Properties

Label 588.4.a.j.1.2
Level $588$
Weight $4$
Character 588.1
Self dual yes
Analytic conductor $34.693$
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] = [588,4,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, 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("588.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(34.6931230834\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 23x^{2} + 18x + 119 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.89590\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} +37.8189 q^{11} -39.9319 q^{13} +24.5082 q^{15} -9.93596 q^{17} +90.4458 q^{19} +118.595 q^{23} -58.2608 q^{25} -27.0000 q^{27} -78.4061 q^{29} -92.0110 q^{31} -113.457 q^{33} +332.435 q^{37} +119.796 q^{39} -71.7451 q^{41} -115.947 q^{43} -73.5246 q^{45} -307.927 q^{47} +29.8079 q^{51} -403.150 q^{53} -308.958 q^{55} -271.337 q^{57} -593.710 q^{59} -333.170 q^{61} +326.220 q^{65} -743.511 q^{67} -355.784 q^{69} -728.272 q^{71} +801.749 q^{73} +174.782 q^{75} +1067.91 q^{79} +81.0000 q^{81} -906.756 q^{83} +81.1709 q^{85} +235.218 q^{87} -1113.27 q^{89} +276.033 q^{93} -738.889 q^{95} -1480.94 q^{97} +340.370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{17} - 192 q^{19} + 192 q^{23} + 324 q^{25} - 108 q^{27} + 96 q^{29} - 48 q^{31} + 256 q^{37} - 1008 q^{41} - 112 q^{43} - 864 q^{47} + 144 q^{51} - 648 q^{53} - 2352 q^{55} + 576 q^{57} - 336 q^{59} - 960 q^{61} - 360 q^{65} + 720 q^{67} - 576 q^{69} - 1344 q^{71} - 672 q^{73} - 972 q^{75} - 1984 q^{79} + 324 q^{81} - 3120 q^{83} + 680 q^{85} - 288 q^{87} - 2160 q^{89} + 144 q^{93} - 3744 q^{95} - 2016 q^{97}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −8.16940 −0.730694 −0.365347 0.930871i \(-0.619050\pi\)
−0.365347 + 0.930871i \(0.619050\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 37.8189 1.03662 0.518310 0.855193i \(-0.326561\pi\)
0.518310 + 0.855193i \(0.326561\pi\)
\(12\) 0 0
\(13\) −39.9319 −0.851933 −0.425966 0.904739i \(-0.640066\pi\)
−0.425966 + 0.904739i \(0.640066\pi\)
\(14\) 0 0
\(15\) 24.5082 0.421866
\(16\) 0 0
\(17\) −9.93596 −0.141754 −0.0708772 0.997485i \(-0.522580\pi\)
−0.0708772 + 0.997485i \(0.522580\pi\)
\(18\) 0 0
\(19\) 90.4458 1.09209 0.546045 0.837756i \(-0.316133\pi\)
0.546045 + 0.837756i \(0.316133\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 118.595 1.07516 0.537580 0.843213i \(-0.319339\pi\)
0.537580 + 0.843213i \(0.319339\pi\)
\(24\) 0 0
\(25\) −58.2608 −0.466087
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −78.4061 −0.502057 −0.251028 0.967980i \(-0.580769\pi\)
−0.251028 + 0.967980i \(0.580769\pi\)
\(30\) 0 0
\(31\) −92.0110 −0.533086 −0.266543 0.963823i \(-0.585881\pi\)
−0.266543 + 0.963823i \(0.585881\pi\)
\(32\) 0 0
\(33\) −113.457 −0.598493
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 332.435 1.47708 0.738541 0.674209i \(-0.235515\pi\)
0.738541 + 0.674209i \(0.235515\pi\)
\(38\) 0 0
\(39\) 119.796 0.491864
\(40\) 0 0
\(41\) −71.7451 −0.273285 −0.136643 0.990620i \(-0.543631\pi\)
−0.136643 + 0.990620i \(0.543631\pi\)
\(42\) 0 0
\(43\) −115.947 −0.411202 −0.205601 0.978636i \(-0.565915\pi\)
−0.205601 + 0.978636i \(0.565915\pi\)
\(44\) 0 0
\(45\) −73.5246 −0.243565
\(46\) 0 0
\(47\) −307.927 −0.955656 −0.477828 0.878453i \(-0.658576\pi\)
−0.477828 + 0.878453i \(0.658576\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 29.8079 0.0818419
\(52\) 0 0
\(53\) −403.150 −1.04485 −0.522424 0.852686i \(-0.674972\pi\)
−0.522424 + 0.852686i \(0.674972\pi\)
\(54\) 0 0
\(55\) −308.958 −0.757451
\(56\) 0 0
\(57\) −271.337 −0.630518
\(58\) 0 0
\(59\) −593.710 −1.31008 −0.655038 0.755596i \(-0.727347\pi\)
−0.655038 + 0.755596i \(0.727347\pi\)
\(60\) 0 0
\(61\) −333.170 −0.699313 −0.349657 0.936878i \(-0.613702\pi\)
−0.349657 + 0.936878i \(0.613702\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 326.220 0.622502
\(66\) 0 0
\(67\) −743.511 −1.35574 −0.677869 0.735183i \(-0.737096\pi\)
−0.677869 + 0.735183i \(0.737096\pi\)
\(68\) 0 0
\(69\) −355.784 −0.620744
\(70\) 0 0
\(71\) −728.272 −1.21732 −0.608662 0.793429i \(-0.708294\pi\)
−0.608662 + 0.793429i \(0.708294\pi\)
\(72\) 0 0
