Properties

Label 736.4.a.e.1.1
Level $736$
Weight $4$
Character 736.1
Self dual yes
Analytic conductor $43.425$
Analytic rank $1$
Dimension $8$
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] = [736,4,Mod(1,736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(736, 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("736.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 736.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: \(43.4254057642\)
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: \( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.70977\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.70977 q^{3} -18.2706 q^{5} -11.7096 q^{7} +48.8601 q^{9} +2.01666 q^{11} -23.3256 q^{13} +159.133 q^{15} +55.7210 q^{17} -105.455 q^{19} +101.988 q^{21} +23.0000 q^{23} +208.814 q^{25} -190.396 q^{27} +275.935 q^{29} +282.019 q^{31} -17.5646 q^{33} +213.941 q^{35} +229.974 q^{37} +203.161 q^{39} -75.0934 q^{41} -125.927 q^{43} -892.702 q^{45} -349.795 q^{47} -205.885 q^{49} -485.317 q^{51} +682.616 q^{53} -36.8455 q^{55} +918.487 q^{57} -604.301 q^{59} -502.787 q^{61} -572.132 q^{63} +426.172 q^{65} -120.360 q^{67} -200.325 q^{69} -381.204 q^{71} +583.989 q^{73} -1818.72 q^{75} -23.6143 q^{77} -714.302 q^{79} +339.085 q^{81} -26.3693 q^{83} -1018.05 q^{85} -2403.33 q^{87} -108.995 q^{89} +273.134 q^{91} -2456.32 q^{93} +1926.72 q^{95} +1365.94 q^{97} +98.5340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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
\(3\) −8.70977 −1.67620 −0.838098 0.545520i \(-0.816332\pi\)
−0.838098 + 0.545520i \(0.816332\pi\)
\(4\) 0 0
\(5\) −18.2706 −1.63417 −0.817085 0.576517i \(-0.804412\pi\)
−0.817085 + 0.576517i \(0.804412\pi\)
\(6\) 0 0
\(7\) −11.7096 −0.632259 −0.316130 0.948716i \(-0.602383\pi\)
−0.316130 + 0.948716i \(0.602383\pi\)
\(8\) 0 0
\(9\) 48.8601 1.80963
\(10\) 0 0
\(11\) 2.01666 0.0552768 0.0276384 0.999618i \(-0.491201\pi\)
0.0276384 + 0.999618i \(0.491201\pi\)
\(12\) 0 0
\(13\) −23.3256 −0.497643 −0.248821 0.968549i \(-0.580043\pi\)
−0.248821 + 0.968549i \(0.580043\pi\)
\(14\) 0 0
\(15\) 159.133 2.73919
\(16\) 0 0
\(17\) 55.7210 0.794960 0.397480 0.917611i \(-0.369885\pi\)
0.397480 + 0.917611i \(0.369885\pi\)
\(18\) 0 0
\(19\) −105.455 −1.27332 −0.636658 0.771146i \(-0.719684\pi\)
−0.636658 + 0.771146i \(0.719684\pi\)
\(20\) 0 0
\(21\) 101.988 1.05979
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 208.814 1.67051
\(26\) 0 0
\(27\) −190.396 −1.35710
\(28\) 0 0
\(29\) 275.935 1.76689 0.883446 0.468532i \(-0.155217\pi\)
0.883446 + 0.468532i \(0.155217\pi\)
\(30\) 0 0
\(31\) 282.019 1.63394 0.816970 0.576680i \(-0.195652\pi\)
0.816970 + 0.576680i \(0.195652\pi\)
\(32\) 0 0
\(33\) −17.5646 −0.0926547
\(34\) 0 0
\(35\) 213.941 1.03322
\(36\) 0 0
\(37\) 229.974 1.02183 0.510913 0.859632i \(-0.329307\pi\)
0.510913 + 0.859632i \(0.329307\pi\)
\(38\) 0 0
\(39\) 203.161 0.834147
\(40\) 0 0
\(41\) −75.0934 −0.286039 −0.143020 0.989720i \(-0.545681\pi\)
−0.143020 + 0.989720i \(0.545681\pi\)
\(42\) 0 0
\(43\) −125.927 −0.446598 −0.223299 0.974750i \(-0.571683\pi\)
−0.223299 + 0.974750i \(0.571683\pi\)
\(44\) 0 0
\(45\) −892.702 −2.95725
\(46\) 0 0
\(47\) −349.795 −1.08559 −0.542796 0.839864i \(-0.682634\pi\)
−0.542796 + 0.839864i \(0.682634\pi\)
\(48\) 0 0
\(49\) −205.885 −0.600248
\(50\) 0 0
\(51\) −485.317 −1.33251
\(52\) 0 0
\(53\) 682.616 1.76914 0.884571 0.466406i \(-0.154451\pi\)
0.884571 + 0.466406i \(0.154451\pi\)
\(54\) 0 0
\(55\) −36.8455 −0.0903317
\(56\) 0 0
\(57\) 918.487 2.13433
\(58\) 0 0
\(59\) −604.301 −1.33345 −0.666723 0.745305i \(-0.732304\pi\)
−0.666723 + 0.745305i \(0.732304\pi\)
\(60\) 0 0
\(61\) −502.787 −1.05533 −0.527666 0.849452i \(-0.676933\pi\)
−0.527666 + 0.849452i \(0.676933\pi\)
\(62\) 0 0
\(63\) −572.132 −1.14416
\(64\) 0 0
\(65\) 426.172 0.813233
\(66\) 0 0
\(67\) −120.360 −0.219467 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(68\) 0 0
\(69\) −200.325 −0.349511
\(70\) 0 0
\(71\) −381.204 −0.637191 −0.318595 0.947891i \(-0.603211\pi\)
−0.318595 + 0.947891i \(0.603211\pi\)
\(72\) 0 0
