Properties

Label 525.2.a.k.1.2
Level 525
Weight 2
Character 525.1
Self dual yes
Analytic conductor 4.192
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of \(x^{3} - x^{2} - 3 x + 1\)
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -1.00000 q^{7} -0.768452 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -1.00000 q^{7} -0.768452 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.96239 q^{12} +1.35026 q^{13} -0.193937 q^{14} +3.77575 q^{16} -3.35026 q^{17} +0.193937 q^{18} +5.35026 q^{19} +1.00000 q^{21} +0.387873 q^{22} +4.96239 q^{23} +0.768452 q^{24} +0.261865 q^{26} -1.00000 q^{27} +1.96239 q^{28} +7.92478 q^{29} +4.57452 q^{31} +2.26916 q^{32} -2.00000 q^{33} -0.649738 q^{34} -1.96239 q^{36} -0.775746 q^{37} +1.03761 q^{38} -1.35026 q^{39} +3.73813 q^{41} +0.193937 q^{42} -12.6253 q^{43} -3.92478 q^{44} +0.962389 q^{46} +9.92478 q^{47} -3.77575 q^{48} +1.00000 q^{49} +3.35026 q^{51} -2.64974 q^{52} +8.57452 q^{53} -0.193937 q^{54} +0.768452 q^{56} -5.35026 q^{57} +1.53690 q^{58} -8.62530 q^{59} -8.70052 q^{61} +0.887166 q^{62} -1.00000 q^{63} -7.11142 q^{64} -0.387873 q^{66} +9.92478 q^{67} +6.57452 q^{68} -4.96239 q^{69} +2.00000 q^{71} -0.768452 q^{72} -9.35026 q^{73} -0.150446 q^{74} -10.4993 q^{76} -2.00000 q^{77} -0.261865 q^{78} +10.7005 q^{79} +1.00000 q^{81} +0.724961 q^{82} +3.22425 q^{83} -1.96239 q^{84} -2.44851 q^{86} -7.92478 q^{87} -1.53690 q^{88} +1.03761 q^{89} -1.35026 q^{91} -9.73813 q^{92} -4.57452 q^{93} +1.92478 q^{94} -2.26916 q^{96} -18.4993 q^{97} +0.193937 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} - 3q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} - 3q^{7} + 9q^{8} + 3q^{9} + 6q^{11} - 5q^{12} - 6q^{13} - q^{14} + 13q^{16} + q^{18} + 6q^{19} + 3q^{21} + 2q^{22} + 4q^{23} - 9q^{24} + 10q^{26} - 3q^{27} - 5q^{28} + 2q^{29} + 2q^{31} + 29q^{32} - 6q^{33} - 12q^{34} + 5q^{36} - 4q^{37} + 14q^{38} + 6q^{39} + 2q^{41} + q^{42} + 4q^{43} + 10q^{44} - 8q^{46} + 8q^{47} - 13q^{48} + 3q^{49} - 18q^{52} + 14q^{53} - q^{54} - 9q^{56} - 6q^{57} - 18q^{58} + 16q^{59} - 6q^{61} - 30q^{62} - 3q^{63} + 13q^{64} - 2q^{66} + 8q^{67} + 8q^{68} - 4q^{69} + 6q^{71} + 9q^{72} - 18q^{73} - 44q^{74} + 2q^{76} - 6q^{77} - 10q^{78} + 12q^{79} + 3q^{81} + 34q^{82} + 8q^{83} + 5q^{84} - 4q^{86} - 2q^{87} + 18q^{88} + 14q^{89} + 6q^{91} - 20q^{92} - 2q^{93} - 16q^{94} - 29q^{96} - 22q^{97} + q^{98} + 6q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.193937 0.137134 0.0685669 0.997647i \(-0.478157\pi\)
0.0685669 + 0.997647i \(0.478157\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.96239 −0.981194
\(5\) 0 0
\(6\) −0.193937 −0.0791743
\(7\) −1.00000 −0.377964
\(8\) −0.768452 −0.271689
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.96239 0.566493
\(13\) 1.35026 0.374495 0.187248 0.982313i \(-0.440043\pi\)
0.187248 + 0.982313i \(0.440043\pi\)
\(14\) −0.193937 −0.0518317
\(15\) 0 0
\(16\) 3.77575 0.943937
\(17\) −3.35026 −0.812558 −0.406279 0.913749i \(-0.633174\pi\)
−0.406279 + 0.913749i \(0.633174\pi\)
\(18\) 0.193937 0.0457113
\(19\) 5.35026 1.22743 0.613717 0.789526i \(-0.289674\pi\)
0.613717 + 0.789526i \(0.289674\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0.387873 0.0826948
\(23\) 4.96239 1.03473 0.517365 0.855765i \(-0.326913\pi\)
0.517365 + 0.855765i \(0.326913\pi\)
\(24\) 0.768452 0.156860
\(25\) 0 0
\(26\) 0.261865 0.0513560
\(27\) −1.00000 −0.192450
\(28\) 1.96239 0.370857
\(29\) 7.92478 1.47159 0.735797 0.677202i \(-0.236808\pi\)
0.735797 + 0.677202i \(0.236808\pi\)
\(30\) 0 0
\(31\) 4.57452 0.821607 0.410804 0.911724i \(-0.365248\pi\)
0.410804 + 0.911724i \(0.365248\pi\)
\(32\) 2.26916 0.401134
\(33\) −2.00000 −0.348155
\(34\) −0.649738 −0.111429
\(35\) 0 0
\(36\) −1.96239 −0.327065
\(37\) −0.775746 −0.127532 −0.0637660 0.997965i \(-0.520311\pi\)
−0.0637660 + 0.997965i \(0.520311\pi\)
\(38\) 1.03761 0.168323
\(39\) −1.35026 −0.216215
\(40\) 0 0
\(41\) 3.73813 0.583799 0.291899 0.956449i \(-0.405713\pi\)
0.291899 + 0.956449i \(0.405713\pi\)
\(42\) 0.193937 0.0299251
\(43\) −12.6253 −1.92534 −0.962670 0.270677i \(-0.912752\pi\)
−0.962670 + 0.270677i \(0.912752\pi\)
\(44\) −3.92478 −0.591682
\(45\) 0 0
\(46\) 0.962389 0.141896
\(47\) 9.92478 1.44768 0.723839 0.689969i \(-0.242376\pi\)
0.723839 + 0.689969i \(0.242376\pi\)
\(48\) −3.77575 −0.544982
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.35026 0.469130
\(52\) −2.64974 −0.367453
\(53\) 8.57452 1.17780 0.588900 0.808206i \(-0.299561\pi\)
0.588900 + 0.808206i \(0.299561\pi\)
\(54\) −0.193937 −0.0263914
\(55\) 0 0
\(56\) 0.768452 0.102689
\(57\) −5.35026 −0.708659
\(58\) 1.53690 0.201805
\(59\) −8.62530 −1.12292 −0.561459 0.827504i \(-0.689760\pi\)
−0.561459 + 0.827504i \(0.689760\pi\)
\(60\) 0 0
\(61\) −8.70052 −1.11399 −0.556994 0.830517i \(-0.688045\pi\)
−0.556994 + 0.830517i \(0.688045\pi\)
\(62\) 0.887166 0.112670
\(63\) −1.00000 −0.125988
\(64\) −7.11142 −0.888927
\(65\) 0 0
\(66\) −0.387873 −0.0477439
