Properties

Label 624.4.a.t.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} -25.8785 q^{11} +13.0000 q^{13} +34.2966 q^{15} -20.3276 q^{17} -154.712 q^{19} -33.7629 q^{21} +180.418 q^{23} +5.69520 q^{25} -27.0000 q^{27} -20.4522 q^{29} -266.424 q^{31} +77.6355 q^{33} -128.661 q^{35} +115.984 q^{37} -39.0000 q^{39} +391.184 q^{41} -151.407 q^{43} -102.890 q^{45} +467.365 q^{47} -216.341 q^{49} +60.9828 q^{51} +79.9842 q^{53} +295.848 q^{55} +464.136 q^{57} +873.710 q^{59} -187.068 q^{61} +101.289 q^{63} -148.619 q^{65} +609.204 q^{67} -541.255 q^{69} -248.038 q^{71} +852.765 q^{73} -17.0856 q^{75} -291.244 q^{77} +331.221 q^{79} +81.0000 q^{81} +435.432 q^{83} +232.389 q^{85} +61.3566 q^{87} +259.233 q^{89} +146.306 q^{91} +799.273 q^{93} +1768.70 q^{95} +1225.17 q^{97} -232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9} + 16 q^{11} + 39 q^{13} - 12 q^{15} - 146 q^{17} - 94 q^{19} + 90 q^{21} + 48 q^{23} + 145 q^{25} - 81 q^{27} - 2 q^{29} - 302 q^{31} - 48 q^{33} - 80 q^{35} + 374 q^{37} - 117 q^{39} + 480 q^{41} + 260 q^{43} + 36 q^{45} + 24 q^{47} + 447 q^{49} + 438 q^{51} - 678 q^{53} + 1552 q^{55} + 282 q^{57} + 1788 q^{59} + 230 q^{61} - 270 q^{63} + 52 q^{65} - 74 q^{67} - 144 q^{69} + 948 q^{71} - 222 q^{73} - 435 q^{75} + 112 q^{77} + 24 q^{79} + 243 q^{81} + 796 q^{83} - 248 q^{85} + 6 q^{87} + 1436 q^{89} - 390 q^{91} + 906 q^{93} + 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −11.4322 −1.02253 −0.511264 0.859424i \(-0.670822\pi\)
−0.511264 + 0.859424i \(0.670822\pi\)
\(6\) 0 0
\(7\) 11.2543 0.607675 0.303838 0.952724i \(-0.401732\pi\)
0.303838 + 0.952724i \(0.401732\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −25.8785 −0.709333 −0.354666 0.934993i \(-0.615406\pi\)
−0.354666 + 0.934993i \(0.615406\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 34.2966 0.590356
\(16\) 0 0
\(17\) −20.3276 −0.290010 −0.145005 0.989431i \(-0.546320\pi\)
−0.145005 + 0.989431i \(0.546320\pi\)
\(18\) 0 0
\(19\) −154.712 −1.86807 −0.934035 0.357181i \(-0.883738\pi\)
−0.934035 + 0.357181i \(0.883738\pi\)
\(20\) 0 0
\(21\) −33.7629 −0.350841
\(22\) 0 0
\(23\) 180.418 1.63565 0.817823 0.575471i \(-0.195181\pi\)
0.817823 + 0.575471i \(0.195181\pi\)
\(24\) 0 0
\(25\) 5.69520 0.0455616
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −20.4522 −0.130961 −0.0654806 0.997854i \(-0.520858\pi\)
−0.0654806 + 0.997854i \(0.520858\pi\)
\(30\) 0 0
\(31\) −266.424 −1.54359 −0.771794 0.635873i \(-0.780640\pi\)
−0.771794 + 0.635873i \(0.780640\pi\)
\(32\) 0 0
\(33\) 77.6355 0.409534
\(34\) 0 0
\(35\) −128.661 −0.621364
\(36\) 0 0
\(37\) 115.984 0.515340 0.257670 0.966233i \(-0.417045\pi\)
0.257670 + 0.966233i \(0.417045\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 391.184 1.49006 0.745032 0.667029i \(-0.232434\pi\)
0.745032 + 0.667029i \(0.232434\pi\)
\(42\) 0 0
\(43\) −151.407 −0.536963 −0.268482 0.963285i \(-0.586522\pi\)
−0.268482 + 0.963285i \(0.586522\pi\)
\(44\) 0 0
\(45\) −102.890 −0.340842
\(46\) 0 0
\(47\) 467.365 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(48\) 0 0
\(49\) −216.341 −0.630731
\(50\) 0 0
\(51\) 60.9828 0.167437
\(52\) 0 0
\(53\) 79.9842 0.207296 0.103648 0.994614i \(-0.466949\pi\)
0.103648 + 0.994614i \(0.466949\pi\)
\(54\) 0 0
\(55\) 295.848 0.725312
\(56\) 0 0
\(57\) 464.136 1.07853
\(58\) 0 0
\(59\) 873.710 1.92792 0.963960 0.266045i \(-0.0857171\pi\)
0.963960 + 0.266045i \(0.0857171\pi\)
\(60\) 0 0
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) 0 0
\(63\) 101.289 0.202558
\(64\) 0 0
\(65\) −148.619 −0.283598
\(66\) 0 0
\(67\) 609.204 1.11084 0.555418 0.831571i \(-0.312558\pi\)
0.555418 + 0.831571i \(0.312558\pi\)
\(68\) 0 0
\(69\) −541.255 −0.944340
\(70\) 0 0
\(71\) −248.038 −0.414601 −0.207301 0.978277i \(-0.566468\pi\)
−0.207301 + 0.978277i \(0.566468\pi\)
\(72\) 0 0
\(73\) 852.765 1.36724 0.683621 0.729838i \(-0.260404\pi\)
0.683621 + 0.729838i \(0.260404\pi\)
\(74\) 0 0
\(75\) −17.0856 −0.0263050
\(76\) 0 0
\(77\) −291.244 −0.431044
\(78\) 0 0
\(79\) 331.221 0.471712 0.235856 0.971788i \(-0.424211\pi\)
