Properties

Label 1521.4.a.bb.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(5.33039\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.33039 q^{2} +20.4131 q^{4} -16.4131 q^{5} -9.67968 q^{7} +66.1667 q^{8} +O(q^{10})\) \(q+5.33039 q^{2} +20.4131 q^{4} -16.4131 q^{5} -9.67968 q^{7} +66.1667 q^{8} -87.4882 q^{10} +27.5882 q^{11} -51.5965 q^{14} +189.390 q^{16} -107.928 q^{17} +2.24723 q^{19} -335.042 q^{20} +147.056 q^{22} -41.8090 q^{23} +144.390 q^{25} -197.592 q^{28} -61.6213 q^{29} -191.932 q^{31} +480.187 q^{32} -575.300 q^{34} +158.874 q^{35} -98.4236 q^{37} +11.9786 q^{38} -1086.00 q^{40} -30.7452 q^{41} +238.325 q^{43} +563.160 q^{44} -222.858 q^{46} -511.482 q^{47} -249.304 q^{49} +769.653 q^{50} -492.825 q^{53} -452.807 q^{55} -640.472 q^{56} -328.466 q^{58} +484.179 q^{59} -444.021 q^{61} -1023.07 q^{62} +1044.47 q^{64} -190.114 q^{67} -2203.15 q^{68} +846.858 q^{70} +484.785 q^{71} +957.780 q^{73} -524.636 q^{74} +45.8729 q^{76} -267.045 q^{77} -375.216 q^{79} -3108.47 q^{80} -163.884 q^{82} -715.765 q^{83} +1771.43 q^{85} +1270.37 q^{86} +1825.42 q^{88} -1038.15 q^{89} -853.451 q^{92} -2726.40 q^{94} -36.8840 q^{95} -65.5636 q^{97} -1328.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8} - 62 q^{10} + 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} - 124 q^{19} - 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} + 144 q^{28} - 194 q^{29} - 26 q^{31} + 654 q^{32} - 1062 q^{34} - 88 q^{35} - 102 q^{37} - 332 q^{38} - 998 q^{40} - 1054 q^{41} + 450 q^{43} - 44 q^{44} + 172 q^{46} - 96 q^{47} + 1070 q^{49} + 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} - 722 q^{58} + 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} + 1134 q^{67} - 1786 q^{68} + 2324 q^{70} + 1064 q^{71} - 952 q^{73} - 1158 q^{74} + 1708 q^{76} - 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 1734 q^{82} - 404 q^{83} + 1394 q^{85} + 3168 q^{86} + 3060 q^{88} + 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.33039 1.88458 0.942289 0.334800i \(-0.108669\pi\)
0.942289 + 0.334800i \(0.108669\pi\)
\(3\) 0 0
\(4\) 20.4131 2.55164
\(5\) −16.4131 −1.46803 −0.734016 0.679132i \(-0.762356\pi\)
−0.734016 + 0.679132i \(0.762356\pi\)
\(6\) 0 0
\(7\) −9.67968 −0.522654 −0.261327 0.965250i \(-0.584160\pi\)
−0.261327 + 0.965250i \(0.584160\pi\)
\(8\) 66.1667 2.92418
\(9\) 0 0
\(10\) −87.4882 −2.76662
\(11\) 27.5882 0.756195 0.378098 0.925766i \(-0.376578\pi\)
0.378098 + 0.925766i \(0.376578\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −51.5965 −0.984982
\(15\) 0 0
\(16\) 189.390 2.95921
\(17\) −107.928 −1.53979 −0.769895 0.638171i \(-0.779691\pi\)
−0.769895 + 0.638171i \(0.779691\pi\)
\(18\) 0 0
\(19\) 2.24723 0.0271342 0.0135671 0.999908i \(-0.495681\pi\)
0.0135671 + 0.999908i \(0.495681\pi\)
\(20\) −335.042 −3.74588
\(21\) 0 0
\(22\) 147.056 1.42511
\(23\) −41.8090 −0.379034 −0.189517 0.981877i \(-0.560692\pi\)
−0.189517 + 0.981877i \(0.560692\pi\)
\(24\) 0 0
\(25\) 144.390 1.15512
\(26\) 0 0
\(27\) 0 0
\(28\) −197.592 −1.33362
\(29\) −61.6213 −0.394579 −0.197289 0.980345i \(-0.563214\pi\)
−0.197289 + 0.980345i \(0.563214\pi\)
\(30\) 0 0
\(31\) −191.932 −1.11200 −0.556000 0.831182i \(-0.687665\pi\)
−0.556000 + 0.831182i \(0.687665\pi\)
\(32\) 480.187 2.65269
\(33\) 0 0
\(34\) −575.300 −2.90185
\(35\) 158.874 0.767272
\(36\) 0 0
\(37\) −98.4236 −0.437317 −0.218659 0.975801i \(-0.570168\pi\)
−0.218659 + 0.975801i \(0.570168\pi\)
\(38\) 11.9786 0.0511366
\(39\) 0 0
\(40\) −1086.00 −4.29279
\(41\) −30.7452 −0.117112 −0.0585561 0.998284i \(-0.518650\pi\)
−0.0585561 + 0.998284i \(0.518650\pi\)
\(42\) 0 0
\(43\) 238.325 0.845216 0.422608 0.906313i \(-0.361115\pi\)
0.422608 + 0.906313i \(0.361115\pi\)
\(44\) 563.160 1.92953
\(45\) 0 0
\(46\) −222.858 −0.714319
\(47\) −511.482 −1.58739 −0.793695 0.608316i \(-0.791845\pi\)
−0.793695 + 0.608316i \(0.791845\pi\)
\(48\) 0 0
\(49\) −249.304 −0.726833
\(50\) 769.653 2.17691
\(51\) 0 0
\(52\) 0 0
\(53\) −492.825 −1.27726 −0.638630 0.769514i \(-0.720498\pi\)
−0.638630 + 0.769514i \(0.720498\pi\)
\(54\) 0 0
\(55\) −452.807 −1.11012
\(56\) −640.472 −1.52833
\(57\) 0 0
\(58\) −328.466 −0.743615
\(59\) 484.179 1.06838 0.534192 0.845363i \(-0.320616\pi\)
0.534192 + 0.845363i \(0.320616\pi\)
\(60\) 0 0
\(61\) −444.021 −0.931985 −0.465993 0.884789i \(-0.654303\pi\)
−0.465993 + 0.884789i \(0.654303\pi\)
\(62\) −1023.07 −2.09565
\(63\) 0 0
\(64\) 1044.47 2.03998
\(65\) 0 0
\(66\) 0 0
