Properties

Label 1856.4.a.be.1.3
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $8$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 19x^{6} + 92x^{4} - 51x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 5 \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.34163\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.89793 q^{3} +15.3330 q^{5} +5.27577 q^{7} -11.8061 q^{9} +O(q^{10})\) \(q-3.89793 q^{3} +15.3330 q^{5} +5.27577 q^{7} -11.8061 q^{9} -28.3477 q^{11} -61.6285 q^{13} -59.7668 q^{15} +39.9279 q^{17} +146.682 q^{19} -20.5646 q^{21} -43.8973 q^{23} +110.099 q^{25} +151.264 q^{27} -29.0000 q^{29} +169.034 q^{31} +110.497 q^{33} +80.8931 q^{35} -69.8939 q^{37} +240.224 q^{39} -229.327 q^{41} -150.534 q^{43} -181.023 q^{45} -237.886 q^{47} -315.166 q^{49} -155.636 q^{51} +218.454 q^{53} -434.654 q^{55} -571.757 q^{57} +115.370 q^{59} -211.782 q^{61} -62.2865 q^{63} -944.947 q^{65} +442.508 q^{67} +171.109 q^{69} +663.263 q^{71} -676.680 q^{73} -429.160 q^{75} -149.556 q^{77} -1319.35 q^{79} -270.849 q^{81} +1266.38 q^{83} +612.212 q^{85} +113.040 q^{87} -1356.44 q^{89} -325.138 q^{91} -658.884 q^{93} +2249.07 q^{95} +823.177 q^{97} +334.678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) \) 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
\(3\) −3.89793 −0.750157 −0.375078 0.926993i \(-0.622384\pi\)
−0.375078 + 0.926993i \(0.622384\pi\)
\(4\) 0 0
\(5\) 15.3330 1.37142 0.685710 0.727874i \(-0.259492\pi\)
0.685710 + 0.727874i \(0.259492\pi\)
\(6\) 0 0
\(7\) 5.27577 0.284865 0.142432 0.989805i \(-0.454508\pi\)
0.142432 + 0.989805i \(0.454508\pi\)
\(8\) 0 0
\(9\) −11.8061 −0.437265
\(10\) 0 0
\(11\) −28.3477 −0.777015 −0.388507 0.921446i \(-0.627009\pi\)
−0.388507 + 0.921446i \(0.627009\pi\)
\(12\) 0 0
\(13\) −61.6285 −1.31482 −0.657411 0.753533i \(-0.728348\pi\)
−0.657411 + 0.753533i \(0.728348\pi\)
\(14\) 0 0
\(15\) −59.7668 −1.02878
\(16\) 0 0
\(17\) 39.9279 0.569643 0.284822 0.958581i \(-0.408066\pi\)
0.284822 + 0.958581i \(0.408066\pi\)
\(18\) 0 0
\(19\) 146.682 1.77112 0.885558 0.464528i \(-0.153776\pi\)
0.885558 + 0.464528i \(0.153776\pi\)
\(20\) 0 0
\(21\) −20.5646 −0.213693
\(22\) 0 0
\(23\) −43.8973 −0.397966 −0.198983 0.980003i \(-0.563764\pi\)
−0.198983 + 0.980003i \(0.563764\pi\)
\(24\) 0 0
\(25\) 110.099 0.880795
\(26\) 0 0
\(27\) 151.264 1.07817
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 169.034 0.979337 0.489669 0.871909i \(-0.337118\pi\)
0.489669 + 0.871909i \(0.337118\pi\)
\(32\) 0 0
\(33\) 110.497 0.582883
\(34\) 0 0
\(35\) 80.8931 0.390670
\(36\) 0 0
\(37\) −69.8939 −0.310554 −0.155277 0.987871i \(-0.549627\pi\)
−0.155277 + 0.987871i \(0.549627\pi\)
\(38\) 0 0
\(39\) 240.224 0.986322
\(40\) 0 0
\(41\) −229.327 −0.873533 −0.436766 0.899575i \(-0.643876\pi\)
−0.436766 + 0.899575i \(0.643876\pi\)
\(42\) 0 0
\(43\) −150.534 −0.533864 −0.266932 0.963715i \(-0.586010\pi\)
−0.266932 + 0.963715i \(0.586010\pi\)
\(44\) 0 0
\(45\) −181.023 −0.599674
\(46\) 0 0
\(47\) −237.886 −0.738282 −0.369141 0.929373i \(-0.620348\pi\)
−0.369141 + 0.929373i \(0.620348\pi\)
\(48\) 0 0
\(49\) −315.166 −0.918852
\(50\) 0 0
\(51\) −155.636 −0.427322
\(52\) 0 0
\(53\) 218.454 0.566169 0.283084 0.959095i \(-0.408642\pi\)
0.283084 + 0.959095i \(0.408642\pi\)
\(54\) 0 0
\(55\) −434.654 −1.06561
\(56\) 0 0
\(57\) −571.757 −1.32862
\(58\) 0 0
\(59\) 115.370 0.254575 0.127287 0.991866i \(-0.459373\pi\)
0.127287 + 0.991866i \(0.459373\pi\)
\(60\) 0 0
\(61\) −211.782 −0.444524 −0.222262 0.974987i \(-0.571344\pi\)
−0.222262 + 0.974987i \(0.571344\pi\)
\(62\) 0 0
\(63\) −62.2865 −0.124561
\(64\) 0 0
\(65\) −944.947 −1.80317
\(66\) 0 0
\(67\) 442.508 0.806880 0.403440 0.915006i \(-0.367814\pi\)
0.403440 + 0.915006i \(0.367814\pi\)
\(68\) 0 0
\(69\) 171.109 0.298537
\(70\) 0 0
\(71\) 663.263 1.10866 0.554330 0.832297i \(-0.312975\pi\)
0.554330 + 0.832297i \(0.312975\pi\)
\(72\) 0 0
\(73\) −676.680 −1.08492 −0.542462 0.840081i \(-0.682508\pi\)
−0.542462 + 0.840081i \(0.682508\pi\)
\(74\) 0 0
\(75\) −429.160 −0.660735
\(76\) 0 0
\(77\) −149.556 −0.221344
\(78\) 0 0
