Properties

Label 624.4.a.u.1.3
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $3$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.36248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 54x - 90 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.70213\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +18.8819 q^{5} +17.9266 q^{7} +9.00000 q^{9} -44.5723 q^{11} +13.0000 q^{13} +56.6456 q^{15} +34.0000 q^{17} +152.835 q^{19} +53.7799 q^{21} -107.234 q^{23} +231.525 q^{25} +27.0000 q^{27} +149.674 q^{29} -0.162885 q^{31} -133.717 q^{33} +338.489 q^{35} -256.342 q^{37} +39.0000 q^{39} +414.723 q^{41} -471.076 q^{43} +169.937 q^{45} -632.710 q^{47} -21.6354 q^{49} +102.000 q^{51} -236.489 q^{53} -841.608 q^{55} +458.505 q^{57} +108.018 q^{59} +888.292 q^{61} +161.340 q^{63} +245.464 q^{65} -637.149 q^{67} -321.702 q^{69} +362.597 q^{71} +723.177 q^{73} +694.576 q^{75} -799.031 q^{77} +964.684 q^{79} +81.0000 q^{81} +431.868 q^{83} +641.984 q^{85} +449.023 q^{87} +117.359 q^{89} +233.046 q^{91} -0.488655 q^{93} +2885.81 q^{95} -1153.38 q^{97} -401.151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 16 q^{5} + 22 q^{7} + 27 q^{9} + 20 q^{11} + 39 q^{13} + 48 q^{15} + 102 q^{17} + 38 q^{19} + 66 q^{21} - 32 q^{23} + 161 q^{25} + 81 q^{27} + 350 q^{29} - 50 q^{31} + 60 q^{33} - 232 q^{35}+ \cdots + 180 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 18.8819 1.68885 0.844423 0.535676i \(-0.179943\pi\)
0.844423 + 0.535676i \(0.179943\pi\)
\(6\) 0 0
\(7\) 17.9266 0.967948 0.483974 0.875082i \(-0.339193\pi\)
0.483974 + 0.875082i \(0.339193\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −44.5723 −1.22173 −0.610866 0.791734i \(-0.709179\pi\)
−0.610866 + 0.791734i \(0.709179\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 56.6456 0.975056
\(16\) 0 0
\(17\) 34.0000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 152.835 1.84541 0.922704 0.385510i \(-0.125974\pi\)
0.922704 + 0.385510i \(0.125974\pi\)
\(20\) 0 0
\(21\) 53.7799 0.558845
\(22\) 0 0
\(23\) −107.234 −0.972168 −0.486084 0.873912i \(-0.661575\pi\)
−0.486084 + 0.873912i \(0.661575\pi\)
\(24\) 0 0
\(25\) 231.525 1.85220
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 149.674 0.958407 0.479204 0.877704i \(-0.340926\pi\)
0.479204 + 0.877704i \(0.340926\pi\)
\(30\) 0 0
\(31\) −0.162885 −0.000943710 0 −0.000471855 1.00000i \(-0.500150\pi\)
−0.000471855 1.00000i \(0.500150\pi\)
\(32\) 0 0
\(33\) −133.717 −0.705367
\(34\) 0 0
\(35\) 338.489 1.63472
\(36\) 0 0
\(37\) −256.342 −1.13898 −0.569491 0.821997i \(-0.692860\pi\)
−0.569491 + 0.821997i \(0.692860\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 414.723 1.57973 0.789865 0.613281i \(-0.210151\pi\)
0.789865 + 0.613281i \(0.210151\pi\)
\(42\) 0 0
\(43\) −471.076 −1.67066 −0.835330 0.549749i \(-0.814723\pi\)
−0.835330 + 0.549749i \(0.814723\pi\)
\(44\) 0 0
\(45\) 169.937 0.562949
\(46\) 0 0
\(47\) −632.710 −1.96362 −0.981812 0.189857i \(-0.939197\pi\)
−0.981812 + 0.189857i \(0.939197\pi\)
\(48\) 0 0
\(49\) −21.6354 −0.0630771
\(50\) 0 0
\(51\) 102.000 0.280056
\(52\) 0 0
\(53\) −236.489 −0.612910 −0.306455 0.951885i \(-0.599143\pi\)
−0.306455 + 0.951885i \(0.599143\pi\)
\(54\) 0 0
\(55\) −841.608 −2.06332
\(56\) 0 0
\(57\) 458.505 1.06545
\(58\) 0 0
\(59\) 108.018 0.238351 0.119176 0.992873i \(-0.461975\pi\)
0.119176 + 0.992873i \(0.461975\pi\)
\(60\) 0 0
\(61\) 888.292 1.86449 0.932247 0.361822i \(-0.117845\pi\)
0.932247 + 0.361822i \(0.117845\pi\)
\(62\) 0 0
\(63\) 161.340 0.322649
\(64\) 0 0
\(65\) 245.464 0.468402
\(66\) 0 0
\(67\) −637.149 −1.16179 −0.580897 0.813977i \(-0.697298\pi\)
−0.580897 + 0.813977i \(0.697298\pi\)
\(68\) 0 0
\(69\) −321.702 −0.561281
\(70\) 0 0
\(71\) 362.597 0.606090 0.303045 0.952976i \(-0.401997\pi\)
0.303045 + 0.952976i \(0.401997\pi\)
\(72\) 0 0
\(73\) 723.177 1.15947 0.579736 0.814804i \(-0.303156\pi\)
0.579736 + 0.814804i \(0.303156\pi\)
\(74\) 0 0
\(75\) 694.576 1.06937
\(76\) 0 0
\(77\) −799.031 −1.18257
\(78\) 0 0
\(79\) 964.684 1.37387 0.686933 0.726721i \(-0.258957\pi\)
