Properties

Label 1344.4.a.f.1.1
Level $1344$
Weight $4$
Character 1344.1
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
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] = [1344,4,Mod(1,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, 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("1344.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -2.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +8.00000 q^{11} +42.0000 q^{13} +6.00000 q^{15} -2.00000 q^{17} +124.000 q^{19} +21.0000 q^{21} +76.0000 q^{23} -121.000 q^{25} -27.0000 q^{27} -254.000 q^{29} -72.0000 q^{31} -24.0000 q^{33} +14.0000 q^{35} -398.000 q^{37} -126.000 q^{39} +462.000 q^{41} -212.000 q^{43} -18.0000 q^{45} -264.000 q^{47} +49.0000 q^{49} +6.00000 q^{51} +162.000 q^{53} -16.0000 q^{55} -372.000 q^{57} +772.000 q^{59} -30.0000 q^{61} -63.0000 q^{63} -84.0000 q^{65} +764.000 q^{67} -228.000 q^{69} -236.000 q^{71} +418.000 q^{73} +363.000 q^{75} -56.0000 q^{77} +552.000 q^{79} +81.0000 q^{81} -1036.00 q^{83} +4.00000 q^{85} +762.000 q^{87} +30.0000 q^{89} -294.000 q^{91} +216.000 q^{93} -248.000 q^{95} -1190.00 q^{97} +72.0000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.178885 −0.0894427 0.995992i \(-0.528509\pi\)
−0.0894427 + 0.995992i \(0.528509\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 8.00000 0.219281 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(12\) 0 0
\(13\) 42.0000 0.896054 0.448027 0.894020i \(-0.352127\pi\)
0.448027 + 0.894020i \(0.352127\pi\)
\(14\) 0 0
\(15\) 6.00000 0.103280
\(16\) 0 0
\(17\) −2.00000 −0.0285336 −0.0142668 0.999898i \(-0.504541\pi\)
−0.0142668 + 0.999898i \(0.504541\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) 76.0000 0.689004 0.344502 0.938786i \(-0.388048\pi\)
0.344502 + 0.938786i \(0.388048\pi\)
\(24\) 0 0
\(25\) −121.000 −0.968000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −254.000 −1.62644 −0.813218 0.581960i \(-0.802286\pi\)
−0.813218 + 0.581960i \(0.802286\pi\)
\(30\) 0 0
\(31\) −72.0000 −0.417148 −0.208574 0.978007i \(-0.566882\pi\)
−0.208574 + 0.978007i \(0.566882\pi\)
\(32\) 0 0
\(33\) −24.0000 −0.126602
\(34\) 0 0
\(35\) 14.0000 0.0676123
\(36\) 0 0
\(37\) −398.000 −1.76840 −0.884200 0.467109i \(-0.845296\pi\)
−0.884200 + 0.467109i \(0.845296\pi\)
\(38\) 0 0
\(39\) −126.000 −0.517337
\(40\) 0 0
\(41\) 462.000 1.75981 0.879906 0.475148i \(-0.157606\pi\)
0.879906 + 0.475148i \(0.157606\pi\)
\(42\) 0 0
\(43\) −212.000 −0.751853 −0.375927 0.926649i \(-0.622676\pi\)
−0.375927 + 0.926649i \(0.622676\pi\)
\(44\) 0 0
\(45\) −18.0000 −0.0596285
\(46\) 0 0
\(47\) −264.000 −0.819327 −0.409663 0.912237i \(-0.634354\pi\)
−0.409663 + 0.912237i \(0.634354\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 6.00000 0.0164739
\(52\) 0 0
\(53\) 162.000 0.419857 0.209928 0.977717i \(-0.432677\pi\)
0.209928 + 0.977717i \(0.432677\pi\)
\(54\) 0 0
\(55\) −16.0000 −0.0392262
\(56\) 0 0
\(57\) −372.000 −0.864432
\(58\) 0 0
\(59\) 772.000 1.70349 0.851744 0.523958i \(-0.175545\pi\)
0.851744 + 0.523958i \(0.175545\pi\)
\(60\) 0 0
\(61\) −30.0000 −0.0629690 −0.0314845 0.999504i \(-0.510023\pi\)
−0.0314845 + 0.999504i \(0.510023\pi\)
\(62\) 0 0
\(63\) −63.0000 −0.125988
\(64\) 0 0
\(65\) −84.0000 −0.160291
\(66\) 0 0
\(67\) 764.000 1.39310 0.696548 0.717510i \(-0.254718\pi\)
0.696548 + 0.717510i \(0.254718\pi\)
\(68\) 0 0
\(69\) −228.000 −0.397797
\(70\) 0 0
\(71\) −236.000 −0.394480 −0.197240 0.980355i \(-0.563198\pi\)
−0.197240 + 0.980355i \(0.563198\pi\)
\(72\) 0 0
\(73\) 418.000 0.670181 0.335090 0.942186i \(-0.391233\pi\)
0.335090 + 0.942186i \(0.391233\pi\)
\(74\) 0 0
\(75\) 363.000 0.558875
\(76\) 0 0
\(77\) −56.0000 −0.0828804
\(78\) 0 0
\(79\) 552.000 0.786137 0.393069 0.919509i \(-0.371413\pi\)
0.393069 + 0.919509i \(0.371413\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1036.00 −1.37007 −0.685035 0.728510i \(-0.740213\pi\)
−0.685035 + 0.728510i \(0.740213\pi\)
\(84\) 0 0
\(85\) 4.00000 0.00510425
\(86\) 0 0
\(87\) 762.000 0.939023
\(88\) 0 0
\(89\) 30.0000 0.0357303 0.0178651 0.999840i \(-0.494313\pi\)
0.0178651 + 0.999840i \(0.494313\pi\)
\(90\) 0 0
\(91\) −294.000 −0.338677
\(92\) 0 0
\(93\) 216.000 0.240840
