Properties

Label 784.4.f.j.783.6
Level $784$
Weight $4$
Character 784.783
Analytic conductor $46.257$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,4,Mod(783,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.783");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 783.6
Character \(\chi\) \(=\) 784.783
Dual form 784.4.f.j.783.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.68961 q^{3} -20.4139i q^{5} +5.37166 q^{9} +O(q^{10})\) \(q-5.68961 q^{3} -20.4139i q^{5} +5.37166 q^{9} +27.4118i q^{11} -33.2667i q^{13} +116.147i q^{15} -51.6329i q^{17} -123.055 q^{19} -203.652i q^{23} -291.726 q^{25} +123.057 q^{27} -27.1005 q^{29} -132.591 q^{31} -155.963i q^{33} -98.1248 q^{37} +189.274i q^{39} -475.182i q^{41} +288.305i q^{43} -109.656i q^{45} +568.917 q^{47} +293.771i q^{51} -133.525 q^{53} +559.581 q^{55} +700.134 q^{57} -307.068 q^{59} +735.787i q^{61} -679.101 q^{65} -130.251i q^{67} +1158.70i q^{69} -708.596i q^{71} -562.738i q^{73} +1659.81 q^{75} +787.673i q^{79} -845.180 q^{81} +300.047 q^{83} -1054.03 q^{85} +154.191 q^{87} +1134.02i q^{89} +754.392 q^{93} +2512.03i q^{95} -234.071i q^{97} +147.247i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −5.68961 −1.09497 −0.547483 0.836817i \(-0.684414\pi\)
−0.547483 + 0.836817i \(0.684414\pi\)
\(4\) 0 0
\(5\) − 20.4139i − 1.82587i −0.408103 0.912936i \(-0.633810\pi\)
0.408103 0.912936i \(-0.366190\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.37166 0.198951
\(10\) 0 0
\(11\) 27.4118i 0.751362i 0.926749 + 0.375681i \(0.122591\pi\)
−0.926749 + 0.375681i \(0.877409\pi\)
\(12\) 0 0
\(13\) − 33.2667i − 0.709732i −0.934917 0.354866i \(-0.884527\pi\)
0.934917 0.354866i \(-0.115473\pi\)
\(14\) 0 0
\(15\) 116.147i 1.99927i
\(16\) 0 0
\(17\) − 51.6329i − 0.736637i −0.929700 0.368318i \(-0.879934\pi\)
0.929700 0.368318i \(-0.120066\pi\)
\(18\) 0 0
\(19\) −123.055 −1.48583 −0.742914 0.669387i \(-0.766557\pi\)
−0.742914 + 0.669387i \(0.766557\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 203.652i − 1.84628i −0.384469 0.923138i \(-0.625615\pi\)
0.384469 0.923138i \(-0.374385\pi\)
\(24\) 0 0
\(25\) −291.726 −2.33381
\(26\) 0 0
\(27\) 123.057 0.877122
\(28\) 0 0
\(29\) −27.1005 −0.173532 −0.0867662 0.996229i \(-0.527653\pi\)
−0.0867662 + 0.996229i \(0.527653\pi\)
\(30\) 0 0
\(31\) −132.591 −0.768196 −0.384098 0.923292i \(-0.625488\pi\)
−0.384098 + 0.923292i \(0.625488\pi\)
\(32\) 0 0
\(33\) − 155.963i − 0.822715i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −98.1248 −0.435990 −0.217995 0.975950i \(-0.569952\pi\)
−0.217995 + 0.975950i \(0.569952\pi\)
\(38\) 0 0
\(39\) 189.274i 0.777132i
\(40\) 0 0
\(41\) − 475.182i − 1.81002i −0.425388 0.905011i \(-0.639862\pi\)
0.425388 0.905011i \(-0.360138\pi\)
\(42\) 0 0
\(43\) 288.305i 1.02247i 0.859442 + 0.511233i \(0.170811\pi\)
−0.859442 + 0.511233i \(0.829189\pi\)
\(44\) 0 0
\(45\) − 109.656i − 0.363258i
\(46\) 0 0
\(47\) 568.917 1.76564 0.882821 0.469710i \(-0.155642\pi\)
0.882821 + 0.469710i \(0.155642\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 293.771i 0.806592i
\(52\) 0 0
\(53\) −133.525 −0.346059 −0.173029 0.984917i \(-0.555356\pi\)
−0.173029 + 0.984917i \(0.555356\pi\)
\(54\) 0 0
\(55\) 559.581 1.37189
\(56\) 0 0
\(57\) 700.134 1.62693
\(58\) 0 0
\(59\) −307.068 −0.677573 −0.338786 0.940863i \(-0.610016\pi\)
−0.338786 + 0.940863i \(0.610016\pi\)
\(60\) 0 0
\(61\) 735.787i 1.54439i 0.635385 + 0.772195i \(0.280841\pi\)
−0.635385 + 0.772195i \(0.719159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −679.101 −1.29588
\(66\) 0 0
\(67\) − 130.251i − 0.237504i −0.992924 0.118752i \(-0.962111\pi\)
0.992924 0.118752i \(-0.0378893\pi\)
\(68\) 0 0
\(69\) 1158.70i 2.02161i
\(70\) 0 0
\(71\) − 708.596i − 1.18443i −0.805778 0.592217i \(-0.798253\pi\)
0.805778 0.592217i \(-0.201747\pi\)
\(72\) 0 0
\(73\) − 562.738i − 0.902240i −0.892463 0.451120i \(-0.851025\pi\)
0.892463 0.451120i \(-0.148975\pi\)
\(74\) 0 0
\(75\) 1659.81 2.55544
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 787.673i 1.12177i 0.827893 + 0.560887i \(0.189540\pi\)
−0.827893 + 0.560887i \(0.810460\pi\)
\(80\) 0 0
