Properties

Label 21.4.a.b.1.1
Level $21$
Weight $4$
Character 21.1
Self dual yes
Analytic conductor $1.239$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 21.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -4.00000 q^{5} -12.0000 q^{6} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -4.00000 q^{5} -12.0000 q^{6} -7.00000 q^{7} +9.00000 q^{9} -16.0000 q^{10} +62.0000 q^{11} -24.0000 q^{12} -62.0000 q^{13} -28.0000 q^{14} +12.0000 q^{15} -64.0000 q^{16} +84.0000 q^{17} +36.0000 q^{18} +100.000 q^{19} -32.0000 q^{20} +21.0000 q^{21} +248.000 q^{22} -42.0000 q^{23} -109.000 q^{25} -248.000 q^{26} -27.0000 q^{27} -56.0000 q^{28} -10.0000 q^{29} +48.0000 q^{30} -48.0000 q^{31} -256.000 q^{32} -186.000 q^{33} +336.000 q^{34} +28.0000 q^{35} +72.0000 q^{36} -246.000 q^{37} +400.000 q^{38} +186.000 q^{39} -248.000 q^{41} +84.0000 q^{42} +68.0000 q^{43} +496.000 q^{44} -36.0000 q^{45} -168.000 q^{46} +324.000 q^{47} +192.000 q^{48} +49.0000 q^{49} -436.000 q^{50} -252.000 q^{51} -496.000 q^{52} +258.000 q^{53} -108.000 q^{54} -248.000 q^{55} -300.000 q^{57} -40.0000 q^{58} +120.000 q^{59} +96.0000 q^{60} +622.000 q^{61} -192.000 q^{62} -63.0000 q^{63} -512.000 q^{64} +248.000 q^{65} -744.000 q^{66} +904.000 q^{67} +672.000 q^{68} +126.000 q^{69} +112.000 q^{70} -678.000 q^{71} -642.000 q^{73} -984.000 q^{74} +327.000 q^{75} +800.000 q^{76} -434.000 q^{77} +744.000 q^{78} +740.000 q^{79} +256.000 q^{80} +81.0000 q^{81} -992.000 q^{82} +468.000 q^{83} +168.000 q^{84} -336.000 q^{85} +272.000 q^{86} +30.0000 q^{87} +200.000 q^{89} -144.000 q^{90} +434.000 q^{91} -336.000 q^{92} +144.000 q^{93} +1296.00 q^{94} -400.000 q^{95} +768.000 q^{96} -1266.00 q^{97} +196.000 q^{98} +558.000 q^{99} +O(q^{100})\)

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 \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.00000 1.00000
\(5\) −4.00000 −0.357771 −0.178885 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) −12.0000 −0.816497
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) −16.0000 −0.505964
\(11\) 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 \(-0.730026\pi\)
−0.661373 + 0.750057i \(0.730026\pi\)
\(14\) −28.0000 −0.534522
\(15\) 12.0000 0.206559
\(16\) −64.0000 −1.00000
\(17\) 84.0000 1.19841 0.599206 0.800595i \(-0.295483\pi\)
0.599206 + 0.800595i \(0.295483\pi\)
\(18\) 36.0000 0.471405
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) −32.0000 −0.357771
\(21\) 21.0000 0.218218
\(22\) 248.000 2.40335
\(23\) −42.0000 −0.380765 −0.190383 0.981710i \(-0.560973\pi\)
−0.190383 + 0.981710i \(0.560973\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) −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 \(0.510193\pi\)
\(30\) 48.0000 0.292119
\(31\) −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
\(35\) 28.0000 0.135225
\(36\) 72.0000 0.333333
\(37\) −246.000 −1.09303 −0.546516 0.837449i \(-0.684046\pi\)
−0.546516 + 0.837449i \(0.684046\pi\)
\(38\) 400.000 1.70759
\(39\) 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 \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) 496.000 1.69943
\(45\) −36.0000 −0.119257
\(46\) −168.000 −0.538484
\(47\) 324.000 1.00554 0.502769 0.864421i \(-0.332315\pi\)
0.502769 + 0.864421i \(0.332315\pi\)
\(48\) 192.000 0.577350
\(49\) 49.0000 0.142857
\(50\) −436.000 −1.23319
\(51\) −252.000 −0.691903
\(52\) −496.000 −1.32275
\(53\) 258.000 0.668661 0.334330 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) −108.000 −0.272166
\(55\) −248.000 −0.608006
\(56\) 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 \(0.457733\pi\)
\(60\) 96.0000 0.206559
\(61\) 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
\(65\) 248.000 0.473240
\(66\) −744.000 −1.38758
\(67\) 904.000 1.64838 0.824188 0.566316i \(-0.191632\pi\)
0.824188 + 0.566316i \(0.191632\pi\)
\(68\) 672.000 1.19841
\(69\) 126.000 0.219835
\(70\) 112.000 0.191237