0.235856 + 0.971788i \(0.424211\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 435.432 0.575842 0.287921 0.957654i \(-0.407036\pi\)
0.287921 + 0.957654i \(0.407036\pi\)
\(84\) 0 0
\(85\) 232.389 0.296543
\(86\) 0 0
\(87\) 61.3566 0.0756105
\(88\) 0 0
\(89\) 259.233 0.308749 0.154375 0.988012i \(-0.450664\pi\)
0.154375 + 0.988012i \(0.450664\pi\)
\(90\) 0 0
\(91\) 146.306 0.168539
\(92\) 0 0
\(93\) 799.273 0.891191
\(94\) 0 0
\(95\) 1768.70 1.91015
\(96\) 0 0
\(97\) 1225.17 1.28245 0.641223 0.767355i \(-0.278428\pi\)
0.641223 + 0.767355i \(0.278428\pi\)
\(98\) 0 0
\(99\) −232.907 −0.236444
\(100\) 0 0
\(101\) 645.416 0.635855 0.317927 0.948115i \(-0.397013\pi\)
0.317927 + 0.948115i \(0.397013\pi\)
\(102\) 0 0
\(103\) 511.137 0.488969 0.244484 0.969653i \(-0.421381\pi\)
0.244484 + 0.969653i \(0.421381\pi\)
\(104\) 0 0
\(105\) 385.984 0.358745
\(106\) 0 0
\(107\) −608.195 −0.549499 −0.274750 0.961516i \(-0.588595\pi\)
−0.274750 + 0.961516i \(0.588595\pi\)
\(108\) 0 0
\(109\) −1300.04 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(110\) 0 0
\(111\) −347.951 −0.297532
\(112\) 0 0
\(113\) 42.1953 0.0351274 0.0175637 0.999846i \(-0.494409\pi\)
0.0175637 + 0.999846i \(0.494409\pi\)
\(114\) 0 0
\(115\) −2062.58 −1.67249
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) −228.773 −0.176232
\(120\) 0 0
\(121\) −661.303 −0.496847
\(122\) 0 0
\(123\) −1173.55 −0.860289
\(124\) 0 0
\(125\) 1363.92 0.975939
\(126\) 0 0
\(127\) 311.018 0.217310 0.108655 0.994080i \(-0.465346\pi\)
0.108655 + 0.994080i \(0.465346\pi\)
\(128\) 0 0
\(129\) 454.222 0.310016
\(130\) 0 0
\(131\) −2000.98 −1.33456 −0.667278 0.744809i \(-0.732541\pi\)
−0.667278 + 0.744809i \(0.732541\pi\)
\(132\) 0 0
\(133\) −1741.17 −1.13518
\(134\) 0 0
\(135\) 308.669 0.196785
\(136\) 0 0
\(137\) 1038.53 0.647644 0.323822 0.946118i \(-0.395032\pi\)
0.323822 + 0.946118i \(0.395032\pi\)
\(138\) 0 0
\(139\) 2858.46 1.74426 0.872128 0.489277i \(-0.162739\pi\)
0.872128 + 0.489277i \(0.162739\pi\)
\(140\) 0 0
\(141\) −1402.09 −0.837430
\(142\) 0 0
\(143\) −336.421 −0.196734
\(144\) 0 0
\(145\) 233.814 0.133911
\(146\) 0 0
\(147\) 649.022 0.364153
\(148\) 0 0
\(149\) 743.479 0.408780 0.204390 0.978890i \(-0.434479\pi\)
0.204390 + 0.978890i \(0.434479\pi\)
\(150\) 0 0
\(151\) −2277.24 −1.22728 −0.613640 0.789586i \(-0.710295\pi\)
−0.613640 + 0.789586i \(0.710295\pi\)
\(152\) 0 0
\(153\) −182.948 −0.0966700
\(154\) 0 0
\(155\) 3045.82 1.57836
\(156\) 0 0
\(157\) 3173.51 1.61321 0.806605 0.591091i \(-0.201303\pi\)
0.806605 + 0.591091i \(0.201303\pi\)
\(158\) 0 0
\(159\) −239.953 −0.119682
\(160\) 0 0
\(161\) 2030.48 0.993941
\(162\) 0 0
\(163\) 2314.65 1.11225 0.556126 0.831098i \(-0.312287\pi\)
0.556126 + 0.831098i \(0.312287\pi\)
\(164\) 0 0
\(165\) −887.545 −0.418759
\(166\) 0 0
\(167\) 2665.65 1.23517 0.617587 0.786502i \(-0.288110\pi\)
0.617587 + 0.786502i \(0.288110\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −1392.41 −0.622690
\(172\) 0 0
\(173\) −165.243 −0.0726198 −0.0363099 0.999341i \(-0.511560\pi\)
−0.0363099 + 0.999341i \(0.511560\pi\)
\(174\) 0 0
\(175\) 64.0954 0.0276866
\(176\) 0 0
\(177\) −2621.13 −1.11309
\(178\) 0 0
\(179\) −712.339 −0.297446 −0.148723 0.988879i \(-0.547516\pi\)
−0.148723 + 0.988879i \(0.547516\pi\)
\(180\) 0 0
\(181\) 2206.53 0.906133 0.453066 0.891477i \(-0.350330\pi\)
0.453066 + 0.891477i \(0.350330\pi\)
\(182\) 0 0
\(183\) 561.204 0.226696
\(184\) 0 0
\(185\) −1325.95 −0.526949
\(186\) 0 0
\(187\) 526.048 0.205714
\(188\) 0 0
\(189\) −303.866 −0.116947
\(190\) 0 0
\(191\) −1470.64 −0.557129 −0.278565 0.960417i \(-0.589859\pi\)
−0.278565 + 0.960417i \(0.589859\pi\)
\(192\) 0 0
\(193\) 369.560 0.137832 0.0689158 0.997622i \(-0.478046\pi\)
0.0689158 + 0.997622i \(0.478046\pi\)
\(194\) 0 0
\(195\) 445.856 0.163735
\(196\) 0 0
\(197\) −4273.41 −1.54552 −0.772761 0.634697i \(-0.781125\pi\)
−0.772761 + 0.634697i \(0.781125\pi\)
\(198\) 0 0
\(199\) −4154.31 −1.47985 −0.739927 0.672687i \(-0.765140\pi\)
−0.739927 + 0.672687i \(0.765140\pi\)
\(200\) 0 0
\(201\) −1827.61 −0.641342
\(202\) 0 0
\(203\) −230.175 −0.0795819
