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Properties

Label 1200.4.a.be.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +8.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +10.0000 q^{13} -18.0000 q^{17} +100.000 q^{19} +24.0000 q^{21} +72.0000 q^{23} +27.0000 q^{27} -234.000 q^{29} +16.0000 q^{31} -108.000 q^{33} +226.000 q^{37} +30.0000 q^{39} +90.0000 q^{41} +452.000 q^{43} +432.000 q^{47} -279.000 q^{49} -54.0000 q^{51} -414.000 q^{53} +300.000 q^{57} +684.000 q^{59} +422.000 q^{61} +72.0000 q^{63} +332.000 q^{67} +216.000 q^{69} +360.000 q^{71} -26.0000 q^{73} -288.000 q^{77} -512.000 q^{79} +81.0000 q^{81} -1188.00 q^{83} -702.000 q^{87} -630.000 q^{89} +80.0000 q^{91} +48.0000 q^{93} +1054.00 q^{97} -324.000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 − 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 10.0000 0.213346 0.106673 − 0.994294i \(-0.465980\pi\)
0.106673 + 0.994294i \(0.465980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −18.0000 −0.256802 −0.128401 − 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) 0 0
\(19\) 100.000 1.20745 0.603726 − 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) 24.0000 0.249392
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 − 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −234.000 −1.49837 −0.749185 − 0.662361i \(-0.769554\pi\)
−0.749185 + 0.662361i \(0.769554\pi\)
\(30\) 0 0
\(31\) 16.0000 0.0926995 0.0463498 − 0.998925i \(-0.485241\pi\)
0.0463498 + 0.998925i \(0.485241\pi\)
\(32\) 0 0
\(33\) −108.000 −0.569709
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 226.000 1.00417 0.502083 − 0.864819i \(-0.332567\pi\)
0.502083 + 0.864819i \(0.332567\pi\)
\(38\) 0 0
\(39\) 30.0000 0.123176
\(40\) 0 0
\(41\) 90.0000 0.342820 0.171410 − 0.985200i \(-0.445168\pi\)
0.171410 + 0.985200i \(0.445168\pi\)
\(42\) 0 0
\(43\) 452.000 1.60301 0.801504 − 0.597989i \(-0.204033\pi\)
0.801504 + 0.597989i \(0.204033\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 432.000 1.34072 0.670358 − 0.742038i \(-0.266140\pi\)
0.670358 + 0.742038i \(0.266140\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −54.0000 −0.148265
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 − 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 300.000 0.697122
\(58\) 0 0
\(59\) 684.000 1.50931 0.754654 − 0.656123i \(-0.227805\pi\)
0.754654 + 0.656123i \(0.227805\pi\)
\(60\) 0 0
\(61\) 422.000 0.885763 0.442882 − 0.896580i \(-0.353956\pi\)
0.442882 + 0.896580i \(0.353956\pi\)
\(62\) 0 0
\(63\) 72.0000 0.143986
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 332.000 0.605377 0.302688 − 0.953090i \(-0.402116\pi\)
0.302688 + 0.953090i \(0.402116\pi\)
\(68\) 0 0
\(69\) 216.000 0.376860
\(70\) 0 0
\(71\) 360.000 0.601748 0.300874 − 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) −26.0000 −0.0416859 −0.0208429 − 0.999783i \(-0.506635\pi\)
−0.0208429 + 0.999783i \(0.506635\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) −512.000 −0.729171 −0.364585 − 0.931170i \(-0.618789\pi\)
−0.364585 + 0.931170i \(0.618789\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1188.00 −1.57108 −0.785542 − 0.618809i \(-0.787616\pi\)
−0.785542 + 0.618809i \(0.787616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −702.000 −0.865084
\(88\) 0 0
\(89\) −630.000 −0.750336 −0.375168 − 0.926957i \(-0.622415\pi\)
−0.375168 + 0.926957i \(0.622415\pi\)
\(90\) 0 0
\(91\) 80.0000 0.0921569
\(92\) 0 0
\(93\) 48.0000 0.0535201
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1054.00 1.10327 0.551637 − 0.834085i \(-0.314004\pi\)
0.551637 + 0.834085i \(0.314004\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 558.000 0.549733 0.274867 − 0.961482i \(-0.411366\pi\)
0.274867 + 0.961482i \(0.411366\pi\)
\(102\) 0 0
\(103\) 8.00000 0.00765304 0.00382652 − 0.999993i \(-0.498782\pi\)
0.00382652 + 0.999993i \(0.498782\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1764.00 1.59376 0.796880 − 0.604138i \(-0.206482\pi\)
0.796880 + 0.604138i \(0.206482\pi\)
\(108\) 0 0
\(109\) 1622.00 1.42532 0.712658 − 0.701512i \(-0.247491\pi\)
0.712658 + 0.701512i \(0.247491\pi\)
\(110\) 0 0