\(73\) 801.749 1.28545 0.642723 0.766098i \(-0.277804\pi\)
0.642723 + 0.766098i \(0.277804\pi\)
\(74\) 0 0
\(75\) 174.782 0.269095
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1067.91 1.52087 0.760435 0.649414i \(-0.224986\pi\)
0.760435 + 0.649414i \(0.224986\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −906.756 −1.19915 −0.599575 0.800319i \(-0.704664\pi\)
−0.599575 + 0.800319i \(0.704664\pi\)
\(84\) 0 0
\(85\) 81.1709 0.103579
\(86\) 0 0
\(87\) 235.218 0.289863
\(88\) 0 0
\(89\) −1113.27 −1.32591 −0.662957 0.748658i \(-0.730699\pi\)
−0.662957 + 0.748658i \(0.730699\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 276.033 0.307777
\(94\) 0 0
\(95\) −738.889 −0.797983
\(96\) 0 0
\(97\) −1480.94 −1.55017 −0.775084 0.631858i \(-0.782293\pi\)
−0.775084 + 0.631858i \(0.782293\pi\)
\(98\) 0 0
\(99\) 340.370 0.345540
\(100\) 0 0
\(101\) 556.316 0.548075 0.274037 0.961719i \(-0.411641\pi\)
0.274037 + 0.961719i \(0.411641\pi\)
\(102\) 0 0
\(103\) −552.435 −0.528476 −0.264238 0.964458i \(-0.585120\pi\)
−0.264238 + 0.964458i \(0.585120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 533.804 0.482288 0.241144 0.970489i \(-0.422477\pi\)
0.241144 + 0.970489i \(0.422477\pi\)
\(108\) 0 0
\(109\) 1094.63 0.961894 0.480947 0.876750i \(-0.340293\pi\)
0.480947 + 0.876750i \(0.340293\pi\)
\(110\) 0 0
\(111\) −997.306 −0.852794
\(112\) 0 0
\(113\) −1425.18 −1.18646 −0.593228 0.805035i \(-0.702147\pi\)
−0.593228 + 0.805035i \(0.702147\pi\)
\(114\) 0 0
\(115\) −968.847 −0.785612
\(116\) 0 0
\(117\) −359.387 −0.283978
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 99.2659 0.0745800
\(122\) 0 0
\(123\) 215.235 0.157781
\(124\) 0 0
\(125\) 1497.13 1.07126
\(126\) 0 0
\(127\) −786.485 −0.549522 −0.274761 0.961513i \(-0.588599\pi\)
−0.274761 + 0.961513i \(0.588599\pi\)
\(128\) 0 0
\(129\) 347.840 0.237407
\(130\) 0 0
\(131\) 27.7303 0.0184947 0.00924735 0.999957i \(-0.497056\pi\)
0.00924735 + 0.999957i \(0.497056\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 220.574 0.140622
\(136\) 0 0
\(137\) −2362.98 −1.47360 −0.736798 0.676113i \(-0.763663\pi\)
−0.736798 + 0.676113i \(0.763663\pi\)
\(138\) 0 0
\(139\) −2513.28 −1.53362 −0.766811 0.641873i \(-0.778158\pi\)
−0.766811 + 0.641873i \(0.778158\pi\)
\(140\) 0 0
\(141\) 923.782 0.551748
\(142\) 0 0
\(143\) −1510.18 −0.883130
\(144\) 0 0
\(145\) 640.531 0.366850
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2265.83 1.24580 0.622900 0.782302i \(-0.285955\pi\)
0.622900 + 0.782302i \(0.285955\pi\)
\(150\) 0 0
\(151\) 283.146 0.152597 0.0762984 0.997085i \(-0.475690\pi\)
0.0762984 + 0.997085i \(0.475690\pi\)
\(152\) 0 0
\(153\) −89.4237 −0.0472515
\(154\) 0 0
\(155\) 751.675 0.389522
\(156\) 0 0
\(157\) −192.581 −0.0978956 −0.0489478 0.998801i \(-0.515587\pi\)
−0.0489478 + 0.998801i \(0.515587\pi\)
\(158\) 0 0
\(159\) 1209.45 0.603243
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −842.833 −0.405005 −0.202502 0.979282i \(-0.564907\pi\)
−0.202502 + 0.979282i \(0.564907\pi\)
\(164\) 0 0
\(165\) 926.873 0.437315
\(166\) 0 0
\(167\) 3859.21 1.78823 0.894116 0.447836i \(-0.147805\pi\)
0.894116 + 0.447836i \(0.147805\pi\)
\(168\) 0 0
\(169\) −602.441 −0.274211
\(170\) 0 0
\(171\) 814.012 0.364030
\(172\) 0 0
\(173\) −1510.66 −0.663891 −0.331946 0.943298i \(-0.607705\pi\)
−0.331946 + 0.943298i \(0.607705\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1781.13 0.756372
\(178\) 0 0
\(179\) −2464.63 −1.02913 −0.514566 0.857451i \(-0.672047\pi\)
−0.514566 + 0.857451i \(0.672047\pi\)
\(180\) 0 0
\(181\) 3297.36 1.35409 0.677047 0.735940i \(-0.263259\pi\)
0.677047 + 0.735940i \(0.263259\pi\)
\(182\) 0 0
\(183\) 999.511 0.403749
\(184\) 0 0
\(185\) −2715.80 −1.07929
\(186\) 0 0
\(187\) −375.767 −0.146945
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4729.71 −1.79178 −0.895889 0.444278i \(-0.853460\pi\)
−0.895889 + 0.444278i \(0.853460\pi\)
\(192\) 0 0
\(193\) 4553.18 1.69816 0.849080 0.528264i \(-0.177157\pi\)
0.849080 + 0.528264i \(0.177157\pi\)
\(194\) 0 0
\(195\) −978.660 −0.359402
\(196\) 0 0
\(197\) −3109.06 −1.12442 −0.562212 0.826993i \(-0.690050\pi\)
−0.562212 + 0.826993i \(0.690050\pi\)
