Properties

Label 538.2.a.e.1.5
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
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: \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.14366\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.65255 q^{3} +1.00000 q^{4} +0.725043 q^{5} +1.65255 q^{6} +1.29978 q^{7} +1.00000 q^{8} -0.269065 q^{9} +0.725043 q^{10} -3.71687 q^{11} +1.65255 q^{12} +2.30856 q^{13} +1.29978 q^{14} +1.19817 q^{15} +1.00000 q^{16} +6.32982 q^{17} -0.269065 q^{18} +0.0212539 q^{19} +0.725043 q^{20} +2.14795 q^{21} -3.71687 q^{22} -1.30511 q^{23} +1.65255 q^{24} -4.47431 q^{25} +2.30856 q^{26} -5.40231 q^{27} +1.29978 q^{28} -2.06303 q^{29} +1.19817 q^{30} -5.82316 q^{31} +1.00000 q^{32} -6.14233 q^{33} +6.32982 q^{34} +0.942394 q^{35} -0.269065 q^{36} -7.25272 q^{37} +0.0212539 q^{38} +3.81503 q^{39} +0.725043 q^{40} +8.77454 q^{41} +2.14795 q^{42} +4.52327 q^{43} -3.71687 q^{44} -0.195084 q^{45} -1.30511 q^{46} +3.85502 q^{47} +1.65255 q^{48} -5.31058 q^{49} -4.47431 q^{50} +10.4604 q^{51} +2.30856 q^{52} -6.86923 q^{53} -5.40231 q^{54} -2.69489 q^{55} +1.29978 q^{56} +0.0351232 q^{57} -2.06303 q^{58} +7.84518 q^{59} +1.19817 q^{60} +3.84958 q^{61} -5.82316 q^{62} -0.349724 q^{63} +1.00000 q^{64} +1.67381 q^{65} -6.14233 q^{66} -9.31541 q^{67} +6.32982 q^{68} -2.15676 q^{69} +0.942394 q^{70} -0.643097 q^{71} -0.269065 q^{72} -2.97920 q^{73} -7.25272 q^{74} -7.39404 q^{75} +0.0212539 q^{76} -4.83110 q^{77} +3.81503 q^{78} +0.418245 q^{79} +0.725043 q^{80} -8.12041 q^{81} +8.77454 q^{82} -15.6322 q^{83} +2.14795 q^{84} +4.58939 q^{85} +4.52327 q^{86} -3.40927 q^{87} -3.71687 q^{88} +10.3879 q^{89} -0.195084 q^{90} +3.00062 q^{91} -1.30511 q^{92} -9.62309 q^{93} +3.85502 q^{94} +0.0154100 q^{95} +1.65255 q^{96} +5.07659 q^{97} -5.31058 q^{98} +1.00008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.65255 0.954103 0.477051 0.878875i \(-0.341706\pi\)
0.477051 + 0.878875i \(0.341706\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.725043 0.324249 0.162125 0.986770i \(-0.448165\pi\)
0.162125 + 0.986770i \(0.448165\pi\)
\(6\) 1.65255 0.674652
\(7\) 1.29978 0.491269 0.245635 0.969362i \(-0.421004\pi\)
0.245635 + 0.969362i \(0.421004\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.269065 −0.0896884
\(10\) 0.725043 0.229279
\(11\) −3.71687 −1.12068 −0.560340 0.828263i \(-0.689329\pi\)
−0.560340 + 0.828263i \(0.689329\pi\)
\(12\) 1.65255 0.477051
\(13\) 2.30856 0.640280 0.320140 0.947370i \(-0.396270\pi\)
0.320140 + 0.947370i \(0.396270\pi\)
\(14\) 1.29978 0.347380
\(15\) 1.19817 0.309367
\(16\) 1.00000 0.250000
\(17\) 6.32982 1.53521 0.767603 0.640925i \(-0.221449\pi\)
0.767603 + 0.640925i \(0.221449\pi\)
\(18\) −0.269065 −0.0634192
\(19\) 0.0212539 0.00487598 0.00243799 0.999997i \(-0.499224\pi\)
0.00243799 + 0.999997i \(0.499224\pi\)
\(20\) 0.725043 0.162125
\(21\) 2.14795 0.468721
\(22\) −3.71687 −0.792440
\(23\) −1.30511 −0.272134 −0.136067 0.990700i \(-0.543446\pi\)
−0.136067 + 0.990700i \(0.543446\pi\)
\(24\) 1.65255 0.337326
\(25\) −4.47431 −0.894863
\(26\) 2.30856 0.452747
\(27\) −5.40231 −1.03967
\(28\) 1.29978 0.245635
\(29\) −2.06303 −0.383095 −0.191548 0.981483i \(-0.561351\pi\)
−0.191548 + 0.981483i \(0.561351\pi\)
\(30\) 1.19817 0.218755
\(31\) −5.82316 −1.04587 −0.522935 0.852372i \(-0.675163\pi\)
−0.522935 + 0.852372i \(0.675163\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.14233 −1.06924
\(34\) 6.32982 1.08555
\(35\) 0.942394 0.159294
\(36\) −0.269065 −0.0448442
\(37\) −7.25272 −1.19234 −0.596170 0.802858i \(-0.703312\pi\)
−0.596170 + 0.802858i \(0.703312\pi\)
\(38\) 0.0212539 0.00344784
\(39\) 3.81503 0.610893
\(40\) 0.725043 0.114639
\(41\) 8.77454 1.37035 0.685176 0.728377i \(-0.259725\pi\)
0.685176 + 0.728377i \(0.259725\pi\)
\(42\) 2.14795 0.331436
\(43\) 4.52327 0.689792 0.344896 0.938641i \(-0.387914\pi\)
0.344896 + 0.938641i \(0.387914\pi\)
\(44\) −3.71687 −0.560340
\(45\) −0.195084 −0.0290814
\(46\) −1.30511 −0.192428
\(47\) 3.85502 0.562313 0.281156 0.959662i \(-0.409282\pi\)
0.281156 + 0.959662i \(0.409282\pi\)
\(48\) 1.65255 0.238526
\(49\) −5.31058 −0.758654
\(50\) −4.47431 −0.632763
\(51\) 10.4604 1.46474
\(52\) 2.30856 0.320140
\(53\) −6.86923 −0.943562 −0.471781 0.881716i \(-0.656389\pi\)
−0.471781 + 0.881716i \(0.656389\pi\)
\(54\) −5.40231 −0.735161
\(55\) −2.69489 −0.363379
\(56\) 1.29978 0.173690
\(57\) 0.0351232 0.00465218
\(58\) −2.06303 −0.270889
\(59\) 7.84518 1.02136 0.510678 0.859772i \(-0.329395\pi\)
0.510678 + 0.859772i \(0.329395\pi\)
\(60\) 1.19817 0.154683
\(61\) 3.84958 0.492888 0.246444 0.969157i \(-0.420738\pi\)
0.246444 + 0.969157i \(0.420738\pi\)
\(62\) −5.82316 −0.739542
\(63\) −0.349724 −0.0440611
\(64\) 1.00000 0.125000
\(65\) 1.67381 0.207610
\(66\) −6.14233 −0.756069
\(67\) −9.31541 −1.13806 −0.569029 0.822317i \(-0.692681\pi\)
−0.569029 + 0.822317i \(0.692681\pi\)
\(68\) 6.32982 0.767603
\(69\) −2.15676 −0.259644
\(70\) 0.942394 0.112638
\(71\) −0.643097 −0.0763215 −0.0381608 0.999272i \(-0.512150\pi\)
−0.0381608 + 0.999272i \(0.512150\pi\)
\(72\) −0.269065 −0.0317096
