↑

Properties

Label 784.4.a.r.1.1
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,4,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,7,0,-7,0,0,0,22,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 784.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000 q^{3} -7.00000 q^{5} +22.0000 q^{9} +5.00000 q^{11} +14.0000 q^{13} -49.0000 q^{15} +21.0000 q^{17} +49.0000 q^{19} +159.000 q^{23} -76.0000 q^{25} -35.0000 q^{27} +58.0000 q^{29} +147.000 q^{31} +35.0000 q^{33} +219.000 q^{37} +98.0000 q^{39} -350.000 q^{41} +124.000 q^{43} -154.000 q^{45} +525.000 q^{47} +147.000 q^{51} +303.000 q^{53} -35.0000 q^{55} +343.000 q^{57} -105.000 q^{59} +413.000 q^{61} -98.0000 q^{65} -415.000 q^{67} +1113.00 q^{69} +432.000 q^{71} +1113.00 q^{73} -532.000 q^{75} +103.000 q^{79} -839.000 q^{81} +1092.00 q^{83} -147.000 q^{85} +406.000 q^{87} +329.000 q^{89} +1029.00 q^{93} -343.000 q^{95} +882.000 q^{97} +110.000 q^{99} +O(q^{100})\)

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
\(3\) 7.00000 1.34715 0.673575 − 0.739119i \(-0.264758\pi\)
0.673575 + 0.739119i \(0.264758\pi\)
\(4\) 0 0
\(5\) −7.00000 −0.626099 −0.313050 − 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) 5.00000 0.137051 0.0685253 − 0.997649i \(-0.478171\pi\)
0.0685253 + 0.997649i \(0.478171\pi\)
\(12\) 0 0
\(13\) 14.0000 0.298685 0.149342 − 0.988786i \(-0.452284\pi\)
0.149342 + 0.988786i \(0.452284\pi\)
\(14\) 0 0
\(15\) −49.0000 −0.843450
\(16\) 0 0
\(17\) 21.0000 0.299603 0.149801 − 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) 0 0
\(19\) 49.0000 0.591651 0.295826 − 0.955242i \(-0.404405\pi\)
0.295826 + 0.955242i \(0.404405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 159.000 1.44147 0.720735 − 0.693211i \(-0.243805\pi\)
0.720735 + 0.693211i \(0.243805\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) −35.0000 −0.249472
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 147.000 0.851677 0.425838 − 0.904799i \(-0.359979\pi\)
0.425838 + 0.904799i \(0.359979\pi\)
\(32\) 0 0
\(33\) 35.0000 0.184628
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 219.000 0.973064 0.486532 − 0.873663i \(-0.338262\pi\)
0.486532 + 0.873663i \(0.338262\pi\)
\(38\) 0 0
\(39\) 98.0000 0.402373
\(40\) 0 0
\(41\) −350.000 −1.33319 −0.666595 − 0.745420i \(-0.732249\pi\)
−0.666595 + 0.745420i \(0.732249\pi\)
\(42\) 0 0
\(43\) 124.000 0.439763 0.219882 − 0.975527i \(-0.429433\pi\)
0.219882 + 0.975527i \(0.429433\pi\)
\(44\) 0 0
\(45\) −154.000 −0.510155
\(46\) 0 0
\(47\) 525.000 1.62934 0.814671 − 0.579923i \(-0.196917\pi\)
0.814671 + 0.579923i \(0.196917\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 147.000 0.403610
\(52\) 0 0
\(53\) 303.000 0.785288 0.392644 − 0.919691i \(-0.371561\pi\)
0.392644 + 0.919691i \(0.371561\pi\)
\(54\) 0 0
\(55\) −35.0000 −0.0858073
\(56\) 0 0
\(57\) 343.000 0.797043
\(58\) 0 0
\(59\) −105.000 −0.231692 −0.115846 − 0.993267i \(-0.536958\pi\)
−0.115846 + 0.993267i \(0.536958\pi\)
\(60\) 0 0
\(61\) 413.000 0.866873 0.433436 − 0.901184i \(-0.357301\pi\)
0.433436 + 0.901184i \(0.357301\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −98.0000 −0.187006
\(66\) 0 0
\(67\) −415.000 −0.756721 −0.378361 − 0.925658i \(-0.623512\pi\)
−0.378361 + 0.925658i \(0.623512\pi\)
\(68\) 0 0
\(69\) 1113.00 1.94188
\(70\) 0 0
\(71\) 432.000 0.722098 0.361049 − 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) 0 0
\(73\) 1113.00 1.78448 0.892238 − 0.451565i \(-0.149134\pi\)
0.892238 + 0.451565i \(0.149134\pi\)
\(74\) 0 0
\(75\) −532.000 −0.819068
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 103.000 0.146689 0.0733443 − 0.997307i \(-0.476633\pi\)
0.0733443 + 0.997307i \(0.476633\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) 1092.00 1.44413 0.722064 − 0.691827i \(-0.243194\pi\)
0.722064 + 0.691827i \(0.243194\pi\)
\(84\) 0 0
\(85\) −147.000 −0.187581
\(86\) 0 0
\(87\) 406.000 0.500319
\(88\) 0 0
\(89\) 329.000 0.391842 0.195921 − 0.980620i \(-0.437230\pi\)
0.195921 + 0.980620i \(0.437230\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1029.00 1.14734
\(94\) 0 0
\(95\) −343.000 −0.370432
\(96\) 0 0
\(97\) 882.000 0.923232 0.461616 − 0.887080i \(-0.347270\pi\)
