Properties

Label 1296.4.a.o.1.2
Level $1296$
Weight $4$
Character 1296.1
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, 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("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1296.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12.1244 q^{5} +22.0000 q^{7} +O(q^{10})\) \(q+12.1244 q^{5} +22.0000 q^{7} +58.8897 q^{11} -49.0000 q^{13} -36.3731 q^{17} +70.0000 q^{19} +24.2487 q^{23} +22.0000 q^{25} +278.860 q^{29} +112.000 q^{31} +266.736 q^{35} +281.000 q^{37} +48.4974 q^{41} -50.0000 q^{43} -242.487 q^{47} +141.000 q^{49} -374.123 q^{53} +714.000 q^{55} +96.9948 q^{59} -679.000 q^{61} -594.093 q^{65} +274.000 q^{67} -446.869 q^{71} -511.000 q^{73} +1295.57 q^{77} +526.000 q^{79} -387.979 q^{83} -441.000 q^{85} +36.3731 q^{89} -1078.00 q^{91} +848.705 q^{95} +1778.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 44 q^{7} - 98 q^{13} + 140 q^{19} + 44 q^{25} + 224 q^{31} + 562 q^{37} - 100 q^{43} + 282 q^{49} + 1428 q^{55} - 1358 q^{61} + 548 q^{67} - 1022 q^{73} + 1052 q^{79} - 882 q^{85} - 2156 q^{91} + 3556 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 12.1244 1.08444 0.542218 0.840238i \(-0.317585\pi\)
0.542218 + 0.840238i \(0.317585\pi\)
\(6\) 0 0
\(7\) 22.0000 1.18789 0.593944 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 58.8897 1.61417 0.807087 0.590432i \(-0.201043\pi\)
0.807087 + 0.590432i \(0.201043\pi\)
\(12\) 0 0
\(13\) −49.0000 −1.04540 −0.522698 0.852518i \(-0.675075\pi\)
−0.522698 + 0.852518i \(0.675075\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −36.3731 −0.518927 −0.259464 0.965753i \(-0.583546\pi\)
−0.259464 + 0.965753i \(0.583546\pi\)
\(18\) 0 0
\(19\) 70.0000 0.845216 0.422608 0.906313i \(-0.361115\pi\)
0.422608 + 0.906313i \(0.361115\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 24.2487 0.219835 0.109918 0.993941i \(-0.464941\pi\)
0.109918 + 0.993941i \(0.464941\pi\)
\(24\) 0 0
\(25\) 22.0000 0.176000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 278.860 1.78562 0.892811 0.450432i \(-0.148730\pi\)
0.892811 + 0.450432i \(0.148730\pi\)
\(30\) 0 0
\(31\) 112.000 0.648897 0.324448 0.945903i \(-0.394821\pi\)
0.324448 + 0.945903i \(0.394821\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 266.736 1.28819
\(36\) 0 0
\(37\) 281.000 1.24854 0.624272 0.781207i \(-0.285396\pi\)
0.624272 + 0.781207i \(0.285396\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 48.4974 0.184732 0.0923662 0.995725i \(-0.470557\pi\)
0.0923662 + 0.995725i \(0.470557\pi\)
\(42\) 0 0
\(43\) −50.0000 −0.177324 −0.0886620 0.996062i \(-0.528259\pi\)
−0.0886620 + 0.996062i \(0.528259\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −242.487 −0.752561 −0.376281 0.926506i \(-0.622797\pi\)
−0.376281 + 0.926506i \(0.622797\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −374.123 −0.969618 −0.484809 0.874620i \(-0.661111\pi\)
−0.484809 + 0.874620i \(0.661111\pi\)
\(54\) 0 0
\(55\) 714.000 1.75047
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 96.9948 0.214028 0.107014 0.994258i \(-0.465871\pi\)
0.107014 + 0.994258i \(0.465871\pi\)
\(60\) 0 0
\(61\) −679.000 −1.42520 −0.712599 0.701572i \(-0.752482\pi\)
−0.712599 + 0.701572i \(0.752482\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −594.093 −1.13366
\(66\) 0 0
\(67\) 274.000 0.499618 0.249809 0.968295i \(-0.419632\pi\)
0.249809 + 0.968295i \(0.419632\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −446.869 −0.746952 −0.373476 0.927640i \(-0.621834\pi\)
−0.373476 + 0.927640i \(0.621834\pi\)
\(72\) 0 0
\(73\) −511.000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1295.57 1.91746
\(78\) 0 0
\(79\) 526.000 0.749109 0.374555 0.927205i \(-0.377796\pi\)
0.374555 + 0.927205i \(0.377796\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −387.979 −0.513088 −0.256544 0.966533i \(-0.582584\pi\)
−0.256544 + 0.966533i \(0.582584\pi\)
\(84\) 0 0
\(85\) −441.000 −0.562743
