Properties

Label 14.4.a.a.1.1
Level $14$
Weight $4$
Character 14.1
Self dual yes
Analytic conductor $0.826$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.826026740080\)
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 \(\chi\) \(=\) 14.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} -16.0000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +37.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +8.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} -16.0000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +37.0000 q^{9} +28.0000 q^{10} -28.0000 q^{11} +32.0000 q^{12} +18.0000 q^{13} +14.0000 q^{14} -112.000 q^{15} +16.0000 q^{16} +74.0000 q^{17} -74.0000 q^{18} +80.0000 q^{19} -56.0000 q^{20} -56.0000 q^{21} +56.0000 q^{22} -112.000 q^{23} -64.0000 q^{24} +71.0000 q^{25} -36.0000 q^{26} +80.0000 q^{27} -28.0000 q^{28} +190.000 q^{29} +224.000 q^{30} +72.0000 q^{31} -32.0000 q^{32} -224.000 q^{33} -148.000 q^{34} +98.0000 q^{35} +148.000 q^{36} -346.000 q^{37} -160.000 q^{38} +144.000 q^{39} +112.000 q^{40} +162.000 q^{41} +112.000 q^{42} -412.000 q^{43} -112.000 q^{44} -518.000 q^{45} +224.000 q^{46} +24.0000 q^{47} +128.000 q^{48} +49.0000 q^{49} -142.000 q^{50} +592.000 q^{51} +72.0000 q^{52} +318.000 q^{53} -160.000 q^{54} +392.000 q^{55} +56.0000 q^{56} +640.000 q^{57} -380.000 q^{58} -200.000 q^{59} -448.000 q^{60} -198.000 q^{61} -144.000 q^{62} -259.000 q^{63} +64.0000 q^{64} -252.000 q^{65} +448.000 q^{66} -716.000 q^{67} +296.000 q^{68} -896.000 q^{69} -196.000 q^{70} +392.000 q^{71} -296.000 q^{72} +538.000 q^{73} +692.000 q^{74} +568.000 q^{75} +320.000 q^{76} +196.000 q^{77} -288.000 q^{78} +240.000 q^{79} -224.000 q^{80} -359.000 q^{81} -324.000 q^{82} -1072.00 q^{83} -224.000 q^{84} -1036.00 q^{85} +824.000 q^{86} +1520.00 q^{87} +224.000 q^{88} +810.000 q^{89} +1036.00 q^{90} -126.000 q^{91} -448.000 q^{92} +576.000 q^{93} -48.0000 q^{94} -1120.00 q^{95} -256.000 q^{96} +1354.00 q^{97} -98.0000 q^{98} -1036.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 8.00000 1.53960 0.769800 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 4.00000 0.500000
\(5\) −14.0000 −1.25220 −0.626099 0.779744i \(-0.715349\pi\)
−0.626099 + 0.779744i \(0.715349\pi\)
\(6\) −16.0000 −1.08866
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 37.0000 1.37037
\(10\) 28.0000 0.885438
\(11\) −28.0000 −0.767483 −0.383742 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(12\) 32.0000 0.769800
\(13\) 18.0000 0.384023 0.192012 0.981393i \(-0.438499\pi\)
0.192012 + 0.981393i \(0.438499\pi\)
\(14\) 14.0000 0.267261
\(15\) −112.000 −1.92789
\(16\) 16.0000 0.250000
\(17\) 74.0000 1.05574 0.527872 0.849324i \(-0.322990\pi\)
0.527872 + 0.849324i \(0.322990\pi\)
\(18\) −74.0000 −0.968998
\(19\) 80.0000 0.965961 0.482980 0.875631i \(-0.339554\pi\)
0.482980 + 0.875631i \(0.339554\pi\)
\(20\) −56.0000 −0.626099
\(21\) −56.0000 −0.581914
\(22\) 56.0000 0.542693
\(23\) −112.000 −1.01537 −0.507687 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) −64.0000 −0.544331
\(25\) 71.0000 0.568000
\(26\) −36.0000 −0.271545
\(27\) 80.0000 0.570222
\(28\) −28.0000 −0.188982
\(29\) 190.000 1.21662 0.608312 0.793698i \(-0.291847\pi\)
0.608312 + 0.793698i \(0.291847\pi\)
\(30\) 224.000 1.36322
\(31\) 72.0000 0.417148 0.208574 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(32\) −32.0000 −0.176777
\(33\) −224.000 −1.18162
\(34\) −148.000 −0.746523
\(35\) 98.0000 0.473286
\(36\) 148.000 0.685185
\(37\) −346.000 −1.53735 −0.768676 0.639638i \(-0.779084\pi\)
−0.768676 + 0.639638i \(0.779084\pi\)
\(38\) −160.000 −0.683038
\(39\) 144.000 0.591242
\(40\) 112.000 0.442719
\(41\) 162.000 0.617077 0.308538 0.951212i \(-0.400160\pi\)
0.308538 + 0.951212i \(0.400160\pi\)
\(42\) 112.000 0.411476
\(43\) −412.000 −1.46115 −0.730575 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) −112.000 −0.383742
\(45\) −518.000 −1.71598
\(46\) 224.000 0.717978
\(47\) 24.0000 0.0744843 0.0372421 0.999306i \(-0.488143\pi\)
0.0372421 + 0.999306i \(0.488143\pi\)
\(48\) 128.000 0.384900
\(49\) 49.0000 0.142857
\(50\) −142.000 −0.401637
\(51\) 592.000 1.62542
\(52\) 72.0000 0.192012
\(53\) 318.000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −160.000 −0.403208
\(55\) 392.000 0.961041
\(56\) 56.0000 0.133631
\(57\) 640.000 1.48719
\(58\) −380.000 −0.860284
\(59\) −200.000 −0.441318 −0.220659 0.975351i \(-0.570821\pi\)
−0.220659 + 0.975351i \(0.570821\pi\)
\(60\) −448.000 −0.963943
\(61\) −198.000 −0.415595 −0.207798 0.978172i \(-0.566630\pi\)
−0.207798 + 0.978172i \(0.566630\pi\)
\(62\) −144.000 −0.294968
\(63\) −259.000 −0.517951
\(64\) 64.0000 0.125000
\(65\) −252.000 −0.480873
\(66\) 448.000 0.835530
\(67\) −716.000 −1.30557 −0.652786 0.757542i \(-0.726400\pi\)
−0.652786 + 0.757542i \(0.726400\pi\)
\(68\) 296.000 0.527872
\(69\) −896.000 −1.56327
\(70\) −196.000 −0.334664
\(71\) 392.000 0.655237 0.327619 0.944810i \(-0.393754\pi\)
0.327619 + 0.944810i \(0.393754\pi\)
\(72\) −296.000 −0.484499
\(73\) 538.000 0.862577 0.431289 0.902214i \(-0.358059\pi\)
0.431289 + 0.902214i \(0.358059\pi\)
\(74\) 692.000 1.08707
\(75\) 568.000 0.874493
\(76\) 320.000 0.482980
