Properties

Label 1521.4.a.b.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +8.00000 q^{4} -17.0000 q^{5} +20.0000 q^{7} +O(q^{10})\) \(q-4.00000 q^{2} +8.00000 q^{4} -17.0000 q^{5} +20.0000 q^{7} +68.0000 q^{10} +32.0000 q^{11} -80.0000 q^{14} -64.0000 q^{16} +13.0000 q^{17} +30.0000 q^{19} -136.000 q^{20} -128.000 q^{22} -78.0000 q^{23} +164.000 q^{25} +160.000 q^{28} -197.000 q^{29} -74.0000 q^{31} +256.000 q^{32} -52.0000 q^{34} -340.000 q^{35} -227.000 q^{37} -120.000 q^{38} +165.000 q^{41} -156.000 q^{43} +256.000 q^{44} +312.000 q^{46} +162.000 q^{47} +57.0000 q^{49} -656.000 q^{50} -93.0000 q^{53} -544.000 q^{55} +788.000 q^{58} +864.000 q^{59} +145.000 q^{61} +296.000 q^{62} -512.000 q^{64} +862.000 q^{67} +104.000 q^{68} +1360.00 q^{70} -654.000 q^{71} +215.000 q^{73} +908.000 q^{74} +240.000 q^{76} +640.000 q^{77} -76.0000 q^{79} +1088.00 q^{80} -660.000 q^{82} -628.000 q^{83} -221.000 q^{85} +624.000 q^{86} +266.000 q^{89} -624.000 q^{92} -648.000 q^{94} -510.000 q^{95} +238.000 q^{97} -228.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 8.00000 1.00000
\(5\) −17.0000 −1.52053 −0.760263 0.649615i \(-0.774930\pi\)
−0.760263 + 0.649615i \(0.774930\pi\)
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 68.0000 2.15035
\(11\) 32.0000 0.877124 0.438562 0.898701i \(-0.355488\pi\)
0.438562 + 0.898701i \(0.355488\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −80.0000 −1.52721
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 13.0000 0.185468 0.0927342 0.995691i \(-0.470439\pi\)
0.0927342 + 0.995691i \(0.470439\pi\)
\(18\) 0 0
\(19\) 30.0000 0.362235 0.181118 0.983461i \(-0.442029\pi\)
0.181118 + 0.983461i \(0.442029\pi\)
\(20\) −136.000 −1.52053
\(21\) 0 0
\(22\) −128.000 −1.24044
\(23\) −78.0000 −0.707136 −0.353568 0.935409i \(-0.615032\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(24\) 0 0
\(25\) 164.000 1.31200
\(26\) 0 0
\(27\) 0 0
\(28\) 160.000 1.07990
\(29\) −197.000 −1.26145 −0.630724 0.776007i \(-0.717242\pi\)
−0.630724 + 0.776007i \(0.717242\pi\)
\(30\) 0 0
\(31\) −74.0000 −0.428735 −0.214368 0.976753i \(-0.568769\pi\)
−0.214368 + 0.976753i \(0.568769\pi\)
\(32\) 256.000 1.41421
\(33\) 0 0
\(34\) −52.0000 −0.262292
\(35\) −340.000 −1.64201
\(36\) 0 0
\(37\) −227.000 −1.00861 −0.504305 0.863526i \(-0.668251\pi\)
−0.504305 + 0.863526i \(0.668251\pi\)
\(38\) −120.000 −0.512278
\(39\) 0 0
\(40\) 0 0
\(41\) 165.000 0.628504 0.314252 0.949340i \(-0.398246\pi\)
0.314252 + 0.949340i \(0.398246\pi\)
\(42\) 0 0
\(43\) −156.000 −0.553251 −0.276625 0.960978i \(-0.589216\pi\)
−0.276625 + 0.960978i \(0.589216\pi\)
\(44\) 256.000 0.877124
\(45\) 0 0
\(46\) 312.000 1.00004
\(47\) 162.000 0.502769 0.251384 0.967887i \(-0.419114\pi\)
0.251384 + 0.967887i \(0.419114\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) −656.000 −1.85545
\(51\) 0 0
\(52\) 0 0
\(53\) −93.0000 −0.241029 −0.120514 0.992712i \(-0.538454\pi\)
−0.120514 + 0.992712i \(0.538454\pi\)
\(54\) 0 0
\(55\) −544.000 −1.33369
\(56\) 0 0
\(57\) 0 0
\(58\) 788.000 1.78396
\(59\) 864.000 1.90650 0.953248 0.302190i \(-0.0977178\pi\)
0.953248 + 0.302190i \(0.0977178\pi\)
\(60\) 0 0
\(61\) 145.000 0.304350 0.152175 0.988354i \(-0.451372\pi\)
0.152175 + 0.988354i \(0.451372\pi\)
\(62\) 296.000 0.606323
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 862.000 1.57179 0.785896 0.618359i \(-0.212202\pi\)
0.785896 + 0.618359i \(0.212202\pi\)
\(68\) 104.000 0.185468
\(69\) 0 0
\(70\) 1360.00 2.32216
\(71\) −654.000 −1.09318 −0.546588 0.837402i \(-0.684074\pi\)
−0.546588 + 0.837402i \(0.684074\pi\)
\(72\) 0 0
\(73\) 215.000 0.344710 0.172355 0.985035i \(-0.444862\pi\)
0.172355 + 0.985035i \(0.444862\pi\)
\(74\) 908.000 1.42639
\(75\) 0 0
\(76\) 240.000 0.362235
\(77\) 640.000 0.947205
\(78\) 0 0
\(79\) −76.0000 −0.108236 −0.0541182 0.998535i \(-0.517235\pi\)
−0.0541182 + 0.998535i \(0.517235\pi\)
\(80\) 1088.00 1.52053
\(81\) 0 0
\(82\) −660.000 −0.888839
\(83\) −628.000 −0.830505 −0.415253 0.909706i \(-0.636307\pi\)
−0.415253 + 0.909706i \(0.636307\pi\)
\(84\) 0 0
\(85\) −221.000 −0.282010
\(86\) 624.000 0.782415
\(87\) 0 0
\(88\) 0 0
\(89\) 266.000 0.316808 0.158404 0.987374i \(-0.449365\pi\)
0.158404 + 0.987374i \(0.449365\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −624.000 −0.707136
\(93\) 0 0
\(94\) −648.000 −0.711022
