Properties

Label 1008.4.a.t.1.1
Level $1008$
Weight $4$
Character 1008.1
Self dual yes
Analytic conductor $59.474$
Analytic rank $1$
Dimension $1$
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] = [1008,4,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, 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("1008.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(59.4739252858\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{5} -7.00000 q^{7} +O(q^{10})\) \(q+16.0000 q^{5} -7.00000 q^{7} -18.0000 q^{11} -54.0000 q^{13} +128.000 q^{17} -52.0000 q^{19} -202.000 q^{23} +131.000 q^{25} -302.000 q^{29} +200.000 q^{31} -112.000 q^{35} -150.000 q^{37} -172.000 q^{41} -164.000 q^{43} -460.000 q^{47} +49.0000 q^{49} +190.000 q^{53} -288.000 q^{55} +96.0000 q^{59} +622.000 q^{61} -864.000 q^{65} -744.000 q^{67} -54.0000 q^{71} +742.000 q^{73} +126.000 q^{77} +92.0000 q^{79} -228.000 q^{83} +2048.00 q^{85} +116.000 q^{89} +378.000 q^{91} -832.000 q^{95} -554.000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 16.0000 1.43108 0.715542 0.698570i \(-0.246180\pi\)
0.715542 + 0.698570i \(0.246180\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −18.0000 −0.493382 −0.246691 0.969094i \(-0.579343\pi\)
−0.246691 + 0.969094i \(0.579343\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 128.000 1.82615 0.913075 0.407791i \(-0.133701\pi\)
0.913075 + 0.407791i \(0.133701\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −202.000 −1.83130 −0.915650 0.401976i \(-0.868324\pi\)
−0.915650 + 0.401976i \(0.868324\pi\)
\(24\) 0 0
\(25\) 131.000 1.04800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −302.000 −1.93379 −0.966896 0.255169i \(-0.917869\pi\)
−0.966896 + 0.255169i \(0.917869\pi\)
\(30\) 0 0
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −112.000 −0.540899
\(36\) 0 0
\(37\) −150.000 −0.666482 −0.333241 0.942842i \(-0.608142\pi\)
−0.333241 + 0.942842i \(0.608142\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −172.000 −0.655168 −0.327584 0.944822i \(-0.606234\pi\)
−0.327584 + 0.944822i \(0.606234\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −460.000 −1.42761 −0.713807 0.700342i \(-0.753031\pi\)
−0.713807 + 0.700342i \(0.753031\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 190.000 0.492425 0.246212 0.969216i \(-0.420814\pi\)
0.246212 + 0.969216i \(0.420814\pi\)
\(54\) 0 0
\(55\) −288.000 −0.706071
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 96.0000 0.211833 0.105916 0.994375i \(-0.466222\pi\)
0.105916 + 0.994375i \(0.466222\pi\)
\(60\) 0 0
\(61\) 622.000 1.30556 0.652778 0.757549i \(-0.273603\pi\)
0.652778 + 0.757549i \(0.273603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −864.000 −1.64871
\(66\) 0 0
\(67\) −744.000 −1.35663 −0.678314 0.734772i \(-0.737289\pi\)
−0.678314 + 0.734772i \(0.737289\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −54.0000 −0.0902623 −0.0451311 0.998981i \(-0.514371\pi\)
−0.0451311 + 0.998981i \(0.514371\pi\)
\(72\) 0 0
\(73\) 742.000 1.18965 0.594826 0.803855i \(-0.297221\pi\)
0.594826 + 0.803855i \(0.297221\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 126.000 0.186481
\(78\) 0 0
\(79\) 92.0000 0.131023 0.0655114 0.997852i \(-0.479132\pi\)
0.0655114 + 0.997852i \(0.479132\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −228.000 −0.301521 −0.150761 0.988570i \(-0.548172\pi\)
−0.150761 + 0.988570i \(0.548172\pi\)
\(84\) 0 0
\(85\) 2048.00 2.61337
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 116.000 0.138157 0.0690785 0.997611i \(-0.477994\pi\)
0.0690785 + 0.997611i \(0.477994\pi\)
\(90\) 0 0
\(91\) 378.000 0.435441
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −832.000 −0.898541
\(96\) 0 0
\(97\) −554.000 −0.579899 −0.289949 0.957042i \(-0.593638\pi\)
−0.289949 + 0.957042i \(0.593638\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 780.000 0.768445 0.384222 0.923241i \(-0.374470\pi\)
