Properties

Label 1380.4.a.a.1.1
Level $1380$
Weight $4$
Character 1380.1
Self dual yes
Analytic conductor $81.423$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(81.4226358079\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1380.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +30.0000 q^{11} -28.0000 q^{13} +15.0000 q^{15} +78.0000 q^{17} -52.0000 q^{19} +48.0000 q^{21} -23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -252.000 q^{29} +200.000 q^{31} -90.0000 q^{33} +80.0000 q^{35} +146.000 q^{37} +84.0000 q^{39} +438.000 q^{41} -46.0000 q^{43} -45.0000 q^{45} +588.000 q^{47} -87.0000 q^{49} -234.000 q^{51} +438.000 q^{53} -150.000 q^{55} +156.000 q^{57} -168.000 q^{59} -586.000 q^{61} -144.000 q^{63} +140.000 q^{65} -94.0000 q^{67} +69.0000 q^{69} -30.0000 q^{71} -466.000 q^{73} -75.0000 q^{75} -480.000 q^{77} -520.000 q^{79} +81.0000 q^{81} +708.000 q^{83} -390.000 q^{85} +756.000 q^{87} -600.000 q^{89} +448.000 q^{91} -600.000 q^{93} +260.000 q^{95} -340.000 q^{97} +270.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 78.0000 1.11281 0.556405 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −252.000 −1.61363 −0.806814 0.590805i \(-0.798810\pi\)
−0.806814 + 0.590805i \(0.798810\pi\)
\(30\) 0 0
\(31\) 200.000 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(32\) 0 0
\(33\) −90.0000 −0.474757
\(34\) 0 0
\(35\) 80.0000 0.386356
\(36\) 0 0
\(37\) 146.000 0.648710 0.324355 0.945936i \(-0.394853\pi\)
0.324355 + 0.945936i \(0.394853\pi\)
\(38\) 0 0
\(39\) 84.0000 0.344891
\(40\) 0 0
\(41\) 438.000 1.66839 0.834196 0.551467i \(-0.185932\pi\)
0.834196 + 0.551467i \(0.185932\pi\)
\(42\) 0 0
\(43\) −46.0000 −0.163138 −0.0815690 0.996668i \(-0.525993\pi\)
−0.0815690 + 0.996668i \(0.525993\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 588.000 1.82486 0.912432 0.409228i \(-0.134202\pi\)
0.912432 + 0.409228i \(0.134202\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) 438.000 1.13517 0.567584 0.823315i \(-0.307878\pi\)
0.567584 + 0.823315i \(0.307878\pi\)
\(54\) 0 0
\(55\) −150.000 −0.367745
\(56\) 0 0
\(57\) 156.000 0.362504
\(58\) 0 0
\(59\) −168.000 −0.370707 −0.185354 0.982672i \(-0.559343\pi\)
−0.185354 + 0.982672i \(0.559343\pi\)
\(60\) 0 0
\(61\) −586.000 −1.22999 −0.614997 0.788530i \(-0.710843\pi\)
−0.614997 + 0.788530i \(0.710843\pi\)
\(62\) 0 0
\(63\) −144.000 −0.287973
\(64\) 0 0
\(65\) 140.000 0.267152
\(66\) 0 0
\(67\) −94.0000 −0.171402 −0.0857010 0.996321i \(-0.527313\pi\)
−0.0857010 + 0.996321i \(0.527313\pi\)
\(68\) 0 0
\(69\) 69.0000 0.120386
\(70\) 0 0
\(71\) −30.0000 −0.0501457 −0.0250729 0.999686i \(-0.507982\pi\)
−0.0250729 + 0.999686i \(0.507982\pi\)
\(72\) 0 0
\(73\) −466.000 −0.747139 −0.373570 0.927602i \(-0.621866\pi\)
−0.373570 + 0.927602i \(0.621866\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) −480.000 −0.710404
\(78\) 0 0
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 708.000 0.936302 0.468151 0.883648i \(-0.344920\pi\)
0.468151 + 0.883648i \(0.344920\pi\)
\(84\) 0 0
\(85\) −390.000 −0.497664
\(86\) 0 0
\(87\) 756.000 0.931629
\(88\) 0 0
\(89\) −600.000 −0.714605 −0.357303 0.933989i \(-0.616304\pi\)
−0.357303 + 0.933989i \(0.616304\pi\)
\(90\) 0 0
\(91\) 448.000 0.516079
\(92\) 0 0
\(93\) −600.000 −0.669001
\(94\) 0 0
\(95\) 260.000 0.280794
\(96\) 0 0
\(97\) −340.000 −0.355895 −0.177947 0.984040i \(-0.556946\pi\)
−0.177947 + 0.984040i \(0.556946\pi\)
\(98\) 0 0
\(99\) 270.000 0.274101
\(100\) 0 0
\(101\) −1164.00 −1.14676 −0.573378 0.819291i \(-0.694367\pi\)
−0.573378 + 0.819291i \(0.694367\pi\)
\(102\) 0 0
\(103\) 656.000 0.627550 0.313775 0.949497i \(-0.398406\pi\)
0.313775 + 0.949497i \(0.398406\pi\)
\(104\) 0 0
\(105\) −240.000 −0.223063
\(106\) 0 0
\(107\) −1992.00 −1.79976 −0.899878 0.436142i \(-0.856345\pi\)
−0.899878 + 0.436142i \(0.856345\pi\)
\(108\) 0 0
\(109\) 1226.00 1.07733 0.538667 0.842518i \(-0.318928\pi\)
0.538667 + 0.842518i \(0.318928\pi\)