\(73\) 583.989 0.936312 0.468156 0.883646i \(-0.344919\pi\)
0.468156 + 0.883646i \(0.344919\pi\)
\(74\) 0 0
\(75\) −1818.72 −2.80011
\(76\) 0 0
\(77\) −23.6143 −0.0349493
\(78\) 0 0
\(79\) −714.302 −1.01728 −0.508641 0.860979i \(-0.669852\pi\)
−0.508641 + 0.860979i \(0.669852\pi\)
\(80\) 0 0
\(81\) 339.085 0.465137
\(82\) 0 0
\(83\) −26.3693 −0.0348724 −0.0174362 0.999848i \(-0.505550\pi\)
−0.0174362 + 0.999848i \(0.505550\pi\)
\(84\) 0 0
\(85\) −1018.05 −1.29910
\(86\) 0 0
\(87\) −2403.33 −2.96166
\(88\) 0 0
\(89\) −108.995 −0.129814 −0.0649069 0.997891i \(-0.520675\pi\)
−0.0649069 + 0.997891i \(0.520675\pi\)
\(90\) 0 0
\(91\) 273.134 0.314639
\(92\) 0 0
\(93\) −2456.32 −2.73880
\(94\) 0 0
\(95\) 1926.72 2.08082
\(96\) 0 0
\(97\) 1365.94 1.42979 0.714897 0.699229i \(-0.246473\pi\)
0.714897 + 0.699229i \(0.246473\pi\)
\(98\) 0 0
\(99\) 98.5340 0.100031
\(100\) 0 0
\(101\) 228.537 0.225151 0.112576 0.993643i \(-0.464090\pi\)
0.112576 + 0.993643i \(0.464090\pi\)
\(102\) 0 0
\(103\) 1580.80 1.51224 0.756121 0.654431i \(-0.227092\pi\)
0.756121 + 0.654431i \(0.227092\pi\)
\(104\) 0 0
\(105\) −1863.38 −1.73188
\(106\) 0 0
\(107\) 319.446 0.288617 0.144309 0.989533i \(-0.453904\pi\)
0.144309 + 0.989533i \(0.453904\pi\)
\(108\) 0 0
\(109\) −554.358 −0.487136 −0.243568 0.969884i \(-0.578318\pi\)
−0.243568 + 0.969884i \(0.578318\pi\)
\(110\) 0 0
\(111\) −2003.02 −1.71278
\(112\) 0 0
\(113\) −636.882 −0.530202 −0.265101 0.964221i \(-0.585405\pi\)
−0.265101 + 0.964221i \(0.585405\pi\)
\(114\) 0 0
\(115\) −420.223 −0.340748
\(116\) 0 0
\(117\) −1139.69 −0.900551
\(118\) 0 0
\(119\) −652.471 −0.502621
\(120\) 0 0
\(121\) −1326.93 −0.996944
\(122\) 0 0
\(123\) 654.046 0.479458
\(124\) 0 0
\(125\) −1531.33 −1.09573
\(126\) 0 0
\(127\) 2634.23 1.84055 0.920277 0.391267i \(-0.127963\pi\)
0.920277 + 0.391267i \(0.127963\pi\)
\(128\) 0 0
\(129\) 1096.80 0.748585
\(130\) 0 0
\(131\) −605.725 −0.403988 −0.201994 0.979387i \(-0.564742\pi\)
−0.201994 + 0.979387i \(0.564742\pi\)
\(132\) 0 0
\(133\) 1234.84 0.805066
\(134\) 0 0
\(135\) 3478.65 2.21774
\(136\) 0 0
\(137\) −262.980 −0.163999 −0.0819996 0.996632i \(-0.526131\pi\)
−0.0819996 + 0.996632i \(0.526131\pi\)
\(138\) 0 0
\(139\) −1428.75 −0.871835 −0.435917 0.899987i \(-0.643576\pi\)
−0.435917 + 0.899987i \(0.643576\pi\)
\(140\) 0 0
\(141\) 3046.63 1.81967
\(142\) 0 0
\(143\) −47.0397 −0.0275081
\(144\) 0 0
\(145\) −5041.50 −2.88740
\(146\) 0 0
\(147\) 1793.21 1.00613
\(148\) 0 0
\(149\) 1255.23 0.690152 0.345076 0.938575i \(-0.387853\pi\)
0.345076 + 0.938575i \(0.387853\pi\)
\(150\) 0 0
\(151\) 552.311 0.297658 0.148829 0.988863i \(-0.452450\pi\)
0.148829 + 0.988863i \(0.452450\pi\)
\(152\) 0 0
\(153\) 2722.53 1.43859
\(154\) 0 0
\(155\) −5152.65 −2.67014
\(156\) 0 0
\(157\) 2800.10 1.42339 0.711695 0.702488i \(-0.247928\pi\)
0.711695 + 0.702488i \(0.247928\pi\)
\(158\) 0 0
\(159\) −5945.43 −2.96543
\(160\) 0 0
\(161\) −269.321 −0.131835
\(162\) 0 0
\(163\) 3701.73 1.77879 0.889393 0.457144i \(-0.151127\pi\)
0.889393 + 0.457144i \(0.151127\pi\)
\(164\) 0 0
\(165\) 320.916 0.151414
\(166\) 0 0
\(167\) −3038.22 −1.40781 −0.703905 0.710294i \(-0.748562\pi\)
−0.703905 + 0.710294i \(0.748562\pi\)
\(168\) 0 0
\(169\) −1652.92 −0.752352
\(170\) 0 0
\(171\) −5152.53 −2.30423
\(172\) 0 0
\(173\) −2984.30 −1.31152 −0.655758 0.754971i \(-0.727651\pi\)
−0.655758 + 0.754971i \(0.727651\pi\)
\(174\) 0 0
\(175\) −2445.13 −1.05620
\(176\) 0 0
\(177\) 5263.33 2.23512
\(178\) 0 0
\(179\) −897.226 −0.374647 −0.187323 0.982298i \(-0.559981\pi\)
−0.187323 + 0.982298i \(0.559981\pi\)
\(180\) 0 0
\(181\) −2049.03 −0.841453 −0.420726 0.907188i \(-0.638225\pi\)
−0.420726 + 0.907188i \(0.638225\pi\)
\(182\) 0 0
\(183\) 4379.16 1.76894
\(184\) 0 0
\(185\) −4201.77 −1.66984
\(186\) 0 0
\(187\) 112.370 0.0439429
\(188\) 0 0
\(189\) 2229.46 0.858041
\(190\) 0 0
\(191\) 1024.04 0.387941 0.193971 0.981007i \(-0.437863\pi\)
0.193971 + 0.981007i \(0.437863\pi\)
\(192\) 0 0
\(193\) −1616.95 −0.603061 −0.301530 0.953457i \(-0.597497\pi\)
−0.301530 + 0.953457i \(0.597497\pi\)
\(194\) 0 0
\(195\) −3711.86 −1.36314
\(196\) 0 0
\(197\) 1786.83 0.646224 0.323112 0.946361i \(-0.395271\pi\)