\(94\) 0 0
\(95\) −248.000 −0.267834
\(96\) 0 0
\(97\) −1190.00 −1.24563 −0.622815 0.782369i \(-0.714011\pi\)
−0.622815 + 0.782369i \(0.714011\pi\)
\(98\) 0 0
\(99\) 72.0000 0.0730937
\(100\) 0 0
\(101\) −1370.00 −1.34970 −0.674852 0.737953i \(-0.735793\pi\)
−0.674852 + 0.737953i \(0.735793\pi\)
\(102\) 0 0
\(103\) 464.000 0.443876 0.221938 0.975061i \(-0.428762\pi\)
0.221938 + 0.975061i \(0.428762\pi\)
\(104\) 0 0
\(105\) −42.0000 −0.0390360
\(106\) 0 0
\(107\) 2136.00 1.92986 0.964930 0.262509i \(-0.0845500\pi\)
0.964930 + 0.262509i \(0.0845500\pi\)
\(108\) 0 0
\(109\) 1226.00 1.07733 0.538667 0.842518i \(-0.318928\pi\)
0.538667 + 0.842518i \(0.318928\pi\)
\(110\) 0 0
\(111\) 1194.00 1.02099
\(112\) 0 0
\(113\) 338.000 0.281384 0.140692 0.990053i \(-0.455067\pi\)
0.140692 + 0.990053i \(0.455067\pi\)
\(114\) 0 0
\(115\) −152.000 −0.123253
\(116\) 0 0
\(117\) 378.000 0.298685
\(118\) 0 0
\(119\) 14.0000 0.0107847
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) 0 0
\(123\) −1386.00 −1.01603
\(124\) 0 0
\(125\) 492.000 0.352047
\(126\) 0 0
\(127\) 2088.00 1.45890 0.729449 0.684035i \(-0.239777\pi\)
0.729449 + 0.684035i \(0.239777\pi\)
\(128\) 0 0
\(129\) 636.000 0.434083
\(130\) 0 0
\(131\) 292.000 0.194749 0.0973747 0.995248i \(-0.468955\pi\)
0.0973747 + 0.995248i \(0.468955\pi\)
\(132\) 0 0
\(133\) −868.000 −0.565903
\(134\) 0 0
\(135\) 54.0000 0.0344265
\(136\) 0 0
\(137\) 818.000 0.510120 0.255060 0.966925i \(-0.417905\pi\)
0.255060 + 0.966925i \(0.417905\pi\)
\(138\) 0 0
\(139\) 2156.00 1.31561 0.657804 0.753189i \(-0.271485\pi\)
0.657804 + 0.753189i \(0.271485\pi\)
\(140\) 0 0
\(141\) 792.000 0.473039
\(142\) 0 0
\(143\) 336.000 0.196488
\(144\) 0 0
\(145\) 508.000 0.290946
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) 0 0
\(149\) 2850.00 1.56699 0.783494 0.621400i \(-0.213436\pi\)
0.783494 + 0.621400i \(0.213436\pi\)
\(150\) 0 0
\(151\) 1672.00 0.901096 0.450548 0.892752i \(-0.351229\pi\)
0.450548 + 0.892752i \(0.351229\pi\)
\(152\) 0 0
\(153\) −18.0000 −0.00951120
\(154\) 0 0
\(155\) 144.000 0.0746217
\(156\) 0 0
\(157\) −446.000 −0.226718 −0.113359 0.993554i \(-0.536161\pi\)
−0.113359 + 0.993554i \(0.536161\pi\)
\(158\) 0 0
\(159\) −486.000 −0.242404
\(160\) 0 0
\(161\) −532.000 −0.260419
\(162\) 0 0
\(163\) −2708.00 −1.30127 −0.650635 0.759391i \(-0.725497\pi\)
−0.650635 + 0.759391i \(0.725497\pi\)
\(164\) 0 0
\(165\) 48.0000 0.0226472
\(166\) 0 0
\(167\) 896.000 0.415177 0.207589 0.978216i \(-0.433439\pi\)
0.207589 + 0.978216i \(0.433439\pi\)
\(168\) 0 0
\(169\) −433.000 −0.197087
\(170\) 0 0
\(171\) 1116.00 0.499080
\(172\) 0 0
\(173\) −4034.00 −1.77283 −0.886414 0.462893i \(-0.846811\pi\)
−0.886414 + 0.462893i \(0.846811\pi\)
\(174\) 0 0
\(175\) 847.000 0.365870
\(176\) 0 0
\(177\) −2316.00 −0.983510
\(178\) 0 0
\(179\) 3480.00 1.45311 0.726557 0.687106i \(-0.241119\pi\)
0.726557 + 0.687106i \(0.241119\pi\)
\(180\) 0 0
\(181\) 2898.00 1.19009 0.595046 0.803692i \(-0.297134\pi\)
0.595046 + 0.803692i \(0.297134\pi\)
\(182\) 0 0
\(183\) 90.0000 0.0363551
\(184\) 0 0
\(185\) 796.000 0.316341
\(186\) 0 0
\(187\) −16.0000 −0.00625688
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 2652.00 1.00467 0.502335 0.864673i \(-0.332474\pi\)
0.502335 + 0.864673i \(0.332474\pi\)
\(192\) 0 0
\(193\) 146.000 0.0544524 0.0272262 0.999629i \(-0.491333\pi\)
0.0272262 + 0.999629i \(0.491333\pi\)
\(194\) 0 0
\(195\) 252.000 0.0925441
\(196\) 0 0
\(197\) 2546.00 0.920787 0.460393 0.887715i \(-0.347708\pi\)
0.460393 + 0.887715i \(0.347708\pi\)
\(198\) 0 0
\(199\) −2536.00 −0.903378 −0.451689 0.892175i \(-0.649178\pi\)
−0.451689 + 0.892175i \(0.649178\pi\)
\(200\) 0 0
\(201\) −2292.00 −0.804305
\(202\) 0 0
\(203\) 1778.00 0.614735
\(204\) 0 0
\(205\) −924.000 −0.314805
\(206\) 0 0
\(207\) 684.000 0.229668
\(208\) 0 0
\(209\) 992.000 0.328316
\(210\) 0 0
\(211\) 1300.00 0.424150 0.212075 0.977253i \(-0.431978\pi\)
0.212075 + 0.977253i \(0.431978\pi\)
\(212\) 0 0
\(213\) 708.000 0.227753
\(214\) 0 0
\(215\) 424.000 0.134496
\(216\) 0 0
\(217\) 504.000 0.157667
\(218\) 0 0
\(219\) −1254.00 −0.386929
\(220\) 0 0
\(221\) −84.0000 −0.0255677
\(222\) 0 0
