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Properties

Label 1600.4.a.by.1.1
Level $1600$
Weight $4$
Character 1600.1
Self dual yes
Analytic conductor $94.403$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1600,4,Mod(1,1600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1600.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,9,0,0,0,-26,0,54,0,-59,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(94.4030560092\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1600.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} -26.0000 q^{7} +54.0000 q^{9} -59.0000 q^{11} -28.0000 q^{13} +5.00000 q^{17} +109.000 q^{19} -234.000 q^{21} +194.000 q^{23} +243.000 q^{27} +32.0000 q^{29} -10.0000 q^{31} -531.000 q^{33} +198.000 q^{37} -252.000 q^{39} +117.000 q^{41} +388.000 q^{43} +68.0000 q^{47} +333.000 q^{49} +45.0000 q^{51} +18.0000 q^{53} +981.000 q^{57} +392.000 q^{59} +710.000 q^{61} -1404.00 q^{63} -253.000 q^{67} +1746.00 q^{69} +612.000 q^{71} -549.000 q^{73} +1534.00 q^{77} -414.000 q^{79} +729.000 q^{81} -121.000 q^{83} +288.000 q^{87} -81.0000 q^{89} +728.000 q^{91} -90.0000 q^{93} -1502.00 q^{97} -3186.00 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −26.0000 −1.40387 −0.701934 − 0.712242i \(-0.747680\pi\)
−0.701934 + 0.712242i \(0.747680\pi\)
\(8\) 0 0
\(9\) 54.0000 2.00000
\(10\) 0 0
\(11\) −59.0000 −1.61720 −0.808599 − 0.588361i \(-0.799774\pi\)
−0.808599 + 0.588361i \(0.799774\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 0.0713340 0.0356670 − 0.999364i \(-0.488644\pi\)
0.0356670 + 0.999364i \(0.488644\pi\)
\(18\) 0 0
\(19\) 109.000 1.31612 0.658061 − 0.752965i \(-0.271377\pi\)
0.658061 + 0.752965i \(0.271377\pi\)
\(20\) 0 0
\(21\) −234.000 −2.43157
\(22\) 0 0
\(23\) 194.000 1.75877 0.879387 − 0.476108i \(-0.157953\pi\)
0.879387 + 0.476108i \(0.157953\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 243.000 1.73205
\(28\) 0 0
\(29\) 32.0000 0.204905 0.102453 − 0.994738i \(-0.467331\pi\)
0.102453 + 0.994738i \(0.467331\pi\)
\(30\) 0 0
\(31\) −10.0000 −0.0579372 −0.0289686 − 0.999580i \(-0.509222\pi\)
−0.0289686 + 0.999580i \(0.509222\pi\)
\(32\) 0 0
\(33\) −531.000 −2.80107
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 198.000 0.879757 0.439878 − 0.898057i \(-0.355022\pi\)
0.439878 + 0.898057i \(0.355022\pi\)
\(38\) 0 0
\(39\) −252.000 −1.03467
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 − 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) 388.000 1.37603 0.688017 − 0.725695i \(-0.258482\pi\)
0.688017 + 0.725695i \(0.258482\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 68.0000 0.211039 0.105519 − 0.994417i \(-0.466350\pi\)
0.105519 + 0.994417i \(0.466350\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) 45.0000 0.123554
\(52\) 0 0
\(53\) 18.0000 0.0466508 0.0233254 − 0.999728i \(-0.492575\pi\)
0.0233254 + 0.999728i \(0.492575\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 981.000 2.27959
\(58\) 0 0
\(59\) 392.000 0.864984 0.432492 − 0.901638i \(-0.357634\pi\)
0.432492 + 0.901638i \(0.357634\pi\)
\(60\) 0 0
\(61\) 710.000 1.49027 0.745133 − 0.666916i \(-0.232386\pi\)
0.745133 + 0.666916i \(0.232386\pi\)
\(62\) 0 0
\(63\) −1404.00 −2.80774
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −253.000 −0.461326 −0.230663 − 0.973034i \(-0.574090\pi\)
−0.230663 + 0.973034i \(0.574090\pi\)
\(68\) 0 0
\(69\) 1746.00 3.04629
\(70\) 0 0
\(71\) 612.000 1.02297 0.511486 − 0.859292i \(-0.329095\pi\)
0.511486 + 0.859292i \(0.329095\pi\)
\(72\) 0 0
\(73\) −549.000 −0.880214 −0.440107 − 0.897945i \(-0.645059\pi\)
−0.440107 + 0.897945i \(0.645059\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1534.00 2.27033
\(78\) 0 0
\(79\) −414.000 −0.589603 −0.294802 − 0.955559i \(-0.595254\pi\)
−0.294802 + 0.955559i \(0.595254\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) −121.000 −0.160018 −0.0800089 − 0.996794i \(-0.525495\pi\)
−0.0800089 + 0.996794i \(0.525495\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 288.000 0.354906
\(88\) 0 0
\(89\) −81.0000 −0.0964717 −0.0482359 − 0.998836i \(-0.515360\pi\)
