Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.23904011012\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 21.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -18.0000 q^{5} +9.00000 q^{6} +7.00000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -18.0000 q^{5} +9.00000 q^{6} +7.00000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +54.0000 q^{10} -36.0000 q^{11} -3.00000 q^{12} -34.0000 q^{13} -21.0000 q^{14} +54.0000 q^{15} -71.0000 q^{16} +42.0000 q^{17} -27.0000 q^{18} -124.000 q^{19} -18.0000 q^{20} -21.0000 q^{21} +108.000 q^{22} -63.0000 q^{24} +199.000 q^{25} +102.000 q^{26} -27.0000 q^{27} +7.00000 q^{28} +102.000 q^{29} -162.000 q^{30} -160.000 q^{31} +45.0000 q^{32} +108.000 q^{33} -126.000 q^{34} -126.000 q^{35} +9.00000 q^{36} +398.000 q^{37} +372.000 q^{38} +102.000 q^{39} -378.000 q^{40} -318.000 q^{41} +63.0000 q^{42} -268.000 q^{43} -36.0000 q^{44} -162.000 q^{45} +240.000 q^{47} +213.000 q^{48} +49.0000 q^{49} -597.000 q^{50} -126.000 q^{51} -34.0000 q^{52} -498.000 q^{53} +81.0000 q^{54} +648.000 q^{55} +147.000 q^{56} +372.000 q^{57} -306.000 q^{58} -132.000 q^{59} +54.0000 q^{60} +398.000 q^{61} +480.000 q^{62} +63.0000 q^{63} +433.000 q^{64} +612.000 q^{65} -324.000 q^{66} +92.0000 q^{67} +42.0000 q^{68} +378.000 q^{70} -720.000 q^{71} +189.000 q^{72} -502.000 q^{73} -1194.00 q^{74} -597.000 q^{75} -124.000 q^{76} -252.000 q^{77} -306.000 q^{78} -1024.00 q^{79} +1278.00 q^{80} +81.0000 q^{81} +954.000 q^{82} -204.000 q^{83} -21.0000 q^{84} -756.000 q^{85} +804.000 q^{86} -306.000 q^{87} -756.000 q^{88} +354.000 q^{89} +486.000 q^{90} -238.000 q^{91} +480.000 q^{93} -720.000 q^{94} +2232.00 q^{95} -135.000 q^{96} -286.000 q^{97} -147.000 q^{98} -324.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.00000 0.125000
\(5\) −18.0000 −1.60997 −0.804984 0.593296i \(-0.797826\pi\)
−0.804984 + 0.593296i \(0.797826\pi\)
\(6\) 9.00000 0.612372
\(7\) 7.00000 0.377964
\(8\) 21.0000 0.928078
\(9\) 9.00000 0.333333
\(10\) 54.0000 1.70763
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) −3.00000 −0.0721688
\(13\) −34.0000 −0.725377 −0.362689 0.931910i \(-0.618141\pi\)
−0.362689 + 0.931910i \(0.618141\pi\)
\(14\) −21.0000 −0.400892
\(15\) 54.0000 0.929516
\(16\) −71.0000 −1.10938
\(17\) 42.0000 0.599206 0.299603 0.954064i \(-0.403146\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(18\) −27.0000 −0.353553
\(19\) −124.000 −1.49724 −0.748620 0.663000i \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) −18.0000 −0.201246
\(21\) −21.0000 −0.218218
\(22\) 108.000 1.04662
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −63.0000 −0.535826
\(25\) 199.000 1.59200
\(26\) 102.000 0.769379
\(27\) −27.0000 −0.192450
\(28\) 7.00000 0.0472456
\(29\) 102.000 0.653135 0.326568 0.945174i \(-0.394108\pi\)
0.326568 + 0.945174i \(0.394108\pi\)
\(30\) −162.000 −0.985901
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 45.0000 0.248592
\(33\) 108.000 0.569709
\(34\) −126.000 −0.635554
\(35\) −126.000 −0.608511
\(36\) 9.00000 0.0416667
\(37\) 398.000 1.76840 0.884200 0.467109i \(-0.154704\pi\)
0.884200 + 0.467109i \(0.154704\pi\)
\(38\) 372.000 1.58806
\(39\) 102.000 0.418797
\(40\) −378.000 −1.49418
\(41\) −318.000 −1.21130 −0.605649 0.795732i \(-0.707087\pi\)
−0.605649 + 0.795732i \(0.707087\pi\)
\(42\) 63.0000 0.231455
\(43\) −268.000 −0.950456 −0.475228 0.879863i \(-0.657634\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(44\) −36.0000 −0.123346
\(45\) −162.000 −0.536656
\(46\) 0 0
\(47\) 240.000 0.744843 0.372421 0.928064i \(-0.378528\pi\)
0.372421 + 0.928064i \(0.378528\pi\)
\(48\) 213.000 0.640498
\(49\) 49.0000 0.142857
\(50\) −597.000 −1.68857
\(51\) −126.000 −0.345952
\(52\) −34.0000 −0.0906721
\(53\) −498.000 −1.29067 −0.645335 0.763899i \(-0.723282\pi\)
−0.645335 + 0.763899i \(0.723282\pi\)
\(54\) 81.0000 0.204124
\(55\) 648.000 1.58866
\(56\) 147.000 0.350780
\(57\) 372.000 0.864432
\(58\) −306.000 −0.692755
\(59\) −132.000 −0.291270 −0.145635 0.989338i \(-0.546523\pi\)
−0.145635 + 0.989338i \(0.546523\pi\)
\(60\) 54.0000 0.116190
\(61\) 398.000 0.835388 0.417694 0.908588i \(-0.362838\pi\)
0.417694 + 0.908588i \(0.362838\pi\)
\(62\) 480.000 0.983227
\(63\) 63.0000 0.125988
\(64\) 433.000 0.845703
\(65\) 612.000 1.16783
\(66\) −324.000 −0.604267
\(67\) 92.0000 0.167755 0.0838775 0.996476i \(-0.473270\pi\)
0.0838775 + 0.996476i \(0.473270\pi\)
\(68\) 42.0000 0.0749007
\(69\) 0 0
\(70\) 378.000 0.645423
\(71\) −720.000 −1.20350 −0.601748 0.798686i \(-0.705529\pi\)
−0.601748 + 0.798686i \(0.705529\pi\)
\(72\) 189.000 0.309359
\(73\) −502.000 −0.804858 −0.402429 0.915451i \(-0.631834\pi\)
−0.402429 + 0.915451i \(0.631834\pi\)
\(74\) −1194.00 −1.87567
\(75\) −597.000 −0.919142
\(76\) −124.000 −0.187155
\(77\) −252.000 −0.372962
\(78\) −306.000 −0.444201
\(79\) −1024.00 −1.45834 −0.729171 0.684332i \(-0.760094\pi\)
−0.729171 + 0.684332i \(0.760094\pi\)
\(80\) 1278.00 1.78606
\(81\) 81.0000 0.111111
\(82\) 954.000 1.28478
\(83\) −204.000 −0.269782 −0.134891 0.990860i \(-0.543068\pi\)
