Properties

Label 462.4.a.b.1.1
Level $462$
Weight $4$
Character 462.1
Self dual yes
Analytic conductor $27.259$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.2588824227\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +28.0000 q^{10} +11.0000 q^{11} -12.0000 q^{12} +38.0000 q^{13} +14.0000 q^{14} +42.0000 q^{15} +16.0000 q^{16} +54.0000 q^{17} -18.0000 q^{18} +40.0000 q^{19} -56.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} +8.00000 q^{23} +24.0000 q^{24} +71.0000 q^{25} -76.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -170.000 q^{29} -84.0000 q^{30} +92.0000 q^{31} -32.0000 q^{32} -33.0000 q^{33} -108.000 q^{34} +98.0000 q^{35} +36.0000 q^{36} +294.000 q^{37} -80.0000 q^{38} -114.000 q^{39} +112.000 q^{40} -258.000 q^{41} -42.0000 q^{42} -52.0000 q^{43} +44.0000 q^{44} -126.000 q^{45} -16.0000 q^{46} -76.0000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -142.000 q^{50} -162.000 q^{51} +152.000 q^{52} -322.000 q^{53} +54.0000 q^{54} -154.000 q^{55} +56.0000 q^{56} -120.000 q^{57} +340.000 q^{58} +260.000 q^{59} +168.000 q^{60} +22.0000 q^{61} -184.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -532.000 q^{65} +66.0000 q^{66} -436.000 q^{67} +216.000 q^{68} -24.0000 q^{69} -196.000 q^{70} -368.000 q^{71} -72.0000 q^{72} -2.00000 q^{73} -588.000 q^{74} -213.000 q^{75} +160.000 q^{76} -77.0000 q^{77} +228.000 q^{78} -200.000 q^{79} -224.000 q^{80} +81.0000 q^{81} +516.000 q^{82} -952.000 q^{83} +84.0000 q^{84} -756.000 q^{85} +104.000 q^{86} +510.000 q^{87} -88.0000 q^{88} -70.0000 q^{89} +252.000 q^{90} -266.000 q^{91} +32.0000 q^{92} -276.000 q^{93} +152.000 q^{94} -560.000 q^{95} +96.0000 q^{96} -1086.00 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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
\(5\) −14.0000 −1.25220 −0.626099 0.779744i \(-0.715349\pi\)
−0.626099 + 0.779744i \(0.715349\pi\)
\(6\) 6.00000 0.408248
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 28.0000 0.885438
\(11\) 11.0000 0.301511
\(12\) −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
\(15\) 42.0000 0.722957
\(16\) 16.0000 0.250000
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) −18.0000 −0.235702
\(19\) 40.0000 0.482980 0.241490 0.970403i \(-0.422364\pi\)
0.241490 + 0.970403i \(0.422364\pi\)
\(20\) −56.0000 −0.626099
\(21\) 21.0000 0.218218
\(22\) −22.0000 −0.213201
\(23\) 8.00000 0.0725268 0.0362634 0.999342i \(-0.488454\pi\)
0.0362634 + 0.999342i \(0.488454\pi\)
\(24\) 24.0000 0.204124
\(25\) 71.0000 0.568000
\(26\) −76.0000 −0.573263
\(27\) −27.0000 −0.192450
\(28\) −28.0000 −0.188982
\(29\) −170.000 −1.08856 −0.544279 0.838904i \(-0.683197\pi\)
−0.544279 + 0.838904i \(0.683197\pi\)
\(30\) −84.0000 −0.511208
\(31\) 92.0000 0.533022 0.266511 0.963832i \(-0.414129\pi\)
0.266511 + 0.963832i \(0.414129\pi\)
\(32\) −32.0000 −0.176777
\(33\) −33.0000 −0.174078
\(34\) −108.000 −0.544760
\(35\) 98.0000 0.473286
\(36\) 36.0000 0.166667
\(37\) 294.000 1.30631 0.653153 0.757226i \(-0.273446\pi\)
0.653153 + 0.757226i \(0.273446\pi\)
\(38\) −80.0000 −0.341519
\(39\) −114.000 −0.468067
\(40\) 112.000 0.442719
\(41\) −258.000 −0.982752 −0.491376 0.870948i \(-0.663506\pi\)
−0.491376 + 0.870948i \(0.663506\pi\)
\(42\) −42.0000 −0.154303
\(43\) −52.0000 −0.184417 −0.0922084 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) 44.0000 0.150756
\(45\) −126.000 −0.417399
\(46\) −16.0000 −0.0512842
\(47\) −76.0000 −0.235867 −0.117933 0.993022i \(-0.537627\pi\)
−0.117933 + 0.993022i \(0.537627\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) −142.000 −0.401637
\(51\) −162.000 −0.444795
\(52\) 152.000 0.405358
\(53\) −322.000 −0.834530 −0.417265 0.908785i \(-0.637011\pi\)
−0.417265 + 0.908785i \(0.637011\pi\)
\(54\) 54.0000 0.136083
\(55\) −154.000 −0.377552
\(56\) 56.0000 0.133631
\(57\) −120.000 −0.278849
\(58\) 340.000 0.769727
\(59\) 260.000 0.573714 0.286857 0.957973i \(-0.407390\pi\)
0.286857 + 0.957973i \(0.407390\pi\)
\(60\) 168.000 0.361478
\(61\) 22.0000 0.0461772 0.0230886 0.999733i \(-0.492650\pi\)
0.0230886 + 0.999733i \(0.492650\pi\)
\(62\) −184.000 −0.376904
\(63\) −63.0000 −0.125988
\(64\) 64.0000 0.125000
\(65\) −532.000 −1.01518
\(66\) 66.0000 0.123091
\(67\) −436.000 −0.795013 −0.397507 0.917599i \(-0.630124\pi\)
−0.397507 + 0.917599i \(0.630124\pi\)
\(68\) 216.000 0.385204
\(69\) −24.0000 −0.0418733
\(70\) −196.000 −0.334664
\(71\) −368.000 −0.615121 −0.307560 0.951529i \(-0.599513\pi\)
−0.307560 + 0.951529i \(0.599513\pi\)
\(72\) −72.0000 −0.117851
\(73\) −2.00000 −0.00320661 −0.00160330 0.999999i \(-0.500510\pi\)
−0.00160330 + 0.999999i \(0.500510\pi\)
\(74\) −588.000 −0.923697
\(75\) −213.000 −0.327935
\(76\) 160.000 0.241490
\(77\) −77.0000 −0.113961
\(78\) 228.000 0.330973
\(79\) −200.000 −0.284832 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(80\) −224.000 −0.313050
\(81\) 81.0000 0.111111
\(82\) 516.000 0.694911
\(83\) −952.000 −1.25898 −0.629491 0.777007i \(-0.716737\pi\)
−0.629491 + 0.777007i \(0.716737\pi\)
\(84\) 84.0000 0.109109
