Properties

Label 1521.2.a.s.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.35690 q^{2} +3.55496 q^{4} +3.69202 q^{5} +0.801938 q^{7} +3.66487 q^{8} +O(q^{10})\) \(q+2.35690 q^{2} +3.55496 q^{4} +3.69202 q^{5} +0.801938 q^{7} +3.66487 q^{8} +8.70171 q^{10} -2.85086 q^{11} +1.89008 q^{14} +1.52781 q^{16} -2.93900 q^{17} +2.44504 q^{19} +13.1250 q^{20} -6.71917 q^{22} +7.78986 q^{23} +8.63102 q^{25} +2.85086 q^{28} -3.85086 q^{29} -2.34481 q^{31} -3.72886 q^{32} -6.92692 q^{34} +2.96077 q^{35} -7.44504 q^{37} +5.76271 q^{38} +13.5308 q^{40} -0.850855 q^{41} -1.61596 q^{43} -10.1347 q^{44} +18.3599 q^{46} +2.44504 q^{47} -6.35690 q^{49} +20.3424 q^{50} +9.96077 q^{53} -10.5254 q^{55} +2.93900 q^{56} -9.07606 q^{58} -5.38404 q^{59} -13.2567 q^{61} -5.52648 q^{62} -11.8442 q^{64} -14.3937 q^{67} -10.4480 q^{68} +6.97823 q^{70} +8.12498 q^{71} +11.8877 q^{73} -17.5472 q^{74} +8.69202 q^{76} -2.28621 q^{77} +5.40581 q^{79} +5.64071 q^{80} -2.00538 q^{82} +7.04892 q^{83} -10.8509 q^{85} -3.80864 q^{86} -10.4480 q^{88} +1.13169 q^{89} +27.6926 q^{92} +5.76271 q^{94} +9.02715 q^{95} +5.94438 q^{97} -14.9825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 11q^{4} + 6q^{5} - 2q^{7} + 12q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 11q^{4} + 6q^{5} - 2q^{7} + 12q^{8} - q^{10} + 5q^{11} + 5q^{14} + 11q^{16} + q^{17} + 7q^{19} + 15q^{20} - 9q^{22} + 11q^{25} - 5q^{28} + 2q^{29} + 16q^{31} + 22q^{32} + 8q^{34} - 4q^{35} - 22q^{37} + 3q^{40} + 11q^{41} - 15q^{43} + 16q^{44} + 7q^{46} + 7q^{47} - 15q^{49} - 3q^{50} + 17q^{53} + 3q^{55} - q^{56} - 12q^{58} - 6q^{59} - 13q^{61} + 2q^{62} - 11q^{67} + 13q^{68} + 24q^{70} - 6q^{73} - 15q^{74} + 21q^{76} - 15q^{77} + 3q^{79} - 20q^{80} - 3q^{82} + 12q^{83} - 19q^{85} - 29q^{86} + 13q^{88} + q^{89} - 7q^{92} + 21q^{95} + 5q^{97} - 29q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35690 1.66658 0.833289 0.552838i \(-0.186455\pi\)
0.833289 + 0.552838i \(0.186455\pi\)
\(3\) 0 0
\(4\) 3.55496 1.77748
\(5\) 3.69202 1.65112 0.825561 0.564313i \(-0.190859\pi\)
0.825561 + 0.564313i \(0.190859\pi\)
\(6\) 0 0
\(7\) 0.801938 0.303104 0.151552 0.988449i \(-0.451573\pi\)
0.151552 + 0.988449i \(0.451573\pi\)
\(8\) 3.66487 1.29573
\(9\) 0 0
\(10\) 8.70171 2.75172
\(11\) −2.85086 −0.859565 −0.429783 0.902932i \(-0.641410\pi\)
−0.429783 + 0.902932i \(0.641410\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.89008 0.505146
\(15\) 0 0
\(16\) 1.52781 0.381953
\(17\) −2.93900 −0.712812 −0.356406 0.934331i \(-0.615998\pi\)
−0.356406 + 0.934331i \(0.615998\pi\)
\(18\) 0 0
\(19\) 2.44504 0.560931 0.280466 0.959864i \(-0.409511\pi\)
0.280466 + 0.959864i \(0.409511\pi\)
\(20\) 13.1250 2.93484
\(21\) 0 0
\(22\) −6.71917 −1.43253
\(23\) 7.78986 1.62430 0.812149 0.583451i \(-0.198298\pi\)
0.812149 + 0.583451i \(0.198298\pi\)
\(24\) 0 0
\(25\) 8.63102 1.72620
\(26\) 0 0
\(27\) 0 0
\(28\) 2.85086 0.538761
\(29\) −3.85086 −0.715086 −0.357543 0.933897i \(-0.616385\pi\)
−0.357543 + 0.933897i \(0.616385\pi\)
\(30\) 0 0
\(31\) −2.34481 −0.421141 −0.210571 0.977579i \(-0.567532\pi\)
−0.210571 + 0.977579i \(0.567532\pi\)
\(32\) −3.72886 −0.659175
\(33\) 0 0
\(34\) −6.92692 −1.18796
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) −7.44504 −1.22396 −0.611979 0.790874i \(-0.709626\pi\)
−0.611979 + 0.790874i \(0.709626\pi\)
\(38\) 5.76271 0.934835
\(39\) 0 0
\(40\) 13.5308 2.13941
\(41\) −0.850855 −0.132881 −0.0664406 0.997790i \(-0.521164\pi\)
−0.0664406 + 0.997790i \(0.521164\pi\)
\(42\) 0 0
\(43\) −1.61596 −0.246431 −0.123216 0.992380i \(-0.539321\pi\)
−0.123216 + 0.992380i \(0.539321\pi\)
\(44\) −10.1347 −1.52786
\(45\) 0 0
\(46\) 18.3599 2.70702
\(47\) 2.44504 0.356646 0.178323 0.983972i \(-0.442933\pi\)
0.178323 + 0.983972i \(0.442933\pi\)
\(48\) 0 0
\(49\) −6.35690 −0.908128
\(50\) 20.3424 2.87685
\(51\) 0 0
\(52\) 0 0
\(53\) 9.96077 1.36822 0.684109 0.729380i \(-0.260191\pi\)
0.684109 + 0.729380i \(0.260191\pi\)
\(54\) 0 0
\(55\) −10.5254 −1.41925
\(56\) 2.93900 0.392741
\(57\) 0 0
\(58\) −9.07606 −1.19175
\(59\) −5.38404 −0.700943 −0.350471 0.936573i \(-0.613979\pi\)
−0.350471 + 0.936573i \(0.613979\pi\)
\(60\) 0 0
\(61\) −13.2567 −1.69734 −0.848671 0.528921i \(-0.822597\pi\)
−0.848671 + 0.528921i \(0.822597\pi\)
\(62\) −5.52648 −0.701864
\(63\) 0 0
\(64\) −11.8442 −1.48052
\(65\) 0 0
\(66\) 0 0
\(67\) −14.3937 −1.75847 −0.879237 0.476384i \(-0.841947\pi\)
−0.879237 + 0.476384i \(0.841947\pi\)
\(68\) −10.4480 −1.26701
\(69\) 0 0
\(70\) 6.97823 0.834058
