Properties

Label 4016.2.a.m.1.23
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $0$
Dimension $23$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.23
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.39399 q^{3} -1.84335 q^{5} +1.40053 q^{7} +8.51918 q^{9} +O(q^{10})\) \(q+3.39399 q^{3} -1.84335 q^{5} +1.40053 q^{7} +8.51918 q^{9} +4.77705 q^{11} -4.38340 q^{13} -6.25633 q^{15} +3.98913 q^{17} +3.05133 q^{19} +4.75340 q^{21} +5.20078 q^{23} -1.60204 q^{25} +18.7321 q^{27} -5.58232 q^{29} +8.38580 q^{31} +16.2133 q^{33} -2.58168 q^{35} -7.29402 q^{37} -14.8772 q^{39} -5.29108 q^{41} -8.60690 q^{43} -15.7039 q^{45} -3.56256 q^{47} -5.03851 q^{49} +13.5391 q^{51} -1.07015 q^{53} -8.80580 q^{55} +10.3562 q^{57} +11.7289 q^{59} +1.87603 q^{61} +11.9314 q^{63} +8.08015 q^{65} -12.5600 q^{67} +17.6514 q^{69} -6.78679 q^{71} +13.3111 q^{73} -5.43732 q^{75} +6.69042 q^{77} +16.8420 q^{79} +38.0189 q^{81} +2.43147 q^{83} -7.35338 q^{85} -18.9464 q^{87} -14.3588 q^{89} -6.13909 q^{91} +28.4613 q^{93} -5.62469 q^{95} -4.75137 q^{97} +40.6966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 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
\(3\) 3.39399 1.95952 0.979761 0.200170i \(-0.0641495\pi\)
0.979761 + 0.200170i \(0.0641495\pi\)
\(4\) 0 0
\(5\) −1.84335 −0.824373 −0.412187 0.911099i \(-0.635235\pi\)
−0.412187 + 0.911099i \(0.635235\pi\)
\(6\) 0 0
\(7\) 1.40053 0.529352 0.264676 0.964337i \(-0.414735\pi\)
0.264676 + 0.964337i \(0.414735\pi\)
\(8\) 0 0
\(9\) 8.51918 2.83973
\(10\) 0 0
\(11\) 4.77705 1.44034 0.720168 0.693800i \(-0.244065\pi\)
0.720168 + 0.693800i \(0.244065\pi\)
\(12\) 0 0
\(13\) −4.38340 −1.21574 −0.607868 0.794038i \(-0.707975\pi\)
−0.607868 + 0.794038i \(0.707975\pi\)
\(14\) 0 0
\(15\) −6.25633 −1.61538
\(16\) 0 0
\(17\) 3.98913 0.967506 0.483753 0.875205i \(-0.339273\pi\)
0.483753 + 0.875205i \(0.339273\pi\)
\(18\) 0 0
\(19\) 3.05133 0.700024 0.350012 0.936745i \(-0.386178\pi\)
0.350012 + 0.936745i \(0.386178\pi\)
\(20\) 0 0
\(21\) 4.75340 1.03728
\(22\) 0 0
\(23\) 5.20078 1.08444 0.542218 0.840238i \(-0.317585\pi\)
0.542218 + 0.840238i \(0.317585\pi\)
\(24\) 0 0
\(25\) −1.60204 −0.320409
\(26\) 0 0
\(27\) 18.7321 3.60499
\(28\) 0 0
\(29\) −5.58232 −1.03661 −0.518306 0.855195i \(-0.673437\pi\)
−0.518306 + 0.855195i \(0.673437\pi\)
\(30\) 0 0
\(31\) 8.38580 1.50613 0.753067 0.657944i \(-0.228574\pi\)
0.753067 + 0.657944i \(0.228574\pi\)
\(32\) 0 0
\(33\) 16.2133 2.82237
\(34\) 0 0
\(35\) −2.58168 −0.436383
\(36\) 0 0
\(37\) −7.29402 −1.19913 −0.599565 0.800326i \(-0.704660\pi\)
−0.599565 + 0.800326i \(0.704660\pi\)
\(38\) 0 0
\(39\) −14.8772 −2.38226
\(40\) 0 0
\(41\) −5.29108 −0.826327 −0.413164 0.910657i \(-0.635576\pi\)
−0.413164 + 0.910657i \(0.635576\pi\)
\(42\) 0 0
\(43\) −8.60690 −1.31254 −0.656270 0.754526i \(-0.727867\pi\)
−0.656270 + 0.754526i \(0.727867\pi\)
\(44\) 0 0
\(45\) −15.7039 −2.34100
\(46\) 0 0
\(47\) −3.56256 −0.519653 −0.259826 0.965655i \(-0.583665\pi\)
−0.259826 + 0.965655i \(0.583665\pi\)
\(48\) 0 0
\(49\) −5.03851 −0.719787
\(50\) 0 0
\(51\) 13.5391 1.89585
\(52\) 0 0
\(53\) −1.07015 −0.146996 −0.0734981 0.997295i \(-0.523416\pi\)
−0.0734981 + 0.997295i \(0.523416\pi\)
\(54\) 0 0
\(55\) −8.80580 −1.18737
\(56\) 0 0
\(57\) 10.3562 1.37171
\(58\) 0 0
\(59\) 11.7289 1.52697 0.763485 0.645826i \(-0.223487\pi\)
0.763485 + 0.645826i \(0.223487\pi\)
\(60\) 0 0
\(61\) 1.87603 0.240200 0.120100 0.992762i \(-0.461678\pi\)
0.120100 + 0.992762i \(0.461678\pi\)
\(62\) 0 0
\(63\) 11.9314 1.50322
\(64\) 0 0
\(65\) 8.08015 1.00222
\(66\) 0 0
\(67\) −12.5600 −1.53445 −0.767227 0.641376i \(-0.778364\pi\)
−0.767227 + 0.641376i \(0.778364\pi\)
\(68\) 0 0
\(69\) 17.6514 2.12498
\(70\) 0 0
\(71\) −6.78679 −0.805444 −0.402722 0.915322i \(-0.631936\pi\)
−0.402722 + 0.915322i \(0.631936\pi\)
\(72\) 0 0
\(73\) 13.3111 1.55794 0.778972 0.627059i \(-0.215741\pi\)
0.778972 + 0.627059i \(0.215741\pi\)
\(74\) 0 0
\(75\) −5.43732 −0.627848
\(76\) 0 0
\(77\) 6.69042 0.762444
\(78\) 0 0
\(79\) 16.8420 1.89487 0.947436 0.319946i \(-0.103665\pi\)