−0.134891 + 0.990860i \(0.543068\pi\)
\(84\) −21.0000 −0.0272772
\(85\) −756.000 −0.964703
\(86\) 804.000 1.00811
\(87\) −306.000 −0.377088
\(88\) −756.000 −0.915794
\(89\) 354.000 0.421617 0.210809 0.977527i \(-0.432390\pi\)
0.210809 + 0.977527i \(0.432390\pi\)
\(90\) 486.000 0.569210
\(91\) −238.000 −0.274167
\(92\) 0 0
\(93\) 480.000 0.535201
\(94\) −720.000 −0.790025
\(95\) 2232.00 2.41051
\(96\) −135.000 −0.143525
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) −147.000 −0.151523
\(99\) −324.000 −0.328921
\(100\) 199.000 0.199000
\(101\) 414.000 0.407867 0.203933 0.978985i \(-0.434627\pi\)
0.203933 + 0.978985i \(0.434627\pi\)
\(102\) 378.000 0.366937
\(103\) 56.0000 0.0535713 0.0267857 0.999641i \(-0.491473\pi\)
0.0267857 + 0.999641i \(0.491473\pi\)
\(104\) −714.000 −0.673206
\(105\) 378.000 0.351324
\(106\) 1494.00 1.36896
\(107\) 12.0000 0.0108419 0.00542095 0.999985i \(-0.498274\pi\)
0.00542095 + 0.999985i \(0.498274\pi\)
\(108\) −27.0000 −0.0240563
\(109\) 1478.00 1.29878 0.649389 0.760457i \(-0.275025\pi\)
0.649389 + 0.760457i \(0.275025\pi\)
\(110\) −1944.00 −1.68503
\(111\) −1194.00 −1.02099
\(112\) −497.000 −0.419304
\(113\) 402.000 0.334664 0.167332 0.985901i \(-0.446485\pi\)
0.167332 + 0.985901i \(0.446485\pi\)
\(114\) −1116.00 −0.916868
\(115\) 0 0
\(116\) 102.000 0.0816419
\(117\) −306.000 −0.241792
\(118\) 396.000 0.308939
\(119\) 294.000 0.226478
\(120\) 1134.00 0.862663
\(121\) −35.0000 −0.0262960
\(122\) −1194.00 −0.886063
\(123\) 954.000 0.699344
\(124\) −160.000 −0.115874
\(125\) −1332.00 −0.953102
\(126\) −189.000 −0.133631
\(127\) 1280.00 0.894344 0.447172 0.894448i \(-0.352431\pi\)
0.447172 + 0.894448i \(0.352431\pi\)
\(128\) −1659.00 −1.14560
\(129\) 804.000 0.548746
\(130\) −1836.00 −1.23868
\(131\) 1764.00 1.17650 0.588250 0.808679i \(-0.299817\pi\)
0.588250 + 0.808679i \(0.299817\pi\)
\(132\) 108.000 0.0712136
\(133\) −868.000 −0.565903
\(134\) −276.000 −0.177931
\(135\) 486.000 0.309839
\(136\) 882.000 0.556109
\(137\) −2358.00 −1.47049 −0.735246 0.677800i \(-0.762934\pi\)
−0.735246 + 0.677800i \(0.762934\pi\)
\(138\) 0 0
\(139\) −52.0000 −0.0317308 −0.0158654 0.999874i \(-0.505050\pi\)
−0.0158654 + 0.999874i \(0.505050\pi\)
\(140\) −126.000 −0.0760639
\(141\) −720.000 −0.430035
\(142\) 2160.00 1.27650
\(143\) 1224.00 0.715776
\(144\) −639.000 −0.369792
\(145\) −1836.00 −1.05153
\(146\) 1506.00 0.853681
\(147\) −147.000 −0.0824786
\(148\) 398.000 0.221050
\(149\) −1746.00 −0.959986 −0.479993 0.877272i \(-0.659361\pi\)
−0.479993 + 0.877272i \(0.659361\pi\)
\(150\) 1791.00 0.974897
\(151\) −232.000 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(152\) −2604.00 −1.38955
\(153\) 378.000 0.199735
\(154\) 756.000 0.395586
\(155\) 2880.00 1.49243
\(156\) 102.000 0.0523496
\(157\) 1694.00 0.861120 0.430560 0.902562i \(-0.358316\pi\)
0.430560 + 0.902562i \(0.358316\pi\)
\(158\) 3072.00 1.54681
\(159\) 1494.00 0.745169
\(160\) −810.000 −0.400226
\(161\) 0 0
\(162\) −243.000 −0.117851
\(163\) −2932.00 −1.40891 −0.704454 0.709750i \(-0.748808\pi\)
−0.704454 + 0.709750i \(0.748808\pi\)
\(164\) −318.000 −0.151412
\(165\) −1944.00 −0.917213
\(166\) 612.000 0.286147
\(167\) 1176.00 0.544920 0.272460 0.962167i \(-0.412163\pi\)
0.272460 + 0.962167i \(0.412163\pi\)
\(168\) −441.000 −0.202523
\(169\) −1041.00 −0.473828
\(170\) 2268.00 1.02322
\(171\) −1116.00 −0.499080
\(172\) −268.000 −0.118807
\(173\) 870.000 0.382340 0.191170 0.981557i \(-0.438772\pi\)
0.191170 + 0.981557i \(0.438772\pi\)
\(174\) 918.000 0.399962
\(175\) 1393.00 0.601719
\(176\) 2556.00 1.09469
\(177\) 396.000 0.168165
\(178\) −1062.00 −0.447193
\(179\) −2316.00 −0.967072 −0.483536 0.875324i \(-0.660648\pi\)
−0.483536 + 0.875324i \(0.660648\pi\)
\(180\) −162.000 −0.0670820
\(181\) −106.000 −0.0435299 −0.0217650 0.999763i \(-0.506929\pi\)
−0.0217650 + 0.999763i \(0.506929\pi\)
\(182\) 714.000 0.290798
\(183\) −1194.00 −0.482312
\(184\) 0 0
\(185\) −7164.00 −2.84707
\(186\) −1440.00 −0.567666
\(187\) −1512.00 −0.591275
\(188\) 240.000 0.0931053
\(189\) −189.000 −0.0727393
\(190\) −6696.00 −2.55673
\(191\) −1128.00 −0.427326 −0.213663 0.976907i \(-0.568539\pi\)
−0.213663 + 0.976907i \(0.568539\pi\)
\(192\) −1299.00 −0.488267
\(193\) 4034.00 1.50453 0.752263 0.658862i \(-0.228962\pi\)
0.752263 + 0.658862i \(0.228962\pi\)
\(194\) 858.000 0.317530
\(195\) −1836.00 −0.674250
\(196\) 49.0000 0.0178571
\(197\) −1314.00 −0.475221 −0.237611 0.971360i \(-0.576364\pi\)
−0.237611 + 0.971360i \(0.576364\pi\)
\(198\) 972.000 0.348874
\(199\) 5096.00 1.81531 0.907653 0.419722i \(-0.137872\pi\)
0.907653 + 0.419722i \(0.137872\pi\)
\(200\) 4179.00 1.47750
\(201\) −276.000 −0.0968534
\(202\) −1242.00 −0.432608
\(203\) 714.000 0.246862
\(204\) −126.000 −0.0432439
\(205\) 5724.00 1.95015
\(206\) −168.000 −0.0568209
\(207\) 0 0
\(208\) 2414.00 0.804715
\(209\) 4464.00 1.47742
\(210\) −1134.00 −0.372635
\(211\) −3076.00 −1.00360 −0.501802 0.864982i \(-0.667330\pi\)
−0.501802 + 0.864982i \(0.667330\pi\)
\(212\) −498.000 −0.161334
\(213\) 2160.00 0.694839