\(71\) 8.12498 0.964258 0.482129 0.876100i \(-0.339864\pi\)
0.482129 + 0.876100i \(0.339864\pi\)
\(72\) 0 0
\(73\) 11.8877 1.39135 0.695674 0.718357i \(-0.255106\pi\)
0.695674 + 0.718357i \(0.255106\pi\)
\(74\) −17.5472 −2.03982
\(75\) 0 0
\(76\) 8.69202 0.997043
\(77\) −2.28621 −0.260538
\(78\) 0 0
\(79\) 5.40581 0.608202 0.304101 0.952640i \(-0.401644\pi\)
0.304101 + 0.952640i \(0.401644\pi\)
\(80\) 5.64071 0.630651
\(81\) 0 0
\(82\) −2.00538 −0.221457
\(83\) 7.04892 0.773719 0.386860 0.922139i \(-0.373560\pi\)
0.386860 + 0.922139i \(0.373560\pi\)
\(84\) 0 0
\(85\) −10.8509 −1.17694
\(86\) −3.80864 −0.410696
\(87\) 0 0
\(88\) −10.4480 −1.11376
\(89\) 1.13169 0.119959 0.0599793 0.998200i \(-0.480897\pi\)
0.0599793 + 0.998200i \(0.480897\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 27.6926 2.88715
\(93\) 0 0
\(94\) 5.76271 0.594378
\(95\) 9.02715 0.926166
\(96\) 0 0
\(97\) 5.94438 0.603560 0.301780 0.953378i \(-0.402419\pi\)
0.301780 + 0.953378i \(0.402419\pi\)
\(98\) −14.9825 −1.51347
\(99\) 0 0
\(100\) 30.6829 3.06829
\(101\) −4.62565 −0.460269 −0.230134 0.973159i \(-0.573917\pi\)
−0.230134 + 0.973159i \(0.573917\pi\)
\(102\) 0 0
\(103\) 1.20775 0.119003 0.0595016 0.998228i \(-0.481049\pi\)
0.0595016 + 0.998228i \(0.481049\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 23.4765 2.28024
\(107\) 9.52111 0.920440 0.460220 0.887805i \(-0.347771\pi\)
0.460220 + 0.887805i \(0.347771\pi\)
\(108\) 0 0
\(109\) 1.78448 0.170922 0.0854611 0.996342i \(-0.472764\pi\)
0.0854611 + 0.996342i \(0.472764\pi\)
\(110\) −24.8073 −2.36528
\(111\) 0 0
\(112\) 1.22521 0.115771
\(113\) −4.95108 −0.465759 −0.232879 0.972506i \(-0.574815\pi\)
−0.232879 + 0.972506i \(0.574815\pi\)
\(114\) 0 0
\(115\) 28.7603 2.68191
\(116\) −13.6896 −1.27105
\(117\) 0 0
\(118\) −12.6896 −1.16817
\(119\) −2.35690 −0.216056
\(120\) 0 0
\(121\) −2.87263 −0.261148
\(122\) −31.2446 −2.82875
\(123\) 0 0
\(124\) −8.33572 −0.748569
\(125\) 13.4058 1.19905
\(126\) 0 0
\(127\) 5.67025 0.503153 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(128\) −20.4577 −1.80822
\(129\) 0 0
\(130\) 0 0
\(131\) −18.2228 −1.59213 −0.796067 0.605208i \(-0.793090\pi\)
−0.796067 + 0.605208i \(0.793090\pi\)
\(132\) 0 0
\(133\) 1.96077 0.170020
\(134\) −33.9245 −2.93063
\(135\) 0 0
\(136\) −10.7711 −0.923612
\(137\) 9.45042 0.807404 0.403702 0.914891i \(-0.367723\pi\)
0.403702 + 0.914891i \(0.367723\pi\)
\(138\) 0 0
\(139\) 4.01507 0.340553 0.170277 0.985396i \(-0.445534\pi\)
0.170277 + 0.985396i \(0.445534\pi\)
\(140\) 10.5254 0.889560
\(141\) 0 0
\(142\) 19.1497 1.60701
\(143\) 0 0
\(144\) 0 0
\(145\) −14.2174 −1.18069
\(146\) 28.0180 2.31879
\(147\) 0 0
\(148\) −26.4668 −2.17556
\(149\) 19.4058 1.58979 0.794893 0.606750i \(-0.207527\pi\)
0.794893 + 0.606750i \(0.207527\pi\)
\(150\) 0 0
\(151\) 12.3623 1.00603 0.503014 0.864278i \(-0.332225\pi\)
0.503014 + 0.864278i \(0.332225\pi\)
\(152\) 8.96077 0.726815
\(153\) 0 0
\(154\) −5.38835 −0.434206
\(155\) −8.65710 −0.695355
\(156\) 0 0
\(157\) −18.6775 −1.49063 −0.745315 0.666712i \(-0.767701\pi\)
−0.745315 + 0.666712i \(0.767701\pi\)
\(158\) 12.7409 1.01361
\(159\) 0 0
\(160\) −13.7670 −1.08838
\(161\) 6.24698 0.492331
\(162\) 0 0
\(163\) −12.3394 −0.966499 −0.483250 0.875483i \(-0.660544\pi\)
−0.483250 + 0.875483i \(0.660544\pi\)
\(164\) −3.02475 −0.236194
\(165\) 0 0
\(166\) 16.6136 1.28946
\(167\) 11.4940 0.889429 0.444715 0.895672i \(-0.353305\pi\)
0.444715 + 0.895672i \(0.353305\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −25.5743 −1.96146
\(171\) 0 0
\(172\) −5.74466 −0.438026
\(173\) −12.1142 −0.921028 −0.460514 0.887653i \(-0.652335\pi\)
−0.460514 + 0.887653i \(0.652335\pi\)
\(174\) 0 0
\(175\) 6.92154 0.523219
\(176\) −4.35557 −0.328313
\(177\) 0 0
\(178\) 2.66727 0.199920
\(179\) 0.538565 0.0402542 0.0201271 0.999797i \(-0.493593\pi\)
0.0201271 + 0.999797i \(0.493593\pi\)
\(180\) 0 0
\(181\) −23.2838 −1.73067 −0.865336 0.501192i \(-0.832895\pi\)
−0.865336 + 0.501192i \(0.832895\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 28.5488 2.10465
\(185\) −27.4873 −2.02090
\(186\) 0 0
\(187\) 8.37867 0.612709
\(188\) 8.69202 0.633931
\(189\) 0 0
\(190\) 21.2760 1.54353
\(191\) −16.7657 −1.21312 −0.606561 0.795037i \(-0.707452\pi\)
−0.606561 + 0.795037i \(0.707452\pi\)
\(192\) 0 0
\(193\) −25.7439 −1.85309 −0.926544 0.376186i \(-0.877235\pi\)
−0.926544 + 0.376186i \(0.877235\pi\)
\(194\) 14.0103 1.00588
\(195\) 0 0
\(196\) −22.5985 −1.61418
