Properties

Label 1200.4.a.br.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
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] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.72015\) of defining polynomial
Character \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} -54.6418 q^{11} +82.7612 q^{13} -100.403 q^{17} +84.1194 q^{19} -65.6418 q^{21} +0.880613 q^{23} +27.0000 q^{27} -99.1194 q^{29} +78.9255 q^{31} -163.926 q^{33} +390.806 q^{37} +248.284 q^{39} +104.642 q^{41} -241.403 q^{43} -512.567 q^{47} +135.761 q^{49} -301.209 q^{51} +284.642 q^{53} +252.358 q^{57} +709.045 q^{59} +470.522 q^{61} -196.926 q^{63} +667.583 q^{67} +2.64184 q^{69} +51.5378 q^{71} +371.284 q^{73} +1195.60 q^{77} +79.3428 q^{79} +81.0000 q^{81} +682.180 q^{83} -297.358 q^{87} +628.478 q^{89} -1810.87 q^{91} +236.777 q^{93} +1519.09 q^{97} -491.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} + 16 q^{11} + 82 q^{13} + 8 q^{17} + 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} - 218 q^{31} + 48 q^{33} + 364 q^{37} + 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} + 24 q^{51} + 444 q^{53} + 630 q^{57} + 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} - 1108 q^{71} + 492 q^{73} + 2600 q^{77} + 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} - 1826 q^{91} - 654 q^{93} + 2370 q^{97} + 144 q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) −21.8806 −1.18144 −0.590721 0.806876i \(-0.701157\pi\)
−0.590721 + 0.806876i \(0.701157\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −54.6418 −1.49774 −0.748870 0.662717i \(-0.769403\pi\)
−0.748870 + 0.662717i \(0.769403\pi\)
\(12\) 0 0
\(13\) 82.7612 1.76568 0.882840 0.469674i \(-0.155629\pi\)
0.882840 + 0.469674i \(0.155629\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −100.403 −1.43243 −0.716215 0.697879i \(-0.754127\pi\)
−0.716215 + 0.697879i \(0.754127\pi\)
\(18\) 0 0
\(19\) 84.1194 1.01570 0.507850 0.861445i \(-0.330440\pi\)
0.507850 + 0.861445i \(0.330440\pi\)
\(20\) 0 0
\(21\) −65.6418 −0.682106
\(22\) 0 0
\(23\) 0.880613 0.00798350 0.00399175 0.999992i \(-0.498729\pi\)
0.00399175 + 0.999992i \(0.498729\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −99.1194 −0.634690 −0.317345 0.948310i \(-0.602791\pi\)
−0.317345 + 0.948310i \(0.602791\pi\)
\(30\) 0 0
\(31\) 78.9255 0.457272 0.228636 0.973512i \(-0.426573\pi\)
0.228636 + 0.973512i \(0.426573\pi\)
\(32\) 0 0
\(33\) −163.926 −0.864720
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 390.806 1.73644 0.868218 0.496183i \(-0.165265\pi\)
0.868218 + 0.496183i \(0.165265\pi\)
\(38\) 0 0
\(39\) 248.284 1.01942
\(40\) 0 0
\(41\) 104.642 0.398593 0.199296 0.979939i \(-0.436134\pi\)
0.199296 + 0.979939i \(0.436134\pi\)
\(42\) 0 0
\(43\) −241.403 −0.856131 −0.428065 0.903748i \(-0.640805\pi\)
−0.428065 + 0.903748i \(0.640805\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −512.567 −1.59076 −0.795379 0.606112i \(-0.792728\pi\)
−0.795379 + 0.606112i \(0.792728\pi\)
\(48\) 0 0
\(49\) 135.761 0.395805
\(50\) 0 0
\(51\) −301.209 −0.827014
\(52\) 0 0
\(53\) 284.642 0.737709 0.368854 0.929487i \(-0.379750\pi\)
0.368854 + 0.929487i \(0.379750\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 252.358 0.586415
\(58\) 0 0
\(59\) 709.045 1.56457 0.782286 0.622919i \(-0.214053\pi\)
0.782286 + 0.622919i \(0.214053\pi\)
\(60\) 0 0
\(61\) 470.522 0.987610 0.493805 0.869573i \(-0.335606\pi\)
0.493805 + 0.869573i \(0.335606\pi\)
\(62\) 0 0
\(63\) −196.926 −0.393814
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 667.583 1.21729 0.608643 0.793444i \(-0.291714\pi\)
0.608643 + 0.793444i \(0.291714\pi\)
\(68\) 0 0
\(69\) 2.64184 0.00460928
\(70\) 0 0
\(71\) 51.5378 0.0861466 0.0430733 0.999072i \(-0.486285\pi\)
0.0430733 + 0.999072i \(0.486285\pi\)
\(72\) 0 0
\(73\) 371.284 0.595280 0.297640 0.954678i \(-0.403800\pi\)
0.297640 + 0.954678i \(0.403800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1195.60 1.76949
\(78\) 0 0
\(79\) 79.3428 0.112997 0.0564985 0.998403i \(-0.482006\pi\)