\(73\) −2.97920 −0.348689 −0.174345 0.984685i \(-0.555781\pi\)
−0.174345 + 0.984685i \(0.555781\pi\)
\(74\) −7.25272 −0.843112
\(75\) −7.39404 −0.853791
\(76\) 0.0212539 0.00243799
\(77\) −4.83110 −0.550555
\(78\) 3.81503 0.431967
\(79\) 0.418245 0.0470563 0.0235281 0.999723i \(-0.492510\pi\)
0.0235281 + 0.999723i \(0.492510\pi\)
\(80\) 0.725043 0.0810623
\(81\) −8.12041 −0.902268
\(82\) 8.77454 0.968985
\(83\) −15.6322 −1.71585 −0.857927 0.513771i \(-0.828248\pi\)
−0.857927 + 0.513771i \(0.828248\pi\)
\(84\) 2.14795 0.234361
\(85\) 4.58939 0.497789
\(86\) 4.52327 0.487757
\(87\) −3.40927 −0.365512
\(88\) −3.71687 −0.396220
\(89\) 10.3879 1.10111 0.550556 0.834798i \(-0.314416\pi\)
0.550556 + 0.834798i \(0.314416\pi\)
\(90\) −0.195084 −0.0205636
\(91\) 3.00062 0.314550
\(92\) −1.30511 −0.136067
\(93\) −9.62309 −0.997868
\(94\) 3.85502 0.397615
\(95\) 0.0154100 0.00158103
\(96\) 1.65255 0.168663
\(97\) 5.07659 0.515450 0.257725 0.966218i \(-0.417027\pi\)
0.257725 + 0.966218i \(0.417027\pi\)
\(98\) −5.31058 −0.536450
\(99\) 1.00008 0.100512
\(100\) −4.47431 −0.447431
\(101\) 12.5321 1.24699 0.623495 0.781827i \(-0.285712\pi\)
0.623495 + 0.781827i \(0.285712\pi\)
\(102\) 10.4604 1.03573
\(103\) 5.43374 0.535403 0.267701 0.963502i \(-0.413736\pi\)
0.267701 + 0.963502i \(0.413736\pi\)
\(104\) 2.30856 0.226373
\(105\) 1.55736 0.151982
\(106\) −6.86923 −0.667199
\(107\) −3.45500 −0.334007 −0.167004 0.985956i \(-0.553409\pi\)
−0.167004 + 0.985956i \(0.553409\pi\)
\(108\) −5.40231 −0.519837
\(109\) −6.80801 −0.652089 −0.326044 0.945354i \(-0.605716\pi\)
−0.326044 + 0.945354i \(0.605716\pi\)
\(110\) −2.69489 −0.256948
\(111\) −11.9855 −1.13762
\(112\) 1.29978 0.122817
\(113\) 0.407582 0.0383421 0.0191710 0.999816i \(-0.493897\pi\)
0.0191710 + 0.999816i \(0.493897\pi\)
\(114\) 0.0351232 0.00328959
\(115\) −0.946259 −0.0882391
\(116\) −2.06303 −0.191548
\(117\) −0.621154 −0.0574257
\(118\) 7.84518 0.722207
\(119\) 8.22735 0.754200
\(120\) 1.19817 0.109378
\(121\) 2.81514 0.255922
\(122\) 3.84958 0.348524
\(123\) 14.5004 1.30746
\(124\) −5.82316 −0.522935
\(125\) −6.86928 −0.614407
\(126\) −0.349724 −0.0311559
\(127\) −9.52589 −0.845286 −0.422643 0.906296i \(-0.638898\pi\)
−0.422643 + 0.906296i \(0.638898\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.47495 0.658132
\(130\) 1.67381 0.146803
\(131\) −21.9325 −1.91625 −0.958127 0.286342i \(-0.907561\pi\)
−0.958127 + 0.286342i \(0.907561\pi\)
\(132\) −6.14233 −0.534621
\(133\) 0.0276253 0.00239542
\(134\) −9.31541 −0.804729
\(135\) −3.91690 −0.337113
\(136\) 6.32982 0.542777
\(137\) −2.48849 −0.212606 −0.106303 0.994334i \(-0.533901\pi\)
−0.106303 + 0.994334i \(0.533901\pi\)
\(138\) −2.15676 −0.183596
\(139\) 21.6244 1.83416 0.917078 0.398708i \(-0.130541\pi\)
0.917078 + 0.398708i \(0.130541\pi\)
\(140\) 0.942394 0.0796468
\(141\) 6.37063 0.536504
\(142\) −0.643097 −0.0539675
\(143\) −8.58064 −0.717549
\(144\) −0.269065 −0.0224221
\(145\) −1.49579 −0.124218
\(146\) −2.97920 −0.246561
\(147\) −8.77602 −0.723834
\(148\) −7.25272 −0.596170
\(149\) 20.9106 1.71306 0.856532 0.516093i \(-0.172614\pi\)
0.856532 + 0.516093i \(0.172614\pi\)
\(150\) −7.39404 −0.603721
\(151\) −2.07209 −0.168624 −0.0843120 0.996439i \(-0.526869\pi\)
−0.0843120 + 0.996439i \(0.526869\pi\)
\(152\) 0.0212539 0.00172392
\(153\) −1.70313 −0.137690
\(154\) −4.83110 −0.389301
\(155\) −4.22204 −0.339123
\(156\) 3.81503 0.305447
\(157\) −2.60975 −0.208281 −0.104140 0.994563i \(-0.533209\pi\)
−0.104140 + 0.994563i \(0.533209\pi\)
\(158\) 0.418245 0.0332738
\(159\) −11.3518 −0.900255
\(160\) 0.725043 0.0573197
\(161\) −1.69635 −0.133691
\(162\) −8.12041 −0.638000
\(163\) −6.63147 −0.519417 −0.259708 0.965687i \(-0.583626\pi\)
−0.259708 + 0.965687i \(0.583626\pi\)
\(164\) 8.77454 0.685176
\(165\) −4.45345 −0.346701
\(166\) −15.6322 −1.21329
\(167\) 22.6369 1.75170 0.875850 0.482584i \(-0.160302\pi\)
0.875850 + 0.482584i \(0.160302\pi\)
\(168\) 2.14795 0.165718
\(169\) −7.67053 −0.590041
\(170\) 4.58939 0.351990
\(171\) −0.00571868 −0.000437318 0
\(172\) 4.52327 0.344896
\(173\) 12.2456 0.931018 0.465509 0.885043i \(-0.345871\pi\)
0.465509 + 0.885043i \(0.345871\pi\)
\(174\) −3.40927 −0.258456
\(175\) −5.81561 −0.439619
\(176\) −3.71687 −0.280170
\(177\) 12.9646 0.974478
\(178\) 10.3879 0.778603
\(179\) −18.0181 −1.34673 −0.673367 0.739308i \(-0.735153\pi\)
−0.673367 + 0.739308i \(0.735153\pi\)
\(180\) −0.195084 −0.0145407
\(181\) 7.77690 0.578052 0.289026 0.957321i \(-0.406669\pi\)
0.289026 + 0.957321i \(0.406669\pi\)
\(182\) 3.00062 0.222421
\(183\) 6.36163 0.470266
\(184\) −1.30511 −0.0962138
\(185\) −5.25854 −0.386615
\(186\) −9.62309 −0.705599
\(187\) −23.5271 −1.72047
\(188\) 3.85502 0.281156
\(189\) −7.02179 −0.510760
\(190\) 0.0154100 0.00111796
\(191\) 0.312243 0.0225931 0.0112965 0.999936i \(-0.496404\pi\)
0.0112965 + 0.999936i \(0.496404\pi\)
\(192\) 1.65255 0.119263
\(193\) −0.336949 −0.0242541 −0.0121271 0.999926i \(-0.503860\pi\)
−0.0121271 + 0.999926i \(0.503860\pi\)
\(194\) 5.07659 0.364478
\(195\) 2.76606 0.198082
\(196\) −5.31058 −0.379327
\(197\) 26.8275 1.91138 0.955691 0.294370i \(-0.0951099\pi\)
