Properties

Label 1620.4.a.h.1.1
Level $1620$
Weight $4$
Character 1620.1
Self dual yes
Analytic conductor $95.583$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,4,Mod(1,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(95.5830942093\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.438516.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.55649\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -32.7265 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} -32.7265 q^{7} -16.1055 q^{11} +18.4043 q^{13} -61.5584 q^{17} +118.317 q^{19} +132.682 q^{23} +25.0000 q^{25} +240.769 q^{29} +36.4265 q^{31} -163.632 q^{35} +65.7676 q^{37} -151.078 q^{41} +293.221 q^{43} -354.799 q^{47} +728.021 q^{49} -708.587 q^{53} -80.5277 q^{55} +220.003 q^{59} -893.735 q^{61} +92.0215 q^{65} +31.0366 q^{67} -196.045 q^{71} -975.957 q^{73} +527.077 q^{77} -844.463 q^{79} -12.8533 q^{83} -307.792 q^{85} +892.671 q^{89} -602.308 q^{91} +591.583 q^{95} -757.518 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} - 13 q^{7} - 57 q^{11} + 14 q^{13} - 3 q^{17} - 31 q^{19} + 69 q^{23} + 100 q^{25} + 69 q^{29} - 58 q^{31} - 65 q^{35} - 388 q^{37} - 396 q^{41} + 371 q^{43} - 129 q^{47} + 111 q^{49} - 1356 q^{53} - 285 q^{55} - 15 q^{59} - 1441 q^{61} + 70 q^{65} + 368 q^{67} - 168 q^{71} - 955 q^{73} - 342 q^{77} - 1408 q^{79} + 789 q^{83} - 15 q^{85} - 1617 q^{89} - 1406 q^{91} - 155 q^{95} - 1495 q^{97}+O(q^{100}) \) 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
\(7\) −32.7265 −1.76706 −0.883531 0.468372i \(-0.844841\pi\)
−0.883531 + 0.468372i \(0.844841\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.1055 −0.441455 −0.220727 0.975336i \(-0.570843\pi\)
−0.220727 + 0.975336i \(0.570843\pi\)
\(12\) 0 0
\(13\) 18.4043 0.392649 0.196324 0.980539i \(-0.437099\pi\)
0.196324 + 0.980539i \(0.437099\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −61.5584 −0.878242 −0.439121 0.898428i \(-0.644710\pi\)
−0.439121 + 0.898428i \(0.644710\pi\)
\(18\) 0 0
\(19\) 118.317 1.42862 0.714308 0.699831i \(-0.246742\pi\)
0.714308 + 0.699831i \(0.246742\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 132.682 1.20287 0.601436 0.798921i \(-0.294595\pi\)
0.601436 + 0.798921i \(0.294595\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 240.769 1.54171 0.770856 0.637009i \(-0.219829\pi\)
0.770856 + 0.637009i \(0.219829\pi\)
\(30\) 0 0
\(31\) 36.4265 0.211045 0.105522 0.994417i \(-0.466348\pi\)
0.105522 + 0.994417i \(0.466348\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −163.632 −0.790254
\(36\) 0 0
\(37\) 65.7676 0.292220 0.146110 0.989268i \(-0.453325\pi\)
0.146110 + 0.989268i \(0.453325\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −151.078 −0.575473 −0.287737 0.957710i \(-0.592903\pi\)
−0.287737 + 0.957710i \(0.592903\pi\)
\(42\) 0 0
\(43\) 293.221 1.03990 0.519951 0.854196i \(-0.325950\pi\)
0.519951 + 0.854196i \(0.325950\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −354.799 −1.10112 −0.550561 0.834795i \(-0.685586\pi\)
−0.550561 + 0.834795i \(0.685586\pi\)
\(48\) 0 0
\(49\) 728.021 2.12251
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −708.587 −1.83645 −0.918225 0.396059i \(-0.870378\pi\)
−0.918225 + 0.396059i \(0.870378\pi\)
\(54\) 0 0
\(55\) −80.5277 −0.197424
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 220.003 0.485457 0.242729 0.970094i \(-0.421958\pi\)
0.242729 + 0.970094i \(0.421958\pi\)
\(60\) 0 0
\(61\) −893.735 −1.87592 −0.937960 0.346744i \(-0.887287\pi\)
−0.937960 + 0.346744i \(0.887287\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 92.0215 0.175598
\(66\) 0 0
\(67\) 31.0366 0.0565929 0.0282965 0.999600i \(-0.490992\pi\)
0.0282965 + 0.999600i \(0.490992\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −196.045 −0.327694 −0.163847 0.986486i \(-0.552390\pi\)
−0.163847 + 0.986486i \(0.552390\pi\)
\(72\) 0 0
\(73\) −975.957 −1.56475 −0.782377 0.622805i \(-0.785993\pi\)
