Properties

Label 1350.4.a.r.1.1
Level $1350$
Weight $4$
Character 1350.1
Self dual yes
Analytic conductor $79.653$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.6525785077\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1350.1

$q$-expansion

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

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
\(5\) 0 0
\(6\) 0 0
\(7\) −14.0000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 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 \(-0.667168\pi\)
−0.501364 + 0.865237i \(0.667168\pi\)
\(14\) −28.0000 −0.534522
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 39.0000 0.556405 0.278203 0.960522i \(-0.410261\pi\)
0.278203 + 0.960522i \(0.410261\pi\)
\(18\) 0 0
\(19\) 32.0000 0.386384 0.193192 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 0.0581456
\(23\) 99.0000 0.897519 0.448759 0.893653i \(-0.351866\pi\)
0.448759 + 0.893653i \(0.351866\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −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
\(35\) 0 0
\(36\) 0 0
\(37\) −314.000 −1.39517 −0.697585 0.716502i \(-0.745742\pi\)
−0.697585 + 0.716502i \(0.745742\pi\)
\(38\) 64.0000 0.273215
\(39\) 0 0
\(40\) 0 0
\(41\) −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 \(-0.677883\pi\)
−0.530199 + 0.847874i \(0.677883\pi\)
\(44\) 12.0000 0.0411152
\(45\) 0 0
\(46\) 198.000 0.634641
\(47\) −531.000 −1.64796 −0.823982 0.566616i \(-0.808252\pi\)
−0.823982 + 0.566616i \(0.808252\pi\)
\(48\) 0 0
\(49\) −147.000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −188.000 −0.501364
\(53\) −564.000 −1.46172 −0.730862 0.682525i \(-0.760882\pi\)
−0.730862 + 0.682525i \(0.760882\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −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
\(65\) 0 0
\(66\) 0 0
\(67\) 268.000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 156.000 0.278203
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(-0.846958\pi\)
−0.886627 + 0.462486i \(0.846958\pi\)
\(74\) −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 \(0.676394\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −216.000 −0.290893
\(83\) −1086.00 −1.43619 −0.718096 0.695944i \(-0.754986\pi\)
−0.718096 + 0.695944i \(0.754986\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −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
\(95\) 0 0
\(96\) 0 0
\(97\) 1642.00 1.71876 0.859381 0.511336i \(-0.170849\pi\)
0.859381 + 0.511336i \(0.170849\pi\)
\(98\) −294.000 −0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 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 \(-0.305771\pi\)
0.573022 + 0.819540i \(0.305771\pi\)
\(104\) −376.000 −0.354518
\(105\) 0 0
\(106\) −1128.00 −1.03359
\(107\) 1542.00 1.39318 0.696592 0.717467i \(-0.254699\pi\)
0.696592 + 0.717467i \(0.254699\pi\)
\(108\) 0 0
\(109\) −556.000 −0.488579 −0.244290 0.969702i \(-0.578555\pi\)
−0.244290 + 0.969702i \(0.578555\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −224.000 −0.188982
\(113\) −1605.00 −1.33616 −0.668078 0.744091i \(-0.732883\pi\)
−0.668078 + 0.744091i \(0.732883\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) −1334.00 −0.932074 −0.466037 0.884765i \(-0.654319\pi\)
−0.466037 + 0.884765i \(0.654319\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −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 \(-0.528025\pi\)
−0.0879302 + 0.996127i \(0.528025\pi\)
\(138\) 0 0
