Properties

Label 648.4.a.h.1.1
Level $648$
Weight $4$
Character 648.1
Self dual yes
Analytic conductor $38.233$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.72153.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 3x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.15756\) 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)
 
\(f(q)\) \(=\) \(q-17.3189 q^{5} -2.37353 q^{7} +O(q^{10})\) \(q-17.3189 q^{5} -2.37353 q^{7} +52.2165 q^{11} +13.6993 q^{13} +82.9217 q^{17} -126.803 q^{19} +54.3661 q^{23} +174.944 q^{25} +212.850 q^{29} -224.122 q^{31} +41.1068 q^{35} -32.2229 q^{37} -500.662 q^{41} -12.8254 q^{43} -209.541 q^{47} -337.366 q^{49} -371.213 q^{53} -904.331 q^{55} -5.81848 q^{59} -604.032 q^{61} -237.256 q^{65} +754.066 q^{67} -43.4780 q^{71} +671.029 q^{73} -123.937 q^{77} -649.202 q^{79} -134.526 q^{83} -1436.11 q^{85} +206.076 q^{89} -32.5156 q^{91} +2196.09 q^{95} -1452.84 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{5} - 3 q^{7} + 16 q^{11} - 29 q^{13} + 17 q^{17} - 109 q^{19} + 37 q^{23} - 97 q^{25} - 3 q^{29} - 331 q^{31} + 171 q^{35} - 366 q^{37} - 378 q^{41} - 506 q^{43} + 171 q^{47} - 829 q^{49} - 410 q^{53} - 1163 q^{55} + 616 q^{59} - 1331 q^{61} - 815 q^{65} - 1162 q^{67} + 344 q^{71} - 1307 q^{73} - 741 q^{77} - 1853 q^{79} + 1421 q^{83} - 2074 q^{85} - 816 q^{89} - 1995 q^{91} + 1292 q^{95} - 2506 q^{97}+O(q^{100}) \) 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
\(5\) −17.3189 −1.54905 −0.774524 0.632545i \(-0.782010\pi\)
−0.774524 + 0.632545i \(0.782010\pi\)
\(6\) 0 0
\(7\) −2.37353 −0.128158 −0.0640792 0.997945i \(-0.520411\pi\)
−0.0640792 + 0.997945i \(0.520411\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 52.2165 1.43126 0.715630 0.698480i \(-0.246140\pi\)
0.715630 + 0.698480i \(0.246140\pi\)
\(12\) 0 0
\(13\) 13.6993 0.292269 0.146135 0.989265i \(-0.453317\pi\)
0.146135 + 0.989265i \(0.453317\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 82.9217 1.18303 0.591514 0.806295i \(-0.298531\pi\)
0.591514 + 0.806295i \(0.298531\pi\)
\(18\) 0 0
\(19\) −126.803 −1.53108 −0.765542 0.643386i \(-0.777529\pi\)
−0.765542 + 0.643386i \(0.777529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 54.3661 0.492875 0.246437 0.969159i \(-0.420740\pi\)
0.246437 + 0.969159i \(0.420740\pi\)
\(24\) 0 0
\(25\) 174.944 1.39955
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 212.850 1.36294 0.681471 0.731845i \(-0.261340\pi\)
0.681471 + 0.731845i \(0.261340\pi\)
\(30\) 0 0
\(31\) −224.122 −1.29850 −0.649250 0.760575i \(-0.724917\pi\)
−0.649250 + 0.760575i \(0.724917\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 41.1068 0.198523
\(36\) 0 0
\(37\) −32.2229 −0.143173 −0.0715866 0.997434i \(-0.522806\pi\)
−0.0715866 + 0.997434i \(0.522806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −500.662 −1.90708 −0.953540 0.301265i \(-0.902591\pi\)
−0.953540 + 0.301265i \(0.902591\pi\)
\(42\) 0 0
\(43\) −12.8254 −0.0454850 −0.0227425 0.999741i \(-0.507240\pi\)
−0.0227425 + 0.999741i \(0.507240\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −209.541 −0.650311 −0.325156 0.945661i \(-0.605417\pi\)
−0.325156 + 0.945661i \(0.605417\pi\)
\(48\) 0 0
\(49\) −337.366 −0.983575
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −371.213 −0.962075 −0.481038 0.876700i \(-0.659740\pi\)
−0.481038 + 0.876700i \(0.659740\pi\)
\(54\) 0 0
\(55\) −904.331 −2.21709
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.81848 −0.0128390 −0.00641950 0.999979i \(-0.502043\pi\)
−0.00641950 + 0.999979i \(0.502043\pi\)
\(60\) 0 0
\(61\) −604.032 −1.26784 −0.633921 0.773398i \(-0.718556\pi\)
−0.633921 + 0.773398i \(0.718556\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −237.256 −0.452739
\(66\) 0 0
\(67\) 754.066 1.37498 0.687491 0.726193i \(-0.258712\pi\)
0.687491 + 0.726193i \(0.258712\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −43.4780 −0.0726744 −0.0363372 0.999340i \(-0.511569\pi\)
−0.0363372 + 0.999340i \(0.511569\pi\)
\(72\) 0 0
\(73\) 671.029 1.07586 0.537931 0.842989i \(-0.319206\pi\)
