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Properties

Label 1728.4.a.m.1.1
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $1$
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] = [1728,4,Mod(1,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -9.00000 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -9.00000 q^{7} +17.0000 q^{11} +44.0000 q^{13} +56.0000 q^{17} +94.0000 q^{19} -50.0000 q^{23} -124.000 q^{25} +30.0000 q^{29} -139.000 q^{31} +9.00000 q^{35} +174.000 q^{37} +318.000 q^{41} +242.000 q^{43} -630.000 q^{47} -262.000 q^{49} -547.000 q^{53} -17.0000 q^{55} +236.000 q^{59} -328.000 q^{61} -44.0000 q^{65} -614.000 q^{67} +296.000 q^{71} +433.000 q^{73} -153.000 q^{77} -56.0000 q^{79} +1225.00 q^{83} -56.0000 q^{85} +1506.00 q^{89} -396.000 q^{91} -94.0000 q^{95} +1391.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −1.00000 −0.0894427 −0.0447214 − 0.998999i \(-0.514240\pi\)
−0.0447214 + 0.998999i \(0.514240\pi\)
\(6\) 0 0
\(7\) −9.00000 −0.485954 −0.242977 − 0.970032i \(-0.578124\pi\)
−0.242977 + 0.970032i \(0.578124\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 17.0000 0.465972 0.232986 − 0.972480i \(-0.425150\pi\)
0.232986 + 0.972480i \(0.425150\pi\)
\(12\) 0 0
\(13\) 44.0000 0.938723 0.469362 − 0.883006i \(-0.344484\pi\)
0.469362 + 0.883006i \(0.344484\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 56.0000 0.798941 0.399470 − 0.916746i \(-0.369194\pi\)
0.399470 + 0.916746i \(0.369194\pi\)
\(18\) 0 0
\(19\) 94.0000 1.13500 0.567502 − 0.823372i \(-0.307910\pi\)
0.567502 + 0.823372i \(0.307910\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −50.0000 −0.453292 −0.226646 − 0.973977i \(-0.572776\pi\)
−0.226646 + 0.973977i \(0.572776\pi\)
\(24\) 0 0
\(25\) −124.000 −0.992000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 − 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) −139.000 −0.805327 −0.402663 − 0.915348i \(-0.631916\pi\)
−0.402663 + 0.915348i \(0.631916\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.00000 0.0434651
\(36\) 0 0
\(37\) 174.000 0.773120 0.386560 − 0.922264i \(-0.373663\pi\)
0.386560 + 0.922264i \(0.373663\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 318.000 1.21130 0.605649 − 0.795732i \(-0.292913\pi\)
0.605649 + 0.795732i \(0.292913\pi\)
\(42\) 0 0
\(43\) 242.000 0.858248 0.429124 − 0.903246i \(-0.358822\pi\)
0.429124 + 0.903246i \(0.358822\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −630.000 −1.95521 −0.977606 − 0.210445i \(-0.932509\pi\)
−0.977606 + 0.210445i \(0.932509\pi\)
\(48\) 0 0
\(49\) −262.000 −0.763848
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −547.000 −1.41766 −0.708832 − 0.705377i \(-0.750778\pi\)
−0.708832 + 0.705377i \(0.750778\pi\)
\(54\) 0 0
\(55\) −17.0000 −0.0416778
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 236.000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −328.000 −0.688461 −0.344230 − 0.938885i \(-0.611860\pi\)
−0.344230 + 0.938885i \(0.611860\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −44.0000 −0.0839620
\(66\) 0 0
\(67\) −614.000 −1.11958 −0.559791 − 0.828634i \(-0.689119\pi\)
−0.559791 + 0.828634i \(0.689119\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 296.000 0.494771 0.247385 − 0.968917i \(-0.420429\pi\)
0.247385 + 0.968917i \(0.420429\pi\)
\(72\) 0 0
\(73\) 433.000 0.694230 0.347115 − 0.937823i \(-0.387161\pi\)
0.347115 + 0.937823i \(0.387161\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −153.000 −0.226441
\(78\) 0 0
\(79\) −56.0000 −0.0797531 −0.0398765 − 0.999205i \(-0.512696\pi\)
−0.0398765 + 0.999205i \(0.512696\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1225.00 1.62001 0.810007 − 0.586420i \(-0.199463\pi\)
0.810007 + 0.586420i \(0.199463\pi\)
\(84\) 0 0
\(85\) −56.0000 −0.0714594
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1506.00 1.79366 0.896830 − 0.442376i \(-0.145864\pi\)
0.896830 + 0.442376i \(0.145864\pi\)
\(90\) 0 0
\(91\) −396.000 −0.456177
