Properties

Label 966.4.a.a.1.1
Level $966$
Weight $4$
Character 966.1
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.9958450655\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +48.0000 q^{11} +12.0000 q^{12} +38.0000 q^{13} -14.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} +114.000 q^{17} -18.0000 q^{18} +56.0000 q^{19} -24.0000 q^{20} +21.0000 q^{21} -96.0000 q^{22} -23.0000 q^{23} -24.0000 q^{24} -89.0000 q^{25} -76.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -162.000 q^{29} +36.0000 q^{30} -16.0000 q^{31} -32.0000 q^{32} +144.000 q^{33} -228.000 q^{34} -42.0000 q^{35} +36.0000 q^{36} -46.0000 q^{37} -112.000 q^{38} +114.000 q^{39} +48.0000 q^{40} -342.000 q^{41} -42.0000 q^{42} +248.000 q^{43} +192.000 q^{44} -54.0000 q^{45} +46.0000 q^{46} -24.0000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +178.000 q^{50} +342.000 q^{51} +152.000 q^{52} +426.000 q^{53} -54.0000 q^{54} -288.000 q^{55} -56.0000 q^{56} +168.000 q^{57} +324.000 q^{58} -852.000 q^{59} -72.0000 q^{60} +338.000 q^{61} +32.0000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -228.000 q^{65} -288.000 q^{66} +488.000 q^{67} +456.000 q^{68} -69.0000 q^{69} +84.0000 q^{70} +336.000 q^{71} -72.0000 q^{72} +362.000 q^{73} +92.0000 q^{74} -267.000 q^{75} +224.000 q^{76} +336.000 q^{77} -228.000 q^{78} +1184.00 q^{79} -96.0000 q^{80} +81.0000 q^{81} +684.000 q^{82} -336.000 q^{83} +84.0000 q^{84} -684.000 q^{85} -496.000 q^{86} -486.000 q^{87} -384.000 q^{88} -78.0000 q^{89} +108.000 q^{90} +266.000 q^{91} -92.0000 q^{92} -48.0000 q^{93} +48.0000 q^{94} -336.000 q^{95} -96.0000 q^{96} +746.000 q^{97} -98.0000 q^{98} +432.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) −6.00000 −0.408248
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 12.0000 0.379473
\(11\) 48.0000 1.31569 0.657843 0.753155i \(-0.271469\pi\)
0.657843 + 0.753155i \(0.271469\pi\)
\(12\) 12.0000 0.288675
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) −14.0000 −0.267261
\(15\) −18.0000 −0.309839
\(16\) 16.0000 0.250000
\(17\) 114.000 1.62642 0.813208 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) −18.0000 −0.235702
\(19\) 56.0000 0.676173 0.338086 0.941115i \(-0.390220\pi\)
0.338086 + 0.941115i \(0.390220\pi\)
\(20\) −24.0000 −0.268328
\(21\) 21.0000 0.218218
\(22\) −96.0000 −0.930330
\(23\) −23.0000 −0.208514
\(24\) −24.0000 −0.204124
\(25\) −89.0000 −0.712000
\(26\) −76.0000 −0.573263
\(27\) 27.0000 0.192450
\(28\) 28.0000 0.188982
\(29\) −162.000 −1.03733 −0.518666 0.854977i \(-0.673571\pi\)
−0.518666 + 0.854977i \(0.673571\pi\)
\(30\) 36.0000 0.219089
\(31\) −16.0000 −0.0926995 −0.0463498 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) −32.0000 −0.176777
\(33\) 144.000 0.759612
\(34\) −228.000 −1.15005
\(35\) −42.0000 −0.202837
\(36\) 36.0000 0.166667
\(37\) −46.0000 −0.204388 −0.102194 0.994764i \(-0.532586\pi\)
−0.102194 + 0.994764i \(0.532586\pi\)
\(38\) −112.000 −0.478126
\(39\) 114.000 0.468067
\(40\) 48.0000 0.189737
\(41\) −342.000 −1.30272 −0.651359 0.758770i \(-0.725801\pi\)
−0.651359 + 0.758770i \(0.725801\pi\)
\(42\) −42.0000 −0.154303
\(43\) 248.000 0.879527 0.439763 0.898114i \(-0.355062\pi\)
0.439763 + 0.898114i \(0.355062\pi\)
\(44\) 192.000 0.657843
\(45\) −54.0000 −0.178885
\(46\) 46.0000 0.147442
\(47\) −24.0000 −0.0744843 −0.0372421 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 48.0000 0.144338
\(49\) 49.0000 0.142857
\(50\) 178.000 0.503460
\(51\) 342.000 0.939011
\(52\) 152.000 0.405358
\(53\) 426.000 1.10407 0.552034 0.833822i \(-0.313852\pi\)
0.552034 + 0.833822i \(0.313852\pi\)
\(54\) −54.0000 −0.136083
\(55\) −288.000 −0.706071
\(56\) −56.0000 −0.133631
\(57\) 168.000 0.390388
\(58\) 324.000 0.733505
\(59\) −852.000 −1.88002 −0.940008 0.341152i \(-0.889183\pi\)
−0.940008 + 0.341152i \(0.889183\pi\)
\(60\) −72.0000 −0.154919
\(61\) 338.000 0.709450 0.354725 0.934971i \(-0.384574\pi\)
0.354725 + 0.934971i \(0.384574\pi\)
\(62\) 32.0000 0.0655485
\(63\) 63.0000 0.125988
\(64\) 64.0000 0.125000
\(65\) −228.000 −0.435076
\(66\) −288.000 −0.537127
\(67\) 488.000 0.889831 0.444916 0.895573i \(-0.353234\pi\)
0.444916 + 0.895573i \(0.353234\pi\)
\(68\) 456.000 0.813208
\(69\) −69.0000 −0.120386
\(70\) 84.0000 0.143427
\(71\) 336.000 0.561632 0.280816 0.959762i \(-0.409395\pi\)
0.280816 + 0.959762i \(0.409395\pi\)
\(72\) −72.0000 −0.117851
\(73\) 362.000 0.580396 0.290198 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) 92.0000 0.144524
\(75\) −267.000 −0.411073
\(76\) 224.000 0.338086
\(77\) 336.000 0.497283
\(78\) −228.000 −0.330973
\(79\) 1184.00 1.68621 0.843104 0.537751i \(-0.180726\pi\)
0.843104 + 0.537751i \(0.180726\pi\)
\(80\) −96.0000 −0.134164
\(81\) 81.0000 0.111111
\(82\) 684.000 0.921161
\(83\) −336.000 −0.444347 −0.222173 0.975007i \(-0.571315\pi\)
−0.222173 + 0.975007i \(0.571315\pi\)
\(84\) 84.0000 0.109109
\(85\) −684.000 −0.872826
\(86\) −496.000 −0.621919
\(87\) −486.000 −0.598904
\(88\) −384.000 −0.465165
