Properties

Label 8470.2.a.cv.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.0970464\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.0970464 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.99058 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.0970464 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.0970464 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.99058 q^{9} -1.00000 q^{10} +0.0970464 q^{12} +4.32646 q^{13} -1.00000 q^{14} +0.0970464 q^{15} +1.00000 q^{16} -2.12295 q^{17} +2.99058 q^{18} -5.51022 q^{19} +1.00000 q^{20} +0.0970464 q^{21} +0.922019 q^{23} -0.0970464 q^{24} +1.00000 q^{25} -4.32646 q^{26} -0.581365 q^{27} +1.00000 q^{28} +8.02820 q^{29} -0.0970464 q^{30} +1.21552 q^{31} -1.00000 q^{32} +2.12295 q^{34} +1.00000 q^{35} -2.99058 q^{36} -5.01306 q^{37} +5.51022 q^{38} +0.419868 q^{39} -1.00000 q^{40} +1.92806 q^{41} -0.0970464 q^{42} -3.75423 q^{43} -2.99058 q^{45} -0.922019 q^{46} -0.865052 q^{47} +0.0970464 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.206025 q^{51} +4.32646 q^{52} -0.729158 q^{53} +0.581365 q^{54} -1.00000 q^{56} -0.534747 q^{57} -8.02820 q^{58} -0.178208 q^{59} +0.0970464 q^{60} +13.0127 q^{61} -1.21552 q^{62} -2.99058 q^{63} +1.00000 q^{64} +4.32646 q^{65} -3.55383 q^{67} -2.12295 q^{68} +0.0894787 q^{69} -1.00000 q^{70} +5.96038 q^{71} +2.99058 q^{72} +14.8292 q^{73} +5.01306 q^{74} +0.0970464 q^{75} -5.51022 q^{76} -0.419868 q^{78} -13.1971 q^{79} +1.00000 q^{80} +8.91533 q^{81} -1.92806 q^{82} -2.63016 q^{83} +0.0970464 q^{84} -2.12295 q^{85} +3.75423 q^{86} +0.779108 q^{87} -18.4678 q^{89} +2.99058 q^{90} +4.32646 q^{91} +0.922019 q^{92} +0.117962 q^{93} +0.865052 q^{94} -5.51022 q^{95} -0.0970464 q^{96} +0.0855815 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9} - 6 q^{10} - q^{12} + 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 21 q^{17} + q^{18} - 3 q^{19} + 6 q^{20} - q^{21} - 10 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} + q^{30} - 4 q^{31} - 6 q^{32} - 21 q^{34} + 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 26 q^{39} - 6 q^{40} + 7 q^{41} + q^{42} + 19 q^{43} - q^{45} + 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 4 q^{51} + 6 q^{52} - 16 q^{53} + 4 q^{54} - 6 q^{56} + 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} - 8 q^{61} + 4 q^{62} - q^{63} + 6 q^{64} + 6 q^{65} - 27 q^{67} + 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} + q^{72} + 13 q^{73} + 2 q^{74} - q^{75} - 3 q^{76} - 26 q^{78} + 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} + 51 q^{83} - q^{84} + 21 q^{85} - 19 q^{86} + 8 q^{87} + q^{89} + q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} - 10 q^{94} - 3 q^{95} + q^{96} + 7 q^{97} - 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.0970464 0.0560298 0.0280149 0.999608i \(-0.491081\pi\)
0.0280149 + 0.999608i \(0.491081\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.0970464 −0.0396190
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.99058 −0.996861
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.0970464 0.0280149
\(13\) 4.32646 1.19994 0.599972 0.800021i \(-0.295178\pi\)
0.599972 + 0.800021i \(0.295178\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.0970464 0.0250573
\(16\) 1.00000 0.250000
\(17\) −2.12295 −0.514891 −0.257445 0.966293i \(-0.582881\pi\)
−0.257445 + 0.966293i \(0.582881\pi\)
\(18\) 2.99058 0.704887
\(19\) −5.51022 −1.26413 −0.632066 0.774915i \(-0.717793\pi\)
−0.632066 + 0.774915i \(0.717793\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.0970464 0.0211773
\(22\) 0 0
\(23\) 0.922019 0.192254 0.0961271 0.995369i \(-0.469354\pi\)
0.0961271 + 0.995369i \(0.469354\pi\)
\(24\) −0.0970464 −0.0198095
\(25\) 1.00000 0.200000
\(26\) −4.32646 −0.848489
\(27\) −0.581365 −0.111884
\(28\) 1.00000 0.188982
\(29\) 8.02820 1.49080 0.745399 0.666618i \(-0.232259\pi\)
0.745399 + 0.666618i \(0.232259\pi\)
\(30\) −0.0970464 −0.0177182
\(31\) 1.21552 0.218314 0.109157 0.994024i \(-0.465185\pi\)
0.109157 + 0.994024i \(0.465185\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.12295 0.364083
\(35\) 1.00000 0.169031
\(36\) −2.99058 −0.498430
\(37\) −5.01306 −0.824142 −0.412071 0.911152i \(-0.635194\pi\)
−0.412071 + 0.911152i \(0.635194\pi\)
\(38\) 5.51022 0.893876
\(39\) 0.419868 0.0672326
\(40\) −1.00000 −0.158114
\(41\) 1.92806 0.301113 0.150556 0.988601i \(-0.451893\pi\)
0.150556 + 0.988601i \(0.451893\pi\)
\(42\) −0.0970464 −0.0149746
\(43\) −3.75423 −0.572514 −0.286257 0.958153i \(-0.592411\pi\)
−0.286257 + 0.958153i \(0.592411\pi\)
\(44\) 0 0
\(45\) −2.99058 −0.445810
\(46\) −0.922019 −0.135944
\(47\) −0.865052 −0.126181 −0.0630904 0.998008i \(-0.520096\pi\)
−0.0630904 + 0.998008i \(0.520096\pi\)
\(48\) 0.0970464 0.0140074
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.206025 −0.0288492
\(52\) 4.32646 0.599972
\(53\) −0.729158 −0.100158 −0.0500788 0.998745i \(-0.515947\pi\)
−0.0500788 + 0.998745i \(0.515947\pi\)
\(54\) 0.581365 0.0791137
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −0.534747 −0.0708290
\(58\) −8.02820 −1.05415