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −94.0000 −0.101518
\(96\) 0 0
\(97\) 1391.00 1.45603 0.728014 − 0.685563i \(-0.240444\pi\)
0.728014 + 0.685563i \(0.240444\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1665.00 1.64033 0.820167 − 0.572124i \(-0.193881\pi\)
0.820167 + 0.572124i \(0.193881\pi\)
\(102\) 0 0
\(103\) 1468.00 1.40433 0.702167 − 0.712013i \(-0.252216\pi\)
0.702167 + 0.712013i \(0.252216\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −627.000 −0.566490 −0.283245 − 0.959048i \(-0.591411\pi\)
−0.283245 + 0.959048i \(0.591411\pi\)
\(108\) 0 0
\(109\) −610.000 −0.536031 −0.268016 − 0.963415i \(-0.586368\pi\)
−0.268016 + 0.963415i \(0.586368\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1134.00 0.944051 0.472025 − 0.881585i \(-0.343523\pi\)
0.472025 + 0.881585i \(0.343523\pi\)
\(114\) 0 0
\(115\) 50.0000 0.0405437
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −504.000 −0.388249
\(120\) 0 0
\(121\) −1042.00 −0.782870
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 249.000 0.178170
\(126\) 0 0
\(127\) 505.000 0.352846 0.176423 − 0.984314i \(-0.443547\pi\)
0.176423 + 0.984314i \(0.443547\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −515.000 −0.343479 −0.171740 − 0.985142i \(-0.554939\pi\)
−0.171740 + 0.985142i \(0.554939\pi\)
\(132\) 0 0
\(133\) −846.000 −0.551560
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.00374171 −0.00187086 − 0.999998i \(-0.500596\pi\)
−0.00187086 + 0.999998i \(0.500596\pi\)
\(138\) 0 0
\(139\) −636.000 −0.388092 −0.194046 − 0.980992i \(-0.562161\pi\)
−0.194046 + 0.980992i \(0.562161\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 748.000 0.437419
\(144\) 0 0
\(145\) −30.0000 −0.0171818
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2239.00 1.23105 0.615524 − 0.788118i \(-0.288945\pi\)
0.615524 + 0.788118i \(0.288945\pi\)
\(150\) 0 0
\(151\) −2695.00 −1.45242 −0.726212 − 0.687471i \(-0.758721\pi\)
−0.726212 + 0.687471i \(0.758721\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 139.000 0.0720306
\(156\) 0 0
\(157\) 1988.00 1.01057 0.505286 − 0.862952i \(-0.331387\pi\)
0.505286 + 0.862952i \(0.331387\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 450.000 0.220279
\(162\) 0 0
\(163\) 2580.00 1.23976 0.619881 − 0.784696i \(-0.287181\pi\)
0.619881 + 0.784696i \(0.287181\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1878.00 0.870204 0.435102 − 0.900381i \(-0.356712\pi\)
0.435102 + 0.900381i \(0.356712\pi\)
\(168\) 0 0
\(169\) −261.000 −0.118798
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2163.00 0.950577 0.475289 − 0.879830i \(-0.342344\pi\)
0.475289 + 0.879830i \(0.342344\pi\)
\(174\) 0 0
\(175\) 1116.00 0.482067
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2181.00 −0.910702 −0.455351 − 0.890312i \(-0.650486\pi\)
−0.455351 + 0.890312i \(0.650486\pi\)
\(180\) 0 0
\(181\) 1488.00 0.611062 0.305531 − 0.952182i \(-0.401166\pi\)
0.305531 + 0.952182i \(0.401166\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −174.000 −0.0691499
\(186\) 0 0
\(187\) 952.000 0.372284
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1764.00 0.668265 0.334132 − 0.942526i \(-0.391557\pi\)
0.334132 + 0.942526i \(0.391557\pi\)
\(192\) 0 0
\(193\) −3547.00 −1.32289 −0.661447 − 0.749992i \(-0.730057\pi\)
−0.661447 + 0.749992i \(0.730057\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4941.00 1.78696 0.893481 − 0.449100i \(-0.148255\pi\)
0.893481 + 0.449100i \(0.148255\pi\)
\(198\) 0 0
\(199\) −1487.00 −0.529702 −0.264851 − 0.964289i \(-0.585323\pi\)
−0.264851 + 0.964289i \(0.585323\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −270.000 −0.0933512
\(204\) 0 0
\(205\) −318.000 −0.108342
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1598.00 0.528880
\(210\) 0 0
\(211\) −442.000 −0.144211 −0.0721055 − 0.997397i \(-0.522972\pi\)
−0.0721055 + 0.997397i \(0.522972\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −242.000 −0.0767640
\(216\) 0 0
\(217\) 1251.00 0.391352
