Properties

Label 648.4.a.j.1.2
Level $648$
Weight $4$
Character 648.1
Self dual yes
Analytic conductor $38.233$
Analytic rank $0$
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] = [648,4,Mod(1,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, 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("648.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.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: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.29952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.39417\) of defining polynomial
Character \(\chi\) \(=\) 648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.80342 q^{5} -26.1228 q^{7} +O(q^{10})\) \(q-5.80342 q^{5} -26.1228 q^{7} -37.2460 q^{11} -30.3373 q^{13} -48.4744 q^{17} +88.1865 q^{19} +84.5259 q^{23} -91.3203 q^{25} +176.724 q^{29} +155.384 q^{31} +151.601 q^{35} -258.310 q^{37} +220.287 q^{41} -47.7615 q^{43} +129.499 q^{47} +339.399 q^{49} +577.029 q^{53} +216.154 q^{55} -243.516 q^{59} -687.632 q^{61} +176.060 q^{65} +378.389 q^{67} +332.826 q^{71} +1072.19 q^{73} +972.968 q^{77} -1260.21 q^{79} +1226.09 q^{83} +281.317 q^{85} -1095.81 q^{89} +792.495 q^{91} -511.783 q^{95} -1128.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{17} + 80 q^{19} + 200 q^{23} + 8 q^{25} + 216 q^{29} + 80 q^{31} + 408 q^{35} - 276 q^{37} + 384 q^{41} + 160 q^{43} + 768 q^{47} - 268 q^{49} + 944 q^{53} + 304 q^{55} + 992 q^{59} - 548 q^{61} + 1328 q^{65} + 464 q^{67} + 1720 q^{71} - 764 q^{73} + 1728 q^{77} + 688 q^{79} + 2128 q^{83} - 1324 q^{85} + 2112 q^{89} + 1776 q^{91} + 2056 q^{95} - 1816 q^{97}+O(q^{100}) \) 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
\(5\) −5.80342 −0.519074 −0.259537 0.965733i \(-0.583570\pi\)
−0.259537 + 0.965733i \(0.583570\pi\)
\(6\) 0 0
\(7\) −26.1228 −1.41050 −0.705249 0.708960i \(-0.749165\pi\)
−0.705249 + 0.708960i \(0.749165\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −37.2460 −1.02092 −0.510458 0.859903i \(-0.670524\pi\)
−0.510458 + 0.859903i \(0.670524\pi\)
\(12\) 0 0
\(13\) −30.3373 −0.647235 −0.323618 0.946188i \(-0.604899\pi\)
−0.323618 + 0.946188i \(0.604899\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −48.4744 −0.691575 −0.345787 0.938313i \(-0.612388\pi\)
−0.345787 + 0.938313i \(0.612388\pi\)
\(18\) 0 0
\(19\) 88.1865 1.06481 0.532404 0.846490i \(-0.321289\pi\)
0.532404 + 0.846490i \(0.321289\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 84.5259 0.766299 0.383149 0.923686i \(-0.374839\pi\)
0.383149 + 0.923686i \(0.374839\pi\)
\(24\) 0 0
\(25\) −91.3203 −0.730562
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 176.724 1.13161 0.565806 0.824538i \(-0.308565\pi\)
0.565806 + 0.824538i \(0.308565\pi\)
\(30\) 0 0
\(31\) 155.384 0.900251 0.450126 0.892965i \(-0.351379\pi\)
0.450126 + 0.892965i \(0.351379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 151.601 0.732152
\(36\) 0 0
\(37\) −258.310 −1.14773 −0.573864 0.818951i \(-0.694556\pi\)
−0.573864 + 0.818951i \(0.694556\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 220.287 0.839098 0.419549 0.907733i \(-0.362188\pi\)
0.419549 + 0.907733i \(0.362188\pi\)
\(42\) 0 0
\(43\) −47.7615 −0.169385 −0.0846925 0.996407i \(-0.526991\pi\)
−0.0846925 + 0.996407i \(0.526991\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 129.499 0.401902 0.200951 0.979601i \(-0.435597\pi\)
0.200951 + 0.979601i \(0.435597\pi\)
\(48\) 0 0
\(49\) 339.399 0.989502
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 577.029 1.49549 0.747746 0.663985i \(-0.231136\pi\)
0.747746 + 0.663985i \(0.231136\pi\)
\(54\) 0 0
\(55\) 216.154 0.529931
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −243.516 −0.537339 −0.268670 0.963232i \(-0.586584\pi\)
−0.268670 + 0.963232i \(0.586584\pi\)
\(60\) 0 0
\(61\) −687.632 −1.44332 −0.721658 0.692250i \(-0.756619\pi\)
−0.721658 + 0.692250i \(0.756619\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 176.060 0.335963
\(66\) 0 0
\(67\) 378.389 0.689964 0.344982 0.938609i \(-0.387885\pi\)