0.947436 + 0.319946i \(0.103665\pi\)
\(80\) 0 0
\(81\) 38.0189 4.22433
\(82\) 0 0
\(83\) 2.43147 0.266889 0.133444 0.991056i \(-0.457396\pi\)
0.133444 + 0.991056i \(0.457396\pi\)
\(84\) 0 0
\(85\) −7.35338 −0.797586
\(86\) 0 0
\(87\) −18.9464 −2.03126
\(88\) 0 0
\(89\) −14.3588 −1.52203 −0.761017 0.648732i \(-0.775300\pi\)
−0.761017 + 0.648732i \(0.775300\pi\)
\(90\) 0 0
\(91\) −6.13909 −0.643552
\(92\) 0 0
\(93\) 28.4613 2.95130
\(94\) 0 0
\(95\) −5.62469 −0.577081
\(96\) 0 0
\(97\) −4.75137 −0.482429 −0.241214 0.970472i \(-0.577546\pi\)
−0.241214 + 0.970472i \(0.577546\pi\)
\(98\) 0 0
\(99\) 40.6966 4.09016
\(100\) 0 0
\(101\) 7.08097 0.704583 0.352291 0.935890i \(-0.385403\pi\)
0.352291 + 0.935890i \(0.385403\pi\)
\(102\) 0 0
\(103\) 12.4588 1.22760 0.613801 0.789461i \(-0.289640\pi\)
0.613801 + 0.789461i \(0.289640\pi\)
\(104\) 0 0
\(105\) −8.76220 −0.855103
\(106\) 0 0
\(107\) −9.90438 −0.957493 −0.478746 0.877953i \(-0.658909\pi\)
−0.478746 + 0.877953i \(0.658909\pi\)
\(108\) 0 0
\(109\) −0.0878728 −0.00841669 −0.00420834 0.999991i \(-0.501340\pi\)
−0.00420834 + 0.999991i \(0.501340\pi\)
\(110\) 0 0
\(111\) −24.7558 −2.34972
\(112\) 0 0
\(113\) 5.46984 0.514559 0.257280 0.966337i \(-0.417174\pi\)
0.257280 + 0.966337i \(0.417174\pi\)
\(114\) 0 0
\(115\) −9.58687 −0.893980
\(116\) 0 0
\(117\) −37.3430 −3.45236
\(118\) 0 0
\(119\) 5.58691 0.512151
\(120\) 0 0
\(121\) 11.8202 1.07457
\(122\) 0 0
\(123\) −17.9579 −1.61921
\(124\) 0 0
\(125\) 12.1699 1.08851
\(126\) 0 0
\(127\) 18.6060 1.65102 0.825509 0.564389i \(-0.190888\pi\)
0.825509 + 0.564389i \(0.190888\pi\)
\(128\) 0 0
\(129\) −29.2117 −2.57195
\(130\) 0 0
\(131\) 2.41585 0.211074 0.105537 0.994415i \(-0.466344\pi\)
0.105537 + 0.994415i \(0.466344\pi\)
\(132\) 0 0
\(133\) 4.27349 0.370559
\(134\) 0 0
\(135\) −34.5298 −2.97186
\(136\) 0 0
\(137\) 20.3205 1.73610 0.868050 0.496477i \(-0.165373\pi\)
0.868050 + 0.496477i \(0.165373\pi\)
\(138\) 0 0
\(139\) −13.5923 −1.15289 −0.576444 0.817137i \(-0.695560\pi\)
−0.576444 + 0.817137i \(0.695560\pi\)
\(140\) 0 0
\(141\) −12.0913 −1.01827
\(142\) 0 0
\(143\) −20.9397 −1.75107
\(144\) 0 0
\(145\) 10.2902 0.854555
\(146\) 0 0
\(147\) −17.1007 −1.41044
\(148\) 0 0
\(149\) −1.86674 −0.152929 −0.0764645 0.997072i \(-0.524363\pi\)
−0.0764645 + 0.997072i \(0.524363\pi\)
\(150\) 0 0
\(151\) −6.63198 −0.539703 −0.269852 0.962902i \(-0.586975\pi\)
−0.269852 + 0.962902i \(0.586975\pi\)
\(152\) 0 0
\(153\) 33.9841 2.74745
\(154\) 0 0
\(155\) −15.4580 −1.24162
\(156\) 0 0
\(157\) −4.99472 −0.398622 −0.199311 0.979936i \(-0.563870\pi\)
−0.199311 + 0.979936i \(0.563870\pi\)
\(158\) 0 0
\(159\) −3.63207 −0.288042
\(160\) 0 0
\(161\) 7.28386 0.574048
\(162\) 0 0
\(163\) −11.3940 −0.892448 −0.446224 0.894921i \(-0.647232\pi\)
−0.446224 + 0.894921i \(0.647232\pi\)
\(164\) 0 0
\(165\) −29.8868 −2.32669
\(166\) 0 0
\(167\) 3.26981 0.253025 0.126513 0.991965i \(-0.459622\pi\)
0.126513 + 0.991965i \(0.459622\pi\)
\(168\) 0 0
\(169\) 6.21416 0.478013
\(170\) 0 0
\(171\) 25.9949 1.98788
\(172\) 0 0
\(173\) 4.17358 0.317311 0.158656 0.987334i \(-0.449284\pi\)
0.158656 + 0.987334i \(0.449284\pi\)
\(174\) 0 0
\(175\) −2.24372 −0.169609
\(176\) 0 0
\(177\) 39.8077 2.99213
\(178\) 0 0
\(179\) −6.86920 −0.513428 −0.256714 0.966487i \(-0.582640\pi\)
−0.256714 + 0.966487i \(0.582640\pi\)
\(180\) 0 0
\(181\) 8.12093 0.603624 0.301812 0.953367i \(-0.402408\pi\)
0.301812 + 0.953367i \(0.402408\pi\)
\(182\) 0 0
\(183\) 6.36722 0.470678
\(184\) 0 0
\(185\) 13.4455 0.988530
\(186\) 0 0
\(187\) 19.0563 1.39353
\(188\) 0 0
\(189\) 26.2349 1.90831
\(190\) 0 0
\(191\) −15.9393 −1.15333 −0.576665 0.816981i \(-0.695646\pi\)
−0.576665 + 0.816981i \(0.695646\pi\)
\(192\) 0 0
\(193\) −13.5715 −0.976895 −0.488448 0.872593i \(-0.662437\pi\)
−0.488448 + 0.872593i \(0.662437\pi\)
\(194\) 0 0
\(195\) 27.4240 1.96387
\(196\) 0 0
\(197\) 19.2258 1.36978 0.684889 0.728647i \(-0.259851\pi\)
0.684889 + 0.728647i \(0.259851\pi\)
\(198\) 0 0
\(199\) 23.5614 1.67022 0.835111 0.550081i \(-0.185403\pi\)