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2464.00 0.749985
\(222\) 0 0
\(223\) −5096.00 −1.53028 −0.765142 − 0.643861i \(-0.777331\pi\)
−0.765142 + 0.643861i \(0.777331\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5484.00 −1.60346 −0.801731 − 0.597685i \(-0.796087\pi\)
−0.801731 + 0.597685i \(0.796087\pi\)
\(228\) 0 0
\(229\) 3530.00 1.01864 0.509321 − 0.860577i \(-0.329897\pi\)
0.509321 + 0.860577i \(0.329897\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1630.00 0.458304 0.229152 − 0.973391i \(-0.426405\pi\)
0.229152 + 0.973391i \(0.426405\pi\)
\(234\) 0 0
\(235\) 630.000 0.174879
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1990.00 −0.538587 −0.269294 − 0.963058i \(-0.586790\pi\)
−0.269294 + 0.963058i \(0.586790\pi\)
\(240\) 0 0
\(241\) 2214.00 0.591769 0.295884 − 0.955224i \(-0.404386\pi\)
0.295884 + 0.955224i \(0.404386\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 262.000 0.0683207
\(246\) 0 0
\(247\) 4136.00 1.06545
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4480.00 1.12659 0.563297 − 0.826254i \(-0.309533\pi\)
0.563297 + 0.826254i \(0.309533\pi\)
\(252\) 0 0
\(253\) −850.000 −0.211222
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2204.00 0.534948 0.267474 − 0.963565i \(-0.413811\pi\)
0.267474 + 0.963565i \(0.413811\pi\)
\(258\) 0 0
\(259\) −1566.00 −0.375701
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1942.00 0.455319 0.227659 − 0.973741i \(-0.426893\pi\)
0.227659 + 0.973741i \(0.426893\pi\)
\(264\) 0 0
\(265\) 547.000 0.126800
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7606.00 1.72396 0.861981 − 0.506940i \(-0.169223\pi\)
0.861981 + 0.506940i \(0.169223\pi\)
\(270\) 0 0
\(271\) 4391.00 0.984259 0.492130 − 0.870522i \(-0.336219\pi\)
0.492130 + 0.870522i \(0.336219\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2108.00 −0.462244
\(276\) 0 0
\(277\) −7000.00 −1.51837 −0.759186 − 0.650873i \(-0.774403\pi\)
−0.759186 + 0.650873i \(0.774403\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8912.00 1.89198 0.945988 − 0.324201i \(-0.105095\pi\)
0.945988 + 0.324201i \(0.105095\pi\)
\(282\) 0 0
\(283\) −3614.00 −0.759117 −0.379558 − 0.925168i \(-0.623924\pi\)
−0.379558 + 0.925168i \(0.623924\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2862.00 −0.588636
\(288\) 0 0
\(289\) −1777.00 −0.361693
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4194.00 0.836232 0.418116 − 0.908394i \(-0.362690\pi\)
0.418116 + 0.908394i \(0.362690\pi\)
\(294\) 0 0
\(295\) −236.000 −0.0465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2200.00 −0.425516
\(300\) 0 0
\(301\) −2178.00 −0.417069
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 328.000 0.0615778
\(306\) 0 0
\(307\) −840.000 −0.156161 −0.0780803 − 0.996947i \(-0.524879\pi\)
−0.0780803 + 0.996947i \(0.524879\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 606.000 0.110492 0.0552462 − 0.998473i \(-0.482406\pi\)
0.0552462 + 0.998473i \(0.482406\pi\)
\(312\) 0 0
\(313\) 2037.00 0.367853 0.183927 − 0.982940i \(-0.441119\pi\)
0.183927 + 0.982940i \(0.441119\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −519.000 −0.0919557 −0.0459778 − 0.998942i \(-0.514640\pi\)
−0.0459778 + 0.998942i \(0.514640\pi\)
\(318\) 0 0
\(319\) 510.000 0.0895126
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5264.00 0.906801
\(324\) 0 0
\(325\) −5456.00 −0.931214
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5670.00 0.950144
\(330\) 0 0
\(331\) −2750.00 −0.456658 −0.228329 − 0.973584i \(-0.573326\pi\)
−0.228329 + 0.973584i \(0.573326\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 614.000 0.100139
\(336\) 0 0
\(337\) 4410.00 0.712843 0.356421 − 0.934325i \(-0.383997\pi\)
0.356421 + 0.934325i \(0.383997\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2363.00 −0.375260
\(342\) 0 0
\(343\) 5445.00 0.857150
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10911.0 1.68799 0.843996 − 0.536350i \(-0.180197\pi\)
0.843996 + 0.536350i \(0.180197\pi\)
\(348\) 0 0
