Properties

Label 8470.2.a.bm.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.73205 q^{12} +2.26795 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} -2.73205 q^{17} +3.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} -5.00000 q^{23} +1.73205 q^{24} +1.00000 q^{25} -2.26795 q^{26} +5.19615 q^{27} -1.00000 q^{28} -8.00000 q^{29} -1.73205 q^{30} +0.732051 q^{31} -1.00000 q^{32} +2.73205 q^{34} +1.00000 q^{35} +6.00000 q^{37} -3.73205 q^{38} -3.92820 q^{39} +1.00000 q^{40} -4.19615 q^{41} -1.73205 q^{42} +8.73205 q^{43} +5.00000 q^{46} -1.26795 q^{47} -1.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.73205 q^{51} +2.26795 q^{52} +4.19615 q^{53} -5.19615 q^{54} +1.00000 q^{56} -6.46410 q^{57} +8.00000 q^{58} -12.4641 q^{59} +1.73205 q^{60} -10.9282 q^{61} -0.732051 q^{62} +1.00000 q^{64} -2.26795 q^{65} +7.66025 q^{67} -2.73205 q^{68} +8.66025 q^{69} -1.00000 q^{70} -6.92820 q^{71} +15.1244 q^{73} -6.00000 q^{74} -1.73205 q^{75} +3.73205 q^{76} +3.92820 q^{78} +7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} +4.19615 q^{82} -5.00000 q^{83} +1.73205 q^{84} +2.73205 q^{85} -8.73205 q^{86} +13.8564 q^{87} +3.66025 q^{89} -2.26795 q^{91} -5.00000 q^{92} -1.26795 q^{93} +1.26795 q^{94} -3.73205 q^{95} +1.73205 q^{96} -2.92820 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + 2q^{10} + 8q^{13} + 2q^{14} + 2q^{16} - 2q^{17} + 4q^{19} - 2q^{20} - 10q^{23} + 2q^{25} - 8q^{26} - 2q^{28} - 16q^{29} - 2q^{31} - 2q^{32} + 2q^{34} + 2q^{35} + 12q^{37} - 4q^{38} + 6q^{39} + 2q^{40} + 2q^{41} + 14q^{43} + 10q^{46} - 6q^{47} + 2q^{49} - 2q^{50} + 6q^{51} + 8q^{52} - 2q^{53} + 2q^{56} - 6q^{57} + 16q^{58} - 18q^{59} - 8q^{61} + 2q^{62} + 2q^{64} - 8q^{65} - 2q^{67} - 2q^{68} - 2q^{70} + 6q^{73} - 12q^{74} + 4q^{76} - 6q^{78} + 14q^{79} - 2q^{80} - 18q^{81} - 2q^{82} - 10q^{83} + 2q^{85} - 14q^{86} - 10q^{89} - 8q^{91} - 10q^{92} - 6q^{93} + 6q^{94} - 4q^{95} + 8q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −1.73205 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.73205 0.707107
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.73205 −0.500000
\(13\) 2.26795 0.629016 0.314508 0.949255i \(-0.398160\pi\)
0.314508 + 0.949255i \(0.398160\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.73205 0.447214
\(16\) 1.00000 0.250000
\(17\) −2.73205 −0.662620 −0.331310 0.943522i \(-0.607491\pi\)
−0.331310 + 0.943522i \(0.607491\pi\)
\(18\) 0 0
\(19\) 3.73205 0.856191 0.428096 0.903733i \(-0.359185\pi\)
0.428096 + 0.903733i \(0.359185\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) −2.26795 −0.444781
\(27\) 5.19615 1.00000
\(28\) −1.00000 −0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.73205 −0.316228
\(31\) 0.732051 0.131480 0.0657401 0.997837i \(-0.479059\pi\)
0.0657401 + 0.997837i \(0.479059\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.73205 0.468543
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −3.73205 −0.605419
\(39\) −3.92820 −0.629016
\(40\) 1.00000 0.158114
\(41\) −4.19615 −0.655329 −0.327664 0.944794i \(-0.606262\pi\)
−0.327664 + 0.944794i \(0.606262\pi\)
\(42\) −1.73205 −0.267261
\(43\) 8.73205 1.33163 0.665813 0.746119i \(-0.268085\pi\)
0.665813 + 0.746119i \(0.268085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) −1.26795 −0.184949 −0.0924747 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) −1.73205 −0.250000
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.73205 0.662620
\(52\) 2.26795 0.314508
\(53\) 4.19615 0.576386 0.288193 0.957572i \(-0.406946\pi\)
0.288193 + 0.957572i \(0.406946\pi\)
\(54\) −5.19615 −0.707107
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.46410 −0.856191
\(58\) 8.00000 1.05045
\(59\) −12.4641 −1.62269 −0.811344 0.584569i \(-0.801264\pi\)
−0.811344 + 0.584569i \(0.801264\pi\)
\(60\) 1.73205 0.223607
\(61\) −10.9282 −1.39921 −0.699607 0.714528i \(-0.746641\pi\)
−0.699607 + 0.714528i \(0.746641\pi\)
\(62\) −0.732051 −0.0929705
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.26795 −0.281304
\(66\) 0 0
\(67\) 7.66025 0.935849 0.467924 0.883768i \(-0.345002\pi\)
0.467924 + 0.883768i \(0.345002\pi\)
\(68\) −2.73205 −0.331310
\(69\) 8.66025 1.04257
\(70\) −1.00000 −0.119523
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) 15.1244 1.77017 0.885086 0.465428i \(-0.154099\pi\)
0.885086 + 0.465428i \(0.154099\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.73205 −0.200000
\(76\) 3.73205 0.428096
\(77\) 0 0
\(78\) 3.92820 0.444781
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) −1.00000 −0.111803
\(81\) −9.00000 −1.00000
\(82\) 4.19615 0.463388
\(83\) −5.00000 −0.548821 −0.274411 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) 1.73205 0.188982
