Properties

Label 8470.2.a.bl.1.2
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 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.73205 q^{12} -1.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} +4.73205 q^{17} +1.73205 q^{19} -1.00000 q^{20} -1.73205 q^{21} -4.46410 q^{23} -1.73205 q^{24} +1.00000 q^{25} +1.73205 q^{26} -5.19615 q^{27} -1.00000 q^{28} +9.46410 q^{29} +1.73205 q^{30} -6.73205 q^{31} -1.00000 q^{32} -4.73205 q^{34} +1.00000 q^{35} -0.535898 q^{37} -1.73205 q^{38} -3.00000 q^{39} +1.00000 q^{40} -3.26795 q^{41} +1.73205 q^{42} +9.66025 q^{43} +4.46410 q^{46} +0.732051 q^{47} +1.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} +8.19615 q^{51} -1.73205 q^{52} -4.73205 q^{53} +5.19615 q^{54} +1.00000 q^{56} +3.00000 q^{57} -9.46410 q^{58} -1.00000 q^{59} -1.73205 q^{60} +4.00000 q^{61} +6.73205 q^{62} +1.00000 q^{64} +1.73205 q^{65} +3.80385 q^{67} +4.73205 q^{68} -7.73205 q^{69} -1.00000 q^{70} -10.9282 q^{71} -7.66025 q^{73} +0.535898 q^{74} +1.73205 q^{75} +1.73205 q^{76} +3.00000 q^{78} -13.9282 q^{79} -1.00000 q^{80} -9.00000 q^{81} +3.26795 q^{82} -1.92820 q^{83} -1.73205 q^{84} -4.73205 q^{85} -9.66025 q^{86} +16.3923 q^{87} -1.26795 q^{89} +1.73205 q^{91} -4.46410 q^{92} -11.6603 q^{93} -0.732051 q^{94} -1.73205 q^{95} -1.73205 q^{96} -5.46410 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 2q^{8} + 2q^{10} + 2q^{14} + 2q^{16} + 6q^{17} - 2q^{20} - 2q^{23} + 2q^{25} - 2q^{28} + 12q^{29} - 10q^{31} - 2q^{32} - 6q^{34} + 2q^{35} - 8q^{37} - 6q^{39} + 2q^{40} - 10q^{41} + 2q^{43} + 2q^{46} - 2q^{47} + 2q^{49} - 2q^{50} + 6q^{51} - 6q^{53} + 2q^{56} + 6q^{57} - 12q^{58} - 2q^{59} + 8q^{61} + 10q^{62} + 2q^{64} + 18q^{67} + 6q^{68} - 12q^{69} - 2q^{70} - 8q^{71} + 2q^{73} + 8q^{74} + 6q^{78} - 14q^{79} - 2q^{80} - 18q^{81} + 10q^{82} + 10q^{83} - 6q^{85} - 2q^{86} + 12q^{87} - 6q^{89} - 2q^{92} - 6q^{93} + 2q^{94} - 4q^{97} - 2q^{98} + O(q^{100}) \)

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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 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
\(13\) −1.73205 −0.480384 −0.240192 0.970725i \(-0.577210\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.73205 −0.447214
\(16\) 1.00000 0.250000
\(17\) 4.73205 1.14769 0.573845 0.818964i \(-0.305451\pi\)
0.573845 + 0.818964i \(0.305451\pi\)
\(18\) 0 0
\(19\) 1.73205 0.397360 0.198680 0.980064i \(-0.436335\pi\)
0.198680 + 0.980064i \(0.436335\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) −4.46410 −0.930830 −0.465415 0.885093i \(-0.654095\pi\)
−0.465415 + 0.885093i \(0.654095\pi\)
\(24\) −1.73205 −0.353553
\(25\) 1.00000 0.200000
\(26\) 1.73205 0.339683
\(27\) −5.19615 −1.00000
\(28\) −1.00000 −0.188982
\(29\) 9.46410 1.75744 0.878720 0.477338i \(-0.158398\pi\)
0.878720 + 0.477338i \(0.158398\pi\)
\(30\) 1.73205 0.316228
\(31\) −6.73205 −1.20911 −0.604556 0.796563i \(-0.706649\pi\)
−0.604556 + 0.796563i \(0.706649\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.73205 −0.811540
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −0.535898 −0.0881012 −0.0440506 0.999029i \(-0.514026\pi\)
−0.0440506 + 0.999029i \(0.514026\pi\)
\(38\) −1.73205 −0.280976
\(39\) −3.00000 −0.480384
\(40\) 1.00000 0.158114
\(41\) −3.26795 −0.510368 −0.255184 0.966893i \(-0.582136\pi\)
−0.255184 + 0.966893i \(0.582136\pi\)
\(42\) 1.73205 0.267261
\(43\) 9.66025 1.47317 0.736587 0.676342i \(-0.236436\pi\)
0.736587 + 0.676342i \(0.236436\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.46410 0.658196
\(47\) 0.732051 0.106781 0.0533903 0.998574i \(-0.482997\pi\)
0.0533903 + 0.998574i \(0.482997\pi\)
\(48\) 1.73205 0.250000
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 8.19615 1.14769
\(52\) −1.73205 −0.240192
\(53\) −4.73205 −0.649997 −0.324999 0.945715i \(-0.605364\pi\)
−0.324999 + 0.945715i \(0.605364\pi\)
\(54\) 5.19615 0.707107
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.00000 0.397360
\(58\) −9.46410 −1.24270
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) −1.73205 −0.223607
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 6.73205 0.854971
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.73205 0.214834
\(66\) 0 0
\(67\) 3.80385 0.464714 0.232357 0.972631i \(-0.425356\pi\)
0.232357 + 0.972631i \(0.425356\pi\)
\(68\) 4.73205 0.573845
\(69\) −7.73205 −0.930830
\(70\) −1.00000 −0.119523
\(71\) −10.9282 −1.29694 −0.648470 0.761241i \(-0.724591\pi\)
−0.648470 + 0.761241i \(0.724591\pi\)
\(72\) 0 0
\(73\) −7.66025 −0.896565 −0.448282 0.893892i \(-0.647964\pi\)
−0.448282 + 0.893892i \(0.647964\pi\)
\(74\) 0.535898 0.0622969
\(75\) 1.73205 0.200000
\(76\) 1.73205 0.198680
