Properties

Label 2496.4.a.l.1.1
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
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] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, 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("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2496.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -4.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -4.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +2.00000 q^{11} +13.0000 q^{13} -12.0000 q^{15} -6.00000 q^{17} -36.0000 q^{19} -12.0000 q^{21} +20.0000 q^{23} -109.000 q^{25} +27.0000 q^{27} +14.0000 q^{29} +152.000 q^{31} +6.00000 q^{33} +16.0000 q^{35} +258.000 q^{37} +39.0000 q^{39} +84.0000 q^{41} -188.000 q^{43} -36.0000 q^{45} -254.000 q^{47} -327.000 q^{49} -18.0000 q^{51} -366.000 q^{53} -8.00000 q^{55} -108.000 q^{57} +550.000 q^{59} +14.0000 q^{61} -36.0000 q^{63} -52.0000 q^{65} +448.000 q^{67} +60.0000 q^{69} -926.000 q^{71} +254.000 q^{73} -327.000 q^{75} -8.00000 q^{77} -1328.00 q^{79} +81.0000 q^{81} +186.000 q^{83} +24.0000 q^{85} +42.0000 q^{87} -336.000 q^{89} -52.0000 q^{91} +456.000 q^{93} +144.000 q^{95} +614.000 q^{97} +18.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −4.00000 −0.357771 −0.178885 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) 0 0
\(7\) −4.00000 −0.215980 −0.107990 0.994152i \(-0.534441\pi\)
−0.107990 + 0.994152i \(0.534441\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.0548202 0.0274101 0.999624i \(-0.491274\pi\)
0.0274101 + 0.999624i \(0.491274\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −12.0000 −0.206559
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) −36.0000 −0.434682 −0.217341 0.976096i \(-0.569738\pi\)
−0.217341 + 0.976096i \(0.569738\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) 20.0000 0.181317 0.0906584 0.995882i \(-0.471103\pi\)
0.0906584 + 0.995882i \(0.471103\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 14.0000 0.0896460 0.0448230 0.998995i \(-0.485728\pi\)
0.0448230 + 0.998995i \(0.485728\pi\)
\(30\) 0 0
\(31\) 152.000 0.880645 0.440323 0.897840i \(-0.354864\pi\)
0.440323 + 0.897840i \(0.354864\pi\)
\(32\) 0 0
\(33\) 6.00000 0.0316505
\(34\) 0 0
\(35\) 16.0000 0.0772712
\(36\) 0 0
\(37\) 258.000 1.14635 0.573175 0.819433i \(-0.305712\pi\)
0.573175 + 0.819433i \(0.305712\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 84.0000 0.319966 0.159983 0.987120i \(-0.448856\pi\)
0.159983 + 0.987120i \(0.448856\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 0 0
\(45\) −36.0000 −0.119257
\(46\) 0 0
\(47\) −254.000 −0.788292 −0.394146 0.919048i \(-0.628960\pi\)
−0.394146 + 0.919048i \(0.628960\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) −18.0000 −0.0494217
\(52\) 0 0
\(53\) −366.000 −0.948565 −0.474283 0.880373i \(-0.657293\pi\)
−0.474283 + 0.880373i \(0.657293\pi\)
\(54\) 0 0
\(55\) −8.00000 −0.0196131
\(56\) 0 0
\(57\) −108.000 −0.250964
\(58\) 0 0
\(59\) 550.000 1.21363 0.606813 0.794845i \(-0.292448\pi\)
0.606813 + 0.794845i \(0.292448\pi\)
\(60\) 0 0
\(61\) 14.0000 0.0293855 0.0146928 0.999892i \(-0.495323\pi\)
0.0146928 + 0.999892i \(0.495323\pi\)
\(62\) 0 0
\(63\) −36.0000 −0.0719932
\(64\) 0 0
\(65\) −52.0000 −0.0992278
\(66\) 0 0
\(67\) 448.000 0.816894 0.408447 0.912782i \(-0.366070\pi\)
0.408447 + 0.912782i \(0.366070\pi\)
\(68\) 0 0
\(69\) 60.0000 0.104683
\(70\) 0 0
\(71\) −926.000 −1.54783 −0.773915 0.633289i \(-0.781704\pi\)
−0.773915 + 0.633289i \(0.781704\pi\)
\(72\) 0 0
\(73\) 254.000 0.407239 0.203620 0.979050i \(-0.434729\pi\)
0.203620 + 0.979050i \(0.434729\pi\)
\(74\) 0 0
\(75\) −327.000 −0.503449
\(76\) 0 0
\(77\) −8.00000 −0.0118401
\(78\) 0 0
\(79\) −1328.00 −1.89129 −0.945644 0.325205i \(-0.894567\pi\)
−0.945644 + 0.325205i \(0.894567\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 186.000 0.245978 0.122989 0.992408i \(-0.460752\pi\)
0.122989 + 0.992408i \(0.460752\pi\)
\(84\) 0 0
\(85\) 24.0000 0.0306255
\(86\) 0 0
\(87\) 42.0000 0.0517572
\(88\) 0 0
\(89\) −336.000 −0.400179 −0.200089 0.979778i \(-0.564123\pi\)
−0.200089 + 0.979778i \(0.564123\pi\)