\(110\) 0 0
\(111\) −438.000 −0.374533
\(112\) 0 0
\(113\) 954.000 0.794202 0.397101 0.917775i \(-0.370016\pi\)
0.397101 + 0.917775i \(0.370016\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) −252.000 −0.199123
\(118\) 0 0
\(119\) −1248.00 −0.961378
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 0 0
\(123\) −1314.00 −0.963247
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 962.000 0.672155 0.336078 0.941834i \(-0.390900\pi\)
0.336078 + 0.941834i \(0.390900\pi\)
\(128\) 0 0
\(129\) 138.000 0.0941878
\(130\) 0 0
\(131\) −2484.00 −1.65670 −0.828351 0.560209i \(-0.810721\pi\)
−0.828351 + 0.560209i \(0.810721\pi\)
\(132\) 0 0
\(133\) 832.000 0.542433
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) −1134.00 −0.707184 −0.353592 0.935400i \(-0.615040\pi\)
−0.353592 + 0.935400i \(0.615040\pi\)
\(138\) 0 0
\(139\) −1636.00 −0.998300 −0.499150 0.866516i \(-0.666354\pi\)
−0.499150 + 0.866516i \(0.666354\pi\)
\(140\) 0 0
\(141\) −1764.00 −1.05359
\(142\) 0 0
\(143\) −840.000 −0.491219
\(144\) 0 0
\(145\) 1260.00 0.721637
\(146\) 0 0
\(147\) 261.000 0.146442
\(148\) 0 0
\(149\) −1566.00 −0.861018 −0.430509 0.902586i \(-0.641666\pi\)
−0.430509 + 0.902586i \(0.641666\pi\)
\(150\) 0 0
\(151\) 788.000 0.424679 0.212340 0.977196i \(-0.431892\pi\)
0.212340 + 0.977196i \(0.431892\pi\)
\(152\) 0 0
\(153\) 702.000 0.370937
\(154\) 0 0
\(155\) −1000.00 −0.518206
\(156\) 0 0
\(157\) 2.00000 0.00101667 0.000508336 1.00000i \(-0.499838\pi\)
0.000508336 1.00000i \(0.499838\pi\)
\(158\) 0 0
\(159\) −1314.00 −0.655390
\(160\) 0 0
\(161\) 368.000 0.180140
\(162\) 0 0
\(163\) −448.000 −0.215276 −0.107638 0.994190i \(-0.534329\pi\)
−0.107638 + 0.994190i \(0.534329\pi\)
\(164\) 0 0
\(165\) 450.000 0.212318
\(166\) 0 0
\(167\) 2904.00 1.34562 0.672809 0.739816i \(-0.265087\pi\)
0.672809 + 0.739816i \(0.265087\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) −468.000 −0.209292
\(172\) 0 0
\(173\) −2322.00 −1.02045 −0.510227 0.860040i \(-0.670438\pi\)
−0.510227 + 0.860040i \(0.670438\pi\)
\(174\) 0 0
\(175\) −400.000 −0.172784
\(176\) 0 0
\(177\) 504.000 0.214028
\(178\) 0 0
\(179\) −96.0000 −0.0400859 −0.0200430 0.999799i \(-0.506380\pi\)
−0.0200430 + 0.999799i \(0.506380\pi\)
\(180\) 0 0
\(181\) −1906.00 −0.782717 −0.391359 0.920238i \(-0.627995\pi\)
−0.391359 + 0.920238i \(0.627995\pi\)
\(182\) 0 0
\(183\) 1758.00 0.710137
\(184\) 0 0
\(185\) −730.000 −0.290112
\(186\) 0 0
\(187\) 2340.00 0.915068
\(188\) 0 0
\(189\) 432.000 0.166261
\(190\) 0 0
\(191\) −2520.00 −0.954664 −0.477332 0.878723i \(-0.658396\pi\)
−0.477332 + 0.878723i \(0.658396\pi\)
\(192\) 0 0
\(193\) −4366.00 −1.62835 −0.814175 0.580620i \(-0.802810\pi\)
−0.814175 + 0.580620i \(0.802810\pi\)
\(194\) 0 0
\(195\) −420.000 −0.154240
\(196\) 0 0
\(197\) 3750.00 1.35623 0.678113 0.734958i \(-0.262798\pi\)
0.678113 + 0.734958i \(0.262798\pi\)
\(198\) 0 0
\(199\) −1576.00 −0.561405 −0.280703 0.959795i \(-0.590567\pi\)
−0.280703 + 0.959795i \(0.590567\pi\)
\(200\) 0 0
\(201\) 282.000 0.0989589
\(202\) 0 0
\(203\) 4032.00 1.39404
\(204\) 0 0
\(205\) −2190.00 −0.746128
\(206\) 0 0
\(207\) −207.000 −0.0695048
\(208\) 0 0
\(209\) −1560.00 −0.516304
\(210\) 0 0
\(211\) 284.000 0.0926605 0.0463303 0.998926i \(-0.485247\pi\)
0.0463303 + 0.998926i \(0.485247\pi\)
\(212\) 0 0
\(213\) 90.0000 0.0289516
\(214\) 0 0
\(215\) 230.000 0.0729575
\(216\) 0 0
\(217\) −3200.00 −1.00106
\(218\) 0 0
\(219\) 1398.00 0.431361
\(220\) 0 0
\(221\) −2184.00 −0.664759
\(222\) 0 0
\(223\) 2294.00 0.688868 0.344434 0.938811i \(-0.388071\pi\)
0.344434 + 0.938811i \(0.388071\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −4584.00 −1.34031 −0.670156 0.742220i \(-0.733773\pi\)
−0.670156 + 0.742220i \(0.733773\pi\)
\(228\) 0 0
\(229\) 4634.00 1.33722 0.668610 0.743613i \(-0.266890\pi\)
0.668610 + 0.743613i \(0.266890\pi\)
\(230\) 0 0
\(231\) 1440.00 0.410152
\(232\) 0 0
