Properties

Label 525.2.a.k.1.1
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 1.1
Root \(0.311108\) of \(x^{3} - x^{2} - 3 x + 1\)
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} -1.00000 q^{7} +0.719004 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} -1.00000 q^{7} +0.719004 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.62222 q^{12} -6.42864 q^{13} +1.90321 q^{14} -4.61285 q^{16} +4.42864 q^{17} -1.90321 q^{18} -2.42864 q^{19} +1.00000 q^{21} -3.80642 q^{22} +1.37778 q^{23} -0.719004 q^{24} +12.2351 q^{26} -1.00000 q^{27} -1.62222 q^{28} +0.755569 q^{29} +5.18421 q^{31} +7.34122 q^{32} -2.00000 q^{33} -8.42864 q^{34} +1.62222 q^{36} +7.61285 q^{37} +4.62222 q^{38} +6.42864 q^{39} -8.23506 q^{41} -1.90321 q^{42} +10.1017 q^{43} +3.24443 q^{44} -2.62222 q^{46} +2.75557 q^{47} +4.61285 q^{48} +1.00000 q^{49} -4.42864 q^{51} -10.4286 q^{52} +9.18421 q^{53} +1.90321 q^{54} -0.719004 q^{56} +2.42864 q^{57} -1.43801 q^{58} +14.1017 q^{59} +6.85728 q^{61} -9.86665 q^{62} -1.00000 q^{63} -4.74620 q^{64} +3.80642 q^{66} +2.75557 q^{67} +7.18421 q^{68} -1.37778 q^{69} +2.00000 q^{71} +0.719004 q^{72} -1.57136 q^{73} -14.4889 q^{74} -3.93978 q^{76} -2.00000 q^{77} -12.2351 q^{78} -4.85728 q^{79} +1.00000 q^{81} +15.6731 q^{82} +11.6128 q^{83} +1.62222 q^{84} -19.2257 q^{86} -0.755569 q^{87} +1.43801 q^{88} +4.62222 q^{89} +6.42864 q^{91} +2.23506 q^{92} -5.18421 q^{93} -5.24443 q^{94} -7.34122 q^{96} -11.9398 q^{97} -1.90321 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} - 3q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - 3q^{3} + 5q^{4} - q^{6} - 3q^{7} + 9q^{8} + 3q^{9} + 6q^{11} - 5q^{12} - 6q^{13} - q^{14} + 13q^{16} + q^{18} + 6q^{19} + 3q^{21} + 2q^{22} + 4q^{23} - 9q^{24} + 10q^{26} - 3q^{27} - 5q^{28} + 2q^{29} + 2q^{31} + 29q^{32} - 6q^{33} - 12q^{34} + 5q^{36} - 4q^{37} + 14q^{38} + 6q^{39} + 2q^{41} + q^{42} + 4q^{43} + 10q^{44} - 8q^{46} + 8q^{47} - 13q^{48} + 3q^{49} - 18q^{52} + 14q^{53} - q^{54} - 9q^{56} - 6q^{57} - 18q^{58} + 16q^{59} - 6q^{61} - 30q^{62} - 3q^{63} + 13q^{64} - 2q^{66} + 8q^{67} + 8q^{68} - 4q^{69} + 6q^{71} + 9q^{72} - 18q^{73} - 44q^{74} + 2q^{76} - 6q^{77} - 10q^{78} + 12q^{79} + 3q^{81} + 34q^{82} + 8q^{83} + 5q^{84} - 4q^{86} - 2q^{87} + 18q^{88} + 14q^{89} + 6q^{91} - 20q^{92} - 2q^{93} - 16q^{94} - 29q^{96} - 22q^{97} + q^{98} + 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90321 −1.34577 −0.672887 0.739745i \(-0.734946\pi\)
−0.672887 + 0.739745i \(0.734946\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.62222 0.811108
\(5\) 0 0
\(6\) 1.90321 0.776983
\(7\) −1.00000 −0.377964
\(8\) 0.719004 0.254206
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.62222 −0.468293
\(13\) −6.42864 −1.78298 −0.891492 0.453037i \(-0.850341\pi\)
−0.891492 + 0.453037i \(0.850341\pi\)
\(14\) 1.90321 0.508655
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) 4.42864 1.07410 0.537051 0.843550i \(-0.319538\pi\)
0.537051 + 0.843550i \(0.319538\pi\)
\(18\) −1.90321 −0.448591
\(19\) −2.42864 −0.557168 −0.278584 0.960412i \(-0.589865\pi\)
−0.278584 + 0.960412i \(0.589865\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −3.80642 −0.811532
\(23\) 1.37778 0.287288 0.143644 0.989629i \(-0.454118\pi\)
0.143644 + 0.989629i \(0.454118\pi\)
\(24\) −0.719004 −0.146766
\(25\) 0 0
\(26\) 12.2351 2.39949
\(27\) −1.00000 −0.192450
\(28\) −1.62222 −0.306570
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) 5.18421 0.931111 0.465556 0.885019i \(-0.345855\pi\)
0.465556 + 0.885019i \(0.345855\pi\)
\(32\) 7.34122 1.29776
\(33\) −2.00000 −0.348155
\(34\) −8.42864 −1.44550
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) 7.61285 1.25154 0.625772 0.780006i \(-0.284784\pi\)
0.625772 + 0.780006i \(0.284784\pi\)
\(38\) 4.62222 0.749822
\(39\) 6.42864 1.02941
\(40\) 0 0
\(41\) −8.23506 −1.28610 −0.643050 0.765824i \(-0.722331\pi\)
−0.643050 + 0.765824i \(0.722331\pi\)
\(42\) −1.90321 −0.293672
\(43\) 10.1017 1.54050 0.770248 0.637744i \(-0.220132\pi\)
0.770248 + 0.637744i \(0.220132\pi\)
\(44\) 3.24443 0.489116
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) 2.75557 0.401941 0.200971 0.979597i \(-0.435590\pi\)
0.200971 + 0.979597i \(0.435590\pi\)
\(48\) 4.61285 0.665807
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.42864 −0.620134
\(52\) −10.4286 −1.44619
\(53\) 9.18421 1.26155 0.630774 0.775967i \(-0.282737\pi\)
0.630774 + 0.775967i \(0.282737\pi\)
\(54\) 1.90321 0.258994
\(55\) 0 0
\(56\) −0.719004 −0.0960809
\(57\) 2.42864 0.321681
\(58\) −1.43801 −0.188820
\(59\) 14.1017 1.83589 0.917943 0.396712i \(-0.129849\pi\)
0.917943 + 0.396712i \(0.129849\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) −9.86665 −1.25307
\(63\) −1.00000 −0.125988
\(64\) −4.74620 −0.593275
\(65\) 0 0
\(66\) 3.80642 0.468538
\(67\) 2.75557 0.336646 0.168323 0.985732i \(-0.446165\pi\)
0.168323 + 0.985732i \(0.446165\pi\)
