Properties

Label 2541.2.a.q.1.2
Level 2541
Weight 2
Character 2541.1
Self dual yes
Analytic conductor 20.290
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\)
Character \(\chi\) = 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +2.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +2.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.38197 q^{10} +1.61803 q^{12} +4.23607 q^{13} -0.618034 q^{14} -2.23607 q^{15} +1.85410 q^{16} -2.00000 q^{17} +0.618034 q^{18} -7.47214 q^{19} -3.61803 q^{20} +1.00000 q^{21} -2.00000 q^{23} +2.23607 q^{24} +2.61803 q^{26} -1.00000 q^{27} +1.61803 q^{28} -5.00000 q^{29} -1.38197 q^{30} +1.52786 q^{31} +5.61803 q^{32} -1.23607 q^{34} -2.23607 q^{35} -1.61803 q^{36} +7.47214 q^{37} -4.61803 q^{38} -4.23607 q^{39} -5.00000 q^{40} +6.47214 q^{41} +0.618034 q^{42} -0.472136 q^{43} +2.23607 q^{45} -1.23607 q^{46} +0.527864 q^{47} -1.85410 q^{48} +1.00000 q^{49} +2.00000 q^{51} -6.85410 q^{52} -2.47214 q^{53} -0.618034 q^{54} +2.23607 q^{56} +7.47214 q^{57} -3.09017 q^{58} -9.94427 q^{59} +3.61803 q^{60} -3.52786 q^{61} +0.944272 q^{62} -1.00000 q^{63} -0.236068 q^{64} +9.47214 q^{65} -13.1803 q^{67} +3.23607 q^{68} +2.00000 q^{69} -1.38197 q^{70} +6.47214 q^{71} -2.23607 q^{72} -2.70820 q^{73} +4.61803 q^{74} +12.0902 q^{76} -2.61803 q^{78} -8.47214 q^{79} +4.14590 q^{80} +1.00000 q^{81} +4.00000 q^{82} -16.4721 q^{83} -1.61803 q^{84} -4.47214 q^{85} -0.291796 q^{86} +5.00000 q^{87} +8.47214 q^{89} +1.38197 q^{90} -4.23607 q^{91} +3.23607 q^{92} -1.52786 q^{93} +0.326238 q^{94} -16.7082 q^{95} -5.61803 q^{96} -17.4164 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + 5q^{10} + q^{12} + 4q^{13} + q^{14} - 3q^{16} - 4q^{17} - q^{18} - 6q^{19} - 5q^{20} + 2q^{21} - 4q^{23} + 3q^{26} - 2q^{27} + q^{28} - 10q^{29} - 5q^{30} + 12q^{31} + 9q^{32} + 2q^{34} - q^{36} + 6q^{37} - 7q^{38} - 4q^{39} - 10q^{40} + 4q^{41} - q^{42} + 8q^{43} + 2q^{46} + 10q^{47} + 3q^{48} + 2q^{49} + 4q^{51} - 7q^{52} + 4q^{53} + q^{54} + 6q^{57} + 5q^{58} - 2q^{59} + 5q^{60} - 16q^{61} - 16q^{62} - 2q^{63} + 4q^{64} + 10q^{65} - 4q^{67} + 2q^{68} + 4q^{69} - 5q^{70} + 4q^{71} + 8q^{73} + 7q^{74} + 13q^{76} - 3q^{78} - 8q^{79} + 15q^{80} + 2q^{81} + 8q^{82} - 24q^{83} - q^{84} - 14q^{86} + 10q^{87} + 8q^{89} + 5q^{90} - 4q^{91} + 2q^{92} - 12q^{93} - 15q^{94} - 20q^{95} - 9q^{96} - 8q^{97} - q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 2.23607 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) −0.618034 −0.252311
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 1.38197 0.437016
\(11\) 0 0
\(12\) 1.61803 0.467086
\(13\) 4.23607 1.17487 0.587437 0.809270i \(-0.300137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(14\) −0.618034 −0.165177
\(15\) −2.23607 −0.577350
\(16\) 1.85410 0.463525
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0.618034 0.145672
\(19\) −7.47214 −1.71423 −0.857113 0.515129i \(-0.827744\pi\)
−0.857113 + 0.515129i \(0.827744\pi\)
\(20\) −3.61803 −0.809017
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 2.61803 0.513439
\(27\) −1.00000 −0.192450
\(28\) 1.61803 0.305780
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −1.38197 −0.252311
\(31\) 1.52786 0.274412 0.137206 0.990543i \(-0.456188\pi\)
0.137206 + 0.990543i \(0.456188\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −1.23607 −0.211984
\(35\) −2.23607 −0.377964
\(36\) −1.61803 −0.269672
\(37\) 7.47214 1.22841 0.614206 0.789146i \(-0.289476\pi\)
0.614206 + 0.789146i \(0.289476\pi\)
\(38\) −4.61803 −0.749144
\(39\) −4.23607 −0.678314
\(40\) −5.00000 −0.790569
\(41\) 6.47214 1.01078 0.505389 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 0.618034 0.0953647
\(43\) −0.472136 −0.0720001 −0.0360000 0.999352i \(-0.511462\pi\)
−0.0360000 + 0.999352i \(0.511462\pi\)
\(44\) 0 0
\(45\) 2.23607 0.333333
\(46\) −1.23607 −0.182248
\(47\) 0.527864 0.0769969 0.0384984 0.999259i \(-0.487743\pi\)
0.0384984 + 0.999259i \(0.487743\pi\)
\(48\) −1.85410 −0.267617
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −6.85410 −0.950493
\(53\) −2.47214 −0.339574 −0.169787 0.985481i \(-0.554308\pi\)
−0.169787 + 0.985481i \(0.554308\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 7.47214 0.989709
\(58\) −3.09017 −0.405759
\(59\) −9.94427 −1.29463 −0.647317 0.762221i \(-0.724109\pi\)
−0.647317 + 0.762221i \(0.724109\pi\)
\(60\) 3.61803 0.467086
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 0.944272 0.119923
\(63\) −1.00000 −0.125988
\(64\) −0.236068 −0.0295085
\(65\) 9.47214 1.17487
\(66\) 0 0
\(67\) −13.1803 −1.61023 −0.805117 0.593115i \(-0.797898\pi\)
−0.805117 + 0.593115i \(0.797898\pi\)
\(68\) 3.23607 0.392431
\(69\) 2.00000 0.240772
\(70\) −1.38197 −0.165177
