Properties

Label 6039.2.a.i.1.9
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.948254\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +O(q^{10})\) \(q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +2.13473 q^{10} -1.00000 q^{11} +5.44016 q^{13} +4.96909 q^{14} -0.586583 q^{16} -4.46012 q^{17} +2.98902 q^{19} -2.47818 q^{20} -0.948254 q^{22} -0.200434 q^{23} +0.0679998 q^{25} +5.15865 q^{26} -5.76854 q^{28} +1.81818 q^{29} +0.728977 q^{31} +5.32449 q^{32} -4.22933 q^{34} +11.7970 q^{35} -3.44866 q^{37} +2.83436 q^{38} -6.61940 q^{40} -2.51230 q^{41} +7.72081 q^{43} +1.10081 q^{44} -0.190062 q^{46} +9.29908 q^{47} +20.4602 q^{49} +0.0644811 q^{50} -5.98860 q^{52} +1.48505 q^{53} -2.25122 q^{55} -15.4082 q^{56} +1.72409 q^{58} -1.65543 q^{59} -1.00000 q^{61} +0.691255 q^{62} +6.22214 q^{64} +12.2470 q^{65} +16.1730 q^{67} +4.90976 q^{68} +11.1865 q^{70} -1.39117 q^{71} -8.30506 q^{73} -3.27021 q^{74} -3.29036 q^{76} -5.24025 q^{77} -11.7081 q^{79} -1.32053 q^{80} -2.38230 q^{82} +1.10951 q^{83} -10.0407 q^{85} +7.32129 q^{86} +2.94036 q^{88} -6.80479 q^{89} +28.5078 q^{91} +0.220640 q^{92} +8.81789 q^{94} +6.72896 q^{95} -9.30845 q^{97} +19.4015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.948254 0.670517 0.335259 0.942126i \(-0.391176\pi\)
0.335259 + 0.942126i \(0.391176\pi\)
\(3\) 0 0
\(4\) −1.10081 −0.550407
\(5\) 2.25122 1.00678 0.503389 0.864060i \(-0.332086\pi\)
0.503389 + 0.864060i \(0.332086\pi\)
\(6\) 0 0
\(7\) 5.24025 1.98063 0.990315 0.138842i \(-0.0443379\pi\)
0.990315 + 0.138842i \(0.0443379\pi\)
\(8\) −2.94036 −1.03957
\(9\) 0 0
\(10\) 2.13473 0.675061
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.44016 1.50883 0.754414 0.656399i \(-0.227921\pi\)
0.754414 + 0.656399i \(0.227921\pi\)
\(14\) 4.96909 1.32805
\(15\) 0 0
\(16\) −0.586583 −0.146646
\(17\) −4.46012 −1.08174 −0.540869 0.841107i \(-0.681905\pi\)
−0.540869 + 0.841107i \(0.681905\pi\)
\(18\) 0 0
\(19\) 2.98902 0.685729 0.342865 0.939385i \(-0.388603\pi\)
0.342865 + 0.939385i \(0.388603\pi\)
\(20\) −2.47818 −0.554137
\(21\) 0 0
\(22\) −0.948254 −0.202169
\(23\) −0.200434 −0.0417934 −0.0208967 0.999782i \(-0.506652\pi\)
−0.0208967 + 0.999782i \(0.506652\pi\)
\(24\) 0 0
\(25\) 0.0679998 0.0136000
\(26\) 5.15865 1.01169
\(27\) 0 0
\(28\) −5.76854 −1.09015
\(29\) 1.81818 0.337627 0.168813 0.985648i \(-0.446006\pi\)
0.168813 + 0.985648i \(0.446006\pi\)
\(30\) 0 0
\(31\) 0.728977 0.130928 0.0654640 0.997855i \(-0.479147\pi\)
0.0654640 + 0.997855i \(0.479147\pi\)
\(32\) 5.32449 0.941246
\(33\) 0 0
\(34\) −4.22933 −0.725324
\(35\) 11.7970 1.99405
\(36\) 0 0
\(37\) −3.44866 −0.566956 −0.283478 0.958979i \(-0.591488\pi\)
−0.283478 + 0.958979i \(0.591488\pi\)
\(38\) 2.83436 0.459793
\(39\) 0 0
\(40\) −6.61940 −1.04662
\(41\) −2.51230 −0.392355 −0.196178 0.980568i \(-0.562853\pi\)
−0.196178 + 0.980568i \(0.562853\pi\)
\(42\) 0 0
\(43\) 7.72081 1.17741 0.588706 0.808347i \(-0.299638\pi\)
0.588706 + 0.808347i \(0.299638\pi\)
\(44\) 1.10081 0.165954
\(45\) 0 0
\(46\) −0.190062 −0.0280232
\(47\) 9.29908 1.35641 0.678205 0.734873i \(-0.262758\pi\)
0.678205 + 0.734873i \(0.262758\pi\)
\(48\) 0 0
\(49\) 20.4602 2.92289
\(50\) 0.0644811 0.00911900
\(51\) 0 0
\(52\) −5.98860 −0.830469
\(53\) 1.48505 0.203987 0.101994 0.994785i \(-0.467478\pi\)
0.101994 + 0.994785i \(0.467478\pi\)
\(54\) 0 0
\(55\) −2.25122 −0.303555
\(56\) −15.4082 −2.05901
\(57\) 0 0
\(58\) 1.72409 0.226385
\(59\) −1.65543 −0.215518 −0.107759 0.994177i \(-0.534368\pi\)
−0.107759 + 0.994177i \(0.534368\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 0.691255 0.0877895
\(63\) 0 0
\(64\) 6.22214 0.777767
\(65\) 12.2470 1.51905
\(66\) 0 0
\(67\) 16.1730 1.97585 0.987923 0.154948i \(-0.0495211\pi\)
0.987923 + 0.154948i \(0.0495211\pi\)
\(68\) 4.90976 0.595396
\(69\) 0 0
\(70\) 11.1865 1.33705
\(71\) −1.39117 −0.165101 −0.0825506 0.996587i \(-0.526307\pi\)
−0.0825506 + 0.996587i \(0.526307\pi\)
\(72\) 0 0
\(73\) −8.30506 −0.972034 −0.486017 0.873949i \(-0.661551\pi\)
−0.486017 + 0.873949i \(0.661551\pi\)
\(74\) −3.27021 −0.380154
\(75\) 0 0
\(76\) −3.29036 −0.377430
\(77\) −5.24025 −0.597182
\(78\) 0 0
\(79\) −11.7081 −1.31726 −0.658630 0.752467i \(-0.728864\pi\)
−0.658630 + 0.752467i \(0.728864\pi\)
\(80\) −1.32053 −0.147640
\(81\) 0 0
\(82\) −2.38230 −0.263081
\(83\) 1.10951 0.121785 0.0608924 0.998144i \(-0.480605\pi\)