0.955691 + 0.294370i \(0.0951099\pi\)
\(198\) 1.00008 0.0710726
\(199\) 4.03560 0.286076 0.143038 0.989717i \(-0.454313\pi\)
0.143038 + 0.989717i \(0.454313\pi\)
\(200\) −4.47431 −0.316382
\(201\) −15.3942 −1.08582
\(202\) 12.5321 0.881755
\(203\) −2.68148 −0.188203
\(204\) 10.4604 0.732372
\(205\) 6.36192 0.444335
\(206\) 5.43374 0.378587
\(207\) 0.351159 0.0244072
\(208\) 2.30856 0.160070
\(209\) −0.0789980 −0.00546440
\(210\) 1.55736 0.107468
\(211\) 9.22688 0.635205 0.317602 0.948224i \(-0.397122\pi\)
0.317602 + 0.948224i \(0.397122\pi\)
\(212\) −6.86923 −0.471781
\(213\) −1.06275 −0.0728186
\(214\) −3.45500 −0.236179
\(215\) 3.27956 0.223664
\(216\) −5.40231 −0.367580
\(217\) −7.56881 −0.513804
\(218\) −6.80801 −0.461096
\(219\) −4.92329 −0.332685
\(220\) −2.69489 −0.181690
\(221\) 14.6128 0.982963
\(222\) −11.9855 −0.804415
\(223\) −9.44842 −0.632713 −0.316357 0.948640i \(-0.602460\pi\)
−0.316357 + 0.948640i \(0.602460\pi\)
\(224\) 1.29978 0.0868450
\(225\) 1.20388 0.0802588
\(226\) 0.407582 0.0271119
\(227\) 15.2061 1.00926 0.504632 0.863335i \(-0.331628\pi\)
0.504632 + 0.863335i \(0.331628\pi\)
\(228\) 0.0351232 0.00232609
\(229\) 28.6411 1.89266 0.946329 0.323205i \(-0.104760\pi\)
0.946329 + 0.323205i \(0.104760\pi\)
\(230\) −0.946259 −0.0623945
\(231\) −7.98366 −0.525286
\(232\) −2.06303 −0.135445
\(233\) 3.14875 0.206281 0.103141 0.994667i \(-0.467111\pi\)
0.103141 + 0.994667i \(0.467111\pi\)
\(234\) −0.621154 −0.0406061
\(235\) 2.79506 0.182329
\(236\) 7.84518 0.510678
\(237\) 0.691173 0.0448965
\(238\) 8.22735 0.533300
\(239\) 12.5059 0.808940 0.404470 0.914551i \(-0.367456\pi\)
0.404470 + 0.914551i \(0.367456\pi\)
\(240\) 1.19817 0.0773417
\(241\) 3.73027 0.240287 0.120144 0.992757i \(-0.461664\pi\)
0.120144 + 0.992757i \(0.461664\pi\)
\(242\) 2.81514 0.180964
\(243\) 2.78751 0.178819
\(244\) 3.84958 0.246444
\(245\) −3.85040 −0.245993
\(246\) 14.5004 0.924511
\(247\) 0.0490660 0.00312199
\(248\) −5.82316 −0.369771
\(249\) −25.8330 −1.63710
\(250\) −6.86928 −0.434452
\(251\) −26.2795 −1.65875 −0.829374 0.558694i \(-0.811303\pi\)
−0.829374 + 0.558694i \(0.811303\pi\)
\(252\) −0.349724 −0.0220306
\(253\) 4.85092 0.304975
\(254\) −9.52589 −0.597708
\(255\) 7.58421 0.474942
\(256\) 1.00000 0.0625000
\(257\) 28.2237 1.76055 0.880273 0.474468i \(-0.157359\pi\)
0.880273 + 0.474468i \(0.157359\pi\)
\(258\) 7.47495 0.465370
\(259\) −9.42692 −0.585760
\(260\) 1.67381 0.103805
\(261\) 0.555090 0.0343592
\(262\) −21.9325 −1.35500
\(263\) 25.3865 1.56540 0.782701 0.622398i \(-0.213842\pi\)
0.782701 + 0.622398i \(0.213842\pi\)
\(264\) −6.14233 −0.378034
\(265\) −4.98049 −0.305949
\(266\) 0.0276253 0.00169382
\(267\) 17.1665 1.05057
\(268\) −9.31541 −0.569029
\(269\) −1.00000 −0.0609711
\(270\) −3.91690 −0.238375
\(271\) −17.9078 −1.08782 −0.543911 0.839143i \(-0.683057\pi\)
−0.543911 + 0.839143i \(0.683057\pi\)
\(272\) 6.32982 0.383802
\(273\) 4.95868 0.300113
\(274\) −2.48849 −0.150335
\(275\) 16.6304 1.00285
\(276\) −2.15676 −0.129822
\(277\) −6.51988 −0.391741 −0.195871 0.980630i \(-0.562753\pi\)
−0.195871 + 0.980630i \(0.562753\pi\)
\(278\) 21.6244 1.29694
\(279\) 1.56681 0.0938024
\(280\) 0.942394 0.0563188
\(281\) −2.61394 −0.155934 −0.0779672 0.996956i \(-0.524843\pi\)
−0.0779672 + 0.996956i \(0.524843\pi\)
\(282\) 6.37063 0.379366
\(283\) −9.65338 −0.573834 −0.286917 0.957955i \(-0.592630\pi\)
−0.286917 + 0.957955i \(0.592630\pi\)
\(284\) −0.643097 −0.0381608
\(285\) 0.0254658 0.00150847
\(286\) −8.58064 −0.507384
\(287\) 11.4049 0.673212
\(288\) −0.269065 −0.0158548
\(289\) 23.0666 1.35686
\(290\) −1.49579 −0.0878356
\(291\) 8.38934 0.491792
\(292\) −2.97920 −0.174345
\(293\) 2.04724 0.119601 0.0598005 0.998210i \(-0.480954\pi\)
0.0598005 + 0.998210i \(0.480954\pi\)
\(294\) −8.77602 −0.511828
\(295\) 5.68809 0.331173
\(296\) −7.25272 −0.421556
\(297\) 20.0797 1.16514
\(298\) 20.9106 1.21132
\(299\) −3.01293 −0.174242
\(300\) −7.39404 −0.426895
\(301\) 5.87924 0.338874
\(302\) −2.07209 −0.119235
\(303\) 20.7100 1.18976
\(304\) 0.0212539 0.00121899
\(305\) 2.79111 0.159818
\(306\) −1.70313 −0.0973616
\(307\) −18.6159 −1.06246 −0.531231 0.847227i \(-0.678271\pi\)
−0.531231 + 0.847227i \(0.678271\pi\)
\(308\) −4.83110 −0.275278
\(309\) 8.97956 0.510829
\(310\) −4.22204 −0.239796
\(311\) −9.88503 −0.560529 −0.280264 0.959923i \(-0.590422\pi\)
−0.280264 + 0.959923i \(0.590422\pi\)
\(312\) 3.81503 0.215983
\(313\) 17.4438 0.985983 0.492991 0.870034i \(-0.335903\pi\)
0.492991 + 0.870034i \(0.335903\pi\)
\(314\) −2.60975 −0.147277
\(315\) −0.253565 −0.0142868
\(316\) 0.418245 0.0235281
\(317\) 2.92828 0.164468 0.0822342 0.996613i \(-0.473794\pi\)
0.0822342 + 0.996613i \(0.473794\pi\)
\(318\) −11.3518 −0.636576
\(319\) 7.66802 0.429327
\(320\) 0.725043 0.0405311
\(321\) −5.70957 −0.318677
\(322\) −1.69635 −0.0945338
\(323\) 0.134533 0.00748563
\(324\) −8.12041 −0.451134
\(325\) −10.3292 −0.572963
\(326\) −6.63147 −0.367283
\(327\) −11.2506 −0.622159
\(328\) 8.77454 0.484493
\(329\) 5.01067 0.276247
\(330\) −4.45345 −0.245155
\(331\) 5.44140 0.299087 0.149543 0.988755i \(-0.452220\pi\)
