Properties

Label 2200.4.a.h.1.1
Level $2200$
Weight $4$
Character 2200.1
Self dual yes
Analytic conductor $129.804$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,4,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(129.804202013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{3} -8.00000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{3} -8.00000 q^{7} -11.0000 q^{9} +11.0000 q^{11} +58.0000 q^{13} -114.000 q^{17} -4.00000 q^{19} -32.0000 q^{21} +152.000 q^{23} -152.000 q^{27} -138.000 q^{29} +208.000 q^{31} +44.0000 q^{33} +226.000 q^{37} +232.000 q^{39} -294.000 q^{41} -276.000 q^{43} +240.000 q^{47} -279.000 q^{49} -456.000 q^{51} +370.000 q^{53} -16.0000 q^{57} -716.000 q^{59} -650.000 q^{61} +88.0000 q^{63} -124.000 q^{67} +608.000 q^{69} +232.000 q^{71} +454.000 q^{73} -88.0000 q^{77} -144.000 q^{79} -311.000 q^{81} +692.000 q^{83} -552.000 q^{87} -1206.00 q^{89} -464.000 q^{91} +832.000 q^{93} +1438.00 q^{97} -121.000 q^{99} +O(q^{100})\)

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
\(3\) 4.00000 0.769800 0.384900 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −114.000 −1.62642 −0.813208 0.581974i \(-0.802281\pi\)
−0.813208 + 0.581974i \(0.802281\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) −32.0000 −0.332522
\(22\) 0 0
\(23\) 152.000 1.37801 0.689004 0.724757i \(-0.258048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −152.000 −1.08342
\(28\) 0 0
\(29\) −138.000 −0.883654 −0.441827 0.897100i \(-0.645669\pi\)
−0.441827 + 0.897100i \(0.645669\pi\)
\(30\) 0 0
\(31\) 208.000 1.20509 0.602547 0.798084i \(-0.294153\pi\)
0.602547 + 0.798084i \(0.294153\pi\)
\(32\) 0 0
\(33\) 44.0000 0.232104
\(34\) 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
\(39\) 232.000 0.952557
\(40\) 0 0
\(41\) −294.000 −1.11988 −0.559940 0.828533i \(-0.689176\pi\)
−0.559940 + 0.828533i \(0.689176\pi\)
\(42\) 0 0
\(43\) −276.000 −0.978828 −0.489414 0.872052i \(-0.662789\pi\)
−0.489414 + 0.872052i \(0.662789\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 240.000 0.744843 0.372421 0.928064i \(-0.378528\pi\)
0.372421 + 0.928064i \(0.378528\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −456.000 −1.25202
\(52\) 0 0
\(53\) 370.000 0.958932 0.479466 0.877560i \(-0.340830\pi\)
0.479466 + 0.877560i \(0.340830\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −16.0000 −0.0371799
\(58\) 0 0
\(59\) −716.000 −1.57992 −0.789960 0.613159i \(-0.789899\pi\)
−0.789960 + 0.613159i \(0.789899\pi\)
\(60\) 0 0
\(61\) −650.000 −1.36433 −0.682164 0.731199i \(-0.738961\pi\)
−0.682164 + 0.731199i \(0.738961\pi\)
\(62\) 0 0
\(63\) 88.0000 0.175983
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −124.000 −0.226105 −0.113052 0.993589i \(-0.536063\pi\)
−0.113052 + 0.993589i \(0.536063\pi\)
\(68\) 0 0
\(69\) 608.000 1.06079
\(70\) 0 0
\(71\) 232.000 0.387793 0.193897 0.981022i \(-0.437887\pi\)
0.193897 + 0.981022i \(0.437887\pi\)
\(72\) 0 0
\(73\) 454.000 0.727900 0.363950 0.931419i \(-0.381428\pi\)
0.363950 + 0.931419i \(0.381428\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −88.0000 −0.130241
\(78\) 0 0
\(79\) −144.000 −0.205079 −0.102540 0.994729i \(-0.532697\pi\)
−0.102540 + 0.994729i \(0.532697\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 692.000 0.915143 0.457571 0.889173i \(-0.348719\pi\)
0.457571 + 0.889173i \(0.348719\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −552.000 −0.680237
\(88\) 0 0
\(89\) −1206.00 −1.43636 −0.718178 0.695859i \(-0.755024\pi\)
−0.718178 + 0.695859i \(0.755024\pi\)
\(90\) 0 0
\(91\) −464.000 −0.534510
\(92\) 0 0
\(93\) 832.000 0.927682
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1438.00 1.50522 0.752612 0.658464i \(-0.228794\pi\)
0.752612 + 0.658464i \(0.228794\pi\)
\(98\) 0 0
\(99\) −121.000 −0.122838
\(100\) 0 0
\(101\) 590.000 0.581259 0.290630 0.956836i \(-0.406135\pi\)
0.290630 + 0.956836i \(0.406135\pi\)