\(67\) 9.92478 1.21250 0.606252 0.795272i \(-0.292672\pi\)
0.606252 + 0.795272i \(0.292672\pi\)
\(68\) 6.57452 0.797277
\(69\) −4.96239 −0.597401
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −0.768452 −0.0905629
\(73\) −9.35026 −1.09437 −0.547183 0.837013i \(-0.684300\pi\)
−0.547183 + 0.837013i \(0.684300\pi\)
\(74\) −0.150446 −0.0174889
\(75\) 0 0
\(76\) −10.4993 −1.20435
\(77\) −2.00000 −0.227921
\(78\) −0.261865 −0.0296504
\(79\) 10.7005 1.20390 0.601951 0.798533i \(-0.294390\pi\)
0.601951 + 0.798533i \(0.294390\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.724961 0.0800586
\(83\) 3.22425 0.353908 0.176954 0.984219i \(-0.443376\pi\)
0.176954 + 0.984219i \(0.443376\pi\)
\(84\) −1.96239 −0.214114
\(85\) 0 0
\(86\) −2.44851 −0.264029
\(87\) −7.92478 −0.849625
\(88\) −1.53690 −0.163835
\(89\) 1.03761 0.109987 0.0549933 0.998487i \(-0.482486\pi\)
0.0549933 + 0.998487i \(0.482486\pi\)
\(90\) 0 0
\(91\) −1.35026 −0.141546
\(92\) −9.73813 −1.01527
\(93\) −4.57452 −0.474355
\(94\) 1.92478 0.198526
\(95\) 0 0
\(96\) −2.26916 −0.231595
\(97\) −18.4993 −1.87832 −0.939159 0.343482i \(-0.888394\pi\)
−0.939159 + 0.343482i \(0.888394\pi\)
\(98\) 0.193937 0.0195906
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −17.6629 −1.75753 −0.878763 0.477259i \(-0.841630\pi\)
−0.878763 + 0.477259i \(0.841630\pi\)
\(102\) 0.649738 0.0643337
\(103\) 6.70052 0.660222 0.330111 0.943942i \(-0.392914\pi\)
0.330111 + 0.943942i \(0.392914\pi\)
\(104\) −1.03761 −0.101746
\(105\) 0 0
\(106\) 1.66291 0.161516
\(107\) 13.7381 1.32812 0.664058 0.747681i \(-0.268833\pi\)
0.664058 + 0.747681i \(0.268833\pi\)
\(108\) 1.96239 0.188831
\(109\) −2.77575 −0.265868 −0.132934 0.991125i \(-0.542440\pi\)
−0.132934 + 0.991125i \(0.542440\pi\)
\(110\) 0 0
\(111\) 0.775746 0.0736306
\(112\) −3.77575 −0.356774
\(113\) 12.0508 1.13364 0.566821 0.823841i \(-0.308173\pi\)
0.566821 + 0.823841i \(0.308173\pi\)
\(114\) −1.03761 −0.0971812
\(115\) 0 0
\(116\) −15.5515 −1.44392
\(117\) 1.35026 0.124832
\(118\) −1.67276 −0.153990
\(119\) 3.35026 0.307118
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −1.68735 −0.152765
\(123\) −3.73813 −0.337056
\(124\) −8.97698 −0.806156
\(125\) 0 0
\(126\) −0.193937 −0.0172772
\(127\) 2.70052 0.239633 0.119816 0.992796i \(-0.461769\pi\)
0.119816 + 0.992796i \(0.461769\pi\)
\(128\) −5.91748 −0.523037
\(129\) 12.6253 1.11160
\(130\) 0 0
\(131\) 20.6253 1.80204 0.901020 0.433777i \(-0.142819\pi\)
0.901020 + 0.433777i \(0.142819\pi\)
\(132\) 3.92478 0.341608
\(133\) −5.35026 −0.463927
\(134\) 1.92478 0.166275
\(135\) 0 0
\(136\) 2.57452 0.220763
\(137\) 22.4993 1.92224 0.961122 0.276124i \(-0.0890499\pi\)
0.961122 + 0.276124i \(0.0890499\pi\)
\(138\) −0.962389 −0.0819240
\(139\) −3.27504 −0.277785 −0.138893 0.990307i \(-0.544354\pi\)
−0.138893 + 0.990307i \(0.544354\pi\)
\(140\) 0 0
\(141\) −9.92478 −0.835817
\(142\) 0.387873 0.0325496
\(143\) 2.70052 0.225829
\(144\) 3.77575 0.314646
\(145\) 0 0
\(146\) −1.81336 −0.150075
\(147\) −1.00000 −0.0824786
\(148\) 1.52232 0.125134
\(149\) −4.44851 −0.364436 −0.182218 0.983258i \(-0.558328\pi\)
−0.182218 + 0.983258i \(0.558328\pi\)
\(150\) 0 0
\(151\) 1.29948 0.105750 0.0528749 0.998601i \(-0.483162\pi\)
0.0528749 + 0.998601i \(0.483162\pi\)
\(152\) −4.11142 −0.333480
\(153\) −3.35026 −0.270853
\(154\) −0.387873 −0.0312557
\(155\) 0 0
\(156\) 2.64974 0.212149
\(157\) −2.64974 −0.211472 −0.105736 0.994394i \(-0.533720\pi\)
−0.105736 + 0.994394i \(0.533720\pi\)
\(158\) 2.07522 0.165096
\(159\) −8.57452 −0.680003
\(160\) 0 0
\(161\) −4.96239 −0.391091
\(162\) 0.193937 0.0152371
\(163\) 5.29948 0.415087 0.207544 0.978226i \(-0.433453\pi\)
0.207544 + 0.978226i \(0.433453\pi\)
\(164\) −7.33567 −0.572820
\(165\) 0 0
\(166\) 0.625301 0.0485327
\(167\) −14.5501 −1.12592 −0.562959 0.826485i \(-0.690337\pi\)
−0.562959 + 0.826485i \(0.690337\pi\)
\(168\) −0.768452 −0.0592874
\(169\) −11.1768 −0.859753
\(170\) 0 0
\(171\) 5.35026 0.409145
\(172\) 24.7757 1.88913
\(173\) −4.49929 −0.342075 −0.171037 0.985265i \(-0.554712\pi\)
−0.171037 + 0.985265i \(0.554712\pi\)
\(174\) −1.53690 −0.116512
\(175\) 0 0
\(176\) 7.55149 0.569215
\(177\) 8.62530 0.648317
\(178\) 0.201231 0.0150829
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 10.6253 0.789772 0.394886 0.918730i \(-0.370784\pi\)
0.394886 + 0.918730i \(0.370784\pi\)
\(182\) −0.261865 −0.0194107
\(183\) 8.70052 0.643161
\(184\) −3.81336 −0.281124
\(185\) 0 0
\(186\) −0.887166 −0.0650502
\(187\) −6.70052 −0.489991
\(188\) −19.4763 −1.42045
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −13.8496 −1.00212 −0.501059 0.865413i \(-0.667056\pi\)
−0.501059 + 0.865413i \(0.667056\pi\)
\(192\) 7.11142 0.513222
\(193\) 15.3258 1.10318 0.551588 0.834116i \(-0.314022\pi\)
0.551588 + 0.834116i \(0.314022\pi\)
\(194\) −3.58769 −0.257581
\(195\) 0 0
\(196\) −1.96239 −0.140171
\(197\) −0.574515 −0.0409325 −0.0204663 0.999791i \(-0.506515\pi\)
