Properties

Label 1764.4.a.bc.1.3
Level $1764$
Weight $4$
Character 1764.1
Self dual yes
Analytic conductor $104.079$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 588)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.89590\) of defining polynomial
Character \(\chi\) \(=\) 1764.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.16940 q^{5} +O(q^{10})\) \(q+8.16940 q^{5} -37.8189 q^{11} -39.9319 q^{13} +9.93596 q^{17} +90.4458 q^{19} -118.595 q^{23} -58.2608 q^{25} +78.4061 q^{29} -92.0110 q^{31} +332.435 q^{37} +71.7451 q^{41} -115.947 q^{43} +307.927 q^{47} +403.150 q^{53} -308.958 q^{55} +593.710 q^{59} -333.170 q^{61} -326.220 q^{65} -743.511 q^{67} +728.272 q^{71} +801.749 q^{73} +1067.91 q^{79} +906.756 q^{83} +81.1709 q^{85} +1113.27 q^{89} +738.889 q^{95} -1480.94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 48q^{17} - 192q^{19} - 192q^{23} + 324q^{25} - 96q^{29} - 48q^{31} + 256q^{37} + 1008q^{41} - 112q^{43} + 864q^{47} + 648q^{53} - 2352q^{55} + 336q^{59} - 960q^{61} + 360q^{65} + 720q^{67} + 1344q^{71} - 672q^{73} - 1984q^{79} + 3120q^{83} + 680q^{85} + 2160q^{89} + 3744q^{95} - 2016q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 8.16940 0.730694 0.365347 0.930871i \(-0.380950\pi\)
0.365347 + 0.930871i \(0.380950\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −37.8189 −1.03662 −0.518310 0.855193i \(-0.673439\pi\)
−0.518310 + 0.855193i \(0.673439\pi\)
\(12\) 0 0
\(13\) −39.9319 −0.851933 −0.425966 0.904739i \(-0.640066\pi\)
−0.425966 + 0.904739i \(0.640066\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.93596 0.141754 0.0708772 0.997485i \(-0.477420\pi\)
0.0708772 + 0.997485i \(0.477420\pi\)
\(18\) 0 0
\(19\) 90.4458 1.09209 0.546045 0.837756i \(-0.316133\pi\)
0.546045 + 0.837756i \(0.316133\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −118.595 −1.07516 −0.537580 0.843213i \(-0.680661\pi\)
−0.537580 + 0.843213i \(0.680661\pi\)
\(24\) 0 0
\(25\) −58.2608 −0.466087
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 78.4061 0.502057 0.251028 0.967980i \(-0.419231\pi\)
0.251028 + 0.967980i \(0.419231\pi\)
\(30\) 0 0
\(31\) −92.0110 −0.533086 −0.266543 0.963823i \(-0.585881\pi\)
−0.266543 + 0.963823i \(0.585881\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 332.435 1.47708 0.738541 0.674209i \(-0.235515\pi\)
0.738541 + 0.674209i \(0.235515\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 71.7451 0.273285 0.136643 0.990620i \(-0.456369\pi\)
0.136643 + 0.990620i \(0.456369\pi\)
\(42\) 0 0
\(43\) −115.947 −0.411202 −0.205601 0.978636i \(-0.565915\pi\)
−0.205601 + 0.978636i \(0.565915\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 307.927 0.955656 0.477828 0.878453i \(-0.341424\pi\)
0.477828 + 0.878453i \(0.341424\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 403.150 1.04485 0.522424 0.852686i \(-0.325028\pi\)
0.522424 + 0.852686i \(0.325028\pi\)
\(54\) 0 0
\(55\) −308.958 −0.757451
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 593.710 1.31008 0.655038 0.755596i \(-0.272653\pi\)
0.655038 + 0.755596i \(0.272653\pi\)
\(60\) 0 0
\(61\) −333.170 −0.699313 −0.349657 0.936878i \(-0.613702\pi\)
−0.349657 + 0.936878i \(0.613702\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −326.220 −0.622502
\(66\) 0 0
\(67\) −743.511 −1.35574 −0.677869 0.735183i \(-0.737096\pi\)
−0.677869 + 0.735183i \(0.737096\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 728.272 1.21732 0.608662 0.793429i \(-0.291706\pi\)
0.608662 + 0.793429i \(0.291706\pi\)
\(72\) 0 0
\(73\) 801.749 1.28545 0.642723 0.766098i \(-0.277804\pi\)
0.642723 + 0.766098i \(0.277804\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1067.91 1.52087 0.760435 0.649414i \(-0.224986\pi\)
