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Properties

Label 2156.4.a.a.1.1
Level $2156$
Weight $4$
Character 2156.1
Self dual yes
Analytic conductor $127.208$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,4,Mod(1,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, 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("2156.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2156.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: \(127.208117972\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 308)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2156.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{3} +12.0000 q^{5} -11.0000 q^{9} +11.0000 q^{11} +10.0000 q^{13} -48.0000 q^{15} -24.0000 q^{17} +94.0000 q^{19} -180.000 q^{23} +19.0000 q^{25} +152.000 q^{27} +30.0000 q^{29} +94.0000 q^{31} -44.0000 q^{33} -214.000 q^{37} -40.0000 q^{39} +48.0000 q^{41} +8.00000 q^{43} -132.000 q^{45} -30.0000 q^{47} +96.0000 q^{51} +54.0000 q^{53} +132.000 q^{55} -376.000 q^{57} -360.000 q^{59} +178.000 q^{61} +120.000 q^{65} -292.000 q^{67} +720.000 q^{69} +312.000 q^{71} -728.000 q^{73} -76.0000 q^{75} -1288.00 q^{79} -311.000 q^{81} -66.0000 q^{83} -288.000 q^{85} -120.000 q^{87} +390.000 q^{89} -376.000 q^{93} +1128.00 q^{95} +1510.00 q^{97} -121.000 q^{99} +O(q^{100})\)

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
\(3\) −4.00000 −0.769800 −0.384900 − 0.922958i \(-0.625764\pi\)
−0.384900 + 0.922958i \(0.625764\pi\)
\(4\) 0 0
\(5\) 12.0000 1.07331 0.536656 − 0.843801i \(-0.319687\pi\)
0.536656 + 0.843801i \(0.319687\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 10.0000 0.213346 0.106673 − 0.994294i \(-0.465980\pi\)
0.106673 + 0.994294i \(0.465980\pi\)
\(14\) 0 0
\(15\) −48.0000 −0.826236
\(16\) 0 0
\(17\) −24.0000 −0.342403 −0.171202 − 0.985236i \(-0.554765\pi\)
−0.171202 + 0.985236i \(0.554765\pi\)
\(18\) 0 0
\(19\) 94.0000 1.13500 0.567502 − 0.823372i \(-0.307910\pi\)
0.567502 + 0.823372i \(0.307910\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −180.000 −1.63185 −0.815926 − 0.578156i \(-0.803772\pi\)
−0.815926 + 0.578156i \(0.803772\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 152.000 1.08342
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 − 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) 94.0000 0.544610 0.272305 − 0.962211i \(-0.412214\pi\)
0.272305 + 0.962211i \(0.412214\pi\)
\(32\) 0 0
\(33\) −44.0000 −0.232104
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −214.000 −0.950848 −0.475424 − 0.879757i \(-0.657705\pi\)
−0.475424 + 0.879757i \(0.657705\pi\)
\(38\) 0 0
\(39\) −40.0000 −0.164234
\(40\) 0 0
\(41\) 48.0000 0.182838 0.0914188 − 0.995813i \(-0.470860\pi\)
0.0914188 + 0.995813i \(0.470860\pi\)
\(42\) 0 0
\(43\) 8.00000 0.0283718 0.0141859 − 0.999899i \(-0.495484\pi\)
0.0141859 + 0.999899i \(0.495484\pi\)
\(44\) 0 0
\(45\) −132.000 −0.437276
\(46\) 0 0
\(47\) −30.0000 −0.0931053 −0.0465527 − 0.998916i \(-0.514824\pi\)
−0.0465527 + 0.998916i \(0.514824\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 96.0000 0.263582
\(52\) 0 0
\(53\) 54.0000 0.139952 0.0699761 − 0.997549i \(-0.477708\pi\)
0.0699761 + 0.997549i \(0.477708\pi\)
\(54\) 0 0
\(55\) 132.000 0.323616
\(56\) 0 0
\(57\) −376.000 −0.873727
\(58\) 0 0
\(59\) −360.000 −0.794373 −0.397187 − 0.917738i \(-0.630013\pi\)
−0.397187 + 0.917738i \(0.630013\pi\)
\(60\) 0 0
\(61\) 178.000 0.373616 0.186808 − 0.982396i \(-0.440186\pi\)
0.186808 + 0.982396i \(0.440186\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 120.000 0.228987
\(66\) 0 0
\(67\) −292.000 −0.532440 −0.266220 − 0.963912i \(-0.585775\pi\)
−0.266220 + 0.963912i \(0.585775\pi\)
\(68\) 0 0
\(69\) 720.000 1.25620
\(70\) 0 0
\(71\) 312.000 0.521515 0.260758 − 0.965404i \(-0.416028\pi\)
0.260758 + 0.965404i \(0.416028\pi\)
\(72\) 0 0
\(73\) −728.000 −1.16720 −0.583602 − 0.812040i \(-0.698357\pi\)
−0.583602 + 0.812040i \(0.698357\pi\)
\(74\) 0 0
\(75\) −76.0000 −0.117010
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1288.00 −1.83432 −0.917160 − 0.398519i \(-0.869524\pi\)
