Properties

Label 63.4.a.a.1.1
Level $63$
Weight $4$
Character 63.1
Self dual yes
Analytic conductor $3.717$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.71712033036\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 63.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +8.00000 q^{4} +4.00000 q^{5} -7.00000 q^{7} +O(q^{10})\) \(q-4.00000 q^{2} +8.00000 q^{4} +4.00000 q^{5} -7.00000 q^{7} -16.0000 q^{10} -62.0000 q^{11} -62.0000 q^{13} +28.0000 q^{14} -64.0000 q^{16} -84.0000 q^{17} +100.000 q^{19} +32.0000 q^{20} +248.000 q^{22} +42.0000 q^{23} -109.000 q^{25} +248.000 q^{26} -56.0000 q^{28} +10.0000 q^{29} -48.0000 q^{31} +256.000 q^{32} +336.000 q^{34} -28.0000 q^{35} -246.000 q^{37} -400.000 q^{38} +248.000 q^{41} +68.0000 q^{43} -496.000 q^{44} -168.000 q^{46} -324.000 q^{47} +49.0000 q^{49} +436.000 q^{50} -496.000 q^{52} -258.000 q^{53} -248.000 q^{55} -40.0000 q^{58} -120.000 q^{59} +622.000 q^{61} +192.000 q^{62} -512.000 q^{64} -248.000 q^{65} +904.000 q^{67} -672.000 q^{68} +112.000 q^{70} +678.000 q^{71} -642.000 q^{73} +984.000 q^{74} +800.000 q^{76} +434.000 q^{77} +740.000 q^{79} -256.000 q^{80} -992.000 q^{82} -468.000 q^{83} -336.000 q^{85} -272.000 q^{86} -200.000 q^{89} +434.000 q^{91} +336.000 q^{92} +1296.00 q^{94} +400.000 q^{95} -1266.00 q^{97} -196.000 q^{98} +O(q^{100})\)

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
\(5\) 4.00000 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) −16.0000 −0.505964
\(11\) −62.0000 −1.69943 −0.849714 0.527244i \(-0.823225\pi\)
−0.849714 + 0.527244i \(0.823225\pi\)
\(12\) 0 0
\(13\) −62.0000 −1.32275 −0.661373 0.750057i \(-0.730026\pi\)
−0.661373 + 0.750057i \(0.730026\pi\)
\(14\) 28.0000 0.534522
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) 0 0
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) 248.000 2.40335
\(23\) 42.0000 0.380765 0.190383 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 248.000 1.87065
\(27\) 0 0
\(28\) −56.0000 −0.377964
\(29\) 10.0000 0.0640329 0.0320164 0.999487i \(-0.489807\pi\)
0.0320164 + 0.999487i \(0.489807\pi\)
\(30\) 0 0
\(31\) −48.0000 −0.278099 −0.139049 0.990285i \(-0.544405\pi\)
−0.139049 + 0.990285i \(0.544405\pi\)
\(32\) 256.000 1.41421
\(33\) 0 0
\(34\) 336.000 1.69481
\(35\) −28.0000 −0.135225
\(36\) 0 0
\(37\) −246.000 −1.09303 −0.546516 0.837449i \(-0.684046\pi\)
−0.546516 + 0.837449i \(0.684046\pi\)
\(38\) −400.000 −1.70759
\(39\) 0 0
\(40\) 0 0
\(41\) 248.000 0.944661 0.472330 0.881422i \(-0.343413\pi\)
0.472330 + 0.881422i \(0.343413\pi\)
\(42\) 0 0
\(43\) 68.0000 0.241161 0.120580 0.992704i \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) −496.000 −1.69943
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) −324.000 −1.00554 −0.502769 0.864421i \(-0.667685\pi\)
−0.502769 + 0.864421i \(0.667685\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 436.000 1.23319
\(51\) 0 0
\(52\) −496.000 −1.32275
\(53\) −258.000 −0.668661 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(54\) 0 0
\(55\) −248.000 −0.608006
\(56\) 0 0
\(57\) 0 0
\(58\) −40.0000 −0.0905562
\(59\) −120.000 −0.264791 −0.132396 0.991197i \(-0.542267\pi\)
−0.132396 + 0.991197i \(0.542267\pi\)
\(60\) 0 0
\(61\) 622.000 1.30556 0.652778 0.757549i \(-0.273603\pi\)
0.652778 + 0.757549i \(0.273603\pi\)
\(62\) 192.000 0.393291
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) −248.000 −0.473240
\(66\) 0 0
\(67\) 904.000 1.64838 0.824188 0.566316i \(-0.191632\pi\)
0.824188 + 0.566316i \(0.191632\pi\)
\(68\) −672.000 −1.19841
\(69\) 0 0
\(70\) 112.000 0.191237
\(71\) 678.000 1.13329 0.566646 0.823961i \(-0.308241\pi\)
0.566646 + 0.823961i \(0.308241\pi\)
\(72\) 0 0
\(73\) −642.000 −1.02932 −0.514660 0.857394i \(-0.672082\pi\)
−0.514660 + 0.857394i \(0.672082\pi\)
\(74\) 984.000 1.54578
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) 434.000 0.642323
\(78\) 0 0
\(79\) 740.000 1.05388 0.526940 0.849903i \(-0.323339\pi\)
0.526940 + 0.849903i \(0.323339\pi\)
\(80\) −256.000 −0.357771
\(81\) 0 0
\(82\) −992.000 −1.33595
\(83\) −468.000 −0.618912 −0.309456 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(84\) 0 0
\(85\) −336.000 −0.428757
\(86\) −272.000 −0.341052
\(87\) 0 0
\(88\) 0 0
\(89\) −200.000 −0.238202 −0.119101 0.992882i \(-0.538001\pi\)
−0.119101 + 0.992882i \(0.538001\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) 336.000 0.380765
\(93\) 0 0
\(94\) 1296.00 1.42204
