Properties

Label 784.4.a.bf.1.2
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,4,Mod(1,784)]
 
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([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("784.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.11692\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.57956 q^{3} -13.4791 q^{5} -14.1868 q^{9} +O(q^{10})\) \(q-3.57956 q^{3} -13.4791 q^{5} -14.1868 q^{9} -0.813227 q^{11} +34.9564 q^{13} +48.2490 q^{15} +117.732 q^{17} +93.2913 q^{19} -120.249 q^{23} +56.6848 q^{25} +147.430 q^{27} +8.56420 q^{29} +82.1070 q^{31} +2.91099 q^{33} +28.8132 q^{37} -125.128 q^{39} -70.5291 q^{41} -417.179 q^{43} +191.224 q^{45} -338.261 q^{47} -421.428 q^{51} +149.121 q^{53} +10.9615 q^{55} -333.942 q^{57} +94.1828 q^{59} -120.525 q^{61} -471.179 q^{65} +792.366 q^{67} +430.438 q^{69} -449.128 q^{71} -469.420 q^{73} -202.907 q^{75} +1019.85 q^{79} -144.693 q^{81} +104.253 q^{83} -1586.91 q^{85} -30.6560 q^{87} -1572.92 q^{89} -293.906 q^{93} -1257.48 q^{95} +550.057 q^{97} +11.5371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 40 q^{9} - 100 q^{11} + 64 q^{15} - 352 q^{23} - 128 q^{25} + 260 q^{29} + 212 q^{37} - 952 q^{39} - 540 q^{43} - 428 q^{51} + 16 q^{53} - 1884 q^{57} - 756 q^{65} + 1944 q^{67} - 2248 q^{71} + 1048 q^{79} - 1256 q^{81} - 3284 q^{85} - 5368 q^{93} - 2192 q^{95} - 3340 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.57956 −0.688886 −0.344443 0.938807i \(-0.611932\pi\)
−0.344443 + 0.938807i \(0.611932\pi\)
\(4\) 0 0
\(5\) −13.4791 −1.20560 −0.602802 0.797891i \(-0.705949\pi\)
−0.602802 + 0.797891i \(0.705949\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −14.1868 −0.525436
\(10\) 0 0
\(11\) −0.813227 −0.0222906 −0.0111453 0.999938i \(-0.503548\pi\)
−0.0111453 + 0.999938i \(0.503548\pi\)
\(12\) 0 0
\(13\) 34.9564 0.745781 0.372891 0.927875i \(-0.378367\pi\)
0.372891 + 0.927875i \(0.378367\pi\)
\(14\) 0 0
\(15\) 48.2490 0.830523
\(16\) 0 0
\(17\) 117.732 1.67966 0.839829 0.542851i \(-0.182655\pi\)
0.839829 + 0.542851i \(0.182655\pi\)
\(18\) 0 0
\(19\) 93.2913 1.12645 0.563224 0.826304i \(-0.309561\pi\)
0.563224 + 0.826304i \(0.309561\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −120.249 −1.09016 −0.545079 0.838384i \(-0.683501\pi\)
−0.545079 + 0.838384i \(0.683501\pi\)
\(24\) 0 0
\(25\) 56.6848 0.453479
\(26\) 0 0
\(27\) 147.430 1.05085
\(28\) 0 0
\(29\) 8.56420 0.0548390 0.0274195 0.999624i \(-0.491271\pi\)
0.0274195 + 0.999624i \(0.491271\pi\)
\(30\) 0 0
\(31\) 82.1070 0.475705 0.237852 0.971301i \(-0.423557\pi\)
0.237852 + 0.971301i \(0.423557\pi\)
\(32\) 0 0
\(33\) 2.91099 0.0153557
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 28.8132 0.128023 0.0640117 0.997949i \(-0.479611\pi\)
0.0640117 + 0.997949i \(0.479611\pi\)
\(38\) 0 0
\(39\) −125.128 −0.513758
\(40\) 0 0
\(41\) −70.5291 −0.268654 −0.134327 0.990937i \(-0.542887\pi\)
−0.134327 + 0.990937i \(0.542887\pi\)
\(42\) 0 0
\(43\) −417.179 −1.47952 −0.739758 0.672873i \(-0.765060\pi\)
−0.739758 + 0.672873i \(0.765060\pi\)
\(44\) 0 0
\(45\) 191.224 0.633467
\(46\) 0 0
\(47\) −338.261 −1.04980 −0.524899 0.851165i \(-0.675897\pi\)
−0.524899 + 0.851165i \(0.675897\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −421.428 −1.15709
\(52\) 0 0
\(53\) 149.121 0.386477 0.193239 0.981152i \(-0.438101\pi\)
0.193239 + 0.981152i \(0.438101\pi\)
\(54\) 0 0
\(55\) 10.9615 0.0268737
\(56\) 0 0
\(57\) −333.942 −0.775994
\(58\) 0 0
\(59\) 94.1828 0.207823 0.103911 0.994587i \(-0.466864\pi\)
0.103911 + 0.994587i \(0.466864\pi\)
\(60\) 0 0
\(61\) −120.525 −0.252977 −0.126488 0.991968i \(-0.540371\pi\)
−0.126488 + 0.991968i \(0.540371\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −471.179 −0.899116
\(66\) 0 0
\(67\) 792.366 1.44482 0.722410 0.691465i \(-0.243035\pi\)
0.722410 + 0.691465i \(0.243035\pi\)
\(68\) 0 0
\(69\) 430.438 0.750995
\(70\) 0 0
\(71\) −449.128 −0.750729 −0.375364 0.926877i \(-0.622482\pi\)
−0.375364 + 0.926877i \(0.622482\pi\)
\(72\) 0 0
\(73\) −469.420 −0.752623 −0.376311 0.926493i \(-0.622808\pi\)
−0.376311 + 0.926493i \(0.622808\pi\)
\(74\) 0 0
\(75\) −202.907 −0.312395
