Properties

Label 8470.2.a.ct.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5225.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(-1.88301\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} -1.00000 q^{10} +3.04678 q^{12} +0.353057 q^{13} +1.00000 q^{14} -3.04678 q^{15} +1.00000 q^{16} +0.810708 q^{17} +6.28284 q^{18} -2.59251 q^{19} -1.00000 q^{20} +3.04678 q^{21} -2.00000 q^{23} +3.04678 q^{24} +1.00000 q^{25} +0.353057 q^{26} +10.0021 q^{27} +1.00000 q^{28} +9.54782 q^{29} -3.04678 q^{30} -5.70820 q^{31} +1.00000 q^{32} +0.810708 q^{34} -1.00000 q^{35} +6.28284 q^{36} +3.56360 q^{37} -2.59251 q^{38} +1.07569 q^{39} -1.00000 q^{40} +8.25732 q^{41} +3.04678 q^{42} +9.49338 q^{43} -6.28284 q^{45} -2.00000 q^{46} -8.60773 q^{47} +3.04678 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.47005 q^{51} +0.353057 q^{52} -2.79967 q^{53} +10.0021 q^{54} +1.00000 q^{56} -7.89879 q^{57} +9.54782 q^{58} +13.2092 q^{59} -3.04678 q^{60} +6.70611 q^{61} -5.70820 q^{62} +6.28284 q^{63} +1.00000 q^{64} -0.353057 q^{65} -1.40749 q^{67} +0.810708 q^{68} -6.09355 q^{69} -1.00000 q^{70} -12.9187 q^{71} +6.28284 q^{72} +5.15008 q^{73} +3.56360 q^{74} +3.04678 q^{75} -2.59251 q^{76} +1.07569 q^{78} +3.64694 q^{79} -1.00000 q^{80} +11.6256 q^{81} +8.25732 q^{82} +2.66819 q^{83} +3.04678 q^{84} -0.810708 q^{85} +9.49338 q^{86} +29.0901 q^{87} +9.63799 q^{89} -6.28284 q^{90} +0.353057 q^{91} -2.00000 q^{92} -17.3916 q^{93} -8.60773 q^{94} +2.59251 q^{95} +3.04678 q^{96} -9.79837 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} - 4 q^{10} + 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} + 6 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} - 8 q^{23} + 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} + 4 q^{28} + 15 q^{29} - 2 q^{30} + 4 q^{31} + 4 q^{32} + 2 q^{34} - 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 4 q^{40} + 11 q^{41} + 2 q^{42} + 7 q^{43} - 6 q^{45} - 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{50} + 18 q^{51} + q^{52} + 10 q^{53} + 14 q^{54} + 4 q^{56} - 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} + 26 q^{61} + 4 q^{62} + 6 q^{63} + 4 q^{64} - q^{65} - 13 q^{67} + 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} + 6 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 3 q^{76} - q^{78} + 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} - 2 q^{83} + 2 q^{84} - 2 q^{85} + 7 q^{86} + 26 q^{87} - 7 q^{89} - 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} + 5 q^{94} + 3 q^{95} + 2 q^{96} + 10 q^{97} + 4 q^{98} + O(q^{100}) \)

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
\(3\) 3.04678 1.75906 0.879528 0.475846i \(-0.157858\pi\)
0.879528 + 0.475846i \(0.157858\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.04678 1.24384
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 6.28284 2.09428
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.04678 0.879528
\(13\) 0.353057 0.0979204 0.0489602 0.998801i \(-0.484409\pi\)
0.0489602 + 0.998801i \(0.484409\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.04678 −0.786674
\(16\) 1.00000 0.250000
\(17\) 0.810708 0.196626 0.0983128 0.995156i \(-0.468655\pi\)
0.0983128 + 0.995156i \(0.468655\pi\)
\(18\) 6.28284 1.48088
\(19\) −2.59251 −0.594762 −0.297381 0.954759i \(-0.596113\pi\)
−0.297381 + 0.954759i \(0.596113\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.04678 0.664861
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 3.04678 0.621921
\(25\) 1.00000 0.200000
\(26\) 0.353057 0.0692401
\(27\) 10.0021 1.92490
\(28\) 1.00000 0.188982
\(29\) 9.54782 1.77299 0.886493 0.462742i \(-0.153134\pi\)
0.886493 + 0.462742i \(0.153134\pi\)
\(30\) −3.04678 −0.556263
\(31\) −5.70820 −1.02522 −0.512612 0.858620i \(-0.671322\pi\)
−0.512612 + 0.858620i \(0.671322\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.810708 0.139035
\(35\) −1.00000 −0.169031
\(36\) 6.28284 1.04714
\(37\) 3.56360 0.585852 0.292926 0.956135i \(-0.405371\pi\)
0.292926 + 0.956135i \(0.405371\pi\)
\(38\) −2.59251 −0.420560
\(39\) 1.07569 0.172247
\(40\) −1.00000 −0.158114
\(41\) 8.25732 1.28958 0.644788 0.764361i \(-0.276946\pi\)
0.644788 + 0.764361i \(0.276946\pi\)
\(42\) 3.04678 0.470128
\(43\) 9.49338 1.44773 0.723864 0.689943i \(-0.242364\pi\)
0.723864 + 0.689943i \(0.242364\pi\)
\(44\) 0 0
\(45\) −6.28284 −0.936591
\(46\) −2.00000 −0.294884
\(47\) −8.60773 −1.25557 −0.627783 0.778388i \(-0.716038\pi\)
−0.627783 + 0.778388i \(0.716038\pi\)
\(48\) 3.04678 0.439764
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.47005 0.345876
\(52\) 0.353057 0.0489602
\(53\) −2.79967 −0.384564 −0.192282 0.981340i \(-0.561589\pi\)
−0.192282 + 0.981340i \(0.561589\pi\)
\(54\) 10.0021 1.36111
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −7.89879 −1.04622
\(58\) 9.54782 1.25369
\(59\) 13.2092 1.71970 0.859849 0.510549i \(-0.170558\pi\)
0.859849 + 0.510549i \(0.170558\pi\)
\(60\) −3.04678 −0.393337
\(61\) 6.70611 0.858630 0.429315 0.903155i \(-0.358755\pi\)
