Properties

Label 525.4.a.b.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -12.0000 q^{6} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -12.0000 q^{6} +7.00000 q^{7} +9.00000 q^{9} +62.0000 q^{11} +24.0000 q^{12} +62.0000 q^{13} -28.0000 q^{14} -64.0000 q^{16} -84.0000 q^{17} -36.0000 q^{18} +100.000 q^{19} +21.0000 q^{21} -248.000 q^{22} +42.0000 q^{23} -248.000 q^{26} +27.0000 q^{27} +56.0000 q^{28} -10.0000 q^{29} -48.0000 q^{31} +256.000 q^{32} +186.000 q^{33} +336.000 q^{34} +72.0000 q^{36} +246.000 q^{37} -400.000 q^{38} +186.000 q^{39} -248.000 q^{41} -84.0000 q^{42} -68.0000 q^{43} +496.000 q^{44} -168.000 q^{46} -324.000 q^{47} -192.000 q^{48} +49.0000 q^{49} -252.000 q^{51} +496.000 q^{52} -258.000 q^{53} -108.000 q^{54} +300.000 q^{57} +40.0000 q^{58} +120.000 q^{59} +622.000 q^{61} +192.000 q^{62} +63.0000 q^{63} -512.000 q^{64} -744.000 q^{66} -904.000 q^{67} -672.000 q^{68} +126.000 q^{69} -678.000 q^{71} +642.000 q^{73} -984.000 q^{74} +800.000 q^{76} +434.000 q^{77} -744.000 q^{78} +740.000 q^{79} +81.0000 q^{81} +992.000 q^{82} -468.000 q^{83} +168.000 q^{84} +272.000 q^{86} -30.0000 q^{87} +200.000 q^{89} +434.000 q^{91} +336.000 q^{92} -144.000 q^{93} +1296.00 q^{94} +768.000 q^{96} +1266.00 q^{97} -196.000 q^{98} +558.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 3.00000 0.577350
\(4\) 8.00000 1.00000
\(5\) 0 0
\(6\) −12.0000 −0.816497
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 62.0000 1.69943 0.849714 0.527244i \(-0.176775\pi\)
0.849714 + 0.527244i \(0.176775\pi\)
\(12\) 24.0000 0.577350
\(13\) 62.0000 1.32275 0.661373 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) −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 \(0.704517\pi\)
\(18\) −36.0000 −0.471405
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) −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
\(25\) 0 0
\(26\) −248.000 −1.87065
\(27\) 27.0000 0.192450
\(28\) 56.0000 0.377964
\(29\) −10.0000 −0.0640329 −0.0320164 0.999487i \(-0.510193\pi\)
−0.0320164 + 0.999487i \(0.510193\pi\)
\(30\) 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
\(33\) 186.000 0.981165
\(34\) 336.000 1.69481
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) 246.000 1.09303 0.546516 0.837449i \(-0.315954\pi\)
0.546516 + 0.837449i \(0.315954\pi\)
\(38\) −400.000 −1.70759
\(39\) 186.000 0.763688
\(40\) 0 0
\(41\) −248.000 −0.944661 −0.472330 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(42\) −84.0000 −0.308607
\(43\) −68.0000 −0.241161 −0.120580 0.992704i \(-0.538476\pi\)
−0.120580 + 0.992704i \(0.538476\pi\)
\(44\) 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 \(0.667685\pi\)
\(48\) −192.000 −0.577350
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −252.000 −0.691903
\(52\) 496.000 1.32275
\(53\) −258.000 −0.668661 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(54\) −108.000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 300.000 0.697122
\(58\) 40.0000 0.0905562
\(59\) 120.000 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(60\) 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
\(63\) 63.0000 0.125988
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) −744.000 −1.38758
\(67\) −904.000 −1.64838 −0.824188 0.566316i \(-0.808368\pi\)
−0.824188 + 0.566316i \(0.808368\pi\)
\(68\) −672.000 −1.19841
\(69\) 126.000 0.219835
\(70\) 0 0
\(71\) −678.000 −1.13329 −0.566646 0.823961i \(-0.691759\pi\)
−0.566646 + 0.823961i \(0.691759\pi\)
\(72\) 0 0
\(73\) 642.000 1.02932 0.514660 0.857394i \(-0.327918\pi\)
0.514660 + 0.857394i \(0.327918\pi\)
\(74\) −984.000 −1.54578
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) 434.000 0.642323
\(78\) −744.000 −1.08002
\(79\) 740.000 1.05388 0.526940 0.849903i \(-0.323339\pi\)
0.526940 + 0.849903i \(0.323339\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 992.000 1.33595
\(83\) −468.000 −0.618912 −0.309456 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(84\) 168.000 0.218218
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) −30.0000 −0.0369694
\(88\) 0 0