0.149543 + 0.988755i \(0.452220\pi\)
\(332\) −15.6322 −0.857927
\(333\) 1.95145 0.106939
\(334\) 22.6369 1.23864
\(335\) −6.75407 −0.369014
\(336\) 2.14795 0.117180
\(337\) −3.73840 −0.203644 −0.101822 0.994803i \(-0.532467\pi\)
−0.101822 + 0.994803i \(0.532467\pi\)
\(338\) −7.67053 −0.417222
\(339\) 0.673551 0.0365823
\(340\) 4.58939 0.248895
\(341\) 21.6439 1.17209
\(342\) −0.00571868 −0.000309231 0
\(343\) −16.0010 −0.863973
\(344\) 4.52327 0.243878
\(345\) −1.56374 −0.0841892
\(346\) 12.2456 0.658329
\(347\) 10.8089 0.580252 0.290126 0.956988i \(-0.406303\pi\)
0.290126 + 0.956988i \(0.406303\pi\)
\(348\) −3.40927 −0.182756
\(349\) 19.8978 1.06510 0.532552 0.846397i \(-0.321233\pi\)
0.532552 + 0.846397i \(0.321233\pi\)
\(350\) −5.81561 −0.310857
\(351\) −12.4716 −0.665683
\(352\) −3.71687 −0.198110
\(353\) −11.8505 −0.630736 −0.315368 0.948969i \(-0.602128\pi\)
−0.315368 + 0.948969i \(0.602128\pi\)
\(354\) 12.9646 0.689060
\(355\) −0.466273 −0.0247472
\(356\) 10.3879 0.550556
\(357\) 13.5961 0.719584
\(358\) −18.0181 −0.952285
\(359\) 28.6632 1.51279 0.756393 0.654118i \(-0.226960\pi\)
0.756393 + 0.654118i \(0.226960\pi\)
\(360\) −0.195084 −0.0102818
\(361\) −18.9995 −0.999976
\(362\) 7.77690 0.408745
\(363\) 4.65217 0.244176
\(364\) 3.00062 0.157275
\(365\) −2.16005 −0.113062
\(366\) 6.36163 0.332528
\(367\) 13.1083 0.684246 0.342123 0.939655i \(-0.388854\pi\)
0.342123 + 0.939655i \(0.388854\pi\)
\(368\) −1.30511 −0.0680335
\(369\) −2.36092 −0.122905
\(370\) −5.25854 −0.273378
\(371\) −8.92847 −0.463543
\(372\) −9.62309 −0.498934
\(373\) −17.2738 −0.894406 −0.447203 0.894432i \(-0.647580\pi\)
−0.447203 + 0.894432i \(0.647580\pi\)
\(374\) −23.5271 −1.21656
\(375\) −11.3519 −0.586208
\(376\) 3.85502 0.198808
\(377\) −4.76264 −0.245288
\(378\) −7.02179 −0.361162
\(379\) 6.30803 0.324022 0.162011 0.986789i \(-0.448202\pi\)
0.162011 + 0.986789i \(0.448202\pi\)
\(380\) 0.0154100 0.000790515 0
\(381\) −15.7421 −0.806490
\(382\) 0.312243 0.0159757
\(383\) −28.2196 −1.44196 −0.720978 0.692958i \(-0.756307\pi\)
−0.720978 + 0.692958i \(0.756307\pi\)
\(384\) 1.65255 0.0843315
\(385\) −3.50276 −0.178517
\(386\) −0.336949 −0.0171502
\(387\) −1.21705 −0.0618663
\(388\) 5.07659 0.257725
\(389\) −3.15029 −0.159726 −0.0798629 0.996806i \(-0.525448\pi\)
−0.0798629 + 0.996806i \(0.525448\pi\)
\(390\) 2.76606 0.140065
\(391\) −8.26110 −0.417782
\(392\) −5.31058 −0.268225
\(393\) −36.2447 −1.82830
\(394\) 26.8275 1.35155
\(395\) 0.303246 0.0152579
\(396\) 1.00008 0.0502559
\(397\) 31.2926 1.57053 0.785265 0.619159i \(-0.212527\pi\)
0.785265 + 0.619159i \(0.212527\pi\)
\(398\) 4.03560 0.202286
\(399\) 0.0456523 0.00228547
\(400\) −4.47431 −0.223716
\(401\) 0.744072 0.0371572 0.0185786 0.999827i \(-0.494086\pi\)
0.0185786 + 0.999827i \(0.494086\pi\)
\(402\) −15.3942 −0.767794
\(403\) −13.4431 −0.669651
\(404\) 12.5321 0.623495
\(405\) −5.88764 −0.292559
\(406\) −2.68148 −0.133080
\(407\) 26.9574 1.33623
\(408\) 10.4604 0.517865
\(409\) 3.60242 0.178128 0.0890641 0.996026i \(-0.471612\pi\)
0.0890641 + 0.996026i \(0.471612\pi\)
\(410\) 6.36192 0.314193
\(411\) −4.11237 −0.202848
\(412\) 5.43374 0.267701
\(413\) 10.1970 0.501760
\(414\) 0.351159 0.0172585
\(415\) −11.3340 −0.556364
\(416\) 2.30856 0.113187
\(417\) 35.7355 1.74997
\(418\) −0.0789980 −0.00386392
\(419\) −17.1323 −0.836966 −0.418483 0.908225i \(-0.637438\pi\)
−0.418483 + 0.908225i \(0.637438\pi\)
\(420\) 1.55736 0.0759912
\(421\) 24.1651 1.17773 0.588866 0.808230i \(-0.299574\pi\)
0.588866 + 0.808230i \(0.299574\pi\)
\(422\) 9.22688 0.449158
\(423\) −1.03725 −0.0504329
\(424\) −6.86923 −0.333600
\(425\) −28.3216 −1.37380
\(426\) −1.06275 −0.0514905
\(427\) 5.00359 0.242141
\(428\) −3.45500 −0.167004
\(429\) −14.1800 −0.684615
\(430\) 3.27956 0.158155
\(431\) 16.8823 0.813193 0.406596 0.913608i \(-0.366716\pi\)
0.406596 + 0.913608i \(0.366716\pi\)
\(432\) −5.40231 −0.259919
\(433\) −13.0644 −0.627834 −0.313917 0.949450i \(-0.601641\pi\)
−0.313917 + 0.949450i \(0.601641\pi\)
\(434\) −7.56881 −0.363314
\(435\) −2.47187 −0.118517
\(436\) −6.80801 −0.326044
\(437\) −0.0277386 −0.00132692
\(438\) −4.92329 −0.235244
\(439\) 3.91599 0.186900 0.0934499 0.995624i \(-0.470211\pi\)
0.0934499 + 0.995624i \(0.470211\pi\)
\(440\) −2.69489 −0.128474
\(441\) 1.42889 0.0680425
\(442\) 14.6128 0.695060
\(443\) 25.2673 1.20049 0.600243 0.799818i \(-0.295071\pi\)
0.600243 + 0.799818i \(0.295071\pi\)
\(444\) −11.9855 −0.568808
\(445\) 7.53165 0.357034
\(446\) −9.44842 −0.447396
\(447\) 34.5559 1.63444
\(448\) 1.29978 0.0614087
\(449\) 27.4706 1.29642 0.648210 0.761462i \(-0.275518\pi\)
0.648210 + 0.761462i \(0.275518\pi\)
\(450\) 1.20388 0.0567515
\(451\) −32.6138 −1.53573
\(452\) 0.407582 0.0191710
\(453\) −3.42423 −0.160885
\(454\) 15.2061 0.713657
\(455\) 2.17558 0.101993
\(456\) 0.0351232 0.00164479
\(457\) −14.5345 −0.679896 −0.339948 0.940444i \(-0.610409\pi\)
−0.339948 + 0.940444i \(0.610409\pi\)
\(458\) 28.6411 1.33831
\(459\) −34.1956 −1.59611
\(460\) −0.946259 −0.0441196
\(461\) −10.4630 −0.487312 −0.243656 0.969862i \(-0.578347\pi\)
