Properties

Label 1728.4.c.j.1727.1
Level $1728$
Weight $4$
Character 1728.1727
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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

$q$-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 \(-0.617723\pi\)
0.361464 0.932386i \(-0.382277\pi\)
\(6\) 0 0
\(7\) − 13.9048i − 0.750791i −0.926865 0.375395i \(-0.877507\pi\)
0.926865 0.375395i \(-0.122493\pi\)
\(8\) 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 \(-0.920916\pi\)
0.969295 0.245902i \(-0.0790840\pi\)
\(18\) 0 0
\(19\) − 120.723i − 1.45767i −0.684691 0.728833i \(-0.740063\pi\)
0.684691 0.728833i \(-0.259937\pi\)
\(20\) 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 \(-0.903656\pi\)
0.954543 0.298075i \(-0.0963444\pi\)
\(30\) 0 0
\(31\) − 111.365i − 0.645217i −0.946532 0.322609i \(-0.895440\pi\)
0.946532 0.322609i \(-0.104560\pi\)
\(32\) 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 \(0.00512011\pi\)
−0.999871 + 0.0160846i \(0.994880\pi\)
\(42\) 0 0
\(43\) − 427.523i − 1.51620i −0.652137 0.758101i \(-0.726127\pi\)
0.652137 0.758101i \(-0.273873\pi\)
\(44\) 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 \(0.123278\pi\)
−0.925937 + 0.377679i \(0.876722\pi\)
\(54\) 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 \(-0.910031\pi\)
0.960321 0.278899i \(-0.0899694\pi\)
\(68\) 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 \(0.0558986\pi\)
−0.984620 + 0.174709i \(0.944101\pi\)
\(80\) 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 \(0.319832\pi\)
−0.536273 + 0.844045i \(0.680168\pi\)
\(90\) 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 \(-0.876658\pi\)
0.925860 0.377866i \(-0.123342\pi\)
\(102\) 0 0
\(103\) − 1202.04i − 1.14991i −0.818187 0.574953i \(-0.805021\pi\)
0.818187 0.574953i \(-0.194979\pi\)
\(104\) 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 \(0.00926407\pi\)
−0.999577 + 0.0290998i \(0.990736\pi\)
\(114\) 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 \(0.351932\pi\)
−0.448574 + 0.893746i \(0.648068\pi\)
\(128\) 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 \(-0.929987\pi\)
0.975908 0.218184i \(-0.0700131\pi\)
\(138\) 0 0
\(139\) − 789.049i − 0.481484i −0.970589 0.240742i \(-0.922609\pi\)
0.970589 0.240742i \(-0.0773908\pi\)
\(140\) 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 \(-0.855699\pi\)
0.898992 0.437965i \(-0.144301\pi\)
\(150\) 0 0
\(151\) 1502.12i 0.809544i 0.914418 + 0.404772i \(0.132649\pi\)
−0.914418 + 0.404772i \(0.867351\pi\)
\(152\) 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 \(0.293156\pi\)
−0.605044 + 0.796192i \(0.706844\pi\)
\(164\) 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 \(-0.874187\pi\)
0.922899 0.385043i \(-0.125813\pi\)
\(174\) 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 \(-0.999036\pi\)
0.999995 0.00302746i \(-0.000963671\pi\)
\(198\) 0 0
\(199\) − 2760.01i − 0.983176i −0.870828 0.491588i \(-0.836417\pi\)
0.870828 0.491588i \(-0.163583\pi\)
\(200\) 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 \(0.149042\pi\)
−0.892368 + 0.451308i \(0.850958\pi\)
\(212\) 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 \(0.00528786\pi\)
−0.999862 + 0.0166115i \(0.994712\pi\)
\(224\) 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 \(0.220676\pi\)
−0.769157 + 0.639060i \(0.779324\pi\)
\(234\) 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 \(0.0829871\pi\)
−0.966207 + 0.257768i \(0.917013\pi\)
\(258\) 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 \(-0.991273\pi\)
0.999624 0.0274125i \(-0.00872675\pi\)
\(270\) 0 0
\(271\) 828.799i 0.185778i 0.995676 + 0.0928892i \(0.0296102\pi\)
−0.995676 + 0.0928892i \(0.970390\pi\)