\(85\) 2.73205 0.296333
\(86\) −8.73205 −0.941601
\(87\) 13.8564 1.48556
\(88\) 0 0
\(89\) 3.66025 0.387986 0.193993 0.981003i \(-0.437856\pi\)
0.193993 + 0.981003i \(0.437856\pi\)
\(90\) 0 0
\(91\) −2.26795 −0.237746
\(92\) −5.00000 −0.521286
\(93\) −1.26795 −0.131480
\(94\) 1.26795 0.130779
\(95\) −3.73205 −0.382900
\(96\) 1.73205 0.176777
\(97\) −2.92820 −0.297314 −0.148657 0.988889i \(-0.547495\pi\)
−0.148657 + 0.988889i \(0.547495\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.4641 1.83725 0.918623 0.395134i \(-0.129302\pi\)
0.918623 + 0.395134i \(0.129302\pi\)
\(102\) −4.73205 −0.468543
\(103\) −7.26795 −0.716132 −0.358066 0.933696i \(-0.616564\pi\)
−0.358066 + 0.933696i \(0.616564\pi\)
\(104\) −2.26795 −0.222391
\(105\) −1.73205 −0.169031
\(106\) −4.19615 −0.407566
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 5.19615 0.500000
\(109\) 18.0526 1.72912 0.864561 0.502528i \(-0.167597\pi\)
0.864561 + 0.502528i \(0.167597\pi\)
\(110\) 0 0
\(111\) −10.3923 −0.986394
\(112\) −1.00000 −0.0944911
\(113\) 10.1244 0.952419 0.476210 0.879332i \(-0.342010\pi\)
0.476210 + 0.879332i \(0.342010\pi\)
\(114\) 6.46410 0.605419
\(115\) 5.00000 0.466252
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 12.4641 1.14741
\(119\) 2.73205 0.250447
\(120\) −1.73205 −0.158114
\(121\) 0 0
\(122\) 10.9282 0.989393
\(123\) 7.26795 0.655329
\(124\) 0.732051 0.0657401
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.1244 −1.60828 −0.804138 0.594442i \(-0.797373\pi\)
−0.804138 + 0.594442i \(0.797373\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.1244 −1.33163
\(130\) 2.26795 0.198912
\(131\) 2.80385 0.244973 0.122487 0.992470i \(-0.460913\pi\)
0.122487 + 0.992470i \(0.460913\pi\)
\(132\) 0 0
\(133\) −3.73205 −0.323610
\(134\) −7.66025 −0.661745
\(135\) −5.19615 −0.447214
\(136\) 2.73205 0.234271
\(137\) −2.66025 −0.227281 −0.113640 0.993522i \(-0.536251\pi\)
−0.113640 + 0.993522i \(0.536251\pi\)
\(138\) −8.66025 −0.737210
\(139\) 6.80385 0.577095 0.288547 0.957466i \(-0.406828\pi\)
0.288547 + 0.957466i \(0.406828\pi\)
\(140\) 1.00000 0.0845154
\(141\) 2.19615 0.184949
\(142\) 6.92820 0.581402
\(143\) 0 0
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) −15.1244 −1.25170
\(147\) −1.73205 −0.142857
\(148\) 6.00000 0.493197
\(149\) −12.5359 −1.02698 −0.513490 0.858095i \(-0.671648\pi\)
−0.513490 + 0.858095i \(0.671648\pi\)
\(150\) 1.73205 0.141421
\(151\) −8.85641 −0.720724 −0.360362 0.932813i \(-0.617347\pi\)
−0.360362 + 0.932813i \(0.617347\pi\)
\(152\) −3.73205 −0.302709
\(153\) 0 0
\(154\) 0 0
\(155\) −0.732051 −0.0587997
\(156\) −3.92820 −0.314508
\(157\) 15.3923 1.22844 0.614220 0.789135i \(-0.289471\pi\)
0.614220 + 0.789135i \(0.289471\pi\)
\(158\) −7.00000 −0.556890
\(159\) −7.26795 −0.576386
\(160\) 1.00000 0.0790569
\(161\) 5.00000 0.394055
\(162\) 9.00000 0.707107
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −4.19615 −0.327664
\(165\) 0 0
\(166\) 5.00000 0.388075
\(167\) 17.6603 1.36659 0.683296 0.730142i \(-0.260546\pi\)
0.683296 + 0.730142i \(0.260546\pi\)
\(168\) −1.73205 −0.133631
\(169\) −7.85641 −0.604339
\(170\) −2.73205 −0.209539
\(171\) 0 0
\(172\) 8.73205 0.665813
\(173\) 19.8564 1.50965 0.754827 0.655924i \(-0.227721\pi\)
0.754827 + 0.655924i \(0.227721\pi\)
\(174\) −13.8564 −1.05045
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 21.5885 1.62269
\(178\) −3.66025 −0.274348
\(179\) −5.80385 −0.433800 −0.216900 0.976194i \(-0.569595\pi\)
−0.216900 + 0.976194i \(0.569595\pi\)
\(180\) 0 0
\(181\) −5.33975 −0.396900 −0.198450 0.980111i \(-0.563591\pi\)
−0.198450 + 0.980111i \(0.563591\pi\)
\(182\) 2.26795 0.168112
\(183\) 18.9282 1.39921
\(184\) 5.00000 0.368605
\(185\) −6.00000 −0.441129
\(186\) 1.26795 0.0929705
\(187\) 0 0
\(188\) −1.26795 −0.0924747
\(189\) −5.19615 −0.377964
\(190\) 3.73205 0.270751
\(191\) 8.66025 0.626634 0.313317 0.949649i \(-0.398560\pi\)
0.313317 + 0.949649i \(0.398560\pi\)
\(192\) −1.73205 −0.125000
\(193\) 9.39230 0.676073 0.338036 0.941133i \(-0.390237\pi\)
0.338036 + 0.941133i \(0.390237\pi\)
\(194\) 2.92820 0.210233
\(195\) 3.92820 0.281304
\(196\) 1.00000 0.0714286
\(197\) 2.53590 0.180675 0.0903376 0.995911i \(-0.471205\pi\)
0.0903376 + 0.995911i \(0.471205\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −13.2679 −0.935849
\(202\) −18.4641 −1.29913
\(203\) 8.00000 0.561490
\(204\) 4.73205 0.331310
\(205\) 4.19615 0.293072
\(206\) 7.26795 0.506382
\(207\) 0 0
\(208\) 2.26795 0.157254
\(209\) 0 0
\(210\) 1.73205 0.119523
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 4.19615 0.288193
\(213\) 12.0000 0.822226