\(81\) −845.180 −1.15937
\(82\) 0 0
\(83\) 300.047 0.396801 0.198400 0.980121i \(-0.436425\pi\)
0.198400 + 0.980121i \(0.436425\pi\)
\(84\) 0 0
\(85\) −1054.03 −1.34500
\(86\) 0 0
\(87\) 154.191 0.190012
\(88\) 0 0
\(89\) 1134.02i 1.35062i 0.737532 + 0.675312i \(0.235991\pi\)
−0.737532 + 0.675312i \(0.764009\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 754.392 0.841149
\(94\) 0 0
\(95\) 2512.03i 2.71293i
\(96\) 0 0
\(97\) − 234.071i − 0.245013i −0.992468 0.122507i \(-0.960907\pi\)
0.992468 0.122507i \(-0.0390933\pi\)
\(98\) 0 0
\(99\) 147.247i 0.149484i
\(100\) 0 0
\(101\) 548.705i 0.540576i 0.962780 + 0.270288i \(0.0871189\pi\)
−0.962780 + 0.270288i \(0.912881\pi\)
\(102\) 0 0
\(103\) 407.887 0.390197 0.195099 0.980784i \(-0.437497\pi\)
0.195099 + 0.980784i \(0.437497\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 431.721i 0.390056i 0.980798 + 0.195028i \(0.0624798\pi\)
−0.980798 + 0.195028i \(0.937520\pi\)
\(108\) 0 0
\(109\) 1417.95 1.24601 0.623004 0.782219i \(-0.285912\pi\)
0.623004 + 0.782219i \(0.285912\pi\)
\(110\) 0 0
\(111\) 558.292 0.477394
\(112\) 0 0
\(113\) −705.689 −0.587483 −0.293742 0.955885i \(-0.594901\pi\)
−0.293742 + 0.955885i \(0.594901\pi\)
\(114\) 0 0
\(115\) −4157.32 −3.37106
\(116\) 0 0
\(117\) − 178.697i − 0.141202i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 579.591 0.435456
\(122\) 0 0
\(123\) 2703.60i 1.98191i
\(124\) 0 0
\(125\) 3403.52i 2.43536i
\(126\) 0 0
\(127\) 1941.75i 1.35671i 0.734735 + 0.678355i \(0.237307\pi\)
−0.734735 + 0.678355i \(0.762693\pi\)
\(128\) 0 0
\(129\) − 1640.34i − 1.11957i
\(130\) 0 0
\(131\) −1042.75 −0.695463 −0.347731 0.937594i \(-0.613048\pi\)
−0.347731 + 0.937594i \(0.613048\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 2512.06i − 1.60151i
\(136\) 0 0
\(137\) −2026.12 −1.26353 −0.631763 0.775162i \(-0.717668\pi\)
−0.631763 + 0.775162i \(0.717668\pi\)
\(138\) 0 0
\(139\) −995.762 −0.607622 −0.303811 0.952732i \(-0.598259\pi\)
−0.303811 + 0.952732i \(0.598259\pi\)
\(140\) 0 0
\(141\) −3236.92 −1.93332
\(142\) 0 0
\(143\) 911.900 0.533265
\(144\) 0 0
\(145\) 553.226i 0.316848i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 925.945 0.509103 0.254551 0.967059i \(-0.418072\pi\)
0.254551 + 0.967059i \(0.418072\pi\)
\(150\) 0 0
\(151\) 2101.38i 1.13250i 0.824233 + 0.566251i \(0.191607\pi\)
−0.824233 + 0.566251i \(0.808393\pi\)
\(152\) 0 0
\(153\) − 277.355i − 0.146554i
\(154\) 0 0
\(155\) 2706.70i 1.40263i
\(156\) 0 0
\(157\) − 2459.33i − 1.25016i −0.780560 0.625081i \(-0.785066\pi\)
0.780560 0.625081i \(-0.214934\pi\)
\(158\) 0 0
\(159\) 759.707 0.378922
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2902.94i 1.39494i 0.716612 + 0.697472i \(0.245692\pi\)
−0.716612 + 0.697472i \(0.754308\pi\)
\(164\) 0 0
\(165\) −3183.80 −1.50217
\(166\) 0 0
\(167\) 1040.40 0.482089 0.241044 0.970514i \(-0.422510\pi\)
0.241044 + 0.970514i \(0.422510\pi\)
\(168\) 0 0
\(169\) 1090.33 0.496281
\(170\) 0 0
\(171\) −661.010 −0.295606
\(172\) 0 0
\(173\) 144.134i 0.0633428i 0.999498 + 0.0316714i \(0.0100830\pi\)
−0.999498 + 0.0316714i \(0.989917\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1747.10 0.741919
\(178\) 0 0
\(179\) 717.436i 0.299574i 0.988718 + 0.149787i \(0.0478587\pi\)
−0.988718 + 0.149787i \(0.952141\pi\)
\(180\) 0 0
\(181\) − 473.409i − 0.194410i −0.995264 0.0972050i \(-0.969010\pi\)
0.995264 0.0972050i \(-0.0309903\pi\)
\(182\) 0 0
\(183\) − 4186.34i − 1.69106i
\(184\) 0 0
\(185\) 2003.11i 0.796061i
\(186\) 0 0
\(187\) 1415.35 0.553481
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 2529.99i − 0.958450i −0.877692 0.479225i \(-0.840918\pi\)
0.877692 0.479225i \(-0.159082\pi\)
\(192\) 0 0
\(193\) 3252.98 1.21324 0.606619 0.794993i \(-0.292525\pi\)
0.606619 + 0.794993i \(0.292525\pi\)
\(194\) 0 0
\(195\) 3863.82 1.41894
\(196\) 0 0
\(197\) 5178.04 1.87269 0.936346 0.351078i \(-0.114185\pi\)
0.936346 + 0.351078i \(0.114185\pi\)
\(198\) 0 0
\(199\) −2039.63 −0.726561 −0.363280 0.931680i \(-0.618343\pi\)
−0.363280 + 0.931680i \(0.618343\pi\)
\(200\) 0 0
\(201\) 741.080i 0.260058i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9700.30 −3.30487