0.686933 + 0.726721i \(0.258957\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 431.868 0.571128 0.285564 0.958360i \(-0.407819\pi\)
0.285564 + 0.958360i \(0.407819\pi\)
\(84\) 0 0
\(85\) 641.984 0.819211
\(86\) 0 0
\(87\) 449.023 0.553337
\(88\) 0 0
\(89\) 117.359 0.139775 0.0698877 0.997555i \(-0.477736\pi\)
0.0698877 + 0.997555i \(0.477736\pi\)
\(90\) 0 0
\(91\) 233.046 0.268460
\(92\) 0 0
\(93\) −0.488655 −0.000544851 0
\(94\) 0 0
\(95\) 2885.81 3.11661
\(96\) 0 0
\(97\) −1153.38 −1.20730 −0.603652 0.797248i \(-0.706288\pi\)
−0.603652 + 0.797248i \(0.706288\pi\)
\(98\) 0 0
\(99\) −401.151 −0.407244
\(100\) 0 0
\(101\) −848.183 −0.835618 −0.417809 0.908535i \(-0.637202\pi\)
−0.417809 + 0.908535i \(0.637202\pi\)
\(102\) 0 0
\(103\) 710.562 0.679745 0.339873 0.940471i \(-0.389616\pi\)
0.339873 + 0.940471i \(0.389616\pi\)
\(104\) 0 0
\(105\) 1015.47 0.943803
\(106\) 0 0
\(107\) −533.245 −0.481782 −0.240891 0.970552i \(-0.577440\pi\)
−0.240891 + 0.970552i \(0.577440\pi\)
\(108\) 0 0
\(109\) −667.057 −0.586170 −0.293085 0.956086i \(-0.594682\pi\)
−0.293085 + 0.956086i \(0.594682\pi\)
\(110\) 0 0
\(111\) −769.026 −0.657592
\(112\) 0 0
\(113\) 302.916 0.252176 0.126088 0.992019i \(-0.459758\pi\)
0.126088 + 0.992019i \(0.459758\pi\)
\(114\) 0 0
\(115\) −2024.78 −1.64184
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 609.506 0.469524
\(120\) 0 0
\(121\) 655.688 0.492628
\(122\) 0 0
\(123\) 1244.17 0.912057
\(124\) 0 0
\(125\) 2011.40 1.43924
\(126\) 0 0
\(127\) 459.472 0.321035 0.160518 0.987033i \(-0.448684\pi\)
0.160518 + 0.987033i \(0.448684\pi\)
\(128\) 0 0
\(129\) −1413.23 −0.964556
\(130\) 0 0
\(131\) −496.848 −0.331373 −0.165686 0.986179i \(-0.552984\pi\)
−0.165686 + 0.986179i \(0.552984\pi\)
\(132\) 0 0
\(133\) 2739.82 1.78626
\(134\) 0 0
\(135\) 509.811 0.325019
\(136\) 0 0
\(137\) −2684.75 −1.67426 −0.837129 0.547006i \(-0.815768\pi\)
−0.837129 + 0.547006i \(0.815768\pi\)
\(138\) 0 0
\(139\) 1728.99 1.05504 0.527522 0.849541i \(-0.323121\pi\)
0.527522 + 0.849541i \(0.323121\pi\)
\(140\) 0 0
\(141\) −1898.13 −1.13370
\(142\) 0 0
\(143\) −579.440 −0.338847
\(144\) 0 0
\(145\) 2826.13 1.61860
\(146\) 0 0
\(147\) −64.9063 −0.0364176
\(148\) 0 0
\(149\) −1132.89 −0.622887 −0.311444 0.950265i \(-0.600812\pi\)
−0.311444 + 0.950265i \(0.600812\pi\)
\(150\) 0 0
\(151\) 1258.52 0.678259 0.339130 0.940740i \(-0.389867\pi\)
0.339130 + 0.940740i \(0.389867\pi\)
\(152\) 0 0
\(153\) 306.000 0.161690
\(154\) 0 0
\(155\) −3.07558 −0.00159378
\(156\) 0 0
\(157\) 2469.89 1.25553 0.627766 0.778402i \(-0.283970\pi\)
0.627766 + 0.778402i \(0.283970\pi\)
\(158\) 0 0
\(159\) −709.466 −0.353864
\(160\) 0 0
\(161\) −1922.35 −0.941007
\(162\) 0 0
\(163\) 733.892 0.352656 0.176328 0.984331i \(-0.443578\pi\)
0.176328 + 0.984331i \(0.443578\pi\)
\(164\) 0 0
\(165\) −2524.83 −1.19126
\(166\) 0 0
\(167\) −2610.39 −1.20957 −0.604784 0.796389i \(-0.706741\pi\)
−0.604784 + 0.796389i \(0.706741\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 1375.51 0.615136
\(172\) 0 0
\(173\) 1164.04 0.511564 0.255782 0.966734i \(-0.417667\pi\)
0.255782 + 0.966734i \(0.417667\pi\)
\(174\) 0 0
\(175\) 4150.47 1.79284
\(176\) 0 0
\(177\) 324.053 0.137612
\(178\) 0 0
\(179\) −911.229 −0.380494 −0.190247 0.981736i \(-0.560929\pi\)
−0.190247 + 0.981736i \(0.560929\pi\)
\(180\) 0 0
\(181\) −4817.82 −1.97848 −0.989242 0.146291i \(-0.953266\pi\)
−0.989242 + 0.146291i \(0.953266\pi\)
\(182\) 0 0
\(183\) 2664.88 1.07647
\(184\) 0 0
\(185\) −4840.22 −1.92357
\(186\) 0 0
\(187\) −1515.46 −0.592627
\(188\) 0 0
\(189\) 484.019 0.186282
\(190\) 0 0
\(191\) −674.312 −0.255453 −0.127726 0.991809i \(-0.540768\pi\)
−0.127726 + 0.991809i \(0.540768\pi\)
\(192\) 0 0
\(193\) −963.634 −0.359398 −0.179699 0.983722i \(-0.557512\pi\)
−0.179699 + 0.983722i \(0.557512\pi\)
\(194\) 0 0
\(195\) 736.393 0.270432
\(196\) 0 0
\(197\) 3995.21 1.44491 0.722454 0.691418i \(-0.243014\pi\)
0.722454 + 0.691418i \(0.243014\pi\)
\(198\) 0 0
\(199\) 1619.07 0.576749 0.288374 0.957518i \(-0.406885\pi\)
0.288374 + 0.957518i \(0.406885\pi\)
\(200\) 0 0
\(201\) −1911.45 −0.670762
\(202\) 0 0
\(203\) 2683.16 0.927688
\(204\) 0 0
\(205\) 7830.76 2.66792
