Properties

Label 1088.4.a.bh.1.2
Level $1088$
Weight $4$
Character 1088.1
Self dual yes
Analytic conductor $64.194$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1088,4,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, 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("1088.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(64.1940780862\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 67x^{5} - 35x^{4} + 893x^{3} + 595x^{2} - 3064x - 2804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 544)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.90461\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.01688 q^{3} +14.1958 q^{5} +9.87960 q^{7} -10.8647 q^{9} +O(q^{10})\) \(q-4.01688 q^{3} +14.1958 q^{5} +9.87960 q^{7} -10.8647 q^{9} -60.2213 q^{11} +51.2168 q^{13} -57.0229 q^{15} -17.0000 q^{17} -34.2361 q^{19} -39.6851 q^{21} -39.8677 q^{23} +76.5216 q^{25} +152.098 q^{27} +196.276 q^{29} +182.486 q^{31} +241.902 q^{33} +140.249 q^{35} +255.616 q^{37} -205.732 q^{39} -194.494 q^{41} +150.979 q^{43} -154.233 q^{45} -350.840 q^{47} -245.394 q^{49} +68.2869 q^{51} -71.9534 q^{53} -854.892 q^{55} +137.522 q^{57} +142.346 q^{59} +610.499 q^{61} -107.339 q^{63} +727.065 q^{65} +368.545 q^{67} +160.144 q^{69} -710.184 q^{71} +1015.72 q^{73} -307.378 q^{75} -594.962 q^{77} +374.294 q^{79} -317.611 q^{81} -773.711 q^{83} -241.329 q^{85} -788.418 q^{87} +1052.83 q^{89} +506.001 q^{91} -733.023 q^{93} -486.010 q^{95} -1596.33 q^{97} +654.287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{7} + 67 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{7} + 67 q^{9} + 108 q^{11} + 34 q^{13} - 128 q^{15} - 119 q^{17} + 124 q^{19} + 296 q^{21} + 6 q^{23} + 197 q^{25} - 248 q^{29} - 50 q^{31} + 512 q^{33} + 640 q^{35} + 484 q^{37} - 1504 q^{39} - 366 q^{41} + 1412 q^{43} + 80 q^{45} - 1012 q^{47} + 1115 q^{49} + 146 q^{53} - 1024 q^{55} + 48 q^{57} + 2332 q^{59} + 548 q^{61} - 2838 q^{63} - 208 q^{65} + 924 q^{67} + 1672 q^{69} - 1286 q^{71} + 870 q^{73} + 3136 q^{75} + 1344 q^{77} - 1818 q^{79} + 3039 q^{81} + 1772 q^{83} - 384 q^{87} + 1706 q^{89} + 588 q^{91} + 5576 q^{93} - 2048 q^{95} + 1802 q^{97} + 5148 q^{99}+O(q^{100}) \) 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
\(3\) −4.01688 −0.773048 −0.386524 0.922279i \(-0.626324\pi\)
−0.386524 + 0.922279i \(0.626324\pi\)
\(4\) 0 0
\(5\) 14.1958 1.26971 0.634857 0.772630i \(-0.281059\pi\)
0.634857 + 0.772630i \(0.281059\pi\)
\(6\) 0 0
\(7\) 9.87960 0.533448 0.266724 0.963773i \(-0.414059\pi\)
0.266724 + 0.963773i \(0.414059\pi\)
\(8\) 0 0
\(9\) −10.8647 −0.402396
\(10\) 0 0
\(11\) −60.2213 −1.65067 −0.825337 0.564640i \(-0.809015\pi\)
−0.825337 + 0.564640i \(0.809015\pi\)
\(12\) 0 0
\(13\) 51.2168 1.09269 0.546346 0.837560i \(-0.316019\pi\)
0.546346 + 0.837560i \(0.316019\pi\)
\(14\) 0 0
\(15\) −57.0229 −0.981550
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) −34.2361 −0.413384 −0.206692 0.978406i \(-0.566270\pi\)
−0.206692 + 0.978406i \(0.566270\pi\)
\(20\) 0 0
\(21\) −39.6851 −0.412381
\(22\) 0 0
\(23\) −39.8677 −0.361435 −0.180717 0.983535i \(-0.557842\pi\)
−0.180717 + 0.983535i \(0.557842\pi\)
\(24\) 0 0
\(25\) 76.5216 0.612173
\(26\) 0 0
\(27\) 152.098 1.08412
\(28\) 0 0
\(29\) 196.276 1.25681 0.628407 0.777885i \(-0.283707\pi\)
0.628407 + 0.777885i \(0.283707\pi\)
\(30\) 0 0
\(31\) 182.486 1.05727 0.528636 0.848849i \(-0.322704\pi\)
0.528636 + 0.848849i \(0.322704\pi\)
\(32\) 0 0
\(33\) 241.902 1.27605
\(34\) 0 0
\(35\) 140.249 0.677326
\(36\) 0 0
\(37\) 255.616 1.13576 0.567878 0.823113i \(-0.307765\pi\)
0.567878 + 0.823113i \(0.307765\pi\)
\(38\) 0 0
\(39\) −205.732 −0.844703
\(40\) 0 0
\(41\) −194.494 −0.740848 −0.370424 0.928863i \(-0.620788\pi\)
−0.370424 + 0.928863i \(0.620788\pi\)
\(42\) 0 0
\(43\) 150.979 0.535443 0.267721 0.963496i \(-0.413729\pi\)
0.267721 + 0.963496i \(0.413729\pi\)
\(44\) 0 0
\(45\) −154.233 −0.510928
\(46\) 0 0
\(47\) −350.840 −1.08884 −0.544418 0.838814i \(-0.683250\pi\)
−0.544418 + 0.838814i \(0.683250\pi\)
\(48\) 0 0
\(49\) −245.394 −0.715433
\(50\) 0 0
\(51\) 68.2869 0.187492
\(52\) 0 0
\(53\) −71.9534 −0.186482 −0.0932412 0.995644i \(-0.529723\pi\)
−0.0932412 + 0.995644i \(0.529723\pi\)