0.384222 + 0.923241i \(0.374470\pi\)
\(102\) 0 0
\(103\) −24.0000 −0.0229591 −0.0114796 0.999934i \(-0.503654\pi\)
−0.0114796 + 0.999934i \(0.503654\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 78.0000 0.0704724 0.0352362 0.999379i \(-0.488782\pi\)
0.0352362 + 0.999379i \(0.488782\pi\)
\(108\) 0 0
\(109\) −1034.00 −0.908617 −0.454308 0.890844i \(-0.650114\pi\)
−0.454308 + 0.890844i \(0.650114\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −186.000 −0.154844 −0.0774222 0.996998i \(-0.524669\pi\)
−0.0774222 + 0.996998i \(0.524669\pi\)
\(114\) 0 0
\(115\) −3232.00 −2.62074
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −896.000 −0.690220
\(120\) 0 0
\(121\) −1007.00 −0.756574
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 96.0000 0.0686920
\(126\) 0 0
\(127\) 316.000 0.220791 0.110396 0.993888i \(-0.464788\pi\)
0.110396 + 0.993888i \(0.464788\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −988.000 −0.658946 −0.329473 0.944165i \(-0.606871\pi\)
−0.329473 + 0.944165i \(0.606871\pi\)
\(132\) 0 0
\(133\) 364.000 0.237314
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −950.000 −0.592438 −0.296219 0.955120i \(-0.595726\pi\)
−0.296219 + 0.955120i \(0.595726\pi\)
\(138\) 0 0
\(139\) 2124.00 1.29608 0.648041 0.761606i \(-0.275589\pi\)
0.648041 + 0.761606i \(0.275589\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 972.000 0.568411
\(144\) 0 0
\(145\) −4832.00 −2.76742
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2242.00 −1.23270 −0.616348 0.787474i \(-0.711389\pi\)
−0.616348 + 0.787474i \(0.711389\pi\)
\(150\) 0 0
\(151\) −2776.00 −1.49608 −0.748039 0.663655i \(-0.769004\pi\)
−0.748039 + 0.663655i \(0.769004\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3200.00 1.65826
\(156\) 0 0
\(157\) −3258.00 −1.65616 −0.828079 0.560612i \(-0.810566\pi\)
−0.828079 + 0.560612i \(0.810566\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1414.00 0.692167
\(162\) 0 0
\(163\) 2432.00 1.16864 0.584322 0.811522i \(-0.301361\pi\)
0.584322 + 0.811522i \(0.301361\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3156.00 −1.46239 −0.731193 0.682170i \(-0.761036\pi\)
−0.731193 + 0.682170i \(0.761036\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3724.00 1.63659 0.818296 0.574797i \(-0.194919\pi\)
0.818296 + 0.574797i \(0.194919\pi\)
\(174\) 0 0
\(175\) −917.000 −0.396107
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2398.00 −1.00131 −0.500656 0.865646i \(-0.666908\pi\)
−0.500656 + 0.865646i \(0.666908\pi\)
\(180\) 0 0
\(181\) 2906.00 1.19338 0.596689 0.802473i \(-0.296483\pi\)
0.596689 + 0.802473i \(0.296483\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2400.00 −0.953792
\(186\) 0 0
\(187\) −2304.00 −0.900990
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1478.00 −0.559918 −0.279959 0.960012i \(-0.590321\pi\)
−0.279959 + 0.960012i \(0.590321\pi\)
\(192\) 0 0
\(193\) −1098.00 −0.409512 −0.204756 0.978813i \(-0.565640\pi\)
−0.204756 + 0.978813i \(0.565640\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5058.00 −1.82928 −0.914639 0.404273i \(-0.867525\pi\)
−0.914639 + 0.404273i \(0.867525\pi\)
\(198\) 0 0
\(199\) 4432.00 1.57877 0.789387 0.613895i \(-0.210398\pi\)
0.789387 + 0.613895i \(0.210398\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2114.00 0.730905
\(204\) 0 0
\(205\) −2752.00 −0.937600
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 936.000 0.309782
\(210\) 0 0
\(211\) −5444.00 −1.77621 −0.888105 0.459640i \(-0.847978\pi\)
−0.888105 + 0.459640i \(0.847978\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2624.00 −0.832350
\(216\) 0 0
\(217\) −1400.00 −0.437964
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6912.00 −2.10385
\(222\) 0 0