\(111\) 678.000 0.579756
\(112\) 0 0
\(113\) 1134.00 0.944051 0.472025 − 0.881585i \(-0.343523\pi\)
0.472025 + 0.881585i \(0.343523\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 90.0000 0.0711154
\(118\) 0 0
\(119\) −144.000 −0.110928
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 270.000 0.197927
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −592.000 −0.413634 −0.206817 − 0.978380i \(-0.566310\pi\)
−0.206817 + 0.978380i \(0.566310\pi\)
\(128\) 0 0
\(129\) 1356.00 0.925497
\(130\) 0 0
\(131\) 1908.00 1.27254 0.636270 − 0.771466i \(-0.280476\pi\)
0.636270 + 0.771466i \(0.280476\pi\)
\(132\) 0 0
\(133\) 800.000 0.521570
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −954.000 −0.594932 −0.297466 − 0.954732i \(-0.596142\pi\)
−0.297466 + 0.954732i \(0.596142\pi\)
\(138\) 0 0
\(139\) −2564.00 −1.56457 −0.782286 − 0.622919i \(-0.785947\pi\)
−0.782286 + 0.622919i \(0.785947\pi\)
\(140\) 0 0
\(141\) 1296.00 0.774063
\(142\) 0 0
\(143\) −360.000 −0.210522
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −837.000 −0.469623
\(148\) 0 0
\(149\) −738.000 −0.405767 −0.202884 − 0.979203i \(-0.565031\pi\)
−0.202884 + 0.979203i \(0.565031\pi\)
\(150\) 0 0
\(151\) 2440.00 1.31500 0.657498 − 0.753456i \(-0.271615\pi\)
0.657498 + 0.753456i \(0.271615\pi\)
\(152\) 0 0
\(153\) −162.000 −0.0856008
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2554.00 1.29829 0.649145 − 0.760665i \(-0.275127\pi\)
0.649145 + 0.760665i \(0.275127\pi\)
\(158\) 0 0
\(159\) −1242.00 −0.619478
\(160\) 0 0
\(161\) 576.000 0.281958
\(162\) 0 0
\(163\) −820.000 −0.394033 −0.197016 − 0.980400i \(-0.563125\pi\)
−0.197016 + 0.980400i \(0.563125\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1944.00 0.900786 0.450393 − 0.892830i \(-0.351284\pi\)
0.450393 + 0.892830i \(0.351284\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) 900.000 0.402484
\(172\) 0 0
\(173\) 1242.00 0.545824 0.272912 − 0.962039i \(-0.412013\pi\)
0.272912 + 0.962039i \(0.412013\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2052.00 0.871400
\(178\) 0 0
\(179\) −1116.00 −0.465999 −0.232999 − 0.972477i \(-0.574854\pi\)
−0.232999 + 0.972477i \(0.574854\pi\)
\(180\) 0 0
\(181\) 1070.00 0.439406 0.219703 − 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(182\) 0 0
\(183\) 1266.00 0.511396
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 648.000 0.253403
\(188\) 0 0
\(189\) 216.000 0.0831306
\(190\) 0 0
\(191\) 576.000 0.218209 0.109104 − 0.994030i \(-0.465202\pi\)
0.109104 + 0.994030i \(0.465202\pi\)
\(192\) 0 0
\(193\) 1342.00 0.500514 0.250257 − 0.968179i \(-0.419485\pi\)
0.250257 + 0.968179i \(0.419485\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1422.00 −0.514281 −0.257140 − 0.966374i \(-0.582780\pi\)
−0.257140 + 0.966374i \(0.582780\pi\)
\(198\) 0 0
\(199\) −872.000 −0.310625 −0.155313 − 0.987865i \(-0.549639\pi\)
−0.155313 + 0.987865i \(0.549639\pi\)
\(200\) 0 0
\(201\) 996.000 0.349515
\(202\) 0 0
\(203\) −1872.00 −0.647235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) −1340.00 −0.437201 −0.218600 − 0.975814i \(-0.570149\pi\)
−0.218600 + 0.975814i \(0.570149\pi\)
\(212\) 0 0
\(213\) 1080.00 0.347420
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 128.000 0.0400424
\(218\) 0 0
\(219\) −78.0000 −0.0240674
\(220\) 0 0
\(221\) −180.000 −0.0547878
\(222\) 0 0
\(223\) 4880.00 1.46542 0.732711 − 0.680540i \(-0.238255\pi\)
0.732711 + 0.680540i \(0.238255\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2700.00 0.789451 0.394725 − 0.918799i \(-0.370840\pi\)
0.394725 + 0.918799i \(0.370840\pi\)
\(228\) 0 0
\(229\) 254.000 0.0732960 0.0366480 − 0.999328i \(-0.488332\pi\)
0.0366480 + 0.999328i \(0.488332\pi\)
\(230\) 0 0
\(231\) −864.000 −0.246091
\(232\) 0 0
\(233\) −4410.00 −1.23995 −0.619976 − 0.784621i \(-0.712858\pi\)
−0.619976 + 0.784621i \(0.712858\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1536.00 −0.420987
\(238\) 0 0
\(239\) 3888.00 1.05228 0.526138 − 0.850399i \(-0.323640\pi\)
0.526138 + 0.850399i \(0.323640\pi\)
\(240\) 0 0
\(241\) 5138.00 1.37331 0.686655 − 0.726984i \(-0.259078\pi\)