\(59\) −0.178208 −0.0232008 −0.0116004 0.999933i \(-0.503693\pi\)
−0.0116004 + 0.999933i \(0.503693\pi\)
\(60\) 0.0970464 0.0125286
\(61\) 13.0127 1.66610 0.833051 0.553195i \(-0.186592\pi\)
0.833051 + 0.553195i \(0.186592\pi\)
\(62\) −1.21552 −0.154372
\(63\) −2.99058 −0.376778
\(64\) 1.00000 0.125000
\(65\) 4.32646 0.536631
\(66\) 0 0
\(67\) −3.55383 −0.434169 −0.217085 0.976153i \(-0.569655\pi\)
−0.217085 + 0.976153i \(0.569655\pi\)
\(68\) −2.12295 −0.257445
\(69\) 0.0894787 0.0107720
\(70\) −1.00000 −0.119523
\(71\) 5.96038 0.707367 0.353683 0.935365i \(-0.384929\pi\)
0.353683 + 0.935365i \(0.384929\pi\)
\(72\) 2.99058 0.352443
\(73\) 14.8292 1.73562 0.867811 0.496895i \(-0.165526\pi\)
0.867811 + 0.496895i \(0.165526\pi\)
\(74\) 5.01306 0.582756
\(75\) 0.0970464 0.0112060
\(76\) −5.51022 −0.632066
\(77\) 0 0
\(78\) −0.419868 −0.0475406
\(79\) −13.1971 −1.48479 −0.742394 0.669964i \(-0.766310\pi\)
−0.742394 + 0.669964i \(0.766310\pi\)
\(80\) 1.00000 0.111803
\(81\) 8.91533 0.990592
\(82\) −1.92806 −0.212919
\(83\) −2.63016 −0.288698 −0.144349 0.989527i \(-0.546109\pi\)
−0.144349 + 0.989527i \(0.546109\pi\)
\(84\) 0.0970464 0.0105886
\(85\) −2.12295 −0.230266
\(86\) 3.75423 0.404829
\(87\) 0.779108 0.0835291
\(88\) 0 0
\(89\) −18.4678 −1.95758 −0.978791 0.204861i \(-0.934326\pi\)
−0.978791 + 0.204861i \(0.934326\pi\)
\(90\) 2.99058 0.315235
\(91\) 4.32646 0.453536
\(92\) 0.922019 0.0961271
\(93\) 0.117962 0.0122321
\(94\) 0.865052 0.0892233
\(95\) −5.51022 −0.565337
\(96\) −0.0970464 −0.00990476
\(97\) 0.0855815 0.00868949 0.00434474 0.999991i \(-0.498617\pi\)
0.00434474 + 0.999991i \(0.498617\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.57523 0.753763 0.376882 0.926261i \(-0.376996\pi\)
0.376882 + 0.926261i \(0.376996\pi\)
\(102\) 0.206025 0.0203995
\(103\) −3.76181 −0.370663 −0.185331 0.982676i \(-0.559336\pi\)
−0.185331 + 0.982676i \(0.559336\pi\)
\(104\) −4.32646 −0.424244
\(105\) 0.0970464 0.00947076
\(106\) 0.729158 0.0708221
\(107\) 13.3165 1.28735 0.643677 0.765297i \(-0.277408\pi\)
0.643677 + 0.765297i \(0.277408\pi\)
\(108\) −0.581365 −0.0559418
\(109\) 8.04563 0.770631 0.385316 0.922785i \(-0.374093\pi\)
0.385316 + 0.922785i \(0.374093\pi\)
\(110\) 0 0
\(111\) −0.486499 −0.0461765
\(112\) 1.00000 0.0944911
\(113\) 9.17493 0.863104 0.431552 0.902088i \(-0.357966\pi\)
0.431552 + 0.902088i \(0.357966\pi\)
\(114\) 0.534747 0.0500837
\(115\) 0.922019 0.0859787
\(116\) 8.02820 0.745399
\(117\) −12.9386 −1.19618
\(118\) 0.178208 0.0164054
\(119\) −2.12295 −0.194610
\(120\) −0.0970464 −0.00885909
\(121\) 0 0
\(122\) −13.0127 −1.17811
\(123\) 0.187112 0.0168713
\(124\) 1.21552 0.109157
\(125\) 1.00000 0.0894427
\(126\) 2.99058 0.266422
\(127\) 8.63204 0.765970 0.382985 0.923755i \(-0.374896\pi\)
0.382985 + 0.923755i \(0.374896\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.364335 −0.0320779
\(130\) −4.32646 −0.379456
\(131\) −5.01748 −0.438379 −0.219189 0.975682i \(-0.570341\pi\)
−0.219189 + 0.975682i \(0.570341\pi\)
\(132\) 0 0
\(133\) −5.51022 −0.477797
\(134\) 3.55383 0.307004
\(135\) −0.581365 −0.0500359
\(136\) 2.12295 0.182041
\(137\) −9.38739 −0.802018 −0.401009 0.916074i \(-0.631340\pi\)
−0.401009 + 0.916074i \(0.631340\pi\)
\(138\) −0.0894787 −0.00761693
\(139\) 6.96159 0.590474 0.295237 0.955424i \(-0.404601\pi\)
0.295237 + 0.955424i \(0.404601\pi\)
\(140\) 1.00000 0.0845154
\(141\) −0.0839502 −0.00706988
\(142\) −5.96038 −0.500184
\(143\) 0 0
\(144\) −2.99058 −0.249215
\(145\) 8.02820 0.666706
\(146\) −14.8292 −1.22727
\(147\) 0.0970464 0.00800426
\(148\) −5.01306 −0.412071
\(149\) 19.8855 1.62908 0.814542 0.580105i \(-0.196989\pi\)
0.814542 + 0.580105i \(0.196989\pi\)
\(150\) −0.0970464 −0.00792381
\(151\) 6.91156 0.562455 0.281227 0.959641i \(-0.409258\pi\)
0.281227 + 0.959641i \(0.409258\pi\)
\(152\) 5.51022 0.446938
\(153\) 6.34886 0.513275
\(154\) 0 0
\(155\) 1.21552 0.0976332
\(156\) 0.419868 0.0336163
\(157\) 9.00126 0.718378 0.359189 0.933265i \(-0.383053\pi\)
0.359189 + 0.933265i \(0.383053\pi\)
\(158\) 13.1971 1.04990
\(159\) −0.0707622 −0.00561181
\(160\) −1.00000 −0.0790569
\(161\) 0.922019 0.0726653
\(162\) −8.91533 −0.700454
\(163\) −7.07107 −0.553849 −0.276924 0.960892i \(-0.589315\pi\)
−0.276924 + 0.960892i \(0.589315\pi\)
\(164\) 1.92806 0.150556
\(165\) 0 0
\(166\) 2.63016 0.204140
\(167\) 19.7421 1.52769 0.763846 0.645398i \(-0.223309\pi\)
0.763846 + 0.645398i \(0.223309\pi\)
\(168\) −0.0970464 −0.00748730
\(169\) 5.71826 0.439866
\(170\) 2.12295 0.162823
\(171\) 16.4788 1.26016
\(172\) −3.75423 −0.286257
\(173\) −13.9142 −1.05787 −0.528937 0.848661i \(-0.677409\pi\)
−0.528937 + 0.848661i \(0.677409\pi\)
\(174\) −0.779108 −0.0590640
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −0.0172945 −0.00129993
\(178\) 18.4678 1.38422
\(179\) −3.61007 −0.269829 −0.134915 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(180\) −2.99058 −0.222905
\(181\) −1.95017 −0.144955 −0.0724776 0.997370i \(-0.523091\pi\)
−0.0724776 + 0.997370i \(0.523091\pi\)
\(182\) −4.32646 −0.320699
\(183\) 1.26283 0.0933514
\(184\) −0.922019 −0.0679722