\(67\) −190.114 −0.346658 −0.173329 0.984864i \(-0.555452\pi\)
−0.173329 + 0.984864i \(0.555452\pi\)
\(68\) −2203.15 −3.92898
\(69\) 0 0
\(70\) 846.858 1.44598
\(71\) 484.785 0.810329 0.405164 0.914244i \(-0.367214\pi\)
0.405164 + 0.914244i \(0.367214\pi\)
\(72\) 0 0
\(73\) 957.780 1.53561 0.767806 0.640683i \(-0.221349\pi\)
0.767806 + 0.640683i \(0.221349\pi\)
\(74\) −524.636 −0.824159
\(75\) 0 0
\(76\) 45.8729 0.0692367
\(77\) −267.045 −0.395228
\(78\) 0 0
\(79\) −375.216 −0.534368 −0.267184 0.963646i \(-0.586093\pi\)
−0.267184 + 0.963646i \(0.586093\pi\)
\(80\) −3108.47 −4.34422
\(81\) 0 0
\(82\) −163.884 −0.220707
\(83\) −715.765 −0.946571 −0.473286 0.880909i \(-0.656932\pi\)
−0.473286 + 0.880909i \(0.656932\pi\)
\(84\) 0 0
\(85\) 1771.43 2.26046
\(86\) 1270.37 1.59288
\(87\) 0 0
\(88\) 1825.42 2.21125
\(89\) −1038.15 −1.23645 −0.618224 0.786002i \(-0.712148\pi\)
−0.618224 + 0.786002i \(0.712148\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −853.451 −0.967157
\(93\) 0 0
\(94\) −2726.40 −2.99156
\(95\) −36.8840 −0.0398339
\(96\) 0 0
\(97\) −65.5636 −0.0686286 −0.0343143 0.999411i \(-0.510925\pi\)
−0.0343143 + 0.999411i \(0.510925\pi\)
\(98\) −1328.89 −1.36977
\(99\) 0 0
\(100\) 2947.44 2.94744
\(101\) −531.798 −0.523920 −0.261960 0.965079i \(-0.584369\pi\)
−0.261960 + 0.965079i \(0.584369\pi\)
\(102\) 0 0
\(103\) −735.984 −0.704064 −0.352032 0.935988i \(-0.614509\pi\)
−0.352032 + 0.935988i \(0.614509\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2626.95 −2.40710
\(107\) 783.265 0.707673 0.353837 0.935307i \(-0.384877\pi\)
0.353837 + 0.935307i \(0.384877\pi\)
\(108\) 0 0
\(109\) 532.339 0.467788 0.233894 0.972262i \(-0.424853\pi\)
0.233894 + 0.972262i \(0.424853\pi\)
\(110\) −2413.64 −2.09210
\(111\) 0 0
\(112\) −1833.23 −1.54664
\(113\) 180.589 0.150340 0.0751699 0.997171i \(-0.476050\pi\)
0.0751699 + 0.997171i \(0.476050\pi\)
\(114\) 0 0
\(115\) 686.215 0.556434
\(116\) −1257.88 −1.00682
\(117\) 0 0
\(118\) 2580.86 2.01346
\(119\) 1044.71 0.804777
\(120\) 0 0
\(121\) −569.893 −0.428169
\(122\) −2366.81 −1.75640
\(123\) 0 0
\(124\) −3917.92 −2.83742
\(125\) −318.242 −0.227716
\(126\) 0 0
\(127\) 1431.63 1.00029 0.500146 0.865941i \(-0.333280\pi\)
0.500146 + 0.865941i \(0.333280\pi\)
\(128\) 1725.94 1.19182
\(129\) 0 0
\(130\) 0 0
\(131\) −2067.32 −1.37880 −0.689400 0.724381i \(-0.742126\pi\)
−0.689400 + 0.724381i \(0.742126\pi\)
\(132\) 0 0
\(133\) −21.7525 −0.0141818
\(134\) −1013.38 −0.653304
\(135\) 0 0
\(136\) −7141.25 −4.50262
\(137\) −387.512 −0.241660 −0.120830 0.992673i \(-0.538556\pi\)
−0.120830 + 0.992673i \(0.538556\pi\)
\(138\) 0 0
\(139\) 752.568 0.459223 0.229611 0.973282i \(-0.426254\pi\)
0.229611 + 0.973282i \(0.426254\pi\)
\(140\) 3243.10 1.95780
\(141\) 0 0
\(142\) 2584.09 1.52713
\(143\) 0 0
\(144\) 0 0
\(145\) 1011.40 0.579254
\(146\) 5105.34 2.89398
\(147\) 0 0
\(148\) −2009.13 −1.11587
\(149\) −2636.72 −1.44972 −0.724862 0.688895i \(-0.758096\pi\)
−0.724862 + 0.688895i \(0.758096\pi\)
\(150\) 0 0
\(151\) 3332.42 1.79595 0.897975 0.440046i \(-0.145038\pi\)
0.897975 + 0.440046i \(0.145038\pi\)
\(152\) 148.692 0.0793454
\(153\) 0 0
\(154\) −1423.45 −0.744839
\(155\) 3150.20 1.63245
\(156\) 0 0
\(157\) −1625.26 −0.826179 −0.413089 0.910690i \(-0.635550\pi\)
−0.413089 + 0.910690i \(0.635550\pi\)
\(158\) −2000.05 −1.00706
\(159\) 0 0
\(160\) −7881.36 −3.89423
\(161\) 404.698 0.198104
\(162\) 0 0
\(163\) 1835.37 0.881944 0.440972 0.897521i \(-0.354634\pi\)
0.440972 + 0.897521i \(0.354634\pi\)
\(164\) −627.605 −0.298828
\(165\) 0 0
\(166\) −3815.31 −1.78389
\(167\) 1945.00 0.901248 0.450624 0.892714i \(-0.351202\pi\)
0.450624 + 0.892714i \(0.351202\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 9442.44 4.26001
\(171\) 0 0
\(172\) 4864.96 2.15668
\(173\) −2531.63 −1.11258 −0.556289 0.830989i \(-0.687775\pi\)
−0.556289 + 0.830989i \(0.687775\pi\)
\(174\) 0 0
\(175\) −1397.65 −0.603726
\(176\) 5224.91 2.23774
\(177\) 0 0
\(178\) −5533.76 −2.33018
\(179\) −4263.01 −1.78007 −0.890035 0.455892i \(-0.849320\pi\)
−0.890035 + 0.455892i \(0.849320\pi\)
\(180\) 0 0
\(181\) 3944.61 1.61989 0.809946 0.586504i \(-0.199496\pi\)
0.809946 + 0.586504i \(0.199496\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2766.36 −1.10836
\(185\) 1615.43 0.641995
\(186\) 0 0
\(187\) −2977.54 −1.16438
\(188\) −10440.9 −4.05044
\(189\) 0 0
\(190\) −196.606 −0.0750701
\(191\) 214.109 0.0811119 0.0405559 0.999177i \(-0.487087\pi\)
0.0405559 + 0.999177i \(0.487087\pi\)