\(79\) −1319.35 −1.87897 −0.939486 0.342588i \(-0.888696\pi\)
−0.939486 + 0.342588i \(0.888696\pi\)
\(80\) 0 0
\(81\) −270.849 −0.371535
\(82\) 0 0
\(83\) 1266.38 1.67473 0.837367 0.546641i \(-0.184094\pi\)
0.837367 + 0.546641i \(0.184094\pi\)
\(84\) 0 0
\(85\) 612.212 0.781221
\(86\) 0 0
\(87\) 113.040 0.139301
\(88\) 0 0
\(89\) −1356.44 −1.61553 −0.807763 0.589507i \(-0.799322\pi\)
−0.807763 + 0.589507i \(0.799322\pi\)
\(90\) 0 0
\(91\) −325.138 −0.374546
\(92\) 0 0
\(93\) −658.884 −0.734656
\(94\) 0 0
\(95\) 2249.07 2.42895
\(96\) 0 0
\(97\) 823.177 0.861659 0.430830 0.902433i \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(98\) 0 0
\(99\) 334.678 0.339761
\(100\) 0 0
\(101\) 895.738 0.882468 0.441234 0.897392i \(-0.354541\pi\)
0.441234 + 0.897392i \(0.354541\pi\)
\(102\) 0 0
\(103\) −774.903 −0.741296 −0.370648 0.928773i \(-0.620864\pi\)
−0.370648 + 0.928773i \(0.620864\pi\)
\(104\) 0 0
\(105\) −315.316 −0.293063
\(106\) 0 0
\(107\) −868.181 −0.784394 −0.392197 0.919881i \(-0.628285\pi\)
−0.392197 + 0.919881i \(0.628285\pi\)
\(108\) 0 0
\(109\) 328.137 0.288347 0.144173 0.989552i \(-0.453948\pi\)
0.144173 + 0.989552i \(0.453948\pi\)
\(110\) 0 0
\(111\) 272.441 0.232964
\(112\) 0 0
\(113\) −1095.47 −0.911971 −0.455986 0.889987i \(-0.650713\pi\)
−0.455986 + 0.889987i \(0.650713\pi\)
\(114\) 0 0
\(115\) −673.076 −0.545779
\(116\) 0 0
\(117\) 727.595 0.574925
\(118\) 0 0
\(119\) 210.650 0.162271
\(120\) 0 0
\(121\) −527.406 −0.396248
\(122\) 0 0
\(123\) 893.900 0.655286
\(124\) 0 0
\(125\) −228.470 −0.163480
\(126\) 0 0
\(127\) 948.925 0.663019 0.331510 0.943452i \(-0.392442\pi\)
0.331510 + 0.943452i \(0.392442\pi\)
\(128\) 0 0
\(129\) 586.769 0.400482
\(130\) 0 0
\(131\) −45.1750 −0.0301294 −0.0150647 0.999887i \(-0.504795\pi\)
−0.0150647 + 0.999887i \(0.504795\pi\)
\(132\) 0 0
\(133\) 773.862 0.504529
\(134\) 0 0
\(135\) 2319.32 1.47863
\(136\) 0 0
\(137\) −1006.30 −0.627550 −0.313775 0.949497i \(-0.601594\pi\)
−0.313775 + 0.949497i \(0.601594\pi\)
\(138\) 0 0
\(139\) 177.789 0.108488 0.0542441 0.998528i \(-0.482725\pi\)
0.0542441 + 0.998528i \(0.482725\pi\)
\(140\) 0 0
\(141\) 927.263 0.553827
\(142\) 0 0
\(143\) 1747.03 1.02164
\(144\) 0 0
\(145\) −444.656 −0.254666
\(146\) 0 0
\(147\) 1228.50 0.689283
\(148\) 0 0
\(149\) 2275.95 1.25136 0.625682 0.780078i \(-0.284821\pi\)
0.625682 + 0.780078i \(0.284821\pi\)
\(150\) 0 0
\(151\) −22.3187 −0.0120283 −0.00601414 0.999982i \(-0.501914\pi\)
−0.00601414 + 0.999982i \(0.501914\pi\)
\(152\) 0 0
\(153\) −471.395 −0.249085
\(154\) 0 0
\(155\) 2591.79 1.34308
\(156\) 0 0
\(157\) −2749.00 −1.39741 −0.698707 0.715408i \(-0.746241\pi\)
−0.698707 + 0.715408i \(0.746241\pi\)
\(158\) 0 0
\(159\) −851.517 −0.424715
\(160\) 0 0
\(161\) −231.592 −0.113367
\(162\) 0 0
\(163\) −1552.50 −0.746018 −0.373009 0.927828i \(-0.621674\pi\)
−0.373009 + 0.927828i \(0.621674\pi\)
\(164\) 0 0
\(165\) 1694.25 0.799378
\(166\) 0 0
\(167\) −2322.26 −1.07606 −0.538029 0.842926i \(-0.680831\pi\)
−0.538029 + 0.842926i \(0.680831\pi\)
\(168\) 0 0
\(169\) 1601.07 0.728755
\(170\) 0 0
\(171\) −1731.75 −0.774447
\(172\) 0 0
\(173\) −363.841 −0.159898 −0.0799490 0.996799i \(-0.525476\pi\)
−0.0799490 + 0.996799i \(0.525476\pi\)
\(174\) 0 0
\(175\) 580.859 0.250908
\(176\) 0 0
\(177\) −449.705 −0.190971
\(178\) 0 0
\(179\) −2532.15 −1.05733 −0.528665 0.848831i \(-0.677307\pi\)
−0.528665 + 0.848831i \(0.677307\pi\)
\(180\) 0 0
\(181\) 1163.94 0.477982 0.238991 0.971022i \(-0.423183\pi\)
0.238991 + 0.971022i \(0.423183\pi\)
\(182\) 0 0
\(183\) 825.512 0.333462
\(184\) 0 0
\(185\) −1071.68 −0.425900
\(186\) 0 0
\(187\) −1131.87 −0.442621
\(188\) 0 0
\(189\) 798.032 0.307134
\(190\) 0 0
\(191\) −1677.77 −0.635599 −0.317799 0.948158i \(-0.602944\pi\)
−0.317799 + 0.948158i \(0.602944\pi\)
\(192\) 0 0
\(193\) 2249.50 0.838978 0.419489 0.907760i \(-0.362209\pi\)
0.419489 + 0.907760i \(0.362209\pi\)
\(194\) 0 0
\(195\) 3683.34 1.35266
\(196\) 0 0
\(197\) −2233.37 −0.807720 −0.403860 0.914821i \(-0.632332\pi\)
−0.403860 + 0.914821i \(0.632332\pi\)
\(198\) 0 0
\(199\) −2986.69 −1.06392 −0.531962 0.846768i \(-0.678545\pi\)