−0.0204663 + 0.999791i \(0.506515\pi\)
\(198\) 0.387873 0.0275649
\(199\) −0.201231 −0.0142649 −0.00713244 0.999975i \(-0.502270\pi\)
−0.00713244 + 0.999975i \(0.502270\pi\)
\(200\) 0 0
\(201\) −9.92478 −0.700040
\(202\) −3.42548 −0.241016
\(203\) −7.92478 −0.556210
\(204\) −6.57452 −0.460308
\(205\) 0 0
\(206\) 1.29948 0.0905388
\(207\) 4.96239 0.344910
\(208\) 5.09825 0.353500
\(209\) 10.7005 0.740171
\(210\) 0 0
\(211\) 6.44851 0.443934 0.221967 0.975054i \(-0.428752\pi\)
0.221967 + 0.975054i \(0.428752\pi\)
\(212\) −16.8265 −1.15565
\(213\) −2.00000 −0.137038
\(214\) 2.66433 0.182130
\(215\) 0 0
\(216\) 0.768452 0.0522865
\(217\) −4.57452 −0.310538
\(218\) −0.538319 −0.0364595
\(219\) 9.35026 0.631832
\(220\) 0 0
\(221\) −4.52373 −0.304299
\(222\) 0.150446 0.0100972
\(223\) 1.55149 0.103896 0.0519478 0.998650i \(-0.483457\pi\)
0.0519478 + 0.998650i \(0.483457\pi\)
\(224\) −2.26916 −0.151615
\(225\) 0 0
\(226\) 2.33709 0.155461
\(227\) −13.1490 −0.872732 −0.436366 0.899769i \(-0.643735\pi\)
−0.436366 + 0.899769i \(0.643735\pi\)
\(228\) 10.4993 0.695333
\(229\) 2.77575 0.183426 0.0917132 0.995785i \(-0.470766\pi\)
0.0917132 + 0.995785i \(0.470766\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −6.08981 −0.399816
\(233\) −0.0507852 −0.00332705 −0.00166353 0.999999i \(-0.500530\pi\)
−0.00166353 + 0.999999i \(0.500530\pi\)
\(234\) 0.261865 0.0171187
\(235\) 0 0
\(236\) 16.9262 1.10180
\(237\) −10.7005 −0.695074
\(238\) 0.649738 0.0421163
\(239\) 5.84955 0.378376 0.189188 0.981941i \(-0.439414\pi\)
0.189188 + 0.981941i \(0.439414\pi\)
\(240\) 0 0
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) −1.35756 −0.0872670
\(243\) −1.00000 −0.0641500
\(244\) 17.0738 1.09304
\(245\) 0 0
\(246\) −0.724961 −0.0462218
\(247\) 7.22425 0.459668
\(248\) −3.51530 −0.223222
\(249\) −3.22425 −0.204329
\(250\) 0 0
\(251\) 19.2243 1.21342 0.606712 0.794922i \(-0.292488\pi\)
0.606712 + 0.794922i \(0.292488\pi\)
\(252\) 1.96239 0.123619
\(253\) 9.92478 0.623965
\(254\) 0.523730 0.0328618
\(255\) 0 0
\(256\) 13.0752 0.817201
\(257\) 7.35026 0.458497 0.229248 0.973368i \(-0.426373\pi\)
0.229248 + 0.973368i \(0.426373\pi\)
\(258\) 2.44851 0.152437
\(259\) 0.775746 0.0482025
\(260\) 0 0
\(261\) 7.92478 0.490531
\(262\) 4.00000 0.247121
\(263\) −12.9624 −0.799295 −0.399648 0.916669i \(-0.630867\pi\)
−0.399648 + 0.916669i \(0.630867\pi\)
\(264\) 1.53690 0.0945899
\(265\) 0 0
\(266\) −1.03761 −0.0636200
\(267\) −1.03761 −0.0635008
\(268\) −19.4763 −1.18970
\(269\) 4.11142 0.250678 0.125339 0.992114i \(-0.459998\pi\)
0.125339 + 0.992114i \(0.459998\pi\)
\(270\) 0 0
\(271\) −16.4241 −0.997691 −0.498846 0.866691i \(-0.666243\pi\)
−0.498846 + 0.866691i \(0.666243\pi\)
\(272\) −12.6497 −0.767003
\(273\) 1.35026 0.0817216
\(274\) 4.36344 0.263605
\(275\) 0 0
\(276\) 9.73813 0.586167
\(277\) −11.0738 −0.665361 −0.332680 0.943040i \(-0.607953\pi\)
−0.332680 + 0.943040i \(0.607953\pi\)
\(278\) −0.635150 −0.0380938
\(279\) 4.57452 0.273869
\(280\) 0 0
\(281\) 14.3733 0.857438 0.428719 0.903438i \(-0.358965\pi\)
0.428719 + 0.903438i \(0.358965\pi\)
\(282\) −1.92478 −0.114619
\(283\) 1.14903 0.0683028 0.0341514 0.999417i \(-0.489127\pi\)
0.0341514 + 0.999417i \(0.489127\pi\)
\(284\) −3.92478 −0.232893
\(285\) 0 0
\(286\) 0.523730 0.0309688
\(287\) −3.73813 −0.220655
\(288\) 2.26916 0.133711
\(289\) −5.77575 −0.339750
\(290\) 0 0
\(291\) 18.4993 1.08445
\(292\) 18.3488 1.07379
\(293\) 0.649738 0.0379581 0.0189791 0.999820i \(-0.493958\pi\)
0.0189791 + 0.999820i \(0.493958\pi\)
\(294\) −0.193937 −0.0113106
\(295\) 0 0
\(296\) 0.596124 0.0346490
\(297\) −2.00000 −0.116052
\(298\) −0.862728 −0.0499765
\(299\) 6.70052 0.387501
\(300\) 0 0
\(301\) 12.6253 0.727710
\(302\) 0.252016 0.0145019
\(303\) 17.6629 1.01471
\(304\) 20.2012 1.15862
\(305\) 0 0
\(306\) −0.649738 −0.0371431
\(307\) 24.1016 1.37555 0.687775 0.725924i \(-0.258588\pi\)
0.687775 + 0.725924i \(0.258588\pi\)
\(308\) 3.92478 0.223635
\(309\) −6.70052 −0.381179
\(310\) 0 0
\(311\) 8.25202 0.467929 0.233964 0.972245i \(-0.424830\pi\)
0.233964 + 0.972245i \(0.424830\pi\)
\(312\) 1.03761 0.0587432
\(313\) −14.9018 −0.842297 −0.421148 0.906992i \(-0.638373\pi\)
−0.421148 + 0.906992i \(0.638373\pi\)
\(314\) −0.513881 −0.0290000
\(315\) 0 0
\(316\) −20.9986 −1.18126
\(317\) 10.1260 0.568733 0.284367 0.958716i \(-0.408217\pi\)
0.284367 + 0.958716i \(0.408217\pi\)
\(318\) −1.66291 −0.0932515
\(319\) 15.8496 0.887405
\(320\) 0 0
\(321\) −13.7381 −0.766788
\(322\) −0.962389 −0.0536318
\(323\) −17.9248 −0.997361
\(324\) −1.96239 −0.109022
\(325\) 0 0
\(326\) 1.02776 0.0569225
\(327\) 2.77575 0.153499
\(328\) −2.87258 −0.158612
\(329\) −9.92478 −0.547171
\(330\) 0 0
\(331\) 27.8496 1.53075 0.765375 0.643585i \(-0.222554\pi\)
0.765375 + 0.643585i \(0.222554\pi\)
\(332\) −6.32724 −0.347252
\(333\) −0.775746 −0.0425106
\(334\) −2.82179 −0.154402
\(335\) 0 0
\(336\) 3.77575 0.205984
\(337\) −3.84955 −0.209699 −0.104849 0.994488i \(-0.533436\pi\)