0.344982 + 0.938609i \(0.387885\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 332.826 0.556327 0.278163 0.960534i \(-0.410274\pi\)
0.278163 + 0.960534i \(0.410274\pi\)
\(72\) 0 0
\(73\) 1072.19 1.71904 0.859520 0.511103i \(-0.170763\pi\)
0.859520 + 0.511103i \(0.170763\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 972.968 1.44000
\(78\) 0 0
\(79\) −1260.21 −1.79474 −0.897370 0.441279i \(-0.854525\pi\)
−0.897370 + 0.441279i \(0.854525\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1226.09 1.62146 0.810731 0.585419i \(-0.199070\pi\)
0.810731 + 0.585419i \(0.199070\pi\)
\(84\) 0 0
\(85\) 281.317 0.358978
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1095.81 −1.30512 −0.652559 0.757738i \(-0.726305\pi\)
−0.652559 + 0.757738i \(0.726305\pi\)
\(90\) 0 0
\(91\) 792.495 0.912923
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −511.783 −0.552714
\(96\) 0 0
\(97\) −1128.60 −1.18136 −0.590681 0.806905i \(-0.701141\pi\)
−0.590681 + 0.806905i \(0.701141\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1235.65 −1.21734 −0.608670 0.793423i \(-0.708297\pi\)
−0.608670 + 0.793423i \(0.708297\pi\)
\(102\) 0 0
\(103\) 1215.92 1.16319 0.581595 0.813479i \(-0.302429\pi\)
0.581595 + 0.813479i \(0.302429\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1779.08 1.60739 0.803694 0.595043i \(-0.202865\pi\)
0.803694 + 0.595043i \(0.202865\pi\)
\(108\) 0 0
\(109\) −974.934 −0.856714 −0.428357 0.903610i \(-0.640907\pi\)
−0.428357 + 0.903610i \(0.640907\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1971.47 1.64124 0.820621 0.571473i \(-0.193628\pi\)
0.820621 + 0.571473i \(0.193628\pi\)
\(114\) 0 0
\(115\) −490.540 −0.397766
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1266.29 0.975464
\(120\) 0 0
\(121\) 56.2621 0.0422705
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1255.40 0.898290
\(126\) 0 0
\(127\) −604.990 −0.422710 −0.211355 0.977409i \(-0.567788\pi\)
−0.211355 + 0.977409i \(0.567788\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1409.70 0.940197 0.470098 0.882614i \(-0.344218\pi\)
0.470098 + 0.882614i \(0.344218\pi\)
\(132\) 0 0
\(133\) −2303.68 −1.50191
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2263.55 1.41159 0.705795 0.708416i \(-0.250590\pi\)
0.705795 + 0.708416i \(0.250590\pi\)
\(138\) 0 0
\(139\) 1540.50 0.940027 0.470013 0.882659i \(-0.344249\pi\)
0.470013 + 0.882659i \(0.344249\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1129.94 0.660773
\(144\) 0 0
\(145\) −1025.60 −0.587390
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 613.034 0.337058 0.168529 0.985697i \(-0.446098\pi\)
0.168529 + 0.985697i \(0.446098\pi\)
\(150\) 0 0
\(151\) 2372.98 1.27888 0.639438 0.768843i \(-0.279167\pi\)
0.639438 + 0.768843i \(0.279167\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −901.759 −0.467297
\(156\) 0 0
\(157\) −1768.06 −0.898769 −0.449385 0.893338i \(-0.648357\pi\)
−0.449385 + 0.893338i \(0.648357\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2208.05 −1.08086
\(162\) 0 0
\(163\) 2657.82 1.27715 0.638577 0.769558i \(-0.279523\pi\)
0.638577 + 0.769558i \(0.279523\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3352.52 −1.55345 −0.776724 0.629842i \(-0.783120\pi\)
−0.776724 + 0.629842i \(0.783120\pi\)
\(168\) 0 0
\(169\) −1276.65 −0.581087
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.43589 0.00107050 0.000535251 1.00000i \(-0.499830\pi\)
0.000535251 1.00000i \(0.499830\pi\)
\(174\) 0 0
\(175\) 2385.54 1.03046
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1084.01 0.452642 0.226321 0.974053i \(-0.427330\pi\)
0.226321 + 0.974053i \(0.427330\pi\)
\(180\) 0 0
\(181\) 869.840 0.357208 0.178604 0.983921i \(-0.442842\pi\)
0.178604 + 0.983921i \(0.442842\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1499.08 0.595756
\(186\) 0 0
\(187\) 1805.48 0.706040
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −425.148 −0.161061 −0.0805304 0.996752i \(-0.525661\pi\)
−0.0805304 + 0.996752i \(0.525661\pi\)
\(192\) 0 0
\(193\) −1.70934 −0.000637520 0 −0.000318760 1.00000i \(-0.500101\pi\)
