Properties

Label 800.4.a.p.1.2
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, 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("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.32456 q^{3} +18.9737 q^{7} +13.0000 q^{9} +O(q^{10})\) \(q+6.32456 q^{3} +18.9737 q^{7} +13.0000 q^{9} +12.6491 q^{11} -38.0000 q^{13} -34.0000 q^{17} +101.193 q^{19} +120.000 q^{21} +82.2192 q^{23} -88.5438 q^{27} +270.000 q^{29} +341.526 q^{31} +80.0000 q^{33} -206.000 q^{37} -240.333 q^{39} -270.000 q^{41} +537.587 q^{43} -132.816 q^{47} +17.0000 q^{49} -215.035 q^{51} +258.000 q^{53} +640.000 q^{57} +75.8947 q^{59} -250.000 q^{61} +246.658 q^{63} -815.868 q^{67} +520.000 q^{69} +645.105 q^{71} +1078.00 q^{73} +240.000 q^{77} +278.280 q^{79} -911.000 q^{81} -1106.80 q^{83} +1707.63 q^{87} +890.000 q^{89} -720.999 q^{91} +2160.00 q^{93} +254.000 q^{97} +164.438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 26 q^{9} - 76 q^{13} - 68 q^{17} + 240 q^{21} + 540 q^{29} + 160 q^{33} - 412 q^{37} - 540 q^{41} + 34 q^{49} + 516 q^{53} + 1280 q^{57} - 500 q^{61} + 1040 q^{69} + 2156 q^{73} + 480 q^{77} - 1822 q^{81} + 1780 q^{89} + 4320 q^{93} + 508 q^{97}+O(q^{100}) \) 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
\(3\) 6.32456 1.21716 0.608581 0.793492i \(-0.291739\pi\)
0.608581 + 0.793492i \(0.291739\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 18.9737 1.02448 0.512241 0.858842i \(-0.328816\pi\)
0.512241 + 0.858842i \(0.328816\pi\)
\(8\) 0 0
\(9\) 13.0000 0.481481
\(10\) 0 0
\(11\) 12.6491 0.346714 0.173357 0.984859i \(-0.444539\pi\)
0.173357 + 0.984859i \(0.444539\pi\)
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 101.193 1.22185 0.610927 0.791687i \(-0.290797\pi\)
0.610927 + 0.791687i \(0.290797\pi\)
\(20\) 0 0
\(21\) 120.000 1.24696
\(22\) 0 0
\(23\) 82.2192 0.745387 0.372693 0.927955i \(-0.378434\pi\)
0.372693 + 0.927955i \(0.378434\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −88.5438 −0.631121
\(28\) 0 0
\(29\) 270.000 1.72889 0.864444 0.502729i \(-0.167671\pi\)
0.864444 + 0.502729i \(0.167671\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) 80.0000 0.422006
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −206.000 −0.915302 −0.457651 0.889132i \(-0.651309\pi\)
−0.457651 + 0.889132i \(0.651309\pi\)
\(38\) 0 0
\(39\) −240.333 −0.986772
\(40\) 0 0
\(41\) −270.000 −1.02846 −0.514231 0.857652i \(-0.671922\pi\)
−0.514231 + 0.857652i \(0.671922\pi\)
\(42\) 0 0
\(43\) 537.587 1.90654 0.953271 0.302117i \(-0.0976935\pi\)
0.953271 + 0.302117i \(0.0976935\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −132.816 −0.412195 −0.206097 0.978531i \(-0.566076\pi\)
−0.206097 + 0.978531i \(0.566076\pi\)
\(48\) 0 0
\(49\) 17.0000 0.0495627
\(50\) 0 0
\(51\) −215.035 −0.590410
\(52\) 0 0
\(53\) 258.000 0.668661 0.334330 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 640.000 1.48719
\(58\) 0 0
\(59\) 75.8947 0.167469 0.0837343 0.996488i \(-0.473315\pi\)
0.0837343 + 0.996488i \(0.473315\pi\)
\(60\) 0 0
\(61\) −250.000 −0.524741 −0.262371 0.964967i \(-0.584504\pi\)
−0.262371 + 0.964967i \(0.584504\pi\)
\(62\) 0 0
\(63\) 246.658 0.493269
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −815.868 −1.48767 −0.743837 0.668362i \(-0.766996\pi\)
−0.743837 + 0.668362i \(0.766996\pi\)
\(68\) 0 0
\(69\) 520.000 0.907256
\(70\) 0 0
\(71\) 645.105 1.07831 0.539154 0.842207i \(-0.318744\pi\)
0.539154 + 0.842207i \(0.318744\pi\)
\(72\) 0 0
\(73\) 1078.00 1.72836 0.864181 0.503182i \(-0.167837\pi\)
0.864181 + 0.503182i \(0.167837\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 240.000 0.355202
\(78\) 0 0
\(79\) 278.280 0.396316 0.198158 0.980170i \(-0.436504\pi\)
0.198158 + 0.980170i \(0.436504\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) −1106.80 −1.46370 −0.731848 0.681468i \(-0.761342\pi\)
−0.731848 + 0.681468i \(0.761342\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1707.63 2.10434
\(88\) 0 0
\(89\) 890.000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −720.999 −0.830563
\(92\) 0 0
\(93\) 2160.00 2.40840
