Properties

Label 5070.2.a.bu.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.13706 q^{11} -1.00000 q^{12} +3.04892 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.69202 q^{17} +1.00000 q^{18} +1.55496 q^{19} +1.00000 q^{20} -3.04892 q^{21} +1.13706 q^{22} +5.40581 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.04892 q^{28} -6.15883 q^{29} -1.00000 q^{30} -0.246980 q^{31} +1.00000 q^{32} -1.13706 q^{33} +2.69202 q^{34} +3.04892 q^{35} +1.00000 q^{36} -0.554958 q^{37} +1.55496 q^{38} +1.00000 q^{40} +8.26875 q^{41} -3.04892 q^{42} +2.03923 q^{43} +1.13706 q^{44} +1.00000 q^{45} +5.40581 q^{46} -6.70171 q^{47} -1.00000 q^{48} +2.29590 q^{49} +1.00000 q^{50} -2.69202 q^{51} +5.77479 q^{53} -1.00000 q^{54} +1.13706 q^{55} +3.04892 q^{56} -1.55496 q^{57} -6.15883 q^{58} +1.36898 q^{59} -1.00000 q^{60} +10.9487 q^{61} -0.246980 q^{62} +3.04892 q^{63} +1.00000 q^{64} -1.13706 q^{66} -2.47219 q^{67} +2.69202 q^{68} -5.40581 q^{69} +3.04892 q^{70} -8.57002 q^{71} +1.00000 q^{72} -14.8334 q^{73} -0.554958 q^{74} -1.00000 q^{75} +1.55496 q^{76} +3.46681 q^{77} +5.33513 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.26875 q^{82} -5.45473 q^{83} -3.04892 q^{84} +2.69202 q^{85} +2.03923 q^{86} +6.15883 q^{87} +1.13706 q^{88} +16.8877 q^{89} +1.00000 q^{90} +5.40581 q^{92} +0.246980 q^{93} -6.70171 q^{94} +1.55496 q^{95} -1.00000 q^{96} +1.93900 q^{97} +2.29590 q^{98} +1.13706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{10} - 2q^{11} - 3q^{12} - 3q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 5q^{19} + 3q^{20} - 2q^{22} + 3q^{23} - 3q^{24} + 3q^{25} - 3q^{27} - 10q^{29} - 3q^{30} + 4q^{31} + 3q^{32} + 2q^{33} + 3q^{34} + 3q^{36} - 2q^{37} + 5q^{38} + 3q^{40} + 17q^{41} + 19q^{43} - 2q^{44} + 3q^{45} + 3q^{46} + 7q^{47} - 3q^{48} - 7q^{49} + 3q^{50} - 3q^{51} + 19q^{53} - 3q^{54} - 2q^{55} - 5q^{57} - 10q^{58} + 19q^{59} - 3q^{60} + q^{61} + 4q^{62} + 3q^{64} + 2q^{66} - q^{67} + 3q^{68} - 3q^{69} - q^{71} + 3q^{72} - 15q^{73} - 2q^{74} - 3q^{75} + 5q^{76} + 7q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 17q^{82} + 6q^{83} + 3q^{85} + 19q^{86} + 10q^{87} - 2q^{88} + 9q^{89} + 3q^{90} + 3q^{92} - 4q^{93} + 7q^{94} + 5q^{95} - 3q^{96} - 4q^{97} - 7q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.04892 1.15238 0.576191 0.817315i \(-0.304538\pi\)
0.576191 + 0.817315i \(0.304538\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.13706 0.342837 0.171419 0.985198i \(-0.445165\pi\)
0.171419 + 0.985198i \(0.445165\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.04892 0.814857
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.69202 0.652911 0.326456 0.945213i \(-0.394146\pi\)
0.326456 + 0.945213i \(0.394146\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.55496 0.356732 0.178366 0.983964i \(-0.442919\pi\)
0.178366 + 0.983964i \(0.442919\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.04892 −0.665328
\(22\) 1.13706 0.242423
\(23\) 5.40581 1.12719 0.563595 0.826051i \(-0.309418\pi\)
0.563595 + 0.826051i \(0.309418\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.04892 0.576191
\(29\) −6.15883 −1.14367 −0.571833 0.820370i \(-0.693768\pi\)
−0.571833 + 0.820370i \(0.693768\pi\)
\(30\) −1.00000 −0.182574
\(31\) −0.246980 −0.0443588 −0.0221794 0.999754i \(-0.507061\pi\)
−0.0221794 + 0.999754i \(0.507061\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.13706 −0.197937
\(34\) 2.69202 0.461678
\(35\) 3.04892 0.515361
\(36\) 1.00000 0.166667
\(37\) −0.554958 −0.0912346 −0.0456173 0.998959i \(-0.514525\pi\)
−0.0456173 + 0.998959i \(0.514525\pi\)
\(38\) 1.55496 0.252248
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 8.26875 1.29136 0.645681 0.763607i \(-0.276574\pi\)
0.645681 + 0.763607i \(0.276574\pi\)
\(42\) −3.04892 −0.470458
\(43\) 2.03923 0.310979 0.155490 0.987838i \(-0.450304\pi\)
0.155490 + 0.987838i \(0.450304\pi\)
\(44\) 1.13706 0.171419
\(45\) 1.00000 0.149071
\(46\) 5.40581 0.797044
\(47\) −6.70171 −0.977545 −0.488772 0.872411i \(-0.662555\pi\)
−0.488772 + 0.872411i \(0.662555\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.29590 0.327985
\(50\) 1.00000 0.141421
\(51\) −2.69202 −0.376958
\(52\) 0 0
\(53\) 5.77479 0.793229 0.396614 0.917985i \(-0.370185\pi\)
0.396614 + 0.917985i \(0.370185\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.13706 0.153322
\(56\) 3.04892 0.407429
\(57\) −1.55496 −0.205959
\(58\) −6.15883 −0.808694
\(59\) 1.36898 0.178226 0.0891128 0.996022i \(-0.471597\pi\)
0.0891128 + 0.996022i \(0.471597\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.9487 1.40184 0.700918 0.713242i \(-0.252774\pi\)
0.700918 + 0.713242i \(0.252774\pi\)
\(62\) −0.246980 −0.0313664
\(63\) 3.04892 0.384127
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.13706 −0.139963
\(67\) −2.47219 −0.302026 −0.151013 0.988532i \(-0.548254\pi\)
−0.151013 + 0.988532i \(0.548254\pi\)
\(68\) 2.69202 0.326456
\(69\) −5.40581 −0.650783
\(70\) 3.04892 0.364415
\(71\) −8.57002 −1.01707 −0.508537 0.861040i \(-0.669814\pi\)