0.429315 + 0.903155i \(0.358755\pi\)
\(62\) −5.70820 −0.724943
\(63\) 6.28284 0.791564
\(64\) 1.00000 0.125000
\(65\) −0.353057 −0.0437913
\(66\) 0 0
\(67\) −1.40749 −0.171953 −0.0859763 0.996297i \(-0.527401\pi\)
−0.0859763 + 0.996297i \(0.527401\pi\)
\(68\) 0.810708 0.0983128
\(69\) −6.09355 −0.733577
\(70\) −1.00000 −0.119523
\(71\) −12.9187 −1.53317 −0.766586 0.642141i \(-0.778046\pi\)
−0.766586 + 0.642141i \(0.778046\pi\)
\(72\) 6.28284 0.740440
\(73\) 5.15008 0.602771 0.301386 0.953502i \(-0.402551\pi\)
0.301386 + 0.953502i \(0.402551\pi\)
\(74\) 3.56360 0.414260
\(75\) 3.04678 0.351811
\(76\) −2.59251 −0.297381
\(77\) 0 0
\(78\) 1.07569 0.121797
\(79\) 3.64694 0.410313 0.205157 0.978729i \(-0.434230\pi\)
0.205157 + 0.978729i \(0.434230\pi\)
\(80\) −1.00000 −0.111803
\(81\) 11.6256 1.29173
\(82\) 8.25732 0.911868
\(83\) 2.66819 0.292872 0.146436 0.989220i \(-0.453220\pi\)
0.146436 + 0.989220i \(0.453220\pi\)
\(84\) 3.04678 0.332431
\(85\) −0.810708 −0.0879336
\(86\) 9.49338 1.02370
\(87\) 29.0901 3.11878
\(88\) 0 0
\(89\) 9.63799 1.02163 0.510813 0.859692i \(-0.329345\pi\)
0.510813 + 0.859692i \(0.329345\pi\)
\(90\) −6.28284 −0.662270
\(91\) 0.353057 0.0370104
\(92\) −2.00000 −0.208514
\(93\) −17.3916 −1.80343
\(94\) −8.60773 −0.887820
\(95\) 2.59251 0.265986
\(96\) 3.04678 0.310960
\(97\) −9.79837 −0.994874 −0.497437 0.867500i \(-0.665725\pi\)
−0.497437 + 0.867500i \(0.665725\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.904358 0.0899870 0.0449935 0.998987i \(-0.485673\pi\)
0.0449935 + 0.998987i \(0.485673\pi\)
\(102\) 2.47005 0.244571
\(103\) 11.2203 1.10557 0.552784 0.833325i \(-0.313565\pi\)
0.552784 + 0.833325i \(0.313565\pi\)
\(104\) 0.353057 0.0346201
\(105\) −3.04678 −0.297335
\(106\) −2.79967 −0.271928
\(107\) −8.89614 −0.860023 −0.430011 0.902823i \(-0.641490\pi\)
−0.430011 + 0.902823i \(0.641490\pi\)
\(108\) 10.0021 0.962452
\(109\) −18.4763 −1.76971 −0.884855 0.465866i \(-0.845743\pi\)
−0.884855 + 0.465866i \(0.845743\pi\)
\(110\) 0 0
\(111\) 10.8575 1.03055
\(112\) 1.00000 0.0944911
\(113\) −12.6902 −1.19380 −0.596899 0.802317i \(-0.703601\pi\)
−0.596899 + 0.802317i \(0.703601\pi\)
\(114\) −7.89879 −0.739789
\(115\) 2.00000 0.186501
\(116\) 9.54782 0.886493
\(117\) 2.21820 0.205073
\(118\) 13.2092 1.21601
\(119\) 0.810708 0.0743175
\(120\) −3.04678 −0.278131
\(121\) 0 0
\(122\) 6.70611 0.607143
\(123\) 25.1582 2.26844
\(124\) −5.70820 −0.512612
\(125\) −1.00000 −0.0894427
\(126\) 6.28284 0.559720
\(127\) −16.2382 −1.44090 −0.720452 0.693505i \(-0.756066\pi\)
−0.720452 + 0.693505i \(0.756066\pi\)
\(128\) 1.00000 0.0883883
\(129\) 28.9242 2.54664
\(130\) −0.353057 −0.0309651
\(131\) 8.08882 0.706723 0.353362 0.935487i \(-0.385039\pi\)
0.353362 + 0.935487i \(0.385039\pi\)
\(132\) 0 0
\(133\) −2.59251 −0.224799
\(134\) −1.40749 −0.121589
\(135\) −10.0021 −0.860843
\(136\) 0.810708 0.0695176
\(137\) 10.4019 0.888696 0.444348 0.895854i \(-0.353435\pi\)
0.444348 + 0.895854i \(0.353435\pi\)
\(138\) −6.09355 −0.518718
\(139\) −0.757168 −0.0642221 −0.0321111 0.999484i \(-0.510223\pi\)
−0.0321111 + 0.999484i \(0.510223\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −26.2258 −2.20861
\(142\) −12.9187 −1.08412
\(143\) 0 0
\(144\) 6.28284 0.523570
\(145\) −9.54782 −0.792903
\(146\) 5.15008 0.426224
\(147\) 3.04678 0.251294
\(148\) 3.56360 0.292926
\(149\) 8.96980 0.734835 0.367417 0.930056i \(-0.380242\pi\)
0.367417 + 0.930056i \(0.380242\pi\)
\(150\) 3.04678 0.248768
\(151\) 21.9731 1.78814 0.894072 0.447923i \(-0.147836\pi\)
0.894072 + 0.447923i \(0.147836\pi\)
\(152\) −2.59251 −0.210280
\(153\) 5.09355 0.411789
\(154\) 0 0
\(155\) 5.70820 0.458494
\(156\) 1.07569 0.0861237
\(157\) 0.0668308 0.00533368 0.00266684 0.999996i \(-0.499151\pi\)
0.00266684 + 0.999996i \(0.499151\pi\)
\(158\) 3.64694 0.290135
\(159\) −8.52995 −0.676469
\(160\) −1.00000 −0.0790569
\(161\) −2.00000 −0.157622
\(162\) 11.6256 0.913393
\(163\) 22.2692 1.74426 0.872128 0.489279i \(-0.162740\pi\)
0.872128 + 0.489279i \(0.162740\pi\)
\(164\) 8.25732 0.644788
\(165\) 0 0
\(166\) 2.66819 0.207092
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 3.04678 0.235064
\(169\) −12.8754 −0.990412
\(170\) −0.810708 −0.0621785
\(171\) −16.2883 −1.24560
\(172\) 9.49338 0.723864
\(173\) −19.8698 −1.51067 −0.755336 0.655338i \(-0.772526\pi\)
−0.755336 + 0.655338i \(0.772526\pi\)
\(174\) 29.0901 2.20531
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 40.2456 3.02505
\(178\) 9.63799 0.722398
\(179\) −8.61246 −0.643726 −0.321863 0.946786i \(-0.604309\pi\)
−0.321863 + 0.946786i \(0.604309\pi\)
\(180\) −6.28284 −0.468296
\(181\) 11.7345 0.872216 0.436108 0.899894i \(-0.356357\pi\)
0.436108 + 0.899894i \(0.356357\pi\)
\(182\) 0.353057 0.0261703
\(183\) 20.4320 1.51038
\(184\) −2.00000 −0.147442
\(185\) −3.56360 −0.262001
\(186\) −17.3916 −1.27522
\(187\) 0 0
\(188\) −8.60773 −0.627783
\(189\) 10.0021 0.727545
\(190\) 2.59251 0.188080
\(191\) −8.15063 −0.589759 −0.294880 0.955534i \(-0.595280\pi\)
