Properties

Label 270.4.a.f.1.1
Level $270$
Weight $4$
Character 270.1
Self dual yes
Analytic conductor $15.931$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 270.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +3.00000 q^{11} +47.0000 q^{13} -28.0000 q^{14} +16.0000 q^{16} -39.0000 q^{17} +32.0000 q^{19} +20.0000 q^{20} -6.00000 q^{22} -99.0000 q^{23} +25.0000 q^{25} -94.0000 q^{26} +56.0000 q^{28} +51.0000 q^{29} +83.0000 q^{31} -32.0000 q^{32} +78.0000 q^{34} +70.0000 q^{35} +314.000 q^{37} -64.0000 q^{38} -40.0000 q^{40} -108.000 q^{41} +299.000 q^{43} +12.0000 q^{44} +198.000 q^{46} +531.000 q^{47} -147.000 q^{49} -50.0000 q^{50} +188.000 q^{52} +564.000 q^{53} +15.0000 q^{55} -112.000 q^{56} -102.000 q^{58} +12.0000 q^{59} +230.000 q^{61} -166.000 q^{62} +64.0000 q^{64} +235.000 q^{65} -268.000 q^{67} -156.000 q^{68} -140.000 q^{70} +120.000 q^{71} +1106.00 q^{73} -628.000 q^{74} +128.000 q^{76} +42.0000 q^{77} -739.000 q^{79} +80.0000 q^{80} +216.000 q^{82} +1086.00 q^{83} -195.000 q^{85} -598.000 q^{86} -24.0000 q^{88} -120.000 q^{89} +658.000 q^{91} -396.000 q^{92} -1062.00 q^{94} +160.000 q^{95} -1642.00 q^{97} +294.000 q^{98} +O(q^{100})\)

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
\(7\) 14.0000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −10.0000 −0.316228
\(11\) 3.00000 0.0822304 0.0411152 0.999154i \(-0.486909\pi\)
0.0411152 + 0.999154i \(0.486909\pi\)
\(12\) 0 0
\(13\) 47.0000 1.00273 0.501364 0.865237i \(-0.332832\pi\)
0.501364 + 0.865237i \(0.332832\pi\)
\(14\) −28.0000 −0.534522
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −39.0000 −0.556405 −0.278203 0.960522i \(-0.589739\pi\)
−0.278203 + 0.960522i \(0.589739\pi\)
\(18\) 0 0
\(19\) 32.0000 0.386384 0.193192 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) −6.00000 −0.0581456
\(23\) −99.0000 −0.897519 −0.448759 0.893653i \(-0.648134\pi\)
−0.448759 + 0.893653i \(0.648134\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −94.0000 −0.709035
\(27\) 0 0
\(28\) 56.0000 0.377964
\(29\) 51.0000 0.326568 0.163284 0.986579i \(-0.447791\pi\)
0.163284 + 0.986579i \(0.447791\pi\)
\(30\) 0 0
\(31\) 83.0000 0.480879 0.240439 0.970664i \(-0.422708\pi\)
0.240439 + 0.970664i \(0.422708\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 78.0000 0.393438
\(35\) 70.0000 0.338062
\(36\) 0 0
\(37\) 314.000 1.39517 0.697585 0.716502i \(-0.254258\pi\)
0.697585 + 0.716502i \(0.254258\pi\)
\(38\) −64.0000 −0.273215
\(39\) 0 0
\(40\) −40.0000 −0.158114
\(41\) −108.000 −0.411385 −0.205692 0.978617i \(-0.565945\pi\)
−0.205692 + 0.978617i \(0.565945\pi\)
\(42\) 0 0
\(43\) 299.000 1.06040 0.530199 0.847874i \(-0.322117\pi\)
0.530199 + 0.847874i \(0.322117\pi\)
\(44\) 12.0000 0.0411152
\(45\) 0 0
\(46\) 198.000 0.634641
\(47\) 531.000 1.64796 0.823982 0.566616i \(-0.191748\pi\)
0.823982 + 0.566616i \(0.191748\pi\)
\(48\) 0 0
\(49\) −147.000 −0.428571
\(50\) −50.0000 −0.141421
\(51\) 0 0
\(52\) 188.000 0.501364
\(53\) 564.000 1.46172 0.730862 0.682525i \(-0.239118\pi\)
0.730862 + 0.682525i \(0.239118\pi\)
\(54\) 0 0
\(55\) 15.0000 0.0367745
\(56\) −112.000 −0.267261
\(57\) 0 0
\(58\) −102.000 −0.230918
\(59\) 12.0000 0.0264791 0.0132396 0.999912i \(-0.495786\pi\)
0.0132396 + 0.999912i \(0.495786\pi\)
\(60\) 0 0
\(61\) 230.000 0.482762 0.241381 0.970430i \(-0.422400\pi\)
0.241381 + 0.970430i \(0.422400\pi\)
\(62\) −166.000 −0.340033
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 235.000 0.448433
\(66\) 0 0
\(67\) −268.000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −156.000 −0.278203
\(69\) 0 0
\(70\) −140.000 −0.239046
\(71\) 120.000 0.200583 0.100291 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 1106.00 1.77325 0.886627 0.462486i \(-0.153042\pi\)
0.886627 + 0.462486i \(0.153042\pi\)
\(74\) −628.000 −0.986534
\(75\) 0 0
\(76\) 128.000 0.193192
\(77\) 42.0000 0.0621603
\(78\) 0 0
\(79\) −739.000 −1.05246 −0.526228 0.850344i \(-0.676394\pi\)
−0.526228 + 0.850344i \(0.676394\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 216.000 0.290893
\(83\) 1086.00 1.43619 0.718096 0.695944i \(-0.245014\pi\)
0.718096 + 0.695944i \(0.245014\pi\)
\(84\) 0 0
\(85\) −195.000 −0.248832
\(86\) −598.000 −0.749814
\(87\) 0 0
\(88\) −24.0000 −0.0290728
\(89\) −120.000 −0.142921 −0.0714605 0.997443i \(-0.522766\pi\)
−0.0714605 + 0.997443i \(0.522766\pi\)
\(90\) 0 0
\(91\) 658.000 0.757991
\(92\) −396.000 −0.448759
\(93\) 0 0
\(94\) −1062.00 −1.16529