\(85\) −756.000 −0.964703
\(86\) 104.000 0.130402
\(87\) 510.000 0.628480
\(88\) −88.0000 −0.106600
\(89\) −70.0000 −0.0833706 −0.0416853 0.999131i \(-0.513273\pi\)
−0.0416853 + 0.999131i \(0.513273\pi\)
\(90\) 252.000 0.295146
\(91\) −266.000 −0.306422
\(92\) 32.0000 0.0362634
\(93\) −276.000 −0.307741
\(94\) 152.000 0.166783
\(95\) −560.000 −0.604787
\(96\) 96.0000 0.102062
\(97\) −1086.00 −1.13677 −0.568385 0.822763i \(-0.692431\pi\)
−0.568385 + 0.822763i \(0.692431\pi\)
\(98\) −98.0000 −0.101015
\(99\) 99.0000 0.100504
\(100\) 284.000 0.284000
\(101\) 102.000 0.100489 0.0502445 0.998737i \(-0.484000\pi\)
0.0502445 + 0.998737i \(0.484000\pi\)
\(102\) 324.000 0.314517
\(103\) 188.000 0.179847 0.0899233 0.995949i \(-0.471338\pi\)
0.0899233 + 0.995949i \(0.471338\pi\)
\(104\) −304.000 −0.286631
\(105\) −294.000 −0.273252
\(106\) 644.000 0.590102
\(107\) −1716.00 −1.55039 −0.775196 0.631721i \(-0.782349\pi\)
−0.775196 + 0.631721i \(0.782349\pi\)
\(108\) −108.000 −0.0962250
\(109\) 310.000 0.272409 0.136205 0.990681i \(-0.456510\pi\)
0.136205 + 0.990681i \(0.456510\pi\)
\(110\) 308.000 0.266970
\(111\) −882.000 −0.754196
\(112\) −112.000 −0.0944911
\(113\) −1822.00 −1.51681 −0.758404 0.651785i \(-0.774021\pi\)
−0.758404 + 0.651785i \(0.774021\pi\)
\(114\) 240.000 0.197176
\(115\) −112.000 −0.0908179
\(116\) −680.000 −0.544279
\(117\) 342.000 0.270239
\(118\) −520.000 −0.405677
\(119\) −378.000 −0.291187
\(120\) −336.000 −0.255604
\(121\) 121.000 0.0909091
\(122\) −44.0000 −0.0326522
\(123\) 774.000 0.567392
\(124\) 368.000 0.266511
\(125\) 756.000 0.540950
\(126\) 126.000 0.0890871
\(127\) −1576.00 −1.10116 −0.550580 0.834782i \(-0.685594\pi\)
−0.550580 + 0.834782i \(0.685594\pi\)
\(128\) −128.000 −0.0883883
\(129\) 156.000 0.106473
\(130\) 1064.00 0.717838
\(131\) 1072.00 0.714970 0.357485 0.933919i \(-0.383634\pi\)
0.357485 + 0.933919i \(0.383634\pi\)
\(132\) −132.000 −0.0870388
\(133\) −280.000 −0.182549
\(134\) 872.000 0.562159
\(135\) 378.000 0.240986
\(136\) −432.000 −0.272380
\(137\) −246.000 −0.153410 −0.0767051 0.997054i \(-0.524440\pi\)
−0.0767051 + 0.997054i \(0.524440\pi\)
\(138\) 48.0000 0.0296089
\(139\) 1640.00 1.00074 0.500370 0.865811i \(-0.333197\pi\)
0.500370 + 0.865811i \(0.333197\pi\)
\(140\) 392.000 0.236643
\(141\) 228.000 0.136178
\(142\) 736.000 0.434956
\(143\) 418.000 0.244440
\(144\) 144.000 0.0833333
\(145\) 2380.00 1.36309
\(146\) 4.00000 0.00226741
\(147\) −147.000 −0.0824786
\(148\) 1176.00 0.653153
\(149\) −850.000 −0.467347 −0.233674 0.972315i \(-0.575075\pi\)
−0.233674 + 0.972315i \(0.575075\pi\)
\(150\) 426.000 0.231885
\(151\) −1448.00 −0.780375 −0.390187 0.920735i \(-0.627590\pi\)
−0.390187 + 0.920735i \(0.627590\pi\)
\(152\) −320.000 −0.170759
\(153\) 486.000 0.256802
\(154\) 154.000 0.0805823
\(155\) −1288.00 −0.667449
\(156\) −456.000 −0.234033
\(157\) −1886.00 −0.958721 −0.479360 0.877618i \(-0.659131\pi\)
−0.479360 + 0.877618i \(0.659131\pi\)
\(158\) 400.000 0.201407
\(159\) 966.000 0.481816
\(160\) 448.000 0.221359
\(161\) −56.0000 −0.0274125
\(162\) −162.000 −0.0785674
\(163\) 228.000 0.109560 0.0547802 0.998498i \(-0.482554\pi\)
0.0547802 + 0.998498i \(0.482554\pi\)
\(164\) −1032.00 −0.491376
\(165\) 462.000 0.217980
\(166\) 1904.00 0.890235
\(167\) 1664.00 0.771043 0.385522 0.922699i \(-0.374022\pi\)
0.385522 + 0.922699i \(0.374022\pi\)
\(168\) −168.000 −0.0771517
\(169\) −753.000 −0.342740
\(170\) 1512.00 0.682148
\(171\) 360.000 0.160993
\(172\) −208.000 −0.0922084
\(173\) 2438.00 1.07143 0.535716 0.844398i \(-0.320042\pi\)
0.535716 + 0.844398i \(0.320042\pi\)
\(174\) −1020.00 −0.444402
\(175\) −497.000 −0.214684
\(176\) 176.000 0.0753778
\(177\) −780.000 −0.331234
\(178\) 140.000 0.0589519
\(179\) 1620.00 0.676450 0.338225 0.941065i \(-0.390174\pi\)
0.338225 + 0.941065i \(0.390174\pi\)
\(180\) −504.000 −0.208700
\(181\) 3602.00 1.47920 0.739598 0.673049i \(-0.235016\pi\)
0.739598 + 0.673049i \(0.235016\pi\)
\(182\) 532.000 0.216673
\(183\) −66.0000 −0.0266604
\(184\) −64.0000 −0.0256421
\(185\) −4116.00 −1.63575
\(186\) 552.000 0.217605
\(187\) 594.000 0.232287
\(188\) −304.000 −0.117933
\(189\) 189.000 0.0727393
\(190\) 1120.00 0.427649
\(191\) 3472.00 1.31531 0.657657 0.753317i \(-0.271547\pi\)
0.657657 + 0.753317i \(0.271547\pi\)
\(192\) −192.000 −0.0721688
\(193\) −222.000 −0.0827975 −0.0413987 0.999143i \(-0.513181\pi\)
−0.0413987 + 0.999143i \(0.513181\pi\)
\(194\) 2172.00 0.803817
\(195\) 1596.00 0.586112
\(196\) 196.000 0.0714286
\(197\) −3906.00 −1.41264 −0.706322 0.707890i \(-0.749647\pi\)
−0.706322 + 0.707890i \(0.749647\pi\)
\(198\) −198.000 −0.0710669
\(199\) 1900.00 0.676821 0.338411 0.940999i \(-0.390111\pi\)
0.338411 + 0.940999i \(0.390111\pi\)
\(200\) −568.000 −0.200818
\(201\) 1308.00 0.459001
\(202\) −204.000 −0.0710564
\(203\) 1190.00 0.411437
\(204\) −648.000 −0.222397
\(205\) 3612.00 1.23060
\(206\) −376.000 −0.127171
\(207\) 72.0000 0.0241756
\(208\) 608.000 0.202679
\(209\) 440.000 0.145624
\(210\) 588.000 0.193218