\(206\) 0 0
\(207\) −965.107 −0.324056
\(208\) 0 0
\(209\) −6812.20 −2.25459
\(210\) 0 0
\(211\) −543.991 −0.177488 −0.0887438 0.996054i \(-0.528285\pi\)
−0.0887438 + 0.996054i \(0.528285\pi\)
\(212\) 0 0
\(213\) 1087.79 0.349926
\(214\) 0 0
\(215\) −8894.79 −2.82149
\(216\) 0 0
\(217\) −2.91998 −0.000913462 0
\(218\) 0 0
\(219\) 2169.53 0.669421
\(220\) 0 0
\(221\) 442.000 0.134535
\(222\) 0 0
\(223\) −2447.45 −0.734949 −0.367474 0.930034i \(-0.619777\pi\)
−0.367474 + 0.930034i \(0.619777\pi\)
\(224\) 0 0
\(225\) 2083.73 0.617401
\(226\) 0 0
\(227\) −467.798 −0.136779 −0.0683896 0.997659i \(-0.521786\pi\)
−0.0683896 + 0.997659i \(0.521786\pi\)
\(228\) 0 0
\(229\) −361.180 −0.104225 −0.0521123 0.998641i \(-0.516595\pi\)
−0.0521123 + 0.998641i \(0.516595\pi\)
\(230\) 0 0
\(231\) −2397.09 −0.682759
\(232\) 0 0
\(233\) −3068.38 −0.862732 −0.431366 0.902177i \(-0.641968\pi\)
−0.431366 + 0.902177i \(0.641968\pi\)
\(234\) 0 0
\(235\) −11946.8 −3.31626
\(236\) 0 0
\(237\) 2894.05 0.793202
\(238\) 0 0
\(239\) −334.108 −0.0904253 −0.0452127 0.998977i \(-0.514397\pi\)
−0.0452127 + 0.998977i \(0.514397\pi\)
\(240\) 0 0
\(241\) 514.961 0.137641 0.0688207 0.997629i \(-0.478076\pi\)
0.0688207 + 0.997629i \(0.478076\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −408.518 −0.106528
\(246\) 0 0
\(247\) 1986.85 0.511824
\(248\) 0 0
\(249\) 1295.60 0.329741
\(250\) 0 0
\(251\) 7028.13 1.76738 0.883688 0.468076i \(-0.155053\pi\)
0.883688 + 0.468076i \(0.155053\pi\)
\(252\) 0 0
\(253\) 4779.67 1.18773
\(254\) 0 0
\(255\) 1925.95 0.472972
\(256\) 0 0
\(257\) −3572.23 −0.867040 −0.433520 0.901144i \(-0.642729\pi\)
−0.433520 + 0.901144i \(0.642729\pi\)
\(258\) 0 0
\(259\) −4595.35 −1.10248
\(260\) 0 0
\(261\) 1347.07 0.319469
\(262\) 0 0
\(263\) −4276.74 −1.00272 −0.501359 0.865239i \(-0.667166\pi\)
−0.501359 + 0.865239i \(0.667166\pi\)
\(264\) 0 0
\(265\) −4465.35 −1.03511
\(266\) 0 0
\(267\) 352.076 0.0806993
\(268\) 0 0
\(269\) 2128.08 0.482347 0.241173 0.970482i \(-0.422468\pi\)
0.241173 + 0.970482i \(0.422468\pi\)
\(270\) 0 0
\(271\) 4386.94 0.983350 0.491675 0.870779i \(-0.336385\pi\)
0.491675 + 0.870779i \(0.336385\pi\)
\(272\) 0 0
\(273\) 699.139 0.154996
\(274\) 0 0
\(275\) −10319.6 −2.26290
\(276\) 0 0
\(277\) 3411.86 0.740067 0.370034 0.929018i \(-0.379346\pi\)
0.370034 + 0.929018i \(0.379346\pi\)
\(278\) 0 0
\(279\) −1.46597 −0.000314570 0
\(280\) 0 0
\(281\) −4466.83 −0.948286 −0.474143 0.880448i \(-0.657242\pi\)
−0.474143 + 0.880448i \(0.657242\pi\)
\(282\) 0 0
\(283\) 1879.30 0.394746 0.197373 0.980329i \(-0.436759\pi\)
0.197373 + 0.980329i \(0.436759\pi\)
\(284\) 0 0
\(285\) 8657.43 1.79938
\(286\) 0 0
\(287\) 7434.60 1.52910
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) −3460.15 −0.697037
\(292\) 0 0
\(293\) −207.996 −0.0414718 −0.0207359 0.999785i \(-0.506601\pi\)
−0.0207359 + 0.999785i \(0.506601\pi\)
\(294\) 0 0
\(295\) 2039.58 0.402539
\(296\) 0 0
\(297\) −1203.45 −0.235122
\(298\) 0 0
\(299\) −1394.04 −0.269631
\(300\) 0 0
\(301\) −8444.80 −1.61711
\(302\) 0 0
\(303\) −2544.55 −0.482444
\(304\) 0 0
\(305\) 16772.6 3.14885
\(306\) 0 0
\(307\) 1743.07 0.324046 0.162023 0.986787i \(-0.448198\pi\)
0.162023 + 0.986787i \(0.448198\pi\)
\(308\) 0 0
\(309\) 2131.69 0.392451
\(310\) 0 0
\(311\) −10027.9 −1.82840 −0.914198 0.405267i \(-0.867178\pi\)
−0.914198 + 0.405267i \(0.867178\pi\)
\(312\) 0 0
\(313\) −2092.80 −0.377929 −0.188965 0.981984i \(-0.560513\pi\)
−0.188965 + 0.981984i \(0.560513\pi\)
\(314\) 0 0
\(315\) 3046.40 0.544905
\(316\) 0 0
\(317\) −10481.8 −1.85715 −0.928573 0.371151i \(-0.878963\pi\)
−0.928573 + 0.371151i \(0.878963\pi\)
\(318\) 0 0
\(319\) −6671.32 −1.17092
\(320\) 0 0
\(321\) −1599.73 −0.278157
\(322\) 0 0
\(323\) 5196.39 0.895154
\(324\) 0 0
\(325\) 3009.83 0.513709
\(326\) 0 0
\(327\) −2001.17 −0.338425
\(328\) 0 0
\(329\) −11342.4 −1.90068
\(330\) 0 0
\(331\) −3954.25 −0.656631 −0.328316 0.944568i \(-0.606481\pi\)
−0.328316 + 0.944568i \(0.606481\pi\)
\(332\) 0 0
\(333\) −2307.08 −0.379661
\(334\) 0 0
\(335\) −12030.6 −1.96209
\(336\) 0 0
\(337\) −7823.04 −1.26454 −0.632268 0.774750i \(-0.717876\pi\)
