Properties

Label 270.4.a.e.1.1
Level $270$
Weight $4$
Character 270.1
Self dual yes
Analytic conductor $15.931$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9305157015\)
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 \(\chi\) \(=\) 270.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -4.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -4.00000 q^{7} -8.00000 q^{8} -10.0000 q^{10} -42.0000 q^{11} +20.0000 q^{13} +8.00000 q^{14} +16.0000 q^{16} -93.0000 q^{17} +59.0000 q^{19} +20.0000 q^{20} +84.0000 q^{22} -9.00000 q^{23} +25.0000 q^{25} -40.0000 q^{26} -16.0000 q^{28} -120.000 q^{29} +47.0000 q^{31} -32.0000 q^{32} +186.000 q^{34} -20.0000 q^{35} -262.000 q^{37} -118.000 q^{38} -40.0000 q^{40} -126.000 q^{41} -178.000 q^{43} -168.000 q^{44} +18.0000 q^{46} -144.000 q^{47} -327.000 q^{49} -50.0000 q^{50} +80.0000 q^{52} -741.000 q^{53} -210.000 q^{55} +32.0000 q^{56} +240.000 q^{58} +444.000 q^{59} +221.000 q^{61} -94.0000 q^{62} +64.0000 q^{64} +100.000 q^{65} -538.000 q^{67} -372.000 q^{68} +40.0000 q^{70} -690.000 q^{71} -1126.00 q^{73} +524.000 q^{74} +236.000 q^{76} +168.000 q^{77} +665.000 q^{79} +80.0000 q^{80} +252.000 q^{82} -75.0000 q^{83} -465.000 q^{85} +356.000 q^{86} +336.000 q^{88} +1086.00 q^{89} -80.0000 q^{91} -36.0000 q^{92} +288.000 q^{94} +295.000 q^{95} +1544.00 q^{97} +654.000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −0.215980 −0.107990 0.994152i \(-0.534441\pi\)
−0.107990 + 0.994152i \(0.534441\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −10.0000 −0.316228
\(11\) −42.0000 −1.15123 −0.575613 0.817723i \(-0.695236\pi\)
−0.575613 + 0.817723i \(0.695236\pi\)
\(12\) 0 0
\(13\) 20.0000 0.426692 0.213346 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 8.00000 0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −93.0000 −1.32681 −0.663406 0.748259i \(-0.730890\pi\)
−0.663406 + 0.748259i \(0.730890\pi\)
\(18\) 0 0
\(19\) 59.0000 0.712396 0.356198 0.934410i \(-0.384073\pi\)
0.356198 + 0.934410i \(0.384073\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 84.0000 0.814039
\(23\) −9.00000 −0.0815926 −0.0407963 0.999167i \(-0.512989\pi\)
−0.0407963 + 0.999167i \(0.512989\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −40.0000 −0.301717
\(27\) 0 0
\(28\) −16.0000 −0.107990
\(29\) −120.000 −0.768395 −0.384197 0.923251i \(-0.625522\pi\)
−0.384197 + 0.923251i \(0.625522\pi\)
\(30\) 0 0
\(31\) 47.0000 0.272305 0.136152 0.990688i \(-0.456526\pi\)
0.136152 + 0.990688i \(0.456526\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 186.000 0.938198
\(35\) −20.0000 −0.0965891
\(36\) 0 0
\(37\) −262.000 −1.16412 −0.582061 0.813145i \(-0.697754\pi\)
−0.582061 + 0.813145i \(0.697754\pi\)
\(38\) −118.000 −0.503740
\(39\) 0 0
\(40\) −40.0000 −0.158114
\(41\) −126.000 −0.479949 −0.239974 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) 0 0
\(43\) −178.000 −0.631273 −0.315637 0.948880i \(-0.602218\pi\)
−0.315637 + 0.948880i \(0.602218\pi\)
\(44\) −168.000 −0.575613
\(45\) 0 0
\(46\) 18.0000 0.0576947
\(47\) −144.000 −0.446906 −0.223453 0.974715i \(-0.571733\pi\)
−0.223453 + 0.974715i \(0.571733\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) −50.0000 −0.141421
\(51\) 0 0
\(52\) 80.0000 0.213346
\(53\) −741.000 −1.92046 −0.960228 0.279217i \(-0.909925\pi\)
−0.960228 + 0.279217i \(0.909925\pi\)
\(54\) 0 0
\(55\) −210.000 −0.514844
\(56\) 32.0000 0.0763604
\(57\) 0 0
\(58\) 240.000 0.543337
\(59\) 444.000 0.979727 0.489863 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(60\) 0 0
\(61\) 221.000 0.463871 0.231936 0.972731i \(-0.425494\pi\)
0.231936 + 0.972731i \(0.425494\pi\)
\(62\) −94.0000 −0.192549
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 100.000 0.190823
\(66\) 0 0
\(67\) −538.000 −0.981002 −0.490501 0.871441i \(-0.663186\pi\)
−0.490501 + 0.871441i \(0.663186\pi\)
\(68\) −372.000 −0.663406
\(69\) 0 0
\(70\) 40.0000 0.0682988
\(71\) −690.000 −1.15335 −0.576676 0.816973i \(-0.695650\pi\)
−0.576676 + 0.816973i \(0.695650\pi\)
\(72\) 0 0
\(73\) −1126.00 −1.80532 −0.902660 0.430355i \(-0.858388\pi\)
−0.902660 + 0.430355i \(0.858388\pi\)
\(74\) 524.000 0.823159
\(75\) 0 0
\(76\) 236.000 0.356198
\(77\) 168.000 0.248641
\(78\) 0 0
\(79\) 665.000 0.947068 0.473534 0.880776i \(-0.342978\pi\)
0.473534 + 0.880776i \(0.342978\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 252.000 0.339375
\(83\) −75.0000 −0.0991846 −0.0495923 0.998770i \(-0.515792\pi\)
−0.0495923 + 0.998770i \(0.515792\pi\)
\(84\) 0 0
