Properties

Label 1521.4.a.r.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} -5.56155 q^{4} -3.56155 q^{5} +27.1771 q^{7} +21.1771 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} -5.56155 q^{4} -3.56155 q^{5} +27.1771 q^{7} +21.1771 q^{8} +5.56155 q^{10} +15.2614 q^{11} -42.4384 q^{14} +11.4233 q^{16} -44.5464 q^{17} -23.9697 q^{19} +19.8078 q^{20} -23.8314 q^{22} -122.739 q^{23} -112.315 q^{25} -151.147 q^{28} +219.909 q^{29} -27.0928 q^{31} -187.255 q^{32} +69.5616 q^{34} -96.7926 q^{35} -94.1922 q^{37} +37.4299 q^{38} -75.4233 q^{40} -160.354 q^{41} -151.302 q^{43} -84.8769 q^{44} +191.663 q^{46} +466.948 q^{47} +395.594 q^{49} +175.386 q^{50} +120.847 q^{53} -54.3542 q^{55} +575.531 q^{56} -343.400 q^{58} -439.633 q^{59} -137.305 q^{61} +42.3068 q^{62} +201.022 q^{64} -512.280 q^{67} +247.747 q^{68} +151.147 q^{70} +410.719 q^{71} +308.004 q^{73} +147.086 q^{74} +133.309 q^{76} +414.759 q^{77} -586.462 q^{79} -40.6847 q^{80} +250.401 q^{82} +1354.20 q^{83} +158.654 q^{85} +236.266 q^{86} +323.191 q^{88} +439.882 q^{89} +682.617 q^{92} -729.164 q^{94} +85.3693 q^{95} +1511.27 q^{97} -617.740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −0.552092 −0.276046 0.961144i \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) 0 0
\(4\) −5.56155 −0.695194
\(5\) −3.56155 −0.318555 −0.159277 0.987234i \(-0.550916\pi\)
−0.159277 + 0.987234i \(0.550916\pi\)
\(6\) 0 0
\(7\) 27.1771 1.46742 0.733712 0.679460i \(-0.237786\pi\)
0.733712 + 0.679460i \(0.237786\pi\)
\(8\) 21.1771 0.935904
\(9\) 0 0
\(10\) 5.56155 0.175872
\(11\) 15.2614 0.418316 0.209158 0.977882i \(-0.432928\pi\)
0.209158 + 0.977882i \(0.432928\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −42.4384 −0.810154
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) −44.5464 −0.635535 −0.317767 0.948169i \(-0.602933\pi\)
−0.317767 + 0.948169i \(0.602933\pi\)
\(18\) 0 0
\(19\) −23.9697 −0.289422 −0.144711 0.989474i \(-0.546225\pi\)
−0.144711 + 0.989474i \(0.546225\pi\)
\(20\) 19.8078 0.221458
\(21\) 0 0
\(22\) −23.8314 −0.230949
\(23\) −122.739 −1.11273 −0.556365 0.830938i \(-0.687804\pi\)
−0.556365 + 0.830938i \(0.687804\pi\)
\(24\) 0 0
\(25\) −112.315 −0.898523
\(26\) 0 0
\(27\) 0 0
\(28\) −151.147 −1.02014
\(29\) 219.909 1.40814 0.704071 0.710130i \(-0.251364\pi\)
0.704071 + 0.710130i \(0.251364\pi\)
\(30\) 0 0
\(31\) −27.0928 −0.156968 −0.0784840 0.996915i \(-0.525008\pi\)
−0.0784840 + 0.996915i \(0.525008\pi\)
\(32\) −187.255 −1.03445
\(33\) 0 0
\(34\) 69.5616 0.350874
\(35\) −96.7926 −0.467455
\(36\) 0 0
\(37\) −94.1922 −0.418516 −0.209258 0.977860i \(-0.567105\pi\)
−0.209258 + 0.977860i \(0.567105\pi\)
\(38\) 37.4299 0.159788
\(39\) 0 0
\(40\) −75.4233 −0.298137
\(41\) −160.354 −0.610808 −0.305404 0.952223i \(-0.598791\pi\)
−0.305404 + 0.952223i \(0.598791\pi\)
\(42\) 0 0
\(43\) −151.302 −0.536589 −0.268295 0.963337i \(-0.586460\pi\)
−0.268295 + 0.963337i \(0.586460\pi\)
\(44\) −84.8769 −0.290811
\(45\) 0 0
\(46\) 191.663 0.614329
\(47\) 466.948 1.44918 0.724589 0.689181i \(-0.242030\pi\)
0.724589 + 0.689181i \(0.242030\pi\)
\(48\) 0 0
\(49\) 395.594 1.15333
\(50\) 175.386 0.496067
\(51\) 0 0
\(52\) 0 0
\(53\) 120.847 0.313199 0.156600 0.987662i \(-0.449947\pi\)
0.156600 + 0.987662i \(0.449947\pi\)
\(54\) 0 0
\(55\) −54.3542 −0.133257
\(56\) 575.531 1.37337
\(57\) 0 0
\(58\) −343.400 −0.777424
\(59\) −439.633 −0.970090 −0.485045 0.874489i \(-0.661197\pi\)
−0.485045 + 0.874489i \(0.661197\pi\)
\(60\) 0 0
\(61\) −137.305 −0.288198 −0.144099 0.989563i \(-0.546028\pi\)
−0.144099 + 0.989563i \(0.546028\pi\)
\(62\) 42.3068 0.0866609
\(63\) 0 0
\(64\) 201.022 0.392621
\(65\) 0 0
\(66\) 0 0
\(67\) −512.280 −0.934104 −0.467052 0.884230i \(-0.654684\pi\)
−0.467052 + 0.884230i \(0.654684\pi\)
\(68\) 247.747 0.441820
\(69\) 0 0
\(70\) 151.147 0.258078
\(71\) 410.719 0.686526 0.343263 0.939239i \(-0.388468\pi\)
0.343263 + 0.939239i \(0.388468\pi\)
\(72\) 0 0
\(73\) 308.004 0.493823 0.246912 0.969038i \(-0.420584\pi\)
0.246912 + 0.969038i \(0.420584\pi\)
\(74\) 147.086 0.231060
\(75\) 0 0
\(76\) 133.309 0.201205
\(77\) 414.759 0.613847
\(78\) 0 0
\(79\) −586.462 −0.835217 −0.417608 0.908627i \(-0.637132\pi\)
−0.417608 + 0.908627i \(0.637132\pi\)
\(80\) −40.6847 −0.0568585
\(81\) 0 0
\(82\) 250.401 0.337222
\(83\) 1354.20 1.79088 0.895440 0.445182i \(-0.146861\pi\)
0.895440 + 0.445182i \(0.146861\pi\)
\(84\) 0 0
\(85\) 158.654 0.202453
\(86\) 236.266 0.296247
\(87\) 0 0
\(88\) 323.191 0.391503
