79583 0.286122
\(177\) −17.5917 −1.32227
\(178\) 14.9789 1.12271
\(179\) −19.5917 −1.46435 −0.732175 0.681117i \(-0.761495\pi\)
−0.732175 + 0.681117i \(0.761495\pi\)
\(180\) −4.09827 −0.305467
\(181\) −3.80953 −0.283160 −0.141580 0.989927i \(-0.545218\pi\)
−0.141580 + 0.989927i \(0.545218\pi\)
\(182\) 0 0
\(183\) 11.7958 0.871973
\(184\) 23.3875 1.72415
\(185\) −11.4581 −0.842415
\(186\) −2.00000 −0.146647
\(187\) −4.82012 −0.352482
\(188\) 2.82843 0.206284
\(189\) 0 0
\(190\) −11.5917 −0.840948
\(191\) −16.2042 −1.17249 −0.586246 0.810133i \(-0.699395\pi\)
−0.586246 + 0.810133i \(0.699395\pi\)
\(192\) −9.89949 −0.714435
\(193\) 22.5917 1.62618 0.813092 0.582136i \(-0.197783\pi\)
0.813092 + 0.582136i \(0.197783\pi\)
\(194\) 4.24264 0.304604
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 3.79583 0.269758
\(199\) −5.07938 −0.360068 −0.180034 0.983660i \(-0.557621\pi\)
−0.180034 + 0.983660i \(0.557621\pi\)
\(200\) −35.3875 −2.50227
\(201\) 5.36812 0.378638
\(202\) 9.75513 0.686368
\(203\) 0 0
\(204\) 1.79583 0.125733
\(205\) 12.1833 0.850920
\(206\) −5.36812 −0.374014
\(207\) 7.79583 0.541848
\(208\) 0 0
\(209\) −10.7362 −0.742641
\(210\) 0 0
\(211\) 7.79583 0.536687 0.268344 0.963323i \(-0.413524\pi\)
0.268344 + 0.963323i \(0.413524\pi\)
\(212\) 12.5917 0.864799
\(213\) −8.48528 −0.581402
\(214\) −6.00000 −0.410152
\(215\) −31.9494 −2.17893
\(216\) −16.9706 −1.15470
\(217\) 0 0
\(218\) 1.59166 0.107801
\(219\) −17.7958 −1.20253
\(220\) −15.5563 −1.04881
\(221\) 0 0
\(222\) −3.95390 −0.265369
\(223\) 5.65685 0.378811 0.189405 0.981899i \(-0.439344\pi\)
0.189405 + 0.981899i \(0.439344\pi\)
\(224\) 0 0
\(225\) −11.7958 −0.786389
\(226\) 2.59166 0.172395
\(227\) −7.35981 −0.488487 −0.244244 0.969714i \(-0.578540\pi\)
−0.244244 + 0.969714i \(0.578540\pi\)
\(228\) 4.00000 0.264906
\(229\) −12.7279 −0.841085 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(230\) 31.9494 2.10668
\(231\) 0 0
\(232\) −2.38749 −0.156747
\(233\) 17.1833 1.12572 0.562859 0.826553i \(-0.309702\pi\)
0.562859 + 0.826553i \(0.309702\pi\)
\(234\) 0 0
\(235\) 11.5917 0.756157
\(236\) −12.4392 −0.809722
\(237\) 3.11716 0.202482
\(238\) 0 0
\(239\) −10.2042 −0.660053 −0.330026 0.943972i \(-0.607058\pi\)
−0.330026 + 0.943972i \(0.607058\pi\)
\(240\) −5.79583 −0.374119
\(241\) −11.1693 −0.719480 −0.359740 0.933053i \(-0.617135\pi\)
−0.359740 + 0.933053i \(0.617135\pi\)
\(242\) 3.40834 0.219096
\(243\) −9.89949 −0.635053
\(244\) 8.34091 0.533972
\(245\) 0 0
\(246\) 4.20417 0.268048
\(247\) 0 0
\(248\) −4.24264 −0.269408
\(249\) −14.0000 −0.887214
\(250\) −27.8512 −1.76146
\(251\) 16.6818 1.05295 0.526474 0.850191i \(-0.323514\pi\)
0.526474 + 0.850191i \(0.323514\pi\)
\(252\) 0 0
\(253\) 29.5917 1.86041
\(254\) −11.5917 −0.727326
\(255\) 7.35981 0.460889
\(256\) −17.0000 −1.06250
\(257\) −8.62965 −0.538303 −0.269151 0.963098i \(-0.586743\pi\)
−0.269151 + 0.963098i \(0.586743\pi\)
\(258\) −11.0250 −0.686385
\(259\) 0 0
\(260\) 0 0
\(261\) −0.795832 −0.0492607
\(262\) −5.36812 −0.331643
\(263\) 3.38749 0.208882 0.104441 0.994531i \(-0.466695\pi\)
0.104441 + 0.994531i \(0.466695\pi\)
\(264\) −16.1043 −0.991154
\(265\) 51.6041 3.17001
\(266\) 0 0
\(267\) −21.1833 −1.29640
\(268\) 3.79583 0.231867
\(269\) −12.4392 −0.758430 −0.379215 0.925308i \(-0.623806\pi\)
−0.379215 + 0.925308i \(0.623806\pi\)
\(270\) −23.1833 −1.41089
\(271\) −17.5186 −1.06418 −0.532088 0.846689i \(-0.678593\pi\)
−0.532088 + 0.846689i \(0.678593\pi\)
\(272\) −1.26984 −0.0769956
\(273\) 0 0
\(274\) −18.7958 −1.13550
\(275\) −44.7750 −2.70003
\(276\) −11.0250 −0.663625
\(277\) 32.1833 1.93371 0.966854 0.255329i \(-0.0821837\pi\)
0.966854 + 0.255329i \(0.0821837\pi\)
