Properties

Label 525.2.a.j.1.3
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
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

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

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 \(-0.265054\pi\)
0.672887 + 0.739745i \(0.265054\pi\)
\(3\) 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 \(-0.149659\pi\)
0.891492 + 0.453037i \(0.149659\pi\)
\(14\) 1.90321 0.508655
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) −4.42864 −1.07410 −0.537051 0.843550i \(-0.680462\pi\)
−0.537051 + 0.843550i \(0.680462\pi\)
\(18\) 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 \(-0.545882\pi\)
−0.143644 + 0.989629i \(0.545882\pi\)
\(24\) −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 \(-0.715216\pi\)
−0.625772 + 0.780006i \(0.715216\pi\)
\(38\) −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 \(-0.779868\pi\)
−0.770248 + 0.637744i \(0.779868\pi\)
\(44\) 3.24443 0.489116
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) −2.75557 −0.401941 −0.200971 0.979597i \(-0.564410\pi\)
−0.200971 + 0.979597i \(0.564410\pi\)
\(48\) −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 \(-0.717263\pi\)
−0.630774 + 0.775967i \(0.717263\pi\)
\(54\) 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 \(-0.553835\pi\)
−0.168323 + 0.985732i \(0.553835\pi\)
\(68\) −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 \(-0.470688\pi\)
0.0919569 + 0.995763i \(0.470688\pi\)
\(74\) −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 \(-0.719964\pi\)
−0.637338 + 0.770585i \(0.719964\pi\)
\(84\) 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 \(-0.292713\pi\)
0.606150 + 0.795350i \(0.292713\pi\)
\(98\) 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 \(-0.356265\pi\)
0.436367 + 0.899769i \(0.356265\pi\)
\(104\) −4.62222 −0.453246
\(105\) 0 0
\(106\) −17.4795 −1.69776
\(107\) −1.76494 −0.170623 −0.0853114 0.996354i \(-0.527189\pi\)
−0.0853114 + 0.996354i \(0.527189\pi\)
\(108\) 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 \(-0.321875\pi\)
0.530845 + 0.847469i \(0.321875\pi\)
\(114\) −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 \(-0.306769\pi\)
0.570450 + 0.821333i \(0.306769\pi\)
\(128\) 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 \(-0.738417\pi\)
−0.680914 + 0.732364i \(0.738417\pi\)
\(138\) −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 \(-0.363380\pi\)
0.416148 + 0.909297i \(0.363380\pi\)
\(158\) −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 \(-0.804273\pi\)
−0.816834 + 0.576873i \(0.804273\pi\)
\(164\) −13.3590 −1.04317
\(165\) 0 0
\(166\) −22.1017 −1.71543
\(167\) −15.3461 −1.18752 −0.593760 0.804642i \(-0.702357\pi\)
−0.593760 + 0.804642i \(0.702357\pi\)
\(168\) −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 \(-0.524955\pi\)
−0.0783179 + 0.996928i \(0.524955\pi\)
\(174\) 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 \(-0.190435\pi\)
0.826312 + 0.563212i \(0.190435\pi\)
\(194\) 22.7239 1.63148
\(195\) 0 0
\(196\) 1.62222 0.115873
\(197\) 1.18421 0.0843713 0.0421857 0.999110i \(-0.486568\pi\)
0.0421857 + 0.999110i \(0.486568\pi\)
\(198\) 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 \(-0.329722\pi\)
0.509794 + 0.860297i \(0.329722\pi\)
\(224\) −7.34122 −0.490506
\(225\) 0 0
\(226\) 21.4795 1.42879
\(227\) 14.3684 0.953665 0.476833 0.878994i \(-0.341785\pi\)
0.476833 + 0.878994i \(0.341785\pi\)
\(228\) −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 \(-0.776154\pi\)
−0.762756 + 0.646687i \(0.776154\pi\)
\(234\) 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 \(-0.495744\pi\)
0.0133689 + 0.999911i \(0.495744\pi\)
\(258\) −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 \(-0.406634\pi\)
0.289129 + 0.957290i \(0.406634\pi\)
\(264\) −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 \(-0.450805\pi\)
0.153935 + 0.988081i \(0.450805\pi\)
\(278\) 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 \(-0.522426\pi\)
−0.0703939 + 0.997519i \(0.522426\pi\)
\(284\) 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 \(-0.579183\pi\)
−0.246203 + 0.969218i \(0.579183\pi\)
\(294\) 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 \(-0.277223\pi\)
0.644121 + 0.764923i \(0.277223\pi\)
\(308\) 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 \(-0.587965\pi\)