\(223\) 5352.00 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3752.00 1.09704 0.548522 0.836136i \(-0.315191\pi\)
0.548522 + 0.836136i \(0.315191\pi\)
\(228\) 0 0
\(229\) −5446.00 −1.57154 −0.785768 0.618521i \(-0.787732\pi\)
−0.785768 + 0.618521i \(0.787732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2602.00 0.731600 0.365800 0.930694i \(-0.380796\pi\)
0.365800 + 0.930694i \(0.380796\pi\)
\(234\) 0 0
\(235\) −7360.00 −2.04304
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1810.00 −0.489871 −0.244935 0.969539i \(-0.578767\pi\)
−0.244935 + 0.969539i \(0.578767\pi\)
\(240\) 0 0
\(241\) 310.000 0.0828583 0.0414292 0.999141i \(-0.486809\pi\)
0.0414292 + 0.999141i \(0.486809\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 784.000 0.204441
\(246\) 0 0
\(247\) 2808.00 0.723355
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −32.0000 −0.00804710 −0.00402355 0.999992i \(-0.501281\pi\)
−0.00402355 + 0.999992i \(0.501281\pi\)
\(252\) 0 0
\(253\) 3636.00 0.903531
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3348.00 −0.812617 −0.406308 0.913736i \(-0.633184\pi\)
−0.406308 + 0.913736i \(0.633184\pi\)
\(258\) 0 0
\(259\) 1050.00 0.251907
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4086.00 0.957998 0.478999 0.877815i \(-0.341000\pi\)
0.478999 + 0.877815i \(0.341000\pi\)
\(264\) 0 0
\(265\) 3040.00 0.704701
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1416.00 −0.320948 −0.160474 0.987040i \(-0.551302\pi\)
−0.160474 + 0.987040i \(0.551302\pi\)
\(270\) 0 0
\(271\) 3856.00 0.864337 0.432168 0.901793i \(-0.357749\pi\)
0.432168 + 0.901793i \(0.357749\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2358.00 −0.517065
\(276\) 0 0
\(277\) −1358.00 −0.294564 −0.147282 0.989095i \(-0.547053\pi\)
−0.147282 + 0.989095i \(0.547053\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1698.00 0.360478 0.180239 0.983623i \(-0.442313\pi\)
0.180239 + 0.983623i \(0.442313\pi\)
\(282\) 0 0
\(283\) 6340.00 1.33171 0.665855 0.746081i \(-0.268067\pi\)
0.665855 + 0.746081i \(0.268067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1204.00 0.247630
\(288\) 0 0
\(289\) 11471.0 2.33483
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9204.00 −1.83517 −0.917583 0.397545i \(-0.869862\pi\)
−0.917583 + 0.397545i \(0.869862\pi\)
\(294\) 0 0
\(295\) 1536.00 0.303150
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10908.0 2.10979
\(300\) 0 0
\(301\) 1148.00 0.219833
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9952.00 1.86836
\(306\) 0 0
\(307\) 2180.00 0.405274 0.202637 0.979254i \(-0.435049\pi\)
0.202637 + 0.979254i \(0.435049\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7212.00 1.31497 0.657484 0.753469i \(-0.271621\pi\)
0.657484 + 0.753469i \(0.271621\pi\)
\(312\) 0 0
\(313\) 1322.00 0.238734 0.119367 0.992850i \(-0.461913\pi\)
0.119367 + 0.992850i \(0.461913\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −474.000 −0.0839826 −0.0419913 0.999118i \(-0.513370\pi\)
−0.0419913 + 0.999118i \(0.513370\pi\)
\(318\) 0 0
\(319\) 5436.00 0.954099
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6656.00 −1.14659
\(324\) 0 0
\(325\) −7074.00 −1.20737
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3220.00 0.539588
\(330\) 0 0
\(331\) 11284.0 1.87379 0.936895 0.349610i \(-0.113686\pi\)
0.936895 + 0.349610i \(0.113686\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11904.0 −1.94145
\(336\) 0 0
\(337\) 6986.00 1.12923 0.564617 0.825353i \(-0.309024\pi\)
0.564617 + 0.825353i \(0.309024\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3600.00 −0.571704
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7926.00 −1.22620 −0.613098 0.790007i \(-0.710077\pi\)
−0.613098 + 0.790007i \(0.710077\pi\)
\(348\) 0 0
\(349\) −3058.00 −0.469029 −0.234514 0.972113i \(-0.575350\pi\)
−0.234514 + 0.972113i \(0.575350\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7008.00 −1.05665 −0.528326 0.849042i \(-0.677180\pi\)