0.835111 + 0.550081i \(0.185403\pi\)
\(200\) 0 0
\(201\) −42.6287 −3.00680
\(202\) 0 0
\(203\) −7.81823 −0.548732
\(204\) 0 0
\(205\) 9.75333 0.681202
\(206\) 0 0
\(207\) 44.3064 3.07950
\(208\) 0 0
\(209\) 14.5764 1.00827
\(210\) 0 0
\(211\) 11.2329 0.773305 0.386652 0.922225i \(-0.373631\pi\)
0.386652 + 0.922225i \(0.373631\pi\)
\(212\) 0 0
\(213\) −23.0343 −1.57828
\(214\) 0 0
\(215\) 15.8656 1.08202
\(216\) 0 0
\(217\) 11.7446 0.797275
\(218\) 0 0
\(219\) 45.1777 3.05283
\(220\) 0 0
\(221\) −17.4859 −1.17623
\(222\) 0 0
\(223\) −11.7261 −0.785236 −0.392618 0.919702i \(-0.628431\pi\)
−0.392618 + 0.919702i \(0.628431\pi\)
\(224\) 0 0
\(225\) −13.6481 −0.909874
\(226\) 0 0
\(227\) −5.27754 −0.350283 −0.175141 0.984543i \(-0.556038\pi\)
−0.175141 + 0.984543i \(0.556038\pi\)
\(228\) 0 0
\(229\) −15.4766 −1.02272 −0.511360 0.859366i \(-0.670858\pi\)
−0.511360 + 0.859366i \(0.670858\pi\)
\(230\) 0 0
\(231\) 22.7072 1.49403
\(232\) 0 0
\(233\) −1.95226 −0.127897 −0.0639484 0.997953i \(-0.520369\pi\)
−0.0639484 + 0.997953i \(0.520369\pi\)
\(234\) 0 0
\(235\) 6.56706 0.428388
\(236\) 0 0
\(237\) 57.1616 3.71304
\(238\) 0 0
\(239\) 3.10139 0.200612 0.100306 0.994957i \(-0.468018\pi\)
0.100306 + 0.994957i \(0.468018\pi\)
\(240\) 0 0
\(241\) −18.9756 −1.22233 −0.611163 0.791505i \(-0.709298\pi\)
−0.611163 + 0.791505i \(0.709298\pi\)
\(242\) 0 0
\(243\) 72.8398 4.67267
\(244\) 0 0
\(245\) 9.28775 0.593373
\(246\) 0 0
\(247\) −13.3752 −0.851044
\(248\) 0 0
\(249\) 8.25240 0.522975
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 24.8444 1.56195
\(254\) 0 0
\(255\) −24.9573 −1.56289
\(256\) 0 0
\(257\) −29.4307 −1.83584 −0.917918 0.396771i \(-0.870131\pi\)
−0.917918 + 0.396771i \(0.870131\pi\)
\(258\) 0 0
\(259\) −10.2155 −0.634761
\(260\) 0 0
\(261\) −47.5568 −2.94369
\(262\) 0 0
\(263\) −13.5204 −0.833702 −0.416851 0.908975i \(-0.636866\pi\)
−0.416851 + 0.908975i \(0.636866\pi\)
\(264\) 0 0
\(265\) 1.97266 0.121180
\(266\) 0 0
\(267\) −48.7338 −2.98246
\(268\) 0 0
\(269\) −29.9875 −1.82837 −0.914184 0.405299i \(-0.867168\pi\)
−0.914184 + 0.405299i \(0.867168\pi\)
\(270\) 0 0
\(271\) −10.2684 −0.623762 −0.311881 0.950121i \(-0.600959\pi\)
−0.311881 + 0.950121i \(0.600959\pi\)
\(272\) 0 0
\(273\) −20.8360 −1.26105
\(274\) 0 0
\(275\) −7.65305 −0.461496
\(276\) 0 0
\(277\) −9.91642 −0.595820 −0.297910 0.954594i \(-0.596290\pi\)
−0.297910 + 0.954594i \(0.596290\pi\)
\(278\) 0 0
\(279\) 71.4402 4.27701
\(280\) 0 0
\(281\) 11.3153 0.675017 0.337508 0.941323i \(-0.390416\pi\)
0.337508 + 0.941323i \(0.390416\pi\)
\(282\) 0 0
\(283\) 12.8104 0.761499 0.380749 0.924678i \(-0.375666\pi\)
0.380749 + 0.924678i \(0.375666\pi\)
\(284\) 0 0
\(285\) −19.0901 −1.13080
\(286\) 0 0
\(287\) −7.41033 −0.437418
\(288\) 0 0
\(289\) −1.08684 −0.0639317
\(290\) 0 0
\(291\) −16.1261 −0.945330
\(292\) 0 0
\(293\) 2.67644 0.156359 0.0781796 0.996939i \(-0.475089\pi\)
0.0781796 + 0.996939i \(0.475089\pi\)
\(294\) 0 0
\(295\) −21.6205 −1.25879
\(296\) 0 0
\(297\) 89.4841 5.19239
\(298\) 0 0
\(299\) −22.7971 −1.31839
\(300\) 0 0
\(301\) −12.0542 −0.694795
\(302\) 0 0
\(303\) 24.0328 1.38065
\(304\) 0 0
\(305\) −3.45818 −0.198015
\(306\) 0 0
\(307\) 0.698323 0.0398554 0.0199277 0.999801i \(-0.493656\pi\)
0.0199277 + 0.999801i \(0.493656\pi\)
\(308\) 0 0
\(309\) 42.2851 2.40551
\(310\) 0 0
\(311\) −3.52864 −0.200091 −0.100046 0.994983i \(-0.531899\pi\)
−0.100046 + 0.994983i \(0.531899\pi\)
\(312\) 0 0
\(313\) −27.1365 −1.53385 −0.766924 0.641738i \(-0.778213\pi\)
−0.766924 + 0.641738i \(0.778213\pi\)
\(314\) 0 0
\(315\) −21.9938 −1.23921
\(316\) 0 0
\(317\) −21.3806 −1.20085 −0.600427 0.799680i \(-0.705003\pi\)
−0.600427 + 0.799680i \(0.705003\pi\)
\(318\) 0 0
\(319\) −26.6671 −1.49307
\(320\) 0 0
\(321\) −33.6154 −1.87623
\(322\) 0 0
\(323\) 12.1722 0.677277
\(324\) 0 0
\(325\) 7.02239 0.389532
\(326\) 0 0
\(327\) −0.298240 −0.0164927
\(328\) 0 0
\(329\) −4.98948 −0.275079
\(330\) 0 0
\(331\) 18.9447 1.04129 0.520646 0.853773i \(-0.325691\pi\)
0.520646 + 0.853773i \(0.325691\pi\)
\(332\) 0 0
\(333\) −62.1391 −3.40520