\(85\) −465.000 −0.593369
\(86\) 356.000 0.446378
\(87\) 0 0
\(88\) 336.000 0.407020
\(89\) 1086.00 1.29344 0.646718 0.762729i \(-0.276141\pi\)
0.646718 + 0.762729i \(0.276141\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) −36.0000 −0.0407963
\(93\) 0 0
\(94\) 288.000 0.316010
\(95\) 295.000 0.318593
\(96\) 0 0
\(97\) 1544.00 1.61618 0.808090 0.589059i \(-0.200501\pi\)
0.808090 + 0.589059i \(0.200501\pi\)
\(98\) 654.000 0.674122
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) 132.000 0.130044 0.0650222 0.997884i \(-0.479288\pi\)
0.0650222 + 0.997884i \(0.479288\pi\)
\(102\) 0 0
\(103\) −892.000 −0.853314 −0.426657 0.904413i \(-0.640309\pi\)
−0.426657 + 0.904413i \(0.640309\pi\)
\(104\) −160.000 −0.150859
\(105\) 0 0
\(106\) 1482.00 1.35797
\(107\) 1140.00 1.02998 0.514990 0.857196i \(-0.327795\pi\)
0.514990 + 0.857196i \(0.327795\pi\)
\(108\) 0 0
\(109\) −1735.00 −1.52461 −0.762307 0.647216i \(-0.775933\pi\)
−0.762307 + 0.647216i \(0.775933\pi\)
\(110\) 420.000 0.364049
\(111\) 0 0
\(112\) −64.0000 −0.0539949
\(113\) 1434.00 1.19380 0.596900 0.802316i \(-0.296399\pi\)
0.596900 + 0.802316i \(0.296399\pi\)
\(114\) 0 0
\(115\) −45.0000 −0.0364893
\(116\) −480.000 −0.384197
\(117\) 0 0
\(118\) −888.000 −0.692771
\(119\) 372.000 0.286565
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) −442.000 −0.328007
\(123\) 0 0
\(124\) 188.000 0.136152
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 686.000 0.479312 0.239656 0.970858i \(-0.422965\pi\)
0.239656 + 0.970858i \(0.422965\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −200.000 −0.134932
\(131\) 114.000 0.0760323 0.0380161 0.999277i \(-0.487896\pi\)
0.0380161 + 0.999277i \(0.487896\pi\)
\(132\) 0 0
\(133\) −236.000 −0.153863
\(134\) 1076.00 0.693673
\(135\) 0 0
\(136\) 744.000 0.469099
\(137\) −159.000 −0.0991554 −0.0495777 0.998770i \(-0.515788\pi\)
−0.0495777 + 0.998770i \(0.515788\pi\)
\(138\) 0 0
\(139\) 2276.00 1.38883 0.694417 0.719573i \(-0.255663\pi\)
0.694417 + 0.719573i \(0.255663\pi\)
\(140\) −80.0000 −0.0482945
\(141\) 0 0
\(142\) 1380.00 0.815542
\(143\) −840.000 −0.491219
\(144\) 0 0
\(145\) −600.000 −0.343636
\(146\) 2252.00 1.27655
\(147\) 0 0
\(148\) −1048.00 −0.582061
\(149\) 1398.00 0.768648 0.384324 0.923198i \(-0.374434\pi\)
0.384324 + 0.923198i \(0.374434\pi\)
\(150\) 0 0
\(151\) 2624.00 1.41416 0.707080 0.707134i \(-0.250012\pi\)
0.707080 + 0.707134i \(0.250012\pi\)
\(152\) −472.000 −0.251870
\(153\) 0 0
\(154\) −336.000 −0.175816
\(155\) 235.000 0.121778
\(156\) 0 0
\(157\) −394.000 −0.200284 −0.100142 0.994973i \(-0.531930\pi\)
−0.100142 + 0.994973i \(0.531930\pi\)
\(158\) −1330.00 −0.669678
\(159\) 0 0
\(160\) −160.000 −0.0790569
\(161\) 36.0000 0.0176223
\(162\) 0 0
\(163\) −3346.00 −1.60785 −0.803923 0.594733i \(-0.797258\pi\)
−0.803923 + 0.594733i \(0.797258\pi\)
\(164\) −504.000 −0.239974
\(165\) 0 0
\(166\) 150.000 0.0701341
\(167\) 1491.00 0.690881 0.345440 0.938441i \(-0.387730\pi\)
0.345440 + 0.938441i \(0.387730\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 930.000 0.419575
\(171\) 0 0
\(172\) −712.000 −0.315637
\(173\) −2403.00 −1.05605 −0.528025 0.849229i \(-0.677067\pi\)
−0.528025 + 0.849229i \(0.677067\pi\)
\(174\) 0 0
\(175\) −100.000 −0.0431959
\(176\) −672.000 −0.287806
\(177\) 0 0
\(178\) −2172.00 −0.914597
\(179\) 2640.00 1.10236 0.551181 0.834386i \(-0.314177\pi\)
0.551181 + 0.834386i \(0.314177\pi\)
\(180\) 0 0
\(181\) 1073.00 0.440638 0.220319 0.975428i \(-0.429290\pi\)
0.220319 + 0.975428i \(0.429290\pi\)
\(182\) 160.000 0.0651648
\(183\) 0 0
\(184\) 72.0000 0.0288473
\(185\) −1310.00 −0.520611
\(186\) 0 0
\(187\) 3906.00 1.52746
\(188\) −576.000 −0.223453
\(189\) 0 0
\(190\) −590.000 −0.225279
\(191\) −1470.00 −0.556887 −0.278444 0.960453i \(-0.589819\pi\)
−0.278444 + 0.960453i \(0.589819\pi\)
\(192\) 0 0
\(193\) −4720.00 −1.76038 −0.880189 0.474623i \(-0.842584\pi\)
−0.880189 + 0.474623i \(0.842584\pi\)
\(194\) −3088.00 −1.14281
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) 765.000 0.276670 0.138335 0.990385i \(-0.455825\pi\)
0.138335 + 0.990385i \(0.455825\pi\)
\(198\) 0 0
\(199\) 668.000 0.237956 0.118978 0.992897i \(-0.462038\pi\)
0.118978 + 0.992897i \(0.462038\pi\)
\(200\) −200.000 −0.0707107
\(201\) 0 0
\(202\) −264.000 −0.0919553
\(203\) 480.000 0.165958
\(204\) 0 0
\(205\) −630.000 −0.214640
\(206\) 1784.00 0.603384
\(207\) 0 0
\(208\) 320.000 0.106673
\(209\) −2478.00 −0.820128
\(210\) 0 0
\(211\) 4601.00 1.50117 0.750583 0.660777i \(-0.229773\pi\)