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) −2.23607 −0.263523
\(73\) −2.70820 −0.316971 −0.158486 0.987361i \(-0.550661\pi\)
−0.158486 + 0.987361i \(0.550661\pi\)
\(74\) 4.61803 0.536836
\(75\) 0 0
\(76\) 12.0902 1.38684
\(77\) 0 0
\(78\) −2.61803 −0.296434
\(79\) −8.47214 −0.953190 −0.476595 0.879123i \(-0.658129\pi\)
−0.476595 + 0.879123i \(0.658129\pi\)
\(80\) 4.14590 0.463525
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −16.4721 −1.80805 −0.904026 0.427478i \(-0.859402\pi\)
−0.904026 + 0.427478i \(0.859402\pi\)
\(84\) −1.61803 −0.176542
\(85\) −4.47214 −0.485071
\(86\) −0.291796 −0.0314652
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 8.47214 0.898045 0.449022 0.893521i \(-0.351772\pi\)
0.449022 + 0.893521i \(0.351772\pi\)
\(90\) 1.38197 0.145672
\(91\) −4.23607 −0.444061
\(92\) 3.23607 0.337383
\(93\) −1.52786 −0.158432
\(94\) 0.326238 0.0336489
\(95\) −16.7082 −1.71423
\(96\) −5.61803 −0.573388
\(97\) −17.4164 −1.76837 −0.884184 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) 0.618034 0.0624309
\(99\) 0 0
\(100\) 0 0
\(101\) −18.4721 −1.83805 −0.919023 0.394204i \(-0.871020\pi\)
−0.919023 + 0.394204i \(0.871020\pi\)
\(102\) 1.23607 0.122389
\(103\) −1.52786 −0.150545 −0.0752725 0.997163i \(-0.523983\pi\)
−0.0752725 + 0.997163i \(0.523983\pi\)
\(104\) −9.47214 −0.928819
\(105\) 2.23607 0.218218
\(106\) −1.52786 −0.148399
\(107\) 11.6525 1.12649 0.563244 0.826291i \(-0.309553\pi\)
0.563244 + 0.826291i \(0.309553\pi\)
\(108\) 1.61803 0.155695
\(109\) 1.52786 0.146343 0.0731714 0.997319i \(-0.476688\pi\)
0.0731714 + 0.997319i \(0.476688\pi\)
\(110\) 0 0
\(111\) −7.47214 −0.709224
\(112\) −1.85410 −0.175196
\(113\) −14.4721 −1.36142 −0.680712 0.732551i \(-0.738329\pi\)
−0.680712 + 0.732551i \(0.738329\pi\)
\(114\) 4.61803 0.432519
\(115\) −4.47214 −0.417029
\(116\) 8.09017 0.751153
\(117\) 4.23607 0.391625
\(118\) −6.14590 −0.565776
\(119\) 2.00000 0.183340
\(120\) 5.00000 0.456435
\(121\) 0 0
\(122\) −2.18034 −0.197399
\(123\) −6.47214 −0.583573
\(124\) −2.47214 −0.222004
\(125\) −11.1803 −1.00000
\(126\) −0.618034 −0.0550588
\(127\) −17.4164 −1.54546 −0.772728 0.634737i \(-0.781108\pi\)
−0.772728 + 0.634737i \(0.781108\pi\)
\(128\) −11.3820 −1.00603
\(129\) 0.472136 0.0415693
\(130\) 5.85410 0.513439
\(131\) 7.41641 0.647975 0.323987 0.946061i \(-0.394976\pi\)
0.323987 + 0.946061i \(0.394976\pi\)
\(132\) 0 0
\(133\) 7.47214 0.647916
\(134\) −8.14590 −0.703698
\(135\) −2.23607 −0.192450
\(136\) 4.47214 0.383482
\(137\) 17.4164 1.48798 0.743992 0.668188i \(-0.232930\pi\)
0.743992 + 0.668188i \(0.232930\pi\)
\(138\) 1.23607 0.105221
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) 3.61803 0.305780
\(141\) −0.527864 −0.0444542
\(142\) 4.00000 0.335673
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) −11.1803 −0.928477
\(146\) −1.67376 −0.138522
\(147\) −1.00000 −0.0824786
\(148\) −12.0902 −0.993806
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 16.7082 1.35521
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 3.41641 0.274412
\(156\) 6.85410 0.548767
\(157\) 12.4721 0.995385 0.497692 0.867354i \(-0.334181\pi\)
0.497692 + 0.867354i \(0.334181\pi\)
\(158\) −5.23607 −0.416559
\(159\) 2.47214 0.196053
\(160\) 12.5623 0.993137
\(161\) 2.00000 0.157622
\(162\) 0.618034 0.0485573
\(163\) −8.23607 −0.645099 −0.322549 0.946553i \(-0.604540\pi\)
−0.322549 + 0.946553i \(0.604540\pi\)
\(164\) −10.4721 −0.817736
\(165\) 0 0
\(166\) −10.1803 −0.790148
\(167\) −21.4164 −1.65725 −0.828626 0.559803i \(-0.810877\pi\)
−0.828626 + 0.559803i \(0.810877\pi\)
\(168\) −2.23607 −0.172516
\(169\) 4.94427 0.380329
\(170\) −2.76393 −0.211984
\(171\) −7.47214 −0.571409
\(172\) 0.763932 0.0582493
\(173\) 16.4721 1.25235 0.626177 0.779681i \(-0.284619\pi\)
0.626177 + 0.779681i \(0.284619\pi\)
\(174\) 3.09017 0.234265
\(175\) 0 0
\(176\) 0 0
\(177\) 9.94427 0.747457
\(178\) 5.23607 0.392460
\(179\) 18.3607 1.37234 0.686171 0.727440i \(-0.259290\pi\)
0.686171 + 0.727440i \(0.259290\pi\)
\(180\) −3.61803 −0.269672
\(181\) −4.94427 −0.367505 −0.183752 0.982973i \(-0.558824\pi\)
−0.183752 + 0.982973i \(0.558824\pi\)
\(182\) −2.61803 −0.194062
\(183\) 3.52786 0.260787
\(184\) 4.47214 0.329690
\(185\) 16.7082 1.22841
\(186\) −0.944272 −0.0692374
\(187\) 0 0
\(188\) −0.854102 −0.0622918
\(189\) 1.00000 0.0727393
\(190\) −10.3262 −0.749144
\(191\) 24.4721 1.77074 0.885371 0.464886i \(-0.153905\pi\)
0.885371 + 0.464886i \(0.153905\pi\)
\(192\) 0.236068 0.0170367
\(193\) −10.4721 −0.753801 −0.376900 0.926254i \(-0.623010\pi\)
−0.376900 + 0.926254i \(0.623010\pi\)
\(194\) −10.7639 −0.772805
\(195\) −9.47214 −0.678314
\(196\) −1.61803 −0.115574
\(197\) −11.8885 −0.847024 −0.423512 0.905891i \(-0.639203\pi\)