\(197\) 21.4209 1.52617 0.763087 0.646296i \(-0.223683\pi\)
0.763087 + 0.646296i \(0.223683\pi\)
\(198\) 0 0
\(199\) 3.52781 0.250080 0.125040 0.992152i \(-0.460094\pi\)
0.125040 + 0.992152i \(0.460094\pi\)
\(200\) 31.6316 2.23669
\(201\) 0 0
\(202\) −10.9022 −0.767074
\(203\) −3.08815 −0.216745
\(204\) 0 0
\(205\) −3.14138 −0.219403
\(206\) 2.84654 0.198328
\(207\) 0 0
\(208\) 0 0
\(209\) −6.97046 −0.482157
\(210\) 0 0
\(211\) 1.21552 0.0836799 0.0418399 0.999124i \(-0.486678\pi\)
0.0418399 + 0.999124i \(0.486678\pi\)
\(212\) 35.4101 2.43198
\(213\) 0 0
\(214\) 22.4403 1.53398
\(215\) −5.96615 −0.406888
\(216\) 0 0
\(217\) −1.88040 −0.127650
\(218\) 4.20583 0.284855
\(219\) 0 0
\(220\) −37.4174 −2.52268
\(221\) 0 0
\(222\) 0 0
\(223\) 17.3884 1.16441 0.582205 0.813042i \(-0.302190\pi\)
0.582205 + 0.813042i \(0.302190\pi\)
\(224\) −2.99031 −0.199799
\(225\) 0 0
\(226\) −11.6692 −0.776223
\(227\) 17.4155 1.15591 0.577954 0.816070i \(-0.303851\pi\)
0.577954 + 0.816070i \(0.303851\pi\)
\(228\) 0 0
\(229\) −18.7603 −1.23972 −0.619858 0.784714i \(-0.712810\pi\)
−0.619858 + 0.784714i \(0.712810\pi\)
\(230\) 67.7851 4.46962
\(231\) 0 0
\(232\) −14.1129 −0.926557
\(233\) −3.95108 −0.258844 −0.129422 0.991590i \(-0.541312\pi\)
−0.129422 + 0.991590i \(0.541312\pi\)
\(234\) 0 0
\(235\) 9.02715 0.588866
\(236\) −19.1400 −1.24591
\(237\) 0 0
\(238\) −5.55496 −0.360074
\(239\) 0.818331 0.0529334 0.0264667 0.999650i \(-0.491574\pi\)
0.0264667 + 0.999650i \(0.491574\pi\)
\(240\) 0 0
\(241\) −6.03252 −0.388589 −0.194295 0.980943i \(-0.562242\pi\)
−0.194295 + 0.980943i \(0.562242\pi\)
\(242\) −6.77048 −0.435223
\(243\) 0 0
\(244\) −47.1269 −3.01699
\(245\) −23.4698 −1.49943
\(246\) 0 0
\(247\) 0 0
\(248\) −8.59345 −0.545685
\(249\) 0 0
\(250\) 31.5961 1.99831
\(251\) 26.8799 1.69665 0.848323 0.529479i \(-0.177613\pi\)
0.848323 + 0.529479i \(0.177613\pi\)
\(252\) 0 0
\(253\) −22.2078 −1.39619
\(254\) 13.3642 0.838544
\(255\) 0 0
\(256\) −24.5284 −1.53303
\(257\) 9.05323 0.564725 0.282362 0.959308i \(-0.408882\pi\)
0.282362 + 0.959308i \(0.408882\pi\)
\(258\) 0 0
\(259\) −5.97046 −0.370986
\(260\) 0 0
\(261\) 0 0
\(262\) −42.9493 −2.65342
\(263\) −23.1511 −1.42756 −0.713778 0.700372i \(-0.753017\pi\)
−0.713778 + 0.700372i \(0.753017\pi\)
\(264\) 0 0
\(265\) 36.7754 2.25909
\(266\) 4.62133 0.283352
\(267\) 0 0
\(268\) −51.1691 −3.12565
\(269\) 2.42088 0.147604 0.0738018 0.997273i \(-0.476487\pi\)
0.0738018 + 0.997273i \(0.476487\pi\)
\(270\) 0 0
\(271\) 21.4450 1.30269 0.651347 0.758780i \(-0.274204\pi\)
0.651347 + 0.758780i \(0.274204\pi\)
\(272\) −4.49024 −0.272261
\(273\) 0 0
\(274\) 22.2737 1.34560
\(275\) −24.6058 −1.48379
\(276\) 0 0
\(277\) −14.8073 −0.889685 −0.444843 0.895609i \(-0.646740\pi\)
−0.444843 + 0.895609i \(0.646740\pi\)
\(278\) 9.46309 0.567558
\(279\) 0 0
\(280\) 10.8509 0.648463
\(281\) 14.5036 0.865215 0.432608 0.901582i \(-0.357594\pi\)
0.432608 + 0.901582i \(0.357594\pi\)
\(282\) 0 0
\(283\) 25.6722 1.52605 0.763026 0.646368i \(-0.223713\pi\)
0.763026 + 0.646368i \(0.223713\pi\)
\(284\) 28.8840 1.71395
\(285\) 0 0
\(286\) 0 0
\(287\) −0.682333 −0.0402768
\(288\) 0 0
\(289\) −8.36227 −0.491898
\(290\) −33.5090 −1.96772
\(291\) 0 0
\(292\) 42.2602 2.47309
\(293\) −26.5230 −1.54949 −0.774746 0.632273i \(-0.782122\pi\)
−0.774746 + 0.632273i \(0.782122\pi\)
\(294\) 0 0
\(295\) −19.8780 −1.15734
\(296\) −27.2851 −1.58592
\(297\) 0 0
\(298\) 45.7375 2.64950
\(299\) 0 0
\(300\) 0 0
\(301\) −1.29590 −0.0746943
\(302\) 29.1366 1.67662
\(303\) 0 0
\(304\) 3.73556 0.214249
\(305\) −48.9439 −2.80252
\(306\) 0 0
\(307\) 8.24698 0.470680 0.235340 0.971913i \(-0.424380\pi\)
0.235340 + 0.971913i \(0.424380\pi\)
\(308\) −8.12737 −0.463100
\(309\) 0 0
\(310\) −20.4039 −1.15886
\(311\) 14.4179 0.817564 0.408782 0.912632i \(-0.365954\pi\)
0.408782 + 0.912632i \(0.365954\pi\)
\(312\) 0 0
\(313\) 14.2338 0.804544 0.402272 0.915520i \(-0.368221\pi\)
0.402272 + 0.915520i \(0.368221\pi\)
\(314\) −44.0210 −2.48425
\(315\) 0 0
\(316\) 19.2174 1.08107
\(317\) −6.84415 −0.384406 −0.192203 0.981355i \(-0.561563\pi\)
−0.192203 + 0.981355i \(0.561563\pi\)
\(318\) 0 0
\(319\) 10.9782 0.614663
\(320\) −43.7289 −2.44452
\(321\) 0 0
\(322\) 14.7235 0.820507
\(323\) −7.18598 −0.399839
\(324\) 0 0
\(325\) 0 0
\(326\) −29.0828 −1.61075
\(327\) 0 0
\(328\) −3.11828 −0.172178
\(329\) 1.96077 0.108101
\(330\) 0 0
\(331\) −9.44265 −0.519015 −0.259507 0.965741i \(-0.583560\pi\)
−0.259507 + 0.965741i \(0.583560\pi\)
\(332\) 25.0586 1.37527
\(333\) 0 0
