Properties

Label 1344.4.b.i.895.14
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $24$
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] = [1344,4,Mod(895,1344)]
 
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([1, 0, 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("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not 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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.14
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.i.895.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +2.26482i q^{5} +(5.56016 - 17.6659i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +2.26482i q^{5} +(5.56016 - 17.6659i) q^{7} +9.00000 q^{9} -56.3635i q^{11} +19.3862i q^{13} -6.79447i q^{15} +127.174i q^{17} +5.09723 q^{19} +(-16.6805 + 52.9978i) q^{21} +94.6187i q^{23} +119.871 q^{25} -27.0000 q^{27} +92.4931 q^{29} +38.9336 q^{31} +169.090i q^{33} +(40.0102 + 12.5928i) q^{35} -290.194 q^{37} -58.1585i q^{39} -118.193i q^{41} +274.296i q^{43} +20.3834i q^{45} +612.725 q^{47} +(-281.169 - 196.451i) q^{49} -381.522i q^{51} -55.9875 q^{53} +127.653 q^{55} -15.2917 q^{57} -701.109 q^{59} -655.954i q^{61} +(50.0414 - 158.993i) q^{63} -43.9062 q^{65} +109.756i q^{67} -283.856i q^{69} -478.781i q^{71} +562.420i q^{73} -359.612 q^{75} +(-995.713 - 313.390i) q^{77} -312.103i q^{79} +81.0000 q^{81} +1038.20 q^{83} -288.027 q^{85} -277.479 q^{87} +743.750i q^{89} +(342.474 + 107.790i) q^{91} -116.801 q^{93} +11.5443i q^{95} +1277.01i q^{97} -507.271i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9} + 56 q^{19} - 60 q^{21} - 432 q^{25} - 648 q^{27} - 464 q^{31} - 568 q^{35} - 504 q^{37} - 560 q^{47} - 128 q^{49} + 784 q^{53} - 424 q^{55} - 168 q^{57} - 800 q^{59} + 180 q^{63} + 560 q^{65} + 1296 q^{75} + 1568 q^{77} + 1944 q^{81} - 1936 q^{83} - 3000 q^{85} + 496 q^{91} + 1392 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 2.26482i 0.202572i 0.994857 + 0.101286i \(0.0322957\pi\)
−0.994857 + 0.101286i \(0.967704\pi\)
\(6\) 0 0
\(7\) 5.56016 17.6659i 0.300220 0.953870i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 56.3635i 1.54493i −0.635058 0.772465i \(-0.719024\pi\)
0.635058 0.772465i \(-0.280976\pi\)
\(12\) 0 0
\(13\) 19.3862i 0.413596i 0.978384 + 0.206798i \(0.0663044\pi\)
−0.978384 + 0.206798i \(0.933696\pi\)
\(14\) 0 0
\(15\) 6.79447i 0.116955i
\(16\) 0 0
\(17\) 127.174i 1.81437i 0.420737 + 0.907183i \(0.361772\pi\)
−0.420737 + 0.907183i \(0.638228\pi\)
\(18\) 0 0
\(19\) 5.09723 0.0615465 0.0307733 0.999526i \(-0.490203\pi\)
0.0307733 + 0.999526i \(0.490203\pi\)
\(20\) 0 0
\(21\) −16.6805 + 52.9978i −0.173332 + 0.550717i
\(22\) 0 0
\(23\) 94.6187i 0.857799i 0.903352 + 0.428899i \(0.141098\pi\)
−0.903352 + 0.428899i \(0.858902\pi\)
\(24\) 0 0
\(25\) 119.871 0.958965
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 92.4931 0.592260 0.296130 0.955148i \(-0.404304\pi\)
0.296130 + 0.955148i \(0.404304\pi\)
\(30\) 0 0
\(31\) 38.9336 0.225571 0.112785 0.993619i \(-0.464023\pi\)
0.112785 + 0.993619i \(0.464023\pi\)
\(32\) 0 0
\(33\) 169.090i 0.891966i
\(34\) 0 0
\(35\) 40.0102 + 12.5928i 0.193227 + 0.0608162i
\(36\) 0 0
\(37\) −290.194 −1.28940 −0.644698 0.764437i \(-0.723017\pi\)
−0.644698 + 0.764437i \(0.723017\pi\)
\(38\) 0 0
\(39\) 58.1585i 0.238790i
\(40\) 0 0
\(41\) 118.193i 0.450212i −0.974334 0.225106i \(-0.927727\pi\)
0.974334 0.225106i \(-0.0722729\pi\)
\(42\) 0 0
\(43\) 274.296i 0.972786i 0.873740 + 0.486393i \(0.161688\pi\)
−0.873740 + 0.486393i \(0.838312\pi\)
\(44\) 0 0
\(45\) 20.3834i 0.0675240i
\(46\) 0 0
\(47\) 612.725 1.90160 0.950799 0.309809i \(-0.100265\pi\)
0.950799 + 0.309809i \(0.100265\pi\)
\(48\) 0 0
\(49\) −281.169 196.451i −0.819736 0.572742i
\(50\) 0 0
\(51\) 381.522i 1.04752i
\(52\) 0 0
\(53\) −55.9875 −0.145103 −0.0725517 0.997365i \(-0.523114\pi\)
−0.0725517 + 0.997365i \(0.523114\pi\)
\(54\) 0 0
\(55\) 127.653 0.312959
\(56\) 0 0
\(57\) −15.2917 −0.0355339
\(58\) 0 0
\(59\) −701.109 −1.54706 −0.773531 0.633758i \(-0.781511\pi\)
−0.773531 + 0.633758i \(0.781511\pi\)
\(60\) 0 0
\(61\) 655.954i 1.37682i −0.725320 0.688412i \(-0.758308\pi\)
0.725320 0.688412i \(-0.241692\pi\)
\(62\) 0 0
\(63\) 50.0414 158.993i 0.100073 0.317957i
\(64\) 0 0
\(65\) −43.9062 −0.0837830
\(66\) 0 0
\(67\) 109.756i 0.200132i 0.994981 + 0.100066i \(0.0319055\pi\)
−0.994981 + 0.100066i \(0.968095\pi\)
\(68\) 0 0
\(69\) 283.856i 0.495250i
\(70\) 0 0
