Properties

Label 525.4.a.g.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 525.1

$q$-expansion

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

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 \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 3.00000 0.577350
\(4\) 1.00000 0.125000
\(5\) 0 0
\(6\) 9.00000 0.612372
\(7\) −7.00000 −0.377964
\(8\) −21.0000 −0.928078
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −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 \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) −21.0000 −0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −42.0000 −0.599206 −0.299603 0.954064i \(-0.596854\pi\)
−0.299603 + 0.954064i \(0.596854\pi\)
\(18\) 27.0000 0.353553
\(19\) −124.000 −1.49724 −0.748620 0.663000i \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) 0 0
\(21\) −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
\(25\) 0 0
\(26\) 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 \(0.394108\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) −398.000 −1.76840 −0.884200 0.467109i \(-0.845296\pi\)
−0.884200 + 0.467109i \(0.845296\pi\)
\(38\) −372.000 −1.58806
\(39\) 102.000 0.418797
\(40\) 0 0
\(41\) −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 \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(44\) −36.0000 −0.123346
\(45\) 0 0
\(46\) 0 0
\(47\) −240.000 −0.744843 −0.372421 0.928064i \(-0.621472\pi\)
−0.372421 + 0.928064i \(0.621472\pi\)
\(48\) −213.000 −0.640498
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −126.000 −0.345952
\(52\) 34.0000 0.0906721
\(53\) 498.000 1.29067 0.645335 0.763899i \(-0.276718\pi\)
0.645335 + 0.763899i \(0.276718\pi\)
\(54\) 81.0000 0.204124
\(55\) 0 0
\(56\) 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 \(0.546523\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −324.000 −0.604267
\(67\) −92.0000 −0.167755 −0.0838775 0.996476i \(-0.526730\pi\)
−0.0838775 + 0.996476i \(0.526730\pi\)
\(68\) −42.0000 −0.0749007
\(69\) 0 0
\(70\) 0 0
\(71\) −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 \(-0.368166\pi\)
0.402429 + 0.915451i \(0.368166\pi\)
\(74\) −1194.00 −1.87567
\(75\) 0 0
\(76\) −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 \(0.760094\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −954.000 −1.28478
\(83\) 204.000 0.269782 0.134891 0.990860i \(-0.456932\pi\)
0.134891 + 0.990860i \(0.456932\pi\)
\(84\) −21.0000 −0.0272772
\(85\) 0 0
\(86\) 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 \(0.432390\pi\)
\(90\) 0 0
\(91\) −238.000 −0.274167
\(92\) 0 0
\(93\) −480.000 −0.535201
\(94\) −720.000 −0.790025
\(95\) 0 0
\(96\) −135.000 −0.143525
\(97\) 286.000 0.299370 0.149685 0.988734i \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 147.000 0.151523
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 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 \(-0.508527\pi\)
−0.0267857 + 0.999641i \(0.508527\pi\)
\(104\) −714.000 −0.673206
\(105\) 0 0
\(106\) 1494.00 1.36896
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 27.0000 0.0240563
\(109\) 1478.00 1.29878 0.649389 0.760457i \(-0.275025\pi\)
0.649389 + 0.760457i \(0.275025\pi\)
\(110\) 0 0
\(111\) −1194.00 −1.02099
\(112\) 497.000 0.419304
\(113\) −402.000 −0.334664 −0.167332 0.985901i \(-0.553515\pi\)
−0.167332 + 0.985901i \(0.553515\pi\)