\(334\) 27.0901 1.48230
\(335\) −53.1420 −2.90346
\(336\) 0 0
\(337\) 2.64310 0.143979 0.0719895 0.997405i \(-0.477065\pi\)
0.0719895 + 0.997405i \(0.477065\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −38.5743 −2.09199
\(341\) 6.68473 0.361998
\(342\) 0 0
\(343\) −10.7114 −0.578361
\(344\) −5.92228 −0.319308
\(345\) 0 0
\(346\) −28.5520 −1.53496
\(347\) 10.1588 0.545355 0.272677 0.962106i \(-0.412091\pi\)
0.272677 + 0.962106i \(0.412091\pi\)
\(348\) 0 0
\(349\) −10.4397 −0.558822 −0.279411 0.960172i \(-0.590139\pi\)
−0.279411 + 0.960172i \(0.590139\pi\)
\(350\) 16.3134 0.871986
\(351\) 0 0
\(352\) 10.6304 0.566604
\(353\) 18.2911 0.973538 0.486769 0.873531i \(-0.338175\pi\)
0.486769 + 0.873531i \(0.338175\pi\)
\(354\) 0 0
\(355\) 29.9976 1.59211
\(356\) 4.02310 0.213224
\(357\) 0 0
\(358\) 1.26934 0.0670867
\(359\) −15.2731 −0.806081 −0.403041 0.915182i \(-0.632047\pi\)
−0.403041 + 0.915182i \(0.632047\pi\)
\(360\) 0 0
\(361\) −13.0218 −0.685356
\(362\) −54.8775 −2.88430
\(363\) 0 0
\(364\) 0 0
\(365\) 43.8896 2.29729
\(366\) 0 0
\(367\) −22.2717 −1.16258 −0.581288 0.813698i \(-0.697451\pi\)
−0.581288 + 0.813698i \(0.697451\pi\)
\(368\) 11.9014 0.620405
\(369\) 0 0
\(370\) −64.7846 −3.36799
\(371\) 7.98792 0.414712
\(372\) 0 0
\(373\) 4.12631 0.213652 0.106826 0.994278i \(-0.465931\pi\)
0.106826 + 0.994278i \(0.465931\pi\)
\(374\) 19.7476 1.02113
\(375\) 0 0
\(376\) 8.96077 0.462116
\(377\) 0 0
\(378\) 0 0
\(379\) 10.7071 0.549986 0.274993 0.961446i \(-0.411324\pi\)
0.274993 + 0.961446i \(0.411324\pi\)
\(380\) 32.0911 1.64624
\(381\) 0 0
\(382\) −39.5150 −2.02176
\(383\) 6.52648 0.333488 0.166744 0.986000i \(-0.446675\pi\)
0.166744 + 0.986000i \(0.446675\pi\)
\(384\) 0 0
\(385\) −8.44073 −0.430179
\(386\) −60.6757 −3.08831
\(387\) 0 0
\(388\) 21.1320 1.07282
\(389\) 11.7922 0.597891 0.298945 0.954270i \(-0.403365\pi\)
0.298945 + 0.954270i \(0.403365\pi\)
\(390\) 0 0
\(391\) −22.8944 −1.15782
\(392\) −23.2972 −1.17669
\(393\) 0 0
\(394\) 50.4868 2.54349
\(395\) 19.9584 1.00422
\(396\) 0 0
\(397\) 12.5429 0.629509 0.314754 0.949173i \(-0.398078\pi\)
0.314754 + 0.949173i \(0.398078\pi\)
\(398\) 8.31468 0.416777
\(399\) 0 0
\(400\) 13.1866 0.659329
\(401\) 17.8702 0.892397 0.446198 0.894934i \(-0.352778\pi\)
0.446198 + 0.894934i \(0.352778\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −16.4440 −0.818118
\(405\) 0 0
\(406\) −7.27844 −0.361223
\(407\) 21.2247 1.05207
\(408\) 0 0
\(409\) 27.9119 1.38015 0.690076 0.723737i \(-0.257577\pi\)
0.690076 + 0.723737i \(0.257577\pi\)
\(410\) −7.40389 −0.365652
\(411\) 0 0
\(412\) 4.29350 0.211526
\(413\) −4.31767 −0.212459
\(414\) 0 0
\(415\) 26.0248 1.27750
\(416\) 0 0
\(417\) 0 0
\(418\) −16.4286 −0.803551
\(419\) 16.4034 0.801360 0.400680 0.916218i \(-0.368774\pi\)
0.400680 + 0.916218i \(0.368774\pi\)
\(420\) 0 0
\(421\) 3.03684 0.148006 0.0740032 0.997258i \(-0.476423\pi\)
0.0740032 + 0.997258i \(0.476423\pi\)
\(422\) 2.86486 0.139459
\(423\) 0 0
\(424\) 36.5050 1.77284
\(425\) −25.3666 −1.23046
\(426\) 0 0
\(427\) −10.6310 −0.514471
\(428\) 33.8471 1.63606
\(429\) 0 0
\(430\) −14.0616 −0.678110
\(431\) −3.33811 −0.160791 −0.0803955 0.996763i \(-0.525618\pi\)
−0.0803955 + 0.996763i \(0.525618\pi\)
\(432\) 0 0
\(433\) 11.9028 0.572010 0.286005 0.958228i \(-0.407673\pi\)
0.286005 + 0.958228i \(0.407673\pi\)
\(434\) −4.43190 −0.212738
\(435\) 0 0
\(436\) 6.34375 0.303810
\(437\) 19.0465 0.911119
\(438\) 0 0
\(439\) 3.71810 0.177455 0.0887277 0.996056i \(-0.471720\pi\)
0.0887277 + 0.996056i \(0.471720\pi\)
\(440\) −38.5743 −1.83896
\(441\) 0 0
\(442\) 0 0
\(443\) −1.45712 −0.0692300 −0.0346150 0.999401i \(-0.511021\pi\)
−0.0346150 + 0.999401i \(0.511021\pi\)
\(444\) 0 0
\(445\) 4.17821 0.198066
\(446\) 40.9825 1.94058
\(447\) 0 0
\(448\) −9.49827 −0.448751
\(449\) −12.1274 −0.572326 −0.286163 0.958181i \(-0.592380\pi\)
−0.286163 + 0.958181i \(0.592380\pi\)
\(450\) 0 0
\(451\) 2.42566 0.114220
\(452\) −17.6009 −0.827876
\(453\) 0 0
\(454\) 41.0465 1.92641
\(455\) 0 0
\(456\) 0 0
\(457\) 3.44803 0.161292 0.0806459 0.996743i \(-0.474302\pi\)
0.0806459 + 0.996743i \(0.474302\pi\)
\(458\) −44.2161 −2.06608
\(459\) 0 0
\(460\) 102.242 4.76704
\(461\) −6.75600 −0.314658 −0.157329 0.987546i \(-0.550288\pi\)
−0.157329 + 0.987546i \(0.550288\pi\)
\(462\) 0 0
\(463\) 7.45175 0.346312 0.173156 0.984894i \(-0.444604\pi\)
0.173156 + 0.984894i \(0.444604\pi\)
\(464\) −5.88338 −0.273129
\(465\) 0 0
\(466\) −9.31229 −0.431384
