Properties

Label 48.4.a.c.1.1
Level $48$
Weight $4$
Character 48.1
Self dual yes
Analytic conductor $2.832$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.83209168028\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +6.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +6.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +38.0000 q^{13} +18.0000 q^{15} -126.000 q^{17} -20.0000 q^{19} +48.0000 q^{21} -168.000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +30.0000 q^{29} +88.0000 q^{31} -36.0000 q^{33} +96.0000 q^{35} +254.000 q^{37} +114.000 q^{39} +42.0000 q^{41} +52.0000 q^{43} +54.0000 q^{45} +96.0000 q^{47} -87.0000 q^{49} -378.000 q^{51} +198.000 q^{53} -72.0000 q^{55} -60.0000 q^{57} +660.000 q^{59} -538.000 q^{61} +144.000 q^{63} +228.000 q^{65} -884.000 q^{67} -504.000 q^{69} -792.000 q^{71} +218.000 q^{73} -267.000 q^{75} -192.000 q^{77} +520.000 q^{79} +81.0000 q^{81} +492.000 q^{83} -756.000 q^{85} +90.0000 q^{87} +810.000 q^{89} +608.000 q^{91} +264.000 q^{93} -120.000 q^{95} +1154.00 q^{97} -108.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) 16.0000 0.863919 0.431959 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) 0 0
\(15\) 18.0000 0.309839
\(16\) 0 0
\(17\) −126.000 −1.79762 −0.898808 0.438342i \(-0.855566\pi\)
−0.898808 + 0.438342i \(0.855566\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) 0 0
\(23\) −168.000 −1.52306 −0.761531 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) 88.0000 0.509847 0.254924 0.966961i \(-0.417950\pi\)
0.254924 + 0.966961i \(0.417950\pi\)
\(32\) 0 0
\(33\) −36.0000 −0.189903
\(34\) 0 0
\(35\) 96.0000 0.463627
\(36\) 0 0
\(37\) 254.000 1.12858 0.564288 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) 0 0
\(39\) 114.000 0.468067
\(40\) 0 0
\(41\) 42.0000 0.159983 0.0799914 0.996796i \(-0.474511\pi\)
0.0799914 + 0.996796i \(0.474511\pi\)
\(42\) 0 0
\(43\) 52.0000 0.184417 0.0922084 0.995740i \(-0.470607\pi\)
0.0922084 + 0.995740i \(0.470607\pi\)
\(44\) 0 0
\(45\) 54.0000 0.178885
\(46\) 0 0
\(47\) 96.0000 0.297937 0.148969 0.988842i \(-0.452405\pi\)
0.148969 + 0.988842i \(0.452405\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) −378.000 −1.03785
\(52\) 0 0
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) 0 0
\(55\) −72.0000 −0.176518
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) 660.000 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −538.000 −1.12924 −0.564622 0.825350i \(-0.690978\pi\)
−0.564622 + 0.825350i \(0.690978\pi\)
\(62\) 0 0
\(63\) 144.000 0.287973
\(64\) 0 0
\(65\) 228.000 0.435076
\(66\) 0 0
\(67\) −884.000 −1.61191 −0.805954 0.591979i \(-0.798347\pi\)
−0.805954 + 0.591979i \(0.798347\pi\)
\(68\) 0 0
\(69\) −504.000 −0.879340
\(70\) 0 0
\(71\) −792.000 −1.32385 −0.661923 0.749572i \(-0.730260\pi\)
−0.661923 + 0.749572i \(0.730260\pi\)
\(72\) 0 0
\(73\) 218.000 0.349520 0.174760 0.984611i \(-0.444085\pi\)
0.174760 + 0.984611i \(0.444085\pi\)
\(74\) 0 0
\(75\) −267.000 −0.411073
\(76\) 0 0
\(77\) −192.000 −0.284161
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 492.000 0.650651 0.325325 0.945602i \(-0.394526\pi\)
0.325325 + 0.945602i \(0.394526\pi\)
\(84\) 0 0
\(85\) −756.000 −0.964703
\(86\) 0 0
\(87\) 90.0000 0.110908
\(88\) 0 0
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 0 0
\(91\) 608.000 0.700393
\(92\) 0 0
\(93\) 264.000 0.294360
\(94\) 0 0
\(95\) −120.000 −0.129597
\(96\) 0 0
\(97\) 1154.00 1.20795 0.603974 0.797004i \(-0.293583\pi\)
0.603974 + 0.797004i \(0.293583\pi\)
\(98\) 0 0
\(99\) −108.000 −0.109640
\(100\) 0 0
\(101\) −618.000 −0.608845 −0.304422 0.952537i \(-0.598463\pi\)
−0.304422 + 0.952537i \(0.598463\pi\)
\(102\) 0 0
\(103\) −128.000 −0.122449 −0.0612243 0.998124i \(-0.519501\pi\)
−0.0612243 + 0.998124i \(0.519501\pi\)
\(104\) 0 0
\(105\) 288.000 0.267675
\(106\) 0 0
\(107\) 1476.00 1.33355 0.666777 0.745257i \(-0.267673\pi\)
0.666777 + 0.745257i \(0.267673\pi\)
\(108\) 0 0
\(109\) 1190.00 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(110\) 0 0
\(111\) 762.000 0.651584
\(112\) 0 0
\(113\) −462.000 −0.384613 −0.192307 0.981335i \(-0.561597\pi\)
−0.192307 + 0.981335i \(0.561597\pi\)