\(90\) 0 0
\(91\) −52.0000 −0.0599020
\(92\) 0 0
\(93\) 456.000 0.508441
\(94\) 0 0
\(95\) 144.000 0.155517
\(96\) 0 0
\(97\) 614.000 0.642704 0.321352 0.946960i \(-0.395863\pi\)
0.321352 + 0.946960i \(0.395863\pi\)
\(98\) 0 0
\(99\) 18.0000 0.0182734
\(100\) 0 0
\(101\) 1606.00 1.58221 0.791104 0.611682i \(-0.209507\pi\)
0.791104 + 0.611682i \(0.209507\pi\)
\(102\) 0 0
\(103\) −208.000 −0.198979 −0.0994896 0.995039i \(-0.531721\pi\)
−0.0994896 + 0.995039i \(0.531721\pi\)
\(104\) 0 0
\(105\) 48.0000 0.0446126
\(106\) 0 0
\(107\) −248.000 −0.224066 −0.112033 0.993704i \(-0.535736\pi\)
−0.112033 + 0.993704i \(0.535736\pi\)
\(108\) 0 0
\(109\) 542.000 0.476277 0.238138 0.971231i \(-0.423463\pi\)
0.238138 + 0.971231i \(0.423463\pi\)
\(110\) 0 0
\(111\) 774.000 0.661845
\(112\) 0 0
\(113\) −2042.00 −1.69996 −0.849979 0.526817i \(-0.823385\pi\)
−0.849979 + 0.526817i \(0.823385\pi\)
\(114\) 0 0
\(115\) −80.0000 −0.0648699
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 24.0000 0.0184880
\(120\) 0 0
\(121\) −1327.00 −0.996995
\(122\) 0 0
\(123\) 252.000 0.184732
\(124\) 0 0
\(125\) 936.000 0.669747
\(126\) 0 0
\(127\) 488.000 0.340968 0.170484 0.985360i \(-0.445467\pi\)
0.170484 + 0.985360i \(0.445467\pi\)
\(128\) 0 0
\(129\) −564.000 −0.384941
\(130\) 0 0
\(131\) 1744.00 1.16316 0.581580 0.813489i \(-0.302435\pi\)
0.581580 + 0.813489i \(0.302435\pi\)
\(132\) 0 0
\(133\) 144.000 0.0938826
\(134\) 0 0
\(135\) −108.000 −0.0688530
\(136\) 0 0
\(137\) −828.000 −0.516356 −0.258178 0.966097i \(-0.583122\pi\)
−0.258178 + 0.966097i \(0.583122\pi\)
\(138\) 0 0
\(139\) −404.000 −0.246524 −0.123262 0.992374i \(-0.539336\pi\)
−0.123262 + 0.992374i \(0.539336\pi\)
\(140\) 0 0
\(141\) −762.000 −0.455120
\(142\) 0 0
\(143\) 26.0000 0.0152044
\(144\) 0 0
\(145\) −56.0000 −0.0320727
\(146\) 0 0
\(147\) −981.000 −0.550418
\(148\) 0 0
\(149\) −2928.00 −1.60987 −0.804937 0.593361i \(-0.797801\pi\)
−0.804937 + 0.593361i \(0.797801\pi\)
\(150\) 0 0
\(151\) −1944.00 −1.04769 −0.523843 0.851815i \(-0.675502\pi\)
−0.523843 + 0.851815i \(0.675502\pi\)
\(152\) 0 0
\(153\) −54.0000 −0.0285336
\(154\) 0 0
\(155\) −608.000 −0.315069
\(156\) 0 0
\(157\) −3590.00 −1.82492 −0.912462 0.409161i \(-0.865822\pi\)
−0.912462 + 0.409161i \(0.865822\pi\)
\(158\) 0 0
\(159\) −1098.00 −0.547654
\(160\) 0 0
\(161\) −80.0000 −0.0391608
\(162\) 0 0
\(163\) −2284.00 −1.09753 −0.548763 0.835978i \(-0.684901\pi\)
−0.548763 + 0.835978i \(0.684901\pi\)
\(164\) 0 0
\(165\) −24.0000 −0.0113236
\(166\) 0 0
\(167\) −3174.00 −1.47073 −0.735364 0.677673i \(-0.762989\pi\)
−0.735364 + 0.677673i \(0.762989\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −324.000 −0.144894
\(172\) 0 0
\(173\) 1358.00 0.596802 0.298401 0.954441i \(-0.403547\pi\)
0.298401 + 0.954441i \(0.403547\pi\)
\(174\) 0 0
\(175\) 436.000 0.188334
\(176\) 0 0
\(177\) 1650.00 0.700687
\(178\) 0 0
\(179\) 708.000 0.295634 0.147817 0.989015i \(-0.452775\pi\)
0.147817 + 0.989015i \(0.452775\pi\)
\(180\) 0 0
\(181\) 546.000 0.224220 0.112110 0.993696i \(-0.464239\pi\)
0.112110 + 0.993696i \(0.464239\pi\)
\(182\) 0 0
\(183\) 42.0000 0.0169657
\(184\) 0 0
\(185\) −1032.00 −0.410131
\(186\) 0 0
\(187\) −12.0000 −0.00469266
\(188\) 0 0
\(189\) −108.000 −0.0415653
\(190\) 0 0
\(191\) 3472.00 1.31531 0.657657 0.753317i \(-0.271547\pi\)
0.657657 + 0.753317i \(0.271547\pi\)
\(192\) 0 0
\(193\) −310.000 −0.115618 −0.0578090 0.998328i \(-0.518411\pi\)
−0.0578090 + 0.998328i \(0.518411\pi\)
\(194\) 0 0
\(195\) −156.000 −0.0572892
\(196\) 0 0
\(197\) −1020.00 −0.368893 −0.184447 0.982843i \(-0.559049\pi\)
−0.184447 + 0.982843i \(0.559049\pi\)
\(198\) 0 0
\(199\) 3256.00 1.15986 0.579929 0.814667i \(-0.303080\pi\)
0.579929 + 0.814667i \(0.303080\pi\)
\(200\) 0 0
\(201\) 1344.00 0.471634
\(202\) 0 0
\(203\) −56.0000 −0.0193617
\(204\) 0 0
\(205\) −336.000 −0.114474
\(206\) 0 0
\(207\) 180.000 0.0604390
\(208\) 0 0
\(209\) −72.0000 −0.0238294
\(210\) 0 0
\(211\) −4564.00 −1.48909 −0.744547 0.667570i \(-0.767334\pi\)
−0.744547 + 0.667570i \(0.767334\pi\)
\(212\) 0 0
\(213\) −2778.00 −0.893640
\(214\) 0 0
\(215\) 752.000 0.238539