−0.243656 + 0.969862i \(0.578347\pi\)
\(462\) −7.98366 −0.371433
\(463\) −27.7519 −1.28974 −0.644871 0.764292i \(-0.723089\pi\)
−0.644871 + 0.764292i \(0.723089\pi\)
\(464\) −2.06303 −0.0957738
\(465\) −6.97715 −0.323558
\(466\) 3.14875 0.145863
\(467\) −12.3322 −0.570665 −0.285333 0.958429i \(-0.592104\pi\)
−0.285333 + 0.958429i \(0.592104\pi\)
\(468\) −0.621154 −0.0287129
\(469\) −12.1080 −0.559093
\(470\) 2.79506 0.128926
\(471\) −4.31276 −0.198721
\(472\) 7.84518 0.361104
\(473\) −16.8124 −0.773036
\(474\) 0.691173 0.0317466
\(475\) −0.0950965 −0.00436333
\(476\) 8.22735 0.377100
\(477\) 1.84827 0.0846265
\(478\) 12.5059 0.572007
\(479\) −2.92366 −0.133586 −0.0667928 0.997767i \(-0.521277\pi\)
−0.0667928 + 0.997767i \(0.521277\pi\)
\(480\) 1.19817 0.0546888
\(481\) −16.7434 −0.763432
\(482\) 3.73027 0.169909
\(483\) −2.80331 −0.127555
\(484\) 2.81514 0.127961
\(485\) 3.68075 0.167134
\(486\) 2.78751 0.126444
\(487\) 10.1739 0.461025 0.230513 0.973069i \(-0.425960\pi\)
0.230513 + 0.973069i \(0.425960\pi\)
\(488\) 3.84958 0.174262
\(489\) −10.9589 −0.495577
\(490\) −3.85040 −0.173943
\(491\) −16.8181 −0.758989 −0.379494 0.925194i \(-0.623902\pi\)
−0.379494 + 0.925194i \(0.623902\pi\)
\(492\) 14.5004 0.653728
\(493\) −13.0586 −0.588130
\(494\) 0.0490660 0.00220758
\(495\) 0.725101 0.0325909
\(496\) −5.82316 −0.261468
\(497\) −0.835882 −0.0374944
\(498\) −25.8330 −1.15761
\(499\) −30.0514 −1.34529 −0.672643 0.739967i \(-0.734841\pi\)
−0.672643 + 0.739967i \(0.734841\pi\)
\(500\) −6.86928 −0.307204
\(501\) 37.4088 1.67130
\(502\) −26.2795 −1.17291
\(503\) −6.83648 −0.304823 −0.152412 0.988317i \(-0.548704\pi\)
−0.152412 + 0.988317i \(0.548704\pi\)
\(504\) −0.349724 −0.0155780
\(505\) 9.08630 0.404335
\(506\) 4.85092 0.215650
\(507\) −12.6760 −0.562960
\(508\) −9.52589 −0.422643
\(509\) −16.9547 −0.751502 −0.375751 0.926721i \(-0.622615\pi\)
−0.375751 + 0.926721i \(0.622615\pi\)
\(510\) 7.58421 0.335835
\(511\) −3.87230 −0.171300
\(512\) 1.00000 0.0441942
\(513\) −0.114820 −0.00506943
\(514\) 28.2237 1.24489
\(515\) 3.93970 0.173604
\(516\) 7.47495 0.329066
\(517\) −14.3286 −0.630172
\(518\) −9.42692 −0.414195
\(519\) 20.2366 0.888287
\(520\) 1.67381 0.0734013
\(521\) 37.1136 1.62597 0.812987 0.582282i \(-0.197840\pi\)
0.812987 + 0.582282i \(0.197840\pi\)
\(522\) 0.555090 0.0242956
\(523\) −9.80481 −0.428734 −0.214367 0.976753i \(-0.568769\pi\)
−0.214367 + 0.976753i \(0.568769\pi\)
\(524\) −21.9325 −0.958127
\(525\) −9.61060 −0.419441
\(526\) 25.3865 1.10691
\(527\) −36.8596 −1.60563
\(528\) −6.14233 −0.267311
\(529\) −21.2967 −0.925943
\(530\) −4.98049 −0.216339
\(531\) −2.11086 −0.0916037
\(532\) 0.0276253 0.00119771
\(533\) 20.2566 0.877410
\(534\) 17.1665 0.742867
\(535\) −2.50502 −0.108302
\(536\) −9.31541 −0.402365
\(537\) −29.7758 −1.28492
\(538\) −1.00000 −0.0431131
\(539\) 19.7388 0.850208
\(540\) −3.91690 −0.168557
\(541\) −15.4647 −0.664878 −0.332439 0.943125i \(-0.607871\pi\)
−0.332439 + 0.943125i \(0.607871\pi\)
\(542\) −17.9078 −0.769206
\(543\) 12.8517 0.551521
\(544\) 6.32982 0.271389
\(545\) −4.93610 −0.211439
\(546\) 4.95868 0.212212
\(547\) −31.5080 −1.34719 −0.673593 0.739103i \(-0.735250\pi\)
−0.673593 + 0.739103i \(0.735250\pi\)
\(548\) −2.48849 −0.106303
\(549\) −1.03579 −0.0442063
\(550\) 16.6304 0.709125
\(551\) −0.0438474 −0.00186796
\(552\) −2.15676 −0.0917979
\(553\) 0.543625 0.0231173
\(554\) −6.51988 −0.277003
\(555\) −8.69001 −0.368871
\(556\) 21.6244 0.917078
\(557\) −5.45649 −0.231199 −0.115599 0.993296i \(-0.536879\pi\)
−0.115599 + 0.993296i \(0.536879\pi\)
\(558\) 1.56681 0.0663283
\(559\) 10.4423 0.441660
\(560\) 0.942394 0.0398234
\(561\) −38.8798 −1.64151
\(562\) −2.61394 −0.110262
\(563\) −5.91106 −0.249121 −0.124561 0.992212i \(-0.539752\pi\)
−0.124561 + 0.992212i \(0.539752\pi\)
\(564\) 6.37063 0.268252
\(565\) 0.295514 0.0124324
\(566\) −9.65338 −0.405762
\(567\) −10.5547 −0.443256
\(568\) −0.643097 −0.0269837
\(569\) 31.9474 1.33930 0.669652 0.742675i \(-0.266443\pi\)
0.669652 + 0.742675i \(0.266443\pi\)
\(570\) 0.0254658 0.00106665
\(571\) −13.7554 −0.575647 −0.287824 0.957683i \(-0.592932\pi\)
−0.287824 + 0.957683i \(0.592932\pi\)
\(572\) −8.58064 −0.358774
\(573\) 0.515998 0.0215561
\(574\) 11.4049 0.476033
\(575\) 5.83946 0.243522
\(576\) −0.269065 −0.0112110
\(577\) −2.93964 −0.122379 −0.0611893 0.998126i \(-0.519489\pi\)
−0.0611893 + 0.998126i \(0.519489\pi\)
\(578\) 23.0666 0.959444
\(579\) −0.556826 −0.0231409
\(580\) −1.49579 −0.0621091
\(581\) −20.3183 −0.842947
\(582\) 8.38934 0.347749
\(583\) 25.5321 1.05743
\(584\) −2.97920 −0.123280
\(585\) −0.450363 −0.0186202
\(586\) 2.04724 0.0845707
\(587\) 15.2876 0.630989 0.315494 0.948927i \(-0.397830\pi\)
0.315494 + 0.948927i \(0.397830\pi\)
\(588\) −8.77602 −0.361917
\(589\) −0.123765 −0.00509964
\(590\) 5.68809 0.234175
\(591\) 44.3340 1.82366
\(592\) −7.25272 −0.298085
\(593\) −33.5005 −1.37570 −0.687850 0.725853i \(-0.741445\pi\)
−0.687850 + 0.725853i \(0.741445\pi\)
\(594\) 20.0797 0.823879
\(595\) 5.96518 0.244549
\(596\) 20.9106 0.856532
\(597\) 6.66904 0.272946
\(598\) −3.01293 −0.123208
