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Properties

Label 1872.4.a.a.1.1
Level $1872$
Weight $4$
Character 1872.1
Self dual yes
Analytic conductor $110.452$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,4,Mod(1,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, 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("1872.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.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: \(110.451575531\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1872.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19.0000 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q-19.0000 q^{5} +3.00000 q^{7} -2.00000 q^{11} -13.0000 q^{13} -77.0000 q^{17} +58.0000 q^{19} +76.0000 q^{23} +236.000 q^{25} +6.00000 q^{29} +292.000 q^{31} -57.0000 q^{35} +207.000 q^{37} -240.000 q^{41} +317.000 q^{43} -375.000 q^{47} -334.000 q^{49} +692.000 q^{53} +38.0000 q^{55} +214.000 q^{59} -488.000 q^{61} +247.000 q^{65} -782.000 q^{67} -1057.00 q^{71} +1174.00 q^{73} -6.00000 q^{77} -892.000 q^{79} +704.000 q^{83} +1463.00 q^{85} -6.00000 q^{89} -39.0000 q^{91} -1102.00 q^{95} +830.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −19.0000 −1.69941 −0.849706 − 0.527257i \(-0.823220\pi\)
−0.849706 + 0.527257i \(0.823220\pi\)
\(6\) 0 0
\(7\) 3.00000 0.161985 0.0809924 − 0.996715i \(-0.474191\pi\)
0.0809924 + 0.996715i \(0.474191\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.0548202 −0.0274101 − 0.999624i \(-0.508726\pi\)
−0.0274101 + 0.999624i \(0.508726\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −77.0000 −1.09854 −0.549272 − 0.835644i \(-0.685095\pi\)
−0.549272 + 0.835644i \(0.685095\pi\)
\(18\) 0 0
\(19\) 58.0000 0.700322 0.350161 − 0.936690i \(-0.386127\pi\)
0.350161 + 0.936690i \(0.386127\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 76.0000 0.689004 0.344502 − 0.938786i \(-0.388048\pi\)
0.344502 + 0.938786i \(0.388048\pi\)
\(24\) 0 0
\(25\) 236.000 1.88800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 − 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 0 0
\(31\) 292.000 1.69177 0.845883 − 0.533368i \(-0.179074\pi\)
0.845883 + 0.533368i \(0.179074\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −57.0000 −0.275279
\(36\) 0 0
\(37\) 207.000 0.919746 0.459873 − 0.887985i \(-0.347895\pi\)
0.459873 + 0.887985i \(0.347895\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −240.000 −0.914188 −0.457094 − 0.889418i \(-0.651110\pi\)
−0.457094 + 0.889418i \(0.651110\pi\)
\(42\) 0 0
\(43\) 317.000 1.12423 0.562117 − 0.827058i \(-0.309987\pi\)
0.562117 + 0.827058i \(0.309987\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −375.000 −1.16382 −0.581908 − 0.813254i \(-0.697694\pi\)
−0.581908 + 0.813254i \(0.697694\pi\)
\(48\) 0 0
\(49\) −334.000 −0.973761
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 692.000 1.79346 0.896731 − 0.442576i \(-0.145935\pi\)
0.896731 + 0.442576i \(0.145935\pi\)
\(54\) 0 0
\(55\) 38.0000 0.0931622
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 214.000 0.472211 0.236105 − 0.971727i \(-0.424129\pi\)
0.236105 + 0.971727i \(0.424129\pi\)
\(60\) 0 0
\(61\) −488.000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 247.000 0.471332
\(66\) 0 0
\(67\) −782.000 −1.42592 −0.712959 − 0.701206i \(-0.752645\pi\)
−0.712959 + 0.701206i \(0.752645\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1057.00 −1.76680 −0.883400 − 0.468619i \(-0.844752\pi\)
−0.883400 + 0.468619i \(0.844752\pi\)
\(72\) 0 0
\(73\) 1174.00 1.88228 0.941139 − 0.338020i \(-0.109757\pi\)
0.941139 + 0.338020i \(0.109757\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 −0.00888004
\(78\) 0 0
\(79\) −892.000 −1.27035 −0.635176 − 0.772367i \(-0.719073\pi\)
−0.635176 + 0.772367i \(0.719073\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 704.000 0.931013 0.465506 − 0.885045i \(-0.345872\pi\)
0.465506 + 0.885045i \(0.345872\pi\)
\(84\) 0 0
\(85\) 1463.00 1.86688