−0.423512 + 0.905891i \(0.639203\pi\)
\(198\) 0 0
\(199\) 23.8885 1.69341 0.846707 0.532059i \(-0.178582\pi\)
0.846707 + 0.532059i \(0.178582\pi\)
\(200\) 0 0
\(201\) 13.1803 0.929669
\(202\) −11.4164 −0.803256
\(203\) 5.00000 0.350931
\(204\) −3.23607 −0.226570
\(205\) 14.4721 1.01078
\(206\) −0.944272 −0.0657905
\(207\) −2.00000 −0.139010
\(208\) 7.85410 0.544584
\(209\) 0 0
\(210\) 1.38197 0.0953647
\(211\) 18.3607 1.26400 0.632001 0.774968i \(-0.282234\pi\)
0.632001 + 0.774968i \(0.282234\pi\)
\(212\) 4.00000 0.274721
\(213\) −6.47214 −0.443463
\(214\) 7.20163 0.492293
\(215\) −1.05573 −0.0720001
\(216\) 2.23607 0.152145
\(217\) −1.52786 −0.103718
\(218\) 0.944272 0.0639542
\(219\) 2.70820 0.183003
\(220\) 0 0
\(221\) −8.47214 −0.569898
\(222\) −4.61803 −0.309942
\(223\) 11.8885 0.796116 0.398058 0.917360i \(-0.369684\pi\)
0.398058 + 0.917360i \(0.369684\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) −8.94427 −0.594964
\(227\) −12.4721 −0.827805 −0.413902 0.910321i \(-0.635835\pi\)
−0.413902 + 0.910321i \(0.635835\pi\)
\(228\) −12.0902 −0.800691
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −2.76393 −0.182248
\(231\) 0 0
\(232\) 11.1803 0.734025
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 2.61803 0.171146
\(235\) 1.18034 0.0769969
\(236\) 16.0902 1.04738
\(237\) 8.47214 0.550324
\(238\) 1.23607 0.0801224
\(239\) −14.1246 −0.913645 −0.456823 0.889558i \(-0.651013\pi\)
−0.456823 + 0.889558i \(0.651013\pi\)
\(240\) −4.14590 −0.267617
\(241\) −3.18034 −0.204864 −0.102432 0.994740i \(-0.532662\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 5.70820 0.365430
\(245\) 2.23607 0.142857
\(246\) −4.00000 −0.255031
\(247\) −31.6525 −2.01400
\(248\) −3.41641 −0.216942
\(249\) 16.4721 1.04388
\(250\) −6.90983 −0.437016
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) 1.61803 0.101927
\(253\) 0 0
\(254\) −10.7639 −0.675389
\(255\) 4.47214 0.280056
\(256\) −6.56231 −0.410144
\(257\) 17.6525 1.10113 0.550566 0.834792i \(-0.314412\pi\)
0.550566 + 0.834792i \(0.314412\pi\)
\(258\) 0.291796 0.0181664
\(259\) −7.47214 −0.464296
\(260\) −15.3262 −0.950493
\(261\) −5.00000 −0.309492
\(262\) 4.58359 0.283175
\(263\) −24.7082 −1.52357 −0.761787 0.647828i \(-0.775678\pi\)
−0.761787 + 0.647828i \(0.775678\pi\)
\(264\) 0 0
\(265\) −5.52786 −0.339574
\(266\) 4.61803 0.283150
\(267\) −8.47214 −0.518486
\(268\) 21.3262 1.30271
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) −1.38197 −0.0841038
\(271\) 28.8885 1.75485 0.877427 0.479710i \(-0.159258\pi\)
0.877427 + 0.479710i \(0.159258\pi\)
\(272\) −3.70820 −0.224843
\(273\) 4.23607 0.256378
\(274\) 10.7639 0.650273
\(275\) 0 0
\(276\) −3.23607 −0.194788
\(277\) 14.3607 0.862850 0.431425 0.902149i \(-0.358011\pi\)
0.431425 + 0.902149i \(0.358011\pi\)
\(278\) −5.52786 −0.331539
\(279\) 1.52786 0.0914708
\(280\) 5.00000 0.298807
\(281\) −2.52786 −0.150800 −0.0753999 0.997153i \(-0.524023\pi\)
−0.0753999 + 0.997153i \(0.524023\pi\)
\(282\) −0.326238 −0.0194272
\(283\) −8.41641 −0.500304 −0.250152 0.968207i \(-0.580481\pi\)
−0.250152 + 0.968207i \(0.580481\pi\)
\(284\) −10.4721 −0.621407
\(285\) 16.7082 0.989709
\(286\) 0 0
\(287\) −6.47214 −0.382038
\(288\) 5.61803 0.331046
\(289\) −13.0000 −0.764706
\(290\) −6.90983 −0.405759
\(291\) 17.4164 1.02097
\(292\) 4.38197 0.256435
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −0.618034 −0.0360445
\(295\) −22.2361 −1.29463
\(296\) −16.7082 −0.971145
\(297\) 0 0
\(298\) −12.9787 −0.751837
\(299\) −8.47214 −0.489956
\(300\) 0 0
\(301\) 0.472136 0.0272135
\(302\) −6.18034 −0.355639
\(303\) 18.4721 1.06120
\(304\) −13.8541 −0.794587
\(305\) −7.88854 −0.451697
\(306\) −1.23607 −0.0706613
\(307\) 5.88854 0.336077 0.168038 0.985780i \(-0.446257\pi\)
0.168038 + 0.985780i \(0.446257\pi\)
\(308\) 0 0
\(309\) 1.52786 0.0869171
\(310\) 2.11146 0.119923
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 9.47214 0.536254
\(313\) −2.94427 −0.166420 −0.0832100 0.996532i \(-0.526517\pi\)
−0.0832100 + 0.996532i \(0.526517\pi\)
\(314\) 7.70820 0.434999
\(315\) −2.23607 −0.125988
\(316\) 13.7082 0.771147
\(317\) −20.9443 −1.17635 −0.588174 0.808735i \(-0.700153\pi\)
−0.588174 + 0.808735i \(0.700153\pi\)
\(318\) 1.52786 0.0856784
\(319\) 0 0
\(320\) −0.527864 −0.0295085
\(321\) −11.6525 −0.650378
\(322\) 1.23607 0.0688834
\(323\) 14.9443 0.831522
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) −5.09017 −0.281918
\(327\) −1.52786 −0.0844911
\(328\) −14.4721 −0.799090
\(329\) −0.527864 −0.0291021
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 26.6525 1.46274
\(333\) 7.47214 0.409471
\(334\) −13.2361 −0.724245
\(335\) −29.4721 −1.61023
\(336\) 1.85410 0.101150
