Properties

Label 6039.2.a.n.1.19
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.07776 q^{2} -0.838436 q^{4} -3.04682 q^{5} +3.44893 q^{7} -3.05915 q^{8} +O(q^{10})\) \(q+1.07776 q^{2} -0.838436 q^{4} -3.04682 q^{5} +3.44893 q^{7} -3.05915 q^{8} -3.28374 q^{10} -1.00000 q^{11} +2.71326 q^{13} +3.71712 q^{14} -1.62015 q^{16} -3.42003 q^{17} -1.81606 q^{19} +2.55457 q^{20} -1.07776 q^{22} +6.08551 q^{23} +4.28313 q^{25} +2.92424 q^{26} -2.89171 q^{28} -0.376637 q^{29} +4.93900 q^{31} +4.37216 q^{32} -3.68597 q^{34} -10.5083 q^{35} +3.50288 q^{37} -1.95727 q^{38} +9.32069 q^{40} -5.87719 q^{41} -8.07962 q^{43} +0.838436 q^{44} +6.55872 q^{46} -6.22226 q^{47} +4.89515 q^{49} +4.61618 q^{50} -2.27489 q^{52} -2.61126 q^{53} +3.04682 q^{55} -10.5508 q^{56} -0.405924 q^{58} +2.54527 q^{59} -1.00000 q^{61} +5.32305 q^{62} +7.95245 q^{64} -8.26682 q^{65} +7.44820 q^{67} +2.86748 q^{68} -11.3254 q^{70} -4.42352 q^{71} +9.20943 q^{73} +3.77526 q^{74} +1.52265 q^{76} -3.44893 q^{77} +5.23667 q^{79} +4.93632 q^{80} -6.33419 q^{82} +8.81625 q^{83} +10.4202 q^{85} -8.70788 q^{86} +3.05915 q^{88} -8.15358 q^{89} +9.35785 q^{91} -5.10231 q^{92} -6.70609 q^{94} +5.53320 q^{95} +1.34982 q^{97} +5.27579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07776 0.762091 0.381045 0.924556i \(-0.375564\pi\)
0.381045 + 0.924556i \(0.375564\pi\)
\(3\) 0 0
\(4\) −0.838436 −0.419218
\(5\) −3.04682 −1.36258 −0.681290 0.732013i \(-0.738581\pi\)
−0.681290 + 0.732013i \(0.738581\pi\)
\(6\) 0 0
\(7\) 3.44893 1.30357 0.651787 0.758402i \(-0.274019\pi\)
0.651787 + 0.758402i \(0.274019\pi\)
\(8\) −3.05915 −1.08157
\(9\) 0 0
\(10\) −3.28374 −1.03841
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.71326 0.752523 0.376261 0.926514i \(-0.377209\pi\)
0.376261 + 0.926514i \(0.377209\pi\)
\(14\) 3.71712 0.993442
\(15\) 0 0
\(16\) −1.62015 −0.405038
\(17\) −3.42003 −0.829480 −0.414740 0.909940i \(-0.636127\pi\)
−0.414740 + 0.909940i \(0.636127\pi\)
\(18\) 0 0
\(19\) −1.81606 −0.416632 −0.208316 0.978062i \(-0.566798\pi\)
−0.208316 + 0.978062i \(0.566798\pi\)
\(20\) 2.55457 0.571218
\(21\) 0 0
\(22\) −1.07776 −0.229779
\(23\) 6.08551 1.26892 0.634459 0.772957i \(-0.281223\pi\)
0.634459 + 0.772957i \(0.281223\pi\)
\(24\) 0 0
\(25\) 4.28313 0.856626
\(26\) 2.92424 0.573491
\(27\) 0 0
\(28\) −2.89171 −0.546482
\(29\) −0.376637 −0.0699397 −0.0349699 0.999388i \(-0.511134\pi\)
−0.0349699 + 0.999388i \(0.511134\pi\)
\(30\) 0 0
\(31\) 4.93900 0.887071 0.443536 0.896257i \(-0.353724\pi\)
0.443536 + 0.896257i \(0.353724\pi\)
\(32\) 4.37216 0.772897
\(33\) 0 0
\(34\) −3.68597 −0.632139
\(35\) −10.5083 −1.77623
\(36\) 0 0
\(37\) 3.50288 0.575870 0.287935 0.957650i \(-0.407031\pi\)
0.287935 + 0.957650i \(0.407031\pi\)
\(38\) −1.95727 −0.317511
\(39\) 0 0
\(40\) 9.32069 1.47373
\(41\) −5.87719 −0.917862 −0.458931 0.888472i \(-0.651768\pi\)
−0.458931 + 0.888472i \(0.651768\pi\)
\(42\) 0 0
\(43\) −8.07962 −1.23213 −0.616065 0.787695i \(-0.711274\pi\)
−0.616065 + 0.787695i \(0.711274\pi\)
\(44\) 0.838436 0.126399
\(45\) 0 0
\(46\) 6.55872 0.967030
\(47\) −6.22226 −0.907609 −0.453805 0.891101i \(-0.649934\pi\)
−0.453805 + 0.891101i \(0.649934\pi\)
\(48\) 0 0
\(49\) 4.89515 0.699307
\(50\) 4.61618 0.652827
\(51\) 0 0
\(52\) −2.27489 −0.315471
\(53\) −2.61126 −0.358684 −0.179342 0.983787i \(-0.557397\pi\)
−0.179342 + 0.983787i \(0.557397\pi\)
\(54\) 0 0
\(55\) 3.04682 0.410834
\(56\) −10.5508 −1.40991
\(57\) 0 0
\(58\) −0.405924 −0.0533004
\(59\) 2.54527 0.331365 0.165683 0.986179i \(-0.447017\pi\)
0.165683 + 0.986179i \(0.447017\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 5.32305 0.676028
\(63\) 0 0
\(64\) 7.95245 0.994056
\(65\) −8.26682 −1.02537
\(66\) 0 0
\(67\) 7.44820 0.909942 0.454971 0.890506i \(-0.349650\pi\)
0.454971 + 0.890506i \(0.349650\pi\)
\(68\) 2.86748 0.347733
\(69\) 0 0
\(70\) −11.3254 −1.35364
\(71\) −4.42352 −0.524975 −0.262488 0.964935i \(-0.584543\pi\)
−0.262488 + 0.964935i \(0.584543\pi\)
\(72\) 0 0
\(73\) 9.20943 1.07788 0.538941 0.842344i \(-0.318825\pi\)
0.538941 + 0.842344i \(0.318825\pi\)
\(74\) 3.77526 0.438865
\(75\) 0 0
\(76\) 1.52265 0.174660
\(77\) −3.44893 −0.393043
\(78\) 0 0
\(79\) 5.23667 0.589171 0.294585 0.955625i \(-0.404818\pi\)
0.294585 + 0.955625i \(0.404818\pi\)
\(80\) 4.93632 0.551898
\(81\) 0 0
\(82\) −6.33419 −0.699494
\(83\) 8.81625 0.967710 0.483855 0.875148i \(-0.339236\pi\)