−0.782377 + 0.622805i \(0.785993\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 527.077 0.780078
\(78\) 0 0
\(79\) −844.463 −1.20265 −0.601326 0.799004i \(-0.705361\pi\)
−0.601326 + 0.799004i \(0.705361\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.8533 −0.0169980 −0.00849900 0.999964i \(-0.502705\pi\)
−0.00849900 + 0.999964i \(0.502705\pi\)
\(84\) 0 0
\(85\) −307.792 −0.392762
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 892.671 1.06318 0.531590 0.847002i \(-0.321595\pi\)
0.531590 + 0.847002i \(0.321595\pi\)
\(90\) 0 0
\(91\) −602.308 −0.693835
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 591.583 0.638897
\(96\) 0 0
\(97\) −757.518 −0.792931 −0.396466 0.918050i \(-0.629763\pi\)
−0.396466 + 0.918050i \(0.629763\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 35.2237 0.0347019 0.0173509 0.999849i \(-0.494477\pi\)
0.0173509 + 0.999849i \(0.494477\pi\)
\(102\) 0 0
\(103\) 610.290 0.583822 0.291911 0.956446i \(-0.405709\pi\)
0.291911 + 0.956446i \(0.405709\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1094.01 −0.988430 −0.494215 0.869340i \(-0.664545\pi\)
−0.494215 + 0.869340i \(0.664545\pi\)
\(108\) 0 0
\(109\) −1818.81 −1.59826 −0.799129 0.601159i \(-0.794706\pi\)
−0.799129 + 0.601159i \(0.794706\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −961.999 −0.800861 −0.400430 0.916327i \(-0.631139\pi\)
−0.400430 + 0.916327i \(0.631139\pi\)
\(114\) 0 0
\(115\) 663.409 0.537941
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2014.59 1.55191
\(120\) 0 0
\(121\) −1071.61 −0.805118
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −833.065 −0.582067 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −562.201 −0.374960 −0.187480 0.982268i \(-0.560032\pi\)
−0.187480 + 0.982268i \(0.560032\pi\)
\(132\) 0 0
\(133\) −3872.09 −2.52445
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1061.62 −0.662045 −0.331022 0.943623i \(-0.607394\pi\)
−0.331022 + 0.943623i \(0.607394\pi\)
\(138\) 0 0
\(139\) −212.848 −0.129882 −0.0649409 0.997889i \(-0.520686\pi\)
−0.0649409 + 0.997889i \(0.520686\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −296.411 −0.173337
\(144\) 0 0
\(145\) 1203.84 0.689475
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1758.09 0.966631 0.483315 0.875446i \(-0.339432\pi\)
0.483315 + 0.875446i \(0.339432\pi\)
\(150\) 0 0
\(151\) 1672.32 0.901268 0.450634 0.892709i \(-0.351198\pi\)
0.450634 + 0.892709i \(0.351198\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 182.133 0.0943822
\(156\) 0 0
\(157\) −2654.46 −1.34936 −0.674679 0.738111i \(-0.735718\pi\)
−0.674679 + 0.738111i \(0.735718\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4342.21 −2.12555
\(162\) 0 0
\(163\) 1825.20 0.877058 0.438529 0.898717i \(-0.355500\pi\)
0.438529 + 0.898717i \(0.355500\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4234.49 1.96213 0.981063 0.193691i \(-0.0620460\pi\)
0.981063 + 0.193691i \(0.0620460\pi\)
\(168\) 0 0
\(169\) −1858.28 −0.845827
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2327.00 −1.02265 −0.511326 0.859387i \(-0.670846\pi\)
−0.511326 + 0.859387i \(0.670846\pi\)
\(174\) 0 0
\(175\) −818.161 −0.353412
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3754.31 1.56766 0.783828 0.620978i \(-0.213264\pi\)
0.783828 + 0.620978i \(0.213264\pi\)
\(180\) 0 0
\(181\) 44.8828 0.0184315 0.00921577 0.999958i \(-0.497066\pi\)
0.00921577 + 0.999958i \(0.497066\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 328.838 0.130685
\(186\) 0 0
\(187\) 991.431 0.387704
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4536.78 1.71869 0.859346 0.511394i \(-0.170871\pi\)
0.859346 + 0.511394i \(0.170871\pi\)
\(192\) 0 0
\(193\) 240.123 0.0895567 0.0447783 0.998997i \(-0.485742\pi\)
0.0447783 + 0.998997i \(0.485742\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3708.98 1.34139 0.670694 0.741734i \(-0.265996\pi\)
0.670694 + 0.741734i \(0.265996\pi\)
\(198\) 0 0
\(199\) −975.782 −0.347595 −0.173797 0.984781i \(-0.555604\pi\)