\(71\) −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 \(-0.672082\pi\)
−0.514660 + 0.857394i \(0.672082\pi\)
\(74\) −984.000 −1.54578
\(75\) 327.000 0.503449
\(76\) 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 \(0.323339\pi\)
\(80\) 256.000 0.357771
\(81\) 81.0000 0.111111
\(82\) −992.000 −1.33595
\(83\) 468.000 0.618912 0.309456 0.950914i \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(84\) 168.000 0.218218
\(85\) −336.000 −0.428757
\(86\) 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 \(0.461999\pi\)
\(90\) −144.000 −0.168655
\(91\) 434.000 0.499951
\(92\) −336.000 −0.380765
\(93\) 144.000 0.160560
\(94\) 1296.00 1.42204
\(95\) −400.000 −0.431991
\(96\) 768.000 0.816497
\(97\) −1266.00 −1.32518 −0.662592 0.748981i \(-0.730544\pi\)
−0.662592 + 0.748981i \(0.730544\pi\)
\(98\) 196.000 0.202031
\(99\) 558.000 0.566476
\(100\) −872.000 −0.872000
\(101\) 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 \(-0.827761\pi\)
−0.857141 + 0.515082i \(0.827761\pi\)
\(104\) 0 0
\(105\) −84.0000 −0.0780720
\(106\) 1032.00 0.945629
\(107\) −1906.00 −1.72206 −0.861028 0.508558i \(-0.830179\pi\)
−0.861028 + 0.508558i \(0.830179\pi\)
\(108\) −216.000 −0.192450
\(109\) −90.0000 −0.0790866 −0.0395433 0.999218i \(-0.512590\pi\)
−0.0395433 + 0.999218i \(0.512590\pi\)
\(110\) −992.000 −0.859850
\(111\) 738.000 0.631062
\(112\) 448.000 0.377964
\(113\) 458.000 0.381283 0.190642 0.981660i \(-0.438943\pi\)
0.190642 + 0.981660i \(0.438943\pi\)
\(114\) −1200.00 −0.985880
\(115\) 168.000 0.136227
\(116\) −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
\(125\) 936.000 0.669747
\(126\) −252.000 −0.178174
\(127\) 804.000 0.561760 0.280880 0.959743i \(-0.409374\pi\)
0.280880 + 0.959743i \(0.409374\pi\)
\(128\) 0 0
\(129\) −204.000 −0.139234
\(130\) 992.000 0.669263
\(131\) 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
\(135\) 108.000 0.0688530
\(136\) 0 0
\(137\) 414.000 0.258178 0.129089 0.991633i \(-0.458795\pi\)
0.129089 + 0.991633i \(0.458795\pi\)
\(138\) 504.000 0.310894
\(139\) −1620.00 −0.988537 −0.494268 0.869309i \(-0.664564\pi\)
−0.494268 + 0.869309i \(0.664564\pi\)
\(140\) 224.000 0.135225
\(141\) −972.000 −0.580547
\(142\) −2712.00 −1.60272
\(143\) −3844.00 −2.24791
\(144\) −576.000 −0.333333
\(145\) 40.0000 0.0229091
\(146\) −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 \(0.274125\pi\)
\(150\) 1308.00 0.711985
\(151\) −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
\(155\) 192.000 0.0994956
\(156\) 1488.00 0.763688
\(157\) −266.000 −0.135217 −0.0676086 0.997712i \(-0.521537\pi\)
−0.0676086 + 0.997712i \(0.521537\pi\)
\(158\) 2960.00 1.49041
\(159\) −774.000 −0.386052
\(160\) 1024.00 0.505964
\(161\) 294.000 0.143916
\(162\) 324.000 0.157135
\(163\) −272.000 −0.130704 −0.0653518 0.997862i \(-0.520817\pi\)
−0.0653518 + 0.997862i \(0.520817\pi\)
\(164\) −1984.00 −0.944661
\(165\) 744.000 0.351032
\(166\) 1872.00 0.875273
\(167\) −1876.00 −0.869277 −0.434638 0.900605i \(-0.643124\pi\)
−0.434638 + 0.900605i \(0.643124\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) −1344.00 −0.606353
\(171\) 900.000 0.402484
\(172\) 544.000 0.241161
\(173\) −152.000 −0.0667997 −0.0333998 0.999442i \(-0.510633\pi\)
−0.0333998 + 0.999442i \(0.510633\pi\)
\(174\) 120.000 0.0522826
\(175\) 763.000 0.329585
\(176\) −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 \(0.459351\pi\)
\(180\) −288.000 −0.119257
\(181\) 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
\(185\) 984.000 0.391055
\(186\) 576.000 0.227067
\(187\) 5208.00 2.03661
\(188\) 2592.00 1.00554
\(189\) 189.000 0.0727393
\(190\) −1600.00 −0.610927
\(191\) −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 \(-0.661262\pi\)
−0.485223 + 0.874390i \(0.661262\pi\)
\(194\) −5064.00 −1.87409
\(195\) −744.000 −0.273225
\(196\) 392.000 0.142857
\(197\) 2354.00 0.851348 0.425674 0.904877i \(-0.360037\pi\)