0.0608924 + 0.998144i \(0.480605\pi\)
\(84\) 0 0
\(85\) −10.0407 −1.08907
\(86\) 7.32129 0.789475
\(87\) 0 0
\(88\) 2.94036 0.313443
\(89\) −6.80479 −0.721306 −0.360653 0.932700i \(-0.617446\pi\)
−0.360653 + 0.932700i \(0.617446\pi\)
\(90\) 0 0
\(91\) 28.5078 2.98843
\(92\) 0.220640 0.0230034
\(93\) 0 0
\(94\) 8.81789 0.909496
\(95\) 6.72896 0.690377
\(96\) 0 0
\(97\) −9.30845 −0.945130 −0.472565 0.881296i \(-0.656672\pi\)
−0.472565 + 0.881296i \(0.656672\pi\)
\(98\) 19.4015 1.95985
\(99\) 0 0
\(100\) −0.0748551 −0.00748551
\(101\) −9.37877 −0.933223 −0.466611 0.884462i \(-0.654525\pi\)
−0.466611 + 0.884462i \(0.654525\pi\)
\(102\) 0 0
\(103\) −9.63463 −0.949328 −0.474664 0.880167i \(-0.657430\pi\)
−0.474664 + 0.880167i \(0.657430\pi\)
\(104\) −15.9960 −1.56854
\(105\) 0 0
\(106\) 1.40820 0.136777
\(107\) −3.02397 −0.292338 −0.146169 0.989260i \(-0.546694\pi\)
−0.146169 + 0.989260i \(0.546694\pi\)
\(108\) 0 0
\(109\) 3.74875 0.359065 0.179533 0.983752i \(-0.442541\pi\)
0.179533 + 0.983752i \(0.442541\pi\)
\(110\) −2.13473 −0.203539
\(111\) 0 0
\(112\) −3.07384 −0.290451
\(113\) 3.50360 0.329591 0.164795 0.986328i \(-0.447304\pi\)
0.164795 + 0.986328i \(0.447304\pi\)
\(114\) 0 0
\(115\) −0.451221 −0.0420766
\(116\) −2.00147 −0.185832
\(117\) 0 0
\(118\) −1.56977 −0.144509
\(119\) −23.3722 −2.14252
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.948254 −0.0858509
\(123\) 0 0
\(124\) −0.802467 −0.0720637
\(125\) −11.1030 −0.993085
\(126\) 0 0
\(127\) 5.45206 0.483792 0.241896 0.970302i \(-0.422231\pi\)
0.241896 + 0.970302i \(0.422231\pi\)
\(128\) −4.74881 −0.419740
\(129\) 0 0
\(130\) 11.6133 1.01855
\(131\) 16.4647 1.43853 0.719265 0.694736i \(-0.244479\pi\)
0.719265 + 0.694736i \(0.244479\pi\)
\(132\) 0 0
\(133\) 15.6632 1.35818
\(134\) 15.3361 1.32484
\(135\) 0 0
\(136\) 13.1144 1.12455
\(137\) −1.13223 −0.0967330 −0.0483665 0.998830i \(-0.515402\pi\)
−0.0483665 + 0.998830i \(0.515402\pi\)
\(138\) 0 0
\(139\) 17.6358 1.49585 0.747923 0.663785i \(-0.231051\pi\)
0.747923 + 0.663785i \(0.231051\pi\)
\(140\) −12.9863 −1.09754
\(141\) 0 0
\(142\) −1.31918 −0.110703
\(143\) −5.44016 −0.454929
\(144\) 0 0
\(145\) 4.09312 0.339915
\(146\) −7.87531 −0.651766
\(147\) 0 0
\(148\) 3.79633 0.312057
\(149\) −18.5492 −1.51961 −0.759804 0.650152i \(-0.774705\pi\)
−0.759804 + 0.650152i \(0.774705\pi\)
\(150\) 0 0
\(151\) −22.4227 −1.82474 −0.912368 0.409371i \(-0.865748\pi\)
−0.912368 + 0.409371i \(0.865748\pi\)
\(152\) −8.78881 −0.712867
\(153\) 0 0
\(154\) −4.96909 −0.400421
\(155\) 1.64109 0.131815
\(156\) 0 0
\(157\) −7.31110 −0.583489 −0.291745 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(158\) −11.1022 −0.883246
\(159\) 0 0
\(160\) 11.9866 0.947625
\(161\) −1.05032 −0.0827772
\(162\) 0 0
\(163\) −1.04410 −0.0817803 −0.0408902 0.999164i \(-0.513019\pi\)
−0.0408902 + 0.999164i \(0.513019\pi\)
\(164\) 2.76557 0.215955
\(165\) 0 0
\(166\) 1.05210 0.0816588
\(167\) 0.242267 0.0187472 0.00937360 0.999956i \(-0.497016\pi\)
0.00937360 + 0.999956i \(0.497016\pi\)
\(168\) 0 0
\(169\) 16.5953 1.27656
\(170\) −9.52116 −0.730240
\(171\) 0 0
\(172\) −8.49917 −0.648056
\(173\) 23.7589 1.80636 0.903179 0.429264i \(-0.141227\pi\)
0.903179 + 0.429264i \(0.141227\pi\)
\(174\) 0 0
\(175\) 0.356336 0.0269365
\(176\) 0.586583 0.0442153
\(177\) 0 0
\(178\) −6.45267 −0.483648
\(179\) 2.79602 0.208984 0.104492 0.994526i \(-0.466678\pi\)
0.104492 + 0.994526i \(0.466678\pi\)
\(180\) 0 0
\(181\) 12.5589 0.933499 0.466750 0.884390i \(-0.345425\pi\)
0.466750 + 0.884390i \(0.345425\pi\)
\(182\) 27.0326 2.00379
\(183\) 0 0
\(184\) 0.589348 0.0434473
\(185\) −7.76370 −0.570799
\(186\) 0 0
\(187\) 4.46012 0.326156
\(188\) −10.2366 −0.746577
\(189\) 0 0
\(190\) 6.38076 0.462909
\(191\) 2.74577 0.198677 0.0993386 0.995054i \(-0.468327\pi\)
0.0993386 + 0.995054i \(0.468327\pi\)
\(192\) 0 0
\(193\) 26.5728 1.91275 0.956377 0.292135i \(-0.0943658\pi\)
0.956377 + 0.292135i \(0.0943658\pi\)
\(194\) −8.82678 −0.633726
\(195\) 0 0
\(196\) −22.5229 −1.60878
\(197\) 8.49601 0.605316 0.302658 0.953099i \(-0.402126\pi\)
0.302658 + 0.953099i \(0.402126\pi\)
\(198\) 0 0
\(199\) −7.89518 −0.559674 −0.279837 0.960047i \(-0.590280\pi\)
−0.279837 + 0.960047i \(0.590280\pi\)
\(200\) −0.199944 −0.0141382
\(201\) 0 0
\(202\) −8.89346 −0.625742
\(203\) 9.52770 0.668714
\(204\) 0 0
\(205\) −5.65574 −0.395014
\(206\) −9.13608 −0.636541
\(207\) 0 0