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 254.000 0.265874 0.132937 0.991124i \(-0.457559\pi\)
0.132937 + 0.991124i \(0.457559\pi\)
\(98\) 0 0
\(99\) 164.438 0.166936
\(100\) 0 0
\(101\) 598.000 0.589141 0.294570 0.955630i \(-0.404823\pi\)
0.294570 + 0.955630i \(0.404823\pi\)
\(102\) 0 0
\(103\) −499.640 −0.477971 −0.238985 0.971023i \(-0.576815\pi\)
−0.238985 + 0.971023i \(0.576815\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 626.131 0.565704 0.282852 0.959164i \(-0.408719\pi\)
0.282852 + 0.959164i \(0.408719\pi\)
\(108\) 0 0
\(109\) 854.000 0.750444 0.375222 0.926935i \(-0.377567\pi\)
0.375222 + 0.926935i \(0.377567\pi\)
\(110\) 0 0
\(111\) −1302.86 −1.11407
\(112\) 0 0
\(113\) −1698.00 −1.41358 −0.706789 0.707424i \(-0.749857\pi\)
−0.706789 + 0.707424i \(0.749857\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −494.000 −0.390345
\(118\) 0 0
\(119\) −645.105 −0.496947
\(120\) 0 0
\(121\) −1171.00 −0.879790
\(122\) 0 0
\(123\) −1707.63 −1.25180
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −234.009 −0.163503 −0.0817516 0.996653i \(-0.526051\pi\)
−0.0817516 + 0.996653i \(0.526051\pi\)
\(128\) 0 0
\(129\) 3400.00 2.32057
\(130\) 0 0
\(131\) −1732.93 −1.15578 −0.577888 0.816116i \(-0.696123\pi\)
−0.577888 + 0.816116i \(0.696123\pi\)
\(132\) 0 0
\(133\) 1920.00 1.25177
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1546.00 −0.964115 −0.482057 0.876140i \(-0.660110\pi\)
−0.482057 + 0.876140i \(0.660110\pi\)
\(138\) 0 0
\(139\) −328.877 −0.200683 −0.100342 0.994953i \(-0.531994\pi\)
−0.100342 + 0.994953i \(0.531994\pi\)
\(140\) 0 0
\(141\) −840.000 −0.501708
\(142\) 0 0
\(143\) −480.666 −0.281086
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 107.517 0.0603258
\(148\) 0 0
\(149\) 3246.00 1.78472 0.892358 0.451328i \(-0.149050\pi\)
0.892358 + 0.451328i \(0.149050\pi\)
\(150\) 0 0
\(151\) 1505.24 0.811225 0.405613 0.914045i \(-0.367058\pi\)
0.405613 + 0.914045i \(0.367058\pi\)
\(152\) 0 0
\(153\) −442.000 −0.233553
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1226.00 0.623219 0.311610 0.950210i \(-0.399132\pi\)
0.311610 + 0.950210i \(0.399132\pi\)
\(158\) 0 0
\(159\) 1631.74 0.813868
\(160\) 0 0
\(161\) 1560.00 0.763635
\(162\) 0 0
\(163\) 1448.32 0.695960 0.347980 0.937502i \(-0.386868\pi\)
0.347980 + 0.937502i \(0.386868\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2333.76 −1.08139 −0.540694 0.841219i \(-0.681838\pi\)
−0.540694 + 0.841219i \(0.681838\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) 1315.51 0.588300
\(172\) 0 0
\(173\) 3098.00 1.36148 0.680742 0.732524i \(-0.261658\pi\)
0.680742 + 0.732524i \(0.261658\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 480.000 0.203836
\(178\) 0 0
\(179\) −2352.73 −0.982411 −0.491206 0.871044i \(-0.663444\pi\)
−0.491206 + 0.871044i \(0.663444\pi\)
\(180\) 0 0
\(181\) 2182.00 0.896060 0.448030 0.894019i \(-0.352126\pi\)
0.448030 + 0.894019i \(0.352126\pi\)
\(182\) 0 0
\(183\) −1581.14 −0.638695
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −430.070 −0.168181
\(188\) 0 0
\(189\) −1680.00 −0.646572
\(190\) 0 0
\(191\) −3023.14 −1.14527 −0.572635 0.819810i \(-0.694079\pi\)
−0.572635 + 0.819810i \(0.694079\pi\)
\(192\) 0 0
\(193\) −1298.00 −0.484104 −0.242052 0.970263i \(-0.577820\pi\)
−0.242052 + 0.970263i \(0.577820\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2846.00 −1.02928 −0.514642 0.857405i \(-0.672075\pi\)
−0.514642 + 0.857405i \(0.672075\pi\)
\(198\) 0 0
\(199\) −3592.35 −1.27967 −0.639836 0.768511i \(-0.720998\pi\)
−0.639836 + 0.768511i \(0.720998\pi\)
\(200\) 0 0
\(201\) −5160.00 −1.81074
\(202\) 0 0
\(203\) 5122.89 1.77121
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1068.85 0.358890
\(208\) 0 0
\(209\) 1280.00 0.423634
\(210\) 0 0
\(211\) 4186.86 1.36604 0.683021 0.730398i \(-0.260666\pi\)
0.683021 + 0.730398i \(0.260666\pi\)
\(212\) 0 0
\(213\) 4080.00 1.31247
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6480.00 2.02715
\(218\) 0 0
\(219\) 6817.87 2.10369
\(220\) 0 0
\(221\) 1292.00 0.393255
\(222\) 0 0