\(278\) 1.70295 0.102136
\(279\) −1.41421 −0.0846668
\(280\) 0 0
\(281\) 24.7958 1.47920 0.739598 0.673049i \(-0.235016\pi\)
0.739598 + 0.673049i \(0.235016\pi\)
\(282\) 4.00000 0.238197
\(283\) 15.8451 0.941893 0.470946 0.882162i \(-0.343913\pi\)
0.470946 + 0.882162i \(0.343913\pi\)
\(284\) −6.00000 −0.356034
\(285\) 16.3931 0.971043
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −15.3875 −0.905147
\(290\) −3.26153 −0.191524
\(291\) −6.00000 −0.351726
\(292\) −12.5836 −0.736397
\(293\) 2.97280 0.173673 0.0868363 0.996223i \(-0.472324\pi\)
0.0868363 + 0.996223i \(0.472324\pi\)
\(294\) 0 0
\(295\) −50.9792 −2.96812
\(296\) −8.38749 −0.487513
\(297\) −21.4725 −1.24596
\(298\) −14.5917 −0.845272
\(299\) 0 0
\(300\) 16.6818 0.963126
\(301\) 0 0
\(302\) 1.59166 0.0915899
\(303\) −13.7958 −0.792550
\(304\) −2.82843 −0.162221
\(305\) 34.1833 1.95733
\(306\) −1.26984 −0.0725922
\(307\) 20.0583 1.14478 0.572392 0.819980i \(-0.306015\pi\)
0.572392 + 0.819980i \(0.306015\pi\)
\(308\) 0 0
\(309\) 7.59166 0.431875
\(310\) −5.79583 −0.329181
\(311\) 28.8323 1.63493 0.817464 0.575980i \(-0.195379\pi\)
0.817464 + 0.575980i \(0.195379\pi\)
\(312\) 0 0
\(313\) −15.2676 −0.862976 −0.431488 0.902119i \(-0.642011\pi\)
−0.431488 + 0.902119i \(0.642011\pi\)
\(314\) 0.144369 0.00814721
\(315\) 0 0
\(316\) 2.20417 0.123994
\(317\) 6.59166 0.370225 0.185112 0.982717i \(-0.440735\pi\)
0.185112 + 0.982717i \(0.440735\pi\)
\(318\) 17.8073 0.998584
\(319\) −3.02084 −0.169135
\(320\) −28.6879 −1.60370
\(321\) 8.48528 0.473602
\(322\) 0 0
\(323\) 3.59166 0.199845
\(324\) 5.00000 0.277778
\(325\) 0 0
\(326\) 15.3875 0.852235
\(327\) −2.25095 −0.124478
\(328\) 8.91839 0.492436
\(329\) 0 0
\(330\) −22.0000 −1.21106
\(331\) 29.3875 1.61528 0.807641 0.589674i \(-0.200744\pi\)
0.807641 + 0.589674i \(0.200744\pi\)
\(332\) −9.89949 −0.543305
\(333\) −2.79583 −0.153211
\(334\) −1.70295 −0.0931814
\(335\) 15.5563 0.849934
\(336\) 0 0
\(337\) 17.9792 0.979387 0.489694 0.871895i \(-0.337109\pi\)
0.489694 + 0.871895i \(0.337109\pi\)
\(338\) 0 0
\(339\) −3.66517 −0.199064
\(340\) 5.20417 0.282236
\(341\) −5.36812 −0.290700
\(342\) −2.82843 −0.152944
\(343\) 0 0
\(344\) −23.3875 −1.26097
\(345\) −45.1833 −2.43259
\(346\) 17.8073 0.957326
\(347\) −32.9792 −1.77041 −0.885207 0.465197i \(-0.845983\pi\)
−0.885207 + 0.465197i \(0.845983\pi\)
\(348\) 1.12548 0.0603318
\(349\) −26.5813 −1.42287 −0.711433 0.702754i \(-0.751953\pi\)
−0.711433 + 0.702754i \(0.751953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −18.9792 −1.01159
\(353\) −5.22375 −0.278032 −0.139016 0.990290i \(-0.544394\pi\)
−0.139016 + 0.990290i \(0.544394\pi\)
\(354\) −17.5917 −0.934986
\(355\) −24.5896 −1.30508
\(356\) −14.9789 −0.793879
\(357\) 0 0
\(358\) −19.5917 −1.03545
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) −12.2948 −0.647994
\(361\) −11.0000 −0.578947
\(362\) −3.80953 −0.200225
\(363\) −4.82012 −0.252990
\(364\) 0 0
\(365\) −51.5708 −2.69934
\(366\) 11.7958 0.616578
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) 7.79583 0.406386
\(369\) 2.97280 0.154758
\(370\) −11.4581 −0.595677
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 6.59166 0.341303 0.170652 0.985331i \(-0.445413\pi\)
0.170652 + 0.985331i \(0.445413\pi\)
\(374\) −4.82012 −0.249242
\(375\) 39.3875 2.03396
\(376\) 8.48528 0.437595
\(377\) 0 0
\(378\) 0 0
\(379\) 25.3875 1.30407 0.652034 0.758190i \(-0.273916\pi\)
0.652034 + 0.758190i \(0.273916\pi\)
\(380\) 11.5917 0.594640
\(381\) 16.3931 0.839843
\(382\) −16.2042 −0.829077
\(383\) −12.1504 −0.620859 −0.310429 0.950596i \(-0.600473\pi\)
−0.310429 + 0.950596i \(0.600473\pi\)
\(384\) 4.24264 0.216506
\(385\) 0 0
\(386\) 22.5917 1.14989