\(211\) 1172.00 0.382388 0.191194 0.981552i \(-0.438764\pi\)
0.191194 + 0.981552i \(0.438764\pi\)
\(212\) −1288.00 −0.417265
\(213\) 1104.00 0.355140
\(214\) 3432.00 1.09629
\(215\) 728.000 0.230926
\(216\) 216.000 0.0680414
\(217\) −644.000 −0.201463
\(218\) −620.000 −0.192622
\(219\) 6.00000 0.00185134
\(220\) −616.000 −0.188776
\(221\) 2052.00 0.624581
\(222\) 1764.00 0.533297
\(223\) −2372.00 −0.712291 −0.356145 0.934431i \(-0.615909\pi\)
−0.356145 + 0.934431i \(0.615909\pi\)
\(224\) 224.000 0.0668153
\(225\) 639.000 0.189333
\(226\) 3644.00 1.07255
\(227\) 1024.00 0.299406 0.149703 0.988731i \(-0.452168\pi\)
0.149703 + 0.988731i \(0.452168\pi\)
\(228\) −480.000 −0.139424
\(229\) 290.000 0.0836845 0.0418422 0.999124i \(-0.486677\pi\)
0.0418422 + 0.999124i \(0.486677\pi\)
\(230\) 224.000 0.0642179
\(231\) 231.000 0.0657952
\(232\) 1360.00 0.384864
\(233\) −5022.00 −1.41203 −0.706013 0.708199i \(-0.749508\pi\)
−0.706013 + 0.708199i \(0.749508\pi\)
\(234\) −684.000 −0.191088
\(235\) 1064.00 0.295352
\(236\) 1040.00 0.286857
\(237\) 600.000 0.164448
\(238\) 756.000 0.205900
\(239\) 200.000 0.0541294 0.0270647 0.999634i \(-0.491384\pi\)
0.0270647 + 0.999634i \(0.491384\pi\)
\(240\) 672.000 0.180739
\(241\) −1498.00 −0.400393 −0.200196 0.979756i \(-0.564158\pi\)
−0.200196 + 0.979756i \(0.564158\pi\)
\(242\) −242.000 −0.0642824
\(243\) −243.000 −0.0641500
\(244\) 88.0000 0.0230886
\(245\) −686.000 −0.178885
\(246\) −1548.00 −0.401207
\(247\) 1520.00 0.391560
\(248\) −736.000 −0.188452
\(249\) 2856.00 0.726874
\(250\) −1512.00 −0.382509
\(251\) −7468.00 −1.87799 −0.938996 0.343928i \(-0.888242\pi\)
−0.938996 + 0.343928i \(0.888242\pi\)
\(252\) −252.000 −0.0629941
\(253\) 88.0000 0.0218676
\(254\) 3152.00 0.778638
\(255\) 2268.00 0.556971
\(256\) 256.000 0.0625000
\(257\) 2194.00 0.532521 0.266261 0.963901i \(-0.414212\pi\)
0.266261 + 0.963901i \(0.414212\pi\)
\(258\) −312.000 −0.0752879
\(259\) −2058.00 −0.493737
\(260\) −2128.00 −0.507588
\(261\) −1530.00 −0.362853
\(262\) −2144.00 −0.505560
\(263\) 3288.00 0.770900 0.385450 0.922729i \(-0.374046\pi\)
0.385450 + 0.922729i \(0.374046\pi\)
\(264\) 264.000 0.0615457
\(265\) 4508.00 1.04500
\(266\) 560.000 0.129082
\(267\) 210.000 0.0481340
\(268\) −1744.00 −0.397507
\(269\) −6710.00 −1.52088 −0.760439 0.649410i \(-0.775016\pi\)
−0.760439 + 0.649410i \(0.775016\pi\)
\(270\) −756.000 −0.170403
\(271\) 4952.00 1.11001 0.555005 0.831847i \(-0.312716\pi\)
0.555005 + 0.831847i \(0.312716\pi\)
\(272\) 864.000 0.192602
\(273\) 798.000 0.176913
\(274\) 492.000 0.108477
\(275\) 781.000 0.171258
\(276\) −96.0000 −0.0209367
\(277\) −3666.00 −0.795193 −0.397597 0.917560i \(-0.630156\pi\)
−0.397597 + 0.917560i \(0.630156\pi\)
\(278\) −3280.00 −0.707631
\(279\) 828.000 0.177674
\(280\) −784.000 −0.167332
\(281\) −6798.00 −1.44318 −0.721592 0.692319i \(-0.756589\pi\)
−0.721592 + 0.692319i \(0.756589\pi\)
\(282\) −456.000 −0.0962922
\(283\) −1992.00 −0.418417 −0.209209 0.977871i \(-0.567089\pi\)
−0.209209 + 0.977871i \(0.567089\pi\)
\(284\) −1472.00 −0.307560
\(285\) 1680.00 0.349174
\(286\) −836.000 −0.172845
\(287\) 1806.00 0.371445
\(288\) −288.000 −0.0589256
\(289\) −1997.00 −0.406473
\(290\) −4760.00 −0.963851
\(291\) 3258.00 0.656314
\(292\) −8.00000 −0.00160330
\(293\) −4002.00 −0.797950 −0.398975 0.916962i \(-0.630634\pi\)
−0.398975 + 0.916962i \(0.630634\pi\)
\(294\) 294.000 0.0583212
\(295\) −3640.00 −0.718403
\(296\) −2352.00 −0.461849
\(297\) −297.000 −0.0580259
\(298\) 1700.00 0.330464
\(299\) 304.000 0.0587986
\(300\) −852.000 −0.163967
\(301\) 364.000 0.0697030
\(302\) 2896.00 0.551808
\(303\) −306.000 −0.0580173
\(304\) 640.000 0.120745
\(305\) −308.000 −0.0578230
\(306\) −972.000 −0.181587
\(307\) 5344.00 0.993479 0.496740 0.867900i \(-0.334530\pi\)
0.496740 + 0.867900i \(0.334530\pi\)
\(308\) −308.000 −0.0569803
\(309\) −564.000 −0.103834
\(310\) 2576.00 0.471958
\(311\) −2268.00 −0.413526 −0.206763 0.978391i \(-0.566293\pi\)
−0.206763 + 0.978391i \(0.566293\pi\)
\(312\) 912.000 0.165487
\(313\) −7422.00 −1.34031 −0.670154 0.742222i \(-0.733772\pi\)
−0.670154 + 0.742222i \(0.733772\pi\)
\(314\) 3772.00 0.677918
\(315\) 882.000 0.157762
\(316\) −800.000 −0.142416
\(317\) −7626.00 −1.35116 −0.675582 0.737285i \(-0.736107\pi\)
−0.675582 + 0.737285i \(0.736107\pi\)
\(318\) −1932.00 −0.340696
\(319\) −1870.00 −0.328213
\(320\) −896.000 −0.156525
\(321\) 5148.00 0.895119
\(322\) 112.000 0.0193836
\(323\) 2160.00 0.372092
\(324\) 324.000 0.0555556
\(325\) 2698.00 0.460487
\(326\) −456.000 −0.0774709
\(327\) −930.000 −0.157276
\(328\) 2064.00 0.347455
\(329\) 532.000 0.0891493
\(330\) −924.000 −0.154135
\(331\) 1492.00 0.247758 0.123879 0.992297i \(-0.460467\pi\)
0.123879 + 0.992297i \(0.460467\pi\)
\(332\) −3808.00 −0.629491
\(333\) 2646.00 0.435435
\(334\) −3328.00 −0.545210
\(335\) 6104.00 0.995514
\(336\) 336.000 0.0545545
\(337\) 74.0000 0.0119615 0.00598077 0.999982i \(-0.498096\pi\)
0.00598077 + 0.999982i \(0.498096\pi\)
\(338\) 1506.00 0.242354