\(71\) 478.781i 0.800294i −0.916451 0.400147i \(-0.868959\pi\)
0.916451 0.400147i \(-0.131041\pi\)
\(72\) 0 0
\(73\) 562.420i 0.901729i 0.892592 + 0.450865i \(0.148884\pi\)
−0.892592 + 0.450865i \(0.851116\pi\)
\(74\) 0 0
\(75\) −359.612 −0.553658
\(76\) 0 0
\(77\) −995.713 313.390i −1.47366 0.463819i
\(78\) 0 0
\(79\) 312.103i 0.444486i −0.974991 0.222243i \(-0.928662\pi\)
0.974991 0.222243i \(-0.0713378\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1038.20 1.37298 0.686488 0.727141i \(-0.259152\pi\)
0.686488 + 0.727141i \(0.259152\pi\)
\(84\) 0 0
\(85\) −288.027 −0.367540
\(86\) 0 0
\(87\) −277.479 −0.341942
\(88\) 0 0
\(89\) 743.750i 0.885813i 0.896568 + 0.442906i \(0.146053\pi\)
−0.896568 + 0.442906i \(0.853947\pi\)
\(90\) 0 0
\(91\) 342.474 + 107.790i 0.394517 + 0.124170i
\(92\) 0 0
\(93\) −116.801 −0.130233
\(94\) 0 0
\(95\) 11.5443i 0.0124676i
\(96\) 0 0
\(97\) 1277.01i 1.33671i 0.743843 + 0.668354i \(0.233001\pi\)
−0.743843 + 0.668354i \(0.766999\pi\)
\(98\) 0 0
\(99\) 507.271i 0.514977i
\(100\) 0 0
\(101\) 1056.67i 1.04101i −0.853858 0.520507i \(-0.825743\pi\)
0.853858 0.520507i \(-0.174257\pi\)
\(102\) 0 0
\(103\) 1919.63 1.83638 0.918190 0.396140i \(-0.129651\pi\)
0.918190 + 0.396140i \(0.129651\pi\)
\(104\) 0 0
\(105\) −120.031 37.7783i −0.111560 0.0351122i
\(106\) 0 0
\(107\) 1062.91i 0.960331i 0.877178 + 0.480165i \(0.159423\pi\)
−0.877178 + 0.480165i \(0.840577\pi\)
\(108\) 0 0
\(109\) 1537.26 1.35085 0.675425 0.737429i \(-0.263960\pi\)
0.675425 + 0.737429i \(0.263960\pi\)
\(110\) 0 0
\(111\) 870.583 0.744433
\(112\) 0 0
\(113\) 190.272 0.158401 0.0792003 0.996859i \(-0.474763\pi\)
0.0792003 + 0.996859i \(0.474763\pi\)
\(114\) 0 0
\(115\) −214.295 −0.173766
\(116\) 0 0
\(117\) 174.475i 0.137865i
\(118\) 0 0
\(119\) 2246.64 + 707.107i 1.73067 + 0.544709i
\(120\) 0 0
\(121\) −1845.84 −1.38681
\(122\) 0 0
\(123\) 354.580i 0.259930i
\(124\) 0 0
\(125\) 554.589i 0.396831i
\(126\) 0 0
\(127\) 697.882i 0.487615i −0.969824 0.243807i \(-0.921604\pi\)
0.969824 0.243807i \(-0.0783964\pi\)
\(128\) 0 0
\(129\) 822.889i 0.561638i
\(130\) 0 0
\(131\) 818.426 0.545849 0.272925 0.962035i \(-0.412009\pi\)
0.272925 + 0.962035i \(0.412009\pi\)
\(132\) 0 0
\(133\) 28.3414 90.0472i 0.0184775 0.0587074i
\(134\) 0 0
\(135\) 61.1502i 0.0389850i
\(136\) 0 0
\(137\) 1451.82 0.905380 0.452690 0.891668i \(-0.350464\pi\)
0.452690 + 0.891668i \(0.350464\pi\)
\(138\) 0 0
\(139\) −1515.70 −0.924893 −0.462446 0.886647i \(-0.653028\pi\)
−0.462446 + 0.886647i \(0.653028\pi\)
\(140\) 0 0
\(141\) −1838.17 −1.09789
\(142\) 0 0
\(143\) 1092.67 0.638977
\(144\) 0 0
\(145\) 209.481i 0.119975i
\(146\) 0 0
\(147\) 843.508 + 589.352i 0.473275 + 0.330673i
\(148\) 0 0
\(149\) 1971.82 1.08415 0.542074 0.840331i \(-0.317639\pi\)
0.542074 + 0.840331i \(0.317639\pi\)
\(150\) 0 0
\(151\) 2864.69i 1.54387i −0.635699 0.771937i \(-0.719288\pi\)
0.635699 0.771937i \(-0.280712\pi\)
\(152\) 0 0
\(153\) 1144.57i 0.604789i
\(154\) 0 0
\(155\) 88.1778i 0.0456943i
\(156\) 0 0
\(157\) 323.010i 0.164198i −0.996624 0.0820988i \(-0.973838\pi\)
0.996624 0.0820988i \(-0.0261623\pi\)
\(158\) 0 0
\(159\) 167.963 0.0837755
\(160\) 0 0
\(161\) 1671.53 + 526.095i 0.818228 + 0.257528i
\(162\) 0 0
\(163\) 1445.22i 0.694468i −0.937778 0.347234i \(-0.887121\pi\)
0.937778 0.347234i \(-0.112879\pi\)
\(164\) 0 0
\(165\) −382.960 −0.180687
\(166\) 0 0
\(167\) 1270.06 0.588505 0.294253 0.955728i \(-0.404929\pi\)
0.294253 + 0.955728i \(0.404929\pi\)
\(168\) 0 0
\(169\) 1821.18 0.828938
\(170\) 0 0
\(171\) 45.8750 0.0205155
\(172\) 0 0
\(173\) 264.158i 0.116090i 0.998314 + 0.0580450i \(0.0184867\pi\)
−0.998314 + 0.0580450i \(0.981513\pi\)
\(174\) 0 0
\(175\) 666.499 2117.62i 0.287901 0.914728i
\(176\) 0 0
\(177\) 2103.33 0.893197
\(178\) 0 0
\(179\) 3334.27i 1.39226i −0.717915 0.696131i \(-0.754903\pi\)
0.717915 0.696131i \(-0.245097\pi\)
\(180\) 0 0
\(181\) 3466.44i 1.42353i −0.702419 0.711763i \(-0.747897\pi\)
0.702419 0.711763i \(-0.252103\pi\)
\(182\) 0 0
\(183\) 1967.86i 0.794910i
\(184\) 0 0
\(185\) 657.239i 0.261195i
\(186\) 0 0
\(187\) 7167.97 2.80307
\(188\) 0 0
\(189\) −150.124 + 476.980i −0.0577774 + 0.183572i
\(190\) 0 0
\(191\) 3189.84i 1.20842i 0.796824 + 0.604212i \(0.206512\pi\)
−0.796824 + 0.604212i \(0.793488\pi\)
\(192\) 0 0
\(193\) −3521.84 −1.31351 −0.656756 0.754103i \(-0.728072\pi\)
−0.656756 + 0.754103i \(0.728072\pi\)
\(194\) 0 0
\(195\) 131.719 0.0483722
\(196\) 0 0