\(223\) 2576.00 0.773550 0.386775 0.922174i \(-0.373589\pi\)
0.386775 + 0.922174i \(0.373589\pi\)
\(224\) 0 0
\(225\) −1089.00 −0.322667
\(226\) 0 0
\(227\) 1836.00 0.536826 0.268413 0.963304i \(-0.413501\pi\)
0.268413 + 0.963304i \(0.413501\pi\)
\(228\) 0 0
\(229\) 1874.00 0.540775 0.270387 0.962752i \(-0.412848\pi\)
0.270387 + 0.962752i \(0.412848\pi\)
\(230\) 0 0
\(231\) 168.000 0.0478510
\(232\) 0 0
\(233\) 3730.00 1.04876 0.524379 0.851485i \(-0.324298\pi\)
0.524379 + 0.851485i \(0.324298\pi\)
\(234\) 0 0
\(235\) 528.000 0.146566
\(236\) 0 0
\(237\) −1656.00 −0.453877
\(238\) 0 0
\(239\) 2004.00 0.542377 0.271188 0.962526i \(-0.412583\pi\)
0.271188 + 0.962526i \(0.412583\pi\)
\(240\) 0 0
\(241\) −646.000 −0.172666 −0.0863330 0.996266i \(-0.527515\pi\)
−0.0863330 + 0.996266i \(0.527515\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −98.0000 −0.0255551
\(246\) 0 0
\(247\) 5208.00 1.34161
\(248\) 0 0
\(249\) 3108.00 0.791010
\(250\) 0 0
\(251\) −1260.00 −0.316855 −0.158427 0.987371i \(-0.550642\pi\)
−0.158427 + 0.987371i \(0.550642\pi\)
\(252\) 0 0
\(253\) 608.000 0.151086
\(254\) 0 0
\(255\) −12.0000 −0.00294694
\(256\) 0 0
\(257\) 5910.00 1.43446 0.717229 0.696838i \(-0.245410\pi\)
0.717229 + 0.696838i \(0.245410\pi\)
\(258\) 0 0
\(259\) 2786.00 0.668392
\(260\) 0 0
\(261\) −2286.00 −0.542145
\(262\) 0 0
\(263\) 2988.00 0.700563 0.350281 0.936645i \(-0.386086\pi\)
0.350281 + 0.936645i \(0.386086\pi\)
\(264\) 0 0
\(265\) −324.000 −0.0751063
\(266\) 0 0
\(267\) −90.0000 −0.0206289
\(268\) 0 0
\(269\) 1318.00 0.298736 0.149368 0.988782i \(-0.452276\pi\)
0.149368 + 0.988782i \(0.452276\pi\)
\(270\) 0 0
\(271\) −5640.00 −1.26423 −0.632114 0.774876i \(-0.717812\pi\)
−0.632114 + 0.774876i \(0.717812\pi\)
\(272\) 0 0
\(273\) 882.000 0.195535
\(274\) 0 0
\(275\) −968.000 −0.212264
\(276\) 0 0
\(277\) −6446.00 −1.39820 −0.699102 0.715022i \(-0.746417\pi\)
−0.699102 + 0.715022i \(0.746417\pi\)
\(278\) 0 0
\(279\) −648.000 −0.139049
\(280\) 0 0
\(281\) 4930.00 1.04662 0.523308 0.852144i \(-0.324698\pi\)
0.523308 + 0.852144i \(0.324698\pi\)
\(282\) 0 0
\(283\) 6260.00 1.31491 0.657453 0.753496i \(-0.271634\pi\)
0.657453 + 0.753496i \(0.271634\pi\)
\(284\) 0 0
\(285\) 744.000 0.154634
\(286\) 0 0
\(287\) −3234.00 −0.665146
\(288\) 0 0
\(289\) −4909.00 −0.999186
\(290\) 0 0
\(291\) 3570.00 0.719165
\(292\) 0 0
\(293\) 2310.00 0.460586 0.230293 0.973121i \(-0.426032\pi\)
0.230293 + 0.973121i \(0.426032\pi\)
\(294\) 0 0
\(295\) −1544.00 −0.304729
\(296\) 0 0
\(297\) −216.000 −0.0422006
\(298\) 0 0
\(299\) 3192.00 0.617385
\(300\) 0 0
\(301\) 1484.00 0.284174
\(302\) 0 0
\(303\) 4110.00 0.779252
\(304\) 0 0
\(305\) 60.0000 0.0112642
\(306\) 0 0
\(307\) −196.000 −0.0364375 −0.0182187 0.999834i \(-0.505800\pi\)
−0.0182187 + 0.999834i \(0.505800\pi\)
\(308\) 0 0
\(309\) −1392.00 −0.256272
\(310\) 0 0
\(311\) −6736.00 −1.22818 −0.614089 0.789237i \(-0.710477\pi\)
−0.614089 + 0.789237i \(0.710477\pi\)
\(312\) 0 0
\(313\) 394.000 0.0711508 0.0355754 0.999367i \(-0.488674\pi\)
0.0355754 + 0.999367i \(0.488674\pi\)
\(314\) 0 0
\(315\) 126.000 0.0225374
\(316\) 0 0
\(317\) 6714.00 1.18958 0.594788 0.803882i \(-0.297236\pi\)
0.594788 + 0.803882i \(0.297236\pi\)
\(318\) 0 0
\(319\) −2032.00 −0.356646
\(320\) 0 0
\(321\) −6408.00 −1.11420
\(322\) 0 0
\(323\) −248.000 −0.0427216
\(324\) 0 0
\(325\) −5082.00 −0.867380
\(326\) 0 0
\(327\) −3678.00 −0.622000
\(328\) 0 0
\(329\) 1848.00 0.309676
\(330\) 0 0
\(331\) −692.000 −0.114912 −0.0574558 0.998348i \(-0.518299\pi\)
−0.0574558 + 0.998348i \(0.518299\pi\)
\(332\) 0 0
\(333\) −3582.00 −0.589467
\(334\) 0 0
\(335\) −1528.00 −0.249205
\(336\) 0 0
\(337\) −1566.00 −0.253132 −0.126566 0.991958i \(-0.540396\pi\)
−0.126566 + 0.991958i \(0.540396\pi\)
\(338\) 0 0
\(339\) −1014.00 −0.162457
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 456.000 0.0711600
\(346\) 0 0
\(347\) 5328.00 0.824271 0.412135 0.911123i \(-0.364783\pi\)
0.412135 + 0.911123i \(0.364783\pi\)
\(348\) 0 0
\(349\) −11326.0 −1.73715 −0.868577 0.495554i \(-0.834965\pi\)
−0.868577 + 0.495554i \(0.834965\pi\)
\(350\) 0 0
\(351\) −1134.00 −0.172446
\(352\) 0 0
\(353\) −2130.00 −0.321157 −0.160579 0.987023i \(-0.551336\pi\)