−0.104849 + 0.994488i \(0.533436\pi\)
\(338\) −2.16759 −0.117901
\(339\) −12.0508 −0.654509
\(340\) 0 0
\(341\) 9.14903 0.495448
\(342\) 1.03761 0.0561076
\(343\) −1.00000 −0.0539949
\(344\) 9.70194 0.523093
\(345\) 0 0
\(346\) −0.872577 −0.0469101
\(347\) 9.58769 0.514694 0.257347 0.966319i \(-0.417152\pi\)
0.257347 + 0.966319i \(0.417152\pi\)
\(348\) 15.5515 0.833648
\(349\) −15.1490 −0.810909 −0.405455 0.914115i \(-0.632887\pi\)
−0.405455 + 0.914115i \(0.632887\pi\)
\(350\) 0 0
\(351\) −1.35026 −0.0720716
\(352\) 4.53832 0.241893
\(353\) −20.3488 −1.08306 −0.541530 0.840681i \(-0.682155\pi\)
−0.541530 + 0.840681i \(0.682155\pi\)
\(354\) 1.67276 0.0889063
\(355\) 0 0
\(356\) −2.03620 −0.107918
\(357\) −3.35026 −0.177315
\(358\) 1.93937 0.102499
\(359\) −31.4010 −1.65728 −0.828642 0.559779i \(-0.810886\pi\)
−0.828642 + 0.559779i \(0.810886\pi\)
\(360\) 0 0
\(361\) 9.62530 0.506595
\(362\) 2.06063 0.108305
\(363\) 7.00000 0.367405
\(364\) 2.64974 0.138884
\(365\) 0 0
\(366\) 1.68735 0.0881992
\(367\) −29.4010 −1.53472 −0.767361 0.641215i \(-0.778431\pi\)
−0.767361 + 0.641215i \(0.778431\pi\)
\(368\) 18.7367 0.976719
\(369\) 3.73813 0.194600
\(370\) 0 0
\(371\) −8.57452 −0.445167
\(372\) 8.97698 0.465435
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) −1.29948 −0.0671943
\(375\) 0 0
\(376\) −7.62672 −0.393318
\(377\) 10.7005 0.551105
\(378\) 0.193937 0.00997502
\(379\) 10.7005 0.549649 0.274824 0.961494i \(-0.411380\pi\)
0.274824 + 0.961494i \(0.411380\pi\)
\(380\) 0 0
\(381\) −2.70052 −0.138352
\(382\) −2.68594 −0.137424
\(383\) −16.7757 −0.857201 −0.428600 0.903494i \(-0.640993\pi\)
−0.428600 + 0.903494i \(0.640993\pi\)
\(384\) 5.91748 0.301975
\(385\) 0 0
\(386\) 2.97224 0.151283
\(387\) −12.6253 −0.641780
\(388\) 36.3028 1.84300
\(389\) −29.3258 −1.48688 −0.743439 0.668804i \(-0.766807\pi\)
−0.743439 + 0.668804i \(0.766807\pi\)
\(390\) 0 0
\(391\) −16.6253 −0.840778
\(392\) −0.768452 −0.0388127
\(393\) −20.6253 −1.04041
\(394\) −0.111420 −0.00561324
\(395\) 0 0
\(396\) −3.92478 −0.197227
\(397\) 18.3488 0.920902 0.460451 0.887685i \(-0.347688\pi\)
0.460451 + 0.887685i \(0.347688\pi\)
\(398\) −0.0390260 −0.00195620
\(399\) 5.35026 0.267848
\(400\) 0 0
\(401\) −37.3258 −1.86396 −0.931981 0.362506i \(-0.881921\pi\)
−0.931981 + 0.362506i \(0.881921\pi\)
\(402\) −1.92478 −0.0959992
\(403\) 6.17679 0.307688
\(404\) 34.6615 1.72447
\(405\) 0 0
\(406\) −1.53690 −0.0762753
\(407\) −1.55149 −0.0769046
\(408\) −2.57452 −0.127458
\(409\) −22.3733 −1.10629 −0.553144 0.833086i \(-0.686572\pi\)
−0.553144 + 0.833086i \(0.686572\pi\)
\(410\) 0 0
\(411\) −22.4993 −1.10981
\(412\) −13.1490 −0.647806
\(413\) 8.62530 0.424423
\(414\) 0.962389 0.0472988
\(415\) 0 0
\(416\) 3.06396 0.150223
\(417\) 3.27504 0.160379
\(418\) 2.07522 0.101502
\(419\) 23.4763 1.14689 0.573445 0.819244i \(-0.305606\pi\)
0.573445 + 0.819244i \(0.305606\pi\)
\(420\) 0 0
\(421\) −25.2243 −1.22935 −0.614677 0.788779i \(-0.710714\pi\)
−0.614677 + 0.788779i \(0.710714\pi\)
\(422\) 1.25060 0.0608783
\(423\) 9.92478 0.482559
\(424\) −6.58910 −0.319995
\(425\) 0 0
\(426\) −0.387873 −0.0187925
\(427\) 8.70052 0.421048
\(428\) −26.9596 −1.30314
\(429\) −2.70052 −0.130383
\(430\) 0 0
\(431\) −19.4010 −0.934516 −0.467258 0.884121i \(-0.654758\pi\)
−0.467258 + 0.884121i \(0.654758\pi\)
\(432\) −3.77575 −0.181661
\(433\) 6.49929 0.312336 0.156168 0.987731i \(-0.450086\pi\)
0.156168 + 0.987731i \(0.450086\pi\)
\(434\) −0.887166 −0.0425853
\(435\) 0 0
\(436\) 5.44709 0.260868
\(437\) 26.5501 1.27006
\(438\) 1.81336 0.0866456
\(439\) −14.6497 −0.699194 −0.349597 0.936900i \(-0.613681\pi\)
−0.349597 + 0.936900i \(0.613681\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −0.877317 −0.0417297
\(443\) 19.1392 0.909330 0.454665 0.890663i \(-0.349759\pi\)
0.454665 + 0.890663i \(0.349759\pi\)
\(444\) −1.52232 −0.0722459
\(445\) 0 0
\(446\) 0.300891 0.0142476
\(447\) 4.44851 0.210407
\(448\) 7.11142 0.335983
\(449\) −32.8021 −1.54803 −0.774013 0.633169i \(-0.781754\pi\)
−0.774013 + 0.633169i \(0.781754\pi\)
\(450\) 0 0
\(451\) 7.47627 0.352044
\(452\) −23.6483 −1.11232
\(453\) −1.29948 −0.0610547
\(454\) −2.55008 −0.119681
\(455\) 0 0
\(456\) 4.11142 0.192535
\(457\) −18.7005 −0.874774 −0.437387 0.899273i \(-0.644096\pi\)
−0.437387 + 0.899273i \(0.644096\pi\)
\(458\) 0.538319 0.0251540
\(459\) 3.35026 0.156377
\(460\) 0 0
\(461\) −6.96239 −0.324271 −0.162135 0.986769i \(-0.551838\pi\)
−0.162135 + 0.986769i \(0.551838\pi\)
\(462\) 0.387873 0.0180455
\(463\) 5.29948 0.246288 0.123144 0.992389i \(-0.460702\pi\)
0.123144 + 0.992389i \(0.460702\pi\)
\(464\) 29.9219 1.38909
\(465\) 0 0
\(466\) −0.00984911 −0.000456251 0
\(467\) −13.1490 −0.608465 −0.304232 0.952598i \(-0.598400\pi\)
−0.304232 + 0.952598i \(0.598400\pi\)
\(468\) −2.64974 −0.122484
\(469\) −9.92478 −0.458284
\(470\) 0 0
\(471\) 2.64974 0.122093
\(472\) 6.62813 0.305084
\(473\) −25.2506 −1.16102
