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Properties

Label 1200.4.a.s.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
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] = [1200,4,Mod(1,1200)] 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("1200.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,32,0,9,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +32.0000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +10.0000 q^{13} +78.0000 q^{17} -140.000 q^{19} -96.0000 q^{21} -192.000 q^{23} -27.0000 q^{27} +6.00000 q^{29} +16.0000 q^{31} +108.000 q^{33} +34.0000 q^{37} -30.0000 q^{39} -390.000 q^{41} -52.0000 q^{43} +408.000 q^{47} +681.000 q^{49} -234.000 q^{51} +114.000 q^{53} +420.000 q^{57} -516.000 q^{59} -58.0000 q^{61} +288.000 q^{63} -892.000 q^{67} +576.000 q^{69} +120.000 q^{71} +646.000 q^{73} -1152.00 q^{77} +1168.00 q^{79} +81.0000 q^{81} -732.000 q^{83} -18.0000 q^{87} -1590.00 q^{89} +320.000 q^{91} -48.0000 q^{93} -194.000 q^{97} -324.000 q^{99} +O(q^{100})\)

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
\(7\) 32.0000 1.72784 0.863919 − 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 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
\(17\) 78.0000 1.11281 0.556405 − 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) −140.000 −1.69043 −0.845216 − 0.534425i \(-0.820528\pi\)
−0.845216 + 0.534425i \(0.820528\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) −192.000 −1.74064 −0.870321 − 0.492485i \(-0.836089\pi\)
−0.870321 + 0.492485i \(0.836089\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 − 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 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
\(37\) 34.0000 0.151069 0.0755347 − 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 0 0
\(39\) −30.0000 −0.123176
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 − 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) −52.0000 −0.184417 −0.0922084 − 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 408.000 1.26623 0.633116 − 0.774057i \(-0.281776\pi\)
0.633116 + 0.774057i \(0.281776\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) 114.000 0.295455 0.147727 − 0.989028i \(-0.452804\pi\)
0.147727 + 0.989028i \(0.452804\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 420.000 0.975971
\(58\) 0 0
\(59\) −516.000 −1.13860 −0.569301 − 0.822129i \(-0.692786\pi\)
−0.569301 + 0.822129i \(0.692786\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 − 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) 288.000 0.575946
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −892.000 −1.62649 −0.813247 − 0.581918i \(-0.802302\pi\)
−0.813247 + 0.581918i \(0.802302\pi\)
\(68\) 0 0
\(69\) 576.000 1.00496
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 646.000 1.03573 0.517867 − 0.855461i \(-0.326726\pi\)
0.517867 + 0.855461i \(0.326726\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1152.00 −1.70497
\(78\) 0 0
\(79\) 1168.00 1.66342 0.831711 − 0.555209i \(-0.187362\pi\)
0.831711 + 0.555209i \(0.187362\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −732.000 −0.968041 −0.484021 − 0.875057i \(-0.660824\pi\)
−0.484021 + 0.875057i \(0.660824\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.0000 −0.0221816
\(88\) 0 0
\(89\) −1590.00 −1.89370 −0.946852 − 0.321669i \(-0.895756\pi\)
−0.946852 + 0.321669i \(0.895756\pi\)
\(90\) 0 0
\(91\) 320.000 0.368628
\(92\) 0 0
\(93\) −48.0000 −0.0535201
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −194.000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 798.000 0.786178 0.393089 − 0.919500i \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(102\) 0 0
\(103\) 272.000 0.260203 0.130102 − 0.991501i \(-0.458470\pi\)
0.130102 + 0.991501i \(0.458470\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 156.000 0.140945 0.0704724 − 0.997514i \(-0.477549\pi\)
0.0704724 + 0.997514i \(0.477549\pi\)
\(108\) 0 0
\(109\) 1622.00 1.42532 0.712658 − 0.701512i \(-0.247491\pi\)