\(95\) 400.000 0.431991
\(96\) 0 0
\(97\) −1266.00 −1.32518 −0.662592 0.748981i \(-0.730544\pi\)
−0.662592 + 0.748981i \(0.730544\pi\)
\(98\) −196.000 −0.202031
\(99\) 0 0
\(100\) −872.000 −0.872000
\(101\) −232.000 −0.228563 −0.114281 0.993448i \(-0.536457\pi\)
−0.114281 + 0.993448i \(0.536457\pi\)
\(102\) 0 0
\(103\) −1792.00 −1.71428 −0.857141 0.515082i \(-0.827761\pi\)
−0.857141 + 0.515082i \(0.827761\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1032.00 0.945629
\(107\) 1906.00 1.72206 0.861028 0.508558i \(-0.169821\pi\)
0.861028 + 0.508558i \(0.169821\pi\)
\(108\) 0 0
\(109\) −90.0000 −0.0790866 −0.0395433 0.999218i \(-0.512590\pi\)
−0.0395433 + 0.999218i \(0.512590\pi\)
\(110\) 992.000 0.859850
\(111\) 0 0
\(112\) 448.000 0.377964
\(113\) −458.000 −0.381283 −0.190642 0.981660i \(-0.561057\pi\)
−0.190642 + 0.981660i \(0.561057\pi\)
\(114\) 0 0
\(115\) 168.000 0.136227
\(116\) 80.0000 0.0640329
\(117\) 0 0
\(118\) 480.000 0.374471
\(119\) 588.000 0.452957
\(120\) 0 0
\(121\) 2513.00 1.88805
\(122\) −2488.00 −1.84634
\(123\) 0 0
\(124\) −384.000 −0.278099
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) 804.000 0.561760 0.280880 0.959743i \(-0.409374\pi\)
0.280880 + 0.959743i \(0.409374\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 992.000 0.669263
\(131\) −812.000 −0.541563 −0.270782 0.962641i \(-0.587282\pi\)
−0.270782 + 0.962641i \(0.587282\pi\)
\(132\) 0 0
\(133\) −700.000 −0.456374
\(134\) −3616.00 −2.33116
\(135\) 0 0
\(136\) 0 0
\(137\) −414.000 −0.258178 −0.129089 0.991633i \(-0.541205\pi\)
−0.129089 + 0.991633i \(0.541205\pi\)
\(138\) 0 0
\(139\) −1620.00 −0.988537 −0.494268 0.869309i \(-0.664564\pi\)
−0.494268 + 0.869309i \(0.664564\pi\)
\(140\) −224.000 −0.135225
\(141\) 0 0
\(142\) −2712.00 −1.60272
\(143\) 3844.00 2.24791
\(144\) 0 0
\(145\) 40.0000 0.0229091
\(146\) 2568.00 1.45568
\(147\) 0 0
\(148\) −1968.00 −1.09303
\(149\) −2370.00 −1.30307 −0.651537 0.758617i \(-0.725875\pi\)
−0.651537 + 0.758617i \(0.725875\pi\)
\(150\) 0 0
\(151\) −568.000 −0.306114 −0.153057 0.988217i \(-0.548912\pi\)
−0.153057 + 0.988217i \(0.548912\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1736.00 −0.908382
\(155\) −192.000 −0.0994956
\(156\) 0 0
\(157\) −266.000 −0.135217 −0.0676086 0.997712i \(-0.521537\pi\)
−0.0676086 + 0.997712i \(0.521537\pi\)
\(158\) −2960.00 −1.49041
\(159\) 0 0
\(160\) 1024.00 0.505964
\(161\) −294.000 −0.143916
\(162\) 0 0
\(163\) −272.000 −0.130704 −0.0653518 0.997862i \(-0.520817\pi\)
−0.0653518 + 0.997862i \(0.520817\pi\)
\(164\) 1984.00 0.944661
\(165\) 0 0
\(166\) 1872.00 0.875273
\(167\) 1876.00 0.869277 0.434638 0.900605i \(-0.356876\pi\)
0.434638 + 0.900605i \(0.356876\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) 1344.00 0.606353
\(171\) 0 0
\(172\) 544.000 0.241161
\(173\) 152.000 0.0667997 0.0333998 0.999442i \(-0.489367\pi\)
0.0333998 + 0.999442i \(0.489367\pi\)
\(174\) 0 0
\(175\) 763.000 0.329585
\(176\) 3968.00 1.69943
\(177\) 0 0
\(178\) 800.000 0.336868
\(179\) −610.000 −0.254713 −0.127356 0.991857i \(-0.540649\pi\)
−0.127356 + 0.991857i \(0.540649\pi\)
\(180\) 0 0
\(181\) 1042.00 0.427907 0.213954 0.976844i \(-0.431366\pi\)
0.213954 + 0.976844i \(0.431366\pi\)
\(182\) −1736.00 −0.707038
\(183\) 0 0
\(184\) 0 0
\(185\) −984.000 −0.391055
\(186\) 0 0
\(187\) 5208.00 2.03661
\(188\) −2592.00 −1.00554
\(189\) 0 0
\(190\) −1600.00 −0.610927
\(191\) 2038.00 0.772065 0.386033 0.922485i \(-0.373845\pi\)
0.386033 + 0.922485i \(0.373845\pi\)
\(192\) 0 0
\(193\) −2602.00 −0.970446 −0.485223 0.874390i \(-0.661262\pi\)
−0.485223 + 0.874390i \(0.661262\pi\)
\(194\) 5064.00 1.87409
\(195\) 0 0
\(196\) 392.000 0.142857
\(197\) −2354.00 −0.851348 −0.425674 0.904877i \(-0.639963\pi\)
−0.425674 + 0.904877i \(0.639963\pi\)
\(198\) 0 0
\(199\) 1680.00 0.598452 0.299226 0.954182i \(-0.403271\pi\)
0.299226 + 0.954182i \(0.403271\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 928.000 0.323237
\(203\) −70.0000 −0.0242022
\(204\) 0 0
\(205\) 992.000 0.337972
\(206\) 7168.00 2.42436
\(207\) 0 0
\(208\) 3968.00 1.32275
\(209\) −6200.00 −2.05198
\(210\) 0 0
\(211\) −668.000 −0.217948 −0.108974 0.994045i \(-0.534757\pi\)
−0.108974 + 0.994045i \(0.534757\pi\)
\(212\) −2064.00 −0.668661
\(213\) 0 0
\(214\) −7624.00 −2.43535
\(215\) 272.000 0.0862802
\(216\) 0 0
\(217\) 336.000 0.105111
\(218\) 360.000 0.111845
\(219\) 0 0
\(220\) −1984.00 −0.608006