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.00714605 −0.00357303 − 0.999994i \(-0.501137\pi\)
−0.00357303 + 0.999994i \(0.501137\pi\)
\(90\) 0 0
\(91\) −39.0000 −0.0449265
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1102.00 −1.19013
\(96\) 0 0
\(97\) 830.000 0.868801 0.434401 − 0.900720i \(-0.356960\pi\)
0.434401 + 0.900720i \(0.356960\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −276.000 −0.271911 −0.135956 − 0.990715i \(-0.543410\pi\)
−0.135956 + 0.990715i \(0.543410\pi\)
\(102\) 0 0
\(103\) 1160.00 1.10969 0.554846 − 0.831953i \(-0.312777\pi\)
0.554846 + 0.831953i \(0.312777\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 324.000 0.292731 0.146366 − 0.989231i \(-0.453242\pi\)
0.146366 + 0.989231i \(0.453242\pi\)
\(108\) 0 0
\(109\) 199.000 0.174869 0.0874346 − 0.996170i \(-0.472133\pi\)
0.0874346 + 0.996170i \(0.472133\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1662.00 1.38361 0.691804 − 0.722085i \(-0.256816\pi\)
0.691804 + 0.722085i \(0.256816\pi\)
\(114\) 0 0
\(115\) −1444.00 −1.17090
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −231.000 −0.177947
\(120\) 0 0
\(121\) −1327.00 −0.996995
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2109.00 −1.50908
\(126\) 0 0
\(127\) −1840.00 −1.28562 −0.642809 − 0.766026i \(-0.722231\pi\)
−0.642809 + 0.766026i \(0.722231\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 95.0000 0.0633602 0.0316801 − 0.999498i \(-0.489914\pi\)
0.0316801 + 0.999498i \(0.489914\pi\)
\(132\) 0 0
\(133\) 174.000 0.113441
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1308.00 −0.815693 −0.407847 − 0.913050i \(-0.633720\pi\)
−0.407847 + 0.913050i \(0.633720\pi\)
\(138\) 0 0
\(139\) −279.000 −0.170248 −0.0851240 − 0.996370i \(-0.527129\pi\)
−0.0851240 + 0.996370i \(0.527129\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 26.0000 0.0152044
\(144\) 0 0
\(145\) −114.000 −0.0652909
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 730.000 0.401369 0.200684 − 0.979656i \(-0.435683\pi\)
0.200684 + 0.979656i \(0.435683\pi\)
\(150\) 0 0
\(151\) −2387.00 −1.28643 −0.643216 − 0.765685i \(-0.722400\pi\)
−0.643216 + 0.765685i \(0.722400\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5548.00 −2.87501
\(156\) 0 0
\(157\) −674.000 −0.342618 −0.171309 − 0.985217i \(-0.554800\pi\)
−0.171309 + 0.985217i \(0.554800\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 228.000 0.111608
\(162\) 0 0
\(163\) 180.000 0.0864950 0.0432475 − 0.999064i \(-0.486230\pi\)
0.0432475 + 0.999064i \(0.486230\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −888.000 −0.411470 −0.205735 − 0.978608i \(-0.565959\pi\)
−0.205735 + 0.978608i \(0.565959\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2868.00 −1.26040 −0.630202 − 0.776431i \(-0.717028\pi\)
−0.630202 + 0.776431i \(0.717028\pi\)
\(174\) 0 0
\(175\) 708.000 0.305827
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4431.00 1.85022 0.925108 − 0.379705i \(-0.123975\pi\)
0.925108 + 0.379705i \(0.123975\pi\)
\(180\) 0 0
\(181\) −672.000 −0.275963 −0.137982 − 0.990435i \(-0.544062\pi\)
−0.137982 + 0.990435i \(0.544062\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3933.00 −1.56303
\(186\) 0 0
\(187\) 154.000 0.0602224
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2094.00 −0.793280 −0.396640 − 0.917974i \(-0.629824\pi\)
−0.396640 + 0.917974i \(0.629824\pi\)
\(192\) 0 0
\(193\) −3712.00 −1.38443 −0.692217 − 0.721690i \(-0.743366\pi\)
−0.692217 + 0.721690i \(0.743366\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3117.00 −1.12729 −0.563647 − 0.826016i \(-0.690602\pi\)
−0.563647 + 0.826016i \(0.690602\pi\)
\(198\) 0 0
\(199\) −4702.00 −1.67495 −0.837477 − 0.546472i \(-0.815970\pi\)
−0.837477 + 0.546472i \(0.815970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.0000 0.00622341
\(204\) 0 0
\(205\) 4560.00 1.55358
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −116.000 −0.0383918
\(210\) 0 0