0.760435 + 0.649414i \(0.224986\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 906.756 1.19915 0.599575 0.800319i \(-0.295336\pi\)
0.599575 + 0.800319i \(0.295336\pi\)
\(84\) 0 0
\(85\) 81.1709 0.103579
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1113.27 1.32591 0.662957 0.748658i \(-0.269301\pi\)
0.662957 + 0.748658i \(0.269301\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 738.889 0.797983
\(96\) 0 0
\(97\) −1480.94 −1.55017 −0.775084 0.631858i \(-0.782293\pi\)
−0.775084 + 0.631858i \(0.782293\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −556.316 −0.548075 −0.274037 0.961719i \(-0.588359\pi\)
−0.274037 + 0.961719i \(0.588359\pi\)
\(102\) 0 0
\(103\) −552.435 −0.528476 −0.264238 0.964458i \(-0.585120\pi\)
−0.264238 + 0.964458i \(0.585120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −533.804 −0.482288 −0.241144 0.970489i \(-0.577523\pi\)
−0.241144 + 0.970489i \(0.577523\pi\)
\(108\) 0 0
\(109\) 1094.63 0.961894 0.480947 0.876750i \(-0.340293\pi\)
0.480947 + 0.876750i \(0.340293\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1425.18 1.18646 0.593228 0.805035i \(-0.297853\pi\)
0.593228 + 0.805035i \(0.297853\pi\)
\(114\) 0 0
\(115\) −968.847 −0.785612
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 99.2659 0.0745800
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1497.13 −1.07126
\(126\) 0 0
\(127\) −786.485 −0.549522 −0.274761 0.961513i \(-0.588599\pi\)
−0.274761 + 0.961513i \(0.588599\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −27.7303 −0.0184947 −0.00924735 0.999957i \(-0.502944\pi\)
−0.00924735 + 0.999957i \(0.502944\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2362.98 1.47360 0.736798 0.676113i \(-0.236337\pi\)
0.736798 + 0.676113i \(0.236337\pi\)
\(138\) 0 0
\(139\) −2513.28 −1.53362 −0.766811 0.641873i \(-0.778158\pi\)
−0.766811 + 0.641873i \(0.778158\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1510.18 0.883130
\(144\) 0 0
\(145\) 640.531 0.366850
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2265.83 −1.24580 −0.622900 0.782302i \(-0.714045\pi\)
−0.622900 + 0.782302i \(0.714045\pi\)
\(150\) 0 0
\(151\) 283.146 0.152597 0.0762984 0.997085i \(-0.475690\pi\)
0.0762984 + 0.997085i \(0.475690\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −751.675 −0.389522
\(156\) 0 0
\(157\) −192.581 −0.0978956 −0.0489478 0.998801i \(-0.515587\pi\)
−0.0489478 + 0.998801i \(0.515587\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −842.833 −0.405005 −0.202502 0.979282i \(-0.564907\pi\)
−0.202502 + 0.979282i \(0.564907\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3859.21 −1.78823 −0.894116 0.447836i \(-0.852195\pi\)
−0.894116 + 0.447836i \(0.852195\pi\)
\(168\) 0 0
\(169\) −602.441 −0.274211
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1510.66 0.663891 0.331946 0.943298i \(-0.392295\pi\)
0.331946 + 0.943298i \(0.392295\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2464.63 1.02913 0.514566 0.857451i \(-0.327953\pi\)
0.514566 + 0.857451i \(0.327953\pi\)
\(180\) 0 0
\(181\) 3297.36 1.35409 0.677047 0.735940i \(-0.263259\pi\)
0.677047 + 0.735940i \(0.263259\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2715.80 1.07929
\(186\) 0 0
\(187\) −375.767 −0.146945
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4729.71 1.79178 0.895889 0.444278i \(-0.146540\pi\)
0.895889 + 0.444278i \(0.146540\pi\)
\(192\) 0 0
\(193\) 4553.18 1.69816 0.849080 0.528264i \(-0.177157\pi\)
0.849080 + 0.528264i \(0.177157\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3109.06 1.12442 0.562212 0.826993i \(-0.309950\pi\)
0.562212 + 0.826993i \(0.309950\pi\)
\(198\) 0 0
\(199\) 443.512 0.157989 0.0789943 0.996875i \(-0.474829\pi\)
0.0789943 + 0.996875i \(0.474829\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 586.114 0.199688