\(185\) −5.01306 −0.368567
\(186\) −0.117962 −0.00864941
\(187\) 0 0
\(188\) −0.865052 −0.0630904
\(189\) −0.581365 −0.0422881
\(190\) 5.51022 0.399753
\(191\) 12.7463 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(192\) 0.0970464 0.00700372
\(193\) 14.5920 1.05035 0.525176 0.850994i \(-0.323999\pi\)
0.525176 + 0.850994i \(0.323999\pi\)
\(194\) −0.0855815 −0.00614439
\(195\) 0.419868 0.0300673
\(196\) 1.00000 0.0714286
\(197\) −24.4598 −1.74269 −0.871346 0.490669i \(-0.836752\pi\)
−0.871346 + 0.490669i \(0.836752\pi\)
\(198\) 0 0
\(199\) −10.4998 −0.744315 −0.372157 0.928170i \(-0.621382\pi\)
−0.372157 + 0.928170i \(0.621382\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −0.344887 −0.0243264
\(202\) −7.57523 −0.532991
\(203\) 8.02820 0.563469
\(204\) −0.206025 −0.0144246
\(205\) 1.92806 0.134662
\(206\) 3.76181 0.262098
\(207\) −2.75737 −0.191651
\(208\) 4.32646 0.299986
\(209\) 0 0
\(210\) −0.0970464 −0.00669684
\(211\) −14.9790 −1.03120 −0.515600 0.856830i \(-0.672431\pi\)
−0.515600 + 0.856830i \(0.672431\pi\)
\(212\) −0.729158 −0.0500788
\(213\) 0.578434 0.0396336
\(214\) −13.3165 −0.910296
\(215\) −3.75423 −0.256036
\(216\) 0.581365 0.0395569
\(217\) 1.21552 0.0825151
\(218\) −8.04563 −0.544919
\(219\) 1.43912 0.0972465
\(220\) 0 0
\(221\) −9.18486 −0.617840
\(222\) 0.486499 0.0326517
\(223\) −6.98647 −0.467848 −0.233924 0.972255i \(-0.575157\pi\)
−0.233924 + 0.972255i \(0.575157\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.99058 −0.199372
\(226\) −9.17493 −0.610307
\(227\) 23.6112 1.56713 0.783565 0.621309i \(-0.213399\pi\)
0.783565 + 0.621309i \(0.213399\pi\)
\(228\) −0.534747 −0.0354145
\(229\) 21.0574 1.39151 0.695755 0.718279i \(-0.255070\pi\)
0.695755 + 0.718279i \(0.255070\pi\)
\(230\) −0.922019 −0.0607961
\(231\) 0 0
\(232\) −8.02820 −0.527077
\(233\) 29.7941 1.95187 0.975937 0.218052i \(-0.0699702\pi\)
0.975937 + 0.218052i \(0.0699702\pi\)
\(234\) 12.9386 0.845825
\(235\) −0.865052 −0.0564298
\(236\) −0.178208 −0.0116004
\(237\) −1.28073 −0.0831923
\(238\) 2.12295 0.137610
\(239\) 18.4084 1.19074 0.595370 0.803452i \(-0.297005\pi\)
0.595370 + 0.803452i \(0.297005\pi\)
\(240\) 0.0970464 0.00626432
\(241\) 2.15635 0.138903 0.0694513 0.997585i \(-0.477875\pi\)
0.0694513 + 0.997585i \(0.477875\pi\)
\(242\) 0 0
\(243\) 2.60929 0.167386
\(244\) 13.0127 0.833051
\(245\) 1.00000 0.0638877
\(246\) −0.187112 −0.0119298
\(247\) −23.8398 −1.51689
\(248\) −1.21552 −0.0771858
\(249\) −0.255248 −0.0161757
\(250\) −1.00000 −0.0632456
\(251\) 19.6172 1.23823 0.619114 0.785301i \(-0.287492\pi\)
0.619114 + 0.785301i \(0.287492\pi\)
\(252\) −2.99058 −0.188389
\(253\) 0 0
\(254\) −8.63204 −0.541622
\(255\) −0.206025 −0.0129018
\(256\) 1.00000 0.0625000
\(257\) −5.06003 −0.315636 −0.157818 0.987468i \(-0.550446\pi\)
−0.157818 + 0.987468i \(0.550446\pi\)
\(258\) 0.364335 0.0226825
\(259\) −5.01306 −0.311496
\(260\) 4.32646 0.268316
\(261\) −24.0090 −1.48612
\(262\) 5.01748 0.309981
\(263\) 16.0394 0.989031 0.494516 0.869169i \(-0.335345\pi\)
0.494516 + 0.869169i \(0.335345\pi\)
\(264\) 0 0
\(265\) −0.729158 −0.0447918
\(266\) 5.51022 0.337853
\(267\) −1.79223 −0.109683
\(268\) −3.55383 −0.217085
\(269\) 7.45344 0.454444 0.227222 0.973843i \(-0.427036\pi\)
0.227222 + 0.973843i \(0.427036\pi\)
\(270\) 0.581365 0.0353807
\(271\) −19.3298 −1.17420 −0.587102 0.809513i \(-0.699731\pi\)
−0.587102 + 0.809513i \(0.699731\pi\)
\(272\) −2.12295 −0.128723
\(273\) 0.419868 0.0254115
\(274\) 9.38739 0.567113
\(275\) 0 0
\(276\) 0.0894787 0.00538598
\(277\) −21.9563 −1.31923 −0.659614 0.751605i \(-0.729280\pi\)
−0.659614 + 0.751605i \(0.729280\pi\)
\(278\) −6.96159 −0.417528
\(279\) −3.63512 −0.217629
\(280\) −1.00000 −0.0597614
\(281\) −0.172162 −0.0102703 −0.00513515 0.999987i \(-0.501635\pi\)
−0.00513515 + 0.999987i \(0.501635\pi\)
\(282\) 0.0839502 0.00499916
\(283\) 14.5989 0.867817 0.433909 0.900957i \(-0.357134\pi\)
0.433909 + 0.900957i \(0.357134\pi\)
\(284\) 5.96038 0.353683
\(285\) −0.534747 −0.0316757
\(286\) 0 0
\(287\) 1.92806 0.113810
\(288\) 2.99058 0.176222
\(289\) −12.4931 −0.734887
\(290\) −8.02820 −0.471432
\(291\) 0.00830538 0.000486870 0
\(292\) 14.8292 0.867811
\(293\) 4.99491 0.291806 0.145903 0.989299i \(-0.453391\pi\)
0.145903 + 0.989299i \(0.453391\pi\)
\(294\) −0.0970464 −0.00565986
\(295\) −0.178208 −0.0103757
\(296\) 5.01306 0.291378
\(297\) 0 0
\(298\) −19.8855 −1.15194
\(299\) 3.98908 0.230694
\(300\) 0.0970464 0.00560298
\(301\) −3.75423 −0.216390
\(302\) −6.91156 −0.397716
\(303\) 0.735149 0.0422332
\(304\) −5.51022 −0.316033
\(305\) 13.0127 0.745104
\(306\) −6.34886 −0.362940
\(307\) −12.1296 −0.692270 −0.346135 0.938185i \(-0.612506\pi\)
−0.346135 + 0.938185i \(0.612506\pi\)
\(308\) 0 0
\(309\) −0.365071 −0.0207681
\(310\) −1.21552 −0.0690371
\(311\) −5.79026 −0.328335 −0.164168 0.986432i \(-0.552494\pi\)
−0.164168 + 0.986432i \(0.552494\pi\)
\(312\) −0.419868 −0.0237703
\(313\) −8.21454 −0.464313 −0.232157 0.972678i \(-0.574578\pi\)
−0.232157 + 0.972678i \(0.574578\pi\)
\(314\) −9.00126 −0.507970
\(315\) −2.99058 −0.168500
\(316\) −13.1971 −0.742394