0.425674 + 0.904877i \(0.360037\pi\)
\(198\) 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
\(205\) 992.000 0.337972
\(206\) −7168.00 −2.42436
\(207\) −378.000 −0.126922
\(208\) 3968.00 1.32275
\(209\) 6200.00 2.05198
\(210\) −336.000 −0.110410
\(211\) −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
\(215\) −272.000 −0.0862802
\(216\) 0 0
\(217\) 336.000 0.105111
\(218\) −360.000 −0.111845
\(219\) 1926.00 0.594279
\(220\) −1984.00 −0.608006
\(221\) −5208.00 −1.58519
\(222\) 2952.00 0.892456
\(223\) −1832.00 −0.550134 −0.275067 0.961425i \(-0.588700\pi\)
−0.275067 + 0.961425i \(0.588700\pi\)
\(224\) 1792.00 0.534522
\(225\) −981.000 −0.290667
\(226\) 1832.00 0.539216
\(227\) 4944.00 1.44557 0.722786 0.691072i \(-0.242861\pi\)
0.722786 + 0.691072i \(0.242861\pi\)
\(228\) −2400.00 −0.697122
\(229\) −5470.00 −1.57846 −0.789231 0.614096i \(-0.789521\pi\)
−0.789231 + 0.614096i \(0.789521\pi\)
\(230\) 672.000 0.192654
\(231\) 1302.00 0.370846
\(232\) 0 0
\(233\) −2802.00 −0.787833 −0.393917 0.919146i \(-0.628880\pi\)
−0.393917 + 0.919146i \(0.628880\pi\)
\(234\) −2232.00 −0.623549
\(235\) −1296.00 −0.359752
\(236\) 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 \(0.550610\pi\)
\(240\) −768.000 −0.206559
\(241\) −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
\(245\) −196.000 −0.0511101
\(246\) 2976.00 0.771312
\(247\) −6200.00 −1.59715
\(248\) 0 0
\(249\) −1404.00 −0.357329
\(250\) 3744.00 0.947165
\(251\) 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
\(255\) 1008.00 0.247543
\(256\) 4096.00 1.00000
\(257\) 7024.00 1.70484 0.852422 0.522854i \(-0.175133\pi\)
0.852422 + 0.522854i \(0.175133\pi\)
\(258\) −816.000 −0.196907
\(259\) 1722.00 0.413127
\(260\) 1984.00 0.473240
\(261\) −90.0000 −0.0213443
\(262\) 3248.00 0.765886
\(263\) 2438.00 0.571610 0.285805 0.958288i \(-0.407739\pi\)
0.285805 + 0.958288i \(0.407739\pi\)
\(264\) 0 0
\(265\) −1032.00 −0.239227
\(266\) −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 \(0.778933\pi\)
\(270\) 432.000 0.0973729
\(271\) −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
\(275\) −6758.00 −1.48190
\(276\) 1008.00 0.219835
\(277\) 5554.00 1.20472 0.602360 0.798224i \(-0.294227\pi\)
0.602360 + 0.798224i \(0.294227\pi\)
\(278\) −6480.00 −1.39800
\(279\) −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 \(-0.330733\pi\)
0.507058 + 0.861912i \(0.330733\pi\)
\(284\) −5424.00 −1.13329
\(285\) 1200.00 0.249410
\(286\) −15376.0 −3.17903
\(287\) 1736.00 0.357048
\(288\) −2304.00 −0.471405
\(289\) 2143.00 0.436190
\(290\) 160.000 0.0323984
\(291\) 3798.00 0.765095
\(292\) −5136.00 −1.02932
\(293\) −6152.00 −1.22663 −0.613317 0.789837i \(-0.710165\pi\)
−0.613317 + 0.789837i \(0.710165\pi\)
\(294\) −588.000 −0.116642
\(295\) −480.000 −0.0947345
\(296\) 0 0
\(297\) −1674.00 −0.327055
\(298\) 9480.00 1.84282
\(299\) 2604.00 0.503656
\(300\) 2616.00 0.503449
\(301\) −476.000 −0.0911501
\(302\) −2272.00 −0.432910
\(303\) −696.000 −0.131961
\(304\) −6400.00 −1.20745
\(305\) −2488.00 −0.467090
\(306\) 3024.00 0.564937
\(307\) 5884.00 1.09387 0.546934 0.837176i \(-0.315795\pi\)
0.546934 + 0.837176i \(0.315795\pi\)
\(308\) −3472.00 −0.642323
\(309\) 5376.00 0.989741
\(310\) 768.000 0.140708
\(311\) 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 \(-0.821670\pi\)
−0.847128 + 0.531389i \(0.821670\pi\)
\(314\) −1064.00 −0.191226
\(315\) 252.000 0.0450749
\(316\) 5920.00 1.05388
\(317\) 3114.00 0.551734 0.275867 0.961196i \(-0.411035\pi\)
0.275867 + 0.961196i \(0.411035\pi\)
\(318\) −3096.00 −0.545959
\(319\) −620.000 −0.108819
\(320\) 2048.00 0.357771
\(321\) 5718.00 0.994229
\(322\) 1176.00 0.203528
\(323\) 8400.00 1.44702
\(324\) 648.000 0.111111
\(325\) 6758.00 1.15344
\(326\) −1088.00 −0.184843
\(327\) 270.000 0.0456607
\(328\) 0 0
\(329\) −2268.00 −0.380057
\(330\) 2976.00 0.496435
\(331\) 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