\(208\) −3.19110 −0.221263
\(209\) −2.98902 −0.206755
\(210\) 0 0
\(211\) 16.1525 1.11199 0.555993 0.831187i \(-0.312338\pi\)
0.555993 + 0.831187i \(0.312338\pi\)
\(212\) −1.63476 −0.112276
\(213\) 0 0
\(214\) −2.86750 −0.196018
\(215\) 17.3813 1.18539
\(216\) 0 0
\(217\) 3.82002 0.259320
\(218\) 3.55477 0.240759
\(219\) 0 0
\(220\) 2.47818 0.167079
\(221\) −24.2638 −1.63216
\(222\) 0 0
\(223\) −22.7987 −1.52672 −0.763358 0.645976i \(-0.776451\pi\)
−0.763358 + 0.645976i \(0.776451\pi\)
\(224\) 27.9017 1.86426
\(225\) 0 0
\(226\) 3.32230 0.220996
\(227\) −15.8784 −1.05389 −0.526943 0.849901i \(-0.676662\pi\)
−0.526943 + 0.849901i \(0.676662\pi\)
\(228\) 0 0
\(229\) 25.2540 1.66883 0.834414 0.551138i \(-0.185806\pi\)
0.834414 + 0.551138i \(0.185806\pi\)
\(230\) −0.427873 −0.0282131
\(231\) 0 0
\(232\) −5.34609 −0.350988
\(233\) 11.6859 0.765568 0.382784 0.923838i \(-0.374965\pi\)
0.382784 + 0.923838i \(0.374965\pi\)
\(234\) 0 0
\(235\) 20.9343 1.36560
\(236\) 1.82232 0.118623
\(237\) 0 0
\(238\) −22.1628 −1.43660
\(239\) 23.1115 1.49496 0.747478 0.664287i \(-0.231265\pi\)
0.747478 + 0.664287i \(0.231265\pi\)
\(240\) 0 0
\(241\) 3.54114 0.228105 0.114052 0.993475i \(-0.463617\pi\)
0.114052 + 0.993475i \(0.463617\pi\)
\(242\) 0.948254 0.0609561
\(243\) 0 0
\(244\) 1.10081 0.0704724
\(245\) 46.0605 2.94270
\(246\) 0 0
\(247\) 16.2608 1.03465
\(248\) −2.14345 −0.136109
\(249\) 0 0
\(250\) −10.5285 −0.665880
\(251\) 19.2885 1.21748 0.608740 0.793370i \(-0.291675\pi\)
0.608740 + 0.793370i \(0.291675\pi\)
\(252\) 0 0
\(253\) 0.200434 0.0126012
\(254\) 5.16994 0.324391
\(255\) 0 0
\(256\) −16.9474 −1.05921
\(257\) −28.5441 −1.78053 −0.890267 0.455440i \(-0.849482\pi\)
−0.890267 + 0.455440i \(0.849482\pi\)
\(258\) 0 0
\(259\) −18.0719 −1.12293
\(260\) −13.4817 −0.836097
\(261\) 0 0
\(262\) 15.6127 0.964559
\(263\) 18.1170 1.11714 0.558570 0.829457i \(-0.311350\pi\)
0.558570 + 0.829457i \(0.311350\pi\)
\(264\) 0 0
\(265\) 3.34317 0.205370
\(266\) 14.8527 0.910680
\(267\) 0 0
\(268\) −17.8034 −1.08752
\(269\) −8.76351 −0.534321 −0.267160 0.963652i \(-0.586085\pi\)
−0.267160 + 0.963652i \(0.586085\pi\)
\(270\) 0 0
\(271\) −21.6248 −1.31361 −0.656806 0.754059i \(-0.728093\pi\)
−0.656806 + 0.754059i \(0.728093\pi\)
\(272\) 2.61623 0.158632
\(273\) 0 0
\(274\) −1.07364 −0.0648612
\(275\) −0.0679998 −0.00410054
\(276\) 0 0
\(277\) 17.1283 1.02914 0.514571 0.857448i \(-0.327951\pi\)
0.514571 + 0.857448i \(0.327951\pi\)
\(278\) 16.7232 1.00299
\(279\) 0 0
\(280\) −34.6873 −2.07297
\(281\) 14.8880 0.888144 0.444072 0.895991i \(-0.353533\pi\)
0.444072 + 0.895991i \(0.353533\pi\)
\(282\) 0 0
\(283\) 19.7803 1.17582 0.587908 0.808928i \(-0.299952\pi\)
0.587908 + 0.808928i \(0.299952\pi\)
\(284\) 1.53142 0.0908728
\(285\) 0 0
\(286\) −5.15865 −0.305037
\(287\) −13.1651 −0.777110
\(288\) 0 0
\(289\) 2.89269 0.170158
\(290\) 3.88132 0.227919
\(291\) 0 0
\(292\) 9.14233 0.535014
\(293\) 16.9333 0.989256 0.494628 0.869105i \(-0.335304\pi\)
0.494628 + 0.869105i \(0.335304\pi\)
\(294\) 0 0
\(295\) −3.72673 −0.216979
\(296\) 10.1403 0.589393
\(297\) 0 0
\(298\) −17.5893 −1.01892
\(299\) −1.09039 −0.0630590
\(300\) 0 0
\(301\) 40.4590 2.33202
\(302\) −21.2625 −1.22352
\(303\) 0 0
\(304\) −1.75331 −0.100559
\(305\) −2.25122 −0.128905
\(306\) 0 0
\(307\) 1.52740 0.0871733 0.0435866 0.999050i \(-0.486122\pi\)
0.0435866 + 0.999050i \(0.486122\pi\)
\(308\) 5.76854 0.328693
\(309\) 0 0
\(310\) 1.55617 0.0883845
\(311\) −31.7654 −1.80125 −0.900626 0.434596i \(-0.856891\pi\)
−0.900626 + 0.434596i \(0.856891\pi\)
\(312\) 0 0
\(313\) −16.9039 −0.955467 −0.477733 0.878505i \(-0.658542\pi\)
−0.477733 + 0.878505i \(0.658542\pi\)
\(314\) −6.93278 −0.391239
\(315\) 0 0
\(316\) 12.8884 0.725029
\(317\) −20.3557 −1.14329 −0.571646 0.820501i \(-0.693695\pi\)
−0.571646 + 0.820501i \(0.693695\pi\)
\(318\) 0 0
\(319\) −1.81818 −0.101798
\(320\) 14.0074 0.783038
\(321\) 0 0
\(322\) −0.995975 −0.0555035
\(323\) −13.3314 −0.741780
\(324\) 0 0
\(325\) 0.369929 0.0205200
\(326\) −0.990074 −0.0548351
\(327\) 0 0
\(328\) 7.38706 0.407882
\(329\) 48.7295 2.68655
\(330\) 0 0
\(331\) 16.6765 0.916621 0.458311 0.888792i \(-0.348455\pi\)
0.458311 + 0.888792i \(0.348455\pi\)
\(332\) −1.22137 −0.0670312
\(333\) 0 0
\(334\) 0.229731 0.0125703
\(335\) 36.4090 1.98924
\(336\) 0 0
\(337\) −6.60838 −0.359981 −0.179991 0.983668i \(-0.557607\pi\)