\(206\) 0 0
\(207\) − 1093.95i − 0.367318i
\(208\) 0 0
\(209\) − 3373.16i − 1.11639i
\(210\) 0 0
\(211\) 1313.78i 0.428647i 0.976763 + 0.214323i \(0.0687546\pi\)
−0.976763 + 0.214323i \(0.931245\pi\)
\(212\) 0 0
\(213\) 4031.63i 1.29692i
\(214\) 0 0
\(215\) 5885.41 1.86689
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3201.76i 0.987922i
\(220\) 0 0
\(221\) −1717.66 −0.522815
\(222\) 0 0
\(223\) 769.389 0.231041 0.115520 0.993305i \(-0.463146\pi\)
0.115520 + 0.993305i \(0.463146\pi\)
\(224\) 0 0
\(225\) −1567.05 −0.464312
\(226\) 0 0
\(227\) 3693.41 1.07991 0.539956 0.841693i \(-0.318441\pi\)
0.539956 + 0.841693i \(0.318441\pi\)
\(228\) 0 0
\(229\) − 224.251i − 0.0647115i −0.999476 0.0323557i \(-0.989699\pi\)
0.999476 0.0323557i \(-0.0103009\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2826.69 0.794775 0.397387 0.917651i \(-0.369917\pi\)
0.397387 + 0.917651i \(0.369917\pi\)
\(234\) 0 0
\(235\) − 11613.8i − 3.22383i
\(236\) 0 0
\(237\) − 4481.55i − 1.22830i
\(238\) 0 0
\(239\) − 635.796i − 0.172076i −0.996292 0.0860381i \(-0.972579\pi\)
0.996292 0.0860381i \(-0.0274207\pi\)
\(240\) 0 0
\(241\) − 6316.97i − 1.68843i −0.536004 0.844216i \(-0.680067\pi\)
0.536004 0.844216i \(-0.319933\pi\)
\(242\) 0 0
\(243\) 1486.21 0.392348
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4093.63i 1.05454i
\(248\) 0 0
\(249\) −1707.15 −0.434483
\(250\) 0 0
\(251\) −2027.17 −0.509775 −0.254888 0.966971i \(-0.582039\pi\)
−0.254888 + 0.966971i \(0.582039\pi\)
\(252\) 0 0
\(253\) 5582.47 1.38722
\(254\) 0 0
\(255\) 5997.01 1.47273
\(256\) 0 0
\(257\) 5246.80i 1.27349i 0.771075 + 0.636744i \(0.219719\pi\)
−0.771075 + 0.636744i \(0.780281\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −145.575 −0.0345244
\(262\) 0 0
\(263\) − 631.404i − 0.148038i −0.997257 0.0740191i \(-0.976417\pi\)
0.997257 0.0740191i \(-0.0235826\pi\)
\(264\) 0 0
\(265\) 2725.77i 0.631859i
\(266\) 0 0
\(267\) − 6452.11i − 1.47889i
\(268\) 0 0
\(269\) 1164.99i 0.264056i 0.991246 + 0.132028i \(0.0421488\pi\)
−0.991246 + 0.132028i \(0.957851\pi\)
\(270\) 0 0
\(271\) −7955.89 −1.78334 −0.891672 0.452683i \(-0.850467\pi\)
−0.891672 + 0.452683i \(0.850467\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 7996.74i − 1.75353i
\(276\) 0 0
\(277\) −3600.97 −0.781087 −0.390544 0.920584i \(-0.627713\pi\)
−0.390544 + 0.920584i \(0.627713\pi\)
\(278\) 0 0
\(279\) −712.235 −0.152833
\(280\) 0 0
\(281\) 1629.95 0.346031 0.173016 0.984919i \(-0.444649\pi\)
0.173016 + 0.984919i \(0.444649\pi\)
\(282\) 0 0
\(283\) −3562.61 −0.748323 −0.374162 0.927364i \(-0.622069\pi\)
−0.374162 + 0.927364i \(0.622069\pi\)
\(284\) 0 0
\(285\) − 14292.4i − 2.97057i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2247.04 0.457366
\(290\) 0 0
\(291\) 1331.77i 0.268281i
\(292\) 0 0
\(293\) − 2256.82i − 0.449982i −0.974361 0.224991i \(-0.927765\pi\)
0.974361 0.224991i \(-0.0722352\pi\)
\(294\) 0 0
\(295\) 6268.44i 1.23716i
\(296\) 0 0
\(297\) 3373.21i 0.659036i
\(298\) 0 0
\(299\) −6774.82 −1.31036
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 3121.92i − 0.591912i
\(304\) 0 0
\(305\) 15020.2 2.81986
\(306\) 0 0
\(307\) −7898.99 −1.46847 −0.734234 0.678897i \(-0.762458\pi\)
−0.734234 + 0.678897i \(0.762458\pi\)
\(308\) 0 0
\(309\) −2320.72 −0.427253
\(310\) 0 0
\(311\) 6032.25 1.09986 0.549932 0.835209i \(-0.314654\pi\)
0.549932 + 0.835209i \(0.314654\pi\)
\(312\) 0 0
\(313\) − 648.749i − 0.117155i −0.998283 0.0585774i \(-0.981344\pi\)
0.998283 0.0585774i \(-0.0186564\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5400.21 −0.956801 −0.478401 0.878142i \(-0.658783\pi\)
−0.478401 + 0.878142i \(0.658783\pi\)
\(318\) 0 0
\(319\) − 742.875i − 0.130386i
\(320\) 0 0
\(321\) − 2456.32i − 0.427098i
\(322\) 0 0
\(323\) 6353.68i 1.09452i
\(324\) 0 0
\(325\) 9704.75i 1.65638i
\(326\) 0 0
\(327\) −8067.57 −1.36434
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 4604.90i − 0.764678i −0.924022 0.382339i \(-0.875119\pi\)
0.924022 0.382339i \(-0.124881\pi\)
\(332\) 0 0
\(333\) −527.093 −0.0867403
\(334\) 0 0
\(335\) −2658.94 −0.433651
\(336\) 0 0