\(68\) 7.18421 0.871213
\(69\) −1.37778 −0.165866
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0.719004 0.0847354
\(73\) −1.57136 −0.183914 −0.0919569 0.995763i \(-0.529312\pi\)
−0.0919569 + 0.995763i \(0.529312\pi\)
\(74\) −14.4889 −1.68430
\(75\) 0 0
\(76\) −3.93978 −0.451923
\(77\) −2.00000 −0.227921
\(78\) −12.2351 −1.38535
\(79\) −4.85728 −0.546487 −0.273243 0.961945i \(-0.588096\pi\)
−0.273243 + 0.961945i \(0.588096\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.6731 1.73080
\(83\) 11.6128 1.27468 0.637338 0.770585i \(-0.280036\pi\)
0.637338 + 0.770585i \(0.280036\pi\)
\(84\) 1.62222 0.176998
\(85\) 0 0
\(86\) −19.2257 −2.07316
\(87\) −0.755569 −0.0810055
\(88\) 1.43801 0.153292
\(89\) 4.62222 0.489954 0.244977 0.969529i \(-0.421220\pi\)
0.244977 + 0.969529i \(0.421220\pi\)
\(90\) 0 0
\(91\) 6.42864 0.673905
\(92\) 2.23506 0.233021
\(93\) −5.18421 −0.537577
\(94\) −5.24443 −0.540922
\(95\) 0 0
\(96\) −7.34122 −0.749260
\(97\) −11.9398 −1.21230 −0.606150 0.795350i \(-0.707287\pi\)
−0.606150 + 0.795350i \(0.707287\pi\)
\(98\) −1.90321 −0.192253
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 1.47949 0.147215 0.0736076 0.997287i \(-0.476549\pi\)
0.0736076 + 0.997287i \(0.476549\pi\)
\(102\) 8.42864 0.834560
\(103\) −8.85728 −0.872734 −0.436367 0.899769i \(-0.643735\pi\)
−0.436367 + 0.899769i \(0.643735\pi\)
\(104\) −4.62222 −0.453246
\(105\) 0 0
\(106\) −17.4795 −1.69776
\(107\) 1.76494 0.170623 0.0853114 0.996354i \(-0.472811\pi\)
0.0853114 + 0.996354i \(0.472811\pi\)
\(108\) −1.62222 −0.156098
\(109\) 5.61285 0.537613 0.268807 0.963194i \(-0.413371\pi\)
0.268807 + 0.963194i \(0.413371\pi\)
\(110\) 0 0
\(111\) −7.61285 −0.722580
\(112\) 4.61285 0.435873
\(113\) −11.2859 −1.06169 −0.530845 0.847469i \(-0.678125\pi\)
−0.530845 + 0.847469i \(0.678125\pi\)
\(114\) −4.62222 −0.432910
\(115\) 0 0
\(116\) 1.22570 0.113803
\(117\) −6.42864 −0.594328
\(118\) −26.8385 −2.47069
\(119\) −4.42864 −0.405973
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −13.0509 −1.18157
\(123\) 8.23506 0.742531
\(124\) 8.40990 0.755232
\(125\) 0 0
\(126\) 1.90321 0.169552
\(127\) −12.8573 −1.14090 −0.570450 0.821333i \(-0.693231\pi\)
−0.570450 + 0.821333i \(0.693231\pi\)
\(128\) −5.64941 −0.499342
\(129\) −10.1017 −0.889406
\(130\) 0 0
\(131\) −2.10171 −0.183627 −0.0918136 0.995776i \(-0.529266\pi\)
−0.0918136 + 0.995776i \(0.529266\pi\)
\(132\) −3.24443 −0.282391
\(133\) 2.42864 0.210590
\(134\) −5.24443 −0.453050
\(135\) 0 0
\(136\) 3.18421 0.273044
\(137\) 15.9398 1.36183 0.680914 0.732364i \(-0.261583\pi\)
0.680914 + 0.732364i \(0.261583\pi\)
\(138\) 2.62222 0.223218
\(139\) 11.6731 0.990097 0.495048 0.868865i \(-0.335150\pi\)
0.495048 + 0.868865i \(0.335150\pi\)
\(140\) 0 0
\(141\) −2.75557 −0.232061
\(142\) −3.80642 −0.319428
\(143\) −12.8573 −1.07518
\(144\) −4.61285 −0.384404
\(145\) 0 0
\(146\) 2.99063 0.247506
\(147\) −1.00000 −0.0824786
\(148\) 12.3497 1.01514
\(149\) −21.2257 −1.73888 −0.869438 0.494041i \(-0.835519\pi\)
−0.869438 + 0.494041i \(0.835519\pi\)
\(150\) 0 0
\(151\) 16.8573 1.37183 0.685913 0.727684i \(-0.259403\pi\)
0.685913 + 0.727684i \(0.259403\pi\)
\(152\) −1.74620 −0.141636
\(153\) 4.42864 0.358034
\(154\) 3.80642 0.306730
\(155\) 0 0
\(156\) 10.4286 0.834959
\(157\) −10.4286 −0.832296 −0.416148 0.909297i \(-0.636620\pi\)
−0.416148 + 0.909297i \(0.636620\pi\)
\(158\) 9.24443 0.735447
\(159\) −9.18421 −0.728355
\(160\) 0 0
\(161\) −1.37778 −0.108585
\(162\) −1.90321 −0.149530
\(163\) 20.8573 1.63367 0.816834 0.576873i \(-0.195727\pi\)
0.816834 + 0.576873i \(0.195727\pi\)
\(164\) −13.3590 −1.04317
\(165\) 0 0
\(166\) −22.1017 −1.71543
\(167\) 15.3461 1.18752 0.593760 0.804642i \(-0.297643\pi\)
0.593760 + 0.804642i \(0.297643\pi\)
\(168\) 0.719004 0.0554723
\(169\) 28.3274 2.17903
\(170\) 0 0
\(171\) −2.42864 −0.185723
\(172\) 16.3872 1.24951
\(173\) 2.06022 0.156636 0.0783179 0.996928i \(-0.475045\pi\)
0.0783179 + 0.996928i \(0.475045\pi\)
\(174\) 1.43801 0.109015
\(175\) 0 0
\(176\) −9.22570 −0.695413
\(177\) −14.1017 −1.05995
\(178\) −8.79706 −0.659367
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −12.1017 −0.899513 −0.449757 0.893151i \(-0.648489\pi\)
−0.449757 + 0.893151i \(0.648489\pi\)
\(182\) −12.2351 −0.906923
\(183\) −6.85728 −0.506905
\(184\) 0.990632 0.0730304
\(185\) 0 0
\(186\) 9.86665 0.723458
\(187\) 8.85728 0.647708
\(188\) 4.47013 0.326017
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0.488863 0.0353729 0.0176864 0.999844i \(-0.494370\pi\)
0.0176864 + 0.999844i \(0.494370\pi\)
\(192\) 4.74620 0.342528
\(193\) −22.9590 −1.65262 −0.826312 0.563212i \(-0.809565\pi\)
−0.826312 + 0.563212i \(0.809565\pi\)
\(194\) 22.7239 1.63148
\(195\) 0 0
\(196\) 1.62222 0.115873
\(197\) −1.18421 −0.0843713 −0.0421857 0.999110i \(-0.513432\pi\)
−0.0421857 + 0.999110i \(0.513432\pi\)
\(198\) −3.80642 −0.270511
