Properties

Label 2160.4.a.be.1.2
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.258712\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} -14.5174 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} -14.5174 q^{7} -49.2845 q^{11} +72.1800 q^{13} -118.017 q^{17} -123.389 q^{19} -91.4883 q^{23} +25.0000 q^{25} -174.400 q^{29} +46.2956 q^{31} +72.5871 q^{35} +154.977 q^{37} +364.203 q^{41} -125.714 q^{43} -221.523 q^{47} -132.244 q^{49} +13.6794 q^{53} +246.423 q^{55} +239.087 q^{59} -54.5457 q^{61} -360.900 q^{65} +76.0558 q^{67} +728.303 q^{71} -501.815 q^{73} +715.485 q^{77} -397.610 q^{79} -1369.46 q^{83} +590.084 q^{85} -1468.13 q^{89} -1047.87 q^{91} +616.945 q^{95} +335.023 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −14.5174 −0.783867 −0.391934 0.919993i \(-0.628194\pi\)
−0.391934 + 0.919993i \(0.628194\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −49.2845 −1.35090 −0.675448 0.737408i \(-0.736050\pi\)
−0.675448 + 0.737408i \(0.736050\pi\)
\(12\) 0 0
\(13\) 72.1800 1.53993 0.769967 0.638084i \(-0.220273\pi\)
0.769967 + 0.638084i \(0.220273\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −118.017 −1.68372 −0.841861 0.539694i \(-0.818540\pi\)
−0.841861 + 0.539694i \(0.818540\pi\)
\(18\) 0 0
\(19\) −123.389 −1.48986 −0.744932 0.667141i \(-0.767518\pi\)
−0.744932 + 0.667141i \(0.767518\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −91.4883 −0.829419 −0.414709 0.909954i \(-0.636117\pi\)
−0.414709 + 0.909954i \(0.636117\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −174.400 −1.11673 −0.558367 0.829594i \(-0.688572\pi\)
−0.558367 + 0.829594i \(0.688572\pi\)
\(30\) 0 0
\(31\) 46.2956 0.268224 0.134112 0.990966i \(-0.457182\pi\)
0.134112 + 0.990966i \(0.457182\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 72.5871 0.350556
\(36\) 0 0
\(37\) 154.977 0.688595 0.344297 0.938861i \(-0.388117\pi\)
0.344297 + 0.938861i \(0.388117\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 364.203 1.38729 0.693645 0.720317i \(-0.256004\pi\)
0.693645 + 0.720317i \(0.256004\pi\)
\(42\) 0 0
\(43\) −125.714 −0.445841 −0.222921 0.974837i \(-0.571559\pi\)
−0.222921 + 0.974837i \(0.571559\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −221.523 −0.687499 −0.343750 0.939061i \(-0.611697\pi\)
−0.343750 + 0.939061i \(0.611697\pi\)
\(48\) 0 0
\(49\) −132.244 −0.385552
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.6794 0.0354530 0.0177265 0.999843i \(-0.494357\pi\)
0.0177265 + 0.999843i \(0.494357\pi\)
\(54\) 0 0
\(55\) 246.423 0.604139
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 239.087 0.527567 0.263784 0.964582i \(-0.415030\pi\)
0.263784 + 0.964582i \(0.415030\pi\)
\(60\) 0 0
\(61\) −54.5457 −0.114490 −0.0572448 0.998360i \(-0.518232\pi\)
−0.0572448 + 0.998360i \(0.518232\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −360.900 −0.688679
\(66\) 0 0
\(67\) 76.0558 0.138682 0.0693410 0.997593i \(-0.477910\pi\)
0.0693410 + 0.997593i \(0.477910\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 728.303 1.21738 0.608688 0.793410i \(-0.291696\pi\)
0.608688 + 0.793410i \(0.291696\pi\)
\(72\) 0 0
\(73\) −501.815 −0.804562 −0.402281 0.915516i \(-0.631782\pi\)
−0.402281 + 0.915516i \(0.631782\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 715.485 1.05892
\(78\) 0 0
\(79\) −397.610 −0.566261 −0.283130 0.959081i \(-0.591373\pi\)
−0.283130 + 0.959081i \(0.591373\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1369.46 −1.81106 −0.905530 0.424283i \(-0.860526\pi\)
−0.905530 + 0.424283i \(0.860526\pi\)
\(84\) 0 0
\(85\) 590.084 0.752984
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1468.13 −1.74856 −0.874278 0.485425i \(-0.838665\pi\)
−0.874278 + 0.485425i \(0.838665\pi\)
\(90\) 0 0
\(91\) −1047.87 −1.20710
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 616.945 0.666287
\(96\) 0 0
\(97\) 335.023 0.350685 0.175343 0.984507i \(-0.443897\pi\)