−0.179991 + 0.983668i \(0.557607\pi\)
\(338\) 15.7366 0.855956
\(339\) 0 0
\(340\) 11.0530 0.599431
\(341\) −0.728977 −0.0394763
\(342\) 0 0
\(343\) 70.5351 3.80854
\(344\) −22.7020 −1.22401
\(345\) 0 0
\(346\) 22.5295 1.21119
\(347\) −16.0359 −0.860851 −0.430425 0.902626i \(-0.641636\pi\)
−0.430425 + 0.902626i \(0.641636\pi\)
\(348\) 0 0
\(349\) 6.89281 0.368964 0.184482 0.982836i \(-0.440939\pi\)
0.184482 + 0.982836i \(0.440939\pi\)
\(350\) 0.337897 0.0180614
\(351\) 0 0
\(352\) −5.32449 −0.283796
\(353\) 13.3187 0.708884 0.354442 0.935078i \(-0.384671\pi\)
0.354442 + 0.935078i \(0.384671\pi\)
\(354\) 0 0
\(355\) −3.13183 −0.166220
\(356\) 7.49080 0.397012
\(357\) 0 0
\(358\) 2.65134 0.140127
\(359\) −35.5183 −1.87458 −0.937291 0.348547i \(-0.886675\pi\)
−0.937291 + 0.348547i \(0.886675\pi\)
\(360\) 0 0
\(361\) −10.0657 −0.529775
\(362\) 11.9091 0.625927
\(363\) 0 0
\(364\) −31.3818 −1.64485
\(365\) −18.6965 −0.978622
\(366\) 0 0
\(367\) −11.8248 −0.617250 −0.308625 0.951184i \(-0.599869\pi\)
−0.308625 + 0.951184i \(0.599869\pi\)
\(368\) 0.117571 0.00612882
\(369\) 0 0
\(370\) −7.36196 −0.382730
\(371\) 7.78203 0.404023
\(372\) 0 0
\(373\) −21.2527 −1.10042 −0.550211 0.835026i \(-0.685453\pi\)
−0.550211 + 0.835026i \(0.685453\pi\)
\(374\) 4.22933 0.218694
\(375\) 0 0
\(376\) −27.3426 −1.41009
\(377\) 9.89116 0.509421
\(378\) 0 0
\(379\) 25.3787 1.30362 0.651808 0.758384i \(-0.274011\pi\)
0.651808 + 0.758384i \(0.274011\pi\)
\(380\) −7.40733 −0.379988
\(381\) 0 0
\(382\) 2.60369 0.133216
\(383\) −8.43965 −0.431246 −0.215623 0.976477i \(-0.569178\pi\)
−0.215623 + 0.976477i \(0.569178\pi\)
\(384\) 0 0
\(385\) −11.7970 −0.601229
\(386\) 25.1978 1.28253
\(387\) 0 0
\(388\) 10.2469 0.520206
\(389\) −24.0747 −1.22063 −0.610317 0.792158i \(-0.708958\pi\)
−0.610317 + 0.792158i \(0.708958\pi\)
\(390\) 0 0
\(391\) 0.893960 0.0452095
\(392\) −60.1605 −3.03856
\(393\) 0 0
\(394\) 8.05638 0.405875
\(395\) −26.3574 −1.32619
\(396\) 0 0
\(397\) −11.1126 −0.557724 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(398\) −7.48664 −0.375271
\(399\) 0 0
\(400\) −0.0398875 −0.00199437
\(401\) 25.1568 1.25627 0.628136 0.778104i \(-0.283818\pi\)
0.628136 + 0.778104i \(0.283818\pi\)
\(402\) 0 0
\(403\) 3.96575 0.197548
\(404\) 10.3243 0.513652
\(405\) 0 0
\(406\) 9.03469 0.448384
\(407\) 3.44866 0.170944
\(408\) 0 0
\(409\) 4.15275 0.205340 0.102670 0.994715i \(-0.467261\pi\)
0.102670 + 0.994715i \(0.467261\pi\)
\(410\) −5.36308 −0.264864
\(411\) 0 0
\(412\) 10.6059 0.522516
\(413\) −8.67485 −0.426862
\(414\) 0 0
\(415\) 2.49776 0.122610
\(416\) 28.9661 1.42018
\(417\) 0 0
\(418\) −2.83436 −0.138633
\(419\) −34.2797 −1.67467 −0.837337 0.546688i \(-0.815889\pi\)
−0.837337 + 0.546688i \(0.815889\pi\)
\(420\) 0 0
\(421\) −36.7527 −1.79122 −0.895609 0.444842i \(-0.853260\pi\)
−0.895609 + 0.444842i \(0.853260\pi\)
\(422\) 15.3167 0.745606
\(423\) 0 0
\(424\) −4.36658 −0.212060
\(425\) −0.303287 −0.0147116
\(426\) 0 0
\(427\) −5.24025 −0.253594
\(428\) 3.32883 0.160905
\(429\) 0 0
\(430\) 16.4818 0.794825
\(431\) −5.91425 −0.284879 −0.142440 0.989803i \(-0.545495\pi\)
−0.142440 + 0.989803i \(0.545495\pi\)
\(432\) 0 0
\(433\) 12.9638 0.623001 0.311500 0.950246i \(-0.399168\pi\)
0.311500 + 0.950246i \(0.399168\pi\)
\(434\) 3.62235 0.173878
\(435\) 0 0
\(436\) −4.12668 −0.197632
\(437\) −0.599102 −0.0286589
\(438\) 0 0
\(439\) −29.2711 −1.39704 −0.698518 0.715593i \(-0.746157\pi\)
−0.698518 + 0.715593i \(0.746157\pi\)
\(440\) 6.61940 0.315568
\(441\) 0 0
\(442\) −23.0082 −1.09439
\(443\) −3.21165 −0.152590 −0.0762951 0.997085i \(-0.524309\pi\)
−0.0762951 + 0.997085i \(0.524309\pi\)
\(444\) 0 0
\(445\) −15.3191 −0.726194
\(446\) −21.6190 −1.02369
\(447\) 0 0
\(448\) 32.6056 1.54047
\(449\) 11.0578 0.521849 0.260924 0.965359i \(-0.415973\pi\)
0.260924 + 0.965359i \(0.415973\pi\)
\(450\) 0 0
\(451\) 2.51230 0.118300
\(452\) −3.85681 −0.181409
\(453\) 0 0
\(454\) −15.0568 −0.706649
\(455\) 64.1774 3.00868
\(456\) 0 0
\(457\) 8.45879 0.395685 0.197843 0.980234i \(-0.436606\pi\)
0.197843 + 0.980234i \(0.436606\pi\)
\(458\) 23.9472 1.11898
\(459\) 0 0
\(460\) 0.496711 0.0231593
\(461\) −8.02284 −0.373661 −0.186830 0.982392i \(-0.559822\pi\)
−0.186830 + 0.982392i \(0.559822\pi\)
\(462\) 0 0
\(463\) 20.0463 0.931630 0.465815 0.884882i \(-0.345761\pi\)
0.465815 + 0.884882i \(0.345761\pi\)
\(464\) −1.06651 −0.0495115
