Properties

Label 450.4.a.f.1.1
Level $450$
Weight $4$
Character 450.1
Self dual yes
Analytic conductor $26.551$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(1,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.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: \(26.5508595026\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +1.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +1.00000 q^{7} -8.00000 q^{8} -42.0000 q^{11} +67.0000 q^{13} -2.00000 q^{14} +16.0000 q^{16} +54.0000 q^{17} -115.000 q^{19} +84.0000 q^{22} -162.000 q^{23} -134.000 q^{26} +4.00000 q^{28} +210.000 q^{29} -193.000 q^{31} -32.0000 q^{32} -108.000 q^{34} +286.000 q^{37} +230.000 q^{38} -12.0000 q^{41} -263.000 q^{43} -168.000 q^{44} +324.000 q^{46} +414.000 q^{47} -342.000 q^{49} +268.000 q^{52} -192.000 q^{53} -8.00000 q^{56} -420.000 q^{58} -690.000 q^{59} -733.000 q^{61} +386.000 q^{62} +64.0000 q^{64} -299.000 q^{67} +216.000 q^{68} +228.000 q^{71} -938.000 q^{73} -572.000 q^{74} -460.000 q^{76} -42.0000 q^{77} -160.000 q^{79} +24.0000 q^{82} -462.000 q^{83} +526.000 q^{86} +336.000 q^{88} +240.000 q^{89} +67.0000 q^{91} -648.000 q^{92} -828.000 q^{94} +511.000 q^{97} +684.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.0539949 0.0269975 0.999636i \(-0.491405\pi\)
0.0269975 + 0.999636i \(0.491405\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −42.0000 −1.15123 −0.575613 0.817723i \(-0.695236\pi\)
−0.575613 + 0.817723i \(0.695236\pi\)
\(12\) 0 0
\(13\) 67.0000 1.42942 0.714710 0.699421i \(-0.246559\pi\)
0.714710 + 0.699421i \(0.246559\pi\)
\(14\) −2.00000 −0.0381802
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) −115.000 −1.38857 −0.694284 0.719701i \(-0.744279\pi\)
−0.694284 + 0.719701i \(0.744279\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 84.0000 0.814039
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −134.000 −1.01075
\(27\) 0 0
\(28\) 4.00000 0.0269975
\(29\) 210.000 1.34469 0.672345 0.740238i \(-0.265287\pi\)
0.672345 + 0.740238i \(0.265287\pi\)
\(30\) 0 0
\(31\) −193.000 −1.11819 −0.559094 0.829104i \(-0.688851\pi\)
−0.559094 + 0.829104i \(0.688851\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −108.000 −0.544760
\(35\) 0 0
\(36\) 0 0
\(37\) 286.000 1.27076 0.635380 0.772200i \(-0.280844\pi\)
0.635380 + 0.772200i \(0.280844\pi\)
\(38\) 230.000 0.981866
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0000 −0.0457094 −0.0228547 0.999739i \(-0.507276\pi\)
−0.0228547 + 0.999739i \(0.507276\pi\)
\(42\) 0 0
\(43\) −263.000 −0.932724 −0.466362 0.884594i \(-0.654436\pi\)
−0.466362 + 0.884594i \(0.654436\pi\)
\(44\) −168.000 −0.575613
\(45\) 0 0
\(46\) 324.000 1.03850
\(47\) 414.000 1.28485 0.642427 0.766347i \(-0.277928\pi\)
0.642427 + 0.766347i \(0.277928\pi\)
\(48\) 0 0
\(49\) −342.000 −0.997085
\(50\) 0 0
\(51\) 0 0
\(52\) 268.000 0.714710
\(53\) −192.000 −0.497608 −0.248804 0.968554i \(-0.580038\pi\)
−0.248804 + 0.968554i \(0.580038\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.00000 −0.0190901
\(57\) 0 0
\(58\) −420.000 −0.950840
\(59\) −690.000 −1.52255 −0.761274 0.648430i \(-0.775426\pi\)
−0.761274 + 0.648430i \(0.775426\pi\)
\(60\) 0 0
\(61\) −733.000 −1.53854 −0.769271 0.638923i \(-0.779380\pi\)
−0.769271 + 0.638923i \(0.779380\pi\)
\(62\) 386.000 0.790678
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −299.000 −0.545204 −0.272602 0.962127i \(-0.587884\pi\)
−0.272602 + 0.962127i \(0.587884\pi\)
\(68\) 216.000 0.385204
\(69\) 0 0
\(70\) 0 0
\(71\) 228.000 0.381107 0.190554 0.981677i \(-0.438972\pi\)
0.190554 + 0.981677i \(0.438972\pi\)
\(72\) 0 0
\(73\) −938.000 −1.50390 −0.751949 0.659221i \(-0.770886\pi\)
−0.751949 + 0.659221i \(0.770886\pi\)
\(74\) −572.000 −0.898563
\(75\) 0 0
\(76\) −460.000 −0.694284
\(77\) −42.0000 −0.0621603
\(78\) 0 0
\(79\) −160.000 −0.227866 −0.113933 0.993488i \(-0.536345\pi\)
−0.113933 + 0.993488i \(0.536345\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 24.0000 0.0323214
\(83\) −462.000 −0.610977 −0.305488 0.952196i \(-0.598820\pi\)
−0.305488 + 0.952196i \(0.598820\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 526.000 0.659535
\(87\) 0 0
\(88\) 336.000 0.407020
\(89\) 240.000 0.285842 0.142921 0.989734i \(-0.454350\pi\)
0.142921 + 0.989734i \(0.454350\pi\)
