Properties

Label 2352.4.a.cq.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
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 1.1
Root \(4.93153\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -19.1403 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -19.1403 q^{5} +9.00000 q^{9} -40.5951 q^{11} +50.4776 q^{13} -57.4210 q^{15} -51.9195 q^{17} +33.1331 q^{19} -62.8499 q^{23} +241.352 q^{25} +27.0000 q^{27} +129.921 q^{29} -242.415 q^{31} -121.785 q^{33} -389.385 q^{37} +151.433 q^{39} -470.110 q^{41} +125.003 q^{43} -172.263 q^{45} +386.624 q^{47} -155.758 q^{51} -611.436 q^{53} +777.004 q^{55} +99.3994 q^{57} -226.432 q^{59} +725.191 q^{61} -966.158 q^{65} -1045.11 q^{67} -188.550 q^{69} -169.839 q^{71} -381.731 q^{73} +724.057 q^{75} +1161.12 q^{79} +81.0000 q^{81} +808.448 q^{83} +993.756 q^{85} +389.764 q^{87} -319.867 q^{89} -727.245 q^{93} -634.179 q^{95} +1133.24 q^{97} -365.356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 36q^{9} - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} + 112q^{43} + 864q^{47} - 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} - 960q^{61} - 360q^{65} - 720q^{67} - 576q^{69} + 1344q^{71} - 672q^{73} + 972q^{75} + 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} - 2160q^{89} + 144q^{93} + 3744q^{95} - 2016q^{97} + O(q^{100}) \)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −19.1403 −1.71196 −0.855982 0.517006i \(-0.827046\pi\)
−0.855982 + 0.517006i \(0.827046\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −40.5951 −1.11272 −0.556359 0.830942i \(-0.687802\pi\)
−0.556359 + 0.830942i \(0.687802\pi\)
\(12\) 0 0
\(13\) 50.4776 1.07692 0.538460 0.842651i \(-0.319006\pi\)
0.538460 + 0.842651i \(0.319006\pi\)
\(14\) 0 0
\(15\) −57.4210 −0.988402
\(16\) 0 0
\(17\) −51.9195 −0.740725 −0.370362 0.928887i \(-0.620766\pi\)
−0.370362 + 0.928887i \(0.620766\pi\)
\(18\) 0 0
\(19\) 33.1331 0.400066 0.200033 0.979789i \(-0.435895\pi\)
0.200033 + 0.979789i \(0.435895\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −62.8499 −0.569787 −0.284894 0.958559i \(-0.591958\pi\)
−0.284894 + 0.958559i \(0.591958\pi\)
\(24\) 0 0
\(25\) 241.352 1.93082
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 129.921 0.831924 0.415962 0.909382i \(-0.363445\pi\)
0.415962 + 0.909382i \(0.363445\pi\)
\(30\) 0 0
\(31\) −242.415 −1.40448 −0.702242 0.711938i \(-0.747818\pi\)
−0.702242 + 0.711938i \(0.747818\pi\)
\(32\) 0 0
\(33\) −121.785 −0.642428
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −389.385 −1.73012 −0.865061 0.501667i \(-0.832720\pi\)
−0.865061 + 0.501667i \(0.832720\pi\)
\(38\) 0 0
\(39\) 151.433 0.621761
\(40\) 0 0
\(41\) −470.110 −1.79071 −0.895353 0.445358i \(-0.853076\pi\)
−0.895353 + 0.445358i \(0.853076\pi\)
\(42\) 0 0
\(43\) 125.003 0.443321 0.221661 0.975124i \(-0.428852\pi\)
0.221661 + 0.975124i \(0.428852\pi\)
\(44\) 0 0
\(45\) −172.263 −0.570654
\(46\) 0 0
\(47\) 386.624 1.19989 0.599946 0.800040i \(-0.295189\pi\)
0.599946 + 0.800040i \(0.295189\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −155.758 −0.427658
\(52\) 0 0
\(53\) −611.436 −1.58466 −0.792332 0.610091i \(-0.791133\pi\)
−0.792332 + 0.610091i \(0.791133\pi\)
\(54\) 0 0
\(55\) 777.004 1.90493
\(56\) 0 0
\(57\) 99.3994 0.230978
\(58\) 0 0
\(59\) −226.432 −0.499644 −0.249822 0.968292i \(-0.580372\pi\)
−0.249822 + 0.968292i \(0.580372\pi\)
\(60\) 0 0
\(61\) 725.191 1.52215 0.761075 0.648664i \(-0.224672\pi\)
0.761075 + 0.648664i \(0.224672\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −966.158 −1.84365
\(66\) 0 0
\(67\) −1045.11 −1.90568 −0.952840 0.303474i \(-0.901853\pi\)
−0.952840 + 0.303474i \(0.901853\pi\)
\(68\) 0 0
\(69\) −188.550 −0.328967
\(70\) 0 0
\(71\) −169.839 −0.283889 −0.141945 0.989875i \(-0.545336\pi\)
−0.141945 + 0.989875i \(0.545336\pi\)
\(72\) 0 0
\(73\) −381.731 −0.612031 −0.306015 0.952027i \(-0.598996\pi\)
−0.306015 + 0.952027i \(0.598996\pi\)
\(74\) 0 0
\(75\) 724.057 1.11476
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1161.12 1.65362 0.826812 0.562479i \(-0.190152\pi\)