\(198\) 0 0
\(199\) 443.512 0.157989 0.0789943 0.996875i \(-0.474829\pi\)
0.0789943 + 0.996875i \(0.474829\pi\)
\(200\) 0 0
\(201\) 2230.53 0.782735
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 586.114 0.199688
\(206\) 0 0
\(207\) 1067.35 0.358387
\(208\) 0 0
\(209\) 3420.56 1.13208
\(210\) 0 0
\(211\) 4653.28 1.51822 0.759112 0.650960i \(-0.225633\pi\)
0.759112 + 0.650960i \(0.225633\pi\)
\(212\) 0 0
\(213\) 2184.82 0.702823
\(214\) 0 0
\(215\) 947.214 0.300463
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2405.25 −0.742153
\(220\) 0 0
\(221\) 396.762 0.120765
\(222\) 0 0
\(223\) −2778.90 −0.834481 −0.417240 0.908796i \(-0.637003\pi\)
−0.417240 + 0.908796i \(0.637003\pi\)
\(224\) 0 0
\(225\) −524.347 −0.155362
\(226\) 0 0
\(227\) −3217.22 −0.940681 −0.470340 0.882485i \(-0.655869\pi\)
−0.470340 + 0.882485i \(0.655869\pi\)
\(228\) 0 0
\(229\) 3728.01 1.07578 0.537891 0.843014i \(-0.319221\pi\)
0.537891 + 0.843014i \(0.319221\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3603.51 −1.01319 −0.506596 0.862183i \(-0.669097\pi\)
−0.506596 + 0.862183i \(0.669097\pi\)
\(234\) 0 0
\(235\) 2515.58 0.698292
\(236\) 0 0
\(237\) −3203.72 −0.878075
\(238\) 0 0
\(239\) 2348.76 0.635684 0.317842 0.948144i \(-0.397042\pi\)
0.317842 + 0.948144i \(0.397042\pi\)
\(240\) 0 0
\(241\) 5093.39 1.36139 0.680693 0.732569i \(-0.261679\pi\)
0.680693 + 0.732569i \(0.261679\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3611.68 −0.930386
\(248\) 0 0
\(249\) 2720.27 0.692329
\(250\) 0 0
\(251\) 5939.02 1.49350 0.746749 0.665106i \(-0.231614\pi\)
0.746749 + 0.665106i \(0.231614\pi\)
\(252\) 0 0
\(253\) 4485.11 1.11453
\(254\) 0 0
\(255\) −243.513 −0.0598014
\(256\) 0 0
\(257\) 1515.89 0.367932 0.183966 0.982933i \(-0.441106\pi\)
0.183966 + 0.982933i \(0.441106\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −705.655 −0.167352
\(262\) 0 0
\(263\) −6433.92 −1.50849 −0.754244 0.656594i \(-0.771997\pi\)
−0.754244 + 0.656594i \(0.771997\pi\)
\(264\) 0 0
\(265\) 3293.50 0.763464
\(266\) 0 0
\(267\) 3339.81 0.765516
\(268\) 0 0
\(269\) 6949.79 1.57523 0.787614 0.616169i \(-0.211316\pi\)
0.787614 + 0.616169i \(0.211316\pi\)
\(270\) 0 0
\(271\) 961.994 0.215635 0.107817 0.994171i \(-0.465614\pi\)
0.107817 + 0.994171i \(0.465614\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2203.36 −0.483154
\(276\) 0 0
\(277\) 760.010 0.164854 0.0824271 0.996597i \(-0.473733\pi\)
0.0824271 + 0.996597i \(0.473733\pi\)
\(278\) 0 0
\(279\) −828.099 −0.177695
\(280\) 0 0
\(281\) −4412.07 −0.936662 −0.468331 0.883553i \(-0.655145\pi\)
−0.468331 + 0.883553i \(0.655145\pi\)
\(282\) 0 0
\(283\) 2602.15 0.546578 0.273289 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(284\) 0 0
\(285\) 2216.67 0.460716
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4814.28 −0.979906
\(290\) 0 0
\(291\) 4442.81 0.894990
\(292\) 0 0
\(293\) 9332.18 1.86072 0.930361 0.366644i \(-0.119493\pi\)
0.930361 + 0.366644i \(0.119493\pi\)
\(294\) 0 0
\(295\) 4850.26 0.957264
\(296\) 0 0
\(297\) −1021.11 −0.199498
\(298\) 0 0
\(299\) −4735.71 −0.915964
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1668.95 −0.316431
\(304\) 0 0
\(305\) 2721.80 0.510984
\(306\) 0 0
\(307\) −4895.51 −0.910103 −0.455051 0.890465i \(-0.650379\pi\)
−0.455051 + 0.890465i \(0.650379\pi\)
\(308\) 0 0
\(309\) 1657.30 0.305116
\(310\) 0 0
\(311\) −7505.28 −1.36844 −0.684221 0.729275i \(-0.739858\pi\)
−0.684221 + 0.729275i \(0.739858\pi\)
\(312\) 0 0
\(313\) 6349.27 1.14659 0.573294 0.819350i \(-0.305665\pi\)
0.573294 + 0.819350i \(0.305665\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7999.60 −1.41736 −0.708679 0.705531i \(-0.750708\pi\)
−0.708679 + 0.705531i \(0.750708\pi\)
\(318\) 0 0
\(319\) −2965.23 −0.520442
\(320\) 0 0
\(321\) −1601.41 −0.278449
\(322\) 0 0
\(323\) −898.666 −0.154808
\(324\) 0 0
\(325\) 2326.47 0.397074
\(326\) 0 0
\(327\) −3283.89 −0.555350
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2121.56 −0.352300 −0.176150 0.984363i \(-0.556364\pi\)
−0.176150 + 0.984363i \(0.556364\pi\)
\(332\) 0 0
\(333\) 2991.92 0.492361
\(334\) 0 0
\(335\) 6074.05 0.990629
\(336\) 0 0