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3420.56 −1.13208
\(210\) 0 0
\(211\) 4653.28 1.51822 0.759112 0.650960i \(-0.225633\pi\)
0.759112 + 0.650960i \(0.225633\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −947.214 −0.300463
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −396.762 −0.120765
\(222\) 0 0
\(223\) −2778.90 −0.834481 −0.417240 0.908796i \(-0.637003\pi\)
−0.417240 + 0.908796i \(0.637003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3217.22 0.940681 0.470340 0.882485i \(-0.344131\pi\)
0.470340 + 0.882485i \(0.344131\pi\)
\(228\) 0 0
\(229\) 3728.01 1.07578 0.537891 0.843014i \(-0.319221\pi\)
0.537891 + 0.843014i \(0.319221\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3603.51 1.01319 0.506596 0.862183i \(-0.330903\pi\)
0.506596 + 0.862183i \(0.330903\pi\)
\(234\) 0 0
\(235\) 2515.58 0.698292
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2348.76 −0.635684 −0.317842 0.948144i \(-0.602958\pi\)
−0.317842 + 0.948144i \(0.602958\pi\)
\(240\) 0 0
\(241\) 5093.39 1.36139 0.680693 0.732569i \(-0.261679\pi\)
0.680693 + 0.732569i \(0.261679\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3611.68 −0.930386
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5939.02 −1.49350 −0.746749 0.665106i \(-0.768386\pi\)
−0.746749 + 0.665106i \(0.768386\pi\)
\(252\) 0 0
\(253\) 4485.11 1.11453
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1515.89 −0.367932 −0.183966 0.982933i \(-0.558894\pi\)
−0.183966 + 0.982933i \(0.558894\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6433.92 1.50849 0.754244 0.656594i \(-0.228003\pi\)
0.754244 + 0.656594i \(0.228003\pi\)
\(264\) 0 0
\(265\) 3293.50 0.763464
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6949.79 −1.57523 −0.787614 0.616169i \(-0.788684\pi\)
−0.787614 + 0.616169i \(0.788684\pi\)
\(270\) 0 0
\(271\) 961.994 0.215635 0.107817 0.994171i \(-0.465614\pi\)
0.107817 + 0.994171i \(0.465614\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2203.36 0.483154
\(276\) 0 0
\(277\) 760.010 0.164854 0.0824271 0.996597i \(-0.473733\pi\)
0.0824271 + 0.996597i \(0.473733\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4412.07 0.936662 0.468331 0.883553i \(-0.344855\pi\)
0.468331 + 0.883553i \(0.344855\pi\)
\(282\) 0 0
\(283\) 2602.15 0.546578 0.273289 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4814.28 −0.979906
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9332.18 −1.86072 −0.930361 0.366644i \(-0.880507\pi\)
−0.930361 + 0.366644i \(0.880507\pi\)
\(294\) 0 0
\(295\) 4850.26 0.957264
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4735.71 0.915964
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2721.80 −0.510984
\(306\) 0 0
\(307\) −4895.51 −0.910103 −0.455051 0.890465i \(-0.650379\pi\)
−0.455051 + 0.890465i \(0.650379\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7505.28 1.36844 0.684221 0.729275i \(-0.260142\pi\)
0.684221 + 0.729275i \(0.260142\pi\)
\(312\) 0 0
\(313\) 6349.27 1.14659 0.573294 0.819350i \(-0.305665\pi\)
0.573294 + 0.819350i \(0.305665\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7999.60 1.41736 0.708679 0.705531i \(-0.249292\pi\)
0.708679 + 0.705531i \(0.249292\pi\)
\(318\) 0 0
\(319\) −2965.23 −0.520442
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 898.666 0.154808
\(324\) 0 0
\(325\) 2326.47 0.397074
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2121.56 −0.352300 −0.176150 0.984363i \(-0.556364\pi\)
−0.176150 + 0.984363i \(0.556364\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6074.05 −0.990629
\(336\) 0 0
\(337\) −9114.19 −1.47324 −0.736620 0.676307i \(-0.763579\pi\)
−0.736620 + 0.676307i \(0.763579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3479.75 0.552607