\(349\) 4650.00 0.713206 0.356603 − 0.934256i \(-0.383935\pi\)
0.356603 + 0.934256i \(0.383935\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9410.00 1.41882 0.709410 − 0.704796i \(-0.248961\pi\)
0.709410 + 0.704796i \(0.248961\pi\)
\(354\) 0 0
\(355\) −296.000 −0.0442537
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2130.00 −0.313140 −0.156570 − 0.987667i \(-0.550044\pi\)
−0.156570 + 0.987667i \(0.550044\pi\)
\(360\) 0 0
\(361\) 1977.00 0.288234
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −433.000 −0.0620939
\(366\) 0 0
\(367\) 1393.00 0.198131 0.0990654 − 0.995081i \(-0.468415\pi\)
0.0990654 + 0.995081i \(0.468415\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4923.00 0.688920
\(372\) 0 0
\(373\) 2920.00 0.405340 0.202670 − 0.979247i \(-0.435038\pi\)
0.202670 + 0.979247i \(0.435038\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1320.00 0.180327
\(378\) 0 0
\(379\) 10780.0 1.46103 0.730516 − 0.682895i \(-0.239279\pi\)
0.730516 + 0.682895i \(0.239279\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3888.00 −0.518714 −0.259357 − 0.965782i \(-0.583511\pi\)
−0.259357 + 0.965782i \(0.583511\pi\)
\(384\) 0 0
\(385\) 153.000 0.0202535
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4009.00 −0.522531 −0.261265 − 0.965267i \(-0.584140\pi\)
−0.261265 + 0.965267i \(0.584140\pi\)
\(390\) 0 0
\(391\) −2800.00 −0.362154
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 56.0000 0.00713333
\(396\) 0 0
\(397\) −860.000 −0.108721 −0.0543604 − 0.998521i \(-0.517312\pi\)
−0.0543604 + 0.998521i \(0.517312\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1716.00 0.213698 0.106849 − 0.994275i \(-0.465924\pi\)
0.106849 + 0.994275i \(0.465924\pi\)
\(402\) 0 0
\(403\) −6116.00 −0.755979
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2958.00 0.360252
\(408\) 0 0
\(409\) −3201.00 −0.386991 −0.193495 − 0.981101i \(-0.561982\pi\)
−0.193495 + 0.981101i \(0.561982\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2124.00 −0.253063
\(414\) 0 0
\(415\) −1225.00 −0.144899
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5700.00 −0.664590 −0.332295 − 0.943175i \(-0.607823\pi\)
−0.332295 + 0.943175i \(0.607823\pi\)
\(420\) 0 0
\(421\) 10312.0 1.19377 0.596884 − 0.802328i \(-0.296405\pi\)
0.596884 + 0.802328i \(0.296405\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6944.00 −0.792549
\(426\) 0 0
\(427\) 2952.00 0.334560
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10354.0 −1.15716 −0.578578 − 0.815627i \(-0.696392\pi\)
−0.578578 + 0.815627i \(0.696392\pi\)
\(432\) 0 0
\(433\) −10787.0 −1.19721 −0.598603 − 0.801046i \(-0.704277\pi\)
−0.598603 + 0.801046i \(0.704277\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4700.00 −0.514489
\(438\) 0 0
\(439\) −14715.0 −1.59979 −0.799896 − 0.600139i \(-0.795112\pi\)
−0.799896 + 0.600139i \(0.795112\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6628.00 0.710848 0.355424 − 0.934705i \(-0.384336\pi\)
0.355424 + 0.934705i \(0.384336\pi\)
\(444\) 0 0
\(445\) −1506.00 −0.160430
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8122.00 0.853677 0.426838 − 0.904328i \(-0.359627\pi\)
0.426838 + 0.904328i \(0.359627\pi\)
\(450\) 0 0
\(451\) 5406.00 0.564431
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 396.000 0.0408017
\(456\) 0 0
\(457\) −6553.00 −0.670758 −0.335379 − 0.942083i \(-0.608864\pi\)
−0.335379 + 0.942083i \(0.608864\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4079.00 0.412100 0.206050 − 0.978541i \(-0.433939\pi\)
0.206050 + 0.978541i \(0.433939\pi\)
\(462\) 0 0
\(463\) −12685.0 −1.27327 −0.636633 − 0.771167i \(-0.719673\pi\)
−0.636633 + 0.771167i \(0.719673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6111.00 −0.605532 −0.302766 − 0.953065i \(-0.597910\pi\)
−0.302766 + 0.953065i \(0.597910\pi\)
\(468\) 0 0
\(469\) 5526.00 0.544066
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4114.00 0.399919
\(474\) 0 0
\(475\) −11656.0 −1.12592