−0.632268 + 0.774750i \(0.717876\pi\)
\(338\) 0 0
\(339\) 908.748 0.145594
\(340\) 0 0
\(341\) 7.26016 0.00115296
\(342\) 0 0
\(343\) −6536.69 −1.02900
\(344\) 0 0
\(345\) −6074.34 −0.947918
\(346\) 0 0
\(347\) −4248.15 −0.657213 −0.328606 0.944467i \(-0.606579\pi\)
−0.328606 + 0.944467i \(0.606579\pi\)
\(348\) 0 0
\(349\) −3202.79 −0.491236 −0.245618 0.969367i \(-0.578991\pi\)
−0.245618 + 0.969367i \(0.578991\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −2667.60 −0.402216 −0.201108 0.979569i \(-0.564454\pi\)
−0.201108 + 0.979569i \(0.564454\pi\)
\(354\) 0 0
\(355\) 6846.52 1.02359
\(356\) 0 0
\(357\) 1828.52 0.271080
\(358\) 0 0
\(359\) −9062.64 −1.33233 −0.666167 0.745803i \(-0.732066\pi\)
−0.666167 + 0.745803i \(0.732066\pi\)
\(360\) 0 0
\(361\) 16499.5 2.40553
\(362\) 0 0
\(363\) 1967.06 0.284419
\(364\) 0 0
\(365\) 13654.9 1.95817
\(366\) 0 0
\(367\) 4141.93 0.589120 0.294560 0.955633i \(-0.404827\pi\)
0.294560 + 0.955633i \(0.404827\pi\)
\(368\) 0 0
\(369\) 3732.51 0.526576
\(370\) 0 0
\(371\) −4239.45 −0.593265
\(372\) 0 0
\(373\) −5350.96 −0.742794 −0.371397 0.928474i \(-0.621121\pi\)
−0.371397 + 0.928474i \(0.621121\pi\)
\(374\) 0 0
\(375\) 6034.20 0.830946
\(376\) 0 0
\(377\) 1945.76 0.265814
\(378\) 0 0
\(379\) 9360.54 1.26865 0.634325 0.773066i \(-0.281278\pi\)
0.634325 + 0.773066i \(0.281278\pi\)
\(380\) 0 0
\(381\) 1378.41 0.185350
\(382\) 0 0
\(383\) −7241.39 −0.966104 −0.483052 0.875592i \(-0.660472\pi\)
−0.483052 + 0.875592i \(0.660472\pi\)
\(384\) 0 0
\(385\) −15087.2 −1.99718
\(386\) 0 0
\(387\) −4239.68 −0.556886
\(388\) 0 0
\(389\) 6258.22 0.815692 0.407846 0.913051i \(-0.366280\pi\)
0.407846 + 0.913051i \(0.366280\pi\)
\(390\) 0 0
\(391\) −3645.96 −0.471571
\(392\) 0 0
\(393\) −1490.54 −0.191318
\(394\) 0 0
\(395\) 18215.0 2.32025
\(396\) 0 0
\(397\) −10114.6 −1.27868 −0.639342 0.768923i \(-0.720793\pi\)
−0.639342 + 0.768923i \(0.720793\pi\)
\(398\) 0 0
\(399\) 8219.45 1.03130
\(400\) 0 0
\(401\) −12414.5 −1.54601 −0.773006 0.634398i \(-0.781248\pi\)
−0.773006 + 0.634398i \(0.781248\pi\)
\(402\) 0 0
\(403\) −2.11751 −0.000261738 0
\(404\) 0 0
\(405\) 1529.43 0.187650
\(406\) 0 0
\(407\) 11425.7 1.39153
\(408\) 0 0
\(409\) −14052.1 −1.69885 −0.849425 0.527709i \(-0.823051\pi\)
−0.849425 + 0.527709i \(0.823051\pi\)
\(410\) 0 0
\(411\) −8054.24 −0.966633
\(412\) 0 0
\(413\) 1936.40 0.230711
\(414\) 0 0
\(415\) 8154.48 0.964548
\(416\) 0 0
\(417\) 5186.98 0.609130
\(418\) 0 0
\(419\) −1003.68 −0.117024 −0.0585119 0.998287i \(-0.518636\pi\)
−0.0585119 + 0.998287i \(0.518636\pi\)
\(420\) 0 0
\(421\) 8376.14 0.969663 0.484832 0.874608i \(-0.338881\pi\)
0.484832 + 0.874608i \(0.338881\pi\)
\(422\) 0 0
\(423\) −5694.39 −0.654541
\(424\) 0 0
\(425\) 7871.86 0.898450
\(426\) 0 0
\(427\) 15924.1 1.80473
\(428\) 0 0
\(429\) −1738.32 −0.195634
\(430\) 0 0
\(431\) 12296.3 1.37423 0.687116 0.726548i \(-0.258877\pi\)
0.687116 + 0.726548i \(0.258877\pi\)
\(432\) 0 0
\(433\) 1010.07 0.112104 0.0560518 0.998428i \(-0.482149\pi\)
0.0560518 + 0.998428i \(0.482149\pi\)
\(434\) 0 0
\(435\) 8478.39 0.934501
\(436\) 0 0
\(437\) −16389.1 −1.79405
\(438\) 0 0
\(439\) −1845.70 −0.200662 −0.100331 0.994954i \(-0.531990\pi\)
−0.100331 + 0.994954i \(0.531990\pi\)
\(440\) 0 0
\(441\) −194.719 −0.0210257
\(442\) 0 0
\(443\) −7313.67 −0.784386 −0.392193 0.919883i \(-0.628283\pi\)
−0.392193 + 0.919883i \(0.628283\pi\)
\(444\) 0 0
\(445\) 2215.95 0.236059
\(446\) 0 0
\(447\) −3398.68 −0.359624
\(448\) 0 0
\(449\) 8443.62 0.887481 0.443740 0.896155i \(-0.353651\pi\)
0.443740 + 0.896155i \(0.353651\pi\)
\(450\) 0 0
\(451\) −18485.2 −1.93001
\(452\) 0 0
\(453\) 3775.57 0.391593
\(454\) 0 0
\(455\) 4400.35 0.453388
\(456\) 0 0
\(457\) 14302.7 1.46401 0.732004 0.681301i \(-0.238585\pi\)
0.732004 + 0.681301i \(0.238585\pi\)
\(458\) 0 0
\(459\) 918.000 0.0933520
\(460\) 0 0
\(461\) −3862.08 −0.390184 −0.195092 0.980785i \(-0.562501\pi\)
−0.195092 + 0.980785i \(0.562501\pi\)
\(462\) 0 0
\(463\) 14364.6 1.44186 0.720929 0.693009i \(-0.243715\pi\)
0.720929 + 0.693009i \(0.243715\pi\)
\(464\) 0 0