−0.0482359 + 0.998836i \(0.515360\pi\)
\(90\) 0 0
\(91\) 728.000 0.838628
\(92\) 0 0
\(93\) −90.0000 −0.100350
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1502.00 −1.57222 −0.786108 − 0.618089i \(-0.787907\pi\)
−0.786108 + 0.618089i \(0.787907\pi\)
\(98\) 0 0
\(99\) −3186.00 −3.23439
\(100\) 0 0
\(101\) 234.000 0.230533 0.115267 − 0.993335i \(-0.463228\pi\)
0.115267 + 0.993335i \(0.463228\pi\)
\(102\) 0 0
\(103\) 1172.00 1.12117 0.560585 − 0.828097i \(-0.310576\pi\)
0.560585 + 0.828097i \(0.310576\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1125.00 1.01643 0.508214 − 0.861231i \(-0.330306\pi\)
0.508214 + 0.861231i \(0.330306\pi\)
\(108\) 0 0
\(109\) 1234.00 1.08436 0.542182 − 0.840261i \(-0.317598\pi\)
0.542182 + 0.840261i \(0.317598\pi\)
\(110\) 0 0
\(111\) 1782.00 1.52378
\(112\) 0 0
\(113\) 567.000 0.472025 0.236013 − 0.971750i \(-0.424159\pi\)
0.236013 + 0.971750i \(0.424159\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1512.00 −1.19474
\(118\) 0 0
\(119\) −130.000 −0.100144
\(120\) 0 0
\(121\) 2150.00 1.61533
\(122\) 0 0
\(123\) 1053.00 0.771917
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2358.00 1.64755 0.823774 − 0.566918i \(-0.191864\pi\)
0.823774 + 0.566918i \(0.191864\pi\)
\(128\) 0 0
\(129\) 3492.00 2.38336
\(130\) 0 0
\(131\) −1692.00 −1.12848 −0.564239 − 0.825611i \(-0.690831\pi\)
−0.564239 + 0.825611i \(0.690831\pi\)
\(132\) 0 0
\(133\) −2834.00 −1.84766
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 229.000 0.142809 0.0714043 − 0.997447i \(-0.477252\pi\)
0.0714043 + 0.997447i \(0.477252\pi\)
\(138\) 0 0
\(139\) 2781.00 1.69699 0.848494 − 0.529205i \(-0.177510\pi\)
0.848494 + 0.529205i \(0.177510\pi\)
\(140\) 0 0
\(141\) 612.000 0.365530
\(142\) 0 0
\(143\) 1652.00 0.966064
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2997.00 1.68155
\(148\) 0 0
\(149\) −1472.00 −0.809335 −0.404668 − 0.914464i \(-0.632613\pi\)
−0.404668 + 0.914464i \(0.632613\pi\)
\(150\) 0 0
\(151\) −1322.00 −0.712469 −0.356235 − 0.934397i \(-0.615940\pi\)
−0.356235 + 0.934397i \(0.615940\pi\)
\(152\) 0 0
\(153\) 270.000 0.142668
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −298.000 −0.151484 −0.0757420 − 0.997127i \(-0.524133\pi\)
−0.0757420 + 0.997127i \(0.524133\pi\)
\(158\) 0 0
\(159\) 162.000 0.0808015
\(160\) 0 0
\(161\) −5044.00 −2.46909
\(162\) 0 0
\(163\) −341.000 −0.163860 −0.0819300 − 0.996638i \(-0.526108\pi\)
−0.0819300 + 0.996638i \(0.526108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −684.000 −0.316943 −0.158472 − 0.987364i \(-0.550657\pi\)
−0.158472 + 0.987364i \(0.550657\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 5886.00 2.63224
\(172\) 0 0
\(173\) 2344.00 1.03012 0.515061 − 0.857154i \(-0.327769\pi\)
0.515061 + 0.857154i \(0.327769\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3528.00 1.49820
\(178\) 0 0
\(179\) −1111.00 −0.463911 −0.231955 − 0.972726i \(-0.574512\pi\)
−0.231955 + 0.972726i \(0.574512\pi\)
\(180\) 0 0
\(181\) −2042.00 −0.838567 −0.419284 − 0.907855i \(-0.637719\pi\)
−0.419284 + 0.907855i \(0.637719\pi\)
\(182\) 0 0
\(183\) 6390.00 2.58122
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −295.000 −0.115361
\(188\) 0 0
\(189\) −6318.00 −2.43157
\(190\) 0 0
\(191\) −5270.00 −1.99646 −0.998230 − 0.0594735i \(-0.981058\pi\)
−0.998230 + 0.0594735i \(0.981058\pi\)
\(192\) 0 0
\(193\) 613.000 0.228625 0.114313 − 0.993445i \(-0.463533\pi\)
0.114313 + 0.993445i \(0.463533\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1174.00 −0.424589 −0.212295 − 0.977206i \(-0.568094\pi\)
−0.212295 + 0.977206i \(0.568094\pi\)
\(198\) 0 0
\(199\) −3428.00 −1.22113 −0.610564 − 0.791967i \(-0.709057\pi\)
−0.610564 + 0.791967i \(0.709057\pi\)
\(200\) 0 0
\(201\) −2277.00 −0.799041
\(202\) 0 0
\(203\) −832.000 −0.287660
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10476.0 3.51755
\(208\) 0 0
\(209\) −6431.00 −2.12843
\(210\) 0 0
\(211\) 2339.00 0.763144 0.381572 − 0.924339i \(-0.375383\pi\)
0.381572 + 0.924339i \(0.375383\pi\)
\(212\) 0 0
\(213\) 5508.00 1.77184
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 260.000 0.0813362
