Properties

Label 1521.2.a.n.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
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: \(1\)
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 \(-0.813545\pi\)
−0.833289 + 0.552838i \(0.813545\pi\)
\(3\) 0 0
\(4\) 3.55496 1.77748
\(5\) −3.69202 −1.65112 −0.825561 0.564313i \(-0.809141\pi\)
−0.825561 + 0.564313i \(0.809141\pi\)
\(6\) 0 0
\(7\) −0.801938 −0.303104 −0.151552 0.988449i \(-0.548427\pi\)
−0.151552 + 0.988449i \(0.548427\pi\)
\(8\) −3.66487 −1.29573
\(9\) 0 0
\(10\) 8.70171 2.75172
\(11\) 2.85086 0.859565 0.429783 0.902932i \(-0.358590\pi\)
0.429783 + 0.902932i \(0.358590\pi\)
\(12\) 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 \(-0.590489\pi\)
−0.280466 + 0.959864i \(0.590489\pi\)
\(20\) −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 \(-0.432468\pi\)
0.210571 + 0.977579i \(0.432468\pi\)
\(32\) 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 \(-0.290374\pi\)
0.611979 + 0.790874i \(0.290374\pi\)
\(38\) 5.76271 0.934835
\(39\) 0 0
\(40\) 13.5308 2.13941
\(41\) 0.850855 0.132881 0.0664406 0.997790i \(-0.478836\pi\)
0.0664406 + 0.997790i \(0.478836\pi\)
\(42\) 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 \(-0.557067\pi\)
−0.178323 + 0.983972i \(0.557067\pi\)
\(48\) 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 \(-0.386021\pi\)
0.350471 + 0.936573i \(0.386021\pi\)
\(60\) 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 \(-0.158053\pi\)
0.879237 + 0.476384i \(0.158053\pi\)
\(68\) −10.4480 −1.26701
\(69\) 0 0
\(70\) −6.97823 −0.834058
\(71\) −8.12498 −0.964258 −0.482129 0.876100i \(-0.660136\pi\)
−0.482129 + 0.876100i \(0.660136\pi\)
\(72\) 0 0
\(73\) −11.8877 −1.39135 −0.695674 0.718357i \(-0.744894\pi\)
−0.695674 + 0.718357i \(0.744894\pi\)
\(74\) −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 \(-0.626440\pi\)
−0.386860 + 0.922139i \(0.626440\pi\)
\(84\) 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 \(-0.519103\pi\)
−0.0599793 + 0.998200i \(0.519103\pi\)
\(90\) 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 \(-0.597581\pi\)
−0.301780 + 0.953378i \(0.597581\pi\)
\(98\) 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 \(-0.527236\pi\)
−0.0854611 + 0.996342i \(0.527236\pi\)
\(110\) 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 \(-0.632277\pi\)
−0.403702 + 0.914891i \(0.632277\pi\)
\(138\) 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 \(-0.792473\pi\)
−0.794893 + 0.606750i \(0.792473\pi\)
\(150\) 0 0
\(151\) −12.3623 −1.00603 −0.503014 0.864278i \(-0.667775\pi\)
−0.503014 + 0.864278i \(0.667775\pi\)
\(152\) 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 \(-0.339456\pi\)
0.483250 + 0.875483i \(0.339456\pi\)
\(164\) 3.02475 0.236194
\(165\) 0 0
\(166\) 16.6136 1.28946
\(167\) −11.4940 −0.889429 −0.444715 0.895672i \(-0.646695\pi\)
−0.444715 + 0.895672i \(0.646695\pi\)
\(168\) 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 \(-0.122765\pi\)
0.926544 + 0.376186i \(0.122765\pi\)
\(194\) 14.0103 1.00588
\(195\) 0 0
\(196\) −22.5985 −1.61418
\(197\) −21.4209 −1.52617 −0.763087 0.646296i \(-0.776317\pi\)
−0.763087 + 0.646296i \(0.776317\pi\)
\(198\) 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 \(-0.697810\pi\)
−0.582205 + 0.813042i \(0.697810\pi\)
\(224\) −2.99031 −0.199799
\(225\) 0 0
\(226\) 11.6692 0.776223
\(227\) −17.4155 −1.15591 −0.577954 0.816070i \(-0.696149\pi\)
−0.577954 + 0.816070i \(0.696149\pi\)
\(228\) 0 0
\(229\) 18.7603 1.23972 0.619858 0.784714i \(-0.287190\pi\)
0.619858 + 0.784714i \(0.287190\pi\)
\(230\) 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 \(-0.508426\pi\)
−0.0264667 + 0.999650i \(0.508426\pi\)
\(240\) 0 0
\(241\) 6.03252 0.388589 0.194295 0.980943i \(-0.437758\pi\)
0.194295 + 0.980943i \(0.437758\pi\)
\(242\) 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 \(-0.725796\pi\)
−0.651347 + 0.758780i \(0.725796\pi\)
\(272\) −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 \(-0.642406\pi\)
−0.432608 + 0.901582i \(0.642406\pi\)