0.826812 + 0.562479i \(0.190152\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 808.448 1.06914 0.534570 0.845124i \(-0.320473\pi\)
0.534570 + 0.845124i \(0.320473\pi\)
\(84\) 0 0
\(85\) 993.756 1.26809
\(86\) 0 0
\(87\) 389.764 0.480311
\(88\) 0 0
\(89\) −319.867 −0.380964 −0.190482 0.981691i \(-0.561005\pi\)
−0.190482 + 0.981691i \(0.561005\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −727.245 −0.810880
\(94\) 0 0
\(95\) −634.179 −0.684899
\(96\) 0 0
\(97\) 1133.24 1.18622 0.593110 0.805121i \(-0.297900\pi\)
0.593110 + 0.805121i \(0.297900\pi\)
\(98\) 0 0
\(99\) −365.356 −0.370906
\(100\) 0 0
\(101\) −1045.74 −1.03025 −0.515126 0.857115i \(-0.672255\pi\)
−0.515126 + 0.857115i \(0.672255\pi\)
\(102\) 0 0
\(103\) 1898.21 1.81589 0.907943 0.419093i \(-0.137652\pi\)
0.907943 + 0.419093i \(0.137652\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 868.785 0.784940 0.392470 0.919765i \(-0.371621\pi\)
0.392470 + 0.919765i \(0.371621\pi\)
\(108\) 0 0
\(109\) 227.649 0.200044 0.100022 0.994985i \(-0.468109\pi\)
0.100022 + 0.994985i \(0.468109\pi\)
\(110\) 0 0
\(111\) −1168.16 −0.998886
\(112\) 0 0
\(113\) 581.462 0.484065 0.242032 0.970268i \(-0.422186\pi\)
0.242032 + 0.970268i \(0.422186\pi\)
\(114\) 0 0
\(115\) 1202.97 0.975455
\(116\) 0 0
\(117\) 454.299 0.358974
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 316.963 0.238139
\(122\) 0 0
\(123\) −1410.33 −1.03386
\(124\) 0 0
\(125\) −2227.02 −1.59353
\(126\) 0 0
\(127\) −1129.10 −0.788908 −0.394454 0.918916i \(-0.629066\pi\)
−0.394454 + 0.918916i \(0.629066\pi\)
\(128\) 0 0
\(129\) 375.010 0.255952
\(130\) 0 0
\(131\) −879.662 −0.586690 −0.293345 0.956007i \(-0.594769\pi\)
−0.293345 + 0.956007i \(0.594769\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −516.789 −0.329467
\(136\) 0 0
\(137\) −323.300 −0.201616 −0.100808 0.994906i \(-0.532143\pi\)
−0.100808 + 0.994906i \(0.532143\pi\)
\(138\) 0 0
\(139\) 1710.64 1.04385 0.521923 0.852993i \(-0.325215\pi\)
0.521923 + 0.852993i \(0.325215\pi\)
\(140\) 0 0
\(141\) 1159.87 0.692758
\(142\) 0 0
\(143\) −2049.14 −1.19831
\(144\) 0 0
\(145\) −2486.74 −1.42422
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −959.778 −0.527705 −0.263853 0.964563i \(-0.584993\pi\)
−0.263853 + 0.964563i \(0.584993\pi\)
\(150\) 0 0
\(151\) 1618.84 0.872449 0.436224 0.899838i \(-0.356315\pi\)
0.436224 + 0.899838i \(0.356315\pi\)
\(152\) 0 0
\(153\) −467.275 −0.246908
\(154\) 0 0
\(155\) 4639.90 2.40443
\(156\) 0 0
\(157\) 557.333 0.283312 0.141656 0.989916i \(-0.454757\pi\)
0.141656 + 0.989916i \(0.454757\pi\)
\(158\) 0 0
\(159\) −1834.31 −0.914906
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 366.034 0.175889 0.0879447 0.996125i \(-0.471970\pi\)
0.0879447 + 0.996125i \(0.471970\pi\)
\(164\) 0 0
\(165\) 2331.01 1.09981
\(166\) 0 0
\(167\) 2834.00 1.31318 0.656591 0.754247i \(-0.271998\pi\)
0.656591 + 0.754247i \(0.271998\pi\)
\(168\) 0 0
\(169\) 350.990 0.159759
\(170\) 0 0
\(171\) 298.198 0.133355
\(172\) 0 0
\(173\) 3570.39 1.56909 0.784543 0.620074i \(-0.212898\pi\)
0.784543 + 0.620074i \(0.212898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −679.297 −0.288470
\(178\) 0 0
\(179\) 3677.38 1.53553 0.767767 0.640729i \(-0.221368\pi\)
0.767767 + 0.640729i \(0.221368\pi\)
\(180\) 0 0
\(181\) 1718.18 0.705588 0.352794 0.935701i \(-0.385232\pi\)
0.352794 + 0.935701i \(0.385232\pi\)
\(182\) 0 0
\(183\) 2175.57 0.878814
\(184\) 0 0
\(185\) 7452.96 2.96191
\(186\) 0 0
\(187\) 2107.68 0.824217
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1959.05 −0.742155 −0.371077 0.928602i \(-0.621012\pi\)
−0.371077 + 0.928602i \(0.621012\pi\)
\(192\) 0 0
\(193\) −2492.46 −0.929591 −0.464795 0.885418i \(-0.653872\pi\)
−0.464795 + 0.885418i \(0.653872\pi\)
\(194\) 0 0
\(195\) −2898.47 −1.06443
\(196\) 0 0
\(197\) 2174.45 0.786410 0.393205 0.919451i \(-0.371366\pi\)
0.393205 + 0.919451i \(0.371366\pi\)
\(198\) 0 0
\(199\) 3864.61 1.37666 0.688329 0.725399i \(-0.258345\pi\)
0.688329 + 0.725399i \(0.258345\pi\)
\(200\) 0 0
\(201\) −3135.33 −1.10024
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8998.07 3.06562
\(206\) 0 0
\(207\) −565.649 −0.189929
\(208\) 0 0