\(335\) −3616.00 −0.589741
\(336\) −1344.00 −0.218218
\(337\) −4166.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(338\) 6588.00 1.06018
\(339\) −1374.00 −0.220134
\(340\) −2688.00 −0.428757
\(341\) −2976.00 −0.472608
\(342\) 3600.00 0.569198
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) −504.000 −0.0786506
\(346\) −608.000 −0.0944690
\(347\) −11366.0 −1.75838 −0.879191 0.476469i \(-0.841917\pi\)
−0.879191 + 0.476469i \(0.841917\pi\)
\(348\) 240.000 0.0369694
\(349\) 9310.00 1.42795 0.713973 0.700174i \(-0.246894\pi\)
0.713973 + 0.700174i \(0.246894\pi\)
\(350\) 3052.00 0.466104
\(351\) 1674.00 0.254563
\(352\) −15872.0 −2.40335
\(353\) −8572.00 −1.29247 −0.646234 0.763139i \(-0.723657\pi\)
−0.646234 + 0.763139i \(0.723657\pi\)
\(354\) −1440.00 −0.216201
\(355\) 2712.00 0.405459
\(356\) 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
\(365\) 2568.00 0.368261
\(366\) −7464.00 −1.06598
\(367\) 5424.00 0.771473 0.385736 0.922609i \(-0.373947\pi\)
0.385736 + 0.922609i \(0.373947\pi\)
\(368\) 2688.00 0.380765
\(369\) −2232.00 −0.314887
\(370\) 3936.00 0.553035
\(371\) −1806.00 −0.252730
\(372\) 1152.00 0.160560
\(373\) 1838.00 0.255142 0.127571 0.991829i \(-0.459282\pi\)
0.127571 + 0.991829i \(0.459282\pi\)
\(374\) 20832.0 2.88021
\(375\) −2808.00 −0.386679
\(376\) 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 \(0.593217\pi\)
\(380\) −3200.00 −0.431991
\(381\) −2412.00 −0.324332
\(382\) −8152.00 −1.09187
\(383\) 9048.00 1.20713 0.603566 0.797313i \(-0.293746\pi\)
0.603566 + 0.797313i \(0.293746\pi\)
\(384\) 0 0
\(385\) 1736.00 0.229805
\(386\) −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 \(0.769369\pi\)
\(390\) −2976.00 −0.386399
\(391\) −3528.00 −0.456314
\(392\) 0 0
\(393\) −2436.00 −0.312672
\(394\) 9416.00 1.20399
\(395\) −2960.00 −0.377048
\(396\) 4464.00 0.566476
\(397\) −1866.00 −0.235899 −0.117949 0.993020i \(-0.537632\pi\)
−0.117949 + 0.993020i \(0.537632\pi\)
\(398\) 6720.00 0.846340
\(399\) 2100.00 0.263487
\(400\) 6976.00 0.872000
\(401\) 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
\(405\) −324.000 −0.0397523
\(406\) 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
\(411\) −1242.00 −0.149059
\(412\) −14336.0 −1.71428
\(413\) −840.000 −0.100082
\(414\) −1512.00 −0.179495
\(415\) −1872.00 −0.221429
\(416\) 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 \(0.367010\pi\)
\(420\) −672.000 −0.0780720
\(421\) 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
\(425\) −9156.00 −1.04501
\(426\) 8136.00 0.925330
\(427\) −4354.00 −0.493454
\(428\) −15248.0 −1.72206
\(429\) 11532.0 1.29783
\(430\) −1088.00 −0.122019
\(431\) 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 \(-0.358723\pi\)
0.429405 + 0.903112i \(0.358723\pi\)
\(434\) 1344.00 0.148650
\(435\) −120.000 −0.0132266
\(436\) −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 \(-0.384519\pi\)
0.354888 + 0.934909i \(0.384519\pi\)
\(444\) 5904.00 0.631062
\(445\) −800.000 −0.0852217
\(446\) −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 \(0.448080\pi\)
\(450\) −3924.00 −0.411065
\(451\) −15376.0 −1.60538
\(452\) 3664.00 0.381283
\(453\) 1704.00 0.176735
\(454\) 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
\(459\) −2268.00 −0.230634
\(460\) 1344.00 0.136227
\(461\) −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 \(-0.518734\pi\)
−0.0588202 + 0.998269i \(0.518734\pi\)
\(464\) 640.000 0.0640329
\(465\) −576.000 −0.0574438
\(466\) −11208.0 −1.11416
\(467\) 5304.00 0.525567 0.262784 0.964855i \(-0.415359\pi\)
0.262784 + 0.964855i \(0.415359\pi\)
\(468\) −4464.00 −0.440916
\(469\) −6328.00 −0.623027
\(470\) −5184.00 −0.508766
\(471\) 798.000 0.0780677
\(472\) 0 0
\(473\) 4216.00 0.409835
\(474\) −8880.00 −0.860489
\(475\) −10900.0 −1.05290
\(476\) −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 \(0.411731\pi\)
\(480\) −3072.00 −0.292119
\(481\) 15252.0 1.44580