0.323112 + 0.946361i \(0.395271\pi\)
\(198\) 0 0
\(199\) 3092.49 1.10161 0.550805 0.834634i \(-0.314321\pi\)
0.550805 + 0.834634i \(0.314321\pi\)
\(200\) 0 0
\(201\) 1048.31 0.367870
\(202\) 0 0
\(203\) −3231.09 −1.11713
\(204\) 0 0
\(205\) 1372.00 0.467437
\(206\) 0 0
\(207\) 1123.78 0.377334
\(208\) 0 0
\(209\) −212.666 −0.0703848
\(210\) 0 0
\(211\) 3733.03 1.21797 0.608987 0.793180i \(-0.291576\pi\)
0.608987 + 0.793180i \(0.291576\pi\)
\(212\) 0 0
\(213\) 3320.19 1.06806
\(214\) 0 0
\(215\) 2300.76 0.729817
\(216\) 0 0
\(217\) −3302.33 −1.03307
\(218\) 0 0
\(219\) −5086.41 −1.56944
\(220\) 0 0
\(221\) −1299.73 −0.395606
\(222\) 0 0
\(223\) −1145.85 −0.344090 −0.172045 0.985089i \(-0.555037\pi\)
−0.172045 + 0.985089i \(0.555037\pi\)
\(224\) 0 0
\(225\) 10202.7 3.02302
\(226\) 0 0
\(227\) −5916.71 −1.72998 −0.864990 0.501789i \(-0.832676\pi\)
−0.864990 + 0.501789i \(0.832676\pi\)
\(228\) 0 0
\(229\) 4894.62 1.41243 0.706214 0.707999i \(-0.250402\pi\)
0.706214 + 0.707999i \(0.250402\pi\)
\(230\) 0 0
\(231\) 205.675 0.0585818
\(232\) 0 0
\(233\) 1921.15 0.540167 0.270083 0.962837i \(-0.412949\pi\)
0.270083 + 0.962837i \(0.412949\pi\)
\(234\) 0 0
\(235\) 6390.96 1.77404
\(236\) 0 0
\(237\) 6221.41 1.70516
\(238\) 0 0
\(239\) 3806.42 1.03020 0.515098 0.857131i \(-0.327756\pi\)
0.515098 + 0.857131i \(0.327756\pi\)
\(240\) 0 0
\(241\) 6887.20 1.84085 0.920423 0.390925i \(-0.127845\pi\)
0.920423 + 0.390925i \(0.127845\pi\)
\(242\) 0 0
\(243\) 2187.35 0.577442
\(244\) 0 0
\(245\) 3761.64 0.980908
\(246\) 0 0
\(247\) 2459.80 0.633657
\(248\) 0 0
\(249\) 229.671 0.0584529
\(250\) 0 0
\(251\) 3961.61 0.996234 0.498117 0.867110i \(-0.334025\pi\)
0.498117 + 0.867110i \(0.334025\pi\)
\(252\) 0 0
\(253\) 46.3831 0.0115260
\(254\) 0 0
\(255\) 8867.02 2.17755
\(256\) 0 0
\(257\) −6821.86 −1.65578 −0.827891 0.560890i \(-0.810459\pi\)
−0.827891 + 0.560890i \(0.810459\pi\)
\(258\) 0 0
\(259\) −2692.91 −0.646059
\(260\) 0 0
\(261\) 13482.2 3.19743
\(262\) 0 0
\(263\) −5109.63 −1.19800 −0.598999 0.800750i \(-0.704435\pi\)
−0.598999 + 0.800750i \(0.704435\pi\)
\(264\) 0 0
\(265\) −12471.8 −2.89108
\(266\) 0 0
\(267\) 949.320 0.217593
\(268\) 0 0
\(269\) −2427.02 −0.550105 −0.275052 0.961429i \(-0.588695\pi\)
−0.275052 + 0.961429i \(0.588695\pi\)
\(270\) 0 0
\(271\) −6618.07 −1.48346 −0.741732 0.670696i \(-0.765996\pi\)
−0.741732 + 0.670696i \(0.765996\pi\)
\(272\) 0 0
\(273\) −2378.93 −0.527397
\(274\) 0 0
\(275\) 421.107 0.0923406
\(276\) 0 0
\(277\) −5142.17 −1.11539 −0.557695 0.830046i \(-0.688314\pi\)
−0.557695 + 0.830046i \(0.688314\pi\)
\(278\) 0 0
\(279\) 13779.5 2.95683
\(280\) 0 0
\(281\) −6603.95 −1.40199 −0.700994 0.713167i \(-0.747260\pi\)
−0.700994 + 0.713167i \(0.747260\pi\)
\(282\) 0 0
\(283\) 6283.79 1.31990 0.659951 0.751309i \(-0.270577\pi\)
0.659951 + 0.751309i \(0.270577\pi\)
\(284\) 0 0
\(285\) −16781.3 −3.48785
\(286\) 0 0
\(287\) 879.314 0.180851
\(288\) 0 0
\(289\) −1808.17 −0.368038
\(290\) 0 0
\(291\) −11897.0 −2.39662
\(292\) 0 0
\(293\) 5960.96 1.18854 0.594272 0.804264i \(-0.297440\pi\)
0.594272 + 0.804264i \(0.297440\pi\)
\(294\) 0 0
\(295\) 11040.9 2.17908
\(296\) 0 0
\(297\) −383.964 −0.0750163
\(298\) 0 0
\(299\) −536.489 −0.103766
\(300\) 0 0
\(301\) 1474.56 0.282366
\(302\) 0 0
\(303\) −1990.51 −0.377398
\(304\) 0 0
\(305\) 9186.21 1.72459
\(306\) 0 0
\(307\) −4006.77 −0.744881 −0.372440 0.928056i \(-0.621479\pi\)
−0.372440 + 0.928056i \(0.621479\pi\)
\(308\) 0 0
\(309\) −13768.4 −2.53481
\(310\) 0 0
\(311\) 3461.38 0.631115 0.315558 0.948906i \(-0.397808\pi\)
0.315558 + 0.948906i \(0.397808\pi\)
\(312\) 0 0
\(313\) −4013.93 −0.724858 −0.362429 0.932011i \(-0.618053\pi\)
−0.362429 + 0.932011i \(0.618053\pi\)
\(314\) 0 0
\(315\) 10453.2 1.86975
\(316\) 0 0
\(317\) −8418.82 −1.49164 −0.745818 0.666150i \(-0.767941\pi\)
−0.745818 + 0.666150i \(0.767941\pi\)
\(318\) 0 0
\(319\) 556.467 0.0976682
\(320\) 0 0
\(321\) −2782.30 −0.483779
\(322\) 0 0
\(323\) −5876.05 −1.01224
\(324\) 0 0
\(325\) −4870.72 −0.831319
\(326\) 0 0
\(327\) 4828.33 0.816535
\(328\) 0 0
\(329\) 4095.96 0.686376
\(330\) 0 0
