Properties

Label 784.5.c.a.97.2
Level $784$
Weight $5$
Character 784.97
Analytic conductor $81.042$
Analytic rank $0$
Dimension $4$
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] = [784,5,Mod(97,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.97");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(81.0420510577\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.2
Root \(1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 784.97
Dual form 784.5.c.a.97.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.03347i q^{3} +0.710974i q^{5} +76.8650 q^{9} +O(q^{10})\) \(q-2.03347i q^{3} +0.710974i q^{5} +76.8650 q^{9} -151.664 q^{11} +260.864i q^{13} +1.44574 q^{15} -385.598i q^{17} -390.140i q^{19} +177.647 q^{23} +624.495 q^{25} -321.013i q^{27} -320.887 q^{29} +1346.37i q^{31} +308.404i q^{33} -797.088 q^{37} +530.459 q^{39} +815.856i q^{41} +2167.70 q^{43} +54.6490i q^{45} +4285.61i q^{47} -784.102 q^{51} -3171.57 q^{53} -107.829i q^{55} -793.337 q^{57} -4706.39i q^{59} -2534.78i q^{61} -185.468 q^{65} +4092.43 q^{67} -361.239i q^{69} +2255.28 q^{71} -4653.19i q^{73} -1269.89i q^{75} +4193.74 q^{79} +5573.29 q^{81} +7799.12i q^{83} +274.150 q^{85} +652.514i q^{87} +9469.11i q^{89} +2737.81 q^{93} +277.379 q^{95} +9945.39i q^{97} -11657.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 196 q^{9} - 24 q^{11} - 888 q^{15} - 104 q^{23} + 948 q^{25} - 1408 q^{29} - 3392 q^{37} - 2200 q^{39} + 2024 q^{43} - 18936 q^{51} - 16680 q^{53} - 6064 q^{57} - 6048 q^{65} + 20816 q^{67} + 1984 q^{71} + 29616 q^{79} + 30852 q^{81} - 29688 q^{85} + 18192 q^{93} - 7240 q^{95} - 72160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.03347i − 0.225941i −0.993598 0.112970i \(-0.963963\pi\)
0.993598 0.112970i \(-0.0360365\pi\)
\(4\) 0 0
\(5\) 0.710974i 0.0284390i 0.999899 + 0.0142195i \(0.00452635\pi\)
−0.999899 + 0.0142195i \(0.995474\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 76.8650 0.948951
\(10\) 0 0
\(11\) −151.664 −1.25342 −0.626711 0.779252i \(-0.715599\pi\)
−0.626711 + 0.779252i \(0.715599\pi\)
\(12\) 0 0
\(13\) 260.864i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(14\) 0 0
\(15\) 1.44574 0.00642552
\(16\) 0 0
\(17\) − 385.598i − 1.33425i −0.744946 0.667125i \(-0.767525\pi\)
0.744946 0.667125i \(-0.232475\pi\)
\(18\) 0 0
\(19\) − 390.140i − 1.08072i −0.841434 0.540360i \(-0.818288\pi\)
0.841434 0.540360i \(-0.181712\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 177.647 0.335816 0.167908 0.985803i \(-0.446299\pi\)
0.167908 + 0.985803i \(0.446299\pi\)
\(24\) 0 0
\(25\) 624.495 0.999191
\(26\) 0 0
\(27\) − 321.013i − 0.440348i
\(28\) 0 0
\(29\) −320.887 −0.381554 −0.190777 0.981633i \(-0.561101\pi\)
−0.190777 + 0.981633i \(0.561101\pi\)
\(30\) 0 0
\(31\) 1346.37i 1.40101i 0.713646 + 0.700506i \(0.247042\pi\)
−0.713646 + 0.700506i \(0.752958\pi\)
\(32\) 0 0
\(33\) 308.404i 0.283199i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −797.088 −0.582241 −0.291121 0.956686i \(-0.594028\pi\)
−0.291121 + 0.956686i \(0.594028\pi\)
\(38\) 0 0
\(39\) 530.459 0.348757
\(40\) 0 0
\(41\) 815.856i 0.485340i 0.970109 + 0.242670i \(0.0780232\pi\)
−0.970109 + 0.242670i \(0.921977\pi\)
\(42\) 0 0
\(43\) 2167.70 1.17236 0.586182 0.810179i \(-0.300630\pi\)
0.586182 + 0.810179i \(0.300630\pi\)
\(44\) 0 0
\(45\) 54.6490i 0.0269872i
\(46\) 0 0
\(47\) 4285.61i 1.94007i 0.242970 + 0.970034i \(0.421878\pi\)
−0.242970 + 0.970034i \(0.578122\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −784.102 −0.301462
\(52\) 0 0
\(53\) −3171.57 −1.12907 −0.564536 0.825408i \(-0.690945\pi\)
−0.564536 + 0.825408i \(0.690945\pi\)
\(54\) 0 0
\(55\) − 107.829i − 0.0356460i
\(56\) 0 0
\(57\) −793.337 −0.244179
\(58\) 0 0
\(59\) − 4706.39i − 1.35202i −0.736891 0.676012i \(-0.763707\pi\)
0.736891 0.676012i \(-0.236293\pi\)
\(60\) 0 0
\(61\) − 2534.78i − 0.681209i −0.940207 0.340604i \(-0.889368\pi\)
0.940207 0.340604i \(-0.110632\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −185.468 −0.0438977
\(66\) 0 0
\(67\) 4092.43 0.911657 0.455828 0.890068i \(-0.349343\pi\)
0.455828 + 0.890068i \(0.349343\pi\)
\(68\) 0 0
\(69\) − 361.239i − 0.0758746i
\(70\) 0 0
\(71\) 2255.28 0.447388 0.223694 0.974659i \(-0.428188\pi\)