\(482\) −9352.00 −0.883759
\(483\) −882.000 −0.0830898
\(484\) 20104.0 1.88805
\(485\) 5064.00 0.474112
\(486\) −972.000 −0.0907218
\(487\) 8944.00 0.832220 0.416110 0.909314i \(-0.363393\pi\)
0.416110 + 0.909314i \(0.363393\pi\)
\(488\) 0 0
\(489\) 816.000 0.0754617
\(490\) −784.000 −0.0722806
\(491\) −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
\(495\) −2232.00 −0.202669
\(496\) 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 \(0.848629\pi\)
\(500\) 7488.00 0.669747
\(501\) 5628.00 0.501877
\(502\) 11168.0 0.992933
\(503\) 1848.00 0.163814 0.0819068 0.996640i \(-0.473899\pi\)
0.0819068 + 0.996640i \(0.473899\pi\)
\(504\) 0 0
\(505\) −928.000 −0.0817732
\(506\) −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 \(0.495288\pi\)
\(510\) 4032.00 0.350078
\(511\) 4494.00 0.389047
\(512\) 16384.0 1.41421
\(513\) −2700.00 −0.232374
\(514\) 28096.0 2.41101
\(515\) 7168.00 0.613320
\(516\) −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 \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(524\) 6496.00 0.541563
\(525\) −2289.00 −0.190286
\(526\) 9752.00 0.808379
\(527\) −4032.00 −0.333276
\(528\) 11904.0 0.981165
\(529\) −10403.0 −0.855018
\(530\) −4128.00 −0.338319
\(531\) 1080.00 0.0882637
\(532\) −5600.00 −0.456374
\(533\) 15376.0 1.24955
\(534\) −2400.00 −0.194491
\(535\) 7624.00 0.616101
\(536\) 0 0
\(537\) −1830.00 −0.147058
\(538\) −27120.0 −2.17328
\(539\) 3038.00 0.242775
\(540\) 864.000 0.0688530
\(541\) −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
\(545\) 360.000 0.0282949
\(546\) −5208.00 −0.408208
\(547\) −3416.00 −0.267016 −0.133508 0.991048i \(-0.542624\pi\)
−0.133508 + 0.991048i \(0.542624\pi\)
\(548\) 3312.00 0.258178
\(549\) 5598.00 0.435185
\(550\) −27032.0 −2.09572
\(551\) −1000.00 −0.0773166
\(552\) 0 0
\(553\) −5180.00 −0.398329
\(554\) 22216.0 1.70373
\(555\) −2952.00 −0.225776
\(556\) −12960.0 −0.988537
\(557\) −526.000 −0.0400132 −0.0200066 0.999800i \(-0.506369\pi\)
−0.0200066 + 0.999800i \(0.506369\pi\)
\(558\) −1728.00 −0.131097
\(559\) −4216.00 −0.318994
\(560\) −1792.00 −0.135225
\(561\) −15624.0 −1.17584
\(562\) 7768.00 0.583049
\(563\) −6712.00 −0.502446 −0.251223 0.967929i \(-0.580833\pi\)
−0.251223 + 0.967929i \(0.580833\pi\)
\(564\) −7776.00 −0.580547
\(565\) −1832.00 −0.136412
\(566\) 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 \(0.450671\pi\)
\(570\) 4800.00 0.352719
\(571\) 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
\(575\) 4578.00 0.332027
\(576\) −4608.00 −0.333333
\(577\) 5434.00 0.392063 0.196032 0.980598i \(-0.437195\pi\)
0.196032 + 0.980598i \(0.437195\pi\)
\(578\) 8572.00 0.616865
\(579\) 7806.00 0.560287
\(580\) 320.000 0.0229091
\(581\) −3276.00 −0.233927
\(582\) 15192.0 1.08201
\(583\) 15996.0 1.13634
\(584\) 0 0
\(585\) 2232.00 0.157747
\(586\) −24608.0 −1.73472
\(587\) 464.000 0.0326258 0.0163129 0.999867i \(-0.494807\pi\)
0.0163129 + 0.999867i \(0.494807\pi\)
\(588\) −1176.00 −0.0824786
\(589\) −4800.00 −0.335790
\(590\) −1920.00 −0.133975
\(591\) −7062.00 −0.491526
\(592\) 15744.0 1.09303
\(593\) 11748.0 0.813546 0.406773 0.913529i \(-0.366654\pi\)
0.406773 + 0.913529i \(0.366654\pi\)
\(594\) −6696.00 −0.462526
\(595\) 2352.00 0.162055
\(596\) 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
\(605\) −10052.0 −0.675491
\(606\) −2784.00 −0.186621
\(607\) 704.000 0.0470749 0.0235375 0.999723i \(-0.492507\pi\)
0.0235375 + 0.999723i \(0.492507\pi\)
\(608\) −25600.0 −1.70759
\(609\) −210.000 −0.0139731
\(610\) −9952.00 −0.660565
\(611\) −20088.0 −1.33007
\(612\) 6048.00 0.399470
\(613\) 24958.0 1.64444 0.822222 0.569167i \(-0.192734\pi\)
0.822222 + 0.569167i \(0.192734\pi\)
\(614\) 23536.0 1.54696
\(615\) −2976.00 −0.195128
\(616\) 0 0
\(617\) −8826.00 −0.575886 −0.287943 0.957648i \(-0.592971\pi\)
−0.287943 + 0.957648i \(0.592971\pi\)
\(618\) 21504.0 1.39971
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 1536.00 0.0994956