\(209\) −1345.04 −0.445161
\(210\) 0 0
\(211\) 127.267 0.0415233 0.0207616 0.999784i \(-0.493391\pi\)
0.0207616 + 0.999784i \(0.493391\pi\)
\(212\) 0 0
\(213\) −509.516 −0.163904
\(214\) 0 0
\(215\) −2392.60 −0.758950
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1145.19 −0.353356
\(220\) 0 0
\(221\) −2620.77 −0.797702
\(222\) 0 0
\(223\) −4071.36 −1.22259 −0.611297 0.791401i \(-0.709352\pi\)
−0.611297 + 0.791401i \(0.709352\pi\)
\(224\) 0 0
\(225\) 2172.17 0.643606
\(226\) 0 0
\(227\) 5282.36 1.54450 0.772252 0.635316i \(-0.219130\pi\)
0.772252 + 0.635316i \(0.219130\pi\)
\(228\) 0 0
\(229\) 4580.19 1.32169 0.660846 0.750522i \(-0.270198\pi\)
0.660846 + 0.750522i \(0.270198\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2156.51 −0.606343 −0.303171 0.952936i \(-0.598045\pi\)
−0.303171 + 0.952936i \(0.598045\pi\)
\(234\) 0 0
\(235\) −7400.11 −2.05417
\(236\) 0 0
\(237\) 3483.36 0.954720
\(238\) 0 0
\(239\) 5755.76 1.55778 0.778889 0.627162i \(-0.215783\pi\)
0.778889 + 0.627162i \(0.215783\pi\)
\(240\) 0 0
\(241\) 2591.55 0.692683 0.346342 0.938108i \(-0.387424\pi\)
0.346342 + 0.938108i \(0.387424\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1672.48 0.430840
\(248\) 0 0
\(249\) 2425.34 0.617269
\(250\) 0 0
\(251\) −3809.47 −0.957974 −0.478987 0.877822i \(-0.658996\pi\)
−0.478987 + 0.877822i \(0.658996\pi\)
\(252\) 0 0
\(253\) 2551.40 0.634012
\(254\) 0 0
\(255\) 2981.27 0.732134
\(256\) 0 0
\(257\) −1357.69 −0.329534 −0.164767 0.986333i \(-0.552687\pi\)
−0.164767 + 0.986333i \(0.552687\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1169.29 0.277308
\(262\) 0 0
\(263\) 4417.73 1.03578 0.517888 0.855448i \(-0.326718\pi\)
0.517888 + 0.855448i \(0.326718\pi\)
\(264\) 0 0
\(265\) 11703.1 2.71289
\(266\) 0 0
\(267\) −959.600 −0.219950
\(268\) 0 0
\(269\) 3627.32 0.822162 0.411081 0.911599i \(-0.365151\pi\)
0.411081 + 0.911599i \(0.365151\pi\)
\(270\) 0 0
\(271\) −5171.03 −1.15911 −0.579553 0.814935i \(-0.696773\pi\)
−0.579553 + 0.814935i \(0.696773\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9797.72 −2.14845
\(276\) 0 0
\(277\) 1794.63 0.389274 0.194637 0.980875i \(-0.437647\pi\)
0.194637 + 0.980875i \(0.437647\pi\)
\(278\) 0 0
\(279\) −2181.74 −0.468162
\(280\) 0 0
\(281\) −9118.63 −1.93584 −0.967921 0.251254i \(-0.919157\pi\)
−0.967921 + 0.251254i \(0.919157\pi\)
\(282\) 0 0
\(283\) 5804.72 1.21927 0.609637 0.792681i \(-0.291315\pi\)
0.609637 + 0.792681i \(0.291315\pi\)
\(284\) 0 0
\(285\) −1902.54 −0.395426
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2217.37 −0.451327
\(290\) 0 0
\(291\) 3399.73 0.684865
\(292\) 0 0
\(293\) −1384.21 −0.275995 −0.137997 0.990433i \(-0.544067\pi\)
−0.137997 + 0.990433i \(0.544067\pi\)
\(294\) 0 0
\(295\) 4333.99 0.855372
\(296\) 0 0
\(297\) −1096.07 −0.214143
\(298\) 0 0
\(299\) −3172.51 −0.613616
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3137.23 −0.594816
\(304\) 0 0
\(305\) −13880.4 −2.60587
\(306\) 0 0
\(307\) −337.112 −0.0626709 −0.0313355 0.999509i \(-0.509976\pi\)
−0.0313355 + 0.999509i \(0.509976\pi\)
\(308\) 0 0
\(309\) 5694.63 1.04840
\(310\) 0 0
\(311\) −1964.69 −0.358224 −0.179112 0.983829i \(-0.557322\pi\)
−0.179112 + 0.983829i \(0.557322\pi\)
\(312\) 0 0
\(313\) 3596.98 0.649563 0.324781 0.945789i \(-0.394709\pi\)
0.324781 + 0.945789i \(0.394709\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −215.450 −0.0381731 −0.0190865 0.999818i \(-0.506076\pi\)
−0.0190865 + 0.999818i \(0.506076\pi\)
\(318\) 0 0
\(319\) −5274.17 −0.925696
\(320\) 0 0
\(321\) 2606.35 0.453185
\(322\) 0 0
\(323\) −1720.25 −0.296339
\(324\) 0 0
\(325\) 12182.9 2.07934
\(326\) 0 0
\(327\) 682.947 0.115496
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1685.35 0.279865 0.139933 0.990161i \(-0.455311\pi\)
0.139933 + 0.990161i \(0.455311\pi\)
\(332\) 0 0
\(333\) −3504.47 −0.576707
\(334\) 0 0
\(335\) 20003.7 3.26245
\(336\) 0 0
\(337\) −7497.87 −1.21197 −0.605987 0.795475i \(-0.707222\pi\)
−0.605987 + 0.795475i \(0.707222\pi\)
\(338\) 0 0
\(339\) 1744.38 0.279475
\(340\) 0 0
\(341\) 9840.87 1.56279
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3608.90 0.563179