\(77\) 196.000 0.290081
\(78\) −288.000 −0.418072
\(79\) 240.000 0.341799 0.170899 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) −224.000 −0.313050
\(81\) −359.000 −0.492455
\(82\) −324.000 −0.436339
\(83\) −1072.00 −1.41768 −0.708839 0.705370i \(-0.750781\pi\)
−0.708839 + 0.705370i \(0.750781\pi\)
\(84\) −224.000 −0.290957
\(85\) −1036.00 −1.32200
\(86\) 824.000 1.03319
\(87\) 1520.00 1.87312
\(88\) 224.000 0.271346
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 1036.00 1.21338
\(91\) −126.000 −0.145147
\(92\) −448.000 −0.507687
\(93\) 576.000 0.642241
\(94\) −48.0000 −0.0526683
\(95\) −1120.00 −1.20957
\(96\) −256.000 −0.272166
\(97\) 1354.00 1.41730 0.708649 0.705561i \(-0.249305\pi\)
0.708649 + 0.705561i \(0.249305\pi\)
\(98\) −98.0000 −0.101015
\(99\) −1036.00 −1.05174
\(100\) 284.000 0.284000
\(101\) −1358.00 −1.33788 −0.668941 0.743316i \(-0.733252\pi\)
−0.668941 + 0.743316i \(0.733252\pi\)
\(102\) −1184.00 −1.14935
\(103\) −832.000 −0.795916 −0.397958 0.917404i \(-0.630281\pi\)
−0.397958 + 0.917404i \(0.630281\pi\)
\(104\) −144.000 −0.135773
\(105\) 784.000 0.728672
\(106\) −636.000 −0.582772
\(107\) 444.000 0.401150 0.200575 0.979678i \(-0.435719\pi\)
0.200575 + 0.979678i \(0.435719\pi\)
\(108\) 320.000 0.285111
\(109\) 1870.00 1.64324 0.821622 0.570033i \(-0.193070\pi\)
0.821622 + 0.570033i \(0.193070\pi\)
\(110\) −784.000 −0.679559
\(111\) −2768.00 −2.36691
\(112\) −112.000 −0.0944911
\(113\) 1378.00 1.14718 0.573590 0.819143i \(-0.305550\pi\)
0.573590 + 0.819143i \(0.305550\pi\)
\(114\) −1280.00 −1.05161
\(115\) 1568.00 1.27145
\(116\) 760.000 0.608312
\(117\) 666.000 0.526254
\(118\) 400.000 0.312059
\(119\) −518.000 −0.399033
\(120\) 896.000 0.681610
\(121\) −547.000 −0.410969
\(122\) 396.000 0.293870
\(123\) 1296.00 0.950052
\(124\) 288.000 0.208574
\(125\) 756.000 0.540950
\(126\) 518.000 0.366247
\(127\) 1944.00 1.35828 0.679142 0.734007i \(-0.262352\pi\)
0.679142 + 0.734007i \(0.262352\pi\)
\(128\) −128.000 −0.0883883
\(129\) −3296.00 −2.24959
\(130\) 504.000 0.340029
\(131\) −848.000 −0.565573 −0.282787 0.959183i \(-0.591259\pi\)
−0.282787 + 0.959183i \(0.591259\pi\)
\(132\) −896.000 −0.590809
\(133\) −560.000 −0.365099
\(134\) 1432.00 0.923179
\(135\) −1120.00 −0.714031
\(136\) −592.000 −0.373262
\(137\) −2966.00 −1.84965 −0.924827 0.380389i \(-0.875790\pi\)
−0.924827 + 0.380389i \(0.875790\pi\)
\(138\) 1792.00 1.10540
\(139\) 2800.00 1.70858 0.854291 0.519795i \(-0.173992\pi\)
0.854291 + 0.519795i \(0.173992\pi\)
\(140\) 392.000 0.236643
\(141\) 192.000 0.114676
\(142\) −784.000 −0.463323
\(143\) −504.000 −0.294731
\(144\) 592.000 0.342593
\(145\) −2660.00 −1.52346
\(146\) −1076.00 −0.609934
\(147\) 392.000 0.219943
\(148\) −1384.00 −0.768676
\(149\) 510.000 0.280408 0.140204 0.990123i \(-0.455224\pi\)
0.140204 + 0.990123i \(0.455224\pi\)
\(150\) −1136.00 −0.618360
\(151\) 592.000 0.319048 0.159524 0.987194i \(-0.449004\pi\)
0.159524 + 0.987194i \(0.449004\pi\)
\(152\) −640.000 −0.341519
\(153\) 2738.00 1.44676
\(154\) −392.000 −0.205119
\(155\) −1008.00 −0.522352
\(156\) 576.000 0.295621
\(157\) −2686.00 −1.36539 −0.682695 0.730704i \(-0.739192\pi\)
−0.682695 + 0.730704i \(0.739192\pi\)
\(158\) −480.000 −0.241688
\(159\) 2544.00 1.26888
\(160\) 448.000 0.221359
\(161\) 784.000 0.383776
\(162\) 718.000 0.348219
\(163\) −1012.00 −0.486294 −0.243147 0.969989i \(-0.578180\pi\)
−0.243147 + 0.969989i \(0.578180\pi\)
\(164\) 648.000 0.308538
\(165\) 3136.00 1.47962
\(166\) 2144.00 1.00245
\(167\) 544.000 0.252072 0.126036 0.992026i \(-0.459775\pi\)
0.126036 + 0.992026i \(0.459775\pi\)
\(168\) 448.000 0.205738
\(169\) −1873.00 −0.852526
\(170\) 2072.00 0.934795
\(171\) 2960.00 1.32372
\(172\) −1648.00 −0.730575
\(173\) 1858.00 0.816538 0.408269 0.912862i \(-0.366132\pi\)
0.408269 + 0.912862i \(0.366132\pi\)
\(174\) −3040.00 −1.32449
\(175\) −497.000 −0.214684
\(176\) −448.000 −0.191871
\(177\) −1600.00 −0.679454
\(178\) −1620.00 −0.682158
\(179\) −300.000 −0.125268 −0.0626342 0.998037i \(-0.519950\pi\)
−0.0626342 + 0.998037i \(0.519950\pi\)
\(180\) −2072.00 −0.857988
\(181\) −2358.00 −0.968336 −0.484168 0.874975i \(-0.660878\pi\)
−0.484168 + 0.874975i \(0.660878\pi\)
\(182\) 252.000 0.102635
\(183\) −1584.00 −0.639851
\(184\) 896.000 0.358989
\(185\) 4844.00 1.92507
\(186\) −1152.00 −0.454133
\(187\) −2072.00 −0.810265
\(188\) 96.0000 0.0372421
\(189\) −560.000 −0.215524
\(190\) 2240.00 0.855298
\(191\) 1392.00 0.527338 0.263669 0.964613i \(-0.415067\pi\)
0.263669 + 0.964613i \(0.415067\pi\)
\(192\) 512.000 0.192450
\(193\) 1778.00 0.663126 0.331563 0.943433i \(-0.392424\pi\)
0.331563 + 0.943433i \(0.392424\pi\)
\(194\) −2708.00 −1.00218
\(195\) −2016.00 −0.740353
\(196\) 196.000 0.0714286
\(197\) 1214.00 0.439055 0.219528 0.975606i \(-0.429548\pi\)
0.219528 + 0.975606i \(0.429548\pi\)
\(198\) 2072.00 0.743690
\(199\) 1040.00 0.370471 0.185235 0.982694i \(-0.440695\pi\)
0.185235 + 0.982694i \(0.440695\pi\)
\(200\) −568.000 −0.200818
\(201\) −5728.00 −2.01006
\(202\) 2716.00 0.946025
\(203\) −1330.00 −0.459841
\(204\) 2368.00 0.812712
\(205\) −2268.00 −0.772702
\(206\) 1664.00 0.562798
\(207\) −4144.00 −1.39144