\(331\) 7515.07 1.24793 0.623966 0.781452i \(-0.285520\pi\)
0.623966 + 0.781452i \(0.285520\pi\)
\(332\) 0 0
\(333\) 11236.6 1.84913
\(334\) 0 0
\(335\) 2199.05 0.358647
\(336\) 0 0
\(337\) 402.803 0.0651100 0.0325550 0.999470i \(-0.489636\pi\)
0.0325550 + 0.999470i \(0.489636\pi\)
\(338\) 0 0
\(339\) 5547.10 0.888723
\(340\) 0 0
\(341\) 568.736 0.0903190
\(342\) 0 0
\(343\) 6427.23 1.01177
\(344\) 0 0
\(345\) 3660.05 0.571161
\(346\) 0 0
\(347\) −9222.48 −1.42677 −0.713384 0.700773i \(-0.752839\pi\)
−0.713384 + 0.700773i \(0.752839\pi\)
\(348\) 0 0
\(349\) −2246.37 −0.344542 −0.172271 0.985050i \(-0.555111\pi\)
−0.172271 + 0.985050i \(0.555111\pi\)
\(350\) 0 0
\(351\) 4441.10 0.675352
\(352\) 0 0
\(353\) 2710.12 0.408626 0.204313 0.978906i \(-0.434504\pi\)
0.204313 + 0.978906i \(0.434504\pi\)
\(354\) 0 0
\(355\) 6964.81 1.04128
\(356\) 0 0
\(357\) 5682.87 0.842491
\(358\) 0 0
\(359\) −325.575 −0.0478641 −0.0239320 0.999714i \(-0.507619\pi\)
−0.0239320 + 0.999714i \(0.507619\pi\)
\(360\) 0 0
\(361\) 4261.73 0.621334
\(362\) 0 0
\(363\) 11557.3 1.67107
\(364\) 0 0
\(365\) −10669.8 −1.53009
\(366\) 0 0
\(367\) −5863.03 −0.833917 −0.416959 0.908926i \(-0.636904\pi\)
−0.416959 + 0.908926i \(0.636904\pi\)
\(368\) 0 0
\(369\) −3669.07 −0.517626
\(370\) 0 0
\(371\) −7993.17 −1.11856
\(372\) 0 0
\(373\) 8236.61 1.14337 0.571683 0.820474i \(-0.306291\pi\)
0.571683 + 0.820474i \(0.306291\pi\)
\(374\) 0 0
\(375\) 13337.6 1.83666
\(376\) 0 0
\(377\) −6436.35 −0.879282
\(378\) 0 0
\(379\) −9815.24 −1.33028 −0.665139 0.746720i \(-0.731628\pi\)
−0.665139 + 0.746720i \(0.731628\pi\)
\(380\) 0 0
\(381\) −22943.6 −3.08513
\(382\) 0 0
\(383\) 9650.81 1.28755 0.643777 0.765213i \(-0.277366\pi\)
0.643777 + 0.765213i \(0.277366\pi\)
\(384\) 0 0
\(385\) 431.446 0.0571131
\(386\) 0 0
\(387\) −6152.81 −0.808178
\(388\) 0 0
\(389\) 4957.68 0.646181 0.323090 0.946368i \(-0.395278\pi\)
0.323090 + 0.946368i \(0.395278\pi\)
\(390\) 0 0
\(391\) 1281.58 0.165761
\(392\) 0 0
\(393\) 5275.72 0.677163
\(394\) 0 0
\(395\) 13050.7 1.66241
\(396\) 0 0
\(397\) −11698.5 −1.47892 −0.739460 0.673201i \(-0.764919\pi\)
−0.739460 + 0.673201i \(0.764919\pi\)
\(398\) 0 0
\(399\) −10755.1 −1.34945
\(400\) 0 0
\(401\) −8186.66 −1.01951 −0.509753 0.860321i \(-0.670263\pi\)
−0.509753 + 0.860321i \(0.670263\pi\)
\(402\) 0 0
\(403\) −6578.27 −0.813118
\(404\) 0 0
\(405\) −6195.27 −0.760113
\(406\) 0 0
\(407\) 463.780 0.0564833
\(408\) 0 0
\(409\) 6292.14 0.760700 0.380350 0.924843i \(-0.375803\pi\)
0.380350 + 0.924843i \(0.375803\pi\)
\(410\) 0 0
\(411\) 2290.49 0.274895
\(412\) 0 0
\(413\) 7076.13 0.843084
\(414\) 0 0
\(415\) 481.783 0.0569874
\(416\) 0 0
\(417\) 12444.1 1.46137
\(418\) 0 0
\(419\) −2333.57 −0.272082 −0.136041 0.990703i \(-0.543438\pi\)
−0.136041 + 0.990703i \(0.543438\pi\)
\(420\) 0 0
\(421\) −636.134 −0.0736420 −0.0368210 0.999322i \(-0.511723\pi\)
−0.0368210 + 0.999322i \(0.511723\pi\)
\(422\) 0 0
\(423\) −17091.0 −1.96452
\(424\) 0 0
\(425\) 11635.3 1.32799
\(426\) 0 0
\(427\) 5887.44 0.667244
\(428\) 0 0
\(429\) 409.705 0.0461090
\(430\) 0 0
\(431\) −15664.1 −1.75061 −0.875307 0.483568i \(-0.839340\pi\)
−0.875307 + 0.483568i \(0.839340\pi\)
\(432\) 0 0
\(433\) −7695.84 −0.854130 −0.427065 0.904221i \(-0.640453\pi\)
−0.427065 + 0.904221i \(0.640453\pi\)
\(434\) 0 0
\(435\) 43910.3 4.83985
\(436\) 0 0
\(437\) −2425.46 −0.265505
\(438\) 0 0
\(439\) −17864.3 −1.94218 −0.971088 0.238723i \(-0.923271\pi\)
−0.971088 + 0.238723i \(0.923271\pi\)
\(440\) 0 0
\(441\) −10059.6 −1.08623
\(442\) 0 0
\(443\) 14239.6 1.52719 0.763593 0.645698i \(-0.223434\pi\)
0.763593 + 0.645698i \(0.223434\pi\)
\(444\) 0 0
\(445\) 1991.40 0.212138
\(446\) 0 0
\(447\) −10932.8 −1.15683
\(448\) 0 0
\(449\) −17765.1 −1.86723 −0.933613 0.358282i \(-0.883363\pi\)
−0.933613 + 0.358282i \(0.883363\pi\)
\(450\) 0 0
\(451\) −151.438 −0.0158113
\(452\) 0 0
\(453\) −4810.50 −0.498934
\(454\) 0 0
\(455\) −4990.31 −0.514174
\(456\) 0 0
\(457\) 16774.3 1.71700 0.858500 0.512814i \(-0.171397\pi\)
0.858500 + 0.512814i \(0.171397\pi\)
\(458\) 0 0
\(459\) −10609.1 −1.07884
\(460\) 0 0