−0.508537 + 0.861040i \(0.669814\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.8334 −1.73612 −0.868059 0.496461i \(-0.834632\pi\)
−0.868059 + 0.496461i \(0.834632\pi\)
\(74\) −0.554958 −0.0645126
\(75\) −1.00000 −0.115470
\(76\) 1.55496 0.178366
\(77\) 3.46681 0.395080
\(78\) 0 0
\(79\) 5.33513 0.600249 0.300124 0.953900i \(-0.402972\pi\)
0.300124 + 0.953900i \(0.402972\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 8.26875 0.913131
\(83\) −5.45473 −0.598734 −0.299367 0.954138i \(-0.596776\pi\)
−0.299367 + 0.954138i \(0.596776\pi\)
\(84\) −3.04892 −0.332664
\(85\) 2.69202 0.291991
\(86\) 2.03923 0.219896
\(87\) 6.15883 0.660296
\(88\) 1.13706 0.121211
\(89\) 16.8877 1.79009 0.895046 0.445974i \(-0.147143\pi\)
0.895046 + 0.445974i \(0.147143\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 5.40581 0.563595
\(93\) 0.246980 0.0256106
\(94\) −6.70171 −0.691229
\(95\) 1.55496 0.159535
\(96\) −1.00000 −0.102062
\(97\) 1.93900 0.196876 0.0984379 0.995143i \(-0.468615\pi\)
0.0984379 + 0.995143i \(0.468615\pi\)
\(98\) 2.29590 0.231921
\(99\) 1.13706 0.114279
\(100\) 1.00000 0.100000
\(101\) −14.4940 −1.44220 −0.721101 0.692830i \(-0.756364\pi\)
−0.721101 + 0.692830i \(0.756364\pi\)
\(102\) −2.69202 −0.266550
\(103\) −16.1836 −1.59462 −0.797308 0.603572i \(-0.793743\pi\)
−0.797308 + 0.603572i \(0.793743\pi\)
\(104\) 0 0
\(105\) −3.04892 −0.297544
\(106\) 5.77479 0.560897
\(107\) 4.92931 0.476535 0.238267 0.971200i \(-0.423421\pi\)
0.238267 + 0.971200i \(0.423421\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.801938 0.0768117 0.0384059 0.999262i \(-0.487772\pi\)
0.0384059 + 0.999262i \(0.487772\pi\)
\(110\) 1.13706 0.108415
\(111\) 0.554958 0.0526743
\(112\) 3.04892 0.288096
\(113\) −11.5211 −1.08381 −0.541907 0.840438i \(-0.682298\pi\)
−0.541907 + 0.840438i \(0.682298\pi\)
\(114\) −1.55496 −0.145635
\(115\) 5.40581 0.504095
\(116\) −6.15883 −0.571833
\(117\) 0 0
\(118\) 1.36898 0.126025
\(119\) 8.20775 0.752403
\(120\) −1.00000 −0.0912871
\(121\) −9.70709 −0.882462
\(122\) 10.9487 0.991248
\(123\) −8.26875 −0.745568
\(124\) −0.246980 −0.0221794
\(125\) 1.00000 0.0894427
\(126\) 3.04892 0.271619
\(127\) 6.22521 0.552398 0.276199 0.961100i \(-0.410925\pi\)
0.276199 + 0.961100i \(0.410925\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.03923 −0.179544
\(130\) 0 0
\(131\) 3.34481 0.292238 0.146119 0.989267i \(-0.453322\pi\)
0.146119 + 0.989267i \(0.453322\pi\)
\(132\) −1.13706 −0.0989687
\(133\) 4.74094 0.411092
\(134\) −2.47219 −0.213565
\(135\) −1.00000 −0.0860663
\(136\) 2.69202 0.230839
\(137\) −9.53079 −0.814271 −0.407135 0.913368i \(-0.633472\pi\)
−0.407135 + 0.913368i \(0.633472\pi\)
\(138\) −5.40581 −0.460173
\(139\) 18.7017 1.58626 0.793129 0.609053i \(-0.208451\pi\)
0.793129 + 0.609053i \(0.208451\pi\)
\(140\) 3.04892 0.257681
\(141\) 6.70171 0.564386
\(142\) −8.57002 −0.719180
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.15883 −0.511463
\(146\) −14.8334 −1.22762
\(147\) −2.29590 −0.189362
\(148\) −0.554958 −0.0456173
\(149\) 16.4209 1.34525 0.672625 0.739983i \(-0.265167\pi\)
0.672625 + 0.739983i \(0.265167\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −9.26636 −0.754085 −0.377043 0.926196i \(-0.623059\pi\)
−0.377043 + 0.926196i \(0.623059\pi\)
\(152\) 1.55496 0.126124
\(153\) 2.69202 0.217637
\(154\) 3.46681 0.279364
\(155\) −0.246980 −0.0198379
\(156\) 0 0
\(157\) 17.0761 1.36282 0.681409 0.731903i \(-0.261367\pi\)
0.681409 + 0.731903i \(0.261367\pi\)
\(158\) 5.33513 0.424440
\(159\) −5.77479 −0.457971
\(160\) 1.00000 0.0790569
\(161\) 16.4819 1.29895
\(162\) 1.00000 0.0785674
\(163\) −9.61894 −0.753414 −0.376707 0.926333i \(-0.622944\pi\)
−0.376707 + 0.926333i \(0.622944\pi\)
\(164\) 8.26875 0.645681
\(165\) −1.13706 −0.0885203
\(166\) −5.45473 −0.423369
\(167\) 21.8509 1.69087 0.845435 0.534078i \(-0.179341\pi\)
0.845435 + 0.534078i \(0.179341\pi\)
\(168\) −3.04892 −0.235229
\(169\) 0 0
\(170\) 2.69202 0.206469
\(171\) 1.55496 0.118911
\(172\) 2.03923 0.155490
\(173\) 16.6853 1.26856 0.634281 0.773103i \(-0.281296\pi\)
0.634281 + 0.773103i \(0.281296\pi\)
\(174\) 6.15883 0.466900
\(175\) 3.04892 0.230476
\(176\) 1.13706 0.0857094
\(177\) −1.36898 −0.102899
\(178\) 16.8877 1.26579
\(179\) −3.94438 −0.294817 −0.147408 0.989076i \(-0.547093\pi\)
−0.147408 + 0.989076i \(0.547093\pi\)
\(180\) 1.00000 0.0745356
\(181\) −20.5308 −1.52604 −0.763021 0.646374i \(-0.776285\pi\)
−0.763021 + 0.646374i \(0.776285\pi\)
\(182\) 0 0
\(183\) −10.9487 −0.809350
\(184\) 5.40581 0.398522
\(185\) −0.554958 −0.0408013
\(186\) 0.246980 0.0181094
\(187\) 3.06100 0.223842
\(188\) −6.70171 −0.488772
\(189\) −3.04892 −0.221776
\(190\) 1.55496 0.112809
\(191\) −6.66487 −0.482253 −0.241127 0.970494i \(-0.577517\pi\)
−0.241127 + 0.970494i \(0.577517\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 16.6558 1.19891 0.599455 0.800409i \(-0.295384\pi\)
0.599455 + 0.800409i \(0.295384\pi\)
\(194\) 1.93900 0.139212
\(195\) 0 0