\(337\) −9114.19 −1.47324 −0.736620 0.676307i \(-0.763579\pi\)
−0.736620 + 0.676307i \(0.763579\pi\)
\(338\) 0 0
\(339\) 4275.53 0.685001
\(340\) 0 0
\(341\) −3479.75 −0.552607
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2906.54 0.453574
\(346\) 0 0
\(347\) −1673.07 −0.258833 −0.129416 0.991590i \(-0.541310\pi\)
−0.129416 + 0.991590i \(0.541310\pi\)
\(348\) 0 0
\(349\) 3467.56 0.531845 0.265923 0.963994i \(-0.414323\pi\)
0.265923 + 0.963994i \(0.414323\pi\)
\(350\) 0 0
\(351\) 1078.16 0.163955
\(352\) 0 0
\(353\) −3984.24 −0.600736 −0.300368 0.953823i \(-0.597109\pi\)
−0.300368 + 0.953823i \(0.597109\pi\)
\(354\) 0 0
\(355\) 5949.55 0.889491
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2170.93 0.319157 0.159579 0.987185i \(-0.448986\pi\)
0.159579 + 0.987185i \(0.448986\pi\)
\(360\) 0 0
\(361\) 1321.45 0.192659
\(362\) 0 0
\(363\) −297.798 −0.0430588
\(364\) 0 0
\(365\) −6549.81 −0.939268
\(366\) 0 0
\(367\) −3592.12 −0.510918 −0.255459 0.966820i \(-0.582227\pi\)
−0.255459 + 0.966820i \(0.582227\pi\)
\(368\) 0 0
\(369\) −645.705 −0.0910951
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7439.80 1.03276 0.516378 0.856361i \(-0.327280\pi\)
0.516378 + 0.856361i \(0.327280\pi\)
\(374\) 0 0
\(375\) −4491.40 −0.618492
\(376\) 0 0
\(377\) 3130.91 0.427719
\(378\) 0 0
\(379\) 11243.9 1.52390 0.761951 0.647635i \(-0.224242\pi\)
0.761951 + 0.647635i \(0.224242\pi\)
\(380\) 0 0
\(381\) 2359.46 0.317267
\(382\) 0 0
\(383\) −3541.25 −0.472452 −0.236226 0.971698i \(-0.575911\pi\)
−0.236226 + 0.971698i \(0.575911\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1043.52 −0.137067
\(388\) 0 0
\(389\) −10558.2 −1.37615 −0.688074 0.725640i \(-0.741544\pi\)
−0.688074 + 0.725640i \(0.741544\pi\)
\(390\) 0 0
\(391\) −1178.35 −0.152409
\(392\) 0 0
\(393\) −83.1908 −0.0106779
\(394\) 0 0
\(395\) −8724.15 −1.11129
\(396\) 0 0
\(397\) 540.574 0.0683391 0.0341696 0.999416i \(-0.489121\pi\)
0.0341696 + 0.999416i \(0.489121\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2167.03 0.269866 0.134933 0.990855i \(-0.456918\pi\)
0.134933 + 0.990855i \(0.456918\pi\)
\(402\) 0 0
\(403\) 3674.17 0.454153
\(404\) 0 0
\(405\) −661.722 −0.0811882
\(406\) 0 0
\(407\) 12572.3 1.53117
\(408\) 0 0
\(409\) 957.690 0.115782 0.0578908 0.998323i \(-0.481562\pi\)
0.0578908 + 0.998323i \(0.481562\pi\)
\(410\) 0 0
\(411\) 7088.93 0.850781
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7407.66 0.876211
\(416\) 0 0
\(417\) 7539.84 0.885437
\(418\) 0 0
\(419\) −6464.42 −0.753718 −0.376859 0.926271i \(-0.622996\pi\)
−0.376859 + 0.926271i \(0.622996\pi\)
\(420\) 0 0
\(421\) −6201.23 −0.717885 −0.358943 0.933360i \(-0.616863\pi\)
−0.358943 + 0.933360i \(0.616863\pi\)
\(422\) 0 0
\(423\) −2771.35 −0.318552
\(424\) 0 0
\(425\) 578.877 0.0660698
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4530.54 0.509875
\(430\) 0 0
\(431\) 1415.59 0.158206 0.0791029 0.996866i \(-0.474794\pi\)
0.0791029 + 0.996866i \(0.474794\pi\)
\(432\) 0 0
\(433\) −8905.73 −0.988411 −0.494206 0.869345i \(-0.664541\pi\)
−0.494206 + 0.869345i \(0.664541\pi\)
\(434\) 0 0
\(435\) −1921.59 −0.211801
\(436\) 0 0
\(437\) 10726.4 1.17417
\(438\) 0 0
\(439\) −10876.5 −1.18248 −0.591238 0.806497i \(-0.701361\pi\)
−0.591238 + 0.806497i \(0.701361\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11403.0 1.22296 0.611479 0.791260i \(-0.290575\pi\)
0.611479 + 0.791260i \(0.290575\pi\)
\(444\) 0 0
\(445\) 9094.74 0.968837
\(446\) 0 0
\(447\) −6797.49 −0.719262
\(448\) 0 0
\(449\) −12689.0 −1.33370 −0.666852 0.745190i \(-0.732359\pi\)
−0.666852 + 0.745190i \(0.732359\pi\)
\(450\) 0 0
\(451\) −2713.32 −0.283293
\(452\) 0 0
\(453\) −849.439 −0.0881019
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2270.05 0.232360 0.116180 0.993228i \(-0.462935\pi\)
0.116180 + 0.993228i \(0.462935\pi\)
\(458\) 0 0
\(459\) 268.271 0.0272806
\(460\) 0 0
\(461\) −10731.2 −1.08417 −0.542085 0.840324i \(-0.682365\pi\)
−0.542085 + 0.840324i \(0.682365\pi\)
\(462\) 0 0
\(463\) −3307.74 −0.332017 −0.166008 0.986124i \(-0.553088\pi\)
−0.166008 + 0.986124i \(0.553088\pi\)
\(464\) 0 0
\(465\) −2255.02 −0.224891
\(466\) 0 0