0.537931 + 0.842989i \(0.319206\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −123.937 −0.183428
\(78\) 0 0
\(79\) −649.202 −0.924569 −0.462284 0.886732i \(-0.652970\pi\)
−0.462284 + 0.886732i \(0.652970\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −134.526 −0.177905 −0.0889526 0.996036i \(-0.528352\pi\)
−0.0889526 + 0.996036i \(0.528352\pi\)
\(84\) 0 0
\(85\) −1436.11 −1.83257
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 206.076 0.245439 0.122719 0.992441i \(-0.460838\pi\)
0.122719 + 0.992441i \(0.460838\pi\)
\(90\) 0 0
\(91\) −32.5156 −0.0374567
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2196.09 2.37172
\(96\) 0 0
\(97\) −1452.84 −1.52076 −0.760379 0.649480i \(-0.774987\pi\)
−0.760379 + 0.649480i \(0.774987\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −203.960 −0.200938 −0.100469 0.994940i \(-0.532034\pi\)
−0.100469 + 0.994940i \(0.532034\pi\)
\(102\) 0 0
\(103\) −1555.55 −1.48808 −0.744042 0.668133i \(-0.767094\pi\)
−0.744042 + 0.668133i \(0.767094\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 584.436 0.528033 0.264016 0.964518i \(-0.414953\pi\)
0.264016 + 0.964518i \(0.414953\pi\)
\(108\) 0 0
\(109\) −782.671 −0.687764 −0.343882 0.939013i \(-0.611742\pi\)
−0.343882 + 0.939013i \(0.611742\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 870.249 0.724479 0.362239 0.932085i \(-0.382012\pi\)
0.362239 + 0.932085i \(0.382012\pi\)
\(114\) 0 0
\(115\) −941.561 −0.763487
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −196.817 −0.151615
\(120\) 0 0
\(121\) 1395.56 1.04851
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −864.968 −0.618921
\(126\) 0 0
\(127\) 994.050 0.694548 0.347274 0.937764i \(-0.387107\pi\)
0.347274 + 0.937764i \(0.387107\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 802.079 0.534946 0.267473 0.963565i \(-0.413811\pi\)
0.267473 + 0.963565i \(0.413811\pi\)
\(132\) 0 0
\(133\) 300.970 0.196221
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1319.87 −0.823096 −0.411548 0.911388i \(-0.635012\pi\)
−0.411548 + 0.911388i \(0.635012\pi\)
\(138\) 0 0
\(139\) 526.793 0.321453 0.160727 0.986999i \(-0.448616\pi\)
0.160727 + 0.986999i \(0.448616\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 715.329 0.418313
\(144\) 0 0
\(145\) −3686.33 −2.11126
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3077.75 −1.69221 −0.846105 0.533017i \(-0.821058\pi\)
−0.846105 + 0.533017i \(0.821058\pi\)
\(150\) 0 0
\(151\) 82.1909 0.0442953 0.0221477 0.999755i \(-0.492950\pi\)
0.0221477 + 0.999755i \(0.492950\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3881.54 2.01144
\(156\) 0 0
\(157\) −2384.18 −1.21196 −0.605981 0.795479i \(-0.707219\pi\)
−0.605981 + 0.795479i \(0.707219\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −129.039 −0.0631660
\(162\) 0 0
\(163\) 1226.77 0.589496 0.294748 0.955575i \(-0.404764\pi\)
0.294748 + 0.955575i \(0.404764\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3896.40 −1.80547 −0.902733 0.430202i \(-0.858442\pi\)
−0.902733 + 0.430202i \(0.858442\pi\)
\(168\) 0 0
\(169\) −2009.33 −0.914579
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −146.404 −0.0643403 −0.0321701 0.999482i \(-0.510242\pi\)
−0.0321701 + 0.999482i \(0.510242\pi\)
\(174\) 0 0
\(175\) −415.233 −0.179364
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1540.84 0.643396 0.321698 0.946842i \(-0.395746\pi\)
0.321698 + 0.946842i \(0.395746\pi\)
\(180\) 0 0
\(181\) 1173.94 0.482090 0.241045 0.970514i \(-0.422510\pi\)
0.241045 + 0.970514i \(0.422510\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 558.065 0.221782
\(186\) 0 0
\(187\) 4329.88 1.69322
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2193.64 0.831026 0.415513 0.909587i \(-0.363602\pi\)
0.415513 + 0.909587i \(0.363602\pi\)
\(192\) 0 0
\(193\) 2463.31 0.918721 0.459360 0.888250i \(-0.348079\pi\)
0.459360 + 0.888250i \(0.348079\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3593.52 −1.29963 −0.649816 0.760091i \(-0.725154\pi\)
−0.649816 + 0.760091i \(0.725154\pi\)