0.483855 + 0.875148i \(0.339236\pi\)
\(84\) 0 0
\(85\) 10.4202 1.13023
\(86\) −8.70788 −0.938995
\(87\) 0 0
\(88\) 3.05915 0.326106
\(89\) −8.15358 −0.864278 −0.432139 0.901807i \(-0.642241\pi\)
−0.432139 + 0.901807i \(0.642241\pi\)
\(90\) 0 0
\(91\) 9.35785 0.980970
\(92\) −5.10231 −0.531953
\(93\) 0 0
\(94\) −6.70609 −0.691681
\(95\) 5.53320 0.567695
\(96\) 0 0
\(97\) 1.34982 0.137054 0.0685269 0.997649i \(-0.478170\pi\)
0.0685269 + 0.997649i \(0.478170\pi\)
\(98\) 5.27579 0.532935
\(99\) 0 0
\(100\) −3.59113 −0.359113
\(101\) −4.97047 −0.494580 −0.247290 0.968942i \(-0.579540\pi\)
−0.247290 + 0.968942i \(0.579540\pi\)
\(102\) 0 0
\(103\) −18.1042 −1.78386 −0.891929 0.452176i \(-0.850648\pi\)
−0.891929 + 0.452176i \(0.850648\pi\)
\(104\) −8.30027 −0.813908
\(105\) 0 0
\(106\) −2.81431 −0.273350
\(107\) −1.38867 −0.134247 −0.0671237 0.997745i \(-0.521382\pi\)
−0.0671237 + 0.997745i \(0.521382\pi\)
\(108\) 0 0
\(109\) −11.4775 −1.09935 −0.549673 0.835380i \(-0.685248\pi\)
−0.549673 + 0.835380i \(0.685248\pi\)
\(110\) 3.28374 0.313092
\(111\) 0 0
\(112\) −5.58780 −0.527998
\(113\) −8.65339 −0.814042 −0.407021 0.913419i \(-0.633432\pi\)
−0.407021 + 0.913419i \(0.633432\pi\)
\(114\) 0 0
\(115\) −18.5415 −1.72900
\(116\) 0.315786 0.0293200
\(117\) 0 0
\(118\) 2.74318 0.252530
\(119\) −11.7955 −1.08129
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.07776 −0.0975757
\(123\) 0 0
\(124\) −4.14104 −0.371876
\(125\) 2.18417 0.195358
\(126\) 0 0
\(127\) −8.67839 −0.770082 −0.385041 0.922899i \(-0.625813\pi\)
−0.385041 + 0.922899i \(0.625813\pi\)
\(128\) −0.173509 −0.0153362
\(129\) 0 0
\(130\) −8.90964 −0.781427
\(131\) −3.54246 −0.309506 −0.154753 0.987953i \(-0.549458\pi\)
−0.154753 + 0.987953i \(0.549458\pi\)
\(132\) 0 0
\(133\) −6.26346 −0.543111
\(134\) 8.02736 0.693458
\(135\) 0 0
\(136\) 10.4624 0.897143
\(137\) −18.0703 −1.54385 −0.771924 0.635715i \(-0.780705\pi\)
−0.771924 + 0.635715i \(0.780705\pi\)
\(138\) 0 0
\(139\) −3.32448 −0.281979 −0.140989 0.990011i \(-0.545028\pi\)
−0.140989 + 0.990011i \(0.545028\pi\)
\(140\) 8.81053 0.744626
\(141\) 0 0
\(142\) −4.76749 −0.400079
\(143\) −2.71326 −0.226894
\(144\) 0 0
\(145\) 1.14755 0.0952986
\(146\) 9.92554 0.821444
\(147\) 0 0
\(148\) −2.93694 −0.241415
\(149\) 1.93908 0.158856 0.0794278 0.996841i \(-0.474691\pi\)
0.0794278 + 0.996841i \(0.474691\pi\)
\(150\) 0 0
\(151\) −8.63107 −0.702386 −0.351193 0.936303i \(-0.614224\pi\)
−0.351193 + 0.936303i \(0.614224\pi\)
\(152\) 5.55559 0.450618
\(153\) 0 0
\(154\) −3.71712 −0.299534
\(155\) −15.0483 −1.20871
\(156\) 0 0
\(157\) −7.40501 −0.590984 −0.295492 0.955345i \(-0.595484\pi\)
−0.295492 + 0.955345i \(0.595484\pi\)
\(158\) 5.64386 0.449002
\(159\) 0 0
\(160\) −13.3212 −1.05313
\(161\) 20.9885 1.65413
\(162\) 0 0
\(163\) −4.33911 −0.339865 −0.169933 0.985456i \(-0.554355\pi\)
−0.169933 + 0.985456i \(0.554355\pi\)
\(164\) 4.92764 0.384784
\(165\) 0 0
\(166\) 9.50180 0.737482
\(167\) −12.1403 −0.939441 −0.469721 0.882815i \(-0.655645\pi\)
−0.469721 + 0.882815i \(0.655645\pi\)
\(168\) 0 0
\(169\) −5.63822 −0.433709
\(170\) 11.2305 0.861340
\(171\) 0 0
\(172\) 6.77424 0.516531
\(173\) 4.34300 0.330192 0.165096 0.986278i \(-0.447207\pi\)
0.165096 + 0.986278i \(0.447207\pi\)
\(174\) 0 0
\(175\) 14.7722 1.11668
\(176\) 1.62015 0.122124
\(177\) 0 0
\(178\) −8.78759 −0.658658
\(179\) −6.46245 −0.483026 −0.241513 0.970398i \(-0.577644\pi\)
−0.241513 + 0.970398i \(0.577644\pi\)
\(180\) 0 0
\(181\) −14.7017 −1.09277 −0.546385 0.837534i \(-0.683996\pi\)
−0.546385 + 0.837534i \(0.683996\pi\)
\(182\) 10.0855 0.747588
\(183\) 0 0
\(184\) −18.6165 −1.37243
\(185\) −10.6727 −0.784669
\(186\) 0 0
\(187\) 3.42003 0.250098
\(188\) 5.21696 0.380486
\(189\) 0 0
\(190\) 5.96346 0.432635
\(191\) −24.4638 −1.77014 −0.885070 0.465459i \(-0.845889\pi\)
−0.885070 + 0.465459i \(0.845889\pi\)
\(192\) 0 0
\(193\) −2.45063 −0.176400 −0.0882001 0.996103i \(-0.528111\pi\)
−0.0882001 + 0.996103i \(0.528111\pi\)
\(194\) 1.45478 0.104447
\(195\) 0 0
\(196\) −4.10427 −0.293162
\(197\) −9.82650 −0.700109 −0.350055 0.936729i \(-0.613837\pi\)
−0.350055 + 0.936729i \(0.613837\pi\)
\(198\) 0 0
\(199\) −18.7230 −1.32724 −0.663619 0.748071i \(-0.730980\pi\)
−0.663619 + 0.748071i \(0.730980\pi\)
\(200\) −13.1027 −0.926503
\(201\) 0 0
\(202\) −5.35697 −0.376915
\(203\) −1.29900 −0.0911717
\(204\) 0 0
\(205\) 17.9067 1.25066
