Properties

Label 2178.4.a.e.1.1
Level $2178$
Weight $4$
Character 2178.1
Self dual yes
Analytic conductor $128.506$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(128.506159993\)
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 \(\chi\) \(=\) 2178.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +16.0000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +16.0000 q^{7} -8.00000 q^{8} +12.0000 q^{10} -38.0000 q^{13} -32.0000 q^{14} +16.0000 q^{16} -126.000 q^{17} -20.0000 q^{19} -24.0000 q^{20} -168.000 q^{23} -89.0000 q^{25} +76.0000 q^{26} +64.0000 q^{28} +30.0000 q^{29} -88.0000 q^{31} -32.0000 q^{32} +252.000 q^{34} -96.0000 q^{35} +254.000 q^{37} +40.0000 q^{38} +48.0000 q^{40} +42.0000 q^{41} +52.0000 q^{43} +336.000 q^{46} +96.0000 q^{47} -87.0000 q^{49} +178.000 q^{50} -152.000 q^{52} -198.000 q^{53} -128.000 q^{56} -60.0000 q^{58} +660.000 q^{59} +538.000 q^{61} +176.000 q^{62} +64.0000 q^{64} +228.000 q^{65} +884.000 q^{67} -504.000 q^{68} +192.000 q^{70} -792.000 q^{71} -218.000 q^{73} -508.000 q^{74} -80.0000 q^{76} +520.000 q^{79} -96.0000 q^{80} -84.0000 q^{82} -492.000 q^{83} +756.000 q^{85} -104.000 q^{86} -810.000 q^{89} -608.000 q^{91} -672.000 q^{92} -192.000 q^{94} +120.000 q^{95} +1154.00 q^{97} +174.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 16.0000 0.863919 0.431959 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 12.0000 0.379473
\(11\) 0 0
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) −32.0000 −0.610883
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −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 \(0.538528\pi\)
\(20\) −24.0000 −0.268328
\(21\) 0 0
\(22\) 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
\(26\) 76.0000 0.573263
\(27\) 0 0
\(28\) 64.0000 0.431959
\(29\) 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 \(-0.582050\pi\)
−0.254924 + 0.966961i \(0.582050\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 252.000 1.27111
\(35\) −96.0000 −0.463627
\(36\) 0 0
\(37\) 254.000 1.12858 0.564288 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) 40.0000 0.170759
\(39\) 0 0
\(40\) 48.0000 0.189737
\(41\) 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
\(45\) 0 0
\(46\) 336.000 1.07697
\(47\) 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
\(50\) 178.000 0.503460
\(51\) 0 0
\(52\) −152.000 −0.405358
\(53\) −198.000 −0.513158 −0.256579 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −128.000 −0.305441
\(57\) 0 0
\(58\) −60.0000 −0.135834
\(59\) 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 \(-0.309022\pi\)
0.564622 + 0.825350i \(0.309022\pi\)
\(62\) 176.000 0.360516
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 228.000 0.435076
\(66\) 0 0
\(67\) 884.000 1.61191 0.805954 0.591979i \(-0.201653\pi\)
0.805954 + 0.591979i \(0.201653\pi\)
\(68\) −504.000 −0.898808
\(69\) 0 0
\(70\) 192.000 0.327834
\(71\) −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 \(-0.555915\pi\)
−0.174760 + 0.984611i \(0.555915\pi\)
\(74\) −508.000 −0.798024
\(75\) 0 0
\(76\) −80.0000 −0.120745
\(77\) 0 0
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) −96.0000 −0.134164
\(81\) 0 0
\(82\) −84.0000 −0.113125
\(83\) −492.000 −0.650651 −0.325325 0.945602i \(-0.605474\pi\)
−0.325325 + 0.945602i \(0.605474\pi\)
\(84\) 0 0
\(85\) 756.000 0.964703
\(86\) −104.000 −0.130402
\(87\) 0 0
\(88\) 0 0
\(89\) −810.000 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) −608.000 −0.700393
\(92\) −672.000 −0.761531
\(93\) 0 0
\(94\) −192.000 −0.210673
\(95\) 120.000 0.129597
\(96\) 0 0
\(97\) 1154.00 1.20795 0.603974 0.797004i \(-0.293583\pi\)
0.603974 + 0.797004i \(0.293583\pi\)
\(98\) 174.000 0.179354
\(99\) 0 0
\(100\) −356.000 −0.356000
\(101\) −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 \(-0.480499\pi\)
0.0612243 + 0.998124i \(0.480499\pi\)
\(104\) 304.000 0.286631
\(105\) 0 0
\(106\) 396.000 0.362858
\(107\) −1476.00 −1.33355 −0.666777 0.745257i \(-0.732327\pi\)
−0.666777 + 0.745257i \(0.732327\pi\)
\(108\) 0 0