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 36.3731 0.0433206 0.0216603 0.999765i \(-0.493105\pi\)
0.0216603 + 0.999765i \(0.493105\pi\)
\(90\) 0 0
\(91\) −1078.00 −1.24181
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 848.705 0.916582
\(96\) 0 0
\(97\) 1778.00 1.86112 0.930560 0.366141i \(-0.119321\pi\)
0.930560 + 0.366141i \(0.119321\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 242.487 0.238895 0.119447 0.992841i \(-0.461888\pi\)
0.119447 + 0.992841i \(0.461888\pi\)
\(102\) 0 0
\(103\) −896.000 −0.857141 −0.428570 0.903508i \(-0.640983\pi\)
−0.428570 + 0.903508i \(0.640983\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −374.123 −0.338017 −0.169009 0.985615i \(-0.554057\pi\)
−0.169009 + 0.985615i \(0.554057\pi\)
\(108\) 0 0
\(109\) −205.000 −0.180142 −0.0900708 0.995935i \(-0.528709\pi\)
−0.0900708 + 0.995935i \(0.528709\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −361.999 −0.301363 −0.150681 0.988582i \(-0.548147\pi\)
−0.150681 + 0.988582i \(0.548147\pi\)
\(114\) 0 0
\(115\) 294.000 0.238397
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −800.207 −0.616428
\(120\) 0 0
\(121\) 2137.00 1.60556
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1248.81 −0.893575
\(126\) 0 0
\(127\) −146.000 −0.102011 −0.0510055 0.998698i \(-0.516243\pi\)
−0.0510055 + 0.998698i \(0.516243\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2643.11 1.76282 0.881411 0.472351i \(-0.156595\pi\)
0.881411 + 0.472351i \(0.156595\pi\)
\(132\) 0 0
\(133\) 1540.00 1.00402
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1965.88 −1.22596 −0.612979 0.790099i \(-0.710029\pi\)
−0.612979 + 0.790099i \(0.710029\pi\)
\(138\) 0 0
\(139\) 3136.00 1.91361 0.956806 0.290727i \(-0.0938973\pi\)
0.956806 + 0.290727i \(0.0938973\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2885.60 −1.68745
\(144\) 0 0
\(145\) 3381.00 1.93639
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 313.501 0.172369 0.0861846 0.996279i \(-0.472533\pi\)
0.0861846 + 0.996279i \(0.472533\pi\)
\(150\) 0 0
\(151\) 2092.00 1.12745 0.563724 0.825963i \(-0.309368\pi\)
0.563724 + 0.825963i \(0.309368\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1357.93 0.703686
\(156\) 0 0
\(157\) 1337.00 0.679645 0.339822 0.940490i \(-0.389633\pi\)
0.339822 + 0.940490i \(0.389633\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 533.472 0.261139
\(162\) 0 0
\(163\) 1744.00 0.838041 0.419020 0.907977i \(-0.362374\pi\)
0.419020 + 0.907977i \(0.362374\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2740.10 −1.26967 −0.634837 0.772646i \(-0.718933\pi\)
−0.634837 + 0.772646i \(0.718933\pi\)
\(168\) 0 0
\(169\) 204.000 0.0928539
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2194.51 −0.964424 −0.482212 0.876055i \(-0.660167\pi\)
−0.482212 + 0.876055i \(0.660167\pi\)
\(174\) 0 0
\(175\) 484.000 0.209068
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2473.37 −1.03278 −0.516392 0.856352i \(-0.672725\pi\)
−0.516392 + 0.856352i \(0.672725\pi\)
\(180\) 0 0
\(181\) −70.0000 −0.0287462 −0.0143731 0.999897i \(-0.504575\pi\)
−0.0143731 + 0.999897i \(0.504575\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3406.94 1.35396
\(186\) 0 0
\(187\) −2142.00 −0.837639
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2075.00 0.786081 0.393041 0.919521i \(-0.371423\pi\)
0.393041 + 0.919521i \(0.371423\pi\)
\(192\) 0 0
\(193\) −121.000 −0.0451283 −0.0225642 0.999745i \(-0.507183\pi\)
−0.0225642 + 0.999745i \(0.507183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2956.61 −1.06929 −0.534644 0.845077i \(-0.679554\pi\)
−0.534644 + 0.845077i \(0.679554\pi\)
\(198\) 0 0
\(199\) 4228.00 1.50611 0.753053 0.657960i \(-0.228581\pi\)
0.753053 + 0.657960i \(0.228581\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6134.92 2.12112
\(204\) 0 0
\(205\) 588.000 0.200330
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4122.28 1.36433
\(210\) 0 0
\(211\) 202.000 0.0659064 0.0329532 0.999457i \(-0.489509\pi\)