0.175343 + 0.984507i \(0.443897\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1206.09 1.18822 0.594109 0.804384i \(-0.297505\pi\)
0.594109 + 0.804384i \(0.297505\pi\)
\(102\) 0 0
\(103\) −1061.11 −1.01509 −0.507545 0.861625i \(-0.669447\pi\)
−0.507545 + 0.861625i \(0.669447\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −475.578 −0.429681 −0.214841 0.976649i \(-0.568923\pi\)
−0.214841 + 0.976649i \(0.568923\pi\)
\(108\) 0 0
\(109\) 1320.42 1.16030 0.580152 0.814508i \(-0.302993\pi\)
0.580152 + 0.814508i \(0.302993\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 68.1750 0.0567555 0.0283777 0.999597i \(-0.490966\pi\)
0.0283777 + 0.999597i \(0.490966\pi\)
\(114\) 0 0
\(115\) 457.442 0.370927
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1713.30 1.31981
\(120\) 0 0
\(121\) 1097.97 0.824918
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 593.009 0.414339 0.207170 0.978305i \(-0.433575\pi\)
0.207170 + 0.978305i \(0.433575\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 338.937 0.226054 0.113027 0.993592i \(-0.463945\pi\)
0.113027 + 0.993592i \(0.463945\pi\)
\(132\) 0 0
\(133\) 1791.29 1.16785
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 811.442 0.506030 0.253015 0.967462i \(-0.418578\pi\)
0.253015 + 0.967462i \(0.418578\pi\)
\(138\) 0 0
\(139\) 3106.13 1.89538 0.947691 0.319189i \(-0.103410\pi\)
0.947691 + 0.319189i \(0.103410\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3557.36 −2.08029
\(144\) 0 0
\(145\) 872.001 0.499419
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2541.01 −1.39710 −0.698550 0.715561i \(-0.746171\pi\)
−0.698550 + 0.715561i \(0.746171\pi\)
\(150\) 0 0
\(151\) 1125.37 0.606499 0.303249 0.952911i \(-0.401928\pi\)
0.303249 + 0.952911i \(0.401928\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −231.478 −0.119953
\(156\) 0 0
\(157\) 3230.05 1.64195 0.820975 0.570963i \(-0.193430\pi\)
0.820975 + 0.570963i \(0.193430\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1328.17 0.650154
\(162\) 0 0
\(163\) 694.054 0.333512 0.166756 0.985998i \(-0.446671\pi\)
0.166756 + 0.985998i \(0.446671\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3216.04 1.49021 0.745103 0.666950i \(-0.232400\pi\)
0.745103 + 0.666950i \(0.232400\pi\)
\(168\) 0 0
\(169\) 3012.95 1.37139
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −297.546 −0.130763 −0.0653816 0.997860i \(-0.520826\pi\)
−0.0653816 + 0.997860i \(0.520826\pi\)
\(174\) 0 0
\(175\) −362.936 −0.156773
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3450.12 −1.44064 −0.720320 0.693642i \(-0.756005\pi\)
−0.720320 + 0.693642i \(0.756005\pi\)
\(180\) 0 0
\(181\) −3089.75 −1.26883 −0.634417 0.772991i \(-0.718760\pi\)
−0.634417 + 0.772991i \(0.718760\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −774.883 −0.307949
\(186\) 0 0
\(187\) 5816.41 2.27453
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1532.11 0.580419 0.290209 0.956963i \(-0.406275\pi\)
0.290209 + 0.956963i \(0.406275\pi\)
\(192\) 0 0
\(193\) −5194.42 −1.93732 −0.968660 0.248389i \(-0.920099\pi\)
−0.968660 + 0.248389i \(0.920099\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2005.61 0.725349 0.362674 0.931916i \(-0.381864\pi\)
0.362674 + 0.931916i \(0.381864\pi\)
\(198\) 0 0
\(199\) 2874.68 1.02402 0.512011 0.858979i \(-0.328901\pi\)
0.512011 + 0.858979i \(0.328901\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2531.84 0.875372
\(204\) 0 0
\(205\) −1821.01 −0.620415
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6081.18 2.01265
\(210\) 0 0
\(211\) 2749.94 0.897220 0.448610 0.893728i \(-0.351919\pi\)
0.448610 + 0.893728i \(0.351919\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 628.569 0.199386
\(216\) 0 0
\(217\) −672.093 −0.210252
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8518.45 −2.59282
\(222\) 0 0
\(223\) 783.727 0.235346 0.117673 0.993052i \(-0.462456\pi\)
0.117673 + 0.993052i \(0.462456\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −145.665 −0.0425909 −0.0212955 0.999773i \(-0.506779\pi\)