\(474\) −2.07522 −0.0953181
\(475\) 0 0
\(476\) −6.57452 −0.301342
\(477\) 8.57452 0.392600
\(478\) 1.13444 0.0518882
\(479\) 5.14903 0.235265 0.117633 0.993057i \(-0.462469\pi\)
0.117633 + 0.993057i \(0.462469\pi\)
\(480\) 0 0
\(481\) −1.04746 −0.0477601
\(482\) −0.0145884 −0.000664486 0
\(483\) 4.96239 0.225797
\(484\) 13.7367 0.624396
\(485\) 0 0
\(486\) −0.193937 −0.00879714
\(487\) −22.1768 −1.00493 −0.502463 0.864599i \(-0.667573\pi\)
−0.502463 + 0.864599i \(0.667573\pi\)
\(488\) 6.68594 0.302658
\(489\) −5.29948 −0.239651
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 7.33567 0.330718
\(493\) −26.5501 −1.19576
\(494\) 1.40105 0.0630361
\(495\) 0 0
\(496\) 17.2722 0.775545
\(497\) −2.00000 −0.0897123
\(498\) −0.625301 −0.0280204
\(499\) −6.55008 −0.293222 −0.146611 0.989194i \(-0.546837\pi\)
−0.146611 + 0.989194i \(0.546837\pi\)
\(500\) 0 0
\(501\) 14.5501 0.650050
\(502\) 3.72829 0.166402
\(503\) 8.77575 0.391291 0.195646 0.980675i \(-0.437320\pi\)
0.195646 + 0.980675i \(0.437320\pi\)
\(504\) 0.768452 0.0342296
\(505\) 0 0
\(506\) 1.92478 0.0855668
\(507\) 11.1768 0.496379
\(508\) −5.29948 −0.235126
\(509\) 13.1392 0.582384 0.291192 0.956665i \(-0.405948\pi\)
0.291192 + 0.956665i \(0.405948\pi\)
\(510\) 0 0
\(511\) 9.35026 0.413631
\(512\) 14.3707 0.635103
\(513\) −5.35026 −0.236220
\(514\) 1.42548 0.0628754
\(515\) 0 0
\(516\) −24.7757 −1.09069
\(517\) 19.8496 0.872982
\(518\) 0.150446 0.00661020
\(519\) 4.49929 0.197497
\(520\) 0 0
\(521\) −37.6629 −1.65004 −0.825021 0.565102i \(-0.808837\pi\)
−0.825021 + 0.565102i \(0.808837\pi\)
\(522\) 1.53690 0.0672685
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −40.4749 −1.76815
\(525\) 0 0
\(526\) −2.51388 −0.109610
\(527\) −15.3258 −0.667603
\(528\) −7.55149 −0.328637
\(529\) 1.62530 0.0706652
\(530\) 0 0
\(531\) −8.62530 −0.374306
\(532\) 10.4993 0.455202
\(533\) 5.04746 0.218630
\(534\) −0.201231 −0.00870811
\(535\) 0 0
\(536\) −7.62672 −0.329424
\(537\) −10.0000 −0.431532
\(538\) 0.797355 0.0343764
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −22.4749 −0.966269 −0.483135 0.875546i \(-0.660502\pi\)
−0.483135 + 0.875546i \(0.660502\pi\)
\(542\) −3.18523 −0.136817
\(543\) −10.6253 −0.455975
\(544\) −7.60228 −0.325945
\(545\) 0 0
\(546\) 0.261865 0.0112068
\(547\) −25.9248 −1.10846 −0.554232 0.832362i \(-0.686988\pi\)
−0.554232 + 0.832362i \(0.686988\pi\)
\(548\) −44.1524 −1.88610
\(549\) −8.70052 −0.371329
\(550\) 0 0
\(551\) 42.3996 1.80629
\(552\) 3.81336 0.162307
\(553\) −10.7005 −0.455033
\(554\) −2.14762 −0.0912435
\(555\) 0 0
\(556\) 6.42690 0.272561
\(557\) −28.5256 −1.20867 −0.604335 0.796730i \(-0.706561\pi\)
−0.604335 + 0.796730i \(0.706561\pi\)
\(558\) 0.887166 0.0375567
\(559\) −17.0475 −0.721031
\(560\) 0 0
\(561\) 6.70052 0.282896
\(562\) 2.78751 0.117584
\(563\) −11.6267 −0.490008 −0.245004 0.969522i \(-0.578789\pi\)
−0.245004 + 0.969522i \(0.578789\pi\)
\(564\) 19.4763 0.820099
\(565\) 0 0
\(566\) 0.222839 0.00936663
\(567\) −1.00000 −0.0419961
\(568\) −1.53690 −0.0644871
\(569\) 9.32582 0.390959 0.195479 0.980708i \(-0.437374\pi\)
0.195479 + 0.980708i \(0.437374\pi\)
\(570\) 0 0
\(571\) −19.6991 −0.824382 −0.412191 0.911097i \(-0.635236\pi\)
−0.412191 + 0.911097i \(0.635236\pi\)
\(572\) −5.29948 −0.221582
\(573\) 13.8496 0.578573
\(574\) −0.724961 −0.0302593
\(575\) 0 0
\(576\) −7.11142 −0.296309
\(577\) 32.7974 1.36537 0.682686 0.730712i \(-0.260812\pi\)
0.682686 + 0.730712i \(0.260812\pi\)
\(578\) −1.12013 −0.0465912
\(579\) −15.3258 −0.636920
\(580\) 0 0
\(581\) −3.22425 −0.133765
\(582\) 3.58769 0.148715
\(583\) 17.1490 0.710240
\(584\) 7.18523 0.297327
\(585\) 0 0
\(586\) 0.126008 0.00520534
\(587\) 18.8218 0.776859 0.388429 0.921479i \(-0.373018\pi\)
0.388429 + 0.921479i \(0.373018\pi\)
\(588\) 1.96239 0.0809275
\(589\) 24.4749 1.00847
\(590\) 0 0
\(591\) 0.574515 0.0236324
\(592\) −2.92902 −0.120382
\(593\) 33.7499 1.38594 0.692971 0.720965i \(-0.256301\pi\)
0.692971 + 0.720965i \(0.256301\pi\)
\(594\) −0.387873 −0.0159146
\(595\) 0 0
\(596\) 8.72970 0.357582
\(597\) 0.201231 0.00823583
\(598\) 1.29948 0.0531395
\(599\) −20.2981 −0.829356 −0.414678 0.909968i \(-0.636106\pi\)
−0.414678 + 0.909968i \(0.636106\pi\)
\(600\) 0 0
\(601\) −13.8496 −0.564935 −0.282468 0.959277i \(-0.591153\pi\)
−0.282468 + 0.959277i \(0.591153\pi\)
\(602\) 2.44851 0.0997937
\(603\) 9.92478 0.404168
\(604\) −2.55008 −0.103761
\(605\) 0 0
\(606\) 3.42548 0.139151
\(607\) −25.2506 −1.02489 −0.512445 0.858720i \(-0.671260\pi\)
−0.512445 + 0.858720i \(0.671260\pi\)
\(608\) 12.1406 0.492366
\(609\) 7.92478 0.321128
\(610\) 0 0
\(611\) 13.4010 0.542148
\(612\) 6.57452 0.265759
\(613\) −9.14903 −0.369526 −0.184763 0.982783i \(-0.559152\pi\)
−0.184763 + 0.982783i \(0.559152\pi\)
\(614\) 4.67418 0.188634
\(615\) 0 0
\(616\) 1.53690 0.0619236
\(617\) 15.9492 0.642091 0.321046 0.947064i \(-0.395966\pi\)
0.321046 + 0.947064i \(0.395966\pi\)
\(618\) −1.29948 −0.0522726