\(95\) −510.000 −0.550788
\(96\) 0 0
\(97\) 238.000 0.249126 0.124563 0.992212i \(-0.460247\pi\)
0.124563 + 0.992212i \(0.460247\pi\)
\(98\) −228.000 −0.235015
\(99\) 0 0
\(100\) 1312.00 1.31200
\(101\) 819.000 0.806867 0.403433 0.915009i \(-0.367817\pi\)
0.403433 + 0.915009i \(0.367817\pi\)
\(102\) 0 0
\(103\) 1638.00 1.56696 0.783480 0.621417i \(-0.213443\pi\)
0.783480 + 0.621417i \(0.213443\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 372.000 0.340866
\(107\) −522.000 −0.471623 −0.235811 0.971799i \(-0.575775\pi\)
−0.235811 + 0.971799i \(0.575775\pi\)
\(108\) 0 0
\(109\) −1634.00 −1.43586 −0.717930 0.696115i \(-0.754910\pi\)
−0.717930 + 0.696115i \(0.754910\pi\)
\(110\) 2176.00 1.88612
\(111\) 0 0
\(112\) −1280.00 −1.07990
\(113\) −327.000 −0.272226 −0.136113 0.990693i \(-0.543461\pi\)
−0.136113 + 0.990693i \(0.543461\pi\)
\(114\) 0 0
\(115\) 1326.00 1.07522
\(116\) −1576.00 −1.26145
\(117\) 0 0
\(118\) −3456.00 −2.69619
\(119\) 260.000 0.200287
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) −580.000 −0.430416
\(123\) 0 0
\(124\) −592.000 −0.428735
\(125\) −663.000 −0.474404
\(126\) 0 0
\(127\) −2158.00 −1.50781 −0.753904 0.656985i \(-0.771831\pi\)
−0.753904 + 0.656985i \(0.771831\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −730.000 −0.486873 −0.243437 0.969917i \(-0.578275\pi\)
−0.243437 + 0.969917i \(0.578275\pi\)
\(132\) 0 0
\(133\) 600.000 0.391177
\(134\) −3448.00 −2.22285
\(135\) 0 0
\(136\) 0 0
\(137\) −1671.00 −1.04207 −0.521033 0.853536i \(-0.674453\pi\)
−0.521033 + 0.853536i \(0.674453\pi\)
\(138\) 0 0
\(139\) 912.000 0.556510 0.278255 0.960507i \(-0.410244\pi\)
0.278255 + 0.960507i \(0.410244\pi\)
\(140\) −2720.00 −1.64201
\(141\) 0 0
\(142\) 2616.00 1.54598
\(143\) 0 0
\(144\) 0 0
\(145\) 3349.00 1.91806
\(146\) −860.000 −0.487494
\(147\) 0 0
\(148\) −1816.00 −1.00861
\(149\) 2115.00 1.16287 0.581435 0.813593i \(-0.302492\pi\)
0.581435 + 0.813593i \(0.302492\pi\)
\(150\) 0 0
\(151\) 514.000 0.277011 0.138506 0.990362i \(-0.455770\pi\)
0.138506 + 0.990362i \(0.455770\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2560.00 −1.33955
\(155\) 1258.00 0.651903
\(156\) 0 0
\(157\) 2901.00 1.47468 0.737341 0.675521i \(-0.236081\pi\)
0.737341 + 0.675521i \(0.236081\pi\)
\(158\) 304.000 0.153069
\(159\) 0 0
\(160\) −4352.00 −2.15035
\(161\) −1560.00 −0.763635
\(162\) 0 0
\(163\) 2360.00 1.13405 0.567023 0.823702i \(-0.308095\pi\)
0.567023 + 0.823702i \(0.308095\pi\)
\(164\) 1320.00 0.628504
\(165\) 0 0
\(166\) 2512.00 1.17451
\(167\) −280.000 −0.129743 −0.0648714 0.997894i \(-0.520664\pi\)
−0.0648714 + 0.997894i \(0.520664\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 884.000 0.398822
\(171\) 0 0
\(172\) −1248.00 −0.553251
\(173\) −1326.00 −0.582739 −0.291370 0.956611i \(-0.594111\pi\)
−0.291370 + 0.956611i \(0.594111\pi\)
\(174\) 0 0
\(175\) 3280.00 1.41683
\(176\) −2048.00 −0.877124
\(177\) 0 0
\(178\) −1064.00 −0.448035
\(179\) −4264.00 −1.78048 −0.890241 0.455490i \(-0.849464\pi\)
−0.890241 + 0.455490i \(0.849464\pi\)
\(180\) 0 0
\(181\) −403.000 −0.165496 −0.0827479 0.996571i \(-0.526370\pi\)
−0.0827479 + 0.996571i \(0.526370\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3859.00 1.53362
\(186\) 0 0
\(187\) 416.000 0.162679
\(188\) 1296.00 0.502769
\(189\) 0 0
\(190\) 2040.00 0.778932
\(191\) 1246.00 0.472028 0.236014 0.971750i \(-0.424159\pi\)
0.236014 + 0.971750i \(0.424159\pi\)
\(192\) 0 0
\(193\) 267.000 0.0995807 0.0497904 0.998760i \(-0.484145\pi\)
0.0497904 + 0.998760i \(0.484145\pi\)
\(194\) −952.000 −0.352318
\(195\) 0 0
\(196\) 456.000 0.166181
\(197\) −1278.00 −0.462202 −0.231101 0.972930i \(-0.574233\pi\)
−0.231101 + 0.972930i \(0.574233\pi\)
\(198\) 0 0
\(199\) 4238.00 1.50967 0.754834 0.655916i \(-0.227717\pi\)
0.754834 + 0.655916i \(0.227717\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −3276.00 −1.14108
\(203\) −3940.00 −1.36224
\(204\) 0 0
\(205\) −2805.00 −0.955657
\(206\) −6552.00 −2.21602
\(207\) 0 0
\(208\) 0 0
\(209\) 960.000 0.317725
\(210\) 0 0
\(211\) 3070.00 1.00165 0.500823 0.865549i \(-0.333031\pi\)
0.500823 + 0.865549i \(0.333031\pi\)
\(212\) −744.000 −0.241029
\(213\) 0 0
\(214\) 2088.00 0.666975
\(215\) 2652.00 0.841232
\(216\) 0 0
\(217\) −1480.00 −0.462991
\(218\) 6536.00 2.03061
\(219\) 0 0
\(220\) −4352.00 −1.33369
