Properties

Label 608.4.a.k.1.1
Level $608$
Weight $4$
Character 608.1
Self dual yes
Analytic conductor $35.873$
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] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, 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("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.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: \(35.8731612835\)
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: \( x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.39349\) of defining polynomial
Character \(\chi\) \(=\) 608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.45545 q^{3} -11.5945 q^{5} +15.9032 q^{7} +62.4055 q^{9} -19.4189 q^{11} -69.4979 q^{13} +109.631 q^{15} -112.461 q^{17} -19.0000 q^{19} -150.372 q^{21} +57.3974 q^{23} +9.43209 q^{25} -334.775 q^{27} -59.8778 q^{29} -223.000 q^{31} +183.614 q^{33} -184.390 q^{35} -258.002 q^{37} +657.133 q^{39} -305.175 q^{41} +322.813 q^{43} -723.559 q^{45} -213.936 q^{47} -90.0873 q^{49} +1063.37 q^{51} +348.651 q^{53} +225.152 q^{55} +179.654 q^{57} -56.3816 q^{59} +87.2763 q^{61} +992.449 q^{63} +805.792 q^{65} +1074.10 q^{67} -542.718 q^{69} +725.349 q^{71} +889.389 q^{73} -89.1846 q^{75} -308.823 q^{77} -1258.90 q^{79} +1480.50 q^{81} +227.650 q^{83} +1303.93 q^{85} +566.171 q^{87} -927.529 q^{89} -1105.24 q^{91} +2108.56 q^{93} +220.295 q^{95} +603.489 q^{97} -1211.84 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) \) 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
\(3\) −9.45545 −1.81970 −0.909851 0.414935i \(-0.863804\pi\)
−0.909851 + 0.414935i \(0.863804\pi\)
\(4\) 0 0
\(5\) −11.5945 −1.03704 −0.518521 0.855065i \(-0.673517\pi\)
−0.518521 + 0.855065i \(0.673517\pi\)
\(6\) 0 0
\(7\) 15.9032 0.858694 0.429347 0.903140i \(-0.358744\pi\)
0.429347 + 0.903140i \(0.358744\pi\)
\(8\) 0 0
\(9\) 62.4055 2.31131
\(10\) 0 0
\(11\) −19.4189 −0.532274 −0.266137 0.963935i \(-0.585747\pi\)
−0.266137 + 0.963935i \(0.585747\pi\)
\(12\) 0 0
\(13\) −69.4979 −1.48271 −0.741355 0.671113i \(-0.765817\pi\)
−0.741355 + 0.671113i \(0.765817\pi\)
\(14\) 0 0
\(15\) 109.631 1.88711
\(16\) 0 0
\(17\) −112.461 −1.60446 −0.802230 0.597015i \(-0.796353\pi\)
−0.802230 + 0.597015i \(0.796353\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) −150.372 −1.56257
\(22\) 0 0
\(23\) 57.3974 0.520356 0.260178 0.965561i \(-0.416219\pi\)
0.260178 + 0.965561i \(0.416219\pi\)
\(24\) 0 0
\(25\) 9.43209 0.0754567
\(26\) 0 0
\(27\) −334.775 −2.38620
\(28\) 0 0
\(29\) −59.8778 −0.383415 −0.191707 0.981452i \(-0.561402\pi\)
−0.191707 + 0.981452i \(0.561402\pi\)
\(30\) 0 0
\(31\) −223.000 −1.29200 −0.645998 0.763339i \(-0.723559\pi\)
−0.645998 + 0.763339i \(0.723559\pi\)
\(32\) 0 0
\(33\) 183.614 0.968580
\(34\) 0 0
\(35\) −184.390 −0.890502
\(36\) 0 0
\(37\) −258.002 −1.14636 −0.573180 0.819429i \(-0.694291\pi\)
−0.573180 + 0.819429i \(0.694291\pi\)
\(38\) 0 0
\(39\) 657.133 2.69809
\(40\) 0 0
\(41\) −305.175 −1.16245 −0.581224 0.813743i \(-0.697426\pi\)
−0.581224 + 0.813743i \(0.697426\pi\)
\(42\) 0 0
\(43\) 322.813 1.14485 0.572424 0.819958i \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(44\) 0 0
\(45\) −723.559 −2.39693
\(46\) 0 0
\(47\) −213.936 −0.663952 −0.331976 0.943288i \(-0.607715\pi\)
−0.331976 + 0.943288i \(0.607715\pi\)
\(48\) 0 0
\(49\) −90.0873 −0.262645
\(50\) 0 0
\(51\) 1063.37 2.91964
\(52\) 0 0
\(53\) 348.651 0.903603 0.451802 0.892118i \(-0.350782\pi\)
0.451802 + 0.892118i \(0.350782\pi\)
\(54\) 0 0
\(55\) 225.152 0.551990
\(56\) 0 0
\(57\) 179.654 0.417468
\(58\) 0 0
\(59\) −56.3816 −0.124411 −0.0622056 0.998063i \(-0.519813\pi\)
−0.0622056 + 0.998063i \(0.519813\pi\)
\(60\) 0 0
\(61\) 87.2763 0.183190 0.0915950 0.995796i \(-0.470803\pi\)
0.0915950 + 0.995796i \(0.470803\pi\)
\(62\) 0 0
\(63\) 992.449 1.98471
\(64\) 0 0
\(65\) 805.792 1.53763
\(66\) 0 0
\(67\) 1074.10 1.95854 0.979272 0.202549i \(-0.0649224\pi\)
0.979272 + 0.202549i \(0.0649224\pi\)
\(68\) 0 0
\(69\) −542.718 −0.946892
\(70\) 0 0
\(71\) 725.349 1.21244 0.606219 0.795298i \(-0.292685\pi\)
0.606219 + 0.795298i \(0.292685\pi\)
\(72\) 0 0
