Properties

Label 6039.2.a.i.1.13
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 1.13
Root \(-2.59890\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +O(q^{10})\) \(q+2.59890 q^{2} +4.75428 q^{4} +3.50938 q^{5} +0.252288 q^{7} +7.15810 q^{8} +9.12053 q^{10} -1.00000 q^{11} +0.314950 q^{13} +0.655671 q^{14} +9.09463 q^{16} -3.13246 q^{17} +0.841231 q^{19} +16.6846 q^{20} -2.59890 q^{22} -1.45222 q^{23} +7.31576 q^{25} +0.818524 q^{26} +1.19945 q^{28} +6.76829 q^{29} -7.07730 q^{31} +9.31982 q^{32} -8.14095 q^{34} +0.885374 q^{35} +8.89628 q^{37} +2.18627 q^{38} +25.1205 q^{40} +8.43604 q^{41} +1.80409 q^{43} -4.75428 q^{44} -3.77418 q^{46} -2.30017 q^{47} -6.93635 q^{49} +19.0129 q^{50} +1.49736 q^{52} +4.41103 q^{53} -3.50938 q^{55} +1.80590 q^{56} +17.5901 q^{58} -4.48036 q^{59} -1.00000 q^{61} -18.3932 q^{62} +6.03203 q^{64} +1.10528 q^{65} -0.0414722 q^{67} -14.8926 q^{68} +2.30100 q^{70} +9.80832 q^{71} +8.90879 q^{73} +23.1205 q^{74} +3.99945 q^{76} -0.252288 q^{77} -8.51680 q^{79} +31.9165 q^{80} +21.9244 q^{82} -11.2151 q^{83} -10.9930 q^{85} +4.68865 q^{86} -7.15810 q^{88} -5.58487 q^{89} +0.0794581 q^{91} -6.90427 q^{92} -5.97790 q^{94} +2.95220 q^{95} +7.54000 q^{97} -18.0269 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59890 1.83770 0.918850 0.394607i \(-0.129119\pi\)
0.918850 + 0.394607i \(0.129119\pi\)
\(3\) 0 0
\(4\) 4.75428 2.37714
\(5\) 3.50938 1.56944 0.784721 0.619849i \(-0.212806\pi\)
0.784721 + 0.619849i \(0.212806\pi\)
\(6\) 0 0
\(7\) 0.252288 0.0953558 0.0476779 0.998863i \(-0.484818\pi\)
0.0476779 + 0.998863i \(0.484818\pi\)
\(8\) 7.15810 2.53077
\(9\) 0 0
\(10\) 9.12053 2.88416
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.314950 0.0873514 0.0436757 0.999046i \(-0.486093\pi\)
0.0436757 + 0.999046i \(0.486093\pi\)
\(14\) 0.655671 0.175235
\(15\) 0 0
\(16\) 9.09463 2.27366
\(17\) −3.13246 −0.759733 −0.379866 0.925041i \(-0.624030\pi\)
−0.379866 + 0.925041i \(0.624030\pi\)
\(18\) 0 0
\(19\) 0.841231 0.192992 0.0964958 0.995333i \(-0.469237\pi\)
0.0964958 + 0.995333i \(0.469237\pi\)
\(20\) 16.6846 3.73079
\(21\) 0 0
\(22\) −2.59890 −0.554087
\(23\) −1.45222 −0.302809 −0.151405 0.988472i \(-0.548380\pi\)
−0.151405 + 0.988472i \(0.548380\pi\)
\(24\) 0 0
\(25\) 7.31576 1.46315
\(26\) 0.818524 0.160526
\(27\) 0 0
\(28\) 1.19945 0.226674
\(29\) 6.76829 1.25684 0.628420 0.777874i \(-0.283702\pi\)
0.628420 + 0.777874i \(0.283702\pi\)
\(30\) 0 0
\(31\) −7.07730 −1.27112 −0.635560 0.772051i \(-0.719231\pi\)
−0.635560 + 0.772051i \(0.719231\pi\)
\(32\) 9.31982 1.64753
\(33\) 0 0
\(34\) −8.14095 −1.39616
\(35\) 0.885374 0.149656
\(36\) 0 0
\(37\) 8.89628 1.46254 0.731270 0.682088i \(-0.238928\pi\)
0.731270 + 0.682088i \(0.238928\pi\)
\(38\) 2.18627 0.354661
\(39\) 0 0
\(40\) 25.1205 3.97190
\(41\) 8.43604 1.31749 0.658744 0.752367i \(-0.271088\pi\)
0.658744 + 0.752367i \(0.271088\pi\)
\(42\) 0 0
\(43\) 1.80409 0.275121 0.137561 0.990493i \(-0.456074\pi\)
0.137561 + 0.990493i \(0.456074\pi\)
\(44\) −4.75428 −0.716735
\(45\) 0 0
\(46\) −3.77418 −0.556473
\(47\) −2.30017 −0.335514 −0.167757 0.985828i \(-0.553652\pi\)
−0.167757 + 0.985828i \(0.553652\pi\)
\(48\) 0 0
\(49\) −6.93635 −0.990907
\(50\) 19.0129 2.68883
\(51\) 0 0
\(52\) 1.49736 0.207647
\(53\) 4.41103 0.605901 0.302951 0.953006i \(-0.402028\pi\)
0.302951 + 0.953006i \(0.402028\pi\)
\(54\) 0 0
\(55\) −3.50938 −0.473205
\(56\) 1.80590 0.241324
\(57\) 0 0
\(58\) 17.5901 2.30969
\(59\) −4.48036 −0.583293 −0.291647 0.956526i \(-0.594203\pi\)
−0.291647 + 0.956526i \(0.594203\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −18.3932 −2.33594
\(63\) 0 0
\(64\) 6.03203 0.754004
\(65\) 1.10528 0.137093
\(66\) 0 0
\(67\) −0.0414722 −0.00506664 −0.00253332 0.999997i \(-0.500806\pi\)
−0.00253332 + 0.999997i \(0.500806\pi\)
\(68\) −14.8926 −1.80599
\(69\) 0 0
\(70\) 2.30100 0.275022
\(71\) 9.80832 1.16403 0.582017 0.813177i \(-0.302264\pi\)
0.582017 + 0.813177i \(0.302264\pi\)
\(72\) 0 0
\(73\) 8.90879 1.04269 0.521347 0.853345i \(-0.325430\pi\)
0.521347 + 0.853345i \(0.325430\pi\)
\(74\) 23.1205 2.68771
\(75\) 0 0
\(76\) 3.99945 0.458768
\(77\) −0.252288 −0.0287509
\(78\) 0 0
\(79\) −8.51680 −0.958215 −0.479108 0.877756i \(-0.659040\pi\)
−0.479108 + 0.877756i \(0.659040\pi\)
\(80\) 31.9165 3.56837
\(81\) 0 0
\(82\) 21.9244 2.42115
\(83\) −11.2151 −1.23102 −0.615509 0.788130i \(-0.711050\pi\)