\(89\) 200.000 0.238202 0.119101 0.992882i \(-0.461999\pi\)
0.119101 + 0.992882i \(0.461999\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) 336.000 0.380765
\(93\) −144.000 −0.160560
\(94\) 1296.00 1.42204
\(95\) 0 0
\(96\) 768.000 0.816497
\(97\) 1266.00 1.32518 0.662592 0.748981i \(-0.269456\pi\)
0.662592 + 0.748981i \(0.269456\pi\)
\(98\) −196.000 −0.202031
\(99\) 558.000 0.566476
\(100\) 0 0
\(101\) 232.000 0.228563 0.114281 0.993448i \(-0.463543\pi\)
0.114281 + 0.993448i \(0.463543\pi\)
\(102\) 1008.00 0.978499
\(103\) 1792.00 1.71428 0.857141 0.515082i \(-0.172239\pi\)
0.857141 + 0.515082i \(0.172239\pi\)
\(104\) 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 \(0.169821\pi\)
\(108\) 216.000 0.192450
\(109\) −90.0000 −0.0790866 −0.0395433 0.999218i \(-0.512590\pi\)
−0.0395433 + 0.999218i \(0.512590\pi\)
\(110\) 0 0
\(111\) 738.000 0.631062
\(112\) −448.000 −0.377964
\(113\) −458.000 −0.381283 −0.190642 0.981660i \(-0.561057\pi\)
−0.190642 + 0.981660i \(0.561057\pi\)
\(114\) −1200.00 −0.985880
\(115\) 0 0
\(116\) −80.0000 −0.0640329
\(117\) 558.000 0.440916
\(118\) −480.000 −0.374471
\(119\) −588.000 −0.452957
\(120\) 0 0
\(121\) 2513.00 1.88805
\(122\) −2488.00 −1.84634
\(123\) −744.000 −0.545400
\(124\) −384.000 −0.278099
\(125\) 0 0
\(126\) −252.000 −0.178174
\(127\) −804.000 −0.561760 −0.280880 0.959743i \(-0.590626\pi\)
−0.280880 + 0.959743i \(0.590626\pi\)
\(128\) 0 0
\(129\) −204.000 −0.139234
\(130\) 0 0
\(131\) 812.000 0.541563 0.270782 0.962641i \(-0.412718\pi\)
0.270782 + 0.962641i \(0.412718\pi\)
\(132\) 1488.00 0.981165
\(133\) 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 \(0.541205\pi\)
\(138\) −504.000 −0.310894
\(139\) −1620.00 −0.988537 −0.494268 0.869309i \(-0.664564\pi\)
−0.494268 + 0.869309i \(0.664564\pi\)
\(140\) 0 0
\(141\) −972.000 −0.580547
\(142\) 2712.00 1.60272
\(143\) 3844.00 2.24791
\(144\) −576.000 −0.333333
\(145\) 0 0
\(146\) −2568.00 −1.45568
\(147\) 147.000 0.0824786
\(148\) 1968.00 1.09303
\(149\) 2370.00 1.30307 0.651537 0.758617i \(-0.274125\pi\)
0.651537 + 0.758617i \(0.274125\pi\)
\(150\) 0 0
\(151\) −568.000 −0.306114 −0.153057 0.988217i \(-0.548912\pi\)
−0.153057 + 0.988217i \(0.548912\pi\)
\(152\) 0 0
\(153\) −756.000 −0.399470
\(154\) −1736.00 −0.908382
\(155\) 0 0
\(156\) 1488.00 0.763688
\(157\) 266.000 0.135217 0.0676086 0.997712i \(-0.478463\pi\)
0.0676086 + 0.997712i \(0.478463\pi\)
\(158\) −2960.00 −1.49041
\(159\) −774.000 −0.386052
\(160\) 0 0
\(161\) 294.000 0.143916
\(162\) −324.000 −0.157135
\(163\) 272.000 0.130704 0.0653518 0.997862i \(-0.479183\pi\)
0.0653518 + 0.997862i \(0.479183\pi\)
\(164\) −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
\(170\) 0 0
\(171\) 900.000 0.402484
\(172\) −544.000 −0.241161
\(173\) 152.000 0.0667997 0.0333998 0.999442i \(-0.489367\pi\)
0.0333998 + 0.999442i \(0.489367\pi\)
\(174\) 120.000 0.0522826
\(175\) 0 0
\(176\) −3968.00 −1.69943
\(177\) 360.000 0.152877
\(178\) −800.000 −0.336868
\(179\) 610.000 0.254713 0.127356 0.991857i \(-0.459351\pi\)
0.127356 + 0.991857i \(0.459351\pi\)
\(180\) 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
\(183\) 1866.00 0.753763
\(184\) 0 0
\(185\) 0 0
\(186\) 576.000 0.227067
\(187\) −5208.00 −2.03661
\(188\) −2592.00 −1.00554
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −2038.00 −0.772065 −0.386033 0.922485i \(-0.626155\pi\)
−0.386033 + 0.922485i \(0.626155\pi\)
\(192\) −1536.00 −0.577350
\(193\) 2602.00 0.970446 0.485223 0.874390i \(-0.338738\pi\)
0.485223 + 0.874390i \(0.338738\pi\)
\(194\) −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 \(0.639963\pi\)
\(198\) −2232.00 −0.801118
\(199\) 1680.00 0.598452 0.299226 0.954182i \(-0.403271\pi\)
0.299226 + 0.954182i \(0.403271\pi\)
\(200\) 0 0
\(201\) −2712.00 −0.951690
\(202\) −928.000 −0.323237
\(203\) −70.0000 −0.0242022
\(204\) −2016.00 −0.691903
\(205\) 0 0
\(206\) −7168.00 −2.42436
\(207\) 378.000 0.126922
\(208\) −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
\(213\) −2034.00 −0.654307
\(214\) −7624.00 −2.43535
\(215\) 0 0