\(89\) −78.0000 −0.0928987 −0.0464493 0.998921i \(-0.514791\pi\)
−0.0464493 + 0.998921i \(0.514791\pi\)
\(90\) 108.000 0.126491
\(91\) 266.000 0.306422
\(92\) −92.0000 −0.104257
\(93\) −48.0000 −0.0535201
\(94\) 48.0000 0.0526683
\(95\) −336.000 −0.362872
\(96\) −96.0000 −0.102062
\(97\) 746.000 0.780874 0.390437 0.920630i \(-0.372324\pi\)
0.390437 + 0.920630i \(0.372324\pi\)
\(98\) −98.0000 −0.101015
\(99\) 432.000 0.438562
\(100\) −356.000 −0.356000
\(101\) 30.0000 0.0295556 0.0147778 0.999891i \(-0.495296\pi\)
0.0147778 + 0.999891i \(0.495296\pi\)
\(102\) −684.000 −0.663981
\(103\) 944.000 0.903059 0.451530 0.892256i \(-0.350879\pi\)
0.451530 + 0.892256i \(0.350879\pi\)
\(104\) −304.000 −0.286631
\(105\) −126.000 −0.117108
\(106\) −852.000 −0.780694
\(107\) 264.000 0.238522 0.119261 0.992863i \(-0.461947\pi\)
0.119261 + 0.992863i \(0.461947\pi\)
\(108\) 108.000 0.0962250
\(109\) 578.000 0.507912 0.253956 0.967216i \(-0.418268\pi\)
0.253956 + 0.967216i \(0.418268\pi\)
\(110\) 576.000 0.499268
\(111\) −138.000 −0.118003
\(112\) 112.000 0.0944911
\(113\) 282.000 0.234764 0.117382 0.993087i \(-0.462550\pi\)
0.117382 + 0.993087i \(0.462550\pi\)
\(114\) −336.000 −0.276046
\(115\) 138.000 0.111901
\(116\) −648.000 −0.518666
\(117\) 342.000 0.270239
\(118\) 1704.00 1.32937
\(119\) 798.000 0.614727
\(120\) 144.000 0.109545
\(121\) 973.000 0.731029
\(122\) −676.000 −0.501657
\(123\) −1026.00 −0.752124
\(124\) −64.0000 −0.0463498
\(125\) 1284.00 0.918756
\(126\) −126.000 −0.0890871
\(127\) −856.000 −0.598092 −0.299046 0.954239i \(-0.596668\pi\)
−0.299046 + 0.954239i \(0.596668\pi\)
\(128\) −128.000 −0.0883883
\(129\) 744.000 0.507795
\(130\) 456.000 0.307645
\(131\) −348.000 −0.232098 −0.116049 0.993243i \(-0.537023\pi\)
−0.116049 + 0.993243i \(0.537023\pi\)
\(132\) 576.000 0.379806
\(133\) 392.000 0.255569
\(134\) −976.000 −0.629206
\(135\) −162.000 −0.103280
\(136\) −912.000 −0.575025
\(137\) −2598.00 −1.62016 −0.810081 0.586318i \(-0.800577\pi\)
−0.810081 + 0.586318i \(0.800577\pi\)
\(138\) 138.000 0.0851257
\(139\) −2692.00 −1.64268 −0.821340 0.570439i \(-0.806773\pi\)
−0.821340 + 0.570439i \(0.806773\pi\)
\(140\) −168.000 −0.101419
\(141\) −72.0000 −0.0430035
\(142\) −672.000 −0.397134
\(143\) 1824.00 1.06665
\(144\) 144.000 0.0833333
\(145\) 972.000 0.556691
\(146\) −724.000 −0.410402
\(147\) 147.000 0.0824786
\(148\) −184.000 −0.102194
\(149\) −966.000 −0.531126 −0.265563 0.964093i \(-0.585558\pi\)
−0.265563 + 0.964093i \(0.585558\pi\)
\(150\) 534.000 0.290673
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −448.000 −0.239063
\(153\) 1026.00 0.542138
\(154\) −672.000 −0.351632
\(155\) 96.0000 0.0497478
\(156\) 456.000 0.234033
\(157\) 362.000 0.184017 0.0920087 0.995758i \(-0.470671\pi\)
0.0920087 + 0.995758i \(0.470671\pi\)
\(158\) −2368.00 −1.19233
\(159\) 1278.00 0.637434
\(160\) 192.000 0.0948683
\(161\) −161.000 −0.0788110
\(162\) −162.000 −0.0785674
\(163\) 3164.00 1.52039 0.760195 0.649695i \(-0.225103\pi\)
0.760195 + 0.649695i \(0.225103\pi\)
\(164\) −1368.00 −0.651359
\(165\) −864.000 −0.407650
\(166\) 672.000 0.314201
\(167\) 2352.00 1.08984 0.544920 0.838488i \(-0.316560\pi\)
0.544920 + 0.838488i \(0.316560\pi\)
\(168\) −168.000 −0.0771517
\(169\) −753.000 −0.342740
\(170\) 1368.00 0.617181
\(171\) 504.000 0.225391
\(172\) 992.000 0.439763
\(173\) −1458.00 −0.640750 −0.320375 0.947291i \(-0.603809\pi\)
−0.320375 + 0.947291i \(0.603809\pi\)
\(174\) 972.000 0.423489
\(175\) −623.000 −0.269111
\(176\) 768.000 0.328921
\(177\) −2556.00 −1.08543
\(178\) 156.000 0.0656893
\(179\) 2076.00 0.866858 0.433429 0.901188i \(-0.357304\pi\)
0.433429 + 0.901188i \(0.357304\pi\)
\(180\) −216.000 −0.0894427
\(181\) −358.000 −0.147016 −0.0735081 0.997295i \(-0.523419\pi\)
−0.0735081 + 0.997295i \(0.523419\pi\)
\(182\) −532.000 −0.216673
\(183\) 1014.00 0.409601
\(184\) 184.000 0.0737210
\(185\) 276.000 0.109686
\(186\) 96.0000 0.0378444
\(187\) 5472.00 2.13985
\(188\) −96.0000 −0.0372421
\(189\) 189.000 0.0727393
\(190\) 672.000 0.256589
\(191\) 3960.00 1.50019 0.750093 0.661332i \(-0.230009\pi\)
0.750093 + 0.661332i \(0.230009\pi\)
\(192\) 192.000 0.0721688
\(193\) −1198.00 −0.446808 −0.223404 0.974726i \(-0.571717\pi\)
−0.223404 + 0.974726i \(0.571717\pi\)
\(194\) −1492.00 −0.552162
\(195\) −684.000 −0.251191
\(196\) 196.000 0.0714286
\(197\) 1446.00 0.522961 0.261480 0.965209i \(-0.415789\pi\)
0.261480 + 0.965209i \(0.415789\pi\)
\(198\) −864.000 −0.310110
\(199\) 4952.00 1.76401 0.882005 0.471240i \(-0.156193\pi\)
0.882005 + 0.471240i \(0.156193\pi\)
\(200\) 712.000 0.251730
\(201\) 1464.00 0.513744
\(202\) −60.0000 −0.0208989
\(203\) −1134.00 −0.392075
\(204\) 1368.00 0.469506
\(205\) 2052.00 0.699112
\(206\) −1888.00 −0.638559
\(207\) −207.000 −0.0695048
\(208\) 608.000 0.202679
\(209\) 2688.00 0.889631
\(210\) 252.000 0.0828079
\(211\) −1084.00 −0.353676 −0.176838 0.984240i \(-0.556587\pi\)
−0.176838 + 0.984240i \(0.556587\pi\)
\(212\) 1704.00 0.552034
