Properties

Label 5054.2.a.p.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.618034 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +1.00000 q^{10} -5.00000 q^{11} -0.618034 q^{12} +4.85410 q^{13} -1.00000 q^{14} -0.618034 q^{15} +1.00000 q^{16} +6.85410 q^{17} -2.61803 q^{18} +1.00000 q^{20} +0.618034 q^{21} -5.00000 q^{22} -3.76393 q^{23} -0.618034 q^{24} -4.00000 q^{25} +4.85410 q^{26} +3.47214 q^{27} -1.00000 q^{28} -8.23607 q^{29} -0.618034 q^{30} -6.70820 q^{31} +1.00000 q^{32} +3.09017 q^{33} +6.85410 q^{34} -1.00000 q^{35} -2.61803 q^{36} +5.94427 q^{37} -3.00000 q^{39} +1.00000 q^{40} -1.09017 q^{41} +0.618034 q^{42} -6.85410 q^{43} -5.00000 q^{44} -2.61803 q^{45} -3.76393 q^{46} +7.94427 q^{47} -0.618034 q^{48} +1.00000 q^{49} -4.00000 q^{50} -4.23607 q^{51} +4.85410 q^{52} -12.7082 q^{53} +3.47214 q^{54} -5.00000 q^{55} -1.00000 q^{56} -8.23607 q^{58} +11.0000 q^{59} -0.618034 q^{60} +7.47214 q^{61} -6.70820 q^{62} +2.61803 q^{63} +1.00000 q^{64} +4.85410 q^{65} +3.09017 q^{66} +7.85410 q^{67} +6.85410 q^{68} +2.32624 q^{69} -1.00000 q^{70} -10.0000 q^{71} -2.61803 q^{72} -14.5623 q^{73} +5.94427 q^{74} +2.47214 q^{75} +5.00000 q^{77} -3.00000 q^{78} -8.85410 q^{79} +1.00000 q^{80} +5.70820 q^{81} -1.09017 q^{82} -15.4164 q^{83} +0.618034 q^{84} +6.85410 q^{85} -6.85410 q^{86} +5.09017 q^{87} -5.00000 q^{88} +4.85410 q^{89} -2.61803 q^{90} -4.85410 q^{91} -3.76393 q^{92} +4.14590 q^{93} +7.94427 q^{94} -0.618034 q^{96} -9.47214 q^{97} +1.00000 q^{98} +13.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9} + 2 q^{10} - 10 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} + 7 q^{17} - 3 q^{18} + 2 q^{20} - q^{21} - 10 q^{22} - 12 q^{23} + q^{24} - 8 q^{25} + 3 q^{26} - 2 q^{27} - 2 q^{28} - 12 q^{29} + q^{30} + 2 q^{32} - 5 q^{33} + 7 q^{34} - 2 q^{35} - 3 q^{36} - 6 q^{37} - 6 q^{39} + 2 q^{40} + 9 q^{41} - q^{42} - 7 q^{43} - 10 q^{44} - 3 q^{45} - 12 q^{46} - 2 q^{47} + q^{48} + 2 q^{49} - 8 q^{50} - 4 q^{51} + 3 q^{52} - 12 q^{53} - 2 q^{54} - 10 q^{55} - 2 q^{56} - 12 q^{58} + 22 q^{59} + q^{60} + 6 q^{61} + 3 q^{63} + 2 q^{64} + 3 q^{65} - 5 q^{66} + 9 q^{67} + 7 q^{68} - 11 q^{69} - 2 q^{70} - 20 q^{71} - 3 q^{72} - 9 q^{73} - 6 q^{74} - 4 q^{75} + 10 q^{77} - 6 q^{78} - 11 q^{79} + 2 q^{80} - 2 q^{81} + 9 q^{82} - 4 q^{83} - q^{84} + 7 q^{85} - 7 q^{86} - q^{87} - 10 q^{88} + 3 q^{89} - 3 q^{90} - 3 q^{91} - 12 q^{92} + 15 q^{93} - 2 q^{94} + q^{96} - 10 q^{97} + 2 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.618034 −0.356822 −0.178411 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −0.618034 −0.252311
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) 1.00000 0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −0.618034 −0.178411
\(13\) 4.85410 1.34629 0.673143 0.739512i \(-0.264944\pi\)
0.673143 + 0.739512i \(0.264944\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.618034 −0.159576
\(16\) 1.00000 0.250000
\(17\) 6.85410 1.66236 0.831182 0.556001i \(-0.187665\pi\)
0.831182 + 0.556001i \(0.187665\pi\)
\(18\) −2.61803 −0.617077
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) 0.618034 0.134866
\(22\) −5.00000 −1.06600
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) −0.618034 −0.126156
\(25\) −4.00000 −0.800000
\(26\) 4.85410 0.951968
\(27\) 3.47214 0.668213
\(28\) −1.00000 −0.188982
\(29\) −8.23607 −1.52940 −0.764700 0.644387i \(-0.777113\pi\)
−0.764700 + 0.644387i \(0.777113\pi\)
\(30\) −0.618034 −0.112837
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.09017 0.537930
\(34\) 6.85410 1.17547
\(35\) −1.00000 −0.169031
\(36\) −2.61803 −0.436339
\(37\) 5.94427 0.977232 0.488616 0.872499i \(-0.337502\pi\)
0.488616 + 0.872499i \(0.337502\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 1.00000 0.158114
\(41\) −1.09017 −0.170256 −0.0851280 0.996370i \(-0.527130\pi\)
−0.0851280 + 0.996370i \(0.527130\pi\)
\(42\) 0.618034 0.0953647
\(43\) −6.85410 −1.04524 −0.522620 0.852566i \(-0.675045\pi\)
−0.522620 + 0.852566i \(0.675045\pi\)
\(44\) −5.00000 −0.753778
\(45\) −2.61803 −0.390273
\(46\) −3.76393 −0.554962
\(47\) 7.94427 1.15879 0.579396 0.815046i \(-0.303289\pi\)
0.579396 + 0.815046i \(0.303289\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) −4.23607 −0.593168
\(52\) 4.85410 0.673143
\(53\) −12.7082 −1.74561 −0.872803 0.488073i \(-0.837700\pi\)
−0.872803 + 0.488073i \(0.837700\pi\)
\(54\) 3.47214 0.472498
\(55\) −5.00000 −0.674200
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −8.23607 −1.08145
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) −0.618034 −0.0797878
\(61\) 7.47214 0.956709 0.478354 0.878167i \(-0.341233\pi\)
0.478354 + 0.878167i \(0.341233\pi\)
\(62\) −6.70820 −0.851943
\(63\) 2.61803 0.329841
\(64\) 1.00000 0.125000
\(65\) 4.85410 0.602077
\(66\) 3.09017 0.380374
\(67\) 7.85410 0.959531 0.479766 0.877397i \(-0.340722\pi\)
0.479766 + 0.877397i \(0.340722\pi\)
\(68\) 6.85410 0.831182
\(69\) 2.32624 0.280046
\(70\) −1.00000 −0.119523