\(272\) 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 \(0.105156\pi\)
−0.945926 + 0.324382i \(0.894844\pi\)
\(282\) 0 0
\(283\) − 196.045i − 0.0411790i −0.999788 0.0205895i \(-0.993446\pi\)
0.999788 0.0205895i \(-0.00655430\pi\)
\(284\) 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 \(0.337320\pi\)
−0.489114 + 0.872220i \(0.662680\pi\)
\(294\) 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 \(0.0624149\pi\)
−0.980837 + 0.194828i \(0.937585\pi\)
\(308\) 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 \(0.126124\pi\)
−0.922522 + 0.385945i \(0.873876\pi\)
\(318\) 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 \(-0.723628\pi\)
0.646164 0.763198i \(-0.276372\pi\)
\(332\) 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 \(0.109154\pi\)
−0.941778 + 0.336236i \(0.890846\pi\)
\(354\) 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 \(-0.973792\pi\)
0.996612 0.0822411i \(-0.0262078\pi\)
\(368\) 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 \(0.155407\pi\)
−0.883167 + 0.469059i \(0.844593\pi\)
\(380\) 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 \(0.0930291\pi\)
−0.957595 + 0.288117i \(0.906971\pi\)
\(390\) 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 \(0.319048\pi\)
−0.538350 + 0.842722i \(0.680952\pi\)
\(402\) 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 \(0.172416\pi\)
−0.856854 + 0.515560i \(0.827584\pi\)
\(440\) 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 \(-0.838029\pi\)
0.873307 0.487170i \(-0.161971\pi\)
\(450\) 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 \(0.181095\pi\)
−0.842479 + 0.538729i \(0.818905\pi\)
\(462\) 0 0
\(463\) − 1963.48i − 0.197086i −0.995133 0.0985428i \(-0.968582\pi\)
0.995133 0.0985428i \(-0.0314181\pi\)
\(464\) 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 \(0.265212\pi\)
−0.672519 + 0.740080i \(0.734788\pi\)
\(488\) 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 \(0.0768660\pi\)
−0.970985 + 0.239142i \(0.923134\pi\)
\(500\) 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 \(-0.785535\pi\)
0.781481 0.623929i \(-0.214465\pi\)
\(510\) 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 \(-0.849624\pi\)
0.890469 0.455043i \(-0.150376\pi\)
\(522\) 0 0
\(523\) − 12111.8i − 1.01265i −0.862344 0.506323i \(-0.831005\pi\)
0.862344 0.506323i \(-0.168995\pi\)
\(524\) 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 \(-0.754363\pi\)
0.716731 0.697350i \(-0.245637\pi\)
\(548\) 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 \(0.136632\pi\)
−0.909281 + 0.416183i \(0.863368\pi\)
\(558\) 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 \(-0.853615\pi\)
0.896105 0.443842i \(-0.146385\pi\)
\(570\) 0 0
\(571\) 24916.7i 1.82615i 0.407792 + 0.913075i \(0.366299\pi\)
−0.407792 + 0.913075i \(0.633701\pi\)
\(572\) 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 \(-0.620412\pi\)
0.369328 0.929299i \(-0.379588\pi\)
\(594\) 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 \(-0.746461\pi\)
0.699202 0.714924i \(-0.253539\pi\)
\(608\) 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 \(-0.971899\pi\)
0.996106 0.0881667i \(-0.0281008\pi\)
\(618\) 0 0
\(619\) − 6967.79i − 0.452438i −0.974076 0.226219i \(-0.927364\pi\)
0.974076 0.226219i \(-0.0726365\pi\)
\(620\) 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 \(-0.792880\pi\)
0.795667 0.605734i \(-0.207120\pi\)
\(632\) 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 \(0.258860\pi\)
−0.687155 + 0.726511i \(0.741140\pi\)
\(642\) 0 0
\(643\) − 25167.2i − 1.54354i −0.635899 0.771772i \(-0.719371\pi\)
0.635899 0.771772i \(-0.280629\pi\)
\(644\) 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 \(-0.890266\pi\)
