Properties

Label 1350.4.a.g.1.1
Level $1350$
Weight $4$
Character 1350.1
Self dual yes
Analytic conductor $79.653$
Analytic rank $0$
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: \(0\)
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

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

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
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 42.0000 1.15123 0.575613 0.817723i \(-0.304764\pi\)
0.575613 + 0.817723i \(0.304764\pi\)
\(12\) 0 0
\(13\) −20.0000 −0.426692 −0.213346 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(14\) −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 \(0.384073\pi\)
\(20\) 0 0
\(21\) 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
\(25\) 0 0
\(26\) 40.0000 0.301717
\(27\) 0 0
\(28\) 16.0000 0.107990
\(29\) 120.000 0.768395 0.384197 0.923251i \(-0.374478\pi\)
0.384197 + 0.923251i \(0.374478\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 262.000 1.16412 0.582061 0.813145i \(-0.302246\pi\)
0.582061 + 0.813145i \(0.302246\pi\)
\(38\) −118.000 −0.503740
\(39\) 0 0
\(40\) 0 0
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) 178.000 0.631273 0.315637 0.948880i \(-0.397782\pi\)
0.315637 + 0.948880i \(0.397782\pi\)
\(44\) 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
\(50\) 0 0
\(51\) 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
\(55\) 0 0
\(56\) −32.0000 −0.0763604
\(57\) 0 0
\(58\) −240.000 −0.543337
\(59\) −444.000 −0.979727 −0.489863 0.871799i \(-0.662953\pi\)
−0.489863 + 0.871799i \(0.662953\pi\)
\(60\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) 538.000 0.981002 0.490501 0.871441i \(-0.336814\pi\)
0.490501 + 0.871441i \(0.336814\pi\)
\(68\) −372.000 −0.663406
\(69\) 0 0
\(70\) 0 0
\(71\) 690.000 1.15335 0.576676 0.816973i \(-0.304350\pi\)
0.576676 + 0.816973i \(0.304350\pi\)
\(72\) 0 0
\(73\) 1126.00 1.80532 0.902660 0.430355i \(-0.141612\pi\)
0.902660 + 0.430355i \(0.141612\pi\)
\(74\) −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 \(0.342978\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) −356.000 −0.446378
\(87\) 0 0
\(88\) −336.000 −0.407020
\(89\) −1086.00 −1.29344 −0.646718 0.762729i \(-0.723859\pi\)
−0.646718 + 0.762729i \(0.723859\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) −36.0000 −0.0407963
\(93\) 0 0
\(94\) 288.000 0.316010
\(95\) 0 0
\(96\) 0 0
\(97\) −1544.00 −1.61618 −0.808090 0.589059i \(-0.799499\pi\)
−0.808090 + 0.589059i \(0.799499\pi\)
\(98\) 654.000 0.674122
\(99\) 0 0
\(100\) 0 0
\(101\) −132.000 −0.130044 −0.0650222 0.997884i \(-0.520712\pi\)
−0.0650222 + 0.997884i \(0.520712\pi\)
\(102\) 0 0
\(103\) 892.000 0.853314 0.426657 0.904413i \(-0.359691\pi\)
0.426657 + 0.904413i \(0.359691\pi\)
\(104\) 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 \(0.775933\pi\)
\(110\) 0 0
\(111\) 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
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) −686.000 −0.479312 −0.239656 0.970858i \(-0.577035\pi\)