\(102\) 0 0
\(103\) 104.000 0.0994896 0.0497448 0.998762i \(-0.484159\pi\)
0.0497448 + 0.998762i \(0.484159\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1364.00 −1.23236 −0.616182 0.787604i \(-0.711321\pi\)
−0.616182 + 0.787604i \(0.711321\pi\)
\(108\) 0 0
\(109\) −826.000 −0.725839 −0.362920 0.931820i \(-0.618220\pi\)
−0.362920 + 0.931820i \(0.618220\pi\)
\(110\) 0 0
\(111\) 904.000 0.773008
\(112\) 0 0
\(113\) −2322.00 −1.93306 −0.966528 0.256560i \(-0.917411\pi\)
−0.966528 + 0.256560i \(0.917411\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −638.000 −0.504129
\(118\) 0 0
\(119\) 912.000 0.702545
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −1176.00 −0.862084
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1296.00 −0.905523 −0.452761 0.891632i \(-0.649561\pi\)
−0.452761 + 0.891632i \(0.649561\pi\)
\(128\) 0 0
\(129\) −1104.00 −0.753502
\(130\) 0 0
\(131\) −2100.00 −1.40059 −0.700297 0.713851i \(-0.746949\pi\)
−0.700297 + 0.713851i \(0.746949\pi\)
\(132\) 0 0
\(133\) 32.0000 0.0208628
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2394.00 −1.49294 −0.746472 0.665417i \(-0.768254\pi\)
−0.746472 + 0.665417i \(0.768254\pi\)
\(138\) 0 0
\(139\) 2340.00 1.42789 0.713943 0.700204i \(-0.246907\pi\)
0.713943 + 0.700204i \(0.246907\pi\)
\(140\) 0 0
\(141\) 960.000 0.573380
\(142\) 0 0
\(143\) 638.000 0.373093
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1116.00 −0.626164
\(148\) 0 0
\(149\) −1826.00 −1.00397 −0.501986 0.864876i \(-0.667397\pi\)
−0.501986 + 0.864876i \(0.667397\pi\)
\(150\) 0 0
\(151\) −3128.00 −1.68578 −0.842891 0.538085i \(-0.819148\pi\)
−0.842891 + 0.538085i \(0.819148\pi\)
\(152\) 0 0
\(153\) 1254.00 0.662614
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 730.000 0.371085 0.185542 0.982636i \(-0.440596\pi\)
0.185542 + 0.982636i \(0.440596\pi\)
\(158\) 0 0
\(159\) 1480.00 0.738186
\(160\) 0 0
\(161\) −1216.00 −0.595244
\(162\) 0 0
\(163\) 2500.00 1.20132 0.600660 0.799505i \(-0.294905\pi\)
0.600660 + 0.799505i \(0.294905\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3560.00 −1.64959 −0.824794 0.565434i \(-0.808709\pi\)
−0.824794 + 0.565434i \(0.808709\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) 44.0000 0.0196770
\(172\) 0 0
\(173\) −2406.00 −1.05737 −0.528684 0.848818i \(-0.677314\pi\)
−0.528684 + 0.848818i \(0.677314\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2864.00 −1.21622
\(178\) 0 0
\(179\) 636.000 0.265569 0.132785 0.991145i \(-0.457608\pi\)
0.132785 + 0.991145i \(0.457608\pi\)
\(180\) 0 0
\(181\) −1842.00 −0.756435 −0.378218 0.925717i \(-0.623463\pi\)
−0.378218 + 0.925717i \(0.623463\pi\)
\(182\) 0 0
\(183\) −2600.00 −1.05026
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1254.00 −0.490383
\(188\) 0 0
\(189\) 1216.00 0.467995
\(190\) 0 0
\(191\) 240.000 0.0909204 0.0454602 0.998966i \(-0.485525\pi\)
0.0454602 + 0.998966i \(0.485525\pi\)
\(192\) 0 0
\(193\) −3778.00 −1.40905 −0.704524 0.709680i \(-0.748840\pi\)
−0.704524 + 0.709680i \(0.748840\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3378.00 1.22169 0.610844 0.791751i \(-0.290830\pi\)
0.610844 + 0.791751i \(0.290830\pi\)
\(198\) 0 0
\(199\) −1784.00 −0.635500 −0.317750 0.948175i \(-0.602927\pi\)
−0.317750 + 0.948175i \(0.602927\pi\)
\(200\) 0 0
\(201\) −496.000 −0.174055
\(202\) 0 0
\(203\) 1104.00 0.381703
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1672.00 −0.561411
\(208\) 0 0
\(209\) −44.0000 −0.0145624
\(210\) 0 0
\(211\) 3804.00 1.24113 0.620564 0.784156i \(-0.286904\pi\)
0.620564 + 0.784156i \(0.286904\pi\)
\(212\) 0 0
\(213\) 928.000 0.298524
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1664.00 −0.520552
\(218\) 0 0
\(219\) 1816.00 0.560337
\(220\) 0 0
\(221\) −6612.00 −2.01254
\(222\) 0 0
\(223\) −3520.00 −1.05703 −0.528513 0.848925i \(-0.677250\pi\)