0.461616 + 0.887080i \(0.347270\pi\)
\(98\) 0 0
\(99\) 110.000 0.111671
\(100\) 0 0
\(101\) −1379.00 −1.35857 −0.679285 − 0.733874i \(-0.737710\pi\)
−0.679285 + 0.733874i \(0.737710\pi\)
\(102\) 0 0
\(103\) −679.000 −0.649552 −0.324776 − 0.945791i \(-0.605289\pi\)
−0.324776 + 0.945791i \(0.605289\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −457.000 −0.412896 −0.206448 − 0.978458i \(-0.566190\pi\)
−0.206448 + 0.978458i \(0.566190\pi\)
\(108\) 0 0
\(109\) −1125.00 −0.988582 −0.494291 − 0.869296i \(-0.664572\pi\)
−0.494291 + 0.869296i \(0.664572\pi\)
\(110\) 0 0
\(111\) 1533.00 1.31086
\(112\) 0 0
\(113\) −1538.00 −1.28038 −0.640190 − 0.768217i \(-0.721144\pi\)
−0.640190 + 0.768217i \(0.721144\pi\)
\(114\) 0 0
\(115\) −1113.00 −0.902502
\(116\) 0 0
\(117\) 308.000 0.243373
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1306.00 −0.981217
\(122\) 0 0
\(123\) −2450.00 −1.79601
\(124\) 0 0
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) −72.0000 −0.0503068 −0.0251534 − 0.999684i \(-0.508007\pi\)
−0.0251534 + 0.999684i \(0.508007\pi\)
\(128\) 0 0
\(129\) 868.000 0.592427
\(130\) 0 0
\(131\) 2149.00 1.43327 0.716637 − 0.697446i \(-0.245680\pi\)
0.716637 + 0.697446i \(0.245680\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 245.000 0.156194
\(136\) 0 0
\(137\) −1125.00 −0.701571 −0.350786 − 0.936456i \(-0.614085\pi\)
−0.350786 + 0.936456i \(0.614085\pi\)
\(138\) 0 0
\(139\) 252.000 0.153772 0.0768862 − 0.997040i \(-0.475502\pi\)
0.0768862 + 0.997040i \(0.475502\pi\)
\(140\) 0 0
\(141\) 3675.00 2.19497
\(142\) 0 0
\(143\) 70.0000 0.0409349
\(144\) 0 0
\(145\) −406.000 −0.232527
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −201.000 −0.110514 −0.0552569 − 0.998472i \(-0.517598\pi\)
−0.0552569 + 0.998472i \(0.517598\pi\)
\(150\) 0 0
\(151\) −1619.00 −0.872532 −0.436266 − 0.899818i \(-0.643699\pi\)
−0.436266 + 0.899818i \(0.643699\pi\)
\(152\) 0 0
\(153\) 462.000 0.244121
\(154\) 0 0
\(155\) −1029.00 −0.533234
\(156\) 0 0
\(157\) −679.000 −0.345160 −0.172580 − 0.984996i \(-0.555210\pi\)
−0.172580 + 0.984996i \(0.555210\pi\)
\(158\) 0 0
\(159\) 2121.00 1.05790
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 467.000 0.224407 0.112203 − 0.993685i \(-0.464209\pi\)
0.112203 + 0.993685i \(0.464209\pi\)
\(164\) 0 0
\(165\) −245.000 −0.115595
\(166\) 0 0
\(167\) 1204.00 0.557894 0.278947 − 0.960306i \(-0.410015\pi\)
0.278947 + 0.960306i \(0.410015\pi\)
\(168\) 0 0
\(169\) −2001.00 −0.910787
\(170\) 0 0
\(171\) 1078.00 0.482086
\(172\) 0 0
\(173\) 2821.00 1.23975 0.619875 − 0.784701i \(-0.287183\pi\)
0.619875 + 0.784701i \(0.287183\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −735.000 −0.312124
\(178\) 0 0
\(179\) 3253.00 1.35833 0.679164 − 0.733987i \(-0.262343\pi\)
0.679164 + 0.733987i \(0.262343\pi\)
\(180\) 0 0
\(181\) −1582.00 −0.649664 −0.324832 − 0.945772i \(-0.605308\pi\)
−0.324832 + 0.945772i \(0.605308\pi\)
\(182\) 0 0
\(183\) 2891.00 1.16781
\(184\) 0 0
\(185\) −1533.00 −0.609235
\(186\) 0 0
\(187\) 105.000 0.0410608
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2557.00 −0.968681 −0.484340 − 0.874880i \(-0.660940\pi\)
−0.484340 + 0.874880i \(0.660940\pi\)
\(192\) 0 0
\(193\) −397.000 −0.148066 −0.0740329 − 0.997256i \(-0.523587\pi\)
−0.0740329 + 0.997256i \(0.523587\pi\)
\(194\) 0 0
\(195\) −686.000 −0.251926
\(196\) 0 0
\(197\) 2914.00 1.05388 0.526939 − 0.849903i \(-0.323340\pi\)
0.526939 + 0.849903i \(0.323340\pi\)
\(198\) 0 0
\(199\) 3339.00 1.18942 0.594712 − 0.803939i \(-0.297266\pi\)
0.594712 + 0.803939i \(0.297266\pi\)
\(200\) 0 0
\(201\) −2905.00 −1.01942
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2450.00 0.834709
\(206\) 0 0
\(207\) 3498.00 1.17453
\(208\) 0 0
\(209\) 245.000 0.0810861
\(210\) 0 0
\(211\) −1780.00 −0.580759 −0.290380 − 0.956911i \(-0.593782\pi\)
−0.290380 + 0.956911i \(0.593782\pi\)
\(212\) 0 0
\(213\) 3024.00 0.972775
\(214\) 0 0
\(215\) −868.000 −0.275335
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7791.00 2.40396
\(220\) 0 0
\(221\) 294.000 0.0894868
\(222\) 0 0
\(223\) −1400.00 −0.420408 −0.210204 − 0.977658i \(-0.567413\pi\)
−0.210204 + 0.977658i \(0.567413\pi\)
\(224\) 0 0
\(225\) −1672.00 −0.495407
\(226\) 0 0