\(387\) −7.79583 −0.396284
\(388\) −4.24264 −0.215387
\(389\) −0.387495 −0.0196468 −0.00982338 0.999952i \(-0.503127\pi\)
−0.00982338 + 0.999952i \(0.503127\pi\)
\(390\) 0 0
\(391\) −9.89949 −0.500639
\(392\) 0 0
\(393\) 7.59166 0.382949
\(394\) −8.00000 −0.403034
\(395\) 9.03328 0.454514
\(396\) −3.79583 −0.190748
\(397\) −7.07107 −0.354887 −0.177443 0.984131i \(-0.556783\pi\)
−0.177443 + 0.984131i \(0.556783\pi\)
\(398\) −5.07938 −0.254606
\(399\) 0 0
\(400\) −11.7958 −0.589792
\(401\) −4.38749 −0.219101 −0.109551 0.993981i \(-0.534941\pi\)
−0.109551 + 0.993981i \(0.534941\pi\)
\(402\) 5.36812 0.267737
\(403\) 0 0
\(404\) −9.75513 −0.485336
\(405\) 20.4914 1.01822
\(406\) 0 0
\(407\) −10.6125 −0.526042
\(408\) 5.38749 0.266721
\(409\) −18.7884 −0.929027 −0.464513 0.885566i \(-0.653771\pi\)
−0.464513 + 0.885566i \(0.653771\pi\)
\(410\) 12.1833 0.601692
\(411\) 26.5813 1.31116
\(412\) 5.36812 0.264468
\(413\) 0 0
\(414\) 7.79583 0.383144
\(415\) −40.5708 −1.99154
\(416\) 0 0
\(417\) −2.40834 −0.117937
\(418\) −10.7362 −0.525126
\(419\) 25.7151 1.25627 0.628133 0.778106i \(-0.283820\pi\)
0.628133 + 0.778106i \(0.283820\pi\)
\(420\) 0 0
\(421\) 12.5917 0.613680 0.306840 0.951761i \(-0.400728\pi\)
0.306840 + 0.951761i \(0.400728\pi\)
\(422\) 7.79583 0.379495
\(423\) 2.82843 0.137523
\(424\) 37.7750 1.83452
\(425\) 14.9789 0.726582
\(426\) −8.48528 −0.411113
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −31.9494 −1.54074
\(431\) 33.1833 1.59838 0.799192 0.601075i \(-0.205261\pi\)
0.799192 + 0.601075i \(0.205261\pi\)
\(432\) −5.65685 −0.272166
\(433\) 20.4914 0.984752 0.492376 0.870383i \(-0.336129\pi\)
0.492376 + 0.870383i \(0.336129\pi\)
\(434\) 0 0
\(435\) 4.61251 0.221153
\(436\) −1.59166 −0.0762268
\(437\) −22.0499 −1.05479
\(438\) −17.7958 −0.850318
\(439\) 20.0583 0.957328 0.478664 0.877998i \(-0.341121\pi\)
0.478664 + 0.877998i \(0.341121\pi\)
\(440\) −46.6690 −2.22486
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000 0.475114 0.237557 0.971374i \(-0.423653\pi\)
0.237557 + 0.971374i \(0.423653\pi\)
\(444\) 3.95390 0.187644
\(445\) −61.3875 −2.91005
\(446\) 5.65685 0.267860
\(447\) 20.6357 0.976036
\(448\) 0 0
\(449\) 23.5917 1.11336 0.556680 0.830727i \(-0.312075\pi\)
0.556680 + 0.830727i \(0.312075\pi\)
\(450\) −11.7958 −0.556061
\(451\) 11.2842 0.531354
\(452\) −2.59166 −0.121902
\(453\) −2.25095 −0.105759
\(454\) −7.35981 −0.345413
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 4.59166 0.214789 0.107394 0.994216i \(-0.465749\pi\)
0.107394 + 0.994216i \(0.465749\pi\)
\(458\) −12.7279 −0.594737
\(459\) 7.18333 0.335289
\(460\) −31.9494 −1.48965
\(461\) −24.1860 −1.12645 −0.563227 0.826302i \(-0.690440\pi\)
−0.563227 + 0.826302i \(0.690440\pi\)
\(462\) 0 0
\(463\) −17.3875 −0.808065 −0.404033 0.914745i \(-0.632392\pi\)
−0.404033 + 0.914745i \(0.632392\pi\)
\(464\) −0.795832 −0.0369456
\(465\) 8.19654 0.380106
\(466\) 17.1833 0.796002
\(467\) 31.9494 1.47844 0.739222 0.673462i \(-0.235194\pi\)
0.739222 + 0.673462i \(0.235194\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 11.5917 0.534684
\(471\) −0.204168 −0.00940759
\(472\) −37.3176 −1.71768
\(473\) −29.5917 −1.36063
\(474\) 3.11716 0.143176
\(475\) 33.3636 1.53083
\(476\) 0 0
\(477\) 12.5917 0.576533
\(478\) −10.2042 −0.466728
\(479\) 20.9245 0.956063 0.478032 0.878343i \(-0.341350\pi\)
0.478032 + 0.878343i \(0.341350\pi\)
\(480\) 28.9792 1.32271
\(481\) 0 0
\(482\) −11.1693 −0.508749
\(483\) 0 0
\(484\) −3.40834 −0.154924
\(485\) −17.3875 −0.789525
\(486\) −9.89949 −0.449050
\(487\) 0.408337 0.0185035 0.00925176 0.999957i \(-0.497055\pi\)
0.00925176 + 0.999957i \(0.497055\pi\)
\(488\) 25.0227 1.13273
\(489\) −21.7612 −0.984076