\(54\) 0 0
\(55\) −854.892 −2.09588
\(56\) 0 0
\(57\) 137.522 0.319566
\(58\) 0 0
\(59\) 142.346 0.314099 0.157049 0.987591i \(-0.449802\pi\)
0.157049 + 0.987591i \(0.449802\pi\)
\(60\) 0 0
\(61\) 610.499 1.28142 0.640708 0.767785i \(-0.278641\pi\)
0.640708 + 0.767785i \(0.278641\pi\)
\(62\) 0 0
\(63\) −107.339 −0.214658
\(64\) 0 0
\(65\) 727.065 1.38741
\(66\) 0 0
\(67\) 368.545 0.672013 0.336007 0.941860i \(-0.390924\pi\)
0.336007 + 0.941860i \(0.390924\pi\)
\(68\) 0 0
\(69\) 160.144 0.279406
\(70\) 0 0
\(71\) −710.184 −1.18709 −0.593544 0.804801i \(-0.702272\pi\)
−0.593544 + 0.804801i \(0.702272\pi\)
\(72\) 0 0
\(73\) 1015.72 1.62851 0.814254 0.580509i \(-0.197146\pi\)
0.814254 + 0.580509i \(0.197146\pi\)
\(74\) 0 0
\(75\) −307.378 −0.473239
\(76\) 0 0
\(77\) −594.962 −0.880549
\(78\) 0 0
\(79\) 374.294 0.533055 0.266528 0.963827i \(-0.414124\pi\)
0.266528 + 0.963827i \(0.414124\pi\)
\(80\) 0 0
\(81\) −317.611 −0.435681
\(82\) 0 0
\(83\) −773.711 −1.02320 −0.511601 0.859223i \(-0.670948\pi\)
−0.511601 + 0.859223i \(0.670948\pi\)
\(84\) 0 0
\(85\) −241.329 −0.307951
\(86\) 0 0
\(87\) −788.418 −0.971578
\(88\) 0 0
\(89\) 1052.83 1.25393 0.626966 0.779046i \(-0.284296\pi\)
0.626966 + 0.779046i \(0.284296\pi\)
\(90\) 0 0
\(91\) 506.001 0.582894
\(92\) 0 0
\(93\) −733.023 −0.817322
\(94\) 0 0
\(95\) −486.010 −0.524880
\(96\) 0 0
\(97\) −1596.33 −1.67096 −0.835479 0.549523i \(-0.814810\pi\)
−0.835479 + 0.549523i \(0.814810\pi\)
\(98\) 0 0
\(99\) 654.287 0.664225
\(100\) 0 0
\(101\) 1355.25 1.33517 0.667587 0.744532i \(-0.267327\pi\)
0.667587 + 0.744532i \(0.267327\pi\)
\(102\) 0 0
\(103\) 1035.23 0.990330 0.495165 0.868799i \(-0.335108\pi\)
0.495165 + 0.868799i \(0.335108\pi\)
\(104\) 0 0
\(105\) −563.363 −0.523606
\(106\) 0 0
\(107\) −154.641 −0.139717 −0.0698585 0.997557i \(-0.522255\pi\)
−0.0698585 + 0.997557i \(0.522255\pi\)
\(108\) 0 0
\(109\) 427.905 0.376017 0.188009 0.982167i \(-0.439797\pi\)
0.188009 + 0.982167i \(0.439797\pi\)
\(110\) 0 0
\(111\) −1026.78 −0.877994
\(112\) 0 0
\(113\) −772.563 −0.643156 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(114\) 0 0
\(115\) −565.956 −0.458918
\(116\) 0 0
\(117\) −556.455 −0.439695
\(118\) 0 0
\(119\) −167.953 −0.129380
\(120\) 0 0
\(121\) 2295.61 1.72473
\(122\) 0 0
\(123\) 781.257 0.572712
\(124\) 0 0
\(125\) −688.191 −0.492429
\(126\) 0 0
\(127\) 1179.19 0.823907 0.411954 0.911205i \(-0.364847\pi\)
0.411954 + 0.911205i \(0.364847\pi\)
\(128\) 0 0
\(129\) −606.463 −0.413923
\(130\) 0 0
\(131\) 2519.21 1.68019 0.840093 0.542442i \(-0.182500\pi\)
0.840093 + 0.542442i \(0.182500\pi\)
\(132\) 0 0
\(133\) −338.239 −0.220519
\(134\) 0 0
\(135\) 2159.16 1.37652
\(136\) 0 0
\(137\) 519.136 0.323743 0.161872 0.986812i \(-0.448247\pi\)
0.161872 + 0.986812i \(0.448247\pi\)
\(138\) 0 0
\(139\) 1381.62 0.843075 0.421537 0.906811i \(-0.361491\pi\)
0.421537 + 0.906811i \(0.361491\pi\)
\(140\) 0 0
\(141\) 1409.28 0.841723
\(142\) 0 0
\(143\) −3084.34 −1.80368
\(144\) 0 0
\(145\) 2786.31 1.59579
\(146\) 0 0
\(147\) 985.716 0.553064
\(148\) 0 0
\(149\) 1914.19 1.05246 0.526230 0.850343i \(-0.323605\pi\)
0.526230 + 0.850343i \(0.323605\pi\)
\(150\) 0 0
\(151\) 1591.12 0.857506 0.428753 0.903422i \(-0.358953\pi\)
0.428753 + 0.903422i \(0.358953\pi\)
\(152\) 0 0
\(153\) 184.700 0.0975954
\(154\) 0 0
\(155\) 2590.54 1.34243
\(156\) 0 0
\(157\) 1017.70 0.517331 0.258665 0.965967i \(-0.416717\pi\)
0.258665 + 0.965967i \(0.416717\pi\)
\(158\) 0 0
\(159\) 289.028 0.144160
\(160\) 0 0
\(161\) −393.877 −0.192807
\(162\) 0 0
\(163\) −3366.72 −1.61780 −0.808900 0.587946i \(-0.799937\pi\)
−0.808900 + 0.587946i \(0.799937\pi\)
\(164\) 0 0
\(165\) 3434.00 1.62022
\(166\) 0 0
\(167\) 2328.20 1.07881 0.539405 0.842046i \(-0.318649\pi\)
0.539405 + 0.842046i \(0.318649\pi\)
\(168\) 0 0
\(169\) 426.162 0.193974
\(170\) 0 0
\(171\) 371.965 0.166344
\(172\) 0 0
\(173\) 2292.81 1.00762 0.503812 0.863813i \(-0.331930\pi\)
0.503812 + 0.863813i \(0.331930\pi\)
\(174\) 0 0
\(175\) 756.003 0.326562
\(176\) 0 0
\(177\) −571.785 −0.242814
\(178\) 0 0
\(179\) −1836.72 −0.766943 −0.383472 0.923553i \(-0.625272\pi\)
−0.383472 + 0.923553i \(0.625272\pi\)
\(180\) 0 0
\(181\) 2746.26 1.12778 0.563889 0.825850i \(-0.309304\pi\)