\(282\) 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 \(-0.217878\pi\)
0.774746 + 0.632273i \(0.217878\pi\)
\(294\) 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 \(-0.575620\pi\)
−0.235340 + 0.971913i \(0.575620\pi\)
\(308\) −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 \(-0.438437\pi\)
0.192203 + 0.981355i \(0.438437\pi\)
\(318\) 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 \(-0.416440\pi\)
0.259507 + 0.965741i \(0.416440\pi\)
\(332\) −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 \(-0.409861\pi\)
0.279411 + 0.960172i \(0.409861\pi\)
\(350\) 16.3134 0.871986
\(351\) 0 0
\(352\) 10.6304 0.566604
\(353\) −18.2911 −0.973538 −0.486769 0.873531i \(-0.661825\pi\)
−0.486769 + 0.873531i \(0.661825\pi\)
\(354\) 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 \(-0.367953\pi\)
0.403041 + 0.915182i \(0.367953\pi\)
\(360\) 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 \(-0.588676\pi\)
−0.274993 + 0.961446i \(0.588676\pi\)
\(380\) 32.0911 1.64624
\(381\) 0 0
\(382\) 39.5150 2.02176
\(383\) −6.52648 −0.333488 −0.166744 0.986000i \(-0.553325\pi\)
−0.166744 + 0.986000i \(0.553325\pi\)
\(384\) 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 \(-0.601922\pi\)
−0.314754 + 0.949173i \(0.601922\pi\)
\(398\) −8.31468 −0.416777
\(399\) 0 0
\(400\) 13.1866 0.659329
\(401\) −17.8702 −0.892397 −0.446198 0.894934i \(-0.647222\pi\)
−0.446198 + 0.894934i \(0.647222\pi\)
\(402\) 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 \(-0.742423\pi\)
−0.690076 + 0.723737i \(0.742423\pi\)
\(410\) 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 \(-0.523577\pi\)
−0.0740032 + 0.997258i \(0.523577\pi\)
\(422\) −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 \(-0.474382\pi\)
0.0803955 + 0.996763i \(0.474382\pi\)
\(432\) 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 \(-0.407620\pi\)
0.286163 + 0.958181i \(0.407620\pi\)
\(450\) 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 \(-0.525698\pi\)
−0.0806459 + 0.996743i \(0.525698\pi\)
\(458\) −44.2161 −2.06608
\(459\) 0 0
\(460\) −102.242 −4.76704
\(461\) 6.75600 0.314658 0.157329 0.987546i \(-0.449712\pi\)
0.157329 + 0.987546i \(0.449712\pi\)
\(462\) 0 0
\(463\) −7.45175 −0.346312 −0.173156 0.984894i \(-0.555396\pi\)
−0.173156 + 0.984894i \(0.555396\pi\)
\(464\) −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 \(-0.520587\pi\)
−0.0646321 + 0.997909i \(0.520587\pi\)
\(480\) 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 \(-0.885135\pi\)
−0.935594 + 0.353079i \(0.885135\pi\)
\(488\) 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 \(-0.368464\pi\)
0.401573 + 0.915827i \(0.368464\pi\)
\(500\) −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 \(-0.461061\pi\)
0.122025 + 0.992527i \(0.461061\pi\)
\(510\) 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 \(-0.370070\pi\)
0.396947 + 0.917842i \(0.370070\pi\)
\(542\) 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 \(-0.562474\pi\)
−0.195009 + 0.980801i \(0.562474\pi\)
\(558\) 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 \(-0.476344\pi\)
0.0742489 + 0.997240i \(0.476344\pi\)
\(578\) 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 \(-0.392069\pi\)
0.332614 + 0.943063i \(0.392069\pi\)
\(588\) 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 \(-0.157510\pi\)
0.880048 + 0.474885i \(0.157510\pi\)
\(594\) 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 \(-0.735311\pi\)
−0.673735 + 0.738973i \(0.735311\pi\)
\(614\) 19.4373 0.784424
\(615\) 0 0
\(616\) 8.37867 0.337586
\(617\) −11.6233 −0.467935 −0.233967 0.972244i \(-0.575171\pi\)
−0.233967 + 0.972244i \(0.575171\pi\)
\(618\) 0 0
\(619\) −16.5381 −0.664722 −0.332361 0.943152i \(-0.607845\pi\)
−0.332361 + 0.943152i \(0.607845\pi\)
\(620\) −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 \(-0.758326\pi\)
−0.725358 + 0.688372i \(0.758326\pi\)
\(632\) −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 \(-0.531873\pi\)
−0.0999632 + 0.994991i \(0.531873\pi\)
\(644\) −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 \(-0.597278\pi\)