\(208\) 288.000 0.0960058
\(209\) −2240.00 −0.741359
\(210\) −1568.00 −0.515249
\(211\) −3868.00 −1.26201 −0.631005 0.775779i \(-0.717357\pi\)
−0.631005 + 0.775779i \(0.717357\pi\)
\(212\) 1272.00 0.412082
\(213\) 3136.00 1.00880
\(214\) −888.000 −0.283656
\(215\) 5768.00 1.82965
\(216\) −640.000 −0.201604
\(217\) −504.000 −0.157667
\(218\) −3740.00 −1.16195
\(219\) 4304.00 1.32802
\(220\) 1568.00 0.480521
\(221\) 1332.00 0.405430
\(222\) 5536.00 1.67366
\(223\) 3968.00 1.19156 0.595778 0.803149i \(-0.296844\pi\)
0.595778 + 0.803149i \(0.296844\pi\)
\(224\) 224.000 0.0668153
\(225\) 2627.00 0.778370
\(226\) −2756.00 −0.811179
\(227\) −3936.00 −1.15084 −0.575422 0.817857i \(-0.695162\pi\)
−0.575422 + 0.817857i \(0.695162\pi\)
\(228\) 2560.00 0.743597
\(229\) 4810.00 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(230\) −3136.00 −0.899051
\(231\) 1568.00 0.446610
\(232\) −1520.00 −0.430142
\(233\) −2182.00 −0.613509 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(234\) −1332.00 −0.372118
\(235\) −336.000 −0.0932690
\(236\) −800.000 −0.220659
\(237\) 1920.00 0.526234
\(238\) 1036.00 0.282159
\(239\) −3000.00 −0.811941 −0.405970 0.913886i \(-0.633066\pi\)
−0.405970 + 0.913886i \(0.633066\pi\)
\(240\) −1792.00 −0.481971
\(241\) 2042.00 0.545796 0.272898 0.962043i \(-0.412018\pi\)
0.272898 + 0.962043i \(0.412018\pi\)
\(242\) 1094.00 0.290599
\(243\) −5032.00 −1.32841
\(244\) −792.000 −0.207798
\(245\) −686.000 −0.178885
\(246\) −2592.00 −0.671788
\(247\) 1440.00 0.370951
\(248\) −576.000 −0.147484
\(249\) −8576.00 −2.18266
\(250\) −1512.00 −0.382509
\(251\) −528.000 −0.132777 −0.0663886 0.997794i \(-0.521148\pi\)
−0.0663886 + 0.997794i \(0.521148\pi\)
\(252\) −1036.00 −0.258976
\(253\) 3136.00 0.779283
\(254\) −3888.00 −0.960452
\(255\) −8288.00 −2.03535
\(256\) 256.000 0.0625000
\(257\) 5634.00 1.36747 0.683734 0.729731i \(-0.260355\pi\)
0.683734 + 0.729731i \(0.260355\pi\)
\(258\) 6592.00 1.59070
\(259\) 2422.00 0.581065
\(260\) −1008.00 −0.240437
\(261\) 7030.00 1.66723
\(262\) 1696.00 0.399921
\(263\) 168.000 0.0393891 0.0196945 0.999806i \(-0.493731\pi\)
0.0196945 + 0.999806i \(0.493731\pi\)
\(264\) 1792.00 0.417765
\(265\) −4452.00 −1.03202
\(266\) 1120.00 0.258164
\(267\) 6480.00 1.48528
\(268\) −2864.00 −0.652786
\(269\) −1310.00 −0.296922 −0.148461 0.988918i \(-0.547432\pi\)
−0.148461 + 0.988918i \(0.547432\pi\)
\(270\) 2240.00 0.504897
\(271\) −2208.00 −0.494932 −0.247466 0.968897i \(-0.579598\pi\)
−0.247466 + 0.968897i \(0.579598\pi\)
\(272\) 1184.00 0.263936
\(273\) −1008.00 −0.223469
\(274\) 5932.00 1.30790
\(275\) −1988.00 −0.435931
\(276\) −3584.00 −0.781636
\(277\) 5294.00 1.14832 0.574162 0.818742i \(-0.305328\pi\)
0.574162 + 0.818742i \(0.305328\pi\)
\(278\) −5600.00 −1.20815
\(279\) 2664.00 0.571647
\(280\) −784.000 −0.167332
\(281\) 3242.00 0.688262 0.344131 0.938922i \(-0.388174\pi\)
0.344131 + 0.938922i \(0.388174\pi\)
\(282\) −384.000 −0.0810882
\(283\) −1592.00 −0.334398 −0.167199 0.985923i \(-0.553472\pi\)
−0.167199 + 0.985923i \(0.553472\pi\)
\(284\) 1568.00 0.327619
\(285\) −8960.00 −1.86226
\(286\) 1008.00 0.208407
\(287\) −1134.00 −0.233233
\(288\) −1184.00 −0.242250
\(289\) 563.000 0.114594
\(290\) 5320.00 1.07725
\(291\) 10832.0 2.18207
\(292\) 2152.00 0.431289
\(293\) −5022.00 −1.00133 −0.500663 0.865642i \(-0.666910\pi\)
−0.500663 + 0.865642i \(0.666910\pi\)
\(294\) −784.000 −0.155523
\(295\) 2800.00 0.552618
\(296\) 2768.00 0.543536
\(297\) −2240.00 −0.437636
\(298\) −1020.00 −0.198279
\(299\) −2016.00 −0.389927
\(300\) 2272.00 0.437247
\(301\) 2884.00 0.552262
\(302\) −1184.00 −0.225601
\(303\) −10864.0 −2.05980
\(304\) 1280.00 0.241490
\(305\) 2772.00 0.520407
\(306\) −5476.00 −1.02301
\(307\) −9536.00 −1.77280 −0.886398 0.462924i \(-0.846800\pi\)
−0.886398 + 0.462924i \(0.846800\pi\)
\(308\) 784.000 0.145041
\(309\) −6656.00 −1.22539
\(310\) 2016.00 0.369358
\(311\) −968.000 −0.176496 −0.0882480 0.996099i \(-0.528127\pi\)
−0.0882480 + 0.996099i \(0.528127\pi\)
\(312\) −1152.00 −0.209036
\(313\) 3058.00 0.552231 0.276116 0.961124i \(-0.410953\pi\)
0.276116 + 0.961124i \(0.410953\pi\)
\(314\) 5372.00 0.965476
\(315\) 3626.00 0.648578
\(316\) 960.000 0.170899
\(317\) −4986.00 −0.883412 −0.441706 0.897160i \(-0.645627\pi\)
−0.441706 + 0.897160i \(0.645627\pi\)
\(318\) −5088.00 −0.897235
\(319\) −5320.00 −0.933739
\(320\) −896.000 −0.156525
\(321\) 3552.00 0.617612
\(322\) −1568.00 −0.271370
\(323\) 5920.00 1.01981
\(324\) −1436.00 −0.246228
\(325\) 1278.00 0.218125
\(326\) 2024.00 0.343862
\(327\) 14960.0 2.52994
\(328\) −1296.00 −0.218170
\(329\) −168.000 −0.0281524
\(330\) −6272.00 −1.04625
\(331\) 8612.00 1.43009 0.715043 0.699081i \(-0.246407\pi\)
0.715043 + 0.699081i \(0.246407\pi\)
\(332\) −4288.00 −0.708839
\(333\) −12802.0 −2.10674
\(334\) −1088.00 −0.178242
\(335\) 10024.0 1.63483
\(336\) −896.000 −0.145479
\(337\) −10206.0 −1.64972 −0.824861 0.565336i \(-0.808747\pi\)
−0.824861 + 0.565336i \(0.808747\pi\)
\(338\) 3746.00 0.602827
\(339\) 11024.0 1.76620
\(340\) −4144.00 −0.661000
\(341\) −2016.00 −0.320154
\(342\) −5920.00 −0.936014
\(343\) −343.000 −0.0539949
\(344\) 3296.00 0.516594