−0.0212955 + 0.999773i \(0.506779\pi\)
\(228\) 0 0
\(229\) −3411.82 −0.984539 −0.492270 0.870443i \(-0.663833\pi\)
−0.492270 + 0.870443i \(0.663833\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −134.977 −0.0379511 −0.0189756 0.999820i \(-0.506040\pi\)
−0.0189756 + 0.999820i \(0.506040\pi\)
\(234\) 0 0
\(235\) 1107.62 0.307459
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2245.32 −0.607690 −0.303845 0.952722i \(-0.598270\pi\)
−0.303845 + 0.952722i \(0.598270\pi\)
\(240\) 0 0
\(241\) 4158.54 1.11151 0.555757 0.831345i \(-0.312428\pi\)
0.555757 + 0.831345i \(0.312428\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 661.222 0.172424
\(246\) 0 0
\(247\) −8906.22 −2.29429
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3946.14 0.992343 0.496171 0.868225i \(-0.334739\pi\)
0.496171 + 0.868225i \(0.334739\pi\)
\(252\) 0 0
\(253\) 4508.96 1.12046
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5695.84 1.38248 0.691239 0.722626i \(-0.257065\pi\)
0.691239 + 0.722626i \(0.257065\pi\)
\(258\) 0 0
\(259\) −2249.86 −0.539767
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2814.06 0.659781 0.329891 0.944019i \(-0.392988\pi\)
0.329891 + 0.944019i \(0.392988\pi\)
\(264\) 0 0
\(265\) −68.3970 −0.0158551
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −200.985 −0.0455548 −0.0227774 0.999741i \(-0.507251\pi\)
−0.0227774 + 0.999741i \(0.507251\pi\)
\(270\) 0 0
\(271\) 2406.05 0.539326 0.269663 0.962955i \(-0.413088\pi\)
0.269663 + 0.962955i \(0.413088\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1232.11 −0.270179
\(276\) 0 0
\(277\) −8429.33 −1.82841 −0.914205 0.405253i \(-0.867184\pi\)
−0.914205 + 0.405253i \(0.867184\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3974.26 −0.843717 −0.421859 0.906662i \(-0.638622\pi\)
−0.421859 + 0.906662i \(0.638622\pi\)
\(282\) 0 0
\(283\) 3072.41 0.645356 0.322678 0.946509i \(-0.395417\pi\)
0.322678 + 0.946509i \(0.395417\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5287.28 −1.08745
\(288\) 0 0
\(289\) 9014.97 1.83492
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3982.21 −0.794004 −0.397002 0.917818i \(-0.629949\pi\)
−0.397002 + 0.917818i \(0.629949\pi\)
\(294\) 0 0
\(295\) −1195.43 −0.235935
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6603.63 −1.27725
\(300\) 0 0
\(301\) 1825.04 0.349480
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 272.729 0.0512013
\(306\) 0 0
\(307\) 2996.06 0.556984 0.278492 0.960439i \(-0.410165\pi\)
0.278492 + 0.960439i \(0.410165\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3079.94 −0.561567 −0.280783 0.959771i \(-0.590594\pi\)
−0.280783 + 0.959771i \(0.590594\pi\)
\(312\) 0 0
\(313\) 7953.65 1.43632 0.718158 0.695880i \(-0.244986\pi\)
0.718158 + 0.695880i \(0.244986\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6832.98 1.21066 0.605328 0.795976i \(-0.293042\pi\)
0.605328 + 0.795976i \(0.293042\pi\)
\(318\) 0 0
\(319\) 8595.23 1.50859
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 14562.0 2.50852
\(324\) 0 0
\(325\) 1804.50 0.307987
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3215.95 0.538908
\(330\) 0 0
\(331\) 2296.57 0.381363 0.190682 0.981652i \(-0.438930\pi\)
0.190682 + 0.981652i \(0.438930\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −380.279 −0.0620205
\(336\) 0 0
\(337\) 7261.48 1.17376 0.586881 0.809673i \(-0.300355\pi\)
0.586881 + 0.809673i \(0.300355\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2281.66 −0.362342
\(342\) 0 0
\(343\) 6899.32 1.08609
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7425.22 1.14872 0.574361 0.818602i \(-0.305251\pi\)
0.574361 + 0.818602i \(0.305251\pi\)
\(348\) 0 0
\(349\) −478.160 −0.0733390 −0.0366695 0.999327i \(-0.511675\pi\)
−0.0366695 + 0.999327i \(0.511675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4993.09 0.752847 0.376424 0.926448i \(-0.377154\pi\)