\(461\) −10129.7 −1.02340 −0.511699 0.859165i \(-0.670984\pi\)
−0.511699 + 0.859165i \(0.670984\pi\)
\(462\) 0 0
\(463\) 6375.12 0.639907 0.319954 0.947433i \(-0.396333\pi\)
0.319954 + 0.947433i \(0.396333\pi\)
\(464\) 0 0
\(465\) 44878.4 4.47567
\(466\) 0 0
\(467\) −287.766 −0.0285144 −0.0142572 0.999898i \(-0.504538\pi\)
−0.0142572 + 0.999898i \(0.504538\pi\)
\(468\) 0 0
\(469\) 1409.37 0.138760
\(470\) 0 0
\(471\) −24388.2 −2.38588
\(472\) 0 0
\(473\) −253.952 −0.0246865
\(474\) 0 0
\(475\) −22020.5 −2.12709
\(476\) 0 0
\(477\) 33352.7 3.20150
\(478\) 0 0
\(479\) 7525.38 0.717836 0.358918 0.933369i \(-0.383146\pi\)
0.358918 + 0.933369i \(0.383146\pi\)
\(480\) 0 0
\(481\) −5364.29 −0.508505
\(482\) 0 0
\(483\) 2345.72 0.220982
\(484\) 0 0
\(485\) −24956.5 −2.33653
\(486\) 0 0
\(487\) −26.7238 −0.00248659 −0.00124330 0.999999i \(-0.500396\pi\)
−0.00124330 + 0.999999i \(0.500396\pi\)
\(488\) 0 0
\(489\) −32241.2 −2.98159
\(490\) 0 0
\(491\) 11096.8 1.01994 0.509970 0.860192i \(-0.329657\pi\)
0.509970 + 0.860192i \(0.329657\pi\)
\(492\) 0 0
\(493\) 15375.4 1.40461
\(494\) 0 0
\(495\) −1800.27 −0.163467
\(496\) 0 0
\(497\) 4463.74 0.402870
\(498\) 0 0
\(499\) 243.921 0.0218826 0.0109413 0.999940i \(-0.496517\pi\)
0.0109413 + 0.999940i \(0.496517\pi\)
\(500\) 0 0
\(501\) 26462.2 2.35977
\(502\) 0 0
\(503\) −5095.04 −0.451643 −0.225822 0.974169i \(-0.572507\pi\)
−0.225822 + 0.974169i \(0.572507\pi\)
\(504\) 0 0
\(505\) −4175.51 −0.367936
\(506\) 0 0
\(507\) 14396.5 1.26109
\(508\) 0 0
\(509\) −434.330 −0.0378219 −0.0189109 0.999821i \(-0.506020\pi\)
−0.0189109 + 0.999821i \(0.506020\pi\)
\(510\) 0 0
\(511\) −6838.28 −0.591992
\(512\) 0 0
\(513\) 20078.2 1.72802
\(514\) 0 0
\(515\) −28882.2 −2.47126
\(516\) 0 0
\(517\) −705.417 −0.0600081
\(518\) 0 0
\(519\) 25992.6 2.19836
\(520\) 0 0
\(521\) 4903.38 0.412325 0.206162 0.978518i \(-0.433902\pi\)
0.206162 + 0.978518i \(0.433902\pi\)
\(522\) 0 0
\(523\) −6138.01 −0.513186 −0.256593 0.966520i \(-0.582600\pi\)
−0.256593 + 0.966520i \(0.582600\pi\)
\(524\) 0 0
\(525\) 21296.5 1.77039
\(526\) 0 0
\(527\) 15714.4 1.29892
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −29526.2 −2.41305
\(532\) 0 0
\(533\) 1751.60 0.142345
\(534\) 0 0
\(535\) −5836.47 −0.471650
\(536\) 0 0
\(537\) 7814.63 0.627982
\(538\) 0 0
\(539\) −415.199 −0.0331798
\(540\) 0 0
\(541\) 1752.95 0.139307 0.0696535 0.997571i \(-0.477811\pi\)
0.0696535 + 0.997571i \(0.477811\pi\)
\(542\) 0 0
\(543\) 17846.5 1.41044
\(544\) 0 0
\(545\) 10128.4 0.796063
\(546\) 0 0
\(547\) −10633.7 −0.831193 −0.415597 0.909549i \(-0.636427\pi\)
−0.415597 + 0.909549i \(0.636427\pi\)
\(548\) 0 0
\(549\) −24566.2 −1.90976
\(550\) 0 0
\(551\) −29098.7 −2.24981
\(552\) 0 0
\(553\) 8364.20 0.643186
\(554\) 0 0
\(555\) 36596.4 2.79898
\(556\) 0 0
\(557\) −4952.02 −0.376704 −0.188352 0.982102i \(-0.560315\pi\)
−0.188352 + 0.982102i \(0.560315\pi\)
\(558\) 0 0
\(559\) 2937.33 0.222246
\(560\) 0 0
\(561\) −978.718 −0.0736568
\(562\) 0 0
\(563\) 3532.30 0.264421 0.132210 0.991222i \(-0.457793\pi\)
0.132210 + 0.991222i \(0.457793\pi\)
\(564\) 0 0
\(565\) 11636.2 0.866441
\(566\) 0 0
\(567\) −3970.55 −0.294087
\(568\) 0 0
\(569\) −16297.4 −1.20074 −0.600372 0.799721i \(-0.704981\pi\)
−0.600372 + 0.799721i \(0.704981\pi\)
\(570\) 0 0
\(571\) 25981.9 1.90422 0.952110 0.305756i \(-0.0989091\pi\)
0.952110 + 0.305756i \(0.0989091\pi\)
\(572\) 0 0
\(573\) −8919.13 −0.650265
\(574\) 0 0
\(575\) 4802.73 0.348326
\(576\) 0 0
\(577\) −22813.0 −1.64596 −0.822981 0.568069i \(-0.807690\pi\)
−0.822981 + 0.568069i \(0.807690\pi\)
\(578\) 0 0
\(579\) 14083.3 1.01085
\(580\) 0 0
\(581\) 308.774 0.0220484
\(582\) 0 0
\(583\) 1376.60 0.0977925
\(584\) 0 0
\(585\) 20822.8 1.47165
\(586\) 0 0
\(587\) −18286.9 −1.28583 −0.642916 0.765937i \(-0.722275\pi\)
−0.642916 + 0.765937i \(0.722275\pi\)
\(588\) 0 0
\(589\) −29740.3 −2.08052
\(590\) 0 0
\(591\) −15562.8 −1.08320
\(592\) 0 0
\(593\) 4978.89 0.344786 0.172393 0.985028i \(-0.444850\pi\)
0.172393 + 0.985028i \(0.444850\pi\)
\(594\) 0 0
\(595\) 11921.0 0.821369
\(596\) 0 0
\(597\) −26934.8 −1.84652
\(598\) 0 0