\(199\) 8.79706 0.623607 0.311803 0.950147i \(-0.399067\pi\)
0.311803 + 0.950147i \(0.399067\pi\)
\(200\) 0 0
\(201\) −2.75557 −0.194363
\(202\) −2.81579 −0.198118
\(203\) −0.755569 −0.0530305
\(204\) −7.18421 −0.502995
\(205\) 0 0
\(206\) 16.8573 1.17450
\(207\) 1.37778 0.0957626
\(208\) 29.6543 2.05616
\(209\) −4.85728 −0.335985
\(210\) 0 0
\(211\) 23.2257 1.59892 0.799461 0.600717i \(-0.205118\pi\)
0.799461 + 0.600717i \(0.205118\pi\)
\(212\) 14.8988 1.02325
\(213\) −2.00000 −0.137038
\(214\) −3.35905 −0.229620
\(215\) 0 0
\(216\) −0.719004 −0.0489220
\(217\) −5.18421 −0.351927
\(218\) −10.6824 −0.723506
\(219\) 1.57136 0.106183
\(220\) 0 0
\(221\) −28.4701 −1.91511
\(222\) 14.4889 0.972429
\(223\) −15.2257 −1.01959 −0.509794 0.860297i \(-0.670278\pi\)
−0.509794 + 0.860297i \(0.670278\pi\)
\(224\) −7.34122 −0.490506
\(225\) 0 0
\(226\) 21.4795 1.42879
\(227\) −14.3684 −0.953665 −0.476833 0.878994i \(-0.658215\pi\)
−0.476833 + 0.878994i \(0.658215\pi\)
\(228\) 3.93978 0.260918
\(229\) −5.61285 −0.370907 −0.185454 0.982653i \(-0.559375\pi\)
−0.185454 + 0.982653i \(0.559375\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0.543257 0.0356666
\(233\) 23.2859 1.52551 0.762756 0.646687i \(-0.223846\pi\)
0.762756 + 0.646687i \(0.223846\pi\)
\(234\) 12.2351 0.799831
\(235\) 0 0
\(236\) 22.8760 1.48910
\(237\) 4.85728 0.315514
\(238\) 8.42864 0.546348
\(239\) −8.48886 −0.549099 −0.274549 0.961573i \(-0.588529\pi\)
−0.274549 + 0.961573i \(0.588529\pi\)
\(240\) 0 0
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 13.3225 0.856402
\(243\) −1.00000 −0.0641500
\(244\) 11.1240 0.712140
\(245\) 0 0
\(246\) −15.6731 −0.999278
\(247\) 15.6128 0.993422
\(248\) 3.72746 0.236694
\(249\) −11.6128 −0.735934
\(250\) 0 0
\(251\) 27.6128 1.74291 0.871454 0.490478i \(-0.163178\pi\)
0.871454 + 0.490478i \(0.163178\pi\)
\(252\) −1.62222 −0.102190
\(253\) 2.75557 0.173241
\(254\) 24.4701 1.53539
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) −0.428639 −0.0267378 −0.0133689 0.999911i \(-0.504256\pi\)
−0.0133689 + 0.999911i \(0.504256\pi\)
\(258\) 19.2257 1.19694
\(259\) −7.61285 −0.473039
\(260\) 0 0
\(261\) 0.755569 0.0467685
\(262\) 4.00000 0.247121
\(263\) −9.37778 −0.578259 −0.289129 0.957290i \(-0.593366\pi\)
−0.289129 + 0.957290i \(0.593366\pi\)
\(264\) −1.43801 −0.0885032
\(265\) 0 0
\(266\) −4.62222 −0.283406
\(267\) −4.62222 −0.282875
\(268\) 4.47013 0.273056
\(269\) 1.74620 0.106468 0.0532339 0.998582i \(-0.483047\pi\)
0.0532339 + 0.998582i \(0.483047\pi\)
\(270\) 0 0
\(271\) −2.69535 −0.163731 −0.0818653 0.996643i \(-0.526088\pi\)
−0.0818653 + 0.996643i \(0.526088\pi\)
\(272\) −20.4286 −1.23867
\(273\) −6.42864 −0.389079
\(274\) −30.3368 −1.83271
\(275\) 0 0
\(276\) −2.23506 −0.134535
\(277\) −5.12399 −0.307870 −0.153935 0.988081i \(-0.549195\pi\)
−0.153935 + 0.988081i \(0.549195\pi\)
\(278\) −22.2163 −1.33245
\(279\) 5.18421 0.310370
\(280\) 0 0
\(281\) 23.9813 1.43060 0.715301 0.698816i \(-0.246290\pi\)
0.715301 + 0.698816i \(0.246290\pi\)
\(282\) 5.24443 0.312301
\(283\) 2.36842 0.140788 0.0703939 0.997519i \(-0.477574\pi\)
0.0703939 + 0.997519i \(0.477574\pi\)
\(284\) 3.24443 0.192522
\(285\) 0 0
\(286\) 24.4701 1.44695
\(287\) 8.23506 0.486100
\(288\) 7.34122 0.432585
\(289\) 2.61285 0.153697
\(290\) 0 0
\(291\) 11.9398 0.699922
\(292\) −2.54909 −0.149174
\(293\) 8.42864 0.492406 0.246203 0.969218i \(-0.420817\pi\)
0.246203 + 0.969218i \(0.420817\pi\)
\(294\) 1.90321 0.110998
\(295\) 0 0
\(296\) 5.47367 0.318150
\(297\) −2.00000 −0.116052
\(298\) 40.3970 2.34014
\(299\) −8.85728 −0.512230
\(300\) 0 0
\(301\) −10.1017 −0.582253
\(302\) −32.0830 −1.84617
\(303\) −1.47949 −0.0849947
\(304\) 11.2029 0.642533
\(305\) 0 0
\(306\) −8.42864 −0.481833
\(307\) −22.5718 −1.28824 −0.644121 0.764923i \(-0.722777\pi\)
−0.644121 + 0.764923i \(0.722777\pi\)
\(308\) −3.24443 −0.184869
\(309\) 8.85728 0.503873
\(310\) 0 0
\(311\) −24.0830 −1.36562 −0.682810 0.730596i \(-0.739242\pi\)
−0.682810 + 0.730596i \(0.739242\pi\)
\(312\) 4.62222 0.261681
\(313\) 9.65433 0.545695 0.272848 0.962057i \(-0.412035\pi\)
0.272848 + 0.962057i \(0.412035\pi\)
\(314\) 19.8479 1.12008
\(315\) 0 0
\(316\) −7.87955 −0.443260
\(317\) −6.04149 −0.339324 −0.169662 0.985502i \(-0.554268\pi\)
−0.169662 + 0.985502i \(0.554268\pi\)
\(318\) 17.4795 0.980201
\(319\) 1.51114 0.0846075
\(320\) 0 0
\(321\) −1.76494 −0.0985092
\(322\) 2.62222 0.146130
\(323\) −10.7556 −0.598456
\(324\) 1.62222 0.0901231
\(325\) 0 0
\(326\) −39.6958 −2.19855
\(327\) −5.61285 −0.310391
\(328\) −5.92104 −0.326935
\(329\) −2.75557 −0.151919
\(330\) 0 0
\(331\) 13.5111 0.742639 0.371320 0.928505i \(-0.378905\pi\)
0.371320 + 0.928505i \(0.378905\pi\)
\(332\) 18.8385 1.03390
\(333\) 7.61285 0.417181
\(334\) −29.2070 −1.59813
\(335\) 0 0
\(336\) −4.61285 −0.251651
\(337\) 10.4889 0.571365 0.285682 0.958324i \(-0.407780\pi\)
0.285682 + 0.958324i \(0.407780\pi\)