\(89\) 439.882 0.523904 0.261952 0.965081i \(-0.415634\pi\)
0.261952 + 0.965081i \(0.415634\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 682.617 0.773563
\(93\) 0 0
\(94\) −729.164 −0.800080
\(95\) 85.3693 0.0921969
\(96\) 0 0
\(97\) 1511.27 1.58192 0.790959 0.611869i \(-0.209582\pi\)
0.790959 + 0.611869i \(0.209582\pi\)
\(98\) −617.740 −0.636747
\(99\) 0 0
\(100\) 624.648 0.624648
\(101\) −336.260 −0.331278 −0.165639 0.986186i \(-0.552969\pi\)
−0.165639 + 0.986186i \(0.552969\pi\)
\(102\) 0 0
\(103\) 322.712 0.308716 0.154358 0.988015i \(-0.450669\pi\)
0.154358 + 0.988015i \(0.450669\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −188.708 −0.172915
\(107\) −1434.62 −1.29617 −0.648083 0.761570i \(-0.724429\pi\)
−0.648083 + 0.761570i \(0.724429\pi\)
\(108\) 0 0
\(109\) −849.147 −0.746179 −0.373089 0.927795i \(-0.621702\pi\)
−0.373089 + 0.927795i \(0.621702\pi\)
\(110\) 84.8769 0.0735699
\(111\) 0 0
\(112\) 310.452 0.261919
\(113\) −1614.53 −1.34409 −0.672044 0.740511i \(-0.734583\pi\)
−0.672044 + 0.740511i \(0.734583\pi\)
\(114\) 0 0
\(115\) 437.140 0.354465
\(116\) −1223.04 −0.978931
\(117\) 0 0
\(118\) 686.509 0.535579
\(119\) −1210.64 −0.932599
\(120\) 0 0
\(121\) −1098.09 −0.825012
\(122\) 214.409 0.159112
\(123\) 0 0
\(124\) 150.678 0.109123
\(125\) 845.211 0.604784
\(126\) 0 0
\(127\) 865.174 0.604502 0.302251 0.953228i \(-0.402262\pi\)
0.302251 + 0.953228i \(0.402262\pi\)
\(128\) 1184.13 0.817683
\(129\) 0 0
\(130\) 0 0
\(131\) 281.400 0.187680 0.0938400 0.995587i \(-0.470086\pi\)
0.0938400 + 0.995587i \(0.470086\pi\)
\(132\) 0 0
\(133\) −651.426 −0.424705
\(134\) 799.953 0.515712
\(135\) 0 0
\(136\) −943.363 −0.594799
\(137\) −2641.43 −1.64725 −0.823624 0.567137i \(-0.808051\pi\)
−0.823624 + 0.567137i \(0.808051\pi\)
\(138\) 0 0
\(139\) −1998.64 −1.21958 −0.609791 0.792562i \(-0.708747\pi\)
−0.609791 + 0.792562i \(0.708747\pi\)
\(140\) 538.317 0.324972
\(141\) 0 0
\(142\) −641.359 −0.379026
\(143\) 0 0
\(144\) 0 0
\(145\) −783.218 −0.448570
\(146\) −480.964 −0.272636
\(147\) 0 0
\(148\) 523.855 0.290950
\(149\) −1752.98 −0.963824 −0.481912 0.876220i \(-0.660058\pi\)
−0.481912 + 0.876220i \(0.660058\pi\)
\(150\) 0 0
\(151\) 2794.64 1.50613 0.753063 0.657949i \(-0.228576\pi\)
0.753063 + 0.657949i \(0.228576\pi\)
\(152\) −507.608 −0.270871
\(153\) 0 0
\(154\) −647.669 −0.338900
\(155\) 96.4924 0.0500030
\(156\) 0 0
\(157\) 3244.87 1.64949 0.824743 0.565508i \(-0.191320\pi\)
0.824743 + 0.565508i \(0.191320\pi\)
\(158\) 915.792 0.461117
\(159\) 0 0
\(160\) 666.918 0.329528
\(161\) −3335.68 −1.63285
\(162\) 0 0
\(163\) −3281.47 −1.57684 −0.788418 0.615139i \(-0.789100\pi\)
−0.788418 + 0.615139i \(0.789100\pi\)
\(164\) 891.818 0.424630
\(165\) 0 0
\(166\) −2114.66 −0.988731
\(167\) −3126.52 −1.44873 −0.724364 0.689418i \(-0.757866\pi\)
−0.724364 + 0.689418i \(0.757866\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −247.747 −0.111773
\(171\) 0 0
\(172\) 841.474 0.373034
\(173\) −97.5698 −0.0428792 −0.0214396 0.999770i \(-0.506825\pi\)
−0.0214396 + 0.999770i \(0.506825\pi\)
\(174\) 0 0
\(175\) −3052.40 −1.31851
\(176\) 174.335 0.0746648
\(177\) 0 0
\(178\) −686.900 −0.289243
\(179\) 34.7150 0.0144956 0.00724782 0.999974i \(-0.497693\pi\)
0.00724782 + 0.999974i \(0.497693\pi\)
\(180\) 0 0
\(181\) −1229.35 −0.504843 −0.252422 0.967617i \(-0.581227\pi\)
−0.252422 + 0.967617i \(0.581227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2599.25 −1.04141
\(185\) 335.471 0.133320
\(186\) 0 0
\(187\) −679.839 −0.265854
\(188\) −2596.96 −1.00746
\(189\) 0 0
\(190\) −133.309 −0.0509012
\(191\) −4280.80 −1.62172 −0.810858 0.585243i \(-0.800999\pi\)
−0.810858 + 0.585243i \(0.800999\pi\)
\(192\) 0 0
\(193\) −472.320 −0.176157 −0.0880786 0.996114i \(-0.528073\pi\)
−0.0880786 + 0.996114i \(0.528073\pi\)
\(194\) −2359.93 −0.873365
\(195\) 0 0
\(196\) −2200.12 −0.801791
\(197\) −4484.37 −1.62182 −0.810908 0.585173i \(-0.801026\pi\)
−0.810908 + 0.585173i \(0.801026\pi\)
\(198\) 0 0
\(199\) −366.240 −0.130463 −0.0652314 0.997870i \(-0.520779\pi\)
−0.0652314 + 0.997870i \(0.520779\pi\)
\(200\) −2378.51 −0.840931
\(201\) 0 0
\(202\) 525.087 0.182896
\(203\) 5976.49 2.06634
\(204\) 0 0
\(205\) 571.110 0.194576
\(206\) −503.932 −0.170440
\(207\) 0 0
\(208\) 0 0
\(209\) −365.810 −0.121070
\(210\) 0 0
\(211\) 2122.55 0.692524 0.346262 0.938138i \(-0.387451\pi\)
0.346262 + 0.938138i \(0.387451\pi\)
\(212\) −672.095 −0.217734
\(213\) 0 0
\(214\) 2240.23 0.715603