\(337\) −8621.08 −1.39353 −0.696766 0.717299i \(-0.745378\pi\)
−0.696766 + 0.717299i \(0.745378\pi\)
\(338\) 0 0
\(339\) 4015.09 0.643274
\(340\) 0 0
\(341\) − 3634.57i − 0.577193i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 23653.5 3.69120
\(346\) 0 0
\(347\) 2071.61i 0.320489i 0.987077 + 0.160244i \(0.0512283\pi\)
−0.987077 + 0.160244i \(0.948772\pi\)
\(348\) 0 0
\(349\) 2795.68i 0.428794i 0.976747 + 0.214397i \(0.0687787\pi\)
−0.976747 + 0.214397i \(0.931221\pi\)
\(350\) 0 0
\(351\) − 4093.69i − 0.622521i
\(352\) 0 0
\(353\) 348.273i 0.0525118i 0.999655 + 0.0262559i \(0.00835848\pi\)
−0.999655 + 0.0262559i \(0.991642\pi\)
\(354\) 0 0
\(355\) −14465.2 −2.16263
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4852.06i 0.713321i 0.934234 + 0.356660i \(0.116085\pi\)
−0.934234 + 0.356660i \(0.883915\pi\)
\(360\) 0 0
\(361\) 8283.51 1.20768
\(362\) 0 0
\(363\) −3297.65 −0.476809
\(364\) 0 0
\(365\) −11487.7 −1.64737
\(366\) 0 0
\(367\) −9356.53 −1.33081 −0.665404 0.746483i \(-0.731741\pi\)
−0.665404 + 0.746483i \(0.731741\pi\)
\(368\) 0 0
\(369\) − 2552.52i − 0.360105i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4573.24 −0.634835 −0.317418 0.948286i \(-0.602816\pi\)
−0.317418 + 0.948286i \(0.602816\pi\)
\(374\) 0 0
\(375\) − 19364.7i − 2.66664i
\(376\) 0 0
\(377\) 901.544i 0.123161i
\(378\) 0 0
\(379\) − 11843.8i − 1.60521i −0.596514 0.802603i \(-0.703448\pi\)
0.596514 0.802603i \(-0.296552\pi\)
\(380\) 0 0
\(381\) − 11047.8i − 1.48555i
\(382\) 0 0
\(383\) −10479.3 −1.39808 −0.699041 0.715082i \(-0.746389\pi\)
−0.699041 + 0.715082i \(0.746389\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1548.68i 0.203420i
\(388\) 0 0
\(389\) −1599.54 −0.208483 −0.104242 0.994552i \(-0.533242\pi\)
−0.104242 + 0.994552i \(0.533242\pi\)
\(390\) 0 0
\(391\) −10515.1 −1.36003
\(392\) 0 0
\(393\) 5932.85 0.761508
\(394\) 0 0
\(395\) 16079.4 2.04821
\(396\) 0 0
\(397\) − 11.6116i − 0.00146793i −1.00000 0.000733966i \(-0.999766\pi\)
1.00000 0.000733966i \(-0.000233629\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7462.34 −0.929306 −0.464653 0.885493i \(-0.653821\pi\)
−0.464653 + 0.885493i \(0.653821\pi\)
\(402\) 0 0
\(403\) 4410.87i 0.545213i
\(404\) 0 0
\(405\) 17253.4i 2.11686i
\(406\) 0 0
\(407\) − 2689.78i − 0.327586i
\(408\) 0 0
\(409\) 5640.51i 0.681920i 0.940078 + 0.340960i \(0.110752\pi\)
−0.940078 + 0.340960i \(0.889248\pi\)
\(410\) 0 0
\(411\) 11527.8 1.38352
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 6125.12i − 0.724507i
\(416\) 0 0
\(417\) 5665.50 0.665325
\(418\) 0 0
\(419\) −8037.03 −0.937076 −0.468538 0.883443i \(-0.655219\pi\)
−0.468538 + 0.883443i \(0.655219\pi\)
\(420\) 0 0
\(421\) 9815.24 1.13626 0.568130 0.822939i \(-0.307667\pi\)
0.568130 + 0.822939i \(0.307667\pi\)
\(422\) 0 0
\(423\) 3056.03 0.351275
\(424\) 0 0
\(425\) 15062.7i 1.71917i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5188.36 −0.583907
\(430\) 0 0
\(431\) − 4053.33i − 0.452998i −0.974011 0.226499i \(-0.927272\pi\)
0.974011 0.226499i \(-0.0727280\pi\)
\(432\) 0 0
\(433\) − 13331.5i − 1.47961i −0.672822 0.739805i \(-0.734918\pi\)
0.672822 0.739805i \(-0.265082\pi\)
\(434\) 0 0
\(435\) − 3147.64i − 0.346938i
\(436\) 0 0
\(437\) 25060.4i 2.74325i
\(438\) 0 0
\(439\) 11361.5 1.23521 0.617604 0.786489i \(-0.288104\pi\)
0.617604 + 0.786489i \(0.288104\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12669.9i 1.35884i 0.733750 + 0.679419i \(0.237768\pi\)
−0.733750 + 0.679419i \(0.762232\pi\)
\(444\) 0 0
\(445\) 23149.7 2.46607
\(446\) 0 0
\(447\) −5268.26 −0.557450
\(448\) 0 0
\(449\) 8911.65 0.936674 0.468337 0.883550i \(-0.344853\pi\)
0.468337 + 0.883550i \(0.344853\pi\)
\(450\) 0 0
\(451\) 13025.6 1.35998
\(452\) 0 0
\(453\) − 11956.0i − 1.24005i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13833.7 1.41600 0.707999 0.706213i \(-0.249598\pi\)
0.707999 + 0.706213i \(0.249598\pi\)
\(458\) 0 0
\(459\) − 6353.78i − 0.646120i
\(460\) 0 0
\(461\) 852.971i 0.0861753i 0.999071 + 0.0430877i \(0.0137195\pi\)
−0.999071 + 0.0430877i \(0.986281\pi\)
\(462\) 0 0
\(463\) − 6647.58i − 0.667255i −0.942705 0.333628i \(-0.891727\pi\)
0.942705 0.333628i \(-0.108273\pi\)
