Properties

Label 2160.4.a.bk.1.3
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1257.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.87370\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} +30.2207 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} +30.2207 q^{7} +67.3185 q^{11} -60.7911 q^{13} -61.1347 q^{17} +27.8771 q^{19} -40.4655 q^{23} +25.0000 q^{25} -212.108 q^{29} -167.318 q^{31} -151.103 q^{35} -366.539 q^{37} +363.385 q^{41} -153.839 q^{43} -434.212 q^{47} +570.291 q^{49} -79.6550 q^{53} -336.593 q^{55} +339.414 q^{59} -525.856 q^{61} +303.956 q^{65} -131.916 q^{67} -296.263 q^{71} -1230.34 q^{73} +2034.41 q^{77} +621.772 q^{79} -76.3372 q^{83} +305.673 q^{85} +192.406 q^{89} -1837.15 q^{91} -139.386 q^{95} -874.771 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53} - 140 q^{55} + 234 q^{59} - 673 q^{61} + 390 q^{65} - 386 q^{67} + 16 q^{71} - 892 q^{73} + 1800 q^{77} - 1263 q^{79} - 1815 q^{83} + 55 q^{85} + 1800 q^{89} - 1284 q^{91} - 355 q^{95} - 840 q^{97}+O(q^{100}) \) 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
\(7\) 30.2207 1.63176 0.815882 0.578218i \(-0.196252\pi\)
0.815882 + 0.578218i \(0.196252\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 67.3185 1.84521 0.922604 0.385748i \(-0.126056\pi\)
0.922604 + 0.385748i \(0.126056\pi\)
\(12\) 0 0
\(13\) −60.7911 −1.29696 −0.648478 0.761234i \(-0.724594\pi\)
−0.648478 + 0.761234i \(0.724594\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −61.1347 −0.872196 −0.436098 0.899899i \(-0.643640\pi\)
−0.436098 + 0.899899i \(0.643640\pi\)
\(18\) 0 0
\(19\) 27.8771 0.336603 0.168301 0.985736i \(-0.446172\pi\)
0.168301 + 0.985736i \(0.446172\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −40.4655 −0.366854 −0.183427 0.983033i \(-0.558719\pi\)
−0.183427 + 0.983033i \(0.558719\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −212.108 −1.35819 −0.679095 0.734051i \(-0.737627\pi\)
−0.679095 + 0.734051i \(0.737627\pi\)
\(30\) 0 0
\(31\) −167.318 −0.969394 −0.484697 0.874682i \(-0.661070\pi\)
−0.484697 + 0.874682i \(0.661070\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −151.103 −0.729747
\(36\) 0 0
\(37\) −366.539 −1.62861 −0.814305 0.580437i \(-0.802882\pi\)
−0.814305 + 0.580437i \(0.802882\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 363.385 1.38418 0.692088 0.721813i \(-0.256691\pi\)
0.692088 + 0.721813i \(0.256691\pi\)
\(42\) 0 0
\(43\) −153.839 −0.545586 −0.272793 0.962073i \(-0.587947\pi\)
−0.272793 + 0.962073i \(0.587947\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −434.212 −1.34758 −0.673791 0.738922i \(-0.735335\pi\)
−0.673791 + 0.738922i \(0.735335\pi\)
\(48\) 0 0
\(49\) 570.291 1.66265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −79.6550 −0.206442 −0.103221 0.994658i \(-0.532915\pi\)
−0.103221 + 0.994658i \(0.532915\pi\)
\(54\) 0 0
\(55\) −336.593 −0.825202
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 339.414 0.748948 0.374474 0.927237i \(-0.377823\pi\)
0.374474 + 0.927237i \(0.377823\pi\)
\(60\) 0 0
\(61\) −525.856 −1.10375 −0.551877 0.833926i \(-0.686088\pi\)
−0.551877 + 0.833926i \(0.686088\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 303.956 0.580016
\(66\) 0 0
\(67\) −131.916 −0.240539 −0.120269 0.992741i \(-0.538376\pi\)
−0.120269 + 0.992741i \(0.538376\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −296.263 −0.495210 −0.247605 0.968861i \(-0.579644\pi\)
−0.247605 + 0.968861i \(0.579644\pi\)
\(72\) 0 0
\(73\) −1230.34 −1.97261 −0.986305 0.164932i \(-0.947259\pi\)
−0.986305 + 0.164932i \(0.947259\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2034.41 3.01095
\(78\) 0 0
\(79\) 621.772 0.885504 0.442752 0.896644i \(-0.354002\pi\)
0.442752 + 0.896644i \(0.354002\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −76.3372 −0.100953 −0.0504765 0.998725i \(-0.516074\pi\)
−0.0504765 + 0.998725i \(0.516074\pi\)
\(84\) 0 0