\(114\) −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
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 1194.00 0.886063
\(123\) −954.000 −0.699344
\(124\) −160.000 −0.115874
\(125\) 0 0
\(126\) −189.000 −0.133631
\(127\) −1280.00 −0.894344 −0.447172 0.894448i \(-0.647569\pi\)
−0.447172 + 0.894448i \(0.647569\pi\)
\(128\) 1659.00 1.14560
\(129\) 804.000 0.548746
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 882.000 0.556109
\(137\) 2358.00 1.47049 0.735246 0.677800i \(-0.237066\pi\)
0.735246 + 0.677800i \(0.237066\pi\)
\(138\) 0 0
\(139\) −52.0000 −0.0317308 −0.0158654 0.999874i \(-0.505050\pi\)
−0.0158654 + 0.999874i \(0.505050\pi\)
\(140\) 0 0
\(141\) −720.000 −0.430035
\(142\) −2160.00 −1.27650
\(143\) −1224.00 −0.715776
\(144\) −639.000 −0.369792
\(145\) 0 0
\(146\) 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 \(0.659361\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) 102.000 0.0523496
\(157\) −1694.00 −0.861120 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(158\) −3072.00 −1.54681
\(159\) 1494.00 0.745169
\(160\) 0 0
\(161\) 0 0
\(162\) 243.000 0.117851
\(163\) 2932.00 1.40891 0.704454 0.709750i \(-0.251192\pi\)
0.704454 + 0.709750i \(0.251192\pi\)
\(164\) −318.000 −0.151412
\(165\) 0 0
\(166\) 612.000 0.286147
\(167\) −1176.00 −0.544920 −0.272460 0.962167i \(-0.587837\pi\)
−0.272460 + 0.962167i \(0.587837\pi\)
\(168\) 441.000 0.202523
\(169\) −1041.00 −0.473828
\(170\) 0 0
\(171\) −1116.00 −0.499080
\(172\) 268.000 0.118807
\(173\) −870.000 −0.382340 −0.191170 0.981557i \(-0.561228\pi\)
−0.191170 + 0.981557i \(0.561228\pi\)
\(174\) 918.000 0.399962
\(175\) 0 0
\(176\) 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 \(0.660648\pi\)
\(180\) 0 0
\(181\) −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
\(185\) 0 0
\(186\) −1440.00 −0.567666
\(187\) 1512.00 0.591275
\(188\) −240.000 −0.0931053
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −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 \(-0.771038\pi\)
−0.752263 + 0.658862i \(0.771038\pi\)
\(194\) 858.000 0.317530
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) 1314.00 0.475221 0.237611 0.971360i \(-0.423636\pi\)
0.237611 + 0.971360i \(0.423636\pi\)
\(198\) −972.000 −0.348874
\(199\) 5096.00 1.81531 0.907653 0.419722i \(-0.137872\pi\)
0.907653 + 0.419722i \(0.137872\pi\)
\(200\) 0 0
\(201\) −276.000 −0.0968534
\(202\) 1242.00 0.432608
\(203\) −714.000 −0.246862
\(204\) −126.000 −0.0432439
\(205\) 0 0
\(206\) −168.000 −0.0568209
\(207\) 0 0
\(208\) −2414.00 −0.804715
\(209\) 4464.00 1.47742
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) −567.000 −0.178609
\(217\) 1120.00 0.350371
\(218\) 4434.00 1.37756
\(219\) 1506.00 0.464685
\(220\) 0 0
\(221\) −1428.00 −0.434650
\(222\) −3582.00 −1.08292
\(223\) 1888.00 0.566950 0.283475 0.958980i \(-0.408513\pi\)
0.283475 + 0.958980i \(0.408513\pi\)
\(224\) 315.000 0.0939590
\(225\) 0 0
\(226\) −1206.00 −0.354964
\(227\) 4716.00 1.37891 0.689454 0.724330i \(-0.257851\pi\)
0.689454 + 0.724330i \(0.257851\pi\)
\(228\) −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 \(-0.506176\pi\)
−0.0194006 + 0.999812i \(0.506176\pi\)
\(234\) 918.000 0.256460
\(235\) 0 0
\(236\) −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 \(0.417406\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) −2862.00 −0.741766