\(216\) 0 0
\(217\) −608.000 −0.190202
\(218\) 0 0
\(219\) 762.000 0.235120
\(220\) 0 0
\(221\) −78.0000 −0.0237414
\(222\) 0 0
\(223\) 72.0000 0.0216210 0.0108105 0.999942i \(-0.496559\pi\)
0.0108105 + 0.999942i \(0.496559\pi\)
\(224\) 0 0
\(225\) −981.000 −0.290667
\(226\) 0 0
\(227\) 2694.00 0.787696 0.393848 0.919176i \(-0.371144\pi\)
0.393848 + 0.919176i \(0.371144\pi\)
\(228\) 0 0
\(229\) −5922.00 −1.70889 −0.854447 0.519538i \(-0.826104\pi\)
−0.854447 + 0.519538i \(0.826104\pi\)
\(230\) 0 0
\(231\) −24.0000 −0.00683586
\(232\) 0 0
\(233\) −5122.00 −1.44014 −0.720072 0.693900i \(-0.755891\pi\)
−0.720072 + 0.693900i \(0.755891\pi\)
\(234\) 0 0
\(235\) 1016.00 0.282028
\(236\) 0 0
\(237\) −3984.00 −1.09194
\(238\) 0 0
\(239\) −5022.00 −1.35919 −0.679595 0.733588i \(-0.737844\pi\)
−0.679595 + 0.733588i \(0.737844\pi\)
\(240\) 0 0
\(241\) −1218.00 −0.325553 −0.162777 0.986663i \(-0.552045\pi\)
−0.162777 + 0.986663i \(0.552045\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1308.00 0.341082
\(246\) 0 0
\(247\) −468.000 −0.120559
\(248\) 0 0
\(249\) 558.000 0.142015
\(250\) 0 0
\(251\) −2112.00 −0.531109 −0.265554 0.964096i \(-0.585555\pi\)
−0.265554 + 0.964096i \(0.585555\pi\)
\(252\) 0 0
\(253\) 40.0000 0.00993984
\(254\) 0 0
\(255\) 72.0000 0.0176816
\(256\) 0 0
\(257\) 2814.00 0.683006 0.341503 0.939881i \(-0.389064\pi\)
0.341503 + 0.939881i \(0.389064\pi\)
\(258\) 0 0
\(259\) −1032.00 −0.247588
\(260\) 0 0
\(261\) 126.000 0.0298820
\(262\) 0 0
\(263\) 4044.00 0.948151 0.474076 0.880484i \(-0.342782\pi\)
0.474076 + 0.880484i \(0.342782\pi\)
\(264\) 0 0
\(265\) 1464.00 0.339369
\(266\) 0 0
\(267\) −1008.00 −0.231043
\(268\) 0 0
\(269\) 1470.00 0.333188 0.166594 0.986026i \(-0.446723\pi\)
0.166594 + 0.986026i \(0.446723\pi\)
\(270\) 0 0
\(271\) 1844.00 0.413340 0.206670 0.978411i \(-0.433737\pi\)
0.206670 + 0.978411i \(0.433737\pi\)
\(272\) 0 0
\(273\) −156.000 −0.0345844
\(274\) 0 0
\(275\) −218.000 −0.0478033
\(276\) 0 0
\(277\) −5766.00 −1.25071 −0.625353 0.780342i \(-0.715045\pi\)
−0.625353 + 0.780342i \(0.715045\pi\)
\(278\) 0 0
\(279\) 1368.00 0.293548
\(280\) 0 0
\(281\) −7468.00 −1.58542 −0.792711 0.609598i \(-0.791331\pi\)
−0.792711 + 0.609598i \(0.791331\pi\)
\(282\) 0 0
\(283\) 1228.00 0.257940 0.128970 0.991648i \(-0.458833\pi\)
0.128970 + 0.991648i \(0.458833\pi\)
\(284\) 0 0
\(285\) 432.000 0.0897876
\(286\) 0 0
\(287\) −336.000 −0.0691061
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 0 0
\(291\) 1842.00 0.371065
\(292\) 0 0
\(293\) −6608.00 −1.31755 −0.658777 0.752338i \(-0.728926\pi\)
−0.658777 + 0.752338i \(0.728926\pi\)
\(294\) 0 0
\(295\) −2200.00 −0.434200
\(296\) 0 0
\(297\) 54.0000 0.0105502
\(298\) 0 0
\(299\) 260.000 0.0502883
\(300\) 0 0
\(301\) 752.000 0.144002
\(302\) 0 0
\(303\) 4818.00 0.913488
\(304\) 0 0
\(305\) −56.0000 −0.0105133
\(306\) 0 0
\(307\) 7664.00 1.42478 0.712390 0.701784i \(-0.247613\pi\)
0.712390 + 0.701784i \(0.247613\pi\)
\(308\) 0 0
\(309\) −624.000 −0.114881
\(310\) 0 0
\(311\) 2340.00 0.426653 0.213327 0.976981i \(-0.431570\pi\)
0.213327 + 0.976981i \(0.431570\pi\)
\(312\) 0 0
\(313\) 6710.00 1.21173 0.605865 0.795567i \(-0.292827\pi\)
0.605865 + 0.795567i \(0.292827\pi\)
\(314\) 0 0
\(315\) 144.000 0.0257571
\(316\) 0 0
\(317\) −4164.00 −0.737771 −0.368886 0.929475i \(-0.620261\pi\)
−0.368886 + 0.929475i \(0.620261\pi\)
\(318\) 0 0
\(319\) 28.0000 0.00491442
\(320\) 0 0
\(321\) −744.000 −0.129365
\(322\) 0 0
\(323\) 216.000 0.0372092
\(324\) 0 0
\(325\) −1417.00 −0.241849
\(326\) 0 0
\(327\) 1626.00 0.274979
\(328\) 0 0
\(329\) 1016.00 0.170255
\(330\) 0 0
\(331\) −10072.0 −1.67253 −0.836265 0.548326i \(-0.815265\pi\)
−0.836265 + 0.548326i \(0.815265\pi\)
\(332\) 0 0
\(333\) 2322.00 0.382117
\(334\) 0 0
\(335\) −1792.00 −0.292261
\(336\) 0 0
\(337\) 2990.00 0.483311 0.241655 0.970362i \(-0.422310\pi\)
0.241655 + 0.970362i \(0.422310\pi\)
\(338\) 0 0
\(339\) −6126.00 −0.981471
\(340\) 0 0
\(341\) 304.000 0.0482772
\(342\) 0 0
\(343\) 2680.00 0.421885
\(344\) 0 0
\(345\) −240.000 −0.0374527
\(346\) 0 0
\(347\) 6564.00 1.01549 0.507743 0.861508i \(-0.330480\pi\)