−0.239656 + 0.970858i \(0.577035\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −114.000 −0.0760323 −0.0380161 0.999277i \(-0.512104\pi\)
−0.0380161 + 0.999277i \(0.512104\pi\)
\(132\) 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 \(0.255663\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1380.00 −0.815542
\(143\) −840.000 −0.491219
\(144\) 0 0
\(145\) 0 0
\(146\) −2252.00 −1.27655
\(147\) 0 0
\(148\) 1048.00 0.582061
\(149\) −1398.00 −0.768648 −0.384324 0.923198i \(-0.625566\pi\)
−0.384324 + 0.923198i \(0.625566\pi\)
\(150\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 394.000 0.200284 0.100142 0.994973i \(-0.468070\pi\)
0.100142 + 0.994973i \(0.468070\pi\)
\(158\) −1330.00 −0.669678
\(159\) 0 0
\(160\) 0 0
\(161\) −36.0000 −0.0176223
\(162\) 0 0
\(163\) 3346.00 1.60785 0.803923 0.594733i \(-0.202742\pi\)
0.803923 + 0.594733i \(0.202742\pi\)
\(164\) 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
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) 672.000 0.287806
\(177\) 0 0
\(178\) 2172.00 0.914597
\(179\) −2640.00 −1.10236 −0.551181 0.834386i \(-0.685823\pi\)
−0.551181 + 0.834386i \(0.685823\pi\)
\(180\) 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
\(185\) 0 0
\(186\) 0 0
\(187\) −3906.00 −1.52746
\(188\) −576.000 −0.223453
\(189\) 0 0
\(190\) 0 0
\(191\) 1470.00 0.556887 0.278444 0.960453i \(-0.410181\pi\)
0.278444 + 0.960453i \(0.410181\pi\)
\(192\) 0 0
\(193\) 4720.00 1.76038 0.880189 0.474623i \(-0.157416\pi\)
0.880189 + 0.474623i \(0.157416\pi\)
\(194\) 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 \(0.462038\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 264.000 0.0919553
\(203\) 480.000 0.165958
\(204\) 0 0
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) 0 0
\(217\) 188.000 0.0588123
\(218\) 3470.00 1.07806
\(219\) 0 0
\(220\) 0 0
\(221\) 1860.00 0.566141
\(222\) 0 0
\(223\) 2158.00 0.648029 0.324014 0.946052i \(-0.394967\pi\)
0.324014 + 0.946052i \(0.394967\pi\)
\(224\) −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 \(0.405525\pi\)
\(230\) 0 0
\(231\) 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
\(235\) 0 0
\(236\) −1776.00 −0.489863
\(237\) 0 0
\(238\) 744.000 0.202632
\(239\) 6414.00 1.73593 0.867965 0.496626i \(-0.165428\pi\)
0.867965 + 0.496626i \(0.165428\pi\)
\(240\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −1180.00 −0.303974
\(248\) −376.000 −0.0962743
\(249\) 0 0
\(250\) 0 0
\(251\) 7308.00 1.83776 0.918878 0.394541i \(-0.129096\pi\)
0.918878 + 0.394541i \(0.129096\pi\)
\(252\) 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
\(260\) 0 0
\(261\) 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
\(265\) 0 0
\(266\) −472.000 −0.108798
\(267\) 0 0
\(268\) 2152.00 0.490501
\(269\) 990.000 0.224392 0.112196 0.993686i \(-0.464212\pi\)
0.112196 + 0.993686i \(0.464212\pi\)
\(270\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 1366.00 0.296300 0.148150 0.988965i \(-0.452668\pi\)
0.148150 + 0.988965i \(0.452668\pi\)
\(278\) −4552.00 −0.982053
\(279\) 0 0