\(221\) 5208.00 1.58519
\(222\) 0 0
\(223\) −1832.00 −0.550134 −0.275067 0.961425i \(-0.588700\pi\)
−0.275067 + 0.961425i \(0.588700\pi\)
\(224\) −1792.00 −0.534522
\(225\) 0 0
\(226\) 1832.00 0.539216
\(227\) −4944.00 −1.44557 −0.722786 0.691072i \(-0.757139\pi\)
−0.722786 + 0.691072i \(0.757139\pi\)
\(228\) 0 0
\(229\) −5470.00 −1.57846 −0.789231 0.614096i \(-0.789521\pi\)
−0.789231 + 0.614096i \(0.789521\pi\)
\(230\) −672.000 −0.192654
\(231\) 0 0
\(232\) 0 0
\(233\) 2802.00 0.787833 0.393917 0.919146i \(-0.371120\pi\)
0.393917 + 0.919146i \(0.371120\pi\)
\(234\) 0 0
\(235\) −1296.00 −0.359752
\(236\) −960.000 −0.264791
\(237\) 0 0
\(238\) −2352.00 −0.640578
\(239\) 1170.00 0.316657 0.158328 0.987386i \(-0.449390\pi\)
0.158328 + 0.987386i \(0.449390\pi\)
\(240\) 0 0
\(241\) −2338.00 −0.624912 −0.312456 0.949932i \(-0.601152\pi\)
−0.312456 + 0.949932i \(0.601152\pi\)
\(242\) −10052.0 −2.67011
\(243\) 0 0
\(244\) 4976.00 1.30556
\(245\) 196.000 0.0511101
\(246\) 0 0
\(247\) −6200.00 −1.59715
\(248\) 0 0
\(249\) 0 0
\(250\) 3744.00 0.947165
\(251\) −2792.00 −0.702109 −0.351055 0.936355i \(-0.614177\pi\)
−0.351055 + 0.936355i \(0.614177\pi\)
\(252\) 0 0
\(253\) −2604.00 −0.647083
\(254\) −3216.00 −0.794448
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) −7024.00 −1.70484 −0.852422 0.522854i \(-0.824867\pi\)
−0.852422 + 0.522854i \(0.824867\pi\)
\(258\) 0 0
\(259\) 1722.00 0.413127
\(260\) −1984.00 −0.473240
\(261\) 0 0
\(262\) 3248.00 0.765886
\(263\) −2438.00 −0.571610 −0.285805 0.958288i \(-0.592261\pi\)
−0.285805 + 0.958288i \(0.592261\pi\)
\(264\) 0 0
\(265\) −1032.00 −0.239227
\(266\) 2800.00 0.645410
\(267\) 0 0
\(268\) 7232.00 1.64838
\(269\) 6780.00 1.53674 0.768372 0.640004i \(-0.221067\pi\)
0.768372 + 0.640004i \(0.221067\pi\)
\(270\) 0 0
\(271\) −1928.00 −0.432168 −0.216084 0.976375i \(-0.569329\pi\)
−0.216084 + 0.976375i \(0.569329\pi\)
\(272\) 5376.00 1.19841
\(273\) 0 0
\(274\) 1656.00 0.365119
\(275\) 6758.00 1.48190
\(276\) 0 0
\(277\) 5554.00 1.20472 0.602360 0.798224i \(-0.294227\pi\)
0.602360 + 0.798224i \(0.294227\pi\)
\(278\) 6480.00 1.39800
\(279\) 0 0
\(280\) 0 0
\(281\) −1942.00 −0.412278 −0.206139 0.978523i \(-0.566090\pi\)
−0.206139 + 0.978523i \(0.566090\pi\)
\(282\) 0 0
\(283\) 4828.00 1.01412 0.507058 0.861912i \(-0.330733\pi\)
0.507058 + 0.861912i \(0.330733\pi\)
\(284\) 5424.00 1.13329
\(285\) 0 0
\(286\) −15376.0 −3.17903
\(287\) −1736.00 −0.357048
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) −160.000 −0.0323984
\(291\) 0 0
\(292\) −5136.00 −1.02932
\(293\) 6152.00 1.22663 0.613317 0.789837i \(-0.289835\pi\)
0.613317 + 0.789837i \(0.289835\pi\)
\(294\) 0 0
\(295\) −480.000 −0.0947345
\(296\) 0 0
\(297\) 0 0
\(298\) 9480.00 1.84282
\(299\) −2604.00 −0.503656
\(300\) 0 0
\(301\) −476.000 −0.0911501
\(302\) 2272.00 0.432910
\(303\) 0 0
\(304\) −6400.00 −1.20745
\(305\) 2488.00 0.467090
\(306\) 0 0
\(307\) 5884.00 1.09387 0.546934 0.837176i \(-0.315795\pi\)
0.546934 + 0.837176i \(0.315795\pi\)
\(308\) 3472.00 0.642323
\(309\) 0 0
\(310\) 768.000 0.140708
\(311\) −9132.00 −1.66504 −0.832521 0.553993i \(-0.813103\pi\)
−0.832521 + 0.553993i \(0.813103\pi\)
\(312\) 0 0
\(313\) −9382.00 −1.69426 −0.847128 0.531389i \(-0.821670\pi\)
−0.847128 + 0.531389i \(0.821670\pi\)
\(314\) 1064.00 0.191226
\(315\) 0 0
\(316\) 5920.00 1.05388
\(317\) −3114.00 −0.551734 −0.275867 0.961196i \(-0.588965\pi\)
−0.275867 + 0.961196i \(0.588965\pi\)
\(318\) 0 0
\(319\) −620.000 −0.108819
\(320\) −2048.00 −0.357771
\(321\) 0 0
\(322\) 1176.00 0.203528
\(323\) −8400.00 −1.44702
\(324\) 0 0
\(325\) 6758.00 1.15344
\(326\) 1088.00 0.184843
\(327\) 0 0
\(328\) 0 0
\(329\) 2268.00 0.380057
\(330\) 0 0
\(331\) 1532.00 0.254400 0.127200 0.991877i \(-0.459401\pi\)
0.127200 + 0.991877i \(0.459401\pi\)
\(332\) −3744.00 −0.618912
\(333\) 0 0
\(334\) −7504.00 −1.22934
\(335\) 3616.00 0.589741
\(336\) 0 0
\(337\) −4166.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(338\) −6588.00 −1.06018
\(339\) 0 0
\(340\) −2688.00 −0.428757
\(341\) 2976.00 0.472608
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −608.000 −0.0944690
\(347\) 11366.0 1.75838 0.879191 0.476469i \(-0.158083\pi\)
0.879191 + 0.476469i \(0.158083\pi\)
\(348\) 0 0
\(349\) 9310.00 1.42795 0.713973 0.700174i \(-0.246894\pi\)
0.713973 + 0.700174i \(0.246894\pi\)
\(350\) −3052.00 −0.466104
\(351\) 0 0
\(352\) −15872.0 −2.40335