−0.000318760 1.00000i \(0.500101\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3079.44 −1.11371 −0.556855 0.830610i \(-0.687992\pi\)
−0.556855 + 0.830610i \(0.687992\pi\)
\(198\) 0 0
\(199\) 3133.50 1.11622 0.558110 0.829767i \(-0.311527\pi\)
0.558110 + 0.829767i \(0.311527\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4616.51 −1.59614
\(204\) 0 0
\(205\) −1278.42 −0.435554
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3284.59 −1.08708
\(210\) 0 0
\(211\) −2787.75 −0.909556 −0.454778 0.890605i \(-0.650281\pi\)
−0.454778 + 0.890605i \(0.650281\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 277.180 0.0879233
\(216\) 0 0
\(217\) −4059.06 −1.26980
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1470.58 0.447611
\(222\) 0 0
\(223\) −1146.60 −0.344315 −0.172157 0.985069i \(-0.555074\pi\)
−0.172157 + 0.985069i \(0.555074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −928.141 −0.271378 −0.135689 0.990751i \(-0.543325\pi\)
−0.135689 + 0.990751i \(0.543325\pi\)
\(228\) 0 0
\(229\) −3517.76 −1.01511 −0.507554 0.861620i \(-0.669450\pi\)
−0.507554 + 0.861620i \(0.669450\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 976.524 0.274567 0.137284 0.990532i \(-0.456163\pi\)
0.137284 + 0.990532i \(0.456163\pi\)
\(234\) 0 0
\(235\) −751.538 −0.208617
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4540.97 −1.22900 −0.614500 0.788917i \(-0.710642\pi\)
−0.614500 + 0.788917i \(0.710642\pi\)
\(240\) 0 0
\(241\) 7220.39 1.92990 0.964951 0.262432i \(-0.0845244\pi\)
0.964951 + 0.262432i \(0.0845244\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1969.68 −0.513625
\(246\) 0 0
\(247\) −2675.34 −0.689182
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1285.44 0.323251 0.161626 0.986852i \(-0.448326\pi\)
0.161626 + 0.986852i \(0.448326\pi\)
\(252\) 0 0
\(253\) −3148.25 −0.782327
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2604.49 0.632155 0.316078 0.948733i \(-0.397634\pi\)
0.316078 + 0.948733i \(0.397634\pi\)
\(258\) 0 0
\(259\) 6747.78 1.61887
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5353.68 1.25522 0.627609 0.778529i \(-0.284034\pi\)
0.627609 + 0.778529i \(0.284034\pi\)
\(264\) 0 0
\(265\) −3348.75 −0.776271
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7141.58 −1.61870 −0.809349 0.587328i \(-0.800180\pi\)
−0.809349 + 0.587328i \(0.800180\pi\)
\(270\) 0 0
\(271\) −6334.52 −1.41991 −0.709953 0.704249i \(-0.751284\pi\)
−0.709953 + 0.704249i \(0.751284\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3401.31 0.745843
\(276\) 0 0
\(277\) 7090.18 1.53793 0.768967 0.639289i \(-0.220771\pi\)
0.768967 + 0.639289i \(0.220771\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6741.01 −1.43109 −0.715543 0.698569i \(-0.753821\pi\)
−0.715543 + 0.698569i \(0.753821\pi\)
\(282\) 0 0
\(283\) 3559.66 0.747703 0.373852 0.927489i \(-0.378037\pi\)
0.373852 + 0.927489i \(0.378037\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5754.50 −1.18354
\(288\) 0 0
\(289\) −2563.23 −0.521725
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2155.41 0.429763 0.214881 0.976640i \(-0.431064\pi\)
0.214881 + 0.976640i \(0.431064\pi\)
\(294\) 0 0
\(295\) 1413.22 0.278919
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2564.29 −0.495975
\(300\) 0 0
\(301\) 1247.66 0.238917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3990.62 0.749187
\(306\) 0 0
\(307\) 1552.85 0.288684 0.144342 0.989528i \(-0.453893\pi\)
0.144342 + 0.989528i \(0.453893\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8414.88 −1.53429 −0.767145 0.641474i \(-0.778323\pi\)
−0.767145 + 0.641474i \(0.778323\pi\)
\(312\) 0 0
\(313\) −5704.25 −1.03011 −0.515053 0.857158i \(-0.672228\pi\)
−0.515053 + 0.857158i \(0.672228\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.363375 6.43823e−5 0 3.21911e−5 1.00000i \(-0.499990\pi\)
3.21911e−5 1.00000i \(0.499990\pi\)
\(318\) 0 0
\(319\) −6582.24 −1.15528
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4274.79 −0.736395
\(324\) 0 0