0.376424 + 0.926448i \(0.377154\pi\)
\(354\) 0 0
\(355\) −3641.52 −0.544427
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6873.09 −1.01044 −0.505219 0.862991i \(-0.668588\pi\)
−0.505219 + 0.862991i \(0.668588\pi\)
\(360\) 0 0
\(361\) 8365.87 1.21969
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2509.08 0.359811
\(366\) 0 0
\(367\) 8688.72 1.23582 0.617912 0.786247i \(-0.287979\pi\)
0.617912 + 0.786247i \(0.287979\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −198.590 −0.0277904
\(372\) 0 0
\(373\) 3494.54 0.485095 0.242548 0.970140i \(-0.422017\pi\)
0.242548 + 0.970140i \(0.422017\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12588.2 −1.71970
\(378\) 0 0
\(379\) 5802.83 0.786468 0.393234 0.919438i \(-0.371356\pi\)
0.393234 + 0.919438i \(0.371356\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3358.56 −0.448080 −0.224040 0.974580i \(-0.571925\pi\)
−0.224040 + 0.974580i \(0.571925\pi\)
\(384\) 0 0
\(385\) −3577.42 −0.473565
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.1370 −0.00249430 −0.00124715 0.999999i \(-0.500397\pi\)
−0.00124715 + 0.999999i \(0.500397\pi\)
\(390\) 0 0
\(391\) 10797.2 1.39651
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1988.05 0.253239
\(396\) 0 0
\(397\) −4348.59 −0.549747 −0.274873 0.961480i \(-0.588636\pi\)
−0.274873 + 0.961480i \(0.588636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8501.61 −1.05873 −0.529364 0.848395i \(-0.677570\pi\)
−0.529364 + 0.848395i \(0.677570\pi\)
\(402\) 0 0
\(403\) 3341.62 0.413047
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7637.95 −0.930219
\(408\) 0 0
\(409\) −2810.67 −0.339801 −0.169900 0.985461i \(-0.554345\pi\)
−0.169900 + 0.985461i \(0.554345\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3470.93 −0.413543
\(414\) 0 0
\(415\) 6847.31 0.809930
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16355.4 −1.90696 −0.953478 0.301461i \(-0.902526\pi\)
−0.953478 + 0.301461i \(0.902526\pi\)
\(420\) 0 0
\(421\) −4510.90 −0.522204 −0.261102 0.965311i \(-0.584086\pi\)
−0.261102 + 0.965311i \(0.584086\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2950.42 −0.336745
\(426\) 0 0
\(427\) 791.864 0.0897447
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5850.47 0.653845 0.326923 0.945051i \(-0.393988\pi\)
0.326923 + 0.945051i \(0.393988\pi\)
\(432\) 0 0
\(433\) −3836.82 −0.425833 −0.212916 0.977070i \(-0.568296\pi\)
−0.212916 + 0.977070i \(0.568296\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11288.7 1.23572
\(438\) 0 0
\(439\) −16227.3 −1.76421 −0.882106 0.471052i \(-0.843875\pi\)
−0.882106 + 0.471052i \(0.843875\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6705.13 0.719120 0.359560 0.933122i \(-0.382927\pi\)
0.359560 + 0.933122i \(0.382927\pi\)
\(444\) 0 0
\(445\) 7340.65 0.781978
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −213.100 −0.0223982 −0.0111991 0.999937i \(-0.503565\pi\)
−0.0111991 + 0.999937i \(0.503565\pi\)
\(450\) 0 0
\(451\) −17949.6 −1.87408
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5239.34 0.539833
\(456\) 0 0
\(457\) 16462.1 1.68504 0.842520 0.538665i \(-0.181071\pi\)
0.842520 + 0.538665i \(0.181071\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1562.06 −0.157814 −0.0789071 0.996882i \(-0.525143\pi\)
−0.0789071 + 0.996882i \(0.525143\pi\)
\(462\) 0 0
\(463\) 5924.27 0.594653 0.297326 0.954776i \(-0.403905\pi\)
0.297326 + 0.954776i \(0.403905\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17905.1 −1.77420 −0.887098 0.461582i \(-0.847282\pi\)
−0.887098 + 0.461582i \(0.847282\pi\)
\(468\) 0 0
\(469\) −1104.13 −0.108708
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6195.75 0.602285
\(474\) 0 0
\(475\) −3084.73 −0.297973
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9915.44 0.945820 0.472910 0.881111i \(-0.343204\pi\)
0.472910 + 0.881111i \(0.343204\pi\)
\(480\) 0 0
\(481\) 11186.2 1.06039
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1675.12 −0.156831