\(337\) 36.4721 1.98676 0.993382 0.114858i \(-0.0366413\pi\)
0.993382 + 0.114858i \(0.0366413\pi\)
\(338\) 3.05573 0.166210
\(339\) 14.4721 0.786019
\(340\) 7.23607 0.392431
\(341\) 0 0
\(342\) −4.61803 −0.249715
\(343\) −1.00000 −0.0539949
\(344\) 1.05573 0.0569210
\(345\) 4.47214 0.240772
\(346\) 10.1803 0.547298
\(347\) −12.9443 −0.694885 −0.347442 0.937701i \(-0.612950\pi\)
−0.347442 + 0.937701i \(0.612950\pi\)
\(348\) −8.09017 −0.433679
\(349\) 22.1246 1.18430 0.592152 0.805827i \(-0.298279\pi\)
0.592152 + 0.805827i \(0.298279\pi\)
\(350\) 0 0
\(351\) −4.23607 −0.226105
\(352\) 0 0
\(353\) 17.1803 0.914417 0.457209 0.889359i \(-0.348849\pi\)
0.457209 + 0.889359i \(0.348849\pi\)
\(354\) 6.14590 0.326651
\(355\) 14.4721 0.768101
\(356\) −13.7082 −0.726533
\(357\) −2.00000 −0.105851
\(358\) 11.3475 0.599735
\(359\) −11.0557 −0.583499 −0.291750 0.956495i \(-0.594237\pi\)
−0.291750 + 0.956495i \(0.594237\pi\)
\(360\) −5.00000 −0.263523
\(361\) 36.8328 1.93857
\(362\) −3.05573 −0.160606
\(363\) 0 0
\(364\) 6.85410 0.359253
\(365\) −6.05573 −0.316971
\(366\) 2.18034 0.113968
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −3.70820 −0.193303
\(369\) 6.47214 0.336926
\(370\) 10.3262 0.536836
\(371\) 2.47214 0.128347
\(372\) 2.47214 0.128174
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 11.1803 0.577350
\(376\) −1.18034 −0.0608714
\(377\) −21.1803 −1.09084
\(378\) 0.618034 0.0317882
\(379\) 16.2361 0.833991 0.416995 0.908909i \(-0.363083\pi\)
0.416995 + 0.908909i \(0.363083\pi\)
\(380\) 27.0344 1.38684
\(381\) 17.4164 0.892270
\(382\) 15.1246 0.773842
\(383\) 16.9443 0.865812 0.432906 0.901439i \(-0.357488\pi\)
0.432906 + 0.901439i \(0.357488\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) −6.47214 −0.329423
\(387\) −0.472136 −0.0240000
\(388\) 28.1803 1.43064
\(389\) −34.4721 −1.74781 −0.873903 0.486100i \(-0.838419\pi\)
−0.873903 + 0.486100i \(0.838419\pi\)
\(390\) −5.85410 −0.296434
\(391\) 4.00000 0.202289
\(392\) −2.23607 −0.112938
\(393\) −7.41641 −0.374108
\(394\) −7.34752 −0.370163
\(395\) −18.9443 −0.953190
\(396\) 0 0
\(397\) 9.05573 0.454494 0.227247 0.973837i \(-0.427028\pi\)
0.227247 + 0.973837i \(0.427028\pi\)
\(398\) 14.7639 0.740049
\(399\) −7.47214 −0.374075
\(400\) 0 0
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) 8.14590 0.406280
\(403\) 6.47214 0.322400
\(404\) 29.8885 1.48701
\(405\) 2.23607 0.111111
\(406\) 3.09017 0.153363
\(407\) 0 0
\(408\) −4.47214 −0.221404
\(409\) 16.4721 0.814495 0.407247 0.913318i \(-0.366489\pi\)
0.407247 + 0.913318i \(0.366489\pi\)
\(410\) 8.94427 0.441726
\(411\) −17.4164 −0.859088
\(412\) 2.47214 0.121793
\(413\) 9.94427 0.489326
\(414\) −1.23607 −0.0607494
\(415\) −36.8328 −1.80805
\(416\) 23.7984 1.16681
\(417\) 8.94427 0.438003
\(418\) 0 0
\(419\) 17.0000 0.830504 0.415252 0.909706i \(-0.363693\pi\)
0.415252 + 0.909706i \(0.363693\pi\)
\(420\) −3.61803 −0.176542
\(421\) 7.36068 0.358738 0.179369 0.983782i \(-0.442594\pi\)
0.179369 + 0.983782i \(0.442594\pi\)
\(422\) 11.3475 0.552389
\(423\) 0.527864 0.0256656
\(424\) 5.52786 0.268457
\(425\) 0 0
\(426\) −4.00000 −0.193801
\(427\) 3.52786 0.170725
\(428\) −18.8541 −0.911347
\(429\) 0 0
\(430\) −0.652476 −0.0314652
\(431\) −15.7639 −0.759322 −0.379661 0.925126i \(-0.623959\pi\)
−0.379661 + 0.925126i \(0.623959\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 33.4164 1.60589 0.802945 0.596053i \(-0.203265\pi\)
0.802945 + 0.596053i \(0.203265\pi\)
\(434\) −0.944272 −0.0453265
\(435\) 11.1803 0.536056
\(436\) −2.47214 −0.118394
\(437\) 14.9443 0.714881
\(438\) 1.67376 0.0799754
\(439\) 8.88854 0.424227 0.212114 0.977245i \(-0.431965\pi\)
0.212114 + 0.977245i \(0.431965\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −5.23607 −0.249054
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 12.0902 0.573774
\(445\) 18.9443 0.898045
\(446\) 7.34752 0.347915
\(447\) 21.0000 0.993266
\(448\) 0.236068 0.0111532
\(449\) −8.47214 −0.399825 −0.199912 0.979814i \(-0.564066\pi\)
−0.199912 + 0.979814i \(0.564066\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 23.4164 1.10142
\(453\) 10.0000 0.469841
\(454\) −7.70820 −0.361764
\(455\) −9.47214 −0.444061
\(456\) −16.7082 −0.782433
\(457\) 38.3607 1.79444 0.897218 0.441587i \(-0.145584\pi\)
0.897218 + 0.441587i \(0.145584\pi\)
\(458\) 8.65248 0.404304
\(459\) 2.00000 0.0933520
\(460\) 7.23607 0.337383
\(461\) −1.05573 −0.0491702 −0.0245851 0.999698i \(-0.507826\pi\)
−0.0245851 + 0.999698i \(0.507826\pi\)
\(462\) 0 0
\(463\) −6.70820 −0.311757 −0.155878 0.987776i \(-0.549821\pi\)
−0.155878 + 0.987776i \(0.549821\pi\)
\(464\) −9.27051 −0.430373
\(465\) −3.41641 −0.158432
\(466\) 1.81966 0.0842941