\(197\) 1263.55 0.456976 0.228488 0.973547i \(-0.426622\pi\)
0.228488 + 0.973547i \(0.426622\pi\)
\(198\) 0 0
\(199\) 2190.59 0.780335 0.390168 0.920744i \(-0.372417\pi\)
0.390168 + 0.920744i \(0.372417\pi\)
\(200\) 0 0
\(201\) 329.269i 0.115546i
\(202\) 0 0
\(203\) 514.276 1633.98i 0.177808 0.564939i
\(204\) 0 0
\(205\) 267.687 0.0912004
\(206\) 0 0
\(207\) 851.569i 0.285933i
\(208\) 0 0
\(209\) 287.297i 0.0950851i
\(210\) 0 0
\(211\) 1235.79i 0.403201i 0.979468 + 0.201601i \(0.0646143\pi\)
−0.979468 + 0.201601i \(0.935386\pi\)
\(212\) 0 0
\(213\) 1436.34i 0.462050i
\(214\) 0 0
\(215\) −621.232 −0.197059
\(216\) 0 0
\(217\) 216.477 687.798i 0.0677208 0.215165i
\(218\) 0 0
\(219\) 1687.26i 0.520614i
\(220\) 0 0
\(221\) −2465.41 −0.750415
\(222\) 0 0
\(223\) −2757.16 −0.827951 −0.413976 0.910288i \(-0.635860\pi\)
−0.413976 + 0.910288i \(0.635860\pi\)
\(224\) 0 0
\(225\) 1078.84 0.319655
\(226\) 0 0
\(227\) 33.7355 0.00986389 0.00493195 0.999988i \(-0.498430\pi\)
0.00493195 + 0.999988i \(0.498430\pi\)
\(228\) 0 0
\(229\) 3958.87i 1.14240i 0.820811 + 0.571199i \(0.193522\pi\)
−0.820811 + 0.571199i \(0.806478\pi\)
\(230\) 0 0
\(231\) 2987.14 + 940.169i 0.850819 + 0.267786i
\(232\) 0 0
\(233\) 5533.39 1.55581 0.777906 0.628381i \(-0.216282\pi\)
0.777906 + 0.628381i \(0.216282\pi\)
\(234\) 0 0
\(235\) 1387.71i 0.385210i
\(236\) 0 0
\(237\) 936.310i 0.256624i
\(238\) 0 0
\(239\) 6534.26i 1.76848i −0.467036 0.884238i \(-0.654678\pi\)
0.467036 0.884238i \(-0.345322\pi\)
\(240\) 0 0
\(241\) 3981.08i 1.06408i −0.846719 0.532041i \(-0.821425\pi\)
0.846719 0.532041i \(-0.178575\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 444.926 636.799i 0.116021 0.166055i
\(246\) 0 0
\(247\) 98.8157i 0.0254554i
\(248\) 0 0
\(249\) −3114.59 −0.792688
\(250\) 0 0
\(251\) 6393.16 1.60770 0.803850 0.594832i \(-0.202782\pi\)
0.803850 + 0.594832i \(0.202782\pi\)
\(252\) 0 0
\(253\) 5333.04 1.32524
\(254\) 0 0
\(255\) 864.080 0.212199
\(256\) 0 0
\(257\) 3544.04i 0.860199i −0.902782 0.430099i \(-0.858479\pi\)
0.902782 0.430099i \(-0.141521\pi\)
\(258\) 0 0
\(259\) −1613.53 + 5126.55i −0.387103 + 1.22992i
\(260\) 0 0
\(261\) 832.438 0.197420
\(262\) 0 0
\(263\) 2458.85i 0.576500i 0.957555 + 0.288250i \(0.0930734\pi\)
−0.957555 + 0.288250i \(0.906927\pi\)
\(264\) 0 0
\(265\) 126.802i 0.0293939i
\(266\) 0 0
\(267\) 2231.25i 0.511424i
\(268\) 0 0
\(269\) 6621.96i 1.50092i 0.660914 + 0.750461i \(0.270169\pi\)
−0.660914 + 0.750461i \(0.729831\pi\)
\(270\) 0 0
\(271\) 914.675 0.205028 0.102514 0.994732i \(-0.467311\pi\)
0.102514 + 0.994732i \(0.467311\pi\)
\(272\) 0 0
\(273\) −1027.42 323.370i −0.227775 0.0716896i
\(274\) 0 0
\(275\) 6756.32i 1.48153i
\(276\) 0 0
\(277\) −2329.10 −0.505206 −0.252603 0.967570i \(-0.581287\pi\)
−0.252603 + 0.967570i \(0.581287\pi\)
\(278\) 0 0
\(279\) 350.403 0.0751902
\(280\) 0 0
\(281\) −1006.08 −0.213586 −0.106793 0.994281i \(-0.534058\pi\)
−0.106793 + 0.994281i \(0.534058\pi\)
\(282\) 0 0
\(283\) 6622.25 1.39100 0.695498 0.718528i \(-0.255184\pi\)
0.695498 + 0.718528i \(0.255184\pi\)
\(284\) 0 0
\(285\) 34.6330i 0.00719817i
\(286\) 0 0
\(287\) −2088.00 657.174i −0.429444 0.135163i
\(288\) 0 0
\(289\) −11260.2 −2.29192
\(290\) 0 0
\(291\) 3831.03i 0.771749i
\(292\) 0 0
\(293\) 5228.26i 1.04245i −0.853419 0.521226i \(-0.825475\pi\)
0.853419 0.521226i \(-0.174525\pi\)
\(294\) 0 0
\(295\) 1587.89i 0.313391i
\(296\) 0 0
\(297\) 1521.81i 0.297322i
\(298\) 0 0
\(299\) −1834.29 −0.354782
\(300\) 0 0
\(301\) 4845.69 + 1525.13i 0.927911 + 0.292050i
\(302\) 0 0
\(303\) 3170.00i 0.601029i
\(304\) 0 0
\(305\) 1485.62 0.278906
\(306\) 0 0
\(307\) 9966.23 1.85278 0.926389 0.376569i \(-0.122896\pi\)
0.926389 + 0.376569i \(0.122896\pi\)
\(308\) 0 0
\(309\) −5758.90 −1.06023
\(310\) 0 0
\(311\) 720.081 0.131293 0.0656463 0.997843i \(-0.479089\pi\)
0.0656463 + 0.997843i \(0.479089\pi\)
\(312\) 0 0
\(313\) 6738.59i 1.21689i 0.793594 + 0.608447i \(0.208207\pi\)
−0.793594 + 0.608447i \(0.791793\pi\)
\(314\) 0 0
\(315\) 360.092 + 113.335i 0.0644091 + 0.0202721i
\(316\) 0 0
\(317\) 6754.62 1.19677 0.598387 0.801207i \(-0.295809\pi\)
0.598387 + 0.801207i \(0.295809\pi\)
\(318\) 0 0
\(319\) 5213.23i 0.915000i
\(320\) 0 0
\(321\) 3188.73i 0.554447i
\(322\) 0 0
\(323\) 648.235i 0.111668i
\(324\) 0 0
\(325\) 2323.83i 0.396624i
\(326\) 0 0
\(327\) −4611.77 −0.779913
\(328\) 0 0
\(329\) 3406.85 10824.3i 0.570898 1.81388i
\(330\) 0 0
\(331\) 9239.33i 1.53426i 0.641492 + 0.767130i \(0.278316\pi\)