\(109\) −1190.00 −1.04570 −0.522850 0.852425i \(-0.675131\pi\)
−0.522850 + 0.852425i \(0.675131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 256.000 0.215980
\(113\) 462.000 0.384613 0.192307 0.981335i \(-0.438403\pi\)
0.192307 + 0.981335i \(0.438403\pi\)
\(114\) 0 0
\(115\) 1008.00 0.817361
\(116\) 120.000 0.0960493
\(117\) 0 0
\(118\) −1320.00 −1.02980
\(119\) −2016.00 −1.55300
\(120\) 0 0
\(121\) 0 0
\(122\) −1076.00 −0.798496
\(123\) 0 0
\(124\) −352.000 −0.254924
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 2536.00 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −456.000 −0.307645
\(131\) 2292.00 1.52865 0.764324 0.644832i \(-0.223073\pi\)
0.764324 + 0.644832i \(0.223073\pi\)
\(132\) 0 0
\(133\) −320.000 −0.208628
\(134\) −1768.00 −1.13979
\(135\) 0 0
\(136\) 1008.00 0.635554
\(137\) 726.000 0.452747 0.226374 0.974041i \(-0.427313\pi\)
0.226374 + 0.974041i \(0.427313\pi\)
\(138\) 0 0
\(139\) −380.000 −0.231879 −0.115939 0.993256i \(-0.536988\pi\)
−0.115939 + 0.993256i \(0.536988\pi\)
\(140\) −384.000 −0.231814
\(141\) 0 0
\(142\) 1584.00 0.936101
\(143\) 0 0
\(144\) 0 0
\(145\) −180.000 −0.103091
\(146\) 436.000 0.247148
\(147\) 0 0
\(148\) 1016.00 0.564288
\(149\) 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 \(0.727476\pi\)
\(152\) 160.000 0.0853797
\(153\) 0 0
\(154\) 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 \(0.450121\pi\)
\(158\) −1040.00 −0.523658
\(159\) 0 0
\(160\) 192.000 0.0948683
\(161\) −2688.00 −1.31580
\(162\) 0 0
\(163\) −1852.00 −0.889938 −0.444969 0.895546i \(-0.646785\pi\)
−0.444969 + 0.895546i \(0.646785\pi\)
\(164\) 168.000 0.0799914
\(165\) 0 0
\(166\) 984.000 0.460080
\(167\) −2136.00 −0.989752 −0.494876 0.868964i \(-0.664787\pi\)
−0.494876 + 0.868964i \(0.664787\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) −1512.00 −0.682148
\(171\) 0 0
\(172\) 208.000 0.0922084
\(173\) 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
\(177\) 0 0
\(178\) 1620.00 0.682158
\(179\) 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 \(0.366588\pi\)
\(182\) 1216.00 0.495252
\(183\) 0 0
\(184\) 1344.00 0.538484
\(185\) −1524.00 −0.605658
\(186\) 0 0
\(187\) 0 0
\(188\) 384.000 0.148969
\(189\) 0 0
\(190\) −240.000 −0.0916391
\(191\) 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 \(-0.358768\pi\)
0.429279 + 0.903172i \(0.358768\pi\)
\(194\) −2308.00 −0.854148
\(195\) 0 0
\(196\) −348.000 −0.126822
\(197\) 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 \(-0.591986\pi\)
−0.284977 + 0.958534i \(0.591986\pi\)
\(200\) 712.000 0.251730
\(201\) 0 0
\(202\) 1236.00 0.430518
\(203\) 480.000 0.165958
\(204\) 0 0
\(205\) −252.000 −0.0858558
\(206\) −256.000 −0.0865843
\(207\) 0 0
\(208\) −608.000 −0.202679
\(209\) 0 0
\(210\) 0 0
\(211\) −3332.00 −1.08713 −0.543565 0.839367i \(-0.682926\pi\)
−0.543565 + 0.839367i \(0.682926\pi\)
\(212\) −792.000 −0.256579
\(213\) 0 0
\(214\) 2952.00 0.942965
\(215\) −312.000 −0.0989685
\(216\) 0 0
\(217\) −1408.00 −0.440467
\(218\) 2380.00 0.739422
\(219\) 0 0
\(220\) 0 0
\(221\) 4788.00 1.45736
\(222\) 0 0
\(223\) 2648.00 0.795171 0.397586 0.917565i \(-0.369848\pi\)
0.397586 + 0.917565i \(0.369848\pi\)
\(224\) −512.000 −0.152721
\(225\) 0 0
\(226\) −924.000 −0.271963
\(227\) 2244.00 0.656121 0.328061 0.944657i \(-0.393605\pi\)
0.328061 + 0.944657i \(0.393605\pi\)
\(228\) 0 0
\(229\) −5650.00 −1.63040 −0.815202 0.579177i \(-0.803374\pi\)
−0.815202 + 0.579177i \(0.803374\pi\)
\(230\) −2016.00 −0.577961
\(231\) 0 0
\(232\) −240.000 −0.0679171
\(233\) 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
\(236\) 2640.00 0.728175
\(237\) 0 0
\(238\) 4032.00 1.09813
\(239\) −1200.00 −0.324776 −0.162388 0.986727i \(-0.551920\pi\)
−0.162388 + 0.986727i \(0.551920\pi\)
\(240\) 0 0
\(241\) 718.000 0.191911 0.0959553 0.995386i \(-0.469409\pi\)
0.0959553 + 0.995386i \(0.469409\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2152.00 0.564622
\(245\) 522.000 0.136120
\(246\) 0 0
\(247\) 760.000 0.195780
\(248\) 704.000 0.180258