\(353\) 8572.00 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(354\) 0 0
\(355\) 2712.00 0.405459
\(356\) −1600.00 −0.238202
\(357\) 0 0
\(358\) 2440.00 0.360218
\(359\) 4790.00 0.704196 0.352098 0.935963i \(-0.385468\pi\)
0.352098 + 0.935963i \(0.385468\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) −4168.00 −0.605153
\(363\) 0 0
\(364\) 3472.00 0.499951
\(365\) −2568.00 −0.368261
\(366\) 0 0
\(367\) 5424.00 0.771473 0.385736 0.922609i \(-0.373947\pi\)
0.385736 + 0.922609i \(0.373947\pi\)
\(368\) −2688.00 −0.380765
\(369\) 0 0
\(370\) 3936.00 0.553035
\(371\) 1806.00 0.252730
\(372\) 0 0
\(373\) 1838.00 0.255142 0.127571 0.991829i \(-0.459282\pi\)
0.127571 + 0.991829i \(0.459282\pi\)
\(374\) −20832.0 −2.88021
\(375\) 0 0
\(376\) 0 0
\(377\) −620.000 −0.0846993
\(378\) 0 0
\(379\) −4260.00 −0.577365 −0.288683 0.957425i \(-0.593217\pi\)
−0.288683 + 0.957425i \(0.593217\pi\)
\(380\) 3200.00 0.431991
\(381\) 0 0
\(382\) −8152.00 −1.09187
\(383\) −9048.00 −1.20713 −0.603566 0.797313i \(-0.706254\pi\)
−0.603566 + 0.797313i \(0.706254\pi\)
\(384\) 0 0
\(385\) 1736.00 0.229805
\(386\) 10408.0 1.37242
\(387\) 0 0
\(388\) −10128.0 −1.32518
\(389\) 11490.0 1.49760 0.748800 0.662796i \(-0.230631\pi\)
0.748800 + 0.662796i \(0.230631\pi\)
\(390\) 0 0
\(391\) −3528.00 −0.456314
\(392\) 0 0
\(393\) 0 0
\(394\) 9416.00 1.20399
\(395\) 2960.00 0.377048
\(396\) 0 0
\(397\) −1866.00 −0.235899 −0.117949 0.993020i \(-0.537632\pi\)
−0.117949 + 0.993020i \(0.537632\pi\)
\(398\) −6720.00 −0.846340
\(399\) 0 0
\(400\) 6976.00 0.872000
\(401\) −13662.0 −1.70137 −0.850683 0.525679i \(-0.823811\pi\)
−0.850683 + 0.525679i \(0.823811\pi\)
\(402\) 0 0
\(403\) 2976.00 0.367854
\(404\) −1856.00 −0.228563
\(405\) 0 0
\(406\) 280.000 0.0342270
\(407\) 15252.0 1.85753
\(408\) 0 0
\(409\) −13210.0 −1.59705 −0.798524 0.601963i \(-0.794385\pi\)
−0.798524 + 0.601963i \(0.794385\pi\)
\(410\) −3968.00 −0.477965
\(411\) 0 0
\(412\) −14336.0 −1.71428
\(413\) 840.000 0.100082
\(414\) 0 0
\(415\) −1872.00 −0.221429
\(416\) −15872.0 −1.87065
\(417\) 0 0
\(418\) 24800.0 2.90193
\(419\) −6960.00 −0.811499 −0.405750 0.913984i \(-0.632990\pi\)
−0.405750 + 0.913984i \(0.632990\pi\)
\(420\) 0 0
\(421\) 8162.00 0.944873 0.472437 0.881365i \(-0.343375\pi\)
0.472437 + 0.881365i \(0.343375\pi\)
\(422\) 2672.00 0.308225
\(423\) 0 0
\(424\) 0 0
\(425\) 9156.00 1.04501
\(426\) 0 0
\(427\) −4354.00 −0.493454
\(428\) 15248.0 1.72206
\(429\) 0 0
\(430\) −1088.00 −0.122019
\(431\) −16602.0 −1.85543 −0.927715 0.373290i \(-0.878230\pi\)
−0.927715 + 0.373290i \(0.878230\pi\)
\(432\) 0 0
\(433\) 7738.00 0.858810 0.429405 0.903112i \(-0.358723\pi\)
0.429405 + 0.903112i \(0.358723\pi\)
\(434\) −1344.00 −0.148650
\(435\) 0 0
\(436\) −720.000 −0.0790866
\(437\) 4200.00 0.459756
\(438\) 0 0
\(439\) −840.000 −0.0913235 −0.0456617 0.998957i \(-0.514540\pi\)
−0.0456617 + 0.998957i \(0.514540\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20832.0 −2.24180
\(443\) −6618.00 −0.709776 −0.354888 0.934909i \(-0.615481\pi\)
−0.354888 + 0.934909i \(0.615481\pi\)
\(444\) 0 0
\(445\) −800.000 −0.0852217
\(446\) 7328.00 0.778006
\(447\) 0 0
\(448\) 3584.00 0.377964
\(449\) −3090.00 −0.324780 −0.162390 0.986727i \(-0.551920\pi\)
−0.162390 + 0.986727i \(0.551920\pi\)
\(450\) 0 0
\(451\) −15376.0 −1.60538
\(452\) −3664.00 −0.381283
\(453\) 0 0
\(454\) 19776.0 2.04435
\(455\) 1736.00 0.178868
\(456\) 0 0
\(457\) 5914.00 0.605351 0.302675 0.953094i \(-0.402120\pi\)
0.302675 + 0.953094i \(0.402120\pi\)
\(458\) 21880.0 2.23228
\(459\) 0 0
\(460\) 1344.00 0.136227
\(461\) 15968.0 1.61324 0.806620 0.591070i \(-0.201294\pi\)
0.806620 + 0.591070i \(0.201294\pi\)
\(462\) 0 0
\(463\) −1172.00 −0.117640 −0.0588202 0.998269i \(-0.518734\pi\)
−0.0588202 + 0.998269i \(0.518734\pi\)
\(464\) −640.000 −0.0640329
\(465\) 0 0
\(466\) −11208.0 −1.11416
\(467\) −5304.00 −0.525567 −0.262784 0.964855i \(-0.584641\pi\)
−0.262784 + 0.964855i \(0.584641\pi\)
\(468\) 0 0
\(469\) −6328.00 −0.623027
\(470\) 5184.00 0.508766
\(471\) 0 0
\(472\) 0 0
\(473\) −4216.00 −0.409835
\(474\) 0 0
\(475\) −10900.0 −1.05290
\(476\) 4704.00 0.452957
\(477\) 0 0
\(478\) −4680.00 −0.447821
\(479\) −5740.00 −0.547531 −0.273765 0.961796i \(-0.588269\pi\)
−0.273765 + 0.961796i \(0.588269\pi\)
\(480\) 0 0
\(481\) 15252.0 1.44580
\(482\) 9352.00 0.883759
\(483\) 0 0
\(484\) 20104.0 1.88805