\(204\) 0 0
\(205\) −4472.09 −1.52363
\(206\) 0 0
\(207\) 1623.77 0.545215
\(208\) 0 0
\(209\) 4003.71 1.32508
\(210\) 0 0
\(211\) −1231.59 −0.401830 −0.200915 0.979609i \(-0.564392\pi\)
−0.200915 + 0.979609i \(0.564392\pi\)
\(212\) 0 0
\(213\) 744.114 0.239370
\(214\) 0 0
\(215\) 1730.92 0.549059
\(216\) 0 0
\(217\) −2998.42 −0.938000
\(218\) 0 0
\(219\) −2558.30 −0.789377
\(220\) 0 0
\(221\) −264.259 −0.0804343
\(222\) 0 0
\(223\) 2187.24 0.656809 0.328404 0.944537i \(-0.393489\pi\)
0.328404 + 0.944537i \(0.393489\pi\)
\(224\) 0 0
\(225\) 51.2568 0.0151872
\(226\) 0 0
\(227\) −4138.67 −1.21010 −0.605051 0.796187i \(-0.706847\pi\)
−0.605051 + 0.796187i \(0.706847\pi\)
\(228\) 0 0
\(229\) −835.354 −0.241056 −0.120528 0.992710i \(-0.538459\pi\)
−0.120528 + 0.992710i \(0.538459\pi\)
\(230\) 0 0
\(231\) 873.733 0.248863
\(232\) 0 0
\(233\) 3685.51 1.03625 0.518124 0.855305i \(-0.326630\pi\)
0.518124 + 0.855305i \(0.326630\pi\)
\(234\) 0 0
\(235\) −5343.01 −1.48315
\(236\) 0 0
\(237\) −993.662 −0.272343
\(238\) 0 0
\(239\) −3026.21 −0.819034 −0.409517 0.912303i \(-0.634303\pi\)
−0.409517 + 0.912303i \(0.634303\pi\)
\(240\) 0 0
\(241\) 3265.58 0.872839 0.436420 0.899743i \(-0.356246\pi\)
0.436420 + 0.899743i \(0.356246\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 2473.25 0.644940
\(246\) 0 0
\(247\) −2011.25 −0.518110
\(248\) 0 0
\(249\) −1306.30 −0.332463
\(250\) 0 0
\(251\) 6363.16 1.60016 0.800078 0.599897i \(-0.204792\pi\)
0.800078 + 0.599897i \(0.204792\pi\)
\(252\) 0 0
\(253\) −4668.96 −1.16022
\(254\) 0 0
\(255\) −697.168 −0.171209
\(256\) 0 0
\(257\) −6085.36 −1.47702 −0.738511 0.674242i \(-0.764471\pi\)
−0.738511 + 0.674242i \(0.764471\pi\)
\(258\) 0 0
\(259\) 1305.31 0.313159
\(260\) 0 0
\(261\) −184.070 −0.0436538
\(262\) 0 0
\(263\) −123.227 −0.0288916 −0.0144458 0.999896i \(-0.504598\pi\)
−0.0144458 + 0.999896i \(0.504598\pi\)
\(264\) 0 0
\(265\) −914.395 −0.211965
\(266\) 0 0
\(267\) −777.700 −0.178256
\(268\) 0 0
\(269\) −1935.79 −0.438763 −0.219381 0.975639i \(-0.570404\pi\)
−0.219381 + 0.975639i \(0.570404\pi\)
\(270\) 0 0
\(271\) 4612.69 1.03395 0.516976 0.856000i \(-0.327058\pi\)
0.516976 + 0.856000i \(0.327058\pi\)
\(272\) 0 0
\(273\) −438.918 −0.0973059
\(274\) 0 0
\(275\) −147.383 −0.0323183
\(276\) 0 0
\(277\) −5834.30 −1.26552 −0.632761 0.774347i \(-0.718078\pi\)
−0.632761 + 0.774347i \(0.718078\pi\)
\(278\) 0 0
\(279\) −2397.82 −0.514529
\(280\) 0 0
\(281\) 4691.91 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(282\) 0 0
\(283\) −3465.60 −0.727945 −0.363973 0.931410i \(-0.618580\pi\)
−0.363973 + 0.931410i \(0.618580\pi\)
\(284\) 0 0
\(285\) −5306.09 −1.10283
\(286\) 0 0
\(287\) 4402.50 0.905475
\(288\) 0 0
\(289\) −4499.79 −0.915894
\(290\) 0 0
\(291\) −3675.51 −0.740420
\(292\) 0 0
\(293\) 2677.31 0.533822 0.266911 0.963721i \(-0.413997\pi\)
0.266911 + 0.963721i \(0.413997\pi\)
\(294\) 0 0
\(295\) −9988.43 −1.97135
\(296\) 0 0
\(297\) 698.720 0.136511
\(298\) 0 0
\(299\) 2345.44 0.453646
\(300\) 0 0
\(301\) −1703.98 −0.326299
\(302\) 0 0
\(303\) −1936.25 −0.367111
\(304\) 0 0
\(305\) 2138.60 0.401494
\(306\) 0 0
\(307\) −471.915 −0.0877316 −0.0438658 0.999037i \(-0.513967\pi\)
−0.0438658 + 0.999037i \(0.513967\pi\)
\(308\) 0 0
\(309\) −1533.41 −0.282306
\(310\) 0 0
\(311\) 1518.52 0.276872 0.138436 0.990371i \(-0.455793\pi\)
0.138436 + 0.990371i \(0.455793\pi\)
\(312\) 0 0
\(313\) 4049.86 0.731348 0.365674 0.930743i \(-0.380839\pi\)
0.365674 + 0.930743i \(0.380839\pi\)
\(314\) 0 0
\(315\) −1157.95 −0.207121
\(316\) 0 0
\(317\) 3253.96 0.576532 0.288266 0.957550i \(-0.406921\pi\)
0.288266 + 0.957550i \(0.406921\pi\)
\(318\) 0 0
\(319\) 529.272 0.0928951
\(320\) 0 0
\(321\) 1824.59 0.317254
\(322\) 0 0
\(323\) 3144.92 0.541759
\(324\) 0 0
\(325\) 74.0375 0.0126365
\(326\) 0 0
\(327\) 3900.11 0.659561
\(328\) 0 0
\(329\) 5259.86 0.881415
\(330\) 0 0
\(331\) 3422.45 0.568322 0.284161 0.958777i \(-0.408285\pi\)
0.284161 + 0.958777i \(0.408285\pi\)
\(332\) 0 0
\(333\) 1043.85 0.171780
\(334\) 0 0
\(335\) −6964.54 −1.13586
\(336\) 0 0