\(465\) 0 0
\(466\) 11.0812 0.513326
\(467\) 9.41646 0.435742 0.217871 0.975978i \(-0.430089\pi\)
0.217871 + 0.975978i \(0.430089\pi\)
\(468\) 0 0
\(469\) 84.7505 3.91342
\(470\) 19.8510 0.915660
\(471\) 0 0
\(472\) 4.86755 0.224047
\(473\) −7.72081 −0.355003
\(474\) 0 0
\(475\) 0.203253 0.00932589
\(476\) 25.7284 1.17926
\(477\) 0 0
\(478\) 21.9155 1.00239
\(479\) −15.6024 −0.712890 −0.356445 0.934316i \(-0.616011\pi\)
−0.356445 + 0.934316i \(0.616011\pi\)
\(480\) 0 0
\(481\) −18.7613 −0.855439
\(482\) 3.35790 0.152948
\(483\) 0 0
\(484\) −1.10081 −0.0500370
\(485\) −20.9554 −0.951535
\(486\) 0 0
\(487\) 28.6801 1.29962 0.649811 0.760096i \(-0.274848\pi\)
0.649811 + 0.760096i \(0.274848\pi\)
\(488\) 2.94036 0.133104
\(489\) 0 0
\(490\) 43.6771 1.97313
\(491\) −0.942339 −0.0425271 −0.0212636 0.999774i \(-0.506769\pi\)
−0.0212636 + 0.999774i \(0.506769\pi\)
\(492\) 0 0
\(493\) −8.10929 −0.365224
\(494\) 15.4193 0.693749
\(495\) 0 0
\(496\) −0.427605 −0.0192000
\(497\) −7.29007 −0.327004
\(498\) 0 0
\(499\) −3.01444 −0.134945 −0.0674725 0.997721i \(-0.521493\pi\)
−0.0674725 + 0.997721i \(0.521493\pi\)
\(500\) 12.2224 0.546601
\(501\) 0 0
\(502\) 18.2904 0.816341
\(503\) −43.9550 −1.95986 −0.979929 0.199346i \(-0.936118\pi\)
−0.979929 + 0.199346i \(0.936118\pi\)
\(504\) 0 0
\(505\) −21.1137 −0.939547
\(506\) 0.190062 0.00844931
\(507\) 0 0
\(508\) −6.00170 −0.266282
\(509\) 23.9124 1.05990 0.529950 0.848029i \(-0.322211\pi\)
0.529950 + 0.848029i \(0.322211\pi\)
\(510\) 0 0
\(511\) −43.5206 −1.92524
\(512\) −6.57278 −0.290479
\(513\) 0 0
\(514\) −27.0671 −1.19388
\(515\) −21.6897 −0.955762
\(516\) 0 0
\(517\) −9.29908 −0.408973
\(518\) −17.1367 −0.752944
\(519\) 0 0
\(520\) −36.0106 −1.57917
\(521\) 37.7693 1.65470 0.827351 0.561685i \(-0.189847\pi\)
0.827351 + 0.561685i \(0.189847\pi\)
\(522\) 0 0
\(523\) −3.21011 −0.140368 −0.0701842 0.997534i \(-0.522359\pi\)
−0.0701842 + 0.997534i \(0.522359\pi\)
\(524\) −18.1246 −0.791776
\(525\) 0 0
\(526\) 17.1795 0.749062
\(527\) −3.25133 −0.141630
\(528\) 0 0
\(529\) −22.9598 −0.998253
\(530\) 3.17018 0.137704
\(531\) 0 0
\(532\) −17.2423 −0.747549
\(533\) −13.6673 −0.591996
\(534\) 0 0
\(535\) −6.80763 −0.294320
\(536\) −47.5544 −2.05404
\(537\) 0 0
\(538\) −8.31004 −0.358271
\(539\) −20.4602 −0.881285
\(540\) 0 0
\(541\) −15.9679 −0.686512 −0.343256 0.939242i \(-0.611530\pi\)
−0.343256 + 0.939242i \(0.611530\pi\)
\(542\) −20.5058 −0.880800
\(543\) 0 0
\(544\) −23.7479 −1.01818
\(545\) 8.43927 0.361499
\(546\) 0 0
\(547\) −13.1472 −0.562132 −0.281066 0.959688i \(-0.590688\pi\)
−0.281066 + 0.959688i \(0.590688\pi\)
\(548\) 1.24638 0.0532425
\(549\) 0 0
\(550\) −0.0644811 −0.00274948
\(551\) 5.43458 0.231521
\(552\) 0 0
\(553\) −61.3532 −2.60900
\(554\) 16.2420 0.690057
\(555\) 0 0
\(556\) −19.4137 −0.823324
\(557\) −0.120019 −0.00508539 −0.00254269 0.999997i \(-0.500809\pi\)
−0.00254269 + 0.999997i \(0.500809\pi\)
\(558\) 0 0
\(559\) 42.0024 1.77651
\(560\) −6.91990 −0.292419
\(561\) 0 0
\(562\) 14.1176 0.595516
\(563\) 22.3237 0.940831 0.470415 0.882445i \(-0.344104\pi\)
0.470415 + 0.882445i \(0.344104\pi\)
\(564\) 0 0
\(565\) 7.88738 0.331825
\(566\) 18.7568 0.788405
\(567\) 0 0
\(568\) 4.09053 0.171635
\(569\) 6.42634 0.269406 0.134703 0.990886i \(-0.456992\pi\)
0.134703 + 0.990886i \(0.456992\pi\)
\(570\) 0 0
\(571\) −21.3502 −0.893479 −0.446740 0.894664i \(-0.647415\pi\)
−0.446740 + 0.894664i \(0.647415\pi\)
\(572\) 5.98860 0.250396
\(573\) 0 0
\(574\) −12.4838 −0.521066
\(575\) −0.0136295 −0.000568388 0
\(576\) 0 0
\(577\) −4.88600 −0.203407 −0.101703 0.994815i \(-0.532429\pi\)
−0.101703 + 0.994815i \(0.532429\pi\)
\(578\) 2.74301 0.114094
\(579\) 0 0
\(580\) −4.50576 −0.187092
\(581\) 5.81413 0.241211
\(582\) 0 0
\(583\) −1.48505 −0.0615044
\(584\) 24.4199 1.01050
\(585\) 0 0
\(586\) 16.0571 0.663313
\(587\) 32.3274 1.33429 0.667147 0.744926i \(-0.267515\pi\)
0.667147 + 0.744926i \(0.267515\pi\)
\(588\) 0 0
\(589\) 2.17893 0.0897812
\(590\) −3.53389 −0.145488
\(591\) 0 0
\(592\) 2.02293 0.0831417
\(593\) −18.0269 −0.740275 −0.370137 0.928977i \(-0.620689\pi\)
−0.370137 + 0.928977i \(0.620689\pi\)
\(594\) 0 0
\(595\) −52.6159 −2.15704
\(596\) 20.4192 0.836402
\(597\) 0 0
\(598\) −1.03397 −0.0422821
\(599\) 6.99770 0.285918 0.142959 0.989729i \(-0.454338\pi\)
0.142959 + 0.989729i \(0.454338\pi\)
\(600\) 0 0
\(601\) 27.9227 1.13899 0.569496 0.821994i \(-0.307138\pi\)