0.223694 + 0.974659i \(0.428188\pi\)
\(72\) 0 0
\(73\) − 4653.19i − 0.873183i −0.899660 0.436591i \(-0.856186\pi\)
0.899660 0.436591i \(-0.143814\pi\)
\(74\) 0 0
\(75\) − 1269.89i − 0.225758i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4193.74 0.671965 0.335983 0.941868i \(-0.390932\pi\)
0.335983 + 0.941868i \(0.390932\pi\)
\(80\) 0 0
\(81\) 5573.29 0.849458
\(82\) 0 0
\(83\) 7799.12i 1.13211i 0.824367 + 0.566056i \(0.191532\pi\)
−0.824367 + 0.566056i \(0.808468\pi\)
\(84\) 0 0
\(85\) 274.150 0.0379447
\(86\) 0 0
\(87\) 652.514i 0.0862088i
\(88\) 0 0
\(89\) 9469.11i 1.19544i 0.801704 + 0.597722i \(0.203927\pi\)
−0.801704 + 0.597722i \(0.796073\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2737.81 0.316546
\(94\) 0 0
\(95\) 277.379 0.0307345
\(96\) 0 0
\(97\) 9945.39i 1.05701i 0.848931 + 0.528504i \(0.177247\pi\)
−0.848931 + 0.528504i \(0.822753\pi\)
\(98\) 0 0
\(99\) −11657.7 −1.18944
\(100\) 0 0
\(101\) 2068.32i 0.202756i 0.994848 + 0.101378i \(0.0323252\pi\)
−0.994848 + 0.101378i \(0.967675\pi\)
\(102\) 0 0
\(103\) − 2504.76i − 0.236098i −0.993008 0.118049i \(-0.962336\pi\)
0.993008 0.118049i \(-0.0376639\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6443.39 −0.562791 −0.281395 0.959592i \(-0.590797\pi\)
−0.281395 + 0.959592i \(0.590797\pi\)
\(108\) 0 0
\(109\) 23546.0 1.98182 0.990912 0.134515i \(-0.0429477\pi\)
0.990912 + 0.134515i \(0.0429477\pi\)
\(110\) 0 0
\(111\) 1620.85i 0.131552i
\(112\) 0 0
\(113\) 19692.2 1.54219 0.771096 0.636719i \(-0.219709\pi\)
0.771096 + 0.636719i \(0.219709\pi\)
\(114\) 0 0
\(115\) 126.302i 0.00955026i
\(116\) 0 0
\(117\) 20051.3i 1.46478i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8360.97 0.571065
\(122\) 0 0
\(123\) 1659.02 0.109658
\(124\) 0 0
\(125\) 888.358i 0.0568549i
\(126\) 0 0
\(127\) 12550.5 0.778135 0.389067 0.921209i \(-0.372797\pi\)
0.389067 + 0.921209i \(0.372797\pi\)
\(128\) 0 0
\(129\) − 4407.95i − 0.264885i
\(130\) 0 0
\(131\) 19323.3i 1.12600i 0.826457 + 0.562999i \(0.190353\pi\)
−0.826457 + 0.562999i \(0.809647\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 228.232 0.0125230
\(136\) 0 0
\(137\) 3708.59 0.197591 0.0987955 0.995108i \(-0.468501\pi\)
0.0987955 + 0.995108i \(0.468501\pi\)
\(138\) 0 0
\(139\) − 1096.09i − 0.0567307i −0.999598 0.0283653i \(-0.990970\pi\)
0.999598 0.0283653i \(-0.00903018\pi\)
\(140\) 0 0
\(141\) 8714.65 0.438341
\(142\) 0 0
\(143\) − 39563.7i − 1.93475i
\(144\) 0 0
\(145\) − 228.143i − 0.0108510i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1233.82 −0.0555748 −0.0277874 0.999614i \(-0.508846\pi\)
−0.0277874 + 0.999614i \(0.508846\pi\)
\(150\) 0 0
\(151\) 42672.8 1.87153 0.935766 0.352622i \(-0.114710\pi\)
0.935766 + 0.352622i \(0.114710\pi\)
\(152\) 0 0
\(153\) − 29639.0i − 1.26614i
\(154\) 0 0
\(155\) −957.236 −0.0398433
\(156\) 0 0
\(157\) − 29884.9i − 1.21242i −0.795305 0.606209i \(-0.792689\pi\)
0.795305 0.606209i \(-0.207311\pi\)
\(158\) 0 0
\(159\) 6449.28i 0.255104i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 26205.2 0.986308 0.493154 0.869942i \(-0.335844\pi\)
0.493154 + 0.869942i \(0.335844\pi\)
\(164\) 0 0
\(165\) −219.267 −0.00805389
\(166\) 0 0
\(167\) 35887.5i 1.28680i 0.765532 + 0.643398i \(0.222476\pi\)
−0.765532 + 0.643398i \(0.777524\pi\)
\(168\) 0 0
\(169\) −39489.2 −1.38263
\(170\) 0 0
\(171\) − 29988.1i − 1.02555i
\(172\) 0 0
\(173\) 10972.7i 0.366625i 0.983055 + 0.183312i \(0.0586820\pi\)
−0.983055 + 0.183312i \(0.941318\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9570.30 −0.305477
\(178\) 0 0
\(179\) 33844.1 1.05628 0.528138 0.849159i \(-0.322891\pi\)
0.528138 + 0.849159i \(0.322891\pi\)
\(180\) 0 0
\(181\) 50094.5i 1.52909i 0.644571 + 0.764545i \(0.277036\pi\)
−0.644571 + 0.764545i \(0.722964\pi\)
\(182\) 0 0
\(183\) −5154.39 −0.153913
\(184\) 0 0
\(185\) − 566.709i − 0.0165583i
\(186\) 0 0
\(187\) 58481.4i 1.67238i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17612.9 −0.482798 −0.241399 0.970426i \(-0.577606\pi\)
−0.241399 + 0.970426i \(0.577606\pi\)
\(192\) 0 0
\(193\) 13264.2 0.356097 0.178048 0.984022i \(-0.443022\pi\)
0.178048 + 0.984022i \(0.443022\pi\)
\(194\) 0 0
\(195\) 377.143i 0.00991828i
\(196\) 0 0
\(197\) 5362.20 0.138169 0.0690844 0.997611i \(-0.477992\pi\)