\(73\) 889.389 1.42596 0.712980 0.701184i \(-0.247345\pi\)
0.712980 + 0.701184i \(0.247345\pi\)
\(74\) 0 0
\(75\) −89.1846 −0.137309
\(76\) 0 0
\(77\) −308.823 −0.457060
\(78\) 0 0
\(79\) −1258.90 −1.79288 −0.896439 0.443168i \(-0.853855\pi\)
−0.896439 + 0.443168i \(0.853855\pi\)
\(80\) 0 0
\(81\) 1480.50 2.03086
\(82\) 0 0
\(83\) 227.650 0.301058 0.150529 0.988606i \(-0.451902\pi\)
0.150529 + 0.988606i \(0.451902\pi\)
\(84\) 0 0
\(85\) 1303.93 1.66389
\(86\) 0 0
\(87\) 566.171 0.697700
\(88\) 0 0
\(89\) −927.529 −1.10470 −0.552348 0.833614i \(-0.686268\pi\)
−0.552348 + 0.833614i \(0.686268\pi\)
\(90\) 0 0
\(91\) −1105.24 −1.27319
\(92\) 0 0
\(93\) 2108.56 2.35105
\(94\) 0 0
\(95\) 220.295 0.237914
\(96\) 0 0
\(97\) 603.489 0.631701 0.315851 0.948809i \(-0.397710\pi\)
0.315851 + 0.948809i \(0.397710\pi\)
\(98\) 0 0
\(99\) −1211.84 −1.23025
\(100\) 0 0
\(101\) −1035.55 −1.02021 −0.510106 0.860112i \(-0.670394\pi\)
−0.510106 + 0.860112i \(0.670394\pi\)
\(102\) 0 0
\(103\) 666.961 0.638035 0.319018 0.947749i \(-0.396647\pi\)
0.319018 + 0.947749i \(0.396647\pi\)
\(104\) 0 0
\(105\) 1743.49 1.62045
\(106\) 0 0
\(107\) −1369.29 −1.23714 −0.618572 0.785728i \(-0.712289\pi\)
−0.618572 + 0.785728i \(0.712289\pi\)
\(108\) 0 0
\(109\) 1326.45 1.16560 0.582800 0.812615i \(-0.301957\pi\)
0.582800 + 0.812615i \(0.301957\pi\)
\(110\) 0 0
\(111\) 2439.53 2.08603
\(112\) 0 0
\(113\) −1204.31 −1.00258 −0.501291 0.865279i \(-0.667141\pi\)
−0.501291 + 0.865279i \(0.667141\pi\)
\(114\) 0 0
\(115\) −665.493 −0.539631
\(116\) 0 0
\(117\) −4337.05 −3.42701
\(118\) 0 0
\(119\) −1788.49 −1.37774
\(120\) 0 0
\(121\) −953.907 −0.716685
\(122\) 0 0
\(123\) 2885.57 2.11531
\(124\) 0 0
\(125\) 1339.95 0.958790
\(126\) 0 0
\(127\) 375.658 0.262475 0.131237 0.991351i \(-0.458105\pi\)
0.131237 + 0.991351i \(0.458105\pi\)
\(128\) 0 0
\(129\) −3052.34 −2.08328
\(130\) 0 0
\(131\) −2063.01 −1.37592 −0.687962 0.725747i \(-0.741495\pi\)
−0.687962 + 0.725747i \(0.741495\pi\)
\(132\) 0 0
\(133\) −302.161 −0.196998
\(134\) 0 0
\(135\) 3881.54 2.47459
\(136\) 0 0
\(137\) 1905.46 1.18828 0.594141 0.804361i \(-0.297492\pi\)
0.594141 + 0.804361i \(0.297492\pi\)
\(138\) 0 0
\(139\) 1869.18 1.14059 0.570293 0.821441i \(-0.306830\pi\)
0.570293 + 0.821441i \(0.306830\pi\)
\(140\) 0 0
\(141\) 2022.86 1.20819
\(142\) 0 0
\(143\) 1349.57 0.789208
\(144\) 0 0
\(145\) 694.252 0.397617
\(146\) 0 0
\(147\) 851.816 0.477936
\(148\) 0 0
\(149\) 2041.70 1.12257 0.561284 0.827623i \(-0.310308\pi\)
0.561284 + 0.827623i \(0.310308\pi\)
\(150\) 0 0
\(151\) 1970.09 1.06175 0.530874 0.847451i \(-0.321864\pi\)
0.530874 + 0.847451i \(0.321864\pi\)
\(152\) 0 0
\(153\) −7018.19 −3.70841
\(154\) 0 0
\(155\) 2585.56 1.33986
\(156\) 0 0
\(157\) 945.030 0.480392 0.240196 0.970724i \(-0.422788\pi\)
0.240196 + 0.970724i \(0.422788\pi\)
\(158\) 0 0
\(159\) −3296.66 −1.64429
\(160\) 0 0
\(161\) 912.803 0.446826
\(162\) 0 0
\(163\) −345.511 −0.166028 −0.0830139 0.996548i \(-0.526455\pi\)
−0.0830139 + 0.996548i \(0.526455\pi\)
\(164\) 0 0
\(165\) −2128.91 −1.00446
\(166\) 0 0
\(167\) 1554.01 0.720079 0.360040 0.932937i \(-0.382763\pi\)
0.360040 + 0.932937i \(0.382763\pi\)
\(168\) 0 0
\(169\) 2632.95 1.19843
\(170\) 0 0
\(171\) −1185.70 −0.530252
\(172\) 0 0
\(173\) −2065.34 −0.907658 −0.453829 0.891089i \(-0.649942\pi\)
−0.453829 + 0.891089i \(0.649942\pi\)
\(174\) 0 0
\(175\) 150.001 0.0647942
\(176\) 0 0
\(177\) 533.113 0.226391
\(178\) 0 0
\(179\) −3565.91 −1.48899 −0.744493 0.667631i \(-0.767309\pi\)
−0.744493 + 0.667631i \(0.767309\pi\)
\(180\) 0 0
\(181\) 1620.30 0.665393 0.332697 0.943034i \(-0.392042\pi\)
0.332697 + 0.943034i \(0.392042\pi\)
\(182\) 0 0
\(183\) −825.237 −0.333351
\(184\) 0 0
\(185\) 2991.41 1.18882
\(186\) 0 0
\(187\) 2183.87 0.854012
\(188\) 0 0
\(189\) −5324.00 −2.04902
\(190\) 0 0
\(191\) −2138.73 −0.810226 −0.405113 0.914267i \(-0.632768\pi\)
−0.405113 + 0.914267i \(0.632768\pi\)
\(192\) 0 0
\(193\) −3704.25 −1.38154 −0.690772 0.723073i \(-0.742729\pi\)
−0.690772 + 0.723073i \(0.742729\pi\)
\(194\) 0 0
\(195\) −7619.12 −2.79803
\(196\) 0 0
\(197\) −520.563 −0.188267 −0.0941334 0.995560i \(-0.530008\pi\)