\(247\) −4216.00 −1.08606
\(248\) 3360.00 0.860323
\(249\) 612.000 0.155759
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) 6822.00 1.65582 0.827908 0.560864i \(-0.189531\pi\)
0.827908 + 0.560864i \(0.189531\pi\)
\(258\) 2412.00 0.582033
\(259\) 2786.00 0.668392
\(260\) 0 0
\(261\) 918.000 0.217712
\(262\) 5292.00 1.24787
\(263\) −2592.00 −0.607717 −0.303858 0.952717i \(-0.598275\pi\)
−0.303858 + 0.952717i \(0.598275\pi\)
\(264\) 2268.00 0.528734
\(265\) 0 0
\(266\) 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 \(0.119037\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) 0 0
\(277\) 6514.00 1.41295 0.706477 0.707736i \(-0.250283\pi\)
0.706477 + 0.707736i \(0.250283\pi\)
\(278\) −156.000 −0.0336556
\(279\) −1440.00 −0.308998
\(280\) 0 0
\(281\) 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 \(-0.611233\pi\)
−0.342380 + 0.939562i \(0.611233\pi\)
\(284\) −720.000 −0.150437
\(285\) 0 0
\(286\) −3672.00 −0.759195
\(287\) 2226.00 0.457828
\(288\) −405.000 −0.0828641
\(289\) −3149.00 −0.640953
\(290\) 0 0
\(291\) 858.000 0.172841
\(292\) 502.000 0.100607
\(293\) −5118.00 −1.02047 −0.510233 0.860036i \(-0.670441\pi\)
−0.510233 + 0.860036i \(0.670441\pi\)
\(294\) 441.000 0.0874818
\(295\) 0 0
\(296\) 8358.00 1.64121
\(297\) −972.000 −0.189903
\(298\) −5238.00 −1.01822
\(299\) 0 0
\(300\) 0 0
\(301\) −1876.00 −0.359239
\(302\) −696.000 −0.132617
\(303\) 1242.00 0.235482
\(304\) 8804.00 1.66100
\(305\) 0 0
\(306\) −1134.00 −0.211851
\(307\) −452.000 −0.0840293 −0.0420147 0.999117i \(-0.513378\pi\)
−0.0420147 + 0.999117i \(0.513378\pi\)
\(308\) 252.000 0.0466202
\(309\) −168.000 −0.0309294
\(310\) 0 0
\(311\) 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 \(-0.662187\pi\)
−0.487762 + 0.872977i \(0.662187\pi\)
\(314\) −5082.00 −0.913356
\(315\) 0 0
\(316\) −1024.00 −0.182293
\(317\) −10086.0 −1.78702 −0.893511 0.449041i \(-0.851766\pi\)
−0.893511 + 0.449041i \(0.851766\pi\)
\(318\) 4482.00 0.790371
\(319\) −3672.00 −0.644491
\(320\) 0 0
\(321\) −36.0000 −0.00625958
\(322\) 0 0
\(323\) 5208.00 0.897154
\(324\) 81.0000 0.0138889
\(325\) 0 0
\(326\) 8796.00 1.49437
\(327\) 4434.00 0.749849
\(328\) 6678.00 1.12418
\(329\) 1680.00 0.281524
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 1491.00 0.242085
\(337\) −4178.00 −0.675342 −0.337671 0.941264i \(-0.609639\pi\)
−0.337671 + 0.941264i \(0.609639\pi\)
\(338\) −3123.00 −0.502570
\(339\) −1206.00 −0.193218
\(340\) 0 0
\(341\) 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 \(-0.503841\pi\)
−0.0120670 + 0.999927i \(0.503841\pi\)
\(348\) 306.000 0.0471360
\(349\) −12418.0 −1.90464 −0.952321 0.305097i \(-0.901311\pi\)
−0.952321 + 0.305097i \(0.901311\pi\)
\(350\) 0 0
\(351\) 918.000 0.139599
\(352\) 1620.00 0.245302
\(353\) 7830.00 1.18059 0.590296 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(354\) −1188.00 −0.178366
\(355\) 0 0
\(356\) 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 \(0.739978\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) −318.000 −0.0461705
\(363\) −105.000 −0.0151820
\(364\) −238.000 −0.0342709
\(365\) 0 0
\(366\) 3582.00 0.511569
\(367\) 3760.00 0.534797 0.267398 0.963586i \(-0.413836\pi\)
0.267398 + 0.963586i \(0.413836\pi\)
\(368\) 0 0
\(369\) −2862.00 −0.403766
\(370\) 0 0
\(371\) −3486.00 −0.487828
\(372\) −480.000 −0.0669001
\(373\) −5870.00 −0.814845 −0.407422 0.913240i \(-0.633572\pi\)