\(85\) 305.673 0.390058
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 192.406 0.229157 0.114579 0.993414i \(-0.463448\pi\)
0.114579 + 0.993414i \(0.463448\pi\)
\(90\) 0 0
\(91\) −1837.15 −2.11633
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −139.386 −0.150533
\(96\) 0 0
\(97\) −874.771 −0.915666 −0.457833 0.889038i \(-0.651374\pi\)
−0.457833 + 0.889038i \(0.651374\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1093.02 1.07682 0.538412 0.842682i \(-0.319025\pi\)
0.538412 + 0.842682i \(0.319025\pi\)
\(102\) 0 0
\(103\) −228.932 −0.219003 −0.109502 0.993987i \(-0.534925\pi\)
−0.109502 + 0.993987i \(0.534925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 479.745 0.433446 0.216723 0.976233i \(-0.430463\pi\)
0.216723 + 0.976233i \(0.430463\pi\)
\(108\) 0 0
\(109\) −116.246 −0.102150 −0.0510751 0.998695i \(-0.516265\pi\)
−0.0510751 + 0.998695i \(0.516265\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −741.222 −0.617065 −0.308532 0.951214i \(-0.599838\pi\)
−0.308532 + 0.951214i \(0.599838\pi\)
\(114\) 0 0
\(115\) 202.327 0.164062
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1847.53 −1.42322
\(120\) 0 0
\(121\) 3200.78 2.40479
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 2595.38 1.81340 0.906702 0.421771i \(-0.138591\pi\)
0.906702 + 0.421771i \(0.138591\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1405.62 0.937476 0.468738 0.883337i \(-0.344709\pi\)
0.468738 + 0.883337i \(0.344709\pi\)
\(132\) 0 0
\(133\) 842.466 0.549256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2457.39 1.53248 0.766239 0.642556i \(-0.222126\pi\)
0.766239 + 0.642556i \(0.222126\pi\)
\(138\) 0 0
\(139\) 585.391 0.357210 0.178605 0.983921i \(-0.442842\pi\)
0.178605 + 0.983921i \(0.442842\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4092.37 −2.39315
\(144\) 0 0
\(145\) 1060.54 0.607401
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −926.505 −0.509411 −0.254705 0.967019i \(-0.581979\pi\)
−0.254705 + 0.967019i \(0.581979\pi\)
\(150\) 0 0
\(151\) −411.900 −0.221987 −0.110993 0.993821i \(-0.535403\pi\)
−0.110993 + 0.993821i \(0.535403\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 836.590 0.433526
\(156\) 0 0
\(157\) −2959.97 −1.50466 −0.752328 0.658789i \(-0.771069\pi\)
−0.752328 + 0.658789i \(0.771069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1222.89 −0.598619
\(162\) 0 0
\(163\) −2504.50 −1.20348 −0.601741 0.798691i \(-0.705526\pi\)
−0.601741 + 0.798691i \(0.705526\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2055.42 0.952413 0.476207 0.879333i \(-0.342011\pi\)
0.476207 + 0.879333i \(0.342011\pi\)
\(168\) 0 0
\(169\) 1498.56 0.682093
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1724.17 −0.757722 −0.378861 0.925454i \(-0.623684\pi\)
−0.378861 + 0.925454i \(0.623684\pi\)
\(174\) 0 0
\(175\) 755.517 0.326353
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3601.35 −1.50379 −0.751893 0.659285i \(-0.770859\pi\)
−0.751893 + 0.659285i \(0.770859\pi\)
\(180\) 0 0
\(181\) −1257.17 −0.516267 −0.258134 0.966109i \(-0.583107\pi\)
−0.258134 + 0.966109i \(0.583107\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1832.69 0.728337
\(186\) 0 0
\(187\) −4115.50 −1.60938
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4677.47 −1.77199 −0.885994 0.463697i \(-0.846523\pi\)
−0.885994 + 0.463697i \(0.846523\pi\)
\(192\) 0 0
\(193\) −2647.88 −0.987556 −0.493778 0.869588i \(-0.664384\pi\)
−0.493778 + 0.869588i \(0.664384\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1368.91 −0.495082 −0.247541 0.968877i \(-0.579622\pi\)
−0.247541 + 0.968877i \(0.579622\pi\)
\(198\) 0 0
\(199\) 5411.43 1.92767 0.963835 0.266501i \(-0.0858675\pi\)
0.963835 + 0.266501i \(0.0858675\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6410.06 −2.21625
\(204\) 0 0
\(205\) −1816.93 −0.619022
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1876.65 0.621102
\(210\) 0 0