−0.0941334 + 0.995560i \(0.530008\pi\)
\(198\) 0 0
\(199\) −67.7611 −0.0241380 −0.0120690 0.999927i \(-0.503842\pi\)
−0.0120690 + 0.999927i \(0.503842\pi\)
\(200\) 0 0
\(201\) −10156.1 −3.56397
\(202\) 0 0
\(203\) −952.250 −0.329236
\(204\) 0 0
\(205\) 3538.35 1.20551
\(206\) 0 0
\(207\) 3581.91 1.20271
\(208\) 0 0
\(209\) 368.959 0.122112
\(210\) 0 0
\(211\) −860.566 −0.280776 −0.140388 0.990097i \(-0.544835\pi\)
−0.140388 + 0.990097i \(0.544835\pi\)
\(212\) 0 0
\(213\) −6858.50 −2.20628
\(214\) 0 0
\(215\) −3742.85 −1.18726
\(216\) 0 0
\(217\) −3546.41 −1.10943
\(218\) 0 0
\(219\) −8409.57 −2.59482
\(220\) 0 0
\(221\) 7815.80 2.37895
\(222\) 0 0
\(223\) 6297.10 1.89096 0.945482 0.325674i \(-0.105591\pi\)
0.945482 + 0.325674i \(0.105591\pi\)
\(224\) 0 0
\(225\) 588.614 0.174404
\(226\) 0 0
\(227\) 10.8392 0.00316928 0.00158464 0.999999i \(-0.499496\pi\)
0.00158464 + 0.999999i \(0.499496\pi\)
\(228\) 0 0
\(229\) 4424.48 1.27676 0.638379 0.769722i \(-0.279605\pi\)
0.638379 + 0.769722i \(0.279605\pi\)
\(230\) 0 0
\(231\) 2920.06 0.831713
\(232\) 0 0
\(233\) −2410.23 −0.677681 −0.338840 0.940844i \(-0.610035\pi\)
−0.338840 + 0.940844i \(0.610035\pi\)
\(234\) 0 0
\(235\) 2480.47 0.688546
\(236\) 0 0
\(237\) 11903.5 3.26250
\(238\) 0 0
\(239\) 2833.55 0.766893 0.383446 0.923563i \(-0.374737\pi\)
0.383446 + 0.923563i \(0.374737\pi\)
\(240\) 0 0
\(241\) 3255.23 0.870074 0.435037 0.900413i \(-0.356735\pi\)
0.435037 + 0.900413i \(0.356735\pi\)
\(242\) 0 0
\(243\) −4959.84 −1.30936
\(244\) 0 0
\(245\) 1044.52 0.272374
\(246\) 0 0
\(247\) 1320.46 0.340157
\(248\) 0 0
\(249\) −2152.53 −0.547836
\(250\) 0 0
\(251\) −4125.96 −1.03756 −0.518782 0.854907i \(-0.673614\pi\)
−0.518782 + 0.854907i \(0.673614\pi\)
\(252\) 0 0
\(253\) −1114.59 −0.276972
\(254\) 0 0
\(255\) −12329.2 −3.02779
\(256\) 0 0
\(257\) 6627.29 1.60856 0.804278 0.594253i \(-0.202552\pi\)
0.804278 + 0.594253i \(0.202552\pi\)
\(258\) 0 0
\(259\) −4103.07 −0.984372
\(260\) 0 0
\(261\) −3736.70 −0.886192
\(262\) 0 0
\(263\) −823.636 −0.193109 −0.0965543 0.995328i \(-0.530782\pi\)
−0.0965543 + 0.995328i \(0.530782\pi\)
\(264\) 0 0
\(265\) −4042.43 −0.937075
\(266\) 0 0
\(267\) 8770.20 2.01022
\(268\) 0 0
\(269\) −5760.24 −1.30561 −0.652803 0.757527i \(-0.726407\pi\)
−0.652803 + 0.757527i \(0.726407\pi\)
\(270\) 0 0
\(271\) 8413.28 1.88587 0.942934 0.332980i \(-0.108054\pi\)
0.942934 + 0.332980i \(0.108054\pi\)
\(272\) 0 0
\(273\) 10450.5 2.31683
\(274\) 0 0
\(275\) −183.161 −0.0401636
\(276\) 0 0
\(277\) −4635.95 −1.00558 −0.502792 0.864407i \(-0.667694\pi\)
−0.502792 + 0.864407i \(0.667694\pi\)
\(278\) 0 0
\(279\) −13916.4 −2.98621
\(280\) 0 0
\(281\) −4108.20 −0.872153 −0.436076 0.899910i \(-0.643632\pi\)
−0.436076 + 0.899910i \(0.643632\pi\)
\(282\) 0 0
\(283\) 5079.96 1.06704 0.533519 0.845788i \(-0.320869\pi\)
0.533519 + 0.845788i \(0.320869\pi\)
\(284\) 0 0
\(285\) −2082.99 −0.432932
\(286\) 0 0
\(287\) −4853.27 −0.998187
\(288\) 0 0
\(289\) 7734.49 1.57429
\(290\) 0 0
\(291\) −5706.26 −1.14951
\(292\) 0 0
\(293\) 7988.86 1.59288 0.796441 0.604716i \(-0.206714\pi\)
0.796441 + 0.604716i \(0.206714\pi\)
\(294\) 0 0
\(295\) 653.716 0.129020
\(296\) 0 0
\(297\) 6500.95 1.27011
\(298\) 0 0
\(299\) −3988.99 −0.771537
\(300\) 0 0
\(301\) 5133.77 0.983074
\(302\) 0 0
\(303\) 9791.62 1.85648
\(304\) 0 0
\(305\) −1011.92 −0.189976
\(306\) 0 0
\(307\) −7003.80 −1.30205 −0.651023 0.759058i \(-0.725660\pi\)
−0.651023 + 0.759058i \(0.725660\pi\)
\(308\) 0 0
\(309\) −6306.42 −1.16103
\(310\) 0 0
\(311\) 2835.42 0.516983 0.258492 0.966013i \(-0.416775\pi\)
0.258492 + 0.966013i \(0.416775\pi\)
\(312\) 0 0
\(313\) 669.394 0.120883 0.0604415 0.998172i \(-0.480749\pi\)
0.0604415 + 0.998172i \(0.480749\pi\)
\(314\) 0 0
\(315\) −11506.9 −2.05823
\(316\) 0 0
\(317\) 5609.44 0.993872 0.496936 0.867787i \(-0.334458\pi\)
0.496936 + 0.867787i \(0.334458\pi\)
\(318\) 0 0
\(319\) 1162.76 0.204082
\(320\) 0 0
\(321\) 12947.3 2.25123
\(322\) 0 0
\(323\) 2136.76 0.368088
\(324\) 0 0
\(325\) −655.510 −0.111880
\(326\) 0 0
\(327\) −12542.1 −2.12105
\(328\) 0 0