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −2.61803 −0.308538
\(73\) −14.5623 −1.70439 −0.852194 0.523225i \(-0.824729\pi\)
−0.852194 + 0.523225i \(0.824729\pi\)
\(74\) 5.94427 0.691008
\(75\) 2.47214 0.285458
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) −3.00000 −0.339683
\(79\) −8.85410 −0.996164 −0.498082 0.867130i \(-0.665962\pi\)
−0.498082 + 0.867130i \(0.665962\pi\)
\(80\) 1.00000 0.111803
\(81\) 5.70820 0.634245
\(82\) −1.09017 −0.120389
\(83\) −15.4164 −1.69217 −0.846085 0.533048i \(-0.821047\pi\)
−0.846085 + 0.533048i \(0.821047\pi\)
\(84\) 0.618034 0.0674330
\(85\) 6.85410 0.743432
\(86\) −6.85410 −0.739097
\(87\) 5.09017 0.545724
\(88\) −5.00000 −0.533002
\(89\) 4.85410 0.514534 0.257267 0.966340i \(-0.417178\pi\)
0.257267 + 0.966340i \(0.417178\pi\)
\(90\) −2.61803 −0.275965
\(91\) −4.85410 −0.508848
\(92\) −3.76393 −0.392417
\(93\) 4.14590 0.429910
\(94\) 7.94427 0.819389
\(95\) 0 0
\(96\) −0.618034 −0.0630778
\(97\) −9.47214 −0.961750 −0.480875 0.876789i \(-0.659681\pi\)
−0.480875 + 0.876789i \(0.659681\pi\)
\(98\) 1.00000 0.101015
\(99\) 13.0902 1.31561
\(100\) −4.00000 −0.400000
\(101\) 4.14590 0.412532 0.206266 0.978496i \(-0.433869\pi\)
0.206266 + 0.978496i \(0.433869\pi\)
\(102\) −4.23607 −0.419433
\(103\) −15.0902 −1.48688 −0.743439 0.668803i \(-0.766807\pi\)
−0.743439 + 0.668803i \(0.766807\pi\)
\(104\) 4.85410 0.475984
\(105\) 0.618034 0.0603139
\(106\) −12.7082 −1.23433
\(107\) −13.9443 −1.34804 −0.674022 0.738711i \(-0.735435\pi\)
−0.674022 + 0.738711i \(0.735435\pi\)
\(108\) 3.47214 0.334106
\(109\) −13.5623 −1.29903 −0.649517 0.760347i \(-0.725029\pi\)
−0.649517 + 0.760347i \(0.725029\pi\)
\(110\) −5.00000 −0.476731
\(111\) −3.67376 −0.348698
\(112\) −1.00000 −0.0944911
\(113\) 3.85410 0.362563 0.181282 0.983431i \(-0.441975\pi\)
0.181282 + 0.983431i \(0.441975\pi\)
\(114\) 0 0
\(115\) −3.76393 −0.350988
\(116\) −8.23607 −0.764700
\(117\) −12.7082 −1.17487
\(118\) 11.0000 1.01263
\(119\) −6.85410 −0.628314
\(120\) −0.618034 −0.0564185
\(121\) 14.0000 1.27273
\(122\) 7.47214 0.676495
\(123\) 0.673762 0.0607511
\(124\) −6.70820 −0.602414
\(125\) −9.00000 −0.804984
\(126\) 2.61803 0.233233
\(127\) −7.32624 −0.650098 −0.325049 0.945697i \(-0.605381\pi\)
−0.325049 + 0.945697i \(0.605381\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.23607 0.372965
\(130\) 4.85410 0.425733
\(131\) −4.38197 −0.382854 −0.191427 0.981507i \(-0.561312\pi\)
−0.191427 + 0.981507i \(0.561312\pi\)
\(132\) 3.09017 0.268965
\(133\) 0 0
\(134\) 7.85410 0.678491
\(135\) 3.47214 0.298834
\(136\) 6.85410 0.587734
\(137\) 21.1246 1.80480 0.902399 0.430902i \(-0.141805\pi\)
0.902399 + 0.430902i \(0.141805\pi\)
\(138\) 2.32624 0.198023
\(139\) −0.236068 −0.0200230 −0.0100115 0.999950i \(-0.503187\pi\)
−0.0100115 + 0.999950i \(0.503187\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −4.90983 −0.413482
\(142\) −10.0000 −0.839181
\(143\) −24.2705 −2.02960
\(144\) −2.61803 −0.218169
\(145\) −8.23607 −0.683968
\(146\) −14.5623 −1.20519
\(147\) −0.618034 −0.0509746
\(148\) 5.94427 0.488616
\(149\) 1.38197 0.113215 0.0566075 0.998397i \(-0.481972\pi\)
0.0566075 + 0.998397i \(0.481972\pi\)
\(150\) 2.47214 0.201849
\(151\) −14.2361 −1.15851 −0.579257 0.815145i \(-0.696657\pi\)
−0.579257 + 0.815145i \(0.696657\pi\)
\(152\) 0 0
\(153\) −17.9443 −1.45071
\(154\) 5.00000 0.402911
\(155\) −6.70820 −0.538816
\(156\) −3.00000 −0.240192
\(157\) 22.3262 1.78183 0.890914 0.454172i \(-0.150065\pi\)
0.890914 + 0.454172i \(0.150065\pi\)
\(158\) −8.85410 −0.704395
\(159\) 7.85410 0.622871
\(160\) 1.00000 0.0790569
\(161\) 3.76393 0.296639
\(162\) 5.70820 0.448479
\(163\) 0.472136 0.0369805 0.0184903 0.999829i \(-0.494114\pi\)
0.0184903 + 0.999829i \(0.494114\pi\)
\(164\) −1.09017 −0.0851280
\(165\) 3.09017 0.240569
\(166\) −15.4164 −1.19655
\(167\) 1.81966 0.140810 0.0704048 0.997519i \(-0.477571\pi\)
0.0704048 + 0.997519i \(0.477571\pi\)
\(168\) 0.618034 0.0476824
\(169\) 10.5623 0.812485
\(170\) 6.85410 0.525686
\(171\) 0 0
\(172\) −6.85410 −0.522620
\(173\) 1.76393 0.134109 0.0670546 0.997749i \(-0.478640\pi\)
0.0670546 + 0.997749i \(0.478640\pi\)
\(174\) 5.09017 0.385885
\(175\) 4.00000 0.302372
\(176\) −5.00000 −0.376889
\(177\) −6.79837 −0.510997
\(178\) 4.85410 0.363830
\(179\) 9.03444 0.675266 0.337633 0.941278i \(-0.390374\pi\)
0.337633 + 0.941278i \(0.390374\pi\)
\(180\) −2.61803 −0.195137
\(181\) −15.7639 −1.17172 −0.585862 0.810411i \(-0.699244\pi\)
−0.585862 + 0.810411i \(0.699244\pi\)
\(182\) −4.85410 −0.359810
\(183\) −4.61803 −0.341375
\(184\) −3.76393 −0.277481
\(185\) 5.94427 0.437032
\(186\) 4.14590 0.303992
\(187\) −34.2705 −2.50611
\(188\) 7.94427 0.579396
\(189\) −3.47214 −0.252561
\(190\) 0 0
\(191\) 2.81966 0.204023 0.102012 0.994783i \(-0.467472\pi\)
0.102012 + 0.994783i \(0.467472\pi\)
\(192\) −0.618034 −0.0446028
\(193\) −10.1803 −0.732797 −0.366398 0.930458i \(-0.619409\pi\)
−0.366398 + 0.930458i \(0.619409\pi\)
\(194\) −9.47214 −0.680060