0.686655 + 0.726984i \(0.259078\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1000.00 0.257605
\(248\) 0 0
\(249\) −3564.00 −0.907066
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 − 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5886.00 1.42863 0.714316 − 0.699823i \(-0.246738\pi\)
0.714316 + 0.699823i \(0.246738\pi\)
\(258\) 0 0
\(259\) 1808.00 0.433759
\(260\) 0 0
\(261\) −2106.00 −0.499456
\(262\) 0 0
\(263\) 2232.00 0.523312 0.261656 − 0.965161i \(-0.415731\pi\)
0.261656 + 0.965161i \(0.415731\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1890.00 −0.433206
\(268\) 0 0
\(269\) −666.000 −0.150954 −0.0754772 − 0.997148i \(-0.524048\pi\)
−0.0754772 + 0.997148i \(0.524048\pi\)
\(270\) 0 0
\(271\) 5536.00 1.24092 0.620458 − 0.784240i \(-0.286947\pi\)
0.620458 + 0.784240i \(0.286947\pi\)
\(272\) 0 0
\(273\) 240.000 0.0532068
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2126.00 −0.461151 −0.230576 − 0.973054i \(-0.574061\pi\)
−0.230576 + 0.973054i \(0.574061\pi\)
\(278\) 0 0
\(279\) 144.000 0.0308998
\(280\) 0 0
\(281\) −2934.00 −0.622875 −0.311437 − 0.950267i \(-0.600810\pi\)
−0.311437 + 0.950267i \(0.600810\pi\)
\(282\) 0 0
\(283\) 2036.00 0.427659 0.213830 − 0.976871i \(-0.431406\pi\)
0.213830 + 0.976871i \(0.431406\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 720.000 0.148085
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 3162.00 0.636975
\(292\) 0 0
\(293\) −2286.00 −0.455800 −0.227900 − 0.973684i \(-0.573186\pi\)
−0.227900 + 0.973684i \(0.573186\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −972.000 −0.189903
\(298\) 0 0
\(299\) 720.000 0.139260
\(300\) 0 0
\(301\) 3616.00 0.692434
\(302\) 0 0
\(303\) 1674.00 0.317389
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1244.00 0.231267 0.115633 − 0.993292i \(-0.463110\pi\)
0.115633 + 0.993292i \(0.463110\pi\)
\(308\) 0 0
\(309\) 24.0000 0.00441849
\(310\) 0 0
\(311\) −1224.00 −0.223173 −0.111586 − 0.993755i \(-0.535593\pi\)
−0.111586 + 0.993755i \(0.535593\pi\)
\(312\) 0 0
\(313\) −1898.00 −0.342752 −0.171376 − 0.985206i \(-0.554821\pi\)
−0.171376 + 0.985206i \(0.554821\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9162.00 1.62331 0.811655 − 0.584137i \(-0.198567\pi\)
0.811655 + 0.584137i \(0.198567\pi\)
\(318\) 0 0
\(319\) 8424.00 1.47854
\(320\) 0 0
\(321\) 5292.00 0.920158
\(322\) 0 0
\(323\) −1800.00 −0.310076
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4866.00 0.822906
\(328\) 0 0
\(329\) 3456.00 0.579135
\(330\) 0 0
\(331\) 4348.00 0.722017 0.361009 − 0.932562i \(-0.382432\pi\)
0.361009 + 0.932562i \(0.382432\pi\)
\(332\) 0 0
\(333\) 2034.00 0.334722
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7154.00 −1.15639 −0.578195 − 0.815899i \(-0.696243\pi\)
−0.578195 + 0.815899i \(0.696243\pi\)
\(338\) 0 0
\(339\) 3402.00 0.545048
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1836.00 −0.284039 −0.142020 − 0.989864i \(-0.545360\pi\)
−0.142020 + 0.989864i \(0.545360\pi\)
\(348\) 0 0
\(349\) 5894.00 0.904007 0.452004 − 0.892016i \(-0.350709\pi\)
0.452004 + 0.892016i \(0.350709\pi\)
\(350\) 0 0
\(351\) 270.000 0.0410585
\(352\) 0 0
\(353\) −11106.0 −1.67454 −0.837270 − 0.546789i \(-0.815850\pi\)
−0.837270 + 0.546789i \(0.815850\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −432.000 −0.0640444
\(358\) 0 0
\(359\) −13176.0 −1.93705 −0.968527 − 0.248907i \(-0.919929\pi\)
−0.968527 + 0.248907i \(0.919929\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6112.00 −0.869329 −0.434665 − 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(368\) 0 0
\(369\) 810.000 0.114273
\(370\) 0 0
\(371\) −3312.00 −0.463478
\(372\) 0 0
\(373\) 13618.0 1.89038 0.945192 − 0.326515i \(-0.105874\pi\)
0.945192 + 0.326515i \(0.105874\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2340.00 −0.319671
\(378\) 0 0
\(379\) −692.000 −0.0937880 −0.0468940 − 0.998900i \(-0.514932\pi\)
−0.0468940 + 0.998900i \(0.514932\pi\)
\(380\) 0 0
\(381\) −1776.00 −0.238812
\(382\) 0 0
\(383\) −8064.00 −1.07585 −0.537926 − 0.842992i \(-0.680792\pi\)
−0.537926 + 0.842992i \(0.680792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4068.00 0.534336
\(388\) 0 0