\(619\) 11.1735 0.449100 0.224550 0.974463i \(-0.427909\pi\)
0.224550 + 0.974463i \(0.427909\pi\)
\(620\) 0 0
\(621\) −4.96239 −0.199134
\(622\) 1.60037 0.0641689
\(623\) −1.03761 −0.0415710
\(624\) −5.09825 −0.204093
\(625\) 0 0
\(626\) −2.89000 −0.115507
\(627\) −10.7005 −0.427338
\(628\) 5.19982 0.207495
\(629\) 2.59895 0.103627
\(630\) 0 0
\(631\) −14.5501 −0.579229 −0.289615 0.957143i \(-0.593527\pi\)
−0.289615 + 0.957143i \(0.593527\pi\)
\(632\) −8.22284 −0.327087
\(633\) −6.44851 −0.256305
\(634\) 1.96380 0.0779926
\(635\) 0 0
\(636\) 16.8265 0.667215
\(637\) 1.35026 0.0534993
\(638\) 3.07381 0.121693
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −38.7269 −1.52962 −0.764810 0.644256i \(-0.777167\pi\)
−0.764810 + 0.644256i \(0.777167\pi\)
\(642\) −2.66433 −0.105153
\(643\) −11.9511 −0.471306 −0.235653 0.971837i \(-0.575723\pi\)
−0.235653 + 0.971837i \(0.575723\pi\)
\(644\) 9.73813 0.383736
\(645\) 0 0
\(646\) −3.47627 −0.136772
\(647\) −14.5501 −0.572023 −0.286011 0.958226i \(-0.592329\pi\)
−0.286011 + 0.958226i \(0.592329\pi\)
\(648\) −0.768452 −0.0301876
\(649\) −17.2506 −0.677145
\(650\) 0 0
\(651\) 4.57452 0.179289
\(652\) −10.3996 −0.407281
\(653\) 49.9756 1.95569 0.977847 0.209319i \(-0.0671247\pi\)
0.977847 + 0.209319i \(0.0671247\pi\)
\(654\) 0.538319 0.0210499
\(655\) 0 0
\(656\) 14.1142 0.551069
\(657\) −9.35026 −0.364788
\(658\) −1.92478 −0.0750356
\(659\) −16.9525 −0.660377 −0.330189 0.943915i \(-0.607112\pi\)
−0.330189 + 0.943915i \(0.607112\pi\)
\(660\) 0 0
\(661\) −15.6531 −0.608834 −0.304417 0.952539i \(-0.598462\pi\)
−0.304417 + 0.952539i \(0.598462\pi\)
\(662\) 5.40105 0.209918
\(663\) 4.52373 0.175687
\(664\) −2.47768 −0.0961528
\(665\) 0 0
\(666\) −0.150446 −0.00582965
\(667\) 39.3258 1.52270
\(668\) 28.5529 1.10475
\(669\) −1.55149 −0.0599842
\(670\) 0 0
\(671\) −17.4010 −0.671760
\(672\) 2.26916 0.0875347
\(673\) 26.0263 1.00324 0.501621 0.865088i \(-0.332737\pi\)
0.501621 + 0.865088i \(0.332737\pi\)
\(674\) −0.746569 −0.0287568
\(675\) 0 0
\(676\) 21.9332 0.843585
\(677\) −35.4518 −1.36252 −0.681262 0.732039i \(-0.738569\pi\)
−0.681262 + 0.732039i \(0.738569\pi\)
\(678\) −2.33709 −0.0897553
\(679\) 18.4993 0.709938
\(680\) 0 0
\(681\) 13.1490 0.503872
\(682\) 1.77433 0.0679427
\(683\) −23.6629 −0.905436 −0.452718 0.891654i \(-0.649546\pi\)
−0.452718 + 0.891654i \(0.649546\pi\)
\(684\) −10.4993 −0.401450
\(685\) 0 0
\(686\) −0.193937 −0.00740453
\(687\) −2.77575 −0.105901
\(688\) −47.6699 −1.81740
\(689\) 11.5778 0.441081
\(690\) 0 0
\(691\) −0.574515 −0.0218556 −0.0109278 0.999940i \(-0.503478\pi\)
−0.0109278 + 0.999940i \(0.503478\pi\)
\(692\) 8.82936 0.335642
\(693\) −2.00000 −0.0759737
\(694\) 1.85940 0.0705820
\(695\) 0 0
\(696\) 6.08981 0.230834
\(697\) −12.5237 −0.474370
\(698\) −2.93795 −0.111203
\(699\) 0.0507852 0.00192087
\(700\) 0 0
\(701\) 42.7269 1.61377 0.806886 0.590707i \(-0.201151\pi\)
0.806886 + 0.590707i \(0.201151\pi\)
\(702\) −0.261865 −0.00988346
\(703\) −4.15045 −0.156537
\(704\) −14.2228 −0.536043
\(705\) 0 0
\(706\) −3.94639 −0.148524
\(707\) 17.6629 0.664282
\(708\) −16.9262 −0.636125
\(709\) 27.2506 1.02342 0.511709 0.859159i \(-0.329013\pi\)
0.511709 + 0.859159i \(0.329013\pi\)
\(710\) 0 0
\(711\) 10.7005 0.401301
\(712\) −0.797355 −0.0298821
\(713\) 22.7005 0.850141
\(714\) −0.649738 −0.0243158
\(715\) 0 0
\(716\) −19.6239 −0.733379
\(717\) −5.84955 −0.218456
\(718\) −6.08981 −0.227270
\(719\) −10.7005 −0.399062 −0.199531 0.979891i \(-0.563942\pi\)
−0.199531 + 0.979891i \(0.563942\pi\)
\(720\) 0 0
\(721\) −6.70052 −0.249541
\(722\) 1.86670 0.0694713
\(723\) 0.0752228 0.00279757
\(724\) −20.8510 −0.774920
\(725\) 0 0
\(726\) 1.35756 0.0503836
\(727\) 39.9511 1.48171 0.740853 0.671668i \(-0.234422\pi\)
0.740853 + 0.671668i \(0.234422\pi\)
\(728\) 1.03761 0.0384564
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.2981 1.56445
\(732\) −17.0738 −0.631066
\(733\) −30.3488 −1.12096 −0.560480 0.828168i \(-0.689383\pi\)
−0.560480 + 0.828168i \(0.689383\pi\)
\(734\) −5.70194 −0.210462
\(735\) 0 0
\(736\) 11.2605 0.415066
\(737\) 19.8496 0.731168
\(738\) 0.724961 0.0266862
\(739\) 37.2506 1.37029 0.685143 0.728409i \(-0.259740\pi\)
0.685143 + 0.728409i \(0.259740\pi\)
\(740\) 0 0
\(741\) −7.22425 −0.265390
\(742\) −1.66291 −0.0610474
\(743\) 26.3634 0.967181 0.483590 0.875294i \(-0.339332\pi\)
0.483590 + 0.875294i \(0.339332\pi\)
\(744\) 3.51530 0.128877
\(745\) 0 0
\(746\) 3.10299 0.113608
\(747\) 3.22425 0.117969
\(748\) 13.1490 0.480776
\(749\) −13.7381 −0.501981
\(750\) 0 0
\(751\) 50.6516 1.84830 0.924152 0.382024i \(-0.124773\pi\)
0.924152 + 0.382024i \(0.124773\pi\)
\(752\) 37.4734 1.36652
\(753\) −19.2243 −0.700571
\(754\) 2.07522 0.0755752
\(755\) 0 0
\(756\) −1.96239 −0.0713714
\(757\) 38.9525 1.41575 0.707877 0.706336i \(-0.249653\pi\)
0.707877 + 0.706336i \(0.249653\pi\)
\(758\) 2.07522 0.0753755
\(759\) −9.92478 −0.360247
\(760\) 0 0