−0.407422 + 0.913240i \(0.633572\pi\)
\(374\) 4536.00 0.627142
\(375\) 0 0
\(376\) 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 \(0.540054\pi\)
\(380\) 0 0
\(381\) −3840.00 −0.516350
\(382\) −3384.00 −0.453247
\(383\) −2160.00 −0.288175 −0.144087 0.989565i \(-0.546025\pi\)
−0.144087 + 0.989565i \(0.546025\pi\)
\(384\) 4977.00 0.661410
\(385\) 0 0
\(386\) −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 \(0.645817\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1029.00 −0.132583
\(393\) 5292.00 0.679252
\(394\) 3942.00 0.504048
\(395\) 0 0
\(396\) −324.000 −0.0411152
\(397\) 6514.00 0.823497 0.411748 0.911298i \(-0.364918\pi\)
0.411748 + 0.911298i \(0.364918\pi\)
\(398\) 15288.0 1.92542
\(399\) 2604.00 0.326724
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.605802\pi\)
\(410\) 0 0
\(411\) 7074.00 0.848990
\(412\) −56.0000 −0.00669641
\(413\) 924.000 0.110090
\(414\) 0 0
\(415\) 0 0
\(416\) −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 \(0.223613\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −6480.00 −0.736988
\(427\) −2786.00 −0.315747
\(428\) −12.0000 −0.00135524
\(429\) −3672.00 −0.413254
\(430\) 0 0
\(431\) 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 \(-0.577095\pi\)
−0.239841 + 0.970812i \(0.577095\pi\)
\(434\) 3360.00 0.371625
\(435\) 0 0
\(436\) 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 \(0.663041\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) −4284.00 −0.461016
\(443\) 5268.00 0.564989 0.282495 0.959269i \(-0.408838\pi\)
0.282495 + 0.959269i \(0.408838\pi\)
\(444\) −1194.00 −0.127623
\(445\) 0 0
\(446\) 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 \(0.590022\pi\)
\(450\) 0 0
\(451\) 11448.0 1.19527
\(452\) −402.000 −0.0418329
\(453\) −696.000 −0.0721875
\(454\) 14148.0 1.46255
\(455\) 0 0
\(456\) 7812.00 0.802260
\(457\) −15770.0 −1.61420 −0.807100 0.590415i \(-0.798964\pi\)
−0.807100 + 0.590415i \(0.798964\pi\)
\(458\) −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 \(-0.446584\pi\)
0.167025 + 0.985953i \(0.446584\pi\)
\(464\) −7242.00 −0.724572
\(465\) 0 0
\(466\) −414.000 −0.0411549
\(467\) −4548.00 −0.450656 −0.225328 0.974283i \(-0.572345\pi\)
−0.225328 + 0.974283i \(0.572345\pi\)
\(468\) 306.000 0.0302240
\(469\) 644.000 0.0634055
\(470\) 0 0
\(471\) −5082.00 −0.497168
\(472\) 2772.00 0.270321
\(473\) −9648.00 −0.937876
\(474\) −9216.00 −0.893048
\(475\) 0 0
\(476\) 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 \(0.625663\pi\)
\(480\) 0 0
\(481\) −13532.0 −1.28276
\(482\) −10794.0 −1.02003
\(483\) 0 0
\(484\) −35.0000 −0.00328700
\(485\) 0 0
\(486\) 729.000 0.0680414
\(487\) −16616.0 −1.54608 −0.773042 0.634355i \(-0.781266\pi\)
−0.773042 + 0.634355i \(0.781266\pi\)
\(488\) −8358.00 −0.775305
\(489\) 8796.00 0.813433
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.634215\pi\)
\(500\) 0 0
\(501\) −3528.00 −0.314610
\(502\) −9180.00 −0.816182
\(503\) 6552.00 0.580794 0.290397 0.956906i \(-0.406213\pi\)
0.290397 + 0.956906i \(0.406213\pi\)
\(504\) 1323.00 0.116927
\(505\) 0 0
\(506\) 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 \(0.461237\pi\)
\(510\) 0 0
\(511\) −3514.00 −0.304208
\(512\) −8733.00 −0.753804
\(513\) −3348.00 −0.288144
\(514\) 20466.0 1.75626
\(515\) 0 0
\(516\) 804.000 0.0685933
\(517\) 8640.00 0.734984
\(518\) 8358.00 0.708937