\(621\) 1134.00 0.0732783
\(622\) 36528.0 2.35473
\(623\) −1400.00 −0.0900318
\(624\) −11904.0 −0.763688
\(625\) 9881.00 0.632384
\(626\) −37528.0 −2.39604
\(627\) −18600.0 −1.18471
\(628\) −2128.00 −0.135217
\(629\) −20664.0 −1.30990
\(630\) 1008.00 0.0637455
\(631\) −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
\(635\) −3216.00 −0.200981
\(636\) −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 \(-0.828919\pi\)
−0.859009 + 0.511961i \(0.828919\pi\)
\(644\) 2352.00 0.143916
\(645\) 816.000 0.0498139
\(646\) 33600.0 2.04640
\(647\) 3844.00 0.233575 0.116788 0.993157i \(-0.462740\pi\)
0.116788 + 0.993157i \(0.462740\pi\)
\(648\) 0 0
\(649\) 7440.00 0.449993
\(650\) 27032.0 1.63120
\(651\) −1008.00 −0.0606861
\(652\) −2176.00 −0.130704
\(653\) −28482.0 −1.70687 −0.853436 0.521198i \(-0.825485\pi\)
−0.853436 + 0.521198i \(0.825485\pi\)
\(654\) 1080.00 0.0645739
\(655\) −3248.00 −0.193756
\(656\) 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 \(0.588928\pi\)
\(660\) 5952.00 0.351032
\(661\) 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
\(665\) 2800.00 0.163277
\(666\) −8856.00 −0.515260
\(667\) 420.000 0.0243815
\(668\) −15008.0 −0.869277
\(669\) 5496.00 0.317620
\(670\) −14464.0 −0.834020
\(671\) 38564.0 2.21870
\(672\) −5376.00 −0.308607
\(673\) −10562.0 −0.604956 −0.302478 0.953156i \(-0.597814\pi\)
−0.302478 + 0.953156i \(0.597814\pi\)
\(674\) −16664.0 −0.952334
\(675\) 2943.00 0.167816
\(676\) 13176.0 0.749659
\(677\) −26016.0 −1.47692 −0.738461 0.674296i \(-0.764447\pi\)
−0.738461 + 0.674296i \(0.764447\pi\)
\(678\) −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 \(-0.419817\pi\)
0.249248 + 0.968440i \(0.419817\pi\)
\(684\) 7200.00 0.402484
\(685\) −1656.00 −0.0923686
\(686\) −1372.00 −0.0763604
\(687\) 16410.0 0.911325
\(688\) −4352.00 −0.241161
\(689\) −15996.0 −0.884469
\(690\) −2016.00 −0.111229
\(691\) 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
\(695\) 6480.00 0.353670
\(696\) 0 0
\(697\) −20832.0 −1.13209
\(698\) 37240.0 2.01942
\(699\) 8406.00 0.454856
\(700\) 6104.00 0.329585
\(701\) −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
\(705\) 3888.00 0.207703
\(706\) −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 \(0.569081\pi\)
\(710\) 10848.0 0.573406
\(711\) 6660.00 0.351293
\(712\) 0 0
\(713\) 2016.00 0.105890
\(714\) 7056.00 0.369838
\(715\) 15376.0 0.804237
\(716\) 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 \(0.756784\pi\)
\(720\) 2304.00 0.119257
\(721\) 12544.0 0.647938
\(722\) 12564.0 0.647623
\(723\) 7014.00 0.360793
\(724\) 8336.00 0.427907
\(725\) 1090.00 0.0558367
\(726\) −30156.0 −1.54159
\(727\) 14624.0 0.746044 0.373022 0.927822i \(-0.378322\pi\)
0.373022 + 0.927822i \(0.378322\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 10272.0 0.520800
\(731\) 5712.00 0.289010
\(732\) −14928.0 −0.753763
\(733\) −20862.0 −1.05124 −0.525618 0.850721i \(-0.676166\pi\)
−0.525618 + 0.850721i \(0.676166\pi\)
\(734\) 21696.0 1.09103
\(735\) 588.000 0.0295084
\(736\) 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 \(0.612614\pi\)
\(740\) 7872.00 0.391055
\(741\) 18600.0 0.922116
\(742\) −7224.00 −0.357414
\(743\) 25578.0 1.26294 0.631471 0.775400i \(-0.282452\pi\)
0.631471 + 0.775400i \(0.282452\pi\)
\(744\) 0 0
\(745\) −9480.00 −0.466202
\(746\) 7352.00 0.360826
\(747\) 4212.00 0.206304
\(748\) 41664.0 2.03661
\(749\) 13342.0 0.650876
\(750\) −11232.0 −0.546846
\(751\) 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
\(755\) 2272.00 0.109519
\(756\) 1512.00 0.0727393
\(757\) 25934.0 1.24516 0.622581 0.782556i \(-0.286084\pi\)
0.622581 + 0.782556i \(0.286084\pi\)
\(758\) −17040.0 −0.816518
\(759\) 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
\(765\) −3024.00 −0.142919
\(766\) 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 \(0.314695\pi\)
\(770\) 6944.00 0.324993