\(334\) 0 0
\(335\) 23.1526 1.26496
\(336\) 0 0
\(337\) −17.5232 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(338\) 0 0
\(339\) 18.5646 1.00829
\(340\) 0 0
\(341\) 40.0594 2.16934
\(342\) 0 0
\(343\) −16.8603 −0.910372
\(344\) 0 0
\(345\) −32.5378 −1.75177
\(346\) 0 0
\(347\) −24.4297 −1.31146 −0.655728 0.754997i \(-0.727638\pi\)
−0.655728 + 0.754997i \(0.727638\pi\)
\(348\) 0 0
\(349\) 9.46353 0.506571 0.253286 0.967392i \(-0.418489\pi\)
0.253286 + 0.967392i \(0.418489\pi\)
\(350\) 0 0
\(351\) −82.1101 −4.38271
\(352\) 0 0
\(353\) −8.34269 −0.444037 −0.222018 0.975043i \(-0.571264\pi\)
−0.222018 + 0.975043i \(0.571264\pi\)
\(354\) 0 0
\(355\) 12.5105 0.663986
\(356\) 0 0
\(357\) 18.9619 1.00357
\(358\) 0 0
\(359\) 14.1164 0.745033 0.372516 0.928026i \(-0.378495\pi\)
0.372516 + 0.928026i \(0.378495\pi\)
\(360\) 0 0
\(361\) −9.68937 −0.509967
\(362\) 0 0
\(363\) 40.1178 2.10564
\(364\) 0 0
\(365\) −24.5370 −1.28433
\(366\) 0 0
\(367\) 30.9280 1.61443 0.807215 0.590258i \(-0.200974\pi\)
0.807215 + 0.590258i \(0.200974\pi\)
\(368\) 0 0
\(369\) −45.0756 −2.34654
\(370\) 0 0
\(371\) −1.49878 −0.0778127
\(372\) 0 0
\(373\) 0.438620 0.0227109 0.0113554 0.999936i \(-0.496385\pi\)
0.0113554 + 0.999936i \(0.496385\pi\)
\(374\) 0 0
\(375\) 41.3046 2.13296
\(376\) 0 0
\(377\) 24.4695 1.26025
\(378\) 0 0
\(379\) −6.13296 −0.315029 −0.157514 0.987517i \(-0.550348\pi\)
−0.157514 + 0.987517i \(0.550348\pi\)
\(380\) 0 0
\(381\) 63.1487 3.23521
\(382\) 0 0
\(383\) −18.0614 −0.922895 −0.461448 0.887167i \(-0.652670\pi\)
−0.461448 + 0.887167i \(0.652670\pi\)
\(384\) 0 0
\(385\) −12.3328 −0.628539
\(386\) 0 0
\(387\) −73.3237 −3.72726
\(388\) 0 0
\(389\) −11.7955 −0.598056 −0.299028 0.954244i \(-0.596662\pi\)
−0.299028 + 0.954244i \(0.596662\pi\)
\(390\) 0 0
\(391\) 20.7466 1.04920
\(392\) 0 0
\(393\) 8.19938 0.413604
\(394\) 0 0
\(395\) −31.0458 −1.56208
\(396\) 0 0
\(397\) 4.72182 0.236982 0.118491 0.992955i \(-0.462194\pi\)
0.118491 + 0.992955i \(0.462194\pi\)
\(398\) 0 0
\(399\) 14.5042 0.726118
\(400\) 0 0
\(401\) 14.1995 0.709087 0.354544 0.935039i \(-0.384636\pi\)
0.354544 + 0.935039i \(0.384636\pi\)
\(402\) 0 0
\(403\) −36.7583 −1.83106
\(404\) 0 0
\(405\) −70.0824 −3.48242
\(406\) 0 0
\(407\) −34.8439 −1.72715
\(408\) 0 0
\(409\) 18.4793 0.913741 0.456871 0.889533i \(-0.348970\pi\)
0.456871 + 0.889533i \(0.348970\pi\)
\(410\) 0 0
\(411\) 68.9677 3.40193
\(412\) 0 0
\(413\) 16.4267 0.808304
\(414\) 0 0
\(415\) −4.48207 −0.220016
\(416\) 0 0
\(417\) −46.1323 −2.25911
\(418\) 0 0
\(419\) 3.75585 0.183485 0.0917427 0.995783i \(-0.470756\pi\)
0.0917427 + 0.995783i \(0.470756\pi\)
\(420\) 0 0
\(421\) 29.4301 1.43433 0.717167 0.696901i \(-0.245438\pi\)
0.717167 + 0.696901i \(0.245438\pi\)
\(422\) 0 0
\(423\) −30.3501 −1.47567
\(424\) 0 0
\(425\) −6.39076 −0.309998
\(426\) 0 0
\(427\) 2.62744 0.127151
\(428\) 0 0
\(429\) −71.0692 −3.43125
\(430\) 0 0
\(431\) −25.5126 −1.22890 −0.614449 0.788957i \(-0.710621\pi\)
−0.614449 + 0.788957i \(0.710621\pi\)
\(432\) 0 0
\(433\) −7.21250 −0.346611 −0.173305 0.984868i \(-0.555445\pi\)
−0.173305 + 0.984868i \(0.555445\pi\)
\(434\) 0 0
\(435\) 34.9249 1.67452
\(436\) 0 0
\(437\) 15.8693 0.759131
\(438\) 0 0
\(439\) 16.6607 0.795173 0.397587 0.917565i \(-0.369848\pi\)
0.397587 + 0.917565i \(0.369848\pi\)
\(440\) 0 0
\(441\) −42.9240 −2.04400
\(442\) 0 0
\(443\) −14.8396 −0.705053 −0.352526 0.935802i \(-0.614677\pi\)
−0.352526 + 0.935802i \(0.614677\pi\)
\(444\) 0 0
\(445\) 26.4684 1.25472
\(446\) 0 0
\(447\) −6.33569 −0.299668
\(448\) 0 0
\(449\) −27.9646 −1.31973 −0.659866 0.751384i \(-0.729387\pi\)
−0.659866 + 0.751384i \(0.729387\pi\)
\(450\) 0 0
\(451\) −25.2757 −1.19019
\(452\) 0 0
\(453\) −22.5089 −1.05756
\(454\) 0 0
\(455\) 11.3165 0.530527
\(456\) 0 0
\(457\) −9.13668 −0.427396 −0.213698 0.976900i \(-0.568551\pi\)
−0.213698 + 0.976900i \(0.568551\pi\)
\(458\) 0 0
\(459\) 74.7247 3.48785
\(460\) 0 0
\(461\) 2.83738 0.132150 0.0660750 0.997815i \(-0.478952\pi\)
0.0660750 + 0.997815i \(0.478952\pi\)
\(462\) 0 0