−0.615509 + 0.788130i \(0.711050\pi\)
\(84\) 0 0
\(85\) −10.9930 −1.19236
\(86\) 4.68865 0.505590
\(87\) 0 0
\(88\) −7.15810 −0.763056
\(89\) −5.58487 −0.591995 −0.295997 0.955189i \(-0.595652\pi\)
−0.295997 + 0.955189i \(0.595652\pi\)
\(90\) 0 0
\(91\) 0.0794581 0.00832947
\(92\) −6.90427 −0.719820
\(93\) 0 0
\(94\) −5.97790 −0.616573
\(95\) 2.95220 0.302889
\(96\) 0 0
\(97\) 7.54000 0.765571 0.382786 0.923837i \(-0.374965\pi\)
0.382786 + 0.923837i \(0.374965\pi\)
\(98\) −18.0269 −1.82099
\(99\) 0 0
\(100\) 34.7812 3.47812
\(101\) 5.85246 0.582342 0.291171 0.956671i \(-0.405955\pi\)
0.291171 + 0.956671i \(0.405955\pi\)
\(102\) 0 0
\(103\) −17.3802 −1.71253 −0.856263 0.516541i \(-0.827219\pi\)
−0.856263 + 0.516541i \(0.827219\pi\)
\(104\) 2.25444 0.221066
\(105\) 0 0
\(106\) 11.4638 1.11346
\(107\) 7.20395 0.696432 0.348216 0.937414i \(-0.386788\pi\)
0.348216 + 0.937414i \(0.386788\pi\)
\(108\) 0 0
\(109\) 3.06602 0.293672 0.146836 0.989161i \(-0.453091\pi\)
0.146836 + 0.989161i \(0.453091\pi\)
\(110\) −9.12053 −0.869608
\(111\) 0 0
\(112\) 2.29446 0.216806
\(113\) 12.0330 1.13197 0.565986 0.824415i \(-0.308496\pi\)
0.565986 + 0.824415i \(0.308496\pi\)
\(114\) 0 0
\(115\) −5.09640 −0.475242
\(116\) 32.1784 2.98769
\(117\) 0 0
\(118\) −11.6440 −1.07192
\(119\) −0.790281 −0.0724450
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.59890 −0.235293
\(123\) 0 0
\(124\) −33.6475 −3.02163
\(125\) 8.12687 0.726889
\(126\) 0 0
\(127\) −7.56103 −0.670933 −0.335466 0.942052i \(-0.608894\pi\)
−0.335466 + 0.942052i \(0.608894\pi\)
\(128\) −2.96300 −0.261895
\(129\) 0 0
\(130\) 2.87251 0.251936
\(131\) −10.0905 −0.881613 −0.440807 0.897602i \(-0.645308\pi\)
−0.440807 + 0.897602i \(0.645308\pi\)
\(132\) 0 0
\(133\) 0.212232 0.0184029
\(134\) −0.107782 −0.00931097
\(135\) 0 0
\(136\) −22.4225 −1.92271
\(137\) −11.2651 −0.962444 −0.481222 0.876599i \(-0.659807\pi\)
−0.481222 + 0.876599i \(0.659807\pi\)
\(138\) 0 0
\(139\) 10.2656 0.870720 0.435360 0.900257i \(-0.356621\pi\)
0.435360 + 0.900257i \(0.356621\pi\)
\(140\) 4.20932 0.355752
\(141\) 0 0
\(142\) 25.4908 2.13914
\(143\) −0.314950 −0.0263374
\(144\) 0 0
\(145\) 23.7525 1.97254
\(146\) 23.1530 1.91616
\(147\) 0 0
\(148\) 42.2954 3.47666
\(149\) −16.9755 −1.39069 −0.695345 0.718676i \(-0.744748\pi\)
−0.695345 + 0.718676i \(0.744748\pi\)
\(150\) 0 0
\(151\) −10.4087 −0.847044 −0.423522 0.905886i \(-0.639206\pi\)
−0.423522 + 0.905886i \(0.639206\pi\)
\(152\) 6.02161 0.488417
\(153\) 0 0
\(154\) −0.655671 −0.0528355
\(155\) −24.8369 −1.99495
\(156\) 0 0
\(157\) −15.3822 −1.22764 −0.613818 0.789448i \(-0.710367\pi\)
−0.613818 + 0.789448i \(0.710367\pi\)
\(158\) −22.1343 −1.76091
\(159\) 0 0
\(160\) 32.7068 2.58570
\(161\) −0.366378 −0.0288746
\(162\) 0 0
\(163\) 0.539069 0.0422232 0.0211116 0.999777i \(-0.493279\pi\)
0.0211116 + 0.999777i \(0.493279\pi\)
\(164\) 40.1073 3.13185
\(165\) 0 0
\(166\) −29.1470 −2.26224
\(167\) −21.2985 −1.64813 −0.824063 0.566498i \(-0.808298\pi\)
−0.824063 + 0.566498i \(0.808298\pi\)
\(168\) 0 0
\(169\) −12.9008 −0.992370
\(170\) −28.5697 −2.19119
\(171\) 0 0
\(172\) 8.57715 0.654002
\(173\) −20.9450 −1.59242 −0.796209 0.605022i \(-0.793164\pi\)
−0.796209 + 0.605022i \(0.793164\pi\)
\(174\) 0 0
\(175\) 1.84568 0.139520
\(176\) −9.09463 −0.685533
\(177\) 0 0
\(178\) −14.5145 −1.08791
\(179\) 2.59129 0.193682 0.0968409 0.995300i \(-0.469126\pi\)
0.0968409 + 0.995300i \(0.469126\pi\)
\(180\) 0 0
\(181\) −10.8521 −0.806634 −0.403317 0.915060i \(-0.632143\pi\)
−0.403317 + 0.915060i \(0.632143\pi\)
\(182\) 0.206504 0.0153071
\(183\) 0 0
\(184\) −10.3952 −0.766341
\(185\) 31.2204 2.29537
\(186\) 0 0
\(187\) 3.13246 0.229068
\(188\) −10.9356 −0.797563
\(189\) 0 0
\(190\) 7.67247 0.556620
\(191\) −14.8806 −1.07672 −0.538362 0.842714i \(-0.680957\pi\)
−0.538362 + 0.842714i \(0.680957\pi\)
\(192\) 0 0
\(193\) −0.222935 −0.0160472 −0.00802362 0.999968i \(-0.502554\pi\)
−0.00802362 + 0.999968i \(0.502554\pi\)
\(194\) 19.5957 1.40689
\(195\) 0 0
\(196\) −32.9774 −2.35553
\(197\) −2.16031 −0.153916 −0.0769580 0.997034i \(-0.524521\pi\)
−0.0769580 + 0.997034i \(0.524521\pi\)
\(198\) 0 0
\(199\) 24.6468 1.74716 0.873581 0.486678i \(-0.161792\pi\)
0.873581 + 0.486678i \(0.161792\pi\)
\(200\) 52.3669 3.70290
\(201\) 0 0
\(202\) 15.2100 1.07017
\(203\) 1.70756 0.119847
\(204\) 0 0
\(205\) 29.6053 2.06772