\(337\) −9301.67 −1.50354 −0.751772 0.659423i \(-0.770801\pi\)
−0.751772 + 0.659423i \(0.770801\pi\)
\(338\) 0 0
\(339\) −126.586 −0.0202808
\(340\) 0 0
\(341\) 6894.66 1.09492
\(342\) 0 0
\(343\) −6294.99 −0.990955
\(344\) 0 0
\(345\) 6187.74 0.965613
\(346\) 0 0
\(347\) −216.898 −0.0335554 −0.0167777 0.999859i \(-0.505341\pi\)
−0.0167777 + 0.999859i \(0.505341\pi\)
\(348\) 0 0
\(349\) −4809.84 −0.737721 −0.368861 0.929485i \(-0.620252\pi\)
−0.368861 + 0.929485i \(0.620252\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) −2859.64 −0.431170 −0.215585 0.976485i \(-0.569166\pi\)
−0.215585 + 0.976485i \(0.569166\pi\)
\(354\) 0 0
\(355\) 2835.62 0.423941
\(356\) 0 0
\(357\) 686.319 0.101747
\(358\) 0 0
\(359\) −3686.04 −0.541899 −0.270949 0.962594i \(-0.587338\pi\)
−0.270949 + 0.962594i \(0.587338\pi\)
\(360\) 0 0
\(361\) 17076.8 2.48969
\(362\) 0 0
\(363\) 1983.91 0.286855
\(364\) 0 0
\(365\) −9748.98 −1.39804
\(366\) 0 0
\(367\) 3470.59 0.493633 0.246816 0.969062i \(-0.420616\pi\)
0.246816 + 0.969062i \(0.420616\pi\)
\(368\) 0 0
\(369\) 3520.65 0.496688
\(370\) 0 0
\(371\) 900.166 0.125968
\(372\) 0 0
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) 0 0
\(375\) −4091.75 −0.563459
\(376\) 0 0
\(377\) −265.879 −0.0363221
\(378\) 0 0
\(379\) −345.604 −0.0468403 −0.0234202 0.999726i \(-0.507456\pi\)
−0.0234202 + 0.999726i \(0.507456\pi\)
\(380\) 0 0
\(381\) −933.055 −0.125464
\(382\) 0 0
\(383\) 3386.40 0.451793 0.225897 0.974151i \(-0.427469\pi\)
0.225897 + 0.974151i \(0.427469\pi\)
\(384\) 0 0
\(385\) 3329.56 0.440754
\(386\) 0 0
\(387\) −1362.67 −0.178988
\(388\) 0 0
\(389\) −1629.88 −0.212438 −0.106219 0.994343i \(-0.533874\pi\)
−0.106219 + 0.994343i \(0.533874\pi\)
\(390\) 0 0
\(391\) −3667.47 −0.474353
\(392\) 0 0
\(393\) 6002.95 0.770506
\(394\) 0 0
\(395\) −3786.58 −0.482338
\(396\) 0 0
\(397\) 7938.94 1.00364 0.501819 0.864973i \(-0.332664\pi\)
0.501819 + 0.864973i \(0.332664\pi\)
\(398\) 0 0
\(399\) 5223.52 0.655396
\(400\) 0 0
\(401\) 214.402 0.0267001 0.0133500 0.999911i \(-0.495750\pi\)
0.0133500 + 0.999911i \(0.495750\pi\)
\(402\) 0 0
\(403\) −3463.52 −0.428114
\(404\) 0 0
\(405\) −926.008 −0.113614
\(406\) 0 0
\(407\) −3001.48 −0.365548
\(408\) 0 0
\(409\) −4783.73 −0.578338 −0.289169 0.957278i \(-0.593379\pi\)
−0.289169 + 0.957278i \(0.593379\pi\)
\(410\) 0 0
\(411\) −3115.58 −0.373917
\(412\) 0 0
\(413\) 9832.99 1.17155
\(414\) 0 0
\(415\) −4977.95 −0.588815
\(416\) 0 0
\(417\) −8575.39 −1.00705
\(418\) 0 0
\(419\) 9903.67 1.15472 0.577358 0.816491i \(-0.304084\pi\)
0.577358 + 0.816491i \(0.304084\pi\)
\(420\) 0 0
\(421\) −12120.6 −1.40314 −0.701572 0.712598i \(-0.747518\pi\)
−0.701572 + 0.712598i \(0.747518\pi\)
\(422\) 0 0
\(423\) 4206.28 0.483491
\(424\) 0 0
\(425\) −115.770 −0.0132133
\(426\) 0 0
\(427\) −2105.32 −0.238603
\(428\) 0 0
\(429\) 1009.26 0.113584
\(430\) 0 0
\(431\) 13672.6 1.52805 0.764023 0.645189i \(-0.223221\pi\)
0.764023 + 0.645189i \(0.223221\pi\)
\(432\) 0 0
\(433\) 7113.10 0.789455 0.394727 0.918798i \(-0.370839\pi\)
0.394727 + 0.918798i \(0.370839\pi\)
\(434\) 0 0
\(435\) −701.441 −0.0773138
\(436\) 0 0
\(437\) −27912.9 −3.05550
\(438\) 0 0
\(439\) 6022.04 0.654707 0.327353 0.944902i \(-0.393843\pi\)
0.327353 + 0.944902i \(0.393843\pi\)
\(440\) 0 0
\(441\) −1947.07 −0.210244
\(442\) 0 0
\(443\) 12994.4 1.39364 0.696821 0.717245i \(-0.254597\pi\)
0.696821 + 0.717245i \(0.254597\pi\)
\(444\) 0 0
\(445\) −2963.61 −0.315704
\(446\) 0 0
\(447\) −2230.44 −0.236009
\(448\) 0 0
\(449\) 10984.3 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(450\) 0 0
\(451\) −10123.2 −1.05695
\(452\) 0 0
\(453\) 6831.72 0.708570
\(454\) 0 0
\(455\) −1672.60 −0.172335
\(456\) 0 0
\(457\) 9834.10 1.00661 0.503304 0.864109i \(-0.332118\pi\)
0.503304 + 0.864109i \(0.332118\pi\)
\(458\) 0 0
\(459\) 548.845 0.0558124
\(460\) 0 0
\(461\) 3401.42 0.343644 0.171822 0.985128i \(-0.445035\pi\)
0.171822 + 0.985128i \(0.445035\pi\)
\(462\) 0 0
\(463\) −1739.42 −0.174596 −0.0872979 0.996182i \(-0.527823\pi\)
−0.0872979 + 0.996182i \(0.527823\pi\)
\(464\) 0 0
\(465\) −9137.45 −0.911267