\(206\) −19.5119 −1.35946
\(207\) 0 0
\(208\) −4.39590 −0.304801
\(209\) 1.81606 0.125619
\(210\) 0 0
\(211\) −12.2966 −0.846531 −0.423265 0.906006i \(-0.639116\pi\)
−0.423265 + 0.906006i \(0.639116\pi\)
\(212\) 2.18938 0.150367
\(213\) 0 0
\(214\) −1.49665 −0.102309
\(215\) 24.6172 1.67888
\(216\) 0 0
\(217\) 17.0343 1.15636
\(218\) −12.3700 −0.837801
\(219\) 0 0
\(220\) −2.55457 −0.172229
\(221\) −9.27944 −0.624203
\(222\) 0 0
\(223\) −0.629713 −0.0421687 −0.0210844 0.999778i \(-0.506712\pi\)
−0.0210844 + 0.999778i \(0.506712\pi\)
\(224\) 15.0793 1.00753
\(225\) 0 0
\(226\) −9.32627 −0.620374
\(227\) 21.8544 1.45053 0.725264 0.688471i \(-0.241718\pi\)
0.725264 + 0.688471i \(0.241718\pi\)
\(228\) 0 0
\(229\) 21.6849 1.43298 0.716489 0.697598i \(-0.245748\pi\)
0.716489 + 0.697598i \(0.245748\pi\)
\(230\) −19.9832 −1.31766
\(231\) 0 0
\(232\) 1.15219 0.0756449
\(233\) −13.8094 −0.904681 −0.452341 0.891845i \(-0.649411\pi\)
−0.452341 + 0.891845i \(0.649411\pi\)
\(234\) 0 0
\(235\) 18.9581 1.23669
\(236\) −2.13404 −0.138914
\(237\) 0 0
\(238\) −12.7127 −0.824040
\(239\) −8.10196 −0.524072 −0.262036 0.965058i \(-0.584394\pi\)
−0.262036 + 0.965058i \(0.584394\pi\)
\(240\) 0 0
\(241\) −5.88395 −0.379018 −0.189509 0.981879i \(-0.560690\pi\)
−0.189509 + 0.981879i \(0.560690\pi\)
\(242\) 1.07776 0.0692810
\(243\) 0 0
\(244\) 0.838436 0.0536754
\(245\) −14.9147 −0.952862
\(246\) 0 0
\(247\) −4.92743 −0.313525
\(248\) −15.1091 −0.959432
\(249\) 0 0
\(250\) 2.35401 0.148881
\(251\) 18.2741 1.15345 0.576726 0.816938i \(-0.304330\pi\)
0.576726 + 0.816938i \(0.304330\pi\)
\(252\) 0 0
\(253\) −6.08551 −0.382593
\(254\) −9.35321 −0.586872
\(255\) 0 0
\(256\) −16.0919 −1.00574
\(257\) 3.47117 0.216526 0.108263 0.994122i \(-0.465471\pi\)
0.108263 + 0.994122i \(0.465471\pi\)
\(258\) 0 0
\(259\) 12.0812 0.750689
\(260\) 6.93120 0.429855
\(261\) 0 0
\(262\) −3.81791 −0.235871
\(263\) −1.56304 −0.0963814 −0.0481907 0.998838i \(-0.515346\pi\)
−0.0481907 + 0.998838i \(0.515346\pi\)
\(264\) 0 0
\(265\) 7.95605 0.488737
\(266\) −6.75050 −0.413900
\(267\) 0 0
\(268\) −6.24483 −0.381464
\(269\) 11.7725 0.717782 0.358891 0.933379i \(-0.383155\pi\)
0.358891 + 0.933379i \(0.383155\pi\)
\(270\) 0 0
\(271\) 20.5162 1.24627 0.623135 0.782114i \(-0.285859\pi\)
0.623135 + 0.782114i \(0.285859\pi\)
\(272\) 5.54098 0.335971
\(273\) 0 0
\(274\) −19.4754 −1.17655
\(275\) −4.28313 −0.258283
\(276\) 0 0
\(277\) −0.101873 −0.00612094 −0.00306047 0.999995i \(-0.500974\pi\)
−0.00306047 + 0.999995i \(0.500974\pi\)
\(278\) −3.58299 −0.214893
\(279\) 0 0
\(280\) 32.1464 1.92112
\(281\) 14.3249 0.854552 0.427276 0.904121i \(-0.359473\pi\)
0.427276 + 0.904121i \(0.359473\pi\)
\(282\) 0 0
\(283\) −26.4737 −1.57370 −0.786850 0.617144i \(-0.788290\pi\)
−0.786850 + 0.617144i \(0.788290\pi\)
\(284\) 3.70884 0.220079
\(285\) 0 0
\(286\) −2.92424 −0.172914
\(287\) −20.2700 −1.19650
\(288\) 0 0
\(289\) −5.30336 −0.311963
\(290\) 1.23678 0.0726261
\(291\) 0 0
\(292\) −7.72151 −0.451867
\(293\) −7.46832 −0.436304 −0.218152 0.975915i \(-0.570003\pi\)
−0.218152 + 0.975915i \(0.570003\pi\)
\(294\) 0 0
\(295\) −7.75498 −0.451512
\(296\) −10.7158 −0.622845
\(297\) 0 0
\(298\) 2.08986 0.121062
\(299\) 16.5116 0.954889
\(300\) 0 0
\(301\) −27.8661 −1.60617
\(302\) −9.30221 −0.535282
\(303\) 0 0
\(304\) 2.94229 0.168752
\(305\) 3.04682 0.174461
\(306\) 0 0
\(307\) 23.2988 1.32973 0.664867 0.746962i \(-0.268488\pi\)
0.664867 + 0.746962i \(0.268488\pi\)
\(308\) 2.89171 0.164770
\(309\) 0 0
\(310\) −16.2184 −0.921143
\(311\) −17.3926 −0.986244 −0.493122 0.869960i \(-0.664144\pi\)
−0.493122 + 0.869960i \(0.664144\pi\)
\(312\) 0 0
\(313\) −1.69958 −0.0960659 −0.0480330 0.998846i \(-0.515295\pi\)
−0.0480330 + 0.998846i \(0.515295\pi\)
\(314\) −7.98081 −0.450383
\(315\) 0 0
\(316\) −4.39061 −0.246991
\(317\) 29.6094 1.66303 0.831515 0.555502i \(-0.187474\pi\)
0.831515 + 0.555502i \(0.187474\pi\)
\(318\) 0 0
\(319\) 0.376637 0.0210876
\(320\) −24.2297 −1.35448
\(321\) 0 0
\(322\) 22.6206 1.26060
\(323\) 6.21098 0.345588
\(324\) 0 0
\(325\) 11.6212 0.644631
\(326\) −4.67651 −0.259008
\(327\) 0 0
\(328\) 17.9792 0.992735
\(329\) −21.4602 −1.18314
\(330\) 0 0
\(331\) −4.47943 −0.246211 −0.123106 0.992394i \(-0.539285\pi\)
−0.123106 + 0.992394i \(0.539285\pi\)
\(332\) −7.39186 −0.405681
\(333\) 0 0
\(334\) −13.0843 −0.715939
\(335\) −22.6933 −1.23987