\(599\) −9230.75 −0.629646 −0.314823 0.949150i \(-0.601945\pi\)
−0.314823 + 0.949150i \(0.601945\pi\)
\(600\) 0 0
\(601\) −2956.20 −0.200642 −0.100321 0.994955i \(-0.531987\pi\)
−0.100321 + 0.994955i \(0.531987\pi\)
\(602\) 0 0
\(603\) −5880.80 −0.397155
\(604\) 0 0
\(605\) 24243.8 1.62918
\(606\) 0 0
\(607\) 13356.1 0.893093 0.446547 0.894760i \(-0.352654\pi\)
0.446547 + 0.894760i \(0.352654\pi\)
\(608\) 0 0
\(609\) 28142.1 1.87254
\(610\) 0 0
\(611\) 8159.18 0.540238
\(612\) 0 0
\(613\) 8588.14 0.565859 0.282930 0.959141i \(-0.408694\pi\)
0.282930 + 0.959141i \(0.408694\pi\)
\(614\) 0 0
\(615\) −11949.8 −0.783516
\(616\) 0 0
\(617\) −6308.36 −0.411612 −0.205806 0.978593i \(-0.565982\pi\)
−0.205806 + 0.978593i \(0.565982\pi\)
\(618\) 0 0
\(619\) 9910.23 0.643499 0.321749 0.946825i \(-0.395729\pi\)
0.321749 + 0.946825i \(0.395729\pi\)
\(620\) 0 0
\(621\) −4379.11 −0.282975
\(622\) 0 0
\(623\) 1276.29 0.0820760
\(624\) 0 0
\(625\) 1876.59 0.120102
\(626\) 0 0
\(627\) 1852.27 0.117979
\(628\) 0 0
\(629\) 12814.4 0.812311
\(630\) 0 0
\(631\) 8551.40 0.539502 0.269751 0.962930i \(-0.413059\pi\)
0.269751 + 0.962930i \(0.413059\pi\)
\(632\) 0 0
\(633\) −32513.8 −2.04156
\(634\) 0 0
\(635\) −48129.0 −3.00778
\(636\) 0 0
\(637\) 4802.39 0.298709
\(638\) 0 0
\(639\) −18625.6 −1.15308
\(640\) 0 0
\(641\) −22921.3 −1.41238 −0.706192 0.708020i \(-0.749589\pi\)
−0.706192 + 0.708020i \(0.749589\pi\)
\(642\) 0 0
\(643\) −29765.5 −1.82556 −0.912782 0.408446i \(-0.866071\pi\)
−0.912782 + 0.408446i \(0.866071\pi\)
\(644\) 0 0
\(645\) −20039.1 −1.22332
\(646\) 0 0
\(647\) 13184.5 0.801139 0.400569 0.916266i \(-0.368812\pi\)
0.400569 + 0.916266i \(0.368812\pi\)
\(648\) 0 0
\(649\) −1218.67 −0.0737087
\(650\) 0 0
\(651\) 28762.6 1.73163
\(652\) 0 0
\(653\) 26389.8 1.58149 0.790744 0.612147i \(-0.209694\pi\)
0.790744 + 0.612147i \(0.209694\pi\)
\(654\) 0 0
\(655\) 11066.9 0.660185
\(656\) 0 0
\(657\) 28533.7 1.69438
\(658\) 0 0
\(659\) −10789.1 −0.637761 −0.318881 0.947795i \(-0.603307\pi\)
−0.318881 + 0.947795i \(0.603307\pi\)
\(660\) 0 0
\(661\) 12018.5 0.707209 0.353605 0.935395i \(-0.384956\pi\)
0.353605 + 0.935395i \(0.384956\pi\)
\(662\) 0 0
\(663\) 11320.3 0.663114
\(664\) 0 0
\(665\) −22561.2 −1.31562
\(666\) 0 0
\(667\) 6346.51 0.368423
\(668\) 0 0
\(669\) 9980.13 0.576762
\(670\) 0 0
\(671\) −1013.95 −0.0583354
\(672\) 0 0
\(673\) −28401.4 −1.62674 −0.813369 0.581747i \(-0.802369\pi\)
−0.813369 + 0.581747i \(0.802369\pi\)
\(674\) 0 0
\(675\) −39757.4 −2.26706
\(676\) 0 0
\(677\) 15573.7 0.884114 0.442057 0.896987i \(-0.354249\pi\)
0.442057 + 0.896987i \(0.354249\pi\)
\(678\) 0 0
\(679\) −15994.6 −0.904001
\(680\) 0 0
\(681\) 51533.1 2.89979
\(682\) 0 0
\(683\) −3646.89 −0.204311 −0.102156 0.994768i \(-0.532574\pi\)
−0.102156 + 0.994768i \(0.532574\pi\)
\(684\) 0 0
\(685\) 4804.79 0.268003
\(686\) 0 0
\(687\) −42631.0 −2.36750
\(688\) 0 0
\(689\) −15922.4 −0.880401
\(690\) 0 0
\(691\) 10621.3 0.584736 0.292368 0.956306i \(-0.405557\pi\)
0.292368 + 0.956306i \(0.405557\pi\)
\(692\) 0 0
\(693\) −1153.79 −0.0632454
\(694\) 0 0
\(695\) 26104.1 1.42473
\(696\) 0 0
\(697\) −4184.28 −0.227390
\(698\) 0 0
\(699\) −16732.8 −0.905425
\(700\) 0 0
\(701\) −11654.1 −0.627919 −0.313959 0.949436i \(-0.601656\pi\)
−0.313959 + 0.949436i \(0.601656\pi\)
\(702\) 0 0
\(703\) −24251.9 −1.30111
\(704\) 0 0
\(705\) −55663.8 −2.97364
\(706\) 0 0
\(707\) −2676.08 −0.142354
\(708\) 0 0
\(709\) −26137.6 −1.38451 −0.692257 0.721652i \(-0.743383\pi\)
−0.692257 + 0.721652i \(0.743383\pi\)
\(710\) 0 0
\(711\) −34900.9 −1.84091
\(712\) 0 0
\(713\) 6486.44 0.340700
\(714\) 0 0
\(715\) 859.443 0.0449529
\(716\) 0 0
\(717\) −33153.0 −1.72681
\(718\) 0 0
\(719\) −3021.94 −0.156744 −0.0783722 0.996924i \(-0.524972\pi\)
−0.0783722 + 0.996924i \(0.524972\pi\)
\(720\) 0 0
\(721\) −18510.6 −0.956130
\(722\) 0 0
\(723\) −59985.9 −3.08562
\(724\) 0 0
\(725\) 57619.2 2.95162
\(726\) 0 0
\(727\) −10162.0 −0.518415 −0.259208 0.965822i \(-0.583461\pi\)
−0.259208 + 0.965822i \(0.583461\pi\)
\(728\) 0 0
\(729\) −28206.6 −1.43304
\(730\) 0 0
\(731\) −7016.78 −0.355028
\(732\) 0 0
\(733\) 12288.4 0.619210 0.309605 0.950865i \(-0.399803\pi\)