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) −13.9282 −1.56705 −0.783523 0.621363i \(-0.786579\pi\)
−0.783523 + 0.621363i \(0.786579\pi\)
\(80\) −1.00000 −0.111803
\(81\) −9.00000 −1.00000
\(82\) 3.26795 0.360885
\(83\) −1.92820 −0.211648 −0.105824 0.994385i \(-0.533748\pi\)
−0.105824 + 0.994385i \(0.533748\pi\)
\(84\) −1.73205 −0.188982
\(85\) −4.73205 −0.513263
\(86\) −9.66025 −1.04169
\(87\) 16.3923 1.75744
\(88\) 0 0
\(89\) −1.26795 −0.134402 −0.0672012 0.997739i \(-0.521407\pi\)
−0.0672012 + 0.997739i \(0.521407\pi\)
\(90\) 0 0
\(91\) 1.73205 0.181568
\(92\) −4.46410 −0.465415
\(93\) −11.6603 −1.20911
\(94\) −0.732051 −0.0755053
\(95\) −1.73205 −0.177705
\(96\) −1.73205 −0.176777
\(97\) −5.46410 −0.554795 −0.277398 0.960755i \(-0.589472\pi\)
−0.277398 + 0.960755i \(0.589472\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.92820 −0.589878 −0.294939 0.955516i \(-0.595299\pi\)
−0.294939 + 0.955516i \(0.595299\pi\)
\(102\) −8.19615 −0.811540
\(103\) 12.1962 1.20172 0.600861 0.799353i \(-0.294824\pi\)
0.600861 + 0.799353i \(0.294824\pi\)
\(104\) 1.73205 0.169842
\(105\) 1.73205 0.169031
\(106\) 4.73205 0.459617
\(107\) 7.46410 0.721582 0.360791 0.932647i \(-0.382507\pi\)
0.360791 + 0.932647i \(0.382507\pi\)
\(108\) −5.19615 −0.500000
\(109\) −10.1962 −0.976614 −0.488307 0.872672i \(-0.662385\pi\)
−0.488307 + 0.872672i \(0.662385\pi\)
\(110\) 0 0
\(111\) −0.928203 −0.0881012
\(112\) −1.00000 −0.0944911
\(113\) 12.6603 1.19098 0.595488 0.803364i \(-0.296959\pi\)
0.595488 + 0.803364i \(0.296959\pi\)
\(114\) −3.00000 −0.280976
\(115\) 4.46410 0.416280
\(116\) 9.46410 0.878720
\(117\) 0 0
\(118\) 1.00000 0.0920575
\(119\) −4.73205 −0.433786
\(120\) 1.73205 0.158114
\(121\) 0 0
\(122\) −4.00000 −0.362143
\(123\) −5.66025 −0.510368
\(124\) −6.73205 −0.604556
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.80385 0.248801 0.124401 0.992232i \(-0.460299\pi\)
0.124401 + 0.992232i \(0.460299\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 16.7321 1.47317
\(130\) −1.73205 −0.151911
\(131\) 7.73205 0.675552 0.337776 0.941226i \(-0.390325\pi\)
0.337776 + 0.941226i \(0.390325\pi\)
\(132\) 0 0
\(133\) −1.73205 −0.150188
\(134\) −3.80385 −0.328602
\(135\) 5.19615 0.447214
\(136\) −4.73205 −0.405770
\(137\) −12.1244 −1.03585 −0.517927 0.855425i \(-0.673296\pi\)
−0.517927 + 0.855425i \(0.673296\pi\)
\(138\) 7.73205 0.658196
\(139\) 18.6603 1.58274 0.791371 0.611336i \(-0.209368\pi\)
0.791371 + 0.611336i \(0.209368\pi\)
\(140\) 1.00000 0.0845154
\(141\) 1.26795 0.106781
\(142\) 10.9282 0.917074
\(143\) 0 0
\(144\) 0 0
\(145\) −9.46410 −0.785951
\(146\) 7.66025 0.633967
\(147\) 1.73205 0.142857
\(148\) −0.535898 −0.0440506
\(149\) −12.9282 −1.05912 −0.529560 0.848273i \(-0.677643\pi\)
−0.529560 + 0.848273i \(0.677643\pi\)
\(150\) −1.73205 −0.141421
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) −1.73205 −0.140488
\(153\) 0 0
\(154\) 0 0
\(155\) 6.73205 0.540731
\(156\) −3.00000 −0.240192
\(157\) −3.53590 −0.282195 −0.141098 0.989996i \(-0.545063\pi\)
−0.141098 + 0.989996i \(0.545063\pi\)
\(158\) 13.9282 1.10807
\(159\) −8.19615 −0.649997
\(160\) 1.00000 0.0790569
\(161\) 4.46410 0.351820
\(162\) 9.00000 0.707107
\(163\) −4.92820 −0.386007 −0.193003 0.981198i \(-0.561823\pi\)
−0.193003 + 0.981198i \(0.561823\pi\)
\(164\) −3.26795 −0.255184
\(165\) 0 0
\(166\) 1.92820 0.149658
\(167\) 3.26795 0.252882 0.126441 0.991974i \(-0.459645\pi\)
0.126441 + 0.991974i \(0.459645\pi\)
\(168\) 1.73205 0.133631
\(169\) −10.0000 −0.769231
\(170\) 4.73205 0.362932
\(171\) 0 0
\(172\) 9.66025 0.736587
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −16.3923 −1.24270
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −1.73205 −0.130189
\(178\) 1.26795 0.0950368
\(179\) 15.1244 1.13045 0.565224 0.824938i \(-0.308790\pi\)
0.565224 + 0.824938i \(0.308790\pi\)
\(180\) 0 0
\(181\) 20.1244 1.49583 0.747916 0.663794i \(-0.231055\pi\)
0.747916 + 0.663794i \(0.231055\pi\)
\(182\) −1.73205 −0.128388
\(183\) 6.92820 0.512148
\(184\) 4.46410 0.329098
\(185\) 0.535898 0.0394000
\(186\) 11.6603 0.854971
\(187\) 0 0
\(188\) 0.732051 0.0533903
\(189\) 5.19615 0.377964
\(190\) 1.73205 0.125656
\(191\) −21.7321 −1.57248 −0.786238 0.617924i \(-0.787974\pi\)
−0.786238 + 0.617924i \(0.787974\pi\)
\(192\) 1.73205 0.125000
\(193\) −19.7846 −1.42413 −0.712064 0.702115i \(-0.752239\pi\)
−0.712064 + 0.702115i \(0.752239\pi\)
\(194\) 5.46410 0.392300
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −12.3923 −0.878467 −0.439234 0.898373i \(-0.644750\pi\)
−0.439234 + 0.898373i \(0.644750\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 6.58846 0.464714
\(202\) 5.92820 0.417107