\(345\) 12544.0 1.95753
\(346\) −3716.00 −0.577380
\(347\) 2004.00 0.310030 0.155015 0.987912i \(-0.450457\pi\)
0.155015 + 0.987912i \(0.450457\pi\)
\(348\) 6080.00 0.936558
\(349\) 1330.00 0.203992 0.101996 0.994785i \(-0.467477\pi\)
0.101996 + 0.994785i \(0.467477\pi\)
\(350\) 994.000 0.151804
\(351\) 1440.00 0.218979
\(352\) 896.000 0.135673
\(353\) 978.000 0.147461 0.0737304 0.997278i \(-0.476510\pi\)
0.0737304 + 0.997278i \(0.476510\pi\)
\(354\) 3200.00 0.480447
\(355\) −5488.00 −0.820487
\(356\) 3240.00 0.482359
\(357\) −4144.00 −0.614352
\(358\) 600.000 0.0885782
\(359\) −9680.00 −1.42309 −0.711547 0.702638i \(-0.752005\pi\)
−0.711547 + 0.702638i \(0.752005\pi\)
\(360\) 4144.00 0.606689
\(361\) −459.000 −0.0669194
\(362\) 4716.00 0.684717
\(363\) −4376.00 −0.632728
\(364\) −504.000 −0.0725736
\(365\) −7532.00 −1.08012
\(366\) 3168.00 0.452443
\(367\) −8656.00 −1.23117 −0.615585 0.788070i \(-0.711080\pi\)
−0.615585 + 0.788070i \(0.711080\pi\)
\(368\) −1792.00 −0.253844
\(369\) 5994.00 0.845624
\(370\) −9688.00 −1.36123
\(371\) −2226.00 −0.311504
\(372\) 2304.00 0.321121
\(373\) 5278.00 0.732666 0.366333 0.930484i \(-0.380613\pi\)
0.366333 + 0.930484i \(0.380613\pi\)
\(374\) 4144.00 0.572944
\(375\) 6048.00 0.832846
\(376\) −192.000 −0.0263342
\(377\) 3420.00 0.467212
\(378\) 1120.00 0.152398
\(379\) 6340.00 0.859272 0.429636 0.903002i \(-0.358642\pi\)
0.429636 + 0.903002i \(0.358642\pi\)
\(380\) −4480.00 −0.604787
\(381\) 15552.0 2.09122
\(382\) −2784.00 −0.372884
\(383\) −6232.00 −0.831437 −0.415718 0.909493i \(-0.636470\pi\)
−0.415718 + 0.909493i \(0.636470\pi\)
\(384\) −1024.00 −0.136083
\(385\) −2744.00 −0.363239
\(386\) −3556.00 −0.468901
\(387\) −15244.0 −2.00232
\(388\) 5416.00 0.708649
\(389\) −14810.0 −1.93033 −0.965163 0.261649i \(-0.915734\pi\)
−0.965163 + 0.261649i \(0.915734\pi\)
\(390\) 4032.00 0.523508
\(391\) −8288.00 −1.07197
\(392\) −392.000 −0.0505076
\(393\) −6784.00 −0.870757
\(394\) −2428.00 −0.310459
\(395\) −3360.00 −0.428000
\(396\) −4144.00 −0.525868
\(397\) 5154.00 0.651566 0.325783 0.945445i \(-0.394372\pi\)
0.325783 + 0.945445i \(0.394372\pi\)
\(398\) −2080.00 −0.261962
\(399\) −4480.00 −0.562107
\(400\) 1136.00 0.142000
\(401\) 3282.00 0.408716 0.204358 0.978896i \(-0.434489\pi\)
0.204358 + 0.978896i \(0.434489\pi\)
\(402\) 11456.0 1.42133
\(403\) 1296.00 0.160194
\(404\) −5432.00 −0.668941
\(405\) 5026.00 0.616652
\(406\) 2660.00 0.325157
\(407\) 9688.00 1.17989
\(408\) −4736.00 −0.574674
\(409\) 5810.00 0.702411 0.351205 0.936298i \(-0.385772\pi\)
0.351205 + 0.936298i \(0.385772\pi\)
\(410\) 4536.00 0.546383
\(411\) −23728.0 −2.84773
\(412\) −3328.00 −0.397958
\(413\) 1400.00 0.166803
\(414\) 8288.00 0.983896
\(415\) 15008.0 1.77521
\(416\) −576.000 −0.0678864
\(417\) 22400.0 2.63053
\(418\) 4480.00 0.524220
\(419\) 13560.0 1.58102 0.790512 0.612446i \(-0.209814\pi\)
0.790512 + 0.612446i \(0.209814\pi\)
\(420\) 3136.00 0.364336
\(421\) −738.000 −0.0854345 −0.0427172 0.999087i \(-0.513601\pi\)
−0.0427172 + 0.999087i \(0.513601\pi\)
\(422\) 7736.00 0.892376
\(423\) 888.000 0.102071
\(424\) −2544.00 −0.291386
\(425\) 5254.00 0.599662
\(426\) −6272.00 −0.713332
\(427\) 1386.00 0.157080
\(428\) 1776.00 0.200575
\(429\) −4032.00 −0.453769
\(430\) −11536.0 −1.29376
\(431\) 1272.00 0.142158 0.0710790 0.997471i \(-0.477356\pi\)
0.0710790 + 0.997471i \(0.477356\pi\)
\(432\) 1280.00 0.142556
\(433\) −5062.00 −0.561811 −0.280906 0.959735i \(-0.590635\pi\)
−0.280906 + 0.959735i \(0.590635\pi\)
\(434\) 1008.00 0.111487
\(435\) −21280.0 −2.34551
\(436\) 7480.00 0.821622
\(437\) −8960.00 −0.980812
\(438\) −8608.00 −0.939055
\(439\) 5640.00 0.613172 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(440\) −3136.00 −0.339779
\(441\) 1813.00 0.195767
\(442\) −2664.00 −0.286682
\(443\) 13388.0 1.43585 0.717927 0.696119i \(-0.245091\pi\)
0.717927 + 0.696119i \(0.245091\pi\)
\(444\) −11072.0 −1.18345
\(445\) −11340.0 −1.20802
\(446\) −7936.00 −0.842557
\(447\) 4080.00 0.431717
\(448\) −448.000 −0.0472456
\(449\) −3230.00 −0.339495 −0.169747 0.985488i \(-0.554295\pi\)
−0.169747 + 0.985488i \(0.554295\pi\)
\(450\) −5254.00 −0.550391
\(451\) −4536.00 −0.473596
\(452\) 5512.00 0.573590
\(453\) 4736.00 0.491207
\(454\) 7872.00 0.813769
\(455\) 1764.00 0.181753
\(456\) −5120.00 −0.525803
\(457\) −10646.0 −1.08971 −0.544857 0.838529i \(-0.683416\pi\)
−0.544857 + 0.838529i \(0.683416\pi\)
\(458\) −9620.00 −0.981470
\(459\) 5920.00 0.602009
\(460\) 6272.00 0.635725
\(461\) 7282.00 0.735698 0.367849 0.929886i \(-0.380094\pi\)
0.367849 + 0.929886i \(0.380094\pi\)
\(462\) −3136.00 −0.315801
\(463\) 12688.0 1.27357 0.636783 0.771043i \(-0.280265\pi\)
0.636783 + 0.771043i \(0.280265\pi\)
\(464\) 3040.00 0.304156
\(465\) −8064.00 −0.804213
\(466\) 4364.00 0.433816
\(467\) −2816.00 −0.279034 −0.139517 0.990220i \(-0.544555\pi\)
−0.139517 + 0.990220i \(0.544555\pi\)
\(468\) 2664.00 0.263127
\(469\) 5012.00 0.493460
\(470\) 672.000 0.0659512
\(471\) −21488.0 −2.10215
\(472\) 1600.00 0.156030
\(473\) 11536.0 1.12141
\(474\) −3840.00 −0.372103
\(475\) 5680.00 0.548666
\(476\) −2072.00 −0.199517
\(477\) 11766.0 1.12941
\(478\) 6000.00 0.574129