0.563889 + 0.825850i \(0.309304\pi\)
\(182\) 0 0
\(183\) −2452.30 −0.990596
\(184\) 0 0
\(185\) 3628.68 1.44208
\(186\) 0 0
\(187\) 1023.76 0.400347
\(188\) 0 0
\(189\) 1502.67 0.578322
\(190\) 0 0
\(191\) −3701.09 −1.40210 −0.701051 0.713111i \(-0.747286\pi\)
−0.701051 + 0.713111i \(0.747286\pi\)
\(192\) 0 0
\(193\) 1389.59 0.518262 0.259131 0.965842i \(-0.416564\pi\)
0.259131 + 0.965842i \(0.416564\pi\)
\(194\) 0 0
\(195\) −2920.53 −1.07253
\(196\) 0 0
\(197\) 3404.70 1.23134 0.615672 0.788003i \(-0.288885\pi\)
0.615672 + 0.788003i \(0.288885\pi\)
\(198\) 0 0
\(199\) 4282.21 1.52542 0.762708 0.646744i \(-0.223870\pi\)
0.762708 + 0.646744i \(0.223870\pi\)
\(200\) 0 0
\(201\) −1480.40 −0.519499
\(202\) 0 0
\(203\) 1939.13 0.670445
\(204\) 0 0
\(205\) −2761.00 −0.940665
\(206\) 0 0
\(207\) 433.151 0.145440
\(208\) 0 0
\(209\) 2061.74 0.682363
\(210\) 0 0
\(211\) −18.1576 −0.00592428 −0.00296214 0.999996i \(-0.500943\pi\)
−0.00296214 + 0.999996i \(0.500943\pi\)
\(212\) 0 0
\(213\) 2852.72 0.917677
\(214\) 0 0
\(215\) 2143.27 0.679859
\(216\) 0 0
\(217\) 1802.89 0.563999
\(218\) 0 0
\(219\) −4080.02 −1.25891
\(220\) 0 0
\(221\) −870.686 −0.265017
\(222\) 0 0
\(223\) 4945.40 1.48506 0.742531 0.669812i \(-0.233625\pi\)
0.742531 + 0.669812i \(0.233625\pi\)
\(224\) 0 0
\(225\) −831.384 −0.246336
\(226\) 0 0
\(227\) −6179.32 −1.80677 −0.903383 0.428835i \(-0.858924\pi\)
−0.903383 + 0.428835i \(0.858924\pi\)
\(228\) 0 0
\(229\) −6838.27 −1.97330 −0.986650 0.162856i \(-0.947929\pi\)
−0.986650 + 0.162856i \(0.947929\pi\)
\(230\) 0 0
\(231\) 2389.89 0.680707
\(232\) 0 0
\(233\) 6168.82 1.73447 0.867237 0.497895i \(-0.165893\pi\)
0.867237 + 0.497895i \(0.165893\pi\)
\(234\) 0 0
\(235\) −4980.47 −1.38251
\(236\) 0 0
\(237\) −1503.49 −0.412077
\(238\) 0 0
\(239\) −632.850 −0.171279 −0.0856394 0.996326i \(-0.527293\pi\)
−0.0856394 + 0.996326i \(0.527293\pi\)
\(240\) 0 0
\(241\) 410.736 0.109783 0.0548917 0.998492i \(-0.482519\pi\)
0.0548917 + 0.998492i \(0.482519\pi\)
\(242\) 0 0
\(243\) −2830.84 −0.747318
\(244\) 0 0
\(245\) −3483.57 −0.908395
\(246\) 0 0
\(247\) −1753.46 −0.451701
\(248\) 0 0
\(249\) 3107.90 0.790985
\(250\) 0 0
\(251\) 3392.00 0.852993 0.426497 0.904489i \(-0.359748\pi\)
0.426497 + 0.904489i \(0.359748\pi\)
\(252\) 0 0
\(253\) 2400.89 0.596611
\(254\) 0 0
\(255\) 969.389 0.238061
\(256\) 0 0
\(257\) −4213.03 −1.02257 −0.511287 0.859410i \(-0.670831\pi\)
−0.511287 + 0.859410i \(0.670831\pi\)
\(258\) 0 0
\(259\) 2525.38 0.605866
\(260\) 0 0
\(261\) −2132.48 −0.505738
\(262\) 0 0
\(263\) 1282.19 0.300620 0.150310 0.988639i \(-0.451973\pi\)
0.150310 + 0.988639i \(0.451973\pi\)
\(264\) 0 0
\(265\) −1021.44 −0.236779
\(266\) 0 0
\(267\) −4229.10 −0.969351
\(268\) 0 0
\(269\) 5951.38 1.34893 0.674465 0.738307i \(-0.264374\pi\)
0.674465 + 0.738307i \(0.264374\pi\)
\(270\) 0 0
\(271\) 6709.65 1.50399 0.751997 0.659167i \(-0.229091\pi\)
0.751997 + 0.659167i \(0.229091\pi\)
\(272\) 0 0
\(273\) −2032.55 −0.450605
\(274\) 0 0
\(275\) −4608.23 −1.01050
\(276\) 0 0
\(277\) −3369.11 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(278\) 0 0
\(279\) −1982.65 −0.425442
\(280\) 0 0
\(281\) −1964.14 −0.416978 −0.208489 0.978025i \(-0.566855\pi\)
−0.208489 + 0.978025i \(0.566855\pi\)
\(282\) 0 0
\(283\) −116.659 −0.0245042 −0.0122521 0.999925i \(-0.503900\pi\)
−0.0122521 + 0.999925i \(0.503900\pi\)
\(284\) 0 0
\(285\) 1952.24 0.405757
\(286\) 0 0
\(287\) −1921.52 −0.395204
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) 6412.27 1.29173
\(292\) 0 0
\(293\) −3801.24 −0.757922 −0.378961 0.925413i \(-0.623718\pi\)
−0.378961 + 0.925413i \(0.623718\pi\)
\(294\) 0 0
\(295\) 2020.72 0.398816
\(296\) 0 0
\(297\) −9159.53 −1.78953
\(298\) 0 0
\(299\) −2041.90 −0.394936
\(300\) 0 0
\(301\) 1491.61 0.285631
\(302\) 0 0
\(303\) −5443.88 −1.03215
\(304\) 0 0
\(305\) 8666.54 1.62703
\(306\) 0 0
\(307\) 4487.26 0.834207 0.417103 0.908859i \(-0.363045\pi\)
0.417103 + 0.908859i \(0.363045\pi\)
\(308\) 0 0
\(309\) −4158.38 −0.765573
\(310\) 0 0
\(311\) 4722.00 0.860965 0.430482 0.902599i \(-0.358343\pi\)
0.430482 + 0.902599i \(0.358343\pi\)
\(312\) 0 0
\(313\) −980.994 −0.177154 −0.0885768 0.996069i \(-0.528232\pi\)