\(325\) 2770.41 0.472846
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3382.87 −0.566881
\(330\) 0 0
\(331\) 8934.34 1.48361 0.741806 0.670615i \(-0.233970\pi\)
0.741806 + 0.670615i \(0.233970\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2195.95 −0.358143
\(336\) 0 0
\(337\) −9524.10 −1.53950 −0.769749 0.638347i \(-0.779618\pi\)
−0.769749 + 0.638347i \(0.779618\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5787.43 −0.919081
\(342\) 0 0
\(343\) 94.0633 0.0148074
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8808.10 −1.36266 −0.681331 0.731976i \(-0.738599\pi\)
−0.681331 + 0.731976i \(0.738599\pi\)
\(348\) 0 0
\(349\) 2311.23 0.354491 0.177246 0.984167i \(-0.443281\pi\)
0.177246 + 0.984167i \(0.443281\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 165.608 0.0249701 0.0124850 0.999922i \(-0.496026\pi\)
0.0124850 + 0.999922i \(0.496026\pi\)
\(354\) 0 0
\(355\) −1931.53 −0.288775
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7007.10 1.03014 0.515071 0.857148i \(-0.327766\pi\)
0.515071 + 0.857148i \(0.327766\pi\)
\(360\) 0 0
\(361\) 917.854 0.133817
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6222.35 −0.892309
\(366\) 0 0
\(367\) 2915.52 0.414683 0.207342 0.978269i \(-0.433519\pi\)
0.207342 + 0.978269i \(0.433519\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15073.6 −2.10939
\(372\) 0 0
\(373\) 4087.11 0.567353 0.283676 0.958920i \(-0.408446\pi\)
0.283676 + 0.958920i \(0.408446\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5361.32 −0.732419
\(378\) 0 0
\(379\) 5785.10 0.784065 0.392032 0.919951i \(-0.371772\pi\)
0.392032 + 0.919951i \(0.371772\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12970.9 1.73050 0.865252 0.501337i \(-0.167158\pi\)
0.865252 + 0.501337i \(0.167158\pi\)
\(384\) 0 0
\(385\) −5646.54 −0.747466
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4556.16 0.593847 0.296923 0.954901i \(-0.404039\pi\)
0.296923 + 0.954901i \(0.404039\pi\)
\(390\) 0 0
\(391\) −4097.34 −0.529953
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7313.52 0.931603
\(396\) 0 0
\(397\) 3193.40 0.403709 0.201854 0.979416i \(-0.435303\pi\)
0.201854 + 0.979416i \(0.435303\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −15032.0 −1.87197 −0.935986 0.352038i \(-0.885489\pi\)
−0.935986 + 0.352038i \(0.885489\pi\)
\(402\) 0 0
\(403\) −4713.93 −0.582674
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9621.01 1.17173
\(408\) 0 0
\(409\) −633.869 −0.0766328 −0.0383164 0.999266i \(-0.512199\pi\)
−0.0383164 + 0.999266i \(0.512199\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6361.30 0.757916
\(414\) 0 0
\(415\) −7115.54 −0.841658
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13989.9 1.63115 0.815574 0.578653i \(-0.196422\pi\)
0.815574 + 0.578653i \(0.196422\pi\)
\(420\) 0 0
\(421\) 8853.43 1.02492 0.512458 0.858712i \(-0.328735\pi\)
0.512458 + 0.858712i \(0.328735\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4426.70 0.505238
\(426\) 0 0
\(427\) 17962.8 2.03579
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16509.1 −1.84504 −0.922522 0.385945i \(-0.873875\pi\)
−0.922522 + 0.385945i \(0.873875\pi\)
\(432\) 0 0
\(433\) 2218.19 0.246188 0.123094 0.992395i \(-0.460718\pi\)
0.123094 + 0.992395i \(0.460718\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7454.04 0.815961
\(438\) 0 0
\(439\) −1445.82 −0.157187 −0.0785936 0.996907i \(-0.525043\pi\)
−0.0785936 + 0.996907i \(0.525043\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −985.625 −0.105708 −0.0528538 0.998602i \(-0.516832\pi\)
−0.0528538 + 0.998602i \(0.516832\pi\)
\(444\) 0 0
\(445\) 6359.44 0.677453
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6682.65 0.702391 0.351196 0.936302i \(-0.385775\pi\)
0.351196 + 0.936302i \(0.385775\pi\)
\(450\) 0 0
\(451\) −8204.79 −0.856649
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4599.18 −0.473875
\(456\) 0 0
\(457\) 4765.44 0.487785 0.243893 0.969802i \(-0.421576\pi\)
0.243893 + 0.969802i \(0.421576\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9834.66 0.993592 0.496796 0.867867i \(-0.334510\pi\)