0.309605 + 0.950865i \(0.399803\pi\)
\(734\) 0 0
\(735\) −32763.0 −1.64419
\(736\) 0 0
\(737\) −242.725 −0.0121315
\(738\) 0 0
\(739\) −16349.4 −0.813834 −0.406917 0.913465i \(-0.633396\pi\)
−0.406917 + 0.913465i \(0.633396\pi\)
\(740\) 0 0
\(741\) −21424.3 −1.06213
\(742\) 0 0
\(743\) −18786.9 −0.927624 −0.463812 0.885934i \(-0.653519\pi\)
−0.463812 + 0.885934i \(0.653519\pi\)
\(744\) 0 0
\(745\) −22933.8 −1.12783
\(746\) 0 0
\(747\) −1288.41 −0.0631062
\(748\) 0 0
\(749\) −3740.59 −0.182481
\(750\) 0 0
\(751\) −32006.4 −1.55517 −0.777584 0.628779i \(-0.783555\pi\)
−0.777584 + 0.628779i \(0.783555\pi\)
\(752\) 0 0
\(753\) −34504.7 −1.66988
\(754\) 0 0
\(755\) −10091.0 −0.486425
\(756\) 0 0
\(757\) 10352.3 0.497042 0.248521 0.968627i \(-0.420055\pi\)
0.248521 + 0.968627i \(0.420055\pi\)
\(758\) 0 0
\(759\) −403.986 −0.0193199
\(760\) 0 0
\(761\) 14560.4 0.693579 0.346790 0.937943i \(-0.387272\pi\)
0.346790 + 0.937943i \(0.387272\pi\)
\(762\) 0 0
\(763\) 6491.31 0.307996
\(764\) 0 0
\(765\) −49742.2 −2.35089
\(766\) 0 0
\(767\) 14095.7 0.663580
\(768\) 0 0
\(769\) −23352.2 −1.09506 −0.547529 0.836786i \(-0.684432\pi\)
−0.547529 + 0.836786i \(0.684432\pi\)
\(770\) 0 0
\(771\) 59416.8 2.77541
\(772\) 0 0
\(773\) 1738.74 0.0809032 0.0404516 0.999181i \(-0.487120\pi\)
0.0404516 + 0.999181i \(0.487120\pi\)
\(774\) 0 0
\(775\) 58889.6 2.72952
\(776\) 0 0
\(777\) 23454.6 1.08292
\(778\) 0 0
\(779\) 7918.96 0.364219
\(780\) 0 0
\(781\) −768.757 −0.0352219
\(782\) 0 0
\(783\) −52537.0 −2.39785
\(784\) 0 0
\(785\) −51159.5 −2.32606
\(786\) 0 0
\(787\) 21512.5 0.974380 0.487190 0.873296i \(-0.338022\pi\)
0.487190 + 0.873296i \(0.338022\pi\)
\(788\) 0 0
\(789\) 44503.7 2.00808
\(790\) 0 0
\(791\) 7457.64 0.335225
\(792\) 0 0
\(793\) 11727.8 0.525178
\(794\) 0 0
\(795\) 108626. 4.84602
\(796\) 0 0
\(797\) −6508.32 −0.289255 −0.144628 0.989486i \(-0.546198\pi\)
−0.144628 + 0.989486i \(0.546198\pi\)
\(798\) 0 0
\(799\) −19490.9 −0.863003
\(800\) 0 0
\(801\) −5325.49 −0.234915
\(802\) 0 0
\(803\) 1177.71 0.0517563
\(804\) 0 0
\(805\) 4920.65 0.215441
\(806\) 0 0
\(807\) 21138.8 0.922083
\(808\) 0 0
\(809\) −3638.30 −0.158116 −0.0790580 0.996870i \(-0.525191\pi\)
−0.0790580 + 0.996870i \(0.525191\pi\)
\(810\) 0 0
\(811\) −45098.6 −1.95268 −0.976341 0.216237i \(-0.930621\pi\)
−0.976341 + 0.216237i \(0.930621\pi\)
\(812\) 0 0
\(813\) 57641.8 2.48658
\(814\) 0 0
\(815\) −67632.8 −2.90684
\(816\) 0 0
\(817\) 13279.6 0.568660
\(818\) 0 0
\(819\) 13345.3 0.569382
\(820\) 0 0
\(821\) −11064.2 −0.470332 −0.235166 0.971955i \(-0.575563\pi\)
−0.235166 + 0.971955i \(0.575563\pi\)
\(822\) 0 0
\(823\) −41270.8 −1.74801 −0.874004 0.485918i \(-0.838485\pi\)
−0.874004 + 0.485918i \(0.838485\pi\)
\(824\) 0 0
\(825\) −3667.74 −0.154781
\(826\) 0 0
\(827\) 23569.3 0.991035 0.495517 0.868598i \(-0.334978\pi\)
0.495517 + 0.868598i \(0.334978\pi\)
\(828\) 0 0
\(829\) 6327.92 0.265112 0.132556 0.991176i \(-0.457682\pi\)
0.132556 + 0.991176i \(0.457682\pi\)
\(830\) 0 0
\(831\) 44787.1 1.86961
\(832\) 0 0
\(833\) −11472.1 −0.477173
\(834\) 0 0
\(835\) 55510.0 2.30060
\(836\) 0 0
\(837\) −53695.4 −2.21742
\(838\) 0 0
\(839\) −45428.3 −1.86932 −0.934659 0.355545i \(-0.884295\pi\)
−0.934659 + 0.355545i \(0.884295\pi\)
\(840\) 0 0
\(841\) 51751.3 2.12191
\(842\) 0 0
\(843\) 57518.9 2.35001
\(844\) 0 0
\(845\) 30199.7 1.22947
\(846\) 0 0
\(847\) 15537.9 0.630328
\(848\) 0 0
\(849\) −54730.3 −2.21241
\(850\) 0 0
\(851\) 5289.41 0.213066
\(852\) 0 0
\(853\) 3650.11 0.146515 0.0732575 0.997313i \(-0.476660\pi\)
0.0732575 + 0.997313i \(0.476660\pi\)
\(854\) 0 0
\(855\) 94139.8 3.76551
\(856\) 0 0
\(857\) 21525.6 0.857993 0.428997 0.903306i \(-0.358867\pi\)
0.428997 + 0.903306i \(0.358867\pi\)
\(858\) 0 0
\(859\) −20494.7 −0.814052 −0.407026 0.913417i \(-0.633434\pi\)
−0.407026 + 0.913417i \(0.633434\pi\)
\(860\) 0 0
\(861\) −7658.62 −0.303142
\(862\) 0 0
\(863\) −34600.9 −1.36481 −0.682403 0.730976i \(-0.739065\pi\)
−0.682403 + 0.730976i \(0.739065\pi\)
\(864\) 0 0
\(865\) 54524.9 2.14324
\(866\) 0 0
\(867\) 15748.8 0.616904
\(868\) 0 0