−0.917160 + 0.398519i \(0.869524\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) −66.0000 −0.0872824 −0.0436412 − 0.999047i \(-0.513896\pi\)
−0.0436412 + 0.999047i \(0.513896\pi\)
\(84\) 0 0
\(85\) −288.000 −0.367506
\(86\) 0 0
\(87\) −120.000 −0.147878
\(88\) 0 0
\(89\) 390.000 0.464493 0.232247 − 0.972657i \(-0.425392\pi\)
0.232247 + 0.972657i \(0.425392\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −376.000 −0.419241
\(94\) 0 0
\(95\) 1128.00 1.21821
\(96\) 0 0
\(97\) 1510.00 1.58059 0.790295 − 0.612726i \(-0.209927\pi\)
0.790295 + 0.612726i \(0.209927\pi\)
\(98\) 0 0
\(99\) −121.000 −0.122838
\(100\) 0 0
\(101\) −90.0000 −0.0886667 −0.0443333 − 0.999017i \(-0.514116\pi\)
−0.0443333 + 0.999017i \(0.514116\pi\)
\(102\) 0 0
\(103\) −302.000 −0.288902 −0.144451 − 0.989512i \(-0.546142\pi\)
−0.144451 + 0.989512i \(0.546142\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 240.000 0.216838 0.108419 − 0.994105i \(-0.465421\pi\)
0.108419 + 0.994105i \(0.465421\pi\)
\(108\) 0 0
\(109\) −166.000 −0.145871 −0.0729354 − 0.997337i \(-0.523237\pi\)
−0.0729354 + 0.997337i \(0.523237\pi\)
\(110\) 0 0
\(111\) 856.000 0.731963
\(112\) 0 0
\(113\) 1662.00 1.38361 0.691804 − 0.722085i \(-0.256816\pi\)
0.691804 + 0.722085i \(0.256816\pi\)
\(114\) 0 0
\(115\) −2160.00 −1.75149
\(116\) 0 0
\(117\) −110.000 −0.0869188
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −192.000 −0.140748
\(124\) 0 0
\(125\) −1272.00 −0.910169
\(126\) 0 0
\(127\) 1544.00 1.07880 0.539401 − 0.842049i \(-0.318651\pi\)
0.539401 + 0.842049i \(0.318651\pi\)
\(128\) 0 0
\(129\) −32.0000 −0.0218406
\(130\) 0 0
\(131\) −2358.00 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1824.00 1.16285
\(136\) 0 0
\(137\) 1206.00 0.752084 0.376042 − 0.926603i \(-0.377285\pi\)
0.376042 + 0.926603i \(0.377285\pi\)
\(138\) 0 0
\(139\) −2630.00 −1.60485 −0.802423 − 0.596755i \(-0.796456\pi\)
−0.802423 + 0.596755i \(0.796456\pi\)
\(140\) 0 0
\(141\) 120.000 0.0716725
\(142\) 0 0
\(143\) 110.000 0.0643263
\(144\) 0 0
\(145\) 360.000 0.206182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −126.000 −0.0692773 −0.0346387 − 0.999400i \(-0.511028\pi\)
−0.0346387 + 0.999400i \(0.511028\pi\)
\(150\) 0 0
\(151\) −1456.00 −0.784686 −0.392343 − 0.919819i \(-0.628335\pi\)
−0.392343 + 0.919819i \(0.628335\pi\)
\(152\) 0 0
\(153\) 264.000 0.139498
\(154\) 0 0
\(155\) 1128.00 0.584536
\(156\) 0 0
\(157\) −992.000 −0.504269 −0.252134 − 0.967692i \(-0.581133\pi\)
−0.252134 + 0.967692i \(0.581133\pi\)
\(158\) 0 0
\(159\) −216.000 −0.107735
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3188.00 1.53192 0.765961 − 0.642887i \(-0.222263\pi\)
0.765961 + 0.642887i \(0.222263\pi\)
\(164\) 0 0
\(165\) −528.000 −0.249120
\(166\) 0 0
\(167\) 252.000 0.116769 0.0583843 − 0.998294i \(-0.481405\pi\)
0.0583843 + 0.998294i \(0.481405\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) −1034.00 −0.462409
\(172\) 0 0
\(173\) 3882.00 1.70603 0.853014 − 0.521887i \(-0.174772\pi\)
0.853014 + 0.521887i \(0.174772\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1440.00 0.611509
\(178\) 0 0
\(179\) −3300.00 −1.37795 −0.688976 − 0.724784i \(-0.741940\pi\)
−0.688976 + 0.724784i \(0.741940\pi\)
\(180\) 0 0
\(181\) 124.000 0.0509218 0.0254609 − 0.999676i \(-0.491895\pi\)
0.0254609 + 0.999676i \(0.491895\pi\)
\(182\) 0 0
\(183\) −712.000 −0.287610
\(184\) 0 0
\(185\) −2568.00 −1.02056
\(186\) 0 0
\(187\) −264.000 −0.103238
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1860.00 0.704633 0.352316 − 0.935881i \(-0.385394\pi\)
0.352316 + 0.935881i \(0.385394\pi\)
\(192\) 0 0
\(193\) −1318.00 −0.491563 −0.245782 − 0.969325i \(-0.579045\pi\)
−0.245782 + 0.969325i \(0.579045\pi\)
\(194\) 0 0
\(195\) −480.000 −0.176274
\(196\) 0 0
\(197\) −1266.00 −0.457862 −0.228931 − 0.973443i \(-0.573523\pi\)
−0.228931 + 0.973443i \(0.573523\pi\)
\(198\) 0 0
\(199\) −830.000 −0.295664 −0.147832 − 0.989012i \(-0.547229\pi\)
−0.147832 + 0.989012i \(0.547229\pi\)
\(200\) 0 0
\(201\) 1168.00 0.409872
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 576.000 0.196242