−0.173797 + 0.984781i \(0.555604\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7879.51 −2.72430
\(204\) 0 0
\(205\) −755.390 −0.257360
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1905.55 −0.630669
\(210\) 0 0
\(211\) −5494.82 −1.79279 −0.896396 0.443255i \(-0.853824\pi\)
−0.896396 + 0.443255i \(0.853824\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1466.11 0.465058
\(216\) 0 0
\(217\) −1192.11 −0.372930
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1132.94 −0.344841
\(222\) 0 0
\(223\) 4116.81 1.23624 0.618121 0.786083i \(-0.287894\pi\)
0.618121 + 0.786083i \(0.287894\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3360.00 −0.982429 −0.491214 0.871039i \(-0.663447\pi\)
−0.491214 + 0.871039i \(0.663447\pi\)
\(228\) 0 0
\(229\) −8.31590 −0.00239969 −0.00119985 0.999999i \(-0.500382\pi\)
−0.00119985 + 0.999999i \(0.500382\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1751.22 0.492386 0.246193 0.969221i \(-0.420820\pi\)
0.246193 + 0.969221i \(0.420820\pi\)
\(234\) 0 0
\(235\) −1773.99 −0.492436
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4681.54 −1.26704 −0.633522 0.773724i \(-0.718392\pi\)
−0.633522 + 0.773724i \(0.718392\pi\)
\(240\) 0 0
\(241\) 5198.33 1.38944 0.694718 0.719282i \(-0.255529\pi\)
0.694718 + 0.719282i \(0.255529\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3640.10 0.949215
\(246\) 0 0
\(247\) 2177.54 0.560945
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3667.47 0.922265 0.461133 0.887331i \(-0.347443\pi\)
0.461133 + 0.887331i \(0.347443\pi\)
\(252\) 0 0
\(253\) −2136.91 −0.531014
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3036.67 −0.737052 −0.368526 0.929617i \(-0.620137\pi\)
−0.368526 + 0.929617i \(0.620137\pi\)
\(258\) 0 0
\(259\) −2152.34 −0.516370
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7370.91 −1.72817 −0.864087 0.503342i \(-0.832103\pi\)
−0.864087 + 0.503342i \(0.832103\pi\)
\(264\) 0 0
\(265\) −3542.93 −0.821286
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1442.57 0.326970 0.163485 0.986546i \(-0.447726\pi\)
0.163485 + 0.986546i \(0.447726\pi\)
\(270\) 0 0
\(271\) −1068.01 −0.239400 −0.119700 0.992810i \(-0.538193\pi\)
−0.119700 + 0.992810i \(0.538193\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −402.638 −0.0882909
\(276\) 0 0
\(277\) −8644.61 −1.87511 −0.937553 0.347843i \(-0.886914\pi\)
−0.937553 + 0.347843i \(0.886914\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1874.83 −0.398017 −0.199008 0.979998i \(-0.563772\pi\)
−0.199008 + 0.979998i \(0.563772\pi\)
\(282\) 0 0
\(283\) −6463.16 −1.35758 −0.678790 0.734332i \(-0.737495\pi\)
−0.678790 + 0.734332i \(0.737495\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4944.25 1.01690
\(288\) 0 0
\(289\) −1123.56 −0.228691
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6949.70 1.38569 0.692843 0.721089i \(-0.256358\pi\)
0.692843 + 0.721089i \(0.256358\pi\)
\(294\) 0 0
\(295\) 1100.02 0.217103
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2441.92 0.472307
\(300\) 0 0
\(301\) −9596.08 −1.83757
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4468.68 −0.838937
\(306\) 0 0
\(307\) −4561.23 −0.847958 −0.423979 0.905672i \(-0.639367\pi\)
−0.423979 + 0.905672i \(0.639367\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4151.60 −0.756964 −0.378482 0.925609i \(-0.623554\pi\)
−0.378482 + 0.925609i \(0.623554\pi\)
\(312\) 0 0
\(313\) −3165.42 −0.571630 −0.285815 0.958285i \(-0.592264\pi\)
−0.285815 + 0.958285i \(0.592264\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6523.81 −1.15588 −0.577939 0.816080i \(-0.696143\pi\)
−0.577939 + 0.816080i \(0.696143\pi\)
\(318\) 0 0
\(319\) −3877.71 −0.680596
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7283.39 −1.25467
\(324\) 0 0
\(325\) 460.108 0.0785298
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11611.3 1.94575
\(330\) 0 0
\(331\) −5420.61 −0.900131 −0.450066 0.892995i \(-0.648599\pi\)
