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Properties

Label 1800.4.a.m.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
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] = [1800,4,Mod(1,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, 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("1800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{7} -14.0000 q^{11} -1.00000 q^{13} +46.0000 q^{17} +19.0000 q^{19} -46.0000 q^{23} -14.0000 q^{29} +133.000 q^{31} -258.000 q^{37} -84.0000 q^{41} +167.000 q^{43} +410.000 q^{47} -318.000 q^{49} +456.000 q^{53} +194.000 q^{59} -17.0000 q^{61} -653.000 q^{67} -828.000 q^{71} -570.000 q^{73} +70.0000 q^{77} -552.000 q^{79} +142.000 q^{83} +1104.00 q^{89} +5.00000 q^{91} -841.000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −5.00000 −0.269975 −0.134987 − 0.990847i \(-0.543099\pi\)
−0.134987 + 0.990847i \(0.543099\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −14.0000 −0.383742 −0.191871 − 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.0213346 −0.0106673 − 0.999943i \(-0.503396\pi\)
−0.0106673 + 0.999943i \(0.503396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46.0000 0.656273 0.328136 − 0.944630i \(-0.393579\pi\)
0.328136 + 0.944630i \(0.393579\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −46.0000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −14.0000 −0.0896460 −0.0448230 − 0.998995i \(-0.514272\pi\)
−0.0448230 + 0.998995i \(0.514272\pi\)
\(30\) 0 0
\(31\) 133.000 0.770565 0.385282 − 0.922799i \(-0.374104\pi\)
0.385282 + 0.922799i \(0.374104\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −258.000 −1.14635 −0.573175 − 0.819433i \(-0.694288\pi\)
−0.573175 + 0.819433i \(0.694288\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −84.0000 −0.319966 −0.159983 − 0.987120i \(-0.551144\pi\)
−0.159983 + 0.987120i \(0.551144\pi\)
\(42\) 0 0
\(43\) 167.000 0.592262 0.296131 − 0.955147i \(-0.404304\pi\)
0.296131 + 0.955147i \(0.404304\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 410.000 1.27244 0.636220 − 0.771508i \(-0.280497\pi\)
0.636220 + 0.771508i \(0.280497\pi\)
\(48\) 0 0
\(49\) −318.000 −0.927114
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 456.000 1.18182 0.590910 − 0.806738i \(-0.298769\pi\)
0.590910 + 0.806738i \(0.298769\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 194.000 0.428079 0.214039 − 0.976825i \(-0.431338\pi\)
0.214039 + 0.976825i \(0.431338\pi\)
\(60\) 0 0
\(61\) −17.0000 −0.0356824 −0.0178412 − 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −653.000 −1.19070 −0.595348 − 0.803468i \(-0.702986\pi\)
−0.595348 + 0.803468i \(0.702986\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −828.000 −1.38402 −0.692011 − 0.721887i \(-0.743275\pi\)
−0.692011 + 0.721887i \(0.743275\pi\)
\(72\) 0 0
\(73\) −570.000 −0.913883 −0.456941 − 0.889497i \(-0.651055\pi\)
−0.456941 + 0.889497i \(0.651055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 70.0000 0.103601
\(78\) 0 0
\(79\) −552.000 −0.786137 −0.393069 − 0.919509i \(-0.628587\pi\)
−0.393069 + 0.919509i \(0.628587\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 142.000 0.187789 0.0938947 − 0.995582i \(-0.470068\pi\)
0.0938947 + 0.995582i \(0.470068\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1104.00 1.31487 0.657437 − 0.753510i \(-0.271641\pi\)
0.657437 + 0.753510i \(0.271641\pi\)
\(90\) 0 0
\(91\) 5.00000 0.00575981
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −841.000 −0.880316 −0.440158 − 0.897920i \(-0.645077\pi\)
−0.440158 + 0.897920i \(0.645077\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −552.000 −0.543822 −0.271911 − 0.962322i \(-0.587656\pi\)