0.0564985 + 0.998403i \(0.482006\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 682.180 0.902156 0.451078 0.892485i \(-0.351040\pi\)
0.451078 + 0.892485i \(0.351040\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −297.358 −0.366438
\(88\) 0 0
\(89\) 628.478 0.748522 0.374261 0.927323i \(-0.377896\pi\)
0.374261 + 0.927323i \(0.377896\pi\)
\(90\) 0 0
\(91\) −1810.87 −2.08605
\(92\) 0 0
\(93\) 236.777 0.264006
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1519.09 1.59011 0.795053 0.606541i \(-0.207443\pi\)
0.795053 + 0.606541i \(0.207443\pi\)
\(98\) 0 0
\(99\) −491.777 −0.499246
\(100\) 0 0
\(101\) 977.777 0.963291 0.481646 0.876366i \(-0.340039\pi\)
0.481646 + 0.876366i \(0.340039\pi\)
\(102\) 0 0
\(103\) −759.194 −0.726268 −0.363134 0.931737i \(-0.618293\pi\)
−0.363134 + 0.931737i \(0.618293\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −945.448 −0.854205 −0.427102 0.904203i \(-0.640466\pi\)
−0.427102 + 0.904203i \(0.640466\pi\)
\(108\) 0 0
\(109\) −688.314 −0.604849 −0.302425 0.953173i \(-0.597796\pi\)
−0.302425 + 0.953173i \(0.597796\pi\)
\(110\) 0 0
\(111\) 1172.42 1.00253
\(112\) 0 0
\(113\) −350.865 −0.292094 −0.146047 0.989278i \(-0.546655\pi\)
−0.146047 + 0.989278i \(0.546655\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 744.851 0.588560
\(118\) 0 0
\(119\) 2196.88 1.69233
\(120\) 0 0
\(121\) 1654.73 1.24322
\(122\) 0 0
\(123\) 313.926 0.230128
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1588.21 −1.10969 −0.554846 0.831953i \(-0.687223\pi\)
−0.554846 + 0.831953i \(0.687223\pi\)
\(128\) 0 0
\(129\) −724.209 −0.494287
\(130\) 0 0
\(131\) 2156.02 1.43795 0.718977 0.695034i \(-0.244611\pi\)
0.718977 + 0.695034i \(0.244611\pi\)
\(132\) 0 0
\(133\) −1840.58 −1.19999
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 316.716 0.197510 0.0987551 0.995112i \(-0.468514\pi\)
0.0987551 + 0.995112i \(0.468514\pi\)
\(138\) 0 0
\(139\) 2624.87 1.60171 0.800857 0.598855i \(-0.204377\pi\)
0.800857 + 0.598855i \(0.204377\pi\)
\(140\) 0 0
\(141\) −1537.70 −0.918425
\(142\) 0 0
\(143\) −4522.23 −2.64453
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 407.284 0.228518
\(148\) 0 0
\(149\) 2782.87 1.53008 0.765038 0.643985i \(-0.222720\pi\)
0.765038 + 0.643985i \(0.222720\pi\)
\(150\) 0 0
\(151\) −1227.25 −0.661407 −0.330704 0.943735i \(-0.607286\pi\)
−0.330704 + 0.943735i \(0.607286\pi\)
\(152\) 0 0
\(153\) −903.628 −0.477477
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3187.33 −1.62023 −0.810117 0.586268i \(-0.800596\pi\)
−0.810117 + 0.586268i \(0.800596\pi\)
\(158\) 0 0
\(159\) 853.926 0.425916
\(160\) 0 0
\(161\) −19.2684 −0.00943204
\(162\) 0 0
\(163\) −2481.64 −1.19250 −0.596249 0.802800i \(-0.703343\pi\)
−0.596249 + 0.802800i \(0.703343\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1904.55 0.882508 0.441254 0.897382i \(-0.354534\pi\)
0.441254 + 0.897382i \(0.354534\pi\)
\(168\) 0 0
\(169\) 4652.42 2.11762
\(170\) 0 0
\(171\) 757.074 0.338567
\(172\) 0 0
\(173\) −3788.88 −1.66511 −0.832553 0.553946i \(-0.813121\pi\)
−0.832553 + 0.553946i \(0.813121\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2127.13 0.903306
\(178\) 0 0
\(179\) 1390.63 0.580672 0.290336 0.956925i \(-0.406233\pi\)
0.290336 + 0.956925i \(0.406233\pi\)
\(180\) 0 0
\(181\) 2512.55 1.03180 0.515902 0.856647i \(-0.327457\pi\)
0.515902 + 0.856647i \(0.327457\pi\)
\(182\) 0 0
\(183\) 1411.57 0.570197
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5486.21 2.14541
\(188\) 0 0
\(189\) −590.777 −0.227369
\(190\) 0 0
\(191\) −2872.82 −1.08833 −0.544163 0.838980i \(-0.683153\pi\)
−0.544163 + 0.838980i \(0.683153\pi\)
\(192\) 0 0
\(193\) 4130.79 1.54063 0.770314 0.637665i \(-0.220100\pi\)
0.770314 + 0.637665i \(0.220100\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2243.52 0.811393 0.405697 0.914008i \(-0.367029\pi\)
0.405697 + 0.914008i \(0.367029\pi\)
\(198\) 0 0
\(199\) −1111.28 −0.395864 −0.197932 0.980216i \(-0.563422\pi\)
−0.197932 + 0.980216i \(0.563422\pi\)
\(200\) 0 0
\(201\) 2002.75 0.702801
\(202\) 0 0
\(203\) 2168.79 0.749849
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.92552 0.00266117