\(317\) −9.85272 −0.553384 −0.276692 0.960959i \(-0.589238\pi\)
−0.276692 + 0.960959i \(0.589238\pi\)
\(318\) 0.0707622 0.00396815
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 1.29232 0.0721301
\(322\) −0.922019 −0.0513821
\(323\) 11.6979 0.650890
\(324\) 8.91533 0.495296
\(325\) 4.32646 0.239989
\(326\) 7.07107 0.391630
\(327\) 0.780799 0.0431783
\(328\) −1.92806 −0.106459
\(329\) −0.865052 −0.0476919
\(330\) 0 0
\(331\) −5.71671 −0.314219 −0.157109 0.987581i \(-0.550218\pi\)
−0.157109 + 0.987581i \(0.550218\pi\)
\(332\) −2.63016 −0.144349
\(333\) 14.9920 0.821554
\(334\) −19.7421 −1.08024
\(335\) −3.55383 −0.194166
\(336\) 0.0970464 0.00529432
\(337\) 29.1366 1.58717 0.793585 0.608459i \(-0.208212\pi\)
0.793585 + 0.608459i \(0.208212\pi\)
\(338\) −5.71826 −0.311032
\(339\) 0.890394 0.0483596
\(340\) −2.12295 −0.115133
\(341\) 0 0
\(342\) −16.4788 −0.891070
\(343\) 1.00000 0.0539949
\(344\) 3.75423 0.202414
\(345\) 0.0894787 0.00481737
\(346\) 13.9142 0.748030
\(347\) −9.51730 −0.510915 −0.255458 0.966820i \(-0.582226\pi\)
−0.255458 + 0.966820i \(0.582226\pi\)
\(348\) 0.779108 0.0417646
\(349\) 8.24510 0.441350 0.220675 0.975347i \(-0.429174\pi\)
0.220675 + 0.975347i \(0.429174\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.51525 −0.134254
\(352\) 0 0
\(353\) 29.6347 1.57730 0.788649 0.614843i \(-0.210781\pi\)
0.788649 + 0.614843i \(0.210781\pi\)
\(354\) 0.0172945 0.000919192 0
\(355\) 5.96038 0.316344
\(356\) −18.4678 −0.978791
\(357\) −0.206025 −0.0109040
\(358\) 3.61007 0.190798
\(359\) 24.7795 1.30781 0.653907 0.756575i \(-0.273129\pi\)
0.653907 + 0.756575i \(0.273129\pi\)
\(360\) 2.99058 0.157618
\(361\) 11.3625 0.598028
\(362\) 1.95017 0.102499
\(363\) 0 0
\(364\) 4.32646 0.226768
\(365\) 14.8292 0.776194
\(366\) −1.26283 −0.0660094
\(367\) 12.8613 0.671352 0.335676 0.941977i \(-0.391035\pi\)
0.335676 + 0.941977i \(0.391035\pi\)
\(368\) 0.922019 0.0480636
\(369\) −5.76603 −0.300167
\(370\) 5.01306 0.260616
\(371\) −0.729158 −0.0378560
\(372\) 0.117962 0.00611606
\(373\) −22.7708 −1.17903 −0.589514 0.807758i \(-0.700681\pi\)
−0.589514 + 0.807758i \(0.700681\pi\)
\(374\) 0 0
\(375\) 0.0970464 0.00501146
\(376\) 0.865052 0.0446117
\(377\) 34.7337 1.78888
\(378\) 0.581365 0.0299022
\(379\) 11.0712 0.568691 0.284346 0.958722i \(-0.408224\pi\)
0.284346 + 0.958722i \(0.408224\pi\)
\(380\) −5.51022 −0.282668
\(381\) 0.837709 0.0429171
\(382\) −12.7463 −0.652156
\(383\) 28.6974 1.46637 0.733184 0.680030i \(-0.238033\pi\)
0.733184 + 0.680030i \(0.238033\pi\)
\(384\) −0.0970464 −0.00495238
\(385\) 0 0
\(386\) −14.5920 −0.742711
\(387\) 11.2273 0.570717
\(388\) 0.0855815 0.00434474
\(389\) 1.68387 0.0853754 0.0426877 0.999088i \(-0.486408\pi\)
0.0426877 + 0.999088i \(0.486408\pi\)
\(390\) −0.419868 −0.0212608
\(391\) −1.95740 −0.0989900
\(392\) −1.00000 −0.0505076
\(393\) −0.486928 −0.0245623
\(394\) 24.4598 1.23227
\(395\) −13.1971 −0.664017
\(396\) 0 0
\(397\) 16.9399 0.850189 0.425095 0.905149i \(-0.360241\pi\)
0.425095 + 0.905149i \(0.360241\pi\)
\(398\) 10.4998 0.526310
\(399\) −0.534747 −0.0267709
\(400\) 1.00000 0.0500000
\(401\) −15.0165 −0.749889 −0.374944 0.927047i \(-0.622338\pi\)
−0.374944 + 0.927047i \(0.622338\pi\)
\(402\) 0.344887 0.0172014
\(403\) 5.25891 0.261965
\(404\) 7.57523 0.376882
\(405\) 8.91533 0.443006
\(406\) −8.02820 −0.398433
\(407\) 0 0
\(408\) 0.206025 0.0101997
\(409\) −33.7143 −1.66707 −0.833533 0.552469i \(-0.813686\pi\)
−0.833533 + 0.552469i \(0.813686\pi\)
\(410\) −1.92806 −0.0952202
\(411\) −0.911012 −0.0449369
\(412\) −3.76181 −0.185331
\(413\) −0.178208 −0.00876907
\(414\) 2.75737 0.135518
\(415\) −2.63016 −0.129109
\(416\) −4.32646 −0.212122
\(417\) 0.675598 0.0330842
\(418\) 0 0
\(419\) −17.7859 −0.868900 −0.434450 0.900696i \(-0.643057\pi\)
−0.434450 + 0.900696i \(0.643057\pi\)
\(420\) 0.0970464 0.00473538
\(421\) 18.2534 0.889619 0.444809 0.895625i \(-0.353271\pi\)
0.444809 + 0.895625i \(0.353271\pi\)
\(422\) 14.9790 0.729168
\(423\) 2.58701 0.125785
\(424\) 0.729158 0.0354111
\(425\) −2.12295 −0.102978
\(426\) −0.578434 −0.0280252
\(427\) 13.0127 0.629728
\(428\) 13.3165 0.643677
\(429\) 0 0
\(430\) 3.75423 0.181045
\(431\) −27.5547 −1.32726 −0.663632 0.748059i \(-0.730986\pi\)
−0.663632 + 0.748059i \(0.730986\pi\)
\(432\) −0.581365 −0.0279709
\(433\) −3.18070 −0.152855 −0.0764273 0.997075i \(-0.524351\pi\)
−0.0764273 + 0.997075i \(0.524351\pi\)
\(434\) −1.21552 −0.0583470
\(435\) 0.779108 0.0373554
\(436\) 8.04563 0.385316
\(437\) −5.08053 −0.243035
\(438\) −1.43912 −0.0687637
\(439\) −39.0123 −1.86196 −0.930978 0.365076i \(-0.881043\pi\)
−0.930978 + 0.365076i \(0.881043\pi\)
\(440\) 0 0
\(441\) −2.99058 −0.142409
\(442\) 9.18486 0.436879
\(443\) 5.00748 0.237912 0.118956 0.992900i \(-0.462045\pi\)
0.118956 + 0.992900i \(0.462045\pi\)
\(444\) −0.486499 −0.0230882
\(445\) −18.4678 −0.875457
\(446\) 6.98647 0.330819
\(447\) 1.92982 0.0912772
\(448\) 1.00000 0.0472456
\(449\) 12.5770 0.593546 0.296773 0.954948i \(-0.404090\pi\)
0.296773 + 0.954948i \(0.404090\pi\)
\(450\) 2.99058 0.140977
\(451\) 0 0