\(214\) −36.0000 −0.0114996
\(215\) 4824.00 1.53020
\(216\) −567.000 −0.178609
\(217\) −1120.00 −0.350371
\(218\) −4434.00 −1.37756
\(219\) 1506.00 0.464685
\(220\) 648.000 0.198583
\(221\) −1428.00 −0.434650
\(222\) 3582.00 1.08292
\(223\) −1888.00 −0.566950 −0.283475 0.958980i \(-0.591487\pi\)
−0.283475 + 0.958980i \(0.591487\pi\)
\(224\) 315.000 0.0939590
\(225\) 1791.00 0.530667
\(226\) −1206.00 −0.354964
\(227\) −4716.00 −1.37891 −0.689454 0.724330i \(-0.742149\pi\)
−0.689454 + 0.724330i \(0.742149\pi\)
\(228\) 372.000 0.108054
\(229\) −1690.00 −0.487678 −0.243839 0.969816i \(-0.578407\pi\)
−0.243839 + 0.969816i \(0.578407\pi\)
\(230\) 0 0
\(231\) 756.000 0.215330
\(232\) 2142.00 0.606160
\(233\) 138.000 0.0388012 0.0194006 0.999812i \(-0.493824\pi\)
0.0194006 + 0.999812i \(0.493824\pi\)
\(234\) 918.000 0.256460
\(235\) −4320.00 −1.19917
\(236\) −132.000 −0.0364088
\(237\) 3072.00 0.841974
\(238\) −882.000 −0.240217
\(239\) 1896.00 0.513147 0.256573 0.966525i \(-0.417406\pi\)
0.256573 + 0.966525i \(0.417406\pi\)
\(240\) −3834.00 −1.03118
\(241\) −3598.00 −0.961691 −0.480846 0.876805i \(-0.659670\pi\)
−0.480846 + 0.876805i \(0.659670\pi\)
\(242\) 105.000 0.0278911
\(243\) −243.000 −0.0641500
\(244\) 398.000 0.104424
\(245\) −882.000 −0.229996
\(246\) −2862.00 −0.741766
\(247\) 4216.00 1.08606
\(248\) −3360.00 −0.860323
\(249\) 612.000 0.155759
\(250\) 3996.00 1.01092
\(251\) −3060.00 −0.769504 −0.384752 0.923020i \(-0.625713\pi\)
−0.384752 + 0.923020i \(0.625713\pi\)
\(252\) 63.0000 0.0157485
\(253\) 0 0
\(254\) −3840.00 −0.948595
\(255\) 2268.00 0.556971
\(256\) 1513.00 0.369385
\(257\) −6822.00 −1.65582 −0.827908 0.560864i \(-0.810469\pi\)
−0.827908 + 0.560864i \(0.810469\pi\)
\(258\) −2412.00 −0.582033
\(259\) 2786.00 0.668392
\(260\) 612.000 0.145979
\(261\) 918.000 0.217712
\(262\) −5292.00 −1.24787
\(263\) 2592.00 0.607717 0.303858 0.952717i \(-0.401725\pi\)
0.303858 + 0.952717i \(0.401725\pi\)
\(264\) 2268.00 0.528734
\(265\) 8964.00 2.07794
\(266\) 2604.00 0.600231
\(267\) −1062.00 −0.243421
\(268\) 92.0000 0.0209694
\(269\) 8214.00 1.86177 0.930886 0.365311i \(-0.119037\pi\)
0.930886 + 0.365311i \(0.119037\pi\)
\(270\) −1458.00 −0.328634
\(271\) −5344.00 −1.19788 −0.598939 0.800795i \(-0.704411\pi\)
−0.598939 + 0.800795i \(0.704411\pi\)
\(272\) −2982.00 −0.664744
\(273\) 714.000 0.158290
\(274\) 7074.00 1.55969
\(275\) −7164.00 −1.57093
\(276\) 0 0
\(277\) −6514.00 −1.41295 −0.706477 0.707736i \(-0.749717\pi\)
−0.706477 + 0.707736i \(0.749717\pi\)
\(278\) 156.000 0.0336556
\(279\) −1440.00 −0.308998
\(280\) −2646.00 −0.564746
\(281\) 6618.00 1.40497 0.702485 0.711698i \(-0.252074\pi\)
0.702485 + 0.711698i \(0.252074\pi\)
\(282\) 2160.00 0.456121
\(283\) 3260.00 0.684759 0.342380 0.939562i \(-0.388767\pi\)
0.342380 + 0.939562i \(0.388767\pi\)
\(284\) −720.000 −0.150437
\(285\) −6696.00 −1.39171
\(286\) −3672.00 −0.759195
\(287\) −2226.00 −0.457828
\(288\) 405.000 0.0828641
\(289\) −3149.00 −0.640953
\(290\) 5508.00 1.11531
\(291\) 858.000 0.172841
\(292\) −502.000 −0.100607
\(293\) 5118.00 1.02047 0.510233 0.860036i \(-0.329559\pi\)
0.510233 + 0.860036i \(0.329559\pi\)
\(294\) 441.000 0.0874818
\(295\) 2376.00 0.468936
\(296\) 8358.00 1.64121
\(297\) 972.000 0.189903
\(298\) 5238.00 1.01822
\(299\) 0 0
\(300\) −597.000 −0.114893
\(301\) −1876.00 −0.359239
\(302\) 696.000 0.132617
\(303\) −1242.00 −0.235482
\(304\) 8804.00 1.66100
\(305\) −7164.00 −1.34495
\(306\) −1134.00 −0.211851
\(307\) 452.000 0.0840293 0.0420147 0.999117i \(-0.486622\pi\)
0.0420147 + 0.999117i \(0.486622\pi\)
\(308\) −252.000 −0.0466202
\(309\) −168.000 −0.0309294
\(310\) −8640.00 −1.58296
\(311\) 5016.00 0.914570 0.457285 0.889320i \(-0.348822\pi\)
0.457285 + 0.889320i \(0.348822\pi\)
\(312\) 2142.00 0.388676
\(313\) 5402.00 0.975524 0.487762 0.872977i \(-0.337813\pi\)
0.487762 + 0.872977i \(0.337813\pi\)
\(314\) −5082.00 −0.913356
\(315\) −1134.00 −0.202837
\(316\) −1024.00 −0.182293
\(317\) 10086.0 1.78702 0.893511 0.449041i \(-0.148234\pi\)
0.893511 + 0.449041i \(0.148234\pi\)
\(318\) −4482.00 −0.790371
\(319\) −3672.00 −0.644491
\(320\) −7794.00 −1.36156
\(321\) −36.0000 −0.00625958
\(322\) 0 0
\(323\) −5208.00 −0.897154
\(324\) 81.0000 0.0138889
\(325\) −6766.00 −1.15480
\(326\) 8796.00 1.49437
\(327\) −4434.00 −0.749849
\(328\) −6678.00 −1.12418
\(329\) 1680.00 0.281524
\(330\) 5832.00 0.972852
\(331\) −8044.00 −1.33577 −0.667883 0.744267i \(-0.732799\pi\)
−0.667883 + 0.744267i \(0.732799\pi\)
\(332\) −204.000 −0.0337228
\(333\) 3582.00 0.589467
\(334\) −3528.00 −0.577975
\(335\) −1656.00 −0.270080
\(336\) 1491.00 0.242085
\(337\) 4178.00 0.675342 0.337671 0.941264i \(-0.390361\pi\)
0.337671 + 0.941264i \(0.390361\pi\)
\(338\) 3123.00 0.502570
\(339\) −1206.00 −0.193218
\(340\) −756.000 −0.120588
\(341\) 5760.00 0.914726
\(342\) 3348.00 0.529354
\(343\) 343.000 0.0539949
\(344\) −5628.00 −0.882097
\(345\) 0 0
\(346\) −2610.00 −0.405533
\(347\) 156.000 0.0241341 0.0120670 0.999927i \(-0.496159\pi\)
0.0120670 + 0.999927i \(0.496159\pi\)