\(227\) −2205.00 −0.644718 −0.322359 − 0.946617i \(-0.604476\pi\)
−0.322359 + 0.946617i \(0.604476\pi\)
\(228\) 0 0
\(229\) −287.000 −0.0828188 −0.0414094 − 0.999142i \(-0.513185\pi\)
−0.0414094 + 0.999142i \(0.513185\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4587.00 1.28972 0.644859 − 0.764301i \(-0.276916\pi\)
0.644859 + 0.764301i \(0.276916\pi\)
\(234\) 0 0
\(235\) −3675.00 −1.02013
\(236\) 0 0
\(237\) 721.000 0.197612
\(238\) 0 0
\(239\) −1668.00 −0.451439 −0.225720 − 0.974192i \(-0.572473\pi\)
−0.225720 + 0.974192i \(0.572473\pi\)
\(240\) 0 0
\(241\) 3409.00 0.911174 0.455587 − 0.890191i \(-0.349429\pi\)
0.455587 + 0.890191i \(0.349429\pi\)
\(242\) 0 0
\(243\) −4928.00 −1.30095
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 686.000 0.176717
\(248\) 0 0
\(249\) 7644.00 1.94546
\(250\) 0 0
\(251\) −4760.00 −1.19701 −0.598503 − 0.801121i \(-0.704238\pi\)
−0.598503 + 0.801121i \(0.704238\pi\)
\(252\) 0 0
\(253\) 795.000 0.197554
\(254\) 0 0
\(255\) −1029.00 −0.252700
\(256\) 0 0
\(257\) 805.000 0.195387 0.0976936 − 0.995217i \(-0.468853\pi\)
0.0976936 + 0.995217i \(0.468853\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1276.00 0.302615
\(262\) 0 0
\(263\) 257.000 0.0602559 0.0301279 − 0.999546i \(-0.490409\pi\)
0.0301279 + 0.999546i \(0.490409\pi\)
\(264\) 0 0
\(265\) −2121.00 −0.491668
\(266\) 0 0
\(267\) 2303.00 0.527870
\(268\) 0 0
\(269\) −3591.00 −0.813930 −0.406965 − 0.913444i \(-0.633413\pi\)
−0.406965 + 0.913444i \(0.633413\pi\)
\(270\) 0 0
\(271\) 1393.00 0.312246 0.156123 − 0.987738i \(-0.450100\pi\)
0.156123 + 0.987738i \(0.450100\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −380.000 −0.0833268
\(276\) 0 0
\(277\) 415.000 0.0900178 0.0450089 − 0.998987i \(-0.485668\pi\)
0.0450089 + 0.998987i \(0.485668\pi\)
\(278\) 0 0
\(279\) 3234.00 0.693959
\(280\) 0 0
\(281\) −4954.00 −1.05171 −0.525856 − 0.850574i \(-0.676255\pi\)
−0.525856 + 0.850574i \(0.676255\pi\)
\(282\) 0 0
\(283\) −4277.00 −0.898379 −0.449190 − 0.893437i \(-0.648287\pi\)
−0.449190 + 0.893437i \(0.648287\pi\)
\(284\) 0 0
\(285\) −2401.00 −0.499028
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4472.00 −0.910238
\(290\) 0 0
\(291\) 6174.00 1.24373
\(292\) 0 0
\(293\) −7742.00 −1.54366 −0.771830 − 0.635829i \(-0.780658\pi\)
−0.771830 + 0.635829i \(0.780658\pi\)
\(294\) 0 0
\(295\) 735.000 0.145062
\(296\) 0 0
\(297\) −175.000 −0.0341903
\(298\) 0 0
\(299\) 2226.00 0.430545
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −9653.00 −1.83020
\(304\) 0 0
\(305\) −2891.00 −0.542748
\(306\) 0 0
\(307\) −7364.00 −1.36901 −0.684504 − 0.729009i \(-0.739981\pi\)
−0.684504 + 0.729009i \(0.739981\pi\)
\(308\) 0 0
\(309\) −4753.00 −0.875044
\(310\) 0 0
\(311\) 9975.00 1.81875 0.909374 − 0.415980i \(-0.136562\pi\)
0.909374 + 0.415980i \(0.136562\pi\)
\(312\) 0 0
\(313\) 4753.00 0.858324 0.429162 − 0.903228i \(-0.358809\pi\)
0.429162 + 0.903228i \(0.358809\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3477.00 −0.616050 −0.308025 − 0.951378i \(-0.599668\pi\)
−0.308025 + 0.951378i \(0.599668\pi\)
\(318\) 0 0
\(319\) 290.000 0.0508993
\(320\) 0 0
\(321\) −3199.00 −0.556233
\(322\) 0 0
\(323\) 1029.00 0.177260
\(324\) 0 0
\(325\) −1064.00 −0.181600
\(326\) 0 0
\(327\) −7875.00 −1.33177
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3341.00 −0.554797 −0.277399 − 0.960755i \(-0.589472\pi\)
−0.277399 + 0.960755i \(0.589472\pi\)
\(332\) 0 0
\(333\) 4818.00 0.792867
\(334\) 0 0
\(335\) 2905.00 0.473782
\(336\) 0 0
\(337\) 7366.00 1.19066 0.595329 − 0.803482i \(-0.297022\pi\)
0.595329 + 0.803482i \(0.297022\pi\)
\(338\) 0 0
\(339\) −10766.0 −1.72486
\(340\) 0 0
\(341\) 735.000 0.116723
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −7791.00 −1.21581
\(346\) 0 0
\(347\) −7415.00 −1.14714 −0.573571 − 0.819156i \(-0.694442\pi\)
−0.573571 + 0.819156i \(0.694442\pi\)
\(348\) 0 0
\(349\) 3878.00 0.594798 0.297399 − 0.954753i \(-0.403881\pi\)
0.297399 + 0.954753i \(0.403881\pi\)
\(350\) 0 0
\(351\) −490.000 −0.0745136
\(352\) 0 0
\(353\) −1267.00 −0.191036 −0.0955179 − 0.995428i \(-0.530451\pi\)
−0.0955179 + 0.995428i \(0.530451\pi\)
\(354\) 0 0
\(355\) −3024.00 −0.452105