0.496796 + 0.867867i \(0.334510\pi\)
\(462\) 0 0
\(463\) −5580.51 −0.560147 −0.280074 0.959978i \(-0.590359\pi\)
−0.280074 + 0.959978i \(0.590359\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4459.02 0.441839 0.220920 0.975292i \(-0.429094\pi\)
0.220920 + 0.975292i \(0.429094\pi\)
\(468\) 0 0
\(469\) −9884.58 −0.973193
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1778.92 0.172928
\(474\) 0 0
\(475\) −8053.21 −0.777909
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10269.3 0.979571 0.489785 0.871843i \(-0.337075\pi\)
0.489785 + 0.871843i \(0.337075\pi\)
\(480\) 0 0
\(481\) 7836.44 0.742850
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6549.75 0.613214
\(486\) 0 0
\(487\) −904.158 −0.0841300 −0.0420650 0.999115i \(-0.513394\pi\)
−0.0420650 + 0.999115i \(0.513394\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20527.1 1.88671 0.943354 0.331789i \(-0.107652\pi\)
0.943354 + 0.331789i \(0.107652\pi\)
\(492\) 0 0
\(493\) −8566.57 −0.782594
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8694.34 −0.784697
\(498\) 0 0
\(499\) −4174.28 −0.374482 −0.187241 0.982314i \(-0.559955\pi\)
−0.187241 + 0.982314i \(0.559955\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4835.95 0.428676 0.214338 0.976760i \(-0.431241\pi\)
0.214338 + 0.976760i \(0.431241\pi\)
\(504\) 0 0
\(505\) 7170.97 0.631889
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5204.54 0.453216 0.226608 0.973986i \(-0.427236\pi\)
0.226608 + 0.973986i \(0.427236\pi\)
\(510\) 0 0
\(511\) −28008.5 −2.42470
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7056.52 −0.603781
\(516\) 0 0
\(517\) −4823.32 −0.410308
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12174.0 1.02371 0.511856 0.859071i \(-0.328958\pi\)
0.511856 + 0.859071i \(0.328958\pi\)
\(522\) 0 0
\(523\) 11583.5 0.968470 0.484235 0.874938i \(-0.339098\pi\)
0.484235 + 0.874938i \(0.339098\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7532.14 −0.622591
\(528\) 0 0
\(529\) −5022.37 −0.412786
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6682.91 −0.543094
\(534\) 0 0
\(535\) −10324.8 −0.834353
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12641.3 −1.01020
\(540\) 0 0
\(541\) 18249.3 1.45027 0.725137 0.688605i \(-0.241777\pi\)
0.725137 + 0.688605i \(0.241777\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5657.96 0.444698
\(546\) 0 0
\(547\) −9610.00 −0.751177 −0.375588 0.926787i \(-0.622559\pi\)
−0.375588 + 0.926787i \(0.622559\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15584.6 1.20495
\(552\) 0 0
\(553\) 32920.1 2.53148
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20247.0 1.54020 0.770101 0.637923i \(-0.220206\pi\)
0.770101 + 0.637923i \(0.220206\pi\)
\(558\) 0 0
\(559\) 1448.95 0.109632
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18344.2 1.37321 0.686603 0.727033i \(-0.259101\pi\)
0.686603 + 0.727033i \(0.259101\pi\)
\(564\) 0 0
\(565\) −11441.3 −0.851926
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23668.6 1.74383 0.871914 0.489658i \(-0.162878\pi\)
0.871914 + 0.489658i \(0.162878\pi\)
\(570\) 0 0
\(571\) 8851.13 0.648701 0.324351 0.945937i \(-0.394854\pi\)
0.324351 + 0.945937i \(0.394854\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7718.93 −0.559829
\(576\) 0 0
\(577\) 16832.2 1.21444 0.607221 0.794533i \(-0.292284\pi\)
0.607221 + 0.794533i \(0.292284\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −32029.0 −2.28707
\(582\) 0 0
\(583\) −21492.0 −1.52677
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23298.5 −1.63822 −0.819109 0.573638i \(-0.805532\pi\)
−0.819109 + 0.573638i \(0.805532\pi\)
\(588\) 0 0
\(589\) 13702.8 0.958595
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8720.33 −0.603881 −0.301940 0.953327i \(-0.597634\pi\)
−0.301940 + 0.953327i \(0.597634\pi\)
\(594\) 0 0
\(595\) −7348.79 −0.506338
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 366.400 0.0249928 0.0124964 0.999922i \(-0.496022\pi\)