0.507743 + 0.861508i \(0.330480\pi\)
\(348\) 0 0
\(349\) 674.000 0.103376 0.0516882 0.998663i \(-0.483540\pi\)
0.0516882 + 0.998663i \(0.483540\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −10732.0 −1.61815 −0.809075 0.587706i \(-0.800031\pi\)
−0.809075 + 0.587706i \(0.800031\pi\)
\(354\) 0 0
\(355\) 3704.00 0.553769
\(356\) 0 0
\(357\) 72.0000 0.0106741
\(358\) 0 0
\(359\) 4842.00 0.711841 0.355921 0.934516i \(-0.384167\pi\)
0.355921 + 0.934516i \(0.384167\pi\)
\(360\) 0 0
\(361\) −5563.00 −0.811051
\(362\) 0 0
\(363\) −3981.00 −0.575615
\(364\) 0 0
\(365\) −1016.00 −0.145698
\(366\) 0 0
\(367\) 6280.00 0.893224 0.446612 0.894728i \(-0.352630\pi\)
0.446612 + 0.894728i \(0.352630\pi\)
\(368\) 0 0
\(369\) 756.000 0.106655
\(370\) 0 0
\(371\) 1464.00 0.204871
\(372\) 0 0
\(373\) −6434.00 −0.893136 −0.446568 0.894750i \(-0.647354\pi\)
−0.446568 + 0.894750i \(0.647354\pi\)
\(374\) 0 0
\(375\) 2808.00 0.386679
\(376\) 0 0
\(377\) 182.000 0.0248633
\(378\) 0 0
\(379\) −9068.00 −1.22900 −0.614501 0.788916i \(-0.710643\pi\)
−0.614501 + 0.788916i \(0.710643\pi\)
\(380\) 0 0
\(381\) 1464.00 0.196858
\(382\) 0 0
\(383\) −3162.00 −0.421855 −0.210928 0.977502i \(-0.567648\pi\)
−0.210928 + 0.977502i \(0.567648\pi\)
\(384\) 0 0
\(385\) 32.0000 0.00423603
\(386\) 0 0
\(387\) −1692.00 −0.222246
\(388\) 0 0
\(389\) 3666.00 0.477824 0.238912 0.971041i \(-0.423209\pi\)
0.238912 + 0.971041i \(0.423209\pi\)
\(390\) 0 0
\(391\) −120.000 −0.0155209
\(392\) 0 0
\(393\) 5232.00 0.671551
\(394\) 0 0
\(395\) 5312.00 0.676647
\(396\) 0 0
\(397\) −11054.0 −1.39744 −0.698721 0.715394i \(-0.746247\pi\)
−0.698721 + 0.715394i \(0.746247\pi\)
\(398\) 0 0
\(399\) 432.000 0.0542031
\(400\) 0 0
\(401\) −5328.00 −0.663510 −0.331755 0.943366i \(-0.607641\pi\)
−0.331755 + 0.943366i \(0.607641\pi\)
\(402\) 0 0
\(403\) 1976.00 0.244247
\(404\) 0 0
\(405\) −324.000 −0.0397523
\(406\) 0 0
\(407\) 516.000 0.0628432
\(408\) 0 0
\(409\) −12074.0 −1.45971 −0.729854 0.683603i \(-0.760412\pi\)
−0.729854 + 0.683603i \(0.760412\pi\)
\(410\) 0 0
\(411\) −2484.00 −0.298118
\(412\) 0 0
\(413\) −2200.00 −0.262118
\(414\) 0 0
\(415\) −744.000 −0.0880037
\(416\) 0 0
\(417\) −1212.00 −0.142331
\(418\) 0 0
\(419\) 13584.0 1.58382 0.791911 0.610636i \(-0.209086\pi\)
0.791911 + 0.610636i \(0.209086\pi\)
\(420\) 0 0
\(421\) 7406.00 0.857355 0.428677 0.903458i \(-0.358980\pi\)
0.428677 + 0.903458i \(0.358980\pi\)
\(422\) 0 0
\(423\) −2286.00 −0.262764
\(424\) 0 0
\(425\) 654.000 0.0746439
\(426\) 0 0
\(427\) −56.0000 −0.00634667
\(428\) 0 0
\(429\) 78.0000 0.00877826
\(430\) 0 0
\(431\) 10134.0 1.13257 0.566285 0.824210i \(-0.308380\pi\)
0.566285 + 0.824210i \(0.308380\pi\)
\(432\) 0 0
\(433\) 9406.00 1.04393 0.521967 0.852966i \(-0.325198\pi\)
0.521967 + 0.852966i \(0.325198\pi\)
\(434\) 0 0
\(435\) −168.000 −0.0185172
\(436\) 0 0
\(437\) −720.000 −0.0788153
\(438\) 0 0
\(439\) −4088.00 −0.444441 −0.222220 0.974996i \(-0.571330\pi\)
−0.222220 + 0.974996i \(0.571330\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) −5328.00 −0.571424 −0.285712 0.958315i \(-0.592230\pi\)
−0.285712 + 0.958315i \(0.592230\pi\)
\(444\) 0 0
\(445\) 1344.00 0.143172
\(446\) 0 0
\(447\) −8784.00 −0.929461
\(448\) 0 0
\(449\) 13160.0 1.38320 0.691602 0.722279i \(-0.256905\pi\)
0.691602 + 0.722279i \(0.256905\pi\)
\(450\) 0 0
\(451\) 168.000 0.0175406
\(452\) 0 0
\(453\) −5832.00 −0.604881
\(454\) 0 0
\(455\) 208.000 0.0214312
\(456\) 0 0
\(457\) −9146.00 −0.936175 −0.468087 0.883682i \(-0.655057\pi\)
−0.468087 + 0.883682i \(0.655057\pi\)
\(458\) 0 0
\(459\) −162.000 −0.0164739
\(460\) 0 0
\(461\) −5580.00 −0.563745 −0.281873 0.959452i \(-0.590956\pi\)
−0.281873 + 0.959452i \(0.590956\pi\)
\(462\) 0 0
\(463\) −14788.0 −1.48436 −0.742178 0.670203i \(-0.766207\pi\)
−0.742178 + 0.670203i \(0.766207\pi\)
\(464\) 0 0
\(465\) −1824.00 −0.181905
\(466\) 0 0
\(467\) 12376.0 1.22632 0.613162 0.789957i \(-0.289897\pi\)
0.613162 + 0.789957i \(0.289897\pi\)
\(468\) 0 0
\(469\) −1792.00 −0.176433
\(470\) 0 0
\(471\) −10770.0 −1.05362
\(472\) 0 0
\(473\) −376.000 −0.0365507
\(474\) 0 0