\(203\) −9.46410 −0.664250
\(204\) 8.19615 0.573845
\(205\) 3.26795 0.228243
\(206\) −12.1962 −0.849746
\(207\) 0 0
\(208\) −1.73205 −0.120096
\(209\) 0 0
\(210\) −1.73205 −0.119523
\(211\) −5.46410 −0.376164 −0.188082 0.982153i \(-0.560227\pi\)
−0.188082 + 0.982153i \(0.560227\pi\)
\(212\) −4.73205 −0.324999
\(213\) −18.9282 −1.29694
\(214\) −7.46410 −0.510235
\(215\) −9.66025 −0.658824
\(216\) 5.19615 0.353553
\(217\) 6.73205 0.457001
\(218\) 10.1962 0.690571
\(219\) −13.2679 −0.896565
\(220\) 0 0
\(221\) −8.19615 −0.551333
\(222\) 0.928203 0.0622969
\(223\) −13.1244 −0.878872 −0.439436 0.898274i \(-0.644822\pi\)
−0.439436 + 0.898274i \(0.644822\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −12.6603 −0.842148
\(227\) 0.535898 0.0355688 0.0177844 0.999842i \(-0.494339\pi\)
0.0177844 + 0.999842i \(0.494339\pi\)
\(228\) 3.00000 0.198680
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) −4.46410 −0.294354
\(231\) 0 0
\(232\) −9.46410 −0.621349
\(233\) −4.85641 −0.318154 −0.159077 0.987266i \(-0.550852\pi\)
−0.159077 + 0.987266i \(0.550852\pi\)
\(234\) 0 0
\(235\) −0.732051 −0.0477537
\(236\) −1.00000 −0.0650945
\(237\) −24.1244 −1.56705
\(238\) 4.73205 0.306733
\(239\) −4.46410 −0.288759 −0.144379 0.989522i \(-0.546119\pi\)
−0.144379 + 0.989522i \(0.546119\pi\)
\(240\) −1.73205 −0.111803
\(241\) −2.92820 −0.188622 −0.0943111 0.995543i \(-0.530065\pi\)
−0.0943111 + 0.995543i \(0.530065\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 5.66025 0.360885
\(247\) −3.00000 −0.190885
\(248\) 6.73205 0.427486
\(249\) −3.33975 −0.211648
\(250\) 1.00000 0.0632456
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.80385 −0.175929
\(255\) −8.19615 −0.513263
\(256\) 1.00000 0.0625000
\(257\) −22.7321 −1.41799 −0.708993 0.705215i \(-0.750850\pi\)
−0.708993 + 0.705215i \(0.750850\pi\)
\(258\) −16.7321 −1.04169
\(259\) 0.535898 0.0332991
\(260\) 1.73205 0.107417
\(261\) 0 0
\(262\) −7.73205 −0.477688
\(263\) 14.2679 0.879799 0.439900 0.898047i \(-0.355014\pi\)
0.439900 + 0.898047i \(0.355014\pi\)
\(264\) 0 0
\(265\) 4.73205 0.290688
\(266\) 1.73205 0.106199
\(267\) −2.19615 −0.134402
\(268\) 3.80385 0.232357
\(269\) −28.5167 −1.73869 −0.869346 0.494204i \(-0.835459\pi\)
−0.869346 + 0.494204i \(0.835459\pi\)
\(270\) −5.19615 −0.316228
\(271\) 11.6077 0.705117 0.352559 0.935790i \(-0.385312\pi\)
0.352559 + 0.935790i \(0.385312\pi\)
\(272\) 4.73205 0.286923
\(273\) 3.00000 0.181568
\(274\) 12.1244 0.732459
\(275\) 0 0
\(276\) −7.73205 −0.465415
\(277\) 26.5885 1.59755 0.798773 0.601633i \(-0.205483\pi\)
0.798773 + 0.601633i \(0.205483\pi\)
\(278\) −18.6603 −1.11917
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −4.26795 −0.254605 −0.127302 0.991864i \(-0.540632\pi\)
−0.127302 + 0.991864i \(0.540632\pi\)
\(282\) −1.26795 −0.0755053
\(283\) −23.7846 −1.41385 −0.706924 0.707289i \(-0.749918\pi\)
−0.706924 + 0.707289i \(0.749918\pi\)
\(284\) −10.9282 −0.648470
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 3.26795 0.192901
\(288\) 0 0
\(289\) 5.39230 0.317194
\(290\) 9.46410 0.555751
\(291\) −9.46410 −0.554795
\(292\) −7.66025 −0.448282
\(293\) 22.6603 1.32383 0.661913 0.749581i \(-0.269745\pi\)
0.661913 + 0.749581i \(0.269745\pi\)
\(294\) −1.73205 −0.101015
\(295\) 1.00000 0.0582223
\(296\) 0.535898 0.0311485
\(297\) 0 0
\(298\) 12.9282 0.748911
\(299\) 7.73205 0.447156
\(300\) 1.73205 0.100000
\(301\) −9.66025 −0.556808
\(302\) 13.0000 0.748066
\(303\) −10.2679 −0.589878
\(304\) 1.73205 0.0993399
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −16.9282 −0.966144 −0.483072 0.875581i \(-0.660479\pi\)
−0.483072 + 0.875581i \(0.660479\pi\)
\(308\) 0 0
\(309\) 21.1244 1.20172
\(310\) −6.73205 −0.382355
\(311\) −21.8038 −1.23638 −0.618191 0.786028i \(-0.712134\pi\)
−0.618191 + 0.786028i \(0.712134\pi\)
\(312\) 3.00000 0.169842
\(313\) −23.4641 −1.32627 −0.663135 0.748500i \(-0.730774\pi\)
−0.663135 + 0.748500i \(0.730774\pi\)
\(314\) 3.53590 0.199542
\(315\) 0 0
\(316\) −13.9282 −0.783523
\(317\) 4.33975 0.243744 0.121872 0.992546i \(-0.461110\pi\)
0.121872 + 0.992546i \(0.461110\pi\)
\(318\) 8.19615 0.459617
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 12.9282 0.721582
\(322\) −4.46410 −0.248775
\(323\) 8.19615 0.456046
\(324\) −9.00000 −0.500000
\(325\) −1.73205 −0.0960769
\(326\) 4.92820 0.272948
\(327\) −17.6603 −0.976614
\(328\) 3.26795 0.180442
\(329\) −0.732051 −0.0403593
\(330\) 0 0
\(331\) −12.1962 −0.670361 −0.335181 0.942154i \(-0.608797\pi\)
−0.335181 + 0.942154i \(0.608797\pi\)
\(332\) −1.92820 −0.105824
\(333\) 0 0
\(334\) −3.26795 −0.178814
\(335\) −3.80385 −0.207826
\(336\) −1.73205 −0.0944911