\(486\) 0 0
\(487\) −11910.8 −1.10828 −0.554138 0.832425i \(-0.686952\pi\)
−0.554138 + 0.832425i \(0.686952\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11063.8 1.01691 0.508453 0.861090i \(-0.330218\pi\)
0.508453 + 0.861090i \(0.330218\pi\)
\(492\) 0 0
\(493\) 20582.2 1.88027
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10573.1 −0.954261
\(498\) 0 0
\(499\) −9347.25 −0.838557 −0.419279 0.907858i \(-0.637717\pi\)
−0.419279 + 0.907858i \(0.637717\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19474.2 −1.72627 −0.863135 0.504973i \(-0.831502\pi\)
−0.863135 + 0.504973i \(0.831502\pi\)
\(504\) 0 0
\(505\) −6030.43 −0.531387
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22164.1 −1.93007 −0.965035 0.262121i \(-0.915578\pi\)
−0.965035 + 0.262121i \(0.915578\pi\)
\(510\) 0 0
\(511\) 7285.06 0.630670
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5305.55 0.453962
\(516\) 0 0
\(517\) 10917.7 0.928740
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 254.564 0.0214062 0.0107031 0.999943i \(-0.496593\pi\)
0.0107031 + 0.999943i \(0.496593\pi\)
\(522\) 0 0
\(523\) −4049.92 −0.338606 −0.169303 0.985564i \(-0.554152\pi\)
−0.169303 + 0.985564i \(0.554152\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5463.66 −0.451614
\(528\) 0 0
\(529\) −3796.89 −0.312064
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26288.1 2.13633
\(534\) 0 0
\(535\) 2377.89 0.192159
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6517.60 0.520841
\(540\) 0 0
\(541\) −4085.88 −0.324705 −0.162353 0.986733i \(-0.551908\pi\)
−0.162353 + 0.986733i \(0.551908\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6602.09 −0.518904
\(546\) 0 0
\(547\) 15392.2 1.20315 0.601575 0.798816i \(-0.294540\pi\)
0.601575 + 0.798816i \(0.294540\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21519.1 1.66378
\(552\) 0 0
\(553\) 5772.27 0.443873
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10897.6 0.828987 0.414493 0.910052i \(-0.363959\pi\)
0.414493 + 0.910052i \(0.363959\pi\)
\(558\) 0 0
\(559\) −9074.02 −0.686566
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1551.69 −0.116156 −0.0580781 0.998312i \(-0.518497\pi\)
−0.0580781 + 0.998312i \(0.518497\pi\)
\(564\) 0 0
\(565\) −340.875 −0.0253818
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1246.95 0.0918715 0.0459357 0.998944i \(-0.485373\pi\)
0.0459357 + 0.998944i \(0.485373\pi\)
\(570\) 0 0
\(571\) −4196.58 −0.307568 −0.153784 0.988104i \(-0.549146\pi\)
−0.153784 + 0.988104i \(0.549146\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2287.21 −0.165884
\(576\) 0 0
\(577\) 20585.1 1.48521 0.742607 0.669728i \(-0.233589\pi\)
0.742607 + 0.669728i \(0.233589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 19881.1 1.41963
\(582\) 0 0
\(583\) −674.183 −0.0478933
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4855.78 −0.341430 −0.170715 0.985320i \(-0.554608\pi\)
−0.170715 + 0.985320i \(0.554608\pi\)
\(588\) 0 0
\(589\) −5712.37 −0.399617
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23965.6 1.65961 0.829804 0.558055i \(-0.188452\pi\)
0.829804 + 0.558055i \(0.188452\pi\)
\(594\) 0 0
\(595\) −8566.50 −0.590239
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14229.1 0.970595 0.485297 0.874349i \(-0.338711\pi\)
0.485297 + 0.874349i \(0.338711\pi\)
\(600\) 0 0
\(601\) −8877.97 −0.602562 −0.301281 0.953535i \(-0.597414\pi\)
−0.301281 + 0.953535i \(0.597414\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5489.83 −0.368915
\(606\) 0 0
\(607\) 10876.7 0.727302 0.363651 0.931535i \(-0.381530\pi\)
0.363651 + 0.931535i \(0.381530\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15989.5 −1.05870
\(612\) 0 0
\(613\) −19544.8 −1.28778 −0.643890 0.765118i \(-0.722680\pi\)
−0.643890 + 0.765118i \(0.722680\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5041.75 −0.328968 −0.164484 0.986380i \(-0.552596\pi\)
−0.164484 + 0.986380i \(0.552596\pi\)