\(490\) 0 0
\(491\) 9.59166 0.432866 0.216433 0.976298i \(-0.430558\pi\)
0.216433 + 0.976298i \(0.430558\pi\)
\(492\) −4.20417 −0.189539
\(493\) 1.01058 0.0455143
\(494\) 0 0
\(495\) −15.5563 −0.699206
\(496\) −1.41421 −0.0635001
\(497\) 0 0
\(498\) −14.0000 −0.627355
\(499\) 25.7958 1.15478 0.577390 0.816468i \(-0.304071\pi\)
0.577390 + 0.816468i \(0.304071\pi\)
\(500\) 27.8512 1.24554
\(501\) 2.40834 0.107597
\(502\) 16.6818 0.744546
\(503\) −25.7151 −1.14658 −0.573290 0.819353i \(-0.694333\pi\)
−0.573290 + 0.819353i \(0.694333\pi\)
\(504\) 0 0
\(505\) −39.9792 −1.77905
\(506\) 29.5917 1.31551
\(507\) 0 0
\(508\) 11.5917 0.514297
\(509\) −6.08996 −0.269933 −0.134966 0.990850i \(-0.543093\pi\)
−0.134966 + 0.990850i \(0.543093\pi\)
\(510\) 7.35981 0.325898
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 16.0000 0.706417
\(514\) −8.62965 −0.380638
\(515\) 22.0000 0.969436
\(516\) 11.0250 0.485347
\(517\) 10.7362 0.472179
\(518\) 0 0
\(519\) −25.1833 −1.10543
\(520\) 0 0
\(521\) −4.67575 −0.204848 −0.102424 0.994741i \(-0.532660\pi\)
−0.102424 + 0.994741i \(0.532660\pi\)
\(522\) −0.795832 −0.0348326
\(523\) −42.1377 −1.84255 −0.921276 0.388910i \(-0.872852\pi\)
−0.921276 + 0.388910i \(0.872852\pi\)
\(524\) 5.36812 0.234507
\(525\) 0 0
\(526\) 3.38749 0.147702
\(527\) 1.79583 0.0782276
\(528\) −5.36812 −0.233617
\(529\) 37.7750 1.64239
\(530\) 51.6041 2.24154
\(531\) −12.4392 −0.539815
\(532\) 0 0
\(533\) 0 0
\(534\) −21.1833 −0.916692
\(535\) 24.5896 1.06310
\(536\) 11.3875 0.491865
\(537\) 27.7068 1.19564
\(538\) −12.4392 −0.536291
\(539\) 0 0
\(540\) 23.1833 0.997652
\(541\) 6.59166 0.283398 0.141699 0.989910i \(-0.454744\pi\)
0.141699 + 0.989910i \(0.454744\pi\)
\(542\) −17.5186 −0.752487
\(543\) 5.38749 0.231200
\(544\) 6.34922 0.272221
\(545\) −6.52307 −0.279418
\(546\) 0 0
\(547\) 10.9792 0.469435 0.234717 0.972064i \(-0.424584\pi\)
0.234717 + 0.972064i \(0.424584\pi\)
\(548\) 18.7958 0.802918
\(549\) 8.34091 0.355981
\(550\) −44.7750 −1.90921
\(551\) 2.25095 0.0958938
\(552\) −33.0749 −1.40776
\(553\) 0 0
\(554\) 32.1833 1.36734
\(555\) 16.2042 0.687829
\(556\) −1.70295 −0.0722212
\(557\) 1.40834 0.0596732 0.0298366 0.999555i \(-0.490501\pi\)
0.0298366 + 0.999555i \(0.490501\pi\)
\(558\) −1.41421 −0.0598684
\(559\) 0 0
\(560\) 0 0
\(561\) 6.81667 0.287800
\(562\) 24.7958 1.04595
\(563\) −12.4392 −0.524249 −0.262125 0.965034i \(-0.584423\pi\)
−0.262125 + 0.965034i \(0.584423\pi\)
\(564\) −4.00000 −0.168430
\(565\) −10.6213 −0.446843
\(566\) 15.8451 0.666019
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) 25.5917 1.07286 0.536429 0.843945i \(-0.319773\pi\)
0.536429 + 0.843945i \(0.319773\pi\)
\(570\) 16.3931 0.686631
\(571\) −23.1833 −0.970192 −0.485096 0.874461i \(-0.661215\pi\)
−0.485096 + 0.874461i \(0.661215\pi\)
\(572\) 0 0
\(573\) 22.9162 0.957336
\(574\) 0 0
\(575\) −91.9583 −3.83493
\(576\) −7.00000 −0.291667
\(577\) −24.7340 −1.02969 −0.514845 0.857283i \(-0.672151\pi\)
−0.514845 + 0.857283i \(0.672151\pi\)
\(578\) −15.3875 −0.640035
\(579\) −31.9494 −1.32777
\(580\) 3.26153 0.135428
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 47.7958 1.97950
\(584\) −37.7507 −1.56213
\(585\) 0 0
\(586\) 2.97280 0.122805
\(587\) −16.6818 −0.688533 −0.344266 0.938872i \(-0.611872\pi\)
−0.344266 + 0.938872i \(0.611872\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) −50.9792 −2.09878
\(591\) 11.3137 0.465384
\(592\) −2.79583 −0.114908
\(593\) 8.34091 0.342520 0.171260 0.985226i \(-0.445216\pi\)
0.171260 + 0.985226i \(0.445216\pi\)
\(594\) −21.4725 −0.881026
\(595\) 0 0
\(596\) 14.5917 0.597698
\(597\) 7.18333 0.293994
\(598\) 0 0
\(599\) −33.5917 −1.37252 −0.686259 0.727357i \(-0.740748\pi\)