\(337\) 25.7846 1.40458 0.702289 0.711892i \(-0.252162\pi\)
0.702289 + 0.711892i \(0.252162\pi\)
\(338\) 10.0000 0.543928
\(339\) 21.9282 1.19098
\(340\) −4.73205 −0.256631
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −9.66025 −0.520846
\(345\) 7.73205 0.416280
\(346\) 10.0000 0.537603
\(347\) −15.4641 −0.830156 −0.415078 0.909786i \(-0.636246\pi\)
−0.415078 + 0.909786i \(0.636246\pi\)
\(348\) 16.3923 0.878720
\(349\) −3.00000 −0.160586 −0.0802932 0.996771i \(-0.525586\pi\)
−0.0802932 + 0.996771i \(0.525586\pi\)
\(350\) 1.00000 0.0534522
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) −9.66025 −0.514163 −0.257082 0.966390i \(-0.582761\pi\)
−0.257082 + 0.966390i \(0.582761\pi\)
\(354\) 1.73205 0.0920575
\(355\) 10.9282 0.580009
\(356\) −1.26795 −0.0672012
\(357\) −8.19615 −0.433786
\(358\) −15.1244 −0.799347
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −16.0000 −0.842105
\(362\) −20.1244 −1.05771
\(363\) 0 0
\(364\) 1.73205 0.0907841
\(365\) 7.66025 0.400956
\(366\) −6.92820 −0.362143
\(367\) −21.6603 −1.13066 −0.565328 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(368\) −4.46410 −0.232707
\(369\) 0 0
\(370\) −0.535898 −0.0278600
\(371\) 4.73205 0.245676
\(372\) −11.6603 −0.604556
\(373\) 0.928203 0.0480605 0.0240303 0.999711i \(-0.492350\pi\)
0.0240303 + 0.999711i \(0.492350\pi\)
\(374\) 0 0
\(375\) −1.73205 −0.0894427
\(376\) −0.732051 −0.0377526
\(377\) −16.3923 −0.844247
\(378\) −5.19615 −0.267261
\(379\) 14.3923 0.739283 0.369642 0.929174i \(-0.379480\pi\)
0.369642 + 0.929174i \(0.379480\pi\)
\(380\) −1.73205 −0.0888523
\(381\) 4.85641 0.248801
\(382\) 21.7321 1.11191
\(383\) 4.58846 0.234459 0.117230 0.993105i \(-0.462599\pi\)
0.117230 + 0.993105i \(0.462599\pi\)
\(384\) −1.73205 −0.0883883
\(385\) 0 0
\(386\) 19.7846 1.00701
\(387\) 0 0
\(388\) −5.46410 −0.277398
\(389\) −35.1244 −1.78088 −0.890438 0.455105i \(-0.849602\pi\)
−0.890438 + 0.455105i \(0.849602\pi\)
\(390\) −3.00000 −0.151911
\(391\) −21.1244 −1.06830
\(392\) −1.00000 −0.0505076
\(393\) 13.3923 0.675552
\(394\) 24.0000 1.20910
\(395\) 13.9282 0.700804
\(396\) 0 0
\(397\) 17.8564 0.896187 0.448094 0.893987i \(-0.352103\pi\)
0.448094 + 0.893987i \(0.352103\pi\)
\(398\) 12.3923 0.621170
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 18.2487 0.911297 0.455649 0.890160i \(-0.349407\pi\)
0.455649 + 0.890160i \(0.349407\pi\)
\(402\) −6.58846 −0.328602
\(403\) 11.6603 0.580839
\(404\) −5.92820 −0.294939
\(405\) 9.00000 0.447214
\(406\) 9.46410 0.469695
\(407\) 0 0
\(408\) −8.19615 −0.405770
\(409\) −37.1769 −1.83828 −0.919140 0.393931i \(-0.871115\pi\)
−0.919140 + 0.393931i \(0.871115\pi\)
\(410\) −3.26795 −0.161393
\(411\) −21.0000 −1.03585
\(412\) 12.1962 0.600861
\(413\) 1.00000 0.0492068
\(414\) 0 0
\(415\) 1.92820 0.0946518
\(416\) 1.73205 0.0849208
\(417\) 32.3205 1.58274
\(418\) 0 0
\(419\) 11.3923 0.556551 0.278275 0.960501i \(-0.410237\pi\)
0.278275 + 0.960501i \(0.410237\pi\)
\(420\) 1.73205 0.0845154
\(421\) 38.2487 1.86413 0.932064 0.362294i \(-0.118006\pi\)
0.932064 + 0.362294i \(0.118006\pi\)
\(422\) 5.46410 0.265988
\(423\) 0 0
\(424\) 4.73205 0.229809
\(425\) 4.73205 0.229538
\(426\) 18.9282 0.917074
\(427\) −4.00000 −0.193574
\(428\) 7.46410 0.360791
\(429\) 0 0
\(430\) 9.66025 0.465859
\(431\) −11.5359 −0.555665 −0.277832 0.960630i \(-0.589616\pi\)
−0.277832 + 0.960630i \(0.589616\pi\)
\(432\) −5.19615 −0.250000
\(433\) 1.80385 0.0866874 0.0433437 0.999060i \(-0.486199\pi\)
0.0433437 + 0.999060i \(0.486199\pi\)
\(434\) −6.73205 −0.323149
\(435\) −16.3923 −0.785951
\(436\) −10.1962 −0.488307
\(437\) −7.73205 −0.369874
\(438\) 13.2679 0.633967
\(439\) −24.1962 −1.15482 −0.577410 0.816455i \(-0.695936\pi\)
−0.577410 + 0.816455i \(0.695936\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.19615 0.389851
\(443\) −5.07180 −0.240968 −0.120484 0.992715i \(-0.538445\pi\)
−0.120484 + 0.992715i \(0.538445\pi\)
\(444\) −0.928203 −0.0440506
\(445\) 1.26795 0.0601066
\(446\) 13.1244 0.621456
\(447\) −22.3923 −1.05912
\(448\) −1.00000 −0.0472456
\(449\) −34.4641 −1.62646 −0.813231 0.581941i \(-0.802293\pi\)
−0.813231 + 0.581941i \(0.802293\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.6603 0.595488
\(453\) −22.5167 −1.05792
\(454\) −0.535898 −0.0251510
\(455\) −1.73205 −0.0811998
\(456\) −3.00000 −0.140488
\(457\) −6.07180 −0.284027 −0.142013 0.989865i \(-0.545358\pi\)
−0.142013 + 0.989865i \(0.545358\pi\)
\(458\) 0 0
\(459\) −24.5885 −1.14769
\(460\) 4.46410 0.208140
\(461\) −10.2487 −0.477330 −0.238665 0.971102i \(-0.576710\pi\)
−0.238665 + 0.971102i \(0.576710\pi\)
\(462\) 0 0
\(463\) 35.0000 1.62659 0.813294 0.581853i \(-0.197672\pi\)
0.813294 + 0.581853i \(0.197672\pi\)