\(452\) 9.17493 0.431552
\(453\) 0.670742 0.0315142
\(454\) −23.6112 −1.10813
\(455\) 4.32646 0.202828
\(456\) 0.534747 0.0250418
\(457\) 36.1549 1.69125 0.845627 0.533775i \(-0.179227\pi\)
0.845627 + 0.533775i \(0.179227\pi\)
\(458\) −21.0574 −0.983946
\(459\) 1.23421 0.0576079
\(460\) 0.922019 0.0429894
\(461\) −26.0248 −1.21209 −0.606047 0.795429i \(-0.707246\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(462\) 0 0
\(463\) 2.55526 0.118753 0.0593766 0.998236i \(-0.481089\pi\)
0.0593766 + 0.998236i \(0.481089\pi\)
\(464\) 8.02820 0.372700
\(465\) 0.117962 0.00547037
\(466\) −29.7941 −1.38018
\(467\) 14.4176 0.667169 0.333585 0.942720i \(-0.391742\pi\)
0.333585 + 0.942720i \(0.391742\pi\)
\(468\) −12.9386 −0.598089
\(469\) −3.55383 −0.164101
\(470\) 0.865052 0.0399019
\(471\) 0.873540 0.0402506
\(472\) 0.178208 0.00820271
\(473\) 0 0
\(474\) 1.28073 0.0588259
\(475\) −5.51022 −0.252826
\(476\) −2.12295 −0.0973052
\(477\) 2.18061 0.0998432
\(478\) −18.4084 −0.841981
\(479\) −32.6066 −1.48984 −0.744918 0.667156i \(-0.767511\pi\)
−0.744918 + 0.667156i \(0.767511\pi\)
\(480\) −0.0970464 −0.00442954
\(481\) −21.6888 −0.988924
\(482\) −2.15635 −0.0982190
\(483\) 0.0894787 0.00407142
\(484\) 0 0
\(485\) 0.0855815 0.00388606
\(486\) −2.60929 −0.118360
\(487\) 16.9041 0.765997 0.382999 0.923749i \(-0.374891\pi\)
0.382999 + 0.923749i \(0.374891\pi\)
\(488\) −13.0127 −0.589056
\(489\) −0.686222 −0.0310320
\(490\) −1.00000 −0.0451754
\(491\) 38.7664 1.74950 0.874752 0.484571i \(-0.161024\pi\)
0.874752 + 0.484571i \(0.161024\pi\)
\(492\) 0.187112 0.00843564
\(493\) −17.0435 −0.767599
\(494\) 23.8398 1.07260
\(495\) 0 0
\(496\) 1.21552 0.0545786
\(497\) 5.96038 0.267360
\(498\) 0.255248 0.0114379
\(499\) −37.6575 −1.68578 −0.842890 0.538086i \(-0.819148\pi\)
−0.842890 + 0.538086i \(0.819148\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.91590 0.0855963
\(502\) −19.6172 −0.875559
\(503\) −28.4963 −1.27059 −0.635294 0.772270i \(-0.719121\pi\)
−0.635294 + 0.772270i \(0.719121\pi\)
\(504\) 2.99058 0.133211
\(505\) 7.57523 0.337093
\(506\) 0 0
\(507\) 0.554937 0.0246456
\(508\) 8.63204 0.382985
\(509\) 21.2554 0.942130 0.471065 0.882099i \(-0.343870\pi\)
0.471065 + 0.882099i \(0.343870\pi\)
\(510\) 0.206025 0.00912293
\(511\) 14.8292 0.656003
\(512\) −1.00000 −0.0441942
\(513\) 3.20345 0.141436
\(514\) 5.06003 0.223189
\(515\) −3.76181 −0.165765
\(516\) −0.364335 −0.0160389
\(517\) 0 0
\(518\) 5.01306 0.220261
\(519\) −1.35032 −0.0592724
\(520\) −4.32646 −0.189728
\(521\) 9.47858 0.415264 0.207632 0.978207i \(-0.433424\pi\)
0.207632 + 0.978207i \(0.433424\pi\)
\(522\) 24.0090 1.05084
\(523\) 21.3021 0.931475 0.465737 0.884923i \(-0.345789\pi\)
0.465737 + 0.884923i \(0.345789\pi\)
\(524\) −5.01748 −0.219189
\(525\) 0.0970464 0.00423545
\(526\) −16.0394 −0.699351
\(527\) −2.58050 −0.112408
\(528\) 0 0
\(529\) −22.1499 −0.963038
\(530\) 0.729158 0.0316726
\(531\) 0.532947 0.0231279
\(532\) −5.51022 −0.238898
\(533\) 8.34169 0.361319
\(534\) 1.79223 0.0775575
\(535\) 13.3165 0.575722
\(536\) 3.55383 0.153502
\(537\) −0.350344 −0.0151185
\(538\) −7.45344 −0.321341
\(539\) 0 0
\(540\) −0.581365 −0.0250180
\(541\) 43.2284 1.85853 0.929267 0.369408i \(-0.120440\pi\)
0.929267 + 0.369408i \(0.120440\pi\)
\(542\) 19.3298 0.830287
\(543\) −0.189257 −0.00812181
\(544\) 2.12295 0.0910207
\(545\) 8.04563 0.344637
\(546\) −0.419868 −0.0179687
\(547\) 22.1833 0.948491 0.474245 0.880393i \(-0.342721\pi\)
0.474245 + 0.880393i \(0.342721\pi\)
\(548\) −9.38739 −0.401009
\(549\) −38.9155 −1.66087
\(550\) 0 0
\(551\) −44.2372 −1.88457
\(552\) −0.0894787 −0.00380847
\(553\) −13.1971 −0.561197
\(554\) 21.9563 0.932835
\(555\) −0.486499 −0.0206507
\(556\) 6.96159 0.295237
\(557\) −3.41806 −0.144828 −0.0724140 0.997375i \(-0.523070\pi\)
−0.0724140 + 0.997375i \(0.523070\pi\)
\(558\) 3.63512 0.153887
\(559\) −16.2425 −0.686985
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 0.172162 0.00726220
\(563\) 16.7614 0.706411 0.353205 0.935546i \(-0.385092\pi\)
0.353205 + 0.935546i \(0.385092\pi\)
\(564\) −0.0839502 −0.00353494
\(565\) 9.17493 0.385992
\(566\) −14.5989 −0.613640
\(567\) 8.91533 0.374409
\(568\) −5.96038 −0.250092
\(569\) −30.7237 −1.28800 −0.644001 0.765024i \(-0.722727\pi\)
−0.644001 + 0.765024i \(0.722727\pi\)
\(570\) 0.534747 0.0223981
\(571\) −26.7892 −1.12109 −0.560546 0.828123i \(-0.689409\pi\)
−0.560546 + 0.828123i \(0.689409\pi\)
\(572\) 0 0
\(573\) 1.23698 0.0516756
\(574\) −1.92806 −0.0804758
\(575\) 0.922019 0.0384509
\(576\) −2.99058 −0.124608
\(577\) 8.18950 0.340933 0.170467 0.985363i \(-0.445472\pi\)
0.170467 + 0.985363i \(0.445472\pi\)
\(578\) 12.4931 0.519644
\(579\) 1.41610 0.0588510
\(580\) 8.02820 0.333353
\(581\) −2.63016 −0.109117
\(582\) −0.00830538 −0.000344269 0
\(583\) 0 0
\(584\) −14.8292 −0.613635
\(585\) −12.9386 −0.534947
\(586\) −4.99491 −0.206338
\(587\) −19.3228 −0.797537 −0.398768 0.917052i \(-0.630562\pi\)
−0.398768 + 0.917052i \(0.630562\pi\)
\(588\) 0.0970464 0.00400213
\(589\) −6.69780 −0.275978
\(590\) 0.178208 0.00733673
\(591\) −2.37374 −0.0976427