0.569496 + 0.821994i \(0.307138\pi\)
\(602\) 38.3654 1.56366
\(603\) 0 0
\(604\) 24.6832 1.00435
\(605\) 2.25122 0.0915252
\(606\) 0 0
\(607\) 47.3972 1.92379 0.961897 0.273412i \(-0.0881522\pi\)
0.961897 + 0.273412i \(0.0881522\pi\)
\(608\) 15.9150 0.645440
\(609\) 0 0
\(610\) −2.13473 −0.0864327
\(611\) 50.5884 2.04659
\(612\) 0 0
\(613\) −36.7435 −1.48406 −0.742028 0.670369i \(-0.766136\pi\)
−0.742028 + 0.670369i \(0.766136\pi\)
\(614\) 1.44836 0.0584512
\(615\) 0 0
\(616\) 15.4082 0.620815
\(617\) −23.6451 −0.951917 −0.475958 0.879468i \(-0.657899\pi\)
−0.475958 + 0.879468i \(0.657899\pi\)
\(618\) 0 0
\(619\) −8.49054 −0.341264 −0.170632 0.985335i \(-0.554581\pi\)
−0.170632 + 0.985335i \(0.554581\pi\)
\(620\) −1.80653 −0.0725521
\(621\) 0 0
\(622\) −30.1217 −1.20777
\(623\) −35.6588 −1.42864
\(624\) 0 0
\(625\) −25.3354 −1.01341
\(626\) −16.0292 −0.640657
\(627\) 0 0
\(628\) 8.04815 0.321156
\(629\) 15.3815 0.613299
\(630\) 0 0
\(631\) 13.5216 0.538288 0.269144 0.963100i \(-0.413259\pi\)
0.269144 + 0.963100i \(0.413259\pi\)
\(632\) 34.4259 1.36939
\(633\) 0 0
\(634\) −19.3024 −0.766597
\(635\) 12.2738 0.487071
\(636\) 0 0
\(637\) 111.307 4.41014
\(638\) −1.72409 −0.0682575
\(639\) 0 0
\(640\) −10.6906 −0.422584
\(641\) 11.5595 0.456574 0.228287 0.973594i \(-0.426688\pi\)
0.228287 + 0.973594i \(0.426688\pi\)
\(642\) 0 0
\(643\) −19.4275 −0.766147 −0.383073 0.923718i \(-0.625134\pi\)
−0.383073 + 0.923718i \(0.625134\pi\)
\(644\) 1.15621 0.0455611
\(645\) 0 0
\(646\) −12.6416 −0.497376
\(647\) 27.7606 1.09138 0.545690 0.837987i \(-0.316268\pi\)
0.545690 + 0.837987i \(0.316268\pi\)
\(648\) 0 0
\(649\) 1.65543 0.0649812
\(650\) 0.350787 0.0137590
\(651\) 0 0
\(652\) 1.14936 0.0450124
\(653\) −23.0651 −0.902608 −0.451304 0.892370i \(-0.649041\pi\)
−0.451304 + 0.892370i \(0.649041\pi\)
\(654\) 0 0
\(655\) 37.0657 1.44828
\(656\) 1.47367 0.0575372
\(657\) 0 0
\(658\) 46.2080 1.80137
\(659\) −7.13729 −0.278029 −0.139015 0.990290i \(-0.544394\pi\)
−0.139015 + 0.990290i \(0.544394\pi\)
\(660\) 0 0
\(661\) −16.0929 −0.625941 −0.312971 0.949763i \(-0.601324\pi\)
−0.312971 + 0.949763i \(0.601324\pi\)
\(662\) 15.8135 0.614610
\(663\) 0 0
\(664\) −3.26237 −0.126604
\(665\) 35.2614 1.36738
\(666\) 0 0
\(667\) −0.364424 −0.0141106
\(668\) −0.266691 −0.0103186
\(669\) 0 0
\(670\) 34.5250 1.33382
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 34.6989 1.33755 0.668773 0.743467i \(-0.266820\pi\)
0.668773 + 0.743467i \(0.266820\pi\)
\(674\) −6.26643 −0.241374
\(675\) 0 0
\(676\) −18.2683 −0.702628
\(677\) 46.9863 1.80583 0.902916 0.429818i \(-0.141422\pi\)
0.902916 + 0.429818i \(0.141422\pi\)
\(678\) 0 0
\(679\) −48.7786 −1.87195
\(680\) 29.5234 1.13217
\(681\) 0 0
\(682\) −0.691255 −0.0264695
\(683\) −1.54777 −0.0592238 −0.0296119 0.999561i \(-0.509427\pi\)
−0.0296119 + 0.999561i \(0.509427\pi\)
\(684\) 0 0
\(685\) −2.54890 −0.0973886
\(686\) 66.8852 2.55369
\(687\) 0 0
\(688\) −4.52889 −0.172662
\(689\) 8.07889 0.307781
\(690\) 0 0
\(691\) 18.0655 0.687245 0.343623 0.939108i \(-0.388346\pi\)
0.343623 + 0.939108i \(0.388346\pi\)
\(692\) −26.1542 −0.994232
\(693\) 0 0
\(694\) −15.2061 −0.577215
\(695\) 39.7020 1.50598
\(696\) 0 0
\(697\) 11.2052 0.424426
\(698\) 6.53614 0.247396
\(699\) 0 0
\(700\) −0.392259 −0.0148260
\(701\) 21.0050 0.793346 0.396673 0.917960i \(-0.370165\pi\)
0.396673 + 0.917960i \(0.370165\pi\)
\(702\) 0 0
\(703\) −10.3081 −0.388779
\(704\) −6.22214 −0.234506
\(705\) 0 0
\(706\) 12.6295 0.475319
\(707\) −49.1471 −1.84837
\(708\) 0 0
\(709\) −12.9485 −0.486291 −0.243146 0.969990i \(-0.578179\pi\)
−0.243146 + 0.969990i \(0.578179\pi\)
\(710\) −2.96977 −0.111453
\(711\) 0 0
\(712\) 20.0085 0.749851
\(713\) −0.146112 −0.00547193
\(714\) 0 0
\(715\) −12.2470 −0.458012
\(716\) −3.07789 −0.115026
\(717\) 0 0
\(718\) −33.6804 −1.25694
\(719\) −11.2061 −0.417917 −0.208958 0.977925i \(-0.567007\pi\)
−0.208958 + 0.977925i \(0.567007\pi\)
\(720\) 0 0
\(721\) −50.4879 −1.88027
\(722\) −9.54487 −0.355223
\(723\) 0 0
\(724\) −13.8251 −0.513804
\(725\) 0.123636 0.00459171
\(726\) 0 0
\(727\) 35.9986 1.33511 0.667557 0.744559i \(-0.267340\pi\)
0.667557 + 0.744559i \(0.267340\pi\)
\(728\) −83.8232 −3.10669
\(729\) 0 0
\(730\) −17.7291 −0.656183
\(731\) −34.4358 −1.27365
\(732\) 0 0
\(733\) 23.6506 0.873554 0.436777 0.899570i \(-0.356120\pi\)
0.436777 + 0.899570i \(0.356120\pi\)