0.712658 + 0.701512i \(0.247491\pi\)
\(110\) 0 0
\(111\) −102.000 −0.0872199
\(112\) 0 0
\(113\) −1074.00 −0.894101 −0.447051 − 0.894509i \(-0.647526\pi\)
−0.447051 + 0.894509i \(0.647526\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 90.0000 0.0711154
\(118\) 0 0
\(119\) 2496.00 1.92276
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 1170.00 0.857686
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1528.00 −1.06762 −0.533811 − 0.845604i \(-0.679241\pi\)
−0.533811 + 0.845604i \(0.679241\pi\)
\(128\) 0 0
\(129\) 156.000 0.106473
\(130\) 0 0
\(131\) −2412.00 −1.60868 −0.804341 − 0.594168i \(-0.797482\pi\)
−0.804341 + 0.594168i \(0.797482\pi\)
\(132\) 0 0
\(133\) −4480.00 −2.92079
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2106.00 −1.31334 −0.656671 − 0.754178i \(-0.728036\pi\)
−0.656671 + 0.754178i \(0.728036\pi\)
\(138\) 0 0
\(139\) 556.000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −1224.00 −0.731060
\(142\) 0 0
\(143\) −360.000 −0.210522
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2043.00 −1.14628
\(148\) 0 0
\(149\) −2418.00 −1.32946 −0.664732 − 0.747081i \(-0.731454\pi\)
−0.664732 + 0.747081i \(0.731454\pi\)
\(150\) 0 0
\(151\) −2840.00 −1.53057 −0.765285 − 0.643692i \(-0.777402\pi\)
−0.765285 + 0.643692i \(0.777402\pi\)
\(152\) 0 0
\(153\) 702.000 0.370937
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2054.00 −1.04412 −0.522061 − 0.852908i \(-0.674837\pi\)
−0.522061 + 0.852908i \(0.674837\pi\)
\(158\) 0 0
\(159\) −342.000 −0.170581
\(160\) 0 0
\(161\) −6144.00 −3.00755
\(162\) 0 0
\(163\) −460.000 −0.221043 −0.110521 − 0.993874i \(-0.535252\pi\)
−0.110521 + 0.993874i \(0.535252\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2016.00 0.934148 0.467074 − 0.884218i \(-0.345308\pi\)
0.467074 + 0.884218i \(0.345308\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) −1260.00 −0.563477
\(172\) 0 0
\(173\) 618.000 0.271593 0.135797 − 0.990737i \(-0.456641\pi\)
0.135797 + 0.990737i \(0.456641\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1548.00 0.657372
\(178\) 0 0
\(179\) 2964.00 1.23765 0.618826 − 0.785528i \(-0.287609\pi\)
0.618826 + 0.785528i \(0.287609\pi\)
\(180\) 0 0
\(181\) −370.000 −0.151944 −0.0759721 − 0.997110i \(-0.524206\pi\)
−0.0759721 + 0.997110i \(0.524206\pi\)
\(182\) 0 0
\(183\) 174.000 0.0702866
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2808.00 −1.09808
\(188\) 0 0
\(189\) −864.000 −0.332522
\(190\) 0 0
\(191\) −1104.00 −0.418234 −0.209117 − 0.977891i \(-0.567059\pi\)
−0.209117 + 0.977891i \(0.567059\pi\)
\(192\) 0 0
\(193\) 2398.00 0.894362 0.447181 − 0.894444i \(-0.352428\pi\)
0.447181 + 0.894444i \(0.352428\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1278.00 −0.462202 −0.231101 − 0.972930i \(-0.574233\pi\)
−0.231101 + 0.972930i \(0.574233\pi\)
\(198\) 0 0
\(199\) −4472.00 −1.59302 −0.796512 − 0.604623i \(-0.793324\pi\)
−0.796512 + 0.604623i \(0.793324\pi\)
\(200\) 0 0
\(201\) 2676.00 0.939057
\(202\) 0 0
\(203\) 192.000 0.0663830
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1728.00 −0.580214
\(208\) 0 0
\(209\) 5040.00 1.66806
\(210\) 0 0
\(211\) −1340.00 −0.437201 −0.218600 − 0.975814i \(-0.570149\pi\)
−0.218600 + 0.975814i \(0.570149\pi\)
\(212\) 0 0
\(213\) −360.000 −0.115807
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 512.000 0.160170
\(218\) 0 0
\(219\) −1938.00 −0.597981
\(220\) 0 0
\(221\) 780.000 0.237414
\(222\) 0 0
\(223\) 2360.00 0.708687 0.354344 − 0.935115i \(-0.384704\pi\)
0.354344 + 0.935115i \(0.384704\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1380.00 0.403497 0.201748 − 0.979437i \(-0.435338\pi\)
0.201748 + 0.979437i \(0.435338\pi\)
\(228\) 0 0
\(229\) 1694.00 0.488833 0.244416 − 0.969670i \(-0.421404\pi\)
0.244416 + 0.969670i \(0.421404\pi\)
\(230\) 0 0
\(231\) 3456.00 0.984364
\(232\) 0 0
\(233\) 5190.00 1.45926 0.729631 − 0.683841i \(-0.239692\pi\)
0.729631 + 0.683841i \(0.239692\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3504.00 −0.960377
\(238\) 0 0