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1019.85 1.45243 0.726217 0.687465i \(-0.241277\pi\)
0.726217 + 0.687465i \(0.241277\pi\)
\(80\) 0 0
\(81\) −144.693 −0.198481
\(82\) 0 0
\(83\) 104.253 0.137870 0.0689352 0.997621i \(-0.478040\pi\)
0.0689352 + 0.997621i \(0.478040\pi\)
\(84\) 0 0
\(85\) −1586.91 −2.02500
\(86\) 0 0
\(87\) −30.6560 −0.0377778
\(88\) 0 0
\(89\) −1572.92 −1.87336 −0.936680 0.350185i \(-0.886119\pi\)
−0.936680 + 0.350185i \(0.886119\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −293.906 −0.327706
\(94\) 0 0
\(95\) −1257.48 −1.35805
\(96\) 0 0
\(97\) 550.057 0.575772 0.287886 0.957665i \(-0.407048\pi\)
0.287886 + 0.957665i \(0.407048\pi\)
\(98\) 0 0
\(99\) 11.5371 0.0117123
\(100\) 0 0
\(101\) −65.7169 −0.0647433 −0.0323717 0.999476i \(-0.510306\pi\)
−0.0323717 + 0.999476i \(0.510306\pi\)
\(102\) 0 0
\(103\) −1829.35 −1.75001 −0.875005 0.484113i \(-0.839142\pi\)
−0.875005 + 0.484113i \(0.839142\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −861.377 −0.778248 −0.389124 0.921185i \(-0.627222\pi\)
−0.389124 + 0.921185i \(0.627222\pi\)
\(108\) 0 0
\(109\) −1620.52 −1.42401 −0.712007 0.702173i \(-0.752213\pi\)
−0.712007 + 0.702173i \(0.752213\pi\)
\(110\) 0 0
\(111\) −103.139 −0.0881935
\(112\) 0 0
\(113\) 380.409 0.316689 0.158344 0.987384i \(-0.449384\pi\)
0.158344 + 0.987384i \(0.449384\pi\)
\(114\) 0 0
\(115\) 1620.84 1.31430
\(116\) 0 0
\(117\) −495.918 −0.391860
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1330.34 −0.999503
\(122\) 0 0
\(123\) 252.463 0.185072
\(124\) 0 0
\(125\) 920.824 0.658888
\(126\) 0 0
\(127\) −958.358 −0.669610 −0.334805 0.942287i \(-0.608671\pi\)
−0.334805 + 0.942287i \(0.608671\pi\)
\(128\) 0 0
\(129\) 1493.32 1.01922
\(130\) 0 0
\(131\) −1152.16 −0.768431 −0.384216 0.923243i \(-0.625528\pi\)
−0.384216 + 0.923243i \(0.625528\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1987.22 −1.26691
\(136\) 0 0
\(137\) 357.377 0.222867 0.111434 0.993772i \(-0.464456\pi\)
0.111434 + 0.993772i \(0.464456\pi\)
\(138\) 0 0
\(139\) 2736.29 1.66970 0.834852 0.550475i \(-0.185553\pi\)
0.834852 + 0.550475i \(0.185553\pi\)
\(140\) 0 0
\(141\) 1210.83 0.723191
\(142\) 0 0
\(143\) −28.4275 −0.0166239
\(144\) 0 0
\(145\) −115.437 −0.0661141
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1409.94 0.775212 0.387606 0.921825i \(-0.373302\pi\)
0.387606 + 0.921825i \(0.373302\pi\)
\(150\) 0 0
\(151\) −2352.35 −1.26776 −0.633879 0.773432i \(-0.718538\pi\)
−0.633879 + 0.773432i \(0.718538\pi\)
\(152\) 0 0
\(153\) −1670.24 −0.882553
\(154\) 0 0
\(155\) −1106.72 −0.573511
\(156\) 0 0
\(157\) 1213.82 0.617029 0.308514 0.951220i \(-0.400168\pi\)
0.308514 + 0.951220i \(0.400168\pi\)
\(158\) 0 0
\(159\) −533.786 −0.266239
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 722.774 0.347313 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(164\) 0 0
\(165\) −39.2374 −0.0185129
\(166\) 0 0
\(167\) −753.016 −0.348923 −0.174462 0.984664i \(-0.555818\pi\)
−0.174462 + 0.984664i \(0.555818\pi\)
\(168\) 0 0
\(169\) −975.051 −0.443810
\(170\) 0 0
\(171\) −1323.50 −0.591876
\(172\) 0 0
\(173\) −1859.14 −0.817038 −0.408519 0.912750i \(-0.633955\pi\)
−0.408519 + 0.912750i \(0.633955\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −337.133 −0.143166
\(178\) 0 0
\(179\) 522.825 0.218312 0.109156 0.994025i \(-0.465185\pi\)
0.109156 + 0.994025i \(0.465185\pi\)
\(180\) 0 0
\(181\) 2901.38 1.19148 0.595740 0.803177i \(-0.296859\pi\)
0.595740 + 0.803177i \(0.296859\pi\)
\(182\) 0 0
\(183\) 431.424 0.174272
\(184\) 0 0
\(185\) −388.375 −0.154345
\(186\) 0 0
\(187\) −95.7427 −0.0374407
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2604.35 −0.986619 −0.493309 0.869854i \(-0.664213\pi\)
−0.493309 + 0.869854i \(0.664213\pi\)
\(192\) 0 0
\(193\) 676.245 0.252214 0.126107 0.992017i \(-0.459752\pi\)
0.126107 + 0.992017i \(0.459752\pi\)
\(194\) 0 0
\(195\) 1686.61 0.619389
\(196\) 0 0
\(197\) −3685.99 −1.33308 −0.666538 0.745471i \(-0.732225\pi\)
−0.666538 + 0.745471i \(0.732225\pi\)
\(198\) 0 0
\(199\) 799.801 0.284907 0.142453 0.989802i \(-0.454501\pi\)
0.142453 + 0.989802i \(0.454501\pi\)
\(200\) 0 0
\(201\) −2836.32 −0.995316