−0.271911 + 0.962322i \(0.587656\pi\)
\(102\) 0 0
\(103\) 308.000 0.294642 0.147321 − 0.989089i \(-0.452935\pi\)
0.147321 + 0.989089i \(0.452935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 984.000 0.889036 0.444518 − 0.895770i \(-0.353375\pi\)
0.444518 + 0.895770i \(0.353375\pi\)
\(108\) 0 0
\(109\) −1843.00 −1.61952 −0.809759 − 0.586763i \(-0.800402\pi\)
−0.809759 + 0.586763i \(0.800402\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 876.000 0.729267 0.364633 − 0.931151i \(-0.381194\pi\)
0.364633 + 0.931151i \(0.381194\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −230.000 −0.177177
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2376.00 −1.66013 −0.830063 − 0.557670i \(-0.811695\pi\)
−0.830063 + 0.557670i \(0.811695\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1056.00 −0.704299 −0.352149 − 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(132\) 0 0
\(133\) −95.0000 −0.0619364
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −778.000 −0.485175 −0.242588 − 0.970129i \(-0.577996\pi\)
−0.242588 + 0.970129i \(0.577996\pi\)
\(138\) 0 0
\(139\) 1692.00 1.03247 0.516236 − 0.856446i \(-0.327333\pi\)
0.516236 + 0.856446i \(0.327333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.0000 0.00818698
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 494.000 0.271611 0.135806 − 0.990736i \(-0.456638\pi\)
0.135806 + 0.990736i \(0.456638\pi\)
\(150\) 0 0
\(151\) −841.000 −0.453242 −0.226621 − 0.973983i \(-0.572768\pi\)
−0.226621 + 0.973983i \(0.572768\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −19.0000 −0.00965838 −0.00482919 − 0.999988i \(-0.501537\pi\)
−0.00482919 + 0.999988i \(0.501537\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 230.000 0.112587
\(162\) 0 0
\(163\) 2261.00 1.08647 0.543237 − 0.839580i \(-0.317199\pi\)
0.543237 + 0.839580i \(0.317199\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2112.00 0.978632 0.489316 − 0.872107i \(-0.337247\pi\)
0.489316 + 0.872107i \(0.337247\pi\)
\(168\) 0 0
\(169\) −2196.00 −0.999545
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 562.000 0.246983 0.123492 − 0.992346i \(-0.460591\pi\)
0.123492 + 0.992346i \(0.460591\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3718.00 −1.55249 −0.776247 − 0.630429i \(-0.782879\pi\)
−0.776247 + 0.630429i \(0.782879\pi\)
\(180\) 0 0
\(181\) −1639.00 −0.673071 −0.336536 − 0.941671i \(-0.609255\pi\)
−0.336536 + 0.941671i \(0.609255\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −644.000 −0.251839
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2410.00 −0.912992 −0.456496 − 0.889725i \(-0.650896\pi\)
−0.456496 + 0.889725i \(0.650896\pi\)
\(192\) 0 0
\(193\) 2621.00 0.977532 0.488766 − 0.872415i \(-0.337447\pi\)
0.488766 + 0.872415i \(0.337447\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4954.00 −1.79166 −0.895832 − 0.444392i \(-0.853420\pi\)
−0.895832 + 0.444392i \(0.853420\pi\)
\(198\) 0 0
\(199\) 1739.00 0.619470 0.309735 − 0.950823i \(-0.399760\pi\)
0.309735 + 0.950823i \(0.399760\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 70.0000 0.0242022
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −266.000 −0.0880364
\(210\) 0 0
\(211\) −4525.00 −1.47637 −0.738184 − 0.674599i \(-0.764317\pi\)
−0.738184 + 0.674599i \(0.764317\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −665.000 −0.208033
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −46.0000 −0.0140013
\(222\) 0 0
\(223\) 3211.00 0.964235 0.482118 − 0.876106i \(-0.339868\pi\)