\(223\) 4762.39 1.43010 0.715052 0.699071i \(-0.246403\pi\)
0.715052 + 0.699071i \(0.246403\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1663.36 0.486348 0.243174 0.969983i \(-0.421811\pi\)
0.243174 + 0.969983i \(0.421811\pi\)
\(228\) 0 0
\(229\) −1050.00 −0.302995 −0.151498 0.988458i \(-0.548410\pi\)
−0.151498 + 0.988458i \(0.548410\pi\)
\(230\) 0 0
\(231\) 1517.89 0.432338
\(232\) 0 0
\(233\) −2778.00 −0.781085 −0.390543 0.920585i \(-0.627713\pi\)
−0.390543 + 0.920585i \(0.627713\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1760.00 0.482381
\(238\) 0 0
\(239\) 2555.12 0.691536 0.345768 0.938320i \(-0.387618\pi\)
0.345768 + 0.938320i \(0.387618\pi\)
\(240\) 0 0
\(241\) −5350.00 −1.42997 −0.714987 0.699138i \(-0.753567\pi\)
−0.714987 + 0.699138i \(0.753567\pi\)
\(242\) 0 0
\(243\) −3370.99 −0.889913
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3845.33 −0.990577
\(248\) 0 0
\(249\) −7000.00 −1.78155
\(250\) 0 0
\(251\) 5881.84 1.47912 0.739558 0.673093i \(-0.235034\pi\)
0.739558 + 0.673093i \(0.235034\pi\)
\(252\) 0 0
\(253\) 1040.00 0.258436
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1074.00 −0.260678 −0.130339 0.991469i \(-0.541607\pi\)
−0.130339 + 0.991469i \(0.541607\pi\)
\(258\) 0 0
\(259\) −3908.58 −0.937711
\(260\) 0 0
\(261\) 3510.00 0.832427
\(262\) 0 0
\(263\) −1486.27 −0.348469 −0.174235 0.984704i \(-0.555745\pi\)
−0.174235 + 0.984704i \(0.555745\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5628.85 1.29019
\(268\) 0 0
\(269\) 406.000 0.0920233 0.0460116 0.998941i \(-0.485349\pi\)
0.0460116 + 0.998941i \(0.485349\pi\)
\(270\) 0 0
\(271\) −392.122 −0.0878957 −0.0439479 0.999034i \(-0.513994\pi\)
−0.0439479 + 0.999034i \(0.513994\pi\)
\(272\) 0 0
\(273\) −4560.00 −1.01093
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5934.00 −1.28715 −0.643573 0.765385i \(-0.722549\pi\)
−0.643573 + 0.765385i \(0.722549\pi\)
\(278\) 0 0
\(279\) 4439.84 0.952710
\(280\) 0 0
\(281\) −1870.00 −0.396992 −0.198496 0.980102i \(-0.563606\pi\)
−0.198496 + 0.980102i \(0.563606\pi\)
\(282\) 0 0
\(283\) 4888.88 1.02690 0.513452 0.858118i \(-0.328366\pi\)
0.513452 + 0.858118i \(0.328366\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5122.89 −1.05364
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) 1606.44 0.323612
\(292\) 0 0
\(293\) −5198.00 −1.03642 −0.518209 0.855254i \(-0.673401\pi\)
−0.518209 + 0.855254i \(0.673401\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1120.00 −0.218818
\(298\) 0 0
\(299\) −3124.33 −0.604297
\(300\) 0 0
\(301\) 10200.0 1.95322
\(302\) 0 0
\(303\) 3782.08 0.717079
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3750.46 −0.697232 −0.348616 0.937266i \(-0.613348\pi\)
−0.348616 + 0.937266i \(0.613348\pi\)
\(308\) 0 0
\(309\) −3160.00 −0.581767
\(310\) 0 0
\(311\) −6261.31 −1.14163 −0.570814 0.821079i \(-0.693372\pi\)
−0.570814 + 0.821079i \(0.693372\pi\)
\(312\) 0 0
\(313\) −2218.00 −0.400539 −0.200270 0.979741i \(-0.564182\pi\)
−0.200270 + 0.979741i \(0.564182\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4134.00 −0.732456 −0.366228 0.930525i \(-0.619351\pi\)
−0.366228 + 0.930525i \(0.619351\pi\)
\(318\) 0 0
\(319\) 3415.26 0.599429
\(320\) 0 0
\(321\) 3960.00 0.688553
\(322\) 0 0
\(323\) −3440.56 −0.592687
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5401.17 0.913411
\(328\) 0 0
\(329\) −2520.00 −0.422286
\(330\) 0 0
\(331\) −11953.4 −1.98495 −0.992476 0.122443i \(-0.960927\pi\)
−0.992476 + 0.122443i \(0.960927\pi\)
\(332\) 0 0
\(333\) −2678.00 −0.440701
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8014.00 1.29540 0.647701 0.761895i \(-0.275731\pi\)
0.647701 + 0.761895i \(0.275731\pi\)
\(338\) 0 0
\(339\) −10739.1 −1.72055
\(340\) 0 0
\(341\) 4320.00 0.686044
\(342\) 0 0
\(343\) −6185.42 −0.973706
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4484.11 −0.693716 −0.346858 0.937918i \(-0.612752\pi\)
−0.346858 + 0.937918i \(0.612752\pi\)
\(348\) 0 0
\(349\) 910.000 0.139574 0.0697868 0.997562i \(-0.477768\pi\)
0.0697868 + 0.997562i \(0.477768\pi\)
\(350\) 0 0
\(351\) 3364.66 0.511659