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 950.665 0.323890
\(206\) 0 0
\(207\) 1705.95 0.572809
\(208\) 0 0
\(209\) −75.8670 −0.0251092
\(210\) 0 0
\(211\) 667.385 0.217747 0.108874 0.994056i \(-0.465276\pi\)
0.108874 + 0.994056i \(0.465276\pi\)
\(212\) 0 0
\(213\) 1607.68 0.517166
\(214\) 0 0
\(215\) 5623.18 1.78371
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1680.32 0.518471
\(220\) 0 0
\(221\) 4115.48 1.25266
\(222\) 0 0
\(223\) −2646.82 −0.794818 −0.397409 0.917642i \(-0.630091\pi\)
−0.397409 + 0.917642i \(0.630091\pi\)
\(224\) 0 0
\(225\) −804.175 −0.238274
\(226\) 0 0
\(227\) −4121.11 −1.20497 −0.602485 0.798131i \(-0.705822\pi\)
−0.602485 + 0.798131i \(0.705822\pi\)
\(228\) 0 0
\(229\) 4066.92 1.17358 0.586790 0.809739i \(-0.300391\pi\)
0.586790 + 0.809739i \(0.300391\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3904.67 −1.09787 −0.548934 0.835865i \(-0.684966\pi\)
−0.548934 + 0.835865i \(0.684966\pi\)
\(234\) 0 0
\(235\) 4559.44 1.26564
\(236\) 0 0
\(237\) −3650.62 −1.00056
\(238\) 0 0
\(239\) −5425.12 −1.46829 −0.734146 0.678991i \(-0.762417\pi\)
−0.734146 + 0.678991i \(0.762417\pi\)
\(240\) 0 0
\(241\) −1602.89 −0.428429 −0.214215 0.976787i \(-0.568719\pi\)
−0.214215 + 0.976787i \(0.568719\pi\)
\(242\) 0 0
\(243\) −3462.69 −0.914121
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3261.13 0.840084
\(248\) 0 0
\(249\) −373.179 −0.0949769
\(250\) 0 0
\(251\) −3805.93 −0.957085 −0.478542 0.878064i \(-0.658835\pi\)
−0.478542 + 0.878064i \(0.658835\pi\)
\(252\) 0 0
\(253\) 97.7897 0.0243003
\(254\) 0 0
\(255\) 5680.45 1.39499
\(256\) 0 0
\(257\) 4589.34 1.11391 0.556956 0.830542i \(-0.311969\pi\)
0.556956 + 0.830542i \(0.311969\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −121.498 −0.0288144
\(262\) 0 0
\(263\) −877.175 −0.205661 −0.102831 0.994699i \(-0.532790\pi\)
−0.102831 + 0.994699i \(0.532790\pi\)
\(264\) 0 0
\(265\) −2010.00 −0.465938
\(266\) 0 0
\(267\) 5630.35 1.29053
\(268\) 0 0
\(269\) −6123.55 −1.38795 −0.693977 0.719997i \(-0.744143\pi\)
−0.693977 + 0.719997i \(0.744143\pi\)
\(270\) 0 0
\(271\) 3489.76 0.782243 0.391122 0.920339i \(-0.372087\pi\)
0.391122 + 0.920339i \(0.372087\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −46.0976 −0.0101083
\(276\) 0 0
\(277\) −4891.70 −1.06106 −0.530530 0.847666i \(-0.678007\pi\)
−0.530530 + 0.847666i \(0.678007\pi\)
\(278\) 0 0
\(279\) −1164.83 −0.249952
\(280\) 0 0
\(281\) 6914.46 1.46791 0.733954 0.679199i \(-0.237673\pi\)
0.733954 + 0.679199i \(0.237673\pi\)
\(282\) 0 0
\(283\) −3559.85 −0.747742 −0.373871 0.927481i \(-0.621970\pi\)
−0.373871 + 0.927481i \(0.621970\pi\)
\(284\) 0 0
\(285\) 4501.22 0.935541
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8947.80 1.82125
\(290\) 0 0
\(291\) −1968.96 −0.396641
\(292\) 0 0
\(293\) 3285.11 0.655011 0.327505 0.944849i \(-0.393792\pi\)
0.327505 + 0.944849i \(0.393792\pi\)
\(294\) 0 0
\(295\) −1269.49 −0.250552
\(296\) 0 0
\(297\) −119.894 −0.0234242
\(298\) 0 0
\(299\) −4203.47 −0.813020
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 235.237 0.0446008
\(304\) 0 0
\(305\) 1624.56 0.304990
\(306\) 0 0
\(307\) 9094.65 1.69075 0.845373 0.534176i \(-0.179378\pi\)
0.845373 + 0.534176i \(0.179378\pi\)
\(308\) 0 0
\(309\) 6548.26 1.20556
\(310\) 0 0
\(311\) 8163.06 1.48838 0.744188 0.667971i \(-0.232837\pi\)
0.744188 + 0.667971i \(0.232837\pi\)
\(312\) 0 0
\(313\) 2979.62 0.538076 0.269038 0.963130i \(-0.413294\pi\)
0.269038 + 0.963130i \(0.413294\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3888.11 −0.688889 −0.344445 0.938807i \(-0.611933\pi\)
−0.344445 + 0.938807i \(0.611933\pi\)
\(318\) 0 0
\(319\) −6.96463 −0.00122240
\(320\) 0 0
\(321\) 3083.35 0.536124
\(322\) 0 0
\(323\) 10983.4 1.89205
\(324\) 0 0
\(325\) 1981.50 0.338196
\(326\) 0 0
\(327\) 5800.74 0.980983
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4893.03 0.812524 0.406262 0.913757i \(-0.366832\pi\)
0.406262 + 0.913757i \(0.366832\pi\)
\(332\) 0 0
\(333\) −408.767 −0.0672681
\(334\) 0 0
\(335\) −10680.3 −1.74188
\(336\) 0 0
\(337\) −1722.10 −0.278364 −0.139182 0.990267i \(-0.544447\pi\)
−0.139182 + 0.990267i \(0.544447\pi\)