\(221\) 0 0
\(222\) 0 0
\(223\) −5378.00 −1.61497 −0.807483 0.589891i \(-0.799171\pi\)
−0.807483 + 0.589891i \(0.799171\pi\)
\(224\) 5120.00 1.52721
\(225\) 0 0
\(226\) 1308.00 0.384986
\(227\) 3974.00 1.16195 0.580977 0.813920i \(-0.302671\pi\)
0.580977 + 0.813920i \(0.302671\pi\)
\(228\) 0 0
\(229\) −6298.00 −1.81740 −0.908698 0.417455i \(-0.862922\pi\)
−0.908698 + 0.417455i \(0.862922\pi\)
\(230\) −5304.00 −1.52059
\(231\) 0 0
\(232\) 0 0
\(233\) −4030.00 −1.13311 −0.566554 0.824025i \(-0.691724\pi\)
−0.566554 + 0.824025i \(0.691724\pi\)
\(234\) 0 0
\(235\) −2754.00 −0.764473
\(236\) 6912.00 1.90650
\(237\) 0 0
\(238\) −1040.00 −0.283249
\(239\) 984.000 0.266317 0.133158 0.991095i \(-0.457488\pi\)
0.133158 + 0.991095i \(0.457488\pi\)
\(240\) 0 0
\(241\) 943.000 0.252050 0.126025 0.992027i \(-0.459778\pi\)
0.126025 + 0.992027i \(0.459778\pi\)
\(242\) 1228.00 0.326194
\(243\) 0 0
\(244\) 1160.00 0.304350
\(245\) −969.000 −0.252682
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 2652.00 0.670909
\(251\) 2730.00 0.686518 0.343259 0.939241i \(-0.388469\pi\)
0.343259 + 0.939241i \(0.388469\pi\)
\(252\) 0 0
\(253\) −2496.00 −0.620246
\(254\) 8632.00 2.13236
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 1885.00 0.457522 0.228761 0.973483i \(-0.426533\pi\)
0.228761 + 0.973483i \(0.426533\pi\)
\(258\) 0 0
\(259\) −4540.00 −1.08920
\(260\) 0 0
\(261\) 0 0
\(262\) 2920.00 0.688543
\(263\) −4032.00 −0.945338 −0.472669 0.881240i \(-0.656709\pi\)
−0.472669 + 0.881240i \(0.656709\pi\)
\(264\) 0 0
\(265\) 1581.00 0.366491
\(266\) −2400.00 −0.553208
\(267\) 0 0
\(268\) 6896.00 1.57179
\(269\) −4006.00 −0.907993 −0.453997 0.891003i \(-0.650002\pi\)
−0.453997 + 0.891003i \(0.650002\pi\)
\(270\) 0 0
\(271\) −4296.00 −0.962965 −0.481482 0.876456i \(-0.659901\pi\)
−0.481482 + 0.876456i \(0.659901\pi\)
\(272\) −832.000 −0.185468
\(273\) 0 0
\(274\) 6684.00 1.47371
\(275\) 5248.00 1.15079
\(276\) 0 0
\(277\) −5551.00 −1.20407 −0.602035 0.798470i \(-0.705643\pi\)
−0.602035 + 0.798470i \(0.705643\pi\)
\(278\) −3648.00 −0.787023
\(279\) 0 0
\(280\) 0 0
\(281\) 5557.00 1.17973 0.589863 0.807504i \(-0.299182\pi\)
0.589863 + 0.807504i \(0.299182\pi\)
\(282\) 0 0
\(283\) 3120.00 0.655352 0.327676 0.944790i \(-0.393734\pi\)
0.327676 + 0.944790i \(0.393734\pi\)
\(284\) −5232.00 −1.09318
\(285\) 0 0
\(286\) 0 0
\(287\) 3300.00 0.678721
\(288\) 0 0
\(289\) −4744.00 −0.965601
\(290\) −13396.0 −2.71255
\(291\) 0 0
\(292\) 1720.00 0.344710
\(293\) −8301.00 −1.65512 −0.827559 0.561379i \(-0.810271\pi\)
−0.827559 + 0.561379i \(0.810271\pi\)
\(294\) 0 0
\(295\) −14688.0 −2.89888
\(296\) 0 0
\(297\) 0 0
\(298\) −8460.00 −1.64455
\(299\) 0 0
\(300\) 0 0
\(301\) −3120.00 −0.597455
\(302\) −2056.00 −0.391753
\(303\) 0 0
\(304\) −1920.00 −0.362235
\(305\) −2465.00 −0.462772
\(306\) 0 0
\(307\) 8678.00 1.61329 0.806644 0.591037i \(-0.201281\pi\)
0.806644 + 0.591037i \(0.201281\pi\)
\(308\) 5120.00 0.947205
\(309\) 0 0
\(310\) −5032.00 −0.921930
\(311\) −8658.00 −1.57862 −0.789309 0.613996i \(-0.789561\pi\)
−0.789309 + 0.613996i \(0.789561\pi\)
\(312\) 0 0
\(313\) −5250.00 −0.948075 −0.474038 0.880505i \(-0.657204\pi\)
−0.474038 + 0.880505i \(0.657204\pi\)
\(314\) −11604.0 −2.08551
\(315\) 0 0
\(316\) −608.000 −0.108236
\(317\) −6413.00 −1.13625 −0.568123 0.822944i \(-0.692330\pi\)
−0.568123 + 0.822944i \(0.692330\pi\)
\(318\) 0 0
\(319\) −6304.00 −1.10645
\(320\) 8704.00 1.52053
\(321\) 0 0
\(322\) 6240.00 1.07994
\(323\) 390.000 0.0671832
\(324\) 0 0
\(325\) 0 0
\(326\) −9440.00 −1.60378
\(327\) 0 0
\(328\) 0 0
\(329\) 3240.00 0.542939
\(330\) 0 0
\(331\) 3488.00 0.579208 0.289604 0.957147i \(-0.406476\pi\)
0.289604 + 0.957147i \(0.406476\pi\)
\(332\) −5024.00 −0.830505
\(333\) 0 0
\(334\) 1120.00 0.183484
\(335\) −14654.0 −2.38995
\(336\) 0 0
\(337\) −1833.00 −0.296290 −0.148145 0.988966i \(-0.547330\pi\)
−0.148145 + 0.988966i \(0.547330\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1768.00 −0.282010
\(341\) −2368.00 −0.376054
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) 0 0
\(346\) 5304.00 0.824118
\(347\) −7230.00 −1.11852 −0.559260 0.828992i \(-0.688915\pi\)
−0.559260 + 0.828992i \(0.688915\pi\)
\(348\) 0 0
\(349\) −5258.00 −0.806459 −0.403230 0.915099i \(-0.632112\pi\)
−0.403230 + 0.915099i \(0.632112\pi\)
\(350\) −13120.0 −2.00370