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1673.07 0.258833 0.129416 0.991590i \(-0.458690\pi\)
0.129416 + 0.991590i \(0.458690\pi\)
\(348\) 0 0
\(349\) 3467.56 0.531845 0.265923 0.963994i \(-0.414323\pi\)
0.265923 + 0.963994i \(0.414323\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3984.24 0.600736 0.300368 0.953823i \(-0.402891\pi\)
0.300368 + 0.953823i \(0.402891\pi\)
\(354\) 0 0
\(355\) 5949.55 0.889491
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2170.93 −0.319157 −0.159579 0.987185i \(-0.551014\pi\)
−0.159579 + 0.987185i \(0.551014\pi\)
\(360\) 0 0
\(361\) 1321.45 0.192659
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6549.81 0.939268
\(366\) 0 0
\(367\) −3592.12 −0.510918 −0.255459 0.966820i \(-0.582227\pi\)
−0.255459 + 0.966820i \(0.582227\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7439.80 1.03276 0.516378 0.856361i \(-0.327280\pi\)
0.516378 + 0.856361i \(0.327280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3130.91 −0.427719
\(378\) 0 0
\(379\) 11243.9 1.52390 0.761951 0.647635i \(-0.224242\pi\)
0.761951 + 0.647635i \(0.224242\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3541.25 0.472452 0.236226 0.971698i \(-0.424089\pi\)
0.236226 + 0.971698i \(0.424089\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10558.2 1.37615 0.688074 0.725640i \(-0.258456\pi\)
0.688074 + 0.725640i \(0.258456\pi\)
\(390\) 0 0
\(391\) −1178.35 −0.152409
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8724.15 1.11129
\(396\) 0 0
\(397\) 540.574 0.0683391 0.0341696 0.999416i \(-0.489121\pi\)
0.0341696 + 0.999416i \(0.489121\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2167.03 −0.269866 −0.134933 0.990855i \(-0.543082\pi\)
−0.134933 + 0.990855i \(0.543082\pi\)
\(402\) 0 0
\(403\) 3674.17 0.454153
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12572.3 −1.53117
\(408\) 0 0
\(409\) 957.690 0.115782 0.0578908 0.998323i \(-0.481562\pi\)
0.0578908 + 0.998323i \(0.481562\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7407.66 0.876211
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6464.42 0.753718 0.376859 0.926271i \(-0.377004\pi\)
0.376859 + 0.926271i \(0.377004\pi\)
\(420\) 0 0
\(421\) −6201.23 −0.717885 −0.358943 0.933360i \(-0.616863\pi\)
−0.358943 + 0.933360i \(0.616863\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −578.877 −0.0660698
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1415.59 −0.158206 −0.0791029 0.996866i \(-0.525206\pi\)
−0.0791029 + 0.996866i \(0.525206\pi\)
\(432\) 0 0
\(433\) −8905.73 −0.988411 −0.494206 0.869345i \(-0.664541\pi\)
−0.494206 + 0.869345i \(0.664541\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10726.4 −1.17417
\(438\) 0 0
\(439\) −10876.5 −1.18248 −0.591238 0.806497i \(-0.701361\pi\)
−0.591238 + 0.806497i \(0.701361\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11403.0 −1.22296 −0.611479 0.791260i \(-0.709425\pi\)
−0.611479 + 0.791260i \(0.709425\pi\)
\(444\) 0 0
\(445\) 9094.74 0.968837
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12689.0 1.33370 0.666852 0.745190i \(-0.267641\pi\)
0.666852 + 0.745190i \(0.267641\pi\)
\(450\) 0 0
\(451\) −2713.32 −0.283293
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2270.05 0.232360 0.116180 0.993228i \(-0.462935\pi\)
0.116180 + 0.993228i \(0.462935\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10731.2 1.08417 0.542085 0.840324i \(-0.317635\pi\)
0.542085 + 0.840324i \(0.317635\pi\)
\(462\) 0 0
\(463\) −3307.74 −0.332017 −0.166008 0.986124i \(-0.553088\pi\)
−0.166008 + 0.986124i \(0.553088\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9246.69 −0.916244 −0.458122 0.888889i \(-0.651478\pi\)
−0.458122 + 0.888889i \(0.651478\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4384.96 0.426260
\(474\) 0 0