\(196\) 2.29590 0.163993
\(197\) 19.4494 1.38571 0.692855 0.721077i \(-0.256353\pi\)
0.692855 + 0.721077i \(0.256353\pi\)
\(198\) 1.13706 0.0808076
\(199\) −21.6558 −1.53514 −0.767569 0.640967i \(-0.778534\pi\)
−0.767569 + 0.640967i \(0.778534\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.47219 0.174375
\(202\) −14.4940 −1.01979
\(203\) −18.7778 −1.31794
\(204\) −2.69202 −0.188479
\(205\) 8.26875 0.577515
\(206\) −16.1836 −1.12756
\(207\) 5.40581 0.375730
\(208\) 0 0
\(209\) 1.76809 0.122301
\(210\) −3.04892 −0.210395
\(211\) 12.1521 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(212\) 5.77479 0.396614
\(213\) 8.57002 0.587208
\(214\) 4.92931 0.336961
\(215\) 2.03923 0.139074
\(216\) −1.00000 −0.0680414
\(217\) −0.753020 −0.0511184
\(218\) 0.801938 0.0543141
\(219\) 14.8334 1.00235
\(220\) 1.13706 0.0766608
\(221\) 0 0
\(222\) 0.554958 0.0372464
\(223\) 1.11960 0.0749743 0.0374871 0.999297i \(-0.488065\pi\)
0.0374871 + 0.999297i \(0.488065\pi\)
\(224\) 3.04892 0.203714
\(225\) 1.00000 0.0666667
\(226\) −11.5211 −0.766373
\(227\) 4.06638 0.269895 0.134947 0.990853i \(-0.456913\pi\)
0.134947 + 0.990853i \(0.456913\pi\)
\(228\) −1.55496 −0.102980
\(229\) −16.5187 −1.09159 −0.545794 0.837920i \(-0.683772\pi\)
−0.545794 + 0.837920i \(0.683772\pi\)
\(230\) 5.40581 0.356449
\(231\) −3.46681 −0.228099
\(232\) −6.15883 −0.404347
\(233\) −6.16421 −0.403831 −0.201915 0.979403i \(-0.564717\pi\)
−0.201915 + 0.979403i \(0.564717\pi\)
\(234\) 0 0
\(235\) −6.70171 −0.437171
\(236\) 1.36898 0.0891128
\(237\) −5.33513 −0.346554
\(238\) 8.20775 0.532029
\(239\) 18.1793 1.17592 0.587960 0.808890i \(-0.299931\pi\)
0.587960 + 0.808890i \(0.299931\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −9.21744 −0.593747 −0.296874 0.954917i \(-0.595944\pi\)
−0.296874 + 0.954917i \(0.595944\pi\)
\(242\) −9.70709 −0.623995
\(243\) −1.00000 −0.0641500
\(244\) 10.9487 0.700918
\(245\) 2.29590 0.146679
\(246\) −8.26875 −0.527196
\(247\) 0 0
\(248\) −0.246980 −0.0156832
\(249\) 5.45473 0.345680
\(250\) 1.00000 0.0632456
\(251\) 17.2054 1.08599 0.542996 0.839735i \(-0.317290\pi\)
0.542996 + 0.839735i \(0.317290\pi\)
\(252\) 3.04892 0.192064
\(253\) 6.14675 0.386443
\(254\) 6.22521 0.390604
\(255\) −2.69202 −0.168581
\(256\) 1.00000 0.0625000
\(257\) −19.5157 −1.21736 −0.608679 0.793417i \(-0.708300\pi\)
−0.608679 + 0.793417i \(0.708300\pi\)
\(258\) −2.03923 −0.126957
\(259\) −1.69202 −0.105137
\(260\) 0 0
\(261\) −6.15883 −0.381222
\(262\) 3.34481 0.206643
\(263\) 12.8834 0.794423 0.397212 0.917727i \(-0.369978\pi\)
0.397212 + 0.917727i \(0.369978\pi\)
\(264\) −1.13706 −0.0699814
\(265\) 5.77479 0.354743
\(266\) 4.74094 0.290686
\(267\) −16.8877 −1.03351
\(268\) −2.47219 −0.151013
\(269\) −30.9138 −1.88485 −0.942423 0.334423i \(-0.891459\pi\)
−0.942423 + 0.334423i \(0.891459\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 7.24027 0.439815 0.219908 0.975521i \(-0.429424\pi\)
0.219908 + 0.975521i \(0.429424\pi\)
\(272\) 2.69202 0.163228
\(273\) 0 0
\(274\) −9.53079 −0.575776
\(275\) 1.13706 0.0685675
\(276\) −5.40581 −0.325392
\(277\) 12.2446 0.735706 0.367853 0.929884i \(-0.380093\pi\)
0.367853 + 0.929884i \(0.380093\pi\)
\(278\) 18.7017 1.12165
\(279\) −0.246980 −0.0147863
\(280\) 3.04892 0.182208
\(281\) −2.74632 −0.163831 −0.0819157 0.996639i \(-0.526104\pi\)
−0.0819157 + 0.996639i \(0.526104\pi\)
\(282\) 6.70171 0.399081
\(283\) 14.9312 0.887570 0.443785 0.896133i \(-0.353635\pi\)
0.443785 + 0.896133i \(0.353635\pi\)
\(284\) −8.57002 −0.508537
\(285\) −1.55496 −0.0921078
\(286\) 0 0
\(287\) 25.2107 1.48814
\(288\) 1.00000 0.0589256
\(289\) −9.75302 −0.573707
\(290\) −6.15883 −0.361659
\(291\) −1.93900 −0.113666
\(292\) −14.8334 −0.868059
\(293\) 18.5810 1.08552 0.542758 0.839889i \(-0.317380\pi\)
0.542758 + 0.839889i \(0.317380\pi\)
\(294\) −2.29590 −0.133899
\(295\) 1.36898 0.0797049
\(296\) −0.554958 −0.0322563
\(297\) −1.13706 −0.0659791
\(298\) 16.4209 0.951236
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 6.21744 0.358367
\(302\) −9.26636 −0.533219
\(303\) 14.4940 0.832656
\(304\) 1.55496 0.0891830
\(305\) 10.9487 0.626920
\(306\) 2.69202 0.153893
\(307\) 32.4359 1.85122 0.925609 0.378482i \(-0.123554\pi\)
0.925609 + 0.378482i \(0.123554\pi\)
\(308\) 3.46681 0.197540
\(309\) 16.1836 0.920652
\(310\) −0.246980 −0.0140275
\(311\) −21.2771 −1.20651 −0.603257 0.797547i \(-0.706131\pi\)
−0.603257 + 0.797547i \(0.706131\pi\)
\(312\) 0 0
\(313\) −18.7060 −1.05733 −0.528663 0.848832i \(-0.677307\pi\)
−0.528663 + 0.848832i \(0.677307\pi\)
\(314\) 17.0761 0.963658
\(315\) 3.04892 0.171787
\(316\) 5.33513 0.300124
\(317\) 17.4776 0.981638 0.490819 0.871262i \(-0.336698\pi\)
0.490819 + 0.871262i \(0.336698\pi\)
\(318\) −5.77479 −0.323834
\(319\) −7.00298 −0.392092
\(320\) 1.00000 0.0559017
\(321\) −4.92931 −0.275127
\(322\) 16.4819 0.918499
\(323\) 4.18598 0.232914
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −9.61894 −0.532744
\(327\) −0.801938 −0.0443473