0.0690844 + 0.997611i \(0.477992\pi\)
\(198\) 0 0
\(199\) 42230.6i 1.06640i 0.845988 + 0.533202i \(0.179011\pi\)
−0.845988 + 0.533202i \(0.820989\pi\)
\(200\) 0 0
\(201\) − 8321.82i − 0.205981i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −580.053 −0.0138026
\(206\) 0 0
\(207\) 13654.8 0.318673
\(208\) 0 0
\(209\) 59170.2i 1.35460i
\(210\) 0 0
\(211\) 77049.0 1.73062 0.865311 0.501236i \(-0.167121\pi\)
0.865311 + 0.501236i \(0.167121\pi\)
\(212\) 0 0
\(213\) − 4586.04i − 0.101083i
\(214\) 0 0
\(215\) 1541.18i 0.0333408i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9462.11 −0.197288
\(220\) 0 0
\(221\) 100589. 2.05951
\(222\) 0 0
\(223\) − 3702.60i − 0.0744555i −0.999307 0.0372278i \(-0.988147\pi\)
0.999307 0.0372278i \(-0.0118527\pi\)
\(224\) 0 0
\(225\) 48001.8 0.948183
\(226\) 0 0
\(227\) 12018.8i 0.233243i 0.993176 + 0.116621i \(0.0372064\pi\)
−0.993176 + 0.116621i \(0.962794\pi\)
\(228\) 0 0
\(229\) 33360.5i 0.636153i 0.948065 + 0.318076i \(0.103037\pi\)
−0.948065 + 0.318076i \(0.896963\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 54138.9 0.997235 0.498618 0.866822i \(-0.333841\pi\)
0.498618 + 0.866822i \(0.333841\pi\)
\(234\) 0 0
\(235\) −3046.96 −0.0551735
\(236\) 0 0
\(237\) − 8527.83i − 0.151824i
\(238\) 0 0
\(239\) −29165.4 −0.510590 −0.255295 0.966863i \(-0.582173\pi\)
−0.255295 + 0.966863i \(0.582173\pi\)
\(240\) 0 0
\(241\) − 100130.i − 1.72398i −0.506929 0.861988i \(-0.669219\pi\)
0.506929 0.861988i \(-0.330781\pi\)
\(242\) 0 0
\(243\) − 37335.2i − 0.632275i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 101774. 1.66817
\(248\) 0 0
\(249\) 15859.3 0.255791
\(250\) 0 0
\(251\) − 81988.2i − 1.30138i −0.759344 0.650690i \(-0.774480\pi\)
0.759344 0.650690i \(-0.225520\pi\)
\(252\) 0 0
\(253\) −26942.6 −0.420919
\(254\) 0 0
\(255\) − 557.476i − 0.00857325i
\(256\) 0 0
\(257\) 73774.8i 1.11697i 0.829515 + 0.558485i \(0.188617\pi\)
−0.829515 + 0.558485i \(0.811383\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −24665.0 −0.362076
\(262\) 0 0
\(263\) −39243.6 −0.567358 −0.283679 0.958919i \(-0.591555\pi\)
−0.283679 + 0.958919i \(0.591555\pi\)
\(264\) 0 0
\(265\) − 2254.90i − 0.0321096i
\(266\) 0 0
\(267\) 19255.1 0.270100
\(268\) 0 0
\(269\) − 10311.6i − 0.142502i −0.997458 0.0712509i \(-0.977301\pi\)
0.997458 0.0712509i \(-0.0226991\pi\)
\(270\) 0 0
\(271\) 36969.3i 0.503388i 0.967807 + 0.251694i \(0.0809876\pi\)
−0.967807 + 0.251694i \(0.919012\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −94713.3 −1.25241
\(276\) 0 0
\(277\) −92405.8 −1.20431 −0.602157 0.798378i \(-0.705692\pi\)
−0.602157 + 0.798378i \(0.705692\pi\)
\(278\) 0 0
\(279\) 103489.i 1.32949i
\(280\) 0 0
\(281\) 21171.6 0.268128 0.134064 0.990973i \(-0.457197\pi\)
0.134064 + 0.990973i \(0.457197\pi\)
\(282\) 0 0
\(283\) − 48065.2i − 0.600147i −0.953916 0.300074i \(-0.902989\pi\)
0.953916 0.300074i \(-0.0970112\pi\)
\(284\) 0 0
\(285\) − 564.042i − 0.00694419i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −65164.9 −0.780222
\(290\) 0 0
\(291\) 20223.6 0.238821
\(292\) 0 0
\(293\) − 26781.1i − 0.311956i −0.987761 0.155978i \(-0.950147\pi\)
0.987761 0.155978i \(-0.0498528\pi\)
\(294\) 0 0
\(295\) 3346.12 0.0384501
\(296\) 0 0
\(297\) 48686.2i 0.551941i
\(298\) 0 0
\(299\) 46341.7i 0.518358i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4205.85 0.0458109
\(304\) 0 0
\(305\) 1802.16 0.0193729
\(306\) 0 0
\(307\) 69996.0i 0.742671i 0.928499 + 0.371336i \(0.121100\pi\)
−0.928499 + 0.371336i \(0.878900\pi\)
\(308\) 0 0
\(309\) −5093.35 −0.0533441
\(310\) 0 0
\(311\) − 86961.6i − 0.899097i −0.893256 0.449549i \(-0.851585\pi\)
0.893256 0.449549i \(-0.148415\pi\)
\(312\) 0 0
\(313\) − 33163.2i − 0.338507i −0.985573 0.169253i \(-0.945864\pi\)
0.985573 0.169253i \(-0.0541357\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29206.0 0.290639 0.145319 0.989385i \(-0.453579\pi\)
0.145319 + 0.989385i \(0.453579\pi\)
\(318\) 0 0
\(319\) 48667.1 0.478249
\(320\) 0 0
\(321\) 13102.4i 0.127157i
\(322\) 0 0
\(323\) −150437. −1.44195
\(324\) 0 0
\(325\) 162908.i 1.54233i
\(326\) 0 0
\(327\) − 47880.1i − 0.447775i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −166993. −1.52420 −0.762102 0.647457i \(-0.775832\pi\)