\(519\) −2610.00 −0.220744
\(520\) 0 0
\(521\) −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 \(-0.764351\pi\)
−0.738258 + 0.674518i \(0.764351\pi\)
\(524\) 1764.00 0.147062
\(525\) 0 0
\(526\) −7776.00 −0.644581
\(527\) 6720.00 0.555461
\(528\) 7668.00 0.632021
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −1188.00 −0.0970900
\(532\) 868.000 0.0707379
\(533\) −10812.0 −0.878649
\(534\) 3186.00 0.258187
\(535\) 0 0
\(536\) 1932.00 0.155690
\(537\) −6948.00 −0.558340
\(538\) 24642.0 1.97471
\(539\) −1764.00 −0.140966
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) −2142.00 −0.167892
\(547\) −20972.0 −1.63930 −0.819651 0.572863i \(-0.805833\pi\)
−0.819651 + 0.572863i \(0.805833\pi\)
\(548\) 2358.00 0.183812
\(549\) 3582.00 0.278463
\(550\) 0 0
\(551\) −12648.0 −0.977900
\(552\) 0 0
\(553\) 7168.00 0.551201
\(554\) 19542.0 1.49866
\(555\) 0 0
\(556\) −52.0000 −0.00396635
\(557\) −21174.0 −1.61072 −0.805360 0.592786i \(-0.798028\pi\)
−0.805360 + 0.592786i \(0.798028\pi\)
\(558\) −4320.00 −0.327742
\(559\) 9112.00 0.689439
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) 19854.0 1.49020
\(563\) 17772.0 1.33037 0.665187 0.746677i \(-0.268352\pi\)
0.665187 + 0.746677i \(0.268352\pi\)
\(564\) −720.000 −0.0537544
\(565\) 0 0
\(566\) −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 \(0.401705\pi\)
\(570\) 0 0
\(571\) 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 \(-0.500023\pi\)
−7.21500e−5 1.00000i \(0.500023\pi\)
\(578\) −9447.00 −0.679833
\(579\) −12102.0 −0.868639
\(580\) 0 0
\(581\) −1428.00 −0.101968
\(582\) 2574.00 0.183326
\(583\) −17928.0 −1.27359
\(584\) −10542.0 −0.746971
\(585\) 0 0
\(586\) −15354.0 −1.08237
\(587\) −26364.0 −1.85376 −0.926881 0.375354i \(-0.877521\pi\)
−0.926881 + 0.375354i \(0.877521\pi\)
\(588\) 147.000 0.0103098
\(589\) 19840.0 1.38793
\(590\) 0 0
\(591\) 3942.00 0.274369
\(592\) 28258.0 1.96182
\(593\) −2298.00 −0.159136 −0.0795679 0.996829i \(-0.525354\pi\)
−0.0795679 + 0.996829i \(0.525354\pi\)
\(594\) −2916.00 −0.201422
\(595\) 0 0
\(596\) −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 \(0.466588\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 3726.00 0.249766
\(607\) −16832.0 −1.12552 −0.562759 0.826621i \(-0.690260\pi\)
−0.562759 + 0.826621i \(0.690260\pi\)
\(608\) 5580.00 0.372202
\(609\) −2142.00 −0.142526
\(610\) 0 0
\(611\) −8160.00 −0.540292
\(612\) −378.000 −0.0249669
\(613\) 2482.00 0.163535 0.0817676 0.996651i \(-0.473943\pi\)
0.0817676 + 0.996651i \(0.473943\pi\)
\(614\) −1356.00 −0.0891266
\(615\) 0 0
\(616\) −5292.00 −0.346138
\(617\) 15798.0 1.03080 0.515400 0.856950i \(-0.327643\pi\)
0.515400 + 0.856950i \(0.327643\pi\)
\(618\) −504.000 −0.0328056
\(619\) −15460.0 −1.00386 −0.501930 0.864908i \(-0.667377\pi\)
−0.501930 + 0.864908i \(0.667377\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15048.0 0.970048
\(623\) −2478.00 −0.159356
\(624\) −7242.00 −0.464603
\(625\) 0 0
\(626\) −16206.0 −1.03470
\(627\) 13392.0 0.852990
\(628\) −1694.00 −0.107640
\(629\) 16716.0 1.05964
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 1494.00 0.0931462
\(637\) 1666.00 0.103625
\(638\) −11016.0 −0.683586
\(639\) −6480.00 −0.401166
\(640\) 0 0
\(641\) −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 \(-0.377734\pi\)
0.374735 + 0.927132i \(0.377734\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 15624.0 0.951576