\(211\) −1987.12 −0.648337 −0.324168 0.945999i \(-0.605084\pi\)
−0.324168 + 0.945999i \(0.605084\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 769.194 0.243993
\(216\) 0 0
\(217\) −5056.47 −1.58182
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3716.45 1.13120
\(222\) 0 0
\(223\) −3687.95 −1.10746 −0.553730 0.832697i \(-0.686796\pi\)
−0.553730 + 0.832697i \(0.686796\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −859.424 −0.251286 −0.125643 0.992076i \(-0.540099\pi\)
−0.125643 + 0.992076i \(0.540099\pi\)
\(228\) 0 0
\(229\) −1862.67 −0.537504 −0.268752 0.963209i \(-0.586611\pi\)
−0.268752 + 0.963209i \(0.586611\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5011.22 1.40900 0.704498 0.709706i \(-0.251172\pi\)
0.704498 + 0.709706i \(0.251172\pi\)
\(234\) 0 0
\(235\) 2171.06 0.602657
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5057.92 −1.36891 −0.684455 0.729055i \(-0.739960\pi\)
−0.684455 + 0.729055i \(0.739960\pi\)
\(240\) 0 0
\(241\) −7276.37 −1.94486 −0.972431 0.233189i \(-0.925084\pi\)
−0.972431 + 0.233189i \(0.925084\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2851.45 −0.743562
\(246\) 0 0
\(247\) −1694.68 −0.436559
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −204.208 −0.0513525 −0.0256762 0.999670i \(-0.508174\pi\)
−0.0256762 + 0.999670i \(0.508174\pi\)
\(252\) 0 0
\(253\) −2724.07 −0.676921
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4591.80 1.11451 0.557254 0.830342i \(-0.311855\pi\)
0.557254 + 0.830342i \(0.311855\pi\)
\(258\) 0 0
\(259\) −11077.1 −2.65751
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7689.01 1.80276 0.901378 0.433033i \(-0.142557\pi\)
0.901378 + 0.433033i \(0.142557\pi\)
\(264\) 0 0
\(265\) 398.275 0.0923239
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5790.73 1.31252 0.656258 0.754536i \(-0.272138\pi\)
0.656258 + 0.754536i \(0.272138\pi\)
\(270\) 0 0
\(271\) −6146.43 −1.37775 −0.688873 0.724882i \(-0.741894\pi\)
−0.688873 + 0.724882i \(0.741894\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1682.96 0.369042
\(276\) 0 0
\(277\) 4811.02 1.04356 0.521781 0.853080i \(-0.325268\pi\)
0.521781 + 0.853080i \(0.325268\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2333.19 0.495325 0.247663 0.968846i \(-0.420338\pi\)
0.247663 + 0.968846i \(0.420338\pi\)
\(282\) 0 0
\(283\) −8697.84 −1.82697 −0.913486 0.406870i \(-0.866620\pi\)
−0.913486 + 0.406870i \(0.866620\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10981.8 2.25865
\(288\) 0 0
\(289\) −1175.55 −0.239273
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1838.13 −0.366501 −0.183251 0.983066i \(-0.558662\pi\)
−0.183251 + 0.983066i \(0.558662\pi\)
\(294\) 0 0
\(295\) −1697.07 −0.334940
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2459.94 0.475793
\(300\) 0 0
\(301\) −4649.12 −0.890268
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2629.28 0.493614
\(306\) 0 0
\(307\) 4152.70 0.772010 0.386005 0.922497i \(-0.373855\pi\)
0.386005 + 0.922497i \(0.373855\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2861.14 −0.521674 −0.260837 0.965383i \(-0.583999\pi\)
−0.260837 + 0.965383i \(0.583999\pi\)
\(312\) 0 0
\(313\) −438.494 −0.0791858 −0.0395929 0.999216i \(-0.512606\pi\)
−0.0395929 + 0.999216i \(0.512606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6279.80 −1.11265 −0.556323 0.830966i \(-0.687788\pi\)
−0.556323 + 0.830966i \(0.687788\pi\)
\(318\) 0 0
\(319\) −14278.8 −2.50614
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1704.26 −0.293584
\(324\) 0 0
\(325\) −1519.78 −0.259391
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −13122.2 −2.19894
\(330\) 0 0
\(331\) 7564.20 1.25609 0.628046 0.778176i \(-0.283855\pi\)
0.628046 + 0.778176i \(0.283855\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 659.580 0.107572
\(336\) 0 0
\(337\) 3468.51 0.560658 0.280329 0.959904i \(-0.409556\pi\)