\(465\) −9.22673 −0.000920171 0
\(466\) 0 0
\(467\) −14181.4 −1.40522 −0.702610 0.711575i \(-0.747982\pi\)
−0.702610 + 0.711575i \(0.747982\pi\)
\(468\) 0 0
\(469\) −11421.9 −1.12455
\(470\) 0 0
\(471\) 7409.66 0.724881
\(472\) 0 0
\(473\) 20996.9 2.04110
\(474\) 0 0
\(475\) 35385.2 3.41807
\(476\) 0 0
\(477\) −2128.40 −0.204303
\(478\) 0 0
\(479\) −1182.57 −0.112804 −0.0564019 0.998408i \(-0.517963\pi\)
−0.0564019 + 0.998408i \(0.517963\pi\)
\(480\) 0 0
\(481\) −3332.45 −0.315897
\(482\) 0 0
\(483\) −5767.04 −0.543291
\(484\) 0 0
\(485\) −21778.1 −2.03895
\(486\) 0 0
\(487\) 16039.2 1.49242 0.746209 0.665712i \(-0.231872\pi\)
0.746209 + 0.665712i \(0.231872\pi\)
\(488\) 0 0
\(489\) 2201.68 0.203606
\(490\) 0 0
\(491\) 11867.2 1.09075 0.545376 0.838191i \(-0.316387\pi\)
0.545376 + 0.838191i \(0.316387\pi\)
\(492\) 0 0
\(493\) 5088.92 0.464896
\(494\) 0 0
\(495\) −7574.48 −0.687772
\(496\) 0 0
\(497\) 6500.15 0.586663
\(498\) 0 0
\(499\) 6945.54 0.623097 0.311548 0.950230i \(-0.399152\pi\)
0.311548 + 0.950230i \(0.399152\pi\)
\(500\) 0 0
\(501\) −7831.17 −0.698345
\(502\) 0 0
\(503\) −5631.84 −0.499227 −0.249614 0.968346i \(-0.580304\pi\)
−0.249614 + 0.968346i \(0.580304\pi\)
\(504\) 0 0
\(505\) −16015.3 −1.41123
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 10395.3 0.905229 0.452614 0.891706i \(-0.350491\pi\)
0.452614 + 0.891706i \(0.350491\pi\)
\(510\) 0 0
\(511\) 12964.1 1.12231
\(512\) 0 0
\(513\) 4126.54 0.355149
\(514\) 0 0
\(515\) 13416.7 1.14799
\(516\) 0 0
\(517\) 28201.3 2.39902
\(518\) 0 0
\(519\) 3492.13 0.295352
\(520\) 0 0
\(521\) 7846.89 0.659844 0.329922 0.944008i \(-0.392978\pi\)
0.329922 + 0.944008i \(0.392978\pi\)
\(522\) 0 0
\(523\) −13670.9 −1.14299 −0.571497 0.820604i \(-0.693637\pi\)
−0.571497 + 0.820604i \(0.693637\pi\)
\(524\) 0 0
\(525\) 12451.4 1.03509
\(526\) 0 0
\(527\) −5.53809 −0.000457767 0
\(528\) 0 0
\(529\) −667.849 −0.0548902
\(530\) 0 0
\(531\) 972.160 0.0794504
\(532\) 0 0
\(533\) 5391.40 0.438138
\(534\) 0 0
\(535\) −10068.7 −0.813657
\(536\) 0 0
\(537\) −2733.69 −0.219678
\(538\) 0 0
\(539\) 964.341 0.0770633
\(540\) 0 0
\(541\) 6194.38 0.492269 0.246134 0.969236i \(-0.420840\pi\)
0.246134 + 0.969236i \(0.420840\pi\)
\(542\) 0 0
\(543\) −14453.4 −1.14228
\(544\) 0 0
\(545\) −12595.3 −0.989951
\(546\) 0 0
\(547\) 1856.88 0.145145 0.0725726 0.997363i \(-0.476879\pi\)
0.0725726 + 0.997363i \(0.476879\pi\)
\(548\) 0 0
\(549\) 7994.63 0.621498
\(550\) 0 0
\(551\) 22875.5 1.76865
\(552\) 0 0
\(553\) 17293.5 1.32983
\(554\) 0 0
\(555\) −14520.7 −1.11057
\(556\) 0 0
\(557\) 15017.2 1.14237 0.571183 0.820822i \(-0.306484\pi\)
0.571183 + 0.820822i \(0.306484\pi\)
\(558\) 0 0
\(559\) −6123.98 −0.463358
\(560\) 0 0
\(561\) −4546.37 −0.342153
\(562\) 0 0
\(563\) −13842.2 −1.03620 −0.518098 0.855321i \(-0.673360\pi\)
−0.518098 + 0.855321i \(0.673360\pi\)
\(564\) 0 0
\(565\) 5719.62 0.425887
\(566\) 0 0
\(567\) 1452.06 0.107550
\(568\) 0 0
\(569\) −17621.0 −1.29826 −0.649131 0.760677i \(-0.724867\pi\)
−0.649131 + 0.760677i \(0.724867\pi\)
\(570\) 0 0
\(571\) −16472.6 −1.20728 −0.603639 0.797258i \(-0.706283\pi\)
−0.603639 + 0.797258i \(0.706283\pi\)
\(572\) 0 0
\(573\) −2022.94 −0.147486
\(574\) 0 0
\(575\) −24827.4 −1.80065
\(576\) 0 0
\(577\) −19638.6 −1.41693 −0.708463 0.705748i \(-0.750611\pi\)
−0.708463 + 0.705748i \(0.750611\pi\)
\(578\) 0 0
\(579\) −2890.90 −0.207499
\(580\) 0 0
\(581\) 7741.94 0.552822
\(582\) 0 0
\(583\) 10540.8 0.748811
\(584\) 0 0
\(585\) 2209.18 0.156134
\(586\) 0 0
\(587\) 15965.5 1.12260 0.561300 0.827612i \(-0.310301\pi\)
0.561300 + 0.827612i \(0.310301\pi\)
\(588\) 0 0
\(589\) −24.8945 −0.00174153
\(590\) 0 0
\(591\) 11985.6 0.834219
\(592\) 0 0
\(593\) −16128.6 −1.11690 −0.558452 0.829537i \(-0.688604\pi\)
−0.558452 + 0.829537i \(0.688604\pi\)
\(594\) 0 0
\(595\) 11508.6 0.792953
\(596\) 0 0
\(597\) 4857.22 0.332986
\(598\) 0 0
\(599\) −18322.6 −1.24982 −0.624910 0.780697i \(-0.714864\pi\)
−0.624910 + 0.780697i \(0.714864\pi\)
\(600\) 0 0
\(601\) 10528.6 0.714594 0.357297 0.933991i \(-0.383698\pi\)
0.357297 + 0.933991i \(0.383698\pi\)
\(602\) 0 0