\(647\) −13560.0 −0.823955 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(648\) −1701.00 −0.103120
\(649\) 4752.00 0.287415
\(650\) 0 0
\(651\) 3360.00 0.202287
\(652\) 2932.00 0.176113
\(653\) −23094.0 −1.38398 −0.691989 0.721908i \(-0.743265\pi\)
−0.691989 + 0.721908i \(0.743265\pi\)
\(654\) 13302.0 0.795335
\(655\) 0 0
\(656\) 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 \(0.267825\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −10746.0 −0.625224
\(667\) 0 0
\(668\) −1176.00 −0.0681150
\(669\) 5664.00 0.327329
\(670\) 0 0
\(671\) −14328.0 −0.824331
\(672\) 945.000 0.0542473
\(673\) 22462.0 1.28655 0.643274 0.765636i \(-0.277576\pi\)
0.643274 + 0.765636i \(0.277576\pi\)
\(674\) −12534.0 −0.716308
\(675\) 0 0
\(676\) −1041.00 −0.0592285
\(677\) 25554.0 1.45069 0.725347 0.688383i \(-0.241679\pi\)
0.725347 + 0.688383i \(0.241679\pi\)
\(678\) −3618.00 −0.204939
\(679\) −2002.00 −0.113151
\(680\) 0 0
\(681\) 14148.0 0.796112
\(682\) 17280.0 0.970213
\(683\) −9276.00 −0.519672 −0.259836 0.965653i \(-0.583669\pi\)
−0.259836 + 0.965653i \(0.583669\pi\)
\(684\) −1116.00 −0.0623850
\(685\) 0 0
\(686\) −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
\(695\) 0 0
\(696\) −6426.00 −0.349967
\(697\) 13356.0 0.725817
\(698\) −37254.0 −2.02018
\(699\) −414.000 −0.0224019
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.552729\pi\)
\(710\) 0 0
\(711\) −9216.00 −0.486114
\(712\) −7434.00 −0.391293
\(713\) 0 0
\(714\) 2646.00 0.138689
\(715\) 0 0
\(716\) −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 \(0.627831\pi\)
\(720\) 0 0
\(721\) 392.000 0.0202480
\(722\) 25551.0 1.31705
\(723\) −10794.0 −0.555233
\(724\) −106.000 −0.00544124
\(725\) 0 0
\(726\) −315.000 −0.0161030
\(727\) 32920.0 1.67942 0.839708 0.543038i \(-0.182726\pi\)
0.839708 + 0.543038i \(0.182726\pi\)
\(728\) 4998.00 0.254448
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −11256.0 −0.569519
\(732\) 1194.00 0.0602889
\(733\) 6946.00 0.350009 0.175004 0.984568i \(-0.444006\pi\)
0.175004 + 0.984568i \(0.444006\pi\)
\(734\) 11280.0 0.567238
\(735\) 0 0
\(736\) 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 \(0.518676\pi\)
\(740\) 0 0
\(741\) −12648.0 −0.627039
\(742\) −10458.0 −0.517419
\(743\) 23520.0 1.16133 0.580663 0.814144i \(-0.302793\pi\)
0.580663 + 0.814144i \(0.302793\pi\)
\(744\) 10080.0 0.496708
\(745\) 0 0
\(746\) −17610.0 −0.864273
\(747\) 1836.00 0.0899273
\(748\) 1512.00 0.0739094
\(749\) 84.0000 0.00409785
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) −189.000 −0.00909241
\(757\) 20770.0 0.997224 0.498612 0.866825i \(-0.333843\pi\)
0.498612 + 0.866825i \(0.333843\pi\)
\(758\) −5556.00 −0.266231
\(759\) 0 0
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.436150\pi\)
\(770\) 0 0
\(771\) 20466.0 0.955986
\(772\) −4034.00 −0.188066
\(773\) 32322.0 1.50393 0.751967 0.659200i \(-0.229105\pi\)
0.751967 + 0.659200i \(0.229105\pi\)
\(774\) 7236.00 0.336037
\(775\) 0 0
\(776\) −6006.00 −0.277839
\(777\) 8358.00 0.385896
\(778\) −20358.0 −0.938136
\(779\) 39432.0 1.81360
\(780\) 0 0
\(781\) 25920.0 1.18757
\(782\) 0 0
\(783\) 2754.00 0.125696
\(784\) −3479.00 −0.158482
\(785\) 0 0
\(786\) 15876.0 0.720456
\(787\) −26228.0 −1.18796 −0.593982 0.804479i \(-0.702445\pi\)
−0.593982 + 0.804479i \(0.702445\pi\)
\(788\) 1314.00 0.0594027