\(329\) −3402.27 −0.570131
\(330\) 0 0
\(331\) −10139.4 −1.68373 −0.841864 0.539689i \(-0.818542\pi\)
−0.841864 + 0.539689i \(0.818542\pi\)
\(332\) 0 0
\(333\) −16100.8 −2.64960
\(334\) 0 0
\(335\) −12453.7 −2.03109
\(336\) 0 0
\(337\) 946.788 0.153041 0.0765205 0.997068i \(-0.475619\pi\)
0.0765205 + 0.997068i \(0.475619\pi\)
\(338\) 0 0
\(339\) 11387.3 1.82440
\(340\) 0 0
\(341\) 4330.40 0.687696
\(342\) 0 0
\(343\) −6887.49 −1.08423
\(344\) 0 0
\(345\) 6292.53 0.981967
\(346\) 0 0
\(347\) 6022.07 0.931648 0.465824 0.884878i \(-0.345758\pi\)
0.465824 + 0.884878i \(0.345758\pi\)
\(348\) 0 0
\(349\) 2271.70 0.348427 0.174214 0.984708i \(-0.444262\pi\)
0.174214 + 0.984708i \(0.444262\pi\)
\(350\) 0 0
\(351\) 23266.1 3.53805
\(352\) 0 0
\(353\) 3519.71 0.530695 0.265348 0.964153i \(-0.414513\pi\)
0.265348 + 0.964153i \(0.414513\pi\)
\(354\) 0 0
\(355\) −8410.05 −1.25735
\(356\) 0 0
\(357\) 16911.0 2.50707
\(358\) 0 0
\(359\) 3485.84 0.512467 0.256234 0.966615i \(-0.417518\pi\)
0.256234 + 0.966615i \(0.417518\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 9019.62 1.30415
\(364\) 0 0
\(365\) −10312.0 −1.47878
\(366\) 0 0
\(367\) 13100.4 1.86331 0.931654 0.363348i \(-0.118366\pi\)
0.931654 + 0.363348i \(0.118366\pi\)
\(368\) 0 0
\(369\) −19044.6 −2.68678
\(370\) 0 0
\(371\) 5544.68 0.775918
\(372\) 0 0
\(373\) −13692.7 −1.90075 −0.950377 0.311101i \(-0.899302\pi\)
−0.950377 + 0.311101i \(0.899302\pi\)
\(374\) 0 0
\(375\) −12669.8 −1.74471
\(376\) 0 0
\(377\) 4161.38 0.568493
\(378\) 0 0
\(379\) −3198.56 −0.433506 −0.216753 0.976226i \(-0.569547\pi\)
−0.216753 + 0.976226i \(0.569547\pi\)
\(380\) 0 0
\(381\) −3552.02 −0.477625
\(382\) 0 0
\(383\) −10724.5 −1.43080 −0.715400 0.698716i \(-0.753755\pi\)
−0.715400 + 0.698716i \(0.753755\pi\)
\(384\) 0 0
\(385\) 3580.64 0.473991
\(386\) 0 0
\(387\) 20145.3 2.64610
\(388\) 0 0
\(389\) −6806.29 −0.887128 −0.443564 0.896243i \(-0.646286\pi\)
−0.443564 + 0.896243i \(0.646286\pi\)
\(390\) 0 0
\(391\) −6454.97 −0.834889
\(392\) 0 0
\(393\) 19506.7 2.50377
\(394\) 0 0
\(395\) 14596.3 1.85929
\(396\) 0 0
\(397\) −6878.98 −0.869637 −0.434819 0.900518i \(-0.643188\pi\)
−0.434819 + 0.900518i \(0.643188\pi\)
\(398\) 0 0
\(399\) 2857.07 0.358477
\(400\) 0 0
\(401\) 10972.1 1.36639 0.683194 0.730237i \(-0.260590\pi\)
0.683194 + 0.730237i \(0.260590\pi\)
\(402\) 0 0
\(403\) 15498.0 1.91566
\(404\) 0 0
\(405\) −17165.6 −2.10609
\(406\) 0 0
\(407\) 5010.12 0.610178
\(408\) 0 0
\(409\) −13952.8 −1.68685 −0.843424 0.537249i \(-0.819464\pi\)
−0.843424 + 0.537249i \(0.819464\pi\)
\(410\) 0 0
\(411\) −18017.0 −2.16232
\(412\) 0 0
\(413\) −896.650 −0.106831
\(414\) 0 0
\(415\) −2639.48 −0.312210
\(416\) 0 0
\(417\) −17673.9 −2.07553
\(418\) 0 0
\(419\) 3901.77 0.454925 0.227463 0.973787i \(-0.426957\pi\)
0.227463 + 0.973787i \(0.426957\pi\)
\(420\) 0 0
\(421\) 5739.67 0.664452 0.332226 0.943200i \(-0.392200\pi\)
0.332226 + 0.943200i \(0.392200\pi\)
\(422\) 0 0
\(423\) −13350.8 −1.53460
\(424\) 0 0
\(425\) −1060.74 −0.121067
\(426\) 0 0
\(427\) 1387.98 0.157304
\(428\) 0 0
\(429\) −12760.8 −1.43612
\(430\) 0 0
\(431\) −4016.82 −0.448917 −0.224459 0.974484i \(-0.572061\pi\)
−0.224459 + 0.974484i \(0.572061\pi\)
\(432\) 0 0
\(433\) −10619.1 −1.17857 −0.589286 0.807925i \(-0.700591\pi\)
−0.589286 + 0.807925i \(0.700591\pi\)
\(434\) 0 0
\(435\) −6564.46 −0.723545
\(436\) 0 0
\(437\) −1090.55 −0.119378
\(438\) 0 0
\(439\) 1594.11 0.173309 0.0866545 0.996238i \(-0.472382\pi\)
0.0866545 + 0.996238i \(0.472382\pi\)
\(440\) 0 0
\(441\) −5621.94 −0.607056
\(442\) 0 0
\(443\) 3512.03 0.376662 0.188331 0.982106i \(-0.439692\pi\)
0.188331 + 0.982106i \(0.439692\pi\)
\(444\) 0 0
\(445\) 10754.2 1.14562
\(446\) 0 0
\(447\) −19305.2 −2.04274
\(448\) 0 0
\(449\) 3239.71 0.340515 0.170258 0.985400i \(-0.445540\pi\)
0.170258 + 0.985400i \(0.445540\pi\)
\(450\) 0 0
\(451\) 5926.16 0.618741
\(452\) 0 0
\(453\) −18628.1 −1.93206
\(454\) 0 0
\(455\) 12814.7 1.32036
\(456\) 0 0
\(457\) −7149.11 −0.731775 −0.365887 0.930659i \(-0.619234\pi\)
−0.365887 + 0.930659i \(0.619234\pi\)
\(458\) 0 0
\(459\) 37649.1 3.82856
\(460\) 0 0