\(771\) −21072.0 −0.984293
\(772\) −20816.0 −0.970446
\(773\) 39568.0 1.84109 0.920545 0.390637i \(-0.127745\pi\)
0.920545 + 0.390637i \(0.127745\pi\)
\(774\) 2448.00 0.113684
\(775\) 5232.00 0.242502
\(776\) 0 0
\(777\) −5166.00 −0.238519
\(778\) −45960.0 −2.11793
\(779\) −24800.0 −1.14063
\(780\) −5952.00 −0.273225
\(781\) −42036.0 −1.92595
\(782\) −14112.0 −0.645325
\(783\) 270.000 0.0123231
\(784\) −3136.00 −0.142857
\(785\) 1064.00 0.0483768
\(786\) −9744.00 −0.442184
\(787\) −12356.0 −0.559649 −0.279825 0.960051i \(-0.590276\pi\)
−0.279825 + 0.960051i \(0.590276\pi\)
\(788\) 18832.0 0.851348
\(789\) −7314.00 −0.330019
\(790\) −11840.0 −0.533226
\(791\) −3206.00 −0.144112
\(792\) 0 0
\(793\) −38564.0 −1.72692
\(794\) −7464.00 −0.333611
\(795\) 3096.00 0.138118
\(796\) 13440.0 0.598452
\(797\) −21736.0 −0.966033 −0.483017 0.875611i \(-0.660459\pi\)
−0.483017 + 0.875611i \(0.660459\pi\)
\(798\) 8400.00 0.372628
\(799\) 27216.0 1.20505
\(800\) 27904.0 1.23319
\(801\) 1800.00 0.0794006
\(802\) 54648.0 2.40609
\(803\) −39804.0 −1.74926
\(804\) −21696.0 −0.951690
\(805\) −1176.00 −0.0514889
\(806\) 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 \(0.813064\pi\)
\(810\) −1296.00 −0.0562183
\(811\) 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
\(815\) 1088.00 0.0467619
\(816\) 16128.0 0.691903
\(817\) 6800.00 0.291190
\(818\) −52840.0 −2.25857
\(819\) 3906.00 0.166650
\(820\) 7936.00 0.337972
\(821\) 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 \(-0.524372\pi\)
−0.0764923 + 0.997070i \(0.524372\pi\)
\(824\) 0 0
\(825\) 20274.0 0.855576
\(826\) −3360.00 −0.141537
\(827\) −27666.0 −1.16329 −0.581645 0.813443i \(-0.697591\pi\)
−0.581645 + 0.813443i \(0.697591\pi\)
\(828\) −3024.00 −0.126922
\(829\) 12890.0 0.540034 0.270017 0.962856i \(-0.412971\pi\)
0.270017 + 0.962856i \(0.412971\pi\)
\(830\) −7488.00 −0.313147
\(831\) −16662.0 −0.695546
\(832\) 31744.0 1.32275
\(833\) 4116.00 0.171202
\(834\) 19440.0 0.807137
\(835\) 7504.00 0.311002
\(836\) 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
\(845\) −6588.00 −0.268206
\(846\) 11664.0 0.474015
\(847\) −17591.0 −0.713617
\(848\) −16512.0 −0.668661
\(849\) −14484.0 −0.585500
\(850\) −36624.0 −1.47787
\(851\) 10332.0 0.416188
\(852\) 16272.0 0.654307
\(853\) −33082.0 −1.32791 −0.663954 0.747773i \(-0.731123\pi\)
−0.663954 + 0.747773i \(0.731123\pi\)
\(854\) −17416.0 −0.697849
\(855\) −3600.00 −0.143997
\(856\) 0 0
\(857\) 7544.00 0.300698 0.150349 0.988633i \(-0.451960\pi\)
0.150349 + 0.988633i \(0.451960\pi\)
\(858\) 46128.0 1.83541
\(859\) 8180.00 0.324910 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(860\) −2176.00 −0.0862802
\(861\) −5208.00 −0.206142
\(862\) 66408.0 2.62397
\(863\) 10518.0 0.414875 0.207437 0.978248i \(-0.433488\pi\)
0.207437 + 0.978248i \(0.433488\pi\)
\(864\) 6912.00 0.272166
\(865\) 608.000 0.0238990
\(866\) 30952.0 1.21454
\(867\) −6429.00 −0.251834
\(868\) 2688.00 0.105111
\(869\) 45880.0 1.79099
\(870\) −480.000 −0.0187052
\(871\) −56048.0 −2.18038
\(872\) 0 0
\(873\) −11394.0 −0.441728
\(874\) −16800.0 −0.650193
\(875\) −6552.00 −0.253141
\(876\) 15408.0 0.594279
\(877\) 14134.0 0.544209 0.272104 0.962268i \(-0.412280\pi\)
0.272104 + 0.962268i \(0.412280\pi\)
\(878\) −3360.00 −0.129151
\(879\) 18456.0 0.708197
\(880\) 15872.0 0.608006
\(881\) 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 \(-0.240233\pi\)
0.728468 + 0.685080i \(0.240233\pi\)
\(884\) −41664.0 −1.58519
\(885\) 1440.00 0.0546950
\(886\) 26472.0 1.00377
\(887\) −43076.0 −1.63061 −0.815305 0.579032i \(-0.803431\pi\)
−0.815305 + 0.579032i \(0.803431\pi\)
\(888\) 0 0
\(889\) −5628.00 −0.212325
\(890\) −3200.00 −0.120522
\(891\) 5022.00 0.188825
\(892\) −14656.0 −0.550134
\(893\) 32400.0 1.21414
\(894\) −28440.0 −1.06396
\(895\) −2440.00 −0.0911287
\(896\) 0 0
\(897\) −7812.00 −0.290786
\(898\) 12360.0 0.459308
\(899\) 480.000 0.0178074