\(467\) −32.6098 −1.50900 −0.754502 0.656298i \(-0.772121\pi\)
−0.754502 + 0.656298i \(0.772121\pi\)
\(468\) 0 0
\(469\) −11.5429 −0.533001
\(470\) 21.2760 0.981391
\(471\) 0 0
\(472\) −19.7318 −0.908232
\(473\) 4.60686 0.211824
\(474\) 0 0
\(475\) 21.1032 0.968282
\(476\) −8.37867 −0.384036
\(477\) 0 0
\(478\) 1.92872 0.0882177
\(479\) 2.82908 0.129264 0.0646321 0.997909i \(-0.479413\pi\)
0.0646321 + 0.997909i \(0.479413\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.2180 −0.647614
\(483\) 0 0
\(484\) −10.2121 −0.464185
\(485\) 21.9468 0.996552
\(486\) 0 0
\(487\) 41.2935 1.87119 0.935594 0.353079i \(-0.114865\pi\)
0.935594 + 0.353079i \(0.114865\pi\)
\(488\) −48.5840 −2.19930
\(489\) 0 0
\(490\) −55.3159 −2.49892
\(491\) 34.6698 1.56463 0.782313 0.622886i \(-0.214040\pi\)
0.782313 + 0.622886i \(0.214040\pi\)
\(492\) 0 0
\(493\) 11.3177 0.509722
\(494\) 0 0
\(495\) 0 0
\(496\) −3.58243 −0.160856
\(497\) 6.51573 0.292270
\(498\) 0 0
\(499\) −17.9409 −0.803146 −0.401573 0.915827i \(-0.631536\pi\)
−0.401573 + 0.915827i \(0.631536\pi\)
\(500\) 47.6571 2.13129
\(501\) 0 0
\(502\) 63.3532 2.82759
\(503\) 26.1812 1.16736 0.583681 0.811983i \(-0.301612\pi\)
0.583681 + 0.811983i \(0.301612\pi\)
\(504\) 0 0
\(505\) −17.0780 −0.759960
\(506\) −52.3414 −2.32686
\(507\) 0 0
\(508\) 20.1575 0.894345
\(509\) −5.50604 −0.244051 −0.122025 0.992527i \(-0.538939\pi\)
−0.122025 + 0.992527i \(0.538939\pi\)
\(510\) 0 0
\(511\) 9.53319 0.421723
\(512\) −16.8955 −0.746681
\(513\) 0 0
\(514\) 21.3375 0.941158
\(515\) 4.45904 0.196489
\(516\) 0 0
\(517\) −6.97046 −0.306560
\(518\) −14.0718 −0.618277
\(519\) 0 0
\(520\) 0 0
\(521\) 26.7211 1.17067 0.585336 0.810791i \(-0.300963\pi\)
0.585336 + 0.810791i \(0.300963\pi\)
\(522\) 0 0
\(523\) 36.5230 1.59704 0.798520 0.601968i \(-0.205617\pi\)
0.798520 + 0.601968i \(0.205617\pi\)
\(524\) −64.7813 −2.82999
\(525\) 0 0
\(526\) −54.5646 −2.37913
\(527\) 6.89141 0.300195
\(528\) 0 0
\(529\) 37.6819 1.63834
\(530\) 86.6757 3.76495
\(531\) 0 0
\(532\) 6.97046 0.302208
\(533\) 0 0
\(534\) 0 0
\(535\) 35.1521 1.51976
\(536\) −52.7512 −2.27851
\(537\) 0 0
\(538\) 5.70576 0.245993
\(539\) 18.1226 0.780595
\(540\) 0 0
\(541\) −18.4655 −0.793893 −0.396947 0.917842i \(-0.629930\pi\)
−0.396947 + 0.917842i \(0.629930\pi\)
\(542\) 50.5437 2.17104
\(543\) 0 0
\(544\) 10.9591 0.469868
\(545\) 6.58834 0.282213
\(546\) 0 0
\(547\) 39.8471 1.70374 0.851870 0.523753i \(-0.175468\pi\)
0.851870 + 0.523753i \(0.175468\pi\)
\(548\) 33.5958 1.43514
\(549\) 0 0
\(550\) −57.9933 −2.47284
\(551\) −9.41550 −0.401114
\(552\) 0 0
\(553\) 4.33513 0.184348
\(554\) −34.8993 −1.48273
\(555\) 0 0
\(556\) 14.2734 0.605327
\(557\) 9.20477 0.390018 0.195009 0.980801i \(-0.437526\pi\)
0.195009 + 0.980801i \(0.437526\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 4.52350 0.191153
\(561\) 0 0
\(562\) 34.1836 1.44195
\(563\) 0.975246 0.0411017 0.0205509 0.999789i \(-0.493458\pi\)
0.0205509 + 0.999789i \(0.493458\pi\)
\(564\) 0 0
\(565\) −18.2795 −0.769024
\(566\) 60.5066 2.54328
\(567\) 0 0
\(568\) 29.7770 1.24942
\(569\) 16.8944 0.708250 0.354125 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(570\) 0 0
\(571\) −44.3226 −1.85484 −0.927421 0.374019i \(-0.877979\pi\)
−0.927421 + 0.374019i \(0.877979\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.60819 −0.0671244
\(575\) 67.2344 2.80387
\(576\) 0 0
\(577\) −3.56704 −0.148498 −0.0742489 0.997240i \(-0.523656\pi\)
−0.0742489 + 0.997240i \(0.523656\pi\)
\(578\) −19.7090 −0.819787
\(579\) 0 0
\(580\) −50.5424 −2.09866
\(581\) 5.65279 0.234517
\(582\) 0 0
\(583\) −28.3967 −1.17607
\(584\) 43.5669 1.80281
\(585\) 0 0
\(586\) −62.5120 −2.58235
\(587\) −16.1172 −0.665229 −0.332614 0.943063i \(-0.607931\pi\)
−0.332614 + 0.943063i \(0.607931\pi\)
\(588\) 0 0
\(589\) −5.73317 −0.236231
\(590\) −46.8504 −1.92880
\(591\) 0 0
\(592\) −11.3746 −0.467494
\(593\) −42.8611 −1.76010 −0.880048 0.474885i \(-0.842490\pi\)
−0.880048 + 0.474885i \(0.842490\pi\)
\(594\) 0 0
\(595\) −8.70171 −0.356735
\(596\) 68.9869 2.82581
\(597\) 0 0
\(598\) 0 0
\(599\) −40.9420 −1.67284 −0.836422 0.548086i \(-0.815357\pi\)
−0.836422 + 0.548086i \(0.815357\pi\)
\(600\) 0 0
\(601\) 1.18705 0.0484206 0.0242103 0.999707i \(-0.492293\pi\)
0.0242103 + 0.999707i \(0.492293\pi\)
\(602\) −3.05429 −0.124484
\(603\) 0 0
\(604\) 43.9474 1.78819
\(605\) −10.6058 −0.431187
\(606\) 0 0
\(607\) 19.9922 0.811460 0.405730 0.913993i \(-0.367017\pi\)
0.405730 + 0.913993i \(0.367017\pi\)
\(608\) −9.11721 −0.369752