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11254.0 −1.07350 −0.536752 − 0.843740i \(-0.680349\pi\)
−0.536752 + 0.843740i \(0.680349\pi\)
\(480\) 0 0
\(481\) 7656.00 0.725745
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1391.00 −0.130231
\(486\) 0 0
\(487\) 13936.0 1.29672 0.648358 − 0.761336i \(-0.275456\pi\)
0.648358 + 0.761336i \(0.275456\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6045.00 −0.555615 −0.277808 − 0.960637i \(-0.589608\pi\)
−0.277808 + 0.960637i \(0.589608\pi\)
\(492\) 0 0
\(493\) 1680.00 0.153475
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2664.00 −0.240436
\(498\) 0 0
\(499\) 19482.0 1.74776 0.873882 − 0.486138i \(-0.161595\pi\)
0.873882 + 0.486138i \(0.161595\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19770.0 1.75249 0.876243 − 0.481869i \(-0.160042\pi\)
0.876243 + 0.481869i \(0.160042\pi\)
\(504\) 0 0
\(505\) −1665.00 −0.146716
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11613.0 1.01127 0.505636 − 0.862747i \(-0.331258\pi\)
0.505636 + 0.862747i \(0.331258\pi\)
\(510\) 0 0
\(511\) −3897.00 −0.337364
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1468.00 −0.125607
\(516\) 0 0
\(517\) −10710.0 −0.911074
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5500.00 −0.462494 −0.231247 − 0.972895i \(-0.574281\pi\)
−0.231247 + 0.972895i \(0.574281\pi\)
\(522\) 0 0
\(523\) 8024.00 0.670870 0.335435 − 0.942063i \(-0.391117\pi\)
0.335435 + 0.942063i \(0.391117\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7784.00 −0.643409
\(528\) 0 0
\(529\) −9667.00 −0.794526
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13992.0 1.13707
\(534\) 0 0
\(535\) 627.000 0.0506684
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4454.00 −0.355932
\(540\) 0 0
\(541\) 9036.00 0.718092 0.359046 − 0.933320i \(-0.383102\pi\)
0.359046 + 0.933320i \(0.383102\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 610.000 0.0479441
\(546\) 0 0
\(547\) −23456.0 −1.83347 −0.916733 − 0.399500i \(-0.869184\pi\)
−0.916733 + 0.399500i \(0.869184\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2820.00 0.218033
\(552\) 0 0
\(553\) 504.000 0.0387563
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18721.0 −1.42412 −0.712059 − 0.702119i \(-0.752237\pi\)
−0.712059 + 0.702119i \(0.752237\pi\)
\(558\) 0 0
\(559\) 10648.0 0.805657
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1417.00 −0.106074 −0.0530368 − 0.998593i \(-0.516890\pi\)
−0.0530368 + 0.998593i \(0.516890\pi\)
\(564\) 0 0
\(565\) −1134.00 −0.0844385
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18596.0 −1.37010 −0.685048 − 0.728498i \(-0.740219\pi\)
−0.685048 + 0.728498i \(0.740219\pi\)
\(570\) 0 0
\(571\) 3708.00 0.271760 0.135880 − 0.990725i \(-0.456614\pi\)
0.135880 + 0.990725i \(0.456614\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6200.00 0.449666
\(576\) 0 0
\(577\) 21494.0 1.55079 0.775396 − 0.631475i \(-0.217550\pi\)
0.775396 + 0.631475i \(0.217550\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11025.0 −0.787253
\(582\) 0 0
\(583\) −9299.00 −0.660592
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16049.0 1.12847 0.564236 − 0.825614i \(-0.309171\pi\)
0.564236 + 0.825614i \(0.309171\pi\)
\(588\) 0 0
\(589\) −13066.0 −0.914049
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4658.00 −0.322565 −0.161283 − 0.986908i \(-0.551563\pi\)
−0.161283 + 0.986908i \(0.551563\pi\)
\(594\) 0 0
\(595\) 504.000 0.0347260
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26538.0 1.81021 0.905103 − 0.425193i \(-0.139794\pi\)
0.905103 + 0.425193i \(0.139794\pi\)
\(600\) 0 0
\(601\) 1147.00 0.0778488 0.0389244 − 0.999242i \(-0.487607\pi\)
0.0389244 + 0.999242i \(0.487607\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1042.00 0.0700220
\(606\) 0 0
\(607\) 26048.0 1.74177 0.870886 − 0.491485i \(-0.163546\pi\)
0.870886 + 0.491485i \(0.163546\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −27720.0 −1.83540
\(612\) 0 0
\(613\) 394.000 0.0259600 0.0129800 − 0.999916i \(-0.495868\pi\)