\(339\) 5466.00 0.875730
\(340\) −3024.00 −0.482351
\(341\) 1012.00 0.160712
\(342\) −720.000 −0.113840
\(343\) −343.000 −0.0539949
\(344\) 416.000 0.0652012
\(345\) 336.000 0.0524337
\(346\) −4876.00 −0.757617
\(347\) 11724.0 1.81377 0.906884 0.421381i \(-0.138454\pi\)
0.906884 + 0.421381i \(0.138454\pi\)
\(348\) 2040.00 0.314240
\(349\) 6350.00 0.973948 0.486974 0.873417i \(-0.338101\pi\)
0.486974 + 0.873417i \(0.338101\pi\)
\(350\) 994.000 0.151804
\(351\) −1026.00 −0.156022
\(352\) −352.000 −0.0533002
\(353\) −5822.00 −0.877829 −0.438915 0.898529i \(-0.644637\pi\)
−0.438915 + 0.898529i \(0.644637\pi\)
\(354\) 1560.00 0.234218
\(355\) 5152.00 0.770253
\(356\) −280.000 −0.0416853
\(357\) 1134.00 0.168117
\(358\) −3240.00 −0.478322
\(359\) 13320.0 1.95822 0.979112 0.203320i \(-0.0651731\pi\)
0.979112 + 0.203320i \(0.0651731\pi\)
\(360\) 1008.00 0.147573
\(361\) −5259.00 −0.766730
\(362\) −7204.00 −1.04595
\(363\) −363.000 −0.0524864
\(364\) −1064.00 −0.153211
\(365\) 28.0000 0.00401531
\(366\) 132.000 0.0188518
\(367\) −11076.0 −1.57537 −0.787687 0.616075i \(-0.788722\pi\)
−0.787687 + 0.616075i \(0.788722\pi\)
\(368\) 128.000 0.0181317
\(369\) −2322.00 −0.327584
\(370\) 8232.00 1.15665
\(371\) 2254.00 0.315423
\(372\) −1104.00 −0.153870
\(373\) −3922.00 −0.544433 −0.272216 0.962236i \(-0.587757\pi\)
−0.272216 + 0.962236i \(0.587757\pi\)
\(374\) −1188.00 −0.164251
\(375\) −2268.00 −0.312317
\(376\) 608.000 0.0833915
\(377\) −6460.00 −0.882512
\(378\) −378.000 −0.0514344
\(379\) −11620.0 −1.57488 −0.787440 0.616392i \(-0.788594\pi\)
−0.787440 + 0.616392i \(0.788594\pi\)
\(380\) −2240.00 −0.302394
\(381\) 4728.00 0.635755
\(382\) −6944.00 −0.930068
\(383\) 3628.00 0.484026 0.242013 0.970273i \(-0.422192\pi\)
0.242013 + 0.970273i \(0.422192\pi\)
\(384\) 384.000 0.0510310
\(385\) 1078.00 0.142701
\(386\) 444.000 0.0585466
\(387\) −468.000 −0.0614723
\(388\) −4344.00 −0.568385
\(389\) 9750.00 1.27081 0.635404 0.772180i \(-0.280833\pi\)
0.635404 + 0.772180i \(0.280833\pi\)
\(390\) −3192.00 −0.414444
\(391\) 432.000 0.0558751
\(392\) −392.000 −0.0505076
\(393\) −3216.00 −0.412788
\(394\) 7812.00 0.998891
\(395\) 2800.00 0.356667
\(396\) 396.000 0.0502519
\(397\) −6606.00 −0.835128 −0.417564 0.908648i \(-0.637116\pi\)
−0.417564 + 0.908648i \(0.637116\pi\)
\(398\) −3800.00 −0.478585
\(399\) 840.000 0.105395
\(400\) 1136.00 0.142000
\(401\) 2.00000 0.000249065 0 0.000124533 1.00000i \(-0.499960\pi\)
0.000124533 1.00000i \(0.499960\pi\)
\(402\) −2616.00 −0.324563
\(403\) 3496.00 0.432129
\(404\) 408.000 0.0502445
\(405\) −1134.00 −0.139133
\(406\) −2380.00 −0.290930
\(407\) 3234.00 0.393866
\(408\) 1296.00 0.157259
\(409\) −13930.0 −1.68409 −0.842047 0.539405i \(-0.818649\pi\)
−0.842047 + 0.539405i \(0.818649\pi\)
\(410\) −7224.00 −0.870166
\(411\) 738.000 0.0885714
\(412\) 752.000 0.0899233
\(413\) −1820.00 −0.216843
\(414\) −144.000 −0.0170947
\(415\) 13328.0 1.57650
\(416\) −1216.00 −0.143316
\(417\) −4920.00 −0.577778
\(418\) −880.000 −0.102972
\(419\) −7740.00 −0.902443 −0.451222 0.892412i \(-0.649012\pi\)
−0.451222 + 0.892412i \(0.649012\pi\)
\(420\) −1176.00 −0.136626
\(421\) −5618.00 −0.650367 −0.325184 0.945651i \(-0.605426\pi\)
−0.325184 + 0.945651i \(0.605426\pi\)
\(422\) −2344.00 −0.270389
\(423\) −684.000 −0.0786223
\(424\) 2576.00 0.295051
\(425\) 3834.00 0.437591
\(426\) −2208.00 −0.251122
\(427\) −154.000 −0.0174534
\(428\) −6864.00 −0.775196
\(429\) −1254.00 −0.141127
\(430\) −1456.00 −0.163290
\(431\) −9008.00 −1.00673 −0.503364 0.864074i \(-0.667905\pi\)
−0.503364 + 0.864074i \(0.667905\pi\)
\(432\) −432.000 −0.0481125
\(433\) −10702.0 −1.18777 −0.593886 0.804549i \(-0.702407\pi\)
−0.593886 + 0.804549i \(0.702407\pi\)
\(434\) 1288.00 0.142456
\(435\) −7140.00 −0.786981
\(436\) 1240.00 0.136205
\(437\) 320.000 0.0350290
\(438\) −12.0000 −0.00130909
\(439\) −4960.00 −0.539243 −0.269622 0.962966i \(-0.586899\pi\)
−0.269622 + 0.962966i \(0.586899\pi\)
\(440\) 1232.00 0.133485
\(441\) 441.000 0.0476190
\(442\) −4104.00 −0.441646
\(443\) 188.000 0.0201629 0.0100814 0.999949i \(-0.496791\pi\)
0.0100814 + 0.999949i \(0.496791\pi\)
\(444\) −3528.00 −0.377098
\(445\) 980.000 0.104397
\(446\) 4744.00 0.503666
\(447\) 2550.00 0.269823
\(448\) −448.000 −0.0472456
\(449\) −3150.00 −0.331086 −0.165543 0.986203i \(-0.552938\pi\)
−0.165543 + 0.986203i \(0.552938\pi\)
\(450\) −1278.00 −0.133879
\(451\) −2838.00 −0.296311
\(452\) −7288.00 −0.758404
\(453\) 4344.00 0.450550
\(454\) −2048.00 −0.211712
\(455\) 3724.00 0.383701
\(456\) 960.000 0.0985880
\(457\) −12806.0 −1.31081 −0.655404 0.755278i \(-0.727502\pi\)
−0.655404 + 0.755278i \(0.727502\pi\)
\(458\) −580.000 −0.0591738
\(459\) −1458.00 −0.148265
\(460\) −448.000 −0.0454089
\(461\) −1778.00 −0.179631 −0.0898153 0.995958i \(-0.528628\pi\)
−0.0898153 + 0.995958i \(0.528628\pi\)
\(462\) −462.000 −0.0465242
\(463\) −15672.0 −1.57309 −0.786544 0.617534i \(-0.788132\pi\)
−0.786544 + 0.617534i \(0.788132\pi\)
\(464\) −2720.00 −0.272140
\(465\) 3864.00 0.385352