−0.641492 + 0.767130i \(0.721684\pi\)
\(332\) 0 0
\(333\) −2611.75 −0.429799
\(334\) 0 0
\(335\) −248.579 −0.0405412
\(336\) 0 0
\(337\) −1886.09 −0.304871 −0.152436 0.988313i \(-0.548712\pi\)
−0.152436 + 0.988313i \(0.548712\pi\)
\(338\) 0 0
\(339\) −570.816 −0.0914527
\(340\) 0 0
\(341\) 2194.44i 0.348491i
\(342\) 0 0
\(343\) −5033.82 + 3874.82i −0.792423 + 0.609973i
\(344\) 0 0
\(345\) 642.884 0.100324
\(346\) 0 0
\(347\) 6243.36i 0.965883i −0.875653 0.482941i \(-0.839568\pi\)
0.875653 0.482941i \(-0.160432\pi\)
\(348\) 0 0
\(349\) 5320.31i 0.816016i 0.912978 + 0.408008i \(0.133776\pi\)
−0.912978 + 0.408008i \(0.866224\pi\)
\(350\) 0 0
\(351\) 523.426i 0.0795967i
\(352\) 0 0
\(353\) 7896.43i 1.19061i −0.803501 0.595304i \(-0.797032\pi\)
0.803501 0.595304i \(-0.202968\pi\)
\(354\) 0 0
\(355\) 1084.35 0.162117
\(356\) 0 0
\(357\) −6739.93 2121.32i −0.999202 0.314488i
\(358\) 0 0
\(359\) 1471.39i 0.216314i −0.994134 0.108157i \(-0.965505\pi\)
0.994134 0.108157i \(-0.0344949\pi\)
\(360\) 0 0
\(361\) −6833.02 −0.996212
\(362\) 0 0
\(363\) 5537.52 0.800674
\(364\) 0 0
\(365\) −1273.78 −0.182665
\(366\) 0 0
\(367\) 8200.26 1.16635 0.583175 0.812347i \(-0.301810\pi\)
0.583175 + 0.812347i \(0.301810\pi\)
\(368\) 0 0
\(369\) 1063.74i 0.150071i
\(370\) 0 0
\(371\) −311.299 + 989.071i −0.0435630 + 0.138410i
\(372\) 0 0
\(373\) −2848.05 −0.395352 −0.197676 0.980267i \(-0.563339\pi\)
−0.197676 + 0.980267i \(0.563339\pi\)
\(374\) 0 0
\(375\) 1663.77i 0.229111i
\(376\) 0 0
\(377\) 1793.09i 0.244957i
\(378\) 0 0
\(379\) 99.8362i 0.0135310i 0.999977 + 0.00676549i \(0.00215354\pi\)
−0.999977 + 0.00676549i \(0.997846\pi\)
\(380\) 0 0
\(381\) 2093.65i 0.281524i
\(382\) 0 0
\(383\) −1922.33 −0.256466 −0.128233 0.991744i \(-0.540931\pi\)
−0.128233 + 0.991744i \(0.540931\pi\)
\(384\) 0 0
\(385\) 709.772 2255.11i 0.0939567 0.298523i
\(386\) 0 0
\(387\) 2468.67i 0.324262i
\(388\) 0 0
\(389\) 9766.34 1.27294 0.636470 0.771302i \(-0.280394\pi\)
0.636470 + 0.771302i \(0.280394\pi\)
\(390\) 0 0
\(391\) −12033.0 −1.55636
\(392\) 0 0
\(393\) −2455.28 −0.315146
\(394\) 0 0
\(395\) 706.859 0.0900403
\(396\) 0 0
\(397\) 2983.13i 0.377125i −0.982061 0.188563i \(-0.939617\pi\)
0.982061 0.188563i \(-0.0603829\pi\)
\(398\) 0 0
\(399\) −85.0241 + 270.142i −0.0106680 + 0.0338947i
\(400\) 0 0
\(401\) 1736.37 0.216235 0.108117 0.994138i \(-0.465518\pi\)
0.108117 + 0.994138i \(0.465518\pi\)
\(402\) 0 0
\(403\) 754.774i 0.0932952i
\(404\) 0 0
\(405\) 183.451i 0.0225080i
\(406\) 0 0
\(407\) 16356.4i 1.99203i
\(408\) 0 0
\(409\) 7946.15i 0.960664i 0.877087 + 0.480332i \(0.159484\pi\)
−0.877087 + 0.480332i \(0.840516\pi\)
\(410\) 0 0
\(411\) −4355.45 −0.522721
\(412\) 0 0
\(413\) −3898.28 + 12385.7i −0.464459 + 1.47570i
\(414\) 0 0
\(415\) 2351.33i 0.278126i
\(416\) 0 0
\(417\) 4547.10 0.533987
\(418\) 0 0
\(419\) 1132.03 0.131989 0.0659945 0.997820i \(-0.478978\pi\)
0.0659945 + 0.997820i \(0.478978\pi\)
\(420\) 0 0
\(421\) 1613.13 0.186744 0.0933719 0.995631i \(-0.470235\pi\)
0.0933719 + 0.995631i \(0.470235\pi\)
\(422\) 0 0
\(423\) 5514.52 0.633866
\(424\) 0 0
\(425\) 15244.4i 1.73991i
\(426\) 0 0
\(427\) −11588.0 3647.20i −1.31331 0.413350i
\(428\) 0 0
\(429\) −3278.01 −0.368914
\(430\) 0 0
\(431\) 1769.22i 0.197728i −0.995101 0.0988638i \(-0.968479\pi\)
0.995101 0.0988638i \(-0.0315208\pi\)
\(432\) 0 0
\(433\) 5778.85i 0.641372i 0.947186 + 0.320686i \(0.103913\pi\)
−0.947186 + 0.320686i \(0.896087\pi\)
\(434\) 0 0
\(435\) 628.442i 0.0692678i
\(436\) 0 0
\(437\) 482.293i 0.0527945i
\(438\) 0 0
\(439\) −4620.52 −0.502336 −0.251168 0.967943i \(-0.580815\pi\)
−0.251168 + 0.967943i \(0.580815\pi\)
\(440\) 0 0
\(441\) −2530.52 1768.05i −0.273245 0.190914i
\(442\) 0 0
\(443\) 9071.81i 0.972945i 0.873696 + 0.486473i \(0.161717\pi\)
−0.873696 + 0.486473i \(0.838283\pi\)
\(444\) 0 0
\(445\) −1684.46 −0.179441
\(446\) 0 0
\(447\) −5915.46 −0.625933
\(448\) 0 0
\(449\) 15112.3 1.58840 0.794201 0.607655i \(-0.207889\pi\)
0.794201 + 0.607655i \(0.207889\pi\)
\(450\) 0 0
\(451\) −6661.79 −0.695546
\(452\) 0 0
\(453\) 8594.06i 0.891356i
\(454\) 0 0
\(455\) −244.125 + 775.644i −0.0251534 + 0.0799181i
\(456\) 0 0
\(457\) −16630.8 −1.70231 −0.851156 0.524913i \(-0.824098\pi\)
−0.851156 + 0.524913i \(0.824098\pi\)
\(458\) 0 0
\(459\) 3433.70i 0.349175i
\(460\) 0 0
\(461\) 11011.6i 1.11250i −0.831016 0.556248i \(-0.812240\pi\)
0.831016 0.556248i \(-0.187760\pi\)
