Properties

Label 8470.2.a.bt.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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.85410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.85410 q^{9} +1.00000 q^{10} +2.61803 q^{12} -2.00000 q^{13} +1.00000 q^{14} -2.61803 q^{15} +1.00000 q^{16} +1.61803 q^{17} -3.85410 q^{18} -6.85410 q^{19} -1.00000 q^{20} -2.61803 q^{21} +6.00000 q^{23} -2.61803 q^{24} +1.00000 q^{25} +2.00000 q^{26} +2.23607 q^{27} -1.00000 q^{28} -3.23607 q^{29} +2.61803 q^{30} -1.23607 q^{31} -1.00000 q^{32} -1.61803 q^{34} +1.00000 q^{35} +3.85410 q^{36} -6.47214 q^{37} +6.85410 q^{38} -5.23607 q^{39} +1.00000 q^{40} +10.3262 q^{41} +2.61803 q^{42} -1.85410 q^{43} -3.85410 q^{45} -6.00000 q^{46} +9.23607 q^{47} +2.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.23607 q^{51} -2.00000 q^{52} +1.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} -17.9443 q^{57} +3.23607 q^{58} -7.61803 q^{59} -2.61803 q^{60} +3.52786 q^{61} +1.23607 q^{62} -3.85410 q^{63} +1.00000 q^{64} +2.00000 q^{65} +6.09017 q^{67} +1.61803 q^{68} +15.7082 q^{69} -1.00000 q^{70} -9.70820 q^{71} -3.85410 q^{72} -13.8541 q^{73} +6.47214 q^{74} +2.61803 q^{75} -6.85410 q^{76} +5.23607 q^{78} +8.47214 q^{79} -1.00000 q^{80} -5.70820 q^{81} -10.3262 q^{82} -10.3262 q^{83} -2.61803 q^{84} -1.61803 q^{85} +1.85410 q^{86} -8.47214 q^{87} +13.0902 q^{89} +3.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} -3.23607 q^{93} -9.23607 q^{94} +6.85410 q^{95} -2.61803 q^{96} -9.56231 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + 2q^{10} + 3q^{12} - 4q^{13} + 2q^{14} - 3q^{15} + 2q^{16} + q^{17} - q^{18} - 7q^{19} - 2q^{20} - 3q^{21} + 12q^{23} - 3q^{24} + 2q^{25} + 4q^{26} - 2q^{28} - 2q^{29} + 3q^{30} + 2q^{31} - 2q^{32} - q^{34} + 2q^{35} + q^{36} - 4q^{37} + 7q^{38} - 6q^{39} + 2q^{40} + 5q^{41} + 3q^{42} + 3q^{43} - q^{45} - 12q^{46} + 14q^{47} + 3q^{48} + 2q^{49} - 2q^{50} + 4q^{51} - 4q^{52} - 2q^{53} + 2q^{56} - 18q^{57} + 2q^{58} - 13q^{59} - 3q^{60} + 16q^{61} - 2q^{62} - q^{63} + 2q^{64} + 4q^{65} + q^{67} + q^{68} + 18q^{69} - 2q^{70} - 6q^{71} - q^{72} - 21q^{73} + 4q^{74} + 3q^{75} - 7q^{76} + 6q^{78} + 8q^{79} - 2q^{80} + 2q^{81} - 5q^{82} - 5q^{83} - 3q^{84} - q^{85} - 3q^{86} - 8q^{87} + 15q^{89} + q^{90} + 4q^{91} + 12q^{92} - 2q^{93} - 14q^{94} + 7q^{95} - 3q^{96} + q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) 2.61803 1.51152 0.755761 0.654847i \(-0.227267\pi\)
0.755761 + 0.654847i \(0.227267\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.61803 −1.06881
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 3.85410 1.28470
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 2.61803 0.755761
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.61803 −0.675973
\(16\) 1.00000 0.250000
\(17\) 1.61803 0.392431 0.196215 0.980561i \(-0.437135\pi\)
0.196215 + 0.980561i \(0.437135\pi\)
\(18\) −3.85410 −0.908421
\(19\) −6.85410 −1.57244 −0.786219 0.617947i \(-0.787964\pi\)
−0.786219 + 0.617947i \(0.787964\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.61803 −0.571302
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −2.61803 −0.534404
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 2.23607 0.430331
\(28\) −1.00000 −0.188982
\(29\) −3.23607 −0.600923 −0.300461 0.953794i \(-0.597141\pi\)
−0.300461 + 0.953794i \(0.597141\pi\)
\(30\) 2.61803 0.477985
\(31\) −1.23607 −0.222004 −0.111002 0.993820i \(-0.535406\pi\)
−0.111002 + 0.993820i \(0.535406\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.61803 −0.277491
\(35\) 1.00000 0.169031
\(36\) 3.85410 0.642350