−0.294880 + 0.955534i \(0.595280\pi\)
\(192\) 3.04678 0.219882
\(193\) −8.74185 −0.629252 −0.314626 0.949216i \(-0.601879\pi\)
−0.314626 + 0.949216i \(0.601879\pi\)
\(194\) −9.79837 −0.703482
\(195\) −1.07569 −0.0770314
\(196\) 1.00000 0.0714286
\(197\) −14.4568 −1.03001 −0.515003 0.857189i \(-0.672209\pi\)
−0.515003 + 0.857189i \(0.672209\pi\)
\(198\) 0 0
\(199\) −11.0446 −0.782930 −0.391465 0.920193i \(-0.628032\pi\)
−0.391465 + 0.920193i \(0.628032\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.28832 −0.302475
\(202\) 0.904358 0.0636304
\(203\) 9.54782 0.670126
\(204\) 2.47005 0.172938
\(205\) −8.25732 −0.576716
\(206\) 11.2203 0.781755
\(207\) −12.5657 −0.873376
\(208\) 0.353057 0.0244801
\(209\) 0 0
\(210\) −3.04678 −0.210248
\(211\) −23.7102 −1.63228 −0.816139 0.577856i \(-0.803889\pi\)
−0.816139 + 0.577856i \(0.803889\pi\)
\(212\) −2.79967 −0.192282
\(213\) −39.3605 −2.69694
\(214\) −8.89614 −0.608128
\(215\) −9.49338 −0.647443
\(216\) 10.0021 0.680556
\(217\) −5.70820 −0.387498
\(218\) −18.4763 −1.25137
\(219\) 15.6911 1.06031
\(220\) 0 0
\(221\) 0.286226 0.0192536
\(222\) 10.8575 0.728707
\(223\) −5.58986 −0.374325 −0.187162 0.982329i \(-0.559929\pi\)
−0.187162 + 0.982329i \(0.559929\pi\)
\(224\) 1.00000 0.0668153
\(225\) 6.28284 0.418856
\(226\) −12.6902 −0.844142
\(227\) 6.32206 0.419610 0.209805 0.977743i \(-0.432717\pi\)
0.209805 + 0.977743i \(0.432717\pi\)
\(228\) −7.89879 −0.523110
\(229\) −3.89113 −0.257133 −0.128566 0.991701i \(-0.541038\pi\)
−0.128566 + 0.991701i \(0.541038\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 9.54782 0.626845
\(233\) −15.1659 −0.993548 −0.496774 0.867880i \(-0.665482\pi\)
−0.496774 + 0.867880i \(0.665482\pi\)
\(234\) 2.21820 0.145008
\(235\) 8.60773 0.561506
\(236\) 13.2092 0.859849
\(237\) 11.1114 0.721764
\(238\) 0.810708 0.0525504
\(239\) 21.0949 1.36452 0.682258 0.731112i \(-0.260998\pi\)
0.682258 + 0.731112i \(0.260998\pi\)
\(240\) −3.04678 −0.196669
\(241\) 11.9416 0.769228 0.384614 0.923078i \(-0.374335\pi\)
0.384614 + 0.923078i \(0.374335\pi\)
\(242\) 0 0
\(243\) 5.41432 0.347329
\(244\) 6.70611 0.429315
\(245\) −1.00000 −0.0638877
\(246\) 25.1582 1.60403
\(247\) −0.915302 −0.0582393
\(248\) −5.70820 −0.362471
\(249\) 8.12938 0.515179
\(250\) −1.00000 −0.0632456
\(251\) 2.11834 0.133709 0.0668543 0.997763i \(-0.478704\pi\)
0.0668543 + 0.997763i \(0.478704\pi\)
\(252\) 6.28284 0.395782
\(253\) 0 0
\(254\) −16.2382 −1.01887
\(255\) −2.47005 −0.154680
\(256\) 1.00000 0.0625000
\(257\) −20.7137 −1.29208 −0.646042 0.763302i \(-0.723577\pi\)
−0.646042 + 0.763302i \(0.723577\pi\)
\(258\) 28.9242 1.80074
\(259\) 3.56360 0.221431
\(260\) −0.353057 −0.0218957
\(261\) 59.9875 3.71313
\(262\) 8.08882 0.499729
\(263\) −3.89531 −0.240195 −0.120097 0.992762i \(-0.538321\pi\)
−0.120097 + 0.992762i \(0.538321\pi\)
\(264\) 0 0
\(265\) 2.79967 0.171982
\(266\) −2.59251 −0.158957
\(267\) 29.3648 1.79710
\(268\) −1.40749 −0.0859763
\(269\) −22.8684 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(270\) −10.0021 −0.608708
\(271\) −20.6082 −1.25186 −0.625929 0.779880i \(-0.715280\pi\)
−0.625929 + 0.779880i \(0.715280\pi\)
\(272\) 0.810708 0.0491564
\(273\) 1.07569 0.0651034
\(274\) 10.4019 0.628403
\(275\) 0 0
\(276\) −6.09355 −0.366789
\(277\) −17.2045 −1.03372 −0.516860 0.856070i \(-0.672899\pi\)
−0.516860 + 0.856070i \(0.672899\pi\)
\(278\) −0.757168 −0.0454119
\(279\) −35.8638 −2.14711
\(280\) −1.00000 −0.0597614
\(281\) −30.4444 −1.81616 −0.908081 0.418795i \(-0.862453\pi\)
−0.908081 + 0.418795i \(0.862453\pi\)
\(282\) −26.2258 −1.56173
\(283\) −9.99517 −0.594151 −0.297076 0.954854i \(-0.596011\pi\)
−0.297076 + 0.954854i \(0.596011\pi\)
\(284\) −12.9187 −0.766586
\(285\) 7.89879 0.467884
\(286\) 0 0
\(287\) 8.25732 0.487414
\(288\) 6.28284 0.370220
\(289\) −16.3428 −0.961338
\(290\) −9.54782 −0.560667
\(291\) −29.8535 −1.75004
\(292\) 5.15008 0.301386
\(293\) 6.46796 0.377862 0.188931 0.981990i \(-0.439498\pi\)
0.188931 + 0.981990i \(0.439498\pi\)
\(294\) 3.04678 0.177692
\(295\) −13.2092 −0.769072
\(296\) 3.56360 0.207130
\(297\) 0 0
\(298\) 8.96980 0.519607
\(299\) −0.706114 −0.0408356
\(300\) 3.04678 0.175906
\(301\) 9.49338 0.547190
\(302\) 21.9731 1.26441
\(303\) 2.75538 0.158292
\(304\) −2.59251 −0.148690
\(305\) −6.70611 −0.383991
\(306\) 5.09355 0.291179
\(307\) 20.6473 1.17840 0.589202 0.807986i \(-0.299442\pi\)
0.589202 + 0.807986i \(0.299442\pi\)
\(308\) 0 0
\(309\) 34.1857 1.94476
\(310\) 5.70820 0.324204
\(311\) 23.4385 1.32907 0.664537 0.747255i \(-0.268629\pi\)
0.664537 + 0.747255i \(0.268629\pi\)
\(312\) 1.07569 0.0608987
\(313\) 11.9059 0.672961 0.336480 0.941690i \(-0.390763\pi\)
0.336480 + 0.941690i \(0.390763\pi\)
\(314\) 0.0668308 0.00377148
\(315\) −6.28284 −0.353998
\(316\) 3.64694 0.205157
\(317\) −34.4185 −1.93314 −0.966568 0.256411i \(-0.917460\pi\)
−0.966568 + 0.256411i \(0.917460\pi\)
\(318\) −8.52995 −0.478336