\(216\) 0 0
\(217\) −336.000 −0.105111
\(218\) 360.000 0.111845
\(219\) 1926.00 0.594279
\(220\) 0 0
\(221\) −5208.00 −1.58519
\(222\) −2952.00 −0.892456
\(223\) 1832.00 0.550134 0.275067 0.961425i \(-0.411300\pi\)
0.275067 + 0.961425i \(0.411300\pi\)
\(224\) 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 \(0.757139\pi\)
\(228\) 2400.00 0.697122
\(229\) −5470.00 −1.57846 −0.789231 0.614096i \(-0.789521\pi\)
−0.789231 + 0.614096i \(0.789521\pi\)
\(230\) 0 0
\(231\) 1302.00 0.370846
\(232\) 0 0
\(233\) 2802.00 0.787833 0.393917 0.919146i \(-0.371120\pi\)
0.393917 + 0.919146i \(0.371120\pi\)
\(234\) −2232.00 −0.623549
\(235\) 0 0
\(236\) 960.000 0.264791
\(237\) 2220.00 0.608458
\(238\) 2352.00 0.640578
\(239\) −1170.00 −0.316657 −0.158328 0.987386i \(-0.550610\pi\)
−0.158328 + 0.987386i \(0.550610\pi\)
\(240\) 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
\(243\) 243.000 0.0641500
\(244\) 4976.00 1.30556
\(245\) 0 0
\(246\) 2976.00 0.771312
\(247\) 6200.00 1.59715
\(248\) 0 0
\(249\) −1404.00 −0.357329
\(250\) 0 0
\(251\) 2792.00 0.702109 0.351055 0.936355i \(-0.385823\pi\)
0.351055 + 0.936355i \(0.385823\pi\)
\(252\) 504.000 0.125988
\(253\) 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 \(0.824867\pi\)
\(258\) 816.000 0.196907
\(259\) 1722.00 0.413127
\(260\) 0 0
\(261\) −90.0000 −0.0213443
\(262\) −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
\(265\) 0 0
\(266\) −2800.00 −0.645410
\(267\) 600.000 0.137526
\(268\) −7232.00 −1.64838
\(269\) −6780.00 −1.53674 −0.768372 0.640004i \(-0.778933\pi\)
−0.768372 + 0.640004i \(0.778933\pi\)
\(270\) 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
\(273\) 1302.00 0.288647
\(274\) 1656.00 0.365119
\(275\) 0 0
\(276\) 1008.00 0.219835
\(277\) −5554.00 −1.20472 −0.602360 0.798224i \(-0.705773\pi\)
−0.602360 + 0.798224i \(0.705773\pi\)
\(278\) 6480.00 1.39800
\(279\) −432.000 −0.0926995
\(280\) 0 0
\(281\) 1942.00 0.412278 0.206139 0.978523i \(-0.433910\pi\)
0.206139 + 0.978523i \(0.433910\pi\)
\(282\) 3888.00 0.821018
\(283\) −4828.00 −1.01412 −0.507058 0.861912i \(-0.669267\pi\)
−0.507058 + 0.861912i \(0.669267\pi\)
\(284\) −5424.00 −1.13329
\(285\) 0 0
\(286\) −15376.0 −3.17903
\(287\) −1736.00 −0.357048
\(288\) 2304.00 0.471405
\(289\) 2143.00 0.436190
\(290\) 0 0
\(291\) 3798.00 0.765095
\(292\) 5136.00 1.02932
\(293\) 6152.00 1.22663 0.613317 0.789837i \(-0.289835\pi\)
0.613317 + 0.789837i \(0.289835\pi\)
\(294\) −588.000 −0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 1674.00 0.327055
\(298\) −9480.00 −1.84282
\(299\) 2604.00 0.503656
\(300\) 0 0
\(301\) −476.000 −0.0911501
\(302\) 2272.00 0.432910
\(303\) 696.000 0.131961
\(304\) −6400.00 −1.20745
\(305\) 0 0
\(306\) 3024.00 0.564937
\(307\) −5884.00 −1.09387 −0.546934 0.837176i \(-0.684205\pi\)
−0.546934 + 0.837176i \(0.684205\pi\)
\(308\) 3472.00 0.642323
\(309\) 5376.00 0.989741
\(310\) 0 0
\(311\) 9132.00 1.66504 0.832521 0.553993i \(-0.186897\pi\)
0.832521 + 0.553993i \(0.186897\pi\)
\(312\) 0 0
\(313\) 9382.00 1.69426 0.847128 0.531389i \(-0.178330\pi\)
0.847128 + 0.531389i \(0.178330\pi\)
\(314\) −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 \(0.588965\pi\)
\(318\) 3096.00 0.545959
\(319\) −620.000 −0.108819
\(320\) 0 0
\(321\) 5718.00 0.994229
\(322\) −1176.00 −0.203528
\(323\) −8400.00 −1.44702
\(324\) 648.000 0.111111
\(325\) 0 0
\(326\) −1088.00 −0.184843
\(327\) −270.000 −0.0456607
\(328\) 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
\(333\) 2214.00 0.364344
\(334\) −7504.00 −1.22934
\(335\) 0 0
\(336\) −1344.00 −0.218218
\(337\) 4166.00 0.673402 0.336701 0.941612i \(-0.390689\pi\)
0.336701 + 0.941612i \(0.390689\pi\)
\(338\) −6588.00 −1.06018
\(339\) −1374.00 −0.220134
\(340\) 0 0
\(341\) −2976.00 −0.472608
\(342\) −3600.00 −0.569198
\(343\) 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 \(0.158083\pi\)
\(348\) −240.000 −0.0369694
\(349\) 9310.00 1.42795 0.713973 0.700174i \(-0.246894\pi\)
0.713973 + 0.700174i \(0.246894\pi\)
\(350\) 0 0
\(351\) 1674.00 0.254563
\(352\) 15872.0 2.40335