\(206\) −45.1695 −3.14711
\(207\) 0 0
\(208\) 2.86435 0.198607
\(209\) −0.841231 −0.0581891
\(210\) 0 0
\(211\) 12.6063 0.867851 0.433926 0.900949i \(-0.357128\pi\)
0.433926 + 0.900949i \(0.357128\pi\)
\(212\) 20.9713 1.44031
\(213\) 0 0
\(214\) 18.7223 1.27983
\(215\) 6.33124 0.431787
\(216\) 0 0
\(217\) −1.78552 −0.121209
\(218\) 7.96828 0.539680
\(219\) 0 0
\(220\) −16.6846 −1.12487
\(221\) −0.986568 −0.0663637
\(222\) 0 0
\(223\) 10.5825 0.708657 0.354329 0.935121i \(-0.384709\pi\)
0.354329 + 0.935121i \(0.384709\pi\)
\(224\) 2.35128 0.157101
\(225\) 0 0
\(226\) 31.2726 2.08023
\(227\) −9.60279 −0.637359 −0.318680 0.947862i \(-0.603239\pi\)
−0.318680 + 0.947862i \(0.603239\pi\)
\(228\) 0 0
\(229\) 4.99502 0.330080 0.165040 0.986287i \(-0.447225\pi\)
0.165040 + 0.986287i \(0.447225\pi\)
\(230\) −13.2450 −0.873352
\(231\) 0 0
\(232\) 48.4481 3.18077
\(233\) −18.4718 −1.21013 −0.605064 0.796177i \(-0.706853\pi\)
−0.605064 + 0.796177i \(0.706853\pi\)
\(234\) 0 0
\(235\) −8.07216 −0.526569
\(236\) −21.3009 −1.38657
\(237\) 0 0
\(238\) −2.05386 −0.133132
\(239\) −6.77945 −0.438526 −0.219263 0.975666i \(-0.570365\pi\)
−0.219263 + 0.975666i \(0.570365\pi\)
\(240\) 0 0
\(241\) 15.6701 1.00940 0.504701 0.863294i \(-0.331603\pi\)
0.504701 + 0.863294i \(0.331603\pi\)
\(242\) 2.59890 0.167064
\(243\) 0 0
\(244\) −4.75428 −0.304362
\(245\) −24.3423 −1.55517
\(246\) 0 0
\(247\) 0.264946 0.0168581
\(248\) −50.6600 −3.21691
\(249\) 0 0
\(250\) 21.1209 1.33580
\(251\) −15.5439 −0.981120 −0.490560 0.871407i \(-0.663208\pi\)
−0.490560 + 0.871407i \(0.663208\pi\)
\(252\) 0 0
\(253\) 1.45222 0.0913004
\(254\) −19.6504 −1.23297
\(255\) 0 0
\(256\) −19.7646 −1.23529
\(257\) −17.3834 −1.08434 −0.542172 0.840267i \(-0.682398\pi\)
−0.542172 + 0.840267i \(0.682398\pi\)
\(258\) 0 0
\(259\) 2.24442 0.139462
\(260\) 5.25481 0.325890
\(261\) 0 0
\(262\) −26.2243 −1.62014
\(263\) 13.4938 0.832062 0.416031 0.909350i \(-0.363421\pi\)
0.416031 + 0.909350i \(0.363421\pi\)
\(264\) 0 0
\(265\) 15.4800 0.950928
\(266\) 0.551571 0.0338190
\(267\) 0 0
\(268\) −0.197171 −0.0120441
\(269\) −17.3064 −1.05519 −0.527594 0.849497i \(-0.676906\pi\)
−0.527594 + 0.849497i \(0.676906\pi\)
\(270\) 0 0
\(271\) −2.53256 −0.153842 −0.0769210 0.997037i \(-0.524509\pi\)
−0.0769210 + 0.997037i \(0.524509\pi\)
\(272\) −28.4885 −1.72737
\(273\) 0 0
\(274\) −29.2769 −1.76868
\(275\) −7.31576 −0.441157
\(276\) 0 0
\(277\) 23.6178 1.41906 0.709529 0.704676i \(-0.248908\pi\)
0.709529 + 0.704676i \(0.248908\pi\)
\(278\) 26.6794 1.60012
\(279\) 0 0
\(280\) 6.33760 0.378744
\(281\) −10.4991 −0.626323 −0.313162 0.949700i \(-0.601388\pi\)
−0.313162 + 0.949700i \(0.601388\pi\)
\(282\) 0 0
\(283\) 7.65586 0.455093 0.227547 0.973767i \(-0.426930\pi\)
0.227547 + 0.973767i \(0.426930\pi\)
\(284\) 46.6315 2.76707
\(285\) 0 0
\(286\) −0.818524 −0.0484003
\(287\) 2.12831 0.125630
\(288\) 0 0
\(289\) −7.18771 −0.422806
\(290\) 61.7304 3.62493
\(291\) 0 0
\(292\) 42.3549 2.47863
\(293\) −13.3177 −0.778026 −0.389013 0.921232i \(-0.627184\pi\)
−0.389013 + 0.921232i \(0.627184\pi\)
\(294\) 0 0
\(295\) −15.7233 −0.915445
\(296\) 63.6805 3.70135
\(297\) 0 0
\(298\) −44.1177 −2.55567
\(299\) −0.457378 −0.0264508
\(300\) 0 0
\(301\) 0.455150 0.0262344
\(302\) −27.0510 −1.55661
\(303\) 0 0
\(304\) 7.65068 0.438797
\(305\) −3.50938 −0.200947
\(306\) 0 0
\(307\) −7.23744 −0.413063 −0.206531 0.978440i \(-0.566218\pi\)
−0.206531 + 0.978440i \(0.566218\pi\)
\(308\) −1.19945 −0.0683449
\(309\) 0 0
\(310\) −64.5487 −3.66612
\(311\) 8.38540 0.475492 0.237746 0.971327i \(-0.423591\pi\)
0.237746 + 0.971327i \(0.423591\pi\)
\(312\) 0 0
\(313\) 8.07210 0.456262 0.228131 0.973630i \(-0.426739\pi\)
0.228131 + 0.973630i \(0.426739\pi\)
\(314\) −39.9769 −2.25603
\(315\) 0 0
\(316\) −40.4913 −2.27781
\(317\) 26.3618 1.48063 0.740314 0.672261i \(-0.234677\pi\)
0.740314 + 0.672261i \(0.234677\pi\)
\(318\) 0 0
\(319\) −6.76829 −0.378952
\(320\) 21.1687 1.18337
\(321\) 0 0
\(322\) −0.952180 −0.0530629
\(323\) −2.63512 −0.146622
\(324\) 0 0
\(325\) 2.30410 0.127808
\(326\) 1.40099 0.0775935
\(327\) 0 0
\(328\) 60.3860 3.33426
\(329\) −0.580304 −0.0319932
\(330\) 0 0
\(331\) 20.6870 1.13706 0.568531 0.822662i \(-0.307512\pi\)
0.568531 + 0.822662i \(0.307512\pi\)
\(332\) −53.3198 −2.92630
\(333\) 0 0
\(334\) −55.3526 −3.02876
\(335\) −0.145542 −0.00795180
\(336\) 0 0