\(215\) 538.870 0.170933
\(216\) 0 0
\(217\) −736.303 −0.230339
\(218\) 1325.99 0.411960
\(219\) 0 0
\(220\) 302.294 0.0926392
\(221\) 0 0
\(222\) 0 0
\(223\) 5926.42 1.77965 0.889826 0.456301i \(-0.150826\pi\)
0.889826 + 0.456301i \(0.150826\pi\)
\(224\) −5089.04 −1.51797
\(225\) 0 0
\(226\) 2521.17 0.742060
\(227\) −895.661 −0.261881 −0.130941 0.991390i \(-0.541800\pi\)
−0.130941 + 0.991390i \(0.541800\pi\)
\(228\) 0 0
\(229\) −627.717 −0.181138 −0.0905692 0.995890i \(-0.528869\pi\)
−0.0905692 + 0.995890i \(0.528869\pi\)
\(230\) −682.617 −0.195698
\(231\) 0 0
\(232\) 4657.03 1.31788
\(233\) −2303.72 −0.647734 −0.323867 0.946103i \(-0.604983\pi\)
−0.323867 + 0.946103i \(0.604983\pi\)
\(234\) 0 0
\(235\) −1663.06 −0.461643
\(236\) 2445.04 0.674401
\(237\) 0 0
\(238\) 1890.48 0.514881
\(239\) 544.622 0.147400 0.0737001 0.997280i \(-0.476519\pi\)
0.0737001 + 0.997280i \(0.476519\pi\)
\(240\) 0 0
\(241\) −5426.10 −1.45031 −0.725157 0.688584i \(-0.758233\pi\)
−0.725157 + 0.688584i \(0.758233\pi\)
\(242\) 1714.73 0.455483
\(243\) 0 0
\(244\) 763.629 0.200354
\(245\) −1408.93 −0.367400
\(246\) 0 0
\(247\) 0 0
\(248\) −573.746 −0.146907
\(249\) 0 0
\(250\) −1319.84 −0.333897
\(251\) 5221.22 1.31299 0.656494 0.754331i \(-0.272039\pi\)
0.656494 + 0.754331i \(0.272039\pi\)
\(252\) 0 0
\(253\) −1873.16 −0.465472
\(254\) −1351.02 −0.333741
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) −658.206 −0.159758 −0.0798789 0.996805i \(-0.525453\pi\)
−0.0798789 + 0.996805i \(0.525453\pi\)
\(258\) 0 0
\(259\) −2559.87 −0.614141
\(260\) 0 0
\(261\) 0 0
\(262\) −439.422 −0.103617
\(263\) −3246.45 −0.761160 −0.380580 0.924748i \(-0.624276\pi\)
−0.380580 + 0.924748i \(0.624276\pi\)
\(264\) 0 0
\(265\) −430.401 −0.0997711
\(266\) 1017.24 0.234477
\(267\) 0 0
\(268\) 2849.07 0.649384
\(269\) 2585.80 0.586093 0.293047 0.956098i \(-0.405331\pi\)
0.293047 + 0.956098i \(0.405331\pi\)
\(270\) 0 0
\(271\) −988.933 −0.221673 −0.110836 0.993839i \(-0.535353\pi\)
−0.110836 + 0.993839i \(0.535353\pi\)
\(272\) −508.867 −0.113436
\(273\) 0 0
\(274\) 4124.74 0.909433
\(275\) −1714.09 −0.375866
\(276\) 0 0
\(277\) 8142.40 1.76617 0.883086 0.469211i \(-0.155462\pi\)
0.883086 + 0.469211i \(0.155462\pi\)
\(278\) 3120.97 0.673322
\(279\) 0 0
\(280\) −2049.78 −0.437493
\(281\) 1534.21 0.325705 0.162853 0.986650i \(-0.447930\pi\)
0.162853 + 0.986650i \(0.447930\pi\)
\(282\) 0 0
\(283\) −6965.00 −1.46299 −0.731495 0.681847i \(-0.761177\pi\)
−0.731495 + 0.681847i \(0.761177\pi\)
\(284\) −2284.23 −0.477269
\(285\) 0 0
\(286\) 0 0
\(287\) −4357.96 −0.896314
\(288\) 0 0
\(289\) −2928.62 −0.596096
\(290\) 1223.04 0.247652
\(291\) 0 0
\(292\) −1712.98 −0.343303
\(293\) 640.029 0.127614 0.0638070 0.997962i \(-0.479676\pi\)
0.0638070 + 0.997962i \(0.479676\pi\)
\(294\) 0 0
\(295\) 1565.77 0.309027
\(296\) −1994.72 −0.391691
\(297\) 0 0
\(298\) 2737.37 0.532120
\(299\) 0 0
\(300\) 0 0
\(301\) −4111.95 −0.787404
\(302\) −4363.99 −0.831520
\(303\) 0 0
\(304\) −273.813 −0.0516587
\(305\) 489.019 0.0918070
\(306\) 0 0
\(307\) 100.406 0.0186660 0.00933299 0.999956i \(-0.497029\pi\)
0.00933299 + 0.999956i \(0.497029\pi\)
\(308\) −2306.71 −0.426743
\(309\) 0 0
\(310\) −150.678 −0.0276062
\(311\) 3878.92 0.707245 0.353623 0.935388i \(-0.384950\pi\)
0.353623 + 0.935388i \(0.384950\pi\)
\(312\) 0 0
\(313\) −3789.39 −0.684311 −0.342155 0.939643i \(-0.611157\pi\)
−0.342155 + 0.939643i \(0.611157\pi\)
\(314\) −5067.04 −0.910668
\(315\) 0 0
\(316\) 3261.64 0.580638
\(317\) 4406.81 0.780791 0.390396 0.920647i \(-0.372338\pi\)
0.390396 + 0.920647i \(0.372338\pi\)
\(318\) 0 0
\(319\) 3356.11 0.589048
\(320\) −715.950 −0.125071
\(321\) 0 0
\(322\) 5208.84 0.901482
\(323\) 1067.76 0.183938
\(324\) 0 0
\(325\) 0 0
\(326\) 5124.19 0.870559
\(327\) 0 0
\(328\) −3395.83 −0.571657
\(329\) 12690.3 2.12656
\(330\) 0 0
\(331\) 4131.49 0.686064 0.343032 0.939324i \(-0.388546\pi\)
0.343032 + 0.939324i \(0.388546\pi\)
\(332\) −7531.47 −1.24501
\(333\) 0 0
\(334\) 4882.23 0.799831
\(335\) 1824.51 0.297564
\(336\) 0 0
\(337\) −4560.82 −0.737221 −0.368611 0.929584i \(-0.620166\pi\)
−0.368611 + 0.929584i \(0.620166\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −882.365 −0.140744
\(341\) −413.473 −0.0656622
\(342\) 0 0
\(343\) 1429.34 0.225007
\(344\) −3204.14 −0.502196
\(345\) 0 0
\(346\) 152.360 0.0236733
\(347\) −10069.4 −1.55779 −0.778896 0.627153i \(-0.784220\pi\)
−0.778896 + 0.627153i \(0.784220\pi\)
\(348\) 0 0
\(349\) −5879.32 −0.901757 −0.450878 0.892585i \(-0.648889\pi\)