0.482118 + 0.876106i \(0.339868\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2484.00 −0.726295 −0.363147 − 0.931732i \(-0.618298\pi\)
−0.363147 + 0.931732i \(0.618298\pi\)
\(228\) 0 0
\(229\) −1847.00 −0.532983 −0.266492 − 0.963837i \(-0.585864\pi\)
−0.266492 + 0.963837i \(0.585864\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1020.00 −0.286792 −0.143396 − 0.989665i \(-0.545802\pi\)
−0.143396 + 0.989665i \(0.545802\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1176.00 −0.318281 −0.159140 − 0.987256i \(-0.550872\pi\)
−0.159140 + 0.987256i \(0.550872\pi\)
\(240\) 0 0
\(241\) −6967.00 −1.86217 −0.931087 − 0.364797i \(-0.881138\pi\)
−0.931087 + 0.364797i \(0.881138\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −19.0000 −0.00489450
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1380.00 −0.347031 −0.173516 − 0.984831i \(-0.555513\pi\)
−0.173516 + 0.984831i \(0.555513\pi\)
\(252\) 0 0
\(253\) 644.000 0.160031
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6924.00 −1.68057 −0.840286 − 0.542143i \(-0.817613\pi\)
−0.840286 + 0.542143i \(0.817613\pi\)
\(258\) 0 0
\(259\) 1290.00 0.309485
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1884.00 0.441720 0.220860 − 0.975305i \(-0.429114\pi\)
0.220860 + 0.975305i \(0.429114\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3610.00 −0.818236 −0.409118 − 0.912481i \(-0.634164\pi\)
−0.409118 + 0.912481i \(0.634164\pi\)
\(270\) 0 0
\(271\) −6072.00 −1.36106 −0.680531 − 0.732719i \(-0.738251\pi\)
−0.680531 + 0.732719i \(0.738251\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2803.00 0.608000 0.304000 − 0.952672i \(-0.401678\pi\)
0.304000 + 0.952672i \(0.401678\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6694.00 1.42111 0.710553 − 0.703644i \(-0.248445\pi\)
0.710553 + 0.703644i \(0.248445\pi\)
\(282\) 0 0
\(283\) 6481.00 1.36133 0.680663 − 0.732596i \(-0.261692\pi\)
0.680663 + 0.732596i \(0.261692\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 420.000 0.0863826
\(288\) 0 0
\(289\) −2797.00 −0.569306
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3014.00 0.600955 0.300477 − 0.953789i \(-0.402854\pi\)
0.300477 + 0.953789i \(0.402854\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 46.0000 0.00889715
\(300\) 0 0
\(301\) −835.000 −0.159896
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5369.00 0.998127 0.499064 − 0.866565i \(-0.333677\pi\)
0.499064 + 0.866565i \(0.333677\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4846.00 −0.883574 −0.441787 − 0.897120i \(-0.645655\pi\)
−0.441787 + 0.897120i \(0.645655\pi\)
\(312\) 0 0
\(313\) 757.000 0.136703 0.0683517 − 0.997661i \(-0.478226\pi\)
0.0683517 + 0.997661i \(0.478226\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7632.00 −1.35223 −0.676113 − 0.736798i \(-0.736337\pi\)
−0.676113 + 0.736798i \(0.736337\pi\)
\(318\) 0 0
\(319\) 196.000 0.0344009
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 874.000 0.150559
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2050.00 −0.343526
\(330\) 0 0
\(331\) −6780.00 −1.12587 −0.562934 − 0.826502i \(-0.690328\pi\)
−0.562934 + 0.826502i \(0.690328\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7849.00 −1.26873 −0.634365 − 0.773033i \(-0.718739\pi\)
−0.634365 + 0.773033i \(0.718739\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1862.00 −0.295698
\(342\) 0 0
\(343\) 3305.00 0.520272
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 634.000 0.0980833 0.0490416 − 0.998797i \(-0.484383\pi\)
0.0490416 + 0.998797i \(0.484383\pi\)
\(348\) 0 0
\(349\) 930.000 0.142641 0.0713206 − 0.997453i \(-0.477279\pi\)