\(192\) 0 0
\(193\) 1207.19 0.450234 0.225117 0.974332i \(-0.427724\pi\)
0.225117 + 0.974332i \(0.427724\pi\)
\(194\) −349.480 −0.129336
\(195\) 0 0
\(196\) −5089.06 −1.85461
\(197\) −927.631 −0.335487 −0.167744 0.985831i \(-0.553648\pi\)
−0.167744 + 0.985831i \(0.553648\pi\)
\(198\) 0 0
\(199\) 478.951 0.170613 0.0853064 0.996355i \(-0.472813\pi\)
0.0853064 + 0.996355i \(0.472813\pi\)
\(200\) 9553.77 3.37777
\(201\) 0 0
\(202\) −2834.69 −0.987368
\(203\) 596.474 0.206228
\(204\) 0 0
\(205\) 504.624 0.171924
\(206\) −3923.08 −1.32686
\(207\) 0 0
\(208\) 0 0
\(209\) 61.9970 0.0205188
\(210\) 0 0
\(211\) −1450.95 −0.473402 −0.236701 0.971583i \(-0.576066\pi\)
−0.236701 + 0.971583i \(0.576066\pi\)
\(212\) −10060.1 −3.25910
\(213\) 0 0
\(214\) 4175.11 1.33367
\(215\) −3911.66 −1.24080
\(216\) 0 0
\(217\) 1857.84 0.581191
\(218\) 2837.58 0.881583
\(219\) 0 0
\(220\) −9243.19 −2.83262
\(221\) 0 0
\(222\) 0 0
\(223\) 2059.79 0.618536 0.309268 0.950975i \(-0.399916\pi\)
0.309268 + 0.950975i \(0.399916\pi\)
\(224\) −4648.06 −1.38644
\(225\) 0 0
\(226\) 962.612 0.283327
\(227\) 4482.46 1.31062 0.655311 0.755359i \(-0.272537\pi\)
0.655311 + 0.755359i \(0.272537\pi\)
\(228\) 0 0
\(229\) 1630.39 0.470477 0.235239 0.971938i \(-0.424413\pi\)
0.235239 + 0.971938i \(0.424413\pi\)
\(230\) 3657.80 1.04864
\(231\) 0 0
\(232\) −4077.27 −1.15382
\(233\) 1903.69 0.535258 0.267629 0.963522i \(-0.413760\pi\)
0.267629 + 0.963522i \(0.413760\pi\)
\(234\) 0 0
\(235\) 8395.00 2.33034
\(236\) 9883.58 2.72613
\(237\) 0 0
\(238\) 5568.72 1.51667
\(239\) 3763.79 1.01866 0.509328 0.860572i \(-0.329894\pi\)
0.509328 + 0.860572i \(0.329894\pi\)
\(240\) 0 0
\(241\) 3614.74 0.966166 0.483083 0.875575i \(-0.339517\pi\)
0.483083 + 0.875575i \(0.339517\pi\)
\(242\) −3037.75 −0.806918
\(243\) 0 0
\(244\) −9063.85 −2.37809
\(245\) 4091.84 1.06701
\(246\) 0 0
\(247\) 0 0
\(248\) −12699.5 −3.25169
\(249\) 0 0
\(250\) −1696.36 −0.429148
\(251\) 5729.77 1.44088 0.720438 0.693520i \(-0.243941\pi\)
0.720438 + 0.693520i \(0.243941\pi\)
\(252\) 0 0
\(253\) −1153.43 −0.286624
\(254\) 7631.18 1.88513
\(255\) 0 0
\(256\) 844.191 0.206101
\(257\) 5525.79 1.34120 0.670602 0.741818i \(-0.266036\pi\)
0.670602 + 0.741818i \(0.266036\pi\)
\(258\) 0 0
\(259\) 952.709 0.228565
\(260\) 0 0
\(261\) 0 0
\(262\) −11019.6 −2.59846
\(263\) −5223.21 −1.22463 −0.612313 0.790615i \(-0.709761\pi\)
−0.612313 + 0.790615i \(0.709761\pi\)
\(264\) 0 0
\(265\) 8088.78 1.87506
\(266\) −115.949 −0.0267267
\(267\) 0 0
\(268\) −3880.81 −0.884545
\(269\) −7203.88 −1.63282 −0.816410 0.577473i \(-0.804039\pi\)
−0.816410 + 0.577473i \(0.804039\pi\)
\(270\) 0 0
\(271\) 8577.69 1.92272 0.961360 0.275293i \(-0.0887750\pi\)
0.961360 + 0.275293i \(0.0887750\pi\)
\(272\) −20440.5 −4.55656
\(273\) 0 0
\(274\) −2065.59 −0.455427
\(275\) 3983.44 0.873493
\(276\) 0 0
\(277\) 7169.19 1.55507 0.777536 0.628838i \(-0.216469\pi\)
0.777536 + 0.628838i \(0.216469\pi\)
\(278\) 4011.48 0.865442
\(279\) 0 0
\(280\) 10512.1 2.24364
\(281\) 849.157 0.180272 0.0901360 0.995929i \(-0.471270\pi\)
0.0901360 + 0.995929i \(0.471270\pi\)
\(282\) 0 0
\(283\) 1115.37 0.234283 0.117141 0.993115i \(-0.462627\pi\)
0.117141 + 0.993115i \(0.462627\pi\)
\(284\) 9895.95 2.06766
\(285\) 0 0
\(286\) 0 0
\(287\) 297.604 0.0612091
\(288\) 0 0
\(289\) 6735.49 1.37095
\(290\) 5391.14 1.09165
\(291\) 0 0
\(292\) 19551.2 3.91832
\(293\) −1863.53 −0.371565 −0.185782 0.982591i \(-0.559482\pi\)
−0.185782 + 0.982591i \(0.559482\pi\)
\(294\) 0 0
\(295\) −7946.87 −1.56842
\(296\) −6512.36 −1.27879
\(297\) 0 0
\(298\) −14054.8 −2.73212
\(299\) 0 0
\(300\) 0 0
\(301\) −2306.91 −0.441755
\(302\) 17763.1 3.38461
\(303\) 0 0
\(304\) 425.602 0.0802959
\(305\) 7287.76 1.36818
\(306\) 0 0
\(307\) 6387.50 1.18747 0.593736 0.804660i \(-0.297652\pi\)
0.593736 + 0.804660i \(0.297652\pi\)
\(308\) −5451.21 −1.00848
\(309\) 0 0
\(310\) 16791.8 3.07648
\(311\) 3492.59 0.636806 0.318403 0.947955i \(-0.396853\pi\)
0.318403 + 0.947955i \(0.396853\pi\)
\(312\) 0 0
\(313\) −5912.01 −1.06762 −0.533812 0.845603i \(-0.679241\pi\)
−0.533812 + 0.845603i \(0.679241\pi\)
\(314\) −8663.29 −1.55700
\(315\) 0 0
\(316\) −7659.31 −1.36351
\(317\) −1677.54 −0.297224 −0.148612 0.988896i \(-0.547481\pi\)
−0.148612 + 0.988896i \(0.547481\pi\)
\(318\) 0 0
\(319\) −1700.02 −0.298378
\(320\) −17143.0 −2.99476
\(321\) 0 0
\(322\) 2157.20 0.373342
\(323\) −242.540 −0.0417810
\(324\) 0 0
\(325\) 0 0
\(326\) 9783.22 1.66209
\(327\) 0 0