\(208\) 0 0
\(209\) −4596.44 −1.52125
\(210\) 0 0
\(211\) 4546.63 1.48343 0.741713 0.670717i \(-0.234014\pi\)
0.741713 + 0.670717i \(0.234014\pi\)
\(212\) 0 0
\(213\) 154.613 0.0497368
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1726.94 −0.540241
\(218\) 0 0
\(219\) 1113.85 0.343685
\(220\) 0 0
\(221\) −8309.48 −2.52921
\(222\) 0 0
\(223\) −3012.93 −0.904755 −0.452378 0.891826i \(-0.649424\pi\)
−0.452378 + 0.891826i \(0.649424\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1515.27 −0.443049 −0.221524 0.975155i \(-0.571103\pi\)
−0.221524 + 0.975155i \(0.571103\pi\)
\(228\) 0 0
\(229\) 2539.98 0.732956 0.366478 0.930427i \(-0.380564\pi\)
0.366478 + 0.930427i \(0.380564\pi\)
\(230\) 0 0
\(231\) 3586.79 1.02162
\(232\) 0 0
\(233\) −4772.91 −1.34199 −0.670996 0.741461i \(-0.734133\pi\)
−0.670996 + 0.741461i \(0.734133\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 238.029 0.0652389
\(238\) 0 0
\(239\) −244.344 −0.0661309 −0.0330655 0.999453i \(-0.510527\pi\)
−0.0330655 + 0.999453i \(0.510527\pi\)
\(240\) 0 0
\(241\) 5573.18 1.48963 0.744813 0.667273i \(-0.232538\pi\)
0.744813 + 0.667273i \(0.232538\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6961.82 1.79340
\(248\) 0 0
\(249\) 2046.54 0.520860
\(250\) 0 0
\(251\) 5786.78 1.45521 0.727606 0.685995i \(-0.240633\pi\)
0.727606 + 0.685995i \(0.240633\pi\)
\(252\) 0 0
\(253\) −48.1183 −0.0119572
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7780.75 −1.88852 −0.944260 0.329200i \(-0.893221\pi\)
−0.944260 + 0.329200i \(0.893221\pi\)
\(258\) 0 0
\(259\) −8551.08 −2.05150
\(260\) 0 0
\(261\) −892.074 −0.211563
\(262\) 0 0
\(263\) 5122.61 1.20104 0.600521 0.799609i \(-0.294960\pi\)
0.600521 + 0.799609i \(0.294960\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1885.43 0.432160
\(268\) 0 0
\(269\) 7125.12 1.61497 0.807484 0.589890i \(-0.200829\pi\)
0.807484 + 0.589890i \(0.200829\pi\)
\(270\) 0 0
\(271\) −1761.58 −0.394865 −0.197433 0.980316i \(-0.563260\pi\)
−0.197433 + 0.980316i \(0.563260\pi\)
\(272\) 0 0
\(273\) −5432.60 −1.20438
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5810.41 1.26034 0.630169 0.776458i \(-0.282986\pi\)
0.630169 + 0.776458i \(0.282986\pi\)
\(278\) 0 0
\(279\) 710.330 0.152424
\(280\) 0 0
\(281\) −4126.19 −0.875972 −0.437986 0.898982i \(-0.644308\pi\)
−0.437986 + 0.898982i \(0.644308\pi\)
\(282\) 0 0
\(283\) −718.330 −0.150884 −0.0754422 0.997150i \(-0.524037\pi\)
−0.0754422 + 0.997150i \(0.524037\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2289.63 −0.470914
\(288\) 0 0
\(289\) 5167.78 1.05186
\(290\) 0 0
\(291\) 4557.27 0.918048
\(292\) 0 0
\(293\) −3756.21 −0.748942 −0.374471 0.927239i \(-0.622176\pi\)
−0.374471 + 0.927239i \(0.622176\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1475.33 −0.288240
\(298\) 0 0
\(299\) 72.8806 0.0140963
\(300\) 0 0
\(301\) 5282.05 1.01147
\(302\) 0 0
\(303\) 2933.33 0.556156
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7599.52 1.41279 0.706397 0.707816i \(-0.250319\pi\)
0.706397 + 0.707816i \(0.250319\pi\)
\(308\) 0 0
\(309\) −2277.58 −0.419311
\(310\) 0 0
\(311\) −2850.03 −0.519648 −0.259824 0.965656i \(-0.583664\pi\)
−0.259824 + 0.965656i \(0.583664\pi\)
\(312\) 0 0
\(313\) 799.684 0.144411 0.0722057 0.997390i \(-0.476996\pi\)
0.0722057 + 0.997390i \(0.476996\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1003.79 −0.177850 −0.0889250 0.996038i \(-0.528343\pi\)
−0.0889250 + 0.996038i \(0.528343\pi\)
\(318\) 0 0
\(319\) 5416.07 0.950600
\(320\) 0 0
\(321\) −2836.34 −0.493175
\(322\) 0 0
\(323\) −8445.84 −1.45492
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2064.94 −0.349210
\(328\) 0 0
\(329\) 11215.3 1.87939
\(330\) 0 0
\(331\) −8367.68 −1.38951 −0.694757 0.719245i \(-0.744488\pi\)
−0.694757 + 0.719245i \(0.744488\pi\)
\(332\) 0 0
\(333\) 3517.26 0.578812
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5034.79 −0.813835 −0.406918 0.913465i \(-0.633396\pi\)
−0.406918 + 0.913465i \(0.633396\pi\)
\(338\) 0 0
\(339\) −1052.60 −0.168641
\(340\) 0 0
\(341\) −4312.64 −0.684875
