Properties

Label 1521.4.a.v.1.1
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 1.1
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 \(-0.891331\pi\)
−0.942289 + 0.334800i \(0.891331\pi\)
\(3\) 0 0
\(4\) 20.4131 2.55164
\(5\) 16.4131 1.46803 0.734016 0.679132i \(-0.237644\pi\)
0.734016 + 0.679132i \(0.237644\pi\)
\(6\) 0 0
\(7\) 9.67968 0.522654 0.261327 0.965250i \(-0.415840\pi\)
0.261327 + 0.965250i \(0.415840\pi\)
\(8\) −66.1667 −2.92418
\(9\) 0 0
\(10\) −87.4882 −2.76662
\(11\) −27.5882 −0.756195 −0.378098 0.925766i \(-0.623422\pi\)
−0.378098 + 0.925766i \(0.623422\pi\)
\(12\) 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 \(-0.504319\pi\)
−0.0135671 + 0.999908i \(0.504319\pi\)
\(20\) 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 \(-0.312335\pi\)
0.556000 + 0.831182i \(0.312335\pi\)
\(32\) −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 \(-0.429832\pi\)
0.218659 + 0.975801i \(0.429832\pi\)
\(38\) 11.9786 0.0511366
\(39\) 0 0
\(40\) −1086.00 −4.29279
\(41\) 30.7452 0.117112 0.0585561 0.998284i \(-0.481350\pi\)
0.0585561 + 0.998284i \(0.481350\pi\)
\(42\) 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 \(-0.208155\pi\)
0.793695 + 0.608316i \(0.208155\pi\)
\(48\) 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 \(-0.679384\pi\)
−0.534192 + 0.845363i \(0.679384\pi\)
\(60\) 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 \(-0.444548\pi\)
0.173329 + 0.984864i \(0.444548\pi\)
\(68\) −2203.15 −3.92898
\(69\) 0 0
\(70\) −846.858 −1.44598
\(71\) −484.785 −0.810329 −0.405164 0.914244i \(-0.632786\pi\)
−0.405164 + 0.914244i \(0.632786\pi\)
\(72\) 0 0
\(73\) −957.780 −1.53561 −0.767806 0.640683i \(-0.778651\pi\)
−0.767806 + 0.640683i \(0.778651\pi\)
\(74\) −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 \(-0.343068\pi\)
0.473286 + 0.880909i \(0.343068\pi\)
\(84\) 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 \(-0.287852\pi\)
0.618224 + 0.786002i \(0.287852\pi\)
\(90\) 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 \(-0.489075\pi\)
0.0343143 + 0.999411i \(0.489075\pi\)
\(98\) 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 \(-0.575147\pi\)
−0.233894 + 0.972262i \(0.575147\pi\)
\(110\) 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 \(-0.461444\pi\)
0.120830 + 0.992673i \(0.461444\pi\)
\(138\) 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 \(-0.241904\pi\)
0.724862 + 0.688895i \(0.241904\pi\)
\(150\) 0 0
\(151\) −3332.42 −1.79595 −0.897975 0.440046i \(-0.854962\pi\)
−0.897975 + 0.440046i \(0.854962\pi\)
\(152\) 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 \(-0.645366\pi\)
−0.440972 + 0.897521i \(0.645366\pi\)
\(164\) 627.605 0.298828
\(165\) 0 0
\(166\) −3815.31 −1.78389
\(167\) −1945.00 −0.901248 −0.450624 0.892714i \(-0.648798\pi\)
−0.450624 + 0.892714i \(0.648798\pi\)
\(168\) 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 \(-0.572276\pi\)
−0.225117 + 0.974332i \(0.572276\pi\)
\(194\) −349.480 −0.129336
\(195\) 0 0
\(196\) −5089.06 −1.85461
\(197\) 927.631 0.335487 0.167744 0.985831i \(-0.446352\pi\)
0.167744 + 0.985831i \(0.446352\pi\)
\(198\) 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 \(-0.600084\pi\)
−0.309268 + 0.950975i \(0.600084\pi\)
\(224\) −4648.06 −1.38644
\(225\) 0 0
\(226\) −962.612 −0.283327
\(227\) −4482.46 −1.31062 −0.655311 0.755359i \(-0.727463\pi\)
−0.655311 + 0.755359i \(0.727463\pi\)
\(228\) 0 0
\(229\) −1630.39 −0.470477 −0.235239 0.971938i \(-0.575587\pi\)
−0.235239 + 0.971938i \(0.575587\pi\)
\(230\) 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 \(-0.670106\pi\)
−0.509328 + 0.860572i \(0.670106\pi\)
\(240\) 0 0
\(241\) −3614.74 −0.966166 −0.483083 0.875575i \(-0.660483\pi\)
−0.483083 + 0.875575i \(0.660483\pi\)
\(242\) 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 \(-0.911225\pi\)
−0.961360 + 0.275293i \(0.911225\pi\)