\(280\) 0 0
\(281\) 5520.00 1.17187 0.585935 0.810358i \(-0.300727\pi\)
0.585935 + 0.810358i \(0.300727\pi\)
\(282\) 0 0
\(283\) −5438.00 −1.14225 −0.571123 0.820865i \(-0.693492\pi\)
−0.571123 + 0.820865i \(0.693492\pi\)
\(284\) 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
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) −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
\(305\) 0 0
\(306\) 0 0
\(307\) −9290.00 −1.72706 −0.863531 0.504295i \(-0.831752\pi\)
−0.863531 + 0.504295i \(0.831752\pi\)
\(308\) 672.000 0.124321
\(309\) 0 0
\(310\) 0 0
\(311\) −8112.00 −1.47907 −0.739533 0.673121i \(-0.764953\pi\)
−0.739533 + 0.673121i \(0.764953\pi\)
\(312\) 0 0
\(313\) 7900.00 1.42663 0.713314 0.700845i \(-0.247193\pi\)
0.713314 + 0.700845i \(0.247193\pi\)
\(314\) −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
\(320\) 0 0
\(321\) 0 0
\(322\) 72.0000 0.0124609
\(323\) −5487.00 −0.945216
\(324\) 0 0
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) 9556.00 1.54465 0.772327 0.635225i \(-0.219093\pi\)
0.772327 + 0.635225i \(0.219093\pi\)
\(338\) 3594.00 0.578366
\(339\) 0 0
\(340\) 0 0
\(341\) 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 \(0.673221\pi\)
\(350\) 0 0
\(351\) 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
\(355\) 0 0
\(356\) −4344.00 −0.646718
\(357\) 0 0
\(358\) 5280.00 0.779488
\(359\) −8778.00 −1.29049 −0.645244 0.763977i \(-0.723244\pi\)
−0.645244 + 0.763977i \(0.723244\pi\)
\(360\) 0 0
\(361\) −3378.00 −0.492492
\(362\) −2146.00 −0.311578
\(363\) 0 0
\(364\) −320.000 −0.0460785
\(365\) 0 0
\(366\) 0 0
\(367\) −956.000 −0.135975 −0.0679875 0.997686i \(-0.521658\pi\)
−0.0679875 + 0.997686i \(0.521658\pi\)
\(368\) −144.000 −0.0203981
\(369\) 0 0
\(370\) 0 0
\(371\) −2964.00 −0.414780
\(372\) 0 0
\(373\) −2300.00 −0.319275 −0.159637 0.987176i \(-0.551032\pi\)
−0.159637 + 0.987176i \(0.551032\pi\)
\(374\) 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 \(0.499374\pi\)
\(380\) 0 0
\(381\) 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
\(385\) 0 0
\(386\) −9440.00 −1.24478
\(387\) 0 0
\(388\) −6176.00 −0.808090
\(389\) 7938.00 1.03463 0.517317 0.855794i \(-0.326931\pi\)
0.517317 + 0.855794i \(0.326931\pi\)
\(390\) 0 0
\(391\) 837.000 0.108258
\(392\) 2616.00 0.337061
\(393\) 0 0
\(394\) −1530.00 −0.195635
\(395\) 0 0
\(396\) 0 0
\(397\) −272.000 −0.0343861 −0.0171931 0.999852i \(-0.505473\pi\)
−0.0171931 + 0.999852i \(0.505473\pi\)
\(398\) −1336.00 −0.168260
\(399\) 0 0
\(400\) 0 0
\(401\) −4554.00 −0.567122 −0.283561 0.958954i \(-0.591516\pi\)
−0.283561 + 0.958954i \(0.591516\pi\)
\(402\) 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 \(0.480728\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3568.00 0.426657
\(413\) −1776.00 −0.211601
\(414\) 0 0
\(415\) 0 0
\(416\) 640.000 0.0754293
\(417\) 0 0
\(418\) −4956.00 −0.579918
\(419\) 1794.00 0.209171 0.104585 0.994516i \(-0.466648\pi\)
0.104585 + 0.994516i \(0.466648\pi\)
\(420\) 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