−0.0885768 + 0.996069i \(0.528232\pi\)
\(314\) 0 0
\(315\) −1523.76 −0.272554
\(316\) 0 0
\(317\) 3171.02 0.561837 0.280919 0.959732i \(-0.409361\pi\)
0.280919 + 0.959732i \(0.409361\pi\)
\(318\) 0 0
\(319\) −11820.0 −2.07459
\(320\) 0 0
\(321\) 621.174 0.108008
\(322\) 0 0
\(323\) 582.014 0.100260
\(324\) 0 0
\(325\) 3919.19 0.668916
\(326\) 0 0
\(327\) −1718.84 −0.290680
\(328\) 0 0
\(329\) −3466.16 −0.580838
\(330\) 0 0
\(331\) 1643.10 0.272850 0.136425 0.990650i \(-0.456439\pi\)
0.136425 + 0.990650i \(0.456439\pi\)
\(332\) 0 0
\(333\) −2777.19 −0.457024
\(334\) 0 0
\(335\) 5231.80 0.853264
\(336\) 0 0
\(337\) −9936.95 −1.60623 −0.803116 0.595822i \(-0.796826\pi\)
−0.803116 + 0.595822i \(0.796826\pi\)
\(338\) 0 0
\(339\) 3103.29 0.497191
\(340\) 0 0
\(341\) −10989.5 −1.74521
\(342\) 0 0
\(343\) −5813.09 −0.915094
\(344\) 0 0
\(345\) 2273.37 0.354766
\(346\) 0 0
\(347\) 11337.7 1.75401 0.877004 0.480484i \(-0.159539\pi\)
0.877004 + 0.480484i \(0.159539\pi\)
\(348\) 0 0
\(349\) 627.001 0.0961678 0.0480839 0.998843i \(-0.484689\pi\)
0.0480839 + 0.998843i \(0.484689\pi\)
\(350\) 0 0
\(351\) 7789.97 1.18461
\(352\) 0 0
\(353\) 5362.12 0.808490 0.404245 0.914651i \(-0.367534\pi\)
0.404245 + 0.914651i \(0.367534\pi\)
\(354\) 0 0
\(355\) −10081.6 −1.50726
\(356\) 0 0
\(357\) 674.647 0.100017
\(358\) 0 0
\(359\) −5063.48 −0.744402 −0.372201 0.928152i \(-0.621397\pi\)
−0.372201 + 0.928152i \(0.621397\pi\)
\(360\) 0 0
\(361\) −5686.89 −0.829113
\(362\) 0 0
\(363\) −9221.18 −1.33330
\(364\) 0 0
\(365\) 14419.0 2.06774
\(366\) 0 0
\(367\) 77.7642 0.0110607 0.00553033 0.999985i \(-0.498240\pi\)
0.00553033 + 0.999985i \(0.498240\pi\)
\(368\) 0 0
\(369\) 2113.11 0.298115
\(370\) 0 0
\(371\) −710.871 −0.0994786
\(372\) 0 0
\(373\) −2917.41 −0.404981 −0.202491 0.979284i \(-0.564904\pi\)
−0.202491 + 0.979284i \(0.564904\pi\)
\(374\) 0 0
\(375\) 2764.38 0.380672
\(376\) 0 0
\(377\) 10052.7 1.37331
\(378\) 0 0
\(379\) −13918.1 −1.88634 −0.943169 0.332313i \(-0.892171\pi\)
−0.943169 + 0.332313i \(0.892171\pi\)
\(380\) 0 0
\(381\) −4736.66 −0.636920
\(382\) 0 0
\(383\) 2374.14 0.316744 0.158372 0.987380i \(-0.449376\pi\)
0.158372 + 0.987380i \(0.449376\pi\)
\(384\) 0 0
\(385\) −8445.99 −1.11804
\(386\) 0 0
\(387\) −1640.34 −0.215460
\(388\) 0 0
\(389\) −3839.71 −0.500465 −0.250232 0.968186i \(-0.580507\pi\)
−0.250232 + 0.968186i \(0.580507\pi\)
\(390\) 0 0
\(391\) 677.751 0.0876608
\(392\) 0 0
\(393\) −10119.4 −1.29887
\(394\) 0 0
\(395\) 5313.41 0.676827
\(396\) 0 0
\(397\) −9335.29 −1.18016 −0.590082 0.807344i \(-0.700904\pi\)
−0.590082 + 0.807344i \(0.700904\pi\)
\(398\) 0 0
\(399\) 1358.66 0.170472
\(400\) 0 0
\(401\) −3648.62 −0.454373 −0.227187 0.973851i \(-0.572953\pi\)
−0.227187 + 0.973851i \(0.572953\pi\)
\(402\) 0 0
\(403\) 9346.34 1.15527
\(404\) 0 0
\(405\) −4508.76 −0.553190
\(406\) 0 0
\(407\) −15393.5 −1.87476
\(408\) 0 0
\(409\) 3796.95 0.459040 0.229520 0.973304i \(-0.426284\pi\)
0.229520 + 0.973304i \(0.426284\pi\)
\(410\) 0 0
\(411\) −2085.31 −0.250269
\(412\) 0 0
\(413\) 1406.32 0.167555
\(414\) 0 0
\(415\) −10983.5 −1.29917
\(416\) 0 0
\(417\) −5549.79 −0.651737
\(418\) 0 0
\(419\) 7116.97 0.829801 0.414901 0.909867i \(-0.363816\pi\)
0.414901 + 0.909867i \(0.363816\pi\)
\(420\) 0 0
\(421\) −4495.19 −0.520385 −0.260192 0.965557i \(-0.583786\pi\)
−0.260192 + 0.965557i \(0.583786\pi\)
\(422\) 0 0
\(423\) 3811.77 0.438144
\(424\) 0 0
\(425\) −1300.87 −0.148474
\(426\) 0 0
\(427\) 6031.48 0.683569
\(428\) 0 0
\(429\) 12389.4 1.39433
\(430\) 0 0
\(431\) −2891.80 −0.323186 −0.161593 0.986857i \(-0.551663\pi\)
−0.161593 + 0.986857i \(0.551663\pi\)
\(432\) 0 0
\(433\) −14298.6 −1.58694 −0.793471 0.608608i \(-0.791728\pi\)
−0.793471 + 0.608608i \(0.791728\pi\)
\(434\) 0 0
\(435\) −11192.3 −1.23363
\(436\) 0 0
\(437\) 1364.92 0.149411
\(438\) 0 0
\(439\) 4699.43 0.510914 0.255457 0.966820i \(-0.417774\pi\)
0.255457 + 0.966820i \(0.417774\pi\)
\(440\) 0 0
\(441\) 2666.13 0.287888
\(442\) 0 0
\(443\) 2195.37 0.235451 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(444\) 0 0
\(445\) 14945.8 1.59214
\(446\) 0 0
\(447\) −7689.06 −0.813602
\(448\) 0 0
\(449\) 1843.65 0.193780 0.0968902 0.995295i \(-0.469110\pi\)