\(734\) −11.2129 −0.413877
\(735\) 0 0
\(736\) −1.06721 −0.0393378
\(737\) −16.1730 −0.595740
\(738\) 0 0
\(739\) −14.8095 −0.544778 −0.272389 0.962187i \(-0.587814\pi\)
−0.272389 + 0.962187i \(0.587814\pi\)
\(740\) 8.54639 0.314171
\(741\) 0 0
\(742\) 7.37934 0.270904
\(743\) −13.3290 −0.488994 −0.244497 0.969650i \(-0.578623\pi\)
−0.244497 + 0.969650i \(0.578623\pi\)
\(744\) 0 0
\(745\) −41.7583 −1.52991
\(746\) −20.1529 −0.737852
\(747\) 0 0
\(748\) −4.90976 −0.179519
\(749\) −15.8464 −0.579014
\(750\) 0 0
\(751\) 1.50326 0.0548546 0.0274273 0.999624i \(-0.491269\pi\)
0.0274273 + 0.999624i \(0.491269\pi\)
\(752\) −5.45468 −0.198912
\(753\) 0 0
\(754\) 9.37934 0.341575
\(755\) −50.4785 −1.83710
\(756\) 0 0
\(757\) −12.6961 −0.461449 −0.230724 0.973019i \(-0.574110\pi\)
−0.230724 + 0.973019i \(0.574110\pi\)
\(758\) 24.0655 0.874097
\(759\) 0 0
\(760\) −19.7856 −0.717698
\(761\) 1.08972 0.0395022 0.0197511 0.999805i \(-0.493713\pi\)
0.0197511 + 0.999805i \(0.493713\pi\)
\(762\) 0 0
\(763\) 19.6444 0.711175
\(764\) −3.02258 −0.109353
\(765\) 0 0
\(766\) −8.00294 −0.289158
\(767\) −9.00578 −0.325180
\(768\) 0 0
\(769\) 17.6529 0.636579 0.318289 0.947994i \(-0.396892\pi\)
0.318289 + 0.947994i \(0.396892\pi\)
\(770\) −11.1865 −0.403135
\(771\) 0 0
\(772\) −29.2517 −1.05279
\(773\) 12.7055 0.456986 0.228493 0.973546i \(-0.426620\pi\)
0.228493 + 0.973546i \(0.426620\pi\)
\(774\) 0 0
\(775\) 0.0495702 0.00178062
\(776\) 27.3702 0.982533
\(777\) 0 0
\(778\) −22.8289 −0.818456
\(779\) −7.50932 −0.269049
\(780\) 0 0
\(781\) 1.39117 0.0497799
\(782\) 0.847702 0.0303138
\(783\) 0 0
\(784\) −12.0016 −0.428630
\(785\) −16.4589 −0.587443
\(786\) 0 0
\(787\) −14.3201 −0.510456 −0.255228 0.966881i \(-0.582151\pi\)
−0.255228 + 0.966881i \(0.582151\pi\)
\(788\) −9.35252 −0.333170
\(789\) 0 0
\(790\) −24.9936 −0.889231
\(791\) 18.3597 0.652797
\(792\) 0 0
\(793\) −5.44016 −0.193186
\(794\) −10.5375 −0.373963
\(795\) 0 0
\(796\) 8.69112 0.308049
\(797\) −55.4418 −1.96385 −0.981924 0.189275i \(-0.939386\pi\)
−0.981924 + 0.189275i \(0.939386\pi\)
\(798\) 0 0
\(799\) −41.4750 −1.46728
\(800\) 0.362064 0.0128009
\(801\) 0 0
\(802\) 23.8551 0.842352
\(803\) 8.30506 0.293079
\(804\) 0 0
\(805\) −2.36451 −0.0833382
\(806\) 3.76054 0.132459
\(807\) 0 0
\(808\) 27.5770 0.970154
\(809\) −37.9294 −1.33353 −0.666763 0.745270i \(-0.732321\pi\)
−0.666763 + 0.745270i \(0.732321\pi\)
\(810\) 0 0
\(811\) −3.23359 −0.113547 −0.0567733 0.998387i \(-0.518081\pi\)
−0.0567733 + 0.998387i \(0.518081\pi\)
\(812\) −10.4882 −0.368065
\(813\) 0 0
\(814\) 3.27021 0.114621
\(815\) −2.35050 −0.0823345
\(816\) 0 0
\(817\) 23.0777 0.807386
\(818\) 3.93786 0.137684
\(819\) 0 0
\(820\) 6.22592 0.217418
\(821\) 44.4292 1.55059 0.775295 0.631599i \(-0.217601\pi\)
0.775295 + 0.631599i \(0.217601\pi\)
\(822\) 0 0
\(823\) 11.0243 0.384284 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(824\) 28.3293 0.986897
\(825\) 0 0
\(826\) −8.22597 −0.286218
\(827\) −38.2151 −1.32887 −0.664435 0.747346i \(-0.731328\pi\)
−0.664435 + 0.747346i \(0.731328\pi\)
\(828\) 0 0
\(829\) 5.10113 0.177170 0.0885848 0.996069i \(-0.471766\pi\)
0.0885848 + 0.996069i \(0.471766\pi\)
\(830\) 2.36851 0.0822122
\(831\) 0 0
\(832\) 33.8494 1.17352
\(833\) −91.2552 −3.16181
\(834\) 0 0
\(835\) 0.545397 0.0188742
\(836\) 3.29036 0.113799
\(837\) 0 0
\(838\) −32.5059 −1.12290
\(839\) −22.4559 −0.775262 −0.387631 0.921815i \(-0.626707\pi\)
−0.387631 + 0.921815i \(0.626707\pi\)
\(840\) 0 0
\(841\) −25.6942 −0.886008
\(842\) −34.8509 −1.20104
\(843\) 0 0
\(844\) −17.7809 −0.612045
\(845\) 37.3597 1.28521
\(846\) 0 0
\(847\) 5.24025 0.180057
\(848\) −0.871104 −0.0299138
\(849\) 0 0
\(850\) −0.287594 −0.00986438
\(851\) 0.691229 0.0236950
\(852\) 0 0
\(853\) 4.17359 0.142901 0.0714505 0.997444i \(-0.477237\pi\)
0.0714505 + 0.997444i \(0.477237\pi\)
\(854\) −4.96909 −0.170039
\(855\) 0 0
\(856\) 8.89157 0.303908
\(857\) −54.4319 −1.85936 −0.929679 0.368371i \(-0.879916\pi\)
−0.929679 + 0.368371i \(0.879916\pi\)
\(858\) 0 0
\(859\) −18.6033 −0.634737 −0.317368 0.948302i \(-0.602799\pi\)
−0.317368 + 0.948302i \(0.602799\pi\)
\(860\) −19.1335 −0.652447
\(861\) 0 0
\(862\) −5.60821 −0.191016
\(863\) −51.6257 −1.75736 −0.878679 0.477412i \(-0.841575\pi\)
−0.878679 + 0.477412i \(0.841575\pi\)
\(864\) 0 0
\(865\) 53.4866 1.81860
\(866\) 12.2930 0.417733
\(867\) 0 0