0.750583 + 0.660777i \(0.229773\pi\)
\(212\) −2964.00 −0.960228
\(213\) 0 0
\(214\) −2280.00 −0.728307
\(215\) −890.000 −0.282314
\(216\) 0 0
\(217\) −188.000 −0.0588123
\(218\) 3470.00 1.07806
\(219\) 0 0
\(220\) −840.000 −0.257422
\(221\) −1860.00 −0.566141
\(222\) 0 0
\(223\) −2158.00 −0.648029 −0.324014 0.946052i \(-0.605033\pi\)
−0.324014 + 0.946052i \(0.605033\pi\)
\(224\) 128.000 0.0381802
\(225\) 0 0
\(226\) −2868.00 −0.844144
\(227\) −3123.00 −0.913131 −0.456566 0.889690i \(-0.650921\pi\)
−0.456566 + 0.889690i \(0.650921\pi\)
\(228\) 0 0
\(229\) 2027.00 0.584925 0.292463 0.956277i \(-0.405525\pi\)
0.292463 + 0.956277i \(0.405525\pi\)
\(230\) 90.0000 0.0258018
\(231\) 0 0
\(232\) 960.000 0.271668
\(233\) 438.000 0.123152 0.0615758 0.998102i \(-0.480387\pi\)
0.0615758 + 0.998102i \(0.480387\pi\)
\(234\) 0 0
\(235\) −720.000 −0.199862
\(236\) 1776.00 0.489863
\(237\) 0 0
\(238\) −744.000 −0.202632
\(239\) −6414.00 −1.73593 −0.867965 0.496626i \(-0.834572\pi\)
−0.867965 + 0.496626i \(0.834572\pi\)
\(240\) 0 0
\(241\) 3431.00 0.917055 0.458527 0.888680i \(-0.348377\pi\)
0.458527 + 0.888680i \(0.348377\pi\)
\(242\) −866.000 −0.230035
\(243\) 0 0
\(244\) 884.000 0.231936
\(245\) −1635.00 −0.426352
\(246\) 0 0
\(247\) 1180.00 0.303974
\(248\) −376.000 −0.0962743
\(249\) 0 0
\(250\) −250.000 −0.0632456
\(251\) −7308.00 −1.83776 −0.918878 0.394541i \(-0.870904\pi\)
−0.918878 + 0.394541i \(0.870904\pi\)
\(252\) 0 0
\(253\) 378.000 0.0939314
\(254\) −1372.00 −0.338925
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 3729.00 0.905092 0.452546 0.891741i \(-0.350516\pi\)
0.452546 + 0.891741i \(0.350516\pi\)
\(258\) 0 0
\(259\) 1048.00 0.251427
\(260\) 400.000 0.0954113
\(261\) 0 0
\(262\) −228.000 −0.0537629
\(263\) 1956.00 0.458601 0.229301 0.973356i \(-0.426356\pi\)
0.229301 + 0.973356i \(0.426356\pi\)
\(264\) 0 0
\(265\) −3705.00 −0.858854
\(266\) 472.000 0.108798
\(267\) 0 0
\(268\) −2152.00 −0.490501
\(269\) −990.000 −0.224392 −0.112196 0.993686i \(-0.535788\pi\)
−0.112196 + 0.993686i \(0.535788\pi\)
\(270\) 0 0
\(271\) 8495.00 1.90419 0.952093 0.305808i \(-0.0989266\pi\)
0.952093 + 0.305808i \(0.0989266\pi\)
\(272\) −1488.00 −0.331703
\(273\) 0 0
\(274\) 318.000 0.0701134
\(275\) −1050.00 −0.230245
\(276\) 0 0
\(277\) −1366.00 −0.296300 −0.148150 0.988965i \(-0.547332\pi\)
−0.148150 + 0.988965i \(0.547332\pi\)
\(278\) −4552.00 −0.982053
\(279\) 0 0
\(280\) 160.000 0.0341494
\(281\) −5520.00 −1.17187 −0.585935 0.810358i \(-0.699273\pi\)
−0.585935 + 0.810358i \(0.699273\pi\)
\(282\) 0 0
\(283\) 5438.00 1.14225 0.571123 0.820865i \(-0.306508\pi\)
0.571123 + 0.820865i \(0.306508\pi\)
\(284\) −2760.00 −0.576676
\(285\) 0 0
\(286\) 1680.00 0.347344
\(287\) 504.000 0.103659
\(288\) 0 0
\(289\) 3736.00 0.760432
\(290\) 1200.00 0.242988
\(291\) 0 0
\(292\) −4504.00 −0.902660
\(293\) 8253.00 1.64555 0.822774 0.568369i \(-0.192425\pi\)
0.822774 + 0.568369i \(0.192425\pi\)
\(294\) 0 0
\(295\) 2220.00 0.438147
\(296\) 2096.00 0.411579
\(297\) 0 0
\(298\) −2796.00 −0.543517
\(299\) −180.000 −0.0348149
\(300\) 0 0
\(301\) 712.000 0.136342
\(302\) −5248.00 −0.999962
\(303\) 0 0
\(304\) 944.000 0.178099
\(305\) 1105.00 0.207450
\(306\) 0 0
\(307\) 9290.00 1.72706 0.863531 0.504295i \(-0.168248\pi\)
0.863531 + 0.504295i \(0.168248\pi\)
\(308\) 672.000 0.124321
\(309\) 0 0
\(310\) −470.000 −0.0861103
\(311\) 8112.00 1.47907 0.739533 0.673121i \(-0.235047\pi\)
0.739533 + 0.673121i \(0.235047\pi\)
\(312\) 0 0
\(313\) −7900.00 −1.42663 −0.713314 0.700845i \(-0.752807\pi\)
−0.713314 + 0.700845i \(0.752807\pi\)
\(314\) 788.000 0.141622
\(315\) 0 0
\(316\) 2660.00 0.473534
\(317\) 4419.00 0.782952 0.391476 0.920188i \(-0.371965\pi\)
0.391476 + 0.920188i \(0.371965\pi\)
\(318\) 0 0
\(319\) 5040.00 0.884595
\(320\) 320.000 0.0559017
\(321\) 0 0
\(322\) −72.0000 −0.0124609
\(323\) −5487.00 −0.945216
\(324\) 0 0
\(325\) 500.000 0.0853385
\(326\) 6692.00 1.13692
\(327\) 0 0
\(328\) 1008.00 0.169687
\(329\) 576.000 0.0965225
\(330\) 0 0
\(331\) −8200.00 −1.36167 −0.680835 0.732437i \(-0.738383\pi\)
−0.680835 + 0.732437i \(0.738383\pi\)
\(332\) −300.000 −0.0495923
\(333\) 0 0
\(334\) −2982.00 −0.488526
\(335\) −2690.00 −0.438718
\(336\) 0 0
\(337\) −9556.00 −1.54465 −0.772327 0.635225i \(-0.780907\pi\)
−0.772327 + 0.635225i \(0.780907\pi\)
\(338\) 3594.00 0.578366
\(339\) 0 0
\(340\) −1860.00 −0.296684
\(341\) −1974.00 −0.313484
\(342\) 0 0
\(343\) 2680.00 0.421885
\(344\) 1424.00 0.223189
\(345\) 0 0