\(462\) 0 0
\(463\) 12523.9i 1.25709i −0.777773 0.628545i \(-0.783651\pi\)
0.777773 0.628545i \(-0.216349\pi\)
\(464\) 0 0
\(465\) 264.533i 0.0263816i
\(466\) 0 0
\(467\) −4373.03 −0.433319 −0.216659 0.976247i \(-0.569516\pi\)
−0.216659 + 0.976247i \(0.569516\pi\)
\(468\) 0 0
\(469\) 1938.95 + 610.262i 0.190900 + 0.0600838i
\(470\) 0 0
\(471\) 969.031i 0.0947995i
\(472\) 0 0
\(473\) 15460.3 1.50289
\(474\) 0 0
\(475\) 611.008 0.0590209
\(476\) 0 0
\(477\) −503.888 −0.0483678
\(478\) 0 0
\(479\) 4027.80 0.384206 0.192103 0.981375i \(-0.438469\pi\)
0.192103 + 0.981375i \(0.438469\pi\)
\(480\) 0 0
\(481\) 5625.75i 0.533290i
\(482\) 0 0
\(483\) −5014.58 1578.28i −0.472404 0.148684i
\(484\) 0 0
\(485\) −2892.20 −0.270780
\(486\) 0 0
\(487\) 4603.46i 0.428342i −0.976796 0.214171i \(-0.931295\pi\)
0.976796 0.214171i \(-0.0687050\pi\)
\(488\) 0 0
\(489\) 4335.66i 0.400951i
\(490\) 0 0
\(491\) 4062.95i 0.373439i −0.982413 0.186719i \(-0.940214\pi\)
0.982413 0.186719i \(-0.0597855\pi\)
\(492\) 0 0
\(493\) 11762.7i 1.07458i
\(494\) 0 0
\(495\) 1148.88 0.104320
\(496\) 0 0
\(497\) −8458.11 2662.10i −0.763377 0.240264i
\(498\) 0 0
\(499\) 6479.08i 0.581249i −0.956837 0.290625i \(-0.906137\pi\)
0.956837 0.290625i \(-0.0938631\pi\)
\(500\) 0 0
\(501\) −3810.19 −0.339774
\(502\) 0 0
\(503\) −4443.44 −0.393883 −0.196942 0.980415i \(-0.563101\pi\)
−0.196942 + 0.980415i \(0.563101\pi\)
\(504\) 0 0
\(505\) 2393.16 0.210880
\(506\) 0 0
\(507\) −5463.53 −0.478588
\(508\) 0 0
\(509\) 1783.20i 0.155282i 0.996981 + 0.0776412i \(0.0247389\pi\)
−0.996981 + 0.0776412i \(0.975261\pi\)
\(510\) 0 0
\(511\) 9935.66 + 3127.14i 0.860132 + 0.270717i
\(512\) 0 0
\(513\) −137.625 −0.0118446
\(514\) 0 0
\(515\) 4347.63i 0.371999i
\(516\) 0 0
\(517\) 34535.3i 2.93783i
\(518\) 0 0
\(519\) 792.474i 0.0670245i
\(520\) 0 0
\(521\) 2198.87i 0.184902i −0.995717 0.0924512i \(-0.970530\pi\)
0.995717 0.0924512i \(-0.0294702\pi\)
\(522\) 0 0
\(523\) −16655.9 −1.39256 −0.696282 0.717768i \(-0.745164\pi\)
−0.696282 + 0.717768i \(0.745164\pi\)
\(524\) 0 0
\(525\) −1999.50 + 6352.87i −0.166219 + 0.528118i
\(526\) 0 0
\(527\) 4951.34i 0.409267i
\(528\) 0 0
\(529\) 3214.30 0.264181
\(530\) 0 0
\(531\) −6309.99 −0.515687
\(532\) 0 0
\(533\) 2291.32 0.186206
\(534\) 0 0
\(535\) −2407.30 −0.194536
\(536\) 0 0
\(537\) 10002.8i 0.803823i
\(538\) 0 0
\(539\) −11072.6 + 15847.7i −0.884846 + 1.26643i
\(540\) 0 0
\(541\) −16221.5 −1.28912 −0.644562 0.764552i \(-0.722960\pi\)
−0.644562 + 0.764552i \(0.722960\pi\)
\(542\) 0 0
\(543\) 10399.3i 0.821873i
\(544\) 0 0
\(545\) 3481.62i 0.273644i
\(546\) 0 0
\(547\) 3649.69i 0.285282i 0.989774 + 0.142641i \(0.0455595\pi\)
−0.989774 + 0.142641i \(0.954441\pi\)
\(548\) 0 0
\(549\) 5903.58i 0.458941i
\(550\) 0 0
\(551\) 471.458 0.0364516
\(552\) 0 0
\(553\) −5513.59 1735.34i −0.423981 0.133444i
\(554\) 0 0
\(555\) 1971.72i 0.150801i
\(556\) 0 0
\(557\) 19277.8 1.46648 0.733238 0.679972i \(-0.238008\pi\)
0.733238 + 0.679972i \(0.238008\pi\)
\(558\) 0 0
\(559\) −5317.55 −0.402341
\(560\) 0 0
\(561\) −21503.9 −1.61835
\(562\) 0 0
\(563\) −19320.9 −1.44632 −0.723160 0.690681i \(-0.757311\pi\)
−0.723160 + 0.690681i \(0.757311\pi\)
\(564\) 0 0
\(565\) 430.932i 0.0320875i
\(566\) 0 0
\(567\) 450.373 1430.94i 0.0333578 0.105986i
\(568\) 0 0
\(569\) 26173.7 1.92840 0.964198 0.265183i \(-0.0854324\pi\)
0.964198 + 0.265183i \(0.0854324\pi\)
\(570\) 0 0
\(571\) 5196.24i 0.380833i −0.981703 0.190417i \(-0.939016\pi\)
0.981703 0.190417i \(-0.0609839\pi\)
\(572\) 0 0
\(573\) 9569.53i 0.697684i
\(574\) 0 0
\(575\) 11342.0i 0.822599i
\(576\) 0 0
\(577\) 11218.6i 0.809421i −0.914445 0.404711i \(-0.867372\pi\)
0.914445 0.404711i \(-0.132628\pi\)
\(578\) 0 0
\(579\) 10565.5 0.758356
\(580\) 0 0
\(581\) 5772.54 18340.7i 0.412195 1.30964i
\(582\) 0 0
\(583\) 3155.65i 0.224174i
\(584\) 0 0
\(585\) −395.156 −0.0279277
\(586\) 0 0
\(587\) 3438.46 0.241772 0.120886 0.992666i \(-0.461426\pi\)
0.120886 + 0.992666i \(0.461426\pi\)
\(588\) 0 0
\(589\) 198.454 0.0138831
\(590\) 0 0
\(591\) −3790.65 −0.263835
\(592\) 0 0
\(593\) 7612.88i 0.527190i −0.964633 0.263595i \(-0.915092\pi\)
0.964633 0.263595i \(-0.0849082\pi\)
\(594\) 0 0
\(595\) −1601.47 + 5088.25i −0.110343 + 0.350585i
\(596\) 0 0
\(597\) −6571.77 −0.450527
\(598\) 0 0
\(599\) 16681.5i 1.13788i −0.822379 0.568940i \(-0.807354\pi\)
0.822379 0.568940i \(-0.192646\pi\)
\(600\) 0 0
\(601\) 5913.34i 0.401348i 0.979658 + 0.200674i \(0.0643132\pi\)