\(206\) 0 0
\(207\) 1980.00 0.664829
\(208\) 0 0
\(209\) 1034.00 0.342217
\(210\) 0 0
\(211\) −4408.00 −1.43820 −0.719098 − 0.694909i \(-0.755445\pi\)
−0.719098 + 0.694909i \(0.755445\pi\)
\(212\) 0 0
\(213\) −1248.00 −0.401463
\(214\) 0 0
\(215\) 96.0000 0.0304518
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2912.00 0.898515
\(220\) 0 0
\(221\) −240.000 −0.0730504
\(222\) 0 0
\(223\) 4246.00 1.27504 0.637518 − 0.770435i \(-0.279961\pi\)
0.637518 + 0.770435i \(0.279961\pi\)
\(224\) 0 0
\(225\) −209.000 −0.0619259
\(226\) 0 0
\(227\) −1710.00 −0.499985 −0.249993 − 0.968248i \(-0.580428\pi\)
−0.249993 + 0.968248i \(0.580428\pi\)
\(228\) 0 0
\(229\) 2044.00 0.589831 0.294916 − 0.955523i \(-0.404708\pi\)
0.294916 + 0.955523i \(0.404708\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 54.0000 0.0151831 0.00759154 − 0.999971i \(-0.497584\pi\)
0.00759154 + 0.999971i \(0.497584\pi\)
\(234\) 0 0
\(235\) −360.000 −0.0999311
\(236\) 0 0
\(237\) 5152.00 1.41206
\(238\) 0 0
\(239\) −4416.00 −1.19518 −0.597588 − 0.801803i \(-0.703874\pi\)
−0.597588 + 0.801803i \(0.703874\pi\)
\(240\) 0 0
\(241\) 808.000 0.215966 0.107983 − 0.994153i \(-0.465561\pi\)
0.107983 + 0.994153i \(0.465561\pi\)
\(242\) 0 0
\(243\) −2860.00 −0.755017
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 940.000 0.242149
\(248\) 0 0
\(249\) 264.000 0.0671900
\(250\) 0 0
\(251\) −5400.00 −1.35795 −0.678974 − 0.734162i \(-0.737575\pi\)
−0.678974 + 0.734162i \(0.737575\pi\)
\(252\) 0 0
\(253\) −1980.00 −0.492022
\(254\) 0 0
\(255\) 1152.00 0.282906
\(256\) 0 0
\(257\) −3882.00 −0.942228 −0.471114 − 0.882072i \(-0.656148\pi\)
−0.471114 + 0.882072i \(0.656148\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −330.000 −0.0782624
\(262\) 0 0
\(263\) −7536.00 −1.76688 −0.883440 − 0.468543i \(-0.844779\pi\)
−0.883440 + 0.468543i \(0.844779\pi\)
\(264\) 0 0
\(265\) 648.000 0.150213
\(266\) 0 0
\(267\) −1560.00 −0.357567
\(268\) 0 0
\(269\) −1596.00 −0.361747 −0.180873 − 0.983506i \(-0.557892\pi\)
−0.180873 + 0.983506i \(0.557892\pi\)
\(270\) 0 0
\(271\) −20.0000 −0.00448308 −0.00224154 − 0.999997i \(-0.500714\pi\)
−0.00224154 + 0.999997i \(0.500714\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 209.000 0.0458297
\(276\) 0 0
\(277\) −5374.00 −1.16568 −0.582838 − 0.812588i \(-0.698058\pi\)
−0.582838 + 0.812588i \(0.698058\pi\)
\(278\) 0 0
\(279\) −1034.00 −0.221878
\(280\) 0 0
\(281\) −786.000 −0.166864 −0.0834321 − 0.996513i \(-0.526588\pi\)
−0.0834321 + 0.996513i \(0.526588\pi\)
\(282\) 0 0
\(283\) −3878.00 −0.814570 −0.407285 − 0.913301i \(-0.633524\pi\)
−0.407285 + 0.913301i \(0.633524\pi\)
\(284\) 0 0
\(285\) −4512.00 −0.937782
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4337.00 −0.882760
\(290\) 0 0
\(291\) −6040.00 −1.21674
\(292\) 0 0
\(293\) 8226.00 1.64016 0.820082 − 0.572246i \(-0.193928\pi\)
0.820082 + 0.572246i \(0.193928\pi\)
\(294\) 0 0
\(295\) −4320.00 −0.852611
\(296\) 0 0
\(297\) 1672.00 0.326664
\(298\) 0 0
\(299\) −1800.00 −0.348149
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 360.000 0.0682556
\(304\) 0 0
\(305\) 2136.00 0.401007
\(306\) 0 0
\(307\) −3926.00 −0.729865 −0.364933 − 0.931034i \(-0.618908\pi\)
−0.364933 + 0.931034i \(0.618908\pi\)
\(308\) 0 0
\(309\) 1208.00 0.222397
\(310\) 0 0
\(311\) −618.000 −0.112680 −0.0563401 − 0.998412i \(-0.517943\pi\)
−0.0563401 + 0.998412i \(0.517943\pi\)
\(312\) 0 0
\(313\) −686.000 −0.123882 −0.0619409 − 0.998080i \(-0.519729\pi\)
−0.0619409 + 0.998080i \(0.519729\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2094.00 0.371012 0.185506 − 0.982643i \(-0.440608\pi\)
0.185506 + 0.982643i \(0.440608\pi\)
\(318\) 0 0
\(319\) 330.000 0.0579199
\(320\) 0 0
\(321\) −960.000 −0.166922
\(322\) 0 0
\(323\) −2256.00 −0.388629
\(324\) 0 0
\(325\) 190.000 0.0324286
\(326\) 0 0
\(327\) 664.000 0.112291
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2444.00 0.405844 0.202922 − 0.979195i \(-0.434956\pi\)
0.202922 + 0.979195i \(0.434956\pi\)
\(332\) 0 0
\(333\) 2354.00 0.387383
\(334\) 0 0