\(475\) 3924.00 0.379043
\(476\) 0 0
\(477\) −3294.00 −0.316188
\(478\) 0 0
\(479\) −834.000 −0.0795541 −0.0397771 0.999209i \(-0.512665\pi\)
−0.0397771 + 0.999209i \(0.512665\pi\)
\(480\) 0 0
\(481\) 3354.00 0.317940
\(482\) 0 0
\(483\) −240.000 −0.0226095
\(484\) 0 0
\(485\) −2456.00 −0.229941
\(486\) 0 0
\(487\) 13192.0 1.22749 0.613744 0.789505i \(-0.289663\pi\)
0.613744 + 0.789505i \(0.289663\pi\)
\(488\) 0 0
\(489\) −6852.00 −0.633657
\(490\) 0 0
\(491\) 16568.0 1.52282 0.761409 0.648272i \(-0.224508\pi\)
0.761409 + 0.648272i \(0.224508\pi\)
\(492\) 0 0
\(493\) −84.0000 −0.00767377
\(494\) 0 0
\(495\) −72.0000 −0.00653770
\(496\) 0 0
\(497\) 3704.00 0.334300
\(498\) 0 0
\(499\) −10136.0 −0.909318 −0.454659 0.890666i \(-0.650239\pi\)
−0.454659 + 0.890666i \(0.650239\pi\)
\(500\) 0 0
\(501\) −9522.00 −0.849125
\(502\) 0 0
\(503\) −10412.0 −0.922959 −0.461479 0.887151i \(-0.652681\pi\)
−0.461479 + 0.887151i \(0.652681\pi\)
\(504\) 0 0
\(505\) −6424.00 −0.566068
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 4180.00 0.363999 0.181999 0.983299i \(-0.441743\pi\)
0.181999 + 0.983299i \(0.441743\pi\)
\(510\) 0 0
\(511\) −1016.00 −0.0879554
\(512\) 0 0
\(513\) −972.000 −0.0836547
\(514\) 0 0
\(515\) 832.000 0.0711889
\(516\) 0 0
\(517\) −508.000 −0.0432143
\(518\) 0 0
\(519\) 4074.00 0.344564
\(520\) 0 0
\(521\) −14610.0 −1.22855 −0.614276 0.789091i \(-0.710552\pi\)
−0.614276 + 0.789091i \(0.710552\pi\)
\(522\) 0 0
\(523\) −2172.00 −0.181596 −0.0907982 0.995869i \(-0.528942\pi\)
−0.0907982 + 0.995869i \(0.528942\pi\)
\(524\) 0 0
\(525\) 1308.00 0.108735
\(526\) 0 0
\(527\) −912.000 −0.0753840
\(528\) 0 0
\(529\) −11767.0 −0.967124
\(530\) 0 0
\(531\) 4950.00 0.404542
\(532\) 0 0
\(533\) 1092.00 0.0887425
\(534\) 0 0
\(535\) 992.000 0.0801643
\(536\) 0 0
\(537\) 2124.00 0.170684
\(538\) 0 0
\(539\) −654.000 −0.0522630
\(540\) 0 0
\(541\) 11758.0 0.934410 0.467205 0.884149i \(-0.345261\pi\)
0.467205 + 0.884149i \(0.345261\pi\)
\(542\) 0 0
\(543\) 1638.00 0.129454
\(544\) 0 0
\(545\) −2168.00 −0.170398
\(546\) 0 0
\(547\) 340.000 0.0265765 0.0132883 0.999912i \(-0.495770\pi\)
0.0132883 + 0.999912i \(0.495770\pi\)
\(548\) 0 0
\(549\) 126.000 0.00979517
\(550\) 0 0
\(551\) −504.000 −0.0389676
\(552\) 0 0
\(553\) 5312.00 0.408480
\(554\) 0 0
\(555\) −3096.00 −0.236789
\(556\) 0 0
\(557\) 3768.00 0.286634 0.143317 0.989677i \(-0.454223\pi\)
0.143317 + 0.989677i \(0.454223\pi\)
\(558\) 0 0
\(559\) −2444.00 −0.184920
\(560\) 0 0
\(561\) −36.0000 −0.00270931
\(562\) 0 0
\(563\) −10172.0 −0.761454 −0.380727 0.924687i \(-0.624326\pi\)
−0.380727 + 0.924687i \(0.624326\pi\)
\(564\) 0 0
\(565\) 8168.00 0.608195
\(566\) 0 0
\(567\) −324.000 −0.0239977
\(568\) 0 0
\(569\) −5506.00 −0.405665 −0.202833 0.979213i \(-0.565015\pi\)
−0.202833 + 0.979213i \(0.565015\pi\)
\(570\) 0 0
\(571\) 2340.00 0.171499 0.0857495 0.996317i \(-0.472672\pi\)
0.0857495 + 0.996317i \(0.472672\pi\)
\(572\) 0 0
\(573\) 10416.0 0.759397
\(574\) 0 0
\(575\) −2180.00 −0.158108
\(576\) 0 0
\(577\) −20094.0 −1.44978 −0.724891 0.688864i \(-0.758110\pi\)
−0.724891 + 0.688864i \(0.758110\pi\)
\(578\) 0 0
\(579\) −930.000 −0.0667521
\(580\) 0 0
\(581\) −744.000 −0.0531262
\(582\) 0 0
\(583\) −732.000 −0.0520006
\(584\) 0 0
\(585\) −468.000 −0.0330759
\(586\) 0 0
\(587\) −7118.00 −0.500496 −0.250248 0.968182i \(-0.580512\pi\)
−0.250248 + 0.968182i \(0.580512\pi\)
\(588\) 0 0
\(589\) −5472.00 −0.382801
\(590\) 0 0
\(591\) −3060.00 −0.212981
\(592\) 0 0
\(593\) −10328.0 −0.715211 −0.357606 0.933873i \(-0.616407\pi\)
−0.357606 + 0.933873i \(0.616407\pi\)
\(594\) 0 0
\(595\) −96.0000 −0.00661448
\(596\) 0 0
\(597\) 9768.00 0.669644
\(598\) 0 0
\(599\) 19732.0 1.34596 0.672978 0.739662i \(-0.265015\pi\)
0.672978 + 0.739662i \(0.265015\pi\)
\(600\) 0 0
\(601\) −12026.0 −0.816224 −0.408112 0.912932i \(-0.633813\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(602\) 0 0
\(603\) 4032.00 0.272298
\(604\) 0 0
\(605\) 5308.00 0.356696
\(606\) 0 0
\(607\) −17016.0 −1.13782 −0.568911 0.822399i \(-0.692635\pi\)
−0.568911 + 0.822399i \(0.692635\pi\)
\(608\) 0 0
\(609\) −168.000 −0.0111785