\(272\) −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 \(-0.528730\pi\)
−0.0901360 + 0.995929i \(0.528730\pi\)
\(282\) 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 \(-0.440518\pi\)
0.185782 + 0.982591i \(0.440518\pi\)
\(294\) 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 \(-0.702348\pi\)
−0.593736 + 0.804660i \(0.702348\pi\)
\(308\) −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 \(-0.452519\pi\)
0.148612 + 0.988896i \(0.452519\pi\)
\(318\) 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 \(-0.446620\pi\)
0.166913 + 0.985972i \(0.446620\pi\)
\(332\) 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 \(-0.114918\pi\)
0.935535 + 0.353235i \(0.114918\pi\)
\(350\) −7450.00 −1.13777
\(351\) 0 0
\(352\) 13247.5 2.00595
\(353\) 10892.3 1.64232 0.821160 0.570698i \(-0.193327\pi\)
0.821160 + 0.570698i \(0.193327\pi\)
\(354\) 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 \(-0.416551\pi\)
0.259169 + 0.965832i \(0.416551\pi\)
\(360\) 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 \(-0.597207\pi\)
−0.300659 + 0.953732i \(0.597207\pi\)
\(380\) −752.917 −0.101642
\(381\) 0 0
\(382\) −1141.28 −0.152862
\(383\) 810.412 0.108120 0.0540602 0.998538i \(-0.482784\pi\)
0.0540602 + 0.998538i \(0.482784\pi\)
\(384\) 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 \(-0.491440\pi\)
0.0268898 + 0.999638i \(0.491440\pi\)
\(398\) −2553.00 −0.321533
\(399\) 0 0
\(400\) 27345.9 3.41823
\(401\) 1186.85 0.147801 0.0739007 0.997266i \(-0.476455\pi\)
0.0739007 + 0.997266i \(0.476455\pi\)
\(402\) 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 \(-0.339170\pi\)
0.484036 + 0.875048i \(0.339170\pi\)
\(410\) −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 \(-0.286470\pi\)
0.621632 + 0.783309i \(0.286470\pi\)
\(422\) 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 \(-0.594115\pi\)
−0.291382 + 0.956607i \(0.594115\pi\)
\(432\) 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 \(-0.673834\pi\)
−0.519372 + 0.854548i \(0.673834\pi\)
\(450\) 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 \(-0.795080\pi\)
−0.799836 + 0.600218i \(0.795080\pi\)
\(458\) 8690.63 0.886652
\(459\) 0 0
\(460\) −14007.8 −1.41982
\(461\) 7747.46 0.782723 0.391361 0.920237i \(-0.372004\pi\)
0.391361 + 0.920237i \(0.372004\pi\)
\(462\) 0 0
\(463\) −333.422 −0.0334675 −0.0167337 0.999860i \(-0.505327\pi\)
−0.0167337 + 0.999860i \(0.505327\pi\)
\(464\) −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 \(-0.400690\pi\)
0.306955 + 0.951724i \(0.400690\pi\)
\(480\) 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 \(-0.377083\pi\)
0.376629 + 0.926364i \(0.377083\pi\)
\(488\) 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 \(-0.800360\pi\)
−0.809682 + 0.586870i \(0.800360\pi\)
\(500\) 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 \(-0.472757\pi\)
0.0854834 + 0.996340i \(0.472757\pi\)
\(510\) 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 \(-0.352466\pi\)
0.447075 + 0.894497i \(0.352466\pi\)
\(542\) 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 \(-0.721369\pi\)
−0.640731 + 0.767766i \(0.721369\pi\)
\(558\) 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 \(-0.177954\pi\)
0.847755 + 0.530388i \(0.177954\pi\)
\(578\) −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 \(-0.552130\pi\)
−0.163041 + 0.986619i \(0.552130\pi\)
\(588\) 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 \(-0.644113\pi\)
−0.437434 + 0.899250i \(0.644113\pi\)
\(594\) 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 \(-0.247373\pi\)
0.712918 + 0.701248i \(0.247373\pi\)
\(614\) 34047.9 2.23788
\(615\) 0 0
\(616\) 17669.5 1.15572
\(617\) 16541.7 1.07933 0.539663 0.841881i \(-0.318552\pi\)
0.539663 + 0.841881i \(0.318552\pi\)
\(618\) 0 0
\(619\) −21138.9 −1.37261 −0.686303 0.727316i \(-0.740767\pi\)
−0.686303 + 0.727316i \(0.740767\pi\)
\(620\) 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 \(-0.444598\pi\)