\(139\) −2494.00 −1.52186 −0.760929 0.648835i \(-0.775257\pi\)
−0.760929 + 0.648835i \(0.775257\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 240.000 0.141833
\(143\) −141.000 −0.0824546
\(144\) 0 0
\(145\) 0 0
\(146\) −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
\(155\) 0 0
\(156\) 0 0
\(157\) 1591.00 0.808762 0.404381 0.914591i \(-0.367487\pi\)
0.404381 + 0.914591i \(0.367487\pi\)
\(158\) −1478.00 −0.744199
\(159\) 0 0
\(160\) 0 0
\(161\) −1386.00 −0.678460
\(162\) 0 0
\(163\) 457.000 0.219601 0.109801 0.993954i \(-0.464979\pi\)
0.109801 + 0.993954i \(0.464979\pi\)
\(164\) −432.000 −0.205692
\(165\) 0 0
\(166\) −2172.00 −1.01554
\(167\) 1164.00 0.539359 0.269680 0.962950i \(-0.413082\pi\)
0.269680 + 0.962950i \(0.413082\pi\)
\(168\) 0 0
\(169\) 12.0000 0.00546199
\(170\) 0 0
\(171\) 0 0
\(172\) −1196.00 −0.530199
\(173\) −3942.00 −1.73240 −0.866199 0.499700i \(-0.833444\pi\)
−0.866199 + 0.499700i \(0.833444\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 0 0
\(187\) 117.000 0.0457534
\(188\) −2124.00 −0.823982
\(189\) 0 0
\(190\) 0 0
\(191\) −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 \(-0.409882\pi\)
0.279348 + 0.960190i \(0.409882\pi\)
\(194\) 3284.00 1.21535
\(195\) 0 0
\(196\) −588.000 −0.214286
\(197\) 2124.00 0.768166 0.384083 0.923299i \(-0.374518\pi\)
0.384083 + 0.923299i \(0.374518\pi\)
\(198\) 0 0
\(199\) −385.000 −0.137145 −0.0685727 0.997646i \(-0.521845\pi\)
−0.0685727 + 0.997646i \(0.521845\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 66.0000 0.0229888
\(203\) −714.000 −0.246862
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 0 0
\(217\) −1162.00 −0.363510
\(218\) −1112.00 −0.345478
\(219\) 0 0
\(220\) 0 0
\(221\) −1833.00 −0.557923
\(222\) 0 0
\(223\) −1388.00 −0.416804 −0.208402 0.978043i \(-0.566826\pi\)
−0.208402 + 0.978043i \(0.566826\pi\)
\(224\) −448.000 −0.133631
\(225\) 0 0
\(226\) −3210.00 −0.944805
\(227\) 4644.00 1.35786 0.678928 0.734205i \(-0.262445\pi\)
0.678928 + 0.734205i \(0.262445\pi\)
\(228\) 0 0
\(229\) 4736.00 1.36665 0.683327 0.730113i \(-0.260532\pi\)
0.683327 + 0.730113i \(0.260532\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 408.000 0.115459
\(233\) −2814.00 −0.791207 −0.395604 0.918421i \(-0.629465\pi\)
−0.395604 + 0.918421i \(0.629465\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −1504.00 −0.387438
\(248\) 664.000 0.170016
\(249\) 0 0
\(250\) 0 0
\(251\) −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 \(-0.520294\pi\)
−0.0637132 + 0.997968i \(0.520294\pi\)
\(258\) 0 0
\(259\) 4396.00 1.05465
\(260\) 0 0
\(261\) 0 0
\(262\) −5766.00 −1.35964
\(263\) −5196.00 −1.21825 −0.609124 0.793075i \(-0.708479\pi\)
−0.609124 + 0.793075i \(0.708479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −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
\(275\) 0 0
\(276\) 0 0
\(277\) 7054.00 1.53009 0.765043 0.643979i \(-0.222718\pi\)
0.765043 + 0.643979i \(0.222718\pi\)
\(278\) −4988.00 −1.07612
\(279\) 0 0
\(280\) 0 0
\(281\) 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 \(-0.533223\pi\)
−0.104184 + 0.994558i \(0.533223\pi\)
\(284\) 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
\(290\) 0 0
\(291\) 0 0
\(292\) −4424.00 −0.886627
\(293\) 4950.00 0.986970 0.493485 0.869754i \(-0.335723\pi\)
0.493485 + 0.869754i \(0.335723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −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
\(305\) 0 0
\(306\) 0 0