\(328\) −2034.31 −0.342457
\(329\) 4950.98 0.829655
\(330\) 0 0
\(331\) −2010.31 −0.333827 −0.166913 0.985972i \(-0.553380\pi\)
−0.166913 + 0.985972i \(0.553380\pi\)
\(332\) −14611.0 −2.41531
\(333\) 0 0
\(334\) 10367.6 1.69847
\(335\) 3120.35 0.508905
\(336\) 0 0
\(337\) 7139.24 1.15400 0.577002 0.816743i \(-0.304222\pi\)
0.577002 + 0.816743i \(0.304222\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 36160.5 5.76787
\(341\) −5295.05 −0.840889
\(342\) 0 0
\(343\) 5733.31 0.902536
\(344\) 15769.2 2.47156
\(345\) 0 0
\(346\) −13494.6 −2.09674
\(347\) −1.13990 −0.000176349 0 −8.81743e−5 1.00000i \(-0.500028\pi\)
−8.81743e−5 1.00000i \(0.500028\pi\)
\(348\) 0 0
\(349\) −12199.1 −1.87107 −0.935535 0.353235i \(-0.885082\pi\)
−0.935535 + 0.353235i \(0.885082\pi\)
\(350\) −7450.00 −1.13777
\(351\) 0 0
\(352\) 13247.5 2.00595
\(353\) −10892.3 −1.64232 −0.821160 0.570698i \(-0.806673\pi\)
−0.821160 + 0.570698i \(0.806673\pi\)
\(354\) 0 0
\(355\) −7956.81 −1.18959
\(356\) −21191.9 −3.15497
\(357\) 0 0
\(358\) −22723.5 −3.35468
\(359\) −3525.78 −0.518339 −0.259169 0.965832i \(-0.583449\pi\)
−0.259169 + 0.965832i \(0.583449\pi\)
\(360\) 0 0
\(361\) −6853.95 −0.999264
\(362\) 21026.3 3.05281
\(363\) 0 0
\(364\) 0 0
\(365\) −15720.1 −2.25433
\(366\) 0 0
\(367\) 2383.75 0.339049 0.169525 0.985526i \(-0.445777\pi\)
0.169525 + 0.985526i \(0.445777\pi\)
\(368\) −7918.19 −1.12164
\(369\) 0 0
\(370\) 8610.90 1.20989
\(371\) 4770.39 0.667564
\(372\) 0 0
\(373\) −13282.2 −1.84377 −0.921885 0.387463i \(-0.873352\pi\)
−0.921885 + 0.387463i \(0.873352\pi\)
\(374\) −15871.5 −2.19437
\(375\) 0 0
\(376\) −33843.1 −4.64181
\(377\) 0 0
\(378\) 0 0
\(379\) 4436.73 0.601318 0.300659 0.953732i \(-0.402793\pi\)
0.300659 + 0.953732i \(0.402793\pi\)
\(380\) −752.917 −0.101642
\(381\) 0 0
\(382\) 1141.28 0.152862
\(383\) −810.412 −0.108120 −0.0540602 0.998538i \(-0.517216\pi\)
−0.0540602 + 0.998538i \(0.517216\pi\)
\(384\) 0 0
\(385\) 4383.03 0.580207
\(386\) 6434.78 0.848501
\(387\) 0 0
\(388\) −1338.35 −0.175115
\(389\) −3463.79 −0.451469 −0.225734 0.974189i \(-0.572478\pi\)
−0.225734 + 0.974189i \(0.572478\pi\)
\(390\) 0 0
\(391\) 4512.37 0.583633
\(392\) −16495.6 −2.12539
\(393\) 0 0
\(394\) −4944.64 −0.632252
\(395\) 6158.45 0.784469
\(396\) 0 0
\(397\) −425.405 −0.0537796 −0.0268898 0.999638i \(-0.508560\pi\)
−0.0268898 + 0.999638i \(0.508560\pi\)
\(398\) 2553.00 0.321533
\(399\) 0 0
\(400\) 27345.9 3.41823
\(401\) −1186.85 −0.147801 −0.0739007 0.997266i \(-0.523545\pi\)
−0.0739007 + 0.997266i \(0.523545\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −10855.6 −1.33685
\(405\) 0 0
\(406\) 3179.44 0.388653
\(407\) −2715.33 −0.330697
\(408\) 0 0
\(409\) −8007.42 −0.968071 −0.484036 0.875048i \(-0.660830\pi\)
−0.484036 + 0.875048i \(0.660830\pi\)
\(410\) 2689.84 0.324005
\(411\) 0 0
\(412\) −15023.7 −1.79652
\(413\) −4686.70 −0.558395
\(414\) 0 0
\(415\) 11747.9 1.38960
\(416\) 0 0
\(417\) 0 0
\(418\) 330.468 0.0386692
\(419\) −6832.46 −0.796629 −0.398314 0.917249i \(-0.630405\pi\)
−0.398314 + 0.917249i \(0.630405\pi\)
\(420\) 0 0
\(421\) −10739.6 −1.24326 −0.621632 0.783309i \(-0.713530\pi\)
−0.621632 + 0.783309i \(0.713530\pi\)
\(422\) −7734.16 −0.892163
\(423\) 0 0
\(424\) −32608.6 −3.73494
\(425\) −15583.7 −1.77864
\(426\) 0 0
\(427\) 4297.99 0.487106
\(428\) 15988.9 1.80573
\(429\) 0 0
\(430\) −20850.7 −2.33839
\(431\) 5214.45 0.582763 0.291382 0.956607i \(-0.405885\pi\)
0.291382 + 0.956607i \(0.405885\pi\)
\(432\) 0 0
\(433\) 8642.24 0.959168 0.479584 0.877496i \(-0.340788\pi\)
0.479584 + 0.877496i \(0.340788\pi\)
\(434\) 9903.02 1.09530
\(435\) 0 0
\(436\) 10866.7 1.19362
\(437\) −93.9545 −0.0102848
\(438\) 0 0
\(439\) −13026.2 −1.41619 −0.708097 0.706116i \(-0.750446\pi\)
−0.708097 + 0.706116i \(0.750446\pi\)
\(440\) −29960.7 −3.24619
\(441\) 0 0
\(442\) 0 0
\(443\) 11533.0 1.23690 0.618450 0.785824i \(-0.287761\pi\)
0.618450 + 0.785824i \(0.287761\pi\)
\(444\) 0 0
\(445\) 17039.3 1.81515
\(446\) 10979.5 1.16568
\(447\) 0 0
\(448\) −10110.2 −1.06621
\(449\) 9882.75 1.03874 0.519372 0.854548i \(-0.326166\pi\)
0.519372 + 0.854548i \(0.326166\pi\)
\(450\) 0 0
\(451\) −848.204 −0.0885596
\(452\) 3686.38 0.383613
\(453\) 0 0
\(454\) 23893.3 2.46997
\(455\) 0 0
\(456\) 0 0
\(457\) 15628.1 1.59967 0.799836 0.600218i \(-0.204920\pi\)
0.799836 + 0.600218i \(0.204920\pi\)
\(458\) 8690.63 0.886652
\(459\) 0 0
\(460\) 14007.8 1.41982
\(461\) −7747.46 −0.782723 −0.391361 0.920237i \(-0.627996\pi\)