−0.272848 + 0.962057i \(0.587965\pi\)
\(314\) 19.8479 1.12008
\(315\) 0 0
\(316\) −7.87955 −0.443260
\(317\) 6.04149 0.339324 0.169662 0.985502i \(-0.445732\pi\)
0.169662 + 0.985502i \(0.445732\pi\)
\(318\) −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 \(-0.592220\pi\)
−0.285682 + 0.958324i \(0.592220\pi\)
\(338\) 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 \(-0.351818\pi\)
0.448894 + 0.893585i \(0.351818\pi\)
\(348\) 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 \(-0.504651\pi\)
−0.0146124 + 0.999893i \(0.504651\pi\)
\(354\) 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 \(-0.514249\pi\)
−0.0447496 + 0.998998i \(0.514249\pi\)
\(368\) 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 \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −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 \(-0.431259\pi\)
0.214282 + 0.976772i \(0.431259\pi\)
\(384\) 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 \(-0.479625\pi\)
0.0639675 + 0.997952i \(0.479625\pi\)
\(398\) 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 \(-0.499539\pi\)
0.00144707 + 0.999999i \(0.499539\pi\)
\(434\) 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 \(-0.307353\pi\)
0.568940 + 0.822379i \(0.307353\pi\)
\(444\) −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 \(-0.476581\pi\)
0.0735051 + 0.997295i \(0.476581\pi\)
\(458\) −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 \(-0.661057\pi\)
−0.484660 + 0.874703i \(0.661057\pi\)
\(464\) −3.48532 −0.161802
\(465\) 0 0
\(466\) −44.3180 −2.05299
\(467\) 14.3684 0.664891 0.332446 0.943122i \(-0.392126\pi\)
0.332446 + 0.943122i \(0.392126\pi\)
\(468\) 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 \(-0.628421\pi\)
−0.392590 + 0.919714i \(0.628421\pi\)
\(488\) −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 \(-0.502747\pi\)
−0.00863113 + 0.999963i \(0.502747\pi\)
\(504\) −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 \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −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 \(-0.368675\pi\)
0.400965 + 0.916093i \(0.368675\pi\)
\(548\) −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 \(-0.735997\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(558\) 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 \(-0.486455\pi\)
0.0425398 + 0.999095i \(0.486455\pi\)
\(564\) −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 \(-0.703606\pi\)
−0.596911 + 0.802307i \(0.703606\pi\)
\(578\) 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 \(-0.882771\pi\)
−0.932945 + 0.360018i \(0.882771\pi\)
\(588\) 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 \(-0.377643\pi\)
0.374999 + 0.927025i \(0.377643\pi\)
\(594\) 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 \(-0.634477\pi\)
−0.410016 + 0.912078i \(0.634477\pi\)
\(608\) 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 \(-0.432853\pi\)
0.209388 + 0.977833i \(0.432853\pi\)
\(614\) 42.9590 1.73368
\(615\) 0 0
\(616\) −1.43801 −0.0579390
\(617\) −39.2859 −1.58159 −0.790796 0.612080i \(-0.790333\pi\)
−0.790796 + 0.612080i \(0.790333\pi\)
\(618\) 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 \(-0.918477\pi\)
−0.967383 + 0.253320i \(0.918477\pi\)
\(644\) −2.23506 −0.0880738
\(645\) 0 0
\(646\) 20.4701 0.805386
\(647\) −15.3461 −0.603319 −0.301660 0.953416i \(-0.597541\pi\)
−0.301660 + 0.953416i \(0.597541\pi\)
\(648\) −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 \(-0.624404\pi\)
−0.380953 + 0.924594i \(0.624404\pi\)
\(654\) 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 \(-0.319890\pi\)
0.536119 + 0.844142i \(0.319890\pi\)
\(674\) −19.9625 −0.768928
\(675\) 0 0
\(676\) 45.9532 1.76743
\(677\) −19.0005 −0.730248 −0.365124 0.930959i \(-0.618973\pi\)
−0.365124 + 0.930959i \(0.618973\pi\)
\(678\) 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 \(-0.472436\pi\)
0.0864862 + 0.996253i \(0.472436\pi\)
\(684\) −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 \(-0.372285\pi\)
0.390549 + 0.920582i \(0.372285\pi\)
\(728\) −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 \(-0.444157\pi\)
0.174539 + 0.984650i \(0.444157\pi\)
\(734\) −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 \(-0.451131\pi\)
0.152923 + 0.988238i \(0.451131\pi\)