0.280329 + 0.959904i \(0.409556\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11263.6 −1.78873
\(342\) 0 0
\(343\) 6868.88 1.08130
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6415.10 0.992452 0.496226 0.868193i \(-0.334719\pi\)
0.496226 + 0.868193i \(0.334719\pi\)
\(348\) 0 0
\(349\) −2248.97 −0.344941 −0.172470 0.985015i \(-0.555175\pi\)
−0.172470 + 0.985015i \(0.555175\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7197.94 −1.08529 −0.542645 0.839962i \(-0.682577\pi\)
−0.542645 + 0.839962i \(0.682577\pi\)
\(354\) 0 0
\(355\) 1481.31 0.221465
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8618.50 1.26704 0.633520 0.773726i \(-0.281609\pi\)
0.633520 + 0.773726i \(0.281609\pi\)
\(360\) 0 0
\(361\) −6081.87 −0.886699
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6151.70 0.882178
\(366\) 0 0
\(367\) 9025.95 1.28379 0.641895 0.766793i \(-0.278149\pi\)
0.641895 + 0.766793i \(0.278149\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2407.23 −0.336865
\(372\) 0 0
\(373\) 7549.69 1.04801 0.524005 0.851715i \(-0.324437\pi\)
0.524005 + 0.851715i \(0.324437\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12894.3 1.76151
\(378\) 0 0
\(379\) −11849.7 −1.60601 −0.803007 0.595970i \(-0.796768\pi\)
−0.803007 + 0.595970i \(0.796768\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8613.06 1.14910 0.574552 0.818468i \(-0.305176\pi\)
0.574552 + 0.818468i \(0.305176\pi\)
\(384\) 0 0
\(385\) −10172.1 −1.34654
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2529.23 −0.329658 −0.164829 0.986322i \(-0.552707\pi\)
−0.164829 + 0.986322i \(0.552707\pi\)
\(390\) 0 0
\(391\) 2473.84 0.319968
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3108.86 −0.396009
\(396\) 0 0
\(397\) −14195.7 −1.79461 −0.897306 0.441410i \(-0.854479\pi\)
−0.897306 + 0.441410i \(0.854479\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11005.5 −1.37054 −0.685270 0.728289i \(-0.740316\pi\)
−0.685270 + 0.728289i \(0.740316\pi\)
\(402\) 0 0
\(403\) 10171.4 1.25726
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −24674.8 −3.00513
\(408\) 0 0
\(409\) 14339.1 1.73356 0.866779 0.498692i \(-0.166186\pi\)
0.866779 + 0.498692i \(0.166186\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10257.3 1.22211
\(414\) 0 0
\(415\) 381.686 0.0451475
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5352.51 −0.624075 −0.312037 0.950070i \(-0.601011\pi\)
−0.312037 + 0.950070i \(0.601011\pi\)
\(420\) 0 0
\(421\) −2683.89 −0.310701 −0.155350 0.987859i \(-0.549651\pi\)
−0.155350 + 0.987859i \(0.549651\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1528.37 −0.174439
\(426\) 0 0
\(427\) −15891.7 −1.80107
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10019.0 1.11972 0.559861 0.828587i \(-0.310855\pi\)
0.559861 + 0.828587i \(0.310855\pi\)
\(432\) 0 0
\(433\) −5292.20 −0.587361 −0.293680 0.955904i \(-0.594880\pi\)
−0.293680 + 0.955904i \(0.594880\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1128.06 −0.123484
\(438\) 0 0
\(439\) −9453.33 −1.02775 −0.513875 0.857865i \(-0.671791\pi\)
−0.513875 + 0.857865i \(0.671791\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2858.23 −0.306543 −0.153272 0.988184i \(-0.548981\pi\)
−0.153272 + 0.988184i \(0.548981\pi\)
\(444\) 0 0
\(445\) −962.030 −0.102482
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9322.81 −0.979890 −0.489945 0.871753i \(-0.662983\pi\)
−0.489945 + 0.871753i \(0.662983\pi\)
\(450\) 0 0
\(451\) 24462.5 2.55409
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9185.75 0.946449
\(456\) 0 0
\(457\) 4960.54 0.507755 0.253878 0.967236i \(-0.418294\pi\)
0.253878 + 0.967236i \(0.418294\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5235.83 −0.528974 −0.264487 0.964389i \(-0.585203\pi\)
−0.264487 + 0.964389i \(0.585203\pi\)
\(462\) 0 0
\(463\) −7121.96 −0.714871 −0.357436 0.933938i \(-0.616349\pi\)