−0.450066 + 0.892995i \(0.648599\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 155.183 0.0253091
\(336\) 0 0
\(337\) 5563.85 0.899354 0.449677 0.893191i \(-0.351539\pi\)
0.449677 + 0.893191i \(0.351539\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −586.668 −0.0931668
\(342\) 0 0
\(343\) −12600.4 −1.98354
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10236.1 −1.58358 −0.791790 0.610793i \(-0.790851\pi\)
−0.791790 + 0.610793i \(0.790851\pi\)
\(348\) 0 0
\(349\) −1500.29 −0.230111 −0.115056 0.993359i \(-0.536705\pi\)
−0.115056 + 0.993359i \(0.536705\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6370.82 0.960580 0.480290 0.877110i \(-0.340531\pi\)
0.480290 + 0.877110i \(0.340531\pi\)
\(354\) 0 0
\(355\) −980.226 −0.146549
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2837.15 0.417100 0.208550 0.978012i \(-0.433126\pi\)
0.208550 + 0.978012i \(0.433126\pi\)
\(360\) 0 0
\(361\) 7139.84 1.04094
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4879.78 −0.699780
\(366\) 0 0
\(367\) −4703.76 −0.669030 −0.334515 0.942390i \(-0.608573\pi\)
−0.334515 + 0.942390i \(0.608573\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 23189.5 3.24512
\(372\) 0 0
\(373\) −5076.15 −0.704647 −0.352323 0.935878i \(-0.614608\pi\)
−0.352323 + 0.935878i \(0.614608\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4431.19 0.605352
\(378\) 0 0
\(379\) 559.935 0.0758890 0.0379445 0.999280i \(-0.487919\pi\)
0.0379445 + 0.999280i \(0.487919\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5773.83 0.770310 0.385155 0.922852i \(-0.374148\pi\)
0.385155 + 0.922852i \(0.374148\pi\)
\(384\) 0 0
\(385\) 2635.38 0.348861
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12668.3 −1.65118 −0.825590 0.564271i \(-0.809157\pi\)
−0.825590 + 0.564271i \(0.809157\pi\)
\(390\) 0 0
\(391\) −8167.69 −1.05641
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4222.32 −0.537843
\(396\) 0 0
\(397\) −1138.65 −0.143947 −0.0719736 0.997407i \(-0.522930\pi\)
−0.0719736 + 0.997407i \(0.522930\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3371.61 0.419875 0.209938 0.977715i \(-0.432674\pi\)
0.209938 + 0.977715i \(0.432674\pi\)
\(402\) 0 0
\(403\) 670.405 0.0828666
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1059.22 −0.129002
\(408\) 0 0
\(409\) 2098.28 0.253676 0.126838 0.991923i \(-0.459517\pi\)
0.126838 + 0.991923i \(0.459517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7199.92 −0.857833
\(414\) 0 0
\(415\) −64.2666 −0.00760174
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1978.06 0.230631 0.115316 0.993329i \(-0.463212\pi\)
0.115316 + 0.993329i \(0.463212\pi\)
\(420\) 0 0
\(421\) −1647.35 −0.190706 −0.0953529 0.995444i \(-0.530398\pi\)
−0.0953529 + 0.995444i \(0.530398\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1538.96 −0.175648
\(426\) 0 0
\(427\) 29248.8 3.31487
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9716.13 −1.08587 −0.542934 0.839775i \(-0.682687\pi\)
−0.542934 + 0.839775i \(0.682687\pi\)
\(432\) 0 0
\(433\) 12751.3 1.41522 0.707610 0.706604i \(-0.249774\pi\)
0.707610 + 0.706604i \(0.249774\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15698.5 1.71844
\(438\) 0 0
\(439\) 1135.01 0.123396 0.0616980 0.998095i \(-0.480348\pi\)
0.0616980 + 0.998095i \(0.480348\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1587.02 0.170207 0.0851033 0.996372i \(-0.472878\pi\)
0.0851033 + 0.996372i \(0.472878\pi\)
\(444\) 0 0
\(445\) 4463.36 0.475468
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11255.5 −1.18302 −0.591512 0.806296i \(-0.701469\pi\)
−0.591512 + 0.806296i \(0.701469\pi\)
\(450\) 0 0
\(451\) 2433.19 0.254045
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3011.54 −0.310293
\(456\) 0 0
\(457\) −13549.3 −1.38689 −0.693444 0.720511i \(-0.743907\pi\)
−0.693444 + 0.720511i \(0.743907\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1628.83 0.164560 0.0822800 0.996609i \(-0.473780\pi\)
0.0822800 + 0.996609i \(0.473780\pi\)
\(462\) 0 0