\(335\) −3504.00 −0.571475
\(336\) 0 0
\(337\) −7378.00 −1.19260 −0.596299 − 0.802763i \(-0.703363\pi\)
−0.596299 + 0.802763i \(0.703363\pi\)
\(338\) 0 0
\(339\) −6648.00 −1.06510
\(340\) 0 0
\(341\) 1034.00 0.164206
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 8640.00 1.34830
\(346\) 0 0
\(347\) −6840.00 −1.05819 −0.529093 − 0.848564i \(-0.677468\pi\)
−0.529093 + 0.848564i \(0.677468\pi\)
\(348\) 0 0
\(349\) 5614.00 0.861062 0.430531 − 0.902576i \(-0.358326\pi\)
0.430531 + 0.902576i \(0.358326\pi\)
\(350\) 0 0
\(351\) 1520.00 0.231144
\(352\) 0 0
\(353\) −9066.00 −1.36695 −0.683477 − 0.729972i \(-0.739533\pi\)
−0.683477 + 0.729972i \(0.739533\pi\)
\(354\) 0 0
\(355\) 3744.00 0.559749
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5256.00 −0.772705 −0.386352 − 0.922351i \(-0.626265\pi\)
−0.386352 + 0.922351i \(0.626265\pi\)
\(360\) 0 0
\(361\) 1977.00 0.288234
\(362\) 0 0
\(363\) −484.000 −0.0699819
\(364\) 0 0
\(365\) −8736.00 −1.25278
\(366\) 0 0
\(367\) 2350.00 0.334248 0.167124 − 0.985936i \(-0.446552\pi\)
0.167124 + 0.985936i \(0.446552\pi\)
\(368\) 0 0
\(369\) −528.000 −0.0744894
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4306.00 −0.597738 −0.298869 − 0.954294i \(-0.596609\pi\)
−0.298869 + 0.954294i \(0.596609\pi\)
\(374\) 0 0
\(375\) 5088.00 0.700649
\(376\) 0 0
\(377\) 300.000 0.0409835
\(378\) 0 0
\(379\) −14380.0 −1.94895 −0.974474 − 0.224502i \(-0.927924\pi\)
−0.974474 + 0.224502i \(0.927924\pi\)
\(380\) 0 0
\(381\) −6176.00 −0.830462
\(382\) 0 0
\(383\) −10746.0 −1.43367 −0.716834 − 0.697244i \(-0.754410\pi\)
−0.716834 + 0.697244i \(0.754410\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −88.0000 −0.0115589
\(388\) 0 0
\(389\) −6870.00 −0.895431 −0.447716 − 0.894176i \(-0.647762\pi\)
−0.447716 + 0.894176i \(0.647762\pi\)
\(390\) 0 0
\(391\) 4320.00 0.558751
\(392\) 0 0
\(393\) 9432.00 1.21064
\(394\) 0 0
\(395\) −15456.0 −1.96880
\(396\) 0 0
\(397\) −15728.0 −1.98833 −0.994163 − 0.107885i \(-0.965592\pi\)
−0.994163 + 0.107885i \(0.965592\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12846.0 1.59975 0.799874 − 0.600168i \(-0.204900\pi\)
0.799874 + 0.600168i \(0.204900\pi\)
\(402\) 0 0
\(403\) 940.000 0.116190
\(404\) 0 0
\(405\) −3732.00 −0.457888
\(406\) 0 0
\(407\) −2354.00 −0.286692
\(408\) 0 0
\(409\) 5740.00 0.693948 0.346974 − 0.937875i \(-0.387209\pi\)
0.346974 + 0.937875i \(0.387209\pi\)
\(410\) 0 0
\(411\) −4824.00 −0.578955
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −792.000 −0.0936813
\(416\) 0 0
\(417\) 10520.0 1.23541
\(418\) 0 0
\(419\) −2292.00 −0.267235 −0.133618 − 0.991033i \(-0.542659\pi\)
−0.133618 + 0.991033i \(0.542659\pi\)
\(420\) 0 0
\(421\) −10834.0 −1.25420 −0.627098 − 0.778940i \(-0.715758\pi\)
−0.627098 + 0.778940i \(0.715758\pi\)
\(422\) 0 0
\(423\) 330.000 0.0379318
\(424\) 0 0
\(425\) −456.000 −0.0520453
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −440.000 −0.0495184
\(430\) 0 0
\(431\) −11400.0 −1.27406 −0.637029 − 0.770840i \(-0.719837\pi\)
−0.637029 + 0.770840i \(0.719837\pi\)
\(432\) 0 0
\(433\) 15958.0 1.77111 0.885557 − 0.464530i \(-0.153777\pi\)
0.885557 + 0.464530i \(0.153777\pi\)
\(434\) 0 0
\(435\) −1440.00 −0.158719
\(436\) 0 0
\(437\) −16920.0 −1.85216
\(438\) 0 0
\(439\) 268.000 0.0291365 0.0145683 − 0.999894i \(-0.495363\pi\)
0.0145683 + 0.999894i \(0.495363\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15324.0 −1.64349 −0.821744 − 0.569857i \(-0.806999\pi\)
−0.821744 + 0.569857i \(0.806999\pi\)
\(444\) 0 0
\(445\) 4680.00 0.498547
\(446\) 0 0
\(447\) 504.000 0.0533297
\(448\) 0 0
\(449\) 2238.00 0.235229 0.117614 − 0.993059i \(-0.462475\pi\)
0.117614 + 0.993059i \(0.462475\pi\)
\(450\) 0 0
\(451\) 528.000 0.0551276
\(452\) 0 0
\(453\) 5824.00 0.604052
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9046.00 −0.925939 −0.462969 − 0.886374i \(-0.653216\pi\)
−0.462969 + 0.886374i \(0.653216\pi\)
\(458\) 0 0
\(459\) −3648.00 −0.370967
\(460\) 0 0
\(461\) −11898.0 −1.20205 −0.601025 − 0.799230i \(-0.705241\pi\)
−0.601025 + 0.799230i \(0.705241\pi\)