\(346\) 0 0
\(347\) 6038.83 0.934241 0.467120 0.884194i \(-0.345291\pi\)
0.467120 + 0.884194i \(0.345291\pi\)
\(348\) 0 0
\(349\) 1992.86 0.305659 0.152830 0.988253i \(-0.451161\pi\)
0.152830 + 0.988253i \(0.451161\pi\)
\(350\) 0 0
\(351\) 1362.90 0.207254
\(352\) 0 0
\(353\) −6336.08 −0.955342 −0.477671 0.878539i \(-0.658519\pi\)
−0.477671 + 0.878539i \(0.658519\pi\)
\(354\) 0 0
\(355\) 3250.77 0.486008
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −904.510 −0.132975 −0.0664877 0.997787i \(-0.521179\pi\)
−0.0664877 + 0.997787i \(0.521179\pi\)
\(360\) 0 0
\(361\) −5761.20 −0.839947
\(362\) 0 0
\(363\) 950.890 0.137490
\(364\) 0 0
\(365\) 7306.46 1.04777
\(366\) 0 0
\(367\) −1252.76 −0.178184 −0.0890920 0.996023i \(-0.528397\pi\)
−0.0890920 + 0.996023i \(0.528397\pi\)
\(368\) 0 0
\(369\) −4230.99 −0.596902
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2565.91 −0.356187 −0.178094 0.984014i \(-0.556993\pi\)
−0.178094 + 0.984014i \(0.556993\pi\)
\(374\) 0 0
\(375\) −6681.06 −0.920022
\(376\) 0 0
\(377\) 6558.12 0.895916
\(378\) 0 0
\(379\) 3900.45 0.528634 0.264317 0.964436i \(-0.414853\pi\)
0.264317 + 0.964436i \(0.414853\pi\)
\(380\) 0 0
\(381\) −3387.29 −0.455476
\(382\) 0 0
\(383\) 1669.11 0.222683 0.111342 0.993782i \(-0.464485\pi\)
0.111342 + 0.993782i \(0.464485\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1125.03 0.147774
\(388\) 0 0
\(389\) 1189.74 0.155070 0.0775351 0.996990i \(-0.475295\pi\)
0.0775351 + 0.996990i \(0.475295\pi\)
\(390\) 0 0
\(391\) 3263.13 0.422056
\(392\) 0 0
\(393\) −2638.99 −0.338726
\(394\) 0 0
\(395\) −22224.2 −2.83094
\(396\) 0 0
\(397\) −802.190 −0.101412 −0.0507062 0.998714i \(-0.516147\pi\)
−0.0507062 + 0.998714i \(0.516147\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10532.2 −1.31161 −0.655803 0.754932i \(-0.727670\pi\)
−0.655803 + 0.754932i \(0.727670\pi\)
\(402\) 0 0
\(403\) −12236.5 −1.51252
\(404\) 0 0
\(405\) −1550.37 −0.190218
\(406\) 0 0
\(407\) 15807.1 1.92514
\(408\) 0 0
\(409\) −3671.52 −0.443875 −0.221937 0.975061i \(-0.571238\pi\)
−0.221937 + 0.975061i \(0.571238\pi\)
\(410\) 0 0
\(411\) −969.899 −0.116403
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −15474.0 −1.83033
\(416\) 0 0
\(417\) 5131.92 0.602664
\(418\) 0 0
\(419\) 1668.29 0.194514 0.0972569 0.995259i \(-0.468993\pi\)
0.0972569 + 0.995259i \(0.468993\pi\)
\(420\) 0 0
\(421\) 16043.7 1.85730 0.928649 0.370961i \(-0.120972\pi\)
0.928649 + 0.370961i \(0.120972\pi\)
\(422\) 0 0
\(423\) 3479.62 0.399964
\(424\) 0 0
\(425\) −12530.9 −1.43020
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6147.43 −0.691844
\(430\) 0 0
\(431\) −12986.7 −1.45138 −0.725692 0.688020i \(-0.758480\pi\)
−0.725692 + 0.688020i \(0.758480\pi\)
\(432\) 0 0
\(433\) 943.959 0.104766 0.0523831 0.998627i \(-0.483318\pi\)
0.0523831 + 0.998627i \(0.483318\pi\)
\(434\) 0 0
\(435\) −7460.21 −0.822275
\(436\) 0 0
\(437\) −2082.41 −0.227953
\(438\) 0 0
\(439\) 6427.32 0.698769 0.349384 0.936980i \(-0.386391\pi\)
0.349384 + 0.936980i \(0.386391\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7254.91 −0.778084 −0.389042 0.921220i \(-0.627194\pi\)
−0.389042 + 0.921220i \(0.627194\pi\)
\(444\) 0 0
\(445\) 6122.35 0.652196
\(446\) 0 0
\(447\) −2879.34 −0.304671
\(448\) 0 0
\(449\) 17381.9 1.82696 0.913478 0.406887i \(-0.133386\pi\)
0.913478 + 0.406887i \(0.133386\pi\)
\(450\) 0 0
\(451\) 19084.2 1.99255
\(452\) 0 0
\(453\) 4856.53 0.503708
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13800.8 −1.41264 −0.706319 0.707894i \(-0.749645\pi\)
−0.706319 + 0.707894i \(0.749645\pi\)
\(458\) 0 0
\(459\) −1401.83 −0.142553
\(460\) 0 0
\(461\) 13.6989 0.00138400 0.000691998 1.00000i \(-0.499780\pi\)
0.000691998 1.00000i \(0.499780\pi\)
\(462\) 0 0
\(463\) −13910.8 −1.39631 −0.698153 0.715949i \(-0.745994\pi\)
−0.698153 + 0.715949i \(0.745994\pi\)
\(464\) 0 0
\(465\) 13919.7 1.38820
\(466\) 0 0
\(467\) 7818.53 0.774729 0.387364 0.921927i \(-0.373386\pi\)
0.387364 + 0.921927i \(0.373386\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1672.00 0.163570
\(472\) 0 0