\(463\) 6960.47 0.698662 0.349331 0.936999i \(-0.386409\pi\)
0.349331 + 0.936999i \(0.386409\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1221.07 0.120994 0.0604970 0.998168i \(-0.480731\pi\)
0.0604970 + 0.998168i \(0.480731\pi\)
\(468\) 0 0
\(469\) −1015.72 −0.100003
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4722.48 −0.459070
\(474\) 0 0
\(475\) 2957.92 0.285723
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16552.1 −1.57888 −0.789442 0.613826i \(-0.789630\pi\)
−0.789442 + 0.613826i \(0.789630\pi\)
\(480\) 0 0
\(481\) 1210.41 0.114740
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3787.59 −0.354610
\(486\) 0 0
\(487\) −3739.16 −0.347921 −0.173960 0.984753i \(-0.555656\pi\)
−0.173960 + 0.984753i \(0.555656\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15633.3 1.43691 0.718453 0.695576i \(-0.244851\pi\)
0.718453 + 0.695576i \(0.244851\pi\)
\(492\) 0 0
\(493\) −14821.4 −1.35400
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6415.86 0.579056
\(498\) 0 0
\(499\) 9221.21 0.827251 0.413625 0.910447i \(-0.364262\pi\)
0.413625 + 0.910447i \(0.364262\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15565.7 1.37980 0.689901 0.723903i \(-0.257654\pi\)
0.689901 + 0.723903i \(0.257654\pi\)
\(504\) 0 0
\(505\) 176.119 0.0155192
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2653.76 −0.231092 −0.115546 0.993302i \(-0.536862\pi\)
−0.115546 + 0.993302i \(0.536862\pi\)
\(510\) 0 0
\(511\) 31939.6 2.76502
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3051.45 0.261093
\(516\) 0 0
\(517\) 5714.22 0.486095
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4708.79 −0.395962 −0.197981 0.980206i \(-0.563438\pi\)
−0.197981 + 0.980206i \(0.563438\pi\)
\(522\) 0 0
\(523\) −2586.36 −0.216240 −0.108120 0.994138i \(-0.534483\pi\)
−0.108120 + 0.994138i \(0.534483\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2242.36 −0.185349
\(528\) 0 0
\(529\) 5437.47 0.446903
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2780.49 −0.225959
\(534\) 0 0
\(535\) −5470.06 −0.442040
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11725.2 −0.936991
\(540\) 0 0
\(541\) −12767.5 −1.01464 −0.507319 0.861758i \(-0.669363\pi\)
−0.507319 + 0.861758i \(0.669363\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9094.04 −0.714763
\(546\) 0 0
\(547\) −1062.83 −0.0830770 −0.0415385 0.999137i \(-0.513226\pi\)
−0.0415385 + 0.999137i \(0.513226\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 28487.0 2.20252
\(552\) 0 0
\(553\) 27636.3 2.12516
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19202.4 1.46074 0.730369 0.683053i \(-0.239348\pi\)
0.730369 + 0.683053i \(0.239348\pi\)
\(558\) 0 0
\(559\) 5396.53 0.408316
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5879.05 0.440093 0.220046 0.975489i \(-0.429379\pi\)
0.220046 + 0.975489i \(0.429379\pi\)
\(564\) 0 0
\(565\) −4809.99 −0.358156
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9008.59 0.663726 0.331863 0.943328i \(-0.392323\pi\)
0.331863 + 0.943328i \(0.392323\pi\)
\(570\) 0 0
\(571\) 9763.61 0.715577 0.357788 0.933803i \(-0.383531\pi\)
0.357788 + 0.933803i \(0.383531\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3317.05 0.240575
\(576\) 0 0
\(577\) −20212.3 −1.45832 −0.729159 0.684344i \(-0.760089\pi\)
−0.729159 + 0.684344i \(0.760089\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 420.643 0.0300365
\(582\) 0 0
\(583\) 11412.2 0.810709
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3458.92 −0.243211 −0.121605 0.992579i \(-0.538804\pi\)
−0.121605 + 0.992579i \(0.538804\pi\)
\(588\) 0 0
\(589\) 4309.86 0.301502
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6880.71 0.476487 0.238243 0.971205i \(-0.423428\pi\)
0.238243 + 0.971205i \(0.423428\pi\)
\(594\) 0 0
\(595\) 10072.9 0.694034
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12564.8 −0.857071 −0.428536 0.903525i \(-0.640970\pi\)
−0.428536 + 0.903525i \(0.640970\pi\)
\(600\) 0 0
