Properties

Label 1521.2.b.k.1351.5
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.5
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.k.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.35690i q^{2} -3.55496 q^{4} +3.69202i q^{5} -0.801938i q^{7} -3.66487i q^{8} +O(q^{10})\) \(q+2.35690i q^{2} -3.55496 q^{4} +3.69202i q^{5} -0.801938i q^{7} -3.66487i q^{8} -8.70171 q^{10} +2.85086i q^{11} +1.89008 q^{14} +1.52781 q^{16} +2.93900 q^{17} +2.44504i q^{19} -13.1250i q^{20} -6.71917 q^{22} -7.78986 q^{23} -8.63102 q^{25} +2.85086i q^{28} -3.85086 q^{29} -2.34481i q^{31} -3.72886i q^{32} +6.92692i q^{34} +2.96077 q^{35} +7.44504i q^{37} -5.76271 q^{38} +13.5308 q^{40} -0.850855i q^{41} +1.61596 q^{43} -10.1347i q^{44} -18.3599i q^{46} -2.44504i q^{47} +6.35690 q^{49} -20.3424i q^{50} +9.96077 q^{53} -10.5254 q^{55} -2.93900 q^{56} -9.07606i q^{58} +5.38404i q^{59} -13.2567 q^{61} +5.52648 q^{62} +11.8442 q^{64} -14.3937i q^{67} -10.4480 q^{68} +6.97823i q^{70} +8.12498i q^{71} -11.8877i q^{73} -17.5472 q^{74} -8.69202i q^{76} +2.28621 q^{77} +5.40581 q^{79} +5.64071i q^{80} +2.00538 q^{82} +7.04892i q^{83} +10.8509i q^{85} +3.80864i q^{86} +10.4480 q^{88} -1.13169i q^{89} +27.6926 q^{92} +5.76271 q^{94} -9.02715 q^{95} +5.94438i q^{97} +14.9825i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 22q^{4} + O(q^{10}) \) \( 6q - 22q^{4} + 2q^{10} + 10q^{14} + 22q^{16} - 2q^{17} - 18q^{22} - 22q^{25} + 4q^{29} - 8q^{35} + 6q^{40} + 30q^{43} + 30q^{49} + 34q^{53} + 6q^{55} + 2q^{56} - 26q^{61} - 4q^{62} + 26q^{68} - 30q^{74} + 30q^{77} + 6q^{79} + 6q^{82} - 26q^{88} - 14q^{92} - 42q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35690i 1.66658i 0.552838 + 0.833289i \(0.313545\pi\)
−0.552838 + 0.833289i \(0.686455\pi\)
\(3\) 0 0
\(4\) −3.55496 −1.77748
\(5\) 3.69202i 1.65112i 0.564313 + 0.825561i \(0.309141\pi\)
−0.564313 + 0.825561i \(0.690859\pi\)
\(6\) 0 0
\(7\) − 0.801938i − 0.303104i −0.988449 0.151552i \(-0.951573\pi\)
0.988449 0.151552i \(-0.0484271\pi\)
\(8\) − 3.66487i − 1.29573i
\(9\) 0 0
\(10\) −8.70171 −2.75172
\(11\) 2.85086i 0.859565i 0.902932 + 0.429783i \(0.141410\pi\)
−0.902932 + 0.429783i \(0.858590\pi\)
\(12\) 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 \(-0.384002\pi\)
0.356406 + 0.934331i \(0.384002\pi\)
\(18\) 0 0
\(19\) 2.44504i 0.560931i 0.959864 + 0.280466i \(0.0904888\pi\)
−0.959864 + 0.280466i \(0.909511\pi\)
\(20\) − 13.1250i − 2.93484i
\(21\) 0 0
\(22\) −6.71917 −1.43253
\(23\) −7.78986 −1.62430 −0.812149 0.583451i \(-0.801702\pi\)
−0.812149 + 0.583451i \(0.801702\pi\)
\(24\) 0 0
\(25\) −8.63102 −1.72620
\(26\) 0 0
\(27\) 0 0
\(28\) 2.85086i 0.538761i
\(29\) −3.85086 −0.715086 −0.357543 0.933897i \(-0.616385\pi\)
−0.357543 + 0.933897i \(0.616385\pi\)
\(30\) 0 0
\(31\) − 2.34481i − 0.421141i −0.977579 0.210571i \(-0.932468\pi\)
0.977579 0.210571i \(-0.0675322\pi\)
\(32\) − 3.72886i − 0.659175i
\(33\) 0 0
\(34\) 6.92692i 1.18796i
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) 7.44504i 1.22396i 0.790874 + 0.611979i \(0.209626\pi\)
−0.790874 + 0.611979i \(0.790374\pi\)
\(38\) −5.76271 −0.934835
\(39\) 0 0
\(40\) 13.5308 2.13941
\(41\) − 0.850855i − 0.132881i −0.997790 0.0664406i \(-0.978836\pi\)
0.997790 0.0664406i \(-0.0211643\pi\)
\(42\) 0 0
\(43\) 1.61596 0.246431 0.123216 0.992380i \(-0.460679\pi\)
0.123216 + 0.992380i \(0.460679\pi\)
\(44\) − 10.1347i − 1.52786i
\(45\) 0 0
\(46\) − 18.3599i − 2.70702i
\(47\) − 2.44504i − 0.356646i −0.983972 0.178323i \(-0.942933\pi\)
0.983972 0.178323i \(-0.0570672\pi\)
\(48\) 0 0
\(49\) 6.35690 0.908128
\(50\) − 20.3424i − 2.87685i
\(51\) 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
\(58\) − 9.07606i − 1.19175i
\(59\) 5.38404i 0.700943i 0.936573 + 0.350471i \(0.113979\pi\)
−0.936573 + 0.350471i \(0.886021\pi\)
\(60\) 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
\(67\) − 14.3937i − 1.75847i −0.476384 0.879237i \(-0.658053\pi\)
0.476384 0.879237i \(-0.341947\pi\)
\(68\) −10.4480 −1.26701
\(69\) 0 0
\(70\) 6.97823i 0.834058i
\(71\) 8.12498i 0.964258i 0.876100 + 0.482129i \(0.160136\pi\)