\(37\) −6.47214 −1.06401 −0.532006 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) 6.85410 1.11188
\(39\) −5.23607 −0.838442
\(40\) 1.00000 0.158114
\(41\) 10.3262 1.61269 0.806344 0.591447i \(-0.201443\pi\)
0.806344 + 0.591447i \(0.201443\pi\)
\(42\) 2.61803 0.403971
\(43\) −1.85410 −0.282748 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) 0 0
\(45\) −3.85410 −0.574536
\(46\) −6.00000 −0.884652
\(47\) 9.23607 1.34722 0.673609 0.739087i \(-0.264743\pi\)
0.673609 + 0.739087i \(0.264743\pi\)
\(48\) 2.61803 0.377881
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.23607 0.593168
\(52\) −2.00000 −0.277350
\(53\) 1.23607 0.169787 0.0848935 0.996390i \(-0.472945\pi\)
0.0848935 + 0.996390i \(0.472945\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −17.9443 −2.37678
\(58\) 3.23607 0.424917
\(59\) −7.61803 −0.991784 −0.495892 0.868384i \(-0.665159\pi\)
−0.495892 + 0.868384i \(0.665159\pi\)
\(60\) −2.61803 −0.337987
\(61\) 3.52786 0.451697 0.225848 0.974162i \(-0.427485\pi\)
0.225848 + 0.974162i \(0.427485\pi\)
\(62\) 1.23607 0.156981
\(63\) −3.85410 −0.485571
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 6.09017 0.744033 0.372016 0.928226i \(-0.378667\pi\)
0.372016 + 0.928226i \(0.378667\pi\)
\(68\) 1.61803 0.196215
\(69\) 15.7082 1.89105
\(70\) −1.00000 −0.119523
\(71\) −9.70820 −1.15215 −0.576076 0.817396i \(-0.695417\pi\)
−0.576076 + 0.817396i \(0.695417\pi\)
\(72\) −3.85410 −0.454210
\(73\) −13.8541 −1.62150 −0.810750 0.585393i \(-0.800940\pi\)
−0.810750 + 0.585393i \(0.800940\pi\)
\(74\) 6.47214 0.752371
\(75\) 2.61803 0.302305
\(76\) −6.85410 −0.786219
\(77\) 0 0
\(78\) 5.23607 0.592868
\(79\) 8.47214 0.953190 0.476595 0.879123i \(-0.341871\pi\)
0.476595 + 0.879123i \(0.341871\pi\)
\(80\) −1.00000 −0.111803
\(81\) −5.70820 −0.634245
\(82\) −10.3262 −1.14034
\(83\) −10.3262 −1.13345 −0.566726 0.823906i \(-0.691790\pi\)
−0.566726 + 0.823906i \(0.691790\pi\)
\(84\) −2.61803 −0.285651
\(85\) −1.61803 −0.175500
\(86\) 1.85410 0.199933
\(87\) −8.47214 −0.908308
\(88\) 0 0
\(89\) 13.0902 1.38756 0.693778 0.720189i \(-0.255945\pi\)
0.693778 + 0.720189i \(0.255945\pi\)
\(90\) 3.85410 0.406258
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) −3.23607 −0.335565
\(94\) −9.23607 −0.952628
\(95\) 6.85410 0.703216
\(96\) −2.61803 −0.267202
\(97\) −9.56231 −0.970905 −0.485453 0.874263i \(-0.661345\pi\)
−0.485453 + 0.874263i \(0.661345\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.94427 −0.889988 −0.444994 0.895533i \(-0.646794\pi\)
−0.444994 + 0.895533i \(0.646794\pi\)
\(102\) −4.23607 −0.419433
\(103\) −16.9443 −1.66957 −0.834784 0.550577i \(-0.814408\pi\)
−0.834784 + 0.550577i \(0.814408\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.61803 0.255494
\(106\) −1.23607 −0.120058
\(107\) −15.3820 −1.48703 −0.743515 0.668719i \(-0.766843\pi\)
−0.743515 + 0.668719i \(0.766843\pi\)
\(108\) 2.23607 0.215166
\(109\) 0.472136 0.0452224 0.0226112 0.999744i \(-0.492802\pi\)
0.0226112 + 0.999744i \(0.492802\pi\)
\(110\) 0 0
\(111\) −16.9443 −1.60828
\(112\) −1.00000 −0.0944911
\(113\) −9.32624 −0.877339 −0.438669 0.898649i \(-0.644550\pi\)
−0.438669 + 0.898649i \(0.644550\pi\)
\(114\) 17.9443 1.68064
\(115\) −6.00000 −0.559503
\(116\) −3.23607 −0.300461
\(117\) −7.70820 −0.712624
\(118\) 7.61803 0.701297
\(119\) −1.61803 −0.148325
\(120\) 2.61803 0.238993
\(121\) 0 0
\(122\) −3.52786 −0.319398
\(123\) 27.0344 2.43761
\(124\) −1.23607 −0.111002
\(125\) −1.00000 −0.0894427
\(126\) 3.85410 0.343351
\(127\) 11.4164 1.01304 0.506521 0.862228i \(-0.330931\pi\)