\(90\) 0 0
\(91\) 67.0000 0.0771814
\(92\) −648.000 −0.734333
\(93\) 0 0
\(94\) −828.000 −0.908529
\(95\) 0 0
\(96\) 0 0
\(97\) 511.000 0.534889 0.267444 0.963573i \(-0.413821\pi\)
0.267444 + 0.963573i \(0.413821\pi\)
\(98\) 684.000 0.705045
\(99\) 0 0
\(100\) 0 0
\(101\) −912.000 −0.898489 −0.449245 0.893409i \(-0.648307\pi\)
−0.449245 + 0.893409i \(0.648307\pi\)
\(102\) 0 0
\(103\) −668.000 −0.639029 −0.319515 0.947581i \(-0.603520\pi\)
−0.319515 + 0.947581i \(0.603520\pi\)
\(104\) −536.000 −0.505376
\(105\) 0 0
\(106\) 384.000 0.351862
\(107\) −1296.00 −1.17093 −0.585463 0.810699i \(-0.699087\pi\)
−0.585463 + 0.810699i \(0.699087\pi\)
\(108\) 0 0
\(109\) −1735.00 −1.52461 −0.762307 0.647216i \(-0.775933\pi\)
−0.762307 + 0.647216i \(0.775933\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 16.0000 0.0134987
\(113\) −1092.00 −0.909086 −0.454543 0.890725i \(-0.650197\pi\)
−0.454543 + 0.890725i \(0.650197\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 840.000 0.672345
\(117\) 0 0
\(118\) 1380.00 1.07660
\(119\) 54.0000 0.0415981
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) 1466.00 1.08791
\(123\) 0 0
\(124\) −772.000 −0.559094
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 0.0111793 0.00558965 0.999984i \(-0.498221\pi\)
0.00558965 + 0.999984i \(0.498221\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −1992.00 −1.32856 −0.664282 0.747482i \(-0.731263\pi\)
−0.664282 + 0.747482i \(0.731263\pi\)
\(132\) 0 0
\(133\) −115.000 −0.0749757
\(134\) 598.000 0.385517
\(135\) 0 0
\(136\) −432.000 −0.272380
\(137\) −2346.00 −1.46301 −0.731505 0.681836i \(-0.761182\pi\)
−0.731505 + 0.681836i \(0.761182\pi\)
\(138\) 0 0
\(139\) 2900.00 1.76960 0.884801 0.465968i \(-0.154294\pi\)
0.884801 + 0.465968i \(0.154294\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −456.000 −0.269484
\(143\) −2814.00 −1.64558
\(144\) 0 0
\(145\) 0 0
\(146\) 1876.00 1.06342
\(147\) 0 0
\(148\) 1144.00 0.635380
\(149\) 2070.00 1.13813 0.569064 0.822293i \(-0.307306\pi\)
0.569064 + 0.822293i \(0.307306\pi\)
\(150\) 0 0
\(151\) 2237.00 1.20559 0.602796 0.797895i \(-0.294053\pi\)
0.602796 + 0.797895i \(0.294053\pi\)
\(152\) 920.000 0.490933
\(153\) 0 0
\(154\) 84.0000 0.0439540
\(155\) 0 0
\(156\) 0 0
\(157\) 241.000 0.122509 0.0612544 0.998122i \(-0.480490\pi\)
0.0612544 + 0.998122i \(0.480490\pi\)
\(158\) 320.000 0.161126
\(159\) 0 0
\(160\) 0 0
\(161\) −162.000 −0.0793006
\(162\) 0 0
\(163\) 3547.00 1.70443 0.852216 0.523190i \(-0.175258\pi\)
0.852216 + 0.523190i \(0.175258\pi\)
\(164\) −48.0000 −0.0228547
\(165\) 0 0
\(166\) 924.000 0.432026
\(167\) 984.000 0.455953 0.227977 0.973667i \(-0.426789\pi\)
0.227977 + 0.973667i \(0.426789\pi\)
\(168\) 0 0
\(169\) 2292.00 1.04324
\(170\) 0 0
\(171\) 0 0
\(172\) −1052.00 −0.466362
\(173\) 3618.00 1.59001 0.795004 0.606604i \(-0.207469\pi\)
0.795004 + 0.606604i \(0.207469\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −672.000 −0.287806
\(177\) 0 0
\(178\) −480.000 −0.202121
\(179\) 150.000 0.0626342 0.0313171 0.999509i \(-0.490030\pi\)
0.0313171 + 0.999509i \(0.490030\pi\)
\(180\) 0 0
\(181\) 197.000 0.0809000 0.0404500 0.999182i \(-0.487121\pi\)
0.0404500 + 0.999182i \(0.487121\pi\)
\(182\) −134.000 −0.0545755
\(183\) 0 0
\(184\) 1296.00 0.519252
\(185\) 0 0
\(186\) 0 0
\(187\) −2268.00 −0.886912
\(188\) 1656.00 0.642427
\(189\) 0 0
\(190\) 0 0
\(191\) −1302.00 −0.493243 −0.246622 0.969112i \(-0.579320\pi\)
−0.246622 + 0.969112i \(0.579320\pi\)
\(192\) 0 0
\(193\) −4163.00 −1.55264 −0.776319 0.630340i \(-0.782916\pi\)
−0.776319 + 0.630340i \(0.782916\pi\)
\(194\) −1022.00 −0.378223
\(195\) 0 0
\(196\) −1368.00 −0.498542
\(197\) 3054.00 1.10451 0.552255 0.833675i \(-0.313767\pi\)
0.552255 + 0.833675i \(0.313767\pi\)
\(198\) 0 0
\(199\) 3425.00 1.22006 0.610030 0.792379i \(-0.291158\pi\)
0.610030 + 0.792379i \(0.291158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1824.00 0.635328
\(203\) 210.000 0.0726065
\(204\) 0 0
\(205\) 0 0
\(206\) 1336.00 0.451862
\(207\) 0 0
\(208\) 1072.00 0.357355
\(209\) 4830.00 1.59856
\(210\) 0 0
\(211\) −2443.00 −0.797076 −0.398538 0.917152i \(-0.630482\pi\)
−0.398538 + 0.917152i \(0.630482\pi\)
\(212\) −768.000 −0.248804
\(213\) 0 0
\(214\) 2592.00 0.827969
\(215\) 0 0
\(216\) 0 0