\(348\) −306.000 −0.0471360
\(349\) −12418.0 −1.90464 −0.952321 0.305097i \(-0.901311\pi\)
−0.952321 + 0.305097i \(0.901311\pi\)
\(350\) −4179.00 −0.638220
\(351\) 918.000 0.139599
\(352\) −1620.00 −0.245302
\(353\) −7830.00 −1.18059 −0.590296 0.807187i \(-0.700989\pi\)
−0.590296 + 0.807187i \(0.700989\pi\)
\(354\) −1188.00 −0.178366
\(355\) 12960.0 1.93759
\(356\) 354.000 0.0527021
\(357\) −882.000 −0.130757
\(358\) 6948.00 1.02574
\(359\) −9312.00 −1.36899 −0.684497 0.729016i \(-0.739978\pi\)
−0.684497 + 0.729016i \(0.739978\pi\)
\(360\) −3402.00 −0.498059
\(361\) 8517.00 1.24173
\(362\) 318.000 0.0461705
\(363\) 105.000 0.0151820
\(364\) −238.000 −0.0342709
\(365\) 9036.00 1.29580
\(366\) 3582.00 0.511569
\(367\) −3760.00 −0.534797 −0.267398 0.963586i \(-0.586164\pi\)
−0.267398 + 0.963586i \(0.586164\pi\)
\(368\) 0 0
\(369\) −2862.00 −0.403766
\(370\) 21492.0 3.01977
\(371\) −3486.00 −0.487828
\(372\) 480.000 0.0669001
\(373\) 5870.00 0.814845 0.407422 0.913240i \(-0.366428\pi\)
0.407422 + 0.913240i \(0.366428\pi\)
\(374\) 4536.00 0.627142
\(375\) 3996.00 0.550273
\(376\) 5040.00 0.691272
\(377\) −3468.00 −0.473769
\(378\) 567.000 0.0771517
\(379\) −1852.00 −0.251005 −0.125502 0.992093i \(-0.540054\pi\)
−0.125502 + 0.992093i \(0.540054\pi\)
\(380\) 2232.00 0.301314
\(381\) −3840.00 −0.516350
\(382\) 3384.00 0.453247
\(383\) 2160.00 0.288175 0.144087 0.989565i \(-0.453975\pi\)
0.144087 + 0.989565i \(0.453975\pi\)
\(384\) 4977.00 0.661410
\(385\) 4536.00 0.600457
\(386\) −12102.0 −1.59579
\(387\) −2412.00 −0.316819
\(388\) −286.000 −0.0374213
\(389\) −6786.00 −0.884483 −0.442241 0.896896i \(-0.645817\pi\)
−0.442241 + 0.896896i \(0.645817\pi\)
\(390\) 5508.00 0.715150
\(391\) 0 0
\(392\) 1029.00 0.132583
\(393\) −5292.00 −0.679252
\(394\) 3942.00 0.504048
\(395\) 18432.0 2.34788
\(396\) −324.000 −0.0411152
\(397\) −6514.00 −0.823497 −0.411748 0.911298i \(-0.635082\pi\)
−0.411748 + 0.911298i \(0.635082\pi\)
\(398\) −15288.0 −1.92542
\(399\) 2604.00 0.326724
\(400\) −14129.0 −1.76612
\(401\) 3330.00 0.414694 0.207347 0.978267i \(-0.433517\pi\)
0.207347 + 0.978267i \(0.433517\pi\)
\(402\) 828.000 0.102729
\(403\) 5440.00 0.672421
\(404\) 414.000 0.0509833
\(405\) −1458.00 −0.178885
\(406\) −2142.00 −0.261837
\(407\) −14328.0 −1.74499
\(408\) −2646.00 −0.321070
\(409\) −5398.00 −0.652601 −0.326301 0.945266i \(-0.605802\pi\)
−0.326301 + 0.945266i \(0.605802\pi\)
\(410\) −17172.0 −2.06845
\(411\) 7074.00 0.848990
\(412\) 56.0000 0.00669641
\(413\) −924.000 −0.110090
\(414\) 0 0
\(415\) 3672.00 0.434341
\(416\) −1530.00 −0.180323
\(417\) 156.000 0.0183198
\(418\) −13392.0 −1.56704
\(419\) 13092.0 1.52646 0.763229 0.646128i \(-0.223613\pi\)
0.763229 + 0.646128i \(0.223613\pi\)
\(420\) 378.000 0.0439155
\(421\) −322.000 −0.0372763 −0.0186381 0.999826i \(-0.505933\pi\)
−0.0186381 + 0.999826i \(0.505933\pi\)
\(422\) 9228.00 1.06448
\(423\) 2160.00 0.248281
\(424\) −10458.0 −1.19784
\(425\) 8358.00 0.953935
\(426\) −6480.00 −0.736988
\(427\) 2786.00 0.315747
\(428\) 12.0000 0.00135524
\(429\) −3672.00 −0.413254
\(430\) −14472.0 −1.62303
\(431\) 2616.00 0.292363 0.146181 0.989258i \(-0.453302\pi\)
0.146181 + 0.989258i \(0.453302\pi\)
\(432\) 1917.00 0.213499
\(433\) 4322.00 0.479681 0.239841 0.970812i \(-0.422905\pi\)
0.239841 + 0.970812i \(0.422905\pi\)
\(434\) 3360.00 0.371625
\(435\) 5508.00 0.607100
\(436\) 1478.00 0.162347
\(437\) 0 0
\(438\) −4518.00 −0.492873
\(439\) −9016.00 −0.980205 −0.490103 0.871665i \(-0.663041\pi\)
−0.490103 + 0.871665i \(0.663041\pi\)
\(440\) 13608.0 1.47440
\(441\) 441.000 0.0476190
\(442\) 4284.00 0.461016
\(443\) −5268.00 −0.564989 −0.282495 0.959269i \(-0.591162\pi\)
−0.282495 + 0.959269i \(0.591162\pi\)
\(444\) −1194.00 −0.127623
\(445\) −6372.00 −0.678790
\(446\) 5664.00 0.601341
\(447\) 5238.00 0.554248
\(448\) 3031.00 0.319646
\(449\) −5310.00 −0.558117 −0.279058 0.960274i \(-0.590022\pi\)
−0.279058 + 0.960274i \(0.590022\pi\)
\(450\) −5373.00 −0.562857
\(451\) 11448.0 1.19527
\(452\) 402.000 0.0418329
\(453\) 696.000 0.0721875
\(454\) 14148.0 1.46255
\(455\) 4284.00 0.441400
\(456\) 7812.00 0.802260
\(457\) 15770.0 1.61420 0.807100 0.590415i \(-0.201036\pi\)
0.807100 + 0.590415i \(0.201036\pi\)
\(458\) 5070.00 0.517261
\(459\) −1134.00 −0.115317
\(460\) 0 0
\(461\) −5370.00 −0.542529 −0.271264 0.962505i \(-0.587442\pi\)
−0.271264 + 0.962505i \(0.587442\pi\)
\(462\) −2268.00 −0.228392
\(463\) −3328.00 −0.334050 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(464\) −7242.00 −0.724572
\(465\) −8640.00 −0.861657
\(466\) −414.000 −0.0411549
\(467\) 4548.00 0.450656 0.225328 0.974283i \(-0.427655\pi\)
0.225328 + 0.974283i \(0.427655\pi\)
\(468\) −306.000 −0.0302240
\(469\) 644.000 0.0634055
\(470\) 12960.0 1.27192
\(471\) −5082.00 −0.497168
\(472\) −2772.00 −0.270321
\(473\) 9648.00 0.937876
\(474\) −9216.00 −0.893048
\(475\) −24676.0 −2.38361
\(476\) 294.000 0.0283098
\(477\) −4482.00 −0.430224
\(478\) −5688.00 −0.544274
\(479\) −8064.00 −0.769214 −0.384607 0.923080i \(-0.625663\pi\)