\(337\) 2.66866 0.145371 0.0726856 0.997355i \(-0.476843\pi\)
0.0726856 + 0.997355i \(0.476843\pi\)
\(338\) −33.5279 −1.82368
\(339\) 0 0
\(340\) −52.2638 −2.83440
\(341\) 7.07730 0.383257
\(342\) 0 0
\(343\) −3.51597 −0.189845
\(344\) 12.9139 0.696269
\(345\) 0 0
\(346\) −54.4339 −2.92638
\(347\) 10.6683 0.572707 0.286354 0.958124i \(-0.407557\pi\)
0.286354 + 0.958124i \(0.407557\pi\)
\(348\) 0 0
\(349\) 25.9644 1.38984 0.694922 0.719085i \(-0.255439\pi\)
0.694922 + 0.719085i \(0.255439\pi\)
\(350\) 4.79673 0.256396
\(351\) 0 0
\(352\) −9.31982 −0.496748
\(353\) 29.9585 1.59453 0.797266 0.603629i \(-0.206279\pi\)
0.797266 + 0.603629i \(0.206279\pi\)
\(354\) 0 0
\(355\) 34.4211 1.82688
\(356\) −26.5520 −1.40725
\(357\) 0 0
\(358\) 6.73449 0.355929
\(359\) −22.7413 −1.20024 −0.600121 0.799909i \(-0.704881\pi\)
−0.600121 + 0.799909i \(0.704881\pi\)
\(360\) 0 0
\(361\) −18.2923 −0.962754
\(362\) −28.2036 −1.48235
\(363\) 0 0
\(364\) 0.377766 0.0198003
\(365\) 31.2643 1.63645
\(366\) 0 0
\(367\) −3.01744 −0.157509 −0.0787545 0.996894i \(-0.525094\pi\)
−0.0787545 + 0.996894i \(0.525094\pi\)
\(368\) −13.2074 −0.688484
\(369\) 0 0
\(370\) 81.1388 4.21821
\(371\) 1.11285 0.0577762
\(372\) 0 0
\(373\) −2.16528 −0.112114 −0.0560570 0.998428i \(-0.517853\pi\)
−0.0560570 + 0.998428i \(0.517853\pi\)
\(374\) 8.14095 0.420958
\(375\) 0 0
\(376\) −16.4648 −0.849108
\(377\) 2.13167 0.109787
\(378\) 0 0
\(379\) 4.56064 0.234264 0.117132 0.993116i \(-0.462630\pi\)
0.117132 + 0.993116i \(0.462630\pi\)
\(380\) 14.0356 0.720010
\(381\) 0 0
\(382\) −38.6732 −1.97869
\(383\) 21.2912 1.08793 0.543964 0.839109i \(-0.316923\pi\)
0.543964 + 0.839109i \(0.316923\pi\)
\(384\) 0 0
\(385\) −0.885374 −0.0451228
\(386\) −0.579386 −0.0294900
\(387\) 0 0
\(388\) 35.8473 1.81987
\(389\) 30.9732 1.57041 0.785203 0.619238i \(-0.212559\pi\)
0.785203 + 0.619238i \(0.212559\pi\)
\(390\) 0 0
\(391\) 4.54903 0.230054
\(392\) −49.6511 −2.50776
\(393\) 0 0
\(394\) −5.61444 −0.282851
\(395\) −29.8887 −1.50386
\(396\) 0 0
\(397\) 23.5944 1.18417 0.592085 0.805875i \(-0.298305\pi\)
0.592085 + 0.805875i \(0.298305\pi\)
\(398\) 64.0545 3.21076
\(399\) 0 0
\(400\) 66.5341 3.32670
\(401\) 20.4153 1.01949 0.509745 0.860326i \(-0.329740\pi\)
0.509745 + 0.860326i \(0.329740\pi\)
\(402\) 0 0
\(403\) −2.22900 −0.111034
\(404\) 27.8243 1.38431
\(405\) 0 0
\(406\) 4.43777 0.220243
\(407\) −8.89628 −0.440972
\(408\) 0 0
\(409\) −17.8248 −0.881379 −0.440689 0.897660i \(-0.645266\pi\)
−0.440689 + 0.897660i \(0.645266\pi\)
\(410\) 76.9411 3.79985
\(411\) 0 0
\(412\) −82.6305 −4.07091
\(413\) −1.13034 −0.0556204
\(414\) 0 0
\(415\) −39.3581 −1.93201
\(416\) 2.93528 0.143914
\(417\) 0 0
\(418\) −2.18627 −0.106934
\(419\) 5.42075 0.264821 0.132410 0.991195i \(-0.457728\pi\)
0.132410 + 0.991195i \(0.457728\pi\)
\(420\) 0 0
\(421\) 38.0113 1.85256 0.926279 0.376840i \(-0.122989\pi\)
0.926279 + 0.376840i \(0.122989\pi\)
\(422\) 32.7624 1.59485
\(423\) 0 0
\(424\) 31.5746 1.53340
\(425\) −22.9163 −1.11160
\(426\) 0 0
\(427\) −0.252288 −0.0122091
\(428\) 34.2496 1.65552
\(429\) 0 0
\(430\) 16.4543 0.793495
\(431\) −14.5523 −0.700961 −0.350480 0.936570i \(-0.613982\pi\)
−0.350480 + 0.936570i \(0.613982\pi\)
\(432\) 0 0
\(433\) −1.86310 −0.0895346 −0.0447673 0.998997i \(-0.514255\pi\)
−0.0447673 + 0.998997i \(0.514255\pi\)
\(434\) −4.64038 −0.222745
\(435\) 0 0
\(436\) 14.5767 0.698099
\(437\) −1.22165 −0.0584396
\(438\) 0 0
\(439\) 0.643741 0.0307241 0.0153620 0.999882i \(-0.495110\pi\)
0.0153620 + 0.999882i \(0.495110\pi\)
\(440\) −25.1205 −1.19757
\(441\) 0 0
\(442\) −2.56399 −0.121957
\(443\) 18.0271 0.856492 0.428246 0.903662i \(-0.359132\pi\)
0.428246 + 0.903662i \(0.359132\pi\)
\(444\) 0 0
\(445\) −19.5994 −0.929102
\(446\) 27.5029 1.30230
\(447\) 0 0
\(448\) 1.52181 0.0718987
\(449\) −27.3932 −1.29277 −0.646383 0.763013i \(-0.723719\pi\)
−0.646383 + 0.763013i \(0.723719\pi\)
\(450\) 0 0
\(451\) −8.43604 −0.397237
\(452\) 57.2084 2.69086
\(453\) 0 0
\(454\) −24.9567 −1.17128
\(455\) 0.278849 0.0130726
\(456\) 0 0
\(457\) −25.1056 −1.17439 −0.587196 0.809445i \(-0.699768\pi\)
−0.587196 + 0.809445i \(0.699768\pi\)
\(458\) 12.9816 0.606589
\(459\) 0 0
\(460\) −24.2297 −1.12972
\(461\) −38.5232 −1.79421 −0.897103 0.441822i \(-0.854332\pi\)
−0.897103 + 0.441822i \(0.854332\pi\)
\(462\) 0 0
\(463\) −3.29366 −0.153070 −0.0765348 0.997067i \(-0.524386\pi\)