−0.357436 + 0.933938i \(0.616349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1991.23 −0.197308 −0.0986541 0.995122i \(-0.531454\pi\)
−0.0986541 + 0.995122i \(0.531454\pi\)
\(468\) 0 0
\(469\) −3986.59 −0.392503
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10356.2 −1.00672
\(474\) 0 0
\(475\) 696.928 0.0673205
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9377.15 0.894474 0.447237 0.894415i \(-0.352408\pi\)
0.447237 + 0.894415i \(0.352408\pi\)
\(480\) 0 0
\(481\) 22282.3 2.11224
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4373.86 0.409498
\(486\) 0 0
\(487\) 4855.98 0.451838 0.225919 0.974146i \(-0.427461\pi\)
0.225919 + 0.974146i \(0.427461\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4048.75 0.372133 0.186067 0.982537i \(-0.440426\pi\)
0.186067 + 0.982537i \(0.440426\pi\)
\(492\) 0 0
\(493\) 12967.2 1.18461
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8953.27 −0.808066
\(498\) 0 0
\(499\) −6259.64 −0.561563 −0.280781 0.959772i \(-0.590594\pi\)
−0.280781 + 0.959772i \(0.590594\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9127.86 0.809128 0.404564 0.914510i \(-0.367423\pi\)
0.404564 + 0.914510i \(0.367423\pi\)
\(504\) 0 0
\(505\) −5465.08 −0.481570
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9495.89 0.826911 0.413456 0.910524i \(-0.364322\pi\)
0.413456 + 0.910524i \(0.364322\pi\)
\(510\) 0 0
\(511\) −37181.8 −3.21883
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1144.66 0.0979412
\(516\) 0 0
\(517\) −29230.5 −2.48657
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12282.6 1.03284 0.516421 0.856335i \(-0.327264\pi\)
0.516421 + 0.856335i \(0.327264\pi\)
\(522\) 0 0
\(523\) −8010.57 −0.669747 −0.334873 0.942263i \(-0.608694\pi\)
−0.334873 + 0.942263i \(0.608694\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10228.9 0.845502
\(528\) 0 0
\(529\) −10529.5 −0.865418
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −22090.6 −1.79521
\(534\) 0 0
\(535\) −2398.73 −0.193843
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 38391.1 3.06794
\(540\) 0 0
\(541\) −15476.8 −1.22995 −0.614974 0.788548i \(-0.710833\pi\)
−0.614974 + 0.788548i \(0.710833\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 581.231 0.0456829
\(546\) 0 0
\(547\) 22837.0 1.78508 0.892540 0.450968i \(-0.148921\pi\)
0.892540 + 0.450968i \(0.148921\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5912.96 −0.457170
\(552\) 0 0
\(553\) 18790.4 1.44493
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15800.3 −1.20194 −0.600971 0.799271i \(-0.705219\pi\)
−0.600971 + 0.799271i \(0.705219\pi\)
\(558\) 0 0
\(559\) 9352.03 0.707601
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13515.7 −1.01176 −0.505879 0.862605i \(-0.668832\pi\)
−0.505879 + 0.862605i \(0.668832\pi\)
\(564\) 0 0
\(565\) 3706.11 0.275960
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19632.0 1.44642 0.723211 0.690627i \(-0.242665\pi\)
0.723211 + 0.690627i \(0.242665\pi\)
\(570\) 0 0
\(571\) 12800.7 0.938166 0.469083 0.883154i \(-0.344584\pi\)
0.469083 + 0.883154i \(0.344584\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1011.64 −0.0733707
\(576\) 0 0
\(577\) −9214.08 −0.664796 −0.332398 0.943139i \(-0.607858\pi\)
−0.332398 + 0.943139i \(0.607858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2306.96 −0.164731
\(582\) 0 0
\(583\) −5362.25 −0.380929
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 275.136 0.0193460 0.00967298 0.999953i \(-0.496921\pi\)
0.00967298 + 0.999953i \(0.496921\pi\)
\(588\) 0 0
\(589\) −4664.34 −0.326300
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15753.5 −1.09093 −0.545464 0.838135i \(-0.683646\pi\)
−0.545464 + 0.838135i \(0.683646\pi\)
\(594\) 0 0
\(595\) 9237.66 0.636483
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13557.3 0.924770 0.462385 0.886679i \(-0.346994\pi\)
0.462385 + 0.886679i \(0.346994\pi\)
\(600\) 0 0
\(601\) 23976.7 1.62734 0.813668 0.581330i \(-0.197467\pi\)