−0.528513 + 0.848925i \(0.677250\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4996.00 1.46078 0.730388 0.683032i \(-0.239339\pi\)
0.730388 + 0.683032i \(0.239339\pi\)
\(228\) 0 0
\(229\) 286.000 0.0825302 0.0412651 0.999148i \(-0.486861\pi\)
0.0412651 + 0.999148i \(0.486861\pi\)
\(230\) 0 0
\(231\) −352.000 −0.100259
\(232\) 0 0
\(233\) 678.000 0.190632 0.0953160 0.995447i \(-0.469614\pi\)
0.0953160 + 0.995447i \(0.469614\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −576.000 −0.157870
\(238\) 0 0
\(239\) 4016.00 1.08692 0.543459 0.839436i \(-0.317114\pi\)
0.543459 + 0.839436i \(0.317114\pi\)
\(240\) 0 0
\(241\) 2866.00 0.766039 0.383019 0.923740i \(-0.374884\pi\)
0.383019 + 0.923740i \(0.374884\pi\)
\(242\) 0 0
\(243\) 2860.00 0.755017
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −232.000 −0.0597644
\(248\) 0 0
\(249\) 2768.00 0.704477
\(250\) 0 0
\(251\) −4428.00 −1.11352 −0.556759 0.830674i \(-0.687955\pi\)
−0.556759 + 0.830674i \(0.687955\pi\)
\(252\) 0 0
\(253\) 1672.00 0.415485
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7042.00 −1.70921 −0.854607 0.519276i \(-0.826202\pi\)
−0.854607 + 0.519276i \(0.826202\pi\)
\(258\) 0 0
\(259\) −1808.00 −0.433759
\(260\) 0 0
\(261\) 1518.00 0.360007
\(262\) 0 0
\(263\) −1288.00 −0.301983 −0.150991 0.988535i \(-0.548247\pi\)
−0.150991 + 0.988535i \(0.548247\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4824.00 −1.10571
\(268\) 0 0
\(269\) 7894.00 1.78924 0.894620 0.446827i \(-0.147446\pi\)
0.894620 + 0.446827i \(0.147446\pi\)
\(270\) 0 0
\(271\) −5104.00 −1.14408 −0.572040 0.820225i \(-0.693848\pi\)
−0.572040 + 0.820225i \(0.693848\pi\)
\(272\) 0 0
\(273\) −1856.00 −0.411466
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4418.00 0.958310 0.479155 0.877730i \(-0.340943\pi\)
0.479155 + 0.877730i \(0.340943\pi\)
\(278\) 0 0
\(279\) −2288.00 −0.490964
\(280\) 0 0
\(281\) −2358.00 −0.500592 −0.250296 0.968169i \(-0.580528\pi\)
−0.250296 + 0.968169i \(0.580528\pi\)
\(282\) 0 0
\(283\) −644.000 −0.135271 −0.0676357 0.997710i \(-0.521546\pi\)
−0.0676357 + 0.997710i \(0.521546\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2352.00 0.483743
\(288\) 0 0
\(289\) 8083.00 1.64523
\(290\) 0 0
\(291\) 5752.00 1.15872
\(292\) 0 0
\(293\) −1678.00 −0.334573 −0.167286 0.985908i \(-0.553500\pi\)
−0.167286 + 0.985908i \(0.553500\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1672.00 −0.326664
\(298\) 0 0
\(299\) 8816.00 1.70516
\(300\) 0 0
\(301\) 2208.00 0.422814
\(302\) 0 0
\(303\) 2360.00 0.447454
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5676.00 −1.05520 −0.527600 0.849493i \(-0.676908\pi\)
−0.527600 + 0.849493i \(0.676908\pi\)
\(308\) 0 0
\(309\) 416.000 0.0765871
\(310\) 0 0
\(311\) 4312.00 0.786209 0.393105 0.919494i \(-0.371401\pi\)
0.393105 + 0.919494i \(0.371401\pi\)
\(312\) 0 0
\(313\) 7190.00 1.29841 0.649206 0.760613i \(-0.275101\pi\)
0.649206 + 0.760613i \(0.275101\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4474.00 0.792697 0.396348 0.918100i \(-0.370277\pi\)
0.396348 + 0.918100i \(0.370277\pi\)
\(318\) 0 0
\(319\) −1518.00 −0.266432
\(320\) 0 0
\(321\) −5456.00 −0.948674
\(322\) 0 0
\(323\) 456.000 0.0785527
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3304.00 −0.558751
\(328\) 0 0
\(329\) −1920.00 −0.321742
\(330\) 0 0
\(331\) −7644.00 −1.26934 −0.634671 0.772782i \(-0.718864\pi\)
−0.634671 + 0.772782i \(0.718864\pi\)
\(332\) 0 0
\(333\) −2486.00 −0.409105
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2926.00 0.472966 0.236483 0.971636i \(-0.424005\pi\)
0.236483 + 0.971636i \(0.424005\pi\)
\(338\) 0 0
\(339\) −9288.00 −1.48807
\(340\) 0 0
\(341\) 2288.00 0.363349
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2140.00 0.331070 0.165535 0.986204i \(-0.447065\pi\)
0.165535 + 0.986204i \(0.447065\pi\)
\(348\) 0 0
\(349\) −8522.00 −1.30708 −0.653542 0.756890i \(-0.726718\pi\)
−0.653542 + 0.756890i \(0.726718\pi\)
\(350\) 0 0