\(475\) −5269.45 −0.509008
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15223.3 1.45213 0.726066 0.687625i \(-0.241347\pi\)
0.726066 + 0.687625i \(0.241347\pi\)
\(480\) 0 0
\(481\) −13274.8 −1.25837
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12098.4 −1.13270
\(486\) 0 0
\(487\) −7758.72 −0.721933 −0.360966 0.932579i \(-0.617553\pi\)
−0.360966 + 0.932579i \(0.617553\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8342.30 0.766767 0.383384 0.923589i \(-0.374759\pi\)
0.383384 + 0.923589i \(0.374759\pi\)
\(492\) 0 0
\(493\) 779.040 0.0711688
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2586.95 −0.232079 −0.116040 0.993245i \(-0.537020\pi\)
−0.116040 + 0.993245i \(0.537020\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 409.682 0.0363157 0.0181578 0.999835i \(-0.494220\pi\)
0.0181578 + 0.999835i \(0.494220\pi\)
\(504\) 0 0
\(505\) −4544.77 −0.400475
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4390.51 0.382330 0.191165 0.981558i \(-0.438774\pi\)
0.191165 + 0.981558i \(0.438774\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4513.06 −0.386154
\(516\) 0 0
\(517\) −11645.5 −0.990652
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −19429.5 −1.63382 −0.816912 0.576762i \(-0.804316\pi\)
−0.816912 + 0.576762i \(0.804316\pi\)
\(522\) 0 0
\(523\) −17948.7 −1.50065 −0.750326 0.661067i \(-0.770104\pi\)
−0.750326 + 0.661067i \(0.770104\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −914.217 −0.0755672
\(528\) 0 0
\(529\) 1897.67 0.155968
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2864.92 −0.232821
\(534\) 0 0
\(535\) −4360.86 −0.352405
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23094.0 1.83528 0.917641 0.397411i \(-0.130091\pi\)
0.917641 + 0.397411i \(0.130091\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8942.47 0.702850
\(546\) 0 0
\(547\) −2266.68 −0.177178 −0.0885889 0.996068i \(-0.528236\pi\)
−0.0885889 + 0.996068i \(0.528236\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7091.50 0.548291
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11038.5 0.839709 0.419854 0.907592i \(-0.362081\pi\)
0.419854 + 0.907592i \(0.362081\pi\)
\(558\) 0 0
\(559\) 4629.97 0.350316
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7059.36 −0.528448 −0.264224 0.964461i \(-0.585116\pi\)
−0.264224 + 0.964461i \(0.585116\pi\)
\(564\) 0 0
\(565\) 11642.9 0.866936
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1352.81 0.0996706 0.0498353 0.998757i \(-0.484130\pi\)
0.0498353 + 0.998757i \(0.484130\pi\)
\(570\) 0 0
\(571\) −20839.2 −1.52731 −0.763655 0.645625i \(-0.776597\pi\)
−0.763655 + 0.645625i \(0.776597\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6909.42 0.501117
\(576\) 0 0
\(577\) −10014.6 −0.722556 −0.361278 0.932458i \(-0.617659\pi\)
−0.361278 + 0.932458i \(0.617659\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −15246.7 −1.08311
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20864.8 1.46709 0.733546 0.679640i \(-0.237864\pi\)
0.733546 + 0.679640i \(0.237864\pi\)
\(588\) 0 0
\(589\) −8322.01 −0.582177
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17547.7 1.21517 0.607586 0.794254i \(-0.292138\pi\)
0.607586 + 0.794254i \(0.292138\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 130.682 0.00891405 0.00445703 0.999990i \(-0.498581\pi\)
0.00445703 + 0.999990i \(0.498581\pi\)
\(600\) 0 0
\(601\) 5964.47 0.404818 0.202409 0.979301i \(-0.435123\pi\)
0.202409 + 0.979301i \(0.435123\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 810.944 0.0544951
\(606\) 0 0
\(607\) 9911.63 0.662769 0.331384 0.943496i \(-0.392484\pi\)
0.331384 + 0.943496i \(0.392484\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12296.1 −0.814155
\(612\) 0 0