\(249\) 0 0
\(250\) −2568.00 −0.649658
\(251\) −6012.00 −1.51185 −0.755924 0.654659i \(-0.772812\pi\)
−0.755924 + 0.654659i \(0.772812\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −5072.00 −1.25294
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2046.00 0.496599 0.248300 0.968683i \(-0.420128\pi\)
0.248300 + 0.968683i \(0.420128\pi\)
\(258\) 0 0
\(259\) 4064.00 0.974999
\(260\) 912.000 0.217538
\(261\) 0 0
\(262\) −4584.00 −1.08092
\(263\) −6072.00 −1.42363 −0.711817 0.702365i \(-0.752127\pi\)
−0.711817 + 0.702365i \(0.752127\pi\)
\(264\) 0 0
\(265\) 1188.00 0.275390
\(266\) 640.000 0.147522
\(267\) 0 0
\(268\) 3536.00 0.805954
\(269\) 6930.00 1.57074 0.785371 0.619025i \(-0.212472\pi\)
0.785371 + 0.619025i \(0.212472\pi\)
\(270\) 0 0
\(271\) −1352.00 −0.303056 −0.151528 0.988453i \(-0.548419\pi\)
−0.151528 + 0.988453i \(0.548419\pi\)
\(272\) −2016.00 −0.449404
\(273\) 0 0
\(274\) −1452.00 −0.320141
\(275\) 0 0
\(276\) 0 0
\(277\) 1186.00 0.257256 0.128628 0.991693i \(-0.458943\pi\)
0.128628 + 0.991693i \(0.458943\pi\)
\(278\) 760.000 0.163963
\(279\) 0 0
\(280\) 768.000 0.163917
\(281\) 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 \(0.595989\pi\)
\(284\) −3168.00 −0.661923
\(285\) 0 0
\(286\) 0 0
\(287\) 672.000 0.138212
\(288\) 0 0
\(289\) 10963.0 2.23143
\(290\) 360.000 0.0728963
\(291\) 0 0
\(292\) −872.000 −0.174760
\(293\) 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
\(296\) −2032.00 −0.399012
\(297\) 0 0
\(298\) −3180.00 −0.618163
\(299\) 6384.00 1.23477
\(300\) 0 0
\(301\) 832.000 0.159321
\(302\) 4864.00 0.926794
\(303\) 0 0
\(304\) −320.000 −0.0603726
\(305\) −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
\(309\) 0 0
\(310\) −1056.00 −0.193473
\(311\) −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 \(0.643390\pi\)
\(314\) −1228.00 −0.220701
\(315\) 0 0
\(316\) 2080.00 0.370282
\(317\) 3426.00 0.607014 0.303507 0.952829i \(-0.401842\pi\)
0.303507 + 0.952829i \(0.401842\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −384.000 −0.0670820
\(321\) 0 0
\(322\) 5376.00 0.930412
\(323\) 2520.00 0.434107
\(324\) 0 0
\(325\) 3382.00 0.577230
\(326\) 3704.00 0.629281
\(327\) 0 0
\(328\) −336.000 −0.0565625
\(329\) 1536.00 0.257393
\(330\) 0 0
\(331\) −2788.00 −0.462968 −0.231484 0.972839i \(-0.574358\pi\)
−0.231484 + 0.972839i \(0.574358\pi\)
\(332\) −1968.00 −0.325325
\(333\) 0 0
\(334\) 4272.00 0.699861
\(335\) −5304.00 −0.865040
\(336\) 0 0
\(337\) −434.000 −0.0701528 −0.0350764 0.999385i \(-0.511167\pi\)
−0.0350764 + 0.999385i \(0.511167\pi\)
\(338\) 1506.00 0.242354
\(339\) 0 0
\(340\) 3024.00 0.482351
\(341\) 0 0
\(342\) 0 0
\(343\) −6880.00 −1.08305
\(344\) −416.000 −0.0652012
\(345\) 0 0
\(346\) −3516.00 −0.546304
\(347\) 6684.00 1.03405 0.517026 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(348\) 0 0
\(349\) −2630.00 −0.403383 −0.201692 0.979449i \(-0.564644\pi\)
−0.201692 + 0.979449i \(0.564644\pi\)
\(350\) 2848.00 0.434949
\(351\) 0 0
\(352\) 0 0
\(353\) 7422.00 1.11907 0.559537 0.828805i \(-0.310979\pi\)
0.559537 + 0.828805i \(0.310979\pi\)
\(354\) 0 0
\(355\) 4752.00 0.710451
\(356\) −3240.00 −0.482359
\(357\) 0 0
\(358\) −1080.00 −0.159441
\(359\) −10440.0 −1.53482 −0.767412 0.641154i \(-0.778456\pi\)
−0.767412 + 0.641154i \(0.778456\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) −3964.00 −0.575534
\(363\) 0 0
\(364\) −2432.00 −0.350196
\(365\) 1308.00 0.187572
\(366\) 0 0
\(367\) 10424.0 1.48264 0.741319 0.671153i \(-0.234200\pi\)
0.741319 + 0.671153i \(0.234200\pi\)
\(368\) −2688.00 −0.380765
\(369\) 0 0
\(370\) 3048.00 0.428265
\(371\) −3168.00 −0.443327
\(372\) 0 0
\(373\) −3278.00 −0.455036 −0.227518 0.973774i \(-0.573061\pi\)
−0.227518 + 0.973774i \(0.573061\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −768.000 −0.105337
\(377\) −1140.00 −0.155737
\(378\) 0 0
\(379\) 6140.00 0.832165 0.416083 0.909327i \(-0.363403\pi\)
0.416083 + 0.909327i \(0.363403\pi\)
\(380\) 480.000 0.0647986
\(381\) 0 0
\(382\) −5376.00 −0.720053