\(217\) −193.000 −0.0603765
\(218\) 3470.00 1.07806
\(219\) 0 0
\(220\) 0 0
\(221\) 3618.00 1.10124
\(222\) 0 0
\(223\) −23.0000 −0.00690670 −0.00345335 0.999994i \(-0.501099\pi\)
−0.00345335 + 0.999994i \(0.501099\pi\)
\(224\) −32.0000 −0.00954504
\(225\) 0 0
\(226\) 2184.00 0.642821
\(227\) −1956.00 −0.571913 −0.285957 0.958243i \(-0.592311\pi\)
−0.285957 + 0.958243i \(0.592311\pi\)
\(228\) 0 0
\(229\) 1805.00 0.520864 0.260432 0.965492i \(-0.416135\pi\)
0.260432 + 0.965492i \(0.416135\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1680.00 −0.475420
\(233\) 3468.00 0.975091 0.487546 0.873098i \(-0.337892\pi\)
0.487546 + 0.873098i \(0.337892\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2760.00 −0.761274
\(237\) 0 0
\(238\) −108.000 −0.0294143
\(239\) −2640.00 −0.714508 −0.357254 0.934007i \(-0.616287\pi\)
−0.357254 + 0.934007i \(0.616287\pi\)
\(240\) 0 0
\(241\) −5383.00 −1.43879 −0.719397 0.694599i \(-0.755582\pi\)
−0.719397 + 0.694599i \(0.755582\pi\)
\(242\) −866.000 −0.230035
\(243\) 0 0
\(244\) −2932.00 −0.769271
\(245\) 0 0
\(246\) 0 0
\(247\) −7705.00 −1.98485
\(248\) 1544.00 0.395339
\(249\) 0 0
\(250\) 0 0
\(251\) 5028.00 1.26440 0.632200 0.774805i \(-0.282152\pi\)
0.632200 + 0.774805i \(0.282152\pi\)
\(252\) 0 0
\(253\) 6804.00 1.69077
\(254\) −32.0000 −0.00790496
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 564.000 0.136892 0.0684462 0.997655i \(-0.478196\pi\)
0.0684462 + 0.997655i \(0.478196\pi\)
\(258\) 0 0
\(259\) 286.000 0.0686146
\(260\) 0 0
\(261\) 0 0
\(262\) 3984.00 0.939436
\(263\) −1812.00 −0.424839 −0.212420 0.977179i \(-0.568134\pi\)
−0.212420 + 0.977179i \(0.568134\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 230.000 0.0530158
\(267\) 0 0
\(268\) −1196.00 −0.272602
\(269\) 5190.00 1.17636 0.588178 0.808731i \(-0.299845\pi\)
0.588178 + 0.808731i \(0.299845\pi\)
\(270\) 0 0
\(271\) 4592.00 1.02931 0.514657 0.857396i \(-0.327919\pi\)
0.514657 + 0.857396i \(0.327919\pi\)
\(272\) 864.000 0.192602
\(273\) 0 0
\(274\) 4692.00 1.03450
\(275\) 0 0
\(276\) 0 0
\(277\) 2191.00 0.475251 0.237625 0.971357i \(-0.423631\pi\)
0.237625 + 0.971357i \(0.423631\pi\)
\(278\) −5800.00 −1.25130
\(279\) 0 0
\(280\) 0 0
\(281\) −7842.00 −1.66482 −0.832410 0.554160i \(-0.813040\pi\)
−0.832410 + 0.554160i \(0.813040\pi\)
\(282\) 0 0
\(283\) 247.000 0.0518821 0.0259410 0.999663i \(-0.491742\pi\)
0.0259410 + 0.999663i \(0.491742\pi\)
\(284\) 912.000 0.190554
\(285\) 0 0
\(286\) 5628.00 1.16360
\(287\) −12.0000 −0.00246808
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) 0 0
\(292\) −3752.00 −0.751949
\(293\) −5442.00 −1.08507 −0.542534 0.840034i \(-0.682535\pi\)
−0.542534 + 0.840034i \(0.682535\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2288.00 −0.449281
\(297\) 0 0
\(298\) −4140.00 −0.804778
\(299\) −10854.0 −2.09934
\(300\) 0 0
\(301\) −263.000 −0.0503624
\(302\) −4474.00 −0.852483
\(303\) 0 0
\(304\) −1840.00 −0.347142
\(305\) 0 0
\(306\) 0 0
\(307\) 3871.00 0.719641 0.359820 0.933022i \(-0.382838\pi\)
0.359820 + 0.933022i \(0.382838\pi\)
\(308\) −168.000 −0.0310802
\(309\) 0 0
\(310\) 0 0
\(311\) 5718.00 1.04257 0.521283 0.853384i \(-0.325454\pi\)
0.521283 + 0.853384i \(0.325454\pi\)
\(312\) 0 0
\(313\) 3637.00 0.656790 0.328395 0.944540i \(-0.393492\pi\)
0.328395 + 0.944540i \(0.393492\pi\)
\(314\) −482.000 −0.0866269
\(315\) 0 0
\(316\) −640.000 −0.113933
\(317\) −1296.00 −0.229623 −0.114812 0.993387i \(-0.536626\pi\)
−0.114812 + 0.993387i \(0.536626\pi\)
\(318\) 0 0
\(319\) −8820.00 −1.54804
\(320\) 0 0
\(321\) 0 0
\(322\) 324.000 0.0560740
\(323\) −6210.00 −1.06976
\(324\) 0 0
\(325\) 0 0
\(326\) −7094.00 −1.20522
\(327\) 0 0
\(328\) 96.0000 0.0161607
\(329\) 414.000 0.0693756
\(330\) 0 0
\(331\) 5132.00 0.852206 0.426103 0.904675i \(-0.359886\pi\)
0.426103 + 0.904675i \(0.359886\pi\)
\(332\) −1848.00 −0.305488
\(333\) 0 0
\(334\) −1968.00 −0.322408
\(335\) 0 0
\(336\) 0 0
\(337\) 6751.00 1.09125 0.545624 0.838030i \(-0.316293\pi\)
0.545624 + 0.838030i \(0.316293\pi\)
\(338\) −4584.00 −0.737683
\(339\) 0 0
\(340\) 0 0
\(341\) 8106.00 1.28729
\(342\) 0 0
\(343\) −685.000 −0.107832
\(344\) 2104.00 0.329768
\(345\) 0 0
\(346\) −7236.00 −1.12431
\(347\) −5226.00 −0.808491 −0.404246 0.914651i \(-0.632466\pi\)
−0.404246 + 0.914651i \(0.632466\pi\)