\(351\) 0 0
\(352\) 8192.00 1.24044
\(353\) −3163.00 −0.476911 −0.238455 0.971153i \(-0.576641\pi\)
−0.238455 + 0.971153i \(0.576641\pi\)
\(354\) 0 0
\(355\) 11118.0 1.66220
\(356\) 2128.00 0.316808
\(357\) 0 0
\(358\) 17056.0 2.51798
\(359\) 10068.0 1.48014 0.740068 0.672532i \(-0.234793\pi\)
0.740068 + 0.672532i \(0.234793\pi\)
\(360\) 0 0
\(361\) −5959.00 −0.868786
\(362\) 1612.00 0.234047
\(363\) 0 0
\(364\) 0 0
\(365\) −3655.00 −0.524141
\(366\) 0 0
\(367\) 7438.00 1.05793 0.528965 0.848644i \(-0.322580\pi\)
0.528965 + 0.848644i \(0.322580\pi\)
\(368\) 4992.00 0.707136
\(369\) 0 0
\(370\) −15436.0 −2.16886
\(371\) −1860.00 −0.260287
\(372\) 0 0
\(373\) −9683.00 −1.34415 −0.672073 0.740485i \(-0.734596\pi\)
−0.672073 + 0.740485i \(0.734596\pi\)
\(374\) −1664.00 −0.230063
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1062.00 −0.143935 −0.0719674 0.997407i \(-0.522928\pi\)
−0.0719674 + 0.997407i \(0.522928\pi\)
\(380\) −4080.00 −0.550788
\(381\) 0 0
\(382\) −4984.00 −0.667549
\(383\) 3532.00 0.471219 0.235609 0.971848i \(-0.424291\pi\)
0.235609 + 0.971848i \(0.424291\pi\)
\(384\) 0 0
\(385\) −10880.0 −1.44025
\(386\) −1068.00 −0.140828
\(387\) 0 0
\(388\) 1904.00 0.249126
\(389\) 11063.0 1.44194 0.720972 0.692964i \(-0.243696\pi\)
0.720972 + 0.692964i \(0.243696\pi\)
\(390\) 0 0
\(391\) −1014.00 −0.131151
\(392\) 0 0
\(393\) 0 0
\(394\) 5112.00 0.653652
\(395\) 1292.00 0.164576
\(396\) 0 0
\(397\) −5986.00 −0.756747 −0.378374 0.925653i \(-0.623517\pi\)
−0.378374 + 0.925653i \(0.623517\pi\)
\(398\) −16952.0 −2.13499
\(399\) 0 0
\(400\) −10496.0 −1.31200
\(401\) −5935.00 −0.739102 −0.369551 0.929211i \(-0.620488\pi\)
−0.369551 + 0.929211i \(0.620488\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6552.00 0.806867
\(405\) 0 0
\(406\) 15760.0 1.92649
\(407\) −7264.00 −0.884676
\(408\) 0 0
\(409\) −15089.0 −1.82421 −0.912106 0.409954i \(-0.865545\pi\)
−0.912106 + 0.409954i \(0.865545\pi\)
\(410\) 11220.0 1.35150
\(411\) 0 0
\(412\) 13104.0 1.56696
\(413\) 17280.0 2.05882
\(414\) 0 0
\(415\) 10676.0 1.26281
\(416\) 0 0
\(417\) 0 0
\(418\) −3840.00 −0.449331
\(419\) 10814.0 1.26086 0.630428 0.776248i \(-0.282880\pi\)
0.630428 + 0.776248i \(0.282880\pi\)
\(420\) 0 0
\(421\) −6535.00 −0.756524 −0.378262 0.925699i \(-0.623478\pi\)
−0.378262 + 0.925699i \(0.623478\pi\)
\(422\) −12280.0 −1.41654
\(423\) 0 0
\(424\) 0 0
\(425\) 2132.00 0.243335
\(426\) 0 0
\(427\) 2900.00 0.328667
\(428\) −4176.00 −0.471623
\(429\) 0 0
\(430\) −10608.0 −1.18968
\(431\) −1980.00 −0.221284 −0.110642 0.993860i \(-0.535291\pi\)
−0.110642 + 0.993860i \(0.535291\pi\)
\(432\) 0 0
\(433\) −6929.00 −0.769022 −0.384511 0.923120i \(-0.625630\pi\)
−0.384511 + 0.923120i \(0.625630\pi\)
\(434\) 5920.00 0.654767
\(435\) 0 0
\(436\) −13072.0 −1.43586
\(437\) −2340.00 −0.256150
\(438\) 0 0
\(439\) −4576.00 −0.497496 −0.248748 0.968568i \(-0.580019\pi\)
−0.248748 + 0.968568i \(0.580019\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8812.00 0.945081 0.472540 0.881309i \(-0.343337\pi\)
0.472540 + 0.881309i \(0.343337\pi\)
\(444\) 0 0
\(445\) −4522.00 −0.481715
\(446\) 21512.0 2.28391
\(447\) 0 0
\(448\) −10240.0 −1.07990
\(449\) −1918.00 −0.201595 −0.100797 0.994907i \(-0.532139\pi\)
−0.100797 + 0.994907i \(0.532139\pi\)
\(450\) 0 0
\(451\) 5280.00 0.551276
\(452\) −2616.00 −0.272226
\(453\) 0 0
\(454\) −15896.0 −1.64325
\(455\) 0 0
\(456\) 0 0
\(457\) −11761.0 −1.20384 −0.601922 0.798555i \(-0.705598\pi\)
−0.601922 + 0.798555i \(0.705598\pi\)
\(458\) 25192.0 2.57019
\(459\) 0 0
\(460\) 10608.0 1.07522
\(461\) −901.000 −0.0910277 −0.0455138 0.998964i \(-0.514493\pi\)
−0.0455138 + 0.998964i \(0.514493\pi\)
\(462\) 0 0
\(463\) 1372.00 0.137715 0.0688577 0.997626i \(-0.478065\pi\)
0.0688577 + 0.997626i \(0.478065\pi\)
\(464\) 12608.0 1.26145
\(465\) 0 0
\(466\) 16120.0 1.60246
\(467\) 6396.00 0.633772 0.316886 0.948464i \(-0.397363\pi\)
0.316886 + 0.948464i \(0.397363\pi\)
\(468\) 0 0
\(469\) 17240.0 1.69738
\(470\) 11016.0 1.08113
\(471\) 0 0
\(472\) 0 0
\(473\) −4992.00 −0.485269
\(474\) 0 0
\(475\) 4920.00 0.475253
\(476\) 2080.00 0.200287
\(477\) 0 0
\(478\) −3936.00 −0.376629
\(479\) −3270.00 −0.311921 −0.155960 0.987763i \(-0.549847\pi\)
−0.155960 + 0.987763i \(0.549847\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3772.00 −0.356452