\(338\) −53.9131 −2.93248
\(339\) 11.2859 0.612967
\(340\) 0 0
\(341\) 10.3684 0.561481
\(342\) 4.62222 0.249941
\(343\) −1.00000 −0.0539949
\(344\) 7.26317 0.391604
\(345\) 0 0
\(346\) −3.92104 −0.210796
\(347\) −16.7239 −0.897787 −0.448894 0.893585i \(-0.648182\pi\)
−0.448894 + 0.893585i \(0.648182\pi\)
\(348\) −1.22570 −0.0657042
\(349\) −16.3684 −0.876181 −0.438091 0.898931i \(-0.644345\pi\)
−0.438091 + 0.898931i \(0.644345\pi\)
\(350\) 0 0
\(351\) 6.42864 0.343135
\(352\) 14.6824 0.782577
\(353\) 0.549086 0.0292249 0.0146124 0.999893i \(-0.495349\pi\)
0.0146124 + 0.999893i \(0.495349\pi\)
\(354\) 26.8385 1.42645
\(355\) 0 0
\(356\) 7.49823 0.397405
\(357\) 4.42864 0.234388
\(358\) −19.0321 −1.00588
\(359\) −0.285442 −0.0150651 −0.00753253 0.999972i \(-0.502398\pi\)
−0.00753253 + 0.999972i \(0.502398\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) 23.0321 1.21054
\(363\) 7.00000 0.367405
\(364\) 10.4286 0.546609
\(365\) 0 0
\(366\) 13.0509 0.682179
\(367\) 1.71456 0.0894992 0.0447496 0.998998i \(-0.485751\pi\)
0.0447496 + 0.998998i \(0.485751\pi\)
\(368\) −6.35551 −0.331304
\(369\) −8.23506 −0.428700
\(370\) 0 0
\(371\) −9.18421 −0.476820
\(372\) −8.40990 −0.436033
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) −16.8573 −0.871669
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) −4.85728 −0.250163
\(378\) −1.90321 −0.0978907
\(379\) −4.85728 −0.249502 −0.124751 0.992188i \(-0.539813\pi\)
−0.124751 + 0.992188i \(0.539813\pi\)
\(380\) 0 0
\(381\) 12.8573 0.658698
\(382\) −0.930409 −0.0476039
\(383\) −8.38715 −0.428563 −0.214282 0.976772i \(-0.568741\pi\)
−0.214282 + 0.976772i \(0.568741\pi\)
\(384\) 5.64941 0.288295
\(385\) 0 0
\(386\) 43.6958 2.22406
\(387\) 10.1017 0.513499
\(388\) −19.3689 −0.983307
\(389\) 8.95899 0.454239 0.227119 0.973867i \(-0.427069\pi\)
0.227119 + 0.973867i \(0.427069\pi\)
\(390\) 0 0
\(391\) 6.10171 0.308577
\(392\) 0.719004 0.0363152
\(393\) 2.10171 0.106017
\(394\) 2.25380 0.113545
\(395\) 0 0
\(396\) 3.24443 0.163039
\(397\) −2.54909 −0.127935 −0.0639675 0.997952i \(-0.520375\pi\)
−0.0639675 + 0.997952i \(0.520375\pi\)
\(398\) −16.7427 −0.839234
\(399\) −2.42864 −0.121584
\(400\) 0 0
\(401\) 0.958989 0.0478896 0.0239448 0.999713i \(-0.492377\pi\)
0.0239448 + 0.999713i \(0.492377\pi\)
\(402\) 5.24443 0.261568
\(403\) −33.3274 −1.66016
\(404\) 2.40006 0.119407
\(405\) 0 0
\(406\) 1.43801 0.0713671
\(407\) 15.2257 0.754710
\(408\) −3.18421 −0.157642
\(409\) −31.9813 −1.58137 −0.790686 0.612222i \(-0.790276\pi\)
−0.790686 + 0.612222i \(0.790276\pi\)
\(410\) 0 0
\(411\) −15.9398 −0.786251
\(412\) −14.3684 −0.707881
\(413\) −14.1017 −0.693900
\(414\) −2.62222 −0.128875
\(415\) 0 0
\(416\) −47.1941 −2.31388
\(417\) −11.6731 −0.571633
\(418\) 9.24443 0.452160
\(419\) −0.470127 −0.0229672 −0.0114836 0.999934i \(-0.503655\pi\)
−0.0114836 + 0.999934i \(0.503655\pi\)
\(420\) 0 0
\(421\) −33.6128 −1.63819 −0.819095 0.573658i \(-0.805524\pi\)
−0.819095 + 0.573658i \(0.805524\pi\)
\(422\) −44.2034 −2.15179
\(423\) 2.75557 0.133980
\(424\) 6.60348 0.320693
\(425\) 0 0
\(426\) 3.80642 0.184422
\(427\) −6.85728 −0.331847
\(428\) 2.86311 0.138394
\(429\) 12.8573 0.620755
\(430\) 0 0
\(431\) 11.7146 0.564270 0.282135 0.959375i \(-0.408957\pi\)
0.282135 + 0.959375i \(0.408957\pi\)
\(432\) 4.61285 0.221936
\(433\) −0.0602231 −0.00289414 −0.00144707 0.999999i \(-0.500461\pi\)
−0.00144707 + 0.999999i \(0.500461\pi\)
\(434\) 9.86665 0.473614
\(435\) 0 0
\(436\) 9.10525 0.436062
\(437\) −3.34614 −0.160068
\(438\) −2.99063 −0.142898
\(439\) −22.4286 −1.07046 −0.535230 0.844706i \(-0.679775\pi\)
−0.535230 + 0.844706i \(0.679775\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 54.1847 2.57730
\(443\) −23.9496 −1.13788 −0.568940 0.822379i \(-0.692647\pi\)
−0.568940 + 0.822379i \(0.692647\pi\)
\(444\) −12.3497 −0.586090
\(445\) 0 0
\(446\) 28.9777 1.37214
\(447\) 21.2257 1.00394
\(448\) 4.74620 0.224237
\(449\) 29.4291 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(450\) 0 0
\(451\) −16.4701 −0.775548
\(452\) −18.3082 −0.861145
\(453\) −16.8573 −0.792024
\(454\) 27.3461 1.28342
\(455\) 0 0
\(456\) 1.74620 0.0817733
\(457\) −3.14272 −0.147010 −0.0735051 0.997295i \(-0.523419\pi\)
−0.0735051 + 0.997295i \(0.523419\pi\)
\(458\) 10.6824 0.499158
\(459\) −4.42864 −0.206711
\(460\) 0 0
\(461\) −3.37778 −0.157319 −0.0786596 0.996902i \(-0.525064\pi\)
−0.0786596 + 0.996902i \(0.525064\pi\)
\(462\) −3.80642 −0.177091
\(463\) 20.8573 0.969320 0.484660 0.874703i \(-0.338943\pi\)
0.484660 + 0.874703i \(0.338943\pi\)
\(464\) −3.48532 −0.161802
\(465\) 0 0
\(466\) −44.3180 −2.05299
\(467\) −14.3684 −0.664891 −0.332446 0.943122i \(-0.607874\pi\)
−0.332446 + 0.943122i \(0.607874\pi\)
\(468\) −10.4286 −0.482064
\(469\) −2.75557 −0.127240
\(470\) 0 0
\(471\) 10.4286 0.480526
\(472\) 10.1392 0.466694
\(473\) 20.2034 0.928954
\(474\) −9.24443 −0.424611
\(475\) 0 0