0.941163 0.337953i \(-0.109734\pi\)
\(654\) 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 \(-0.895368\pi\)
0.946459 0.322824i \(-0.104632\pi\)
\(678\) 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 \(-0.918071\pi\)
0.967059 0.254554i \(-0.0819286\pi\)
\(692\) 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 \(0.127257\pi\)
−0.921143 + 0.389224i \(0.872743\pi\)
\(702\) 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 \(0.0538899\pi\)
−0.985703 + 0.168492i \(0.946110\pi\)
\(728\) 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 \(-0.675740\pi\)
0.524479 0.851423i \(-0.324260\pi\)
\(740\) 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 \(-0.764253\pi\)
0.738049 0.674747i \(-0.235747\pi\)
\(752\) 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 \(-0.858452\pi\)
0.902746 0.430175i \(-0.141548\pi\)
\(762\) 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 \(0.0635029\pi\)
−0.980166 + 0.198179i \(0.936497\pi\)
\(774\) 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 \(-0.909609\pi\)
0.959951 0.280169i \(-0.0903906\pi\)
\(788\) 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 \(-0.684184\pi\)
0.546877 0.837213i \(-0.315816\pi\)
\(798\) 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 \(0.362761\pi\)
−0.417914 + 0.908487i \(0.637239\pi\)
\(810\) 0 0
\(811\) − 19357.2i − 0.838129i −0.907956 0.419065i \(-0.862358\pi\)
0.907956 0.419065i \(-0.137642\pi\)
\(812\) 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 \(-0.823480\pi\)
0.850135 0.526565i \(-0.176520\pi\)
\(822\) 0 0
\(823\) − 10955.1i − 0.463997i −0.972716 0.231999i \(-0.925474\pi\)
0.972716 0.231999i \(-0.0745265\pi\)
\(824\) 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 \(0.0844473\pi\)
−0.965014 + 0.262198i \(0.915553\pi\)
\(858\) 0 0
\(859\) 33557.9i 1.33292i 0.745539 + 0.666462i \(0.232192\pi\)
−0.745539 + 0.666462i \(0.767808\pi\)
\(860\) 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 \(0.358738\pi\)
−0.429362 + 0.903132i \(0.641262\pi\)
\(882\) 0 0
\(883\) 12998.5i 0.495394i 0.968838 + 0.247697i \(0.0796738\pi\)
−0.968838 + 0.247697i \(0.920326\pi\)
\(884\) 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 \(-0.722643\pi\)
0.643800 0.765194i \(-0.277357\pi\)
\(908\) 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 \(-0.890802\pi\)
0.941731 0.336367i \(-0.109198\pi\)
\(920\) 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 \(0.0511002\pi\)
−0.987142 + 0.159847i \(0.948900\pi\)
\(930\) 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 \(0.0637814\pi\)
−0.979992 + 0.199037i \(0.936219\pi\)
\(942\) 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 \(-0.946204\pi\)
0.985753 0.168201i \(-0.0537959\pi\)
\(954\) 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 \(0.0900034\pi\)
−0.960291 + 0.279001i \(0.909997\pi\)
\(968\) 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 \(0.0495764\pi\)
−0.987896 + 0.155120i \(0.950424\pi\)
\(978\) 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 \(0.214565\pi\)
−0.781285 + 0.624175i \(0.785435\pi\)
\(992\) 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.1727.1 12
3.2 odd 2 inner 1728.4.c.j.1727.11 12
4.3 odd 2 inner 1728.4.c.j.1727.2 12
8.3 odd 2 108.4.b.b.107.11 yes 12
8.5 even 2 108.4.b.b.107.1 12
12.11 even 2 inner 1728.4.c.j.1727.12 12
24.5 odd 2 108.4.b.b.107.12 yes 12
24.11 even 2 108.4.b.b.107.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.1 12 8.5 even 2
108.4.b.b.107.2 yes 12 24.11 even 2
108.4.b.b.107.11 yes 12 8.3 odd 2
108.4.b.b.107.12 yes 12 24.5 odd 2
1728.4.c.j.1727.1 12 1.1 even 1 trivial
1728.4.c.j.1727.2 12 4.3 odd 2 inner
1728.4.c.j.1727.11 12 3.2 odd 2 inner
1728.4.c.j.1727.12 12 12.11 even 2 inner