\(346\) 4806.00 0.746740
\(347\) 10116.0 1.56500 0.782500 0.622650i \(-0.213944\pi\)
0.782500 + 0.622650i \(0.213944\pi\)
\(348\) 0 0
\(349\) −6751.00 −1.03545 −0.517726 0.855546i \(-0.673221\pi\)
−0.517726 + 0.855546i \(0.673221\pi\)
\(350\) 200.000 0.0305441
\(351\) 0 0
\(352\) 1344.00 0.203510
\(353\) 4062.00 0.612460 0.306230 0.951958i \(-0.400932\pi\)
0.306230 + 0.951958i \(0.400932\pi\)
\(354\) 0 0
\(355\) −3450.00 −0.515794
\(356\) 4344.00 0.646718
\(357\) 0 0
\(358\) −5280.00 −0.779488
\(359\) 8778.00 1.29049 0.645244 0.763977i \(-0.276756\pi\)
0.645244 + 0.763977i \(0.276756\pi\)
\(360\) 0 0
\(361\) −3378.00 −0.492492
\(362\) −2146.00 −0.311578
\(363\) 0 0
\(364\) −320.000 −0.0460785
\(365\) −5630.00 −0.807363
\(366\) 0 0
\(367\) 956.000 0.135975 0.0679875 0.997686i \(-0.478342\pi\)
0.0679875 + 0.997686i \(0.478342\pi\)
\(368\) −144.000 −0.0203981
\(369\) 0 0
\(370\) 2620.00 0.368128
\(371\) 2964.00 0.414780
\(372\) 0 0
\(373\) 2300.00 0.319275 0.159637 0.987176i \(-0.448968\pi\)
0.159637 + 0.987176i \(0.448968\pi\)
\(374\) −7812.00 −1.08008
\(375\) 0 0
\(376\) 1152.00 0.158005
\(377\) −2400.00 −0.327868
\(378\) 0 0
\(379\) 29.0000 0.00393042 0.00196521 0.999998i \(-0.499374\pi\)
0.00196521 + 0.999998i \(0.499374\pi\)
\(380\) 1180.00 0.159297
\(381\) 0 0
\(382\) 2940.00 0.393779
\(383\) 8127.00 1.08426 0.542128 0.840296i \(-0.317619\pi\)
0.542128 + 0.840296i \(0.317619\pi\)
\(384\) 0 0
\(385\) 840.000 0.111196
\(386\) 9440.00 1.24478
\(387\) 0 0
\(388\) 6176.00 0.808090
\(389\) −7938.00 −1.03463 −0.517317 0.855794i \(-0.673069\pi\)
−0.517317 + 0.855794i \(0.673069\pi\)
\(390\) 0 0
\(391\) 837.000 0.108258
\(392\) 2616.00 0.337061
\(393\) 0 0
\(394\) −1530.00 −0.195635
\(395\) 3325.00 0.423542
\(396\) 0 0
\(397\) 272.000 0.0343861 0.0171931 0.999852i \(-0.494527\pi\)
0.0171931 + 0.999852i \(0.494527\pi\)
\(398\) −1336.00 −0.168260
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) 4554.00 0.567122 0.283561 0.958954i \(-0.408484\pi\)
0.283561 + 0.958954i \(0.408484\pi\)
\(402\) 0 0
\(403\) 940.000 0.116190
\(404\) 528.000 0.0650222
\(405\) 0 0
\(406\) −960.000 −0.117350
\(407\) 11004.0 1.34017
\(408\) 0 0
\(409\) 1001.00 0.121018 0.0605089 0.998168i \(-0.480728\pi\)
0.0605089 + 0.998168i \(0.480728\pi\)
\(410\) 1260.00 0.151773
\(411\) 0 0
\(412\) −3568.00 −0.426657
\(413\) −1776.00 −0.211601
\(414\) 0 0
\(415\) −375.000 −0.0443567
\(416\) −640.000 −0.0754293
\(417\) 0 0
\(418\) 4956.00 0.579918
\(419\) −1794.00 −0.209171 −0.104585 0.994516i \(-0.533352\pi\)
−0.104585 + 0.994516i \(0.533352\pi\)
\(420\) 0 0
\(421\) −16129.0 −1.86717 −0.933586 0.358354i \(-0.883338\pi\)
−0.933586 + 0.358354i \(0.883338\pi\)
\(422\) −9202.00 −1.06148
\(423\) 0 0
\(424\) 5928.00 0.678984
\(425\) −2325.00 −0.265363
\(426\) 0 0
\(427\) −884.000 −0.100187
\(428\) 4560.00 0.514990
\(429\) 0 0
\(430\) 1780.00 0.199626
\(431\) −13356.0 −1.49266 −0.746329 0.665577i \(-0.768186\pi\)
−0.746329 + 0.665577i \(0.768186\pi\)
\(432\) 0 0
\(433\) −11500.0 −1.27634 −0.638169 0.769896i \(-0.720308\pi\)
−0.638169 + 0.769896i \(0.720308\pi\)
\(434\) 376.000 0.0415866
\(435\) 0 0
\(436\) −6940.00 −0.762307
\(437\) −531.000 −0.0581263
\(438\) 0 0
\(439\) −11149.0 −1.21210 −0.606051 0.795426i \(-0.707247\pi\)
−0.606051 + 0.795426i \(0.707247\pi\)
\(440\) 1680.00 0.182025
\(441\) 0 0
\(442\) 3720.00 0.400322
\(443\) 3849.00 0.412803 0.206401 0.978467i \(-0.433825\pi\)
0.206401 + 0.978467i \(0.433825\pi\)
\(444\) 0 0
\(445\) 5430.00 0.578442
\(446\) 4316.00 0.458225
\(447\) 0 0
\(448\) −256.000 −0.0269975
\(449\) 18048.0 1.89697 0.948483 0.316828i \(-0.102618\pi\)
0.948483 + 0.316828i \(0.102618\pi\)
\(450\) 0 0
\(451\) 5292.00 0.552529
\(452\) 5736.00 0.596900
\(453\) 0 0
\(454\) 6246.00 0.645681
\(455\) −400.000 −0.0412138
\(456\) 0 0
\(457\) −4264.00 −0.436458 −0.218229 0.975898i \(-0.570028\pi\)
−0.218229 + 0.975898i \(0.570028\pi\)
\(458\) −4054.00 −0.413605
\(459\) 0 0
\(460\) −180.000 −0.0182447
\(461\) −10242.0 −1.03475 −0.517373 0.855760i \(-0.673090\pi\)
−0.517373 + 0.855760i \(0.673090\pi\)
\(462\) 0 0
\(463\) 3302.00 0.331441 0.165720 0.986173i \(-0.447005\pi\)
0.165720 + 0.986173i \(0.447005\pi\)
\(464\) −1920.00 −0.192099
\(465\) 0 0
\(466\) −876.000 −0.0870814
\(467\) −1923.00 −0.190548 −0.0952739 0.995451i \(-0.530373\pi\)
−0.0952739 + 0.995451i \(0.530373\pi\)
\(468\) 0 0
\(469\) 2152.00 0.211877
\(470\) 1440.00 0.141324
\(471\) 0 0
\(472\) −3552.00 −0.346386
\(473\) 7476.00 0.726738
\(474\) 0 0