\(198\) 0 0
\(199\) −4599.36 −1.63839 −0.819197 0.573513i \(-0.805580\pi\)
−0.819197 + 0.573513i \(0.805580\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −505.206 −0.174672
\(204\) 0 0
\(205\) 8670.91 2.95416
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6621.20 −2.19138
\(210\) 0 0
\(211\) −2332.24 −0.760938 −0.380469 0.924794i \(-0.624237\pi\)
−0.380469 + 0.924794i \(0.624237\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 222.122 0.0704584
\(216\) 0 0
\(217\) 531.959 0.166414
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1135.97 0.345763
\(222\) 0 0
\(223\) −4057.10 −1.21831 −0.609156 0.793050i \(-0.708492\pi\)
−0.609156 + 0.793050i \(0.708492\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1297.25 −0.379303 −0.189651 0.981852i \(-0.560736\pi\)
−0.189651 + 0.981852i \(0.560736\pi\)
\(228\) 0 0
\(229\) 3111.96 0.898010 0.449005 0.893529i \(-0.351779\pi\)
0.449005 + 0.893529i \(0.351779\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2034.74 −0.572103 −0.286052 0.958214i \(-0.592343\pi\)
−0.286052 + 0.958214i \(0.592343\pi\)
\(234\) 0 0
\(235\) 3629.01 1.00736
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1468.46 −0.397434 −0.198717 0.980057i \(-0.563677\pi\)
−0.198717 + 0.980057i \(0.563677\pi\)
\(240\) 0 0
\(241\) 659.163 0.176184 0.0880922 0.996112i \(-0.471923\pi\)
0.0880922 + 0.996112i \(0.471923\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5842.81 1.52361
\(246\) 0 0
\(247\) −1737.11 −0.447489
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5509.13 1.38539 0.692696 0.721230i \(-0.256423\pi\)
0.692696 + 0.721230i \(0.256423\pi\)
\(252\) 0 0
\(253\) 2838.81 0.705432
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3667.63 −0.890197 −0.445098 0.895482i \(-0.646831\pi\)
−0.445098 + 0.895482i \(0.646831\pi\)
\(258\) 0 0
\(259\) 76.4819 0.0183489
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4992.77 1.17060 0.585299 0.810818i \(-0.300977\pi\)
0.585299 + 0.810818i \(0.300977\pi\)
\(264\) 0 0
\(265\) 6428.99 1.49030
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −88.5142 −0.0200625 −0.0100312 0.999950i \(-0.503193\pi\)
−0.0100312 + 0.999950i \(0.503193\pi\)
\(270\) 0 0
\(271\) 5226.56 1.17155 0.585777 0.810472i \(-0.300789\pi\)
0.585777 + 0.810472i \(0.300789\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9134.94 2.00312
\(276\) 0 0
\(277\) −2112.25 −0.458170 −0.229085 0.973406i \(-0.573573\pi\)
−0.229085 + 0.973406i \(0.573573\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1725.14 −0.366239 −0.183120 0.983091i \(-0.558620\pi\)
−0.183120 + 0.983091i \(0.558620\pi\)
\(282\) 0 0
\(283\) 3341.79 0.701940 0.350970 0.936387i \(-0.385852\pi\)
0.350970 + 0.936387i \(0.385852\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1188.33 0.244408
\(288\) 0 0
\(289\) 1963.01 0.399554
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5258.42 1.04846 0.524232 0.851575i \(-0.324352\pi\)
0.524232 + 0.851575i \(0.324352\pi\)
\(294\) 0 0
\(295\) 100.770 0.0198882
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 744.778 0.144052
\(300\) 0 0
\(301\) 30.4414 0.00582928
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10461.2 1.96395
\(306\) 0 0
\(307\) 2582.56 0.480112 0.240056 0.970759i \(-0.422834\pi\)
0.240056 + 0.970759i \(0.422834\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5088.53 0.927794 0.463897 0.885889i \(-0.346451\pi\)
0.463897 + 0.885889i \(0.346451\pi\)
\(312\) 0 0
\(313\) −2931.06 −0.529308 −0.264654 0.964343i \(-0.585258\pi\)
−0.264654 + 0.964343i \(0.585258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3599.57 −0.637766 −0.318883 0.947794i \(-0.603308\pi\)
−0.318883 + 0.947794i \(0.603308\pi\)
\(318\) 0 0
\(319\) 11114.3 1.95073
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10514.7 −1.81131
\(324\) 0 0
\(325\) 2396.60 0.409045
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 497.350 0.0833428
\(330\) 0 0
\(331\) 3677.81 0.610727 0.305364 0.952236i \(-0.401222\pi\)
0.305364 + 0.952236i \(0.401222\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −13059.6 −2.12991