\(473\) −5074.52 −0.493291
\(474\) 0 0
\(475\) 7996.75 0.772455
\(476\) 0 0
\(477\) −5502.92 −0.528221
\(478\) 0 0
\(479\) 18025.4 1.71942 0.859709 0.510784i \(-0.170645\pi\)
0.859709 + 0.510784i \(0.170645\pi\)
\(480\) 0 0
\(481\) −19655.2 −1.86320
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21690.6 −2.03077
\(486\) 0 0
\(487\) 17431.4 1.62195 0.810977 0.585078i \(-0.198936\pi\)
0.810977 + 0.585078i \(0.198936\pi\)
\(488\) 0 0
\(489\) 1098.10 0.101550
\(490\) 0 0
\(491\) −7107.17 −0.653243 −0.326621 0.945155i \(-0.605910\pi\)
−0.326621 + 0.945155i \(0.605910\pi\)
\(492\) 0 0
\(493\) −6745.45 −0.616226
\(494\) 0 0
\(495\) 6993.03 0.634977
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3499.61 −0.313956 −0.156978 0.987602i \(-0.550175\pi\)
−0.156978 + 0.987602i \(0.550175\pi\)
\(500\) 0 0
\(501\) 8501.99 0.758166
\(502\) 0 0
\(503\) 9115.24 0.808009 0.404005 0.914757i \(-0.367618\pi\)
0.404005 + 0.914757i \(0.367618\pi\)
\(504\) 0 0
\(505\) 20015.9 1.76375
\(506\) 0 0
\(507\) 1052.97 0.0922367
\(508\) 0 0
\(509\) 16049.7 1.39762 0.698811 0.715306i \(-0.253713\pi\)
0.698811 + 0.715306i \(0.253713\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 894.594 0.0769928
\(514\) 0 0
\(515\) −36332.4 −3.10873
\(516\) 0 0
\(517\) −15695.1 −1.33514
\(518\) 0 0
\(519\) 10711.2 0.905912
\(520\) 0 0
\(521\) 5574.48 0.468757 0.234379 0.972145i \(-0.424694\pi\)
0.234379 + 0.972145i \(0.424694\pi\)
\(522\) 0 0
\(523\) −458.890 −0.0383669 −0.0191834 0.999816i \(-0.506107\pi\)
−0.0191834 + 0.999816i \(0.506107\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12586.1 1.04034
\(528\) 0 0
\(529\) −8216.89 −0.675342
\(530\) 0 0
\(531\) −2037.89 −0.166548
\(532\) 0 0
\(533\) −23730.1 −1.92845
\(534\) 0 0
\(535\) −16628.8 −1.34379
\(536\) 0 0
\(537\) 11032.1 0.886541
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10972.7 −0.871999 −0.436000 0.899947i \(-0.643605\pi\)
−0.436000 + 0.899947i \(0.643605\pi\)
\(542\) 0 0
\(543\) 5154.54 0.407371
\(544\) 0 0
\(545\) −4357.28 −0.342468
\(546\) 0 0
\(547\) 19276.8 1.50679 0.753396 0.657567i \(-0.228414\pi\)
0.753396 + 0.657567i \(0.228414\pi\)
\(548\) 0 0
\(549\) 6526.72 0.507383
\(550\) 0 0
\(551\) 4304.70 0.332825
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 22358.9 1.71006
\(556\) 0 0
\(557\) −12191.4 −0.927410 −0.463705 0.885990i \(-0.653480\pi\)
−0.463705 + 0.885990i \(0.653480\pi\)
\(558\) 0 0
\(559\) 6309.87 0.477422
\(560\) 0 0
\(561\) 6323.03 0.475862
\(562\) 0 0
\(563\) 3236.06 0.242244 0.121122 0.992638i \(-0.461351\pi\)
0.121122 + 0.992638i \(0.461351\pi\)
\(564\) 0 0
\(565\) −11129.4 −0.828701
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12167.8 0.896488 0.448244 0.893911i \(-0.352050\pi\)
0.448244 + 0.893911i \(0.352050\pi\)
\(570\) 0 0
\(571\) 24412.3 1.78918 0.894592 0.446884i \(-0.147466\pi\)
0.894592 + 0.446884i \(0.147466\pi\)
\(572\) 0 0
\(573\) −5877.14 −0.428483
\(574\) 0 0
\(575\) −15169.0 −1.10016
\(576\) 0 0
\(577\) −4346.85 −0.313625 −0.156812 0.987628i \(-0.550122\pi\)
−0.156812 + 0.987628i \(0.550122\pi\)
\(578\) 0 0
\(579\) −7477.37 −0.536700
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24821.3 1.76328
\(584\) 0 0
\(585\) −8695.42 −0.614550
\(586\) 0 0
\(587\) 8752.61 0.615432 0.307716 0.951478i \(-0.400435\pi\)
0.307716 + 0.951478i \(0.400435\pi\)
\(588\) 0 0
\(589\) −8031.97 −0.561887
\(590\) 0 0
\(591\) 6523.34 0.454034
\(592\) 0 0
\(593\) 7788.77 0.539370 0.269685 0.962949i \(-0.413080\pi\)
0.269685 + 0.962949i \(0.413080\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11593.8 0.794814
\(598\) 0 0
\(599\) 1144.02 0.0780354 0.0390177 0.999239i \(-0.487577\pi\)
0.0390177 + 0.999239i \(0.487577\pi\)
\(600\) 0 0
\(601\) −24673.4 −1.67463 −0.837314 0.546723i \(-0.815875\pi\)
−0.837314 + 0.546723i \(0.815875\pi\)
\(602\) 0 0
\(603\) −9405.99 −0.635226
\(604\) 0 0
\(605\) −6066.78 −0.407686
\(606\) 0 0
\(607\) −6325.62 −0.422980 −0.211490 0.977380i \(-0.567832\pi\)
−0.211490 + 0.977380i \(0.567832\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19515.9 1.29219
\(612\) 0 0
\(613\) 22976.2 1.51387 0.756935 0.653491i \(-0.226696\pi\)