\(466\) 0 0
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) 0 0
\(469\) 6856.16 0.675028
\(470\) 0 0
\(471\) −9520.54 −0.931387
\(472\) 0 0
\(473\) 3918.20 0.380886
\(474\) 0 0
\(475\) −881.114 −0.0851122
\(476\) 0 0
\(477\) 719.858 0.0690986
\(478\) 0 0
\(479\) −8431.98 −0.804315 −0.402158 0.915570i \(-0.631740\pi\)
−0.402158 + 0.915570i \(0.631740\pi\)
\(480\) 0 0
\(481\) 1507.79 0.142930
\(482\) 0 0
\(483\) −6091.45 −0.573852
\(484\) 0 0
\(485\) −14006.4 −1.31133
\(486\) 0 0
\(487\) 11684.7 1.08723 0.543617 0.839334i \(-0.317055\pi\)
0.543617 + 0.839334i \(0.317055\pi\)
\(488\) 0 0
\(489\) −6943.94 −0.642159
\(490\) 0 0
\(491\) 3954.70 0.363489 0.181745 0.983346i \(-0.441826\pi\)
0.181745 + 0.983346i \(0.441826\pi\)
\(492\) 0 0
\(493\) 415.744 0.0379801
\(494\) 0 0
\(495\) 2662.63 0.241771
\(496\) 0 0
\(497\) −2791.49 −0.251943
\(498\) 0 0
\(499\) 5690.37 0.510493 0.255246 0.966876i \(-0.417843\pi\)
0.255246 + 0.966876i \(0.417843\pi\)
\(500\) 0 0
\(501\) −7996.95 −0.713128
\(502\) 0 0
\(503\) −10859.1 −0.962595 −0.481298 0.876557i \(-0.659834\pi\)
−0.481298 + 0.876557i \(0.659834\pi\)
\(504\) 0 0
\(505\) −7378.53 −0.650178
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) −18558.6 −1.61610 −0.808049 0.589115i \(-0.799476\pi\)
−0.808049 + 0.589115i \(0.799476\pi\)
\(510\) 0 0
\(511\) 9597.27 0.830838
\(512\) 0 0
\(513\) 4177.22 0.359510
\(514\) 0 0
\(515\) −5843.42 −0.499984
\(516\) 0 0
\(517\) −12094.7 −1.02887
\(518\) 0 0
\(519\) 495.730 0.0419271
\(520\) 0 0
\(521\) 17297.5 1.45454 0.727271 0.686350i \(-0.240788\pi\)
0.727271 + 0.686350i \(0.240788\pi\)
\(522\) 0 0
\(523\) 5016.11 0.419386 0.209693 0.977767i \(-0.432753\pi\)
0.209693 + 0.977767i \(0.432753\pi\)
\(524\) 0 0
\(525\) −192.286 −0.0159849
\(526\) 0 0
\(527\) 5415.77 0.447656
\(528\) 0 0
\(529\) 20383.8 1.67533
\(530\) 0 0
\(531\) 7863.39 0.642640
\(532\) 0 0
\(533\) 5085.39 0.413269
\(534\) 0 0
\(535\) 6953.01 0.561878
\(536\) 0 0
\(537\) 2137.02 0.171730
\(538\) 0 0
\(539\) 5598.57 0.447398
\(540\) 0 0
\(541\) 17642.3 1.40204 0.701018 0.713144i \(-0.252729\pi\)
0.701018 + 0.713144i \(0.252729\pi\)
\(542\) 0 0
\(543\) −6619.59 −0.523156
\(544\) 0 0
\(545\) 14862.3 1.16813
\(546\) 0 0
\(547\) 18414.9 1.43943 0.719713 0.694271i \(-0.244273\pi\)
0.719713 + 0.694271i \(0.244273\pi\)
\(548\) 0 0
\(549\) −1683.61 −0.130883
\(550\) 0 0
\(551\) 3164.20 0.244645
\(552\) 0 0
\(553\) 3727.66 0.286648
\(554\) 0 0
\(555\) 3977.84 0.304234
\(556\) 0 0
\(557\) 8179.15 0.622193 0.311096 0.950378i \(-0.399304\pi\)
0.311096 + 0.950378i \(0.399304\pi\)
\(558\) 0 0
\(559\) −1968.30 −0.148927
\(560\) 0 0
\(561\) −1578.14 −0.118769
\(562\) 0 0
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) 0 0
\(565\) −482.385 −0.0359187
\(566\) 0 0
\(567\) 911.598 0.0675194
\(568\) 0 0
\(569\) 10118.3 0.745485 0.372743 0.927935i \(-0.378417\pi\)
0.372743 + 0.927935i \(0.378417\pi\)
\(570\) 0 0
\(571\) −23428.9 −1.71711 −0.858555 0.512721i \(-0.828638\pi\)
−0.858555 + 0.512721i \(0.828638\pi\)
\(572\) 0 0
\(573\) 4411.92 0.321659
\(574\) 0 0
\(575\) 1027.52 0.0745225
\(576\) 0 0
\(577\) 20508.1 1.47966 0.739831 0.672793i \(-0.234906\pi\)
0.739831 + 0.672793i \(0.234906\pi\)
\(578\) 0 0
\(579\) −1108.68 −0.0795771
\(580\) 0 0
\(581\) 4900.49 0.349925
\(582\) 0 0
\(583\) −2069.87 −0.147042
\(584\) 0 0
\(585\) −1337.57 −0.0945327
\(586\) 0 0
\(587\) 5968.43 0.419665 0.209833 0.977737i \(-0.432708\pi\)
0.209833 + 0.977737i \(0.432708\pi\)
\(588\) 0 0
\(589\) 41219.0 2.88353
\(590\) 0 0
\(591\) 12820.2 0.892308
\(592\) 0 0
\(593\) −14659.5 −1.01517 −0.507584 0.861602i \(-0.669461\pi\)
−0.507584 + 0.861602i \(0.669461\pi\)
\(594\) 0 0
\(595\) 2615.38 0.180202
\(596\) 0 0
\(597\) 12462.9 0.854394
\(598\) 0 0
\(599\) −23635.9 −1.61225 −0.806125 0.591746i \(-0.798439\pi\)
−0.806125 + 0.591746i \(0.798439\pi\)
\(600\) 0 0
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) 0 0
\(603\) 5482.83 0.370279
\(604\) 0 0
\(605\) 7560.15 0.508039
\(606\) 0 0
\(607\) 5098.56 0.340930 0.170465 0.985364i \(-0.445473\pi\)
0.170465 + 0.985364i \(0.445473\pi\)