−0.384607 + 0.923080i \(0.625663\pi\)
\(480\) 2430.00 0.231070
\(481\) −13532.0 −1.28276
\(482\) 10794.0 1.02003
\(483\) 0 0
\(484\) −35.0000 −0.00328700
\(485\) 5148.00 0.481977
\(486\) 729.000 0.0680414
\(487\) 16616.0 1.54608 0.773042 0.634355i \(-0.218734\pi\)
0.773042 + 0.634355i \(0.218734\pi\)
\(488\) 8358.00 0.775305
\(489\) 8796.00 0.813433
\(490\) 2646.00 0.243947
\(491\) −7140.00 −0.656260 −0.328130 0.944633i \(-0.606418\pi\)
−0.328130 + 0.944633i \(0.606418\pi\)
\(492\) 954.000 0.0874180
\(493\) 4284.00 0.391362
\(494\) −12648.0 −1.15194
\(495\) 5832.00 0.529553
\(496\) 11360.0 1.02839
\(497\) −5040.00 −0.454879
\(498\) −1836.00 −0.165207
\(499\) −9124.00 −0.818530 −0.409265 0.912416i \(-0.634215\pi\)
−0.409265 + 0.912416i \(0.634215\pi\)
\(500\) −1332.00 −0.119138
\(501\) −3528.00 −0.314610
\(502\) 9180.00 0.816182
\(503\) −6552.00 −0.580794 −0.290397 0.956906i \(-0.593787\pi\)
−0.290397 + 0.956906i \(0.593787\pi\)
\(504\) 1323.00 0.116927
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) 3123.00 0.273565
\(508\) 1280.00 0.111793
\(509\) 2790.00 0.242956 0.121478 0.992594i \(-0.461237\pi\)
0.121478 + 0.992594i \(0.461237\pi\)
\(510\) −6804.00 −0.590757
\(511\) −3514.00 −0.304208
\(512\) 8733.00 0.753804
\(513\) 3348.00 0.288144
\(514\) 20466.0 1.75626
\(515\) −1008.00 −0.0862481
\(516\) 804.000 0.0685933
\(517\) −8640.00 −0.734984
\(518\) −8358.00 −0.708937
\(519\) −2610.00 −0.220744
\(520\) 12852.0 1.08384
\(521\) −14862.0 −1.24974 −0.624871 0.780728i \(-0.714849\pi\)
−0.624871 + 0.780728i \(0.714849\pi\)
\(522\) −2754.00 −0.230918
\(523\) 17660.0 1.47652 0.738258 0.674518i \(-0.235649\pi\)
0.738258 + 0.674518i \(0.235649\pi\)
\(524\) 1764.00 0.147062
\(525\) −4179.00 −0.347403
\(526\) −7776.00 −0.644581
\(527\) −6720.00 −0.555461
\(528\) −7668.00 −0.632021
\(529\) −12167.0 −1.00000
\(530\) −26892.0 −2.20399
\(531\) −1188.00 −0.0970900
\(532\) −868.000 −0.0707379
\(533\) 10812.0 0.878649
\(534\) 3186.00 0.258187
\(535\) −216.000 −0.0174551
\(536\) 1932.00 0.155690
\(537\) 6948.00 0.558340
\(538\) −24642.0 −1.97471
\(539\) −1764.00 −0.140966
\(540\) 486.000 0.0387298
\(541\) −19834.0 −1.57621 −0.788106 0.615540i \(-0.788938\pi\)
−0.788106 + 0.615540i \(0.788938\pi\)
\(542\) 16032.0 1.27054
\(543\) 318.000 0.0251320
\(544\) 1890.00 0.148958
\(545\) −26604.0 −2.09099
\(546\) −2142.00 −0.167892
\(547\) 20972.0 1.63930 0.819651 0.572863i \(-0.194167\pi\)
0.819651 + 0.572863i \(0.194167\pi\)
\(548\) −2358.00 −0.183812
\(549\) 3582.00 0.278463
\(550\) 21492.0 1.66622
\(551\) −12648.0 −0.977900
\(552\) 0 0
\(553\) −7168.00 −0.551201
\(554\) 19542.0 1.49866
\(555\) 21492.0 1.64376
\(556\) −52.0000 −0.00396635
\(557\) 21174.0 1.61072 0.805360 0.592786i \(-0.201972\pi\)
0.805360 + 0.592786i \(0.201972\pi\)
\(558\) 4320.00 0.327742
\(559\) 9112.00 0.689439
\(560\) 8946.00 0.675067
\(561\) 4536.00 0.341373
\(562\) −19854.0 −1.49020
\(563\) −17772.0 −1.33037 −0.665187 0.746677i \(-0.731648\pi\)
−0.665187 + 0.746677i \(0.731648\pi\)
\(564\) −720.000 −0.0537544
\(565\) −7236.00 −0.538798
\(566\) −9780.00 −0.726297
\(567\) 567.000 0.0419961
\(568\) −15120.0 −1.11694
\(569\) 8250.00 0.607835 0.303917 0.952698i \(-0.401705\pi\)
0.303917 + 0.952698i \(0.401705\pi\)
\(570\) 20088.0 1.47613
\(571\) 20756.0 1.52121 0.760606 0.649214i \(-0.224902\pi\)
0.760606 + 0.649214i \(0.224902\pi\)
\(572\) 1224.00 0.0894720
\(573\) 3384.00 0.246717
\(574\) 6678.00 0.485600
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) 2.00000 0.000144300 0 7.21500e−5 1.00000i \(-0.499977\pi\)
7.21500e−5 1.00000i \(0.499977\pi\)
\(578\) 9447.00 0.679833
\(579\) −12102.0 −0.868639
\(580\) −1836.00 −0.131441
\(581\) −1428.00 −0.101968
\(582\) −2574.00 −0.183326
\(583\) 17928.0 1.27359
\(584\) −10542.0 −0.746971
\(585\) 5508.00 0.389278
\(586\) −15354.0 −1.08237
\(587\) 26364.0 1.85376 0.926881 0.375354i \(-0.122479\pi\)
0.926881 + 0.375354i \(0.122479\pi\)
\(588\) −147.000 −0.0103098
\(589\) 19840.0 1.38793
\(590\) −7128.00 −0.497382
\(591\) 3942.00 0.274369
\(592\) −28258.0 −1.96182
\(593\) 2298.00 0.159136 0.0795679 0.996829i \(-0.474646\pi\)
0.0795679 + 0.996829i \(0.474646\pi\)
\(594\) −2916.00 −0.201422
\(595\) −5292.00 −0.364623
\(596\) −1746.00 −0.119998
\(597\) −15288.0 −1.04807
\(598\) 0 0
\(599\) 3072.00 0.209547 0.104773 0.994496i \(-0.466588\pi\)
0.104773 + 0.994496i \(0.466588\pi\)
\(600\) −12537.0 −0.853035
\(601\) 24554.0 1.66652 0.833260 0.552881i \(-0.186472\pi\)
0.833260 + 0.552881i \(0.186472\pi\)
\(602\) 5628.00 0.381030
\(603\) 828.000 0.0559184
\(604\) −232.000 −0.0156290
\(605\) 630.000 0.0423358
\(606\) 3726.00 0.249766
\(607\) 16832.0 1.12552 0.562759 0.826621i \(-0.309740\pi\)
0.562759 + 0.826621i \(0.309740\pi\)
\(608\) −5580.00 −0.372202
\(609\) −2142.00 −0.142526
\(610\) 21492.0 1.42653
\(611\) −8160.00 −0.540292
\(612\) 378.000 0.0249669
\(613\) −2482.00 −0.163535 −0.0817676 0.996651i \(-0.526057\pi\)
−0.0817676 + 0.996651i \(0.526057\pi\)
\(614\) −1356.00 −0.0891266
\(615\) −17172.0 −1.12592
\(616\) −5292.00 −0.346138