\(307\) 4777.00 0.888071 0.444035 0.896009i \(-0.353546\pi\)
0.444035 + 0.896009i \(0.353546\pi\)
\(308\) −168.000 −0.0310802
\(309\) 0 0
\(310\) 0 0
\(311\) −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 \(-0.414714\pi\)
0.264739 + 0.964320i \(0.414714\pi\)
\(314\) 3182.00 0.571881
\(315\) 0 0
\(316\) −2956.00 −0.526228
\(317\) −8352.00 −1.47980 −0.739898 0.672720i \(-0.765126\pi\)
−0.739898 + 0.672720i \(0.765126\pi\)
\(318\) 0 0
\(319\) 153.000 0.0268538
\(320\) 0 0
\(321\) 0 0
\(322\) −2772.00 −0.479744
\(323\) 1248.00 0.214986
\(324\) 0 0
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) 1672.00 0.270266 0.135133 0.990827i \(-0.456854\pi\)
0.135133 + 0.990827i \(0.456854\pi\)
\(338\) 24.0000 0.00386221
\(339\) 0 0
\(340\) 0 0
\(341\) 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 \(-0.371561\pi\)
0.392643 + 0.919691i \(0.371561\pi\)
\(348\) 0 0
\(349\) 8594.00 1.31813 0.659063 0.752087i \(-0.270953\pi\)
0.659063 + 0.752087i \(0.270953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 96.0000 0.0145364
\(353\) −12711.0 −1.91654 −0.958269 0.285866i \(-0.907719\pi\)
−0.958269 + 0.285866i \(0.907719\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −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
\(365\) 0 0
\(366\) 0 0
\(367\) 7630.00 1.08524 0.542620 0.839979i \(-0.317433\pi\)
0.542620 + 0.839979i \(0.317433\pi\)
\(368\) 1584.00 0.224380
\(369\) 0 0
\(370\) 0 0
\(371\) 7896.00 1.10496
\(372\) 0 0
\(373\) 3883.00 0.539019 0.269510 0.962998i \(-0.413138\pi\)
0.269510 + 0.962998i \(0.413138\pi\)
\(374\) 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 \(0.882820\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3876.00 −0.519145
\(383\) 14139.0 1.88634 0.943171 0.332307i \(-0.107827\pi\)
0.943171 + 0.332307i \(0.107827\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) 6685.00 0.845115 0.422557 0.906336i \(-0.361133\pi\)
0.422557 + 0.906336i \(0.361133\pi\)
\(398\) −770.000 −0.0969764
\(399\) 0 0
\(400\) 0 0
\(401\) 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 \(0.500481\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4792.00 0.573022
\(413\) −168.000 −0.0200163
\(414\) 0 0
\(415\) 0 0
\(416\) −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
\(425\) 0 0
\(426\) 0 0
\(427\) −3220.00 −0.364934
\(428\) 6168.00 0.696592
\(429\) 0 0
\(430\) 0 0
\(431\) 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 \(-0.386353\pi\)
0.349495 + 0.936938i \(0.386353\pi\)
\(434\) −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 \(0.609568\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3666.00 −0.394511
\(443\) 3360.00 0.360358 0.180179 0.983634i \(-0.442332\pi\)
0.180179 + 0.983634i \(0.442332\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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
\(455\) 0 0
\(456\) 0 0
\(457\) 916.000 0.0937608 0.0468804 0.998901i \(-0.485072\pi\)
0.0468804 + 0.998901i \(0.485072\pi\)
\(458\) 9472.00 0.966370
\(459\) 0 0
\(460\) 0 0
\(461\) 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 \(-0.560007\pi\)
−0.187401 + 0.982283i \(0.560007\pi\)
\(464\) 816.000 0.0816419
\(465\) 0 0
\(466\) −5628.00 −0.559468
\(467\) −9840.00 −0.975034 −0.487517 0.873113i \(-0.662097\pi\)
−0.487517 + 0.873113i \(0.662097\pi\)
\(468\) 0 0
\(469\) −3752.00 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 96.0000 0.00936178
\(473\) −897.000 −0.0871968
\(474\) 0 0
\(475\) 0 0