\(464\) 9.46410 0.439360
\(465\) 11.6603 0.540731
\(466\) 4.85641 0.224969
\(467\) −30.1244 −1.39399 −0.696994 0.717077i \(-0.745480\pi\)
−0.696994 + 0.717077i \(0.745480\pi\)
\(468\) 0 0
\(469\) −3.80385 −0.175645
\(470\) 0.732051 0.0337670
\(471\) −6.12436 −0.282195
\(472\) 1.00000 0.0460287
\(473\) 0 0
\(474\) 24.1244 1.10807
\(475\) 1.73205 0.0794719
\(476\) −4.73205 −0.216893
\(477\) 0 0
\(478\) 4.46410 0.204183
\(479\) −18.7321 −0.855889 −0.427945 0.903805i \(-0.640762\pi\)
−0.427945 + 0.903805i \(0.640762\pi\)
\(480\) 1.73205 0.0790569
\(481\) 0.928203 0.0423224
\(482\) 2.92820 0.133376
\(483\) 7.73205 0.351820
\(484\) 0 0
\(485\) 5.46410 0.248112
\(486\) 0 0
\(487\) −32.7128 −1.48236 −0.741180 0.671307i \(-0.765733\pi\)
−0.741180 + 0.671307i \(0.765733\pi\)
\(488\) −4.00000 −0.181071
\(489\) −8.53590 −0.386007
\(490\) 1.00000 0.0451754
\(491\) 27.5167 1.24181 0.620905 0.783886i \(-0.286765\pi\)
0.620905 + 0.783886i \(0.286765\pi\)
\(492\) −5.66025 −0.255184
\(493\) 44.7846 2.01700
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) −6.73205 −0.302278
\(497\) 10.9282 0.490197
\(498\) 3.33975 0.149658
\(499\) 22.3923 1.00242 0.501209 0.865326i \(-0.332889\pi\)
0.501209 + 0.865326i \(0.332889\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 5.66025 0.252882
\(502\) −10.3923 −0.463831
\(503\) 22.9808 1.02466 0.512331 0.858788i \(-0.328782\pi\)
0.512331 + 0.858788i \(0.328782\pi\)
\(504\) 0 0
\(505\) 5.92820 0.263802
\(506\) 0 0
\(507\) −17.3205 −0.769231
\(508\) 2.80385 0.124401
\(509\) −24.8038 −1.09941 −0.549706 0.835358i \(-0.685260\pi\)
−0.549706 + 0.835358i \(0.685260\pi\)
\(510\) 8.19615 0.362932
\(511\) 7.66025 0.338870
\(512\) −1.00000 −0.0441942
\(513\) −9.00000 −0.397360
\(514\) 22.7321 1.00267
\(515\) −12.1962 −0.537427
\(516\) 16.7321 0.736587
\(517\) 0 0
\(518\) −0.535898 −0.0235460
\(519\) −17.3205 −0.760286
\(520\) −1.73205 −0.0759555
\(521\) −21.4641 −0.940359 −0.470180 0.882571i \(-0.655811\pi\)
−0.470180 + 0.882571i \(0.655811\pi\)
\(522\) 0 0
\(523\) −17.2487 −0.754233 −0.377117 0.926166i \(-0.623084\pi\)
−0.377117 + 0.926166i \(0.623084\pi\)
\(524\) 7.73205 0.337776
\(525\) −1.73205 −0.0755929
\(526\) −14.2679 −0.622112
\(527\) −31.8564 −1.38769
\(528\) 0 0
\(529\) −3.07180 −0.133556
\(530\) −4.73205 −0.205547
\(531\) 0 0
\(532\) −1.73205 −0.0750939
\(533\) 5.66025 0.245173
\(534\) 2.19615 0.0950368
\(535\) −7.46410 −0.322701
\(536\) −3.80385 −0.164301
\(537\) 26.1962 1.13045
\(538\) 28.5167 1.22944
\(539\) 0 0
\(540\) 5.19615 0.223607
\(541\) 24.2487 1.04253 0.521267 0.853394i \(-0.325460\pi\)
0.521267 + 0.853394i \(0.325460\pi\)
\(542\) −11.6077 −0.498593
\(543\) 34.8564 1.49583
\(544\) −4.73205 −0.202885
\(545\) 10.1962 0.436755
\(546\) −3.00000 −0.128388
\(547\) −4.58846 −0.196188 −0.0980941 0.995177i \(-0.531275\pi\)
−0.0980941 + 0.995177i \(0.531275\pi\)
\(548\) −12.1244 −0.517927
\(549\) 0 0
\(550\) 0 0
\(551\) 16.3923 0.698336
\(552\) 7.73205 0.329098
\(553\) 13.9282 0.592287
\(554\) −26.5885 −1.12964
\(555\) 0.928203 0.0394000
\(556\) 18.6603 0.791371
\(557\) −1.94744 −0.0825157 −0.0412579 0.999149i \(-0.513137\pi\)
−0.0412579 + 0.999149i \(0.513137\pi\)
\(558\) 0 0
\(559\) −16.7321 −0.707690
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 4.26795 0.180033
\(563\) 35.5359 1.49766 0.748830 0.662762i \(-0.230616\pi\)
0.748830 + 0.662762i \(0.230616\pi\)
\(564\) 1.26795 0.0533903
\(565\) −12.6603 −0.532621
\(566\) 23.7846 0.999742
\(567\) 9.00000 0.377964
\(568\) 10.9282 0.458537
\(569\) −10.1244 −0.424435 −0.212218 0.977222i \(-0.568069\pi\)
−0.212218 + 0.977222i \(0.568069\pi\)
\(570\) 3.00000 0.125656
\(571\) 35.3205 1.47812 0.739059 0.673641i \(-0.235271\pi\)
0.739059 + 0.673641i \(0.235271\pi\)
\(572\) 0 0
\(573\) −37.6410 −1.57248
\(574\) −3.26795 −0.136402
\(575\) −4.46410 −0.186166
\(576\) 0 0
\(577\) 29.8038 1.24075 0.620375 0.784305i \(-0.286980\pi\)
0.620375 + 0.784305i \(0.286980\pi\)
\(578\) −5.39230 −0.224290
\(579\) −34.2679 −1.42413
\(580\) −9.46410 −0.392975
\(581\) 1.92820 0.0799953
\(582\) 9.46410 0.392300
\(583\) 0 0
\(584\) 7.66025 0.316984
\(585\) 0 0
\(586\) −22.6603 −0.936086
\(587\) −11.8756 −0.490160 −0.245080 0.969503i \(-0.578814\pi\)
−0.245080 + 0.969503i \(0.578814\pi\)
\(588\) 1.73205 0.0714286
\(589\) −11.6603 −0.480452
\(590\) −1.00000 −0.0411693
\(591\) −41.5692 −1.70993
\(592\) −0.535898 −0.0220253
\(593\) 10.9282 0.448768 0.224384 0.974501i \(-0.427963\pi\)
0.224384 + 0.974501i \(0.427963\pi\)
\(594\) 0 0
\(595\) 4.73205 0.193995
\(596\) −12.9282 −0.529560
\(597\) −21.4641 −0.878467
\(598\) −7.73205 −0.316187
\(599\) −14.6603 −0.599002 −0.299501 0.954096i \(-0.596820\pi\)