\(336\) 0 0
\(337\) 6.91513 0.376691 0.188345 0.982103i \(-0.439688\pi\)
0.188345 + 0.982103i \(0.439688\pi\)
\(338\) −6.07664 −0.330526
\(339\) 0 0
\(340\) −8.73670 −0.473814
\(341\) −4.93900 −0.267462
\(342\) 0 0
\(343\) −7.25949 −0.391976
\(344\) 24.7168 1.33264
\(345\) 0 0
\(346\) 4.68070 0.251636
\(347\) −18.5560 −0.996140 −0.498070 0.867137i \(-0.665958\pi\)
−0.498070 + 0.867137i \(0.665958\pi\)
\(348\) 0 0
\(349\) −6.98021 −0.373642 −0.186821 0.982394i \(-0.559818\pi\)
−0.186821 + 0.982394i \(0.559818\pi\)
\(350\) 15.9209 0.851008
\(351\) 0 0
\(352\) −4.37216 −0.233037
\(353\) −25.0961 −1.33573 −0.667867 0.744281i \(-0.732792\pi\)
−0.667867 + 0.744281i \(0.732792\pi\)
\(354\) 0 0
\(355\) 13.4777 0.715321
\(356\) 6.83625 0.362321
\(357\) 0 0
\(358\) −6.96496 −0.368110
\(359\) 26.6233 1.40513 0.702563 0.711621i \(-0.252039\pi\)
0.702563 + 0.711621i \(0.252039\pi\)
\(360\) 0 0
\(361\) −15.7019 −0.826418
\(362\) −15.8449 −0.832790
\(363\) 0 0
\(364\) −7.84596 −0.411240
\(365\) −28.0595 −1.46870
\(366\) 0 0
\(367\) 2.88012 0.150341 0.0751705 0.997171i \(-0.476050\pi\)
0.0751705 + 0.997171i \(0.476050\pi\)
\(368\) −9.85947 −0.513960
\(369\) 0 0
\(370\) −11.5025 −0.597989
\(371\) −9.00607 −0.467572
\(372\) 0 0
\(373\) −5.86690 −0.303777 −0.151888 0.988398i \(-0.548535\pi\)
−0.151888 + 0.988398i \(0.548535\pi\)
\(374\) 3.68597 0.190597
\(375\) 0 0
\(376\) 19.0348 0.981646
\(377\) −1.02191 −0.0526313
\(378\) 0 0
\(379\) 25.8822 1.32948 0.664739 0.747076i \(-0.268543\pi\)
0.664739 + 0.747076i \(0.268543\pi\)
\(380\) −4.63924 −0.237988
\(381\) 0 0
\(382\) −26.3661 −1.34901
\(383\) 18.2813 0.934133 0.467066 0.884222i \(-0.345311\pi\)
0.467066 + 0.884222i \(0.345311\pi\)
\(384\) 0 0
\(385\) 10.5083 0.535552
\(386\) −2.64119 −0.134433
\(387\) 0 0
\(388\) −1.13174 −0.0574554
\(389\) −6.60365 −0.334818 −0.167409 0.985887i \(-0.553540\pi\)
−0.167409 + 0.985887i \(0.553540\pi\)
\(390\) 0 0
\(391\) −20.8127 −1.05254
\(392\) −14.9750 −0.756351
\(393\) 0 0
\(394\) −10.5906 −0.533547
\(395\) −15.9552 −0.802793
\(396\) 0 0
\(397\) 5.31584 0.266795 0.133397 0.991063i \(-0.457411\pi\)
0.133397 + 0.991063i \(0.457411\pi\)
\(398\) −20.1789 −1.01148
\(399\) 0 0
\(400\) −6.93933 −0.346967
\(401\) 23.5518 1.17612 0.588060 0.808817i \(-0.299892\pi\)
0.588060 + 0.808817i \(0.299892\pi\)
\(402\) 0 0
\(403\) 13.4008 0.667541
\(404\) 4.16742 0.207337
\(405\) 0 0
\(406\) −1.40001 −0.0694811
\(407\) −3.50288 −0.173631
\(408\) 0 0
\(409\) 29.8085 1.47393 0.736967 0.675929i \(-0.236258\pi\)
0.736967 + 0.675929i \(0.236258\pi\)
\(410\) 19.2992 0.953117
\(411\) 0 0
\(412\) 15.1792 0.747825
\(413\) 8.77846 0.431960
\(414\) 0 0
\(415\) −26.8616 −1.31858
\(416\) 11.8628 0.581622
\(417\) 0 0
\(418\) 1.95727 0.0957333
\(419\) 22.0112 1.07532 0.537660 0.843162i \(-0.319309\pi\)
0.537660 + 0.843162i \(0.319309\pi\)
\(420\) 0 0
\(421\) −2.64031 −0.128681 −0.0643404 0.997928i \(-0.520494\pi\)
−0.0643404 + 0.997928i \(0.520494\pi\)
\(422\) −13.2527 −0.645133
\(423\) 0 0
\(424\) 7.98824 0.387943
\(425\) −14.6485 −0.710554
\(426\) 0 0
\(427\) −3.44893 −0.166906
\(428\) 1.16431 0.0562790
\(429\) 0 0
\(430\) 26.5314 1.27946
\(431\) −11.5029 −0.554076 −0.277038 0.960859i \(-0.589353\pi\)
−0.277038 + 0.960859i \(0.589353\pi\)
\(432\) 0 0
\(433\) 19.3604 0.930403 0.465201 0.885205i \(-0.345982\pi\)
0.465201 + 0.885205i \(0.345982\pi\)
\(434\) 18.3589 0.881254
\(435\) 0 0
\(436\) 9.62316 0.460866
\(437\) −11.0516 −0.528671
\(438\) 0 0
\(439\) −13.8560 −0.661312 −0.330656 0.943751i \(-0.607270\pi\)
−0.330656 + 0.943751i \(0.607270\pi\)
\(440\) −9.32069 −0.444346
\(441\) 0 0
\(442\) −10.0010 −0.475699
\(443\) −13.1908 −0.626714 −0.313357 0.949635i \(-0.601454\pi\)
−0.313357 + 0.949635i \(0.601454\pi\)
\(444\) 0 0
\(445\) 24.8425 1.17765
\(446\) −0.678679 −0.0321364
\(447\) 0 0
\(448\) 27.4275 1.29583
\(449\) −10.4842 −0.494782 −0.247391 0.968916i \(-0.579573\pi\)
−0.247391 + 0.968916i \(0.579573\pi\)
\(450\) 0 0
\(451\) 5.87719 0.276746
\(452\) 7.25531 0.341261
\(453\) 0 0
\(454\) 23.5538 1.10543
\(455\) −28.5117 −1.33665
\(456\) 0 0
\(457\) 5.55221 0.259722 0.129861 0.991532i \(-0.458547\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(458\) 23.3711 1.09206
\(459\) 0 0
\(460\) 15.5458 0.724829
\(461\) 14.2444 0.663430 0.331715 0.943380i \(-0.392373\pi\)
0.331715 + 0.943380i \(0.392373\pi\)
\(462\) 0 0