\(464\) 0 0
\(465\) − 15400.1i − 1.53583i
\(466\) 0 0
\(467\) 7396.09 0.732870 0.366435 0.930444i \(-0.380578\pi\)
0.366435 + 0.930444i \(0.380578\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13992.6i 1.36889i
\(472\) 0 0
\(473\) −7902.96 −0.768242
\(474\) 0 0
\(475\) 35898.3 3.46764
\(476\) 0 0
\(477\) −717.253 −0.0688485
\(478\) 0 0
\(479\) −10779.1 −1.02820 −0.514102 0.857729i \(-0.671875\pi\)
−0.514102 + 0.857729i \(0.671875\pi\)
\(480\) 0 0
\(481\) 3264.28i 0.309436i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4778.29 −0.447363
\(486\) 0 0
\(487\) − 5643.59i − 0.525124i −0.964915 0.262562i \(-0.915433\pi\)
0.964915 0.262562i \(-0.0845675\pi\)
\(488\) 0 0
\(489\) − 16516.6i − 1.52742i
\(490\) 0 0
\(491\) − 2800.93i − 0.257442i −0.991681 0.128721i \(-0.958913\pi\)
0.991681 0.128721i \(-0.0410872\pi\)
\(492\) 0 0
\(493\) 1399.28i 0.127830i
\(494\) 0 0
\(495\) 3005.88 0.272938
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 17704.6i − 1.58831i −0.607712 0.794157i \(-0.707913\pi\)
0.607712 0.794157i \(-0.292087\pi\)
\(500\) 0 0
\(501\) −5919.49 −0.527871
\(502\) 0 0
\(503\) −8247.73 −0.731110 −0.365555 0.930790i \(-0.619121\pi\)
−0.365555 + 0.930790i \(0.619121\pi\)
\(504\) 0 0
\(505\) 11201.2 0.987022
\(506\) 0 0
\(507\) −6203.55 −0.543410
\(508\) 0 0
\(509\) 409.106i 0.0356253i 0.999841 + 0.0178127i \(0.00567025\pi\)
−0.999841 + 0.0178127i \(0.994330\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −15142.7 −1.30325
\(514\) 0 0
\(515\) − 8326.55i − 0.712450i
\(516\) 0 0
\(517\) 15595.1i 1.32664i
\(518\) 0 0
\(519\) − 820.066i − 0.0693582i
\(520\) 0 0
\(521\) − 10656.7i − 0.896123i −0.894003 0.448062i \(-0.852114\pi\)
0.894003 0.448062i \(-0.147886\pi\)
\(522\) 0 0
\(523\) −3241.19 −0.270989 −0.135494 0.990778i \(-0.543262\pi\)
−0.135494 + 0.990778i \(0.543262\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6846.07i 0.565882i
\(528\) 0 0
\(529\) −29307.1 −2.40874
\(530\) 0 0
\(531\) −1649.46 −0.134803
\(532\) 0 0
\(533\) −15807.7 −1.28463
\(534\) 0 0
\(535\) 8813.09 0.712193
\(536\) 0 0
\(537\) − 4081.93i − 0.328023i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −15351.8 −1.22001 −0.610004 0.792399i \(-0.708832\pi\)
−0.610004 + 0.792399i \(0.708832\pi\)
\(542\) 0 0
\(543\) 2693.51i 0.212872i
\(544\) 0 0
\(545\) − 28945.8i − 2.27505i
\(546\) 0 0
\(547\) 3418.07i 0.267178i 0.991037 + 0.133589i \(0.0426502\pi\)
−0.991037 + 0.133589i \(0.957350\pi\)
\(548\) 0 0
\(549\) 3952.40i 0.307257i
\(550\) 0 0
\(551\) 3334.85 0.257839
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 11396.9i − 0.871660i
\(556\) 0 0
\(557\) −13943.5 −1.06069 −0.530345 0.847782i \(-0.677938\pi\)
−0.530345 + 0.847782i \(0.677938\pi\)
\(558\) 0 0
\(559\) 9590.94 0.725677
\(560\) 0 0
\(561\) −8052.81 −0.606042
\(562\) 0 0
\(563\) −10616.9 −0.794761 −0.397380 0.917654i \(-0.630081\pi\)
−0.397380 + 0.917654i \(0.630081\pi\)
\(564\) 0 0
\(565\) 14405.8i 1.07267i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 421.063 0.0310227 0.0155113 0.999880i \(-0.495062\pi\)
0.0155113 + 0.999880i \(0.495062\pi\)
\(570\) 0 0
\(571\) 25824.9i 1.89271i 0.323126 + 0.946356i \(0.395266\pi\)
−0.323126 + 0.946356i \(0.604734\pi\)
\(572\) 0 0
\(573\) 14394.7i 1.04947i
\(574\) 0 0
\(575\) 59410.5i 4.30885i
\(576\) 0 0
\(577\) − 8500.59i − 0.613318i −0.951820 0.306659i \(-0.900789\pi\)
0.951820 0.306659i \(-0.0992110\pi\)
\(578\) 0 0
\(579\) −18508.2 −1.32845
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 3660.17i − 0.260015i
\(584\) 0 0
\(585\) −3647.90 −0.257816
\(586\) 0 0
\(587\) −20752.1 −1.45916 −0.729582 0.683893i \(-0.760285\pi\)
−0.729582 + 0.683893i \(0.760285\pi\)
\(588\) 0 0
\(589\) 16316.0 1.14141
\(590\) 0 0
\(591\) −29461.1 −2.05053
\(592\) 0 0
\(593\) − 4423.58i − 0.306332i −0.988201 0.153166i \(-0.951053\pi\)
0.988201 0.153166i \(-0.0489469\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11604.7 0.795559
\(598\) 0 0
\(599\) 19047.4i 1.29926i 0.760250 + 0.649630i \(0.225076\pi\)
−0.760250 + 0.649630i \(0.774924\pi\)
\(600\) 0 0
\(601\) − 13205.5i − 0.896282i −0.893963 0.448141i \(-0.852086\pi\)