\(463\) −6.80612 −0.316307 −0.158154 0.987415i \(-0.550554\pi\)
−0.158154 + 0.987415i \(0.550554\pi\)
\(464\) 0 0
\(465\) −52.4643 −2.43298
\(466\) 0 0
\(467\) −8.10149 −0.374892 −0.187446 0.982275i \(-0.560021\pi\)
−0.187446 + 0.982275i \(0.560021\pi\)
\(468\) 0 0
\(469\) −17.5908 −0.812266
\(470\) 0 0
\(471\) −16.9520 −0.781109
\(472\) 0 0
\(473\) −41.1156 −1.89050
\(474\) 0 0
\(475\) −4.88837 −0.224294
\(476\) 0 0
\(477\) −9.11679 −0.417429
\(478\) 0 0
\(479\) −34.3545 −1.56970 −0.784849 0.619687i \(-0.787260\pi\)
−0.784849 + 0.619687i \(0.787260\pi\)
\(480\) 0 0
\(481\) 31.9726 1.45782
\(482\) 0 0
\(483\) 24.7214 1.12486
\(484\) 0 0
\(485\) 8.75846 0.397701
\(486\) 0 0
\(487\) −35.2753 −1.59848 −0.799239 0.601014i \(-0.794764\pi\)
−0.799239 + 0.601014i \(0.794764\pi\)
\(488\) 0 0
\(489\) −38.6712 −1.74877
\(490\) 0 0
\(491\) −10.7936 −0.487107 −0.243554 0.969887i \(-0.578313\pi\)
−0.243554 + 0.969887i \(0.578313\pi\)
\(492\) 0 0
\(493\) −22.2686 −1.00293
\(494\) 0 0
\(495\) −75.0182 −3.37182
\(496\) 0 0
\(497\) −9.50512 −0.426363
\(498\) 0 0
\(499\) 24.8451 1.11222 0.556109 0.831109i \(-0.312294\pi\)
0.556109 + 0.831109i \(0.312294\pi\)
\(500\) 0 0
\(501\) 11.0977 0.495809
\(502\) 0 0
\(503\) −10.5262 −0.469339 −0.234669 0.972075i \(-0.575401\pi\)
−0.234669 + 0.972075i \(0.575401\pi\)
\(504\) 0 0
\(505\) −13.0527 −0.580839
\(506\) 0 0
\(507\) 21.0908 0.936676
\(508\) 0 0
\(509\) −23.8969 −1.05921 −0.529606 0.848244i \(-0.677660\pi\)
−0.529606 + 0.848244i \(0.677660\pi\)
\(510\) 0 0
\(511\) 18.6426 0.824700
\(512\) 0 0
\(513\) 57.1578 2.52358
\(514\) 0 0
\(515\) −22.9660 −1.01200
\(516\) 0 0
\(517\) −17.0185 −0.748474
\(518\) 0 0
\(519\) 14.1651 0.621779
\(520\) 0 0
\(521\) −6.35855 −0.278573 −0.139287 0.990252i \(-0.544481\pi\)
−0.139287 + 0.990252i \(0.544481\pi\)
\(522\) 0 0
\(523\) 18.6614 0.816008 0.408004 0.912980i \(-0.366225\pi\)
0.408004 + 0.912980i \(0.366225\pi\)
\(524\) 0 0
\(525\) −7.61515 −0.332353
\(526\) 0 0
\(527\) 33.4521 1.45719
\(528\) 0 0
\(529\) 4.04806 0.176003
\(530\) 0 0
\(531\) 99.9204 4.33618
\(532\) 0 0
\(533\) 23.1929 1.00460
\(534\) 0 0
\(535\) 18.2573 0.789331
\(536\) 0 0
\(537\) −23.3140 −1.00607
\(538\) 0 0
\(539\) −24.0692 −1.03673
\(540\) 0 0
\(541\) 21.0074 0.903180 0.451590 0.892226i \(-0.350857\pi\)
0.451590 + 0.892226i \(0.350857\pi\)
\(542\) 0 0
\(543\) 27.5624 1.18281
\(544\) 0 0
\(545\) 0.161981 0.00693849
\(546\) 0 0
\(547\) 40.7191 1.74102 0.870512 0.492147i \(-0.163788\pi\)
0.870512 + 0.492147i \(0.163788\pi\)
\(548\) 0 0
\(549\) 15.9822 0.682104
\(550\) 0 0
\(551\) −17.0335 −0.725652
\(552\) 0 0
\(553\) 23.5878 1.00305
\(554\) 0 0
\(555\) 45.6338 1.93705
\(556\) 0 0
\(557\) −42.4706 −1.79954 −0.899769 0.436367i \(-0.856265\pi\)
−0.899769 + 0.436367i \(0.856265\pi\)
\(558\) 0 0
\(559\) 37.7274 1.59570
\(560\) 0 0
\(561\) 64.6769 2.73066
\(562\) 0 0
\(563\) 19.7100 0.830676 0.415338 0.909667i \(-0.363663\pi\)
0.415338 + 0.909667i \(0.363663\pi\)
\(564\) 0 0
\(565\) −10.0829 −0.424189
\(566\) 0 0
\(567\) 53.2468 2.23615
\(568\) 0 0
\(569\) 37.0950 1.55510 0.777551 0.628820i \(-0.216462\pi\)
0.777551 + 0.628820i \(0.216462\pi\)
\(570\) 0 0
\(571\) −27.6347 −1.15648 −0.578238 0.815868i \(-0.696260\pi\)
−0.578238 + 0.815868i \(0.696260\pi\)
\(572\) 0 0
\(573\) −54.0980 −2.25998
\(574\) 0 0
\(575\) −8.33187 −0.347463
\(576\) 0 0
\(577\) 1.64302 0.0683999 0.0341999 0.999415i \(-0.489112\pi\)
0.0341999 + 0.999415i \(0.489112\pi\)
\(578\) 0 0
\(579\) −46.0614 −1.91425
\(580\) 0 0
\(581\) 3.40536 0.141278
\(582\) 0 0
\(583\) −5.11215 −0.211724
\(584\) 0 0
\(585\) 68.8363 2.84603
\(586\) 0 0
\(587\) −15.2520 −0.629516 −0.314758 0.949172i \(-0.601923\pi\)
−0.314758 + 0.949172i \(0.601923\pi\)
\(588\) 0 0
\(589\) 25.5879 1.05433
\(590\) 0 0
\(591\) 65.2521 2.68411
\(592\) 0 0
\(593\) −2.20855 −0.0906943 −0.0453472 0.998971i \(-0.514439\pi\)
−0.0453472 + 0.998971i \(0.514439\pi\)
\(594\) 0 0
\(595\) −10.2987 −0.422204
\(596\) 0 0
\(597\) 79.9672 3.27284
\(598\) 0 0
\(599\) −32.3973 −1.32372 −0.661859 0.749628i \(-0.730232\pi\)