\(383\) 3072.00 0.409848 0.204924 0.978778i \(-0.434305\pi\)
0.204924 + 0.978778i \(0.434305\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4604.00 −0.607092
\(387\) 0 0
\(388\) 4616.00 0.603974
\(389\) −6150.00 −0.801587 −0.400794 0.916168i \(-0.631266\pi\)
−0.400794 + 0.916168i \(0.631266\pi\)
\(390\) 0 0
\(391\) 21168.0 2.73788
\(392\) 696.000 0.0896768
\(393\) 0 0
\(394\) −8748.00 −1.11857
\(395\) −3120.00 −0.397428
\(396\) 0 0
\(397\) −106.000 −0.0134005 −0.00670024 0.999978i \(-0.502133\pi\)
−0.00670024 + 0.999978i \(0.502133\pi\)
\(398\) 3200.00 0.403019
\(399\) 0 0
\(400\) −1424.00 −0.178000
\(401\) 1758.00 0.218929 0.109464 0.993991i \(-0.465086\pi\)
0.109464 + 0.993991i \(0.465086\pi\)
\(402\) 0 0
\(403\) 3344.00 0.413341
\(404\) −2472.00 −0.304422
\(405\) 0 0
\(406\) −960.000 −0.117350
\(407\) 0 0
\(408\) 0 0
\(409\) 3670.00 0.443691 0.221846 0.975082i \(-0.428792\pi\)
0.221846 + 0.975082i \(0.428792\pi\)
\(410\) 504.000 0.0607092
\(411\) 0 0
\(412\) 512.000 0.0612243
\(413\) 10560.0 1.25817
\(414\) 0 0
\(415\) 2952.00 0.349176
\(416\) 1216.00 0.143316
\(417\) 0 0
\(418\) 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 \(0.337069\pi\)
\(422\) 6664.00 0.768717
\(423\) 0 0
\(424\) 1584.00 0.181429
\(425\) 11214.0 1.27990
\(426\) 0 0
\(427\) 8608.00 0.975575
\(428\) −5904.00 −0.666777
\(429\) 0 0
\(430\) 624.000 0.0699813
\(431\) 9792.00 1.09435 0.547174 0.837019i \(-0.315704\pi\)
0.547174 + 0.837019i \(0.315704\pi\)
\(432\) 0 0
\(433\) −7342.00 −0.814859 −0.407430 0.913237i \(-0.633575\pi\)
−0.407430 + 0.913237i \(0.633575\pi\)
\(434\) 2816.00 0.311457
\(435\) 0 0
\(436\) −4760.00 −0.522850
\(437\) 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
\(441\) 0 0
\(442\) −9576.00 −1.03051
\(443\) 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
\(446\) −5296.00 −0.562271
\(447\) 0 0
\(448\) 1024.00 0.107990
\(449\) 1710.00 0.179732 0.0898662 0.995954i \(-0.471356\pi\)
0.0898662 + 0.995954i \(0.471356\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1848.00 0.192307
\(453\) 0 0
\(454\) −4488.00 −0.463948
\(455\) 3648.00 0.375870
\(456\) 0 0
\(457\) 646.000 0.0661239 0.0330619 0.999453i \(-0.489474\pi\)
0.0330619 + 0.999453i \(0.489474\pi\)
\(458\) 11300.0 1.15287
\(459\) 0 0
\(460\) 4032.00 0.408680
\(461\) −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 \(-0.609365\pi\)
−0.336861 + 0.941554i \(0.609365\pi\)
\(464\) 480.000 0.0480247
\(465\) 0 0
\(466\) −9396.00 −0.934037
\(467\) −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
\(470\) 1152.00 0.113059
\(471\) 0 0
\(472\) −5280.00 −0.514898
\(473\) 0 0
\(474\) 0 0
\(475\) 1780.00 0.171941
\(476\) −8064.00 −0.776498
\(477\) 0 0
\(478\) 2400.00 0.229652
\(479\) 9840.00 0.938624 0.469312 0.883032i \(-0.344502\pi\)
0.469312 + 0.883032i \(0.344502\pi\)
\(480\) 0 0
\(481\) −9652.00 −0.914955
\(482\) −1436.00 −0.135701
\(483\) 0 0
\(484\) 0 0
\(485\) −6924.00 −0.648253
\(486\) 0 0
\(487\) 1424.00 0.132500 0.0662501 0.997803i \(-0.478896\pi\)
0.0662501 + 0.997803i \(0.478896\pi\)
\(488\) −4304.00 −0.399248
\(489\) 0 0
\(490\) −1044.00 −0.0962513
\(491\) −4548.00 −0.418021 −0.209011 0.977913i \(-0.567024\pi\)
−0.209011 + 0.977913i \(0.567024\pi\)
\(492\) 0 0
\(493\) −3780.00 −0.345320
\(494\) −1520.00 −0.138437
\(495\) 0 0
\(496\) −1408.00 −0.127462
\(497\) −12672.0 −1.14370
\(498\) 0 0
\(499\) 6500.00 0.583126 0.291563 0.956552i \(-0.405825\pi\)
0.291563 + 0.956552i \(0.405825\pi\)
\(500\) 5136.00 0.459378
\(501\) 0 0
\(502\) 12024.0 1.06904
\(503\) 12168.0 1.07862 0.539308 0.842108i \(-0.318686\pi\)
0.539308 + 0.842108i \(0.318686\pi\)
\(504\) 0 0
\(505\) 3708.00 0.326740
\(506\) 0 0
\(507\) 0 0
\(508\) 10144.0 0.885959
\(509\) 21090.0 1.83654 0.918269 0.395957i \(-0.129587\pi\)
0.918269 + 0.395957i \(0.129587\pi\)
\(510\) 0 0
\(511\) −3488.00 −0.301957
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −4092.00 −0.351149
\(515\) −768.000 −0.0657129
\(516\) 0 0
\(517\) 0 0
\(518\) −8128.00 −0.689428
\(519\) 0 0
\(520\) −1824.00 −0.153822