0.0968902 + 0.995295i \(0.469110\pi\)
\(450\) 0 0
\(451\) 11712.7 1.22290
\(452\) 0 0
\(453\) −6391.33 −0.662894
\(454\) 0 0
\(455\) 7183.11 0.740109
\(456\) 0 0
\(457\) 5769.52 0.590562 0.295281 0.955410i \(-0.404587\pi\)
0.295281 + 0.955410i \(0.404587\pi\)
\(458\) 0 0
\(459\) −2585.66 −0.262938
\(460\) 0 0
\(461\) −7335.80 −0.741133 −0.370567 0.928806i \(-0.620836\pi\)
−0.370567 + 0.928806i \(0.620836\pi\)
\(462\) 0 0
\(463\) −2659.54 −0.266953 −0.133477 0.991052i \(-0.542614\pi\)
−0.133477 + 0.991052i \(0.542614\pi\)
\(464\) 0 0
\(465\) −10405.9 −1.03776
\(466\) 0 0
\(467\) 5657.03 0.560549 0.280274 0.959920i \(-0.409575\pi\)
0.280274 + 0.959920i \(0.409575\pi\)
\(468\) 0 0
\(469\) 3641.07 0.358484
\(470\) 0 0
\(471\) −4087.96 −0.399922
\(472\) 0 0
\(473\) −9092.14 −0.883842
\(474\) 0 0
\(475\) −2619.80 −0.253063
\(476\) 0 0
\(477\) 781.753 0.0750398
\(478\) 0 0
\(479\) 1924.95 0.183618 0.0918090 0.995777i \(-0.470735\pi\)
0.0918090 + 0.995777i \(0.470735\pi\)
\(480\) 0 0
\(481\) 13091.8 1.24103
\(482\) 0 0
\(483\) 1582.16 0.149049
\(484\) 0 0
\(485\) −22661.2 −2.12164
\(486\) 0 0
\(487\) −9457.20 −0.879973 −0.439986 0.898004i \(-0.645017\pi\)
−0.439986 + 0.898004i \(0.645017\pi\)
\(488\) 0 0
\(489\) 13523.7 1.25064
\(490\) 0 0
\(491\) 17148.2 1.57615 0.788074 0.615580i \(-0.211078\pi\)
0.788074 + 0.615580i \(0.211078\pi\)
\(492\) 0 0
\(493\) −3336.70 −0.304822
\(494\) 0 0
\(495\) 9288.14 0.843376
\(496\) 0 0
\(497\) −7016.33 −0.633250
\(498\) 0 0
\(499\) 19203.5 1.72278 0.861389 0.507946i \(-0.169595\pi\)
0.861389 + 0.507946i \(0.169595\pi\)
\(500\) 0 0
\(501\) −9352.08 −0.833973
\(502\) 0 0
\(503\) 7853.63 0.696175 0.348087 0.937462i \(-0.386831\pi\)
0.348087 + 0.937462i \(0.386831\pi\)
\(504\) 0 0
\(505\) 19238.9 1.69529
\(506\) 0 0
\(507\) −1711.84 −0.149952
\(508\) 0 0
\(509\) 6822.18 0.594082 0.297041 0.954865i \(-0.404000\pi\)
0.297041 + 0.954865i \(0.404000\pi\)
\(510\) 0 0
\(511\) 10034.9 0.868724
\(512\) 0 0
\(513\) −5207.24 −0.448158
\(514\) 0 0
\(515\) 14695.9 1.25744
\(516\) 0 0
\(517\) 21128.1 1.79731
\(518\) 0 0
\(519\) −9209.93 −0.778942
\(520\) 0 0
\(521\) −6072.25 −0.510614 −0.255307 0.966860i \(-0.582177\pi\)
−0.255307 + 0.966860i \(0.582177\pi\)
\(522\) 0 0
\(523\) 13825.4 1.15591 0.577955 0.816069i \(-0.303851\pi\)
0.577955 + 0.816069i \(0.303851\pi\)
\(524\) 0 0
\(525\) −3036.77 −0.252449
\(526\) 0 0
\(527\) −3102.26 −0.256426
\(528\) 0 0
\(529\) −10577.6 −0.869365
\(530\) 0 0
\(531\) −1546.54 −0.126392
\(532\) 0 0
\(533\) −9961.34 −0.809519
\(534\) 0 0
\(535\) −2195.26 −0.177401
\(536\) 0 0
\(537\) 7377.87 0.592884
\(538\) 0 0
\(539\) 14777.9 1.18095
\(540\) 0 0
\(541\) −17020.5 −1.35262 −0.676312 0.736615i \(-0.736423\pi\)
−0.676312 + 0.736615i \(0.736423\pi\)
\(542\) 0 0
\(543\) −11031.4 −0.871827
\(544\) 0 0
\(545\) 6074.47 0.477435
\(546\) 0 0
\(547\) −21180.6 −1.65560 −0.827802 0.561020i \(-0.810409\pi\)
−0.827802 + 0.561020i \(0.810409\pi\)
\(548\) 0 0
\(549\) −6632.89 −0.515637
\(550\) 0 0
\(551\) −6719.74 −0.519547
\(552\) 0 0
\(553\) 3697.87 0.284357
\(554\) 0 0
\(555\) −14575.9 −1.11480
\(556\) 0 0
\(557\) −12123.8 −0.922263 −0.461131 0.887332i \(-0.652556\pi\)
−0.461131 + 0.887332i \(0.652556\pi\)
\(558\) 0 0
\(559\) 7732.65 0.585074
\(560\) 0 0
\(561\) −4112.33 −0.309488
\(562\) 0 0
\(563\) 3995.69 0.299108 0.149554 0.988754i \(-0.452216\pi\)
0.149554 + 0.988754i \(0.452216\pi\)
\(564\) 0 0
\(565\) −10967.2 −0.816624
\(566\) 0 0
\(567\) −3137.87 −0.232413
\(568\) 0 0
\(569\) −19168.8 −1.41230 −0.706148 0.708064i \(-0.749569\pi\)
−0.706148 + 0.708064i \(0.749569\pi\)
\(570\) 0 0
\(571\) 19127.7 1.40187 0.700937 0.713223i \(-0.252765\pi\)
0.700937 + 0.713223i \(0.252765\pi\)
\(572\) 0 0
\(573\) 14866.8 1.08389
\(574\) 0 0
\(575\) −3050.74 −0.221260
\(576\) 0 0
\(577\) 11272.7 0.813328 0.406664 0.913578i \(-0.366692\pi\)
0.406664 + 0.913578i \(0.366692\pi\)
\(578\) 0 0
\(579\) −5581.80 −0.400642
\(580\) 0 0
\(581\) −7643.95 −0.545825
\(582\) 0 0
\(583\) 4333.13 0.307822
\(584\) 0 0
\(585\) −7899.34 −0.558287
\(586\) 0 0
\(587\) 22029.2 1.54896 0.774482 0.632596i \(-0.218011\pi\)
0.774482 + 0.632596i \(0.218011\pi\)
\(588\) 0 0
\(589\) −6247.60 −0.437059
\(590\) 0 0