\(342\) 0 0
\(343\) 4534.51 0.713821
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2105.53 0.325737 0.162868 0.986648i \(-0.447925\pi\)
0.162868 + 0.986648i \(0.447925\pi\)
\(348\) 0 0
\(349\) −8813.47 −1.35179 −0.675894 0.736999i \(-0.736242\pi\)
−0.675894 + 0.736999i \(0.736242\pi\)
\(350\) 0 0
\(351\) 2234.55 0.339805
\(352\) 0 0
\(353\) 9348.98 1.40962 0.704810 0.709396i \(-0.251032\pi\)
0.704810 + 0.709396i \(0.251032\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6590.64 0.977069
\(358\) 0 0
\(359\) 9233.33 1.35743 0.678714 0.734403i \(-0.262538\pi\)
0.678714 + 0.734403i \(0.262538\pi\)
\(360\) 0 0
\(361\) 217.071 0.0316477
\(362\) 0 0
\(363\) 4964.19 0.717775
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6996.60 0.995148 0.497574 0.867421i \(-0.334224\pi\)
0.497574 + 0.867421i \(0.334224\pi\)
\(368\) 0 0
\(369\) 941.777 0.132864
\(370\) 0 0
\(371\) −6228.14 −0.871560
\(372\) 0 0
\(373\) 2945.48 0.408877 0.204438 0.978879i \(-0.434463\pi\)
0.204438 + 0.978879i \(0.434463\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8203.24 −1.12066
\(378\) 0 0
\(379\) −8499.83 −1.15200 −0.575998 0.817451i \(-0.695387\pi\)
−0.575998 + 0.817451i \(0.695387\pi\)
\(380\) 0 0
\(381\) −4764.63 −0.640681
\(382\) 0 0
\(383\) 6426.01 0.857320 0.428660 0.903466i \(-0.358986\pi\)
0.428660 + 0.903466i \(0.358986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2172.63 −0.285377
\(388\) 0 0
\(389\) −675.805 −0.0880840 −0.0440420 0.999030i \(-0.514024\pi\)
−0.0440420 + 0.999030i \(0.514024\pi\)
\(390\) 0 0
\(391\) −88.4162 −0.0114358
\(392\) 0 0
\(393\) 6468.05 0.830203
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −9331.43 −1.17967 −0.589837 0.807522i \(-0.700808\pi\)
−0.589837 + 0.807522i \(0.700808\pi\)
\(398\) 0 0
\(399\) −5521.75 −0.692815
\(400\) 0 0
\(401\) −427.213 −0.0532021 −0.0266010 0.999646i \(-0.508468\pi\)
−0.0266010 + 0.999646i \(0.508468\pi\)
\(402\) 0 0
\(403\) 6531.97 0.807396
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21354.4 −2.60073
\(408\) 0 0
\(409\) 1243.18 0.150297 0.0751484 0.997172i \(-0.476057\pi\)
0.0751484 + 0.997172i \(0.476057\pi\)
\(410\) 0 0
\(411\) 950.149 0.114033
\(412\) 0 0
\(413\) −15514.3 −1.84845
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7874.60 0.924750
\(418\) 0 0
\(419\) 3802.17 0.443313 0.221656 0.975125i \(-0.428854\pi\)
0.221656 + 0.975125i \(0.428854\pi\)
\(420\) 0 0
\(421\) −12785.6 −1.48012 −0.740059 0.672542i \(-0.765203\pi\)
−0.740059 + 0.672542i \(0.765203\pi\)
\(422\) 0 0
\(423\) −4613.11 −0.530253
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10295.3 −1.16680
\(428\) 0 0
\(429\) −13566.7 −1.52682
\(430\) 0 0
\(431\) 9588.11 1.07156 0.535781 0.844357i \(-0.320017\pi\)
0.535781 + 0.844357i \(0.320017\pi\)
\(432\) 0 0
\(433\) 8493.83 0.942697 0.471348 0.881947i \(-0.343767\pi\)
0.471348 + 0.881947i \(0.343767\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 74.0766 0.00810885
\(438\) 0 0
\(439\) −5167.40 −0.561792 −0.280896 0.959738i \(-0.590632\pi\)
−0.280896 + 0.959738i \(0.590632\pi\)
\(440\) 0 0
\(441\) 1221.85 0.131935
\(442\) 0 0
\(443\) −8192.86 −0.878679 −0.439339 0.898321i \(-0.644787\pi\)
−0.439339 + 0.898321i \(0.644787\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8348.60 0.883390
\(448\) 0 0
\(449\) −18252.3 −1.91844 −0.959218 0.282666i \(-0.908781\pi\)
−0.959218 + 0.282666i \(0.908781\pi\)
\(450\) 0 0
\(451\) −5717.82 −0.596988
\(452\) 0 0
\(453\) −3681.76 −0.381864
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −330.123 −0.0337910 −0.0168955 0.999857i \(-0.505378\pi\)
−0.0168955 + 0.999857i \(0.505378\pi\)
\(458\) 0 0
\(459\) −2710.88 −0.275671
\(460\) 0 0
\(461\) −4773.18 −0.482232 −0.241116 0.970496i \(-0.577513\pi\)
−0.241116 + 0.970496i \(0.577513\pi\)
\(462\) 0 0
\(463\) −7860.46 −0.788999 −0.394499 0.918896i \(-0.629082\pi\)
−0.394499 + 0.918896i \(0.629082\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10520.8 −1.04250 −0.521249 0.853404i \(-0.674534\pi\)
−0.521249 + 0.853404i \(0.674534\pi\)
\(468\) 0 0