\(352\) 0 0
\(353\) −12962.0 −1.95438 −0.977192 0.212357i \(-0.931886\pi\)
−0.977192 + 0.212357i \(0.931886\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4080.00 −0.604864
\(358\) 0 0
\(359\) −12193.7 −1.79265 −0.896325 0.443398i \(-0.853773\pi\)
−0.896325 + 0.443398i \(0.853773\pi\)
\(360\) 0 0
\(361\) 3381.00 0.492929
\(362\) 0 0
\(363\) −7406.05 −1.07085
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3434.23 0.488462 0.244231 0.969717i \(-0.421464\pi\)
0.244231 + 0.969717i \(0.421464\pi\)
\(368\) 0 0
\(369\) −3510.00 −0.495185
\(370\) 0 0
\(371\) 4895.21 0.685031
\(372\) 0 0
\(373\) −4622.00 −0.641603 −0.320802 0.947146i \(-0.603952\pi\)
−0.320802 + 0.947146i \(0.603952\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10260.0 −1.40164
\(378\) 0 0
\(379\) 8449.61 1.14519 0.572595 0.819838i \(-0.305937\pi\)
0.572595 + 0.819838i \(0.305937\pi\)
\(380\) 0 0
\(381\) −1480.00 −0.199010
\(382\) 0 0
\(383\) −1815.15 −0.242166 −0.121083 0.992642i \(-0.538637\pi\)
−0.121083 + 0.992642i \(0.538637\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6988.63 0.917964
\(388\) 0 0
\(389\) −11106.0 −1.44755 −0.723774 0.690037i \(-0.757594\pi\)
−0.723774 + 0.690037i \(0.757594\pi\)
\(390\) 0 0
\(391\) −2795.45 −0.361566
\(392\) 0 0
\(393\) −10960.0 −1.40677
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5754.00 0.727418 0.363709 0.931513i \(-0.381510\pi\)
0.363709 + 0.931513i \(0.381510\pi\)
\(398\) 0 0
\(399\) 12143.1 1.52360
\(400\) 0 0
\(401\) −1118.00 −0.139228 −0.0696138 0.997574i \(-0.522177\pi\)
−0.0696138 + 0.997574i \(0.522177\pi\)
\(402\) 0 0
\(403\) −12978.0 −1.60417
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2605.72 −0.317348
\(408\) 0 0
\(409\) −11374.0 −1.37508 −0.687540 0.726146i \(-0.741310\pi\)
−0.687540 + 0.726146i \(0.741310\pi\)
\(410\) 0 0
\(411\) −9777.76 −1.17348
\(412\) 0 0
\(413\) 1440.00 0.171568
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2080.00 −0.244264
\(418\) 0 0
\(419\) 12674.4 1.47777 0.738885 0.673832i \(-0.235353\pi\)
0.738885 + 0.673832i \(0.235353\pi\)
\(420\) 0 0
\(421\) 1150.00 0.133130 0.0665648 0.997782i \(-0.478796\pi\)
0.0665648 + 0.997782i \(0.478796\pi\)
\(422\) 0 0
\(423\) −1726.60 −0.198464
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4743.42 −0.537588
\(428\) 0 0
\(429\) −3040.00 −0.342127
\(430\) 0 0
\(431\) −1353.45 −0.151261 −0.0756307 0.997136i \(-0.524097\pi\)
−0.0756307 + 0.997136i \(0.524097\pi\)
\(432\) 0 0
\(433\) 7918.00 0.878787 0.439394 0.898295i \(-0.355193\pi\)
0.439394 + 0.898295i \(0.355193\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8320.00 0.910754
\(438\) 0 0
\(439\) −14217.6 −1.54572 −0.772858 0.634579i \(-0.781173\pi\)
−0.772858 + 0.634579i \(0.781173\pi\)
\(440\) 0 0
\(441\) 221.000 0.0238635
\(442\) 0 0
\(443\) 10581.0 1.13480 0.567401 0.823441i \(-0.307949\pi\)
0.567401 + 0.823441i \(0.307949\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 20529.5 2.17229
\(448\) 0 0
\(449\) 4474.00 0.470247 0.235124 0.971965i \(-0.424450\pi\)
0.235124 + 0.971965i \(0.424450\pi\)
\(450\) 0 0
\(451\) −3415.26 −0.356582
\(452\) 0 0
\(453\) 9520.00 0.987392
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4154.00 −0.425199 −0.212599 0.977139i \(-0.568193\pi\)
−0.212599 + 0.977139i \(0.568193\pi\)
\(458\) 0 0
\(459\) 3010.49 0.306138
\(460\) 0 0
\(461\) −11282.0 −1.13982 −0.569908 0.821709i \(-0.693021\pi\)
−0.569908 + 0.821709i \(0.693021\pi\)
\(462\) 0 0
\(463\) −5458.09 −0.547860 −0.273930 0.961750i \(-0.588324\pi\)
−0.273930 + 0.961750i \(0.588324\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3775.76 −0.374136 −0.187068 0.982347i \(-0.559898\pi\)
−0.187068 + 0.982347i \(0.559898\pi\)
\(468\) 0 0
\(469\) −15480.0 −1.52409
\(470\) 0 0
\(471\) 7753.90 0.758559
\(472\) 0 0
\(473\) 6800.00 0.661024
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3354.00 0.321948
\(478\) 0 0
\(479\) −8930.27 −0.851847 −0.425923 0.904759i \(-0.640051\pi\)
−0.425923 + 0.904759i \(0.640051\pi\)
\(480\) 0 0
\(481\) 7828.00 0.742050
\(482\) 0 0
\(483\) 9866.31 0.929467