\(218\) 0 0
\(219\) −4941.00 −1.52457
\(220\) 0 0
\(221\) −140.000 −0.0426128
\(222\) 0 0
\(223\) −3932.00 −1.18075 −0.590373 − 0.807131i \(-0.701019\pi\)
−0.590373 + 0.807131i \(0.701019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6084.00 1.77890 0.889448 − 0.457037i \(-0.151089\pi\)
0.889448 + 0.457037i \(0.151089\pi\)
\(228\) 0 0
\(229\) −4996.00 −1.44168 −0.720841 − 0.693101i \(-0.756244\pi\)
−0.720841 + 0.693101i \(0.756244\pi\)
\(230\) 0 0
\(231\) 13806.0 3.93233
\(232\) 0 0
\(233\) 3222.00 0.905924 0.452962 − 0.891530i \(-0.350367\pi\)
0.452962 + 0.891530i \(0.350367\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3726.00 −1.02122
\(238\) 0 0
\(239\) −2736.00 −0.740490 −0.370245 − 0.928934i \(-0.620726\pi\)
−0.370245 + 0.928934i \(0.620726\pi\)
\(240\) 0 0
\(241\) 1673.00 0.447168 0.223584 − 0.974685i \(-0.428224\pi\)
0.223584 + 0.974685i \(0.428224\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3052.00 −0.786211
\(248\) 0 0
\(249\) −1089.00 −0.277159
\(250\) 0 0
\(251\) 5355.00 1.34663 0.673316 − 0.739355i \(-0.264869\pi\)
0.673316 + 0.739355i \(0.264869\pi\)
\(252\) 0 0
\(253\) −11446.0 −2.84428
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5490.00 −1.33252 −0.666258 − 0.745721i \(-0.732105\pi\)
−0.666258 + 0.745721i \(0.732105\pi\)
\(258\) 0 0
\(259\) −5148.00 −1.23506
\(260\) 0 0
\(261\) 1728.00 0.409810
\(262\) 0 0
\(263\) −3150.00 −0.738545 −0.369272 − 0.929321i \(-0.620393\pi\)
−0.369272 + 0.929321i \(0.620393\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −729.000 −0.167094
\(268\) 0 0
\(269\) −176.000 −0.0398919 −0.0199459 − 0.999801i \(-0.506349\pi\)
−0.0199459 + 0.999801i \(0.506349\pi\)
\(270\) 0 0
\(271\) −2394.00 −0.536624 −0.268312 − 0.963332i \(-0.586466\pi\)
−0.268312 + 0.963332i \(0.586466\pi\)
\(272\) 0 0
\(273\) 6552.00 1.45255
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6256.00 1.35699 0.678496 − 0.734604i \(-0.262632\pi\)
0.678496 + 0.734604i \(0.262632\pi\)
\(278\) 0 0
\(279\) −540.000 −0.115874
\(280\) 0 0
\(281\) −4802.00 −1.01944 −0.509721 − 0.860340i \(-0.670251\pi\)
−0.509721 + 0.860340i \(0.670251\pi\)
\(282\) 0 0
\(283\) 2123.00 0.445934 0.222967 − 0.974826i \(-0.428426\pi\)
0.222967 + 0.974826i \(0.428426\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3042.00 −0.625657
\(288\) 0 0
\(289\) −4888.00 −0.994911
\(290\) 0 0
\(291\) −13518.0 −2.72316
\(292\) 0 0
\(293\) 8834.00 1.76139 0.880696 − 0.473682i \(-0.157075\pi\)
0.880696 + 0.473682i \(0.157075\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −14337.0 −2.80107
\(298\) 0 0
\(299\) −5432.00 −1.05064
\(300\) 0 0
\(301\) −10088.0 −1.93177
\(302\) 0 0
\(303\) 2106.00 0.399296
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1369.00 0.254505 0.127252 − 0.991870i \(-0.459384\pi\)
0.127252 + 0.991870i \(0.459384\pi\)
\(308\) 0 0
\(309\) 10548.0 1.94192
\(310\) 0 0
\(311\) 10426.0 1.90098 0.950489 − 0.310758i \(-0.100583\pi\)
0.950489 + 0.310758i \(0.100583\pi\)
\(312\) 0 0
\(313\) −3574.00 −0.645413 −0.322707 − 0.946499i \(-0.604593\pi\)
−0.322707 + 0.946499i \(0.604593\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9036.00 −1.60099 −0.800493 − 0.599343i \(-0.795429\pi\)
−0.800493 + 0.599343i \(0.795429\pi\)
\(318\) 0 0
\(319\) −1888.00 −0.331372
\(320\) 0 0
\(321\) 10125.0 1.76051
\(322\) 0 0
\(323\) 545.000 0.0938842
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11106.0 1.87817
\(328\) 0 0
\(329\) −1768.00 −0.296271
\(330\) 0 0
\(331\) 10233.0 1.69926 0.849632 − 0.527376i \(-0.176824\pi\)
0.849632 + 0.527376i \(0.176824\pi\)
\(332\) 0 0
\(333\) 10692.0 1.75951
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4627.00 0.747919 0.373960 − 0.927445i \(-0.378000\pi\)
0.373960 + 0.927445i \(0.378000\pi\)
\(338\) 0 0
\(339\) 5103.00 0.817572
\(340\) 0 0
\(341\) 590.000 0.0936959
\(342\) 0 0
\(343\) 260.000 0.0409291
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4901.00 0.758212 0.379106 − 0.925353i \(-0.376232\pi\)
0.379106 + 0.925353i \(0.376232\pi\)
\(348\) 0 0
\(349\) 4482.00 0.687438 0.343719 − 0.939072i \(-0.388313\pi\)