\(95\) 160.000 0.172796
\(96\) 0 0
\(97\) −1642.00 −1.71876 −0.859381 0.511336i \(-0.829151\pi\)
−0.859381 + 0.511336i \(0.829151\pi\)
\(98\) 294.000 0.303046
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) 33.0000 0.0325111 0.0162556 0.999868i \(-0.494825\pi\)
0.0162556 + 0.999868i \(0.494825\pi\)
\(102\) 0 0
\(103\) −1198.00 −1.14604 −0.573022 0.819540i \(-0.694229\pi\)
−0.573022 + 0.819540i \(0.694229\pi\)
\(104\) −376.000 −0.354518
\(105\) 0 0
\(106\) −1128.00 −1.03359
\(107\) −1542.00 −1.39318 −0.696592 0.717467i \(-0.745301\pi\)
−0.696592 + 0.717467i \(0.745301\pi\)
\(108\) 0 0
\(109\) −556.000 −0.488579 −0.244290 0.969702i \(-0.578555\pi\)
−0.244290 + 0.969702i \(0.578555\pi\)
\(110\) −30.0000 −0.0260035
\(111\) 0 0
\(112\) 224.000 0.188982
\(113\) 1605.00 1.33616 0.668078 0.744091i \(-0.267117\pi\)
0.668078 + 0.744091i \(0.267117\pi\)
\(114\) 0 0
\(115\) −495.000 −0.401383
\(116\) 204.000 0.163284
\(117\) 0 0
\(118\) −24.0000 −0.0187236
\(119\) −546.000 −0.420603
\(120\) 0 0
\(121\) −1322.00 −0.993238
\(122\) −460.000 −0.341364
\(123\) 0 0
\(124\) 332.000 0.240439
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1334.00 0.932074 0.466037 0.884765i \(-0.345681\pi\)
0.466037 + 0.884765i \(0.345681\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −470.000 −0.317090
\(131\) −2883.00 −1.92282 −0.961408 0.275127i \(-0.911280\pi\)
−0.961408 + 0.275127i \(0.911280\pi\)
\(132\) 0 0
\(133\) 448.000 0.292079
\(134\) 536.000 0.345547
\(135\) 0 0
\(136\) 312.000 0.196719
\(137\) 282.000 0.175860 0.0879302 0.996127i \(-0.471975\pi\)
0.0879302 + 0.996127i \(0.471975\pi\)
\(138\) 0 0
\(139\) −2494.00 −1.52186 −0.760929 0.648835i \(-0.775257\pi\)
−0.760929 + 0.648835i \(0.775257\pi\)
\(140\) 280.000 0.169031
\(141\) 0 0
\(142\) −240.000 −0.141833
\(143\) 141.000 0.0824546
\(144\) 0 0
\(145\) 255.000 0.146045
\(146\) −2212.00 −1.25388
\(147\) 0 0
\(148\) 1256.00 0.697585
\(149\) 2595.00 1.42678 0.713392 0.700766i \(-0.247158\pi\)
0.713392 + 0.700766i \(0.247158\pi\)
\(150\) 0 0
\(151\) 1229.00 0.662348 0.331174 0.943570i \(-0.392555\pi\)
0.331174 + 0.943570i \(0.392555\pi\)
\(152\) −256.000 −0.136608
\(153\) 0 0
\(154\) −84.0000 −0.0439540
\(155\) 415.000 0.215055
\(156\) 0 0
\(157\) −1591.00 −0.808762 −0.404381 0.914591i \(-0.632513\pi\)
−0.404381 + 0.914591i \(0.632513\pi\)
\(158\) 1478.00 0.744199
\(159\) 0 0
\(160\) −160.000 −0.0790569
\(161\) −1386.00 −0.678460
\(162\) 0 0
\(163\) −457.000 −0.219601 −0.109801 0.993954i \(-0.535021\pi\)
−0.109801 + 0.993954i \(0.535021\pi\)
\(164\) −432.000 −0.205692
\(165\) 0 0
\(166\) −2172.00 −1.01554
\(167\) −1164.00 −0.539359 −0.269680 0.962950i \(-0.586918\pi\)
−0.269680 + 0.962950i \(0.586918\pi\)
\(168\) 0 0
\(169\) 12.0000 0.00546199
\(170\) 390.000 0.175951
\(171\) 0 0
\(172\) 1196.00 0.530199
\(173\) 3942.00 1.73240 0.866199 0.499700i \(-0.166556\pi\)
0.866199 + 0.499700i \(0.166556\pi\)
\(174\) 0 0
\(175\) 350.000 0.151186
\(176\) 48.0000 0.0205576
\(177\) 0 0
\(178\) 240.000 0.101060
\(179\) −1212.00 −0.506085 −0.253042 0.967455i \(-0.581431\pi\)
−0.253042 + 0.967455i \(0.581431\pi\)
\(180\) 0 0
\(181\) 2288.00 0.939590 0.469795 0.882776i \(-0.344328\pi\)
0.469795 + 0.882776i \(0.344328\pi\)
\(182\) −1316.00 −0.535980
\(183\) 0 0
\(184\) 792.000 0.317321
\(185\) 1570.00 0.623939
\(186\) 0 0
\(187\) −117.000 −0.0457534
\(188\) 2124.00 0.823982
\(189\) 0 0
\(190\) −320.000 −0.122185
\(191\) −1938.00 −0.734182 −0.367091 0.930185i \(-0.619646\pi\)
−0.367091 + 0.930185i \(0.619646\pi\)
\(192\) 0 0
\(193\) −1498.00 −0.558696 −0.279348 0.960190i \(-0.590118\pi\)
−0.279348 + 0.960190i \(0.590118\pi\)
\(194\) 3284.00 1.21535
\(195\) 0 0
\(196\) −588.000 −0.214286
\(197\) −2124.00 −0.768166 −0.384083 0.923299i \(-0.625482\pi\)
−0.384083 + 0.923299i \(0.625482\pi\)
\(198\) 0 0
\(199\) −385.000 −0.137145 −0.0685727 0.997646i \(-0.521845\pi\)
−0.0685727 + 0.997646i \(0.521845\pi\)
\(200\) −200.000 −0.0707107
\(201\) 0 0
\(202\) −66.0000 −0.0229888
\(203\) 714.000 0.246862
\(204\) 0 0
\(205\) −540.000 −0.183977
\(206\) 2396.00 0.810375
\(207\) 0 0
\(208\) 752.000 0.250682
\(209\) 96.0000 0.0317725
\(210\) 0 0
\(211\) 3170.00 1.03427 0.517137 0.855903i \(-0.326998\pi\)
0.517137 + 0.855903i \(0.326998\pi\)
\(212\) 2256.00 0.730862
\(213\) 0 0
\(214\) 3084.00 0.985130
\(215\) 1495.00 0.474224
\(216\) 0 0
\(217\) 1162.00 0.363510
\(218\) 1112.00 0.345478
\(219\) 0 0
\(220\) 60.0000 0.0183873