0.0329532 + 0.999457i \(0.489509\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −606.218 −0.192296
\(216\) 0 0
\(217\) 2464.00 0.770817
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1782.28 0.542485
\(222\) 0 0
\(223\) 238.000 0.0714693 0.0357347 0.999361i \(-0.488623\pi\)
0.0357347 + 0.999361i \(0.488623\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5552.95 −1.62362 −0.811812 0.583919i \(-0.801518\pi\)
−0.811812 + 0.583919i \(0.801518\pi\)
\(228\) 0 0
\(229\) 2975.00 0.858487 0.429244 0.903189i \(-0.358780\pi\)
0.429244 + 0.903189i \(0.358780\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 181.865 0.0511347 0.0255674 0.999673i \(-0.491861\pi\)
0.0255674 + 0.999673i \(0.491861\pi\)
\(234\) 0 0
\(235\) −2940.00 −0.816104
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5417.85 1.46633 0.733163 0.680053i \(-0.238043\pi\)
0.733163 + 0.680053i \(0.238043\pi\)
\(240\) 0 0
\(241\) −2443.00 −0.652977 −0.326489 0.945201i \(-0.605865\pi\)
−0.326489 + 0.945201i \(0.605865\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1709.53 0.445788
\(246\) 0 0
\(247\) −3430.00 −0.883586
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5310.47 1.33543 0.667717 0.744416i \(-0.267272\pi\)
0.667717 + 0.744416i \(0.267272\pi\)
\(252\) 0 0
\(253\) 1428.00 0.354852
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7795.96 1.89221 0.946106 0.323856i \(-0.104979\pi\)
0.946106 + 0.323856i \(0.104979\pi\)
\(258\) 0 0
\(259\) 6182.00 1.48313
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2487.22 −0.583152 −0.291576 0.956548i \(-0.594180\pi\)
−0.291576 + 0.956548i \(0.594180\pi\)
\(264\) 0 0
\(265\) −4536.00 −1.05149
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 109.119 0.0247328 0.0123664 0.999924i \(-0.496064\pi\)
0.0123664 + 0.999924i \(0.496064\pi\)
\(270\) 0 0
\(271\) 5614.00 1.25840 0.629200 0.777244i \(-0.283383\pi\)
0.629200 + 0.777244i \(0.283383\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1295.57 0.284095
\(276\) 0 0
\(277\) 6026.00 1.30710 0.653551 0.756882i \(-0.273279\pi\)
0.653551 + 0.756882i \(0.273279\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4657.48 −0.988762 −0.494381 0.869245i \(-0.664605\pi\)
−0.494381 + 0.869245i \(0.664605\pi\)
\(282\) 0 0
\(283\) −140.000 −0.0294068 −0.0147034 0.999892i \(-0.504680\pi\)
−0.0147034 + 0.999892i \(0.504680\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1066.94 0.219441
\(288\) 0 0
\(289\) −3590.00 −0.730714
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2485.49 0.495577 0.247788 0.968814i \(-0.420296\pi\)
0.247788 + 0.968814i \(0.420296\pi\)
\(294\) 0 0
\(295\) 1176.00 0.232100
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1188.19 −0.229815
\(300\) 0 0
\(301\) −1100.00 −0.210641
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8232.44 −1.54553
\(306\) 0 0
\(307\) 2968.00 0.551768 0.275884 0.961191i \(-0.411029\pi\)
0.275884 + 0.961191i \(0.411029\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5286.22 −0.963839 −0.481920 0.876215i \(-0.660060\pi\)
−0.481920 + 0.876215i \(0.660060\pi\)
\(312\) 0 0
\(313\) −5593.00 −1.01002 −0.505008 0.863115i \(-0.668510\pi\)
−0.505008 + 0.863115i \(0.668510\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7504.98 −1.32972 −0.664860 0.746968i \(-0.731509\pi\)
−0.664860 + 0.746968i \(0.731509\pi\)
\(318\) 0 0
\(319\) 16422.0 2.88231
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2546.11 −0.438606
\(324\) 0 0
\(325\) −1078.00 −0.183990
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5334.72 −0.893959
\(330\) 0 0
\(331\) −7754.00 −1.28761 −0.643804 0.765190i \(-0.722645\pi\)
−0.643804 + 0.765190i \(0.722645\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3322.07 0.541804
\(336\) 0 0
\(337\) 2570.00 0.415421 0.207710 0.978190i \(-0.433399\pi\)
0.207710 + 0.978190i \(0.433399\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6595.65 1.04743
\(342\) 0 0
\(343\) −4444.00 −0.699573