−0.876100 + 0.482129i \(0.839864\pi\)
\(72\) 0 0
\(73\) − 11.8877i − 1.39135i −0.718357 0.695674i \(-0.755106\pi\)
0.718357 0.695674i \(-0.244894\pi\)
\(74\) −17.5472 −2.03982
\(75\) 0 0
\(76\) − 8.69202i − 0.997043i
\(77\) 2.28621 0.260538
\(78\) 0 0
\(79\) 5.40581 0.608202 0.304101 0.952640i \(-0.401644\pi\)
0.304101 + 0.952640i \(0.401644\pi\)
\(80\) 5.64071i 0.630651i
\(81\) 0 0
\(82\) 2.00538 0.221457
\(83\) 7.04892i 0.773719i 0.922139 + 0.386860i \(0.126440\pi\)
−0.922139 + 0.386860i \(0.873560\pi\)
\(84\) 0 0
\(85\) 10.8509i 1.17694i
\(86\) 3.80864i 0.410696i
\(87\) 0 0
\(88\) 10.4480 1.11376
\(89\) − 1.13169i − 0.119959i −0.998200 0.0599793i \(-0.980897\pi\)
0.998200 0.0599793i \(-0.0191035\pi\)
\(90\) 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
\(97\) 5.94438i 0.603560i 0.953378 + 0.301780i \(0.0975808\pi\)
−0.953378 + 0.301780i \(0.902419\pi\)
\(98\) 14.9825i 1.51347i
\(99\) 0 0
\(100\) 30.6829 3.06829
\(101\) 4.62565 0.460269 0.230134 0.973159i \(-0.426083\pi\)
0.230134 + 0.973159i \(0.426083\pi\)
\(102\) 0 0
\(103\) −1.20775 −0.119003 −0.0595016 0.998228i \(-0.518951\pi\)
−0.0595016 + 0.998228i \(0.518951\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 23.4765i 2.28024i
\(107\) 9.52111 0.920440 0.460220 0.887805i \(-0.347771\pi\)
0.460220 + 0.887805i \(0.347771\pi\)
\(108\) 0 0
\(109\) 1.78448i 0.170922i 0.996342 + 0.0854611i \(0.0272363\pi\)
−0.996342 + 0.0854611i \(0.972764\pi\)
\(110\) − 24.8073i − 2.36528i
\(111\) 0 0
\(112\) − 1.22521i − 0.115771i
\(113\) −4.95108 −0.465759 −0.232879 0.972506i \(-0.574815\pi\)
−0.232879 + 0.972506i \(0.574815\pi\)
\(114\) 0 0
\(115\) − 28.7603i − 2.68191i
\(116\) 13.6896 1.27105
\(117\) 0 0
\(118\) −12.6896 −1.16817
\(119\) − 2.35690i − 0.216056i
\(120\) 0 0
\(121\) 2.87263 0.261148
\(122\) − 31.2446i − 2.82875i
\(123\) 0 0
\(124\) 8.33572i 0.748569i
\(125\) − 13.4058i − 1.19905i
\(126\) 0 0
\(127\) −5.67025 −0.503153 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(128\) 20.4577i 1.80822i
\(129\) 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
\(136\) − 10.7711i − 0.923612i
\(137\) − 9.45042i − 0.807404i −0.914891 0.403702i \(-0.867723\pi\)
0.914891 0.403702i \(-0.132277\pi\)
\(138\) 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
\(145\) − 14.2174i − 1.18069i
\(146\) 28.0180 2.31879
\(147\) 0 0
\(148\) − 26.4668i − 2.17556i
\(149\) 19.4058i 1.58979i 0.606750 + 0.794893i \(0.292473\pi\)
−0.606750 + 0.794893i \(0.707527\pi\)
\(150\) 0 0
\(151\) − 12.3623i − 1.00603i −0.864278 0.503014i \(-0.832225\pi\)
0.864278 0.503014i \(-0.167775\pi\)
\(152\) 8.96077 0.726815
\(153\) 0 0
\(154\) 5.38835i 0.434206i
\(155\) 8.65710 0.695355
\(156\) 0 0
\(157\) −18.6775 −1.49063 −0.745315 0.666712i \(-0.767701\pi\)
−0.745315 + 0.666712i \(0.767701\pi\)
\(158\) 12.7409i 1.01361i
\(159\) 0 0
\(160\) 13.7670 1.08838
\(161\) 6.24698i 0.492331i
\(162\) 0 0
\(163\) 12.3394i 0.966499i 0.875483 + 0.483250i \(0.160544\pi\)
−0.875483 + 0.483250i \(0.839456\pi\)
\(164\) 3.02475i 0.236194i
\(165\) 0 0
\(166\) −16.6136 −1.28946
\(167\) − 11.4940i − 0.889429i −0.895672 0.444715i \(-0.853305\pi\)
0.895672 0.444715i \(-0.146695\pi\)
\(168\) 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 \(-0.347665\pi\)
0.460514 + 0.887653i \(0.347665\pi\)
\(174\) 0 0
\(175\) 6.92154i 0.523219i
\(176\) 4.35557i 0.328313i
\(177\) 0 0
\(178\) 2.66727 0.199920
\(179\) −0.538565 −0.0402542 −0.0201271 0.999797i \(-0.506407\pi\)
−0.0201271 + 0.999797i \(0.506407\pi\)
\(180\) 0 0
\(181\) 23.2838 1.73067 0.865336 0.501192i \(-0.167105\pi\)
0.865336 + 0.501192i \(0.167105\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 28.5488i 2.10465i
\(185\) −27.4873 −2.02090
\(186\) 0 0
\(187\) 8.37867i 0.612709i
\(188\) 8.69202i 0.633931i
\(189\) 0 0
\(190\) − 21.2760i − 1.54353i
\(191\) −16.7657 −1.21312 −0.606561 0.795037i \(-0.707452\pi\)
−0.606561 + 0.795037i \(0.707452\pi\)
\(192\) 0 0
\(193\) 25.7439i 1.85309i 0.376186 + 0.926544i \(0.377235\pi\)
−0.376186 + 0.926544i \(0.622765\pi\)
\(194\) −14.0103 −1.00588
\(195\) 0 0