\(461\) −1452.36 −0.146731 −0.0733657 0.997305i \(-0.523374\pi\)
−0.0733657 + 0.997305i \(0.523374\pi\)
\(462\) 0 0
\(463\) 4887.26 0.490562 0.245281 0.969452i \(-0.421120\pi\)
0.245281 + 0.969452i \(0.421120\pi\)
\(464\) 0 0
\(465\) −24447.7 −2.43814
\(466\) 0 0
\(467\) −15576.4 −1.54344 −0.771722 0.635960i \(-0.780604\pi\)
−0.771722 + 0.635960i \(0.780604\pi\)
\(468\) 0 0
\(469\) 17081.7 1.68179
\(470\) 0 0
\(471\) −8935.68 −0.874171
\(472\) 0 0
\(473\) −6268.66 −0.609373
\(474\) 0 0
\(475\) −179.210 −0.0173110
\(476\) 0 0
\(477\) 21757.8 2.08851
\(478\) 0 0
\(479\) −9155.87 −0.873366 −0.436683 0.899616i \(-0.643847\pi\)
−0.436683 + 0.899616i \(0.643847\pi\)
\(480\) 0 0
\(481\) 17930.6 1.69972
\(482\) 0 0
\(483\) −8630.97 −0.813090
\(484\) 0 0
\(485\) −6997.14 −0.655101
\(486\) 0 0
\(487\) −4855.86 −0.451827 −0.225914 0.974147i \(-0.572537\pi\)
−0.225914 + 0.974147i \(0.572537\pi\)
\(488\) 0 0
\(489\) 3266.96 0.302121
\(490\) 0 0
\(491\) 13319.0 1.22419 0.612094 0.790785i \(-0.290327\pi\)
0.612094 + 0.790785i \(0.290327\pi\)
\(492\) 0 0
\(493\) 6733.92 0.615173
\(494\) 0 0
\(495\) 14050.7 1.27582
\(496\) 0 0
\(497\) 11535.4 1.04111
\(498\) 0 0
\(499\) −9859.86 −0.884545 −0.442273 0.896881i \(-0.645828\pi\)
−0.442273 + 0.896881i \(0.645828\pi\)
\(500\) 0 0
\(501\) −14693.9 −1.31033
\(502\) 0 0
\(503\) 9994.26 0.885929 0.442964 0.896539i \(-0.353927\pi\)
0.442964 + 0.896539i \(0.353927\pi\)
\(504\) 0 0
\(505\) 12006.7 1.05800
\(506\) 0 0
\(507\) −24895.8 −2.18079
\(508\) 0 0
\(509\) 9489.87 0.826387 0.413194 0.910643i \(-0.364413\pi\)
0.413194 + 0.910643i \(0.364413\pi\)
\(510\) 0 0
\(511\) 14144.2 1.22446
\(512\) 0 0
\(513\) 6360.72 0.547432
\(514\) 0 0
\(515\) −7733.07 −0.661670
\(516\) 0 0
\(517\) 4154.39 0.353404
\(518\) 0 0
\(519\) 19528.7 1.65167
\(520\) 0 0
\(521\) −14621.5 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(522\) 0 0
\(523\) 7500.48 0.627099 0.313550 0.949572i \(-0.398482\pi\)
0.313550 + 0.949572i \(0.398482\pi\)
\(524\) 0 0
\(525\) −1418.32 −0.117906
\(526\) 0 0
\(527\) 25078.8 2.07296
\(528\) 0 0
\(529\) −8872.54 −0.729230
\(530\) 0 0
\(531\) −3518.52 −0.287553
\(532\) 0 0
\(533\) 21209.0 1.72358
\(534\) 0 0
\(535\) 15876.2 1.28297
\(536\) 0 0
\(537\) 33717.2 2.70951
\(538\) 0 0
\(539\) 1749.39 0.139799
\(540\) 0 0
\(541\) −2413.87 −0.191831 −0.0959153 0.995390i \(-0.530578\pi\)
−0.0959153 + 0.995390i \(0.530578\pi\)
\(542\) 0 0
\(543\) −15320.7 −1.21082
\(544\) 0 0
\(545\) −15379.5 −1.20878
\(546\) 0 0
\(547\) 21120.6 1.65092 0.825458 0.564464i \(-0.190917\pi\)
0.825458 + 0.564464i \(0.190917\pi\)
\(548\) 0 0
\(549\) 5446.52 0.423410
\(550\) 0 0
\(551\) 1137.68 0.0879614
\(552\) 0 0
\(553\) −20020.6 −1.53953
\(554\) 0 0
\(555\) −28285.1 −2.16331
\(556\) 0 0
\(557\) 16059.4 1.22165 0.610823 0.791767i \(-0.290839\pi\)
0.610823 + 0.791767i \(0.290839\pi\)
\(558\) 0 0
\(559\) −22434.8 −1.69748
\(560\) 0 0
\(561\) −20649.4 −1.55405
\(562\) 0 0
\(563\) −19306.8 −1.44527 −0.722634 0.691231i \(-0.757069\pi\)
−0.722634 + 0.691231i \(0.757069\pi\)
\(564\) 0 0
\(565\) 13963.3 1.03972
\(566\) 0 0
\(567\) 23544.7 1.74389
\(568\) 0 0
\(569\) −11379.0 −0.838373 −0.419187 0.907900i \(-0.637685\pi\)
−0.419187 + 0.907900i \(0.637685\pi\)
\(570\) 0 0
\(571\) 16788.6 1.23044 0.615219 0.788356i \(-0.289068\pi\)
0.615219 + 0.788356i \(0.289068\pi\)
\(572\) 0 0
\(573\) 20222.7 1.47437
\(574\) 0 0
\(575\) 541.377 0.0392643
\(576\) 0 0
\(577\) 4084.31 0.294683 0.147342 0.989086i \(-0.452928\pi\)
0.147342 + 0.989086i \(0.452928\pi\)
\(578\) 0 0
\(579\) 35025.4 2.51400
\(580\) 0 0
\(581\) 3620.37 0.258516
\(582\) 0 0
\(583\) −6770.42 −0.480964
\(584\) 0 0
\(585\) 50285.8 3.55396
\(586\) 0 0
\(587\) −10687.2 −0.751458 −0.375729 0.926730i \(-0.622608\pi\)
−0.375729 + 0.926730i \(0.622608\pi\)
\(588\) 0 0
\(589\) 4236.99 0.296404
\(590\) 0 0
\(591\) 4922.15 0.342590
\(592\) 0 0
\(593\) −18697.2 −1.29477 −0.647387 0.762162i \(-0.724138\pi\)
−0.647387 + 0.762162i \(0.724138\pi\)
\(594\) 0 0
\(595\) 20736.7 1.42877
\(596\) 0 0
\(597\) 640.712 0.0439239
\(598\) 0 0
\(599\) −3455.17 −0.235683 −0.117842 0.993032i \(-0.537598\pi\)