\(900\) −7848.00 −0.290667
\(901\) 21672.0 0.801331
\(902\) −61504.0 −2.27035
\(903\) 1428.00 0.0526255
\(904\) 0 0
\(905\) −4168.00 −0.153093
\(906\) 6816.00 0.249941
\(907\) −32236.0 −1.18013 −0.590065 0.807355i \(-0.700898\pi\)
−0.590065 + 0.807355i \(0.700898\pi\)
\(908\) 39552.0 1.44557
\(909\) 2088.00 0.0761877
\(910\) −6944.00 −0.252958
\(911\) −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
\(915\) 7464.00 0.269675
\(916\) −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
\(925\) 26814.0 0.953123
\(926\) −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 \(0.460554\pi\)
\(930\) −2304.00 −0.0812378
\(931\) 4900.00 0.172493
\(932\) −22416.0 −0.787833
\(933\) −27396.0 −0.961313
\(934\) 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
\(939\) 28146.0 0.978179
\(940\) −10368.0 −0.359752
\(941\) −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
\(945\) −756.000 −0.0260240
\(946\) 16864.0 0.579594
\(947\) 42694.0 1.46501 0.732507 0.680759i \(-0.238350\pi\)
0.732507 + 0.680759i \(0.238350\pi\)
\(948\) −17760.0 −0.608458
\(949\) 39804.0 1.36153
\(950\) −43600.0 −1.48902
\(951\) −9342.00 −0.318544
\(952\) 0 0
\(953\) −16742.0 −0.569073 −0.284537 0.958665i \(-0.591840\pi\)
−0.284537 + 0.958665i \(0.591840\pi\)
\(954\) 9288.00 0.315210
\(955\) 8152.00 0.276223
\(956\) −9360.00 −0.316657
\(957\) 1860.00 0.0628268
\(958\) 22960.0 0.774326
\(959\) −2898.00 −0.0975822
\(960\) −6144.00 −0.206559
\(961\) −27487.0 −0.922661
\(962\) 61008.0 2.04467
\(963\) −17154.0 −0.574019
\(964\) −18704.0 −0.624912
\(965\) 10408.0 0.347197
\(966\) −3528.00 −0.117507
\(967\) −9956.00 −0.331089 −0.165545 0.986202i \(-0.552938\pi\)
−0.165545 + 0.986202i \(0.552938\pi\)
\(968\) 0 0
\(969\) −25200.0 −0.835439
\(970\) 20256.0 0.670496
\(971\) −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
\(975\) −20274.0 −0.665936
\(976\) −39808.0 −1.30556
\(977\) −786.000 −0.0257383 −0.0128692 0.999917i \(-0.504096\pi\)
−0.0128692 + 0.999917i \(0.504096\pi\)
\(978\) 3264.00 0.106719
\(979\) 12400.0 0.404807
\(980\) −1568.00 −0.0511101
\(981\) −810.000 −0.0263622
\(982\) −22232.0 −0.722456
\(983\) 51888.0 1.68359 0.841796 0.539796i \(-0.181499\pi\)
0.841796 + 0.539796i \(0.181499\pi\)
\(984\) 0 0
\(985\) −9416.00 −0.304588
\(986\) −3360.00 −0.108524
\(987\) 6804.00 0.219426
\(988\) −49600.0 −1.59715
\(989\) −2856.00 −0.0918256
\(990\) −8928.00 −0.286617
\(991\) −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
\(995\) −6720.00 −0.214109
\(996\) −11232.0 −0.357329
\(997\) −386.000 −0.0122615 −0.00613076 0.999981i \(-0.501951\pi\)
−0.00613076 + 0.999981i \(0.501951\pi\)
\(998\) −79280.0 −2.51459
\(999\) 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 21.4.a.b.1.1 1
3.2 odd 2 63.4.a.a.1.1 1
4.3 odd 2 336.4.a.h.1.1 1
5.2 odd 4 525.4.d.b.274.2 2
5.3 odd 4 525.4.d.b.274.1 2
5.4 even 2 525.4.a.b.1.1 1
7.2 even 3 147.4.e.c.67.1 2
7.3 odd 6 147.4.e.b.79.1 2
7.4 even 3 147.4.e.c.79.1 2
7.5 odd 6 147.4.e.b.67.1 2
7.6 odd 2 147.4.a.g.1.1 1
8.3 odd 2 1344.4.a.i.1.1 1
8.5 even 2 1344.4.a.w.1.1 1
12.11 even 2 1008.4.a.m.1.1 1
15.14 odd 2 1575.4.a.k.1.1 1
21.2 odd 6 441.4.e.m.361.1 2
21.5 even 6 441.4.e.n.361.1 2
21.11 odd 6 441.4.e.m.226.1 2
21.17 even 6 441.4.e.n.226.1 2
21.20 even 2 441.4.a.b.1.1 1
28.27 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 1.1 even 1 trivial
63.4.a.a.1.1 1 3.2 odd 2
147.4.a.g.1.1 1 7.6 odd 2
147.4.e.b.67.1 2 7.5 odd 6
147.4.e.b.79.1 2 7.3 odd 6
147.4.e.c.67.1 2 7.2 even 3
147.4.e.c.79.1 2 7.4 even 3
336.4.a.h.1.1 1 4.3 odd 2
441.4.a.b.1.1 1 21.20 even 2
441.4.e.m.226.1 2 21.11 odd 6
441.4.e.m.361.1 2 21.2 odd 6
441.4.e.n.226.1 2 21.17 even 6
441.4.e.n.361.1 2 21.5 even 6
525.4.a.b.1.1 1 5.4 even 2
525.4.d.b.274.1 2 5.3 odd 4
525.4.d.b.274.2 2 5.2 odd 4
1008.4.a.m.1.1 1 12.11 even 2
1344.4.a.i.1.1 1 8.3 odd 2
1344.4.a.w.1.1 1 8.5 even 2
1575.4.a.k.1.1 1 15.14 odd 2
2352.4.a.l.1.1 1 28.27 even 2