−0.661859 + 0.749628i \(0.730232\pi\)
\(600\) 0 0
\(601\) 36.2203 1.47746 0.738729 0.674003i \(-0.235426\pi\)
0.738729 + 0.674003i \(0.235426\pi\)
\(602\) 0 0
\(603\) −107.001 −4.35743
\(604\) 0 0
\(605\) −21.7889 −0.885844
\(606\) 0 0
\(607\) 25.2944 1.02667 0.513335 0.858188i \(-0.328410\pi\)
0.513335 + 0.858188i \(0.328410\pi\)
\(608\) 0 0
\(609\) −26.5350 −1.07525
\(610\) 0 0
\(611\) 15.6161 0.631760
\(612\) 0 0
\(613\) −18.4143 −0.743748 −0.371874 0.928283i \(-0.621285\pi\)
−0.371874 + 0.928283i \(0.621285\pi\)
\(614\) 0 0
\(615\) 33.1027 1.33483
\(616\) 0 0
\(617\) 1.35977 0.0547422 0.0273711 0.999625i \(-0.491286\pi\)
0.0273711 + 0.999625i \(0.491286\pi\)
\(618\) 0 0
\(619\) −25.5085 −1.02527 −0.512636 0.858606i \(-0.671331\pi\)
−0.512636 + 0.858606i \(0.671331\pi\)
\(620\) 0 0
\(621\) 97.4213 3.90938
\(622\) 0 0
\(623\) −20.1100 −0.805692
\(624\) 0 0
\(625\) −14.4232 −0.576929
\(626\) 0 0
\(627\) 49.4721 1.97573
\(628\) 0 0
\(629\) −29.0968 −1.16017
\(630\) 0 0
\(631\) 14.7112 0.585642 0.292821 0.956167i \(-0.405406\pi\)
0.292821 + 0.956167i \(0.405406\pi\)
\(632\) 0 0
\(633\) 38.1244 1.51531
\(634\) 0 0
\(635\) −34.2975 −1.36105
\(636\) 0 0
\(637\) 22.0858 0.875070
\(638\) 0 0
\(639\) −57.8179 −2.28724
\(640\) 0 0
\(641\) 23.7342 0.937444 0.468722 0.883346i \(-0.344715\pi\)
0.468722 + 0.883346i \(0.344715\pi\)
\(642\) 0 0
\(643\) −16.6792 −0.657764 −0.328882 0.944371i \(-0.606672\pi\)
−0.328882 + 0.944371i \(0.606672\pi\)
\(644\) 0 0
\(645\) 53.8476 2.12025
\(646\) 0 0
\(647\) −24.1697 −0.950208 −0.475104 0.879930i \(-0.657590\pi\)
−0.475104 + 0.879930i \(0.657590\pi\)
\(648\) 0 0
\(649\) 56.0295 2.19935
\(650\) 0 0
\(651\) 39.8611 1.56228
\(652\) 0 0
\(653\) −36.4713 −1.42723 −0.713615 0.700538i \(-0.752944\pi\)
−0.713615 + 0.700538i \(0.752944\pi\)
\(654\) 0 0
\(655\) −4.45327 −0.174004
\(656\) 0 0
\(657\) 113.400 4.42414
\(658\) 0 0
\(659\) −3.31362 −0.129080 −0.0645401 0.997915i \(-0.520558\pi\)
−0.0645401 + 0.997915i \(0.520558\pi\)
\(660\) 0 0
\(661\) 33.9178 1.31925 0.659626 0.751594i \(-0.270715\pi\)
0.659626 + 0.751594i \(0.270715\pi\)
\(662\) 0 0
\(663\) −59.3471 −2.30485
\(664\) 0 0
\(665\) −7.87756 −0.305479
\(666\) 0 0
\(667\) −29.0324 −1.12414
\(668\) 0 0
\(669\) −39.7982 −1.53869
\(670\) 0 0
\(671\) 8.96187 0.345969
\(672\) 0 0
\(673\) 49.5453 1.90983 0.954915 0.296879i \(-0.0959459\pi\)
0.954915 + 0.296879i \(0.0959459\pi\)
\(674\) 0 0
\(675\) −30.0096 −1.15507
\(676\) 0 0
\(677\) −19.1096 −0.734442 −0.367221 0.930134i \(-0.619691\pi\)
−0.367221 + 0.930134i \(0.619691\pi\)
\(678\) 0 0
\(679\) −6.65445 −0.255375
\(680\) 0 0
\(681\) −17.9119 −0.686387
\(682\) 0 0
\(683\) 6.83998 0.261724 0.130862 0.991401i \(-0.458225\pi\)
0.130862 + 0.991401i \(0.458225\pi\)
\(684\) 0 0
\(685\) −37.4579 −1.43119
\(686\) 0 0
\(687\) −52.5274 −2.00404
\(688\) 0 0
\(689\) 4.69088 0.178708
\(690\) 0 0
\(691\) 43.2778 1.64637 0.823183 0.567776i \(-0.192196\pi\)
0.823183 + 0.567776i \(0.192196\pi\)
\(692\) 0 0
\(693\) 56.9969 2.16513
\(694\) 0 0
\(695\) 25.0555 0.950410
\(696\) 0 0
\(697\) −21.1068 −0.799477
\(698\) 0 0
\(699\) −6.62595 −0.250617
\(700\) 0 0
\(701\) −17.7344 −0.669817 −0.334909 0.942251i \(-0.608705\pi\)
−0.334909 + 0.942251i \(0.608705\pi\)
\(702\) 0 0
\(703\) −22.2565 −0.839419
\(704\) 0 0
\(705\) 22.2885 0.839435
\(706\) 0 0
\(707\) 9.91713 0.372972
\(708\) 0 0
\(709\) 27.9542 1.04984 0.524921 0.851151i \(-0.324095\pi\)
0.524921 + 0.851151i \(0.324095\pi\)
\(710\) 0 0
\(711\) 143.480 5.38092
\(712\) 0 0
\(713\) 43.6127 1.63331
\(714\) 0 0
\(715\) 38.5993 1.44353
\(716\) 0 0
\(717\) 10.5261 0.393104
\(718\) 0 0
\(719\) −7.64866 −0.285247 −0.142623 0.989777i \(-0.545554\pi\)
−0.142623 + 0.989777i \(0.545554\pi\)
\(720\) 0 0
\(721\) 17.4490 0.649833
\(722\) 0 0
\(723\) −64.4030 −2.39517
\(724\) 0 0
\(725\) 8.94313 0.332139
\(726\) 0 0
\(727\) 7.82864 0.290348 0.145174 0.989406i \(-0.453626\pi\)
0.145174 + 0.989406i \(0.453626\pi\)
\(728\) 0 0
\(729\) 133.161 4.93188
\(730\) 0 0
\(731\) −34.3340 −1.26989
\(732\) 0 0
\(733\) −27.9936 −1.03397 −0.516983 0.855996i \(-0.672945\pi\)