0.343719 + 0.939072i \(0.388313\pi\)
\(350\) 0 0
\(351\) −6804.00 −1.03467
\(352\) 0 0
\(353\) −1210.00 −0.182441 −0.0912207 − 0.995831i \(-0.529077\pi\)
−0.0912207 + 0.995831i \(0.529077\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1170.00 −0.173454
\(358\) 0 0
\(359\) 9882.00 1.45279 0.726396 − 0.687277i \(-0.241194\pi\)
0.726396 + 0.687277i \(0.241194\pi\)
\(360\) 0 0
\(361\) 5022.00 0.732177
\(362\) 0 0
\(363\) 19350.0 2.79783
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11260.0 1.60155 0.800773 − 0.598968i \(-0.204422\pi\)
0.800773 + 0.598968i \(0.204422\pi\)
\(368\) 0 0
\(369\) 6318.00 0.891333
\(370\) 0 0
\(371\) −468.000 −0.0654915
\(372\) 0 0
\(373\) 3230.00 0.448373 0.224186 − 0.974546i \(-0.428028\pi\)
0.224186 + 0.974546i \(0.428028\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −896.000 −0.122404
\(378\) 0 0
\(379\) 11575.0 1.56878 0.784390 − 0.620268i \(-0.212976\pi\)
0.784390 + 0.620268i \(0.212976\pi\)
\(380\) 0 0
\(381\) 21222.0 2.85364
\(382\) 0 0
\(383\) 18.0000 0.00240145 0.00120073 − 0.999999i \(-0.499618\pi\)
0.00120073 + 0.999999i \(0.499618\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20952.0 2.75207
\(388\) 0 0
\(389\) −10710.0 −1.39593 −0.697967 − 0.716130i \(-0.745912\pi\)
−0.697967 + 0.716130i \(0.745912\pi\)
\(390\) 0 0
\(391\) 970.000 0.125460
\(392\) 0 0
\(393\) −15228.0 −1.95458
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3788.00 0.478877 0.239439 − 0.970912i \(-0.423037\pi\)
0.239439 + 0.970912i \(0.423037\pi\)
\(398\) 0 0
\(399\) −25506.0 −3.20024
\(400\) 0 0
\(401\) −10539.0 −1.31245 −0.656225 − 0.754565i \(-0.727848\pi\)
−0.656225 + 0.754565i \(0.727848\pi\)
\(402\) 0 0
\(403\) 280.000 0.0346099
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11682.0 −1.42274
\(408\) 0 0
\(409\) 5581.00 0.674725 0.337363 − 0.941375i \(-0.390465\pi\)
0.337363 + 0.941375i \(0.390465\pi\)
\(410\) 0 0
\(411\) 2061.00 0.247352
\(412\) 0 0
\(413\) −10192.0 −1.21432
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 25029.0 2.93927
\(418\) 0 0
\(419\) 5193.00 0.605476 0.302738 − 0.953074i \(-0.402099\pi\)
0.302738 + 0.953074i \(0.402099\pi\)
\(420\) 0 0
\(421\) −4788.00 −0.554282 −0.277141 − 0.960829i \(-0.589387\pi\)
−0.277141 + 0.960829i \(0.589387\pi\)
\(422\) 0 0
\(423\) 3672.00 0.422077
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18460.0 −2.09214
\(428\) 0 0
\(429\) 14868.0 1.67327
\(430\) 0 0
\(431\) −8006.00 −0.894746 −0.447373 − 0.894348i \(-0.647640\pi\)
−0.447373 + 0.894348i \(0.647640\pi\)
\(432\) 0 0
\(433\) −2395.00 −0.265811 −0.132906 − 0.991129i \(-0.542431\pi\)
−0.132906 + 0.991129i \(0.542431\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21146.0 2.31476
\(438\) 0 0
\(439\) −1864.00 −0.202651 −0.101326 − 0.994853i \(-0.532308\pi\)
−0.101326 + 0.994853i \(0.532308\pi\)
\(440\) 0 0
\(441\) 17982.0 1.94169
\(442\) 0 0
\(443\) 5463.00 0.585903 0.292951 − 0.956127i \(-0.405363\pi\)
0.292951 + 0.956127i \(0.405363\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13248.0 −1.40181
\(448\) 0 0
\(449\) 12969.0 1.36313 0.681565 − 0.731758i \(-0.261300\pi\)
0.681565 + 0.731758i \(0.261300\pi\)
\(450\) 0 0
\(451\) −6903.00 −0.720731
\(452\) 0 0
\(453\) −11898.0 −1.23403
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18313.0 1.87450 0.937249 − 0.348659i \(-0.113363\pi\)
0.937249 + 0.348659i \(0.113363\pi\)
\(458\) 0 0
\(459\) 1215.00 0.123554
\(460\) 0 0
\(461\) −12492.0 −1.26206 −0.631031 − 0.775758i \(-0.717368\pi\)
−0.631031 + 0.775758i \(0.717368\pi\)
\(462\) 0 0
\(463\) −4428.00 −0.444464 −0.222232 − 0.974994i \(-0.571334\pi\)
−0.222232 + 0.974994i \(0.571334\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1084.00 −0.107412 −0.0537061 − 0.998557i \(-0.517103\pi\)
−0.0537061 + 0.998557i \(0.517103\pi\)
\(468\) 0 0
\(469\) 6578.00 0.647641
\(470\) 0 0
\(471\) −2682.00 −0.262378
\(472\) 0 0
\(473\) −22892.0 −2.22532
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 972.000 0.0933015
\(478\) 0 0
\(479\) 13082.0 1.24787 0.623937 − 0.781474i \(-0.285532\pi\)
0.623937 + 0.781474i \(0.285532\pi\)
\(480\) 0 0