\(221\) −1833.00 −0.557923
\(222\) 0 0
\(223\) 1388.00 0.416804 0.208402 0.978043i \(-0.433174\pi\)
0.208402 + 0.978043i \(0.433174\pi\)
\(224\) −448.000 −0.133631
\(225\) 0 0
\(226\) −3210.00 −0.944805
\(227\) −4644.00 −1.35786 −0.678928 0.734205i \(-0.737555\pi\)
−0.678928 + 0.734205i \(0.737555\pi\)
\(228\) 0 0
\(229\) 4736.00 1.36665 0.683327 0.730113i \(-0.260532\pi\)
0.683327 + 0.730113i \(0.260532\pi\)
\(230\) 990.000 0.283820
\(231\) 0 0
\(232\) −408.000 −0.115459
\(233\) 2814.00 0.791207 0.395604 0.918421i \(-0.370535\pi\)
0.395604 + 0.918421i \(0.370535\pi\)
\(234\) 0 0
\(235\) 2655.00 0.736992
\(236\) 48.0000 0.0132396
\(237\) 0 0
\(238\) 1092.00 0.297411
\(239\) −2202.00 −0.595965 −0.297982 0.954571i \(-0.596314\pi\)
−0.297982 + 0.954571i \(0.596314\pi\)
\(240\) 0 0
\(241\) 3485.00 0.931488 0.465744 0.884920i \(-0.345787\pi\)
0.465744 + 0.884920i \(0.345787\pi\)
\(242\) 2644.00 0.702325
\(243\) 0 0
\(244\) 920.000 0.241381
\(245\) −735.000 −0.191663
\(246\) 0 0
\(247\) 1504.00 0.387438
\(248\) −664.000 −0.170016
\(249\) 0 0
\(250\) −250.000 −0.0632456
\(251\) −6345.00 −1.59559 −0.797795 0.602929i \(-0.794000\pi\)
−0.797795 + 0.602929i \(0.794000\pi\)
\(252\) 0 0
\(253\) −297.000 −0.0738033
\(254\) −2668.00 −0.659076
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 525.000 0.127426 0.0637132 0.997968i \(-0.479706\pi\)
0.0637132 + 0.997968i \(0.479706\pi\)
\(258\) 0 0
\(259\) 4396.00 1.05465
\(260\) 940.000 0.224217
\(261\) 0 0
\(262\) 5766.00 1.35964
\(263\) 5196.00 1.21825 0.609124 0.793075i \(-0.291521\pi\)
0.609124 + 0.793075i \(0.291521\pi\)
\(264\) 0 0
\(265\) 2820.00 0.653703
\(266\) −896.000 −0.206531
\(267\) 0 0
\(268\) −1072.00 −0.244339
\(269\) −7479.00 −1.69518 −0.847589 0.530654i \(-0.821946\pi\)
−0.847589 + 0.530654i \(0.821946\pi\)
\(270\) 0 0
\(271\) −856.000 −0.191876 −0.0959378 0.995387i \(-0.530585\pi\)
−0.0959378 + 0.995387i \(0.530585\pi\)
\(272\) −624.000 −0.139101
\(273\) 0 0
\(274\) −564.000 −0.124352
\(275\) 75.0000 0.0164461
\(276\) 0 0
\(277\) −7054.00 −1.53009 −0.765043 0.643979i \(-0.777282\pi\)
−0.765043 + 0.643979i \(0.777282\pi\)
\(278\) 4988.00 1.07612
\(279\) 0 0
\(280\) −560.000 −0.119523
\(281\) 1014.00 0.215268 0.107634 0.994191i \(-0.465673\pi\)
0.107634 + 0.994191i \(0.465673\pi\)
\(282\) 0 0
\(283\) 992.000 0.208368 0.104184 0.994558i \(-0.466777\pi\)
0.104184 + 0.994558i \(0.466777\pi\)
\(284\) 480.000 0.100291
\(285\) 0 0
\(286\) −282.000 −0.0583042
\(287\) −1512.00 −0.310977
\(288\) 0 0
\(289\) −3392.00 −0.690413
\(290\) −510.000 −0.103270
\(291\) 0 0
\(292\) 4424.00 0.886627
\(293\) −4950.00 −0.986970 −0.493485 0.869754i \(-0.664277\pi\)
−0.493485 + 0.869754i \(0.664277\pi\)
\(294\) 0 0
\(295\) 60.0000 0.0118418
\(296\) −2512.00 −0.493267
\(297\) 0 0
\(298\) −5190.00 −1.00889
\(299\) −4653.00 −0.899966
\(300\) 0 0
\(301\) 4186.00 0.801585
\(302\) −2458.00 −0.468351
\(303\) 0 0
\(304\) 512.000 0.0965961
\(305\) 1150.00 0.215898
\(306\) 0 0
\(307\) −4777.00 −0.888071 −0.444035 0.896009i \(-0.646454\pi\)
−0.444035 + 0.896009i \(0.646454\pi\)
\(308\) 168.000 0.0310802
\(309\) 0 0
\(310\) −830.000 −0.152067
\(311\) −7692.00 −1.40249 −0.701243 0.712922i \(-0.747371\pi\)
−0.701243 + 0.712922i \(0.747371\pi\)
\(312\) 0 0
\(313\) −2932.00 −0.529477 −0.264739 0.964320i \(-0.585286\pi\)
−0.264739 + 0.964320i \(0.585286\pi\)
\(314\) 3182.00 0.571881
\(315\) 0 0
\(316\) −2956.00 −0.526228
\(317\) 8352.00 1.47980 0.739898 0.672720i \(-0.234874\pi\)
0.739898 + 0.672720i \(0.234874\pi\)
\(318\) 0 0
\(319\) 153.000 0.0268538
\(320\) 320.000 0.0559017
\(321\) 0 0
\(322\) 2772.00 0.479744
\(323\) −1248.00 −0.214986
\(324\) 0 0
\(325\) 1175.00 0.200545
\(326\) 914.000 0.155282
\(327\) 0 0
\(328\) 864.000 0.145446
\(329\) 7434.00 1.24574
\(330\) 0 0
\(331\) −3070.00 −0.509796 −0.254898 0.966968i \(-0.582042\pi\)
−0.254898 + 0.966968i \(0.582042\pi\)
\(332\) 4344.00 0.718096
\(333\) 0 0
\(334\) 2328.00 0.381385
\(335\) −1340.00 −0.218543
\(336\) 0 0
\(337\) −1672.00 −0.270266 −0.135133 0.990827i \(-0.543146\pi\)
−0.135133 + 0.990827i \(0.543146\pi\)
\(338\) −24.0000 −0.00386221
\(339\) 0 0
\(340\) −780.000 −0.124416
\(341\) 249.000 0.0395428
\(342\) 0 0
\(343\) −6860.00 −1.07990
\(344\) −2392.00 −0.374907
\(345\) 0 0
\(346\) −7884.00 −1.22499
\(347\) −5076.00 −0.785285 −0.392643 0.919691i \(-0.628439\pi\)
−0.392643 + 0.919691i \(0.628439\pi\)
\(348\) 0 0
\(349\) 8594.00 1.31813 0.659063 0.752087i \(-0.270953\pi\)