\(610\) 0 0
\(611\) −3302.00 −0.218633
\(612\) 0 0
\(613\) −11654.0 −0.767864 −0.383932 0.923361i \(-0.625430\pi\)
−0.383932 + 0.923361i \(0.625430\pi\)
\(614\) 0 0
\(615\) −1008.00 −0.0660918
\(616\) 0 0
\(617\) 11612.0 0.757669 0.378834 0.925465i \(-0.376325\pi\)
0.378834 + 0.925465i \(0.376325\pi\)
\(618\) 0 0
\(619\) 4024.00 0.261290 0.130645 0.991429i \(-0.458295\pi\)
0.130645 + 0.991429i \(0.458295\pi\)
\(620\) 0 0
\(621\) 540.000 0.0348945
\(622\) 0 0
\(623\) 1344.00 0.0864305
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) −216.000 −0.0137579
\(628\) 0 0
\(629\) −1548.00 −0.0981285
\(630\) 0 0
\(631\) 1088.00 0.0686412 0.0343206 0.999411i \(-0.489073\pi\)
0.0343206 + 0.999411i \(0.489073\pi\)
\(632\) 0 0
\(633\) −13692.0 −0.859729
\(634\) 0 0
\(635\) −1952.00 −0.121989
\(636\) 0 0
\(637\) −4251.00 −0.264412
\(638\) 0 0
\(639\) −8334.00 −0.515944
\(640\) 0 0
\(641\) −7078.00 −0.436138 −0.218069 0.975933i \(-0.569976\pi\)
−0.218069 + 0.975933i \(0.569976\pi\)
\(642\) 0 0
\(643\) 8336.00 0.511259 0.255630 0.966775i \(-0.417717\pi\)
0.255630 + 0.966775i \(0.417717\pi\)
\(644\) 0 0
\(645\) 2256.00 0.137721
\(646\) 0 0
\(647\) −32.0000 −0.00194444 −0.000972218 1.00000i \(-0.500309\pi\)
−0.000972218 1.00000i \(0.500309\pi\)
\(648\) 0 0
\(649\) 1100.00 0.0665312
\(650\) 0 0
\(651\) −1824.00 −0.109813
\(652\) 0 0
\(653\) 15822.0 0.948182 0.474091 0.880476i \(-0.342777\pi\)
0.474091 + 0.880476i \(0.342777\pi\)
\(654\) 0 0
\(655\) −6976.00 −0.416145
\(656\) 0 0
\(657\) 2286.00 0.135746
\(658\) 0 0
\(659\) 21540.0 1.27326 0.636631 0.771169i \(-0.280328\pi\)
0.636631 + 0.771169i \(0.280328\pi\)
\(660\) 0 0
\(661\) −8270.00 −0.486635 −0.243317 0.969947i \(-0.578236\pi\)
−0.243317 + 0.969947i \(0.578236\pi\)
\(662\) 0 0
\(663\) −234.000 −0.0137071
\(664\) 0 0
\(665\) −576.000 −0.0335885
\(666\) 0 0
\(667\) 280.000 0.0162543
\(668\) 0 0
\(669\) 216.000 0.0124829
\(670\) 0 0
\(671\) 28.0000 0.00161092
\(672\) 0 0
\(673\) 8482.00 0.485820 0.242910 0.970049i \(-0.421898\pi\)
0.242910 + 0.970049i \(0.421898\pi\)
\(674\) 0 0
\(675\) −2943.00 −0.167816
\(676\) 0 0
\(677\) −2550.00 −0.144763 −0.0723814 0.997377i \(-0.523060\pi\)
−0.0723814 + 0.997377i \(0.523060\pi\)
\(678\) 0 0
\(679\) −2456.00 −0.138811
\(680\) 0 0
\(681\) 8082.00 0.454777
\(682\) 0 0
\(683\) −31534.0 −1.76664 −0.883320 0.468771i \(-0.844697\pi\)
−0.883320 + 0.468771i \(0.844697\pi\)
\(684\) 0 0
\(685\) 3312.00 0.184737
\(686\) 0 0
\(687\) −17766.0 −0.986631
\(688\) 0 0
\(689\) −4758.00 −0.263085
\(690\) 0 0
\(691\) 33832.0 1.86256 0.931281 0.364302i \(-0.118693\pi\)
0.931281 + 0.364302i \(0.118693\pi\)
\(692\) 0 0
\(693\) −72.0000 −0.00394669
\(694\) 0 0
\(695\) 1616.00 0.0881991
\(696\) 0 0
\(697\) −504.000 −0.0273893
\(698\) 0 0
\(699\) −15366.0 −0.831467
\(700\) 0 0
\(701\) −19422.0 −1.04645 −0.523223 0.852196i \(-0.675271\pi\)
−0.523223 + 0.852196i \(0.675271\pi\)
\(702\) 0 0
\(703\) −9288.00 −0.498298
\(704\) 0 0
\(705\) 3048.00 0.162829
\(706\) 0 0
\(707\) −6424.00 −0.341725
\(708\) 0 0
\(709\) 1894.00 0.100325 0.0501627 0.998741i \(-0.484026\pi\)
0.0501627 + 0.998741i \(0.484026\pi\)
\(710\) 0 0
\(711\) −11952.0 −0.630429
\(712\) 0 0
\(713\) 3040.00 0.159676
\(714\) 0 0
\(715\) −104.000 −0.00543969
\(716\) 0 0
\(717\) −15066.0 −0.784728
\(718\) 0 0
\(719\) 20156.0 1.04547 0.522734 0.852496i \(-0.324912\pi\)
0.522734 + 0.852496i \(0.324912\pi\)
\(720\) 0 0
\(721\) 832.000 0.0429754
\(722\) 0 0
\(723\) −3654.00 −0.187958
\(724\) 0 0
\(725\) −1526.00 −0.0781713
\(726\) 0 0
\(727\) −11128.0 −0.567696 −0.283848 0.958869i \(-0.591611\pi\)
−0.283848 + 0.958869i \(0.591611\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1128.00 0.0570733
\(732\) 0 0
\(733\) −16202.0 −0.816418 −0.408209 0.912888i \(-0.633847\pi\)
−0.408209 + 0.912888i \(0.633847\pi\)
\(734\) 0 0
\(735\) 3924.00 0.196924
\(736\) 0 0
\(737\) 896.000 0.0447823
\(738\) 0 0
\(739\) −5328.00 −0.265215 −0.132607 0.991169i \(-0.542335\pi\)
−0.132607 + 0.991169i \(0.542335\pi\)
\(740\) 0 0
\(741\) −1404.00 −0.0696049
\(742\) 0 0
\(743\) 20482.0 1.01132 0.505661 0.862732i \(-0.331249\pi\)