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5587.60 −0.864432 −0.432216 0.901770i \(-0.642268\pi\)
−0.432216 + 0.901770i \(0.642268\pi\)
\(348\) 0 0
\(349\) −6790.00 −1.04143 −0.520717 0.853729i \(-0.674335\pi\)
−0.520717 + 0.853729i \(0.674335\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8244.56 1.24310 0.621549 0.783375i \(-0.286504\pi\)
0.621549 + 0.783375i \(0.286504\pi\)
\(354\) 0 0
\(355\) −5418.00 −0.810021
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7409.71 −1.08933 −0.544665 0.838653i \(-0.683343\pi\)
−0.544665 + 0.838653i \(0.683343\pi\)
\(360\) 0 0
\(361\) −1959.00 −0.285610
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6195.55 −0.888465
\(366\) 0 0
\(367\) 9436.00 1.34211 0.671056 0.741407i \(-0.265841\pi\)
0.671056 + 0.741407i \(0.265841\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8230.71 −1.15180
\(372\) 0 0
\(373\) 1742.00 0.241816 0.120908 0.992664i \(-0.461419\pi\)
0.120908 + 0.992664i \(0.461419\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13664.1 −1.86668
\(378\) 0 0
\(379\) −8408.00 −1.13955 −0.569776 0.821800i \(-0.692970\pi\)
−0.569776 + 0.821800i \(0.692970\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2497.62 0.333217 0.166609 0.986023i \(-0.446718\pi\)
0.166609 + 0.986023i \(0.446718\pi\)
\(384\) 0 0
\(385\) 15708.0 2.07936
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5334.72 0.695324 0.347662 0.937620i \(-0.386976\pi\)
0.347662 + 0.937620i \(0.386976\pi\)
\(390\) 0 0
\(391\) −882.000 −0.114078
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6377.41 0.812360
\(396\) 0 0
\(397\) 8813.00 1.11414 0.557068 0.830467i \(-0.311926\pi\)
0.557068 + 0.830467i \(0.311926\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8145.83 −1.01442 −0.507211 0.861822i \(-0.669324\pi\)
−0.507211 + 0.861822i \(0.669324\pi\)
\(402\) 0 0
\(403\) −5488.00 −0.678354
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16548.0 2.01537
\(408\) 0 0
\(409\) 6125.00 0.740493 0.370247 0.928933i \(-0.379273\pi\)
0.370247 + 0.928933i \(0.379273\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2133.89 0.254241
\(414\) 0 0
\(415\) −4704.00 −0.556410
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5189.22 0.605036 0.302518 0.953144i \(-0.402173\pi\)
0.302518 + 0.953144i \(0.402173\pi\)
\(420\) 0 0
\(421\) 3695.00 0.427751 0.213876 0.976861i \(-0.431391\pi\)
0.213876 + 0.976861i \(0.431391\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −800.207 −0.0913312
\(426\) 0 0
\(427\) −14938.0 −1.69298
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2618.86 −0.292682 −0.146341 0.989234i \(-0.546750\pi\)
−0.146341 + 0.989234i \(0.546750\pi\)
\(432\) 0 0
\(433\) −15757.0 −1.74881 −0.874403 0.485200i \(-0.838747\pi\)
−0.874403 + 0.485200i \(0.838747\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1697.41 0.185808
\(438\) 0 0
\(439\) 868.000 0.0943676 0.0471838 0.998886i \(-0.484975\pi\)
0.0471838 + 0.998886i \(0.484975\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5771.19 −0.618956 −0.309478 0.950907i \(-0.600154\pi\)
−0.309478 + 0.950907i \(0.600154\pi\)
\(444\) 0 0
\(445\) 441.000 0.0469784
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18560.7 1.95085 0.975425 0.220332i \(-0.0707142\pi\)
0.975425 + 0.220332i \(0.0707142\pi\)
\(450\) 0 0
\(451\) 2856.00 0.298190
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −13070.1 −1.34667
\(456\) 0 0
\(457\) −2659.00 −0.272172 −0.136086 0.990697i \(-0.543452\pi\)
−0.136086 + 0.990697i \(0.543452\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 387.979 0.0391974 0.0195987 0.999808i \(-0.493761\pi\)
0.0195987 + 0.999808i \(0.493761\pi\)
\(462\) 0 0
\(463\) 8968.00 0.900169 0.450085 0.892986i \(-0.351394\pi\)
0.450085 + 0.892986i \(0.351394\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10620.9 −1.05242 −0.526208 0.850356i \(-0.676387\pi\)
−0.526208 + 0.850356i \(0.676387\pi\)
\(468\) 0 0
\(469\) 6028.00 0.593491