\(476\) −7.18421 −0.329288
\(477\) 9.18421 0.420516
\(478\) 16.1561 0.738963
\(479\) 6.36842 0.290980 0.145490 0.989360i \(-0.453524\pi\)
0.145490 + 0.989360i \(0.453524\pi\)
\(480\) 0 0
\(481\) −48.9403 −2.23148
\(482\) 13.7877 0.628012
\(483\) 1.37778 0.0626914
\(484\) −11.3555 −0.516160
\(485\) 0 0
\(486\) 1.90321 0.0863314
\(487\) 17.3274 0.785180 0.392590 0.919714i \(-0.371579\pi\)
0.392590 + 0.919714i \(0.371579\pi\)
\(488\) 4.93041 0.223189
\(489\) −20.8573 −0.943199
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 13.3590 0.602272
\(493\) 3.34614 0.150703
\(494\) −29.7146 −1.33692
\(495\) 0 0
\(496\) −23.9140 −1.07377
\(497\) −2.00000 −0.0897123
\(498\) 22.1017 0.990401
\(499\) 23.3461 1.04512 0.522558 0.852603i \(-0.324978\pi\)
0.522558 + 0.852603i \(0.324978\pi\)
\(500\) 0 0
\(501\) −15.3461 −0.685615
\(502\) −52.5531 −2.34556
\(503\) 0.387152 0.0172623 0.00863113 0.999963i \(-0.497253\pi\)
0.00863113 + 0.999963i \(0.497253\pi\)
\(504\) −0.719004 −0.0320270
\(505\) 0 0
\(506\) −5.24443 −0.233143
\(507\) −28.3274 −1.25806
\(508\) −20.8573 −0.925392
\(509\) −29.9496 −1.32749 −0.663747 0.747957i \(-0.731035\pi\)
−0.663747 + 0.747957i \(0.731035\pi\)
\(510\) 0 0
\(511\) 1.57136 0.0695129
\(512\) −27.2306 −1.20343
\(513\) 2.42864 0.107227
\(514\) 0.815792 0.0359830
\(515\) 0 0
\(516\) −16.3872 −0.721404
\(517\) 5.51114 0.242380
\(518\) 14.4889 0.636604
\(519\) −2.06022 −0.0904338
\(520\) 0 0
\(521\) −18.5205 −0.811398 −0.405699 0.914007i \(-0.632972\pi\)
−0.405699 + 0.914007i \(0.632972\pi\)
\(522\) −1.43801 −0.0629399
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −3.40943 −0.148942
\(525\) 0 0
\(526\) 17.8479 0.778206
\(527\) 22.9590 1.00011
\(528\) 9.22570 0.401497
\(529\) −21.1017 −0.917466
\(530\) 0 0
\(531\) 14.1017 0.611962
\(532\) 3.93978 0.170811
\(533\) 52.9403 2.29310
\(534\) 8.79706 0.380686
\(535\) 0 0
\(536\) 1.98126 0.0855776
\(537\) −10.0000 −0.431532
\(538\) −3.32339 −0.143282
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 14.5906 0.627298 0.313649 0.949539i \(-0.398449\pi\)
0.313649 + 0.949539i \(0.398449\pi\)
\(542\) 5.12981 0.220344
\(543\) 12.1017 0.519334
\(544\) 32.5116 1.39392
\(545\) 0 0
\(546\) 12.2351 0.523612
\(547\) −18.7556 −0.801930 −0.400965 0.916093i \(-0.631325\pi\)
−0.400965 + 0.916093i \(0.631325\pi\)
\(548\) 25.8578 1.10459
\(549\) 6.85728 0.292662
\(550\) 0 0
\(551\) −1.83500 −0.0781738
\(552\) −0.990632 −0.0421641
\(553\) 4.85728 0.206553
\(554\) 9.75203 0.414324
\(555\) 0 0
\(556\) 18.9362 0.803075
\(557\) 31.8765 1.35065 0.675325 0.737520i \(-0.264003\pi\)
0.675325 + 0.737520i \(0.264003\pi\)
\(558\) −9.86665 −0.417688
\(559\) −64.9403 −2.74668
\(560\) 0 0
\(561\) −8.85728 −0.373955
\(562\) −45.6414 −1.92527
\(563\) −2.01874 −0.0850796 −0.0425398 0.999095i \(-0.513545\pi\)
−0.0425398 + 0.999095i \(0.513545\pi\)
\(564\) −4.47013 −0.188226
\(565\) 0 0
\(566\) −4.50760 −0.189468
\(567\) −1.00000 −0.0419961
\(568\) 1.43801 0.0603375
\(569\) −28.9590 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(570\) 0 0
\(571\) 8.97773 0.375706 0.187853 0.982197i \(-0.439847\pi\)
0.187853 + 0.982197i \(0.439847\pi\)
\(572\) −20.8573 −0.872087
\(573\) −0.488863 −0.0204225
\(574\) −15.6731 −0.654181
\(575\) 0 0
\(576\) −4.74620 −0.197758
\(577\) 28.6766 1.19382 0.596911 0.802307i \(-0.296394\pi\)
0.596911 + 0.802307i \(0.296394\pi\)
\(578\) −4.97280 −0.206841
\(579\) 22.9590 0.954143
\(580\) 0 0
\(581\) −11.6128 −0.481782
\(582\) −22.7239 −0.941937
\(583\) 18.3684 0.760742
\(584\) −1.12981 −0.0467520
\(585\) 0 0
\(586\) −16.0415 −0.662668
\(587\) 45.2070 1.86589 0.932945 0.360018i \(-0.117229\pi\)
0.932945 + 0.360018i \(0.117229\pi\)
\(588\) −1.62222 −0.0668990
\(589\) −12.5906 −0.518786
\(590\) 0 0
\(591\) 1.18421 0.0487118
\(592\) −35.1169 −1.44330
\(593\) −18.2636 −0.749998 −0.374999 0.927025i \(-0.622357\pi\)
−0.374999 + 0.927025i \(0.622357\pi\)
\(594\) 3.80642 0.156179
\(595\) 0 0
\(596\) −34.4327 −1.41042
\(597\) −8.79706 −0.360040
\(598\) 16.8573 0.689345
\(599\) −22.7368 −0.929002 −0.464501 0.885573i \(-0.653766\pi\)
−0.464501 + 0.885573i \(0.653766\pi\)
\(600\) 0 0
\(601\) 0.488863 0.0199411 0.00997056 0.999950i \(-0.496826\pi\)
0.00997056 + 0.999950i \(0.496826\pi\)
\(602\) 19.2257 0.783581
\(603\) 2.75557 0.112215
\(604\) 27.3461 1.11270
\(605\) 0 0
\(606\) 2.81579 0.114384
\(607\) 20.2034 0.820032 0.410016 0.912078i \(-0.365523\pi\)
0.410016 + 0.912078i \(0.365523\pi\)
\(608\) −17.8292 −0.723069
\(609\) 0.755569 0.0306172
\(610\) 0 0
\(611\) −17.7146 −0.716654
\(612\) 7.18421 0.290404
\(613\) −10.3684 −0.418776 −0.209388 0.977833i \(-0.567147\pi\)
−0.209388 + 0.977833i \(0.567147\pi\)
\(614\) 42.9590 1.73368
\(615\) 0 0
\(616\) −1.43801 −0.0579390
\(617\) 39.2859 1.58159 0.790796 0.612080i \(-0.209667\pi\)
0.790796 + 0.612080i \(0.209667\pi\)
\(618\) −16.8573 −0.678099
\(619\) 42.8988 1.72425 0.862123 0.506698i \(-0.169134\pi\)