−0.979658 + 0.200674i \(0.935687\pi\)
\(602\) 0 0
\(603\) 987.807i 0.0667108i
\(604\) 0 0
\(605\) 4180.50i 0.280928i
\(606\) 0 0
\(607\) −26481.0 −1.77073 −0.885363 0.464900i \(-0.846090\pi\)
−0.885363 + 0.464900i \(0.846090\pi\)
\(608\) 0 0
\(609\) −1542.83 + 4901.93i −0.102658 + 0.326168i
\(610\) 0 0
\(611\) 11878.4i 0.786494i
\(612\) 0 0
\(613\) −25016.9 −1.64832 −0.824162 0.566355i \(-0.808353\pi\)
−0.824162 + 0.566355i \(0.808353\pi\)
\(614\) 0 0
\(615\) −803.061 −0.0526546
\(616\) 0 0
\(617\) 10545.5 0.688078 0.344039 0.938955i \(-0.388205\pi\)
0.344039 + 0.938955i \(0.388205\pi\)
\(618\) 0 0
\(619\) −18079.7 −1.17397 −0.586984 0.809598i \(-0.699685\pi\)
−0.586984 + 0.809598i \(0.699685\pi\)
\(620\) 0 0
\(621\) 2554.71i 0.165083i
\(622\) 0 0
\(623\) 13139.0 + 4135.37i 0.844950 + 0.265939i
\(624\) 0 0
\(625\) 13727.8 0.878578
\(626\) 0 0
\(627\) 861.892i 0.0548974i
\(628\) 0 0
\(629\) 36905.2i 2.33944i
\(630\) 0 0
\(631\) 14856.1i 0.937263i −0.883394 0.468632i \(-0.844747\pi\)
0.883394 0.468632i \(-0.155253\pi\)
\(632\) 0 0
\(633\) 3707.38i 0.232788i
\(634\) 0 0
\(635\) 1580.58 0.0987770
\(636\) 0 0
\(637\) 3808.42 5450.79i 0.236884 0.339040i
\(638\) 0 0
\(639\) 4309.03i 0.266765i
\(640\) 0 0
\(641\) −8403.54 −0.517815 −0.258908 0.965902i \(-0.583363\pi\)
−0.258908 + 0.965902i \(0.583363\pi\)
\(642\) 0 0
\(643\) −10196.8 −0.625383 −0.312692 0.949855i \(-0.601231\pi\)
−0.312692 + 0.949855i \(0.601231\pi\)
\(644\) 0 0
\(645\) 1863.70 0.113772
\(646\) 0 0
\(647\) 7696.83 0.467687 0.233844 0.972274i \(-0.424870\pi\)
0.233844 + 0.972274i \(0.424870\pi\)
\(648\) 0 0
\(649\) 39517.0i 2.39010i
\(650\) 0 0
\(651\) −649.431 + 2063.40i −0.0390986 + 0.124226i
\(652\) 0 0
\(653\) 16048.6 0.961759 0.480879 0.876787i \(-0.340317\pi\)
0.480879 + 0.876787i \(0.340317\pi\)
\(654\) 0 0
\(655\) 1853.59i 0.110574i
\(656\) 0 0
\(657\) 5061.78i 0.300576i
\(658\) 0 0
\(659\) 5087.65i 0.300739i −0.988630 0.150369i \(-0.951954\pi\)
0.988630 0.150369i \(-0.0480463\pi\)
\(660\) 0 0
\(661\) 14786.6i 0.870095i 0.900408 + 0.435047i \(0.143268\pi\)
−0.900408 + 0.435047i \(0.856732\pi\)
\(662\) 0 0
\(663\) 7396.24 0.433252
\(664\) 0 0
\(665\) 203.941 + 64.1882i 0.0118925 + 0.00374302i
\(666\) 0 0
\(667\) 8751.58i 0.508040i
\(668\) 0 0
\(669\) 8271.48 0.478018
\(670\) 0 0
\(671\) −36971.8 −2.12710
\(672\) 0 0
\(673\) −11709.5 −0.670680 −0.335340 0.942097i \(-0.608851\pi\)
−0.335340 + 0.942097i \(0.608851\pi\)
\(674\) 0 0
\(675\) −3236.51 −0.184553
\(676\) 0 0
\(677\) 6581.03i 0.373603i 0.982398 + 0.186802i \(0.0598122\pi\)
−0.982398 + 0.186802i \(0.940188\pi\)
\(678\) 0 0
\(679\) 22559.6 + 7100.37i 1.27505 + 0.401307i
\(680\) 0 0
\(681\) −101.207 −0.00569492
\(682\) 0 0
\(683\) 1412.04i 0.0791070i −0.999217 0.0395535i \(-0.987406\pi\)
0.999217 0.0395535i \(-0.0125936\pi\)
\(684\) 0 0
\(685\) 3288.11i 0.183405i
\(686\) 0 0
\(687\) 11876.6i 0.659564i
\(688\) 0 0
\(689\) 1085.38i 0.0600142i
\(690\) 0 0
\(691\) −5861.32 −0.322684 −0.161342 0.986899i \(-0.551582\pi\)
−0.161342 + 0.986899i \(0.551582\pi\)
\(692\) 0 0
\(693\) −8961.41 2820.51i −0.491221 0.154606i
\(694\) 0 0
\(695\) 3432.80i 0.187357i
\(696\) 0 0
\(697\) 15031.1 0.816850
\(698\) 0 0
\(699\) −16600.2 −0.898248
\(700\) 0 0
\(701\) 20729.3 1.11688 0.558442 0.829544i \(-0.311399\pi\)
0.558442 + 0.829544i \(0.311399\pi\)
\(702\) 0 0
\(703\) −1479.19 −0.0793578
\(704\) 0 0
\(705\) 4163.14i 0.222401i
\(706\) 0 0
\(707\) −18667.0 5875.24i −0.992991 0.312533i
\(708\) 0 0
\(709\) 15204.3 0.805371 0.402685 0.915338i \(-0.368077\pi\)
0.402685 + 0.915338i \(0.368077\pi\)
\(710\) 0 0
\(711\) 2808.93i 0.148162i
\(712\) 0 0
\(713\) 3683.85i 0.193494i
\(714\) 0 0
\(715\) 2474.71i 0.129439i
\(716\) 0 0
\(717\) 19602.8i 1.02103i
\(718\) 0 0
\(719\) −9047.94 −0.469306 −0.234653 0.972079i \(-0.575395\pi\)
−0.234653 + 0.972079i \(0.575395\pi\)
\(720\) 0 0
\(721\) 10673.5 33912.1i 0.551318 1.75167i
\(722\) 0 0
\(723\) 11943.2i 0.614348i
\(724\) 0 0
\(725\) 11087.2 0.567956
\(726\) 0 0
\(727\) 1540.38 0.0785824 0.0392912 0.999228i \(-0.487490\pi\)
0.0392912 + 0.999228i \(0.487490\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −34883.3 −1.76499
\(732\) 0 0
\(733\) 32338.6i 1.62954i 0.579784 + 0.814770i \(0.303137\pi\)
−0.579784 + 0.814770i \(0.696863\pi\)
\(734\) 0 0
\(735\) −1334.78 + 1910.40i −0.0669850 + 0.0958722i
\(736\) 0 0
\(737\) 6186.25 0.309190
\(738\) 0 0
\(739\) 23307.1i 1.16017i 0.814555 + 0.580086i \(0.196981\pi\)