\(114\) 0 0
\(115\) −1008.00 −0.817361
\(116\) 0 0
\(117\) 342.000 0.270239
\(118\) 0 0
\(119\) −2016.00 −1.55300
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 0 0
\(123\) 126.000 0.0923662
\(124\) 0 0
\(125\) −1284.00 −0.918756
\(126\) 0 0
\(127\) 2536.00 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(128\) 0 0
\(129\) 156.000 0.106473
\(130\) 0 0
\(131\) −2292.00 −1.52865 −0.764324 0.644832i \(-0.776927\pi\)
−0.764324 + 0.644832i \(0.776927\pi\)
\(132\) 0 0
\(133\) −320.000 −0.208628
\(134\) 0 0
\(135\) 162.000 0.103280
\(136\) 0 0
\(137\) −726.000 −0.452747 −0.226374 0.974041i \(-0.572687\pi\)
−0.226374 + 0.974041i \(0.572687\pi\)
\(138\) 0 0
\(139\) −380.000 −0.231879 −0.115939 0.993256i \(-0.536988\pi\)
−0.115939 + 0.993256i \(0.536988\pi\)
\(140\) 0 0
\(141\) 288.000 0.172014
\(142\) 0 0
\(143\) −456.000 −0.266662
\(144\) 0 0
\(145\) 180.000 0.103091
\(146\) 0 0
\(147\) −261.000 −0.146442
\(148\) 0 0
\(149\) 1590.00 0.874214 0.437107 0.899410i \(-0.356003\pi\)
0.437107 + 0.899410i \(0.356003\pi\)
\(150\) 0 0
\(151\) −2432.00 −1.31068 −0.655342 0.755332i \(-0.727476\pi\)
−0.655342 + 0.755332i \(0.727476\pi\)
\(152\) 0 0
\(153\) −1134.00 −0.599206
\(154\) 0 0
\(155\) 528.000 0.273613
\(156\) 0 0
\(157\) 614.000 0.312118 0.156059 0.987748i \(-0.450121\pi\)
0.156059 + 0.987748i \(0.450121\pi\)
\(158\) 0 0
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) −2688.00 −1.31580
\(162\) 0 0
\(163\) 1852.00 0.889938 0.444969 0.895546i \(-0.353215\pi\)
0.444969 + 0.895546i \(0.353215\pi\)
\(164\) 0 0
\(165\) −216.000 −0.101913
\(166\) 0 0
\(167\) 2136.00 0.989752 0.494876 0.868964i \(-0.335213\pi\)
0.494876 + 0.868964i \(0.335213\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) 0 0
\(173\) 1758.00 0.772591 0.386296 0.922375i \(-0.373754\pi\)
0.386296 + 0.922375i \(0.373754\pi\)
\(174\) 0 0
\(175\) −1424.00 −0.615110
\(176\) 0 0
\(177\) 1980.00 0.840824
\(178\) 0 0
\(179\) 540.000 0.225483 0.112742 0.993624i \(-0.464037\pi\)
0.112742 + 0.993624i \(0.464037\pi\)
\(180\) 0 0
\(181\) 1982.00 0.813928 0.406964 0.913444i \(-0.366588\pi\)
0.406964 + 0.913444i \(0.366588\pi\)
\(182\) 0 0
\(183\) −1614.00 −0.651969
\(184\) 0 0
\(185\) 1524.00 0.605658
\(186\) 0 0
\(187\) 1512.00 0.591275
\(188\) 0 0
\(189\) 432.000 0.166261
\(190\) 0 0
\(191\) 2688.00 1.01831 0.509154 0.860675i \(-0.329958\pi\)
0.509154 + 0.860675i \(0.329958\pi\)
\(192\) 0 0
\(193\) −2302.00 −0.858557 −0.429279 0.903172i \(-0.641232\pi\)
−0.429279 + 0.903172i \(0.641232\pi\)
\(194\) 0 0
\(195\) 684.000 0.251191
\(196\) 0 0
\(197\) 4374.00 1.58190 0.790951 0.611880i \(-0.209586\pi\)
0.790951 + 0.611880i \(0.209586\pi\)
\(198\) 0 0
\(199\) 1600.00 0.569955 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(200\) 0 0
\(201\) −2652.00 −0.930635
\(202\) 0 0
\(203\) 480.000 0.165958
\(204\) 0 0
\(205\) 252.000 0.0858558
\(206\) 0 0
\(207\) −1512.00 −0.507687
\(208\) 0 0
\(209\) 240.000 0.0794313
\(210\) 0 0
\(211\) −3332.00 −1.08713 −0.543565 0.839367i \(-0.682926\pi\)
−0.543565 + 0.839367i \(0.682926\pi\)
\(212\) 0 0
\(213\) −2376.00 −0.764323
\(214\) 0 0
\(215\) 312.000 0.0989685
\(216\) 0 0
\(217\) 1408.00 0.440467
\(218\) 0 0
\(219\) 654.000 0.201796
\(220\) 0 0
\(221\) −4788.00 −1.45736
\(222\) 0 0
\(223\) −2648.00 −0.795171 −0.397586 0.917565i \(-0.630152\pi\)
−0.397586 + 0.917565i \(0.630152\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) −2244.00 −0.656121 −0.328061 0.944657i \(-0.606395\pi\)
−0.328061 + 0.944657i \(0.606395\pi\)
\(228\) 0 0
\(229\) −5650.00 −1.63040 −0.815202 0.579177i \(-0.803374\pi\)
−0.815202 + 0.579177i \(0.803374\pi\)
\(230\) 0 0
\(231\) −576.000 −0.164061
\(232\) 0 0
\(233\) 4698.00 1.32093 0.660464 0.750858i \(-0.270360\pi\)
0.660464 + 0.750858i \(0.270360\pi\)
\(234\) 0 0
\(235\) 576.000 0.159890
\(236\) 0 0
\(237\) 1560.00 0.427565
\(238\) 0 0
\(239\) 1200.00 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(240\) 0 0
\(241\) −718.000 −0.191911 −0.0959553 0.995386i \(-0.530591\pi\)
−0.0959553 + 0.995386i \(0.530591\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −522.000 −0.136120
\(246\) 0 0
\(247\) −760.000 −0.195780