\(211\) −2743.00 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6023.00 −1.91054
\(216\) 0 0
\(217\) 876.000 0.274040
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1001.00 0.304681
\(222\) 0 0
\(223\) −4303.00 −1.29215 −0.646077 − 0.763273i \(-0.723591\pi\)
−0.646077 + 0.763273i \(0.723591\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2136.00 0.624543 0.312272 − 0.949993i \(-0.398910\pi\)
0.312272 + 0.949993i \(0.398910\pi\)
\(228\) 0 0
\(229\) 205.000 0.0591563 0.0295781 − 0.999562i \(-0.490584\pi\)
0.0295781 + 0.999562i \(0.490584\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −37.0000 −0.0104032 −0.00520161 − 0.999986i \(-0.501656\pi\)
−0.00520161 + 0.999986i \(0.501656\pi\)
\(234\) 0 0
\(235\) 7125.00 1.97780
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4099.00 −1.10938 −0.554691 − 0.832056i \(-0.687164\pi\)
−0.554691 + 0.832056i \(0.687164\pi\)
\(240\) 0 0
\(241\) 6258.00 1.67267 0.836334 − 0.548220i \(-0.184694\pi\)
0.836334 + 0.548220i \(0.184694\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6346.00 1.65482
\(246\) 0 0
\(247\) −754.000 −0.194234
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3992.00 1.00388 0.501938 − 0.864904i \(-0.332621\pi\)
0.501938 + 0.864904i \(0.332621\pi\)
\(252\) 0 0
\(253\) −152.000 −0.0377714
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4097.00 −0.994412 −0.497206 − 0.867633i \(-0.665641\pi\)
−0.497206 + 0.867633i \(0.665641\pi\)
\(258\) 0 0
\(259\) 621.000 0.148985
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6308.00 −1.47897 −0.739483 − 0.673175i \(-0.764930\pi\)
−0.739483 + 0.673175i \(0.764930\pi\)
\(264\) 0 0
\(265\) −13148.0 −3.04783
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4792.00 1.08615 0.543073 − 0.839685i \(-0.317261\pi\)
0.543073 + 0.839685i \(0.317261\pi\)
\(270\) 0 0
\(271\) −2585.00 −0.579438 −0.289719 − 0.957112i \(-0.593562\pi\)
−0.289719 + 0.957112i \(0.593562\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −472.000 −0.103501
\(276\) 0 0
\(277\) −5156.00 −1.11839 −0.559195 − 0.829036i \(-0.688890\pi\)
−0.559195 + 0.829036i \(0.688890\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2650.00 0.562583 0.281291 − 0.959622i \(-0.409237\pi\)
0.281291 + 0.959622i \(0.409237\pi\)
\(282\) 0 0
\(283\) 2596.00 0.545287 0.272643 − 0.962115i \(-0.412102\pi\)
0.272643 + 0.962115i \(0.412102\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −720.000 −0.148085
\(288\) 0 0
\(289\) 1016.00 0.206798
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −435.000 −0.0867337 −0.0433668 − 0.999059i \(-0.513808\pi\)
−0.0433668 + 0.999059i \(0.513808\pi\)
\(294\) 0 0
\(295\) −4066.00 −0.802480
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −988.000 −0.191095
\(300\) 0 0
\(301\) 951.000 0.182109
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9272.00 1.74070
\(306\) 0 0
\(307\) 3090.00 0.574448 0.287224 − 0.957863i \(-0.407268\pi\)
0.287224 + 0.957863i \(0.407268\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2682.00 0.489011 0.244505 − 0.969648i \(-0.421374\pi\)
0.244505 + 0.969648i \(0.421374\pi\)
\(312\) 0 0
\(313\) −801.000 −0.144649 −0.0723246 − 0.997381i \(-0.523042\pi\)
−0.0723246 + 0.997381i \(0.523042\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9750.00 −1.72749 −0.863745 − 0.503929i \(-0.831887\pi\)
−0.863745 + 0.503929i \(0.831887\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.00210618
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4466.00 −0.769334
\(324\) 0 0
\(325\) −3068.00 −0.523637
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1125.00 −0.188521
\(330\) 0 0
\(331\) −6620.00 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14858.0 2.42322
\(336\) 0 0
\(337\) 399.000 0.0644953 0.0322476 − 0.999480i \(-0.489733\pi\)
0.0322476 + 0.999480i \(0.489733\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −584.000 −0.0927430
\(342\) 0 0
\(343\) −2031.00 −0.319719