\(196\) −22.5985 −1.61418
\(197\) 21.4209i 1.52617i 0.646296 + 0.763087i \(0.276317\pi\)
−0.646296 + 0.763087i \(0.723683\pi\)
\(198\) 0 0
\(199\) −3.52781 −0.250080 −0.125040 0.992152i \(-0.539906\pi\)
−0.125040 + 0.992152i \(0.539906\pi\)
\(200\) 31.6316i 2.23669i
\(201\) 0 0
\(202\) 10.9022i 0.767074i
\(203\) 3.08815i 0.216745i
\(204\) 0 0
\(205\) 3.14138 0.219403
\(206\) − 2.84654i − 0.198328i
\(207\) 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
\(214\) 22.4403i 1.53398i
\(215\) 5.96615i 0.406888i
\(216\) 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
\(223\) 17.3884i 1.16441i 0.813042 + 0.582205i \(0.197810\pi\)
−0.813042 + 0.582205i \(0.802190\pi\)
\(224\) −2.99031 −0.199799
\(225\) 0 0
\(226\) − 11.6692i − 0.776223i
\(227\) 17.4155i 1.15591i 0.816070 + 0.577954i \(0.196149\pi\)
−0.816070 + 0.577954i \(0.803851\pi\)
\(228\) 0 0
\(229\) 18.7603i 1.23972i 0.784714 + 0.619858i \(0.212810\pi\)
−0.784714 + 0.619858i \(0.787190\pi\)
\(230\) 67.7851 4.46962
\(231\) 0 0
\(232\) 14.1129i 0.926557i
\(233\) 3.95108 0.258844 0.129422 0.991590i \(-0.458688\pi\)
0.129422 + 0.991590i \(0.458688\pi\)
\(234\) 0 0
\(235\) 9.02715 0.588866
\(236\) − 19.1400i − 1.24591i
\(237\) 0 0
\(238\) 5.55496 0.360074
\(239\) 0.818331i 0.0529334i 0.999650 + 0.0264667i \(0.00842560\pi\)
−0.999650 + 0.0264667i \(0.991574\pi\)
\(240\) 0 0
\(241\) 6.03252i 0.388589i 0.980943 + 0.194295i \(0.0622417\pi\)
−0.980943 + 0.194295i \(0.937758\pi\)
\(242\) 6.77048i 0.435223i
\(243\) 0 0
\(244\) 47.1269 3.01699
\(245\) 23.4698i 1.49943i
\(246\) 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 \(-0.822387\pi\)
−0.848323 + 0.529479i \(0.822387\pi\)
\(252\) 0 0
\(253\) − 22.2078i − 1.39619i
\(254\) − 13.3642i − 0.838544i
\(255\) 0 0
\(256\) −24.5284 −1.53303
\(257\) −9.05323 −0.564725 −0.282362 0.959308i \(-0.591118\pi\)
−0.282362 + 0.959308i \(0.591118\pi\)
\(258\) 0 0
\(259\) 5.97046 0.370986
\(260\) 0 0
\(261\) 0 0
\(262\) − 42.9493i − 2.65342i
\(263\) −23.1511 −1.42756 −0.713778 0.700372i \(-0.753017\pi\)
−0.713778 + 0.700372i \(0.753017\pi\)
\(264\) 0 0
\(265\) 36.7754i 2.25909i
\(266\) 4.62133i 0.283352i
\(267\) 0 0
\(268\) 51.1691i 3.12565i
\(269\) 2.42088 0.147604 0.0738018 0.997273i \(-0.476487\pi\)
0.0738018 + 0.997273i \(0.476487\pi\)
\(270\) 0 0
\(271\) − 21.4450i − 1.30269i −0.758780 0.651347i \(-0.774204\pi\)
0.758780 0.651347i \(-0.225796\pi\)
\(272\) 4.49024 0.272261
\(273\) 0 0
\(274\) 22.2737 1.34560
\(275\) − 24.6058i − 1.48379i
\(276\) 0 0
\(277\) 14.8073 0.889685 0.444843 0.895609i \(-0.353260\pi\)
0.444843 + 0.895609i \(0.353260\pi\)
\(278\) 9.46309i 0.567558i
\(279\) 0 0
\(280\) − 10.8509i − 0.648463i
\(281\) − 14.5036i − 0.865215i −0.901582 0.432608i \(-0.857594\pi\)
0.901582 0.432608i \(-0.142406\pi\)
\(282\) 0 0
\(283\) −25.6722 −1.52605 −0.763026 0.646368i \(-0.776287\pi\)
−0.763026 + 0.646368i \(0.776287\pi\)
\(284\) − 28.8840i − 1.71395i
\(285\) 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
\(292\) 42.2602i 2.47309i
\(293\) 26.5230i 1.54949i 0.632273 + 0.774746i \(0.282122\pi\)
−0.632273 + 0.774746i \(0.717878\pi\)
\(294\) 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
\(301\) − 1.29590i − 0.0746943i
\(302\) 29.1366 1.67662
\(303\) 0 0
\(304\) 3.73556i 0.214249i
\(305\) − 48.9439i − 2.80252i
\(306\) 0 0
\(307\) − 8.24698i − 0.470680i −0.971913 0.235340i \(-0.924380\pi\)
0.971913 0.235340i \(-0.0756204\pi\)
\(308\) −8.12737 −0.463100
\(309\) 0 0
\(310\) 20.4039i 1.15886i
\(311\) −14.4179 −0.817564 −0.408782 0.912632i \(-0.634046\pi\)
−0.408782 + 0.912632i \(0.634046\pi\)
\(312\) 0 0
\(313\) 14.2338 0.804544 0.402272 0.915520i \(-0.368221\pi\)
0.402272 + 0.915520i \(0.368221\pi\)
\(314\) − 44.0210i − 2.48425i
\(315\) 0 0
\(316\) −19.2174 −1.08107
\(317\) − 6.84415i − 0.384406i −0.981355 0.192203i \(-0.938437\pi\)
0.981355 0.192203i \(-0.0615632\pi\)
\(318\) 0 0
\(319\) − 10.9782i − 0.614663i
\(320\) 43.7289i 2.44452i
\(321\) 0 0
\(322\) −14.7235 −0.820507
\(323\) 7.18598i 0.399839i
\(324\) 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