−0.528326 + 0.849042i \(0.677180\pi\)
\(354\) 0 0
\(355\) −864.000 −0.129173
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11286.0 −1.65920 −0.829599 0.558359i \(-0.811431\pi\)
−0.829599 + 0.558359i \(0.811431\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11872.0 1.70249
\(366\) 0 0
\(367\) 2752.00 0.391426 0.195713 0.980661i \(-0.437298\pi\)
0.195713 + 0.980661i \(0.437298\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1330.00 −0.186119
\(372\) 0 0
\(373\) 206.000 0.0285959 0.0142980 0.999898i \(-0.495449\pi\)
0.0142980 + 0.999898i \(0.495449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16308.0 2.22786
\(378\) 0 0
\(379\) −684.000 −0.0927037 −0.0463519 0.998925i \(-0.514760\pi\)
−0.0463519 + 0.998925i \(0.514760\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9048.00 1.20713 0.603566 0.797313i \(-0.293746\pi\)
0.603566 + 0.797313i \(0.293746\pi\)
\(384\) 0 0
\(385\) 2016.00 0.266870
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3242.00 0.422560 0.211280 0.977426i \(-0.432237\pi\)
0.211280 + 0.977426i \(0.432237\pi\)
\(390\) 0 0
\(391\) −25856.0 −3.34423
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1472.00 0.187505
\(396\) 0 0
\(397\) 1702.00 0.215166 0.107583 0.994196i \(-0.465689\pi\)
0.107583 + 0.994196i \(0.465689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8742.00 −1.08866 −0.544332 0.838870i \(-0.683217\pi\)
−0.544332 + 0.838870i \(0.683217\pi\)
\(402\) 0 0
\(403\) −10800.0 −1.33495
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2700.00 0.328831
\(408\) 0 0
\(409\) 510.000 0.0616574 0.0308287 0.999525i \(-0.490185\pi\)
0.0308287 + 0.999525i \(0.490185\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −672.000 −0.0800653
\(414\) 0 0
\(415\) −3648.00 −0.431502
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5144.00 0.599763 0.299882 0.953976i \(-0.403053\pi\)
0.299882 + 0.953976i \(0.403053\pi\)
\(420\) 0 0
\(421\) 514.000 0.0595032 0.0297516 0.999557i \(-0.490528\pi\)
0.0297516 + 0.999557i \(0.490528\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16768.0 1.91381
\(426\) 0 0
\(427\) −4354.00 −0.493454
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13626.0 1.52283 0.761417 0.648263i \(-0.224504\pi\)
0.761417 + 0.648263i \(0.224504\pi\)
\(432\) 0 0
\(433\) 8794.00 0.976011 0.488005 0.872841i \(-0.337725\pi\)
0.488005 + 0.872841i \(0.337725\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10504.0 1.14983
\(438\) 0 0
\(439\) −2616.00 −0.284407 −0.142204 0.989837i \(-0.545419\pi\)
−0.142204 + 0.989837i \(0.545419\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −838.000 −0.0898749 −0.0449375 0.998990i \(-0.514309\pi\)
−0.0449375 + 0.998990i \(0.514309\pi\)
\(444\) 0 0
\(445\) 1856.00 0.197714
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9662.00 1.01554 0.507771 0.861492i \(-0.330470\pi\)
0.507771 + 0.861492i \(0.330470\pi\)
\(450\) 0 0
\(451\) 3096.00 0.323248
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6048.00 0.623153
\(456\) 0 0
\(457\) −3590.00 −0.367469 −0.183734 0.982976i \(-0.558819\pi\)
−0.183734 + 0.982976i \(0.558819\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11628.0 −1.17477 −0.587386 0.809307i \(-0.699843\pi\)
−0.587386 + 0.809307i \(0.699843\pi\)
\(462\) 0 0
\(463\) 2116.00 0.212395 0.106197 0.994345i \(-0.466132\pi\)
0.106197 + 0.994345i \(0.466132\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5088.00 −0.504164 −0.252082 0.967706i \(-0.581115\pi\)
−0.252082 + 0.967706i \(0.581115\pi\)
\(468\) 0 0
\(469\) 5208.00 0.512757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2952.00 0.286962
\(474\) 0 0
\(475\) −6812.00 −0.658013
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5068.00 0.483430 0.241715 0.970347i \(-0.422290\pi\)
0.241715 + 0.970347i \(0.422290\pi\)