\(475\) 1475.00 0.142479
\(476\) 1488.00 0.143282
\(477\) 0 0
\(478\) 12828.0 1.22749
\(479\) 15246.0 1.45430 0.727148 0.686481i \(-0.240846\pi\)
0.727148 + 0.686481i \(0.240846\pi\)
\(480\) 0 0
\(481\) −5240.00 −0.496722
\(482\) −6862.00 −0.648455
\(483\) 0 0
\(484\) 1732.00 0.162660
\(485\) 7720.00 0.722778
\(486\) 0 0
\(487\) −8206.00 −0.763551 −0.381776 0.924255i \(-0.624687\pi\)
−0.381776 + 0.924255i \(0.624687\pi\)
\(488\) −1768.00 −0.164003
\(489\) 0 0
\(490\) 3270.00 0.301477
\(491\) −16806.0 −1.54469 −0.772346 0.635202i \(-0.780917\pi\)
−0.772346 + 0.635202i \(0.780917\pi\)
\(492\) 0 0
\(493\) 11160.0 1.01952
\(494\) −2360.00 −0.214942
\(495\) 0 0
\(496\) 752.000 0.0680762
\(497\) 2760.00 0.249100
\(498\) 0 0
\(499\) −5425.00 −0.486686 −0.243343 0.969940i \(-0.578244\pi\)
−0.243343 + 0.969940i \(0.578244\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) 14616.0 1.29949
\(503\) −19665.0 −1.74318 −0.871589 0.490236i \(-0.836910\pi\)
−0.871589 + 0.490236i \(0.836910\pi\)
\(504\) 0 0
\(505\) 660.000 0.0581577
\(506\) −756.000 −0.0664196
\(507\) 0 0
\(508\) 2744.00 0.239656
\(509\) −14724.0 −1.28218 −0.641090 0.767466i \(-0.721518\pi\)
−0.641090 + 0.767466i \(0.721518\pi\)
\(510\) 0 0
\(511\) 4504.00 0.389912
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −7458.00 −0.639997
\(515\) −4460.00 −0.381614
\(516\) 0 0
\(517\) 6048.00 0.514489
\(518\) −2096.00 −0.177786
\(519\) 0 0
\(520\) −800.000 −0.0674660
\(521\) 2058.00 0.173057 0.0865284 0.996249i \(-0.472423\pi\)
0.0865284 + 0.996249i \(0.472423\pi\)
\(522\) 0 0
\(523\) 11912.0 0.995938 0.497969 0.867195i \(-0.334079\pi\)
0.497969 + 0.867195i \(0.334079\pi\)
\(524\) 456.000 0.0380161
\(525\) 0 0
\(526\) −3912.00 −0.324280
\(527\) −4371.00 −0.361297
\(528\) 0 0
\(529\) −12086.0 −0.993343
\(530\) 7410.00 0.607302
\(531\) 0 0
\(532\) −944.000 −0.0769316
\(533\) −2520.00 −0.204790
\(534\) 0 0
\(535\) 5700.00 0.460621
\(536\) 4304.00 0.346837
\(537\) 0 0
\(538\) 1980.00 0.158669
\(539\) 13734.0 1.09752
\(540\) 0 0
\(541\) −5170.00 −0.410861 −0.205430 0.978672i \(-0.565859\pi\)
−0.205430 + 0.978672i \(0.565859\pi\)
\(542\) −16990.0 −1.34646
\(543\) 0 0
\(544\) 2976.00 0.234550
\(545\) −8675.00 −0.681828
\(546\) 0 0
\(547\) −4186.00 −0.327204 −0.163602 0.986526i \(-0.552311\pi\)
−0.163602 + 0.986526i \(0.552311\pi\)
\(548\) −636.000 −0.0495777
\(549\) 0 0
\(550\) 2100.00 0.162808
\(551\) −7080.00 −0.547401
\(552\) 0 0
\(553\) −2660.00 −0.204547
\(554\) 2732.00 0.209515
\(555\) 0 0
\(556\) 9104.00 0.694417
\(557\) 13026.0 0.990896 0.495448 0.868637i \(-0.335004\pi\)
0.495448 + 0.868637i \(0.335004\pi\)
\(558\) 0 0
\(559\) −3560.00 −0.269359
\(560\) −320.000 −0.0241473
\(561\) 0 0
\(562\) 11040.0 0.828638
\(563\) −10668.0 −0.798584 −0.399292 0.916824i \(-0.630744\pi\)
−0.399292 + 0.916824i \(0.630744\pi\)
\(564\) 0 0
\(565\) 7170.00 0.533883
\(566\) −10876.0 −0.807690
\(567\) 0 0
\(568\) 5520.00 0.407771
\(569\) −15372.0 −1.13256 −0.566281 0.824212i \(-0.691618\pi\)
−0.566281 + 0.824212i \(0.691618\pi\)
\(570\) 0 0
\(571\) −14989.0 −1.09855 −0.549273 0.835643i \(-0.685095\pi\)
−0.549273 + 0.835643i \(0.685095\pi\)
\(572\) −3360.00 −0.245610
\(573\) 0 0
\(574\) −1008.00 −0.0732981
\(575\) −225.000 −0.0163185
\(576\) 0 0
\(577\) −1066.00 −0.0769119 −0.0384559 0.999260i \(-0.512244\pi\)
−0.0384559 + 0.999260i \(0.512244\pi\)
\(578\) −7472.00 −0.537706
\(579\) 0 0
\(580\) −2400.00 −0.171818
\(581\) 300.000 0.0214219
\(582\) 0 0
\(583\) 31122.0 2.21088
\(584\) 9008.00 0.638277
\(585\) 0 0
\(586\) −16506.0 −1.16358
\(587\) −621.000 −0.0436651 −0.0218325 0.999762i \(-0.506950\pi\)
−0.0218325 + 0.999762i \(0.506950\pi\)
\(588\) 0 0
\(589\) 2773.00 0.193989
\(590\) −4440.00 −0.309817
\(591\) 0 0
\(592\) −4192.00 −0.291031
\(593\) −20187.0 −1.39794 −0.698972 0.715149i \(-0.746359\pi\)
−0.698972 + 0.715149i \(0.746359\pi\)
\(594\) 0 0
\(595\) 1860.00 0.128156
\(596\) 5592.00 0.384324
\(597\) 0 0
\(598\) 360.000 0.0246179
\(599\) −18228.0 −1.24337 −0.621683 0.783269i \(-0.713551\pi\)
−0.621683 + 0.783269i \(0.713551\pi\)
\(600\) 0 0
\(601\) −11743.0 −0.797017 −0.398508 0.917165i \(-0.630472\pi\)
−0.398508 + 0.917165i \(0.630472\pi\)
\(602\) −1424.00 −0.0964085
\(603\) 0 0
\(604\) 10496.0 0.707080
\(605\) 2165.00 0.145487
\(606\) 0 0
\(607\) −24418.0 −1.63278 −0.816389 0.577503i \(-0.804027\pi\)
−0.816389 + 0.577503i \(0.804027\pi\)
\(608\) −1888.00 −0.125935
\(609\) 0 0
\(610\) −2210.00 −0.146689
\(611\) −2880.00 −0.190691
\(612\) 0 0