−0.0765348 + 0.997067i \(0.524386\pi\)
\(464\) 61.5551 2.85762
\(465\) 0 0
\(466\) −48.0064 −2.22385
\(467\) 33.9437 1.57073 0.785364 0.619035i \(-0.212476\pi\)
0.785364 + 0.619035i \(0.212476\pi\)
\(468\) 0 0
\(469\) −0.0104629 −0.000483134 0
\(470\) −20.9787 −0.967676
\(471\) 0 0
\(472\) −32.0709 −1.47618
\(473\) −1.80409 −0.0829522
\(474\) 0 0
\(475\) 6.15424 0.282376
\(476\) −3.75722 −0.172212
\(477\) 0 0
\(478\) −17.6191 −0.805879
\(479\) −10.0038 −0.457084 −0.228542 0.973534i \(-0.573396\pi\)
−0.228542 + 0.973534i \(0.573396\pi\)
\(480\) 0 0
\(481\) 2.80188 0.127755
\(482\) 40.7251 1.85498
\(483\) 0 0
\(484\) 4.75428 0.216104
\(485\) 26.4607 1.20152
\(486\) 0 0
\(487\) −10.7252 −0.486005 −0.243002 0.970026i \(-0.578132\pi\)
−0.243002 + 0.970026i \(0.578132\pi\)
\(488\) −7.15810 −0.324032
\(489\) 0 0
\(490\) −63.2632 −2.85794
\(491\) −8.35754 −0.377170 −0.188585 0.982057i \(-0.560390\pi\)
−0.188585 + 0.982057i \(0.560390\pi\)
\(492\) 0 0
\(493\) −21.2014 −0.954863
\(494\) 0.688567 0.0309801
\(495\) 0 0
\(496\) −64.3654 −2.89009
\(497\) 2.47452 0.110997
\(498\) 0 0
\(499\) 11.9000 0.532715 0.266358 0.963874i \(-0.414180\pi\)
0.266358 + 0.963874i \(0.414180\pi\)
\(500\) 38.6374 1.72792
\(501\) 0 0
\(502\) −40.3970 −1.80300
\(503\) 21.4919 0.958278 0.479139 0.877739i \(-0.340949\pi\)
0.479139 + 0.877739i \(0.340949\pi\)
\(504\) 0 0
\(505\) 20.5385 0.913952
\(506\) 3.77418 0.167783
\(507\) 0 0
\(508\) −35.9473 −1.59490
\(509\) 37.0220 1.64097 0.820485 0.571668i \(-0.193703\pi\)
0.820485 + 0.571668i \(0.193703\pi\)
\(510\) 0 0
\(511\) 2.24758 0.0994270
\(512\) −45.4402 −2.00819
\(513\) 0 0
\(514\) −45.1776 −1.99270
\(515\) −60.9939 −2.68771
\(516\) 0 0
\(517\) 2.30017 0.101161
\(518\) 5.83303 0.256289
\(519\) 0 0
\(520\) 7.91170 0.346951
\(521\) −25.1196 −1.10051 −0.550256 0.834996i \(-0.685470\pi\)
−0.550256 + 0.834996i \(0.685470\pi\)
\(522\) 0 0
\(523\) 29.0802 1.27159 0.635794 0.771859i \(-0.280673\pi\)
0.635794 + 0.771859i \(0.280673\pi\)
\(524\) −47.9732 −2.09572
\(525\) 0 0
\(526\) 35.0690 1.52908
\(527\) 22.1693 0.965712
\(528\) 0 0
\(529\) −20.8911 −0.908307
\(530\) 40.2309 1.74752
\(531\) 0 0
\(532\) 1.00901 0.0437462
\(533\) 2.65693 0.115084
\(534\) 0 0
\(535\) 25.2814 1.09301
\(536\) −0.296863 −0.0128225
\(537\) 0 0
\(538\) −44.9775 −1.93912
\(539\) 6.93635 0.298770
\(540\) 0 0
\(541\) −0.392586 −0.0168786 −0.00843929 0.999964i \(-0.502686\pi\)
−0.00843929 + 0.999964i \(0.502686\pi\)
\(542\) −6.58186 −0.282715
\(543\) 0 0
\(544\) −29.1939 −1.25168
\(545\) 10.7598 0.460901
\(546\) 0 0
\(547\) 15.1084 0.645990 0.322995 0.946401i \(-0.395310\pi\)
0.322995 + 0.946401i \(0.395310\pi\)
\(548\) −53.5575 −2.28786
\(549\) 0 0
\(550\) −19.0129 −0.810713
\(551\) 5.69370 0.242560
\(552\) 0 0
\(553\) −2.14869 −0.0913714
\(554\) 61.3804 2.60780
\(555\) 0 0
\(556\) 48.8057 2.06982
\(557\) 44.8252 1.89931 0.949653 0.313304i \(-0.101436\pi\)
0.949653 + 0.313304i \(0.101436\pi\)
\(558\) 0 0
\(559\) 0.568198 0.0240322
\(560\) 8.05215 0.340265
\(561\) 0 0
\(562\) −27.2861 −1.15099
\(563\) −18.8989 −0.796492 −0.398246 0.917279i \(-0.630381\pi\)
−0.398246 + 0.917279i \(0.630381\pi\)
\(564\) 0 0
\(565\) 42.2285 1.77657
\(566\) 19.8968 0.836325
\(567\) 0 0
\(568\) 70.2089 2.94590
\(569\) −2.20824 −0.0925744 −0.0462872 0.998928i \(-0.514739\pi\)
−0.0462872 + 0.998928i \(0.514739\pi\)
\(570\) 0 0
\(571\) −12.1493 −0.508431 −0.254216 0.967148i \(-0.581817\pi\)
−0.254216 + 0.967148i \(0.581817\pi\)
\(572\) −1.49736 −0.0626078
\(573\) 0 0
\(574\) 5.53126 0.230870
\(575\) −10.6241 −0.443056
\(576\) 0 0
\(577\) −11.7039 −0.487238 −0.243619 0.969871i \(-0.578335\pi\)
−0.243619 + 0.969871i \(0.578335\pi\)
\(578\) −18.6801 −0.776991
\(579\) 0 0
\(580\) 112.926 4.68900
\(581\) −2.82944 −0.117385
\(582\) 0 0
\(583\) −4.41103 −0.182686
\(584\) 63.7700 2.63882
\(585\) 0 0
\(586\) −34.6113 −1.42978
\(587\) −32.9310 −1.35921 −0.679604 0.733580i \(-0.737848\pi\)
−0.679604 + 0.733580i \(0.737848\pi\)
\(588\) 0 0
\(589\) −5.95364 −0.245315
\(590\) −40.8632 −1.68231
\(591\) 0 0
\(592\) 80.9084 3.32531
\(593\) −35.1041 −1.44155 −0.720776 0.693168i \(-0.756214\pi\)
−0.720776 + 0.693168i \(0.756214\pi\)
\(594\) 0 0
\(595\) −2.77340 −0.113698
\(596\) −80.7064 −3.30586
\(597\) 0 0
\(598\) −1.18868 −0.0486087
\(599\) 35.7935 1.46248 0.731241 0.682119i \(-0.238941\pi\)
0.731241 + 0.682119i \(0.238941\pi\)
\(600\) 0 0