\(239\) −2352.00 −0.636562 −0.318281 − 0.947996i \(-0.603105\pi\)
−0.318281 + 0.947996i \(0.603105\pi\)
\(240\) 0 0
\(241\) −3502.00 −0.936032 −0.468016 − 0.883720i \(-0.655031\pi\)
−0.468016 + 0.883720i \(0.655031\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1400.00 −0.360647
\(248\) 0 0
\(249\) 2196.00 0.558899
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 − 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) 6912.00 1.71760
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1506.00 −0.365532 −0.182766 − 0.983156i \(-0.558505\pi\)
−0.182766 + 0.983156i \(0.558505\pi\)
\(258\) 0 0
\(259\) 1088.00 0.261023
\(260\) 0 0
\(261\) 54.0000 0.0128066
\(262\) 0 0
\(263\) −432.000 −0.101286 −0.0506431 − 0.998717i \(-0.516127\pi\)
−0.0506431 + 0.998717i \(0.516127\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4770.00 1.09333
\(268\) 0 0
\(269\) 54.0000 0.0122395 0.00611977 − 0.999981i \(-0.498052\pi\)
0.00611977 + 0.999981i \(0.498052\pi\)
\(270\) 0 0
\(271\) 6496.00 1.45610 0.728051 − 0.685522i \(-0.240426\pi\)
0.728051 + 0.685522i \(0.240426\pi\)
\(272\) 0 0
\(273\) −960.000 −0.212827
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 466.000 0.101080 0.0505401 − 0.998722i \(-0.483906\pi\)
0.0505401 + 0.998722i \(0.483906\pi\)
\(278\) 0 0
\(279\) 144.000 0.0308998
\(280\) 0 0
\(281\) −4854.00 −1.03048 −0.515241 − 0.857045i \(-0.672298\pi\)
−0.515241 + 0.857045i \(0.672298\pi\)
\(282\) 0 0
\(283\) −4516.00 −0.948581 −0.474290 − 0.880368i \(-0.657295\pi\)
−0.474290 + 0.880368i \(0.657295\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12480.0 −2.56680
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 582.000 0.117242
\(292\) 0 0
\(293\) −8574.00 −1.70955 −0.854775 − 0.518998i \(-0.826305\pi\)
−0.854775 + 0.518998i \(0.826305\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 972.000 0.189903
\(298\) 0 0
\(299\) −1920.00 −0.371359
\(300\) 0 0
\(301\) −1664.00 −0.318642
\(302\) 0 0
\(303\) −2394.00 −0.453900
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3476.00 0.646208 0.323104 − 0.946363i \(-0.395274\pi\)
0.323104 + 0.946363i \(0.395274\pi\)
\(308\) 0 0
\(309\) −816.000 −0.150229
\(310\) 0 0
\(311\) −2424.00 −0.441969 −0.220985 − 0.975277i \(-0.570927\pi\)
−0.220985 + 0.975277i \(0.570927\pi\)
\(312\) 0 0
\(313\) 1558.00 0.281353 0.140676 − 0.990056i \(-0.455072\pi\)
0.140676 + 0.990056i \(0.455072\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8538.00 1.51275 0.756375 − 0.654138i \(-0.226968\pi\)
0.756375 + 0.654138i \(0.226968\pi\)
\(318\) 0 0
\(319\) −216.000 −0.0379112
\(320\) 0 0
\(321\) −468.000 −0.0813745
\(322\) 0 0
\(323\) −10920.0 −1.88113
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4866.00 −0.822906
\(328\) 0 0
\(329\) 13056.0 2.18784
\(330\) 0 0
\(331\) 988.000 0.164065 0.0820323 − 0.996630i \(-0.473859\pi\)
0.0820323 + 0.996630i \(0.473859\pi\)
\(332\) 0 0
\(333\) 306.000 0.0503564
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2546.00 −0.411541 −0.205771 − 0.978600i \(-0.565970\pi\)
−0.205771 + 0.978600i \(0.565970\pi\)
\(338\) 0 0
\(339\) 3222.00 0.516209
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) 10816.0 1.70265
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8556.00 1.32366 0.661830 − 0.749654i \(-0.269780\pi\)
0.661830 + 0.749654i \(0.269780\pi\)
\(348\) 0 0
\(349\) −3706.00 −0.568417 −0.284209 − 0.958762i \(-0.591731\pi\)
−0.284209 + 0.958762i \(0.591731\pi\)
\(350\) 0 0
\(351\) −270.000 −0.0410585
\(352\) 0 0
\(353\) −11394.0 −1.71796 −0.858982 − 0.512005i \(-0.828903\pi\)
−0.858982 + 0.512005i \(0.828903\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7488.00 −1.11010
\(358\) 0 0
\(359\) 264.000 0.0388117 0.0194058 − 0.999812i \(-0.493823\pi\)
0.0194058 + 0.999812i \(0.493823\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) 0 0
\(363\) 105.000 0.0151820
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10232.0 1.45533 0.727665 − 0.685933i \(-0.240606\pi\)
0.727665 + 0.685933i \(0.240606\pi\)
\(368\) 0 0