\(328\) 8.26875 0.456565
\(329\) −20.4330 −1.12651
\(330\) −1.13706 −0.0625933
\(331\) −23.1890 −1.27458 −0.637290 0.770624i \(-0.719945\pi\)
−0.637290 + 0.770624i \(0.719945\pi\)
\(332\) −5.45473 −0.299367
\(333\) −0.554958 −0.0304115
\(334\) 21.8509 1.19563
\(335\) −2.47219 −0.135070
\(336\) −3.04892 −0.166332
\(337\) −26.1400 −1.42394 −0.711970 0.702210i \(-0.752197\pi\)
−0.711970 + 0.702210i \(0.752197\pi\)
\(338\) 0 0
\(339\) 11.5211 0.625741
\(340\) 2.69202 0.145995
\(341\) −0.280831 −0.0152079
\(342\) 1.55496 0.0840825
\(343\) −14.3424 −0.774418
\(344\) 2.03923 0.109948
\(345\) −5.40581 −0.291039
\(346\) 16.6853 0.897008
\(347\) 15.2959 0.821127 0.410563 0.911832i \(-0.365332\pi\)
0.410563 + 0.911832i \(0.365332\pi\)
\(348\) 6.15883 0.330148
\(349\) −19.2107 −1.02833 −0.514164 0.857692i \(-0.671898\pi\)
−0.514164 + 0.857692i \(0.671898\pi\)
\(350\) 3.04892 0.162971
\(351\) 0 0
\(352\) 1.13706 0.0606057
\(353\) −5.35450 −0.284991 −0.142496 0.989795i \(-0.545513\pi\)
−0.142496 + 0.989795i \(0.545513\pi\)
\(354\) −1.36898 −0.0727603
\(355\) −8.57002 −0.454850
\(356\) 16.8877 0.895046
\(357\) −8.20775 −0.434400
\(358\) −3.94438 −0.208467
\(359\) 8.49934 0.448578 0.224289 0.974523i \(-0.427994\pi\)
0.224289 + 0.974523i \(0.427994\pi\)
\(360\) 1.00000 0.0527046
\(361\) −16.5821 −0.872742
\(362\) −20.5308 −1.07907
\(363\) 9.70709 0.509490
\(364\) 0 0
\(365\) −14.8334 −0.776415
\(366\) −10.9487 −0.572297
\(367\) 4.76032 0.248486 0.124243 0.992252i \(-0.460350\pi\)
0.124243 + 0.992252i \(0.460350\pi\)
\(368\) 5.40581 0.281797
\(369\) 8.26875 0.430454
\(370\) −0.554958 −0.0288509
\(371\) 17.6069 0.914103
\(372\) 0.246980 0.0128053
\(373\) −20.2064 −1.04625 −0.523124 0.852256i \(-0.675234\pi\)
−0.523124 + 0.852256i \(0.675234\pi\)
\(374\) 3.06100 0.158280
\(375\) −1.00000 −0.0516398
\(376\) −6.70171 −0.345614
\(377\) 0 0
\(378\) −3.04892 −0.156819
\(379\) 24.4088 1.25380 0.626898 0.779101i \(-0.284324\pi\)
0.626898 + 0.779101i \(0.284324\pi\)
\(380\) 1.55496 0.0797677
\(381\) −6.22521 −0.318927
\(382\) −6.66487 −0.341005
\(383\) 32.0237 1.63633 0.818167 0.574981i \(-0.194990\pi\)
0.818167 + 0.574981i \(0.194990\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 3.46681 0.176685
\(386\) 16.6558 0.847757
\(387\) 2.03923 0.103660
\(388\) 1.93900 0.0984379
\(389\) 21.1879 1.07427 0.537135 0.843497i \(-0.319507\pi\)
0.537135 + 0.843497i \(0.319507\pi\)
\(390\) 0 0
\(391\) 14.5526 0.735955
\(392\) 2.29590 0.115960
\(393\) −3.34481 −0.168724
\(394\) 19.4494 0.979844
\(395\) 5.33513 0.268439
\(396\) 1.13706 0.0571396
\(397\) 10.9608 0.550105 0.275053 0.961429i \(-0.411305\pi\)
0.275053 + 0.961429i \(0.411305\pi\)
\(398\) −21.6558 −1.08551
\(399\) −4.74094 −0.237344
\(400\) 1.00000 0.0500000
\(401\) −29.2301 −1.45968 −0.729841 0.683617i \(-0.760406\pi\)
−0.729841 + 0.683617i \(0.760406\pi\)
\(402\) 2.47219 0.123302
\(403\) 0 0
\(404\) −14.4940 −0.721101
\(405\) 1.00000 0.0496904
\(406\) −18.7778 −0.931925
\(407\) −0.631023 −0.0312786
\(408\) −2.69202 −0.133275
\(409\) 36.3163 1.79573 0.897864 0.440274i \(-0.145119\pi\)
0.897864 + 0.440274i \(0.145119\pi\)
\(410\) 8.26875 0.408364
\(411\) 9.53079 0.470119
\(412\) −16.1836 −0.797308
\(413\) 4.17390 0.205384
\(414\) 5.40581 0.265681
\(415\) −5.45473 −0.267762
\(416\) 0 0
\(417\) −18.7017 −0.915827
\(418\) 1.76809 0.0864799
\(419\) 11.0382 0.539250 0.269625 0.962965i \(-0.413100\pi\)
0.269625 + 0.962965i \(0.413100\pi\)
\(420\) −3.04892 −0.148772
\(421\) 4.58642 0.223528 0.111764 0.993735i \(-0.464350\pi\)
0.111764 + 0.993735i \(0.464350\pi\)
\(422\) 12.1521 0.591556
\(423\) −6.70171 −0.325848
\(424\) 5.77479 0.280449
\(425\) 2.69202 0.130582
\(426\) 8.57002 0.415219
\(427\) 33.3817 1.61545
\(428\) 4.92931 0.238267
\(429\) 0 0
\(430\) 2.03923 0.0983403
\(431\) 19.3642 0.932740 0.466370 0.884590i \(-0.345561\pi\)
0.466370 + 0.884590i \(0.345561\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.41119 0.211988 0.105994 0.994367i \(-0.466198\pi\)
0.105994 + 0.994367i \(0.466198\pi\)
\(434\) −0.753020 −0.0361461
\(435\) 6.15883 0.295293
\(436\) 0.801938 0.0384059
\(437\) 8.40581 0.402105
\(438\) 14.8334 0.708767
\(439\) 39.5260 1.88647 0.943237 0.332121i \(-0.107764\pi\)
0.943237 + 0.332121i \(0.107764\pi\)
\(440\) 1.13706 0.0542074
\(441\) 2.29590 0.109328
\(442\) 0 0
\(443\) −7.16182 −0.340268 −0.170134 0.985421i \(-0.554420\pi\)
−0.170134 + 0.985421i \(0.554420\pi\)
\(444\) 0.554958 0.0263371
\(445\) 16.8877 0.800553
\(446\) 1.11960 0.0530148
\(447\) −16.4209 −0.776681
\(448\) 3.04892 0.144048
\(449\) −4.46011 −0.210485 −0.105243 0.994447i \(-0.533562\pi\)
−0.105243 + 0.994447i \(0.533562\pi\)
\(450\) 1.00000 0.0471405
\(451\) 9.40209 0.442727
\(452\) −11.5211 −0.541907
\(453\) 9.26636 0.435371
\(454\) 4.06638 0.190844
\(455\) 0 0
\(456\) −1.55496 −0.0728176
\(457\) −8.04593 −0.376373 −0.188186 0.982133i \(-0.560261\pi\)
−0.188186 + 0.982133i \(0.560261\pi\)
\(458\) −16.5187 −0.771869
\(459\) −2.69202 −0.125653
\(460\) 5.40581 0.252047