0.0124964 + 0.999922i \(0.496022\pi\)
\(600\) 0 0
\(601\) 5886.95 0.399557 0.199779 0.979841i \(-0.435978\pi\)
0.199779 + 0.979841i \(0.435978\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −326.513 −0.0219415
\(606\) 0 0
\(607\) −18187.1 −1.21613 −0.608067 0.793886i \(-0.708055\pi\)
−0.608067 + 0.793886i \(0.708055\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3928.65 −0.260125
\(612\) 0 0
\(613\) 1998.47 0.131676 0.0658379 0.997830i \(-0.479028\pi\)
0.0658379 + 0.997830i \(0.479028\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15800.3 1.03095 0.515474 0.856905i \(-0.327616\pi\)
0.515474 + 0.856905i \(0.327616\pi\)
\(618\) 0 0
\(619\) 9766.86 0.634189 0.317095 0.948394i \(-0.397293\pi\)
0.317095 + 0.948394i \(0.397293\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 28625.6 1.84087
\(624\) 0 0
\(625\) 4129.43 0.264283
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12521.4 0.793739
\(630\) 0 0
\(631\) −9966.80 −0.628799 −0.314399 0.949291i \(-0.601803\pi\)
−0.314399 + 0.949291i \(0.601803\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3511.01 0.219418
\(636\) 0 0
\(637\) −10296.5 −0.640440
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12710.5 0.783206 0.391603 0.920134i \(-0.371921\pi\)
0.391603 + 0.920134i \(0.371921\pi\)
\(642\) 0 0
\(643\) 12811.4 0.785742 0.392871 0.919594i \(-0.371482\pi\)
0.392871 + 0.919594i \(0.371482\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3729.11 0.226594 0.113297 0.993561i \(-0.463859\pi\)
0.113297 + 0.993561i \(0.463859\pi\)
\(648\) 0 0
\(649\) 9069.97 0.548579
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3306.70 −0.198164 −0.0990820 0.995079i \(-0.531591\pi\)
−0.0990820 + 0.995079i \(0.531591\pi\)
\(654\) 0 0
\(655\) −8181.07 −0.488032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17957.0 −1.06146 −0.530732 0.847539i \(-0.678083\pi\)
−0.530732 + 0.847539i \(0.678083\pi\)
\(660\) 0 0
\(661\) 9695.14 0.570495 0.285247 0.958454i \(-0.407924\pi\)
0.285247 + 0.958454i \(0.407924\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13369.2 0.779602
\(666\) 0 0
\(667\) 14937.7 0.867153
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 25611.5 1.47350
\(672\) 0 0
\(673\) 21587.2 1.23644 0.618222 0.786003i \(-0.287853\pi\)
0.618222 + 0.786003i \(0.287853\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20672.4 1.17356 0.586782 0.809745i \(-0.300394\pi\)
0.586782 + 0.809745i \(0.300394\pi\)
\(678\) 0 0
\(679\) 29482.2 1.66631
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24403.9 1.36719 0.683594 0.729862i \(-0.260416\pi\)
0.683594 + 0.729862i \(0.260416\pi\)
\(684\) 0 0
\(685\) −13136.3 −0.732719
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17505.5 −0.967935
\(690\) 0 0
\(691\) −5858.07 −0.322506 −0.161253 0.986913i \(-0.551553\pi\)
−0.161253 + 0.986913i \(0.551553\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8940.19 −0.487943
\(696\) 0 0
\(697\) −10678.3 −0.580299
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10530.8 0.567393 0.283697 0.958914i \(-0.408439\pi\)
0.283697 + 0.958914i \(0.408439\pi\)
\(702\) 0 0
\(703\) −22779.5 −1.22211
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32278.5 1.71705
\(708\) 0 0
\(709\) −3776.78 −0.200057 −0.100028 0.994985i \(-0.531893\pi\)
−0.100028 + 0.994985i \(0.531893\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13134.0 0.689861
\(714\) 0 0
\(715\) −6557.53 −0.342990
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7109.15 0.368743 0.184372 0.982857i \(-0.440975\pi\)
0.184372 + 0.982857i \(0.440975\pi\)
\(720\) 0 0
\(721\) −31763.3 −1.64068
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −16138.4 −0.826713
\(726\) 0 0
\(727\) 3660.36 0.186733 0.0933667 0.995632i \(-0.470237\pi\)
0.0933667 + 0.995632i \(0.470237\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2315.21 0.117142
\(732\) 0 0
\(733\) −36559.4 −1.84223 −0.921113 0.389296i \(-0.872718\pi\)
−0.921113 + 0.389296i \(0.872718\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14093.5 −0.704396