−0.686259 + 0.727357i \(0.740748\pi\)
\(600\) 50.0455 2.04310
\(601\) 24.1860 0.986567 0.493284 0.869868i \(-0.335796\pi\)
0.493284 + 0.869868i \(0.335796\pi\)
\(602\) 0 0
\(603\) 3.79583 0.154578
\(604\) −1.59166 −0.0647639
\(605\) −13.9683 −0.567892
\(606\) −13.7958 −0.560417
\(607\) −4.79064 −0.194446 −0.0972231 0.995263i \(-0.530996\pi\)
−0.0972231 + 0.995263i \(0.530996\pi\)
\(608\) 14.1421 0.573539
\(609\) 0 0
\(610\) 34.1833 1.38404
\(611\) 0 0
\(612\) 1.26984 0.0513304
\(613\) 41.9792 1.69552 0.847761 0.530378i \(-0.177950\pi\)
0.847761 + 0.530378i \(0.177950\pi\)
\(614\) 20.0583 0.809485
\(615\) −17.2298 −0.694774
\(616\) 0 0
\(617\) 4.38749 0.176634 0.0883169 0.996092i \(-0.471851\pi\)
0.0883169 + 0.996092i \(0.471851\pi\)
\(618\) 7.59166 0.305381
\(619\) 33.9116 1.36302 0.681512 0.731807i \(-0.261323\pi\)
0.681512 + 0.731807i \(0.261323\pi\)
\(620\) 5.79583 0.232766
\(621\) −44.0999 −1.76967
\(622\) 28.8323 1.15607
\(623\) 0 0
\(624\) 0 0
\(625\) 55.1625 2.20650
\(626\) −15.2676 −0.610216
\(627\) 15.1833 0.606364
\(628\) −0.144369 −0.00576095
\(629\) 3.55027 0.141559
\(630\) 0 0
\(631\) −39.5917 −1.57612 −0.788060 0.615599i \(-0.788914\pi\)
−0.788060 + 0.615599i \(0.788914\pi\)
\(632\) 6.61251 0.263031
\(633\) −11.0250 −0.438203
\(634\) 6.59166 0.261788
\(635\) 47.5058 1.88521
\(636\) −17.8073 −0.706105
\(637\) 0 0
\(638\) −3.02084 −0.119596
\(639\) −6.00000 −0.237356
\(640\) 12.2948 0.485995
\(641\) −4.79583 −0.189424 −0.0947120 0.995505i \(-0.530193\pi\)
−0.0947120 + 0.995505i \(0.530193\pi\)
\(642\) 8.48528 0.334887
\(643\) −5.65685 −0.223085 −0.111542 0.993760i \(-0.535579\pi\)
−0.111542 + 0.993760i \(0.535579\pi\)
\(644\) 0 0
\(645\) 45.1833 1.77909
\(646\) 3.59166 0.141312
\(647\) −38.7318 −1.52270 −0.761351 0.648339i \(-0.775464\pi\)
−0.761351 + 0.648339i \(0.775464\pi\)
\(648\) 15.0000 0.589256
\(649\) −47.2170 −1.85343
\(650\) 0 0
\(651\) 0 0
\(652\) −15.3875 −0.602621
\(653\) −13.1833 −0.515903 −0.257952 0.966158i \(-0.583048\pi\)
−0.257952 + 0.966158i \(0.583048\pi\)
\(654\) −2.25095 −0.0880192
\(655\) 22.0000 0.859611
\(656\) 2.97280 0.116068
\(657\) −12.5836 −0.490931
\(658\) 0 0
\(659\) −18.4083 −0.717087 −0.358543 0.933513i \(-0.616727\pi\)
−0.358543 + 0.933513i \(0.616727\pi\)
\(660\) 22.0000 0.856349
\(661\) −7.50417 −0.291879 −0.145939 0.989294i \(-0.546620\pi\)
−0.145939 + 0.989294i \(0.546620\pi\)
\(662\) 29.3875 1.14218
\(663\) 0 0
\(664\) −29.6985 −1.15252
\(665\) 0 0
\(666\) −2.79583 −0.108336
\(667\) −6.20417 −0.240226
\(668\) 1.70295 0.0658892
\(669\) −8.00000 −0.309298
\(670\) 15.5563 0.600994
\(671\) 31.6607 1.22225
\(672\) 0 0
\(673\) −23.9792 −0.924329 −0.462164 0.886794i \(-0.652927\pi\)
−0.462164 + 0.886794i \(0.652927\pi\)
\(674\) 17.9792 0.692531
\(675\) 66.7273 2.56833
\(676\) 0 0
\(677\) −7.64854 −0.293957 −0.146979 0.989140i \(-0.546955\pi\)
−0.146979 + 0.989140i \(0.546955\pi\)
\(678\) −3.66517 −0.140760
\(679\) 0 0
\(680\) 15.6125 0.598712
\(681\) 10.4083 0.398848
\(682\) −5.36812 −0.205556
\(683\) −6.77499 −0.259238 −0.129619 0.991564i \(-0.541375\pi\)
−0.129619 + 0.991564i \(0.541375\pi\)
\(684\) 2.82843 0.108148
\(685\) 77.0304 2.94318
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) −7.79583 −0.297213
\(689\) 0 0
\(690\) −45.1833 −1.72010
\(691\) 2.53969 0.0966143 0.0483072 0.998833i \(-0.484617\pi\)
0.0483072 + 0.998833i \(0.484617\pi\)
\(692\) −17.8073 −0.676932
\(693\) 0 0
\(694\) −32.9792 −1.25187
\(695\) −6.97916 −0.264735
\(696\) 3.37643 0.127983
\(697\) −3.77499 −0.142988
\(698\) −26.5813 −1.00612
\(699\) −24.3009 −0.919144
\(700\) 0 0
\(701\) −29.5917 −1.11766 −0.558831 0.829282i \(-0.688750\pi\)
−0.558831 + 0.829282i \(0.688750\pi\)