\(348\) 0 0
\(349\) −6190.00 −0.949407 −0.474704 0.880146i \(-0.657445\pi\)
−0.474704 + 0.880146i \(0.657445\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1344.00 0.203510
\(353\) 6618.00 0.997849 0.498924 0.866646i \(-0.333729\pi\)
0.498924 + 0.866646i \(0.333729\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 960.000 0.142921
\(357\) 0 0
\(358\) −300.000 −0.0442891
\(359\) 3420.00 0.502787 0.251394 0.967885i \(-0.419111\pi\)
0.251394 + 0.967885i \(0.419111\pi\)
\(360\) 0 0
\(361\) 6366.00 0.928124
\(362\) −394.000 −0.0572049
\(363\) 0 0
\(364\) 268.000 0.0385907
\(365\) 0 0
\(366\) 0 0
\(367\) 871.000 0.123885 0.0619425 0.998080i \(-0.480270\pi\)
0.0619425 + 0.998080i \(0.480270\pi\)
\(368\) −2592.00 −0.367167
\(369\) 0 0
\(370\) 0 0
\(371\) −192.000 −0.0268683
\(372\) 0 0
\(373\) −6383.00 −0.886057 −0.443028 0.896508i \(-0.646096\pi\)
−0.443028 + 0.896508i \(0.646096\pi\)
\(374\) 4536.00 0.627142
\(375\) 0 0
\(376\) −3312.00 −0.454264
\(377\) 14070.0 1.92213
\(378\) 0 0
\(379\) −9865.00 −1.33702 −0.668511 0.743703i \(-0.733068\pi\)
−0.668511 + 0.743703i \(0.733068\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2604.00 0.348775
\(383\) 9828.00 1.31119 0.655597 0.755111i \(-0.272417\pi\)
0.655597 + 0.755111i \(0.272417\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8326.00 1.09788
\(387\) 0 0
\(388\) 2044.00 0.267444
\(389\) −12540.0 −1.63446 −0.817228 0.576315i \(-0.804490\pi\)
−0.817228 + 0.576315i \(0.804490\pi\)
\(390\) 0 0
\(391\) −8748.00 −1.13147
\(392\) 2736.00 0.352523
\(393\) 0 0
\(394\) −6108.00 −0.781007
\(395\) 0 0
\(396\) 0 0
\(397\) 1381.00 0.174585 0.0872927 0.996183i \(-0.472178\pi\)
0.0872927 + 0.996183i \(0.472178\pi\)
\(398\) −6850.00 −0.862712
\(399\) 0 0
\(400\) 0 0
\(401\) −14232.0 −1.77235 −0.886175 0.463351i \(-0.846647\pi\)
−0.886175 + 0.463351i \(0.846647\pi\)
\(402\) 0 0
\(403\) −12931.0 −1.59836
\(404\) −3648.00 −0.449245
\(405\) 0 0
\(406\) −420.000 −0.0513405
\(407\) −12012.0 −1.46293
\(408\) 0 0
\(409\) 2645.00 0.319772 0.159886 0.987135i \(-0.448887\pi\)
0.159886 + 0.987135i \(0.448887\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2672.00 −0.319515
\(413\) −690.000 −0.0822099
\(414\) 0 0
\(415\) 0 0
\(416\) −2144.00 −0.252688
\(417\) 0 0
\(418\) −9660.00 −1.13035
\(419\) −3000.00 −0.349784 −0.174892 0.984588i \(-0.555958\pi\)
−0.174892 + 0.984588i \(0.555958\pi\)
\(420\) 0 0
\(421\) −11338.0 −1.31254 −0.656271 0.754525i \(-0.727867\pi\)
−0.656271 + 0.754525i \(0.727867\pi\)
\(422\) 4886.00 0.563618
\(423\) 0 0
\(424\) 1536.00 0.175931
\(425\) 0 0
\(426\) 0 0
\(427\) −733.000 −0.0830734
\(428\) −5184.00 −0.585463
\(429\) 0 0
\(430\) 0 0
\(431\) 3258.00 0.364112 0.182056 0.983288i \(-0.441725\pi\)
0.182056 + 0.983288i \(0.441725\pi\)
\(432\) 0 0
\(433\) −1163.00 −0.129077 −0.0645384 0.997915i \(-0.520557\pi\)
−0.0645384 + 0.997915i \(0.520557\pi\)
\(434\) 386.000 0.0426926
\(435\) 0 0
\(436\) −6940.00 −0.762307
\(437\) 18630.0 2.03934
\(438\) 0 0
\(439\) 6695.00 0.727870 0.363935 0.931424i \(-0.381433\pi\)
0.363935 + 0.931424i \(0.381433\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7236.00 −0.778691
\(443\) 16368.0 1.75546 0.877728 0.479159i \(-0.159058\pi\)
0.877728 + 0.479159i \(0.159058\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 46.0000 0.00488377
\(447\) 0 0
\(448\) 64.0000 0.00674937
\(449\) −16380.0 −1.72165 −0.860824 0.508903i \(-0.830051\pi\)
−0.860824 + 0.508903i \(0.830051\pi\)
\(450\) 0 0
\(451\) 504.000 0.0526218
\(452\) −4368.00 −0.454543
\(453\) 0 0
\(454\) 3912.00 0.404404
\(455\) 0 0
\(456\) 0 0
\(457\) 13786.0 1.41112 0.705560 0.708650i \(-0.250696\pi\)
0.705560 + 0.708650i \(0.250696\pi\)
\(458\) −3610.00 −0.368306
\(459\) 0 0
\(460\) 0 0
\(461\) −11832.0 −1.19538 −0.597691 0.801726i \(-0.703915\pi\)
−0.597691 + 0.801726i \(0.703915\pi\)
\(462\) 0 0
\(463\) −3008.00 −0.301930 −0.150965 0.988539i \(-0.548238\pi\)
−0.150965 + 0.988539i \(0.548238\pi\)
\(464\) 3360.00 0.336173
\(465\) 0 0
\(466\) −6936.00 −0.689494
\(467\) 4434.00 0.439360 0.219680 0.975572i \(-0.429499\pi\)
0.219680 + 0.975572i \(0.429499\pi\)
\(468\) 0 0
\(469\) −299.000 −0.0294382
\(470\) 0 0
\(471\) 0 0
\(472\) 5520.00 0.538302
\(473\) 11046.0 1.07378
\(474\) 0 0
\(475\) 0 0
\(476\) 216.000 0.0207990