\(331\) − 9.44265i − 0.519015i −0.965741 0.259507i \(-0.916440\pi\)
0.965741 0.259507i \(-0.0835602\pi\)
\(332\) − 25.0586i − 1.37527i
\(333\) 0 0
\(334\) 27.0901 1.48230
\(335\) 53.1420 2.90346
\(336\) 0 0
\(337\) −2.64310 −0.143979 −0.0719895 0.997405i \(-0.522935\pi\)
−0.0719895 + 0.997405i \(0.522935\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 38.5743i − 2.09199i
\(341\) 6.68473 0.361998
\(342\) 0 0
\(343\) − 10.7114i − 0.578361i
\(344\) − 5.92228i − 0.319308i
\(345\) 0 0
\(346\) 28.5520i 1.53496i
\(347\) 10.1588 0.545355 0.272677 0.962106i \(-0.412091\pi\)
0.272677 + 0.962106i \(0.412091\pi\)
\(348\) 0 0
\(349\) 10.4397i 0.558822i 0.960172 + 0.279411i \(0.0901393\pi\)
−0.960172 + 0.279411i \(0.909861\pi\)
\(350\) −16.3134 −0.871986
\(351\) 0 0
\(352\) 10.6304 0.566604
\(353\) 18.2911i 0.973538i 0.873531 + 0.486769i \(0.161825\pi\)
−0.873531 + 0.486769i \(0.838175\pi\)
\(354\) 0 0
\(355\) −29.9976 −1.59211
\(356\) 4.02310i 0.213224i
\(357\) 0 0
\(358\) − 1.26934i − 0.0670867i
\(359\) 15.2731i 0.806081i 0.915182 + 0.403041i \(0.132047\pi\)
−0.915182 + 0.403041i \(0.867953\pi\)
\(360\) 0 0
\(361\) 13.0218 0.685356
\(362\) 54.8775i 2.88430i
\(363\) 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
\(370\) − 64.7846i − 3.36799i
\(371\) − 7.98792i − 0.414712i
\(372\) 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
\(379\) 10.7071i 0.549986i 0.961446 + 0.274993i \(0.0886756\pi\)
−0.961446 + 0.274993i \(0.911324\pi\)
\(380\) 32.0911 1.64624
\(381\) 0 0
\(382\) − 39.5150i − 2.02176i
\(383\) 6.52648i 0.333488i 0.986000 + 0.166744i \(0.0533253\pi\)
−0.986000 + 0.166744i \(0.946675\pi\)
\(384\) 0 0
\(385\) 8.44073i 0.430179i
\(386\) −60.6757 −3.08831
\(387\) 0 0
\(388\) − 21.1320i − 1.07282i
\(389\) −11.7922 −0.597891 −0.298945 0.954270i \(-0.596635\pi\)
−0.298945 + 0.954270i \(0.596635\pi\)
\(390\) 0 0
\(391\) −22.8944 −1.15782
\(392\) − 23.2972i − 1.17669i
\(393\) 0 0
\(394\) −50.4868 −2.54349
\(395\) 19.9584i 1.00422i
\(396\) 0 0
\(397\) − 12.5429i − 0.629509i −0.949173 0.314754i \(-0.898078\pi\)
0.949173 0.314754i \(-0.101922\pi\)
\(398\) − 8.31468i − 0.416777i
\(399\) 0 0
\(400\) −13.1866 −0.659329
\(401\) − 17.8702i − 0.892397i −0.894934 0.446198i \(-0.852778\pi\)
0.894934 0.446198i \(-0.147222\pi\)
\(402\) 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
\(409\) 27.9119i 1.38015i 0.723737 + 0.690076i \(0.242423\pi\)
−0.723737 + 0.690076i \(0.757577\pi\)
\(410\) 7.40389i 0.365652i
\(411\) 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
\(418\) − 16.4286i − 0.803551i
\(419\) 16.4034 0.801360 0.400680 0.916218i \(-0.368774\pi\)
0.400680 + 0.916218i \(0.368774\pi\)
\(420\) 0 0
\(421\) 3.03684i 0.148006i 0.997258 + 0.0740032i \(0.0235775\pi\)
−0.997258 + 0.0740032i \(0.976423\pi\)
\(422\) 2.86486i 0.139459i
\(423\) 0 0
\(424\) − 36.5050i − 1.77284i
\(425\) −25.3666 −1.23046
\(426\) 0 0
\(427\) 10.6310i 0.514471i
\(428\) −33.8471 −1.63606
\(429\) 0 0
\(430\) −14.0616 −0.678110
\(431\) − 3.33811i − 0.160791i −0.996763 0.0803955i \(-0.974382\pi\)
0.996763 0.0803955i \(-0.0256183\pi\)
\(432\) 0 0
\(433\) −11.9028 −0.572010 −0.286005 0.958228i \(-0.592327\pi\)
−0.286005 + 0.958228i \(0.592327\pi\)
\(434\) − 4.43190i − 0.212738i
\(435\) 0 0
\(436\) − 6.34375i − 0.303810i
\(437\) − 19.0465i − 0.911119i
\(438\) 0 0
\(439\) −3.71810 −0.177455 −0.0887277 0.996056i \(-0.528280\pi\)
−0.0887277 + 0.996056i \(0.528280\pi\)
\(440\) 38.5743i 1.83896i
\(441\) 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
\(448\) − 9.49827i − 0.448751i
\(449\) 12.1274i 0.572326i 0.958181 + 0.286163i \(0.0923799\pi\)
−0.958181 + 0.286163i \(0.907620\pi\)
\(450\) 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
\(457\) 3.44803i 0.161292i 0.996743 + 0.0806459i \(0.0256983\pi\)
−0.996743 + 0.0806459i \(0.974302\pi\)
\(458\) −44.2161 −2.06608
\(459\) 0 0
\(460\) 102.242i 4.76704i
\(461\) − 6.75600i − 0.314658i −0.987546 0.157329i \(-0.949712\pi\)
0.987546 0.157329i \(-0.0502884\pi\)
\(462\) 0 0
\(463\) − 7.45175i − 0.346312i −0.984894 0.173156i \(-0.944604\pi\)