\(213\) 1008.00 0.324258
\(214\) −528.000 −0.168660
\(215\) −1488.00 −0.472004
\(216\) −216.000 −0.0680414
\(217\) −112.000 −0.0350371
\(218\) −1156.00 −0.359148
\(219\) 1086.00 0.335092
\(220\) −1152.00 −0.353036
\(221\) 4332.00 1.31856
\(222\) 276.000 0.0834410
\(223\) 1856.00 0.557341 0.278670 0.960387i \(-0.410106\pi\)
0.278670 + 0.960387i \(0.410106\pi\)
\(224\) −224.000 −0.0668153
\(225\) −801.000 −0.237333
\(226\) −564.000 −0.166003
\(227\) −168.000 −0.0491214 −0.0245607 0.999698i \(-0.507819\pi\)
−0.0245607 + 0.999698i \(0.507819\pi\)
\(228\) 672.000 0.195194
\(229\) 2858.00 0.824725 0.412362 0.911020i \(-0.364704\pi\)
0.412362 + 0.911020i \(0.364704\pi\)
\(230\) −276.000 −0.0791257
\(231\) 1008.00 0.287106
\(232\) 1296.00 0.366752
\(233\) −3942.00 −1.10836 −0.554182 0.832395i \(-0.686969\pi\)
−0.554182 + 0.832395i \(0.686969\pi\)
\(234\) −684.000 −0.191088
\(235\) 144.000 0.0399724
\(236\) −3408.00 −0.940008
\(237\) 3552.00 0.973532
\(238\) −1596.00 −0.434678
\(239\) 6048.00 1.63687 0.818436 0.574597i \(-0.194841\pi\)
0.818436 + 0.574597i \(0.194841\pi\)
\(240\) −288.000 −0.0774597
\(241\) −598.000 −0.159836 −0.0799182 0.996801i \(-0.525466\pi\)
−0.0799182 + 0.996801i \(0.525466\pi\)
\(242\) −1946.00 −0.516916
\(243\) 243.000 0.0641500
\(244\) 1352.00 0.354725
\(245\) −294.000 −0.0766652
\(246\) 2052.00 0.531832
\(247\) 2128.00 0.548184
\(248\) 128.000 0.0327742
\(249\) −1008.00 −0.256544
\(250\) −2568.00 −0.649658
\(251\) −600.000 −0.150883 −0.0754416 0.997150i \(-0.524037\pi\)
−0.0754416 + 0.997150i \(0.524037\pi\)
\(252\) 252.000 0.0629941
\(253\) −1104.00 −0.274339
\(254\) 1712.00 0.422915
\(255\) −2052.00 −0.503926
\(256\) 256.000 0.0625000
\(257\) 882.000 0.214076 0.107038 0.994255i \(-0.465863\pi\)
0.107038 + 0.994255i \(0.465863\pi\)
\(258\) −1488.00 −0.359065
\(259\) −322.000 −0.0772514
\(260\) −912.000 −0.217538
\(261\) −1458.00 −0.345778
\(262\) 696.000 0.164118
\(263\) −4920.00 −1.15354 −0.576768 0.816908i \(-0.695687\pi\)
−0.576768 + 0.816908i \(0.695687\pi\)
\(264\) −1152.00 −0.268563
\(265\) −2556.00 −0.592505
\(266\) −784.000 −0.180715
\(267\) −234.000 −0.0536351
\(268\) 1952.00 0.444916
\(269\) 942.000 0.213512 0.106756 0.994285i \(-0.465954\pi\)
0.106756 + 0.994285i \(0.465954\pi\)
\(270\) 324.000 0.0730297
\(271\) 824.000 0.184703 0.0923514 0.995726i \(-0.470562\pi\)
0.0923514 + 0.995726i \(0.470562\pi\)
\(272\) 1824.00 0.406604
\(273\) 798.000 0.176913
\(274\) 5196.00 1.14563
\(275\) −4272.00 −0.936768
\(276\) −276.000 −0.0601929
\(277\) −4906.00 −1.06416 −0.532081 0.846693i \(-0.678590\pi\)
−0.532081 + 0.846693i \(0.678590\pi\)
\(278\) 5384.00 1.16155
\(279\) −144.000 −0.0308998
\(280\) 336.000 0.0717137
\(281\) −3822.00 −0.811393 −0.405696 0.914008i \(-0.632971\pi\)
−0.405696 + 0.914008i \(0.632971\pi\)
\(282\) 144.000 0.0304081
\(283\) 2408.00 0.505798 0.252899 0.967493i \(-0.418616\pi\)
0.252899 + 0.967493i \(0.418616\pi\)
\(284\) 1344.00 0.280816
\(285\) −1008.00 −0.209504
\(286\) −3648.00 −0.754233
\(287\) −2394.00 −0.492381
\(288\) −288.000 −0.0589256
\(289\) 8083.00 1.64523
\(290\) −1944.00 −0.393640
\(291\) 2238.00 0.450838
\(292\) 1448.00 0.290198
\(293\) −1014.00 −0.202179 −0.101090 0.994877i \(-0.532233\pi\)
−0.101090 + 0.994877i \(0.532233\pi\)
\(294\) −294.000 −0.0583212
\(295\) 5112.00 1.00892
\(296\) 368.000 0.0722620
\(297\) 1296.00 0.253204
\(298\) 1932.00 0.375563
\(299\) −874.000 −0.169046
\(300\) −1068.00 −0.205537
\(301\) 1736.00 0.332430
\(302\) −2416.00 −0.460348
\(303\) 90.0000 0.0170639
\(304\) 896.000 0.169043
\(305\) −2028.00 −0.380731
\(306\) −2052.00 −0.383350
\(307\) −3988.00 −0.741391 −0.370696 0.928754i \(-0.620881\pi\)
−0.370696 + 0.928754i \(0.620881\pi\)
\(308\) 1344.00 0.248641
\(309\) 2832.00 0.521381
\(310\) −192.000 −0.0351770
\(311\) 4680.00 0.853307 0.426653 0.904415i \(-0.359692\pi\)
0.426653 + 0.904415i \(0.359692\pi\)
\(312\) −912.000 −0.165487
\(313\) 2450.00 0.442435 0.221218 0.975224i \(-0.428997\pi\)
0.221218 + 0.975224i \(0.428997\pi\)
\(314\) −724.000 −0.130120
\(315\) −378.000 −0.0676123
\(316\) 4736.00 0.843104
\(317\) 6414.00 1.13642 0.568212 0.822883i \(-0.307636\pi\)
0.568212 + 0.822883i \(0.307636\pi\)
\(318\) −2556.00 −0.450734
\(319\) −7776.00 −1.36480
\(320\) −384.000 −0.0670820
\(321\) 792.000 0.137711
\(322\) 322.000 0.0557278
\(323\) 6384.00 1.09974
\(324\) 324.000 0.0555556
\(325\) −3382.00 −0.577230
\(326\) −6328.00 −1.07508
\(327\) 1734.00 0.293243
\(328\) 2736.00 0.460580
\(329\) −168.000 −0.0281524
\(330\) 1728.00 0.288252
\(331\) −7540.00 −1.25207 −0.626036 0.779794i \(-0.715324\pi\)
−0.626036 + 0.779794i \(0.715324\pi\)
\(332\) −1344.00 −0.222173
\(333\) −414.000 −0.0681293
\(334\) −4704.00 −0.770633
\(335\) −2928.00 −0.477534
\(336\) 336.000 0.0545545
\(337\) −2014.00 −0.325548 −0.162774 0.986663i \(-0.552044\pi\)
−0.162774 + 0.986663i \(0.552044\pi\)
\(338\) 1506.00 0.242354
\(339\) 846.000 0.135541
\(340\) −2736.00 −0.436413
\(341\) −768.000 −0.121963