\(476\) −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
\(485\) 0 0
\(486\) 0 0
\(487\) 4588.00 0.426904 0.213452 0.976954i \(-0.431529\pi\)
0.213452 + 0.976954i \(0.431529\pi\)
\(488\) 1840.00 0.170682
\(489\) 0 0
\(490\) 0 0
\(491\) 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 \(0.676133\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −12690.0 −1.12825
\(503\) −4653.00 −0.412459 −0.206230 0.978504i \(-0.566119\pi\)
−0.206230 + 0.978504i \(0.566119\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 0 0
\(517\) −1593.00 −0.135513
\(518\) 8792.00 0.745750
\(519\) 0 0
\(520\) 0 0
\(521\) 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 \(-0.764056\pi\)
−0.737631 + 0.675204i \(0.764056\pi\)
\(524\) −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
\(530\) 0 0
\(531\) 0 0
\(532\) −1792.00 −0.146040
\(533\) 5076.00 0.412507
\(534\) 0 0
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 4033.00 0.315244 0.157622 0.987499i \(-0.449617\pi\)
0.157622 + 0.987499i \(0.449617\pi\)
\(548\) −1128.00 −0.0879302
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.488375\pi\)
0.0365139 + 0.999333i \(0.488375\pi\)
\(558\) 0 0
\(559\) 14053.0 1.06329
\(560\) 0 0
\(561\) 0 0
\(562\) 2028.00 0.152217
\(563\) 23754.0 1.77817 0.889087 0.457739i \(-0.151340\pi\)
0.889087 + 0.457739i \(0.151340\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −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
\(575\) 0 0
\(576\) 0 0
\(577\) −17168.0 −1.23867 −0.619336 0.785126i \(-0.712598\pi\)
−0.619336 + 0.785126i \(0.712598\pi\)
\(578\) −6784.00 −0.488196
\(579\) 0 0
\(580\) 0 0
\(581\) 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 \(-0.585423\pi\)
−0.265155 + 0.964206i \(0.585423\pi\)
\(588\) 0 0
\(589\) 2656.00 0.185804
\(590\) 0 0
\(591\) 0 0
\(592\) −5024.00 −0.348792
\(593\) 15543.0 1.07635 0.538174 0.842834i \(-0.319114\pi\)
0.538174 + 0.842834i \(0.319114\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 8074.00 0.539891 0.269945 0.962876i \(-0.412994\pi\)
0.269945 + 0.962876i \(0.412994\pi\)
\(608\) 1024.00 0.0683038
\(609\) 0 0
\(610\) 0 0
\(611\) 24957.0 1.65246
\(612\) 0 0
\(613\) −26855.0 −1.76943 −0.884717 0.466128i \(-0.845649\pi\)
−0.884717 + 0.466128i \(0.845649\pi\)
\(614\) 9554.00 0.627961
\(615\) 0 0
\(616\) −336.000 −0.0219770
\(617\) −24447.0 −1.59514 −0.797568 0.603229i \(-0.793881\pi\)
−0.797568 + 0.603229i \(0.793881\pi\)
\(618\) 0 0
\(619\) 1850.00 0.120126 0.0600628 0.998195i \(-0.480870\pi\)
0.0600628 + 0.998195i \(0.480870\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −15384.0 −0.991708
\(623\) 1680.00 0.108038
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 0 0
\(637\) 6909.00 0.429740
\(638\) 306.000 0.0189885
\(639\) 0 0
\(640\) 0 0
\(641\) 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 \(-0.604589\pi\)
−0.322696 + 0.946503i \(0.604589\pi\)
\(644\) −5544.00 −0.339230
\(645\) 0 0
\(646\) 2496.00 0.152018
\(647\) −5484.00 −0.333228 −0.166614 0.986022i \(-0.553283\pi\)
−0.166614 + 0.986022i \(0.553283\pi\)
\(648\) 0 0
\(649\) 36.0000 0.00217739
\(650\) 0 0
\(651\) 0 0
\(652\) 1828.00 0.109801
\(653\) 26784.0 1.60511 0.802557 0.596576i \(-0.203473\pi\)
0.802557 + 0.596576i \(0.203473\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −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
\(665\) 0 0
\(666\) 0 0
\(667\) 5049.00 0.293101
\(668\) 4656.00 0.269680
\(669\) 0 0
\(670\) 0 0