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2422.30 0.225390 0.112695 0.993630i \(-0.464052\pi\)
0.112695 + 0.993630i \(0.464052\pi\)
\(488\) 0 0
\(489\) 9160.00 0.847095
\(490\) 0 0
\(491\) −8993.52 −0.826623 −0.413311 0.910590i \(-0.635628\pi\)
−0.413311 + 0.910590i \(0.635628\pi\)
\(492\) 0 0
\(493\) −9180.00 −0.838634
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12240.0 1.10471
\(498\) 0 0
\(499\) −3541.75 −0.317737 −0.158868 0.987300i \(-0.550785\pi\)
−0.158868 + 0.987300i \(0.550785\pi\)
\(500\) 0 0
\(501\) −14760.0 −1.31622
\(502\) 0 0
\(503\) 2384.36 0.211358 0.105679 0.994400i \(-0.466298\pi\)
0.105679 + 0.994400i \(0.466298\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4762.39 −0.417170
\(508\) 0 0
\(509\) 2350.00 0.204640 0.102320 0.994752i \(-0.467373\pi\)
0.102320 + 0.994752i \(0.467373\pi\)
\(510\) 0 0
\(511\) 20453.6 1.77067
\(512\) 0 0
\(513\) −8960.00 −0.771138
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1680.00 −0.142914
\(518\) 0 0
\(519\) 19593.5 1.65714
\(520\) 0 0
\(521\) 858.000 0.0721491 0.0360745 0.999349i \(-0.488515\pi\)
0.0360745 + 0.999349i \(0.488515\pi\)
\(522\) 0 0
\(523\) 5799.62 0.484894 0.242447 0.970165i \(-0.422050\pi\)
0.242447 + 0.970165i \(0.422050\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11611.9 −0.959813
\(528\) 0 0
\(529\) −5407.00 −0.444399
\(530\) 0 0
\(531\) 986.631 0.0806330
\(532\) 0 0
\(533\) 10260.0 0.833790
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14880.0 −1.19575
\(538\) 0 0
\(539\) 215.035 0.0171841
\(540\) 0 0
\(541\) 20478.0 1.62739 0.813695 0.581292i \(-0.197453\pi\)
0.813695 + 0.581292i \(0.197453\pi\)
\(542\) 0 0
\(543\) 13800.2 1.09065
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −10429.2 −0.815210 −0.407605 0.913158i \(-0.633636\pi\)
−0.407605 + 0.913158i \(0.633636\pi\)
\(548\) 0 0
\(549\) −3250.00 −0.252653
\(550\) 0 0
\(551\) 27322.1 2.11245
\(552\) 0 0
\(553\) 5280.00 0.406019
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13194.0 1.00368 0.501838 0.864962i \(-0.332657\pi\)
0.501838 + 0.864962i \(0.332657\pi\)
\(558\) 0 0
\(559\) −20428.3 −1.54566
\(560\) 0 0
\(561\) −2720.00 −0.204703
\(562\) 0 0
\(563\) 9771.44 0.731469 0.365734 0.930719i \(-0.380818\pi\)
0.365734 + 0.930719i \(0.380818\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −17285.0 −1.28025
\(568\) 0 0
\(569\) 4594.00 0.338472 0.169236 0.985576i \(-0.445870\pi\)
0.169236 + 0.985576i \(0.445870\pi\)
\(570\) 0 0
\(571\) 4389.24 0.321688 0.160844 0.986980i \(-0.448578\pi\)
0.160844 + 0.986980i \(0.448578\pi\)
\(572\) 0 0
\(573\) −19120.0 −1.39398
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14926.0 1.07691 0.538455 0.842654i \(-0.319008\pi\)
0.538455 + 0.842654i \(0.319008\pi\)
\(578\) 0 0
\(579\) −8209.27 −0.589233
\(580\) 0 0
\(581\) −21000.0 −1.49953
\(582\) 0 0
\(583\) 3263.47 0.231834
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8101.76 −0.569668 −0.284834 0.958577i \(-0.591939\pi\)
−0.284834 + 0.958577i \(0.591939\pi\)
\(588\) 0 0
\(589\) 34560.0 2.41769
\(590\) 0 0
\(591\) −17999.7 −1.25281
\(592\) 0 0
\(593\) 26958.0 1.86683 0.933417 0.358794i \(-0.116812\pi\)
0.933417 + 0.358794i \(0.116812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22720.0 −1.55757
\(598\) 0 0
\(599\) 6349.85 0.433135 0.216568 0.976268i \(-0.430514\pi\)
0.216568 + 0.976268i \(0.430514\pi\)
\(600\) 0 0
\(601\) 21970.0 1.49114 0.745570 0.666427i \(-0.232177\pi\)
0.745570 + 0.666427i \(0.232177\pi\)
\(602\) 0 0
\(603\) −10606.3 −0.716287
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3876.95 −0.259243 −0.129622 0.991564i \(-0.541376\pi\)
−0.129622 + 0.991564i \(0.541376\pi\)
\(608\) 0 0
\(609\) 32400.0 2.15585
\(610\) 0 0
\(611\) 5047.00 0.334173
\(612\) 0 0
\(613\) −2878.00 −0.189627 −0.0948135 0.995495i \(-0.530225\pi\)
−0.0948135 + 0.995495i \(0.530225\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27354.0 −1.78481 −0.892407 0.451231i \(-0.850985\pi\)
−0.892407 + 0.451231i \(0.850985\pi\)
\(618\) 0 0
\(619\) 12547.9 0.814771 0.407386 0.913256i \(-0.366441\pi\)