−0.299501 + 0.954096i \(0.596820\pi\)
\(600\) −1.73205 −0.0707107
\(601\) 36.5885 1.49247 0.746237 0.665680i \(-0.231858\pi\)
0.746237 + 0.665680i \(0.231858\pi\)
\(602\) 9.66025 0.393723
\(603\) 0 0
\(604\) −13.0000 −0.528962
\(605\) 0 0
\(606\) 10.2679 0.417107
\(607\) 43.4641 1.76415 0.882077 0.471106i \(-0.156145\pi\)
0.882077 + 0.471106i \(0.156145\pi\)
\(608\) −1.73205 −0.0702439
\(609\) −16.3923 −0.664250
\(610\) 4.00000 0.161955
\(611\) −1.26795 −0.0512957
\(612\) 0 0
\(613\) 34.9808 1.41286 0.706430 0.707783i \(-0.250305\pi\)
0.706430 + 0.707783i \(0.250305\pi\)
\(614\) 16.9282 0.683167
\(615\) 5.66025 0.228243
\(616\) 0 0
\(617\) −18.3923 −0.740446 −0.370223 0.928943i \(-0.620719\pi\)
−0.370223 + 0.928943i \(0.620719\pi\)
\(618\) −21.1244 −0.849746
\(619\) −21.3923 −0.859829 −0.429915 0.902870i \(-0.641456\pi\)
−0.429915 + 0.902870i \(0.641456\pi\)
\(620\) 6.73205 0.270366
\(621\) 23.1962 0.930830
\(622\) 21.8038 0.874255
\(623\) 1.26795 0.0507993
\(624\) −3.00000 −0.120096
\(625\) 1.00000 0.0400000
\(626\) 23.4641 0.937814
\(627\) 0 0
\(628\) −3.53590 −0.141098
\(629\) −2.53590 −0.101113
\(630\) 0 0
\(631\) 13.4641 0.535997 0.267999 0.963419i \(-0.413638\pi\)
0.267999 + 0.963419i \(0.413638\pi\)
\(632\) 13.9282 0.554034
\(633\) −9.46410 −0.376164
\(634\) −4.33975 −0.172353
\(635\) −2.80385 −0.111267
\(636\) −8.19615 −0.324999
\(637\) −1.73205 −0.0686264
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 14.3205 0.565626 0.282813 0.959175i \(-0.408732\pi\)
0.282813 + 0.959175i \(0.408732\pi\)
\(642\) −12.9282 −0.510235
\(643\) 8.24871 0.325297 0.162649 0.986684i \(-0.447996\pi\)
0.162649 + 0.986684i \(0.447996\pi\)
\(644\) 4.46410 0.175910
\(645\) −16.7321 −0.658824
\(646\) −8.19615 −0.322473
\(647\) 30.4449 1.19691 0.598456 0.801156i \(-0.295781\pi\)
0.598456 + 0.801156i \(0.295781\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) 1.73205 0.0679366
\(651\) 11.6603 0.457001
\(652\) −4.92820 −0.193003
\(653\) 34.0526 1.33258 0.666290 0.745693i \(-0.267881\pi\)
0.666290 + 0.745693i \(0.267881\pi\)
\(654\) 17.6603 0.690571
\(655\) −7.73205 −0.302116
\(656\) −3.26795 −0.127592
\(657\) 0 0
\(658\) 0.732051 0.0285383
\(659\) −39.1244 −1.52407 −0.762034 0.647537i \(-0.775799\pi\)
−0.762034 + 0.647537i \(0.775799\pi\)
\(660\) 0 0
\(661\) 25.5885 0.995276 0.497638 0.867385i \(-0.334201\pi\)
0.497638 + 0.867385i \(0.334201\pi\)
\(662\) 12.1962 0.474017
\(663\) −14.1962 −0.551333
\(664\) 1.92820 0.0748288
\(665\) 1.73205 0.0671660
\(666\) 0 0
\(667\) −42.2487 −1.63588
\(668\) 3.26795 0.126441
\(669\) −22.7321 −0.878872
\(670\) 3.80385 0.146955
\(671\) 0 0
\(672\) 1.73205 0.0668153
\(673\) −7.78461 −0.300075 −0.150037 0.988680i \(-0.547939\pi\)
−0.150037 + 0.988680i \(0.547939\pi\)
\(674\) −25.7846 −0.993186
\(675\) −5.19615 −0.200000
\(676\) −10.0000 −0.384615
\(677\) −22.2679 −0.855827 −0.427913 0.903820i \(-0.640751\pi\)
−0.427913 + 0.903820i \(0.640751\pi\)
\(678\) −21.9282 −0.842148
\(679\) 5.46410 0.209693
\(680\) 4.73205 0.181466
\(681\) 0.928203 0.0355688
\(682\) 0 0
\(683\) −16.3923 −0.627234 −0.313617 0.949550i \(-0.601541\pi\)
−0.313617 + 0.949550i \(0.601541\pi\)
\(684\) 0 0
\(685\) 12.1244 0.463248
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 9.66025 0.368294
\(689\) 8.19615 0.312249
\(690\) −7.73205 −0.294354
\(691\) 50.6410 1.92648 0.963238 0.268651i \(-0.0865779\pi\)
0.963238 + 0.268651i \(0.0865779\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 15.4641 0.587009
\(695\) −18.6603 −0.707824
\(696\) −16.3923 −0.621349
\(697\) −15.4641 −0.585745
\(698\) 3.00000 0.113552
\(699\) −8.41154 −0.318154
\(700\) −1.00000 −0.0377964
\(701\) −38.5885 −1.45747 −0.728733 0.684798i \(-0.759890\pi\)
−0.728733 + 0.684798i \(0.759890\pi\)
\(702\) −9.00000 −0.339683
\(703\) −0.928203 −0.0350078
\(704\) 0 0
\(705\) −1.26795 −0.0477537
\(706\) 9.66025 0.363568
\(707\) 5.92820 0.222953
\(708\) −1.73205 −0.0650945
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −10.9282 −0.410128
\(711\) 0 0
\(712\) 1.26795 0.0475184
\(713\) 30.0526 1.12548
\(714\) 8.19615 0.306733
\(715\) 0 0
\(716\) 15.1244 0.565224
\(717\) −7.73205 −0.288759
\(718\) 14.0000 0.522475
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −12.1962 −0.454208
\(722\) 16.0000 0.595458
\(723\) −5.07180 −0.188622
\(724\) 20.1244 0.747916
\(725\) 9.46410 0.351488
\(726\) 0 0
\(727\) −3.71281 −0.137701 −0.0688503 0.997627i \(-0.521933\pi\)
−0.0688503 + 0.997627i \(0.521933\pi\)
\(728\) −1.73205 −0.0641941
\(729\) 27.0000 1.00000
\(730\) −7.66025 −0.283519
\(731\) 45.7128 1.69075
\(732\) 6.92820 0.256074
\(733\) 18.5167 0.683928 0.341964 0.939713i \(-0.388908\pi\)
