Properties

Label 177.4.a.d.1.5
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.04902\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.04902 q^{2} +3.00000 q^{3} -3.80150 q^{4} +16.1855 q^{5} +6.14707 q^{6} -1.13960 q^{7} -24.1816 q^{8} +9.00000 q^{9} +33.1645 q^{10} +36.3258 q^{11} -11.4045 q^{12} +78.1440 q^{13} -2.33506 q^{14} +48.5565 q^{15} -19.1366 q^{16} -43.6811 q^{17} +18.4412 q^{18} -18.4272 q^{19} -61.5292 q^{20} -3.41879 q^{21} +74.4325 q^{22} -4.45159 q^{23} -72.5447 q^{24} +136.971 q^{25} +160.119 q^{26} +27.0000 q^{27} +4.33218 q^{28} -161.921 q^{29} +99.4936 q^{30} +245.150 q^{31} +154.241 q^{32} +108.977 q^{33} -89.5037 q^{34} -18.4450 q^{35} -34.2135 q^{36} -173.192 q^{37} -37.7579 q^{38} +234.432 q^{39} -391.391 q^{40} +128.039 q^{41} -7.00519 q^{42} -229.176 q^{43} -138.092 q^{44} +145.670 q^{45} -9.12142 q^{46} -138.962 q^{47} -57.4099 q^{48} -341.701 q^{49} +280.657 q^{50} -131.043 q^{51} -297.064 q^{52} -168.587 q^{53} +55.3237 q^{54} +587.952 q^{55} +27.5572 q^{56} -55.2817 q^{57} -331.780 q^{58} +59.0000 q^{59} -184.588 q^{60} -878.522 q^{61} +502.318 q^{62} -10.2564 q^{63} +469.137 q^{64} +1264.80 q^{65} +223.297 q^{66} +361.058 q^{67} +166.054 q^{68} -13.3548 q^{69} -37.7942 q^{70} -761.904 q^{71} -217.634 q^{72} -404.740 q^{73} -354.874 q^{74} +410.913 q^{75} +70.0511 q^{76} -41.3968 q^{77} +480.357 q^{78} -493.825 q^{79} -309.736 q^{80} +81.0000 q^{81} +262.356 q^{82} +398.428 q^{83} +12.9965 q^{84} -707.002 q^{85} -469.588 q^{86} -485.762 q^{87} -878.415 q^{88} -1327.94 q^{89} +298.481 q^{90} -89.0527 q^{91} +16.9227 q^{92} +735.449 q^{93} -284.736 q^{94} -298.254 q^{95} +462.723 q^{96} -1462.81 q^{97} -700.154 q^{98} +326.932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + 21q^{10} + 67q^{11} + 102q^{12} + 33q^{13} + 79q^{14} + 126q^{15} - 30q^{16} + 139q^{17} + 54q^{18} + 64q^{19} + 117q^{20} + 159q^{21} - 84q^{22} + 226q^{23} + 153q^{24} + 96q^{25} + 24q^{26} + 216q^{27} + 34q^{28} + 456q^{29} + 63q^{30} + 124q^{31} + 174q^{32} + 201q^{33} - 114q^{34} + 556q^{35} + 306q^{36} + 127q^{37} + 237q^{38} + 99q^{39} - 188q^{40} + 425q^{41} + 237q^{42} - 115q^{43} + 510q^{44} + 378q^{45} - 711q^{46} + 420q^{47} - 90q^{48} + 171q^{49} - 137q^{50} + 417q^{51} - 922q^{52} + 98q^{53} + 162q^{54} - 616q^{55} - 412q^{56} + 192q^{57} - 1548q^{58} + 472q^{59} + 351q^{60} - 1254q^{61} - 766q^{62} + 477q^{63} - 2019q^{64} - 734q^{65} - 252q^{66} - 1010q^{67} - 503q^{68} + 678q^{69} - 2956q^{70} - 17q^{71} + 459q^{72} - 1180q^{73} - 1228q^{74} + 288q^{75} - 2008q^{76} + 441q^{77} + 72q^{78} - 873q^{79} - 865q^{80} + 648q^{81} - 3645q^{82} + 759q^{83} + 102q^{84} - 850q^{85} - 1226q^{86} + 1368q^{87} - 3047q^{88} + 988q^{89} + 189q^{90} - 2111q^{91} - 1062q^{92} + 372q^{93} - 2240q^{94} + 1822q^{95} + 522q^{96} - 668q^{97} - 1368q^{98} + 603q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04902 0.724440 0.362220 0.932093i \(-0.382019\pi\)
0.362220 + 0.932093i \(0.382019\pi\)
\(3\) 3.00000 0.577350
\(4\) −3.80150 −0.475187
\(5\) 16.1855 1.44768 0.723838 0.689970i \(-0.242376\pi\)
0.723838 + 0.689970i \(0.242376\pi\)
\(6\) 6.14707 0.418255
\(7\) −1.13960 −0.0615325 −0.0307662 0.999527i \(-0.509795\pi\)
−0.0307662 + 0.999527i \(0.509795\pi\)
\(8\) −24.1816 −1.06868
\(9\) 9.00000 0.333333
\(10\) 33.1645 1.04875
\(11\) 36.3258 0.995695 0.497847 0.867265i \(-0.334124\pi\)
0.497847 + 0.867265i \(0.334124\pi\)
\(12\) −11.4045 −0.274349
\(13\) 78.1440 1.66717 0.833586 0.552389i \(-0.186284\pi\)
0.833586 + 0.552389i \(0.186284\pi\)
\(14\) −2.33506 −0.0445766
\(15\) 48.5565 0.835816
\(16\) −19.1366 −0.299010
\(17\) −43.6811 −0.623190 −0.311595 0.950215i \(-0.600863\pi\)
−0.311595 + 0.950215i \(0.600863\pi\)
\(18\) 18.4412 0.241480
\(19\) −18.4272 −0.222500 −0.111250 0.993792i \(-0.535485\pi\)
−0.111250 + 0.993792i \(0.535485\pi\)
\(20\) −61.5292 −0.687917
\(21\) −3.41879 −0.0355258
\(22\) 74.4325 0.721321
\(23\) −4.45159 −0.0403574 −0.0201787 0.999796i \(-0.506424\pi\)
−0.0201787 + 0.999796i \(0.506424\pi\)
\(24\) −72.5447 −0.617005
\(25\) 136.971 1.09577
\(26\) 160.119 1.20777
\(27\) 27.0000 0.192450
\(28\) 4.33218 0.0292394
\(29\) −161.921 −1.03682 −0.518412 0.855131i \(-0.673477\pi\)
−0.518412 + 0.855131i \(0.673477\pi\)
\(30\) 99.4936 0.605499
\(31\) 245.150 1.42033 0.710164 0.704036i \(-0.248621\pi\)
0.710164 + 0.704036i \(0.248621\pi\)
\(32\) 154.241 0.852069
\(33\) 108.977 0.574865
\(34\) −89.5037 −0.451464
\(35\) −18.4450 −0.0890791
\(36\) −34.2135 −0.158396
\(37\) −173.192 −0.769528 −0.384764 0.923015i \(-0.625717\pi\)
−0.384764 + 0.923015i \(0.625717\pi\)
\(38\) −37.7579 −0.161188
\(39\) 234.432 0.962542
\(40\) −391.391 −1.54711
\(41\) 128.039 0.487716 0.243858 0.969811i \(-0.421587\pi\)
0.243858 + 0.969811i \(0.421587\pi\)
\(42\) −7.00519 −0.0257363
\(43\) −229.176 −0.812769 −0.406384 0.913702i \(-0.633211\pi\)
−0.406384 + 0.913702i \(0.633211\pi\)
\(44\) −138.092 −0.473141
\(45\) 145.670 0.482559
\(46\) −9.12142 −0.0292365
\(47\) −138.962 −0.431269 −0.215635 0.976474i \(-0.569182\pi\)
−0.215635 + 0.976474i \(0.569182\pi\)
\(48\) −57.4099 −0.172634
\(49\) −341.701 −0.996214
\(50\) 280.657 0.793817
\(51\) −131.043 −0.359799
