Properties

Label 1728.4.c.j.1727.12
Level $1728$
Weight $4$
Character 1728.1727
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.12
Root \(0.886307 + 1.60260i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.4.c.j.1727.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.8488i q^{5} +13.9048i q^{7} +O(q^{10})\) \(q+20.8488i q^{5} +13.9048i q^{7} +34.5116 q^{11} +31.3361 q^{13} +34.4719i q^{17} +120.723i q^{19} -137.155 q^{23} -309.672 q^{25} +93.1005i q^{29} +111.365i q^{31} -289.899 q^{35} +146.680 q^{37} -8.44531i q^{41} +427.523i q^{43} +318.826 q^{47} +149.655 q^{49} -291.451i q^{53} +719.525i q^{55} +364.665 q^{59} +289.983 q^{61} +653.320i q^{65} +305.907i q^{67} +102.802 q^{71} +442.688 q^{73} +479.878i q^{77} -245.350i q^{79} -478.981 q^{83} -718.697 q^{85} -1417.36i q^{89} +435.723i q^{91} -2516.92 q^{95} +1153.69 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 20.8488i 1.86477i 0.361464 + 0.932386i \(0.382277\pi\)
−0.361464 + 0.932386i \(0.617723\pi\)
\(6\) 0 0
\(7\) 13.9048i 0.750791i 0.926865 + 0.375395i \(0.122493\pi\)
−0.926865 + 0.375395i \(0.877507\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 34.5116 0.945967 0.472984 0.881071i \(-0.343177\pi\)
0.472984 + 0.881071i \(0.343177\pi\)
\(12\) 0 0
\(13\) 31.3361 0.668544 0.334272 0.942477i \(-0.391510\pi\)
0.334272 + 0.942477i \(0.391510\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 34.4719i 0.491803i 0.969295 + 0.245902i \(0.0790840\pi\)
−0.969295 + 0.245902i \(0.920916\pi\)
\(18\) 0 0
\(19\) 120.723i 1.45767i 0.684691 + 0.728833i \(0.259937\pi\)
−0.684691 + 0.728833i \(0.740063\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −137.155 −1.24343 −0.621714 0.783244i \(-0.713564\pi\)
−0.621714 + 0.783244i \(0.713564\pi\)
\(24\) 0 0
\(25\) −309.672 −2.47738
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 93.1005i 0.596149i 0.954543 + 0.298075i \(0.0963444\pi\)
−0.954543 + 0.298075i \(0.903656\pi\)
\(30\) 0 0
\(31\) 111.365i 0.645217i 0.946532 + 0.322609i \(0.104560\pi\)
−0.946532 + 0.322609i \(0.895440\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −289.899 −1.40005
\(36\) 0 0
\(37\) 146.680 0.651733 0.325867 0.945416i \(-0.394344\pi\)
0.325867 + 0.945416i \(0.394344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.44531i − 0.0321692i −0.999871 0.0160846i \(-0.994880\pi\)
0.999871 0.0160846i \(-0.00512011\pi\)
\(42\) 0 0
\(43\) 427.523i 1.51620i 0.652137 + 0.758101i \(0.273873\pi\)
−0.652137 + 0.758101i \(0.726127\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 318.826 0.989481 0.494740 0.869041i \(-0.335263\pi\)
0.494740 + 0.869041i \(0.335263\pi\)
\(48\) 0 0
\(49\) 149.655 0.436313
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 291.451i − 0.755357i −0.925937 0.377679i \(-0.876722\pi\)
0.925937 0.377679i \(-0.123278\pi\)
\(54\) 0 0
\(55\) 719.525i 1.76401i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 364.665 0.804666 0.402333 0.915493i \(-0.368199\pi\)
0.402333 + 0.915493i \(0.368199\pi\)
\(60\) 0 0
\(61\) 289.983 0.608664 0.304332 0.952566i \(-0.401567\pi\)
0.304332 + 0.952566i \(0.401567\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 653.320i 1.24668i
\(66\) 0 0
\(67\) 305.907i 0.557797i 0.960321 + 0.278899i \(0.0899694\pi\)
−0.960321 + 0.278899i \(0.910031\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 102.802 0.171836 0.0859178 0.996302i \(-0.472618\pi\)
0.0859178 + 0.996302i \(0.472618\pi\)
\(72\) 0 0
\(73\) 442.688 0.709764 0.354882 0.934911i \(-0.384521\pi\)
0.354882 + 0.934911i \(0.384521\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 479.878i 0.710224i
\(78\) 0 0
\(79\) − 245.350i − 0.349419i −0.984620 0.174709i \(-0.944101\pi\)
0.984620 0.174709i \(-0.0558986\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −478.981 −0.633433 −0.316717 0.948520i \(-0.602580\pi\)
−0.316717 + 0.948520i \(0.602580\pi\)
\(84\) 0 0
\(85\) −718.697 −0.917101
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1417.36i − 1.68809i −0.536273 0.844045i \(-0.680168\pi\)