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4685.00 −0.688760 −0.344380 − 0.938830i \(-0.611911\pi\)
−0.344380 + 0.938830i \(0.611911\pi\)
\(360\) 0 0
\(361\) −4458.00 −0.649949
\(362\) 0 0
\(363\) −9142.00 −1.32185
\(364\) 0 0
\(365\) −7791.00 −1.11726
\(366\) 0 0
\(367\) −4641.00 −0.660104 −0.330052 − 0.943963i \(-0.607066\pi\)
−0.330052 + 0.943963i \(0.607066\pi\)
\(368\) 0 0
\(369\) −7700.00 −1.08630
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −8797.00 −1.22116 −0.610578 − 0.791956i \(-0.709063\pi\)
−0.610578 + 0.791956i \(0.709063\pi\)
\(374\) 0 0
\(375\) 9849.00 1.35627
\(376\) 0 0
\(377\) 812.000 0.110929
\(378\) 0 0
\(379\) −13680.0 −1.85407 −0.927037 − 0.374969i \(-0.877653\pi\)
−0.927037 + 0.374969i \(0.877653\pi\)
\(380\) 0 0
\(381\) −504.000 −0.0677709
\(382\) 0 0
\(383\) 9765.00 1.30279 0.651395 − 0.758739i \(-0.274184\pi\)
0.651395 + 0.758739i \(0.274184\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2728.00 0.358326
\(388\) 0 0
\(389\) 1731.00 0.225617 0.112809 − 0.993617i \(-0.464015\pi\)
0.112809 + 0.993617i \(0.464015\pi\)
\(390\) 0 0
\(391\) 3339.00 0.431868
\(392\) 0 0
\(393\) 15043.0 1.93084
\(394\) 0 0
\(395\) −721.000 −0.0918416
\(396\) 0 0
\(397\) −10983.0 −1.38847 −0.694233 − 0.719750i \(-0.744256\pi\)
−0.694233 + 0.719750i \(0.744256\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6603.00 0.822289 0.411145 − 0.911570i \(-0.365129\pi\)
0.411145 + 0.911570i \(0.365129\pi\)
\(402\) 0 0
\(403\) 2058.00 0.254383
\(404\) 0 0
\(405\) 5873.00 0.720572
\(406\) 0 0
\(407\) 1095.00 0.133359
\(408\) 0 0
\(409\) −10955.0 −1.32443 −0.662213 − 0.749316i \(-0.730382\pi\)
−0.662213 + 0.749316i \(0.730382\pi\)
\(410\) 0 0
\(411\) −7875.00 −0.945122
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −7644.00 −0.904167
\(416\) 0 0
\(417\) 1764.00 0.207155
\(418\) 0 0
\(419\) 6636.00 0.773723 0.386861 − 0.922138i \(-0.373559\pi\)
0.386861 + 0.922138i \(0.373559\pi\)
\(420\) 0 0
\(421\) −16630.0 −1.92517 −0.962585 − 0.270980i \(-0.912652\pi\)
−0.962585 + 0.270980i \(0.912652\pi\)
\(422\) 0 0
\(423\) 11550.0 1.32761
\(424\) 0 0
\(425\) −1596.00 −0.182159
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 490.000 0.0551455
\(430\) 0 0
\(431\) −4923.00 −0.550192 −0.275096 − 0.961417i \(-0.588710\pi\)
−0.275096 + 0.961417i \(0.588710\pi\)
\(432\) 0 0
\(433\) −8974.00 −0.995988 −0.497994 − 0.867180i \(-0.665930\pi\)
−0.497994 + 0.867180i \(0.665930\pi\)
\(434\) 0 0
\(435\) −2842.00 −0.313249
\(436\) 0 0
\(437\) 7791.00 0.852847
\(438\) 0 0
\(439\) −4179.00 −0.454334 −0.227167 − 0.973856i \(-0.572946\pi\)
−0.227167 + 0.973856i \(0.572946\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12927.0 1.38641 0.693206 − 0.720740i \(-0.256198\pi\)
0.693206 + 0.720740i \(0.256198\pi\)
\(444\) 0 0
\(445\) −2303.00 −0.245332
\(446\) 0 0
\(447\) −1407.00 −0.148879
\(448\) 0 0
\(449\) −2826.00 −0.297032 −0.148516 − 0.988910i \(-0.547450\pi\)
−0.148516 + 0.988910i \(0.547450\pi\)
\(450\) 0 0
\(451\) −1750.00 −0.182715
\(452\) 0 0
\(453\) −11333.0 −1.17543
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8479.00 0.867901 0.433951 − 0.900937i \(-0.357119\pi\)
0.433951 + 0.900937i \(0.357119\pi\)
\(458\) 0 0
\(459\) −735.000 −0.0747426
\(460\) 0 0
\(461\) −9338.00 −0.943414 −0.471707 − 0.881755i \(-0.656362\pi\)
−0.471707 + 0.881755i \(0.656362\pi\)
\(462\) 0 0
\(463\) 4016.00 0.403109 0.201554 − 0.979477i \(-0.435401\pi\)
0.201554 + 0.979477i \(0.435401\pi\)
\(464\) 0 0
\(465\) −7203.00 −0.718347
\(466\) 0 0
\(467\) −5859.00 −0.580561 −0.290281 − 0.956942i \(-0.593749\pi\)
−0.290281 + 0.956942i \(0.593749\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4753.00 −0.464982
\(472\) 0 0
\(473\) 620.000 0.0602698
\(474\) 0 0
\(475\) −3724.00 −0.359724
\(476\) 0 0
\(477\) 6666.00 0.639864
\(478\) 0 0
\(479\) 6503.00 0.620312 0.310156 − 0.950686i \(-0.399619\pi\)
0.310156 + 0.950686i \(0.399619\pi\)
\(480\) 0 0
\(481\) 3066.00 0.290639
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6174.00 −0.578035
\(486\) 0 0
\(487\) 16049.0 1.49333 0.746663 − 0.665203i \(-0.231655\pi\)
0.746663 + 0.665203i \(0.231655\pi\)
\(488\) 0 0
\(489\) 3269.00 0.302309
\(490\) 0 0
\(491\) −8864.00 −0.814718 −0.407359 − 0.913268i \(-0.633550\pi\)