\(342\) −1008.00 −0.159375
\(343\) 343.000 0.0539949
\(344\) −1984.00 −0.310960
\(345\) 414.000 0.0646058
\(346\) 2916.00 0.453078
\(347\) −4428.00 −0.685036 −0.342518 0.939511i \(-0.611280\pi\)
−0.342518 + 0.939511i \(0.611280\pi\)
\(348\) −1944.00 −0.299452
\(349\) 5582.00 0.856154 0.428077 0.903742i \(-0.359191\pi\)
0.428077 + 0.903742i \(0.359191\pi\)
\(350\) 1246.00 0.190290
\(351\) 1026.00 0.156022
\(352\) −1536.00 −0.232583
\(353\) −5934.00 −0.894717 −0.447358 0.894355i \(-0.647635\pi\)
−0.447358 + 0.894355i \(0.647635\pi\)
\(354\) 5112.00 0.767513
\(355\) −2016.00 −0.301403
\(356\) −312.000 −0.0464493
\(357\) 2394.00 0.354913
\(358\) −4152.00 −0.612961
\(359\) 2256.00 0.331663 0.165832 0.986154i \(-0.446969\pi\)
0.165832 + 0.986154i \(0.446969\pi\)
\(360\) 432.000 0.0632456
\(361\) −3723.00 −0.542790
\(362\) 716.000 0.103956
\(363\) 2919.00 0.422060
\(364\) 1064.00 0.153211
\(365\) −2172.00 −0.311473
\(366\) −2028.00 −0.289632
\(367\) 3296.00 0.468801 0.234400 0.972140i \(-0.424687\pi\)
0.234400 + 0.972140i \(0.424687\pi\)
\(368\) −368.000 −0.0521286
\(369\) −3078.00 −0.434239
\(370\) −552.000 −0.0775598
\(371\) 2982.00 0.417298
\(372\) −192.000 −0.0267600
\(373\) 13250.0 1.83930 0.919650 0.392739i \(-0.128472\pi\)
0.919650 + 0.392739i \(0.128472\pi\)
\(374\) −10944.0 −1.51310
\(375\) 3852.00 0.530444
\(376\) 192.000 0.0263342
\(377\) −6156.00 −0.840982
\(378\) −378.000 −0.0514344
\(379\) −8128.00 −1.10160 −0.550801 0.834636i \(-0.685678\pi\)
−0.550801 + 0.834636i \(0.685678\pi\)
\(380\) −1344.00 −0.181436
\(381\) −2568.00 −0.345309
\(382\) −7920.00 −1.06079
\(383\) −8976.00 −1.19753 −0.598763 0.800927i \(-0.704341\pi\)
−0.598763 + 0.800927i \(0.704341\pi\)
\(384\) −384.000 −0.0510310
\(385\) −2016.00 −0.266870
\(386\) 2396.00 0.315941
\(387\) 2232.00 0.293176
\(388\) 2984.00 0.390437
\(389\) −14478.0 −1.88705 −0.943527 0.331297i \(-0.892514\pi\)
−0.943527 + 0.331297i \(0.892514\pi\)
\(390\) 1368.00 0.177619
\(391\) −2622.00 −0.339131
\(392\) −392.000 −0.0505076
\(393\) −1044.00 −0.134002
\(394\) −2892.00 −0.369789
\(395\) −7104.00 −0.904914
\(396\) 1728.00 0.219281
\(397\) −14122.0 −1.78530 −0.892648 0.450754i \(-0.851155\pi\)
−0.892648 + 0.450754i \(0.851155\pi\)
\(398\) −9904.00 −1.24734
\(399\) 1176.00 0.147553
\(400\) −1424.00 −0.178000
\(401\) 1266.00 0.157658 0.0788292 0.996888i \(-0.474882\pi\)
0.0788292 + 0.996888i \(0.474882\pi\)
\(402\) −2928.00 −0.363272
\(403\) −608.000 −0.0751529
\(404\) 120.000 0.0147778
\(405\) −486.000 −0.0596285
\(406\) 2268.00 0.277239
\(407\) −2208.00 −0.268910
\(408\) −2736.00 −0.331991
\(409\) 6554.00 0.792358 0.396179 0.918173i \(-0.370336\pi\)
0.396179 + 0.918173i \(0.370336\pi\)
\(410\) −4104.00 −0.494347
\(411\) −7794.00 −0.935401
\(412\) 3776.00 0.451530
\(413\) −5964.00 −0.710579
\(414\) 414.000 0.0491473
\(415\) 2016.00 0.238462
\(416\) −1216.00 −0.143316
\(417\) −8076.00 −0.948401
\(418\) −5376.00 −0.629064
\(419\) −2016.00 −0.235055 −0.117527 0.993070i \(-0.537497\pi\)
−0.117527 + 0.993070i \(0.537497\pi\)
\(420\) −504.000 −0.0585540
\(421\) 4322.00 0.500336 0.250168 0.968202i \(-0.419514\pi\)
0.250168 + 0.968202i \(0.419514\pi\)
\(422\) 2168.00 0.250087
\(423\) −216.000 −0.0248281
\(424\) −3408.00 −0.390347
\(425\) −10146.0 −1.15801
\(426\) −2016.00 −0.229285
\(427\) 2366.00 0.268147
\(428\) 1056.00 0.119261
\(429\) 5472.00 0.615829
\(430\) 2976.00 0.333757
\(431\) −11112.0 −1.24187 −0.620935 0.783862i \(-0.713247\pi\)
−0.620935 + 0.783862i \(0.713247\pi\)
\(432\) 432.000 0.0481125
\(433\) 12554.0 1.39332 0.696659 0.717402i \(-0.254669\pi\)
0.696659 + 0.717402i \(0.254669\pi\)
\(434\) 224.000 0.0247750
\(435\) 2916.00 0.321406
\(436\) 2312.00 0.253956
\(437\) −1288.00 −0.140992
\(438\) −2172.00 −0.236946
\(439\) 416.000 0.0452269 0.0226134 0.999744i \(-0.492801\pi\)
0.0226134 + 0.999744i \(0.492801\pi\)
\(440\) 2304.00 0.249634
\(441\) 441.000 0.0476190
\(442\) −8664.00 −0.932363
\(443\) −10044.0 −1.07721 −0.538606 0.842558i \(-0.681049\pi\)
−0.538606 + 0.842558i \(0.681049\pi\)
\(444\) −552.000 −0.0590017
\(445\) 468.000 0.0498547
\(446\) −3712.00 −0.394099
\(447\) −2898.00 −0.306646
\(448\) 448.000 0.0472456
\(449\) −14190.0 −1.49146 −0.745732 0.666246i \(-0.767900\pi\)
−0.745732 + 0.666246i \(0.767900\pi\)
\(450\) 1602.00 0.167820
\(451\) −16416.0 −1.71397
\(452\) 1128.00 0.117382
\(453\) 3624.00 0.375873
\(454\) 336.000 0.0347341
\(455\) −1596.00 −0.164443
\(456\) −1344.00 −0.138023
\(457\) 3794.00 0.388350 0.194175 0.980967i \(-0.437797\pi\)
0.194175 + 0.980967i \(0.437797\pi\)
\(458\) −5716.00 −0.583168
\(459\) 3078.00 0.313004
\(460\) 552.000 0.0559503
\(461\) 6366.00 0.643154 0.321577 0.946883i \(-0.395787\pi\)
0.321577 + 0.946883i \(0.395787\pi\)
\(462\) −2016.00 −0.203015
\(463\) 2936.00 0.294703 0.147352 0.989084i \(-0.452925\pi\)
0.147352 + 0.989084i \(0.452925\pi\)
\(464\) −2592.00 −0.259333
\(465\) 288.000 0.0287219
\(466\) 7884.00 0.783732
\(467\) 11904.0 1.17955 0.589777 0.807566i \(-0.299216\pi\)