\(214\) 2.00000 0.136717
\(215\) −8.73205 −0.595521
\(216\) −5.19615 −0.353553
\(217\) −0.732051 −0.0496948
\(218\) −18.0526 −1.22267
\(219\) −26.1962 −1.77017
\(220\) 0 0
\(221\) −6.19615 −0.416798
\(222\) 10.3923 0.697486
\(223\) −14.7321 −0.986531 −0.493266 0.869879i \(-0.664197\pi\)
−0.493266 + 0.869879i \(0.664197\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −10.1244 −0.673462
\(227\) 17.3205 1.14960 0.574801 0.818293i \(-0.305079\pi\)
0.574801 + 0.818293i \(0.305079\pi\)
\(228\) −6.46410 −0.428096
\(229\) −21.8564 −1.44431 −0.722156 0.691730i \(-0.756849\pi\)
−0.722156 + 0.691730i \(0.756849\pi\)
\(230\) −5.00000 −0.329690
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) −12.4641 −0.816550 −0.408275 0.912859i \(-0.633870\pi\)
−0.408275 + 0.912859i \(0.633870\pi\)
\(234\) 0 0
\(235\) 1.26795 0.0827119
\(236\) −12.4641 −0.811344
\(237\) −12.1244 −0.787562
\(238\) −2.73205 −0.177093
\(239\) −2.46410 −0.159389 −0.0796947 0.996819i \(-0.525395\pi\)
−0.0796947 + 0.996819i \(0.525395\pi\)
\(240\) 1.73205 0.111803
\(241\) −10.5359 −0.678677 −0.339338 0.940664i \(-0.610203\pi\)
−0.339338 + 0.940664i \(0.610203\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −10.9282 −0.699607
\(245\) −1.00000 −0.0638877
\(246\) −7.26795 −0.463388
\(247\) 8.46410 0.538558
\(248\) −0.732051 −0.0464853
\(249\) 8.66025 0.548821
\(250\) 1.00000 0.0632456
\(251\) 14.3923 0.908434 0.454217 0.890891i \(-0.349919\pi\)
0.454217 + 0.890891i \(0.349919\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 18.1244 1.13722
\(255\) −4.73205 −0.296333
\(256\) 1.00000 0.0625000
\(257\) −2.19615 −0.136992 −0.0684961 0.997651i \(-0.521820\pi\)
−0.0684961 + 0.997651i \(0.521820\pi\)
\(258\) 15.1244 0.941601
\(259\) −6.00000 −0.372822
\(260\) −2.26795 −0.140652
\(261\) 0 0
\(262\) −2.80385 −0.173222
\(263\) 14.1244 0.870945 0.435473 0.900202i \(-0.356581\pi\)
0.435473 + 0.900202i \(0.356581\pi\)
\(264\) 0 0
\(265\) −4.19615 −0.257768
\(266\) 3.73205 0.228827
\(267\) −6.33975 −0.387986
\(268\) 7.66025 0.467924
\(269\) −18.2679 −1.11382 −0.556908 0.830574i \(-0.688013\pi\)
−0.556908 + 0.830574i \(0.688013\pi\)
\(270\) 5.19615 0.316228
\(271\) −14.5359 −0.882993 −0.441496 0.897263i \(-0.645552\pi\)
−0.441496 + 0.897263i \(0.645552\pi\)
\(272\) −2.73205 −0.165655
\(273\) 3.92820 0.237746
\(274\) 2.66025 0.160712
\(275\) 0 0
\(276\) 8.66025 0.521286
\(277\) −12.1962 −0.732796 −0.366398 0.930458i \(-0.619409\pi\)
−0.366398 + 0.930458i \(0.619409\pi\)
\(278\) −6.80385 −0.408068
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −4.26795 −0.254605 −0.127302 0.991864i \(-0.540632\pi\)
−0.127302 + 0.991864i \(0.540632\pi\)
\(282\) −2.19615 −0.130779
\(283\) 15.7846 0.938298 0.469149 0.883119i \(-0.344561\pi\)
0.469149 + 0.883119i \(0.344561\pi\)
\(284\) −6.92820 −0.411113
\(285\) 6.46410 0.382900
\(286\) 0 0
\(287\) 4.19615 0.247691
\(288\) 0 0
\(289\) −9.53590 −0.560935
\(290\) −8.00000 −0.469776
\(291\) 5.07180 0.297314
\(292\) 15.1244 0.885086
\(293\) −8.26795 −0.483019 −0.241509 0.970398i \(-0.577642\pi\)
−0.241509 + 0.970398i \(0.577642\pi\)
\(294\) 1.73205 0.101015
\(295\) 12.4641 0.725688
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 12.5359 0.726185
\(299\) −11.3397 −0.655794
\(300\) −1.73205 −0.100000
\(301\) −8.73205 −0.503307
\(302\) 8.85641 0.509629
\(303\) −31.9808 −1.83725
\(304\) 3.73205 0.214048
\(305\) 10.9282 0.625747
\(306\) 0 0
\(307\) 20.9282 1.19444 0.597218 0.802079i \(-0.296273\pi\)
0.597218 + 0.802079i \(0.296273\pi\)
\(308\) 0 0
\(309\) 12.5885 0.716132
\(310\) 0.732051 0.0415777
\(311\) 9.26795 0.525537 0.262769 0.964859i \(-0.415364\pi\)
0.262769 + 0.964859i \(0.415364\pi\)
\(312\) 3.92820 0.222391
\(313\) −26.3923 −1.49178 −0.745891 0.666068i \(-0.767976\pi\)
−0.745891 + 0.666068i \(0.767976\pi\)
\(314\) −15.3923 −0.868638
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) −19.1244 −1.07413 −0.537065 0.843541i \(-0.680467\pi\)
−0.537065 + 0.843541i \(0.680467\pi\)
\(318\) 7.26795 0.407566
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.46410 0.193347
\(322\) −5.00000 −0.278639
\(323\) −10.1962 −0.567329
\(324\) −9.00000 −0.500000
\(325\) 2.26795 0.125803
\(326\) −10.0000 −0.553849
\(327\) −31.2679 −1.72912
\(328\) 4.19615 0.231694
\(329\) 1.26795 0.0699043
\(330\) 0 0
\(331\) −4.05256 −0.222749 −0.111374 0.993779i \(-0.535525\pi\)
−0.111374 + 0.993779i \(0.535525\pi\)
\(332\) −5.00000 −0.274411
\(333\) 0 0
\(334\) −17.6603 −0.966326
\(335\) −7.66025 −0.418524
\(336\) 1.73205 0.0944911
\(337\) −11.3923 −0.620578 −0.310289 0.950642i \(-0.600426\pi\)
−0.310289 + 0.950642i \(0.600426\pi\)
\(338\) 7.85641 0.427332