\(461\) 4.00538 0.186549 0.0932745 0.995640i \(-0.470267\pi\)
0.0932745 + 0.995640i \(0.470267\pi\)
\(462\) −3.46681 −0.161291
\(463\) −37.6142 −1.74808 −0.874039 0.485856i \(-0.838508\pi\)
−0.874039 + 0.485856i \(0.838508\pi\)
\(464\) −6.15883 −0.285917
\(465\) 0.246980 0.0114534
\(466\) −6.16421 −0.285552
\(467\) −32.8079 −1.51817 −0.759084 0.650992i \(-0.774353\pi\)
−0.759084 + 0.650992i \(0.774353\pi\)
\(468\) 0 0
\(469\) −7.53750 −0.348049
\(470\) −6.70171 −0.309127
\(471\) −17.0761 −0.786824
\(472\) 1.36898 0.0630123
\(473\) 2.31873 0.106615
\(474\) −5.33513 −0.245050
\(475\) 1.55496 0.0713464
\(476\) 8.20775 0.376202
\(477\) 5.77479 0.264410
\(478\) 18.1793 0.831501
\(479\) −9.88530 −0.451671 −0.225835 0.974165i \(-0.572511\pi\)
−0.225835 + 0.974165i \(0.572511\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −9.21744 −0.419843
\(483\) −16.4819 −0.749951
\(484\) −9.70709 −0.441231
\(485\) 1.93900 0.0880455
\(486\) −1.00000 −0.0453609
\(487\) 20.2610 0.918113 0.459056 0.888407i \(-0.348188\pi\)
0.459056 + 0.888407i \(0.348188\pi\)
\(488\) 10.9487 0.495624
\(489\) 9.61894 0.434984
\(490\) 2.29590 0.103718
\(491\) −22.4940 −1.01514 −0.507569 0.861611i \(-0.669456\pi\)
−0.507569 + 0.861611i \(0.669456\pi\)
\(492\) −8.26875 −0.372784
\(493\) −16.5797 −0.746713
\(494\) 0 0
\(495\) 1.13706 0.0511072
\(496\) −0.246980 −0.0110897
\(497\) −26.1293 −1.17206
\(498\) 5.45473 0.244432
\(499\) −15.2728 −0.683704 −0.341852 0.939754i \(-0.611054\pi\)
−0.341852 + 0.939754i \(0.611054\pi\)
\(500\) 1.00000 0.0447214
\(501\) −21.8509 −0.976224
\(502\) 17.2054 0.767913
\(503\) −2.95838 −0.131908 −0.0659538 0.997823i \(-0.521009\pi\)
−0.0659538 + 0.997823i \(0.521009\pi\)
\(504\) 3.04892 0.135810
\(505\) −14.4940 −0.644973
\(506\) 6.14675 0.273256
\(507\) 0 0
\(508\) 6.22521 0.276199
\(509\) −4.34614 −0.192639 −0.0963197 0.995350i \(-0.530707\pi\)
−0.0963197 + 0.995350i \(0.530707\pi\)
\(510\) −2.69202 −0.119205
\(511\) −45.2258 −2.00067
\(512\) 1.00000 0.0441942
\(513\) −1.55496 −0.0686531
\(514\) −19.5157 −0.860802
\(515\) −16.1836 −0.713134
\(516\) −2.03923 −0.0897720
\(517\) −7.62027 −0.335139
\(518\) −1.69202 −0.0743432
\(519\) −16.6853 −0.732404
\(520\) 0 0
\(521\) 9.13228 0.400092 0.200046 0.979786i \(-0.435891\pi\)
0.200046 + 0.979786i \(0.435891\pi\)
\(522\) −6.15883 −0.269565
\(523\) 10.2828 0.449633 0.224817 0.974401i \(-0.427822\pi\)
0.224817 + 0.974401i \(0.427822\pi\)
\(524\) 3.34481 0.146119
\(525\) −3.04892 −0.133066
\(526\) 12.8834 0.561742
\(527\) −0.664874 −0.0289624
\(528\) −1.13706 −0.0494843
\(529\) 6.22282 0.270557
\(530\) 5.77479 0.250841
\(531\) 1.36898 0.0594086
\(532\) 4.74094 0.205546
\(533\) 0 0
\(534\) −16.8877 −0.730802
\(535\) 4.92931 0.213113
\(536\) −2.47219 −0.106782
\(537\) 3.94438 0.170212
\(538\) −30.9138 −1.33279
\(539\) 2.61058 0.112446
\(540\) −1.00000 −0.0430331
\(541\) 1.47517 0.0634226 0.0317113 0.999497i \(-0.489904\pi\)
0.0317113 + 0.999497i \(0.489904\pi\)
\(542\) 7.24027 0.310996
\(543\) 20.5308 0.881061
\(544\) 2.69202 0.115419
\(545\) 0.801938 0.0343512
\(546\) 0 0
\(547\) −4.82908 −0.206477 −0.103238 0.994657i \(-0.532920\pi\)
−0.103238 + 0.994657i \(0.532920\pi\)
\(548\) −9.53079 −0.407135
\(549\) 10.9487 0.467279
\(550\) 1.13706 0.0484845
\(551\) −9.57673 −0.407982
\(552\) −5.40581 −0.230087
\(553\) 16.2664 0.691716
\(554\) 12.2446 0.520223
\(555\) 0.554958 0.0235567
\(556\) 18.7017 0.793129
\(557\) 3.26145 0.138192 0.0690961 0.997610i \(-0.477988\pi\)
0.0690961 + 0.997610i \(0.477988\pi\)
\(558\) −0.246980 −0.0104555
\(559\) 0 0
\(560\) 3.04892 0.128840
\(561\) −3.06100 −0.129235
\(562\) −2.74632 −0.115846
\(563\) 18.1551 0.765147 0.382573 0.923925i \(-0.375038\pi\)
0.382573 + 0.923925i \(0.375038\pi\)
\(564\) 6.70171 0.282193
\(565\) −11.5211 −0.484697
\(566\) 14.9312 0.627606
\(567\) 3.04892 0.128042
\(568\) −8.57002 −0.359590
\(569\) −39.2669 −1.64616 −0.823078 0.567928i \(-0.807745\pi\)
−0.823078 + 0.567928i \(0.807745\pi\)
\(570\) −1.55496 −0.0651300
\(571\) −36.8649 −1.54275 −0.771373 0.636383i \(-0.780430\pi\)
−0.771373 + 0.636383i \(0.780430\pi\)
\(572\) 0 0
\(573\) 6.66487 0.278429
\(574\) 25.2107 1.05228
\(575\) 5.40581 0.225438
\(576\) 1.00000 0.0416667
\(577\) −20.3043 −0.845277 −0.422639 0.906298i \(-0.638896\pi\)
−0.422639 + 0.906298i \(0.638896\pi\)
\(578\) −9.75302 −0.405672
\(579\) −16.6558 −0.692190
\(580\) −6.15883 −0.255732
\(581\) −16.6310 −0.689971
\(582\) −1.93900 −0.0803742
\(583\) 6.56630 0.271948
\(584\) −14.8334 −0.613810
\(585\) 0 0
\(586\) 18.5810 0.767576
\(587\) 24.5472 1.01317 0.506585 0.862190i \(-0.330907\pi\)
0.506585 + 0.862190i \(0.330907\pi\)
\(588\) −2.29590 −0.0946812
\(589\) −0.384043 −0.0158242
\(590\) 1.36898 0.0563599
\(591\) −19.4494 −0.800040
\(592\) −0.554958 −0.0228086
\(593\) −24.0844 −0.989029 −0.494514 0.869169i \(-0.664654\pi\)
−0.494514 + 0.869169i \(0.664654\pi\)
\(594\) −1.13706 −0.0466543
\(595\) 8.20775 0.336485
\(596\) 16.4209 0.672625
\(597\) 21.6558 0.886312
\(598\) 0 0