\(480\) 0 0
\(481\) 8100.00 0.767834
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8864.00 −0.829884
\(486\) 0 0
\(487\) −11008.0 −1.02427 −0.512136 0.858905i \(-0.671145\pi\)
−0.512136 + 0.858905i \(0.671145\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11466.0 1.05388 0.526938 0.849904i \(-0.323340\pi\)
0.526938 + 0.849904i \(0.323340\pi\)
\(492\) 0 0
\(493\) −38656.0 −3.53140
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 378.000 0.0341159
\(498\) 0 0
\(499\) 11900.0 1.06757 0.533785 0.845620i \(-0.320769\pi\)
0.533785 + 0.845620i \(0.320769\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1832.00 −0.162395 −0.0811977 0.996698i \(-0.525875\pi\)
−0.0811977 + 0.996698i \(0.525875\pi\)
\(504\) 0 0
\(505\) 12480.0 1.09971
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16968.0 −1.47759 −0.738795 0.673930i \(-0.764605\pi\)
−0.738795 + 0.673930i \(0.764605\pi\)
\(510\) 0 0
\(511\) −5194.00 −0.449646
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −384.000 −0.0328564
\(516\) 0 0
\(517\) 8280.00 0.704360
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14976.0 1.25933 0.629665 0.776867i \(-0.283192\pi\)
0.629665 + 0.776867i \(0.283192\pi\)
\(522\) 0 0
\(523\) −9812.00 −0.820361 −0.410181 0.912004i \(-0.634534\pi\)
−0.410181 + 0.912004i \(0.634534\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25600.0 2.11604
\(528\) 0 0
\(529\) 28637.0 2.35366
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9288.00 0.754799
\(534\) 0 0
\(535\) 1248.00 0.100852
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −882.000 −0.0704832
\(540\) 0 0
\(541\) 21370.0 1.69828 0.849139 0.528170i \(-0.177122\pi\)
0.849139 + 0.528170i \(0.177122\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16544.0 −1.30031
\(546\) 0 0
\(547\) 8120.00 0.634710 0.317355 0.948307i \(-0.397205\pi\)
0.317355 + 0.948307i \(0.397205\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15704.0 1.21418
\(552\) 0 0
\(553\) −644.000 −0.0495220
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19934.0 1.51639 0.758196 0.652026i \(-0.226081\pi\)
0.758196 + 0.652026i \(0.226081\pi\)
\(558\) 0 0
\(559\) 8856.00 0.670070
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 352.000 0.0263500 0.0131750 0.999913i \(-0.495806\pi\)
0.0131750 + 0.999913i \(0.495806\pi\)
\(564\) 0 0
\(565\) −2976.00 −0.221595
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10698.0 0.788196 0.394098 0.919068i \(-0.371057\pi\)
0.394098 + 0.919068i \(0.371057\pi\)
\(570\) 0 0
\(571\) 14824.0 1.08645 0.543227 0.839586i \(-0.317202\pi\)
0.543227 + 0.839586i \(0.317202\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −26462.0 −1.91920
\(576\) 0 0
\(577\) −15318.0 −1.10519 −0.552597 0.833449i \(-0.686363\pi\)
−0.552597 + 0.833449i \(0.686363\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1596.00 0.113964
\(582\) 0 0
\(583\) −3420.00 −0.242954
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5672.00 −0.398822 −0.199411 0.979916i \(-0.563903\pi\)
−0.199411 + 0.979916i \(0.563903\pi\)
\(588\) 0 0
\(589\) −10400.0 −0.727546
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25968.0 1.79828 0.899138 0.437665i \(-0.144194\pi\)
0.899138 + 0.437665i \(0.144194\pi\)
\(594\) 0 0
\(595\) −14336.0 −0.987763
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20846.0 −1.42194 −0.710972 0.703220i \(-0.751745\pi\)
−0.710972 + 0.703220i \(0.751745\pi\)
\(600\) 0 0
\(601\) −4430.00 −0.300671 −0.150336 0.988635i \(-0.548035\pi\)
−0.150336 + 0.988635i \(0.548035\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16112.0 −1.08272
\(606\) 0 0
\(607\) 23744.0 1.58771 0.793854 0.608108i \(-0.208071\pi\)
0.793854 + 0.608108i \(0.208071\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24840.0 1.64471
\(612\) 0 0
\(613\) 9982.00 0.657699 0.328849 0.944382i \(-0.393339\pi\)