−0.391361 + 0.920237i \(0.627996\pi\)
\(462\) 0 0
\(463\) 333.422 0.0334675 0.0167337 0.999860i \(-0.494673\pi\)
0.0167337 + 0.999860i \(0.494673\pi\)
\(464\) −11670.4 −1.16764
\(465\) 0 0
\(466\) 10147.4 1.00874
\(467\) −8198.33 −0.812363 −0.406182 0.913792i \(-0.633140\pi\)
−0.406182 + 0.913792i \(0.633140\pi\)
\(468\) 0 0
\(469\) 1840.24 0.181182
\(470\) 44748.7 4.39171
\(471\) 0 0
\(472\) 32036.5 3.12415
\(473\) 6574.96 0.639148
\(474\) 0 0
\(475\) 324.477 0.0313432
\(476\) 21325.8 2.05350
\(477\) 0 0
\(478\) 20062.5 1.91974
\(479\) −6435.88 −0.613910 −0.306955 0.951724i \(-0.599310\pi\)
−0.306955 + 0.951724i \(0.599310\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 19268.0 1.82082
\(483\) 0 0
\(484\) −11633.3 −1.09253
\(485\) 1076.10 0.100749
\(486\) 0 0
\(487\) −8095.37 −0.753257 −0.376629 0.926364i \(-0.622917\pi\)
−0.376629 + 0.926364i \(0.622917\pi\)
\(488\) −29379.4 −2.72529
\(489\) 0 0
\(490\) 21811.1 2.01087
\(491\) −5116.46 −0.470270 −0.235135 0.971963i \(-0.575553\pi\)
−0.235135 + 0.971963i \(0.575553\pi\)
\(492\) 0 0
\(493\) 6650.67 0.607568
\(494\) 0 0
\(495\) 0 0
\(496\) −36349.9 −3.29064
\(497\) −4692.56 −0.423521
\(498\) 0 0
\(499\) 18050.7 1.61936 0.809682 0.586870i \(-0.199640\pi\)
0.809682 + 0.586870i \(0.199640\pi\)
\(500\) −6496.31 −0.581047
\(501\) 0 0
\(502\) 30541.9 2.71544
\(503\) −10531.1 −0.933512 −0.466756 0.884386i \(-0.654577\pi\)
−0.466756 + 0.884386i \(0.654577\pi\)
\(504\) 0 0
\(505\) 8728.45 0.769131
\(506\) −6148.25 −0.540165
\(507\) 0 0
\(508\) 29224.1 2.55238
\(509\) −1963.31 −0.170967 −0.0854834 0.996340i \(-0.527243\pi\)
−0.0854834 + 0.996340i \(0.527243\pi\)
\(510\) 0 0
\(511\) −9271.00 −0.802593
\(512\) −9307.69 −0.803409
\(513\) 0 0
\(514\) 29454.6 2.52760
\(515\) 12079.8 1.03359
\(516\) 0 0
\(517\) −14110.9 −1.20038
\(518\) 5078.31 0.430750
\(519\) 0 0
\(520\) 0 0
\(521\) 7044.93 0.592407 0.296203 0.955125i \(-0.404279\pi\)
0.296203 + 0.955125i \(0.404279\pi\)
\(522\) 0 0
\(523\) −3213.29 −0.268657 −0.134328 0.990937i \(-0.542888\pi\)
−0.134328 + 0.990937i \(0.542888\pi\)
\(524\) −42200.4 −3.51819
\(525\) 0 0
\(526\) −27841.7 −2.30790
\(527\) 20714.9 1.71225
\(528\) 0 0
\(529\) −10419.0 −0.856333
\(530\) 43116.4 3.53369
\(531\) 0 0
\(532\) −444.036 −0.0361868
\(533\) 0 0
\(534\) 0 0
\(535\) −12855.8 −1.03889
\(536\) −12579.2 −1.01369
\(537\) 0 0
\(538\) −38399.5 −3.07718
\(539\) −6877.83 −0.549627
\(540\) 0 0
\(541\) −11251.4 −0.894150 −0.447075 0.894497i \(-0.647534\pi\)
−0.447075 + 0.894497i \(0.647534\pi\)
\(542\) 45722.4 3.62352
\(543\) 0 0
\(544\) −51825.8 −4.08458
\(545\) −8737.33 −0.686727
\(546\) 0 0
\(547\) 1533.54 0.119871 0.0599353 0.998202i \(-0.480911\pi\)
0.0599353 + 0.998202i \(0.480911\pi\)
\(548\) −7910.31 −0.616628
\(549\) 0 0
\(550\) 21233.3 1.64617
\(551\) −138.477 −0.0107066
\(552\) 0 0
\(553\) 3631.97 0.279289
\(554\) 38214.6 2.93066
\(555\) 0 0
\(556\) 15362.2 1.17177
\(557\) 16845.7 1.28146 0.640731 0.767766i \(-0.278631\pi\)
0.640731 + 0.767766i \(0.278631\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 30089.0 2.27052
\(561\) 0 0
\(562\) 4526.34 0.339737
\(563\) 20820.1 1.55855 0.779273 0.626685i \(-0.215589\pi\)
0.779273 + 0.626685i \(0.215589\pi\)
\(564\) 0 0
\(565\) −2964.03 −0.220704
\(566\) 5945.37 0.441524
\(567\) 0 0
\(568\) 32076.6 2.36955
\(569\) −23636.6 −1.74147 −0.870735 0.491752i \(-0.836357\pi\)
−0.870735 + 0.491752i \(0.836357\pi\)
\(570\) 0 0
\(571\) −26955.1 −1.97554 −0.987771 0.155913i \(-0.950168\pi\)
−0.987771 + 0.155913i \(0.950168\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1586.35 0.115353
\(575\) −6036.78 −0.437828
\(576\) 0 0
\(577\) −23499.8 −1.69551 −0.847755 0.530388i \(-0.822046\pi\)
−0.847755 + 0.530388i \(0.822046\pi\)
\(578\) 35902.8 2.58367
\(579\) 0 0
\(580\) 20645.7 1.47805
\(581\) 6928.38 0.494729
\(582\) 0 0
\(583\) −13596.1 −0.965857
\(584\) 63373.1 4.49040
\(585\) 0 0
\(586\) −9933.33 −0.700243
\(587\) 4637.50 0.326082 0.163041 0.986619i \(-0.447870\pi\)
0.163041 + 0.986619i \(0.447870\pi\)
\(588\) 0 0
\(589\) −431.316 −0.0301733
\(590\) −42359.9 −2.95582
\(591\) 0 0
\(592\) −18640.4 −1.29411
\(593\) 12633.5 0.874869 0.437434 0.899250i \(-0.355887\pi\)
0.437434 + 0.899250i \(0.355887\pi\)
\(594\) 0 0
\(595\) −17146.9 −1.18144
\(596\) −53823.7 −3.69917
\(597\) 0 0
\(598\) 0 0
\(599\) 18757.1 1.27946 0.639730 0.768600i \(-0.279046\pi\)
0.639730 + 0.768600i \(0.279046\pi\)
\(600\) 0 0
\(601\) −3632.98 −0.246576 −0.123288 0.992371i \(-0.539344\pi\)