0.0713206 + 0.997453i \(0.477279\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4286.00 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4236.00 0.622751 0.311375 − 0.950287i \(-0.399210\pi\)
0.311375 + 0.950287i \(0.399210\pi\)
\(360\) 0 0
\(361\) −6498.00 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1451.00 −0.206380 −0.103190 − 0.994662i \(-0.532905\pi\)
−0.103190 + 0.994662i \(0.532905\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2280.00 −0.319061
\(372\) 0 0
\(373\) −3115.00 −0.432409 −0.216205 − 0.976348i \(-0.569368\pi\)
−0.216205 + 0.976348i \(0.569368\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0000 0.00191256
\(378\) 0 0
\(379\) −1415.00 −0.191777 −0.0958887 − 0.995392i \(-0.530569\pi\)
−0.0958887 + 0.995392i \(0.530569\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −180.000 −0.0240145 −0.0120073 − 0.999928i \(-0.503822\pi\)
−0.0120073 + 0.999928i \(0.503822\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12372.0 −1.61256 −0.806279 − 0.591535i \(-0.798522\pi\)
−0.806279 + 0.591535i \(0.798522\pi\)
\(390\) 0 0
\(391\) −2116.00 −0.273685
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5767.00 −0.729062 −0.364531 − 0.931191i \(-0.618771\pi\)
−0.364531 + 0.931191i \(0.618771\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3120.00 0.388542 0.194271 − 0.980948i \(-0.437766\pi\)
0.194271 + 0.980948i \(0.437766\pi\)
\(402\) 0 0
\(403\) −133.000 −0.0164397
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3612.00 0.439902
\(408\) 0 0
\(409\) 1501.00 0.181466 0.0907331 − 0.995875i \(-0.471079\pi\)
0.0907331 + 0.995875i \(0.471079\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −970.000 −0.115570
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9072.00 1.05775 0.528874 − 0.848701i \(-0.322614\pi\)
0.528874 + 0.848701i \(0.322614\pi\)
\(420\) 0 0
\(421\) 7350.00 0.850872 0.425436 − 0.904989i \(-0.360121\pi\)
0.425436 + 0.904989i \(0.360121\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 85.0000 0.00963334
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5962.00 −0.666310 −0.333155 − 0.942872i \(-0.608113\pi\)
−0.333155 + 0.942872i \(0.608113\pi\)
\(432\) 0 0
\(433\) 10093.0 1.12018 0.560091 − 0.828431i \(-0.310766\pi\)
0.560091 + 0.828431i \(0.310766\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −874.000 −0.0956730
\(438\) 0 0
\(439\) −2555.00 −0.277776 −0.138888 − 0.990308i \(-0.544353\pi\)
−0.138888 + 0.990308i \(0.544353\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6240.00 0.669236 0.334618 − 0.942354i \(-0.391393\pi\)
0.334618 + 0.942354i \(0.391393\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3324.00 −0.349375 −0.174687 − 0.984624i \(-0.555891\pi\)
−0.174687 + 0.984624i \(0.555891\pi\)
\(450\) 0 0
\(451\) 1176.00 0.122784
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16774.0 −1.71697 −0.858484 − 0.512840i \(-0.828593\pi\)
−0.858484 + 0.512840i \(0.828593\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14304.0 −1.44513 −0.722564 − 0.691304i \(-0.757036\pi\)
−0.722564 + 0.691304i \(0.757036\pi\)
\(462\) 0 0
\(463\) −6936.00 −0.696206 −0.348103 − 0.937456i \(-0.613174\pi\)
−0.348103 + 0.937456i \(0.613174\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15622.0 1.54797 0.773983 − 0.633207i \(-0.218262\pi\)
0.773983 + 0.633207i \(0.218262\pi\)
\(468\) 0 0
\(469\) 3265.00 0.321458
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2338.00 −0.227276
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13354.0 1.27382 0.636910 − 0.770938i \(-0.280212\pi\)
0.636910 + 0.770938i \(0.280212\pi\)
\(480\) 0 0
\(481\) 258.000 0.0244569