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2944.49 −0.286232
\(474\) 0 0
\(475\) 1540.00 0.148758
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15252.4 1.45491 0.727455 0.686156i \(-0.240703\pi\)
0.727455 + 0.686156i \(0.240703\pi\)
\(480\) 0 0
\(481\) −13769.0 −1.30522
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21557.1 2.01826
\(486\) 0 0
\(487\) 3688.00 0.343161 0.171580 0.985170i \(-0.445113\pi\)
0.171580 + 0.985170i \(0.445113\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10596.7 −0.973975 −0.486988 0.873409i \(-0.661904\pi\)
−0.486988 + 0.873409i \(0.661904\pi\)
\(492\) 0 0
\(493\) −10143.0 −0.926608
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9831.12 −0.887296
\(498\) 0 0
\(499\) −18086.0 −1.62253 −0.811263 0.584681i \(-0.801220\pi\)
−0.811263 + 0.584681i \(0.801220\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21096.4 1.87006 0.935031 0.354566i \(-0.115371\pi\)
0.935031 + 0.354566i \(0.115371\pi\)
\(504\) 0 0
\(505\) 2940.00 0.259066
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2424.87 −0.211160 −0.105580 0.994411i \(-0.533670\pi\)
−0.105580 + 0.994411i \(0.533670\pi\)
\(510\) 0 0
\(511\) −11242.0 −0.973223
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10863.4 −0.929514
\(516\) 0 0
\(517\) −14280.0 −1.21477
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15858.7 1.33355 0.666776 0.745258i \(-0.267674\pi\)
0.666776 + 0.745258i \(0.267674\pi\)
\(522\) 0 0
\(523\) −7238.00 −0.605154 −0.302577 0.953125i \(-0.597847\pi\)
−0.302577 + 0.953125i \(0.597847\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4073.78 −0.336730
\(528\) 0 0
\(529\) −11579.0 −0.951673
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2376.37 −0.193119
\(534\) 0 0
\(535\) −4536.00 −0.366558
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8303.45 0.663553
\(540\) 0 0
\(541\) −15541.0 −1.23505 −0.617523 0.786553i \(-0.711864\pi\)
−0.617523 + 0.786553i \(0.711864\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2485.49 −0.195352
\(546\) 0 0
\(547\) −12956.0 −1.01272 −0.506361 0.862322i \(-0.669010\pi\)
−0.506361 + 0.862322i \(0.669010\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19520.2 1.50924
\(552\) 0 0
\(553\) 11572.0 0.889858
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18960.8 −1.44236 −0.721179 0.692749i \(-0.756399\pi\)
−0.721179 + 0.692749i \(0.756399\pi\)
\(558\) 0 0
\(559\) 2450.00 0.185374
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18574.5 1.39045 0.695224 0.718793i \(-0.255305\pi\)
0.695224 + 0.718793i \(0.255305\pi\)
\(564\) 0 0
\(565\) −4389.00 −0.326808
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2305.36 0.169852 0.0849259 0.996387i \(-0.472935\pi\)
0.0849259 + 0.996387i \(0.472935\pi\)
\(570\) 0 0
\(571\) −968.000 −0.0709449 −0.0354725 0.999371i \(-0.511294\pi\)
−0.0354725 + 0.999371i \(0.511294\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 533.472 0.0386910
\(576\) 0 0
\(577\) 18011.0 1.29949 0.649747 0.760151i \(-0.274875\pi\)
0.649747 + 0.760151i \(0.274875\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8535.55 −0.609491
\(582\) 0 0
\(583\) −22032.0 −1.56513
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23594.0 −1.65899 −0.829496 0.558512i \(-0.811372\pi\)
−0.829496 + 0.558512i \(0.811372\pi\)
\(588\) 0 0
\(589\) 7840.00 0.548458
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2727.98 −0.188912 −0.0944559 0.995529i \(-0.530111\pi\)
−0.0944559 + 0.995529i \(0.530111\pi\)
\(594\) 0 0
\(595\) −9702.00 −0.668476
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15990.3 −1.09073 −0.545364 0.838200i \(-0.683608\pi\)
−0.545364 + 0.838200i \(0.683608\pi\)
\(600\) 0 0
\(601\) −21217.0 −1.44003 −0.720016 0.693957i \(-0.755866\pi\)
−0.720016 + 0.693957i \(0.755866\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25909.7 1.74113
\(606\) 0 0
\(607\) −2786.00 −0.186294 −0.0931468 0.995652i \(-0.529693\pi\)