−0.450878 + 0.892585i \(0.648889\pi\)
\(350\) 4766.49 0.727942
\(351\) 0 0
\(352\) −2857.76 −0.432725
\(353\) −9142.56 −1.37850 −0.689249 0.724525i \(-0.742059\pi\)
−0.689249 + 0.724525i \(0.742059\pi\)
\(354\) 0 0
\(355\) −1462.80 −0.218696
\(356\) −2446.43 −0.364215
\(357\) 0 0
\(358\) −54.2093 −0.00800293
\(359\) −2754.32 −0.404924 −0.202462 0.979290i \(-0.564894\pi\)
−0.202462 + 0.979290i \(0.564894\pi\)
\(360\) 0 0
\(361\) −6284.45 −0.916235
\(362\) 1919.69 0.278720
\(363\) 0 0
\(364\) 0 0
\(365\) −1096.97 −0.157310
\(366\) 0 0
\(367\) 3040.19 0.432416 0.216208 0.976347i \(-0.430631\pi\)
0.216208 + 0.976347i \(0.430631\pi\)
\(368\) −1402.08 −0.198610
\(369\) 0 0
\(370\) −523.855 −0.0736052
\(371\) 3284.26 0.459596
\(372\) 0 0
\(373\) −5384.72 −0.747481 −0.373740 0.927533i \(-0.621925\pi\)
−0.373740 + 0.927533i \(0.621925\pi\)
\(374\) 1061.60 0.146776
\(375\) 0 0
\(376\) 9888.59 1.35629
\(377\) 0 0
\(378\) 0 0
\(379\) 3424.27 0.464097 0.232049 0.972704i \(-0.425457\pi\)
0.232049 + 0.972704i \(0.425457\pi\)
\(380\) −474.786 −0.0640948
\(381\) 0 0
\(382\) 6684.69 0.895336
\(383\) −382.985 −0.0510956 −0.0255478 0.999674i \(-0.508133\pi\)
−0.0255478 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) −1477.19 −0.195544
\(386\) 737.553 0.0972551
\(387\) 0 0
\(388\) −8405.00 −1.09974
\(389\) −8588.34 −1.11940 −0.559699 0.828696i \(-0.689083\pi\)
−0.559699 + 0.828696i \(0.689083\pi\)
\(390\) 0 0
\(391\) 5467.56 0.707178
\(392\) 8377.52 1.07941
\(393\) 0 0
\(394\) 7002.57 0.895392
\(395\) 2088.72 0.266063
\(396\) 0 0
\(397\) 7239.16 0.915171 0.457586 0.889166i \(-0.348714\pi\)
0.457586 + 0.889166i \(0.348714\pi\)
\(398\) 571.904 0.0720275
\(399\) 0 0
\(400\) −1283.01 −0.160376
\(401\) 4269.62 0.531708 0.265854 0.964013i \(-0.414346\pi\)
0.265854 + 0.964013i \(0.414346\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1870.12 0.230302
\(405\) 0 0
\(406\) −9332.60 −1.14081
\(407\) −1437.50 −0.175072
\(408\) 0 0
\(409\) −13562.5 −1.63967 −0.819834 0.572602i \(-0.805934\pi\)
−0.819834 + 0.572602i \(0.805934\pi\)
\(410\) −891.818 −0.107424
\(411\) 0 0
\(412\) −1794.78 −0.214618
\(413\) −11947.9 −1.42353
\(414\) 0 0
\(415\) −4823.06 −0.570494
\(416\) 0 0
\(417\) 0 0
\(418\) 571.232 0.0668418
\(419\) 14576.9 1.69959 0.849794 0.527114i \(-0.176726\pi\)
0.849794 + 0.527114i \(0.176726\pi\)
\(420\) 0 0
\(421\) −15848.4 −1.83469 −0.917343 0.398099i \(-0.869670\pi\)
−0.917343 + 0.398099i \(0.869670\pi\)
\(422\) −3314.48 −0.382337
\(423\) 0 0
\(424\) 2559.18 0.293124
\(425\) 5003.24 0.571042
\(426\) 0 0
\(427\) −3731.55 −0.422909
\(428\) 7978.70 0.901087
\(429\) 0 0
\(430\) −841.474 −0.0943709
\(431\) 10694.7 1.19524 0.597618 0.801781i \(-0.296114\pi\)
0.597618 + 0.801781i \(0.296114\pi\)
\(432\) 0 0
\(433\) −16079.0 −1.78454 −0.892272 0.451498i \(-0.850890\pi\)
−0.892272 + 0.451498i \(0.850890\pi\)
\(434\) 1149.78 0.127168
\(435\) 0 0
\(436\) 4722.57 0.518739
\(437\) 2942.01 0.322049
\(438\) 0 0
\(439\) 6035.80 0.656203 0.328101 0.944643i \(-0.393591\pi\)
0.328101 + 0.944643i \(0.393591\pi\)
\(440\) −1151.06 −0.124715
\(441\) 0 0
\(442\) 0 0
\(443\) −10201.3 −1.09409 −0.547043 0.837105i \(-0.684247\pi\)
−0.547043 + 0.837105i \(0.684247\pi\)
\(444\) 0 0
\(445\) −1566.66 −0.166892
\(446\) −9254.41 −0.982532
\(447\) 0 0
\(448\) 5463.19 0.576141
\(449\) −5822.54 −0.611988 −0.305994 0.952033i \(-0.598989\pi\)
−0.305994 + 0.952033i \(0.598989\pi\)
\(450\) 0 0
\(451\) −2447.22 −0.255511
\(452\) 8979.27 0.934402
\(453\) 0 0
\(454\) 1398.62 0.144583
\(455\) 0 0
\(456\) 0 0
\(457\) −4621.60 −0.473062 −0.236531 0.971624i \(-0.576010\pi\)
−0.236531 + 0.971624i \(0.576010\pi\)
\(458\) 980.213 0.100005
\(459\) 0 0
\(460\) −2431.18 −0.246422
\(461\) 5127.77 0.518056 0.259028 0.965870i \(-0.416598\pi\)
0.259028 + 0.965870i \(0.416598\pi\)
\(462\) 0 0
\(463\) −6486.27 −0.651064 −0.325532 0.945531i \(-0.605543\pi\)
−0.325532 + 0.945531i \(0.605543\pi\)
\(464\) 2512.09 0.251338
\(465\) 0 0
\(466\) 3597.39 0.357609
\(467\) −12978.0 −1.28598 −0.642990 0.765875i \(-0.722306\pi\)
−0.642990 + 0.765875i \(0.722306\pi\)
\(468\) 0 0
\(469\) −13922.3 −1.37073
\(470\) 2596.96 0.254869
\(471\) 0 0
\(472\) −9310.13 −0.907910
\(473\) −2309.08 −0.224464
\(474\) 0 0
\(475\) 2692.16 0.260053
\(476\) 6733.04 0.648337
\(477\) 0 0
\(478\) −850.456 −0.0813786
\(479\) −5808.96 −0.554109 −0.277055 0.960854i \(-0.589358\pi\)
−0.277055 + 0.960854i \(0.589358\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 8473.14 0.800707