\(483\) 0 0
\(484\) −2456.00 −0.230654
\(485\) −4046.00 −0.378803
\(486\) 0 0
\(487\) 19920.0 1.85351 0.926757 0.375661i \(-0.122584\pi\)
0.926757 + 0.375661i \(0.122584\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 3876.00 0.357347
\(491\) −6552.00 −0.602215 −0.301108 0.953590i \(-0.597356\pi\)
−0.301108 + 0.953590i \(0.597356\pi\)
\(492\) 0 0
\(493\) −2561.00 −0.233959
\(494\) 0 0
\(495\) 0 0
\(496\) 4736.00 0.428735
\(497\) −13080.0 −1.18052
\(498\) 0 0
\(499\) 1746.00 0.156637 0.0783183 0.996928i \(-0.475045\pi\)
0.0783183 + 0.996928i \(0.475045\pi\)
\(500\) −5304.00 −0.474404
\(501\) 0 0
\(502\) −10920.0 −0.970883
\(503\) −14692.0 −1.30235 −0.651177 0.758926i \(-0.725724\pi\)
−0.651177 + 0.758926i \(0.725724\pi\)
\(504\) 0 0
\(505\) −13923.0 −1.22686
\(506\) 9984.00 0.877160
\(507\) 0 0
\(508\) −17264.0 −1.50781
\(509\) −8077.00 −0.703353 −0.351677 0.936122i \(-0.614388\pi\)
−0.351677 + 0.936122i \(0.614388\pi\)
\(510\) 0 0
\(511\) 4300.00 0.372252
\(512\) −16384.0 −1.41421
\(513\) 0 0
\(514\) −7540.00 −0.647033
\(515\) −27846.0 −2.38260
\(516\) 0 0
\(517\) 5184.00 0.440990
\(518\) 18160.0 1.54036
\(519\) 0 0
\(520\) 0 0
\(521\) −11247.0 −0.945758 −0.472879 0.881127i \(-0.656785\pi\)
−0.472879 + 0.881127i \(0.656785\pi\)
\(522\) 0 0
\(523\) 2732.00 0.228417 0.114208 0.993457i \(-0.463567\pi\)
0.114208 + 0.993457i \(0.463567\pi\)
\(524\) −5840.00 −0.486873
\(525\) 0 0
\(526\) 16128.0 1.33691
\(527\) −962.000 −0.0795168
\(528\) 0 0
\(529\) −6083.00 −0.499959
\(530\) −6324.00 −0.518296
\(531\) 0 0
\(532\) 4800.00 0.391177
\(533\) 0 0
\(534\) 0 0
\(535\) 8874.00 0.717115
\(536\) 0 0
\(537\) 0 0
\(538\) 16024.0 1.28410
\(539\) 1824.00 0.145761
\(540\) 0 0
\(541\) −18375.0 −1.46026 −0.730132 0.683306i \(-0.760542\pi\)
−0.730132 + 0.683306i \(0.760542\pi\)
\(542\) 17184.0 1.36184
\(543\) 0 0
\(544\) 3328.00 0.262292
\(545\) 27778.0 2.18326
\(546\) 0 0
\(547\) −10346.0 −0.808708 −0.404354 0.914603i \(-0.632504\pi\)
−0.404354 + 0.914603i \(0.632504\pi\)
\(548\) −13368.0 −1.04207
\(549\) 0 0
\(550\) −20992.0 −1.62746
\(551\) −5910.00 −0.456941
\(552\) 0 0
\(553\) −1520.00 −0.116884
\(554\) 22204.0 1.70281
\(555\) 0 0
\(556\) 7296.00 0.556510
\(557\) −345.000 −0.0262444 −0.0131222 0.999914i \(-0.504177\pi\)
−0.0131222 + 0.999914i \(0.504177\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 21760.0 1.64201
\(561\) 0 0
\(562\) −22228.0 −1.66838
\(563\) 8580.00 0.642280 0.321140 0.947032i \(-0.395934\pi\)
0.321140 + 0.947032i \(0.395934\pi\)
\(564\) 0 0
\(565\) 5559.00 0.413927
\(566\) −12480.0 −0.926808
\(567\) 0 0
\(568\) 0 0
\(569\) 19682.0 1.45011 0.725055 0.688691i \(-0.241814\pi\)
0.725055 + 0.688691i \(0.241814\pi\)
\(570\) 0 0
\(571\) 26624.0 1.95128 0.975639 0.219382i \(-0.0704042\pi\)
0.975639 + 0.219382i \(0.0704042\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −13200.0 −0.959856
\(575\) −12792.0 −0.927762
\(576\) 0 0
\(577\) −14101.0 −1.01739 −0.508694 0.860948i \(-0.669871\pi\)
−0.508694 + 0.860948i \(0.669871\pi\)
\(578\) 18976.0 1.36557
\(579\) 0 0
\(580\) 26792.0 1.91806
\(581\) −12560.0 −0.896862
\(582\) 0 0
\(583\) −2976.00 −0.211412
\(584\) 0 0
\(585\) 0 0
\(586\) 33204.0 2.34069
\(587\) −1408.00 −0.0990023 −0.0495012 0.998774i \(-0.515763\pi\)
−0.0495012 + 0.998774i \(0.515763\pi\)
\(588\) 0 0
\(589\) −2220.00 −0.155303
\(590\) 58752.0 4.09963
\(591\) 0 0
\(592\) 14528.0 1.00861
\(593\) 1241.00 0.0859389 0.0429694 0.999076i \(-0.486318\pi\)
0.0429694 + 0.999076i \(0.486318\pi\)
\(594\) 0 0
\(595\) −4420.00 −0.304542
\(596\) 16920.0 1.16287
\(597\) 0 0
\(598\) 0 0
\(599\) −11078.0 −0.755651 −0.377825 0.925877i \(-0.623328\pi\)
−0.377825 + 0.925877i \(0.623328\pi\)
\(600\) 0 0
\(601\) −13817.0 −0.937782 −0.468891 0.883256i \(-0.655346\pi\)
−0.468891 + 0.883256i \(0.655346\pi\)
\(602\) 12480.0 0.844928
\(603\) 0 0
\(604\) 4112.00 0.277011
\(605\) 5219.00 0.350715
\(606\) 0 0
\(607\) 8270.00 0.552997 0.276498 0.961014i \(-0.410826\pi\)
0.276498 + 0.961014i \(0.410826\pi\)
\(608\) 7680.00 0.512278
\(609\) 0 0
\(610\) 9860.00 0.654459
\(611\) 0 0
\(612\) 0 0
\(613\) 22273.0 1.46753 0.733767 0.679402i \(-0.237761\pi\)
0.733767 + 0.679402i \(0.237761\pi\)
\(614\) −34712.0 −2.28153
\(615\) 0 0
\(616\) 0 0
\(617\) 18989.0 1.23901 0.619504 0.784993i \(-0.287334\pi\)
0.619504 + 0.784993i \(0.287334\pi\)