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 461.000 0.0428951 0.0214475 − 0.999770i \(-0.493173\pi\)
0.0214475 + 0.999770i \(0.493173\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3768.00 −0.346329 −0.173164 − 0.984893i \(-0.555399\pi\)
−0.173164 + 0.984893i \(0.555399\pi\)
\(492\) 0 0
\(493\) −644.000 −0.0588323
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4140.00 0.373651
\(498\) 0 0
\(499\) −14317.0 −1.28440 −0.642201 − 0.766536i \(-0.721979\pi\)
−0.642201 + 0.766536i \(0.721979\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9228.00 0.818004 0.409002 − 0.912533i \(-0.365877\pi\)
0.409002 + 0.912533i \(0.365877\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4574.00 −0.398308 −0.199154 − 0.979968i \(-0.563819\pi\)
−0.199154 + 0.979968i \(0.563819\pi\)
\(510\) 0 0
\(511\) 2850.00 0.246725
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5740.00 −0.488288
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8494.00 0.714259 0.357129 − 0.934055i \(-0.383755\pi\)
0.357129 + 0.934055i \(0.383755\pi\)
\(522\) 0 0
\(523\) −8263.00 −0.690852 −0.345426 − 0.938446i \(-0.612266\pi\)
−0.345426 + 0.938446i \(0.612266\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6118.00 0.505701
\(528\) 0 0
\(529\) −10051.0 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 84.0000 0.00682635
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4452.00 0.355772
\(540\) 0 0
\(541\) 21157.0 1.68135 0.840675 − 0.541540i \(-0.182158\pi\)
0.840675 + 0.541540i \(0.182158\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4048.00 0.316417 0.158208 − 0.987406i \(-0.449428\pi\)
0.158208 + 0.987406i \(0.449428\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −266.000 −0.0205662
\(552\) 0 0
\(553\) 2760.00 0.212237
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6758.00 −0.514086 −0.257043 − 0.966400i \(-0.582748\pi\)
−0.257043 + 0.966400i \(0.582748\pi\)
\(558\) 0 0
\(559\) −167.000 −0.0126357
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24506.0 1.83447 0.917233 − 0.398350i \(-0.130417\pi\)
0.917233 + 0.398350i \(0.130417\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1430.00 0.105358 0.0526790 − 0.998611i \(-0.483224\pi\)
0.0526790 + 0.998611i \(0.483224\pi\)
\(570\) 0 0
\(571\) 3691.00 0.270514 0.135257 − 0.990811i \(-0.456814\pi\)
0.135257 + 0.990811i \(0.456814\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4571.00 −0.329798 −0.164899 − 0.986310i \(-0.552730\pi\)
−0.164899 + 0.986310i \(0.552730\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −710.000 −0.0506984
\(582\) 0 0
\(583\) −6384.00 −0.453513
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14808.0 1.04121 0.520606 − 0.853797i \(-0.325706\pi\)
0.520606 + 0.853797i \(0.325706\pi\)
\(588\) 0 0
\(589\) 2527.00 0.176780
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −24588.0 −1.70271 −0.851356 − 0.524588i \(-0.824219\pi\)
−0.851356 + 0.524588i \(0.824219\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27564.0 1.88019 0.940096 − 0.340911i \(-0.110735\pi\)
0.940096 + 0.340911i \(0.110735\pi\)
\(600\) 0 0
\(601\) 10987.0 0.745706 0.372853 − 0.927891i \(-0.378380\pi\)
0.372853 + 0.927891i \(0.378380\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13200.0 0.882655 0.441327 − 0.897346i \(-0.354508\pi\)
0.441327 + 0.897346i \(0.354508\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −410.000 −0.0271470
\(612\) 0 0
\(613\) −21066.0 −1.38801 −0.694003 − 0.719972i \(-0.744155\pi\)
−0.694003 + 0.719972i \(0.744155\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12336.0 0.804909 0.402454 − 0.915440i \(-0.368157\pi\)