\(702\) 0 0
\(703\) 7.90781 0.298249
\(704\) −26.5708 −1.00143
\(705\) −16.3931 −0.617399
\(706\) −5.22375 −0.196598
\(707\) 0 0
\(708\) 17.5917 0.661135
\(709\) −9.20417 −0.345670 −0.172835 0.984951i \(-0.555293\pi\)
−0.172835 + 0.984951i \(0.555293\pi\)
\(710\) −24.5896 −0.922832
\(711\) 2.20417 0.0826628
\(712\) −44.9366 −1.68407
\(713\) −11.0250 −0.412888
\(714\) 0 0
\(715\) 0 0
\(716\) 19.5917 0.732175
\(717\) 14.4309 0.538931
\(718\) −4.00000 −0.149279
\(719\) −1.99169 −0.0742775 −0.0371387 0.999310i \(-0.511824\pi\)
−0.0371387 + 0.999310i \(0.511824\pi\)
\(720\) −4.09827 −0.152734
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 15.7958 0.587453
\(724\) 3.80953 0.141580
\(725\) 9.38749 0.348643
\(726\) −4.82012 −0.178891
\(727\) −32.4974 −1.20526 −0.602632 0.798020i \(-0.705881\pi\)
−0.602632 + 0.798020i \(0.705881\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) −51.5708 −1.90872
\(731\) 9.89949 0.366146
\(732\) −11.7958 −0.435986
\(733\) 33.2193 1.22698 0.613491 0.789702i \(-0.289765\pi\)
0.613491 + 0.789702i \(0.289765\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) −38.9792 −1.43679
\(737\) 14.4083 0.530738
\(738\) 2.97280 0.109430
\(739\) −23.1833 −0.852812 −0.426406 0.904532i \(-0.640221\pi\)
−0.426406 + 0.904532i \(0.640221\pi\)
\(740\) 11.4581 0.421207
\(741\) 0 0
\(742\) 0 0
\(743\) −29.1833 −1.07063 −0.535316 0.844652i \(-0.679808\pi\)
−0.535316 + 0.844652i \(0.679808\pi\)
\(744\) 6.00000 0.219971
\(745\) 59.8006 2.19092
\(746\) 6.59166 0.241338
\(747\) −9.89949 −0.362204
\(748\) 4.82012 0.176241
\(749\) 0 0
\(750\) 39.3875 1.43823
\(751\) −7.79583 −0.284474 −0.142237 0.989833i \(-0.545430\pi\)
−0.142237 + 0.989833i \(0.545430\pi\)
\(752\) 2.82843 0.103142
\(753\) −23.5917 −0.859728
\(754\) 0 0
\(755\) −6.52307 −0.237399
\(756\) 0 0
\(757\) 13.1833 0.479156 0.239578 0.970877i \(-0.422991\pi\)
0.239578 + 0.970877i \(0.422991\pi\)
\(758\) 25.3875 0.922115
\(759\) −41.8489 −1.51902
\(760\) 34.7750 1.26142
\(761\) 17.8073 0.645514 0.322757 0.946482i \(-0.395390\pi\)
0.322757 + 0.946482i \(0.395390\pi\)
\(762\) 16.3931 0.593859
\(763\) 0 0
\(764\) 16.2042 0.586246
\(765\) 5.20417 0.188157
\(766\) −12.1504 −0.439013
\(767\) 0 0
\(768\) 24.0416 0.867528
\(769\) −4.24264 −0.152994 −0.0764968 0.997070i \(-0.524373\pi\)
−0.0764968 + 0.997070i \(0.524373\pi\)
\(770\) 0 0
\(771\) 12.2042 0.439522
\(772\) −22.5917 −0.813092
\(773\) −34.7779 −1.25087 −0.625436 0.780275i \(-0.715079\pi\)
−0.625436 + 0.780275i \(0.715079\pi\)
\(774\) −7.79583 −0.280215
\(775\) 16.6818 0.599229
\(776\) −12.7279 −0.456906
\(777\) 0 0
\(778\) −0.387495 −0.0138924
\(779\) −8.40834 −0.301260
\(780\) 0 0
\(781\) −22.7750 −0.814953
\(782\) −9.89949 −0.354005
\(783\) 4.50190 0.160885
\(784\) 0 0
\(785\) −0.591663 −0.0211174
\(786\) 7.59166 0.270786
\(787\) 42.3969 1.51129 0.755644 0.654983i \(-0.227324\pi\)
0.755644 + 0.654983i \(0.227324\pi\)
\(788\) 8.00000 0.284988
\(789\) −4.79064 −0.170551
\(790\) 9.03328 0.321390
\(791\) 0 0
\(792\) −11.3875 −0.404637
\(793\) 0 0
\(794\) −7.07107 −0.250943
\(795\) −72.9792 −2.58830
\(796\) 5.07938 0.180034
\(797\) −0.836738 −0.0296388 −0.0148194 0.999890i \(-0.504717\pi\)
−0.0148194 + 0.999890i \(0.504717\pi\)
\(798\) 0 0
\(799\) −3.59166 −0.127064
\(800\) 58.9792 2.08523
\(801\) −14.9789 −0.529252
\(802\) −4.38749 −0.154928
\(803\) −47.7650 −1.68559
\(804\) −5.36812 −0.189319
\(805\) 0 0
\(806\) 0 0
\(807\) 17.5917 0.619256
\(808\) −29.2654 −1.02955
\(809\) −37.3667 −1.31374 −0.656871 0.754003i \(-0.728120\pi\)
−0.656871 + 0.754003i \(0.728120\pi\)
\(810\) 20.4914 0.719993
\(811\) 52.0372 1.82727 0.913636 0.406533i \(-0.133262\pi\)
0.913636 + 0.406533i \(0.133262\pi\)