−0.117842 + 0.993032i \(0.537598\pi\)
\(600\) 0 0
\(601\) −15030.6 −1.02015 −0.510075 0.860130i \(-0.670382\pi\)
−0.510075 + 0.860130i \(0.670382\pi\)
\(602\) 0 0
\(603\) 67029.9 4.52681
\(604\) 0 0
\(605\) 11060.1 0.743232
\(606\) 0 0
\(607\) 2229.47 0.149080 0.0745398 0.997218i \(-0.476251\pi\)
0.0745398 + 0.997218i \(0.476251\pi\)
\(608\) 0 0
\(609\) 9003.95 0.599111
\(610\) 0 0
\(611\) 14868.1 0.984448
\(612\) 0 0
\(613\) 14821.0 0.976534 0.488267 0.872694i \(-0.337629\pi\)
0.488267 + 0.872694i \(0.337629\pi\)
\(614\) 0 0
\(615\) −33456.7 −2.19367
\(616\) 0 0
\(617\) −19301.7 −1.25941 −0.629706 0.776833i \(-0.716825\pi\)
−0.629706 + 0.776833i \(0.716825\pi\)
\(618\) 0 0
\(619\) 20550.4 1.33440 0.667198 0.744880i \(-0.267493\pi\)
0.667198 + 0.744880i \(0.267493\pi\)
\(620\) 0 0
\(621\) −19215.2 −1.24167
\(622\) 0 0
\(623\) −14750.7 −0.948595
\(624\) 0 0
\(625\) −16715.0 −1.06976
\(626\) 0 0
\(627\) −3488.67 −0.222207
\(628\) 0 0
\(629\) 29015.2 1.83929
\(630\) 0 0
\(631\) −6019.78 −0.379784 −0.189892 0.981805i \(-0.560814\pi\)
−0.189892 + 0.981805i \(0.560814\pi\)
\(632\) 0 0
\(633\) 8137.04 0.510929
\(634\) 0 0
\(635\) −4355.56 −0.272197
\(636\) 0 0
\(637\) 6260.88 0.389427
\(638\) 0 0
\(639\) 45265.8 2.80233
\(640\) 0 0
\(641\) 554.836 0.0341883 0.0170941 0.999854i \(-0.494559\pi\)
0.0170941 + 0.999854i \(0.494559\pi\)
\(642\) 0 0
\(643\) 25135.5 1.54160 0.770800 0.637077i \(-0.219857\pi\)
0.770800 + 0.637077i \(0.219857\pi\)
\(644\) 0 0
\(645\) 35390.3 2.16045
\(646\) 0 0
\(647\) 17081.4 1.03792 0.518962 0.854797i \(-0.326318\pi\)
0.518962 + 0.854797i \(0.326318\pi\)
\(648\) 0 0
\(649\) 1094.87 0.0662208
\(650\) 0 0
\(651\) 33532.9 2.01883
\(652\) 0 0
\(653\) −17375.8 −1.04130 −0.520649 0.853771i \(-0.674310\pi\)
−0.520649 + 0.853771i \(0.674310\pi\)
\(654\) 0 0
\(655\) 23919.5 1.42689
\(656\) 0 0
\(657\) 55502.8 3.29584
\(658\) 0 0
\(659\) −14944.2 −0.883377 −0.441688 0.897169i \(-0.645620\pi\)
−0.441688 + 0.897169i \(0.645620\pi\)
\(660\) 0 0
\(661\) −2829.77 −0.166513 −0.0832565 0.996528i \(-0.526532\pi\)
−0.0832565 + 0.996528i \(0.526532\pi\)
\(662\) 0 0
\(663\) −73901.9 −4.32898
\(664\) 0 0
\(665\) 3503.41 0.204295
\(666\) 0 0
\(667\) −3436.83 −0.199512
\(668\) 0 0
\(669\) −59541.9 −3.44099
\(670\) 0 0
\(671\) −1694.81 −0.0975072
\(672\) 0 0
\(673\) −18325.9 −1.04965 −0.524823 0.851211i \(-0.675869\pi\)
−0.524823 + 0.851211i \(0.675869\pi\)
\(674\) 0 0
\(675\) −3157.62 −0.180055
\(676\) 0 0
\(677\) 12887.9 0.731645 0.365823 0.930685i \(-0.380788\pi\)
0.365823 + 0.930685i \(0.380788\pi\)
\(678\) 0 0
\(679\) 9597.42 0.542438
\(680\) 0 0
\(681\) −102.490 −0.00576714
\(682\) 0 0
\(683\) −21490.8 −1.20399 −0.601993 0.798501i \(-0.705626\pi\)
−0.601993 + 0.798501i \(0.705626\pi\)
\(684\) 0 0
\(685\) −22092.8 −1.23230
\(686\) 0 0
\(687\) −41835.4 −2.32332
\(688\) 0 0
\(689\) −24230.5 −1.33978
\(690\) 0 0
\(691\) −7795.43 −0.429164 −0.214582 0.976706i \(-0.568839\pi\)
−0.214582 + 0.976706i \(0.568839\pi\)
\(692\) 0 0
\(693\) −19272.2 −1.05641
\(694\) 0 0
\(695\) −21672.1 −1.18284
\(696\) 0 0
\(697\) 34320.3 1.86510
\(698\) 0 0
\(699\) 22789.8 1.23318
\(700\) 0 0
\(701\) −4462.67 −0.240446 −0.120223 0.992747i \(-0.538361\pi\)
−0.120223 + 0.992747i \(0.538361\pi\)
\(702\) 0 0
\(703\) 4902.05 0.262993
\(704\) 0 0
\(705\) −23454.0 −1.25295
\(706\) 0 0
\(707\) −16468.6 −0.876050
\(708\) 0 0
\(709\) −27093.8 −1.43516 −0.717581 0.696475i \(-0.754751\pi\)
−0.717581 + 0.696475i \(0.754751\pi\)
\(710\) 0 0
\(711\) −78562.3 −4.14390
\(712\) 0 0
\(713\) −12799.6 −0.672298
\(714\) 0 0
\(715\) −15647.6 −0.818442
\(716\) 0 0
\(717\) −26792.5 −1.39552
\(718\) 0 0
\(719\) −5743.87 −0.297928 −0.148964 0.988843i \(-0.547594\pi\)
−0.148964 + 0.988843i \(0.547594\pi\)
\(720\) 0 0
\(721\) 10606.8 0.547877
\(722\) 0 0
\(723\) −30779.6 −1.58327
\(724\) 0 0
\(725\) −564.772 −0.0289312
\(726\) 0 0
\(727\) −33173.9 −1.69237 −0.846185 0.532889i \(-0.821106\pi\)
−0.846185 + 0.532889i \(0.821106\pi\)
\(728\) 0 0
\(729\) 6924.11 0.351781
\(730\) 0 0
\(731\) −36303.9 −1.83686
\(732\) 0 0
\(733\) 9406.08 0.473972 0.236986 0.971513i \(-0.423840\pi\)