\(52\) −297.064 −0.792219
\(53\) −168.587 −0.436929 −0.218464 0.975845i \(-0.570105\pi\)
−0.218464 + 0.975845i \(0.570105\pi\)
\(54\) 55.3237 0.139418
\(55\) 587.952 1.44144
\(56\) 27.5572 0.0657588
\(57\) −55.2817 −0.128460
\(58\) −331.780 −0.751117
\(59\) 59.0000 0.130189
\(60\) −184.588 −0.397169
\(61\) −878.522 −1.84399 −0.921994 0.387205i \(-0.873441\pi\)
−0.921994 + 0.387205i \(0.873441\pi\)
\(62\) 502.318 1.02894
\(63\) −10.2564 −0.0205108
\(64\) 469.137 0.916283
\(65\) 1264.80 2.41353
\(66\) 223.297 0.416455
\(67\) 361.058 0.658362 0.329181 0.944267i \(-0.393227\pi\)
0.329181 + 0.944267i \(0.393227\pi\)
\(68\) 166.054 0.296132
\(69\) −13.3548 −0.0233004
\(70\) −37.7942 −0.0645324
\(71\) −761.904 −1.27354 −0.636771 0.771053i \(-0.719730\pi\)
−0.636771 + 0.771053i \(0.719730\pi\)
\(72\) −217.634 −0.356228
\(73\) −404.740 −0.648921 −0.324460 0.945899i \(-0.605183\pi\)
−0.324460 + 0.945899i \(0.605183\pi\)
\(74\) −354.874 −0.557477
\(75\) 410.913 0.632641
\(76\) 70.0511 0.105729
\(77\) −41.3968 −0.0612676
\(78\) 480.357 0.697304
\(79\) −493.825 −0.703286 −0.351643 0.936134i \(-0.614377\pi\)
−0.351643 + 0.936134i \(0.614377\pi\)
\(80\) −309.736 −0.432870
\(81\) 81.0000 0.111111
\(82\) 262.356 0.353321
\(83\) 398.428 0.526906 0.263453 0.964672i \(-0.415139\pi\)
0.263453 + 0.964672i \(0.415139\pi\)
\(84\) 12.9965 0.0168814
\(85\) −707.002 −0.902177
\(86\) −469.588 −0.588802
\(87\) −485.762 −0.598611
\(88\) −878.415 −1.06408
\(89\) −1327.94 −1.58159 −0.790796 0.612080i \(-0.790333\pi\)
−0.790796 + 0.612080i \(0.790333\pi\)
\(90\) 298.481 0.349585
\(91\) −89.0527 −0.102585
\(92\) 16.9227 0.0191773
\(93\) 735.449 0.820027
\(94\) −284.736 −0.312429
\(95\) −298.254 −0.322108
\(96\) 462.723 0.491942
\(97\) −1462.81 −1.53120 −0.765598 0.643320i \(-0.777557\pi\)
−0.765598 + 0.643320i \(0.777557\pi\)
\(98\) −700.154 −0.721697
\(99\) 326.932 0.331898
\(100\) −520.694 −0.520694
\(101\) 228.118 0.224738 0.112369 0.993667i \(-0.464156\pi\)
0.112369 + 0.993667i \(0.464156\pi\)
\(102\) −268.511 −0.260653
\(103\) 764.688 0.731524 0.365762 0.930708i \(-0.380808\pi\)
0.365762 + 0.930708i \(0.380808\pi\)
\(104\) −1889.64 −1.78168
\(105\) −55.3349 −0.0514299
\(106\) −345.439 −0.316529
\(107\) 1557.21 1.40692 0.703462 0.710733i \(-0.251637\pi\)
0.703462 + 0.710733i \(0.251637\pi\)
\(108\) −102.640 −0.0914498
\(109\) 1589.34 1.39662 0.698310 0.715796i \(-0.253936\pi\)
0.698310 + 0.715796i \(0.253936\pi\)
\(110\) 1204.73 1.04424
\(111\) −519.575 −0.444287
\(112\) 21.8081 0.0183988
\(113\) 1409.48 1.17339 0.586694 0.809809i \(-0.300429\pi\)
0.586694 + 0.809809i \(0.300429\pi\)
\(114\) −113.274 −0.0930618
\(115\) −72.0513 −0.0584245
\(116\) 615.541 0.492686
\(117\) 703.296 0.555724
\(118\) 120.892 0.0943140
\(119\) 49.7789 0.0383464
\(120\) −1174.17 −0.893224
\(121\) −11.4358 −0.00859188
\(122\) −1800.11 −1.33586
\(123\) 384.118 0.281583
\(124\) −931.935 −0.674921
\(125\) 193.755 0.138640
\(126\) −21.0156 −0.0148589
\(127\) −1270.35 −0.887603 −0.443802 0.896125i \(-0.646370\pi\)
−0.443802 + 0.896125i \(0.646370\pi\)
\(128\) −272.655 −0.188278
\(129\) −687.529 −0.469252
\(130\) 2591.61 1.74845
\(131\) 2502.45 1.66901 0.834504 0.551002i \(-0.185754\pi\)
0.834504 + 0.551002i \(0.185754\pi\)
\(132\) −414.277 −0.273168
\(133\) 20.9996 0.0136910
\(134\) 739.817 0.476944
\(135\) 437.009 0.278605
\(136\) 1056.28 0.665993
\(137\) 458.017 0.285628 0.142814 0.989750i \(-0.454385\pi\)
0.142814 + 0.989750i \(0.454385\pi\)
\(138\) −27.3643 −0.0168797
\(139\) 286.953 0.175101 0.0875505 0.996160i \(-0.472096\pi\)
0.0875505 + 0.996160i \(0.472096\pi\)
\(140\) 70.1185 0.0423293
\(141\) −416.885 −0.248993
\(142\) −1561.16 −0.922604
\(143\) 2838.64 1.65999
\(144\) −172.230 −0.0996700
\(145\) −2620.77 −1.50099
\(146\) −829.322 −0.470104
\(147\) −1025.10 −0.575164
\(148\) 658.388 0.365670
\(149\) 2607.82 1.43383 0.716915 0.697160i \(-0.245553\pi\)
0.716915 + 0.697160i \(0.245553\pi\)
\(150\) 841.970 0.458311
\(151\) −20.5789 −0.0110907 −0.00554533 0.999985i \(-0.501765\pi\)
−0.00554533 + 0.999985i \(0.501765\pi\)
\(152\) 445.600 0.237782
\(153\) −393.130 −0.207730
\(154\) −84.8231 −0.0443847
\(155\) 3967.87 2.05617
\(156\) −891.192 −0.457388
\(157\) −1903.56 −0.967649 −0.483825 0.875165i \(-0.660753\pi\)
−0.483825 + 0.875165i \(0.660753\pi\)
\(158\) −1011.86 −0.509488
\(159\) −505.762 −0.252261
\(160\) 2496.47 1.23352
\(161\) 5.07302 0.00248329
\(162\) 165.971 0.0804933
\(163\) −195.798 −0.0940862 −0.0470431 0.998893i \(-0.514980\pi\)
−0.0470431 + 0.998893i \(0.514980\pi\)
\(164\) −486.741 −0.231757
\(165\) 1763.86 0.832218
\(166\) 816.390 0.381712
\(167\) −2098.12 −0.972198 −0.486099 0.873904i \(-0.661581\pi\)
−0.486099 + 0.873904i \(0.661581\pi\)
\(168\) 82.6717 0.0379659
\(169\) 3909.48 1.77946
\(170\) −1448.66 −0.653573
\(171\) −165.845 −0.0741667
\(172\) 871.213 0.386217
\(173\) 1996.80 0.877537 0.438769 0.898600i \(-0.355415\pi\)
0.438769 + 0.898600i \(0.355415\pi\)
\(174\) −995.339 −0.433658
\(175\) −156.092 −0.0674253
\(176\) −695.154 −0.297723
\(177\) 177.000 0.0751646
\(178\) −2720.99 −1.14577
\(179\) −2474.76 −1.03336 −0.516682 0.856177i \(-0.672833\pi\)
−0.516682 + 0.856177i \(0.672833\pi\)