0.984894 0.173156i \(-0.0553965\pi\)
\(464\) −5.88338 −0.273129
\(465\) 0 0
\(466\) 9.31229i 0.431384i
\(467\) 32.6098 1.50900 0.754502 0.656298i \(-0.227879\pi\)
0.754502 + 0.656298i \(0.227879\pi\)
\(468\) 0 0
\(469\) −11.5429 −0.533001
\(470\) 21.2760i 0.981391i
\(471\) 0 0
\(472\) 19.7318 0.908232
\(473\) 4.60686i 0.211824i
\(474\) 0 0
\(475\) − 21.1032i − 0.968282i
\(476\) 8.37867i 0.384036i
\(477\) 0 0
\(478\) −1.92872 −0.0882177
\(479\) − 2.82908i − 0.129264i −0.997909 0.0646321i \(-0.979413\pi\)
0.997909 0.0646321i \(-0.0205874\pi\)
\(480\) 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
\(487\) 41.2935i 1.87119i 0.353079 + 0.935594i \(0.385135\pi\)
−0.353079 + 0.935594i \(0.614865\pi\)
\(488\) 48.5840i 2.19930i
\(489\) 0 0
\(490\) −55.3159 −2.49892
\(491\) −34.6698 −1.56463 −0.782313 0.622886i \(-0.785960\pi\)
−0.782313 + 0.622886i \(0.785960\pi\)
\(492\) 0 0
\(493\) −11.3177 −0.509722
\(494\) 0 0
\(495\) 0 0
\(496\) − 3.58243i − 0.160856i
\(497\) 6.51573 0.292270
\(498\) 0 0
\(499\) − 17.9409i − 0.803146i −0.915827 0.401573i \(-0.868464\pi\)
0.915827 0.401573i \(-0.131536\pi\)
\(500\) 47.6571i 2.13129i
\(501\) 0 0
\(502\) − 63.3532i − 2.82759i
\(503\) 26.1812 1.16736 0.583681 0.811983i \(-0.301612\pi\)
0.583681 + 0.811983i \(0.301612\pi\)
\(504\) 0 0
\(505\) 17.0780i 0.759960i
\(506\) 52.3414 2.32686
\(507\) 0 0
\(508\) 20.1575 0.894345
\(509\) − 5.50604i − 0.244051i −0.992527 0.122025i \(-0.961061\pi\)
0.992527 0.122025i \(-0.0389390\pi\)
\(510\) 0 0
\(511\) −9.53319 −0.421723
\(512\) − 16.8955i − 0.746681i
\(513\) 0 0
\(514\) − 21.3375i − 0.941158i
\(515\) − 4.45904i − 0.196489i
\(516\) 0 0
\(517\) 6.97046 0.306560
\(518\) 14.0718i 0.618277i
\(519\) 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
\(526\) − 54.5646i − 2.37913i
\(527\) − 6.89141i − 0.300195i
\(528\) 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
\(535\) 35.1521i 1.51976i
\(536\) −52.7512 −2.27851
\(537\) 0 0
\(538\) 5.70576i 0.245993i
\(539\) 18.1226i 0.780595i
\(540\) 0 0
\(541\) 18.4655i 0.793893i 0.917842 + 0.396947i \(0.129930\pi\)
−0.917842 + 0.396947i \(0.870070\pi\)
\(542\) 50.5437 2.17104
\(543\) 0 0
\(544\) − 10.9591i − 0.469868i
\(545\) −6.58834 −0.282213
\(546\) 0 0
\(547\) 39.8471 1.70374 0.851870 0.523753i \(-0.175468\pi\)
0.851870 + 0.523753i \(0.175468\pi\)
\(548\) 33.5958i 1.43514i
\(549\) 0 0
\(550\) 57.9933 2.47284
\(551\) − 9.41550i − 0.401114i
\(552\) 0 0
\(553\) − 4.33513i − 0.184348i
\(554\) 34.8993i 1.48273i
\(555\) 0 0
\(556\) −14.2734 −0.605327
\(557\) − 9.20477i − 0.390018i −0.980801 0.195009i \(-0.937526\pi\)
0.980801 0.195009i \(-0.0624737\pi\)
\(558\) 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 \(-0.506542\pi\)
−0.0205509 + 0.999789i \(0.506542\pi\)
\(564\) 0 0
\(565\) − 18.2795i − 0.769024i
\(566\) − 60.5066i − 2.54328i
\(567\) 0 0
\(568\) 29.7770 1.24942
\(569\) −16.8944 −0.708250 −0.354125 0.935198i \(-0.615221\pi\)
−0.354125 + 0.935198i \(0.615221\pi\)
\(570\) 0 0
\(571\) 44.3226 1.85484 0.927421 0.374019i \(-0.122021\pi\)
0.927421 + 0.374019i \(0.122021\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 1.60819i − 0.0671244i
\(575\) 67.2344 2.80387
\(576\) 0 0
\(577\) − 3.56704i − 0.148498i −0.997240 0.0742489i \(-0.976344\pi\)
0.997240 0.0742489i \(-0.0236559\pi\)
\(578\) − 19.7090i − 0.819787i
\(579\) 0 0
\(580\) 50.5424i 2.09866i
\(581\) 5.65279 0.234517
\(582\) 0 0
\(583\) 28.3967i 1.17607i
\(584\) −43.5669 −1.80281
\(585\) 0 0
\(586\) −62.5120 −2.58235
\(587\) − 16.1172i − 0.665229i −0.943063 0.332614i \(-0.892069\pi\)
0.943063 0.332614i \(-0.107931\pi\)
\(588\) 0 0
\(589\) 5.73317 0.236231
\(590\) − 46.8504i − 1.92880i
\(591\) 0 0
\(592\) 11.3746i 0.467494i
\(593\) 42.8611i 1.76010i 0.474885 + 0.880048i \(0.342490\pi\)
−0.474885 + 0.880048i \(0.657510\pi\)
\(594\) 0 0
\(595\) 8.70171 0.356735
\(596\) − 68.9869i − 2.82581i
\(597\) 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
\(604\) 43.9474i 1.78819i
\(605\) 10.6058i 0.431187i
\(606\) 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
\(613\) 33.3618i 1.34747i 0.738973 + 0.673735i \(0.235311\pi\)