−0.407359 + 0.913268i \(0.633550\pi\)
\(492\) 0 0
\(493\) 1218.00 0.111270
\(494\) 0 0
\(495\) −770.000 −0.0699170
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10211.0 0.916046 0.458023 − 0.888940i \(-0.348558\pi\)
0.458023 + 0.888940i \(0.348558\pi\)
\(500\) 0 0
\(501\) 8428.00 0.751567
\(502\) 0 0
\(503\) −1680.00 −0.148921 −0.0744607 − 0.997224i \(-0.523724\pi\)
−0.0744607 + 0.997224i \(0.523724\pi\)
\(504\) 0 0
\(505\) 9653.00 0.850600
\(506\) 0 0
\(507\) −14007.0 −1.22697
\(508\) 0 0
\(509\) 9457.00 0.823525 0.411762 − 0.911291i \(-0.364913\pi\)
0.411762 + 0.911291i \(0.364913\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1715.00 −0.147601
\(514\) 0 0
\(515\) 4753.00 0.406684
\(516\) 0 0
\(517\) 2625.00 0.223302
\(518\) 0 0
\(519\) 19747.0 1.67013
\(520\) 0 0
\(521\) 18081.0 1.52043 0.760214 − 0.649673i \(-0.225094\pi\)
0.760214 + 0.649673i \(0.225094\pi\)
\(522\) 0 0
\(523\) 20377.0 1.70368 0.851839 − 0.523803i \(-0.175487\pi\)
0.851839 + 0.523803i \(0.175487\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3087.00 0.255165
\(528\) 0 0
\(529\) 13114.0 1.07783
\(530\) 0 0
\(531\) −2310.00 −0.188786
\(532\) 0 0
\(533\) −4900.00 −0.398204
\(534\) 0 0
\(535\) 3199.00 0.258514
\(536\) 0 0
\(537\) 22771.0 1.82987
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6193.00 −0.492159 −0.246079 − 0.969250i \(-0.579142\pi\)
−0.246079 + 0.969250i \(0.579142\pi\)
\(542\) 0 0
\(543\) −11074.0 −0.875195
\(544\) 0 0
\(545\) 7875.00 0.618950
\(546\) 0 0
\(547\) 18464.0 1.44326 0.721630 − 0.692279i \(-0.243393\pi\)
0.721630 + 0.692279i \(0.243393\pi\)
\(548\) 0 0
\(549\) 9086.00 0.706341
\(550\) 0 0
\(551\) 2842.00 0.219734
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10731.0 −0.820731
\(556\) 0 0
\(557\) −9413.00 −0.716053 −0.358027 − 0.933711i \(-0.616550\pi\)
−0.358027 + 0.933711i \(0.616550\pi\)
\(558\) 0 0
\(559\) 1736.00 0.131351
\(560\) 0 0
\(561\) 735.000 0.0553150
\(562\) 0 0
\(563\) 3199.00 0.239470 0.119735 − 0.992806i \(-0.461795\pi\)
0.119735 + 0.992806i \(0.461795\pi\)
\(564\) 0 0
\(565\) 10766.0 0.801644
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21583.0 1.59017 0.795085 − 0.606498i \(-0.207426\pi\)
0.795085 + 0.606498i \(0.207426\pi\)
\(570\) 0 0
\(571\) −20267.0 −1.48537 −0.742686 − 0.669640i \(-0.766449\pi\)
−0.742686 + 0.669640i \(0.766449\pi\)
\(572\) 0 0
\(573\) −17899.0 −1.30496
\(574\) 0 0
\(575\) −12084.0 −0.876413
\(576\) 0 0
\(577\) −13951.0 −1.00656 −0.503282 − 0.864122i \(-0.667874\pi\)
−0.503282 + 0.864122i \(0.667874\pi\)
\(578\) 0 0
\(579\) −2779.00 −0.199467
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1515.00 0.107624
\(584\) 0 0
\(585\) −2156.00 −0.152375
\(586\) 0 0
\(587\) −20972.0 −1.47463 −0.737314 − 0.675550i \(-0.763906\pi\)
−0.737314 + 0.675550i \(0.763906\pi\)
\(588\) 0 0
\(589\) 7203.00 0.503895
\(590\) 0 0
\(591\) 20398.0 1.41973
\(592\) 0 0
\(593\) 189.000 0.0130882 0.00654410 − 0.999979i \(-0.497917\pi\)
0.00654410 + 0.999979i \(0.497917\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23373.0 1.60233
\(598\) 0 0
\(599\) 10281.0 0.701286 0.350643 − 0.936509i \(-0.385963\pi\)
0.350643 + 0.936509i \(0.385963\pi\)
\(600\) 0 0
\(601\) 6090.00 0.413338 0.206669 − 0.978411i \(-0.433738\pi\)
0.206669 + 0.978411i \(0.433738\pi\)
\(602\) 0 0
\(603\) −9130.00 −0.616588
\(604\) 0 0
\(605\) 9142.00 0.614339
\(606\) 0 0
\(607\) 4949.00 0.330929 0.165464 − 0.986216i \(-0.447088\pi\)
0.165464 + 0.986216i \(0.447088\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7350.00 0.486660
\(612\) 0 0
\(613\) −15797.0 −1.04084 −0.520420 − 0.853910i \(-0.674225\pi\)
−0.520420 + 0.853910i \(0.674225\pi\)
\(614\) 0 0
\(615\) 17150.0 1.12448
\(616\) 0 0
\(617\) −9378.00 −0.611903 −0.305951 − 0.952047i \(-0.598975\pi\)
−0.305951 + 0.952047i \(0.598975\pi\)
\(618\) 0 0
\(619\) −24353.0 −1.58131 −0.790654 − 0.612263i \(-0.790259\pi\)
−0.790654 + 0.612263i \(0.790259\pi\)
\(620\) 0 0
\(621\) −5565.00 −0.359607
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 0 0
\(627\) 1715.00 0.109235
\(628\) 0 0
\(629\) 4599.00 0.291533
\(630\) 0 0
\(631\) 12640.0 0.797449 0.398725 − 0.917071i \(-0.369453\pi\)
0.398725 + 0.917071i \(0.369453\pi\)
\(632\) 0 0