−0.123288 + 0.992371i \(0.539344\pi\)
\(602\) −12296.8 −0.832522
\(603\) 0 0
\(604\) 68025.0 4.58261
\(605\) 9353.71 0.628566
\(606\) 0 0
\(607\) −12700.0 −0.849219 −0.424610 0.905377i \(-0.639589\pi\)
−0.424610 + 0.905377i \(0.639589\pi\)
\(608\) 1079.09 0.0719786
\(609\) 0 0
\(610\) 38846.6 2.57845
\(611\) 0 0
\(612\) 0 0
\(613\) −21640.1 −1.42584 −0.712918 0.701248i \(-0.752627\pi\)
−0.712918 + 0.701248i \(0.752627\pi\)
\(614\) 34047.9 2.23788
\(615\) 0 0
\(616\) −17669.5 −1.15572
\(617\) −16541.7 −1.07933 −0.539663 0.841881i \(-0.681448\pi\)
−0.539663 + 0.841881i \(0.681448\pi\)
\(618\) 0 0
\(619\) 21138.9 1.37261 0.686303 0.727316i \(-0.259233\pi\)
0.686303 + 0.727316i \(0.259233\pi\)
\(620\) 64305.2 4.16542
\(621\) 0 0
\(622\) 18616.9 1.20011
\(623\) 10049.0 0.646235
\(624\) 0 0
\(625\) −12825.4 −0.820823
\(626\) −31513.3 −2.01202
\(627\) 0 0
\(628\) −33176.6 −2.10811
\(629\) 10622.7 0.673377
\(630\) 0 0
\(631\) −5489.80 −0.346348 −0.173174 0.984891i \(-0.555402\pi\)
−0.173174 + 0.984891i \(0.555402\pi\)
\(632\) −24826.8 −1.56259
\(633\) 0 0
\(634\) −8941.94 −0.560141
\(635\) −23497.5 −1.46846
\(636\) 0 0
\(637\) 0 0
\(638\) −9061.76 −0.562318
\(639\) 0 0
\(640\) −28328.1 −1.74963
\(641\) −4297.04 −0.264778 −0.132389 0.991198i \(-0.542265\pi\)
−0.132389 + 0.991198i \(0.542265\pi\)
\(642\) 0 0
\(643\) −25696.9 −1.57603 −0.788016 0.615655i \(-0.788892\pi\)
−0.788016 + 0.615655i \(0.788892\pi\)
\(644\) 8261.14 0.505488
\(645\) 0 0
\(646\) −1292.83 −0.0787396
\(647\) −2174.98 −0.132160 −0.0660798 0.997814i \(-0.521049\pi\)
−0.0660798 + 0.997814i \(0.521049\pi\)
\(648\) 0 0
\(649\) 13357.6 0.807907
\(650\) 0 0
\(651\) 0 0
\(652\) 37465.5 2.25040
\(653\) 15454.5 0.926160 0.463080 0.886316i \(-0.346744\pi\)
0.463080 + 0.886316i \(0.346744\pi\)
\(654\) 0 0
\(655\) 33931.1 2.02412
\(656\) −5822.82 −0.346560
\(657\) 0 0
\(658\) 26390.7 1.56355
\(659\) 3148.77 0.186129 0.0930643 0.995660i \(-0.470334\pi\)
0.0930643 + 0.995660i \(0.470334\pi\)
\(660\) 0 0
\(661\) 2099.70 0.123553 0.0617767 0.998090i \(-0.480323\pi\)
0.0617767 + 0.998090i \(0.480323\pi\)
\(662\) −10715.7 −0.629122
\(663\) 0 0
\(664\) −47359.8 −2.76795
\(665\) 357.026 0.0208193
\(666\) 0 0
\(667\) 2576.32 0.149559
\(668\) 39703.4 2.29966
\(669\) 0 0
\(670\) 16632.7 0.959071
\(671\) −12249.7 −0.704763
\(672\) 0 0
\(673\) 30970.8 1.77390 0.886950 0.461865i \(-0.152819\pi\)
0.886950 + 0.461865i \(0.152819\pi\)
\(674\) 38055.0 2.17481
\(675\) 0 0
\(676\) 0 0
\(677\) −14640.6 −0.831141 −0.415570 0.909561i \(-0.636418\pi\)
−0.415570 + 0.909561i \(0.636418\pi\)
\(678\) 0 0
\(679\) 634.635 0.0358690
\(680\) 117210. 6.60999
\(681\) 0 0
\(682\) −28224.7 −1.58472
\(683\) −6685.83 −0.374563 −0.187281 0.982306i \(-0.559968\pi\)
−0.187281 + 0.982306i \(0.559968\pi\)
\(684\) 0 0
\(685\) 6360.27 0.354764
\(686\) 30560.8 1.70090
\(687\) 0 0
\(688\) 45136.3 2.50117
\(689\) 0 0
\(690\) 0 0
\(691\) 30194.1 1.66228 0.831141 0.556062i \(-0.187688\pi\)
0.831141 + 0.556062i \(0.187688\pi\)
\(692\) −51678.4 −2.83890
\(693\) 0 0
\(694\) −6.07611 −0.000332343 0
\(695\) −12352.0 −0.674154
\(696\) 0 0
\(697\) 3318.28 0.180328
\(698\) −65026.0 −3.52618
\(699\) 0 0
\(700\) −28530.3 −1.54049
\(701\) 30300.9 1.63260 0.816298 0.577631i \(-0.196023\pi\)
0.816298 + 0.577631i \(0.196023\pi\)
\(702\) 0 0
\(703\) −221.181 −0.0118663
\(704\) 28815.1 1.54263
\(705\) 0 0
\(706\) −58060.3 −3.09508
\(707\) 5147.64 0.273829
\(708\) 0 0
\(709\) −26123.2 −1.38375 −0.691875 0.722017i \(-0.743215\pi\)
−0.691875 + 0.722017i \(0.743215\pi\)
\(710\) −42412.9 −2.24187
\(711\) 0 0
\(712\) −68691.1 −3.61560
\(713\) 8024.48 0.421486
\(714\) 0 0
\(715\) 0 0
\(716\) −87021.3 −4.54209
\(717\) 0 0
\(718\) −18793.8 −0.976850
\(719\) −19325.7 −1.00240 −0.501200 0.865331i \(-0.667108\pi\)
−0.501200 + 0.865331i \(0.667108\pi\)
\(720\) 0 0
\(721\) 7124.09 0.367982
\(722\) −36534.2 −1.88319
\(723\) 0 0
\(724\) 80521.7 4.13338
\(725\) −8897.47 −0.455784
\(726\) 0 0
\(727\) 26065.8 1.32975 0.664875 0.746954i \(-0.268485\pi\)
0.664875 + 0.746954i \(0.268485\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −83794.4 −4.24845
\(731\) −25722.0 −1.30145
\(732\) 0 0
\(733\) −1055.45 −0.0531843 −0.0265921 0.999646i \(-0.508466\pi\)
−0.0265921 + 0.999646i \(0.508466\pi\)
\(734\) 12706.4 0.638965
\(735\) 0 0
\(736\) −20076.2 −1.00546
\(737\) −5244.89 −0.262141
\(738\) 0 0
\(739\) −9410.40 −0.468426 −0.234213 0.972185i \(-0.575251\pi\)