−0.814555 + 0.580086i \(0.803019\pi\)
\(740\) 0 0
\(741\) 296.447i 0.0146967i
\(742\) 0 0
\(743\) 537.576i 0.0265434i 0.999912 + 0.0132717i \(0.00422464\pi\)
−0.999912 + 0.0132717i \(0.995775\pi\)
\(744\) 0 0
\(745\) 4465.83i 0.219618i
\(746\) 0 0
\(747\) 9343.78 0.457658
\(748\) 0 0
\(749\) 18777.3 + 5909.94i 0.916030 + 0.288311i
\(750\) 0 0
\(751\) 8344.36i 0.405446i 0.979236 + 0.202723i \(0.0649791\pi\)
−0.979236 + 0.202723i \(0.935021\pi\)
\(752\) 0 0
\(753\) −19179.5 −0.928206
\(754\) 0 0
\(755\) 6488.01 0.312746
\(756\) 0 0
\(757\) −15987.8 −0.767617 −0.383808 0.923413i \(-0.625388\pi\)
−0.383808 + 0.923413i \(0.625388\pi\)
\(758\) 0 0
\(759\) −15999.1 −0.765127
\(760\) 0 0
\(761\) 30206.1i 1.43886i 0.694567 + 0.719428i \(0.255596\pi\)
−0.694567 + 0.719428i \(0.744404\pi\)
\(762\) 0 0
\(763\) 8547.39 27157.1i 0.405552 1.28853i
\(764\) 0 0
\(765\) −2592.24 −0.122513
\(766\) 0 0
\(767\) 13591.8i 0.639859i
\(768\) 0 0
\(769\) 21943.5i 1.02900i 0.857490 + 0.514501i \(0.172023\pi\)
−0.857490 + 0.514501i \(0.827977\pi\)
\(770\) 0 0
\(771\) 10632.1i 0.496636i
\(772\) 0 0
\(773\) 7922.07i 0.368612i 0.982869 + 0.184306i \(0.0590037\pi\)
−0.982869 + 0.184306i \(0.940996\pi\)
\(774\) 0 0
\(775\) 4667.00 0.216314
\(776\) 0 0
\(777\) 4840.58 15379.6i 0.223494 0.710092i
\(778\) 0 0
\(779\) 602.459i 0.0277090i
\(780\) 0 0
\(781\) −26985.8 −1.23640
\(782\) 0 0
\(783\) −2497.31 −0.113981
\(784\) 0 0
\(785\) 731.561 0.0332618
\(786\) 0 0
\(787\) −26633.6 −1.20634 −0.603168 0.797614i \(-0.706095\pi\)
−0.603168 + 0.797614i \(0.706095\pi\)
\(788\) 0 0
\(789\) 7376.56i 0.332842i
\(790\) 0 0
\(791\) 1057.94 3361.33i 0.0475551 0.151094i
\(792\) 0 0
\(793\) 12716.4 0.569449
\(794\) 0 0
\(795\) 380.405i 0.0169706i
\(796\) 0 0
\(797\) 12638.0i 0.561684i 0.959754 + 0.280842i \(0.0906138\pi\)
−0.959754 + 0.280842i \(0.909386\pi\)
\(798\) 0 0
\(799\) 77922.6i 3.45019i
\(800\) 0 0
\(801\) 6693.75i 0.295271i
\(802\) 0 0
\(803\) 31699.9 1.39311
\(804\) 0 0
\(805\) −1191.51 + 3785.71i −0.0521680 + 0.165750i
\(806\) 0 0
\(807\) 19865.9i 0.866558i
\(808\) 0 0
\(809\) −3290.23 −0.142989 −0.0714945 0.997441i \(-0.522777\pi\)
−0.0714945 + 0.997441i \(0.522777\pi\)
\(810\) 0 0
\(811\) 42757.5 1.85132 0.925658 0.378361i \(-0.123512\pi\)
0.925658 + 0.378361i \(0.123512\pi\)
\(812\) 0 0
\(813\) −2744.03 −0.118373
\(814\) 0 0
\(815\) 3273.17 0.140680
\(816\) 0 0
\(817\) 1398.15i 0.0598716i
\(818\) 0 0
\(819\) 3082.27 + 970.111i 0.131506 + 0.0413900i
\(820\) 0 0
\(821\) 3209.29 0.136425 0.0682126 0.997671i \(-0.478270\pi\)
0.0682126 + 0.997671i \(0.478270\pi\)
\(822\) 0 0
\(823\) 10327.4i 0.437412i 0.975791 + 0.218706i \(0.0701836\pi\)
−0.975791 + 0.218706i \(0.929816\pi\)
\(824\) 0 0
\(825\) 20269.0i 0.855363i
\(826\) 0 0
\(827\) 9065.80i 0.381195i 0.981668 + 0.190598i \(0.0610426\pi\)
−0.981668 + 0.190598i \(0.938957\pi\)
\(828\) 0 0
\(829\) 6037.66i 0.252951i 0.991970 + 0.126476i \(0.0403666\pi\)
−0.991970 + 0.126476i \(0.959633\pi\)
\(830\) 0 0
\(831\) 6987.30 0.291681
\(832\) 0 0
\(833\) 24983.4 35757.4i 1.03916 1.48730i
\(834\) 0 0
\(835\) 2876.47i 0.119215i
\(836\) 0 0
\(837\) −1051.21 −0.0434111
\(838\) 0 0
\(839\) −42840.9 −1.76285 −0.881425 0.472323i \(-0.843416\pi\)
−0.881425 + 0.472323i \(0.843416\pi\)
\(840\) 0 0
\(841\) −15834.0 −0.649228
\(842\) 0 0
\(843\) 3018.24 0.123314
\(844\) 0 0
\(845\) 4124.64i 0.167920i
\(846\) 0 0
\(847\) −10263.2 + 32608.5i −0.416348 + 1.32283i
\(848\) 0 0
\(849\) −19866.8 −0.803092
\(850\) 0 0
\(851\) 27457.8i 1.10604i
\(852\) 0 0
\(853\) 46668.7i 1.87328i −0.350296 0.936639i \(-0.613919\pi\)
0.350296 0.936639i \(-0.386081\pi\)
\(854\) 0 0
\(855\) 103.899i 0.00415587i
\(856\) 0 0
\(857\) 23.9058i 0.000952868i 1.00000 0.000476434i \(0.000151654\pi\)
−1.00000 0.000476434i \(0.999848\pi\)
\(858\) 0 0
\(859\) −3773.28 −0.149875 −0.0749375 0.997188i \(-0.523876\pi\)
−0.0749375 + 0.997188i \(0.523876\pi\)
\(860\) 0 0
\(861\) 6263.99 + 1971.52i 0.247940 + 0.0780363i
\(862\) 0 0
\(863\) 26099.3i 1.02947i −0.857351 0.514733i \(-0.827891\pi\)
0.857351 0.514733i \(-0.172109\pi\)
\(864\) 0 0
\(865\) −598.271 −0.0235166
\(866\) 0 0
\(867\) 33780.6 1.32324
\(868\) 0 0
\(869\) −17591.2 −0.686699
\(870\) 0 0
\(871\) −2127.75 −0.0827740
\(872\) 0 0
\(873\) 11493.1i 0.445569i
\(874\) 0 0
\(875\) 9797.32 + 3083.60i 0.378525 + 0.119137i
\(876\) 0 0
\(877\) −11202.2 −0.431324 −0.215662 0.976468i \(-0.569191\pi\)