\(180\) −553.763 −0.229306
\(181\) 618.103 0.253830 0.126915 0.991914i \(-0.459492\pi\)
0.126915 + 0.991914i \(0.459492\pi\)
\(182\) −182.471 −0.0743168
\(183\) −2635.57 −1.06463
\(184\) 107.646 0.0431294
\(185\) −2803.20 −1.11403
\(186\) 1506.95 0.594060
\(187\) −1586.75 −0.620507
\(188\) 528.263 0.204934
\(189\) −30.7691 −0.0118419
\(190\) −611.131 −0.233348
\(191\) 552.196 0.209191 0.104596 0.994515i \(-0.466645\pi\)
0.104596 + 0.994515i \(0.466645\pi\)
\(192\) 1407.41 0.529016
\(193\) 4002.75 1.49287 0.746436 0.665457i \(-0.231763\pi\)
0.746436 + 0.665457i \(0.231763\pi\)
\(194\) −2997.34 −1.10926
\(195\) 3794.40 1.39345
\(196\) 1298.98 0.473388
\(197\) 1380.23 0.499173 0.249586 0.968353i \(-0.419705\pi\)
0.249586 + 0.968353i \(0.419705\pi\)
\(198\) 669.892 0.240440
\(199\) 2132.94 0.759800 0.379900 0.925028i \(-0.375958\pi\)
0.379900 + 0.925028i \(0.375958\pi\)
\(200\) −3312.17 −1.17103
\(201\) 1083.17 0.380106
\(202\) 467.419 0.162809
\(203\) 184.524 0.0637984
\(204\) 498.161 0.170972
\(205\) 2072.38 0.706055
\(206\) 1566.86 0.529945
\(207\) −40.0643 −0.0134525
\(208\) −1495.41 −0.498501
\(209\) −669.385 −0.221542
\(210\) −113.383 −0.0372578
\(211\) 6000.66 1.95783 0.978916 0.204265i \(-0.0654806\pi\)
0.978916 + 0.204265i \(0.0654806\pi\)
\(212\) 640.884 0.207623
\(213\) −2285.71 −0.735279
\(214\) 3190.76 1.01923
\(215\) −3709.34 −1.17663
\(216\) −652.902 −0.205668
\(217\) −279.372 −0.0873963
\(218\) 3256.60 1.01177
\(219\) −1214.22 −0.374655
\(220\) −2235.10 −0.684956
\(221\) −3413.42 −1.03897
\(222\) −1064.62 −0.321859
\(223\) −4919.66 −1.47733 −0.738666 0.674071i \(-0.764544\pi\)
−0.738666 + 0.674071i \(0.764544\pi\)
\(224\) −175.773 −0.0524299
\(225\) 1232.74 0.365256
\(226\) 2888.06 0.850048
\(227\) 1021.59 0.298701 0.149350 0.988784i \(-0.452282\pi\)
0.149350 + 0.988784i \(0.452282\pi\)
\(228\) 210.153 0.0610427
\(229\) 1745.58 0.503716 0.251858 0.967764i \(-0.418958\pi\)
0.251858 + 0.967764i \(0.418958\pi\)
\(230\) −147.635 −0.0423250
\(231\) −124.190 −0.0353728
\(232\) 3915.50 1.10804
\(233\) −1961.28 −0.551450 −0.275725 0.961237i \(-0.588918\pi\)
−0.275725 + 0.961237i \(0.588918\pi\)
\(234\) 1441.07 0.402589
\(235\) −2249.17 −0.624339
\(236\) −224.288 −0.0618641
\(237\) −1481.47 −0.406042
\(238\) 101.998 0.0277797
\(239\) 1822.98 0.493385 0.246692 0.969094i \(-0.420656\pi\)
0.246692 + 0.969094i \(0.420656\pi\)
\(240\) −929.209 −0.249917
\(241\) −6154.85 −1.64510 −0.822549 0.568695i \(-0.807449\pi\)
−0.822549 + 0.568695i \(0.807449\pi\)
\(242\) −23.4322 −0.00622430
\(243\) 243.000 0.0641500
\(244\) 3339.70 0.876239
\(245\) −5530.61 −1.44220
\(246\) 787.067 0.203990
\(247\) −1439.98 −0.370946
\(248\) −5928.10 −1.51788
\(249\) 1195.29 0.304209
\(250\) 397.009 0.100436
\(251\) 4896.77 1.23140 0.615700 0.787981i \(-0.288873\pi\)
0.615700 + 0.787981i \(0.288873\pi\)
\(252\) 38.9896 0.00974648
\(253\) −161.708 −0.0401837
\(254\) −2602.98 −0.643015
\(255\) −2121.00 −0.520872
\(256\) −4311.77 −1.05268
\(257\) −6238.24 −1.51413 −0.757063 0.653341i \(-0.773367\pi\)
−0.757063 + 0.653341i \(0.773367\pi\)
\(258\) −1408.76 −0.339945
\(259\) 197.369 0.0473510
\(260\) −4808.14 −1.14688
\(261\) −1457.29 −0.345608
\(262\) 5127.58 1.20910
\(263\) −5359.90 −1.25667 −0.628337 0.777941i \(-0.716264\pi\)
−0.628337 + 0.777941i \(0.716264\pi\)
\(264\) −2635.24 −0.614349
\(265\) −2728.67 −0.632532
\(266\) 43.0288 0.00991829
\(267\) −3983.83 −0.913132
\(268\) −1372.56 −0.312845
\(269\) −7201.10 −1.63219 −0.816094 0.577919i \(-0.803865\pi\)
−0.816094 + 0.577919i \(0.803865\pi\)
\(270\) 895.442 0.201833
\(271\) −1135.65 −0.254560 −0.127280 0.991867i \(-0.540625\pi\)
−0.127280 + 0.991867i \(0.540625\pi\)
\(272\) 835.910 0.186340
\(273\) −267.158 −0.0592276
\(274\) 938.489 0.206920
\(275\) 4975.58 1.09105
\(276\) 50.7682 0.0110720
\(277\) 1044.58 0.226580 0.113290 0.993562i \(-0.463861\pi\)
0.113290 + 0.993562i \(0.463861\pi\)
\(278\) 587.974 0.126850
\(279\) 2206.35 0.473443
\(280\) 446.028 0.0951974
\(281\) 6506.31 1.38126 0.690630 0.723209i \(-0.257333\pi\)
0.690630 + 0.723209i \(0.257333\pi\)
\(282\) −854.208 −0.180381
\(283\) −8390.09 −1.76233 −0.881164 0.472811i \(-0.843239\pi\)
−0.881164 + 0.472811i \(0.843239\pi\)
\(284\) 2896.38 0.605170
\(285\) −894.763 −0.185969
\(286\) 5816.45 1.20257
\(287\) −145.913 −0.0300104
\(288\) 1388.17 0.284023
\(289\) −3004.96 −0.611634
\(290\) −5370.02 −1.08737
\(291\) −4388.43 −0.884036
\(292\) 1538.62 0.308359
\(293\) 3643.33 0.726436 0.363218 0.931704i \(-0.381678\pi\)
0.363218 + 0.931704i \(0.381678\pi\)
\(294\) −2100.46 −0.416672
\(295\) 954.945 0.188471
\(296\) 4188.05 0.822383
\(297\) 980.797 0.191622
\(298\) 5343.48 1.03872
\(299\) −347.865 −0.0672828
\(300\) −1562.08 −0.300623
\(301\) 261.169 0.0500117
\(302\) −42.1667 −0.00803451
\(303\) 684.354 0.129753
\(304\) 352.636 0.0665297
\(305\) −14219.3 −2.66950
\(306\) −805.533 −0.150488
\(307\) −521.575 −0.0969637 −0.0484819 0.998824i \(-0.515438\pi\)
−0.0484819 + 0.998824i \(0.515438\pi\)
\(308\) 157.370 0.0291136
\(309\) 2294.06 0.422345
\(310\) 8130.27 1.48957
\(311\) 5344.76 0.974513 0.487256 0.873259i \(-0.337998\pi\)
0.487256 + 0.873259i \(0.337998\pi\)
\(312\) −5668.93 −1.02865