\(463\) 5.17058 0.240297 0.120149 0.992756i \(-0.461663\pi\)
0.120149 + 0.992756i \(0.461663\pi\)
\(464\) 0.610210 0.0283283
\(465\) 0 0
\(466\) −14.8832 −0.689449
\(467\) −36.3845 −1.68368 −0.841838 0.539731i \(-0.818526\pi\)
−0.841838 + 0.539731i \(0.818526\pi\)
\(468\) 0 0
\(469\) 25.6883 1.18618
\(470\) 20.4323 0.942471
\(471\) 0 0
\(472\) −7.78635 −0.358396
\(473\) 8.07962 0.371501
\(474\) 0 0
\(475\) −7.77841 −0.356898
\(476\) 9.88975 0.453296
\(477\) 0 0
\(478\) −8.73196 −0.399390
\(479\) −37.5482 −1.71562 −0.857810 0.513967i \(-0.828175\pi\)
−0.857810 + 0.513967i \(0.828175\pi\)
\(480\) 0 0
\(481\) 9.50422 0.433355
\(482\) −6.34148 −0.288846
\(483\) 0 0
\(484\) −0.838436 −0.0381107
\(485\) −4.11267 −0.186747
\(486\) 0 0
\(487\) 11.4763 0.520042 0.260021 0.965603i \(-0.416270\pi\)
0.260021 + 0.965603i \(0.416270\pi\)
\(488\) 3.05915 0.138481
\(489\) 0 0
\(490\) −16.0744 −0.726167
\(491\) −15.7233 −0.709581 −0.354791 0.934946i \(-0.615448\pi\)
−0.354791 + 0.934946i \(0.615448\pi\)
\(492\) 0 0
\(493\) 1.28811 0.0580136
\(494\) −5.31058 −0.238934
\(495\) 0 0
\(496\) −8.00194 −0.359298
\(497\) −15.2564 −0.684345
\(498\) 0 0
\(499\) −31.2829 −1.40041 −0.700207 0.713940i \(-0.746909\pi\)
−0.700207 + 0.713940i \(0.746909\pi\)
\(500\) −1.83129 −0.0818977
\(501\) 0 0
\(502\) 19.6951 0.879035
\(503\) −22.4470 −1.00086 −0.500431 0.865777i \(-0.666825\pi\)
−0.500431 + 0.865777i \(0.666825\pi\)
\(504\) 0 0
\(505\) 15.1441 0.673905
\(506\) −6.55872 −0.291570
\(507\) 0 0
\(508\) 7.27627 0.322832
\(509\) −7.38076 −0.327146 −0.163573 0.986531i \(-0.552302\pi\)
−0.163573 + 0.986531i \(0.552302\pi\)
\(510\) 0 0
\(511\) 31.7627 1.40510
\(512\) −16.9962 −0.751131
\(513\) 0 0
\(514\) 3.74108 0.165012
\(515\) 55.1602 2.43065
\(516\) 0 0
\(517\) 6.22226 0.273655
\(518\) 13.0206 0.572093
\(519\) 0 0
\(520\) 25.2894 1.10902
\(521\) −27.5979 −1.20909 −0.604543 0.796572i \(-0.706644\pi\)
−0.604543 + 0.796572i \(0.706644\pi\)
\(522\) 0 0
\(523\) 21.3348 0.932904 0.466452 0.884547i \(-0.345532\pi\)
0.466452 + 0.884547i \(0.345532\pi\)
\(524\) 2.97012 0.129750
\(525\) 0 0
\(526\) −1.68458 −0.0734513
\(527\) −16.8916 −0.735808
\(528\) 0 0
\(529\) 14.0335 0.610151
\(530\) 8.57471 0.372462
\(531\) 0 0
\(532\) 5.25151 0.227682
\(533\) −15.9463 −0.690712
\(534\) 0 0
\(535\) 4.23102 0.182923
\(536\) −22.7851 −0.984168
\(537\) 0 0
\(538\) 12.6879 0.547015
\(539\) −4.89515 −0.210849
\(540\) 0 0
\(541\) 2.69913 0.116045 0.0580224 0.998315i \(-0.481521\pi\)
0.0580224 + 0.998315i \(0.481521\pi\)
\(542\) 22.1115 0.949770
\(543\) 0 0
\(544\) −14.9530 −0.641102
\(545\) 34.9699 1.49795
\(546\) 0 0
\(547\) 6.51857 0.278714 0.139357 0.990242i \(-0.455496\pi\)
0.139357 + 0.990242i \(0.455496\pi\)
\(548\) 15.1508 0.647208
\(549\) 0 0
\(550\) −4.61618 −0.196835
\(551\) 0.683994 0.0291391
\(552\) 0 0
\(553\) 18.0609 0.768028
\(554\) −0.109794 −0.00466471
\(555\) 0 0
\(556\) 2.78736 0.118210
\(557\) 1.33512 0.0565708 0.0282854 0.999600i \(-0.490995\pi\)
0.0282854 + 0.999600i \(0.490995\pi\)
\(558\) 0 0
\(559\) −21.9221 −0.927206
\(560\) 17.0250 0.719440
\(561\) 0 0
\(562\) 15.4388 0.651246
\(563\) 9.26909 0.390646 0.195323 0.980739i \(-0.437425\pi\)
0.195323 + 0.980739i \(0.437425\pi\)
\(564\) 0 0
\(565\) 26.3653 1.10920
\(566\) −28.5323 −1.19930
\(567\) 0 0
\(568\) 13.5322 0.567799
\(569\) 24.4489 1.02495 0.512475 0.858702i \(-0.328729\pi\)
0.512475 + 0.858702i \(0.328729\pi\)
\(570\) 0 0
\(571\) −13.5222 −0.565888 −0.282944 0.959136i \(-0.591311\pi\)
−0.282944 + 0.959136i \(0.591311\pi\)
\(572\) 2.27489 0.0951181
\(573\) 0 0
\(574\) −21.8462 −0.911843
\(575\) 26.0651 1.08699
\(576\) 0 0
\(577\) 14.4217 0.600383 0.300192 0.953879i \(-0.402949\pi\)
0.300192 + 0.953879i \(0.402949\pi\)
\(578\) −5.71575 −0.237744
\(579\) 0 0
\(580\) −0.962144 −0.0399509
\(581\) 30.4067 1.26148
\(582\) 0 0
\(583\) 2.61126 0.108147
\(584\) −28.1730 −1.16581
\(585\) 0 0
\(586\) −8.04905 −0.332503
\(587\) 32.9640 1.36057 0.680285 0.732947i \(-0.261856\pi\)
0.680285 + 0.732947i \(0.261856\pi\)
\(588\) 0 0
\(589\) −8.96951 −0.369582
\(590\) −8.35799 −0.344093
\(591\) 0 0
\(592\) −5.67520 −0.233249
\(593\) −44.9564 −1.84614 −0.923069 0.384633i \(-0.874328\pi\)
−0.923069 + 0.384633i \(0.874328\pi\)
\(594\) 0 0
\(595\) 35.9387 1.47334
\(596\) −1.62579 −0.0665951
\(597\) 0 0
\(598\) 17.7955 0.727712
\(599\) 10.6539 0.435306 0.217653 0.976026i \(-0.430160\pi\)
0.217653 + 0.976026i \(0.430160\pi\)