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12201.0 −1.88756 −0.943781 − 0.330571i \(-0.892759\pi\)
−0.943781 + 0.330571i \(0.892759\pi\)
\(348\) 0 0
\(349\) −189.000 −0.0289884 −0.0144942 − 0.999895i \(-0.504614\pi\)
−0.0144942 + 0.999895i \(0.504614\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10444.0 1.57473 0.787363 − 0.616490i \(-0.211446\pi\)
0.787363 + 0.616490i \(0.211446\pi\)
\(354\) 0 0
\(355\) 20083.0 3.00252
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9792.00 1.43956 0.719780 − 0.694202i \(-0.244243\pi\)
0.719780 + 0.694202i \(0.244243\pi\)
\(360\) 0 0
\(361\) −3495.00 −0.509549
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −22306.0 −3.19877
\(366\) 0 0
\(367\) 11906.0 1.69343 0.846714 − 0.532048i \(-0.178577\pi\)
0.846714 + 0.532048i \(0.178577\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2076.00 0.290514
\(372\) 0 0
\(373\) −6628.00 −0.920067 −0.460033 − 0.887902i \(-0.652163\pi\)
−0.460033 + 0.887902i \(0.652163\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −78.0000 −0.0106557
\(378\) 0 0
\(379\) 3760.00 0.509600 0.254800 − 0.966994i \(-0.417990\pi\)
0.254800 + 0.966994i \(0.417990\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10065.0 −1.34281 −0.671407 − 0.741089i \(-0.734310\pi\)
−0.671407 + 0.741089i \(0.734310\pi\)
\(384\) 0 0
\(385\) 114.000 0.0150909
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3994.00 −0.520575 −0.260288 − 0.965531i \(-0.583817\pi\)
−0.260288 + 0.965531i \(0.583817\pi\)
\(390\) 0 0
\(391\) −5852.00 −0.756901
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16948.0 2.15885
\(396\) 0 0
\(397\) 8930.00 1.12893 0.564463 − 0.825458i \(-0.309083\pi\)
0.564463 + 0.825458i \(0.309083\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5592.00 −0.696387 −0.348193 − 0.937423i \(-0.613205\pi\)
−0.348193 + 0.937423i \(0.613205\pi\)
\(402\) 0 0
\(403\) −3796.00 −0.469211
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −414.000 −0.0504207
\(408\) 0 0
\(409\) 6276.00 0.758749 0.379374 − 0.925243i \(-0.376139\pi\)
0.379374 + 0.925243i \(0.376139\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 642.000 0.0764909
\(414\) 0 0
\(415\) −13376.0 −1.58217
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 429.000 0.0500191 0.0250096 − 0.999687i \(-0.492038\pi\)
0.0250096 + 0.999687i \(0.492038\pi\)
\(420\) 0 0
\(421\) −97.0000 −0.0112292 −0.00561460 − 0.999984i \(-0.501787\pi\)
−0.00561460 + 0.999984i \(0.501787\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18172.0 −2.07405
\(426\) 0 0
\(427\) −1464.00 −0.165920
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12293.0 1.37386 0.686929 − 0.726724i \(-0.258958\pi\)
0.686929 + 0.726724i \(0.258958\pi\)
\(432\) 0 0
\(433\) −13553.0 −1.50419 −0.752097 − 0.659053i \(-0.770957\pi\)
−0.752097 + 0.659053i \(0.770957\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4408.00 0.482525
\(438\) 0 0
\(439\) 7182.00 0.780816 0.390408 − 0.920642i \(-0.372334\pi\)
0.390408 + 0.920642i \(0.372334\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8815.00 −0.945402 −0.472701 − 0.881223i \(-0.656721\pi\)
−0.472701 + 0.881223i \(0.656721\pi\)
\(444\) 0 0
\(445\) 114.000 0.0121441
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6550.00 −0.688449 −0.344225 − 0.938887i \(-0.611858\pi\)
−0.344225 + 0.938887i \(0.611858\pi\)
\(450\) 0 0
\(451\) 480.000 0.0501160
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 741.000 0.0763486
\(456\) 0 0
\(457\) −934.000 −0.0956032 −0.0478016 − 0.998857i \(-0.515222\pi\)
−0.0478016 + 0.998857i \(0.515222\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3809.00 0.384822 0.192411 − 0.981314i \(-0.438369\pi\)
0.192411 + 0.981314i \(0.438369\pi\)
\(462\) 0 0
\(463\) 17368.0 1.74332 0.871662 − 0.490107i \(-0.163042\pi\)
0.871662 + 0.490107i \(0.163042\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 348.000 0.0344829 0.0172415 − 0.999851i \(-0.494512\pi\)
0.0172415 + 0.999851i \(0.494512\pi\)