\(248\) 0 0
\(249\) 1476.00 0.375653
\(250\) 0 0
\(251\) −6012.00 −1.51185 −0.755924 0.654659i \(-0.772812\pi\)
−0.755924 + 0.654659i \(0.772812\pi\)
\(252\) 0 0
\(253\) 2016.00 0.500968
\(254\) 0 0
\(255\) −2268.00 −0.556971
\(256\) 0 0
\(257\) −2046.00 −0.496599 −0.248300 0.968683i \(-0.579872\pi\)
−0.248300 + 0.968683i \(0.579872\pi\)
\(258\) 0 0
\(259\) 4064.00 0.974999
\(260\) 0 0
\(261\) 270.000 0.0640329
\(262\) 0 0
\(263\) 6072.00 1.42363 0.711817 0.702365i \(-0.247873\pi\)
0.711817 + 0.702365i \(0.247873\pi\)
\(264\) 0 0
\(265\) 1188.00 0.275390
\(266\) 0 0
\(267\) 2430.00 0.556980
\(268\) 0 0
\(269\) −6930.00 −1.57074 −0.785371 0.619025i \(-0.787528\pi\)
−0.785371 + 0.619025i \(0.787528\pi\)
\(270\) 0 0
\(271\) −1352.00 −0.303056 −0.151528 0.988453i \(-0.548419\pi\)
−0.151528 + 0.988453i \(0.548419\pi\)
\(272\) 0 0
\(273\) 1824.00 0.404372
\(274\) 0 0
\(275\) 1068.00 0.234192
\(276\) 0 0
\(277\) −1186.00 −0.257256 −0.128628 0.991693i \(-0.541057\pi\)
−0.128628 + 0.991693i \(0.541057\pi\)
\(278\) 0 0
\(279\) 792.000 0.169949
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) 0 0
\(283\) −2828.00 −0.594018 −0.297009 0.954875i \(-0.595989\pi\)
−0.297009 + 0.954875i \(0.595989\pi\)
\(284\) 0 0
\(285\) −360.000 −0.0748230
\(286\) 0 0
\(287\) 672.000 0.138212
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 0 0
\(291\) 3462.00 0.697409
\(292\) 0 0
\(293\) 4758.00 0.948687 0.474344 0.880340i \(-0.342685\pi\)
0.474344 + 0.880340i \(0.342685\pi\)
\(294\) 0 0
\(295\) 3960.00 0.781560
\(296\) 0 0
\(297\) −324.000 −0.0633010
\(298\) 0 0
\(299\) −6384.00 −1.23477
\(300\) 0 0
\(301\) 832.000 0.159321
\(302\) 0 0
\(303\) −1854.00 −0.351517
\(304\) 0 0
\(305\) −3228.00 −0.606016
\(306\) 0 0
\(307\) 8476.00 1.57574 0.787868 0.615844i \(-0.211185\pi\)
0.787868 + 0.615844i \(0.211185\pi\)
\(308\) 0 0
\(309\) −384.000 −0.0706958
\(310\) 0 0
\(311\) −4632.00 −0.844555 −0.422278 0.906467i \(-0.638769\pi\)
−0.422278 + 0.906467i \(0.638769\pi\)
\(312\) 0 0
\(313\) −4822.00 −0.870785 −0.435392 0.900241i \(-0.643390\pi\)
−0.435392 + 0.900241i \(0.643390\pi\)
\(314\) 0 0
\(315\) 864.000 0.154542
\(316\) 0 0
\(317\) −3426.00 −0.607014 −0.303507 0.952829i \(-0.598158\pi\)
−0.303507 + 0.952829i \(0.598158\pi\)
\(318\) 0 0
\(319\) −360.000 −0.0631854
\(320\) 0 0
\(321\) 4428.00 0.769928
\(322\) 0 0
\(323\) 2520.00 0.434107
\(324\) 0 0
\(325\) −3382.00 −0.577230
\(326\) 0 0
\(327\) 3570.00 0.603735
\(328\) 0 0
\(329\) 1536.00 0.257393
\(330\) 0 0
\(331\) 2788.00 0.462968 0.231484 0.972839i \(-0.425642\pi\)
0.231484 + 0.972839i \(0.425642\pi\)
\(332\) 0 0
\(333\) 2286.00 0.376192
\(334\) 0 0
\(335\) −5304.00 −0.865040
\(336\) 0 0
\(337\) 434.000 0.0701528 0.0350764 0.999385i \(-0.488833\pi\)
0.0350764 + 0.999385i \(0.488833\pi\)
\(338\) 0 0
\(339\) −1386.00 −0.222057
\(340\) 0 0
\(341\) −1056.00 −0.167700
\(342\) 0 0
\(343\) −6880.00 −1.08305
\(344\) 0 0
\(345\) −3024.00 −0.471903
\(346\) 0 0
\(347\) −6684.00 −1.03405 −0.517026 0.855970i \(-0.672961\pi\)
−0.517026 + 0.855970i \(0.672961\pi\)
\(348\) 0 0
\(349\) 2630.00 0.403383 0.201692 0.979449i \(-0.435356\pi\)
0.201692 + 0.979449i \(0.435356\pi\)
\(350\) 0 0
\(351\) 1026.00 0.156022
\(352\) 0 0
\(353\) −7422.00 −1.11907 −0.559537 0.828805i \(-0.689021\pi\)
−0.559537 + 0.828805i \(0.689021\pi\)
\(354\) 0 0
\(355\) −4752.00 −0.710451
\(356\) 0 0
\(357\) −6048.00 −0.896622
\(358\) 0 0
\(359\) 10440.0 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) −3561.00 −0.514887
\(364\) 0 0
\(365\) 1308.00 0.187572
\(366\) 0 0
\(367\) −10424.0 −1.48264 −0.741319 0.671153i \(-0.765800\pi\)
−0.741319 + 0.671153i \(0.765800\pi\)
\(368\) 0 0
\(369\) 378.000 0.0533276
\(370\) 0 0
\(371\) 3168.00 0.443327
\(372\) 0 0
\(373\) 3278.00 0.455036 0.227518 0.973774i \(-0.426939\pi\)
0.227518 + 0.973774i \(0.426939\pi\)
\(374\) 0 0
\(375\) −3852.00 −0.530444
\(376\) 0 0
\(377\) 1140.00 0.155737
\(378\) 0 0
\(379\) −6140.00 −0.832165 −0.416083 0.909327i \(-0.636597\pi\)
−0.416083 + 0.909327i \(0.636597\pi\)
\(380\) 0 0
\(381\) 7608.00 1.02302
\(382\) 0 0
\(383\) 3072.00 0.409848 0.204924 0.978778i \(-0.434305\pi\)
0.204924 + 0.978778i \(0.434305\pi\)