\(233\) −5082.00 −1.42890 −0.714448 0.699688i \(-0.753322\pi\)
−0.714448 + 0.699688i \(0.753322\pi\)
\(234\) 0 0
\(235\) −2940.00 −0.816104
\(236\) 0 0
\(237\) 1560.00 0.427565
\(238\) 0 0
\(239\) −606.000 −0.164012 −0.0820060 0.996632i \(-0.526133\pi\)
−0.0820060 + 0.996632i \(0.526133\pi\)
\(240\) 0 0
\(241\) −3454.00 −0.923202 −0.461601 0.887088i \(-0.652725\pi\)
−0.461601 + 0.887088i \(0.652725\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 435.000 0.113433
\(246\) 0 0
\(247\) 1456.00 0.375073
\(248\) 0 0
\(249\) −2124.00 −0.540574
\(250\) 0 0
\(251\) −1758.00 −0.442088 −0.221044 0.975264i \(-0.570946\pi\)
−0.221044 + 0.975264i \(0.570946\pi\)
\(252\) 0 0
\(253\) −690.000 −0.171462
\(254\) 0 0
\(255\) 1170.00 0.287326
\(256\) 0 0
\(257\) −3546.00 −0.860675 −0.430337 0.902668i \(-0.641605\pi\)
−0.430337 + 0.902668i \(0.641605\pi\)
\(258\) 0 0
\(259\) −2336.00 −0.560432
\(260\) 0 0
\(261\) −2268.00 −0.537876
\(262\) 0 0
\(263\) −168.000 −0.0393891 −0.0196945 0.999806i \(-0.506269\pi\)
−0.0196945 + 0.999806i \(0.506269\pi\)
\(264\) 0 0
\(265\) −2190.00 −0.507663
\(266\) 0 0
\(267\) 1800.00 0.412578
\(268\) 0 0
\(269\) −360.000 −0.0815970 −0.0407985 0.999167i \(-0.512990\pi\)
−0.0407985 + 0.999167i \(0.512990\pi\)
\(270\) 0 0
\(271\) 7208.00 1.61570 0.807850 0.589388i \(-0.200631\pi\)
0.807850 + 0.589388i \(0.200631\pi\)
\(272\) 0 0
\(273\) −1344.00 −0.297958
\(274\) 0 0
\(275\) 750.000 0.164461
\(276\) 0 0
\(277\) 2012.00 0.436424 0.218212 0.975901i \(-0.429978\pi\)
0.218212 + 0.975901i \(0.429978\pi\)
\(278\) 0 0
\(279\) 1800.00 0.386248
\(280\) 0 0
\(281\) −5520.00 −1.17187 −0.585935 0.810358i \(-0.699273\pi\)
−0.585935 + 0.810358i \(0.699273\pi\)
\(282\) 0 0
\(283\) 998.000 0.209629 0.104814 0.994492i \(-0.466575\pi\)
0.104814 + 0.994492i \(0.466575\pi\)
\(284\) 0 0
\(285\) −780.000 −0.162117
\(286\) 0 0
\(287\) −7008.00 −1.44136
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 1020.00 0.205476
\(292\) 0 0
\(293\) −6834.00 −1.36262 −0.681308 0.731997i \(-0.738589\pi\)
−0.681308 + 0.731997i \(0.738589\pi\)
\(294\) 0 0
\(295\) 840.000 0.165785
\(296\) 0 0
\(297\) −810.000 −0.158252
\(298\) 0 0
\(299\) 644.000 0.124560
\(300\) 0 0
\(301\) 736.000 0.140938
\(302\) 0 0
\(303\) 3492.00 0.662080
\(304\) 0 0
\(305\) 2930.00 0.550070
\(306\) 0 0
\(307\) −5380.00 −1.00017 −0.500086 0.865976i \(-0.666698\pi\)
−0.500086 + 0.865976i \(0.666698\pi\)
\(308\) 0 0
\(309\) −1968.00 −0.362316
\(310\) 0 0
\(311\) 4650.00 0.847837 0.423919 0.905700i \(-0.360654\pi\)
0.423919 + 0.905700i \(0.360654\pi\)
\(312\) 0 0
\(313\) −9232.00 −1.66717 −0.833584 0.552393i \(-0.813715\pi\)
−0.833584 + 0.552393i \(0.813715\pi\)
\(314\) 0 0
\(315\) 720.000 0.128785
\(316\) 0 0
\(317\) −4566.00 −0.808997 −0.404499 0.914539i \(-0.632554\pi\)
−0.404499 + 0.914539i \(0.632554\pi\)
\(318\) 0 0
\(319\) −7560.00 −1.32689
\(320\) 0 0
\(321\) 5976.00 1.03909
\(322\) 0 0
\(323\) −4056.00 −0.698706
\(324\) 0 0
\(325\) −700.000 −0.119474
\(326\) 0 0
\(327\) −3678.00 −0.622000
\(328\) 0 0
\(329\) −9408.00 −1.57653
\(330\) 0 0
\(331\) 3140.00 0.521420 0.260710 0.965417i \(-0.416043\pi\)
0.260710 + 0.965417i \(0.416043\pi\)
\(332\) 0 0
\(333\) 1314.00 0.216237
\(334\) 0 0
\(335\) 470.000 0.0766533
\(336\) 0 0
\(337\) 6896.00 1.11469 0.557343 0.830282i \(-0.311821\pi\)
0.557343 + 0.830282i \(0.311821\pi\)
\(338\) 0 0
\(339\) −2862.00 −0.458532
\(340\) 0 0
\(341\) 6000.00 0.952839
\(342\) 0 0
\(343\) 6880.00 1.08305
\(344\) 0 0
\(345\) −345.000 −0.0538382
\(346\) 0 0
\(347\) −3084.00 −0.477112 −0.238556 0.971129i \(-0.576674\pi\)
−0.238556 + 0.971129i \(0.576674\pi\)
\(348\) 0 0
\(349\) 1226.00 0.188041 0.0940205 0.995570i \(-0.470028\pi\)
0.0940205 + 0.995570i \(0.470028\pi\)
\(350\) 0 0
\(351\) 756.000 0.114964
\(352\) 0 0
\(353\) 8394.00 1.26563 0.632815 0.774303i \(-0.281899\pi\)
0.632815 + 0.774303i \(0.281899\pi\)
\(354\) 0 0
\(355\) 150.000 0.0224258
\(356\) 0 0