\(603\) −5734.34 −0.387264
\(604\) 0 0
\(605\) 12380.6 0.831974
\(606\) 0 0
\(607\) −16560.5 −1.10737 −0.553683 0.832728i \(-0.686778\pi\)
−0.553683 + 0.832728i \(0.686778\pi\)
\(608\) 0 0
\(609\) 8049.47 0.535601
\(610\) 0 0
\(611\) −8225.24 −0.544611
\(612\) 0 0
\(613\) 23354.0 1.53876 0.769379 0.638793i \(-0.220566\pi\)
0.769379 + 0.638793i \(0.220566\pi\)
\(614\) 0 0
\(615\) 23492.3 1.54032
\(616\) 0 0
\(617\) −23424.3 −1.52840 −0.764202 0.644977i \(-0.776867\pi\)
−0.764202 + 0.644977i \(0.776867\pi\)
\(618\) 0 0
\(619\) 3275.77 0.212705 0.106352 0.994328i \(-0.466083\pi\)
0.106352 + 0.994328i \(0.466083\pi\)
\(620\) 0 0
\(621\) −2895.32 −0.187094
\(622\) 0 0
\(623\) 2103.85 0.135295
\(624\) 0 0
\(625\) 9038.33 0.578453
\(626\) 0 0
\(627\) −20436.6 −1.30169
\(628\) 0 0
\(629\) −8715.63 −0.552488
\(630\) 0 0
\(631\) 24146.4 1.52338 0.761691 0.647941i \(-0.224369\pi\)
0.761691 + 0.647941i \(0.224369\pi\)
\(632\) 0 0
\(633\) −1631.97 −0.102473
\(634\) 0 0
\(635\) 8675.69 0.542180
\(636\) 0 0
\(637\) −281.261 −0.0174944
\(638\) 0 0
\(639\) 3263.37 0.202030
\(640\) 0 0
\(641\) −23506.8 −1.44846 −0.724230 0.689558i \(-0.757805\pi\)
−0.724230 + 0.689558i \(0.757805\pi\)
\(642\) 0 0
\(643\) −5273.79 −0.323449 −0.161725 0.986836i \(-0.551706\pi\)
−0.161725 + 0.986836i \(0.551706\pi\)
\(644\) 0 0
\(645\) −26684.4 −1.62899
\(646\) 0 0
\(647\) 7106.47 0.431815 0.215907 0.976414i \(-0.430729\pi\)
0.215907 + 0.976414i \(0.430729\pi\)
\(648\) 0 0
\(649\) −4814.60 −0.291201
\(650\) 0 0
\(651\) −8.75995 −0.000527388 0
\(652\) 0 0
\(653\) −8474.11 −0.507837 −0.253918 0.967226i \(-0.581719\pi\)
−0.253918 + 0.967226i \(0.581719\pi\)
\(654\) 0 0
\(655\) −9381.43 −0.559638
\(656\) 0 0
\(657\) 6508.59 0.386491
\(658\) 0 0
\(659\) 18444.7 1.09030 0.545148 0.838340i \(-0.316474\pi\)
0.545148 + 0.838340i \(0.316474\pi\)
\(660\) 0 0
\(661\) 29184.5 1.71732 0.858659 0.512547i \(-0.171298\pi\)
0.858659 + 0.512547i \(0.171298\pi\)
\(662\) 0 0
\(663\) 1326.00 0.0776736
\(664\) 0 0
\(665\) 51732.9 3.01672
\(666\) 0 0
\(667\) −16050.2 −0.931732
\(668\) 0 0
\(669\) −7342.36 −0.424323
\(670\) 0 0
\(671\) −39593.2 −2.27791
\(672\) 0 0
\(673\) −28477.5 −1.63110 −0.815548 0.578689i \(-0.803564\pi\)
−0.815548 + 0.578689i \(0.803564\pi\)
\(674\) 0 0
\(675\) 6251.18 0.356457
\(676\) 0 0
\(677\) 6422.80 0.364621 0.182310 0.983241i \(-0.441642\pi\)
0.182310 + 0.983241i \(0.441642\pi\)
\(678\) 0 0
\(679\) −20676.3 −1.16861
\(680\) 0 0
\(681\) −1403.40 −0.0789695
\(682\) 0 0
\(683\) 54.3801 0.00304655 0.00152328 0.999999i \(-0.499515\pi\)
0.00152328 + 0.999999i \(0.499515\pi\)
\(684\) 0 0
\(685\) −50693.1 −2.82756
\(686\) 0 0
\(687\) −1083.54 −0.0601741
\(688\) 0 0
\(689\) −3074.35 −0.169991
\(690\) 0 0
\(691\) −14382.3 −0.791794 −0.395897 0.918295i \(-0.629566\pi\)
−0.395897 + 0.918295i \(0.629566\pi\)
\(692\) 0 0
\(693\) −7191.28 −0.394191
\(694\) 0 0
\(695\) 32646.6 1.78181
\(696\) 0 0
\(697\) 14100.6 0.766281
\(698\) 0 0
\(699\) −9205.15 −0.498098
\(700\) 0 0
\(701\) 32081.3 1.72852 0.864262 0.503042i \(-0.167786\pi\)
0.864262 + 0.503042i \(0.167786\pi\)
\(702\) 0 0
\(703\) −39178.0 −2.10189
\(704\) 0 0
\(705\) −35840.3 −1.91464
\(706\) 0 0
\(707\) −15205.1 −0.808835
\(708\) 0 0
\(709\) 11549.8 0.611793 0.305896 0.952065i \(-0.401044\pi\)
0.305896 + 0.952065i \(0.401044\pi\)
\(710\) 0 0
\(711\) 8682.16 0.457955
\(712\) 0 0
\(713\) 17.4668 0.000917445 0
\(714\) 0 0
\(715\) −10940.9 −0.572261
\(716\) 0 0
\(717\) −1002.32 −0.0522071
\(718\) 0 0
\(719\) 33235.7 1.72390 0.861949 0.506995i \(-0.169244\pi\)
0.861949 + 0.506995i \(0.169244\pi\)
\(720\) 0 0
\(721\) 12738.0 0.657958
\(722\) 0 0
\(723\) 1544.88 0.0794672
\(724\) 0 0
\(725\) 34653.4 1.77516
\(726\) 0 0
\(727\) −13863.5 −0.707249 −0.353625 0.935387i \(-0.615051\pi\)
−0.353625 + 0.935387i \(0.615051\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −16016.6 −0.810389
\(732\) 0 0
\(733\) 13448.6 0.677675 0.338838 0.940845i \(-0.389966\pi\)
0.338838 + 0.940845i \(0.389966\pi\)
\(734\) 0 0
\(735\) −1225.55 −0.0615037
\(736\) 0 0
\(737\) 28399.2 1.41940
\(738\) 0 0
\(739\) 36402.4 1.81202 0.906012 0.423252i \(-0.139112\pi\)