\(369\) −3510.00 −0.495185
\(370\) 0 0
\(371\) 3648.00 0.510498
\(372\) 0 0
\(373\) 562.000 0.0780141 0.0390070 − 0.999239i \(-0.487581\pi\)
0.0390070 + 0.999239i \(0.487581\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60.0000 0.00819670
\(378\) 0 0
\(379\) 7228.00 0.979624 0.489812 − 0.871828i \(-0.337065\pi\)
0.489812 + 0.871828i \(0.337065\pi\)
\(380\) 0 0
\(381\) 4584.00 0.616392
\(382\) 0 0
\(383\) −5736.00 −0.765263 −0.382632 − 0.923901i \(-0.624982\pi\)
−0.382632 + 0.923901i \(0.624982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −468.000 −0.0614723
\(388\) 0 0
\(389\) −9186.00 −1.19730 −0.598649 − 0.801012i \(-0.704295\pi\)
−0.598649 + 0.801012i \(0.704295\pi\)
\(390\) 0 0
\(391\) −14976.0 −1.93700
\(392\) 0 0
\(393\) 7236.00 0.928773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 394.000 0.0498093 0.0249047 − 0.999690i \(-0.492072\pi\)
0.0249047 + 0.999690i \(0.492072\pi\)
\(398\) 0 0
\(399\) 13440.0 1.68632
\(400\) 0 0
\(401\) −1614.00 −0.200996 −0.100498 − 0.994937i \(-0.532044\pi\)
−0.100498 + 0.994937i \(0.532044\pi\)
\(402\) 0 0
\(403\) 160.000 0.0197771
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1224.00 −0.149070
\(408\) 0 0
\(409\) 1034.00 0.125007 0.0625037 − 0.998045i \(-0.480091\pi\)
0.0625037 + 0.998045i \(0.480091\pi\)
\(410\) 0 0
\(411\) 6318.00 0.758258
\(412\) 0 0
\(413\) −16512.0 −1.96732
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1668.00 −0.195881
\(418\) 0 0
\(419\) −3708.00 −0.432333 −0.216167 − 0.976356i \(-0.569355\pi\)
−0.216167 + 0.976356i \(0.569355\pi\)
\(420\) 0 0
\(421\) −4930.00 −0.570721 −0.285360 − 0.958420i \(-0.592113\pi\)
−0.285360 + 0.958420i \(0.592113\pi\)
\(422\) 0 0
\(423\) 3672.00 0.422077
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1856.00 −0.210347
\(428\) 0 0
\(429\) 1080.00 0.121545
\(430\) 0 0
\(431\) 2592.00 0.289680 0.144840 − 0.989455i \(-0.453733\pi\)
0.144840 + 0.989455i \(0.453733\pi\)
\(432\) 0 0
\(433\) −2162.00 −0.239952 −0.119976 − 0.992777i \(-0.538282\pi\)
−0.119976 + 0.992777i \(0.538282\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26880.0 2.94244
\(438\) 0 0
\(439\) −1352.00 −0.146987 −0.0734937 − 0.997296i \(-0.523415\pi\)
−0.0734937 + 0.997296i \(0.523415\pi\)
\(440\) 0 0
\(441\) 6129.00 0.661808
\(442\) 0 0
\(443\) 5532.00 0.593303 0.296652 − 0.954986i \(-0.404130\pi\)
0.296652 + 0.954986i \(0.404130\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7254.00 0.767567
\(448\) 0 0
\(449\) −3198.00 −0.336131 −0.168066 − 0.985776i \(-0.553752\pi\)
−0.168066 + 0.985776i \(0.553752\pi\)
\(450\) 0 0
\(451\) 14040.0 1.46589
\(452\) 0 0
\(453\) 8520.00 0.883674
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1510.00 0.154562 0.0772810 − 0.997009i \(-0.475376\pi\)
0.0772810 + 0.997009i \(0.475376\pi\)
\(458\) 0 0
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) 16086.0 1.62516 0.812581 − 0.582848i \(-0.198062\pi\)
0.812581 + 0.582848i \(0.198062\pi\)
\(462\) 0 0
\(463\) 5384.00 0.540423 0.270211 − 0.962801i \(-0.412906\pi\)
0.270211 + 0.962801i \(0.412906\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2604.00 −0.258027 −0.129014 − 0.991643i \(-0.541181\pi\)
−0.129014 + 0.991643i \(0.541181\pi\)
\(468\) 0 0
\(469\) −28544.0 −2.81032
\(470\) 0 0
\(471\) 6162.00 0.602824
\(472\) 0 0
\(473\) 1872.00 0.181976
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1026.00 0.0984849
\(478\) 0 0
\(479\) −11136.0 −1.06225 −0.531124 − 0.847294i \(-0.678230\pi\)
−0.531124 + 0.847294i \(0.678230\pi\)
\(480\) 0 0
\(481\) 340.000 0.0322301
\(482\) 0 0
\(483\) 18432.0 1.73641
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14624.0 1.36073 0.680366 − 0.732872i \(-0.261821\pi\)
0.680366 + 0.732872i \(0.261821\pi\)
\(488\) 0 0
\(489\) 1380.00 0.127619
\(490\) 0 0
\(491\) −11844.0 −1.08862 −0.544310 − 0.838884i \(-0.683208\pi\)
−0.544310 + 0.838884i \(0.683208\pi\)
\(492\) 0 0
\(493\) 468.000 0.0427539
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3840.00 0.346575
\(498\) 0 0