0.536273 0.844045i \(-0.319832\pi\)
\(90\) 0 0
\(91\) 435.723i 0.501937i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2516.92 −2.71822
\(96\) 0 0
\(97\) 1153.69 1.20762 0.603811 0.797128i \(-0.293648\pi\)
0.603811 + 0.797128i \(0.293648\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 767.096i 0.755732i 0.925860 + 0.377866i \(0.123342\pi\)
−0.925860 + 0.377866i \(0.876658\pi\)
\(102\) 0 0
\(103\) 1202.04i 1.14991i 0.818187 + 0.574953i \(0.194979\pi\)
−0.818187 + 0.574953i \(0.805021\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1309.28 −1.18292 −0.591462 0.806333i \(-0.701449\pi\)
−0.591462 + 0.806333i \(0.701449\pi\)
\(108\) 0 0
\(109\) −1102.04 −0.968407 −0.484204 0.874955i \(-0.660891\pi\)
−0.484204 + 0.874955i \(0.660891\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 69.9098i − 0.0581997i −0.999577 0.0290998i \(-0.990736\pi\)
0.999577 0.0290998i \(-0.00926407\pi\)
\(114\) 0 0
\(115\) − 2859.52i − 2.31871i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −479.326 −0.369241
\(120\) 0 0
\(121\) −139.949 −0.105146
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 3850.19i − 2.75497i
\(126\) 0 0
\(127\) − 2558.29i − 1.78749i −0.448574 0.893746i \(-0.648068\pi\)
0.448574 0.893746i \(-0.351932\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −86.6249 −0.0577744 −0.0288872 0.999583i \(-0.509196\pi\)
−0.0288872 + 0.999583i \(0.509196\pi\)
\(132\) 0 0
\(133\) −1678.63 −1.09440
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 699.734i 0.436367i 0.975908 + 0.218184i \(0.0700131\pi\)
−0.975908 + 0.218184i \(0.929987\pi\)
\(138\) 0 0
\(139\) 789.049i 0.481484i 0.970589 + 0.240742i \(0.0773908\pi\)
−0.970589 + 0.240742i \(0.922609\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1081.46 0.632421
\(144\) 0 0
\(145\) −1941.03 −1.11168
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1593.12i 0.875930i 0.898992 + 0.437965i \(0.144301\pi\)
−0.898992 + 0.437965i \(0.855699\pi\)
\(150\) 0 0
\(151\) − 1502.12i − 0.809544i −0.914418 0.404772i \(-0.867351\pi\)
0.914418 0.404772i \(-0.132649\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2321.82 −1.20318
\(156\) 0 0
\(157\) 3596.08 1.82802 0.914008 0.405696i \(-0.132971\pi\)
0.914008 + 0.405696i \(0.132971\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 1907.12i − 0.933555i
\(162\) 0 0
\(163\) − 3313.82i − 1.59238i −0.605044 0.796192i \(-0.706844\pi\)
0.605044 0.796192i \(-0.293156\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −487.046 −0.225681 −0.112841 0.993613i \(-0.535995\pi\)
−0.112841 + 0.993613i \(0.535995\pi\)
\(168\) 0 0
\(169\) −1215.05 −0.553049
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1752.30i 0.770086i 0.922899 + 0.385043i \(0.125813\pi\)
−0.922899 + 0.385043i \(0.874187\pi\)
\(174\) 0 0
\(175\) − 4305.94i − 1.85999i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2807.62 −1.17236 −0.586178 0.810182i \(-0.699368\pi\)
−0.586178 + 0.810182i \(0.699368\pi\)
\(180\) 0 0
\(181\) 2307.37 0.947546 0.473773 0.880647i \(-0.342892\pi\)
0.473773 + 0.880647i \(0.342892\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3058.11i 1.21533i
\(186\) 0 0
\(187\) 1189.68i 0.465230i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3375.74 −1.27885 −0.639425 0.768854i \(-0.720827\pi\)
−0.639425 + 0.768854i \(0.720827\pi\)
\(192\) 0 0
\(193\) 561.917 0.209573 0.104787 0.994495i \(-0.466584\pi\)
0.104787 + 0.994495i \(0.466584\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.7420i 0.00605491i 0.999995 + 0.00302746i \(0.000963671\pi\)
−0.999995 + 0.00302746i \(0.999036\pi\)
\(198\) 0 0
\(199\) 2760.01i 0.983176i 0.870828 + 0.491588i \(0.163583\pi\)
−0.870828 + 0.491588i \(0.836417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1294.55 −0.447583
\(204\) 0 0
\(205\) 176.075 0.0599882
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4166.33i 1.37890i
\(210\) 0 0
\(211\) − 2766.47i − 0.902615i −0.892368 0.451308i \(-0.850958\pi\)