−0.531962 + 0.846768i \(0.678545\pi\)
\(200\) 0 0
\(201\) −1724.86 −0.605286
\(202\) 0 0
\(203\) −152.997 −0.0528981
\(204\) 0 0
\(205\) −3516.26 −1.19798
\(206\) 0 0
\(207\) 518.259 0.174017
\(208\) 0 0
\(209\) −4158.11 −1.37618
\(210\) 0 0
\(211\) −5179.02 −1.68975 −0.844877 0.534960i \(-0.820327\pi\)
−0.844877 + 0.534960i \(0.820327\pi\)
\(212\) 0 0
\(213\) −2585.35 −0.831669
\(214\) 0 0
\(215\) −2308.13 −0.732153
\(216\) 0 0
\(217\) 891.786 0.278979
\(218\) 0 0
\(219\) 2637.65 0.813863
\(220\) 0 0
\(221\) −2460.70 −0.748979
\(222\) 0 0
\(223\) 5527.03 1.65972 0.829859 0.557973i \(-0.188421\pi\)
0.829859 + 0.557973i \(0.188421\pi\)
\(224\) 0 0
\(225\) −1299.85 −0.385141
\(226\) 0 0
\(227\) −2788.28 −0.815262 −0.407631 0.913147i \(-0.633645\pi\)
−0.407631 + 0.913147i \(0.633645\pi\)
\(228\) 0 0
\(229\) 2527.23 0.729274 0.364637 0.931150i \(-0.381193\pi\)
0.364637 + 0.931150i \(0.381193\pi\)
\(230\) 0 0
\(231\) 582.959 0.166043
\(232\) 0 0
\(233\) −5606.98 −1.57650 −0.788252 0.615352i \(-0.789014\pi\)
−0.788252 + 0.615352i \(0.789014\pi\)
\(234\) 0 0
\(235\) −3647.49 −1.01249
\(236\) 0 0
\(237\) 5142.74 1.40952
\(238\) 0 0
\(239\) −5924.09 −1.60334 −0.801668 0.597769i \(-0.796054\pi\)
−0.801668 + 0.597769i \(0.796054\pi\)
\(240\) 0 0
\(241\) −876.073 −0.234161 −0.117081 0.993122i \(-0.537354\pi\)
−0.117081 + 0.993122i \(0.537354\pi\)
\(242\) 0 0
\(243\) −3028.37 −0.799465
\(244\) 0 0
\(245\) −4832.43 −1.26013
\(246\) 0 0
\(247\) −9039.81 −2.32870
\(248\) 0 0
\(249\) −4936.25 −1.25631
\(250\) 0 0
\(251\) 5914.86 1.48742 0.743711 0.668502i \(-0.233064\pi\)
0.743711 + 0.668502i \(0.233064\pi\)
\(252\) 0 0
\(253\) 1244.39 0.309226
\(254\) 0 0
\(255\) −2386.36 −0.586038
\(256\) 0 0
\(257\) −1589.22 −0.385730 −0.192865 0.981225i \(-0.561778\pi\)
−0.192865 + 0.981225i \(0.561778\pi\)
\(258\) 0 0
\(259\) −368.744 −0.0884658
\(260\) 0 0
\(261\) 342.378 0.0811980
\(262\) 0 0
\(263\) 7777.46 1.82349 0.911747 0.410752i \(-0.134734\pi\)
0.911747 + 0.410752i \(0.134734\pi\)
\(264\) 0 0
\(265\) 3349.54 0.776456
\(266\) 0 0
\(267\) 5287.29 1.21190
\(268\) 0 0
\(269\) 925.953 0.209875 0.104937 0.994479i \(-0.466536\pi\)
0.104937 + 0.994479i \(0.466536\pi\)
\(270\) 0 0
\(271\) 2235.53 0.501103 0.250551 0.968103i \(-0.419388\pi\)
0.250551 + 0.968103i \(0.419388\pi\)
\(272\) 0 0
\(273\) 1267.36 0.280968
\(274\) 0 0
\(275\) −3121.07 −0.684391
\(276\) 0 0
\(277\) −1810.59 −0.392736 −0.196368 0.980530i \(-0.562915\pi\)
−0.196368 + 0.980530i \(0.562915\pi\)
\(278\) 0 0
\(279\) −1995.64 −0.428230
\(280\) 0 0
\(281\) −7795.69 −1.65499 −0.827495 0.561474i \(-0.810235\pi\)
−0.827495 + 0.561474i \(0.810235\pi\)
\(282\) 0 0
\(283\) 7693.60 1.61603 0.808016 0.589160i \(-0.200541\pi\)
0.808016 + 0.589160i \(0.200541\pi\)
\(284\) 0 0
\(285\) −8766.72 −1.82209
\(286\) 0 0
\(287\) −1209.88 −0.248839
\(288\) 0 0
\(289\) −3318.76 −0.675507
\(290\) 0 0
\(291\) −3208.69 −0.646380
\(292\) 0 0
\(293\) 2093.31 0.417381 0.208691 0.977982i \(-0.433080\pi\)
0.208691 + 0.977982i \(0.433080\pi\)
\(294\) 0 0
\(295\) 1768.97 0.349129
\(296\) 0 0
\(297\) −4287.98 −0.837757
\(298\) 0 0
\(299\) 2705.33 0.523255
\(300\) 0 0
\(301\) −794.181 −0.152079
\(302\) 0 0
\(303\) −3491.52 −0.661989
\(304\) 0 0
\(305\) −3247.25 −0.609629
\(306\) 0 0
\(307\) −3700.08 −0.687865 −0.343933 0.938994i \(-0.611759\pi\)
−0.343933 + 0.938994i \(0.611759\pi\)
\(308\) 0 0
\(309\) 3020.52 0.556088
\(310\) 0 0
\(311\) −758.661 −0.138327 −0.0691635 0.997605i \(-0.522033\pi\)
−0.0691635 + 0.997605i \(0.522033\pi\)
\(312\) 0 0
\(313\) 2441.99 0.440989 0.220494 0.975388i \(-0.429233\pi\)
0.220494 + 0.975388i \(0.429233\pi\)
\(314\) 0 0
\(315\) −955.036 −0.170826
\(316\) 0 0
\(317\) −5664.85 −1.00369 −0.501845 0.864958i \(-0.667345\pi\)
−0.501845 + 0.864958i \(0.667345\pi\)
\(318\) 0 0
\(319\) 822.084 0.144288
\(320\) 0 0
\(321\) 3384.11 0.588419
\(322\) 0 0
\(323\) 5856.71 1.00890
\(324\) 0 0
\(325\) −6785.26 −1.15809
\(326\) 0 0
\(327\) −1279.05 −0.216305
\(328\) 0 0
\(329\) −1255.03 −0.210310
\(330\) 0 0
\(331\) −5046.15 −0.837950 −0.418975 0.907998i \(-0.637611\pi\)
−0.418975 + 0.907998i \(0.637611\pi\)