\(467\) 27.9443 1.29311 0.646553 0.762869i \(-0.276210\pi\)
0.646553 + 0.762869i \(0.276210\pi\)
\(468\) −6.85410 −0.316831
\(469\) 13.1803 0.608612
\(470\) 0.729490 0.0336489
\(471\) −12.4721 −0.574686
\(472\) 22.2361 1.02350
\(473\) 0 0
\(474\) 5.23607 0.240501
\(475\) 0 0
\(476\) −3.23607 −0.148325
\(477\) −2.47214 −0.113191
\(478\) −8.72949 −0.399278
\(479\) 0.111456 0.00509256 0.00254628 0.999997i \(-0.499189\pi\)
0.00254628 + 0.999997i \(0.499189\pi\)
\(480\) −12.5623 −0.573388
\(481\) 31.6525 1.44323
\(482\) −1.96556 −0.0895287
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) −38.9443 −1.76837
\(486\) −0.618034 −0.0280346
\(487\) −29.8885 −1.35438 −0.677190 0.735809i \(-0.736802\pi\)
−0.677190 + 0.735809i \(0.736802\pi\)
\(488\) 7.88854 0.357098
\(489\) 8.23607 0.372448
\(490\) 1.38197 0.0624309
\(491\) −28.1246 −1.26925 −0.634623 0.772822i \(-0.718845\pi\)
−0.634623 + 0.772822i \(0.718845\pi\)
\(492\) 10.4721 0.472120
\(493\) 10.0000 0.450377
\(494\) −19.5623 −0.880150
\(495\) 0 0
\(496\) 2.83282 0.127197
\(497\) −6.47214 −0.290315
\(498\) 10.1803 0.456192
\(499\) 24.2361 1.08496 0.542478 0.840070i \(-0.317486\pi\)
0.542478 + 0.840070i \(0.317486\pi\)
\(500\) 18.0902 0.809017
\(501\) 21.4164 0.956815
\(502\) −6.79837 −0.303426
\(503\) 25.8885 1.15431 0.577157 0.816634i \(-0.304162\pi\)
0.577157 + 0.816634i \(0.304162\pi\)
\(504\) 2.23607 0.0996024
\(505\) −41.3050 −1.83805
\(506\) 0 0
\(507\) −4.94427 −0.219583
\(508\) 28.1803 1.25030
\(509\) 6.58359 0.291813 0.145906 0.989298i \(-0.453390\pi\)
0.145906 + 0.989298i \(0.453390\pi\)
\(510\) 2.76393 0.122389
\(511\) 2.70820 0.119804
\(512\) 18.7082 0.826794
\(513\) 7.47214 0.329903
\(514\) 10.9098 0.481212
\(515\) −3.41641 −0.150545
\(516\) −0.763932 −0.0336302
\(517\) 0 0
\(518\) −4.61803 −0.202905
\(519\) −16.4721 −0.723047
\(520\) −21.1803 −0.928819
\(521\) −30.1246 −1.31978 −0.659892 0.751361i \(-0.729398\pi\)
−0.659892 + 0.751361i \(0.729398\pi\)
\(522\) −3.09017 −0.135253
\(523\) 28.4164 1.24256 0.621281 0.783588i \(-0.286612\pi\)
0.621281 + 0.783588i \(0.286612\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −15.2705 −0.665826
\(527\) −3.05573 −0.133110
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −3.41641 −0.148399
\(531\) −9.94427 −0.431545
\(532\) −12.0902 −0.524175
\(533\) 27.4164 1.18754
\(534\) −5.23607 −0.226587
\(535\) 26.0557 1.12649
\(536\) 29.4721 1.27300
\(537\) −18.3607 −0.792322
\(538\) −8.29180 −0.357485
\(539\) 0 0
\(540\) 3.61803 0.155695
\(541\) −20.3607 −0.875374 −0.437687 0.899127i \(-0.644202\pi\)
−0.437687 + 0.899127i \(0.644202\pi\)
\(542\) 17.8541 0.766899
\(543\) 4.94427 0.212179
\(544\) −11.2361 −0.481742
\(545\) 3.41641 0.146343
\(546\) 2.61803 0.112042
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −28.1803 −1.20380
\(549\) −3.52786 −0.150566
\(550\) 0 0
\(551\) 37.3607 1.59162
\(552\) −4.47214 −0.190347
\(553\) 8.47214 0.360272
\(554\) 8.87539 0.377079
\(555\) −16.7082 −0.709224
\(556\) 14.4721 0.613755
\(557\) 37.8328 1.60303 0.801514 0.597976i \(-0.204028\pi\)
0.801514 + 0.597976i \(0.204028\pi\)
\(558\) 0.944272 0.0399742
\(559\) −2.00000 −0.0845910
\(560\) −4.14590 −0.175196
\(561\) 0 0
\(562\) −1.56231 −0.0659019
\(563\) 13.8885 0.585332 0.292666 0.956215i \(-0.405458\pi\)
0.292666 + 0.956215i \(0.405458\pi\)
\(564\) 0.854102 0.0359642
\(565\) −32.3607 −1.36142
\(566\) −5.20163 −0.218641
\(567\) −1.00000 −0.0419961
\(568\) −14.4721 −0.607237
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 10.3262 0.432519
\(571\) 8.83282 0.369642 0.184821 0.982772i \(-0.440829\pi\)
0.184821 + 0.982772i \(0.440829\pi\)
\(572\) 0 0
\(573\) −24.4721 −1.02234
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 17.5279 0.729695 0.364847 0.931067i \(-0.381121\pi\)
0.364847 + 0.931067i \(0.381121\pi\)
\(578\) −8.03444 −0.334189
\(579\) 10.4721 0.435207
\(580\) 18.0902 0.751153
\(581\) 16.4721 0.683379
\(582\) 10.7639 0.446179
\(583\) 0 0
\(584\) 6.05573 0.250588
\(585\) 9.47214 0.391625
\(586\) 0 0
\(587\) 36.7771 1.51795 0.758976 0.651118i \(-0.225700\pi\)
0.758976 + 0.651118i \(0.225700\pi\)
\(588\) 1.61803 0.0667266
\(589\) −11.4164 −0.470405
\(590\) −13.7426 −0.565776
\(591\) 11.8885 0.489029
\(592\) 13.8541 0.569400
\(593\) −39.8885 −1.63803 −0.819013 0.573775i \(-0.805478\pi\)
−0.819013 + 0.573775i \(0.805478\pi\)
\(594\) 0 0
\(595\) 4.47214 0.183340
\(596\) 33.9787 1.39182
\(597\) −23.8885 −0.977693
\(598\) −5.23607 −0.214119
\(599\) −4.47214 −0.182727 −0.0913633 0.995818i \(-0.529122\pi\)
−0.0913633 + 0.995818i \(0.529122\pi\)
\(600\) 0 0
\(601\) 5.29180 0.215857 0.107928 0.994159i \(-0.465578\pi\)
0.107928 + 0.994159i \(0.465578\pi\)
\(602\) 0.291796 0.0118927
\(603\) −13.1803 −0.536745
\(604\) 16.1803 0.658369