−0.738973 + 0.673735i \(0.764689\pi\)
\(614\) 19.4373 0.784424
\(615\) 0 0
\(616\) − 8.37867i − 0.337586i
\(617\) 11.6233i 0.467935i 0.972244 + 0.233967i \(0.0751709\pi\)
−0.972244 + 0.233967i \(0.924829\pi\)
\(618\) 0 0
\(619\) − 16.5381i − 0.664722i −0.943152 0.332361i \(-0.892155\pi\)
0.943152 0.332361i \(-0.107845\pi\)
\(620\) −30.7756 −1.23598
\(621\) 0 0
\(622\) − 33.9815i − 1.36253i
\(623\) −0.907542 −0.0363599
\(624\) 0 0
\(625\) 6.33944 0.253577
\(626\) 33.5477i 1.34083i
\(627\) 0 0
\(628\) 66.3979 2.64956
\(629\) 21.8810i 0.872452i
\(630\) 0 0
\(631\) − 36.4416i − 1.45072i −0.688372 0.725358i \(-0.741674\pi\)
0.688372 0.725358i \(-0.258326\pi\)
\(632\) − 19.8116i − 0.788064i
\(633\) 0 0
\(634\) 16.1309 0.640642
\(635\) − 20.9347i − 0.830768i
\(636\) 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 \(-0.319444\pi\)
0.537300 + 0.843391i \(0.319444\pi\)
\(642\) 0 0
\(643\) 5.06962i 0.199926i 0.994991 + 0.0999632i \(0.0318725\pi\)
−0.994991 + 0.0999632i \(0.968127\pi\)
\(644\) − 22.2078i − 0.875108i
\(645\) 0 0
\(646\) −16.9366 −0.666362
\(647\) 19.3207 0.759573 0.379787 0.925074i \(-0.375997\pi\)
0.379787 + 0.925074i \(0.375997\pi\)
\(648\) 0 0
\(649\) −15.3491 −0.602506
\(650\) 0 0
\(651\) 0 0
\(652\) − 43.8662i − 1.71793i
\(653\) −35.2355 −1.37887 −0.689436 0.724347i \(-0.742141\pi\)
−0.689436 + 0.724347i \(0.742141\pi\)
\(654\) 0 0
\(655\) − 67.2790i − 2.62881i
\(656\) − 1.29995i − 0.0507544i
\(657\) 0 0
\(658\) − 4.62133i − 0.180158i
\(659\) 4.36168 0.169907 0.0849535 0.996385i \(-0.472926\pi\)
0.0849535 + 0.996385i \(0.472926\pi\)
\(660\) 0 0
\(661\) − 15.4709i − 0.601747i −0.953664 0.300873i \(-0.902722\pi\)
0.953664 0.300873i \(-0.0972782\pi\)
\(662\) 22.2553 0.864978
\(663\) 0 0
\(664\) 25.8334 1.00253
\(665\) 7.23921i 0.280725i
\(666\) 0 0
\(667\) 29.9976 1.16151
\(668\) 40.8605i 1.58094i
\(669\) 0 0
\(670\) 125.250i 4.83883i
\(671\) − 37.7928i − 1.45898i
\(672\) 0 0
\(673\) 11.7409 0.452580 0.226290 0.974060i \(-0.427340\pi\)
0.226290 + 0.974060i \(0.427340\pi\)
\(674\) − 6.22952i − 0.239952i
\(675\) 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
\(682\) 15.7552i 0.603298i
\(683\) − 20.4058i − 0.780807i −0.920644 0.390403i \(-0.872336\pi\)
0.920644 0.390403i \(-0.127664\pi\)
\(684\) 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
\(691\) − 27.8039i − 1.05771i −0.848713 0.528854i \(-0.822622\pi\)
0.848713 0.528854i \(-0.177378\pi\)
\(692\) −43.0656 −1.63711
\(693\) 0 0
\(694\) 23.9433i 0.908876i
\(695\) 14.8237i 0.562295i
\(696\) 0 0
\(697\) − 2.50066i − 0.0947194i
\(698\) −24.6052 −0.931321
\(699\) 0 0
\(700\) − 24.6058i − 0.930012i
\(701\) 11.9715 0.452158 0.226079 0.974109i \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(702\) 0 0
\(703\) −18.2034 −0.686556
\(704\) 33.7660i 1.27260i
\(705\) 0 0
\(706\) −43.1102 −1.62248
\(707\) − 3.70948i − 0.139509i
\(708\) 0 0
\(709\) 32.2664i 1.21179i 0.795545 + 0.605894i \(0.207185\pi\)
−0.795545 + 0.605894i \(0.792815\pi\)
\(710\) − 70.7012i − 2.65337i
\(711\) 0 0
\(712\) −4.14749 −0.155434
\(713\) 18.2658i 0.684058i
\(714\) 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 \(-0.427505\pi\)
0.225787 + 0.974177i \(0.427505\pi\)
\(720\) 0 0
\(721\) 0.968541i 0.0360704i
\(722\) 30.6910i 1.14220i
\(723\) 0 0
\(724\) −82.7730 −3.07623
\(725\) 33.2368 1.23438
\(726\) 0 0
\(727\) 16.6200 0.616402 0.308201 0.951321i \(-0.400273\pi\)
0.308201 + 0.951321i \(0.400273\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 103.443i 3.82861i
\(731\) 4.74930 0.175659
\(732\) 0 0
\(733\) − 17.7912i − 0.657132i −0.944481 0.328566i \(-0.893435\pi\)
0.944481 0.328566i \(-0.106565\pi\)
\(734\) − 52.4922i − 1.93752i
\(735\) 0 0
\(736\) 29.0473i 1.07070i
\(737\) 41.0344 1.51152
\(738\) 0 0
\(739\) − 27.3618i − 1.00652i −0.864135 0.503260i \(-0.832134\pi\)
0.864135 0.503260i \(-0.167866\pi\)
\(740\) 97.7160 3.59211
\(741\) 0 0
\(742\) 18.8267 0.691150
\(743\) 8.38596i 0.307651i 0.988098 + 0.153826i \(0.0491594\pi\)
−0.988098 + 0.153826i \(0.950841\pi\)
\(744\) 0 0
\(745\) −71.6467 −2.62493
