↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

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

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

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

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
\(5\) 0 0
\(6\) 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 \(-0.517459\pi\)
−0.0548202 + 0.998496i \(0.517459\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 − 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) 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
\(19\) −44.0000 −0.531279 −0.265639 − 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 96.0000 0.870321 0.435161 − 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 134.000 0.858041 0.429020 − 0.903295i \(-0.358859\pi\)
0.429020 + 0.903295i \(0.358859\pi\)
\(30\) 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
\(45\) 0 0
\(46\) 0 0
\(47\) −200.000 −0.620702 −0.310351 − 0.950622i \(-0.600447\pi\)
−0.310351 + 0.950622i \(0.600447\pi\)
\(48\) 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
\(55\) 0 0
\(56\) 0 0
\(57\) 132.000 0.306734
\(58\) 0 0
\(59\) −36.0000 −0.0794373 −0.0397187 − 0.999211i \(-0.512646\pi\)
−0.0397187 + 0.999211i \(0.512646\pi\)
\(60\) 0 0
\(61\) −442.000 −0.927743 −0.463871 − 0.885903i \(-0.653540\pi\)
−0.463871 + 0.885903i \(0.653540\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 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 \(-0.398785\pi\)
0.312644 + 0.949870i \(0.398785\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 1726.00 1.80669 0.903344 − 0.428917i \(-0.141105\pi\)
0.903344 + 0.428917i \(0.141105\pi\)
\(98\) 0 0
\(99\) −36.0000 −0.0365468
\(100\) 0 0
\(101\) 1182.00 1.16449 0.582245 − 0.813014i \(-0.302175\pi\)
0.582245 + 0.813014i \(0.302175\pi\)
\(102\) 0 0
\(103\) 1968.00 1.88265 0.941324 − 0.337503i \(-0.109582\pi\)
0.941324 + 0.337503i \(0.109582\pi\)
\(104\) 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 \(-0.451987\pi\)
0.150264 + 0.988646i \(0.451987\pi\)
\(110\) 0 0
\(111\) −294.000 −0.251399
\(112\) 0 0
\(113\) −114.000 −0.0949046 −0.0474523 − 0.998874i \(-0.515110\pi\)
−0.0474523 + 0.998874i \(0.515110\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 2344.00 1.63777 0.818883 − 0.573960i \(-0.194594\pi\)
0.818883 + 0.573960i \(0.194594\pi\)
\(128\) 0 0
\(129\) −36.0000 −0.0245707
\(130\) 0 0
\(131\) 2164.00 1.44328 0.721640 − 0.692269i \(-0.243389\pi\)
0.721640 + 0.692269i \(0.243389\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2822.00 1.75985 0.879926 − 0.475111i \(-0.157592\pi\)
0.879926 + 0.475111i \(0.157592\pi\)
\(138\) 0 0
\(139\) −1972.00 −1.20333 −0.601665 − 0.798749i \(-0.705496\pi\)
−0.601665 + 0.798749i \(0.705496\pi\)
\(140\) 0 0
\(141\) 600.000 0.358363
\(142\) 0 0
\(143\) 216.000 0.126313
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1029.00 0.577350
\(148\) 0 0
\(149\) −1394.00 −0.766449 −0.383225 − 0.923655i \(-0.625186\pi\)
−0.383225 + 0.923655i \(0.625186\pi\)
\(150\) 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
\(155\) 0 0
\(156\) 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
\(165\) 0 0
\(166\) 0 0
\(167\) −832.000 −0.385522 −0.192761 − 0.981246i \(-0.561744\pi\)
−0.192761 + 0.981246i \(0.561744\pi\)
\(168\) 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 \(-0.262433\pi\)
0.678955 + 0.734180i \(0.262433\pi\)
\(180\) 0 0
\(181\) 3086.00 1.26730 0.633648 − 0.773621i \(-0.281557\pi\)
0.633648 + 0.773621i \(0.281557\pi\)
\(182\) 0 0
\(183\) 1326.00 0.535632
\(184\) 0 0
\(185\) 0 0
\(186\) 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 \(-0.335197\pi\)
0.494920 + 0.868939i \(0.335197\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 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
\(205\) 0 0
\(206\) 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 \(-0.427721\pi\)
0.225126 + 0.974330i \(0.427721\pi\)
\(212\) 0 0
\(213\) −1896.00 −0.609914
\(214\) 0 0
\(215\) 0 0
\(216\) 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 \(-0.787009\pi\)