\(195\) −3.00000 −0.214834
\(196\) 1.00000 0.0714286
\(197\) −7.56231 −0.538792 −0.269396 0.963029i \(-0.586824\pi\)
−0.269396 + 0.963029i \(0.586824\pi\)
\(198\) 13.0902 0.930278
\(199\) −4.90983 −0.348049 −0.174024 0.984741i \(-0.555677\pi\)
−0.174024 + 0.984741i \(0.555677\pi\)
\(200\) −4.00000 −0.282843
\(201\) −4.85410 −0.342382
\(202\) 4.14590 0.291704
\(203\) 8.23607 0.578059
\(204\) −4.23607 −0.296584
\(205\) −1.09017 −0.0761408
\(206\) −15.0902 −1.05138
\(207\) 9.85410 0.684907
\(208\) 4.85410 0.336571
\(209\) 0 0
\(210\) 0.618034 0.0426484
\(211\) −4.05573 −0.279208 −0.139604 0.990207i \(-0.544583\pi\)
−0.139604 + 0.990207i \(0.544583\pi\)
\(212\) −12.7082 −0.872803
\(213\) 6.18034 0.423470
\(214\) −13.9443 −0.953211
\(215\) −6.85410 −0.467446
\(216\) 3.47214 0.236249
\(217\) 6.70820 0.455383
\(218\) −13.5623 −0.918555
\(219\) 9.00000 0.608164
\(220\) −5.00000 −0.337100
\(221\) 33.2705 2.23802
\(222\) −3.67376 −0.246567
\(223\) 5.65248 0.378518 0.189259 0.981927i \(-0.439391\pi\)
0.189259 + 0.981927i \(0.439391\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 10.4721 0.698142
\(226\) 3.85410 0.256371
\(227\) −14.2361 −0.944881 −0.472441 0.881363i \(-0.656627\pi\)
−0.472441 + 0.881363i \(0.656627\pi\)
\(228\) 0 0
\(229\) −10.0902 −0.666777 −0.333389 0.942790i \(-0.608192\pi\)
−0.333389 + 0.942790i \(0.608192\pi\)
\(230\) −3.76393 −0.248186
\(231\) −3.09017 −0.203318
\(232\) −8.23607 −0.540724
\(233\) −11.0344 −0.722890 −0.361445 0.932393i \(-0.617717\pi\)
−0.361445 + 0.932393i \(0.617717\pi\)
\(234\) −12.7082 −0.830761
\(235\) 7.94427 0.518227
\(236\) 11.0000 0.716039
\(237\) 5.47214 0.355453
\(238\) −6.85410 −0.444285
\(239\) −11.1459 −0.720968 −0.360484 0.932765i \(-0.617388\pi\)
−0.360484 + 0.932765i \(0.617388\pi\)
\(240\) −0.618034 −0.0398939
\(241\) 7.90983 0.509517 0.254758 0.967005i \(-0.418004\pi\)
0.254758 + 0.967005i \(0.418004\pi\)
\(242\) 14.0000 0.899954
\(243\) −13.9443 −0.894525
\(244\) 7.47214 0.478354
\(245\) 1.00000 0.0638877
\(246\) 0.673762 0.0429575
\(247\) 0 0
\(248\) −6.70820 −0.425971
\(249\) 9.52786 0.603804
\(250\) −9.00000 −0.569210
\(251\) 6.47214 0.408518 0.204259 0.978917i \(-0.434522\pi\)
0.204259 + 0.978917i \(0.434522\pi\)
\(252\) 2.61803 0.164921
\(253\) 18.8197 1.18318
\(254\) −7.32624 −0.459689
\(255\) −4.23607 −0.265273
\(256\) 1.00000 0.0625000
\(257\) 8.05573 0.502503 0.251251 0.967922i \(-0.419158\pi\)
0.251251 + 0.967922i \(0.419158\pi\)
\(258\) 4.23607 0.263726
\(259\) −5.94427 −0.369359
\(260\) 4.85410 0.301039
\(261\) 21.5623 1.33467
\(262\) −4.38197 −0.270719
\(263\) −25.1803 −1.55269 −0.776343 0.630311i \(-0.782928\pi\)
−0.776343 + 0.630311i \(0.782928\pi\)
\(264\) 3.09017 0.190187
\(265\) −12.7082 −0.780659
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 7.85410 0.479766
\(269\) 9.18034 0.559735 0.279868 0.960039i \(-0.409709\pi\)
0.279868 + 0.960039i \(0.409709\pi\)
\(270\) 3.47214 0.211307
\(271\) 1.65248 0.100381 0.0501904 0.998740i \(-0.484017\pi\)
0.0501904 + 0.998740i \(0.484017\pi\)
\(272\) 6.85410 0.415591
\(273\) 3.00000 0.181568
\(274\) 21.1246 1.27618
\(275\) 20.0000 1.20605
\(276\) 2.32624 0.140023
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −0.236068 −0.0141584
\(279\) 17.5623 1.05143
\(280\) −1.00000 −0.0597614
\(281\) −12.2361 −0.729943 −0.364971 0.931019i \(-0.618921\pi\)
−0.364971 + 0.931019i \(0.618921\pi\)
\(282\) −4.90983 −0.292376
\(283\) 6.43769 0.382681 0.191341 0.981524i \(-0.438716\pi\)
0.191341 + 0.981524i \(0.438716\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −24.2705 −1.43515
\(287\) 1.09017 0.0643507
\(288\) −2.61803 −0.154269
\(289\) 29.9787 1.76345
\(290\) −8.23607 −0.483639
\(291\) 5.85410 0.343174
\(292\) −14.5623 −0.852194
\(293\) 16.2361 0.948521 0.474261 0.880385i \(-0.342715\pi\)
0.474261 + 0.880385i \(0.342715\pi\)
\(294\) −0.618034 −0.0360445
\(295\) 11.0000 0.640445
\(296\) 5.94427 0.345504
\(297\) −17.3607 −1.00737
\(298\) 1.38197 0.0800551
\(299\) −18.2705 −1.05661
\(300\) 2.47214 0.142729
\(301\) 6.85410 0.395064
\(302\) −14.2361 −0.819194
\(303\) −2.56231 −0.147201
\(304\) 0 0
\(305\) 7.47214 0.427853
\(306\) −17.9443 −1.02581
\(307\) 4.94427 0.282185 0.141092 0.989996i \(-0.454939\pi\)
0.141092 + 0.989996i \(0.454939\pi\)
\(308\) 5.00000 0.284901
\(309\) 9.32624 0.530551
\(310\) −6.70820 −0.381000
\(311\) 19.7639 1.12071 0.560355 0.828253i \(-0.310665\pi\)
0.560355 + 0.828253i \(0.310665\pi\)
\(312\) −3.00000 −0.169842
\(313\) 18.9098 1.06885 0.534423 0.845217i \(-0.320529\pi\)
0.534423 + 0.845217i \(0.320529\pi\)
\(314\) 22.3262 1.25994
\(315\) 2.61803 0.147510
\(316\) −8.85410 −0.498082
\(317\) 8.41641 0.472713 0.236356 0.971666i \(-0.424047\pi\)
0.236356 + 0.971666i \(0.424047\pi\)
\(318\) 7.85410 0.440436
\(319\) 41.1803 2.30566
\(320\) 1.00000 0.0559017
\(321\) 8.61803 0.481012
\(322\) 3.76393 0.209756
\(323\) 0 0
\(324\) 5.70820 0.317122
\(325\) −19.4164 −1.07703
\(326\) 0.472136 0.0261492
\(327\) 8.38197 0.463524