0.893963 0.448141i \(-0.147914\pi\)
\(602\) 0 0
\(603\) − 699.667i − 0.0472515i
\(604\) 0 0
\(605\) − 11831.7i − 0.795086i
\(606\) 0 0
\(607\) −8202.94 −0.548512 −0.274256 0.961657i \(-0.588432\pi\)
−0.274256 + 0.961657i \(0.588432\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 18926.0i − 1.25313i
\(612\) 0 0
\(613\) −13977.1 −0.920932 −0.460466 0.887677i \(-0.652318\pi\)
−0.460466 + 0.887677i \(0.652318\pi\)
\(614\) 0 0
\(615\) 55190.9 3.61872
\(616\) 0 0
\(617\) 17469.6 1.13987 0.569935 0.821690i \(-0.306968\pi\)
0.569935 + 0.821690i \(0.306968\pi\)
\(618\) 0 0
\(619\) 11475.8 0.745156 0.372578 0.928001i \(-0.378474\pi\)
0.372578 + 0.928001i \(0.378474\pi\)
\(620\) 0 0
\(621\) − 25060.7i − 1.61941i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 33013.2 2.11285
\(626\) 0 0
\(627\) 19192.0i 1.22241i
\(628\) 0 0
\(629\) 5066.47i 0.321166i
\(630\) 0 0
\(631\) − 25620.4i − 1.61638i −0.588924 0.808188i \(-0.700448\pi\)
0.588924 0.808188i \(-0.299552\pi\)
\(632\) 0 0
\(633\) − 7474.90i − 0.469353i
\(634\) 0 0
\(635\) 39638.5 2.47718
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 3806.34i − 0.235644i
\(640\) 0 0
\(641\) 13393.0 0.825258 0.412629 0.910899i \(-0.364611\pi\)
0.412629 + 0.910899i \(0.364611\pi\)
\(642\) 0 0
\(643\) 6636.01 0.406997 0.203498 0.979075i \(-0.434769\pi\)
0.203498 + 0.979075i \(0.434769\pi\)
\(644\) 0 0
\(645\) −33485.7 −2.04418
\(646\) 0 0
\(647\) −10046.4 −0.610458 −0.305229 0.952279i \(-0.598733\pi\)
−0.305229 + 0.952279i \(0.598733\pi\)
\(648\) 0 0
\(649\) − 8417.29i − 0.509102i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8981.76 −0.538260 −0.269130 0.963104i \(-0.586736\pi\)
−0.269130 + 0.963104i \(0.586736\pi\)
\(654\) 0 0
\(655\) 21286.6i 1.26983i
\(656\) 0 0
\(657\) − 3022.84i − 0.179501i
\(658\) 0 0
\(659\) 30917.7i 1.82759i 0.406177 + 0.913794i \(0.366862\pi\)
−0.406177 + 0.913794i \(0.633138\pi\)
\(660\) 0 0
\(661\) − 3506.27i − 0.206321i −0.994665 0.103160i \(-0.967105\pi\)
0.994665 0.103160i \(-0.0328955\pi\)
\(662\) 0 0
\(663\) 9772.79 0.572464
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5519.07i 0.320389i
\(668\) 0 0
\(669\) −4377.52 −0.252982
\(670\) 0 0
\(671\) −20169.3 −1.16040
\(672\) 0 0
\(673\) −3063.54 −0.175469 −0.0877345 0.996144i \(-0.527963\pi\)
−0.0877345 + 0.996144i \(0.527963\pi\)
\(674\) 0 0
\(675\) −35898.8 −2.04703
\(676\) 0 0
\(677\) 6003.94i 0.340842i 0.985371 + 0.170421i \(0.0545128\pi\)
−0.985371 + 0.170421i \(0.945487\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −21014.0 −1.18247
\(682\) 0 0
\(683\) − 1856.95i − 0.104033i −0.998646 0.0520164i \(-0.983435\pi\)
0.998646 0.0520164i \(-0.0165648\pi\)
\(684\) 0 0
\(685\) 41360.9i 2.30703i
\(686\) 0 0
\(687\) 1275.90i 0.0708569i
\(688\) 0 0
\(689\) 4441.94i 0.245609i
\(690\) 0 0
\(691\) 9303.94 0.512212 0.256106 0.966649i \(-0.417560\pi\)
0.256106 + 0.966649i \(0.417560\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20327.4i 1.10944i
\(696\) 0 0
\(697\) −24535.0 −1.33333
\(698\) 0 0
\(699\) −16082.8 −0.870251
\(700\) 0 0
\(701\) −28304.2 −1.52502 −0.762508 0.646979i \(-0.776032\pi\)
−0.762508 + 0.646979i \(0.776032\pi\)
\(702\) 0 0
\(703\) 12074.7 0.647806
\(704\) 0 0
\(705\) 66078.0i 3.52999i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5950.28 −0.315187 −0.157593 0.987504i \(-0.550374\pi\)
−0.157593 + 0.987504i \(0.550374\pi\)
\(710\) 0 0
\(711\) 4231.11i 0.223177i
\(712\) 0 0
\(713\) 27002.4i 1.41830i
\(714\) 0 0
\(715\) − 18615.4i − 0.973674i
\(716\) 0 0
\(717\) 3617.43i 0.188418i
\(718\) 0 0
\(719\) 11142.8 0.577966 0.288983 0.957334i \(-0.406683\pi\)
0.288983 + 0.957334i \(0.406683\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 35941.1i 1.84877i
\(724\) 0 0
\(725\) 7905.92 0.404991
\(726\) 0 0
\(727\) −5517.58 −0.281480 −0.140740 0.990047i \(-0.544948\pi\)
−0.140740 + 0.990047i \(0.544948\pi\)
\(728\) 0 0
\(729\) 14363.9 0.729762
\(730\) 0 0
\(731\) 14886.0 0.753186
\(732\) 0 0
\(733\) 20181.2i 1.01693i 0.861083 + 0.508464i \(0.169787\pi\)
−0.861083 + 0.508464i \(0.830213\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3570.43 0.178451
\(738\) 0 0