\(338\) 0 0
\(339\) −1361.69 −0.218163
\(340\) 0 0
\(341\) −66.7716 −0.0106038
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5801.90 −0.905402
\(346\) 0 0
\(347\) 238.058 0.0368289 0.0184145 0.999830i \(-0.494138\pi\)
0.0184145 + 0.999830i \(0.494138\pi\)
\(348\) 0 0
\(349\) −10053.1 −1.54192 −0.770959 0.636884i \(-0.780223\pi\)
−0.770959 + 0.636884i \(0.780223\pi\)
\(350\) 0 0
\(351\) 5153.63 0.783706
\(352\) 0 0
\(353\) −3470.16 −0.523224 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(354\) 0 0
\(355\) 6053.82 0.905081
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1407.54 −0.206928 −0.103464 0.994633i \(-0.532993\pi\)
−0.103464 + 0.994633i \(0.532993\pi\)
\(360\) 0 0
\(361\) 1844.27 0.268884
\(362\) 0 0
\(363\) 4762.02 0.688544
\(364\) 0 0
\(365\) 6327.34 0.907364
\(366\) 0 0
\(367\) −11133.0 −1.58348 −0.791742 0.610855i \(-0.790826\pi\)
−0.791742 + 0.610855i \(0.790826\pi\)
\(368\) 0 0
\(369\) 1000.58 0.141160
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9025.94 1.25294 0.626468 0.779447i \(-0.284500\pi\)
0.626468 + 0.779447i \(0.284500\pi\)
\(374\) 0 0
\(375\) −3296.14 −0.453899
\(376\) 0 0
\(377\) 299.373 0.0408979
\(378\) 0 0
\(379\) 5855.75 0.793640 0.396820 0.917896i \(-0.370114\pi\)
0.396820 + 0.917896i \(0.370114\pi\)
\(380\) 0 0
\(381\) 3430.50 0.461285
\(382\) 0 0
\(383\) 7788.03 1.03903 0.519517 0.854460i \(-0.326112\pi\)
0.519517 + 0.854460i \(0.326112\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5918.42 0.777391
\(388\) 0 0
\(389\) −3814.05 −0.497121 −0.248560 0.968616i \(-0.579957\pi\)
−0.248560 + 0.968616i \(0.579957\pi\)
\(390\) 0 0
\(391\) −14157.1 −1.83109
\(392\) 0 0
\(393\) 4124.21 0.529362
\(394\) 0 0
\(395\) −13746.6 −1.75106
\(396\) 0 0
\(397\) −10165.3 −1.28509 −0.642545 0.766248i \(-0.722121\pi\)
−0.642545 + 0.766248i \(0.722121\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11502.5 1.43244 0.716222 0.697873i \(-0.245870\pi\)
0.716222 + 0.697873i \(0.245870\pi\)
\(402\) 0 0
\(403\) 2870.16 0.354772
\(404\) 0 0
\(405\) 1950.32 0.239289
\(406\) 0 0
\(407\) −23.4317 −0.00285372
\(408\) 0 0
\(409\) 3266.27 0.394882 0.197441 0.980315i \(-0.436737\pi\)
0.197441 + 0.980315i \(0.436737\pi\)
\(410\) 0 0
\(411\) −1279.25 −0.153530
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1405.23 −0.166217
\(416\) 0 0
\(417\) −9794.69 −1.15024
\(418\) 0 0
\(419\) 6822.93 0.795518 0.397759 0.917490i \(-0.369788\pi\)
0.397759 + 0.917490i \(0.369788\pi\)
\(420\) 0 0
\(421\) 1431.63 0.165733 0.0828665 0.996561i \(-0.473592\pi\)
0.0828665 + 0.996561i \(0.473592\pi\)
\(422\) 0 0
\(423\) 4798.83 0.551601
\(424\) 0 0
\(425\) 6673.61 0.761689
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 101.758 0.0114520
\(430\) 0 0
\(431\) −15142.2 −1.69228 −0.846141 0.532959i \(-0.821080\pi\)
−0.846141 + 0.532959i \(0.821080\pi\)
\(432\) 0 0
\(433\) −5475.65 −0.607721 −0.303860 0.952717i \(-0.598276\pi\)
−0.303860 + 0.952717i \(0.598276\pi\)
\(434\) 0 0
\(435\) 413.214 0.0455451
\(436\) 0 0
\(437\) −11218.2 −1.22801
\(438\) 0 0
\(439\) −1780.54 −0.193578 −0.0967890 0.995305i \(-0.530857\pi\)
−0.0967890 + 0.995305i \(0.530857\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3259.64 0.349594 0.174797 0.984605i \(-0.444073\pi\)
0.174797 + 0.984605i \(0.444073\pi\)
\(444\) 0 0
\(445\) 21201.5 2.25853
\(446\) 0 0
\(447\) −5046.95 −0.534033
\(448\) 0 0
\(449\) −6826.19 −0.717478 −0.358739 0.933438i \(-0.616793\pi\)
−0.358739 + 0.933438i \(0.616793\pi\)
\(450\) 0 0
\(451\) 57.3562 0.00598846
\(452\) 0 0
\(453\) 8420.37 0.873341
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3700.03 0.378731 0.189365 0.981907i \(-0.439357\pi\)
0.189365 + 0.981907i \(0.439357\pi\)
\(458\) 0 0
\(459\) 17357.3 1.76507
\(460\) 0 0
\(461\) −9400.80 −0.949759 −0.474880 0.880051i \(-0.657508\pi\)
−0.474880 + 0.880051i \(0.657508\pi\)
\(462\) 0 0
\(463\) −15483.9 −1.55420 −0.777102 0.629374i \(-0.783311\pi\)
−0.777102 + 0.629374i \(0.783311\pi\)
\(464\) 0 0
\(465\) 3961.58 0.395084
\(466\) 0 0
\(467\) 2205.62 0.218552 0.109276 0.994011i \(-0.465147\pi\)
0.109276 + 0.994011i \(0.465147\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4344.94 −0.425062