−0.516983 + 0.855996i \(0.672945\pi\)
\(734\) 0 0
\(735\) 31.5226 1.16273
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) 46.1441 1.69744 0.848719 0.528844i \(-0.177374\pi\)
0.848719 + 0.528844i \(0.177374\pi\)
\(740\) 0 0
\(741\) −45.3953 −1.66764
\(742\) 0 0
\(743\) 23.7107 0.869860 0.434930 0.900464i \(-0.356773\pi\)
0.434930 + 0.900464i \(0.356773\pi\)
\(744\) 0 0
\(745\) 3.44106 0.126071
\(746\) 0 0
\(747\) 20.7142 0.757892
\(748\) 0 0
\(749\) −13.8714 −0.506850
\(750\) 0 0
\(751\) 36.9780 1.34935 0.674673 0.738116i \(-0.264285\pi\)
0.674673 + 0.738116i \(0.264285\pi\)
\(752\) 0 0
\(753\) 3.39399 0.123684
\(754\) 0 0
\(755\) 12.2251 0.444917
\(756\) 0 0
\(757\) −48.4471 −1.76084 −0.880420 0.474196i \(-0.842739\pi\)
−0.880420 + 0.474196i \(0.842739\pi\)
\(758\) 0 0
\(759\) 84.3216 3.06068
\(760\) 0 0
\(761\) −9.34501 −0.338756 −0.169378 0.985551i \(-0.554176\pi\)
−0.169378 + 0.985551i \(0.554176\pi\)
\(762\) 0 0
\(763\) −0.123069 −0.00445539
\(764\) 0 0
\(765\) −62.6448 −2.26493
\(766\) 0 0
\(767\) −51.4123 −1.85639
\(768\) 0 0
\(769\) 26.0335 0.938794 0.469397 0.882987i \(-0.344471\pi\)
0.469397 + 0.882987i \(0.344471\pi\)
\(770\) 0 0
\(771\) −99.8875 −3.59736
\(772\) 0 0
\(773\) −13.7858 −0.495842 −0.247921 0.968780i \(-0.579747\pi\)
−0.247921 + 0.968780i \(0.579747\pi\)
\(774\) 0 0
\(775\) −13.4344 −0.482579
\(776\) 0 0
\(777\) −34.6714 −1.24383
\(778\) 0 0
\(779\) −16.1448 −0.578449
\(780\) 0 0
\(781\) −32.4208 −1.16011
\(782\) 0 0
\(783\) −104.568 −3.73697
\(784\) 0 0
\(785\) 9.20704 0.328613
\(786\) 0 0
\(787\) −15.7457 −0.561274 −0.280637 0.959814i \(-0.590546\pi\)
−0.280637 + 0.959814i \(0.590546\pi\)
\(788\) 0 0
\(789\) −45.8881 −1.63366
\(790\) 0 0
\(791\) 7.66069 0.272383
\(792\) 0 0
\(793\) −8.22336 −0.292020
\(794\) 0 0
\(795\) 6.69520 0.237454
\(796\) 0 0
\(797\) 29.7717 1.05457 0.527284 0.849689i \(-0.323210\pi\)
0.527284 + 0.849689i \(0.323210\pi\)
\(798\) 0 0
\(799\) −14.2115 −0.502767
\(800\) 0 0
\(801\) −122.326 −4.32216
\(802\) 0 0
\(803\) 63.5877 2.24396
\(804\) 0 0
\(805\) −13.4267 −0.473230
\(806\) 0 0
\(807\) −101.777 −3.58273
\(808\) 0 0
\(809\) 16.1292 0.567073 0.283536 0.958961i \(-0.408492\pi\)
0.283536 + 0.958961i \(0.408492\pi\)
\(810\) 0 0
\(811\) −36.8744 −1.29484 −0.647418 0.762135i \(-0.724151\pi\)
−0.647418 + 0.762135i \(0.724151\pi\)
\(812\) 0 0
\(813\) −34.8510 −1.22228
\(814\) 0 0
\(815\) 21.0032 0.735710
\(816\) 0 0
\(817\) −26.2625 −0.918809
\(818\) 0 0
\(819\) −52.3001 −1.82751
\(820\) 0 0
\(821\) −15.2812 −0.533319 −0.266659 0.963791i \(-0.585920\pi\)
−0.266659 + 0.963791i \(0.585920\pi\)
\(822\) 0 0
\(823\) 22.0565 0.768842 0.384421 0.923158i \(-0.374401\pi\)
0.384421 + 0.923158i \(0.374401\pi\)
\(824\) 0 0
\(825\) −25.9744 −0.904312
\(826\) 0 0
\(827\) 51.3239 1.78471 0.892353 0.451337i \(-0.149053\pi\)
0.892353 + 0.451337i \(0.149053\pi\)
\(828\) 0 0
\(829\) −55.0672 −1.91256 −0.956282 0.292447i \(-0.905531\pi\)
−0.956282 + 0.292447i \(0.905531\pi\)
\(830\) 0 0
\(831\) −33.6563 −1.16752
\(832\) 0 0
\(833\) −20.0993 −0.696398
\(834\) 0 0
\(835\) −6.02741 −0.208587
\(836\) 0 0
\(837\) 157.083 5.42960
\(838\) 0 0
\(839\) 17.1112 0.590744 0.295372 0.955382i \(-0.404556\pi\)
0.295372 + 0.955382i \(0.404556\pi\)
\(840\) 0 0
\(841\) 2.16233 0.0745632
\(842\) 0 0
\(843\) 38.4042 1.32271
\(844\) 0 0
\(845\) −11.4549 −0.394061
\(846\) 0 0
\(847\) 16.5546 0.568824
\(848\) 0 0
\(849\) 43.4784 1.49217
\(850\) 0 0
\(851\) −37.9346 −1.30038
\(852\) 0 0
\(853\) 19.7633 0.676684 0.338342 0.941023i \(-0.390134\pi\)
0.338342 + 0.941023i \(0.390134\pi\)
\(854\) 0 0
\(855\) −47.9177 −1.63875
\(856\) 0 0
\(857\) 24.4967 0.836791 0.418396 0.908265i \(-0.362593\pi\)
0.418396 + 0.908265i \(0.362593\pi\)
\(858\) 0 0
\(859\) 20.7099 0.706612 0.353306 0.935508i \(-0.385058\pi\)
0.353306 + 0.935508i \(0.385058\pi\)
\(860\) 0 0
\(861\) −25.1506 −0.857130
\(862\) 0 0
\(863\) −24.1636 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(864\) 0 0
\(865\) −7.69339 −0.261583
\(866\) 0 0
\(867\) −3.68872 −0.125276