\(739\) 16392.3i 0.815968i 0.912989 + 0.407984i \(0.133768\pi\)
−0.912989 + 0.407984i \(0.866232\pi\)
\(740\) 0 0
\(741\) − 23291.1i − 1.15468i
\(742\) 0 0
\(743\) − 20914.6i − 1.03268i −0.856383 0.516341i \(-0.827294\pi\)
0.856383 0.516341i \(-0.172706\pi\)
\(744\) 0 0
\(745\) − 18902.1i − 0.929556i
\(746\) 0 0
\(747\) 1611.75 0.0789437
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 18466.3i − 0.897266i −0.893716 0.448633i \(-0.851911\pi\)
0.893716 0.448633i \(-0.148089\pi\)
\(752\) 0 0
\(753\) 11533.8 0.558187
\(754\) 0 0
\(755\) 42897.3 2.06780
\(756\) 0 0
\(757\) 21663.1 1.04010 0.520051 0.854135i \(-0.325913\pi\)
0.520051 + 0.854135i \(0.325913\pi\)
\(758\) 0 0
\(759\) −31762.1 −1.51896
\(760\) 0 0
\(761\) 26900.4i 1.28139i 0.767795 + 0.640696i \(0.221354\pi\)
−0.767795 + 0.640696i \(0.778646\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5661.88 −0.267589
\(766\) 0 0
\(767\) 10215.1i 0.480895i
\(768\) 0 0
\(769\) 30388.7i 1.42502i 0.701660 + 0.712512i \(0.252442\pi\)
−0.701660 + 0.712512i \(0.747558\pi\)
\(770\) 0 0
\(771\) − 29852.3i − 1.39443i
\(772\) 0 0
\(773\) − 801.174i − 0.0372784i −0.999826 0.0186392i \(-0.994067\pi\)
0.999826 0.0186392i \(-0.00593339\pi\)
\(774\) 0 0
\(775\) 38680.3 1.79282
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 58473.5i 2.68938i
\(780\) 0 0
\(781\) 19423.9 0.889939
\(782\) 0 0
\(783\) −3334.90 −0.152209
\(784\) 0 0
\(785\) −50204.3 −2.28264
\(786\) 0 0
\(787\) −33591.9 −1.52150 −0.760752 0.649043i \(-0.775169\pi\)
−0.760752 + 0.649043i \(0.775169\pi\)
\(788\) 0 0
\(789\) 3592.44i 0.162097i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24477.2 1.09610
\(794\) 0 0
\(795\) − 15508.5i − 0.691864i
\(796\) 0 0
\(797\) − 2563.22i − 0.113919i −0.998376 0.0569597i \(-0.981859\pi\)
0.998376 0.0569597i \(-0.0181406\pi\)
\(798\) 0 0
\(799\) − 29374.9i − 1.30064i
\(800\) 0 0
\(801\) 6091.56i 0.268707i
\(802\) 0 0
\(803\) 15425.7 0.677908
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6628.37i − 0.289132i
\(808\) 0 0
\(809\) 34904.7 1.51692 0.758458 0.651722i \(-0.225953\pi\)
0.758458 + 0.651722i \(0.225953\pi\)
\(810\) 0 0
\(811\) 21556.7 0.933363 0.466682 0.884425i \(-0.345449\pi\)
0.466682 + 0.884425i \(0.345449\pi\)
\(812\) 0 0
\(813\) 45265.9 1.95270
\(814\) 0 0
\(815\) 59260.2 2.54699
\(816\) 0 0
\(817\) − 35477.3i − 1.51921i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25349.0 −1.07757 −0.538785 0.842443i \(-0.681117\pi\)
−0.538785 + 0.842443i \(0.681117\pi\)
\(822\) 0 0
\(823\) 16954.9i 0.718116i 0.933315 + 0.359058i \(0.116902\pi\)
−0.933315 + 0.359058i \(0.883098\pi\)
\(824\) 0 0
\(825\) 45498.3i 1.92006i
\(826\) 0 0
\(827\) 38797.2i 1.63133i 0.578523 + 0.815666i \(0.303629\pi\)
−0.578523 + 0.815666i \(0.696371\pi\)
\(828\) 0 0
\(829\) 29945.2i 1.25457i 0.778789 + 0.627285i \(0.215834\pi\)
−0.778789 + 0.627285i \(0.784166\pi\)
\(830\) 0 0
\(831\) 20488.1 0.855264
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 21238.7i − 0.880232i
\(836\) 0 0
\(837\) −16316.3 −0.673802
\(838\) 0 0
\(839\) −36918.6 −1.51916 −0.759578 0.650416i \(-0.774595\pi\)
−0.759578 + 0.650416i \(0.774595\pi\)
\(840\) 0 0
\(841\) −23654.6 −0.969887
\(842\) 0 0
\(843\) −9273.79 −0.378893
\(844\) 0 0
\(845\) − 22257.8i − 0.906145i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20269.9 0.819388
\(850\) 0 0
\(851\) 19983.3i 0.804957i
\(852\) 0 0
\(853\) − 21921.6i − 0.879930i −0.898015 0.439965i \(-0.854991\pi\)
0.898015 0.439965i \(-0.145009\pi\)
\(854\) 0 0
\(855\) 13493.8i 0.539739i
\(856\) 0 0
\(857\) 7836.91i 0.312373i 0.987728 + 0.156187i \(0.0499201\pi\)
−0.987728 + 0.156187i \(0.950080\pi\)
\(858\) 0 0
\(859\) 5866.31 0.233010 0.116505 0.993190i \(-0.462831\pi\)
0.116505 + 0.993190i \(0.462831\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 472.119i 0.0186224i 0.999957 + 0.00931120i \(0.00296389\pi\)
−0.999957 + 0.00931120i \(0.997036\pi\)
\(864\) 0 0
\(865\) 2942.33 0.115656
\(866\) 0 0
\(867\) −12784.8 −0.500801
\(868\) 0 0
\(869\) −21591.6 −0.842858
\(870\) 0 0
\(871\) −4333.03 −0.168564
\(872\) 0 0