\(469\) −14607.1 −1.43815
\(470\) 0 0
\(471\) −9561.99 −0.935442
\(472\) 0 0
\(473\) 13190.7 1.28226
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2561.78 0.245903
\(478\) 0 0
\(479\) 3013.64 0.287467 0.143734 0.989616i \(-0.454089\pi\)
0.143734 + 0.989616i \(0.454089\pi\)
\(480\) 0 0
\(481\) 32343.6 3.06599
\(482\) 0 0
\(483\) −57.8051 −0.00544559
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7048.81 −0.655876 −0.327938 0.944699i \(-0.606354\pi\)
−0.327938 + 0.944699i \(0.606354\pi\)
\(488\) 0 0
\(489\) −7444.93 −0.688489
\(490\) 0 0
\(491\) −6338.68 −0.582608 −0.291304 0.956630i \(-0.594089\pi\)
−0.291304 + 0.956630i \(0.594089\pi\)
\(492\) 0 0
\(493\) 9951.89 0.909149
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1127.68 −0.101777
\(498\) 0 0
\(499\) 5402.66 0.484682 0.242341 0.970191i \(-0.422085\pi\)
0.242341 + 0.970191i \(0.422085\pi\)
\(500\) 0 0
\(501\) 5713.66 0.509516
\(502\) 0 0
\(503\) 7770.85 0.688837 0.344419 0.938816i \(-0.388076\pi\)
0.344419 + 0.938816i \(0.388076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 13957.3 1.22261
\(508\) 0 0
\(509\) 729.019 0.0634837 0.0317418 0.999496i \(-0.489895\pi\)
0.0317418 + 0.999496i \(0.489895\pi\)
\(510\) 0 0
\(511\) −8123.91 −0.703289
\(512\) 0 0
\(513\) 2271.22 0.195472
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 28007.6 2.38254
\(518\) 0 0
\(519\) −11366.6 −0.961349
\(520\) 0 0
\(521\) 14783.8 1.24317 0.621585 0.783346i \(-0.286489\pi\)
0.621585 + 0.783346i \(0.286489\pi\)
\(522\) 0 0
\(523\) −395.434 −0.0330614 −0.0165307 0.999863i \(-0.505262\pi\)
−0.0165307 + 0.999863i \(0.505262\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7924.36 −0.655011
\(528\) 0 0
\(529\) −12166.2 −0.999936
\(530\) 0 0
\(531\) 6381.40 0.521524
\(532\) 0 0
\(533\) 8660.29 0.703787
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4171.88 0.335251
\(538\) 0 0
\(539\) −7418.24 −0.592813
\(540\) 0 0
\(541\) −16507.9 −1.31188 −0.655942 0.754811i \(-0.727728\pi\)
−0.655942 + 0.754811i \(0.727728\pi\)
\(542\) 0 0
\(543\) 7537.66 0.595713
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9242.35 −0.722440 −0.361220 0.932481i \(-0.617640\pi\)
−0.361220 + 0.932481i \(0.617640\pi\)
\(548\) 0 0
\(549\) 4234.70 0.329203
\(550\) 0 0
\(551\) −8337.86 −0.644655
\(552\) 0 0
\(553\) −1736.07 −0.133499
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10449.5 −0.794898 −0.397449 0.917624i \(-0.630104\pi\)
−0.397449 + 0.917624i \(0.630104\pi\)
\(558\) 0 0
\(559\) −19978.8 −1.51165
\(560\) 0 0
\(561\) 16458.6 1.23865
\(562\) 0 0
\(563\) 13670.1 1.02331 0.511656 0.859191i \(-0.329032\pi\)
0.511656 + 0.859191i \(0.329032\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1772.33 −0.131271
\(568\) 0 0
\(569\) 13616.7 1.00323 0.501617 0.865090i \(-0.332739\pi\)
0.501617 + 0.865090i \(0.332739\pi\)
\(570\) 0 0
\(571\) 11833.4 0.867270 0.433635 0.901089i \(-0.357231\pi\)
0.433635 + 0.901089i \(0.357231\pi\)
\(572\) 0 0
\(573\) −8618.47 −0.628345
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7131.30 0.514523 0.257262 0.966342i \(-0.417180\pi\)
0.257262 + 0.966342i \(0.417180\pi\)
\(578\) 0 0
\(579\) 12392.4 0.889482
\(580\) 0 0
\(581\) −14926.5 −1.06584
\(582\) 0 0
\(583\) −15553.4 −1.10490
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25214.3 −1.77292 −0.886461 0.462804i \(-0.846843\pi\)
−0.886461 + 0.462804i \(0.846843\pi\)
\(588\) 0 0
\(589\) 6639.17 0.464452
\(590\) 0 0
\(591\) 6730.57 0.468458
\(592\) 0 0
\(593\) 4218.04 0.292098 0.146049 0.989277i \(-0.453344\pi\)
0.146049 + 0.989277i \(0.453344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3333.85 −0.228552
\(598\) 0 0
\(599\) −5956.57 −0.406308 −0.203154 0.979147i \(-0.565119\pi\)
−0.203154 + 0.979147i \(0.565119\pi\)
\(600\) 0 0
\(601\) −5182.29 −0.351731 −0.175865 0.984414i \(-0.556272\pi\)
−0.175865 + 0.984414i \(0.556272\pi\)
\(602\) 0 0
\(603\) 6008.24 0.405762
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −10142.0 −0.678171 −0.339086 0.940755i \(-0.610118\pi\)
−0.339086 + 0.940755i \(0.610118\pi\)
\(608\) 0 0
\(609\) 6506.38 0.432926