\(599\) 0.150988 0.00616920 0.00308460 0.999995i \(-0.499018\pi\)
0.00308460 + 0.999995i \(0.499018\pi\)
\(600\) −7.39404 −0.301861
\(601\) −34.9546 −1.42583 −0.712915 0.701251i \(-0.752625\pi\)
−0.712915 + 0.701251i \(0.752625\pi\)
\(602\) 5.87924 0.239620
\(603\) 2.50645 0.102071
\(604\) −2.07209 −0.0843120
\(605\) 2.04110 0.0829824
\(606\) 20.7100 0.841285
\(607\) 10.5301 0.427404 0.213702 0.976899i \(-0.431448\pi\)
0.213702 + 0.976899i \(0.431448\pi\)
\(608\) 0.0212539 0.000861959 0
\(609\) −4.43129 −0.179565
\(610\) 2.79111 0.113009
\(611\) 8.89957 0.360038
\(612\) −1.70313 −0.0688451
\(613\) −20.2271 −0.816965 −0.408483 0.912766i \(-0.633942\pi\)
−0.408483 + 0.912766i \(0.633942\pi\)
\(614\) −18.6159 −0.751275
\(615\) 10.5134 0.423942
\(616\) −4.83110 −0.194651
\(617\) 7.82969 0.315211 0.157606 0.987502i \(-0.449622\pi\)
0.157606 + 0.987502i \(0.449622\pi\)
\(618\) 8.97956 0.361211
\(619\) −23.6992 −0.952549 −0.476275 0.879297i \(-0.658013\pi\)
−0.476275 + 0.879297i \(0.658013\pi\)
\(620\) −4.22204 −0.169561
\(621\) 7.05059 0.282931
\(622\) −9.88503 −0.396354
\(623\) 13.5019 0.540942
\(624\) 3.81503 0.152723
\(625\) 17.3910 0.695642
\(626\) 17.4438 0.697195
\(627\) −0.130548 −0.00521360
\(628\) −2.60975 −0.104140
\(629\) −45.9084 −1.83049
\(630\) −0.253565 −0.0101023
\(631\) −31.8687 −1.26867 −0.634336 0.773058i \(-0.718726\pi\)
−0.634336 + 0.773058i \(0.718726\pi\)
\(632\) 0.418245 0.0166369
\(633\) 15.2479 0.606050
\(634\) 2.92828 0.116297
\(635\) −6.90668 −0.274083
\(636\) −11.3518 −0.450127
\(637\) −12.2598 −0.485752
\(638\) 7.66802 0.303580
\(639\) 0.173035 0.00684515
\(640\) 0.725043 0.0286598
\(641\) 10.3578 0.409108 0.204554 0.978855i \(-0.434426\pi\)
0.204554 + 0.978855i \(0.434426\pi\)
\(642\) −5.70957 −0.225339
\(643\) −21.9761 −0.866653 −0.433326 0.901237i \(-0.642660\pi\)
−0.433326 + 0.901237i \(0.642660\pi\)
\(644\) −1.69635 −0.0668455
\(645\) 5.41966 0.213399
\(646\) 0.134533 0.00529314
\(647\) 9.10376 0.357906 0.178953 0.983858i \(-0.442729\pi\)
0.178953 + 0.983858i \(0.442729\pi\)
\(648\) −8.12041 −0.319000
\(649\) −29.1595 −1.14461
\(650\) −10.3292 −0.405146
\(651\) −12.5079 −0.490222
\(652\) −6.63147 −0.259708
\(653\) 6.84892 0.268019 0.134010 0.990980i \(-0.457215\pi\)
0.134010 + 0.990980i \(0.457215\pi\)
\(654\) −11.2506 −0.439933
\(655\) −15.9020 −0.621344
\(656\) 8.77454 0.342588
\(657\) 0.801599 0.0312734
\(658\) 5.01067 0.195336
\(659\) −10.9102 −0.424999 −0.212500 0.977161i \(-0.568160\pi\)
−0.212500 + 0.977161i \(0.568160\pi\)
\(660\) −4.45345 −0.173350
\(661\) −9.26687 −0.360439 −0.180220 0.983626i \(-0.557681\pi\)
−0.180220 + 0.983626i \(0.557681\pi\)
\(662\) 5.44140 0.211486
\(663\) 24.1484 0.937847
\(664\) −15.6322 −0.606646
\(665\) 0.0200295 0.000776712 0
\(666\) 1.95145 0.0756173
\(667\) 2.69248 0.104253
\(668\) 22.6369 0.875850
\(669\) −15.6140 −0.603673
\(670\) −6.75407 −0.260933
\(671\) −14.3084 −0.552369
\(672\) 2.14795 0.0828590
\(673\) −15.0272 −0.579255 −0.289627 0.957139i \(-0.593531\pi\)
−0.289627 + 0.957139i \(0.593531\pi\)
\(674\) −3.73840 −0.143998
\(675\) 24.1716 0.930366
\(676\) −7.67053 −0.295020
\(677\) −46.3947 −1.78309 −0.891547 0.452929i \(-0.850379\pi\)
−0.891547 + 0.452929i \(0.850379\pi\)
\(678\) 0.673551 0.0258676
\(679\) 6.59843 0.253225
\(680\) 4.58939 0.175995
\(681\) 25.1289 0.962941
\(682\) 21.6439 0.828790
\(683\) −34.2506 −1.31056 −0.655281 0.755385i \(-0.727450\pi\)
−0.655281 + 0.755385i \(0.727450\pi\)
\(684\) −0.00571868 −0.000218659 0
\(685\) −1.80426 −0.0689373
\(686\) −16.0010 −0.610921
\(687\) 47.3310 1.80579
\(688\) 4.52327 0.172448
\(689\) −15.8581 −0.604144
\(690\) −1.56374 −0.0595307
\(691\) −1.19208 −0.0453489 −0.0226744 0.999743i \(-0.507218\pi\)
−0.0226744 + 0.999743i \(0.507218\pi\)
\(692\) 12.2456 0.465509
\(693\) 1.29988 0.0493784
\(694\) 10.8089 0.410300
\(695\) 15.6786 0.594723
\(696\) −3.40927 −0.129228
\(697\) 55.5412 2.10377
\(698\) 19.8978 0.753142
\(699\) 5.20348 0.196814
\(700\) −5.81561 −0.219809
\(701\) −24.6698 −0.931767 −0.465883 0.884846i \(-0.654263\pi\)
−0.465883 + 0.884846i \(0.654263\pi\)
\(702\) −12.4716 −0.470709
\(703\) −0.154149 −0.00581382
\(704\) −3.71687 −0.140085
\(705\) 4.61898 0.173961
\(706\) −11.8505 −0.445998
\(707\) 16.2889 0.612608
\(708\) 12.9646 0.487239
\(709\) 23.3891 0.878398 0.439199 0.898390i \(-0.355262\pi\)
0.439199 + 0.898390i \(0.355262\pi\)
\(710\) −0.466273 −0.0174989
\(711\) −0.112535 −0.00422040
\(712\) 10.3879 0.389302
\(713\) 7.59986 0.284617
\(714\) 13.5961 0.508823
\(715\) −6.22133 −0.232665
\(716\) −18.0181 −0.673367
\(717\) 20.6667 0.771812
\(718\) 28.6632 1.06970
\(719\) −1.52136 −0.0567371 −0.0283686 0.999598i \(-0.509031\pi\)
−0.0283686 + 0.999598i \(0.509031\pi\)
\(720\) −0.195084 −0.00727034
\(721\) 7.06265 0.263027
\(722\) −18.9995 −0.707090
\(723\) 6.16446 0.229259
\(724\) 7.77690 0.289026
\(725\) 9.23065 0.342818
\(726\) 4.65217 0.172658
\(727\) 36.3855 1.34946 0.674732 0.738063i \(-0.264259\pi\)
0.674732 + 0.738063i \(0.264259\pi\)
\(728\) 3.00062 0.111210
\(729\) 28.9677 1.07288
\(730\) −2.16005 −0.0799470
\(731\) 28.6315 1.05897
\(732\) 6.36163 0.235133