0.505661 + 0.862732i \(0.331249\pi\)
\(744\) 0 0
\(745\) 11712.0 0.575966
\(746\) 0 0
\(747\) 1674.00 0.0819926
\(748\) 0 0
\(749\) 992.000 0.0483937
\(750\) 0 0
\(751\) −8040.00 −0.390657 −0.195329 0.980738i \(-0.562577\pi\)
−0.195329 + 0.980738i \(0.562577\pi\)
\(752\) 0 0
\(753\) −6336.00 −0.306636
\(754\) 0 0
\(755\) 7776.00 0.374831
\(756\) 0 0
\(757\) 15822.0 0.759657 0.379829 0.925057i \(-0.375983\pi\)
0.379829 + 0.925057i \(0.375983\pi\)
\(758\) 0 0
\(759\) 120.000 0.00573877
\(760\) 0 0
\(761\) −1452.00 −0.0691655 −0.0345828 0.999402i \(-0.511010\pi\)
−0.0345828 + 0.999402i \(0.511010\pi\)
\(762\) 0 0
\(763\) −2168.00 −0.102866
\(764\) 0 0
\(765\) 216.000 0.0102085
\(766\) 0 0
\(767\) 7150.00 0.336599
\(768\) 0 0
\(769\) 32298.0 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) 8442.00 0.394334
\(772\) 0 0
\(773\) −18736.0 −0.871781 −0.435891 0.900000i \(-0.643567\pi\)
−0.435891 + 0.900000i \(0.643567\pi\)
\(774\) 0 0
\(775\) −16568.0 −0.767923
\(776\) 0 0
\(777\) −3096.00 −0.142945
\(778\) 0 0
\(779\) −3024.00 −0.139083
\(780\) 0 0
\(781\) −1852.00 −0.0848525
\(782\) 0 0
\(783\) 378.000 0.0172524
\(784\) 0 0
\(785\) 14360.0 0.652905
\(786\) 0 0
\(787\) −40816.0 −1.84871 −0.924354 0.381536i \(-0.875395\pi\)
−0.924354 + 0.381536i \(0.875395\pi\)
\(788\) 0 0
\(789\) 12132.0 0.547415
\(790\) 0 0
\(791\) 8168.00 0.367156
\(792\) 0 0
\(793\) 182.000 0.00815008
\(794\) 0 0
\(795\) 4392.00 0.195935
\(796\) 0 0
\(797\) −4518.00 −0.200798 −0.100399 0.994947i \(-0.532012\pi\)
−0.100399 + 0.994947i \(0.532012\pi\)
\(798\) 0 0
\(799\) 1524.00 0.0674784
\(800\) 0 0
\(801\) −3024.00 −0.133393
\(802\) 0 0
\(803\) 508.000 0.0223249
\(804\) 0 0
\(805\) 320.000 0.0140106
\(806\) 0 0
\(807\) 4410.00 0.192366
\(808\) 0 0
\(809\) −5058.00 −0.219814 −0.109907 0.993942i \(-0.535055\pi\)
−0.109907 + 0.993942i \(0.535055\pi\)
\(810\) 0 0
\(811\) −22564.0 −0.976978 −0.488489 0.872570i \(-0.662452\pi\)
−0.488489 + 0.872570i \(0.662452\pi\)
\(812\) 0 0
\(813\) 5532.00 0.238642
\(814\) 0 0
\(815\) 9136.00 0.392663
\(816\) 0 0
\(817\) 6768.00 0.289819
\(818\) 0 0
\(819\) −468.000 −0.0199673
\(820\) 0 0
\(821\) −32584.0 −1.38513 −0.692564 0.721357i \(-0.743519\pi\)
−0.692564 + 0.721357i \(0.743519\pi\)
\(822\) 0 0
\(823\) 9288.00 0.393389 0.196695 0.980465i \(-0.436979\pi\)
0.196695 + 0.980465i \(0.436979\pi\)
\(824\) 0 0
\(825\) −654.000 −0.0275992
\(826\) 0 0
\(827\) 20586.0 0.865593 0.432796 0.901492i \(-0.357527\pi\)
0.432796 + 0.901492i \(0.357527\pi\)
\(828\) 0 0
\(829\) 46118.0 1.93214 0.966070 0.258280i \(-0.0831556\pi\)
0.966070 + 0.258280i \(0.0831556\pi\)
\(830\) 0 0
\(831\) −17298.0 −0.722095
\(832\) 0 0
\(833\) 1962.00 0.0816078
\(834\) 0 0
\(835\) 12696.0 0.526183
\(836\) 0 0
\(837\) 4104.00 0.169480
\(838\) 0 0
\(839\) 39230.0 1.61427 0.807133 0.590369i \(-0.201018\pi\)
0.807133 + 0.590369i \(0.201018\pi\)
\(840\) 0 0
\(841\) −24193.0 −0.991964
\(842\) 0 0
\(843\) −22404.0 −0.915344
\(844\) 0 0
\(845\) −676.000 −0.0275208
\(846\) 0 0
\(847\) 5308.00 0.215331
\(848\) 0 0
\(849\) 3684.00 0.148922
\(850\) 0 0
\(851\) 5160.00 0.207853
\(852\) 0 0
\(853\) 18674.0 0.749573 0.374786 0.927111i \(-0.377716\pi\)
0.374786 + 0.927111i \(0.377716\pi\)
\(854\) 0 0
\(855\) 1296.00 0.0518389
\(856\) 0 0
\(857\) 41678.0 1.66125 0.830626 0.556830i \(-0.187983\pi\)
0.830626 + 0.556830i \(0.187983\pi\)
\(858\) 0 0
\(859\) −14740.0 −0.585474 −0.292737 0.956193i \(-0.594566\pi\)
−0.292737 + 0.956193i \(0.594566\pi\)
\(860\) 0 0
\(861\) −1008.00 −0.0398984
\(862\) 0 0
\(863\) 24982.0 0.985396 0.492698 0.870200i \(-0.336011\pi\)
0.492698 + 0.870200i \(0.336011\pi\)
\(864\) 0 0
\(865\) −5432.00 −0.213519
\(866\) 0 0
\(867\) −14631.0 −0.573120
\(868\) 0 0
\(869\) −2656.00 −0.103681
\(870\) 0 0
\(871\) 5824.00 0.226566
\(872\) 0 0
\(873\) 5526.00 0.214235
\(874\) 0 0
\(875\) −3744.00 −0.144652
\(876\) 0 0
\(877\) −1134.00 −0.0436630 −0.0218315 0.999762i \(-0.506950\pi\)
−0.0218315 + 0.999762i \(0.506950\pi\)
\(878\) 0 0
\(879\) −19824.0 −0.760690
\(880\) 0 0
\(881\) 34950.0 1.33654 0.668272 0.743917i \(-0.267034\pi\)
0.668272 + 0.743917i \(0.267034\pi\)