\(485\) −5064.00 −0.474112
\(486\) 0 0
\(487\) 8944.00 0.832220 0.416110 0.909314i \(-0.363393\pi\)
0.416110 + 0.909314i \(0.363393\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −784.000 −0.0722806
\(491\) 5558.00 0.510853 0.255427 0.966828i \(-0.417784\pi\)
0.255427 + 0.966828i \(0.417784\pi\)
\(492\) 0 0
\(493\) −840.000 −0.0767377
\(494\) 24800.0 2.25871
\(495\) 0 0
\(496\) 3072.00 0.278099
\(497\) −4746.00 −0.428344
\(498\) 0 0
\(499\) −19820.0 −1.77809 −0.889043 0.457823i \(-0.848629\pi\)
−0.889043 + 0.457823i \(0.848629\pi\)
\(500\) −7488.00 −0.669747
\(501\) 0 0
\(502\) 11168.0 0.992933
\(503\) −1848.00 −0.163814 −0.0819068 0.996640i \(-0.526101\pi\)
−0.0819068 + 0.996640i \(0.526101\pi\)
\(504\) 0 0
\(505\) −928.000 −0.0817732
\(506\) 10416.0 0.915114
\(507\) 0 0
\(508\) 6432.00 0.561760
\(509\) −340.000 −0.0296075 −0.0148038 0.999890i \(-0.504712\pi\)
−0.0148038 + 0.999890i \(0.504712\pi\)
\(510\) 0 0
\(511\) 4494.00 0.389047
\(512\) −16384.0 −1.41421
\(513\) 0 0
\(514\) 28096.0 2.41101
\(515\) −7168.00 −0.613320
\(516\) 0 0
\(517\) 20088.0 1.70884
\(518\) −6888.00 −0.584250
\(519\) 0 0
\(520\) 0 0
\(521\) −10212.0 −0.858725 −0.429363 0.903132i \(-0.641262\pi\)
−0.429363 + 0.903132i \(0.641262\pi\)
\(522\) 0 0
\(523\) −9332.00 −0.780229 −0.390115 0.920766i \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(524\) −6496.00 −0.541563
\(525\) 0 0
\(526\) 9752.00 0.808379
\(527\) 4032.00 0.333276
\(528\) 0 0
\(529\) −10403.0 −0.855018
\(530\) 4128.00 0.338319
\(531\) 0 0
\(532\) −5600.00 −0.456374
\(533\) −15376.0 −1.24955
\(534\) 0 0
\(535\) 7624.00 0.616101
\(536\) 0 0
\(537\) 0 0
\(538\) −27120.0 −2.17328
\(539\) −3038.00 −0.242775
\(540\) 0 0
\(541\) −8998.00 −0.715073 −0.357536 0.933899i \(-0.616383\pi\)
−0.357536 + 0.933899i \(0.616383\pi\)
\(542\) 7712.00 0.611179
\(543\) 0 0
\(544\) −21504.0 −1.69481
\(545\) −360.000 −0.0282949
\(546\) 0 0
\(547\) −3416.00 −0.267016 −0.133508 0.991048i \(-0.542624\pi\)
−0.133508 + 0.991048i \(0.542624\pi\)
\(548\) −3312.00 −0.258178
\(549\) 0 0
\(550\) −27032.0 −2.09572
\(551\) 1000.00 0.0773166
\(552\) 0 0
\(553\) −5180.00 −0.398329
\(554\) −22216.0 −1.70373
\(555\) 0 0
\(556\) −12960.0 −0.988537
\(557\) 526.000 0.0400132 0.0200066 0.999800i \(-0.493631\pi\)
0.0200066 + 0.999800i \(0.493631\pi\)
\(558\) 0 0
\(559\) −4216.00 −0.318994
\(560\) 1792.00 0.135225
\(561\) 0 0
\(562\) 7768.00 0.583049
\(563\) 6712.00 0.502446 0.251223 0.967929i \(-0.419167\pi\)
0.251223 + 0.967929i \(0.419167\pi\)
\(564\) 0 0
\(565\) −1832.00 −0.136412
\(566\) −19312.0 −1.43418
\(567\) 0 0
\(568\) 0 0
\(569\) −4190.00 −0.308706 −0.154353 0.988016i \(-0.549329\pi\)
−0.154353 + 0.988016i \(0.549329\pi\)
\(570\) 0 0
\(571\) 3032.00 0.222216 0.111108 0.993808i \(-0.464560\pi\)
0.111108 + 0.993808i \(0.464560\pi\)
\(572\) 30752.0 2.24791
\(573\) 0 0
\(574\) 6944.00 0.504942
\(575\) −4578.00 −0.332027
\(576\) 0 0
\(577\) 5434.00 0.392063 0.196032 0.980598i \(-0.437195\pi\)
0.196032 + 0.980598i \(0.437195\pi\)
\(578\) −8572.00 −0.616865
\(579\) 0 0
\(580\) 320.000 0.0229091
\(581\) 3276.00 0.233927
\(582\) 0 0
\(583\) 15996.0 1.13634
\(584\) 0 0
\(585\) 0 0
\(586\) −24608.0 −1.73472
\(587\) −464.000 −0.0326258 −0.0163129 0.999867i \(-0.505193\pi\)
−0.0163129 + 0.999867i \(0.505193\pi\)
\(588\) 0 0
\(589\) −4800.00 −0.335790
\(590\) 1920.00 0.133975
\(591\) 0 0
\(592\) 15744.0 1.09303
\(593\) −11748.0 −0.813546 −0.406773 0.913529i \(-0.633346\pi\)
−0.406773 + 0.913529i \(0.633346\pi\)
\(594\) 0 0
\(595\) 2352.00 0.162055
\(596\) −18960.0 −1.30307
\(597\) 0 0
\(598\) 10416.0 0.712277
\(599\) −7650.00 −0.521821 −0.260910 0.965363i \(-0.584023\pi\)
−0.260910 + 0.965363i \(0.584023\pi\)
\(600\) 0 0
\(601\) −22878.0 −1.55277 −0.776384 0.630261i \(-0.782948\pi\)
−0.776384 + 0.630261i \(0.782948\pi\)
\(602\) 1904.00 0.128906
\(603\) 0 0
\(604\) −4544.00 −0.306114
\(605\) 10052.0 0.675491
\(606\) 0 0
\(607\) 704.000 0.0470749 0.0235375 0.999723i \(-0.492507\pi\)
0.0235375 + 0.999723i \(0.492507\pi\)
\(608\) 25600.0 1.70759
\(609\) 0 0
\(610\) −9952.00 −0.660565
\(611\) 20088.0 1.33007
\(612\) 0 0
\(613\) 24958.0 1.64444 0.822222 0.569167i \(-0.192734\pi\)
0.822222 + 0.569167i \(0.192734\pi\)
\(614\) −23536.0 −1.54696
\(615\) 0 0
\(616\) 0 0
\(617\) 8826.00 0.575886 0.287943 0.957648i \(-0.407029\pi\)
0.287943 + 0.957648i \(0.407029\pi\)
\(618\) 0 0
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) −1536.00 −0.0994956