\(477\) 0 0
\(478\) 5280.00 0.505233
\(479\) −7410.00 −0.706830 −0.353415 0.935467i \(-0.614980\pi\)
−0.353415 + 0.935467i \(0.614980\pi\)
\(480\) 0 0
\(481\) 19162.0 1.81645
\(482\) 10766.0 1.01738
\(483\) 0 0
\(484\) 1732.00 0.162660
\(485\) 0 0
\(486\) 0 0
\(487\) 8671.00 0.806818 0.403409 0.915020i \(-0.367825\pi\)
0.403409 + 0.915020i \(0.367825\pi\)
\(488\) 5864.00 0.543957
\(489\) 0 0
\(490\) 0 0
\(491\) 19368.0 1.78017 0.890087 0.455790i \(-0.150643\pi\)
0.890087 + 0.455790i \(0.150643\pi\)
\(492\) 0 0
\(493\) 11340.0 1.03596
\(494\) 15410.0 1.40350
\(495\) 0 0
\(496\) −3088.00 −0.279547
\(497\) 228.000 0.0205779
\(498\) 0 0
\(499\) −8875.00 −0.796192 −0.398096 0.917344i \(-0.630329\pi\)
−0.398096 + 0.917344i \(0.630329\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −10056.0 −0.894066
\(503\) −10452.0 −0.926504 −0.463252 0.886227i \(-0.653318\pi\)
−0.463252 + 0.886227i \(0.653318\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −13608.0 −1.19555
\(507\) 0 0
\(508\) 64.0000 0.00558965
\(509\) 19770.0 1.72159 0.860796 0.508951i \(-0.169967\pi\)
0.860796 + 0.508951i \(0.169967\pi\)
\(510\) 0 0
\(511\) −938.000 −0.0812029
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −1128.00 −0.0967976
\(515\) 0 0
\(516\) 0 0
\(517\) −17388.0 −1.47916
\(518\) −572.000 −0.0485178
\(519\) 0 0
\(520\) 0 0
\(521\) 11238.0 0.945001 0.472501 0.881330i \(-0.343351\pi\)
0.472501 + 0.881330i \(0.343351\pi\)
\(522\) 0 0
\(523\) 7447.00 0.622628 0.311314 0.950307i \(-0.399231\pi\)
0.311314 + 0.950307i \(0.399231\pi\)
\(524\) −7968.00 −0.664282
\(525\) 0 0
\(526\) 3624.00 0.300407
\(527\) −10422.0 −0.861460
\(528\) 0 0
\(529\) 14077.0 1.15698
\(530\) 0 0
\(531\) 0 0
\(532\) −460.000 −0.0374878
\(533\) −804.000 −0.0653379
\(534\) 0 0
\(535\) 0 0
\(536\) 2392.00 0.192759
\(537\) 0 0
\(538\) −10380.0 −0.831810
\(539\) 14364.0 1.14787
\(540\) 0 0
\(541\) −17623.0 −1.40050 −0.700251 0.713896i \(-0.746929\pi\)
−0.700251 + 0.713896i \(0.746929\pi\)
\(542\) −9184.00 −0.727835
\(543\) 0 0
\(544\) −1728.00 −0.136190
\(545\) 0 0
\(546\) 0 0
\(547\) 10096.0 0.789166 0.394583 0.918860i \(-0.370889\pi\)
0.394583 + 0.918860i \(0.370889\pi\)
\(548\) −9384.00 −0.731505
\(549\) 0 0
\(550\) 0 0
\(551\) −24150.0 −1.86720
\(552\) 0 0
\(553\) −160.000 −0.0123036
\(554\) −4382.00 −0.336053
\(555\) 0 0
\(556\) 11600.0 0.884801
\(557\) 14514.0 1.10409 0.552045 0.833814i \(-0.313848\pi\)
0.552045 + 0.833814i \(0.313848\pi\)
\(558\) 0 0
\(559\) −17621.0 −1.33325
\(560\) 0 0
\(561\) 0 0
\(562\) 15684.0 1.17721
\(563\) −10242.0 −0.766694 −0.383347 0.923604i \(-0.625229\pi\)
−0.383347 + 0.923604i \(0.625229\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −494.000 −0.0366862
\(567\) 0 0
\(568\) −1824.00 −0.134742
\(569\) 6750.00 0.497319 0.248660 0.968591i \(-0.420010\pi\)
0.248660 + 0.968591i \(0.420010\pi\)
\(570\) 0 0
\(571\) 17117.0 1.25451 0.627254 0.778815i \(-0.284179\pi\)
0.627254 + 0.778815i \(0.284179\pi\)
\(572\) −11256.0 −0.822792
\(573\) 0 0
\(574\) 24.0000 0.00174519
\(575\) 0 0
\(576\) 0 0
\(577\) 301.000 0.0217171 0.0108586 0.999941i \(-0.496544\pi\)
0.0108586 + 0.999941i \(0.496544\pi\)
\(578\) 3994.00 0.287420
\(579\) 0 0
\(580\) 0 0
\(581\) −462.000 −0.0329897
\(582\) 0 0
\(583\) 8064.00 0.572859
\(584\) 7504.00 0.531708
\(585\) 0 0
\(586\) 10884.0 0.767259
\(587\) −15456.0 −1.08678 −0.543388 0.839482i \(-0.682859\pi\)
−0.543388 + 0.839482i \(0.682859\pi\)
\(588\) 0 0
\(589\) 22195.0 1.55268
\(590\) 0 0
\(591\) 0 0
\(592\) 4576.00 0.317690
\(593\) −9492.00 −0.657318 −0.328659 0.944449i \(-0.606597\pi\)
−0.328659 + 0.944449i \(0.606597\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8280.00 0.569064
\(597\) 0 0
\(598\) 21708.0 1.48446
\(599\) −1500.00 −0.102318 −0.0511589 0.998691i \(-0.516291\pi\)
−0.0511589 + 0.998691i \(0.516291\pi\)
\(600\) 0 0
\(601\) 14627.0 0.992758 0.496379 0.868106i \(-0.334663\pi\)
0.496379 + 0.868106i \(0.334663\pi\)
\(602\) 526.000 0.0356116
\(603\) 0 0
\(604\) 8948.00 0.602796
\(605\) 0 0
\(606\) 0 0
\(607\) −16184.0 −1.08219 −0.541094 0.840962i \(-0.681990\pi\)
−0.541094 + 0.840962i \(0.681990\pi\)
\(608\) 3680.00 0.245467
\(609\) 0 0
\(610\) 0 0
\(611\) 27738.0 1.83659
\(612\) 0 0
\(613\) 18502.0 1.21907 0.609534 0.792760i \(-0.291357\pi\)