0.589777 + 0.807566i \(0.299216\pi\)
\(468\) 1368.00 0.135119
\(469\) 3416.00 0.336325
\(470\) −288.000 −0.0282648
\(471\) 1086.00 0.106243
\(472\) 6816.00 0.664686
\(473\) 11904.0 1.15718
\(474\) −7104.00 −0.688391
\(475\) −4984.00 −0.481435
\(476\) 3192.00 0.307364
\(477\) 3834.00 0.368023
\(478\) −12096.0 −1.15744
\(479\) −15120.0 −1.44228 −0.721138 0.692791i \(-0.756381\pi\)
−0.721138 + 0.692791i \(0.756381\pi\)
\(480\) 576.000 0.0547723
\(481\) −1748.00 −0.165700
\(482\) 1196.00 0.113021
\(483\) −483.000 −0.0455016
\(484\) 3892.00 0.365515
\(485\) −4476.00 −0.419061
\(486\) −486.000 −0.0453609
\(487\) −5344.00 −0.497248 −0.248624 0.968600i \(-0.579978\pi\)
−0.248624 + 0.968600i \(0.579978\pi\)
\(488\) −2704.00 −0.250829
\(489\) 9492.00 0.877798
\(490\) 588.000 0.0542105
\(491\) 13908.0 1.27833 0.639164 0.769070i \(-0.279280\pi\)
0.639164 + 0.769070i \(0.279280\pi\)
\(492\) −4104.00 −0.376062
\(493\) −18468.0 −1.68713
\(494\) −4256.00 −0.387624
\(495\) −2592.00 −0.235357
\(496\) −256.000 −0.0231749
\(497\) 2352.00 0.212277
\(498\) 2016.00 0.181404
\(499\) −7780.00 −0.697957 −0.348979 0.937131i \(-0.613471\pi\)
−0.348979 + 0.937131i \(0.613471\pi\)
\(500\) 5136.00 0.459378
\(501\) 7056.00 0.629219
\(502\) 1200.00 0.106690
\(503\) 5808.00 0.514843 0.257421 0.966299i \(-0.417127\pi\)
0.257421 + 0.966299i \(0.417127\pi\)
\(504\) −504.000 −0.0445435
\(505\) −180.000 −0.0158612
\(506\) 2208.00 0.193987
\(507\) −2259.00 −0.197881
\(508\) −3424.00 −0.299046
\(509\) 8910.00 0.775892 0.387946 0.921682i \(-0.373185\pi\)
0.387946 + 0.921682i \(0.373185\pi\)
\(510\) 4104.00 0.356330
\(511\) 2534.00 0.219369
\(512\) −512.000 −0.0441942
\(513\) 1512.00 0.130129
\(514\) −1764.00 −0.151375
\(515\) −5664.00 −0.484632
\(516\) 2976.00 0.253897
\(517\) −1152.00 −0.0979979
\(518\) 644.000 0.0546250
\(519\) −4374.00 −0.369937
\(520\) 1824.00 0.153822
\(521\) 19002.0 1.59787 0.798937 0.601414i \(-0.205396\pi\)
0.798937 + 0.601414i \(0.205396\pi\)
\(522\) 2916.00 0.244502
\(523\) −4000.00 −0.334432 −0.167216 0.985920i \(-0.553478\pi\)
−0.167216 + 0.985920i \(0.553478\pi\)
\(524\) −1392.00 −0.116049
\(525\) −1869.00 −0.155371
\(526\) 9840.00 0.815674
\(527\) −1824.00 −0.150768
\(528\) 2304.00 0.189903
\(529\) 529.000 0.0434783
\(530\) 5112.00 0.418964
\(531\) −7668.00 −0.626672
\(532\) 1568.00 0.127785
\(533\) −12996.0 −1.05613
\(534\) 468.000 0.0379257
\(535\) −1584.00 −0.128004
\(536\) −3904.00 −0.314603
\(537\) 6228.00 0.500481
\(538\) −1884.00 −0.150976
\(539\) 2352.00 0.187955
\(540\) −648.000 −0.0516398
\(541\) 5150.00 0.409271 0.204636 0.978838i \(-0.434399\pi\)
0.204636 + 0.978838i \(0.434399\pi\)
\(542\) −1648.00 −0.130605
\(543\) −1074.00 −0.0848798
\(544\) −3648.00 −0.287512
\(545\) −3468.00 −0.272574
\(546\) −1596.00 −0.125096
\(547\) −2476.00 −0.193540 −0.0967698 0.995307i \(-0.530851\pi\)
−0.0967698 + 0.995307i \(0.530851\pi\)
\(548\) −10392.0 −0.810081
\(549\) 3042.00 0.236483
\(550\) 8544.00 0.662395
\(551\) −9072.00 −0.701416
\(552\) 552.000 0.0425628
\(553\) 8288.00 0.637327
\(554\) 9812.00 0.752476
\(555\) 828.000 0.0633273
\(556\) −10768.0 −0.821340
\(557\) −9798.00 −0.745340 −0.372670 0.927964i \(-0.621558\pi\)
−0.372670 + 0.927964i \(0.621558\pi\)
\(558\) 288.000 0.0218495
\(559\) 9424.00 0.713046
\(560\) −672.000 −0.0507093
\(561\) 16416.0 1.23544
\(562\) 7644.00 0.573741
\(563\) −1464.00 −0.109592 −0.0547960 0.998498i \(-0.517451\pi\)
−0.0547960 + 0.998498i \(0.517451\pi\)
\(564\) −288.000 −0.0215018
\(565\) −1692.00 −0.125988
\(566\) −4816.00 −0.357653
\(567\) 567.000 0.0419961
\(568\) −2688.00 −0.198567
\(569\) 9306.00 0.685638 0.342819 0.939402i \(-0.388618\pi\)
0.342819 + 0.939402i \(0.388618\pi\)
\(570\) 2016.00 0.148142
\(571\) −23056.0 −1.68978 −0.844889 0.534941i \(-0.820334\pi\)
−0.844889 + 0.534941i \(0.820334\pi\)
\(572\) 7296.00 0.533324
\(573\) 11880.0 0.866133
\(574\) 4788.00 0.348166
\(575\) 2047.00 0.148462
\(576\) 576.000 0.0416667
\(577\) −526.000 −0.0379509 −0.0189754 0.999820i \(-0.506040\pi\)
−0.0189754 + 0.999820i \(0.506040\pi\)
\(578\) −16166.0 −1.16335
\(579\) −3594.00 −0.257965
\(580\) 3888.00 0.278346
\(581\) −2352.00 −0.167947
\(582\) −4476.00 −0.318791
\(583\) 20448.0 1.45261
\(584\) −2896.00 −0.205201
\(585\) −2052.00 −0.145025
\(586\) 2028.00 0.142962
\(587\) 18876.0 1.32725 0.663625 0.748065i \(-0.269017\pi\)
0.663625 + 0.748065i \(0.269017\pi\)
\(588\) 588.000 0.0412393
\(589\) −896.000 −0.0626809
\(590\) −10224.0 −0.713416
\(591\) 4338.00 0.301931
\(592\) −736.000 −0.0510970
\(593\) −17214.0 −1.19206 −0.596032 0.802960i \(-0.703257\pi\)
−0.596032 + 0.802960i \(0.703257\pi\)
\(594\) −2592.00 −0.179042
\(595\) −4788.00 −0.329897
\(596\) −3864.00 −0.265563
\(597\) 14856.0 1.01845
\(598\) 1748.00 0.119534
\(599\) 2808.00 0.191539 0.0957694 0.995404i \(-0.469469\pi\)
0.0957694 + 0.995404i \(0.469469\pi\)
\(600\) 2136.00 0.145336
\(601\) 19898.0 1.35051 0.675255 0.737584i \(-0.264034\pi\)