\(481\) −5544.00 −0.525540
\(482\) 0 0
\(483\) −45396.0 −4.27658
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3014.00 0.280446 0.140223 − 0.990120i \(-0.455218\pi\)
0.140223 + 0.990120i \(0.455218\pi\)
\(488\) 0 0
\(489\) −3069.00 −0.283814
\(490\) 0 0
\(491\) −3564.00 −0.327579 −0.163789 − 0.986495i \(-0.552372\pi\)
−0.163789 + 0.986495i \(0.552372\pi\)
\(492\) 0 0
\(493\) 160.000 0.0146167
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15912.0 −1.43612
\(498\) 0 0
\(499\) −15796.0 −1.41709 −0.708543 − 0.705667i \(-0.750647\pi\)
−0.708543 + 0.705667i \(0.750647\pi\)
\(500\) 0 0
\(501\) −6156.00 −0.548962
\(502\) 0 0
\(503\) 10908.0 0.966926 0.483463 − 0.875365i \(-0.339379\pi\)
0.483463 + 0.875365i \(0.339379\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12717.0 −1.11397
\(508\) 0 0
\(509\) −21946.0 −1.91108 −0.955540 − 0.294863i \(-0.904726\pi\)
−0.955540 + 0.294863i \(0.904726\pi\)
\(510\) 0 0
\(511\) 14274.0 1.23570
\(512\) 0 0
\(513\) 26487.0 2.27959
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4012.00 −0.341291
\(518\) 0 0
\(519\) 21096.0 1.78422
\(520\) 0 0
\(521\) −6395.00 −0.537754 −0.268877 − 0.963174i \(-0.586653\pi\)
−0.268877 + 0.963174i \(0.586653\pi\)
\(522\) 0 0
\(523\) −5615.00 −0.469459 −0.234729 − 0.972061i \(-0.575420\pi\)
−0.234729 + 0.972061i \(0.575420\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −50.0000 −0.00413289
\(528\) 0 0
\(529\) 25469.0 2.09329
\(530\) 0 0
\(531\) 21168.0 1.72997
\(532\) 0 0
\(533\) −3276.00 −0.266228
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9999.00 −0.803517
\(538\) 0 0
\(539\) −19647.0 −1.57005
\(540\) 0 0
\(541\) 4112.00 0.326781 0.163391 − 0.986561i \(-0.447757\pi\)
0.163391 + 0.986561i \(0.447757\pi\)
\(542\) 0 0
\(543\) −18378.0 −1.45244
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2167.00 −0.169386 −0.0846931 − 0.996407i \(-0.526991\pi\)
−0.0846931 + 0.996407i \(0.526991\pi\)
\(548\) 0 0
\(549\) 38340.0 2.98053
\(550\) 0 0
\(551\) 3488.00 0.269680
\(552\) 0 0
\(553\) 10764.0 0.827725
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19444.0 1.47912 0.739559 − 0.673092i \(-0.235034\pi\)
0.739559 + 0.673092i \(0.235034\pi\)
\(558\) 0 0
\(559\) −10864.0 −0.822000
\(560\) 0 0
\(561\) −2655.00 −0.199811
\(562\) 0 0
\(563\) 20416.0 1.52830 0.764149 − 0.645040i \(-0.223159\pi\)
0.764149 + 0.645040i \(0.223159\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −18954.0 −1.40387
\(568\) 0 0
\(569\) −3127.00 −0.230388 −0.115194 − 0.993343i \(-0.536749\pi\)
−0.115194 + 0.993343i \(0.536749\pi\)
\(570\) 0 0
\(571\) −22580.0 −1.65489 −0.827446 − 0.561545i \(-0.810207\pi\)
−0.827446 + 0.561545i \(0.810207\pi\)
\(572\) 0 0
\(573\) −47430.0 −3.45797
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −829.000 −0.0598123 −0.0299062 − 0.999553i \(-0.509521\pi\)
−0.0299062 + 0.999553i \(0.509521\pi\)
\(578\) 0 0
\(579\) 5517.00 0.395991
\(580\) 0 0
\(581\) 3146.00 0.224644
\(582\) 0 0
\(583\) −1062.00 −0.0754435
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7119.00 0.500567 0.250283 − 0.968173i \(-0.419476\pi\)
0.250283 + 0.968173i \(0.419476\pi\)
\(588\) 0 0
\(589\) −1090.00 −0.0762524
\(590\) 0 0
\(591\) −10566.0 −0.735410
\(592\) 0 0
\(593\) −8217.00 −0.569025 −0.284512 − 0.958672i \(-0.591832\pi\)
−0.284512 + 0.958672i \(0.591832\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30852.0 −2.11506
\(598\) 0 0
\(599\) 90.0000 0.00613907 0.00306953 − 0.999995i \(-0.499023\pi\)
0.00306953 + 0.999995i \(0.499023\pi\)
\(600\) 0 0
\(601\) −17117.0 −1.16176 −0.580879 − 0.813990i \(-0.697291\pi\)
−0.580879 + 0.813990i \(0.697291\pi\)
\(602\) 0 0
\(603\) −13662.0 −0.922653
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15120.0 1.01104 0.505520 − 0.862815i \(-0.331301\pi\)
0.505520 + 0.862815i \(0.331301\pi\)
\(608\) 0 0
\(609\) −7488.00 −0.498241
\(610\) 0 0
\(611\) −1904.00 −0.126068
\(612\) 0 0
\(613\) 6570.00 0.432887 0.216444 − 0.976295i \(-0.430554\pi\)
0.216444 + 0.976295i \(0.430554\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18846.0 1.22968 0.614839 − 0.788653i \(-0.289221\pi\)