\(389\) 12654.0 1.64931 0.824657 − 0.565633i \(-0.191368\pi\)
0.824657 + 0.565633i \(0.191368\pi\)
\(390\) 0 0
\(391\) −1296.00 −0.167625
\(392\) 0 0
\(393\) 5724.00 0.734701
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 106.000 0.0134005 0.00670024 − 0.999978i \(-0.497867\pi\)
0.00670024 + 0.999978i \(0.497867\pi\)
\(398\) 0 0
\(399\) 2400.00 0.301129
\(400\) 0 0
\(401\) −4014.00 −0.499874 −0.249937 − 0.968262i \(-0.580410\pi\)
−0.249937 + 0.968262i \(0.580410\pi\)
\(402\) 0 0
\(403\) 160.000 0.0197771
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8136.00 −0.990876
\(408\) 0 0
\(409\) 3914.00 0.473190 0.236595 − 0.971608i \(-0.423968\pi\)
0.236595 + 0.971608i \(0.423968\pi\)
\(410\) 0 0
\(411\) −2862.00 −0.343484
\(412\) 0 0
\(413\) 5472.00 0.651960
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7692.00 −0.903307
\(418\) 0 0
\(419\) −4428.00 −0.516282 −0.258141 − 0.966107i \(-0.583110\pi\)
−0.258141 + 0.966107i \(0.583110\pi\)
\(420\) 0 0
\(421\) −15490.0 −1.79320 −0.896599 − 0.442843i \(-0.853970\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(422\) 0 0
\(423\) 3888.00 0.446906
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3376.00 0.382614
\(428\) 0 0
\(429\) −1080.00 −0.121545
\(430\) 0 0
\(431\) −6768.00 −0.756388 −0.378194 − 0.925726i \(-0.623455\pi\)
−0.378194 + 0.925726i \(0.623455\pi\)
\(432\) 0 0
\(433\) −1298.00 −0.144060 −0.0720299 − 0.997402i \(-0.522948\pi\)
−0.0720299 + 0.997402i \(0.522948\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7200.00 0.788153
\(438\) 0 0
\(439\) 2248.00 0.244399 0.122200 − 0.992506i \(-0.461005\pi\)
0.122200 + 0.992506i \(0.461005\pi\)
\(440\) 0 0
\(441\) −2511.00 −0.271137
\(442\) 0 0
\(443\) −9612.00 −1.03088 −0.515440 − 0.856926i \(-0.672372\pi\)
−0.515440 + 0.856926i \(0.672372\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2214.00 −0.234270
\(448\) 0 0
\(449\) 162.000 0.0170273 0.00851364 − 0.999964i \(-0.497290\pi\)
0.00851364 + 0.999964i \(0.497290\pi\)
\(450\) 0 0
\(451\) −3240.00 −0.338283
\(452\) 0 0
\(453\) 7320.00 0.759213
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1370.00 −0.140232 −0.0701159 − 0.997539i \(-0.522337\pi\)
−0.0701159 + 0.997539i \(0.522337\pi\)
\(458\) 0 0
\(459\) −486.000 −0.0494217
\(460\) 0 0
\(461\) −15354.0 −1.55121 −0.775604 − 0.631220i \(-0.782555\pi\)
−0.775604 + 0.631220i \(0.782555\pi\)
\(462\) 0 0
\(463\) −13024.0 −1.30729 −0.653646 − 0.756800i \(-0.726762\pi\)
−0.653646 + 0.756800i \(0.726762\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14436.0 −1.43045 −0.715223 − 0.698896i \(-0.753675\pi\)
−0.715223 + 0.698896i \(0.753675\pi\)
\(468\) 0 0
\(469\) 2656.00 0.261498
\(470\) 0 0
\(471\) 7662.00 0.749568
\(472\) 0 0
\(473\) −16272.0 −1.58179
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3726.00 −0.357656
\(478\) 0 0
\(479\) −12096.0 −1.15382 −0.576911 − 0.816807i \(-0.695742\pi\)
−0.576911 + 0.816807i \(0.695742\pi\)
\(480\) 0 0
\(481\) 2260.00 0.214235
\(482\) 0 0
\(483\) 1728.00 0.162788
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6056.00 0.563498 0.281749 − 0.959488i \(-0.409085\pi\)
0.281749 + 0.959488i \(0.409085\pi\)
\(488\) 0 0
\(489\) −2460.00 −0.227495
\(490\) 0 0
\(491\) −7524.00 −0.691555 −0.345777 − 0.938317i \(-0.612385\pi\)
−0.345777 + 0.938317i \(0.612385\pi\)
\(492\) 0 0
\(493\) 4212.00 0.384785
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2880.00 0.259931
\(498\) 0 0
\(499\) −5276.00 −0.473319 −0.236660 − 0.971593i \(-0.576053\pi\)
−0.236660 + 0.971593i \(0.576053\pi\)
\(500\) 0 0
\(501\) 5832.00 0.520069
\(502\) 0 0
\(503\) 4968.00 0.440382 0.220191 − 0.975457i \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6291.00 −0.551071
\(508\) 0 0
\(509\) 10998.0 0.957717 0.478858 − 0.877892i \(-0.341051\pi\)
0.478858 + 0.877892i \(0.341051\pi\)
\(510\) 0 0
\(511\) −208.000 −0.0180066
\(512\) 0 0
\(513\) 2700.00 0.232374
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −15552.0 −1.32297
\(518\) 0 0
\(519\) 3726.00 0.315131
\(520\) 0 0
\(521\) −8838.00 −0.743186 −0.371593 − 0.928396i \(-0.621188\pi\)
−0.371593 + 0.928396i \(0.621188\pi\)
\(522\) 0 0