0.675255 + 0.737584i \(0.264034\pi\)
\(602\) −3472.00 −0.235063
\(603\) 4392.00 0.296610
\(604\) 4832.00 0.325515
\(605\) −5838.00 −0.392311
\(606\) −180.000 −0.0120660
\(607\) −23344.0 −1.56096 −0.780481 0.625180i \(-0.785026\pi\)
−0.780481 + 0.625180i \(0.785026\pi\)
\(608\) −1792.00 −0.119532
\(609\) −3402.00 −0.226365
\(610\) 4056.00 0.269217
\(611\) −912.000 −0.0603855
\(612\) 4104.00 0.271069
\(613\) 2282.00 0.150357 0.0751787 0.997170i \(-0.476047\pi\)
0.0751787 + 0.997170i \(0.476047\pi\)
\(614\) 7976.00 0.524243
\(615\) 6156.00 0.403632
\(616\) −2688.00 −0.175816
\(617\) −28926.0 −1.88739 −0.943693 0.330823i \(-0.892674\pi\)
−0.943693 + 0.330823i \(0.892674\pi\)
\(618\) −5664.00 −0.368672
\(619\) −7288.00 −0.473230 −0.236615 0.971603i \(-0.576038\pi\)
−0.236615 + 0.971603i \(0.576038\pi\)
\(620\) 384.000 0.0248739
\(621\) −621.000 −0.0401286
\(622\) −9360.00 −0.603379
\(623\) −546.000 −0.0351124
\(624\) 1824.00 0.117017
\(625\) 3421.00 0.218944
\(626\) −4900.00 −0.312849
\(627\) 8064.00 0.513629
\(628\) 1448.00 0.0920087
\(629\) −5244.00 −0.332420
\(630\) 756.000 0.0478091
\(631\) 18128.0 1.14368 0.571842 0.820364i \(-0.306229\pi\)
0.571842 + 0.820364i \(0.306229\pi\)
\(632\) −9472.00 −0.596164
\(633\) −3252.00 −0.204195
\(634\) −12828.0 −0.803572
\(635\) 5136.00 0.320970
\(636\) 5112.00 0.318717
\(637\) 1862.00 0.115817
\(638\) 15552.0 0.965062
\(639\) 3024.00 0.187211
\(640\) 768.000 0.0474342
\(641\) 16626.0 1.02447 0.512237 0.858844i \(-0.328817\pi\)
0.512237 + 0.858844i \(0.328817\pi\)
\(642\) −1584.00 −0.0973762
\(643\) 14288.0 0.876304 0.438152 0.898901i \(-0.355633\pi\)
0.438152 + 0.898901i \(0.355633\pi\)
\(644\) −644.000 −0.0394055
\(645\) −4464.00 −0.272511
\(646\) −12768.0 −0.777632
\(647\) 27192.0 1.65228 0.826142 0.563462i \(-0.190531\pi\)
0.826142 + 0.563462i \(0.190531\pi\)
\(648\) −648.000 −0.0392837
\(649\) −40896.0 −2.47351
\(650\) 6764.00 0.408163
\(651\) −336.000 −0.0202287
\(652\) 12656.0 0.760195
\(653\) −9066.00 −0.543308 −0.271654 0.962395i \(-0.587571\pi\)
−0.271654 + 0.962395i \(0.587571\pi\)
\(654\) −3468.00 −0.207354
\(655\) 2088.00 0.124557
\(656\) −5472.00 −0.325679
\(657\) 3258.00 0.193465
\(658\) 336.000 0.0199068
\(659\) 9504.00 0.561796 0.280898 0.959738i \(-0.409368\pi\)
0.280898 + 0.959738i \(0.409368\pi\)
\(660\) −3456.00 −0.203825
\(661\) 16778.0 0.987275 0.493637 0.869668i \(-0.335667\pi\)
0.493637 + 0.869668i \(0.335667\pi\)
\(662\) 15080.0 0.885349
\(663\) 12996.0 0.761271
\(664\) 2688.00 0.157100
\(665\) −2352.00 −0.137153
\(666\) 828.000 0.0481747
\(667\) 3726.00 0.216299
\(668\) 9408.00 0.544920
\(669\) 5568.00 0.321781
\(670\) 5856.00 0.337667
\(671\) 16224.0 0.933414
\(672\) −672.000 −0.0385758
\(673\) −15982.0 −0.915395 −0.457697 0.889108i \(-0.651326\pi\)
−0.457697 + 0.889108i \(0.651326\pi\)
\(674\) 4028.00 0.230197
\(675\) −2403.00 −0.137024
\(676\) −3012.00 −0.171370
\(677\) 7674.00 0.435651 0.217826 0.975988i \(-0.430104\pi\)
0.217826 + 0.975988i \(0.430104\pi\)
\(678\) −1692.00 −0.0958420
\(679\) 5222.00 0.295143
\(680\) 5472.00 0.308591
\(681\) −504.000 −0.0283602
\(682\) 1536.00 0.0862412
\(683\) −12804.0 −0.717323 −0.358661 0.933468i \(-0.616767\pi\)
−0.358661 + 0.933468i \(0.616767\pi\)
\(684\) 2016.00 0.112695
\(685\) 15588.0 0.869470
\(686\) −686.000 −0.0381802
\(687\) 8574.00 0.476155
\(688\) 3968.00 0.219882
\(689\) 16188.0 0.895085
\(690\) −828.000 −0.0456832
\(691\) −6508.00 −0.358287 −0.179143 0.983823i \(-0.557333\pi\)
−0.179143 + 0.983823i \(0.557333\pi\)
\(692\) −5832.00 −0.320375
\(693\) 3024.00 0.165761
\(694\) 8856.00 0.484394
\(695\) 16152.0 0.881554
\(696\) 3888.00 0.211745
\(697\) −38988.0 −2.11876
\(698\) −11164.0 −0.605392
\(699\) −11826.0 −0.639915
\(700\) −2492.00 −0.134555
\(701\) −29094.0 −1.56757 −0.783784 0.621033i \(-0.786713\pi\)
−0.783784 + 0.621033i \(0.786713\pi\)
\(702\) −2052.00 −0.110324
\(703\) −2576.00 −0.138202
\(704\) 3072.00 0.164461
\(705\) 432.000 0.0230781
\(706\) 11868.0 0.632660
\(707\) 210.000 0.0111710
\(708\) −10224.0 −0.542714
\(709\) 14834.0 0.785758 0.392879 0.919590i \(-0.371479\pi\)
0.392879 + 0.919590i \(0.371479\pi\)
\(710\) 4032.00 0.213124
\(711\) 10656.0 0.562069
\(712\) 624.000 0.0328446
\(713\) 368.000 0.0193292
\(714\) −4788.00 −0.250961
\(715\) −10944.0 −0.572423
\(716\) 8304.00 0.433429
\(717\) 18144.0 0.945049
\(718\) −4512.00 −0.234521
\(719\) −37200.0 −1.92952 −0.964761 0.263129i \(-0.915246\pi\)
−0.964761 + 0.263129i \(0.915246\pi\)
\(720\) −864.000 −0.0447214
\(721\) 6608.00 0.341324
\(722\) 7446.00 0.383811
\(723\) −1794.00 −0.0922816
\(724\) −1432.00 −0.0735081
\(725\) 14418.0 0.738581
\(726\) −5838.00 −0.298441
\(727\) −29608.0 −1.51045 −0.755227 0.655463i \(-0.772473\pi\)
−0.755227 + 0.655463i \(0.772473\pi\)
\(728\) −2128.00 −0.108336
\(729\) 729.000 0.0370370
\(730\) 4344.00 0.220245
\(731\) 28272.0 1.43048
\(732\) 4056.00 0.204801
\(733\) 5930.00 0.298812 0.149406 0.988776i \(-0.452264\pi\)
0.149406 + 0.988776i \(0.452264\pi\)