0.906012 + 0.423252i \(0.139112\pi\)
\(740\) 0 0
\(741\) 5960.56 0.295502
\(742\) 0 0
\(743\) −23695.5 −1.16999 −0.584996 0.811036i \(-0.698904\pi\)
−0.584996 + 0.811036i \(0.698904\pi\)
\(744\) 0 0
\(745\) −21391.1 −1.05196
\(746\) 0 0
\(747\) 3886.81 0.190376
\(748\) 0 0
\(749\) −9559.29 −0.466340
\(750\) 0 0
\(751\) 4052.82 0.196923 0.0984617 0.995141i \(-0.468608\pi\)
0.0984617 + 0.995141i \(0.468608\pi\)
\(752\) 0 0
\(753\) 21084.4 1.02040
\(754\) 0 0
\(755\) 23763.3 1.14548
\(756\) 0 0
\(757\) 15508.4 0.744603 0.372301 0.928112i \(-0.378569\pi\)
0.372301 + 0.928112i \(0.378569\pi\)
\(758\) 0 0
\(759\) 14339.0 0.685735
\(760\) 0 0
\(761\) 29272.1 1.39436 0.697182 0.716894i \(-0.254437\pi\)
0.697182 + 0.716894i \(0.254437\pi\)
\(762\) 0 0
\(763\) −11958.1 −0.567382
\(764\) 0 0
\(765\) 5777.86 0.273070
\(766\) 0 0
\(767\) 1404.23 0.0661067
\(768\) 0 0
\(769\) 35088.4 1.64541 0.822706 0.568468i \(-0.192464\pi\)
0.822706 + 0.568468i \(0.192464\pi\)
\(770\) 0 0
\(771\) −10716.7 −0.500586
\(772\) 0 0
\(773\) −11475.6 −0.533956 −0.266978 0.963703i \(-0.586025\pi\)
−0.266978 + 0.963703i \(0.586025\pi\)
\(774\) 0 0
\(775\) −37.7120 −0.00174794
\(776\) 0 0
\(777\) −13786.1 −0.636515
\(778\) 0 0
\(779\) 63384.2 2.91524
\(780\) 0 0
\(781\) −16161.8 −0.740479
\(782\) 0 0
\(783\) 4041.20 0.184446
\(784\) 0 0
\(785\) 46636.1 2.12040
\(786\) 0 0
\(787\) 17215.7 0.779761 0.389881 0.920865i \(-0.372516\pi\)
0.389881 + 0.920865i \(0.372516\pi\)
\(788\) 0 0
\(789\) −12830.2 −0.578920
\(790\) 0 0
\(791\) 5430.27 0.244094
\(792\) 0 0
\(793\) 11547.8 0.517118
\(794\) 0 0
\(795\) −13396.1 −0.597622
\(796\) 0 0
\(797\) 43951.6 1.95338 0.976692 0.214646i \(-0.0688599\pi\)
0.976692 + 0.214646i \(0.0688599\pi\)
\(798\) 0 0
\(799\) −21512.2 −0.952497
\(800\) 0 0
\(801\) 1056.23 0.0465918
\(802\) 0 0
\(803\) −32233.6 −1.41656
\(804\) 0 0
\(805\) −36297.5 −1.58922
\(806\) 0 0
\(807\) 6384.24 0.278483
\(808\) 0 0
\(809\) −27914.7 −1.21314 −0.606569 0.795031i \(-0.707455\pi\)
−0.606569 + 0.795031i \(0.707455\pi\)
\(810\) 0 0
\(811\) −38016.3 −1.64603 −0.823017 0.568016i \(-0.807711\pi\)
−0.823017 + 0.568016i \(0.807711\pi\)
\(812\) 0 0
\(813\) 13160.8 0.567737
\(814\) 0 0
\(815\) 13857.3 0.595581
\(816\) 0 0
\(817\) −71996.8 −3.08305
\(818\) 0 0
\(819\) 2097.42 0.0894868
\(820\) 0 0
\(821\) −15367.7 −0.653273 −0.326637 0.945150i \(-0.605915\pi\)
−0.326637 + 0.945150i \(0.605915\pi\)
\(822\) 0 0
\(823\) −377.068 −0.0159705 −0.00798527 0.999968i \(-0.502542\pi\)
−0.00798527 + 0.999968i \(0.502542\pi\)
\(824\) 0 0
\(825\) −30958.8 −1.30648
\(826\) 0 0
\(827\) 5466.15 0.229839 0.114919 0.993375i \(-0.463339\pi\)
0.114919 + 0.993375i \(0.463339\pi\)
\(828\) 0 0
\(829\) −18760.0 −0.785962 −0.392981 0.919547i \(-0.628556\pi\)
−0.392981 + 0.919547i \(0.628556\pi\)
\(830\) 0 0
\(831\) 10235.6 0.427278
\(832\) 0 0
\(833\) −735.605 −0.0305969
\(834\) 0 0
\(835\) −49289.1 −2.04278
\(836\) 0 0
\(837\) −4.39790 −0.000181617 0
\(838\) 0 0
\(839\) 46530.4 1.91467 0.957334 0.288983i \(-0.0933170\pi\)
0.957334 + 0.288983i \(0.0933170\pi\)
\(840\) 0 0
\(841\) −1986.62 −0.0814558
\(842\) 0 0
\(843\) −13400.5 −0.547493
\(844\) 0 0
\(845\) 3191.04 0.129911
\(846\) 0 0
\(847\) 11754.3 0.476839
\(848\) 0 0
\(849\) 5637.91 0.227906
\(850\) 0 0
\(851\) 27488.6 1.10728
\(852\) 0 0
\(853\) −11378.2 −0.456719 −0.228359 0.973577i \(-0.573336\pi\)
−0.228359 + 0.973577i \(0.573336\pi\)
\(854\) 0 0
\(855\) 25972.3 1.03887
\(856\) 0 0
\(857\) 16515.4 0.658289 0.329145 0.944280i \(-0.393240\pi\)
0.329145 + 0.944280i \(0.393240\pi\)
\(858\) 0 0
\(859\) −1477.37 −0.0586811 −0.0293406 0.999569i \(-0.509341\pi\)
−0.0293406 + 0.999569i \(0.509341\pi\)
\(860\) 0 0
\(861\) 22303.8 0.882824
\(862\) 0 0
\(863\) −14215.0 −0.560702 −0.280351 0.959898i \(-0.590451\pi\)
−0.280351 + 0.959898i \(0.590451\pi\)
\(864\) 0 0
\(865\) 21979.3 0.863953
\(866\) 0 0
\(867\) −11271.0 −0.441503
\(868\) 0 0
\(869\) −42998.2 −1.67850
\(870\) 0 0
\(871\) −8282.94 −0.322223
\(872\) 0 0
\(873\) −10380.5 −0.402434
\(874\) 0 0
\(875\) 36057.6 1.39311
\(876\) 0 0