\(479\) −3160.00 −0.301428 −0.150714 0.988577i \(-0.548157\pi\)
−0.150714 + 0.988577i \(0.548157\pi\)
\(480\) 3584.00 0.340805
\(481\) −6228.00 −0.590379
\(482\) −4084.00 −0.385936
\(483\) 6272.00 0.590861
\(484\) −2188.00 −0.205485
\(485\) −18956.0 −1.77474
\(486\) 10064.0 0.939326
\(487\) −14176.0 −1.31905 −0.659523 0.751684i \(-0.729242\pi\)
−0.659523 + 0.751684i \(0.729242\pi\)
\(488\) 1584.00 0.146935
\(489\) −8096.00 −0.748699
\(490\) 1372.00 0.126491
\(491\) −11268.0 −1.03568 −0.517839 0.855478i \(-0.673263\pi\)
−0.517839 + 0.855478i \(0.673263\pi\)
\(492\) 5184.00 0.475026
\(493\) 14060.0 1.28444
\(494\) −2880.00 −0.262302
\(495\) 14504.0 1.31698
\(496\) 1152.00 0.104287
\(497\) −2744.00 −0.247656
\(498\) 17152.0 1.54337
\(499\) −4460.00 −0.400114 −0.200057 0.979784i \(-0.564113\pi\)
−0.200057 + 0.979784i \(0.564113\pi\)
\(500\) 3024.00 0.270475
\(501\) 4352.00 0.388090
\(502\) 1056.00 0.0938876
\(503\) −1512.00 −0.134029 −0.0670147 0.997752i \(-0.521347\pi\)
−0.0670147 + 0.997752i \(0.521347\pi\)
\(504\) 2072.00 0.183123
\(505\) 19012.0 1.67529
\(506\) −6272.00 −0.551036
\(507\) −14984.0 −1.31255
\(508\) 7776.00 0.679142
\(509\) −11790.0 −1.02668 −0.513342 0.858184i \(-0.671593\pi\)
−0.513342 + 0.858184i \(0.671593\pi\)
\(510\) 16576.0 1.43921
\(511\) −3766.00 −0.326024
\(512\) −512.000 −0.0441942
\(513\) 6400.00 0.550813
\(514\) −11268.0 −0.966946
\(515\) 11648.0 0.996645
\(516\) −13184.0 −1.12479
\(517\) −672.000 −0.0571654
\(518\) −4844.00 −0.410875
\(519\) 14864.0 1.25714
\(520\) 2016.00 0.170014
\(521\) 1362.00 0.114530 0.0572652 0.998359i \(-0.481762\pi\)
0.0572652 + 0.998359i \(0.481762\pi\)
\(522\) −14060.0 −1.17891
\(523\) 6968.00 0.582580 0.291290 0.956635i \(-0.405916\pi\)
0.291290 + 0.956635i \(0.405916\pi\)
\(524\) −3392.00 −0.282787
\(525\) −3976.00 −0.330527
\(526\) −336.000 −0.0278523
\(527\) 5328.00 0.440401
\(528\) −3584.00 −0.295405
\(529\) 377.000 0.0309855
\(530\) 8904.00 0.729745
\(531\) −7400.00 −0.604770
\(532\) −2240.00 −0.182549
\(533\) 2916.00 0.236972
\(534\) −12960.0 −1.05025
\(535\) −6216.00 −0.502320
\(536\) 5728.00 0.461589
\(537\) −2400.00 −0.192863
\(538\) 2620.00 0.209956
\(539\) −1372.00 −0.109640
\(540\) −4480.00 −0.357016
\(541\) 7062.00 0.561218 0.280609 0.959822i \(-0.409464\pi\)
0.280609 + 0.959822i \(0.409464\pi\)
\(542\) 4416.00 0.349969
\(543\) −18864.0 −1.49085
\(544\) −2368.00 −0.186631
\(545\) −26180.0 −2.05767
\(546\) 2016.00 0.158016
\(547\) −8196.00 −0.640650 −0.320325 0.947308i \(-0.603792\pi\)
−0.320325 + 0.947308i \(0.603792\pi\)
\(548\) −11864.0 −0.924827
\(549\) −7326.00 −0.569519
\(550\) 3976.00 0.308249
\(551\) 15200.0 1.17521
\(552\) 7168.00 0.552700
\(553\) −1680.00 −0.129188
\(554\) −10588.0 −0.811987
\(555\) 38752.0 2.96384
\(556\) 11200.0 0.854291
\(557\) −7466.00 −0.567944 −0.283972 0.958833i \(-0.591652\pi\)
−0.283972 + 0.958833i \(0.591652\pi\)
\(558\) −5328.00 −0.404215
\(559\) −7416.00 −0.561115
\(560\) 1568.00 0.118322
\(561\) −16576.0 −1.24749
\(562\) −6484.00 −0.486674
\(563\) 24968.0 1.86905 0.934526 0.355896i \(-0.115824\pi\)
0.934526 + 0.355896i \(0.115824\pi\)
\(564\) 768.000 0.0573380
\(565\) −19292.0 −1.43650
\(566\) 3184.00 0.236455
\(567\) 2513.00 0.186131
\(568\) −3136.00 −0.231661
\(569\) 14250.0 1.04990 0.524948 0.851134i \(-0.324085\pi\)
0.524948 + 0.851134i \(0.324085\pi\)
\(570\) 17920.0 1.31682
\(571\) 6372.00 0.467005 0.233503 0.972356i \(-0.424981\pi\)
0.233503 + 0.972356i \(0.424981\pi\)
\(572\) −2016.00 −0.147366
\(573\) 11136.0 0.811890
\(574\) 2268.00 0.164921
\(575\) −7952.00 −0.576733
\(576\) 2368.00 0.171296
\(577\) −8366.00 −0.603607 −0.301803 0.953370i \(-0.597589\pi\)
−0.301803 + 0.953370i \(0.597589\pi\)
\(578\) −1126.00 −0.0810301
\(579\) 14224.0 1.02095
\(580\) −10640.0 −0.761728
\(581\) 7504.00 0.535832
\(582\) −21664.0 −1.54296
\(583\) −8904.00 −0.632532
\(584\) −4304.00 −0.304967
\(585\) −9324.00 −0.658974
\(586\) 10044.0 0.708044
\(587\) 20384.0 1.43328 0.716642 0.697441i \(-0.245678\pi\)
0.716642 + 0.697441i \(0.245678\pi\)
\(588\) 1568.00 0.109971
\(589\) 5760.00 0.402948
\(590\) −5600.00 −0.390760
\(591\) 9712.00 0.675970
\(592\) −5536.00 −0.384338
\(593\) 9378.00 0.649424 0.324712 0.945813i \(-0.394733\pi\)
0.324712 + 0.945813i \(0.394733\pi\)
\(594\) 4480.00 0.309456
\(595\) 7252.00 0.499669
\(596\) 2040.00 0.140204
\(597\) 8320.00 0.570377
\(598\) 4032.00 0.275720
\(599\) −9000.00 −0.613907 −0.306953 0.951725i \(-0.599310\pi\)
−0.306953 + 0.951725i \(0.599310\pi\)
\(600\) −4544.00 −0.309180
\(601\) 7562.00 0.513245 0.256623 0.966512i \(-0.417390\pi\)
0.256623 + 0.966512i \(0.417390\pi\)
\(602\) −5768.00 −0.390509
\(603\) −26492.0 −1.78912
\(604\) 2368.00 0.159524
\(605\) 7658.00 0.514615
\(606\) 21728.0 1.45650
\(607\) −2976.00 −0.198999 −0.0994993 0.995038i \(-0.531724\pi\)
−0.0994993 + 0.995038i \(0.531724\pi\)
\(608\) −2560.00 −0.170759
\(609\) −10640.0 −0.707971
\(610\) −5544.00 −0.367984
\(611\) 432.000 0.0286037
\(612\) 10952.0 0.723380
\(613\) 4278.00 0.281871 0.140935 0.990019i \(-0.454989\pi\)
0.140935 + 0.990019i \(0.454989\pi\)
\(614\) 19072.0 1.25356
\(615\) −18144.0 −1.18965
\(616\) −1568.00 −0.102559