\(339\) −17.5359 −0.952419
\(340\) 2.73205 0.148166
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −8.73205 −0.470801
\(345\) −8.66025 −0.466252
\(346\) −19.8564 −1.06749
\(347\) 8.53590 0.458231 0.229116 0.973399i \(-0.426417\pi\)
0.229116 + 0.973399i \(0.426417\pi\)
\(348\) 13.8564 0.742781
\(349\) 6.46410 0.346015 0.173008 0.984920i \(-0.444651\pi\)
0.173008 + 0.984920i \(0.444651\pi\)
\(350\) 1.00000 0.0534522
\(351\) 11.7846 0.629016
\(352\) 0 0
\(353\) 1.80385 0.0960091 0.0480046 0.998847i \(-0.484714\pi\)
0.0480046 + 0.998847i \(0.484714\pi\)
\(354\) −21.5885 −1.14741
\(355\) 6.92820 0.367711
\(356\) 3.66025 0.193993
\(357\) −4.73205 −0.250447
\(358\) 5.80385 0.306743
\(359\) 33.7128 1.77929 0.889647 0.456649i \(-0.150950\pi\)
0.889647 + 0.456649i \(0.150950\pi\)
\(360\) 0 0
\(361\) −5.07180 −0.266937
\(362\) 5.33975 0.280651
\(363\) 0 0
\(364\) −2.26795 −0.118873
\(365\) −15.1244 −0.791645
\(366\) −18.9282 −0.989393
\(367\) −20.7321 −1.08220 −0.541102 0.840957i \(-0.681993\pi\)
−0.541102 + 0.840957i \(0.681993\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) −4.19615 −0.217853
\(372\) −1.26795 −0.0657401
\(373\) 13.3205 0.689710 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(374\) 0 0
\(375\) 1.73205 0.0894427
\(376\) 1.26795 0.0653895
\(377\) −18.1436 −0.934443
\(378\) 5.19615 0.267261
\(379\) 34.7846 1.78677 0.893383 0.449297i \(-0.148325\pi\)
0.893383 + 0.449297i \(0.148325\pi\)
\(380\) −3.73205 −0.191450
\(381\) 31.3923 1.60828
\(382\) −8.66025 −0.443097
\(383\) 8.33975 0.426141 0.213071 0.977037i \(-0.431654\pi\)
0.213071 + 0.977037i \(0.431654\pi\)
\(384\) 1.73205 0.0883883
\(385\) 0 0
\(386\) −9.39230 −0.478056
\(387\) 0 0
\(388\) −2.92820 −0.148657
\(389\) 1.12436 0.0570071 0.0285035 0.999594i \(-0.490926\pi\)
0.0285035 + 0.999594i \(0.490926\pi\)
\(390\) −3.92820 −0.198912
\(391\) 13.6603 0.690829
\(392\) −1.00000 −0.0505076
\(393\) −4.85641 −0.244973
\(394\) −2.53590 −0.127757
\(395\) −7.00000 −0.352208
\(396\) 0 0
\(397\) −28.7846 −1.44466 −0.722329 0.691549i \(-0.756928\pi\)
−0.722329 + 0.691549i \(0.756928\pi\)
\(398\) 4.00000 0.200502
\(399\) 6.46410 0.323610
\(400\) 1.00000 0.0500000
\(401\) −25.4641 −1.27162 −0.635808 0.771847i \(-0.719333\pi\)
−0.635808 + 0.771847i \(0.719333\pi\)
\(402\) 13.2679 0.661745
\(403\) 1.66025 0.0827031
\(404\) 18.4641 0.918623
\(405\) 9.00000 0.447214
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) −4.73205 −0.234271
\(409\) 19.7128 0.974736 0.487368 0.873197i \(-0.337957\pi\)
0.487368 + 0.873197i \(0.337957\pi\)
\(410\) −4.19615 −0.207233
\(411\) 4.60770 0.227281
\(412\) −7.26795 −0.358066
\(413\) 12.4641 0.613318
\(414\) 0 0
\(415\) 5.00000 0.245440
\(416\) −2.26795 −0.111195
\(417\) −11.7846 −0.577095
\(418\) 0 0
\(419\) 2.07180 0.101214 0.0506069 0.998719i \(-0.483884\pi\)
0.0506069 + 0.998719i \(0.483884\pi\)
\(420\) −1.73205 −0.0845154
\(421\) −28.3923 −1.38376 −0.691878 0.722014i \(-0.743216\pi\)
−0.691878 + 0.722014i \(0.743216\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −4.19615 −0.203783
\(425\) −2.73205 −0.132524
\(426\) −12.0000 −0.581402
\(427\) 10.9282 0.528853
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 8.73205 0.421097
\(431\) 27.5359 1.32636 0.663179 0.748461i \(-0.269207\pi\)
0.663179 + 0.748461i \(0.269207\pi\)
\(432\) 5.19615 0.250000
\(433\) 18.0526 0.867551 0.433775 0.901021i \(-0.357181\pi\)
0.433775 + 0.901021i \(0.357181\pi\)
\(434\) 0.732051 0.0351396
\(435\) −13.8564 −0.664364
\(436\) 18.0526 0.864561
\(437\) −18.6603 −0.892641
\(438\) 26.1962 1.25170
\(439\) 11.2679 0.537790 0.268895 0.963170i \(-0.413342\pi\)
0.268895 + 0.963170i \(0.413342\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.19615 0.294721
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −10.3923 −0.493197
\(445\) −3.66025 −0.173513
\(446\) 14.7321 0.697583
\(447\) 21.7128 1.02698
\(448\) −1.00000 −0.0472456
\(449\) 10.6077 0.500608 0.250304 0.968167i \(-0.419469\pi\)
0.250304 + 0.968167i \(0.419469\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.1244 0.476210
\(453\) 15.3397 0.720724
\(454\) −17.3205 −0.812892
\(455\) 2.26795 0.106323
\(456\) 6.46410 0.302709
\(457\) −17.3923 −0.813578 −0.406789 0.913522i \(-0.633352\pi\)
−0.406789 + 0.913522i \(0.633352\pi\)
\(458\) 21.8564 1.02128
\(459\) −14.1962 −0.662620
\(460\) 5.00000 0.233126
\(461\) −17.4641 −0.813384 −0.406692 0.913565i \(-0.633318\pi\)
−0.406692 + 0.913565i \(0.633318\pi\)
\(462\) 0 0
\(463\) −36.1769 −1.68128 −0.840642 0.541591i \(-0.817822\pi\)
−0.840642 + 0.541591i \(0.817822\pi\)
\(464\) −8.00000 −0.371391
\(465\) 1.26795 0.0587997
\(466\) 12.4641 0.577388