\(483\) 0 0
\(484\) 6107.09 0.573543
\(485\) −5382.46 −0.503928
\(486\) 0 0
\(487\) 5387.14 0.501262 0.250631 0.968083i \(-0.419362\pi\)
0.250631 + 0.968083i \(0.419362\pi\)
\(488\) −2907.72 −0.269726
\(489\) 0 0
\(490\) 2200.12 0.202839
\(491\) −15259.1 −1.40251 −0.701255 0.712911i \(-0.747376\pi\)
−0.701255 + 0.712911i \(0.747376\pi\)
\(492\) 0 0
\(493\) −9796.16 −0.894922
\(494\) 0 0
\(495\) 0 0
\(496\) −309.489 −0.0280171
\(497\) 11162.1 1.00742
\(498\) 0 0
\(499\) −1856.04 −0.166509 −0.0832544 0.996528i \(-0.526531\pi\)
−0.0832544 + 0.996528i \(0.526531\pi\)
\(500\) −4700.69 −0.420442
\(501\) 0 0
\(502\) −8153.20 −0.724891
\(503\) −1049.46 −0.0930283 −0.0465142 0.998918i \(-0.514811\pi\)
−0.0465142 + 0.998918i \(0.514811\pi\)
\(504\) 0 0
\(505\) 1197.61 0.105530
\(506\) 2925.04 0.256984
\(507\) 0 0
\(508\) −4811.71 −0.420246
\(509\) −551.106 −0.0479909 −0.0239954 0.999712i \(-0.507639\pi\)
−0.0239954 + 0.999712i \(0.507639\pi\)
\(510\) 0 0
\(511\) 8370.64 0.724649
\(512\) −4074.36 −0.351686
\(513\) 0 0
\(514\) 1027.82 0.0882010
\(515\) −1149.36 −0.0983431
\(516\) 0 0
\(517\) 7126.26 0.606214
\(518\) 3997.37 0.339063
\(519\) 0 0
\(520\) 0 0
\(521\) 8995.30 0.756413 0.378206 0.925721i \(-0.376541\pi\)
0.378206 + 0.925721i \(0.376541\pi\)
\(522\) 0 0
\(523\) 2663.91 0.222724 0.111362 0.993780i \(-0.464479\pi\)
0.111362 + 0.993780i \(0.464479\pi\)
\(524\) −1565.02 −0.130474
\(525\) 0 0
\(526\) 5069.51 0.420230
\(527\) 1206.89 0.0997586
\(528\) 0 0
\(529\) 2897.77 0.238167
\(530\) 672.095 0.0550829
\(531\) 0 0
\(532\) 3622.94 0.295253
\(533\) 0 0
\(534\) 0 0
\(535\) 5109.47 0.412900
\(536\) −10848.6 −0.874232
\(537\) 0 0
\(538\) −4037.86 −0.323577
\(539\) 6037.30 0.482458
\(540\) 0 0
\(541\) 6169.23 0.490270 0.245135 0.969489i \(-0.421168\pi\)
0.245135 + 0.969489i \(0.421168\pi\)
\(542\) 1544.27 0.122384
\(543\) 0 0
\(544\) 8341.52 0.657426
\(545\) 3024.28 0.237699
\(546\) 0 0
\(547\) 5140.42 0.401807 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(548\) 14690.5 1.14516
\(549\) 0 0
\(550\) 2676.64 0.207513
\(551\) −5271.15 −0.407547
\(552\) 0 0
\(553\) −15938.3 −1.22562
\(554\) −12714.8 −0.975090
\(555\) 0 0
\(556\) 11115.5 0.847847
\(557\) 2778.56 0.211367 0.105683 0.994400i \(-0.466297\pi\)
0.105683 + 0.994400i \(0.466297\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1105.69 −0.0834356
\(561\) 0 0
\(562\) −2395.75 −0.179819
\(563\) −4906.14 −0.367263 −0.183632 0.982995i \(-0.558785\pi\)
−0.183632 + 0.982995i \(0.558785\pi\)
\(564\) 0 0
\(565\) 5750.22 0.428166
\(566\) 10876.2 0.807706
\(567\) 0 0
\(568\) 8697.82 0.642522
\(569\) 9363.15 0.689849 0.344924 0.938631i \(-0.387905\pi\)
0.344924 + 0.938631i \(0.387905\pi\)
\(570\) 0 0
\(571\) 7199.32 0.527640 0.263820 0.964572i \(-0.415018\pi\)
0.263820 + 0.964572i \(0.415018\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6805.18 0.494848
\(575\) 13785.4 0.999813
\(576\) 0 0
\(577\) 11449.6 0.826086 0.413043 0.910711i \(-0.364466\pi\)
0.413043 + 0.910711i \(0.364466\pi\)
\(578\) 4573.19 0.329100
\(579\) 0 0
\(580\) 4355.91 0.311843
\(581\) 36803.3 2.62798
\(582\) 0 0
\(583\) 1844.28 0.131016
\(584\) 6522.62 0.462171
\(585\) 0 0
\(586\) −999.439 −0.0704547
\(587\) −5439.39 −0.382466 −0.191233 0.981545i \(-0.561249\pi\)
−0.191233 + 0.981545i \(0.561249\pi\)
\(588\) 0 0
\(589\) 649.406 0.0454301
\(590\) −2445.04 −0.170611
\(591\) 0 0
\(592\) −1075.99 −0.0747006
\(593\) −28405.8 −1.96709 −0.983547 0.180651i \(-0.942180\pi\)
−0.983547 + 0.180651i \(0.942180\pi\)
\(594\) 0 0
\(595\) 4311.76 0.297084
\(596\) 9749.30 0.670045
\(597\) 0 0
\(598\) 0 0
\(599\) 10482.3 0.715020 0.357510 0.933909i \(-0.383626\pi\)
0.357510 + 0.933909i \(0.383626\pi\)
\(600\) 0 0
\(601\) 3199.54 0.217158 0.108579 0.994088i \(-0.465370\pi\)
0.108579 + 0.994088i \(0.465370\pi\)
\(602\) 6421.02 0.434720
\(603\) 0 0
\(604\) −15542.6 −1.04705
\(605\) 3910.91 0.262812
\(606\) 0 0
\(607\) 11342.8 0.758468 0.379234 0.925301i \(-0.376188\pi\)
0.379234 + 0.925301i \(0.376188\pi\)
\(608\) 4488.44 0.299392
\(609\) 0 0
\(610\) −763.629 −0.0506859
\(611\) 0 0
\(612\) 0 0
\(613\) −14385.4 −0.947831 −0.473916 0.880570i \(-0.657160\pi\)
−0.473916 + 0.880570i \(0.657160\pi\)
\(614\) −156.789 −0.0103053
\(615\) 0 0
\(616\) 8783.39 0.574502
\(617\) 22056.8 1.43918 0.719588 0.694401i \(-0.244331\pi\)
0.719588 + 0.694401i \(0.244331\pi\)
\(618\) 0 0
\(619\) −13621.4 −0.884477 −0.442238 0.896898i \(-0.645815\pi\)
