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Properties

Label 960.4.a.e.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, 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("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} +9.00000 q^{9} +4.00000 q^{11} -54.0000 q^{13} +15.0000 q^{15} +114.000 q^{17} +44.0000 q^{19} -96.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -134.000 q^{29} +272.000 q^{31} -12.0000 q^{33} +98.0000 q^{37} +162.000 q^{39} -6.00000 q^{41} +12.0000 q^{43} -45.0000 q^{45} +200.000 q^{47} -343.000 q^{49} -342.000 q^{51} -654.000 q^{53} -20.0000 q^{55} -132.000 q^{57} +36.0000 q^{59} +442.000 q^{61} +270.000 q^{65} -188.000 q^{67} +288.000 q^{69} +632.000 q^{71} -390.000 q^{73} -75.0000 q^{75} -688.000 q^{79} +81.0000 q^{81} +1188.00 q^{83} -570.000 q^{85} +402.000 q^{87} -694.000 q^{89} -816.000 q^{93} -220.000 q^{95} -1726.00 q^{97} +36.0000 q^{99} +O(q^{100})\)

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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 0.109640 0.0548202 − 0.998496i \(-0.482541\pi\)
0.0548202 + 0.998496i \(0.482541\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 − 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 114.000 1.62642 0.813208 − 0.581974i \(-0.197719\pi\)
0.813208 + 0.581974i \(0.197719\pi\)
\(18\) 0 0
\(19\) 44.0000 0.531279 0.265639 − 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 −0.870321 −0.435161 − 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −134.000 −0.858041 −0.429020 − 0.903295i \(-0.641141\pi\)
−0.429020 + 0.903295i \(0.641141\pi\)
\(30\) 0 0
\(31\) 272.000 1.57589 0.787946 − 0.615745i \(-0.211145\pi\)
0.787946 + 0.615745i \(0.211145\pi\)
\(32\) 0 0
\(33\) −12.0000 −0.0633010
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 98.0000 0.435435 0.217718 − 0.976012i \(-0.430139\pi\)
0.217718 + 0.976012i \(0.430139\pi\)
\(38\) 0 0
\(39\) 162.000 0.665148
\(40\) 0 0
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) 0 0
\(43\) 12.0000 0.0425577 0.0212789 − 0.999774i \(-0.493226\pi\)
0.0212789 + 0.999774i \(0.493226\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 200.000 0.620702 0.310351 − 0.950622i \(-0.399553\pi\)
0.310351 + 0.950622i \(0.399553\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) −342.000 −0.939011
\(52\) 0 0
\(53\) −654.000 −1.69498 −0.847489 − 0.530813i \(-0.821887\pi\)
−0.847489 + 0.530813i \(0.821887\pi\)
\(54\) 0 0
\(55\) −20.0000 −0.0490327
\(56\) 0 0
\(57\) −132.000 −0.306734
\(58\) 0 0
\(59\) 36.0000 0.0794373 0.0397187 − 0.999211i \(-0.487354\pi\)
0.0397187 + 0.999211i \(0.487354\pi\)
\(60\) 0 0
\(61\) 442.000 0.927743 0.463871 − 0.885903i \(-0.346460\pi\)
0.463871 + 0.885903i \(0.346460\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 270.000 0.515221
\(66\) 0 0
\(67\) −188.000 −0.342804 −0.171402 − 0.985201i \(-0.554830\pi\)
−0.171402 + 0.985201i \(0.554830\pi\)
\(68\) 0 0
\(69\) 288.000 0.502480
\(70\) 0 0
\(71\) 632.000 1.05640 0.528201 − 0.849119i \(-0.322867\pi\)
0.528201 + 0.849119i \(0.322867\pi\)
\(72\) 0 0
\(73\) −390.000 −0.625288 −0.312644 − 0.949870i \(-0.601215\pi\)
−0.312644 + 0.949870i \(0.601215\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −688.000 −0.979823 −0.489912 − 0.871772i \(-0.662971\pi\)
−0.489912 + 0.871772i \(0.662971\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1188.00 1.57108 0.785542 − 0.618809i \(-0.212384\pi\)
0.785542 + 0.618809i \(0.212384\pi\)
\(84\) 0 0
\(85\) −570.000 −0.727355
\(86\) 0 0
\(87\) 402.000 0.495390
\(88\) 0 0
\(89\) −694.000 −0.826560 −0.413280 − 0.910604i \(-0.635617\pi\)
−0.413280 + 0.910604i \(0.635617\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −816.000 −0.909841
\(94\) 0 0
\(95\) −220.000 −0.237595
\(96\) 0 0
\(97\) −1726.00 −1.80669 −0.903344 − 0.428917i \(-0.858895\pi\)
−0.903344 + 0.428917i \(0.858895\pi\)
\(98\) 0 0
\(99\) 36.0000 0.0365468
\(100\) 0 0
\(101\) −1182.00 −1.16449 −0.582245 − 0.813014i \(-0.697825\pi\)
−0.582245 + 0.813014i \(0.697825\pi\)
\(102\) 0 0