\(617\) 18794.0 1.22629 0.613143 0.789972i \(-0.289905\pi\)
0.613143 + 0.789972i \(0.289905\pi\)
\(618\) 13312.0 0.866484
\(619\) 18040.0 1.17139 0.585694 0.810532i \(-0.300822\pi\)
0.585694 + 0.810532i \(0.300822\pi\)
\(620\) −4032.00 −0.261176
\(621\) −8960.00 −0.578989
\(622\) 1936.00 0.124801
\(623\) −5670.00 −0.364629
\(624\) 2304.00 0.147811
\(625\) −19459.0 −1.24538
\(626\) −6116.00 −0.390486
\(627\) −17920.0 −1.14140
\(628\) −10744.0 −0.682695
\(629\) −25604.0 −1.62305
\(630\) −7252.00 −0.458614
\(631\) −21688.0 −1.36828 −0.684141 0.729350i \(-0.739823\pi\)
−0.684141 + 0.729350i \(0.739823\pi\)
\(632\) −1920.00 −0.120844
\(633\) −30944.0 −1.94299
\(634\) 9972.00 0.624667
\(635\) −27216.0 −1.70084
\(636\) 10176.0 0.634441
\(637\) 882.000 0.0548605
\(638\) 10640.0 0.660253
\(639\) 14504.0 0.897918
\(640\) 1792.00 0.110680
\(641\) −10558.0 −0.650571 −0.325285 0.945616i \(-0.605460\pi\)
−0.325285 + 0.945616i \(0.605460\pi\)
\(642\) −7104.00 −0.436717
\(643\) −26152.0 −1.60394 −0.801971 0.597363i \(-0.796215\pi\)
−0.801971 + 0.597363i \(0.796215\pi\)
\(644\) 3136.00 0.191888
\(645\) 46144.0 2.81693
\(646\) −11840.0 −0.721112
\(647\) 25584.0 1.55458 0.777288 0.629145i \(-0.216595\pi\)
0.777288 + 0.629145i \(0.216595\pi\)
\(648\) 2872.00 0.174109
\(649\) 5600.00 0.338705
\(650\) −2556.00 −0.154238
\(651\) −4032.00 −0.242744
\(652\) −4048.00 −0.243147
\(653\) 15198.0 0.910787 0.455393 0.890290i \(-0.349499\pi\)
0.455393 + 0.890290i \(0.349499\pi\)
\(654\) −29920.0 −1.78894
\(655\) 11872.0 0.708210
\(656\) 2592.00 0.154269
\(657\) 19906.0 1.18205
\(658\) 336.000 0.0199068
\(659\) −6100.00 −0.360580 −0.180290 0.983613i \(-0.557704\pi\)
−0.180290 + 0.983613i \(0.557704\pi\)
\(660\) 12544.0 0.739810
\(661\) −2318.00 −0.136399 −0.0681995 0.997672i \(-0.521725\pi\)
−0.0681995 + 0.997672i \(0.521725\pi\)
\(662\) −17224.0 −1.01122
\(663\) 10656.0 0.624200
\(664\) 8576.00 0.501225
\(665\) 7840.00 0.457176
\(666\) 25604.0 1.48969
\(667\) −21280.0 −1.23533
\(668\) 2176.00 0.126036
\(669\) 31744.0 1.83452
\(670\) −20048.0 −1.15600
\(671\) 5544.00 0.318962
\(672\) 1792.00 0.102869
\(673\) −10222.0 −0.585482 −0.292741 0.956192i \(-0.594567\pi\)
−0.292741 + 0.956192i \(0.594567\pi\)
\(674\) 20412.0 1.16653
\(675\) 5680.00 0.323886
\(676\) −7492.00 −0.426263
\(677\) 25434.0 1.44388 0.721941 0.691955i \(-0.243250\pi\)
0.721941 + 0.691955i \(0.243250\pi\)
\(678\) −22048.0 −1.24889
\(679\) −9478.00 −0.535688
\(680\) 8288.00 0.467397
\(681\) −31488.0 −1.77184
\(682\) 4032.00 0.226383
\(683\) −8532.00 −0.477991 −0.238996 0.971021i \(-0.576818\pi\)
−0.238996 + 0.971021i \(0.576818\pi\)
\(684\) 11840.0 0.661862
\(685\) 41524.0 2.31613
\(686\) 686.000 0.0381802
\(687\) 38480.0 2.13698
\(688\) −6592.00 −0.365287
\(689\) 5724.00 0.316498
\(690\) −25088.0 −1.38418
\(691\) 20672.0 1.13806 0.569030 0.822317i \(-0.307319\pi\)
0.569030 + 0.822317i \(0.307319\pi\)
\(692\) 7432.00 0.408269
\(693\) 7252.00 0.397519
\(694\) −4008.00 −0.219224
\(695\) −39200.0 −2.13948
\(696\) −12160.0 −0.662247
\(697\) 11988.0 0.651475
\(698\) −2660.00 −0.144244
\(699\) −17456.0 −0.944559
\(700\) −1988.00 −0.107342
\(701\) −21458.0 −1.15614 −0.578072 0.815985i \(-0.696195\pi\)
−0.578072 + 0.815985i \(0.696195\pi\)
\(702\) −2880.00 −0.154841
\(703\) −27680.0 −1.48502
\(704\) −1792.00 −0.0959354
\(705\) −2688.00 −0.143597
\(706\) −1956.00 −0.104271
\(707\) 9506.00 0.505672
\(708\) −6400.00 −0.339727
\(709\) −9850.00 −0.521755 −0.260878 0.965372i \(-0.584012\pi\)
−0.260878 + 0.965372i \(0.584012\pi\)
\(710\) 10976.0 0.580172
\(711\) 8880.00 0.468391
\(712\) −6480.00 −0.341079
\(713\) −8064.00 −0.423561
\(714\) 8288.00 0.434413
\(715\) 7056.00 0.369062
\(716\) −1200.00 −0.0626342
\(717\) −24000.0 −1.25006
\(718\) 19360.0 1.00628
\(719\) −18840.0 −0.977209 −0.488605 0.872505i \(-0.662494\pi\)
−0.488605 + 0.872505i \(0.662494\pi\)
\(720\) −8288.00 −0.428994
\(721\) 5824.00 0.300828
\(722\) 918.000 0.0473191
\(723\) 16336.0 0.840308
\(724\) −9432.00 −0.484168
\(725\) 13490.0 0.691043
\(726\) 8752.00 0.447407
\(727\) 37504.0 1.91327 0.956634 0.291291i \(-0.0940849\pi\)
0.956634 + 0.291291i \(0.0940849\pi\)
\(728\) 1008.00 0.0513173
\(729\) −30563.0 −1.55276
\(730\) 15064.0 0.763758
\(731\) −30488.0 −1.54260
\(732\) −6336.00 −0.319925
\(733\) 13338.0 0.672101 0.336051 0.941844i \(-0.390909\pi\)
0.336051 + 0.941844i \(0.390909\pi\)
\(734\) 17312.0 0.870569
\(735\) −5488.00 −0.275412
\(736\) 3584.00 0.179495
\(737\) 20048.0 1.00200
\(738\) −11988.0 −0.597946
\(739\) 17100.0 0.851196 0.425598 0.904912i \(-0.360064\pi\)
0.425598 + 0.904912i \(0.360064\pi\)
\(740\) 19376.0 0.962535
\(741\) 11520.0 0.571117
\(742\) 4452.00 0.220267
\(743\) −19632.0 −0.969352 −0.484676 0.874694i \(-0.661062\pi\)
−0.484676 + 0.874694i \(0.661062\pi\)
\(744\) −4608.00 −0.227067
\(745\) −7140.00 −0.351127
\(746\) −10556.0 −0.518073
\(747\) −39664.0 −1.94274
\(748\) −8288.00 −0.405133
\(749\) −3108.00 −0.151621
\(750\) −12096.0 −0.588911
\(751\) 33912.0 1.64776 0.823879 0.566766i \(-0.191805\pi\)
0.823879 + 0.566766i \(0.191805\pi\)
\(752\) 384.000 0.0186211
\(753\) −4224.00 −0.204424
\(754\) −6840.00 −0.330369
\(755\) −8288.00 −0.399512
\(756\) −2240.00 −0.107762