0.609534 + 0.792760i \(0.291357\pi\)
\(614\) −7742.00 −0.508863
\(615\) 0 0
\(616\) 336.000 0.0219770
\(617\) −13896.0 −0.906697 −0.453348 0.891333i \(-0.649771\pi\)
−0.453348 + 0.891333i \(0.649771\pi\)
\(618\) 0 0
\(619\) −9895.00 −0.642510 −0.321255 0.946993i \(-0.604105\pi\)
−0.321255 + 0.946993i \(0.604105\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11436.0 −0.737206
\(623\) 240.000 0.0154340
\(624\) 0 0
\(625\) 0 0
\(626\) −7274.00 −0.464421
\(627\) 0 0
\(628\) 964.000 0.0612544
\(629\) 15444.0 0.979003
\(630\) 0 0
\(631\) 467.000 0.0294627 0.0147314 0.999891i \(-0.495311\pi\)
0.0147314 + 0.999891i \(0.495311\pi\)
\(632\) 1280.00 0.0805628
\(633\) 0 0
\(634\) 2592.00 0.162368
\(635\) 0 0
\(636\) 0 0
\(637\) −22914.0 −1.42525
\(638\) 17640.0 1.09463
\(639\) 0 0
\(640\) 0 0
\(641\) −30612.0 −1.88627 −0.943137 0.332405i \(-0.892140\pi\)
−0.943137 + 0.332405i \(0.892140\pi\)
\(642\) 0 0
\(643\) 1852.00 0.113586 0.0567930 0.998386i \(-0.481913\pi\)
0.0567930 + 0.998386i \(0.481913\pi\)
\(644\) −648.000 −0.0396503
\(645\) 0 0
\(646\) 12420.0 0.756437
\(647\) −21156.0 −1.28551 −0.642757 0.766070i \(-0.722210\pi\)
−0.642757 + 0.766070i \(0.722210\pi\)
\(648\) 0 0
\(649\) 28980.0 1.75280
\(650\) 0 0
\(651\) 0 0
\(652\) 14188.0 0.852216
\(653\) −9702.00 −0.581422 −0.290711 0.956811i \(-0.593892\pi\)
−0.290711 + 0.956811i \(0.593892\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −192.000 −0.0114273
\(657\) 0 0
\(658\) −828.000 −0.0490559
\(659\) −1980.00 −0.117041 −0.0585204 0.998286i \(-0.518638\pi\)
−0.0585204 + 0.998286i \(0.518638\pi\)
\(660\) 0 0
\(661\) −20158.0 −1.18617 −0.593083 0.805142i \(-0.702089\pi\)
−0.593083 + 0.805142i \(0.702089\pi\)
\(662\) −10264.0 −0.602601
\(663\) 0 0
\(664\) 3696.00 0.216013
\(665\) 0 0
\(666\) 0 0
\(667\) −34020.0 −1.97490
\(668\) 3936.00 0.227977
\(669\) 0 0
\(670\) 0 0
\(671\) 30786.0 1.77121
\(672\) 0 0
\(673\) 16882.0 0.966944 0.483472 0.875360i \(-0.339376\pi\)
0.483472 + 0.875360i \(0.339376\pi\)
\(674\) −13502.0 −0.771628
\(675\) 0 0
\(676\) 9168.00 0.521620
\(677\) 20934.0 1.18842 0.594209 0.804311i \(-0.297465\pi\)
0.594209 + 0.804311i \(0.297465\pi\)
\(678\) 0 0
\(679\) 511.000 0.0288813
\(680\) 0 0
\(681\) 0 0
\(682\) −16212.0 −0.910249
\(683\) −8712.00 −0.488075 −0.244038 0.969766i \(-0.578472\pi\)
−0.244038 + 0.969766i \(0.578472\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1370.00 0.0762490
\(687\) 0 0
\(688\) −4208.00 −0.233181
\(689\) −12864.0 −0.711291
\(690\) 0 0
\(691\) −14128.0 −0.777792 −0.388896 0.921282i \(-0.627144\pi\)
−0.388896 + 0.921282i \(0.627144\pi\)
\(692\) 14472.0 0.795004
\(693\) 0 0
\(694\) 10452.0 0.571689
\(695\) 0 0
\(696\) 0 0
\(697\) −648.000 −0.0352148
\(698\) 12380.0 0.671332
\(699\) 0 0
\(700\) 0 0
\(701\) 28278.0 1.52360 0.761801 0.647811i \(-0.224315\pi\)
0.761801 + 0.647811i \(0.224315\pi\)
\(702\) 0 0
\(703\) −32890.0 −1.76454
\(704\) −2688.00 −0.143903
\(705\) 0 0
\(706\) −13236.0 −0.705586
\(707\) −912.000 −0.0485138
\(708\) 0 0
\(709\) 8885.00 0.470639 0.235320 0.971918i \(-0.424386\pi\)
0.235320 + 0.971918i \(0.424386\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1920.00 −0.101060
\(713\) 31266.0 1.64225
\(714\) 0 0
\(715\) 0 0
\(716\) 600.000 0.0313171
\(717\) 0 0
\(718\) −6840.00 −0.355524
\(719\) 7530.00 0.390572 0.195286 0.980746i \(-0.437436\pi\)
0.195286 + 0.980746i \(0.437436\pi\)
\(720\) 0 0
\(721\) −668.000 −0.0345043
\(722\) −12732.0 −0.656283
\(723\) 0 0
\(724\) 788.000 0.0404500
\(725\) 0 0
\(726\) 0 0
\(727\) 1801.00 0.0918781 0.0459391 0.998944i \(-0.485372\pi\)
0.0459391 + 0.998944i \(0.485372\pi\)
\(728\) −536.000 −0.0272877
\(729\) 0 0
\(730\) 0 0
\(731\) −14202.0 −0.718577
\(732\) 0 0
\(733\) 7882.00 0.397174 0.198587 0.980083i \(-0.436365\pi\)
0.198587 + 0.980083i \(0.436365\pi\)
\(734\) −1742.00 −0.0876000
\(735\) 0 0
\(736\) 5184.00 0.259626
\(737\) 12558.0 0.627652
\(738\) 0 0
\(739\) 33860.0 1.68547 0.842734 0.538331i \(-0.180945\pi\)
0.842734 + 0.538331i \(0.180945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 384.000 0.0189988
\(743\) −20652.0 −1.01972 −0.509858 0.860259i \(-0.670302\pi\)
−0.509858 + 0.860259i \(0.670302\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12766.0 0.626537
\(747\) 0 0
\(748\) −9072.00 −0.443456