\(600\) 0 0
\(601\) 0.426911 0.0174140 0.00870702 0.999962i \(-0.497228\pi\)
0.00870702 + 0.999962i \(0.497228\pi\)
\(602\) −30.0329 −1.22405
\(603\) 0 0
\(604\) 7.23659 0.294453
\(605\) −3.04682 −0.123871
\(606\) 0 0
\(607\) −37.0498 −1.50381 −0.751903 0.659274i \(-0.770864\pi\)
−0.751903 + 0.659274i \(0.770864\pi\)
\(608\) −7.94010 −0.322013
\(609\) 0 0
\(610\) 3.28374 0.132955
\(611\) −16.8826 −0.682997
\(612\) 0 0
\(613\) 1.90829 0.0770750 0.0385375 0.999257i \(-0.487730\pi\)
0.0385375 + 0.999257i \(0.487730\pi\)
\(614\) 25.1105 1.01338
\(615\) 0 0
\(616\) 10.5508 0.425104
\(617\) −36.7736 −1.48045 −0.740226 0.672359i \(-0.765281\pi\)
−0.740226 + 0.672359i \(0.765281\pi\)
\(618\) 0 0
\(619\) −25.8507 −1.03903 −0.519513 0.854462i \(-0.673887\pi\)
−0.519513 + 0.854462i \(0.673887\pi\)
\(620\) 12.6170 0.506711
\(621\) 0 0
\(622\) −18.7450 −0.751608
\(623\) −28.1212 −1.12665
\(624\) 0 0
\(625\) −28.0704 −1.12282
\(626\) −1.83174 −0.0732110
\(627\) 0 0
\(628\) 6.20862 0.247751
\(629\) −11.9800 −0.477673
\(630\) 0 0
\(631\) −44.0002 −1.75162 −0.875810 0.482657i \(-0.839672\pi\)
−0.875810 + 0.482657i \(0.839672\pi\)
\(632\) −16.0197 −0.637231
\(633\) 0 0
\(634\) 31.9118 1.26738
\(635\) 26.4415 1.04930
\(636\) 0 0
\(637\) 13.2818 0.526244
\(638\) 0.405924 0.0160707
\(639\) 0 0
\(640\) 0.528651 0.0208968
\(641\) 45.9375 1.81442 0.907211 0.420677i \(-0.138207\pi\)
0.907211 + 0.420677i \(0.138207\pi\)
\(642\) 0 0
\(643\) −10.9831 −0.433132 −0.216566 0.976268i \(-0.569486\pi\)
−0.216566 + 0.976268i \(0.569486\pi\)
\(644\) −17.5975 −0.693440
\(645\) 0 0
\(646\) 6.69393 0.263369
\(647\) 9.24470 0.363447 0.181723 0.983350i \(-0.441832\pi\)
0.181723 + 0.983350i \(0.441832\pi\)
\(648\) 0 0
\(649\) −2.54527 −0.0999104
\(650\) 12.5249 0.491267
\(651\) 0 0
\(652\) 3.63806 0.142478
\(653\) 36.3448 1.42228 0.711140 0.703050i \(-0.248179\pi\)
0.711140 + 0.703050i \(0.248179\pi\)
\(654\) 0 0
\(655\) 10.7932 0.421727
\(656\) 9.52194 0.371769
\(657\) 0 0
\(658\) −23.1289 −0.901657
\(659\) −12.8729 −0.501458 −0.250729 0.968057i \(-0.580670\pi\)
−0.250729 + 0.968057i \(0.580670\pi\)
\(660\) 0 0
\(661\) 24.6328 0.958105 0.479053 0.877786i \(-0.340980\pi\)
0.479053 + 0.877786i \(0.340980\pi\)
\(662\) −4.82774 −0.187635
\(663\) 0 0
\(664\) −26.9702 −1.04665
\(665\) 19.0837 0.740032
\(666\) 0 0
\(667\) −2.29203 −0.0887478
\(668\) 10.1788 0.393831
\(669\) 0 0
\(670\) −24.4579 −0.944893
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 28.8608 1.11250 0.556250 0.831015i \(-0.312240\pi\)
0.556250 + 0.831015i \(0.312240\pi\)
\(674\) 7.45284 0.287073
\(675\) 0 0
\(676\) 4.72729 0.181819
\(677\) −27.3298 −1.05037 −0.525184 0.850989i \(-0.676004\pi\)
−0.525184 + 0.850989i \(0.676004\pi\)
\(678\) 0 0
\(679\) 4.65545 0.178660
\(680\) −31.8771 −1.22243
\(681\) 0 0
\(682\) −5.32305 −0.203830
\(683\) 4.55326 0.174226 0.0871128 0.996198i \(-0.472236\pi\)
0.0871128 + 0.996198i \(0.472236\pi\)
\(684\) 0 0
\(685\) 55.0569 2.10362
\(686\) −7.82398 −0.298721
\(687\) 0 0
\(688\) 13.0902 0.499060
\(689\) −7.08503 −0.269918
\(690\) 0 0
\(691\) 6.69036 0.254513 0.127257 0.991870i \(-0.459383\pi\)
0.127257 + 0.991870i \(0.459383\pi\)
\(692\) −3.64132 −0.138422
\(693\) 0 0
\(694\) −19.9989 −0.759149
\(695\) 10.1291 0.384219
\(696\) 0 0
\(697\) 20.1002 0.761348
\(698\) −7.52298 −0.284749
\(699\) 0 0
\(700\) −12.3856 −0.468131
\(701\) 3.64599 0.137707 0.0688536 0.997627i \(-0.478066\pi\)
0.0688536 + 0.997627i \(0.478066\pi\)
\(702\) 0 0
\(703\) −6.36143 −0.239926
\(704\) −7.95245 −0.299719
\(705\) 0 0
\(706\) −27.0476 −1.01795
\(707\) −17.1428 −0.644722
\(708\) 0 0
\(709\) 8.31591 0.312310 0.156155 0.987733i \(-0.450090\pi\)
0.156155 + 0.987733i \(0.450090\pi\)
\(710\) 14.5257 0.545140
\(711\) 0 0
\(712\) 24.9430 0.934779
\(713\) 30.0564 1.12562
\(714\) 0 0
\(715\) 8.26682 0.309162
\(716\) 5.41835 0.202493
\(717\) 0 0
\(718\) 28.6935 1.07083
\(719\) 10.6667 0.397800 0.198900 0.980020i \(-0.436263\pi\)
0.198900 + 0.980020i \(0.436263\pi\)
\(720\) 0 0
\(721\) −62.4401 −2.32539
\(722\) −16.9229 −0.629805
\(723\) 0 0
\(724\) 12.3264 0.458109
\(725\) −1.61319 −0.0599122
\(726\) 0 0
\(727\) −8.80084 −0.326405 −0.163203 0.986593i \(-0.552182\pi\)
−0.163203 + 0.986593i \(0.552182\pi\)
\(728\) −28.6271 −1.06099
\(729\) 0 0
\(730\) −30.2414 −1.11928
\(731\) 27.6326 1.02203
\(732\) 0 0
\(733\) −4.17577 −0.154236 −0.0771179 0.997022i \(-0.524572\pi\)