0.813668 + 0.581330i \(0.197467\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16003.9 −1.07546
\(606\) 0 0
\(607\) 12766.6 0.853677 0.426839 0.904328i \(-0.359627\pi\)
0.426839 + 0.904328i \(0.359627\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 26396.2 1.74775
\(612\) 0 0
\(613\) −16776.3 −1.10537 −0.552684 0.833391i \(-0.686396\pi\)
−0.552684 + 0.833391i \(0.686396\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17941.4 −1.17065 −0.585327 0.810798i \(-0.699034\pi\)
−0.585327 + 0.810798i \(0.699034\pi\)
\(618\) 0 0
\(619\) 987.096 0.0640949 0.0320475 0.999486i \(-0.489797\pi\)
0.0320475 + 0.999486i \(0.489797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5814.64 0.373930
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22408.2 1.42047
\(630\) 0 0
\(631\) −18504.9 −1.16746 −0.583730 0.811948i \(-0.698407\pi\)
−0.583730 + 0.811948i \(0.698407\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12976.9 −0.810979
\(636\) 0 0
\(637\) −34668.6 −2.15639
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10260.2 −0.632219 −0.316110 0.948723i \(-0.602377\pi\)
−0.316110 + 0.948723i \(0.602377\pi\)
\(642\) 0 0
\(643\) −6221.29 −0.381561 −0.190781 0.981633i \(-0.561102\pi\)
−0.190781 + 0.981633i \(0.561102\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21446.7 −1.30318 −0.651590 0.758572i \(-0.725898\pi\)
−0.651590 + 0.758572i \(0.725898\pi\)
\(648\) 0 0
\(649\) 22848.8 1.38196
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25051.0 1.50126 0.750628 0.660725i \(-0.229751\pi\)
0.750628 + 0.660725i \(0.229751\pi\)
\(654\) 0 0
\(655\) −7028.09 −0.419252
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.7125 0.00122435 0.000612174 1.00000i \(-0.499805\pi\)
0.000612174 1.00000i \(0.499805\pi\)
\(660\) 0 0
\(661\) 23294.7 1.37074 0.685371 0.728194i \(-0.259640\pi\)
0.685371 + 0.728194i \(0.259640\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4212.33 −0.245635
\(666\) 0 0
\(667\) 8583.05 0.498257
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −35399.9 −2.03666
\(672\) 0 0
\(673\) 25764.3 1.47569 0.737845 0.674970i \(-0.235843\pi\)
0.737845 + 0.674970i \(0.235843\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26667.3 −1.51390 −0.756948 0.653475i \(-0.773311\pi\)
−0.756948 + 0.653475i \(0.773311\pi\)
\(678\) 0 0
\(679\) −26436.2 −1.49415
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14.8616 0.000832599 0 0.000416299 1.00000i \(-0.499867\pi\)
0.000416299 1.00000i \(0.499867\pi\)
\(684\) 0 0
\(685\) −12287.0 −0.685345
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4842.31 0.267747
\(690\) 0 0
\(691\) −26645.5 −1.46692 −0.733460 0.679733i \(-0.762096\pi\)
−0.733460 + 0.679733i \(0.762096\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2926.96 −0.159749
\(696\) 0 0
\(697\) −22215.4 −1.20727
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 431.952 0.0232733 0.0116367 0.999932i \(-0.496296\pi\)
0.0116367 + 0.999932i \(0.496296\pi\)
\(702\) 0 0
\(703\) −10218.0 −0.548195
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 33031.7 1.75712
\(708\) 0 0
\(709\) 35463.0 1.87848 0.939240 0.343261i \(-0.111532\pi\)
0.939240 + 0.343261i \(0.111532\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6770.60 0.355626
\(714\) 0 0
\(715\) 20461.8 1.07025
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −26340.5 −1.36625 −0.683125 0.730302i \(-0.739379\pi\)
−0.683125 + 0.730302i \(0.739379\pi\)
\(720\) 0 0
\(721\) −6918.48 −0.357362
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5302.70 −0.271638
\(726\) 0 0
\(727\) 38279.1 1.95281 0.976404 0.215952i \(-0.0692853\pi\)
0.976404 + 0.215952i \(0.0692853\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9404.89 0.475858
\(732\) 0 0
\(733\) −16445.7 −0.828700 −0.414350 0.910118i \(-0.635991\pi\)
−0.414350 + 0.910118i \(0.635991\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8880.39 −0.443844