\(617\) −15798.0 −1.03080 −0.515400 0.856950i \(-0.672357\pi\)
−0.515400 + 0.856950i \(0.672357\pi\)
\(618\) 504.000 0.0328056
\(619\) −15460.0 −1.00386 −0.501930 0.864908i \(-0.667377\pi\)
−0.501930 + 0.864908i \(0.667377\pi\)
\(620\) 2880.00 0.186554
\(621\) 0 0
\(622\) −15048.0 −0.970048
\(623\) 2478.00 0.159356
\(624\) −7242.00 −0.464603
\(625\) −899.000 −0.0575360
\(626\) −16206.0 −1.03470
\(627\) −13392.0 −0.852990
\(628\) 1694.00 0.107640
\(629\) 16716.0 1.05964
\(630\) 3402.00 0.215141
\(631\) −7720.00 −0.487050 −0.243525 0.969895i \(-0.578304\pi\)
−0.243525 + 0.969895i \(0.578304\pi\)
\(632\) −21504.0 −1.35345
\(633\) 9228.00 0.579431
\(634\) −30258.0 −1.89542
\(635\) −23040.0 −1.43987
\(636\) 1494.00 0.0931462
\(637\) −1666.00 −0.103625
\(638\) 11016.0 0.683586
\(639\) −6480.00 −0.401166
\(640\) 29862.0 1.84437
\(641\) −17262.0 −1.06366 −0.531832 0.846850i \(-0.678496\pi\)
−0.531832 + 0.846850i \(0.678496\pi\)
\(642\) 108.000 0.00663928
\(643\) −12220.0 −0.749471 −0.374735 0.927132i \(-0.622266\pi\)
−0.374735 + 0.927132i \(0.622266\pi\)
\(644\) 0 0
\(645\) −14472.0 −0.883464
\(646\) 15624.0 0.951576
\(647\) 13560.0 0.823955 0.411977 0.911194i \(-0.364838\pi\)
0.411977 + 0.911194i \(0.364838\pi\)
\(648\) 1701.00 0.103120
\(649\) 4752.00 0.287415
\(650\) 20298.0 1.22485
\(651\) 3360.00 0.202287
\(652\) −2932.00 −0.176113
\(653\) 23094.0 1.38398 0.691989 0.721908i \(-0.256735\pi\)
0.691989 + 0.721908i \(0.256735\pi\)
\(654\) 13302.0 0.795335
\(655\) −31752.0 −1.89413
\(656\) 22578.0 1.34378
\(657\) −4518.00 −0.268286
\(658\) −5040.00 −0.298601
\(659\) 22548.0 1.33285 0.666423 0.745574i \(-0.267825\pi\)
0.666423 + 0.745574i \(0.267825\pi\)
\(660\) −1944.00 −0.114652
\(661\) 17462.0 1.02752 0.513762 0.857933i \(-0.328252\pi\)
0.513762 + 0.857933i \(0.328252\pi\)
\(662\) 24132.0 1.41679
\(663\) 4284.00 0.250945
\(664\) −4284.00 −0.250379
\(665\) 15624.0 0.911087
\(666\) −10746.0 −0.625224
\(667\) 0 0
\(668\) 1176.00 0.0681150
\(669\) 5664.00 0.327329
\(670\) 4968.00 0.286464
\(671\) −14328.0 −0.824331
\(672\) −945.000 −0.0542473
\(673\) −22462.0 −1.28655 −0.643274 0.765636i \(-0.722424\pi\)
−0.643274 + 0.765636i \(0.722424\pi\)
\(674\) −12534.0 −0.716308
\(675\) −5373.00 −0.306381
\(676\) −1041.00 −0.0592285
\(677\) −25554.0 −1.45069 −0.725347 0.688383i \(-0.758321\pi\)
−0.725347 + 0.688383i \(0.758321\pi\)
\(678\) 3618.00 0.204939
\(679\) −2002.00 −0.113151
\(680\) −15876.0 −0.895319
\(681\) 14148.0 0.796112
\(682\) −17280.0 −0.970213
\(683\) 9276.00 0.519672 0.259836 0.965653i \(-0.416331\pi\)
0.259836 + 0.965653i \(0.416331\pi\)
\(684\) −1116.00 −0.0623850
\(685\) 42444.0 2.36745
\(686\) −1029.00 −0.0572703
\(687\) 5070.00 0.281561
\(688\) 19028.0 1.05441
\(689\) 16932.0 0.936223
\(690\) 0 0
\(691\) 27380.0 1.50736 0.753679 0.657243i \(-0.228277\pi\)
0.753679 + 0.657243i \(0.228277\pi\)
\(692\) 870.000 0.0477925
\(693\) −2268.00 −0.124321
\(694\) −468.000 −0.0255980
\(695\) 936.000 0.0510856
\(696\) −6426.00 −0.349967
\(697\) −13356.0 −0.725817
\(698\) 37254.0 2.02018
\(699\) −414.000 −0.0224019
\(700\) 1393.00 0.0752149
\(701\) 25830.0 1.39171 0.695853 0.718184i \(-0.255027\pi\)
0.695853 + 0.718184i \(0.255027\pi\)
\(702\) −2754.00 −0.148067
\(703\) −49352.0 −2.64772
\(704\) −15588.0 −0.834510
\(705\) 12960.0 0.692343
\(706\) 23490.0 1.25221
\(707\) 2898.00 0.154159
\(708\) 396.000 0.0210206
\(709\) −6226.00 −0.329792 −0.164896 0.986311i \(-0.552729\pi\)
−0.164896 + 0.986311i \(0.552729\pi\)
\(710\) −38880.0 −2.05513
\(711\) −9216.00 −0.486114
\(712\) 7434.00 0.391293
\(713\) 0 0
\(714\) 2646.00 0.138689
\(715\) −22032.0 −1.15238
\(716\) −2316.00 −0.120884
\(717\) −5688.00 −0.296265
\(718\) 27936.0 1.45204
\(719\) −15072.0 −0.781767 −0.390884 0.920440i \(-0.627831\pi\)
−0.390884 + 0.920440i \(0.627831\pi\)
\(720\) 11502.0 0.595353
\(721\) 392.000 0.0202480
\(722\) −25551.0 −1.31705
\(723\) 10794.0 0.555233
\(724\) −106.000 −0.00544124
\(725\) 20298.0 1.03979
\(726\) −315.000 −0.0161030
\(727\) −32920.0 −1.67942 −0.839708 0.543038i \(-0.817274\pi\)
−0.839708 + 0.543038i \(0.817274\pi\)
\(728\) −4998.00 −0.254448
\(729\) 729.000 0.0370370
\(730\) −27108.0 −1.37440
\(731\) −11256.0 −0.569519
\(732\) −1194.00 −0.0602889
\(733\) −6946.00 −0.350009 −0.175004 0.984568i \(-0.555994\pi\)
−0.175004 + 0.984568i \(0.555994\pi\)
\(734\) 11280.0 0.567238
\(735\) 2646.00 0.132788
\(736\) 0 0
\(737\) −3312.00 −0.165535
\(738\) 8586.00 0.428259
\(739\) −2356.00 −0.117276 −0.0586379 0.998279i \(-0.518676\pi\)
−0.0586379 + 0.998279i \(0.518676\pi\)
\(740\) −7164.00 −0.355884
\(741\) −12648.0 −0.627039
\(742\) 10458.0 0.517419
\(743\) −23520.0 −1.16133 −0.580663 0.814144i \(-0.697207\pi\)
−0.580663 + 0.814144i \(0.697207\pi\)
\(744\) 10080.0 0.496708
\(745\) 31428.0 1.54555
\(746\) −17610.0 −0.864273
\(747\) −1836.00 −0.0899273
\(748\) −1512.00 −0.0739094
\(749\) 84.0000 0.00409785
\(750\) −11988.0 −0.583653
\(751\) 3008.00 0.146156 0.0730782 0.997326i \(-0.476718\pi\)
0.0730782 + 0.997326i \(0.476718\pi\)
\(752\) −17040.0 −0.826310
\(753\) 9180.00 0.444273