0.341964 + 0.939713i \(0.388908\pi\)
\(734\) 21.6603 0.799495
\(735\) −1.73205 −0.0638877
\(736\) 4.46410 0.164549
\(737\) 0 0
\(738\) 0 0
\(739\) 25.6603 0.943928 0.471964 0.881618i \(-0.343545\pi\)
0.471964 + 0.881618i \(0.343545\pi\)
\(740\) 0.535898 0.0197000
\(741\) −5.19615 −0.190885
\(742\) −4.73205 −0.173719
\(743\) −39.1769 −1.43726 −0.718631 0.695392i \(-0.755231\pi\)
−0.718631 + 0.695392i \(0.755231\pi\)
\(744\) 11.6603 0.427486
\(745\) 12.9282 0.473653
\(746\) −0.928203 −0.0339839
\(747\) 0 0
\(748\) 0 0
\(749\) −7.46410 −0.272732
\(750\) 1.73205 0.0632456
\(751\) −13.3397 −0.486774 −0.243387 0.969929i \(-0.578259\pi\)
−0.243387 + 0.969929i \(0.578259\pi\)
\(752\) 0.732051 0.0266951
\(753\) 18.0000 0.655956
\(754\) 16.3923 0.596973
\(755\) 13.0000 0.473118
\(756\) 5.19615 0.188982
\(757\) 24.9808 0.907941 0.453971 0.891017i \(-0.350007\pi\)
0.453971 + 0.891017i \(0.350007\pi\)
\(758\) −14.3923 −0.522752
\(759\) 0 0
\(760\) 1.73205 0.0628281
\(761\) 23.5167 0.852478 0.426239 0.904611i \(-0.359838\pi\)
0.426239 + 0.904611i \(0.359838\pi\)
\(762\) −4.85641 −0.175929
\(763\) 10.1962 0.369126
\(764\) −21.7321 −0.786238
\(765\) 0 0
\(766\) −4.58846 −0.165788
\(767\) 1.73205 0.0625407
\(768\) 1.73205 0.0625000
\(769\) −9.60770 −0.346462 −0.173231 0.984881i \(-0.555421\pi\)
−0.173231 + 0.984881i \(0.555421\pi\)
\(770\) 0 0
\(771\) −39.3731 −1.41799
\(772\) −19.7846 −0.712064
\(773\) −27.3923 −0.985233 −0.492616 0.870247i \(-0.663959\pi\)
−0.492616 + 0.870247i \(0.663959\pi\)
\(774\) 0 0
\(775\) −6.73205 −0.241822
\(776\) 5.46410 0.196150
\(777\) 0.928203 0.0332991
\(778\) 35.1244 1.25927
\(779\) −5.66025 −0.202800
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) 21.1244 0.755405
\(783\) −49.1769 −1.75744
\(784\) 1.00000 0.0357143
\(785\) 3.53590 0.126202
\(786\) −13.3923 −0.477688
\(787\) 29.7128 1.05915 0.529574 0.848264i \(-0.322352\pi\)
0.529574 + 0.848264i \(0.322352\pi\)
\(788\) −24.0000 −0.854965
\(789\) 24.7128 0.879799
\(790\) −13.9282 −0.495543
\(791\) −12.6603 −0.450147
\(792\) 0 0
\(793\) −6.92820 −0.246028
\(794\) −17.8564 −0.633700
\(795\) 8.19615 0.290688
\(796\) −12.3923 −0.439234
\(797\) −46.3205 −1.64076 −0.820378 0.571821i \(-0.806237\pi\)
−0.820378 + 0.571821i \(0.806237\pi\)
\(798\) 3.00000 0.106199
\(799\) 3.46410 0.122551
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −18.2487 −0.644384
\(803\) 0 0
\(804\) 6.58846 0.232357
\(805\) −4.46410 −0.157339
\(806\) −11.6603 −0.410715
\(807\) −49.3923 −1.73869
\(808\) 5.92820 0.208553
\(809\) −23.8564 −0.838747 −0.419373 0.907814i \(-0.637750\pi\)
−0.419373 + 0.907814i \(0.637750\pi\)
\(810\) −9.00000 −0.316228
\(811\) −27.3205 −0.959353 −0.479676 0.877445i \(-0.659246\pi\)
−0.479676 + 0.877445i \(0.659246\pi\)
\(812\) −9.46410 −0.332125
\(813\) 20.1051 0.705117
\(814\) 0 0
\(815\) 4.92820 0.172627
\(816\) 8.19615 0.286923
\(817\) 16.7321 0.585380
\(818\) 37.1769 1.29986
\(819\) 0 0
\(820\) 3.26795 0.114122
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 21.0000 0.732459
\(823\) 5.71281 0.199136 0.0995681 0.995031i \(-0.468254\pi\)
0.0995681 + 0.995031i \(0.468254\pi\)
\(824\) −12.1962 −0.424873
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) 14.9808 0.520932 0.260466 0.965483i \(-0.416124\pi\)
0.260466 + 0.965483i \(0.416124\pi\)
\(828\) 0 0
\(829\) −34.3731 −1.19383 −0.596913 0.802306i \(-0.703606\pi\)
−0.596913 + 0.802306i \(0.703606\pi\)
\(830\) −1.92820 −0.0669289
\(831\) 46.0526 1.59755
\(832\) −1.73205 −0.0600481
\(833\) 4.73205 0.163956
\(834\) −32.3205 −1.11917
\(835\) −3.26795 −0.113092
\(836\) 0 0
\(837\) 34.9808 1.20911
\(838\) −11.3923 −0.393541
\(839\) −11.6603 −0.402557 −0.201278 0.979534i \(-0.564510\pi\)
−0.201278 + 0.979534i \(0.564510\pi\)
\(840\) −1.73205 −0.0597614
\(841\) 60.5692 2.08859
\(842\) −38.2487 −1.31814
\(843\) −7.39230 −0.254605
\(844\) −5.46410 −0.188082
\(845\) 10.0000 0.344010
\(846\) 0 0
\(847\) 0 0
\(848\) −4.73205 −0.162499
\(849\) −41.1962 −1.41385
\(850\) −4.73205 −0.162308
\(851\) 2.39230 0.0820072
\(852\) −18.9282 −0.648470
\(853\) −42.2679 −1.44723 −0.723614 0.690205i \(-0.757520\pi\)
−0.723614 + 0.690205i \(0.757520\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −7.46410 −0.255118
\(857\) −7.60770 −0.259874 −0.129937 0.991522i \(-0.541477\pi\)
−0.129937 + 0.991522i \(0.541477\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) −9.66025 −0.329412
\(861\) 5.66025 0.192901
\(862\) 11.5359 0.392914
\(863\) 30.6410 1.04303 0.521516 0.853241i \(-0.325367\pi\)
0.521516 + 0.853241i \(0.325367\pi\)
\(864\) 5.19615 0.176777
\(865\) 10.0000 0.340010
\(866\) −1.80385 −0.0612972
\(867\) 9.33975 0.317194