\(336\) 0 0
\(337\) −5556.04 −0.898091 −0.449046 0.893509i \(-0.648236\pi\)
−0.449046 + 0.893509i \(0.648236\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11702.9 −1.85849
\(342\) 0 0
\(343\) 1614.87 0.254212
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3069.12 0.474810 0.237405 0.971411i \(-0.423703\pi\)
0.237405 + 0.971411i \(0.423703\pi\)
\(348\) 0 0
\(349\) −1191.43 −0.182739 −0.0913694 0.995817i \(-0.529124\pi\)
−0.0913694 + 0.995817i \(0.529124\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6431.00 −0.969653 −0.484827 0.874610i \(-0.661117\pi\)
−0.484827 + 0.874610i \(0.661117\pi\)
\(354\) 0 0
\(355\) 752.989 0.112576
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4236.73 −0.622858 −0.311429 0.950270i \(-0.600808\pi\)
−0.311429 + 0.950270i \(0.600808\pi\)
\(360\) 0 0
\(361\) 9219.99 1.34422
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11621.5 −1.66656
\(366\) 0 0
\(367\) 7372.78 1.04865 0.524327 0.851517i \(-0.324317\pi\)
0.524327 + 0.851517i \(0.324317\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 881.083 0.123298
\(372\) 0 0
\(373\) −11985.0 −1.66370 −0.831852 0.554998i \(-0.812719\pi\)
−0.831852 + 0.554998i \(0.812719\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2915.90 0.398346
\(378\) 0 0
\(379\) −3872.02 −0.524782 −0.262391 0.964962i \(-0.584511\pi\)
−0.262391 + 0.964962i \(0.584511\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7242.66 −0.966274 −0.483137 0.875545i \(-0.660503\pi\)
−0.483137 + 0.875545i \(0.660503\pi\)
\(384\) 0 0
\(385\) 2146.45 0.284139
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9732.33 1.26851 0.634253 0.773126i \(-0.281308\pi\)
0.634253 + 0.773126i \(0.281308\pi\)
\(390\) 0 0
\(391\) 4508.13 0.583085
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11243.5 1.43220
\(396\) 0 0
\(397\) −1393.66 −0.176186 −0.0880930 0.996112i \(-0.528077\pi\)
−0.0880930 + 0.996112i \(0.528077\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1225.66 0.152634 0.0763171 0.997084i \(-0.475684\pi\)
0.0763171 + 0.997084i \(0.475684\pi\)
\(402\) 0 0
\(403\) −3070.31 −0.379512
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1682.57 −0.204918
\(408\) 0 0
\(409\) 8162.83 0.986860 0.493430 0.869785i \(-0.335743\pi\)
0.493430 + 0.869785i \(0.335743\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.8103 0.00164543
\(414\) 0 0
\(415\) 2329.84 0.275584
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2821.60 −0.328984 −0.164492 0.986378i \(-0.552598\pi\)
−0.164492 + 0.986378i \(0.552598\pi\)
\(420\) 0 0
\(421\) −4639.90 −0.537138 −0.268569 0.963260i \(-0.586551\pi\)
−0.268569 + 0.963260i \(0.586551\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14506.6 1.65571
\(426\) 0 0
\(427\) 1433.69 0.162485
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5998.72 −0.670413 −0.335207 0.942145i \(-0.608806\pi\)
−0.335207 + 0.942145i \(0.608806\pi\)
\(432\) 0 0
\(433\) −10187.5 −1.13067 −0.565334 0.824862i \(-0.691253\pi\)
−0.565334 + 0.824862i \(0.691253\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6893.79 −0.754633
\(438\) 0 0
\(439\) 15745.4 1.71182 0.855910 0.517125i \(-0.172998\pi\)
0.855910 + 0.517125i \(0.172998\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2433.66 −0.261009 −0.130504 0.991448i \(-0.541660\pi\)
−0.130504 + 0.991448i \(0.541660\pi\)
\(444\) 0 0
\(445\) −3569.01 −0.380197
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7394.75 −0.777238 −0.388619 0.921399i \(-0.627048\pi\)
−0.388619 + 0.921399i \(0.627048\pi\)
\(450\) 0 0
\(451\) −26142.8 −2.72953
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 563.134 0.0580223
\(456\) 0 0
\(457\) 2129.91 0.218015 0.109008 0.994041i \(-0.465233\pi\)
0.109008 + 0.994041i \(0.465233\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8313.78 0.839938 0.419969 0.907538i \(-0.362041\pi\)
0.419969 + 0.907538i \(0.362041\pi\)
\(462\) 0 0
\(463\) −1845.59 −0.185253 −0.0926263 0.995701i \(-0.529526\pi\)