\(499\) 11284.0 1.01231 0.506154 − 0.862443i \(-0.331067\pi\)
0.506154 + 0.862443i \(0.331067\pi\)
\(500\) 0 0
\(501\) −6048.00 −0.539331
\(502\) 0 0
\(503\) 4032.00 0.357412 0.178706 − 0.983903i \(-0.442809\pi\)
0.178706 + 0.983903i \(0.442809\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6291.00 0.551071
\(508\) 0 0
\(509\) −17562.0 −1.52932 −0.764658 − 0.644436i \(-0.777092\pi\)
−0.764658 + 0.644436i \(0.777092\pi\)
\(510\) 0 0
\(511\) 20672.0 1.78958
\(512\) 0 0
\(513\) 3780.00 0.325324
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −14688.0 −1.24947
\(518\) 0 0
\(519\) −1854.00 −0.156805
\(520\) 0 0
\(521\) 3162.00 0.265892 0.132946 − 0.991123i \(-0.457556\pi\)
0.132946 + 0.991123i \(0.457556\pi\)
\(522\) 0 0
\(523\) 6764.00 0.565524 0.282762 − 0.959190i \(-0.408749\pi\)
0.282762 + 0.959190i \(0.408749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1248.00 0.103157
\(528\) 0 0
\(529\) 24697.0 2.02983
\(530\) 0 0
\(531\) −4644.00 −0.379534
\(532\) 0 0
\(533\) −3900.00 −0.316938
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8892.00 −0.714559
\(538\) 0 0
\(539\) −24516.0 −1.95914
\(540\) 0 0
\(541\) 17798.0 1.41441 0.707205 − 0.707009i \(-0.249956\pi\)
0.707205 + 0.707009i \(0.249956\pi\)
\(542\) 0 0
\(543\) 1110.00 0.0877250
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19996.0 −1.56301 −0.781506 − 0.623898i \(-0.785548\pi\)
−0.781506 + 0.623898i \(0.785548\pi\)
\(548\) 0 0
\(549\) −522.000 −0.0405800
\(550\) 0 0
\(551\) −840.000 −0.0649459
\(552\) 0 0
\(553\) 37376.0 2.87412
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11094.0 −0.843928 −0.421964 − 0.906613i \(-0.638659\pi\)
−0.421964 + 0.906613i \(0.638659\pi\)
\(558\) 0 0
\(559\) −520.000 −0.0393446
\(560\) 0 0
\(561\) 8424.00 0.633978
\(562\) 0 0
\(563\) 900.000 0.0673721 0.0336860 − 0.999432i \(-0.489275\pi\)
0.0336860 + 0.999432i \(0.489275\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2592.00 0.191982
\(568\) 0 0
\(569\) 7914.00 0.583079 0.291540 − 0.956559i \(-0.405832\pi\)
0.291540 + 0.956559i \(0.405832\pi\)
\(570\) 0 0
\(571\) 2380.00 0.174431 0.0872153 − 0.996189i \(-0.472203\pi\)
0.0872153 + 0.996189i \(0.472203\pi\)
\(572\) 0 0
\(573\) 3312.00 0.241467
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25726.0 1.85613 0.928065 − 0.372417i \(-0.121471\pi\)
0.928065 + 0.372417i \(0.121471\pi\)
\(578\) 0 0
\(579\) −7194.00 −0.516360
\(580\) 0 0
\(581\) −23424.0 −1.67262
\(582\) 0 0
\(583\) −4104.00 −0.291544
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3612.00 0.253975 0.126987 − 0.991904i \(-0.459469\pi\)
0.126987 + 0.991904i \(0.459469\pi\)
\(588\) 0 0
\(589\) −2240.00 −0.156702
\(590\) 0 0
\(591\) 3834.00 0.266852
\(592\) 0 0
\(593\) −2898.00 −0.200686 −0.100343 − 0.994953i \(-0.531994\pi\)
−0.100343 + 0.994953i \(0.531994\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13416.0 0.919733
\(598\) 0 0
\(599\) −2664.00 −0.181716 −0.0908582 − 0.995864i \(-0.528961\pi\)
−0.0908582 + 0.995864i \(0.528961\pi\)
\(600\) 0 0
\(601\) −502.000 −0.0340716 −0.0170358 − 0.999855i \(-0.505423\pi\)
−0.0170358 + 0.999855i \(0.505423\pi\)
\(602\) 0 0
\(603\) −8028.00 −0.542165
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7976.00 0.533337 0.266669 − 0.963788i \(-0.414077\pi\)
0.266669 + 0.963788i \(0.414077\pi\)
\(608\) 0 0
\(609\) −576.000 −0.0383263
\(610\) 0 0
\(611\) 4080.00 0.270146
\(612\) 0 0
\(613\) −20414.0 −1.34505 −0.672523 − 0.740076i \(-0.734790\pi\)
−0.672523 + 0.740076i \(0.734790\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6342.00 0.413808 0.206904 − 0.978361i \(-0.433661\pi\)
0.206904 + 0.978361i \(0.433661\pi\)
\(618\) 0 0
\(619\) −22676.0 −1.47242 −0.736208 − 0.676755i \(-0.763385\pi\)
−0.736208 + 0.676755i \(0.763385\pi\)
\(620\) 0 0
\(621\) 5184.00 0.334987
\(622\) 0 0
\(623\) −50880.0 −3.27201
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −15120.0 −0.963054
\(628\) 0 0
\(629\) 2652.00 0.168112
\(630\) 0 0
\(631\) 7048.00 0.444654 0.222327 − 0.974972i \(-0.428635\pi\)