\(618\) 0 0
\(619\) 72.0000 0.00467516 0.00233758 0.999997i \(-0.499256\pi\)
0.00233758 + 0.999997i \(0.499256\pi\)
\(620\) 10064.0 0.651903
\(621\) 0 0
\(622\) 34632.0 2.23250
\(623\) 5320.00 0.342121
\(624\) 0 0
\(625\) −9229.00 −0.590656
\(626\) 21000.0 1.34078
\(627\) 0 0
\(628\) 23208.0 1.47468
\(629\) −2951.00 −0.187065
\(630\) 0 0
\(631\) −23380.0 −1.47503 −0.737514 0.675331i \(-0.764001\pi\)
−0.737514 + 0.675331i \(0.764001\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 25652.0 1.60689
\(635\) 36686.0 2.29266
\(636\) 0 0
\(637\) 0 0
\(638\) 25216.0 1.56475
\(639\) 0 0
\(640\) 0 0
\(641\) −6383.00 −0.393313 −0.196656 0.980472i \(-0.563008\pi\)
−0.196656 + 0.980472i \(0.563008\pi\)
\(642\) 0 0
\(643\) −17104.0 −1.04901 −0.524507 0.851406i \(-0.675750\pi\)
−0.524507 + 0.851406i \(0.675750\pi\)
\(644\) −12480.0 −0.763635
\(645\) 0 0
\(646\) −1560.00 −0.0950114
\(647\) −6994.00 −0.424981 −0.212490 0.977163i \(-0.568157\pi\)
−0.212490 + 0.977163i \(0.568157\pi\)
\(648\) 0 0
\(649\) 27648.0 1.67223
\(650\) 0 0
\(651\) 0 0
\(652\) 18880.0 1.13405
\(653\) 5250.00 0.314622 0.157311 0.987549i \(-0.449717\pi\)
0.157311 + 0.987549i \(0.449717\pi\)
\(654\) 0 0
\(655\) 12410.0 0.740304
\(656\) −10560.0 −0.628504
\(657\) 0 0
\(658\) −12960.0 −0.767832
\(659\) 4340.00 0.256544 0.128272 0.991739i \(-0.459057\pi\)
0.128272 + 0.991739i \(0.459057\pi\)
\(660\) 0 0
\(661\) −4179.00 −0.245907 −0.122953 0.992412i \(-0.539237\pi\)
−0.122953 + 0.992412i \(0.539237\pi\)
\(662\) −13952.0 −0.819124
\(663\) 0 0
\(664\) 0 0
\(665\) −10200.0 −0.594796
\(666\) 0 0
\(667\) 15366.0 0.892015
\(668\) −2240.00 −0.129743
\(669\) 0 0
\(670\) 58616.0 3.37990
\(671\) 4640.00 0.266953
\(672\) 0 0
\(673\) 22867.0 1.30974 0.654872 0.755740i \(-0.272722\pi\)
0.654872 + 0.755740i \(0.272722\pi\)
\(674\) 7332.00 0.419018
\(675\) 0 0
\(676\) 0 0
\(677\) −5410.00 −0.307124 −0.153562 0.988139i \(-0.549075\pi\)
−0.153562 + 0.988139i \(0.549075\pi\)
\(678\) 0 0
\(679\) 4760.00 0.269031
\(680\) 0 0
\(681\) 0 0
\(682\) 9472.00 0.531821
\(683\) 13578.0 0.760685 0.380342 0.924846i \(-0.375806\pi\)
0.380342 + 0.924846i \(0.375806\pi\)
\(684\) 0 0
\(685\) 28407.0 1.58449
\(686\) 22880.0 1.27341
\(687\) 0 0
\(688\) 9984.00 0.553251
\(689\) 0 0
\(690\) 0 0
\(691\) 12744.0 0.701599 0.350799 0.936451i \(-0.385910\pi\)
0.350799 + 0.936451i \(0.385910\pi\)
\(692\) −10608.0 −0.582739
\(693\) 0 0
\(694\) 28920.0 1.58183
\(695\) −15504.0 −0.846187
\(696\) 0 0
\(697\) 2145.00 0.116568
\(698\) 21032.0 1.14051
\(699\) 0 0
\(700\) 26240.0 1.41683
\(701\) −16406.0 −0.883946 −0.441973 0.897028i \(-0.645721\pi\)
−0.441973 + 0.897028i \(0.645721\pi\)
\(702\) 0 0
\(703\) −6810.00 −0.365354
\(704\) −16384.0 −0.877124
\(705\) 0 0
\(706\) 12652.0 0.674454
\(707\) 16380.0 0.871334
\(708\) 0 0
\(709\) 709.000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) −44472.0 −2.35071
\(711\) 0 0
\(712\) 0 0
\(713\) 5772.00 0.303174
\(714\) 0 0
\(715\) 0 0
\(716\) −34112.0 −1.78048
\(717\) 0 0
\(718\) −40272.0 −2.09323
\(719\) 7644.00 0.396486 0.198243 0.980153i \(-0.436477\pi\)
0.198243 + 0.980153i \(0.436477\pi\)
\(720\) 0 0
\(721\) 32760.0 1.69216
\(722\) 23836.0 1.22865
\(723\) 0 0
\(724\) −3224.00 −0.165496
\(725\) −32308.0 −1.65502
\(726\) 0 0
\(727\) −15808.0 −0.806446 −0.403223 0.915102i \(-0.632110\pi\)
−0.403223 + 0.915102i \(0.632110\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14620.0 0.741247
\(731\) −2028.00 −0.102611
\(732\) 0 0
\(733\) −2583.00 −0.130157 −0.0650786 0.997880i \(-0.520730\pi\)
−0.0650786 + 0.997880i \(0.520730\pi\)
\(734\) −29752.0 −1.49614
\(735\) 0 0
\(736\) −19968.0 −1.00004
\(737\) 27584.0 1.37866
\(738\) 0 0
\(739\) 4076.00 0.202893 0.101447 0.994841i \(-0.467653\pi\)
0.101447 + 0.994841i \(0.467653\pi\)
\(740\) 30872.0 1.53362
\(741\) 0 0
\(742\) 7440.00 0.368101
\(743\) 34056.0 1.68155 0.840776 0.541383i \(-0.182099\pi\)
0.840776 + 0.541383i \(0.182099\pi\)
\(744\) 0 0
\(745\) −35955.0 −1.76817
\(746\) 38732.0 1.90091
\(747\) 0 0
\(748\) 3328.00 0.162679
\(749\) −10440.0 −0.509305
\(750\) 0 0
\(751\) −364.000 −0.0176865 −0.00884324 0.999961i \(-0.502815\pi\)
−0.00884324 + 0.999961i \(0.502815\pi\)
\(752\) −10368.0 −0.502769
\(753\) 0 0
\(754\) 0 0
\(755\) −8738.00 −0.421203
\(756\) 0 0
\(757\) −6914.00 −0.331960 −0.165980 0.986129i \(-0.553079\pi\)