−0.160579 + 0.987023i \(0.551336\pi\)
\(354\) 0 0
\(355\) 472.000 0.0705666
\(356\) 0 0
\(357\) −42.0000 −0.00622654
\(358\) 0 0
\(359\) 3044.00 0.447510 0.223755 0.974645i \(-0.428168\pi\)
0.223755 + 0.974645i \(0.428168\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) 3801.00 0.549589
\(364\) 0 0
\(365\) −836.000 −0.119886
\(366\) 0 0
\(367\) 12416.0 1.76597 0.882984 0.469404i \(-0.155531\pi\)
0.882984 + 0.469404i \(0.155531\pi\)
\(368\) 0 0
\(369\) 4158.00 0.586604
\(370\) 0 0
\(371\) −1134.00 −0.158691
\(372\) 0 0
\(373\) 7442.00 1.03306 0.516531 0.856268i \(-0.327223\pi\)
0.516531 + 0.856268i \(0.327223\pi\)
\(374\) 0 0
\(375\) −1476.00 −0.203254
\(376\) 0 0
\(377\) −10668.0 −1.45737
\(378\) 0 0
\(379\) −100.000 −0.0135532 −0.00677659 0.999977i \(-0.502157\pi\)
−0.00677659 + 0.999977i \(0.502157\pi\)
\(380\) 0 0
\(381\) −6264.00 −0.842295
\(382\) 0 0
\(383\) 8080.00 1.07799 0.538993 0.842310i \(-0.318805\pi\)
0.538993 + 0.842310i \(0.318805\pi\)
\(384\) 0 0
\(385\) 112.000 0.0148261
\(386\) 0 0
\(387\) −1908.00 −0.250618
\(388\) 0 0
\(389\) 5482.00 0.714520 0.357260 0.934005i \(-0.383711\pi\)
0.357260 + 0.934005i \(0.383711\pi\)
\(390\) 0 0
\(391\) −152.000 −0.0196598
\(392\) 0 0
\(393\) −876.000 −0.112439
\(394\) 0 0
\(395\) −1104.00 −0.140629
\(396\) 0 0
\(397\) −10446.0 −1.32058 −0.660289 0.751011i \(-0.729566\pi\)
−0.660289 + 0.751011i \(0.729566\pi\)
\(398\) 0 0
\(399\) 2604.00 0.326724
\(400\) 0 0
\(401\) −11334.0 −1.41145 −0.705727 0.708484i \(-0.749379\pi\)
−0.705727 + 0.708484i \(0.749379\pi\)
\(402\) 0 0
\(403\) −3024.00 −0.373787
\(404\) 0 0
\(405\) −162.000 −0.0198762
\(406\) 0 0
\(407\) −3184.00 −0.387776
\(408\) 0 0
\(409\) 8594.00 1.03899 0.519494 0.854474i \(-0.326121\pi\)
0.519494 + 0.854474i \(0.326121\pi\)
\(410\) 0 0
\(411\) −2454.00 −0.294518
\(412\) 0 0
\(413\) −5404.00 −0.643858
\(414\) 0 0
\(415\) 2072.00 0.245085
\(416\) 0 0
\(417\) −6468.00 −0.759567
\(418\) 0 0
\(419\) −10500.0 −1.22424 −0.612122 0.790763i \(-0.709684\pi\)
−0.612122 + 0.790763i \(0.709684\pi\)
\(420\) 0 0
\(421\) 12066.0 1.39682 0.698410 0.715698i \(-0.253891\pi\)
0.698410 + 0.715698i \(0.253891\pi\)
\(422\) 0 0
\(423\) −2376.00 −0.273109
\(424\) 0 0
\(425\) 242.000 0.0276205
\(426\) 0 0
\(427\) 210.000 0.0238000
\(428\) 0 0
\(429\) −1008.00 −0.113442
\(430\) 0 0
\(431\) 4332.00 0.484142 0.242071 0.970259i \(-0.422173\pi\)
0.242071 + 0.970259i \(0.422173\pi\)
\(432\) 0 0
\(433\) −1918.00 −0.212871 −0.106436 0.994320i \(-0.533944\pi\)
−0.106436 + 0.994320i \(0.533944\pi\)
\(434\) 0 0
\(435\) −1524.00 −0.167977
\(436\) 0 0
\(437\) 9424.00 1.03160
\(438\) 0 0
\(439\) −7992.00 −0.868878 −0.434439 0.900701i \(-0.643053\pi\)
−0.434439 + 0.900701i \(0.643053\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −3184.00 −0.341482 −0.170741 0.985316i \(-0.554616\pi\)
−0.170741 + 0.985316i \(0.554616\pi\)
\(444\) 0 0
\(445\) −60.0000 −0.00639162
\(446\) 0 0
\(447\) −8550.00 −0.904700
\(448\) 0 0
\(449\) 11426.0 1.20095 0.600475 0.799644i \(-0.294978\pi\)
0.600475 + 0.799644i \(0.294978\pi\)
\(450\) 0 0
\(451\) 3696.00 0.385893
\(452\) 0 0
\(453\) −5016.00 −0.520248
\(454\) 0 0
\(455\) 588.000 0.0605843
\(456\) 0 0
\(457\) −16934.0 −1.73335 −0.866673 0.498877i \(-0.833746\pi\)
−0.866673 + 0.498877i \(0.833746\pi\)
\(458\) 0 0
\(459\) 54.0000 0.00549129
\(460\) 0 0
\(461\) 17038.0 1.72134 0.860671 0.509161i \(-0.170044\pi\)
0.860671 + 0.509161i \(0.170044\pi\)
\(462\) 0 0
\(463\) −13592.0 −1.36431 −0.682153 0.731209i \(-0.738956\pi\)
−0.682153 + 0.731209i \(0.738956\pi\)
\(464\) 0 0
\(465\) −432.000 −0.0430828
\(466\) 0 0
\(467\) −8612.00 −0.853353 −0.426676 0.904404i \(-0.640316\pi\)
−0.426676 + 0.904404i \(0.640316\pi\)
\(468\) 0 0
\(469\) −5348.00 −0.526541
\(470\) 0 0
\(471\) 1338.00 0.130896
\(472\) 0 0
\(473\) −1696.00 −0.164867
\(474\) 0 0
\(475\) −15004.0 −1.44933
\(476\) 0 0
\(477\) 1458.00 0.139952
\(478\) 0 0
\(479\) 7432.00 0.708928 0.354464 0.935070i \(-0.384663\pi\)
0.354464 + 0.935070i \(0.384663\pi\)
\(480\) 0 0
\(481\) −16716.0 −1.58458
\(482\) 0 0
\(483\) 1596.00 0.150353
\(484\) 0 0
\(485\) 2380.00 0.222825
\(486\) 0 0
\(487\) −6616.00 −0.615605 −0.307802 0.951450i \(-0.599594\pi\)