\(601\) 13392.7 0.908986 0.454493 0.890750i \(-0.349821\pi\)
0.454493 + 0.890750i \(0.349821\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5358.06 −0.360060
\(606\) 0 0
\(607\) −20209.4 −1.35136 −0.675679 0.737196i \(-0.736149\pi\)
−0.675679 + 0.737196i \(0.736149\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6529.82 −0.432354
\(612\) 0 0
\(613\) 16892.6 1.11303 0.556513 0.830839i \(-0.312139\pi\)
0.556513 + 0.830839i \(0.312139\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26113.6 1.70388 0.851939 0.523640i \(-0.175426\pi\)
0.851939 + 0.523640i \(0.175426\pi\)
\(618\) 0 0
\(619\) −8267.32 −0.536820 −0.268410 0.963305i \(-0.586498\pi\)
−0.268410 + 0.963305i \(0.586498\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −29214.0 −1.87870
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4048.55 −0.256639
\(630\) 0 0
\(631\) 29318.5 1.84968 0.924841 0.380353i \(-0.124198\pi\)
0.924841 + 0.380353i \(0.124198\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4165.32 −0.260308
\(636\) 0 0
\(637\) 13398.7 0.833401
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13322.0 0.820884 0.410442 0.911887i \(-0.365374\pi\)
0.410442 + 0.911887i \(0.365374\pi\)
\(642\) 0 0
\(643\) −13836.6 −0.848617 −0.424308 0.905518i \(-0.639483\pi\)
−0.424308 + 0.905518i \(0.639483\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2485.83 −0.151048 −0.0755239 0.997144i \(-0.524063\pi\)
−0.0755239 + 0.997144i \(0.524063\pi\)
\(648\) 0 0
\(649\) −3543.27 −0.214307
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28397.2 −1.70179 −0.850894 0.525338i \(-0.823939\pi\)
−0.850894 + 0.525338i \(0.823939\pi\)
\(654\) 0 0
\(655\) −2811.01 −0.167687
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21563.6 −1.27465 −0.637327 0.770593i \(-0.719960\pi\)
−0.637327 + 0.770593i \(0.719960\pi\)
\(660\) 0 0
\(661\) −29325.9 −1.72564 −0.862819 0.505513i \(-0.831303\pi\)
−0.862819 + 0.505513i \(0.831303\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −19360.4 −1.12897
\(666\) 0 0
\(667\) 31945.7 1.85448
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14394.1 0.828133
\(672\) 0 0
\(673\) −3823.52 −0.218998 −0.109499 0.993987i \(-0.534925\pi\)
−0.109499 + 0.993987i \(0.534925\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4502.69 −0.255617 −0.127808 0.991799i \(-0.540794\pi\)
−0.127808 + 0.991799i \(0.540794\pi\)
\(678\) 0 0
\(679\) 24790.9 1.40116
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25841.5 −1.44773 −0.723863 0.689944i \(-0.757635\pi\)
−0.723863 + 0.689944i \(0.757635\pi\)
\(684\) 0 0
\(685\) −5308.09 −0.296075
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −13041.0 −0.721080
\(690\) 0 0
\(691\) −25894.2 −1.42556 −0.712780 0.701387i \(-0.752564\pi\)
−0.712780 + 0.701387i \(0.752564\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1064.24 −0.0580849
\(696\) 0 0
\(697\) 9300.12 0.505405
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30603.6 1.64890 0.824452 0.565932i \(-0.191484\pi\)
0.824452 + 0.565932i \(0.191484\pi\)
\(702\) 0 0
\(703\) 7781.40 0.417470
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1152.75 −0.0613204
\(708\) 0 0
\(709\) −12245.5 −0.648646 −0.324323 0.945946i \(-0.605136\pi\)
−0.324323 + 0.945946i \(0.605136\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4833.14 0.253860
\(714\) 0 0
\(715\) −1482.06 −0.0775185
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16585.7 0.860282 0.430141 0.902762i \(-0.358464\pi\)
0.430141 + 0.902762i \(0.358464\pi\)
\(720\) 0 0
\(721\) −19972.6 −1.03165
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6019.22 0.308343
\(726\) 0 0
\(727\) 8074.43 0.411918 0.205959 0.978561i \(-0.433969\pi\)
0.205959 + 0.978561i \(0.433969\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −18050.2 −0.913286
\(732\) 0 0
\(733\) −12009.3 −0.605147 −0.302573 0.953126i \(-0.597846\pi\)
−0.302573 + 0.953126i \(0.597846\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −499.861 −0.0249832
\(738\) 0 0