\(608\) 0 0
\(609\) 690.525 0.0459466
\(610\) 0 0
\(611\) 6075.74 0.402288
\(612\) 0 0
\(613\) 1516.39 0.0999128 0.0499564 0.998751i \(-0.484092\pi\)
0.0499564 + 0.998751i \(0.484092\pi\)
\(614\) 0 0
\(615\) 13416.3 0.879668
\(616\) 0 0
\(617\) 18539.3 1.20966 0.604832 0.796353i \(-0.293240\pi\)
0.604832 + 0.796353i \(0.293240\pi\)
\(618\) 0 0
\(619\) −25684.9 −1.66779 −0.833897 0.551920i \(-0.813895\pi\)
−0.833897 + 0.551920i \(0.813895\pi\)
\(620\) 0 0
\(621\) −4871.30 −0.314780
\(622\) 0 0
\(623\) 2917.49 0.187619
\(624\) 0 0
\(625\) −16304.5 −1.04349
\(626\) 0 0
\(627\) −12011.1 −0.765038
\(628\) 0 0
\(629\) −2357.67 −0.149454
\(630\) 0 0
\(631\) 22410.9 1.41389 0.706945 0.707269i \(-0.250073\pi\)
0.706945 + 0.707269i \(0.250073\pi\)
\(632\) 0 0
\(633\) 3694.77 0.231997
\(634\) 0 0
\(635\) −3555.62 −0.222206
\(636\) 0 0
\(637\) −2812.43 −0.174933
\(638\) 0 0
\(639\) −2232.34 −0.138200
\(640\) 0 0
\(641\) 6827.81 0.420721 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(642\) 0 0
\(643\) 23264.3 1.42684 0.713418 0.700738i \(-0.247146\pi\)
0.713418 + 0.700738i \(0.247146\pi\)
\(644\) 0 0
\(645\) −5192.76 −0.316999
\(646\) 0 0
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) 0 0
\(649\) −22610.3 −1.36754
\(650\) 0 0
\(651\) 8995.26 0.541554
\(652\) 0 0
\(653\) 10909.0 0.653755 0.326878 0.945067i \(-0.394003\pi\)
0.326878 + 0.945067i \(0.394003\pi\)
\(654\) 0 0
\(655\) 22875.7 1.36462
\(656\) 0 0
\(657\) 7674.89 0.455747
\(658\) 0 0
\(659\) 4182.99 0.247263 0.123631 0.992328i \(-0.460546\pi\)
0.123631 + 0.992328i \(0.460546\pi\)
\(660\) 0 0
\(661\) 2224.23 0.130881 0.0654406 0.997856i \(-0.479155\pi\)
0.0654406 + 0.997856i \(0.479155\pi\)
\(662\) 0 0
\(663\) 792.776 0.0464387
\(664\) 0 0
\(665\) 19905.4 1.16075
\(666\) 0 0
\(667\) −3689.95 −0.214206
\(668\) 0 0
\(669\) −6561.72 −0.379209
\(670\) 0 0
\(671\) 4841.04 0.278519
\(672\) 0 0
\(673\) −24152.5 −1.38337 −0.691687 0.722197i \(-0.743132\pi\)
−0.691687 + 0.722197i \(0.743132\pi\)
\(674\) 0 0
\(675\) −153.770 −0.00876833
\(676\) 0 0
\(677\) −15310.7 −0.869187 −0.434593 0.900627i \(-0.643108\pi\)
−0.434593 + 0.900627i \(0.643108\pi\)
\(678\) 0 0
\(679\) 13788.4 0.779310
\(680\) 0 0
\(681\) 12416.0 0.698652
\(682\) 0 0
\(683\) −11399.6 −0.638646 −0.319323 0.947646i \(-0.603455\pi\)
−0.319323 + 0.947646i \(0.603455\pi\)
\(684\) 0 0
\(685\) −11872.6 −0.662233
\(686\) 0 0
\(687\) 2506.06 0.139174
\(688\) 0 0
\(689\) 1039.79 0.0574935
\(690\) 0 0
\(691\) 3323.23 0.182955 0.0914773 0.995807i \(-0.470841\pi\)
0.0914773 + 0.995807i \(0.470841\pi\)
\(692\) 0 0
\(693\) −2621.20 −0.143681
\(694\) 0 0
\(695\) −32678.5 −1.78355
\(696\) 0 0
\(697\) −7951.82 −0.432133
\(698\) 0 0
\(699\) −11056.5 −0.598279
\(700\) 0 0
\(701\) −12670.4 −0.682673 −0.341336 0.939941i \(-0.610880\pi\)
−0.341336 + 0.939941i \(0.610880\pi\)
\(702\) 0 0
\(703\) −17944.0 −0.962692
\(704\) 0 0
\(705\) 16029.0 0.856295
\(706\) 0 0
\(707\) 7263.71 0.386393
\(708\) 0 0
\(709\) 13075.2 0.692594 0.346297 0.938125i \(-0.387439\pi\)
0.346297 + 0.938125i \(0.387439\pi\)
\(710\) 0 0
\(711\) 2980.99 0.157237
\(712\) 0 0
\(713\) −48067.8 −2.52476
\(714\) 0 0
\(715\) 3846.03 0.201165
\(716\) 0 0
\(717\) 9078.62 0.472869
\(718\) 0 0
\(719\) 2988.41 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(720\) 0 0
\(721\) 5752.48 0.297134
\(722\) 0 0
\(723\) −9796.73 −0.503934
\(724\) 0 0
\(725\) −116.479 −0.00596680
\(726\) 0 0
\(727\) 5507.46 0.280963 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 3077.75 0.155725
\(732\) 0 0
\(733\) −36585.2 −1.84353 −0.921764 0.387751i \(-0.873252\pi\)
−0.921764 + 0.387751i \(0.873252\pi\)
\(734\) 0 0
\(735\) −7419.75 −0.372356
\(736\) 0 0
\(737\) −15765.3 −0.787953
\(738\) 0 0
\(739\) −6425.89 −0.319865 −0.159933 0.987128i \(-0.551128\pi\)
−0.159933 + 0.987128i \(0.551128\pi\)
\(740\) 0 0
\(741\) 6033.76 0.299131
\(742\) 0 0
\(743\) −20411.0 −1.00782 −0.503908 0.863757i \(-0.668105\pi\)
−0.503908 + 0.863757i \(0.668105\pi\)
\(744\) 0 0
\(745\) −8499.60 −0.417988
\(746\) 0 0
\(747\) 3918.89 0.191947
\(748\) 0 0
\(749\) −6844.81 −0.333917