0.402454 + 0.915440i \(0.368157\pi\)
\(618\) 0 0
\(619\) −1441.00 −0.0935681 −0.0467841 − 0.998905i \(-0.514897\pi\)
−0.0467841 + 0.998905i \(0.514897\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5520.00 −0.354983
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −11868.0 −0.752318
\(630\) 0 0
\(631\) −9839.00 −0.620736 −0.310368 − 0.950616i \(-0.600452\pi\)
−0.310368 + 0.950616i \(0.600452\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 318.000 0.0197796
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21564.0 1.32875 0.664373 − 0.747401i \(-0.268698\pi\)
0.664373 + 0.747401i \(0.268698\pi\)
\(642\) 0 0
\(643\) −8604.00 −0.527696 −0.263848 − 0.964564i \(-0.584992\pi\)
−0.263848 + 0.964564i \(0.584992\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3444.00 0.209270 0.104635 − 0.994511i \(-0.466633\pi\)
0.104635 + 0.994511i \(0.466633\pi\)
\(648\) 0 0
\(649\) −2716.00 −0.164272
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3518.00 −0.210827 −0.105413 − 0.994428i \(-0.533617\pi\)
−0.105413 + 0.994428i \(0.533617\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12612.0 0.745514 0.372757 − 0.927929i \(-0.378412\pi\)
0.372757 + 0.927929i \(0.378412\pi\)
\(660\) 0 0
\(661\) 27090.0 1.59407 0.797034 − 0.603935i \(-0.206401\pi\)
0.797034 + 0.603935i \(0.206401\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 644.000 0.0373850
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 238.000 0.0136928
\(672\) 0 0
\(673\) 5442.00 0.311699 0.155850 − 0.987781i \(-0.450188\pi\)
0.155850 + 0.987781i \(0.450188\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15226.0 −0.864376 −0.432188 − 0.901783i \(-0.642258\pi\)
−0.432188 + 0.901783i \(0.642258\pi\)
\(678\) 0 0
\(679\) 4205.00 0.237663
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −552.000 −0.0309249 −0.0154624 − 0.999880i \(-0.504922\pi\)
−0.0154624 + 0.999880i \(0.504922\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −456.000 −0.0252137
\(690\) 0 0
\(691\) 9776.00 0.538201 0.269100 − 0.963112i \(-0.413274\pi\)
0.269100 + 0.963112i \(0.413274\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3864.00 −0.209985
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13066.0 −0.703989 −0.351994 − 0.936002i \(-0.614496\pi\)
−0.351994 + 0.936002i \(0.614496\pi\)
\(702\) 0 0
\(703\) −4902.00 −0.262991
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2760.00 0.146818
\(708\) 0 0
\(709\) 28985.0 1.53534 0.767669 − 0.640847i \(-0.221417\pi\)
0.767669 + 0.640847i \(0.221417\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6118.00 −0.321348
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15722.0 −0.815482 −0.407741 − 0.913098i \(-0.633683\pi\)
−0.407741 + 0.913098i \(0.633683\pi\)
\(720\) 0 0
\(721\) −1540.00 −0.0795459
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32611.0 1.66365 0.831826 − 0.555036i \(-0.187296\pi\)
0.831826 + 0.555036i \(0.187296\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7682.00 0.388685
\(732\) 0 0
\(733\) −8358.00 −0.421159 −0.210580 − 0.977577i \(-0.567535\pi\)
−0.210580 + 0.977577i \(0.567535\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9142.00 0.456920
\(738\) 0 0
\(739\) 20604.0 1.02562 0.512808 − 0.858503i \(-0.328605\pi\)
0.512808 + 0.858503i \(0.328605\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19476.0 −0.961649 −0.480824 − 0.876817i \(-0.659663\pi\)
−0.480824 + 0.876817i \(0.659663\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4920.00 −0.240017
\(750\) 0 0
\(751\) 3864.00 0.187749 0.0938744 − 0.995584i \(-0.470075\pi\)