\(357\) 3744.00 0.555052
\(358\) 0 0
\(359\) −12480.0 −1.83473 −0.917367 0.398043i \(-0.869689\pi\)
−0.917367 + 0.398043i \(0.869689\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) 1293.00 0.186956
\(364\) 0 0
\(365\) 2330.00 0.334131
\(366\) 0 0
\(367\) 5312.00 0.755543 0.377771 0.925899i \(-0.376691\pi\)
0.377771 + 0.925899i \(0.376691\pi\)
\(368\) 0 0
\(369\) 3942.00 0.556131
\(370\) 0 0
\(371\) −7008.00 −0.980693
\(372\) 0 0
\(373\) 1490.00 0.206835 0.103417 0.994638i \(-0.467022\pi\)
0.103417 + 0.994638i \(0.467022\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 7056.00 0.963932
\(378\) 0 0
\(379\) 6968.00 0.944386 0.472193 0.881495i \(-0.343463\pi\)
0.472193 + 0.881495i \(0.343463\pi\)
\(380\) 0 0
\(381\) −2886.00 −0.388069
\(382\) 0 0
\(383\) 1512.00 0.201722 0.100861 0.994901i \(-0.467840\pi\)
0.100861 + 0.994901i \(0.467840\pi\)
\(384\) 0 0
\(385\) 2400.00 0.317702
\(386\) 0 0
\(387\) −414.000 −0.0543793
\(388\) 0 0
\(389\) −2514.00 −0.327673 −0.163837 0.986487i \(-0.552387\pi\)
−0.163837 + 0.986487i \(0.552387\pi\)
\(390\) 0 0
\(391\) −1794.00 −0.232037
\(392\) 0 0
\(393\) 7452.00 0.956498
\(394\) 0 0
\(395\) 2600.00 0.331190
\(396\) 0 0
\(397\) −2548.00 −0.322117 −0.161059 0.986945i \(-0.551491\pi\)
−0.161059 + 0.986945i \(0.551491\pi\)
\(398\) 0 0
\(399\) −2496.00 −0.313174
\(400\) 0 0
\(401\) 15324.0 1.90834 0.954170 0.299266i \(-0.0967419\pi\)
0.954170 + 0.299266i \(0.0967419\pi\)
\(402\) 0 0
\(403\) −5600.00 −0.692198
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 4380.00 0.533436
\(408\) 0 0
\(409\) −14362.0 −1.73632 −0.868160 0.496284i \(-0.834698\pi\)
−0.868160 + 0.496284i \(0.834698\pi\)
\(410\) 0 0
\(411\) 3402.00 0.408293
\(412\) 0 0
\(413\) 2688.00 0.320261
\(414\) 0 0
\(415\) −3540.00 −0.418727
\(416\) 0 0
\(417\) 4908.00 0.576369
\(418\) 0 0
\(419\) 6774.00 0.789813 0.394906 0.918721i \(-0.370777\pi\)
0.394906 + 0.918721i \(0.370777\pi\)
\(420\) 0 0
\(421\) 15662.0 1.81311 0.906555 0.422088i \(-0.138703\pi\)
0.906555 + 0.422088i \(0.138703\pi\)
\(422\) 0 0
\(423\) 5292.00 0.608288
\(424\) 0 0
\(425\) 1950.00 0.222562
\(426\) 0 0
\(427\) 9376.00 1.06261
\(428\) 0 0
\(429\) 2520.00 0.283605
\(430\) 0 0
\(431\) −11688.0 −1.30624 −0.653122 0.757253i \(-0.726541\pi\)
−0.653122 + 0.757253i \(0.726541\pi\)
\(432\) 0 0
\(433\) −1588.00 −0.176246 −0.0881229 0.996110i \(-0.528087\pi\)
−0.0881229 + 0.996110i \(0.528087\pi\)
\(434\) 0 0
\(435\) −3780.00 −0.416637
\(436\) 0 0
\(437\) 1196.00 0.130921
\(438\) 0 0
\(439\) 8300.00 0.902363 0.451182 0.892432i \(-0.351003\pi\)
0.451182 + 0.892432i \(0.351003\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) 2988.00 0.320461 0.160230 0.987080i \(-0.448776\pi\)
0.160230 + 0.987080i \(0.448776\pi\)
\(444\) 0 0
\(445\) 3000.00 0.319581
\(446\) 0 0
\(447\) 4698.00 0.497109
\(448\) 0 0
\(449\) −9546.00 −1.00335 −0.501674 0.865057i \(-0.667282\pi\)
−0.501674 + 0.865057i \(0.667282\pi\)
\(450\) 0 0
\(451\) 13140.0 1.37193
\(452\) 0 0
\(453\) −2364.00 −0.245189
\(454\) 0 0
\(455\) −2240.00 −0.230797
\(456\) 0 0
\(457\) −10540.0 −1.07886 −0.539432 0.842029i \(-0.681361\pi\)
−0.539432 + 0.842029i \(0.681361\pi\)
\(458\) 0 0
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) 11988.0 1.21114 0.605571 0.795791i \(-0.292945\pi\)
0.605571 + 0.795791i \(0.292945\pi\)
\(462\) 0 0
\(463\) 19382.0 1.94548 0.972741 0.231895i \(-0.0744927\pi\)
0.972741 + 0.231895i \(0.0744927\pi\)
\(464\) 0 0
\(465\) 3000.00 0.299186
\(466\) 0 0
\(467\) −9180.00 −0.909635 −0.454818 0.890585i \(-0.650296\pi\)
−0.454818 + 0.890585i \(0.650296\pi\)
\(468\) 0 0
\(469\) 1504.00 0.148077
\(470\) 0 0
\(471\) −6.00000 −0.000586975 0
\(472\) 0 0
\(473\) −1380.00 −0.134149
\(474\) 0 0
\(475\) −1300.00 −0.125575
\(476\) 0 0
\(477\) 3942.00 0.378389
\(478\) 0 0
\(479\) −192.000 −0.0183146 −0.00915731 0.999958i \(-0.502915\pi\)
−0.00915731 + 0.999958i \(0.502915\pi\)
\(480\) 0 0
\(481\) −4088.00 −0.387519