\(618\) 0 0
\(619\) 5208.05 0.338173 0.169087 0.985601i \(-0.445918\pi\)
0.169087 + 0.985601i \(0.445918\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 21313.5 1.37064
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18289.9 −1.15940
\(630\) 0 0
\(631\) 20284.6 1.27974 0.639872 0.768482i \(-0.278987\pi\)
0.639872 + 0.768482i \(0.278987\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2965.05 −0.185298
\(636\) 0 0
\(637\) −9545.40 −0.593724
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20852.4 1.28490 0.642449 0.766329i \(-0.277919\pi\)
0.642449 + 0.766329i \(0.277919\pi\)
\(642\) 0 0
\(643\) 2187.22 0.134146 0.0670729 0.997748i \(-0.478634\pi\)
0.0670729 + 0.997748i \(0.478634\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17044.1 1.03566 0.517831 0.855483i \(-0.326740\pi\)
0.517831 + 0.855483i \(0.326740\pi\)
\(648\) 0 0
\(649\) −11783.3 −0.712688
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8474.26 −0.507846 −0.253923 0.967224i \(-0.581721\pi\)
−0.253923 + 0.967224i \(0.581721\pi\)
\(654\) 0 0
\(655\) −1694.69 −0.101094
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 25560.2 1.51090 0.755450 0.655207i \(-0.227419\pi\)
0.755450 + 0.655207i \(0.227419\pi\)
\(660\) 0 0
\(661\) 1209.59 0.0711766 0.0355883 0.999367i \(-0.488670\pi\)
0.0355883 + 0.999367i \(0.488670\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8956.46 −0.522281
\(666\) 0 0
\(667\) 15955.6 0.926241
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2688.26 0.154664
\(672\) 0 0
\(673\) −8698.21 −0.498204 −0.249102 0.968477i \(-0.580135\pi\)
−0.249102 + 0.968477i \(0.580135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8424.49 0.478256 0.239128 0.970988i \(-0.423138\pi\)
0.239128 + 0.970988i \(0.423138\pi\)
\(678\) 0 0
\(679\) −4863.68 −0.274891
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17828.6 −0.998817 −0.499408 0.866367i \(-0.666449\pi\)
−0.499408 + 0.866367i \(0.666449\pi\)
\(684\) 0 0
\(685\) −4057.21 −0.226304
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 987.378 0.0545952
\(690\) 0 0
\(691\) −14525.1 −0.799652 −0.399826 0.916591i \(-0.630930\pi\)
−0.399826 + 0.916591i \(0.630930\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15530.6 −0.847641
\(696\) 0 0
\(697\) −42982.0 −2.33581
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18815.5 −1.01377 −0.506883 0.862015i \(-0.669202\pi\)
−0.506883 + 0.862015i \(0.669202\pi\)
\(702\) 0 0
\(703\) −19122.4 −1.02591
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17509.3 −0.931405
\(708\) 0 0
\(709\) −12934.4 −0.685137 −0.342569 0.939493i \(-0.611297\pi\)
−0.342569 + 0.939493i \(0.611297\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4235.51 −0.222470
\(714\) 0 0
\(715\) 17786.8 0.930333
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8471.10 −0.439386 −0.219693 0.975569i \(-0.570506\pi\)
−0.219693 + 0.975569i \(0.570506\pi\)
\(720\) 0 0
\(721\) 15404.6 0.795695
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4360.00 −0.223347
\(726\) 0 0
\(727\) −24369.5 −1.24321 −0.621605 0.783331i \(-0.713519\pi\)
−0.621605 + 0.783331i \(0.713519\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14836.3 0.750673
\(732\) 0 0
\(733\) −35411.8 −1.78440 −0.892199 0.451642i \(-0.850838\pi\)
−0.892199 + 0.451642i \(0.850838\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3748.37 −0.187345
\(738\) 0 0
\(739\) 24447.0 1.21691 0.608456 0.793588i \(-0.291789\pi\)
0.608456 + 0.793588i \(0.291789\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24125.9 −1.19125 −0.595623 0.803264i \(-0.703095\pi\)
−0.595623 + 0.803264i \(0.703095\pi\)
\(744\) 0 0
\(745\) 12705.1 0.624802
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6904.17 0.336813
\(750\) 0 0
\(751\) −11882.4 −0.577356 −0.288678 0.957426i \(-0.593216\pi\)
−0.288678 + 0.957426i \(0.593216\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5626.85 −0.271234
\(756\) 0 0