−0.0771179 + 0.997022i \(0.524572\pi\)
\(734\) 3.10407 0.114573
\(735\) 0 0
\(736\) 26.6069 0.980742
\(737\) −7.44820 −0.274358
\(738\) 0 0
\(739\) 54.2009 1.99381 0.996906 0.0786054i \(-0.0250467\pi\)
0.996906 + 0.0786054i \(0.0250467\pi\)
\(740\) 8.94834 0.328947
\(741\) 0 0
\(742\) −9.70637 −0.356332
\(743\) −22.4243 −0.822666 −0.411333 0.911485i \(-0.634937\pi\)
−0.411333 + 0.911485i \(0.634937\pi\)
\(744\) 0 0
\(745\) −5.90803 −0.216453
\(746\) −6.32310 −0.231505
\(747\) 0 0
\(748\) −2.86748 −0.104845
\(749\) −4.78942 −0.175002
\(750\) 0 0
\(751\) −46.3856 −1.69263 −0.846316 0.532681i \(-0.821185\pi\)
−0.846316 + 0.532681i \(0.821185\pi\)
\(752\) 10.0810 0.367617
\(753\) 0 0
\(754\) −1.10138 −0.0401098
\(755\) 26.2973 0.957058
\(756\) 0 0
\(757\) −2.80886 −0.102090 −0.0510448 0.998696i \(-0.516255\pi\)
−0.0510448 + 0.998696i \(0.516255\pi\)
\(758\) 27.8947 1.01318
\(759\) 0 0
\(760\) −16.9269 −0.614003
\(761\) −18.9731 −0.687773 −0.343887 0.939011i \(-0.611744\pi\)
−0.343887 + 0.939011i \(0.611744\pi\)
\(762\) 0 0
\(763\) −39.5852 −1.43308
\(764\) 20.5113 0.742074
\(765\) 0 0
\(766\) 19.7029 0.711894
\(767\) 6.90597 0.249360
\(768\) 0 0
\(769\) 27.9821 1.00906 0.504530 0.863394i \(-0.331666\pi\)
0.504530 + 0.863394i \(0.331666\pi\)
\(770\) 11.3254 0.408139
\(771\) 0 0
\(772\) 2.05470 0.0739501
\(773\) 17.6472 0.634727 0.317363 0.948304i \(-0.397203\pi\)
0.317363 + 0.948304i \(0.397203\pi\)
\(774\) 0 0
\(775\) 21.1544 0.759888
\(776\) −4.12931 −0.148234
\(777\) 0 0
\(778\) −7.11714 −0.255162
\(779\) 10.6733 0.382411
\(780\) 0 0
\(781\) 4.42352 0.158286
\(782\) −22.4310 −0.802132
\(783\) 0 0
\(784\) −7.93089 −0.283246
\(785\) 22.5617 0.805263
\(786\) 0 0
\(787\) −50.6943 −1.80706 −0.903528 0.428528i \(-0.859032\pi\)
−0.903528 + 0.428528i \(0.859032\pi\)
\(788\) 8.23889 0.293498
\(789\) 0 0
\(790\) −17.1959 −0.611801
\(791\) −29.8450 −1.06116
\(792\) 0 0
\(793\) −2.71326 −0.0963507
\(794\) 5.72920 0.203322
\(795\) 0 0
\(796\) 15.6980 0.556402
\(797\) 13.5778 0.480951 0.240476 0.970655i \(-0.422697\pi\)
0.240476 + 0.970655i \(0.422697\pi\)
\(798\) 0 0
\(799\) 21.2803 0.752844
\(800\) 18.7266 0.662084
\(801\) 0 0
\(802\) 25.3831 0.896310
\(803\) −9.20943 −0.324994
\(804\) 0 0
\(805\) −63.9484 −2.25388
\(806\) 14.4428 0.508727
\(807\) 0 0
\(808\) 15.2054 0.534924
\(809\) 44.4156 1.56157 0.780785 0.624799i \(-0.214819\pi\)
0.780785 + 0.624799i \(0.214819\pi\)
\(810\) 0 0
\(811\) 41.7978 1.46772 0.733859 0.679301i \(-0.237717\pi\)
0.733859 + 0.679301i \(0.237717\pi\)
\(812\) 1.08913 0.0382208
\(813\) 0 0
\(814\) −3.77526 −0.132323
\(815\) 13.2205 0.463094
\(816\) 0 0
\(817\) 14.6730 0.513345
\(818\) 32.1263 1.12327
\(819\) 0 0
\(820\) −15.0137 −0.524300
\(821\) −6.14459 −0.214448 −0.107224 0.994235i \(-0.534196\pi\)
−0.107224 + 0.994235i \(0.534196\pi\)
\(822\) 0 0
\(823\) −10.4844 −0.365464 −0.182732 0.983163i \(-0.558494\pi\)
−0.182732 + 0.983163i \(0.558494\pi\)
\(824\) 55.3834 1.92937
\(825\) 0 0
\(826\) 9.46106 0.329192
\(827\) 1.30033 0.0452168 0.0226084 0.999744i \(-0.492803\pi\)
0.0226084 + 0.999744i \(0.492803\pi\)
\(828\) 0 0
\(829\) 51.8433 1.80059 0.900296 0.435279i \(-0.143350\pi\)
0.900296 + 0.435279i \(0.143350\pi\)
\(830\) −28.9503 −1.00488
\(831\) 0 0
\(832\) 21.5770 0.748050
\(833\) −16.7416 −0.580061
\(834\) 0 0
\(835\) 36.9892 1.28006
\(836\) −1.52265 −0.0526618
\(837\) 0 0
\(838\) 23.7228 0.819491
\(839\) 8.55057 0.295198 0.147599 0.989047i \(-0.452845\pi\)
0.147599 + 0.989047i \(0.452845\pi\)
\(840\) 0 0
\(841\) −28.8581 −0.995108
\(842\) −2.84562 −0.0980665
\(843\) 0 0
\(844\) 10.3099 0.354881
\(845\) 17.1787 0.590964
\(846\) 0 0
\(847\) 3.44893 0.118507
\(848\) 4.23065 0.145281
\(849\) 0 0
\(850\) −15.7875 −0.541507
\(851\) 21.3168 0.730731
\(852\) 0 0
\(853\) −3.34288 −0.114458 −0.0572289 0.998361i \(-0.518226\pi\)
−0.0572289 + 0.998361i \(0.518226\pi\)
\(854\) −3.71712 −0.127197
\(855\) 0 0
\(856\) 4.24814 0.145198
\(857\) 39.8478 1.36117 0.680587 0.732667i \(-0.261725\pi\)
0.680587 + 0.732667i \(0.261725\pi\)
\(858\) 0 0
\(859\) −9.75337 −0.332781 −0.166390 0.986060i \(-0.553211\pi\)
−0.166390 + 0.986060i \(0.553211\pi\)
\(860\) −20.6399 −0.703815
\(861\) 0 0
\(862\) −12.3974 −0.422256
\(863\) −13.2225 −0.450098 −0.225049 0.974347i \(-0.572254\pi\)
−0.225049 + 0.974347i \(0.572254\pi\)
\(864\) 0 0
\(865\) −13.2323 −0.449913