\(467\) 9246.69 0.916244 0.458122 0.888889i \(-0.348522\pi\)
0.458122 + 0.888889i \(0.348522\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 577.742 0.0565201
\(472\) 0 0
\(473\) −4384.96 −0.426260
\(474\) 0 0
\(475\) −5269.45 −0.509008
\(476\) 0 0
\(477\) −3628.35 −0.348283
\(478\) 0 0
\(479\) −15223.3 −1.45213 −0.726066 0.687625i \(-0.758653\pi\)
−0.726066 + 0.687625i \(0.758653\pi\)
\(480\) 0 0
\(481\) −13274.8 −1.25837
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12098.4 1.13270
\(486\) 0 0
\(487\) −7758.72 −0.721933 −0.360966 0.932579i \(-0.617553\pi\)
−0.360966 + 0.932579i \(0.617553\pi\)
\(488\) 0 0
\(489\) 2528.50 0.233830
\(490\) 0 0
\(491\) −8342.30 −0.766767 −0.383384 0.923589i \(-0.625241\pi\)
−0.383384 + 0.923589i \(0.625241\pi\)
\(492\) 0 0
\(493\) 779.040 0.0711688
\(494\) 0 0
\(495\) −2780.62 −0.252484
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2586.95 −0.232079 −0.116040 0.993245i \(-0.537020\pi\)
−0.116040 + 0.993245i \(0.537020\pi\)
\(500\) 0 0
\(501\) −11577.6 −1.03244
\(502\) 0 0
\(503\) −409.682 −0.0363157 −0.0181578 0.999835i \(-0.505780\pi\)
−0.0181578 + 0.999835i \(0.505780\pi\)
\(504\) 0 0
\(505\) −4544.77 −0.400475
\(506\) 0 0
\(507\) 1807.32 0.158316
\(508\) 0 0
\(509\) −4390.51 −0.382330 −0.191165 0.981558i \(-0.561226\pi\)
−0.191165 + 0.981558i \(0.561226\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2442.04 −0.210173
\(514\) 0 0
\(515\) 4513.06 0.386154
\(516\) 0 0
\(517\) −11645.5 −0.990652
\(518\) 0 0
\(519\) 4531.98 0.383298
\(520\) 0 0
\(521\) 19429.5 1.63382 0.816912 0.576762i \(-0.195684\pi\)
0.816912 + 0.576762i \(0.195684\pi\)
\(522\) 0 0
\(523\) −17948.7 −1.50065 −0.750326 0.661067i \(-0.770104\pi\)
−0.750326 + 0.661067i \(0.770104\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 914.217 0.0755672
\(528\) 0 0
\(529\) 1897.67 0.155968
\(530\) 0 0
\(531\) −5343.39 −0.436692
\(532\) 0 0
\(533\) 2864.92 0.232821
\(534\) 0 0
\(535\) −4360.86 −0.352405
\(536\) 0 0
\(537\) 7393.88 0.594170
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23094.0 1.83528 0.917641 0.397411i \(-0.130091\pi\)
0.917641 + 0.397411i \(0.130091\pi\)
\(542\) 0 0
\(543\) −9892.09 −0.781787
\(544\) 0 0
\(545\) −8942.47 −0.702850
\(546\) 0 0
\(547\) −2266.68 −0.177178 −0.0885889 0.996068i \(-0.528236\pi\)
−0.0885889 + 0.996068i \(0.528236\pi\)
\(548\) 0 0
\(549\) −2998.53 −0.233104
\(550\) 0 0
\(551\) −7091.50 −0.548291
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8147.40 0.623131
\(556\) 0 0
\(557\) −11038.5 −0.839709 −0.419854 0.907592i \(-0.637919\pi\)
−0.419854 + 0.907592i \(0.637919\pi\)
\(558\) 0 0
\(559\) 4629.97 0.350316
\(560\) 0 0
\(561\) 1127.30 0.0848390
\(562\) 0 0
\(563\) 7059.36 0.528448 0.264224 0.964461i \(-0.414884\pi\)
0.264224 + 0.964461i \(0.414884\pi\)
\(564\) 0 0
\(565\) 11642.9 0.866936
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1352.81 −0.0996706 −0.0498353 0.998757i \(-0.515870\pi\)
−0.0498353 + 0.998757i \(0.515870\pi\)
\(570\) 0 0
\(571\) −20839.2 −1.52731 −0.763655 0.645625i \(-0.776597\pi\)
−0.763655 + 0.645625i \(0.776597\pi\)
\(572\) 0 0
\(573\) 14189.1 1.03448
\(574\) 0 0
\(575\) −6909.42 −0.501117
\(576\) 0 0
\(577\) −10014.6 −0.722556 −0.361278 0.932458i \(-0.617659\pi\)
−0.361278 + 0.932458i \(0.617659\pi\)
\(578\) 0 0
\(579\) −13659.5 −0.980433
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −15246.7 −1.08311
\(584\) 0 0
\(585\) 2935.98 0.207501
\(586\) 0 0
\(587\) −20864.8 −1.46709 −0.733546 0.679640i \(-0.762136\pi\)
−0.733546 + 0.679640i \(0.762136\pi\)
\(588\) 0 0
\(589\) −8322.01 −0.582177
\(590\) 0 0
\(591\) 9327.19 0.649187
\(592\) 0 0
\(593\) −17547.7 −1.21517 −0.607586 0.794254i \(-0.707862\pi\)
−0.607586 + 0.794254i \(0.707862\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1330.54 −0.0912147
\(598\) 0 0
\(599\) −130.682 −0.00891405 −0.00445703 0.999990i \(-0.501419\pi\)
−0.00445703 + 0.999990i \(0.501419\pi\)
\(600\) 0 0
\(601\) 5964.47 0.404818 0.202409 0.979301i \(-0.435123\pi\)
0.202409 + 0.979301i \(0.435123\pi\)
\(602\) 0 0
\(603\) −6691.60 −0.451912
\(604\) 0 0
\(605\) −810.944 −0.0544951
\(606\) 0 0
\(607\) 9911.63 0.662769 0.331384 0.943496i \(-0.392484\pi\)