\(671\) 690.000 0.0396977
\(672\) 0 0
\(673\) 11062.0 0.633594 0.316797 0.948493i \(-0.397393\pi\)
0.316797 + 0.948493i \(0.397393\pi\)
\(674\) 3344.00 0.191107
\(675\) 0 0
\(676\) 48.0000 0.00273100
\(677\) 9348.00 0.530684 0.265342 0.964154i \(-0.414515\pi\)
0.265342 + 0.964154i \(0.414515\pi\)
\(678\) 0 0
\(679\) −22988.0 −1.29926
\(680\) 0 0
\(681\) 0 0
\(682\) 498.000 0.0279610
\(683\) −19248.0 −1.07834 −0.539169 0.842198i \(-0.681261\pi\)
−0.539169 + 0.842198i \(0.681261\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) −4212.00 −0.228897
\(698\) 17188.0 0.932056
\(699\) 0 0
\(700\) 0 0
\(701\) −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 \(0.672907\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −960.000 −0.0505302
\(713\) 8217.00 0.431598
\(714\) 0 0
\(715\) 0 0
\(716\) −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
\(725\) 0 0
\(726\) 0 0
\(727\) −24428.0 −1.24620 −0.623098 0.782144i \(-0.714126\pi\)
−0.623098 + 0.782144i \(0.714126\pi\)
\(728\) 5264.00 0.267990
\(729\) 0 0
\(730\) 0 0
\(731\) −11661.0 −0.590010
\(732\) 0 0
\(733\) 21418.0 1.07925 0.539626 0.841905i \(-0.318566\pi\)
0.539626 + 0.841905i \(0.318566\pi\)
\(734\) 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 \(0.505261\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15792.0 0.781324
\(743\) 34209.0 1.68911 0.844553 0.535471i \(-0.179866\pi\)
0.844553 + 0.535471i \(0.179866\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 23719.0 1.13881 0.569407 0.822056i \(-0.307173\pi\)
0.569407 + 0.822056i \(0.307173\pi\)
\(758\) −27536.0 −1.31946
\(759\) 0 0
\(760\) 0 0
\(761\) −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 \(0.536268\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5992.00 0.279348
\(773\) 36258.0 1.68708 0.843538 0.537070i \(-0.180469\pi\)
0.843538 + 0.537070i \(0.180469\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 18877.0 0.855009 0.427505 0.904013i \(-0.359393\pi\)
0.427505 + 0.904013i \(0.359393\pi\)
\(788\) 8496.00 0.384083
\(789\) 0 0
\(790\) 0 0
\(791\) 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 \(-0.382778\pi\)
0.359996 + 0.932954i \(0.382778\pi\)
\(798\) 0 0
\(799\) −20709.0 −0.916936
\(800\) 0 0
\(801\) 0 0
\(802\) 9144.00 0.402601
\(803\) −3318.00 −0.145815
\(804\) 0 0
\(805\) 0 0
\(806\) −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
\(815\) 0 0
\(816\) 0 0
\(817\) −9568.00 −0.409721
\(818\) −50.0000 −0.00213717
\(819\) 0 0
\(820\) 0 0
\(821\) −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 \(-0.180019\pi\)
0.844296 + 0.535877i \(0.180019\pi\)
\(824\) 9584.00 0.405187
\(825\) 0 0
\(826\) −336.000 −0.0141537
\(827\) −32394.0 −1.36209 −0.681046 0.732241i \(-0.738475\pi\)
−0.681046 + 0.732241i \(0.738475\pi\)
\(828\) 0 0
\(829\) 34820.0 1.45880 0.729402 0.684085i \(-0.239798\pi\)
0.729402 + 0.684085i \(0.239798\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3008.00 −0.125341
\(833\) −5733.00 −0.238459
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 0 0
\(847\) 18508.0 0.750817
\(848\) −9024.00 −0.365431
\(849\) 0 0
\(850\) 0 0
\(851\) −31086.0 −1.25219
\(852\) 0 0
\(853\) 19393.0 0.778433 0.389217 0.921146i \(-0.372746\pi\)
0.389217 + 0.921146i \(0.372746\pi\)
\(854\) −6440.00 −0.258047
\(855\) 0 0
\(856\) 12336.0 0.492565
\(857\) 8430.00 0.336013 0.168007 0.985786i \(-0.446267\pi\)
0.168007 + 0.985786i \(0.446267\pi\)
\(858\) 0 0
\(859\) 15470.0 0.614470 0.307235 0.951634i \(-0.400596\pi\)