\(313\) 4368.71 0.788927 0.394463 0.918912i \(-0.370931\pi\)
0.394463 + 0.918912i \(0.370931\pi\)
\(314\) −3900.45 −0.701004
\(315\) −166.005 −0.0296930
\(316\) 1877.27 0.334193
\(317\) 5679.69 1.00632 0.503160 0.864193i \(-0.332171\pi\)
0.503160 + 0.864193i \(0.332171\pi\)
\(318\) −1036.32 −0.182748
\(319\) −5881.90 −1.03236
\(320\) 7593.22 1.32648
\(321\) 4671.62 0.812288
\(322\) 10.3948 0.00179900
\(323\) 804.923 0.138660
\(324\) −307.921 −0.0527986
\(325\) 10703.5 1.82683
\(326\) −401.194 −0.0681598
\(327\) 4768.03 0.806339
\(328\) −3096.19 −0.521215
\(329\) 158.361 0.0265371
\(330\) 3614.18 0.602892
\(331\) 3620.41 0.601196 0.300598 0.953751i \(-0.402814\pi\)
0.300598 + 0.953751i \(0.402814\pi\)
\(332\) −1514.62 −0.250379
\(333\) −1558.73 −0.256509
\(334\) −4299.09 −0.704299
\(335\) 5843.91 0.953096
\(336\) 65.4242 0.0106226
\(337\) −4951.57 −0.800383 −0.400191 0.916432i \(-0.631056\pi\)
−0.400191 + 0.916432i \(0.631056\pi\)
\(338\) 8010.63 1.28911
\(339\) 4228.44 0.677455
\(340\) 2687.66 0.428703
\(341\) 8905.26 1.41421
\(342\) −339.821 −0.0537293
\(343\) 780.284 0.122832
\(344\) 5541.84 0.868593
\(345\) −216.154 −0.0337314
\(346\) 4091.49 0.635723
\(347\) −8903.74 −1.37746 −0.688729 0.725019i \(-0.741831\pi\)
−0.688729 + 0.725019i \(0.741831\pi\)
\(348\) 1846.62 0.284452
\(349\) −10124.5 −1.55287 −0.776433 0.630200i \(-0.782973\pi\)
−0.776433 + 0.630200i \(0.782973\pi\)
\(350\) −319.836 −0.0488455
\(351\) 2109.89 0.320847
\(352\) 5602.93 0.848401
\(353\) 10278.4 1.54975 0.774877 0.632112i \(-0.217812\pi\)
0.774877 + 0.632112i \(0.217812\pi\)
\(354\) 362.677 0.0544522
\(355\) −12331.8 −1.84368
\(356\) 5048.17 0.751552
\(357\) 149.337 0.0221393
\(358\) −5070.84 −0.748610
\(359\) 11976.1 1.76065 0.880324 0.474374i \(-0.157325\pi\)
0.880324 + 0.474374i \(0.157325\pi\)
\(360\) −3522.52 −0.515703
\(361\) −6519.44 −0.950494
\(362\) 1266.51 0.183885
\(363\) −34.3074 −0.00496052
\(364\) 338.534 0.0487472
\(365\) −6550.92 −0.939427
\(366\) −5400.34 −0.771258
\(367\) 11740.3 1.66987 0.834933 0.550352i \(-0.185507\pi\)
0.834933 + 0.550352i \(0.185507\pi\)
\(368\) 85.1885 0.0120673
\(369\) 1152.35 0.162572
\(370\) −5743.82 −0.807046
\(371\) 192.122 0.0268853
\(372\) −2795.81 −0.389666
\(373\) −4055.41 −0.562952 −0.281476 0.959568i \(-0.590824\pi\)
−0.281476 + 0.959568i \(0.590824\pi\)
\(374\) −3251.29 −0.449520
\(375\) 581.265 0.0800437
\(376\) 3360.31 0.460891
\(377\) −12653.1 −1.72857
\(378\) −63.0467 −0.00857876
\(379\) −1798.03 −0.243691 −0.121845 0.992549i \(-0.538881\pi\)
−0.121845 + 0.992549i \(0.538881\pi\)
\(380\) 1133.81 0.153062
\(381\) −3811.06 −0.512458
\(382\) 1131.46 0.151546
\(383\) 7620.06 1.01662 0.508312 0.861173i \(-0.330270\pi\)
0.508312 + 0.861173i \(0.330270\pi\)
\(384\) −817.966 −0.108702
\(385\) −670.028 −0.0886956
\(386\) 8201.73 1.08150
\(387\) −2062.59 −0.270923
\(388\) 5560.87 0.727604
\(389\) −978.747 −0.127569 −0.0637846 0.997964i \(-0.520317\pi\)
−0.0637846 + 0.997964i \(0.520317\pi\)
\(390\) 7774.82 1.00947
\(391\) 194.451 0.0251504
\(392\) 8262.87 1.06464
\(393\) 7507.35 0.963602
\(394\) 2828.12 0.361621
\(395\) −7992.80 −1.01813
\(396\) −1242.83 −0.157714
\(397\) −2554.43 −0.322930 −0.161465 0.986878i \(-0.551622\pi\)
−0.161465 + 0.986878i \(0.551622\pi\)
\(398\) 4370.45 0.550429
\(399\) 62.9989 0.00790449
\(400\) −2621.16 −0.327645
\(401\) −12779.6 −1.59148 −0.795740 0.605639i \(-0.792918\pi\)
−0.795740 + 0.605639i \(0.792918\pi\)
\(402\) 2219.45 0.275364
\(403\) 19157.0 2.36793
\(404\) −867.190 −0.106793
\(405\) 1311.03 0.160853
\(406\) 378.095 0.0462181
\(407\) −6291.33 −0.766215
\(408\) 3168.83 0.384511
\(409\) −14112.8 −1.70620 −0.853099 0.521749i \(-0.825280\pi\)
−0.853099 + 0.521749i \(0.825280\pi\)
\(410\) 4246.36 0.511495
\(411\) 1374.05 0.164908
\(412\) −2906.96 −0.347611
\(413\) −67.2362 −0.00801085
\(414\) −82.0928 −0.00974551
\(415\) 6448.77 0.762789
\(416\) 12053.0 1.42055
\(417\) 860.859 0.101095
\(418\) −1371.59 −0.160494
\(419\) 3577.00 0.417060 0.208530 0.978016i \(-0.433132\pi\)
0.208530 + 0.978016i \(0.433132\pi\)
\(420\) 210.356 0.0244388
\(421\) −10454.2 −1.21023 −0.605116 0.796137i \(-0.706873\pi\)
−0.605116 + 0.796137i \(0.706873\pi\)
\(422\) 12295.5 1.41833
\(423\) −1250.66 −0.143756
\(424\) 4076.70 0.466939
\(425\) −5983.04 −0.682871
\(426\) −4683.48 −0.532666
\(427\) 1001.16 0.113465
\(428\) −5919.72 −0.668552
\(429\) 8515.93 0.958398
\(430\) −7600.52 −0.852395
\(431\) −12521.0 −1.39934 −0.699670 0.714466i \(-0.746670\pi\)
−0.699670 + 0.714466i \(0.746670\pi\)
\(432\) −516.689 −0.0575445
\(433\) 13582.0 1.50741 0.753707 0.657210i \(-0.228264\pi\)
0.753707 + 0.657210i \(0.228264\pi\)
\(434\) −572.440 −0.0633133
\(435\) −7862.31 −0.866595
\(436\) −6041.89 −0.663656
\(437\) 82.0306 0.00897953
\(438\) −2487.97 −0.271415
\(439\) 1232.27 0.133970 0.0669851 0.997754i \(-0.478662\pi\)
0.0669851 + 0.997754i \(0.478662\pi\)
\(440\) −14217.6 −1.54045
\(441\) −3075.31 −0.332071
\(442\) −6994.18 −0.752668
\(443\) 12543.1 1.34524 0.672622 0.739986i \(-0.265168\pi\)
0.672622 + 0.739986i \(0.265168\pi\)
\(444\) 1975.16 0.211120
\(445\) −21493.4 −2.28963
\(446\) −10080.5 −1.07024
\(447\) 7823.45 0.827822
\(448\) −534.627 −0.0563812