\(754\) 10404.0 0.502508
\(755\) 4176.00 0.201298
\(756\) −189.000 −0.00909241
\(757\) −20770.0 −0.997224 −0.498612 0.866825i \(-0.666157\pi\)
−0.498612 + 0.866825i \(0.666157\pi\)
\(758\) 5556.00 0.266231
\(759\) 0 0
\(760\) 46872.0 2.23714
\(761\) 11538.0 0.549609 0.274804 0.961500i \(-0.411387\pi\)
0.274804 + 0.961500i \(0.411387\pi\)
\(762\) 11520.0 0.547671
\(763\) 10346.0 0.490892
\(764\) −1128.00 −0.0534157
\(765\) −6804.00 −0.321568
\(766\) −6480.00 −0.305655
\(767\) 4488.00 0.211281
\(768\) −4539.00 −0.213264
\(769\) 8498.00 0.398499 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(770\) −13608.0 −0.636881
\(771\) 20466.0 0.955986
\(772\) 4034.00 0.188066
\(773\) −32322.0 −1.50393 −0.751967 0.659200i \(-0.770895\pi\)
−0.751967 + 0.659200i \(0.770895\pi\)
\(774\) 7236.00 0.336037
\(775\) −31840.0 −1.47578
\(776\) −6006.00 −0.277839
\(777\) −8358.00 −0.385896
\(778\) 20358.0 0.938136
\(779\) 39432.0 1.81360
\(780\) −1836.00 −0.0842812
\(781\) 25920.0 1.18757
\(782\) 0 0
\(783\) −2754.00 −0.125696
\(784\) −3479.00 −0.158482
\(785\) −30492.0 −1.38638
\(786\) 15876.0 0.720456
\(787\) 26228.0 1.18796 0.593982 0.804479i \(-0.297555\pi\)
0.593982 + 0.804479i \(0.297555\pi\)
\(788\) −1314.00 −0.0594027
\(789\) −7776.00 −0.350866
\(790\) −55296.0 −2.49031
\(791\) 2814.00 0.126491
\(792\) −6804.00 −0.305265
\(793\) −13532.0 −0.605972
\(794\) 19542.0 0.873450
\(795\) −26892.0 −1.19970
\(796\) 5096.00 0.226913
\(797\) −43338.0 −1.92611 −0.963056 0.269302i \(-0.913207\pi\)
−0.963056 + 0.269302i \(0.913207\pi\)
\(798\) −7812.00 −0.346544
\(799\) 10080.0 0.446314
\(800\) 8955.00 0.395759
\(801\) 3186.00 0.140539
\(802\) −9990.00 −0.439849
\(803\) 18072.0 0.794206
\(804\) −276.000 −0.0121067
\(805\) 0 0
\(806\) −16320.0 −0.713210
\(807\) −24642.0 −1.07489
\(808\) 8694.00 0.378532
\(809\) −28902.0 −1.25604 −0.628022 0.778195i \(-0.716135\pi\)
−0.628022 + 0.778195i \(0.716135\pi\)
\(810\) 4374.00 0.189737
\(811\) 27164.0 1.17615 0.588075 0.808807i \(-0.299886\pi\)
0.588075 + 0.808807i \(0.299886\pi\)
\(812\) 714.000 0.0308577
\(813\) 16032.0 0.691595
\(814\) 42984.0 1.85085
\(815\) 52776.0 2.26830
\(816\) 8946.00 0.383790
\(817\) 33232.0 1.42306
\(818\) 16194.0 0.692188
\(819\) −2142.00 −0.0913889
\(820\) 5724.00 0.243769
\(821\) −17202.0 −0.731247 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(822\) −21222.0 −0.900489
\(823\) −5992.00 −0.253789 −0.126894 0.991916i \(-0.540501\pi\)
−0.126894 + 0.991916i \(0.540501\pi\)
\(824\) 1176.00 0.0497183
\(825\) 21492.0 0.906976
\(826\) 2772.00 0.116768
\(827\) 25884.0 1.08836 0.544181 0.838968i \(-0.316841\pi\)
0.544181 + 0.838968i \(0.316841\pi\)
\(828\) 0 0
\(829\) −1474.00 −0.0617541 −0.0308770 0.999523i \(-0.509830\pi\)
−0.0308770 + 0.999523i \(0.509830\pi\)
\(830\) −11016.0 −0.460688
\(831\) 19542.0 0.815770
\(832\) −14722.0 −0.613454
\(833\) 2058.00 0.0856008
\(834\) −468.000 −0.0194311
\(835\) −21168.0 −0.877304
\(836\) 4464.00 0.184678
\(837\) 4320.00 0.178400
\(838\) −39276.0 −1.61905
\(839\) 33528.0 1.37964 0.689818 0.723983i \(-0.257690\pi\)
0.689818 + 0.723983i \(0.257690\pi\)
\(840\) 7938.00 0.326056
\(841\) −13985.0 −0.573414
\(842\) 966.000 0.0395375
\(843\) −19854.0 −0.811160
\(844\) −3076.00 −0.125451
\(845\) 18738.0 0.762848
\(846\) −6480.00 −0.263342
\(847\) −245.000 −0.00993896
\(848\) 35358.0 1.43184
\(849\) −9780.00 −0.395346
\(850\) −25074.0 −1.01180
\(851\) 0 0
\(852\) 2160.00 0.0868549
\(853\) 1190.00 0.0477665 0.0238832 0.999715i \(-0.492397\pi\)
0.0238832 + 0.999715i \(0.492397\pi\)
\(854\) −8358.00 −0.334900
\(855\) 20088.0 0.803503
\(856\) 252.000 0.0100621
\(857\) 34578.0 1.37825 0.689126 0.724642i \(-0.257995\pi\)
0.689126 + 0.724642i \(0.257995\pi\)
\(858\) 11016.0 0.438322
\(859\) −44404.0 −1.76373 −0.881865 0.471501i \(-0.843712\pi\)
−0.881865 + 0.471501i \(0.843712\pi\)
\(860\) 4824.00 0.191276
\(861\) 6678.00 0.264327
\(862\) −7848.00 −0.310097
\(863\) −38328.0 −1.51182 −0.755910 0.654676i \(-0.772805\pi\)
−0.755910 + 0.654676i \(0.772805\pi\)
\(864\) −1215.00 −0.0478416
\(865\) −15660.0 −0.615556
\(866\) −12966.0 −0.508779
\(867\) 9447.00 0.370054
\(868\) −1120.00 −0.0437964
\(869\) 36864.0 1.43904
\(870\) −16524.0 −0.643927
\(871\) −3128.00 −0.121686
\(872\) 31038.0 1.20537
\(873\) −2574.00 −0.0997900
\(874\) 0 0
\(875\) −9324.00 −0.360239
\(876\) 1506.00 0.0580856
\(877\) −38842.0 −1.49555 −0.747777 0.663950i \(-0.768879\pi\)
−0.747777 + 0.663950i \(0.768879\pi\)
\(878\) 27048.0 1.03966
\(879\) −15354.0 −0.589167
\(880\) −46008.0 −1.76242
\(881\) −35046.0 −1.34022 −0.670108 0.742264i \(-0.733752\pi\)
−0.670108 + 0.742264i \(0.733752\pi\)
\(882\) −1323.00 −0.0505076
\(883\) 14204.0 0.541339 0.270670 0.962672i \(-0.412755\pi\)
0.270670 + 0.962672i \(0.412755\pi\)
\(884\) −1428.00 −0.0543313
\(885\) −7128.00 −0.270740
\(886\) 15804.0 0.599262
\(887\) −26136.0 −0.989359 −0.494679 0.869076i \(-0.664714\pi\)
−0.494679 + 0.869076i \(0.664714\pi\)
\(888\) −25074.0 −0.947554
\(889\) 8960.00 0.338030
\(890\) 19116.0 0.719966
\(891\) −2916.00 −0.109640