0.614839 + 0.788653i \(0.289221\pi\)
\(618\) 0 0
\(619\) 16316.0 1.05944 0.529722 − 0.848172i \(-0.322296\pi\)
0.529722 + 0.848172i \(0.322296\pi\)
\(620\) 0 0
\(621\) 47142.0 3.04629
\(622\) 0 0
\(623\) 2106.00 0.135434
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −57879.0 −3.68655
\(628\) 0 0
\(629\) 990.000 0.0627566
\(630\) 0 0
\(631\) −20170.0 −1.27251 −0.636256 − 0.771478i \(-0.719518\pi\)
−0.636256 + 0.771478i \(0.719518\pi\)
\(632\) 0 0
\(633\) 21051.0 1.32180
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9324.00 −0.579953
\(638\) 0 0
\(639\) 33048.0 2.04594
\(640\) 0 0
\(641\) −12726.0 −0.784160 −0.392080 − 0.919931i \(-0.628244\pi\)
−0.392080 + 0.919931i \(0.628244\pi\)
\(642\) 0 0
\(643\) 2196.00 0.134684 0.0673420 − 0.997730i \(-0.478548\pi\)
0.0673420 + 0.997730i \(0.478548\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16884.0 1.02593 0.512966 − 0.858409i \(-0.328547\pi\)
0.512966 + 0.858409i \(0.328547\pi\)
\(648\) 0 0
\(649\) −23128.0 −1.39885
\(650\) 0 0
\(651\) 2340.00 0.140878
\(652\) 0 0
\(653\) 4018.00 0.240791 0.120395 − 0.992726i \(-0.461584\pi\)
0.120395 + 0.992726i \(0.461584\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −29646.0 −1.76043
\(658\) 0 0
\(659\) 19071.0 1.12732 0.563658 − 0.826009i \(-0.309394\pi\)
0.563658 + 0.826009i \(0.309394\pi\)
\(660\) 0 0
\(661\) −17424.0 −1.02529 −0.512644 − 0.858601i \(-0.671334\pi\)
−0.512644 + 0.858601i \(0.671334\pi\)
\(662\) 0 0
\(663\) −1260.00 −0.0738075
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6208.00 0.360382
\(668\) 0 0
\(669\) −35388.0 −2.04511
\(670\) 0 0
\(671\) −41890.0 −2.41005
\(672\) 0 0
\(673\) −5382.00 −0.308263 −0.154131 − 0.988050i \(-0.549258\pi\)
−0.154131 + 0.988050i \(0.549258\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4496.00 0.255237 0.127618 − 0.991823i \(-0.459267\pi\)
0.127618 + 0.991823i \(0.459267\pi\)
\(678\) 0 0
\(679\) 39052.0 2.20718
\(680\) 0 0
\(681\) 54756.0 3.08114
\(682\) 0 0
\(683\) 3249.00 0.182020 0.0910099 − 0.995850i \(-0.470991\pi\)
0.0910099 + 0.995850i \(0.470991\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −44964.0 −2.49706
\(688\) 0 0
\(689\) −504.000 −0.0278677
\(690\) 0 0
\(691\) −13399.0 −0.737658 −0.368829 − 0.929497i \(-0.620241\pi\)
−0.368829 + 0.929497i \(0.620241\pi\)
\(692\) 0 0
\(693\) 82836.0 4.54066
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 585.000 0.0317912
\(698\) 0 0
\(699\) 28998.0 1.56911
\(700\) 0 0
\(701\) 18148.0 0.977804 0.488902 − 0.872339i \(-0.337398\pi\)
0.488902 + 0.872339i \(0.337398\pi\)
\(702\) 0 0
\(703\) 21582.0 1.15787
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6084.00 −0.323638
\(708\) 0 0
\(709\) −4868.00 −0.257858 −0.128929 − 0.991654i \(-0.541154\pi\)
−0.128929 + 0.991654i \(0.541154\pi\)
\(710\) 0 0
\(711\) −22356.0 −1.17921
\(712\) 0 0
\(713\) −1940.00 −0.101898
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24624.0 −1.28257
\(718\) 0 0
\(719\) 17366.0 0.900755 0.450377 − 0.892838i \(-0.351289\pi\)
0.450377 + 0.892838i \(0.351289\pi\)
\(720\) 0 0
\(721\) −30472.0 −1.57398
\(722\) 0 0
\(723\) 15057.0 0.774517
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21824.0 −1.11335 −0.556676 − 0.830729i \(-0.687924\pi\)
−0.556676 + 0.830729i \(0.687924\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 1940.00 0.0981580
\(732\) 0 0
\(733\) 31428.0 1.58366 0.791828 − 0.610744i \(-0.209130\pi\)
0.791828 + 0.610744i \(0.209130\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14927.0 0.746056
\(738\) 0 0
\(739\) −14292.0 −0.711420 −0.355710 − 0.934596i \(-0.615761\pi\)
−0.355710 + 0.934596i \(0.615761\pi\)
\(740\) 0 0
\(741\) −27468.0 −1.36176
\(742\) 0 0
\(743\) −13950.0 −0.688797 −0.344398 − 0.938824i \(-0.611917\pi\)
−0.344398 + 0.938824i \(0.611917\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6534.00 −0.320036
\(748\) 0 0
\(749\) −29250.0 −1.42693
\(750\) 0 0
\(751\) 38736.0 1.88215 0.941076 − 0.338194i \(-0.109816\pi\)
0.941076 + 0.338194i \(0.109816\pi\)
\(752\) 0 0
\(753\) 48195.0 2.33243
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3664.00 0.175919 0.0879593 − 0.996124i \(-0.471965\pi\)