0.236986 + 0.971513i \(0.423840\pi\)
\(734\) 0 0
\(735\) −9876.37 −0.495640
\(736\) 0 0
\(737\) −20857.9 −1.04248
\(738\) 0 0
\(739\) 11407.9 0.567858 0.283929 0.958845i \(-0.408362\pi\)
0.283929 + 0.958845i \(0.408362\pi\)
\(740\) 0 0
\(741\) −12485.5 −0.618985
\(742\) 0 0
\(743\) 25469.4 1.25758 0.628790 0.777575i \(-0.283551\pi\)
0.628790 + 0.777575i \(0.283551\pi\)
\(744\) 0 0
\(745\) −23672.5 −1.16415
\(746\) 0 0
\(747\) 14206.6 0.695839
\(748\) 0 0
\(749\) −21776.2 −1.06233
\(750\) 0 0
\(751\) 23819.5 1.15737 0.578686 0.815550i \(-0.303566\pi\)
0.578686 + 0.815550i \(0.303566\pi\)
\(752\) 0 0
\(753\) 39012.8 1.88806
\(754\) 0 0
\(755\) −22842.2 −1.10108
\(756\) 0 0
\(757\) 6611.66 0.317444 0.158722 0.987323i \(-0.449263\pi\)
0.158722 + 0.987323i \(0.449263\pi\)
\(758\) 0 0
\(759\) 10539.0 0.504006
\(760\) 0 0
\(761\) 29643.4 1.41205 0.706026 0.708185i \(-0.250486\pi\)
0.706026 + 0.708185i \(0.250486\pi\)
\(762\) 0 0
\(763\) 21094.8 1.00089
\(764\) 0 0
\(765\) 81372.3 3.84578
\(766\) 0 0
\(767\) 3918.40 0.184466
\(768\) 0 0
\(769\) −6329.72 −0.296821 −0.148411 0.988926i \(-0.547416\pi\)
−0.148411 + 0.988926i \(0.547416\pi\)
\(770\) 0 0
\(771\) −62664.0 −2.92709
\(772\) 0 0
\(773\) −24504.8 −1.14020 −0.570101 0.821575i \(-0.693096\pi\)
−0.570101 + 0.821575i \(0.693096\pi\)
\(774\) 0 0
\(775\) −2103.35 −0.0974898
\(776\) 0 0
\(777\) 38796.4 1.79126
\(778\) 0 0
\(779\) 5798.33 0.266684
\(780\) 0 0
\(781\) −14085.5 −0.645349
\(782\) 0 0
\(783\) 20045.6 0.914904
\(784\) 0 0
\(785\) −10957.1 −0.498187
\(786\) 0 0
\(787\) 33851.3 1.53325 0.766627 0.642093i \(-0.221934\pi\)
0.766627 + 0.642093i \(0.221934\pi\)
\(788\) 0 0
\(789\) 7787.84 0.351400
\(790\) 0 0
\(791\) −19152.4 −0.860910
\(792\) 0 0
\(793\) −6065.52 −0.271618
\(794\) 0 0
\(795\) 38223.0 1.70520
\(796\) 0 0
\(797\) −18282.7 −0.812556 −0.406278 0.913750i \(-0.633174\pi\)
−0.406278 + 0.913750i \(0.633174\pi\)
\(798\) 0 0
\(799\) 24059.4 1.06528
\(800\) 0 0
\(801\) −57882.9 −2.55330
\(802\) 0 0
\(803\) −17270.9 −0.759002
\(804\) 0 0
\(805\) −10583.5 −0.463377
\(806\) 0 0
\(807\) 54465.7 2.37582
\(808\) 0 0
\(809\) −12733.6 −0.553386 −0.276693 0.960958i \(-0.589238\pi\)
−0.276693 + 0.960958i \(0.589238\pi\)
\(810\) 0 0
\(811\) 10126.7 0.438469 0.219234 0.975672i \(-0.429644\pi\)
0.219234 + 0.975672i \(0.429644\pi\)
\(812\) 0 0
\(813\) −79551.3 −3.43172
\(814\) 0 0
\(815\) 4006.03 0.172178
\(816\) 0 0
\(817\) −6133.44 −0.262646
\(818\) 0 0
\(819\) −68973.1 −2.94275
\(820\) 0 0
\(821\) −1944.98 −0.0826802 −0.0413401 0.999145i \(-0.513163\pi\)
−0.0413401 + 0.999145i \(0.513163\pi\)
\(822\) 0 0
\(823\) 31712.7 1.34318 0.671589 0.740924i \(-0.265612\pi\)
0.671589 + 0.740924i \(0.265612\pi\)
\(824\) 0 0
\(825\) 1731.86 0.0730858
\(826\) 0 0
\(827\) −3728.06 −0.156756 −0.0783781 0.996924i \(-0.524974\pi\)
−0.0783781 + 0.996924i \(0.524974\pi\)
\(828\) 0 0
\(829\) 13200.4 0.553037 0.276518 0.961009i \(-0.410819\pi\)
0.276518 + 0.961009i \(0.410819\pi\)
\(830\) 0 0
\(831\) 43834.9 1.82986
\(832\) 0 0
\(833\) 10131.3 0.421404
\(834\) 0 0
\(835\) −18018.0 −0.746753
\(836\) 0 0
\(837\) 74654.6 3.08296
\(838\) 0 0
\(839\) −11445.0 −0.470949 −0.235475 0.971881i \(-0.575664\pi\)
−0.235475 + 0.971881i \(0.575664\pi\)
\(840\) 0 0
\(841\) −20803.7 −0.852993
\(842\) 0 0
\(843\) 38844.9 1.58706
\(844\) 0 0
\(845\) −30527.7 −1.24282
\(846\) 0 0
\(847\) −15170.2 −0.615413
\(848\) 0 0
\(849\) −48033.2 −1.94169
\(850\) 0 0
\(851\) −14808.7 −0.596515
\(852\) 0 0
\(853\) −8701.48 −0.349276 −0.174638 0.984633i \(-0.555876\pi\)
−0.174638 + 0.984633i \(0.555876\pi\)
\(854\) 0 0
\(855\) 13747.6 0.549894
\(856\) 0 0
\(857\) −720.765 −0.0287291 −0.0143646 0.999897i \(-0.504573\pi\)
−0.0143646 + 0.999897i \(0.504573\pi\)
\(858\) 0 0
\(859\) 4678.63 0.185836 0.0929179 0.995674i \(-0.470381\pi\)
0.0929179 + 0.995674i \(0.470381\pi\)
\(860\) 0 0
\(861\) 45889.9 1.81640
\(862\) 0 0
\(863\) 7009.92 0.276501 0.138250 0.990397i \(-0.455852\pi\)
0.138250 + 0.990397i \(0.455852\pi\)
\(864\) 0 0
\(865\) 23946.5 0.941280
\(866\) 0 0
\(867\) −73133.0 −2.86474
\(868\) 0 0
\(869\) 24446.4 0.954302