\(384\) 0 0
\(385\) −1152.00 −0.152497
\(386\) 0 0
\(387\) 468.000 0.0614723
\(388\) 0 0
\(389\) 6150.00 0.801587 0.400794 0.916168i \(-0.368734\pi\)
0.400794 + 0.916168i \(0.368734\pi\)
\(390\) 0 0
\(391\) 21168.0 2.73788
\(392\) 0 0
\(393\) −6876.00 −0.882566
\(394\) 0 0
\(395\) 3120.00 0.397428
\(396\) 0 0
\(397\) −106.000 −0.0134005 −0.00670024 0.999978i \(-0.502133\pi\)
−0.00670024 + 0.999978i \(0.502133\pi\)
\(398\) 0 0
\(399\) −960.000 −0.120451
\(400\) 0 0
\(401\) −1758.00 −0.218929 −0.109464 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(402\) 0 0
\(403\) 3344.00 0.413341
\(404\) 0 0
\(405\) 486.000 0.0596285
\(406\) 0 0
\(407\) −3048.00 −0.371213
\(408\) 0 0
\(409\) −3670.00 −0.443691 −0.221846 0.975082i \(-0.571208\pi\)
−0.221846 + 0.975082i \(0.571208\pi\)
\(410\) 0 0
\(411\) −2178.00 −0.261394
\(412\) 0 0
\(413\) 10560.0 1.25817
\(414\) 0 0
\(415\) 2952.00 0.349176
\(416\) 0 0
\(417\) −1140.00 −0.133875
\(418\) 0 0
\(419\) 9660.00 1.12631 0.563153 0.826353i \(-0.309588\pi\)
0.563153 + 0.826353i \(0.309588\pi\)
\(420\) 0 0
\(421\) 8462.00 0.979602 0.489801 0.871834i \(-0.337069\pi\)
0.489801 + 0.871834i \(0.337069\pi\)
\(422\) 0 0
\(423\) 864.000 0.0993123
\(424\) 0 0
\(425\) 11214.0 1.27990
\(426\) 0 0
\(427\) −8608.00 −0.975575
\(428\) 0 0
\(429\) −1368.00 −0.153957
\(430\) 0 0
\(431\) −9792.00 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(432\) 0 0
\(433\) −7342.00 −0.814859 −0.407430 0.913237i \(-0.633575\pi\)
−0.407430 + 0.913237i \(0.633575\pi\)
\(434\) 0 0
\(435\) 540.000 0.0595196
\(436\) 0 0
\(437\) 3360.00 0.367805
\(438\) 0 0
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) 17412.0 1.86742 0.933712 0.358024i \(-0.116549\pi\)
0.933712 + 0.358024i \(0.116549\pi\)
\(444\) 0 0
\(445\) 4860.00 0.517722
\(446\) 0 0
\(447\) 4770.00 0.504728
\(448\) 0 0
\(449\) −1710.00 −0.179732 −0.0898662 0.995954i \(-0.528644\pi\)
−0.0898662 + 0.995954i \(0.528644\pi\)
\(450\) 0 0
\(451\) −504.000 −0.0526218
\(452\) 0 0
\(453\) −7296.00 −0.756724
\(454\) 0 0
\(455\) 3648.00 0.375870
\(456\) 0 0
\(457\) −646.000 −0.0661239 −0.0330619 0.999453i \(-0.510526\pi\)
−0.0330619 + 0.999453i \(0.510526\pi\)
\(458\) 0 0
\(459\) −3402.00 −0.345952
\(460\) 0 0
\(461\) −6018.00 −0.607996 −0.303998 0.952673i \(-0.598322\pi\)
−0.303998 + 0.952673i \(0.598322\pi\)
\(462\) 0 0
\(463\) 6712.00 0.673722 0.336861 0.941554i \(-0.390635\pi\)
0.336861 + 0.941554i \(0.390635\pi\)
\(464\) 0 0
\(465\) 1584.00 0.157970
\(466\) 0 0
\(467\) −5364.00 −0.531512 −0.265756 0.964040i \(-0.585622\pi\)
−0.265756 + 0.964040i \(0.585622\pi\)
\(468\) 0 0
\(469\) −14144.0 −1.39256
\(470\) 0 0
\(471\) 1842.00 0.180201
\(472\) 0 0
\(473\) −624.000 −0.0606587
\(474\) 0 0
\(475\) 1780.00 0.171941
\(476\) 0 0
\(477\) 1782.00 0.171053
\(478\) 0 0
\(479\) −9840.00 −0.938624 −0.469312 0.883032i \(-0.655498\pi\)
−0.469312 + 0.883032i \(0.655498\pi\)
\(480\) 0 0
\(481\) 9652.00 0.914955
\(482\) 0 0
\(483\) −8064.00 −0.759678
\(484\) 0 0
\(485\) 6924.00 0.648253
\(486\) 0 0
\(487\) −1424.00 −0.132500 −0.0662501 0.997803i \(-0.521104\pi\)
−0.0662501 + 0.997803i \(0.521104\pi\)
\(488\) 0 0
\(489\) 5556.00 0.513806
\(490\) 0 0
\(491\) 4548.00 0.418021 0.209011 0.977913i \(-0.432976\pi\)
0.209011 + 0.977913i \(0.432976\pi\)
\(492\) 0 0
\(493\) −3780.00 −0.345320
\(494\) 0 0
\(495\) −648.000 −0.0588393
\(496\) 0 0
\(497\) −12672.0 −1.14370
\(498\) 0 0
\(499\) −6500.00 −0.583126 −0.291563 0.956552i \(-0.594175\pi\)
−0.291563 + 0.956552i \(0.594175\pi\)
\(500\) 0 0
\(501\) 6408.00 0.571434
\(502\) 0 0
\(503\) −12168.0 −1.07862 −0.539308 0.842108i \(-0.681314\pi\)
−0.539308 + 0.842108i \(0.681314\pi\)
\(504\) 0 0
\(505\) −3708.00 −0.326740
\(506\) 0 0
\(507\) −2259.00 −0.197881
\(508\) 0 0
\(509\) −21090.0 −1.83654 −0.918269 0.395957i \(-0.870413\pi\)
−0.918269 + 0.395957i \(0.870413\pi\)
\(510\) 0 0
\(511\) 3488.00 0.301957
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) −768.000 −0.0657129
\(516\) 0 0
\(517\) −1152.00 −0.0979979
\(518\) 0 0
\(519\) 5274.00 0.446056
\(520\) 0 0
\(521\) −5238.00 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(522\) 0 0