\(462\) 0 0
\(463\) 6512.00 0.653646 0.326823 − 0.945085i \(-0.394022\pi\)
0.326823 + 0.945085i \(0.394022\pi\)
\(464\) 0 0
\(465\) −4512.00 −0.449976
\(466\) 0 0
\(467\) −11676.0 −1.15696 −0.578481 − 0.815696i \(-0.696354\pi\)
−0.578481 + 0.815696i \(0.696354\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3968.00 0.388186
\(472\) 0 0
\(473\) 88.0000 0.00855443
\(474\) 0 0
\(475\) 1786.00 0.172521
\(476\) 0 0
\(477\) −594.000 −0.0570176
\(478\) 0 0
\(479\) −16920.0 −1.61398 −0.806988 − 0.590568i \(-0.798904\pi\)
−0.806988 + 0.590568i \(0.798904\pi\)
\(480\) 0 0
\(481\) −2140.00 −0.202860
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18120.0 1.69647
\(486\) 0 0
\(487\) −5188.00 −0.482732 −0.241366 − 0.970434i \(-0.577596\pi\)
−0.241366 + 0.970434i \(0.577596\pi\)
\(488\) 0 0
\(489\) −12752.0 −1.17927
\(490\) 0 0
\(491\) 21036.0 1.93349 0.966743 − 0.255751i \(-0.0823226\pi\)
0.966743 + 0.255751i \(0.0823226\pi\)
\(492\) 0 0
\(493\) −720.000 −0.0657752
\(494\) 0 0
\(495\) −1452.00 −0.131844
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2284.00 −0.204902 −0.102451 − 0.994738i \(-0.532668\pi\)
−0.102451 + 0.994738i \(0.532668\pi\)
\(500\) 0 0
\(501\) −1008.00 −0.0898885
\(502\) 0 0
\(503\) 11004.0 0.975436 0.487718 − 0.873001i \(-0.337830\pi\)
0.487718 + 0.873001i \(0.337830\pi\)
\(504\) 0 0
\(505\) −1080.00 −0.0951671
\(506\) 0 0
\(507\) 8388.00 0.734762
\(508\) 0 0
\(509\) 5244.00 0.456653 0.228326 − 0.973585i \(-0.426675\pi\)
0.228326 + 0.973585i \(0.426675\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 14288.0 1.22969
\(514\) 0 0
\(515\) −3624.00 −0.310083
\(516\) 0 0
\(517\) −330.000 −0.0280723
\(518\) 0 0
\(519\) −15528.0 −1.31330
\(520\) 0 0
\(521\) 1902.00 0.159939 0.0799694 − 0.996797i \(-0.474518\pi\)
0.0799694 + 0.996797i \(0.474518\pi\)
\(522\) 0 0
\(523\) 8194.00 0.685083 0.342542 − 0.939503i \(-0.388712\pi\)
0.342542 + 0.939503i \(0.388712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2256.00 −0.186476
\(528\) 0 0
\(529\) 20233.0 1.66294
\(530\) 0 0
\(531\) 3960.00 0.323633
\(532\) 0 0
\(533\) 480.000 0.0390077
\(534\) 0 0
\(535\) 2880.00 0.232735
\(536\) 0 0
\(537\) 13200.0 1.06075
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10406.0 0.826967 0.413483 − 0.910512i \(-0.364312\pi\)
0.413483 + 0.910512i \(0.364312\pi\)
\(542\) 0 0
\(543\) −496.000 −0.0391996
\(544\) 0 0
\(545\) −1992.00 −0.156565
\(546\) 0 0
\(547\) 20900.0 1.63367 0.816837 − 0.576869i \(-0.195726\pi\)
0.816837 + 0.576869i \(0.195726\pi\)
\(548\) 0 0
\(549\) −1958.00 −0.152214
\(550\) 0 0
\(551\) 2820.00 0.218033
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10272.0 0.785625
\(556\) 0 0
\(557\) 10038.0 0.763597 0.381799 − 0.924246i \(-0.375305\pi\)
0.381799 + 0.924246i \(0.375305\pi\)
\(558\) 0 0
\(559\) 80.0000 0.00605302
\(560\) 0 0
\(561\) 1056.00 0.0794730
\(562\) 0 0
\(563\) 8154.00 0.610391 0.305195 − 0.952290i \(-0.401278\pi\)
0.305195 + 0.952290i \(0.401278\pi\)
\(564\) 0 0
\(565\) 19944.0 1.48504
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2730.00 −0.201138 −0.100569 − 0.994930i \(-0.532066\pi\)
−0.100569 + 0.994930i \(0.532066\pi\)
\(570\) 0 0
\(571\) 3692.00 0.270587 0.135294 − 0.990806i \(-0.456802\pi\)
0.135294 + 0.990806i \(0.456802\pi\)
\(572\) 0 0
\(573\) −7440.00 −0.542427
\(574\) 0 0
\(575\) −3420.00 −0.248041
\(576\) 0 0
\(577\) 3946.00 0.284704 0.142352 − 0.989816i \(-0.454534\pi\)
0.142352 + 0.989816i \(0.454534\pi\)
\(578\) 0 0
\(579\) 5272.00 0.378406
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 594.000 0.0421972
\(584\) 0 0
\(585\) −1320.00 −0.0932911
\(586\) 0 0
\(587\) 3312.00 0.232881 0.116440 − 0.993198i \(-0.462852\pi\)
0.116440 + 0.993198i \(0.462852\pi\)
\(588\) 0 0
\(589\) 8836.00 0.618134
\(590\) 0 0
\(591\) 5064.00 0.352462
\(592\) 0 0
\(593\) 10920.0 0.756207 0.378103 − 0.925763i \(-0.376576\pi\)
0.378103 + 0.925763i \(0.376576\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3320.00 0.227602
\(598\) 0 0
\(599\) −25800.0 −1.75987 −0.879933 − 0.475098i \(-0.842413\pi\)
−0.879933 + 0.475098i \(0.842413\pi\)