\(468\) 0 0
\(469\) −2346.00 −0.230977
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −634.000 −0.0616308
\(474\) 0 0
\(475\) 13688.0 1.32221
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19111.0 −1.82297 −0.911486 − 0.411330i \(-0.865064\pi\)
−0.911486 + 0.411330i \(0.865064\pi\)
\(480\) 0 0
\(481\) −2691.00 −0.255092
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15770.0 −1.47645
\(486\) 0 0
\(487\) 3608.00 0.335717 0.167858 − 0.985811i \(-0.446315\pi\)
0.167858 + 0.985811i \(0.446315\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6291.00 0.578226 0.289113 − 0.957295i \(-0.406640\pi\)
0.289113 + 0.957295i \(0.406640\pi\)
\(492\) 0 0
\(493\) −462.000 −0.0422057
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3171.00 −0.286195
\(498\) 0 0
\(499\) −14736.0 −1.32199 −0.660996 − 0.750389i \(-0.729866\pi\)
−0.660996 + 0.750389i \(0.729866\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2982.00 −0.264336 −0.132168 − 0.991227i \(-0.542194\pi\)
−0.132168 + 0.991227i \(0.542194\pi\)
\(504\) 0 0
\(505\) 5244.00 0.462089
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3078.00 0.268035 0.134018 − 0.990979i \(-0.457212\pi\)
0.134018 + 0.990979i \(0.457212\pi\)
\(510\) 0 0
\(511\) 3522.00 0.304900
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −22040.0 −1.88582
\(516\) 0 0
\(517\) 750.000 0.0638007
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3815.00 −0.320803 −0.160401 − 0.987052i \(-0.551279\pi\)
−0.160401 + 0.987052i \(0.551279\pi\)
\(522\) 0 0
\(523\) 5180.00 0.433089 0.216545 − 0.976273i \(-0.430521\pi\)
0.216545 + 0.976273i \(0.430521\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22484.0 −1.85848
\(528\) 0 0
\(529\) −6391.00 −0.525273
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3120.00 0.253550
\(534\) 0 0
\(535\) −6156.00 −0.497471
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 668.000 0.0533818
\(540\) 0 0
\(541\) 717.000 0.0569801 0.0284901 − 0.999594i \(-0.490930\pi\)
0.0284901 + 0.999594i \(0.490930\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3781.00 −0.297175
\(546\) 0 0
\(547\) −13949.0 −1.09034 −0.545170 − 0.838325i \(-0.683535\pi\)
−0.545170 + 0.838325i \(0.683535\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 348.000 0.0269062
\(552\) 0 0
\(553\) −2676.00 −0.205778
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22499.0 1.71151 0.855757 − 0.517378i \(-0.173092\pi\)
0.855757 + 0.517378i \(0.173092\pi\)
\(558\) 0 0
\(559\) −4121.00 −0.311806
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9715.00 0.727244 0.363622 − 0.931547i \(-0.381540\pi\)
0.363622 + 0.931547i \(0.381540\pi\)
\(564\) 0 0
\(565\) −31578.0 −2.35132
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13481.0 0.993239 0.496619 − 0.867968i \(-0.334574\pi\)
0.496619 + 0.867968i \(0.334574\pi\)
\(570\) 0 0
\(571\) −17361.0 −1.27239 −0.636195 − 0.771528i \(-0.719493\pi\)
−0.636195 + 0.771528i \(0.719493\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 17936.0 1.30084
\(576\) 0 0
\(577\) −15790.0 −1.13925 −0.569624 − 0.821905i \(-0.692911\pi\)
−0.569624 + 0.821905i \(0.692911\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2112.00 0.150810
\(582\) 0 0
\(583\) −1384.00 −0.0983181
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10084.0 0.709048 0.354524 − 0.935047i \(-0.384643\pi\)
0.354524 + 0.935047i \(0.384643\pi\)
\(588\) 0 0
\(589\) 16936.0 1.18478
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19622.0 1.35882 0.679409 − 0.733760i \(-0.262236\pi\)
0.679409 + 0.733760i \(0.262236\pi\)
\(594\) 0 0
\(595\) 4389.00 0.302406
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10746.0 −0.733004 −0.366502 − 0.930417i \(-0.619445\pi\)
−0.366502 + 0.930417i \(0.619445\pi\)
\(600\) 0 0
\(601\) 16453.0 1.11669 0.558346 − 0.829608i \(-0.311436\pi\)
0.558346 + 0.829608i \(0.311436\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25213.0 1.69430
\(606\) 0 0