\(609\) 0 0
\(610\) −115.356 −4.67062
\(611\) 0 0
\(612\) 0 0
\(613\) 33.3618 1.34747 0.673735 0.738973i \(-0.264689\pi\)
0.673735 + 0.738973i \(0.264689\pi\)
\(614\) 19.4373 0.784424
\(615\) 0 0
\(616\) −8.37867 −0.337586
\(617\) 11.6233 0.467935 0.233967 0.972244i \(-0.424829\pi\)
0.233967 + 0.972244i \(0.424829\pi\)
\(618\) 0 0
\(619\) 16.5381 0.664722 0.332361 0.943152i \(-0.392155\pi\)
0.332361 + 0.943152i \(0.392155\pi\)
\(620\) −30.7756 −1.23598
\(621\) 0 0
\(622\) 33.9815 1.36253
\(623\) 0.907542 0.0363599
\(624\) 0 0
\(625\) 6.33944 0.253577
\(626\) 33.5477 1.34083
\(627\) 0 0
\(628\) −66.3979 −2.64956
\(629\) 21.8810 0.872452
\(630\) 0 0
\(631\) 36.4416 1.45072 0.725358 0.688372i \(-0.241674\pi\)
0.725358 + 0.688372i \(0.241674\pi\)
\(632\) 19.8116 0.788064
\(633\) 0 0
\(634\) −16.1309 −0.640642
\(635\) 20.9347 0.830768
\(636\) 0 0
\(637\) 0 0
\(638\) 25.8745 1.02438
\(639\) 0 0
\(640\) −75.5303 −2.98560
\(641\) −27.2067 −1.07460 −0.537300 0.843391i \(-0.680556\pi\)
−0.537300 + 0.843391i \(0.680556\pi\)
\(642\) 0 0
\(643\) 5.06962 0.199926 0.0999632 0.994991i \(-0.468127\pi\)
0.0999632 + 0.994991i \(0.468127\pi\)
\(644\) 22.2078 0.875108
\(645\) 0 0
\(646\) −16.9366 −0.666362
\(647\) −19.3207 −0.759573 −0.379787 0.925074i \(-0.624003\pi\)
−0.379787 + 0.925074i \(0.624003\pi\)
\(648\) 0 0
\(649\) 15.3491 0.602506
\(650\) 0 0
\(651\) 0 0
\(652\) −43.8662 −1.71793
\(653\) −35.2355 −1.37887 −0.689436 0.724347i \(-0.742141\pi\)
−0.689436 + 0.724347i \(0.742141\pi\)
\(654\) 0 0
\(655\) −67.2790 −2.62881
\(656\) −1.29995 −0.0507544
\(657\) 0 0
\(658\) 4.62133 0.180158
\(659\) 4.36168 0.169907 0.0849535 0.996385i \(-0.472926\pi\)
0.0849535 + 0.996385i \(0.472926\pi\)
\(660\) 0 0
\(661\) 15.4709 0.601747 0.300873 0.953664i \(-0.402722\pi\)
0.300873 + 0.953664i \(0.402722\pi\)
\(662\) −22.2553 −0.864978
\(663\) 0 0
\(664\) 25.8334 1.00253
\(665\) 7.23921 0.280725
\(666\) 0 0
\(667\) −29.9976 −1.16151
\(668\) 40.8605 1.58094
\(669\) 0 0
\(670\) −125.250 −4.83883
\(671\) 37.7928 1.45898
\(672\) 0 0
\(673\) −11.7409 −0.452580 −0.226290 0.974060i \(-0.572660\pi\)
−0.226290 + 0.974060i \(0.572660\pi\)
\(674\) 6.22952 0.239952
\(675\) 0 0
\(676\) 0 0
\(677\) −3.44504 −0.132404 −0.0662019 0.997806i \(-0.521088\pi\)
−0.0662019 + 0.997806i \(0.521088\pi\)
\(678\) 0 0
\(679\) 4.76702 0.182941
\(680\) −39.7670 −1.52500
\(681\) 0 0
\(682\) 15.7552 0.603298
\(683\) 20.4058 0.780807 0.390403 0.920644i \(-0.372336\pi\)
0.390403 + 0.920644i \(0.372336\pi\)
\(684\) 0 0
\(685\) 34.8911 1.33312
\(686\) −25.2457 −0.963883
\(687\) 0 0
\(688\) −2.46888 −0.0941251
\(689\) 0 0
\(690\) 0 0
\(691\) −27.8039 −1.05771 −0.528854 0.848713i \(-0.677378\pi\)
−0.528854 + 0.848713i \(0.677378\pi\)
\(692\) −43.0656 −1.63711
\(693\) 0 0
\(694\) 23.9433 0.908876
\(695\) 14.8237 0.562295
\(696\) 0 0
\(697\) 2.50066 0.0947194
\(698\) −24.6052 −0.931321
\(699\) 0 0
\(700\) 24.6058 0.930012
\(701\) −11.9715 −0.452158 −0.226079 0.974109i \(-0.572591\pi\)
−0.226079 + 0.974109i \(0.572591\pi\)
\(702\) 0 0
\(703\) −18.2034 −0.686556
\(704\) 33.7660 1.27260
\(705\) 0 0
\(706\) 43.1102 1.62248
\(707\) −3.70948 −0.139509
\(708\) 0 0
\(709\) −32.2664 −1.21179 −0.605894 0.795545i \(-0.707185\pi\)
−0.605894 + 0.795545i \(0.707185\pi\)
\(710\) 70.7012 2.65337
\(711\) 0 0
\(712\) 4.14749 0.155434
\(713\) −18.2658 −0.684058
\(714\) 0 0
\(715\) 0 0
\(716\) 1.91457 0.0715510
\(717\) 0 0
\(718\) −35.9970 −1.34340
\(719\) −12.1086 −0.451574 −0.225787 0.974177i \(-0.572495\pi\)
−0.225787 + 0.974177i \(0.572495\pi\)
\(720\) 0 0
\(721\) 0.968541 0.0360704
\(722\) −30.6910 −1.14220
\(723\) 0 0
\(724\) −82.7730 −3.07623
\(725\) −33.2368 −1.23438
\(726\) 0 0
\(727\) −16.6200 −0.616402 −0.308201 0.951321i \(-0.599727\pi\)
−0.308201 + 0.951321i \(0.599727\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 103.443 3.82861
\(731\) 4.74930 0.175659
\(732\) 0 0
\(733\) −17.7912 −0.657132 −0.328566 0.944481i \(-0.606565\pi\)
−0.328566 + 0.944481i \(0.606565\pi\)
\(734\) −52.4922 −1.93752
\(735\) 0 0
\(736\) −29.0473 −1.07070
\(737\) 41.0344 1.51152
\(738\) 0 0
\(739\) 27.3618 1.00652 0.503260 0.864135i \(-0.332134\pi\)
0.503260 + 0.864135i \(0.332134\pi\)
\(740\) −97.7160 −3.59211
\(741\) 0 0
\(742\) 18.8267 0.691150
\(743\) 8.38596 0.307651 0.153826 0.988098i \(-0.450841\pi\)
0.153826 + 0.988098i \(0.450841\pi\)
\(744\) 0 0
\(745\) 71.6467 2.62493
\(746\) 9.72528 0.356068
\(747\) 0 0
\(748\) 29.7858 1.08908
\(749\) 7.63533 0.278989
\(750\) 0 0