\(332\) 0 0
\(333\) 825.178 0.135794
\(334\) 0 0
\(335\) 6784.95 1.10657
\(336\) 0 0
\(337\) −2064.77 −0.333754 −0.166877 0.985978i \(-0.553368\pi\)
−0.166877 + 0.985978i \(0.553368\pi\)
\(338\) 0 0
\(339\) 4270.05 0.684121
\(340\) 0 0
\(341\) −4791.74 −0.760959
\(342\) 0 0
\(343\) −3472.33 −0.546613
\(344\) 0 0
\(345\) 2623.60 0.409420
\(346\) 0 0
\(347\) −10261.9 −1.58757 −0.793785 0.608199i \(-0.791892\pi\)
−0.793785 + 0.608199i \(0.791892\pi\)
\(348\) 0 0
\(349\) −9965.45 −1.52848 −0.764238 0.644934i \(-0.776885\pi\)
−0.764238 + 0.644934i \(0.776885\pi\)
\(350\) 0 0
\(351\) −9322.15 −1.41761
\(352\) 0 0
\(353\) −2978.90 −0.449153 −0.224576 0.974457i \(-0.572100\pi\)
−0.224576 + 0.974457i \(0.572100\pi\)
\(354\) 0 0
\(355\) 10169.8 1.52044
\(356\) 0 0
\(357\) −821.100 −0.121729
\(358\) 0 0
\(359\) −8156.81 −1.19916 −0.599582 0.800313i \(-0.704667\pi\)
−0.599582 + 0.800313i \(0.704667\pi\)
\(360\) 0 0
\(361\) 14656.7 2.13686
\(362\) 0 0
\(363\) 2055.79 0.297248
\(364\) 0 0
\(365\) −10375.5 −1.48789
\(366\) 0 0
\(367\) 6202.75 0.882237 0.441119 0.897449i \(-0.354582\pi\)
0.441119 + 0.897449i \(0.354582\pi\)
\(368\) 0 0
\(369\) 2707.47 0.381965
\(370\) 0 0
\(371\) 1152.51 0.161282
\(372\) 0 0
\(373\) −5940.53 −0.824635 −0.412317 0.911040i \(-0.635281\pi\)
−0.412317 + 0.911040i \(0.635281\pi\)
\(374\) 0 0
\(375\) 890.559 0.122635
\(376\) 0 0
\(377\) 1787.23 0.244156
\(378\) 0 0
\(379\) −3283.69 −0.445045 −0.222522 0.974928i \(-0.571429\pi\)
−0.222522 + 0.974928i \(0.571429\pi\)
\(380\) 0 0
\(381\) −3698.84 −0.497369
\(382\) 0 0
\(383\) 6651.64 0.887423 0.443712 0.896170i \(-0.353661\pi\)
0.443712 + 0.896170i \(0.353661\pi\)
\(384\) 0 0
\(385\) −2293.14 −0.303556
\(386\) 0 0
\(387\) 1777.22 0.233440
\(388\) 0 0
\(389\) −10607.5 −1.38257 −0.691285 0.722582i \(-0.742955\pi\)
−0.691285 + 0.722582i \(0.742955\pi\)
\(390\) 0 0
\(391\) −1752.73 −0.226699
\(392\) 0 0
\(393\) 176.089 0.0226018
\(394\) 0 0
\(395\) −20229.6 −2.57686
\(396\) 0 0
\(397\) −4945.27 −0.625178 −0.312589 0.949888i \(-0.601196\pi\)
−0.312589 + 0.949888i \(0.601196\pi\)
\(398\) 0 0
\(399\) −3016.46 −0.378476
\(400\) 0 0
\(401\) 14041.2 1.74859 0.874295 0.485396i \(-0.161324\pi\)
0.874295 + 0.485396i \(0.161324\pi\)
\(402\) 0 0
\(403\) −10417.3 −1.28765
\(404\) 0 0
\(405\) −4152.91 −0.509531
\(406\) 0 0
\(407\) 1981.33 0.241305
\(408\) 0 0
\(409\) −11566.1 −1.39830 −0.699150 0.714975i \(-0.746438\pi\)
−0.699150 + 0.714975i \(0.746438\pi\)
\(410\) 0 0
\(411\) 3922.50 0.470761
\(412\) 0 0
\(413\) 608.666 0.0725194
\(414\) 0 0
\(415\) 19417.3 2.29677
\(416\) 0 0
\(417\) −693.008 −0.0813831
\(418\) 0 0
\(419\) −10233.3 −1.19314 −0.596572 0.802560i \(-0.703471\pi\)
−0.596572 + 0.802560i \(0.703471\pi\)
\(420\) 0 0
\(421\) 11582.5 1.34085 0.670423 0.741979i \(-0.266113\pi\)
0.670423 + 0.741979i \(0.266113\pi\)
\(422\) 0 0
\(423\) 2808.52 0.322825
\(424\) 0 0
\(425\) 4396.04 0.501739
\(426\) 0 0
\(427\) −1117.31 −0.126629
\(428\) 0 0
\(429\) −6809.79 −0.766387
\(430\) 0 0
\(431\) −1754.30 −0.196060 −0.0980299 0.995183i \(-0.531254\pi\)
−0.0980299 + 0.995183i \(0.531254\pi\)
\(432\) 0 0
\(433\) 1060.29 0.117677 0.0588387 0.998268i \(-0.481260\pi\)
0.0588387 + 0.998268i \(0.481260\pi\)
\(434\) 0 0
\(435\) 1733.24 0.191040
\(436\) 0 0
\(437\) −6438.96 −0.704845
\(438\) 0 0
\(439\) 11608.1 1.26202 0.631008 0.775776i \(-0.282641\pi\)
0.631008 + 0.775776i \(0.282641\pi\)
\(440\) 0 0
\(441\) 3720.90 0.401782
\(442\) 0 0
\(443\) 4090.97 0.438754 0.219377 0.975640i \(-0.429598\pi\)
0.219377 + 0.975640i \(0.429598\pi\)
\(444\) 0 0
\(445\) −20798.2 −2.21557
\(446\) 0 0
\(447\) −8871.50 −0.938720
\(448\) 0 0
\(449\) −16360.6 −1.71961 −0.859807 0.510620i \(-0.829416\pi\)
−0.859807 + 0.510620i \(0.829416\pi\)
\(450\) 0 0
\(451\) 6500.90 0.678748
\(452\) 0 0
\(453\) 86.9967 0.00902309
\(454\) 0 0
\(455\) −4985.32 −0.513661
\(456\) 0 0
\(457\) 11693.7 1.19696 0.598479 0.801138i \(-0.295772\pi\)
0.598479 + 0.801138i \(0.295772\pi\)
\(458\) 0 0
\(459\) 6039.64 0.614175
\(460\) 0 0
\(461\) 14730.0 1.48817 0.744084 0.668086i \(-0.232886\pi\)
0.744084 + 0.668086i \(0.232886\pi\)