−0.0931468 + 0.995652i \(0.529693\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11881.9 0.786725
\(612\) 0 0
\(613\) −9178.00 −0.604724 −0.302362 0.953193i \(-0.597775\pi\)
−0.302362 + 0.953193i \(0.597775\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15628.3 −1.01973 −0.509863 0.860255i \(-0.670304\pi\)
−0.509863 + 0.860255i \(0.670304\pi\)
\(618\) 0 0
\(619\) −13748.0 −0.892696 −0.446348 0.894859i \(-0.647276\pi\)
−0.446348 + 0.894859i \(0.647276\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 800.207 0.0514601
\(624\) 0 0
\(625\) −17891.0 −1.14502
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10220.8 −0.647903
\(630\) 0 0
\(631\) −8948.00 −0.564523 −0.282262 0.959337i \(-0.591085\pi\)
−0.282262 + 0.959337i \(0.591085\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1770.16 −0.110624
\(636\) 0 0
\(637\) −6909.00 −0.429740
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5745.21 0.354013 0.177006 0.984210i \(-0.443359\pi\)
0.177006 + 0.984210i \(0.443359\pi\)
\(642\) 0 0
\(643\) −3920.00 −0.240419 −0.120210 0.992749i \(-0.538357\pi\)
−0.120210 + 0.992749i \(0.538357\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2837.10 −0.172392 −0.0861962 0.996278i \(-0.527471\pi\)
−0.0861962 + 0.996278i \(0.527471\pi\)
\(648\) 0 0
\(649\) 5712.00 0.345479
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20673.8 −1.23894 −0.619469 0.785021i \(-0.712652\pi\)
−0.619469 + 0.785021i \(0.712652\pi\)
\(654\) 0 0
\(655\) 32046.0 1.91167
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3308.22 −0.195554 −0.0977768 0.995208i \(-0.531173\pi\)
−0.0977768 + 0.995208i \(0.531173\pi\)
\(660\) 0 0
\(661\) −32431.0 −1.90835 −0.954175 0.299248i \(-0.903264\pi\)
−0.954175 + 0.299248i \(0.903264\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18671.5 1.08880
\(666\) 0 0
\(667\) 6762.00 0.392542
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −39986.1 −2.30052
\(672\) 0 0
\(673\) 20399.0 1.16839 0.584193 0.811615i \(-0.301411\pi\)
0.584193 + 0.811615i \(0.301411\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 387.979 0.0220255 0.0110127 0.999939i \(-0.496494\pi\)
0.0110127 + 0.999939i \(0.496494\pi\)
\(678\) 0 0
\(679\) 39116.0 2.21080
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25669.0 1.43806 0.719031 0.694978i \(-0.244586\pi\)
0.719031 + 0.694978i \(0.244586\pi\)
\(684\) 0 0
\(685\) −23835.0 −1.32947
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18332.0 1.01364
\(690\) 0 0
\(691\) 5782.00 0.318318 0.159159 0.987253i \(-0.449122\pi\)
0.159159 + 0.987253i \(0.449122\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38022.0 2.07519
\(696\) 0 0
\(697\) −1764.00 −0.0958626
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13047.5 −0.702994 −0.351497 0.936189i \(-0.614327\pi\)
−0.351497 + 0.936189i \(0.614327\pi\)
\(702\) 0 0
\(703\) 19670.0 1.05529
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5334.72 0.283780
\(708\) 0 0
\(709\) 17879.0 0.947052 0.473526 0.880780i \(-0.342981\pi\)
0.473526 + 0.880780i \(0.342981\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2715.86 0.142650
\(714\) 0 0
\(715\) −34986.0 −1.82993
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10548.2 −0.547123 −0.273561 0.961855i \(-0.588202\pi\)
−0.273561 + 0.961855i \(0.588202\pi\)
\(720\) 0 0
\(721\) −19712.0 −1.01819
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6134.92 0.314269
\(726\) 0 0
\(727\) 32746.0 1.67054 0.835270 0.549841i \(-0.185312\pi\)
0.835270 + 0.549841i \(0.185312\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1818.65 0.0920182
\(732\) 0 0
\(733\) 27482.0 1.38482 0.692408 0.721506i \(-0.256550\pi\)
0.692408 + 0.721506i \(0.256550\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16135.8 0.806471
\(738\) 0 0
\(739\) −24716.0 −1.23030 −0.615151 0.788410i \(-0.710905\pi\)
−0.615151 + 0.788410i \(0.710905\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31849.0 −1.57258 −0.786288 0.617860i \(-0.788000\pi\)