\(328\) −1.09017 −0.0601946
\(329\) −7.94427 −0.437982
\(330\) 3.09017 0.170108
\(331\) 12.4721 0.685531 0.342765 0.939421i \(-0.388636\pi\)
0.342765 + 0.939421i \(0.388636\pi\)
\(332\) −15.4164 −0.846085
\(333\) −15.5623 −0.852809
\(334\) 1.81966 0.0995674
\(335\) 7.85410 0.429115
\(336\) 0.618034 0.0337165
\(337\) −33.4508 −1.82218 −0.911092 0.412203i \(-0.864759\pi\)
−0.911092 + 0.412203i \(0.864759\pi\)
\(338\) 10.5623 0.574514
\(339\) −2.38197 −0.129371
\(340\) 6.85410 0.371716
\(341\) 33.5410 1.81635
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −6.85410 −0.369548
\(345\) 2.32624 0.125240
\(346\) 1.76393 0.0948296
\(347\) −5.05573 −0.271406 −0.135703 0.990750i \(-0.543329\pi\)
−0.135703 + 0.990750i \(0.543329\pi\)
\(348\) 5.09017 0.272862
\(349\) −6.70820 −0.359082 −0.179541 0.983750i \(-0.557461\pi\)
−0.179541 + 0.983750i \(0.557461\pi\)
\(350\) 4.00000 0.213809
\(351\) 16.8541 0.899605
\(352\) −5.00000 −0.266501
\(353\) 13.0902 0.696719 0.348360 0.937361i \(-0.386739\pi\)
0.348360 + 0.937361i \(0.386739\pi\)
\(354\) −6.79837 −0.361329
\(355\) −10.0000 −0.530745
\(356\) 4.85410 0.257267
\(357\) 4.23607 0.224196
\(358\) 9.03444 0.477485
\(359\) 27.9443 1.47484 0.737421 0.675433i \(-0.236043\pi\)
0.737421 + 0.675433i \(0.236043\pi\)
\(360\) −2.61803 −0.137983
\(361\) 0 0
\(362\) −15.7639 −0.828534
\(363\) −8.65248 −0.454137
\(364\) −4.85410 −0.254424
\(365\) −14.5623 −0.762226
\(366\) −4.61803 −0.241389
\(367\) 9.67376 0.504966 0.252483 0.967601i \(-0.418753\pi\)
0.252483 + 0.967601i \(0.418753\pi\)
\(368\) −3.76393 −0.196209
\(369\) 2.85410 0.148579
\(370\) 5.94427 0.309028
\(371\) 12.7082 0.659777
\(372\) 4.14590 0.214955
\(373\) −6.85410 −0.354892 −0.177446 0.984131i \(-0.556784\pi\)
−0.177446 + 0.984131i \(0.556784\pi\)
\(374\) −34.2705 −1.77209
\(375\) 5.56231 0.287236
\(376\) 7.94427 0.409695
\(377\) −39.9787 −2.05901
\(378\) −3.47214 −0.178587
\(379\) 12.0344 0.618168 0.309084 0.951035i \(-0.399978\pi\)
0.309084 + 0.951035i \(0.399978\pi\)
\(380\) 0 0
\(381\) 4.52786 0.231970
\(382\) 2.81966 0.144266
\(383\) 5.94427 0.303738 0.151869 0.988401i \(-0.451471\pi\)
0.151869 + 0.988401i \(0.451471\pi\)
\(384\) −0.618034 −0.0315389
\(385\) 5.00000 0.254824
\(386\) −10.1803 −0.518166
\(387\) 17.9443 0.912159
\(388\) −9.47214 −0.480875
\(389\) 11.3820 0.577089 0.288544 0.957467i \(-0.406829\pi\)
0.288544 + 0.957467i \(0.406829\pi\)
\(390\) −3.00000 −0.151911
\(391\) −25.7984 −1.30468
\(392\) 1.00000 0.0505076
\(393\) 2.70820 0.136611
\(394\) −7.56231 −0.380983
\(395\) −8.85410 −0.445498
\(396\) 13.0902 0.657806
\(397\) −32.2705 −1.61961 −0.809805 0.586699i \(-0.800427\pi\)
−0.809805 + 0.586699i \(0.800427\pi\)
\(398\) −4.90983 −0.246108
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −6.58359 −0.328769 −0.164384 0.986396i \(-0.552564\pi\)
−0.164384 + 0.986396i \(0.552564\pi\)
\(402\) −4.85410 −0.242101
\(403\) −32.5623 −1.62204
\(404\) 4.14590 0.206266
\(405\) 5.70820 0.283643
\(406\) 8.23607 0.408749
\(407\) −29.7214 −1.47323
\(408\) −4.23607 −0.209717
\(409\) 17.2148 0.851216 0.425608 0.904908i \(-0.360060\pi\)
0.425608 + 0.904908i \(0.360060\pi\)
\(410\) −1.09017 −0.0538397
\(411\) −13.0557 −0.643992
\(412\) −15.0902 −0.743439
\(413\) −11.0000 −0.541275
\(414\) 9.85410 0.484303
\(415\) −15.4164 −0.756762
\(416\) 4.85410 0.237992
\(417\) 0.145898 0.00714466
\(418\) 0 0
\(419\) −25.7426 −1.25761 −0.628805 0.777563i \(-0.716456\pi\)
−0.628805 + 0.777563i \(0.716456\pi\)
\(420\) 0.618034 0.0301570
\(421\) −8.05573 −0.392612 −0.196306 0.980543i \(-0.562895\pi\)
−0.196306 + 0.980543i \(0.562895\pi\)
\(422\) −4.05573 −0.197430
\(423\) −20.7984 −1.01125
\(424\) −12.7082 −0.617165
\(425\) −27.4164 −1.32989
\(426\) 6.18034 0.299438
\(427\) −7.47214 −0.361602
\(428\) −13.9443 −0.674022
\(429\) 15.0000 0.724207
\(430\) −6.85410 −0.330534
\(431\) 2.94427 0.141821 0.0709103 0.997483i \(-0.477410\pi\)
0.0709103 + 0.997483i \(0.477410\pi\)
\(432\) 3.47214 0.167053
\(433\) −27.6180 −1.32724 −0.663619 0.748071i \(-0.730980\pi\)
−0.663619 + 0.748071i \(0.730980\pi\)
\(434\) 6.70820 0.322004
\(435\) 5.09017 0.244055
\(436\) −13.5623 −0.649517
\(437\) 0 0
\(438\) 9.00000 0.430037
\(439\) −22.0557 −1.05266 −0.526331 0.850280i \(-0.676433\pi\)
−0.526331 + 0.850280i \(0.676433\pi\)
\(440\) −5.00000 −0.238366
\(441\) −2.61803 −0.124668
\(442\) 33.2705 1.58252
\(443\) 17.2148 0.817899 0.408949 0.912557i \(-0.365895\pi\)
0.408949 + 0.912557i \(0.365895\pi\)
\(444\) −3.67376 −0.174349
\(445\) 4.85410 0.230107
\(446\) 5.65248 0.267652
\(447\) −0.854102 −0.0403976
\(448\) −1.00000 −0.0472456
\(449\) −22.1803 −1.04675 −0.523377 0.852101i \(-0.675328\pi\)
−0.523377 + 0.852101i \(0.675328\pi\)
\(450\) 10.4721 0.493661
\(451\) 5.45085 0.256670
\(452\) 3.85410 0.181282
\(453\) 8.79837 0.413384
\(454\) −14.2361 −0.668132
\(455\) −4.85410 −0.227564
\(456\) 0 0
\(457\) 29.8885 1.39813 0.699064 0.715060i \(-0.253600\pi\)
0.699064 + 0.715060i \(0.253600\pi\)
\(458\) −10.0902 −0.471483