\(633\) −12460.0 −0.782371
\(634\) 0 0
\(635\) 504.000 0.0314971
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 9504.00 0.588376
\(640\) 0 0
\(641\) −1041.00 −0.0641451 −0.0320726 − 0.999486i \(-0.510211\pi\)
−0.0320726 + 0.999486i \(0.510211\pi\)
\(642\) 0 0
\(643\) 9548.00 0.585593 0.292797 − 0.956175i \(-0.405414\pi\)
0.292797 + 0.956175i \(0.405414\pi\)
\(644\) 0 0
\(645\) −6076.00 −0.370918
\(646\) 0 0
\(647\) −3241.00 −0.196935 −0.0984674 − 0.995140i \(-0.531394\pi\)
−0.0984674 + 0.995140i \(0.531394\pi\)
\(648\) 0 0
\(649\) −525.000 −0.0317535
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8853.00 −0.530543 −0.265272 − 0.964174i \(-0.585462\pi\)
−0.265272 + 0.964174i \(0.585462\pi\)
\(654\) 0 0
\(655\) −15043.0 −0.897372
\(656\) 0 0
\(657\) 24486.0 1.45402
\(658\) 0 0
\(659\) −7044.00 −0.416381 −0.208191 − 0.978088i \(-0.566757\pi\)
−0.208191 + 0.978088i \(0.566757\pi\)
\(660\) 0 0
\(661\) 12089.0 0.711358 0.355679 − 0.934608i \(-0.384250\pi\)
0.355679 + 0.934608i \(0.384250\pi\)
\(662\) 0 0
\(663\) 2058.00 0.120552
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9222.00 0.535348
\(668\) 0 0
\(669\) −9800.00 −0.566353
\(670\) 0 0
\(671\) 2065.00 0.118805
\(672\) 0 0
\(673\) 982.000 0.0562456 0.0281228 − 0.999604i \(-0.491047\pi\)
0.0281228 + 0.999604i \(0.491047\pi\)
\(674\) 0 0
\(675\) 2660.00 0.151679
\(676\) 0 0
\(677\) 30513.0 1.73222 0.866108 − 0.499857i \(-0.166614\pi\)
0.866108 + 0.499857i \(0.166614\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −15435.0 −0.868532
\(682\) 0 0
\(683\) −11475.0 −0.642868 −0.321434 − 0.946932i \(-0.604165\pi\)
−0.321434 + 0.946932i \(0.604165\pi\)
\(684\) 0 0
\(685\) 7875.00 0.439253
\(686\) 0 0
\(687\) −2009.00 −0.111569
\(688\) 0 0
\(689\) 4242.00 0.234553
\(690\) 0 0
\(691\) −28315.0 −1.55883 −0.779416 − 0.626506i \(-0.784484\pi\)
−0.779416 + 0.626506i \(0.784484\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1764.00 −0.0962767
\(696\) 0 0
\(697\) −7350.00 −0.399428
\(698\) 0 0
\(699\) 32109.0 1.73744
\(700\) 0 0
\(701\) 10614.0 0.571876 0.285938 − 0.958248i \(-0.407695\pi\)
0.285938 + 0.958248i \(0.407695\pi\)
\(702\) 0 0
\(703\) 10731.0 0.575715
\(704\) 0 0
\(705\) −25725.0 −1.37427
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10299.0 0.545539 0.272769 − 0.962079i \(-0.412060\pi\)
0.272769 + 0.962079i \(0.412060\pi\)
\(710\) 0 0
\(711\) 2266.00 0.119524
\(712\) 0 0
\(713\) 23373.0 1.22767
\(714\) 0 0
\(715\) −490.000 −0.0256293
\(716\) 0 0
\(717\) −11676.0 −0.608156
\(718\) 0 0
\(719\) 32529.0 1.68724 0.843621 − 0.536939i \(-0.180420\pi\)
0.843621 + 0.536939i \(0.180420\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 23863.0 1.22749
\(724\) 0 0
\(725\) −4408.00 −0.225806
\(726\) 0 0
\(727\) 29456.0 1.50270 0.751350 − 0.659904i \(-0.229403\pi\)
0.751350 + 0.659904i \(0.229403\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) 2604.00 0.131754
\(732\) 0 0
\(733\) −27867.0 −1.40422 −0.702109 − 0.712070i \(-0.747758\pi\)
−0.702109 + 0.712070i \(0.747758\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2075.00 −0.103709
\(738\) 0 0
\(739\) −19539.0 −0.972603 −0.486302 − 0.873791i \(-0.661654\pi\)
−0.486302 + 0.873791i \(0.661654\pi\)
\(740\) 0 0
\(741\) 4802.00 0.238065
\(742\) 0 0
\(743\) −1248.00 −0.0616214 −0.0308107 − 0.999525i \(-0.509809\pi\)
−0.0308107 + 0.999525i \(0.509809\pi\)
\(744\) 0 0
\(745\) 1407.00 0.0691926
\(746\) 0 0
\(747\) 24024.0 1.17670
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28093.0 −1.36502 −0.682509 − 0.730877i \(-0.739111\pi\)
−0.682509 + 0.730877i \(0.739111\pi\)
\(752\) 0 0
\(753\) −33320.0 −1.61255
\(754\) 0 0
\(755\) 11333.0 0.546292
\(756\) 0 0
\(757\) 35954.0 1.72625 0.863124 − 0.504991i \(-0.168504\pi\)
0.863124 + 0.504991i \(0.168504\pi\)
\(758\) 0 0
\(759\) 5565.00 0.266135
\(760\) 0 0
\(761\) 861.000 0.0410134 0.0205067 − 0.999790i \(-0.493472\pi\)
0.0205067 + 0.999790i \(0.493472\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3234.00 −0.152844
\(766\) 0 0
\(767\) −1470.00 −0.0692029
\(768\) 0 0
\(769\) −24710.0 −1.15873 −0.579366 − 0.815067i \(-0.696700\pi\)
−0.579366 + 0.815067i \(0.696700\pi\)
\(770\) 0 0
\(771\) 5635.00 0.263216
\(772\) 0 0