−0.762102 + 0.647457i \(0.775832\pi\)
\(332\) 0 0
\(333\) −61268.2 −0.552518
\(334\) 0 0
\(335\) 2909.61i 0.0259266i
\(336\) 0 0
\(337\) 16126.0 0.141993 0.0709964 0.997477i \(-0.477382\pi\)
0.0709964 + 0.997477i \(0.477382\pi\)
\(338\) 0 0
\(339\) − 40043.5i − 0.348444i
\(340\) 0 0
\(341\) − 204196.i − 1.75606i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 256.832 0.00215779
\(346\) 0 0
\(347\) −10071.3 −0.0836428 −0.0418214 0.999125i \(-0.513316\pi\)
−0.0418214 + 0.999125i \(0.513316\pi\)
\(348\) 0 0
\(349\) − 97674.1i − 0.801915i −0.916097 0.400958i \(-0.868677\pi\)
0.916097 0.400958i \(-0.131323\pi\)
\(350\) 0 0
\(351\) 83740.9 0.679710
\(352\) 0 0
\(353\) 85255.3i 0.684182i 0.939667 + 0.342091i \(0.111135\pi\)
−0.939667 + 0.342091i \(0.888865\pi\)
\(354\) 0 0
\(355\) 1603.45i 0.0127232i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 142240. 1.10365 0.551826 0.833959i \(-0.313931\pi\)
0.551826 + 0.833959i \(0.313931\pi\)
\(360\) 0 0
\(361\) −21888.1 −0.167955
\(362\) 0 0
\(363\) − 17001.8i − 0.129027i
\(364\) 0 0
\(365\) 3308.30 0.0248324
\(366\) 0 0
\(367\) 177841.i 1.32038i 0.751098 + 0.660191i \(0.229525\pi\)
−0.751098 + 0.660191i \(0.770475\pi\)
\(368\) 0 0
\(369\) 62710.8i 0.460564i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −21670.8 −0.155761 −0.0778803 0.996963i \(-0.524815\pi\)
−0.0778803 + 0.996963i \(0.524815\pi\)
\(374\) 0 0
\(375\) 1806.45 0.0128459
\(376\) 0 0
\(377\) − 83708.0i − 0.588958i
\(378\) 0 0
\(379\) −125199. −0.871608 −0.435804 0.900042i \(-0.643536\pi\)
−0.435804 + 0.900042i \(0.643536\pi\)
\(380\) 0 0
\(381\) − 25521.1i − 0.175812i
\(382\) 0 0
\(383\) − 211410.i − 1.44121i −0.693345 0.720605i \(-0.743864\pi\)
0.693345 0.720605i \(-0.256136\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 166620. 1.11252
\(388\) 0 0
\(389\) −39160.8 −0.258793 −0.129396 0.991593i \(-0.541304\pi\)
−0.129396 + 0.991593i \(0.541304\pi\)
\(390\) 0 0
\(391\) − 68500.3i − 0.448063i
\(392\) 0 0
\(393\) 39293.2 0.254409
\(394\) 0 0
\(395\) 2981.64i 0.0191100i
\(396\) 0 0
\(397\) 196525.i 1.24691i 0.781858 + 0.623456i \(0.214272\pi\)
−0.781858 + 0.623456i \(0.785728\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 213650. 1.32866 0.664330 0.747439i \(-0.268717\pi\)
0.664330 + 0.747439i \(0.268717\pi\)
\(402\) 0 0
\(403\) −351221. −2.16257
\(404\) 0 0
\(405\) 3962.47i 0.0241577i
\(406\) 0 0
\(407\) 120890. 0.729794
\(408\) 0 0
\(409\) − 35604.1i − 0.212840i −0.994321 0.106420i \(-0.966061\pi\)
0.994321 0.106420i \(-0.0339388\pi\)
\(410\) 0 0
\(411\) − 7541.29i − 0.0446439i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5544.97 −0.0321961
\(416\) 0 0
\(417\) −2228.87 −0.0128178
\(418\) 0 0
\(419\) − 254947.i − 1.45219i −0.687597 0.726093i \(-0.741334\pi\)
0.687597 0.726093i \(-0.258666\pi\)
\(420\) 0 0
\(421\) 107186. 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(422\) 0 0
\(423\) 329413.i 1.84103i
\(424\) 0 0
\(425\) − 240804.i − 1.33317i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −80451.6 −0.437139
\(430\) 0 0
\(431\) −119958. −0.645768 −0.322884 0.946439i \(-0.604652\pi\)
−0.322884 + 0.946439i \(0.604652\pi\)
\(432\) 0 0
\(433\) − 110159.i − 0.587547i −0.955875 0.293773i \(-0.905089\pi\)
0.955875 0.293773i \(-0.0949111\pi\)
\(434\) 0 0
\(435\) −463.921 −0.00245169
\(436\) 0 0
\(437\) − 69307.1i − 0.362923i
\(438\) 0 0
\(439\) − 352295.i − 1.82801i −0.405708 0.914003i \(-0.632975\pi\)
0.405708 0.914003i \(-0.367025\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −285732. −1.45597 −0.727984 0.685594i \(-0.759543\pi\)
−0.727984 + 0.685594i \(0.759543\pi\)
\(444\) 0 0
\(445\) −6732.29 −0.0339972
\(446\) 0 0
\(447\) 2508.92i 0.0125566i
\(448\) 0 0
\(449\) −342184. −1.69733 −0.848667 0.528927i \(-0.822594\pi\)
−0.848667 + 0.528927i \(0.822594\pi\)
\(450\) 0 0
\(451\) − 123736.i − 0.608335i
\(452\) 0 0
\(453\) − 86773.8i − 0.422856i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −277624. −1.32931 −0.664653 0.747152i \(-0.731421\pi\)
−0.664653 + 0.747152i \(0.731421\pi\)
\(458\) 0 0
\(459\) −123782. −0.587534
\(460\) 0 0
\(461\) 361934.i 1.70305i 0.524315 + 0.851524i \(0.324322\pi\)
−0.524315 + 0.851524i \(0.675678\pi\)
\(462\) 0 0
\(463\) −146876. −0.685153 −0.342577 0.939490i \(-0.611300\pi\)