0.407386 + 0.913256i \(0.366441\pi\)
\(620\) 0 0
\(621\) −7280.00 −0.470429
\(622\) 0 0
\(623\) 16886.6 1.08595
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8095.43 0.515631
\(628\) 0 0
\(629\) 7004.00 0.443987
\(630\) 0 0
\(631\) −30876.5 −1.94798 −0.973988 0.226598i \(-0.927240\pi\)
−0.973988 + 0.226598i \(0.927240\pi\)
\(632\) 0 0
\(633\) 26480.0 1.66269
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −646.000 −0.0401812
\(638\) 0 0
\(639\) 8386.36 0.519185
\(640\) 0 0
\(641\) −9430.00 −0.581065 −0.290532 0.956865i \(-0.593832\pi\)
−0.290532 + 0.956865i \(0.593832\pi\)
\(642\) 0 0
\(643\) 9847.33 0.603952 0.301976 0.953316i \(-0.402354\pi\)
0.301976 + 0.953316i \(0.402354\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30048.0 1.82582 0.912911 0.408158i \(-0.133829\pi\)
0.912911 + 0.408158i \(0.133829\pi\)
\(648\) 0 0
\(649\) 960.000 0.0580636
\(650\) 0 0
\(651\) 40983.1 2.46737
\(652\) 0 0
\(653\) −18742.0 −1.12317 −0.561586 0.827418i \(-0.689809\pi\)
−0.561586 + 0.827418i \(0.689809\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14014.0 0.832174
\(658\) 0 0
\(659\) −8323.11 −0.491992 −0.245996 0.969271i \(-0.579115\pi\)
−0.245996 + 0.969271i \(0.579115\pi\)
\(660\) 0 0
\(661\) 7630.00 0.448975 0.224488 0.974477i \(-0.427929\pi\)
0.224488 + 0.974477i \(0.427929\pi\)
\(662\) 0 0
\(663\) 8171.33 0.478655
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22199.2 1.28869
\(668\) 0 0
\(669\) 30120.0 1.74067
\(670\) 0 0
\(671\) −3162.28 −0.181935
\(672\) 0 0
\(673\) 10878.0 0.623055 0.311528 0.950237i \(-0.399159\pi\)
0.311528 + 0.950237i \(0.399159\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −126.000 −0.00715299 −0.00357649 0.999994i \(-0.501138\pi\)
−0.00357649 + 0.999994i \(0.501138\pi\)
\(678\) 0 0
\(679\) 4819.31 0.272383
\(680\) 0 0
\(681\) 10520.0 0.591964
\(682\) 0 0
\(683\) −16412.2 −0.919467 −0.459734 0.888057i \(-0.652055\pi\)
−0.459734 + 0.888057i \(0.652055\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6640.78 −0.368794
\(688\) 0 0
\(689\) −9804.00 −0.542094
\(690\) 0 0
\(691\) −13193.0 −0.726319 −0.363159 0.931727i \(-0.618302\pi\)
−0.363159 + 0.931727i \(0.618302\pi\)
\(692\) 0 0
\(693\) 3120.00 0.171023
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9180.00 0.498877
\(698\) 0 0
\(699\) −17569.6 −0.950707
\(700\) 0 0
\(701\) −22010.0 −1.18589 −0.592943 0.805244i \(-0.702034\pi\)
−0.592943 + 0.805244i \(0.702034\pi\)
\(702\) 0 0
\(703\) −20845.7 −1.11837
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11346.3 0.603564
\(708\) 0 0
\(709\) 550.000 0.0291335 0.0145668 0.999894i \(-0.495363\pi\)
0.0145668 + 0.999894i \(0.495363\pi\)
\(710\) 0 0
\(711\) 3617.65 0.190819
\(712\) 0 0
\(713\) 28080.0 1.47490
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16160.0 0.841710
\(718\) 0 0
\(719\) 17936.4 0.930343 0.465171 0.885221i \(-0.345993\pi\)
0.465171 + 0.885221i \(0.345993\pi\)
\(720\) 0 0
\(721\) −9480.00 −0.489672
\(722\) 0 0
\(723\) −33836.4 −1.74051
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −16728.4 −0.853403 −0.426701 0.904393i \(-0.640324\pi\)
−0.426701 + 0.904393i \(0.640324\pi\)
\(728\) 0 0
\(729\) 3277.00 0.166489
\(730\) 0 0
\(731\) −18278.0 −0.924808
\(732\) 0 0
\(733\) −2422.00 −0.122044 −0.0610222 0.998136i \(-0.519436\pi\)
−0.0610222 + 0.998136i \(0.519436\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10320.0 −0.515797
\(738\) 0 0
\(739\) 19555.5 0.973426 0.486713 0.873562i \(-0.338196\pi\)
0.486713 + 0.873562i \(0.338196\pi\)
\(740\) 0 0
\(741\) −24320.0 −1.20569
\(742\) 0 0
\(743\) −31059.9 −1.53362 −0.766808 0.641876i \(-0.778156\pi\)
−0.766808 + 0.641876i \(0.778156\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −14388.4 −0.704743
\(748\) 0 0
\(749\) 11880.0 0.579554
\(750\) 0 0
\(751\) −12155.8 −0.590641 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(752\) 0 0
\(753\) 37200.0 1.80032
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19346.0 0.928854 0.464427 0.885611i \(-0.346260\pi\)
0.464427 + 0.885611i \(0.346260\pi\)
\(758\) 0 0