\(761\) 48.2130 1.74772 0.873860 0.486178i \(-0.161609\pi\)
0.873860 + 0.486178i \(0.161609\pi\)
\(762\) −0.523730 −0.0189727
\(763\) 2.77575 0.100489
\(764\) 27.1782 0.983273
\(765\) 0 0
\(766\) −3.25343 −0.117551
\(767\) −11.6464 −0.420528
\(768\) −13.0752 −0.471811
\(769\) 4.44851 0.160417 0.0802086 0.996778i \(-0.474441\pi\)
0.0802086 + 0.996778i \(0.474441\pi\)
\(770\) 0 0
\(771\) −7.35026 −0.264713
\(772\) −30.0752 −1.08243
\(773\) 39.3014 1.41357 0.706786 0.707427i \(-0.250144\pi\)
0.706786 + 0.707427i \(0.250144\pi\)
\(774\) −2.44851 −0.0880098
\(775\) 0 0
\(776\) 14.2158 0.510318
\(777\) −0.775746 −0.0278297
\(778\) −5.68735 −0.203901
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −3.22425 −0.115299
\(783\) −7.92478 −0.283208
\(784\) 3.77575 0.134848
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 0.897015 0.0319751 0.0159876 0.999872i \(-0.494911\pi\)
0.0159876 + 0.999872i \(0.494911\pi\)
\(788\) 1.12742 0.0401628
\(789\) 12.9624 0.461473
\(790\) 0 0
\(791\) −12.0508 −0.428477
\(792\) −1.53690 −0.0546115
\(793\) −11.7480 −0.417183
\(794\) 3.55851 0.126287
\(795\) 0 0
\(796\) 0.394893 0.0139966
\(797\) 3.19982 0.113343 0.0566717 0.998393i \(-0.481951\pi\)
0.0566717 + 0.998393i \(0.481951\pi\)
\(798\) 1.03761 0.0367310
\(799\) −33.2506 −1.17632
\(800\) 0 0
\(801\) 1.03761 0.0366622
\(802\) −7.23884 −0.255612
\(803\) −18.7005 −0.659927
\(804\) 19.4763 0.686875
\(805\) 0 0
\(806\) 1.19791 0.0421944
\(807\) −4.11142 −0.144729
\(808\) 13.5731 0.477500
\(809\) 4.44851 0.156401 0.0782006 0.996938i \(-0.475083\pi\)
0.0782006 + 0.996938i \(0.475083\pi\)
\(810\) 0 0
\(811\) 37.6747 1.32294 0.661468 0.749973i \(-0.269934\pi\)
0.661468 + 0.749973i \(0.269934\pi\)
\(812\) 15.5515 0.545750
\(813\) 16.4241 0.576017
\(814\) −0.300891 −0.0105462
\(815\) 0 0
\(816\) 12.6497 0.442829
\(817\) −67.5487 −2.36323
\(818\) −4.33900 −0.151710
\(819\) −1.35026 −0.0471820
\(820\) 0 0
\(821\) −0.749399 −0.0261542 −0.0130771 0.999914i \(-0.504163\pi\)
−0.0130771 + 0.999914i \(0.504163\pi\)
\(822\) −4.36344 −0.152192
\(823\) −26.3996 −0.920233 −0.460117 0.887858i \(-0.652192\pi\)
−0.460117 + 0.887858i \(0.652192\pi\)
\(824\) −5.14903 −0.179375
\(825\) 0 0
\(826\) 1.67276 0.0582028
\(827\) 5.43724 0.189071 0.0945357 0.995521i \(-0.469863\pi\)
0.0945357 + 0.995521i \(0.469863\pi\)
\(828\) −9.73813 −0.338424
\(829\) 22.7757 0.791034 0.395517 0.918459i \(-0.370565\pi\)
0.395517 + 0.918459i \(0.370565\pi\)
\(830\) 0 0
\(831\) 11.0738 0.384146
\(832\) −9.60228 −0.332899
\(833\) −3.35026 −0.116080
\(834\) 0.635150 0.0219934
\(835\) 0 0
\(836\) −20.9986 −0.726251
\(837\) −4.57452 −0.158118
\(838\) 4.55291 0.157278
\(839\) 15.8496 0.547187 0.273594 0.961845i \(-0.411788\pi\)
0.273594 + 0.961845i \(0.411788\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) −4.89191 −0.168586
\(843\) −14.3733 −0.495042
\(844\) −12.6545 −0.435585
\(845\) 0 0
\(846\) 1.92478 0.0661752
\(847\) 7.00000 0.240523
\(848\) 32.3752 1.11177
\(849\) −1.14903 −0.0394346
\(850\) 0 0
\(851\) −3.84955 −0.131961
\(852\) 3.92478 0.134461
\(853\) −21.0494 −0.720717 −0.360358 0.932814i \(-0.617346\pi\)
−0.360358 + 0.932814i \(0.617346\pi\)
\(854\) 1.68735 0.0577399
\(855\) 0 0
\(856\) −10.5571 −0.360834
\(857\) 50.1524 1.71317 0.856586 0.516004i \(-0.172581\pi\)
0.856586 + 0.516004i \(0.172581\pi\)
\(858\) −0.523730 −0.0178799
\(859\) −5.35026 −0.182549 −0.0912743 0.995826i \(-0.529094\pi\)
−0.0912743 + 0.995826i \(0.529094\pi\)
\(860\) 0 0
\(861\) 3.73813 0.127395
\(862\) −3.76257 −0.128154
\(863\) −33.6893 −1.14680 −0.573398 0.819277i \(-0.694375\pi\)
−0.573398 + 0.819277i \(0.694375\pi\)
\(864\) −2.26916 −0.0771984
\(865\) 0 0
\(866\) 1.26045 0.0428319
\(867\) 5.77575 0.196155
\(868\) 8.97698 0.304698
\(869\) 21.4010 0.725981
\(870\) 0 0
\(871\) 13.4010 0.454077
\(872\) 2.13303 0.0722334
\(873\) −18.4993 −0.626106
\(874\) 5.14903 0.174169
\(875\) 0 0
\(876\) −18.3488 −0.619950
\(877\) −21.5026 −0.726092 −0.363046 0.931771i \(-0.618263\pi\)
−0.363046 + 0.931771i \(0.618263\pi\)
\(878\) −2.84112 −0.0958832
\(879\) −0.649738 −0.0219151
\(880\) 0 0
\(881\) 32.3634 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(882\) 0.193937 0.00653018
\(883\) −2.59895 −0.0874617 −0.0437309 0.999043i \(-0.513924\pi\)
−0.0437309 + 0.999043i \(0.513924\pi\)
\(884\) 8.87732 0.298576
\(885\) 0 0
\(886\) 3.71179 0.124700
\(887\) 38.2784 1.28526 0.642631 0.766176i \(-0.277843\pi\)
0.642631 + 0.766176i \(0.277843\pi\)
\(888\) −0.596124 −0.0200046
\(889\) −2.70052 −0.0905727
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −3.04463 −0.101942
\(893\) 53.1002 1.77693
\(894\) 0.862728 0.0288539
\(895\) 0 0
\(896\) 5.91748 0.197689
\(897\) −6.70052 −0.223724
\(898\) −6.36153 −0.212287
\(899\) 36.2520 1.20907
\(900\) 0 0
\(901\) −28.7269 −0.957031
\(902\) 1.44992 0.0482771
\(903\) −12.6253 −0.420144
\(904\) −9.26045 −0.307998
\(905\) 0 0
\(906\) −0.252016 −0.00837267
\(907\) −49.9972 −1.66013 −0.830064 0.557668i \(-0.811696\pi\)