\(351\) −8816.00 −1.34064
\(352\) 0 0
\(353\) −12834.0 −1.93508 −0.967542 0.252709i \(-0.918678\pi\)
−0.967542 + 0.252709i \(0.918678\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3648.00 0.540820
\(358\) 0 0
\(359\) −4264.00 −0.626867 −0.313434 0.949610i \(-0.601479\pi\)
−0.313434 + 0.949610i \(0.601479\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 484.000 0.0699819
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9840.00 −1.39957 −0.699787 0.714351i \(-0.746722\pi\)
−0.699787 + 0.714351i \(0.746722\pi\)
\(368\) 0 0
\(369\) 3234.00 0.456247
\(370\) 0 0
\(371\) −2960.00 −0.414220
\(372\) 0 0
\(373\) −1502.00 −0.208500 −0.104250 0.994551i \(-0.533244\pi\)
−0.104250 + 0.994551i \(0.533244\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8004.00 −1.09344
\(378\) 0 0
\(379\) −10700.0 −1.45019 −0.725095 0.688649i \(-0.758204\pi\)
−0.725095 + 0.688649i \(0.758204\pi\)
\(380\) 0 0
\(381\) −5184.00 −0.697072
\(382\) 0 0
\(383\) −4000.00 −0.533657 −0.266828 0.963744i \(-0.585976\pi\)
−0.266828 + 0.963744i \(0.585976\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3036.00 0.398782
\(388\) 0 0
\(389\) 286.000 0.0372771 0.0186385 0.999826i \(-0.494067\pi\)
0.0186385 + 0.999826i \(0.494067\pi\)
\(390\) 0 0
\(391\) −17328.0 −2.24121
\(392\) 0 0
\(393\) −8400.00 −1.07818
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2230.00 −0.281916 −0.140958 0.990016i \(-0.545018\pi\)
−0.140958 + 0.990016i \(0.545018\pi\)
\(398\) 0 0
\(399\) 128.000 0.0160602
\(400\) 0 0
\(401\) 3186.00 0.396761 0.198381 0.980125i \(-0.436432\pi\)
0.198381 + 0.980125i \(0.436432\pi\)
\(402\) 0 0
\(403\) 12064.0 1.49119
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2486.00 0.302768
\(408\) 0 0
\(409\) 5290.00 0.639544 0.319772 0.947494i \(-0.396394\pi\)
0.319772 + 0.947494i \(0.396394\pi\)
\(410\) 0 0
\(411\) −9576.00 −1.14927
\(412\) 0 0
\(413\) 5728.00 0.682461
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 9360.00 1.09919
\(418\) 0 0
\(419\) 3372.00 0.393157 0.196579 0.980488i \(-0.437017\pi\)
0.196579 + 0.980488i \(0.437017\pi\)
\(420\) 0 0
\(421\) −1410.00 −0.163228 −0.0816142 0.996664i \(-0.526008\pi\)
−0.0816142 + 0.996664i \(0.526008\pi\)
\(422\) 0 0
\(423\) −2640.00 −0.303454
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5200.00 0.589334
\(428\) 0 0
\(429\) 2552.00 0.287207
\(430\) 0 0
\(431\) −7728.00 −0.863677 −0.431838 0.901951i \(-0.642135\pi\)
−0.431838 + 0.901951i \(0.642135\pi\)
\(432\) 0 0
\(433\) −466.000 −0.0517195 −0.0258597 0.999666i \(-0.508232\pi\)
−0.0258597 + 0.999666i \(0.508232\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −608.000 −0.0665551
\(438\) 0 0
\(439\) 8296.00 0.901928 0.450964 0.892542i \(-0.351080\pi\)
0.450964 + 0.892542i \(0.351080\pi\)
\(440\) 0 0
\(441\) 3069.00 0.331390
\(442\) 0 0
\(443\) 5692.00 0.610463 0.305231 0.952278i \(-0.401266\pi\)
0.305231 + 0.952278i \(0.401266\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7304.00 −0.772858
\(448\) 0 0
\(449\) 5698.00 0.598898 0.299449 0.954112i \(-0.403197\pi\)
0.299449 + 0.954112i \(0.403197\pi\)
\(450\) 0 0
\(451\) −3234.00 −0.337657
\(452\) 0 0
\(453\) −12512.0 −1.29772
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5990.00 0.613130 0.306565 0.951850i \(-0.400820\pi\)
0.306565 + 0.951850i \(0.400820\pi\)
\(458\) 0 0
\(459\) 17328.0 1.76210
\(460\) 0 0
\(461\) −8186.00 −0.827028 −0.413514 0.910498i \(-0.635699\pi\)
−0.413514 + 0.910498i \(0.635699\pi\)
\(462\) 0 0
\(463\) 8272.00 0.830308 0.415154 0.909751i \(-0.363728\pi\)
0.415154 + 0.909751i \(0.363728\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −44.0000 −0.00435991 −0.00217995 0.999998i \(-0.500694\pi\)
−0.00217995 + 0.999998i \(0.500694\pi\)
\(468\) 0 0
\(469\) 992.000 0.0976680
\(470\) 0 0
\(471\) 2920.00 0.285661
\(472\) 0 0
\(473\) −3036.00 −0.295128
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4070.00 −0.390676