\(523\) 22436.0 1.87583 0.937914 − 0.346869i \(-0.112755\pi\)
0.937914 + 0.346869i \(0.112755\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −288.000 −0.0238055
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 6156.00 0.503103
\(532\) 0 0
\(533\) 900.000 0.0731395
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3348.00 −0.269044
\(538\) 0 0
\(539\) 10044.0 0.802645
\(540\) 0 0
\(541\) −4762.00 −0.378437 −0.189218 − 0.981935i \(-0.560595\pi\)
−0.189218 + 0.981935i \(0.560595\pi\)
\(542\) 0 0
\(543\) 3210.00 0.253691
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6004.00 −0.469310 −0.234655 − 0.972079i \(-0.575396\pi\)
−0.234655 + 0.972079i \(0.575396\pi\)
\(548\) 0 0
\(549\) 3798.00 0.295254
\(550\) 0 0
\(551\) −23400.0 −1.80921
\(552\) 0 0
\(553\) −4096.00 −0.314972
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5274.00 0.401197 0.200598 − 0.979674i \(-0.435711\pi\)
0.200598 + 0.979674i \(0.435711\pi\)
\(558\) 0 0
\(559\) 4520.00 0.341996
\(560\) 0 0
\(561\) 1944.00 0.146303
\(562\) 0 0
\(563\) −12420.0 −0.929735 −0.464867 − 0.885380i \(-0.653898\pi\)
−0.464867 + 0.885380i \(0.653898\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 648.000 0.0479955
\(568\) 0 0
\(569\) −21366.0 −1.57418 −0.787091 − 0.616837i \(-0.788414\pi\)
−0.787091 + 0.616837i \(0.788414\pi\)
\(570\) 0 0
\(571\) −21140.0 −1.54935 −0.774677 − 0.632357i \(-0.782088\pi\)
−0.774677 + 0.632357i \(0.782088\pi\)
\(572\) 0 0
\(573\) 1728.00 0.125983
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3266.00 −0.235642 −0.117821 − 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(578\) 0 0
\(579\) 4026.00 0.288972
\(580\) 0 0
\(581\) −9504.00 −0.678644
\(582\) 0 0
\(583\) 14904.0 1.05877
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17028.0 1.19731 0.598655 − 0.801007i \(-0.295702\pi\)
0.598655 + 0.801007i \(0.295702\pi\)
\(588\) 0 0
\(589\) 1600.00 0.111930
\(590\) 0 0
\(591\) −4266.00 −0.296920
\(592\) 0 0
\(593\) −9522.00 −0.659396 −0.329698 − 0.944086i \(-0.606947\pi\)
−0.329698 + 0.944086i \(0.606947\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2616.00 −0.179340
\(598\) 0 0
\(599\) 10296.0 0.702309 0.351155 − 0.936318i \(-0.385789\pi\)
0.351155 + 0.936318i \(0.385789\pi\)
\(600\) 0 0
\(601\) −3382.00 −0.229542 −0.114771 − 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(602\) 0 0
\(603\) 2988.00 0.201792
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20656.0 −1.38122 −0.690611 − 0.723227i \(-0.742658\pi\)
−0.690611 + 0.723227i \(0.742658\pi\)
\(608\) 0 0
\(609\) −5616.00 −0.373681
\(610\) 0 0
\(611\) 4320.00 0.286037
\(612\) 0 0
\(613\) 22114.0 1.45706 0.728529 − 0.685015i \(-0.240205\pi\)
0.728529 + 0.685015i \(0.240205\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19962.0 −1.30250 −0.651248 − 0.758865i \(-0.725754\pi\)
−0.651248 + 0.758865i \(0.725754\pi\)
\(618\) 0 0
\(619\) 604.000 0.0392194 0.0196097 − 0.999808i \(-0.493758\pi\)
0.0196097 + 0.999808i \(0.493758\pi\)
\(620\) 0 0
\(621\) 1944.00 0.125620
\(622\) 0 0
\(623\) −5040.00 −0.324115
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10800.0 −0.687895
\(628\) 0 0
\(629\) −4068.00 −0.257872
\(630\) 0 0
\(631\) −152.000 −0.00958958 −0.00479479 − 0.999989i \(-0.501526\pi\)
−0.00479479 + 0.999989i \(0.501526\pi\)
\(632\) 0 0
\(633\) −4020.00 −0.252418
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2790.00 −0.173538
\(638\) 0 0
\(639\) 3240.00 0.200583
\(640\) 0 0
\(641\) 4194.00 0.258429 0.129215 − 0.991617i \(-0.458754\pi\)
0.129215 + 0.991617i \(0.458754\pi\)
\(642\) 0 0
\(643\) −7252.00 −0.444776 −0.222388 − 0.974958i \(-0.571385\pi\)
−0.222388 + 0.974958i \(0.571385\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6696.00 −0.406873 −0.203437 − 0.979088i \(-0.565211\pi\)
−0.203437 + 0.979088i \(0.565211\pi\)
\(648\) 0 0
\(649\) −24624.0 −1.48933
\(650\) 0 0
\(651\) 384.000 0.0231185
\(652\) 0 0
\(653\) −28422.0 −1.70328 −0.851638 − 0.524131i \(-0.824390\pi\)
−0.851638 + 0.524131i \(0.824390\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −234.000 −0.0138953
\(658\) 0 0
\(659\) 19908.0 1.17679 0.588396 − 0.808573i \(-0.299760\pi\)
0.588396 + 0.808573i \(0.299760\pi\)