\(425\) 0 0
\(426\) 0 0
\(427\) 884.000 0.100187
\(428\) 4560.00 0.514990
\(429\) 0 0
\(430\) 0 0
\(431\) 13356.0 1.49266 0.746329 0.665577i \(-0.231814\pi\)
0.746329 + 0.665577i \(0.231814\pi\)
\(432\) 0 0
\(433\) 11500.0 1.27634 0.638169 0.769896i \(-0.279692\pi\)
0.638169 + 0.769896i \(0.279692\pi\)
\(434\) −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 \(0.707247\pi\)
\(440\) 0 0
\(441\) 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
\(445\) 0 0
\(446\) −4316.00 −0.458225
\(447\) 0 0
\(448\) 256.000 0.0269975
\(449\) −18048.0 −1.89697 −0.948483 0.316828i \(-0.897382\pi\)
−0.948483 + 0.316828i \(0.897382\pi\)
\(450\) 0 0
\(451\) 5292.00 0.552529
\(452\) 5736.00 0.596900
\(453\) 0 0
\(454\) 6246.00 0.645681
\(455\) 0 0
\(456\) 0 0
\(457\) 4264.00 0.436458 0.218229 0.975898i \(-0.429972\pi\)
0.218229 + 0.975898i \(0.429972\pi\)
\(458\) −4054.00 −0.413605
\(459\) 0 0
\(460\) 0 0
\(461\) 10242.0 1.03475 0.517373 0.855760i \(-0.326910\pi\)
0.517373 + 0.855760i \(0.326910\pi\)
\(462\) 0 0
\(463\) −3302.00 −0.331441 −0.165720 0.986173i \(-0.552995\pi\)
−0.165720 + 0.986173i \(0.552995\pi\)
\(464\) 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
\(470\) 0 0
\(471\) 0 0
\(472\) 3552.00 0.346386
\(473\) 7476.00 0.726738
\(474\) 0 0
\(475\) 0 0
\(476\) −1488.00 −0.143282
\(477\) 0 0
\(478\) −12828.0 −1.22749
\(479\) −15246.0 −1.45430 −0.727148 0.686481i \(-0.759154\pi\)
−0.727148 + 0.686481i \(0.759154\pi\)
\(480\) 0 0
\(481\) −5240.00 −0.496722
\(482\) −6862.00 −0.648455
\(483\) 0 0
\(484\) 1732.00 0.162660
\(485\) 0 0
\(486\) 0 0
\(487\) 8206.00 0.763551 0.381776 0.924255i \(-0.375313\pi\)
0.381776 + 0.924255i \(0.375313\pi\)
\(488\) −1768.00 −0.164003
\(489\) 0 0
\(490\) 0 0
\(491\) 16806.0 1.54469 0.772346 0.635202i \(-0.219083\pi\)
0.772346 + 0.635202i \(0.219083\pi\)
\(492\) 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 \(0.578244\pi\)
\(500\) 0 0
\(501\) 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
\(505\) 0 0
\(506\) 756.000 0.0664196
\(507\) 0 0
\(508\) −2744.00 −0.239656
\(509\) 14724.0 1.28218 0.641090 0.767466i \(-0.278482\pi\)
0.641090 + 0.767466i \(0.278482\pi\)
\(510\) 0 0
\(511\) 4504.00 0.389912
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −7458.00 −0.639997
\(515\) 0 0
\(516\) 0 0
\(517\) −6048.00 −0.514489
\(518\) −2096.00 −0.177786
\(519\) 0 0
\(520\) 0 0
\(521\) −2058.00 −0.173057 −0.0865284 0.996249i \(-0.527577\pi\)
−0.0865284 + 0.996249i \(0.527577\pi\)
\(522\) 0 0
\(523\) −11912.0 −0.995938 −0.497969 0.867195i \(-0.665921\pi\)
−0.497969 + 0.867195i \(0.665921\pi\)
\(524\) −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
\(530\) 0 0
\(531\) 0 0
\(532\) 944.000 0.0769316
\(533\) −2520.00 −0.204790
\(534\) 0 0
\(535\) 0 0
\(536\) −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
\(545\) 0 0
\(546\) 0 0
\(547\) 4186.00 0.327204 0.163602 0.986526i \(-0.447689\pi\)
0.163602 + 0.986526i \(0.447689\pi\)
\(548\) −636.000 −0.0495777
\(549\) 0 0
\(550\) 0 0
\(551\) 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