\(466\) 10044.0 0.998453
\(467\) 4.00000 0.000396355 0 0.000198178 1.00000i \(-0.499937\pi\)
0.000198178 1.00000i \(0.499937\pi\)
\(468\) 1368.00 0.135119
\(469\) 3052.00 0.300487
\(470\) −2128.00 −0.208845
\(471\) 5658.00 0.553518
\(472\) −2080.00 −0.202838
\(473\) −572.000 −0.0556038
\(474\) −1200.00 −0.116282
\(475\) 2840.00 0.274333
\(476\) −1512.00 −0.145593
\(477\) −2898.00 −0.278177
\(478\) −400.000 −0.0382753
\(479\) 14640.0 1.39649 0.698245 0.715859i \(-0.253965\pi\)
0.698245 + 0.715859i \(0.253965\pi\)
\(480\) −1344.00 −0.127802
\(481\) 11172.0 1.05904
\(482\) 2996.00 0.283120
\(483\) 168.000 0.0158266
\(484\) 484.000 0.0454545
\(485\) 15204.0 1.42346
\(486\) 486.000 0.0453609
\(487\) −19896.0 −1.85128 −0.925640 0.378404i \(-0.876473\pi\)
−0.925640 + 0.378404i \(0.876473\pi\)
\(488\) −176.000 −0.0163261
\(489\) −684.000 −0.0632547
\(490\) 1372.00 0.126491
\(491\) 7972.00 0.732732 0.366366 0.930471i \(-0.380602\pi\)
0.366366 + 0.930471i \(0.380602\pi\)
\(492\) 3096.00 0.283696
\(493\) −9180.00 −0.838634
\(494\) −3040.00 −0.276875
\(495\) −1386.00 −0.125851
\(496\) 1472.00 0.133256
\(497\) 2576.00 0.232494
\(498\) −5712.00 −0.513978
\(499\) −2260.00 −0.202748 −0.101374 0.994848i \(-0.532324\pi\)
−0.101374 + 0.994848i \(0.532324\pi\)
\(500\) 3024.00 0.270475
\(501\) −4992.00 −0.445162
\(502\) 14936.0 1.32794
\(503\) 6288.00 0.557392 0.278696 0.960379i \(-0.410098\pi\)
0.278696 + 0.960379i \(0.410098\pi\)
\(504\) 504.000 0.0445435
\(505\) −1428.00 −0.125832
\(506\) −176.000 −0.0154628
\(507\) 2259.00 0.197881
\(508\) −6304.00 −0.550580
\(509\) 12010.0 1.04584 0.522921 0.852381i \(-0.324842\pi\)
0.522921 + 0.852381i \(0.324842\pi\)
\(510\) −4536.00 −0.393838
\(511\) 14.0000 0.00121198
\(512\) −512.000 −0.0441942
\(513\) −1080.00 −0.0929496
\(514\) −4388.00 −0.376549
\(515\) −2632.00 −0.225203
\(516\) 624.000 0.0532366
\(517\) −836.000 −0.0711165
\(518\) 4116.00 0.349125
\(519\) −7314.00 −0.618591
\(520\) 4256.00 0.358919
\(521\) 10842.0 0.911702 0.455851 0.890056i \(-0.349335\pi\)
0.455851 + 0.890056i \(0.349335\pi\)
\(522\) 3060.00 0.256576
\(523\) 18808.0 1.57250 0.786249 0.617910i \(-0.212020\pi\)
0.786249 + 0.617910i \(0.212020\pi\)
\(524\) 4288.00 0.357485
\(525\) 1491.00 0.123948
\(526\) −6576.00 −0.545109
\(527\) 4968.00 0.410644
\(528\) −528.000 −0.0435194
\(529\) −12103.0 −0.994740
\(530\) −9016.00 −0.738925
\(531\) 2340.00 0.191238
\(532\) −1120.00 −0.0912747
\(533\) −9804.00 −0.796732
\(534\) −420.000 −0.0340359
\(535\) 24024.0 1.94140
\(536\) 3488.00 0.281080
\(537\) −4860.00 −0.390548
\(538\) 13420.0 1.07542
\(539\) 539.000 0.0430730
\(540\) 1512.00 0.120493
\(541\) 15622.0 1.24148 0.620741 0.784015i \(-0.286832\pi\)
0.620741 + 0.784015i \(0.286832\pi\)
\(542\) −9904.00 −0.784895
\(543\) −10806.0 −0.854014
\(544\) −1728.00 −0.136190
\(545\) −4340.00 −0.341110
\(546\) −1596.00 −0.125096
\(547\) 2284.00 0.178532 0.0892658 0.996008i \(-0.471548\pi\)
0.0892658 + 0.996008i \(0.471548\pi\)
\(548\) −984.000 −0.0767051
\(549\) 198.000 0.0153924
\(550\) −1562.00 −0.121098
\(551\) −6800.00 −0.525753
\(552\) 192.000 0.0148045
\(553\) 1400.00 0.107657
\(554\) 7332.00 0.562287
\(555\) 12348.0 0.944403
\(556\) 6560.00 0.500370
\(557\) 16854.0 1.28209 0.641047 0.767501i \(-0.278500\pi\)
0.641047 + 0.767501i \(0.278500\pi\)
\(558\) −1656.00 −0.125635
\(559\) −1976.00 −0.149510
\(560\) 1568.00 0.118322
\(561\) −1782.00 −0.134111
\(562\) 13596.0 1.02049
\(563\) 528.000 0.0395250 0.0197625 0.999805i \(-0.493709\pi\)
0.0197625 + 0.999805i \(0.493709\pi\)
\(564\) 912.000 0.0680889
\(565\) 25508.0 1.89934
\(566\) 3984.00 0.295866
\(567\) −567.000 −0.0419961
\(568\) 2944.00 0.217478
\(569\) 8050.00 0.593099 0.296550 0.955017i \(-0.404164\pi\)
0.296550 + 0.955017i \(0.404164\pi\)
\(570\) −3360.00 −0.246903
\(571\) −11308.0 −0.828765 −0.414383 0.910103i \(-0.636002\pi\)
−0.414383 + 0.910103i \(0.636002\pi\)
\(572\) 1672.00 0.122220
\(573\) −10416.0 −0.759397
\(574\) −3612.00 −0.262652
\(575\) 568.000 0.0411952
\(576\) 576.000 0.0416667
\(577\) 5274.00 0.380519 0.190260 0.981734i \(-0.439067\pi\)
0.190260 + 0.981734i \(0.439067\pi\)
\(578\) 3994.00 0.287420
\(579\) 666.000 0.0478031
\(580\) 9520.00 0.681546
\(581\) 6664.00 0.475851
\(582\) −6516.00 −0.464084
\(583\) −3542.00 −0.251620
\(584\) 16.0000 0.00113371
\(585\) −4788.00 −0.338392
\(586\) 8004.00 0.564236
\(587\) 11604.0 0.815926 0.407963 0.912999i \(-0.366239\pi\)
0.407963 + 0.912999i \(0.366239\pi\)
\(588\) −588.000 −0.0412393
\(589\) 3680.00 0.257439
\(590\) 7280.00 0.507988
\(591\) 11718.0 0.815591
\(592\) 4704.00 0.326576
\(593\) −13002.0 −0.900385 −0.450192 0.892932i \(-0.648645\pi\)
−0.450192 + 0.892932i \(0.648645\pi\)
\(594\) 594.000 0.0410305
\(595\) 5292.00 0.364623
\(596\) −3400.00 −0.233674
\(597\) −5700.00 −0.390763
\(598\) −608.000 −0.0415769
\(599\) 16320.0 1.11322 0.556609 0.830775i \(-0.312102\pi\)
0.556609 + 0.830775i \(0.312102\pi\)
\(600\) 1704.00 0.115943
\(601\) −9258.00 −0.628356 −0.314178 0.949364i \(-0.601729\pi\)