\(523\) −8588.00 −0.718025 −0.359012 0.933333i \(-0.616886\pi\)
−0.359012 + 0.933333i \(0.616886\pi\)
\(524\) 0 0
\(525\) −4272.00 −0.355134
\(526\) 0 0
\(527\) −11088.0 −0.916510
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) 5940.00 0.485450
\(532\) 0 0
\(533\) 1596.00 0.129701
\(534\) 0 0
\(535\) 8856.00 0.715660
\(536\) 0 0
\(537\) 1620.00 0.130183
\(538\) 0 0
\(539\) 1044.00 0.0834291
\(540\) 0 0
\(541\) 3062.00 0.243338 0.121669 0.992571i \(-0.461175\pi\)
0.121669 + 0.992571i \(0.461175\pi\)
\(542\) 0 0
\(543\) 5946.00 0.469921
\(544\) 0 0
\(545\) 7140.00 0.561182
\(546\) 0 0
\(547\) 8476.00 0.662537 0.331268 0.943537i \(-0.392523\pi\)
0.331268 + 0.943537i \(0.392523\pi\)
\(548\) 0 0
\(549\) −4842.00 −0.376414
\(550\) 0 0
\(551\) −600.000 −0.0463899
\(552\) 0 0
\(553\) 8320.00 0.639787
\(554\) 0 0
\(555\) 4572.00 0.349677
\(556\) 0 0
\(557\) −12546.0 −0.954383 −0.477191 0.878799i \(-0.658345\pi\)
−0.477191 + 0.878799i \(0.658345\pi\)
\(558\) 0 0
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) 0 0
\(563\) 12.0000 0.000898294 0 0.000449147 1.00000i \(-0.499857\pi\)
0.000449147 1.00000i \(0.499857\pi\)
\(564\) 0 0
\(565\) −2772.00 −0.206405
\(566\) 0 0
\(567\) 1296.00 0.0959910
\(568\) 0 0
\(569\) 19290.0 1.42123 0.710614 0.703582i \(-0.248417\pi\)
0.710614 + 0.703582i \(0.248417\pi\)
\(570\) 0 0
\(571\) 12148.0 0.890329 0.445165 0.895449i \(-0.353145\pi\)
0.445165 + 0.895449i \(0.353145\pi\)
\(572\) 0 0
\(573\) 8064.00 0.587920
\(574\) 0 0
\(575\) 14952.0 1.08442
\(576\) 0 0
\(577\) −10366.0 −0.747907 −0.373953 0.927447i \(-0.621998\pi\)
−0.373953 + 0.927447i \(0.621998\pi\)
\(578\) 0 0
\(579\) −6906.00 −0.495688
\(580\) 0 0
\(581\) 7872.00 0.562109
\(582\) 0 0
\(583\) −2376.00 −0.168789
\(584\) 0 0
\(585\) 2052.00 0.145025
\(586\) 0 0
\(587\) −7644.00 −0.537482 −0.268741 0.963213i \(-0.586607\pi\)
−0.268741 + 0.963213i \(0.586607\pi\)
\(588\) 0 0
\(589\) −1760.00 −0.123123
\(590\) 0 0
\(591\) 13122.0 0.913311
\(592\) 0 0
\(593\) 8658.00 0.599564 0.299782 0.954008i \(-0.403086\pi\)
0.299782 + 0.954008i \(0.403086\pi\)
\(594\) 0 0
\(595\) −12096.0 −0.833425
\(596\) 0 0
\(597\) 4800.00 0.329064
\(598\) 0 0
\(599\) −25800.0 −1.75987 −0.879933 0.475098i \(-0.842413\pi\)
−0.879933 + 0.475098i \(0.842413\pi\)
\(600\) 0 0
\(601\) 16202.0 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(602\) 0 0
\(603\) −7956.00 −0.537302
\(604\) 0 0
\(605\) −7122.00 −0.478596
\(606\) 0 0
\(607\) 24136.0 1.61392 0.806960 0.590605i \(-0.201111\pi\)
0.806960 + 0.590605i \(0.201111\pi\)
\(608\) 0 0
\(609\) 1440.00 0.0958157
\(610\) 0 0
\(611\) 3648.00 0.241542
\(612\) 0 0
\(613\) −4642.00 −0.305854 −0.152927 0.988237i \(-0.548870\pi\)
−0.152927 + 0.988237i \(0.548870\pi\)
\(614\) 0 0
\(615\) 756.000 0.0495689
\(616\) 0 0
\(617\) −6726.00 −0.438863 −0.219432 0.975628i \(-0.570420\pi\)
−0.219432 + 0.975628i \(0.570420\pi\)
\(618\) 0 0
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 0 0
\(623\) 12960.0 0.833437
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) 720.000 0.0458597
\(628\) 0 0
\(629\) −32004.0 −2.02875
\(630\) 0 0
\(631\) −29792.0 −1.87956 −0.939779 0.341783i \(-0.888969\pi\)
−0.939779 + 0.341783i \(0.888969\pi\)
\(632\) 0 0
\(633\) −9996.00 −0.627655
\(634\) 0 0
\(635\) 15216.0 0.950911
\(636\) 0 0
\(637\) −3306.00 −0.205633
\(638\) 0 0
\(639\) −7128.00 −0.441282
\(640\) 0 0
\(641\) −10158.0 −0.625923 −0.312962 0.949766i \(-0.601321\pi\)
−0.312962 + 0.949766i \(0.601321\pi\)
\(642\) 0 0
\(643\) −29828.0 −1.82940 −0.914698 0.404138i \(-0.867571\pi\)
−0.914698 + 0.404138i \(0.867571\pi\)
\(644\) 0 0
\(645\) 936.000 0.0571395
\(646\) 0 0
\(647\) −1944.00 −0.118124 −0.0590622 0.998254i \(-0.518811\pi\)
−0.0590622 + 0.998254i \(0.518811\pi\)
\(648\) 0 0
\(649\) −7920.00 −0.479025
\(650\) 0 0
\(651\) 4224.00 0.254304
\(652\) 0 0
\(653\) 26718.0 1.60116 0.800579 0.599227i \(-0.204525\pi\)
0.800579 + 0.599227i \(0.204525\pi\)
\(654\) 0 0
\(655\) −13752.0 −0.820359
\(656\) 0 0
\(657\) 1962.00 0.116507
\(658\) 0 0
\(659\) −4260.00 −0.251815 −0.125907 0.992042i \(-0.540184\pi\)
−0.125907 + 0.992042i \(0.540184\pi\)
\(660\) 0 0