\(462\) 0 0
\(463\) −12020.1 −1.20653 −0.603263 0.797542i \(-0.706133\pi\)
−0.603263 + 0.797542i \(0.706133\pi\)
\(464\) 0 0
\(465\) −10102.6 −1.00752
\(466\) 0 0
\(467\) 5973.63 0.591920 0.295960 0.955200i \(-0.404360\pi\)
0.295960 + 0.955200i \(0.404360\pi\)
\(468\) 0 0
\(469\) 2334.57 0.229852
\(470\) 0 0
\(471\) 10715.4 1.04828
\(472\) 0 0
\(473\) 4267.29 0.414820
\(474\) 0 0
\(475\) 16149.6 1.55999
\(476\) 0 0
\(477\) −2579.10 −0.247566
\(478\) 0 0
\(479\) 17413.9 1.66109 0.830545 0.556951i \(-0.188029\pi\)
0.830545 + 0.556951i \(0.188029\pi\)
\(480\) 0 0
\(481\) 4307.46 0.408322
\(482\) 0 0
\(483\) 902.730 0.0850427
\(484\) 0 0
\(485\) 12621.7 1.18170
\(486\) 0 0
\(487\) −3885.79 −0.361565 −0.180782 0.983523i \(-0.557863\pi\)
−0.180782 + 0.983523i \(0.557863\pi\)
\(488\) 0 0
\(489\) 6051.53 0.559631
\(490\) 0 0
\(491\) 14369.1 1.32071 0.660354 0.750954i \(-0.270406\pi\)
0.660354 + 0.750954i \(0.270406\pi\)
\(492\) 0 0
\(493\) −1157.91 −0.105780
\(494\) 0 0
\(495\) 5131.60 0.465956
\(496\) 0 0
\(497\) 3499.22 0.315818
\(498\) 0 0
\(499\) −466.364 −0.0418383 −0.0209191 0.999781i \(-0.506659\pi\)
−0.0209191 + 0.999781i \(0.506659\pi\)
\(500\) 0 0
\(501\) 9052.00 0.807212
\(502\) 0 0
\(503\) 3145.80 0.278856 0.139428 0.990232i \(-0.455474\pi\)
0.139428 + 0.990232i \(0.455474\pi\)
\(504\) 0 0
\(505\) 13734.3 1.21024
\(506\) 0 0
\(507\) −6240.87 −0.546680
\(508\) 0 0
\(509\) 1033.27 0.0899779 0.0449889 0.998987i \(-0.485675\pi\)
0.0449889 + 0.998987i \(0.485675\pi\)
\(510\) 0 0
\(511\) −3570.01 −0.309056
\(512\) 0 0
\(513\) 22187.7 1.90957
\(514\) 0 0
\(515\) −11881.6 −1.01663
\(516\) 0 0
\(517\) 6743.53 0.573656
\(518\) 0 0
\(519\) 1418.23 0.119949
\(520\) 0 0
\(521\) −11615.8 −0.976767 −0.488384 0.872629i \(-0.662413\pi\)
−0.488384 + 0.872629i \(0.662413\pi\)
\(522\) 0 0
\(523\) 14249.8 1.19140 0.595700 0.803207i \(-0.296875\pi\)
0.595700 + 0.803207i \(0.296875\pi\)
\(524\) 0 0
\(525\) −2264.15 −0.188220
\(526\) 0 0
\(527\) 6749.18 0.557873
\(528\) 0 0
\(529\) −10240.0 −0.841623
\(530\) 0 0
\(531\) −1362.08 −0.111317
\(532\) 0 0
\(533\) 14133.1 1.14854
\(534\) 0 0
\(535\) −13311.8 −1.07573
\(536\) 0 0
\(537\) 9870.15 0.793163
\(538\) 0 0
\(539\) 8934.25 0.713962
\(540\) 0 0
\(541\) 1221.09 0.0970404 0.0485202 0.998822i \(-0.484549\pi\)
0.0485202 + 0.998822i \(0.484549\pi\)
\(542\) 0 0
\(543\) −4536.95 −0.358562
\(544\) 0 0
\(545\) 5031.30 0.395445
\(546\) 0 0
\(547\) −23448.2 −1.83285 −0.916427 0.400201i \(-0.868940\pi\)
−0.916427 + 0.400201i \(0.868940\pi\)
\(548\) 0 0
\(549\) 2500.33 0.194375
\(550\) 0 0
\(551\) −4253.79 −0.328888
\(552\) 0 0
\(553\) −6960.60 −0.535253
\(554\) 0 0
\(555\) 4177.33 0.319492
\(556\) 0 0
\(557\) 14174.7 1.07828 0.539140 0.842216i \(-0.318749\pi\)
0.539140 + 0.842216i \(0.318749\pi\)
\(558\) 0 0
\(559\) 9277.16 0.701936
\(560\) 0 0
\(561\) 4411.93 0.332035
\(562\) 0 0
\(563\) 3348.85 0.250688 0.125344 0.992113i \(-0.459997\pi\)
0.125344 + 0.992113i \(0.459997\pi\)
\(564\) 0 0
\(565\) −16796.7 −1.25070
\(566\) 0 0
\(567\) −1428.94 −0.105837
\(568\) 0 0
\(569\) 10533.7 0.776091 0.388045 0.921640i \(-0.373150\pi\)
0.388045 + 0.921640i \(0.373150\pi\)
\(570\) 0 0
\(571\) −19267.0 −1.41208 −0.706041 0.708171i \(-0.749520\pi\)
−0.706041 + 0.708171i \(0.749520\pi\)
\(572\) 0 0
\(573\) 6539.84 0.476799
\(574\) 0 0
\(575\) −4833.07 −0.350527
\(576\) 0 0
\(577\) 4747.42 0.342526 0.171263 0.985225i \(-0.445215\pi\)
0.171263 + 0.985225i \(0.445215\pi\)
\(578\) 0 0
\(579\) −8768.41 −0.629365
\(580\) 0 0
\(581\) 6681.11 0.477073
\(582\) 0 0
\(583\) −6192.67 −0.439921
\(584\) 0 0
\(585\) 11156.2 0.788464
\(586\) 0 0
\(587\) −7603.14 −0.534608 −0.267304 0.963612i \(-0.586133\pi\)
−0.267304 + 0.963612i \(0.586133\pi\)
\(588\) 0 0
\(589\) 24794.3 1.73452
\(590\) 0 0
\(591\) 8705.51 0.605917
\(592\) 0 0
\(593\) 1892.76 0.131073 0.0655366 0.997850i \(-0.479124\pi\)
0.0655366 + 0.997850i \(0.479124\pi\)
\(594\) 0 0
\(595\) 3229.89 0.222542
\(596\) 0 0
\(597\) 11641.9 0.798110
\(598\) 0 0
\(599\) −18998.8 −1.29594 −0.647970 0.761665i \(-0.724382\pi\)
−0.647970 + 0.761665i \(0.724382\pi\)
\(600\) 0 0