\(449\) −1370.68 −0.144068 −0.0720340 0.997402i \(-0.522949\pi\)
−0.0720340 + 0.997402i \(0.522949\pi\)
\(450\) 2525.91 0.264606
\(451\) 4651.13 0.485617
\(452\) −5358.14 −0.557579
\(453\) −61.7368 −0.00640319
\(454\) 2093.25 0.216391
\(455\) −1441.36 −0.148510
\(456\) 1336.80 0.137284
\(457\) −7052.45 −0.721881 −0.360941 0.932589i \(-0.617544\pi\)
−0.360941 + 0.932589i \(0.617544\pi\)
\(458\) 3576.73 0.364912
\(459\) −1179.39 −0.119933
\(460\) 273.903 0.0277626
\(461\) 5082.44 0.513476 0.256738 0.966481i \(-0.417352\pi\)
0.256738 + 0.966481i \(0.417352\pi\)
\(462\) −254.469 −0.0256255
\(463\) −5699.92 −0.572133 −0.286067 0.958210i \(-0.592348\pi\)
−0.286067 + 0.958210i \(0.592348\pi\)
\(464\) 3098.62 0.310021
\(465\) 11903.6 1.18713
\(466\) −4018.71 −0.399492
\(467\) 16957.5 1.68030 0.840149 0.542355i \(-0.182467\pi\)
0.840149 + 0.542355i \(0.182467\pi\)
\(468\) −2673.58 −0.264073
\(469\) −411.461 −0.0405107
\(470\) −4608.60 −0.452296
\(471\) −5710.69 −0.558673
\(472\) −1426.71 −0.139131
\(473\) −8325.02 −0.809270
\(474\) −3035.58 −0.294153
\(475\) −2524.00 −0.243808
\(476\) −189.234 −0.0182217
\(477\) −1517.28 −0.145643
\(478\) 3735.34 0.357427
\(479\) 15755.7 1.50291 0.751456 0.659783i \(-0.229352\pi\)
0.751456 + 0.659783i \(0.229352\pi\)
\(480\) 7489.41 0.712174
\(481\) −13533.9 −1.28294
\(482\) −12611.4 −1.19177
\(483\) 15.2191 0.00143373
\(484\) 43.4731 0.00408275
\(485\) −23676.4 −2.21668
\(486\) 497.913 0.0464728
\(487\) −6390.04 −0.594580 −0.297290 0.954787i \(-0.596083\pi\)
−0.297290 + 0.954787i \(0.596083\pi\)
\(488\) 21244.0 1.97064
\(489\) −587.393 −0.0543207
\(490\) −11332.4 −1.04478
\(491\) 15901.1 1.46152 0.730760 0.682635i \(-0.239166\pi\)
0.730760 + 0.682635i \(0.239166\pi\)
\(492\) −1460.22 −0.133805
\(493\) 7072.88 0.646139
\(494\) −2950.55 −0.268728
\(495\) 5291.57 0.480481
\(496\) −4691.34 −0.424692
\(497\) 868.264 0.0783641
\(498\) 2449.17 0.220381
\(499\) 3902.49 0.350100 0.175050 0.984560i \(-0.443991\pi\)
0.175050 + 0.984560i \(0.443991\pi\)
\(500\) −736.559 −0.0658799
\(501\) −6294.35 −0.561299
\(502\) 10033.6 0.892075
\(503\) −3985.69 −0.353306 −0.176653 0.984273i \(-0.556527\pi\)
−0.176653 + 0.984273i \(0.556527\pi\)
\(504\) 248.015 0.0219196
\(505\) 3692.21 0.325349
\(506\) −331.343 −0.0291107
\(507\) 11728.4 1.02737
\(508\) 4829.24 0.421778
\(509\) −4706.40 −0.409838 −0.204919 0.978779i \(-0.565693\pi\)
−0.204919 + 0.978779i \(0.565693\pi\)
\(510\) −4345.99 −0.377341
\(511\) 461.240 0.0399297
\(512\) −6653.69 −0.574325
\(513\) −497.536 −0.0428201
\(514\) −12782.3 −1.09689
\(515\) 12376.9 1.05901
\(516\) 2613.64 0.222983
\(517\) −5047.90 −0.429413
\(518\) 404.414 0.0343029
\(519\) 5990.40 0.506646
\(520\) −30584.9 −2.57930
\(521\) 3269.28 0.274913 0.137456 0.990508i \(-0.456107\pi\)
0.137456 + 0.990508i \(0.456107\pi\)
\(522\) −2986.02 −0.250372
\(523\) 6507.07 0.544043 0.272021 0.962291i \(-0.412308\pi\)
0.272021 + 0.962291i \(0.412308\pi\)
\(524\) −9513.06 −0.793091
\(525\) −468.275 −0.0389280
\(526\) −10982.6 −0.910385
\(527\) −10708.4 −0.885134
\(528\) −2085.46 −0.171890
\(529\) −12147.2 −0.998371
\(530\) −5591.11 −0.458231
\(531\) 531.000 0.0433963
\(532\) −79.8301 −0.00650578
\(533\) 10005.5 0.813107
\(534\) −8162.96 −0.661509
\(535\) 25204.2 2.03677
\(536\) −8730.95 −0.703581
\(537\) −7424.28 −0.596613
\(538\) −14755.2 −1.18242
\(539\) −12412.6 −0.991925
\(540\) −1661.29 −0.132390
\(541\) 17118.2 1.36039 0.680193 0.733033i \(-0.261896\pi\)
0.680193 + 0.733033i \(0.261896\pi\)
\(542\) −2326.97 −0.184413
\(543\) 1854.31 0.146549
\(544\) −6737.42 −0.531001
\(545\) 25724.3 2.02185
\(546\) −547.413 −0.0429068
\(547\) 18217.3 1.42398 0.711988 0.702192i \(-0.247795\pi\)
0.711988 + 0.702192i \(0.247795\pi\)
\(548\) −1741.15 −0.135727
\(549\) −7906.70 −0.614662
\(550\) 10195.1 0.790400
\(551\) 2983.75 0.230694
\(552\) 322.939 0.0249007
\(553\) 562.761 0.0432749
\(554\) 2140.37 0.164144
\(555\) −8409.59 −0.643184
\(556\) −1090.85 −0.0832057
\(557\) 11664.8 0.887347 0.443673 0.896189i \(-0.353675\pi\)
0.443673 + 0.896189i \(0.353675\pi\)
\(558\) 4520.86 0.342981
\(559\) −17908.8 −1.35503
\(560\) 352.975 0.0266355
\(561\) −4760.26 −0.358250
\(562\) 13331.6 1.00064
\(563\) 11698.6 0.875733 0.437867 0.899040i \(-0.355734\pi\)
0.437867 + 0.899040i \(0.355734\pi\)
\(564\) 1584.79 0.118319
\(565\) 22813.2 1.69869
\(566\) −17191.5 −1.27670
\(567\) −92.3074 −0.00683694
\(568\) 18424.0 1.36101
\(569\) 3094.45 0.227990 0.113995 0.993481i \(-0.463635\pi\)
0.113995 + 0.993481i \(0.463635\pi\)
\(570\) −1833.39 −0.134723
\(571\) 20401.7 1.49525 0.747623 0.664123i \(-0.231195\pi\)
0.747623 + 0.664123i \(0.231195\pi\)
\(572\) −10791.1 −0.788808
\(573\) 1656.59 0.120777
\(574\) −298.980 −0.0217407
\(575\) −609.739 −0.0442224
\(576\) 4222.23 0.305428
\(577\) −20917.5 −1.50920 −0.754599 0.656187i \(-0.772168\pi\)
−0.754599 + 0.656187i \(0.772168\pi\)
\(578\) −6157.24 −0.443092
\(579\) 12008.3 0.861910
\(580\) 9962.85 0.713250
\(581\) −454.048 −0.0324218
\(582\) −8992.01 −0.640431
\(583\) −6124.07 −0.435048
\(584\) 9787.24 0.693491
\(585\) 11383.2 0.804509
\(586\) 7465.28 0.526259
\(587\) −18046.9 −1.26895 −0.634475 0.772943i \(-0.718784\pi\)