−0.784361 + 0.620305i \(0.787009\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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 \(-0.304283\pi\)
0.576846 + 0.816853i \(0.304283\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3590.00 1.00939 0.504697 − 0.863297i \(-0.331604\pi\)
0.504697 + 0.863297i \(0.331604\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 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 \(-0.758972\pi\)
−0.726754 + 0.686898i \(0.758972\pi\)
\(252\) 0 0
\(253\) −384.000 −0.0954224
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2594.00 −0.629608 −0.314804 − 0.949157i \(-0.601939\pi\)
−0.314804 + 0.949157i \(0.601939\pi\)
\(258\) 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 \(-0.357356\pi\)
0.433280 + 0.901260i \(0.357356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2082.00 0.477215
\(268\) 0 0
\(269\) −2250.00 −0.509981 −0.254991 − 0.966944i \(-0.582072\pi\)
−0.254991 + 0.966944i \(0.582072\pi\)
\(270\) 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
\(275\) 0 0
\(276\) 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
\(285\) 0 0
\(286\) 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
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 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 \(-0.149363\pi\)
0.891913 + 0.452207i \(0.149363\pi\)
\(314\) 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
\(325\) 0 0
\(326\) 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 \(-0.319797\pi\)
0.536365 + 0.843986i \(0.319797\pi\)
\(332\) 0 0
\(333\) 882.000 0.145145
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −626.000 −0.101188 −0.0505941 − 0.998719i \(-0.516111\pi\)
−0.0505941 + 0.998719i \(0.516111\pi\)
\(338\) 0 0
\(339\) 342.000 0.0547932
\(340\) 0 0
\(341\) −1088.00 −0.172782
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 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 \(-0.772550\pi\)
−0.755385 + 0.655282i \(0.772550\pi\)
\(350\) 0 0
\(351\) 1458.00 0.221716
\(352\) 0 0
\(353\) 8894.00 1.34102 0.670510 − 0.741901i \(-0.266075\pi\)
0.670510 + 0.741901i \(0.266075\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) −7016.00 −0.997908 −0.498954 − 0.866628i \(-0.666282\pi\)
−0.498954 + 0.866628i \(0.666282\pi\)
\(368\) 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
\(375\) 0 0
\(376\) 0 0
\(377\) −7236.00 −0.988522
\(378\) 0 0
\(379\) −4900.00 −0.664106 −0.332053 − 0.943261i \(-0.607741\pi\)
−0.332053 + 0.943261i \(0.607741\pi\)
\(380\) 0 0
\(381\) −7032.00 −0.945565
\(382\) 0 0
\(383\) 7800.00 1.04063 0.520315 − 0.853974i \(-0.325814\pi\)
0.520315 + 0.853974i \(0.325814\pi\)
\(384\) 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 \(-0.794558\pi\)
−0.798850 + 0.601530i \(0.794558\pi\)
\(390\) 0 0
\(391\) −10944.0 −1.41550
\(392\) 0 0
\(393\) −6492.00 −0.833278
\(394\) 0 0
\(395\) 0 0
\(396\) 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
\(405\) 0 0
\(406\) 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
\(415\) 0 0
\(416\) 0 0
\(417\) 5916.00 0.694743
\(418\) 0 0
\(419\) 5412.00 0.631011 0.315505 − 0.948924i \(-0.397826\pi\)
0.315505 + 0.948924i \(0.397826\pi\)
\(420\) 0 0
\(421\) −10690.0 −1.23753 −0.618763 − 0.785577i \(-0.712366\pi\)
−0.618763 + 0.785577i \(0.712366\pi\)
\(422\) 0 0
\(423\) −1800.00 −0.206901
\(424\) 0 0
\(425\) 0 0
\(426\) 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 \(-0.947743\pi\)
−0.986554 + 0.163433i \(0.947743\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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
\(445\) 0 0
\(446\) 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 \(-0.563399\pi\)
−0.197860 + 0.980230i \(0.563399\pi\)
\(458\) 0 0
\(459\) 3078.00 0.313004
\(460\) 0 0
\(461\) −1706.00 −0.172356 −0.0861782 − 0.996280i \(-0.527465\pi\)
−0.0861782 + 0.996280i \(0.527465\pi\)
\(462\) 0 0
\(463\) 3944.00 0.395882 0.197941 − 0.980214i \(-0.436575\pi\)
0.197941 + 0.980214i \(0.436575\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 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