\(482\) 0 0
\(483\) −1104.00 −0.104004
\(484\) 0 0
\(485\) 1700.00 0.159161
\(486\) 0 0
\(487\) 3446.00 0.320643 0.160322 0.987065i \(-0.448747\pi\)
0.160322 + 0.987065i \(0.448747\pi\)
\(488\) 0 0
\(489\) 1344.00 0.124290
\(490\) 0 0
\(491\) 10068.0 0.925382 0.462691 0.886520i \(-0.346884\pi\)
0.462691 + 0.886520i \(0.346884\pi\)
\(492\) 0 0
\(493\) −19656.0 −1.79566
\(494\) 0 0
\(495\) −1350.00 −0.122582
\(496\) 0 0
\(497\) 480.000 0.0433218
\(498\) 0 0
\(499\) −14020.0 −1.25776 −0.628879 0.777503i \(-0.716486\pi\)
−0.628879 + 0.777503i \(0.716486\pi\)
\(500\) 0 0
\(501\) −8712.00 −0.776893
\(502\) 0 0
\(503\) −14952.0 −1.32540 −0.662701 0.748885i \(-0.730590\pi\)
−0.662701 + 0.748885i \(0.730590\pi\)
\(504\) 0 0
\(505\) 5820.00 0.512845
\(506\) 0 0
\(507\) 4239.00 0.371323
\(508\) 0 0
\(509\) −13692.0 −1.19231 −0.596156 0.802868i \(-0.703306\pi\)
−0.596156 + 0.802868i \(0.703306\pi\)
\(510\) 0 0
\(511\) 7456.00 0.645468
\(512\) 0 0
\(513\) 1404.00 0.120835
\(514\) 0 0
\(515\) −3280.00 −0.280649
\(516\) 0 0
\(517\) 17640.0 1.50059
\(518\) 0 0
\(519\) 6966.00 0.589159
\(520\) 0 0
\(521\) −13704.0 −1.15237 −0.576183 0.817320i \(-0.695459\pi\)
−0.576183 + 0.817320i \(0.695459\pi\)
\(522\) 0 0
\(523\) −17026.0 −1.42351 −0.711754 0.702429i \(-0.752099\pi\)
−0.711754 + 0.702429i \(0.752099\pi\)
\(524\) 0 0
\(525\) 1200.00 0.0997567
\(526\) 0 0
\(527\) 15600.0 1.28946
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −1512.00 −0.123569
\(532\) 0 0
\(533\) −12264.0 −0.996647
\(534\) 0 0
\(535\) 9960.00 0.804875
\(536\) 0 0
\(537\) 288.000 0.0231436
\(538\) 0 0
\(539\) −2610.00 −0.208573
\(540\) 0 0
\(541\) −1942.00 −0.154331 −0.0771655 0.997018i \(-0.524587\pi\)
−0.0771655 + 0.997018i \(0.524587\pi\)
\(542\) 0 0
\(543\) 5718.00 0.451902
\(544\) 0 0
\(545\) −6130.00 −0.481799
\(546\) 0 0
\(547\) 13988.0 1.09339 0.546694 0.837332i \(-0.315886\pi\)
0.546694 + 0.837332i \(0.315886\pi\)
\(548\) 0 0
\(549\) −5274.00 −0.409998
\(550\) 0 0
\(551\) 13104.0 1.01316
\(552\) 0 0
\(553\) 8320.00 0.639787
\(554\) 0 0
\(555\) 2190.00 0.167496
\(556\) 0 0
\(557\) −11382.0 −0.865836 −0.432918 0.901433i \(-0.642516\pi\)
−0.432918 + 0.901433i \(0.642516\pi\)
\(558\) 0 0
\(559\) 1288.00 0.0974537
\(560\) 0 0
\(561\) −7020.00 −0.528315
\(562\) 0 0
\(563\) 20844.0 1.56034 0.780169 0.625569i \(-0.215133\pi\)
0.780169 + 0.625569i \(0.215133\pi\)
\(564\) 0 0
\(565\) −4770.00 −0.355178
\(566\) 0 0
\(567\) −1296.00 −0.0959910
\(568\) 0 0
\(569\) 5676.00 0.418190 0.209095 0.977895i \(-0.432948\pi\)
0.209095 + 0.977895i \(0.432948\pi\)
\(570\) 0 0
\(571\) 11936.0 0.874792 0.437396 0.899269i \(-0.355901\pi\)
0.437396 + 0.899269i \(0.355901\pi\)
\(572\) 0 0
\(573\) 7560.00 0.551175
\(574\) 0 0
\(575\) −575.000 −0.0417029
\(576\) 0 0
\(577\) −7594.00 −0.547907 −0.273954 0.961743i \(-0.588331\pi\)
−0.273954 + 0.961743i \(0.588331\pi\)
\(578\) 0 0
\(579\) 13098.0 0.940128
\(580\) 0 0
\(581\) −11328.0 −0.808889
\(582\) 0 0
\(583\) 13140.0 0.933453
\(584\) 0 0
\(585\) 1260.00 0.0890506
\(586\) 0 0
\(587\) 26172.0 1.84026 0.920131 0.391610i \(-0.128082\pi\)
0.920131 + 0.391610i \(0.128082\pi\)
\(588\) 0 0
\(589\) −10400.0 −0.727546
\(590\) 0 0
\(591\) −11250.0 −0.783017
\(592\) 0 0
\(593\) −15786.0 −1.09318 −0.546588 0.837402i \(-0.684074\pi\)
−0.546588 + 0.837402i \(0.684074\pi\)
\(594\) 0 0
\(595\) 6240.00 0.429941
\(596\) 0 0
\(597\) 4728.00 0.324128
\(598\) 0 0
\(599\) −15714.0 −1.07188 −0.535940 0.844256i \(-0.680043\pi\)
−0.535940 + 0.844256i \(0.680043\pi\)
\(600\) 0 0
\(601\) −16714.0 −1.13441 −0.567203 0.823578i \(-0.691975\pi\)
−0.567203 + 0.823578i \(0.691975\pi\)
\(602\) 0 0
\(603\) −846.000 −0.0571340
\(604\) 0 0
\(605\) 2155.00 0.144815
\(606\) 0 0
\(607\) 8606.00 0.575464 0.287732 0.957711i \(-0.407099\pi\)
0.287732 + 0.957711i \(0.407099\pi\)
\(608\) 0 0
\(609\) −12096.0 −0.804852
\(610\) 0 0
\(611\) −16464.0 −1.09012
\(612\) 0 0