\(613\) 2672.00 0.176054 0.0880270 0.996118i \(-0.471944\pi\)
0.0880270 + 0.996118i \(0.471944\pi\)
\(614\) −18580.0 −1.22122
\(615\) 0 0
\(616\) −1344.00 −0.0879080
\(617\) −8601.00 −0.561205 −0.280602 0.959824i \(-0.590534\pi\)
−0.280602 + 0.959824i \(0.590534\pi\)
\(618\) 0 0
\(619\) 21308.0 1.38359 0.691794 0.722095i \(-0.256821\pi\)
0.691794 + 0.722095i \(0.256821\pi\)
\(620\) 940.000 0.0608892
\(621\) 0 0
\(622\) −16224.0 −1.04586
\(623\) −4344.00 −0.279356
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 15800.0 1.00878
\(627\) 0 0
\(628\) −1576.00 −0.100142
\(629\) 24366.0 1.54457
\(630\) 0 0
\(631\) −19015.0 −1.19964 −0.599822 0.800134i \(-0.704762\pi\)
−0.599822 + 0.800134i \(0.704762\pi\)
\(632\) −5320.00 −0.334839
\(633\) 0 0
\(634\) −8838.00 −0.553631
\(635\) 3430.00 0.214355
\(636\) 0 0
\(637\) −6540.00 −0.406788
\(638\) −10080.0 −0.625503
\(639\) 0 0
\(640\) −640.000 −0.0395285
\(641\) −4416.00 −0.272108 −0.136054 0.990701i \(-0.543442\pi\)
−0.136054 + 0.990701i \(0.543442\pi\)
\(642\) 0 0
\(643\) 7580.00 0.464893 0.232446 0.972609i \(-0.425327\pi\)
0.232446 + 0.972609i \(0.425327\pi\)
\(644\) 144.000 0.00881117
\(645\) 0 0
\(646\) 10974.0 0.668369
\(647\) −14901.0 −0.905439 −0.452719 0.891653i \(-0.649546\pi\)
−0.452719 + 0.891653i \(0.649546\pi\)
\(648\) 0 0
\(649\) −18648.0 −1.12789
\(650\) −1000.00 −0.0603434
\(651\) 0 0
\(652\) −13384.0 −0.803923
\(653\) 12915.0 0.773971 0.386985 0.922086i \(-0.373516\pi\)
0.386985 + 0.922086i \(0.373516\pi\)
\(654\) 0 0
\(655\) 570.000 0.0340027
\(656\) −2016.00 −0.119987
\(657\) 0 0
\(658\) −1152.00 −0.0682517
\(659\) 28128.0 1.66269 0.831344 0.555758i \(-0.187572\pi\)
0.831344 + 0.555758i \(0.187572\pi\)
\(660\) 0 0
\(661\) −8362.00 −0.492049 −0.246024 0.969264i \(-0.579124\pi\)
−0.246024 + 0.969264i \(0.579124\pi\)
\(662\) 16400.0 0.962846
\(663\) 0 0
\(664\) 600.000 0.0350670
\(665\) −1180.00 −0.0688097
\(666\) 0 0
\(667\) 1080.00 0.0626953
\(668\) 5964.00 0.345440
\(669\) 0 0
\(670\) 5380.00 0.310220
\(671\) −9282.00 −0.534020
\(672\) 0 0
\(673\) 29708.0 1.70157 0.850787 0.525511i \(-0.176126\pi\)
0.850787 + 0.525511i \(0.176126\pi\)
\(674\) 19112.0 1.09224
\(675\) 0 0
\(676\) −7188.00 −0.408967
\(677\) 6762.00 0.383877 0.191939 0.981407i \(-0.438523\pi\)
0.191939 + 0.981407i \(0.438523\pi\)
\(678\) 0 0
\(679\) −6176.00 −0.349062
\(680\) 3720.00 0.209787
\(681\) 0 0
\(682\) 3948.00 0.221667
\(683\) −19155.0 −1.07313 −0.536563 0.843860i \(-0.680278\pi\)
−0.536563 + 0.843860i \(0.680278\pi\)
\(684\) 0 0
\(685\) −795.000 −0.0443436
\(686\) −5360.00 −0.298317
\(687\) 0 0
\(688\) −2848.00 −0.157818
\(689\) −14820.0 −0.819444
\(690\) 0 0
\(691\) −22975.0 −1.26485 −0.632424 0.774622i \(-0.717940\pi\)
−0.632424 + 0.774622i \(0.717940\pi\)
\(692\) −9612.00 −0.528025
\(693\) 0 0
\(694\) −20232.0 −1.10662
\(695\) 11380.0 0.621105
\(696\) 0 0
\(697\) 11718.0 0.636802
\(698\) 13502.0 0.732175
\(699\) 0 0
\(700\) −400.000 −0.0215980
\(701\) −6450.00 −0.347522 −0.173761 0.984788i \(-0.555592\pi\)
−0.173761 + 0.984788i \(0.555592\pi\)
\(702\) 0 0
\(703\) −15458.0 −0.829317
\(704\) −2688.00 −0.143903
\(705\) 0 0
\(706\) −8124.00 −0.433075
\(707\) −528.000 −0.0280870
\(708\) 0 0
\(709\) 34538.0 1.82948 0.914740 0.404042i \(-0.132395\pi\)
0.914740 + 0.404042i \(0.132395\pi\)
\(710\) 6900.00 0.364722
\(711\) 0 0
\(712\) −8688.00 −0.457299
\(713\) −423.000 −0.0222181
\(714\) 0 0
\(715\) −4200.00 −0.219680
\(716\) 10560.0 0.551181
\(717\) 0 0
\(718\) −17556.0 −0.912513
\(719\) 27114.0 1.40637 0.703186 0.711006i \(-0.251760\pi\)
0.703186 + 0.711006i \(0.251760\pi\)
\(720\) 0 0
\(721\) 3568.00 0.184299
\(722\) 6756.00 0.348244
\(723\) 0 0
\(724\) 4292.00 0.220319
\(725\) −3000.00 −0.153679
\(726\) 0 0
\(727\) 236.000 0.0120396 0.00601978 0.999982i \(-0.498084\pi\)
0.00601978 + 0.999982i \(0.498084\pi\)
\(728\) 640.000 0.0325824
\(729\) 0 0
\(730\) 11260.0 0.570892
\(731\) 16554.0 0.837581
\(732\) 0 0
\(733\) 27128.0 1.36698 0.683489 0.729960i \(-0.260462\pi\)
0.683489 + 0.729960i \(0.260462\pi\)
\(734\) −1912.00 −0.0961488
\(735\) 0 0
\(736\) 288.000 0.0144237
\(737\) 22596.0 1.12935
\(738\) 0 0
\(739\) 5249.00 0.261282 0.130641 0.991430i \(-0.458296\pi\)
0.130641 + 0.991430i \(0.458296\pi\)
\(740\) −5240.00 −0.260306
\(741\) 0 0
\(742\) −5928.00 −0.293293
\(743\) 13896.0 0.686130 0.343065 0.939312i \(-0.388535\pi\)
0.343065 + 0.939312i \(0.388535\pi\)
\(744\) 0 0
\(745\) 6990.00 0.343750
\(746\) −4600.00 −0.225761
\(747\) 0 0
\(748\) 15624.0 0.763730