\(738\) 0 0
\(739\) 5150.65 0.256387 0.128193 0.991749i \(-0.459082\pi\)
0.128193 + 0.991749i \(0.459082\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8844.91 −0.436727 −0.218364 0.975868i \(-0.570072\pi\)
−0.218364 + 0.975868i \(0.570072\pi\)
\(744\) 0 0
\(745\) 4632.52 0.227816
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 14498.2 0.707282
\(750\) 0 0
\(751\) −5774.81 −0.280594 −0.140297 0.990109i \(-0.544806\pi\)
−0.140297 + 0.990109i \(0.544806\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2059.50 0.0992754
\(756\) 0 0
\(757\) 14267.1 0.685003 0.342501 0.939517i \(-0.388726\pi\)
0.342501 + 0.939517i \(0.388726\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21823.0 1.03953 0.519765 0.854309i \(-0.326020\pi\)
0.519765 + 0.854309i \(0.326020\pi\)
\(762\) 0 0
\(763\) −3513.04 −0.166685
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20633.3 −0.971352
\(768\) 0 0
\(769\) −3080.07 −0.144435 −0.0722173 0.997389i \(-0.523008\pi\)
−0.0722173 + 0.997389i \(0.523008\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8834.10 0.411049 0.205524 0.978652i \(-0.434110\pi\)
0.205524 + 0.978652i \(0.434110\pi\)
\(774\) 0 0
\(775\) −4182.95 −0.193879
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10130.1 0.465917
\(780\) 0 0
\(781\) −19944.0 −0.913766
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14799.8 0.672903
\(786\) 0 0
\(787\) −11060.1 −0.500952 −0.250476 0.968123i \(-0.580587\pi\)
−0.250476 + 0.968123i \(0.580587\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −22400.3 −1.00690
\(792\) 0 0
\(793\) 31967.4 1.43152
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3379.95 −0.150218 −0.0751091 0.997175i \(-0.523930\pi\)
−0.0751091 + 0.997175i \(0.523930\pi\)
\(798\) 0 0
\(799\) 26545.4 1.17536
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −82824.7 −3.63988
\(804\) 0 0
\(805\) 6114.47 0.267710
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38358.8 1.66703 0.833513 0.552499i \(-0.186326\pi\)
0.833513 + 0.552499i \(0.186326\pi\)
\(810\) 0 0
\(811\) 9110.42 0.394464 0.197232 0.980357i \(-0.436805\pi\)
0.197232 + 0.980357i \(0.436805\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12522.5 0.538214
\(816\) 0 0
\(817\) −4288.58 −0.183646
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8177.12 −0.347605 −0.173802 0.984781i \(-0.555605\pi\)
−0.173802 + 0.984781i \(0.555605\pi\)
\(822\) 0 0
\(823\) 4238.40 0.179515 0.0897577 0.995964i \(-0.471391\pi\)
0.0897577 + 0.995964i \(0.471391\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12021.9 −0.505492 −0.252746 0.967533i \(-0.581334\pi\)
−0.252746 + 0.967533i \(0.581334\pi\)
\(828\) 0 0
\(829\) 6564.14 0.275009 0.137504 0.990501i \(-0.456092\pi\)
0.137504 + 0.990501i \(0.456092\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −34864.5 −1.45016
\(834\) 0 0
\(835\) −10277.1 −0.425932
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3645.41 −0.150004 −0.0750021 0.997183i \(-0.523896\pi\)
−0.0750021 + 0.997183i \(0.523896\pi\)
\(840\) 0 0
\(841\) 20600.9 0.844679
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7492.79 −0.305041
\(846\) 0 0
\(847\) 96729.9 3.92406
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14832.2 0.597462
\(852\) 0 0
\(853\) −45697.7 −1.83430 −0.917151 0.398540i \(-0.869517\pi\)
−0.917151 + 0.398540i \(0.869517\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13766.3 0.548716 0.274358 0.961628i \(-0.411535\pi\)
0.274358 + 0.961628i \(0.411535\pi\)
\(858\) 0 0
\(859\) −15422.0 −0.612563 −0.306282 0.951941i \(-0.599085\pi\)
−0.306282 + 0.951941i \(0.599085\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7382.54 0.291199 0.145599 0.989344i \(-0.453489\pi\)
0.145599 + 0.989344i \(0.453489\pi\)
\(864\) 0 0
\(865\) 8620.83 0.338863
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 41856.8 1.63394
\(870\) 0 0
\(871\) 8019.32 0.311968