\(103\) −1968.00 −1.88265 −0.941324 − 0.337503i \(-0.890418\pi\)
−0.941324 + 0.337503i \(0.890418\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 796.000 0.719180 0.359590 − 0.933110i \(-0.382917\pi\)
0.359590 + 0.933110i \(0.382917\pi\)
\(108\) 0 0
\(109\) −342.000 −0.300529 −0.150264 − 0.988646i \(-0.548013\pi\)
−0.150264 + 0.988646i \(0.548013\pi\)
\(110\) 0 0
\(111\) −294.000 −0.251399
\(112\) 0 0
\(113\) 114.000 0.0949046 0.0474523 − 0.998874i \(-0.484890\pi\)
0.0474523 + 0.998874i \(0.484890\pi\)
\(114\) 0 0
\(115\) 480.000 0.389219
\(116\) 0 0
\(117\) −486.000 −0.384023
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1315.00 −0.987979
\(122\) 0 0
\(123\) 18.0000 0.0131952
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −2344.00 −1.63777 −0.818883 − 0.573960i \(-0.805406\pi\)
−0.818883 + 0.573960i \(0.805406\pi\)
\(128\) 0 0
\(129\) −36.0000 −0.0245707
\(130\) 0 0
\(131\) −2164.00 −1.44328 −0.721640 − 0.692269i \(-0.756611\pi\)
−0.721640 + 0.692269i \(0.756611\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) −2822.00 −1.75985 −0.879926 − 0.475111i \(-0.842408\pi\)
−0.879926 + 0.475111i \(0.842408\pi\)
\(138\) 0 0
\(139\) 1972.00 1.20333 0.601665 − 0.798749i \(-0.294504\pi\)
0.601665 + 0.798749i \(0.294504\pi\)
\(140\) 0 0
\(141\) −600.000 −0.358363
\(142\) 0 0
\(143\) −216.000 −0.126313
\(144\) 0 0
\(145\) 670.000 0.383727
\(146\) 0 0
\(147\) 1029.00 0.577350
\(148\) 0 0
\(149\) 1394.00 0.766449 0.383225 − 0.923655i \(-0.374814\pi\)
0.383225 + 0.923655i \(0.374814\pi\)
\(150\) 0 0
\(151\) 2216.00 1.19427 0.597137 − 0.802139i \(-0.296305\pi\)
0.597137 + 0.802139i \(0.296305\pi\)
\(152\) 0 0
\(153\) 1026.00 0.542138
\(154\) 0 0
\(155\) −1360.00 −0.704760
\(156\) 0 0
\(157\) 954.000 0.484952 0.242476 − 0.970157i \(-0.422040\pi\)
0.242476 + 0.970157i \(0.422040\pi\)
\(158\) 0 0
\(159\) 1962.00 0.978596
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3404.00 −1.63572 −0.817858 − 0.575419i \(-0.804839\pi\)
−0.817858 + 0.575419i \(0.804839\pi\)
\(164\) 0 0
\(165\) 60.0000 0.0283091
\(166\) 0 0
\(167\) 832.000 0.385522 0.192761 − 0.981246i \(-0.438256\pi\)
0.192761 + 0.981246i \(0.438256\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 396.000 0.177093
\(172\) 0 0
\(173\) 362.000 0.159089 0.0795444 − 0.996831i \(-0.474653\pi\)
0.0795444 + 0.996831i \(0.474653\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −108.000 −0.0458631
\(178\) 0 0
\(179\) −3252.00 −1.35791 −0.678955 − 0.734180i \(-0.737567\pi\)
−0.678955 + 0.734180i \(0.737567\pi\)
\(180\) 0 0
\(181\) −3086.00 −1.26730 −0.633648 − 0.773621i \(-0.718443\pi\)
−0.633648 + 0.773621i \(0.718443\pi\)
\(182\) 0 0
\(183\) −1326.00 −0.535632
\(184\) 0 0
\(185\) −490.000 −0.194733
\(186\) 0 0
\(187\) 456.000 0.178321
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4080.00 1.54565 0.772823 − 0.634621i \(-0.218844\pi\)
0.772823 + 0.634621i \(0.218844\pi\)
\(192\) 0 0
\(193\) −2654.00 −0.989840 −0.494920 − 0.868939i \(-0.664803\pi\)
−0.494920 + 0.868939i \(0.664803\pi\)
\(194\) 0 0
\(195\) −810.000 −0.297463
\(196\) 0 0
\(197\) −1534.00 −0.554787 −0.277393 − 0.960756i \(-0.589471\pi\)
−0.277393 + 0.960756i \(0.589471\pi\)
\(198\) 0 0
\(199\) −4344.00 −1.54743 −0.773714 − 0.633536i \(-0.781603\pi\)
−0.773714 + 0.633536i \(0.781603\pi\)
\(200\) 0 0
\(201\) 564.000 0.197918
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 30.0000 0.0102209
\(206\) 0 0
\(207\) −864.000 −0.290107
\(208\) 0 0
\(209\) 176.000 0.0582496
\(210\) 0 0
\(211\) −1380.00 −0.450252 −0.225126 − 0.974330i \(-0.572279\pi\)
−0.225126 + 0.974330i \(0.572279\pi\)
\(212\) 0 0
\(213\) −1896.00 −0.609914
\(214\) 0 0
\(215\) −60.0000 −0.0190324
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1170.00 0.361010
\(220\) 0 0
\(221\) −6156.00 −1.87374
\(222\) 0 0
\(223\) 5224.00 1.56872 0.784361 − 0.620305i \(-0.212991\pi\)
0.784361 + 0.620305i \(0.212991\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 3364.00 0.983597 0.491799 − 0.870709i \(-0.336340\pi\)
0.491799 + 0.870709i \(0.336340\pi\)