0.892368 0.451308i \(-0.149042\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8913.34 −2.82737
\(216\) 0 0
\(217\) −1548.51 −0.484423
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1080.21i 0.328792i
\(222\) 0 0
\(223\) − 110.636i − 0.0332231i −0.999862 0.0166115i \(-0.994712\pi\)
0.999862 0.0166115i \(-0.00528786\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1800.57 −0.526468 −0.263234 0.964732i \(-0.584789\pi\)
−0.263234 + 0.964732i \(0.584789\pi\)
\(228\) 0 0
\(229\) 1491.95 0.430528 0.215264 0.976556i \(-0.430939\pi\)
0.215264 + 0.976556i \(0.430939\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 4545.75i − 1.27812i −0.769157 0.639060i \(-0.779324\pi\)
0.769157 0.639060i \(-0.220676\pi\)
\(234\) 0 0
\(235\) 6647.14i 1.84516i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3305.97 0.894751 0.447376 0.894346i \(-0.352359\pi\)
0.447376 + 0.894346i \(0.352359\pi\)
\(240\) 0 0
\(241\) −2337.95 −0.624898 −0.312449 0.949934i \(-0.601149\pi\)
−0.312449 + 0.949934i \(0.601149\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3120.13i 0.813624i
\(246\) 0 0
\(247\) 3782.97i 0.974514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3625.36 0.911675 0.455838 0.890063i \(-0.349340\pi\)
0.455838 + 0.890063i \(0.349340\pi\)
\(252\) 0 0
\(253\) −4733.45 −1.17624
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2124.02i − 0.515537i −0.966207 0.257768i \(-0.917013\pi\)
0.966207 0.257768i \(-0.0829871\pi\)
\(258\) 0 0
\(259\) 2039.57i 0.489315i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4785.87 −1.12209 −0.561044 0.827786i \(-0.689600\pi\)
−0.561044 + 0.827786i \(0.689600\pi\)
\(264\) 0 0
\(265\) 6076.41 1.40857
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 241.884i 0.0548249i 0.999624 + 0.0274125i \(0.00872675\pi\)
−0.999624 + 0.0274125i \(0.991273\pi\)
\(270\) 0 0
\(271\) − 828.799i − 0.185778i −0.995676 0.0928892i \(-0.970390\pi\)
0.995676 0.0928892i \(-0.0296102\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10687.3 −2.34352
\(276\) 0 0
\(277\) 5897.19 1.27916 0.639581 0.768724i \(-0.279108\pi\)
0.639581 + 0.768724i \(0.279108\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 3055.95i − 0.648763i −0.945926 0.324382i \(-0.894844\pi\)
0.945926 0.324382i \(-0.105156\pi\)
\(282\) 0 0
\(283\) 196.045i 0.0411790i 0.999788 + 0.0205895i \(0.00655430\pi\)
−0.999788 + 0.0205895i \(0.993446\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 117.431 0.0241523
\(288\) 0 0
\(289\) 3724.69 0.758130
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 8748.98i − 1.74444i −0.489114 0.872220i \(-0.662680\pi\)
0.489114 0.872220i \(-0.337320\pi\)
\(294\) 0 0
\(295\) 7602.82i 1.50052i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4297.91 −0.831287
\(300\) 0 0
\(301\) −5944.65 −1.13835
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6045.79i 1.13502i
\(306\) 0 0
\(307\) − 2095.99i − 0.389656i −0.980837 0.194828i \(-0.937585\pi\)
0.980837 0.194828i \(-0.0624149\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3393.74 −0.618783 −0.309391 0.950935i \(-0.600125\pi\)
−0.309391 + 0.950935i \(0.600125\pi\)
\(312\) 0 0
\(313\) 3579.49 0.646405 0.323203 0.946330i \(-0.395240\pi\)
0.323203 + 0.946330i \(0.395240\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 4356.56i − 0.771889i −0.922522 0.385945i \(-0.873876\pi\)
0.922522 0.385945i \(-0.126124\pi\)
\(318\) 0 0
\(319\) 3213.05i 0.563937i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4161.53 −0.716885
\(324\) 0 0
\(325\) −9703.91 −1.65623
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4433.23i 0.742893i
\(330\) 0 0
\(331\) 9191.99i 1.52640i 0.646164 + 0.763198i \(0.276372\pi\)
−0.646164 + 0.763198i \(0.723628\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6377.78 −1.04017
\(336\) 0 0
\(337\) 1663.33 0.268865 0.134432 0.990923i \(-0.457079\pi\)
0.134432 + 0.990923i \(0.457079\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3843.38i 0.610354i
\(342\) 0 0
\(343\) 6850.30i 1.07837i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9668.26 −1.49573 −0.747866 0.663850i \(-0.768922\pi\)