\(892\) −1888.00 −0.0708687
\(893\) −29760.0 −1.11521
\(894\) −15714.0 −0.587869
\(895\) 41688.0 1.55696
\(896\) −11613.0 −0.432995
\(897\) 0 0
\(898\) 15930.0 0.591972
\(899\) −16320.0 −0.605453
\(900\) 1791.00 0.0663333
\(901\) −20916.0 −0.773377
\(902\) −34344.0 −1.26777
\(903\) 5628.00 0.207407
\(904\) 8442.00 0.310594
\(905\) 1908.00 0.0700818
\(906\) −2088.00 −0.0765664
\(907\) −9052.00 −0.331386 −0.165693 0.986177i \(-0.552986\pi\)
−0.165693 + 0.986177i \(0.552986\pi\)
\(908\) −4716.00 −0.172363
\(909\) 3726.00 0.135956
\(910\) −12852.0 −0.468175
\(911\) 5016.00 0.182423 0.0912116 0.995832i \(-0.470926\pi\)
0.0912116 + 0.995832i \(0.470926\pi\)
\(912\) −26412.0 −0.958979
\(913\) 7344.00 0.266211
\(914\) −47310.0 −1.71212
\(915\) 21492.0 0.776507
\(916\) −1690.00 −0.0609598
\(917\) 12348.0 0.444675
\(918\) 3402.00 0.122312
\(919\) 44552.0 1.59917 0.799584 0.600555i \(-0.205054\pi\)
0.799584 + 0.600555i \(0.205054\pi\)
\(920\) 0 0
\(921\) −1356.00 −0.0485144
\(922\) 16110.0 0.575439
\(923\) 24480.0 0.872989
\(924\) 756.000 0.0269162
\(925\) 79202.0 2.81529
\(926\) 9984.00 0.354314
\(927\) 504.000 0.0178571
\(928\) 4590.00 0.162364
\(929\) 24234.0 0.855858 0.427929 0.903812i \(-0.359243\pi\)
0.427929 + 0.903812i \(0.359243\pi\)
\(930\) 25920.0 0.913925
\(931\) −6076.00 −0.213891
\(932\) 138.000 0.00485015
\(933\) −15048.0 −0.528027
\(934\) −13644.0 −0.477993
\(935\) 27216.0 0.951934
\(936\) −6426.00 −0.224402
\(937\) −13894.0 −0.484415 −0.242208 0.970224i \(-0.577872\pi\)
−0.242208 + 0.970224i \(0.577872\pi\)
\(938\) −1932.00 −0.0672516
\(939\) −16206.0 −0.563219
\(940\) −4320.00 −0.149897
\(941\) 46758.0 1.61984 0.809919 0.586542i \(-0.199511\pi\)
0.809919 + 0.586542i \(0.199511\pi\)
\(942\) 15246.0 0.527326
\(943\) 0 0
\(944\) 9372.00 0.323128
\(945\) 3402.00 0.117108
\(946\) −28944.0 −0.994768
\(947\) 13812.0 0.473949 0.236974 0.971516i \(-0.423844\pi\)
0.236974 + 0.971516i \(0.423844\pi\)
\(948\) 3072.00 0.105247
\(949\) 17068.0 0.583826
\(950\) 74028.0 2.52820
\(951\) −30258.0 −1.03174
\(952\) 6174.00 0.210190
\(953\) −58518.0 −1.98907 −0.994535 0.104402i \(-0.966707\pi\)
−0.994535 + 0.104402i \(0.966707\pi\)
\(954\) 13446.0 0.456321
\(955\) 20304.0 0.687981
\(956\) 1896.00 0.0641433
\(957\) 11016.0 0.372097
\(958\) 24192.0 0.815875
\(959\) −16506.0 −0.555794
\(960\) 23382.0 0.786095
\(961\) −4191.00 −0.140680
\(962\) 40596.0 1.36057
\(963\) 108.000 0.00361397
\(964\) −3598.00 −0.120211
\(965\) −72612.0 −2.42224
\(966\) 0 0
\(967\) 19640.0 0.653133 0.326567 0.945174i \(-0.394108\pi\)
0.326567 + 0.945174i \(0.394108\pi\)
\(968\) −735.000 −0.0244047
\(969\) 15624.0 0.517972
\(970\) −15444.0 −0.511213
\(971\) −58308.0 −1.92708 −0.963539 0.267568i \(-0.913780\pi\)
−0.963539 + 0.267568i \(0.913780\pi\)
\(972\) −243.000 −0.00801875
\(973\) −364.000 −0.0119931
\(974\) −49848.0 −1.63987
\(975\) 20298.0 0.666724
\(976\) −28258.0 −0.926759
\(977\) −23550.0 −0.771168 −0.385584 0.922673i \(-0.626000\pi\)
−0.385584 + 0.922673i \(0.626000\pi\)
\(978\) −26388.0 −0.862776
\(979\) −12744.0 −0.416037
\(980\) −882.000 −0.0287494
\(981\) 13302.0 0.432926
\(982\) 21420.0 0.696069
\(983\) 15768.0 0.511619 0.255809 0.966727i \(-0.417658\pi\)
0.255809 + 0.966727i \(0.417658\pi\)
\(984\) 20034.0 0.649045
\(985\) 23652.0 0.765092
\(986\) −12852.0 −0.415102
\(987\) −5040.00 −0.162538
\(988\) 4216.00 0.135758
\(989\) 0 0
\(990\) −17496.0 −0.561676
\(991\) 35264.0 1.13037 0.565186 0.824964i \(-0.308805\pi\)
0.565186 + 0.824964i \(0.308805\pi\)
\(992\) −7200.00 −0.230444
\(993\) 24132.0 0.771204
\(994\) 15120.0 0.482472
\(995\) −91728.0 −2.92259
\(996\) 612.000 0.0194698
\(997\) −29338.0 −0.931940 −0.465970 0.884801i \(-0.654294\pi\)
−0.465970 + 0.884801i \(0.654294\pi\)
\(998\) 27372.0 0.868182
\(999\) −10746.0 −0.340329
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 21.4.a.a.1.1 1
3.2 odd 2 63.4.a.c.1.1 1
4.3 odd 2 336.4.a.f.1.1 1
5.2 odd 4 525.4.d.c.274.1 2
5.3 odd 4 525.4.d.c.274.2 2
5.4 even 2 525.4.a.g.1.1 1
7.2 even 3 147.4.e.i.67.1 2
7.3 odd 6 147.4.e.g.79.1 2
7.4 even 3 147.4.e.i.79.1 2
7.5 odd 6 147.4.e.g.67.1 2
7.6 odd 2 147.4.a.c.1.1 1
8.3 odd 2 1344.4.a.n.1.1 1
8.5 even 2 1344.4.a.ba.1.1 1
12.11 even 2 1008.4.a.v.1.1 1
15.14 odd 2 1575.4.a.b.1.1 1
21.2 odd 6 441.4.e.b.361.1 2
21.5 even 6 441.4.e.d.361.1 2
21.11 odd 6 441.4.e.b.226.1 2
21.17 even 6 441.4.e.d.226.1 2
21.20 even 2 441.4.a.j.1.1 1
28.27 even 2 2352.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 1.1 even 1 trivial
63.4.a.c.1.1 1 3.2 odd 2
147.4.a.c.1.1 1 7.6 odd 2
147.4.e.g.67.1 2 7.5 odd 6
147.4.e.g.79.1 2 7.3 odd 6
147.4.e.i.67.1 2 7.2 even 3
147.4.e.i.79.1 2 7.4 even 3
336.4.a.f.1.1 1 4.3 odd 2
441.4.a.j.1.1 1 21.20 even 2
441.4.e.b.226.1 2 21.11 odd 6
441.4.e.b.361.1 2 21.2 odd 6
441.4.e.d.226.1 2 21.17 even 6
441.4.e.d.361.1 2 21.5 even 6
525.4.a.g.1.1 1 5.4 even 2
525.4.d.c.274.1 2 5.2 odd 4
525.4.d.c.274.2 2 5.3 odd 4
1008.4.a.v.1.1 1 12.11 even 2
1344.4.a.n.1.1 1 8.3 odd 2
1344.4.a.ba.1.1 1 8.5 even 2
1575.4.a.b.1.1 1 15.14 odd 2
2352.4.a.r.1.1 1 28.27 even 2