\(599\) 1.84415 0.0753499 0.0376750 0.999290i \(-0.488005\pi\)
0.0376750 + 0.999290i \(0.488005\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −15.6770 −0.639476 −0.319738 0.947506i \(-0.603595\pi\)
−0.319738 + 0.947506i \(0.603595\pi\)
\(602\) 6.21744 0.253404
\(603\) −2.47219 −0.100675
\(604\) −9.26636 −0.377043
\(605\) −9.70709 −0.394649
\(606\) 14.4940 0.588777
\(607\) −29.6866 −1.20494 −0.602472 0.798140i \(-0.705817\pi\)
−0.602472 + 0.798140i \(0.705817\pi\)
\(608\) 1.55496 0.0630619
\(609\) 18.7778 0.760914
\(610\) 10.9487 0.443299
\(611\) 0 0
\(612\) 2.69202 0.108819
\(613\) −42.5163 −1.71722 −0.858609 0.512631i \(-0.828671\pi\)
−0.858609 + 0.512631i \(0.828671\pi\)
\(614\) 32.4359 1.30901
\(615\) −8.26875 −0.333428
\(616\) 3.46681 0.139682
\(617\) 27.3666 1.10174 0.550869 0.834592i \(-0.314297\pi\)
0.550869 + 0.834592i \(0.314297\pi\)
\(618\) 16.1836 0.650999
\(619\) 27.9420 1.12308 0.561542 0.827449i \(-0.310208\pi\)
0.561542 + 0.827449i \(0.310208\pi\)
\(620\) −0.246980 −0.00991894
\(621\) −5.40581 −0.216928
\(622\) −21.2771 −0.853134
\(623\) 51.4892 2.06287
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −18.7060 −0.747643
\(627\) −1.76809 −0.0706105
\(628\) 17.0761 0.681409
\(629\) −1.49396 −0.0595681
\(630\) 3.04892 0.121472
\(631\) 7.43594 0.296020 0.148010 0.988986i \(-0.452713\pi\)
0.148010 + 0.988986i \(0.452713\pi\)
\(632\) 5.33513 0.212220
\(633\) −12.1521 −0.483004
\(634\) 17.4776 0.694123
\(635\) 6.22521 0.247040
\(636\) −5.77479 −0.228985
\(637\) 0 0
\(638\) −7.00298 −0.277251
\(639\) −8.57002 −0.339025
\(640\) 1.00000 0.0395285
\(641\) −24.9444 −0.985244 −0.492622 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(642\) −4.92931 −0.194544
\(643\) 30.1661 1.18964 0.594818 0.803860i \(-0.297224\pi\)
0.594818 + 0.803860i \(0.297224\pi\)
\(644\) 16.4819 0.649477
\(645\) −2.03923 −0.0802946
\(646\) 4.18598 0.164695
\(647\) −44.4053 −1.74575 −0.872877 0.487940i \(-0.837748\pi\)
−0.872877 + 0.487940i \(0.837748\pi\)
\(648\) 1.00000 0.0392837
\(649\) 1.55661 0.0611024
\(650\) 0 0
\(651\) 0.753020 0.0295132
\(652\) −9.61894 −0.376707
\(653\) −29.2573 −1.14493 −0.572463 0.819931i \(-0.694012\pi\)
−0.572463 + 0.819931i \(0.694012\pi\)
\(654\) −0.801938 −0.0313582
\(655\) 3.34481 0.130693
\(656\) 8.26875 0.322840
\(657\) −14.8334 −0.578706
\(658\) −20.4330 −0.796560
\(659\) −11.4808 −0.447229 −0.223614 0.974678i \(-0.571786\pi\)
−0.223614 + 0.974678i \(0.571786\pi\)
\(660\) −1.13706 −0.0442601
\(661\) 26.1129 1.01567 0.507837 0.861453i \(-0.330445\pi\)
0.507837 + 0.861453i \(0.330445\pi\)
\(662\) −23.1890 −0.901265
\(663\) 0 0
\(664\) −5.45473 −0.211685
\(665\) 4.74094 0.183846
\(666\) −0.554958 −0.0215042
\(667\) −33.2935 −1.28913
\(668\) 21.8509 0.845435
\(669\) −1.11960 −0.0432864
\(670\) −2.47219 −0.0955090
\(671\) 12.4494 0.480602
\(672\) −3.04892 −0.117615
\(673\) −12.4190 −0.478716 −0.239358 0.970931i \(-0.576937\pi\)
−0.239358 + 0.970931i \(0.576937\pi\)
\(674\) −26.1400 −1.00688
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 12.7646 0.490585 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(678\) 11.5211 0.442465
\(679\) 5.91185 0.226876
\(680\) 2.69202 0.103234
\(681\) −4.06638 −0.155824
\(682\) −0.280831 −0.0107536
\(683\) 31.6142 1.20968 0.604841 0.796346i \(-0.293237\pi\)
0.604841 + 0.796346i \(0.293237\pi\)
\(684\) 1.55496 0.0594553
\(685\) −9.53079 −0.364153
\(686\) −14.3424 −0.547596
\(687\) 16.5187 0.630228
\(688\) 2.03923 0.0777449
\(689\) 0 0
\(690\) −5.40581 −0.205796
\(691\) 2.06962 0.0787322 0.0393661 0.999225i \(-0.487466\pi\)
0.0393661 + 0.999225i \(0.487466\pi\)
\(692\) 16.6853 0.634281
\(693\) 3.46681 0.131693
\(694\) 15.2959 0.580624
\(695\) 18.7017 0.709396
\(696\) 6.15883 0.233450
\(697\) 22.2597 0.843144
\(698\) −19.2107 −0.727137
\(699\) 6.16421 0.233152
\(700\) 3.04892 0.115238
\(701\) −23.0030 −0.868811 −0.434405 0.900717i \(-0.643041\pi\)
−0.434405 + 0.900717i \(0.643041\pi\)
\(702\) 0 0
\(703\) −0.862937 −0.0325463
\(704\) 1.13706 0.0428547
\(705\) 6.70171 0.252401
\(706\) −5.35450 −0.201519
\(707\) −44.1909 −1.66197
\(708\) −1.36898 −0.0514493
\(709\) −32.4373 −1.21821 −0.609104 0.793091i \(-0.708471\pi\)
−0.609104 + 0.793091i \(0.708471\pi\)
\(710\) −8.57002 −0.321627
\(711\) 5.33513 0.200083
\(712\) 16.8877 0.632893
\(713\) −1.33513 −0.0500008
\(714\) −8.20775 −0.307167
\(715\) 0 0
\(716\) −3.94438 −0.147408
\(717\) −18.1793 −0.678917
\(718\) 8.49934 0.317192
\(719\) −13.7845 −0.514074 −0.257037 0.966402i \(-0.582746\pi\)
−0.257037 + 0.966402i \(0.582746\pi\)
\(720\) 1.00000 0.0372678
\(721\) −49.3424 −1.83761
\(722\) −16.5821 −0.617122
\(723\) 9.21744 0.342800
\(724\) −20.5308 −0.763021
\(725\) −6.15883 −0.228733
\(726\) 9.70709 0.360264
\(727\) 4.64848 0.172403 0.0862013 0.996278i \(-0.472527\pi\)
0.0862013 + 0.996278i \(0.472527\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.8334 −0.549009
\(731\) 5.48965 0.203042
\(732\) −10.9487 −0.404675
\(733\) −27.4717 −1.01469 −0.507345 0.861743i \(-0.669373\pi\)