0.331384 + 0.943496i \(0.392484\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12296.1 0.814155
\(612\) 0 0
\(613\) −6482.53 −0.427124 −0.213562 0.976930i \(-0.568506\pi\)
−0.213562 + 0.976930i \(0.568506\pi\)
\(614\) 0 0
\(615\) −1758.34 −0.115290
\(616\) 0 0
\(617\) 7189.36 0.469097 0.234548 0.972104i \(-0.424639\pi\)
0.234548 + 0.972104i \(0.424639\pi\)
\(618\) 0 0
\(619\) −1861.39 −0.120865 −0.0604327 0.998172i \(-0.519248\pi\)
−0.0604327 + 0.998172i \(0.519248\pi\)
\(620\) 0 0
\(621\) −3202.05 −0.206915
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4948.07 −0.316677
\(626\) 0 0
\(627\) −10261.7 −0.653607
\(628\) 0 0
\(629\) −3303.06 −0.209383
\(630\) 0 0
\(631\) −29032.0 −1.83161 −0.915803 0.401627i \(-0.868445\pi\)
−0.915803 + 0.401627i \(0.868445\pi\)
\(632\) 0 0
\(633\) −13959.9 −0.876547
\(634\) 0 0
\(635\) 6425.12 0.401532
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6554.45 −0.405775
\(640\) 0 0
\(641\) 24457.7 1.50705 0.753527 0.657417i \(-0.228351\pi\)
0.753527 + 0.657417i \(0.228351\pi\)
\(642\) 0 0
\(643\) −10968.2 −0.672695 −0.336348 0.941738i \(-0.609192\pi\)
−0.336348 + 0.941738i \(0.609192\pi\)
\(644\) 0 0
\(645\) −2841.64 −0.173472
\(646\) 0 0
\(647\) −5484.74 −0.333272 −0.166636 0.986018i \(-0.553291\pi\)
−0.166636 + 0.986018i \(0.553291\pi\)
\(648\) 0 0
\(649\) −22453.4 −1.35805
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10330.0 −0.619055 −0.309527 0.950891i \(-0.600171\pi\)
−0.309527 + 0.950891i \(0.600171\pi\)
\(654\) 0 0
\(655\) −226.540 −0.0135140
\(656\) 0 0
\(657\) 7215.74 0.428482
\(658\) 0 0
\(659\) 26822.2 1.58550 0.792751 0.609546i \(-0.208648\pi\)
0.792751 + 0.609546i \(0.208648\pi\)
\(660\) 0 0
\(661\) −2498.76 −0.147035 −0.0735177 0.997294i \(-0.523423\pi\)
−0.0735177 + 0.997294i \(0.523423\pi\)
\(662\) 0 0
\(663\) −1190.29 −0.0697238
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9298.54 −0.539791
\(668\) 0 0
\(669\) 8336.71 0.481788
\(670\) 0 0
\(671\) −12600.1 −0.724922
\(672\) 0 0
\(673\) 10092.1 0.578044 0.289022 0.957322i \(-0.406670\pi\)
0.289022 + 0.957322i \(0.406670\pi\)
\(674\) 0 0
\(675\) 1573.04 0.0896984
\(676\) 0 0
\(677\) 26366.4 1.49681 0.748406 0.663241i \(-0.230820\pi\)
0.748406 + 0.663241i \(0.230820\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9651.67 0.543102
\(682\) 0 0
\(683\) 22570.8 1.26449 0.632245 0.774769i \(-0.282134\pi\)
0.632245 + 0.774769i \(0.282134\pi\)
\(684\) 0 0
\(685\) 19304.1 1.07675
\(686\) 0 0
\(687\) −11184.0 −0.621103
\(688\) 0 0
\(689\) 16098.6 0.890140
\(690\) 0 0
\(691\) 11931.5 0.656867 0.328434 0.944527i \(-0.393479\pi\)
0.328434 + 0.944527i \(0.393479\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20532.0 1.12061
\(696\) 0 0
\(697\) 712.856 0.0387394
\(698\) 0 0
\(699\) 10810.5 0.584967
\(700\) 0 0
\(701\) 16592.6 0.893997 0.446999 0.894535i \(-0.352493\pi\)
0.446999 + 0.894535i \(0.352493\pi\)
\(702\) 0 0
\(703\) 30067.4 1.61311
\(704\) 0 0
\(705\) −7546.75 −0.403159
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18714.0 0.991283 0.495641 0.868527i \(-0.334933\pi\)
0.495641 + 0.868527i \(0.334933\pi\)
\(710\) 0 0
\(711\) 9611.15 0.506957
\(712\) 0 0
\(713\) −10912.0 −0.573152
\(714\) 0 0
\(715\) 12337.3 0.645298
\(716\) 0 0
\(717\) −7046.27 −0.367012
\(718\) 0 0
\(719\) 24854.8 1.28919 0.644594 0.764525i \(-0.277026\pi\)
0.644594 + 0.764525i \(0.277026\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −15280.2 −0.785997
\(724\) 0 0
\(725\) 4568.00 0.234002
\(726\) 0 0
\(727\) 22506.0 1.14814 0.574071 0.818805i \(-0.305363\pi\)
0.574071 + 0.818805i \(0.305363\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1152.04 0.0582897
\(732\) 0 0
\(733\) 23792.0 1.19888 0.599440 0.800420i \(-0.295390\pi\)
0.599440 + 0.800420i \(0.295390\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −28118.8 −1.40538
\(738\) 0 0
\(739\) 4803.24 0.239093 0.119547 0.992829i \(-0.461856\pi\)
0.119547 + 0.992829i \(0.461856\pi\)
\(740\) 0 0
\(741\) 10835.0 0.537159
\(742\) 0 0
\(743\) 5076.10 0.250638 0.125319 0.992116i \(-0.460005\pi\)
0.125319 + 0.992116i \(0.460005\pi\)
\(744\) 0 0
\(745\) −18510.5 −0.910298