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) 0 0
\(663\) −14364.0 −0.841405
\(664\) 0 0
\(665\) −1920.00 −0.111962
\(666\) 0 0
\(667\) −5040.00 −0.292578
\(668\) 0 0
\(669\) −7944.00 −0.459092
\(670\) 0 0
\(671\) 6456.00 0.371432
\(672\) 0 0
\(673\) −32542.0 −1.86390 −0.931948 0.362592i \(-0.881892\pi\)
−0.931948 + 0.362592i \(0.881892\pi\)
\(674\) 0 0
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) 14214.0 0.806925 0.403463 0.914996i \(-0.367807\pi\)
0.403463 + 0.914996i \(0.367807\pi\)
\(678\) 0 0
\(679\) 18464.0 1.04357
\(680\) 0 0
\(681\) −6732.00 −0.378812
\(682\) 0 0
\(683\) 7092.00 0.397317 0.198659 0.980069i \(-0.436341\pi\)
0.198659 + 0.980069i \(0.436341\pi\)
\(684\) 0 0
\(685\) −4356.00 −0.242970
\(686\) 0 0
\(687\) −16950.0 −0.941314
\(688\) 0 0
\(689\) 7524.00 0.416026
\(690\) 0 0
\(691\) 13228.0 0.728244 0.364122 0.931351i \(-0.381369\pi\)
0.364122 + 0.931351i \(0.381369\pi\)
\(692\) 0 0
\(693\) −1728.00 −0.0947205
\(694\) 0 0
\(695\) −2280.00 −0.124439
\(696\) 0 0
\(697\) −5292.00 −0.287588
\(698\) 0 0
\(699\) 14094.0 0.762638
\(700\) 0 0
\(701\) 28062.0 1.51196 0.755982 0.654592i \(-0.227160\pi\)
0.755982 + 0.654592i \(0.227160\pi\)
\(702\) 0 0
\(703\) −5080.00 −0.272540
\(704\) 0 0
\(705\) 1728.00 0.0923124
\(706\) 0 0
\(707\) −9888.00 −0.525992
\(708\) 0 0
\(709\) −27250.0 −1.44343 −0.721717 0.692188i \(-0.756647\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(710\) 0 0
\(711\) 4680.00 0.246855
\(712\) 0 0
\(713\) −14784.0 −0.776529
\(714\) 0 0
\(715\) −2736.00 −0.143106
\(716\) 0 0
\(717\) 3600.00 0.187510
\(718\) 0 0
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) 0 0
\(721\) −2048.00 −0.105786
\(722\) 0 0
\(723\) −2154.00 −0.110800
\(724\) 0 0
\(725\) −2670.00 −0.136774
\(726\) 0 0
\(727\) −17984.0 −0.917455 −0.458727 0.888577i \(-0.651695\pi\)
−0.458727 + 0.888577i \(0.651695\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6552.00 −0.331511
\(732\) 0 0
\(733\) 16598.0 0.836373 0.418186 0.908361i \(-0.362666\pi\)
0.418186 + 0.908361i \(0.362666\pi\)
\(734\) 0 0
\(735\) −1566.00 −0.0785888
\(736\) 0 0
\(737\) 10608.0 0.530191
\(738\) 0 0
\(739\) −1460.00 −0.0726752 −0.0363376 0.999340i \(-0.511569\pi\)
−0.0363376 + 0.999340i \(0.511569\pi\)
\(740\) 0 0
\(741\) −2280.00 −0.113034
\(742\) 0 0
\(743\) 30072.0 1.48484 0.742419 0.669936i \(-0.233678\pi\)
0.742419 + 0.669936i \(0.233678\pi\)
\(744\) 0 0
\(745\) 9540.00 0.469152
\(746\) 0 0
\(747\) 4428.00 0.216884
\(748\) 0 0
\(749\) 23616.0 1.15208
\(750\) 0 0
\(751\) 18088.0 0.878882 0.439441 0.898271i \(-0.355177\pi\)
0.439441 + 0.898271i \(0.355177\pi\)
\(752\) 0 0
\(753\) −18036.0 −0.872866
\(754\) 0 0
\(755\) −14592.0 −0.703387
\(756\) 0 0
\(757\) 24734.0 1.18755 0.593773 0.804633i \(-0.297638\pi\)
0.593773 + 0.804633i \(0.297638\pi\)
\(758\) 0 0
\(759\) 6048.00 0.289234
\(760\) 0 0
\(761\) −22278.0 −1.06120 −0.530602 0.847621i \(-0.678034\pi\)
−0.530602 + 0.847621i \(0.678034\pi\)
\(762\) 0 0
\(763\) 19040.0 0.903400
\(764\) 0 0
\(765\) −6804.00 −0.321568
\(766\) 0 0
\(767\) 25080.0 1.18069
\(768\) 0 0
\(769\) 16130.0 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(770\) 0 0
\(771\) −6138.00 −0.286712
\(772\) 0 0
\(773\) 29718.0 1.38277 0.691386 0.722486i \(-0.257001\pi\)
0.691386 + 0.722486i \(0.257001\pi\)
\(774\) 0 0
\(775\) −7832.00 −0.363011
\(776\) 0 0
\(777\) 12192.0 0.562916
\(778\) 0 0
\(779\) −840.000 −0.0386343
\(780\) 0 0
\(781\) 9504.00 0.435442
\(782\) 0 0
\(783\) 810.000 0.0369694
\(784\) 0 0
\(785\) 3684.00 0.167500
\(786\) 0 0
\(787\) −9524.00 −0.431377 −0.215689 0.976462i \(-0.569200\pi\)
−0.215689 + 0.976462i \(0.569200\pi\)
\(788\) 0 0
\(789\) 18216.0 0.821935
\(790\) 0 0
\(791\) −7392.00 −0.332275
\(792\) 0 0
\(793\) −20444.0 −0.915495
\(794\) 0 0
\(795\) 3564.00 0.158996
\(796\) 0 0
\(797\) −33906.0 −1.50692 −0.753458 0.657496i \(-0.771616\pi\)
−0.753458 + 0.657496i \(0.771616\pi\)
\(798\) 0 0
\(799\) −12096.0 −0.535577
\(800\) 0 0
\(801\) 7290.00 0.321572
\(802\) 0 0
\(803\) −2616.00 −0.114965
\(804\) 0 0
\(805\) −16128.0 −0.706133
\(806\) 0 0
\(807\) −20790.0 −0.906868
\(808\) 0 0
\(809\) −630.000 −0.0273790 −0.0136895 0.999906i \(-0.504358\pi\)