−0.507345 + 0.861743i \(0.669373\pi\)
\(734\) 4.76032 0.175706
\(735\) −2.29590 −0.0846854
\(736\) 5.40581 0.199261
\(737\) −2.81104 −0.103546
\(738\) 8.26875 0.304377
\(739\) −48.0170 −1.76633 −0.883167 0.469059i \(-0.844593\pi\)
−0.883167 + 0.469059i \(0.844593\pi\)
\(740\) −0.554958 −0.0204007
\(741\) 0 0
\(742\) 17.6069 0.646368
\(743\) −23.6517 −0.867698 −0.433849 0.900986i \(-0.642845\pi\)
−0.433849 + 0.900986i \(0.642845\pi\)
\(744\) 0.246980 0.00905471
\(745\) 16.4209 0.601614
\(746\) −20.2064 −0.739810
\(747\) −5.45473 −0.199578
\(748\) 3.06100 0.111921
\(749\) 15.0291 0.549150
\(750\) −1.00000 −0.0365148
\(751\) −39.5133 −1.44186 −0.720931 0.693007i \(-0.756286\pi\)
−0.720931 + 0.693007i \(0.756286\pi\)
\(752\) −6.70171 −0.244386
\(753\) −17.2054 −0.626998
\(754\) 0 0
\(755\) −9.26636 −0.337237
\(756\) −3.04892 −0.110888
\(757\) −11.2239 −0.407939 −0.203969 0.978977i \(-0.565384\pi\)
−0.203969 + 0.978977i \(0.565384\pi\)
\(758\) 24.4088 0.886567
\(759\) −6.14675 −0.223113
\(760\) 1.55496 0.0564043
\(761\) −5.80864 −0.210563 −0.105282 0.994442i \(-0.533574\pi\)
−0.105282 + 0.994442i \(0.533574\pi\)
\(762\) −6.22521 −0.225516
\(763\) 2.44504 0.0885165
\(764\) −6.66487 −0.241127
\(765\) 2.69202 0.0973302
\(766\) 32.0237 1.15706
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −31.8316 −1.14788 −0.573938 0.818899i \(-0.694585\pi\)
−0.573938 + 0.818899i \(0.694585\pi\)
\(770\) 3.46681 0.124935
\(771\) 19.5157 0.702842
\(772\) 16.6558 0.599455
\(773\) 39.7434 1.42947 0.714736 0.699394i \(-0.246547\pi\)
0.714736 + 0.699394i \(0.246547\pi\)
\(774\) 2.03923 0.0732986
\(775\) −0.246980 −0.00887177
\(776\) 1.93900 0.0696061
\(777\) 1.69202 0.0607009
\(778\) 21.1879 0.759623
\(779\) 12.8576 0.460670
\(780\) 0 0
\(781\) −9.74466 −0.348691
\(782\) 14.5526 0.520399
\(783\) 6.15883 0.220099
\(784\) 2.29590 0.0819963
\(785\) 17.0761 0.609471
\(786\) −3.34481 −0.119306
\(787\) 19.3183 0.688622 0.344311 0.938856i \(-0.388113\pi\)
0.344311 + 0.938856i \(0.388113\pi\)
\(788\) 19.4494 0.692855
\(789\) −12.8834 −0.458660
\(790\) 5.33513 0.189815
\(791\) −35.1269 −1.24897
\(792\) 1.13706 0.0404038
\(793\) 0 0
\(794\) 10.9608 0.388983
\(795\) −5.77479 −0.204811
\(796\) −21.6558 −0.767569
\(797\) 29.4946 1.04475 0.522375 0.852716i \(-0.325046\pi\)
0.522375 + 0.852716i \(0.325046\pi\)
\(798\) −4.74094 −0.167827
\(799\) −18.0411 −0.638250
\(800\) 1.00000 0.0353553
\(801\) 16.8877 0.596697
\(802\) −29.2301 −1.03215
\(803\) −16.8665 −0.595206
\(804\) 2.47219 0.0871874
\(805\) 16.4819 0.580910
\(806\) 0 0
\(807\) 30.9138 1.08822
\(808\) −14.4940 −0.509896
\(809\) 14.0901 0.495380 0.247690 0.968839i \(-0.420329\pi\)
0.247690 + 0.968839i \(0.420329\pi\)
\(810\) 1.00000 0.0351364
\(811\) −45.5060 −1.59793 −0.798967 0.601375i \(-0.794620\pi\)
−0.798967 + 0.601375i \(0.794620\pi\)
\(812\) −18.7778 −0.658971
\(813\) −7.24027 −0.253928
\(814\) −0.631023 −0.0221173
\(815\) −9.61894 −0.336937
\(816\) −2.69202 −0.0942396
\(817\) 3.17092 0.110936
\(818\) 36.3163 1.26977
\(819\) 0 0
\(820\) 8.26875 0.288757
\(821\) −14.6853 −0.512521 −0.256261 0.966608i \(-0.582491\pi\)
−0.256261 + 0.966608i \(0.582491\pi\)
\(822\) 9.53079 0.332425
\(823\) −32.0551 −1.11737 −0.558686 0.829379i \(-0.688694\pi\)
−0.558686 + 0.829379i \(0.688694\pi\)
\(824\) −16.1836 −0.563782
\(825\) −1.13706 −0.0395875
\(826\) 4.17390 0.145229
\(827\) 8.09352 0.281439 0.140720 0.990049i \(-0.455058\pi\)
0.140720 + 0.990049i \(0.455058\pi\)
\(828\) 5.40581 0.187865
\(829\) −23.8708 −0.829068 −0.414534 0.910034i \(-0.636055\pi\)
−0.414534 + 0.910034i \(0.636055\pi\)
\(830\) −5.45473 −0.189336
\(831\) −12.2446 −0.424760
\(832\) 0 0
\(833\) 6.18060 0.214145
\(834\) −18.7017 −0.647587
\(835\) 21.8509 0.756180
\(836\) 1.76809 0.0611505
\(837\) 0.246980 0.00853686
\(838\) 11.0382 0.381307
\(839\) −17.5569 −0.606131 −0.303065 0.952970i \(-0.598010\pi\)
−0.303065 + 0.952970i \(0.598010\pi\)
\(840\) −3.04892 −0.105198
\(841\) 8.93123 0.307973
\(842\) 4.58642 0.158058
\(843\) 2.74632 0.0945881
\(844\) 12.1521 0.418294
\(845\) 0 0
\(846\) −6.70171 −0.230410
\(847\) −29.5961 −1.01693
\(848\) 5.77479 0.198307
\(849\) −14.9312 −0.512439
\(850\) 2.69202 0.0923356
\(851\) −3.00000 −0.102839
\(852\) 8.57002 0.293604
\(853\) −40.2892 −1.37948 −0.689738 0.724059i \(-0.742274\pi\)
−0.689738 + 0.724059i \(0.742274\pi\)
\(854\) 33.3817 1.14230
\(855\) 1.55496 0.0531784
\(856\) 4.92931 0.168480
\(857\) 44.5913 1.52321 0.761605 0.648041i \(-0.224412\pi\)
0.761605 + 0.648041i \(0.224412\pi\)
\(858\) 0 0
\(859\) −40.6467 −1.38685 −0.693423 0.720530i \(-0.743898\pi\)
−0.693423 + 0.720530i \(0.743898\pi\)
\(860\) 2.03923 0.0695371
\(861\) −25.2107 −0.859180
\(862\) 19.3642 0.659547
\(863\) −31.9748 −1.08843 −0.544217 0.838945i \(-0.683173\pi\)
−0.544217 + 0.838945i \(0.683173\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 16.6853 0.567318
\(866\) 4.41119 0.149898
\(867\) 9.75302 0.331230
\(868\) −0.753020 −0.0255592
\(869\) 6.06638 0.205788