\(459\) 23.7984 1.11081
\(460\) −3.76393 −0.175494
\(461\) 42.5410 1.98133 0.990666 0.136309i \(-0.0435239\pi\)
0.990666 + 0.136309i \(0.0435239\pi\)
\(462\) −3.09017 −0.143768
\(463\) −13.5623 −0.630294 −0.315147 0.949043i \(-0.602054\pi\)
−0.315147 + 0.949043i \(0.602054\pi\)
\(464\) −8.23607 −0.382350
\(465\) 4.14590 0.192261
\(466\) −11.0344 −0.511161
\(467\) 33.4721 1.54891 0.774453 0.632632i \(-0.218025\pi\)
0.774453 + 0.632632i \(0.218025\pi\)
\(468\) −12.7082 −0.587437
\(469\) −7.85410 −0.362669
\(470\) 7.94427 0.366442
\(471\) −13.7984 −0.635796
\(472\) 11.0000 0.506316
\(473\) 34.2705 1.57576
\(474\) 5.47214 0.251344
\(475\) 0 0
\(476\) −6.85410 −0.314157
\(477\) 33.2705 1.52335
\(478\) −11.1459 −0.509802
\(479\) −0.180340 −0.00823994 −0.00411997 0.999992i \(-0.501311\pi\)
−0.00411997 + 0.999992i \(0.501311\pi\)
\(480\) −0.618034 −0.0282093
\(481\) 28.8541 1.31563
\(482\) 7.90983 0.360283
\(483\) −2.32624 −0.105847
\(484\) 14.0000 0.636364
\(485\) −9.47214 −0.430108
\(486\) −13.9443 −0.632525
\(487\) 11.1803 0.506630 0.253315 0.967384i \(-0.418479\pi\)
0.253315 + 0.967384i \(0.418479\pi\)
\(488\) 7.47214 0.338248
\(489\) −0.291796 −0.0131955
\(490\) 1.00000 0.0451754
\(491\) −19.5967 −0.884389 −0.442194 0.896919i \(-0.645800\pi\)
−0.442194 + 0.896919i \(0.645800\pi\)
\(492\) 0.673762 0.0303755
\(493\) −56.4508 −2.54242
\(494\) 0 0
\(495\) 13.0902 0.588359
\(496\) −6.70820 −0.301207
\(497\) 10.0000 0.448561
\(498\) 9.52786 0.426954
\(499\) 16.4164 0.734899 0.367450 0.930043i \(-0.380231\pi\)
0.367450 + 0.930043i \(0.380231\pi\)
\(500\) −9.00000 −0.402492
\(501\) −1.12461 −0.0502439
\(502\) 6.47214 0.288866
\(503\) 31.7771 1.41687 0.708435 0.705776i \(-0.249401\pi\)
0.708435 + 0.705776i \(0.249401\pi\)
\(504\) 2.61803 0.116617
\(505\) 4.14590 0.184490
\(506\) 18.8197 0.836636
\(507\) −6.52786 −0.289913
\(508\) −7.32624 −0.325049
\(509\) 1.41641 0.0627812 0.0313906 0.999507i \(-0.490006\pi\)
0.0313906 + 0.999507i \(0.490006\pi\)
\(510\) −4.23607 −0.187576
\(511\) 14.5623 0.644198
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.05573 0.355323
\(515\) −15.0902 −0.664952
\(516\) 4.23607 0.186482
\(517\) −39.7214 −1.74694
\(518\) −5.94427 −0.261176
\(519\) −1.09017 −0.0478531
\(520\) 4.85410 0.212866
\(521\) 25.6525 1.12386 0.561928 0.827186i \(-0.310060\pi\)
0.561928 + 0.827186i \(0.310060\pi\)
\(522\) 21.5623 0.943756
\(523\) −35.1591 −1.53740 −0.768699 0.639611i \(-0.779096\pi\)
−0.768699 + 0.639611i \(0.779096\pi\)
\(524\) −4.38197 −0.191427
\(525\) −2.47214 −0.107893
\(526\) −25.1803 −1.09791
\(527\) −45.9787 −2.00286
\(528\) 3.09017 0.134482
\(529\) −8.83282 −0.384035
\(530\) −12.7082 −0.552009
\(531\) −28.7984 −1.24974
\(532\) 0 0
\(533\) −5.29180 −0.229213
\(534\) −3.00000 −0.129823
\(535\) −13.9443 −0.602863
\(536\) 7.85410 0.339246
\(537\) −5.58359 −0.240950
\(538\) 9.18034 0.395793
\(539\) −5.00000 −0.215365
\(540\) 3.47214 0.149417
\(541\) −30.3050 −1.30291 −0.651456 0.758687i \(-0.725841\pi\)
−0.651456 + 0.758687i \(0.725841\pi\)
\(542\) 1.65248 0.0709799
\(543\) 9.74265 0.418097
\(544\) 6.85410 0.293867
\(545\) −13.5623 −0.580945
\(546\) 3.00000 0.128388
\(547\) 9.18034 0.392523 0.196261 0.980552i \(-0.437120\pi\)
0.196261 + 0.980552i \(0.437120\pi\)
\(548\) 21.1246 0.902399
\(549\) −19.5623 −0.834899
\(550\) 20.0000 0.852803
\(551\) 0 0
\(552\) 2.32624 0.0990113
\(553\) 8.85410 0.376515
\(554\) −8.00000 −0.339887
\(555\) −3.67376 −0.155943
\(556\) −0.236068 −0.0100115
\(557\) −27.7639 −1.17640 −0.588198 0.808717i \(-0.700162\pi\)
−0.588198 + 0.808717i \(0.700162\pi\)
\(558\) 17.5623 0.743472
\(559\) −33.2705 −1.40719
\(560\) −1.00000 −0.0422577
\(561\) 21.1803 0.894235
\(562\) −12.2361 −0.516147
\(563\) 36.0689 1.52012 0.760061 0.649852i \(-0.225169\pi\)
0.760061 + 0.649852i \(0.225169\pi\)
\(564\) −4.90983 −0.206741
\(565\) 3.85410 0.162143
\(566\) 6.43769 0.270596
\(567\) −5.70820 −0.239722
\(568\) −10.0000 −0.419591
\(569\) −30.2705 −1.26901 −0.634503 0.772920i \(-0.718795\pi\)
−0.634503 + 0.772920i \(0.718795\pi\)
\(570\) 0 0
\(571\) 31.0132 1.29786 0.648930 0.760848i \(-0.275217\pi\)
0.648930 + 0.760848i \(0.275217\pi\)
\(572\) −24.2705 −1.01480
\(573\) −1.74265 −0.0728001
\(574\) 1.09017 0.0455028
\(575\) 15.0557 0.627867
\(576\) −2.61803 −0.109085
\(577\) −3.29180 −0.137039 −0.0685196 0.997650i \(-0.521828\pi\)
−0.0685196 + 0.997650i \(0.521828\pi\)
\(578\) 29.9787 1.24695
\(579\) 6.29180 0.261478
\(580\) −8.23607 −0.341984
\(581\) 15.4164 0.639580
\(582\) 5.85410 0.242660
\(583\) 63.5410 2.63160
\(584\) −14.5623 −0.602593
\(585\) −12.7082 −0.525420
\(586\) 16.2361 0.670706
\(587\) −41.9443 −1.73123 −0.865613 0.500714i \(-0.833071\pi\)
−0.865613 + 0.500714i \(0.833071\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 0 0
\(590\) 11.0000 0.452863
\(591\) 4.67376 0.192253
\(592\) 5.94427 0.244308
\(593\) 5.43769 0.223299 0.111650 0.993748i \(-0.464387\pi\)
0.111650 + 0.993748i \(0.464387\pi\)
\(594\) −17.3607 −0.712317