0.328849 + 0.944382i \(0.393339\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21090.0 1.37610 0.688048 0.725665i \(-0.258468\pi\)
0.688048 + 0.725665i \(0.258468\pi\)
\(618\) 0 0
\(619\) 12900.0 0.837633 0.418816 0.908071i \(-0.362445\pi\)
0.418816 + 0.908071i \(0.362445\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −812.000 −0.0522184
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −19200.0 −1.21710
\(630\) 0 0
\(631\) −140.000 −0.00883251 −0.00441625 0.999990i \(-0.501406\pi\)
−0.00441625 + 0.999990i \(0.501406\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5056.00 0.315970
\(636\) 0 0
\(637\) −2646.00 −0.164581
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2642.00 0.162797 0.0813984 0.996682i \(-0.474061\pi\)
0.0813984 + 0.996682i \(0.474061\pi\)
\(642\) 0 0
\(643\) 1388.00 0.0851281 0.0425641 0.999094i \(-0.486447\pi\)
0.0425641 + 0.999094i \(0.486447\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11196.0 −0.680309 −0.340155 0.940369i \(-0.610479\pi\)
−0.340155 + 0.940369i \(0.610479\pi\)
\(648\) 0 0
\(649\) −1728.00 −0.104515
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9398.00 −0.563204 −0.281602 0.959531i \(-0.590866\pi\)
−0.281602 + 0.959531i \(0.590866\pi\)
\(654\) 0 0
\(655\) −15808.0 −0.943007
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14050.0 −0.830516 −0.415258 0.909704i \(-0.636309\pi\)
−0.415258 + 0.909704i \(0.636309\pi\)
\(660\) 0 0
\(661\) 22382.0 1.31703 0.658517 0.752566i \(-0.271184\pi\)
0.658517 + 0.752566i \(0.271184\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5824.00 0.339617
\(666\) 0 0
\(667\) 61004.0 3.54136
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11196.0 −0.644138
\(672\) 0 0
\(673\) −6050.00 −0.346524 −0.173262 0.984876i \(-0.555431\pi\)
−0.173262 + 0.984876i \(0.555431\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7500.00 −0.425773 −0.212887 0.977077i \(-0.568286\pi\)
−0.212887 + 0.977077i \(0.568286\pi\)
\(678\) 0 0
\(679\) 3878.00 0.219181
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −26430.0 −1.48070 −0.740348 0.672223i \(-0.765339\pi\)
−0.740348 + 0.672223i \(0.765339\pi\)
\(684\) 0 0
\(685\) −15200.0 −0.847828
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10260.0 −0.567308
\(690\) 0 0
\(691\) −14228.0 −0.783298 −0.391649 0.920115i \(-0.628095\pi\)
−0.391649 + 0.920115i \(0.628095\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33984.0 1.85480
\(696\) 0 0
\(697\) −22016.0 −1.19644
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12338.0 0.664764 0.332382 0.943145i \(-0.392148\pi\)
0.332382 + 0.943145i \(0.392148\pi\)
\(702\) 0 0
\(703\) 7800.00 0.418467
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5460.00 −0.290445
\(708\) 0 0
\(709\) 7054.00 0.373651 0.186825 0.982393i \(-0.440180\pi\)
0.186825 + 0.982393i \(0.440180\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −40400.0 −2.12201
\(714\) 0 0
\(715\) 15552.0 0.813443
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8640.00 0.448147 0.224073 0.974572i \(-0.428064\pi\)
0.224073 + 0.974572i \(0.428064\pi\)
\(720\) 0 0
\(721\) 168.000 0.00867774
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −39562.0 −2.02661
\(726\) 0 0
\(727\) −20120.0 −1.02642 −0.513211 0.858262i \(-0.671544\pi\)
−0.513211 + 0.858262i \(0.671544\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20992.0 −1.06213
\(732\) 0 0
\(733\) 14554.0 0.733376 0.366688 0.930344i \(-0.380492\pi\)
0.366688 + 0.930344i \(0.380492\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13392.0 0.669336
\(738\) 0 0
\(739\) 23600.0 1.17475 0.587375 0.809315i \(-0.300161\pi\)
0.587375 + 0.809315i \(0.300161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28970.0 1.43043 0.715213 0.698907i \(-0.246330\pi\)
0.715213 + 0.698907i \(0.246330\pi\)
\(744\) 0 0
\(745\) −35872.0 −1.76409