\(746\) 0 0
\(747\) −8160.80 −0.399716
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −9158.72 −0.445015 −0.222508 0.974931i \(-0.571424\pi\)
−0.222508 + 0.974931i \(0.571424\pi\)
\(752\) 0 0
\(753\) −17817.1 −0.862271
\(754\) 0 0
\(755\) −2313.14 −0.111502
\(756\) 0 0
\(757\) 33682.2 1.61717 0.808587 0.588377i \(-0.200233\pi\)
0.808587 + 0.588377i \(0.200233\pi\)
\(758\) 0 0
\(759\) −13455.3 −0.643475
\(760\) 0 0
\(761\) −13346.3 −0.635745 −0.317873 0.948133i \(-0.602968\pi\)
−0.317873 + 0.948133i \(0.602968\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 730.538 0.0345264
\(766\) 0 0
\(767\) 23708.0 1.11610
\(768\) 0 0
\(769\) 15530.6 0.728281 0.364140 0.931344i \(-0.381363\pi\)
0.364140 + 0.931344i \(0.381363\pi\)
\(770\) 0 0
\(771\) −4547.66 −0.212425
\(772\) 0 0
\(773\) 11015.2 0.512536 0.256268 0.966606i \(-0.417507\pi\)
0.256268 + 0.966606i \(0.417507\pi\)
\(774\) 0 0
\(775\) 5360.63 0.248464
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6489.04 −0.298452
\(780\) 0 0
\(781\) −27542.4 −1.26190
\(782\) 0 0
\(783\) 2116.96 0.0966209
\(784\) 0 0
\(785\) 1573.27 0.0715317
\(786\) 0 0
\(787\) −21192.6 −0.959892 −0.479946 0.877298i \(-0.659344\pi\)
−0.479946 + 0.877298i \(0.659344\pi\)
\(788\) 0 0
\(789\) 19301.8 0.870927
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13304.1 0.595768
\(794\) 0 0
\(795\) −9880.49 −0.440786
\(796\) 0 0
\(797\) −18189.9 −0.808432 −0.404216 0.914663i \(-0.632456\pi\)
−0.404216 + 0.914663i \(0.632456\pi\)
\(798\) 0 0
\(799\) 3059.56 0.135468
\(800\) 0 0
\(801\) −10019.4 −0.441971
\(802\) 0 0
\(803\) 30321.2 1.33252
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20849.4 −0.909458
\(808\) 0 0
\(809\) 45516.5 1.97809 0.989045 0.147611i \(-0.0471584\pi\)
0.989045 + 0.147611i \(0.0471584\pi\)
\(810\) 0 0
\(811\) 42099.9 1.82285 0.911423 0.411472i \(-0.134985\pi\)
0.911423 + 0.411472i \(0.134985\pi\)
\(812\) 0 0
\(813\) −2885.98 −0.124497
\(814\) 0 0
\(815\) 6885.45 0.295935
\(816\) 0 0
\(817\) −10486.9 −0.449069
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26555.9 −1.12888 −0.564439 0.825475i \(-0.690907\pi\)
−0.564439 + 0.825475i \(0.690907\pi\)
\(822\) 0 0
\(823\) −1554.25 −0.0658295 −0.0329147 0.999458i \(-0.510479\pi\)
−0.0329147 + 0.999458i \(0.510479\pi\)
\(824\) 0 0
\(825\) 6610.07 0.278949
\(826\) 0 0
\(827\) −10548.3 −0.443531 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(828\) 0 0
\(829\) −4331.26 −0.181461 −0.0907303 0.995875i \(-0.528920\pi\)
−0.0907303 + 0.995875i \(0.528920\pi\)
\(830\) 0 0
\(831\) −2280.03 −0.0951786
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −31527.5 −1.30665
\(836\) 0 0
\(837\) 2484.30 0.102592
\(838\) 0 0
\(839\) 23557.7 0.969369 0.484685 0.874689i \(-0.338934\pi\)
0.484685 + 0.874689i \(0.338934\pi\)
\(840\) 0 0
\(841\) −18241.5 −0.747939
\(842\) 0 0
\(843\) 13236.2 0.540782
\(844\) 0 0
\(845\) 4921.59 0.200364
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7806.45 −0.315567
\(850\) 0 0
\(851\) 39425.0 1.58810
\(852\) 0 0
\(853\) 11493.6 0.461350 0.230675 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(854\) 0 0
\(855\) −6650.00 −0.265994
\(856\) 0 0
\(857\) 2534.23 0.101012 0.0505062 0.998724i \(-0.483917\pi\)
0.0505062 + 0.998724i \(0.483917\pi\)
\(858\) 0 0
\(859\) −3127.09 −0.124208 −0.0621041 0.998070i \(-0.519781\pi\)
−0.0621041 + 0.998070i \(0.519781\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15309.9 0.603887 0.301943 0.953326i \(-0.402365\pi\)
0.301943 + 0.953326i \(0.402365\pi\)
\(864\) 0 0
\(865\) 12341.2 0.485101
\(866\) 0 0
\(867\) 14442.8 0.565749
\(868\) 0 0
\(869\) 40387.0 1.57656
\(870\) 0 0
\(871\) 29689.8 1.15500
\(872\) 0 0
\(873\) −13328.4 −0.516723
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −42420.7 −1.63335 −0.816673 0.577100i \(-0.804184\pi\)
−0.816673 + 0.577100i \(0.804184\pi\)
\(878\) 0 0
\(879\) −27996.5 −1.07429
\(880\) 0 0
\(881\) −10060.8 −0.384740 −0.192370 0.981322i \(-0.561617\pi\)
−0.192370 + 0.981322i \(0.561617\pi\)
\(882\) 0 0
\(883\) 17437.8 0.664585 0.332293 0.943176i \(-0.392178\pi\)
0.332293 + 0.943176i \(0.392178\pi\)
\(884\) 0 0
\(885\) −14550.8 −0.552677