\(868\) −4.20513 −0.142731
\(869\) 11.7081 0.397169
\(870\) 0 0
\(871\) 87.9836 2.98121
\(872\) −11.0227 −0.373275
\(873\) 0 0
\(874\) −0.568101 −0.0192163
\(875\) −58.1827 −1.96693
\(876\) 0 0
\(877\) −49.5572 −1.67343 −0.836715 0.547639i \(-0.815527\pi\)
−0.836715 + 0.547639i \(0.815527\pi\)
\(878\) −27.7565 −0.936736
\(879\) 0 0
\(880\) 1.32053 0.0445150
\(881\) −48.5126 −1.63443 −0.817216 0.576332i \(-0.804484\pi\)
−0.817216 + 0.576332i \(0.804484\pi\)
\(882\) 0 0
\(883\) −26.0469 −0.876549 −0.438275 0.898841i \(-0.644410\pi\)
−0.438275 + 0.898841i \(0.644410\pi\)
\(884\) 26.7099 0.898350
\(885\) 0 0
\(886\) −3.04546 −0.102314
\(887\) −46.1043 −1.54803 −0.774015 0.633168i \(-0.781754\pi\)
−0.774015 + 0.633168i \(0.781754\pi\)
\(888\) 0 0
\(889\) 28.5702 0.958213
\(890\) −14.5264 −0.486926
\(891\) 0 0
\(892\) 25.0971 0.840315
\(893\) 27.7952 0.930130
\(894\) 0 0
\(895\) 6.29445 0.210400
\(896\) −24.8850 −0.831348
\(897\) 0 0
\(898\) 10.4856 0.349909
\(899\) 1.32541 0.0442048
\(900\) 0 0
\(901\) −6.62350 −0.220661
\(902\) 2.38230 0.0793219
\(903\) 0 0
\(904\) −10.3018 −0.342634
\(905\) 28.2730 0.939826
\(906\) 0 0
\(907\) 4.84142 0.160757 0.0803783 0.996764i \(-0.474387\pi\)
0.0803783 + 0.996764i \(0.474387\pi\)
\(908\) 17.4791 0.580066
\(909\) 0 0
\(910\) 60.8565 2.01737
\(911\) −37.1866 −1.23205 −0.616024 0.787727i \(-0.711258\pi\)
−0.616024 + 0.787727i \(0.711258\pi\)
\(912\) 0 0
\(913\) −1.10951 −0.0367195
\(914\) 8.02108 0.265314
\(915\) 0 0
\(916\) −27.7999 −0.918534
\(917\) 86.2793 2.84919
\(918\) 0 0
\(919\) −42.7495 −1.41018 −0.705088 0.709120i \(-0.749092\pi\)
−0.705088 + 0.709120i \(0.749092\pi\)
\(920\) 1.32675 0.0437418
\(921\) 0 0
\(922\) −7.60769 −0.250546
\(923\) −7.56817 −0.249109
\(924\) 0 0
\(925\) −0.234508 −0.00771058
\(926\) 19.0090 0.624674
\(927\) 0 0
\(928\) 9.68087 0.317790
\(929\) −22.1672 −0.727284 −0.363642 0.931539i \(-0.618467\pi\)
−0.363642 + 0.931539i \(0.618467\pi\)
\(930\) 0 0
\(931\) 61.1562 2.00431
\(932\) −12.8640 −0.421374
\(933\) 0 0
\(934\) 8.92920 0.292172
\(935\) 10.0407 0.328367
\(936\) 0 0
\(937\) 25.4112 0.830148 0.415074 0.909788i \(-0.363756\pi\)
0.415074 + 0.909788i \(0.363756\pi\)
\(938\) 80.3651 2.62401
\(939\) 0 0
\(940\) −23.0447 −0.751637
\(941\) −35.3840 −1.15348 −0.576742 0.816926i \(-0.695676\pi\)
−0.576742 + 0.816926i \(0.695676\pi\)
\(942\) 0 0
\(943\) 0.503550 0.0163978
\(944\) 0.971045 0.0316048
\(945\) 0 0
\(946\) −7.32129 −0.238036
\(947\) −15.1601 −0.492638 −0.246319 0.969189i \(-0.579221\pi\)
−0.246319 + 0.969189i \(0.579221\pi\)
\(948\) 0 0
\(949\) −45.1808 −1.46663
\(950\) 0.192736 0.00625317
\(951\) 0 0
\(952\) 68.7226 2.22731
\(953\) 50.7429 1.64372 0.821862 0.569687i \(-0.192935\pi\)
0.821862 + 0.569687i \(0.192935\pi\)
\(954\) 0 0
\(955\) 6.18134 0.200024
\(956\) −25.4414 −0.822834
\(957\) 0 0
\(958\) −14.7950 −0.478005
\(959\) −5.93318 −0.191592
\(960\) 0 0
\(961\) −30.4686 −0.982858
\(962\) −17.7904 −0.573587
\(963\) 0 0
\(964\) −3.89814 −0.125550
\(965\) 59.8213 1.92572
\(966\) 0 0
\(967\) 1.89879 0.0610611 0.0305305 0.999534i \(-0.490280\pi\)
0.0305305 + 0.999534i \(0.490280\pi\)
\(968\) −2.94036 −0.0945068
\(969\) 0 0
\(970\) −19.8710 −0.638021
\(971\) −28.5211 −0.915284 −0.457642 0.889136i \(-0.651306\pi\)
−0.457642 + 0.889136i \(0.651306\pi\)
\(972\) 0 0
\(973\) 92.4159 2.96272
\(974\) 27.1961 0.871418
\(975\) 0 0
\(976\) 0.586583 0.0187761
\(977\) 44.8572 1.43511 0.717554 0.696502i \(-0.245261\pi\)
0.717554 + 0.696502i \(0.245261\pi\)
\(978\) 0 0
\(979\) 6.80479 0.217482
\(980\) −50.7041 −1.61968
\(981\) 0 0
\(982\) −0.893577 −0.0285152
\(983\) −15.8302 −0.504904 −0.252452 0.967609i \(-0.581237\pi\)
−0.252452 + 0.967609i \(0.581237\pi\)
\(984\) 0 0
\(985\) 19.1264 0.609418
\(986\) −7.68967 −0.244889
\(987\) 0 0
\(988\) −17.9001 −0.569477
\(989\) −1.54751 −0.0492080
\(990\) 0 0
\(991\) −12.4145 −0.394359 −0.197180 0.980367i \(-0.563178\pi\)
−0.197180 + 0.980367i \(0.563178\pi\)
\(992\) 3.88143 0.123236
\(993\) 0 0
\(994\) −6.91284 −0.219262
\(995\) −17.7738 −0.563467
\(996\) 0 0
\(997\) −16.4628 −0.521383 −0.260691 0.965422i \(-0.583950\pi\)
−0.260691 + 0.965422i \(0.583950\pi\)
\(998\) −2.85846 −0.0904829
\(999\) 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.i.1.9 13
3.2 odd 2 2013.2.a.e.1.5 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.5 13 3.2 odd 2
6039.2.a.i.1.9 13 1.1 even 1 trivial