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −27.1046 −1.51283
\(322\) −2.00000 −0.111456
\(323\) −2.10177 −0.116945
\(324\) 11.6256 0.645866
\(325\) 0.353057 0.0195841
\(326\) 22.2692 1.23337
\(327\) −56.2932 −3.11302
\(328\) 8.25732 0.455934
\(329\) −8.60773 −0.474559
\(330\) 0 0
\(331\) −11.6770 −0.641829 −0.320914 0.947108i \(-0.603990\pi\)
−0.320914 + 0.947108i \(0.603990\pi\)
\(332\) 2.66819 0.146436
\(333\) 22.3895 1.22694
\(334\) 12.0000 0.656611
\(335\) 1.40749 0.0768996
\(336\) 3.04678 0.166215
\(337\) 10.3337 0.562915 0.281457 0.959574i \(-0.409182\pi\)
0.281457 + 0.959574i \(0.409182\pi\)
\(338\) −12.8754 −0.700327
\(339\) −38.6643 −2.09996
\(340\) −0.810708 −0.0439668
\(341\) 0 0
\(342\) −16.2883 −0.880771
\(343\) 1.00000 0.0539949
\(344\) 9.49338 0.511849
\(345\) 6.09355 0.328066
\(346\) −19.8698 −1.06821
\(347\) 18.3202 0.983480 0.491740 0.870742i \(-0.336361\pi\)
0.491740 + 0.870742i \(0.336361\pi\)
\(348\) 29.0901 1.55939
\(349\) 29.9216 1.60167 0.800833 0.598888i \(-0.204391\pi\)
0.800833 + 0.598888i \(0.204391\pi\)
\(350\) 1.00000 0.0534522
\(351\) 3.53131 0.188487
\(352\) 0 0
\(353\) −8.93024 −0.475309 −0.237654 0.971350i \(-0.576379\pi\)
−0.237654 + 0.971350i \(0.576379\pi\)
\(354\) 40.2456 2.13903
\(355\) 12.9187 0.685656
\(356\) 9.63799 0.510813
\(357\) 2.47005 0.130729
\(358\) −8.61246 −0.455183
\(359\) 29.7915 1.57234 0.786169 0.618012i \(-0.212062\pi\)
0.786169 + 0.618012i \(0.212062\pi\)
\(360\) −6.28284 −0.331135
\(361\) −12.2789 −0.646258
\(362\) 11.7345 0.616750
\(363\) 0 0
\(364\) 0.353057 0.0185052
\(365\) −5.15008 −0.269567
\(366\) 20.4320 1.06800
\(367\) −24.7535 −1.29212 −0.646062 0.763285i \(-0.723585\pi\)
−0.646062 + 0.763285i \(0.723585\pi\)
\(368\) −2.00000 −0.104257
\(369\) 51.8794 2.70074
\(370\) −3.56360 −0.185263
\(371\) −2.79967 −0.145351
\(372\) −17.3916 −0.901713
\(373\) 9.75070 0.504872 0.252436 0.967614i \(-0.418768\pi\)
0.252436 + 0.967614i \(0.418768\pi\)
\(374\) 0 0
\(375\) −3.04678 −0.157335
\(376\) −8.60773 −0.443910
\(377\) 3.37092 0.173611
\(378\) 10.0021 0.514452
\(379\) 21.0454 1.08103 0.540514 0.841335i \(-0.318230\pi\)
0.540514 + 0.841335i \(0.318230\pi\)
\(380\) 2.59251 0.132993
\(381\) −49.4740 −2.53463
\(382\) −8.15063 −0.417023
\(383\) −34.8828 −1.78243 −0.891214 0.453582i \(-0.850146\pi\)
−0.891214 + 0.453582i \(0.850146\pi\)
\(384\) 3.04678 0.155480
\(385\) 0 0
\(386\) −8.74185 −0.444948
\(387\) 59.6455 3.03195
\(388\) −9.79837 −0.497437
\(389\) −12.7069 −0.644263 −0.322131 0.946695i \(-0.604399\pi\)
−0.322131 + 0.946695i \(0.604399\pi\)
\(390\) −1.07569 −0.0544694
\(391\) −1.62142 −0.0819985
\(392\) 1.00000 0.0505076
\(393\) 24.6448 1.24317
\(394\) −14.4568 −0.728324
\(395\) −3.64694 −0.183498
\(396\) 0 0
\(397\) 3.33036 0.167146 0.0835729 0.996502i \(-0.473367\pi\)
0.0835729 + 0.996502i \(0.473367\pi\)
\(398\) −11.0446 −0.553615
\(399\) −7.89879 −0.395434
\(400\) 1.00000 0.0500000
\(401\) −9.27248 −0.463046 −0.231523 0.972829i \(-0.574371\pi\)
−0.231523 + 0.972829i \(0.574371\pi\)
\(402\) −4.28832 −0.213882
\(403\) −2.01532 −0.100390
\(404\) 0.904358 0.0449935
\(405\) −11.6256 −0.577681
\(406\) 9.54782 0.473850
\(407\) 0 0
\(408\) 2.47005 0.122285
\(409\) 0.349614 0.0172873 0.00864366 0.999963i \(-0.497249\pi\)
0.00864366 + 0.999963i \(0.497249\pi\)
\(410\) −8.25732 −0.407800
\(411\) 31.6923 1.56327
\(412\) 11.2203 0.552784
\(413\) 13.2092 0.649985
\(414\) −12.5657 −0.617570
\(415\) −2.66819 −0.130976
\(416\) 0.353057 0.0173100
\(417\) −2.30692 −0.112970
\(418\) 0 0
\(419\) 27.9340 1.36466 0.682332 0.731043i \(-0.260966\pi\)
0.682332 + 0.731043i \(0.260966\pi\)
\(420\) −3.04678 −0.148667
\(421\) 8.07221 0.393415 0.196708 0.980462i \(-0.436975\pi\)
0.196708 + 0.980462i \(0.436975\pi\)
\(422\) −23.7102 −1.15419
\(423\) −54.0810 −2.62951
\(424\) −2.79967 −0.135964
\(425\) 0.810708 0.0393251
\(426\) −39.3605 −1.90702
\(427\) 6.70611 0.324532
\(428\) −8.89614 −0.430011
\(429\) 0 0
\(430\) −9.49338 −0.457812
\(431\) −20.0088 −0.963791 −0.481895 0.876229i \(-0.660051\pi\)
−0.481895 + 0.876229i \(0.660051\pi\)
\(432\) 10.0021 0.481226
\(433\) −38.7447 −1.86195 −0.930977 0.365079i \(-0.881042\pi\)
−0.930977 + 0.365079i \(0.881042\pi\)
\(434\) −5.70820 −0.274003
\(435\) −29.0901 −1.39476
\(436\) −18.4763 −0.884855
\(437\) 5.18501 0.248033
\(438\) 15.6911 0.749751
\(439\) −5.62560 −0.268495 −0.134248 0.990948i \(-0.542862\pi\)
−0.134248 + 0.990948i \(0.542862\pi\)
\(440\) 0 0
\(441\) 6.28284 0.299183
\(442\) 0.286226 0.0136144
\(443\) −15.6763 −0.744804 −0.372402 0.928072i \(-0.621466\pi\)
−0.372402 + 0.928072i \(0.621466\pi\)
\(444\) 10.8575 0.515273
\(445\) −9.63799 −0.456885
\(446\) −5.58986 −0.264688
\(447\) 27.3290 1.29262
\(448\) 1.00000 0.0472456
\(449\) −26.1032 −1.23189 −0.615943 0.787791i \(-0.711225\pi\)
−0.615943 + 0.787791i \(0.711225\pi\)
\(450\) 6.28284 0.296176
\(451\) 0 0
\(452\) −12.6902 −0.596899
\(453\) 66.9471 3.14545
\(454\) 6.32206 0.296709
\(455\) −0.353057 −0.0165516
\(456\) −7.89879 −0.369895