−0.307802 + 0.951450i \(0.599594\pi\)
\(488\) 0 0
\(489\) 8124.00 0.751288
\(490\) 0 0
\(491\) −17040.0 −1.56620 −0.783100 0.621896i \(-0.786363\pi\)
−0.783100 + 0.621896i \(0.786363\pi\)
\(492\) 0 0
\(493\) 508.000 0.0464081
\(494\) 0 0
\(495\) −144.000 −0.0130754
\(496\) 0 0
\(497\) 1652.00 0.149099
\(498\) 0 0
\(499\) 2948.00 0.264470 0.132235 0.991218i \(-0.457785\pi\)
0.132235 + 0.991218i \(0.457785\pi\)
\(500\) 0 0
\(501\) −2688.00 −0.239703
\(502\) 0 0
\(503\) 17304.0 1.53389 0.766946 0.641712i \(-0.221776\pi\)
0.766946 + 0.641712i \(0.221776\pi\)
\(504\) 0 0
\(505\) 2740.00 0.241442
\(506\) 0 0
\(507\) 1299.00 0.113788
\(508\) 0 0
\(509\) −4650.00 −0.404927 −0.202463 0.979290i \(-0.564895\pi\)
−0.202463 + 0.979290i \(0.564895\pi\)
\(510\) 0 0
\(511\) −2926.00 −0.253305
\(512\) 0 0
\(513\) −3348.00 −0.288144
\(514\) 0 0
\(515\) −928.000 −0.0794030
\(516\) 0 0
\(517\) −2112.00 −0.179663
\(518\) 0 0
\(519\) 12102.0 1.02354
\(520\) 0 0
\(521\) 16854.0 1.41725 0.708625 0.705585i \(-0.249316\pi\)
0.708625 + 0.705585i \(0.249316\pi\)
\(522\) 0 0
\(523\) 124.000 0.0103674 0.00518369 0.999987i \(-0.498350\pi\)
0.00518369 + 0.999987i \(0.498350\pi\)
\(524\) 0 0
\(525\) −2541.00 −0.211235
\(526\) 0 0
\(527\) 144.000 0.0119027
\(528\) 0 0
\(529\) −6391.00 −0.525273
\(530\) 0 0
\(531\) 6948.00 0.567830
\(532\) 0 0
\(533\) 19404.0 1.57689
\(534\) 0 0
\(535\) −4272.00 −0.345224
\(536\) 0 0
\(537\) −10440.0 −0.838956
\(538\) 0 0
\(539\) 392.000 0.0313259
\(540\) 0 0
\(541\) −5382.00 −0.427708 −0.213854 0.976866i \(-0.568602\pi\)
−0.213854 + 0.976866i \(0.568602\pi\)
\(542\) 0 0
\(543\) −8694.00 −0.687100
\(544\) 0 0
\(545\) −2452.00 −0.192720
\(546\) 0 0
\(547\) −17460.0 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(548\) 0 0
\(549\) −270.000 −0.0209897
\(550\) 0 0
\(551\) −31496.0 −2.43516
\(552\) 0 0
\(553\) −3864.00 −0.297132
\(554\) 0 0
\(555\) −2388.00 −0.182640
\(556\) 0 0
\(557\) 9514.00 0.723736 0.361868 0.932229i \(-0.382139\pi\)
0.361868 + 0.932229i \(0.382139\pi\)
\(558\) 0 0
\(559\) −8904.00 −0.673701
\(560\) 0 0
\(561\) 48.0000 0.00361241
\(562\) 0 0
\(563\) −3988.00 −0.298533 −0.149267 0.988797i \(-0.547691\pi\)
−0.149267 + 0.988797i \(0.547691\pi\)
\(564\) 0 0
\(565\) −676.000 −0.0503355
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) 11346.0 0.835939 0.417969 0.908461i \(-0.362742\pi\)
0.417969 + 0.908461i \(0.362742\pi\)
\(570\) 0 0
\(571\) 8436.00 0.618276 0.309138 0.951017i \(-0.399959\pi\)
0.309138 + 0.951017i \(0.399959\pi\)
\(572\) 0 0
\(573\) −7956.00 −0.580047
\(574\) 0 0
\(575\) −9196.00 −0.666956
\(576\) 0 0
\(577\) 2098.00 0.151371 0.0756853 0.997132i \(-0.475886\pi\)
0.0756853 + 0.997132i \(0.475886\pi\)
\(578\) 0 0
\(579\) −438.000 −0.0314381
\(580\) 0 0
\(581\) 7252.00 0.517838
\(582\) 0 0
\(583\) 1296.00 0.0920666
\(584\) 0 0
\(585\) −756.000 −0.0534303
\(586\) 0 0
\(587\) −9436.00 −0.663484 −0.331742 0.943370i \(-0.607636\pi\)
−0.331742 + 0.943370i \(0.607636\pi\)
\(588\) 0 0
\(589\) −8928.00 −0.624570
\(590\) 0 0
\(591\) −7638.00 −0.531616
\(592\) 0 0
\(593\) −1314.00 −0.0909941 −0.0454971 0.998964i \(-0.514487\pi\)
−0.0454971 + 0.998964i \(0.514487\pi\)
\(594\) 0 0
\(595\) −28.0000 −0.00192922
\(596\) 0 0
\(597\) 7608.00 0.521566
\(598\) 0 0
\(599\) −8940.00 −0.609814 −0.304907 0.952382i \(-0.598625\pi\)
−0.304907 + 0.952382i \(0.598625\pi\)
\(600\) 0 0
\(601\) 16058.0 1.08988 0.544941 0.838474i \(-0.316552\pi\)
0.544941 + 0.838474i \(0.316552\pi\)
\(602\) 0 0
\(603\) 6876.00 0.464365
\(604\) 0 0
\(605\) 2534.00 0.170284
\(606\) 0 0
\(607\) 3936.00 0.263192 0.131596 0.991303i \(-0.457990\pi\)
0.131596 + 0.991303i \(0.457990\pi\)
\(608\) 0 0
\(609\) −5334.00 −0.354917
\(610\) 0 0
\(611\) −11088.0 −0.734161
\(612\) 0 0
\(613\) −174.000 −0.0114646 −0.00573230 0.999984i \(-0.501825\pi\)
−0.00573230 + 0.999984i \(0.501825\pi\)
\(614\) 0 0
\(615\) 2772.00 0.181753
\(616\) 0 0
\(617\) 16018.0 1.04515 0.522577 0.852592i \(-0.324971\pi\)
0.522577 + 0.852592i \(0.324971\pi\)
\(618\) 0 0
\(619\) 3068.00 0.199214 0.0996069 0.995027i \(-0.468241\pi\)
0.0996069 + 0.995027i \(0.468241\pi\)
\(620\) 0 0
\(621\) −2052.00 −0.132599