\(759\) 6577.54 0.314558
\(760\) 0 0
\(761\) −33078.0 −1.57566 −0.787830 0.615893i \(-0.788795\pi\)
−0.787830 + 0.615893i \(0.788795\pi\)
\(762\) 0 0
\(763\) 16203.5 0.768816
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2884.00 −0.135769
\(768\) 0 0
\(769\) 32530.0 1.52544 0.762719 0.646730i \(-0.223864\pi\)
0.762719 + 0.646730i \(0.223864\pi\)
\(770\) 0 0
\(771\) −6792.57 −0.317287
\(772\) 0 0
\(773\) 12002.0 0.558450 0.279225 0.960226i \(-0.409922\pi\)
0.279225 + 0.960226i \(0.409922\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −24720.0 −1.14134
\(778\) 0 0
\(779\) −27322.1 −1.25663
\(780\) 0 0
\(781\) 8160.00 0.373864
\(782\) 0 0
\(783\) −23906.8 −1.09114
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19954.0 0.903789 0.451895 0.892071i \(-0.350748\pi\)
0.451895 + 0.892071i \(0.350748\pi\)
\(788\) 0 0
\(789\) −9400.00 −0.424143
\(790\) 0 0
\(791\) −32217.3 −1.44819
\(792\) 0 0
\(793\) 9500.00 0.425416
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32666.0 1.45181 0.725903 0.687797i \(-0.241422\pi\)
0.725903 + 0.687797i \(0.241422\pi\)
\(798\) 0 0
\(799\) 4515.73 0.199944
\(800\) 0 0
\(801\) 11570.0 0.510369
\(802\) 0 0
\(803\) 13635.7 0.599246
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2567.77 0.112007
\(808\) 0 0
\(809\) −23110.0 −1.00433 −0.502166 0.864771i \(-0.667463\pi\)
−0.502166 + 0.864771i \(0.667463\pi\)
\(810\) 0 0
\(811\) 35632.5 1.54282 0.771411 0.636338i \(-0.219552\pi\)
0.771411 + 0.636338i \(0.219552\pi\)
\(812\) 0 0
\(813\) −2480.00 −0.106983
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 54400.0 2.32952
\(818\) 0 0
\(819\) −9372.99 −0.399901
\(820\) 0 0
\(821\) −8850.00 −0.376208 −0.188104 0.982149i \(-0.560234\pi\)
−0.188104 + 0.982149i \(0.560234\pi\)
\(822\) 0 0
\(823\) 13895.0 0.588519 0.294259 0.955726i \(-0.404927\pi\)
0.294259 + 0.955726i \(0.404927\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35841.3 1.50704 0.753520 0.657425i \(-0.228354\pi\)
0.753520 + 0.657425i \(0.228354\pi\)
\(828\) 0 0
\(829\) −23034.0 −0.965023 −0.482511 0.875890i \(-0.660275\pi\)
−0.482511 + 0.875890i \(0.660275\pi\)
\(830\) 0 0
\(831\) −37529.9 −1.56666
\(832\) 0 0
\(833\) −578.000 −0.0240414
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −30240.0 −1.24880
\(838\) 0 0
\(839\) −36960.7 −1.52089 −0.760444 0.649403i \(-0.775019\pi\)
−0.760444 + 0.649403i \(0.775019\pi\)
\(840\) 0 0
\(841\) 48511.0 1.98905
\(842\) 0 0
\(843\) −11826.9 −0.483204
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22218.2 −0.901328
\(848\) 0 0
\(849\) 30920.0 1.24991
\(850\) 0 0
\(851\) −16937.2 −0.682254
\(852\) 0 0
\(853\) 19122.0 0.767555 0.383778 0.923425i \(-0.374623\pi\)
0.383778 + 0.923425i \(0.374623\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −17786.0 −0.708936 −0.354468 0.935068i \(-0.615338\pi\)
−0.354468 + 0.935068i \(0.615338\pi\)
\(858\) 0 0
\(859\) −28713.5 −1.14050 −0.570251 0.821470i \(-0.693154\pi\)
−0.570251 + 0.821470i \(0.693154\pi\)
\(860\) 0 0
\(861\) −32400.0 −1.28245
\(862\) 0 0
\(863\) −23748.7 −0.936750 −0.468375 0.883530i \(-0.655160\pi\)
−0.468375 + 0.883530i \(0.655160\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −23761.4 −0.930770
\(868\) 0 0
\(869\) 3520.00 0.137408
\(870\) 0 0
\(871\) 31003.0 1.20608
\(872\) 0 0
\(873\) 3302.00 0.128013
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7706.00 0.296708 0.148354 0.988934i \(-0.452602\pi\)
0.148354 + 0.988934i \(0.452602\pi\)
\(878\) 0 0
\(879\) −32875.0 −1.26149
\(880\) 0 0
\(881\) 10410.0 0.398095 0.199048 0.979990i \(-0.436215\pi\)
0.199048 + 0.979990i \(0.436215\pi\)
\(882\) 0 0
\(883\) 26822.4 1.02225 0.511125 0.859506i \(-0.329229\pi\)
0.511125 + 0.859506i \(0.329229\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21130.3 −0.799873 −0.399937 0.916543i \(-0.630968\pi\)
−0.399937 + 0.916543i \(0.630968\pi\)
\(888\) 0 0
\(889\) −4440.00 −0.167506
\(890\) 0 0
\(891\) −11523.3 −0.433273
\(892\) 0 0
\(893\) −13440.0 −0.503642
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −19760.0 −0.735526
\(898\) 0 0