\(738\) 0 0
\(739\) 15735.7 0.783285 0.391642 0.920118i \(-0.371907\pi\)
0.391642 + 0.920118i \(0.371907\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −23694.3 −1.16993 −0.584966 0.811058i \(-0.698892\pi\)
−0.584966 + 0.811058i \(0.698892\pi\)
\(744\) 0 0
\(745\) −3557.69 −0.174958
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −46474.6 −2.26722
\(750\) 0 0
\(751\) −17287.0 −0.839962 −0.419981 0.907533i \(-0.637963\pi\)
−0.419981 + 0.907533i \(0.637963\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13771.4 −0.663831
\(756\) 0 0
\(757\) 18565.8 0.891394 0.445697 0.895184i \(-0.352956\pi\)
0.445697 + 0.895184i \(0.352956\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5937.47 −0.282829 −0.141415 0.989950i \(-0.545165\pi\)
−0.141415 + 0.989950i \(0.545165\pi\)
\(762\) 0 0
\(763\) 25468.0 1.20839
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7387.61 0.347785
\(768\) 0 0
\(769\) 2489.54 0.116743 0.0583713 0.998295i \(-0.481409\pi\)
0.0583713 + 0.998295i \(0.481409\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16109.8 −0.749584 −0.374792 0.927109i \(-0.622286\pi\)
−0.374792 + 0.927109i \(0.622286\pi\)
\(774\) 0 0
\(775\) −14189.7 −0.657690
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19426.3 0.893479
\(780\) 0 0
\(781\) −12396.4 −0.567963
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10260.8 0.466528
\(786\) 0 0
\(787\) 15810.0 0.716095 0.358047 0.933703i \(-0.383443\pi\)
0.358047 + 0.933703i \(0.383443\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −51500.2 −2.31497
\(792\) 0 0
\(793\) 20860.9 0.934164
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4185.16 −0.186005 −0.0930024 0.995666i \(-0.529646\pi\)
−0.0930024 + 0.995666i \(0.529646\pi\)
\(798\) 0 0
\(799\) −6277.39 −0.277945
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −39934.6 −1.75500
\(804\) 0 0
\(805\) 12814.3 0.561047
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −35154.2 −1.52776 −0.763878 0.645360i \(-0.776707\pi\)
−0.763878 + 0.645360i \(0.776707\pi\)
\(810\) 0 0
\(811\) −3028.93 −0.131147 −0.0655734 0.997848i \(-0.520888\pi\)
−0.0655734 + 0.997848i \(0.520888\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15424.4 −0.662938
\(816\) 0 0
\(817\) −4211.91 −0.180363
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4797.98 −0.203959 −0.101980 0.994786i \(-0.532518\pi\)
−0.101980 + 0.994786i \(0.532518\pi\)
\(822\) 0 0
\(823\) −40058.7 −1.69667 −0.848335 0.529460i \(-0.822394\pi\)
−0.848335 + 0.529460i \(0.822394\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22168.6 0.932139 0.466069 0.884748i \(-0.345670\pi\)
0.466069 + 0.884748i \(0.345670\pi\)
\(828\) 0 0
\(829\) 13091.9 0.548494 0.274247 0.961659i \(-0.411571\pi\)
0.274247 + 0.961659i \(0.411571\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −16452.2 −0.684314
\(834\) 0 0
\(835\) 19456.1 0.806354
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34790.3 1.43158 0.715790 0.698316i \(-0.246067\pi\)
0.715790 + 0.698316i \(0.246067\pi\)
\(840\) 0 0
\(841\) 6842.23 0.280546
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7408.93 0.301627
\(846\) 0 0
\(847\) −1469.72 −0.0596225
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21833.9 −0.879502
\(852\) 0 0
\(853\) 47980.7 1.92594 0.962971 0.269604i \(-0.0868928\pi\)
0.962971 + 0.269604i \(0.0868928\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32015.4 −1.27611 −0.638055 0.769991i \(-0.720261\pi\)
−0.638055 + 0.769991i \(0.720261\pi\)
\(858\) 0 0
\(859\) −32513.5 −1.29144 −0.645719 0.763575i \(-0.723442\pi\)
−0.645719 + 0.763575i \(0.723442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25867.1 −1.02031 −0.510154 0.860083i \(-0.670411\pi\)
−0.510154 + 0.860083i \(0.670411\pi\)
\(864\) 0 0
\(865\) −14.1365 −0.000555670 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 46937.7 1.83228
\(870\) 0 0
\(871\) −11479.3 −0.446569
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −32794.5 −1.26704
\(876\) 0 0
\(877\) −574.703 −0.0221281 −0.0110641 0.999939i \(-0.503522\pi\)