0.222327 + 0.974972i \(0.428635\pi\)
\(632\) 0 0
\(633\) 4020.00 0.252418
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6810.00 0.423582
\(638\) 0 0
\(639\) 1080.00 0.0668609
\(640\) 0 0
\(641\) −20286.0 −1.25000 −0.624999 − 0.780625i \(-0.714901\pi\)
−0.624999 + 0.780625i \(0.714901\pi\)
\(642\) 0 0
\(643\) −16108.0 −0.987928 −0.493964 − 0.869482i \(-0.664452\pi\)
−0.493964 + 0.869482i \(0.664452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27456.0 1.66833 0.834163 − 0.551518i \(-0.185951\pi\)
0.834163 + 0.551518i \(0.185951\pi\)
\(648\) 0 0
\(649\) 18576.0 1.12353
\(650\) 0 0
\(651\) −1536.00 −0.0924740
\(652\) 0 0
\(653\) 12522.0 0.750419 0.375210 − 0.926940i \(-0.377571\pi\)
0.375210 + 0.926940i \(0.377571\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5814.00 0.345245
\(658\) 0 0
\(659\) 16308.0 0.963990 0.481995 − 0.876174i \(-0.339912\pi\)
0.481995 + 0.876174i \(0.339912\pi\)
\(660\) 0 0
\(661\) 32078.0 1.88758 0.943789 − 0.330547i \(-0.107233\pi\)
0.943789 + 0.330547i \(0.107233\pi\)
\(662\) 0 0
\(663\) −2340.00 −0.137071
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1152.00 −0.0668750
\(668\) 0 0
\(669\) −7080.00 −0.409161
\(670\) 0 0
\(671\) 2088.00 0.120129
\(672\) 0 0
\(673\) −4610.00 −0.264045 −0.132023 − 0.991247i \(-0.542147\pi\)
−0.132023 + 0.991247i \(0.542147\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10782.0 −0.612091 −0.306046 − 0.952017i \(-0.599006\pi\)
−0.306046 + 0.952017i \(0.599006\pi\)
\(678\) 0 0
\(679\) −6208.00 −0.350871
\(680\) 0 0
\(681\) −4140.00 −0.232959
\(682\) 0 0
\(683\) 2892.00 0.162019 0.0810097 − 0.996713i \(-0.474186\pi\)
0.0810097 + 0.996713i \(0.474186\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5082.00 −0.282228
\(688\) 0 0
\(689\) 1140.00 0.0630342
\(690\) 0 0
\(691\) 29572.0 1.62803 0.814017 − 0.580841i \(-0.197276\pi\)
0.814017 + 0.580841i \(0.197276\pi\)
\(692\) 0 0
\(693\) −10368.0 −0.568323
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −30420.0 −1.65314
\(698\) 0 0
\(699\) −15570.0 −0.842506
\(700\) 0 0
\(701\) 5766.00 0.310669 0.155334 − 0.987862i \(-0.450354\pi\)
0.155334 + 0.987862i \(0.450354\pi\)
\(702\) 0 0
\(703\) −4760.00 −0.255372
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25536.0 1.35839
\(708\) 0 0
\(709\) 3326.00 0.176178 0.0880892 − 0.996113i \(-0.471924\pi\)
0.0880892 + 0.996113i \(0.471924\pi\)
\(710\) 0 0
\(711\) 10512.0 0.554474
\(712\) 0 0
\(713\) −3072.00 −0.161357
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7056.00 0.367519
\(718\) 0 0
\(719\) 7728.00 0.400843 0.200421 − 0.979710i \(-0.435769\pi\)
0.200421 + 0.979710i \(0.435769\pi\)
\(720\) 0 0
\(721\) 8704.00 0.449589
\(722\) 0 0
\(723\) 10506.0 0.540418
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21616.0 −1.10274 −0.551371 − 0.834260i \(-0.685895\pi\)
−0.551371 + 0.834260i \(0.685895\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4056.00 −0.205221
\(732\) 0 0
\(733\) −10118.0 −0.509846 −0.254923 − 0.966961i \(-0.582050\pi\)
−0.254923 + 0.966961i \(0.582050\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32112.0 1.60497
\(738\) 0 0
\(739\) −10460.0 −0.520673 −0.260336 − 0.965518i \(-0.583833\pi\)
−0.260336 + 0.965518i \(0.583833\pi\)
\(740\) 0 0
\(741\) 4200.00 0.208220
\(742\) 0 0
\(743\) 17232.0 0.850849 0.425424 − 0.904994i \(-0.360125\pi\)
0.425424 + 0.904994i \(0.360125\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6588.00 −0.322680
\(748\) 0 0
\(749\) 4992.00 0.243530
\(750\) 0 0
\(751\) −26912.0 −1.30763 −0.653817 − 0.756653i \(-0.726833\pi\)
−0.653817 + 0.756653i \(0.726833\pi\)
\(752\) 0 0
\(753\) 14364.0 0.695157
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −13838.0 −0.664400 −0.332200 − 0.943209i \(-0.607791\pi\)
−0.332200 + 0.943209i \(0.607791\pi\)
\(758\) 0 0
\(759\) −20736.0 −0.991659
\(760\) 0 0
\(761\) −17238.0 −0.821126 −0.410563 − 0.911832i \(-0.634668\pi\)
−0.410563 + 0.911832i \(0.634668\pi\)
\(762\) 0 0
\(763\) 51904.0 2.46271
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5160.00 −0.242916