\(607\) −14078.0 −0.941365 −0.470682 − 0.882303i \(-0.655992\pi\)
−0.470682 + 0.882303i \(0.655992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4875.00 0.322785
\(612\) 0 0
\(613\) 194.000 0.0127824 0.00639118 − 0.999980i \(-0.497966\pi\)
0.00639118 + 0.999980i \(0.497966\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21792.0 −1.42190 −0.710950 − 0.703242i \(-0.751735\pi\)
−0.710950 + 0.703242i \(0.751735\pi\)
\(618\) 0 0
\(619\) 5276.00 0.342585 0.171293 − 0.985220i \(-0.445206\pi\)
0.171293 + 0.985220i \(0.445206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18.0000 −0.00115755
\(624\) 0 0
\(625\) 10571.0 0.676544
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15939.0 −1.01038
\(630\) 0 0
\(631\) 3305.00 0.208510 0.104255 − 0.994551i \(-0.466754\pi\)
0.104255 + 0.994551i \(0.466754\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34960.0 2.18480
\(636\) 0 0
\(637\) 4342.00 0.270073
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21122.0 1.30151 0.650756 − 0.759287i \(-0.274452\pi\)
0.650756 + 0.759287i \(0.274452\pi\)
\(642\) 0 0
\(643\) 4098.00 0.251336 0.125668 − 0.992072i \(-0.459893\pi\)
0.125668 + 0.992072i \(0.459893\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15454.0 −0.939041 −0.469520 − 0.882922i \(-0.655573\pi\)
−0.469520 + 0.882922i \(0.655573\pi\)
\(648\) 0 0
\(649\) −428.000 −0.0258867
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26136.0 −1.56628 −0.783140 − 0.621846i \(-0.786383\pi\)
−0.783140 + 0.621846i \(0.786383\pi\)
\(654\) 0 0
\(655\) −1805.00 −0.107675
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7884.00 0.466035 0.233017 − 0.972473i \(-0.425140\pi\)
0.233017 + 0.972473i \(0.425140\pi\)
\(660\) 0 0
\(661\) −20818.0 −1.22500 −0.612501 − 0.790470i \(-0.709836\pi\)
−0.612501 + 0.790470i \(0.709836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3306.00 −0.192784
\(666\) 0 0
\(667\) 456.000 0.0264714
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 976.000 0.0561521
\(672\) 0 0
\(673\) −7595.00 −0.435016 −0.217508 − 0.976059i \(-0.569793\pi\)
−0.217508 + 0.976059i \(0.569793\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8428.00 −0.478455 −0.239228 − 0.970963i \(-0.576894\pi\)
−0.239228 + 0.970963i \(0.576894\pi\)
\(678\) 0 0
\(679\) 2490.00 0.140733
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12676.0 0.710152 0.355076 − 0.934837i \(-0.384455\pi\)
0.355076 + 0.934837i \(0.384455\pi\)
\(684\) 0 0
\(685\) 24852.0 1.38620
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8996.00 −0.497417
\(690\) 0 0
\(691\) 9736.00 0.535998 0.267999 − 0.963419i \(-0.413638\pi\)
0.267999 + 0.963419i \(0.413638\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5301.00 0.289321
\(696\) 0 0
\(697\) 18480.0 1.00428
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11052.0 0.595475 0.297738 − 0.954648i \(-0.403768\pi\)
0.297738 + 0.954648i \(0.403768\pi\)
\(702\) 0 0
\(703\) 12006.0 0.644118
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −828.000 −0.0440455
\(708\) 0 0
\(709\) 13214.0 0.699947 0.349973 − 0.936760i \(-0.386191\pi\)
0.349973 + 0.936760i \(0.386191\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 22192.0 1.16563
\(714\) 0 0
\(715\) −494.000 −0.0258385
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5150.00 −0.267125 −0.133562 − 0.991040i \(-0.542642\pi\)
−0.133562 + 0.991040i \(0.542642\pi\)
\(720\) 0 0
\(721\) 3480.00 0.179753
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1416.00 0.0725364
\(726\) 0 0
\(727\) −5178.00 −0.264156 −0.132078 − 0.991239i \(-0.542165\pi\)
−0.132078 + 0.991239i \(0.542165\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24409.0 −1.23502
\(732\) 0 0
\(733\) 26777.0 1.34929 0.674646 − 0.738141i \(-0.264296\pi\)
0.674646 + 0.738141i \(0.264296\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1564.00 0.0781692
\(738\) 0 0
\(739\) 2500.00 0.124444 0.0622219 − 0.998062i \(-0.480181\pi\)