\(478\) 0 0
\(479\) −12864.0 −1.22708 −0.613540 0.789664i \(-0.710255\pi\)
−0.613540 + 0.789664i \(0.710255\pi\)
\(480\) 0 0
\(481\) 13108.0 1.24256
\(482\) 0 0
\(483\) −4864.00 −0.458219
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5672.00 0.527768 0.263884 0.964554i \(-0.414996\pi\)
0.263884 + 0.964554i \(0.414996\pi\)
\(488\) 0 0
\(489\) 10000.0 0.924776
\(490\) 0 0
\(491\) 11268.0 1.03568 0.517839 0.855478i \(-0.326737\pi\)
0.517839 + 0.855478i \(0.326737\pi\)
\(492\) 0 0
\(493\) 15732.0 1.43719
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1856.00 −0.167511
\(498\) 0 0
\(499\) −13348.0 −1.19747 −0.598736 0.800946i \(-0.704330\pi\)
−0.598736 + 0.800946i \(0.704330\pi\)
\(500\) 0 0
\(501\) −14240.0 −1.26985
\(502\) 0 0
\(503\) 14504.0 1.28569 0.642844 0.765997i \(-0.277754\pi\)
0.642844 + 0.765997i \(0.277754\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4668.00 0.408902
\(508\) 0 0
\(509\) 4134.00 0.359993 0.179996 0.983667i \(-0.442391\pi\)
0.179996 + 0.983667i \(0.442391\pi\)
\(510\) 0 0
\(511\) −3632.00 −0.314423
\(512\) 0 0
\(513\) 608.000 0.0523272
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2640.00 0.224578
\(518\) 0 0
\(519\) −9624.00 −0.813963
\(520\) 0 0
\(521\) 16410.0 1.37991 0.689957 0.723850i \(-0.257629\pi\)
0.689957 + 0.723850i \(0.257629\pi\)
\(522\) 0 0
\(523\) 4748.00 0.396970 0.198485 0.980104i \(-0.436398\pi\)
0.198485 + 0.980104i \(0.436398\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −23712.0 −1.95998
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) 7876.00 0.643671
\(532\) 0 0
\(533\) −17052.0 −1.38575
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2544.00 0.204435
\(538\) 0 0
\(539\) −3069.00 −0.245253
\(540\) 0 0
\(541\) −5930.00 −0.471258 −0.235629 0.971843i \(-0.575715\pi\)
−0.235629 + 0.971843i \(0.575715\pi\)
\(542\) 0 0
\(543\) −7368.00 −0.582304
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −15836.0 −1.23784 −0.618920 0.785454i \(-0.712429\pi\)
−0.618920 + 0.785454i \(0.712429\pi\)
\(548\) 0 0
\(549\) 7150.00 0.555837
\(550\) 0 0
\(551\) 552.000 0.0426787
\(552\) 0 0
\(553\) 1152.00 0.0885859
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10554.0 0.802850 0.401425 0.915892i \(-0.368515\pi\)
0.401425 + 0.915892i \(0.368515\pi\)
\(558\) 0 0
\(559\) −16008.0 −1.21121
\(560\) 0 0
\(561\) −5016.00 −0.377497
\(562\) 0 0
\(563\) −24332.0 −1.82144 −0.910721 0.413023i \(-0.864473\pi\)
−0.910721 + 0.413023i \(0.864473\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2488.00 0.184279
\(568\) 0 0
\(569\) 18570.0 1.36818 0.684090 0.729397i \(-0.260199\pi\)
0.684090 + 0.729397i \(0.260199\pi\)
\(570\) 0 0
\(571\) −2092.00 −0.153323 −0.0766615 0.997057i \(-0.524426\pi\)
−0.0766615 + 0.997057i \(0.524426\pi\)
\(572\) 0 0
\(573\) 960.000 0.0699905
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −16898.0 −1.21919 −0.609595 0.792713i \(-0.708668\pi\)
−0.609595 + 0.792713i \(0.708668\pi\)
\(578\) 0 0
\(579\) −15112.0 −1.08469
\(580\) 0 0
\(581\) −5536.00 −0.395305
\(582\) 0 0
\(583\) 4070.00 0.289129
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26996.0 −1.89820 −0.949101 0.314973i \(-0.898005\pi\)
−0.949101 + 0.314973i \(0.898005\pi\)
\(588\) 0 0
\(589\) −832.000 −0.0582037
\(590\) 0 0
\(591\) 13512.0 0.940456
\(592\) 0 0
\(593\) 5934.00 0.410928 0.205464 0.978665i \(-0.434130\pi\)
0.205464 + 0.978665i \(0.434130\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7136.00 −0.489208
\(598\) 0 0
\(599\) 24120.0 1.64527 0.822635 0.568570i \(-0.192503\pi\)
0.822635 + 0.568570i \(0.192503\pi\)
\(600\) 0 0
\(601\) −6070.00 −0.411981 −0.205990 0.978554i \(-0.566042\pi\)
−0.205990 + 0.978554i \(0.566042\pi\)
\(602\) 0 0
\(603\) 1364.00 0.0921167
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1424.00 0.0952197 0.0476099 0.998866i \(-0.484840\pi\)
0.0476099 + 0.998866i \(0.484840\pi\)
\(608\) 0 0