\(744\) −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 \(-0.448054\pi\)
0.162470 + 0.986714i \(0.448054\pi\)
\(758\) −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 \(-0.322068\pi\)
0.530329 + 0.847792i \(0.322068\pi\)
\(774\) −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 \(-0.710453\pi\)
−0.614030 + 0.789283i \(0.710453\pi\)
\(788\) 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 \(-0.391248\pi\)
0.335046 + 0.942202i \(0.391248\pi\)
\(798\) −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 \(-0.600612\pi\)
−0.310845 + 0.950461i \(0.600612\pi\)
\(824\) −6.36842 −0.221854
\(825\) 0 0
\(826\) 26.8385 0.933832
\(827\) 35.2128 1.22447 0.612234 0.790676i \(-0.290271\pi\)
0.612234 + 0.790676i \(0.290271\pi\)
\(828\) −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 \(-0.584960\pi\)
−0.263752 + 0.964591i \(0.584960\pi\)
\(854\) 13.0509 0.446591
\(855\) 0 0
\(856\) 1.26900 0.0433734
\(857\) 19.8578 0.678328 0.339164 0.940727i \(-0.389856\pi\)
0.339164 + 0.940727i \(0.389856\pi\)
\(858\) 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 \(-0.733201\pi\)
−0.668822 + 0.743423i \(0.733201\pi\)
\(864\) −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 \(-0.899253\pi\)
−0.950328 + 0.311249i \(0.899253\pi\)
\(878\) −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 \(-0.307991\pi\)
0.567293 + 0.823516i \(0.307991\pi\)
\(884\) −46.1847 −1.55336
\(885\) 0 0
\(886\) 45.5812 1.53133
\(887\) 47.8992 1.60830 0.804150 0.594427i \(-0.202621\pi\)
0.804150 + 0.594427i \(0.202621\pi\)
\(888\) 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 \(-0.370934\pi\)
0.394454 + 0.918916i \(0.370934\pi\)
\(908\) 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 \(-0.593768\pi\)
−0.290338 + 0.956924i \(0.593768\pi\)
\(938\) −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 \(-0.334953\pi\)
0.495587 + 0.868558i \(0.334953\pi\)
\(948\) −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 \(-0.810923\pi\)
−0.828706 + 0.559684i \(0.810923\pi\)
\(954\) −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 \(-0.220172\pi\)
0.770168 + 0.637841i \(0.220172\pi\)
\(968\) 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 \(-0.644926\pi\)
−0.439731 + 0.898130i \(0.644926\pi\)
\(978\) −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 \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 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 \(-0.650039\pi\)
−0.454099 + 0.890951i \(0.650039\pi\)
\(998\) 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.j.1.3 3
3.2 odd 2 1575.2.a.x.1.1 3
4.3 odd 2 8400.2.a.dg.1.3 3
5.2 odd 4 105.2.d.b.64.5 yes 6
5.3 odd 4 105.2.d.b.64.2 6
5.4 even 2 525.2.a.k.1.1 3
7.6 odd 2 3675.2.a.bi.1.3 3
15.2 even 4 315.2.d.e.64.2 6
15.8 even 4 315.2.d.e.64.5 6
15.14 odd 2 1575.2.a.w.1.3 3
20.3 even 4 1680.2.t.k.1009.2 6
20.7 even 4 1680.2.t.k.1009.5 6
20.19 odd 2 8400.2.a.dj.1.1 3
35.2 odd 12 735.2.q.e.214.5 12
35.3 even 12 735.2.q.f.79.5 12
35.12 even 12 735.2.q.f.214.5 12
35.13 even 4 735.2.d.b.589.2 6
35.17 even 12 735.2.q.f.79.2 12
35.18 odd 12 735.2.q.e.79.5 12
35.23 odd 12 735.2.q.e.214.2 12
35.27 even 4 735.2.d.b.589.5 6
35.32 odd 12 735.2.q.e.79.2 12
35.33 even 12 735.2.q.f.214.2 12
35.34 odd 2 3675.2.a.bj.1.1 3
60.23 odd 4 5040.2.t.v.1009.4 6
60.47 odd 4 5040.2.t.v.1009.3 6
105.62 odd 4 2205.2.d.l.1324.2 6
105.83 odd 4 2205.2.d.l.1324.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 5.3 odd 4
105.2.d.b.64.5 yes 6 5.2 odd 4
315.2.d.e.64.2 6 15.2 even 4
315.2.d.e.64.5 6 15.8 even 4
525.2.a.j.1.3 3 1.1 even 1 trivial
525.2.a.k.1.1 3 5.4 even 2
735.2.d.b.589.2 6 35.13 even 4
735.2.d.b.589.5 6 35.27 even 4
735.2.q.e.79.2 12 35.32 odd 12
735.2.q.e.79.5 12 35.18 odd 12
735.2.q.e.214.2 12 35.23 odd 12
735.2.q.e.214.5 12 35.2 odd 12
735.2.q.f.79.2 12 35.17 even 12
735.2.q.f.79.5 12 35.3 even 12
735.2.q.f.214.2 12 35.33 even 12
735.2.q.f.214.5 12 35.12 even 12
1575.2.a.w.1.3 3 15.14 odd 2
1575.2.a.x.1.1 3 3.2 odd 2
1680.2.t.k.1009.2 6 20.3 even 4
1680.2.t.k.1009.5 6 20.7 even 4
2205.2.d.l.1324.2 6 105.62 odd 4
2205.2.d.l.1324.5 6 105.83 odd 4
3675.2.a.bi.1.3 3 7.6 odd 2
3675.2.a.bj.1.1 3 35.34 odd 2
5040.2.t.v.1009.3 6 60.47 odd 4
5040.2.t.v.1009.4 6 60.23 odd 4
8400.2.a.dg.1.3 3 4.3 odd 2
8400.2.a.dj.1.1 3 20.19 odd 2