−0.342577 + 0.939490i \(0.611300\pi\)
\(464\) 0 0
\(465\) 1946.51i 0.00900224i
\(466\) 0 0
\(467\) 268410.i 1.23073i 0.788241 + 0.615367i \(0.210992\pi\)
−0.788241 + 0.615367i \(0.789008\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −60770.0 −0.273935
\(472\) 0 0
\(473\) −328762. −1.46947
\(474\) 0 0
\(475\) − 243640.i − 1.07985i
\(476\) 0 0
\(477\) −243782. −1.07143
\(478\) 0 0
\(479\) 155285.i 0.676797i 0.941003 + 0.338399i \(0.109885\pi\)
−0.941003 + 0.338399i \(0.890115\pi\)
\(480\) 0 0
\(481\) − 207932.i − 0.898733i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7070.91 −0.0300602
\(486\) 0 0
\(487\) 155628. 0.656190 0.328095 0.944645i \(-0.393593\pi\)
0.328095 + 0.944645i \(0.393593\pi\)
\(488\) 0 0
\(489\) − 53287.5i − 0.222847i
\(490\) 0 0
\(491\) −37550.7 −0.155760 −0.0778798 0.996963i \(-0.524815\pi\)
−0.0778798 + 0.996963i \(0.524815\pi\)
\(492\) 0 0
\(493\) 123734.i 0.509089i
\(494\) 0 0
\(495\) − 8288.29i − 0.0338263i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −167820. −0.673975 −0.336987 0.941509i \(-0.609408\pi\)
−0.336987 + 0.941509i \(0.609408\pi\)
\(500\) 0 0
\(501\) 72976.0 0.290740
\(502\) 0 0
\(503\) 133035.i 0.525810i 0.964822 + 0.262905i \(0.0846806\pi\)
−0.964822 + 0.262905i \(0.915319\pi\)
\(504\) 0 0
\(505\) −1470.52 −0.00576617
\(506\) 0 0
\(507\) 80299.9i 0.312392i
\(508\) 0 0
\(509\) − 227994.i − 0.880011i −0.897995 0.440005i \(-0.854976\pi\)
0.897995 0.440005i \(-0.145024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −125240. −0.475892
\(514\) 0 0
\(515\) 1780.82 0.00671437
\(516\) 0 0
\(517\) − 649973.i − 2.43172i
\(518\) 0 0
\(519\) 22312.7 0.0828355
\(520\) 0 0
\(521\) 341411.i 1.25777i 0.777498 + 0.628885i \(0.216489\pi\)
−0.777498 + 0.628885i \(0.783511\pi\)
\(522\) 0 0
\(523\) − 62912.0i − 0.230001i −0.993365 0.115001i \(-0.963313\pi\)
0.993365 0.115001i \(-0.0366870\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 519159. 1.86930
\(528\) 0 0
\(529\) −248283. −0.887228
\(530\) 0 0
\(531\) − 361757.i − 1.28300i
\(532\) 0 0
\(533\) −212828. −0.749159
\(534\) 0 0
\(535\) − 4581.08i − 0.0160052i
\(536\) 0 0
\(537\) − 68820.9i − 0.238656i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 251203. 0.858282 0.429141 0.903237i \(-0.358816\pi\)
0.429141 + 0.903237i \(0.358816\pi\)
\(542\) 0 0
\(543\) 101866. 0.345484
\(544\) 0 0
\(545\) 16740.6i 0.0563610i
\(546\) 0 0
\(547\) 153431. 0.512789 0.256395 0.966572i \(-0.417465\pi\)
0.256395 + 0.966572i \(0.417465\pi\)
\(548\) 0 0
\(549\) − 194836.i − 0.646434i
\(550\) 0 0
\(551\) 125191.i 0.412353i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1152.38 −0.00374121
\(556\) 0 0
\(557\) −475151. −1.53152 −0.765758 0.643129i \(-0.777636\pi\)
−0.765758 + 0.643129i \(0.777636\pi\)
\(558\) 0 0
\(559\) 565476.i 1.80963i
\(560\) 0 0
\(561\) 118920. 0.377858
\(562\) 0 0
\(563\) 39487.0i 0.124577i 0.998058 + 0.0622884i \(0.0198398\pi\)
−0.998058 + 0.0622884i \(0.980160\pi\)
\(564\) 0 0
\(565\) 14000.7i 0.0438583i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −85307.7 −0.263490 −0.131745 0.991284i \(-0.542058\pi\)
−0.131745 + 0.991284i \(0.542058\pi\)
\(570\) 0 0
\(571\) −89099.0 −0.273276 −0.136638 0.990621i \(-0.543630\pi\)
−0.136638 + 0.990621i \(0.543630\pi\)
\(572\) 0 0
\(573\) 35815.4i 0.109084i
\(574\) 0 0
\(575\) 110939. 0.335545
\(576\) 0 0
\(577\) 311076.i 0.934362i 0.884162 + 0.467181i \(0.154730\pi\)
−0.884162 + 0.467181i \(0.845270\pi\)
\(578\) 0 0
\(579\) − 26972.4i − 0.0804568i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 481012. 1.41520
\(584\) 0 0
\(585\) −14256.0 −0.0416567
\(586\) 0 0
\(587\) − 62466.3i − 0.181288i −0.995883 0.0906440i \(-0.971107\pi\)
0.995883 0.0906440i \(-0.0288926\pi\)
\(588\) 0 0
\(589\) 525274. 1.51410
\(590\) 0 0
\(591\) − 10903.9i − 0.0312180i
\(592\) 0 0
\(593\) − 47567.3i − 0.135269i −0.997710 0.0676346i \(-0.978455\pi\)
0.997710 0.0676346i \(-0.0215452\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 85874.6 0.240944
\(598\) 0 0
\(599\) −564035. −1.57200 −0.785999 0.618227i \(-0.787851\pi\)
−0.785999 + 0.618227i \(0.787851\pi\)
\(600\) 0 0
\(601\) 255637.i 0.707740i 0.935295 + 0.353870i \(0.115135\pi\)
−0.935295 + 0.353870i \(0.884865\pi\)