\(601\) 2034.70 0.138098 0.0690491 0.997613i \(-0.478003\pi\)
0.0690491 + 0.997613i \(0.478003\pi\)
\(602\) 0 0
\(603\) −5224.31 −0.352820
\(604\) 0 0
\(605\) −8086.69 −0.543423
\(606\) 0 0
\(607\) −1691.66 −0.113117 −0.0565587 0.998399i \(-0.518013\pi\)
−0.0565587 + 0.998399i \(0.518013\pi\)
\(608\) 0 0
\(609\) 596.373 0.0396818
\(610\) 0 0
\(611\) 14660.6 0.970708
\(612\) 0 0
\(613\) 2051.86 0.135194 0.0675969 0.997713i \(-0.478467\pi\)
0.0675969 + 0.997713i \(0.478467\pi\)
\(614\) 0 0
\(615\) 13706.1 0.898674
\(616\) 0 0
\(617\) −21839.0 −1.42497 −0.712483 0.701690i \(-0.752429\pi\)
−0.712483 + 0.701690i \(0.752429\pi\)
\(618\) 0 0
\(619\) 5707.58 0.370609 0.185305 0.982681i \(-0.440673\pi\)
0.185305 + 0.982681i \(0.440673\pi\)
\(620\) 0 0
\(621\) −6640.07 −0.429077
\(622\) 0 0
\(623\) −7156.24 −0.460207
\(624\) 0 0
\(625\) −17265.5 −1.10499
\(626\) 0 0
\(627\) 16208.0 1.03235
\(628\) 0 0
\(629\) −2790.72 −0.176905
\(630\) 0 0
\(631\) 848.735 0.0535461 0.0267731 0.999642i \(-0.491477\pi\)
0.0267731 + 0.999642i \(0.491477\pi\)
\(632\) 0 0
\(633\) 20187.4 1.26758
\(634\) 0 0
\(635\) 14549.8 0.909279
\(636\) 0 0
\(637\) 19423.2 1.20813
\(638\) 0 0
\(639\) −7830.59 −0.484778
\(640\) 0 0
\(641\) −25537.5 −1.57359 −0.786796 0.617213i \(-0.788262\pi\)
−0.786796 + 0.617213i \(0.788262\pi\)
\(642\) 0 0
\(643\) −4902.36 −0.300669 −0.150334 0.988635i \(-0.548035\pi\)
−0.150334 + 0.988635i \(0.548035\pi\)
\(644\) 0 0
\(645\) 8996.91 0.549229
\(646\) 0 0
\(647\) −3920.19 −0.238205 −0.119103 0.992882i \(-0.538002\pi\)
−0.119103 + 0.992882i \(0.538002\pi\)
\(648\) 0 0
\(649\) −3270.48 −0.197808
\(650\) 0 0
\(651\) −3476.12 −0.209278
\(652\) 0 0
\(653\) 3640.59 0.218174 0.109087 0.994032i \(-0.465207\pi\)
0.109087 + 0.994032i \(0.465207\pi\)
\(654\) 0 0
\(655\) −692.666 −0.0413201
\(656\) 0 0
\(657\) 7988.99 0.474399
\(658\) 0 0
\(659\) −14826.8 −0.876437 −0.438219 0.898868i \(-0.644390\pi\)
−0.438219 + 0.898868i \(0.644390\pi\)
\(660\) 0 0
\(661\) −25302.1 −1.48886 −0.744432 0.667698i \(-0.767280\pi\)
−0.744432 + 0.667698i \(0.767280\pi\)
\(662\) 0 0
\(663\) 9591.62 0.561852
\(664\) 0 0
\(665\) 11865.6 0.691921
\(666\) 0 0
\(667\) 1273.02 0.0739005
\(668\) 0 0
\(669\) −21544.0 −1.24505
\(670\) 0 0
\(671\) 6003.55 0.345402
\(672\) 0 0
\(673\) −18983.4 −1.08730 −0.543652 0.839311i \(-0.682959\pi\)
−0.543652 + 0.839311i \(0.682959\pi\)
\(674\) 0 0
\(675\) 16654.0 0.949651
\(676\) 0 0
\(677\) −16033.8 −0.910236 −0.455118 0.890431i \(-0.650403\pi\)
−0.455118 + 0.890431i \(0.650403\pi\)
\(678\) 0 0
\(679\) 4342.89 0.245456
\(680\) 0 0
\(681\) 10868.5 0.611574
\(682\) 0 0
\(683\) −1451.21 −0.0813017 −0.0406509 0.999173i \(-0.512943\pi\)
−0.0406509 + 0.999173i \(0.512943\pi\)
\(684\) 0 0
\(685\) −15429.6 −0.860635
\(686\) 0 0
\(687\) −9850.95 −0.547070
\(688\) 0 0
\(689\) −13463.0 −0.744410
\(690\) 0 0
\(691\) 8126.55 0.447393 0.223697 0.974659i \(-0.428188\pi\)
0.223697 + 0.974659i \(0.428188\pi\)
\(692\) 0 0
\(693\) 1765.68 0.0967860
\(694\) 0 0
\(695\) 2726.03 0.148783
\(696\) 0 0
\(697\) −9156.54 −0.497602
\(698\) 0 0
\(699\) 21855.6 1.18263
\(700\) 0 0
\(701\) −1475.94 −0.0795229 −0.0397614 0.999209i \(-0.512660\pi\)
−0.0397614 + 0.999209i \(0.512660\pi\)
\(702\) 0 0
\(703\) −10252.2 −0.550027
\(704\) 0 0
\(705\) 14217.7 0.759530
\(706\) 0 0
\(707\) 4725.71 0.251384
\(708\) 0 0
\(709\) 228.875 0.0121235 0.00606177 0.999982i \(-0.498070\pi\)
0.00606177 + 0.999982i \(0.498070\pi\)
\(710\) 0 0
\(711\) 15576.5 0.821608
\(712\) 0 0
\(713\) −7420.15 −0.389743
\(714\) 0 0
\(715\) 26787.1 1.40109
\(716\) 0 0
\(717\) 23091.7 1.20275
\(718\) 0 0
\(719\) 15696.4 0.814155 0.407077 0.913394i \(-0.366548\pi\)
0.407077 + 0.913394i \(0.366548\pi\)
\(720\) 0 0
\(721\) −4088.21 −0.211169
\(722\) 0 0
\(723\) 3414.87 0.175658
\(724\) 0 0
\(725\) −3192.88 −0.163560
\(726\) 0 0
\(727\) 14329.5 0.731020 0.365510 0.930807i \(-0.380895\pi\)
0.365510 + 0.930807i \(0.380895\pi\)
\(728\) 0 0
\(729\) 19117.3 0.971259
\(730\) 0 0
\(731\) −6010.49 −0.304112
\(732\) 0 0
\(733\) 15912.1 0.801810 0.400905 0.916120i \(-0.368696\pi\)
0.400905 + 0.916120i \(0.368696\pi\)