\(228\) 0 0
\(229\) −3998.00 −1.15369 −0.576846 − 0.816853i \(-0.695717\pi\)
−0.576846 + 0.816853i \(0.695717\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3590.00 −1.00939 −0.504697 − 0.863297i \(-0.668396\pi\)
−0.504697 + 0.863297i \(0.668396\pi\)
\(234\) 0 0
\(235\) −1000.00 −0.277586
\(236\) 0 0
\(237\) 2064.00 0.565701
\(238\) 0 0
\(239\) 1104.00 0.298794 0.149397 − 0.988777i \(-0.452267\pi\)
0.149397 + 0.988777i \(0.452267\pi\)
\(240\) 0 0
\(241\) 1618.00 0.432467 0.216233 − 0.976342i \(-0.430623\pi\)
0.216233 + 0.976342i \(0.430623\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1715.00 0.447214
\(246\) 0 0
\(247\) −2376.00 −0.612070
\(248\) 0 0
\(249\) −3564.00 −0.907066
\(250\) 0 0
\(251\) 5780.00 1.45351 0.726754 − 0.686898i \(-0.241028\pi\)
0.726754 + 0.686898i \(0.241028\pi\)
\(252\) 0 0
\(253\) −384.000 −0.0954224
\(254\) 0 0
\(255\) 1710.00 0.419939
\(256\) 0 0
\(257\) 2594.00 0.629608 0.314804 − 0.949157i \(-0.398061\pi\)
0.314804 + 0.949157i \(0.398061\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1206.00 −0.286014
\(262\) 0 0
\(263\) −3696.00 −0.866559 −0.433280 − 0.901260i \(-0.642644\pi\)
−0.433280 + 0.901260i \(0.642644\pi\)
\(264\) 0 0
\(265\) 3270.00 0.758017
\(266\) 0 0
\(267\) 2082.00 0.477215
\(268\) 0 0
\(269\) 2250.00 0.509981 0.254991 − 0.966944i \(-0.417928\pi\)
0.254991 + 0.966944i \(0.417928\pi\)
\(270\) 0 0
\(271\) −2208.00 −0.494932 −0.247466 − 0.968897i \(-0.579598\pi\)
−0.247466 + 0.968897i \(0.579598\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 100.000 0.0219281
\(276\) 0 0
\(277\) 1682.00 0.364843 0.182422 − 0.983220i \(-0.441606\pi\)
0.182422 + 0.983220i \(0.441606\pi\)
\(278\) 0 0
\(279\) 2448.00 0.525297
\(280\) 0 0
\(281\) 7306.00 1.55103 0.775515 − 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −8164.00 −1.71484 −0.857419 − 0.514618i \(-0.827934\pi\)
−0.857419 + 0.514618i \(0.827934\pi\)
\(284\) 0 0
\(285\) 660.000 0.137176
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8083.00 1.64523
\(290\) 0 0
\(291\) 5178.00 1.04309
\(292\) 0 0
\(293\) 514.000 0.102485 0.0512427 − 0.998686i \(-0.483682\pi\)
0.0512427 + 0.998686i \(0.483682\pi\)
\(294\) 0 0
\(295\) −180.000 −0.0355254
\(296\) 0 0
\(297\) −108.000 −0.0211003
\(298\) 0 0
\(299\) 5184.00 1.00267
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3546.00 0.672318
\(304\) 0 0
\(305\) −2210.00 −0.414899
\(306\) 0 0
\(307\) −2476.00 −0.460302 −0.230151 − 0.973155i \(-0.573922\pi\)
−0.230151 + 0.973155i \(0.573922\pi\)
\(308\) 0 0
\(309\) 5904.00 1.08695
\(310\) 0 0
\(311\) −2296.00 −0.418631 −0.209315 − 0.977848i \(-0.567124\pi\)
−0.209315 + 0.977848i \(0.567124\pi\)
\(312\) 0 0
\(313\) −9878.00 −1.78383 −0.891913 − 0.452207i \(-0.850637\pi\)
−0.891913 + 0.452207i \(0.850637\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2138.00 0.378808 0.189404 − 0.981899i \(-0.439344\pi\)
0.189404 + 0.981899i \(0.439344\pi\)
\(318\) 0 0
\(319\) −536.000 −0.0940760
\(320\) 0 0
\(321\) −2388.00 −0.415219
\(322\) 0 0
\(323\) 5016.00 0.864080
\(324\) 0 0
\(325\) −1350.00 −0.230414
\(326\) 0 0
\(327\) 1026.00 0.173510
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6460.00 −1.07273 −0.536365 − 0.843986i \(-0.680203\pi\)
−0.536365 + 0.843986i \(0.680203\pi\)
\(332\) 0 0
\(333\) 882.000 0.145145
\(334\) 0 0
\(335\) 940.000 0.153307
\(336\) 0 0
\(337\) 626.000 0.101188 0.0505941 − 0.998719i \(-0.483889\pi\)
0.0505941 + 0.998719i \(0.483889\pi\)
\(338\) 0 0
\(339\) −342.000 −0.0547932
\(340\) 0 0
\(341\) 1088.00 0.172782
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1440.00 −0.224716
\(346\) 0 0
\(347\) 876.000 0.135522 0.0677610 − 0.997702i \(-0.478414\pi\)
0.0677610 + 0.997702i \(0.478414\pi\)
\(348\) 0 0
\(349\) 9850.00 1.51077 0.755385 − 0.655282i \(-0.227450\pi\)
0.755385 + 0.655282i \(0.227450\pi\)
\(350\) 0 0
\(351\) 1458.00 0.221716
\(352\) 0 0
\(353\) −8894.00 −1.34102 −0.670510 − 0.741901i \(-0.733925\pi\)
−0.670510 + 0.741901i \(0.733925\pi\)
\(354\) 0 0
\(355\) −3160.00 −0.472438