\(591\) −13676.3 −0.951888
\(592\) 0 0
\(593\) −20575.4 −1.42484 −0.712421 0.701753i \(-0.752401\pi\)
−0.712421 + 0.701753i \(0.752401\pi\)
\(594\) 0 0
\(595\) −2384.23 −0.164276
\(596\) 0 0
\(597\) −17201.1 −1.17922
\(598\) 0 0
\(599\) −21007.3 −1.43295 −0.716473 0.697615i \(-0.754245\pi\)
−0.716473 + 0.697615i \(0.754245\pi\)
\(600\) 0 0
\(601\) 20499.1 1.39131 0.695653 0.718378i \(-0.255115\pi\)
0.695653 + 0.718378i \(0.255115\pi\)
\(602\) 0 0
\(603\) −4004.13 −0.270416
\(604\) 0 0
\(605\) 32588.1 2.18991
\(606\) 0 0
\(607\) 2031.26 0.135826 0.0679131 0.997691i \(-0.478366\pi\)
0.0679131 + 0.997691i \(0.478366\pi\)
\(608\) 0 0
\(609\) −7789.25 −0.518287
\(610\) 0 0
\(611\) −17968.9 −1.18976
\(612\) 0 0
\(613\) −7817.91 −0.515110 −0.257555 0.966264i \(-0.582917\pi\)
−0.257555 + 0.966264i \(0.582917\pi\)
\(614\) 0 0
\(615\) 11090.6 0.727180
\(616\) 0 0
\(617\) −18880.6 −1.23194 −0.615968 0.787771i \(-0.711235\pi\)
−0.615968 + 0.787771i \(0.711235\pi\)
\(618\) 0 0
\(619\) −16860.2 −1.09478 −0.547389 0.836879i \(-0.684378\pi\)
−0.547389 + 0.836879i \(0.684378\pi\)
\(620\) 0 0
\(621\) −6063.80 −0.391839
\(622\) 0 0
\(623\) 10401.6 0.668908
\(624\) 0 0
\(625\) −19334.6 −1.23742
\(626\) 0 0
\(627\) −8281.77 −0.527499
\(628\) 0 0
\(629\) −4345.47 −0.275461
\(630\) 0 0
\(631\) 5789.26 0.365241 0.182620 0.983184i \(-0.441542\pi\)
0.182620 + 0.983184i \(0.441542\pi\)
\(632\) 0 0
\(633\) 72.9370 0.00457976
\(634\) 0 0
\(635\) 16739.6 1.04613
\(636\) 0 0
\(637\) −12568.3 −0.781748
\(638\) 0 0
\(639\) 7715.93 0.477680
\(640\) 0 0
\(641\) −4747.96 −0.292563 −0.146282 0.989243i \(-0.546731\pi\)
−0.146282 + 0.989243i \(0.546731\pi\)
\(642\) 0 0
\(643\) −24608.1 −1.50925 −0.754625 0.656157i \(-0.772181\pi\)
−0.754625 + 0.656157i \(0.772181\pi\)
\(644\) 0 0
\(645\) −8609.25 −0.525564
\(646\) 0 0
\(647\) 15126.4 0.919134 0.459567 0.888143i \(-0.348005\pi\)
0.459567 + 0.888143i \(0.348005\pi\)
\(648\) 0 0
\(649\) −8572.25 −0.518475
\(650\) 0 0
\(651\) −7241.97 −0.435999
\(652\) 0 0
\(653\) 17608.3 1.05523 0.527615 0.849484i \(-0.323086\pi\)
0.527615 + 0.849484i \(0.323086\pi\)
\(654\) 0 0
\(655\) 35762.3 2.13336
\(656\) 0 0
\(657\) −11035.5 −0.655305
\(658\) 0 0
\(659\) 4983.53 0.294584 0.147292 0.989093i \(-0.452944\pi\)
0.147292 + 0.989093i \(0.452944\pi\)
\(660\) 0 0
\(661\) 12178.7 0.716636 0.358318 0.933600i \(-0.383350\pi\)
0.358318 + 0.933600i \(0.383350\pi\)
\(662\) 0 0
\(663\) 3497.44 0.204871
\(664\) 0 0
\(665\) −4801.58 −0.279996
\(666\) 0 0
\(667\) −7825.10 −0.454256
\(668\) 0 0
\(669\) −19865.1 −1.14802
\(670\) 0 0
\(671\) −36765.1 −2.11520
\(672\) 0 0
\(673\) −13407.2 −0.767917 −0.383959 0.923350i \(-0.625439\pi\)
−0.383959 + 0.923350i \(0.625439\pi\)
\(674\) 0 0
\(675\) 11638.8 0.663669
\(676\) 0 0
\(677\) −2154.14 −0.122290 −0.0611450 0.998129i \(-0.519475\pi\)
−0.0611450 + 0.998129i \(0.519475\pi\)
\(678\) 0 0
\(679\) −15771.1 −0.891369
\(680\) 0 0
\(681\) 24821.6 1.39672
\(682\) 0 0
\(683\) −27829.8 −1.55912 −0.779558 0.626330i \(-0.784556\pi\)
−0.779558 + 0.626330i \(0.784556\pi\)
\(684\) 0 0
\(685\) 7369.57 0.411061
\(686\) 0 0
\(687\) 27468.5 1.52546
\(688\) 0 0
\(689\) −3685.23 −0.203768
\(690\) 0 0
\(691\) −2940.02 −0.161857 −0.0809287 0.996720i \(-0.525789\pi\)
−0.0809287 + 0.996720i \(0.525789\pi\)
\(692\) 0 0
\(693\) 6464.09 0.354330
\(694\) 0 0
\(695\) 19613.2 1.07046
\(696\) 0 0
\(697\) 3306.39 0.179682
\(698\) 0 0
\(699\) −24779.4 −1.34083
\(700\) 0 0
\(701\) −31363.2 −1.68983 −0.844915 0.534900i \(-0.820349\pi\)
−0.844915 + 0.534900i \(0.820349\pi\)
\(702\) 0 0
\(703\) −8751.28 −0.469503
\(704\) 0 0
\(705\) 20005.9 1.06875
\(706\) 0 0
\(707\) 13389.3 0.712246
\(708\) 0 0
\(709\) 29484.9 1.56182 0.780910 0.624644i \(-0.214756\pi\)
0.780910 + 0.624644i \(0.214756\pi\)
\(710\) 0 0
\(711\) −4066.59 −0.214499
\(712\) 0 0
\(713\) −7275.29 −0.382134
\(714\) 0 0
\(715\) −43784.8 −2.29015
\(716\) 0 0
\(717\) 2542.08 0.132407
\(718\) 0 0
\(719\) −2019.45 −0.104746 −0.0523732 0.998628i \(-0.516679\pi\)
−0.0523732 + 0.998628i \(0.516679\pi\)
\(720\) 0 0
\(721\) 10227.6 0.528290
\(722\) 0 0
\(723\) −1649.87 −0.0848679
\(724\) 0 0
\(725\) 15019.4 0.769388
\(726\) 0 0
\(727\) 15239.9 0.777465 0.388733 0.921351i \(-0.372913\pi\)