\(605\) 0 0
\(606\) 11.4164 0.463760
\(607\) 10.0557 0.408149 0.204075 0.978955i \(-0.434581\pi\)
0.204075 + 0.978955i \(0.434581\pi\)
\(608\) −41.9787 −1.70246
\(609\) −5.00000 −0.202610
\(610\) −4.87539 −0.197399
\(611\) 2.23607 0.0904616
\(612\) 3.23607 0.130810
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 3.63932 0.146871
\(615\) −14.4721 −0.583573
\(616\) 0 0
\(617\) −26.8328 −1.08025 −0.540124 0.841585i \(-0.681623\pi\)
−0.540124 + 0.841585i \(0.681623\pi\)
\(618\) 0.944272 0.0379842
\(619\) −29.8885 −1.20132 −0.600661 0.799504i \(-0.705096\pi\)
−0.600661 + 0.799504i \(0.705096\pi\)
\(620\) −5.52786 −0.222004
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) −8.47214 −0.339429
\(624\) −7.85410 −0.314416
\(625\) −25.0000 −1.00000
\(626\) −1.81966 −0.0727282
\(627\) 0 0
\(628\) −20.1803 −0.805283
\(629\) −14.9443 −0.595867
\(630\) −1.38197 −0.0550588
\(631\) −26.8328 −1.06820 −0.534099 0.845422i \(-0.679349\pi\)
−0.534099 + 0.845422i \(0.679349\pi\)
\(632\) 18.9443 0.753563
\(633\) −18.3607 −0.729772
\(634\) −12.9443 −0.514083
\(635\) −38.9443 −1.54546
\(636\) −4.00000 −0.158610
\(637\) 4.23607 0.167839
\(638\) 0 0
\(639\) 6.47214 0.256034
\(640\) −25.4508 −1.00603
\(641\) −11.4164 −0.450921 −0.225460 0.974252i \(-0.572389\pi\)
−0.225460 + 0.974252i \(0.572389\pi\)
\(642\) −7.20163 −0.284226
\(643\) 33.7771 1.33204 0.666019 0.745935i \(-0.267997\pi\)
0.666019 + 0.745935i \(0.267997\pi\)
\(644\) −3.23607 −0.127519
\(645\) 1.05573 0.0415693
\(646\) 9.23607 0.363388
\(647\) −12.3050 −0.483758 −0.241879 0.970306i \(-0.577764\pi\)
−0.241879 + 0.970306i \(0.577764\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) 1.52786 0.0598817
\(652\) 13.3262 0.521896
\(653\) 19.0557 0.745708 0.372854 0.927890i \(-0.378379\pi\)
0.372854 + 0.927890i \(0.378379\pi\)
\(654\) −0.944272 −0.0369240
\(655\) 16.5836 0.647975
\(656\) 12.0000 0.468521
\(657\) −2.70820 −0.105657
\(658\) −0.326238 −0.0127181
\(659\) −16.1246 −0.628126 −0.314063 0.949402i \(-0.601690\pi\)
−0.314063 + 0.949402i \(0.601690\pi\)
\(660\) 0 0
\(661\) 37.3050 1.45099 0.725497 0.688225i \(-0.241610\pi\)
0.725497 + 0.688225i \(0.241610\pi\)
\(662\) −9.88854 −0.384329
\(663\) 8.47214 0.329030
\(664\) 36.8328 1.42939
\(665\) 16.7082 0.647916
\(666\) 4.61803 0.178945
\(667\) 10.0000 0.387202
\(668\) 34.6525 1.34074
\(669\) −11.8885 −0.459638
\(670\) −18.2148 −0.703698
\(671\) 0 0
\(672\) 5.61803 0.216720
\(673\) −7.05573 −0.271978 −0.135989 0.990710i \(-0.543421\pi\)
−0.135989 + 0.990710i \(0.543421\pi\)
\(674\) 22.5410 0.868248
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) −26.4721 −1.01741 −0.508703 0.860942i \(-0.669875\pi\)
−0.508703 + 0.860942i \(0.669875\pi\)
\(678\) 8.94427 0.343503
\(679\) 17.4164 0.668380
\(680\) 10.0000 0.383482
\(681\) 12.4721 0.477933
\(682\) 0 0
\(683\) 27.0557 1.03526 0.517629 0.855605i \(-0.326815\pi\)
0.517629 + 0.855605i \(0.326815\pi\)
\(684\) 12.0902 0.462279
\(685\) 38.9443 1.48798
\(686\) −0.618034 −0.0235966
\(687\) −14.0000 −0.534133
\(688\) −0.875388 −0.0333739
\(689\) −10.4721 −0.398957
\(690\) 2.76393 0.105221
\(691\) 38.8328 1.47727 0.738635 0.674106i \(-0.235471\pi\)
0.738635 + 0.674106i \(0.235471\pi\)
\(692\) −26.6525 −1.01318
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) −20.0000 −0.758643
\(696\) −11.1803 −0.423790
\(697\) −12.9443 −0.490299
\(698\) 13.6738 0.517560
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) 47.8885 1.80873 0.904363 0.426765i \(-0.140347\pi\)
0.904363 + 0.426765i \(0.140347\pi\)
\(702\) −2.61803 −0.0988113
\(703\) −55.8328 −2.10577
\(704\) 0 0
\(705\) −1.18034 −0.0444542
\(706\) 10.6180 0.399615
\(707\) 18.4721 0.694716
\(708\) −16.0902 −0.604706
\(709\) −43.4721 −1.63263 −0.816315 0.577607i \(-0.803987\pi\)
−0.816315 + 0.577607i \(0.803987\pi\)
\(710\) 8.94427 0.335673
\(711\) −8.47214 −0.317730
\(712\) −18.9443 −0.709967
\(713\) −3.05573 −0.114438
\(714\) −1.23607 −0.0462587
\(715\) 0 0
\(716\) −29.7082 −1.11025
\(717\) 14.1246 0.527493
\(718\) −6.83282 −0.254998
\(719\) −23.3607 −0.871206 −0.435603 0.900139i \(-0.643465\pi\)
−0.435603 + 0.900139i \(0.643465\pi\)
\(720\) 4.14590 0.154508
\(721\) 1.52786 0.0569006
\(722\) 22.7639 0.847186
\(723\) 3.18034 0.118278
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) −12.4721 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(728\) 9.47214 0.351061
\(729\) 1.00000 0.0370370
\(730\) −3.74265 −0.138522
\(731\) 0.944272 0.0349252
\(732\) −5.70820 −0.210981
\(733\) −18.3607 −0.678167 −0.339084 0.940756i \(-0.610117\pi\)
−0.339084 + 0.940756i \(0.610117\pi\)
\(734\) −9.88854 −0.364993
\(735\) −2.23607 −0.0824786
\(736\) −11.2361 −0.414167
\(737\) 0 0
\(738\) 4.00000 0.147242