\(734\) −6592.00 −0.331492
\(735\) −882.000 −0.0442627
\(736\) 736.000 0.0368605
\(737\) 23424.0 1.17074
\(738\) 6156.00 0.307054
\(739\) 19196.0 0.955529 0.477765 0.878488i \(-0.341447\pi\)
0.477765 + 0.878488i \(0.341447\pi\)
\(740\) 1104.00 0.0548430
\(741\) 6384.00 0.316494
\(742\) −5964.00 −0.295075
\(743\) 1512.00 0.0746567 0.0373283 0.999303i \(-0.488115\pi\)
0.0373283 + 0.999303i \(0.488115\pi\)
\(744\) 384.000 0.0189222
\(745\) 5796.00 0.285032
\(746\) −26500.0 −1.30058
\(747\) −3024.00 −0.148116
\(748\) 21888.0 1.06993
\(749\) 1848.00 0.0901528
\(750\) −7704.00 −0.375080
\(751\) 7400.00 0.359560 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(752\) −384.000 −0.0186211
\(753\) −1800.00 −0.0871124
\(754\) 12312.0 0.594664
\(755\) −7248.00 −0.349380
\(756\) 756.000 0.0363696
\(757\) −7894.00 −0.379012 −0.189506 0.981880i \(-0.560689\pi\)
−0.189506 + 0.981880i \(0.560689\pi\)
\(758\) 16256.0 0.778951
\(759\) −3312.00 −0.158390
\(760\) 2688.00 0.128295
\(761\) 7818.00 0.372408 0.186204 0.982511i \(-0.440382\pi\)
0.186204 + 0.982511i \(0.440382\pi\)
\(762\) 5136.00 0.244170
\(763\) 4046.00 0.191973
\(764\) 15840.0 0.750093
\(765\) −6156.00 −0.290942
\(766\) 17952.0 0.846778
\(767\) −32376.0 −1.52416
\(768\) 768.000 0.0360844
\(769\) −25990.0 −1.21876 −0.609378 0.792880i \(-0.708581\pi\)
−0.609378 + 0.792880i \(0.708581\pi\)
\(770\) 4032.00 0.188705
\(771\) 2646.00 0.123597
\(772\) −4792.00 −0.223404
\(773\) 20466.0 0.952278 0.476139 0.879370i \(-0.342036\pi\)
0.476139 + 0.879370i \(0.342036\pi\)
\(774\) −4464.00 −0.207306
\(775\) 1424.00 0.0660021
\(776\) −5968.00 −0.276081
\(777\) −966.000 −0.0446011
\(778\) 28956.0 1.33435
\(779\) −19152.0 −0.880862
\(780\) −2736.00 −0.125596
\(781\) 16128.0 0.738931
\(782\) 5244.00 0.239802
\(783\) −4374.00 −0.199635
\(784\) 784.000 0.0357143
\(785\) −2172.00 −0.0987541
\(786\) 2088.00 0.0947538
\(787\) −5248.00 −0.237701 −0.118851 0.992912i \(-0.537921\pi\)
−0.118851 + 0.992912i \(0.537921\pi\)
\(788\) 5784.00 0.261480
\(789\) −14760.0 −0.665995
\(790\) 14208.0 0.639871
\(791\) 1974.00 0.0887324
\(792\) −3456.00 −0.155055
\(793\) 12844.0 0.575162
\(794\) 28244.0 1.26240
\(795\) −7668.00 −0.342083
\(796\) 19808.0 0.882005
\(797\) 6234.00 0.277063 0.138532 0.990358i \(-0.455762\pi\)
0.138532 + 0.990358i \(0.455762\pi\)
\(798\) −2352.00 −0.104336
\(799\) −2736.00 −0.121142
\(800\) 2848.00 0.125865
\(801\) −702.000 −0.0309662
\(802\) −2532.00 −0.111481
\(803\) 17376.0 0.763619
\(804\) 5856.00 0.256872
\(805\) 966.000 0.0422944
\(806\) 1216.00 0.0531412
\(807\) 2826.00 0.123271
\(808\) −240.000 −0.0104495
\(809\) 14970.0 0.650577 0.325289 0.945615i \(-0.394539\pi\)
0.325289 + 0.945615i \(0.394539\pi\)
\(810\) 972.000 0.0421637
\(811\) −35620.0 −1.54228 −0.771139 0.636667i \(-0.780313\pi\)
−0.771139 + 0.636667i \(0.780313\pi\)
\(812\) −4536.00 −0.196037
\(813\) 2472.00 0.106638
\(814\) 4416.00 0.190148
\(815\) −18984.0 −0.815927
\(816\) 5472.00 0.234753
\(817\) 13888.0 0.594712
\(818\) −13108.0 −0.560282
\(819\) 2394.00 0.102141
\(820\) 8208.00 0.349556
\(821\) −25458.0 −1.08221 −0.541103 0.840957i \(-0.681993\pi\)
−0.541103 + 0.840957i \(0.681993\pi\)
\(822\) 15588.0 0.661428
\(823\) −32248.0 −1.36585 −0.682925 0.730488i \(-0.739292\pi\)
−0.682925 + 0.730488i \(0.739292\pi\)
\(824\) −7552.00 −0.319280
\(825\) −12816.0 −0.540843
\(826\) 11928.0 0.502455
\(827\) 31704.0 1.33308 0.666539 0.745470i \(-0.267775\pi\)
0.666539 + 0.745470i \(0.267775\pi\)
\(828\) −828.000 −0.0347524
\(829\) −33010.0 −1.38297 −0.691487 0.722389i \(-0.743044\pi\)
−0.691487 + 0.722389i \(0.743044\pi\)
\(830\) −4032.00 −0.168618
\(831\) −14718.0 −0.614394
\(832\) 2432.00 0.101339
\(833\) 5586.00 0.232345
\(834\) 16152.0 0.670621
\(835\) −14112.0 −0.584869
\(836\) 10752.0 0.444815
\(837\) −432.000 −0.0178400
\(838\) 4032.00 0.166209
\(839\) −264.000 −0.0108633 −0.00543164 0.999985i \(-0.501729\pi\)
−0.00543164 + 0.999985i \(0.501729\pi\)
\(840\) 1008.00 0.0414039
\(841\) 1855.00 0.0760589
\(842\) −8644.00 −0.353791
\(843\) −11466.0 −0.468458
\(844\) −4336.00 −0.176838
\(845\) 4518.00 0.183934
\(846\) 432.000 0.0175561
\(847\) 6811.00 0.276303
\(848\) 6816.00 0.276017
\(849\) 7224.00 0.292022
\(850\) 20292.0 0.818835
\(851\) 1058.00 0.0426178
\(852\) 4032.00 0.162129
\(853\) 21710.0 0.871438 0.435719 0.900083i \(-0.356494\pi\)
0.435719 + 0.900083i \(0.356494\pi\)
\(854\) −4732.00 −0.189609
\(855\) −3024.00 −0.120957
\(856\) −2112.00 −0.0843302
\(857\) 9546.00 0.380496 0.190248 0.981736i \(-0.439071\pi\)
0.190248 + 0.981736i \(0.439071\pi\)
\(858\) −10944.0 −0.435457
\(859\) −40156.0 −1.59500 −0.797500 0.603319i \(-0.793845\pi\)
−0.797500 + 0.603319i \(0.793845\pi\)
\(860\) −5952.00 −0.236002
\(861\) −7182.00 −0.284276
\(862\) 22224.0 0.878135
\(863\) −36096.0 −1.42378 −0.711890 0.702291i \(-0.752161\pi\)
−0.711890 + 0.702291i \(0.752161\pi\)
\(864\) −864.000 −0.0340207
\(865\) 8748.00 0.343862
\(866\) −25108.0 −0.985225