0.659063 + 0.752087i \(0.270953\pi\)
\(350\) −700.000 −0.106904
\(351\) 0 0
\(352\) −96.0000 −0.0145364
\(353\) 12711.0 1.91654 0.958269 0.285866i \(-0.0922813\pi\)
0.958269 + 0.285866i \(0.0922813\pi\)
\(354\) 0 0
\(355\) 600.000 0.0897034
\(356\) −480.000 −0.0714605
\(357\) 0 0
\(358\) 2424.00 0.357856
\(359\) −1464.00 −0.215228 −0.107614 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(360\) 0 0
\(361\) −5835.00 −0.850707
\(362\) −4576.00 −0.664390
\(363\) 0 0
\(364\) 2632.00 0.378995
\(365\) 5530.00 0.793023
\(366\) 0 0
\(367\) −7630.00 −1.08524 −0.542620 0.839979i \(-0.682567\pi\)
−0.542620 + 0.839979i \(0.682567\pi\)
\(368\) −1584.00 −0.224380
\(369\) 0 0
\(370\) −3140.00 −0.441191
\(371\) 7896.00 1.10496
\(372\) 0 0
\(373\) −3883.00 −0.539019 −0.269510 0.962998i \(-0.586862\pi\)
−0.269510 + 0.962998i \(0.586862\pi\)
\(374\) 234.000 0.0323525
\(375\) 0 0
\(376\) −4248.00 −0.582643
\(377\) 2397.00 0.327458
\(378\) 0 0
\(379\) −13768.0 −1.86600 −0.933001 0.359874i \(-0.882820\pi\)
−0.933001 + 0.359874i \(0.882820\pi\)
\(380\) 640.000 0.0863982
\(381\) 0 0
\(382\) 3876.00 0.519145
\(383\) −14139.0 −1.88634 −0.943171 0.332307i \(-0.892173\pi\)
−0.943171 + 0.332307i \(0.892173\pi\)
\(384\) 0 0
\(385\) 210.000 0.0277989
\(386\) 2996.00 0.395058
\(387\) 0 0
\(388\) −6568.00 −0.859381
\(389\) 567.000 0.0739024 0.0369512 0.999317i \(-0.488235\pi\)
0.0369512 + 0.999317i \(0.488235\pi\)
\(390\) 0 0
\(391\) 3861.00 0.499384
\(392\) 1176.00 0.151523
\(393\) 0 0
\(394\) 4248.00 0.543176
\(395\) −3695.00 −0.470672
\(396\) 0 0
\(397\) −6685.00 −0.845115 −0.422557 0.906336i \(-0.638867\pi\)
−0.422557 + 0.906336i \(0.638867\pi\)
\(398\) 770.000 0.0969764
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) 4572.00 0.569364 0.284682 0.958622i \(-0.408112\pi\)
0.284682 + 0.958622i \(0.408112\pi\)
\(402\) 0 0
\(403\) 3901.00 0.482190
\(404\) 132.000 0.0162556
\(405\) 0 0
\(406\) −1428.00 −0.174558
\(407\) 942.000 0.114725
\(408\) 0 0
\(409\) −25.0000 −0.00302242 −0.00151121 0.999999i \(-0.500481\pi\)
−0.00151121 + 0.999999i \(0.500481\pi\)
\(410\) 1080.00 0.130091
\(411\) 0 0
\(412\) −4792.00 −0.573022
\(413\) 168.000 0.0200163
\(414\) 0 0
\(415\) 5430.00 0.642285
\(416\) −1504.00 −0.177259
\(417\) 0 0
\(418\) −192.000 −0.0224666
\(419\) 12453.0 1.45195 0.725977 0.687719i \(-0.241388\pi\)
0.725977 + 0.687719i \(0.241388\pi\)
\(420\) 0 0
\(421\) 5048.00 0.584381 0.292191 0.956360i \(-0.405616\pi\)
0.292191 + 0.956360i \(0.405616\pi\)
\(422\) −6340.00 −0.731342
\(423\) 0 0
\(424\) −4512.00 −0.516797
\(425\) −975.000 −0.111281
\(426\) 0 0
\(427\) 3220.00 0.364934
\(428\) −6168.00 −0.696592
\(429\) 0 0
\(430\) −2990.00 −0.335327
\(431\) 5400.00 0.603501 0.301750 0.953387i \(-0.402429\pi\)
0.301750 + 0.953387i \(0.402429\pi\)
\(432\) 0 0
\(433\) −6298.00 −0.698990 −0.349495 0.936938i \(-0.613647\pi\)
−0.349495 + 0.936938i \(0.613647\pi\)
\(434\) −2324.00 −0.257040
\(435\) 0 0
\(436\) −2224.00 −0.244290
\(437\) −3168.00 −0.346787
\(438\) 0 0
\(439\) −6208.00 −0.674924 −0.337462 0.941339i \(-0.609568\pi\)
−0.337462 + 0.941339i \(0.609568\pi\)
\(440\) −120.000 −0.0130018
\(441\) 0 0
\(442\) 3666.00 0.394511
\(443\) −3360.00 −0.360358 −0.180179 0.983634i \(-0.557668\pi\)
−0.180179 + 0.983634i \(0.557668\pi\)
\(444\) 0 0
\(445\) −600.000 −0.0639162
\(446\) −2776.00 −0.294725
\(447\) 0 0
\(448\) 896.000 0.0944911
\(449\) 14394.0 1.51291 0.756453 0.654048i \(-0.226931\pi\)
0.756453 + 0.654048i \(0.226931\pi\)
\(450\) 0 0
\(451\) −324.000 −0.0338283
\(452\) 6420.00 0.668078
\(453\) 0 0
\(454\) 9288.00 0.960149
\(455\) 3290.00 0.338984
\(456\) 0 0
\(457\) −916.000 −0.0937608 −0.0468804 0.998901i \(-0.514928\pi\)
−0.0468804 + 0.998901i \(0.514928\pi\)
\(458\) −9472.00 −0.966370
\(459\) 0 0
\(460\) −1980.00 −0.200691
\(461\) 8550.00 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(462\) 0 0
\(463\) 3734.00 0.374803 0.187401 0.982283i \(-0.439993\pi\)
0.187401 + 0.982283i \(0.439993\pi\)
\(464\) 816.000 0.0816419
\(465\) 0 0
\(466\) −5628.00 −0.559468
\(467\) 9840.00 0.975034 0.487517 0.873113i \(-0.337903\pi\)
0.487517 + 0.873113i \(0.337903\pi\)
\(468\) 0 0
\(469\) −3752.00 −0.369406
\(470\) −5310.00 −0.521132
\(471\) 0 0
\(472\) −96.0000 −0.00936178
\(473\) 897.000 0.0871968
\(474\) 0 0
\(475\) 800.000 0.0772769
\(476\) −2184.00 −0.210301
\(477\) 0 0
\(478\) 4404.00 0.421411
\(479\) −17280.0 −1.64832 −0.824158 0.566360i \(-0.808351\pi\)