\(613\) 24038.0 1.58383 0.791913 0.610634i \(-0.209085\pi\)
0.791913 + 0.610634i \(0.209085\pi\)
\(614\) 0 0
\(615\) 6570.00 0.430777
\(616\) 0 0
\(617\) 13134.0 0.856977 0.428489 0.903547i \(-0.359046\pi\)
0.428489 + 0.903547i \(0.359046\pi\)
\(618\) 0 0
\(619\) 11660.0 0.757116 0.378558 0.925578i \(-0.376420\pi\)
0.378558 + 0.925578i \(0.376420\pi\)
\(620\) 0 0
\(621\) 621.000 0.0401286
\(622\) 0 0
\(623\) 9600.00 0.617361
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 4680.00 0.298088
\(628\) 0 0
\(629\) 11388.0 0.721891
\(630\) 0 0
\(631\) 24680.0 1.55704 0.778522 0.627617i \(-0.215970\pi\)
0.778522 + 0.627617i \(0.215970\pi\)
\(632\) 0 0
\(633\) −852.000 −0.0534976
\(634\) 0 0
\(635\) −4810.00 −0.300597
\(636\) 0 0
\(637\) 2436.00 0.151519
\(638\) 0 0
\(639\) −270.000 −0.0167152
\(640\) 0 0
\(641\) 8184.00 0.504288 0.252144 0.967690i \(-0.418864\pi\)
0.252144 + 0.967690i \(0.418864\pi\)
\(642\) 0 0
\(643\) 3998.00 0.245203 0.122602 0.992456i \(-0.460876\pi\)
0.122602 + 0.992456i \(0.460876\pi\)
\(644\) 0 0
\(645\) −690.000 −0.0421221
\(646\) 0 0
\(647\) 25476.0 1.54801 0.774007 0.633177i \(-0.218250\pi\)
0.774007 + 0.633177i \(0.218250\pi\)
\(648\) 0 0
\(649\) −5040.00 −0.304834
\(650\) 0 0
\(651\) 9600.00 0.577963
\(652\) 0 0
\(653\) 10146.0 0.608030 0.304015 0.952667i \(-0.401673\pi\)
0.304015 + 0.952667i \(0.401673\pi\)
\(654\) 0 0
\(655\) 12420.0 0.740900
\(656\) 0 0
\(657\) −4194.00 −0.249046
\(658\) 0 0
\(659\) −25986.0 −1.53607 −0.768036 0.640407i \(-0.778766\pi\)
−0.768036 + 0.640407i \(0.778766\pi\)
\(660\) 0 0
\(661\) 4034.00 0.237374 0.118687 0.992932i \(-0.462131\pi\)
0.118687 + 0.992932i \(0.462131\pi\)
\(662\) 0 0
\(663\) 6552.00 0.383799
\(664\) 0 0
\(665\) −4160.00 −0.242583
\(666\) 0 0
\(667\) 5796.00 0.336465
\(668\) 0 0
\(669\) −6882.00 −0.397718
\(670\) 0 0
\(671\) −17580.0 −1.01143
\(672\) 0 0
\(673\) −13954.0 −0.799238 −0.399619 0.916681i \(-0.630858\pi\)
−0.399619 + 0.916681i \(0.630858\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) −14814.0 −0.840987 −0.420494 0.907296i \(-0.638143\pi\)
−0.420494 + 0.907296i \(0.638143\pi\)
\(678\) 0 0
\(679\) 5440.00 0.307464
\(680\) 0 0
\(681\) 13752.0 0.773829
\(682\) 0 0
\(683\) −20388.0 −1.14220 −0.571102 0.820879i \(-0.693484\pi\)
−0.571102 + 0.820879i \(0.693484\pi\)
\(684\) 0 0
\(685\) 5670.00 0.316262
\(686\) 0 0
\(687\) −13902.0 −0.772044
\(688\) 0 0
\(689\) −12264.0 −0.678115
\(690\) 0 0
\(691\) −9508.00 −0.523446 −0.261723 0.965143i \(-0.584291\pi\)
−0.261723 + 0.965143i \(0.584291\pi\)
\(692\) 0 0
\(693\) −4320.00 −0.236801
\(694\) 0 0
\(695\) 8180.00 0.446453
\(696\) 0 0
\(697\) 34164.0 1.85661
\(698\) 0 0
\(699\) 15246.0 0.824974
\(700\) 0 0
\(701\) 5790.00 0.311962 0.155981 0.987760i \(-0.450146\pi\)
0.155981 + 0.987760i \(0.450146\pi\)
\(702\) 0 0
\(703\) −7592.00 −0.407308
\(704\) 0 0
\(705\) 8820.00 0.471178
\(706\) 0 0
\(707\) 18624.0 0.990704
\(708\) 0 0
\(709\) −6238.00 −0.330427 −0.165214 0.986258i \(-0.552831\pi\)
−0.165214 + 0.986258i \(0.552831\pi\)
\(710\) 0 0
\(711\) −4680.00 −0.246855
\(712\) 0 0
\(713\) −4600.00 −0.241615
\(714\) 0 0
\(715\) 4200.00 0.219680
\(716\) 0 0
\(717\) 1818.00 0.0946924
\(718\) 0 0
\(719\) −5010.00 −0.259863 −0.129931 0.991523i \(-0.541476\pi\)
−0.129931 + 0.991523i \(0.541476\pi\)
\(720\) 0 0
\(721\) −10496.0 −0.542152
\(722\) 0 0
\(723\) 10362.0 0.533011
\(724\) 0 0
\(725\) −6300.00 −0.322726
\(726\) 0 0
\(727\) −13876.0 −0.707885 −0.353942 0.935267i \(-0.615159\pi\)
−0.353942 + 0.935267i \(0.615159\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −3588.00 −0.181542
\(732\) 0 0
\(733\) −9766.00 −0.492108 −0.246054 0.969256i \(-0.579134\pi\)
−0.246054 + 0.969256i \(0.579134\pi\)
\(734\) 0 0
\(735\) −1305.00 −0.0654907
\(736\) 0 0
\(737\) −2820.00 −0.140944
\(738\) 0 0
\(739\) −8740.00 −0.435056 −0.217528 0.976054i \(-0.569799\pi\)
−0.217528 + 0.976054i \(0.569799\pi\)
\(740\) 0 0
\(741\) −4368.00 −0.216549