−0.234213 + 0.972185i \(0.575251\pi\)
\(740\) 32976.0 1.63814
\(741\) 0 0
\(742\) 25428.1 1.25808
\(743\) −7523.70 −0.371491 −0.185746 0.982598i \(-0.559470\pi\)
−0.185746 + 0.982598i \(0.559470\pi\)
\(744\) 0 0
\(745\) 43276.8 2.12824
\(746\) −70799.4 −3.47473
\(747\) 0 0
\(748\) −60780.8 −2.97108
\(749\) −7581.76 −0.369868
\(750\) 0 0
\(751\) −12984.1 −0.630886 −0.315443 0.948945i \(-0.602153\pi\)
−0.315443 + 0.948945i \(0.602153\pi\)
\(752\) −96869.3 −4.69742
\(753\) 0 0
\(754\) 0 0
\(755\) −54695.3 −2.63651
\(756\) 0 0
\(757\) −27934.6 −1.34122 −0.670609 0.741811i \(-0.733967\pi\)
−0.670609 + 0.741811i \(0.733967\pi\)
\(758\) 23649.5 1.13323
\(759\) 0 0
\(760\) −2440.49 −0.116482
\(761\) −15519.3 −0.739255 −0.369627 0.929180i \(-0.620515\pi\)
−0.369627 + 0.929180i \(0.620515\pi\)
\(762\) 0 0
\(763\) −5152.88 −0.244491
\(764\) 4370.62 0.206968
\(765\) 0 0
\(766\) −4319.82 −0.203761
\(767\) 0 0
\(768\) 0 0
\(769\) −12885.2 −0.604228 −0.302114 0.953272i \(-0.597692\pi\)
−0.302114 + 0.953272i \(0.597692\pi\)
\(770\) 23363.3 1.09345
\(771\) 0 0
\(772\) 24642.4 1.14883
\(773\) 5892.04 0.274155 0.137078 0.990560i \(-0.456229\pi\)
0.137078 + 0.990560i \(0.456229\pi\)
\(774\) 0 0
\(775\) −27713.0 −1.28449
\(776\) −4338.12 −0.200682
\(777\) 0 0
\(778\) −18463.4 −0.850828
\(779\) −69.0916 −0.00317775
\(780\) 0 0
\(781\) 13374.3 0.612766
\(782\) 24052.7 1.09990
\(783\) 0 0
\(784\) −47215.5 −2.15085
\(785\) 26675.6 1.21286
\(786\) 0 0
\(787\) 21020.4 0.952091 0.476045 0.879421i \(-0.342070\pi\)
0.476045 + 0.879421i \(0.342070\pi\)
\(788\) −18935.8 −0.856042
\(789\) 0 0
\(790\) 32827.0 1.47839
\(791\) −1748.05 −0.0785757
\(792\) 0 0
\(793\) 0 0
\(794\) −2267.58 −0.101352
\(795\) 0 0
\(796\) 9776.87 0.435342
\(797\) −31355.6 −1.39356 −0.696782 0.717283i \(-0.745386\pi\)
−0.696782 + 0.717283i \(0.745386\pi\)
\(798\) 0 0
\(799\) 55203.3 2.44425
\(800\) 69334.0 3.06416
\(801\) 0 0
\(802\) −6326.37 −0.278543
\(803\) 26423.4 1.16122
\(804\) 0 0
\(805\) −6642.34 −0.290822
\(806\) 0 0
\(807\) 0 0
\(808\) −35187.3 −1.53204
\(809\) 18132.5 0.788017 0.394009 0.919107i \(-0.371088\pi\)
0.394009 + 0.919107i \(0.371088\pi\)
\(810\) 0 0
\(811\) 24755.3 1.07186 0.535928 0.844263i \(-0.319962\pi\)
0.535928 + 0.844263i \(0.319962\pi\)
\(812\) 12175.9 0.526219
\(813\) 0 0
\(814\) −14473.8 −0.623225
\(815\) −30124.0 −1.29472
\(816\) 0 0
\(817\) 535.572 0.0229343
\(818\) −42682.7 −1.82441
\(819\) 0 0
\(820\) 10300.9 0.438688
\(821\) 4082.65 0.173551 0.0867755 0.996228i \(-0.472344\pi\)
0.0867755 + 0.996228i \(0.472344\pi\)
\(822\) 0 0
\(823\) −34327.0 −1.45391 −0.726954 0.686687i \(-0.759065\pi\)
−0.726954 + 0.686687i \(0.759065\pi\)
\(824\) −48697.6 −2.05881
\(825\) 0 0
\(826\) −24981.9 −1.05234
\(827\) 3228.87 0.135767 0.0678833 0.997693i \(-0.478375\pi\)
0.0678833 + 0.997693i \(0.478375\pi\)
\(828\) 0 0
\(829\) −10452.4 −0.437908 −0.218954 0.975735i \(-0.570265\pi\)
−0.218954 + 0.975735i \(0.570265\pi\)
\(830\) 62621.0 2.61880
\(831\) 0 0
\(832\) 0 0
\(833\) 26906.9 1.11917
\(834\) 0 0
\(835\) −31923.4 −1.32306
\(836\) 1265.55 0.0523564
\(837\) 0 0
\(838\) −36419.7 −1.50131
\(839\) −28289.0 −1.16406 −0.582028 0.813169i \(-0.697741\pi\)
−0.582028 + 0.813169i \(0.697741\pi\)
\(840\) 0 0
\(841\) −20591.8 −0.844308
\(842\) −57246.1 −2.34303
\(843\) 0 0
\(844\) −29618.5 −1.20795
\(845\) 0 0
\(846\) 0 0
\(847\) 5516.39 0.223784
\(848\) −93335.9 −3.77968
\(849\) 0 0
\(850\) −83067.2 −3.35198
\(851\) 4114.99 0.165758
\(852\) 0 0
\(853\) 26631.8 1.06900 0.534498 0.845170i \(-0.320501\pi\)
0.534498 + 0.845170i \(0.320501\pi\)
\(854\) 22910.0 0.917989
\(855\) 0 0
\(856\) 51826.0 2.06936
\(857\) −11796.7 −0.470209 −0.235104 0.971970i \(-0.575543\pi\)
−0.235104 + 0.971970i \(0.575543\pi\)
\(858\) 0 0
\(859\) −22672.8 −0.900567 −0.450283 0.892886i \(-0.648677\pi\)
−0.450283 + 0.892886i \(0.648677\pi\)
\(860\) −79849.0 −3.16608
\(861\) 0 0
\(862\) 27795.0 1.09826
\(863\) 21421.1 0.844940 0.422470 0.906377i \(-0.361163\pi\)
0.422470 + 0.906377i \(0.361163\pi\)
\(864\) 0 0
\(865\) 41551.9 1.63330
\(866\) 46066.6 1.80763
\(867\) 0 0
\(868\) 37924.3 1.48299
\(869\) −10351.5 −0.404086
\(870\) 0 0
\(871\) 0 0
\(872\) 35223.1 1.36790
\(873\) 0 0
\(874\) −500.814 −0.0193825
\(875\) 3080.48 0.119016
\(876\) 0 0
\(877\) 5155.20 0.198493 0.0992466 0.995063i \(-0.468357\pi\)
0.0992466 + 0.995063i \(0.468357\pi\)
\(878\) −69435.0 −2.66893
\(879\) 0 0
\(880\) −85756.9 −3.28507
\(881\) −23692.2 −0.906027 −0.453013 0.891504i \(-0.649651\pi\)