−0.442238 + 0.896898i \(0.645815\pi\)
\(620\) −536.648 −0.0347618
\(621\) 0 0
\(622\) −6057.14 −0.390465
\(623\) 11954.7 0.768789
\(624\) 0 0
\(625\) 11029.2 0.705866
\(626\) 5917.34 0.377803
\(627\) 0 0
\(628\) −18046.5 −1.14671
\(629\) 4195.92 0.265982
\(630\) 0 0
\(631\) 18737.5 1.18214 0.591068 0.806622i \(-0.298707\pi\)
0.591068 + 0.806622i \(0.298707\pi\)
\(632\) −12419.6 −0.781683
\(633\) 0 0
\(634\) −6881.46 −0.431069
\(635\) −3081.36 −0.192567
\(636\) 0 0
\(637\) 0 0
\(638\) −5240.75 −0.325209
\(639\) 0 0
\(640\) −4217.35 −0.260477
\(641\) −29798.7 −1.83616 −0.918081 0.396394i \(-0.870261\pi\)
−0.918081 + 0.396394i \(0.870261\pi\)
\(642\) 0 0
\(643\) 22983.5 1.40961 0.704807 0.709399i \(-0.251034\pi\)
0.704807 + 0.709399i \(0.251034\pi\)
\(644\) 18551.5 1.13515
\(645\) 0 0
\(646\) −1667.37 −0.101551
\(647\) 24905.4 1.51334 0.756672 0.653794i \(-0.226824\pi\)
0.756672 + 0.653794i \(0.226824\pi\)
\(648\) 0 0
\(649\) −6709.39 −0.405804
\(650\) 0 0
\(651\) 0 0
\(652\) 18250.1 1.09621
\(653\) −10077.8 −0.603946 −0.301973 0.953316i \(-0.597645\pi\)
−0.301973 + 0.953316i \(0.597645\pi\)
\(654\) 0 0
\(655\) −1002.22 −0.0597864
\(656\) −1831.77 −0.109022
\(657\) 0 0
\(658\) −19816.5 −1.17406
\(659\) −12334.6 −0.729116 −0.364558 0.931181i \(-0.618780\pi\)
−0.364558 + 0.931181i \(0.618780\pi\)
\(660\) 0 0
\(661\) 12749.1 0.750202 0.375101 0.926984i \(-0.377608\pi\)
0.375101 + 0.926984i \(0.377608\pi\)
\(662\) −6451.54 −0.378771
\(663\) 0 0
\(664\) 28678.1 1.67609
\(665\) 2320.09 0.135292
\(666\) 0 0
\(667\) −26991.3 −1.56688
\(668\) 17388.3 1.00715
\(669\) 0 0
\(670\) −2849.07 −0.164283
\(671\) −2095.46 −0.120558
\(672\) 0 0
\(673\) −13618.2 −0.780007 −0.390004 0.920813i \(-0.627526\pi\)
−0.390004 + 0.920813i \(0.627526\pi\)
\(674\) 7121.96 0.407014
\(675\) 0 0
\(676\) 0 0
\(677\) −9655.67 −0.548150 −0.274075 0.961708i \(-0.588372\pi\)
−0.274075 + 0.961708i \(0.588372\pi\)
\(678\) 0 0
\(679\) 41071.9 2.32135
\(680\) 3359.84 0.189476
\(681\) 0 0
\(682\) 645.660 0.0362516
\(683\) 16316.8 0.914119 0.457060 0.889436i \(-0.348903\pi\)
0.457060 + 0.889436i \(0.348903\pi\)
\(684\) 0 0
\(685\) 9407.61 0.524739
\(686\) −2232.00 −0.124225
\(687\) 0 0
\(688\) −1728.37 −0.0957753
\(689\) 0 0
\(690\) 0 0
\(691\) −2350.84 −0.129421 −0.0647106 0.997904i \(-0.520612\pi\)
−0.0647106 + 0.997904i \(0.520612\pi\)
\(692\) 542.640 0.0298093
\(693\) 0 0
\(694\) 15723.9 0.860045
\(695\) 7118.24 0.388504
\(696\) 0 0
\(697\) 7143.20 0.388189
\(698\) 9180.88 0.497853
\(699\) 0 0
\(700\) 16976.1 0.916623
\(701\) 8076.90 0.435179 0.217589 0.976040i \(-0.430181\pi\)
0.217589 + 0.976040i \(0.430181\pi\)
\(702\) 0 0
\(703\) 2257.76 0.121128
\(704\) 3067.87 0.164239
\(705\) 0 0
\(706\) 14276.6 0.761058
\(707\) −9138.55 −0.486125
\(708\) 0 0
\(709\) 13624.9 0.721712 0.360856 0.932622i \(-0.382485\pi\)
0.360856 + 0.932622i \(0.382485\pi\)
\(710\) 2284.23 0.120741
\(711\) 0 0
\(712\) 9315.43 0.490324
\(713\) 3325.33 0.174663
\(714\) 0 0
\(715\) 0 0
\(716\) −193.069 −0.0100773
\(717\) 0 0
\(718\) 4301.02 0.223555
\(719\) −16235.8 −0.842131 −0.421066 0.907030i \(-0.638344\pi\)
−0.421066 + 0.907030i \(0.638344\pi\)
\(720\) 0 0
\(721\) 8770.37 0.453018
\(722\) 9813.51 0.505846
\(723\) 0 0
\(724\) 6837.08 0.350964
\(725\) −24699.2 −1.26525
\(726\) 0 0
\(727\) 24181.2 1.23361 0.616803 0.787118i \(-0.288428\pi\)
0.616803 + 0.787118i \(0.288428\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1712.98 0.0868496
\(731\) 6739.96 0.341021
\(732\) 0 0
\(733\) −3053.70 −0.153876 −0.0769379 0.997036i \(-0.524514\pi\)
−0.0769379 + 0.997036i \(0.524514\pi\)
\(734\) −4747.41 −0.238733
\(735\) 0 0
\(736\) 22983.4 1.15106
\(737\) −7818.10 −0.390751
\(738\) 0 0
\(739\) 8033.62 0.399894 0.199947 0.979807i \(-0.435923\pi\)
0.199947 + 0.979807i \(0.435923\pi\)
\(740\) −1865.74 −0.0926836
\(741\) 0 0
\(742\) −5128.54 −0.253739
\(743\) −16139.6 −0.796912 −0.398456 0.917187i \(-0.630454\pi\)
−0.398456 + 0.917187i \(0.630454\pi\)
\(744\) 0 0
\(745\) 6243.33 0.307031
\(746\) 8408.53 0.412678
\(747\) 0 0
\(748\) 3780.96 0.184820
\(749\) −38988.7 −1.90202
\(750\) 0 0
\(751\) −18491.1 −0.898469 −0.449235 0.893414i \(-0.648303\pi\)
−0.449235 + 0.893414i \(0.648303\pi\)
\(752\) 5334.08 0.258662
\(753\) 0 0
\(754\) 0 0
\(755\) −9953.28 −0.479784
\(756\) 0 0
\(757\) 160.630 0.00771227 0.00385613 0.999993i \(-0.498773\pi\)
0.00385613 + 0.999993i \(0.498773\pi\)
\(758\) −5347.17 −0.256224
\(759\) 0 0
\(760\) 1807.87 0.0862874