\(750\) 0 0
\(751\) 24259.5 1.17875 0.589375 0.807860i \(-0.299374\pi\)
0.589375 + 0.807860i \(0.299374\pi\)
\(752\) 0 0
\(753\) −19089.5 −0.923850
\(754\) 0 0
\(755\) 26033.9 1.25493
\(756\) 0 0
\(757\) 9295.39 0.446297 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(758\) 0 0
\(759\) 14006.9 0.669851
\(760\) 0 0
\(761\) −21974.7 −1.04676 −0.523378 0.852101i \(-0.675328\pi\)
−0.523378 + 0.852101i \(0.675328\pi\)
\(762\) 0 0
\(763\) −14631.0 −0.694204
\(764\) 0 0
\(765\) 2091.50 0.0988476
\(766\) 0 0
\(767\) 11358.2 0.534709
\(768\) 0 0
\(769\) 22987.4 1.07795 0.538977 0.842320i \(-0.318811\pi\)
0.538977 + 0.842320i \(0.318811\pi\)
\(770\) 0 0
\(771\) 18256.1 0.852759
\(772\) 0 0
\(773\) 31970.9 1.48760 0.743799 0.668404i \(-0.233022\pi\)
0.743799 + 0.668404i \(0.233022\pi\)
\(774\) 0 0
\(775\) −1517.34 −0.0703283
\(776\) 0 0
\(777\) −3915.94 −0.180803
\(778\) 0 0
\(779\) −60520.8 −2.78354
\(780\) 0 0
\(781\) 6418.85 0.294090
\(782\) 0 0
\(783\) 552.209 0.0252035
\(784\) 0 0
\(785\) −36280.2 −1.64955
\(786\) 0 0
\(787\) −6087.26 −0.275715 −0.137857 0.990452i \(-0.544022\pi\)
−0.137857 + 0.990452i \(0.544022\pi\)
\(788\) 0 0
\(789\) 369.680 0.0166805
\(790\) 0 0
\(791\) 474.878 0.0213460
\(792\) 0 0
\(793\) −2431.88 −0.108901
\(794\) 0 0
\(795\) 2743.19 0.122378
\(796\) 0 0
\(797\) 23080.0 1.02577 0.512883 0.858458i \(-0.328577\pi\)
0.512883 + 0.858458i \(0.328577\pi\)
\(798\) 0 0
\(799\) −9500.41 −0.420651
\(800\) 0 0
\(801\) 2333.10 0.102916
\(802\) 0 0
\(803\) −22068.3 −0.969829
\(804\) 0 0
\(805\) −23212.9 −1.01633
\(806\) 0 0
\(807\) 5807.37 0.253320
\(808\) 0 0
\(809\) −32377.8 −1.40710 −0.703550 0.710646i \(-0.748403\pi\)
−0.703550 + 0.710646i \(0.748403\pi\)
\(810\) 0 0
\(811\) −26352.8 −1.14103 −0.570513 0.821288i \(-0.693256\pi\)
−0.570513 + 0.821288i \(0.693256\pi\)
\(812\) 0 0
\(813\) −13838.1 −0.596952
\(814\) 0 0
\(815\) −26461.5 −1.13731
\(816\) 0 0
\(817\) 23424.5 1.00308
\(818\) 0 0
\(819\) 1316.75 0.0561796
\(820\) 0 0
\(821\) 35355.3 1.50294 0.751468 0.659770i \(-0.229346\pi\)
0.751468 + 0.659770i \(0.229346\pi\)
\(822\) 0 0
\(823\) 12663.3 0.536347 0.268173 0.963371i \(-0.413580\pi\)
0.268173 + 0.963371i \(0.413580\pi\)
\(824\) 0 0
\(825\) 442.149 0.0186590
\(826\) 0 0
\(827\) 16295.2 0.685176 0.342588 0.939486i \(-0.388697\pi\)
0.342588 + 0.939486i \(0.388697\pi\)
\(828\) 0 0
\(829\) 13638.9 0.571411 0.285705 0.958318i \(-0.407772\pi\)
0.285705 + 0.958318i \(0.407772\pi\)
\(830\) 0 0
\(831\) 17502.9 0.730649
\(832\) 0 0
\(833\) 4397.69 0.182918
\(834\) 0 0
\(835\) −30474.2 −1.26300
\(836\) 0 0
\(837\) 7193.46 0.297064
\(838\) 0 0
\(839\) 1890.31 0.0777838 0.0388919 0.999243i \(-0.487617\pi\)
0.0388919 + 0.999243i \(0.487617\pi\)
\(840\) 0 0
\(841\) −23970.7 −0.982849
\(842\) 0 0
\(843\) −14075.7 −0.575082
\(844\) 0 0
\(845\) −1932.04 −0.0786559
\(846\) 0 0
\(847\) −7442.50 −0.301921
\(848\) 0 0
\(849\) 10396.8 0.420279
\(850\) 0 0
\(851\) 20925.6 0.842914
\(852\) 0 0
\(853\) 1620.21 0.0650351 0.0325175 0.999471i \(-0.489648\pi\)
0.0325175 + 0.999471i \(0.489648\pi\)
\(854\) 0 0
\(855\) 15918.3 0.636718
\(856\) 0 0
\(857\) −14508.4 −0.578292 −0.289146 0.957285i \(-0.593371\pi\)
−0.289146 + 0.957285i \(0.593371\pi\)
\(858\) 0 0
\(859\) −29639.8 −1.17730 −0.588648 0.808389i \(-0.700340\pi\)
−0.588648 + 0.808389i \(0.700340\pi\)
\(860\) 0 0
\(861\) −13207.5 −0.522776
\(862\) 0 0
\(863\) −21528.8 −0.849186 −0.424593 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(864\) 0 0
\(865\) 1889.10 0.0742557
\(866\) 0 0
\(867\) 13499.4 0.528792
\(868\) 0 0
\(869\) −8571.50 −0.334601
\(870\) 0 0
\(871\) 7919.65 0.308091
\(872\) 0 0
\(873\) 11026.5 0.427482
\(874\) 0 0
\(875\) 15349.9 0.593054
\(876\) 0 0
\(877\) 14865.3 0.572366 0.286183 0.958175i \(-0.407613\pi\)
0.286183 + 0.958175i \(0.407613\pi\)
\(878\) 0 0
\(879\) −8031.92 −0.308202
\(880\) 0 0
\(881\) −21336.0 −0.815921 −0.407961 0.913000i \(-0.633760\pi\)
−0.407961 + 0.913000i \(0.633760\pi\)
\(882\) 0 0
\(883\) −37538.2 −1.43065 −0.715323 0.698794i \(-0.753720\pi\)
−0.715323 + 0.698794i \(0.753720\pi\)
\(884\) 0 0