\(773\) −16499.0 −0.767694 −0.383847 − 0.923397i \(-0.625401\pi\)
−0.383847 + 0.923397i \(0.625401\pi\)
\(774\) 0 0
\(775\) −11172.0 −0.517819
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17150.0 −0.788784
\(780\) 0 0
\(781\) 2160.00 0.0989640
\(782\) 0 0
\(783\) −2030.00 −0.0926517
\(784\) 0 0
\(785\) 4753.00 0.216104
\(786\) 0 0
\(787\) 16471.0 0.746033 0.373016 − 0.927825i \(-0.378324\pi\)
0.373016 + 0.927825i \(0.378324\pi\)
\(788\) 0 0
\(789\) 1799.00 0.0811738
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5782.00 0.258922
\(794\) 0 0
\(795\) −14847.0 −0.662351
\(796\) 0 0
\(797\) 36470.0 1.62087 0.810435 − 0.585828i \(-0.199231\pi\)
0.810435 + 0.585828i \(0.199231\pi\)
\(798\) 0 0
\(799\) 11025.0 0.488156
\(800\) 0 0
\(801\) 7238.00 0.319279
\(802\) 0 0
\(803\) 5565.00 0.244564
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25137.0 −1.09649
\(808\) 0 0
\(809\) 35751.0 1.55369 0.776847 − 0.629690i \(-0.216818\pi\)
0.776847 + 0.629690i \(0.216818\pi\)
\(810\) 0 0
\(811\) −16492.0 −0.714072 −0.357036 − 0.934091i \(-0.616213\pi\)
−0.357036 + 0.934091i \(0.616213\pi\)
\(812\) 0 0
\(813\) 9751.00 0.420643
\(814\) 0 0
\(815\) −3269.00 −0.140501
\(816\) 0 0
\(817\) 6076.00 0.260186
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −41473.0 −1.76299 −0.881497 − 0.472190i \(-0.843464\pi\)
−0.881497 + 0.472190i \(0.843464\pi\)
\(822\) 0 0
\(823\) 25065.0 1.06162 0.530809 − 0.847492i \(-0.321888\pi\)
0.530809 + 0.847492i \(0.321888\pi\)
\(824\) 0 0
\(825\) −2660.00 −0.112254
\(826\) 0 0
\(827\) −9732.00 −0.409208 −0.204604 − 0.978845i \(-0.565591\pi\)
−0.204604 + 0.978845i \(0.565591\pi\)
\(828\) 0 0
\(829\) −27755.0 −1.16281 −0.581406 − 0.813614i \(-0.697497\pi\)
−0.581406 + 0.813614i \(0.697497\pi\)
\(830\) 0 0
\(831\) 2905.00 0.121268
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −8428.00 −0.349297
\(836\) 0 0
\(837\) −5145.00 −0.212470
\(838\) 0 0
\(839\) 21112.0 0.868733 0.434367 − 0.900736i \(-0.356972\pi\)
0.434367 + 0.900736i \(0.356972\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 0 0
\(843\) −34678.0 −1.41681
\(844\) 0 0
\(845\) 14007.0 0.570243
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −29939.0 −1.21025
\(850\) 0 0
\(851\) 34821.0 1.40264
\(852\) 0 0
\(853\) 21238.0 0.852492 0.426246 − 0.904607i \(-0.359836\pi\)
0.426246 + 0.904607i \(0.359836\pi\)
\(854\) 0 0
\(855\) −7546.00 −0.301834
\(856\) 0 0
\(857\) 35609.0 1.41935 0.709673 − 0.704531i \(-0.248842\pi\)
0.709673 + 0.704531i \(0.248842\pi\)
\(858\) 0 0
\(859\) 2177.00 0.0864706 0.0432353 − 0.999065i \(-0.486233\pi\)
0.0432353 + 0.999065i \(0.486233\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32247.0 1.27196 0.635980 − 0.771706i \(-0.280596\pi\)
0.635980 + 0.771706i \(0.280596\pi\)
\(864\) 0 0
\(865\) −19747.0 −0.776206
\(866\) 0 0
\(867\) −31304.0 −1.22623
\(868\) 0 0
\(869\) 515.000 0.0201038
\(870\) 0 0
\(871\) −5810.00 −0.226021
\(872\) 0 0
\(873\) 19404.0 0.752263
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27631.0 1.06389 0.531946 − 0.846779i \(-0.321461\pi\)
0.531946 + 0.846779i \(0.321461\pi\)
\(878\) 0 0
\(879\) −54194.0 −2.07954
\(880\) 0 0
\(881\) −24402.0 −0.933172 −0.466586 − 0.884476i \(-0.654516\pi\)
−0.466586 + 0.884476i \(0.654516\pi\)
\(882\) 0 0
\(883\) 19612.0 0.747448 0.373724 − 0.927540i \(-0.378081\pi\)
0.373724 + 0.927540i \(0.378081\pi\)
\(884\) 0 0
\(885\) 5145.00 0.195421
\(886\) 0 0
\(887\) 2261.00 0.0855884 0.0427942 − 0.999084i \(-0.486374\pi\)
0.0427942 + 0.999084i \(0.486374\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4195.00 −0.157730
\(892\) 0 0
\(893\) 25725.0 0.964003
\(894\) 0 0
\(895\) −22771.0 −0.850448
\(896\) 0 0
\(897\) 15582.0 0.580009
\(898\) 0 0
\(899\) 8526.00 0.316305
\(900\) 0 0
\(901\) 6363.00 0.235274
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11074.0 0.406754
\(906\) 0 0
\(907\) 23833.0 0.872505 0.436252 − 0.899824i \(-0.356305\pi\)
0.436252 + 0.899824i \(0.356305\pi\)
\(908\) 0 0
\(909\) −30338.0 −1.10698
\(910\) 0 0
\(911\) −31824.0 −1.15738 −0.578692 − 0.815546i \(-0.696437\pi\)
−0.578692 + 0.815546i \(0.696437\pi\)
\(912\) 0 0
\(913\) 5460.00 0.197919
\(914\) 0 0
\(915\) −20237.0 −0.731163