\(457\) 35.5911 1.66488 0.832441 0.554114i \(-0.186943\pi\)
0.832441 + 0.554114i \(0.186943\pi\)
\(458\) −3.89113 −0.181820
\(459\) 8.10877 0.378485
\(460\) 2.00000 0.0932505
\(461\) 9.66571 0.450177 0.225088 0.974338i \(-0.427733\pi\)
0.225088 + 0.974338i \(0.427733\pi\)
\(462\) 0 0
\(463\) −29.9726 −1.39295 −0.696473 0.717583i \(-0.745248\pi\)
−0.696473 + 0.717583i \(0.745248\pi\)
\(464\) 9.54782 0.443246
\(465\) 17.3916 0.806517
\(466\) −15.1659 −0.702545
\(467\) 30.5086 1.41177 0.705885 0.708326i \(-0.250550\pi\)
0.705885 + 0.708326i \(0.250550\pi\)
\(468\) 2.21820 0.102536
\(469\) −1.40749 −0.0649920
\(470\) 8.60773 0.397045
\(471\) 0.203619 0.00938225
\(472\) 13.2092 0.608005
\(473\) 0 0
\(474\) 11.1114 0.510364
\(475\) −2.59251 −0.118952
\(476\) 0.810708 0.0371587
\(477\) −17.5899 −0.805384
\(478\) 21.0949 0.964858
\(479\) 38.7127 1.76883 0.884414 0.466704i \(-0.154559\pi\)
0.884414 + 0.466704i \(0.154559\pi\)
\(480\) −3.04678 −0.139066
\(481\) 1.25815 0.0573668
\(482\) 11.9416 0.543926
\(483\) −6.09355 −0.277266
\(484\) 0 0
\(485\) 9.79837 0.444921
\(486\) 5.41432 0.245598
\(487\) 33.8375 1.53332 0.766662 0.642052i \(-0.221916\pi\)
0.766662 + 0.642052i \(0.221916\pi\)
\(488\) 6.70611 0.303572
\(489\) 67.8491 3.06824
\(490\) −1.00000 −0.0451754
\(491\) 18.5755 0.838301 0.419150 0.907917i \(-0.362328\pi\)
0.419150 + 0.907917i \(0.362328\pi\)
\(492\) 25.1582 1.13422
\(493\) 7.74050 0.348614
\(494\) −0.915302 −0.0411814
\(495\) 0 0
\(496\) −5.70820 −0.256306
\(497\) −12.9187 −0.579485
\(498\) 8.12938 0.364286
\(499\) −14.7069 −0.658372 −0.329186 0.944265i \(-0.606774\pi\)
−0.329186 + 0.944265i \(0.606774\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 36.5613 1.63344
\(502\) 2.11834 0.0945462
\(503\) −5.50234 −0.245337 −0.122669 0.992448i \(-0.539145\pi\)
−0.122669 + 0.992448i \(0.539145\pi\)
\(504\) 6.28284 0.279860
\(505\) −0.904358 −0.0402434
\(506\) 0 0
\(507\) −39.2283 −1.74219
\(508\) −16.2382 −0.720452
\(509\) 24.1045 1.06841 0.534207 0.845354i \(-0.320610\pi\)
0.534207 + 0.845354i \(0.320610\pi\)
\(510\) −2.47005 −0.109375
\(511\) 5.15008 0.227826
\(512\) 1.00000 0.0441942
\(513\) −25.9305 −1.14486
\(514\) −20.7137 −0.913641
\(515\) −11.2203 −0.494425
\(516\) 28.9242 1.27332
\(517\) 0 0
\(518\) 3.56360 0.156575
\(519\) −60.5388 −2.65736
\(520\) −0.353057 −0.0154826
\(521\) 8.73293 0.382597 0.191298 0.981532i \(-0.438730\pi\)
0.191298 + 0.981532i \(0.438730\pi\)
\(522\) 59.9875 2.62558
\(523\) −44.2475 −1.93481 −0.967405 0.253235i \(-0.918505\pi\)
−0.967405 + 0.253235i \(0.918505\pi\)
\(524\) 8.08882 0.353362
\(525\) 3.04678 0.132972
\(526\) −3.89531 −0.169843
\(527\) −4.62769 −0.201585
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 2.79967 0.121610
\(531\) 82.9917 3.60153
\(532\) −2.59251 −0.112399
\(533\) 2.91530 0.126276
\(534\) 29.3648 1.27074
\(535\) 8.89614 0.384614
\(536\) −1.40749 −0.0607944
\(537\) −26.2402 −1.13235
\(538\) −22.8684 −0.985928
\(539\) 0 0
\(540\) −10.0021 −0.430421
\(541\) −14.0407 −0.603655 −0.301827 0.953363i \(-0.597597\pi\)
−0.301827 + 0.953363i \(0.597597\pi\)
\(542\) −20.6082 −0.885197
\(543\) 35.7523 1.53428
\(544\) 0.810708 0.0347588
\(545\) 18.4763 0.791438
\(546\) 1.07569 0.0460351
\(547\) −6.48093 −0.277105 −0.138552 0.990355i \(-0.544245\pi\)
−0.138552 + 0.990355i \(0.544245\pi\)
\(548\) 10.4019 0.444348
\(549\) 42.1335 1.79821
\(550\) 0 0
\(551\) −24.7528 −1.05450
\(552\) −6.09355 −0.259359
\(553\) 3.64694 0.155084
\(554\) −17.2045 −0.730950
\(555\) −10.8575 −0.460875
\(556\) −0.757168 −0.0321111
\(557\) −10.0109 −0.424177 −0.212089 0.977250i \(-0.568027\pi\)
−0.212089 + 0.977250i \(0.568027\pi\)
\(558\) −35.8638 −1.51823
\(559\) 3.35170 0.141762
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −30.4444 −1.28422
\(563\) −3.32206 −0.140008 −0.0700040 0.997547i \(-0.522301\pi\)
−0.0700040 + 0.997547i \(0.522301\pi\)
\(564\) −26.2258 −1.10431
\(565\) 12.6902 0.533882
\(566\) −9.99517 −0.420128
\(567\) 11.6256 0.488229
\(568\) −12.9187 −0.542058
\(569\) −30.5120 −1.27913 −0.639564 0.768738i \(-0.720885\pi\)
−0.639564 + 0.768738i \(0.720885\pi\)
\(570\) 7.89879 0.330844
\(571\) −2.37904 −0.0995597 −0.0497799 0.998760i \(-0.515852\pi\)
−0.0497799 + 0.998760i \(0.515852\pi\)
\(572\) 0 0
\(573\) −24.8332 −1.03742
\(574\) 8.25732 0.344654
\(575\) −2.00000 −0.0834058
\(576\) 6.28284 0.261785
\(577\) −12.5384 −0.521981 −0.260990 0.965341i \(-0.584049\pi\)
−0.260990 + 0.965341i \(0.584049\pi\)
\(578\) −16.3428 −0.679769
\(579\) −26.6345 −1.10689
\(580\) −9.54782 −0.396452
\(581\) 2.66819 0.110695
\(582\) −29.8535 −1.23747
\(583\) 0 0
\(584\) 5.15008 0.213112
\(585\) −2.21820 −0.0917113
\(586\) 6.46796 0.267189
\(587\) 43.5800 1.79874 0.899370 0.437189i \(-0.144026\pi\)
0.899370 + 0.437189i \(0.144026\pi\)
\(588\) 3.04678 0.125647
\(589\) 14.7986 0.609764
\(590\) −13.2092 −0.543816
\(591\) −44.0467 −1.81184
\(592\) 3.56360 0.146463
\(593\) −0.0535401 −0.00219863 −0.00109931 0.999999i \(-0.500350\pi\)