\(870\) 6.15883 0.208804
\(871\) 0 0
\(872\) 0.801938 0.0271570
\(873\) 1.93900 0.0656252
\(874\) 8.40581 0.284331
\(875\) 3.04892 0.103072
\(876\) 14.8334 0.501174
\(877\) −39.2476 −1.32530 −0.662648 0.748931i \(-0.730567\pi\)
−0.662648 + 0.748931i \(0.730567\pi\)
\(878\) 39.5260 1.33394
\(879\) −18.5810 −0.626723
\(880\) 1.13706 0.0383304
\(881\) −24.3153 −0.819202 −0.409601 0.912265i \(-0.634332\pi\)
−0.409601 + 0.912265i \(0.634332\pi\)
\(882\) 2.29590 0.0773069
\(883\) 51.8756 1.74575 0.872877 0.487941i \(-0.162252\pi\)
0.872877 + 0.487941i \(0.162252\pi\)
\(884\) 0 0
\(885\) −1.36898 −0.0460177
\(886\) −7.16182 −0.240606
\(887\) −13.4383 −0.451215 −0.225608 0.974218i \(-0.572437\pi\)
−0.225608 + 0.974218i \(0.572437\pi\)
\(888\) 0.554958 0.0186232
\(889\) 18.9801 0.636574
\(890\) 16.8877 0.566077
\(891\) 1.13706 0.0380931
\(892\) 1.11960 0.0374871
\(893\) −10.4209 −0.348721
\(894\) −16.4209 −0.549196
\(895\) −3.94438 −0.131846
\(896\) 3.04892 0.101857
\(897\) 0 0
\(898\) −4.46011 −0.148836
\(899\) 1.52111 0.0507317
\(900\) 1.00000 0.0333333
\(901\) 15.5459 0.517908
\(902\) 9.40209 0.313055
\(903\) −6.21744 −0.206903
\(904\) −11.5211 −0.383186
\(905\) −20.5308 −0.682467
\(906\) 9.26636 0.307854
\(907\) −6.58881 −0.218778 −0.109389 0.993999i \(-0.534889\pi\)
−0.109389 + 0.993999i \(0.534889\pi\)
\(908\) 4.06638 0.134947
\(909\) −14.4940 −0.480734
\(910\) 0 0
\(911\) 51.2646 1.69847 0.849235 0.528015i \(-0.177063\pi\)
0.849235 + 0.528015i \(0.177063\pi\)
\(912\) −1.55496 −0.0514898
\(913\) −6.20237 −0.205269
\(914\) −8.04593 −0.266136
\(915\) −10.9487 −0.361953
\(916\) −16.5187 −0.545794
\(917\) 10.1981 0.336770
\(918\) −2.69202 −0.0888499
\(919\) 50.7066 1.67266 0.836328 0.548229i \(-0.184698\pi\)
0.836328 + 0.548229i \(0.184698\pi\)
\(920\) 5.40581 0.178224
\(921\) −32.4359 −1.06880
\(922\) 4.00538 0.131910
\(923\) 0 0
\(924\) −3.46681 −0.114050
\(925\) −0.554958 −0.0182469
\(926\) −37.6142 −1.23608
\(927\) −16.1836 −0.531539
\(928\) −6.15883 −0.202174
\(929\) −18.2903 −0.600084 −0.300042 0.953926i \(-0.597001\pi\)
−0.300042 + 0.953926i \(0.597001\pi\)
\(930\) 0.246980 0.00809878
\(931\) 3.57002 0.117003
\(932\) −6.16421 −0.201915
\(933\) 21.2771 0.696581
\(934\) −32.8079 −1.07351
\(935\) 3.06100 0.100105
\(936\) 0 0
\(937\) −59.6679 −1.94926 −0.974632 0.223814i \(-0.928149\pi\)
−0.974632 + 0.223814i \(0.928149\pi\)
\(938\) −7.53750 −0.246108
\(939\) 18.7060 0.610448
\(940\) −6.70171 −0.218586
\(941\) −36.2513 −1.18176 −0.590879 0.806760i \(-0.701219\pi\)
−0.590879 + 0.806760i \(0.701219\pi\)
\(942\) −17.0761 −0.556368
\(943\) 44.6993 1.45561
\(944\) 1.36898 0.0445564
\(945\) −3.04892 −0.0991813
\(946\) 2.31873 0.0753885
\(947\) 7.02475 0.228274 0.114137 0.993465i \(-0.463590\pi\)
0.114137 + 0.993465i \(0.463590\pi\)
\(948\) −5.33513 −0.173277
\(949\) 0 0
\(950\) 1.55496 0.0504495
\(951\) −17.4776 −0.566749
\(952\) 8.20775 0.266015
\(953\) 12.5730 0.407280 0.203640 0.979046i \(-0.434723\pi\)
0.203640 + 0.979046i \(0.434723\pi\)
\(954\) 5.77479 0.186966
\(955\) −6.66487 −0.215670
\(956\) 18.1793 0.587960
\(957\) 7.00298 0.226374
\(958\) −9.88530 −0.319379
\(959\) −29.0586 −0.938351
\(960\) −1.00000 −0.0322749
\(961\) −30.9390 −0.998032
\(962\) 0 0
\(963\) 4.92931 0.158845
\(964\) −9.21744 −0.296874
\(965\) 16.6558 0.536168
\(966\) −16.4819 −0.530296
\(967\) −23.1371 −0.744038 −0.372019 0.928225i \(-0.621334\pi\)
−0.372019 + 0.928225i \(0.621334\pi\)
\(968\) −9.70709 −0.311998
\(969\) −4.18598 −0.134473
\(970\) 1.93900 0.0622576
\(971\) −26.5394 −0.851690 −0.425845 0.904796i \(-0.640023\pi\)
−0.425845 + 0.904796i \(0.640023\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 57.0200 1.82798
\(974\) 20.2610 0.649204
\(975\) 0 0
\(976\) 10.9487 0.350459
\(977\) 19.5670 0.626005 0.313003 0.949752i \(-0.398665\pi\)
0.313003 + 0.949752i \(0.398665\pi\)
\(978\) 9.61894 0.307580
\(979\) 19.2024 0.613711
\(980\) 2.29590 0.0733397
\(981\) 0.801938 0.0256039
\(982\) −22.4940 −0.717811
\(983\) −8.82610 −0.281509 −0.140754 0.990045i \(-0.544953\pi\)
−0.140754 + 0.990045i \(0.544953\pi\)
\(984\) −8.26875 −0.263598
\(985\) 19.4494 0.619708
\(986\) −16.5797 −0.528006
\(987\) 20.4330 0.650388
\(988\) 0 0
\(989\) 11.0237 0.350533
\(990\) 1.13706 0.0361382
\(991\) 0.601483 0.0191067 0.00955336 0.999954i \(-0.496959\pi\)
0.00955336 + 0.999954i \(0.496959\pi\)
\(992\) −0.246980 −0.00784161
\(993\) 23.1890 0.735880
\(994\) −26.1293 −0.828771
\(995\) −21.6558 −0.686534
\(996\) 5.45473 0.172840
\(997\) 13.4101 0.424703 0.212351 0.977193i \(-0.431888\pi\)
0.212351 + 0.977193i \(0.431888\pi\)
\(998\) −15.2728 −0.483452
\(999\) 0.554958 0.0175581
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 5070.2.a.bu.1.3 yes 3
13.5 odd 4 5070.2.b.t.1351.3 6
13.8 odd 4 5070.2.b.t.1351.4 6
13.12 even 2 5070.2.a.bj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.1 3 13.12 even 2
5070.2.a.bu.1.3 yes 3 1.1 even 1 trivial
5070.2.b.t.1351.3 6 13.5 odd 4
5070.2.b.t.1351.4 6 13.8 odd 4