\(592\) −5.01306 −0.206035
\(593\) 46.6962 1.91758 0.958792 0.284109i \(-0.0916979\pi\)
0.958792 + 0.284109i \(0.0916979\pi\)
\(594\) 0 0
\(595\) −2.12295 −0.0870325
\(596\) 19.8855 0.814542
\(597\) −1.01897 −0.0417038
\(598\) −3.98908 −0.163126
\(599\) −20.0821 −0.820533 −0.410267 0.911966i \(-0.634564\pi\)
−0.410267 + 0.911966i \(0.634564\pi\)
\(600\) −0.0970464 −0.00396190
\(601\) 19.0105 0.775456 0.387728 0.921774i \(-0.373260\pi\)
0.387728 + 0.921774i \(0.373260\pi\)
\(602\) 3.75423 0.153011
\(603\) 10.6280 0.432806
\(604\) 6.91156 0.281227
\(605\) 0 0
\(606\) −0.735149 −0.0298634
\(607\) −27.4450 −1.11396 −0.556979 0.830526i \(-0.688040\pi\)
−0.556979 + 0.830526i \(0.688040\pi\)
\(608\) 5.51022 0.223469
\(609\) 0.779108 0.0315711
\(610\) −13.0127 −0.526868
\(611\) −3.74261 −0.151410
\(612\) 6.34886 0.256637
\(613\) −9.50074 −0.383731 −0.191866 0.981421i \(-0.561454\pi\)
−0.191866 + 0.981421i \(0.561454\pi\)
\(614\) 12.1296 0.489509
\(615\) 0.187112 0.00754507
\(616\) 0 0
\(617\) 33.6314 1.35395 0.676974 0.736007i \(-0.263291\pi\)
0.676974 + 0.736007i \(0.263291\pi\)
\(618\) 0.365071 0.0146853
\(619\) 27.2183 1.09400 0.546998 0.837134i \(-0.315770\pi\)
0.546998 + 0.837134i \(0.315770\pi\)
\(620\) 1.21552 0.0488166
\(621\) −0.536029 −0.0215101
\(622\) 5.79026 0.232168
\(623\) −18.4678 −0.739896
\(624\) 0.419868 0.0168082
\(625\) 1.00000 0.0400000
\(626\) 8.21454 0.328319
\(627\) 0 0
\(628\) 9.00126 0.359189
\(629\) 10.6425 0.424343
\(630\) 2.99058 0.119148
\(631\) 28.1766 1.12169 0.560846 0.827920i \(-0.310476\pi\)
0.560846 + 0.827920i \(0.310476\pi\)
\(632\) 13.1971 0.524952
\(633\) −1.45366 −0.0577779
\(634\) 9.85272 0.391301
\(635\) 8.63204 0.342552
\(636\) −0.0707622 −0.00280590
\(637\) 4.32646 0.171421
\(638\) 0 0
\(639\) −17.8250 −0.705146
\(640\) −1.00000 −0.0395285
\(641\) 9.40813 0.371599 0.185799 0.982588i \(-0.440513\pi\)
0.185799 + 0.982588i \(0.440513\pi\)
\(642\) −1.29232 −0.0510037
\(643\) −41.8999 −1.65237 −0.826185 0.563398i \(-0.809494\pi\)
−0.826185 + 0.563398i \(0.809494\pi\)
\(644\) 0.922019 0.0363326
\(645\) −0.364335 −0.0143457
\(646\) −11.6979 −0.460249
\(647\) 17.5282 0.689105 0.344553 0.938767i \(-0.388031\pi\)
0.344553 + 0.938767i \(0.388031\pi\)
\(648\) −8.91533 −0.350227
\(649\) 0 0
\(650\) −4.32646 −0.169698
\(651\) 0.117962 0.00462330
\(652\) −7.07107 −0.276924
\(653\) 47.2120 1.84755 0.923775 0.382937i \(-0.125087\pi\)
0.923775 + 0.382937i \(0.125087\pi\)
\(654\) −0.780799 −0.0305317
\(655\) −5.01748 −0.196049
\(656\) 1.92806 0.0752782
\(657\) −44.3478 −1.73017
\(658\) 0.865052 0.0337232
\(659\) 4.53445 0.176637 0.0883185 0.996092i \(-0.471851\pi\)
0.0883185 + 0.996092i \(0.471851\pi\)
\(660\) 0 0
\(661\) −33.2581 −1.29359 −0.646794 0.762664i \(-0.723891\pi\)
−0.646794 + 0.762664i \(0.723891\pi\)
\(662\) 5.71671 0.222186
\(663\) −0.891358 −0.0346175
\(664\) 2.63016 0.102070
\(665\) −5.51022 −0.213677
\(666\) −14.9920 −0.580927
\(667\) 7.40215 0.286612
\(668\) 19.7421 0.763846
\(669\) −0.678012 −0.0262134
\(670\) 3.55383 0.137296
\(671\) 0 0
\(672\) −0.0970464 −0.00374365
\(673\) −3.32610 −0.128212 −0.0641059 0.997943i \(-0.520420\pi\)
−0.0641059 + 0.997943i \(0.520420\pi\)
\(674\) −29.1366 −1.12230
\(675\) −0.581365 −0.0223767
\(676\) 5.71826 0.219933
\(677\) −8.90346 −0.342188 −0.171094 0.985255i \(-0.554730\pi\)
−0.171094 + 0.985255i \(0.554730\pi\)
\(678\) −0.890394 −0.0341954
\(679\) 0.0855815 0.00328432
\(680\) 2.12295 0.0814114
\(681\) 2.29138 0.0878060
\(682\) 0 0
\(683\) 26.5881 1.01736 0.508682 0.860954i \(-0.330133\pi\)
0.508682 + 0.860954i \(0.330133\pi\)
\(684\) 16.4788 0.630081
\(685\) −9.38739 −0.358674
\(686\) −1.00000 −0.0381802
\(687\) 2.04354 0.0779660
\(688\) −3.75423 −0.143129
\(689\) −3.15468 −0.120184
\(690\) −0.0894787 −0.00340639
\(691\) −28.8459 −1.09735 −0.548674 0.836036i \(-0.684867\pi\)
−0.548674 + 0.836036i \(0.684867\pi\)
\(692\) −13.9142 −0.528937
\(693\) 0 0
\(694\) 9.51730 0.361272
\(695\) 6.96159 0.264068
\(696\) −0.779108 −0.0295320
\(697\) −4.09318 −0.155040
\(698\) −8.24510 −0.312082
\(699\) 2.89141 0.109363
\(700\) 1.00000 0.0377964
\(701\) −10.9136 −0.412203 −0.206101 0.978531i \(-0.566078\pi\)
−0.206101 + 0.978531i \(0.566078\pi\)
\(702\) 2.51525 0.0949320
\(703\) 27.6231 1.04182
\(704\) 0 0
\(705\) −0.0839502 −0.00316175
\(706\) −29.6347 −1.11532
\(707\) 7.57523 0.284896
\(708\) −0.0172945 −0.000649967 0
\(709\) −35.0191 −1.31517 −0.657584 0.753381i \(-0.728422\pi\)
−0.657584 + 0.753381i \(0.728422\pi\)
\(710\) −5.96038 −0.223689
\(711\) 39.4670 1.48013
\(712\) 18.4678 0.692110
\(713\) 1.12074 0.0419719
\(714\) 0.206025 0.00771028
\(715\) 0 0
\(716\) −3.61007 −0.134915
\(717\) 1.78647 0.0667169
\(718\) −24.7795 −0.924765
\(719\) 43.4663 1.62102 0.810510 0.585725i \(-0.199190\pi\)
0.810510 + 0.585725i \(0.199190\pi\)
\(720\) −2.99058 −0.111452
\(721\) −3.76181 −0.140097
\(722\) −11.3625 −0.422870
\(723\) 0.209266 0.00778269
\(724\) −1.95017 −0.0724776
\(725\) 8.02820 0.298160
\(726\) 0 0
\(727\) −52.4475 −1.94517 −0.972585 0.232548i \(-0.925294\pi\)
−0.972585 + 0.232548i \(0.925294\pi\)