0.0938744 + 0.995584i \(0.470075\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 18871.0 0.906048 0.453024 − 0.891498i \(-0.350345\pi\)
0.453024 + 0.891498i \(0.350345\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36372.0 1.73257 0.866284 − 0.499552i \(-0.166502\pi\)
0.866284 + 0.499552i \(0.166502\pi\)
\(762\) 0 0
\(763\) 9215.00 0.437229
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −194.000 −0.00913290
\(768\) 0 0
\(769\) 4603.00 0.215850 0.107925 − 0.994159i \(-0.465579\pi\)
0.107925 + 0.994159i \(0.465579\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −36.0000 −0.00167507 −0.000837536 − 1.00000i \(-0.500267\pi\)
−0.000837536 1.00000i \(0.500267\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1596.00 −0.0734052
\(780\) 0 0
\(781\) 11592.0 0.531107
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20281.0 0.918602 0.459301 − 0.888281i \(-0.348100\pi\)
0.459301 + 0.888281i \(0.348100\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4380.00 −0.196884
\(792\) 0 0
\(793\) 17.0000 0.000761271 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −37524.0 −1.66771 −0.833857 − 0.551980i \(-0.813872\pi\)
−0.833857 + 0.551980i \(0.813872\pi\)
\(798\) 0 0
\(799\) 18860.0 0.835067
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7980.00 0.350695
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31224.0 −1.35696 −0.678478 − 0.734621i \(-0.737360\pi\)
−0.678478 + 0.734621i \(0.737360\pi\)
\(810\) 0 0
\(811\) 32579.0 1.41061 0.705304 − 0.708905i \(-0.250810\pi\)
0.705304 + 0.708905i \(0.250810\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3173.00 0.135874
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19810.0 0.842112 0.421056 − 0.907035i \(-0.361660\pi\)
0.421056 + 0.907035i \(0.361660\pi\)
\(822\) 0 0
\(823\) −10273.0 −0.435108 −0.217554 − 0.976048i \(-0.569808\pi\)
−0.217554 + 0.976048i \(0.569808\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16656.0 0.700346 0.350173 − 0.936685i \(-0.386123\pi\)
0.350173 + 0.936685i \(0.386123\pi\)
\(828\) 0 0
\(829\) −4790.00 −0.200680 −0.100340 − 0.994953i \(-0.531993\pi\)
−0.100340 + 0.994953i \(0.531993\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14628.0 −0.608440
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7414.00 −0.305077 −0.152539 − 0.988298i \(-0.548745\pi\)
−0.152539 + 0.988298i \(0.548745\pi\)
\(840\) 0 0
\(841\) −24193.0 −0.991964
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5675.00 0.230219
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11868.0 0.478061
\(852\) 0 0
\(853\) −30155.0 −1.21042 −0.605210 − 0.796066i \(-0.706911\pi\)
−0.605210 + 0.796066i \(0.706911\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8244.00 −0.328599 −0.164300 − 0.986410i \(-0.552536\pi\)
−0.164300 + 0.986410i \(0.552536\pi\)
\(858\) 0 0
\(859\) 17552.0 0.697167 0.348584 − 0.937278i \(-0.386663\pi\)
0.348584 + 0.937278i \(0.386663\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −34104.0 −1.34521 −0.672604 − 0.740003i \(-0.734824\pi\)
−0.672604 + 0.740003i \(0.734824\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7728.00 0.301674
\(870\) 0 0
\(871\) 653.000 0.0254031
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46229.0 1.77998 0.889990 − 0.455980i \(-0.150711\pi\)
0.889990 + 0.455980i \(0.150711\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22440.0 0.858142 0.429071 − 0.903271i \(-0.358841\pi\)
0.429071 + 0.903271i \(0.358841\pi\)
\(882\) 0 0
\(883\) 17143.0 0.653350 0.326675 − 0.945137i \(-0.394072\pi\)
0.326675 + 0.945137i \(0.394072\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23626.0 −0.894344 −0.447172 − 0.894448i \(-0.647569\pi\)