\(749\) −1296.00 −0.0632240
\(750\) 0 0
\(751\) 7472.00 0.363059 0.181529 0.983386i \(-0.441895\pi\)
0.181529 + 0.983386i \(0.441895\pi\)
\(752\) 6624.00 0.321213
\(753\) 0 0
\(754\) −28140.0 −1.35915
\(755\) 0 0
\(756\) 0 0
\(757\) 32251.0 1.54846 0.774229 0.632906i \(-0.218138\pi\)
0.774229 + 0.632906i \(0.218138\pi\)
\(758\) 19730.0 0.945417
\(759\) 0 0
\(760\) 0 0
\(761\) −16812.0 −0.800834 −0.400417 0.916333i \(-0.631135\pi\)
−0.400417 + 0.916333i \(0.631135\pi\)
\(762\) 0 0
\(763\) −1735.00 −0.0823214
\(764\) −5208.00 −0.246622
\(765\) 0 0
\(766\) −19656.0 −0.927154
\(767\) −46230.0 −2.17636
\(768\) 0 0
\(769\) −34645.0 −1.62462 −0.812309 0.583228i \(-0.801789\pi\)
−0.812309 + 0.583228i \(0.801789\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16652.0 −0.776319
\(773\) −8412.00 −0.391408 −0.195704 0.980663i \(-0.562699\pi\)
−0.195704 + 0.980663i \(0.562699\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4088.00 −0.189112
\(777\) 0 0
\(778\) 25080.0 1.15573
\(779\) 1380.00 0.0634706
\(780\) 0 0
\(781\) −9576.00 −0.438740
\(782\) 17496.0 0.800071
\(783\) 0 0
\(784\) −5472.00 −0.249271
\(785\) 0 0
\(786\) 0 0
\(787\) −18329.0 −0.830188 −0.415094 0.909778i \(-0.636251\pi\)
−0.415094 + 0.909778i \(0.636251\pi\)
\(788\) 12216.0 0.552255
\(789\) 0 0
\(790\) 0 0
\(791\) −1092.00 −0.0490860
\(792\) 0 0
\(793\) −49111.0 −2.19922
\(794\) −2762.00 −0.123451
\(795\) 0 0
\(796\) 13700.0 0.610030
\(797\) 16044.0 0.713059 0.356529 0.934284i \(-0.383960\pi\)
0.356529 + 0.934284i \(0.383960\pi\)
\(798\) 0 0
\(799\) 22356.0 0.989860
\(800\) 0 0
\(801\) 0 0
\(802\) 28464.0 1.25324
\(803\) 39396.0 1.73133
\(804\) 0 0
\(805\) 0 0
\(806\) 25862.0 1.13021
\(807\) 0 0
\(808\) 7296.00 0.317664
\(809\) 24000.0 1.04301 0.521505 0.853248i \(-0.325371\pi\)
0.521505 + 0.853248i \(0.325371\pi\)
\(810\) 0 0
\(811\) 5117.00 0.221556 0.110778 0.993845i \(-0.464666\pi\)
0.110778 + 0.993845i \(0.464666\pi\)
\(812\) 840.000 0.0363032
\(813\) 0 0
\(814\) 24024.0 1.03445
\(815\) 0 0
\(816\) 0 0
\(817\) 30245.0 1.29515
\(818\) −5290.00 −0.226113
\(819\) 0 0
\(820\) 0 0
\(821\) −13542.0 −0.575663 −0.287831 0.957681i \(-0.592934\pi\)
−0.287831 + 0.957681i \(0.592934\pi\)
\(822\) 0 0
\(823\) −1283.00 −0.0543409 −0.0271705 0.999631i \(-0.508650\pi\)
−0.0271705 + 0.999631i \(0.508650\pi\)
\(824\) 5344.00 0.225931
\(825\) 0 0
\(826\) 1380.00 0.0581312
\(827\) 16344.0 0.687227 0.343613 0.939111i \(-0.388349\pi\)
0.343613 + 0.939111i \(0.388349\pi\)
\(828\) 0 0
\(829\) −790.000 −0.0330975 −0.0165488 0.999863i \(-0.505268\pi\)
−0.0165488 + 0.999863i \(0.505268\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4288.00 0.178677
\(833\) −18468.0 −0.768161
\(834\) 0 0
\(835\) 0 0
\(836\) 19320.0 0.799278
\(837\) 0 0
\(838\) 6000.00 0.247335
\(839\) 9990.00 0.411076 0.205538 0.978649i \(-0.434106\pi\)
0.205538 + 0.978649i \(0.434106\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 22676.0 0.928108
\(843\) 0 0
\(844\) −9772.00 −0.398538
\(845\) 0 0
\(846\) 0 0
\(847\) 433.000 0.0175656
\(848\) −3072.00 −0.124402
\(849\) 0 0
\(850\) 0 0
\(851\) −46332.0 −1.86632
\(852\) 0 0
\(853\) −24743.0 −0.993182 −0.496591 0.867985i \(-0.665415\pi\)
−0.496591 + 0.867985i \(0.665415\pi\)
\(854\) 1466.00 0.0587418
\(855\) 0 0
\(856\) 10368.0 0.413985
\(857\) −23556.0 −0.938924 −0.469462 0.882953i \(-0.655552\pi\)
−0.469462 + 0.882953i \(0.655552\pi\)
\(858\) 0 0
\(859\) −34000.0 −1.35048 −0.675242 0.737597i \(-0.735961\pi\)
−0.675242 + 0.737597i \(0.735961\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6516.00 −0.257466
\(863\) −37032.0 −1.46070 −0.730350 0.683073i \(-0.760643\pi\)
−0.730350 + 0.683073i \(0.760643\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2326.00 0.0912710
\(867\) 0 0
\(868\) −772.000 −0.0301882
\(869\) 6720.00 0.262325
\(870\) 0 0
\(871\) −20033.0 −0.779325
\(872\) 13880.0 0.539032
\(873\) 0 0
\(874\) −37260.0 −1.44203
\(875\) 0 0
\(876\) 0 0
\(877\) −2519.00 −0.0969904 −0.0484952 0.998823i \(-0.515443\pi\)
−0.0484952 + 0.998823i \(0.515443\pi\)
\(878\) −13390.0 −0.514682
\(879\) 0 0
\(880\) 0 0
\(881\) −43992.0 −1.68232 −0.841162 0.540783i \(-0.818128\pi\)
−0.841162 + 0.540783i \(0.818128\pi\)
\(882\) 0 0
\(883\) 19177.0 0.730869 0.365435 0.930837i \(-0.380920\pi\)