\(353\) 8572.00 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(354\) −1440.00 −0.216201
\(355\) 0 0
\(356\) 1600.00 0.238202
\(357\) −1764.00 −0.261515
\(358\) −2440.00 −0.360218
\(359\) −4790.00 −0.704196 −0.352098 0.935963i \(-0.614532\pi\)
−0.352098 + 0.935963i \(0.614532\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) −4168.00 −0.605153
\(363\) 7539.00 1.09007
\(364\) 3472.00 0.499951
\(365\) 0 0
\(366\) −7464.00 −1.06598
\(367\) −5424.00 −0.771473 −0.385736 0.922609i \(-0.626053\pi\)
−0.385736 + 0.922609i \(0.626053\pi\)
\(368\) −2688.00 −0.380765
\(369\) −2232.00 −0.314887
\(370\) 0 0
\(371\) −1806.00 −0.252730
\(372\) −1152.00 −0.160560
\(373\) −1838.00 −0.255142 −0.127571 0.991829i \(-0.540718\pi\)
−0.127571 + 0.991829i \(0.540718\pi\)
\(374\) 20832.0 2.88021
\(375\) 0 0
\(376\) 0 0
\(377\) −620.000 −0.0846993
\(378\) −756.000 −0.102869
\(379\) −4260.00 −0.577365 −0.288683 0.957425i \(-0.593217\pi\)
−0.288683 + 0.957425i \(0.593217\pi\)
\(380\) 0 0
\(381\) −2412.00 −0.324332
\(382\) 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
\(385\) 0 0
\(386\) −10408.0 −1.37242
\(387\) −612.000 −0.0803868
\(388\) 10128.0 1.32518
\(389\) −11490.0 −1.49760 −0.748800 0.662796i \(-0.769369\pi\)
−0.748800 + 0.662796i \(0.769369\pi\)
\(390\) 0 0
\(391\) −3528.00 −0.456314
\(392\) 0 0
\(393\) 2436.00 0.312672
\(394\) 9416.00 1.20399
\(395\) 0 0
\(396\) 4464.00 0.566476
\(397\) 1866.00 0.235899 0.117949 0.993020i \(-0.462368\pi\)
0.117949 + 0.993020i \(0.462368\pi\)
\(398\) −6720.00 −0.846340
\(399\) 2100.00 0.263487
\(400\) 0 0
\(401\) 13662.0 1.70137 0.850683 0.525679i \(-0.176189\pi\)
0.850683 + 0.525679i \(0.176189\pi\)
\(402\) 10848.0 1.34589
\(403\) −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 \(0.794385\pi\)
\(410\) 0 0
\(411\) −1242.00 −0.149059
\(412\) 14336.0 1.71428
\(413\) 840.000 0.100082
\(414\) −1512.00 −0.179495
\(415\) 0 0
\(416\) 15872.0 1.87065
\(417\) −4860.00 −0.570732
\(418\) −24800.0 −2.90193
\(419\) 6960.00 0.811499 0.405750 0.913984i \(-0.367010\pi\)
0.405750 + 0.913984i \(0.367010\pi\)
\(420\) 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
\(423\) −2916.00 −0.335179
\(424\) 0 0
\(425\) 0 0
\(426\) 8136.00 0.925330
\(427\) 4354.00 0.493454
\(428\) 15248.0 1.72206
\(429\) 11532.0 1.29783
\(430\) 0 0
\(431\) 16602.0 1.85543 0.927715 0.373290i \(-0.121770\pi\)
0.927715 + 0.373290i \(0.121770\pi\)
\(432\) −1728.00 −0.192450
\(433\) −7738.00 −0.858810 −0.429405 0.903112i \(-0.641277\pi\)
−0.429405 + 0.903112i \(0.641277\pi\)
\(434\) 1344.00 0.148650
\(435\) 0 0
\(436\) −720.000 −0.0790866
\(437\) 4200.00 0.459756
\(438\) −7704.00 −0.840437
\(439\) −840.000 −0.0913235 −0.0456617 0.998957i \(-0.514540\pi\)
−0.0456617 + 0.998957i \(0.514540\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 20832.0 2.24180
\(443\) −6618.00 −0.709776 −0.354888 0.934909i \(-0.615481\pi\)
−0.354888 + 0.934909i \(0.615481\pi\)
\(444\) 5904.00 0.631062
\(445\) 0 0
\(446\) −7328.00 −0.778006
\(447\) 7110.00 0.752330
\(448\) −3584.00 −0.377964
\(449\) 3090.00 0.324780 0.162390 0.986727i \(-0.448080\pi\)
0.162390 + 0.986727i \(0.448080\pi\)
\(450\) 0 0
\(451\) −15376.0 −1.60538
\(452\) −3664.00 −0.381283
\(453\) −1704.00 −0.176735
\(454\) 19776.0 2.04435
\(455\) 0 0
\(456\) 0 0
\(457\) −5914.00 −0.605351 −0.302675 0.953094i \(-0.597880\pi\)
−0.302675 + 0.953094i \(0.597880\pi\)
\(458\) 21880.0 2.23228
\(459\) −2268.00 −0.230634
\(460\) 0 0
\(461\) −15968.0 −1.61324 −0.806620 0.591070i \(-0.798706\pi\)
−0.806620 + 0.591070i \(0.798706\pi\)
\(462\) −5208.00 −0.524455
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) 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 \(0.584641\pi\)
\(468\) 4464.00 0.440916
\(469\) −6328.00 −0.623027
\(470\) 0 0
\(471\) 798.000 0.0780677
\(472\) 0 0
\(473\) −4216.00 −0.409835
\(474\) −8880.00 −0.860489
\(475\) 0 0
\(476\) −4704.00 −0.452957
\(477\) −2322.00 −0.222887
\(478\) 4680.00 0.447821
\(479\) 5740.00 0.547531 0.273765 0.961796i \(-0.411731\pi\)
0.273765 + 0.961796i \(0.411731\pi\)