0.862123 + 0.506698i \(0.169134\pi\)
\(620\) 0 0
\(621\) −1.37778 −0.0552886
\(622\) 45.8350 1.83782
\(623\) −4.62222 −0.185185
\(624\) −29.6543 −1.18712
\(625\) 0 0
\(626\) −18.3742 −0.734383
\(627\) 4.85728 0.193981
\(628\) −16.9175 −0.675082
\(629\) 33.7146 1.34429
\(630\) 0 0
\(631\) 15.3461 0.610920 0.305460 0.952205i \(-0.401190\pi\)
0.305460 + 0.952205i \(0.401190\pi\)
\(632\) −3.49240 −0.138920
\(633\) −23.2257 −0.923139
\(634\) 11.4982 0.456653
\(635\) 0 0
\(636\) −14.8988 −0.590775
\(637\) −6.42864 −0.254712
\(638\) −2.87601 −0.113863
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 30.6735 1.21153 0.605766 0.795643i \(-0.292867\pi\)
0.605766 + 0.795643i \(0.292867\pi\)
\(642\) 3.35905 0.132571
\(643\) 49.0607 1.93477 0.967383 0.253320i \(-0.0815225\pi\)
0.967383 + 0.253320i \(0.0815225\pi\)
\(644\) −2.23506 −0.0880738
\(645\) 0 0
\(646\) 20.4701 0.805386
\(647\) 15.3461 0.603319 0.301660 0.953416i \(-0.402459\pi\)
0.301660 + 0.953416i \(0.402459\pi\)
\(648\) 0.719004 0.0282451
\(649\) 28.2034 1.10708
\(650\) 0 0
\(651\) 5.18421 0.203185
\(652\) 33.8350 1.32508
\(653\) 19.4697 0.761906 0.380953 0.924594i \(-0.375596\pi\)
0.380953 + 0.924594i \(0.375596\pi\)
\(654\) 10.6824 0.417716
\(655\) 0 0
\(656\) 37.9871 1.48315
\(657\) −1.57136 −0.0613046
\(658\) 5.24443 0.204449
\(659\) 30.9403 1.20526 0.602631 0.798020i \(-0.294119\pi\)
0.602631 + 0.798020i \(0.294119\pi\)
\(660\) 0 0
\(661\) 47.7975 1.85911 0.929554 0.368685i \(-0.120192\pi\)
0.929554 + 0.368685i \(0.120192\pi\)
\(662\) −25.7146 −0.999425
\(663\) 28.4701 1.10569
\(664\) 8.34968 0.324030
\(665\) 0 0
\(666\) −14.4889 −0.561432
\(667\) 1.04101 0.0403081
\(668\) 24.8948 0.963207
\(669\) 15.2257 0.588659
\(670\) 0 0
\(671\) 13.7146 0.529445
\(672\) 7.34122 0.283194
\(673\) −27.8163 −1.07224 −0.536119 0.844142i \(-0.680110\pi\)
−0.536119 + 0.844142i \(0.680110\pi\)
\(674\) −19.9625 −0.768928
\(675\) 0 0
\(676\) 45.9532 1.76743
\(677\) 19.0005 0.730248 0.365124 0.930959i \(-0.381027\pi\)
0.365124 + 0.930959i \(0.381027\pi\)
\(678\) −21.4795 −0.824915
\(679\) 11.9398 0.458207
\(680\) 0 0
\(681\) 14.3684 0.550599
\(682\) −19.7333 −0.755627
\(683\) −4.52051 −0.172972 −0.0864862 0.996253i \(-0.527564\pi\)
−0.0864862 + 0.996253i \(0.527564\pi\)
\(684\) −3.93978 −0.150641
\(685\) 0 0
\(686\) 1.90321 0.0726650
\(687\) 5.61285 0.214143
\(688\) −46.5977 −1.77652
\(689\) −59.0420 −2.24932
\(690\) 0 0
\(691\) −1.18421 −0.0450494 −0.0225247 0.999746i \(-0.507170\pi\)
−0.0225247 + 0.999746i \(0.507170\pi\)
\(692\) 3.34213 0.127049
\(693\) −2.00000 −0.0759737
\(694\) 31.8292 1.20822
\(695\) 0 0
\(696\) −0.543257 −0.0205921
\(697\) −36.4701 −1.38140
\(698\) 31.1526 1.17914
\(699\) −23.2859 −0.880754
\(700\) 0 0
\(701\) −26.6735 −1.00745 −0.503723 0.863865i \(-0.668037\pi\)
−0.503723 + 0.863865i \(0.668037\pi\)
\(702\) −12.2351 −0.461783
\(703\) −18.4889 −0.697321
\(704\) −9.49240 −0.357758
\(705\) 0 0
\(706\) −1.04503 −0.0393301
\(707\) −1.47949 −0.0556421
\(708\) −22.8760 −0.859733
\(709\) −18.2034 −0.683644 −0.341822 0.939765i \(-0.611044\pi\)
−0.341822 + 0.939765i \(0.611044\pi\)
\(710\) 0 0
\(711\) −4.85728 −0.182162
\(712\) 3.32339 0.124549
\(713\) 7.14272 0.267497
\(714\) −8.42864 −0.315434
\(715\) 0 0
\(716\) 16.2222 0.606250
\(717\) 8.48886 0.317022
\(718\) 0.543257 0.0202742
\(719\) 4.85728 0.181146 0.0905730 0.995890i \(-0.471130\pi\)
0.0905730 + 0.995890i \(0.471130\pi\)
\(720\) 0 0
\(721\) 8.85728 0.329862
\(722\) 24.9353 0.927997
\(723\) 7.24443 0.269423
\(724\) −19.6316 −0.729602
\(725\) 0 0
\(726\) −13.3225 −0.494444
\(727\) −21.0607 −0.781098 −0.390549 0.920582i \(-0.627715\pi\)
−0.390549 + 0.920582i \(0.627715\pi\)
\(728\) 4.62222 0.171311
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 44.7368 1.65465
\(732\) −11.1240 −0.411154
\(733\) −9.45091 −0.349077 −0.174539 0.984650i \(-0.555843\pi\)
−0.174539 + 0.984650i \(0.555843\pi\)
\(734\) −3.26317 −0.120446
\(735\) 0 0
\(736\) 10.1146 0.372830
\(737\) 5.51114 0.203005
\(738\) 15.6731 0.576934
\(739\) −8.20342 −0.301768 −0.150884 0.988551i \(-0.548212\pi\)
−0.150884 + 0.988551i \(0.548212\pi\)
\(740\) 0 0
\(741\) −15.6128 −0.573552
\(742\) 17.4795 0.641692
\(743\) −8.33677 −0.305847 −0.152923 0.988238i \(-0.548869\pi\)
−0.152923 + 0.988238i \(0.548869\pi\)
\(744\) −3.72746 −0.136655
\(745\) 0 0
\(746\) −30.4514 −1.11490
\(747\) 11.6128 0.424892
\(748\) 14.3684 0.525361
\(749\) −1.76494 −0.0644894
\(750\) 0 0
\(751\) −25.9180 −0.945760 −0.472880 0.881127i \(-0.656786\pi\)
−0.472880 + 0.881127i \(0.656786\pi\)
\(752\) −12.7110 −0.463523
\(753\) −27.6128 −1.00627
\(754\) 9.24443 0.336662
\(755\) 0 0
\(756\) 1.62222 0.0589994
\(757\) −8.94025 −0.324939 −0.162470 0.986714i \(-0.551946\pi\)
−0.162470 + 0.986714i \(0.551946\pi\)
\(758\) 9.24443 0.335773
\(759\) −2.75557 −0.100021
\(760\) 0 0
\(761\) −0.825636 −0.0299293 −0.0149646 0.999888i \(-0.504764\pi\)