\(812\) 0 0
\(813\) 24.7750 0.868897
\(814\) −10.6125 −0.371968
\(815\) −63.0621 −2.20897
\(816\) 1.79583 0.0628667
\(817\) 22.0499 0.771430
\(818\) −18.7884 −0.656921
\(819\) 0 0
\(820\) −12.1833 −0.425460
\(821\) 36.3667 1.26920 0.634602 0.772839i \(-0.281164\pi\)
0.634602 + 0.772839i \(0.281164\pi\)
\(822\) 26.5813 0.927130
\(823\) −20.7750 −0.724171 −0.362085 0.932145i \(-0.617935\pi\)
−0.362085 + 0.932145i \(0.617935\pi\)
\(824\) 16.1043 0.561022
\(825\) 63.3214 2.20457
\(826\) 0 0
\(827\) −20.9792 −0.729517 −0.364758 0.931102i \(-0.618848\pi\)
−0.364758 + 0.931102i \(0.618848\pi\)
\(828\) −7.79583 −0.270924
\(829\) 27.2737 0.947254 0.473627 0.880725i \(-0.342944\pi\)
0.473627 + 0.880725i \(0.342944\pi\)
\(830\) −40.5708 −1.40823
\(831\) −45.5141 −1.57887
\(832\) 0 0
\(833\) 0 0
\(834\) −2.40834 −0.0833939
\(835\) 6.97916 0.241524
\(836\) 10.7362 0.371320
\(837\) 8.00000 0.276520
\(838\) 25.7151 0.888314
\(839\) −5.65685 −0.195296 −0.0976481 0.995221i \(-0.531132\pi\)
−0.0976481 + 0.995221i \(0.531132\pi\)
\(840\) 0 0
\(841\) −28.3667 −0.978160
\(842\) 12.5917 0.433937
\(843\) −35.0666 −1.20776
\(844\) −7.79583 −0.268344
\(845\) 0 0
\(846\) 2.82843 0.0972433
\(847\) 0 0
\(848\) 12.5917 0.432399
\(849\) −22.4083 −0.769052
\(850\) 14.9789 0.513771
\(851\) −21.7958 −0.747151
\(852\) 8.48528 0.290701
\(853\) 33.5080 1.14729 0.573646 0.819103i \(-0.305528\pi\)
0.573646 + 0.819103i \(0.305528\pi\)
\(854\) 0 0
\(855\) 11.5917 0.396427
\(856\) 18.0000 0.615227
\(857\) 17.4037 0.594498 0.297249 0.954800i \(-0.403931\pi\)
0.297249 + 0.954800i \(0.403931\pi\)
\(858\) 0 0
\(859\) −10.1882 −0.347618 −0.173809 0.984779i \(-0.555608\pi\)
−0.173809 + 0.984779i \(0.555608\pi\)
\(860\) 31.9494 1.08947
\(861\) 0 0
\(862\) 33.1833 1.13023
\(863\) 8.20417 0.279273 0.139637 0.990203i \(-0.455407\pi\)
0.139637 + 0.990203i \(0.455407\pi\)
\(864\) 28.2843 0.962250
\(865\) −72.9792 −2.48137
\(866\) 20.4914 0.696325
\(867\) 21.7612 0.739049
\(868\) 0 0
\(869\) 8.36665 0.283819
\(870\) 4.61251 0.156379
\(871\) 0 0
\(872\) −4.77499 −0.161702
\(873\) −4.24264 −0.143592
\(874\) −22.0499 −0.745850
\(875\) 0 0
\(876\) 17.7958 0.601265
\(877\) 4.79583 0.161944 0.0809719 0.996716i \(-0.474198\pi\)
0.0809719 + 0.996716i \(0.474198\pi\)
\(878\) 20.0583 0.676933
\(879\) −4.20417 −0.141803
\(880\) −15.5563 −0.524404
\(881\) −10.5919 −0.356849 −0.178424 0.983954i \(-0.557100\pi\)
−0.178424 + 0.983954i \(0.557100\pi\)
\(882\) 0 0
\(883\) −45.3875 −1.52741 −0.763705 0.645565i \(-0.776622\pi\)
−0.763705 + 0.645565i \(0.776622\pi\)
\(884\) 0 0
\(885\) 72.0954 2.42346
\(886\) 10.0000 0.335957
\(887\) 42.1377 1.41484 0.707422 0.706791i \(-0.249858\pi\)
0.707422 + 0.706791i \(0.249858\pi\)
\(888\) 11.8617 0.398053
\(889\) 0 0
\(890\) −61.3875 −2.05771
\(891\) 18.9792 0.635826
\(892\) −5.65685 −0.189405
\(893\) −8.00000 −0.267710
\(894\) 20.6357 0.690162
\(895\) 80.2920 2.68386
\(896\) 0 0
\(897\) 0 0
\(898\) 23.5917 0.787264
\(899\) 1.12548 0.0375367
\(900\) 11.7958 0.393194
\(901\) −15.9895 −0.532686
\(902\) 11.2842 0.375724
\(903\) 0 0
\(904\) −7.77499 −0.258592
\(905\) 15.6125 0.518977
\(906\) −2.25095 −0.0747829
\(907\) 20.6125 0.684427 0.342214 0.939622i \(-0.388823\pi\)
0.342214 + 0.939622i \(0.388823\pi\)
\(908\) 7.35981 0.244244
\(909\) −9.75513 −0.323557
\(910\) 0 0
\(911\) −4.81667 −0.159584 −0.0797918 0.996812i \(-0.525426\pi\)
−0.0797918 + 0.996812i \(0.525426\pi\)
\(912\) 4.00000 0.132453
\(913\) −37.5768 −1.24361
\(914\) 4.59166 0.151879
\(915\) −48.3425 −1.59815
\(916\) 12.7279 0.420542
\(917\) 0 0
\(918\) 7.18333 0.237085
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) −95.8483 −3.16003
\(921\) −28.3667 −0.934713