−0.747866 + 0.663850i \(0.768922\pi\)
\(348\) 0 0
\(349\) −9928.24 −1.52277 −0.761384 0.648301i \(-0.775480\pi\)
−0.761384 + 0.648301i \(0.775480\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 4460.01i − 0.672471i −0.941778 0.336236i \(-0.890846\pi\)
0.941778 0.336236i \(-0.109154\pi\)
\(354\) 0 0
\(355\) 2143.29i 0.320434i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2158.32 −0.317303 −0.158652 0.987335i \(-0.550715\pi\)
−0.158652 + 0.987335i \(0.550715\pi\)
\(360\) 0 0
\(361\) −7714.95 −1.12479
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9229.52i 1.32355i
\(366\) 0 0
\(367\) 1156.43i 0.164482i 0.996612 + 0.0822411i \(0.0262078\pi\)
−0.996612 + 0.0822411i \(0.973792\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4052.59 0.567115
\(372\) 0 0
\(373\) 6286.39 0.872647 0.436323 0.899790i \(-0.356280\pi\)
0.436323 + 0.899790i \(0.356280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2917.41i 0.398552i
\(378\) 0 0
\(379\) − 6921.76i − 0.938119i −0.883167 0.469059i \(-0.844593\pi\)
0.883167 0.469059i \(-0.155407\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1802.67 0.240502 0.120251 0.992744i \(-0.461630\pi\)
0.120251 + 0.992744i \(0.461630\pi\)
\(384\) 0 0
\(385\) −10004.9 −1.32441
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 4421.02i − 0.576233i −0.957595 0.288117i \(-0.906971\pi\)
0.957595 0.288117i \(-0.0930291\pi\)
\(390\) 0 0
\(391\) − 4728.00i − 0.611522i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5115.26 0.651586
\(396\) 0 0
\(397\) 13769.8 1.74077 0.870383 0.492375i \(-0.163871\pi\)
0.870383 + 0.492375i \(0.163871\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 13534.1i − 1.68544i −0.538350 0.842722i \(-0.680952\pi\)
0.538350 0.842722i \(-0.319048\pi\)
\(402\) 0 0
\(403\) 3489.74i 0.431356i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5062.18 0.616518
\(408\) 0 0
\(409\) 7230.24 0.874114 0.437057 0.899434i \(-0.356021\pi\)
0.437057 + 0.899434i \(0.356021\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5070.61i 0.604136i
\(414\) 0 0
\(415\) − 9986.16i − 1.18121i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9066.85 1.05715 0.528574 0.848887i \(-0.322727\pi\)
0.528574 + 0.848887i \(0.322727\pi\)
\(420\) 0 0
\(421\) −6017.63 −0.696630 −0.348315 0.937378i \(-0.613246\pi\)
−0.348315 + 0.937378i \(0.613246\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 10675.0i − 1.21838i
\(426\) 0 0
\(427\) 4032.17i 0.456979i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10533.7 −1.17724 −0.588619 0.808410i \(-0.700328\pi\)
−0.588619 + 0.808410i \(0.700328\pi\)
\(432\) 0 0
\(433\) −79.2056 −0.00879071 −0.00439536 0.999990i \(-0.501399\pi\)
−0.00439536 + 0.999990i \(0.501399\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 16557.7i − 1.81250i
\(438\) 0 0
\(439\) − 9484.31i − 1.03112i −0.856854 0.515560i \(-0.827584\pi\)
0.856854 0.515560i \(-0.172416\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9734.58 1.04403 0.522013 0.852937i \(-0.325181\pi\)
0.522013 + 0.852937i \(0.325181\pi\)
\(444\) 0 0
\(445\) 29550.3 3.14790
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9270.01i 0.974340i 0.873307 + 0.487170i \(0.161971\pi\)
−0.873307 + 0.487170i \(0.838029\pi\)
\(450\) 0 0
\(451\) − 291.461i − 0.0304310i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9084.31 −0.935997
\(456\) 0 0
\(457\) −2542.86 −0.260285 −0.130142 0.991495i \(-0.541543\pi\)
−0.130142 + 0.991495i \(0.541543\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 10664.8i − 1.07746i −0.842479 0.538729i \(-0.818905\pi\)
0.842479 0.538729i \(-0.181095\pi\)
\(462\) 0 0
\(463\) 1963.48i 0.197086i 0.995133 + 0.0985428i \(0.0314181\pi\)
−0.995133 + 0.0985428i \(0.968582\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19778.8 −1.95986 −0.979928 0.199350i \(-0.936117\pi\)
−0.979928 + 0.199350i \(0.936117\pi\)
\(468\) 0 0
\(469\) −4253.58 −0.418789
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14754.5i 1.43428i