\(560\) 0 0
\(561\) 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
\(565\) 0 0
\(566\) 10876.0 0.807690
\(567\) 0 0
\(568\) −5520.00 −0.407771
\(569\) 15372.0 1.13256 0.566281 0.824212i \(-0.308382\pi\)
0.566281 + 0.824212i \(0.308382\pi\)
\(570\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) 1066.00 0.0769119 0.0384559 0.999260i \(-0.487756\pi\)
0.0384559 + 0.999260i \(0.487756\pi\)
\(578\) −7472.00 −0.537706
\(579\) 0 0
\(580\) 0 0
\(581\) −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
\(590\) 0 0
\(591\) 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
\(595\) 0 0
\(596\) −5592.00 −0.384324
\(597\) 0 0
\(598\) −360.000 −0.0246179
\(599\) 18228.0 1.24337 0.621683 0.783269i \(-0.286449\pi\)
0.621683 + 0.783269i \(0.286449\pi\)
\(600\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 24418.0 1.63278 0.816389 0.577503i \(-0.195973\pi\)
0.816389 + 0.577503i \(0.195973\pi\)
\(608\) −1888.00 −0.125935
\(609\) 0 0
\(610\) 0 0
\(611\) 2880.00 0.190691
\(612\) 0 0
\(613\) −2672.00 −0.176054 −0.0880270 0.996118i \(-0.528056\pi\)
−0.0880270 + 0.996118i \(0.528056\pi\)
\(614\) 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 \(0.256821\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16224.0 1.04586
\(623\) −4344.00 −0.279356
\(624\) 0 0
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) 0 0
\(637\) 6540.00 0.406788
\(638\) −10080.0 −0.625503
\(639\) 0 0
\(640\) 0 0
\(641\) 4416.00 0.272108 0.136054 0.990701i \(-0.456558\pi\)
0.136054 + 0.990701i \(0.456558\pi\)
\(642\) 0 0
\(643\) −7580.00 −0.464893 −0.232446 0.972609i \(-0.574673\pi\)
−0.232446 + 0.972609i \(0.574673\pi\)
\(644\) −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
\(650\) 0 0
\(651\) 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
\(655\) 0 0
\(656\) 2016.00 0.119987
\(657\) 0 0
\(658\) 1152.00 0.0682517
\(659\) −28128.0 −1.66269 −0.831344 0.555758i \(-0.812428\pi\)
−0.831344 + 0.555758i \(0.812428\pi\)
\(660\) 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
\(665\) 0 0
\(666\) 0 0
\(667\) −1080.00 −0.0626953
\(668\) 5964.00 0.345440
\(669\) 0 0
\(670\) 0 0
\(671\) 9282.00 0.534020
\(672\) 0 0
\(673\) −29708.0 −1.70157 −0.850787 0.525511i \(-0.823874\pi\)
−0.850787 + 0.525511i \(0.823874\pi\)
\(674\) −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
\(680\) 0 0
\(681\) 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
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) −11718.0 −0.636802
\(698\) 13502.0 0.732175
\(699\) 0 0
\(700\) 0 0
\(701\) 6450.00 0.347522 0.173761 0.984788i \(-0.444408\pi\)
0.173761 + 0.984788i \(0.444408\pi\)
\(702\) 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 \(0.132395\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8688.00 0.457299
\(713\) −423.000 −0.0222181
\(714\) 0 0
\(715\) 0 0
\(716\) −10560.0 −0.551181
\(717\) 0 0
\(718\) 17556.0 0.912513
\(719\) −27114.0 −1.40637 −0.703186 0.711006i \(-0.748240\pi\)
−0.703186 + 0.711006i \(0.748240\pi\)
\(720\) 0 0