−0.00109931 + 0.999999i \(0.500350\pi\)
\(594\) 0 0
\(595\) −0.810708 −0.0332358
\(596\) 8.96980 0.367417
\(597\) −33.6504 −1.37722
\(598\) −0.706114 −0.0288751
\(599\) 28.9504 1.18288 0.591440 0.806349i \(-0.298560\pi\)
0.591440 + 0.806349i \(0.298560\pi\)
\(600\) 3.04678 0.124384
\(601\) −31.0406 −1.26617 −0.633085 0.774082i \(-0.718212\pi\)
−0.633085 + 0.774082i \(0.718212\pi\)
\(602\) 9.49338 0.386921
\(603\) −8.84306 −0.360117
\(604\) 21.9731 0.894072
\(605\) 0 0
\(606\) 2.75538 0.111929
\(607\) −8.20288 −0.332945 −0.166472 0.986046i \(-0.553238\pi\)
−0.166472 + 0.986046i \(0.553238\pi\)
\(608\) −2.59251 −0.105140
\(609\) 29.0901 1.17879
\(610\) −6.70611 −0.271523
\(611\) −3.03902 −0.122946
\(612\) 5.09355 0.205895
\(613\) 21.6035 0.872558 0.436279 0.899812i \(-0.356296\pi\)
0.436279 + 0.899812i \(0.356296\pi\)
\(614\) 20.6473 0.833257
\(615\) −25.1582 −1.01448
\(616\) 0 0
\(617\) 16.5772 0.667372 0.333686 0.942684i \(-0.391707\pi\)
0.333686 + 0.942684i \(0.391707\pi\)
\(618\) 34.1857 1.37515
\(619\) −17.8133 −0.715975 −0.357988 0.933726i \(-0.616537\pi\)
−0.357988 + 0.933726i \(0.616537\pi\)
\(620\) 5.70820 0.229247
\(621\) −20.0042 −0.802740
\(622\) 23.4385 0.939798
\(623\) 9.63799 0.386138
\(624\) 1.07569 0.0430619
\(625\) 1.00000 0.0400000
\(626\) 11.9059 0.475855
\(627\) 0 0
\(628\) 0.0668308 0.00266684
\(629\) 2.88904 0.115193
\(630\) −6.28284 −0.250314
\(631\) 13.5121 0.537908 0.268954 0.963153i \(-0.413322\pi\)
0.268954 + 0.963153i \(0.413322\pi\)
\(632\) 3.64694 0.145068
\(633\) −72.2397 −2.87127
\(634\) −34.4185 −1.36693
\(635\) 16.2382 0.644392
\(636\) −8.52995 −0.338235
\(637\) 0.353057 0.0139886
\(638\) 0 0
\(639\) −81.1665 −3.21090
\(640\) −1.00000 −0.0395285
\(641\) 44.3116 1.75020 0.875102 0.483939i \(-0.160794\pi\)
0.875102 + 0.483939i \(0.160794\pi\)
\(642\) −27.1046 −1.06973
\(643\) 34.2852 1.35208 0.676039 0.736866i \(-0.263695\pi\)
0.676039 + 0.736866i \(0.263695\pi\)
\(644\) −2.00000 −0.0788110
\(645\) −28.9242 −1.13889
\(646\) −2.10177 −0.0826929
\(647\) 27.8742 1.09585 0.547925 0.836528i \(-0.315418\pi\)
0.547925 + 0.836528i \(0.315418\pi\)
\(648\) 11.6256 0.456697
\(649\) 0 0
\(650\) 0.353057 0.0138480
\(651\) −17.3916 −0.681631
\(652\) 22.2692 0.872128
\(653\) −27.5831 −1.07941 −0.539705 0.841854i \(-0.681464\pi\)
−0.539705 + 0.841854i \(0.681464\pi\)
\(654\) −56.2932 −2.20124
\(655\) −8.08882 −0.316056
\(656\) 8.25732 0.322394
\(657\) 32.3571 1.26237
\(658\) −8.60773 −0.335564
\(659\) −2.49969 −0.0973742 −0.0486871 0.998814i \(-0.515504\pi\)
−0.0486871 + 0.998814i \(0.515504\pi\)
\(660\) 0 0
\(661\) 10.1499 0.394785 0.197393 0.980325i \(-0.436753\pi\)
0.197393 + 0.980325i \(0.436753\pi\)
\(662\) −11.6770 −0.453841
\(663\) 0.872067 0.0338683
\(664\) 2.66819 0.103546
\(665\) 2.59251 0.100533
\(666\) 22.3895 0.867577
\(667\) −19.0956 −0.739386
\(668\) 12.0000 0.464294
\(669\) −17.0311 −0.658459
\(670\) 1.40749 0.0543762
\(671\) 0 0
\(672\) 3.04678 0.117532
\(673\) −36.9488 −1.42427 −0.712136 0.702042i \(-0.752272\pi\)
−0.712136 + 0.702042i \(0.752272\pi\)
\(674\) 10.3337 0.398041
\(675\) 10.0021 0.384981
\(676\) −12.8754 −0.495206
\(677\) −8.83753 −0.339654 −0.169827 0.985474i \(-0.554321\pi\)
−0.169827 + 0.985474i \(0.554321\pi\)
\(678\) −38.6643 −1.48489
\(679\) −9.79837 −0.376027
\(680\) −0.810708 −0.0310892
\(681\) 19.2619 0.738117
\(682\) 0 0
\(683\) 33.0803 1.26578 0.632892 0.774240i \(-0.281868\pi\)
0.632892 + 0.774240i \(0.281868\pi\)
\(684\) −16.2883 −0.622799
\(685\) −10.4019 −0.397437
\(686\) 1.00000 0.0381802
\(687\) −11.8554 −0.452311
\(688\) 9.49338 0.361932
\(689\) −0.988441 −0.0376566
\(690\) 6.09355 0.231978
\(691\) −9.22130 −0.350795 −0.175397 0.984498i \(-0.556121\pi\)
−0.175397 + 0.984498i \(0.556121\pi\)
\(692\) −19.8698 −0.755336
\(693\) 0 0
\(694\) 18.3202 0.695426
\(695\) 0.757168 0.0287210
\(696\) 29.0901 1.10266
\(697\) 6.69427 0.253564
\(698\) 29.9216 1.13255
\(699\) −46.2070 −1.74771
\(700\) 1.00000 0.0377964
\(701\) −26.2969 −0.993220 −0.496610 0.867974i \(-0.665422\pi\)
−0.496610 + 0.867974i \(0.665422\pi\)
\(702\) 3.53131 0.133281
\(703\) −9.23865 −0.348442
\(704\) 0 0
\(705\) 26.2258 0.987722
\(706\) −8.93024 −0.336094
\(707\) 0.904358 0.0340119
\(708\) 40.2456 1.51252
\(709\) −34.6228 −1.30029 −0.650143 0.759812i \(-0.725291\pi\)
−0.650143 + 0.759812i \(0.725291\pi\)
\(710\) 12.9187 0.484832
\(711\) 22.9132 0.859311
\(712\) 9.63799 0.361199
\(713\) 11.4164 0.427548
\(714\) 2.47005 0.0924391
\(715\) 0 0
\(716\) −8.61246 −0.321863
\(717\) 64.2714 2.40026
\(718\) 29.7915 1.11181
\(719\) −37.1839 −1.38673 −0.693363 0.720589i \(-0.743872\pi\)
−0.693363 + 0.720589i \(0.743872\pi\)
\(720\) −6.28284 −0.234148
\(721\) 11.2203 0.417865
\(722\) −12.2789 −0.456974
\(723\) 36.3835 1.35312
\(724\) 11.7345 0.436108
\(725\) 9.54782 0.354597
\(726\) 0 0
\(727\) 16.8666 0.625547 0.312773 0.949828i \(-0.398742\pi\)
0.312773 + 0.949828i \(0.398742\pi\)
\(728\) 0.353057 0.0130852
\(729\) −18.3806 −0.680762
\(730\) −5.15008 −0.190613