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −546.000 −0.0266361
\(750\) 0 0
\(751\) −18848.0 −0.915810 −0.457905 0.889001i \(-0.651400\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −44416.0 −2.14101
\(756\) 0 0
\(757\) −26594.0 −1.27685 −0.638425 0.769684i \(-0.720414\pi\)
−0.638425 + 0.769684i \(0.720414\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3012.00 0.143476 0.0717378 0.997424i \(-0.477146\pi\)
0.0717378 + 0.997424i \(0.477146\pi\)
\(762\) 0 0
\(763\) 7238.00 0.343425
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5184.00 −0.244046
\(768\) 0 0
\(769\) −12614.0 −0.591512 −0.295756 0.955264i \(-0.595571\pi\)
−0.295756 + 0.955264i \(0.595571\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −40684.0 −1.89302 −0.946508 0.322680i \(-0.895416\pi\)
−0.946508 + 0.322680i \(0.895416\pi\)
\(774\) 0 0
\(775\) 26200.0 1.21436
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8944.00 0.411363
\(780\) 0 0
\(781\) 972.000 0.0445338
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −52128.0 −2.37010
\(786\) 0 0
\(787\) −28564.0 −1.29377 −0.646885 0.762588i \(-0.723929\pi\)
−0.646885 + 0.762588i \(0.723929\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1302.00 0.0585257
\(792\) 0 0
\(793\) −33588.0 −1.50409
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7140.00 −0.317330 −0.158665 0.987332i \(-0.550719\pi\)
−0.158665 + 0.987332i \(0.550719\pi\)
\(798\) 0 0
\(799\) −58880.0 −2.60704
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −13356.0 −0.586953
\(804\) 0 0
\(805\) 22624.0 0.990548
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26682.0 −1.15957 −0.579783 0.814771i \(-0.696863\pi\)
−0.579783 + 0.814771i \(0.696863\pi\)
\(810\) 0 0
\(811\) −18012.0 −0.779885 −0.389943 0.920839i \(-0.627505\pi\)
−0.389943 + 0.920839i \(0.627505\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 38912.0 1.67243
\(816\) 0 0
\(817\) 8528.00 0.365186
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3270.00 0.139006 0.0695029 0.997582i \(-0.477859\pi\)
0.0695029 + 0.997582i \(0.477859\pi\)
\(822\) 0 0
\(823\) 41692.0 1.76585 0.882923 0.469517i \(-0.155572\pi\)
0.882923 + 0.469517i \(0.155572\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12078.0 0.507852 0.253926 0.967224i \(-0.418278\pi\)
0.253926 + 0.967224i \(0.418278\pi\)
\(828\) 0 0
\(829\) −24046.0 −1.00742 −0.503711 0.863872i \(-0.668032\pi\)
−0.503711 + 0.863872i \(0.668032\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6272.00 0.260879
\(834\) 0 0
\(835\) −50496.0 −2.09280
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12204.0 −0.502180 −0.251090 0.967964i \(-0.580789\pi\)
−0.251090 + 0.967964i \(0.580789\pi\)
\(840\) 0 0
\(841\) 66815.0 2.73955
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11504.0 0.468343
\(846\) 0 0
\(847\) 7049.00 0.285958
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 30300.0 1.22053
\(852\) 0 0
\(853\) −5930.00 −0.238030 −0.119015 0.992892i \(-0.537974\pi\)
−0.119015 + 0.992892i \(0.537974\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5268.00 0.209978 0.104989 0.994473i \(-0.466519\pi\)
0.104989 + 0.994473i \(0.466519\pi\)
\(858\) 0 0
\(859\) 10028.0 0.398313 0.199157 0.979968i \(-0.436180\pi\)
0.199157 + 0.979968i \(0.436180\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −37306.0 −1.47151 −0.735754 0.677249i \(-0.763172\pi\)
−0.735754 + 0.677249i \(0.763172\pi\)
\(864\) 0 0
\(865\) 59584.0 2.34210
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1656.00 −0.0646444
\(870\) 0 0
\(871\) 40176.0 1.56293
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −672.000 −0.0259631
\(876\) 0 0
\(877\) −39178.0 −1.50849 −0.754246 0.656592i \(-0.771997\pi\)
−0.754246 + 0.656592i \(0.771997\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36872.0 1.41004 0.705022 0.709185i \(-0.250937\pi\)