\(742\) 0 0
\(743\) −5952.00 −0.293887 −0.146943 0.989145i \(-0.546943\pi\)
−0.146943 + 0.989145i \(0.546943\pi\)
\(744\) 0 0
\(745\) 7830.00 0.385059
\(746\) 0 0
\(747\) 6372.00 0.312101
\(748\) 0 0
\(749\) 31872.0 1.55484
\(750\) 0 0
\(751\) −6784.00 −0.329629 −0.164815 0.986325i \(-0.552703\pi\)
−0.164815 + 0.986325i \(0.552703\pi\)
\(752\) 0 0
\(753\) 5274.00 0.255239
\(754\) 0 0
\(755\) −3940.00 −0.189922
\(756\) 0 0
\(757\) 10550.0 0.506534 0.253267 0.967396i \(-0.418495\pi\)
0.253267 + 0.967396i \(0.418495\pi\)
\(758\) 0 0
\(759\) 2070.00 0.0989937
\(760\) 0 0
\(761\) −38430.0 −1.83060 −0.915300 0.402773i \(-0.868046\pi\)
−0.915300 + 0.402773i \(0.868046\pi\)
\(762\) 0 0
\(763\) −19616.0 −0.930730
\(764\) 0 0
\(765\) −3510.00 −0.165888
\(766\) 0 0
\(767\) 4704.00 0.221449
\(768\) 0 0
\(769\) 19358.0 0.907760 0.453880 0.891063i \(-0.350039\pi\)
0.453880 + 0.891063i \(0.350039\pi\)
\(770\) 0 0
\(771\) 10638.0 0.496911
\(772\) 0 0
\(773\) 7182.00 0.334177 0.167088 0.985942i \(-0.446563\pi\)
0.167088 + 0.985942i \(0.446563\pi\)
\(774\) 0 0
\(775\) 5000.00 0.231749
\(776\) 0 0
\(777\) 7008.00 0.323566
\(778\) 0 0
\(779\) −22776.0 −1.04754
\(780\) 0 0
\(781\) −900.000 −0.0412350
\(782\) 0 0
\(783\) 6804.00 0.310543
\(784\) 0 0
\(785\) −10.0000 −0.000454669 0
\(786\) 0 0
\(787\) 28190.0 1.27683 0.638415 0.769692i \(-0.279590\pi\)
0.638415 + 0.769692i \(0.279590\pi\)
\(788\) 0 0
\(789\) 504.000 0.0227413
\(790\) 0 0
\(791\) −15264.0 −0.686126
\(792\) 0 0
\(793\) 16408.0 0.734761
\(794\) 0 0
\(795\) 6570.00 0.293099
\(796\) 0 0
\(797\) −29310.0 −1.30265 −0.651326 0.758798i \(-0.725787\pi\)
−0.651326 + 0.758798i \(0.725787\pi\)
\(798\) 0 0
\(799\) 45864.0 2.03073
\(800\) 0 0
\(801\) −5400.00 −0.238202
\(802\) 0 0
\(803\) −13980.0 −0.614375
\(804\) 0 0
\(805\) −1840.00 −0.0805608
\(806\) 0 0
\(807\) 1080.00 0.0471100
\(808\) 0 0
\(809\) 30330.0 1.31810 0.659052 0.752097i \(-0.270958\pi\)
0.659052 + 0.752097i \(0.270958\pi\)
\(810\) 0 0
\(811\) 28508.0 1.23434 0.617171 0.786829i \(-0.288279\pi\)
0.617171 + 0.786829i \(0.288279\pi\)
\(812\) 0 0
\(813\) −21624.0 −0.932825
\(814\) 0 0
\(815\) 2240.00 0.0962746
\(816\) 0 0
\(817\) 2392.00 0.102430
\(818\) 0 0
\(819\) 4032.00 0.172026
\(820\) 0 0
\(821\) −8616.00 −0.366261 −0.183131 0.983089i \(-0.558623\pi\)
−0.183131 + 0.983089i \(0.558623\pi\)
\(822\) 0 0
\(823\) −16702.0 −0.707406 −0.353703 0.935358i \(-0.615078\pi\)
−0.353703 + 0.935358i \(0.615078\pi\)
\(824\) 0 0
\(825\) −2250.00 −0.0949514
\(826\) 0 0
\(827\) 27648.0 1.16253 0.581267 0.813713i \(-0.302557\pi\)
0.581267 + 0.813713i \(0.302557\pi\)
\(828\) 0 0
\(829\) −40678.0 −1.70423 −0.852114 0.523356i \(-0.824680\pi\)
−0.852114 + 0.523356i \(0.824680\pi\)
\(830\) 0 0
\(831\) −6036.00 −0.251969
\(832\) 0 0
\(833\) −6786.00 −0.282258
\(834\) 0 0
\(835\) −14520.0 −0.601779
\(836\) 0 0
\(837\) −5400.00 −0.223000
\(838\) 0 0
\(839\) 6372.00 0.262200 0.131100 0.991369i \(-0.458149\pi\)
0.131100 + 0.991369i \(0.458149\pi\)
\(840\) 0 0
\(841\) 39115.0 1.60380
\(842\) 0 0
\(843\) 16560.0 0.676580
\(844\) 0 0
\(845\) 7065.00 0.287625
\(846\) 0 0
\(847\) 6896.00 0.279751
\(848\) 0 0
\(849\) −2994.00 −0.121029
\(850\) 0 0
\(851\) −3358.00 −0.135265
\(852\) 0 0
\(853\) −42844.0 −1.71975 −0.859877 0.510501i \(-0.829460\pi\)
−0.859877 + 0.510501i \(0.829460\pi\)
\(854\) 0 0
\(855\) 2340.00 0.0935980
\(856\) 0 0
\(857\) −16686.0 −0.665091 −0.332545 0.943087i \(-0.607907\pi\)
−0.332545 + 0.943087i \(0.607907\pi\)
\(858\) 0 0
\(859\) −27196.0 −1.08023 −0.540114 0.841592i \(-0.681619\pi\)
−0.540114 + 0.841592i \(0.681619\pi\)
\(860\) 0 0
\(861\) 21024.0 0.832167
\(862\) 0 0
\(863\) 24828.0 0.979322 0.489661 0.871913i \(-0.337121\pi\)
0.489661 + 0.871913i \(0.337121\pi\)
\(864\) 0 0
\(865\) 11610.0 0.456361
\(866\) 0 0
\(867\) −3513.00 −0.137610
\(868\) 0 0
\(869\) −15600.0 −0.608969
\(870\) 0 0
\(871\) 2632.00 0.102390
\(872\) 0 0