\(731\) 7.69636 0.284660
\(732\) 20.4320 0.755189
\(733\) −35.0947 −1.29625 −0.648126 0.761533i \(-0.724447\pi\)
−0.648126 + 0.761533i \(0.724447\pi\)
\(734\) −24.7535 −0.913669
\(735\) −3.04678 −0.112382
\(736\) −2.00000 −0.0737210
\(737\) 0 0
\(738\) 51.8794 1.90971
\(739\) 8.98641 0.330570 0.165285 0.986246i \(-0.447146\pi\)
0.165285 + 0.986246i \(0.447146\pi\)
\(740\) −3.56360 −0.131000
\(741\) −2.78872 −0.102446
\(742\) −2.79967 −0.102779
\(743\) 9.59663 0.352066 0.176033 0.984384i \(-0.443673\pi\)
0.176033 + 0.984384i \(0.443673\pi\)
\(744\) −17.3916 −0.637608
\(745\) −8.96980 −0.328628
\(746\) 9.75070 0.356999
\(747\) 16.7638 0.613357
\(748\) 0 0
\(749\) −8.89614 −0.325058
\(750\) −3.04678 −0.111253
\(751\) −14.1491 −0.516309 −0.258154 0.966104i \(-0.583114\pi\)
−0.258154 + 0.966104i \(0.583114\pi\)
\(752\) −8.60773 −0.313892
\(753\) 6.45411 0.235201
\(754\) 3.37092 0.122762
\(755\) −21.9731 −0.799682
\(756\) 10.0021 0.363773
\(757\) 45.6765 1.66014 0.830070 0.557660i \(-0.188301\pi\)
0.830070 + 0.557660i \(0.188301\pi\)
\(758\) 21.0454 0.764403
\(759\) 0 0
\(760\) 2.59251 0.0940401
\(761\) 16.5237 0.598985 0.299493 0.954099i \(-0.403183\pi\)
0.299493 + 0.954099i \(0.403183\pi\)
\(762\) −49.4740 −1.79226
\(763\) −18.4763 −0.668888
\(764\) −8.15063 −0.294880
\(765\) −5.09355 −0.184158
\(766\) −34.8828 −1.26037
\(767\) 4.66362 0.168393
\(768\) 3.04678 0.109941
\(769\) −26.5929 −0.958963 −0.479482 0.877552i \(-0.659175\pi\)
−0.479482 + 0.877552i \(0.659175\pi\)
\(770\) 0 0
\(771\) −63.1099 −2.27285
\(772\) −8.74185 −0.314626
\(773\) 35.4777 1.27604 0.638022 0.770019i \(-0.279753\pi\)
0.638022 + 0.770019i \(0.279753\pi\)
\(774\) 59.6455 2.14391
\(775\) −5.70820 −0.205045
\(776\) −9.79837 −0.351741
\(777\) 10.8575 0.389510
\(778\) −12.7069 −0.455563
\(779\) −21.4072 −0.766991
\(780\) −1.07569 −0.0385157
\(781\) 0 0
\(782\) −1.62142 −0.0579817
\(783\) 95.4982 3.41283
\(784\) 1.00000 0.0357143
\(785\) −0.0668308 −0.00238529
\(786\) 24.6448 0.879051
\(787\) 16.0990 0.573868 0.286934 0.957950i \(-0.407364\pi\)
0.286934 + 0.957950i \(0.407364\pi\)
\(788\) −14.4568 −0.515003
\(789\) −11.8681 −0.422517
\(790\) −3.64694 −0.129752
\(791\) −12.6902 −0.451213
\(792\) 0 0
\(793\) 2.36764 0.0840773
\(794\) 3.33036 0.118190
\(795\) 8.52995 0.302526
\(796\) −11.0446 −0.391465
\(797\) 38.0457 1.34765 0.673825 0.738891i \(-0.264650\pi\)
0.673825 + 0.738891i \(0.264650\pi\)
\(798\) −7.89879 −0.279614
\(799\) −6.97836 −0.246876
\(800\) 1.00000 0.0353553
\(801\) 60.5540 2.13957
\(802\) −9.27248 −0.327423
\(803\) 0 0
\(804\) −4.28832 −0.151237
\(805\) 2.00000 0.0704907
\(806\) −2.01532 −0.0709866
\(807\) −69.6750 −2.45268
\(808\) 0.904358 0.0318152
\(809\) −14.7984 −0.520283 −0.260142 0.965570i \(-0.583769\pi\)
−0.260142 + 0.965570i \(0.583769\pi\)
\(810\) −11.6256 −0.408482
\(811\) 0.702276 0.0246602 0.0123301 0.999924i \(-0.496075\pi\)
0.0123301 + 0.999924i \(0.496075\pi\)
\(812\) 9.54782 0.335063
\(813\) −62.7885 −2.20209
\(814\) 0 0
\(815\) −22.2692 −0.780055
\(816\) 2.47005 0.0864689
\(817\) −24.6117 −0.861053
\(818\) 0.349614 0.0122240
\(819\) 2.21820 0.0775102
\(820\) −8.25732 −0.288358
\(821\) 1.50680 0.0525877 0.0262938 0.999654i \(-0.491629\pi\)
0.0262938 + 0.999654i \(0.491629\pi\)
\(822\) 31.6923 1.10540
\(823\) −30.8989 −1.07707 −0.538534 0.842604i \(-0.681021\pi\)
−0.538534 + 0.842604i \(0.681021\pi\)
\(824\) 11.2203 0.390877
\(825\) 0 0
\(826\) 13.2092 0.459609
\(827\) 52.7658 1.83485 0.917423 0.397914i \(-0.130266\pi\)
0.917423 + 0.397914i \(0.130266\pi\)
\(828\) −12.5657 −0.436688
\(829\) −27.7575 −0.964057 −0.482028 0.876156i \(-0.660100\pi\)
−0.482028 + 0.876156i \(0.660100\pi\)
\(830\) −2.66819 −0.0926143
\(831\) −52.4183 −1.81837
\(832\) 0.353057 0.0122400
\(833\) 0.810708 0.0280894
\(834\) −2.30692 −0.0798821
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −57.0940 −1.97346
\(838\) 27.9340 0.964963
\(839\) −48.0237 −1.65796 −0.828981 0.559277i \(-0.811079\pi\)
−0.828981 + 0.559277i \(0.811079\pi\)
\(840\) −3.04678 −0.105124
\(841\) 62.1609 2.14348
\(842\) 8.07221 0.278187
\(843\) −92.7573 −3.19473
\(844\) −23.7102 −0.816139
\(845\) 12.8754 0.442926
\(846\) −54.0810 −1.85934
\(847\) 0 0
\(848\) −2.79967 −0.0961409
\(849\) −30.4530 −1.04515
\(850\) 0.810708 0.0278071
\(851\) −7.12720 −0.244317
\(852\) −39.3605 −1.34847
\(853\) −36.0186 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(854\) 6.70611 0.229478
\(855\) 16.2883 0.557049
\(856\) −8.89614 −0.304064
\(857\) 5.73142 0.195782 0.0978908 0.995197i \(-0.468790\pi\)
0.0978908 + 0.995197i \(0.468790\pi\)
\(858\) 0 0
\(859\) 33.4569 1.14153 0.570767 0.821112i \(-0.306646\pi\)
0.570767 + 0.821112i \(0.306646\pi\)
\(860\) −9.49338 −0.323722
\(861\) 25.1582 0.857389
\(862\) −20.0088 −0.681503
\(863\) 10.8522 0.369413 0.184707 0.982794i \(-0.440867\pi\)
0.184707 + 0.982794i \(0.440867\pi\)
\(864\) 10.0021 0.340278
\(865\) 19.8698 0.675593
\(866\) −38.7447 −1.31660
\(867\) −49.7927 −1.69105