−0.634475 + 0.772943i \(0.718784\pi\)
\(588\) 3896.93 0.273311
\(589\) −4517.43 −0.316023
\(590\) 1956.71 0.136536
\(591\) 4140.68 0.288198
\(592\) 3314.31 0.230097
\(593\) 5854.76 0.405441 0.202720 0.979237i \(-0.435022\pi\)
0.202720 + 0.979237i \(0.435022\pi\)
\(594\) 2009.68 0.138818
\(595\) 805.697 0.0555132
\(596\) −9913.61 −0.681338
\(597\) 6398.82 0.438671
\(598\) −712.784 −0.0487423
\(599\) −5850.74 −0.399090 −0.199545 0.979889i \(-0.563946\pi\)
−0.199545 + 0.979889i \(0.563946\pi\)
\(600\) −9936.51 −0.676094
\(601\) −5057.35 −0.343251 −0.171625 0.985162i \(-0.554902\pi\)
−0.171625 + 0.985162i \(0.554902\pi\)
\(602\) 535.141 0.0362305
\(603\) 3249.52 0.219454
\(604\) 78.2307 0.00527014
\(605\) −185.094 −0.0124383
\(606\) 1402.26 0.0939981
\(607\) 7535.66 0.503893 0.251946 0.967741i \(-0.418929\pi\)
0.251946 + 0.967741i \(0.418929\pi\)
\(608\) −2842.24 −0.189585
\(609\) 553.573 0.0368340
\(610\) −29135.8 −1.93389
\(611\) −10859.0 −0.719000
\(612\) 1494.48 0.0987106
\(613\) −9172.53 −0.604364 −0.302182 0.953250i \(-0.597715\pi\)
−0.302182 + 0.953250i \(0.597715\pi\)
\(614\) −1068.72 −0.0702444
\(615\) 6217.14 0.407641
\(616\) 1001.04 0.0654757
\(617\) 6795.93 0.443426 0.221713 0.975112i \(-0.428835\pi\)
0.221713 + 0.975112i \(0.428835\pi\)
\(618\) 4700.59 0.305964
\(619\) −18841.8 −1.22345 −0.611726 0.791070i \(-0.709524\pi\)
−0.611726 + 0.791070i \(0.709524\pi\)
\(620\) −15083.9 −0.977068
\(621\) −120.193 −0.00776679
\(622\) 10951.5 0.705976
\(623\) 1513.32 0.0973192
\(624\) −4486.24 −0.287810
\(625\) −13985.3 −0.895062
\(626\) 8951.59 0.571530
\(627\) −2008.15 −0.127907
\(628\) 7236.39 0.459815
\(629\) 7565.21 0.479562
\(630\) −340.148 −0.0215108
\(631\) 4816.45 0.303867 0.151933 0.988391i \(-0.451450\pi\)
0.151933 + 0.988391i \(0.451450\pi\)
\(632\) 11941.4 0.751591
\(633\) 18002.0 1.13035
\(634\) 11637.8 0.729018
\(635\) −20561.3 −1.28496
\(636\) 1922.65 0.119871
\(637\) −26701.9 −1.66086
\(638\) −12052.2 −0.747883
\(639\) −6857.14 −0.424514
\(640\) −4413.07 −0.272565
\(641\) −4109.33 −0.253212 −0.126606 0.991953i \(-0.540408\pi\)
−0.126606 + 0.991953i \(0.540408\pi\)
\(642\) 9572.27 0.588454
\(643\) −20097.1 −1.23259 −0.616293 0.787517i \(-0.711366\pi\)
−0.616293 + 0.787517i \(0.711366\pi\)
\(644\) −19.2851 −0.00118003
\(645\) −11128.0 −0.679326
\(646\) 1649.31 0.100451
\(647\) −11607.1 −0.705290 −0.352645 0.935757i \(-0.614718\pi\)
−0.352645 + 0.935757i \(0.614718\pi\)
\(648\) −1958.71 −0.118743
\(649\) 2143.22 0.129628
\(650\) 21931.6 1.32343
\(651\) −838.115 −0.0504583
\(652\) 744.324 0.0447085
\(653\) 28649.4 1.71691 0.858453 0.512893i \(-0.171426\pi\)
0.858453 + 0.512893i \(0.171426\pi\)
\(654\) 9769.81 0.584144
\(655\) 40503.4 2.41618
\(656\) −2450.24 −0.145832
\(657\) −3642.66 −0.216307
\(658\) 324.485 0.0192245
\(659\) −2146.45 −0.126880 −0.0634398 0.997986i \(-0.520207\pi\)
−0.0634398 + 0.997986i \(0.520207\pi\)
\(660\) −6705.29 −0.395459
\(661\) 7235.64 0.425769 0.212885 0.977077i \(-0.431714\pi\)
0.212885 + 0.977077i \(0.431714\pi\)
\(662\) 7418.32 0.435530
\(663\) −10240.3 −0.599847
\(664\) −9634.62 −0.563096
\(665\) 339.890 0.0198201
\(666\) −3193.87 −0.185826
\(667\) 720.805 0.0418436
\(668\) 7975.98 0.461976
\(669\) −14759.0 −0.852938
\(670\) 11974.3 0.690460
\(671\) −31913.0 −1.83605
\(672\) −527.318 −0.0302704
\(673\) 16869.8 0.966247 0.483124 0.875552i \(-0.339502\pi\)
0.483124 + 0.875552i \(0.339502\pi\)
\(674\) −10145.9 −0.579829
\(675\) 3698.21 0.210880
\(676\) −14861.9 −0.845578
\(677\) 269.724 0.0153121 0.00765607 0.999971i \(-0.497563\pi\)
0.00765607 + 0.999971i \(0.497563\pi\)
\(678\) 8664.18 0.490776
\(679\) 1667.02 0.0942183
\(680\) 17096.4 0.964143
\(681\) 3064.76 0.172455
\(682\) 18247.1 1.02451
\(683\) −2809.90 −0.157420 −0.0787099 0.996898i \(-0.525080\pi\)
−0.0787099 + 0.996898i \(0.525080\pi\)
\(684\) 630.460 0.0352430
\(685\) 7413.25 0.413497
\(686\) 1598.82 0.0889844
\(687\) 5236.73 0.290820
\(688\) 4385.67 0.243026
\(689\) −13174.1 −0.728436
\(690\) −442.905 −0.0244364
\(691\) 24053.0 1.32420 0.662098 0.749418i \(-0.269666\pi\)
0.662098 + 0.749418i \(0.269666\pi\)
\(692\) −7590.83 −0.416994
\(693\) −372.571 −0.0204225
\(694\) −18244.0 −0.997885
\(695\) 4644.48 0.253490
\(696\) 11746.5 0.639726
\(697\) −5592.90 −0.303940
\(698\) −20745.3 −1.12496
\(699\) −5883.84 −0.318380
\(700\) 593.382 0.0320396
\(701\) 32736.0 1.76379 0.881897 0.471441i \(-0.156266\pi\)
0.881897 + 0.471441i \(0.156266\pi\)
\(702\) 4323.21 0.232435
\(703\) 3191.45 0.171220
\(704\) 17041.8 0.912338
\(705\) −6747.50 −0.360462
\(706\) 21060.7 1.12270
\(707\) −259.963 −0.0138287
\(708\) −672.865 −0.0357173
\(709\) −20629.3 −1.09273 −0.546367 0.837546i \(-0.683989\pi\)
−0.546367 + 0.837546i \(0.683989\pi\)
\(710\) −25268.2 −1.33563
\(711\) −4444.42 −0.234429
\(712\) 32111.7 1.69022
\(713\) −1091.31 −0.0573208
\(714\) 305.995 0.0160386
\(715\) 45944.9 2.40314
\(716\) 9407.79 0.491041
\(717\) 5468.95 0.284856
\(718\) 24539.2 1.27548
\(719\) 6316.05 0.327606 0.163803 0.986493i \(-0.447624\pi\)
0.163803 + 0.986493i \(0.447624\pi\)
\(720\) −2787.63 −0.144290
\(721\) −871.437 −0.0450125
\(722\) −13358.5 −0.688575
\(723\) −18464.5 −0.949798
\(724\) −2349.72 −0.120617