\(728\) −4.32646 −0.160349
\(729\) −26.4928 −0.981213
\(730\) −14.8292 −0.548852
\(731\) 7.97004 0.294783
\(732\) 1.26283 0.0466757
\(733\) 26.2363 0.969059 0.484529 0.874775i \(-0.338991\pi\)
0.484529 + 0.874775i \(0.338991\pi\)
\(734\) −12.8613 −0.474718
\(735\) 0.0970464 0.00357961
\(736\) −0.922019 −0.0339861
\(737\) 0 0
\(738\) 5.76603 0.212250
\(739\) 29.8531 1.09816 0.549081 0.835769i \(-0.314978\pi\)
0.549081 + 0.835769i \(0.314978\pi\)
\(740\) −5.01306 −0.184284
\(741\) −2.31356 −0.0849909
\(742\) 0.729158 0.0267682
\(743\) 39.8331 1.46133 0.730667 0.682734i \(-0.239209\pi\)
0.730667 + 0.682734i \(0.239209\pi\)
\(744\) −0.117962 −0.00432471
\(745\) 19.8855 0.728548
\(746\) 22.7708 0.833698
\(747\) 7.86571 0.287791
\(748\) 0 0
\(749\) 13.3165 0.486574
\(750\) −0.0970464 −0.00354363
\(751\) 48.2801 1.76177 0.880884 0.473333i \(-0.156949\pi\)
0.880884 + 0.473333i \(0.156949\pi\)
\(752\) −0.865052 −0.0315452
\(753\) 1.90378 0.0693776
\(754\) −34.7337 −1.26493
\(755\) 6.91156 0.251537
\(756\) −0.581365 −0.0211440
\(757\) −8.61332 −0.313056 −0.156528 0.987673i \(-0.550030\pi\)
−0.156528 + 0.987673i \(0.550030\pi\)
\(758\) −11.0712 −0.402125
\(759\) 0 0
\(760\) 5.51022 0.199877
\(761\) 0.634416 0.0229975 0.0114988 0.999934i \(-0.496340\pi\)
0.0114988 + 0.999934i \(0.496340\pi\)
\(762\) −0.837709 −0.0303470
\(763\) 8.04563 0.291271
\(764\) 12.7463 0.461144
\(765\) 6.34886 0.229543
\(766\) −28.6974 −1.03688
\(767\) −0.771012 −0.0278396
\(768\) 0.0970464 0.00350186
\(769\) 34.4997 1.24409 0.622046 0.782981i \(-0.286302\pi\)
0.622046 + 0.782981i \(0.286302\pi\)
\(770\) 0 0
\(771\) −0.491058 −0.0176850
\(772\) 14.5920 0.525176
\(773\) −32.9878 −1.18649 −0.593244 0.805023i \(-0.702153\pi\)
−0.593244 + 0.805023i \(0.702153\pi\)
\(774\) −11.2273 −0.403558
\(775\) 1.21552 0.0436629
\(776\) −0.0855815 −0.00307220
\(777\) −0.486499 −0.0174531
\(778\) −1.68387 −0.0603695
\(779\) −10.6241 −0.380646
\(780\) 0.419868 0.0150337
\(781\) 0 0
\(782\) 1.95740 0.0699965
\(783\) −4.66731 −0.166796
\(784\) 1.00000 0.0357143
\(785\) 9.00126 0.321269
\(786\) 0.486928 0.0173682
\(787\) 33.6231 1.19853 0.599266 0.800550i \(-0.295459\pi\)
0.599266 + 0.800550i \(0.295459\pi\)
\(788\) −24.4598 −0.871346
\(789\) 1.55657 0.0554152
\(790\) 13.1971 0.469531
\(791\) 9.17493 0.326223
\(792\) 0 0
\(793\) 56.2988 1.99923
\(794\) −16.9399 −0.601175
\(795\) −0.0707622 −0.00250968
\(796\) −10.4998 −0.372157
\(797\) 45.0594 1.59609 0.798043 0.602600i \(-0.205869\pi\)
0.798043 + 0.602600i \(0.205869\pi\)
\(798\) 0.534747 0.0189299
\(799\) 1.83646 0.0649694
\(800\) −1.00000 −0.0353553
\(801\) 55.2294 1.95144
\(802\) 15.0165 0.530252
\(803\) 0 0
\(804\) −0.344887 −0.0121632
\(805\) 0.922019 0.0324969
\(806\) −5.25891 −0.185237
\(807\) 0.723330 0.0254624
\(808\) −7.57523 −0.266496
\(809\) 12.1005 0.425432 0.212716 0.977114i \(-0.431769\pi\)
0.212716 + 0.977114i \(0.431769\pi\)
\(810\) −8.91533 −0.313253
\(811\) 19.6540 0.690147 0.345073 0.938576i \(-0.387854\pi\)
0.345073 + 0.938576i \(0.387854\pi\)
\(812\) 8.02820 0.281735
\(813\) −1.87589 −0.0657904
\(814\) 0 0
\(815\) −7.07107 −0.247689
\(816\) −0.206025 −0.00721231
\(817\) 20.6866 0.723734
\(818\) 33.7143 1.17879
\(819\) −12.9386 −0.452113
\(820\) 1.92806 0.0673309
\(821\) −5.45071 −0.190231 −0.0951156 0.995466i \(-0.530322\pi\)
−0.0951156 + 0.995466i \(0.530322\pi\)
\(822\) 0.911012 0.0317752
\(823\) 34.8782 1.21578 0.607889 0.794022i \(-0.292016\pi\)
0.607889 + 0.794022i \(0.292016\pi\)
\(824\) 3.76181 0.131049
\(825\) 0 0
\(826\) 0.178208 0.00620067
\(827\) 16.7849 0.583670 0.291835 0.956469i \(-0.405734\pi\)
0.291835 + 0.956469i \(0.405734\pi\)
\(828\) −2.75737 −0.0958254
\(829\) −7.98157 −0.277211 −0.138606 0.990348i \(-0.544262\pi\)
−0.138606 + 0.990348i \(0.544262\pi\)
\(830\) 2.63016 0.0912942
\(831\) −2.13078 −0.0739160
\(832\) 4.32646 0.149993
\(833\) −2.12295 −0.0735558
\(834\) −0.675598 −0.0233940
\(835\) 19.7421 0.683205
\(836\) 0 0
\(837\) −0.706662 −0.0244258
\(838\) 17.7859 0.614405
\(839\) −6.47669 −0.223600 −0.111800 0.993731i \(-0.535662\pi\)
−0.111800 + 0.993731i \(0.535662\pi\)
\(840\) −0.0970464 −0.00334842
\(841\) 35.4520 1.22248
\(842\) −18.2534 −0.629055
\(843\) −0.0167077 −0.000575443 0
\(844\) −14.9790 −0.515600
\(845\) 5.71826 0.196714
\(846\) −2.58701 −0.0889432
\(847\) 0 0
\(848\) −0.729158 −0.0250394
\(849\) 1.41678 0.0486236
\(850\) 2.12295 0.0728166
\(851\) −4.62213 −0.158445
\(852\) 0.578434 0.0198168
\(853\) 11.1007 0.380081 0.190040 0.981776i \(-0.439138\pi\)
0.190040 + 0.981776i \(0.439138\pi\)
\(854\) −13.0127 −0.445285
\(855\) 16.4788 0.563562
\(856\) −13.3165 −0.455148
\(857\) 21.2871 0.727154 0.363577 0.931564i \(-0.381555\pi\)
0.363577 + 0.931564i \(0.381555\pi\)
\(858\) 0 0
\(859\) 14.8904 0.508053 0.254027 0.967197i \(-0.418245\pi\)
0.254027 + 0.967197i \(0.418245\pi\)
\(860\) −3.75423 −0.128018
\(861\) 0.187112 0.00637675
\(862\) 27.5547 0.938518
\(863\) −47.6345 −1.62150 −0.810748 0.585395i \(-0.800939\pi\)
−0.810748 + 0.585395i \(0.800939\pi\)
\(864\) 0.581365 0.0197784
\(865\) −13.9142 −0.473095
\(866\) 3.18070 0.108084