\(751\) −38.7778 −1.41502 −0.707511 0.706703i \(-0.750182\pi\)
−0.707511 + 0.706703i \(0.750182\pi\)
\(752\) 3.73556 0.136222
\(753\) 0 0
\(754\) 0 0
\(755\) 45.6418 1.66107
\(756\) 0 0
\(757\) 12.9729 0.471506 0.235753 0.971813i \(-0.424244\pi\)
0.235753 + 0.971813i \(0.424244\pi\)
\(758\) 25.2355 0.916594
\(759\) 0 0
\(760\) 33.0834 1.20006
\(761\) −5.15585 −0.186899 −0.0934497 0.995624i \(-0.529789\pi\)
−0.0934497 + 0.995624i \(0.529789\pi\)
\(762\) 0 0
\(763\) 1.43104 0.0518072
\(764\) −59.6013 −2.15630
\(765\) 0 0
\(766\) 15.3822 0.555783
\(767\) 0 0
\(768\) 0 0
\(769\) 35.5013 1.28021 0.640104 0.768288i \(-0.278891\pi\)
0.640104 + 0.768288i \(0.278891\pi\)
\(770\) −19.8939 −0.716927
\(771\) 0 0
\(772\) −91.5186 −3.29383
\(773\) −6.15585 −0.221411 −0.110705 0.993853i \(-0.535311\pi\)
−0.110705 + 0.993853i \(0.535311\pi\)
\(774\) 0 0
\(775\) −20.2381 −0.726976
\(776\) 21.7854 0.782050
\(777\) 0 0
\(778\) 27.7931 0.996431
\(779\) −2.08038 −0.0745372
\(780\) 0 0
\(781\) −23.1631 −0.828843
\(782\) −53.9597 −1.92960
\(783\) 0 0
\(784\) −9.71214 −0.346862
\(785\) −68.9579 −2.46121
\(786\) 0 0
\(787\) −14.2107 −0.506558 −0.253279 0.967393i \(-0.581509\pi\)
−0.253279 + 0.967393i \(0.581509\pi\)
\(788\) 76.1503 2.71274
\(789\) 0 0
\(790\) 47.0398 1.67360
\(791\) −3.97046 −0.141173
\(792\) 0 0
\(793\) 0 0
\(794\) 29.5623 1.04913
\(795\) 0 0
\(796\) 12.5412 0.444512
\(797\) −11.9022 −0.421596 −0.210798 0.977530i \(-0.567606\pi\)
−0.210798 + 0.977530i \(0.567606\pi\)
\(798\) 0 0
\(799\) −7.18598 −0.254222
\(800\) −32.1839 −1.13787
\(801\) 0 0
\(802\) 42.1183 1.48725
\(803\) −33.8901 −1.19596
\(804\) 0 0
\(805\) 23.0640 0.812899
\(806\) 0 0
\(807\) 0 0
\(808\) −16.9524 −0.596384
\(809\) −1.61596 −0.0568140 −0.0284070 0.999596i \(-0.509043\pi\)
−0.0284070 + 0.999596i \(0.509043\pi\)
\(810\) 0 0
\(811\) 51.9657 1.82476 0.912381 0.409342i \(-0.134242\pi\)
0.912381 + 0.409342i \(0.134242\pi\)
\(812\) −10.9782 −0.385260
\(813\) 0 0
\(814\) 50.0245 1.75336
\(815\) −45.5575 −1.59581
\(816\) 0 0
\(817\) −3.95108 −0.138231
\(818\) 65.7853 2.30013
\(819\) 0 0
\(820\) −11.1675 −0.389985
\(821\) 54.8327 1.91367 0.956837 0.290627i \(-0.0938638\pi\)
0.956837 + 0.290627i \(0.0938638\pi\)
\(822\) 0 0
\(823\) −46.8514 −1.63314 −0.816569 0.577247i \(-0.804127\pi\)
−0.816569 + 0.577247i \(0.804127\pi\)
\(824\) 4.42626 0.154196
\(825\) 0 0
\(826\) −10.1763 −0.354078
\(827\) 21.2021 0.737270 0.368635 0.929574i \(-0.379825\pi\)
0.368635 + 0.929574i \(0.379825\pi\)
\(828\) 0 0
\(829\) 18.2972 0.635489 0.317744 0.948176i \(-0.397075\pi\)
0.317744 + 0.948176i \(0.397075\pi\)
\(830\) 61.3376 2.12906
\(831\) 0 0
\(832\) 0 0
\(833\) 18.6829 0.647325
\(834\) 0 0
\(835\) 42.4359 1.46856
\(836\) −24.7797 −0.857024
\(837\) 0 0
\(838\) 38.6612 1.33553
\(839\) 21.1414 0.729881 0.364941 0.931031i \(-0.381089\pi\)
0.364941 + 0.931031i \(0.381089\pi\)
\(840\) 0 0
\(841\) −14.1709 −0.488652
\(842\) 7.15751 0.246664
\(843\) 0 0
\(844\) 4.32113 0.148739
\(845\) 0 0
\(846\) 0 0
\(847\) −2.30367 −0.0791549
\(848\) 15.2182 0.522594
\(849\) 0 0
\(850\) −59.7864 −2.05066
\(851\) −57.9958 −1.98807
\(852\) 0 0
\(853\) 7.13036 0.244139 0.122069 0.992522i \(-0.461047\pi\)
0.122069 + 0.992522i \(0.461047\pi\)
\(854\) −25.0562 −0.857406
\(855\) 0 0
\(856\) 34.8937 1.19264
\(857\) 44.7741 1.52945 0.764726 0.644355i \(-0.222874\pi\)
0.764726 + 0.644355i \(0.222874\pi\)
\(858\) 0 0
\(859\) 57.3782 1.95772 0.978859 0.204535i \(-0.0655681\pi\)
0.978859 + 0.204535i \(0.0655681\pi\)
\(860\) −21.2094 −0.723235
\(861\) 0 0
\(862\) −7.86758 −0.267971
\(863\) −6.67563 −0.227241 −0.113621 0.993524i \(-0.536245\pi\)
−0.113621 + 0.993524i \(0.536245\pi\)
\(864\) 0 0
\(865\) −44.7260 −1.52073
\(866\) 28.0536 0.953299
\(867\) 0 0
\(868\) −6.68473 −0.226894
\(869\) −15.4112 −0.522789
\(870\) 0 0
\(871\) 0 0
\(872\) 6.53989 0.221469
\(873\) 0 0
\(874\) 44.8907 1.51845
\(875\) 10.7506 0.363438
\(876\) 0 0
\(877\) −25.2983 −0.854263 −0.427131 0.904190i \(-0.640476\pi\)
−0.427131 + 0.904190i \(0.640476\pi\)
\(878\) 8.76318 0.295743
\(879\) 0 0
\(880\) −16.0809 −0.542085
\(881\) 35.3787 1.19194 0.595969 0.803008i \(-0.296768\pi\)
0.595969 + 0.803008i \(0.296768\pi\)
\(882\) 0 0
\(883\) −11.5851 −0.389869 −0.194935 0.980816i \(-0.562449\pi\)
−0.194935 + 0.980816i \(0.562449\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.43429 −0.115377
\(887\) 27.2892 0.916281 0.458141 0.888880i \(-0.348516\pi\)
0.458141 + 0.888880i \(0.348516\pi\)
\(888\) 0 0
\(889\) 4.54719 0.152508