−0.0926263 + 0.995701i \(0.529526\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15653.2 1.55106 0.775529 0.631312i \(-0.217483\pi\)
0.775529 + 0.631312i \(0.217483\pi\)
\(468\) 0 0
\(469\) −1789.79 −0.176215
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −669.697 −0.0651009
\(474\) 0 0
\(475\) −22183.4 −2.14283
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1973.95 −0.188292 −0.0941462 0.995558i \(-0.530012\pi\)
−0.0941462 + 0.995558i \(0.530012\pi\)
\(480\) 0 0
\(481\) −441.431 −0.0418451
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 25161.6 2.35573
\(486\) 0 0
\(487\) −1753.49 −0.163158 −0.0815792 0.996667i \(-0.525996\pi\)
−0.0815792 + 0.996667i \(0.525996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6222.18 −0.571900 −0.285950 0.958245i \(-0.592309\pi\)
−0.285950 + 0.958245i \(0.592309\pi\)
\(492\) 0 0
\(493\) 17649.9 1.61240
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 103.196 0.00931383
\(498\) 0 0
\(499\) −17293.8 −1.55145 −0.775727 0.631069i \(-0.782616\pi\)
−0.775727 + 0.631069i \(0.782616\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20578.6 −1.82416 −0.912082 0.410007i \(-0.865526\pi\)
−0.912082 + 0.410007i \(0.865526\pi\)
\(504\) 0 0
\(505\) 3532.35 0.311263
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18555.1 1.61580 0.807900 0.589319i \(-0.200604\pi\)
0.807900 + 0.589319i \(0.200604\pi\)
\(510\) 0 0
\(511\) −1592.70 −0.137881
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26940.3 2.30511
\(516\) 0 0
\(517\) −10941.5 −0.930764
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5321.51 0.447485 0.223742 0.974648i \(-0.428173\pi\)
0.223742 + 0.974648i \(0.428173\pi\)
\(522\) 0 0
\(523\) 7766.94 0.649378 0.324689 0.945821i \(-0.394740\pi\)
0.324689 + 0.945821i \(0.394740\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18584.6 −1.53616
\(528\) 0 0
\(529\) −9211.32 −0.757074
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6858.72 −0.557381
\(534\) 0 0
\(535\) −10121.8 −0.817948
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17616.1 −1.40775
\(540\) 0 0
\(541\) −1292.25 −0.102695 −0.0513476 0.998681i \(-0.516352\pi\)
−0.0513476 + 0.998681i \(0.516352\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13555.0 1.06538
\(546\) 0 0
\(547\) 7532.95 0.588822 0.294411 0.955679i \(-0.404876\pi\)
0.294411 + 0.955679i \(0.404876\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −26990.1 −2.08678
\(552\) 0 0
\(553\) 1540.90 0.118491
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9856.22 0.749769 0.374884 0.927072i \(-0.377682\pi\)
0.374884 + 0.927072i \(0.377682\pi\)
\(558\) 0 0
\(559\) −175.699 −0.0132939
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10282.2 0.769704 0.384852 0.922978i \(-0.374253\pi\)
0.384852 + 0.922978i \(0.374253\pi\)
\(564\) 0 0
\(565\) −15071.7 −1.12225
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1322.64 0.0974483 0.0487241 0.998812i \(-0.484484\pi\)
0.0487241 + 0.998812i \(0.484484\pi\)
\(570\) 0 0
\(571\) −4254.58 −0.311819 −0.155909 0.987771i \(-0.549831\pi\)
−0.155909 + 0.987771i \(0.549831\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9511.01 0.689803
\(576\) 0 0
\(577\) −13268.4 −0.957312 −0.478656 0.878003i \(-0.658876\pi\)
−0.478656 + 0.878003i \(0.658876\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 319.301 0.0228000
\(582\) 0 0
\(583\) −19383.4 −1.37698
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4710.50 −0.331215 −0.165608 0.986192i \(-0.552959\pi\)
−0.165608 + 0.986192i \(0.552959\pi\)
\(588\) 0 0
\(589\) 28419.3 1.98811
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −16303.6 −1.12902 −0.564510 0.825426i \(-0.690935\pi\)
−0.564510 + 0.825426i \(0.690935\pi\)
\(594\) 0 0
\(595\) 3408.65 0.234859
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10558.9 0.720240 0.360120 0.932906i \(-0.382736\pi\)
0.360120 + 0.932906i \(0.382736\pi\)
\(600\) 0 0
\(601\) −1872.35 −0.127080 −0.0635399 0.997979i \(-0.520239\pi\)
−0.0635399 + 0.997979i \(0.520239\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24169.5 −1.62418