\(610\) 0 0
\(611\) −42420.7 −2.80877
\(612\) 0 0
\(613\) −4022.95 −0.265066 −0.132533 0.991179i \(-0.542311\pi\)
−0.132533 + 0.991179i \(0.542311\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27074.4 −1.76657 −0.883285 0.468837i \(-0.844673\pi\)
−0.883285 + 0.468837i \(0.844673\pi\)
\(618\) 0 0
\(619\) 17245.7 1.11981 0.559905 0.828557i \(-0.310838\pi\)
0.559905 + 0.828557i \(0.310838\pi\)
\(620\) 0 0
\(621\) 23.7766 0.00153643
\(622\) 0 0
\(623\) −13751.5 −0.884336
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −13789.3 −0.878297
\(628\) 0 0
\(629\) −39238.1 −2.48732
\(630\) 0 0
\(631\) 14254.4 0.899302 0.449651 0.893204i \(-0.351548\pi\)
0.449651 + 0.893204i \(0.351548\pi\)
\(632\) 0 0
\(633\) 13639.9 0.856456
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 11235.8 0.698865
\(638\) 0 0
\(639\) 463.840 0.0287155
\(640\) 0 0
\(641\) 2611.61 0.160924 0.0804621 0.996758i \(-0.474360\pi\)
0.0804621 + 0.996758i \(0.474360\pi\)
\(642\) 0 0
\(643\) 27414.4 1.68136 0.840682 0.541529i \(-0.182154\pi\)
0.840682 + 0.541529i \(0.182154\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12820.8 −0.779041 −0.389521 0.921018i \(-0.627359\pi\)
−0.389521 + 0.921018i \(0.627359\pi\)
\(648\) 0 0
\(649\) −38743.5 −2.34332
\(650\) 0 0
\(651\) −5180.82 −0.311908
\(652\) 0 0
\(653\) −15786.4 −0.946049 −0.473025 0.881049i \(-0.656838\pi\)
−0.473025 + 0.881049i \(0.656838\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3341.55 0.198427
\(658\) 0 0
\(659\) −4822.61 −0.285072 −0.142536 0.989790i \(-0.545526\pi\)
−0.142536 + 0.989790i \(0.545526\pi\)
\(660\) 0 0
\(661\) −23624.9 −1.39017 −0.695084 0.718929i \(-0.744633\pi\)
−0.695084 + 0.718929i \(0.744633\pi\)
\(662\) 0 0
\(663\) −24928.4 −1.46024
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −87.2858 −0.00506705
\(668\) 0 0
\(669\) −9038.78 −0.522361
\(670\) 0 0
\(671\) −25710.2 −1.47918
\(672\) 0 0
\(673\) −8903.65 −0.509971 −0.254985 0.966945i \(-0.582071\pi\)
−0.254985 + 0.966945i \(0.582071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9034.56 −0.512890 −0.256445 0.966559i \(-0.582551\pi\)
−0.256445 + 0.966559i \(0.582551\pi\)
\(678\) 0 0
\(679\) −33238.6 −1.87862
\(680\) 0 0
\(681\) −4545.81 −0.255794
\(682\) 0 0
\(683\) −28676.8 −1.60657 −0.803284 0.595596i \(-0.796916\pi\)
−0.803284 + 0.595596i \(0.796916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7619.95 0.423172
\(688\) 0 0
\(689\) 23557.3 1.30256
\(690\) 0 0
\(691\) 20653.1 1.13702 0.568509 0.822677i \(-0.307520\pi\)
0.568509 + 0.822677i \(0.307520\pi\)
\(692\) 0 0
\(693\) 10760.4 0.589831
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10506.4 −0.570957
\(698\) 0 0
\(699\) −14318.7 −0.774799
\(700\) 0 0
\(701\) −11000.1 −0.592678 −0.296339 0.955083i \(-0.595766\pi\)
−0.296339 + 0.955083i \(0.595766\pi\)
\(702\) 0 0
\(703\) 32874.4 1.76370
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −21394.4 −1.13807
\(708\) 0 0
\(709\) −372.560 −0.0197345 −0.00986725 0.999951i \(-0.503141\pi\)
−0.00986725 + 0.999951i \(0.503141\pi\)
\(710\) 0 0
\(711\) 714.086 0.0376657
\(712\) 0 0
\(713\) 69.5028 0.00365063
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −733.032 −0.0381807
\(718\) 0 0
\(719\) 7207.32 0.373835 0.186918 0.982376i \(-0.440150\pi\)
0.186918 + 0.982376i \(0.440150\pi\)
\(720\) 0 0
\(721\) 16611.6 0.858043
\(722\) 0 0
\(723\) 16719.5 0.860036
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23423.6 1.19496 0.597479 0.801885i \(-0.296169\pi\)
0.597479 + 0.801885i \(0.296169\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 24237.6 1.22635
\(732\) 0 0
\(733\) −20368.0 −1.02634 −0.513172 0.858286i \(-0.671530\pi\)
−0.513172 + 0.858286i \(0.671530\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36477.9 −1.82318
\(738\) 0 0
\(739\) −2603.64 −0.129603 −0.0648014 0.997898i \(-0.520641\pi\)
−0.0648014 + 0.997898i \(0.520641\pi\)
\(740\) 0 0
\(741\) 20885.5 1.03542
\(742\) 0 0
\(743\) −8627.83 −0.426009 −0.213004 0.977051i \(-0.568325\pi\)
−0.213004 + 0.977051i \(0.568325\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6139.62 0.300719