\(886\) 0 0
\(887\) 16016.7 0.606299 0.303149 0.952943i \(-0.401962\pi\)
0.303149 + 0.952943i \(0.401962\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3063.33 0.115180
\(892\) 0 0
\(893\) −27850.8 −1.04366
\(894\) 0 0
\(895\) 20134.5 0.751981
\(896\) 0 0
\(897\) 14207.1 0.528832
\(898\) 0 0
\(899\) 7214.22 0.267639
\(900\) 0 0
\(901\) 4005.69 0.148112
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −26937.5 −0.989428
\(906\) 0 0
\(907\) −10131.0 −0.370887 −0.185443 0.982655i \(-0.559372\pi\)
−0.185443 + 0.982655i \(0.559372\pi\)
\(908\) 0 0
\(909\) 5006.85 0.182692
\(910\) 0 0
\(911\) 1320.58 0.0480273 0.0240137 0.999712i \(-0.492355\pi\)
0.0240137 + 0.999712i \(0.492355\pi\)
\(912\) 0 0
\(913\) −34292.5 −1.24306
\(914\) 0 0
\(915\) −8165.41 −0.295017
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −7285.76 −0.261518 −0.130759 0.991414i \(-0.541741\pi\)
−0.130759 + 0.991414i \(0.541741\pi\)
\(920\) 0 0
\(921\) 14686.5 0.525448
\(922\) 0 0
\(923\) 29081.3 1.03708
\(924\) 0 0
\(925\) −19368.0 −0.688448
\(926\) 0 0
\(927\) −4971.91 −0.176159
\(928\) 0 0
\(929\) −23708.0 −0.837280 −0.418640 0.908152i \(-0.637493\pi\)
−0.418640 + 0.908152i \(0.637493\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 22515.8 0.790070
\(934\) 0 0
\(935\) 3069.79 0.107372
\(936\) 0 0
\(937\) 8853.60 0.308682 0.154341 0.988018i \(-0.450675\pi\)
0.154341 + 0.988018i \(0.450675\pi\)
\(938\) 0 0
\(939\) −19047.8 −0.661983
\(940\) 0 0
\(941\) 53971.6 1.86974 0.934869 0.354994i \(-0.115517\pi\)
0.934869 + 0.354994i \(0.115517\pi\)
\(942\) 0 0
\(943\) −8508.57 −0.293825
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8019.80 0.275194 0.137597 0.990488i \(-0.456062\pi\)
0.137597 + 0.990488i \(0.456062\pi\)
\(948\) 0 0
\(949\) −32015.4 −1.09511
\(950\) 0 0
\(951\) 23998.8 0.818312
\(952\) 0 0
\(953\) 42628.0 1.44896 0.724479 0.689297i \(-0.242080\pi\)
0.724479 + 0.689297i \(0.242080\pi\)
\(954\) 0 0
\(955\) 38638.9 1.30924
\(956\) 0 0
\(957\) 8895.69 0.300477
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21325.0 −0.715820
\(962\) 0 0
\(963\) 4804.24 0.160763
\(964\) 0 0
\(965\) −37196.8 −1.24084
\(966\) 0 0
\(967\) −43161.7 −1.43535 −0.717676 0.696377i \(-0.754794\pi\)
−0.717676 + 0.696377i \(0.754794\pi\)
\(968\) 0 0
\(969\) 2696.00 0.0893787
\(970\) 0 0
\(971\) 39775.4 1.31458 0.657288 0.753640i \(-0.271704\pi\)
0.657288 + 0.753640i \(0.271704\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6979.40 −0.229251
\(976\) 0 0
\(977\) −26658.1 −0.872945 −0.436472 0.899718i \(-0.643772\pi\)
−0.436472 + 0.899718i \(0.643772\pi\)
\(978\) 0 0
\(979\) −42102.6 −1.37447
\(980\) 0 0
\(981\) 9851.66 0.320631
\(982\) 0 0
\(983\) 2942.43 0.0954719 0.0477359 0.998860i \(-0.484799\pi\)
0.0477359 + 0.998860i \(0.484799\pi\)
\(984\) 0 0
\(985\) 25399.2 0.821610
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13750.6 −0.442108
\(990\) 0 0
\(991\) 44969.8 1.44149 0.720743 0.693203i \(-0.243801\pi\)
0.720743 + 0.693203i \(0.243801\pi\)
\(992\) 0 0
\(993\) 6364.68 0.203401
\(994\) 0 0
\(995\) −3623.23 −0.115441
\(996\) 0 0
\(997\) −27116.4 −0.861370 −0.430685 0.902502i \(-0.641728\pi\)
−0.430685 + 0.902502i \(0.641728\pi\)
\(998\) 0 0
\(999\) −8975.75 −0.284265
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 588.4.a.j.1.2 4
3.2 odd 2 1764.4.a.bc.1.3 4
4.3 odd 2 2352.4.a.cq.1.2 4
7.2 even 3 588.4.i.l.361.3 8
7.3 odd 6 588.4.i.k.373.2 8
7.4 even 3 588.4.i.l.373.3 8
7.5 odd 6 588.4.i.k.361.2 8
7.6 odd 2 588.4.a.k.1.3 yes 4
21.2 odd 6 1764.4.k.bb.361.2 8
21.5 even 6 1764.4.k.bd.361.3 8
21.11 odd 6 1764.4.k.bb.1549.2 8
21.17 even 6 1764.4.k.bd.1549.3 8
21.20 even 2 1764.4.a.ba.1.2 4
28.27 even 2 2352.4.a.cl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.2 4 1.1 even 1 trivial
588.4.a.k.1.3 yes 4 7.6 odd 2
588.4.i.k.361.2 8 7.5 odd 6
588.4.i.k.373.2 8 7.3 odd 6
588.4.i.l.361.3 8 7.2 even 3
588.4.i.l.373.3 8 7.4 even 3
1764.4.a.ba.1.2 4 21.20 even 2
1764.4.a.bc.1.3 4 3.2 odd 2
1764.4.k.bb.361.2 8 21.2 odd 6
1764.4.k.bb.1549.2 8 21.11 odd 6
1764.4.k.bd.361.3 8 21.5 even 6
1764.4.k.bd.1549.3 8 21.17 even 6
2352.4.a.cl.1.3 4 28.27 even 2
2352.4.a.cq.1.2 4 4.3 odd 2