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1464.00 −0.215228 −0.107614 − 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) 3945.00 0.570410
\(364\) 0 0
\(365\) 1950.00 0.279637
\(366\) 0 0
\(367\) 7016.00 0.997908 0.498954 − 0.866628i \(-0.333718\pi\)
0.498954 + 0.866628i \(0.333718\pi\)
\(368\) 0 0
\(369\) −54.0000 −0.00761823
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1010.00 0.140203 0.0701016 − 0.997540i \(-0.477668\pi\)
0.0701016 + 0.997540i \(0.477668\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 7236.00 0.988522
\(378\) 0 0
\(379\) 4900.00 0.664106 0.332053 − 0.943261i \(-0.392259\pi\)
0.332053 + 0.943261i \(0.392259\pi\)
\(380\) 0 0
\(381\) 7032.00 0.945565
\(382\) 0 0
\(383\) −7800.00 −1.04063 −0.520315 − 0.853974i \(-0.674186\pi\)
−0.520315 + 0.853974i \(0.674186\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 108.000 0.0141859
\(388\) 0 0
\(389\) 12258.0 1.59770 0.798850 − 0.601530i \(-0.205442\pi\)
0.798850 + 0.601530i \(0.205442\pi\)
\(390\) 0 0
\(391\) −10944.0 −1.41550
\(392\) 0 0
\(393\) 6492.00 0.833278
\(394\) 0 0
\(395\) 3440.00 0.438190
\(396\) 0 0
\(397\) −5558.00 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1970.00 0.245329 0.122665 − 0.992448i \(-0.460856\pi\)
0.122665 + 0.992448i \(0.460856\pi\)
\(402\) 0 0
\(403\) −14688.0 −1.81554
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 392.000 0.0477413
\(408\) 0 0
\(409\) 15626.0 1.88913 0.944567 − 0.328318i \(-0.106482\pi\)
0.944567 + 0.328318i \(0.106482\pi\)
\(410\) 0 0
\(411\) 8466.00 1.01605
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5940.00 −0.702610
\(416\) 0 0
\(417\) −5916.00 −0.694743
\(418\) 0 0
\(419\) −5412.00 −0.631011 −0.315505 − 0.948924i \(-0.602174\pi\)
−0.315505 + 0.948924i \(0.602174\pi\)
\(420\) 0 0
\(421\) 10690.0 1.23753 0.618763 − 0.785577i \(-0.287634\pi\)
0.618763 + 0.785577i \(0.287634\pi\)
\(422\) 0 0
\(423\) 1800.00 0.206901
\(424\) 0 0
\(425\) 2850.00 0.325283
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 648.000 0.0729271
\(430\) 0 0
\(431\) 14048.0 1.57000 0.784998 − 0.619498i \(-0.212664\pi\)
0.784998 + 0.619498i \(0.212664\pi\)
\(432\) 0 0
\(433\) 17778.0 1.97311 0.986554 − 0.163433i \(-0.0522567\pi\)
0.986554 + 0.163433i \(0.0522567\pi\)
\(434\) 0 0
\(435\) −2010.00 −0.221545
\(436\) 0 0
\(437\) −4224.00 −0.462383
\(438\) 0 0
\(439\) −7240.00 −0.787122 −0.393561 − 0.919299i \(-0.628757\pi\)
−0.393561 + 0.919299i \(0.628757\pi\)
\(440\) 0 0
\(441\) −3087.00 −0.333333
\(442\) 0 0
\(443\) 11740.0 1.25911 0.629553 − 0.776957i \(-0.283238\pi\)
0.629553 + 0.776957i \(0.283238\pi\)
\(444\) 0 0
\(445\) 3470.00 0.369649
\(446\) 0 0
\(447\) −4182.00 −0.442510
\(448\) 0 0
\(449\) 15234.0 1.60120 0.800598 − 0.599202i \(-0.204515\pi\)
0.800598 + 0.599202i \(0.204515\pi\)
\(450\) 0 0
\(451\) −24.0000 −0.00250580
\(452\) 0 0
\(453\) −6648.00 −0.689515
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3866.00 0.395720 0.197860 − 0.980230i \(-0.436601\pi\)
0.197860 + 0.980230i \(0.436601\pi\)
\(458\) 0 0
\(459\) −3078.00 −0.313004
\(460\) 0 0
\(461\) 1706.00 0.172356 0.0861782 − 0.996280i \(-0.472535\pi\)
0.0861782 + 0.996280i \(0.472535\pi\)
\(462\) 0 0
\(463\) −3944.00 −0.395882 −0.197941 − 0.980214i \(-0.563425\pi\)
−0.197941 + 0.980214i \(0.563425\pi\)
\(464\) 0 0
\(465\) 4080.00 0.406893
\(466\) 0 0
\(467\) −9452.00 −0.936588 −0.468294 − 0.883573i \(-0.655131\pi\)
−0.468294 + 0.883573i \(0.655131\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2862.00 −0.279987
\(472\) 0 0
\(473\) 48.0000 0.00466605
\(474\) 0 0
\(475\) 1100.00 0.106256
\(476\) 0 0
\(477\) −5886.00 −0.564993
\(478\) 0 0
\(479\) 12544.0 1.19656 0.598278 − 0.801289i \(-0.295852\pi\)
0.598278 + 0.801289i \(0.295852\pi\)
\(480\) 0 0
\(481\) −5292.00 −0.501652
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8630.00 0.807975
\(486\) 0 0
\(487\) −7936.00 −0.738428 −0.369214 − 0.929344i \(-0.620373\pi\)
−0.369214 + 0.929344i \(0.620373\pi\)
\(488\) 0 0
\(489\) 10212.0 0.944382
\(490\) 0 0