\(621\) 0 0
\(622\) 36528.0 2.35473
\(623\) 1400.00 0.0900318
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 37528.0 2.39604
\(627\) 0 0
\(628\) −2128.00 −0.135217
\(629\) 20664.0 1.30990
\(630\) 0 0
\(631\) −3268.00 −0.206176 −0.103088 0.994672i \(-0.532872\pi\)
−0.103088 + 0.994672i \(0.532872\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 12456.0 0.780270
\(635\) 3216.00 0.200981
\(636\) 0 0
\(637\) −3038.00 −0.188964
\(638\) 2480.00 0.153894
\(639\) 0 0
\(640\) 0 0
\(641\) −13062.0 −0.804864 −0.402432 0.915450i \(-0.631835\pi\)
−0.402432 + 0.915450i \(0.631835\pi\)
\(642\) 0 0
\(643\) −28012.0 −1.71802 −0.859009 0.511961i \(-0.828919\pi\)
−0.859009 + 0.511961i \(0.828919\pi\)
\(644\) −2352.00 −0.143916
\(645\) 0 0
\(646\) 33600.0 2.04640
\(647\) −3844.00 −0.233575 −0.116788 0.993157i \(-0.537260\pi\)
−0.116788 + 0.993157i \(0.537260\pi\)
\(648\) 0 0
\(649\) 7440.00 0.449993
\(650\) −27032.0 −1.63120
\(651\) 0 0
\(652\) −2176.00 −0.130704
\(653\) 28482.0 1.70687 0.853436 0.521198i \(-0.174515\pi\)
0.853436 + 0.521198i \(0.174515\pi\)
\(654\) 0 0
\(655\) −3248.00 −0.193756
\(656\) −15872.0 −0.944661
\(657\) 0 0
\(658\) −9072.00 −0.537482
\(659\) 9330.00 0.551510 0.275755 0.961228i \(-0.411072\pi\)
0.275755 + 0.961228i \(0.411072\pi\)
\(660\) 0 0
\(661\) 8782.00 0.516763 0.258381 0.966043i \(-0.416811\pi\)
0.258381 + 0.966043i \(0.416811\pi\)
\(662\) −6128.00 −0.359776
\(663\) 0 0
\(664\) 0 0
\(665\) −2800.00 −0.163277
\(666\) 0 0
\(667\) 420.000 0.0243815
\(668\) 15008.0 0.869277
\(669\) 0 0
\(670\) −14464.0 −0.834020
\(671\) −38564.0 −2.21870
\(672\) 0 0
\(673\) −10562.0 −0.604956 −0.302478 0.953156i \(-0.597814\pi\)
−0.302478 + 0.953156i \(0.597814\pi\)
\(674\) 16664.0 0.952334
\(675\) 0 0
\(676\) 13176.0 0.749659
\(677\) 26016.0 1.47692 0.738461 0.674296i \(-0.235553\pi\)
0.738461 + 0.674296i \(0.235553\pi\)
\(678\) 0 0
\(679\) 8862.00 0.500872
\(680\) 0 0
\(681\) 0 0
\(682\) −11904.0 −0.668369
\(683\) −8898.00 −0.498496 −0.249248 0.968440i \(-0.580183\pi\)
−0.249248 + 0.968440i \(0.580183\pi\)
\(684\) 0 0
\(685\) −1656.00 −0.0923686
\(686\) 1372.00 0.0763604
\(687\) 0 0
\(688\) −4352.00 −0.241161
\(689\) 15996.0 0.884469
\(690\) 0 0
\(691\) 30572.0 1.68309 0.841544 0.540189i \(-0.181647\pi\)
0.841544 + 0.540189i \(0.181647\pi\)
\(692\) 1216.00 0.0667997
\(693\) 0 0
\(694\) −45464.0 −2.48673
\(695\) −6480.00 −0.353670
\(696\) 0 0
\(697\) −20832.0 −1.13209
\(698\) −37240.0 −2.01942
\(699\) 0 0
\(700\) 6104.00 0.329585
\(701\) 30618.0 1.64968 0.824840 0.565366i \(-0.191265\pi\)
0.824840 + 0.565366i \(0.191265\pi\)
\(702\) 0 0
\(703\) −24600.0 −1.31978
\(704\) 31744.0 1.69943
\(705\) 0 0
\(706\) −34288.0 −1.82783
\(707\) 1624.00 0.0863887
\(708\) 0 0
\(709\) −8130.00 −0.430647 −0.215323 0.976543i \(-0.569081\pi\)
−0.215323 + 0.976543i \(0.569081\pi\)
\(710\) −10848.0 −0.573406
\(711\) 0 0
\(712\) 0 0
\(713\) −2016.00 −0.105890
\(714\) 0 0
\(715\) 15376.0 0.804237
\(716\) −4880.00 −0.254713
\(717\) 0 0
\(718\) −19160.0 −0.995884
\(719\) 27840.0 1.44403 0.722014 0.691878i \(-0.243216\pi\)
0.722014 + 0.691878i \(0.243216\pi\)
\(720\) 0 0
\(721\) 12544.0 0.647938
\(722\) −12564.0 −0.647623
\(723\) 0 0
\(724\) 8336.00 0.427907
\(725\) −1090.00 −0.0558367
\(726\) 0 0
\(727\) 14624.0 0.746044 0.373022 0.927822i \(-0.378322\pi\)
0.373022 + 0.927822i \(0.378322\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10272.0 0.520800
\(731\) −5712.00 −0.289010
\(732\) 0 0
\(733\) −20862.0 −1.05124 −0.525618 0.850721i \(-0.676166\pi\)
−0.525618 + 0.850721i \(0.676166\pi\)
\(734\) −21696.0 −1.09103
\(735\) 0 0
\(736\) 10752.0 0.538484
\(737\) −56048.0 −2.80130
\(738\) 0 0
\(739\) −13920.0 −0.692903 −0.346452 0.938068i \(-0.612614\pi\)
−0.346452 + 0.938068i \(0.612614\pi\)
\(740\) −7872.00 −0.391055
\(741\) 0 0
\(742\) −7224.00 −0.357414
\(743\) −25578.0 −1.26294 −0.631471 0.775400i \(-0.717548\pi\)
−0.631471 + 0.775400i \(0.717548\pi\)
\(744\) 0 0
\(745\) −9480.00 −0.466202
\(746\) −7352.00 −0.360826
\(747\) 0 0
\(748\) 41664.0 2.03661
\(749\) −13342.0 −0.650876
\(750\) 0 0
\(751\) 33472.0 1.62638 0.813189 0.581999i \(-0.197729\pi\)
0.813189 + 0.581999i \(0.197729\pi\)
\(752\) 20736.0 1.00554
\(753\) 0 0
\(754\) 2480.00 0.119783
\(755\) −2272.00 −0.109519
\(756\) 0 0
\(757\) 25934.0 1.24516 0.622581 0.782556i \(-0.286084\pi\)
0.622581 + 0.782556i \(0.286084\pi\)