\(739\) −16.1115 −0.592669 −0.296335 0.955084i \(-0.595764\pi\)
−0.296335 + 0.955084i \(0.595764\pi\)
\(740\) −27.0344 −0.993806
\(741\) 31.6525 1.16278
\(742\) 1.52786 0.0560897
\(743\) −18.5967 −0.682249 −0.341124 0.940018i \(-0.610808\pi\)
−0.341124 + 0.940018i \(0.610808\pi\)
\(744\) 3.41641 0.125252
\(745\) −46.9574 −1.72039
\(746\) −3.70820 −0.135767
\(747\) −16.4721 −0.602684
\(748\) 0 0
\(749\) −11.6525 −0.425772
\(750\) 6.90983 0.252311
\(751\) 15.1803 0.553938 0.276969 0.960879i \(-0.410670\pi\)
0.276969 + 0.960879i \(0.410670\pi\)
\(752\) 0.978714 0.0356900
\(753\) 11.0000 0.400862
\(754\) −13.0902 −0.476716
\(755\) −22.3607 −0.813788
\(756\) −1.61803 −0.0588473
\(757\) −42.4164 −1.54165 −0.770825 0.637047i \(-0.780156\pi\)
−0.770825 + 0.637047i \(0.780156\pi\)
\(758\) 10.0344 0.364467
\(759\) 0 0
\(760\) 37.3607 1.35521
\(761\) −15.5279 −0.562885 −0.281442 0.959578i \(-0.590813\pi\)
−0.281442 + 0.959578i \(0.590813\pi\)
\(762\) 10.7639 0.389936
\(763\) −1.52786 −0.0553124
\(764\) −39.5967 −1.43256
\(765\) −4.47214 −0.161690
\(766\) 10.4721 0.378374
\(767\) −42.1246 −1.52103
\(768\) 6.56231 0.236797
\(769\) −7.18034 −0.258930 −0.129465 0.991584i \(-0.541326\pi\)
−0.129465 + 0.991584i \(0.541326\pi\)
\(770\) 0 0
\(771\) −17.6525 −0.635738
\(772\) 16.9443 0.609838
\(773\) 3.18034 0.114389 0.0571944 0.998363i \(-0.481785\pi\)
0.0571944 + 0.998363i \(0.481785\pi\)
\(774\) −0.291796 −0.0104884
\(775\) 0 0
\(776\) 38.9443 1.39802
\(777\) 7.47214 0.268061
\(778\) −21.3050 −0.763820
\(779\) −48.3607 −1.73270
\(780\) 15.3262 0.548767
\(781\) 0 0
\(782\) 2.47214 0.0884034
\(783\) 5.00000 0.178685
\(784\) 1.85410 0.0662179
\(785\) 27.8885 0.995385
\(786\) −4.58359 −0.163491
\(787\) 35.2492 1.25650 0.628250 0.778012i \(-0.283772\pi\)
0.628250 + 0.778012i \(0.283772\pi\)
\(788\) 19.2361 0.685257
\(789\) 24.7082 0.879635
\(790\) −11.7082 −0.416559
\(791\) 14.4721 0.514570
\(792\) 0 0
\(793\) −14.9443 −0.530687
\(794\) 5.59675 0.198621
\(795\) 5.52786 0.196053
\(796\) −38.6525 −1.37000
\(797\) −42.2361 −1.49608 −0.748039 0.663655i \(-0.769004\pi\)
−0.748039 + 0.663655i \(0.769004\pi\)
\(798\) −4.61803 −0.163477
\(799\) −1.05573 −0.0373490
\(800\) 0 0
\(801\) 8.47214 0.299348
\(802\) −16.0689 −0.567412
\(803\) 0 0
\(804\) −21.3262 −0.752118
\(805\) 4.47214 0.157622
\(806\) 4.00000 0.140894
\(807\) 13.4164 0.472280
\(808\) 41.3050 1.45310
\(809\) −4.63932 −0.163110 −0.0815549 0.996669i \(-0.525989\pi\)
−0.0815549 + 0.996669i \(0.525989\pi\)
\(810\) 1.38197 0.0485573
\(811\) −49.4721 −1.73720 −0.868601 0.495512i \(-0.834980\pi\)
−0.868601 + 0.495512i \(0.834980\pi\)
\(812\) −8.09017 −0.283909
\(813\) −28.8885 −1.01317
\(814\) 0 0
\(815\) −18.4164 −0.645099
\(816\) 3.70820 0.129813
\(817\) 3.52786 0.123424
\(818\) 10.1803 0.355947
\(819\) −4.23607 −0.148020
\(820\) −23.4164 −0.817736
\(821\) 51.8328 1.80898 0.904489 0.426497i \(-0.140253\pi\)
0.904489 + 0.426497i \(0.140253\pi\)
\(822\) −10.7639 −0.375435
\(823\) 25.0689 0.873846 0.436923 0.899499i \(-0.356068\pi\)
0.436923 + 0.899499i \(0.356068\pi\)
\(824\) 3.41641 0.119016
\(825\) 0 0
\(826\) 6.14590 0.213843
\(827\) 21.7639 0.756806 0.378403 0.925641i \(-0.376473\pi\)
0.378403 + 0.925641i \(0.376473\pi\)
\(828\) 3.23607 0.112461
\(829\) −3.05573 −0.106130 −0.0530649 0.998591i \(-0.516899\pi\)
−0.0530649 + 0.998591i \(0.516899\pi\)
\(830\) −22.7639 −0.790148
\(831\) −14.3607 −0.498166
\(832\) −1.00000 −0.0346688
\(833\) −2.00000 −0.0692959
\(834\) 5.52786 0.191414
\(835\) −47.8885 −1.65725
\(836\) 0 0
\(837\) −1.52786 −0.0528107
\(838\) 10.5066 0.362944
\(839\) −51.2492 −1.76932 −0.884660 0.466237i \(-0.845609\pi\)
−0.884660 + 0.466237i \(0.845609\pi\)
\(840\) −5.00000 −0.172516
\(841\) −4.00000 −0.137931
\(842\) 4.54915 0.156774
\(843\) 2.52786 0.0870643
\(844\) −29.7082 −1.02260
\(845\) 11.0557 0.380329
\(846\) 0.326238 0.0112163
\(847\) 0 0
\(848\) −4.58359 −0.157401
\(849\) 8.41641 0.288850
\(850\) 0 0
\(851\) −14.9443 −0.512283
\(852\) 10.4721 0.358769
\(853\) 2.58359 0.0884605 0.0442303 0.999021i \(-0.485916\pi\)
0.0442303 + 0.999021i \(0.485916\pi\)
\(854\) 2.18034 0.0746097
\(855\) −16.7082 −0.571409
\(856\) −26.0557 −0.890566
\(857\) 41.4164 1.41476 0.707379 0.706835i \(-0.249878\pi\)
0.707379 + 0.706835i \(0.249878\pi\)
\(858\) 0 0
\(859\) 26.9443 0.919327 0.459663 0.888093i \(-0.347970\pi\)
0.459663 + 0.888093i \(0.347970\pi\)
\(860\) 1.70820 0.0582493
\(861\) 6.47214 0.220570
\(862\) −9.74265 −0.331836
\(863\) −9.88854 −0.336610 −0.168305 0.985735i \(-0.553829\pi\)
−0.168305 + 0.985735i \(0.553829\pi\)
\(864\) −5.61803 −0.191129
\(865\) 36.8328 1.25235
\(866\) 20.6525 0.701800
\(867\) 13.0000 0.441503
\(868\) 2.47214 0.0839098
\(869\) 0 0