0.0129800 + 0.999916i \(0.495868\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27822.0 −1.81535 −0.907675 − 0.419673i \(-0.862145\pi\)
−0.907675 + 0.419673i \(0.862145\pi\)
\(618\) 0 0
\(619\) −21580.0 −1.40125 −0.700625 − 0.713530i \(-0.747095\pi\)
−0.700625 + 0.713530i \(0.747095\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13554.0 −0.871637
\(624\) 0 0
\(625\) 15251.0 0.976064
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9744.00 0.617677
\(630\) 0 0
\(631\) 7855.00 0.495567 0.247783 − 0.968815i \(-0.420298\pi\)
0.247783 + 0.968815i \(0.420298\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −505.000 −0.0315595
\(636\) 0 0
\(637\) −11528.0 −0.717042
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30770.0 −1.89601 −0.948005 − 0.318257i \(-0.896903\pi\)
−0.948005 + 0.318257i \(0.896903\pi\)
\(642\) 0 0
\(643\) 18420.0 1.12973 0.564863 − 0.825185i \(-0.308929\pi\)
0.564863 + 0.825185i \(0.308929\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10448.0 0.634858 0.317429 − 0.948282i \(-0.397180\pi\)
0.317429 + 0.948282i \(0.397180\pi\)
\(648\) 0 0
\(649\) 4012.00 0.242658
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19221.0 −1.15188 −0.575939 − 0.817493i \(-0.695363\pi\)
−0.575939 + 0.817493i \(0.695363\pi\)
\(654\) 0 0
\(655\) 515.000 0.0307217
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −29649.0 −1.75260 −0.876298 − 0.481769i \(-0.839994\pi\)
−0.876298 + 0.481769i \(0.839994\pi\)
\(660\) 0 0
\(661\) −4478.00 −0.263501 −0.131750 − 0.991283i \(-0.542060\pi\)
−0.131750 + 0.991283i \(0.542060\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 846.000 0.0493330
\(666\) 0 0
\(667\) −1500.00 −0.0870768
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5576.00 −0.320803
\(672\) 0 0
\(673\) −14099.0 −0.807543 −0.403772 − 0.914860i \(-0.632301\pi\)
−0.403772 + 0.914860i \(0.632301\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14782.0 −0.839170 −0.419585 − 0.907716i \(-0.637824\pi\)
−0.419585 + 0.907716i \(0.637824\pi\)
\(678\) 0 0
\(679\) −12519.0 −0.707563
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24620.0 1.37929 0.689647 − 0.724145i \(-0.257766\pi\)
0.689647 + 0.724145i \(0.257766\pi\)
\(684\) 0 0
\(685\) 6.00000 0.000334669 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24068.0 −1.33080
\(690\) 0 0
\(691\) −30716.0 −1.69102 −0.845508 − 0.533963i \(-0.820702\pi\)
−0.845508 + 0.533963i \(0.820702\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 636.000 0.0347120
\(696\) 0 0
\(697\) 17808.0 0.967756
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3147.00 −0.169559 −0.0847793 − 0.996400i \(-0.527019\pi\)
−0.0847793 + 0.996400i \(0.527019\pi\)
\(702\) 0 0
\(703\) 16356.0 0.877494
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14985.0 −0.797127
\(708\) 0 0
\(709\) 22660.0 1.20030 0.600151 − 0.799887i \(-0.295107\pi\)
0.600151 + 0.799887i \(0.295107\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6950.00 0.365048
\(714\) 0 0
\(715\) −748.000 −0.0391239
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4828.00 0.250423 0.125211 − 0.992130i \(-0.460039\pi\)
0.125211 + 0.992130i \(0.460039\pi\)
\(720\) 0 0
\(721\) −13212.0 −0.682442
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3720.00 −0.190562
\(726\) 0 0
\(727\) −3849.00 −0.196357 −0.0981785 − 0.995169i \(-0.531302\pi\)
−0.0981785 + 0.995169i \(0.531302\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 13552.0 0.685689
\(732\) 0 0
\(733\) −11738.0 −0.591477 −0.295739 − 0.955269i \(-0.595566\pi\)
−0.295739 + 0.955269i \(0.595566\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10438.0 −0.521694
\(738\) 0 0
\(739\) 26824.0 1.33523 0.667616 − 0.744506i \(-0.267315\pi\)
0.667616 + 0.744506i \(0.267315\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −68.0000 −0.00335757 −0.00167879 − 0.999999i \(-0.500534\pi\)
−0.00167879 + 0.999999i \(0.500534\pi\)
\(744\) 0 0
\(745\) −2239.00 −0.110108
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5643.00 0.275288