\(866\) 20.8659 0.709051
\(867\) 0 0
\(868\) −14.2822 −0.484768
\(869\) −5.23667 −0.177642
\(870\) 0 0
\(871\) 20.2089 0.684752
\(872\) 35.1114 1.18902
\(873\) 0 0
\(874\) −11.9110 −0.402896
\(875\) 7.53307 0.254664
\(876\) 0 0
\(877\) −13.0314 −0.440039 −0.220020 0.975495i \(-0.570612\pi\)
−0.220020 + 0.975495i \(0.570612\pi\)
\(878\) −14.9335 −0.503980
\(879\) 0 0
\(880\) −4.93632 −0.166403
\(881\) 16.8445 0.567507 0.283753 0.958897i \(-0.408420\pi\)
0.283753 + 0.958897i \(0.408420\pi\)
\(882\) 0 0
\(883\) −11.2257 −0.377775 −0.188888 0.981999i \(-0.560488\pi\)
−0.188888 + 0.981999i \(0.560488\pi\)
\(884\) 7.78022 0.261677
\(885\) 0 0
\(886\) −14.2165 −0.477613
\(887\) −18.9067 −0.634823 −0.317412 0.948288i \(-0.602814\pi\)
−0.317412 + 0.948288i \(0.602814\pi\)
\(888\) 0 0
\(889\) −29.9312 −1.00386
\(890\) 26.7742 0.897475
\(891\) 0 0
\(892\) 0.527974 0.0176779
\(893\) 11.3000 0.378139
\(894\) 0 0
\(895\) 19.6899 0.658162
\(896\) −0.598421 −0.0199918
\(897\) 0 0
\(898\) −11.2995 −0.377068
\(899\) −1.86021 −0.0620415
\(900\) 0 0
\(901\) 8.93061 0.297522
\(902\) 6.33419 0.210905
\(903\) 0 0
\(904\) 26.4720 0.880446
\(905\) 44.7935 1.48899
\(906\) 0 0
\(907\) 27.4334 0.910911 0.455456 0.890259i \(-0.349476\pi\)
0.455456 + 0.890259i \(0.349476\pi\)
\(908\) −18.3235 −0.608088
\(909\) 0 0
\(910\) −30.7288 −1.01865
\(911\) −15.5416 −0.514916 −0.257458 0.966289i \(-0.582885\pi\)
−0.257458 + 0.966289i \(0.582885\pi\)
\(912\) 0 0
\(913\) −8.81625 −0.291775
\(914\) 5.98395 0.197931
\(915\) 0 0
\(916\) −18.1814 −0.600730
\(917\) −12.2177 −0.403464
\(918\) 0 0
\(919\) 50.8268 1.67662 0.838310 0.545193i \(-0.183544\pi\)
0.838310 + 0.545193i \(0.183544\pi\)
\(920\) 56.7212 1.87004
\(921\) 0 0
\(922\) 15.3521 0.505593
\(923\) −12.0022 −0.395056
\(924\) 0 0
\(925\) 15.0033 0.493305
\(926\) 5.57264 0.183128
\(927\) 0 0
\(928\) −1.64672 −0.0540562
\(929\) 8.78485 0.288221 0.144111 0.989562i \(-0.453968\pi\)
0.144111 + 0.989562i \(0.453968\pi\)
\(930\) 0 0
\(931\) −8.88987 −0.291354
\(932\) 11.5783 0.379259
\(933\) 0 0
\(934\) −39.2138 −1.28311
\(935\) −10.4202 −0.340778
\(936\) 0 0
\(937\) 56.6855 1.85184 0.925918 0.377725i \(-0.123293\pi\)
0.925918 + 0.377725i \(0.123293\pi\)
\(938\) 27.6858 0.903975
\(939\) 0 0
\(940\) −15.8952 −0.518443
\(941\) 41.0121 1.33696 0.668479 0.743731i \(-0.266946\pi\)
0.668479 + 0.743731i \(0.266946\pi\)
\(942\) 0 0
\(943\) −35.7657 −1.16469
\(944\) −4.12372 −0.134216
\(945\) 0 0
\(946\) 8.70788 0.283118
\(947\) 9.18122 0.298349 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(948\) 0 0
\(949\) 24.9876 0.811131
\(950\) −8.38325 −0.271988
\(951\) 0 0
\(952\) 36.0841 1.16949
\(953\) 17.0299 0.551653 0.275826 0.961207i \(-0.411049\pi\)
0.275826 + 0.961207i \(0.411049\pi\)
\(954\) 0 0
\(955\) 74.5369 2.41196
\(956\) 6.79297 0.219700
\(957\) 0 0
\(958\) −40.4679 −1.30746
\(959\) −62.3232 −2.01252
\(960\) 0 0
\(961\) −6.60625 −0.213105
\(962\) 10.2433 0.330256
\(963\) 0 0
\(964\) 4.93331 0.158891
\(965\) 7.46663 0.240359
\(966\) 0 0
\(967\) −16.5606 −0.532555 −0.266277 0.963896i \(-0.585794\pi\)
−0.266277 + 0.963896i \(0.585794\pi\)
\(968\) −3.05915 −0.0983248
\(969\) 0 0
\(970\) −4.43247 −0.142318
\(971\) 12.9086 0.414256 0.207128 0.978314i \(-0.433588\pi\)
0.207128 + 0.978314i \(0.433588\pi\)
\(972\) 0 0
\(973\) −11.4659 −0.367580
\(974\) 12.3687 0.396319
\(975\) 0 0
\(976\) 1.62015 0.0518599
\(977\) −6.73089 −0.215340 −0.107670 0.994187i \(-0.534339\pi\)
−0.107670 + 0.994187i \(0.534339\pi\)
\(978\) 0 0
\(979\) 8.15358 0.260590
\(980\) 12.5050 0.399457
\(981\) 0 0
\(982\) −16.9459 −0.540765
\(983\) 3.89460 0.124219 0.0621093 0.998069i \(-0.480217\pi\)
0.0621093 + 0.998069i \(0.480217\pi\)
\(984\) 0 0
\(985\) 29.9396 0.953955
\(986\) 1.38827 0.0442116
\(987\) 0 0
\(988\) 4.13134 0.131435
\(989\) −49.1686 −1.56347
\(990\) 0 0
\(991\) 47.0696 1.49522 0.747608 0.664141i \(-0.231202\pi\)
0.747608 + 0.664141i \(0.231202\pi\)
\(992\) 21.5941 0.685614
\(993\) 0 0
\(994\) −16.4428 −0.521533
\(995\) 57.0457 1.80847
\(996\) 0 0
\(997\) −35.0684 −1.11063 −0.555314 0.831641i \(-0.687402\pi\)
−0.555314 + 0.831641i \(0.687402\pi\)
\(998\) −33.7154 −1.06724
\(999\) 0 0
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 6039.2.a.n.1.19 25
3.2 odd 2 6039.2.a.o.1.7 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6039.2.a.n.1.19 25 1.1 even 1 trivial
6039.2.a.o.1.7 yes 25 3.2 odd 2