\(882\) 0 0
\(883\) −3068.00 −0.116927 −0.0584634 0.998290i \(-0.518620\pi\)
−0.0584634 + 0.998290i \(0.518620\pi\)
\(884\) 0 0
\(885\) −6600.00 −0.250685
\(886\) 0 0
\(887\) 14080.0 0.532988 0.266494 0.963837i \(-0.414135\pi\)
0.266494 + 0.963837i \(0.414135\pi\)
\(888\) 0 0
\(889\) −1952.00 −0.0736423
\(890\) 0 0
\(891\) 162.000 0.00609114
\(892\) 0 0
\(893\) 9144.00 0.342657
\(894\) 0 0
\(895\) −2832.00 −0.105769
\(896\) 0 0
\(897\) 780.000 0.0290339
\(898\) 0 0
\(899\) 2128.00 0.0789464
\(900\) 0 0
\(901\) 2196.00 0.0811980
\(902\) 0 0
\(903\) 2256.00 0.0831395
\(904\) 0 0
\(905\) −2184.00 −0.0802195
\(906\) 0 0
\(907\) −24876.0 −0.910688 −0.455344 0.890316i \(-0.650484\pi\)
−0.455344 + 0.890316i \(0.650484\pi\)
\(908\) 0 0
\(909\) 14454.0 0.527403
\(910\) 0 0
\(911\) −51456.0 −1.87136 −0.935682 0.352843i \(-0.885215\pi\)
−0.935682 + 0.352843i \(0.885215\pi\)
\(912\) 0 0
\(913\) 372.000 0.0134846
\(914\) 0 0
\(915\) −168.000 −0.00606985
\(916\) 0 0
\(917\) −6976.00 −0.251219
\(918\) 0 0
\(919\) 31032.0 1.11388 0.556938 0.830554i \(-0.311976\pi\)
0.556938 + 0.830554i \(0.311976\pi\)
\(920\) 0 0
\(921\) 22992.0 0.822597
\(922\) 0 0
\(923\) −12038.0 −0.429291
\(924\) 0 0
\(925\) −28122.0 −0.999617
\(926\) 0 0
\(927\) −1872.00 −0.0663264
\(928\) 0 0
\(929\) 50820.0 1.79478 0.897390 0.441239i \(-0.145461\pi\)
0.897390 + 0.441239i \(0.145461\pi\)
\(930\) 0 0
\(931\) 11772.0 0.414406
\(932\) 0 0
\(933\) 7020.00 0.246328
\(934\) 0 0
\(935\) 48.0000 0.00167890
\(936\) 0 0
\(937\) 5982.00 0.208563 0.104281 0.994548i \(-0.466746\pi\)
0.104281 + 0.994548i \(0.466746\pi\)
\(938\) 0 0
\(939\) 20130.0 0.699593
\(940\) 0 0
\(941\) 20224.0 0.700620 0.350310 0.936634i \(-0.386076\pi\)
0.350310 + 0.936634i \(0.386076\pi\)
\(942\) 0 0
\(943\) 1680.00 0.0580152
\(944\) 0 0
\(945\) 432.000 0.0148709
\(946\) 0 0
\(947\) 8478.00 0.290917 0.145458 0.989364i \(-0.453534\pi\)
0.145458 + 0.989364i \(0.453534\pi\)
\(948\) 0 0
\(949\) 3302.00 0.112948
\(950\) 0 0
\(951\) −12492.0 −0.425953
\(952\) 0 0
\(953\) 40918.0 1.39083 0.695417 0.718607i \(-0.255220\pi\)
0.695417 + 0.718607i \(0.255220\pi\)
\(954\) 0 0
\(955\) −13888.0 −0.470581
\(956\) 0 0
\(957\) 84.0000 0.00283734
\(958\) 0 0
\(959\) 3312.00 0.111522
\(960\) 0 0
\(961\) −6687.00 −0.224464
\(962\) 0 0
\(963\) −2232.00 −0.0746887
\(964\) 0 0
\(965\) 1240.00 0.0413648
\(966\) 0 0
\(967\) 4624.00 0.153772 0.0768862 0.997040i \(-0.475502\pi\)
0.0768862 + 0.997040i \(0.475502\pi\)
\(968\) 0 0
\(969\) 648.000 0.0214827
\(970\) 0 0
\(971\) 15300.0 0.505665 0.252832 0.967510i \(-0.418638\pi\)
0.252832 + 0.967510i \(0.418638\pi\)
\(972\) 0 0
\(973\) 1616.00 0.0532442
\(974\) 0 0
\(975\) −4251.00 −0.139632
\(976\) 0 0
\(977\) 19584.0 0.641298 0.320649 0.947198i \(-0.396099\pi\)
0.320649 + 0.947198i \(0.396099\pi\)
\(978\) 0 0
\(979\) −672.000 −0.0219379
\(980\) 0 0
\(981\) 4878.00 0.158759
\(982\) 0 0
\(983\) 17582.0 0.570477 0.285238 0.958457i \(-0.407927\pi\)
0.285238 + 0.958457i \(0.407927\pi\)
\(984\) 0 0
\(985\) 4080.00 0.131979
\(986\) 0 0
\(987\) 3048.00 0.0982968
\(988\) 0 0
\(989\) −3760.00 −0.120891
\(990\) 0 0
\(991\) −47904.0 −1.53554 −0.767770 0.640725i \(-0.778634\pi\)
−0.767770 + 0.640725i \(0.778634\pi\)
\(992\) 0 0
\(993\) −30216.0 −0.965635
\(994\) 0 0
\(995\) −13024.0 −0.414963
\(996\) 0 0
\(997\) 44578.0 1.41605 0.708024 0.706189i \(-0.249587\pi\)
0.708024 + 0.706189i \(0.249587\pi\)
\(998\) 0 0
\(999\) 6966.00 0.220615
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 2496.4.a.l.1.1 1
4.3 odd 2 2496.4.a.c.1.1 1
8.3 odd 2 78.4.a.f.1.1 1
8.5 even 2 624.4.a.c.1.1 1
24.5 odd 2 1872.4.a.f.1.1 1
24.11 even 2 234.4.a.c.1.1 1
40.19 odd 2 1950.4.a.a.1.1 1
104.51 odd 2 1014.4.a.e.1.1 1
104.83 even 4 1014.4.b.g.337.1 2
104.99 even 4 1014.4.b.g.337.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 8.3 odd 2
234.4.a.c.1.1 1 24.11 even 2
624.4.a.c.1.1 1 8.5 even 2
1014.4.a.e.1.1 1 104.51 odd 2
1014.4.b.g.337.1 2 104.83 even 4
1014.4.b.g.337.2 2 104.99 even 4
1872.4.a.f.1.1 1 24.5 odd 2
1950.4.a.a.1.1 1 40.19 odd 2
2496.4.a.c.1.1 1 4.3 odd 2
2496.4.a.l.1.1 1 1.1 even 1 trivial