\(789\) −7776.00 −0.350866
\(790\) 0 0
\(791\) 2814.00 0.126491
\(792\) 6804.00 0.305265
\(793\) 13532.0 0.605972
\(794\) 19542.0 0.873450
\(795\) 0 0
\(796\) 5096.00 0.226913
\(797\) 43338.0 1.92611 0.963056 0.269302i \(-0.0867931\pi\)
0.963056 + 0.269302i \(0.0867931\pi\)
\(798\) 7812.00 0.346544
\(799\) 10080.0 0.446314
\(800\) 0 0
\(801\) 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 \(0.716135\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 8946.00 0.383790
\(817\) −33232.0 −1.42306
\(818\) −16194.0 −0.692188
\(819\) −2142.00 −0.0913889
\(820\) 0 0
\(821\) −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 \(-0.459499\pi\)
0.126894 + 0.991916i \(0.459499\pi\)
\(824\) 1176.00 0.0497183
\(825\) 0 0
\(826\) 2772.00 0.116768
\(827\) −25884.0 −1.08836 −0.544181 0.838968i \(-0.683159\pi\)
−0.544181 + 0.838968i \(0.683159\pi\)
\(828\) 0 0
\(829\) −1474.00 −0.0617541 −0.0308770 0.999523i \(-0.509830\pi\)
−0.0308770 + 0.999523i \(0.509830\pi\)
\(830\) 0 0
\(831\) 19542.0 0.815770
\(832\) 14722.0 0.613454
\(833\) −2058.00 −0.0856008
\(834\) −468.000 −0.0194311
\(835\) 0 0
\(836\) 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 \(0.257690\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) −966.000 −0.0395375
\(843\) 19854.0 0.811160
\(844\) −3076.00 −0.125451
\(845\) 0 0
\(846\) −6480.00 −0.263342
\(847\) 245.000 0.00993896
\(848\) −35358.0 −1.43184
\(849\) −9780.00 −0.395346
\(850\) 0 0
\(851\) 0 0
\(852\) −2160.00 −0.0868549
\(853\) −1190.00 −0.0477665 −0.0238832 0.999715i \(-0.507603\pi\)
−0.0238832 + 0.999715i \(0.507603\pi\)
\(854\) −8358.00 −0.334900
\(855\) 0 0
\(856\) 252.000 0.0100621
\(857\) −34578.0 −1.37825 −0.689126 0.724642i \(-0.742005\pi\)
−0.689126 + 0.724642i \(0.742005\pi\)
\(858\) −11016.0 −0.438322
\(859\) −44404.0 −1.76373 −0.881865 0.471501i \(-0.843712\pi\)
−0.881865 + 0.471501i \(0.843712\pi\)
\(860\) 0 0
\(861\) 6678.00 0.264327
\(862\) 7848.00 0.310097
\(863\) 38328.0 1.51182 0.755910 0.654676i \(-0.227195\pi\)
0.755910 + 0.654676i \(0.227195\pi\)
\(864\) −1215.00 −0.0478416
\(865\) 0 0
\(866\) −12966.0 −0.508779
\(867\) −9447.00 −0.370054
\(868\) 1120.00 0.0437964
\(869\) 36864.0 1.43904
\(870\) 0 0
\(871\) −3128.00 −0.121686
\(872\) −31038.0 −1.20537
\(873\) 2574.00 0.0997900
\(874\) 0 0
\(875\) 0 0
\(876\) 1506.00 0.0580856
\(877\) 38842.0 1.49555 0.747777 0.663950i \(-0.231121\pi\)
0.747777 + 0.663950i \(0.231121\pi\)
\(878\) −27048.0 −1.03966
\(879\) −15354.0 −0.589167
\(880\) 0 0
\(881\) −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 \(-0.587245\pi\)
−0.270670 + 0.962672i \(0.587245\pi\)
\(884\) −1428.00 −0.0543313
\(885\) 0 0
\(886\) 15804.0 0.599262
\(887\) 26136.0 0.989359 0.494679 0.869076i \(-0.335286\pi\)
0.494679 + 0.869076i \(0.335286\pi\)
\(888\) 25074.0 0.947554
\(889\) 8960.00 0.338030
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 1888.00 0.0708687
\(893\) 29760.0 1.11521
\(894\) −15714.0 −0.587869
\(895\) 0 0
\(896\) −11613.0 −0.432995
\(897\) 0 0
\(898\) −15930.0 −0.591972
\(899\) −16320.0 −0.605453
\(900\) 0 0
\(901\) −20916.0 −0.773377
\(902\) 34344.0 1.26777
\(903\) −5628.00 −0.207407
\(904\) 8442.00 0.310594
\(905\) 0 0
\(906\) −2088.00 −0.0765664
\(907\) 9052.00 0.331386 0.165693 0.986177i \(-0.447014\pi\)
0.165693 + 0.986177i \(0.447014\pi\)
\(908\) 4716.00 0.172363
\(909\) 3726.00 0.135956