\(868\) 0 0
\(869\) 80.4551 2.72925
\(870\) 0 0
\(871\) 55.0557 1.86549
\(872\) 0 0
\(873\) −40.4778 −1.36997
\(874\) 0 0
\(875\) 17.0444 0.576205
\(876\) 0 0
\(877\) −5.99030 −0.202278 −0.101139 0.994872i \(-0.532249\pi\)
−0.101139 + 0.994872i \(0.532249\pi\)
\(878\) 0 0
\(879\) 9.08382 0.306390
\(880\) 0 0
\(881\) −0.901329 −0.0303666 −0.0151833 0.999885i \(-0.504833\pi\)
−0.0151833 + 0.999885i \(0.504833\pi\)
\(882\) 0 0
\(883\) 38.5894 1.29864 0.649318 0.760517i \(-0.275054\pi\)
0.649318 + 0.760517i \(0.275054\pi\)
\(884\) 0 0
\(885\) −73.3797 −2.46663
\(886\) 0 0
\(887\) 3.31707 0.111376 0.0556881 0.998448i \(-0.482265\pi\)
0.0556881 + 0.998448i \(0.482265\pi\)
\(888\) 0 0
\(889\) 26.0584 0.873969
\(890\) 0 0
\(891\) 181.618 6.08445
\(892\) 0 0
\(893\) −10.8706 −0.363769
\(894\) 0 0
\(895\) 12.6624 0.423256
\(896\) 0 0
\(897\) −77.3730 −2.58341
\(898\) 0 0
\(899\) −46.8122 −1.56128
\(900\) 0 0
\(901\) −4.26896 −0.142220
\(902\) 0 0
\(903\) −40.9120 −1.36147
\(904\) 0 0
\(905\) −14.9698 −0.497612
\(906\) 0 0
\(907\) −31.9616 −1.06127 −0.530633 0.847602i \(-0.678046\pi\)
−0.530633 + 0.847602i \(0.678046\pi\)
\(908\) 0 0
\(909\) 60.3241 2.00082
\(910\) 0 0
\(911\) 12.6487 0.419072 0.209536 0.977801i \(-0.432805\pi\)
0.209536 + 0.977801i \(0.432805\pi\)
\(912\) 0 0
\(913\) 11.6153 0.384410
\(914\) 0 0
\(915\) −11.7370 −0.388014
\(916\) 0 0
\(917\) 3.38348 0.111732
\(918\) 0 0
\(919\) −3.78557 −0.124874 −0.0624371 0.998049i \(-0.519887\pi\)
−0.0624371 + 0.998049i \(0.519887\pi\)
\(920\) 0 0
\(921\) 2.37010 0.0780976
\(922\) 0 0
\(923\) 29.7492 0.979206
\(924\) 0 0
\(925\) 11.6853 0.384212
\(926\) 0 0
\(927\) 106.139 3.48606
\(928\) 0 0
\(929\) −42.6045 −1.39781 −0.698905 0.715215i \(-0.746329\pi\)
−0.698905 + 0.715215i \(0.746329\pi\)
\(930\) 0 0
\(931\) −15.3742 −0.503868
\(932\) 0 0
\(933\) −11.9762 −0.392083
\(934\) 0 0
\(935\) −35.1275 −1.14879
\(936\) 0 0
\(937\) 29.5011 0.963759 0.481880 0.876237i \(-0.339954\pi\)
0.481880 + 0.876237i \(0.339954\pi\)
\(938\) 0 0
\(939\) −92.1012 −3.00561
\(940\) 0 0
\(941\) −9.54909 −0.311291 −0.155646 0.987813i \(-0.549746\pi\)
−0.155646 + 0.987813i \(0.549746\pi\)
\(942\) 0 0
\(943\) −27.5177 −0.896099
\(944\) 0 0
\(945\) −48.3602 −1.57316
\(946\) 0 0
\(947\) −31.4278 −1.02127 −0.510634 0.859798i \(-0.670589\pi\)
−0.510634 + 0.859798i \(0.670589\pi\)
\(948\) 0 0
\(949\) −58.3477 −1.89405
\(950\) 0 0
\(951\) −72.5655 −2.35310
\(952\) 0 0
\(953\) 47.1158 1.52623 0.763116 0.646262i \(-0.223669\pi\)
0.763116 + 0.646262i \(0.223669\pi\)
\(954\) 0 0
\(955\) 29.3818 0.950774
\(956\) 0 0
\(957\) −90.5078 −2.92570
\(958\) 0 0
\(959\) 28.4596 0.919007
\(960\) 0 0
\(961\) 39.3216 1.26844
\(962\) 0 0
\(963\) −84.3772 −2.71902
\(964\) 0 0
\(965\) 25.0170 0.805326
\(966\) 0 0
\(967\) −42.3038 −1.36040 −0.680200 0.733027i \(-0.738107\pi\)
−0.680200 + 0.733027i \(0.738107\pi\)
\(968\) 0 0
\(969\) 41.3122 1.32714
\(970\) 0 0
\(971\) −2.26896 −0.0728143 −0.0364072 0.999337i \(-0.511591\pi\)
−0.0364072 + 0.999337i \(0.511591\pi\)
\(972\) 0 0
\(973\) −19.0365 −0.610283
\(974\) 0 0
\(975\) 23.8340 0.763297
\(976\) 0 0
\(977\) 39.6079 1.26717 0.633584 0.773674i \(-0.281583\pi\)
0.633584 + 0.773674i \(0.281583\pi\)
\(978\) 0 0
\(979\) −68.5929 −2.19224
\(980\) 0 0
\(981\) −0.748604 −0.0239011
\(982\) 0 0
\(983\) 14.2362 0.454063 0.227032 0.973887i \(-0.427098\pi\)
0.227032 + 0.973887i \(0.427098\pi\)
\(984\) 0 0
\(985\) −35.4399 −1.12921
\(986\) 0 0
\(987\) −16.9343 −0.539024
\(988\) 0 0
\(989\) −44.7625 −1.42337
\(990\) 0 0
\(991\) 60.6156 1.92552 0.962758 0.270363i \(-0.0871438\pi\)
0.962758 + 0.270363i \(0.0871438\pi\)
\(992\) 0 0
\(993\) 64.2980 2.04044
\(994\) 0 0
\(995\) −43.4320 −1.37689
\(996\) 0 0
\(997\) −22.2190 −0.703682 −0.351841 0.936060i \(-0.614444\pi\)
−0.351841 + 0.936060i \(0.614444\pi\)
\(998\) 0 0
\(999\) −136.632 −4.32285
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 4016.2.a.m.1.23 23
4.3 odd 2 2008.2.a.d.1.1 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.1 23 4.3 odd 2
4016.2.a.m.1.23 23 1.1 even 1 trivial