−0.786288 + 0.617860i \(0.788000\pi\)
\(744\) 0 0
\(745\) 3801.00 0.186923
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −8230.71 −0.401527
\(750\) 0 0
\(751\) −7358.00 −0.357520 −0.178760 0.983893i \(-0.557209\pi\)
−0.178760 + 0.983893i \(0.557209\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25364.2 1.22264
\(756\) 0 0
\(757\) −30886.0 −1.48292 −0.741460 0.670997i \(-0.765866\pi\)
−0.741460 + 0.670997i \(0.765866\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23775.9 −1.13255 −0.566277 0.824215i \(-0.691617\pi\)
−0.566277 + 0.824215i \(0.691617\pi\)
\(762\) 0 0
\(763\) −4510.00 −0.213988
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4752.75 −0.223744
\(768\) 0 0
\(769\) −7609.00 −0.356811 −0.178405 0.983957i \(-0.557094\pi\)
−0.178405 + 0.983957i \(0.557094\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −10293.6 −0.478958 −0.239479 0.970902i \(-0.576977\pi\)
−0.239479 + 0.970902i \(0.576977\pi\)
\(774\) 0 0
\(775\) 2464.00 0.114206
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3394.82 0.156139
\(780\) 0 0
\(781\) −26316.0 −1.20571
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 16210.3 0.737031
\(786\) 0 0
\(787\) 40054.0 1.81419 0.907097 0.420921i \(-0.138293\pi\)
0.907097 + 0.420921i \(0.138293\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7963.97 −0.357985
\(792\) 0 0
\(793\) 33271.0 1.48990
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10099.6 −0.448865 −0.224433 0.974490i \(-0.572053\pi\)
−0.224433 + 0.974490i \(0.572053\pi\)
\(798\) 0 0
\(799\) 8820.00 0.390525
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30092.7 −1.32247
\(804\) 0 0
\(805\) 6468.00 0.283189
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9773.96 −0.424764 −0.212382 0.977187i \(-0.568122\pi\)
−0.212382 + 0.977187i \(0.568122\pi\)
\(810\) 0 0
\(811\) −35840.0 −1.55180 −0.775902 0.630854i \(-0.782705\pi\)
−0.775902 + 0.630854i \(0.782705\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21144.9 0.908801
\(816\) 0 0
\(817\) −3500.00 −0.149877
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8024.59 −0.341121 −0.170560 0.985347i \(-0.554558\pi\)
−0.170560 + 0.985347i \(0.554558\pi\)
\(822\) 0 0
\(823\) −14828.0 −0.628034 −0.314017 0.949417i \(-0.601675\pi\)
−0.314017 + 0.949417i \(0.601675\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38981.5 −1.63908 −0.819541 0.573021i \(-0.805772\pi\)
−0.819541 + 0.573021i \(0.805772\pi\)
\(828\) 0 0
\(829\) 27146.0 1.13730 0.568649 0.822580i \(-0.307466\pi\)
0.568649 + 0.822580i \(0.307466\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5128.60 −0.213320
\(834\) 0 0
\(835\) −33222.0 −1.37688
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24176.0 −0.994812 −0.497406 0.867518i \(-0.665714\pi\)
−0.497406 + 0.867518i \(0.665714\pi\)
\(840\) 0 0
\(841\) 53374.0 2.18845
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2473.37 0.100694
\(846\) 0 0
\(847\) 47014.0 1.90723
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6813.89 0.274474
\(852\) 0 0
\(853\) 19670.0 0.789552 0.394776 0.918777i \(-0.370822\pi\)
0.394776 + 0.918777i \(0.370822\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 133.368 0.00531594 0.00265797 0.999996i \(-0.499154\pi\)
0.00265797 + 0.999996i \(0.499154\pi\)
\(858\) 0 0
\(859\) −14756.0 −0.586110 −0.293055 0.956096i \(-0.594672\pi\)
−0.293055 + 0.956096i \(0.594672\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28215.1 −1.11292 −0.556462 0.830873i \(-0.687842\pi\)
−0.556462 + 0.830873i \(0.687842\pi\)
\(864\) 0 0
\(865\) −26607.0 −1.04586
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 30976.0 1.20919
\(870\) 0 0
\(871\) −13426.0 −0.522299
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −27473.8 −1.06147
\(876\) 0 0
\(877\) −31963.0 −1.23069 −0.615344 0.788258i \(-0.710983\pi\)
−0.615344 + 0.788258i \(0.710983\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31571.8 −1.20736 −0.603679 0.797228i \(-0.706299\pi\)