\(870\) 0 0
\(871\) −74647.8 −2.90396
\(872\) 0 0
\(873\) 37661.0 1.46006
\(874\) 0 0
\(875\) 21309.5 0.823307
\(876\) 0 0
\(877\) −22737.0 −0.875453 −0.437727 0.899108i \(-0.644216\pi\)
−0.437727 + 0.899108i \(0.644216\pi\)
\(878\) 0 0
\(879\) −75538.3 −2.89857
\(880\) 0 0
\(881\) 7697.02 0.294346 0.147173 0.989111i \(-0.452983\pi\)
0.147173 + 0.989111i \(0.452983\pi\)
\(882\) 0 0
\(883\) −7842.91 −0.298907 −0.149454 0.988769i \(-0.547751\pi\)
−0.149454 + 0.988769i \(0.547751\pi\)
\(884\) 0 0
\(885\) −6181.18 −0.234777
\(886\) 0 0
\(887\) 41079.7 1.55504 0.777521 0.628857i \(-0.216477\pi\)
0.777521 + 0.628857i \(0.216477\pi\)
\(888\) 0 0
\(889\) 5974.18 0.225385
\(890\) 0 0
\(891\) −28749.6 −1.08097
\(892\) 0 0
\(893\) 4064.78 0.152321
\(894\) 0 0
\(895\) 41344.9 1.54414
\(896\) 0 0
\(897\) 37717.7 1.40397
\(898\) 0 0
\(899\) 13352.7 0.495370
\(900\) 0 0
\(901\) −39209.7 −1.44979
\(902\) 0 0
\(903\) −48542.0 −1.78890
\(904\) 0 0
\(905\) −18786.6 −0.690041
\(906\) 0 0
\(907\) −38639.4 −1.41455 −0.707277 0.706937i \(-0.750076\pi\)
−0.707277 + 0.706937i \(0.750076\pi\)
\(908\) 0 0
\(909\) −64624.2 −2.35803
\(910\) 0 0
\(911\) 18398.0 0.669103 0.334551 0.942377i \(-0.391415\pi\)
0.334551 + 0.942377i \(0.391415\pi\)
\(912\) 0 0
\(913\) −4420.70 −0.160245
\(914\) 0 0
\(915\) 9568.19 0.345699
\(916\) 0 0
\(917\) −32808.5 −1.18150
\(918\) 0 0
\(919\) 24502.3 0.879495 0.439748 0.898121i \(-0.355068\pi\)
0.439748 + 0.898121i \(0.355068\pi\)
\(920\) 0 0
\(921\) 66224.1 2.36934
\(922\) 0 0
\(923\) −50410.2 −1.79770
\(924\) 0 0
\(925\) −2433.50 −0.0865006
\(926\) 0 0
\(927\) 41622.0 1.47470
\(928\) 0 0
\(929\) −47616.6 −1.68165 −0.840823 0.541310i \(-0.817928\pi\)
−0.840823 + 0.541310i \(0.817928\pi\)
\(930\) 0 0
\(931\) 1711.66 0.0602549
\(932\) 0 0
\(933\) −26810.1 −0.940755
\(934\) 0 0
\(935\) −25320.8 −0.885646
\(936\) 0 0
\(937\) 21360.6 0.744739 0.372369 0.928085i \(-0.378545\pi\)
0.372369 + 0.928085i \(0.378545\pi\)
\(938\) 0 0
\(939\) −6329.42 −0.219971
\(940\) 0 0
\(941\) 23923.3 0.828776 0.414388 0.910100i \(-0.363996\pi\)
0.414388 + 0.910100i \(0.363996\pi\)
\(942\) 0 0
\(943\) −17516.3 −0.604887
\(944\) 0 0
\(945\) 61729.0 2.12492
\(946\) 0 0
\(947\) −23330.5 −0.800569 −0.400285 0.916391i \(-0.631089\pi\)
−0.400285 + 0.916391i \(0.631089\pi\)
\(948\) 0 0
\(949\) −61810.7 −2.11429
\(950\) 0 0
\(951\) −53039.8 −1.80855
\(952\) 0 0
\(953\) −25756.2 −0.875472 −0.437736 0.899104i \(-0.644220\pi\)
−0.437736 + 0.899104i \(0.644220\pi\)
\(954\) 0 0
\(955\) 24797.5 0.840238
\(956\) 0 0
\(957\) −10994.4 −0.371368
\(958\) 0 0
\(959\) 30303.0 1.02037
\(960\) 0 0
\(961\) 19937.8 0.669255
\(962\) 0 0
\(963\) −85451.3 −2.85943
\(964\) 0 0
\(965\) 42948.9 1.43272
\(966\) 0 0
\(967\) 2488.74 0.0827635 0.0413818 0.999143i \(-0.486824\pi\)
0.0413818 + 0.999143i \(0.486824\pi\)
\(968\) 0 0
\(969\) −20204.0 −0.669811
\(970\) 0 0
\(971\) −4909.90 −0.162272 −0.0811360 0.996703i \(-0.525855\pi\)
−0.0811360 + 0.996703i \(0.525855\pi\)
\(972\) 0 0
\(973\) 29726.0 0.979415
\(974\) 0 0
\(975\) 6198.14 0.203589
\(976\) 0 0
\(977\) 47452.4 1.55388 0.776938 0.629578i \(-0.216772\pi\)
0.776938 + 0.629578i \(0.216772\pi\)
\(978\) 0 0
\(979\) 18011.6 0.588000
\(980\) 0 0
\(981\) 82777.5 2.69407
\(982\) 0 0
\(983\) −27802.2 −0.902088 −0.451044 0.892502i \(-0.648948\pi\)
−0.451044 + 0.892502i \(0.648948\pi\)
\(984\) 0 0
\(985\) 6035.66 0.195241
\(986\) 0 0
\(987\) 32170.0 1.03747
\(988\) 0 0
\(989\) 18528.6 0.595728
\(990\) 0 0
\(991\) −26200.5 −0.839845 −0.419922 0.907560i \(-0.637943\pi\)
−0.419922 + 0.907560i \(0.637943\pi\)
\(992\) 0 0
\(993\) 95873.0 3.06388
\(994\) 0 0
\(995\) 785.655 0.0250321
\(996\) 0 0
\(997\) 25254.6 0.802227 0.401113 0.916028i \(-0.368623\pi\)
0.401113 + 0.916028i \(0.368623\pi\)
\(998\) 0 0
\(999\) 86372.7 2.73545
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 608.4.a.k.1.1 yes 7
4.3 odd 2 608.4.a.j.1.7 7
8.3 odd 2 1216.4.a.bg.1.1 7
8.5 even 2 1216.4.a.bf.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.j.1.7 7 4.3 odd 2
608.4.a.k.1.1 yes 7 1.1 even 1 trivial
1216.4.a.bf.1.7 7 8.5 even 2
1216.4.a.bg.1.1 7 8.3 odd 2