\(595\) −6.85410 −0.280991
\(596\) 1.38197 0.0566075
\(597\) 3.03444 0.124191
\(598\) −18.2705 −0.747137
\(599\) 17.0000 0.694601 0.347301 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(600\) 2.47214 0.100925
\(601\) 14.1246 0.576155 0.288077 0.957607i \(-0.406984\pi\)
0.288077 + 0.957607i \(0.406984\pi\)
\(602\) 6.85410 0.279352
\(603\) −20.5623 −0.837362
\(604\) −14.2361 −0.579257
\(605\) 14.0000 0.569181
\(606\) −2.56231 −0.104087
\(607\) 16.9443 0.687747 0.343873 0.939016i \(-0.388261\pi\)
0.343873 + 0.939016i \(0.388261\pi\)
\(608\) 0 0
\(609\) −5.09017 −0.206264
\(610\) 7.47214 0.302538
\(611\) 38.5623 1.56006
\(612\) −17.9443 −0.725354
\(613\) −10.6525 −0.430249 −0.215125 0.976587i \(-0.569016\pi\)
−0.215125 + 0.976587i \(0.569016\pi\)
\(614\) 4.94427 0.199535
\(615\) 0.673762 0.0271687
\(616\) 5.00000 0.201456
\(617\) −24.3262 −0.979337 −0.489669 0.871909i \(-0.662882\pi\)
−0.489669 + 0.871909i \(0.662882\pi\)
\(618\) 9.32624 0.375156
\(619\) −34.7984 −1.39866 −0.699332 0.714797i \(-0.746519\pi\)
−0.699332 + 0.714797i \(0.746519\pi\)
\(620\) −6.70820 −0.269408
\(621\) −13.0689 −0.524436
\(622\) 19.7639 0.792461
\(623\) −4.85410 −0.194475
\(624\) −3.00000 −0.120096
\(625\) 11.0000 0.440000
\(626\) 18.9098 0.755789
\(627\) 0 0
\(628\) 22.3262 0.890914
\(629\) 40.7426 1.62452
\(630\) 2.61803 0.104305
\(631\) 12.9443 0.515303 0.257652 0.966238i \(-0.417051\pi\)
0.257652 + 0.966238i \(0.417051\pi\)
\(632\) −8.85410 −0.352197
\(633\) 2.50658 0.0996275
\(634\) 8.41641 0.334258
\(635\) −7.32624 −0.290733
\(636\) 7.85410 0.311435
\(637\) 4.85410 0.192327
\(638\) 41.1803 1.63035
\(639\) 26.1803 1.03568
\(640\) 1.00000 0.0395285
\(641\) 18.9098 0.746893 0.373447 0.927652i \(-0.378176\pi\)
0.373447 + 0.927652i \(0.378176\pi\)
\(642\) 8.61803 0.340127
\(643\) 22.6525 0.893326 0.446663 0.894702i \(-0.352612\pi\)
0.446663 + 0.894702i \(0.352612\pi\)
\(644\) 3.76393 0.148320
\(645\) 4.23607 0.166795
\(646\) 0 0
\(647\) −41.1033 −1.61594 −0.807969 0.589225i \(-0.799433\pi\)
−0.807969 + 0.589225i \(0.799433\pi\)
\(648\) 5.70820 0.224239
\(649\) −55.0000 −2.15894
\(650\) −19.4164 −0.761574
\(651\) −4.14590 −0.162491
\(652\) 0.472136 0.0184903
\(653\) 23.6869 0.926941 0.463470 0.886112i \(-0.346604\pi\)
0.463470 + 0.886112i \(0.346604\pi\)
\(654\) 8.38197 0.327761
\(655\) −4.38197 −0.171218
\(656\) −1.09017 −0.0425640
\(657\) 38.1246 1.48738
\(658\) −7.94427 −0.309700
\(659\) 33.5410 1.30657 0.653286 0.757111i \(-0.273390\pi\)
0.653286 + 0.757111i \(0.273390\pi\)
\(660\) 3.09017 0.120285
\(661\) −11.8328 −0.460243 −0.230122 0.973162i \(-0.573912\pi\)
−0.230122 + 0.973162i \(0.573912\pi\)
\(662\) 12.4721 0.484743
\(663\) −20.5623 −0.798574
\(664\) −15.4164 −0.598273
\(665\) 0 0
\(666\) −15.5623 −0.603027
\(667\) 31.0000 1.20032
\(668\) 1.81966 0.0704048
\(669\) −3.49342 −0.135064
\(670\) 7.85410 0.303430
\(671\) −37.3607 −1.44229
\(672\) 0.618034 0.0238412
\(673\) 20.9098 0.806015 0.403007 0.915197i \(-0.367965\pi\)
0.403007 + 0.915197i \(0.367965\pi\)
\(674\) −33.4508 −1.28848
\(675\) −13.8885 −0.534570
\(676\) 10.5623 0.406243
\(677\) 3.70820 0.142518 0.0712589 0.997458i \(-0.477298\pi\)
0.0712589 + 0.997458i \(0.477298\pi\)
\(678\) −2.38197 −0.0914789
\(679\) 9.47214 0.363507
\(680\) 6.85410 0.262843
\(681\) 8.79837 0.337154
\(682\) 33.5410 1.28435
\(683\) −20.2705 −0.775630 −0.387815 0.921737i \(-0.626770\pi\)
−0.387815 + 0.921737i \(0.626770\pi\)
\(684\) 0 0
\(685\) 21.1246 0.807130
\(686\) −1.00000 −0.0381802
\(687\) 6.23607 0.237921
\(688\) −6.85410 −0.261310
\(689\) −61.6869 −2.35008
\(690\) 2.32624 0.0885584
\(691\) 9.47214 0.360337 0.180169 0.983636i \(-0.442336\pi\)
0.180169 + 0.983636i \(0.442336\pi\)
\(692\) 1.76393 0.0670546
\(693\) −13.0902 −0.497254
\(694\) −5.05573 −0.191913
\(695\) −0.236068 −0.00895457
\(696\) 5.09017 0.192942
\(697\) −7.47214 −0.283027
\(698\) −6.70820 −0.253909
\(699\) 6.81966 0.257943
\(700\) 4.00000 0.151186
\(701\) 1.72949 0.0653219 0.0326610 0.999466i \(-0.489602\pi\)
0.0326610 + 0.999466i \(0.489602\pi\)
\(702\) 16.8541 0.636117
\(703\) 0 0
\(704\) −5.00000 −0.188445
\(705\) −4.90983 −0.184915
\(706\) 13.0902 0.492655
\(707\) −4.14590 −0.155923
\(708\) −6.79837 −0.255499
\(709\) 39.6525 1.48918 0.744590 0.667522i \(-0.232645\pi\)
0.744590 + 0.667522i \(0.232645\pi\)
\(710\) −10.0000 −0.375293
\(711\) 23.1803 0.869331
\(712\) 4.85410 0.181915
\(713\) 25.2492 0.945591
\(714\) 4.23607 0.158531
\(715\) −24.2705 −0.907666
\(716\) 9.03444 0.337633
\(717\) 6.88854 0.257257
\(718\) 27.9443 1.04287
\(719\) 39.8328 1.48551 0.742757 0.669561i \(-0.233518\pi\)
0.742757 + 0.669561i \(0.233518\pi\)
\(720\) −2.61803 −0.0975684
\(721\) 15.0902 0.561987
\(722\) 0 0
\(723\) −4.88854 −0.181807
\(724\) −15.7639 −0.585862
\(725\) 32.9443 1.22352
\(726\) −8.65248 −0.321123
\(727\) 7.87539 0.292082 0.146041 0.989279i \(-0.453347\pi\)
0.146041 + 0.989279i \(0.453347\pi\)
\(728\) −4.85410 −0.179905
\(729\) −8.50658 −0.315058
\(730\) −14.5623 −0.538975