\(873\) − 1257.35i − 0.0487455i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23962.5 0.922640 0.461320 0.887234i \(-0.347376\pi\)
0.461320 + 0.887234i \(0.347376\pi\)
\(878\) 0 0
\(879\) 12840.4i 0.492715i
\(880\) 0 0
\(881\) − 24460.0i − 0.935390i −0.883890 0.467695i \(-0.845085\pi\)
0.883890 0.467695i \(-0.154915\pi\)
\(882\) 0 0
\(883\) 17519.7i 0.667708i 0.942625 + 0.333854i \(0.108349\pi\)
−0.942625 + 0.333854i \(0.891651\pi\)
\(884\) 0 0
\(885\) − 35665.0i − 1.35465i
\(886\) 0 0
\(887\) −7452.05 −0.282092 −0.141046 0.990003i \(-0.545046\pi\)
−0.141046 + 0.990003i \(0.545046\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 23167.9i − 0.871106i
\(892\) 0 0
\(893\) −70008.1 −2.62344
\(894\) 0 0
\(895\) 14645.6 0.546983
\(896\) 0 0
\(897\) 38546.1 1.43480
\(898\) 0 0
\(899\) 3593.29 0.133307
\(900\) 0 0
\(901\) 6894.30i 0.254919i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9664.11 −0.354968
\(906\) 0 0
\(907\) − 37014.8i − 1.35508i −0.735486 0.677540i \(-0.763046\pi\)
0.735486 0.677540i \(-0.236954\pi\)
\(908\) 0 0
\(909\) 2947.46i 0.107548i
\(910\) 0 0
\(911\) 12342.5i 0.448876i 0.974488 + 0.224438i \(0.0720546\pi\)
−0.974488 + 0.224438i \(0.927945\pi\)
\(912\) 0 0
\(913\) 8224.84i 0.298141i
\(914\) 0 0
\(915\) −85459.3 −3.08765
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 16482.6i − 0.591633i −0.955245 0.295817i \(-0.904408\pi\)
0.955245 0.295817i \(-0.0955918\pi\)
\(920\) 0 0
\(921\) 44942.2 1.60792
\(922\) 0 0
\(923\) −23572.6 −0.840631
\(924\) 0 0
\(925\) 28625.5 1.01752
\(926\) 0 0
\(927\) 2191.03 0.0776299
\(928\) 0 0
\(929\) 21750.4i 0.768145i 0.923303 + 0.384073i \(0.125479\pi\)
−0.923303 + 0.384073i \(0.874521\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −34321.2 −1.20431
\(934\) 0 0
\(935\) − 28892.8i − 1.01058i
\(936\) 0 0
\(937\) − 20698.0i − 0.721638i −0.932636 0.360819i \(-0.882497\pi\)
0.932636 0.360819i \(-0.117503\pi\)
\(938\) 0 0
\(939\) 3691.13i 0.128281i
\(940\) 0 0
\(941\) − 40305.4i − 1.39630i −0.715951 0.698151i \(-0.754006\pi\)
0.715951 0.698151i \(-0.245994\pi\)
\(942\) 0 0
\(943\) −96771.7 −3.34180
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7361.90i − 0.252618i −0.991991 0.126309i \(-0.959687\pi\)
0.991991 0.126309i \(-0.0403131\pi\)
\(948\) 0 0
\(949\) −18720.4 −0.640348
\(950\) 0 0
\(951\) 30725.1 1.04767
\(952\) 0 0
\(953\) −38052.9 −1.29345 −0.646723 0.762725i \(-0.723861\pi\)
−0.646723 + 0.762725i \(0.723861\pi\)
\(954\) 0 0
\(955\) −51647.0 −1.75001
\(956\) 0 0
\(957\) 4226.67i 0.142768i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −12210.6 −0.409874
\(962\) 0 0
\(963\) 2319.06i 0.0776019i
\(964\) 0 0
\(965\) − 66405.9i − 2.21522i
\(966\) 0 0
\(967\) − 6095.61i − 0.202711i −0.994850 0.101356i \(-0.967682\pi\)
0.994850 0.101356i \(-0.0323180\pi\)
\(968\) 0 0
\(969\) − 36150.0i − 1.19846i
\(970\) 0 0
\(971\) 22135.7 0.731585 0.365792 0.930697i \(-0.380798\pi\)
0.365792 + 0.930697i \(0.380798\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 55216.2i − 1.81368i
\(976\) 0 0
\(977\) −49659.8 −1.62616 −0.813080 0.582153i \(-0.802211\pi\)
−0.813080 + 0.582153i \(0.802211\pi\)
\(978\) 0 0
\(979\) −31085.5 −1.01481
\(980\) 0 0
\(981\) 7616.74 0.247894
\(982\) 0 0
\(983\) 50074.5 1.62475 0.812375 0.583135i \(-0.198174\pi\)
0.812375 + 0.583135i \(0.198174\pi\)
\(984\) 0 0
\(985\) − 105704.i − 3.41930i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 58713.8 1.88776
\(990\) 0 0
\(991\) − 6833.25i − 0.219037i −0.993985 0.109518i \(-0.965069\pi\)
0.993985 0.109518i \(-0.0349308\pi\)
\(992\) 0 0
\(993\) 26200.1i 0.837297i
\(994\) 0 0
\(995\) 41636.7i 1.32661i
\(996\) 0 0
\(997\) 29362.7i 0.932724i 0.884594 + 0.466362i \(0.154436\pi\)
−0.884594 + 0.466362i \(0.845564\pi\)
\(998\) 0 0
\(999\) −12074.9 −0.382416
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 784.4.f.j.783.6 yes 24
4.3 odd 2 inner 784.4.f.j.783.20 yes 24
7.6 odd 2 inner 784.4.f.j.783.19 yes 24
28.27 even 2 inner 784.4.f.j.783.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
784.4.f.j.783.5 24 28.27 even 2 inner
784.4.f.j.783.6 yes 24 1.1 even 1 trivial
784.4.f.j.783.19 yes 24 7.6 odd 2 inner
784.4.f.j.783.20 yes 24 4.3 odd 2 inner