\(472\) 0 0
\(473\) 339.261 0.0329794
\(474\) 0 0
\(475\) 5288.20 0.510820
\(476\) 0 0
\(477\) −2115.54 −0.203069
\(478\) 0 0
\(479\) −2349.32 −0.224098 −0.112049 0.993703i \(-0.535741\pi\)
−0.112049 + 0.993703i \(0.535741\pi\)
\(480\) 0 0
\(481\) 1007.21 0.0954775
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7414.25 −0.694152
\(486\) 0 0
\(487\) −10394.3 −0.967167 −0.483583 0.875298i \(-0.660665\pi\)
−0.483583 + 0.875298i \(0.660665\pi\)
\(488\) 0 0
\(489\) −2587.21 −0.239259
\(490\) 0 0
\(491\) −12586.7 −1.15689 −0.578444 0.815722i \(-0.696340\pi\)
−0.578444 + 0.815722i \(0.696340\pi\)
\(492\) 0 0
\(493\) 1008.28 0.0921108
\(494\) 0 0
\(495\) −155.509 −0.0141204
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −10627.9 −0.953450 −0.476725 0.879053i \(-0.658176\pi\)
−0.476725 + 0.879053i \(0.658176\pi\)
\(500\) 0 0
\(501\) 2695.46 0.240368
\(502\) 0 0
\(503\) 6719.02 0.595599 0.297800 0.954628i \(-0.403747\pi\)
0.297800 + 0.954628i \(0.403747\pi\)
\(504\) 0 0
\(505\) 885.802 0.0780548
\(506\) 0 0
\(507\) 3490.25 0.305735
\(508\) 0 0
\(509\) 3904.33 0.339993 0.169997 0.985445i \(-0.445624\pi\)
0.169997 + 0.985445i \(0.445624\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13754.0 1.18373
\(514\) 0 0
\(515\) 24657.9 2.10982
\(516\) 0 0
\(517\) 275.083 0.0234007
\(518\) 0 0
\(519\) 6654.89 0.562846
\(520\) 0 0
\(521\) −15699.7 −1.32018 −0.660092 0.751184i \(-0.729483\pi\)
−0.660092 + 0.751184i \(0.729483\pi\)
\(522\) 0 0
\(523\) −10152.1 −0.848794 −0.424397 0.905476i \(-0.639514\pi\)
−0.424397 + 0.905476i \(0.639514\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9666.61 0.799021
\(528\) 0 0
\(529\) 2292.83 0.188447
\(530\) 0 0
\(531\) −1336.15 −0.109198
\(532\) 0 0
\(533\) −2465.44 −0.200357
\(534\) 0 0
\(535\) 11610.6 0.938258
\(536\) 0 0
\(537\) −1871.48 −0.150392
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19846.6 −1.57722 −0.788608 0.614896i \(-0.789198\pi\)
−0.788608 + 0.614896i \(0.789198\pi\)
\(542\) 0 0
\(543\) −10385.7 −0.820794
\(544\) 0 0
\(545\) 21843.0 1.71679
\(546\) 0 0
\(547\) 22798.9 1.78210 0.891052 0.453901i \(-0.149968\pi\)
0.891052 + 0.453901i \(0.149968\pi\)
\(548\) 0 0
\(549\) 1709.85 0.132923
\(550\) 0 0
\(551\) 798.965 0.0617733
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1390.21 0.106326
\(556\) 0 0
\(557\) −17998.3 −1.36914 −0.684570 0.728947i \(-0.740010\pi\)
−0.684570 + 0.728947i \(0.740010\pi\)
\(558\) 0 0
\(559\) −14583.1 −1.10340
\(560\) 0 0
\(561\) 342.717 0.0257923
\(562\) 0 0
\(563\) −195.636 −0.0146449 −0.00732246 0.999973i \(-0.502331\pi\)
−0.00732246 + 0.999973i \(0.502331\pi\)
\(564\) 0 0
\(565\) −5127.55 −0.381801
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19660.4 −1.44852 −0.724260 0.689527i \(-0.757818\pi\)
−0.724260 + 0.689527i \(0.757818\pi\)
\(570\) 0 0
\(571\) 15764.5 1.15538 0.577691 0.816255i \(-0.303954\pi\)
0.577691 + 0.816255i \(0.303954\pi\)
\(572\) 0 0
\(573\) 9322.42 0.679668
\(574\) 0 0
\(575\) −6816.30 −0.494364
\(576\) 0 0
\(577\) 22306.4 1.60941 0.804704 0.593676i \(-0.202324\pi\)
0.804704 + 0.593676i \(0.202324\pi\)
\(578\) 0 0
\(579\) −2420.66 −0.173746
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −121.269 −0.00861483
\(584\) 0 0
\(585\) 6684.51 0.472428
\(586\) 0 0
\(587\) 15953.2 1.12173 0.560866 0.827906i \(-0.310468\pi\)
0.560866 + 0.827906i \(0.310468\pi\)
\(588\) 0 0
\(589\) 7659.87 0.535856
\(590\) 0 0
\(591\) 13194.2 0.918338
\(592\) 0 0
\(593\) −3155.68 −0.218530 −0.109265 0.994013i \(-0.534850\pi\)
−0.109265 + 0.994013i \(0.534850\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2862.93 −0.196268
\(598\) 0 0
\(599\) −25456.3 −1.73642 −0.868212 0.496194i \(-0.834730\pi\)
−0.868212 + 0.496194i \(0.834730\pi\)
\(600\) 0 0
\(601\) 5580.96 0.378789 0.189395 0.981901i \(-0.439347\pi\)
0.189395 + 0.981901i \(0.439347\pi\)
\(602\) 0 0
\(603\) −11241.1 −0.759160
\(604\) 0 0
\(605\) 17931.7 1.20500
\(606\) 0 0
\(607\) −381.133 −0.0254855 −0.0127427 0.999919i \(-0.504056\pi\)
−0.0127427 + 0.999919i \(0.504056\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11824.4 −0.782919
\(612\) 0 0