0.365435 + 0.930837i \(0.380920\pi\)
\(884\) 14472.0 0.550618
\(885\) 0 0
\(886\) −32736.0 −1.24130
\(887\) 44994.0 1.70321 0.851607 0.524181i \(-0.175628\pi\)
0.851607 + 0.524181i \(0.175628\pi\)
\(888\) 0 0
\(889\) 16.0000 0.000603625 0
\(890\) 0 0
\(891\) 0 0
\(892\) −92.0000 −0.00345335
\(893\) −47610.0 −1.78411
\(894\) 0 0
\(895\) 0 0
\(896\) −128.000 −0.00477252
\(897\) 0 0
\(898\) 32760.0 1.21739
\(899\) −40530.0 −1.50362
\(900\) 0 0
\(901\) −10368.0 −0.383361
\(902\) −1008.00 −0.0372092
\(903\) 0 0
\(904\) 8736.00 0.321410
\(905\) 0 0
\(906\) 0 0
\(907\) 52396.0 1.91817 0.959085 0.283117i \(-0.0913686\pi\)
0.959085 + 0.283117i \(0.0913686\pi\)
\(908\) −7824.00 −0.285957
\(909\) 0 0
\(910\) 0 0
\(911\) −7242.00 −0.263379 −0.131689 0.991291i \(-0.542040\pi\)
−0.131689 + 0.991291i \(0.542040\pi\)
\(912\) 0 0
\(913\) 19404.0 0.703372
\(914\) −27572.0 −0.997813
\(915\) 0 0
\(916\) 7220.00 0.260432
\(917\) −1992.00 −0.0717357
\(918\) 0 0
\(919\) 4085.00 0.146629 0.0733143 0.997309i \(-0.476642\pi\)
0.0733143 + 0.997309i \(0.476642\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 23664.0 0.845263
\(923\) 15276.0 0.544762
\(924\) 0 0
\(925\) 0 0
\(926\) 6016.00 0.213497
\(927\) 0 0
\(928\) −6720.00 −0.237710
\(929\) 3030.00 0.107009 0.0535043 0.998568i \(-0.482961\pi\)
0.0535043 + 0.998568i \(0.482961\pi\)
\(930\) 0 0
\(931\) 39330.0 1.38452
\(932\) 13872.0 0.487546
\(933\) 0 0
\(934\) −8868.00 −0.310674
\(935\) 0 0
\(936\) 0 0
\(937\) −5759.00 −0.200788 −0.100394 0.994948i \(-0.532010\pi\)
−0.100394 + 0.994948i \(0.532010\pi\)
\(938\) 598.000 0.0208160
\(939\) 0 0
\(940\) 0 0
\(941\) 258.000 0.00893790 0.00446895 0.999990i \(-0.498577\pi\)
0.00446895 + 0.999990i \(0.498577\pi\)
\(942\) 0 0
\(943\) 1944.00 0.0671319
\(944\) −11040.0 −0.380637
\(945\) 0 0
\(946\) −22092.0 −0.759274
\(947\) 1374.00 0.0471478 0.0235739 0.999722i \(-0.492495\pi\)
0.0235739 + 0.999722i \(0.492495\pi\)
\(948\) 0 0
\(949\) −62846.0 −2.14970
\(950\) 0 0
\(951\) 0 0
\(952\) −432.000 −0.0147071
\(953\) 9288.00 0.315706 0.157853 0.987463i \(-0.449543\pi\)
0.157853 + 0.987463i \(0.449543\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −10560.0 −0.357254
\(957\) 0 0
\(958\) 14820.0 0.499804
\(959\) −2346.00 −0.0789951
\(960\) 0 0
\(961\) 7458.00 0.250344
\(962\) −38324.0 −1.28442
\(963\) 0 0
\(964\) −21532.0 −0.719397
\(965\) 0 0
\(966\) 0 0
\(967\) 21616.0 0.718846 0.359423 0.933175i \(-0.382974\pi\)
0.359423 + 0.933175i \(0.382974\pi\)
\(968\) −3464.00 −0.115018
\(969\) 0 0
\(970\) 0 0
\(971\) 19098.0 0.631188 0.315594 0.948894i \(-0.397796\pi\)
0.315594 + 0.948894i \(0.397796\pi\)
\(972\) 0 0
\(973\) 2900.00 0.0955496
\(974\) −17342.0 −0.570507
\(975\) 0 0
\(976\) −11728.0 −0.384635
\(977\) −18246.0 −0.597483 −0.298742 0.954334i \(-0.596567\pi\)
−0.298742 + 0.954334i \(0.596567\pi\)
\(978\) 0 0
\(979\) −10080.0 −0.329069
\(980\) 0 0
\(981\) 0 0
\(982\) −38736.0 −1.25877
\(983\) −38772.0 −1.25802 −0.629011 0.777397i \(-0.716540\pi\)
−0.629011 + 0.777397i \(0.716540\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −22680.0 −0.732534
\(987\) 0 0
\(988\) −30820.0 −0.992424
\(989\) 42606.0 1.36986
\(990\) 0 0
\(991\) −23053.0 −0.738953 −0.369477 0.929240i \(-0.620463\pi\)
−0.369477 + 0.929240i \(0.620463\pi\)
\(992\) 6176.00 0.197670
\(993\) 0 0
\(994\) −456.000 −0.0145507
\(995\) 0 0
\(996\) 0 0
\(997\) 10366.0 0.329282 0.164641 0.986354i \(-0.447353\pi\)
0.164641 + 0.986354i \(0.447353\pi\)
\(998\) 17750.0 0.562992
\(999\) 0 0
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 450.4.a.f.1.1 1
3.2 odd 2 150.4.a.h.1.1 yes 1
5.2 odd 4 450.4.c.a.199.1 2
5.3 odd 4 450.4.c.a.199.2 2
5.4 even 2 450.4.a.o.1.1 1
12.11 even 2 1200.4.a.i.1.1 1
15.2 even 4 150.4.c.e.49.2 2
15.8 even 4 150.4.c.e.49.1 2
15.14 odd 2 150.4.a.a.1.1 1
60.23 odd 4 1200.4.f.c.49.1 2
60.47 odd 4 1200.4.f.c.49.2 2
60.59 even 2 1200.4.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.4.a.a.1.1 1 15.14 odd 2
150.4.a.h.1.1 yes 1 3.2 odd 2
150.4.c.e.49.1 2 15.8 even 4
150.4.c.e.49.2 2 15.2 even 4
450.4.a.f.1.1 1 1.1 even 1 trivial
450.4.a.o.1.1 1 5.4 even 2
450.4.c.a.199.1 2 5.2 odd 4
450.4.c.a.199.2 2 5.3 odd 4
1200.4.a.i.1.1 1 12.11 even 2
1200.4.a.bb.1.1 1 60.59 even 2
1200.4.f.c.49.1 2 60.23 odd 4
1200.4.f.c.49.2 2 60.47 odd 4