\(622\) 0 0
\(623\) −210.000 −0.0135048
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) 0 0
\(627\) −2976.00 −0.189553
\(628\) 0 0
\(629\) 796.000 0.0504588
\(630\) 0 0
\(631\) 24656.0 1.55553 0.777765 0.628555i \(-0.216353\pi\)
0.777765 + 0.628555i \(0.216353\pi\)
\(632\) 0 0
\(633\) −3900.00 −0.244883
\(634\) 0 0
\(635\) −4176.00 −0.260976
\(636\) 0 0
\(637\) 2058.00 0.128008
\(638\) 0 0
\(639\) −2124.00 −0.131493
\(640\) 0 0
\(641\) 7594.00 0.467933 0.233966 0.972245i \(-0.424829\pi\)
0.233966 + 0.972245i \(0.424829\pi\)
\(642\) 0 0
\(643\) 3724.00 0.228398 0.114199 0.993458i \(-0.463570\pi\)
0.114199 + 0.993458i \(0.463570\pi\)
\(644\) 0 0
\(645\) −1272.00 −0.0776511
\(646\) 0 0
\(647\) 3792.00 0.230416 0.115208 0.993341i \(-0.463247\pi\)
0.115208 + 0.993341i \(0.463247\pi\)
\(648\) 0 0
\(649\) 6176.00 0.373543
\(650\) 0 0
\(651\) −1512.00 −0.0910291
\(652\) 0 0
\(653\) −24702.0 −1.48034 −0.740171 0.672418i \(-0.765256\pi\)
−0.740171 + 0.672418i \(0.765256\pi\)
\(654\) 0 0
\(655\) −584.000 −0.0348378
\(656\) 0 0
\(657\) 3762.00 0.223394
\(658\) 0 0
\(659\) 20144.0 1.19074 0.595371 0.803451i \(-0.297005\pi\)
0.595371 + 0.803451i \(0.297005\pi\)
\(660\) 0 0
\(661\) 2522.00 0.148403 0.0742015 0.997243i \(-0.476359\pi\)
0.0742015 + 0.997243i \(0.476359\pi\)
\(662\) 0 0
\(663\) 252.000 0.0147615
\(664\) 0 0
\(665\) 1736.00 0.101232
\(666\) 0 0
\(667\) −19304.0 −1.12062
\(668\) 0 0
\(669\) −7728.00 −0.446609
\(670\) 0 0
\(671\) −240.000 −0.0138079
\(672\) 0 0
\(673\) −10414.0 −0.596479 −0.298239 0.954491i \(-0.596399\pi\)
−0.298239 + 0.954491i \(0.596399\pi\)
\(674\) 0 0
\(675\) 3267.00 0.186292
\(676\) 0 0
\(677\) 22230.0 1.26199 0.630996 0.775786i \(-0.282647\pi\)
0.630996 + 0.775786i \(0.282647\pi\)
\(678\) 0 0
\(679\) 8330.00 0.470804
\(680\) 0 0
\(681\) −5508.00 −0.309937
\(682\) 0 0
\(683\) −18192.0 −1.01918 −0.509588 0.860418i \(-0.670202\pi\)
−0.509588 + 0.860418i \(0.670202\pi\)
\(684\) 0 0
\(685\) −1636.00 −0.0912531
\(686\) 0 0
\(687\) −5622.00 −0.312216
\(688\) 0 0
\(689\) 6804.00 0.376214
\(690\) 0 0
\(691\) −8108.00 −0.446372 −0.223186 0.974776i \(-0.571646\pi\)
−0.223186 + 0.974776i \(0.571646\pi\)
\(692\) 0 0
\(693\) −504.000 −0.0276268
\(694\) 0 0
\(695\) −4312.00 −0.235343
\(696\) 0 0
\(697\) −924.000 −0.0502138
\(698\) 0 0
\(699\) −11190.0 −0.605500
\(700\) 0 0
\(701\) 5794.00 0.312177 0.156089 0.987743i \(-0.450111\pi\)
0.156089 + 0.987743i \(0.450111\pi\)
\(702\) 0 0
\(703\) −49352.0 −2.64772
\(704\) 0 0
\(705\) −1584.00 −0.0846197
\(706\) 0 0
\(707\) 9590.00 0.510140
\(708\) 0 0
\(709\) 1954.00 0.103504 0.0517518 0.998660i \(-0.483520\pi\)
0.0517518 + 0.998660i \(0.483520\pi\)
\(710\) 0 0
\(711\) 4968.00 0.262046
\(712\) 0 0
\(713\) −5472.00 −0.287417
\(714\) 0 0
\(715\) −672.000 −0.0351488
\(716\) 0 0
\(717\) −6012.00 −0.313141
\(718\) 0 0
\(719\) −32016.0 −1.66063 −0.830317 0.557292i \(-0.811840\pi\)
−0.830317 + 0.557292i \(0.811840\pi\)
\(720\) 0 0
\(721\) −3248.00 −0.167770
\(722\) 0 0
\(723\) 1938.00 0.0996888
\(724\) 0 0
\(725\) 30734.0 1.57439
\(726\) 0 0
\(727\) −23072.0 −1.17702 −0.588510 0.808490i \(-0.700285\pi\)
−0.588510 + 0.808490i \(0.700285\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 424.000 0.0214531
\(732\) 0 0
\(733\) −31782.0 −1.60149 −0.800747 0.599003i \(-0.795564\pi\)
−0.800747 + 0.599003i \(0.795564\pi\)
\(734\) 0 0
\(735\) 294.000 0.0147542
\(736\) 0 0
\(737\) 6112.00 0.305480
\(738\) 0 0
\(739\) 24396.0 1.21437 0.607186 0.794559i \(-0.292298\pi\)
0.607186 + 0.794559i \(0.292298\pi\)
\(740\) 0 0
\(741\) −15624.0 −0.774578
\(742\) 0 0
\(743\) −32604.0 −1.60986 −0.804929 0.593371i \(-0.797797\pi\)
−0.804929 + 0.593371i \(0.797797\pi\)
\(744\) 0 0
\(745\) −5700.00 −0.280311
\(746\) 0 0
\(747\) −9324.00 −0.456690
\(748\) 0 0
\(749\) −14952.0 −0.729418
\(750\) 0 0
\(751\) −7680.00 −0.373165 −0.186583 0.982439i \(-0.559741\pi\)
−0.186583 + 0.982439i \(0.559741\pi\)
\(752\) 0 0
\(753\) 3780.00 0.182936
\(754\) 0 0
\(755\) −3344.00 −0.161193
\(756\) 0 0
\(757\) −366.000 −0.0175727 −0.00878633 0.999961i \(-0.502797\pi\)
−0.00878633 + 0.999961i \(0.502797\pi\)
\(758\) 0 0
\(759\) −1824.00 −0.0872293
\(760\) 0 0