−0.314178 + 0.949364i \(0.601729\pi\)
\(602\) −728.000 −0.0492875
\(603\) −3924.00 −0.265004
\(604\) −5792.00 −0.390187
\(605\) −1694.00 −0.113836
\(606\) 612.000 0.0410244
\(607\) 25824.0 1.72679 0.863397 0.504525i \(-0.168332\pi\)
0.863397 + 0.504525i \(0.168332\pi\)
\(608\) −1280.00 −0.0853797
\(609\) −3570.00 −0.237543
\(610\) 616.000 0.0408871
\(611\) −2888.00 −0.191221
\(612\) 1944.00 0.128401
\(613\) 15518.0 1.02246 0.511228 0.859445i \(-0.329191\pi\)
0.511228 + 0.859445i \(0.329191\pi\)
\(614\) −10688.0 −0.702496
\(615\) −10836.0 −0.710487
\(616\) 616.000 0.0402911
\(617\) −14486.0 −0.945194 −0.472597 0.881279i \(-0.656683\pi\)
−0.472597 + 0.881279i \(0.656683\pi\)
\(618\) 1128.00 0.0734220
\(619\) 12460.0 0.809062 0.404531 0.914524i \(-0.367435\pi\)
0.404531 + 0.914524i \(0.367435\pi\)
\(620\) −5152.00 −0.333725
\(621\) −216.000 −0.0139578
\(622\) 4536.00 0.292407
\(623\) 490.000 0.0315111
\(624\) −1824.00 −0.117017
\(625\) −19459.0 −1.24538
\(626\) 14844.0 0.947741
\(627\) −1320.00 −0.0840761
\(628\) −7544.00 −0.479360
\(629\) 15876.0 1.00639
\(630\) −1764.00 −0.111555
\(631\) −16648.0 −1.05031 −0.525156 0.851006i \(-0.675993\pi\)
−0.525156 + 0.851006i \(0.675993\pi\)
\(632\) 1600.00 0.100703
\(633\) −3516.00 −0.220772
\(634\) 15252.0 0.955417
\(635\) 22064.0 1.37887
\(636\) 3864.00 0.240908
\(637\) 1862.00 0.115817
\(638\) 3740.00 0.232082
\(639\) −3312.00 −0.205040
\(640\) 1792.00 0.110680
\(641\) −10638.0 −0.655500 −0.327750 0.944764i \(-0.606290\pi\)
−0.327750 + 0.944764i \(0.606290\pi\)
\(642\) −10296.0 −0.632945
\(643\) 4588.00 0.281389 0.140694 0.990053i \(-0.455066\pi\)
0.140694 + 0.990053i \(0.455066\pi\)
\(644\) −224.000 −0.0137063
\(645\) −2184.00 −0.133325
\(646\) −4320.00 −0.263109
\(647\) −8796.00 −0.534477 −0.267238 0.963630i \(-0.586111\pi\)
−0.267238 + 0.963630i \(0.586111\pi\)
\(648\) −648.000 −0.0392837
\(649\) 2860.00 0.172981
\(650\) −5396.00 −0.325613
\(651\) 1932.00 0.116315
\(652\) 912.000 0.0547802
\(653\) 18878.0 1.13132 0.565661 0.824638i \(-0.308621\pi\)
0.565661 + 0.824638i \(0.308621\pi\)
\(654\) 1860.00 0.111211
\(655\) −15008.0 −0.895284
\(656\) −4128.00 −0.245688
\(657\) −18.0000 −0.00106887
\(658\) −1064.00 −0.0630381
\(659\) −20780.0 −1.22834 −0.614168 0.789175i \(-0.710508\pi\)
−0.614168 + 0.789175i \(0.710508\pi\)
\(660\) 1848.00 0.108990
\(661\) 10402.0 0.612089 0.306045 0.952017i \(-0.400994\pi\)
0.306045 + 0.952017i \(0.400994\pi\)
\(662\) −2984.00 −0.175191
\(663\) −6156.00 −0.360602
\(664\) 7616.00 0.445118
\(665\) 3920.00 0.228588
\(666\) −5292.00 −0.307899
\(667\) −1360.00 −0.0789496
\(668\) 6656.00 0.385522
\(669\) 7116.00 0.411241
\(670\) −12208.0 −0.703935
\(671\) 242.000 0.0139230
\(672\) −672.000 −0.0385758
\(673\) 23018.0 1.31839 0.659197 0.751971i \(-0.270896\pi\)
0.659197 + 0.751971i \(0.270896\pi\)
\(674\) −148.000 −0.00845808
\(675\) −1917.00 −0.109312
\(676\) −3012.00 −0.171370
\(677\) −16866.0 −0.957479 −0.478739 0.877957i \(-0.658906\pi\)
−0.478739 + 0.877957i \(0.658906\pi\)
\(678\) −10932.0 −0.619234
\(679\) 7602.00 0.429658
\(680\) 6048.00 0.341074
\(681\) −3072.00 −0.172862
\(682\) −2024.00 −0.113641
\(683\) 4668.00 0.261517 0.130758 0.991414i \(-0.458259\pi\)
0.130758 + 0.991414i \(0.458259\pi\)
\(684\) 1440.00 0.0804967
\(685\) 3444.00 0.192100
\(686\) 686.000 0.0381802
\(687\) −870.000 −0.0483152
\(688\) −832.000 −0.0461042
\(689\) −12236.0 −0.676567
\(690\) −672.000 −0.0370762
\(691\) −33108.0 −1.82270 −0.911351 0.411629i \(-0.864960\pi\)
−0.911351 + 0.411629i \(0.864960\pi\)
\(692\) 9752.00 0.535716
\(693\) −693.000 −0.0379869
\(694\) −23448.0 −1.28253
\(695\) −22960.0 −1.25313
\(696\) −4080.00 −0.222201
\(697\) −13932.0 −0.757119
\(698\) −12700.0 −0.688685
\(699\) 15066.0 0.815234
\(700\) −1988.00 −0.107342
\(701\) 7302.00 0.393428 0.196714 0.980461i \(-0.436973\pi\)
0.196714 + 0.980461i \(0.436973\pi\)
\(702\) 2052.00 0.110324
\(703\) 11760.0 0.630920
\(704\) 704.000 0.0376889
\(705\) −3192.00 −0.170522
\(706\) 11644.0 0.620719
\(707\) −714.000 −0.0379812
\(708\) −3120.00 −0.165617
\(709\) −10570.0 −0.559894 −0.279947 0.960015i \(-0.590317\pi\)
−0.279947 + 0.960015i \(0.590317\pi\)
\(710\) −10304.0 −0.544651
\(711\) −1800.00 −0.0949441
\(712\) 560.000 0.0294760
\(713\) 736.000 0.0386584
\(714\) −2268.00 −0.118876
\(715\) −5852.00 −0.306087
\(716\) 6480.00 0.338225
\(717\) −600.000 −0.0312516
\(718\) −26640.0 −1.38467
\(719\) 20220.0 1.04879 0.524394 0.851476i \(-0.324292\pi\)
0.524394 + 0.851476i \(0.324292\pi\)
\(720\) −2016.00 −0.104350
\(721\) −1316.00 −0.0679756
\(722\) 10518.0 0.542160
\(723\) 4494.00 0.231167
\(724\) 14408.0 0.739598
\(725\) −12070.0 −0.618301
\(726\) 726.000 0.0371135
\(727\) −29996.0 −1.53025 −0.765124 0.643883i \(-0.777322\pi\)
−0.765124 + 0.643883i \(0.777322\pi\)
\(728\) 2128.00 0.108336
\(729\) 729.000 0.0370370
\(730\) −56.0000 −0.00283925
\(731\) −2808.00 −0.142076
\(732\) −264.000 −0.0133302
\(733\) −2282.00 −0.114990 −0.0574949 0.998346i \(-0.518311\pi\)
−0.0574949 + 0.998346i \(0.518311\pi\)