\(734\) 0 0
\(735\) 18836.5 0.945297
\(736\) 0 0
\(737\) −12544.1 −0.626958
\(738\) 0 0
\(739\) 12354.6 0.614983 0.307492 0.951551i \(-0.400510\pi\)
0.307492 + 0.951551i \(0.400510\pi\)
\(740\) 0 0
\(741\) 35236.5 1.74689
\(742\) 0 0
\(743\) −5080.77 −0.250869 −0.125434 0.992102i \(-0.540032\pi\)
−0.125434 + 0.992102i \(0.540032\pi\)
\(744\) 0 0
\(745\) 34897.1 1.71615
\(746\) 0 0
\(747\) −14951.0 −0.732302
\(748\) 0 0
\(749\) −4580.32 −0.223446
\(750\) 0 0
\(751\) −34750.4 −1.68850 −0.844248 0.535952i \(-0.819953\pi\)
−0.844248 + 0.535952i \(0.819953\pi\)
\(752\) 0 0
\(753\) −23055.7 −1.11580
\(754\) 0 0
\(755\) −342.212 −0.0164958
\(756\) 0 0
\(757\) 2602.27 0.124942 0.0624709 0.998047i \(-0.480102\pi\)
0.0624709 + 0.998047i \(0.480102\pi\)
\(758\) 0 0
\(759\) −4850.54 −0.231968
\(760\) 0 0
\(761\) 26480.5 1.26139 0.630694 0.776031i \(-0.282770\pi\)
0.630694 + 0.776031i \(0.282770\pi\)
\(762\) 0 0
\(763\) 1731.17 0.0821398
\(764\) 0 0
\(765\) −7227.87 −0.341600
\(766\) 0 0
\(767\) −7110.09 −0.334720
\(768\) 0 0
\(769\) −7464.68 −0.350043 −0.175022 0.984565i \(-0.556000\pi\)
−0.175022 + 0.984565i \(0.556000\pi\)
\(770\) 0 0
\(771\) 6194.65 0.289358
\(772\) 0 0
\(773\) 39751.4 1.84962 0.924812 0.380424i \(-0.124222\pi\)
0.924812 + 0.380424i \(0.124222\pi\)
\(774\) 0 0
\(775\) 18610.6 0.862596
\(776\) 0 0
\(777\) 1437.34 0.0663632
\(778\) 0 0
\(779\) −33638.2 −1.54713
\(780\) 0 0
\(781\) −18802.0 −0.861446
\(782\) 0 0
\(783\) −4386.64 −0.200212
\(784\) 0 0
\(785\) −42150.3 −1.91644
\(786\) 0 0
\(787\) −16132.1 −0.730681 −0.365341 0.930874i \(-0.619048\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(788\) 0 0
\(789\) −30316.0 −1.36791
\(790\) 0 0
\(791\) −5779.42 −0.259788
\(792\) 0 0
\(793\) 13051.8 0.584469
\(794\) 0 0
\(795\) −13056.3 −0.582463
\(796\) 0 0
\(797\) 30649.4 1.36218 0.681089 0.732201i \(-0.261507\pi\)
0.681089 + 0.732201i \(0.261507\pi\)
\(798\) 0 0
\(799\) −9498.28 −0.420557
\(800\) 0 0
\(801\) 16014.3 0.706413
\(802\) 0 0
\(803\) 19182.3 0.843002
\(804\) 0 0
\(805\) −3550.99 −0.155473
\(806\) 0 0
\(807\) −3609.30 −0.157439
\(808\) 0 0
\(809\) 31988.2 1.39017 0.695083 0.718929i \(-0.255367\pi\)
0.695083 + 0.718929i \(0.255367\pi\)
\(810\) 0 0
\(811\) −39086.8 −1.69238 −0.846192 0.532877i \(-0.821111\pi\)
−0.846192 + 0.532877i \(0.821111\pi\)
\(812\) 0 0
\(813\) −8713.94 −0.375906
\(814\) 0 0
\(815\) −23804.4 −1.02311
\(816\) 0 0
\(817\) −22080.6 −0.945536
\(818\) 0 0
\(819\) 3838.63 0.163776
\(820\) 0 0
\(821\) 20056.3 0.852582 0.426291 0.904586i \(-0.359820\pi\)
0.426291 + 0.904586i \(0.359820\pi\)
\(822\) 0 0
\(823\) 24931.0 1.05594 0.527970 0.849263i \(-0.322953\pi\)
0.527970 + 0.849263i \(0.322953\pi\)
\(824\) 0 0
\(825\) 12165.7 0.513401
\(826\) 0 0
\(827\) −40104.5 −1.68630 −0.843150 0.537678i \(-0.819302\pi\)
−0.843150 + 0.537678i \(0.819302\pi\)
\(828\) 0 0
\(829\) 35722.2 1.49660 0.748300 0.663360i \(-0.230870\pi\)
0.748300 + 0.663360i \(0.230870\pi\)
\(830\) 0 0
\(831\) 7057.56 0.294614
\(832\) 0 0
\(833\) −12583.9 −0.523418
\(834\) 0 0
\(835\) −35607.1 −1.47573
\(836\) 0 0
\(837\) 25568.7 1.05590
\(838\) 0 0
\(839\) −185.813 −0.00764598 −0.00382299 0.999993i \(-0.501217\pi\)
−0.00382299 + 0.999993i \(0.501217\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 30387.1 1.24150
\(844\) 0 0
\(845\) 24549.2 0.999429
\(846\) 0 0
\(847\) −2782.47 −0.112877
\(848\) 0 0
\(849\) −29989.1 −1.21228
\(850\) 0 0
\(851\) 3068.16 0.123590
\(852\) 0 0
\(853\) 32958.5 1.32295 0.661475 0.749967i \(-0.269931\pi\)
0.661475 + 0.749967i \(0.269931\pi\)
\(854\) 0 0
\(855\) −26552.9 −1.06209
\(856\) 0 0
\(857\) −36681.9 −1.46211 −0.731056 0.682318i \(-0.760972\pi\)
−0.731056 + 0.682318i \(0.760972\pi\)
\(858\) 0 0
\(859\) 25084.0 0.996340 0.498170 0.867079i \(-0.334006\pi\)
0.498170 + 0.867079i \(0.334006\pi\)
\(860\) 0 0
\(861\) 4716.01 0.186668
\(862\) 0 0
\(863\) 6552.57 0.258461 0.129231 0.991615i \(-0.458749\pi\)
0.129231 + 0.991615i \(0.458749\pi\)
\(864\) 0 0
\(865\) −5578.76 −0.219287
\(866\) 0 0
\(867\) 12936.3 0.506736
\(868\) 0 0
\(869\) 37400.6 1.45999
\(870\) 0 0
\(871\) −27271.1 −1.06090