0.0622219 + 0.998062i \(0.480181\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20267.0 −1.00071 −0.500353 − 0.865822i \(-0.666796\pi\)
−0.500353 + 0.865822i \(0.666796\pi\)
\(744\) 0 0
\(745\) −13870.0 −0.682091
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 972.000 0.0474180
\(750\) 0 0
\(751\) 14280.0 0.693854 0.346927 − 0.937892i \(-0.387225\pi\)
0.346927 + 0.937892i \(0.387225\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 45353.0 2.18618
\(756\) 0 0
\(757\) 3332.00 0.159978 0.0799892 − 0.996796i \(-0.474511\pi\)
0.0799892 + 0.996796i \(0.474511\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33178.0 1.58042 0.790211 − 0.612835i \(-0.209971\pi\)
0.790211 + 0.612835i \(0.209971\pi\)
\(762\) 0 0
\(763\) 597.000 0.0283261
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2782.00 −0.130968
\(768\) 0 0
\(769\) 10784.0 0.505697 0.252848 − 0.967506i \(-0.418633\pi\)
0.252848 + 0.967506i \(0.418633\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23169.0 1.07805 0.539024 − 0.842290i \(-0.318793\pi\)
0.539024 + 0.842290i \(0.318793\pi\)
\(774\) 0 0
\(775\) 68912.0 3.19405
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13920.0 −0.640226
\(780\) 0 0
\(781\) 2114.00 0.0968564
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12806.0 0.582249
\(786\) 0 0
\(787\) −17776.0 −0.805141 −0.402571 − 0.915389i \(-0.631883\pi\)
−0.402571 + 0.915389i \(0.631883\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4986.00 0.224124
\(792\) 0 0
\(793\) 6344.00 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17858.0 0.793680 0.396840 − 0.917888i \(-0.370107\pi\)
0.396840 + 0.917888i \(0.370107\pi\)
\(798\) 0 0
\(799\) 28875.0 1.27850
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2348.00 −0.103187
\(804\) 0 0
\(805\) −4332.00 −0.189668
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21623.0 −0.939709 −0.469854 − 0.882744i \(-0.655694\pi\)
−0.469854 + 0.882744i \(0.655694\pi\)
\(810\) 0 0
\(811\) −39652.0 −1.71686 −0.858428 − 0.512934i \(-0.828558\pi\)
−0.858428 + 0.512934i \(0.828558\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3420.00 −0.146991
\(816\) 0 0
\(817\) 18386.0 0.787325
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7707.00 −0.327620 −0.163810 − 0.986492i \(-0.552378\pi\)
−0.163810 + 0.986492i \(0.552378\pi\)
\(822\) 0 0
\(823\) −1358.00 −0.0575175 −0.0287588 − 0.999586i \(-0.509155\pi\)
−0.0287588 + 0.999586i \(0.509155\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29774.0 1.25193 0.625963 − 0.779852i \(-0.284706\pi\)
0.625963 + 0.779852i \(0.284706\pi\)
\(828\) 0 0
\(829\) 35170.0 1.47347 0.736734 − 0.676183i \(-0.236367\pi\)
0.736734 + 0.676183i \(0.236367\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25718.0 1.06972
\(834\) 0 0
\(835\) 16872.0 0.699257
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12480.0 0.513537 0.256768 − 0.966473i \(-0.417342\pi\)
0.256768 + 0.966473i \(0.417342\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3211.00 −0.130724
\(846\) 0 0
\(847\) −3981.00 −0.161498
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15732.0 0.633709
\(852\) 0 0
\(853\) 37133.0 1.49052 0.745258 − 0.666777i \(-0.232326\pi\)
0.745258 + 0.666777i \(0.232326\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3618.00 0.144211 0.0721053 − 0.997397i \(-0.477028\pi\)
0.0721053 + 0.997397i \(0.477028\pi\)
\(858\) 0 0
\(859\) −46140.0 −1.83269 −0.916343 − 0.400395i \(-0.868873\pi\)
−0.916343 + 0.400395i \(0.868873\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1753.00 0.0691458 0.0345729 − 0.999402i \(-0.488993\pi\)
0.0345729 + 0.999402i \(0.488993\pi\)
\(864\) 0 0
\(865\) 54492.0 2.14195
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1784.00 0.0696410
\(870\) 0 0
\(871\) 10166.0 0.395478
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6327.00 −0.244448
\(876\) 0 0
\(877\) 19485.0 0.750241 0.375121 − 0.926976i \(-0.377601\pi\)