\(613\) −6482.53 −0.427124 −0.213562 0.976930i \(-0.568506\pi\)
−0.213562 + 0.976930i \(0.568506\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7189.36 −0.469097 −0.234548 0.972104i \(-0.575361\pi\)
−0.234548 + 0.972104i \(0.575361\pi\)
\(618\) 0 0
\(619\) −1861.39 −0.120865 −0.0604327 0.998172i \(-0.519248\pi\)
−0.0604327 + 0.998172i \(0.519248\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4948.07 −0.316677
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3303.06 0.209383
\(630\) 0 0
\(631\) −29032.0 −1.83161 −0.915803 0.401627i \(-0.868445\pi\)
−0.915803 + 0.401627i \(0.868445\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6425.12 −0.401532
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24457.7 −1.50705 −0.753527 0.657417i \(-0.771649\pi\)
−0.753527 + 0.657417i \(0.771649\pi\)
\(642\) 0 0
\(643\) −10968.2 −0.672695 −0.336348 0.941738i \(-0.609192\pi\)
−0.336348 + 0.941738i \(0.609192\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5484.74 0.333272 0.166636 0.986018i \(-0.446709\pi\)
0.166636 + 0.986018i \(0.446709\pi\)
\(648\) 0 0
\(649\) −22453.4 −1.35805
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10330.0 0.619055 0.309527 0.950891i \(-0.399829\pi\)
0.309527 + 0.950891i \(0.399829\pi\)
\(654\) 0 0
\(655\) −226.540 −0.0135140
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26822.2 −1.58550 −0.792751 0.609546i \(-0.791352\pi\)
−0.792751 + 0.609546i \(0.791352\pi\)
\(660\) 0 0
\(661\) −2498.76 −0.147035 −0.0735177 0.997294i \(-0.523423\pi\)
−0.0735177 + 0.997294i \(0.523423\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9298.54 −0.539791
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12600.1 0.724922
\(672\) 0 0
\(673\) 10092.1 0.578044 0.289022 0.957322i \(-0.406670\pi\)
0.289022 + 0.957322i \(0.406670\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26366.4 −1.49681 −0.748406 0.663241i \(-0.769180\pi\)
−0.748406 + 0.663241i \(0.769180\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22570.8 −1.26449 −0.632245 0.774769i \(-0.717866\pi\)
−0.632245 + 0.774769i \(0.717866\pi\)
\(684\) 0 0
\(685\) 19304.1 1.07675
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16098.6 −0.890140
\(690\) 0 0
\(691\) 11931.5 0.656867 0.328434 0.944527i \(-0.393479\pi\)
0.328434 + 0.944527i \(0.393479\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20532.0 −1.12061
\(696\) 0 0
\(697\) 712.856 0.0387394
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −16592.6 −0.893997 −0.446999 0.894535i \(-0.647507\pi\)
−0.446999 + 0.894535i \(0.647507\pi\)
\(702\) 0 0
\(703\) 30067.4 1.61311
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18714.0 0.991283 0.495641 0.868527i \(-0.334933\pi\)
0.495641 + 0.868527i \(0.334933\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10912.0 0.573152
\(714\) 0 0
\(715\) 12337.3 0.645298
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24854.8 −1.28919 −0.644594 0.764525i \(-0.722974\pi\)
−0.644594 + 0.764525i \(0.722974\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4568.00 −0.234002
\(726\) 0 0
\(727\) 22506.0 1.14814 0.574071 0.818805i \(-0.305363\pi\)
0.574071 + 0.818805i \(0.305363\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1152.04 −0.0582897
\(732\) 0 0
\(733\) 23792.0 1.19888 0.599440 0.800420i \(-0.295390\pi\)
0.599440 + 0.800420i \(0.295390\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28118.8 1.40538
\(738\) 0 0
\(739\) 4803.24 0.239093 0.119547 0.992829i \(-0.461856\pi\)
0.119547 + 0.992829i \(0.461856\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5076.10 −0.250638 −0.125319 0.992116i \(-0.539995\pi\)
−0.125319 + 0.992116i \(0.539995\pi\)
\(744\) 0 0
\(745\) −18510.5 −0.910298
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −9158.72 −0.445015 −0.222508 0.974931i \(-0.571424\pi\)