\(734\) 22152.0 1.11396
\(735\) 2058.00 0.103280
\(736\) −256.000 −0.0128210
\(737\) −4796.00 −0.239705
\(738\) 4644.00 0.231637
\(739\) 3260.00 0.162275 0.0811374 0.996703i \(-0.474145\pi\)
0.0811374 + 0.996703i \(0.474145\pi\)
\(740\) −16464.0 −0.817877
\(741\) −4560.00 −0.226067
\(742\) −4508.00 −0.223038
\(743\) −14512.0 −0.716546 −0.358273 0.933617i \(-0.616634\pi\)
−0.358273 + 0.933617i \(0.616634\pi\)
\(744\) 2208.00 0.108803
\(745\) 11900.0 0.585211
\(746\) 7844.00 0.384972
\(747\) −8568.00 −0.419661
\(748\) 2376.00 0.116143
\(749\) 12012.0 0.585993
\(750\) 4536.00 0.220842
\(751\) −20128.0 −0.978004 −0.489002 0.872283i \(-0.662639\pi\)
−0.489002 + 0.872283i \(0.662639\pi\)
\(752\) −1216.00 −0.0589667
\(753\) 22404.0 1.08426
\(754\) 12920.0 0.624030
\(755\) 20272.0 0.977184
\(756\) 756.000 0.0363696
\(757\) −3066.00 −0.147207 −0.0736035 0.997288i \(-0.523450\pi\)
−0.0736035 + 0.997288i \(0.523450\pi\)
\(758\) 23240.0 1.11361
\(759\) −264.000 −0.0126253
\(760\) 4480.00 0.213825
\(761\) 26982.0 1.28528 0.642639 0.766169i \(-0.277839\pi\)
0.642639 + 0.766169i \(0.277839\pi\)
\(762\) −9456.00 −0.449547
\(763\) −2170.00 −0.102961
\(764\) 13888.0 0.657657
\(765\) −6804.00 −0.321568
\(766\) −7256.00 −0.342258
\(767\) 9880.00 0.465119
\(768\) −768.000 −0.0360844
\(769\) 3550.00 0.166471 0.0832355 0.996530i \(-0.473475\pi\)
0.0832355 + 0.996530i \(0.473475\pi\)
\(770\) −2156.00 −0.100905
\(771\) −6582.00 −0.307451
\(772\) −888.000 −0.0413987
\(773\) −33062.0 −1.53837 −0.769183 0.639028i \(-0.779337\pi\)
−0.769183 + 0.639028i \(0.779337\pi\)
\(774\) 936.000 0.0434675
\(775\) 6532.00 0.302757
\(776\) 8688.00 0.401909
\(777\) 6174.00 0.285059
\(778\) −19500.0 −0.898598
\(779\) −10320.0 −0.474650
\(780\) 6384.00 0.293056
\(781\) −4048.00 −0.185466
\(782\) −864.000 −0.0395097
\(783\) 4590.00 0.209493
\(784\) 784.000 0.0357143
\(785\) 26404.0 1.20051
\(786\) 6432.00 0.291885
\(787\) 18824.0 0.852609 0.426304 0.904580i \(-0.359815\pi\)
0.426304 + 0.904580i \(0.359815\pi\)
\(788\) −15624.0 −0.706322
\(789\) −9864.00 −0.445079
\(790\) −5600.00 −0.252201
\(791\) 12754.0 0.573300
\(792\) −792.000 −0.0355335
\(793\) 836.000 0.0374366
\(794\) 13212.0 0.590524
\(795\) −13524.0 −0.603329
\(796\) 7600.00 0.338411
\(797\) 41314.0 1.83616 0.918078 0.396399i \(-0.129740\pi\)
0.918078 + 0.396399i \(0.129740\pi\)
\(798\) −1680.00 −0.0745255
\(799\) −4104.00 −0.181713
\(800\) −2272.00 −0.100409
\(801\) −630.000 −0.0277902
\(802\) −4.00000 −0.000176116 0
\(803\) −22.0000 −0.000966828 0
\(804\) 5232.00 0.229501
\(805\) 784.000 0.0343259
\(806\) −6992.00 −0.305562
\(807\) 20130.0 0.878079
\(808\) −816.000 −0.0355282
\(809\) 16090.0 0.699251 0.349626 0.936889i \(-0.386309\pi\)
0.349626 + 0.936889i \(0.386309\pi\)
\(810\) 2268.00 0.0983820
\(811\) 4472.00 0.193629 0.0968145 0.995302i \(-0.469135\pi\)
0.0968145 + 0.995302i \(0.469135\pi\)
\(812\) 4760.00 0.205718
\(813\) −14856.0 −0.640864
\(814\) −6468.00 −0.278505
\(815\) −3192.00 −0.137191
\(816\) −2592.00 −0.111199
\(817\) −2080.00 −0.0890698
\(818\) 27860.0 1.19083
\(819\) −2394.00 −0.102141
\(820\) 14448.0 0.615300
\(821\) −44338.0 −1.88478 −0.942392 0.334512i \(-0.891429\pi\)
−0.942392 + 0.334512i \(0.891429\pi\)
\(822\) −1476.00 −0.0626295
\(823\) −19432.0 −0.823034 −0.411517 0.911402i \(-0.635001\pi\)
−0.411517 + 0.911402i \(0.635001\pi\)
\(824\) −1504.00 −0.0635853
\(825\) −2343.00 −0.0988761
\(826\) 3640.00 0.153331
\(827\) 15764.0 0.662839 0.331420 0.943483i \(-0.392472\pi\)
0.331420 + 0.943483i \(0.392472\pi\)
\(828\) 288.000 0.0120878
\(829\) 16570.0 0.694210 0.347105 0.937826i \(-0.387165\pi\)
0.347105 + 0.937826i \(0.387165\pi\)
\(830\) −26656.0 −1.11475
\(831\) 10998.0 0.459105
\(832\) 2432.00 0.101339
\(833\) 2646.00 0.110058
\(834\) 9840.00 0.408551
\(835\) −23296.0 −0.965499
\(836\) 1760.00 0.0728120
\(837\) −2484.00 −0.102580
\(838\) 15480.0 0.638124
\(839\) −5580.00 −0.229610 −0.114805 0.993388i \(-0.536624\pi\)
−0.114805 + 0.993388i \(0.536624\pi\)
\(840\) 2352.00 0.0966092
\(841\) 4511.00 0.184960
\(842\) 11236.0 0.459879
\(843\) 20394.0 0.833223
\(844\) 4688.00 0.191194
\(845\) 10542.0 0.429178
\(846\) 1368.00 0.0555943
\(847\) −847.000 −0.0343604
\(848\) −5152.00 −0.208633
\(849\) 5976.00 0.241573
\(850\) −7668.00 −0.309424
\(851\) 2352.00 0.0947421
\(852\) 4416.00 0.177570
\(853\) −17522.0 −0.703332 −0.351666 0.936126i \(-0.614385\pi\)
−0.351666 + 0.936126i \(0.614385\pi\)
\(854\) 308.000 0.0123414
\(855\) −5040.00 −0.201596
\(856\) 13728.0 0.548146
\(857\) −37146.0 −1.48061 −0.740305 0.672271i \(-0.765319\pi\)
−0.740305 + 0.672271i \(0.765319\pi\)
\(858\) 2508.00 0.0997922
\(859\) 46820.0 1.85969 0.929847 0.367945i \(-0.119939\pi\)
0.929847 + 0.367945i \(0.119939\pi\)
\(860\) 2912.00 0.115463
\(861\) −5418.00 −0.214454
\(862\) 18016.0 0.711865
\(863\) 25808.0 1.01798 0.508989 0.860773i \(-0.330019\pi\)
0.508989 + 0.860773i \(0.330019\pi\)
\(864\) 864.000 0.0340207
\(865\) −34132.0 −1.34164
\(866\) 21404.0 0.839882
\(867\) 5991.00 0.234677
\(868\) −2576.00 −0.100732