−0.0149646 + 0.999888i \(0.504764\pi\)
\(762\) −24.4701 −0.886459
\(763\) −5.61285 −0.203199
\(764\) 0.793040 0.0286912
\(765\) 0 0
\(766\) 15.9625 0.576750
\(767\) −90.6548 −3.27336
\(768\) −20.2444 −0.730508
\(769\) 21.2257 0.765418 0.382709 0.923869i \(-0.374991\pi\)
0.382709 + 0.923869i \(0.374991\pi\)
\(770\) 0 0
\(771\) 0.428639 0.0154371
\(772\) −37.2444 −1.34046
\(773\) −29.4893 −1.06066 −0.530329 0.847792i \(-0.677932\pi\)
−0.530329 + 0.847792i \(0.677932\pi\)
\(774\) −19.2257 −0.691053
\(775\) 0 0
\(776\) −8.58474 −0.308174
\(777\) 7.61285 0.273109
\(778\) −17.0509 −0.611303
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −11.6128 −0.415275
\(783\) −0.755569 −0.0270018
\(784\) −4.61285 −0.164745
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 34.4514 1.22806 0.614030 0.789283i \(-0.289547\pi\)
0.614030 + 0.789283i \(0.289547\pi\)
\(788\) −1.92104 −0.0684343
\(789\) 9.37778 0.333858
\(790\) 0 0
\(791\) 11.2859 0.401281
\(792\) 1.43801 0.0510974
\(793\) −44.0830 −1.56543
\(794\) 4.85145 0.172172
\(795\) 0 0
\(796\) 14.2707 0.505812
\(797\) −18.9175 −0.670092 −0.335046 0.942202i \(-0.608752\pi\)
−0.335046 + 0.942202i \(0.608752\pi\)
\(798\) 4.62222 0.163625
\(799\) 12.2034 0.431726
\(800\) 0 0
\(801\) 4.62222 0.163318
\(802\) −1.82516 −0.0644486
\(803\) −3.14272 −0.110904
\(804\) −4.47013 −0.157649
\(805\) 0 0
\(806\) 63.4291 2.23420
\(807\) −1.74620 −0.0614692
\(808\) 1.06376 0.0374230
\(809\) 21.2257 0.746256 0.373128 0.927780i \(-0.378285\pi\)
0.373128 + 0.927780i \(0.378285\pi\)
\(810\) 0 0
\(811\) −21.5081 −0.755251 −0.377625 0.925958i \(-0.623259\pi\)
−0.377625 + 0.925958i \(0.623259\pi\)
\(812\) −1.22570 −0.0430135
\(813\) 2.69535 0.0945299
\(814\) −28.9777 −1.01567
\(815\) 0 0
\(816\) 20.4286 0.715145
\(817\) −24.5334 −0.858315
\(818\) 60.8671 2.12817
\(819\) 6.42864 0.224635
\(820\) 0 0
\(821\) −46.2034 −1.61251 −0.806255 0.591568i \(-0.798509\pi\)
−0.806255 + 0.591568i \(0.798509\pi\)
\(822\) 30.3368 1.05812
\(823\) 17.8350 0.621689 0.310845 0.950461i \(-0.399388\pi\)
0.310845 + 0.950461i \(0.399388\pi\)
\(824\) −6.36842 −0.221854
\(825\) 0 0
\(826\) 26.8385 0.933832
\(827\) −35.2128 −1.22447 −0.612234 0.790676i \(-0.709729\pi\)
−0.612234 + 0.790676i \(0.709729\pi\)
\(828\) 2.23506 0.0776738
\(829\) 14.3872 0.499686 0.249843 0.968286i \(-0.419621\pi\)
0.249843 + 0.968286i \(0.419621\pi\)
\(830\) 0 0
\(831\) 5.12399 0.177749
\(832\) 30.5116 1.05780
\(833\) 4.42864 0.153443
\(834\) 22.2163 0.769289
\(835\) 0 0
\(836\) −7.87955 −0.272520
\(837\) −5.18421 −0.179192
\(838\) 0.894751 0.0309087
\(839\) 1.51114 0.0521703 0.0260851 0.999660i \(-0.491696\pi\)
0.0260851 + 0.999660i \(0.491696\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 63.9724 2.20463
\(843\) −23.9813 −0.825959
\(844\) 37.6771 1.29690
\(845\) 0 0
\(846\) −5.24443 −0.180307
\(847\) 7.00000 0.240523
\(848\) −42.3654 −1.45483
\(849\) −2.36842 −0.0812838
\(850\) 0 0
\(851\) 10.4889 0.359554
\(852\) −3.24443 −0.111152
\(853\) 15.4064 0.527504 0.263752 0.964591i \(-0.415040\pi\)
0.263752 + 0.964591i \(0.415040\pi\)
\(854\) 13.0509 0.446591
\(855\) 0 0
\(856\) 1.26900 0.0433734
\(857\) −19.8578 −0.678328 −0.339164 0.940727i \(-0.610144\pi\)
−0.339164 + 0.940727i \(0.610144\pi\)
\(858\) −24.4701 −0.835396
\(859\) 2.42864 0.0828641 0.0414321 0.999141i \(-0.486808\pi\)
0.0414321 + 0.999141i \(0.486808\pi\)
\(860\) 0 0
\(861\) −8.23506 −0.280650
\(862\) −22.2953 −0.759380
\(863\) 39.2958 1.33764 0.668822 0.743423i \(-0.266799\pi\)
0.668822 + 0.743423i \(0.266799\pi\)
\(864\) −7.34122 −0.249753
\(865\) 0 0
\(866\) 0.114617 0.00389485
\(867\) −2.61285 −0.0887370
\(868\) −8.40990 −0.285451
\(869\) −9.71456 −0.329544
\(870\) 0 0
\(871\) −17.7146 −0.600235
\(872\) 4.03566 0.136665
\(873\) −11.9398 −0.404100
\(874\) 6.36842 0.215415
\(875\) 0 0
\(876\) 2.54909 0.0861256
\(877\) 56.2864 1.90066 0.950328 0.311249i \(-0.100747\pi\)
0.950328 + 0.311249i \(0.100747\pi\)
\(878\) 42.6865 1.44060
\(879\) −8.42864 −0.284291
\(880\) 0 0
\(881\) −2.33677 −0.0787279 −0.0393640 0.999225i \(-0.512533\pi\)
−0.0393640 + 0.999225i \(0.512533\pi\)
\(882\) −1.90321 −0.0640845
\(883\) −33.7146 −1.13459 −0.567293 0.823516i \(-0.692009\pi\)
−0.567293 + 0.823516i \(0.692009\pi\)
\(884\) −46.1847 −1.55336
\(885\) 0 0
\(886\) 45.5812 1.53133
\(887\) −47.8992 −1.60830 −0.804150 0.594427i \(-0.797379\pi\)
−0.804150 + 0.594427i \(0.797379\pi\)
\(888\) −5.47367 −0.183684
\(889\) 12.8573 0.431219
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −24.6994 −0.826996
\(893\) −6.69228 −0.223949
\(894\) −40.3970 −1.35108
\(895\) 0 0
\(896\) 5.64941 0.188734
\(897\) 8.85728 0.295736
\(898\) −56.0098 −1.86907
\(899\) 3.91703 0.130640
\(900\) 0 0
\(901\) 40.6735 1.35503
\(902\) 31.3461 1.04371
\(903\) 10.1017 0.336164
\(904\) −8.11462 −0.269888
\(905\) 0 0
\(906\) 32.0830 1.06589
\(907\) −23.7591 −0.788908 −0.394454 0.918916i \(-0.629066\pi\)