−0.215662 + 0.976468i \(0.569191\pi\)
\(878\) 0 0
\(879\) 15684.8i 0.601860i
\(880\) 0 0
\(881\) 45579.1i 1.74302i −0.490378 0.871510i \(-0.663141\pi\)
0.490378 0.871510i \(-0.336859\pi\)
\(882\) 0 0
\(883\) 38399.2i 1.46346i −0.681595 0.731730i \(-0.738713\pi\)
0.681595 0.731730i \(-0.261287\pi\)
\(884\) 0 0
\(885\) 4763.67i 0.180937i
\(886\) 0 0
\(887\) 20414.4 0.772772 0.386386 0.922337i \(-0.373723\pi\)
0.386386 + 0.922337i \(0.373723\pi\)
\(888\) 0 0
\(889\) −12328.7 3880.33i −0.465121 0.146392i
\(890\) 0 0
\(891\) 4565.44i 0.171659i
\(892\) 0 0
\(893\) 3123.20 0.117037
\(894\) 0 0
\(895\) 7551.53 0.282033
\(896\) 0 0
\(897\) 5502.88 0.204834
\(898\) 0 0
\(899\) 3601.09 0.133596
\(900\) 0 0
\(901\) 7120.15i 0.263271i
\(902\) 0 0
\(903\) −14537.1 4575.39i −0.535730 0.168615i
\(904\) 0 0
\(905\) 7850.87 0.288367
\(906\) 0 0
\(907\) 15236.6i 0.557798i −0.960320 0.278899i \(-0.910030\pi\)
0.960320 0.278899i \(-0.0899695\pi\)
\(908\) 0 0
\(909\) 9510.01i 0.347004i
\(910\) 0 0
\(911\) 37590.2i 1.36709i −0.729908 0.683545i \(-0.760437\pi\)
0.729908 0.683545i \(-0.239563\pi\)
\(912\) 0 0
\(913\) 58516.4i 2.12115i
\(914\) 0 0
\(915\) −4456.86 −0.161026
\(916\) 0 0
\(917\) 4550.58 14458.3i 0.163875 0.520669i
\(918\) 0 0
\(919\) 31369.0i 1.12597i 0.826466 + 0.562986i \(0.190348\pi\)
−0.826466 + 0.562986i \(0.809652\pi\)
\(920\) 0 0
\(921\) −29898.7 −1.06970
\(922\) 0 0
\(923\) 9281.73 0.330999
\(924\) 0 0
\(925\) −34785.8 −1.23649
\(926\) 0 0
\(927\) 17276.7 0.612127
\(928\) 0 0
\(929\) 12287.4i 0.433947i 0.976177 + 0.216974i \(0.0696186\pi\)
−0.976177 + 0.216974i \(0.930381\pi\)
\(930\) 0 0
\(931\) −1433.18 1001.35i −0.0504519 0.0352503i
\(932\) 0 0
\(933\) −2160.24 −0.0758019
\(934\) 0 0
\(935\) 16234.2i 0.567823i
\(936\) 0 0
\(937\) 45545.1i 1.58793i −0.607962 0.793966i \(-0.708013\pi\)
0.607962 0.793966i \(-0.291987\pi\)
\(938\) 0 0
\(939\) 20215.8i 0.702574i
\(940\) 0 0
\(941\) 40159.5i 1.39125i 0.718406 + 0.695624i \(0.244872\pi\)
−0.718406 + 0.695624i \(0.755128\pi\)
\(942\) 0 0
\(943\) 11183.3 0.386192
\(944\) 0 0
\(945\) −1080.27 340.005i −0.0371866 0.0117041i
\(946\) 0 0
\(947\) 28786.1i 0.987774i 0.869526 + 0.493887i \(0.164424\pi\)
−0.869526 + 0.493887i \(0.835576\pi\)
\(948\) 0 0
\(949\) −10903.2 −0.372952
\(950\) 0 0
\(951\) −20263.9 −0.690958
\(952\) 0 0
\(953\) −41034.9 −1.39481 −0.697404 0.716679i \(-0.745661\pi\)
−0.697404 + 0.716679i \(0.745661\pi\)
\(954\) 0 0
\(955\) −7224.43 −0.244793
\(956\) 0 0
\(957\) 15639.7i 0.528276i
\(958\) 0 0
\(959\) 8072.32 25647.7i 0.271813 0.863615i
\(960\) 0 0
\(961\) −28275.2 −0.949118
\(962\) 0 0
\(963\) 9566.19i 0.320110i
\(964\) 0 0
\(965\) 7976.35i 0.266081i
\(966\) 0 0
\(967\) 50246.8i 1.67097i −0.549514 0.835485i \(-0.685187\pi\)
0.549514 0.835485i \(-0.314813\pi\)
\(968\) 0 0
\(969\) 1944.70i 0.0644715i
\(970\) 0 0
\(971\) 9184.38 0.303544 0.151772 0.988416i \(-0.451502\pi\)
0.151772 + 0.988416i \(0.451502\pi\)
\(972\) 0 0
\(973\) −8427.54 + 26776.3i −0.277672 + 0.882228i
\(974\) 0 0
\(975\) 6971.49i 0.228991i
\(976\) 0 0
\(977\) −8572.75 −0.280723 −0.140362 0.990100i \(-0.544827\pi\)
−0.140362 + 0.990100i \(0.544827\pi\)
\(978\) 0 0
\(979\) 41920.3 1.36852
\(980\) 0 0
\(981\) 13835.3 0.450283
\(982\) 0 0
\(983\) −24208.5 −0.785484 −0.392742 0.919649i \(-0.628473\pi\)
−0.392742 + 0.919649i \(0.628473\pi\)
\(984\) 0 0
\(985\) 2861.72i 0.0925706i
\(986\) 0 0
\(987\) −10220.5 + 32473.0i −0.329608 + 1.04724i
\(988\) 0 0
\(989\) −25953.6 −0.834454
\(990\) 0 0
\(991\) 45932.1i 1.47233i −0.676801 0.736166i \(-0.736634\pi\)
0.676801 0.736166i \(-0.263366\pi\)
\(992\) 0 0
\(993\) 27718.0i 0.885805i
\(994\) 0 0
\(995\) 4961.30i 0.158074i
\(996\) 0 0
\(997\) 11888.2i 0.377637i 0.982012 + 0.188819i \(0.0604658\pi\)
−0.982012 + 0.188819i \(0.939534\pi\)
\(998\) 0 0
\(999\) 7835.25 0.248144
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.b.i.895.14 24
4.3 odd 2 1344.4.b.j.895.14 24
7.6 odd 2 1344.4.b.j.895.11 24
8.3 odd 2 672.4.b.a.223.11 24
8.5 even 2 672.4.b.b.223.11 yes 24
28.27 even 2 inner 1344.4.b.i.895.11 24
56.13 odd 2 672.4.b.a.223.14 yes 24
56.27 even 2 672.4.b.b.223.14 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.b.a.223.11 24 8.3 odd 2
672.4.b.a.223.14 yes 24 56.13 odd 2
672.4.b.b.223.11 yes 24 8.5 even 2
672.4.b.b.223.14 yes 24 56.27 even 2
1344.4.b.i.895.11 24 28.27 even 2 inner
1344.4.b.i.895.14 24 1.1 even 1 trivial
1344.4.b.j.895.11 24 7.6 odd 2
1344.4.b.j.895.14 24 4.3 odd 2