\(609\) 4416.00 0.293835
\(610\) 0 0
\(611\) 13920.0 0.921674
\(612\) 0 0
\(613\) −1742.00 −0.114778 −0.0573888 0.998352i \(-0.518277\pi\)
−0.0573888 + 0.998352i \(0.518277\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16518.0 1.07778 0.538889 0.842376i \(-0.318844\pi\)
0.538889 + 0.842376i \(0.318844\pi\)
\(618\) 0 0
\(619\) −1084.00 −0.0703871 −0.0351936 0.999381i \(-0.511205\pi\)
−0.0351936 + 0.999381i \(0.511205\pi\)
\(620\) 0 0
\(621\) −23104.0 −1.49297
\(622\) 0 0
\(623\) 9648.00 0.620448
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −176.000 −0.0112101
\(628\) 0 0
\(629\) −25764.0 −1.63319
\(630\) 0 0
\(631\) 248.000 0.0156462 0.00782308 0.999969i \(-0.497510\pi\)
0.00782308 + 0.999969i \(0.497510\pi\)
\(632\) 0 0
\(633\) 15216.0 0.955421
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −16182.0 −1.00652
\(638\) 0 0
\(639\) −2552.00 −0.157990
\(640\) 0 0
\(641\) 21378.0 1.31729 0.658643 0.752456i \(-0.271131\pi\)
0.658643 + 0.752456i \(0.271131\pi\)
\(642\) 0 0
\(643\) −18364.0 −1.12629 −0.563146 0.826358i \(-0.690409\pi\)
−0.563146 + 0.826358i \(0.690409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −31064.0 −1.88756 −0.943780 0.330573i \(-0.892758\pi\)
−0.943780 + 0.330573i \(0.892758\pi\)
\(648\) 0 0
\(649\) −7876.00 −0.476364
\(650\) 0 0
\(651\) −6656.00 −0.400721
\(652\) 0 0
\(653\) −13398.0 −0.802916 −0.401458 0.915877i \(-0.631496\pi\)
−0.401458 + 0.915877i \(0.631496\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4994.00 −0.296552
\(658\) 0 0
\(659\) −14724.0 −0.870358 −0.435179 0.900344i \(-0.643315\pi\)
−0.435179 + 0.900344i \(0.643315\pi\)
\(660\) 0 0
\(661\) 23182.0 1.36411 0.682054 0.731302i \(-0.261087\pi\)
0.682054 + 0.731302i \(0.261087\pi\)
\(662\) 0 0
\(663\) −26448.0 −1.54925
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20976.0 −1.21768
\(668\) 0 0
\(669\) −14080.0 −0.813698
\(670\) 0 0
\(671\) −7150.00 −0.411360
\(672\) 0 0
\(673\) 7902.00 0.452600 0.226300 0.974058i \(-0.427337\pi\)
0.226300 + 0.974058i \(0.427337\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27826.0 1.57968 0.789838 0.613316i \(-0.210165\pi\)
0.789838 + 0.613316i \(0.210165\pi\)
\(678\) 0 0
\(679\) −11504.0 −0.650196
\(680\) 0 0
\(681\) 19984.0 1.12451
\(682\) 0 0
\(683\) −5780.00 −0.323815 −0.161907 0.986806i \(-0.551765\pi\)
−0.161907 + 0.986806i \(0.551765\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1144.00 0.0635318
\(688\) 0 0
\(689\) 21460.0 1.18659
\(690\) 0 0
\(691\) 10044.0 0.552955 0.276477 0.961020i \(-0.410833\pi\)
0.276477 + 0.961020i \(0.410833\pi\)
\(692\) 0 0
\(693\) 968.000 0.0530610
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 33516.0 1.82139
\(698\) 0 0
\(699\) 2712.00 0.146749
\(700\) 0 0
\(701\) −5322.00 −0.286746 −0.143373 0.989669i \(-0.545795\pi\)
−0.143373 + 0.989669i \(0.545795\pi\)
\(702\) 0 0
\(703\) −904.000 −0.0484993
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4720.00 −0.251080
\(708\) 0 0
\(709\) 19902.0 1.05421 0.527105 0.849800i \(-0.323277\pi\)
0.527105 + 0.849800i \(0.323277\pi\)
\(710\) 0 0
\(711\) 1584.00 0.0835508
\(712\) 0 0
\(713\) 31616.0 1.66063
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16064.0 0.836710
\(718\) 0 0
\(719\) −18688.0 −0.969325 −0.484663 0.874701i \(-0.661058\pi\)
−0.484663 + 0.874701i \(0.661058\pi\)
\(720\) 0 0
\(721\) −832.000 −0.0429754
\(722\) 0 0
\(723\) 11464.0 0.589697
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6760.00 −0.344862 −0.172431 0.985022i \(-0.555162\pi\)
−0.172431 + 0.985022i \(0.555162\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) 31464.0 1.59198
\(732\) 0 0
\(733\) 25162.0 1.26791 0.633956 0.773369i \(-0.281430\pi\)
0.633956 + 0.773369i \(0.281430\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1364.00 −0.0681731
\(738\) 0 0
\(739\) −14548.0 −0.724164 −0.362082 0.932146i \(-0.617934\pi\)
−0.362082 + 0.932146i \(0.617934\pi\)