0.756935 + 0.653491i \(0.226696\pi\)
\(614\) 0 0
\(615\) 26994.2 1.76994
\(616\) 0 0
\(617\) 8229.26 0.536949 0.268475 0.963287i \(-0.413480\pi\)
0.268475 + 0.963287i \(0.413480\pi\)
\(618\) 0 0
\(619\) 16852.8 1.09430 0.547150 0.837035i \(-0.315713\pi\)
0.547150 + 0.837035i \(0.315713\pi\)
\(620\) 0 0
\(621\) −1696.95 −0.109656
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 12456.9 0.797239
\(626\) 0 0
\(627\) −4035.13 −0.257014
\(628\) 0 0
\(629\) 20216.7 1.28154
\(630\) 0 0
\(631\) 8408.46 0.530484 0.265242 0.964182i \(-0.414548\pi\)
0.265242 + 0.964182i \(0.414548\pi\)
\(632\) 0 0
\(633\) 381.801 0.0239735
\(634\) 0 0
\(635\) 21611.3 1.35058
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1528.55 −0.0946298
\(640\) 0 0
\(641\) 9122.52 0.562118 0.281059 0.959690i \(-0.409314\pi\)
0.281059 + 0.959690i \(0.409314\pi\)
\(642\) 0 0
\(643\) −19279.4 −1.18243 −0.591217 0.806513i \(-0.701352\pi\)
−0.591217 + 0.806513i \(0.701352\pi\)
\(644\) 0 0
\(645\) −7177.81 −0.438180
\(646\) 0 0
\(647\) −2177.86 −0.132335 −0.0661673 0.997809i \(-0.521077\pi\)
−0.0661673 + 0.997809i \(0.521077\pi\)
\(648\) 0 0
\(649\) 9192.05 0.555962
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6288.85 −0.376879 −0.188439 0.982085i \(-0.560343\pi\)
−0.188439 + 0.982085i \(0.560343\pi\)
\(654\) 0 0
\(655\) 16837.0 1.00439
\(656\) 0 0
\(657\) −3435.58 −0.204010
\(658\) 0 0
\(659\) −20346.5 −1.20271 −0.601357 0.798980i \(-0.705373\pi\)
−0.601357 + 0.798980i \(0.705373\pi\)
\(660\) 0 0
\(661\) 27669.2 1.62815 0.814074 0.580761i \(-0.197245\pi\)
0.814074 + 0.580761i \(0.197245\pi\)
\(662\) 0 0
\(663\) −7862.31 −0.460553
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8165.54 −0.474020
\(668\) 0 0
\(669\) −12214.1 −0.705865
\(670\) 0 0
\(671\) −29439.2 −1.69372
\(672\) 0 0
\(673\) 11862.6 0.679450 0.339725 0.940525i \(-0.389666\pi\)
0.339725 + 0.940525i \(0.389666\pi\)
\(674\) 0 0
\(675\) 6516.51 0.371586
\(676\) 0 0
\(677\) −13526.6 −0.767901 −0.383950 0.923354i \(-0.625437\pi\)
−0.383950 + 0.923354i \(0.625437\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 15847.1 0.891720
\(682\) 0 0
\(683\) 713.863 0.0399930 0.0199965 0.999800i \(-0.493634\pi\)
0.0199965 + 0.999800i \(0.493634\pi\)
\(684\) 0 0
\(685\) 6188.06 0.345159
\(686\) 0 0
\(687\) 13740.6 0.763079
\(688\) 0 0
\(689\) −30863.8 −1.70656
\(690\) 0 0
\(691\) −15816.1 −0.870727 −0.435364 0.900255i \(-0.643380\pi\)
−0.435364 + 0.900255i \(0.643380\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −32742.2 −1.78702
\(696\) 0 0
\(697\) 24407.9 1.32642
\(698\) 0 0
\(699\) −6469.54 −0.350072
\(700\) 0 0
\(701\) 28556.9 1.53863 0.769314 0.638871i \(-0.220598\pi\)
0.769314 + 0.638871i \(0.220598\pi\)
\(702\) 0 0
\(703\) −12901.5 −0.692163
\(704\) 0 0
\(705\) −22200.3 −1.18598
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1934.72 −0.102482 −0.0512411 0.998686i \(-0.516318\pi\)
−0.0512411 + 0.998686i \(0.516318\pi\)
\(710\) 0 0
\(711\) 10450.1 0.551208
\(712\) 0 0
\(713\) 15235.8 0.800258
\(714\) 0 0
\(715\) 39221.3 2.05146
\(716\) 0 0
\(717\) 17267.3 0.899384
\(718\) 0 0
\(719\) 18666.7 0.968218 0.484109 0.875008i \(-0.339144\pi\)
0.484109 + 0.875008i \(0.339144\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 7774.66 0.399921
\(724\) 0 0
\(725\) 31356.8 1.60629
\(726\) 0 0
\(727\) −25955.3 −1.32411 −0.662055 0.749455i \(-0.730315\pi\)
−0.662055 + 0.749455i \(0.730315\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6490.10 −0.328379
\(732\) 0 0
\(733\) 2083.79 0.105002 0.0525010 0.998621i \(-0.483281\pi\)
0.0525010 + 0.998621i \(0.483281\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42426.4 2.12048
\(738\) 0 0
\(739\) −28857.3 −1.43645 −0.718223 0.695813i \(-0.755044\pi\)
−0.718223 + 0.695813i \(0.755044\pi\)
\(740\) 0 0
\(741\) 5017.44 0.248745
\(742\) 0 0
\(743\) 899.017 0.0443900 0.0221950 0.999754i \(-0.492935\pi\)
0.0221950 + 0.999754i \(0.492935\pi\)
\(744\) 0 0
\(745\) 18370.5 0.903412
\(746\) 0 0
\(747\) 7276.03 0.356380
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29009.3 1.40954 0.704770 0.709436i \(-0.251050\pi\)