0.705022 + 0.709185i \(0.250937\pi\)
\(882\) 0 0
\(883\) 19964.0 0.760863 0.380432 0.924809i \(-0.375775\pi\)
0.380432 + 0.924809i \(0.375775\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12924.0 0.489228 0.244614 0.969621i \(-0.421339\pi\)
0.244614 + 0.969621i \(0.421339\pi\)
\(888\) 0 0
\(889\) −2212.00 −0.0834512
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 23920.0 0.896363
\(894\) 0 0
\(895\) −38368.0 −1.43296
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −60400.0 −2.24077
\(900\) 0 0
\(901\) 24320.0 0.899242
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 46496.0 1.70782
\(906\) 0 0
\(907\) −44804.0 −1.64023 −0.820117 0.572196i \(-0.806092\pi\)
−0.820117 + 0.572196i \(0.806092\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −26102.0 −0.949284 −0.474642 0.880179i \(-0.657422\pi\)
−0.474642 + 0.880179i \(0.657422\pi\)
\(912\) 0 0
\(913\) 4104.00 0.148765
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6916.00 0.249058
\(918\) 0 0
\(919\) 176.000 0.00631741 0.00315871 0.999995i \(-0.498995\pi\)
0.00315871 + 0.999995i \(0.498995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2916.00 0.103988
\(924\) 0 0
\(925\) −19650.0 −0.698474
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −23116.0 −0.816374 −0.408187 0.912898i \(-0.633839\pi\)
−0.408187 + 0.912898i \(0.633839\pi\)
\(930\) 0 0
\(931\) −2548.00 −0.0896964
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −36864.0 −1.28939
\(936\) 0 0
\(937\) −20142.0 −0.702252 −0.351126 0.936328i \(-0.614201\pi\)
−0.351126 + 0.936328i \(0.614201\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −19896.0 −0.689257 −0.344629 0.938739i \(-0.611995\pi\)
−0.344629 + 0.938739i \(0.611995\pi\)
\(942\) 0 0
\(943\) 34744.0 1.19981
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7658.00 −0.262779 −0.131389 0.991331i \(-0.541944\pi\)
−0.131389 + 0.991331i \(0.541944\pi\)
\(948\) 0 0
\(949\) −40068.0 −1.37056
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −12826.0 −0.435965 −0.217983 0.975953i \(-0.569948\pi\)
−0.217983 + 0.975953i \(0.569948\pi\)
\(954\) 0 0
\(955\) −23648.0 −0.801289
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6650.00 0.223920
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −17568.0 −0.586046
\(966\) 0 0
\(967\) −53148.0 −1.76745 −0.883725 0.468006i \(-0.844972\pi\)
−0.883725 + 0.468006i \(0.844972\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 30340.0 1.00274 0.501368 0.865234i \(-0.332830\pi\)
0.501368 + 0.865234i \(0.332830\pi\)
\(972\) 0 0
\(973\) −14868.0 −0.489873
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6714.00 0.219857 0.109928 0.993940i \(-0.464938\pi\)
0.109928 + 0.993940i \(0.464938\pi\)
\(978\) 0 0
\(979\) −2088.00 −0.0681642
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22128.0 0.717979 0.358990 0.933342i \(-0.383121\pi\)
0.358990 + 0.933342i \(0.383121\pi\)
\(984\) 0 0
\(985\) −80928.0 −2.61785
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 33128.0 1.06513
\(990\) 0 0
\(991\) −41928.0 −1.34398 −0.671991 0.740559i \(-0.734561\pi\)
−0.671991 + 0.740559i \(0.734561\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 70912.0 2.25936
\(996\) 0 0
\(997\) 28894.0 0.917836 0.458918 0.888479i \(-0.348237\pi\)
0.458918 + 0.888479i \(0.348237\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1008.4.a.t.1.1 1
3.2 odd 2 336.4.a.a.1.1 1
4.3 odd 2 504.4.a.h.1.1 1
12.11 even 2 168.4.a.d.1.1 1
21.20 even 2 2352.4.a.bj.1.1 1
24.5 odd 2 1344.4.a.z.1.1 1
24.11 even 2 1344.4.a.l.1.1 1
84.83 odd 2 1176.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.d.1.1 1 12.11 even 2
336.4.a.a.1.1 1 3.2 odd 2
504.4.a.h.1.1 1 4.3 odd 2
1008.4.a.t.1.1 1 1.1 even 1 trivial
1176.4.a.h.1.1 1 84.83 odd 2
1344.4.a.l.1.1 1 24.11 even 2
1344.4.a.z.1.1 1 24.5 odd 2
2352.4.a.bj.1.1 1 21.20 even 2