−0.824158 + 0.566360i \(0.808351\pi\)
\(480\) 0 0
\(481\) 14758.0 1.39897
\(482\) −6970.00 −0.658661
\(483\) 0 0
\(484\) −5288.00 −0.496619
\(485\) −8210.00 −0.768653
\(486\) 0 0
\(487\) −4588.00 −0.426904 −0.213452 0.976954i \(-0.568471\pi\)
−0.213452 + 0.976954i \(0.568471\pi\)
\(488\) −1840.00 −0.170682
\(489\) 0 0
\(490\) 1470.00 0.135526
\(491\) 636.000 0.0584568 0.0292284 0.999573i \(-0.490695\pi\)
0.0292284 + 0.999573i \(0.490695\pi\)
\(492\) 0 0
\(493\) −1989.00 −0.181704
\(494\) −3008.00 −0.273960
\(495\) 0 0
\(496\) 1328.00 0.120220
\(497\) 1680.00 0.151626
\(498\) 0 0
\(499\) −11716.0 −1.05106 −0.525531 0.850774i \(-0.676133\pi\)
−0.525531 + 0.850774i \(0.676133\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) 12690.0 1.12825
\(503\) 4653.00 0.412459 0.206230 0.978504i \(-0.433881\pi\)
0.206230 + 0.978504i \(0.433881\pi\)
\(504\) 0 0
\(505\) 165.000 0.0145394
\(506\) 594.000 0.0521868
\(507\) 0 0
\(508\) 5336.00 0.466037
\(509\) 16479.0 1.43501 0.717504 0.696555i \(-0.245285\pi\)
0.717504 + 0.696555i \(0.245285\pi\)
\(510\) 0 0
\(511\) 15484.0 1.34045
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −1050.00 −0.0901041
\(515\) −5990.00 −0.512526
\(516\) 0 0
\(517\) 1593.00 0.135513
\(518\) −8792.00 −0.745750
\(519\) 0 0
\(520\) −1880.00 −0.158545
\(521\) 3120.00 0.262360 0.131180 0.991359i \(-0.458123\pi\)
0.131180 + 0.991359i \(0.458123\pi\)
\(522\) 0 0
\(523\) 17645.0 1.47526 0.737631 0.675204i \(-0.235944\pi\)
0.737631 + 0.675204i \(0.235944\pi\)
\(524\) −11532.0 −0.961408
\(525\) 0 0
\(526\) −10392.0 −0.861431
\(527\) −3237.00 −0.267563
\(528\) 0 0
\(529\) −2366.00 −0.194460
\(530\) −5640.00 −0.462238
\(531\) 0 0
\(532\) 1792.00 0.146040
\(533\) −5076.00 −0.412507
\(534\) 0 0
\(535\) −7710.00 −0.623051
\(536\) 2144.00 0.172774
\(537\) 0 0
\(538\) 14958.0 1.19867
\(539\) −441.000 −0.0352416
\(540\) 0 0
\(541\) −2182.00 −0.173404 −0.0867019 0.996234i \(-0.527633\pi\)
−0.0867019 + 0.996234i \(0.527633\pi\)
\(542\) 1712.00 0.135677
\(543\) 0 0
\(544\) 1248.00 0.0983595
\(545\) −2780.00 −0.218499
\(546\) 0 0
\(547\) −4033.00 −0.315244 −0.157622 0.987499i \(-0.550383\pi\)
−0.157622 + 0.987499i \(0.550383\pi\)
\(548\) 1128.00 0.0879302
\(549\) 0 0
\(550\) −150.000 −0.0116291
\(551\) 1632.00 0.126181
\(552\) 0 0
\(553\) −10346.0 −0.795582
\(554\) 14108.0 1.08193
\(555\) 0 0
\(556\) −9976.00 −0.760929
\(557\) −960.000 −0.0730278 −0.0365139 0.999333i \(-0.511625\pi\)
−0.0365139 + 0.999333i \(0.511625\pi\)
\(558\) 0 0
\(559\) 14053.0 1.06329
\(560\) 1120.00 0.0845154
\(561\) 0 0
\(562\) −2028.00 −0.152217
\(563\) −23754.0 −1.77817 −0.889087 0.457739i \(-0.848660\pi\)
−0.889087 + 0.457739i \(0.848660\pi\)
\(564\) 0 0
\(565\) 8025.00 0.597547
\(566\) −1984.00 −0.147339
\(567\) 0 0
\(568\) −960.000 −0.0709167
\(569\) −22536.0 −1.66038 −0.830192 0.557478i \(-0.811769\pi\)
−0.830192 + 0.557478i \(0.811769\pi\)
\(570\) 0 0
\(571\) 17726.0 1.29914 0.649571 0.760301i \(-0.274949\pi\)
0.649571 + 0.760301i \(0.274949\pi\)
\(572\) 564.000 0.0412273
\(573\) 0 0
\(574\) 3024.00 0.219894
\(575\) −2475.00 −0.179504
\(576\) 0 0
\(577\) 17168.0 1.23867 0.619336 0.785126i \(-0.287402\pi\)
0.619336 + 0.785126i \(0.287402\pi\)
\(578\) 6784.00 0.488196
\(579\) 0 0
\(580\) 1020.00 0.0730227
\(581\) 15204.0 1.08566
\(582\) 0 0
\(583\) 1692.00 0.120198
\(584\) −8848.00 −0.626940
\(585\) 0 0
\(586\) 9900.00 0.697893
\(587\) 7542.00 0.530309 0.265155 0.964206i \(-0.414577\pi\)
0.265155 + 0.964206i \(0.414577\pi\)
\(588\) 0 0
\(589\) 2656.00 0.185804
\(590\) −120.000 −0.00837343
\(591\) 0 0
\(592\) 5024.00 0.348792
\(593\) −15543.0 −1.07635 −0.538174 0.842834i \(-0.680886\pi\)
−0.538174 + 0.842834i \(0.680886\pi\)
\(594\) 0 0
\(595\) −2730.00 −0.188099
\(596\) 10380.0 0.713392
\(597\) 0 0
\(598\) 9306.00 0.636372
\(599\) 16026.0 1.09316 0.546581 0.837406i \(-0.315929\pi\)
0.546581 + 0.837406i \(0.315929\pi\)
\(600\) 0 0
\(601\) 10469.0 0.710548 0.355274 0.934762i \(-0.384388\pi\)
0.355274 + 0.934762i \(0.384388\pi\)
\(602\) −8372.00 −0.566806
\(603\) 0 0
\(604\) 4916.00 0.331174
\(605\) −6610.00 −0.444190
\(606\) 0 0
\(607\) −8074.00 −0.539891 −0.269945 0.962876i \(-0.587006\pi\)
−0.269945 + 0.962876i \(0.587006\pi\)
\(608\) −1024.00 −0.0683038
\(609\) 0 0
\(610\) −2300.00 −0.152663
\(611\) 24957.0 1.65246
\(612\) 0 0
\(613\) 26855.0 1.76943 0.884717 0.466128i \(-0.154351\pi\)
0.884717 + 0.466128i \(0.154351\pi\)
\(614\) 9554.00 0.627961
\(615\) 0 0