−0.394454 + 0.918916i \(0.629066\pi\)
\(908\) −23.3087 −0.773525
\(909\) 1.47949 0.0490717
\(910\) 0 0
\(911\) 22.9403 0.760045 0.380022 0.924977i \(-0.375916\pi\)
0.380022 + 0.924977i \(0.375916\pi\)
\(912\) −11.2029 −0.370967
\(913\) 23.2257 0.768658
\(914\) 5.98126 0.197843
\(915\) 0 0
\(916\) −9.10525 −0.300846
\(917\) 2.10171 0.0694046
\(918\) 8.42864 0.278187
\(919\) 16.9777 0.560043 0.280022 0.959994i \(-0.409658\pi\)
0.280022 + 0.959994i \(0.409658\pi\)
\(920\) 0 0
\(921\) 22.5718 0.743767
\(922\) 6.42864 0.211716
\(923\) −12.8573 −0.423202
\(924\) 3.24443 0.106734
\(925\) 0 0
\(926\) −39.6958 −1.30449
\(927\) −8.85728 −0.290911
\(928\) 5.54680 0.182082
\(929\) 39.3403 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(930\) 0 0
\(931\) −2.42864 −0.0795954
\(932\) 37.7748 1.23735
\(933\) 24.0830 0.788441
\(934\) 27.3461 0.894793
\(935\) 0 0
\(936\) −4.62222 −0.151082
\(937\) 17.7748 0.580677 0.290338 0.956924i \(-0.406232\pi\)
0.290338 + 0.956924i \(0.406232\pi\)
\(938\) 5.24443 0.171237
\(939\) −9.65433 −0.315057
\(940\) 0 0
\(941\) 35.5812 1.15991 0.579957 0.814647i \(-0.303069\pi\)
0.579957 + 0.814647i \(0.303069\pi\)
\(942\) −19.8479 −0.646680
\(943\) −11.3461 −0.369481
\(944\) −65.0490 −2.11717
\(945\) 0 0
\(946\) −38.4514 −1.25016
\(947\) −30.5018 −0.991174 −0.495587 0.868558i \(-0.665047\pi\)
−0.495587 + 0.868558i \(0.665047\pi\)
\(948\) 7.87955 0.255916
\(949\) 10.1017 0.327915
\(950\) 0 0
\(951\) 6.04149 0.195909
\(952\) −3.18421 −0.103201
\(953\) 51.1655 1.65741 0.828706 0.559684i \(-0.189077\pi\)
0.828706 + 0.559684i \(0.189077\pi\)
\(954\) −17.4795 −0.565920
\(955\) 0 0
\(956\) −13.7708 −0.445378
\(957\) −1.51114 −0.0488481
\(958\) −12.1204 −0.391594
\(959\) −15.9398 −0.514722
\(960\) 0 0
\(961\) −4.12399 −0.133032
\(962\) 93.1437 3.00307
\(963\) 1.76494 0.0568743
\(964\) −11.7520 −0.378507
\(965\) 0 0
\(966\) −2.62222 −0.0843684
\(967\) −47.8992 −1.54034 −0.770168 0.637841i \(-0.779828\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(968\) −5.03303 −0.161768
\(969\) 10.7556 0.345519
\(970\) 0 0
\(971\) 40.6735 1.30528 0.652638 0.757670i \(-0.273662\pi\)
0.652638 + 0.757670i \(0.273662\pi\)
\(972\) −1.62222 −0.0520326
\(973\) −11.6731 −0.374221
\(974\) −32.9777 −1.05667
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) 27.4893 0.879462 0.439731 0.898130i \(-0.355074\pi\)
0.439731 + 0.898130i \(0.355074\pi\)
\(978\) 39.6958 1.26933
\(979\) 9.24443 0.295453
\(980\) 0 0
\(981\) 5.61285 0.179204
\(982\) −3.80642 −0.121468
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 5.92104 0.188756
\(985\) 0 0
\(986\) −6.36842 −0.202812
\(987\) 2.75557 0.0877107
\(988\) 25.3274 0.805772
\(989\) 13.9180 0.442566
\(990\) 0 0
\(991\) −34.6923 −1.10204 −0.551018 0.834493i \(-0.685761\pi\)
−0.551018 + 0.834493i \(0.685761\pi\)
\(992\) 38.0584 1.20836
\(993\) −13.5111 −0.428763
\(994\) 3.80642 0.120732
\(995\) 0 0
\(996\) −18.8385 −0.596922
\(997\) 28.6766 0.908197 0.454099 0.890951i \(-0.349961\pi\)
0.454099 + 0.890951i \(0.349961\pi\)
\(998\) −44.4327 −1.40649
\(999\) −7.61285 −0.240860
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.1.1 3
3.2 odd 2 1575.2.a.w.1.3 3
4.3 odd 2 8400.2.a.dj.1.1 3
5.2 odd 4 105.2.d.b.64.2 6
5.3 odd 4 105.2.d.b.64.5 yes 6
5.4 even 2 525.2.a.j.1.3 3
7.6 odd 2 3675.2.a.bj.1.1 3
15.2 even 4 315.2.d.e.64.5 6
15.8 even 4 315.2.d.e.64.2 6
15.14 odd 2 1575.2.a.x.1.1 3
20.3 even 4 1680.2.t.k.1009.5 6
20.7 even 4 1680.2.t.k.1009.2 6
20.19 odd 2 8400.2.a.dg.1.3 3
35.2 odd 12 735.2.q.e.214.2 12
35.3 even 12 735.2.q.f.79.2 12
35.12 even 12 735.2.q.f.214.2 12
35.13 even 4 735.2.d.b.589.5 6
35.17 even 12 735.2.q.f.79.5 12
35.18 odd 12 735.2.q.e.79.2 12
35.23 odd 12 735.2.q.e.214.5 12
35.27 even 4 735.2.d.b.589.2 6
35.32 odd 12 735.2.q.e.79.5 12
35.33 even 12 735.2.q.f.214.5 12
35.34 odd 2 3675.2.a.bi.1.3 3
60.23 odd 4 5040.2.t.v.1009.3 6
60.47 odd 4 5040.2.t.v.1009.4 6
105.62 odd 4 2205.2.d.l.1324.5 6
105.83 odd 4 2205.2.d.l.1324.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 5.2 odd 4
105.2.d.b.64.5 yes 6 5.3 odd 4
315.2.d.e.64.2 6 15.8 even 4
315.2.d.e.64.5 6 15.2 even 4
525.2.a.j.1.3 3 5.4 even 2
525.2.a.k.1.1 3 1.1 even 1 trivial
735.2.d.b.589.2 6 35.27 even 4
735.2.d.b.589.5 6 35.13 even 4
735.2.q.e.79.2 12 35.18 odd 12
735.2.q.e.79.5 12 35.32 odd 12
735.2.q.e.214.2 12 35.2 odd 12
735.2.q.e.214.5 12 35.23 odd 12
735.2.q.f.79.2 12 35.3 even 12
735.2.q.f.79.5 12 35.17 even 12
735.2.q.f.214.2 12 35.12 even 12
735.2.q.f.214.5 12 35.33 even 12
1575.2.a.w.1.3 3 3.2 odd 2
1575.2.a.x.1.1 3 15.14 odd 2
1680.2.t.k.1009.2 6 20.7 even 4
1680.2.t.k.1009.5 6 20.3 even 4
2205.2.d.l.1324.2 6 105.83 odd 4
2205.2.d.l.1324.5 6 105.62 odd 4
3675.2.a.bi.1.3 3 35.34 odd 2
3675.2.a.bj.1.1 3 7.6 odd 2
5040.2.t.v.1009.3 6 60.23 odd 4
5040.2.t.v.1009.4 6 60.47 odd 4
8400.2.a.dg.1.3 3 20.19 odd 2
8400.2.a.dj.1.1 3 4.3 odd 2