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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
\(925\) 0 0
\(926\) 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 \(0.359243\pi\)
\(930\) 0 0
\(931\) −6076.00 −0.213891
\(932\) −138.000 −0.00485015
\(933\) 15048.0 0.528027
\(934\) −13644.0 −0.477993
\(935\) 0 0
\(936\) −6426.00 −0.224402
\(937\) 13894.0 0.484415 0.242208 0.970224i \(-0.422128\pi\)
0.242208 + 0.970224i \(0.422128\pi\)
\(938\) 1932.00 0.0672516
\(939\) −16206.0 −0.563219
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) −28944.0 −0.994768
\(947\) −13812.0 −0.473949 −0.236974 0.971516i \(-0.576156\pi\)
−0.236974 + 0.971516i \(0.576156\pi\)
\(948\) −3072.00 −0.105247
\(949\) 17068.0 0.583826
\(950\) 0 0
\(951\) −30258.0 −1.03174
\(952\) −6174.00 −0.210190
\(953\) 58518.0 1.98907 0.994535 0.104402i \(-0.0332930\pi\)
0.994535 + 0.104402i \(0.0332930\pi\)
\(954\) 13446.0 0.456321
\(955\) 0 0
\(956\) 1896.00 0.0641433
\(957\) −11016.0 −0.372097
\(958\) −24192.0 −0.815875
\(959\) −16506.0 −0.555794
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) −40596.0 −1.36057
\(963\) −108.000 −0.00361397
\(964\) −3598.00 −0.120211
\(965\) 0 0
\(966\) 0 0
\(967\) −19640.0 −0.653133 −0.326567 0.945174i \(-0.605892\pi\)
−0.326567 + 0.945174i \(0.605892\pi\)
\(968\) 735.000 0.0244047
\(969\) 15624.0 0.517972
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −28258.0 −0.926759
\(977\) 23550.0 0.771168 0.385584 0.922673i \(-0.374000\pi\)
0.385584 + 0.922673i \(0.374000\pi\)
\(978\) 26388.0 0.862776
\(979\) −12744.0 −0.416037
\(980\) 0 0
\(981\) 13302.0 0.432926
\(982\) −21420.0 −0.696069
\(983\) −15768.0 −0.511619 −0.255809 0.966727i \(-0.582342\pi\)
−0.255809 + 0.966727i \(0.582342\pi\)
\(984\) 20034.0 0.649045
\(985\) 0 0
\(986\) −12852.0 −0.415102
\(987\) 5040.00 0.162538
\(988\) −4216.00 −0.135758
\(989\) 0 0
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) 612.000 0.0194698
\(997\) 29338.0 0.931940 0.465970 0.884801i \(-0.345706\pi\)
0.465970 + 0.884801i \(0.345706\pi\)
\(998\) −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 525.4.a.g.1.1 1
3.2 odd 2 1575.4.a.b.1.1 1
5.2 odd 4 525.4.d.c.274.2 2
5.3 odd 4 525.4.d.c.274.1 2
5.4 even 2 21.4.a.a.1.1 1
15.14 odd 2 63.4.a.c.1.1 1
20.19 odd 2 336.4.a.f.1.1 1
35.4 even 6 147.4.e.i.79.1 2
35.9 even 6 147.4.e.i.67.1 2
35.19 odd 6 147.4.e.g.67.1 2
35.24 odd 6 147.4.e.g.79.1 2
35.34 odd 2 147.4.a.c.1.1 1
40.19 odd 2 1344.4.a.n.1.1 1
40.29 even 2 1344.4.a.ba.1.1 1
60.59 even 2 1008.4.a.v.1.1 1
105.44 odd 6 441.4.e.b.361.1 2
105.59 even 6 441.4.e.d.226.1 2
105.74 odd 6 441.4.e.b.226.1 2
105.89 even 6 441.4.e.d.361.1 2
105.104 even 2 441.4.a.j.1.1 1
140.139 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 5.4 even 2
63.4.a.c.1.1 1 15.14 odd 2
147.4.a.c.1.1 1 35.34 odd 2
147.4.e.g.67.1 2 35.19 odd 6
147.4.e.g.79.1 2 35.24 odd 6
147.4.e.i.67.1 2 35.9 even 6
147.4.e.i.79.1 2 35.4 even 6
336.4.a.f.1.1 1 20.19 odd 2
441.4.a.j.1.1 1 105.104 even 2
441.4.e.b.226.1 2 105.74 odd 6
441.4.e.b.361.1 2 105.44 odd 6
441.4.e.d.226.1 2 105.59 even 6
441.4.e.d.361.1 2 105.89 even 6
525.4.a.g.1.1 1 1.1 even 1 trivial
525.4.d.c.274.1 2 5.3 odd 4
525.4.d.c.274.2 2 5.2 odd 4
1008.4.a.v.1.1 1 60.59 even 2
1344.4.a.n.1.1 1 40.19 odd 2
1344.4.a.ba.1.1 1 40.29 even 2
1575.4.a.b.1.1 1 3.2 odd 2
2352.4.a.r.1.1 1 140.139 even 2