\(768\) 0 0
\(769\) 21698.0 1.01749 0.508745 − 0.860917i \(-0.330110\pi\)
0.508745 + 0.860917i \(0.330110\pi\)
\(770\) 0 0
\(771\) 4518.00 0.211040
\(772\) 0 0
\(773\) −18366.0 −0.854565 −0.427283 − 0.904118i \(-0.640529\pi\)
−0.427283 + 0.904118i \(0.640529\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3264.00 −0.150702
\(778\) 0 0
\(779\) 54600.0 2.51123
\(780\) 0 0
\(781\) −4320.00 −0.197928
\(782\) 0 0
\(783\) −162.000 −0.00739388
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −30316.0 −1.37312 −0.686562 − 0.727071i \(-0.740881\pi\)
−0.686562 + 0.727071i \(0.740881\pi\)
\(788\) 0 0
\(789\) 1296.00 0.0584776
\(790\) 0 0
\(791\) −34368.0 −1.54486
\(792\) 0 0
\(793\) −580.000 −0.0259728
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7494.00 −0.333063 −0.166531 − 0.986036i \(-0.553257\pi\)
−0.166531 + 0.986036i \(0.553257\pi\)
\(798\) 0 0
\(799\) 31824.0 1.40908
\(800\) 0 0
\(801\) −14310.0 −0.631235
\(802\) 0 0
\(803\) −23256.0 −1.02203
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −162.000 −0.00706651
\(808\) 0 0
\(809\) −11526.0 −0.500906 −0.250453 − 0.968129i \(-0.580579\pi\)
−0.250453 + 0.968129i \(0.580579\pi\)
\(810\) 0 0
\(811\) 33820.0 1.46434 0.732171 − 0.681121i \(-0.238507\pi\)
0.732171 + 0.681121i \(0.238507\pi\)
\(812\) 0 0
\(813\) −19488.0 −0.840681
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7280.00 0.311744
\(818\) 0 0
\(819\) 2880.00 0.122876
\(820\) 0 0
\(821\) 19566.0 0.831739 0.415870 − 0.909424i \(-0.363477\pi\)
0.415870 + 0.909424i \(0.363477\pi\)
\(822\) 0 0
\(823\) −40096.0 −1.69825 −0.849124 − 0.528193i \(-0.822870\pi\)
−0.849124 + 0.528193i \(0.822870\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31884.0 1.34065 0.670324 − 0.742069i \(-0.266155\pi\)
0.670324 + 0.742069i \(0.266155\pi\)
\(828\) 0 0
\(829\) −24442.0 −1.02401 −0.512006 − 0.858982i \(-0.671097\pi\)
−0.512006 + 0.858982i \(0.671097\pi\)
\(830\) 0 0
\(831\) −1398.00 −0.0583587
\(832\) 0 0
\(833\) 53118.0 2.20940
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −432.000 −0.0178400
\(838\) 0 0
\(839\) 31944.0 1.31446 0.657228 − 0.753691i \(-0.271729\pi\)
0.657228 + 0.753691i \(0.271729\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 14562.0 0.594949
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1120.00 −0.0454352
\(848\) 0 0
\(849\) 13548.0 0.547663
\(850\) 0 0
\(851\) −6528.00 −0.262958
\(852\) 0 0
\(853\) −17486.0 −0.701887 −0.350943 − 0.936397i \(-0.614139\pi\)
−0.350943 + 0.936397i \(0.614139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43434.0 −1.73125 −0.865623 − 0.500697i \(-0.833077\pi\)
−0.865623 + 0.500697i \(0.833077\pi\)
\(858\) 0 0
\(859\) −10820.0 −0.429771 −0.214886 − 0.976639i \(-0.568938\pi\)
−0.214886 + 0.976639i \(0.568938\pi\)
\(860\) 0 0
\(861\) 37440.0 1.48194
\(862\) 0 0
\(863\) 29976.0 1.18238 0.591191 − 0.806532i \(-0.298658\pi\)
0.591191 + 0.806532i \(0.298658\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3513.00 −0.137610
\(868\) 0 0
\(869\) −42048.0 −1.64140
\(870\) 0 0
\(871\) −8920.00 −0.347007
\(872\) 0 0
\(873\) −1746.00 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40522.0 1.56024 0.780120 − 0.625630i \(-0.215158\pi\)
0.780120 + 0.625630i \(0.215158\pi\)
\(878\) 0 0
\(879\) 25722.0 0.987010
\(880\) 0 0
\(881\) 15570.0 0.595422 0.297711 − 0.954656i \(-0.403777\pi\)
0.297711 + 0.954656i \(0.403777\pi\)
\(882\) 0 0
\(883\) −1084.00 −0.0413131 −0.0206566 − 0.999787i \(-0.506576\pi\)
−0.0206566 + 0.999787i \(0.506576\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8208.00 −0.310708 −0.155354 − 0.987859i \(-0.549652\pi\)
−0.155354 + 0.987859i \(0.549652\pi\)
\(888\) 0 0
\(889\) −48896.0 −1.84468
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 0 0
\(893\) −57120.0 −2.14048
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5760.00 0.214404
\(898\) 0 0
\(899\) 96.0000 0.00356149
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) 0 0
\(903\) 4992.00 0.183968