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 7936.00 0.738428 0.369214 − 0.929344i \(-0.379627\pi\)
0.369214 + 0.929344i \(0.379627\pi\)
\(488\) 0 0
\(489\) 10212.0 0.944382
\(490\) 0 0
\(491\) 8412.00 0.773174 0.386587 − 0.922253i \(-0.373654\pi\)
0.386587 + 0.922253i \(0.373654\pi\)
\(492\) 0 0
\(493\) −15276.0 −1.39553
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 15092.0 1.35393 0.676965 − 0.736016i \(-0.263295\pi\)
0.676965 + 0.736016i \(0.263295\pi\)
\(500\) 0 0
\(501\) 2496.00 0.222581
\(502\) 0 0
\(503\) 6112.00 0.541790 0.270895 − 0.962609i \(-0.412680\pi\)
0.270895 + 0.962609i \(0.412680\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2157.00 −0.188946
\(508\) 0 0
\(509\) 2534.00 0.220663 0.110332 − 0.993895i \(-0.464809\pi\)
0.110332 + 0.993895i \(0.464809\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1188.00 0.102245
\(514\) 0 0
\(515\) 0 0
\(516\) 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
\(535\) 0 0
\(536\) 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 \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) 0 0
\(543\) −9258.00 −0.731674
\(544\) 0 0
\(545\) 0 0
\(546\) 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
\(555\) 0 0
\(556\) 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
\(565\) 0 0
\(566\) 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 \(-0.559556\pi\)
−0.186010 + 0.982548i \(0.559556\pi\)
\(572\) 0 0
\(573\) −12240.0 −0.892379
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6526.00 0.470851 0.235425 − 0.971892i \(-0.424352\pi\)
0.235425 + 0.971892i \(0.424352\pi\)
\(578\) 0 0
\(579\) −7962.00 −0.571484
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2616.00 0.185838
\(584\) 0 0
\(585\) 0 0
\(586\) 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 \(-0.392352\pi\)
0.331775 + 0.943358i \(0.392352\pi\)
\(594\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 13000.0 0.869281 0.434641 − 0.900604i \(-0.356875\pi\)
0.434641 + 0.900604i \(0.356875\pi\)
\(608\) 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
\(615\) 0 0
\(616\) 0 0
\(617\) −4474.00 −0.291923 −0.145961 − 0.989290i \(-0.546628\pi\)
−0.145961 + 0.989290i \(0.546628\pi\)
\(618\) 0 0
\(619\) 12556.0 0.815296 0.407648 − 0.913139i \(-0.366349\pi\)
0.407648 + 0.913139i \(0.366349\pi\)
\(620\) 0 0
\(621\) −2592.00 −0.167493
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 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
\(645\) 0 0
\(646\) 0 0
\(647\) −2784.00 −0.169166 −0.0845829 − 0.996416i \(-0.526956\pi\)
−0.0845829 + 0.996416i \(0.526956\pi\)
\(648\) 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
\(655\) 0 0
\(656\) 0 0
\(657\) 3510.00 0.208429
\(658\) 0 0
\(659\) −8108.00 −0.479276 −0.239638 − 0.970862i \(-0.577029\pi\)
−0.239638 + 0.970862i \(0.577029\pi\)
\(660\) 0 0
\(661\) 1230.00 0.0723774 0.0361887 − 0.999345i \(-0.488478\pi\)
0.0361887 + 0.999345i \(0.488478\pi\)
\(662\) 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 \(-0.367908\pi\)
0.403170 + 0.915125i \(0.367908\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(685\) 0 0
\(686\) 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 \(-0.675560\pi\)
−0.523997 + 0.851720i \(0.675560\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 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 \(-0.217233\pi\)
0.776025 + 0.630702i \(0.217233\pi\)
\(702\) 0 0
\(703\) −4312.00 −0.231337
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −25090.0 −1.32902 −0.664510 − 0.747280i \(-0.731360\pi\)
−0.664510 + 0.747280i \(0.731360\pi\)
\(710\) 0 0
\(711\) −6192.00 −0.326608
\(712\) 0 0
\(713\) 26112.0 1.37153
\(714\) 0 0
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 21616.0 1.10274 0.551371 − 0.834260i \(-0.314105\pi\)
0.551371 + 0.834260i \(0.314105\pi\)
\(728\) 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
\(735\) 0 0
\(736\) 0 0
\(737\) 752.000 0.0375852
\(738\) 0 0
\(739\) −764.000 −0.0380300 −0.0190150 − 0.999819i \(-0.506053\pi\)
−0.0190150 + 0.999819i \(0.506053\pi\)