0.704770 + 0.709436i \(0.251050\pi\)
\(752\) 0 0
\(753\) −11428.4 −0.553087
\(754\) 0 0
\(755\) −30985.2 −1.49360
\(756\) 0 0
\(757\) −10932.4 −0.524892 −0.262446 0.964947i \(-0.584529\pi\)
−0.262446 + 0.964947i \(0.584529\pi\)
\(758\) 0 0
\(759\) 7654.20 0.366047
\(760\) 0 0
\(761\) 13212.8 0.629387 0.314693 0.949193i \(-0.398098\pi\)
0.314693 + 0.949193i \(0.398098\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 8943.80 0.422698
\(766\) 0 0
\(767\) −11429.8 −0.538077
\(768\) 0 0
\(769\) −30129.7 −1.41288 −0.706440 0.707773i \(-0.749700\pi\)
−0.706440 + 0.707773i \(0.749700\pi\)
\(770\) 0 0
\(771\) −4073.06 −0.190257
\(772\) 0 0
\(773\) −11840.7 −0.550947 −0.275474 0.961309i \(-0.588835\pi\)
−0.275474 + 0.961309i \(0.588835\pi\)
\(774\) 0 0
\(775\) −58507.4 −2.71180
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −15576.2 −0.716401
\(780\) 0 0
\(781\) 6894.62 0.315889
\(782\) 0 0
\(783\) 3507.88 0.160104
\(784\) 0 0
\(785\) −10667.5 −0.485020
\(786\) 0 0
\(787\) 6529.10 0.295727 0.147863 0.989008i \(-0.452760\pi\)
0.147863 + 0.989008i \(0.452760\pi\)
\(788\) 0 0
\(789\) 13253.2 0.598006
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36605.9 1.63924
\(794\) 0 0
\(795\) 35109.2 1.56628
\(796\) 0 0
\(797\) 41987.6 1.86609 0.933046 0.359757i \(-0.117140\pi\)
0.933046 + 0.359757i \(0.117140\pi\)
\(798\) 0 0
\(799\) −20073.3 −0.888790
\(800\) 0 0
\(801\) −2878.80 −0.126988
\(802\) 0 0
\(803\) 15496.4 0.681017
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10882.0 0.474676
\(808\) 0 0
\(809\) −13634.1 −0.592522 −0.296261 0.955107i \(-0.595740\pi\)
−0.296261 + 0.955107i \(0.595740\pi\)
\(810\) 0 0
\(811\) 20093.9 0.870025 0.435013 0.900424i \(-0.356744\pi\)
0.435013 + 0.900424i \(0.356744\pi\)
\(812\) 0 0
\(813\) −15513.1 −0.669210
\(814\) 0 0
\(815\) −7006.01 −0.301116
\(816\) 0 0
\(817\) 4141.75 0.177358
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7636.29 −0.324615 −0.162307 0.986740i \(-0.551894\pi\)
−0.162307 + 0.986740i \(0.551894\pi\)
\(822\) 0 0
\(823\) 28788.4 1.21932 0.609660 0.792663i \(-0.291306\pi\)
0.609660 + 0.792663i \(0.291306\pi\)
\(824\) 0 0
\(825\) −29393.2 −1.24041
\(826\) 0 0
\(827\) 20515.4 0.862623 0.431312 0.902203i \(-0.358051\pi\)
0.431312 + 0.902203i \(0.358051\pi\)
\(828\) 0 0
\(829\) 16264.4 0.681407 0.340704 0.940171i \(-0.389335\pi\)
0.340704 + 0.940171i \(0.389335\pi\)
\(830\) 0 0
\(831\) 5383.89 0.224748
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −54243.7 −2.24812
\(836\) 0 0
\(837\) −6545.21 −0.270293
\(838\) 0 0
\(839\) 13770.2 0.566629 0.283314 0.959027i \(-0.408566\pi\)
0.283314 + 0.959027i \(0.408566\pi\)
\(840\) 0 0
\(841\) −7509.45 −0.307903
\(842\) 0 0
\(843\) −27355.9 −1.11766
\(844\) 0 0
\(845\) −6718.06 −0.273501
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 17414.2 0.703948
\(850\) 0 0
\(851\) 24472.8 0.985802
\(852\) 0 0
\(853\) −8031.48 −0.322383 −0.161191 0.986923i \(-0.551534\pi\)
−0.161191 + 0.986923i \(0.551534\pi\)
\(854\) 0 0
\(855\) −5707.61 −0.228300
\(856\) 0 0
\(857\) −42764.0 −1.70454 −0.852270 0.523102i \(-0.824775\pi\)
−0.852270 + 0.523102i \(0.824775\pi\)
\(858\) 0 0
\(859\) −8155.05 −0.323919 −0.161960 0.986797i \(-0.551781\pi\)
−0.161960 + 0.986797i \(0.551781\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40660.0 −1.60380 −0.801901 0.597457i \(-0.796178\pi\)
−0.801901 + 0.597457i \(0.796178\pi\)
\(864\) 0 0
\(865\) −68338.5 −2.68622
\(866\) 0 0
\(867\) −6652.11 −0.260574
\(868\) 0 0
\(869\) −47135.8 −1.84002
\(870\) 0 0
\(871\) −52754.7 −2.05227
\(872\) 0 0
\(873\) 10199.2 0.395407
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28260.6 −1.08813 −0.544067 0.839042i \(-0.683116\pi\)
−0.544067 + 0.839042i \(0.683116\pi\)
\(878\) 0 0
\(879\) −4152.64 −0.159346
\(880\) 0 0
\(881\) 15951.2 0.609998 0.304999 0.952353i \(-0.401344\pi\)
0.304999 + 0.952353i \(0.401344\pi\)
\(882\) 0 0
\(883\) 1750.48 0.0667137 0.0333569 0.999444i \(-0.489380\pi\)
0.0333569 + 0.999444i \(0.489380\pi\)
\(884\) 0 0
\(885\) 13002.0 0.493849
\(886\) 0 0
\(887\) −2007.02 −0.0759741 −0.0379871 0.999278i \(-0.512095\pi\)