\(474\) 0 0
\(475\) − 37384.4i − 3.61119i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16062.6 1.53219 0.766095 0.642728i \(-0.222197\pi\)
0.766095 + 0.642728i \(0.222197\pi\)
\(480\) 0 0
\(481\) 4596.39 0.435712
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24053.0i 2.25194i
\(486\) 0 0
\(487\) − 15907.5i − 1.48016i −0.672519 0.740080i \(-0.734788\pi\)
0.672519 0.740080i \(-0.265212\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5924.49 0.544538 0.272269 0.962221i \(-0.412226\pi\)
0.272269 + 0.962221i \(0.412226\pi\)
\(492\) 0 0
\(493\) −3209.35 −0.293188
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1429.44i 0.129013i
\(498\) 0 0
\(499\) − 5331.33i − 0.478283i −0.970985 0.239142i \(-0.923134\pi\)
0.970985 0.239142i \(-0.0768660\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2144.13 0.190064 0.0950319 0.995474i \(-0.469705\pi\)
0.0950319 + 0.995474i \(0.469705\pi\)
\(504\) 0 0
\(505\) −15993.0 −1.40927
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14329.9i 1.24786i 0.781481 + 0.623929i \(0.214465\pi\)
−0.781481 + 0.623929i \(0.785535\pi\)
\(510\) 0 0
\(511\) 6155.51i 0.532884i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25061.0 −2.14431
\(516\) 0 0
\(517\) 11003.2 0.936016
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10822.8i 0.910087i 0.890469 + 0.455043i \(0.150376\pi\)
−0.890469 + 0.455043i \(0.849624\pi\)
\(522\) 0 0
\(523\) 12111.8i 1.01265i 0.862344 + 0.506323i \(0.168995\pi\)
−0.862344 + 0.506323i \(0.831005\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3838.96 −0.317320
\(528\) 0 0
\(529\) 6644.58 0.546115
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 264.643i − 0.0215065i
\(534\) 0 0
\(535\) − 27296.9i − 2.20588i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5164.85 0.412738
\(540\) 0 0
\(541\) −22063.1 −1.75336 −0.876681 0.481073i \(-0.840247\pi\)
−0.876681 + 0.481073i \(0.840247\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 22976.2i − 1.80586i
\(546\) 0 0
\(547\) 17842.7i 1.39470i 0.716731 + 0.697350i \(0.245637\pi\)
−0.716731 + 0.697350i \(0.754363\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11239.3 −0.868987
\(552\) 0 0
\(553\) 3411.56 0.262340
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 10942.0i − 0.832365i −0.909281 0.416183i \(-0.863368\pi\)
0.909281 0.416183i \(-0.136632\pi\)
\(558\) 0 0
\(559\) 13396.9i 1.01365i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1991.56 0.149084 0.0745418 0.997218i \(-0.476251\pi\)
0.0745418 + 0.997218i \(0.476251\pi\)
\(564\) 0 0
\(565\) 1457.53 0.108529
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12048.3i 0.887684i 0.896105 + 0.443842i \(0.146385\pi\)
−0.896105 + 0.443842i \(0.853615\pi\)
\(570\) 0 0
\(571\) − 24916.7i − 1.82615i −0.407792 0.913075i \(-0.633701\pi\)
0.407792 0.913075i \(-0.366299\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 42473.2 3.08044
\(576\) 0 0
\(577\) −17705.3 −1.27743 −0.638717 0.769441i \(-0.720535\pi\)
−0.638717 + 0.769441i \(0.720535\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 6660.15i − 0.475576i
\(582\) 0 0
\(583\) − 10058.5i − 0.714543i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22453.6 1.57880 0.789402 0.613876i \(-0.210391\pi\)
0.789402 + 0.613876i \(0.210391\pi\)
\(588\) 0 0
\(589\) −13444.3 −0.940511
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26839.1i 1.85860i 0.369328 + 0.929299i \(0.379588\pi\)
−0.369328 + 0.929299i \(0.620412\pi\)
\(594\) 0 0
\(595\) − 9993.36i − 0.688551i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9254.44 −0.631262 −0.315631 0.948882i \(-0.602216\pi\)
−0.315631 + 0.948882i \(0.602216\pi\)
\(600\) 0 0
\(601\) 1102.45 0.0748250 0.0374125 0.999300i \(-0.488088\pi\)
0.0374125 + 0.999300i \(0.488088\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 2917.77i − 0.196073i
\(606\) 0 0
\(607\) 21383.2i 1.42985i 0.699202 + 0.714924i \(0.253539\pi\)
−0.699202 + 0.714924i \(0.746461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9990.77 0.661511