\(757\) −14601.3 −0.701049 −0.350525 0.936554i \(-0.613997\pi\)
−0.350525 + 0.936554i \(0.613997\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20296.3 0.966809 0.483404 0.875397i \(-0.339400\pi\)
0.483404 + 0.875397i \(0.339400\pi\)
\(762\) 0 0
\(763\) −19169.1 −0.909524
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 17257.3 0.812418
\(768\) 0 0
\(769\) 36322.0 1.70326 0.851629 0.524146i \(-0.175615\pi\)
0.851629 + 0.524146i \(0.175615\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28930.9 −1.34615 −0.673073 0.739576i \(-0.735026\pi\)
−0.673073 + 0.739576i \(0.735026\pi\)
\(774\) 0 0
\(775\) 1157.39 0.0536448
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −44938.6 −2.06687
\(780\) 0 0
\(781\) −35894.1 −1.64455
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −16150.3 −0.734303
\(786\) 0 0
\(787\) −21128.3 −0.956978 −0.478489 0.878094i \(-0.658815\pi\)
−0.478489 + 0.878094i \(0.658815\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −989.726 −0.0444887
\(792\) 0 0
\(793\) −3937.11 −0.176306
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2765.87 0.122926 0.0614632 0.998109i \(-0.480423\pi\)
0.0614632 + 0.998109i \(0.480423\pi\)
\(798\) 0 0
\(799\) 26143.5 1.15756
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 24731.7 1.08688
\(804\) 0 0
\(805\) −6640.87 −0.290758
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16756.6 −0.728220 −0.364110 0.931356i \(-0.618627\pi\)
−0.364110 + 0.931356i \(0.618627\pi\)
\(810\) 0 0
\(811\) −17829.6 −0.771987 −0.385993 0.922502i \(-0.626141\pi\)
−0.385993 + 0.922502i \(0.626141\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3470.27 −0.149151
\(816\) 0 0
\(817\) 15511.7 0.664243
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6757.48 0.287256 0.143628 0.989632i \(-0.454123\pi\)
0.143628 + 0.989632i \(0.454123\pi\)
\(822\) 0 0
\(823\) −7121.28 −0.301619 −0.150809 0.988563i \(-0.548188\pi\)
−0.150809 + 0.988563i \(0.548188\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1171.74 −0.0492688 −0.0246344 0.999697i \(-0.507842\pi\)
−0.0246344 + 0.999697i \(0.507842\pi\)
\(828\) 0 0
\(829\) 23617.8 0.989483 0.494742 0.869040i \(-0.335263\pi\)
0.494742 + 0.869040i \(0.335263\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 15607.1 0.649163
\(834\) 0 0
\(835\) −16080.2 −0.666440
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −35054.3 −1.44244 −0.721222 0.692704i \(-0.756419\pi\)
−0.721222 + 0.692704i \(0.756419\pi\)
\(840\) 0 0
\(841\) 6026.42 0.247096
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −15064.8 −0.613306
\(846\) 0 0
\(847\) −15939.6 −0.646627
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −14178.6 −0.571133
\(852\) 0 0
\(853\) −32772.3 −1.31548 −0.657740 0.753245i \(-0.728487\pi\)
−0.657740 + 0.753245i \(0.728487\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3503.93 0.139664 0.0698319 0.997559i \(-0.477754\pi\)
0.0698319 + 0.997559i \(0.477754\pi\)
\(858\) 0 0
\(859\) −31044.1 −1.23307 −0.616537 0.787326i \(-0.711465\pi\)
−0.616537 + 0.787326i \(0.711465\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26333.6 1.03871 0.519354 0.854559i \(-0.326173\pi\)
0.519354 + 0.854559i \(0.326173\pi\)
\(864\) 0 0
\(865\) 1487.73 0.0584790
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19596.0 0.764959
\(870\) 0 0
\(871\) 5489.71 0.213561
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1814.68 0.0701112
\(876\) 0 0
\(877\) −40977.3 −1.57777 −0.788886 0.614540i \(-0.789342\pi\)
−0.788886 + 0.614540i \(0.789342\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −37022.4 −1.41579 −0.707897 0.706315i \(-0.750356\pi\)
−0.707897 + 0.706315i \(0.750356\pi\)
\(882\) 0 0
\(883\) 36037.9 1.37347 0.686734 0.726909i \(-0.259044\pi\)
0.686734 + 0.726909i \(0.259044\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1465.05 0.0554584 0.0277292 0.999615i \(-0.491172\pi\)
0.0277292 + 0.999615i \(0.491172\pi\)
\(888\) 0 0
\(889\) −8608.97 −0.324787