\(748\) 0 0
\(749\) 20687.0 1.00919
\(750\) 0 0
\(751\) 18735.5 0.910343 0.455172 0.890404i \(-0.349578\pi\)
0.455172 + 0.890404i \(0.349578\pi\)
\(752\) 0 0
\(753\) 17360.3 0.840167
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10787.3 0.517926 0.258963 0.965887i \(-0.416619\pi\)
0.258963 + 0.965887i \(0.416619\pi\)
\(758\) 0 0
\(759\) −144.355 −0.00690349
\(760\) 0 0
\(761\) −2175.95 −0.103651 −0.0518253 0.998656i \(-0.516504\pi\)
−0.0518253 + 0.998656i \(0.516504\pi\)
\(762\) 0 0
\(763\) 15060.7 0.714594
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 58681.4 2.76253
\(768\) 0 0
\(769\) 18705.8 0.877176 0.438588 0.898688i \(-0.355479\pi\)
0.438588 + 0.898688i \(0.355479\pi\)
\(770\) 0 0
\(771\) −23342.2 −1.09034
\(772\) 0 0
\(773\) 8243.60 0.383573 0.191786 0.981437i \(-0.438572\pi\)
0.191786 + 0.981437i \(0.438572\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −25653.2 −1.18443
\(778\) 0 0
\(779\) 8802.41 0.404851
\(780\) 0 0
\(781\) −2816.12 −0.129025
\(782\) 0 0
\(783\) −2676.22 −0.122146
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29442.1 1.33354 0.666770 0.745263i \(-0.267676\pi\)
0.666770 + 0.745263i \(0.267676\pi\)
\(788\) 0 0
\(789\) 15367.8 0.693422
\(790\) 0 0
\(791\) 7677.15 0.345092
\(792\) 0 0
\(793\) 38941.0 1.74380
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42045.4 1.86866 0.934332 0.356403i \(-0.115997\pi\)
0.934332 + 0.356403i \(0.115997\pi\)
\(798\) 0 0
\(799\) 51463.3 2.27865
\(800\) 0 0
\(801\) 5656.30 0.249507
\(802\) 0 0
\(803\) −20287.6 −0.891575
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 21375.4 0.932402
\(808\) 0 0
\(809\) 43757.6 1.90165 0.950825 0.309727i \(-0.100238\pi\)
0.950825 + 0.309727i \(0.100238\pi\)
\(810\) 0 0
\(811\) −3273.00 −0.141715 −0.0708574 0.997486i \(-0.522574\pi\)
−0.0708574 + 0.997486i \(0.522574\pi\)
\(812\) 0 0
\(813\) −5284.74 −0.227976
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20306.7 −0.869572
\(818\) 0 0
\(819\) −16297.8 −0.695349
\(820\) 0 0
\(821\) −40442.6 −1.71919 −0.859595 0.510976i \(-0.829284\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(822\) 0 0
\(823\) 27236.8 1.15360 0.576802 0.816884i \(-0.304300\pi\)
0.576802 + 0.816884i \(0.304300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33344.5 1.40206 0.701028 0.713133i \(-0.252725\pi\)
0.701028 + 0.713133i \(0.252725\pi\)
\(828\) 0 0
\(829\) −32085.0 −1.34422 −0.672110 0.740452i \(-0.734612\pi\)
−0.672110 + 0.740452i \(0.734612\pi\)
\(830\) 0 0
\(831\) 17431.2 0.727656
\(832\) 0 0
\(833\) −13630.8 −0.566964
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2130.99 0.0880021
\(838\) 0 0
\(839\) 7072.24 0.291014 0.145507 0.989357i \(-0.453519\pi\)
0.145507 + 0.989357i \(0.453519\pi\)
\(840\) 0 0
\(841\) −14564.3 −0.597169
\(842\) 0 0
\(843\) −12378.6 −0.505743
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −36206.5 −1.46880
\(848\) 0 0
\(849\) −2154.99 −0.0871131
\(850\) 0 0
\(851\) 344.149 0.0138628
\(852\) 0 0
\(853\) −10225.0 −0.410431 −0.205215 0.978717i \(-0.565789\pi\)
−0.205215 + 0.978717i \(0.565789\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9783.87 0.389977 0.194989 0.980805i \(-0.437533\pi\)
0.194989 + 0.980805i \(0.437533\pi\)
\(858\) 0 0
\(859\) 22931.8 0.910853 0.455427 0.890273i \(-0.349487\pi\)
0.455427 + 0.890273i \(0.349487\pi\)
\(860\) 0 0
\(861\) −6868.88 −0.271883
\(862\) 0 0
\(863\) 23506.0 0.927175 0.463588 0.886051i \(-0.346562\pi\)
0.463588 + 0.886051i \(0.346562\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 15503.3 0.607290
\(868\) 0 0
\(869\) −4335.44 −0.169240
\(870\) 0 0
\(871\) 55250.0 2.14934
\(872\) 0 0
\(873\) 13671.8 0.530035
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34672.0 1.33499 0.667497 0.744613i \(-0.267366\pi\)
0.667497 + 0.744613i \(0.267366\pi\)
\(878\) 0 0
\(879\) −11268.6 −0.432402
\(880\) 0 0
\(881\) −2920.92 −0.111701 −0.0558504 0.998439i \(-0.517787\pi\)
−0.0558504 + 0.998439i \(0.517787\pi\)
\(882\) 0 0
\(883\) −7123.74 −0.271498 −0.135749 0.990743i \(-0.543344\pi\)