\(870\) 6.90983 0.234265
\(871\) −55.8328 −1.89182
\(872\) −3.41641 −0.115694
\(873\) −17.4164 −0.589456
\(874\) 9.23607 0.312415
\(875\) 11.1803 0.377964
\(876\) −4.38197 −0.148053
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) 5.49342 0.185394
\(879\) 0 0
\(880\) 0 0
\(881\) 35.7639 1.20492 0.602459 0.798150i \(-0.294188\pi\)
0.602459 + 0.798150i \(0.294188\pi\)
\(882\) 0.618034 0.0208103
\(883\) −2.81966 −0.0948891 −0.0474446 0.998874i \(-0.515108\pi\)
−0.0474446 + 0.998874i \(0.515108\pi\)
\(884\) 13.7082 0.461057
\(885\) 22.2361 0.747457
\(886\) 8.65248 0.290686
\(887\) 0.944272 0.0317055 0.0158528 0.999874i \(-0.494954\pi\)
0.0158528 + 0.999874i \(0.494954\pi\)
\(888\) 16.7082 0.560691
\(889\) 17.4164 0.584128
\(890\) 11.7082 0.392460
\(891\) 0 0
\(892\) −19.2361 −0.644071
\(893\) −3.94427 −0.131990
\(894\) 12.9787 0.434073
\(895\) 41.0557 1.37234
\(896\) 11.3820 0.380245
\(897\) 8.47214 0.282876
\(898\) −5.23607 −0.174730
\(899\) −7.63932 −0.254786
\(900\) 0 0
\(901\) 4.94427 0.164718
\(902\) 0 0
\(903\) −0.472136 −0.0157117
\(904\) 32.3607 1.07630
\(905\) −11.0557 −0.367505
\(906\) 6.18034 0.205328
\(907\) −15.7771 −0.523870 −0.261935 0.965086i \(-0.584361\pi\)
−0.261935 + 0.965086i \(0.584361\pi\)
\(908\) 20.1803 0.669708
\(909\) −18.4721 −0.612682
\(910\) −5.85410 −0.194062
\(911\) 26.2492 0.869676 0.434838 0.900509i \(-0.356806\pi\)
0.434838 + 0.900509i \(0.356806\pi\)
\(912\) 13.8541 0.458755
\(913\) 0 0
\(914\) 23.7082 0.784198
\(915\) 7.88854 0.260787
\(916\) −22.6525 −0.748459
\(917\) −7.41641 −0.244911
\(918\) 1.23607 0.0407963
\(919\) −13.5279 −0.446243 −0.223122 0.974791i \(-0.571625\pi\)
−0.223122 + 0.974791i \(0.571625\pi\)
\(920\) 10.0000 0.329690
\(921\) −5.88854 −0.194034
\(922\) −0.652476 −0.0214881
\(923\) 27.4164 0.902422
\(924\) 0 0
\(925\) 0 0
\(926\) −4.14590 −0.136243
\(927\) −1.52786 −0.0501816
\(928\) −28.0902 −0.922105
\(929\) −0.708204 −0.0232354 −0.0116177 0.999933i \(-0.503698\pi\)
−0.0116177 + 0.999933i \(0.503698\pi\)
\(930\) −2.11146 −0.0692374
\(931\) −7.47214 −0.244889
\(932\) −4.76393 −0.156048
\(933\) 0 0
\(934\) 17.2705 0.565108
\(935\) 0 0
\(936\) −9.47214 −0.309606
\(937\) −6.36068 −0.207794 −0.103897 0.994588i \(-0.533131\pi\)
−0.103897 + 0.994588i \(0.533131\pi\)
\(938\) 8.14590 0.265973
\(939\) 2.94427 0.0960827
\(940\) −1.90983 −0.0622918
\(941\) −53.4164 −1.74133 −0.870663 0.491881i \(-0.836310\pi\)
−0.870663 + 0.491881i \(0.836310\pi\)
\(942\) −7.70820 −0.251147
\(943\) −12.9443 −0.421523
\(944\) −18.4377 −0.600096
\(945\) 2.23607 0.0727393
\(946\) 0 0
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) −13.7082 −0.445222
\(949\) −11.4721 −0.372401
\(950\) 0 0
\(951\) 20.9443 0.679165
\(952\) −4.47214 −0.144943
\(953\) 18.4164 0.596566 0.298283 0.954477i \(-0.403586\pi\)
0.298283 + 0.954477i \(0.403586\pi\)
\(954\) −1.52786 −0.0494664
\(955\) 54.7214 1.77074
\(956\) 22.8541 0.739154
\(957\) 0 0
\(958\) 0.0688837 0.00222553
\(959\) −17.4164 −0.562405
\(960\) 0.527864 0.0170367
\(961\) −28.6656 −0.924698
\(962\) 19.5623 0.630714
\(963\) 11.6525 0.375496
\(964\) 5.14590 0.165738
\(965\) −23.4164 −0.753801
\(966\) −1.23607 −0.0397698
\(967\) 49.3050 1.58554 0.792770 0.609521i \(-0.208638\pi\)
0.792770 + 0.609521i \(0.208638\pi\)
\(968\) 0 0
\(969\) −14.9443 −0.480079
\(970\) −24.0689 −0.772805
\(971\) 29.9443 0.960957 0.480479 0.877006i \(-0.340463\pi\)
0.480479 + 0.877006i \(0.340463\pi\)
\(972\) 1.61803 0.0518985
\(973\) 8.94427 0.286740
\(974\) −18.4721 −0.591885
\(975\) 0 0
\(976\) −6.54102 −0.209373
\(977\) 7.52786 0.240838 0.120419 0.992723i \(-0.461576\pi\)
0.120419 + 0.992723i \(0.461576\pi\)
\(978\) 5.09017 0.162766
\(979\) 0 0
\(980\) −3.61803 −0.115574
\(981\) 1.52786 0.0487809
\(982\) −17.3820 −0.554681
\(983\) −40.0000 −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 14.4721 0.461355
\(985\) −26.5836 −0.847024
\(986\) 6.18034 0.196822
\(987\) 0.527864 0.0168021
\(988\) 51.2148 1.62936
\(989\) 0.944272 0.0300261
\(990\) 0 0
\(991\) 11.1803 0.355155 0.177578 0.984107i \(-0.443174\pi\)
0.177578 + 0.984107i \(0.443174\pi\)
\(992\) 8.58359 0.272529
\(993\) 16.0000 0.507745
\(994\) −4.00000 −0.126872
\(995\) 53.4164 1.69341
\(996\) −26.6525 −0.844516
\(997\) −41.1935 −1.30461 −0.652306 0.757956i \(-0.726198\pi\)
−0.652306 + 0.757956i \(0.726198\pi\)
\(998\) 14.9787 0.474143
\(999\) −7.47214 −0.236408
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 2541.2.a.q.1.2 2
3.2 odd 2 7623.2.a.bn.1.1 2
11.10 odd 2 2541.2.a.y.1.1 yes 2
33.32 even 2 7623.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.q.1.2 2 1.1 even 1 trivial
2541.2.a.y.1.1 yes 2 11.10 odd 2
7623.2.a.y.1.2 2 33.32 even 2
7623.2.a.bn.1.1 2 3.2 odd 2