−0.165980 + 0.986129i \(0.553079\pi\)
\(758\) 4248.00 0.203554
\(759\) 0 0
\(760\) 0 0
\(761\) −13982.0 −0.666028 −0.333014 0.942922i \(-0.608066\pi\)
−0.333014 + 0.942922i \(0.608066\pi\)
\(762\) 0 0
\(763\) −32680.0 −1.55058
\(764\) 9968.00 0.472028
\(765\) 0 0
\(766\) −14128.0 −0.666404
\(767\) 0 0
\(768\) 0 0
\(769\) −18066.0 −0.847174 −0.423587 0.905855i \(-0.639229\pi\)
−0.423587 + 0.905855i \(0.639229\pi\)
\(770\) 43520.0 2.03682
\(771\) 0 0
\(772\) 2136.00 0.0995807
\(773\) −14434.0 −0.671610 −0.335805 0.941931i \(-0.609008\pi\)
−0.335805 + 0.941931i \(0.609008\pi\)
\(774\) 0 0
\(775\) −12136.0 −0.562501
\(776\) 0 0
\(777\) 0 0
\(778\) −44252.0 −2.03922
\(779\) 4950.00 0.227666
\(780\) 0 0
\(781\) −20928.0 −0.958851
\(782\) 4056.00 0.185476
\(783\) 0 0
\(784\) −3648.00 −0.166181
\(785\) −49317.0 −2.24229
\(786\) 0 0
\(787\) −15398.0 −0.697433 −0.348716 0.937228i \(-0.613382\pi\)
−0.348716 + 0.937228i \(0.613382\pi\)
\(788\) −10224.0 −0.462202
\(789\) 0 0
\(790\) −5168.00 −0.232746
\(791\) −6540.00 −0.293977
\(792\) 0 0
\(793\) 0 0
\(794\) 23944.0 1.07020
\(795\) 0 0
\(796\) 33904.0 1.50967
\(797\) 36842.0 1.63740 0.818702 0.574219i \(-0.194694\pi\)
0.818702 + 0.574219i \(0.194694\pi\)
\(798\) 0 0
\(799\) 2106.00 0.0932477
\(800\) 41984.0 1.85545
\(801\) 0 0
\(802\) 23740.0 1.04525
\(803\) 6880.00 0.302354
\(804\) 0 0
\(805\) 26520.0 1.16113
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −41511.0 −1.80402 −0.902008 0.431719i \(-0.857907\pi\)
−0.902008 + 0.431719i \(0.857907\pi\)
\(810\) 0 0
\(811\) 23066.0 0.998714 0.499357 0.866396i \(-0.333570\pi\)
0.499357 + 0.866396i \(0.333570\pi\)
\(812\) −31520.0 −1.36224
\(813\) 0 0
\(814\) 29056.0 1.25112
\(815\) −40120.0 −1.72435
\(816\) 0 0
\(817\) −4680.00 −0.200407
\(818\) 60356.0 2.57983
\(819\) 0 0
\(820\) −22440.0 −0.955657
\(821\) −28838.0 −1.22589 −0.612943 0.790127i \(-0.710015\pi\)
−0.612943 + 0.790127i \(0.710015\pi\)
\(822\) 0 0
\(823\) 27456.0 1.16289 0.581443 0.813587i \(-0.302488\pi\)
0.581443 + 0.813587i \(0.302488\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −69120.0 −2.91161
\(827\) 33572.0 1.41162 0.705812 0.708399i \(-0.250582\pi\)
0.705812 + 0.708399i \(0.250582\pi\)
\(828\) 0 0
\(829\) −45799.0 −1.91878 −0.959388 0.282090i \(-0.908972\pi\)
−0.959388 + 0.282090i \(0.908972\pi\)
\(830\) −42704.0 −1.78588
\(831\) 0 0
\(832\) 0 0
\(833\) 741.000 0.0308213
\(834\) 0 0
\(835\) 4760.00 0.197277
\(836\) 7680.00 0.317725
\(837\) 0 0
\(838\) −43256.0 −1.78312
\(839\) −32286.0 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(840\) 0 0
\(841\) 14420.0 0.591250
\(842\) 26140.0 1.06989
\(843\) 0 0
\(844\) 24560.0 1.00165
\(845\) 0 0
\(846\) 0 0
\(847\) −6140.00 −0.249083
\(848\) 5952.00 0.241029
\(849\) 0 0
\(850\) −8528.00 −0.344127
\(851\) 17706.0 0.713224
\(852\) 0 0
\(853\) 20937.0 0.840409 0.420205 0.907429i \(-0.361958\pi\)
0.420205 + 0.907429i \(0.361958\pi\)
\(854\) −11600.0 −0.464805
\(855\) 0 0
\(856\) 0 0
\(857\) 7189.00 0.286548 0.143274 0.989683i \(-0.454237\pi\)
0.143274 + 0.989683i \(0.454237\pi\)
\(858\) 0 0
\(859\) −32498.0 −1.29082 −0.645412 0.763835i \(-0.723314\pi\)
−0.645412 + 0.763835i \(0.723314\pi\)
\(860\) 21216.0 0.841232
\(861\) 0 0
\(862\) 7920.00 0.312942
\(863\) −8428.00 −0.332436 −0.166218 0.986089i \(-0.553156\pi\)
−0.166218 + 0.986089i \(0.553156\pi\)
\(864\) 0 0
\(865\) 22542.0 0.886071
\(866\) 27716.0 1.08756
\(867\) 0 0
\(868\) −11840.0 −0.462991
\(869\) −2432.00 −0.0949367
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 9360.00 0.362250
\(875\) −13260.0 −0.512308
\(876\) 0 0
\(877\) −6847.00 −0.263634 −0.131817 0.991274i \(-0.542081\pi\)
−0.131817 + 0.991274i \(0.542081\pi\)
\(878\) 18304.0 0.703565
\(879\) 0 0
\(880\) 34816.0 1.33369
\(881\) −29731.0 −1.13696 −0.568481 0.822697i \(-0.692469\pi\)
−0.568481 + 0.822697i \(0.692469\pi\)
\(882\) 0 0
\(883\) −23738.0 −0.904697 −0.452348 0.891841i \(-0.649414\pi\)
−0.452348 + 0.891841i \(0.649414\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −35248.0 −1.33655
\(887\) −27588.0 −1.04432 −0.522161 0.852847i \(-0.674874\pi\)
−0.522161 + 0.852847i \(0.674874\pi\)
\(888\) 0 0
\(889\) −43160.0 −1.62828
\(890\) 18088.0 0.681248
\(891\) 0 0
\(892\) −43024.0 −1.61497
\(893\) 4860.00 0.182121
\(894\) 0 0
\(895\) 72488.0 2.70727
\(896\) 0 0
\(897\) 0 0