0.173174 + 0.984891i \(0.444598\pi\)
\(632\) 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 \(-0.211108\pi\)
0.788016 + 0.615655i \(0.211108\pi\)
\(644\) −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 \(-0.519677\pi\)
−0.0617767 + 0.998090i \(0.519677\pi\)
\(662\) −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 \(-0.440032\pi\)
0.187281 + 0.982306i \(0.440032\pi\)
\(684\) 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 \(-0.812312\pi\)
−0.831141 + 0.556062i \(0.812312\pi\)
\(692\) −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 \(-0.256785\pi\)
0.691875 + 0.722017i \(0.256785\pi\)
\(710\) 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 \(-0.491534\pi\)
0.0265921 + 0.999646i \(0.491534\pi\)
\(734\) −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 \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(740\) 32976.0 1.63814
\(741\) 0 0
\(742\) 25428.1 1.25808
\(743\) 7523.70 0.371491 0.185746 0.982598i \(-0.440530\pi\)
0.185746 + 0.982598i \(0.440530\pi\)
\(744\) 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 \(-0.379485\pi\)
0.369627 + 0.929180i \(0.379485\pi\)
\(762\) 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 \(-0.402308\pi\)
0.302114 + 0.953272i \(0.402308\pi\)
\(770\) 23363.3 1.09345
\(771\) 0 0
\(772\) −24642.4 −1.14883
\(773\) −5892.04 −0.274155 −0.137078 0.990560i \(-0.543771\pi\)
−0.137078 + 0.990560i \(0.543771\pi\)
\(774\) 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 \(-0.657930\pi\)
−0.476045 + 0.879421i \(0.657930\pi\)
\(788\) 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 \(-0.680038\pi\)
−0.535928 + 0.844263i \(0.680038\pi\)
\(812\) −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 \(-0.527656\pi\)
−0.0867755 + 0.996228i \(0.527656\pi\)
\(822\) 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 \(-0.521625\pi\)
−0.0678833 + 0.997693i \(0.521625\pi\)
\(828\) 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 \(-0.302259\pi\)
0.582028 + 0.813169i \(0.302259\pi\)
\(840\) 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 \(-0.679499\pi\)
−0.534498 + 0.845170i \(0.679499\pi\)
\(854\) 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 \(-0.638837\pi\)
−0.422470 + 0.906377i \(0.638837\pi\)
\(864\) 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 \(-0.531643\pi\)
−0.0992466 + 0.995063i \(0.531643\pi\)
\(878\) 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 \(-0.449568\pi\)
0.157774 + 0.987475i \(0.449568\pi\)
\(930\) 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 \(-0.377379\pi\)
0.375768 + 0.926714i \(0.377379\pi\)
\(942\) 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 \(-0.174014\pi\)
0.854254 + 0.519856i \(0.174014\pi\)
\(948\) 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 \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(968\) 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 \(-0.499383\pi\)
0.00193970 + 0.999998i \(0.499383\pi\)
\(978\) 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 \(-0.638620\pi\)
−0.421852 + 0.906665i \(0.638620\pi\)
\(984\) 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.v.1.1 4
3.2 odd 2 507.4.a.m.1.4 4
13.3 even 3 117.4.g.e.100.4 8
13.9 even 3 117.4.g.e.55.4 8
13.12 even 2 1521.4.a.bb.1.4 4
39.5 even 4 507.4.b.h.337.1 8
39.8 even 4 507.4.b.h.337.8 8
39.29 odd 6 39.4.e.c.22.1 yes 8
39.35 odd 6 39.4.e.c.16.1 8
39.38 odd 2 507.4.a.i.1.1 4
156.35 even 6 624.4.q.i.289.1 8
156.107 even 6 624.4.q.i.529.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 39.35 odd 6
39.4.e.c.22.1 yes 8 39.29 odd 6
117.4.g.e.55.4 8 13.9 even 3
117.4.g.e.100.4 8 13.3 even 3
507.4.a.i.1.1 4 39.38 odd 2
507.4.a.m.1.4 4 3.2 odd 2
507.4.b.h.337.1 8 39.5 even 4
507.4.b.h.337.8 8 39.8 even 4
624.4.q.i.289.1 8 156.35 even 6
624.4.q.i.529.1 8 156.107 even 6
1521.4.a.v.1.1 4 1.1 even 1 trivial
1521.4.a.bb.1.4 4 13.12 even 2