\(869\) −2200.00 −0.0858802
\(870\) 14280.0 0.556480
\(871\) −16568.0 −0.644530
\(872\) −2480.00 −0.0963112
\(873\) −9774.00 −0.378923
\(874\) −640.000 −0.0247692
\(875\) −5292.00 −0.204460
\(876\) 24.0000 0.000925668 0
\(877\) −43546.0 −1.67667 −0.838337 0.545152i \(-0.816472\pi\)
−0.838337 + 0.545152i \(0.816472\pi\)
\(878\) 9920.00 0.381303
\(879\) 12006.0 0.460697
\(880\) −2464.00 −0.0943880
\(881\) −10278.0 −0.393047 −0.196524 0.980499i \(-0.562965\pi\)
−0.196524 + 0.980499i \(0.562965\pi\)
\(882\) −882.000 −0.0336718
\(883\) 20708.0 0.789218 0.394609 0.918849i \(-0.370880\pi\)
0.394609 + 0.918849i \(0.370880\pi\)
\(884\) 8208.00 0.312291
\(885\) 10920.0 0.414770
\(886\) −376.000 −0.0142573
\(887\) −30296.0 −1.14683 −0.573416 0.819264i \(-0.694382\pi\)
−0.573416 + 0.819264i \(0.694382\pi\)
\(888\) 7056.00 0.266648
\(889\) 11032.0 0.416200
\(890\) −1960.00 −0.0738195
\(891\) 891.000 0.0335013
\(892\) −9488.00 −0.356145
\(893\) −3040.00 −0.113919
\(894\) −5100.00 −0.190794
\(895\) −22680.0 −0.847049
\(896\) 896.000 0.0334077
\(897\) −912.000 −0.0339474
\(898\) 6300.00 0.234113
\(899\) −15640.0 −0.580226
\(900\) 2556.00 0.0946667
\(901\) −17388.0 −0.642928
\(902\) 5676.00 0.209523
\(903\) −1092.00 −0.0402431
\(904\) 14576.0 0.536273
\(905\) −50428.0 −1.85225
\(906\) −8688.00 −0.318587
\(907\) 29604.0 1.08378 0.541888 0.840451i \(-0.317710\pi\)
0.541888 + 0.840451i \(0.317710\pi\)
\(908\) 4096.00 0.149703
\(909\) 918.000 0.0334963
\(910\) −7448.00 −0.271317
\(911\) 12112.0 0.440492 0.220246 0.975444i \(-0.429314\pi\)
0.220246 + 0.975444i \(0.429314\pi\)
\(912\) −1920.00 −0.0697122
\(913\) −10472.0 −0.379598
\(914\) 25612.0 0.926881
\(915\) 924.000 0.0333842
\(916\) 1160.00 0.0418422
\(917\) −7504.00 −0.270233
\(918\) 2916.00 0.104839
\(919\) −33320.0 −1.19600 −0.598001 0.801496i \(-0.704038\pi\)
−0.598001 + 0.801496i \(0.704038\pi\)
\(920\) 896.000 0.0321090
\(921\) −16032.0 −0.573586
\(922\) 3556.00 0.127018
\(923\) −13984.0 −0.498688
\(924\) 924.000 0.0328976
\(925\) 20874.0 0.741982
\(926\) 31344.0 1.11234
\(927\) 1692.00 0.0599488
\(928\) 5440.00 0.192432
\(929\) −45950.0 −1.62279 −0.811394 0.584499i \(-0.801291\pi\)
−0.811394 + 0.584499i \(0.801291\pi\)
\(930\) −7728.00 −0.272485
\(931\) 1960.00 0.0689972
\(932\) −20088.0 −0.706013
\(933\) 6804.00 0.238749
\(934\) −8.00000 −0.000280266 0
\(935\) −8316.00 −0.290869
\(936\) −2736.00 −0.0955438
\(937\) 11054.0 0.385399 0.192699 0.981258i \(-0.438276\pi\)
0.192699 + 0.981258i \(0.438276\pi\)
\(938\) −6104.00 −0.212476
\(939\) 22266.0 0.773827
\(940\) 4256.00 0.147676
\(941\) −55818.0 −1.93370 −0.966852 0.255339i \(-0.917813\pi\)
−0.966852 + 0.255339i \(0.917813\pi\)
\(942\) −11316.0 −0.391396
\(943\) −2064.00 −0.0712758
\(944\) 4160.00 0.143428
\(945\) −2646.00 −0.0910840
\(946\) 1144.00 0.0393178
\(947\) −33636.0 −1.15420 −0.577098 0.816675i \(-0.695815\pi\)
−0.577098 + 0.816675i \(0.695815\pi\)
\(948\) 2400.00 0.0822240
\(949\) −76.0000 −0.00259965
\(950\) −5680.00 −0.193983
\(951\) 22878.0 0.780095
\(952\) 3024.00 0.102950
\(953\) 31418.0 1.06792 0.533961 0.845509i \(-0.320703\pi\)
0.533961 + 0.845509i \(0.320703\pi\)
\(954\) 5796.00 0.196701
\(955\) −48608.0 −1.64703
\(956\) 800.000 0.0270647
\(957\) 5610.00 0.189494
\(958\) −29280.0 −0.987467
\(959\) 1722.00 0.0579836
\(960\) 2688.00 0.0903696
\(961\) −21327.0 −0.715887
\(962\) −22344.0 −0.748856
\(963\) −15444.0 −0.516797
\(964\) −5992.00 −0.200196
\(965\) 3108.00 0.103679
\(966\) −336.000 −0.0111911
\(967\) −13176.0 −0.438171 −0.219086 0.975706i \(-0.570307\pi\)
−0.219086 + 0.975706i \(0.570307\pi\)
\(968\) −968.000 −0.0321412
\(969\) −6480.00 −0.214827
\(970\) −30408.0 −1.00654
\(971\) 27132.0 0.896712 0.448356 0.893855i \(-0.352010\pi\)
0.448356 + 0.893855i \(0.352010\pi\)
\(972\) −972.000 −0.0320750
\(973\) −11480.0 −0.378245
\(974\) 39792.0 1.30905
\(975\) −8094.00 −0.265862
\(976\) 352.000 0.0115443
\(977\) 27154.0 0.889185 0.444592 0.895733i \(-0.353349\pi\)
0.444592 + 0.895733i \(0.353349\pi\)
\(978\) 1368.00 0.0447278
\(979\) −770.000 −0.0251372
\(980\) −2744.00 −0.0894427
\(981\) 2790.00 0.0908031
\(982\) −15944.0 −0.518120
\(983\) 35068.0 1.13784 0.568919 0.822393i \(-0.307362\pi\)
0.568919 + 0.822393i \(0.307362\pi\)
\(984\) −6192.00 −0.200603
\(985\) 54684.0 1.76891
\(986\) 18360.0 0.593004
\(987\) −1596.00 −0.0514704
\(988\) 6080.00 0.195780
\(989\) −416.000 −0.0133752
\(990\) 2772.00 0.0889898
\(991\) 26072.0 0.835726 0.417863 0.908510i \(-0.362779\pi\)
0.417863 + 0.908510i \(0.362779\pi\)
\(992\) −2944.00 −0.0942259
\(993\) −4476.00 −0.143043
\(994\) −5152.00 −0.164398
\(995\) −26600.0 −0.847514
\(996\) 11424.0 0.363437
\(997\) −19866.0 −0.631056 −0.315528 0.948916i \(-0.602182\pi\)
−0.315528 + 0.948916i \(0.602182\pi\)
\(998\) 4520.00 0.143365
\(999\) −7938.00 −0.251399
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 462.4.a.b.1.1 1
3.2 odd 2 1386.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.4.a.b.1.1 1 1.1 even 1 trivial
1386.4.a.m.1.1 1 3.2 odd 2