0.388733 + 0.921351i \(0.372913\pi\)
\(728\) 0 0
\(729\) 19946.6 1.01339
\(730\) 0 0
\(731\) −2566.64 −0.129864
\(732\) 0 0
\(733\) 21343.6 1.07550 0.537752 0.843103i \(-0.319274\pi\)
0.537752 + 0.843103i \(0.319274\pi\)
\(734\) 0 0
\(735\) 13993.1 0.702233
\(736\) 0 0
\(737\) −22194.2 −1.10927
\(738\) 0 0
\(739\) 10706.6 0.532949 0.266475 0.963842i \(-0.414141\pi\)
0.266475 + 0.963842i \(0.414141\pi\)
\(740\) 0 0
\(741\) 7043.45 0.349187
\(742\) 0 0
\(743\) −29819.2 −1.47235 −0.736177 0.676789i \(-0.763371\pi\)
−0.736177 + 0.676789i \(0.763371\pi\)
\(744\) 0 0
\(745\) 27173.5 1.33632
\(746\) 0 0
\(747\) 8406.14 0.411733
\(748\) 0 0
\(749\) −1527.79 −0.0745318
\(750\) 0 0
\(751\) 17903.8 0.869934 0.434967 0.900447i \(-0.356760\pi\)
0.434967 + 0.900447i \(0.356760\pi\)
\(752\) 0 0
\(753\) −13625.3 −0.659405
\(754\) 0 0
\(755\) 22587.3 1.08879
\(756\) 0 0
\(757\) 3664.20 0.175928 0.0879641 0.996124i \(-0.471964\pi\)
0.0879641 + 0.996124i \(0.471964\pi\)
\(758\) 0 0
\(759\) −9644.07 −0.461209
\(760\) 0 0
\(761\) −603.032 −0.0287252 −0.0143626 0.999897i \(-0.504572\pi\)
−0.0143626 + 0.999897i \(0.504572\pi\)
\(762\) 0 0
\(763\) 4227.53 0.200586
\(764\) 0 0
\(765\) 2621.97 0.123918
\(766\) 0 0
\(767\) 7290.49 0.343213
\(768\) 0 0
\(769\) −12780.9 −0.599338 −0.299669 0.954043i \(-0.596876\pi\)
−0.299669 + 0.954043i \(0.596876\pi\)
\(770\) 0 0
\(771\) 16923.2 0.790499
\(772\) 0 0
\(773\) −15919.8 −0.740743 −0.370372 0.928884i \(-0.620770\pi\)
−0.370372 + 0.928884i \(0.620770\pi\)
\(774\) 0 0
\(775\) 13964.1 0.647233
\(776\) 0 0
\(777\) −10144.1 −0.468364
\(778\) 0 0
\(779\) 6658.70 0.306255
\(780\) 0 0
\(781\) 42768.2 1.95950
\(782\) 0 0
\(783\) 29853.2 1.36254
\(784\) 0 0
\(785\) 14447.0 0.656862
\(786\) 0 0
\(787\) 14004.2 0.634300 0.317150 0.948375i \(-0.397274\pi\)
0.317150 + 0.948375i \(0.397274\pi\)
\(788\) 0 0
\(789\) −5150.38 −0.232393
\(790\) 0 0
\(791\) −7632.61 −0.343090
\(792\) 0 0
\(793\) 31267.8 1.40019
\(794\) 0 0
\(795\) 4102.99 0.183042
\(796\) 0 0
\(797\) 35890.2 1.59510 0.797551 0.603252i \(-0.206129\pi\)
0.797551 + 0.603252i \(0.206129\pi\)
\(798\) 0 0
\(799\) 5964.28 0.264082
\(800\) 0 0
\(801\) −11438.7 −0.504578
\(802\) 0 0
\(803\) −61168.0 −2.68814
\(804\) 0 0
\(805\) −5591.41 −0.244809
\(806\) 0 0
\(807\) −23906.0 −1.04279
\(808\) 0 0
\(809\) 3780.80 0.164309 0.0821544 0.996620i \(-0.473820\pi\)
0.0821544 + 0.996620i \(0.473820\pi\)
\(810\) 0 0
\(811\) −31627.6 −1.36941 −0.684707 0.728818i \(-0.740070\pi\)
−0.684707 + 0.728818i \(0.740070\pi\)
\(812\) 0 0
\(813\) −26951.8 −1.16266
\(814\) 0 0
\(815\) −47793.3 −2.05414
\(816\) 0 0
\(817\) −5168.92 −0.221344
\(818\) 0 0
\(819\) −5497.55 −0.234554
\(820\) 0 0
\(821\) −36041.9 −1.53212 −0.766060 0.642769i \(-0.777786\pi\)
−0.766060 + 0.642769i \(0.777786\pi\)
\(822\) 0 0
\(823\) 5132.22 0.217373 0.108686 0.994076i \(-0.465336\pi\)
0.108686 + 0.994076i \(0.465336\pi\)
\(824\) 0 0
\(825\) 18510.7 0.781164
\(826\) 0 0
\(827\) −43411.1 −1.82533 −0.912667 0.408703i \(-0.865981\pi\)
−0.912667 + 0.408703i \(0.865981\pi\)
\(828\) 0 0
\(829\) 19608.0 0.821488 0.410744 0.911751i \(-0.365269\pi\)
0.410744 + 0.911751i \(0.365269\pi\)
\(830\) 0 0
\(831\) 13533.3 0.564939
\(832\) 0 0
\(833\) 4171.69 0.173518
\(834\) 0 0
\(835\) 33050.7 1.36978
\(836\) 0 0
\(837\) 27755.7 1.14621
\(838\) 0 0
\(839\) −39957.8 −1.64422 −0.822109 0.569331i \(-0.807202\pi\)
−0.822109 + 0.569331i \(0.807202\pi\)
\(840\) 0 0
\(841\) 14135.4 0.579583
\(842\) 0 0
\(843\) 7889.72 0.322344
\(844\) 0 0
\(845\) 6049.72 0.246292
\(846\) 0 0
\(847\) 22679.7 0.920051
\(848\) 0 0
\(849\) 468.607 0.0189429
\(850\) 0 0
\(851\) −10190.8 −0.410501
\(852\) 0 0
\(853\) 15291.9 0.613814 0.306907 0.951739i \(-0.400706\pi\)
0.306907 + 0.951739i \(0.400706\pi\)
\(854\) 0 0
\(855\) 5280.35 0.211210
\(856\) 0 0
\(857\) 46263.3 1.84402 0.922010 0.387166i \(-0.126546\pi\)
0.922010 + 0.387166i \(0.126546\pi\)
\(858\) 0 0
\(859\) −38492.2 −1.52891 −0.764457 0.644675i \(-0.776993\pi\)
−0.764457 + 0.644675i \(0.776993\pi\)
\(860\) 0 0
\(861\) 7718.50 0.305512
\(862\) 0 0
\(863\) −33188.8 −1.30911 −0.654553 0.756016i \(-0.727143\pi\)
−0.654553 + 0.756016i \(0.727143\pi\)
\(864\) 0 0
\(865\) 32548.3 1.27939