\(601\) −8.14840 −0.332380 −0.166190 0.986094i \(-0.553147\pi\)
−0.166190 + 0.986094i \(0.553147\pi\)
\(602\) 1.18289 0.0482110
\(603\) 0 0
\(604\) −49.4857 −2.01354
\(605\) 3.50938 0.142677
\(606\) 0 0
\(607\) 23.1424 0.939320 0.469660 0.882847i \(-0.344377\pi\)
0.469660 + 0.882847i \(0.344377\pi\)
\(608\) 7.84012 0.317959
\(609\) 0 0
\(610\) −9.12053 −0.369279
\(611\) −0.724437 −0.0293076
\(612\) 0 0
\(613\) −4.11938 −0.166380 −0.0831901 0.996534i \(-0.526511\pi\)
−0.0831901 + 0.996534i \(0.526511\pi\)
\(614\) −18.8094 −0.759085
\(615\) 0 0
\(616\) −1.80590 −0.0727619
\(617\) 25.0455 1.00829 0.504146 0.863618i \(-0.331807\pi\)
0.504146 + 0.863618i \(0.331807\pi\)
\(618\) 0 0
\(619\) 42.8615 1.72275 0.861374 0.507972i \(-0.169605\pi\)
0.861374 + 0.507972i \(0.169605\pi\)
\(620\) −118.082 −4.74228
\(621\) 0 0
\(622\) 21.7928 0.873812
\(623\) −1.40899 −0.0564502
\(624\) 0 0
\(625\) −8.05850 −0.322340
\(626\) 20.9786 0.838473
\(627\) 0 0
\(628\) −73.1314 −2.91826
\(629\) −27.8672 −1.11114
\(630\) 0 0
\(631\) −13.0422 −0.519204 −0.259602 0.965716i \(-0.583591\pi\)
−0.259602 + 0.965716i \(0.583591\pi\)
\(632\) −60.9641 −2.42502
\(633\) 0 0
\(634\) 68.5118 2.72095
\(635\) −26.5345 −1.05299
\(636\) 0 0
\(637\) −2.18460 −0.0865572
\(638\) −17.5901 −0.696399
\(639\) 0 0
\(640\) −10.3983 −0.411028
\(641\) −41.7027 −1.64716 −0.823580 0.567201i \(-0.808026\pi\)
−0.823580 + 0.567201i \(0.808026\pi\)
\(642\) 0 0
\(643\) 18.6948 0.737249 0.368625 0.929578i \(-0.379829\pi\)
0.368625 + 0.929578i \(0.379829\pi\)
\(644\) −1.74186 −0.0686391
\(645\) 0 0
\(646\) −6.84841 −0.269447
\(647\) −37.1905 −1.46211 −0.731055 0.682319i \(-0.760972\pi\)
−0.731055 + 0.682319i \(0.760972\pi\)
\(648\) 0 0
\(649\) 4.48036 0.175869
\(650\) 5.98812 0.234873
\(651\) 0 0
\(652\) 2.56289 0.100370
\(653\) −35.5250 −1.39020 −0.695100 0.718914i \(-0.744640\pi\)
−0.695100 + 0.718914i \(0.744640\pi\)
\(654\) 0 0
\(655\) −35.4115 −1.38364
\(656\) 76.7226 2.99551
\(657\) 0 0
\(658\) −1.50815 −0.0587939
\(659\) 25.4043 0.989613 0.494806 0.869003i \(-0.335239\pi\)
0.494806 + 0.869003i \(0.335239\pi\)
\(660\) 0 0
\(661\) −22.8515 −0.888819 −0.444409 0.895824i \(-0.646586\pi\)
−0.444409 + 0.895824i \(0.646586\pi\)
\(662\) 53.7635 2.08958
\(663\) 0 0
\(664\) −80.2789 −3.11543
\(665\) 0.744804 0.0288823
\(666\) 0 0
\(667\) −9.82906 −0.380583
\(668\) −101.259 −3.91783
\(669\) 0 0
\(670\) −0.378249 −0.0146130
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 24.2548 0.934954 0.467477 0.884005i \(-0.345163\pi\)
0.467477 + 0.884005i \(0.345163\pi\)
\(674\) 6.93558 0.267148
\(675\) 0 0
\(676\) −61.3341 −2.35900
\(677\) 17.7813 0.683390 0.341695 0.939811i \(-0.388999\pi\)
0.341695 + 0.939811i \(0.388999\pi\)
\(678\) 0 0
\(679\) 1.90225 0.0730017
\(680\) −78.6889 −3.01758
\(681\) 0 0
\(682\) 18.3932 0.704312
\(683\) −40.4477 −1.54769 −0.773844 0.633376i \(-0.781669\pi\)
−0.773844 + 0.633376i \(0.781669\pi\)
\(684\) 0 0
\(685\) −39.5336 −1.51050
\(686\) −9.13766 −0.348877
\(687\) 0 0
\(688\) 16.4075 0.625531
\(689\) 1.38925 0.0529264
\(690\) 0 0
\(691\) 49.4541 1.88132 0.940661 0.339349i \(-0.110207\pi\)
0.940661 + 0.339349i \(0.110207\pi\)
\(692\) −99.5783 −3.78540
\(693\) 0 0
\(694\) 27.7260 1.05246
\(695\) 36.0260 1.36654
\(696\) 0 0
\(697\) −26.4255 −1.00094
\(698\) 67.4789 2.55411
\(699\) 0 0
\(700\) 8.77486 0.331659
\(701\) 8.75971 0.330850 0.165425 0.986222i \(-0.447100\pi\)
0.165425 + 0.986222i \(0.447100\pi\)
\(702\) 0 0
\(703\) 7.48383 0.282258
\(704\) −6.03203 −0.227341
\(705\) 0 0
\(706\) 77.8592 2.93027
\(707\) 1.47651 0.0555297
\(708\) 0 0
\(709\) 1.59624 0.0599480 0.0299740 0.999551i \(-0.490458\pi\)
0.0299740 + 0.999551i \(0.490458\pi\)
\(710\) 89.4571 3.35726
\(711\) 0 0
\(712\) −39.9770 −1.49820
\(713\) 10.2778 0.384907
\(714\) 0 0
\(715\) −1.10528 −0.0413351
\(716\) 12.3197 0.460409
\(717\) 0 0
\(718\) −59.1025 −2.20568
\(719\) −24.2156 −0.903091 −0.451545 0.892248i \(-0.649127\pi\)
−0.451545 + 0.892248i \(0.649127\pi\)
\(720\) 0 0
\(721\) −4.38482 −0.163299
\(722\) −47.5399 −1.76925
\(723\) 0 0
\(724\) −51.5942 −1.91748
\(725\) 49.5152 1.83895
\(726\) 0 0
\(727\) 6.33810 0.235067 0.117534 0.993069i \(-0.462501\pi\)
0.117534 + 0.993069i \(0.462501\pi\)
\(728\) 0.568769 0.0210800
\(729\) 0 0
\(730\) 81.2529 3.00730
\(731\) −5.65124 −0.209019
\(732\) 0 0
\(733\) −23.5758 −0.870790 −0.435395 0.900239i \(-0.643391\pi\)