0.375121 + 0.926976i \(0.377601\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3067.00 0.117287 0.0586435 − 0.998279i \(-0.481322\pi\)
0.0586435 + 0.998279i \(0.481322\pi\)
\(882\) 0 0
\(883\) −38321.0 −1.46048 −0.730240 − 0.683190i \(-0.760592\pi\)
−0.730240 + 0.683190i \(0.760592\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20936.0 0.792516 0.396258 − 0.918139i \(-0.370309\pi\)
0.396258 + 0.918139i \(0.370309\pi\)
\(888\) 0 0
\(889\) −5520.00 −0.208251
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −21750.0 −0.815046
\(894\) 0 0
\(895\) −84189.0 −3.14428
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1752.00 0.0649972
\(900\) 0 0
\(901\) −53284.0 −1.97020
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12768.0 0.468975
\(906\) 0 0
\(907\) 32321.0 1.18324 0.591621 − 0.806216i \(-0.298488\pi\)
0.591621 + 0.806216i \(0.298488\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22766.0 −0.827960 −0.413980 − 0.910286i \(-0.635862\pi\)
−0.413980 + 0.910286i \(0.635862\pi\)
\(912\) 0 0
\(913\) −1408.00 −0.0510383
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 285.000 0.0102634
\(918\) 0 0
\(919\) −3112.00 −0.111703 −0.0558517 − 0.998439i \(-0.517787\pi\)
−0.0558517 + 0.998439i \(0.517787\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13741.0 0.490022
\(924\) 0 0
\(925\) 48852.0 1.73648
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8980.00 0.317141 0.158571 − 0.987348i \(-0.449311\pi\)
0.158571 + 0.987348i \(0.449311\pi\)
\(930\) 0 0
\(931\) −19372.0 −0.681946
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2926.00 −0.102343
\(936\) 0 0
\(937\) −33306.0 −1.16122 −0.580608 − 0.814183i \(-0.697185\pi\)
−0.580608 + 0.814183i \(0.697185\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 55299.0 1.91572 0.957862 − 0.287230i \(-0.0927342\pi\)
0.957862 + 0.287230i \(0.0927342\pi\)
\(942\) 0 0
\(943\) −18240.0 −0.629879
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28578.0 −0.980634 −0.490317 − 0.871544i \(-0.663119\pi\)
−0.490317 + 0.871544i \(0.663119\pi\)
\(948\) 0 0
\(949\) −15262.0 −0.522050
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −10623.0 −0.361084 −0.180542 − 0.983567i \(-0.557785\pi\)
−0.180542 + 0.983567i \(0.557785\pi\)
\(954\) 0 0
\(955\) 39786.0 1.34811
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3924.00 −0.132130
\(960\) 0 0
\(961\) 55473.0 1.86207
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 70528.0 2.35272
\(966\) 0 0
\(967\) 11605.0 0.385927 0.192964 − 0.981206i \(-0.438190\pi\)
0.192964 + 0.981206i \(0.438190\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6057.00 0.200184 0.100092 − 0.994978i \(-0.468086\pi\)
0.100092 + 0.994978i \(0.468086\pi\)
\(972\) 0 0
\(973\) −837.000 −0.0275776
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5166.00 0.169166 0.0845829 − 0.996416i \(-0.473044\pi\)
0.0845829 + 0.996416i \(0.473044\pi\)
\(978\) 0 0
\(979\) 12.0000 0.000391748 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21075.0 −0.683813 −0.341906 − 0.939734i \(-0.611073\pi\)
−0.341906 + 0.939734i \(0.611073\pi\)
\(984\) 0 0
\(985\) 59223.0 1.91574
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24092.0 0.774602
\(990\) 0 0
\(991\) −29050.0 −0.931184 −0.465592 − 0.884999i \(-0.654159\pi\)
−0.465592 + 0.884999i \(0.654159\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 89338.0 2.84644
\(996\) 0 0
\(997\) −56450.0 −1.79317 −0.896584 − 0.442873i \(-0.853959\pi\)
−0.896584 + 0.442873i \(0.853959\pi\)
\(998\) 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 1872.4.a.a.1.1 1
3.2 odd 2 208.4.a.a.1.1 1
4.3 odd 2 936.4.a.a.1.1 1
12.11 even 2 104.4.a.b.1.1 ✓ 1
24.5 odd 2 832.4.a.p.1.1 1
24.11 even 2 832.4.a.b.1.1 1
156.155 even 2 1352.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.4.a.b.1.1 ✓ 1 12.11 even 2
208.4.a.a.1.1 1 3.2 odd 2
832.4.a.b.1.1 1 24.11 even 2
832.4.a.p.1.1 1 24.5 odd 2
936.4.a.a.1.1 1 4.3 odd 2
1352.4.a.c.1.1 1 156.155 even 2
1872.4.a.a.1.1 1 1.1 even 1 trivial