\(725\) −22178.4 −1.13612
\(726\) −70.2966 −0.00359360
\(727\) 36426.5 1.85830 0.929150 0.369703i \(-0.120541\pi\)
0.929150 + 0.369703i \(0.120541\pi\)
\(728\) 2153.43 0.109631
\(729\) 729.000 0.0370370
\(730\) −13423.0 −0.680558
\(731\) 10010.7 0.506509
\(732\) 10019.1 0.505897
\(733\) −14013.9 −0.706161 −0.353081 0.935593i \(-0.614866\pi\)
−0.353081 + 0.935593i \(0.614866\pi\)
\(734\) 24056.2 1.20972
\(735\) −16591.8 −0.832652
\(736\) −686.618 −0.0343873
\(737\) 13115.7 0.655528
\(738\) 2361.20 0.117774
\(739\) −23407.6 −1.16517 −0.582587 0.812768i \(-0.697960\pi\)
−0.582587 + 0.812768i \(0.697960\pi\)
\(740\) 10656.3 0.529372
\(741\) −4319.94 −0.214166
\(742\) 393.662 0.0194768
\(743\) 9526.11 0.470362 0.235181 0.971952i \(-0.424432\pi\)
0.235181 + 0.971952i \(0.424432\pi\)
\(744\) −17784.3 −0.876349
\(745\) 42208.9 2.07572
\(746\) −8309.64 −0.407825
\(747\) 3585.86 0.175635
\(748\) 6032.03 0.294857
\(749\) −1774.59 −0.0865715
\(750\) 1191.03 0.0579868
\(751\) −17166.3 −0.834098 −0.417049 0.908884i \(-0.636936\pi\)
−0.417049 + 0.908884i \(0.636936\pi\)
\(752\) 2659.26 0.128954
\(753\) 14690.3 0.710949
\(754\) −25926.6 −1.25224
\(755\) −333.080 −0.0160557
\(756\) 116.969 0.00562713
\(757\) 9502.71 0.456251 0.228125 0.973632i \(-0.426740\pi\)
0.228125 + 0.973632i \(0.426740\pi\)
\(758\) −3684.21 −0.176539
\(759\) −485.123 −0.0232001
\(760\) 7212.26 0.344232
\(761\) 23314.4 1.11057 0.555287 0.831659i \(-0.312608\pi\)
0.555287 + 0.831659i \(0.312608\pi\)
\(762\) −7808.95 −0.371245
\(763\) −1811.21 −0.0859375
\(764\) −2099.17 −0.0994049
\(765\) −6363.01 −0.300726
\(766\) 15613.7 0.736483
\(767\) 4610.50 0.217047
\(768\) −12935.3 −0.607764
\(769\) 36770.4 1.72428 0.862142 0.506667i \(-0.169123\pi\)
0.862142 + 0.506667i \(0.169123\pi\)
\(770\) −1372.90 −0.0642546
\(771\) −18714.7 −0.874182
\(772\) −15216.4 −0.709393
\(773\) 30333.2 1.41140 0.705698 0.708513i \(-0.250634\pi\)
0.705698 + 0.708513i \(0.250634\pi\)
\(774\) −4226.29 −0.196267
\(775\) 33578.4 1.55635
\(776\) 35373.1 1.63636
\(777\) 592.107 0.0273381
\(778\) −2005.48 −0.0924162
\(779\) −2359.41 −0.108517
\(780\) −14424.4 −0.662150
\(781\) −27676.8 −1.26806
\(782\) 398.434 0.0182199
\(783\) −4371.86 −0.199537
\(784\) 6539.02 0.297878
\(785\) −30810.2 −1.40084
\(786\) 15382.7 0.698072
\(787\) −10130.6 −0.458851 −0.229426 0.973326i \(-0.573685\pi\)
−0.229426 + 0.973326i \(0.573685\pi\)
\(788\) −5246.93 −0.237201
\(789\) −16079.7 −0.725541
\(790\) −16377.5 −0.737574
\(791\) −1606.24 −0.0722014
\(792\) −7905.73 −0.354694
\(793\) −68651.2 −3.07424
\(794\) −5234.09 −0.233943
\(795\) −8186.01 −0.365192
\(796\) −8108.37 −0.361047
\(797\) −24869.8 −1.10531 −0.552655 0.833410i \(-0.686385\pi\)
−0.552655 + 0.833410i \(0.686385\pi\)
\(798\) 129.086 0.00572633
\(799\) 6070.01 0.268763
\(800\) 21126.5 0.933670
\(801\) −11951.5 −0.527197
\(802\) −26185.7 −1.15293
\(803\) −14702.5 −0.646127
\(804\) −4117.68 −0.180621
\(805\) 82.1095 0.00359501
\(806\) 39253.1 1.71542
\(807\) −21603.3 −0.942344
\(808\) −5516.25 −0.240174
\(809\) −28593.0 −1.24262 −0.621308 0.783567i \(-0.713398\pi\)
−0.621308 + 0.783567i \(0.713398\pi\)
\(810\) 2686.33 0.116528
\(811\) −28737.4 −1.24428 −0.622138 0.782908i \(-0.713736\pi\)
−0.622138 + 0.782908i \(0.713736\pi\)
\(812\) −701.469 −0.0303162
\(813\) −3406.94 −0.146970
\(814\) −12891.1 −0.555077
\(815\) −3169.08 −0.136206
\(816\) 2507.73 0.107583
\(817\) 4223.09 0.180841
\(818\) −28917.6 −1.23604
\(819\) −801.474 −0.0341951
\(820\) −7878.15 −0.335508
\(821\) 28082.4 1.19377 0.596884 0.802328i \(-0.296405\pi\)
0.596884 + 0.802328i \(0.296405\pi\)
\(822\) 2815.47 0.119466
\(823\) 16565.4 0.701621 0.350810 0.936447i \(-0.385906\pi\)
0.350810 + 0.936447i \(0.385906\pi\)
\(824\) −18491.3 −0.781768
\(825\) 14926.7 0.629918
\(826\) −137.769 −0.00580338
\(827\) −25792.9 −1.08453 −0.542265 0.840207i \(-0.682433\pi\)
−0.542265 + 0.840207i \(0.682433\pi\)
\(828\) 152.304 0.00639245
\(829\) 34787.4 1.45744 0.728720 0.684812i \(-0.240116\pi\)
0.728720 + 0.684812i \(0.240116\pi\)
\(830\) 13213.7 0.552595
\(831\) 3133.74 0.130816
\(832\) 36660.2 1.52760
\(833\) 14925.9 0.620830
\(834\) 1763.92 0.0732369
\(835\) −33959.1 −1.40743
\(836\) 2544.66 0.105274
\(837\) 6619.04 0.273342
\(838\) 7329.37 0.302135
\(839\) −40608.0 −1.67097 −0.835485 0.549513i \(-0.814813\pi\)
−0.835485 + 0.549513i \(0.814813\pi\)
\(840\) 1338.08 0.0549623
\(841\) 1829.31 0.0750056
\(842\) −21421.0 −0.876740
\(843\) 19518.9 0.797470
\(844\) −22811.5 −0.930336
\(845\) 63277.0 2.57609
\(846\) −2562.63 −0.104143
\(847\) 13.0322 0.000528680 0
\(848\) 3226.19 0.130646
\(849\) −25170.3 −1.01748
\(850\) −12259.4 −0.494699
\(851\) 770.979 0.0310562
\(852\) 8689.13 0.349395
\(853\) 18149.1 0.728505 0.364252 0.931300i \(-0.381325\pi\)
0.364252 + 0.931300i \(0.381325\pi\)
\(854\) 2051.40 0.0821986
\(855\) −2684.29 −0.107369
\(856\) −37655.7 −1.50356
\(857\) −4806.87 −0.191598 −0.0957990 0.995401i \(-0.530541\pi\)
−0.0957990 + 0.995401i \(0.530541\pi\)
\(858\) 17449.4 0.694302
\(859\) 15908.4 0.631885 0.315942 0.948778i \(-0.397679\pi\)
0.315942 + 0.948778i \(0.397679\pi\)
\(860\) 14101.0 0.559118
\(861\) −437.740 −0.0173265
\(862\) −25655.8 −1.01374