\(606\) 0 0
\(607\) −23071.7 −1.54276 −0.771378 0.636377i \(-0.780432\pi\)
−0.771378 + 0.636377i \(0.780432\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2870.56 −0.190066
\(612\) 0 0
\(613\) −24292.0 −1.60056 −0.800281 0.599625i \(-0.795317\pi\)
−0.800281 + 0.599625i \(0.795317\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7773.95 0.507240 0.253620 0.967304i \(-0.418379\pi\)
0.253620 + 0.967304i \(0.418379\pi\)
\(618\) 0 0
\(619\) −30575.1 −1.98533 −0.992664 0.120905i \(-0.961421\pi\)
−0.992664 + 0.120905i \(0.961421\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −489.128 −0.0314550
\(624\) 0 0
\(625\) −6887.68 −0.440811
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2671.98 −0.169378
\(630\) 0 0
\(631\) 14474.9 0.913209 0.456605 0.889670i \(-0.349065\pi\)
0.456605 + 0.889670i \(0.349065\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −17215.8 −1.07589
\(636\) 0 0
\(637\) −4621.68 −0.287469
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13683.8 0.843179 0.421589 0.906787i \(-0.361472\pi\)
0.421589 + 0.906787i \(0.361472\pi\)
\(642\) 0 0
\(643\) 16015.5 0.982255 0.491128 0.871088i \(-0.336585\pi\)
0.491128 + 0.871088i \(0.336585\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12224.1 −0.742783 −0.371392 0.928476i \(-0.621119\pi\)
−0.371392 + 0.928476i \(0.621119\pi\)
\(648\) 0 0
\(649\) −303.821 −0.0183760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16129.9 0.966636 0.483318 0.875445i \(-0.339431\pi\)
0.483318 + 0.875445i \(0.339431\pi\)
\(654\) 0 0
\(655\) −13891.1 −0.828657
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7158.63 −0.423157 −0.211579 0.977361i \(-0.567860\pi\)
−0.211579 + 0.977361i \(0.567860\pi\)
\(660\) 0 0
\(661\) 9520.06 0.560193 0.280096 0.959972i \(-0.409634\pi\)
0.280096 + 0.959972i \(0.409634\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5212.46 −0.303956
\(666\) 0 0
\(667\) 11571.9 0.671760
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −31540.4 −1.81461
\(672\) 0 0
\(673\) −7603.64 −0.435511 −0.217755 0.976003i \(-0.569874\pi\)
−0.217755 + 0.976003i \(0.569874\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4951.96 0.281122 0.140561 0.990072i \(-0.455109\pi\)
0.140561 + 0.990072i \(0.455109\pi\)
\(678\) 0 0
\(679\) 3448.35 0.194898
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28231.4 1.58162 0.790809 0.612063i \(-0.209660\pi\)
0.790809 + 0.612063i \(0.209660\pi\)
\(684\) 0 0
\(685\) 22858.7 1.27502
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5085.35 −0.281185
\(690\) 0 0
\(691\) 25510.3 1.40442 0.702212 0.711968i \(-0.252196\pi\)
0.702212 + 0.711968i \(0.252196\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9123.47 −0.497947
\(696\) 0 0
\(697\) −41515.8 −2.25613
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17524.1 −0.944191 −0.472095 0.881547i \(-0.656502\pi\)
−0.472095 + 0.881547i \(0.656502\pi\)
\(702\) 0 0
\(703\) 4085.96 0.219210
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 484.103 0.0257519
\(708\) 0 0
\(709\) −34981.6 −1.85298 −0.926490 0.376319i \(-0.877190\pi\)
−0.926490 + 0.376319i \(0.877190\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12184.6 −0.639998
\(714\) 0 0
\(715\) −12388.7 −0.647987
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30485.5 −1.58125 −0.790623 0.612303i \(-0.790243\pi\)
−0.790623 + 0.612303i \(0.790243\pi\)
\(720\) 0 0
\(721\) 3692.13 0.190710
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 37236.8 1.90751
\(726\) 0 0
\(727\) 26697.8 1.36199 0.680994 0.732289i \(-0.261548\pi\)
0.680994 + 0.732289i \(0.261548\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1063.50 −0.0538100
\(732\) 0 0
\(733\) 30516.9 1.53775 0.768874 0.639401i \(-0.220818\pi\)
0.768874 + 0.639401i \(0.220818\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39374.7 1.96796
\(738\) 0 0
\(739\) −20020.0 −0.996544 −0.498272 0.867021i \(-0.666032\pi\)
−0.498272 + 0.867021i \(0.666032\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16034.7 −0.791731 −0.395866 0.918308i \(-0.629555\pi\)