\(480\) 0 0
\(481\) 15252.0 1.44580
\(482\) 9352.00 0.883759
\(483\) 882.000 0.0830898
\(484\) 20104.0 1.88805
\(485\) 0 0
\(486\) −972.000 −0.0907218
\(487\) −8944.00 −0.832220 −0.416110 0.909314i \(-0.636607\pi\)
−0.416110 + 0.909314i \(0.636607\pi\)
\(488\) 0 0
\(489\) 816.000 0.0754617
\(490\) 0 0
\(491\) −5558.00 −0.510853 −0.255427 0.966828i \(-0.582216\pi\)
−0.255427 + 0.966828i \(0.582216\pi\)
\(492\) −5952.00 −0.545400
\(493\) 840.000 0.0767377
\(494\) −24800.0 −2.25871
\(495\) 0 0
\(496\) 3072.00 0.278099
\(497\) −4746.00 −0.428344
\(498\) 5616.00 0.505339
\(499\) −19820.0 −1.77809 −0.889043 0.457823i \(-0.848629\pi\)
−0.889043 + 0.457823i \(0.848629\pi\)
\(500\) 0 0
\(501\) 5628.00 0.501877
\(502\) −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
\(505\) 0 0
\(506\) −10416.0 −0.915114
\(507\) 4941.00 0.432816
\(508\) −6432.00 −0.561760
\(509\) 340.000 0.0296075 0.0148038 0.999890i \(-0.495288\pi\)
0.0148038 + 0.999890i \(0.495288\pi\)
\(510\) 0 0
\(511\) 4494.00 0.389047
\(512\) −16384.0 −1.41421
\(513\) 2700.00 0.232374
\(514\) 28096.0 2.41101
\(515\) 0 0
\(516\) −1632.00 −0.139234
\(517\) −20088.0 −1.70884
\(518\) −6888.00 −0.584250
\(519\) 456.000 0.0385668
\(520\) 0 0
\(521\) 10212.0 0.858725 0.429363 0.903132i \(-0.358738\pi\)
0.429363 + 0.903132i \(0.358738\pi\)
\(522\) 360.000 0.0301854
\(523\) 9332.00 0.780229 0.390115 0.920766i \(-0.372435\pi\)
0.390115 + 0.920766i \(0.372435\pi\)
\(524\) 6496.00 0.541563
\(525\) 0 0
\(526\) 9752.00 0.808379
\(527\) 4032.00 0.333276
\(528\) −11904.0 −0.981165
\(529\) −10403.0 −0.855018
\(530\) 0 0
\(531\) 1080.00 0.0882637
\(532\) 5600.00 0.456374
\(533\) −15376.0 −1.24955
\(534\) −2400.00 −0.194491
\(535\) 0 0
\(536\) 0 0
\(537\) 1830.00 0.147058
\(538\) 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
\(543\) 3126.00 0.247052
\(544\) −21504.0 −1.69481
\(545\) 0 0
\(546\) −5208.00 −0.408208
\(547\) 3416.00 0.267016 0.133508 0.991048i \(-0.457376\pi\)
0.133508 + 0.991048i \(0.457376\pi\)
\(548\) −3312.00 −0.258178
\(549\) 5598.00 0.435185
\(550\) 0 0
\(551\) −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 \(0.493631\pi\)
\(558\) 1728.00 0.131097
\(559\) −4216.00 −0.318994
\(560\) 0 0
\(561\) −15624.0 −1.17584
\(562\) −7768.00 −0.583049
\(563\) 6712.00 0.502446 0.251223 0.967929i \(-0.419167\pi\)
0.251223 + 0.967929i \(0.419167\pi\)
\(564\) −7776.00 −0.580547
\(565\) 0 0
\(566\) 19312.0 1.43418
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) 4190.00 0.308706 0.154353 0.988016i \(-0.450671\pi\)
0.154353 + 0.988016i \(0.450671\pi\)
\(570\) 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
\(573\) −6114.00 −0.445752
\(574\) 6944.00 0.504942
\(575\) 0 0
\(576\) −4608.00 −0.333333
\(577\) −5434.00 −0.392063 −0.196032 0.980598i \(-0.562805\pi\)
−0.196032 + 0.980598i \(0.562805\pi\)
\(578\) −8572.00 −0.616865
\(579\) 7806.00 0.560287
\(580\) 0 0
\(581\) −3276.00 −0.233927
\(582\) −15192.0 −1.08201
\(583\) −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 \(0.505193\pi\)
\(588\) 1176.00 0.0824786
\(589\) −4800.00 −0.335790
\(590\) 0 0
\(591\) −7062.00 −0.491526
\(592\) −15744.0 −1.09303
\(593\) −11748.0 −0.813546 −0.406773 0.913529i \(-0.633346\pi\)
−0.406773 + 0.913529i \(0.633346\pi\)
\(594\) −6696.00 −0.462526
\(595\) 0 0
\(596\) 18960.0 1.30307
\(597\) 5040.00 0.345517
\(598\) −10416.0 −0.712277
\(599\) 7650.00 0.521821 0.260910 0.965363i \(-0.415977\pi\)
0.260910 + 0.965363i \(0.415977\pi\)
\(600\) 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
\(603\) −8136.00 −0.549459
\(604\) −4544.00 −0.306114
\(605\) 0 0
\(606\) −2784.00 −0.186621
\(607\) −704.000 −0.0470749 −0.0235375 0.999723i \(-0.507493\pi\)
−0.0235375 + 0.999723i \(0.507493\pi\)
\(608\) 25600.0 1.70759
\(609\) −210.000 −0.0139731
\(610\) 0 0
\(611\) −20088.0 −1.33007
\(612\) −6048.00 −0.399470
\(613\) −24958.0 −1.64444 −0.822222 0.569167i \(-0.807266\pi\)
−0.822222 + 0.569167i \(0.807266\pi\)
\(614\) 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 \(0.407029\pi\)
\(618\) −21504.0 −1.39971