\(761\) 26799.1 1.27656 0.638282 0.769803i \(-0.279645\pi\)
0.638282 + 0.769803i \(0.279645\pi\)
\(762\) 0 0
\(763\) −23077.3 −1.09496
\(764\) 23807.9 1.12741
\(765\) 0 0
\(766\) 598.052 0.0282095
\(767\) 0 0
\(768\) 0 0
\(769\) 5145.82 0.241304 0.120652 0.992695i \(-0.461501\pi\)
0.120652 + 0.992695i \(0.461501\pi\)
\(770\) 2306.71 0.107958
\(771\) 0 0
\(772\) 2626.83 0.122463
\(773\) 12810.6 0.596072 0.298036 0.954555i \(-0.403668\pi\)
0.298036 + 0.954555i \(0.403668\pi\)
\(774\) 0 0
\(775\) 3042.94 0.141039
\(776\) 32004.3 1.48052
\(777\) 0 0
\(778\) 13411.1 0.618011
\(779\) 3843.64 0.176781
\(780\) 0 0
\(781\) 6268.13 0.287185
\(782\) −8537.89 −0.390428
\(783\) 0 0
\(784\) 4518.98 0.205857
\(785\) −11556.8 −0.525452
\(786\) 0 0
\(787\) −28073.0 −1.27153 −0.635764 0.771883i \(-0.719315\pi\)
−0.635764 + 0.771883i \(0.719315\pi\)
\(788\) 24940.0 1.12748
\(789\) 0 0
\(790\) −3261.64 −0.146891
\(791\) −43878.1 −1.97235
\(792\) 0 0
\(793\) 0 0
\(794\) −11304.3 −0.505259
\(795\) 0 0
\(796\) 2036.87 0.0906970
\(797\) −30093.1 −1.33746 −0.668729 0.743507i \(-0.733161\pi\)
−0.668729 + 0.743507i \(0.733161\pi\)
\(798\) 0 0
\(799\) −20800.8 −0.921003
\(800\) 21031.6 0.929473
\(801\) 0 0
\(802\) −6667.24 −0.293552
\(803\) 4700.56 0.206574
\(804\) 0 0
\(805\) 11880.2 0.520151
\(806\) 0 0
\(807\) 0 0
\(808\) −7120.99 −0.310044
\(809\) 24337.1 1.05766 0.528831 0.848727i \(-0.322631\pi\)
0.528831 + 0.848727i \(0.322631\pi\)
\(810\) 0 0
\(811\) −19078.7 −0.826071 −0.413035 0.910715i \(-0.635531\pi\)
−0.413035 + 0.910715i \(0.635531\pi\)
\(812\) −33238.5 −1.43651
\(813\) 0 0
\(814\) 2244.74 0.0966559
\(815\) 11687.1 0.502309
\(816\) 0 0
\(817\) 3626.66 0.155301
\(818\) 21178.6 0.905248
\(819\) 0 0
\(820\) −3176.26 −0.135268
\(821\) 2013.92 0.0856104 0.0428052 0.999083i \(-0.486371\pi\)
0.0428052 + 0.999083i \(0.486371\pi\)
\(822\) 0 0
\(823\) −7692.10 −0.325795 −0.162898 0.986643i \(-0.552084\pi\)
−0.162898 + 0.986643i \(0.552084\pi\)
\(824\) 6834.10 0.288929
\(825\) 0 0
\(826\) 18657.3 0.785922
\(827\) −4762.76 −0.200263 −0.100131 0.994974i \(-0.531926\pi\)
−0.100131 + 0.994974i \(0.531926\pi\)
\(828\) 0 0
\(829\) 19977.7 0.836976 0.418488 0.908222i \(-0.362560\pi\)
0.418488 + 0.908222i \(0.362560\pi\)
\(830\) 7531.47 0.314965
\(831\) 0 0
\(832\) 0 0
\(833\) −17622.3 −0.732984
\(834\) 0 0
\(835\) 11135.3 0.461499
\(836\) 2034.47 0.0841671
\(837\) 0 0
\(838\) −22762.6 −0.938330
\(839\) −30615.8 −1.25980 −0.629901 0.776676i \(-0.716905\pi\)
−0.629901 + 0.776676i \(0.716905\pi\)
\(840\) 0 0
\(841\) 23971.0 0.982861
\(842\) 24748.1 1.01292
\(843\) 0 0
\(844\) −11804.7 −0.481439
\(845\) 0 0
\(846\) 0 0
\(847\) −29842.9 −1.21064
\(848\) 1380.47 0.0559026
\(849\) 0 0
\(850\) −7812.83 −0.315268
\(851\) 11561.0 0.465696
\(852\) 0 0
\(853\) −5660.88 −0.227227 −0.113614 0.993525i \(-0.536243\pi\)
−0.113614 + 0.993525i \(0.536243\pi\)
\(854\) 5827.01 0.233485
\(855\) 0 0
\(856\) −30381.0 −1.21309
\(857\) −41346.1 −1.64802 −0.824012 0.566572i \(-0.808269\pi\)
−0.824012 + 0.566572i \(0.808269\pi\)
\(858\) 0 0
\(859\) −34810.5 −1.38268 −0.691339 0.722530i \(-0.742979\pi\)
−0.691339 + 0.722530i \(0.742979\pi\)
\(860\) −2996.96 −0.118832
\(861\) 0 0
\(862\) −16700.4 −0.659880
\(863\) −8360.51 −0.329774 −0.164887 0.986312i \(-0.552726\pi\)
−0.164887 + 0.986312i \(0.552726\pi\)
\(864\) 0 0
\(865\) 347.500 0.0136594
\(866\) 25108.2 0.985233
\(867\) 0 0
\(868\) 4094.99 0.160130
\(869\) −8950.21 −0.349385
\(870\) 0 0
\(871\) 0 0
\(872\) −17982.4 −0.698352
\(873\) 0 0
\(874\) −4594.10 −0.177801
\(875\) 22970.4 0.887475
\(876\) 0 0
\(877\) 40579.3 1.56245 0.781223 0.624251i \(-0.214596\pi\)
0.781223 + 0.624251i \(0.214596\pi\)
\(878\) −9425.22 −0.362285
\(879\) 0 0
\(880\) −620.903 −0.0237848
\(881\) −10445.2 −0.399442 −0.199721 0.979853i \(-0.564004\pi\)
−0.199721 + 0.979853i \(0.564004\pi\)
\(882\) 0 0
\(883\) 18227.6 0.694685 0.347343 0.937738i \(-0.387084\pi\)
0.347343 + 0.937738i \(0.387084\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 15929.9 0.604036
\(887\) 23517.7 0.890245 0.445122 0.895470i \(-0.353160\pi\)
0.445122 + 0.895470i \(0.353160\pi\)
\(888\) 0 0
\(889\) 23512.9 0.887061
\(890\) 2446.43 0.0921399
\(891\) 0 0
\(892\) −32960.1 −1.23720
\(893\) −11192.6 −0.419424
\(894\) 0 0
\(895\) −123.639 −0.00461766
\(896\) 32181.2 1.19989
\(897\) 0 0
\(898\) 9092.21 0.337874
\(899\) −5957.95 −0.221033
\(900\) 0 0
\(901\) −5383.28 −0.199049
\(902\) 3821.47 0.141065
\(903\) 0 0
\(904\) −34191.0 −1.25794