\(602\) 0 0
\(603\) 314565. 0.865118
\(604\) 0 0
\(605\) 5944.43i 0.0162405i
\(606\) 0 0
\(607\) − 572699.i − 1.55435i −0.629284 0.777176i \(-0.716652\pi\)
0.629284 0.777176i \(-0.283348\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.11796e6 −2.99464
\(612\) 0 0
\(613\) 283002. 0.753127 0.376563 0.926391i \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(614\) 0 0
\(615\) 1179.52i 0.00311856i
\(616\) 0 0
\(617\) 248170. 0.651896 0.325948 0.945388i \(-0.394317\pi\)
0.325948 + 0.945388i \(0.394317\pi\)
\(618\) 0 0
\(619\) − 56588.1i − 0.147688i −0.997270 0.0738438i \(-0.976473\pi\)
0.997270 0.0738438i \(-0.0235266\pi\)
\(620\) 0 0
\(621\) − 57027.0i − 0.147876i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 389677. 0.997574
\(626\) 0 0
\(627\) 120321. 0.306059
\(628\) 0 0
\(629\) 307356.i 0.776855i
\(630\) 0 0
\(631\) −681448. −1.71149 −0.855744 0.517399i \(-0.826900\pi\)
−0.855744 + 0.517399i \(0.826900\pi\)
\(632\) 0 0
\(633\) − 156677.i − 0.391018i
\(634\) 0 0
\(635\) 8923.10i 0.0221293i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 173352. 0.424549
\(640\) 0 0
\(641\) 678114. 1.65039 0.825195 0.564848i \(-0.191065\pi\)
0.825195 + 0.564848i \(0.191065\pi\)
\(642\) 0 0
\(643\) − 345520.i − 0.835701i −0.908516 0.417851i \(-0.862784\pi\)
0.908516 0.417851i \(-0.137216\pi\)
\(644\) 0 0
\(645\) 3133.94 0.00753305
\(646\) 0 0
\(647\) 7757.50i 0.0185316i 0.999957 + 0.00926581i \(0.00294944\pi\)
−0.999957 + 0.00926581i \(0.997051\pi\)
\(648\) 0 0
\(649\) 713791.i 1.69466i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −448473. −1.05174 −0.525872 0.850564i \(-0.676261\pi\)
−0.525872 + 0.850564i \(0.676261\pi\)
\(654\) 0 0
\(655\) −13738.3 −0.0320222
\(656\) 0 0
\(657\) − 357667.i − 0.828607i
\(658\) 0 0
\(659\) −583660. −1.34397 −0.671984 0.740565i \(-0.734558\pi\)
−0.671984 + 0.740565i \(0.734558\pi\)
\(660\) 0 0
\(661\) 382886.i 0.876327i 0.898895 + 0.438164i \(0.144371\pi\)
−0.898895 + 0.438164i \(0.855629\pi\)
\(662\) 0 0
\(663\) − 204544.i − 0.465329i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −57004.6 −0.128132
\(668\) 0 0
\(669\) −7529.12 −0.0168226
\(670\) 0 0
\(671\) 384435.i 0.853842i
\(672\) 0 0
\(673\) −323802. −0.714907 −0.357453 0.933931i \(-0.616355\pi\)
−0.357453 + 0.933931i \(0.616355\pi\)
\(674\) 0 0
\(675\) − 200471.i − 0.439992i
\(676\) 0 0
\(677\) 374803.i 0.817760i 0.912588 + 0.408880i \(0.134081\pi\)
−0.912588 + 0.408880i \(0.865919\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24439.8 0.0526990
\(682\) 0 0
\(683\) −97364.5 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(684\) 0 0
\(685\) 2636.71i 0.00561928i
\(686\) 0 0
\(687\) 67837.5 0.143733
\(688\) 0 0
\(689\) − 827348.i − 1.74281i
\(690\) 0 0
\(691\) 787116.i 1.64848i 0.566244 + 0.824238i \(0.308396\pi\)
−0.566244 + 0.824238i \(0.691604\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 779.294 0.00161336
\(696\) 0 0
\(697\) 314593. 0.647564
\(698\) 0 0
\(699\) − 110090.i − 0.225316i
\(700\) 0 0
\(701\) −232522. −0.473182 −0.236591 0.971609i \(-0.576030\pi\)
−0.236591 + 0.971609i \(0.576030\pi\)
\(702\) 0 0
\(703\) 310976.i 0.629240i
\(704\) 0 0
\(705\) 6195.89i 0.0124659i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 151049. 0.300487 0.150243 0.988649i \(-0.451994\pi\)
0.150243 + 0.988649i \(0.451994\pi\)
\(710\) 0 0
\(711\) 322351. 0.637662
\(712\) 0 0
\(713\) 239179.i 0.470483i
\(714\) 0 0
\(715\) 28128.8 0.0550223
\(716\) 0 0
\(717\) 59306.9i 0.115363i
\(718\) 0 0
\(719\) − 712110.i − 1.37749i −0.725002 0.688746i \(-0.758161\pi\)
0.725002 0.688746i \(-0.241839\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −203612. −0.389517
\(724\) 0 0
\(725\) −200392. −0.381246
\(726\) 0 0
\(727\) 693160.i 1.31149i 0.754983 + 0.655744i \(0.227645\pi\)
−0.754983 + 0.655744i \(0.772355\pi\)
\(728\) 0 0
\(729\) 375517. 0.706601
\(730\) 0 0
\(731\) − 835861.i − 1.56423i
\(732\) 0 0
\(733\) − 269107.i − 0.500861i −0.968135 0.250431i \(-0.919428\pi\)
0.968135 0.250431i \(-0.0805722\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −620674. −1.14269
\(738\) 0 0
\(739\) −86185.1 −0.157813 −0.0789066 0.996882i \(-0.525143\pi\)
−0.0789066 + 0.996882i \(0.525143\pi\)
\(740\) 0 0
\(741\) − 206953.i − 0.376908i