\(467\) −17.5885 −0.813897 −0.406948 0.913451i \(-0.633407\pi\)
−0.406948 + 0.913451i \(0.633407\pi\)
\(468\) 0 0
\(469\) −7.66025 −0.353718
\(470\) −1.26795 −0.0584861
\(471\) −26.6603 −1.22844
\(472\) 12.4641 0.573707
\(473\) 0 0
\(474\) 12.1244 0.556890
\(475\) 3.73205 0.171238
\(476\) 2.73205 0.125223
\(477\) 0 0
\(478\) 2.46410 0.112705
\(479\) 22.9808 1.05002 0.525009 0.851097i \(-0.324062\pi\)
0.525009 + 0.851097i \(0.324062\pi\)
\(480\) −1.73205 −0.0790569
\(481\) 13.6077 0.620457
\(482\) 10.5359 0.479897
\(483\) −8.66025 −0.394055
\(484\) 0 0
\(485\) 2.92820 0.132963
\(486\) 0 0
\(487\) −3.39230 −0.153720 −0.0768600 0.997042i \(-0.524489\pi\)
−0.0768600 + 0.997042i \(0.524489\pi\)
\(488\) 10.9282 0.494697
\(489\) −17.3205 −0.783260
\(490\) 1.00000 0.0451754
\(491\) 7.66025 0.345702 0.172851 0.984948i \(-0.444702\pi\)
0.172851 + 0.984948i \(0.444702\pi\)
\(492\) 7.26795 0.327664
\(493\) 21.8564 0.984363
\(494\) −8.46410 −0.380818
\(495\) 0 0
\(496\) 0.732051 0.0328701
\(497\) 6.92820 0.310772
\(498\) −8.66025 −0.388075
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −30.5885 −1.36659
\(502\) −14.3923 −0.642360
\(503\) −7.80385 −0.347956 −0.173978 0.984750i \(-0.555662\pi\)
−0.173978 + 0.984750i \(0.555662\pi\)
\(504\) 0 0
\(505\) −18.4641 −0.821642
\(506\) 0 0
\(507\) 13.6077 0.604339
\(508\) −18.1244 −0.804138
\(509\) −15.3397 −0.679922 −0.339961 0.940439i \(-0.610414\pi\)
−0.339961 + 0.940439i \(0.610414\pi\)
\(510\) 4.73205 0.209539
\(511\) −15.1244 −0.669062
\(512\) −1.00000 −0.0441942
\(513\) 19.3923 0.856191
\(514\) 2.19615 0.0968681
\(515\) 7.26795 0.320264
\(516\) −15.1244 −0.665813
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) −34.3923 −1.50965
\(520\) 2.26795 0.0994562
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) 2.60770 0.114027 0.0570133 0.998373i \(-0.481842\pi\)
0.0570133 + 0.998373i \(0.481842\pi\)
\(524\) 2.80385 0.122487
\(525\) 1.73205 0.0755929
\(526\) −14.1244 −0.615851
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 4.19615 0.182269
\(531\) 0 0
\(532\) −3.73205 −0.161805
\(533\) −9.51666 −0.412212
\(534\) 6.33975 0.274348
\(535\) 2.00000 0.0864675
\(536\) −7.66025 −0.330873
\(537\) 10.0526 0.433800
\(538\) 18.2679 0.787587
\(539\) 0 0
\(540\) −5.19615 −0.223607
\(541\) −9.60770 −0.413067 −0.206534 0.978440i \(-0.566218\pi\)
−0.206534 + 0.978440i \(0.566218\pi\)
\(542\) 14.5359 0.624370
\(543\) 9.24871 0.396900
\(544\) 2.73205 0.117136
\(545\) −18.0526 −0.773287
\(546\) −3.92820 −0.168112
\(547\) −37.5167 −1.60410 −0.802048 0.597259i \(-0.796256\pi\)
−0.802048 + 0.597259i \(0.796256\pi\)
\(548\) −2.66025 −0.113640
\(549\) 0 0
\(550\) 0 0
\(551\) −29.8564 −1.27193
\(552\) −8.66025 −0.368605
\(553\) −7.00000 −0.297670
\(554\) 12.1962 0.518165
\(555\) 10.3923 0.441129
\(556\) 6.80385 0.288547
\(557\) −27.3731 −1.15983 −0.579917 0.814676i \(-0.696915\pi\)
−0.579917 + 0.814676i \(0.696915\pi\)
\(558\) 0 0
\(559\) 19.8038 0.837614
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 4.26795 0.180033
\(563\) 1.53590 0.0647304 0.0323652 0.999476i \(-0.489696\pi\)
0.0323652 + 0.999476i \(0.489696\pi\)
\(564\) 2.19615 0.0924747
\(565\) −10.1244 −0.425935
\(566\) −15.7846 −0.663477
\(567\) 9.00000 0.377964
\(568\) 6.92820 0.290701
\(569\) −38.1244 −1.59826 −0.799128 0.601161i \(-0.794705\pi\)
−0.799128 + 0.601161i \(0.794705\pi\)
\(570\) −6.46410 −0.270751
\(571\) −13.4641 −0.563455 −0.281728 0.959494i \(-0.590907\pi\)
−0.281728 + 0.959494i \(0.590907\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) −4.19615 −0.175144
\(575\) −5.00000 −0.208514
\(576\) 0 0
\(577\) 18.4449 0.767870 0.383935 0.923360i \(-0.374569\pi\)
0.383935 + 0.923360i \(0.374569\pi\)
\(578\) 9.53590 0.396641
\(579\) −16.2679 −0.676073
\(580\) 8.00000 0.332182
\(581\) 5.00000 0.207435
\(582\) −5.07180 −0.210233
\(583\) 0 0
\(584\) −15.1244 −0.625850
\(585\) 0 0
\(586\) 8.26795 0.341546
\(587\) 14.5167 0.599167 0.299583 0.954070i \(-0.403152\pi\)
0.299583 + 0.954070i \(0.403152\pi\)
\(588\) −1.73205 −0.0714286
\(589\) 2.73205 0.112572
\(590\) −12.4641 −0.513139
\(591\) −4.39230 −0.180675
\(592\) 6.00000 0.246598
\(593\) 29.8564 1.22606 0.613028 0.790061i \(-0.289951\pi\)
0.613028 + 0.790061i \(0.289951\pi\)
\(594\) 0 0
\(595\) −2.73205 −0.112003
\(596\) −12.5359 −0.513490
\(597\) 6.92820 0.283552
\(598\) 11.3397 0.463717
\(599\) −31.9808 −1.30670 −0.653349 0.757057i \(-0.726637\pi\)
−0.653349 + 0.757057i \(0.726637\pi\)
\(600\) 1.73205 0.0707107
\(601\) −3.66025 −0.149305 −0.0746524 0.997210i \(-0.523785\pi\)
−0.0746524 + 0.997210i \(0.523785\pi\)
\(602\) 8.73205 0.355892
\(603\) 0 0
\(604\) −8.85641 −0.360362
\(605\) 0 0