0.0879593 + 0.996124i \(0.471965\pi\)
\(758\) 0 0
\(759\) −103014. −4.92644
\(760\) 0 0
\(761\) 19557.0 0.931591 0.465795 − 0.884892i \(-0.345768\pi\)
0.465795 + 0.884892i \(0.345768\pi\)
\(762\) 0 0
\(763\) −32084.0 −1.52231
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10976.0 −0.516715
\(768\) 0 0
\(769\) −13283.0 −0.622883 −0.311442 − 0.950265i \(-0.600812\pi\)
−0.311442 + 0.950265i \(0.600812\pi\)
\(770\) 0 0
\(771\) −49410.0 −2.30799
\(772\) 0 0
\(773\) −24840.0 −1.15580 −0.577900 − 0.816108i \(-0.696127\pi\)
−0.577900 + 0.816108i \(0.696127\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −46332.0 −2.13919
\(778\) 0 0
\(779\) 12753.0 0.586552
\(780\) 0 0
\(781\) −36108.0 −1.65435
\(782\) 0 0
\(783\) 7776.00 0.354906
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18044.0 0.817280 0.408640 − 0.912696i \(-0.366003\pi\)
0.408640 + 0.912696i \(0.366003\pi\)
\(788\) 0 0
\(789\) −28350.0 −1.27920
\(790\) 0 0
\(791\) −14742.0 −0.662661
\(792\) 0 0
\(793\) −19880.0 −0.890239
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6174.00 0.274397 0.137198 − 0.990544i \(-0.456190\pi\)
0.137198 + 0.990544i \(0.456190\pi\)
\(798\) 0 0
\(799\) 340.000 0.0150542
\(800\) 0 0
\(801\) −4374.00 −0.192943
\(802\) 0 0
\(803\) 32391.0 1.42348
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1584.00 −0.0690947
\(808\) 0 0
\(809\) −1998.00 −0.0868306 −0.0434153 − 0.999057i \(-0.513824\pi\)
−0.0434153 + 0.999057i \(0.513824\pi\)
\(810\) 0 0
\(811\) −7156.00 −0.309841 −0.154921 − 0.987927i \(-0.549512\pi\)
−0.154921 + 0.987927i \(0.549512\pi\)
\(812\) 0 0
\(813\) −21546.0 −0.929460
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 42292.0 1.81103
\(818\) 0 0
\(819\) 39312.0 1.67726
\(820\) 0 0
\(821\) −27922.0 −1.18695 −0.593474 − 0.804853i \(-0.702244\pi\)
−0.593474 + 0.804853i \(0.702244\pi\)
\(822\) 0 0
\(823\) −22636.0 −0.958738 −0.479369 − 0.877613i \(-0.659134\pi\)
−0.479369 + 0.877613i \(0.659134\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26559.0 1.11674 0.558372 − 0.829591i \(-0.311426\pi\)
0.558372 + 0.829591i \(0.311426\pi\)
\(828\) 0 0
\(829\) −12580.0 −0.527046 −0.263523 − 0.964653i \(-0.584885\pi\)
−0.263523 + 0.964653i \(0.584885\pi\)
\(830\) 0 0
\(831\) 56304.0 2.35038
\(832\) 0 0
\(833\) 1665.00 0.0692543
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2430.00 −0.100350
\(838\) 0 0
\(839\) 11344.0 0.466792 0.233396 − 0.972382i \(-0.425016\pi\)
0.233396 + 0.972382i \(0.425016\pi\)
\(840\) 0 0
\(841\) −23365.0 −0.958014
\(842\) 0 0
\(843\) −43218.0 −1.76573
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −55900.0 −2.26771
\(848\) 0 0
\(849\) 19107.0 0.772380
\(850\) 0 0
\(851\) 38412.0 1.54729
\(852\) 0 0
\(853\) −14786.0 −0.593509 −0.296754 − 0.954954i \(-0.595904\pi\)
−0.296754 + 0.954954i \(0.595904\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29259.0 −1.16624 −0.583120 − 0.812386i \(-0.698168\pi\)
−0.583120 + 0.812386i \(0.698168\pi\)
\(858\) 0 0
\(859\) −13651.0 −0.542219 −0.271109 − 0.962549i \(-0.587391\pi\)
−0.271109 + 0.962549i \(0.587391\pi\)
\(860\) 0 0
\(861\) −27378.0 −1.08367
\(862\) 0 0
\(863\) −29016.0 −1.14451 −0.572257 − 0.820074i \(-0.693932\pi\)
−0.572257 + 0.820074i \(0.693932\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −43992.0 −1.72324
\(868\) 0 0
\(869\) 24426.0 0.953504
\(870\) 0 0
\(871\) 7084.00 0.275582
\(872\) 0 0
\(873\) −81108.0 −3.14443
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −21412.0 −0.824438 −0.412219 − 0.911085i \(-0.635246\pi\)
−0.412219 + 0.911085i \(0.635246\pi\)
\(878\) 0 0
\(879\) 79506.0 3.05082
\(880\) 0 0
\(881\) −1170.00 −0.0447427 −0.0223713 − 0.999750i \(-0.507122\pi\)
−0.0223713 + 0.999750i \(0.507122\pi\)
\(882\) 0 0
\(883\) −12655.0 −0.482304 −0.241152 − 0.970487i \(-0.577525\pi\)
−0.241152 + 0.970487i \(0.577525\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32764.0 −1.24026 −0.620128 − 0.784500i \(-0.712919\pi\)
−0.620128 + 0.784500i \(0.712919\pi\)
\(888\) 0 0
\(889\) −61308.0 −2.31294
\(890\) 0 0
\(891\) −43011.0 −1.61720