\(616\) −336.000 −0.0219770
\(617\) 24447.0 1.59514 0.797568 0.603229i \(-0.206119\pi\)
0.797568 + 0.603229i \(0.206119\pi\)
\(618\) 0 0
\(619\) 1850.00 0.120126 0.0600628 0.998195i \(-0.480870\pi\)
0.0600628 + 0.998195i \(0.480870\pi\)
\(620\) 1660.00 0.107528
\(621\) 0 0
\(622\) 15384.0 0.991708
\(623\) −1680.00 −0.108038
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 5864.00 0.374397
\(627\) 0 0
\(628\) −6364.00 −0.404381
\(629\) −12246.0 −0.776280
\(630\) 0 0
\(631\) 21728.0 1.37081 0.685403 0.728164i \(-0.259626\pi\)
0.685403 + 0.728164i \(0.259626\pi\)
\(632\) 5912.00 0.372099
\(633\) 0 0
\(634\) −16704.0 −1.04637
\(635\) 6670.00 0.416836
\(636\) 0 0
\(637\) −6909.00 −0.429740
\(638\) −306.000 −0.0189885
\(639\) 0 0
\(640\) −640.000 −0.0395285
\(641\) 23862.0 1.47035 0.735173 0.677879i \(-0.237101\pi\)
0.735173 + 0.677879i \(0.237101\pi\)
\(642\) 0 0
\(643\) 10523.0 0.645391 0.322696 0.946503i \(-0.395411\pi\)
0.322696 + 0.946503i \(0.395411\pi\)
\(644\) −5544.00 −0.339230
\(645\) 0 0
\(646\) 2496.00 0.152018
\(647\) 5484.00 0.333228 0.166614 0.986022i \(-0.446717\pi\)
0.166614 + 0.986022i \(0.446717\pi\)
\(648\) 0 0
\(649\) 36.0000 0.00217739
\(650\) −2350.00 −0.141807
\(651\) 0 0
\(652\) −1828.00 −0.109801
\(653\) −26784.0 −1.60511 −0.802557 0.596576i \(-0.796527\pi\)
−0.802557 + 0.596576i \(0.796527\pi\)
\(654\) 0 0
\(655\) −14415.0 −0.859909
\(656\) −1728.00 −0.102846
\(657\) 0 0
\(658\) −14868.0 −0.880874
\(659\) −12120.0 −0.716431 −0.358216 0.933639i \(-0.616615\pi\)
−0.358216 + 0.933639i \(0.616615\pi\)
\(660\) 0 0
\(661\) −18226.0 −1.07248 −0.536240 0.844066i \(-0.680156\pi\)
−0.536240 + 0.844066i \(0.680156\pi\)
\(662\) 6140.00 0.360480
\(663\) 0 0
\(664\) −8688.00 −0.507771
\(665\) 2240.00 0.130622
\(666\) 0 0
\(667\) −5049.00 −0.293101
\(668\) −4656.00 −0.269680
\(669\) 0 0
\(670\) 2680.00 0.154533
\(671\) 690.000 0.0396977
\(672\) 0 0
\(673\) −11062.0 −0.633594 −0.316797 0.948493i \(-0.602607\pi\)
−0.316797 + 0.948493i \(0.602607\pi\)
\(674\) 3344.00 0.191107
\(675\) 0 0
\(676\) 48.0000 0.00273100
\(677\) −9348.00 −0.530684 −0.265342 0.964154i \(-0.585485\pi\)
−0.265342 + 0.964154i \(0.585485\pi\)
\(678\) 0 0
\(679\) −22988.0 −1.29926
\(680\) 1560.00 0.0879754
\(681\) 0 0
\(682\) −498.000 −0.0279610
\(683\) 19248.0 1.07834 0.539169 0.842198i \(-0.318739\pi\)
0.539169 + 0.842198i \(0.318739\pi\)
\(684\) 0 0
\(685\) 1410.00 0.0786472
\(686\) 13720.0 0.763604
\(687\) 0 0
\(688\) 4784.00 0.265099
\(689\) 26508.0 1.46571
\(690\) 0 0
\(691\) −17710.0 −0.974993 −0.487496 0.873125i \(-0.662090\pi\)
−0.487496 + 0.873125i \(0.662090\pi\)
\(692\) 15768.0 0.866199
\(693\) 0 0
\(694\) 10152.0 0.555280
\(695\) −12470.0 −0.680596
\(696\) 0 0
\(697\) 4212.00 0.228897
\(698\) −17188.0 −0.932056
\(699\) 0 0
\(700\) 1400.00 0.0755929
\(701\) −19437.0 −1.04725 −0.523627 0.851947i \(-0.675422\pi\)
−0.523627 + 0.851947i \(0.675422\pi\)
\(702\) 0 0
\(703\) 10048.0 0.539072
\(704\) 192.000 0.0102788
\(705\) 0 0
\(706\) −25422.0 −1.35520
\(707\) 462.000 0.0245761
\(708\) 0 0
\(709\) −19516.0 −1.03376 −0.516882 0.856057i \(-0.672907\pi\)
−0.516882 + 0.856057i \(0.672907\pi\)
\(710\) −1200.00 −0.0634299
\(711\) 0 0
\(712\) 960.000 0.0505302
\(713\) −8217.00 −0.431598
\(714\) 0 0
\(715\) 705.000 0.0368748
\(716\) −4848.00 −0.253042
\(717\) 0 0
\(718\) 2928.00 0.152189
\(719\) 17358.0 0.900340 0.450170 0.892943i \(-0.351363\pi\)
0.450170 + 0.892943i \(0.351363\pi\)
\(720\) 0 0
\(721\) −16772.0 −0.866327
\(722\) 11670.0 0.601541
\(723\) 0 0
\(724\) 9152.00 0.469795
\(725\) 1275.00 0.0653135
\(726\) 0 0
\(727\) 24428.0 1.24620 0.623098 0.782144i \(-0.285874\pi\)
0.623098 + 0.782144i \(0.285874\pi\)
\(728\) −5264.00 −0.267990
\(729\) 0 0
\(730\) −11060.0 −0.560752
\(731\) −11661.0 −0.590010
\(732\) 0 0
\(733\) −21418.0 −1.07925 −0.539626 0.841905i \(-0.681434\pi\)
−0.539626 + 0.841905i \(0.681434\pi\)
\(734\) 15260.0 0.767380
\(735\) 0 0
\(736\) 3168.00 0.158660
\(737\) −804.000 −0.0401842
\(738\) 0 0
\(739\) −664.000 −0.0330523 −0.0165261 0.999863i \(-0.505261\pi\)
−0.0165261 + 0.999863i \(0.505261\pi\)
\(740\) 6280.00 0.311969
\(741\) 0 0
\(742\) −15792.0 −0.781324
\(743\) −34209.0 −1.68911 −0.844553 0.535471i \(-0.820134\pi\)
−0.844553 + 0.535471i \(0.820134\pi\)
\(744\) 0 0
\(745\) 12975.0 0.638077
\(746\) 7766.00 0.381144
\(747\) 0 0
\(748\) −468.000 −0.0228767
\(749\) −21588.0 −1.05315
\(750\) 0 0
\(751\) 6857.00 0.333176 0.166588 0.986027i \(-0.446725\pi\)