\(750\) 0 0
\(751\) −36039.0 −1.75111 −0.875554 − 0.483121i \(-0.839503\pi\)
−0.875554 + 0.483121i \(0.839503\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2695.00 0.129909
\(756\) 0 0
\(757\) −18650.0 −0.895437 −0.447718 − 0.894175i \(-0.647763\pi\)
−0.447718 + 0.894175i \(0.647763\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18848.0 0.897818 0.448909 − 0.893577i \(-0.351813\pi\)
0.448909 + 0.893577i \(0.351813\pi\)
\(762\) 0 0
\(763\) 5490.00 0.260487
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10384.0 0.488846
\(768\) 0 0
\(769\) −29215.0 −1.36999 −0.684993 − 0.728549i \(-0.740195\pi\)
−0.684993 + 0.728549i \(0.740195\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26586.0 −1.23704 −0.618520 − 0.785769i \(-0.712267\pi\)
−0.618520 + 0.785769i \(0.712267\pi\)
\(774\) 0 0
\(775\) 17236.0 0.798884
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29892.0 1.37483
\(780\) 0 0
\(781\) 5032.00 0.230549
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1988.00 −0.0903882
\(786\) 0 0
\(787\) −15520.0 −0.702958 −0.351479 − 0.936196i \(-0.614321\pi\)
−0.351479 + 0.936196i \(0.614321\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10206.0 −0.458766
\(792\) 0 0
\(793\) −14432.0 −0.646274
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7477.00 −0.332307 −0.166154 − 0.986100i \(-0.553135\pi\)
−0.166154 + 0.986100i \(0.553135\pi\)
\(798\) 0 0
\(799\) −35280.0 −1.56210
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7361.00 0.323492
\(804\) 0 0
\(805\) −450.000 −0.0197024
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −25282.0 −1.09872 −0.549362 − 0.835584i \(-0.685129\pi\)
−0.549362 + 0.835584i \(0.685129\pi\)
\(810\) 0 0
\(811\) 35582.0 1.54063 0.770316 − 0.637662i \(-0.220098\pi\)
0.770316 + 0.637662i \(0.220098\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2580.00 −0.110888
\(816\) 0 0
\(817\) 22748.0 0.974115
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37506.0 −1.59436 −0.797179 − 0.603743i \(-0.793675\pi\)
−0.797179 + 0.603743i \(0.793675\pi\)
\(822\) 0 0
\(823\) 12185.0 0.516090 0.258045 − 0.966133i \(-0.416922\pi\)
0.258045 + 0.966133i \(0.416922\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8128.00 0.341763 0.170882 − 0.985292i \(-0.445338\pi\)
0.170882 + 0.985292i \(0.445338\pi\)
\(828\) 0 0
\(829\) 39542.0 1.65664 0.828318 − 0.560259i \(-0.189298\pi\)
0.828318 + 0.560259i \(0.189298\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14672.0 −0.610270
\(834\) 0 0
\(835\) −1878.00 −0.0778334
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40368.0 1.66109 0.830547 − 0.556948i \(-0.188028\pi\)
0.830547 + 0.556948i \(0.188028\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 261.000 0.0106256
\(846\) 0 0
\(847\) 9378.00 0.380439
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8700.00 −0.350449
\(852\) 0 0
\(853\) 20194.0 0.810585 0.405293 − 0.914187i \(-0.367170\pi\)
0.405293 + 0.914187i \(0.367170\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8664.00 0.345340 0.172670 − 0.984980i \(-0.444761\pi\)
0.172670 + 0.984980i \(0.444761\pi\)
\(858\) 0 0
\(859\) 15672.0 0.622493 0.311247 − 0.950329i \(-0.399253\pi\)
0.311247 + 0.950329i \(0.399253\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −46076.0 −1.81743 −0.908717 − 0.417413i \(-0.862937\pi\)
−0.908717 + 0.417413i \(0.862937\pi\)
\(864\) 0 0
\(865\) −2163.00 −0.0850222
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −952.000 −0.0371627
\(870\) 0 0
\(871\) −27016.0 −1.05098
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2241.00 −0.0865824
\(876\) 0 0
\(877\) 28926.0 1.11375 0.556877 − 0.830595i \(-0.312000\pi\)
0.556877 + 0.830595i \(0.312000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −10372.0 −0.396642 −0.198321 − 0.980137i \(-0.563549\pi\)
−0.198321 + 0.980137i \(0.563549\pi\)
\(882\) 0 0
\(883\) 15502.0 0.590808 0.295404 − 0.955372i \(-0.404546\pi\)