\(868\) 6.73205 0.228501
\(869\) 0 0
\(870\) 16.3923 0.555751
\(871\) −6.58846 −0.223241
\(872\) 10.1962 0.345285
\(873\) 0 0
\(874\) 7.73205 0.261541
\(875\) 1.00000 0.0338062
\(876\) −13.2679 −0.448282
\(877\) 5.12436 0.173037 0.0865186 0.996250i \(-0.472426\pi\)
0.0865186 + 0.996250i \(0.472426\pi\)
\(878\) 24.1962 0.816581
\(879\) 39.2487 1.32383
\(880\) 0 0
\(881\) 1.32051 0.0444890 0.0222445 0.999753i \(-0.492919\pi\)
0.0222445 + 0.999753i \(0.492919\pi\)
\(882\) 0 0
\(883\) 13.6077 0.457935 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(884\) −8.19615 −0.275666
\(885\) 1.73205 0.0582223
\(886\) 5.07180 0.170390
\(887\) −38.9808 −1.30885 −0.654423 0.756129i \(-0.727088\pi\)
−0.654423 + 0.756129i \(0.727088\pi\)
\(888\) 0.928203 0.0311485
\(889\) −2.80385 −0.0940380
\(890\) −1.26795 −0.0425018
\(891\) 0 0
\(892\) −13.1244 −0.439436
\(893\) 1.26795 0.0424303
\(894\) 22.3923 0.748911
\(895\) −15.1244 −0.505551
\(896\) 1.00000 0.0334077
\(897\) 13.3923 0.447156
\(898\) 34.4641 1.15008
\(899\) −63.7128 −2.12494
\(900\) 0 0
\(901\) −22.3923 −0.745996
\(902\) 0 0
\(903\) −16.7321 −0.556808
\(904\) −12.6603 −0.421074
\(905\) −20.1244 −0.668956
\(906\) 22.5167 0.748066
\(907\) −1.07180 −0.0355884 −0.0177942 0.999842i \(-0.505664\pi\)
−0.0177942 + 0.999842i \(0.505664\pi\)
\(908\) 0.535898 0.0177844
\(909\) 0 0
\(910\) 1.73205 0.0574169
\(911\) −46.2679 −1.53293 −0.766463 0.642289i \(-0.777985\pi\)
−0.766463 + 0.642289i \(0.777985\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) 6.07180 0.200837
\(915\) −6.92820 −0.229039
\(916\) 0 0
\(917\) −7.73205 −0.255335
\(918\) 24.5885 0.811540
\(919\) −49.7128 −1.63987 −0.819937 0.572453i \(-0.805992\pi\)
−0.819937 + 0.572453i \(0.805992\pi\)
\(920\) −4.46410 −0.147177
\(921\) −29.3205 −0.966144
\(922\) 10.2487 0.337523
\(923\) 18.9282 0.623029
\(924\) 0 0
\(925\) −0.535898 −0.0176202
\(926\) −35.0000 −1.15017
\(927\) 0 0
\(928\) −9.46410 −0.310674
\(929\) −3.32051 −0.108942 −0.0544712 0.998515i \(-0.517347\pi\)
−0.0544712 + 0.998515i \(0.517347\pi\)
\(930\) −11.6603 −0.382355
\(931\) 1.73205 0.0567657
\(932\) −4.85641 −0.159077
\(933\) −37.7654 −1.23638
\(934\) 30.1244 0.985699
\(935\) 0 0
\(936\) 0 0
\(937\) 18.6410 0.608975 0.304488 0.952516i \(-0.401515\pi\)
0.304488 + 0.952516i \(0.401515\pi\)
\(938\) 3.80385 0.124200
\(939\) −40.6410 −1.32627
\(940\) −0.732051 −0.0238769
\(941\) 57.7128 1.88138 0.940692 0.339262i \(-0.110177\pi\)
0.940692 + 0.339262i \(0.110177\pi\)
\(942\) 6.12436 0.199542
\(943\) 14.5885 0.475066
\(944\) −1.00000 −0.0325472
\(945\) −5.19615 −0.169031
\(946\) 0 0
\(947\) 6.73205 0.218762 0.109381 0.994000i \(-0.465113\pi\)
0.109381 + 0.994000i \(0.465113\pi\)
\(948\) −24.1244 −0.783523
\(949\) 13.2679 0.430696
\(950\) −1.73205 −0.0561951
\(951\) 7.51666 0.243744
\(952\) 4.73205 0.153367
\(953\) −19.9282 −0.645538 −0.322769 0.946478i \(-0.604614\pi\)
−0.322769 + 0.946478i \(0.604614\pi\)
\(954\) 0 0
\(955\) 21.7321 0.703233
\(956\) −4.46410 −0.144379
\(957\) 0 0
\(958\) 18.7321 0.605205
\(959\) 12.1244 0.391516
\(960\) −1.73205 −0.0559017
\(961\) 14.3205 0.461952
\(962\) −0.928203 −0.0299265
\(963\) 0 0
\(964\) −2.92820 −0.0943111
\(965\) 19.7846 0.636889
\(966\) −7.73205 −0.248775
\(967\) 44.4974 1.43094 0.715470 0.698643i \(-0.246212\pi\)
0.715470 + 0.698643i \(0.246212\pi\)
\(968\) 0 0
\(969\) 14.1962 0.456046
\(970\) −5.46410 −0.175442
\(971\) −27.3923 −0.879061 −0.439530 0.898228i \(-0.644855\pi\)
−0.439530 + 0.898228i \(0.644855\pi\)
\(972\) 0 0
\(973\) −18.6603 −0.598220
\(974\) 32.7128 1.04819
\(975\) −3.00000 −0.0960769
\(976\) 4.00000 0.128037
\(977\) 4.51666 0.144501 0.0722504 0.997387i \(-0.476982\pi\)
0.0722504 + 0.997387i \(0.476982\pi\)
\(978\) 8.53590 0.272948
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −27.5167 −0.878092
\(983\) 36.9282 1.17783 0.588913 0.808196i \(-0.299556\pi\)
0.588913 + 0.808196i \(0.299556\pi\)
\(984\) 5.66025 0.180442
\(985\) 24.0000 0.764704
\(986\) −44.7846 −1.42623
\(987\) −1.26795 −0.0403593
\(988\) −3.00000 −0.0954427
\(989\) −43.1244 −1.37127
\(990\) 0 0
\(991\) −20.5167 −0.651733 −0.325867 0.945416i \(-0.605656\pi\)
−0.325867 + 0.945416i \(0.605656\pi\)
\(992\) 6.73205 0.213743
\(993\) −21.1244 −0.670361
\(994\) −10.9282 −0.346622
\(995\) 12.3923 0.392862
\(996\) −3.33975 −0.105824
\(997\) −7.19615 −0.227904 −0.113952 0.993486i \(-0.536351\pi\)
−0.113952 + 0.993486i \(0.536351\pi\)
\(998\) −22.3923 −0.708816
\(999\) 2.78461 0.0881012
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.bl.1.2 2
11.10 odd 2 8470.2.a.ca.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bl.1.2 2 1.1 even 1 trivial
8470.2.a.ca.1.2 yes 2 11.10 odd 2