\(612\) 0 0
\(613\) −19769.0 −1.30255 −0.651274 0.758843i \(-0.725765\pi\)
−0.651274 + 0.758843i \(0.725765\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2702.48i 0.176333i 0.996106 + 0.0881667i \(0.0281008\pi\)
−0.996106 + 0.0881667i \(0.971899\pi\)
\(618\) 0 0
\(619\) 6967.79i 0.452438i 0.974076 + 0.226219i \(0.0726365\pi\)
−0.974076 + 0.226219i \(0.927364\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 19708.2 1.26740
\(624\) 0 0
\(625\) 41562.7 2.66001
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5056.35i 0.320524i
\(630\) 0 0
\(631\) 19202.4i 1.21147i 0.795667 + 0.605734i \(0.207120\pi\)
−0.795667 + 0.605734i \(0.792880\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 53337.2 3.33326
\(636\) 0 0
\(637\) 4689.61 0.291694
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 23580.8i − 1.45302i −0.687155 0.726511i \(-0.741140\pi\)
0.687155 0.726511i \(-0.258860\pi\)
\(642\) 0 0
\(643\) 25167.2i 1.54354i 0.635899 + 0.771772i \(0.280629\pi\)
−0.635899 + 0.771772i \(0.719371\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2247.67 0.136577 0.0682883 0.997666i \(-0.478246\pi\)
0.0682883 + 0.997666i \(0.478246\pi\)
\(648\) 0 0
\(649\) 12585.2 0.761188
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11278.6i 0.675906i 0.941163 + 0.337953i \(0.109734\pi\)
−0.941163 + 0.337953i \(0.890266\pi\)
\(654\) 0 0
\(655\) − 1806.02i − 0.107736i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −658.636 −0.0389329 −0.0194665 0.999811i \(-0.506197\pi\)
−0.0194665 + 0.999811i \(0.506197\pi\)
\(660\) 0 0
\(661\) −1740.41 −0.102412 −0.0512060 0.998688i \(-0.516306\pi\)
−0.0512060 + 0.998688i \(0.516306\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 34997.4i − 2.04081i
\(666\) 0 0
\(667\) − 12769.2i − 0.741269i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10007.8 0.575776
\(672\) 0 0
\(673\) −273.146 −0.0156449 −0.00782243 0.999969i \(-0.502490\pi\)
−0.00782243 + 0.999969i \(0.502490\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11373.1i 0.645649i 0.946459 + 0.322824i \(0.104632\pi\)
−0.946459 + 0.322824i \(0.895368\pi\)
\(678\) 0 0
\(679\) 16041.8i 0.906671i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1223.56 −0.0685480 −0.0342740 0.999412i \(-0.510912\pi\)
−0.0342740 + 0.999412i \(0.510912\pi\)
\(684\) 0 0
\(685\) −14588.6 −0.813725
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 9132.95i − 0.504989i
\(690\) 0 0
\(691\) 9247.55i 0.509108i 0.967059 + 0.254554i \(0.0819286\pi\)
−0.967059 + 0.254554i \(0.918071\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16450.7 −0.897858
\(696\) 0 0
\(697\) 291.126 0.0158209
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 14448.0i − 0.778447i −0.921143 0.389224i \(-0.872743\pi\)
0.921143 0.389224i \(-0.127257\pi\)
\(702\) 0 0
\(703\) 17707.6i 0.950009i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10666.4 −0.567397
\(708\) 0 0
\(709\) −7819.28 −0.414188 −0.207094 0.978321i \(-0.566401\pi\)
−0.207094 + 0.978321i \(0.566401\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 15274.3i − 0.802281i
\(714\) 0 0
\(715\) 22547.1i 1.17932i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7251.47 0.376126 0.188063 0.982157i \(-0.439779\pi\)
0.188063 + 0.982157i \(0.439779\pi\)
\(720\) 0 0
\(721\) −16714.1 −0.863338
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 28830.6i − 1.47689i
\(726\) 0 0
\(727\) − 6605.59i − 0.336985i −0.985703 0.168492i \(-0.946110\pi\)
0.985703 0.168492i \(-0.0538899\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −14737.5 −0.745673
\(732\) 0 0
\(733\) 37147.9 1.87188 0.935942 0.352155i \(-0.114551\pi\)
0.935942 + 0.352155i \(0.114551\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10557.3i 0.527658i
\(738\) 0 0
\(739\) 34209.1i 1.70285i 0.524479 + 0.851423i \(0.324260\pi\)
−0.524479 + 0.851423i \(0.675740\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15746.4 0.777494 0.388747 0.921345i \(-0.372908\pi\)
0.388747 + 0.921345i \(0.372908\pi\)