0.166588 + 0.986027i \(0.446725\pi\)
\(752\) 8496.00 0.411991
\(753\) 0 0
\(754\) −4794.00 −0.231548
\(755\) 6145.00 0.296211
\(756\) 0 0
\(757\) −23719.0 −1.13881 −0.569407 0.822056i \(-0.692827\pi\)
−0.569407 + 0.822056i \(0.692827\pi\)
\(758\) 27536.0 1.31946
\(759\) 0 0
\(760\) −1280.00 −0.0610927
\(761\) −14418.0 −0.686796 −0.343398 0.939190i \(-0.611578\pi\)
−0.343398 + 0.939190i \(0.611578\pi\)
\(762\) 0 0
\(763\) −7784.00 −0.369331
\(764\) −7752.00 −0.367091
\(765\) 0 0
\(766\) 28278.0 1.33385
\(767\) 564.000 0.0265513
\(768\) 0 0
\(769\) −4849.00 −0.227385 −0.113693 0.993516i \(-0.536268\pi\)
−0.113693 + 0.993516i \(0.536268\pi\)
\(770\) −420.000 −0.0196568
\(771\) 0 0
\(772\) −5992.00 −0.279348
\(773\) −36258.0 −1.68708 −0.843538 0.537070i \(-0.819531\pi\)
−0.843538 + 0.537070i \(0.819531\pi\)
\(774\) 0 0
\(775\) 2075.00 0.0961757
\(776\) 13136.0 0.607674
\(777\) 0 0
\(778\) −1134.00 −0.0522569
\(779\) −3456.00 −0.158953
\(780\) 0 0
\(781\) 360.000 0.0164940
\(782\) −7722.00 −0.353118
\(783\) 0 0
\(784\) −2352.00 −0.107143
\(785\) −7955.00 −0.361689
\(786\) 0 0
\(787\) −18877.0 −0.855009 −0.427505 0.904013i \(-0.640607\pi\)
−0.427505 + 0.904013i \(0.640607\pi\)
\(788\) −8496.00 −0.384083
\(789\) 0 0
\(790\) 7390.00 0.332816
\(791\) 22470.0 1.01004
\(792\) 0 0
\(793\) 10810.0 0.484079
\(794\) 13370.0 0.597586
\(795\) 0 0
\(796\) −1540.00 −0.0685727
\(797\) −16200.0 −0.719992 −0.359996 0.932954i \(-0.617222\pi\)
−0.359996 + 0.932954i \(0.617222\pi\)
\(798\) 0 0
\(799\) −20709.0 −0.916936
\(800\) −800.000 −0.0353553
\(801\) 0 0
\(802\) −9144.00 −0.402601
\(803\) 3318.00 0.145815
\(804\) 0 0
\(805\) −6930.00 −0.303417
\(806\) −7802.00 −0.340960
\(807\) 0 0
\(808\) −264.000 −0.0114944
\(809\) −26760.0 −1.16296 −0.581478 0.813562i \(-0.697525\pi\)
−0.581478 + 0.813562i \(0.697525\pi\)
\(810\) 0 0
\(811\) −10510.0 −0.455063 −0.227531 0.973771i \(-0.573065\pi\)
−0.227531 + 0.973771i \(0.573065\pi\)
\(812\) 2856.00 0.123431
\(813\) 0 0
\(814\) −1884.00 −0.0811231
\(815\) −2285.00 −0.0982087
\(816\) 0 0
\(817\) 9568.00 0.409721
\(818\) 50.0000 0.00213717
\(819\) 0 0
\(820\) −2160.00 −0.0919884
\(821\) −28230.0 −1.20004 −0.600021 0.799985i \(-0.704841\pi\)
−0.600021 + 0.799985i \(0.704841\pi\)
\(822\) 0 0
\(823\) −39868.0 −1.68859 −0.844296 0.535877i \(-0.819981\pi\)
−0.844296 + 0.535877i \(0.819981\pi\)
\(824\) 9584.00 0.405187
\(825\) 0 0
\(826\) −336.000 −0.0141537
\(827\) 32394.0 1.36209 0.681046 0.732241i \(-0.261525\pi\)
0.681046 + 0.732241i \(0.261525\pi\)
\(828\) 0 0
\(829\) 34820.0 1.45880 0.729402 0.684085i \(-0.239798\pi\)
0.729402 + 0.684085i \(0.239798\pi\)
\(830\) −10860.0 −0.454164
\(831\) 0 0
\(832\) 3008.00 0.125341
\(833\) 5733.00 0.238459
\(834\) 0 0
\(835\) −5820.00 −0.241209
\(836\) 384.000 0.0158863
\(837\) 0 0
\(838\) −24906.0 −1.02669
\(839\) −1146.00 −0.0471565 −0.0235783 0.999722i \(-0.507506\pi\)
−0.0235783 + 0.999722i \(0.507506\pi\)
\(840\) 0 0
\(841\) −21788.0 −0.893354
\(842\) −10096.0 −0.413220
\(843\) 0 0
\(844\) 12680.0 0.517137
\(845\) 60.0000 0.00244268
\(846\) 0 0
\(847\) −18508.0 −0.750817
\(848\) 9024.00 0.365431
\(849\) 0 0
\(850\) 1950.00 0.0786876
\(851\) −31086.0 −1.25219
\(852\) 0 0
\(853\) −19393.0 −0.778433 −0.389217 0.921146i \(-0.627254\pi\)
−0.389217 + 0.921146i \(0.627254\pi\)
\(854\) −6440.00 −0.258047
\(855\) 0 0
\(856\) 12336.0 0.492565
\(857\) −8430.00 −0.336013 −0.168007 0.985786i \(-0.553733\pi\)
−0.168007 + 0.985786i \(0.553733\pi\)
\(858\) 0 0
\(859\) 15470.0 0.614470 0.307235 0.951634i \(-0.400596\pi\)
0.307235 + 0.951634i \(0.400596\pi\)
\(860\) 5980.00 0.237112
\(861\) 0 0
\(862\) −10800.0 −0.426740
\(863\) 5871.00 0.231577 0.115789 0.993274i \(-0.463060\pi\)
0.115789 + 0.993274i \(0.463060\pi\)
\(864\) 0 0
\(865\) 19710.0 0.774752
\(866\) 12596.0 0.494260
\(867\) 0 0
\(868\) 4648.00 0.181755
\(869\) −2217.00 −0.0865438
\(870\) 0 0
\(871\) −12596.0 −0.490011
\(872\) 4448.00 0.172739
\(873\) 0 0
\(874\) 6336.00 0.245216
\(875\) 1750.00 0.0676123
\(876\) 0 0
\(877\) −11299.0 −0.435051 −0.217526 0.976055i \(-0.569799\pi\)
−0.217526 + 0.976055i \(0.569799\pi\)
\(878\) 12416.0 0.477243
\(879\) 0 0
\(880\) 240.000 0.00919363
\(881\) 29682.0 1.13509 0.567544 0.823343i \(-0.307894\pi\)
0.567544 + 0.823343i \(0.307894\pi\)
\(882\) 0 0
\(883\) 40316.0 1.53651 0.768257 0.640142i \(-0.221124\pi\)
0.768257 + 0.640142i \(0.221124\pi\)
\(884\) −7332.00 −0.278961
\(885\) 0 0