−0.0136895 + 0.999906i \(0.504358\pi\)
\(810\) 0 0
\(811\) 20788.0 0.900081 0.450040 0.893008i \(-0.351410\pi\)
0.450040 + 0.893008i \(0.351410\pi\)
\(812\) 0 0
\(813\) −4056.00 −0.174969
\(814\) 0 0
\(815\) 11112.0 0.477591
\(816\) 0 0
\(817\) −1040.00 −0.0445349
\(818\) 0 0
\(819\) 5472.00 0.233464
\(820\) 0 0
\(821\) −43098.0 −1.83207 −0.916036 0.401097i \(-0.868629\pi\)
−0.916036 + 0.401097i \(0.868629\pi\)
\(822\) 0 0
\(823\) 14272.0 0.604484 0.302242 0.953231i \(-0.402265\pi\)
0.302242 + 0.953231i \(0.402265\pi\)
\(824\) 0 0
\(825\) 3204.00 0.135211
\(826\) 0 0
\(827\) −13644.0 −0.573698 −0.286849 0.957976i \(-0.592608\pi\)
−0.286849 + 0.957976i \(0.592608\pi\)
\(828\) 0 0
\(829\) −2410.00 −0.100968 −0.0504842 0.998725i \(-0.516076\pi\)
−0.0504842 + 0.998725i \(0.516076\pi\)
\(830\) 0 0
\(831\) −3558.00 −0.148527
\(832\) 0 0
\(833\) 10962.0 0.455955
\(834\) 0 0
\(835\) 12816.0 0.531157
\(836\) 0 0
\(837\) 2376.00 0.0981202
\(838\) 0 0
\(839\) −23160.0 −0.953006 −0.476503 0.879173i \(-0.658096\pi\)
−0.476503 + 0.879173i \(0.658096\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 0 0
\(843\) 7326.00 0.299313
\(844\) 0 0
\(845\) −4518.00 −0.183934
\(846\) 0 0
\(847\) −18992.0 −0.770452
\(848\) 0 0
\(849\) −8484.00 −0.342957
\(850\) 0 0
\(851\) −42672.0 −1.71889
\(852\) 0 0
\(853\) 32078.0 1.28761 0.643804 0.765190i \(-0.277355\pi\)
0.643804 + 0.765190i \(0.277355\pi\)
\(854\) 0 0
\(855\) −1080.00 −0.0431991
\(856\) 0 0
\(857\) −14406.0 −0.574212 −0.287106 0.957899i \(-0.592693\pi\)
−0.287106 + 0.957899i \(0.592693\pi\)
\(858\) 0 0
\(859\) −30620.0 −1.21623 −0.608115 0.793849i \(-0.708074\pi\)
−0.608115 + 0.793849i \(0.708074\pi\)
\(860\) 0 0
\(861\) 2016.00 0.0797969
\(862\) 0 0
\(863\) −17568.0 −0.692957 −0.346478 0.938058i \(-0.612623\pi\)
−0.346478 + 0.938058i \(0.612623\pi\)
\(864\) 0 0
\(865\) 10548.0 0.414616
\(866\) 0 0
\(867\) 32889.0 1.28831
\(868\) 0 0
\(869\) −6240.00 −0.243587
\(870\) 0 0
\(871\) −33592.0 −1.30680
\(872\) 0 0
\(873\) 10386.0 0.402649
\(874\) 0 0
\(875\) −20544.0 −0.793730
\(876\) 0 0
\(877\) −21706.0 −0.835758 −0.417879 0.908503i \(-0.637226\pi\)
−0.417879 + 0.908503i \(0.637226\pi\)
\(878\) 0 0
\(879\) 14274.0 0.547725
\(880\) 0 0
\(881\) −14958.0 −0.572018 −0.286009 0.958227i \(-0.592329\pi\)
−0.286009 + 0.958227i \(0.592329\pi\)
\(882\) 0 0
\(883\) 32812.0 1.25052 0.625261 0.780415i \(-0.284992\pi\)
0.625261 + 0.780415i \(0.284992\pi\)
\(884\) 0 0
\(885\) 11880.0 0.451234
\(886\) 0 0
\(887\) 38856.0 1.47086 0.735432 0.677598i \(-0.236979\pi\)
0.735432 + 0.677598i \(0.236979\pi\)
\(888\) 0 0
\(889\) 40576.0 1.53079
\(890\) 0 0
\(891\) −972.000 −0.0365468
\(892\) 0 0
\(893\) −1920.00 −0.0719489
\(894\) 0 0
\(895\) 3240.00 0.121007
\(896\) 0 0
\(897\) −19152.0 −0.712895
\(898\) 0 0
\(899\) 2640.00 0.0979410
\(900\) 0 0
\(901\) −24948.0 −0.922462
\(902\) 0 0
\(903\) 2496.00 0.0919841
\(904\) 0 0
\(905\) 11892.0 0.436799
\(906\) 0 0
\(907\) 28276.0 1.03516 0.517579 0.855635i \(-0.326833\pi\)
0.517579 + 0.855635i \(0.326833\pi\)
\(908\) 0 0
\(909\) −5562.00 −0.202948
\(910\) 0 0
\(911\) −8112.00 −0.295019 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(912\) 0 0
\(913\) −5904.00 −0.214013
\(914\) 0 0
\(915\) −9684.00 −0.349883
\(916\) 0 0
\(917\) −36672.0 −1.32063
\(918\) 0 0
\(919\) 26080.0 0.936126 0.468063 0.883695i \(-0.344952\pi\)
0.468063 + 0.883695i \(0.344952\pi\)
\(920\) 0 0
\(921\) 25428.0 0.909751
\(922\) 0 0
\(923\) −30096.0 −1.07326
\(924\) 0 0
\(925\) −22606.0 −0.803547
\(926\) 0 0
\(927\) −1152.00 −0.0408162
\(928\) 0 0
\(929\) 49170.0 1.73651 0.868254 0.496120i \(-0.165243\pi\)
0.868254 + 0.496120i \(0.165243\pi\)
\(930\) 0 0
\(931\) 1740.00 0.0612526
\(932\) 0 0
\(933\) −13896.0 −0.487604
\(934\) 0 0
\(935\) 9072.00 0.317311
\(936\) 0 0
\(937\) 48314.0 1.68447 0.842236 0.539110i \(-0.181239\pi\)
0.842236 + 0.539110i \(0.181239\pi\)
\(938\) 0 0
\(939\) −14466.0 −0.502748
\(940\) 0 0
\(941\) 34782.0 1.20495 0.602477 0.798137i \(-0.294181\pi\)
0.602477 + 0.798137i \(0.294181\pi\)
\(942\) 0 0
\(943\) −7056.00 −0.243664
\(944\) 0 0
\(945\) 2592.00 0.0892251
\(946\) 0 0
\(947\) 25116.0 0.861838 0.430919 0.902391i \(-0.358190\pi\)