\(521\) 5238.00 0.440462 0.220231 0.975448i \(-0.429319\pi\)
0.220231 + 0.975448i \(0.429319\pi\)
\(522\) 0 0
\(523\) −8588.00 −0.718025 −0.359012 0.933333i \(-0.616886\pi\)
−0.359012 + 0.933333i \(0.616886\pi\)
\(524\) 9168.00 0.764324
\(525\) 0 0
\(526\) 12144.0 1.00666
\(527\) 11088.0 0.916510
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) −2376.00 −0.194730
\(531\) 0 0
\(532\) −1280.00 −0.104314
\(533\) −1596.00 −0.129701
\(534\) 0 0
\(535\) 8856.00 0.715660
\(536\) −7072.00 −0.569895
\(537\) 0 0
\(538\) −13860.0 −1.11068
\(539\) 0 0
\(540\) 0 0
\(541\) −3062.00 −0.243338 −0.121669 0.992571i \(-0.538825\pi\)
−0.121669 + 0.992571i \(0.538825\pi\)
\(542\) 2704.00 0.214293
\(543\) 0 0
\(544\) 4032.00 0.317777
\(545\) 7140.00 0.561182
\(546\) 0 0
\(547\) 8476.00 0.662537 0.331268 0.943537i \(-0.392523\pi\)
0.331268 + 0.943537i \(0.392523\pi\)
\(548\) 2904.00 0.226374
\(549\) 0 0
\(550\) 0 0
\(551\) −600.000 −0.0463899
\(552\) 0 0
\(553\) 8320.00 0.639787
\(554\) −2372.00 −0.181907
\(555\) 0 0
\(556\) −1520.00 −0.115939
\(557\) −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
\(560\) −1536.00 −0.115907
\(561\) 0 0
\(562\) −4884.00 −0.366582
\(563\) −12.0000 −0.000898294 0 −0.000449147 1.00000i \(-0.500143\pi\)
−0.000449147 1.00000i \(0.500143\pi\)
\(564\) 0 0
\(565\) −2772.00 −0.206405
\(566\) 5656.00 0.420034
\(567\) 0 0
\(568\) 6336.00 0.468050
\(569\) 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
\(573\) 0 0
\(574\) −1344.00 −0.0977308
\(575\) 14952.0 1.08442
\(576\) 0 0
\(577\) −10366.0 −0.747907 −0.373953 0.927447i \(-0.621998\pi\)
−0.373953 + 0.927447i \(0.621998\pi\)
\(578\) −21926.0 −1.57786
\(579\) 0 0
\(580\) −720.000 −0.0515455
\(581\) −7872.00 −0.562109
\(582\) 0 0
\(583\) 0 0
\(584\) 1744.00 0.123574
\(585\) 0 0
\(586\) −9516.00 −0.670823
\(587\) −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
\(590\) 7920.00 0.552646
\(591\) 0 0
\(592\) 4064.00 0.282144
\(593\) 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
\(596\) 6360.00 0.437107
\(597\) 0 0
\(598\) −12768.0 −0.873114
\(599\) −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 \(-0.685307\pi\)
−0.549828 + 0.835278i \(0.685307\pi\)
\(602\) −1664.00 −0.112657
\(603\) 0 0
\(604\) −9728.00 −0.655342
\(605\) 0 0
\(606\) 0 0
\(607\) 24136.0 1.61392 0.806960 0.590605i \(-0.201111\pi\)
0.806960 + 0.590605i \(0.201111\pi\)
\(608\) 640.000 0.0426898
\(609\) 0 0
\(610\) 6456.00 0.428518
\(611\) −3648.00 −0.241542
\(612\) 0 0
\(613\) 4642.00 0.305854 0.152927 0.988237i \(-0.451130\pi\)
0.152927 + 0.988237i \(0.451130\pi\)
\(614\) −16952.0 −1.11421
\(615\) 0 0
\(616\) 0 0
\(617\) 6726.00 0.438863 0.219432 0.975628i \(-0.429580\pi\)
0.219432 + 0.975628i \(0.429580\pi\)
\(618\) 0 0
\(619\) −21220.0 −1.37787 −0.688937 0.724821i \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(620\) 2112.00 0.136806
\(621\) 0 0
\(622\) 9264.00 0.597191
\(623\) −12960.0 −0.833437
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 9644.00 0.615738
\(627\) 0 0
\(628\) 2456.00 0.156059
\(629\) −32004.0 −2.02875
\(630\) 0 0
\(631\) 29792.0 1.87956 0.939779 0.341783i \(-0.111031\pi\)
0.939779 + 0.341783i \(0.111031\pi\)
\(632\) −4160.00 −0.261829
\(633\) 0 0
\(634\) −6852.00 −0.429223
\(635\) −15216.0 −0.950911
\(636\) 0 0
\(637\) 3306.00 0.205633
\(638\) 0 0
\(639\) 0 0
\(640\) 768.000 0.0474342
\(641\) 10158.0 0.625923 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(642\) 0 0
\(643\) 29828.0 1.82940 0.914698 0.404138i \(-0.132429\pi\)
0.914698 + 0.404138i \(0.132429\pi\)
\(644\) −10752.0 −0.657901
\(645\) 0 0
\(646\) −5040.00 −0.306960
\(647\) −1944.00 −0.118124 −0.0590622 0.998254i \(-0.518811\pi\)
−0.0590622 + 0.998254i \(0.518811\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −6764.00 −0.408163
\(651\) 0 0
\(652\) −7408.00 −0.444969
\(653\) −26718.0 −1.60116 −0.800579 0.599227i \(-0.795475\pi\)
−0.800579 + 0.599227i \(0.795475\pi\)
\(654\) 0 0
\(655\) −13752.0 −0.820359
\(656\) 672.000 0.0399957
\(657\) 0 0
\(658\) −3072.00 −0.182005