\(733\) −18.7869 −0.693908 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(734\) 13.1083 0.483835
\(735\) −6.36299 −0.234703
\(736\) −1.30511 −0.0481069
\(737\) 34.6242 1.27540
\(738\) −2.36092 −0.0869067
\(739\) −38.7776 −1.42646 −0.713228 0.700932i \(-0.752767\pi\)
−0.713228 + 0.700932i \(0.752767\pi\)
\(740\) −5.25854 −0.193308
\(741\) 0.0810841 0.00297870
\(742\) −8.92847 −0.327774
\(743\) −6.80190 −0.249537 −0.124769 0.992186i \(-0.539819\pi\)
−0.124769 + 0.992186i \(0.539819\pi\)
\(744\) −9.62309 −0.352800
\(745\) 15.1611 0.555460
\(746\) −17.2738 −0.632441
\(747\) 4.20607 0.153892
\(748\) −23.5271 −0.860237
\(749\) −4.49073 −0.164088
\(750\) −11.3519 −0.414511
\(751\) 10.5672 0.385601 0.192801 0.981238i \(-0.438243\pi\)
0.192801 + 0.981238i \(0.438243\pi\)
\(752\) 3.85502 0.140578
\(753\) −43.4283 −1.58262
\(754\) −4.76264 −0.173445
\(755\) −1.50235 −0.0546762
\(756\) −7.02179 −0.255380
\(757\) −22.6389 −0.822825 −0.411412 0.911449i \(-0.634964\pi\)
−0.411412 + 0.911449i \(0.634964\pi\)
\(758\) 6.30803 0.229118
\(759\) 8.01641 0.290977
\(760\) 0.0154100 0.000558979 0
\(761\) 1.83584 0.0665490 0.0332745 0.999446i \(-0.489406\pi\)
0.0332745 + 0.999446i \(0.489406\pi\)
\(762\) −15.7421 −0.570274
\(763\) −8.84889 −0.320351
\(764\) 0.312243 0.0112965
\(765\) −1.23484 −0.0446459
\(766\) −28.2196 −1.01962
\(767\) 18.1111 0.653954
\(768\) 1.65255 0.0596314
\(769\) 50.0137 1.80354 0.901770 0.432215i \(-0.142268\pi\)
0.901770 + 0.432215i \(0.142268\pi\)
\(770\) −3.50276 −0.126231
\(771\) 46.6412 1.67974
\(772\) −0.336949 −0.0121271
\(773\) 40.2988 1.44945 0.724724 0.689039i \(-0.241967\pi\)
0.724724 + 0.689039i \(0.241967\pi\)
\(774\) −1.21705 −0.0437461
\(775\) 26.0546 0.935911
\(776\) 5.07659 0.182239
\(777\) −15.5785 −0.558875
\(778\) −3.15029 −0.112943
\(779\) 0.186493 0.00668180
\(780\) 2.76606 0.0990408
\(781\) 2.39031 0.0855319
\(782\) −8.26110 −0.295416
\(783\) 11.1451 0.398294
\(784\) −5.31058 −0.189664
\(785\) −1.89218 −0.0675349
\(786\) −36.2447 −1.29281
\(787\) 38.2500 1.36347 0.681733 0.731601i \(-0.261227\pi\)
0.681733 + 0.731601i \(0.261227\pi\)
\(788\) 26.8275 0.955691
\(789\) 41.9526 1.49355
\(790\) 0.303246 0.0107890
\(791\) 0.529765 0.0188363
\(792\) 1.00008 0.0355363
\(793\) 8.88700 0.315586
\(794\) 31.2926 1.11053
\(795\) −8.23053 −0.291907
\(796\) 4.03560 0.143038
\(797\) −33.4247 −1.18397 −0.591983 0.805951i \(-0.701655\pi\)
−0.591983 + 0.805951i \(0.701655\pi\)
\(798\) 0.0456523 0.00161607
\(799\) 24.4016 0.863266
\(800\) −4.47431 −0.158191
\(801\) −2.79501 −0.0987569
\(802\) 0.744072 0.0262741
\(803\) 11.0733 0.390769
\(804\) −15.3942 −0.542912
\(805\) −1.22993 −0.0433492
\(806\) −13.4431 −0.473514
\(807\) −1.65255 −0.0581727
\(808\) 12.5321 0.440877
\(809\) −39.8841 −1.40225 −0.701125 0.713038i \(-0.747319\pi\)
−0.701125 + 0.713038i \(0.747319\pi\)
\(810\) −5.88764 −0.206871
\(811\) 34.7237 1.21931 0.609657 0.792665i \(-0.291307\pi\)
0.609657 + 0.792665i \(0.291307\pi\)
\(812\) −2.68148 −0.0941015
\(813\) −29.5936 −1.03789
\(814\) 26.9574 0.944858
\(815\) −4.80810 −0.168420
\(816\) 10.4604 0.366186
\(817\) 0.0961370 0.00336341
\(818\) 3.60242 0.125956
\(819\) −0.807361 −0.0282115
\(820\) 6.36192 0.222168
\(821\) −48.5169 −1.69325 −0.846626 0.532189i \(-0.821370\pi\)
−0.846626 + 0.532189i \(0.821370\pi\)
\(822\) −4.11237 −0.143435
\(823\) 19.5287 0.680728 0.340364 0.940294i \(-0.389450\pi\)
0.340364 + 0.940294i \(0.389450\pi\)
\(824\) 5.43374 0.189293
\(825\) 27.4827 0.956825
\(826\) 10.1970 0.354798
\(827\) 18.5400 0.644698 0.322349 0.946621i \(-0.395528\pi\)
0.322349 + 0.946621i \(0.395528\pi\)
\(828\) 0.351159 0.0122036
\(829\) −20.2344 −0.702769 −0.351385 0.936231i \(-0.614289\pi\)
−0.351385 + 0.936231i \(0.614289\pi\)
\(830\) −11.3340 −0.393409
\(831\) −10.7744 −0.373762
\(832\) 2.30856 0.0800351
\(833\) −33.6150 −1.16469
\(834\) 35.7355 1.23742
\(835\) 16.4128 0.567987
\(836\) −0.0789980 −0.00273220
\(837\) 31.4585 1.08737
\(838\) −17.1323 −0.591824
\(839\) 46.3931 1.60167 0.800833 0.598887i \(-0.204390\pi\)
0.800833 + 0.598887i \(0.204390\pi\)
\(840\) 1.55736 0.0537339
\(841\) −24.7439 −0.853238
\(842\) 24.1651 0.832783
\(843\) −4.31967 −0.148777
\(844\) 9.22688 0.317602
\(845\) −5.56146 −0.191320
\(846\) −1.03725 −0.0356615
\(847\) 3.65905 0.125727
\(848\) −6.86923 −0.235890
\(849\) −15.9527 −0.547496
\(850\) −28.3216 −0.971422
\(851\) 9.46559 0.324476
\(852\) −1.06275 −0.0364093
\(853\) −0.133230 −0.00456172 −0.00228086 0.999997i \(-0.500726\pi\)
−0.00228086 + 0.999997i \(0.500726\pi\)
\(854\) 5.00359 0.171219
\(855\) −0.00414629 −0.000141800 0
\(856\) −3.45500 −0.118089
\(857\) −10.2942 −0.351642 −0.175821 0.984422i \(-0.556258\pi\)
−0.175821 + 0.984422i \(0.556258\pi\)
\(858\) −14.1800 −0.484096
\(859\) 35.5814 1.21402 0.607011 0.794694i \(-0.292368\pi\)
0.607011 + 0.794694i \(0.292368\pi\)
\(860\) 3.27956 0.111832
\(861\) 18.8473 0.642313
\(862\) 16.8823 0.575014
\(863\) 17.9896 0.612372 0.306186 0.951972i \(-0.400947\pi\)
0.306186 + 0.951972i \(0.400947\pi\)
\(864\) −5.40231 −0.183790
\(865\) 8.87861 0.301882
\(866\) −13.0644 −0.443946
\(867\) 38.1188 1.29458