\(867\) −1.21241 −0.0411756
\(868\) 1.21552 0.0412576
\(869\) 0 0
\(870\) −0.779108 −0.0264142
\(871\) −15.3755 −0.520979
\(872\) −8.04563 −0.272459
\(873\) −0.255939 −0.00866221
\(874\) 5.08053 0.171851
\(875\) 1.00000 0.0338062
\(876\) 1.43912 0.0486233
\(877\) −36.4415 −1.23054 −0.615270 0.788316i \(-0.710953\pi\)
−0.615270 + 0.788316i \(0.710953\pi\)
\(878\) 39.0123 1.31660
\(879\) 0.484738 0.0163498
\(880\) 0 0
\(881\) 52.4724 1.76784 0.883920 0.467639i \(-0.154895\pi\)
0.883920 + 0.467639i \(0.154895\pi\)
\(882\) 2.99058 0.100698
\(883\) −6.85399 −0.230655 −0.115328 0.993328i \(-0.536792\pi\)
−0.115328 + 0.993328i \(0.536792\pi\)
\(884\) −9.18486 −0.308920
\(885\) −0.0172945 −0.000581348 0
\(886\) −5.00748 −0.168229
\(887\) 23.4947 0.788876 0.394438 0.918922i \(-0.370939\pi\)
0.394438 + 0.918922i \(0.370939\pi\)
\(888\) 0.486499 0.0163258
\(889\) 8.63204 0.289509
\(890\) 18.4678 0.619042
\(891\) 0 0
\(892\) −6.98647 −0.233924
\(893\) 4.76663 0.159509
\(894\) −1.92982 −0.0645427
\(895\) −3.61007 −0.120671
\(896\) −1.00000 −0.0334077
\(897\) 0.387126 0.0129258
\(898\) −12.5770 −0.419700
\(899\) 9.75846 0.325463
\(900\) −2.99058 −0.0996861
\(901\) 1.54797 0.0515702
\(902\) 0 0
\(903\) −0.364335 −0.0121243
\(904\) −9.17493 −0.305153
\(905\) −1.95017 −0.0648260
\(906\) −0.670742 −0.0222839
\(907\) 33.9046 1.12578 0.562892 0.826530i \(-0.309689\pi\)
0.562892 + 0.826530i \(0.309689\pi\)
\(908\) 23.6112 0.783565
\(909\) −22.6543 −0.751397
\(910\) −4.32646 −0.143421
\(911\) −48.8133 −1.61726 −0.808628 0.588320i \(-0.799790\pi\)
−0.808628 + 0.588320i \(0.799790\pi\)
\(912\) −0.534747 −0.0177073
\(913\) 0 0
\(914\) −36.1549 −1.19590
\(915\) 1.26283 0.0417480
\(916\) 21.0574 0.695755
\(917\) −5.01748 −0.165692
\(918\) −1.23421 −0.0407349
\(919\) −16.4073 −0.541228 −0.270614 0.962688i \(-0.587227\pi\)
−0.270614 + 0.962688i \(0.587227\pi\)
\(920\) −0.922019 −0.0303981
\(921\) −1.17713 −0.0387878
\(922\) 26.0248 0.857080
\(923\) 25.7873 0.848801
\(924\) 0 0
\(925\) −5.01306 −0.164828
\(926\) −2.55526 −0.0839711
\(927\) 11.2500 0.369499
\(928\) −8.02820 −0.263539
\(929\) −18.9935 −0.623157 −0.311579 0.950220i \(-0.600858\pi\)
−0.311579 + 0.950220i \(0.600858\pi\)
\(930\) −0.117962 −0.00386813
\(931\) −5.51022 −0.180590
\(932\) 29.7941 0.975937
\(933\) −0.561924 −0.0183966
\(934\) −14.4176 −0.471760
\(935\) 0 0
\(936\) 12.9386 0.422913
\(937\) −5.26363 −0.171955 −0.0859777 0.996297i \(-0.527401\pi\)
−0.0859777 + 0.996297i \(0.527401\pi\)
\(938\) 3.55383 0.116037
\(939\) −0.797192 −0.0260154
\(940\) −0.865052 −0.0282149
\(941\) −36.6208 −1.19380 −0.596901 0.802315i \(-0.703602\pi\)
−0.596901 + 0.802315i \(0.703602\pi\)
\(942\) −0.873540 −0.0284615
\(943\) 1.77771 0.0578902
\(944\) −0.178208 −0.00580019
\(945\) −0.581365 −0.0189118
\(946\) 0 0
\(947\) 39.3320 1.27812 0.639060 0.769157i \(-0.279324\pi\)
0.639060 + 0.769157i \(0.279324\pi\)
\(948\) −1.28073 −0.0415962
\(949\) 64.1578 2.08265
\(950\) 5.51022 0.178775
\(951\) −0.956171 −0.0310060
\(952\) 2.12295 0.0688052
\(953\) 30.1849 0.977784 0.488892 0.872344i \(-0.337401\pi\)
0.488892 + 0.872344i \(0.337401\pi\)
\(954\) −2.18061 −0.0705998
\(955\) 12.7463 0.412460
\(956\) 18.4084 0.595370
\(957\) 0 0
\(958\) 32.6066 1.05347
\(959\) −9.38739 −0.303134
\(960\) 0.0970464 0.00313216
\(961\) −29.5225 −0.952339
\(962\) 21.6888 0.699275
\(963\) −39.8241 −1.28331
\(964\) 2.15635 0.0694513
\(965\) 14.5920 0.469732
\(966\) −0.0894787 −0.00287893
\(967\) 52.3303 1.68283 0.841414 0.540391i \(-0.181724\pi\)
0.841414 + 0.540391i \(0.181724\pi\)
\(968\) 0 0
\(969\) 1.13524 0.0364692
\(970\) −0.0855815 −0.00274786
\(971\) 34.1707 1.09659 0.548294 0.836285i \(-0.315277\pi\)
0.548294 + 0.836285i \(0.315277\pi\)
\(972\) 2.60929 0.0836932
\(973\) 6.96159 0.223178
\(974\) −16.9041 −0.541642
\(975\) 0.419868 0.0134465
\(976\) 13.0127 0.416526
\(977\) −43.2069 −1.38231 −0.691156 0.722706i \(-0.742898\pi\)
−0.691156 + 0.722706i \(0.742898\pi\)
\(978\) 0.686222 0.0219430
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −24.0611 −0.768212
\(982\) −38.7664 −1.23709
\(983\) 43.9672 1.40233 0.701167 0.712997i \(-0.252663\pi\)
0.701167 + 0.712997i \(0.252663\pi\)
\(984\) −0.187112 −0.00596490
\(985\) −24.4598 −0.779356
\(986\) 17.0435 0.542774
\(987\) −0.0839502 −0.00267216
\(988\) −23.8398 −0.758444
\(989\) −3.46147 −0.110068
\(990\) 0 0
\(991\) 11.0881 0.352224 0.176112 0.984370i \(-0.443648\pi\)
0.176112 + 0.984370i \(0.443648\pi\)
\(992\) −1.21552 −0.0385929
\(993\) −0.554786 −0.0176056
\(994\) −5.96038 −0.189052
\(995\) −10.4998 −0.332868
\(996\) −0.255248 −0.00808783
\(997\) −14.3365 −0.454042 −0.227021 0.973890i \(-0.572899\pi\)
−0.227021 + 0.973890i \(0.572899\pi\)
\(998\) 37.6575 1.19203
\(999\) 2.91441 0.0922080
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 8470.2.a.cv.1.4 6
11.7 odd 10 770.2.n.g.71.2 12
11.8 odd 10 770.2.n.g.141.2 yes 12
11.10 odd 2 8470.2.a.db.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.71.2 12 11.7 odd 10
770.2.n.g.141.2 yes 12 11.8 odd 10
8470.2.a.cv.1.4 6 1.1 even 1 trivial
8470.2.a.db.1.4 6 11.10 odd 2