\(863\) 8799.82 0.347102 0.173551 0.984825i \(-0.444476\pi\)
0.173551 + 0.984825i \(0.444476\pi\)
\(864\) 4164.51 0.163981
\(865\) 32319.2 1.27039
\(866\) 27829.9 1.09203
\(867\) −9014.88 −0.353127
\(868\) 1062.03 0.0415296
\(869\) −17938.6 −0.700258
\(870\) −16110.1 −0.627796
\(871\) 28214.5 1.09760
\(872\) −38432.8 −1.49255
\(873\) −13165.3 −0.510399
\(874\) 168.083 0.00650513
\(875\) −220.803 −0.00853085
\(876\) 4615.85 0.178031
\(877\) 8805.63 0.339048 0.169524 0.985526i \(-0.445777\pi\)
0.169524 + 0.985526i \(0.445777\pi\)
\(878\) 2524.95 0.0970533
\(879\) 10930.0 0.419408
\(880\) −11251.4 −0.431006
\(881\) −37620.6 −1.43867 −0.719335 0.694663i \(-0.755553\pi\)
−0.719335 + 0.694663i \(0.755553\pi\)
\(882\) −6301.39 −0.240566
\(883\) 15013.0 0.572173 0.286086 0.958204i \(-0.407646\pi\)
0.286086 + 0.958204i \(0.407646\pi\)
\(884\) 12976.1 0.493703
\(885\) 2864.84 0.108814
\(886\) 25701.2 0.974548
\(887\) 32943.4 1.24705 0.623525 0.781804i \(-0.285700\pi\)
0.623525 + 0.781804i \(0.285700\pi\)
\(888\) 12564.1 0.474803
\(889\) 1447.69 0.0546164
\(890\) −44040.6 −1.65870
\(891\) 2942.39 0.110633
\(892\) 18702.1 0.702009
\(893\) 2560.68 0.0959574
\(894\) 16030.4 0.599707
\(895\) −40055.3 −1.49598
\(896\) 310.717 0.0115852
\(897\) −1043.60 −0.0388458
\(898\) −2808.56 −0.104369
\(899\) −39694.8 −1.47263
\(900\) −4686.25 −0.173565
\(901\) 7364.08 0.272290
\(902\) 9530.28 0.351800
\(903\) 783.506 0.0288743
\(904\) −34083.4 −1.25398
\(905\) 10004.3 0.367464
\(906\) −126.500 −0.00463873
\(907\) 46052.7 1.68595 0.842973 0.537955i \(-0.180803\pi\)
0.842973 + 0.537955i \(0.180803\pi\)
\(908\) −3883.55 −0.141939
\(909\) 2053.06 0.0749128
\(910\) −2953.39 −0.107587
\(911\) −20657.9 −0.751290 −0.375645 0.926764i \(-0.622579\pi\)
−0.375645 + 0.926764i \(0.622579\pi\)
\(912\) 1057.91 0.0384110
\(913\) 14473.2 0.524638
\(914\) −14450.7 −0.522960
\(915\) −42658.0 −1.54123
\(916\) −6635.80 −0.239359
\(917\) −2851.79 −0.102698
\(918\) −2416.60 −0.0868842
\(919\) −47631.7 −1.70971 −0.854855 0.518867i \(-0.826354\pi\)
−0.854855 + 0.518867i \(0.826354\pi\)
\(920\) 1742.31 0.0624374
\(921\) −1564.73 −0.0559820
\(922\) 10414.0 0.371983
\(923\) −59538.3 −2.12321
\(924\) 472.109 0.0168087
\(925\) −23722.2 −0.843224
\(926\) −11679.3 −0.414476
\(927\) 6882.19 0.243841
\(928\) −24974.8 −0.883447
\(929\) −27546.0 −0.972827 −0.486413 0.873729i \(-0.661695\pi\)
−0.486413 + 0.873729i \(0.661695\pi\)
\(930\) 24390.8 0.860006
\(931\) 6296.61 0.221658
\(932\) 7455.81 0.262042
\(933\) 16034.3 0.562635
\(934\) 34746.3 1.21727
\(935\) −25682.4 −0.898293
\(936\) −17006.8 −0.593894
\(937\) −8405.09 −0.293044 −0.146522 0.989207i \(-0.546808\pi\)
−0.146522 + 0.989207i \(0.546808\pi\)
\(938\) −843.094 −0.0293475
\(939\) 13106.1 0.455487
\(940\) 8550.21 0.296678
\(941\) −12903.8 −0.447027 −0.223514 0.974701i \(-0.571753\pi\)
−0.223514 + 0.974701i \(0.571753\pi\)
\(942\) −11701.3 −0.404725
\(943\) −569.978 −0.0196830
\(944\) −1129.06 −0.0389278
\(945\) −498.014 −0.0171433
\(946\) −17058.2 −0.586267
\(947\) −26496.9 −0.909223 −0.454612 0.890690i \(-0.650222\pi\)
−0.454612 + 0.890690i \(0.650222\pi\)
\(948\) 5631.82 0.192946
\(949\) −31628.0 −1.08186
\(950\) −5171.73 −0.176624
\(951\) 17039.1 0.580999
\(952\) −1203.73 −0.0409802
\(953\) 3951.58 0.134317 0.0671585 0.997742i \(-0.478607\pi\)
0.0671585 + 0.997742i \(0.478607\pi\)
\(954\) −3108.95 −0.105510
\(955\) 8937.57 0.302841
\(956\) −6930.06 −0.234450
\(957\) −17645.7 −0.596034
\(958\) 32283.8 1.08877
\(959\) −521.955 −0.0175754
\(960\) 22779.7 0.765844
\(961\) 30307.3 1.01733
\(962\) −27731.3 −0.929410
\(963\) 14014.9 0.468975
\(964\) 23397.6 0.781729
\(965\) 64786.6 2.16120
\(966\) 31.1843 0.00103865
\(967\) 46296.1 1.53959 0.769793 0.638293i \(-0.220359\pi\)
0.769793 + 0.638293i \(0.220359\pi\)
\(968\) 276.535 0.00918200
\(969\) 2414.77 0.0800553
\(970\) −48513.4 −1.60585
\(971\) −49931.5 −1.65023 −0.825117 0.564961i \(-0.808891\pi\)
−0.825117 + 0.564961i \(0.808891\pi\)
\(972\) −923.764 −0.0304833
\(973\) −327.011 −0.0107744
\(974\) −13093.4 −0.430737
\(975\) 32110.4 1.05472
\(976\) 16812.0 0.551371
\(977\) −13238.2 −0.433497 −0.216748 0.976227i \(-0.569545\pi\)
−0.216748 + 0.976227i \(0.569545\pi\)
\(978\) −1203.58 −0.0393521
\(979\) −48238.6 −1.57478
\(980\) 21024.6 0.685313
\(981\) 14304.1 0.465540
\(982\) 32581.7 1.05878
\(983\) 30187.5 0.979483 0.489742 0.871868i \(-0.337091\pi\)
0.489742 + 0.871868i \(0.337091\pi\)
\(984\) −9288.56 −0.300923
\(985\) 22339.7 0.722641
\(986\) 14492.5 0.468089
\(987\) 475.082 0.0153212
\(988\) 5474.07 0.176269
\(989\) 1020.20 0.0328013
\(990\) 10842.6 0.348080
\(991\) −57834.2 −1.85385 −0.926924 0.375250i \(-0.877557\pi\)
−0.926924 + 0.375250i \(0.877557\pi\)
\(992\) 37812.1 1.21022
\(993\) 10861.2 0.347101
\(994\) 1779.10 0.0567701
\(995\) 34522.7 1.09994
\(996\) −4543.87 −0.144556
\(997\) 22267.8 0.707351 0.353675 0.935368i \(-0.384932\pi\)
0.353675 + 0.935368i \(0.384932\pi\)
\(998\) 7996.31 0.253626
\(999\) −4676.18 −0.148096
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 177.4.a.d.1.5 8
3.2 odd 2 531.4.a.e.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.d.1.5 8 1.1 even 1 trivial
531.4.a.e.1.4 8 3.2 odd 2