\(659\) 4260.00 0.251815 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(660\) 0 0
\(661\) 22862.0 1.34528 0.672639 0.739971i \(-0.265161\pi\)
0.672639 + 0.739971i \(0.265161\pi\)
\(662\) 5576.00 0.327368
\(663\) 0 0
\(664\) 3936.00 0.230040
\(665\) 1920.00 0.111962
\(666\) 0 0
\(667\) −5040.00 −0.292578
\(668\) −8544.00 −0.494876
\(669\) 0 0
\(670\) 10608.0 0.611676
\(671\) 0 0
\(672\) 0 0
\(673\) 32542.0 1.86390 0.931948 0.362592i \(-0.118108\pi\)
0.931948 + 0.362592i \(0.118108\pi\)
\(674\) 868.000 0.0496055
\(675\) 0 0
\(676\) −3012.00 −0.171370
\(677\) 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
\(680\) −6048.00 −0.341074
\(681\) 0 0
\(682\) 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
\(686\) 13760.0 0.765830
\(687\) 0 0
\(688\) 832.000 0.0461042
\(689\) 7524.00 0.416026
\(690\) 0 0
\(691\) −13228.0 −0.728244 −0.364122 0.931351i \(-0.618631\pi\)
−0.364122 + 0.931351i \(0.618631\pi\)
\(692\) 7032.00 0.386296
\(693\) 0 0
\(694\) −13368.0 −0.731185
\(695\) 2280.00 0.124439
\(696\) 0 0
\(697\) −5292.00 −0.287588
\(698\) 5260.00 0.285235
\(699\) 0 0
\(700\) −5696.00 −0.307555
\(701\) 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
\(705\) 0 0
\(706\) −14844.0 −0.791305
\(707\) −9888.00 −0.525992
\(708\) 0 0
\(709\) −27250.0 −1.44343 −0.721717 0.692188i \(-0.756647\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(710\) −9504.00 −0.502364
\(711\) 0 0
\(712\) 6480.00 0.341079
\(713\) 14784.0 0.776529
\(714\) 0 0
\(715\) 0 0
\(716\) 2160.00 0.112742
\(717\) 0 0
\(718\) 20880.0 1.08529
\(719\) 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
\(722\) 12918.0 0.665870
\(723\) 0 0
\(724\) 7928.00 0.406964
\(725\) −2670.00 −0.136774
\(726\) 0 0
\(727\) 17984.0 0.917455 0.458727 0.888577i \(-0.348305\pi\)
0.458727 + 0.888577i \(0.348305\pi\)
\(728\) 4864.00 0.247626
\(729\) 0 0
\(730\) −2616.00 −0.132634
\(731\) −6552.00 −0.331511
\(732\) 0 0
\(733\) −16598.0 −0.836373 −0.418186 0.908361i \(-0.637334\pi\)
−0.418186 + 0.908361i \(0.637334\pi\)
\(734\) −20848.0 −1.04838
\(735\) 0 0
\(736\) 5376.00 0.269242
\(737\) 0 0
\(738\) 0 0
\(739\) −1460.00 −0.0726752 −0.0363376 0.999340i \(-0.511569\pi\)
−0.0363376 + 0.999340i \(0.511569\pi\)
\(740\) −6096.00 −0.302829
\(741\) 0 0
\(742\) 6336.00 0.313480
\(743\) −30072.0 −1.48484 −0.742419 0.669936i \(-0.766322\pi\)
−0.742419 + 0.669936i \(0.766322\pi\)
\(744\) 0 0
\(745\) −9540.00 −0.469152
\(746\) 6556.00 0.321759
\(747\) 0 0
\(748\) 0 0
\(749\) −23616.0 −1.15208
\(750\) 0 0
\(751\) −18088.0 −0.878882 −0.439441 0.898271i \(-0.644823\pi\)
−0.439441 + 0.898271i \(0.644823\pi\)
\(752\) 1536.00 0.0744843
\(753\) 0 0
\(754\) 2280.00 0.110123
\(755\) 14592.0 0.703387
\(756\) 0 0
\(757\) 24734.0 1.18755 0.593773 0.804633i \(-0.297638\pi\)
0.593773 + 0.804633i \(0.297638\pi\)
\(758\) −12280.0 −0.588430
\(759\) 0 0
\(760\) −960.000 −0.0458196
\(761\) −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
\(764\) 10752.0 0.509154
\(765\) 0 0
\(766\) −6144.00 −0.289806
\(767\) −25080.0 −1.18069
\(768\) 0 0
\(769\) −16130.0 −0.756388 −0.378194 0.925726i \(-0.623455\pi\)
−0.378194 + 0.925726i \(0.623455\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9208.00 0.429279
\(773\) −29718.0 −1.38277 −0.691386 0.722486i \(-0.742999\pi\)
−0.691386 + 0.722486i \(0.742999\pi\)
\(774\) 0 0
\(775\) 7832.00 0.363011
\(776\) −9232.00 −0.427074
\(777\) 0 0
\(778\) 12300.0 0.566808
\(779\) −840.000 −0.0386343
\(780\) 0 0
\(781\) 0 0
\(782\) −42336.0 −1.93597
\(783\) 0 0
\(784\) −1392.00 −0.0634111
\(785\) −3684.00 −0.167500
\(786\) 0 0
\(787\) −9524.00 −0.431377 −0.215689 0.976462i \(-0.569200\pi\)
−0.215689 + 0.976462i \(0.569200\pi\)
\(788\) 17496.0 0.790951
\(789\) 0 0
\(790\) 6240.00 0.281024
\(791\) 7392.00 0.332275
\(792\) 0 0
\(793\) −20444.0 −0.915495
\(794\) 212.000 0.00947556
\(795\) 0 0
\(796\) −6400.00 −0.284977
\(797\) 33906.0 1.50692 0.753458 0.657496i \(-0.228384\pi\)
0.753458 + 0.657496i \(0.228384\pi\)
\(798\) 0 0
\(799\) −12096.0 −0.535577