\(491\) −8412.00 −0.773174 −0.386587 − 0.922253i \(-0.626346\pi\)
−0.386587 + 0.922253i \(0.626346\pi\)
\(492\) 0 0
\(493\) −15276.0 −1.39553
\(494\) 0 0
\(495\) −180.000 −0.0163442
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15092.0 −1.35393 −0.676965 − 0.736016i \(-0.736705\pi\)
−0.676965 + 0.736016i \(0.736705\pi\)
\(500\) 0 0
\(501\) −2496.00 −0.222581
\(502\) 0 0
\(503\) −6112.00 −0.541790 −0.270895 − 0.962609i \(-0.587320\pi\)
−0.270895 + 0.962609i \(0.587320\pi\)
\(504\) 0 0
\(505\) 5910.00 0.520775
\(506\) 0 0
\(507\) −2157.00 −0.188946
\(508\) 0 0
\(509\) −2534.00 −0.220663 −0.110332 − 0.993895i \(-0.535191\pi\)
−0.110332 + 0.993895i \(0.535191\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1188.00 −0.102245
\(514\) 0 0
\(515\) 9840.00 0.841946
\(516\) 0 0
\(517\) 800.000 0.0680541
\(518\) 0 0
\(519\) −1086.00 −0.0918499
\(520\) 0 0
\(521\) −9894.00 −0.831985 −0.415992 − 0.909368i \(-0.636566\pi\)
−0.415992 + 0.909368i \(0.636566\pi\)
\(522\) 0 0
\(523\) 16172.0 1.35211 0.676054 − 0.736852i \(-0.263689\pi\)
0.676054 + 0.736852i \(0.263689\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31008.0 2.56305
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) 324.000 0.0264791
\(532\) 0 0
\(533\) 324.000 0.0263302
\(534\) 0 0
\(535\) −3980.00 −0.321627
\(536\) 0 0
\(537\) 9756.00 0.783990
\(538\) 0 0
\(539\) −1372.00 −0.109640
\(540\) 0 0
\(541\) 6138.00 0.487788 0.243894 − 0.969802i \(-0.421575\pi\)
0.243894 + 0.969802i \(0.421575\pi\)
\(542\) 0 0
\(543\) 9258.00 0.731674
\(544\) 0 0
\(545\) 1710.00 0.134401
\(546\) 0 0
\(547\) −21852.0 −1.70809 −0.854044 − 0.520201i \(-0.825857\pi\)
−0.854044 + 0.520201i \(0.825857\pi\)
\(548\) 0 0
\(549\) 3978.00 0.309248
\(550\) 0 0
\(551\) −5896.00 −0.455859
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1470.00 0.112429
\(556\) 0 0
\(557\) 1962.00 0.149251 0.0746253 − 0.997212i \(-0.476224\pi\)
0.0746253 + 0.997212i \(0.476224\pi\)
\(558\) 0 0
\(559\) −648.000 −0.0490295
\(560\) 0 0
\(561\) −1368.00 −0.102954
\(562\) 0 0
\(563\) −10876.0 −0.814154 −0.407077 − 0.913394i \(-0.633452\pi\)
−0.407077 + 0.913394i \(0.633452\pi\)
\(564\) 0 0
\(565\) −570.000 −0.0424426
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5610.00 0.413328 0.206664 − 0.978412i \(-0.433739\pi\)
0.206664 + 0.978412i \(0.433739\pi\)
\(570\) 0 0
\(571\) 5076.00 0.372021 0.186010 − 0.982548i \(-0.440444\pi\)
0.186010 + 0.982548i \(0.440444\pi\)
\(572\) 0 0
\(573\) −12240.0 −0.892379
\(574\) 0 0
\(575\) −2400.00 −0.174064
\(576\) 0 0
\(577\) −6526.00 −0.470851 −0.235425 − 0.971892i \(-0.575648\pi\)
−0.235425 + 0.971892i \(0.575648\pi\)
\(578\) 0 0
\(579\) 7962.00 0.571484
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2616.00 −0.185838
\(584\) 0 0
\(585\) 2430.00 0.171740
\(586\) 0 0
\(587\) 2332.00 0.163973 0.0819863 − 0.996633i \(-0.473874\pi\)
0.0819863 + 0.996633i \(0.473874\pi\)
\(588\) 0 0
\(589\) 11968.0 0.837237
\(590\) 0 0
\(591\) 4602.00 0.320306
\(592\) 0 0
\(593\) −9582.00 −0.663551 −0.331775 − 0.943358i \(-0.607648\pi\)
−0.331775 + 0.943358i \(0.607648\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13032.0 0.893407
\(598\) 0 0
\(599\) 17624.0 1.20217 0.601083 − 0.799187i \(-0.294736\pi\)
0.601083 + 0.799187i \(0.294736\pi\)
\(600\) 0 0
\(601\) −21238.0 −1.44146 −0.720729 − 0.693217i \(-0.756193\pi\)
−0.720729 + 0.693217i \(0.756193\pi\)
\(602\) 0 0
\(603\) −1692.00 −0.114268
\(604\) 0 0
\(605\) 6575.00 0.441838
\(606\) 0 0
\(607\) −13000.0 −0.869281 −0.434641 − 0.900604i \(-0.643125\pi\)
−0.434641 + 0.900604i \(0.643125\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10800.0 −0.715092
\(612\) 0 0
\(613\) −9214.00 −0.607096 −0.303548 − 0.952816i \(-0.598171\pi\)
−0.303548 + 0.952816i \(0.598171\pi\)
\(614\) 0 0
\(615\) −90.0000 −0.00590106
\(616\) 0 0
\(617\) 4474.00 0.291923 0.145961 − 0.989290i \(-0.453372\pi\)
0.145961 + 0.989290i \(0.453372\pi\)
\(618\) 0 0
\(619\) −12556.0 −0.815296 −0.407648 − 0.913139i \(-0.633651\pi\)
−0.407648 + 0.913139i \(0.633651\pi\)
\(620\) 0 0