\(746\) 9.72528i 0.356068i
\(747\) 0 0
\(748\) − 29.7858i − 1.08908i
\(749\) − 7.63533i − 0.278989i
\(750\) 0 0
\(751\) 38.7778 1.41502 0.707511 0.706703i \(-0.249818\pi\)
0.707511 + 0.706703i \(0.249818\pi\)
\(752\) − 3.73556i − 0.136222i
\(753\) 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
\(760\) 33.0834i 1.20006i
\(761\) 5.15585i 0.186899i 0.995624 + 0.0934497i \(0.0297894\pi\)
−0.995624 + 0.0934497i \(0.970211\pi\)
\(762\) 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
\(769\) 35.5013i 1.28021i 0.768288 + 0.640104i \(0.221109\pi\)
−0.768288 + 0.640104i \(0.778891\pi\)
\(770\) −19.8939 −0.716927
\(771\) 0 0
\(772\) − 91.5186i − 3.29383i
\(773\) − 6.15585i − 0.221411i −0.993853 0.110705i \(-0.964689\pi\)
0.993853 0.110705i \(-0.0353110\pi\)
\(774\) 0 0
\(775\) 20.2381i 0.726976i
\(776\) 21.7854 0.782050
\(777\) 0 0
\(778\) − 27.7931i − 0.996431i
\(779\) 2.08038 0.0745372
\(780\) 0 0
\(781\) −23.1631 −0.828843
\(782\) − 53.9597i − 1.92960i
\(783\) 0 0
\(784\) 9.71214 0.346862
\(785\) − 68.9579i − 2.46121i
\(786\) 0 0
\(787\) 14.2107i 0.506558i 0.967393 + 0.253279i \(0.0815091\pi\)
−0.967393 + 0.253279i \(0.918491\pi\)
\(788\) − 76.1503i − 2.71274i
\(789\) 0 0
\(790\) −47.0398 −1.67360
\(791\) 3.97046i 0.141173i
\(792\) 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 \(-0.432394\pi\)
0.210798 + 0.977530i \(0.432394\pi\)
\(798\) 0 0
\(799\) − 7.18598i − 0.254222i
\(800\) 32.1839i 1.13787i
\(801\) 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
\(808\) − 16.9524i − 0.596384i
\(809\) −1.61596 −0.0568140 −0.0284070 0.999596i \(-0.509043\pi\)
−0.0284070 + 0.999596i \(0.509043\pi\)
\(810\) 0 0
\(811\) 51.9657i 1.82476i 0.409342 + 0.912381i \(0.365758\pi\)
−0.409342 + 0.912381i \(0.634242\pi\)
\(812\) − 10.9782i − 0.385260i
\(813\) 0 0
\(814\) − 50.0245i − 1.75336i
\(815\) −45.5575 −1.59581
\(816\) 0 0
\(817\) 3.95108i 0.138231i
\(818\) −65.7853 −2.30013
\(819\) 0 0
\(820\) −11.1675 −0.389985
\(821\) 54.8327i 1.91367i 0.290627 + 0.956837i \(0.406136\pi\)
−0.290627 + 0.956837i \(0.593864\pi\)
\(822\) 0 0
\(823\) 46.8514 1.63314 0.816569 0.577247i \(-0.195873\pi\)
0.816569 + 0.577247i \(0.195873\pi\)
\(824\) 4.42626i 0.154196i
\(825\) 0 0
\(826\) 10.1763i 0.354078i
\(827\) − 21.2021i − 0.737270i −0.929574 0.368635i \(-0.879825\pi\)
0.929574 0.368635i \(-0.120175\pi\)
\(828\) 0 0
\(829\) −18.2972 −0.635489 −0.317744 0.948176i \(-0.602925\pi\)
−0.317744 + 0.948176i \(0.602925\pi\)
\(830\) − 61.3376i − 2.12906i
\(831\) 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
\(838\) 38.6612i 1.33553i
\(839\) − 21.1414i − 0.729881i −0.931031 0.364941i \(-0.881089\pi\)
0.931031 0.364941i \(-0.118911\pi\)
\(840\) 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
\(847\) − 2.30367i − 0.0791549i
\(848\) 15.2182 0.522594
\(849\) 0 0
\(850\) − 59.7864i − 2.05066i
\(851\) − 57.9958i − 1.98807i
\(852\) 0 0
\(853\) − 7.13036i − 0.244139i −0.992522 0.122069i \(-0.961047\pi\)
0.992522 0.122069i \(-0.0389531\pi\)
\(854\) −25.0562 −0.857406
\(855\) 0 0
\(856\) − 34.8937i − 1.19264i
\(857\) −44.7741 −1.52945 −0.764726 0.644355i \(-0.777126\pi\)
−0.764726 + 0.644355i \(0.777126\pi\)
\(858\) 0 0
\(859\) 57.3782 1.95772 0.978859 0.204535i \(-0.0655681\pi\)
0.978859 + 0.204535i \(0.0655681\pi\)
\(860\) − 21.2094i − 0.723235i
\(861\) 0 0
\(862\) 7.86758 0.267971
\(863\) − 6.67563i − 0.227241i −0.993524 0.113621i \(-0.963755\pi\)
0.993524 0.113621i \(-0.0362448\pi\)
\(864\) 0 0
\(865\) 44.7260i 1.52073i
\(866\) − 28.0536i − 0.953299i
\(867\) 0 0
\(868\) 6.68473 0.226894
\(869\) 15.4112i 0.522789i
\(870\) 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
\(877\) − 25.2983i − 0.854263i −0.904190 0.427131i \(-0.859524\pi\)
0.904190 0.427131i \(-0.140476\pi\)
\(878\) − 8.76318i − 0.295743i
\(879\) 0 0
\(880\) −16.0809 −0.542085
\(881\) −35.3787 −1.19194 −0.595969 0.803008i \(-0.703232\pi\)
−0.595969 + 0.803008i \(0.703232\pi\)
\(882\) 0 0
\(883\) 11.5851 0.389869 0.194935 0.980816i \(-0.437551\pi\)
0.194935 + 0.980816i \(0.437551\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 3.43429i − 0.115377i