\(800\) 2848.00 0.125865
\(801\) 0 0
\(802\) −3516.00 −0.154806
\(803\) 0 0
\(804\) 0 0
\(805\) 16128.0 0.706133
\(806\) −6688.00 −0.292276
\(807\) 0 0
\(808\) 4944.00 0.215259
\(809\) −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 \(0.351410\pi\)
\(812\) 1920.00 0.0829788
\(813\) 0 0
\(814\) 0 0
\(815\) 11112.0 0.477591
\(816\) 0 0
\(817\) −1040.00 −0.0445349
\(818\) −7340.00 −0.313737
\(819\) 0 0
\(820\) −1008.00 −0.0429279
\(821\) −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 \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(824\) −1024.00 −0.0432921
\(825\) 0 0
\(826\) −21120.0 −0.889660
\(827\) 13644.0 0.573698 0.286849 0.957976i \(-0.407392\pi\)
0.286849 + 0.957976i \(0.407392\pi\)
\(828\) 0 0
\(829\) −2410.00 −0.100968 −0.0504842 0.998725i \(-0.516076\pi\)
−0.0504842 + 0.998725i \(0.516076\pi\)
\(830\) −5904.00 −0.246905
\(831\) 0 0
\(832\) −2432.00 −0.101339
\(833\) 10962.0 0.455955
\(834\) 0 0
\(835\) 12816.0 0.531157
\(836\) 0 0
\(837\) 0 0
\(838\) −19320.0 −0.796418
\(839\) −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
\(842\) −16924.0 −0.692684
\(843\) 0 0
\(844\) −13328.0 −0.543565
\(845\) 4518.00 0.183934
\(846\) 0 0
\(847\) 0 0
\(848\) −3168.00 −0.128290
\(849\) 0 0
\(850\) −22428.0 −0.905028
\(851\) −42672.0 −1.71889
\(852\) 0 0
\(853\) −32078.0 −1.28761 −0.643804 0.765190i \(-0.722645\pi\)
−0.643804 + 0.765190i \(0.722645\pi\)
\(854\) −17216.0 −0.689835
\(855\) 0 0
\(856\) 11808.0 0.471483
\(857\) −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 \(-0.291926\pi\)
0.608115 + 0.793849i \(0.291926\pi\)
\(860\) −1248.00 −0.0494842
\(861\) 0 0
\(862\) −19584.0 −0.773821
\(863\) −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
\(866\) 14684.0 0.576192
\(867\) 0 0
\(868\) −5632.00 −0.220233
\(869\) 0 0
\(870\) 0 0
\(871\) −33592.0 −1.30680
\(872\) 9520.00 0.369711
\(873\) 0 0
\(874\) −6720.00 −0.260077
\(875\) 20544.0 0.793730
\(876\) 0 0
\(877\) 21706.0 0.835758 0.417879 0.908503i \(-0.362774\pi\)
0.417879 + 0.908503i \(0.362774\pi\)
\(878\) 21280.0 0.817956
\(879\) 0 0
\(880\) 0 0
\(881\) 14958.0 0.572018 0.286009 0.958227i \(-0.407671\pi\)
0.286009 + 0.958227i \(0.407671\pi\)
\(882\) 0 0
\(883\) −32812.0 −1.25052 −0.625261 0.780415i \(-0.715008\pi\)
−0.625261 + 0.780415i \(0.715008\pi\)
\(884\) 19152.0 0.728678
\(885\) 0 0
\(886\) −34824.0 −1.32047
\(887\) −38856.0 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(888\) 0 0
\(889\) 40576.0 1.53079
\(890\) −9720.00 −0.366084
\(891\) 0 0
\(892\) 10592.0 0.397586
\(893\) −1920.00 −0.0719489
\(894\) 0 0
\(895\) −3240.00 −0.121007
\(896\) −2048.00 −0.0763604
\(897\) 0 0
\(898\) −3420.00 −0.127090
\(899\) −2640.00 −0.0979410
\(900\) 0 0
\(901\) 24948.0 0.922462
\(902\) 0 0
\(903\) 0 0
\(904\) −3696.00 −0.135981
\(905\) −11892.0 −0.436799
\(906\) 0 0
\(907\) −28276.0 −1.03516 −0.517579 0.855635i \(-0.673167\pi\)
−0.517579 + 0.855635i \(0.673167\pi\)
\(908\) 8976.00 0.328061
\(909\) 0 0
\(910\) −7296.00 −0.265780
\(911\) −8112.00 −0.295019 −0.147510 0.989061i \(-0.547126\pi\)
−0.147510 + 0.989061i \(0.547126\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1292.00 −0.0467566
\(915\) 0 0
\(916\) −22600.0 −0.815202
\(917\) 36672.0 1.32063
\(918\) 0 0
\(919\) 26080.0 0.936126 0.468063 0.883695i \(-0.344952\pi\)
0.468063 + 0.883695i \(0.344952\pi\)
\(920\) −8064.00 −0.288981
\(921\) 0 0
\(922\) 12036.0 0.429918
\(923\) 30096.0 1.07326
\(924\) 0 0
\(925\) −22606.0 −0.803547
\(926\) 13424.0 0.476393
\(927\) 0 0
\(928\) −960.000 −0.0339586
\(929\) −49170.0 −1.73651 −0.868254 0.496120i \(-0.834757\pi\)
−0.868254 + 0.496120i \(0.834757\pi\)
\(930\) 0 0
\(931\) 1740.00 0.0612526
\(932\) 18792.0 0.660464
\(933\) 0 0
\(934\) 10728.0 0.375836
\(935\) 0 0
\(936\) 0 0
\(937\) −48314.0 −1.68447 −0.842236 0.539110i \(-0.818761\pi\)
−0.842236 + 0.539110i \(0.818761\pi\)
\(938\) −28288.0 −0.984687
\(939\) 0 0
\(940\) −2304.00 −0.0799449
\(941\) 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
\(944\) 10560.0 0.364088
\(945\) 0 0