\(621\) 2592.00 0.167493
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −528.000 −0.0336304
\(628\) 0 0
\(629\) 11172.0 0.708198
\(630\) 0 0
\(631\) −26936.0 −1.69937 −0.849687 − 0.527287i \(-0.823209\pi\)
−0.849687 + 0.527287i \(0.823209\pi\)
\(632\) 0 0
\(633\) 4140.00 0.259953
\(634\) 0 0
\(635\) 11720.0 0.732432
\(636\) 0 0
\(637\) 18522.0 1.15207
\(638\) 0 0
\(639\) 5688.00 0.352134
\(640\) 0 0
\(641\) −19134.0 −1.17901 −0.589507 − 0.807764i \(-0.700678\pi\)
−0.589507 + 0.807764i \(0.700678\pi\)
\(642\) 0 0
\(643\) 12436.0 0.762718 0.381359 − 0.924427i \(-0.375456\pi\)
0.381359 + 0.924427i \(0.375456\pi\)
\(644\) 0 0
\(645\) 180.000 0.0109884
\(646\) 0 0
\(647\) 2784.00 0.169166 0.0845829 − 0.996416i \(-0.473044\pi\)
0.0845829 + 0.996416i \(0.473044\pi\)
\(648\) 0 0
\(649\) 144.000 0.00870954
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7318.00 −0.438554 −0.219277 − 0.975663i \(-0.570370\pi\)
−0.219277 + 0.975663i \(0.570370\pi\)
\(654\) 0 0
\(655\) 10820.0 0.645454
\(656\) 0 0
\(657\) −3510.00 −0.208429
\(658\) 0 0
\(659\) 8108.00 0.479276 0.239638 − 0.970862i \(-0.422971\pi\)
0.239638 + 0.970862i \(0.422971\pi\)
\(660\) 0 0
\(661\) −1230.00 −0.0723774 −0.0361887 − 0.999345i \(-0.511522\pi\)
−0.0361887 + 0.999345i \(0.511522\pi\)
\(662\) 0 0
\(663\) 18468.0 1.08181
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12864.0 0.746771
\(668\) 0 0
\(669\) −15672.0 −0.905702
\(670\) 0 0
\(671\) 1768.00 0.101718
\(672\) 0 0
\(673\) −14078.0 −0.806340 −0.403170 − 0.915125i \(-0.632092\pi\)
−0.403170 + 0.915125i \(0.632092\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) −25246.0 −1.43321 −0.716605 − 0.697480i \(-0.754305\pi\)
−0.716605 + 0.697480i \(0.754305\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −10092.0 −0.567880
\(682\) 0 0
\(683\) 24332.0 1.36316 0.681580 − 0.731744i \(-0.261293\pi\)
0.681580 + 0.731744i \(0.261293\pi\)
\(684\) 0 0
\(685\) 14110.0 0.787030
\(686\) 0 0
\(687\) 11994.0 0.666084
\(688\) 0 0
\(689\) 35316.0 1.95273
\(690\) 0 0
\(691\) 19036.0 1.04799 0.523997 − 0.851720i \(-0.324440\pi\)
0.523997 + 0.851720i \(0.324440\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9860.00 −0.538145
\(696\) 0 0
\(697\) −684.000 −0.0371712
\(698\) 0 0
\(699\) 10770.0 0.582774
\(700\) 0 0
\(701\) −28806.0 −1.55205 −0.776025 − 0.630702i \(-0.782767\pi\)
−0.776025 + 0.630702i \(0.782767\pi\)
\(702\) 0 0
\(703\) 4312.00 0.231337
\(704\) 0 0
\(705\) 3000.00 0.160265
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 25090.0 1.32902 0.664510 − 0.747280i \(-0.268640\pi\)
0.664510 + 0.747280i \(0.268640\pi\)
\(710\) 0 0
\(711\) −6192.00 −0.326608
\(712\) 0 0
\(713\) −26112.0 −1.37153
\(714\) 0 0
\(715\) 1080.00 0.0564891
\(716\) 0 0
\(717\) −3312.00 −0.172509
\(718\) 0 0
\(719\) −36432.0 −1.88969 −0.944843 − 0.327523i \(-0.893786\pi\)
−0.944843 + 0.327523i \(0.893786\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −4854.00 −0.249685
\(724\) 0 0
\(725\) −3350.00 −0.171608
\(726\) 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
\(731\) 1368.00 0.0692166
\(732\) 0 0
\(733\) −28102.0 −1.41606 −0.708029 − 0.706183i \(-0.750416\pi\)
−0.708029 + 0.706183i \(0.750416\pi\)
\(734\) 0 0
\(735\) −5145.00 −0.258199
\(736\) 0 0
\(737\) −752.000 −0.0375852
\(738\) 0 0
\(739\) 764.000 0.0380300 0.0190150 − 0.999819i \(-0.493947\pi\)
0.0190150 + 0.999819i \(0.493947\pi\)
\(740\) 0 0
\(741\) 7128.00 0.353379
\(742\) 0 0
\(743\) −6256.00 −0.308897 −0.154448 − 0.988001i \(-0.549360\pi\)
−0.154448 + 0.988001i \(0.549360\pi\)
\(744\) 0 0
\(745\) −6970.00 −0.342766
\(746\) 0 0
\(747\) 10692.0 0.523695
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1184.00 −0.0575297 −0.0287648 − 0.999586i \(-0.509157\pi\)
−0.0287648 + 0.999586i \(0.509157\pi\)
\(752\) 0 0
\(753\) −17340.0 −0.839183
\(754\) 0 0
\(755\) −11080.0 −0.534096
\(756\) 0 0
\(757\) −26446.0 −1.26974 −0.634872 − 0.772617i \(-0.718947\pi\)
−0.634872 + 0.772617i \(0.718947\pi\)
\(758\) 0 0
\(759\) 1152.00 0.0550922
\(760\) 0 0