\(749\) −4560.00 −0.222455
\(750\) 0 0
\(751\) 27665.0 1.34422 0.672111 0.740451i \(-0.265388\pi\)
0.672111 + 0.740451i \(0.265388\pi\)
\(752\) −2304.00 −0.111726
\(753\) 0 0
\(754\) 4800.00 0.231838
\(755\) 13120.0 0.632431
\(756\) 0 0
\(757\) −8122.00 −0.389959 −0.194980 0.980807i \(-0.562464\pi\)
−0.194980 + 0.980807i \(0.562464\pi\)
\(758\) −58.0000 −0.00277923
\(759\) 0 0
\(760\) −2360.00 −0.112640
\(761\) 10584.0 0.504165 0.252083 0.967706i \(-0.418885\pi\)
0.252083 + 0.967706i \(0.418885\pi\)
\(762\) 0 0
\(763\) 6940.00 0.329286
\(764\) −5880.00 −0.278444
\(765\) 0 0
\(766\) −16254.0 −0.766685
\(767\) 8880.00 0.418042
\(768\) 0 0
\(769\) −18619.0 −0.873106 −0.436553 0.899679i \(-0.643801\pi\)
−0.436553 + 0.899679i \(0.643801\pi\)
\(770\) −1680.00 −0.0786273
\(771\) 0 0
\(772\) −18880.0 −0.880189
\(773\) 22251.0 1.03533 0.517667 0.855582i \(-0.326801\pi\)
0.517667 + 0.855582i \(0.326801\pi\)
\(774\) 0 0
\(775\) 1175.00 0.0544610
\(776\) −12352.0 −0.571406
\(777\) 0 0
\(778\) 15876.0 0.731597
\(779\) −7434.00 −0.341914
\(780\) 0 0
\(781\) 28980.0 1.32777
\(782\) −1674.00 −0.0765500
\(783\) 0 0
\(784\) −5232.00 −0.238338
\(785\) −1970.00 −0.0895698
\(786\) 0 0
\(787\) 24854.0 1.12573 0.562865 0.826549i \(-0.309699\pi\)
0.562865 + 0.826549i \(0.309699\pi\)
\(788\) 3060.00 0.138335
\(789\) 0 0
\(790\) −6650.00 −0.299489
\(791\) −5736.00 −0.257837
\(792\) 0 0
\(793\) 4420.00 0.197930
\(794\) −544.000 −0.0243147
\(795\) 0 0
\(796\) 2672.00 0.118978
\(797\) 3681.00 0.163598 0.0817991 0.996649i \(-0.473933\pi\)
0.0817991 + 0.996649i \(0.473933\pi\)
\(798\) 0 0
\(799\) 13392.0 0.592960
\(800\) −800.000 −0.0353553
\(801\) 0 0
\(802\) −9108.00 −0.401016
\(803\) 47292.0 2.07833
\(804\) 0 0
\(805\) 180.000 0.00788095
\(806\) −1880.00 −0.0821590
\(807\) 0 0
\(808\) −1056.00 −0.0459777
\(809\) −5142.00 −0.223465 −0.111732 0.993738i \(-0.535640\pi\)
−0.111732 + 0.993738i \(0.535640\pi\)
\(810\) 0 0
\(811\) −18484.0 −0.800322 −0.400161 0.916445i \(-0.631046\pi\)
−0.400161 + 0.916445i \(0.631046\pi\)
\(812\) 1920.00 0.0829788
\(813\) 0 0
\(814\) −22008.0 −0.947641
\(815\) −16730.0 −0.719051
\(816\) 0 0
\(817\) −10502.0 −0.449717
\(818\) −2002.00 −0.0855725
\(819\) 0 0
\(820\) −2520.00 −0.107320
\(821\) 25014.0 1.06333 0.531665 0.846954i \(-0.321566\pi\)
0.531665 + 0.846954i \(0.321566\pi\)
\(822\) 0 0
\(823\) −32146.0 −1.36153 −0.680765 0.732502i \(-0.738352\pi\)
−0.680765 + 0.732502i \(0.738352\pi\)
\(824\) 7136.00 0.301692
\(825\) 0 0
\(826\) 3552.00 0.149625
\(827\) −10977.0 −0.461557 −0.230779 0.973006i \(-0.574127\pi\)
−0.230779 + 0.973006i \(0.574127\pi\)
\(828\) 0 0
\(829\) 36602.0 1.53346 0.766731 0.641969i \(-0.221882\pi\)
0.766731 + 0.641969i \(0.221882\pi\)
\(830\) 750.000 0.0313649
\(831\) 0 0
\(832\) 1280.00 0.0533366
\(833\) 30411.0 1.26492
\(834\) 0 0
\(835\) 7455.00 0.308971
\(836\) −9912.00 −0.410064
\(837\) 0 0
\(838\) 3588.00 0.147906
\(839\) 11076.0 0.455764 0.227882 0.973689i \(-0.426820\pi\)
0.227882 + 0.973689i \(0.426820\pi\)
\(840\) 0 0
\(841\) −9989.00 −0.409570
\(842\) 32258.0 1.32029
\(843\) 0 0
\(844\) 18404.0 0.750583
\(845\) −8985.00 −0.365791
\(846\) 0 0
\(847\) −1732.00 −0.0702624
\(848\) −11856.0 −0.480114
\(849\) 0 0
\(850\) 4650.00 0.187640
\(851\) 2358.00 0.0949838
\(852\) 0 0
\(853\) 36848.0 1.47908 0.739538 0.673115i \(-0.235044\pi\)
0.739538 + 0.673115i \(0.235044\pi\)
\(854\) 1768.00 0.0708428
\(855\) 0 0
\(856\) −9120.00 −0.364153
\(857\) −26961.0 −1.07464 −0.537322 0.843377i \(-0.680564\pi\)
−0.537322 + 0.843377i \(0.680564\pi\)
\(858\) 0 0
\(859\) −415.000 −0.0164838 −0.00824192 0.999966i \(-0.502624\pi\)
−0.00824192 + 0.999966i \(0.502624\pi\)
\(860\) −3560.00 −0.141157
\(861\) 0 0
\(862\) 26712.0 1.05547
\(863\) −45501.0 −1.79475 −0.897377 0.441265i \(-0.854530\pi\)
−0.897377 + 0.441265i \(0.854530\pi\)
\(864\) 0 0
\(865\) −12015.0 −0.472280
\(866\) 23000.0 0.902508
\(867\) 0 0
\(868\) −752.000 −0.0294062
\(869\) −27930.0 −1.09029
\(870\) 0 0
\(871\) −10760.0 −0.418586
\(872\) 13880.0 0.539032
\(873\) 0 0
\(874\) 1062.00 0.0411015
\(875\) −500.000 −0.0193178
\(876\) 0 0
\(877\) 35042.0 1.34924 0.674620 0.738165i \(-0.264307\pi\)
0.674620 + 0.738165i \(0.264307\pi\)
\(878\) 22298.0 0.857085
\(879\) 0 0
\(880\) −3360.00 −0.128711
\(881\) 1080.00 0.0413009 0.0206505 0.999787i \(-0.493426\pi\)
0.0206505 + 0.999787i \(0.493426\pi\)
\(882\) 0 0
\(883\) −20164.0 −0.768485 −0.384243 0.923232i \(-0.625537\pi\)
−0.384243 + 0.923232i \(0.625537\pi\)
\(884\) −7440.00 −0.283070