\(739\) 8208.53 0.408600 0.204300 0.978908i \(-0.434508\pi\)
0.204300 + 0.978908i \(0.434508\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1904.13 0.0940185 0.0470093 0.998894i \(-0.485031\pi\)
0.0470093 + 0.998894i \(0.485031\pi\)
\(744\) 0 0
\(745\) 8790.43 0.432290
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 35803.1 1.74662
\(750\) 0 0
\(751\) −8188.19 −0.397858 −0.198929 0.980014i \(-0.563746\pi\)
−0.198929 + 0.980014i \(0.563746\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8361.60 0.403059
\(756\) 0 0
\(757\) 15107.8 0.725367 0.362684 0.931912i \(-0.381861\pi\)
0.362684 + 0.931912i \(0.381861\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −24723.7 −1.17770 −0.588851 0.808241i \(-0.700420\pi\)
−0.588851 + 0.808241i \(0.700420\pi\)
\(762\) 0 0
\(763\) 59523.1 2.82422
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4049.01 0.190614
\(768\) 0 0
\(769\) −34299.9 −1.60844 −0.804218 0.594335i \(-0.797415\pi\)
−0.804218 + 0.594335i \(0.797415\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8485.24 −0.394816 −0.197408 0.980321i \(-0.563252\pi\)
−0.197408 + 0.980321i \(0.563252\pi\)
\(774\) 0 0
\(775\) 910.663 0.0422090
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17875.0 −0.822131
\(780\) 0 0
\(781\) 3157.41 0.144662
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −13272.3 −0.603451
\(786\) 0 0
\(787\) −5230.11 −0.236891 −0.118446 0.992961i \(-0.537791\pi\)
−0.118446 + 0.992961i \(0.537791\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 31482.8 1.41517
\(792\) 0 0
\(793\) −16448.6 −0.736578
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6834.31 0.303744 0.151872 0.988400i \(-0.451470\pi\)
0.151872 + 0.988400i \(0.451470\pi\)
\(798\) 0 0
\(799\) 21840.8 0.967051
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15718.3 0.690768
\(804\) 0 0
\(805\) −21711.0 −0.950575
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32358.8 1.40627 0.703137 0.711055i \(-0.251782\pi\)
0.703137 + 0.711055i \(0.251782\pi\)
\(810\) 0 0
\(811\) 14804.3 0.640997 0.320499 0.947249i \(-0.396150\pi\)
0.320499 + 0.947249i \(0.396150\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9125.99 0.392232
\(816\) 0 0
\(817\) 34692.9 1.48562
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19769.7 −0.840398 −0.420199 0.907432i \(-0.638040\pi\)
−0.420199 + 0.907432i \(0.638040\pi\)
\(822\) 0 0
\(823\) 34322.1 1.45370 0.726848 0.686798i \(-0.240984\pi\)
0.726848 + 0.686798i \(0.240984\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9870.43 0.415028 0.207514 0.978232i \(-0.433463\pi\)
0.207514 + 0.978232i \(0.433463\pi\)
\(828\) 0 0
\(829\) −3911.55 −0.163876 −0.0819382 0.996637i \(-0.526111\pi\)
−0.0819382 + 0.996637i \(0.526111\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −44815.8 −1.86408
\(834\) 0 0
\(835\) 21172.5 0.877489
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8347.07 −0.343472 −0.171736 0.985143i \(-0.554938\pi\)
−0.171736 + 0.985143i \(0.554938\pi\)
\(840\) 0 0
\(841\) 33580.7 1.37688
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9291.41 −0.378265
\(846\) 0 0
\(847\) 35070.1 1.42269
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8726.16 0.351503
\(852\) 0 0
\(853\) −5782.37 −0.232104 −0.116052 0.993243i \(-0.537024\pi\)
−0.116052 + 0.993243i \(0.537024\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3636.13 −0.144933 −0.0724666 0.997371i \(-0.523087\pi\)
−0.0724666 + 0.997371i \(0.523087\pi\)
\(858\) 0 0
\(859\) 34374.1 1.36534 0.682671 0.730726i \(-0.260818\pi\)
0.682671 + 0.730726i \(0.260818\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28082.0 −1.10767 −0.553836 0.832626i \(-0.686837\pi\)
−0.553836 + 0.832626i \(0.686837\pi\)
\(864\) 0 0
\(865\) −11635.0 −0.457344
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13600.5 0.530916
\(870\) 0 0
\(871\) 571.207 0.0222212
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4090.81 −0.158051
\(876\) 0 0