−0.447172 + 0.894448i \(0.647569\pi\)
\(888\) 0 0
\(889\) 11880.0 0.448192
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7790.00 0.291918
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1862.00 −0.0690781
\(900\) 0 0
\(901\) 20976.0 0.775596
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11268.0 0.412511 0.206256 − 0.978498i \(-0.433872\pi\)
0.206256 + 0.978498i \(0.433872\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10046.0 −0.365355 −0.182678 − 0.983173i \(-0.558476\pi\)
−0.182678 + 0.983173i \(0.558476\pi\)
\(912\) 0 0
\(913\) −1988.00 −0.0720626
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5280.00 0.190143
\(918\) 0 0
\(919\) 15359.0 0.551302 0.275651 − 0.961258i \(-0.411107\pi\)
0.275651 + 0.961258i \(0.411107\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 828.000 0.0295276
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39790.0 1.40524 0.702620 − 0.711565i \(-0.252014\pi\)
0.702620 + 0.711565i \(0.252014\pi\)
\(930\) 0 0
\(931\) −6042.00 −0.212694
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17009.0 0.593020 0.296510 − 0.955030i \(-0.404177\pi\)
0.296510 + 0.955030i \(0.404177\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11674.0 0.404422 0.202211 − 0.979342i \(-0.435187\pi\)
0.202211 + 0.979342i \(0.435187\pi\)
\(942\) 0 0
\(943\) 3864.00 0.133435
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6026.00 0.206778 0.103389 − 0.994641i \(-0.467031\pi\)
0.103389 + 0.994641i \(0.467031\pi\)
\(948\) 0 0
\(949\) 570.000 0.0194973
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14088.0 −0.478862 −0.239431 − 0.970913i \(-0.576961\pi\)
−0.239431 + 0.970913i \(0.576961\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3890.00 0.130985
\(960\) 0 0
\(961\) −12102.0 −0.406230
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 11208.0 0.372725 0.186362 − 0.982481i \(-0.440330\pi\)
0.186362 + 0.982481i \(0.440330\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26054.0 0.861084 0.430542 − 0.902571i \(-0.358322\pi\)
0.430542 + 0.902571i \(0.358322\pi\)
\(972\) 0 0
\(973\) −8460.00 −0.278741
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26870.0 −0.879885 −0.439942 − 0.898026i \(-0.645001\pi\)
−0.439942 + 0.898026i \(0.645001\pi\)
\(978\) 0 0
\(979\) −15456.0 −0.504572
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −23388.0 −0.758862 −0.379431 − 0.925220i \(-0.623880\pi\)
−0.379431 + 0.925220i \(0.623880\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7682.00 −0.246990
\(990\) 0 0
\(991\) 17345.0 0.555986 0.277993 − 0.960583i \(-0.410331\pi\)
0.277993 + 0.960583i \(0.410331\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25998.0 0.825842 0.412921 − 0.910767i \(-0.364508\pi\)
0.412921 + 0.910767i \(0.364508\pi\)
\(998\) 0 0
\(999\) 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 1800.4.a.m.1.1 1
3.2 odd 2 600.4.a.e.1.1 ✓ 1
5.2 odd 4 1800.4.f.l.649.1 2
5.3 odd 4 1800.4.f.l.649.2 2
5.4 even 2 1800.4.a.v.1.1 1
12.11 even 2 1200.4.a.bd.1.1 1
15.2 even 4 600.4.f.e.49.2 2
15.8 even 4 600.4.f.e.49.1 2
15.14 odd 2 600.4.a.n.1.1 yes 1
60.23 odd 4 1200.4.f.i.49.2 2
60.47 odd 4 1200.4.f.i.49.1 2
60.59 even 2 1200.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.e.1.1 ✓ 1 3.2 odd 2
600.4.a.n.1.1 yes 1 15.14 odd 2
600.4.f.e.49.1 2 15.8 even 4
600.4.f.e.49.2 2 15.2 even 4
1200.4.a.g.1.1 1 60.59 even 2
1200.4.a.bd.1.1 1 12.11 even 2
1200.4.f.i.49.1 2 60.47 odd 4
1200.4.f.i.49.2 2 60.23 odd 4
1800.4.a.m.1.1 1 1.1 even 1 trivial
1800.4.a.v.1.1 1 5.4 even 2
1800.4.f.l.649.1 2 5.2 odd 4
1800.4.f.l.649.2 2 5.3 odd 4