−0.453013 + 0.891504i \(0.649651\pi\)
\(882\) 0 0
\(883\) −14591.5 −0.556108 −0.278054 0.960565i \(-0.589689\pi\)
−0.278054 + 0.960565i \(0.589689\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 61475.2 2.33104
\(887\) 9722.26 0.368029 0.184014 0.982924i \(-0.441091\pi\)
0.184014 + 0.982924i \(0.441091\pi\)
\(888\) 0 0
\(889\) −13857.8 −0.522806
\(890\) 90826.1 3.42078
\(891\) 0 0
\(892\) 42046.6 1.57828
\(893\) −1149.42 −0.0430726
\(894\) 0 0
\(895\) 69969.2 2.61320
\(896\) −16706.6 −0.622911
\(897\) 0 0
\(898\) 52678.9 1.95759
\(899\) 11827.1 0.438771
\(900\) 0 0
\(901\) 53189.7 1.96671
\(902\) −4521.26 −0.166898
\(903\) 0 0
\(904\) 11949.0 0.439621
\(905\) −64743.2 −2.37805
\(906\) 0 0
\(907\) 11799.0 0.431951 0.215975 0.976399i \(-0.430707\pi\)
0.215975 + 0.976399i \(0.430707\pi\)
\(908\) 91500.9 3.34423
\(909\) 0 0
\(910\) 0 0
\(911\) 43012.4 1.56429 0.782143 0.623099i \(-0.214127\pi\)
0.782143 + 0.623099i \(0.214127\pi\)
\(912\) 0 0
\(913\) −19746.6 −0.715793
\(914\) 83303.8 3.01471
\(915\) 0 0
\(916\) 33281.3 1.20049
\(917\) 20011.0 0.720635
\(918\) 0 0
\(919\) −4951.41 −0.177728 −0.0888639 0.996044i \(-0.528324\pi\)
−0.0888639 + 0.996044i \(0.528324\pi\)
\(920\) 45404.5 1.62711
\(921\) 0 0
\(922\) −41297.0 −1.47510
\(923\) 0 0
\(924\) 0 0
\(925\) −14211.3 −0.505152
\(926\) 1777.27 0.0630721
\(927\) 0 0
\(928\) −29589.8 −1.04669
\(929\) −8934.86 −0.315547 −0.157774 0.987475i \(-0.550432\pi\)
−0.157774 + 0.987475i \(0.550432\pi\)
\(930\) 0 0
\(931\) −560.243 −0.0197221
\(932\) 38860.2 1.36578
\(933\) 0 0
\(934\) −43700.3 −1.53096
\(935\) 48870.6 1.70935
\(936\) 0 0
\(937\) −13182.8 −0.459620 −0.229810 0.973235i \(-0.573811\pi\)
−0.229810 + 0.973235i \(0.573811\pi\)
\(938\) 9809.20 0.341452
\(939\) 0 0
\(940\) 171368. 5.94618
\(941\) −21693.7 −0.751536 −0.375768 0.926714i \(-0.622621\pi\)
−0.375768 + 0.926714i \(0.622621\pi\)
\(942\) 0 0
\(943\) 1285.43 0.0443895
\(944\) 91698.4 3.16158
\(945\) 0 0
\(946\) 35047.1 1.20452
\(947\) −49790.0 −1.70851 −0.854254 0.519856i \(-0.825986\pi\)
−0.854254 + 0.519856i \(0.825986\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1729.59 0.0590687
\(951\) 0 0
\(952\) 69125.0 2.35331
\(953\) −4217.93 −0.143371 −0.0716853 0.997427i \(-0.522838\pi\)
−0.0716853 + 0.997427i \(0.522838\pi\)
\(954\) 0 0
\(955\) −3514.19 −0.119075
\(956\) 76830.5 2.59924
\(957\) 0 0
\(958\) −34305.8 −1.15696
\(959\) 3750.99 0.126304
\(960\) 0 0
\(961\) 7046.87 0.236544
\(962\) 0 0
\(963\) 0 0
\(964\) 73788.1 2.46530
\(965\) −19813.7 −0.660958
\(966\) 0 0
\(967\) 40927.9 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(968\) −37707.9 −1.25204
\(969\) 0 0
\(970\) 5736.04 0.189869
\(971\) 17114.8 0.565645 0.282822 0.959172i \(-0.408729\pi\)
0.282822 + 0.959172i \(0.408729\pi\)
\(972\) 0 0
\(973\) −7284.62 −0.240015
\(974\) −43151.5 −1.41957
\(975\) 0 0
\(976\) −84093.0 −2.75794
\(977\) −118.470 −0.00387940 −0.00193970 0.999998i \(-0.500617\pi\)
−0.00193970 + 0.999998i \(0.500617\pi\)
\(978\) 0 0
\(979\) −28640.7 −0.934996
\(980\) 83527.2 2.72263
\(981\) 0 0
\(982\) −27272.8 −0.886261
\(983\) 26002.8 0.843705 0.421852 0.906665i \(-0.361380\pi\)
0.421852 + 0.906665i \(0.361380\pi\)
\(984\) 0 0
\(985\) 15225.3 0.492506
\(986\) 35450.7 1.14501
\(987\) 0 0
\(988\) 0 0
\(989\) −9964.14 −0.320365
\(990\) 0 0
\(991\) −16062.0 −0.514860 −0.257430 0.966297i \(-0.582876\pi\)
−0.257430 + 0.966297i \(0.582876\pi\)
\(992\) −92163.3 −2.94979
\(993\) 0 0
\(994\) −25013.2 −0.798159
\(995\) −7861.07 −0.250465
\(996\) 0 0
\(997\) 1361.54 0.0432503 0.0216251 0.999766i \(-0.493116\pi\)
0.0216251 + 0.999766i \(0.493116\pi\)
\(998\) 96217.5 3.05182
\(999\) 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.bb.1.4 4
3.2 odd 2 507.4.a.i.1.1 4
13.4 even 6 117.4.g.e.55.4 8
13.10 even 6 117.4.g.e.100.4 8
13.12 even 2 1521.4.a.v.1.1 4
39.5 even 4 507.4.b.h.337.8 8
39.8 even 4 507.4.b.h.337.1 8
39.17 odd 6 39.4.e.c.16.1 8
39.23 odd 6 39.4.e.c.22.1 yes 8
39.38 odd 2 507.4.a.m.1.4 4
156.23 even 6 624.4.q.i.529.1 8
156.95 even 6 624.4.q.i.289.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 39.17 odd 6
39.4.e.c.22.1 yes 8 39.23 odd 6
117.4.g.e.55.4 8 13.4 even 6
117.4.g.e.100.4 8 13.10 even 6
507.4.a.i.1.1 4 3.2 odd 2
507.4.a.m.1.4 4 39.38 odd 2
507.4.b.h.337.1 8 39.8 even 4
507.4.b.h.337.8 8 39.5 even 4
624.4.q.i.289.1 8 156.95 even 6
624.4.q.i.529.1 8 156.23 even 6
1521.4.a.v.1.1 4 13.12 even 2
1521.4.a.bb.1.4 4 1.1 even 1 trivial