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3777.59 −0.145949
\(876\) 0 0
\(877\) −12079.7 −0.465112 −0.232556 0.972583i \(-0.574709\pi\)
−0.232556 + 0.972583i \(0.574709\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 25501.7 0.975227 0.487613 0.873060i \(-0.337868\pi\)
0.487613 + 0.873060i \(0.337868\pi\)
\(882\) 0 0
\(883\) 29353.2 1.11870 0.559351 0.828931i \(-0.311050\pi\)
0.559351 + 0.828931i \(0.311050\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17185.8 −0.650556 −0.325278 0.945618i \(-0.605458\pi\)
−0.325278 + 0.945618i \(0.605458\pi\)
\(888\) 0 0
\(889\) 78434.1 2.95905
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12104.6 −0.453600
\(894\) 0 0
\(895\) 18006.8 0.672513
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 35489.5 1.31662
\(900\) 0 0
\(901\) 4869.68 0.180058
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6285.83 0.230882
\(906\) 0 0
\(907\) 38511.6 1.40988 0.704938 0.709269i \(-0.250975\pi\)
0.704938 + 0.709269i \(0.250975\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11451.4 0.416467 0.208233 0.978079i \(-0.433229\pi\)
0.208233 + 0.978079i \(0.433229\pi\)
\(912\) 0 0
\(913\) −5138.90 −0.186279
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 42478.8 1.52974
\(918\) 0 0
\(919\) −14096.0 −0.505968 −0.252984 0.967470i \(-0.581412\pi\)
−0.252984 + 0.967470i \(0.581412\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18010.1 0.642265
\(924\) 0 0
\(925\) −9163.47 −0.325722
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15020.3 −0.530462 −0.265231 0.964185i \(-0.585448\pi\)
−0.265231 + 0.964185i \(0.585448\pi\)
\(930\) 0 0
\(931\) 15898.1 0.559654
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 20577.5 0.719739
\(936\) 0 0
\(937\) 18430.5 0.642582 0.321291 0.946981i \(-0.395883\pi\)
0.321291 + 0.946981i \(0.395883\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 55141.3 1.91026 0.955131 0.296184i \(-0.0957141\pi\)
0.955131 + 0.296184i \(0.0957141\pi\)
\(942\) 0 0
\(943\) −14704.5 −0.507790
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −24231.3 −0.831480 −0.415740 0.909484i \(-0.636477\pi\)
−0.415740 + 0.909484i \(0.636477\pi\)
\(948\) 0 0
\(949\) 74793.8 2.55839
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19107.5 −0.649479 −0.324739 0.945804i \(-0.605277\pi\)
−0.324739 + 0.945804i \(0.605277\pi\)
\(954\) 0 0
\(955\) 23387.3 0.792457
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 74264.2 2.50064
\(960\) 0 0
\(961\) −1795.67 −0.0602756
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 13239.4 0.441649
\(966\) 0 0
\(967\) −21993.6 −0.731401 −0.365701 0.930732i \(-0.619171\pi\)
−0.365701 + 0.930732i \(0.619171\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19409.3 −0.641477 −0.320738 0.947168i \(-0.603931\pi\)
−0.320738 + 0.947168i \(0.603931\pi\)
\(972\) 0 0
\(973\) 17690.9 0.582883
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36384.1 1.19143 0.595717 0.803194i \(-0.296868\pi\)
0.595717 + 0.803194i \(0.296868\pi\)
\(978\) 0 0
\(979\) 12952.5 0.422843
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4202.43 0.136355 0.0681774 0.997673i \(-0.478282\pi\)
0.0681774 + 0.997673i \(0.478282\pi\)
\(984\) 0 0
\(985\) 6844.57 0.221407
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6225.16 0.200150
\(990\) 0 0
\(991\) 17753.0 0.569064 0.284532 0.958666i \(-0.408162\pi\)
0.284532 + 0.958666i \(0.408162\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −27057.2 −0.862080
\(996\) 0 0
\(997\) 12519.0 0.397673 0.198837 0.980033i \(-0.436284\pi\)
0.198837 + 0.980033i \(0.436284\pi\)
\(998\) 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.bk.1.3 3
3.2 odd 2 2160.4.a.bs.1.3 3
4.3 odd 2 1080.4.a.d.1.1 3
12.11 even 2 1080.4.a.j.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.1 3 4.3 odd 2
1080.4.a.j.1.1 yes 3 12.11 even 2
2160.4.a.bk.1.3 3 1.1 even 1 trivial
2160.4.a.bs.1.3 3 3.2 odd 2