\(731\) −46.9787 −1.73757
\(732\) −4.61803 −0.170687
\(733\) 12.3475 0.456066 0.228033 0.973653i \(-0.426771\pi\)
0.228033 + 0.973653i \(0.426771\pi\)
\(734\) 9.67376 0.357065
\(735\) −0.618034 −0.0227965
\(736\) −3.76393 −0.138740
\(737\) −39.2705 −1.44655
\(738\) 2.85410 0.105061
\(739\) 11.4164 0.419959 0.209980 0.977706i \(-0.432660\pi\)
0.209980 + 0.977706i \(0.432660\pi\)
\(740\) 5.94427 0.218516
\(741\) 0 0
\(742\) 12.7082 0.466533
\(743\) −50.5066 −1.85291 −0.926453 0.376410i \(-0.877159\pi\)
−0.926453 + 0.376410i \(0.877159\pi\)
\(744\) 4.14590 0.151996
\(745\) 1.38197 0.0506313
\(746\) −6.85410 −0.250947
\(747\) 40.3607 1.47672
\(748\) −34.2705 −1.25305
\(749\) 13.9443 0.509513
\(750\) 5.56231 0.203107
\(751\) 12.1803 0.444467 0.222233 0.974993i \(-0.428665\pi\)
0.222233 + 0.974993i \(0.428665\pi\)
\(752\) 7.94427 0.289698
\(753\) −4.00000 −0.145768
\(754\) −39.9787 −1.45594
\(755\) −14.2361 −0.518104
\(756\) −3.47214 −0.126280
\(757\) −5.88854 −0.214023 −0.107011 0.994258i \(-0.534128\pi\)
−0.107011 + 0.994258i \(0.534128\pi\)
\(758\) 12.0344 0.437111
\(759\) −11.6312 −0.422185
\(760\) 0 0
\(761\) 13.8328 0.501439 0.250720 0.968060i \(-0.419333\pi\)
0.250720 + 0.968060i \(0.419333\pi\)
\(762\) 4.52786 0.164027
\(763\) 13.5623 0.490988
\(764\) 2.81966 0.102012
\(765\) −17.9443 −0.648777
\(766\) 5.94427 0.214775
\(767\) 53.3951 1.92799
\(768\) −0.618034 −0.0223014
\(769\) 39.4164 1.42139 0.710696 0.703499i \(-0.248380\pi\)
0.710696 + 0.703499i \(0.248380\pi\)
\(770\) 5.00000 0.180187
\(771\) −4.97871 −0.179304
\(772\) −10.1803 −0.366398
\(773\) 0.437694 0.0157428 0.00787138 0.999969i \(-0.497494\pi\)
0.00787138 + 0.999969i \(0.497494\pi\)
\(774\) 17.9443 0.644994
\(775\) 26.8328 0.963863
\(776\) −9.47214 −0.340030
\(777\) 3.67376 0.131795
\(778\) 11.3820 0.408063
\(779\) 0 0
\(780\) −3.00000 −0.107417
\(781\) 50.0000 1.78914
\(782\) −25.7984 −0.922548
\(783\) −28.5967 −1.02196
\(784\) 1.00000 0.0357143
\(785\) 22.3262 0.796858
\(786\) 2.70820 0.0965984
\(787\) −18.6738 −0.665648 −0.332824 0.942989i \(-0.608001\pi\)
−0.332824 + 0.942989i \(0.608001\pi\)
\(788\) −7.56231 −0.269396
\(789\) 15.5623 0.554033
\(790\) −8.85410 −0.315015
\(791\) −3.85410 −0.137036
\(792\) 13.0902 0.465139
\(793\) 36.2705 1.28800
\(794\) −32.2705 −1.14524
\(795\) 7.85410 0.278556
\(796\) −4.90983 −0.174024
\(797\) 12.7984 0.453342 0.226671 0.973971i \(-0.427216\pi\)
0.226671 + 0.973971i \(0.427216\pi\)
\(798\) 0 0
\(799\) 54.4508 1.92633
\(800\) −4.00000 −0.141421
\(801\) −12.7082 −0.449022
\(802\) −6.58359 −0.232475
\(803\) 72.8115 2.56946
\(804\) −4.85410 −0.171191
\(805\) 3.76393 0.132661
\(806\) −32.5623 −1.14696
\(807\) −5.67376 −0.199726
\(808\) 4.14590 0.145852
\(809\) −26.1459 −0.919241 −0.459620 0.888115i \(-0.652015\pi\)
−0.459620 + 0.888115i \(0.652015\pi\)
\(810\) 5.70820 0.200566
\(811\) −40.5410 −1.42359 −0.711794 0.702388i \(-0.752117\pi\)
−0.711794 + 0.702388i \(0.752117\pi\)
\(812\) 8.23607 0.289029
\(813\) −1.02129 −0.0358181
\(814\) −29.7214 −1.04173
\(815\) 0.472136 0.0165382
\(816\) −4.23607 −0.148292
\(817\) 0 0
\(818\) 17.2148 0.601901
\(819\) 12.7082 0.444061
\(820\) −1.09017 −0.0380704
\(821\) 9.76393 0.340764 0.170382 0.985378i \(-0.445500\pi\)
0.170382 + 0.985378i \(0.445500\pi\)
\(822\) −13.0557 −0.455371
\(823\) −33.7639 −1.17694 −0.588468 0.808520i \(-0.700269\pi\)
−0.588468 + 0.808520i \(0.700269\pi\)
\(824\) −15.0902 −0.525691
\(825\) −12.3607 −0.430344
\(826\) −11.0000 −0.382739
\(827\) 13.3820 0.465337 0.232668 0.972556i \(-0.425254\pi\)
0.232668 + 0.972556i \(0.425254\pi\)
\(828\) 9.85410 0.342454
\(829\) 6.88854 0.239249 0.119625 0.992819i \(-0.461831\pi\)
0.119625 + 0.992819i \(0.461831\pi\)
\(830\) −15.4164 −0.535111
\(831\) 4.94427 0.171515
\(832\) 4.85410 0.168286
\(833\) 6.85410 0.237481
\(834\) 0.145898 0.00505204
\(835\) 1.81966 0.0629719
\(836\) 0 0
\(837\) −23.2918 −0.805082
\(838\) −25.7426 −0.889265
\(839\) 41.1803 1.42170 0.710852 0.703342i \(-0.248310\pi\)
0.710852 + 0.703342i \(0.248310\pi\)
\(840\) 0.618034 0.0213242
\(841\) 38.8328 1.33906
\(842\) −8.05573 −0.277619
\(843\) 7.56231 0.260460
\(844\) −4.05573 −0.139604
\(845\) 10.5623 0.363354
\(846\) −20.7984 −0.715063
\(847\) −14.0000 −0.481046
\(848\) −12.7082 −0.436402
\(849\) −3.97871 −0.136549
\(850\) −27.4164 −0.940375
\(851\) −22.3738 −0.766965
\(852\) 6.18034 0.211735
\(853\) 38.9574 1.33388 0.666938 0.745113i \(-0.267604\pi\)
0.666938 + 0.745113i \(0.267604\pi\)
\(854\) −7.47214 −0.255691
\(855\) 0 0
\(856\) −13.9443 −0.476605
\(857\) 17.7426 0.606077 0.303039 0.952978i \(-0.401999\pi\)
0.303039 + 0.952978i \(0.401999\pi\)
\(858\) 15.0000 0.512092
\(859\) −5.06888 −0.172948 −0.0864740 0.996254i \(-0.527560\pi\)
−0.0864740 + 0.996254i \(0.527560\pi\)
\(860\) −6.85410 −0.233723
\(861\) −0.673762 −0.0229618
\(862\) 2.94427 0.100282
\(863\) −7.29180 −0.248216 −0.124108 0.992269i \(-0.539607\pi\)
−0.124108 + 0.992269i \(0.539607\pi\)
\(864\) 3.47214 0.118124