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 0 0
\(621\) 1134.00 0.0732783
\(622\) −36528.0 −2.35473
\(623\) 1400.00 0.0900318
\(624\) −11904.0 −0.763688
\(625\) 0 0
\(626\) −37528.0 −2.39604
\(627\) 18600.0 1.18471
\(628\) 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
\(633\) −2004.00 −0.125832
\(634\) 12456.0 0.780270
\(635\) 0 0
\(636\) −6192.00 −0.386052
\(637\) 3038.00 0.188964
\(638\) 2480.00 0.153894
\(639\) −6102.00 −0.377764
\(640\) 0 0
\(641\) 13062.0 0.804864 0.402432 0.915450i \(-0.368165\pi\)
0.402432 + 0.915450i \(0.368165\pi\)
\(642\) −22872.0 −1.40605
\(643\) 28012.0 1.71802 0.859009 0.511961i \(-0.171081\pi\)
0.859009 + 0.511961i \(0.171081\pi\)
\(644\) 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
\(650\) 0 0
\(651\) −1008.00 −0.0606861
\(652\) 2176.00 0.130704
\(653\) 28482.0 1.70687 0.853436 0.521198i \(-0.174515\pi\)
0.853436 + 0.521198i \(0.174515\pi\)
\(654\) 1080.00 0.0645739
\(655\) 0 0
\(656\) 15872.0 0.944661
\(657\) 5778.00 0.343107
\(658\) 9072.00 0.537482
\(659\) −9330.00 −0.551510 −0.275755 0.961228i \(-0.588928\pi\)
−0.275755 + 0.961228i \(0.588928\pi\)
\(660\) 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
\(663\) −15624.0 −0.915212
\(664\) 0 0
\(665\) 0 0
\(666\) −8856.00 −0.515260
\(667\) −420.000 −0.0243815
\(668\) 15008.0 0.869277
\(669\) 5496.00 0.317620
\(670\) 0 0
\(671\) 38564.0 2.21870
\(672\) 5376.00 0.308607
\(673\) 10562.0 0.604956 0.302478 0.953156i \(-0.402186\pi\)
0.302478 + 0.953156i \(0.402186\pi\)
\(674\) −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 \(0.235553\pi\)
\(678\) 5496.00 0.311317
\(679\) 8862.00 0.500872
\(680\) 0 0
\(681\) −14832.0 −0.834601
\(682\) 11904.0 0.668369
\(683\) −8898.00 −0.498496 −0.249248 0.968440i \(-0.580183\pi\)
−0.249248 + 0.968440i \(0.580183\pi\)
\(684\) 7200.00 0.402484
\(685\) 0 0
\(686\) −1372.00 −0.0763604
\(687\) −16410.0 −0.911325
\(688\) 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
\(693\) 3906.00 0.214108
\(694\) −45464.0 −2.48673
\(695\) 0 0
\(696\) 0 0
\(697\) 20832.0 1.13209
\(698\) −37240.0 −2.01942
\(699\) 8406.00 0.454856
\(700\) 0 0
\(701\) −30618.0 −1.64968 −0.824840 0.565366i \(-0.808735\pi\)
−0.824840 + 0.565366i \(0.808735\pi\)
\(702\) −6696.00 −0.360006
\(703\) 24600.0 1.31978
\(704\) −31744.0 −1.69943
\(705\) 0 0
\(706\) −34288.0 −1.82783
\(707\) 1624.00 0.0863887
\(708\) 2880.00 0.152877
\(709\) −8130.00 −0.430647 −0.215323 0.976543i \(-0.569081\pi\)
−0.215323 + 0.976543i \(0.569081\pi\)
\(710\) 0 0
\(711\) 6660.00 0.351293
\(712\) 0 0
\(713\) −2016.00 −0.105890
\(714\) 7056.00 0.369838
\(715\) 0 0
\(716\) 4880.00 0.254713
\(717\) −3510.00 −0.182822
\(718\) 19160.0 0.995884
\(719\) −27840.0 −1.44403 −0.722014 0.691878i \(-0.756784\pi\)
−0.722014 + 0.691878i \(0.756784\pi\)
\(720\) 0 0
\(721\) 12544.0 0.647938
\(722\) −12564.0 −0.647623
\(723\) −7014.00 −0.360793
\(724\) 8336.00 0.427907
\(725\) 0 0
\(726\) −30156.0 −1.54159
\(727\) −14624.0 −0.746044 −0.373022 0.927822i \(-0.621678\pi\)
−0.373022 + 0.927822i \(0.621678\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 5712.00 0.289010
\(732\) 14928.0 0.753763
\(733\) 20862.0 1.05124 0.525618 0.850721i \(-0.323834\pi\)
0.525618 + 0.850721i \(0.323834\pi\)
\(734\) 21696.0 1.09103
\(735\) 0 0
\(736\) 10752.0 0.538484
\(737\) −56048.0 −2.80130
\(738\) 8928.00 0.445317
\(739\) −13920.0 −0.692903 −0.346452 0.938068i \(-0.612614\pi\)
−0.346452 + 0.938068i \(0.612614\pi\)
\(740\) 0 0
\(741\) 18600.0 0.922116
\(742\) 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
\(745\) 0 0
\(746\) 7352.00 0.360826
\(747\) −4212.00 −0.206304
\(748\) −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
\(753\) 8376.00 0.405363
\(754\) 2480.00 0.119783
\(755\) 0 0
\(756\) 1512.00 0.0727393
\(757\) −25934.0 −1.24516 −0.622581 0.782556i \(-0.713916\pi\)
−0.622581 + 0.782556i \(0.713916\pi\)
\(758\) 17040.0 0.816518
\(759\) 7812.00 0.373594
\(760\) 0 0
\(761\) 26952.0 1.28385 0.641925 0.766768i \(-0.278136\pi\)
0.641925 + 0.766768i \(0.278136\pi\)
\(762\) 9648.00 0.458675
\(763\) −630.000 −0.0298919