−0.395866 + 0.918308i \(0.629555\pi\)
\(744\) 0 0
\(745\) 53303.2 2.62131
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1387.17 −0.0676718
\(750\) 0 0
\(751\) 24725.1 1.20137 0.600686 0.799485i \(-0.294894\pi\)
0.600686 + 0.799485i \(0.294894\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1423.45 −0.0686156
\(756\) 0 0
\(757\) −13567.6 −0.651419 −0.325709 0.945470i \(-0.605603\pi\)
−0.325709 + 0.945470i \(0.605603\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2217.39 0.105625 0.0528124 0.998604i \(-0.483181\pi\)
0.0528124 + 0.998604i \(0.483181\pi\)
\(762\) 0 0
\(763\) 1857.69 0.0881427
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −79.7091 −0.00375245
\(768\) 0 0
\(769\) −34899.0 −1.63653 −0.818263 0.574844i \(-0.805063\pi\)
−0.818263 + 0.574844i \(0.805063\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 10244.1 0.476656 0.238328 0.971185i \(-0.423401\pi\)
0.238328 + 0.971185i \(0.423401\pi\)
\(774\) 0 0
\(775\) −39208.7 −1.81732
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63485.5 2.91990
\(780\) 0 0
\(781\) −2270.27 −0.104016
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 41291.3 1.87739
\(786\) 0 0
\(787\) −10861.5 −0.491959 −0.245979 0.969275i \(-0.579110\pi\)
−0.245979 + 0.969275i \(0.579110\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2065.56 −0.0928480
\(792\) 0 0
\(793\) −8274.81 −0.370551
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −35604.8 −1.58242 −0.791209 0.611546i \(-0.790548\pi\)
−0.791209 + 0.611546i \(0.790548\pi\)
\(798\) 0 0
\(799\) −17375.5 −0.769336
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 35038.7 1.53984
\(804\) 0 0
\(805\) 2234.82 0.0978472
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13929.4 −0.605355 −0.302678 0.953093i \(-0.597881\pi\)
−0.302678 + 0.953093i \(0.597881\pi\)
\(810\) 0 0
\(811\) 41996.6 1.81837 0.909187 0.416388i \(-0.136704\pi\)
0.909187 + 0.416388i \(0.136704\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −21246.2 −0.913157
\(816\) 0 0
\(817\) 1626.30 0.0696414
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21080.1 −0.896104 −0.448052 0.894007i \(-0.647882\pi\)
−0.448052 + 0.894007i \(0.647882\pi\)
\(822\) 0 0
\(823\) 34365.1 1.45552 0.727760 0.685832i \(-0.240561\pi\)
0.727760 + 0.685832i \(0.240561\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32925.7 1.38445 0.692225 0.721682i \(-0.256631\pi\)
0.692225 + 0.721682i \(0.256631\pi\)
\(828\) 0 0
\(829\) 21195.7 0.888008 0.444004 0.896025i \(-0.353558\pi\)
0.444004 + 0.896025i \(0.353558\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −27975.0 −1.16360
\(834\) 0 0
\(835\) 67481.4 2.79675
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 23736.8 0.976740 0.488370 0.872637i \(-0.337592\pi\)
0.488370 + 0.872637i \(0.337592\pi\)
\(840\) 0 0
\(841\) 20916.3 0.857613
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 34799.3 1.41673
\(846\) 0 0
\(847\) −3312.40 −0.134375
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1751.83 −0.0705665
\(852\) 0 0
\(853\) 21928.5 0.880208 0.440104 0.897947i \(-0.354942\pi\)
0.440104 + 0.897947i \(0.354942\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 40332.7 1.60763 0.803814 0.594881i \(-0.202801\pi\)
0.803814 + 0.594881i \(0.202801\pi\)
\(858\) 0 0
\(859\) 22945.1 0.911382 0.455691 0.890138i \(-0.349392\pi\)
0.455691 + 0.890138i \(0.349392\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40453.2 1.59565 0.797824 0.602890i \(-0.205984\pi\)
0.797824 + 0.602890i \(0.205984\pi\)
\(864\) 0 0
\(865\) 2535.55 0.0996662
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −33899.1 −1.32330
\(870\) 0 0
\(871\) 10330.2 0.401865
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2053.02 0.0793198
\(876\) 0 0
\(877\) 4561.41 0.175631 0.0878153 0.996137i \(-0.472011\pi\)
0.0878153 + 0.996137i \(0.472011\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31553.9 −1.20667 −0.603336 0.797487i \(-0.706162\pi\)
−0.603336 + 0.797487i \(0.706162\pi\)
\(882\) 0 0