−0.0379871 + 0.999278i \(0.512095\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3288.20 −0.123635
\(892\) 0 0
\(893\) 12810.1 0.480036
\(894\) 0 0
\(895\) −70386.3 −2.62878
\(896\) 0 0
\(897\) −9517.54 −0.354271
\(898\) 0 0
\(899\) −31494.9 −1.16842
\(900\) 0 0
\(901\) 31745.4 1.17380
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −32886.6 −1.20794
\(906\) 0 0
\(907\) 12069.7 0.441861 0.220930 0.975290i \(-0.429091\pi\)
0.220930 + 0.975290i \(0.429091\pi\)
\(908\) 0 0
\(909\) −9411.69 −0.343417
\(910\) 0 0
\(911\) −40167.1 −1.46081 −0.730404 0.683016i \(-0.760668\pi\)
−0.730404 + 0.683016i \(0.760668\pi\)
\(912\) 0 0
\(913\) −32819.0 −1.18965
\(914\) 0 0
\(915\) −41641.2 −1.50450
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 13503.5 0.484699 0.242349 0.970189i \(-0.422082\pi\)
0.242349 + 0.970189i \(0.422082\pi\)
\(920\) 0 0
\(921\) −1011.33 −0.0361831
\(922\) 0 0
\(923\) −8573.05 −0.305726
\(924\) 0 0
\(925\) −93978.9 −3.34055
\(926\) 0 0
\(927\) 17083.9 0.605295
\(928\) 0 0
\(929\) 31001.9 1.09488 0.547438 0.836846i \(-0.315603\pi\)
0.547438 + 0.836846i \(0.315603\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −5894.08 −0.206821
\(934\) 0 0
\(935\) −40341.6 −1.41103
\(936\) 0 0
\(937\) −34998.0 −1.22021 −0.610104 0.792322i \(-0.708872\pi\)
−0.610104 + 0.792322i \(0.708872\pi\)
\(938\) 0 0
\(939\) 10790.9 0.375025
\(940\) 0 0
\(941\) −11652.5 −0.403679 −0.201839 0.979419i \(-0.564692\pi\)
−0.201839 + 0.979419i \(0.564692\pi\)
\(942\) 0 0
\(943\) 29546.4 1.02032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2921.85 −0.100261 −0.0501306 0.998743i \(-0.515964\pi\)
−0.0501306 + 0.998743i \(0.515964\pi\)
\(948\) 0 0
\(949\) −19268.9 −0.659109
\(950\) 0 0
\(951\) −646.350 −0.0220392
\(952\) 0 0
\(953\) 24262.5 0.824702 0.412351 0.911025i \(-0.364708\pi\)
0.412351 + 0.911025i \(0.364708\pi\)
\(954\) 0 0
\(955\) 37496.8 1.27054
\(956\) 0 0
\(957\) −15822.5 −0.534451
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 28974.1 0.972577
\(962\) 0 0
\(963\) 7819.06 0.261647
\(964\) 0 0
\(965\) 47706.5 1.59143
\(966\) 0 0
\(967\) −10258.0 −0.341131 −0.170565 0.985346i \(-0.554559\pi\)
−0.170565 + 0.985346i \(0.554559\pi\)
\(968\) 0 0
\(969\) −5160.76 −0.171091
\(970\) 0 0
\(971\) −16635.7 −0.549809 −0.274905 0.961472i \(-0.588646\pi\)
−0.274905 + 0.961472i \(0.588646\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 36548.7 1.20051
\(976\) 0 0
\(977\) 12850.2 0.420791 0.210395 0.977616i \(-0.432525\pi\)
0.210395 + 0.977616i \(0.432525\pi\)
\(978\) 0 0
\(979\) 12985.0 0.423905
\(980\) 0 0
\(981\) 2048.84 0.0666814
\(982\) 0 0
\(983\) −42272.8 −1.37161 −0.685805 0.727785i \(-0.740550\pi\)
−0.685805 + 0.727785i \(0.740550\pi\)
\(984\) 0 0
\(985\) −41619.6 −1.34631
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7856.44 −0.252599
\(990\) 0 0
\(991\) 11504.2 0.368761 0.184380 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(992\) 0 0
\(993\) 5056.06 0.161580
\(994\) 0 0
\(995\) −73969.9 −2.35679
\(996\) 0 0
\(997\) −17180.9 −0.545763 −0.272881 0.962048i \(-0.587977\pi\)
−0.272881 + 0.962048i \(0.587977\pi\)
\(998\) 0 0
\(999\) −10513.4 −0.332962
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 2352.4.a.cq.1.1 4
4.3 odd 2 588.4.a.j.1.1 4
7.6 odd 2 2352.4.a.cl.1.4 4
12.11 even 2 1764.4.a.bc.1.4 4
28.3 even 6 588.4.i.k.373.1 8
28.11 odd 6 588.4.i.l.373.4 8
28.19 even 6 588.4.i.k.361.1 8
28.23 odd 6 588.4.i.l.361.4 8
28.27 even 2 588.4.a.k.1.4 yes 4
84.11 even 6 1764.4.k.bb.1549.1 8
84.23 even 6 1764.4.k.bb.361.1 8
84.47 odd 6 1764.4.k.bd.361.4 8
84.59 odd 6 1764.4.k.bd.1549.4 8
84.83 odd 2 1764.4.a.ba.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.1 4 4.3 odd 2
588.4.a.k.1.4 yes 4 28.27 even 2
588.4.i.k.361.1 8 28.19 even 6
588.4.i.k.373.1 8 28.3 even 6
588.4.i.l.361.4 8 28.23 odd 6
588.4.i.l.373.4 8 28.11 odd 6
1764.4.a.ba.1.1 4 84.83 odd 2
1764.4.a.bc.1.4 4 12.11 even 2
1764.4.k.bb.361.1 8 84.23 even 6
1764.4.k.bb.1549.1 8 84.11 even 6
1764.4.k.bd.361.4 8 84.47 odd 6
1764.4.k.bd.1549.4 8 84.59 odd 6
2352.4.a.cl.1.4 4 7.6 odd 2
2352.4.a.cq.1.1 4 1.1 even 1 trivial