\(740\) 0 0
\(741\) −928.000 −0.0460067
\(742\) 0 0
\(743\) 2168.00 0.107047 0.0535237 0.998567i \(-0.482955\pi\)
0.0535237 + 0.998567i \(0.482955\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −7612.00 −0.372836
\(748\) 0 0
\(749\) 10912.0 0.532331
\(750\) 0 0
\(751\) −8768.00 −0.426030 −0.213015 0.977049i \(-0.568328\pi\)
−0.213015 + 0.977049i \(0.568328\pi\)
\(752\) 0 0
\(753\) −17712.0 −0.857186
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19794.0 0.950363 0.475182 0.879888i \(-0.342382\pi\)
0.475182 + 0.879888i \(0.342382\pi\)
\(758\) 0 0
\(759\) 6688.00 0.319841
\(760\) 0 0
\(761\) −6262.00 −0.298288 −0.149144 0.988815i \(-0.547652\pi\)
−0.149144 + 0.988815i \(0.547652\pi\)
\(762\) 0 0
\(763\) 6608.00 0.313533
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41528.0 −1.95501
\(768\) 0 0
\(769\) −39454.0 −1.85013 −0.925063 0.379813i \(-0.875988\pi\)
−0.925063 + 0.379813i \(0.875988\pi\)
\(770\) 0 0
\(771\) −28168.0 −1.31575
\(772\) 0 0
\(773\) 20034.0 0.932177 0.466089 0.884738i \(-0.345663\pi\)
0.466089 + 0.884738i \(0.345663\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −7232.00 −0.333908
\(778\) 0 0
\(779\) 1176.00 0.0540880
\(780\) 0 0
\(781\) 2552.00 0.116924
\(782\) 0 0
\(783\) 20976.0 0.957370
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 35572.0 1.61119 0.805594 0.592468i \(-0.201846\pi\)
0.805594 + 0.592468i \(0.201846\pi\)
\(788\) 0 0
\(789\) −5152.00 −0.232466
\(790\) 0 0
\(791\) 18576.0 0.835002
\(792\) 0 0
\(793\) −37700.0 −1.68823
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35994.0 1.59972 0.799858 0.600190i \(-0.204908\pi\)
0.799858 + 0.600190i \(0.204908\pi\)
\(798\) 0 0
\(799\) −27360.0 −1.21142
\(800\) 0 0
\(801\) 13266.0 0.585182
\(802\) 0 0
\(803\) 4994.00 0.219470
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 31576.0 1.37736
\(808\) 0 0
\(809\) −19110.0 −0.830497 −0.415248 0.909708i \(-0.636305\pi\)
−0.415248 + 0.909708i \(0.636305\pi\)
\(810\) 0 0
\(811\) 3300.00 0.142884 0.0714418 0.997445i \(-0.477240\pi\)
0.0714418 + 0.997445i \(0.477240\pi\)
\(812\) 0 0
\(813\) −20416.0 −0.880714
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1104.00 0.0472755
\(818\) 0 0
\(819\) 5104.00 0.217763
\(820\) 0 0
\(821\) 4862.00 0.206681 0.103340 0.994646i \(-0.467047\pi\)
0.103340 + 0.994646i \(0.467047\pi\)
\(822\) 0 0
\(823\) −7176.00 −0.303936 −0.151968 0.988385i \(-0.548561\pi\)
−0.151968 + 0.988385i \(0.548561\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10276.0 −0.432082 −0.216041 0.976384i \(-0.569314\pi\)
−0.216041 + 0.976384i \(0.569314\pi\)
\(828\) 0 0
\(829\) 40646.0 1.70289 0.851444 0.524446i \(-0.175727\pi\)
0.851444 + 0.524446i \(0.175727\pi\)
\(830\) 0 0
\(831\) 17672.0 0.737707
\(832\) 0 0
\(833\) 31806.0 1.32294
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −31616.0 −1.30563
\(838\) 0 0
\(839\) 8808.00 0.362439 0.181219 0.983443i \(-0.441996\pi\)
0.181219 + 0.983443i \(0.441996\pi\)
\(840\) 0 0
\(841\) −5345.00 −0.219156
\(842\) 0 0
\(843\) −9432.00 −0.385356
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −968.000 −0.0392690
\(848\) 0 0
\(849\) −2576.00 −0.104132
\(850\) 0 0
\(851\) 34352.0 1.38375
\(852\) 0 0
\(853\) 18658.0 0.748931 0.374465 0.927241i \(-0.377826\pi\)
0.374465 + 0.927241i \(0.377826\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22410.0 −0.893245 −0.446623 0.894722i \(-0.647373\pi\)
−0.446623 + 0.894722i \(0.647373\pi\)
\(858\) 0 0
\(859\) −4780.00 −0.189862 −0.0949310 0.995484i \(-0.530263\pi\)
−0.0949310 + 0.995484i \(0.530263\pi\)
\(860\) 0 0
\(861\) 9408.00 0.372385
\(862\) 0 0
\(863\) −5088.00 −0.200692 −0.100346 0.994953i \(-0.531995\pi\)
−0.100346 + 0.994953i \(0.531995\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 32332.0 1.26650
\(868\) 0 0
\(869\) −1584.00 −0.0618337
\(870\) 0 0
\(871\) −7192.00 −0.279784
\(872\) 0 0
\(873\) −15818.0 −0.613240