\(758\) 17040.0 0.816518
\(759\) 0 0
\(760\) 0 0
\(761\) −26952.0 −1.28385 −0.641925 0.766768i \(-0.721864\pi\)
−0.641925 + 0.766768i \(0.721864\pi\)
\(762\) 0 0
\(763\) 630.000 0.0298919
\(764\) 16304.0 0.772065
\(765\) 0 0
\(766\) 36192.0 1.70714
\(767\) 7440.00 0.350251
\(768\) 0 0
\(769\) 23450.0 1.09965 0.549824 0.835281i \(-0.314695\pi\)
0.549824 + 0.835281i \(0.314695\pi\)
\(770\) −6944.00 −0.324993
\(771\) 0 0
\(772\) −20816.0 −0.970446
\(773\) −39568.0 −1.84109 −0.920545 0.390637i \(-0.872255\pi\)
−0.920545 + 0.390637i \(0.872255\pi\)
\(774\) 0 0
\(775\) 5232.00 0.242502
\(776\) 0 0
\(777\) 0 0
\(778\) −45960.0 −2.11793
\(779\) 24800.0 1.14063
\(780\) 0 0
\(781\) −42036.0 −1.92595
\(782\) 14112.0 0.645325
\(783\) 0 0
\(784\) −3136.00 −0.142857
\(785\) −1064.00 −0.0483768
\(786\) 0 0
\(787\) −12356.0 −0.559649 −0.279825 0.960051i \(-0.590276\pi\)
−0.279825 + 0.960051i \(0.590276\pi\)
\(788\) −18832.0 −0.851348
\(789\) 0 0
\(790\) −11840.0 −0.533226
\(791\) 3206.00 0.144112
\(792\) 0 0
\(793\) −38564.0 −1.72692
\(794\) 7464.00 0.333611
\(795\) 0 0
\(796\) 13440.0 0.598452
\(797\) 21736.0 0.966033 0.483017 0.875611i \(-0.339541\pi\)
0.483017 + 0.875611i \(0.339541\pi\)
\(798\) 0 0
\(799\) 27216.0 1.20505
\(800\) −27904.0 −1.23319
\(801\) 0 0
\(802\) 54648.0 2.40609
\(803\) 39804.0 1.74926
\(804\) 0 0
\(805\) −1176.00 −0.0514889
\(806\) −11904.0 −0.520224
\(807\) 0 0
\(808\) 0 0
\(809\) 38310.0 1.66490 0.832452 0.554097i \(-0.186936\pi\)
0.832452 + 0.554097i \(0.186936\pi\)
\(810\) 0 0
\(811\) 2132.00 0.0923115 0.0461558 0.998934i \(-0.485303\pi\)
0.0461558 + 0.998934i \(0.485303\pi\)
\(812\) −560.000 −0.0242022
\(813\) 0 0
\(814\) −61008.0 −2.62694
\(815\) −1088.00 −0.0467619
\(816\) 0 0
\(817\) 6800.00 0.291190
\(818\) 52840.0 2.25857
\(819\) 0 0
\(820\) 7936.00 0.337972
\(821\) −5002.00 −0.212632 −0.106316 0.994332i \(-0.533906\pi\)
−0.106316 + 0.994332i \(0.533906\pi\)
\(822\) 0 0
\(823\) −3612.00 −0.152985 −0.0764923 0.997070i \(-0.524372\pi\)
−0.0764923 + 0.997070i \(0.524372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −3360.00 −0.141537
\(827\) 27666.0 1.16329 0.581645 0.813443i \(-0.302409\pi\)
0.581645 + 0.813443i \(0.302409\pi\)
\(828\) 0 0
\(829\) 12890.0 0.540034 0.270017 0.962856i \(-0.412971\pi\)
0.270017 + 0.962856i \(0.412971\pi\)
\(830\) 7488.00 0.313147
\(831\) 0 0
\(832\) 31744.0 1.32275
\(833\) −4116.00 −0.171202
\(834\) 0 0
\(835\) 7504.00 0.311002
\(836\) −49600.0 −2.05198
\(837\) 0 0
\(838\) 27840.0 1.14763
\(839\) 9340.00 0.384330 0.192165 0.981363i \(-0.438449\pi\)
0.192165 + 0.981363i \(0.438449\pi\)
\(840\) 0 0
\(841\) −24289.0 −0.995900
\(842\) −32648.0 −1.33625
\(843\) 0 0
\(844\) −5344.00 −0.217948
\(845\) 6588.00 0.268206
\(846\) 0 0
\(847\) −17591.0 −0.713617
\(848\) 16512.0 0.668661
\(849\) 0 0
\(850\) −36624.0 −1.47787
\(851\) −10332.0 −0.416188
\(852\) 0 0
\(853\) −33082.0 −1.32791 −0.663954 0.747773i \(-0.731123\pi\)
−0.663954 + 0.747773i \(0.731123\pi\)
\(854\) 17416.0 0.697849
\(855\) 0 0
\(856\) 0 0
\(857\) −7544.00 −0.300698 −0.150349 0.988633i \(-0.548040\pi\)
−0.150349 + 0.988633i \(0.548040\pi\)
\(858\) 0 0
\(859\) 8180.00 0.324910 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(860\) 2176.00 0.0862802
\(861\) 0 0
\(862\) 66408.0 2.62397
\(863\) −10518.0 −0.414875 −0.207437 0.978248i \(-0.566512\pi\)
−0.207437 + 0.978248i \(0.566512\pi\)
\(864\) 0 0
\(865\) 608.000 0.0238990
\(866\) −30952.0 −1.21454
\(867\) 0 0
\(868\) 2688.00 0.105111
\(869\) −45880.0 −1.79099
\(870\) 0 0
\(871\) −56048.0 −2.18038
\(872\) 0 0
\(873\) 0 0
\(874\) −16800.0 −0.650193
\(875\) 6552.00 0.253141
\(876\) 0 0
\(877\) 14134.0 0.544209 0.272104 0.962268i \(-0.412280\pi\)
0.272104 + 0.962268i \(0.412280\pi\)
\(878\) 3360.00 0.129151
\(879\) 0 0
\(880\) 15872.0 0.608006
\(881\) −6492.00 −0.248265 −0.124132 0.992266i \(-0.539615\pi\)
−0.124132 + 0.992266i \(0.539615\pi\)
\(882\) 0 0
\(883\) 38228.0 1.45694 0.728468 0.685080i \(-0.240233\pi\)
0.728468 + 0.685080i \(0.240233\pi\)
\(884\) 41664.0 1.58519
\(885\) 0 0
\(886\) 26472.0 1.00377
\(887\) 43076.0 1.63061 0.815305 0.579032i \(-0.196569\pi\)
0.815305 + 0.579032i \(0.196569\pi\)
\(888\) 0 0
\(889\) −5628.00 −0.212325
\(890\) 3200.00 0.120522
\(891\) 0 0
\(892\) −14656.0 −0.550134
\(893\) −32400.0 −1.21414
\(894\) 0 0
\(895\) −2440.00 −0.0911287
\(896\) 0 0
\(897\) 0 0
\(898\) 12360.0 0.459308
\(899\) −480.000 −0.0178074