\(606\) 31.9808 1.29913
\(607\) 0.248711 0.0100949 0.00504744 0.999987i \(-0.498393\pi\)
0.00504744 + 0.999987i \(0.498393\pi\)
\(608\) −3.73205 −0.151355
\(609\) −13.8564 −0.561490
\(610\) −10.9282 −0.442470
\(611\) −2.87564 −0.116336
\(612\) 0 0
\(613\) −37.2679 −1.50524 −0.752619 0.658456i \(-0.771210\pi\)
−0.752619 + 0.658456i \(0.771210\pi\)
\(614\) −20.9282 −0.844594
\(615\) −7.26795 −0.293072
\(616\) 0 0
\(617\) 13.6077 0.547825 0.273913 0.961755i \(-0.411682\pi\)
0.273913 + 0.961755i \(0.411682\pi\)
\(618\) −12.5885 −0.506382
\(619\) 32.7128 1.31484 0.657419 0.753525i \(-0.271648\pi\)
0.657419 + 0.753525i \(0.271648\pi\)
\(620\) −0.732051 −0.0293999
\(621\) −25.9808 −1.04257
\(622\) −9.26795 −0.371611
\(623\) −3.66025 −0.146645
\(624\) −3.92820 −0.157254
\(625\) 1.00000 0.0400000
\(626\) 26.3923 1.05485
\(627\) 0 0
\(628\) 15.3923 0.614220
\(629\) −16.3923 −0.653604
\(630\) 0 0
\(631\) −39.3205 −1.56532 −0.782662 0.622446i \(-0.786139\pi\)
−0.782662 + 0.622446i \(0.786139\pi\)
\(632\) −7.00000 −0.278445
\(633\) −13.8564 −0.550743
\(634\) 19.1244 0.759525
\(635\) 18.1244 0.719243
\(636\) −7.26795 −0.288193
\(637\) 2.26795 0.0898594
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −39.5359 −1.56157 −0.780787 0.624797i \(-0.785182\pi\)
−0.780787 + 0.624797i \(0.785182\pi\)
\(642\) −3.46410 −0.136717
\(643\) 23.1769 0.914008 0.457004 0.889465i \(-0.348922\pi\)
0.457004 + 0.889465i \(0.348922\pi\)
\(644\) 5.00000 0.197028
\(645\) 15.1244 0.595521
\(646\) 10.1962 0.401162
\(647\) −21.8038 −0.857198 −0.428599 0.903495i \(-0.640993\pi\)
−0.428599 + 0.903495i \(0.640993\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) −2.26795 −0.0889563
\(651\) 1.26795 0.0496948
\(652\) 10.0000 0.391630
\(653\) −2.19615 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(654\) 31.2679 1.22267
\(655\) −2.80385 −0.109555
\(656\) −4.19615 −0.163832
\(657\) 0 0
\(658\) −1.26795 −0.0494298
\(659\) 32.4449 1.26387 0.631936 0.775020i \(-0.282260\pi\)
0.631936 + 0.775020i \(0.282260\pi\)
\(660\) 0 0
\(661\) 3.33975 0.129901 0.0649505 0.997888i \(-0.479311\pi\)
0.0649505 + 0.997888i \(0.479311\pi\)
\(662\) 4.05256 0.157507
\(663\) 10.7321 0.416798
\(664\) 5.00000 0.194038
\(665\) 3.73205 0.144723
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 17.6603 0.683296
\(669\) 25.5167 0.986531
\(670\) 7.66025 0.295941
\(671\) 0 0
\(672\) −1.73205 −0.0668153
\(673\) −47.1051 −1.81577 −0.907884 0.419221i \(-0.862303\pi\)
−0.907884 + 0.419221i \(0.862303\pi\)
\(674\) 11.3923 0.438815
\(675\) 5.19615 0.200000
\(676\) −7.85641 −0.302169
\(677\) −45.9808 −1.76718 −0.883592 0.468257i \(-0.844882\pi\)
−0.883592 + 0.468257i \(0.844882\pi\)
\(678\) 17.5359 0.673462
\(679\) 2.92820 0.112374
\(680\) −2.73205 −0.104769
\(681\) −30.0000 −1.14960
\(682\) 0 0
\(683\) −38.9282 −1.48955 −0.744773 0.667318i \(-0.767442\pi\)
−0.744773 + 0.667318i \(0.767442\pi\)
\(684\) 0 0
\(685\) 2.66025 0.101643
\(686\) 1.00000 0.0381802
\(687\) 37.8564 1.44431
\(688\) 8.73205 0.332906
\(689\) 9.51666 0.362556
\(690\) 8.66025 0.329690
\(691\) 1.85641 0.0706210 0.0353105 0.999376i \(-0.488758\pi\)
0.0353105 + 0.999376i \(0.488758\pi\)
\(692\) 19.8564 0.754827
\(693\) 0 0
\(694\) −8.53590 −0.324018
\(695\) −6.80385 −0.258085
\(696\) −13.8564 −0.525226
\(697\) 11.4641 0.434234
\(698\) −6.46410 −0.244670
\(699\) 21.5885 0.816550
\(700\) −1.00000 −0.0377964
\(701\) −47.9090 −1.80950 −0.904748 0.425947i \(-0.859941\pi\)
−0.904748 + 0.425947i \(0.859941\pi\)
\(702\) −11.7846 −0.444781
\(703\) 22.3923 0.844542
\(704\) 0 0
\(705\) −2.19615 −0.0827119
\(706\) −1.80385 −0.0678887
\(707\) −18.4641 −0.694414
\(708\) 21.5885 0.811344
\(709\) 1.32051 0.0495927 0.0247964 0.999693i \(-0.492106\pi\)
0.0247964 + 0.999693i \(0.492106\pi\)
\(710\) −6.92820 −0.260011
\(711\) 0 0
\(712\) −3.66025 −0.137174
\(713\) −3.66025 −0.137078
\(714\) 4.73205 0.177093
\(715\) 0 0
\(716\) −5.80385 −0.216900
\(717\) 4.26795 0.159389
\(718\) −33.7128 −1.25815
\(719\) −34.9282 −1.30260 −0.651301 0.758819i \(-0.725777\pi\)
−0.651301 + 0.758819i \(0.725777\pi\)
\(720\) 0 0
\(721\) 7.26795 0.270673
\(722\) 5.07180 0.188753
\(723\) 18.2487 0.678677
\(724\) −5.33975 −0.198450
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −15.3205 −0.568206 −0.284103 0.958794i \(-0.591696\pi\)
−0.284103 + 0.958794i \(0.591696\pi\)
\(728\) 2.26795 0.0840558
\(729\) 27.0000 1.00000
\(730\) 15.1244 0.559778
\(731\) −23.8564 −0.882361
\(732\) 18.9282 0.699607
\(733\) 23.5885 0.871260 0.435630 0.900126i \(-0.356526\pi\)
0.435630 + 0.900126i \(0.356526\pi\)
\(734\) 20.7321 0.765234
\(735\) 1.73205 0.0638877
\(736\) 5.00000 0.184302