−0.222508 + 0.974931i \(0.571424\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2313.14 0.111502
\(756\) 0 0
\(757\) 33682.2 1.61717 0.808587 0.588377i \(-0.200233\pi\)
0.808587 + 0.588377i \(0.200233\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13346.3 0.635745 0.317873 0.948133i \(-0.397032\pi\)
0.317873 + 0.948133i \(0.397032\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23708.0 −1.11610
\(768\) 0 0
\(769\) 15530.6 0.728281 0.364140 0.931344i \(-0.381363\pi\)
0.364140 + 0.931344i \(0.381363\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11015.2 −0.512536 −0.256268 0.966606i \(-0.582493\pi\)
−0.256268 + 0.966606i \(0.582493\pi\)
\(774\) 0 0
\(775\) 5360.63 0.248464
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6489.04 0.298452
\(780\) 0 0
\(781\) −27542.4 −1.26190
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1573.27 −0.0715317
\(786\) 0 0
\(787\) −21192.6 −0.959892 −0.479946 0.877298i \(-0.659344\pi\)
−0.479946 + 0.877298i \(0.659344\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13304.1 0.595768
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18189.9 0.808432 0.404216 0.914663i \(-0.367544\pi\)
0.404216 + 0.914663i \(0.367544\pi\)
\(798\) 0 0
\(799\) 3059.56 0.135468
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30321.2 −1.33252
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −45516.5 −1.97809 −0.989045 0.147611i \(-0.952842\pi\)
−0.989045 + 0.147611i \(0.952842\pi\)
\(810\) 0 0
\(811\) 42099.9 1.82285 0.911423 0.411472i \(-0.134985\pi\)
0.911423 + 0.411472i \(0.134985\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6885.45 −0.295935
\(816\) 0 0
\(817\) −10486.9 −0.449069
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26555.9 1.12888 0.564439 0.825475i \(-0.309093\pi\)
0.564439 + 0.825475i \(0.309093\pi\)
\(822\) 0 0
\(823\) −1554.25 −0.0658295 −0.0329147 0.999458i \(-0.510479\pi\)
−0.0329147 + 0.999458i \(0.510479\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10548.3 0.443531 0.221766 0.975100i \(-0.428818\pi\)
0.221766 + 0.975100i \(0.428818\pi\)
\(828\) 0 0
\(829\) −4331.26 −0.181461 −0.0907303 0.995875i \(-0.528920\pi\)
−0.0907303 + 0.995875i \(0.528920\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −31527.5 −1.30665
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23557.7 −0.969369 −0.484685 0.874689i \(-0.661066\pi\)
−0.484685 + 0.874689i \(0.661066\pi\)
\(840\) 0 0
\(841\) −18241.5 −0.747939
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4921.59 −0.200364
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −39425.0 −1.58810
\(852\) 0 0
\(853\) 11493.6 0.461350 0.230675 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2534.23 −0.101012 −0.0505062 0.998724i \(-0.516083\pi\)
−0.0505062 + 0.998724i \(0.516083\pi\)
\(858\) 0 0
\(859\) −3127.09 −0.124208 −0.0621041 0.998070i \(-0.519781\pi\)
−0.0621041 + 0.998070i \(0.519781\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15309.9 −0.603887 −0.301943 0.953326i \(-0.597635\pi\)
−0.301943 + 0.953326i \(0.597635\pi\)
\(864\) 0 0
\(865\) 12341.2 0.485101
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −40387.0 −1.57656
\(870\) 0 0
\(871\) 29689.8 1.15500
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −42420.7 −1.63335 −0.816673 0.577100i \(-0.804184\pi\)
−0.816673 + 0.577100i \(0.804184\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10060.8 0.384740 0.192370 0.981322i \(-0.438383\pi\)
0.192370 + 0.981322i \(0.438383\pi\)
\(882\) 0 0
\(883\) 17437.8 0.664585 0.332293 0.943176i \(-0.392178\pi\)
0.332293 + 0.943176i \(0.392178\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16016.7 −0.606299 −0.303149 0.952943i \(-0.598038\pi\)
−0.303149 + 0.952943i \(0.598038\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27850.8 1.04366