\(872\) 0 0
\(873\) −9718.55 −0.376773
\(874\) 0 0
\(875\) −1205.35 −0.0465696
\(876\) 0 0
\(877\) 39373.9 1.51603 0.758017 0.652234i \(-0.226168\pi\)
0.758017 + 0.652234i \(0.226168\pi\)
\(878\) 0 0
\(879\) −8159.59 −0.313101
\(880\) 0 0
\(881\) −8182.51 −0.312912 −0.156456 0.987685i \(-0.550007\pi\)
−0.156456 + 0.987685i \(0.550007\pi\)
\(882\) 0 0
\(883\) −7461.63 −0.284376 −0.142188 0.989840i \(-0.545414\pi\)
−0.142188 + 0.989840i \(0.545414\pi\)
\(884\) 0 0
\(885\) −6895.30 −0.261902
\(886\) 0 0
\(887\) 29745.2 1.12598 0.562990 0.826463i \(-0.309651\pi\)
0.562990 + 0.826463i \(0.309651\pi\)
\(888\) 0 0
\(889\) 5006.31 0.188871
\(890\) 0 0
\(891\) 7677.95 0.288688
\(892\) 0 0
\(893\) −34893.7 −1.30758
\(894\) 0 0
\(895\) −38825.4 −1.45004
\(896\) 0 0
\(897\) −10545.2 −0.392523
\(898\) 0 0
\(899\) −4901.99 −0.181858
\(900\) 0 0
\(901\) 8722.40 0.322514
\(902\) 0 0
\(903\) 3095.66 0.114083
\(904\) 0 0
\(905\) 17846.6 0.655515
\(906\) 0 0
\(907\) −34626.6 −1.26765 −0.633825 0.773476i \(-0.718516\pi\)
−0.633825 + 0.773476i \(0.718516\pi\)
\(908\) 0 0
\(909\) −10575.2 −0.385872
\(910\) 0 0
\(911\) −6172.29 −0.224476 −0.112238 0.993681i \(-0.535802\pi\)
−0.112238 + 0.993681i \(0.535802\pi\)
\(912\) 0 0
\(913\) −35898.9 −1.30129
\(914\) 0 0
\(915\) 12657.5 0.457317
\(916\) 0 0
\(917\) −238.333 −0.00858281
\(918\) 0 0
\(919\) −15779.0 −0.566376 −0.283188 0.959064i \(-0.591392\pi\)
−0.283188 + 0.959064i \(0.591392\pi\)
\(920\) 0 0
\(921\) 14422.6 0.516007
\(922\) 0 0
\(923\) −40875.9 −1.45769
\(924\) 0 0
\(925\) −7695.28 −0.273534
\(926\) 0 0
\(927\) 9148.62 0.324143
\(928\) 0 0
\(929\) −9692.85 −0.342316 −0.171158 0.985244i \(-0.554751\pi\)
−0.171158 + 0.985244i \(0.554751\pi\)
\(930\) 0 0
\(931\) −46229.3 −1.62739
\(932\) 0 0
\(933\) 2957.21 0.103767
\(934\) 0 0
\(935\) −17354.8 −0.607020
\(936\) 0 0
\(937\) 17253.8 0.601554 0.300777 0.953694i \(-0.402754\pi\)
0.300777 + 0.953694i \(0.402754\pi\)
\(938\) 0 0
\(939\) −9518.70 −0.330811
\(940\) 0 0
\(941\) −6249.78 −0.216511 −0.108256 0.994123i \(-0.534526\pi\)
−0.108256 + 0.994123i \(0.534526\pi\)
\(942\) 0 0
\(943\) 10066.8 0.347637
\(944\) 0 0
\(945\) 12236.2 0.421210
\(946\) 0 0
\(947\) −38713.7 −1.32843 −0.664217 0.747540i \(-0.731235\pi\)
−0.664217 + 0.747540i \(0.731235\pi\)
\(948\) 0 0
\(949\) 41702.8 1.42648
\(950\) 0 0
\(951\) 22081.2 0.752924
\(952\) 0 0
\(953\) 28936.5 0.983573 0.491787 0.870716i \(-0.336344\pi\)
0.491787 + 0.870716i \(0.336344\pi\)
\(954\) 0 0
\(955\) −25725.2 −0.871673
\(956\) 0 0
\(957\) −3204.43 −0.108239
\(958\) 0 0
\(959\) −5309.03 −0.178767
\(960\) 0 0
\(961\) −1218.42 −0.0408989
\(962\) 0 0
\(963\) 10249.9 0.342988
\(964\) 0 0
\(965\) 34491.5 1.15059
\(966\) 0 0
\(967\) −8697.74 −0.289245 −0.144623 0.989487i \(-0.546197\pi\)
−0.144623 + 0.989487i \(0.546197\pi\)
\(968\) 0 0
\(969\) −22829.1 −0.756837
\(970\) 0 0
\(971\) 41062.0 1.35710 0.678549 0.734555i \(-0.262609\pi\)
0.678549 + 0.734555i \(0.262609\pi\)
\(972\) 0 0
\(973\) 937.973 0.0309045
\(974\) 0 0
\(975\) 26448.5 0.868748
\(976\) 0 0
\(977\) 35397.4 1.15912 0.579561 0.814929i \(-0.303224\pi\)
0.579561 + 0.814929i \(0.303224\pi\)
\(978\) 0 0
\(979\) 38451.9 1.25529
\(980\) 0 0
\(981\) −3874.03 −0.126084
\(982\) 0 0
\(983\) 51453.4 1.66949 0.834746 0.550636i \(-0.185615\pi\)
0.834746 + 0.550636i \(0.185615\pi\)
\(984\) 0 0
\(985\) −34244.1 −1.10772
\(986\) 0 0
\(987\) 4892.02 0.157766
\(988\) 0 0
\(989\) 6608.03 0.212460
\(990\) 0 0
\(991\) 43563.5 1.39641 0.698205 0.715898i \(-0.253983\pi\)
0.698205 + 0.715898i \(0.253983\pi\)
\(992\) 0 0
\(993\) 19669.5 0.628594
\(994\) 0 0
\(995\) −45794.8 −1.45909
\(996\) 0 0
\(997\) 45472.8 1.44447 0.722236 0.691647i \(-0.243114\pi\)
0.722236 + 0.691647i \(0.243114\pi\)
\(998\) 0 0
\(999\) −10572.4 −0.334831
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 1856.4.a.be.1.3 8
4.3 odd 2 inner 1856.4.a.be.1.6 8
8.3 odd 2 928.4.a.c.1.3 8
8.5 even 2 928.4.a.c.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.c.1.3 8 8.3 odd 2
928.4.a.c.1.6 yes 8 8.5 even 2
1856.4.a.be.1.3 8 1.1 even 1 trivial
1856.4.a.be.1.6 8 4.3 odd 2 inner