\(877\) −26099.9 −1.00494 −0.502469 0.864595i \(-0.667575\pi\)
−0.502469 + 0.864595i \(0.667575\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29982.7 1.14659 0.573294 0.819350i \(-0.305665\pi\)
0.573294 + 0.819350i \(0.305665\pi\)
\(882\) 0 0
\(883\) 124.860 0.00475865 0.00237932 0.999997i \(-0.499243\pi\)
0.00237932 + 0.999997i \(0.499243\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2631.59 −0.0996167 −0.0498084 0.998759i \(-0.515861\pi\)
−0.0498084 + 0.998759i \(0.515861\pi\)
\(888\) 0 0
\(889\) 27263.3 1.02855
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −41978.6 −1.57308
\(894\) 0 0
\(895\) 18771.6 0.701077
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8770.37 0.325371
\(900\) 0 0
\(901\) 43619.5 1.61285
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 224.414 0.00824284
\(906\) 0 0
\(907\) 29239.7 1.07044 0.535219 0.844713i \(-0.320229\pi\)
0.535219 + 0.844713i \(0.320229\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 46512.2 1.69157 0.845783 0.533527i \(-0.179134\pi\)
0.845783 + 0.533527i \(0.179134\pi\)
\(912\) 0 0
\(913\) 207.009 0.00750385
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18398.8 0.662577
\(918\) 0 0
\(919\) 16873.5 0.605664 0.302832 0.953044i \(-0.402068\pi\)
0.302832 + 0.953044i \(0.402068\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3608.08 −0.128669
\(924\) 0 0
\(925\) 1644.19 0.0584439
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 32397.2 1.14415 0.572076 0.820201i \(-0.306138\pi\)
0.572076 + 0.820201i \(0.306138\pi\)
\(930\) 0 0
\(931\) 86137.0 3.03225
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4957.16 0.173386
\(936\) 0 0
\(937\) −5339.74 −0.186170 −0.0930852 0.995658i \(-0.529673\pi\)
−0.0930852 + 0.995658i \(0.529673\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24451.1 0.847058 0.423529 0.905883i \(-0.360791\pi\)
0.423529 + 0.905883i \(0.360791\pi\)
\(942\) 0 0
\(943\) −20045.3 −0.692221
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23200.1 −0.796096 −0.398048 0.917365i \(-0.630312\pi\)
−0.398048 + 0.917365i \(0.630312\pi\)
\(948\) 0 0
\(949\) −17961.8 −0.614399
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22349.2 −0.759666 −0.379833 0.925055i \(-0.624019\pi\)
−0.379833 + 0.925055i \(0.624019\pi\)
\(954\) 0 0
\(955\) 22683.9 0.768623
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 34743.0 1.16987
\(960\) 0 0
\(961\) −28464.1 −0.955460
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1200.62 0.0400510
\(966\) 0 0
\(967\) −2505.06 −0.0833063 −0.0416532 0.999132i \(-0.513262\pi\)
−0.0416532 + 0.999132i \(0.513262\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 50594.9 1.67216 0.836080 0.548607i \(-0.184842\pi\)
0.836080 + 0.548607i \(0.184842\pi\)
\(972\) 0 0
\(973\) 6965.77 0.229509
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19735.5 −0.646259 −0.323130 0.946355i \(-0.604735\pi\)
−0.323130 + 0.946355i \(0.604735\pi\)
\(978\) 0 0
\(979\) −14376.9 −0.469345
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −54357.7 −1.76373 −0.881863 0.471506i \(-0.843711\pi\)
−0.881863 + 0.471506i \(0.843711\pi\)
\(984\) 0 0
\(985\) 18544.9 0.599887
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 38905.1 1.25087
\(990\) 0 0
\(991\) −37794.5 −1.21148 −0.605742 0.795661i \(-0.707124\pi\)
−0.605742 + 0.795661i \(0.707124\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4878.91 −0.155449
\(996\) 0 0
\(997\) −21129.2 −0.671182 −0.335591 0.942008i \(-0.608936\pi\)
−0.335591 + 0.942008i \(0.608936\pi\)
\(998\) 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 1620.4.a.h.1.1 4
3.2 odd 2 1620.4.a.g.1.1 4
9.2 odd 6 180.4.i.b.121.2 yes 8
9.4 even 3 540.4.i.b.181.4 8
9.5 odd 6 180.4.i.b.61.2 8
9.7 even 3 540.4.i.b.361.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.4.i.b.61.2 8 9.5 odd 6
180.4.i.b.121.2 yes 8 9.2 odd 6
540.4.i.b.181.4 8 9.4 even 3
540.4.i.b.361.4 8 9.7 even 3
1620.4.a.g.1.1 4 3.2 odd 2
1620.4.a.h.1.1 4 1.1 even 1 trivial