\(613\) −8235.98 −0.542656 −0.271328 0.962487i \(-0.587463\pi\)
−0.271328 + 0.962487i \(0.587463\pi\)
\(614\) 0 0
\(615\) −3402.96 −0.223123
\(616\) 0 0
\(617\) −27419.8 −1.78911 −0.894555 0.446958i \(-0.852507\pi\)
−0.894555 + 0.446958i \(0.852507\pi\)
\(618\) 0 0
\(619\) 16373.4 1.06317 0.531585 0.847005i \(-0.321597\pi\)
0.531585 + 0.847005i \(0.321597\pi\)
\(620\) 0 0
\(621\) −17728.4 −1.14560
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19497.4 −1.24784
\(626\) 0 0
\(627\) 271.570 0.0172974
\(628\) 0 0
\(629\) 3392.24 0.215035
\(630\) 0 0
\(631\) −4059.60 −0.256118 −0.128059 0.991767i \(-0.540875\pi\)
−0.128059 + 0.991767i \(0.540875\pi\)
\(632\) 0 0
\(633\) −2388.94 −0.150003
\(634\) 0 0
\(635\) 12917.8 0.807284
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6371.68 0.394460
\(640\) 0 0
\(641\) −6388.63 −0.393660 −0.196830 0.980438i \(-0.563065\pi\)
−0.196830 + 0.980438i \(0.563065\pi\)
\(642\) 0 0
\(643\) 18308.0 1.12286 0.561428 0.827525i \(-0.310252\pi\)
0.561428 + 0.827525i \(0.310252\pi\)
\(644\) 0 0
\(645\) −20128.5 −1.22877
\(646\) 0 0
\(647\) −3303.90 −0.200757 −0.100379 0.994949i \(-0.532005\pi\)
−0.100379 + 0.994949i \(0.532005\pi\)
\(648\) 0 0
\(649\) −76.5919 −0.00463251
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4371.27 0.261961 0.130981 0.991385i \(-0.458187\pi\)
0.130981 + 0.991385i \(0.458187\pi\)
\(654\) 0 0
\(655\) 15530.0 0.926423
\(656\) 0 0
\(657\) 6659.55 0.395455
\(658\) 0 0
\(659\) −6259.75 −0.370023 −0.185012 0.982736i \(-0.559232\pi\)
−0.185012 + 0.982736i \(0.559232\pi\)
\(660\) 0 0
\(661\) −14845.7 −0.873574 −0.436787 0.899565i \(-0.643884\pi\)
−0.436787 + 0.899565i \(0.643884\pi\)
\(662\) 0 0
\(663\) −14731.6 −0.862938
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1029.84 −0.0597832
\(668\) 0 0
\(669\) 9474.46 0.547539
\(670\) 0 0
\(671\) 98.0138 0.00563902
\(672\) 0 0
\(673\) 9409.13 0.538923 0.269462 0.963011i \(-0.413154\pi\)
0.269462 + 0.963011i \(0.413154\pi\)
\(674\) 0 0
\(675\) 8357.07 0.476539
\(676\) 0 0
\(677\) 2950.63 0.167507 0.0837533 0.996487i \(-0.473309\pi\)
0.0837533 + 0.996487i \(0.473309\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14751.8 0.830086
\(682\) 0 0
\(683\) −6280.85 −0.351874 −0.175937 0.984401i \(-0.556296\pi\)
−0.175937 + 0.984401i \(0.556296\pi\)
\(684\) 0 0
\(685\) −4817.11 −0.268689
\(686\) 0 0
\(687\) −14557.8 −0.808463
\(688\) 0 0
\(689\) 5212.72 0.288228
\(690\) 0 0
\(691\) −32763.2 −1.80372 −0.901861 0.432027i \(-0.857798\pi\)
−0.901861 + 0.432027i \(0.857798\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −36882.5 −2.01300
\(696\) 0 0
\(697\) −8303.53 −0.451246
\(698\) 0 0
\(699\) 13977.0 0.756306
\(700\) 0 0
\(701\) −1775.97 −0.0956883 −0.0478442 0.998855i \(-0.515235\pi\)
−0.0478442 + 0.998855i \(0.515235\pi\)
\(702\) 0 0
\(703\) 2688.02 0.144212
\(704\) 0 0
\(705\) −16320.8 −0.871881
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8862.43 0.469444 0.234722 0.972063i \(-0.424582\pi\)
0.234722 + 0.972063i \(0.424582\pi\)
\(710\) 0 0
\(711\) −14468.4 −0.763162
\(712\) 0 0
\(713\) −9873.28 −0.518594
\(714\) 0 0
\(715\) 383.175 0.0200419
\(716\) 0 0
\(717\) 19419.5 1.01149
\(718\) 0 0
\(719\) −27499.2 −1.42635 −0.713177 0.700984i \(-0.752744\pi\)
−0.713177 + 0.700984i \(0.752744\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5737.65 0.295139
\(724\) 0 0
\(725\) 485.460 0.0248683
\(726\) 0 0
\(727\) −25434.9 −1.29756 −0.648781 0.760975i \(-0.724721\pi\)
−0.648781 + 0.760975i \(0.724721\pi\)
\(728\) 0 0
\(729\) 16301.6 0.828206
\(730\) 0 0
\(731\) −49115.3 −2.48508
\(732\) 0 0
\(733\) −24155.6 −1.21720 −0.608600 0.793477i \(-0.708269\pi\)
−0.608600 + 0.793477i \(0.708269\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −644.373 −0.0322060
\(738\) 0 0
\(739\) −27512.9 −1.36952 −0.684762 0.728767i \(-0.740094\pi\)
−0.684762 + 0.728767i \(0.740094\pi\)
\(740\) 0 0
\(741\) −11673.4 −0.578722
\(742\) 0 0
\(743\) −5995.09 −0.296014 −0.148007 0.988986i \(-0.547286\pi\)
−0.148007 + 0.988986i \(0.547286\pi\)
\(744\) 0 0
\(745\) −19004.6 −0.934598
\(746\) 0 0
\(747\) −1479.01 −0.0724420
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1545.09 −0.0750747 −0.0375373 0.999295i \(-0.511951\pi\)