\(660\) 0 0
\(661\) 14318.0 0.842520 0.421260 − 0.906940i \(-0.361588\pi\)
0.421260 + 0.906940i \(0.361588\pi\)
\(662\) 0 0
\(663\) −540.000 −0.0316318
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16848.0 −0.978047
\(668\) 0 0
\(669\) 14640.0 0.846061
\(670\) 0 0
\(671\) −15192.0 −0.874040
\(672\) 0 0
\(673\) −30050.0 −1.72116 −0.860581 − 0.509313i \(-0.829899\pi\)
−0.860581 + 0.509313i \(0.829899\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22158.0 −1.25790 −0.628952 − 0.777444i \(-0.716516\pi\)
−0.628952 + 0.777444i \(0.716516\pi\)
\(678\) 0 0
\(679\) 8432.00 0.476569
\(680\) 0 0
\(681\) 8100.00 0.455790
\(682\) 0 0
\(683\) −3132.00 −0.175465 −0.0877325 − 0.996144i \(-0.527962\pi\)
−0.0877325 + 0.996144i \(0.527962\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 762.000 0.0423175
\(688\) 0 0
\(689\) −4140.00 −0.228914
\(690\) 0 0
\(691\) 20932.0 1.15237 0.576187 − 0.817318i \(-0.304540\pi\)
0.576187 + 0.817318i \(0.304540\pi\)
\(692\) 0 0
\(693\) −2592.00 −0.142081
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1620.00 −0.0880371
\(698\) 0 0
\(699\) −13230.0 −0.715886
\(700\) 0 0
\(701\) −21834.0 −1.17640 −0.588202 − 0.808714i \(-0.700164\pi\)
−0.588202 + 0.808714i \(0.700164\pi\)
\(702\) 0 0
\(703\) 22600.0 1.21248
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4464.00 0.237463
\(708\) 0 0
\(709\) 12446.0 0.659266 0.329633 − 0.944109i \(-0.393075\pi\)
0.329633 + 0.944109i \(0.393075\pi\)
\(710\) 0 0
\(711\) −4608.00 −0.243057
\(712\) 0 0
\(713\) 1152.00 0.0605088
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11664.0 0.607531
\(718\) 0 0
\(719\) 12528.0 0.649813 0.324907 − 0.945746i \(-0.394667\pi\)
0.324907 + 0.945746i \(0.394667\pi\)
\(720\) 0 0
\(721\) 64.0000 0.00330580
\(722\) 0 0
\(723\) 15414.0 0.792881
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11576.0 0.590550 0.295275 − 0.955412i \(-0.404589\pi\)
0.295275 + 0.955412i \(0.404589\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −8136.00 −0.411656
\(732\) 0 0
\(733\) 29338.0 1.47834 0.739170 − 0.673519i \(-0.235218\pi\)
0.739170 + 0.673519i \(0.235218\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11952.0 −0.597364
\(738\) 0 0
\(739\) −2540.00 −0.126435 −0.0632175 − 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(740\) 0 0
\(741\) 3000.00 0.148728
\(742\) 0 0
\(743\) −18792.0 −0.927876 −0.463938 − 0.885868i \(-0.653564\pi\)
−0.463938 + 0.885868i \(0.653564\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −10692.0 −0.523695
\(748\) 0 0
\(749\) 14112.0 0.688440
\(750\) 0 0
\(751\) −4832.00 −0.234783 −0.117392 − 0.993086i \(-0.537453\pi\)
−0.117392 + 0.993086i \(0.537453\pi\)
\(752\) 0 0
\(753\) −14364.0 −0.695157
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20818.0 0.999529 0.499764 − 0.866161i \(-0.333420\pi\)
0.499764 + 0.866161i \(0.333420\pi\)
\(758\) 0 0
\(759\) −7776.00 −0.371872
\(760\) 0 0
\(761\) 12042.0 0.573617 0.286808 − 0.957988i \(-0.407406\pi\)
0.286808 + 0.957988i \(0.407406\pi\)
\(762\) 0 0
\(763\) 12976.0 0.615679
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6840.00 0.322005
\(768\) 0 0
\(769\) 13058.0 0.612332 0.306166 − 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(770\) 0 0
\(771\) 17658.0 0.824821
\(772\) 0 0
\(773\) 11826.0 0.550261 0.275130 − 0.961407i \(-0.411279\pi\)
0.275130 + 0.961407i \(0.411279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5424.00 0.250431
\(778\) 0 0
\(779\) 9000.00 0.413939
\(780\) 0 0
\(781\) −12960.0 −0.593784
\(782\) 0 0
\(783\) −6318.00 −0.288361
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 11996.0 0.543343 0.271672 − 0.962390i \(-0.412424\pi\)
0.271672 + 0.962390i \(0.412424\pi\)
\(788\) 0 0
\(789\) 6696.00 0.302134
\(790\) 0 0
\(791\) 9072.00 0.407792
\(792\) 0 0
\(793\) 4220.00 0.188974
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6966.00 −0.309596 −0.154798 − 0.987946i \(-0.549473\pi\)
−0.154798 + 0.987946i \(0.549473\pi\)
\(798\) 0 0
\(799\) −7776.00 −0.344299
\(800\) 0 0
\(801\) −5670.00 −0.250112
\(802\) 0 0
\(803\) 936.000 0.0411342
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1998.00 −0.0871536