−0.135749 + 0.990743i \(0.543344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7923.42 0.299935 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(888\) 0 0
\(889\) 34751.0 1.31104
\(890\) 0 0
\(891\) −4425.99 −0.166415
\(892\) 0 0
\(893\) −43116.9 −1.61573
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 218.642 0.00813850
\(898\) 0 0
\(899\) −7823.05 −0.290226
\(900\) 0 0
\(901\) −28578.9 −1.05672
\(902\) 0 0
\(903\) 15846.1 0.583972
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19771.7 −0.723825 −0.361912 0.932212i \(-0.617876\pi\)
−0.361912 + 0.932212i \(0.617876\pi\)
\(908\) 0 0
\(909\) 8799.99 0.321097
\(910\) 0 0
\(911\) −38222.7 −1.39009 −0.695046 0.718966i \(-0.744616\pi\)
−0.695046 + 0.718966i \(0.744616\pi\)
\(912\) 0 0
\(913\) −37275.5 −1.35119
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −47174.9 −1.69886
\(918\) 0 0
\(919\) −39237.3 −1.40840 −0.704199 0.710003i \(-0.748694\pi\)
−0.704199 + 0.710003i \(0.748694\pi\)
\(920\) 0 0
\(921\) 22798.6 0.815677
\(922\) 0 0
\(923\) 4265.33 0.152107
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6832.74 −0.242089
\(928\) 0 0
\(929\) 20024.5 0.707194 0.353597 0.935398i \(-0.384958\pi\)
0.353597 + 0.935398i \(0.384958\pi\)
\(930\) 0 0
\(931\) 11420.2 0.402020
\(932\) 0 0
\(933\) −8550.09 −0.300019
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −8717.10 −0.303922 −0.151961 0.988386i \(-0.548559\pi\)
−0.151961 + 0.988386i \(0.548559\pi\)
\(938\) 0 0
\(939\) 2399.05 0.0833760
\(940\) 0 0
\(941\) 39276.2 1.36065 0.680323 0.732913i \(-0.261840\pi\)
0.680323 + 0.732913i \(0.261840\pi\)
\(942\) 0 0
\(943\) 92.1490 0.00318217
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20080.8 0.689060 0.344530 0.938775i \(-0.388038\pi\)
0.344530 + 0.938775i \(0.388038\pi\)
\(948\) 0 0
\(949\) 30727.9 1.05107
\(950\) 0 0
\(951\) −3011.37 −0.102682
\(952\) 0 0
\(953\) 38549.9 1.31034 0.655170 0.755481i \(-0.272597\pi\)
0.655170 + 0.755481i \(0.272597\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 16248.2 0.548829
\(958\) 0 0
\(959\) −6929.95 −0.233347
\(960\) 0 0
\(961\) −23561.8 −0.790902
\(962\) 0 0
\(963\) −8509.03 −0.284735
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24108.2 0.801724 0.400862 0.916138i \(-0.368711\pi\)
0.400862 + 0.916138i \(0.368711\pi\)
\(968\) 0 0
\(969\) −25337.5 −0.839999
\(970\) 0 0
\(971\) 41760.1 1.38017 0.690084 0.723729i \(-0.257573\pi\)
0.690084 + 0.723729i \(0.257573\pi\)
\(972\) 0 0
\(973\) −57433.7 −1.89233
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4532.52 0.148422 0.0742109 0.997243i \(-0.476356\pi\)
0.0742109 + 0.997243i \(0.476356\pi\)
\(978\) 0 0
\(979\) −34341.2 −1.12109
\(980\) 0 0
\(981\) −6194.83 −0.201616
\(982\) 0 0
\(983\) 48822.8 1.58414 0.792068 0.610432i \(-0.209004\pi\)
0.792068 + 0.610432i \(0.209004\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 33645.9 1.08507
\(988\) 0 0
\(989\) −212.583 −0.00683492
\(990\) 0 0
\(991\) 26937.6 0.863473 0.431737 0.902000i \(-0.357901\pi\)
0.431737 + 0.902000i \(0.357901\pi\)
\(992\) 0 0
\(993\) −25103.0 −0.802236
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9191.03 −0.291959 −0.145979 0.989288i \(-0.546633\pi\)
−0.145979 + 0.989288i \(0.546633\pi\)
\(998\) 0 0
\(999\) 10551.8 0.334177
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 1200.4.a.br.1.1 2
4.3 odd 2 600.4.a.s.1.2 2
5.2 odd 4 1200.4.f.x.49.1 4
5.3 odd 4 1200.4.f.x.49.4 4
5.4 even 2 1200.4.a.bp.1.2 2
12.11 even 2 1800.4.a.bo.1.2 2
20.3 even 4 600.4.f.j.49.1 4
20.7 even 4 600.4.f.j.49.4 4
20.19 odd 2 600.4.a.u.1.1 yes 2
60.23 odd 4 1800.4.f.z.649.1 4
60.47 odd 4 1800.4.f.z.649.4 4
60.59 even 2 1800.4.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.s.1.2 2 4.3 odd 2
600.4.a.u.1.1 yes 2 20.19 odd 2
600.4.f.j.49.1 4 20.3 even 4
600.4.f.j.49.4 4 20.7 even 4
1200.4.a.bp.1.2 2 5.4 even 2
1200.4.a.br.1.1 2 1.1 even 1 trivial
1200.4.f.x.49.1 4 5.2 odd 4
1200.4.f.x.49.4 4 5.3 odd 4
1800.4.a.bm.1.1 2 60.59 even 2
1800.4.a.bo.1.2 2 12.11 even 2
1800.4.f.z.649.1 4 60.23 odd 4
1800.4.f.z.649.4 4 60.47 odd 4