−0.435395 + 0.900239i \(0.643391\pi\)
\(734\) −7.84202 −0.289454
\(735\) 0 0
\(736\) −13.5345 −0.498887
\(737\) 0.0414722 0.00152765
\(738\) 0 0
\(739\) 35.2908 1.29819 0.649096 0.760706i \(-0.275147\pi\)
0.649096 + 0.760706i \(0.275147\pi\)
\(740\) 148.431 5.45642
\(741\) 0 0
\(742\) 2.89218 0.106175
\(743\) −37.4102 −1.37245 −0.686224 0.727390i \(-0.740733\pi\)
−0.686224 + 0.727390i \(0.740733\pi\)
\(744\) 0 0
\(745\) −59.5736 −2.18261
\(746\) −5.62735 −0.206032
\(747\) 0 0
\(748\) 14.8926 0.544527
\(749\) 1.81747 0.0664089
\(750\) 0 0
\(751\) −4.73838 −0.172906 −0.0864530 0.996256i \(-0.527553\pi\)
−0.0864530 + 0.996256i \(0.527553\pi\)
\(752\) −20.9191 −0.762843
\(753\) 0 0
\(754\) 5.54001 0.201755
\(755\) −36.5279 −1.32939
\(756\) 0 0
\(757\) 6.11168 0.222133 0.111066 0.993813i \(-0.464573\pi\)
0.111066 + 0.993813i \(0.464573\pi\)
\(758\) 11.8526 0.430507
\(759\) 0 0
\(760\) 21.1321 0.766543
\(761\) 33.3959 1.21060 0.605300 0.795998i \(-0.293053\pi\)
0.605300 + 0.795998i \(0.293053\pi\)
\(762\) 0 0
\(763\) 0.773520 0.0280033
\(764\) −70.7466 −2.55952
\(765\) 0 0
\(766\) 55.3336 1.99928
\(767\) −1.41109 −0.0509515
\(768\) 0 0
\(769\) −5.52203 −0.199130 −0.0995648 0.995031i \(-0.531745\pi\)
−0.0995648 + 0.995031i \(0.531745\pi\)
\(770\) −2.30100 −0.0829222
\(771\) 0 0
\(772\) −1.05990 −0.0381465
\(773\) −35.4489 −1.27501 −0.637504 0.770447i \(-0.720033\pi\)
−0.637504 + 0.770447i \(0.720033\pi\)
\(774\) 0 0
\(775\) −51.7758 −1.85984
\(776\) 53.9721 1.93749
\(777\) 0 0
\(778\) 80.4964 2.88593
\(779\) 7.09665 0.254264
\(780\) 0 0
\(781\) −9.80832 −0.350969
\(782\) 11.8225 0.422770
\(783\) 0 0
\(784\) −63.0835 −2.25298
\(785\) −53.9821 −1.92670
\(786\) 0 0
\(787\) 18.2222 0.649551 0.324776 0.945791i \(-0.394711\pi\)
0.324776 + 0.945791i \(0.394711\pi\)
\(788\) −10.2707 −0.365880
\(789\) 0 0
\(790\) −77.6778 −2.76365
\(791\) 3.03579 0.107940
\(792\) 0 0
\(793\) −0.314950 −0.0111842
\(794\) 61.3196 2.17615
\(795\) 0 0
\(796\) 117.178 4.15325
\(797\) 40.8082 1.44550 0.722751 0.691109i \(-0.242877\pi\)
0.722751 + 0.691109i \(0.242877\pi\)
\(798\) 0 0
\(799\) 7.20517 0.254901
\(800\) 68.1815 2.41058
\(801\) 0 0
\(802\) 53.0572 1.87352
\(803\) −8.90879 −0.314384
\(804\) 0 0
\(805\) −1.28576 −0.0453171
\(806\) −5.79294 −0.204047
\(807\) 0 0
\(808\) 41.8925 1.47377
\(809\) 47.5619 1.67219 0.836094 0.548586i \(-0.184834\pi\)
0.836094 + 0.548586i \(0.184834\pi\)
\(810\) 0 0
\(811\) −10.5129 −0.369157 −0.184578 0.982818i \(-0.559092\pi\)
−0.184578 + 0.982818i \(0.559092\pi\)
\(812\) 8.11821 0.284893
\(813\) 0 0
\(814\) −23.1205 −0.810375
\(815\) 1.89180 0.0662668
\(816\) 0 0
\(817\) 1.51766 0.0530961
\(818\) −46.3248 −1.61971
\(819\) 0 0
\(820\) 140.752 4.91526
\(821\) −21.5569 −0.752339 −0.376170 0.926551i \(-0.622759\pi\)
−0.376170 + 0.926551i \(0.622759\pi\)
\(822\) 0 0
\(823\) −5.18282 −0.180662 −0.0903309 0.995912i \(-0.528792\pi\)
−0.0903309 + 0.995912i \(0.528792\pi\)
\(824\) −124.409 −4.33401
\(825\) 0 0
\(826\) −2.93764 −0.102214
\(827\) −8.46918 −0.294502 −0.147251 0.989099i \(-0.547043\pi\)
−0.147251 + 0.989099i \(0.547043\pi\)
\(828\) 0 0
\(829\) 23.7804 0.825927 0.412964 0.910748i \(-0.364494\pi\)
0.412964 + 0.910748i \(0.364494\pi\)
\(830\) −102.288 −3.55046
\(831\) 0 0
\(832\) 1.89979 0.0658633
\(833\) 21.7278 0.752825
\(834\) 0 0
\(835\) −74.7445 −2.58664
\(836\) −3.99945 −0.138324
\(837\) 0 0
\(838\) 14.0880 0.486661
\(839\) −0.710743 −0.0245376 −0.0122688 0.999925i \(-0.503905\pi\)
−0.0122688 + 0.999925i \(0.503905\pi\)
\(840\) 0 0
\(841\) 16.8098 0.579647
\(842\) 98.7875 3.40444
\(843\) 0 0
\(844\) 59.9337 2.06300
\(845\) −45.2738 −1.55747
\(846\) 0 0
\(847\) 0.252288 0.00866871
\(848\) 40.1167 1.37761
\(849\) 0 0
\(850\) −59.5572 −2.04279
\(851\) −12.9194 −0.442871
\(852\) 0 0
\(853\) 24.6568 0.844231 0.422116 0.906542i \(-0.361288\pi\)
0.422116 + 0.906542i \(0.361288\pi\)
\(854\) −0.655671 −0.0224366
\(855\) 0 0
\(856\) 51.5666 1.76251
\(857\) 3.62125 0.123699 0.0618497 0.998085i \(-0.480300\pi\)
0.0618497 + 0.998085i \(0.480300\pi\)
\(858\) 0 0
\(859\) 14.1354 0.482295 0.241147 0.970488i \(-0.422476\pi\)
0.241147 + 0.970488i \(0.422476\pi\)
\(860\) 30.1005 1.02642
\(861\) 0 0
\(862\) −37.8200 −1.28816
\(863\) 10.6439 0.362324 0.181162 0.983453i \(-0.442014\pi\)
0.181162 + 0.983453i \(0.442014\pi\)
\(864\) 0 0
\(865\) −73.5039 −2.49921
\(866\) −4.84200 −0.164538