\(808\) 0 0
\(809\) −40806.0 −1.77338 −0.886689 − 0.462367i \(-0.847000\pi\)
−0.886689 + 0.462367i \(0.847000\pi\)
\(810\) 0 0
\(811\) 17980.0 0.778500 0.389250 − 0.921132i \(-0.372734\pi\)
0.389250 + 0.921132i \(0.372734\pi\)
\(812\) 0 0
\(813\) 16608.0 0.716443
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 45200.0 1.93555
\(818\) 0 0
\(819\) 720.000 0.0307190
\(820\) 0 0
\(821\) −12834.0 −0.545566 −0.272783 − 0.962076i \(-0.587944\pi\)
−0.272783 + 0.962076i \(0.587944\pi\)
\(822\) 0 0
\(823\) −37864.0 −1.60371 −0.801857 − 0.597516i \(-0.796154\pi\)
−0.801857 + 0.597516i \(0.796154\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42516.0 1.78770 0.893849 − 0.448368i \(-0.147995\pi\)
0.893849 + 0.448368i \(0.147995\pi\)
\(828\) 0 0
\(829\) 45638.0 1.91203 0.956015 − 0.293317i \(-0.0947591\pi\)
0.956015 + 0.293317i \(0.0947591\pi\)
\(830\) 0 0
\(831\) −6378.00 −0.266246
\(832\) 0 0
\(833\) 5022.00 0.208886
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 432.000 0.0178400
\(838\) 0 0
\(839\) −17496.0 −0.719939 −0.359970 − 0.932964i \(-0.617213\pi\)
−0.359970 + 0.932964i \(0.617213\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) −8802.00 −0.359617
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −280.000 −0.0113588
\(848\) 0 0
\(849\) 6108.00 0.246909
\(850\) 0 0
\(851\) 16272.0 0.655461
\(852\) 0 0
\(853\) −32174.0 −1.29146 −0.645731 − 0.763565i \(-0.723447\pi\)
−0.645731 + 0.763565i \(0.723447\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38934.0 1.55188 0.775939 − 0.630807i \(-0.217276\pi\)
0.775939 + 0.630807i \(0.217276\pi\)
\(858\) 0 0
\(859\) −29780.0 −1.18286 −0.591432 − 0.806355i \(-0.701437\pi\)
−0.591432 + 0.806355i \(0.701437\pi\)
\(860\) 0 0
\(861\) 2160.00 0.0854966
\(862\) 0 0
\(863\) −48096.0 −1.89711 −0.948556 − 0.316611i \(-0.897455\pi\)
−0.948556 + 0.316611i \(0.897455\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −13767.0 −0.539275
\(868\) 0 0
\(869\) 18432.0 0.719520
\(870\) 0 0
\(871\) 3320.00 0.129155
\(872\) 0 0
\(873\) 9486.00 0.367758
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −21302.0 −0.820202 −0.410101 − 0.912040i \(-0.634507\pi\)
−0.410101 + 0.912040i \(0.634507\pi\)
\(878\) 0 0
\(879\) −6858.00 −0.263157
\(880\) 0 0
\(881\) −7470.00 −0.285665 −0.142832 − 0.989747i \(-0.545621\pi\)
−0.142832 + 0.989747i \(0.545621\pi\)
\(882\) 0 0
\(883\) 764.000 0.0291174 0.0145587 − 0.999894i \(-0.495366\pi\)
0.0145587 + 0.999894i \(0.495366\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32328.0 1.22375 0.611876 − 0.790954i \(-0.290415\pi\)
0.611876 + 0.790954i \(0.290415\pi\)
\(888\) 0 0
\(889\) −4736.00 −0.178673
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 0 0
\(893\) 43200.0 1.61885
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2160.00 0.0804017
\(898\) 0 0
\(899\) −3744.00 −0.138898
\(900\) 0 0
\(901\) 7452.00 0.275541
\(902\) 0 0
\(903\) 10848.0 0.399777
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −36316.0 −1.32950 −0.664748 − 0.747068i \(-0.731461\pi\)
−0.664748 + 0.747068i \(0.731461\pi\)
\(908\) 0 0
\(909\) 5022.00 0.183244
\(910\) 0 0
\(911\) 13392.0 0.487044 0.243522 − 0.969895i \(-0.421697\pi\)
0.243522 + 0.969895i \(0.421697\pi\)
\(912\) 0 0
\(913\) 42768.0 1.55029
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15264.0 0.549686
\(918\) 0 0
\(919\) −38072.0 −1.36657 −0.683286 − 0.730151i \(-0.739450\pi\)
−0.683286 + 0.730151i \(0.739450\pi\)
\(920\) 0 0
\(921\) 3732.00 0.133522
\(922\) 0 0
\(923\) 3600.00 0.128381
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 72.0000 0.00255101
\(928\) 0 0
\(929\) −12798.0 −0.451979 −0.225990 − 0.974130i \(-0.572562\pi\)
−0.225990 + 0.974130i \(0.572562\pi\)
\(930\) 0 0
\(931\) −27900.0 −0.982154
\(932\) 0 0
\(933\) −3672.00 −0.128849
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −34874.0 −1.21588 −0.607942 − 0.793981i \(-0.708005\pi\)
−0.607942 + 0.793981i \(0.708005\pi\)
\(938\) 0 0
\(939\) −5694.00 −0.197888
\(940\) 0 0
\(941\) 17190.0 0.595513 0.297757 − 0.954642i \(-0.403762\pi\)
0.297757 + 0.954642i \(0.403762\pi\)
\(942\) 0 0
\(943\) 6480.00 0.223773