\(761\) 36778.0 1.75191 0.875954 − 0.482395i \(-0.160233\pi\)
0.875954 + 0.482395i \(0.160233\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5130.00 −0.242452
\(766\) 0 0
\(767\) −1944.00 −0.0915173
\(768\) 0 0
\(769\) −10302.0 −0.483094 −0.241547 − 0.970389i \(-0.577655\pi\)
−0.241547 + 0.970389i \(0.577655\pi\)
\(770\) 0 0
\(771\) −7782.00 −0.363504
\(772\) 0 0
\(773\) 4674.00 0.217480 0.108740 − 0.994070i \(-0.465318\pi\)
0.108740 + 0.994070i \(0.465318\pi\)
\(774\) 0 0
\(775\) 6800.00 0.315178
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −264.000 −0.0121422
\(780\) 0 0
\(781\) 2528.00 0.115825
\(782\) 0 0
\(783\) 3618.00 0.165130
\(784\) 0 0
\(785\) −4770.00 −0.216877
\(786\) 0 0
\(787\) −23084.0 −1.04556 −0.522780 − 0.852468i \(-0.675105\pi\)
−0.522780 + 0.852468i \(0.675105\pi\)
\(788\) 0 0
\(789\) 11088.0 0.500308
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −23868.0 −1.06882
\(794\) 0 0
\(795\) −9810.00 −0.437641
\(796\) 0 0
\(797\) −10694.0 −0.475283 −0.237642 − 0.971353i \(-0.576374\pi\)
−0.237642 + 0.971353i \(0.576374\pi\)
\(798\) 0 0
\(799\) 22800.0 1.00952
\(800\) 0 0
\(801\) −6246.00 −0.275520
\(802\) 0 0
\(803\) −1560.00 −0.0685569
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6750.00 −0.294438
\(808\) 0 0
\(809\) 9594.00 0.416943 0.208472 − 0.978028i \(-0.433151\pi\)
0.208472 + 0.978028i \(0.433151\pi\)
\(810\) 0 0
\(811\) 10244.0 0.443546 0.221773 − 0.975098i \(-0.428816\pi\)
0.221773 + 0.975098i \(0.428816\pi\)
\(812\) 0 0
\(813\) 6624.00 0.285749
\(814\) 0 0
\(815\) 17020.0 0.731515
\(816\) 0 0
\(817\) 528.000 0.0226100
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1390.00 −0.0590881 −0.0295441 − 0.999563i \(-0.509406\pi\)
−0.0295441 + 0.999563i \(0.509406\pi\)
\(822\) 0 0
\(823\) 8448.00 0.357811 0.178906 − 0.983866i \(-0.442744\pi\)
0.178906 + 0.983866i \(0.442744\pi\)
\(824\) 0 0
\(825\) −300.000 −0.0126602
\(826\) 0 0
\(827\) 41484.0 1.74430 0.872152 − 0.489234i \(-0.162724\pi\)
0.872152 + 0.489234i \(0.162724\pi\)
\(828\) 0 0
\(829\) 31610.0 1.32432 0.662160 − 0.749363i \(-0.269640\pi\)
0.662160 + 0.749363i \(0.269640\pi\)
\(830\) 0 0
\(831\) −5046.00 −0.210642
\(832\) 0 0
\(833\) −39102.0 −1.62642
\(834\) 0 0
\(835\) −4160.00 −0.172410
\(836\) 0 0
\(837\) −7344.00 −0.303280
\(838\) 0 0
\(839\) −38264.0 −1.57452 −0.787259 − 0.616623i \(-0.788500\pi\)
−0.787259 + 0.616623i \(0.788500\pi\)
\(840\) 0 0
\(841\) −6433.00 −0.263766
\(842\) 0 0
\(843\) −21918.0 −0.895488
\(844\) 0 0
\(845\) −3595.00 −0.146357
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 24492.0 0.990063
\(850\) 0 0
\(851\) −9408.00 −0.378968
\(852\) 0 0
\(853\) −30350.0 −1.21825 −0.609123 − 0.793076i \(-0.708479\pi\)
−0.609123 + 0.793076i \(0.708479\pi\)
\(854\) 0 0
\(855\) −1980.00 −0.0791983
\(856\) 0 0
\(857\) −12566.0 −0.500871 −0.250435 − 0.968133i \(-0.580574\pi\)
−0.250435 + 0.968133i \(0.580574\pi\)
\(858\) 0 0
\(859\) 11812.0 0.469174 0.234587 − 0.972095i \(-0.424626\pi\)
0.234587 + 0.972095i \(0.424626\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31496.0 1.24234 0.621168 − 0.783677i \(-0.286658\pi\)
0.621168 + 0.783677i \(0.286658\pi\)
\(864\) 0 0
\(865\) −1810.00 −0.0711466
\(866\) 0 0
\(867\) −24249.0 −0.949872
\(868\) 0 0
\(869\) −2752.00 −0.107428
\(870\) 0 0
\(871\) 10152.0 0.394934
\(872\) 0 0
\(873\) −15534.0 −0.602229
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7414.00 −0.285465 −0.142733 − 0.989761i \(-0.545589\pi\)
−0.142733 + 0.989761i \(0.545589\pi\)
\(878\) 0 0
\(879\) −1542.00 −0.0591699
\(880\) 0 0
\(881\) −22190.0 −0.848581 −0.424291 − 0.905526i \(-0.639477\pi\)
−0.424291 + 0.905526i \(0.639477\pi\)
\(882\) 0 0
\(883\) −10172.0 −0.387673 −0.193836 − 0.981034i \(-0.562093\pi\)
−0.193836 + 0.981034i \(0.562093\pi\)
\(884\) 0 0
\(885\) 540.000 0.0205106
\(886\) 0 0
\(887\) 20784.0 0.786763 0.393381 − 0.919375i \(-0.371305\pi\)
0.393381 + 0.919375i \(0.371305\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 324.000 0.0121823
\(892\) 0 0
\(893\) 8800.00 0.329766
\(894\) 0 0