\(868\) −7.56881 −0.256902
\(869\) −1.55456 −0.0527350
\(870\) −2.47187 −0.0838042
\(871\) −21.5052 −0.728677
\(872\) −6.80801 −0.230548
\(873\) −1.36593 −0.0462298
\(874\) −0.0277386 −0.000938273 0
\(875\) −8.92853 −0.301839
\(876\) −4.92329 −0.166343
\(877\) 47.6131 1.60778 0.803890 0.594778i \(-0.202760\pi\)
0.803890 + 0.594778i \(0.202760\pi\)
\(878\) 3.91599 0.132158
\(879\) 3.38318 0.114112
\(880\) −2.69489 −0.0908448
\(881\) −15.7287 −0.529914 −0.264957 0.964260i \(-0.585358\pi\)
−0.264957 + 0.964260i \(0.585358\pi\)
\(882\) 1.42889 0.0481133
\(883\) −20.9111 −0.703713 −0.351856 0.936054i \(-0.614449\pi\)
−0.351856 + 0.936054i \(0.614449\pi\)
\(884\) 14.6128 0.491481
\(885\) 9.39988 0.315973
\(886\) 25.2673 0.848871
\(887\) 3.47543 0.116694 0.0583468 0.998296i \(-0.481417\pi\)
0.0583468 + 0.998296i \(0.481417\pi\)
\(888\) −11.9855 −0.402208
\(889\) −12.3815 −0.415263
\(890\) 7.53165 0.252461
\(891\) 30.1825 1.01115
\(892\) −9.44842 −0.316357
\(893\) 0.0819342 0.00274182
\(894\) 34.5559 1.15572
\(895\) −13.0639 −0.436677
\(896\) 1.29978 0.0434225
\(897\) −4.97902 −0.166245
\(898\) 27.4706 0.916707
\(899\) 12.0134 0.400668
\(900\) 1.20388 0.0401294
\(901\) −43.4810 −1.44856
\(902\) −32.6138 −1.08592
\(903\) 9.71576 0.323320
\(904\) 0.407582 0.0135560
\(905\) 5.63859 0.187433
\(906\) −3.42423 −0.113763
\(907\) 5.99370 0.199018 0.0995088 0.995037i \(-0.468273\pi\)
0.0995088 + 0.995037i \(0.468273\pi\)
\(908\) 15.2061 0.504632
\(909\) −3.37195 −0.111840
\(910\) 2.17558 0.0721196
\(911\) 3.66404 0.121395 0.0606976 0.998156i \(-0.480667\pi\)
0.0606976 + 0.998156i \(0.480667\pi\)
\(912\) 0.0351232 0.00116305
\(913\) 58.1028 1.92292
\(914\) −14.5345 −0.480759
\(915\) 4.61246 0.152483
\(916\) 28.6411 0.946329
\(917\) −28.5074 −0.941397
\(918\) −34.1956 −1.12862
\(919\) 44.8323 1.47888 0.739441 0.673222i \(-0.235090\pi\)
0.739441 + 0.673222i \(0.235090\pi\)
\(920\) −0.946259 −0.0311972
\(921\) −30.7637 −1.01370
\(922\) −10.4630 −0.344581
\(923\) −1.48463 −0.0488672
\(924\) −7.98366 −0.262643
\(925\) 32.4510 1.06698
\(926\) −27.7519 −0.911985
\(927\) −1.46203 −0.0480194
\(928\) −2.06303 −0.0677223
\(929\) −35.1823 −1.15429 −0.577146 0.816641i \(-0.695834\pi\)
−0.577146 + 0.816641i \(0.695834\pi\)
\(930\) −6.97715 −0.228790
\(931\) −0.112870 −0.00369918
\(932\) 3.14875 0.103141
\(933\) −16.3355 −0.534802
\(934\) −12.3322 −0.403521
\(935\) −17.0582 −0.557862
\(936\) −0.621154 −0.0203031
\(937\) 37.7329 1.23268 0.616339 0.787481i \(-0.288615\pi\)
0.616339 + 0.787481i \(0.288615\pi\)
\(938\) −12.1080 −0.395339
\(939\) 28.8268 0.940729
\(940\) 2.79506 0.0911647
\(941\) −31.0321 −1.01162 −0.505809 0.862646i \(-0.668806\pi\)
−0.505809 + 0.862646i \(0.668806\pi\)
\(942\) −4.31276 −0.140517
\(943\) −11.4517 −0.372919
\(944\) 7.84518 0.255339
\(945\) −5.09110 −0.165613
\(946\) −16.8124 −0.546619
\(947\) 19.5388 0.634924 0.317462 0.948271i \(-0.397169\pi\)
0.317462 + 0.948271i \(0.397169\pi\)
\(948\) 0.691173 0.0224482
\(949\) −6.87768 −0.223259
\(950\) −0.0950965 −0.00308534
\(951\) 4.83914 0.156920
\(952\) 8.22735 0.266650
\(953\) 15.9894 0.517948 0.258974 0.965884i \(-0.416616\pi\)
0.258974 + 0.965884i \(0.416616\pi\)
\(954\) 1.84827 0.0598400
\(955\) 0.226389 0.00732579
\(956\) 12.5059 0.404470
\(957\) 12.6718 0.409622
\(958\) −2.92366 −0.0944592
\(959\) −3.23448 −0.104447
\(960\) 1.19817 0.0386709
\(961\) 2.90921 0.0938456
\(962\) −16.7434 −0.539828
\(963\) 0.929619 0.0299566
\(964\) 3.73027 0.120144
\(965\) −0.244302 −0.00786437
\(966\) −2.80331 −0.0901950
\(967\) 38.1787 1.22774 0.613872 0.789406i \(-0.289611\pi\)
0.613872 + 0.789406i \(0.289611\pi\)
\(968\) 2.81514 0.0904820
\(969\) 0.222323 0.00714206
\(970\) 3.68075 0.118182
\(971\) 2.16411 0.0694497 0.0347248 0.999397i \(-0.488945\pi\)
0.0347248 + 0.999397i \(0.488945\pi\)
\(972\) 2.78751 0.0894093
\(973\) 28.1069 0.901064
\(974\) 10.1739 0.325994
\(975\) −17.0696 −0.546665
\(976\) 3.84958 0.123222
\(977\) −14.7297 −0.471243 −0.235622 0.971845i \(-0.575713\pi\)
−0.235622 + 0.971845i \(0.575713\pi\)
\(978\) −10.9589 −0.350426
\(979\) −38.6104 −1.23399
\(980\) −3.85040 −0.122996
\(981\) 1.83180 0.0584848
\(982\) −16.8181 −0.536686
\(983\) 36.4217 1.16167 0.580836 0.814020i \(-0.302726\pi\)
0.580836 + 0.814020i \(0.302726\pi\)
\(984\) 14.5004 0.462256
\(985\) 19.4511 0.619764
\(986\) −13.0586 −0.415871
\(987\) 8.28040 0.263568
\(988\) 0.0490660 0.00156100
\(989\) −5.90336 −0.187716
\(990\) 0.725101 0.0230452
\(991\) −24.5429 −0.779632 −0.389816 0.920893i \(-0.627461\pi\)
−0.389816 + 0.920893i \(0.627461\pi\)
\(992\) −5.82316 −0.184886
\(993\) 8.99221 0.285359
\(994\) −0.835882 −0.0265126
\(995\) 2.92598 0.0927598
\(996\) −25.8330 −0.818551
\(997\) −52.7413 −1.67033 −0.835167 0.549996i \(-0.814629\pi\)
−0.835167 + 0.549996i \(0.814629\pi\)
\(998\) −30.0514 −0.951260
\(999\) 39.1814 1.23965
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 538.2.a.e.1.5 7
3.2 odd 2 4842.2.a.n.1.4 7
4.3 odd 2 4304.2.a.h.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.e.1.5 7 1.1 even 1 trivial
4304.2.a.h.1.3 7 4.3 odd 2
4842.2.a.n.1.4 7 3.2 odd 2