\(744\) 0 0
\(745\) −33214.6 −1.63341
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18205.3i − 0.888128i
\(750\) 0 0
\(751\) 27773.5i 1.34949i 0.738049 + 0.674747i \(0.235747\pi\)
−0.738049 + 0.674747i \(0.764253\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 31317.5 1.50961
\(756\) 0 0
\(757\) 12694.0 0.609471 0.304736 0.952437i \(-0.401432\pi\)
0.304736 + 0.952437i \(0.401432\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18061.4i 0.860349i 0.902746 + 0.430175i \(0.141548\pi\)
−0.902746 + 0.430175i \(0.858452\pi\)
\(762\) 0 0
\(763\) − 15323.7i − 0.727071i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11427.2 0.537955
\(768\) 0 0
\(769\) 32615.8 1.52946 0.764731 0.644350i \(-0.222872\pi\)
0.764731 + 0.644350i \(0.222872\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8518.40i − 0.396359i −0.980166 0.198179i \(-0.936497\pi\)
0.980166 0.198179i \(-0.0635029\pi\)
\(774\) 0 0
\(775\) − 34486.6i − 1.59844i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1019.54 0.0468919
\(780\) 0 0
\(781\) 3547.86 0.162551
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 74973.9i 3.40883i
\(786\) 0 0
\(787\) 12371.2i 0.560338i 0.959951 + 0.280169i \(0.0903906\pi\)
−0.959951 + 0.280169i \(0.909609\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 972.085 0.0436958
\(792\) 0 0
\(793\) 9086.93 0.406919
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37675.0i 1.67443i 0.546877 + 0.837213i \(0.315816\pi\)
−0.546877 + 0.837213i \(0.684184\pi\)
\(798\) 0 0
\(799\) 10990.5i 0.486630i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15277.9 0.671413
\(804\) 0 0
\(805\) 39761.2 1.74087
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 41809.2i − 1.81697i −0.417914 0.908487i \(-0.637239\pi\)
0.417914 0.908487i \(-0.362761\pi\)
\(810\) 0 0
\(811\) 19357.2i 0.838129i 0.907956 + 0.419065i \(0.137642\pi\)
−0.907956 + 0.419065i \(0.862358\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 69089.2 2.96943
\(816\) 0 0
\(817\) −51611.7 −2.21012
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24774.0i 1.05313i 0.850135 + 0.526565i \(0.176520\pi\)
−0.850135 + 0.526565i \(0.823480\pi\)
\(822\) 0 0
\(823\) 10955.1i 0.463997i 0.972716 + 0.231999i \(0.0745265\pi\)
−0.972716 + 0.231999i \(0.925474\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15446.2 0.649475 0.324738 0.945804i \(-0.394724\pi\)
0.324738 + 0.945804i \(0.394724\pi\)
\(828\) 0 0
\(829\) −4599.69 −0.192707 −0.0963533 0.995347i \(-0.530718\pi\)
−0.0963533 + 0.995347i \(0.530718\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5158.90i 0.214580i
\(834\) 0 0
\(835\) − 10154.3i − 0.420844i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26299.3 1.08219 0.541093 0.840963i \(-0.318011\pi\)
0.541093 + 0.840963i \(0.318011\pi\)
\(840\) 0 0
\(841\) 15721.3 0.644606
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 25332.3i − 1.03131i
\(846\) 0 0
\(847\) − 1945.97i − 0.0789426i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −20118.0 −0.810384
\(852\) 0 0
\(853\) −26223.5 −1.05261 −0.526305 0.850296i \(-0.676423\pi\)
−0.526305 + 0.850296i \(0.676423\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 13156.2i − 0.524396i −0.965014 0.262198i \(-0.915553\pi\)
0.965014 0.262198i \(-0.0844473\pi\)
\(858\) 0 0
\(859\) − 33557.9i − 1.33292i −0.745539 0.666462i \(-0.767808\pi\)
0.745539 0.666462i \(-0.232192\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16298.7 −0.642889 −0.321445 0.946928i \(-0.604168\pi\)
−0.321445 + 0.946928i \(0.604168\pi\)
\(864\) 0 0
\(865\) −36533.3 −1.43603
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 8467.44i − 0.330539i
\(870\) 0 0
\(871\) 9585.92i 0.372912i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 53536.2 2.06841
\(876\) 0 0
\(877\) 28884.4 1.11215 0.556075 0.831132i \(-0.312307\pi\)
0.556075 + 0.831132i \(0.312307\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 47233.0i − 1.80626i −0.429362 0.903132i \(-0.641262\pi\)
0.429362 0.903132i \(-0.358738\pi\)