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27333.5 1.02428
\(894\) 0 0
\(895\) 17250.6 0.644273
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8073.96 −0.299535
\(900\) 0 0
\(901\) −1614.40 −0.0596930
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15448.7 0.567440
\(906\) 0 0
\(907\) −33660.8 −1.23229 −0.616146 0.787632i \(-0.711307\pi\)
−0.616146 + 0.787632i \(0.711307\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25992.7 0.945311 0.472655 0.881247i \(-0.343296\pi\)
0.472655 + 0.881247i \(0.343296\pi\)
\(912\) 0 0
\(913\) 67493.3 2.44655
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4920.50 −0.177196
\(918\) 0 0
\(919\) −1149.54 −0.0412620 −0.0206310 0.999787i \(-0.506568\pi\)
−0.0206310 + 0.999787i \(0.506568\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 52568.9 1.87468
\(924\) 0 0
\(925\) 3874.42 0.137719
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1923.20 −0.0679204 −0.0339602 0.999423i \(-0.510812\pi\)
−0.0339602 + 0.999423i \(0.510812\pi\)
\(930\) 0 0
\(931\) 16317.5 0.574420
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −29082.0 −1.01720
\(936\) 0 0
\(937\) 3511.90 0.122443 0.0612213 0.998124i \(-0.480500\pi\)
0.0612213 + 0.998124i \(0.480500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6848.16 0.237241 0.118620 0.992940i \(-0.462153\pi\)
0.118620 + 0.992940i \(0.462153\pi\)
\(942\) 0 0
\(943\) −33320.3 −1.15064
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 48357.3 1.65935 0.829673 0.558250i \(-0.188527\pi\)
0.829673 + 0.558250i \(0.188527\pi\)
\(948\) 0 0
\(949\) −36221.0 −1.23897
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −38701.1 −1.31548 −0.657740 0.753245i \(-0.728488\pi\)
−0.657740 + 0.753245i \(0.728488\pi\)
\(954\) 0 0
\(955\) −7660.57 −0.259571
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11780.0 −0.396660
\(960\) 0 0
\(961\) −27647.7 −0.928056
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 25972.1 0.866396
\(966\) 0 0
\(967\) 24312.7 0.808526 0.404263 0.914643i \(-0.367528\pi\)
0.404263 + 0.914643i \(0.367528\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −37464.3 −1.23820 −0.619098 0.785314i \(-0.712501\pi\)
−0.619098 + 0.785314i \(0.712501\pi\)
\(972\) 0 0
\(973\) −45092.9 −1.48573
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3186.09 −0.104332 −0.0521659 0.998638i \(-0.516612\pi\)
−0.0521659 + 0.998638i \(0.516612\pi\)
\(978\) 0 0
\(979\) 72356.1 2.36212
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 30345.6 0.984614 0.492307 0.870422i \(-0.336154\pi\)
0.492307 + 0.870422i \(0.336154\pi\)
\(984\) 0 0
\(985\) −10028.0 −0.324386
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11501.3 0.369789
\(990\) 0 0
\(991\) −3443.75 −0.110388 −0.0551940 0.998476i \(-0.517578\pi\)
−0.0551940 + 0.998476i \(0.517578\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −14373.4 −0.457957
\(996\) 0 0
\(997\) −4567.89 −0.145102 −0.0725510 0.997365i \(-0.523114\pi\)
−0.0725510 + 0.997365i \(0.523114\pi\)
\(998\) 0 0
\(999\) 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 2160.4.a.be.1.2 3
3.2 odd 2 2160.4.a.bm.1.2 3
4.3 odd 2 135.4.a.g.1.2 yes 3
12.11 even 2 135.4.a.f.1.2 3
20.3 even 4 675.4.b.k.649.3 6
20.7 even 4 675.4.b.k.649.4 6
20.19 odd 2 675.4.a.q.1.2 3
36.7 odd 6 405.4.e.r.271.2 6
36.11 even 6 405.4.e.t.271.2 6
36.23 even 6 405.4.e.t.136.2 6
36.31 odd 6 405.4.e.r.136.2 6
60.23 odd 4 675.4.b.l.649.4 6
60.47 odd 4 675.4.b.l.649.3 6
60.59 even 2 675.4.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.2 3 12.11 even 2
135.4.a.g.1.2 yes 3 4.3 odd 2
405.4.e.r.136.2 6 36.31 odd 6
405.4.e.r.271.2 6 36.7 odd 6
405.4.e.t.136.2 6 36.23 even 6
405.4.e.t.271.2 6 36.11 even 6
675.4.a.q.1.2 3 20.19 odd 2
675.4.a.r.1.2 3 60.59 even 2
675.4.b.k.649.3 6 20.3 even 4
675.4.b.k.649.4 6 20.7 even 4
675.4.b.l.649.3 6 60.47 odd 4
675.4.b.l.649.4 6 60.23 odd 4
2160.4.a.be.1.2 3 1.1 even 1 trivial
2160.4.a.bm.1.2 3 3.2 odd 2