\(764\) −16304.0 −0.772065
\(765\) 0 0
\(766\) 36192.0 1.70714
\(767\) 7440.00 0.350251
\(768\) 12288.0 0.577350
\(769\) 23450.0 1.09965 0.549824 0.835281i \(-0.314695\pi\)
0.549824 + 0.835281i \(0.314695\pi\)
\(770\) 0 0
\(771\) −21072.0 −0.984293
\(772\) 20816.0 0.970446
\(773\) −39568.0 −1.84109 −0.920545 0.390637i \(-0.872255\pi\)
−0.920545 + 0.390637i \(0.872255\pi\)
\(774\) 2448.00 0.113684
\(775\) 0 0
\(776\) 0 0
\(777\) 5166.00 0.238519
\(778\) 45960.0 2.11793
\(779\) −24800.0 −1.14063
\(780\) 0 0
\(781\) −42036.0 −1.92595
\(782\) 14112.0 0.645325
\(783\) −270.000 −0.0123231
\(784\) −3136.00 −0.142857
\(785\) 0 0
\(786\) −9744.00 −0.442184
\(787\) 12356.0 0.559649 0.279825 0.960051i \(-0.409724\pi\)
0.279825 + 0.960051i \(0.409724\pi\)
\(788\) −18832.0 −0.851348
\(789\) −7314.00 −0.330019
\(790\) 0 0
\(791\) −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 \(0.339541\pi\)
\(798\) −8400.00 −0.372628
\(799\) 27216.0 1.20505
\(800\) 0 0
\(801\) 1800.00 0.0794006
\(802\) −54648.0 −2.40609
\(803\) 39804.0 1.74926
\(804\) −21696.0 −0.951690
\(805\) 0 0
\(806\) 11904.0 0.520224
\(807\) −20340.0 −0.887239
\(808\) 0 0
\(809\) −38310.0 −1.66490 −0.832452 0.554097i \(-0.813064\pi\)
−0.832452 + 0.554097i \(0.813064\pi\)
\(810\) 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
\(813\) −5784.00 −0.249513
\(814\) −61008.0 −2.62694
\(815\) 0 0
\(816\) 16128.0 0.691903
\(817\) −6800.00 −0.291190
\(818\) 52840.0 2.25857
\(819\) 3906.00 0.166650
\(820\) 0 0
\(821\) 5002.00 0.212632 0.106316 0.994332i \(-0.466094\pi\)
0.106316 + 0.994332i \(0.466094\pi\)
\(822\) 4968.00 0.210802
\(823\) 3612.00 0.152985 0.0764923 0.997070i \(-0.475628\pi\)
0.0764923 + 0.997070i \(0.475628\pi\)
\(824\) 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 \(0.302409\pi\)
\(828\) 3024.00 0.126922
\(829\) 12890.0 0.540034 0.270017 0.962856i \(-0.412971\pi\)
0.270017 + 0.962856i \(0.412971\pi\)
\(830\) 0 0
\(831\) −16662.0 −0.695546
\(832\) −31744.0 −1.32275
\(833\) −4116.00 −0.171202
\(834\) 19440.0 0.807137
\(835\) 0 0
\(836\) 49600.0 2.05198
\(837\) −1296.00 −0.0535201
\(838\) −27840.0 −1.14763
\(839\) −9340.00 −0.384330 −0.192165 0.981363i \(-0.561551\pi\)
−0.192165 + 0.981363i \(0.561551\pi\)
\(840\) 0 0
\(841\) −24289.0 −0.995900
\(842\) −32648.0 −1.33625
\(843\) 5826.00 0.238029
\(844\) −5344.00 −0.217948
\(845\) 0 0
\(846\) 11664.0 0.474015
\(847\) 17591.0 0.713617
\(848\) 16512.0 0.668661
\(849\) −14484.0 −0.585500
\(850\) 0 0
\(851\) 10332.0 0.416188
\(852\) −16272.0 −0.654307
\(853\) 33082.0 1.32791 0.663954 0.747773i \(-0.268877\pi\)
0.663954 + 0.747773i \(0.268877\pi\)
\(854\) −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 \(0.548040\pi\)
\(858\) −46128.0 −1.83541
\(859\) 8180.00 0.324910 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(860\) 0 0
\(861\) −5208.00 −0.206142
\(862\) −66408.0 −2.62397
\(863\) −10518.0 −0.414875 −0.207437 0.978248i \(-0.566512\pi\)
−0.207437 + 0.978248i \(0.566512\pi\)
\(864\) 6912.00 0.272166
\(865\) 0 0
\(866\) 30952.0 1.21454
\(867\) 6429.00 0.251834
\(868\) −2688.00 −0.105111
\(869\) 45880.0 1.79099
\(870\) 0 0
\(871\) −56048.0 −2.18038
\(872\) 0 0
\(873\) 11394.0 0.441728
\(874\) −16800.0 −0.650193
\(875\) 0 0
\(876\) 15408.0 0.594279
\(877\) −14134.0 −0.544209 −0.272104 0.962268i \(-0.587720\pi\)
−0.272104 + 0.962268i \(0.587720\pi\)
\(878\) 3360.00 0.129151
\(879\) 18456.0 0.708197
\(880\) 0 0
\(881\) 6492.00 0.248265 0.124132 0.992266i \(-0.460385\pi\)
0.124132 + 0.992266i \(0.460385\pi\)
\(882\) −1764.00 −0.0673435
\(883\) −38228.0 −1.45694 −0.728468 0.685080i \(-0.759767\pi\)
−0.728468 + 0.685080i \(0.759767\pi\)
\(884\) −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
\(890\) 0 0
\(891\) 5022.00 0.188825
\(892\) 14656.0 0.550134
\(893\) −32400.0 −1.21414
\(894\) −28440.0 −1.06396
\(895\) 0 0
\(896\) 0 0
\(897\) 7812.00 0.290786
\(898\) −12360.0 −0.459308
\(899\) 480.000 0.0178074
\(900\) 0 0
\(901\) 21672.0 0.801331
\(902\) 61504.0 2.27035
\(903\) −1428.00 −0.0526255
\(904\) 0 0
\(905\) 0 0