\(905\) 4378.38 0.160820
\(906\) 0 0
\(907\) −30564.6 −1.11894 −0.559471 0.828850i \(-0.688996\pi\)
−0.559471 + 0.828850i \(0.688996\pi\)
\(908\) 4981.26 0.182058
\(909\) 0 0
\(910\) 0 0
\(911\) 32766.5 1.19166 0.595831 0.803110i \(-0.296823\pi\)
0.595831 + 0.803110i \(0.296823\pi\)
\(912\) 0 0
\(913\) 20667.0 0.749154
\(914\) 7216.87 0.261174
\(915\) 0 0
\(916\) 3491.08 0.125926
\(917\) 7647.64 0.275406
\(918\) 0 0
\(919\) 20686.7 0.742538 0.371269 0.928525i \(-0.378923\pi\)
0.371269 + 0.928525i \(0.378923\pi\)
\(920\) 9257.35 0.331745
\(921\) 0 0
\(922\) −8007.28 −0.286015
\(923\) 0 0
\(924\) 0 0
\(925\) 10579.2 0.376047
\(926\) 10128.6 0.359447
\(927\) 0 0
\(928\) −41179.0 −1.45665
\(929\) 45632.2 1.61156 0.805782 0.592212i \(-0.201745\pi\)
0.805782 + 0.592212i \(0.201745\pi\)
\(930\) 0 0
\(931\) −9482.26 −0.333801
\(932\) 12812.3 0.450301
\(933\) 0 0
\(934\) 20265.9 0.709979
\(935\) 2421.28 0.0846892
\(936\) 0 0
\(937\) −17761.4 −0.619253 −0.309626 0.950858i \(-0.600204\pi\)
−0.309626 + 0.950858i \(0.600204\pi\)
\(938\) 21740.4 0.756768
\(939\) 0 0
\(940\) 9249.19 0.320931
\(941\) −44888.3 −1.55507 −0.777534 0.628841i \(-0.783530\pi\)
−0.777534 + 0.628841i \(0.783530\pi\)
\(942\) 0 0
\(943\) 19681.7 0.679664
\(944\) −5022.05 −0.173150
\(945\) 0 0
\(946\) 3605.74 0.123925
\(947\) 16069.6 0.551415 0.275708 0.961242i \(-0.411088\pi\)
0.275708 + 0.961242i \(0.411088\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4203.96 −0.143573
\(951\) 0 0
\(952\) −25637.8 −0.872823
\(953\) −3512.03 −0.119377 −0.0596883 0.998217i \(-0.519011\pi\)
−0.0596883 + 0.998217i \(0.519011\pi\)
\(954\) 0 0
\(955\) 15246.3 0.516605
\(956\) −3028.94 −0.102472
\(957\) 0 0
\(958\) 9071.00 0.305919
\(959\) −71786.5 −2.41721
\(960\) 0 0
\(961\) −29057.0 −0.975361
\(962\) 0 0
\(963\) 0 0
\(964\) 30177.5 1.00825
\(965\) 1682.19 0.0561158
\(966\) 0 0
\(967\) 37011.9 1.23084 0.615421 0.788199i \(-0.288986\pi\)
0.615421 + 0.788199i \(0.288986\pi\)
\(968\) −23254.4 −0.772132
\(969\) 0 0
\(970\) 8405.00 0.278215
\(971\) −19532.3 −0.645542 −0.322771 0.946477i \(-0.604614\pi\)
−0.322771 + 0.946477i \(0.604614\pi\)
\(972\) 0 0
\(973\) −54317.1 −1.78965
\(974\) −8412.30 −0.276743
\(975\) 0 0
\(976\) −1568.47 −0.0514402
\(977\) 30201.2 0.988970 0.494485 0.869186i \(-0.335357\pi\)
0.494485 + 0.869186i \(0.335357\pi\)
\(978\) 0 0
\(979\) 6713.21 0.219157
\(980\) 7835.83 0.255415
\(981\) 0 0
\(982\) 23827.8 0.774315
\(983\) −38774.9 −1.25812 −0.629058 0.777359i \(-0.716559\pi\)
−0.629058 + 0.777359i \(0.716559\pi\)
\(984\) 0 0
\(985\) 15971.3 0.516638
\(986\) 15297.2 0.494080
\(987\) 0 0
\(988\) 0 0
\(989\) 18570.6 0.597079
\(990\) 0 0
\(991\) −27728.9 −0.888838 −0.444419 0.895819i \(-0.646590\pi\)
−0.444419 + 0.895819i \(0.646590\pi\)
\(992\) 5073.25 0.162375
\(993\) 0 0
\(994\) −17430.3 −0.556192
\(995\) 1304.38 0.0415596
\(996\) 0 0
\(997\) −48918.2 −1.55392 −0.776958 0.629552i \(-0.783239\pi\)
−0.776958 + 0.629552i \(0.783239\pi\)
\(998\) 2898.31 0.0919283
\(999\) 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 1521.4.a.r.1.1 2
3.2 odd 2 169.4.a.g.1.2 2
13.12 even 2 117.4.a.d.1.2 2
39.2 even 12 169.4.e.f.147.3 8
39.5 even 4 169.4.b.f.168.2 4
39.8 even 4 169.4.b.f.168.3 4
39.11 even 12 169.4.e.f.147.2 8
39.17 odd 6 169.4.c.g.146.2 4
39.20 even 12 169.4.e.f.23.3 8
39.23 odd 6 169.4.c.g.22.2 4
39.29 odd 6 169.4.c.j.22.1 4
39.32 even 12 169.4.e.f.23.2 8
39.35 odd 6 169.4.c.j.146.1 4
39.38 odd 2 13.4.a.b.1.1 2
52.51 odd 2 1872.4.a.bb.1.2 2
156.155 even 2 208.4.a.h.1.1 2
195.38 even 4 325.4.b.e.274.3 4
195.77 even 4 325.4.b.e.274.2 4
195.194 odd 2 325.4.a.f.1.2 2
273.272 even 2 637.4.a.b.1.1 2
312.77 odd 2 832.4.a.s.1.1 2
312.155 even 2 832.4.a.z.1.2 2
429.428 even 2 1573.4.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.1 2 39.38 odd 2
117.4.a.d.1.2 2 13.12 even 2
169.4.a.g.1.2 2 3.2 odd 2
169.4.b.f.168.2 4 39.5 even 4
169.4.b.f.168.3 4 39.8 even 4
169.4.c.g.22.2 4 39.23 odd 6
169.4.c.g.146.2 4 39.17 odd 6
169.4.c.j.22.1 4 39.29 odd 6
169.4.c.j.146.1 4 39.35 odd 6
169.4.e.f.23.2 8 39.32 even 12
169.4.e.f.23.3 8 39.20 even 12
169.4.e.f.147.2 8 39.11 even 12
169.4.e.f.147.3 8 39.2 even 12
208.4.a.h.1.1 2 156.155 even 2
325.4.a.f.1.2 2 195.194 odd 2
325.4.b.e.274.2 4 195.77 even 4
325.4.b.e.274.3 4 195.38 even 4
637.4.a.b.1.1 2 273.272 even 2
832.4.a.s.1.1 2 312.77 odd 2
832.4.a.z.1.2 2 312.155 even 2
1521.4.a.r.1.1 2 1.1 even 1 trivial
1573.4.a.b.1.2 2 429.428 even 2
1872.4.a.bb.1.2 2 52.51 odd 2