\(895\) 16260.0 0.607276
\(896\) 0 0
\(897\) −15552.0 −0.578892
\(898\) 0 0
\(899\) −36448.0 −1.35218
\(900\) 0 0
\(901\) −74556.0 −2.75674
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15430.0 0.566752
\(906\) 0 0
\(907\) −7652.00 −0.280133 −0.140066 − 0.990142i \(-0.544732\pi\)
−0.140066 + 0.990142i \(0.544732\pi\)
\(908\) 0 0
\(909\) −10638.0 −0.388163
\(910\) 0 0
\(911\) −19296.0 −0.701762 −0.350881 − 0.936420i \(-0.614118\pi\)
−0.350881 + 0.936420i \(0.614118\pi\)
\(912\) 0 0
\(913\) 4752.00 0.172254
\(914\) 0 0
\(915\) 6630.00 0.239542
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 35896.0 1.28847 0.644233 − 0.764830i \(-0.277177\pi\)
0.644233 + 0.764830i \(0.277177\pi\)
\(920\) 0 0
\(921\) 7428.00 0.265756
\(922\) 0 0
\(923\) −34128.0 −1.21705
\(924\) 0 0
\(925\) 2450.00 0.0870870
\(926\) 0 0
\(927\) −17712.0 −0.627550
\(928\) 0 0
\(929\) −16350.0 −0.577423 −0.288712 − 0.957416i \(-0.593227\pi\)
−0.288712 + 0.957416i \(0.593227\pi\)
\(930\) 0 0
\(931\) −15092.0 −0.531279
\(932\) 0 0
\(933\) 6888.00 0.241697
\(934\) 0 0
\(935\) −2280.00 −0.0797476
\(936\) 0 0
\(937\) −19686.0 −0.686354 −0.343177 − 0.939271i \(-0.611503\pi\)
−0.343177 + 0.939271i \(0.611503\pi\)
\(938\) 0 0
\(939\) 29634.0 1.02989
\(940\) 0 0
\(941\) −56246.0 −1.94853 −0.974265 − 0.225405i \(-0.927630\pi\)
−0.974265 + 0.225405i \(0.927630\pi\)
\(942\) 0 0
\(943\) 576.000 0.0198909
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11436.0 −0.392418 −0.196209 − 0.980562i \(-0.562863\pi\)
−0.196209 + 0.980562i \(0.562863\pi\)
\(948\) 0 0
\(949\) 21060.0 0.720376
\(950\) 0 0
\(951\) −6414.00 −0.218705
\(952\) 0 0
\(953\) −22582.0 −0.767579 −0.383789 − 0.923421i \(-0.625381\pi\)
−0.383789 + 0.923421i \(0.625381\pi\)
\(954\) 0 0
\(955\) −20400.0 −0.691234
\(956\) 0 0
\(957\) 1608.00 0.0543148
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 44193.0 1.48343
\(962\) 0 0
\(963\) 7164.00 0.239727
\(964\) 0 0
\(965\) 13270.0 0.442670
\(966\) 0 0
\(967\) 2112.00 0.0702351 0.0351175 − 0.999383i \(-0.488819\pi\)
0.0351175 + 0.999383i \(0.488819\pi\)
\(968\) 0 0
\(969\) −15048.0 −0.498877
\(970\) 0 0
\(971\) −47964.0 −1.58521 −0.792605 − 0.609736i \(-0.791275\pi\)
−0.792605 + 0.609736i \(0.791275\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4050.00 0.133030
\(976\) 0 0
\(977\) −10510.0 −0.344160 −0.172080 − 0.985083i \(-0.555049\pi\)
−0.172080 + 0.985083i \(0.555049\pi\)
\(978\) 0 0
\(979\) −2776.00 −0.0906245
\(980\) 0 0
\(981\) −3078.00 −0.100176
\(982\) 0 0
\(983\) −11488.0 −0.372747 −0.186373 − 0.982479i \(-0.559673\pi\)
−0.186373 + 0.982479i \(0.559673\pi\)
\(984\) 0 0
\(985\) 7670.00 0.248108
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1152.00 −0.0370389
\(990\) 0 0
\(991\) 23120.0 0.741101 0.370550 − 0.928812i \(-0.379169\pi\)
0.370550 + 0.928812i \(0.379169\pi\)
\(992\) 0 0
\(993\) 19380.0 0.619341
\(994\) 0 0
\(995\) 21720.0 0.692030
\(996\) 0 0
\(997\) −30078.0 −0.955446 −0.477723 − 0.878510i \(-0.658538\pi\)
−0.477723 + 0.878510i \(0.658538\pi\)
\(998\) 0 0
\(999\) −2646.00 −0.0837995
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 960.4.a.e.1.1 1
4.3 odd 2 960.4.a.x.1.1 1
8.3 odd 2 120.4.a.d.1.1 ✓ 1
8.5 even 2 240.4.a.k.1.1 1
24.5 odd 2 720.4.a.i.1.1 1
24.11 even 2 360.4.a.c.1.1 1
40.3 even 4 600.4.f.d.49.1 2
40.13 odd 4 1200.4.f.l.49.2 2
40.19 odd 2 600.4.a.m.1.1 1
40.27 even 4 600.4.f.d.49.2 2
40.29 even 2 1200.4.a.j.1.1 1
40.37 odd 4 1200.4.f.l.49.1 2
120.59 even 2 1800.4.a.s.1.1 1
120.83 odd 4 1800.4.f.m.649.2 2
120.107 odd 4 1800.4.f.m.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.d.1.1 ✓ 1 8.3 odd 2
240.4.a.k.1.1 1 8.5 even 2
360.4.a.c.1.1 1 24.11 even 2
600.4.a.m.1.1 1 40.19 odd 2
600.4.f.d.49.1 2 40.3 even 4
600.4.f.d.49.2 2 40.27 even 4
720.4.a.i.1.1 1 24.5 odd 2
960.4.a.e.1.1 1 1.1 even 1 trivial
960.4.a.x.1.1 1 4.3 odd 2
1200.4.a.j.1.1 1 40.29 even 2
1200.4.f.l.49.1 2 40.37 odd 4
1200.4.f.l.49.2 2 40.13 odd 4
1800.4.a.s.1.1 1 120.59 even 2
1800.4.f.m.649.1 2 120.107 odd 4
1800.4.f.m.649.2 2 120.83 odd 4