\(742\) 0 0
\(743\) 457439. 0.828620 0.414310 0.910136i \(-0.364023\pi\)
0.414310 + 0.910136i \(0.364023\pi\)
\(744\) 0 0
\(745\) − 877.210i − 0.00158049i
\(746\) 0 0
\(747\) 599480.i 1.07432i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 581442. 1.03092 0.515462 0.856913i \(-0.327620\pi\)
0.515462 + 0.856913i \(0.327620\pi\)
\(752\) 0 0
\(753\) −166720. −0.294035
\(754\) 0 0
\(755\) 30339.2i 0.0532244i
\(756\) 0 0
\(757\) 398071. 0.694655 0.347327 0.937744i \(-0.387089\pi\)
0.347327 + 0.937744i \(0.387089\pi\)
\(758\) 0 0
\(759\) 54787.0i 0.0951029i
\(760\) 0 0
\(761\) 323965.i 0.559408i 0.960086 + 0.279704i \(0.0902363\pi\)
−0.960086 + 0.279704i \(0.909764\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 21072.6 0.0360076
\(766\) 0 0
\(767\) 1.22773e6 2.08695
\(768\) 0 0
\(769\) 318602.i 0.538760i 0.963034 + 0.269380i \(0.0868187\pi\)
−0.963034 + 0.269380i \(0.913181\pi\)
\(770\) 0 0
\(771\) 150019. 0.252369
\(772\) 0 0
\(773\) − 512015.i − 0.856887i −0.903569 0.428443i \(-0.859062\pi\)
0.903569 0.428443i \(-0.140938\pi\)
\(774\) 0 0
\(775\) 840803.i 1.39988i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 318298. 0.524516
\(780\) 0 0
\(781\) −342045. −0.560765
\(782\) 0 0
\(783\) 103009.i 0.168017i
\(784\) 0 0
\(785\) 21247.4 0.0344799
\(786\) 0 0
\(787\) − 644919.i − 1.04125i −0.853785 0.520626i \(-0.825699\pi\)
0.853785 0.520626i \(-0.174301\pi\)
\(788\) 0 0
\(789\) 79800.5i 0.128189i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 661233. 1.05150
\(794\) 0 0
\(795\) −4585.27 −0.00725488
\(796\) 0 0
\(797\) − 278844.i − 0.438980i −0.975615 0.219490i \(-0.929561\pi\)
0.975615 0.219490i \(-0.0704393\pi\)
\(798\) 0 0
\(799\) 1.65252e6 2.58853
\(800\) 0 0
\(801\) 727843.i 1.13442i
\(802\) 0 0
\(803\) 705721.i 1.09447i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20968.2 −0.0321970
\(808\) 0 0
\(809\) 662520. 1.01228 0.506142 0.862450i \(-0.331071\pi\)
0.506142 + 0.862450i \(0.331071\pi\)
\(810\) 0 0
\(811\) − 586078.i − 0.891073i −0.895264 0.445537i \(-0.853013\pi\)
0.895264 0.445537i \(-0.146987\pi\)
\(812\) 0 0
\(813\) 75175.9 0.113736
\(814\) 0 0
\(815\) 18631.2i 0.0280496i
\(816\) 0 0
\(817\) − 845706.i − 1.26700i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 790534. 1.17283 0.586414 0.810012i \(-0.300539\pi\)
0.586414 + 0.810012i \(0.300539\pi\)
\(822\) 0 0
\(823\) 1.15645e6 1.70737 0.853684 0.520791i \(-0.174363\pi\)
0.853684 + 0.520791i \(0.174363\pi\)
\(824\) 0 0
\(825\) 192597.i 0.282970i
\(826\) 0 0
\(827\) −390408. −0.570832 −0.285416 0.958404i \(-0.592132\pi\)
−0.285416 + 0.958404i \(0.592132\pi\)
\(828\) 0 0
\(829\) − 531226.i − 0.772984i −0.922293 0.386492i \(-0.873687\pi\)
0.922293 0.386492i \(-0.126313\pi\)
\(830\) 0 0
\(831\) 187904.i 0.272104i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −25515.1 −0.0365951
\(836\) 0 0
\(837\) 432204. 0.616933
\(838\) 0 0
\(839\) 263842.i 0.374817i 0.982282 + 0.187409i \(0.0600089\pi\)
−0.982282 + 0.187409i \(0.939991\pi\)
\(840\) 0 0
\(841\) −604312. −0.854416
\(842\) 0 0
\(843\) − 43051.9i − 0.0605811i
\(844\) 0 0
\(845\) − 28075.8i − 0.0393204i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −97739.0 −0.135598
\(850\) 0 0
\(851\) −141600. −0.195526
\(852\) 0 0
\(853\) 1.17089e6i 1.60923i 0.593798 + 0.804614i \(0.297628\pi\)
−0.593798 + 0.804614i \(0.702372\pi\)
\(854\) 0 0
\(855\) 21320.8 0.0291656
\(856\) 0 0
\(857\) − 424840.i − 0.578447i −0.957262 0.289223i \(-0.906603\pi\)
0.957262 0.289223i \(-0.0933970\pi\)
\(858\) 0 0
\(859\) 408175.i 0.553172i 0.960989 + 0.276586i \(0.0892030\pi\)
−0.960989 + 0.276586i \(0.910797\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 882741. 1.18525 0.592627 0.805477i \(-0.298091\pi\)
0.592627 + 0.805477i \(0.298091\pi\)
\(864\) 0 0
\(865\) −7801.31 −0.0104264
\(866\) 0 0
\(867\) 132511.i 0.176284i
\(868\) 0 0
\(869\) −636039. −0.842256
\(870\) 0 0
\(871\) 1.06757e6i 1.40721i
\(872\) 0 0
\(873\) 764452.i 1.00305i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −595392. −0.774112 −0.387056 0.922056i \(-0.626508\pi\)
−0.387056 + 0.922056i \(0.626508\pi\)
\(878\) 0 0
\(879\) −54458.5 −0.0704835
\(880\) 0 0
\(881\) 238444.i 0.307209i 0.988132 + 0.153605i \(0.0490882\pi\)