\(885\) 0 0
\(886\) −7698.00 −0.291895
\(887\) 20067.0 0.759621 0.379811 0.925064i \(-0.375989\pi\)
0.379811 + 0.925064i \(0.375989\pi\)
\(888\) 0 0
\(889\) −2744.00 −0.103522
\(890\) −10860.0 −0.409020
\(891\) 0 0
\(892\) −8632.00 −0.324014
\(893\) −8496.00 −0.318374
\(894\) 0 0
\(895\) 13200.0 0.492991
\(896\) 512.000 0.0190901
\(897\) 0 0
\(898\) −36096.0 −1.34136
\(899\) −5640.00 −0.209238
\(900\) 0 0
\(901\) 68913.0 2.54809
\(902\) −10584.0 −0.390697
\(903\) 0 0
\(904\) −11472.0 −0.422072
\(905\) 5365.00 0.197059
\(906\) 0 0
\(907\) −26524.0 −0.971020 −0.485510 0.874231i \(-0.661366\pi\)
−0.485510 + 0.874231i \(0.661366\pi\)
\(908\) −12492.0 −0.456566
\(909\) 0 0
\(910\) 800.000 0.0291426
\(911\) 35568.0 1.29355 0.646773 0.762683i \(-0.276118\pi\)
0.646773 + 0.762683i \(0.276118\pi\)
\(912\) 0 0
\(913\) 3150.00 0.114184
\(914\) 8528.00 0.308623
\(915\) 0 0
\(916\) 8108.00 0.292463
\(917\) −456.000 −0.0164214
\(918\) 0 0
\(919\) −23704.0 −0.850841 −0.425420 0.904996i \(-0.639874\pi\)
−0.425420 + 0.904996i \(0.639874\pi\)
\(920\) 360.000 0.0129009
\(921\) 0 0
\(922\) 20484.0 0.731675
\(923\) −13800.0 −0.492126
\(924\) 0 0
\(925\) −6550.00 −0.232825
\(926\) −6604.00 −0.234364
\(927\) 0 0
\(928\) 3840.00 0.135834
\(929\) −40590.0 −1.43349 −0.716746 0.697334i \(-0.754369\pi\)
−0.716746 + 0.697334i \(0.754369\pi\)
\(930\) 0 0
\(931\) −19293.0 −0.679165
\(932\) 1752.00 0.0615758
\(933\) 0 0
\(934\) 3846.00 0.134738
\(935\) 19530.0 0.683101
\(936\) 0 0
\(937\) −12964.0 −0.451991 −0.225995 0.974128i \(-0.572563\pi\)
−0.225995 + 0.974128i \(0.572563\pi\)
\(938\) −4304.00 −0.149819
\(939\) 0 0
\(940\) −2880.00 −0.0999311
\(941\) 29922.0 1.03659 0.518294 0.855203i \(-0.326567\pi\)
0.518294 + 0.855203i \(0.326567\pi\)
\(942\) 0 0
\(943\) 1134.00 0.0391603
\(944\) 7104.00 0.244932
\(945\) 0 0
\(946\) −14952.0 −0.513881
\(947\) 5241.00 0.179841 0.0899206 0.995949i \(-0.471339\pi\)
0.0899206 + 0.995949i \(0.471339\pi\)
\(948\) 0 0
\(949\) −22520.0 −0.770316
\(950\) −2950.00 −0.100748
\(951\) 0 0
\(952\) −2976.00 −0.101316
\(953\) 26214.0 0.891033 0.445517 0.895274i \(-0.353020\pi\)
0.445517 + 0.895274i \(0.353020\pi\)
\(954\) 0 0
\(955\) −7350.00 −0.249048
\(956\) −25656.0 −0.867965
\(957\) 0 0
\(958\) −30492.0 −1.02834
\(959\) 636.000 0.0214155
\(960\) 0 0
\(961\) −27582.0 −0.925850
\(962\) 10480.0 0.351236
\(963\) 0 0
\(964\) 13724.0 0.458527
\(965\) −23600.0 −0.787265
\(966\) 0 0
\(967\) 18278.0 0.607840 0.303920 0.952698i \(-0.401705\pi\)
0.303920 + 0.952698i \(0.401705\pi\)
\(968\) −3464.00 −0.115018
\(969\) 0 0
\(970\) −15440.0 −0.511081
\(971\) −24942.0 −0.824333 −0.412166 0.911109i \(-0.635228\pi\)
−0.412166 + 0.911109i \(0.635228\pi\)
\(972\) 0 0
\(973\) −9104.00 −0.299960
\(974\) 16412.0 0.539912
\(975\) 0 0
\(976\) 3536.00 0.115968
\(977\) −11226.0 −0.367607 −0.183803 0.982963i \(-0.558841\pi\)
−0.183803 + 0.982963i \(0.558841\pi\)
\(978\) 0 0
\(979\) −45612.0 −1.48904
\(980\) −6540.00 −0.213176
\(981\) 0 0
\(982\) 33612.0 1.09226
\(983\) −23073.0 −0.748641 −0.374321 0.927299i \(-0.622124\pi\)
−0.374321 + 0.927299i \(0.622124\pi\)
\(984\) 0 0
\(985\) 3825.00 0.123731
\(986\) −22320.0 −0.720906
\(987\) 0 0
\(988\) 4720.00 0.151987
\(989\) 1602.00 0.0515072
\(990\) 0 0
\(991\) 22037.0 0.706386 0.353193 0.935551i \(-0.385096\pi\)
0.353193 + 0.935551i \(0.385096\pi\)
\(992\) −1504.00 −0.0481371
\(993\) 0 0
\(994\) −5520.00 −0.176141
\(995\) 3340.00 0.106417
\(996\) 0 0
\(997\) 19082.0 0.606151 0.303076 0.952966i \(-0.401986\pi\)
0.303076 + 0.952966i \(0.401986\pi\)
\(998\) 10850.0 0.344139
\(999\) 0 0
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 270.4.a.e.1.1 1
3.2 odd 2 270.4.a.i.1.1 yes 1
4.3 odd 2 2160.4.a.o.1.1 1
5.2 odd 4 1350.4.c.d.649.1 2
5.3 odd 4 1350.4.c.d.649.2 2
5.4 even 2 1350.4.a.u.1.1 1
9.2 odd 6 810.4.e.h.271.1 2
9.4 even 3 810.4.e.q.541.1 2
9.5 odd 6 810.4.e.h.541.1 2
9.7 even 3 810.4.e.q.271.1 2
12.11 even 2 2160.4.a.e.1.1 1
15.2 even 4 1350.4.c.q.649.2 2
15.8 even 4 1350.4.c.q.649.1 2
15.14 odd 2 1350.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.e.1.1 1 1.1 even 1 trivial
270.4.a.i.1.1 yes 1 3.2 odd 2
810.4.e.h.271.1 2 9.2 odd 6
810.4.e.h.541.1 2 9.5 odd 6
810.4.e.q.271.1 2 9.7 even 3
810.4.e.q.541.1 2 9.4 even 3
1350.4.a.g.1.1 1 15.14 odd 2
1350.4.a.u.1.1 1 5.4 even 2
1350.4.c.d.649.1 2 5.2 odd 4
1350.4.c.d.649.2 2 5.3 odd 4
1350.4.c.q.649.1 2 15.8 even 4
1350.4.c.q.649.2 2 15.2 even 4
2160.4.a.e.1.1 1 12.11 even 2
2160.4.a.o.1.1 1 4.3 odd 2