0.430919 + 0.902391i \(0.358190\pi\)
\(948\) 0 0
\(949\) 8284.00 0.283361
\(950\) 0 0
\(951\) −10278.0 −0.350460
\(952\) 0 0
\(953\) −15462.0 −0.525565 −0.262782 0.964855i \(-0.584640\pi\)
−0.262782 + 0.964855i \(0.584640\pi\)
\(954\) 0 0
\(955\) 16128.0 0.546481
\(956\) 0 0
\(957\) −1080.00 −0.0364801
\(958\) 0 0
\(959\) −11616.0 −0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 0 0
\(963\) 13284.0 0.444518
\(964\) 0 0
\(965\) −13812.0 −0.460750
\(966\) 0 0
\(967\) 736.000 0.0244759 0.0122379 0.999925i \(-0.496104\pi\)
0.0122379 + 0.999925i \(0.496104\pi\)
\(968\) 0 0
\(969\) 7560.00 0.250632
\(970\) 0 0
\(971\) 29268.0 0.967307 0.483653 0.875260i \(-0.339310\pi\)
0.483653 + 0.875260i \(0.339310\pi\)
\(972\) 0 0
\(973\) −6080.00 −0.200325
\(974\) 0 0
\(975\) −10146.0 −0.333264
\(976\) 0 0
\(977\) 16674.0 0.546007 0.273003 0.962013i \(-0.411983\pi\)
0.273003 + 0.962013i \(0.411983\pi\)
\(978\) 0 0
\(979\) −9720.00 −0.317316
\(980\) 0 0
\(981\) 10710.0 0.348567
\(982\) 0 0
\(983\) 31272.0 1.01467 0.507336 0.861749i \(-0.330630\pi\)
0.507336 + 0.861749i \(0.330630\pi\)
\(984\) 0 0
\(985\) 26244.0 0.848937
\(986\) 0 0
\(987\) 4608.00 0.148606
\(988\) 0 0
\(989\) −8736.00 −0.280878
\(990\) 0 0
\(991\) 15928.0 0.510565 0.255282 0.966867i \(-0.417832\pi\)
0.255282 + 0.966867i \(0.417832\pi\)
\(992\) 0 0
\(993\) 8364.00 0.267295
\(994\) 0 0
\(995\) 9600.00 0.305870
\(996\) 0 0
\(997\) 42014.0 1.33460 0.667300 0.744789i \(-0.267450\pi\)
0.667300 + 0.744789i \(0.267450\pi\)
\(998\) 0 0
\(999\) 6858.00 0.217195
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 48.4.a.c.1.1 1
3.2 odd 2 144.4.a.c.1.1 1
4.3 odd 2 6.4.a.a.1.1 1
5.2 odd 4 1200.4.f.j.49.1 2
5.3 odd 4 1200.4.f.j.49.2 2
5.4 even 2 1200.4.a.b.1.1 1
7.6 odd 2 2352.4.a.e.1.1 1
8.3 odd 2 192.4.a.i.1.1 1
8.5 even 2 192.4.a.c.1.1 1
12.11 even 2 18.4.a.a.1.1 1
16.3 odd 4 768.4.d.n.385.1 2
16.5 even 4 768.4.d.c.385.1 2
16.11 odd 4 768.4.d.n.385.2 2
16.13 even 4 768.4.d.c.385.2 2
20.3 even 4 150.4.c.d.49.2 2
20.7 even 4 150.4.c.d.49.1 2
20.19 odd 2 150.4.a.i.1.1 1
24.5 odd 2 576.4.a.r.1.1 1
24.11 even 2 576.4.a.q.1.1 1
28.3 even 6 294.4.e.g.79.1 2
28.11 odd 6 294.4.e.h.79.1 2
28.19 even 6 294.4.e.g.67.1 2
28.23 odd 6 294.4.e.h.67.1 2
28.27 even 2 294.4.a.e.1.1 1
36.7 odd 6 162.4.c.f.109.1 2
36.11 even 6 162.4.c.c.109.1 2
36.23 even 6 162.4.c.c.55.1 2
36.31 odd 6 162.4.c.f.55.1 2
44.43 even 2 726.4.a.f.1.1 1
52.31 even 4 1014.4.b.d.337.2 2
52.47 even 4 1014.4.b.d.337.1 2
52.51 odd 2 1014.4.a.g.1.1 1
60.23 odd 4 450.4.c.e.199.1 2
60.47 odd 4 450.4.c.e.199.2 2
60.59 even 2 450.4.a.h.1.1 1
68.67 odd 2 1734.4.a.d.1.1 1
76.75 even 2 2166.4.a.i.1.1 1
84.11 even 6 882.4.g.i.667.1 2
84.23 even 6 882.4.g.i.361.1 2
84.47 odd 6 882.4.g.f.361.1 2
84.59 odd 6 882.4.g.f.667.1 2
84.83 odd 2 882.4.a.n.1.1 1
132.131 odd 2 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 4.3 odd 2
18.4.a.a.1.1 1 12.11 even 2
48.4.a.c.1.1 1 1.1 even 1 trivial
144.4.a.c.1.1 1 3.2 odd 2
150.4.a.i.1.1 1 20.19 odd 2
150.4.c.d.49.1 2 20.7 even 4
150.4.c.d.49.2 2 20.3 even 4
162.4.c.c.55.1 2 36.23 even 6
162.4.c.c.109.1 2 36.11 even 6
162.4.c.f.55.1 2 36.31 odd 6
162.4.c.f.109.1 2 36.7 odd 6
192.4.a.c.1.1 1 8.5 even 2
192.4.a.i.1.1 1 8.3 odd 2
294.4.a.e.1.1 1 28.27 even 2
294.4.e.g.67.1 2 28.19 even 6
294.4.e.g.79.1 2 28.3 even 6
294.4.e.h.67.1 2 28.23 odd 6
294.4.e.h.79.1 2 28.11 odd 6
450.4.a.h.1.1 1 60.59 even 2
450.4.c.e.199.1 2 60.23 odd 4
450.4.c.e.199.2 2 60.47 odd 4
576.4.a.q.1.1 1 24.11 even 2
576.4.a.r.1.1 1 24.5 odd 2
726.4.a.f.1.1 1 44.43 even 2
768.4.d.c.385.1 2 16.5 even 4
768.4.d.c.385.2 2 16.13 even 4
768.4.d.n.385.1 2 16.3 odd 4
768.4.d.n.385.2 2 16.11 odd 4
882.4.a.n.1.1 1 84.83 odd 2
882.4.g.f.361.1 2 84.47 odd 6
882.4.g.f.667.1 2 84.59 odd 6
882.4.g.i.361.1 2 84.23 even 6
882.4.g.i.667.1 2 84.11 even 6
1014.4.a.g.1.1 1 52.51 odd 2
1014.4.b.d.337.1 2 52.47 even 4
1014.4.b.d.337.2 2 52.31 even 4
1200.4.a.b.1.1 1 5.4 even 2
1200.4.f.j.49.1 2 5.2 odd 4
1200.4.f.j.49.2 2 5.3 odd 4
1734.4.a.d.1.1 1 68.67 odd 2
2166.4.a.i.1.1 1 76.75 even 2
2178.4.a.e.1.1 1 132.131 odd 2
2352.4.a.e.1.1 1 7.6 odd 2