\(887\) 27.2892 0.916281 0.458141 0.888880i \(-0.348516\pi\)
0.458141 + 0.888880i \(0.348516\pi\)
\(888\) 0 0
\(889\) 4.54719i 0.152508i
\(890\) 9.84761i 0.330093i
\(891\) 0 0
\(892\) − 61.8149i − 2.06972i
\(893\) 5.97823 0.200054
\(894\) 0 0
\(895\) − 1.98839i − 0.0664646i
\(896\) 16.4058 0.548080
\(897\) 0 0
\(898\) −28.5830 −0.953826
\(899\) 9.02954i 0.301152i
\(900\) 0 0
\(901\) 29.2747 0.975282
\(902\) 5.71704i 0.190357i
\(903\) 0 0
\(904\) 18.1451i 0.603497i
\(905\) 85.9643i 2.85755i
\(906\) 0 0
\(907\) 30.4219 1.01014 0.505072 0.863077i \(-0.331466\pi\)
0.505072 + 0.863077i \(0.331466\pi\)
\(908\) − 61.9114i − 2.05460i
\(909\) 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
\(916\) − 66.6921i − 2.20357i
\(917\) 14.6136i 0.482582i
\(918\) 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
\(925\) − 64.2583i − 2.11280i
\(926\) 17.5630 0.577156
\(927\) 0 0
\(928\) 14.3593i 0.471367i
\(929\) − 13.3478i − 0.437927i −0.975733 0.218964i \(-0.929732\pi\)
0.975733 0.218964i \(-0.0702676\pi\)
\(930\) 0 0
\(931\) 15.5429i 0.509397i
\(932\) −14.0459 −0.460090
\(933\) 0 0
\(934\) 76.8580i 2.51487i
\(935\) −30.9342 −1.01166
\(936\) 0 0
\(937\) 38.6872 1.26386 0.631928 0.775027i \(-0.282264\pi\)
0.631928 + 0.775027i \(0.282264\pi\)
\(938\) − 27.2054i − 0.888286i
\(939\) 0 0
\(940\) −32.0911 −1.04670
\(941\) 35.7275i 1.16468i 0.812944 + 0.582342i \(0.197864\pi\)
−0.812944 + 0.582342i \(0.802136\pi\)
\(942\) 0 0
\(943\) 6.62804i 0.215839i
\(944\) 8.22580i 0.267727i
\(945\) 0 0
\(946\) −10.8579 −0.353020
\(947\) 17.8436i 0.579838i 0.957051 + 0.289919i \(0.0936283\pi\)
−0.957051 + 0.289919i \(0.906372\pi\)
\(948\) 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 \(-0.394073\pi\)
0.326671 + 0.945138i \(0.394073\pi\)
\(954\) 0 0
\(955\) − 61.8993i − 2.00301i
\(956\) − 2.90913i − 0.0940881i
\(957\) 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
\(964\) − 21.4454i − 0.690709i
\(965\) −95.0471 −3.05967
\(966\) 0 0
\(967\) 32.7894i 1.05444i 0.849730 + 0.527218i \(0.176765\pi\)
−0.849730 + 0.527218i \(0.823235\pi\)
\(968\) − 10.5278i − 0.338377i
\(969\) 0 0
\(970\) − 51.7263i − 1.66083i
\(971\) 26.9845 0.865973 0.432986 0.901401i \(-0.357460\pi\)
0.432986 + 0.901401i \(0.357460\pi\)
\(972\) 0 0
\(973\) − 3.21983i − 0.103223i
\(974\) −97.3245 −3.11848
\(975\) 0 0
\(976\) −20.2537 −0.648305
\(977\) − 16.9571i − 0.542504i −0.962508 0.271252i \(-0.912562\pi\)
0.962508 0.271252i \(-0.0874377\pi\)
\(978\) 0 0
\(979\) 3.22627 0.103112
\(980\) − 83.4341i − 2.66521i
\(981\) 0 0
\(982\) − 81.7131i − 2.60757i
\(983\) 32.6631i 1.04179i 0.853621 + 0.520895i \(0.174402\pi\)
−0.853621 + 0.520895i \(0.825598\pi\)
\(984\) 0 0
\(985\) −79.0863 −2.51990
\(986\) − 26.6746i − 0.849491i
\(987\) 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
\(994\) 15.3569i 0.487091i
\(995\) − 13.0248i − 0.412912i
\(996\) 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.b.k.1351.5 6
3.2 odd 2 507.2.b.f.337.2 6
13.5 odd 4 1521.2.a.s.1.2 3
13.8 odd 4 1521.2.a.n.1.2 3
13.12 even 2 inner 1521.2.b.k.1351.2 6
39.2 even 12 507.2.e.l.22.2 6
39.5 even 4 507.2.a.i.1.2 3
39.8 even 4 507.2.a.l.1.2 yes 3
39.11 even 12 507.2.e.i.22.2 6
39.17 odd 6 507.2.j.i.361.5 12
39.20 even 12 507.2.e.i.484.2 6
39.23 odd 6 507.2.j.i.316.2 12
39.29 odd 6 507.2.j.i.316.5 12
39.32 even 12 507.2.e.l.484.2 6
39.35 odd 6 507.2.j.i.361.2 12
39.38 odd 2 507.2.b.f.337.5 6
156.47 odd 4 8112.2.a.cp.1.3 3
156.83 odd 4 8112.2.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 39.5 even 4
507.2.a.l.1.2 yes 3 39.8 even 4
507.2.b.f.337.2 6 3.2 odd 2
507.2.b.f.337.5 6 39.38 odd 2
507.2.e.i.22.2 6 39.11 even 12
507.2.e.i.484.2 6 39.20 even 12
507.2.e.l.22.2 6 39.2 even 12
507.2.e.l.484.2 6 39.32 even 12
507.2.j.i.316.2 12 39.23 odd 6
507.2.j.i.316.5 12 39.29 odd 6
507.2.j.i.361.2 12 39.35 odd 6
507.2.j.i.361.5 12 39.17 odd 6
1521.2.a.n.1.2 3 13.8 odd 4
1521.2.a.s.1.2 3 13.5 odd 4
1521.2.b.k.1351.2 6 13.12 even 2 inner
1521.2.b.k.1351.5 6 1.1 even 1 trivial
8112.2.a.cg.1.1 3 156.83 odd 4
8112.2.a.cp.1.3 3 156.47 odd 4