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22714.0 0.874569 0.437285 0.899323i \(-0.355940\pi\)
0.437285 + 0.899323i \(0.355940\pi\)
\(878\) 0 0
\(879\) −6712.00 −0.257554
\(880\) 0 0
\(881\) 38354.0 1.46672 0.733359 0.679841i \(-0.237951\pi\)
0.733359 + 0.679841i \(0.237951\pi\)
\(882\) 0 0
\(883\) −10892.0 −0.415113 −0.207557 0.978223i \(-0.566551\pi\)
−0.207557 + 0.978223i \(0.566551\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32104.0 1.21527 0.607636 0.794215i \(-0.292118\pi\)
0.607636 + 0.794215i \(0.292118\pi\)
\(888\) 0 0
\(889\) 10368.0 0.391149
\(890\) 0 0
\(891\) −3421.00 −0.128628
\(892\) 0 0
\(893\) −960.000 −0.0359744
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 35264.0 1.31263
\(898\) 0 0
\(899\) −28704.0 −1.06489
\(900\) 0 0
\(901\) −42180.0 −1.55962
\(902\) 0 0
\(903\) 8832.00 0.325482
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20396.0 0.746679 0.373340 0.927695i \(-0.378213\pi\)
0.373340 + 0.927695i \(0.378213\pi\)
\(908\) 0 0
\(909\) −6490.00 −0.236809
\(910\) 0 0
\(911\) −10432.0 −0.379394 −0.189697 0.981843i \(-0.560751\pi\)
−0.189697 + 0.981843i \(0.560751\pi\)
\(912\) 0 0
\(913\) 7612.00 0.275926
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16800.0 0.605000
\(918\) 0 0
\(919\) 3080.00 0.110555 0.0552774 0.998471i \(-0.482396\pi\)
0.0552774 + 0.998471i \(0.482396\pi\)
\(920\) 0 0
\(921\) −22704.0 −0.812293
\(922\) 0 0
\(923\) 13456.0 0.479859
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1144.00 −0.0405328
\(928\) 0 0
\(929\) −15582.0 −0.550300 −0.275150 0.961401i \(-0.588728\pi\)
−0.275150 + 0.961401i \(0.588728\pi\)
\(930\) 0 0
\(931\) 1116.00 0.0392862
\(932\) 0 0
\(933\) 17248.0 0.605224
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21862.0 0.762220 0.381110 0.924530i \(-0.375542\pi\)
0.381110 + 0.924530i \(0.375542\pi\)
\(938\) 0 0
\(939\) 28760.0 0.999518
\(940\) 0 0
\(941\) −33850.0 −1.17267 −0.586333 0.810070i \(-0.699429\pi\)
−0.586333 + 0.810070i \(0.699429\pi\)
\(942\) 0 0
\(943\) −44688.0 −1.54320
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26836.0 0.920858 0.460429 0.887696i \(-0.347696\pi\)
0.460429 + 0.887696i \(0.347696\pi\)
\(948\) 0 0
\(949\) 26332.0 0.900709
\(950\) 0 0
\(951\) 17896.0 0.610218
\(952\) 0 0
\(953\) −25194.0 −0.856363 −0.428181 0.903693i \(-0.640846\pi\)
−0.428181 + 0.903693i \(0.640846\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6072.00 −0.205099
\(958\) 0 0
\(959\) 19152.0 0.644891
\(960\) 0 0
\(961\) 13473.0 0.452251
\(962\) 0 0
\(963\) 15004.0 0.502074
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 10296.0 0.342396 0.171198 0.985237i \(-0.445236\pi\)
0.171198 + 0.985237i \(0.445236\pi\)
\(968\) 0 0
\(969\) 1824.00 0.0604699
\(970\) 0 0
\(971\) 46116.0 1.52413 0.762066 0.647499i \(-0.224185\pi\)
0.762066 + 0.647499i \(0.224185\pi\)
\(972\) 0 0
\(973\) −18720.0 −0.616789
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37070.0 1.21389 0.606947 0.794742i \(-0.292394\pi\)
0.606947 + 0.794742i \(0.292394\pi\)
\(978\) 0 0
\(979\) −13266.0 −0.433078
\(980\) 0 0
\(981\) 9086.00 0.295712
\(982\) 0 0
\(983\) −13896.0 −0.450879 −0.225439 0.974257i \(-0.572382\pi\)
−0.225439 + 0.974257i \(0.572382\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −7680.00 −0.247677
\(988\) 0 0
\(989\) −41952.0 −1.34883
\(990\) 0 0
\(991\) −39856.0 −1.27757 −0.638783 0.769387i \(-0.720562\pi\)
−0.638783 + 0.769387i \(0.720562\pi\)
\(992\) 0 0
\(993\) −30576.0 −0.977140
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 13138.0 0.417337 0.208668 0.977986i \(-0.433087\pi\)
0.208668 + 0.977986i \(0.433087\pi\)
\(998\) 0 0
\(999\) −34352.0 −1.08794
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 2200.4.a.h.1.1 1
5.4 even 2 440.4.a.b.1.1 1
20.19 odd 2 880.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.4.a.b.1.1 1 5.4 even 2
880.4.a.l.1.1 1 20.19 odd 2
2200.4.a.h.1.1 1 1.1 even 1 trivial