−0.988132 + 0.153605i \(0.950912\pi\)
\(882\) 0 0
\(883\) −189523. −0.243076 −0.121538 0.992587i \(-0.538783\pi\)
−0.121538 + 0.992587i \(0.538783\pi\)
\(884\) 0 0
\(885\) − 6804.24i − 0.00868746i
\(886\) 0 0
\(887\) − 819553.i − 1.04167i −0.853658 0.520834i \(-0.825621\pi\)
0.853658 0.520834i \(-0.174379\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −845268. −1.06473
\(892\) 0 0
\(893\) 1.67199e6 2.09667
\(894\) 0 0
\(895\) 24062.3i 0.0300394i
\(896\) 0 0
\(897\) 94234.3 0.117118
\(898\) 0 0
\(899\) − 432034.i − 0.534563i
\(900\) 0 0
\(901\) 1.22295e6i 1.50646i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −35615.9 −0.0434857
\(906\) 0 0
\(907\) 657619. 0.799392 0.399696 0.916648i \(-0.369116\pi\)
0.399696 + 0.916648i \(0.369116\pi\)
\(908\) 0 0
\(909\) 158981.i 0.192406i
\(910\) 0 0
\(911\) −1.32632e6 −1.59812 −0.799061 0.601250i \(-0.794670\pi\)
−0.799061 + 0.601250i \(0.794670\pi\)
\(912\) 0 0
\(913\) − 1.18285e6i − 1.41901i
\(914\) 0 0
\(915\) − 3664.64i − 0.00437712i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 236310. 0.279803 0.139901 0.990165i \(-0.455321\pi\)
0.139901 + 0.990165i \(0.455321\pi\)
\(920\) 0 0
\(921\) 142335. 0.167800
\(922\) 0 0
\(923\) 588322.i 0.690577i
\(924\) 0 0
\(925\) −497777. −0.581770
\(926\) 0 0
\(927\) − 192528.i − 0.224045i
\(928\) 0 0
\(929\) 352762.i 0.408743i 0.978893 + 0.204372i \(0.0655151\pi\)
−0.978893 + 0.204372i \(0.934485\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −176834. −0.203143
\(934\) 0 0
\(935\) −41578.7 −0.0475607
\(936\) 0 0
\(937\) − 113445.i − 0.129213i −0.997911 0.0646066i \(-0.979421\pi\)
0.997911 0.0646066i \(-0.0205793\pi\)
\(938\) 0 0
\(939\) −67436.3 −0.0764826
\(940\) 0 0
\(941\) − 1.17351e6i − 1.32528i −0.748937 0.662642i \(-0.769435\pi\)
0.748937 0.662642i \(-0.230565\pi\)
\(942\) 0 0
\(943\) 144934.i 0.162985i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −271958. −0.303250 −0.151625 0.988438i \(-0.548451\pi\)
−0.151625 + 0.988438i \(0.548451\pi\)
\(948\) 0 0
\(949\) 1.21385e6 1.34782
\(950\) 0 0
\(951\) − 59389.5i − 0.0656672i
\(952\) 0 0
\(953\) 465476. 0.512520 0.256260 0.966608i \(-0.417510\pi\)
0.256260 + 0.966608i \(0.417510\pi\)
\(954\) 0 0
\(955\) − 12522.3i − 0.0137303i
\(956\) 0 0
\(957\) − 98962.9i − 0.108056i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −889199. −0.962836
\(962\) 0 0
\(963\) −495271. −0.534061
\(964\) 0 0
\(965\) 9430.53i 0.0101270i
\(966\) 0 0
\(967\) 1.49573e6 1.59956 0.799782 0.600291i \(-0.204949\pi\)
0.799782 + 0.600291i \(0.204949\pi\)
\(968\) 0 0
\(969\) 305909.i 0.325795i
\(970\) 0 0
\(971\) − 404524.i − 0.429048i −0.976719 0.214524i \(-0.931180\pi\)
0.976719 0.214524i \(-0.0688201\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 331269. 0.348475
\(976\) 0 0
\(977\) −1.65194e6 −1.73063 −0.865315 0.501229i \(-0.832882\pi\)
−0.865315 + 0.501229i \(0.832882\pi\)
\(978\) 0 0
\(979\) − 1.43612e6i − 1.49839i
\(980\) 0 0
\(981\) 1.80987e6 1.88065
\(982\) 0 0
\(983\) − 978562.i − 1.01270i −0.862328 0.506350i \(-0.830994\pi\)
0.862328 0.506350i \(-0.169006\pi\)
\(984\) 0 0
\(985\) 3812.38i 0.00392938i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 385085. 0.393699
\(990\) 0 0
\(991\) −1.26854e6 −1.29168 −0.645841 0.763472i \(-0.723493\pi\)
−0.645841 + 0.763472i \(0.723493\pi\)
\(992\) 0 0
\(993\) 339575.i 0.344380i
\(994\) 0 0
\(995\) −30024.9 −0.0303274
\(996\) 0 0
\(997\) 2071.33i 0.00208382i 0.999999 + 0.00104191i \(0.000331650\pi\)
−0.999999 + 0.00104191i \(0.999668\pi\)
\(998\) 0 0
\(999\) 255876.i 0.256389i
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 784.5.c.a.97.2 4
4.3 odd 2 98.5.b.a.97.4 yes 4
7.6 odd 2 inner 784.5.c.a.97.3 4
12.11 even 2 882.5.c.c.685.1 4
28.3 even 6 98.5.d.c.19.1 8
28.11 odd 6 98.5.d.c.19.2 8
28.19 even 6 98.5.d.c.31.2 8
28.23 odd 6 98.5.d.c.31.1 8
28.27 even 2 98.5.b.a.97.3 4
84.83 odd 2 882.5.c.c.685.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.5.b.a.97.3 4 28.27 even 2
98.5.b.a.97.4 yes 4 4.3 odd 2
98.5.d.c.19.1 8 28.3 even 6
98.5.d.c.19.2 8 28.11 odd 6
98.5.d.c.31.1 8 28.23 odd 6
98.5.d.c.31.2 8 28.19 even 6
784.5.c.a.97.2 4 1.1 even 1 trivial
784.5.c.a.97.3 4 7.6 odd 2 inner
882.5.c.c.685.1 4 12.11 even 2
882.5.c.c.685.2 4 84.83 odd 2