\(740\) 0 0
\(741\) −7128.00 −0.353379
\(742\) 0 0
\(743\) 6256.00 0.308897 0.154448 − 0.988001i \(-0.450640\pi\)
0.154448 + 0.988001i \(0.450640\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 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
\(765\) 0 0
\(766\) 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
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 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
\(795\) 0 0
\(796\) 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 \(-0.571184\pi\)
−0.221773 + 0.975098i \(0.571184\pi\)
\(812\) 0 0
\(813\) 6624.00 0.285749
\(814\) 0 0
\(815\) 0 0
\(816\) 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 \(-0.490594\pi\)
0.0295441 + 0.999563i \(0.490594\pi\)
\(822\) 0 0
\(823\) −8448.00 −0.357811 −0.178906 − 0.983866i \(-0.557256\pi\)
−0.178906 + 0.983866i \(0.557256\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 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 \(-0.730360\pi\)
−0.662160 + 0.749363i \(0.730360\pi\)
\(830\) 0 0
\(831\) −5046.00 −0.210642
\(832\) 0 0
\(833\) 39102.0 1.62642
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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
\(855\) 0 0
\(856\) 0 0
\(857\) 12566.0 0.500871 0.250435 − 0.968133i \(-0.419426\pi\)
0.250435 + 0.968133i \(0.419426\pi\)
\(858\) 0 0
\(859\) −11812.0 −0.469174 −0.234587 − 0.972095i \(-0.575374\pi\)
−0.234587 + 0.972095i \(0.575374\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31496.0 −1.24234 −0.621168 − 0.783677i \(-0.713342\pi\)
−0.621168 + 0.783677i \(0.713342\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(885\) 0 0
\(886\) 0 0
\(887\) −20784.0 −0.786763 −0.393381 − 0.919375i \(-0.628695\pi\)
−0.393381 + 0.919375i \(0.628695\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −324.000 −0.0121823
\(892\) 0 0
\(893\) 8800.00 0.329766
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 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
\(915\) 0 0
\(916\) 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
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) 19686.0 0.686354 0.343177 − 0.939271i \(-0.388497\pi\)
0.343177 + 0.939271i \(0.388497\pi\)
\(938\) 0 0
\(939\) −29634.0 −1.02989
\(940\) 0 0
\(941\) 56246.0 1.94853 0.974265 − 0.225405i \(-0.0723704\pi\)
0.974265 + 0.225405i \(0.0723704\pi\)
\(942\) 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 \(-0.374619\pi\)
0.383789 + 0.923421i \(0.374619\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) −2112.00 −0.0702351 −0.0351175 − 0.999383i \(-0.511181\pi\)
−0.0351175 + 0.999383i \(0.511181\pi\)
\(968\) 0 0
\(969\) −15048.0 −0.498877
\(970\) 0 0
\(971\) 47964.0 1.58521 0.792605 − 0.609736i \(-0.208725\pi\)
0.792605 + 0.609736i \(0.208725\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10510.0 0.344160 0.172080 − 0.985083i \(-0.444951\pi\)
0.172080 + 0.985083i \(0.444951\pi\)
\(978\) 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 \(-0.440327\pi\)
0.186373 + 0.982479i \(0.440327\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 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 1200.4.a.j.1.1 1
4.3 odd 2 600.4.a.m.1.1 1
5.2 odd 4 1200.4.f.l.49.2 2
5.3 odd 4 1200.4.f.l.49.1 2
5.4 even 2 240.4.a.k.1.1 1
12.11 even 2 1800.4.a.s.1.1 1
15.14 odd 2 720.4.a.i.1.1 1
20.3 even 4 600.4.f.d.49.2 2
20.7 even 4 600.4.f.d.49.1 2
20.19 odd 2 120.4.a.d.1.1 ✓ 1
40.19 odd 2 960.4.a.x.1.1 1
40.29 even 2 960.4.a.e.1.1 1
60.23 odd 4 1800.4.f.m.649.1 2
60.47 odd 4 1800.4.f.m.649.2 2
60.59 even 2 360.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.d.1.1 ✓ 1 20.19 odd 2
240.4.a.k.1.1 1 5.4 even 2
360.4.a.c.1.1 1 60.59 even 2
600.4.a.m.1.1 1 4.3 odd 2
600.4.f.d.49.1 2 20.7 even 4
600.4.f.d.49.2 2 20.3 even 4
720.4.a.i.1.1 1 15.14 odd 2
960.4.a.e.1.1 1 40.29 even 2
960.4.a.x.1.1 1 40.19 odd 2
1200.4.a.j.1.1 1 1.1 even 1 trivial
1200.4.f.l.49.1 2 5.3 odd 4
1200.4.f.l.49.2 2 5.2 odd 4
1800.4.a.s.1.1 1 12.11 even 2
1800.4.f.m.649.1 2 60.23 odd 4
1800.4.f.m.649.2 2 60.47 odd 4