\(886\) 6720.00 0.254811
\(887\) −21945.0 −0.830711 −0.415356 0.909659i \(-0.636343\pi\)
−0.415356 + 0.909659i \(0.636343\pi\)
\(888\) 0 0
\(889\) 18676.0 0.704581
\(890\) 1200.00 0.0451956
\(891\) 0 0
\(892\) 5552.00 0.208402
\(893\) 16992.0 0.636748
\(894\) 0 0
\(895\) −6060.00 −0.226328
\(896\) −1792.00 −0.0668153
\(897\) 0 0
\(898\) −28788.0 −1.06979
\(899\) 4233.00 0.157039
\(900\) 0 0
\(901\) −21996.0 −0.813311
\(902\) 648.000 0.0239202
\(903\) 0 0
\(904\) −12840.0 −0.472403
\(905\) 11440.0 0.420197
\(906\) 0 0
\(907\) 24911.0 0.911969 0.455985 0.889988i \(-0.349287\pi\)
0.455985 + 0.889988i \(0.349287\pi\)
\(908\) −18576.0 −0.678928
\(909\) 0 0
\(910\) −6580.00 −0.239698
\(911\) 33264.0 1.20975 0.604877 0.796319i \(-0.293222\pi\)
0.604877 + 0.796319i \(0.293222\pi\)
\(912\) 0 0
\(913\) 3258.00 0.118099
\(914\) 1832.00 0.0662989
\(915\) 0 0
\(916\) 18944.0 0.683327
\(917\) −40362.0 −1.45351
\(918\) 0 0
\(919\) −23191.0 −0.832427 −0.416214 0.909267i \(-0.636643\pi\)
−0.416214 + 0.909267i \(0.636643\pi\)
\(920\) 3960.00 0.141910
\(921\) 0 0
\(922\) −17100.0 −0.610801
\(923\) 5640.00 0.201130
\(924\) 0 0
\(925\) 7850.00 0.279034
\(926\) −7468.00 −0.265026
\(927\) 0 0
\(928\) −1632.00 −0.0577296
\(929\) 2160.00 0.0762834 0.0381417 0.999272i \(-0.487856\pi\)
0.0381417 + 0.999272i \(0.487856\pi\)
\(930\) 0 0
\(931\) −4704.00 −0.165593
\(932\) 11256.0 0.395604
\(933\) 0 0
\(934\) −19680.0 −0.689453
\(935\) −585.000 −0.0204615
\(936\) 0 0
\(937\) 2066.00 0.0720312 0.0360156 0.999351i \(-0.488533\pi\)
0.0360156 + 0.999351i \(0.488533\pi\)
\(938\) 7504.00 0.261209
\(939\) 0 0
\(940\) 10620.0 0.368496
\(941\) −22233.0 −0.770218 −0.385109 0.922871i \(-0.625836\pi\)
−0.385109 + 0.922871i \(0.625836\pi\)
\(942\) 0 0
\(943\) 10692.0 0.369225
\(944\) 192.000 0.00661978
\(945\) 0 0
\(946\) −1794.00 −0.0616575
\(947\) −17754.0 −0.609216 −0.304608 0.952478i \(-0.598525\pi\)
−0.304608 + 0.952478i \(0.598525\pi\)
\(948\) 0 0
\(949\) 51982.0 1.77809
\(950\) −1600.00 −0.0546430
\(951\) 0 0
\(952\) 4368.00 0.148706
\(953\) 33891.0 1.15198 0.575990 0.817457i \(-0.304617\pi\)
0.575990 + 0.817457i \(0.304617\pi\)
\(954\) 0 0
\(955\) −9690.00 −0.328336
\(956\) −8808.00 −0.297982
\(957\) 0 0
\(958\) 34560.0 1.16554
\(959\) 3948.00 0.132938
\(960\) 0 0
\(961\) −22902.0 −0.768756
\(962\) −29516.0 −0.989225
\(963\) 0 0
\(964\) 13940.0 0.465744
\(965\) −7490.00 −0.249857
\(966\) 0 0
\(967\) 51074.0 1.69848 0.849239 0.528008i \(-0.177061\pi\)
0.849239 + 0.528008i \(0.177061\pi\)
\(968\) 10576.0 0.351163
\(969\) 0 0
\(970\) 16420.0 0.543520
\(971\) 20967.0 0.692959 0.346479 0.938058i \(-0.387377\pi\)
0.346479 + 0.938058i \(0.387377\pi\)
\(972\) 0 0
\(973\) −34916.0 −1.15042
\(974\) 9176.00 0.301867
\(975\) 0 0
\(976\) 3680.00 0.120691
\(977\) 31749.0 1.03965 0.519826 0.854272i \(-0.325997\pi\)
0.519826 + 0.854272i \(0.325997\pi\)
\(978\) 0 0
\(979\) −360.000 −0.0117525
\(980\) −2940.00 −0.0958315
\(981\) 0 0
\(982\) −1272.00 −0.0413352
\(983\) 47325.0 1.53554 0.767769 0.640727i \(-0.221367\pi\)
0.767769 + 0.640727i \(0.221367\pi\)
\(984\) 0 0
\(985\) −10620.0 −0.343534
\(986\) 3978.00 0.128484
\(987\) 0 0
\(988\) 6016.00 0.193719
\(989\) −29601.0 −0.951726
\(990\) 0 0
\(991\) 2363.00 0.0757449 0.0378724 0.999283i \(-0.487942\pi\)
0.0378724 + 0.999283i \(0.487942\pi\)
\(992\) −2656.00 −0.0850081
\(993\) 0 0
\(994\) −3360.00 −0.107216
\(995\) −1925.00 −0.0613333
\(996\) 0 0
\(997\) 45569.0 1.44753 0.723764 0.690048i \(-0.242411\pi\)
0.723764 + 0.690048i \(0.242411\pi\)
\(998\) 23432.0 0.743213
\(999\) 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.f.1.1 1
3.2 odd 2 270.4.a.j.1.1 yes 1
4.3 odd 2 2160.4.a.l.1.1 1
5.2 odd 4 1350.4.c.k.649.1 2
5.3 odd 4 1350.4.c.k.649.2 2
5.4 even 2 1350.4.a.r.1.1 1
9.2 odd 6 810.4.e.f.271.1 2
9.4 even 3 810.4.e.n.541.1 2
9.5 odd 6 810.4.e.f.541.1 2
9.7 even 3 810.4.e.n.271.1 2
12.11 even 2 2160.4.a.b.1.1 1
15.2 even 4 1350.4.c.j.649.2 2
15.8 even 4 1350.4.c.j.649.1 2
15.14 odd 2 1350.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.f.1.1 1 1.1 even 1 trivial
270.4.a.j.1.1 yes 1 3.2 odd 2
810.4.e.f.271.1 2 9.2 odd 6
810.4.e.f.541.1 2 9.5 odd 6
810.4.e.n.271.1 2 9.7 even 3
810.4.e.n.541.1 2 9.4 even 3
1350.4.a.e.1.1 1 15.14 odd 2
1350.4.a.r.1.1 1 5.4 even 2
1350.4.c.j.649.1 2 15.8 even 4
1350.4.c.j.649.2 2 15.2 even 4
1350.4.c.k.649.1 2 5.2 odd 4
1350.4.c.k.649.2 2 5.3 odd 4
2160.4.a.b.1.1 1 12.11 even 2
2160.4.a.l.1.1 1 4.3 odd 2