\(892\) 0 0
\(893\) 7412.00 0.277753
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −48888.0 −1.81976
\(898\) 0 0
\(899\) −320.000 −0.0118716
\(900\) 0 0
\(901\) 90.0000 0.00332779
\(902\) 0 0
\(903\) −90792.0 −3.34592
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −29844.0 −1.09256 −0.546281 − 0.837602i \(-0.683957\pi\)
−0.546281 + 0.837602i \(0.683957\pi\)
\(908\) 0 0
\(909\) 12636.0 0.461067
\(910\) 0 0
\(911\) −15628.0 −0.568363 −0.284182 − 0.958770i \(-0.591722\pi\)
−0.284182 + 0.958770i \(0.591722\pi\)
\(912\) 0 0
\(913\) 7139.00 0.258780
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43992.0 1.58424
\(918\) 0 0
\(919\) −42974.0 −1.54253 −0.771263 − 0.636517i \(-0.780375\pi\)
−0.771263 + 0.636517i \(0.780375\pi\)
\(920\) 0 0
\(921\) 12321.0 0.440815
\(922\) 0 0
\(923\) −17136.0 −0.611092
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 63288.0 2.24234
\(928\) 0 0
\(929\) 13342.0 0.471191 0.235596 − 0.971851i \(-0.424296\pi\)
0.235596 + 0.971851i \(0.424296\pi\)
\(930\) 0 0
\(931\) 36297.0 1.27775
\(932\) 0 0
\(933\) 93834.0 3.29259
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3005.00 −0.104770 −0.0523848 − 0.998627i \(-0.516682\pi\)
−0.0523848 + 0.998627i \(0.516682\pi\)
\(938\) 0 0
\(939\) −32166.0 −1.11789
\(940\) 0 0
\(941\) −16204.0 −0.561355 −0.280678 − 0.959802i \(-0.590559\pi\)
−0.280678 + 0.959802i \(0.590559\pi\)
\(942\) 0 0
\(943\) 22698.0 0.783827
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30200.0 −1.03629 −0.518146 − 0.855292i \(-0.673378\pi\)
−0.518146 + 0.855292i \(0.673378\pi\)
\(948\) 0 0
\(949\) 15372.0 0.525813
\(950\) 0 0
\(951\) −81324.0 −2.77299
\(952\) 0 0
\(953\) −29583.0 −1.00555 −0.502774 − 0.864418i \(-0.667687\pi\)
−0.502774 + 0.864418i \(0.667687\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −16992.0 −0.573953
\(958\) 0 0
\(959\) −5954.00 −0.200485
\(960\) 0 0
\(961\) −29691.0 −0.996643
\(962\) 0 0
\(963\) 60750.0 2.03286
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6480.00 0.215494 0.107747 − 0.994178i \(-0.465636\pi\)
0.107747 + 0.994178i \(0.465636\pi\)
\(968\) 0 0
\(969\) 4905.00 0.162612
\(970\) 0 0
\(971\) −40171.0 −1.32765 −0.663825 − 0.747888i \(-0.731068\pi\)
−0.663825 + 0.747888i \(0.731068\pi\)
\(972\) 0 0
\(973\) −72306.0 −2.38235
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50801.0 1.66353 0.831765 − 0.555129i \(-0.187331\pi\)
0.831765 + 0.555129i \(0.187331\pi\)
\(978\) 0 0
\(979\) 4779.00 0.156014
\(980\) 0 0
\(981\) 66636.0 2.16873
\(982\) 0 0
\(983\) 58338.0 1.89287 0.946436 − 0.322891i \(-0.104655\pi\)
0.946436 + 0.322891i \(0.104655\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −15912.0 −0.513156
\(988\) 0 0
\(989\) 75272.0 2.42013
\(990\) 0 0
\(991\) 51202.0 1.64126 0.820628 − 0.571462i \(-0.193624\pi\)
0.820628 + 0.571462i \(0.193624\pi\)
\(992\) 0 0
\(993\) 92097.0 2.94321
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1764.00 −0.0560345 −0.0280173 − 0.999607i \(-0.508919\pi\)
−0.0280173 + 0.999607i \(0.508919\pi\)
\(998\) 0 0
\(999\) 48114.0 1.52378
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 1600.4.a.by.1.1 1
4.3 odd 2 1600.4.a.c.1.1 1
5.4 even 2 1600.4.a.b.1.1 1
8.3 odd 2 200.4.a.j.1.1 yes 1
8.5 even 2 400.4.a.a.1.1 1
20.19 odd 2 1600.4.a.bz.1.1 1
24.11 even 2 1800.4.a.bh.1.1 1
40.3 even 4 200.4.c.b.49.2 2
40.13 odd 4 400.4.c.b.49.1 2
40.19 odd 2 200.4.a.b.1.1 ✓ 1
40.27 even 4 200.4.c.b.49.1 2
40.29 even 2 400.4.a.t.1.1 1
40.37 odd 4 400.4.c.b.49.2 2
120.59 even 2 1800.4.a.c.1.1 1
120.83 odd 4 1800.4.f.w.649.1 2
120.107 odd 4 1800.4.f.w.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.b.1.1 ✓ 1 40.19 odd 2
200.4.a.j.1.1 yes 1 8.3 odd 2
200.4.c.b.49.1 2 40.27 even 4
200.4.c.b.49.2 2 40.3 even 4
400.4.a.a.1.1 1 8.5 even 2
400.4.a.t.1.1 1 40.29 even 2
400.4.c.b.49.1 2 40.13 odd 4
400.4.c.b.49.2 2 40.37 odd 4
1600.4.a.b.1.1 1 5.4 even 2
1600.4.a.c.1.1 1 4.3 odd 2
1600.4.a.by.1.1 1 1.1 even 1 trivial
1600.4.a.bz.1.1 1 20.19 odd 2
1800.4.a.c.1.1 1 120.59 even 2
1800.4.a.bh.1.1 1 24.11 even 2
1800.4.f.w.649.1 2 120.83 odd 4
1800.4.f.w.649.2 2 120.107 odd 4