\(757\) −31386.0 −1.50693 −0.753463 0.657490i \(-0.771618\pi\)
−0.753463 + 0.657490i \(0.771618\pi\)
\(758\) −12680.0 −0.607597
\(759\) 25088.0 1.19978
\(760\) 8960.00 0.427649
\(761\) −34558.0 −1.64616 −0.823079 0.567927i \(-0.807746\pi\)
−0.823079 + 0.567927i \(0.807746\pi\)
\(762\) −31104.0 −1.47871
\(763\) −13090.0 −0.621088
\(764\) 5568.00 0.263669
\(765\) −38332.0 −1.81163
\(766\) 12464.0 0.587915
\(767\) −3600.00 −0.169476
\(768\) 2048.00 0.0962250
\(769\) 39130.0 1.83493 0.917467 0.397812i \(-0.130231\pi\)
0.917467 + 0.397812i \(0.130231\pi\)
\(770\) 5488.00 0.256849
\(771\) 45072.0 2.10535
\(772\) 7112.00 0.331563
\(773\) −25982.0 −1.20894 −0.604468 0.796629i \(-0.706614\pi\)
−0.604468 + 0.796629i \(0.706614\pi\)
\(774\) 30488.0 1.41585
\(775\) 5112.00 0.236940
\(776\) −10832.0 −0.501090
\(777\) 19376.0 0.894608
\(778\) 29620.0 1.36495
\(779\) 12960.0 0.596072
\(780\) −8064.00 −0.370176
\(781\) −10976.0 −0.502884
\(782\) 16576.0 0.758001
\(783\) 15200.0 0.693747
\(784\) 784.000 0.0357143
\(785\) 37604.0 1.70974
\(786\) 13568.0 0.615718
\(787\) 35424.0 1.60448 0.802242 0.596999i \(-0.203640\pi\)
0.802242 + 0.596999i \(0.203640\pi\)
\(788\) 4856.00 0.219528
\(789\) 1344.00 0.0606434
\(790\) 6720.00 0.302642
\(791\) −9646.00 −0.433593
\(792\) 8288.00 0.371845
\(793\) −3564.00 −0.159598
\(794\) −10308.0 −0.460727
\(795\) −35616.0 −1.58889
\(796\) 4160.00 0.185235
\(797\) −30606.0 −1.36025 −0.680126 0.733096i \(-0.738075\pi\)
−0.680126 + 0.733096i \(0.738075\pi\)
\(798\) 8960.00 0.397469
\(799\) 1776.00 0.0786362
\(800\) −2272.00 −0.100409
\(801\) 29970.0 1.32202
\(802\) −6564.00 −0.289006
\(803\) −15064.0 −0.662014
\(804\) −22912.0 −1.00503
\(805\) −10976.0 −0.480563
\(806\) −2592.00 −0.113275
\(807\) −10480.0 −0.457142
\(808\) 10864.0 0.473013
\(809\) 16810.0 0.730542 0.365271 0.930901i \(-0.380976\pi\)
0.365271 + 0.930901i \(0.380976\pi\)
\(810\) −10052.0 −0.436039
\(811\) −9368.00 −0.405616 −0.202808 0.979218i \(-0.565007\pi\)
−0.202808 + 0.979218i \(0.565007\pi\)
\(812\) −5320.00 −0.229920
\(813\) −17664.0 −0.761997
\(814\) −19376.0 −0.834310
\(815\) 14168.0 0.608937
\(816\) 9472.00 0.406356
\(817\) −32960.0 −1.41141
\(818\) −11620.0 −0.496679
\(819\) −4662.00 −0.198905
\(820\) −9072.00 −0.386351
\(821\) 34382.0 1.46156 0.730780 0.682614i \(-0.239157\pi\)
0.730780 + 0.682614i \(0.239157\pi\)
\(822\) 47456.0 2.01365
\(823\) −4472.00 −0.189410 −0.0947048 0.995505i \(-0.530191\pi\)
−0.0947048 + 0.995505i \(0.530191\pi\)
\(824\) 6656.00 0.281399
\(825\) −15904.0 −0.671159
\(826\) −2800.00 −0.117947
\(827\) −1716.00 −0.0721538 −0.0360769 0.999349i \(-0.511486\pi\)
−0.0360769 + 0.999349i \(0.511486\pi\)
\(828\) −16576.0 −0.695720
\(829\) −7910.00 −0.331394 −0.165697 0.986177i \(-0.552987\pi\)
−0.165697 + 0.986177i \(0.552987\pi\)
\(830\) −30016.0 −1.25527
\(831\) 42352.0 1.76796
\(832\) 1152.00 0.0480029
\(833\) 3626.00 0.150820
\(834\) −44800.0 −1.86007
\(835\) −7616.00 −0.315644
\(836\) −8960.00 −0.370680
\(837\) 5760.00 0.237867
\(838\) −27120.0 −1.11795
\(839\) −19360.0 −0.796641 −0.398320 0.917246i \(-0.630407\pi\)
−0.398320 + 0.917246i \(0.630407\pi\)
\(840\) −6272.00 −0.257624
\(841\) 11711.0 0.480175
\(842\) 1476.00 0.0604113
\(843\) 25936.0 1.05965
\(844\) −15472.0 −0.631005
\(845\) 26222.0 1.06753
\(846\) −1776.00 −0.0721751
\(847\) 3829.00 0.155332
\(848\) 5088.00 0.206041
\(849\) −12736.0 −0.514839
\(850\) −10508.0 −0.424025
\(851\) 38752.0 1.56099
\(852\) 12544.0 0.504402
\(853\) 698.000 0.0280177 0.0140088 0.999902i \(-0.495541\pi\)
0.0140088 + 0.999902i \(0.495541\pi\)
\(854\) −2772.00 −0.111072
\(855\) −41440.0 −1.65757
\(856\) −3552.00 −0.141828
\(857\) −23406.0 −0.932945 −0.466472 0.884536i \(-0.654475\pi\)
−0.466472 + 0.884536i \(0.654475\pi\)
\(858\) 8064.00 0.320863
\(859\) 7280.00 0.289162 0.144581 0.989493i \(-0.453817\pi\)
0.144581 + 0.989493i \(0.453817\pi\)
\(860\) 23072.0 0.914824
\(861\) −9072.00 −0.359086
\(862\) −2544.00 −0.100521
\(863\) 9808.00 0.386869 0.193435 0.981113i \(-0.438037\pi\)
0.193435 + 0.981113i \(0.438037\pi\)
\(864\) −2560.00 −0.100802
\(865\) −26012.0 −1.02247
\(866\) 10124.0 0.397260
\(867\) 4504.00 0.176429
\(868\) −2016.00 −0.0788335
\(869\) −6720.00 −0.262325
\(870\) 42560.0 1.65853
\(871\) −12888.0 −0.501370
\(872\) −14960.0 −0.580974
\(873\) 50098.0 1.94222
\(874\) 17920.0 0.693539
\(875\) −5292.00 −0.204460
\(876\) 17216.0 0.664012
\(877\) −8066.00 −0.310570 −0.155285 0.987870i \(-0.549630\pi\)
−0.155285 + 0.987870i \(0.549630\pi\)
\(878\) −11280.0 −0.433578
\(879\) −40176.0 −1.54164
\(880\) 6272.00 0.240260
\(881\) 25842.0 0.988240 0.494120 0.869394i \(-0.335490\pi\)
0.494120 + 0.869394i \(0.335490\pi\)
\(882\) −3626.00 −0.138428
\(883\) −5692.00 −0.216932 −0.108466 0.994100i \(-0.534594\pi\)
−0.108466 + 0.994100i \(0.534594\pi\)
\(884\) 5328.00 0.202715
\(885\) 22400.0 0.850811
\(886\) −26776.0 −1.01530
\(887\) −13536.0 −0.512395 −0.256198 0.966624i \(-0.582470\pi\)
−0.256198 + 0.966624i \(0.582470\pi\)
\(888\) 22144.0 0.836829
\(889\) −13608.0 −0.513383
\(890\) 22680.0 0.854197
\(891\) 10052.0 0.377951
\(892\) 15872.0 0.595778
\(893\) 1920.00 0.0719489
\(894\) −8160.00 −0.305270