\(869\) −1440.50 −0.0562321
\(870\) 0 0
\(871\) 2807.47 0.109216
\(872\) 0 0
\(873\) 66739.9 2.58740
\(874\) 0 0
\(875\) 17931.3 0.692788
\(876\) 0 0
\(877\) 4815.58 0.185417 0.0927085 0.995693i \(-0.470448\pi\)
0.0927085 + 0.995693i \(0.470448\pi\)
\(878\) 0 0
\(879\) −51918.6 −1.99223
\(880\) 0 0
\(881\) 45060.2 1.72318 0.861588 0.507609i \(-0.169470\pi\)
0.861588 + 0.507609i \(0.169470\pi\)
\(882\) 0 0
\(883\) −30384.6 −1.15801 −0.579005 0.815324i \(-0.696559\pi\)
−0.579005 + 0.815324i \(0.696559\pi\)
\(884\) 0 0
\(885\) −96164.0 −3.65256
\(886\) 0 0
\(887\) 22951.5 0.868810 0.434405 0.900718i \(-0.356959\pi\)
0.434405 + 0.900718i \(0.356959\pi\)
\(888\) 0 0
\(889\) −30845.8 −1.16371
\(890\) 0 0
\(891\) 683.817 0.0257113
\(892\) 0 0
\(893\) 36887.6 1.38230
\(894\) 0 0
\(895\) 16392.8 0.612237
\(896\) 0 0
\(897\) 4672.69 0.173932
\(898\) 0 0
\(899\) 77819.0 2.88700
\(900\) 0 0
\(901\) 38036.0 1.40640
\(902\) 0 0
\(903\) −12843.1 −0.473300
\(904\) 0 0
\(905\) 37436.9 1.37508
\(906\) 0 0
\(907\) 1091.19 0.0399474 0.0199737 0.999801i \(-0.493642\pi\)
0.0199737 + 0.999801i \(0.493642\pi\)
\(908\) 0 0
\(909\) 11166.3 0.407441
\(910\) 0 0
\(911\) −10998.3 −0.399988 −0.199994 0.979797i \(-0.564092\pi\)
−0.199994 + 0.979797i \(0.564092\pi\)
\(912\) 0 0
\(913\) −53.1778 −0.00192763
\(914\) 0 0
\(915\) −80009.7 −2.89075
\(916\) 0 0
\(917\) 7092.80 0.255425
\(918\) 0 0
\(919\) 30936.1 1.11043 0.555217 0.831706i \(-0.312635\pi\)
0.555217 + 0.831706i \(0.312635\pi\)
\(920\) 0 0
\(921\) 34898.0 1.24857
\(922\) 0 0
\(923\) 8891.80 0.317093
\(924\) 0 0
\(925\) 48021.9 1.70697
\(926\) 0 0
\(927\) 77238.1 2.73660
\(928\) 0 0
\(929\) 28736.7 1.01488 0.507438 0.861688i \(-0.330593\pi\)
0.507438 + 0.861688i \(0.330593\pi\)
\(930\) 0 0
\(931\) 21711.6 0.764305
\(932\) 0 0
\(933\) −30147.8 −1.05787
\(934\) 0 0
\(935\) −2053.07 −0.0718101
\(936\) 0 0
\(937\) 6636.94 0.231397 0.115699 0.993284i \(-0.463089\pi\)
0.115699 + 0.993284i \(0.463089\pi\)
\(938\) 0 0
\(939\) 34960.4 1.21500
\(940\) 0 0
\(941\) −24771.1 −0.858145 −0.429073 0.903270i \(-0.641160\pi\)
−0.429073 + 0.903270i \(0.641160\pi\)
\(942\) 0 0
\(943\) −1727.15 −0.0596433
\(944\) 0 0
\(945\) −40733.6 −1.40218
\(946\) 0 0
\(947\) −44391.1 −1.52325 −0.761625 0.648018i \(-0.775598\pi\)
−0.761625 + 0.648018i \(0.775598\pi\)
\(948\) 0 0
\(949\) −13621.9 −0.465949
\(950\) 0 0
\(951\) 73326.0 2.50027
\(952\) 0 0
\(953\) 43327.0 1.47272 0.736359 0.676591i \(-0.236543\pi\)
0.736359 + 0.676591i \(0.236543\pi\)
\(954\) 0 0
\(955\) −18709.8 −0.633962
\(956\) 0 0
\(957\) −4846.70 −0.163711
\(958\) 0 0
\(959\) 3079.39 0.103690
\(960\) 0 0
\(961\) 49743.8 1.66976
\(962\) 0 0
\(963\) 15608.2 0.522291
\(964\) 0 0
\(965\) 29542.6 0.985504
\(966\) 0 0
\(967\) 51071.7 1.69840 0.849201 0.528070i \(-0.177084\pi\)
0.849201 + 0.528070i \(0.177084\pi\)
\(968\) 0 0
\(969\) 51179.0 1.69671
\(970\) 0 0
\(971\) 20649.3 0.682459 0.341229 0.939980i \(-0.389157\pi\)
0.341229 + 0.939980i \(0.389157\pi\)
\(972\) 0 0
\(973\) 16730.1 0.551226
\(974\) 0 0
\(975\) 42422.8 1.39345
\(976\) 0 0
\(977\) −38114.1 −1.24809 −0.624043 0.781390i \(-0.714511\pi\)
−0.624043 + 0.781390i \(0.714511\pi\)
\(978\) 0 0
\(979\) −219.805 −0.00717569
\(980\) 0 0
\(981\) −27086.0 −0.881537
\(982\) 0 0
\(983\) −47799.2 −1.55092 −0.775461 0.631395i \(-0.782483\pi\)
−0.775461 + 0.631395i \(0.782483\pi\)
\(984\) 0 0
\(985\) −32646.4 −1.05604
\(986\) 0 0
\(987\) −35674.9 −1.15050
\(988\) 0 0
\(989\) −2896.32 −0.0931221
\(990\) 0 0
\(991\) 43820.0 1.40463 0.702314 0.711867i \(-0.252150\pi\)
0.702314 + 0.711867i \(0.252150\pi\)
\(992\) 0 0
\(993\) −65454.5 −2.09178
\(994\) 0 0
\(995\) −56501.5 −1.80022
\(996\) 0 0
\(997\) 11484.4 0.364809 0.182405 0.983224i \(-0.441612\pi\)
0.182405 + 0.983224i \(0.441612\pi\)
\(998\) 0 0
\(999\) −43786.3 −1.38672
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 736.4.a.e.1.1 8
4.3 odd 2 736.4.a.f.1.8 yes 8
8.3 odd 2 1472.4.a.bg.1.1 8
8.5 even 2 1472.4.a.bh.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.4.a.e.1.1 8 1.1 even 1 trivial
736.4.a.f.1.8 yes 8 4.3 odd 2
1472.4.a.bg.1.1 8 8.3 odd 2
1472.4.a.bh.1.8 8 8.5 even 2