\(894\) 0 0
\(895\) 20134.5 0.751981
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7214.22 −0.267639
\(900\) 0 0
\(901\) 4005.69 0.148112
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26937.5 0.989428
\(906\) 0 0
\(907\) −10131.0 −0.370887 −0.185443 0.982655i \(-0.559372\pi\)
−0.185443 + 0.982655i \(0.559372\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1320.58 −0.0480273 −0.0240137 0.999712i \(-0.507645\pi\)
−0.0240137 + 0.999712i \(0.507645\pi\)
\(912\) 0 0
\(913\) −34292.5 −1.24306
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −7285.76 −0.261518 −0.130759 0.991414i \(-0.541741\pi\)
−0.130759 + 0.991414i \(0.541741\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −29081.3 −1.03708
\(924\) 0 0
\(925\) −19368.0 −0.688448
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23708.0 0.837280 0.418640 0.908152i \(-0.362507\pi\)
0.418640 + 0.908152i \(0.362507\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3069.79 −0.107372
\(936\) 0 0
\(937\) 8853.60 0.308682 0.154341 0.988018i \(-0.450675\pi\)
0.154341 + 0.988018i \(0.450675\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −53971.6 −1.86974 −0.934869 0.354994i \(-0.884483\pi\)
−0.934869 + 0.354994i \(0.884483\pi\)
\(942\) 0 0
\(943\) −8508.57 −0.293825
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8019.80 −0.275194 −0.137597 0.990488i \(-0.543938\pi\)
−0.137597 + 0.990488i \(0.543938\pi\)
\(948\) 0 0
\(949\) −32015.4 −1.09511
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −42628.0 −1.44896 −0.724479 0.689297i \(-0.757920\pi\)
−0.724479 + 0.689297i \(0.757920\pi\)
\(954\) 0 0
\(955\) 38638.9 1.30924
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21325.0 −0.715820
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37196.8 1.24084
\(966\) 0 0
\(967\) −43161.7 −1.43535 −0.717676 0.696377i \(-0.754794\pi\)
−0.717676 + 0.696377i \(0.754794\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −39775.4 −1.31458 −0.657288 0.753640i \(-0.728296\pi\)
−0.657288 + 0.753640i \(0.728296\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26658.1 0.872945 0.436472 0.899718i \(-0.356228\pi\)
0.436472 + 0.899718i \(0.356228\pi\)
\(978\) 0 0
\(979\) −42102.6 −1.37447
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2942.43 −0.0954719 −0.0477359 0.998860i \(-0.515201\pi\)
−0.0477359 + 0.998860i \(0.515201\pi\)
\(984\) 0 0
\(985\) 25399.2 0.821610
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13750.6 0.442108
\(990\) 0 0
\(991\) 44969.8 1.44149 0.720743 0.693203i \(-0.243801\pi\)
0.720743 + 0.693203i \(0.243801\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3623.23 0.115441
\(996\) 0 0
\(997\) −27116.4 −0.861370 −0.430685 0.902502i \(-0.641728\pi\)
−0.430685 + 0.902502i \(0.641728\pi\)
\(998\) 0 0
\(999\) 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 1764.4.a.bc.1.3 4
3.2 odd 2 588.4.a.j.1.2 4
7.2 even 3 1764.4.k.bb.361.2 8
7.3 odd 6 1764.4.k.bd.1549.3 8
7.4 even 3 1764.4.k.bb.1549.2 8
7.5 odd 6 1764.4.k.bd.361.3 8
7.6 odd 2 1764.4.a.ba.1.2 4
12.11 even 2 2352.4.a.cq.1.2 4
21.2 odd 6 588.4.i.l.361.3 8
21.5 even 6 588.4.i.k.361.2 8
21.11 odd 6 588.4.i.l.373.3 8
21.17 even 6 588.4.i.k.373.2 8
21.20 even 2 588.4.a.k.1.3 yes 4
84.83 odd 2 2352.4.a.cl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.2 4 3.2 odd 2
588.4.a.k.1.3 yes 4 21.20 even 2
588.4.i.k.361.2 8 21.5 even 6
588.4.i.k.373.2 8 21.17 even 6
588.4.i.l.361.3 8 21.2 odd 6
588.4.i.l.373.3 8 21.11 odd 6
1764.4.a.ba.1.2 4 7.6 odd 2
1764.4.a.bc.1.3 4 1.1 even 1 trivial
1764.4.k.bb.361.2 8 7.2 even 3
1764.4.k.bb.1549.2 8 7.4 even 3
1764.4.k.bd.361.3 8 7.5 odd 6
1764.4.k.bd.1549.3 8 7.3 odd 6
2352.4.a.cl.1.3 4 84.83 odd 2
2352.4.a.cq.1.2 4 12.11 even 2