\(890\) 9.84761 0.330093
\(891\) 0 0
\(892\) 61.8149 2.06972
\(893\) 5.97823 0.200054
\(894\) 0 0
\(895\) 1.98839 0.0664646
\(896\) −16.4058 −0.548080
\(897\) 0 0
\(898\) −28.5830 −0.953826
\(899\) 9.02954 0.301152
\(900\) 0 0
\(901\) −29.2747 −0.975282
\(902\) 5.71704 0.190357
\(903\) 0 0
\(904\) −18.1451 −0.603497
\(905\) −85.9643 −2.85755
\(906\) 0 0
\(907\) −30.4219 −1.01014 −0.505072 0.863077i \(-0.668534\pi\)
−0.505072 + 0.863077i \(0.668534\pi\)
\(908\) 61.9114 2.05460
\(909\) 0 0
\(910\) 0 0
\(911\) −53.5719 −1.77492 −0.887459 0.460887i \(-0.847531\pi\)
−0.887459 + 0.460887i \(0.847531\pi\)
\(912\) 0 0
\(913\) −20.0954 −0.665062
\(914\) 8.12664 0.268805
\(915\) 0 0
\(916\) −66.6921 −2.20357
\(917\) −14.6136 −0.482582
\(918\) 0 0
\(919\) −36.1672 −1.19305 −0.596523 0.802596i \(-0.703451\pi\)
−0.596523 + 0.802596i \(0.703451\pi\)
\(920\) 105.403 3.47503
\(921\) 0 0
\(922\) −15.9232 −0.524403
\(923\) 0 0
\(924\) 0 0
\(925\) −64.2583 −2.11280
\(926\) 17.5630 0.577156
\(927\) 0 0
\(928\) 14.3593 0.471367
\(929\) −13.3478 −0.437927 −0.218964 0.975733i \(-0.570268\pi\)
−0.218964 + 0.975733i \(0.570268\pi\)
\(930\) 0 0
\(931\) −15.5429 −0.509397
\(932\) −14.0459 −0.460090
\(933\) 0 0
\(934\) −76.8580 −2.51487
\(935\) 30.9342 1.01166
\(936\) 0 0
\(937\) 38.6872 1.26386 0.631928 0.775027i \(-0.282264\pi\)
0.631928 + 0.775027i \(0.282264\pi\)
\(938\) −27.2054 −0.888286
\(939\) 0 0
\(940\) 32.0911 1.04670
\(941\) 35.7275 1.16468 0.582342 0.812944i \(-0.302136\pi\)
0.582342 + 0.812944i \(0.302136\pi\)
\(942\) 0 0
\(943\) −6.62804 −0.215839
\(944\) −8.22580 −0.267727
\(945\) 0 0
\(946\) 10.8579 0.353020
\(947\) −17.8436 −0.579838 −0.289919 0.957051i \(-0.593628\pi\)
−0.289919 + 0.957051i \(0.593628\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 49.7381 1.61372
\(951\) 0 0
\(952\) −8.63773 −0.279950
\(953\) −20.1691 −0.653342 −0.326671 0.945138i \(-0.605927\pi\)
−0.326671 + 0.945138i \(0.605927\pi\)
\(954\) 0 0
\(955\) −61.8993 −2.00301
\(956\) 2.90913 0.0940881
\(957\) 0 0
\(958\) 6.66786 0.215429
\(959\) 7.57865 0.244727
\(960\) 0 0
\(961\) −25.5018 −0.822640
\(962\) 0 0
\(963\) 0 0
\(964\) −21.4454 −0.690709
\(965\) −95.0471 −3.05967
\(966\) 0 0
\(967\) 32.7894 1.05444 0.527218 0.849730i \(-0.323235\pi\)
0.527218 + 0.849730i \(0.323235\pi\)
\(968\) −10.5278 −0.338377
\(969\) 0 0
\(970\) 51.7263 1.66083
\(971\) 26.9845 0.865973 0.432986 0.901401i \(-0.357460\pi\)
0.432986 + 0.901401i \(0.357460\pi\)
\(972\) 0 0
\(973\) 3.21983 0.103223
\(974\) 97.3245 3.11848
\(975\) 0 0
\(976\) −20.2537 −0.648305
\(977\) −16.9571 −0.542504 −0.271252 0.962508i \(-0.587438\pi\)
−0.271252 + 0.962508i \(0.587438\pi\)
\(978\) 0 0
\(979\) −3.22627 −0.103112
\(980\) −83.4341 −2.66521
\(981\) 0 0
\(982\) 81.7131 2.60757
\(983\) −32.6631 −1.04179 −0.520895 0.853621i \(-0.674402\pi\)
−0.520895 + 0.853621i \(0.674402\pi\)
\(984\) 0 0
\(985\) 79.0863 2.51990
\(986\) 26.6746 0.849491
\(987\) 0 0
\(988\) 0 0
\(989\) −12.5881 −0.400277
\(990\) 0 0
\(991\) 7.64310 0.242791 0.121396 0.992604i \(-0.461263\pi\)
0.121396 + 0.992604i \(0.461263\pi\)
\(992\) 8.74348 0.277606
\(993\) 0 0
\(994\) 15.3569 0.487091
\(995\) 13.0248 0.412912
\(996\) 0 0
\(997\) 36.1256 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(998\) −42.2849 −1.33850
\(999\) 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 1521.2.a.s.1.2 3
3.2 odd 2 507.2.a.i.1.2 3
12.11 even 2 8112.2.a.cg.1.1 3
13.5 odd 4 1521.2.b.k.1351.2 6
13.8 odd 4 1521.2.b.k.1351.5 6
13.12 even 2 1521.2.a.n.1.2 3
39.2 even 12 507.2.j.i.316.2 12
39.5 even 4 507.2.b.f.337.5 6
39.8 even 4 507.2.b.f.337.2 6
39.11 even 12 507.2.j.i.316.5 12
39.17 odd 6 507.2.e.i.484.2 6
39.20 even 12 507.2.j.i.361.2 12
39.23 odd 6 507.2.e.i.22.2 6
39.29 odd 6 507.2.e.l.22.2 6
39.32 even 12 507.2.j.i.361.5 12
39.35 odd 6 507.2.e.l.484.2 6
39.38 odd 2 507.2.a.l.1.2 yes 3
156.155 even 2 8112.2.a.cp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 3.2 odd 2
507.2.a.l.1.2 yes 3 39.38 odd 2
507.2.b.f.337.2 6 39.8 even 4
507.2.b.f.337.5 6 39.5 even 4
507.2.e.i.22.2 6 39.23 odd 6
507.2.e.i.484.2 6 39.17 odd 6
507.2.e.l.22.2 6 39.29 odd 6
507.2.e.l.484.2 6 39.35 odd 6
507.2.j.i.316.2 12 39.2 even 12
507.2.j.i.316.5 12 39.11 even 12
507.2.j.i.361.2 12 39.20 even 12
507.2.j.i.361.5 12 39.32 even 12
1521.2.a.n.1.2 3 13.12 even 2
1521.2.a.s.1.2 3 1.1 even 1 trivial
1521.2.b.k.1351.2 6 13.5 odd 4
1521.2.b.k.1351.5 6 13.8 odd 4
8112.2.a.cg.1.1 3 12.11 even 2
8112.2.a.cp.1.3 3 156.155 even 2