\(867\) 24249.0 0.949872
\(868\) −448.000 −0.0175186
\(869\) 56832.0 2.21852
\(870\) −5832.00 −0.227268
\(871\) 18544.0 0.721400
\(872\) −4624.00 −0.179574
\(873\) 6714.00 0.260291
\(874\) 2576.00 0.0996962
\(875\) 8988.00 0.347257
\(876\) 4344.00 0.167546
\(877\) 22094.0 0.850697 0.425349 0.905030i \(-0.360152\pi\)
0.425349 + 0.905030i \(0.360152\pi\)
\(878\) −832.000 −0.0319802
\(879\) −3042.00 −0.116728
\(880\) −4608.00 −0.176518
\(881\) 19866.0 0.759708 0.379854 0.925046i \(-0.375974\pi\)
0.379854 + 0.925046i \(0.375974\pi\)
\(882\) −882.000 −0.0336718
\(883\) 41348.0 1.57584 0.787922 0.615775i \(-0.211157\pi\)
0.787922 + 0.615775i \(0.211157\pi\)
\(884\) 17328.0 0.659280
\(885\) 15336.0 0.582502
\(886\) 20088.0 0.761704
\(887\) −44448.0 −1.68255 −0.841273 0.540611i \(-0.818193\pi\)
−0.841273 + 0.540611i \(0.818193\pi\)
\(888\) 1104.00 0.0417205
\(889\) −5992.00 −0.226058
\(890\) −936.000 −0.0352526
\(891\) 3888.00 0.146187
\(892\) 7424.00 0.278670
\(893\) −1344.00 −0.0503642
\(894\) 5796.00 0.216831
\(895\) −12456.0 −0.465205
\(896\) −896.000 −0.0334077
\(897\) −2622.00 −0.0975987
\(898\) 28380.0 1.05462
\(899\) 2592.00 0.0961602
\(900\) −3204.00 −0.118667
\(901\) 48564.0 1.79567
\(902\) 32832.0 1.21196
\(903\) 5208.00 0.191928
\(904\) −2256.00 −0.0830016
\(905\) 2148.00 0.0788972
\(906\) −7248.00 −0.265782
\(907\) −1144.00 −0.0418808 −0.0209404 0.999781i \(-0.506666\pi\)
−0.0209404 + 0.999781i \(0.506666\pi\)
\(908\) −672.000 −0.0245607
\(909\) 270.000 0.00985185
\(910\) 3192.00 0.116279
\(911\) −25848.0 −0.940047 −0.470023 0.882654i \(-0.655754\pi\)
−0.470023 + 0.882654i \(0.655754\pi\)
\(912\) 2688.00 0.0975971
\(913\) −16128.0 −0.584621
\(914\) −7588.00 −0.274605
\(915\) −6084.00 −0.219815
\(916\) 11432.0 0.412362
\(917\) −2436.00 −0.0877250
\(918\) −6156.00 −0.221327
\(919\) −30808.0 −1.10583 −0.552917 0.833236i \(-0.686485\pi\)
−0.552917 + 0.833236i \(0.686485\pi\)
\(920\) −1104.00 −0.0395628
\(921\) −11964.0 −0.428043
\(922\) −12732.0 −0.454779
\(923\) 12768.0 0.455324
\(924\) 4032.00 0.143553
\(925\) 4094.00 0.145524
\(926\) −5872.00 −0.208386
\(927\) 8496.00 0.301020
\(928\) 5184.00 0.183376
\(929\) 21810.0 0.770251 0.385125 0.922864i \(-0.374158\pi\)
0.385125 + 0.922864i \(0.374158\pi\)
\(930\) −576.000 −0.0203094
\(931\) 2744.00 0.0965961
\(932\) −15768.0 −0.554182
\(933\) 14040.0 0.492657
\(934\) −23808.0 −0.834070
\(935\) −32832.0 −1.14836
\(936\) −2736.00 −0.0955438
\(937\) −23734.0 −0.827488 −0.413744 0.910393i \(-0.635779\pi\)
−0.413744 + 0.910393i \(0.635779\pi\)
\(938\) −6832.00 −0.237817
\(939\) 7350.00 0.255440
\(940\) 576.000 0.0199862
\(941\) 34050.0 1.17959 0.589797 0.807551i \(-0.299208\pi\)
0.589797 + 0.807551i \(0.299208\pi\)
\(942\) −2172.00 −0.0751248
\(943\) 7866.00 0.271635
\(944\) −13632.0 −0.470004
\(945\) −1134.00 −0.0390360
\(946\) −23808.0 −0.818250
\(947\) 20700.0 0.710306 0.355153 0.934808i \(-0.384429\pi\)
0.355153 + 0.934808i \(0.384429\pi\)
\(948\) 14208.0 0.486766
\(949\) 13756.0 0.470536
\(950\) 9968.00 0.340426
\(951\) 19242.0 0.656114
\(952\) −6384.00 −0.217339
\(953\) 52314.0 1.77819 0.889096 0.457721i \(-0.151334\pi\)
0.889096 + 0.457721i \(0.151334\pi\)
\(954\) −7668.00 −0.260231
\(955\) −23760.0 −0.805084
\(956\) 24192.0 0.818436
\(957\) −23328.0 −0.787970
\(958\) 30240.0 1.01984
\(959\) −18186.0 −0.612363
\(960\) −1152.00 −0.0387298
\(961\) −29535.0 −0.991407
\(962\) 3496.00 0.117168
\(963\) 2376.00 0.0795073
\(964\) −2392.00 −0.0799182
\(965\) 7188.00 0.239782
\(966\) 966.000 0.0321745
\(967\) −32968.0 −1.09636 −0.548180 0.836361i \(-0.684679\pi\)
−0.548180 + 0.836361i \(0.684679\pi\)
\(968\) −7784.00 −0.258458
\(969\) 19152.0 0.634934
\(970\) 8952.00 0.296321
\(971\) 2160.00 0.0713879 0.0356940 0.999363i \(-0.488636\pi\)
0.0356940 + 0.999363i \(0.488636\pi\)
\(972\) 972.000 0.0320750
\(973\) −18844.0 −0.620875
\(974\) 10688.0 0.351607
\(975\) −10146.0 −0.333264
\(976\) 5408.00 0.177363
\(977\) −46446.0 −1.52092 −0.760460 0.649385i \(-0.775027\pi\)
−0.760460 + 0.649385i \(0.775027\pi\)
\(978\) −18984.0 −0.620697
\(979\) −3744.00 −0.122225
\(980\) −1176.00 −0.0383326
\(981\) 5202.00 0.169304
\(982\) −27816.0 −0.903915
\(983\) 24432.0 0.792736 0.396368 0.918092i \(-0.370270\pi\)
0.396368 + 0.918092i \(0.370270\pi\)
\(984\) 8208.00 0.265916
\(985\) −8676.00 −0.280650
\(986\) 36936.0 1.19298
\(987\) −504.000 −0.0162538
\(988\) 8512.00 0.274092
\(989\) −5704.00 −0.183394
\(990\) 5184.00 0.166423
\(991\) 3920.00 0.125654 0.0628269 0.998024i \(-0.479988\pi\)
0.0628269 + 0.998024i \(0.479988\pi\)
\(992\) 512.000 0.0163871
\(993\) −22620.0 −0.722884
\(994\) −4704.00 −0.150102
\(995\) −29712.0 −0.946667
\(996\) −4032.00 −0.128272
\(997\) 5126.00 0.162831 0.0814153 0.996680i \(-0.474056\pi\)
0.0814153 + 0.996680i \(0.474056\pi\)
\(998\) 15560.0 0.493530
\(999\) −1242.00 −0.0393345
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 966.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.4.a.a.1.1 1 1.1 even 1 trivial