\(600\) 0 0
\(601\) 24700.0 1.67643 0.838215 − 0.545340i \(-0.183600\pi\)
0.838215 + 0.545340i \(0.183600\pi\)
\(602\) 0 0
\(603\) 3212.00 0.216920
\(604\) 0 0
\(605\) 1452.00 0.0975739
\(606\) 0 0
\(607\) 9016.00 0.602880 0.301440 − 0.953485i \(-0.402533\pi\)
0.301440 + 0.953485i \(0.402533\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −300.000 −0.0198637
\(612\) 0 0
\(613\) −1618.00 −0.106608 −0.0533038 − 0.998578i \(-0.516975\pi\)
−0.0533038 + 0.998578i \(0.516975\pi\)
\(614\) 0 0
\(615\) −2304.00 −0.151067
\(616\) 0 0
\(617\) −11310.0 −0.737963 −0.368982 − 0.929437i \(-0.620294\pi\)
−0.368982 + 0.929437i \(0.620294\pi\)
\(618\) 0 0
\(619\) 24700.0 1.60384 0.801920 − 0.597432i \(-0.203812\pi\)
0.801920 + 0.597432i \(0.203812\pi\)
\(620\) 0 0
\(621\) −27360.0 −1.76799
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) −4136.00 −0.263438
\(628\) 0 0
\(629\) 5136.00 0.325573
\(630\) 0 0
\(631\) −2500.00 −0.157723 −0.0788617 − 0.996886i \(-0.525129\pi\)
−0.0788617 + 0.996886i \(0.525129\pi\)
\(632\) 0 0
\(633\) 17632.0 1.10712
\(634\) 0 0
\(635\) 18528.0 1.15789
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3432.00 −0.212469
\(640\) 0 0
\(641\) 3450.00 0.212585 0.106292 − 0.994335i \(-0.466102\pi\)
0.106292 + 0.994335i \(0.466102\pi\)
\(642\) 0 0
\(643\) −18416.0 −1.12948 −0.564740 − 0.825269i \(-0.691024\pi\)
−0.564740 + 0.825269i \(0.691024\pi\)
\(644\) 0 0
\(645\) −384.000 −0.0234418
\(646\) 0 0
\(647\) 7014.00 0.426196 0.213098 − 0.977031i \(-0.431645\pi\)
0.213098 + 0.977031i \(0.431645\pi\)
\(648\) 0 0
\(649\) −3960.00 −0.239512
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5070.00 −0.303835 −0.151918 − 0.988393i \(-0.548545\pi\)
−0.151918 + 0.988393i \(0.548545\pi\)
\(654\) 0 0
\(655\) −28296.0 −1.68796
\(656\) 0 0
\(657\) 8008.00 0.475528
\(658\) 0 0
\(659\) 30600.0 1.80881 0.904406 − 0.426673i \(-0.140315\pi\)
0.904406 + 0.426673i \(0.140315\pi\)
\(660\) 0 0
\(661\) −22364.0 −1.31597 −0.657987 − 0.753029i \(-0.728592\pi\)
−0.657987 + 0.753029i \(0.728592\pi\)
\(662\) 0 0
\(663\) 960.000 0.0562343
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5400.00 −0.313477
\(668\) 0 0
\(669\) −16984.0 −0.981524
\(670\) 0 0
\(671\) 1958.00 0.112649
\(672\) 0 0
\(673\) −31522.0 −1.80547 −0.902737 − 0.430193i \(-0.858445\pi\)
−0.902737 + 0.430193i \(0.858445\pi\)
\(674\) 0 0
\(675\) 2888.00 0.164680
\(676\) 0 0
\(677\) 30210.0 1.71501 0.857507 − 0.514472i \(-0.172012\pi\)
0.857507 + 0.514472i \(0.172012\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6840.00 0.384889
\(682\) 0 0
\(683\) 12492.0 0.699843 0.349922 − 0.936779i \(-0.386208\pi\)
0.349922 + 0.936779i \(0.386208\pi\)
\(684\) 0 0
\(685\) 14472.0 0.807221
\(686\) 0 0
\(687\) −8176.00 −0.454052
\(688\) 0 0
\(689\) 540.000 0.0298583
\(690\) 0 0
\(691\) −21608.0 −1.18959 −0.594795 − 0.803877i \(-0.702767\pi\)
−0.594795 + 0.803877i \(0.702767\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −31560.0 −1.72250
\(696\) 0 0
\(697\) −1152.00 −0.0626042
\(698\) 0 0
\(699\) −216.000 −0.0116879
\(700\) 0 0
\(701\) 28266.0 1.52296 0.761478 − 0.648191i \(-0.224474\pi\)
0.761478 + 0.648191i \(0.224474\pi\)
\(702\) 0 0
\(703\) −20116.0 −1.07922
\(704\) 0 0
\(705\) 1440.00 0.0769270
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5978.00 0.316655 0.158328 − 0.987387i \(-0.449390\pi\)
0.158328 + 0.987387i \(0.449390\pi\)
\(710\) 0 0
\(711\) 14168.0 0.747316
\(712\) 0 0
\(713\) −16920.0 −0.888722
\(714\) 0 0
\(715\) 1320.00 0.0690422
\(716\) 0 0
\(717\) 17664.0 0.920048
\(718\) 0 0
\(719\) −6330.00 −0.328330 −0.164165 − 0.986433i \(-0.552493\pi\)
−0.164165 + 0.986433i \(0.552493\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3232.00 −0.166251
\(724\) 0 0
\(725\) 570.000 0.0291990
\(726\) 0 0
\(727\) 8710.00 0.444341 0.222171 − 0.975008i \(-0.428686\pi\)
0.222171 + 0.975008i \(0.428686\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) −192.000 −0.00971461
\(732\) 0 0
\(733\) 2506.00 0.126277 0.0631386 − 0.998005i \(-0.479889\pi\)