0.295404 + 0.955372i \(0.404546\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29524.0 −1.11761 −0.558804 − 0.829300i \(-0.688740\pi\)
−0.558804 + 0.829300i \(0.688740\pi\)
\(888\) 0 0
\(889\) −4545.00 −0.171467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −59220.0 −2.21917
\(894\) 0 0
\(895\) 2181.00 0.0814556
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4170.00 −0.154702
\(900\) 0 0
\(901\) −30632.0 −1.13263
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1488.00 −0.0546550
\(906\) 0 0
\(907\) −27226.0 −0.996719 −0.498360 − 0.866970i \(-0.666064\pi\)
−0.498360 + 0.866970i \(0.666064\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −53510.0 −1.94607 −0.973033 − 0.230668i \(-0.925909\pi\)
−0.973033 + 0.230668i \(0.925909\pi\)
\(912\) 0 0
\(913\) 20825.0 0.754882
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4635.00 0.166915
\(918\) 0 0
\(919\) −443.000 −0.0159012 −0.00795061 − 0.999968i \(-0.502531\pi\)
−0.00795061 + 0.999968i \(0.502531\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13024.0 0.464453
\(924\) 0 0
\(925\) −21576.0 −0.766935
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 27260.0 0.962725 0.481363 − 0.876522i \(-0.340142\pi\)
0.481363 + 0.876522i \(0.340142\pi\)
\(930\) 0 0
\(931\) −24628.0 −0.866971
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −952.000 −0.0332981
\(936\) 0 0
\(937\) 21075.0 0.734781 0.367391 − 0.930067i \(-0.380251\pi\)
0.367391 + 0.930067i \(0.380251\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −22749.0 −0.788094 −0.394047 − 0.919090i \(-0.628925\pi\)
−0.394047 + 0.919090i \(0.628925\pi\)
\(942\) 0 0
\(943\) −15900.0 −0.549072
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36201.0 −1.24221 −0.621106 − 0.783727i \(-0.713316\pi\)
−0.621106 + 0.783727i \(0.713316\pi\)
\(948\) 0 0
\(949\) 19052.0 0.651690
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4428.00 −0.150511 −0.0752555 − 0.997164i \(-0.523977\pi\)
−0.0752555 + 0.997164i \(0.523977\pi\)
\(954\) 0 0
\(955\) −1764.00 −0.0597714
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 54.0000 0.00181830
\(960\) 0 0
\(961\) −10470.0 −0.351448
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3547.00 0.118323
\(966\) 0 0
\(967\) 26927.0 0.895464 0.447732 − 0.894168i \(-0.352232\pi\)
0.447732 + 0.894168i \(0.352232\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 47961.0 1.58511 0.792555 − 0.609800i \(-0.208750\pi\)
0.792555 + 0.609800i \(0.208750\pi\)
\(972\) 0 0
\(973\) 5724.00 0.188595
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16746.0 0.548364 0.274182 − 0.961678i \(-0.411593\pi\)
0.274182 + 0.961678i \(0.411593\pi\)
\(978\) 0 0
\(979\) 25602.0 0.835795
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 37146.0 1.20526 0.602631 − 0.798020i \(-0.294119\pi\)
0.602631 + 0.798020i \(0.294119\pi\)
\(984\) 0 0
\(985\) −4941.00 −0.159831
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12100.0 −0.389037
\(990\) 0 0
\(991\) 42079.0 1.34882 0.674411 − 0.738356i \(-0.264397\pi\)
0.674411 + 0.738356i \(0.264397\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1487.00 0.0473780
\(996\) 0 0
\(997\) −46492.0 −1.47685 −0.738423 − 0.674337i \(-0.764429\pi\)
−0.738423 + 0.674337i \(0.764429\pi\)
\(998\) 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.a.m.1.1 1
3.2 odd 2 1728.4.a.s.1.1 1
4.3 odd 2 1728.4.a.n.1.1 1
8.3 odd 2 432.4.a.i.1.1 1
8.5 even 2 216.4.a.c.1.1 yes 1
12.11 even 2 1728.4.a.t.1.1 1
24.5 odd 2 216.4.a.b.1.1 ✓ 1
24.11 even 2 432.4.a.f.1.1 1
72.5 odd 6 648.4.i.g.217.1 2
72.13 even 6 648.4.i.f.217.1 2
72.29 odd 6 648.4.i.g.433.1 2
72.61 even 6 648.4.i.f.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.b.1.1 ✓ 1 24.5 odd 2
216.4.a.c.1.1 yes 1 8.5 even 2
432.4.a.f.1.1 1 24.11 even 2
432.4.a.i.1.1 1 8.3 odd 2
648.4.i.f.217.1 2 72.13 even 6
648.4.i.f.433.1 2 72.61 even 6
648.4.i.g.217.1 2 72.5 odd 6
648.4.i.g.433.1 2 72.29 odd 6
1728.4.a.m.1.1 1 1.1 even 1 trivial
1728.4.a.n.1.1 1 4.3 odd 2
1728.4.a.s.1.1 1 3.2 odd 2
1728.4.a.t.1.1 1 12.11 even 2