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16819.0 0.603708 0.301854 − 0.953354i \(-0.402394\pi\)
0.301854 + 0.953354i \(0.402394\pi\)
\(920\) 0 0
\(921\) −51548.0 −1.84426
\(922\) 0 0
\(923\) 6048.00 0.215680
\(924\) 0 0
\(925\) −16644.0 −0.591623
\(926\) 0 0
\(927\) −14938.0 −0.529265
\(928\) 0 0
\(929\) −1799.00 −0.0635342 −0.0317671 − 0.999495i \(-0.510113\pi\)
−0.0317671 + 0.999495i \(0.510113\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 69825.0 2.45013
\(934\) 0 0
\(935\) −735.000 −0.0257081
\(936\) 0 0
\(937\) −14154.0 −0.493480 −0.246740 − 0.969082i \(-0.579359\pi\)
−0.246740 + 0.969082i \(0.579359\pi\)
\(938\) 0 0
\(939\) 33271.0 1.15629
\(940\) 0 0
\(941\) −12047.0 −0.417344 −0.208672 − 0.977986i \(-0.566914\pi\)
−0.208672 + 0.977986i \(0.566914\pi\)
\(942\) 0 0
\(943\) −55650.0 −1.92175
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24379.0 0.836548 0.418274 − 0.908321i \(-0.362635\pi\)
0.418274 + 0.908321i \(0.362635\pi\)
\(948\) 0 0
\(949\) 15582.0 0.532996
\(950\) 0 0
\(951\) −24339.0 −0.829912
\(952\) 0 0
\(953\) −52330.0 −1.77874 −0.889368 − 0.457192i \(-0.848855\pi\)
−0.889368 + 0.457192i \(0.848855\pi\)
\(954\) 0 0
\(955\) 17899.0 0.606490
\(956\) 0 0
\(957\) 2030.00 0.0685690
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8182.00 −0.274647
\(962\) 0 0
\(963\) −10054.0 −0.336434
\(964\) 0 0
\(965\) 2779.00 0.0927038
\(966\) 0 0
\(967\) 12416.0 0.412897 0.206449 − 0.978457i \(-0.433809\pi\)
0.206449 + 0.978457i \(0.433809\pi\)
\(968\) 0 0
\(969\) 7203.00 0.238796
\(970\) 0 0
\(971\) 36813.0 1.21667 0.608334 − 0.793681i \(-0.291838\pi\)
0.608334 + 0.793681i \(0.291838\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7448.00 −0.244643
\(976\) 0 0
\(977\) 34995.0 1.14595 0.572973 − 0.819574i \(-0.305790\pi\)
0.572973 + 0.819574i \(0.305790\pi\)
\(978\) 0 0
\(979\) 1645.00 0.0537022
\(980\) 0 0
\(981\) −24750.0 −0.805511
\(982\) 0 0
\(983\) −14301.0 −0.464019 −0.232010 − 0.972713i \(-0.574530\pi\)
−0.232010 + 0.972713i \(0.574530\pi\)
\(984\) 0 0
\(985\) −20398.0 −0.659832
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19716.0 0.633905
\(990\) 0 0
\(991\) 2665.00 0.0854253 0.0427127 − 0.999087i \(-0.486400\pi\)
0.0427127 + 0.999087i \(0.486400\pi\)
\(992\) 0 0
\(993\) −23387.0 −0.747396
\(994\) 0 0
\(995\) −23373.0 −0.744697
\(996\) 0 0
\(997\) −24871.0 −0.790043 −0.395021 − 0.918672i \(-0.629263\pi\)
−0.395021 + 0.918672i \(0.629263\pi\)
\(998\) 0 0
\(999\) −7665.00 −0.242753
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 784.4.a.r.1.1 1
4.3 odd 2 49.4.a.c.1.1 1
7.3 odd 6 112.4.i.c.65.1 2
7.5 odd 6 112.4.i.c.81.1 2
7.6 odd 2 784.4.a.b.1.1 1
12.11 even 2 441.4.a.e.1.1 1
20.19 odd 2 1225.4.a.d.1.1 1
28.3 even 6 7.4.c.a.2.1 ✓ 2
28.11 odd 6 49.4.c.a.30.1 2
28.19 even 6 7.4.c.a.4.1 yes 2
28.23 odd 6 49.4.c.a.18.1 2
28.27 even 2 49.4.a.d.1.1 1
56.3 even 6 448.4.i.f.65.1 2
56.5 odd 6 448.4.i.a.193.1 2
56.19 even 6 448.4.i.f.193.1 2
56.45 odd 6 448.4.i.a.65.1 2
84.11 even 6 441.4.e.k.226.1 2
84.23 even 6 441.4.e.k.361.1 2
84.47 odd 6 63.4.e.b.46.1 2
84.59 odd 6 63.4.e.b.37.1 2
84.83 odd 2 441.4.a.d.1.1 1
140.3 odd 12 175.4.k.a.149.2 4
140.19 even 6 175.4.e.a.151.1 2
140.47 odd 12 175.4.k.a.74.2 4
140.59 even 6 175.4.e.a.51.1 2
140.87 odd 12 175.4.k.a.149.1 4
140.103 odd 12 175.4.k.a.74.1 4
140.139 even 2 1225.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.c.a.2.1 ✓ 2 28.3 even 6
7.4.c.a.4.1 yes 2 28.19 even 6
49.4.a.c.1.1 1 4.3 odd 2
49.4.a.d.1.1 1 28.27 even 2
49.4.c.a.18.1 2 28.23 odd 6
49.4.c.a.30.1 2 28.11 odd 6
63.4.e.b.37.1 2 84.59 odd 6
63.4.e.b.46.1 2 84.47 odd 6
112.4.i.c.65.1 2 7.3 odd 6
112.4.i.c.81.1 2 7.5 odd 6
175.4.e.a.51.1 2 140.59 even 6
175.4.e.a.151.1 2 140.19 even 6
175.4.k.a.74.1 4 140.103 odd 12
175.4.k.a.74.2 4 140.47 odd 12
175.4.k.a.149.1 4 140.87 odd 12
175.4.k.a.149.2 4 140.3 odd 12
441.4.a.d.1.1 1 84.83 odd 2
441.4.a.e.1.1 1 12.11 even 2
441.4.e.k.226.1 2 84.11 even 6
441.4.e.k.361.1 2 84.23 even 6
448.4.i.a.65.1 2 56.45 odd 6
448.4.i.a.193.1 2 56.5 odd 6
448.4.i.f.65.1 2 56.3 even 6
448.4.i.f.193.1 2 56.19 even 6
784.4.a.b.1.1 1 7.6 odd 2
784.4.a.r.1.1 1 1.1 even 1 trivial
1225.4.a.c.1.1 1 140.139 even 2
1225.4.a.d.1.1 1 20.19 odd 2