0.0631386 + 0.998005i \(0.479889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3212.00 −0.160537
\(738\) 0 0
\(739\) 17060.0 0.849205 0.424602 − 0.905380i \(-0.360414\pi\)
0.424602 + 0.905380i \(0.360414\pi\)
\(740\) 0 0
\(741\) −3760.00 −0.186406
\(742\) 0 0
\(743\) 22392.0 1.10563 0.552815 − 0.833304i \(-0.313554\pi\)
0.552815 + 0.833304i \(0.313554\pi\)
\(744\) 0 0
\(745\) −1512.00 −0.0743562
\(746\) 0 0
\(747\) 726.000 0.0355595
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16828.0 −0.817660 −0.408830 − 0.912611i \(-0.634063\pi\)
−0.408830 + 0.912611i \(0.634063\pi\)
\(752\) 0 0
\(753\) 21600.0 1.04535
\(754\) 0 0
\(755\) −17472.0 −0.842213
\(756\) 0 0
\(757\) 10514.0 0.504806 0.252403 − 0.967622i \(-0.418779\pi\)
0.252403 + 0.967622i \(0.418779\pi\)
\(758\) 0 0
\(759\) 7920.00 0.378759
\(760\) 0 0
\(761\) −12084.0 −0.575617 −0.287809 − 0.957688i \(-0.592927\pi\)
−0.287809 + 0.957688i \(0.592927\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3168.00 0.149725
\(766\) 0 0
\(767\) −3600.00 −0.169476
\(768\) 0 0
\(769\) −1940.00 −0.0909729 −0.0454865 − 0.998965i \(-0.514484\pi\)
−0.0454865 + 0.998965i \(0.514484\pi\)
\(770\) 0 0
\(771\) 15528.0 0.725327
\(772\) 0 0
\(773\) 10800.0 0.502521 0.251261 − 0.967919i \(-0.419155\pi\)
0.251261 + 0.967919i \(0.419155\pi\)
\(774\) 0 0
\(775\) 1786.00 0.0827807
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4512.00 0.207521
\(780\) 0 0
\(781\) 3432.00 0.157243
\(782\) 0 0
\(783\) 4560.00 0.208124
\(784\) 0 0
\(785\) −11904.0 −0.541238
\(786\) 0 0
\(787\) −11378.0 −0.515352 −0.257676 − 0.966231i \(-0.582957\pi\)
−0.257676 + 0.966231i \(0.582957\pi\)
\(788\) 0 0
\(789\) 30144.0 1.36015
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1780.00 0.0797095
\(794\) 0 0
\(795\) −2592.00 −0.115634
\(796\) 0 0
\(797\) −15528.0 −0.690125 −0.345063 − 0.938580i \(-0.612142\pi\)
−0.345063 + 0.938580i \(0.612142\pi\)
\(798\) 0 0
\(799\) 720.000 0.0318796
\(800\) 0 0
\(801\) −4290.00 −0.189238
\(802\) 0 0
\(803\) −8008.00 −0.351926
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6384.00 0.278473
\(808\) 0 0
\(809\) 5082.00 0.220857 0.110429 − 0.993884i \(-0.464778\pi\)
0.110429 + 0.993884i \(0.464778\pi\)
\(810\) 0 0
\(811\) −24086.0 −1.04288 −0.521439 − 0.853289i \(-0.674605\pi\)
−0.521439 + 0.853289i \(0.674605\pi\)
\(812\) 0 0
\(813\) 80.0000 0.00345107
\(814\) 0 0
\(815\) 38256.0 1.64423
\(816\) 0 0
\(817\) 752.000 0.0322021
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6210.00 0.263984 0.131992 − 0.991251i \(-0.457863\pi\)
0.131992 + 0.991251i \(0.457863\pi\)
\(822\) 0 0
\(823\) 20612.0 0.873012 0.436506 − 0.899701i \(-0.356216\pi\)
0.436506 + 0.899701i \(0.356216\pi\)
\(824\) 0 0
\(825\) −836.000 −0.0352797
\(826\) 0 0
\(827\) −3948.00 −0.166004 −0.0830021 − 0.996549i \(-0.526451\pi\)
−0.0830021 + 0.996549i \(0.526451\pi\)
\(828\) 0 0
\(829\) 5692.00 0.238470 0.119235 − 0.992866i \(-0.461956\pi\)
0.119235 + 0.992866i \(0.461956\pi\)
\(830\) 0 0
\(831\) 21496.0 0.897338
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3024.00 0.125329
\(836\) 0 0
\(837\) 14288.0 0.590042
\(838\) 0 0
\(839\) 29070.0 1.19620 0.598098 − 0.801423i \(-0.295923\pi\)
0.598098 + 0.801423i \(0.295923\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 0 0
\(843\) 3144.00 0.128452
\(844\) 0 0
\(845\) −25164.0 −1.02446
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15512.0 0.627056
\(850\) 0 0
\(851\) 38520.0 1.55164
\(852\) 0 0
\(853\) −20582.0 −0.826160 −0.413080 − 0.910695i \(-0.635547\pi\)
−0.413080 + 0.910695i \(0.635547\pi\)
\(854\) 0 0
\(855\) −12408.0 −0.496310
\(856\) 0 0
\(857\) 36156.0 1.44115 0.720575 − 0.693377i \(-0.243878\pi\)
0.720575 + 0.693377i \(0.243878\pi\)
\(858\) 0 0
\(859\) −19784.0 −0.785822 −0.392911 − 0.919576i \(-0.628532\pi\)
−0.392911 + 0.919576i \(0.628532\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34068.0 −1.34379 −0.671894 − 0.740648i \(-0.734519\pi\)
−0.671894 + 0.740648i \(0.734519\pi\)
\(864\) 0 0
\(865\) 46584.0 1.83110
\(866\) 0 0
\(867\) 17348.0 0.679549
\(868\) 0 0
\(869\) −14168.0 −0.553068
\(870\) 0 0