−0.603679 + 0.797228i \(0.706299\pi\)
\(882\) 0 0
\(883\) −34112.0 −1.30007 −0.650034 0.759905i \(-0.725245\pi\)
−0.650034 + 0.759905i \(0.725245\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36688.3 1.38881 0.694404 0.719585i \(-0.255668\pi\)
0.694404 + 0.719585i \(0.255668\pi\)
\(888\) 0 0
\(889\) −3212.00 −0.121178
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16974.1 −0.636077
\(894\) 0 0
\(895\) −29988.0 −1.11999
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 31232.3 1.15868
\(900\) 0 0
\(901\) 13608.0 0.503161
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −848.705 −0.0311734
\(906\) 0 0
\(907\) −36716.0 −1.34414 −0.672070 0.740488i \(-0.734595\pi\)
−0.672070 + 0.740488i \(0.734595\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 54535.4 1.98336 0.991678 0.128745i \(-0.0410949\pi\)
0.991678 + 0.128745i \(0.0410949\pi\)
\(912\) 0 0
\(913\) −22848.0 −0.828213
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 58148.4 2.09403
\(918\) 0 0
\(919\) 4318.00 0.154992 0.0774960 0.996993i \(-0.475307\pi\)
0.0774960 + 0.996993i \(0.475307\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 21896.6 0.780861
\(924\) 0 0
\(925\) 6182.00 0.219744
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −33402.6 −1.17966 −0.589830 0.807528i \(-0.700805\pi\)
−0.589830 + 0.807528i \(0.700805\pi\)
\(930\) 0 0
\(931\) 9870.00 0.347450
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −25970.4 −0.908366
\(936\) 0 0
\(937\) −13489.0 −0.470295 −0.235148 0.971960i \(-0.575557\pi\)
−0.235148 + 0.971960i \(0.575557\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20599.3 0.713621 0.356810 0.934177i \(-0.383864\pi\)
0.356810 + 0.934177i \(0.383864\pi\)
\(942\) 0 0
\(943\) 1176.00 0.0406106
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15543.4 0.533362 0.266681 0.963785i \(-0.414073\pi\)
0.266681 + 0.963785i \(0.414073\pi\)
\(948\) 0 0
\(949\) 25039.0 0.856481
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2582.49 −0.0877807 −0.0438903 0.999036i \(-0.513975\pi\)
−0.0438903 + 0.999036i \(0.513975\pi\)
\(954\) 0 0
\(955\) 25158.0 0.852454
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −43249.3 −1.45630
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1467.05 −0.0489388
\(966\) 0 0
\(967\) 41494.0 1.37989 0.689947 0.723860i \(-0.257634\pi\)
0.689947 + 0.723860i \(0.257634\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2837.10 0.0937661 0.0468830 0.998900i \(-0.485071\pi\)
0.0468830 + 0.998900i \(0.485071\pi\)
\(972\) 0 0
\(973\) 68992.0 2.27316
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47479.0 1.55475 0.777373 0.629040i \(-0.216552\pi\)
0.777373 + 0.629040i \(0.216552\pi\)
\(978\) 0 0
\(979\) 2142.00 0.0699271
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17798.6 0.577503 0.288752 0.957404i \(-0.406760\pi\)
0.288752 + 0.957404i \(0.406760\pi\)
\(984\) 0 0
\(985\) −35847.0 −1.15957
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1212.44 −0.0389820
\(990\) 0 0
\(991\) 21706.0 0.695776 0.347888 0.937536i \(-0.386899\pi\)
0.347888 + 0.937536i \(0.386899\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 51261.8 1.63327
\(996\) 0 0
\(997\) −25375.0 −0.806052 −0.403026 0.915188i \(-0.632042\pi\)
−0.403026 + 0.915188i \(0.632042\pi\)
\(998\) 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 1296.4.a.o.1.2 2
3.2 odd 2 inner 1296.4.a.o.1.1 2
4.3 odd 2 81.4.a.c.1.1 2
12.11 even 2 81.4.a.c.1.2 yes 2
20.19 odd 2 2025.4.a.k.1.2 2
36.7 odd 6 81.4.c.f.28.2 4
36.11 even 6 81.4.c.f.28.1 4
36.23 even 6 81.4.c.f.55.1 4
36.31 odd 6 81.4.c.f.55.2 4
60.59 even 2 2025.4.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.4.a.c.1.1 2 4.3 odd 2
81.4.a.c.1.2 yes 2 12.11 even 2
81.4.c.f.28.1 4 36.11 even 6
81.4.c.f.28.2 4 36.7 odd 6
81.4.c.f.55.1 4 36.23 even 6
81.4.c.f.55.2 4 36.31 odd 6
1296.4.a.o.1.1 2 3.2 odd 2 inner
1296.4.a.o.1.2 2 1.1 even 1 trivial
2025.4.a.k.1.1 2 60.59 even 2
2025.4.a.k.1.2 2 20.19 odd 2