\(866\) 0 0
\(867\) −1160.88 −0.0454734
\(868\) 0 0
\(869\) −22540.5 −0.879900
\(870\) 0 0
\(871\) 18875.7 0.734303
\(872\) 0 0
\(873\) 17343.7 0.672387
\(874\) 0 0
\(875\) −6799.05 −0.262686
\(876\) 0 0
\(877\) −8448.30 −0.325289 −0.162645 0.986685i \(-0.552002\pi\)
−0.162645 + 0.986685i \(0.552002\pi\)
\(878\) 0 0
\(879\) 15269.1 0.585910
\(880\) 0 0
\(881\) −11721.8 −0.448262 −0.224131 0.974559i \(-0.571954\pi\)
−0.224131 + 0.974559i \(0.571954\pi\)
\(882\) 0 0
\(883\) −20348.6 −0.775519 −0.387760 0.921761i \(-0.626751\pi\)
−0.387760 + 0.921761i \(0.626751\pi\)
\(884\) 0 0
\(885\) −8116.96 −0.308304
\(886\) 0 0
\(887\) 41522.0 1.57178 0.785892 0.618363i \(-0.212204\pi\)
0.785892 + 0.618363i \(0.212204\pi\)
\(888\) 0 0
\(889\) 11649.9 0.439512
\(890\) 0 0
\(891\) 19127.0 0.719167
\(892\) 0 0
\(893\) 12011.4 0.450108
\(894\) 0 0
\(895\) −26073.8 −0.973798
\(896\) 0 0
\(897\) 8202.05 0.305305
\(898\) 0 0
\(899\) 35817.7 1.32879
\(900\) 0 0
\(901\) 1223.21 0.0452286
\(902\) 0 0
\(903\) −5991.61 −0.220806
\(904\) 0 0
\(905\) 38985.5 1.43196
\(906\) 0 0
\(907\) 41036.0 1.50229 0.751145 0.660138i \(-0.229502\pi\)
0.751145 + 0.660138i \(0.229502\pi\)
\(908\) 0 0
\(909\) −14724.4 −0.537269
\(910\) 0 0
\(911\) 6771.68 0.246274 0.123137 0.992390i \(-0.460705\pi\)
0.123137 + 0.992390i \(0.460705\pi\)
\(912\) 0 0
\(913\) 46593.9 1.68897
\(914\) 0 0
\(915\) −34812.4 −1.25777
\(916\) 0 0
\(917\) 24888.8 0.896292
\(918\) 0 0
\(919\) 3589.93 0.128858 0.0644292 0.997922i \(-0.479477\pi\)
0.0644292 + 0.997922i \(0.479477\pi\)
\(920\) 0 0
\(921\) −18024.8 −0.644882
\(922\) 0 0
\(923\) −36373.3 −1.29712
\(924\) 0 0
\(925\) 19560.1 0.695279
\(926\) 0 0
\(927\) −11247.4 −0.398505
\(928\) 0 0
\(929\) −38282.3 −1.35199 −0.675996 0.736906i \(-0.736286\pi\)
−0.675996 + 0.736906i \(0.736286\pi\)
\(930\) 0 0
\(931\) 8401.32 0.295749
\(932\) 0 0
\(933\) −18967.7 −0.665568
\(934\) 0 0
\(935\) 14533.2 0.508326
\(936\) 0 0
\(937\) −26099.5 −0.909961 −0.454980 0.890501i \(-0.650354\pi\)
−0.454980 + 0.890501i \(0.650354\pi\)
\(938\) 0 0
\(939\) 3940.53 0.136948
\(940\) 0 0
\(941\) −33481.9 −1.15991 −0.579956 0.814648i \(-0.696930\pi\)
−0.579956 + 0.814648i \(0.696930\pi\)
\(942\) 0 0
\(943\) 7754.02 0.267768
\(944\) 0 0
\(945\) 21331.6 0.734303
\(946\) 0 0
\(947\) 25045.8 0.859430 0.429715 0.902964i \(-0.358614\pi\)
0.429715 + 0.902964i \(0.358614\pi\)
\(948\) 0 0
\(949\) 52021.9 1.77946
\(950\) 0 0
\(951\) −12737.6 −0.434327
\(952\) 0 0
\(953\) −3465.20 −0.117785 −0.0588924 0.998264i \(-0.518757\pi\)
−0.0588924 + 0.998264i \(0.518757\pi\)
\(954\) 0 0
\(955\) −52540.1 −1.78027
\(956\) 0 0
\(957\) 47479.6 1.60376
\(958\) 0 0
\(959\) 5128.86 0.172700
\(960\) 0 0
\(961\) 3510.05 0.117822
\(962\) 0 0
\(963\) 1680.13 0.0562216
\(964\) 0 0
\(965\) 19726.3 0.658045
\(966\) 0 0
\(967\) 45361.5 1.50851 0.754254 0.656583i \(-0.227999\pi\)
0.754254 + 0.656583i \(0.227999\pi\)
\(968\) 0 0
\(969\) −2337.88 −0.0775061
\(970\) 0 0
\(971\) 50036.4 1.65370 0.826851 0.562421i \(-0.190130\pi\)
0.826851 + 0.562421i \(0.190130\pi\)
\(972\) 0 0
\(973\) 13649.8 0.449737
\(974\) 0 0
\(975\) −15742.9 −0.517104
\(976\) 0 0
\(977\) −6689.64 −0.219059 −0.109530 0.993984i \(-0.534934\pi\)
−0.109530 + 0.993984i \(0.534934\pi\)
\(978\) 0 0
\(979\) −63403.0 −2.06983
\(980\) 0 0
\(981\) −4649.06 −0.151308
\(982\) 0 0
\(983\) 31585.2 1.02483 0.512416 0.858737i \(-0.328751\pi\)
0.512416 + 0.858737i \(0.328751\pi\)
\(984\) 0 0
\(985\) 48332.5 1.56345
\(986\) 0 0
\(987\) 13923.1 0.449016
\(988\) 0 0
\(989\) −6019.18 −0.193528
\(990\) 0 0
\(991\) −30182.2 −0.967476 −0.483738 0.875213i \(-0.660721\pi\)
−0.483738 + 0.875213i \(0.660721\pi\)
\(992\) 0 0
\(993\) −6600.15 −0.210926
\(994\) 0 0
\(995\) 60789.5 1.93684
\(996\) 0 0
\(997\) 24670.2 0.783663 0.391832 0.920037i \(-0.371842\pi\)
0.391832 + 0.920037i \(0.371842\pi\)
\(998\) 0 0
\(999\) 38878.6 1.23130
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 1088.4.a.bh.1.2 7
4.3 odd 2 1088.4.a.bg.1.6 7
8.3 odd 2 544.4.a.k.1.2 7
8.5 even 2 544.4.a.l.1.6 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
544.4.a.k.1.2 7 8.3 odd 2
544.4.a.l.1.6 yes 7 8.5 even 2
1088.4.a.bg.1.6 7 4.3 odd 2
1088.4.a.bh.1.2 7 1.1 even 1 trivial