\(761\) 29374.0 1.39922 0.699610 0.714525i \(-0.253357\pi\)
0.699610 + 0.714525i \(0.253357\pi\)
\(762\) 0 0
\(763\) −8582.00 −0.407194
\(764\) 0 0
\(765\) 36.0000 0.00170142
\(766\) 0 0
\(767\) 32424.0 1.52642
\(768\) 0 0
\(769\) −38990.0 −1.82837 −0.914184 0.405299i \(-0.867167\pi\)
−0.914184 + 0.405299i \(0.867167\pi\)
\(770\) 0 0
\(771\) −17730.0 −0.828185
\(772\) 0 0
\(773\) 20470.0 0.952464 0.476232 0.879320i \(-0.342002\pi\)
0.476232 + 0.879320i \(0.342002\pi\)
\(774\) 0 0
\(775\) 8712.00 0.403799
\(776\) 0 0
\(777\) −8358.00 −0.385896
\(778\) 0 0
\(779\) 57288.0 2.63486
\(780\) 0 0
\(781\) −1888.00 −0.0865019
\(782\) 0 0
\(783\) 6858.00 0.313008
\(784\) 0 0
\(785\) 892.000 0.0405565
\(786\) 0 0
\(787\) −29916.0 −1.35501 −0.677503 0.735520i \(-0.736938\pi\)
−0.677503 + 0.735520i \(0.736938\pi\)
\(788\) 0 0
\(789\) −8964.00 −0.404470
\(790\) 0 0
\(791\) −2366.00 −0.106353
\(792\) 0 0
\(793\) −1260.00 −0.0564236
\(794\) 0 0
\(795\) 972.000 0.0433626
\(796\) 0 0
\(797\) −4914.00 −0.218398 −0.109199 0.994020i \(-0.534828\pi\)
−0.109199 + 0.994020i \(0.534828\pi\)
\(798\) 0 0
\(799\) 528.000 0.0233783
\(800\) 0 0
\(801\) 270.000 0.0119101
\(802\) 0 0
\(803\) 3344.00 0.146958
\(804\) 0 0
\(805\) 1064.00 0.0465852
\(806\) 0 0
\(807\) −3954.00 −0.172475
\(808\) 0 0
\(809\) 34250.0 1.48846 0.744231 0.667922i \(-0.232816\pi\)
0.744231 + 0.667922i \(0.232816\pi\)
\(810\) 0 0
\(811\) 41804.0 1.81003 0.905017 0.425376i \(-0.139858\pi\)
0.905017 + 0.425376i \(0.139858\pi\)
\(812\) 0 0
\(813\) 16920.0 0.729902
\(814\) 0 0
\(815\) 5416.00 0.232778
\(816\) 0 0
\(817\) −26288.0 −1.12570
\(818\) 0 0
\(819\) −2646.00 −0.112892
\(820\) 0 0
\(821\) −30862.0 −1.31193 −0.655963 0.754793i \(-0.727737\pi\)
−0.655963 + 0.754793i \(0.727737\pi\)
\(822\) 0 0
\(823\) 10576.0 0.447942 0.223971 0.974596i \(-0.428098\pi\)
0.223971 + 0.974596i \(0.428098\pi\)
\(824\) 0 0
\(825\) 2904.00 0.122551
\(826\) 0 0
\(827\) 10680.0 0.449069 0.224534 0.974466i \(-0.427914\pi\)
0.224534 + 0.974466i \(0.427914\pi\)
\(828\) 0 0
\(829\) 1178.00 0.0493530 0.0246765 0.999695i \(-0.492144\pi\)
0.0246765 + 0.999695i \(0.492144\pi\)
\(830\) 0 0
\(831\) 19338.0 0.807254
\(832\) 0 0
\(833\) −98.0000 −0.00407623
\(834\) 0 0
\(835\) −1792.00 −0.0742691
\(836\) 0 0
\(837\) 1944.00 0.0802801
\(838\) 0 0
\(839\) −5600.00 −0.230433 −0.115217 0.993340i \(-0.536756\pi\)
−0.115217 + 0.993340i \(0.536756\pi\)
\(840\) 0 0
\(841\) 40127.0 1.64529
\(842\) 0 0
\(843\) −14790.0 −0.604264
\(844\) 0 0
\(845\) 866.000 0.0352560
\(846\) 0 0
\(847\) 8869.00 0.359790
\(848\) 0 0
\(849\) −18780.0 −0.759161
\(850\) 0 0
\(851\) −30248.0 −1.21843
\(852\) 0 0
\(853\) 826.000 0.0331556 0.0165778 0.999863i \(-0.494723\pi\)
0.0165778 + 0.999863i \(0.494723\pi\)
\(854\) 0 0
\(855\) −2232.00 −0.0892781
\(856\) 0 0
\(857\) 45918.0 1.83026 0.915128 0.403164i \(-0.132090\pi\)
0.915128 + 0.403164i \(0.132090\pi\)
\(858\) 0 0
\(859\) −42380.0 −1.68334 −0.841669 0.539994i \(-0.818426\pi\)
−0.841669 + 0.539994i \(0.818426\pi\)
\(860\) 0 0
\(861\) 9702.00 0.384022
\(862\) 0 0
\(863\) −26524.0 −1.04622 −0.523110 0.852265i \(-0.675228\pi\)
−0.523110 + 0.852265i \(0.675228\pi\)
\(864\) 0 0
\(865\) 8068.00 0.317133
\(866\) 0 0
\(867\) 14727.0 0.576880
\(868\) 0 0
\(869\) 4416.00 0.172385
\(870\) 0 0
\(871\) 32088.0 1.24829
\(872\) 0 0
\(873\) −10710.0 −0.415210
\(874\) 0 0
\(875\) −3444.00 −0.133061
\(876\) 0 0
\(877\) −20614.0 −0.793712 −0.396856 0.917881i \(-0.629899\pi\)
−0.396856 + 0.917881i \(0.629899\pi\)
\(878\) 0 0
\(879\) −6930.00 −0.265919
\(880\) 0 0
\(881\) −23730.0 −0.907473 −0.453737 0.891136i \(-0.649909\pi\)
−0.453737 + 0.891136i \(0.649909\pi\)
\(882\) 0 0
\(883\) 9028.00 0.344073 0.172036 0.985091i \(-0.444965\pi\)
0.172036 + 0.985091i \(0.444965\pi\)
\(884\) 0 0
\(885\) 4632.00 0.175936
\(886\) 0 0
\(887\) 37200.0 1.40818 0.704089 0.710112i \(-0.251356\pi\)
0.704089 + 0.710112i \(0.251356\pi\)
\(888\) 0 0
\(889\) −14616.0 −0.551412
\(890\) 0 0
\(891\) 648.000 0.0243646
\(892\) 0 0
\(893\) −32736.0 −1.22673
\(894\) 0 0
\(895\) −6960.00 −0.259941
\(896\) 0 0
\(897\) −9576.00 −0.356447
\(898\) 0 0
\(899\) 18288.0 0.678464
\(900\) 0 0
\(901\) −324.000 −0.0119800
\(902\) 0 0