\(900\) 0 0
\(901\) 21672.0 0.801331
\(902\) 61504.0 2.27035
\(903\) 0 0
\(904\) 0 0
\(905\) 4168.00 0.153093
\(906\) 0 0
\(907\) −32236.0 −1.18013 −0.590065 0.807355i \(-0.700898\pi\)
−0.590065 + 0.807355i \(0.700898\pi\)
\(908\) −39552.0 −1.44557
\(909\) 0 0
\(910\) −6944.00 −0.252958
\(911\) 46518.0 1.69178 0.845889 0.533359i \(-0.179070\pi\)
0.845889 + 0.533359i \(0.179070\pi\)
\(912\) 0 0
\(913\) 29016.0 1.05180
\(914\) −23656.0 −0.856095
\(915\) 0 0
\(916\) −43760.0 −1.57846
\(917\) 5684.00 0.204692
\(918\) 0 0
\(919\) 17840.0 0.640356 0.320178 0.947357i \(-0.396257\pi\)
0.320178 + 0.947357i \(0.396257\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −63872.0 −2.28147
\(923\) −42036.0 −1.49906
\(924\) 0 0
\(925\) 26814.0 0.953123
\(926\) 4688.00 0.166369
\(927\) 0 0
\(928\) 2560.00 0.0905562
\(929\) −7000.00 −0.247215 −0.123607 0.992331i \(-0.539446\pi\)
−0.123607 + 0.992331i \(0.539446\pi\)
\(930\) 0 0
\(931\) 4900.00 0.172493
\(932\) 22416.0 0.787833
\(933\) 0 0
\(934\) 21216.0 0.743264
\(935\) 20832.0 0.728641
\(936\) 0 0
\(937\) 36114.0 1.25912 0.629559 0.776953i \(-0.283236\pi\)
0.629559 + 0.776953i \(0.283236\pi\)
\(938\) 25312.0 0.881094
\(939\) 0 0
\(940\) −10368.0 −0.359752
\(941\) 4748.00 0.164485 0.0822425 0.996612i \(-0.473792\pi\)
0.0822425 + 0.996612i \(0.473792\pi\)
\(942\) 0 0
\(943\) 10416.0 0.359694
\(944\) 7680.00 0.264791
\(945\) 0 0
\(946\) 16864.0 0.579594
\(947\) −42694.0 −1.46501 −0.732507 0.680759i \(-0.761650\pi\)
−0.732507 + 0.680759i \(0.761650\pi\)
\(948\) 0 0
\(949\) 39804.0 1.36153
\(950\) 43600.0 1.48902
\(951\) 0 0
\(952\) 0 0
\(953\) 16742.0 0.569073 0.284537 0.958665i \(-0.408160\pi\)
0.284537 + 0.958665i \(0.408160\pi\)
\(954\) 0 0
\(955\) 8152.00 0.276223
\(956\) 9360.00 0.316657
\(957\) 0 0
\(958\) 22960.0 0.774326
\(959\) 2898.00 0.0975822
\(960\) 0 0
\(961\) −27487.0 −0.922661
\(962\) −61008.0 −2.04467
\(963\) 0 0
\(964\) −18704.0 −0.624912
\(965\) −10408.0 −0.347197
\(966\) 0 0
\(967\) −9956.00 −0.331089 −0.165545 0.986202i \(-0.552938\pi\)
−0.165545 + 0.986202i \(0.552938\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 20256.0 0.670496
\(971\) 26388.0 0.872123 0.436061 0.899917i \(-0.356373\pi\)
0.436061 + 0.899917i \(0.356373\pi\)
\(972\) 0 0
\(973\) 11340.0 0.373632
\(974\) −35776.0 −1.17694
\(975\) 0 0
\(976\) −39808.0 −1.30556
\(977\) 786.000 0.0257383 0.0128692 0.999917i \(-0.495904\pi\)
0.0128692 + 0.999917i \(0.495904\pi\)
\(978\) 0 0
\(979\) 12400.0 0.404807
\(980\) 1568.00 0.0511101
\(981\) 0 0
\(982\) −22232.0 −0.722456
\(983\) −51888.0 −1.68359 −0.841796 0.539796i \(-0.818501\pi\)
−0.841796 + 0.539796i \(0.818501\pi\)
\(984\) 0 0
\(985\) −9416.00 −0.304588
\(986\) 3360.00 0.108524
\(987\) 0 0
\(988\) −49600.0 −1.59715
\(989\) 2856.00 0.0918256
\(990\) 0 0
\(991\) −51928.0 −1.66453 −0.832264 0.554379i \(-0.812956\pi\)
−0.832264 + 0.554379i \(0.812956\pi\)
\(992\) −12288.0 −0.393291
\(993\) 0 0
\(994\) 18984.0 0.605771
\(995\) 6720.00 0.214109
\(996\) 0 0
\(997\) −386.000 −0.0122615 −0.00613076 0.999981i \(-0.501951\pi\)
−0.00613076 + 0.999981i \(0.501951\pi\)
\(998\) 79280.0 2.51459
\(999\) 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 63.4.a.a.1.1 1
3.2 odd 2 21.4.a.b.1.1 1
4.3 odd 2 1008.4.a.m.1.1 1
5.4 even 2 1575.4.a.k.1.1 1
7.2 even 3 441.4.e.m.361.1 2
7.3 odd 6 441.4.e.n.226.1 2
7.4 even 3 441.4.e.m.226.1 2
7.5 odd 6 441.4.e.n.361.1 2
7.6 odd 2 441.4.a.b.1.1 1
12.11 even 2 336.4.a.h.1.1 1
15.2 even 4 525.4.d.b.274.2 2
15.8 even 4 525.4.d.b.274.1 2
15.14 odd 2 525.4.a.b.1.1 1
21.2 odd 6 147.4.e.c.67.1 2
21.5 even 6 147.4.e.b.67.1 2
21.11 odd 6 147.4.e.c.79.1 2
21.17 even 6 147.4.e.b.79.1 2
21.20 even 2 147.4.a.g.1.1 1
24.5 odd 2 1344.4.a.w.1.1 1
24.11 even 2 1344.4.a.i.1.1 1
84.83 odd 2 2352.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 3.2 odd 2
63.4.a.a.1.1 1 1.1 even 1 trivial
147.4.a.g.1.1 1 21.20 even 2
147.4.e.b.67.1 2 21.5 even 6
147.4.e.b.79.1 2 21.17 even 6
147.4.e.c.67.1 2 21.2 odd 6
147.4.e.c.79.1 2 21.11 odd 6
336.4.a.h.1.1 1 12.11 even 2
441.4.a.b.1.1 1 7.6 odd 2
441.4.e.m.226.1 2 7.4 even 3
441.4.e.m.361.1 2 7.2 even 3
441.4.e.n.226.1 2 7.3 odd 6
441.4.e.n.361.1 2 7.5 odd 6
525.4.a.b.1.1 1 15.14 odd 2
525.4.d.b.274.1 2 15.8 even 4
525.4.d.b.274.2 2 15.2 even 4
1008.4.a.m.1.1 1 4.3 odd 2
1344.4.a.i.1.1 1 24.11 even 2
1344.4.a.w.1.1 1 24.5 odd 2
1575.4.a.k.1.1 1 5.4 even 2
2352.4.a.l.1.1 1 84.83 odd 2