−0.0375373 + 0.999295i \(0.511951\pi\)
\(752\) 0 0
\(753\) 13623.6 0.659322
\(754\) 0 0
\(755\) 31707.5 1.52841
\(756\) 0 0
\(757\) −5157.82 −0.247641 −0.123820 0.992305i \(-0.539515\pi\)
−0.123820 + 0.992305i \(0.539515\pi\)
\(758\) 0 0
\(759\) −350.044 −0.0167402
\(760\) 0 0
\(761\) −3289.96 −0.156716 −0.0783581 0.996925i \(-0.524968\pi\)
−0.0783581 + 0.996925i \(0.524968\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 22513.2 1.06401
\(766\) 0 0
\(767\) 3292.29 0.154990
\(768\) 0 0
\(769\) −11146.5 −0.522697 −0.261348 0.965245i \(-0.584167\pi\)
−0.261348 + 0.965245i \(0.584167\pi\)
\(770\) 0 0
\(771\) −16427.8 −0.767358
\(772\) 0 0
\(773\) −15830.2 −0.736576 −0.368288 0.929712i \(-0.620056\pi\)
−0.368288 + 0.929712i \(0.620056\pi\)
\(774\) 0 0
\(775\) 4654.22 0.215722
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6579.75 −0.302624
\(780\) 0 0
\(781\) 365.243 0.0167342
\(782\) 0 0
\(783\) 1262.62 0.0576277
\(784\) 0 0
\(785\) −16361.2 −0.743892
\(786\) 0 0
\(787\) 15163.4 0.686809 0.343404 0.939188i \(-0.388420\pi\)
0.343404 + 0.939188i \(0.388420\pi\)
\(788\) 0 0
\(789\) 3139.90 0.141677
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4213.10 −0.188665
\(794\) 0 0
\(795\) 7194.93 0.320978
\(796\) 0 0
\(797\) 29398.3 1.30658 0.653289 0.757109i \(-0.273389\pi\)
0.653289 + 0.757109i \(0.273389\pi\)
\(798\) 0 0
\(799\) −39824.1 −1.76330
\(800\) 0 0
\(801\) 22314.6 0.984331
\(802\) 0 0
\(803\) 381.745 0.0167764
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 21919.6 0.956142
\(808\) 0 0
\(809\) −20712.9 −0.900155 −0.450078 0.892989i \(-0.648604\pi\)
−0.450078 + 0.892989i \(0.648604\pi\)
\(810\) 0 0
\(811\) −27369.9 −1.18506 −0.592532 0.805547i \(-0.701872\pi\)
−0.592532 + 0.805547i \(0.701872\pi\)
\(812\) 0 0
\(813\) −12491.8 −0.538876
\(814\) 0 0
\(815\) −9742.31 −0.418722
\(816\) 0 0
\(817\) −38919.2 −1.66660
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −35362.7 −1.50325 −0.751623 0.659592i \(-0.770729\pi\)
−0.751623 + 0.659592i \(0.770729\pi\)
\(822\) 0 0
\(823\) 29190.4 1.23635 0.618174 0.786042i \(-0.287873\pi\)
0.618174 + 0.786042i \(0.287873\pi\)
\(824\) 0 0
\(825\) 165.009 0.00696349
\(826\) 0 0
\(827\) 7302.08 0.307035 0.153518 0.988146i \(-0.450940\pi\)
0.153518 + 0.988146i \(0.450940\pi\)
\(828\) 0 0
\(829\) 4250.77 0.178088 0.0890442 0.996028i \(-0.471619\pi\)
0.0890442 + 0.996028i \(0.471619\pi\)
\(830\) 0 0
\(831\) 17510.1 0.730949
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10149.9 0.420663
\(836\) 0 0
\(837\) 12105.1 0.499895
\(838\) 0 0
\(839\) 39527.7 1.62652 0.813258 0.581903i \(-0.197692\pi\)
0.813258 + 0.581903i \(0.197692\pi\)
\(840\) 0 0
\(841\) −24315.7 −0.996993
\(842\) 0 0
\(843\) −24750.7 −1.01122
\(844\) 0 0
\(845\) 13142.8 0.535059
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 12742.7 0.515109
\(850\) 0 0
\(851\) −3464.76 −0.139566
\(852\) 0 0
\(853\) 31656.1 1.27067 0.635337 0.772235i \(-0.280861\pi\)
0.635337 + 0.772235i \(0.280861\pi\)
\(854\) 0 0
\(855\) 17839.6 0.713568
\(856\) 0 0
\(857\) 1193.16 0.0475583 0.0237792 0.999717i \(-0.492430\pi\)
0.0237792 + 0.999717i \(0.492430\pi\)
\(858\) 0 0
\(859\) −29060.0 −1.15427 −0.577134 0.816650i \(-0.695829\pi\)
−0.577134 + 0.816650i \(0.695829\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23063.0 −0.909702 −0.454851 0.890567i \(-0.650308\pi\)
−0.454851 + 0.890567i \(0.650308\pi\)
\(864\) 0 0
\(865\) 25059.4 0.985024
\(866\) 0 0
\(867\) −32029.2 −1.25463
\(868\) 0 0
\(869\) −829.371 −0.0323757
\(870\) 0 0
\(871\) 27698.2 1.07752
\(872\) 0 0
\(873\) −7803.54 −0.302531
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 33871.0 1.30415 0.652077 0.758153i \(-0.273898\pi\)
0.652077 + 0.758153i \(0.273898\pi\)
\(878\) 0 0
\(879\) −11759.2 −0.451228
\(880\) 0 0
\(881\) −43331.1 −1.65705 −0.828525 0.559953i \(-0.810819\pi\)
−0.828525 + 0.559953i \(0.810819\pi\)
\(882\) 0 0
\(883\) 40897.3 1.55867 0.779334 0.626609i \(-0.215558\pi\)
0.779334 + 0.626609i \(0.215558\pi\)
\(884\) 0 0
\(885\) 4544.23 0.172602
\(886\) 0 0
\(887\) 45065.8 1.70593 0.852965 0.521968i \(-0.174802\pi\)