−0.830064 + 0.557668i \(0.811696\pi\)
\(908\) 25.8035 0.856320
\(909\) −17.6629 −0.585842
\(910\) 0 0
\(911\) −24.9525 −0.826715 −0.413357 0.910569i \(-0.635644\pi\)
−0.413357 + 0.910569i \(0.635644\pi\)
\(912\) −20.2012 −0.668930
\(913\) 6.44851 0.213414
\(914\) −3.62672 −0.119961
\(915\) 0 0
\(916\) −5.44709 −0.179977
\(917\) −20.6253 −0.681107
\(918\) 0.649738 0.0214446
\(919\) −11.6991 −0.385918 −0.192959 0.981207i \(-0.561808\pi\)
−0.192959 + 0.981207i \(0.561808\pi\)
\(920\) 0 0
\(921\) −24.1016 −0.794174
\(922\) −1.35026 −0.0444685
\(923\) 2.70052 0.0888888
\(924\) −3.92478 −0.129116
\(925\) 0 0
\(926\) 1.02776 0.0337744
\(927\) 6.70052 0.220074
\(928\) 17.9826 0.590307
\(929\) 23.7090 0.777866 0.388933 0.921266i \(-0.372844\pi\)
0.388933 + 0.921266i \(0.372844\pi\)
\(930\) 0 0
\(931\) 5.35026 0.175348
\(932\) 0.0996603 0.00326448
\(933\) −8.25202 −0.270159
\(934\) −2.55008 −0.0834411
\(935\) 0 0
\(936\) −1.03761 −0.0339154
\(937\) −19.9003 −0.650116 −0.325058 0.945694i \(-0.605384\pi\)
−0.325058 + 0.945694i \(0.605384\pi\)
\(938\) −1.92478 −0.0628462
\(939\) 14.9018 0.486300
\(940\) 0 0
\(941\) −6.28821 −0.204990 −0.102495 0.994734i \(-0.532683\pi\)
−0.102495 + 0.994734i \(0.532683\pi\)
\(942\) 0.513881 0.0167432
\(943\) 18.5501 0.604074
\(944\) −32.5669 −1.05996
\(945\) 0 0
\(946\) −4.89701 −0.159216
\(947\) −40.0362 −1.30100 −0.650501 0.759506i \(-0.725441\pi\)
−0.650501 + 0.759506i \(0.725441\pi\)
\(948\) 20.9986 0.682002
\(949\) −12.6253 −0.409835
\(950\) 0 0
\(951\) −10.1260 −0.328358
\(952\) −2.57452 −0.0834405
\(953\) 40.9478 1.32643 0.663215 0.748429i \(-0.269192\pi\)
0.663215 + 0.748429i \(0.269192\pi\)
\(954\) 1.66291 0.0538388
\(955\) 0 0
\(956\) −11.4791 −0.371261
\(957\) −15.8496 −0.512343
\(958\) 0.998585 0.0322628
\(959\) −22.4993 −0.726540
\(960\) 0 0
\(961\) −10.0738 −0.324962
\(962\) −0.203141 −0.00654953
\(963\) 13.7381 0.442705
\(964\) 0.147616 0.00475440
\(965\) 0 0
\(966\) 0.962389 0.0309643
\(967\) 38.2784 1.23095 0.615475 0.788157i \(-0.288964\pi\)
0.615475 + 0.788157i \(0.288964\pi\)
\(968\) 5.37916 0.172893
\(969\) 17.9248 0.575827
\(970\) 0 0
\(971\) −28.7269 −0.921889 −0.460945 0.887429i \(-0.652489\pi\)
−0.460945 + 0.887429i \(0.652489\pi\)
\(972\) 1.96239 0.0629436
\(973\) 3.27504 0.104993
\(974\) −4.30089 −0.137809
\(975\) 0 0
\(976\) −32.8510 −1.05153
\(977\) −41.3014 −1.32135 −0.660674 0.750673i \(-0.729730\pi\)
−0.660674 + 0.750673i \(0.729730\pi\)
\(978\) −1.02776 −0.0328642
\(979\) 2.07522 0.0663244
\(980\) 0 0
\(981\) −2.77575 −0.0886228
\(982\) 0.387873 0.0123775
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 2.87258 0.0915744
\(985\) 0 0
\(986\) −5.14903 −0.163979
\(987\) 9.92478 0.315909
\(988\) −14.1768 −0.451024
\(989\) −62.6516 −1.99221
\(990\) 0 0
\(991\) 25.1002 0.797333 0.398666 0.917096i \(-0.369473\pi\)
0.398666 + 0.917096i \(0.369473\pi\)
\(992\) 10.3803 0.329575
\(993\) −27.8496 −0.883779
\(994\) −0.387873 −0.0123026
\(995\) 0 0
\(996\) 6.32724 0.200486
\(997\) 32.7974 1.03870 0.519351 0.854561i \(-0.326174\pi\)
0.519351 + 0.854561i \(0.326174\pi\)
\(998\) −1.27030 −0.0402106
\(999\) 0.775746 0.0245435
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 525.2.a.k.1.2 3
3.2 odd 2 1575.2.a.w.1.2 3
4.3 odd 2 8400.2.a.dj.1.3 3
5.2 odd 4 105.2.d.b.64.4 yes 6
5.3 odd 4 105.2.d.b.64.3 6
5.4 even 2 525.2.a.j.1.2 3
7.6 odd 2 3675.2.a.bj.1.2 3
15.2 even 4 315.2.d.e.64.3 6
15.8 even 4 315.2.d.e.64.4 6
15.14 odd 2 1575.2.a.x.1.2 3
20.3 even 4 1680.2.t.k.1009.4 6
20.7 even 4 1680.2.t.k.1009.1 6
20.19 odd 2 8400.2.a.dg.1.1 3
35.2 odd 12 735.2.q.e.214.4 12
35.3 even 12 735.2.q.f.79.4 12
35.12 even 12 735.2.q.f.214.4 12
35.13 even 4 735.2.d.b.589.3 6
35.17 even 12 735.2.q.f.79.3 12
35.18 odd 12 735.2.q.e.79.4 12
35.23 odd 12 735.2.q.e.214.3 12
35.27 even 4 735.2.d.b.589.4 6
35.32 odd 12 735.2.q.e.79.3 12
35.33 even 12 735.2.q.f.214.3 12
35.34 odd 2 3675.2.a.bi.1.2 3
60.23 odd 4 5040.2.t.v.1009.6 6
60.47 odd 4 5040.2.t.v.1009.5 6
105.62 odd 4 2205.2.d.l.1324.3 6
105.83 odd 4 2205.2.d.l.1324.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.3 6 5.3 odd 4
105.2.d.b.64.4 yes 6 5.2 odd 4
315.2.d.e.64.3 6 15.2 even 4
315.2.d.e.64.4 6 15.8 even 4
525.2.a.j.1.2 3 5.4 even 2
525.2.a.k.1.2 3 1.1 even 1 trivial
735.2.d.b.589.3 6 35.13 even 4
735.2.d.b.589.4 6 35.27 even 4
735.2.q.e.79.3 12 35.32 odd 12
735.2.q.e.79.4 12 35.18 odd 12
735.2.q.e.214.3 12 35.23 odd 12
735.2.q.e.214.4 12 35.2 odd 12
735.2.q.f.79.3 12 35.17 even 12
735.2.q.f.79.4 12 35.3 even 12
735.2.q.f.214.3 12 35.33 even 12
735.2.q.f.214.4 12 35.12 even 12
1575.2.a.w.1.2 3 3.2 odd 2
1575.2.a.x.1.2 3 15.14 odd 2
1680.2.t.k.1009.1 6 20.7 even 4
1680.2.t.k.1009.4 6 20.3 even 4
2205.2.d.l.1324.3 6 105.62 odd 4
2205.2.d.l.1324.4 6 105.83 odd 4
3675.2.a.bi.1.2 3 35.34 odd 2
3675.2.a.bj.1.2 3 7.6 odd 2
5040.2.t.v.1009.5 6 60.47 odd 4
5040.2.t.v.1009.6 6 60.23 odd 4
8400.2.a.dg.1.1 3 20.19 odd 2
8400.2.a.dj.1.3 3 4.3 odd 2