\(737\) 0 0
\(738\) 0 0
\(739\) −26.9808 −0.992503 −0.496252 0.868179i \(-0.665291\pi\)
−0.496252 + 0.868179i \(0.665291\pi\)
\(740\) −6.00000 −0.220564
\(741\) −14.6603 −0.538558
\(742\) 4.19615 0.154046
\(743\) −12.5359 −0.459898 −0.229949 0.973203i \(-0.573856\pi\)
−0.229949 + 0.973203i \(0.573856\pi\)
\(744\) 1.26795 0.0464853
\(745\) 12.5359 0.459280
\(746\) −13.3205 −0.487698
\(747\) 0 0
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) −1.73205 −0.0632456
\(751\) −42.3731 −1.54622 −0.773108 0.634275i \(-0.781299\pi\)
−0.773108 + 0.634275i \(0.781299\pi\)
\(752\) −1.26795 −0.0462373
\(753\) −24.9282 −0.908434
\(754\) 18.1436 0.660751
\(755\) 8.85641 0.322318
\(756\) −5.19615 −0.188982
\(757\) −25.9090 −0.941677 −0.470839 0.882219i \(-0.656049\pi\)
−0.470839 + 0.882219i \(0.656049\pi\)
\(758\) −34.7846 −1.26343
\(759\) 0 0
\(760\) 3.73205 0.135376
\(761\) 40.4449 1.46613 0.733063 0.680161i \(-0.238090\pi\)
0.733063 + 0.680161i \(0.238090\pi\)
\(762\) −31.3923 −1.13722
\(763\) −18.0526 −0.653547
\(764\) 8.66025 0.313317
\(765\) 0 0
\(766\) −8.33975 −0.301327
\(767\) −28.2679 −1.02070
\(768\) −1.73205 −0.0625000
\(769\) −34.7846 −1.25437 −0.627183 0.778872i \(-0.715792\pi\)
−0.627183 + 0.778872i \(0.715792\pi\)
\(770\) 0 0
\(771\) 3.80385 0.136992
\(772\) 9.39230 0.338036
\(773\) 27.5359 0.990397 0.495199 0.868780i \(-0.335095\pi\)
0.495199 + 0.868780i \(0.335095\pi\)
\(774\) 0 0
\(775\) 0.732051 0.0262960
\(776\) 2.92820 0.105116
\(777\) 10.3923 0.372822
\(778\) −1.12436 −0.0403101
\(779\) −15.6603 −0.561087
\(780\) 3.92820 0.140652
\(781\) 0 0
\(782\) −13.6603 −0.488490
\(783\) −41.5692 −1.48556
\(784\) 1.00000 0.0357143
\(785\) −15.3923 −0.549375
\(786\) 4.85641 0.173222
\(787\) −46.4974 −1.65745 −0.828727 0.559653i \(-0.810934\pi\)
−0.828727 + 0.559653i \(0.810934\pi\)
\(788\) 2.53590 0.0903376
\(789\) −24.4641 −0.870945
\(790\) 7.00000 0.249049
\(791\) −10.1244 −0.359981
\(792\) 0 0
\(793\) −24.7846 −0.880127
\(794\) 28.7846 1.02153
\(795\) 7.26795 0.257768
\(796\) −4.00000 −0.141776
\(797\) 34.1769 1.21061 0.605304 0.795994i \(-0.293051\pi\)
0.605304 + 0.795994i \(0.293051\pi\)
\(798\) −6.46410 −0.228827
\(799\) 3.46410 0.122551
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 25.4641 0.899169
\(803\) 0 0
\(804\) −13.2679 −0.467924
\(805\) −5.00000 −0.176227
\(806\) −1.66025 −0.0584800
\(807\) 31.6410 1.11382
\(808\) −18.4641 −0.649565
\(809\) 9.21539 0.323996 0.161998 0.986791i \(-0.448206\pi\)
0.161998 + 0.986791i \(0.448206\pi\)
\(810\) −9.00000 −0.316228
\(811\) 30.5359 1.07226 0.536130 0.844135i \(-0.319886\pi\)
0.536130 + 0.844135i \(0.319886\pi\)
\(812\) 8.00000 0.280745
\(813\) 25.1769 0.882993
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 4.73205 0.165655
\(817\) 32.5885 1.14013
\(818\) −19.7128 −0.689242
\(819\) 0 0
\(820\) 4.19615 0.146536
\(821\) −43.8564 −1.53060 −0.765300 0.643674i \(-0.777409\pi\)
−0.765300 + 0.643674i \(0.777409\pi\)
\(822\) −4.60770 −0.160712
\(823\) 42.7846 1.49138 0.745689 0.666294i \(-0.232121\pi\)
0.745689 + 0.666294i \(0.232121\pi\)
\(824\) 7.26795 0.253191
\(825\) 0 0
\(826\) −12.4641 −0.433682
\(827\) −19.9090 −0.692303 −0.346151 0.938179i \(-0.612512\pi\)
−0.346151 + 0.938179i \(0.612512\pi\)
\(828\) 0 0
\(829\) −0.124356 −0.00431905 −0.00215953 0.999998i \(-0.500687\pi\)
−0.00215953 + 0.999998i \(0.500687\pi\)
\(830\) −5.00000 −0.173553
\(831\) 21.1244 0.732796
\(832\) 2.26795 0.0786270
\(833\) −2.73205 −0.0946600
\(834\) 11.7846 0.408068
\(835\) −17.6603 −0.611158
\(836\) 0 0
\(837\) 3.80385 0.131480
\(838\) −2.07180 −0.0715690
\(839\) 12.1962 0.421058 0.210529 0.977588i \(-0.432481\pi\)
0.210529 + 0.977588i \(0.432481\pi\)
\(840\) 1.73205 0.0597614
\(841\) 35.0000 1.20690
\(842\) 28.3923 0.978463
\(843\) 7.39230 0.254605
\(844\) 8.00000 0.275371
\(845\) 7.85641 0.270269
\(846\) 0 0
\(847\) 0 0
\(848\) 4.19615 0.144096
\(849\) −27.3397 −0.938298
\(850\) 2.73205 0.0937086
\(851\) −30.0000 −1.02839
\(852\) 12.0000 0.411113
\(853\) −43.8372 −1.50096 −0.750478 0.660895i \(-0.770177\pi\)
−0.750478 + 0.660895i \(0.770177\pi\)
\(854\) −10.9282 −0.373955
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 15.6077 0.533149 0.266574 0.963814i \(-0.414108\pi\)
0.266574 + 0.963814i \(0.414108\pi\)
\(858\) 0 0
\(859\) 4.14359 0.141378 0.0706888 0.997498i \(-0.477480\pi\)
0.0706888 + 0.997498i \(0.477480\pi\)
\(860\) −8.73205 −0.297760
\(861\) −7.26795 −0.247691
\(862\) −27.5359 −0.937876
\(863\) −10.1436 −0.345292 −0.172646 0.984984i \(-0.555232\pi\)
−0.172646 + 0.984984i \(0.555232\pi\)
\(864\) −5.19615 −0.176777
\(865\) −19.8564 −0.675138
\(866\) −18.0526 −0.613451