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 34076.0 1.24749 0.623746 − 0.781627i \(-0.285610\pi\)
0.623746 + 0.781627i \(0.285610\pi\)
\(908\) 0 0
\(909\) 7182.00 0.262059
\(910\) 0 0
\(911\) 15072.0 0.548142 0.274071 − 0.961709i \(-0.411630\pi\)
0.274071 + 0.961709i \(0.411630\pi\)
\(912\) 0 0
\(913\) 26352.0 0.955229
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −77184.0 −2.77954
\(918\) 0 0
\(919\) −24392.0 −0.875536 −0.437768 − 0.899088i \(-0.644231\pi\)
−0.437768 + 0.899088i \(0.644231\pi\)
\(920\) 0 0
\(921\) −10428.0 −0.373088
\(922\) 0 0
\(923\) 1200.00 0.0427936
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2448.00 0.0867345
\(928\) 0 0
\(929\) 13602.0 0.480374 0.240187 − 0.970727i \(-0.422791\pi\)
0.240187 + 0.970727i \(0.422791\pi\)
\(930\) 0 0
\(931\) −95340.0 −3.35622
\(932\) 0 0
\(933\) 7272.00 0.255171
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47974.0 1.67262 0.836309 − 0.548259i \(-0.184709\pi\)
0.836309 + 0.548259i \(0.184709\pi\)
\(938\) 0 0
\(939\) −4674.00 −0.162439
\(940\) 0 0
\(941\) −48330.0 −1.67430 −0.837148 − 0.546976i \(-0.815779\pi\)
−0.837148 + 0.546976i \(0.815779\pi\)
\(942\) 0 0
\(943\) 74880.0 2.58582
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9324.00 −0.319946 −0.159973 − 0.987121i \(-0.551141\pi\)
−0.159973 + 0.987121i \(0.551141\pi\)
\(948\) 0 0
\(949\) 6460.00 0.220970
\(950\) 0 0
\(951\) −25614.0 −0.873387
\(952\) 0 0
\(953\) 14838.0 0.504355 0.252177 − 0.967681i \(-0.418853\pi\)
0.252177 + 0.967681i \(0.418853\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 648.000 0.0218881
\(958\) 0 0
\(959\) −67392.0 −2.26924
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) 1404.00 0.0469816
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 11360.0 0.377780 0.188890 − 0.981998i \(-0.439511\pi\)
0.188890 + 0.981998i \(0.439511\pi\)
\(968\) 0 0
\(969\) 32760.0 1.08607
\(970\) 0 0
\(971\) 6972.00 0.230424 0.115212 − 0.993341i \(-0.463245\pi\)
0.115212 + 0.993341i \(0.463245\pi\)
\(972\) 0 0
\(973\) 17792.0 0.586213
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41166.0 1.34802 0.674011 − 0.738722i \(-0.264570\pi\)
0.674011 + 0.738722i \(0.264570\pi\)
\(978\) 0 0
\(979\) 57240.0 1.86864
\(980\) 0 0
\(981\) 14598.0 0.475105
\(982\) 0 0
\(983\) 22464.0 0.728881 0.364441 − 0.931227i \(-0.381260\pi\)
0.364441 + 0.931227i \(0.381260\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −39168.0 −1.26315
\(988\) 0 0
\(989\) 9984.00 0.321004
\(990\) 0 0
\(991\) 10192.0 0.326700 0.163350 − 0.986568i \(-0.447770\pi\)
0.163350 + 0.986568i \(0.447770\pi\)
\(992\) 0 0
\(993\) −2964.00 −0.0947228
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 322.000 0.0102285 0.00511426 − 0.999987i \(-0.498372\pi\)
0.00511426 + 0.999987i \(0.498372\pi\)
\(998\) 0 0
\(999\) −918.000 −0.0290733
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.s.1.1 1
4.3 odd 2 300.4.a.e.1.1 1
5.2 odd 4 1200.4.f.e.49.2 2
5.3 odd 4 1200.4.f.e.49.1 2
5.4 even 2 240.4.a.j.1.1 1
12.11 even 2 900.4.a.b.1.1 1
15.14 odd 2 720.4.a.c.1.1 1
20.3 even 4 300.4.d.d.49.2 2
20.7 even 4 300.4.d.d.49.1 2
20.19 odd 2 60.4.a.b.1.1 ✓ 1
40.19 odd 2 960.4.a.bb.1.1 1
40.29 even 2 960.4.a.a.1.1 1
60.23 odd 4 900.4.d.b.649.2 2
60.47 odd 4 900.4.d.b.649.1 2
60.59 even 2 180.4.a.c.1.1 1
180.59 even 6 1620.4.i.g.541.1 2
180.79 odd 6 1620.4.i.a.1081.1 2
180.119 even 6 1620.4.i.g.1081.1 2
180.139 odd 6 1620.4.i.a.541.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 ✓ 1 20.19 odd 2
180.4.a.c.1.1 1 60.59 even 2
240.4.a.j.1.1 1 5.4 even 2
300.4.a.e.1.1 1 4.3 odd 2
300.4.d.d.49.1 2 20.7 even 4
300.4.d.d.49.2 2 20.3 even 4
720.4.a.c.1.1 1 15.14 odd 2
900.4.a.b.1.1 1 12.11 even 2
900.4.d.b.649.1 2 60.47 odd 4
900.4.d.b.649.2 2 60.23 odd 4
960.4.a.a.1.1 1 40.29 even 2
960.4.a.bb.1.1 1 40.19 odd 2
1200.4.a.s.1.1 1 1.1 even 1 trivial
1200.4.f.e.49.1 2 5.3 odd 4
1200.4.f.e.49.2 2 5.2 odd 4
1620.4.i.a.541.1 2 180.139 odd 6
1620.4.i.a.1081.1 2 180.79 odd 6
1620.4.i.g.541.1 2 180.59 even 6
1620.4.i.g.1081.1 2 180.119 even 6