\(871\) −2920.00 −0.113594
\(872\) 0 0
\(873\) −16610.0 −0.643944
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19510.0 −0.751204 −0.375602 − 0.926781i \(-0.622564\pi\)
−0.375602 + 0.926781i \(0.622564\pi\)
\(878\) 0 0
\(879\) −32904.0 −1.26260
\(880\) 0 0
\(881\) 24474.0 0.935925 0.467963 − 0.883748i \(-0.344988\pi\)
0.467963 + 0.883748i \(0.344988\pi\)
\(882\) 0 0
\(883\) 49388.0 1.88226 0.941132 − 0.338040i \(-0.109764\pi\)
0.941132 + 0.338040i \(0.109764\pi\)
\(884\) 0 0
\(885\) 17280.0 0.656340
\(886\) 0 0
\(887\) 30216.0 1.14380 0.571902 − 0.820322i \(-0.306206\pi\)
0.571902 + 0.820322i \(0.306206\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3421.00 −0.128628
\(892\) 0 0
\(893\) −2820.00 −0.105675
\(894\) 0 0
\(895\) −39600.0 −1.47897
\(896\) 0 0
\(897\) 7200.00 0.268006
\(898\) 0 0
\(899\) 2820.00 0.104619
\(900\) 0 0
\(901\) −1296.00 −0.0479201
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1488.00 0.0546550
\(906\) 0 0
\(907\) −4492.00 −0.164448 −0.0822240 − 0.996614i \(-0.526202\pi\)
−0.0822240 + 0.996614i \(0.526202\pi\)
\(908\) 0 0
\(909\) 990.000 0.0361235
\(910\) 0 0
\(911\) 28932.0 1.05221 0.526103 − 0.850421i \(-0.323653\pi\)
0.526103 + 0.850421i \(0.323653\pi\)
\(912\) 0 0
\(913\) −726.000 −0.0263166
\(914\) 0 0
\(915\) −8544.00 −0.308695
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32056.0 −1.15063 −0.575315 − 0.817932i \(-0.695121\pi\)
−0.575315 + 0.817932i \(0.695121\pi\)
\(920\) 0 0
\(921\) 15704.0 0.561851
\(922\) 0 0
\(923\) 3120.00 0.111263
\(924\) 0 0
\(925\) −4066.00 −0.144529
\(926\) 0 0
\(927\) 3322.00 0.117701
\(928\) 0 0
\(929\) 22182.0 0.783388 0.391694 − 0.920095i \(-0.371889\pi\)
0.391694 + 0.920095i \(0.371889\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2472.00 0.0867413
\(934\) 0 0
\(935\) −3168.00 −0.110807
\(936\) 0 0
\(937\) −29792.0 −1.03870 −0.519350 − 0.854562i \(-0.673826\pi\)
−0.519350 + 0.854562i \(0.673826\pi\)
\(938\) 0 0
\(939\) 2744.00 0.0953643
\(940\) 0 0
\(941\) −30702.0 −1.06361 −0.531805 − 0.846867i \(-0.678486\pi\)
−0.531805 + 0.846867i \(0.678486\pi\)
\(942\) 0 0
\(943\) −8640.00 −0.298364
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5244.00 0.179944 0.0899721 − 0.995944i \(-0.471322\pi\)
0.0899721 + 0.995944i \(0.471322\pi\)
\(948\) 0 0
\(949\) −7280.00 −0.249019
\(950\) 0 0
\(951\) −8376.00 −0.285605
\(952\) 0 0
\(953\) −24150.0 −0.820876 −0.410438 − 0.911888i \(-0.634624\pi\)
−0.410438 + 0.911888i \(0.634624\pi\)
\(954\) 0 0
\(955\) 22320.0 0.756291
\(956\) 0 0
\(957\) −1320.00 −0.0445868
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −20955.0 −0.703400
\(962\) 0 0
\(963\) −2640.00 −0.0883414
\(964\) 0 0
\(965\) −15816.0 −0.527601
\(966\) 0 0
\(967\) 15512.0 0.515856 0.257928 − 0.966164i \(-0.416960\pi\)
0.257928 + 0.966164i \(0.416960\pi\)
\(968\) 0 0
\(969\) 9024.00 0.299167
\(970\) 0 0
\(971\) −52404.0 −1.73195 −0.865975 − 0.500086i \(-0.833302\pi\)
−0.865975 + 0.500086i \(0.833302\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −760.000 −0.0249636
\(976\) 0 0
\(977\) −5070.00 −0.166022 −0.0830111 − 0.996549i \(-0.526454\pi\)
−0.0830111 + 0.996549i \(0.526454\pi\)
\(978\) 0 0
\(979\) 4290.00 0.140050
\(980\) 0 0
\(981\) 1826.00 0.0594288
\(982\) 0 0
\(983\) −18702.0 −0.606817 −0.303409 − 0.952861i \(-0.598125\pi\)
−0.303409 + 0.952861i \(0.598125\pi\)
\(984\) 0 0
\(985\) −15192.0 −0.491429
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1440.00 −0.0462986
\(990\) 0 0
\(991\) 40496.0 1.29808 0.649040 − 0.760754i \(-0.275171\pi\)
0.649040 + 0.760754i \(0.275171\pi\)
\(992\) 0 0
\(993\) −9776.00 −0.312419
\(994\) 0 0
\(995\) −9960.00 −0.317340
\(996\) 0 0
\(997\) 23686.0 0.752400 0.376200 − 0.926538i \(-0.377230\pi\)
0.376200 + 0.926538i \(0.377230\pi\)
\(998\) 0 0
\(999\) −32528.0 −1.03017
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 2156.4.a.a.1.1 1
7.6 odd 2 308.4.a.b.1.1 ✓ 1
28.27 even 2 1232.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
308.4.a.b.1.1 ✓ 1 7.6 odd 2
1232.4.a.c.1.1 1 28.27 even 2
2156.4.a.a.1.1 1 1.1 even 1 trivial