\(873\) −3060.00 −0.118632
\(874\) 0 0
\(875\) 2000.00 0.0772712
\(876\) 0 0
\(877\) 4340.00 0.167105 0.0835527 0.996503i \(-0.473373\pi\)
0.0835527 + 0.996503i \(0.473373\pi\)
\(878\) 0 0
\(879\) 20502.0 0.786707
\(880\) 0 0
\(881\) −21240.0 −0.812252 −0.406126 0.913817i \(-0.633121\pi\)
−0.406126 + 0.913817i \(0.633121\pi\)
\(882\) 0 0
\(883\) 12944.0 0.493319 0.246659 0.969102i \(-0.420667\pi\)
0.246659 + 0.969102i \(0.420667\pi\)
\(884\) 0 0
\(885\) −2520.00 −0.0957162
\(886\) 0 0
\(887\) 1308.00 0.0495134 0.0247567 0.999694i \(-0.492119\pi\)
0.0247567 + 0.999694i \(0.492119\pi\)
\(888\) 0 0
\(889\) −15392.0 −0.580687
\(890\) 0 0
\(891\) 2430.00 0.0913671
\(892\) 0 0
\(893\) −30576.0 −1.14579
\(894\) 0 0
\(895\) 480.000 0.0179270
\(896\) 0 0
\(897\) −1932.00 −0.0719148
\(898\) 0 0
\(899\) −50400.0 −1.86978
\(900\) 0 0
\(901\) 34164.0 1.26323
\(902\) 0 0
\(903\) −2208.00 −0.0813706
\(904\) 0 0
\(905\) 9530.00 0.350042
\(906\) 0 0
\(907\) −14710.0 −0.538520 −0.269260 0.963068i \(-0.586779\pi\)
−0.269260 + 0.963068i \(0.586779\pi\)
\(908\) 0 0
\(909\) −10476.0 −0.382252
\(910\) 0 0
\(911\) 7548.00 0.274508 0.137254 0.990536i \(-0.456172\pi\)
0.137254 + 0.990536i \(0.456172\pi\)
\(912\) 0 0
\(913\) 21240.0 0.769925
\(914\) 0 0
\(915\) −8790.00 −0.317583
\(916\) 0 0
\(917\) 39744.0 1.43126
\(918\) 0 0
\(919\) −17728.0 −0.636336 −0.318168 0.948034i \(-0.603068\pi\)
−0.318168 + 0.948034i \(0.603068\pi\)
\(920\) 0 0
\(921\) 16140.0 0.577450
\(922\) 0 0
\(923\) 840.000 0.0299555
\(924\) 0 0
\(925\) 3650.00 0.129742
\(926\) 0 0
\(927\) 5904.00 0.209183
\(928\) 0 0
\(929\) 19110.0 0.674896 0.337448 0.941344i \(-0.390436\pi\)
0.337448 + 0.941344i \(0.390436\pi\)
\(930\) 0 0
\(931\) 4524.00 0.159257
\(932\) 0 0
\(933\) −13950.0 −0.489499
\(934\) 0 0
\(935\) −11700.0 −0.409231
\(936\) 0 0
\(937\) 16004.0 0.557981 0.278990 0.960294i \(-0.410000\pi\)
0.278990 + 0.960294i \(0.410000\pi\)
\(938\) 0 0
\(939\) 27696.0 0.962540
\(940\) 0 0
\(941\) −30510.0 −1.05696 −0.528479 0.848946i \(-0.677237\pi\)
−0.528479 + 0.848946i \(0.677237\pi\)
\(942\) 0 0
\(943\) −10074.0 −0.347884
\(944\) 0 0
\(945\) −2160.00 −0.0743543
\(946\) 0 0
\(947\) 37044.0 1.27114 0.635569 0.772044i \(-0.280765\pi\)
0.635569 + 0.772044i \(0.280765\pi\)
\(948\) 0 0
\(949\) 13048.0 0.446318
\(950\) 0 0
\(951\) 13698.0 0.467075
\(952\) 0 0
\(953\) −34998.0 −1.18961 −0.594804 0.803871i \(-0.702770\pi\)
−0.594804 + 0.803871i \(0.702770\pi\)
\(954\) 0 0
\(955\) 12600.0 0.426939
\(956\) 0 0
\(957\) 22680.0 0.766082
\(958\) 0 0
\(959\) 18144.0 0.610949
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 0 0
\(963\) −17928.0 −0.599919
\(964\) 0 0
\(965\) 21830.0 0.728220
\(966\) 0 0
\(967\) 13454.0 0.447416 0.223708 0.974656i \(-0.428184\pi\)
0.223708 + 0.974656i \(0.428184\pi\)
\(968\) 0 0
\(969\) 12168.0 0.403398
\(970\) 0 0
\(971\) −57222.0 −1.89119 −0.945593 0.325352i \(-0.894517\pi\)
−0.945593 + 0.325352i \(0.894517\pi\)
\(972\) 0 0
\(973\) 26176.0 0.862450
\(974\) 0 0
\(975\) 2100.00 0.0689783
\(976\) 0 0
\(977\) −5874.00 −0.192350 −0.0961750 0.995364i \(-0.530661\pi\)
−0.0961750 + 0.995364i \(0.530661\pi\)
\(978\) 0 0
\(979\) −18000.0 −0.587623
\(980\) 0 0
\(981\) 11034.0 0.359112
\(982\) 0 0
\(983\) −22944.0 −0.744456 −0.372228 0.928141i \(-0.621406\pi\)
−0.372228 + 0.928141i \(0.621406\pi\)
\(984\) 0 0
\(985\) −18750.0 −0.606523
\(986\) 0 0
\(987\) 28224.0 0.910213
\(988\) 0 0
\(989\) 1058.00 0.0340166
\(990\) 0 0
\(991\) −14956.0 −0.479408 −0.239704 0.970846i \(-0.577050\pi\)
−0.239704 + 0.970846i \(0.577050\pi\)
\(992\) 0 0
\(993\) −9420.00 −0.301042
\(994\) 0 0
\(995\) 7880.00 0.251068
\(996\) 0 0
\(997\) −34396.0 −1.09261 −0.546305 0.837586i \(-0.683966\pi\)
−0.546305 + 0.837586i \(0.683966\pi\)
\(998\) 0 0
\(999\) −3942.00 −0.124844
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 1380.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.4.a.a.1.1 1 1.1 even 1 trivial