\(877\) 15634.9 0.602001 0.301000 0.953624i \(-0.402679\pi\)
0.301000 + 0.953624i \(0.402679\pi\)
\(878\) 0 0
\(879\) −623.987 −0.0239438
\(880\) 0 0
\(881\) −35793.2 −1.36879 −0.684395 0.729111i \(-0.739934\pi\)
−0.684395 + 0.729111i \(0.739934\pi\)
\(882\) 0 0
\(883\) 18937.7 0.721749 0.360875 0.932614i \(-0.382478\pi\)
0.360875 + 0.932614i \(0.382478\pi\)
\(884\) 0 0
\(885\) 6118.74 0.232406
\(886\) 0 0
\(887\) 16591.7 0.628066 0.314033 0.949412i \(-0.398320\pi\)
0.314033 + 0.949412i \(0.398320\pi\)
\(888\) 0 0
\(889\) 8236.78 0.310746
\(890\) 0 0
\(891\) −3610.35 −0.135748
\(892\) 0 0
\(893\) −96700.3 −3.62369
\(894\) 0 0
\(895\) −17205.7 −0.642596
\(896\) 0 0
\(897\) −4182.13 −0.155671
\(898\) 0 0
\(899\) −24.3797 −0.000904459 0
\(900\) 0 0
\(901\) −8040.62 −0.297305
\(902\) 0 0
\(903\) −25334.4 −0.933639
\(904\) 0 0
\(905\) −90969.4 −3.34135
\(906\) 0 0
\(907\) −28947.7 −1.05975 −0.529875 0.848076i \(-0.677761\pi\)
−0.529875 + 0.848076i \(0.677761\pi\)
\(908\) 0 0
\(909\) −7633.65 −0.278539
\(910\) 0 0
\(911\) −13179.1 −0.479302 −0.239651 0.970859i \(-0.577033\pi\)
−0.239651 + 0.970859i \(0.577033\pi\)
\(912\) 0 0
\(913\) −19249.3 −0.697766
\(914\) 0 0
\(915\) 50317.9 1.81799
\(916\) 0 0
\(917\) −8906.82 −0.320751
\(918\) 0 0
\(919\) −29617.7 −1.06311 −0.531555 0.847024i \(-0.678392\pi\)
−0.531555 + 0.847024i \(0.678392\pi\)
\(920\) 0 0
\(921\) 5229.21 0.187088
\(922\) 0 0
\(923\) 4713.76 0.168099
\(924\) 0 0
\(925\) −59349.7 −2.10963
\(926\) 0 0
\(927\) 6395.06 0.226582
\(928\) 0 0
\(929\) −36633.5 −1.29376 −0.646882 0.762590i \(-0.723927\pi\)
−0.646882 + 0.762590i \(0.723927\pi\)
\(930\) 0 0
\(931\) −3306.65 −0.116403
\(932\) 0 0
\(933\) −30083.8 −1.05563
\(934\) 0 0
\(935\) −28614.7 −1.00086
\(936\) 0 0
\(937\) 3252.46 0.113397 0.0566986 0.998391i \(-0.481943\pi\)
0.0566986 + 0.998391i \(0.481943\pi\)
\(938\) 0 0
\(939\) −6278.39 −0.218198
\(940\) 0 0
\(941\) −20638.9 −0.714995 −0.357497 0.933914i \(-0.616370\pi\)
−0.357497 + 0.933914i \(0.616370\pi\)
\(942\) 0 0
\(943\) −44472.5 −1.53576
\(944\) 0 0
\(945\) 9139.20 0.314601
\(946\) 0 0
\(947\) 21790.8 0.747737 0.373869 0.927482i \(-0.378031\pi\)
0.373869 + 0.927482i \(0.378031\pi\)
\(948\) 0 0
\(949\) 9401.30 0.321580
\(950\) 0 0
\(951\) −31445.3 −1.07222
\(952\) 0 0
\(953\) 31780.0 1.08023 0.540114 0.841592i \(-0.318381\pi\)
0.540114 + 0.841592i \(0.318381\pi\)
\(954\) 0 0
\(955\) −12732.3 −0.431421
\(956\) 0 0
\(957\) −20014.0 −0.676029
\(958\) 0 0
\(959\) −48128.5 −1.62059
\(960\) 0 0
\(961\) −29791.0 −0.999999
\(962\) 0 0
\(963\) −4799.20 −0.160594
\(964\) 0 0
\(965\) −18195.2 −0.606969
\(966\) 0 0
\(967\) 38528.7 1.28128 0.640640 0.767841i \(-0.278669\pi\)
0.640640 + 0.767841i \(0.278669\pi\)
\(968\) 0 0
\(969\) 15589.2 0.516818
\(970\) 0 0
\(971\) −22579.5 −0.746252 −0.373126 0.927781i \(-0.621714\pi\)
−0.373126 + 0.927781i \(0.621714\pi\)
\(972\) 0 0
\(973\) 30995.0 1.02123
\(974\) 0 0
\(975\) 9029.49 0.296590
\(976\) 0 0
\(977\) 30254.4 0.990712 0.495356 0.868690i \(-0.335038\pi\)
0.495356 + 0.868690i \(0.335038\pi\)
\(978\) 0 0
\(979\) −5230.95 −0.170768
\(980\) 0 0
\(981\) −6003.51 −0.195390
\(982\) 0 0
\(983\) 7071.61 0.229450 0.114725 0.993397i \(-0.463401\pi\)
0.114725 + 0.993397i \(0.463401\pi\)
\(984\) 0 0
\(985\) 75437.1 2.44023
\(986\) 0 0
\(987\) −34027.1 −1.09736
\(988\) 0 0
\(989\) 50515.4 1.62416
\(990\) 0 0
\(991\) −2808.28 −0.0900180 −0.0450090 0.998987i \(-0.514332\pi\)
−0.0450090 + 0.998987i \(0.514332\pi\)
\(992\) 0 0
\(993\) −11862.7 −0.379106
\(994\) 0 0
\(995\) 30571.1 0.974040
\(996\) 0 0
\(997\) −15225.5 −0.483648 −0.241824 0.970320i \(-0.577746\pi\)
−0.241824 + 0.970320i \(0.577746\pi\)
\(998\) 0 0
\(999\) −6921.23 −0.219197
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 624.4.a.u.1.3 3
3.2 odd 2 1872.4.a.bj.1.1 3
4.3 odd 2 312.4.a.g.1.3 3
8.3 odd 2 2496.4.a.bo.1.1 3
8.5 even 2 2496.4.a.bk.1.1 3
12.11 even 2 936.4.a.k.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.g.1.3 3 4.3 odd 2
624.4.a.u.1.3 3 1.1 even 1 trivial
936.4.a.k.1.1 3 12.11 even 2
1872.4.a.bj.1.1 3 3.2 odd 2
2496.4.a.bk.1.1 3 8.5 even 2
2496.4.a.bo.1.1 3 8.3 odd 2