\(946\) 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
\(950\) −3560.00 −0.121581
\(951\) 0 0
\(952\) 16128.0 0.549067
\(953\) −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
\(956\) −4800.00 −0.162388
\(957\) 0 0
\(958\) −19680.0 −0.663708
\(959\) 11616.0 0.391137
\(960\) 0 0
\(961\) −22047.0 −0.740056
\(962\) 19304.0 0.646971
\(963\) 0 0
\(964\) 2872.00 0.0959553
\(965\) −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
\(969\) 0 0
\(970\) 13848.0 0.458384
\(971\) 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
\(974\) −2848.00 −0.0936918
\(975\) 0 0
\(976\) 8608.00 0.282311
\(977\) −16674.0 −0.546007 −0.273003 0.962013i \(-0.588017\pi\)
−0.273003 + 0.962013i \(0.588017\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2088.00 0.0680599
\(981\) 0 0
\(982\) 9096.00 0.295586
\(983\) 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
\(986\) 7560.00 0.244178
\(987\) 0 0
\(988\) 3040.00 0.0978900
\(989\) −8736.00 −0.280878
\(990\) 0 0
\(991\) −15928.0 −0.510565 −0.255282 0.966867i \(-0.582168\pi\)
−0.255282 + 0.966867i \(0.582168\pi\)
\(992\) 2816.00 0.0901291
\(993\) 0 0
\(994\) 25344.0 0.808715
\(995\) 9600.00 0.305870
\(996\) 0 0
\(997\) −42014.0 −1.33460 −0.667300 0.744789i \(-0.732550\pi\)
−0.667300 + 0.744789i \(0.732550\pi\)
\(998\) −13000.0 −0.412332
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2178.4.a.e.1.1 1
3.2 odd 2 726.4.a.f.1.1 1
11.10 odd 2 18.4.a.a.1.1 1
33.32 even 2 6.4.a.a.1.1 1
44.43 even 2 144.4.a.c.1.1 1
55.32 even 4 450.4.c.e.199.2 2
55.43 even 4 450.4.c.e.199.1 2
55.54 odd 2 450.4.a.h.1.1 1
77.10 even 6 882.4.g.f.667.1 2
77.32 odd 6 882.4.g.i.667.1 2
77.54 even 6 882.4.g.f.361.1 2
77.65 odd 6 882.4.g.i.361.1 2
77.76 even 2 882.4.a.n.1.1 1
88.21 odd 2 576.4.a.q.1.1 1
88.43 even 2 576.4.a.r.1.1 1
99.32 even 6 162.4.c.f.55.1 2
99.43 odd 6 162.4.c.c.109.1 2
99.65 even 6 162.4.c.f.109.1 2
99.76 odd 6 162.4.c.c.55.1 2
132.131 odd 2 48.4.a.c.1.1 1
165.32 odd 4 150.4.c.d.49.1 2
165.98 odd 4 150.4.c.d.49.2 2
165.164 even 2 150.4.a.i.1.1 1
231.32 even 6 294.4.e.h.79.1 2
231.65 even 6 294.4.e.h.67.1 2
231.131 odd 6 294.4.e.g.67.1 2
231.164 odd 6 294.4.e.g.79.1 2
231.230 odd 2 294.4.a.e.1.1 1
264.131 odd 2 192.4.a.c.1.1 1
264.197 even 2 192.4.a.i.1.1 1
429.164 odd 4 1014.4.b.d.337.1 2
429.395 odd 4 1014.4.b.d.337.2 2
429.428 even 2 1014.4.a.g.1.1 1
528.131 odd 4 768.4.d.c.385.2 2
528.197 even 4 768.4.d.n.385.2 2
528.395 odd 4 768.4.d.c.385.1 2
528.461 even 4 768.4.d.n.385.1 2
561.560 even 2 1734.4.a.d.1.1 1
627.626 odd 2 2166.4.a.i.1.1 1
660.263 even 4 1200.4.f.j.49.2 2
660.527 even 4 1200.4.f.j.49.1 2
660.659 odd 2 1200.4.a.b.1.1 1
924.923 even 2 2352.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 33.32 even 2
18.4.a.a.1.1 1 11.10 odd 2
48.4.a.c.1.1 1 132.131 odd 2
144.4.a.c.1.1 1 44.43 even 2
150.4.a.i.1.1 1 165.164 even 2
150.4.c.d.49.1 2 165.32 odd 4
150.4.c.d.49.2 2 165.98 odd 4
162.4.c.c.55.1 2 99.76 odd 6
162.4.c.c.109.1 2 99.43 odd 6
162.4.c.f.55.1 2 99.32 even 6
162.4.c.f.109.1 2 99.65 even 6
192.4.a.c.1.1 1 264.131 odd 2
192.4.a.i.1.1 1 264.197 even 2
294.4.a.e.1.1 1 231.230 odd 2
294.4.e.g.67.1 2 231.131 odd 6
294.4.e.g.79.1 2 231.164 odd 6
294.4.e.h.67.1 2 231.65 even 6
294.4.e.h.79.1 2 231.32 even 6
450.4.a.h.1.1 1 55.54 odd 2
450.4.c.e.199.1 2 55.43 even 4
450.4.c.e.199.2 2 55.32 even 4
576.4.a.q.1.1 1 88.21 odd 2
576.4.a.r.1.1 1 88.43 even 2
726.4.a.f.1.1 1 3.2 odd 2
768.4.d.c.385.1 2 528.395 odd 4
768.4.d.c.385.2 2 528.131 odd 4
768.4.d.n.385.1 2 528.461 even 4
768.4.d.n.385.2 2 528.197 even 4
882.4.a.n.1.1 1 77.76 even 2
882.4.g.f.361.1 2 77.54 even 6
882.4.g.f.667.1 2 77.10 even 6
882.4.g.i.361.1 2 77.65 odd 6
882.4.g.i.667.1 2 77.32 odd 6
1014.4.a.g.1.1 1 429.428 even 2
1014.4.b.d.337.1 2 429.164 odd 4
1014.4.b.d.337.2 2 429.395 odd 4
1200.4.a.b.1.1 1 660.659 odd 2
1200.4.f.j.49.1 2 660.527 even 4
1200.4.f.j.49.2 2 660.263 even 4
1734.4.a.d.1.1 1 561.560 even 2
2166.4.a.i.1.1 1 627.626 odd 2
2178.4.a.e.1.1 1 1.1 even 1 trivial
2352.4.a.e.1.1 1 924.923 even 2