\(885\) 29965.3 1.13816
\(886\) 0 0
\(887\) −34575.0 −1.30881 −0.654406 0.756144i \(-0.727081\pi\)
−0.654406 + 0.756144i \(0.727081\pi\)
\(888\) 0 0
\(889\) 3500.29 0.132054
\(890\) 0 0
\(891\) −2096.16 −0.0788148
\(892\) 0 0
\(893\) −72306.9 −2.70958
\(894\) 0 0
\(895\) 8143.61 0.304146
\(896\) 0 0
\(897\) −7036.32 −0.261913
\(898\) 0 0
\(899\) 5448.96 0.202150
\(900\) 0 0
\(901\) −1625.89 −0.0601178
\(902\) 0 0
\(903\) 5111.95 0.188389
\(904\) 0 0
\(905\) −25225.5 −0.926545
\(906\) 0 0
\(907\) 10424.8 0.381641 0.190820 0.981625i \(-0.438885\pi\)
0.190820 + 0.981625i \(0.438885\pi\)
\(908\) 0 0
\(909\) 5808.75 0.211952
\(910\) 0 0
\(911\) −10961.8 −0.398661 −0.199331 0.979932i \(-0.563877\pi\)
−0.199331 + 0.979932i \(0.563877\pi\)
\(912\) 0 0
\(913\) −11268.3 −0.408464
\(914\) 0 0
\(915\) −6415.80 −0.231803
\(916\) 0 0
\(917\) −22519.7 −0.810976
\(918\) 0 0
\(919\) 10779.2 0.386914 0.193457 0.981109i \(-0.438030\pi\)
0.193457 + 0.981109i \(0.438030\pi\)
\(920\) 0 0
\(921\) 1415.74 0.0506519
\(922\) 0 0
\(923\) −3224.49 −0.114990
\(924\) 0 0
\(925\) 660.549 0.0234797
\(926\) 0 0
\(927\) 4600.23 0.162990
\(928\) 0 0
\(929\) 5429.07 0.191735 0.0958675 0.995394i \(-0.469437\pi\)
0.0958675 + 0.995394i \(0.469437\pi\)
\(930\) 0 0
\(931\) 33470.5 1.17825
\(932\) 0 0
\(933\) −4555.55 −0.159852
\(934\) 0 0
\(935\) −6013.88 −0.210348
\(936\) 0 0
\(937\) −21300.1 −0.742631 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(938\) 0 0
\(939\) −12149.6 −0.422244
\(940\) 0 0
\(941\) 26851.2 0.930207 0.465103 0.885256i \(-0.346017\pi\)
0.465103 + 0.885256i \(0.346017\pi\)
\(942\) 0 0
\(943\) 70576.7 2.43722
\(944\) 0 0
\(945\) 3473.86 0.119582
\(946\) 0 0
\(947\) 8021.68 0.275258 0.137629 0.990484i \(-0.456052\pi\)
0.137629 + 0.990484i \(0.456052\pi\)
\(948\) 0 0
\(949\) 11085.9 0.379204
\(950\) 0 0
\(951\) −9761.88 −0.332861
\(952\) 0 0
\(953\) 35715.0 1.21398 0.606990 0.794709i \(-0.292377\pi\)
0.606990 + 0.794709i \(0.292377\pi\)
\(954\) 0 0
\(955\) 16812.6 0.569680
\(956\) 0 0
\(957\) −1587.82 −0.0536330
\(958\) 0 0
\(959\) 11687.9 0.393557
\(960\) 0 0
\(961\) 41190.9 1.38266
\(962\) 0 0
\(963\) −5473.76 −0.183166
\(964\) 0 0
\(965\) −4224.88 −0.140936
\(966\) 0 0
\(967\) −53338.8 −1.77380 −0.886898 0.461965i \(-0.847145\pi\)
−0.886898 + 0.461965i \(0.847145\pi\)
\(968\) 0 0
\(969\) −9434.77 −0.312785
\(970\) 0 0
\(971\) −23112.9 −0.763882 −0.381941 0.924187i \(-0.624744\pi\)
−0.381941 + 0.924187i \(0.624744\pi\)
\(972\) 0 0
\(973\) 32170.0 1.05994
\(974\) 0 0
\(975\) −222.113 −0.00729569
\(976\) 0 0
\(977\) −52874.6 −1.73143 −0.865715 0.500538i \(-0.833136\pi\)
−0.865715 + 0.500538i \(0.833136\pi\)
\(978\) 0 0
\(979\) −6708.57 −0.219006
\(980\) 0 0
\(981\) −11700.3 −0.380798
\(982\) 0 0
\(983\) −45173.1 −1.46572 −0.732858 0.680381i \(-0.761814\pi\)
−0.732858 + 0.680381i \(0.761814\pi\)
\(984\) 0 0
\(985\) 48854.5 1.58034
\(986\) 0 0
\(987\) −15779.6 −0.508885
\(988\) 0 0
\(989\) −27316.7 −0.878281
\(990\) 0 0
\(991\) −60485.6 −1.93884 −0.969418 0.245414i \(-0.921076\pi\)
−0.969418 + 0.245414i \(0.921076\pi\)
\(992\) 0 0
\(993\) −10267.3 −0.328121
\(994\) 0 0
\(995\) 47492.8 1.51319
\(996\) 0 0
\(997\) −18108.1 −0.575214 −0.287607 0.957749i \(-0.592860\pi\)
−0.287607 + 0.957749i \(0.592860\pi\)
\(998\) 0 0
\(999\) −3131.56 −0.0991773
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 624.4.a.t.1.1 3
3.2 odd 2 1872.4.a.bk.1.3 3
4.3 odd 2 39.4.a.c.1.3 3
8.3 odd 2 2496.4.a.bl.1.3 3
8.5 even 2 2496.4.a.bp.1.3 3
12.11 even 2 117.4.a.f.1.1 3
20.19 odd 2 975.4.a.l.1.1 3
28.27 even 2 1911.4.a.k.1.3 3
52.31 even 4 507.4.b.g.337.1 6
52.47 even 4 507.4.b.g.337.6 6
52.51 odd 2 507.4.a.h.1.1 3
156.155 even 2 1521.4.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 4.3 odd 2
117.4.a.f.1.1 3 12.11 even 2
507.4.a.h.1.1 3 52.51 odd 2
507.4.b.g.337.1 6 52.31 even 4
507.4.b.g.337.6 6 52.47 even 4
624.4.a.t.1.1 3 1.1 even 1 trivial
975.4.a.l.1.1 3 20.19 odd 2
1521.4.a.u.1.3 3 156.155 even 2
1872.4.a.bk.1.3 3 3.2 odd 2
1911.4.a.k.1.3 3 28.27 even 2
2496.4.a.bl.1.3 3 8.3 odd 2
2496.4.a.bp.1.3 3 8.5 even 2