−0.0110641 + 0.999939i \(0.503522\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 28334.1 1.08354 0.541771 0.840526i \(-0.317754\pi\)
0.541771 + 0.840526i \(0.317754\pi\)
\(882\) 0 0
\(883\) 13424.9 0.511646 0.255823 0.966724i \(-0.417654\pi\)
0.255823 + 0.966724i \(0.417654\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43763.1 −1.65662 −0.828310 0.560271i \(-0.810697\pi\)
−0.828310 + 0.560271i \(0.810697\pi\)
\(888\) 0 0
\(889\) 15804.0 0.596232
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11420.1 0.427948
\(894\) 0 0
\(895\) −6290.99 −0.234955
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 27460.0 1.01874
\(900\) 0 0
\(901\) −27971.2 −1.03424
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5048.05 −0.185417
\(906\) 0 0
\(907\) 42797.1 1.56677 0.783383 0.621540i \(-0.213493\pi\)
0.783383 + 0.621540i \(0.213493\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2007.15 0.0729964 0.0364982 0.999334i \(-0.488380\pi\)
0.0364982 + 0.999334i \(0.488380\pi\)
\(912\) 0 0
\(913\) −45667.1 −1.65538
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −36825.2 −1.32614
\(918\) 0 0
\(919\) −43558.3 −1.56350 −0.781750 0.623592i \(-0.785673\pi\)
−0.781750 + 0.623592i \(0.785673\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10097.0 −0.360074
\(924\) 0 0
\(925\) 23589.0 0.838487
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −31508.1 −1.11275 −0.556376 0.830931i \(-0.687808\pi\)
−0.556376 + 0.830931i \(0.687808\pi\)
\(930\) 0 0
\(931\) 29930.4 1.05363
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10477.9 −0.366487
\(936\) 0 0
\(937\) −20675.3 −0.720846 −0.360423 0.932789i \(-0.617368\pi\)
−0.360423 + 0.932789i \(0.617368\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1425.54 −0.0493849 −0.0246925 0.999695i \(-0.507861\pi\)
−0.0246925 + 0.999695i \(0.507861\pi\)
\(942\) 0 0
\(943\) 18619.9 0.643000
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35661.7 1.22371 0.611853 0.790971i \(-0.290424\pi\)
0.611853 + 0.790971i \(0.290424\pi\)
\(948\) 0 0
\(949\) −32527.2 −1.11262
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10384.9 0.352991 0.176496 0.984301i \(-0.443524\pi\)
0.176496 + 0.984301i \(0.443524\pi\)
\(954\) 0 0
\(955\) 2467.31 0.0836024
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −59130.1 −1.99104
\(960\) 0 0
\(961\) −5646.82 −0.189548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.92005 0.000330920 0
\(966\) 0 0
\(967\) −8443.39 −0.280787 −0.140394 0.990096i \(-0.544837\pi\)
−0.140394 + 0.990096i \(0.544837\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6938.43 −0.229315 −0.114657 0.993405i \(-0.536577\pi\)
−0.114657 + 0.993405i \(0.536577\pi\)
\(972\) 0 0
\(973\) −40242.2 −1.32590
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38777.5 1.26981 0.634903 0.772591i \(-0.281040\pi\)
0.634903 + 0.772591i \(0.281040\pi\)
\(978\) 0 0
\(979\) 40814.5 1.33242
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7592.55 −0.246353 −0.123176 0.992385i \(-0.539308\pi\)
−0.123176 + 0.992385i \(0.539308\pi\)
\(984\) 0 0
\(985\) 17871.3 0.578098
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4037.08 −0.129799
\(990\) 0 0
\(991\) 5.96741 0.000191282 0 9.56412e−5 1.00000i \(-0.499970\pi\)
9.56412e−5 1.00000i \(0.499970\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −18185.0 −0.579401
\(996\) 0 0
\(997\) 34771.7 1.10454 0.552272 0.833664i \(-0.313761\pi\)
0.552272 + 0.833664i \(0.313761\pi\)
\(998\) 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 648.4.a.j.1.2 yes 4
3.2 odd 2 648.4.a.g.1.3 4
4.3 odd 2 1296.4.a.bb.1.2 4
9.2 odd 6 648.4.i.v.433.2 8
9.4 even 3 648.4.i.u.217.3 8
9.5 odd 6 648.4.i.v.217.2 8
9.7 even 3 648.4.i.u.433.3 8
12.11 even 2 1296.4.a.x.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.4.a.g.1.3 4 3.2 odd 2
648.4.a.j.1.2 yes 4 1.1 even 1 trivial
648.4.i.u.217.3 8 9.4 even 3
648.4.i.u.433.3 8 9.7 even 3
648.4.i.v.217.2 8 9.5 odd 6
648.4.i.v.433.2 8 9.2 odd 6
1296.4.a.x.1.3 4 12.11 even 2
1296.4.a.bb.1.2 4 4.3 odd 2