\(867\) 0 0
\(868\) −8.48885 −0.288130
\(869\) 8.51680 0.288913
\(870\) 0 0
\(871\) −0.0130617 −0.000442578 0
\(872\) 21.9469 0.743216
\(873\) 0 0
\(874\) −3.17496 −0.107395
\(875\) 2.05031 0.0693131
\(876\) 0 0
\(877\) 7.29331 0.246278 0.123139 0.992389i \(-0.460704\pi\)
0.123139 + 0.992389i \(0.460704\pi\)
\(878\) 1.67302 0.0564616
\(879\) 0 0
\(880\) −31.9165 −1.07591
\(881\) 12.8614 0.433311 0.216655 0.976248i \(-0.430485\pi\)
0.216655 + 0.976248i \(0.430485\pi\)
\(882\) 0 0
\(883\) 21.2952 0.716639 0.358319 0.933599i \(-0.383350\pi\)
0.358319 + 0.933599i \(0.383350\pi\)
\(884\) −4.69042 −0.157756
\(885\) 0 0
\(886\) 46.8505 1.57397
\(887\) −57.3228 −1.92471 −0.962356 0.271793i \(-0.912383\pi\)
−0.962356 + 0.271793i \(0.912383\pi\)
\(888\) 0 0
\(889\) −1.90756 −0.0639774
\(890\) −50.9370 −1.70741
\(891\) 0 0
\(892\) 50.3122 1.68458
\(893\) −1.93497 −0.0647513
\(894\) 0 0
\(895\) 9.09381 0.303973
\(896\) −0.747528 −0.0249732
\(897\) 0 0
\(898\) −71.1922 −2.37572
\(899\) −47.9012 −1.59759
\(900\) 0 0
\(901\) −13.8174 −0.460323
\(902\) −21.9244 −0.730003
\(903\) 0 0
\(904\) 86.1337 2.86476
\(905\) −38.0843 −1.26597
\(906\) 0 0
\(907\) 19.1793 0.636838 0.318419 0.947950i \(-0.396848\pi\)
0.318419 + 0.947950i \(0.396848\pi\)
\(908\) −45.6543 −1.51509
\(909\) 0 0
\(910\) 0.724700 0.0240236
\(911\) 10.8028 0.357912 0.178956 0.983857i \(-0.442728\pi\)
0.178956 + 0.983857i \(0.442728\pi\)
\(912\) 0 0
\(913\) 11.2151 0.371166
\(914\) −65.2470 −2.15818
\(915\) 0 0
\(916\) 23.7477 0.784648
\(917\) −2.54572 −0.0840670
\(918\) 0 0
\(919\) 18.6484 0.615155 0.307578 0.951523i \(-0.400482\pi\)
0.307578 + 0.951523i \(0.400482\pi\)
\(920\) −36.4806 −1.20273
\(921\) 0 0
\(922\) −100.118 −3.29721
\(923\) 3.08913 0.101680
\(924\) 0 0
\(925\) 65.0830 2.13992
\(926\) −8.55990 −0.281296
\(927\) 0 0
\(928\) 63.0793 2.07068
\(929\) 47.6052 1.56188 0.780938 0.624608i \(-0.214741\pi\)
0.780938 + 0.624608i \(0.214741\pi\)
\(930\) 0 0
\(931\) −5.83507 −0.191237
\(932\) −87.8201 −2.87664
\(933\) 0 0
\(934\) 88.2163 2.88652
\(935\) 10.9930 0.359509
\(936\) 0 0
\(937\) 9.10229 0.297359 0.148680 0.988885i \(-0.452498\pi\)
0.148680 + 0.988885i \(0.452498\pi\)
\(938\) −0.0271921 −0.000887855 0
\(939\) 0 0
\(940\) −38.3773 −1.25173
\(941\) 34.9570 1.13956 0.569782 0.821796i \(-0.307028\pi\)
0.569782 + 0.821796i \(0.307028\pi\)
\(942\) 0 0
\(943\) −12.2510 −0.398947
\(944\) −40.7472 −1.32621
\(945\) 0 0
\(946\) −4.68865 −0.152441
\(947\) 19.1383 0.621910 0.310955 0.950425i \(-0.399351\pi\)
0.310955 + 0.950425i \(0.399351\pi\)
\(948\) 0 0
\(949\) 2.80582 0.0910809
\(950\) 15.9943 0.518922
\(951\) 0 0
\(952\) −5.65691 −0.183342
\(953\) −45.7828 −1.48305 −0.741525 0.670925i \(-0.765897\pi\)
−0.741525 + 0.670925i \(0.765897\pi\)
\(954\) 0 0
\(955\) −52.2217 −1.68986
\(956\) −32.2314 −1.04244
\(957\) 0 0
\(958\) −25.9988 −0.839982
\(959\) −2.84205 −0.0917747
\(960\) 0 0
\(961\) 19.0881 0.615746
\(962\) 7.28182 0.234775
\(963\) 0 0
\(964\) 74.5002 2.39949
\(965\) −0.782365 −0.0251852
\(966\) 0 0
\(967\) −9.33277 −0.300122 −0.150061 0.988677i \(-0.547947\pi\)
−0.150061 + 0.988677i \(0.547947\pi\)
\(968\) 7.15810 0.230070
\(969\) 0 0
\(970\) 68.7688 2.20803
\(971\) −2.32628 −0.0746540 −0.0373270 0.999303i \(-0.511884\pi\)
−0.0373270 + 0.999303i \(0.511884\pi\)
\(972\) 0 0
\(973\) 2.58989 0.0830282
\(974\) −27.8737 −0.893131
\(975\) 0 0
\(976\) −9.09463 −0.291112
\(977\) 38.7073 1.23836 0.619178 0.785251i \(-0.287466\pi\)
0.619178 + 0.785251i \(0.287466\pi\)
\(978\) 0 0
\(979\) 5.58487 0.178493
\(980\) −115.730 −3.69686
\(981\) 0 0
\(982\) −21.7204 −0.693126
\(983\) 43.4665 1.38637 0.693184 0.720761i \(-0.256207\pi\)
0.693184 + 0.720761i \(0.256207\pi\)
\(984\) 0 0
\(985\) −7.58136 −0.241562
\(986\) −55.1003 −1.75475
\(987\) 0 0
\(988\) 1.25963 0.0400740
\(989\) −2.61994 −0.0833093
\(990\) 0 0
\(991\) 50.1426 1.59283 0.796417 0.604748i \(-0.206726\pi\)
0.796417 + 0.604748i \(0.206726\pi\)
\(992\) −65.9591 −2.09420
\(993\) 0 0
\(994\) 6.43103 0.203980
\(995\) 86.4949 2.74207
\(996\) 0 0
\(997\) −46.5501 −1.47426 −0.737128 0.675753i \(-0.763818\pi\)
−0.737128 + 0.675753i \(0.763818\pi\)
\(998\) 30.9268 0.978971
\(999\) 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.1.13 13
3.2 odd 2 2013.2.a.e.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.1 13 3.2 odd 2
6039.2.a.i.1.13 13 1.1 even 1 trivial