−0.300873 + 0.953664i \(0.597278\pi\)
\(662\) −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 \(-0.627664\pi\)
−0.390403 + 0.920644i \(0.627664\pi\)
\(684\) 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 \(-0.322622\pi\)
0.528854 + 0.848713i \(0.322622\pi\)
\(692\) −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 \(-0.292815\pi\)
0.605894 + 0.795545i \(0.292815\pi\)
\(710\) −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 \(-0.393435\pi\)
0.328566 + 0.944481i \(0.393435\pi\)
\(734\) 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 \(-0.667866\pi\)
−0.503260 + 0.864135i \(0.667866\pi\)
\(740\) −97.7160 −3.59211
\(741\) 0 0
\(742\) 18.8267 0.691150
\(743\) −8.38596 −0.307651 −0.153826 0.988098i \(-0.549159\pi\)
−0.153826 + 0.988098i \(0.549159\pi\)
\(744\) 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 \(-0.470211\pi\)
0.0934497 + 0.995624i \(0.470211\pi\)
\(762\) 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 \(-0.721109\pi\)
−0.640104 + 0.768288i \(0.721109\pi\)
\(770\) −19.8939 −0.716927
\(771\) 0 0
\(772\) 91.5186 3.29383
\(773\) 6.15585 0.221411 0.110705 0.993853i \(-0.464689\pi\)
0.110705 + 0.993853i \(0.464689\pi\)
\(774\) 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 \(-0.418491\pi\)
0.253279 + 0.967393i \(0.418491\pi\)
\(788\) −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 \(-0.865758\pi\)
−0.912381 + 0.409342i \(0.865758\pi\)
\(812\) 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 \(-0.906136\pi\)
−0.956837 + 0.290627i \(0.906136\pi\)
\(822\) 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 \(-0.620175\pi\)
−0.368635 + 0.929574i \(0.620175\pi\)
\(828\) 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 \(-0.618911\pi\)
−0.364941 + 0.931031i \(0.618911\pi\)
\(840\) 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 \(-0.538953\pi\)
−0.122069 + 0.992522i \(0.538953\pi\)
\(854\) −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 \(-0.463755\pi\)
0.113621 + 0.993524i \(0.463755\pi\)
\(864\) 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 \(-0.359524\pi\)
0.427131 + 0.904190i \(0.359524\pi\)
\(878\) −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 \(-0.429732\pi\)
0.218964 + 0.975733i \(0.429732\pi\)
\(930\) 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 \(-0.697864\pi\)
−0.582342 + 0.812944i \(0.697864\pi\)
\(942\) 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 \(-0.406372\pi\)
0.289919 + 0.957051i \(0.406372\pi\)
\(948\) 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 \(-0.676765\pi\)
−0.527218 + 0.849730i \(0.676765\pi\)
\(968\) 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 \(-0.412562\pi\)
0.271252 + 0.962508i \(0.412562\pi\)
\(978\) 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 \(-0.325598\pi\)
0.520895 + 0.853621i \(0.325598\pi\)
\(984\) 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.n.1.2 3
3.2 odd 2 507.2.a.l.1.2 yes 3
12.11 even 2 8112.2.a.cp.1.3 3
13.5 odd 4 1521.2.b.k.1351.5 6
13.8 odd 4 1521.2.b.k.1351.2 6
13.12 even 2 1521.2.a.s.1.2 3
39.2 even 12 507.2.j.i.316.5 12
39.5 even 4 507.2.b.f.337.2 6
39.8 even 4 507.2.b.f.337.5 6
39.11 even 12 507.2.j.i.316.2 12
39.17 odd 6 507.2.e.l.484.2 6
39.20 even 12 507.2.j.i.361.5 12
39.23 odd 6 507.2.e.l.22.2 6
39.29 odd 6 507.2.e.i.22.2 6
39.32 even 12 507.2.j.i.361.2 12
39.35 odd 6 507.2.e.i.484.2 6
39.38 odd 2 507.2.a.i.1.2 3
156.155 even 2 8112.2.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 39.38 odd 2
507.2.a.l.1.2 yes 3 3.2 odd 2
507.2.b.f.337.2 6 39.5 even 4
507.2.b.f.337.5 6 39.8 even 4
507.2.e.i.22.2 6 39.29 odd 6
507.2.e.i.484.2 6 39.35 odd 6
507.2.e.l.22.2 6 39.23 odd 6
507.2.e.l.484.2 6 39.17 odd 6
507.2.j.i.316.2 12 39.11 even 12
507.2.j.i.316.5 12 39.2 even 12
507.2.j.i.361.2 12 39.32 even 12
507.2.j.i.361.5 12 39.20 even 12
1521.2.a.n.1.2 3 1.1 even 1 trivial
1521.2.a.s.1.2 3 13.12 even 2
1521.2.b.k.1351.2 6 13.8 odd 4
1521.2.b.k.1351.5 6 13.5 odd 4
8112.2.a.cg.1.1 3 156.155 even 2
8112.2.a.cp.1.3 3 12.11 even 2