0.852965 + 0.521968i \(0.174802\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 117.668 0.00442427
\(892\) 0 0
\(893\) −31556.8 −1.18254
\(894\) 0 0
\(895\) −7047.19 −0.263197
\(896\) 0 0
\(897\) 15046.6 0.560078
\(898\) 0 0
\(899\) 703.180 0.0260872
\(900\) 0 0
\(901\) 17556.3 0.649150
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −39107.9 −1.43645
\(906\) 0 0
\(907\) −25282.5 −0.925570 −0.462785 0.886471i \(-0.653150\pi\)
−0.462785 + 0.886471i \(0.653150\pi\)
\(908\) 0 0
\(909\) 932.311 0.0340185
\(910\) 0 0
\(911\) 41646.1 1.51460 0.757298 0.653070i \(-0.226519\pi\)
0.757298 + 0.653070i \(0.226519\pi\)
\(912\) 0 0
\(913\) −84.7812 −0.00307322
\(914\) 0 0
\(915\) −5815.19 −0.210103
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26112.5 0.937292 0.468646 0.883386i \(-0.344742\pi\)
0.468646 + 0.883386i \(0.344742\pi\)
\(920\) 0 0
\(921\) −32554.8 −1.16473
\(922\) 0 0
\(923\) −15699.9 −0.559879
\(924\) 0 0
\(925\) 1633.27 0.0580559
\(926\) 0 0
\(927\) 25952.6 0.919519
\(928\) 0 0
\(929\) 32357.0 1.14273 0.571366 0.820695i \(-0.306414\pi\)
0.571366 + 0.820695i \(0.306414\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −29220.1 −1.02532
\(934\) 0 0
\(935\) 1290.52 0.0451386
\(936\) 0 0
\(937\) 32947.0 1.14870 0.574350 0.818610i \(-0.305255\pi\)
0.574350 + 0.818610i \(0.305255\pi\)
\(938\) 0 0
\(939\) −10665.7 −0.370673
\(940\) 0 0
\(941\) −501.602 −0.0173770 −0.00868849 0.999962i \(-0.502766\pi\)
−0.00868849 + 0.999962i \(0.502766\pi\)
\(942\) 0 0
\(943\) 8481.06 0.292875
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6436.90 0.220878 0.110439 0.993883i \(-0.464774\pi\)
0.110439 + 0.993883i \(0.464774\pi\)
\(948\) 0 0
\(949\) −16409.2 −0.561292
\(950\) 0 0
\(951\) 13917.7 0.474566
\(952\) 0 0
\(953\) 47511.2 1.61494 0.807470 0.589908i \(-0.200836\pi\)
0.807470 + 0.589908i \(0.200836\pi\)
\(954\) 0 0
\(955\) 35104.2 1.18947
\(956\) 0 0
\(957\) 24.9303 0.000842092 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23049.4 −0.773705
\(962\) 0 0
\(963\) 12220.2 0.408919
\(964\) 0 0
\(965\) −9115.15 −0.304069
\(966\) 0 0
\(967\) −7817.32 −0.259967 −0.129984 0.991516i \(-0.541492\pi\)
−0.129984 + 0.991516i \(0.541492\pi\)
\(968\) 0 0
\(969\) −39315.6 −1.30340
\(970\) 0 0
\(971\) −1503.50 −0.0496905 −0.0248453 0.999691i \(-0.507909\pi\)
−0.0248453 + 0.999691i \(0.507909\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7092.88 −0.232978
\(976\) 0 0
\(977\) 33389.1 1.09336 0.546680 0.837342i \(-0.315892\pi\)
0.546680 + 0.837342i \(0.315892\pi\)
\(978\) 0 0
\(979\) 1279.14 0.0417584
\(980\) 0 0
\(981\) 22989.9 0.748228
\(982\) 0 0
\(983\) 5451.02 0.176867 0.0884337 0.996082i \(-0.471814\pi\)
0.0884337 + 0.996082i \(0.471814\pi\)
\(984\) 0 0
\(985\) 49683.7 1.60716
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 50165.4 1.61291
\(990\) 0 0
\(991\) 46530.0 1.49150 0.745750 0.666226i \(-0.232092\pi\)
0.745750 + 0.666226i \(0.232092\pi\)
\(992\) 0 0
\(993\) −17514.9 −0.559736
\(994\) 0 0
\(995\) −10780.6 −0.343484
\(996\) 0 0
\(997\) 11410.0 0.362444 0.181222 0.983442i \(-0.441995\pi\)
0.181222 + 0.983442i \(0.441995\pi\)
\(998\) 0 0
\(999\) 4247.95 0.134534
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.a.bf.1.2 4
4.3 odd 2 49.4.a.e.1.4 yes 4
7.6 odd 2 inner 784.4.a.bf.1.3 4
12.11 even 2 441.4.a.u.1.2 4
20.19 odd 2 1225.4.a.bb.1.1 4
28.3 even 6 49.4.c.e.30.2 8
28.11 odd 6 49.4.c.e.30.1 8
28.19 even 6 49.4.c.e.18.2 8
28.23 odd 6 49.4.c.e.18.1 8
28.27 even 2 49.4.a.e.1.3 4
84.11 even 6 441.4.e.y.226.3 8
84.23 even 6 441.4.e.y.361.3 8
84.47 odd 6 441.4.e.y.361.4 8
84.59 odd 6 441.4.e.y.226.4 8
84.83 odd 2 441.4.a.u.1.1 4
140.139 even 2 1225.4.a.bb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.3 4 28.27 even 2
49.4.a.e.1.4 yes 4 4.3 odd 2
49.4.c.e.18.1 8 28.23 odd 6
49.4.c.e.18.2 8 28.19 even 6
49.4.c.e.30.1 8 28.11 odd 6
49.4.c.e.30.2 8 28.3 even 6
441.4.a.u.1.1 4 84.83 odd 2
441.4.a.u.1.2 4 12.11 even 2
441.4.e.y.226.3 8 84.11 even 6
441.4.e.y.226.4 8 84.59 odd 6
441.4.e.y.361.3 8 84.23 even 6
441.4.e.y.361.4 8 84.47 odd 6
784.4.a.bf.1.2 4 1.1 even 1 trivial
784.4.a.bf.1.3 4 7.6 odd 2 inner
1225.4.a.bb.1.1 4 20.19 odd 2
1225.4.a.bb.1.2 4 140.139 even 2