\(882\) 0 0
\(883\) − 12998.5i − 0.495394i −0.968838 0.247697i \(-0.920326\pi\)
0.968838 0.247697i \(-0.0796738\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16177.5 −0.612386 −0.306193 0.951970i \(-0.599055\pi\)
−0.306193 + 0.951970i \(0.599055\pi\)
\(888\) 0 0
\(889\) 35572.6 1.34203
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 38489.5i 1.44233i
\(894\) 0 0
\(895\) − 58535.6i − 2.18618i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10368.1 −0.384646
\(900\) 0 0
\(901\) 10046.9 0.371487
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48106.0i 1.76696i
\(906\) 0 0
\(907\) 41803.5i 1.53039i 0.643800 + 0.765194i \(0.277357\pi\)
−0.643800 + 0.765194i \(0.722643\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25834.2 0.939543 0.469772 0.882788i \(-0.344336\pi\)
0.469772 + 0.882788i \(0.344336\pi\)
\(912\) 0 0
\(913\) −16530.4 −0.599207
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1204.51i − 0.0433765i
\(918\) 0 0
\(919\) 18742.0i 0.672734i 0.941731 + 0.336367i \(0.109198\pi\)
−0.941731 + 0.336367i \(0.890802\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3221.41 0.114880
\(924\) 0 0
\(925\) −45422.8 −1.61459
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 9052.31i − 0.319695i −0.987142 0.159847i \(-0.948900\pi\)
0.987142 0.159847i \(-0.0511002\pi\)
\(930\) 0 0
\(931\) 18066.8i 0.635999i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −24803.4 −0.867547
\(936\) 0 0
\(937\) −28189.2 −0.982819 −0.491409 0.870929i \(-0.663518\pi\)
−0.491409 + 0.870929i \(0.663518\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 11490.8i − 0.398074i −0.979992 0.199037i \(-0.936219\pi\)
0.979992 0.199037i \(-0.0637814\pi\)
\(942\) 0 0
\(943\) 1158.32i 0.0400001i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15309.8 0.525344 0.262672 0.964885i \(-0.415396\pi\)
0.262672 + 0.964885i \(0.415396\pi\)
\(948\) 0 0
\(949\) 13872.1 0.474508
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9896.90i 0.336403i 0.985753 + 0.168201i \(0.0537959\pi\)
−0.985753 + 0.168201i \(0.946204\pi\)
\(954\) 0 0
\(955\) − 70380.1i − 2.38476i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9729.69 −0.327620
\(960\) 0 0
\(961\) 17388.9 0.583695
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11715.3i 0.390807i
\(966\) 0 0
\(967\) − 16779.4i − 0.558003i −0.960291 0.279001i \(-0.909997\pi\)
0.960291 0.279001i \(-0.0900034\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53400.1 1.76487 0.882437 0.470431i \(-0.155902\pi\)
0.882437 + 0.470431i \(0.155902\pi\)
\(972\) 0 0
\(973\) −10971.6 −0.361494
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 9474.13i − 0.310240i −0.987896 0.155120i \(-0.950424\pi\)
0.987896 0.155120i \(-0.0495764\pi\)
\(978\) 0 0
\(979\) − 48915.4i − 1.59688i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −56693.6 −1.83952 −0.919758 0.392485i \(-0.871615\pi\)
−0.919758 + 0.392485i \(0.871615\pi\)
\(984\) 0 0
\(985\) −349.050 −0.0112910
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 58637.1i − 1.88529i
\(990\) 0 0
\(991\) − 38944.6i − 1.24835i −0.781285 0.624175i \(-0.785435\pi\)
0.781285 0.624175i \(-0.214565\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −57542.9 −1.83340
\(996\) 0 0
\(997\) 33995.1 1.07987 0.539937 0.841705i \(-0.318448\pi\)
0.539937 + 0.841705i \(0.318448\pi\)
\(998\) 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 1728.4.c.j.1727.12 12
3.2 odd 2 inner 1728.4.c.j.1727.2 12
4.3 odd 2 inner 1728.4.c.j.1727.11 12
8.3 odd 2 108.4.b.b.107.12 yes 12
8.5 even 2 108.4.b.b.107.2 yes 12
12.11 even 2 inner 1728.4.c.j.1727.1 12
24.5 odd 2 108.4.b.b.107.11 yes 12
24.11 even 2 108.4.b.b.107.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.1 12 24.11 even 2
108.4.b.b.107.2 yes 12 8.5 even 2
108.4.b.b.107.11 yes 12 24.5 odd 2
108.4.b.b.107.12 yes 12 8.3 odd 2
1728.4.c.j.1727.1 12 12.11 even 2 inner
1728.4.c.j.1727.2 12 3.2 odd 2 inner
1728.4.c.j.1727.11 12 4.3 odd 2 inner
1728.4.c.j.1727.12 12 1.1 even 1 trivial