\(898\) 7672.00 0.285098
\(899\) 14578.0 0.540827
\(900\) 0 0
\(901\) −1209.00 −0.0447033
\(902\) −21120.0 −0.779622
\(903\) 0 0
\(904\) 0 0
\(905\) 6851.00 0.251641
\(906\) 0 0
\(907\) −37128.0 −1.35922 −0.679611 0.733572i \(-0.737852\pi\)
−0.679611 + 0.733572i \(0.737852\pi\)
\(908\) 31792.0 1.16195
\(909\) 0 0
\(910\) 0 0
\(911\) −20516.0 −0.746131 −0.373066 0.927805i \(-0.621693\pi\)
−0.373066 + 0.927805i \(0.621693\pi\)
\(912\) 0 0
\(913\) −20096.0 −0.728456
\(914\) 47044.0 1.70249
\(915\) 0 0
\(916\) −50384.0 −1.81740
\(917\) −14600.0 −0.525774
\(918\) 0 0
\(919\) −21006.0 −0.753998 −0.376999 0.926214i \(-0.623044\pi\)
−0.376999 + 0.926214i \(0.623044\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3604.00 0.128733
\(923\) 0 0
\(924\) 0 0
\(925\) −37228.0 −1.32330
\(926\) −5488.00 −0.194759
\(927\) 0 0
\(928\) −50432.0 −1.78396
\(929\) −20427.0 −0.721408 −0.360704 0.932680i \(-0.617464\pi\)
−0.360704 + 0.932680i \(0.617464\pi\)
\(930\) 0 0
\(931\) 1710.00 0.0601965
\(932\) −32240.0 −1.13311
\(933\) 0 0
\(934\) −25584.0 −0.896289
\(935\) −7072.00 −0.247357
\(936\) 0 0
\(937\) 33191.0 1.15721 0.578603 0.815609i \(-0.303598\pi\)
0.578603 + 0.815609i \(0.303598\pi\)
\(938\) −68960.0 −2.40045
\(939\) 0 0
\(940\) −22032.0 −0.764473
\(941\) 36422.0 1.26177 0.630884 0.775877i \(-0.282692\pi\)
0.630884 + 0.775877i \(0.282692\pi\)
\(942\) 0 0
\(943\) −12870.0 −0.444438
\(944\) −55296.0 −1.90650
\(945\) 0 0
\(946\) 19968.0 0.686275
\(947\) 39630.0 1.35988 0.679938 0.733270i \(-0.262007\pi\)
0.679938 + 0.733270i \(0.262007\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −19680.0 −0.672109
\(951\) 0 0
\(952\) 0 0
\(953\) −57642.0 −1.95929 −0.979647 0.200727i \(-0.935670\pi\)
−0.979647 + 0.200727i \(0.935670\pi\)
\(954\) 0 0
\(955\) −21182.0 −0.717731
\(956\) 7872.00 0.266317
\(957\) 0 0
\(958\) 13080.0 0.441123
\(959\) −33420.0 −1.12533
\(960\) 0 0
\(961\) −24315.0 −0.816186
\(962\) 0 0
\(963\) 0 0
\(964\) 7544.00 0.252050
\(965\) −4539.00 −0.151415
\(966\) 0 0
\(967\) 2162.00 0.0718979 0.0359489 0.999354i \(-0.488555\pi\)
0.0359489 + 0.999354i \(0.488555\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 16184.0 0.535708
\(971\) 19758.0 0.653001 0.326501 0.945197i \(-0.394130\pi\)
0.326501 + 0.945197i \(0.394130\pi\)
\(972\) 0 0
\(973\) 18240.0 0.600974
\(974\) −79680.0 −2.62126
\(975\) 0 0
\(976\) −9280.00 −0.304350
\(977\) 12489.0 0.408965 0.204482 0.978870i \(-0.434449\pi\)
0.204482 + 0.978870i \(0.434449\pi\)
\(978\) 0 0
\(979\) 8512.00 0.277880
\(980\) −7752.00 −0.252682
\(981\) 0 0
\(982\) 26208.0 0.851661
\(983\) 28658.0 0.929856 0.464928 0.885349i \(-0.346080\pi\)
0.464928 + 0.885349i \(0.346080\pi\)
\(984\) 0 0
\(985\) 21726.0 0.702790
\(986\) 10244.0 0.330868
\(987\) 0 0
\(988\) 0 0
\(989\) 12168.0 0.391223
\(990\) 0 0
\(991\) −42794.0 −1.37174 −0.685871 0.727723i \(-0.740579\pi\)
−0.685871 + 0.727723i \(0.740579\pi\)
\(992\) −18944.0 −0.606323
\(993\) 0 0
\(994\) 52320.0 1.66951
\(995\) −72046.0 −2.29549
\(996\) 0 0
\(997\) −52583.0 −1.67033 −0.835166 0.549998i \(-0.814628\pi\)
−0.835166 + 0.549998i \(0.814628\pi\)
\(998\) −6984.00 −0.221518
\(999\) 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 1521.4.a.b.1.1 1
3.2 odd 2 169.4.a.d.1.1 1
13.3 even 3 117.4.g.c.100.1 2
13.9 even 3 117.4.g.c.55.1 2
13.12 even 2 1521.4.a.k.1.1 1
39.2 even 12 169.4.e.c.147.2 4
39.5 even 4 169.4.b.c.168.1 2
39.8 even 4 169.4.b.c.168.2 2
39.11 even 12 169.4.e.c.147.1 4
39.17 odd 6 169.4.c.d.146.1 2
39.20 even 12 169.4.e.c.23.2 4
39.23 odd 6 169.4.c.d.22.1 2
39.29 odd 6 13.4.c.a.9.1 yes 2
39.32 even 12 169.4.e.c.23.1 4
39.35 odd 6 13.4.c.a.3.1 2
39.38 odd 2 169.4.a.a.1.1 1
156.35 even 6 208.4.i.b.81.1 2
156.107 even 6 208.4.i.b.113.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.a.3.1 2 39.35 odd 6
13.4.c.a.9.1 yes 2 39.29 odd 6
117.4.g.c.55.1 2 13.9 even 3
117.4.g.c.100.1 2 13.3 even 3
169.4.a.a.1.1 1 39.38 odd 2
169.4.a.d.1.1 1 3.2 odd 2
169.4.b.c.168.1 2 39.5 even 4
169.4.b.c.168.2 2 39.8 even 4
169.4.c.d.22.1 2 39.23 odd 6
169.4.c.d.146.1 2 39.17 odd 6
169.4.e.c.23.1 4 39.32 even 12
169.4.e.c.23.2 4 39.20 even 12
169.4.e.c.147.1 4 39.11 even 12
169.4.e.c.147.2 4 39.2 even 12
208.4.i.b.81.1 2 156.35 even 6
208.4.i.b.113.1 2 156.107 even 6
1521.4.a.b.1.1 1 1.1 even 1 trivial
1521.4.a.k.1.1 1 13.12 even 2