\(867\) 16.5167 0.560935
\(868\) −0.732051 −0.0248474
\(869\) 0 0
\(870\) 13.8564 0.469776
\(871\) 17.3731 0.588664
\(872\) −18.0526 −0.611337
\(873\) 0 0
\(874\) 18.6603 0.631193
\(875\) 1.00000 0.0338062
\(876\) −26.1962 −0.885086
\(877\) −9.94744 −0.335901 −0.167951 0.985795i \(-0.553715\pi\)
−0.167951 + 0.985795i \(0.553715\pi\)
\(878\) −11.2679 −0.380275
\(879\) 14.3205 0.483019
\(880\) 0 0
\(881\) −53.4256 −1.79996 −0.899978 0.435936i \(-0.856417\pi\)
−0.899978 + 0.435936i \(0.856417\pi\)
\(882\) 0 0
\(883\) −32.6410 −1.09846 −0.549229 0.835672i \(-0.685078\pi\)
−0.549229 + 0.835672i \(0.685078\pi\)
\(884\) −6.19615 −0.208399
\(885\) −21.5885 −0.725688
\(886\) −4.00000 −0.134383
\(887\) −2.33975 −0.0785610 −0.0392805 0.999228i \(-0.512507\pi\)
−0.0392805 + 0.999228i \(0.512507\pi\)
\(888\) 10.3923 0.348743
\(889\) 18.1244 0.607871
\(890\) 3.66025 0.122692
\(891\) 0 0
\(892\) −14.7321 −0.493266
\(893\) −4.73205 −0.158352
\(894\) −21.7128 −0.726185
\(895\) 5.80385 0.194001
\(896\) 1.00000 0.0334077
\(897\) 19.6410 0.655794
\(898\) −10.6077 −0.353983
\(899\) −5.85641 −0.195322
\(900\) 0 0
\(901\) −11.4641 −0.381925
\(902\) 0 0
\(903\) 15.1244 0.503307
\(904\) −10.1244 −0.336731
\(905\) 5.33975 0.177499
\(906\) −15.3397 −0.509629
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 17.3205 0.574801
\(909\) 0 0
\(910\) −2.26795 −0.0751818
\(911\) 29.9808 0.993307 0.496653 0.867949i \(-0.334562\pi\)
0.496653 + 0.867949i \(0.334562\pi\)
\(912\) −6.46410 −0.214048
\(913\) 0 0
\(914\) 17.3923 0.575286
\(915\) −18.9282 −0.625747
\(916\) −21.8564 −0.722156
\(917\) −2.80385 −0.0925912
\(918\) 14.1962 0.468543
\(919\) −12.1436 −0.400580 −0.200290 0.979737i \(-0.564188\pi\)
−0.200290 + 0.979737i \(0.564188\pi\)
\(920\) −5.00000 −0.164845
\(921\) −36.2487 −1.19444
\(922\) 17.4641 0.575150
\(923\) −15.7128 −0.517194
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 36.1769 1.18885
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −47.3205 −1.55254 −0.776268 0.630403i \(-0.782890\pi\)
−0.776268 + 0.630403i \(0.782890\pi\)
\(930\) −1.26795 −0.0415777
\(931\) 3.73205 0.122313
\(932\) −12.4641 −0.408275
\(933\) −16.0526 −0.525537
\(934\) 17.5885 0.575512
\(935\) 0 0
\(936\) 0 0
\(937\) −10.5359 −0.344193 −0.172096 0.985080i \(-0.555054\pi\)
−0.172096 + 0.985080i \(0.555054\pi\)
\(938\) 7.66025 0.250116
\(939\) 45.7128 1.49178
\(940\) 1.26795 0.0413559
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 26.6603 0.868638
\(943\) 20.9808 0.683228
\(944\) −12.4641 −0.405672
\(945\) 5.19615 0.169031
\(946\) 0 0
\(947\) −60.0526 −1.95145 −0.975723 0.219008i \(-0.929718\pi\)
−0.975723 + 0.219008i \(0.929718\pi\)
\(948\) −12.1244 −0.393781
\(949\) 34.3013 1.11347
\(950\) −3.73205 −0.121084
\(951\) 33.1244 1.07413
\(952\) −2.73205 −0.0885463
\(953\) 2.60770 0.0844715 0.0422358 0.999108i \(-0.486552\pi\)
0.0422358 + 0.999108i \(0.486552\pi\)
\(954\) 0 0
\(955\) −8.66025 −0.280239
\(956\) −2.46410 −0.0796947
\(957\) 0 0
\(958\) −22.9808 −0.742475
\(959\) 2.66025 0.0859041
\(960\) 1.73205 0.0559017
\(961\) −30.4641 −0.982713
\(962\) −13.6077 −0.438730
\(963\) 0 0
\(964\) −10.5359 −0.339338
\(965\) −9.39230 −0.302349
\(966\) 8.66025 0.278639
\(967\) −12.7846 −0.411125 −0.205563 0.978644i \(-0.565902\pi\)
−0.205563 + 0.978644i \(0.565902\pi\)
\(968\) 0 0
\(969\) 17.6603 0.567329
\(970\) −2.92820 −0.0940189
\(971\) −4.71281 −0.151241 −0.0756207 0.997137i \(-0.524094\pi\)
−0.0756207 + 0.997137i \(0.524094\pi\)
\(972\) 0 0
\(973\) −6.80385 −0.218121
\(974\) 3.39230 0.108696
\(975\) −3.92820 −0.125803
\(976\) −10.9282 −0.349803
\(977\) −17.7321 −0.567299 −0.283649 0.958928i \(-0.591545\pi\)
−0.283649 + 0.958928i \(0.591545\pi\)
\(978\) 17.3205 0.553849
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −7.66025 −0.244449
\(983\) 58.3923 1.86243 0.931213 0.364476i \(-0.118752\pi\)
0.931213 + 0.364476i \(0.118752\pi\)
\(984\) −7.26795 −0.231694
\(985\) −2.53590 −0.0808004
\(986\) −21.8564 −0.696050
\(987\) −2.19615 −0.0699043
\(988\) 8.46410 0.269279
\(989\) −43.6603 −1.38832
\(990\) 0 0
\(991\) −26.1244 −0.829868 −0.414934 0.909852i \(-0.636195\pi\)
−0.414934 + 0.909852i \(0.636195\pi\)
\(992\) −0.732051 −0.0232426
\(993\) 7.01924 0.222749
\(994\) −6.92820 −0.219749
\(995\) 4.00000 0.126809
\(996\) 8.66025 0.274411
\(997\) −21.3397 −0.675837 −0.337918 0.941175i \(-0.609723\pi\)
−0.337918 + 0.941175i \(0.609723\pi\)
\(998\) 22.0000 0.696398
\(999\) 31.1769 0.986394
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.bm.1.1 2
11.10 odd 2 8470.2.a.bz.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bm.1.1 2 1.1 even 1 trivial
8470.2.a.bz.1.1 yes 2 11.10 odd 2