\(895\) 4200.00 0.156861
\(896\) 896.000 0.0334077
\(897\) −16128.0 −0.600332
\(898\) 6460.00 0.240059
\(899\) 13680.0 0.507512
\(900\) 10508.0 0.389185
\(901\) 23532.0 0.870105
\(902\) 9072.00 0.334883
\(903\) 23072.0 0.850264
\(904\) −11024.0 −0.405589
\(905\) 33012.0 1.21255
\(906\) −9472.00 −0.347336
\(907\) 17004.0 0.622501 0.311251 0.950328i \(-0.399252\pi\)
0.311251 + 0.950328i \(0.399252\pi\)
\(908\) −15744.0 −0.575422
\(909\) −50246.0 −1.83339
\(910\) −3528.00 −0.128519
\(911\) −14568.0 −0.529813 −0.264906 0.964274i \(-0.585341\pi\)
−0.264906 + 0.964274i \(0.585341\pi\)
\(912\) 10240.0 0.371799
\(913\) 30016.0 1.08804
\(914\) 21292.0 0.770544
\(915\) 22176.0 0.801220
\(916\) 19240.0 0.694004
\(917\) 5936.00 0.213767
\(918\) −11840.0 −0.425684
\(919\) −1400.00 −0.0502522 −0.0251261 0.999684i \(-0.507999\pi\)
−0.0251261 + 0.999684i \(0.507999\pi\)
\(920\) −12544.0 −0.449525
\(921\) −76288.0 −2.72940
\(922\) −14564.0 −0.520217
\(923\) 7056.00 0.251626
\(924\) 6272.00 0.223305
\(925\) −24566.0 −0.873216
\(926\) −25376.0 −0.900548
\(927\) −30784.0 −1.09070
\(928\) −6080.00 −0.215071
\(929\) −13830.0 −0.488426 −0.244213 0.969722i \(-0.578530\pi\)
−0.244213 + 0.969722i \(0.578530\pi\)
\(930\) 16128.0 0.568664
\(931\) 3920.00 0.137994
\(932\) −8728.00 −0.306754
\(933\) −7744.00 −0.271733
\(934\) 5632.00 0.197307
\(935\) 29008.0 1.01461
\(936\) −5328.00 −0.186059
\(937\) −24166.0 −0.842549 −0.421275 0.906933i \(-0.638417\pi\)
−0.421275 + 0.906933i \(0.638417\pi\)
\(938\) −10024.0 −0.348929
\(939\) 24464.0 0.850216
\(940\) −1344.00 −0.0466345
\(941\) −10838.0 −0.375461 −0.187730 0.982221i \(-0.560113\pi\)
−0.187730 + 0.982221i \(0.560113\pi\)
\(942\) 42976.0 1.48645
\(943\) −18144.0 −0.626564
\(944\) −3200.00 −0.110330
\(945\) 7840.00 0.269879
\(946\) −23072.0 −0.792955
\(947\) −40916.0 −1.40400 −0.702002 0.712175i \(-0.747710\pi\)
−0.702002 + 0.712175i \(0.747710\pi\)
\(948\) 7680.00 0.263117
\(949\) 9684.00 0.331250
\(950\) −11360.0 −0.387965
\(951\) −39888.0 −1.36010
\(952\) 4144.00 0.141080
\(953\) 56618.0 1.92449 0.962244 0.272189i \(-0.0877475\pi\)
0.962244 + 0.272189i \(0.0877475\pi\)
\(954\) −23532.0 −0.798613
\(955\) −19488.0 −0.660332
\(956\) −12000.0 −0.405970
\(957\) −42560.0 −1.43759
\(958\) 6320.00 0.213142
\(959\) 20762.0 0.699103
\(960\) −7168.00 −0.240986
\(961\) −24607.0 −0.825988
\(962\) 12456.0 0.417461
\(963\) 16428.0 0.549725
\(964\) 8168.00 0.272898
\(965\) −24892.0 −0.830365
\(966\) −12544.0 −0.417802
\(967\) 17504.0 0.582100 0.291050 0.956708i \(-0.405995\pi\)
0.291050 + 0.956708i \(0.405995\pi\)
\(968\) 4376.00 0.145300
\(969\) 47360.0 1.57010
\(970\) 37912.0 1.25493
\(971\) 23112.0 0.763851 0.381926 0.924193i \(-0.375261\pi\)
0.381926 + 0.924193i \(0.375261\pi\)
\(972\) −20128.0 −0.664204
\(973\) −19600.0 −0.645783
\(974\) 28352.0 0.932707
\(975\) 10224.0 0.335826
\(976\) −3168.00 −0.103899
\(977\) 23874.0 0.781778 0.390889 0.920438i \(-0.372168\pi\)
0.390889 + 0.920438i \(0.372168\pi\)
\(978\) 16192.0 0.529410
\(979\) −22680.0 −0.740404
\(980\) −2744.00 −0.0894427
\(981\) 69190.0 2.25185
\(982\) 22536.0 0.732335
\(983\) −15312.0 −0.496823 −0.248411 0.968655i \(-0.579909\pi\)
−0.248411 + 0.968655i \(0.579909\pi\)
\(984\) −10368.0 −0.335894
\(985\) −16996.0 −0.549784
\(986\) −28120.0 −0.908239
\(987\) −1344.00 −0.0433435
\(988\) 5760.00 0.185476
\(989\) 46144.0 1.48361
\(990\) −29008.0 −0.931247
\(991\) −16528.0 −0.529797 −0.264899 0.964276i \(-0.585339\pi\)
−0.264899 + 0.964276i \(0.585339\pi\)
\(992\) −2304.00 −0.0737420
\(993\) 68896.0 2.20176
\(994\) 5488.00 0.175120
\(995\) −14560.0 −0.463903
\(996\) −34304.0 −1.09133
\(997\) −28606.0 −0.908687 −0.454344 0.890827i \(-0.650126\pi\)
−0.454344 + 0.890827i \(0.650126\pi\)
\(998\) 8920.00 0.282924
\(999\) −27680.0 −0.876633
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 14.4.a.a.1.1 1
3.2 odd 2 126.4.a.h.1.1 1
4.3 odd 2 112.4.a.a.1.1 1
5.2 odd 4 350.4.c.b.99.1 2
5.3 odd 4 350.4.c.b.99.2 2
5.4 even 2 350.4.a.l.1.1 1
7.2 even 3 98.4.c.d.67.1 2
7.3 odd 6 98.4.c.f.79.1 2
7.4 even 3 98.4.c.d.79.1 2
7.5 odd 6 98.4.c.f.67.1 2
7.6 odd 2 98.4.a.a.1.1 1
8.3 odd 2 448.4.a.o.1.1 1
8.5 even 2 448.4.a.b.1.1 1
11.10 odd 2 1694.4.a.g.1.1 1
12.11 even 2 1008.4.a.s.1.1 1
13.12 even 2 2366.4.a.h.1.1 1
21.2 odd 6 882.4.g.b.361.1 2
21.5 even 6 882.4.g.k.361.1 2
21.11 odd 6 882.4.g.b.667.1 2
21.17 even 6 882.4.g.k.667.1 2
21.20 even 2 882.4.a.i.1.1 1
28.27 even 2 784.4.a.s.1.1 1
35.34 odd 2 2450.4.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.4.a.a.1.1 1 1.1 even 1 trivial
98.4.a.a.1.1 1 7.6 odd 2
98.4.c.d.67.1 2 7.2 even 3
98.4.c.d.79.1 2 7.4 even 3
98.4.c.f.67.1 2 7.5 odd 6
98.4.c.f.79.1 2 7.3 odd 6
112.4.a.a.1.1 1 4.3 odd 2
126.4.a.h.1.1 1 3.2 odd 2
350.4.a.l.1.1 1 5.4 even 2
350.4.c.b.99.1 2 5.2 odd 4
350.4.c.b.99.2 2 5.3 odd 4
448.4.a.b.1.1 1 8.5 even 2
448.4.a.o.1.1 1 8.3 odd 2
784.4.a.s.1.1 1 28.27 even 2
882.4.a.i.1.1 1 21.20 even 2
882.4.g.b.361.1 2 21.2 odd 6
882.4.g.b.667.1 2 21.11 odd 6
882.4.g.k.361.1 2 21.5 even 6
882.4.g.k.667.1 2 21.17 even 6
1008.4.a.s.1.1 1 12.11 even 2
1694.4.a.g.1.1 1 11.10 odd 2
2366.4.a.h.1.1 1 13.12 even 2
2450.4.a.bo.1.1 1 35.34 odd 2