\(865\) 1.76393 0.0599755
\(866\) −27.6180 −0.938499
\(867\) −18.5279 −0.629239
\(868\) 6.70820 0.227691
\(869\) 44.2705 1.50177
\(870\) 5.09017 0.172573
\(871\) 38.1246 1.29180
\(872\) −13.5623 −0.459278
\(873\) 24.7984 0.839298
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 9.00000 0.304082
\(877\) 5.32624 0.179854 0.0899271 0.995948i \(-0.471337\pi\)
0.0899271 + 0.995948i \(0.471337\pi\)
\(878\) −22.0557 −0.744345
\(879\) −10.0344 −0.338453
\(880\) −5.00000 −0.168550
\(881\) 14.1459 0.476587 0.238294 0.971193i \(-0.423412\pi\)
0.238294 + 0.971193i \(0.423412\pi\)
\(882\) −2.61803 −0.0881538
\(883\) −11.0213 −0.370896 −0.185448 0.982654i \(-0.559374\pi\)
−0.185448 + 0.982654i \(0.559374\pi\)
\(884\) 33.2705 1.11901
\(885\) −6.79837 −0.228525
\(886\) 17.2148 0.578342
\(887\) 20.8328 0.699497 0.349749 0.936844i \(-0.386267\pi\)
0.349749 + 0.936844i \(0.386267\pi\)
\(888\) −3.67376 −0.123283
\(889\) 7.32624 0.245714
\(890\) 4.85410 0.162710
\(891\) −28.5410 −0.956160
\(892\) 5.65248 0.189259
\(893\) 0 0
\(894\) −0.854102 −0.0285654
\(895\) 9.03444 0.301988
\(896\) −1.00000 −0.0334077
\(897\) 11.2918 0.377022
\(898\) −22.1803 −0.740168
\(899\) 55.2492 1.84266
\(900\) 10.4721 0.349071
\(901\) −87.1033 −2.90183
\(902\) 5.45085 0.181493
\(903\) −4.23607 −0.140968
\(904\) 3.85410 0.128186
\(905\) −15.7639 −0.524011
\(906\) 8.79837 0.292306
\(907\) 16.6738 0.553643 0.276822 0.960921i \(-0.410719\pi\)
0.276822 + 0.960921i \(0.410719\pi\)
\(908\) −14.2361 −0.472441
\(909\) −10.8541 −0.360008
\(910\) −4.85410 −0.160912
\(911\) 46.1591 1.52932 0.764659 0.644435i \(-0.222907\pi\)
0.764659 + 0.644435i \(0.222907\pi\)
\(912\) 0 0
\(913\) 77.0820 2.55104
\(914\) 29.8885 0.988625
\(915\) −4.61803 −0.152667
\(916\) −10.0902 −0.333389
\(917\) 4.38197 0.144705
\(918\) 23.7984 0.785463
\(919\) 10.0344 0.331006 0.165503 0.986209i \(-0.447075\pi\)
0.165503 + 0.986209i \(0.447075\pi\)
\(920\) −3.76393 −0.124093
\(921\) −3.05573 −0.100690
\(922\) 42.5410 1.40101
\(923\) −48.5410 −1.59775
\(924\) −3.09017 −0.101659
\(925\) −23.7771 −0.781786
\(926\) −13.5623 −0.445685
\(927\) 39.5066 1.29757
\(928\) −8.23607 −0.270362
\(929\) −26.8885 −0.882185 −0.441092 0.897462i \(-0.645409\pi\)
−0.441092 + 0.897462i \(0.645409\pi\)
\(930\) 4.14590 0.135949
\(931\) 0 0
\(932\) −11.0344 −0.361445
\(933\) −12.2148 −0.399894
\(934\) 33.4721 1.09524
\(935\) −34.2705 −1.12077
\(936\) −12.7082 −0.415381
\(937\) 40.3951 1.31965 0.659826 0.751419i \(-0.270630\pi\)
0.659826 + 0.751419i \(0.270630\pi\)
\(938\) −7.85410 −0.256446
\(939\) −11.6869 −0.381388
\(940\) 7.94427 0.259114
\(941\) 27.2148 0.887177 0.443588 0.896231i \(-0.353705\pi\)
0.443588 + 0.896231i \(0.353705\pi\)
\(942\) −13.7984 −0.449575
\(943\) 4.10333 0.133623
\(944\) 11.0000 0.358020
\(945\) −3.47214 −0.112949
\(946\) 34.2705 1.11423
\(947\) 49.8885 1.62116 0.810580 0.585628i \(-0.199152\pi\)
0.810580 + 0.585628i \(0.199152\pi\)
\(948\) 5.47214 0.177727
\(949\) −70.6869 −2.29459
\(950\) 0 0
\(951\) −5.20163 −0.168674
\(952\) −6.85410 −0.222143
\(953\) −30.7771 −0.996968 −0.498484 0.866899i \(-0.666110\pi\)
−0.498484 + 0.866899i \(0.666110\pi\)
\(954\) 33.2705 1.07717
\(955\) 2.81966 0.0912421
\(956\) −11.1459 −0.360484
\(957\) −25.4508 −0.822709
\(958\) −0.180340 −0.00582652
\(959\) −21.1246 −0.682149
\(960\) −0.618034 −0.0199470
\(961\) 14.0000 0.451613
\(962\) 28.8541 0.930294
\(963\) 36.5066 1.17641
\(964\) 7.90983 0.254758
\(965\) −10.1803 −0.327717
\(966\) −2.32624 −0.0748455
\(967\) −55.2705 −1.77738 −0.888690 0.458509i \(-0.848384\pi\)
−0.888690 + 0.458509i \(0.848384\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) −9.47214 −0.304132
\(971\) −34.4164 −1.10448 −0.552238 0.833687i \(-0.686226\pi\)
−0.552238 + 0.833687i \(0.686226\pi\)
\(972\) −13.9443 −0.447263
\(973\) 0.236068 0.00756799
\(974\) 11.1803 0.358241
\(975\) 12.0000 0.384308
\(976\) 7.47214 0.239177
\(977\) −48.7771 −1.56052 −0.780259 0.625457i \(-0.784913\pi\)
−0.780259 + 0.625457i \(0.784913\pi\)
\(978\) −0.291796 −0.00933061
\(979\) −24.2705 −0.775689
\(980\) 1.00000 0.0319438
\(981\) 35.5066 1.13364
\(982\) −19.5967 −0.625357
\(983\) 9.49342 0.302793 0.151397 0.988473i \(-0.451623\pi\)
0.151397 + 0.988473i \(0.451623\pi\)
\(984\) 0.673762 0.0214788
\(985\) −7.56231 −0.240955
\(986\) −56.4508 −1.79776
\(987\) 4.90983 0.156282
\(988\) 0 0
\(989\) 25.7984 0.820341
\(990\) 13.0902 0.416033
\(991\) −25.4164 −0.807379 −0.403689 0.914896i \(-0.632272\pi\)
−0.403689 + 0.914896i \(0.632272\pi\)
\(992\) −6.70820 −0.212986
\(993\) −7.70820 −0.244612
\(994\) 10.0000 0.317181
\(995\) −4.90983 −0.155652
\(996\) 9.52786 0.301902
\(997\) 31.0344 0.982871 0.491435 0.870914i \(-0.336472\pi\)
0.491435 + 0.870914i \(0.336472\pi\)
\(998\) 16.4164 0.519652
\(999\) 20.6393 0.652999
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 5054.2.a.p.1.1 yes 2
19.18 odd 2 5054.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.f.1.2 2 19.18 odd 2
5054.2.a.p.1.1 yes 2 1.1 even 1 trivial