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40284.0 1.38232 0.691158 − 0.722703i \(-0.257101\pi\)
0.691158 + 0.722703i \(0.257101\pi\)
\(948\) 0 0
\(949\) −260.000 −0.00889353
\(950\) 0 0
\(951\) 27486.0 0.937218
\(952\) 0 0
\(953\) −15498.0 −0.526789 −0.263394 − 0.964688i \(-0.584842\pi\)
−0.263394 + 0.964688i \(0.584842\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 25272.0 0.853634
\(958\) 0 0
\(959\) −7632.00 −0.256987
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) 15876.0 0.531253
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37160.0 1.23577 0.617883 − 0.786270i \(-0.287991\pi\)
0.617883 + 0.786270i \(0.287991\pi\)
\(968\) 0 0
\(969\) −5400.00 −0.179023
\(970\) 0 0
\(971\) −18468.0 −0.610367 −0.305183 − 0.952294i \(-0.598718\pi\)
−0.305183 + 0.952294i \(0.598718\pi\)
\(972\) 0 0
\(973\) −20512.0 −0.675832
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10386.0 −0.340100 −0.170050 − 0.985435i \(-0.554393\pi\)
−0.170050 + 0.985435i \(0.554393\pi\)
\(978\) 0 0
\(979\) 22680.0 0.740404
\(980\) 0 0
\(981\) 14598.0 0.475105
\(982\) 0 0
\(983\) 44136.0 1.43206 0.716032 − 0.698067i \(-0.245956\pi\)
0.716032 + 0.698067i \(0.245956\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 10368.0 0.334364
\(988\) 0 0
\(989\) 32544.0 1.04635
\(990\) 0 0
\(991\) 28432.0 0.911375 0.455687 − 0.890140i \(-0.349394\pi\)
0.455687 + 0.890140i \(0.349394\pi\)
\(992\) 0 0
\(993\) 13044.0 0.416857
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 39778.0 1.26357 0.631786 − 0.775143i \(-0.282322\pi\)
0.631786 + 0.775143i \(0.282322\pi\)
\(998\) 0 0
\(999\) 6102.00 0.193252
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 1200.4.a.be.1.1 1
4.3 odd 2 300.4.a.b.1.1 1
5.2 odd 4 1200.4.f.d.49.1 2
5.3 odd 4 1200.4.f.d.49.2 2
5.4 even 2 48.4.a.a.1.1 1
12.11 even 2 900.4.a.g.1.1 1
15.14 odd 2 144.4.a.g.1.1 1
20.3 even 4 300.4.d.e.49.1 2
20.7 even 4 300.4.d.e.49.2 2
20.19 odd 2 12.4.a.a.1.1 ✓ 1
35.34 odd 2 2352.4.a.bk.1.1 1
40.19 odd 2 192.4.a.f.1.1 1
40.29 even 2 192.4.a.l.1.1 1
60.23 odd 4 900.4.d.c.649.2 2
60.47 odd 4 900.4.d.c.649.1 2
60.59 even 2 36.4.a.a.1.1 1
80.19 odd 4 768.4.d.g.385.2 2
80.29 even 4 768.4.d.j.385.1 2
80.59 odd 4 768.4.d.g.385.1 2
80.69 even 4 768.4.d.j.385.2 2
120.29 odd 2 576.4.a.a.1.1 1
120.59 even 2 576.4.a.b.1.1 1
140.19 even 6 588.4.i.e.361.1 2
140.39 odd 6 588.4.i.d.373.1 2
140.59 even 6 588.4.i.e.373.1 2
140.79 odd 6 588.4.i.d.361.1 2
140.139 even 2 588.4.a.c.1.1 1
180.59 even 6 324.4.e.a.217.1 2
180.79 odd 6 324.4.e.h.109.1 2
180.119 even 6 324.4.e.a.109.1 2
180.139 odd 6 324.4.e.h.217.1 2
220.219 even 2 1452.4.a.d.1.1 1
260.99 even 4 2028.4.b.c.337.1 2
260.239 even 4 2028.4.b.c.337.2 2
260.259 odd 2 2028.4.a.c.1.1 1
420.59 odd 6 1764.4.k.o.1549.1 2
420.179 even 6 1764.4.k.b.1549.1 2
420.299 odd 6 1764.4.k.o.361.1 2
420.359 even 6 1764.4.k.b.361.1 2
420.419 odd 2 1764.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 ✓ 1 20.19 odd 2
36.4.a.a.1.1 1 60.59 even 2
48.4.a.a.1.1 1 5.4 even 2
144.4.a.g.1.1 1 15.14 odd 2
192.4.a.f.1.1 1 40.19 odd 2
192.4.a.l.1.1 1 40.29 even 2
300.4.a.b.1.1 1 4.3 odd 2
300.4.d.e.49.1 2 20.3 even 4
300.4.d.e.49.2 2 20.7 even 4
324.4.e.a.109.1 2 180.119 even 6
324.4.e.a.217.1 2 180.59 even 6
324.4.e.h.109.1 2 180.79 odd 6
324.4.e.h.217.1 2 180.139 odd 6
576.4.a.a.1.1 1 120.29 odd 2
576.4.a.b.1.1 1 120.59 even 2
588.4.a.c.1.1 1 140.139 even 2
588.4.i.d.361.1 2 140.79 odd 6
588.4.i.d.373.1 2 140.39 odd 6
588.4.i.e.361.1 2 140.19 even 6
588.4.i.e.373.1 2 140.59 even 6
768.4.d.g.385.1 2 80.59 odd 4
768.4.d.g.385.2 2 80.19 odd 4
768.4.d.j.385.1 2 80.29 even 4
768.4.d.j.385.2 2 80.69 even 4
900.4.a.g.1.1 1 12.11 even 2
900.4.d.c.649.1 2 60.47 odd 4
900.4.d.c.649.2 2 60.23 odd 4
1200.4.a.be.1.1 1 1.1 even 1 trivial
1200.4.f.d.49.1 2 5.2 odd 4
1200.4.f.d.49.2 2 5.3 odd 4
1452.4.a.d.1.1 1 220.219 even 2
1764.4.a.b.1.1 1 420.419 odd 2
1764.4.k.b.361.1 2 420.359 even 6
1764.4.k.b.1549.1 2 420.179 even 6
1764.4.k.o.361.1 2 420.299 odd 6
1764.4.k.o.1549.1 2 420.59 odd 6
2028.4.a.c.1.1 1 260.259 odd 2
2028.4.b.c.337.1 2 260.99 even 4
2028.4.b.c.337.2 2 260.239 even 4
2352.4.a.bk.1.1 1 35.34 odd 2