\(922\) −24.1860 −0.796523
\(923\) 0 0
\(924\) 0 0
\(925\) 32.9792 1.08435
\(926\) −17.3875 −0.571389
\(927\) 5.36812 0.176312
\(928\) 3.97916 0.130622
\(929\) 21.3576 0.700719 0.350360 0.936615i \(-0.386059\pi\)
0.350360 + 0.936615i \(0.386059\pi\)
\(930\) 8.19654 0.268775
\(931\) 0 0
\(932\) −17.1833 −0.562859
\(933\) −40.7750 −1.33491
\(934\) 31.9494 1.04542
\(935\) 19.7541 0.646030
\(936\) 0 0
\(937\) 13.7090 0.447854 0.223927 0.974606i \(-0.428112\pi\)
0.223927 + 0.974606i \(0.428112\pi\)
\(938\) 0 0
\(939\) 21.5917 0.704617
\(940\) −11.5917 −0.378078
\(941\) −3.66517 −0.119481 −0.0597405 0.998214i \(-0.519027\pi\)
−0.0597405 + 0.998214i \(0.519027\pi\)
\(942\) −0.204168 −0.00665217
\(943\) 23.1754 0.754695
\(944\) −12.4392 −0.404861
\(945\) 0 0
\(946\) −29.5917 −0.962108
\(947\) 37.7958 1.22820 0.614100 0.789228i \(-0.289519\pi\)
0.614100 + 0.789228i \(0.289519\pi\)
\(948\) −3.11716 −0.101241
\(949\) 0 0
\(950\) 33.3636 1.08246
\(951\) −9.32202 −0.302287
\(952\) 0 0
\(953\) −43.1833 −1.39885 −0.699423 0.714708i \(-0.746559\pi\)
−0.699423 + 0.714708i \(0.746559\pi\)
\(954\) 12.5917 0.407670
\(955\) 66.4091 2.14895
\(956\) 10.2042 0.330026
\(957\) 4.27212 0.138098
\(958\) 20.9245 0.676039
\(959\) 0 0
\(960\) 40.5708 1.30942
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 11.1693 0.359740
\(965\) −92.5868 −2.98047
\(966\) 0 0
\(967\) −17.3875 −0.559144 −0.279572 0.960125i \(-0.590193\pi\)
−0.279572 + 0.960125i \(0.590193\pi\)
\(968\) −10.2250 −0.328644
\(969\) −5.07938 −0.163173
\(970\) −17.3875 −0.558279
\(971\) 22.6274 0.726148 0.363074 0.931760i \(-0.381727\pi\)
0.363074 + 0.931760i \(0.381727\pi\)
\(972\) 9.89949 0.317526
\(973\) 0 0
\(974\) 0.408337 0.0130840
\(975\) 0 0
\(976\) 8.34091 0.266986
\(977\) −28.3875 −0.908196 −0.454098 0.890952i \(-0.650038\pi\)
−0.454098 + 0.890952i \(0.650038\pi\)
\(978\) −21.7612 −0.695847
\(979\) −56.8573 −1.81717
\(980\) 0 0
\(981\) −1.59166 −0.0508179
\(982\) 9.59166 0.306082
\(983\) −0.866213 −0.0276279 −0.0138140 0.999905i \(-0.504397\pi\)
−0.0138140 + 0.999905i \(0.504397\pi\)
\(984\) −12.6125 −0.402072
\(985\) 32.7862 1.04465
\(986\) 1.01058 0.0321835
\(987\) 0 0
\(988\) 0 0
\(989\) −60.7750 −1.93253
\(990\) −15.5563 −0.494413
\(991\) 27.7958 0.882964 0.441482 0.897270i \(-0.354453\pi\)
0.441482 + 0.897270i \(0.354453\pi\)
\(992\) 7.07107 0.224507
\(993\) −41.5602 −1.31887
\(994\) 0 0
\(995\) 20.8167 0.659933
\(996\) 14.0000 0.443607
\(997\) −1.26984 −0.0402164 −0.0201082 0.999798i \(-0.506401\pi\)
−0.0201082 + 0.999798i \(0.506401\pi\)
\(998\) 25.7958 0.816553
\(999\) 15.8156 0.500384
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 8281.2.a.bv.1.1 4
7.6 odd 2 inner 8281.2.a.bv.1.4 4
13.3 even 3 637.2.f.g.295.3 yes 8
13.9 even 3 637.2.f.g.393.3 yes 8
13.12 even 2 8281.2.a.bn.1.2 4
91.3 odd 6 637.2.g.h.373.3 8
91.9 even 3 637.2.g.h.263.2 8
91.16 even 3 637.2.h.k.165.4 8
91.48 odd 6 637.2.f.g.393.2 yes 8
91.55 odd 6 637.2.f.g.295.2 8
91.61 odd 6 637.2.g.h.263.3 8
91.68 odd 6 637.2.h.k.165.1 8
91.74 even 3 637.2.h.k.471.4 8
91.81 even 3 637.2.g.h.373.2 8
91.87 odd 6 637.2.h.k.471.1 8
91.90 odd 2 8281.2.a.bn.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.f.g.295.2 8 91.55 odd 6
637.2.f.g.295.3 yes 8 13.3 even 3
637.2.f.g.393.2 yes 8 91.48 odd 6
637.2.f.g.393.3 yes 8 13.9 even 3
637.2.g.h.263.2 8 91.9 even 3
637.2.g.h.263.3 8 91.61 odd 6
637.2.g.h.373.2 8 91.81 even 3
637.2.g.h.373.3 8 91.3 odd 6
637.2.h.k.165.1 8 91.68 odd 6
637.2.h.k.165.4 8 91.16 even 3
637.2.h.k.471.1 8 91.87 odd 6
637.2.h.k.471.4 8 91.74 even 3
8281.2.a.bn.1.2 4 13.12 even 2
8281.2.a.bn.1.3 4 91.90 odd 2
8281.2.a.bv.1.1 4 1.1 even 1 trivial
8281.2.a.bv.1.4 4 7.6 odd 2 inner