\(903\) −4452.00 −0.164068
\(904\) 0 0
\(905\) −5796.00 −0.212890
\(906\) 0 0
\(907\) −23988.0 −0.878179 −0.439090 0.898443i \(-0.644699\pi\)
−0.439090 + 0.898443i \(0.644699\pi\)
\(908\) 0 0
\(909\) −12330.0 −0.449901
\(910\) 0 0
\(911\) 15276.0 0.555561 0.277781 0.960645i \(-0.410401\pi\)
0.277781 + 0.960645i \(0.410401\pi\)
\(912\) 0 0
\(913\) −8288.00 −0.300430
\(914\) 0 0
\(915\) −180.000 −0.00650341
\(916\) 0 0
\(917\) −2044.00 −0.0736083
\(918\) 0 0
\(919\) −10760.0 −0.386224 −0.193112 0.981177i \(-0.561858\pi\)
−0.193112 + 0.981177i \(0.561858\pi\)
\(920\) 0 0
\(921\) 588.000 0.0210372
\(922\) 0 0
\(923\) −9912.00 −0.353475
\(924\) 0 0
\(925\) 48158.0 1.71181
\(926\) 0 0
\(927\) 4176.00 0.147959
\(928\) 0 0
\(929\) −52890.0 −1.86788 −0.933942 0.357424i \(-0.883655\pi\)
−0.933942 + 0.357424i \(0.883655\pi\)
\(930\) 0 0
\(931\) 6076.00 0.213891
\(932\) 0 0
\(933\) 20208.0 0.709089
\(934\) 0 0
\(935\) 32.0000 0.00111926
\(936\) 0 0
\(937\) −6118.00 −0.213305 −0.106652 0.994296i \(-0.534013\pi\)
−0.106652 + 0.994296i \(0.534013\pi\)
\(938\) 0 0
\(939\) −1182.00 −0.0410789
\(940\) 0 0
\(941\) 32230.0 1.11654 0.558272 0.829658i \(-0.311465\pi\)
0.558272 + 0.829658i \(0.311465\pi\)
\(942\) 0 0
\(943\) 35112.0 1.21252
\(944\) 0 0
\(945\) −378.000 −0.0130120
\(946\) 0 0
\(947\) 18544.0 0.636324 0.318162 0.948036i \(-0.396934\pi\)
0.318162 + 0.948036i \(0.396934\pi\)
\(948\) 0 0
\(949\) 17556.0 0.600518
\(950\) 0 0
\(951\) −20142.0 −0.686802
\(952\) 0 0
\(953\) 25930.0 0.881380 0.440690 0.897659i \(-0.354734\pi\)
0.440690 + 0.897659i \(0.354734\pi\)
\(954\) 0 0
\(955\) −5304.00 −0.179721
\(956\) 0 0
\(957\) 6096.00 0.205910
\(958\) 0 0
\(959\) −5726.00 −0.192807
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) 19224.0 0.643286
\(964\) 0 0
\(965\) −292.000 −0.00974074
\(966\) 0 0
\(967\) 8192.00 0.272427 0.136214 0.990680i \(-0.456507\pi\)
0.136214 + 0.990680i \(0.456507\pi\)
\(968\) 0 0
\(969\) 744.000 0.0246653
\(970\) 0 0
\(971\) 54444.0 1.79937 0.899686 0.436537i \(-0.143795\pi\)
0.899686 + 0.436537i \(0.143795\pi\)
\(972\) 0 0
\(973\) −15092.0 −0.497253
\(974\) 0 0
\(975\) 15246.0 0.500782
\(976\) 0 0
\(977\) −25446.0 −0.833255 −0.416627 0.909077i \(-0.636788\pi\)
−0.416627 + 0.909077i \(0.636788\pi\)
\(978\) 0 0
\(979\) 240.000 0.00783497
\(980\) 0 0
\(981\) 11034.0 0.359112
\(982\) 0 0
\(983\) 33192.0 1.07697 0.538484 0.842635i \(-0.318997\pi\)
0.538484 + 0.842635i \(0.318997\pi\)
\(984\) 0 0
\(985\) −5092.00 −0.164715
\(986\) 0 0
\(987\) −5544.00 −0.178792
\(988\) 0 0
\(989\) −16112.0 −0.518030
\(990\) 0 0
\(991\) 11024.0 0.353369 0.176685 0.984268i \(-0.443463\pi\)
0.176685 + 0.984268i \(0.443463\pi\)
\(992\) 0 0
\(993\) 2076.00 0.0663443
\(994\) 0 0
\(995\) 5072.00 0.161601
\(996\) 0 0
\(997\) 40714.0 1.29331 0.646653 0.762785i \(-0.276168\pi\)
0.646653 + 0.762785i \(0.276168\pi\)
\(998\) 0 0
\(999\) 10746.0 0.340329
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 1344.4.a.f.1.1 1
4.3 odd 2 1344.4.a.t.1.1 1
8.3 odd 2 336.4.a.d.1.1 1
8.5 even 2 42.4.a.b.1.1 1
24.5 odd 2 126.4.a.c.1.1 1
24.11 even 2 1008.4.a.j.1.1 1
40.13 odd 4 1050.4.g.n.799.1 2
40.29 even 2 1050.4.a.d.1.1 1
40.37 odd 4 1050.4.g.n.799.2 2
56.5 odd 6 294.4.e.d.67.1 2
56.13 odd 2 294.4.a.h.1.1 1
56.27 even 2 2352.4.a.ba.1.1 1
56.37 even 6 294.4.e.a.67.1 2
56.45 odd 6 294.4.e.d.79.1 2
56.53 even 6 294.4.e.a.79.1 2
168.5 even 6 882.4.g.r.361.1 2
168.53 odd 6 882.4.g.s.667.1 2
168.101 even 6 882.4.g.r.667.1 2
168.125 even 2 882.4.a.d.1.1 1
168.149 odd 6 882.4.g.s.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.a.b.1.1 1 8.5 even 2
126.4.a.c.1.1 1 24.5 odd 2
294.4.a.h.1.1 1 56.13 odd 2
294.4.e.a.67.1 2 56.37 even 6
294.4.e.a.79.1 2 56.53 even 6
294.4.e.d.67.1 2 56.5 odd 6
294.4.e.d.79.1 2 56.45 odd 6
336.4.a.d.1.1 1 8.3 odd 2
882.4.a.d.1.1 1 168.125 even 2
882.4.g.r.361.1 2 168.5 even 6
882.4.g.r.667.1 2 168.101 even 6
882.4.g.s.361.1 2 168.149 odd 6
882.4.g.s.667.1 2 168.53 odd 6
1008.4.a.j.1.1 1 24.11 even 2
1050.4.a.d.1.1 1 40.29 even 2
1050.4.g.n.799.1 2 40.13 odd 4
1050.4.g.n.799.2 2 40.37 odd 4
1344.4.a.f.1.1 1 1.1 even 1 trivial
1344.4.a.t.1.1 1 4.3 odd 2
2352.4.a.ba.1.1 1 56.27 even 2