0.307235 + 0.951634i \(0.400596\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10800.0 0.426740
\(863\) −5871.00 −0.231577 −0.115789 0.993274i \(-0.536940\pi\)
−0.115789 + 0.993274i \(0.536940\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) 11299.0 0.435051 0.217526 0.976055i \(-0.430201\pi\)
0.217526 + 0.976055i \(0.430201\pi\)
\(878\) −12416.0 −0.477243
\(879\) 0 0
\(880\) 0 0
\(881\) 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 \(-0.778876\pi\)
−0.768257 + 0.640142i \(0.778876\pi\)
\(884\) −7332.00 −0.278961
\(885\) 0 0
\(886\) 6720.00 0.254811
\(887\) 21945.0 0.830711 0.415356 0.909659i \(-0.363657\pi\)
0.415356 + 0.909659i \(0.363657\pi\)
\(888\) 0 0
\(889\) 18676.0 0.704581
\(890\) 0 0
\(891\) 0 0
\(892\) −5552.00 −0.208402
\(893\) −16992.0 −0.636748
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) −24911.0 −0.911969 −0.455985 0.889988i \(-0.650713\pi\)
−0.455985 + 0.889988i \(0.650713\pi\)
\(908\) 18576.0 0.678928
\(909\) 0 0
\(910\) 0 0
\(911\) 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 \(0.636643\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17100.0 0.610801
\(923\) −5640.00 −0.201130
\(924\) 0 0
\(925\) 0 0
\(926\) −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
\(935\) 0 0
\(936\) 0 0
\(937\) −2066.00 −0.0720312 −0.0360156 0.999351i \(-0.511467\pi\)
−0.0360156 + 0.999351i \(0.511467\pi\)
\(938\) −7504.00 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −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 \(-0.401475\pi\)
0.304608 + 0.952478i \(0.401475\pi\)
\(948\) 0 0
\(949\) 51982.0 1.77809
\(950\) 0 0
\(951\) 0 0
\(952\) −4368.00 −0.148706
\(953\) −33891.0 −1.15198 −0.575990 0.817457i \(-0.695383\pi\)
−0.575990 + 0.817457i \(0.695383\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −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
\(965\) 0 0
\(966\) 0 0
\(967\) −51074.0 −1.69848 −0.849239 0.528008i \(-0.822939\pi\)
−0.849239 + 0.528008i \(0.822939\pi\)
\(968\) −10576.0 −0.351163
\(969\) 0 0
\(970\) 0 0
\(971\) 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 \(-0.674003\pi\)
−0.519826 + 0.854272i \(0.674003\pi\)
\(978\) 0 0
\(979\) −360.000 −0.0117525
\(980\) 0 0
\(981\) 0 0
\(982\) 1272.00 0.0413352
\(983\) −47325.0 −1.53554 −0.767769 0.640727i \(-0.778633\pi\)
−0.767769 + 0.640727i \(0.778633\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) −45569.0 −1.44753 −0.723764 0.690048i \(-0.757589\pi\)
−0.723764 + 0.690048i \(0.757589\pi\)
\(998\) −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 1350.4.a.r.1.1 1
3.2 odd 2 1350.4.a.e.1.1 1
5.2 odd 4 1350.4.c.k.649.2 2
5.3 odd 4 1350.4.c.k.649.1 2
5.4 even 2 270.4.a.f.1.1 1
15.2 even 4 1350.4.c.j.649.1 2
15.8 even 4 1350.4.c.j.649.2 2
15.14 odd 2 270.4.a.j.1.1 yes 1
20.19 odd 2 2160.4.a.l.1.1 1
45.4 even 6 810.4.e.n.541.1 2
45.14 odd 6 810.4.e.f.541.1 2
45.29 odd 6 810.4.e.f.271.1 2
45.34 even 6 810.4.e.n.271.1 2
60.59 even 2 2160.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.f.1.1 1 5.4 even 2
270.4.a.j.1.1 yes 1 15.14 odd 2
810.4.e.f.271.1 2 45.29 odd 6
810.4.e.f.541.1 2 45.14 odd 6
810.4.e.n.271.1 2 45.34 even 6
810.4.e.n.541.1 2 45.4 even 6
1350.4.a.e.1.1 1 3.2 odd 2
1350.4.a.r.1.1 1 1.1 even 1 trivial
1350.4.c.j.649.1 2 15.2 even 4
1350.4.c.j.649.2 2 15.8 even 4
1350.4.c.k.649.1 2 5.3 odd 4
1350.4.c.k.649.2 2 5.2 odd 4
2160.4.a.b.1.1 1 60.59 even 2
2160.4.a.l.1.1 1 20.19 odd 2