\(906\) 6816.00 0.249941
\(907\) 32236.0 1.18013 0.590065 0.807355i \(-0.299102\pi\)
0.590065 + 0.807355i \(0.299102\pi\)
\(908\) −39552.0 −1.44557
\(909\) 2088.00 0.0761877
\(910\) 0 0
\(911\) −46518.0 −1.69178 −0.845889 0.533359i \(-0.820930\pi\)
−0.845889 + 0.533359i \(0.820930\pi\)
\(912\) −19200.0 −0.697122
\(913\) −29016.0 −1.05180
\(914\) 23656.0 0.856095
\(915\) 0 0
\(916\) −43760.0 −1.57846
\(917\) 5684.00 0.204692
\(918\) 9072.00 0.326166
\(919\) 17840.0 0.640356 0.320178 0.947357i \(-0.396257\pi\)
0.320178 + 0.947357i \(0.396257\pi\)
\(920\) 0 0
\(921\) −17652.0 −0.631545
\(922\) 63872.0 2.28147
\(923\) −42036.0 −1.49906
\(924\) 10416.0 0.370846
\(925\) 0 0
\(926\) −4688.00 −0.166369
\(927\) 16128.0 0.571427
\(928\) −2560.00 −0.0905562
\(929\) 7000.00 0.247215 0.123607 0.992331i \(-0.460554\pi\)
0.123607 + 0.992331i \(0.460554\pi\)
\(930\) 0 0
\(931\) 4900.00 0.172493
\(932\) 22416.0 0.787833
\(933\) 27396.0 0.961313
\(934\) 21216.0 0.743264
\(935\) 0 0
\(936\) 0 0
\(937\) −36114.0 −1.25912 −0.629559 0.776953i \(-0.716764\pi\)
−0.629559 + 0.776953i \(0.716764\pi\)
\(938\) 25312.0 0.881094
\(939\) 28146.0 0.978179
\(940\) 0 0
\(941\) −4748.00 −0.164485 −0.0822425 0.996612i \(-0.526208\pi\)
−0.0822425 + 0.996612i \(0.526208\pi\)
\(942\) −3192.00 −0.110404
\(943\) −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 \(0.761650\pi\)
\(948\) 17760.0 0.608458
\(949\) 39804.0 1.36153
\(950\) 0 0
\(951\) −9342.00 −0.318544
\(952\) 0 0
\(953\) 16742.0 0.569073 0.284537 0.958665i \(-0.408160\pi\)
0.284537 + 0.958665i \(0.408160\pi\)
\(954\) 9288.00 0.315210
\(955\) 0 0
\(956\) −9360.00 −0.316657
\(957\) −1860.00 −0.0628268
\(958\) −22960.0 −0.774326
\(959\) −2898.00 −0.0975822
\(960\) 0 0
\(961\) −27487.0 −0.922661
\(962\) −61008.0 −2.04467
\(963\) 17154.0 0.574019
\(964\) −18704.0 −0.624912
\(965\) 0 0
\(966\) −3528.00 −0.117507
\(967\) 9956.00 0.331089 0.165545 0.986202i \(-0.447062\pi\)
0.165545 + 0.986202i \(0.447062\pi\)
\(968\) 0 0
\(969\) −25200.0 −0.835439
\(970\) 0 0
\(971\) −26388.0 −0.872123 −0.436061 0.899917i \(-0.643627\pi\)
−0.436061 + 0.899917i \(0.643627\pi\)
\(972\) 1944.00 0.0641500
\(973\) −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 \(0.495904\pi\)
\(978\) −3264.00 −0.106719
\(979\) 12400.0 0.404807
\(980\) 0 0
\(981\) −810.000 −0.0263622
\(982\) 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
\(985\) 0 0
\(986\) −3360.00 −0.108524
\(987\) −6804.00 −0.219426
\(988\) 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
\(993\) 4596.00 0.146878
\(994\) 18984.0 0.605771
\(995\) 0 0
\(996\) −11232.0 −0.357329
\(997\) 386.000 0.0122615 0.00613076 0.999981i \(-0.498049\pi\)
0.00613076 + 0.999981i \(0.498049\pi\)
\(998\) 79280.0 2.51459
\(999\) 6642.00 0.210354
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 525.4.a.b.1.1 1
3.2 odd 2 1575.4.a.k.1.1 1
5.2 odd 4 525.4.d.b.274.1 2
5.3 odd 4 525.4.d.b.274.2 2
5.4 even 2 21.4.a.b.1.1 1
15.14 odd 2 63.4.a.a.1.1 1
20.19 odd 2 336.4.a.h.1.1 1
35.4 even 6 147.4.e.c.79.1 2
35.9 even 6 147.4.e.c.67.1 2
35.19 odd 6 147.4.e.b.67.1 2
35.24 odd 6 147.4.e.b.79.1 2
35.34 odd 2 147.4.a.g.1.1 1
40.19 odd 2 1344.4.a.i.1.1 1
40.29 even 2 1344.4.a.w.1.1 1
60.59 even 2 1008.4.a.m.1.1 1
105.44 odd 6 441.4.e.m.361.1 2
105.59 even 6 441.4.e.n.226.1 2
105.74 odd 6 441.4.e.m.226.1 2
105.89 even 6 441.4.e.n.361.1 2
105.104 even 2 441.4.a.b.1.1 1
140.139 even 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 5.4 even 2
63.4.a.a.1.1 1 15.14 odd 2
147.4.a.g.1.1 1 35.34 odd 2
147.4.e.b.67.1 2 35.19 odd 6
147.4.e.b.79.1 2 35.24 odd 6
147.4.e.c.67.1 2 35.9 even 6
147.4.e.c.79.1 2 35.4 even 6
336.4.a.h.1.1 1 20.19 odd 2
441.4.a.b.1.1 1 105.104 even 2
441.4.e.m.226.1 2 105.74 odd 6
441.4.e.m.361.1 2 105.44 odd 6
441.4.e.n.226.1 2 105.59 even 6
441.4.e.n.361.1 2 105.89 even 6
525.4.a.b.1.1 1 1.1 even 1 trivial
525.4.d.b.274.1 2 5.2 odd 4
525.4.d.b.274.2 2 5.3 odd 4
1008.4.a.m.1.1 1 60.59 even 2
1344.4.a.i.1.1 1 40.19 odd 2
1344.4.a.w.1.1 1 40.29 even 2
1575.4.a.k.1.1 1 3.2 odd 2
2352.4.a.l.1.1 1 140.139 even 2