\(899\) 92212.0 3.42096
\(900\) 0 0
\(901\) −8772.00 −0.324348
\(902\) 0 0
\(903\) 64510.5 2.37738
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19220.3 0.703639 0.351819 0.936068i \(-0.385563\pi\)
0.351819 + 0.936068i \(0.385563\pi\)
\(908\) 0 0
\(909\) 7774.00 0.283660
\(910\) 0 0
\(911\) 39402.0 1.43298 0.716491 0.697597i \(-0.245747\pi\)
0.716491 + 0.697597i \(0.245747\pi\)
\(912\) 0 0
\(913\) −14000.0 −0.507483
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −32880.0 −1.18407
\(918\) 0 0
\(919\) 42931.1 1.54099 0.770493 0.637449i \(-0.220010\pi\)
0.770493 + 0.637449i \(0.220010\pi\)
\(920\) 0 0
\(921\) −23720.0 −0.848643
\(922\) 0 0
\(923\) −24514.0 −0.874201
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6495.32 −0.230134
\(928\) 0 0
\(929\) 34746.0 1.22710 0.613552 0.789654i \(-0.289740\pi\)
0.613552 + 0.789654i \(0.289740\pi\)
\(930\) 0 0
\(931\) 1720.28 0.0605584
\(932\) 0 0
\(933\) −39600.0 −1.38955
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −21594.0 −0.752876 −0.376438 0.926442i \(-0.622851\pi\)
−0.376438 + 0.926442i \(0.622851\pi\)
\(938\) 0 0
\(939\) −14027.9 −0.487521
\(940\) 0 0
\(941\) −20018.0 −0.693484 −0.346742 0.937961i \(-0.612712\pi\)
−0.346742 + 0.937961i \(0.612712\pi\)
\(942\) 0 0
\(943\) −22199.2 −0.766601
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46150.3 −1.58361 −0.791807 0.610771i \(-0.790859\pi\)
−0.791807 + 0.610771i \(0.790859\pi\)
\(948\) 0 0
\(949\) −40964.0 −1.40121
\(950\) 0 0
\(951\) −26145.7 −0.891517
\(952\) 0 0
\(953\) 342.000 0.0116248 0.00581242 0.999983i \(-0.498150\pi\)
0.00581242 + 0.999983i \(0.498150\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 21600.0 0.729602
\(958\) 0 0
\(959\) −29333.3 −0.987718
\(960\) 0 0
\(961\) 86849.0 2.91528
\(962\) 0 0
\(963\) 8139.70 0.272376
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 51728.5 1.72025 0.860123 0.510087i \(-0.170387\pi\)
0.860123 + 0.510087i \(0.170387\pi\)
\(968\) 0 0
\(969\) −21760.0 −0.721395
\(970\) 0 0
\(971\) −3099.03 −0.102423 −0.0512115 0.998688i \(-0.516308\pi\)
−0.0512115 + 0.998688i \(0.516308\pi\)
\(972\) 0 0
\(973\) −6240.00 −0.205596
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26226.0 −0.858796 −0.429398 0.903115i \(-0.641274\pi\)
−0.429398 + 0.903115i \(0.641274\pi\)
\(978\) 0 0
\(979\) 11257.7 0.367516
\(980\) 0 0
\(981\) 11102.0 0.361325
\(982\) 0 0
\(983\) −11049.0 −0.358503 −0.179251 0.983803i \(-0.557368\pi\)
−0.179251 + 0.983803i \(0.557368\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −15937.9 −0.513990
\(988\) 0 0
\(989\) 44200.0 1.42111
\(990\) 0 0
\(991\) 45928.9 1.47223 0.736115 0.676856i \(-0.236658\pi\)
0.736115 + 0.676856i \(0.236658\pi\)
\(992\) 0 0
\(993\) −75600.0 −2.41601
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 31026.0 0.985560 0.492780 0.870154i \(-0.335981\pi\)
0.492780 + 0.870154i \(0.335981\pi\)
\(998\) 0 0
\(999\) 18240.0 0.577666
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 800.4.a.p.1.2 2
4.3 odd 2 inner 800.4.a.p.1.1 2
5.2 odd 4 800.4.c.j.449.2 4
5.3 odd 4 800.4.c.j.449.4 4
5.4 even 2 160.4.a.f.1.1 2
8.3 odd 2 1600.4.a.ch.1.2 2
8.5 even 2 1600.4.a.ch.1.1 2
15.14 odd 2 1440.4.a.v.1.1 2
20.3 even 4 800.4.c.j.449.1 4
20.7 even 4 800.4.c.j.449.3 4
20.19 odd 2 160.4.a.f.1.2 yes 2
40.19 odd 2 320.4.a.p.1.1 2
40.29 even 2 320.4.a.p.1.2 2
60.59 even 2 1440.4.a.v.1.2 2
80.19 odd 4 1280.4.d.u.641.3 4
80.29 even 4 1280.4.d.u.641.1 4
80.59 odd 4 1280.4.d.u.641.2 4
80.69 even 4 1280.4.d.u.641.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 2 5.4 even 2
160.4.a.f.1.2 yes 2 20.19 odd 2
320.4.a.p.1.1 2 40.19 odd 2
320.4.a.p.1.2 2 40.29 even 2
800.4.a.p.1.1 2 4.3 odd 2 inner
800.4.a.p.1.2 2 1.1 even 1 trivial
800.4.c.j.449.1 4 20.3 even 4
800.4.c.j.449.2 4 5.2 odd 4
800.4.c.j.449.3 4 20.7 even 4
800.4.c.j.449.4 4 5.3 odd 4
1280.4.d.u.641.1 4 80.29 even 4
1280.4.d.u.641.2 4 80.59 odd 4
1280.4.d.u.641.3 4 80.19 odd 4
1280.4.d.u.641.4 4 80.69 even 4
1440.4.a.v.1.1 2 15.14 odd 2
1440.4.a.v.1.2 2 60.59 even 2
1600.4.a.ch.1.1 2 8.5 even 2
1600.4.a.ch.1.2 2 8.3 odd 2