\(721\) 3568.00 0.184299
\(722\) 6756.00 0.348244
\(723\) 0 0
\(724\) 4292.00 0.220319
\(725\) 0 0
\(726\) 0 0
\(727\) −236.000 −0.0120396 −0.00601978 0.999982i \(-0.501916\pi\)
−0.00601978 + 0.999982i \(0.501916\pi\)
\(728\) 640.000 0.0325824
\(729\) 0 0
\(730\) 0 0
\(731\) −16554.0 −0.837581
\(732\) 0 0
\(733\) −27128.0 −1.36698 −0.683489 0.729960i \(-0.739538\pi\)
−0.683489 + 0.729960i \(0.739538\pi\)
\(734\) 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 \(0.458296\pi\)
\(740\) 0 0
\(741\) 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
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 8122.00 0.389959 0.194980 0.980807i \(-0.437536\pi\)
0.194980 + 0.980807i \(0.437536\pi\)
\(758\) −58.0000 −0.00277923
\(759\) 0 0
\(760\) 0 0
\(761\) −10584.0 −0.504165 −0.252083 0.967706i \(-0.581115\pi\)
−0.252083 + 0.967706i \(0.581115\pi\)
\(762\) 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 \(0.643801\pi\)
\(770\) 0 0
\(771\) 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
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) −24854.0 −1.12573 −0.562865 0.826549i \(-0.690301\pi\)
−0.562865 + 0.826549i \(0.690301\pi\)
\(788\) 3060.00 0.138335
\(789\) 0 0
\(790\) 0 0
\(791\) 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
\(800\) 0 0
\(801\) 0 0
\(802\) 9108.00 0.401016
\(803\) 47292.0 2.07833
\(804\) 0 0
\(805\) 0 0
\(806\) 1880.00 0.0821590
\(807\) 0 0
\(808\) 1056.00 0.0459777
\(809\) 5142.00 0.223465 0.111732 0.993738i \(-0.464360\pi\)
0.111732 + 0.993738i \(0.464360\pi\)
\(810\) 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
\(815\) 0 0
\(816\) 0 0
\(817\) 10502.0 0.449717
\(818\) −2002.00 −0.0855725
\(819\) 0 0
\(820\) 0 0
\(821\) −25014.0 −1.06333 −0.531665 0.846954i \(-0.678434\pi\)
−0.531665 + 0.846954i \(0.678434\pi\)
\(822\) 0 0
\(823\) 32146.0 1.36153 0.680765 0.732502i \(-0.261648\pi\)
0.680765 + 0.732502i \(0.261648\pi\)
\(824\) −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 \(0.221882\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1280.00 −0.0533366
\(833\) 30411.0 1.26492
\(834\) 0 0
\(835\) 0 0
\(836\) 9912.00 0.410064
\(837\) 0 0
\(838\) −3588.00 −0.147906
\(839\) −11076.0 −0.455764 −0.227882 0.973689i \(-0.573180\pi\)
−0.227882 + 0.973689i \(0.573180\pi\)
\(840\) 0 0
\(841\) −9989.00 −0.409570
\(842\) 32258.0 1.32029
\(843\) 0 0
\(844\) 18404.0 0.750583
\(845\) 0 0
\(846\) 0 0
\(847\) 1732.00 0.0702624
\(848\) −11856.0 −0.480114
\(849\) 0 0
\(850\) 0 0
\(851\) −2358.00 −0.0949838
\(852\) 0 0
\(853\) −36848.0 −1.47908 −0.739538 0.673115i \(-0.764956\pi\)
−0.739538 + 0.673115i \(0.764956\pi\)
\(854\) −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 \(0.502624\pi\)
\(860\) 0 0
\(861\) 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
\(865\) 0 0
\(866\) −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
\(875\) 0 0
\(876\) 0 0
\(877\) −35042.0 −1.34924 −0.674620 0.738165i \(-0.735693\pi\)
−0.674620 + 0.738165i \(0.735693\pi\)
\(878\) 22298.0 0.857085
\(879\) 0 0
\(880\) 0 0