\(883\) 22726.0 0.866128 0.433064 0.901363i \(-0.357432\pi\)
0.433064 + 0.901363i \(0.357432\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1634.36 −0.0618675 −0.0309337 0.999521i \(-0.509848\pi\)
−0.0309337 + 0.999521i \(0.509848\pi\)
\(888\) 0 0
\(889\) −2359.40 −0.0890121
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 26570.4 0.995681
\(894\) 0 0
\(895\) −26685.6 −0.996651
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −47704.5 −1.76978
\(900\) 0 0
\(901\) −30781.6 −1.13816
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20331.3 −0.746781
\(906\) 0 0
\(907\) −6535.33 −0.239253 −0.119626 0.992819i \(-0.538170\pi\)
−0.119626 + 0.992819i \(0.538170\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −46086.7 −1.67609 −0.838046 0.545600i \(-0.816302\pi\)
−0.838046 + 0.545600i \(0.816302\pi\)
\(912\) 0 0
\(913\) −7024.47 −0.254629
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1903.75 −0.0685578
\(918\) 0 0
\(919\) 1857.94 0.0666898 0.0333449 0.999444i \(-0.489384\pi\)
0.0333449 + 0.999444i \(0.489384\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −595.617 −0.0212405
\(924\) 0 0
\(925\) −5637.19 −0.200378
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24301.6 −0.858246 −0.429123 0.903246i \(-0.641177\pi\)
−0.429123 + 0.903246i \(0.641177\pi\)
\(930\) 0 0
\(931\) 42779.1 1.50594
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −74988.7 −2.62288
\(936\) 0 0
\(937\) 52069.9 1.81542 0.907710 0.419597i \(-0.137829\pi\)
0.907710 + 0.419597i \(0.137829\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15709.0 −0.544206 −0.272103 0.962268i \(-0.587719\pi\)
−0.272103 + 0.962268i \(0.587719\pi\)
\(942\) 0 0
\(943\) −27219.1 −0.939952
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52488.7 −1.80111 −0.900556 0.434740i \(-0.856840\pi\)
−0.900556 + 0.434740i \(0.856840\pi\)
\(948\) 0 0
\(949\) 9192.62 0.314441
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2447.37 −0.0831880 −0.0415940 0.999135i \(-0.513244\pi\)
−0.0415940 + 0.999135i \(0.513244\pi\)
\(954\) 0 0
\(955\) −37991.3 −1.28730
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3132.75 0.105487
\(960\) 0 0
\(961\) 20439.7 0.686103
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −42661.8 −1.42314
\(966\) 0 0
\(967\) −29392.2 −0.977447 −0.488723 0.872439i \(-0.662537\pi\)
−0.488723 + 0.872439i \(0.662537\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 32763.9 1.08285 0.541423 0.840751i \(-0.317886\pi\)
0.541423 + 0.840751i \(0.317886\pi\)
\(972\) 0 0
\(973\) −1250.36 −0.0411969
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −669.381 −0.0219195 −0.0109598 0.999940i \(-0.503489\pi\)
−0.0109598 + 0.999940i \(0.503489\pi\)
\(978\) 0 0
\(979\) 10760.6 0.351287
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −23791.0 −0.771938 −0.385969 0.922512i \(-0.626133\pi\)
−0.385969 + 0.922512i \(0.626133\pi\)
\(984\) 0 0
\(985\) 62235.7 2.01319
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −697.267 −0.0224184
\(990\) 0 0
\(991\) 7069.41 0.226607 0.113303 0.993560i \(-0.463857\pi\)
0.113303 + 0.993560i \(0.463857\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 79655.8 2.53795
\(996\) 0 0
\(997\) 34324.7 1.09034 0.545172 0.838324i \(-0.316464\pi\)
0.545172 + 0.838324i \(0.316464\pi\)
\(998\) 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.h.1.1 4
3.2 odd 2 648.4.a.i.1.4 4
4.3 odd 2 1296.4.a.y.1.1 4
9.2 odd 6 72.4.i.a.49.2 yes 8
9.4 even 3 216.4.i.a.73.4 8
9.5 odd 6 72.4.i.a.25.2 8
9.7 even 3 216.4.i.a.145.4 8
12.11 even 2 1296.4.a.ba.1.4 4
36.7 odd 6 432.4.i.e.145.4 8
36.11 even 6 144.4.i.e.49.3 8
36.23 even 6 144.4.i.e.97.3 8
36.31 odd 6 432.4.i.e.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.i.a.25.2 8 9.5 odd 6
72.4.i.a.49.2 yes 8 9.2 odd 6
144.4.i.e.49.3 8 36.11 even 6
144.4.i.e.97.3 8 36.23 even 6
216.4.i.a.73.4 8 9.4 even 3
216.4.i.a.145.4 8 9.7 even 3
432.4.i.e.145.4 8 36.7 odd 6
432.4.i.e.289.4 8 36.31 odd 6
648.4.a.h.1.1 4 1.1 even 1 trivial
648.4.a.i.1.4 4 3.2 odd 2
1296.4.a.y.1.1 4 4.3 odd 2
1296.4.a.ba.1.4 4 12.11 even 2