\(868\) −5.70820 −0.193749
\(869\) 0 0
\(870\) −29.0901 −0.986246
\(871\) −0.496925 −0.0168377
\(872\) −18.4763 −0.625687
\(873\) −61.5617 −2.08355
\(874\) 5.18501 0.175386
\(875\) −1.00000 −0.0338062
\(876\) 15.6911 0.530154
\(877\) −21.1287 −0.713465 −0.356732 0.934207i \(-0.616109\pi\)
−0.356732 + 0.934207i \(0.616109\pi\)
\(878\) −5.62560 −0.189855
\(879\) 19.7064 0.664681
\(880\) 0 0
\(881\) 24.5805 0.828137 0.414069 0.910246i \(-0.364107\pi\)
0.414069 + 0.910246i \(0.364107\pi\)
\(882\) 6.28284 0.211554
\(883\) −15.4728 −0.520700 −0.260350 0.965514i \(-0.583838\pi\)
−0.260350 + 0.965514i \(0.583838\pi\)
\(884\) 0.286226 0.00962682
\(885\) −40.2456 −1.35284
\(886\) −15.6763 −0.526656
\(887\) −20.4358 −0.686165 −0.343083 0.939305i \(-0.611471\pi\)
−0.343083 + 0.939305i \(0.611471\pi\)
\(888\) 10.8575 0.364353
\(889\) −16.2382 −0.544610
\(890\) −9.63799 −0.323066
\(891\) 0 0
\(892\) −5.58986 −0.187162
\(893\) 22.3156 0.746763
\(894\) 27.3290 0.914017
\(895\) 8.61246 0.287883
\(896\) 1.00000 0.0334077
\(897\) −2.15137 −0.0718322
\(898\) −26.1032 −0.871075
\(899\) −54.5009 −1.81771
\(900\) 6.28284 0.209428
\(901\) −2.26971 −0.0756151
\(902\) 0 0
\(903\) 28.9242 0.962538
\(904\) −12.6902 −0.422071
\(905\) −11.7345 −0.390067
\(906\) 66.9471 2.22417
\(907\) 30.9777 1.02860 0.514298 0.857611i \(-0.328052\pi\)
0.514298 + 0.857611i \(0.328052\pi\)
\(908\) 6.32206 0.209805
\(909\) 5.68194 0.188458
\(910\) −0.353057 −0.0117037
\(911\) −29.0687 −0.963090 −0.481545 0.876421i \(-0.659924\pi\)
−0.481545 + 0.876421i \(0.659924\pi\)
\(912\) −7.89879 −0.261555
\(913\) 0 0
\(914\) 35.5911 1.17725
\(915\) −20.4320 −0.675462
\(916\) −3.89113 −0.128566
\(917\) 8.08882 0.267116
\(918\) 8.10877 0.267629
\(919\) −38.2375 −1.26134 −0.630669 0.776052i \(-0.717220\pi\)
−0.630669 + 0.776052i \(0.717220\pi\)
\(920\) 2.00000 0.0659380
\(921\) 62.9077 2.07288
\(922\) 9.66571 0.318323
\(923\) −4.56105 −0.150129
\(924\) 0 0
\(925\) 3.56360 0.117170
\(926\) −29.9726 −0.984962
\(927\) 70.4953 2.31537
\(928\) 9.54782 0.313423
\(929\) 38.7575 1.27159 0.635797 0.771857i \(-0.280672\pi\)
0.635797 + 0.771857i \(0.280672\pi\)
\(930\) 17.3916 0.570294
\(931\) −2.59251 −0.0849660
\(932\) −15.1659 −0.496774
\(933\) 71.4118 2.33792
\(934\) 30.5086 0.998272
\(935\) 0 0
\(936\) 2.21820 0.0725042
\(937\) −10.4997 −0.343010 −0.171505 0.985183i \(-0.554863\pi\)
−0.171505 + 0.985183i \(0.554863\pi\)
\(938\) −1.40749 −0.0459563
\(939\) 36.2746 1.18378
\(940\) 8.60773 0.280753
\(941\) 10.1957 0.332369 0.166185 0.986095i \(-0.446855\pi\)
0.166185 + 0.986095i \(0.446855\pi\)
\(942\) 0.203619 0.00663425
\(943\) −16.5146 −0.537790
\(944\) 13.2092 0.429924
\(945\) −10.0021 −0.325368
\(946\) 0 0
\(947\) −47.2023 −1.53387 −0.766934 0.641726i \(-0.778219\pi\)
−0.766934 + 0.641726i \(0.778219\pi\)
\(948\) 11.1114 0.360882
\(949\) 1.81827 0.0590236
\(950\) −2.59251 −0.0841120
\(951\) −104.865 −3.40050
\(952\) 0.810708 0.0262752
\(953\) −5.18794 −0.168054 −0.0840269 0.996463i \(-0.526778\pi\)
−0.0840269 + 0.996463i \(0.526778\pi\)
\(954\) −17.5899 −0.569493
\(955\) 8.15063 0.263748
\(956\) 21.0949 0.682258
\(957\) 0 0
\(958\) 38.7127 1.25075
\(959\) 10.4019 0.335896
\(960\) −3.04678 −0.0983343
\(961\) 1.58359 0.0510836
\(962\) 1.25815 0.0405645
\(963\) −55.8931 −1.80113
\(964\) 11.9416 0.384614
\(965\) 8.74185 0.281410
\(966\) −6.09355 −0.196057
\(967\) −18.9841 −0.610486 −0.305243 0.952274i \(-0.598738\pi\)
−0.305243 + 0.952274i \(0.598738\pi\)
\(968\) 0 0
\(969\) −6.40361 −0.205714
\(970\) 9.79837 0.314607
\(971\) −41.9080 −1.34489 −0.672447 0.740146i \(-0.734757\pi\)
−0.672447 + 0.740146i \(0.734757\pi\)
\(972\) 5.41432 0.173664
\(973\) −0.757168 −0.0242737
\(974\) 33.8375 1.08422
\(975\) 1.07569 0.0344495
\(976\) 6.70611 0.214657
\(977\) 8.12073 0.259805 0.129903 0.991527i \(-0.458534\pi\)
0.129903 + 0.991527i \(0.458534\pi\)
\(978\) 67.8491 2.16958
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −116.084 −3.70627
\(982\) 18.5755 0.592768
\(983\) −38.8143 −1.23799 −0.618993 0.785397i \(-0.712459\pi\)
−0.618993 + 0.785397i \(0.712459\pi\)
\(984\) 25.1582 0.802014
\(985\) 14.4568 0.460632
\(986\) 7.74050 0.246508
\(987\) −26.2258 −0.834777
\(988\) −0.915302 −0.0291196
\(989\) −18.9868 −0.603744
\(990\) 0 0
\(991\) 1.30139 0.0413400 0.0206700 0.999786i \(-0.493420\pi\)
0.0206700 + 0.999786i \(0.493420\pi\)
\(992\) −5.70820 −0.181236
\(993\) −35.5773 −1.12901
\(994\) −12.9187 −0.409758
\(995\) 11.0446 0.350137
\(996\) 8.12938 0.257589
\(997\) −14.8018 −0.468777 −0.234389 0.972143i \(-0.575309\pi\)
−0.234389 + 0.972143i \(0.575309\pi\)
\(998\) −14.7069 −0.465539
\(999\) 35.6434 1.12771
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.ct.1.4 4
11.2 odd 10 770.2.n.e.631.2 yes 8
11.6 odd 10 770.2.n.e.421.2 8
11.10 odd 2 8470.2.a.cq.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.e.421.2 8 11.6 odd 10
770.2.n.e.631.2 yes 8 11.2 odd 10
8470.2.a.cq.1.4 4 11.10 odd 2
8470.2.a.ct.1.4 4 1.1 even 1 trivial