\(881\) −1080.00 −0.0413009 −0.0206505 0.999787i \(-0.506574\pi\)
−0.0206505 + 0.999787i \(0.506574\pi\)
\(882\) 0 0
\(883\) 20164.0 0.768485 0.384243 0.923232i \(-0.374463\pi\)
0.384243 + 0.923232i \(0.374463\pi\)
\(884\) 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
\(890\) 0 0
\(891\) 0 0
\(892\) 8632.00 0.324014
\(893\) −8496.00 −0.318374
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) 26524.0 0.971020 0.485510 0.874231i \(-0.338634\pi\)
0.485510 + 0.874231i \(0.338634\pi\)
\(908\) −12492.0 −0.456566
\(909\) 0 0
\(910\) 0 0
\(911\) −35568.0 −1.29355 −0.646773 0.762683i \(-0.723882\pi\)
−0.646773 + 0.762683i \(0.723882\pi\)
\(912\) 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 \(0.639874\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −20484.0 −0.731675
\(923\) −13800.0 −0.492126
\(924\) 0 0
\(925\) 0 0
\(926\) 6604.00 0.234364
\(927\) 0 0
\(928\) −3840.00 −0.135834
\(929\) 40590.0 1.43349 0.716746 0.697334i \(-0.245631\pi\)
0.716746 + 0.697334i \(0.245631\pi\)
\(930\) 0 0
\(931\) −19293.0 −0.679165
\(932\) 1752.00 0.0615758
\(933\) 0 0
\(934\) 3846.00 0.134738
\(935\) 0 0
\(936\) 0 0
\(937\) 12964.0 0.451991 0.225995 0.974128i \(-0.427437\pi\)
0.225995 + 0.974128i \(0.427437\pi\)
\(938\) −4304.00 −0.149819
\(939\) 0 0
\(940\) 0 0
\(941\) −29922.0 −1.03659 −0.518294 0.855203i \(-0.673433\pi\)
−0.518294 + 0.855203i \(0.673433\pi\)
\(942\) 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
\(950\) 0 0
\(951\) 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
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −18278.0 −0.607840 −0.303920 0.952698i \(-0.598295\pi\)
−0.303920 + 0.952698i \(0.598295\pi\)
\(968\) −3464.00 −0.115018
\(969\) 0 0
\(970\) 0 0
\(971\) 24942.0 0.824333 0.412166 0.911109i \(-0.364772\pi\)
0.412166 + 0.911109i \(0.364772\pi\)
\(972\) 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
\(980\) 0 0
\(981\) 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
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) −19082.0 −0.606151 −0.303076 0.952966i \(-0.598014\pi\)
−0.303076 + 0.952966i \(0.598014\pi\)
\(998\) 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 1350.4.a.g.1.1 1
3.2 odd 2 1350.4.a.u.1.1 1
5.2 odd 4 1350.4.c.q.649.1 2
5.3 odd 4 1350.4.c.q.649.2 2
5.4 even 2 270.4.a.i.1.1 yes 1
15.2 even 4 1350.4.c.d.649.2 2
15.8 even 4 1350.4.c.d.649.1 2
15.14 odd 2 270.4.a.e.1.1 1
20.19 odd 2 2160.4.a.e.1.1 1
45.4 even 6 810.4.e.h.541.1 2
45.14 odd 6 810.4.e.q.541.1 2
45.29 odd 6 810.4.e.q.271.1 2
45.34 even 6 810.4.e.h.271.1 2
60.59 even 2 2160.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.e.1.1 1 15.14 odd 2
270.4.a.i.1.1 yes 1 5.4 even 2
810.4.e.h.271.1 2 45.34 even 6
810.4.e.h.541.1 2 45.4 even 6
810.4.e.q.271.1 2 45.29 odd 6
810.4.e.q.541.1 2 45.14 odd 6
1350.4.a.g.1.1 1 1.1 even 1 trivial
1350.4.a.u.1.1 1 3.2 odd 2
1350.4.c.d.649.1 2 15.8 even 4
1350.4.c.d.649.2 2 15.2 even 4
1350.4.c.q.649.1 2 5.2 odd 4
1350.4.c.q.649.2 2 5.3 odd 4
2160.4.a.e.1.1 1 20.19 odd 2
2160.4.a.o.1.1 1 60.59 even 2