0.506521 + 0.862228i \(0.330931\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.85410 −0.427380
\(130\) −2.00000 −0.175412
\(131\) 7.79837 0.681347 0.340674 0.940182i \(-0.389345\pi\)
0.340674 + 0.940182i \(0.389345\pi\)
\(132\) 0 0
\(133\) 6.85410 0.594326
\(134\) −6.09017 −0.526111
\(135\) −2.23607 −0.192450
\(136\) −1.61803 −0.138745
\(137\) −1.09017 −0.0931395 −0.0465698 0.998915i \(-0.514829\pi\)
−0.0465698 + 0.998915i \(0.514829\pi\)
\(138\) −15.7082 −1.33717
\(139\) −18.4721 −1.56679 −0.783393 0.621527i \(-0.786513\pi\)
−0.783393 + 0.621527i \(0.786513\pi\)
\(140\) 1.00000 0.0845154
\(141\) 24.1803 2.03635
\(142\) 9.70820 0.814694
\(143\) 0 0
\(144\) 3.85410 0.321175
\(145\) 3.23607 0.268741
\(146\) 13.8541 1.14657
\(147\) 2.61803 0.215932
\(148\) −6.47214 −0.532006
\(149\) −24.1803 −1.98093 −0.990465 0.137762i \(-0.956009\pi\)
−0.990465 + 0.137762i \(0.956009\pi\)
\(150\) −2.61803 −0.213762
\(151\) −2.29180 −0.186504 −0.0932519 0.995643i \(-0.529726\pi\)
−0.0932519 + 0.995643i \(0.529726\pi\)
\(152\) 6.85410 0.555941
\(153\) 6.23607 0.504156
\(154\) 0 0
\(155\) 1.23607 0.0992834
\(156\) −5.23607 −0.419221
\(157\) 0.291796 0.0232879 0.0116439 0.999932i \(-0.496294\pi\)
0.0116439 + 0.999932i \(0.496294\pi\)
\(158\) −8.47214 −0.674007
\(159\) 3.23607 0.256637
\(160\) 1.00000 0.0790569
\(161\) −6.00000 −0.472866
\(162\) 5.70820 0.448479
\(163\) 2.85410 0.223551 0.111775 0.993734i \(-0.464346\pi\)
0.111775 + 0.993734i \(0.464346\pi\)
\(164\) 10.3262 0.806344
\(165\) 0 0
\(166\) 10.3262 0.801471
\(167\) −6.47214 −0.500829 −0.250414 0.968139i \(-0.580567\pi\)
−0.250414 + 0.968139i \(0.580567\pi\)
\(168\) 2.61803 0.201986
\(169\) −9.00000 −0.692308
\(170\) 1.61803 0.124098
\(171\) −26.4164 −2.02011
\(172\) −1.85410 −0.141374
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 8.47214 0.642271
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −19.9443 −1.49910
\(178\) −13.0902 −0.981150
\(179\) −6.38197 −0.477011 −0.238505 0.971141i \(-0.576657\pi\)
−0.238505 + 0.971141i \(0.576657\pi\)
\(180\) −3.85410 −0.287268
\(181\) 3.52786 0.262224 0.131112 0.991368i \(-0.458145\pi\)
0.131112 + 0.991368i \(0.458145\pi\)
\(182\) −2.00000 −0.148250
\(183\) 9.23607 0.682750
\(184\) −6.00000 −0.442326
\(185\) 6.47214 0.475841
\(186\) 3.23607 0.237280
\(187\) 0 0
\(188\) 9.23607 0.673609
\(189\) −2.23607 −0.162650
\(190\) −6.85410 −0.497249
\(191\) −1.52786 −0.110552 −0.0552762 0.998471i \(-0.517604\pi\)
−0.0552762 + 0.998471i \(0.517604\pi\)
\(192\) 2.61803 0.188940
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 9.56231 0.686534
\(195\) 5.23607 0.374963
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −19.4164 −1.37639 −0.688196 0.725525i \(-0.741597\pi\)
−0.688196 + 0.725525i \(0.741597\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 15.9443 1.12462
\(202\) 8.94427 0.629317
\(203\) 3.23607 0.227127
\(204\) 4.23607 0.296584
\(205\) −10.3262 −0.721216
\(206\) 16.9443 1.18056
\(207\) 23.1246 1.60727
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.61803 −0.180662
\(211\) −11.2705 −0.775894 −0.387947 0.921682i \(-0.626816\pi\)
−0.387947 + 0.921682i \(0.626816\pi\)
\(212\) 1.23607 0.0848935
\(213\) −25.4164 −1.74150
\(214\) 15.3820 1.05149
\(215\) 1.85410 0.126449
\(216\) −2.23607 −0.152145
\(217\) 1.23607 0.0839098
\(218\) −0.472136 −0.0319771
\(219\) −36.2705 −2.45093
\(220\) 0 0
\(221\) −3.23607 −0.217681
\(222\) 16.9443 1.13723
\(223\) 6.29180 0.421330 0.210665 0.977558i \(-0.432437\pi\)
0.210665 + 0.977558i \(0.432437\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.85410 0.256940
\(226\) 9.32624 0.620372
\(227\) 24.2705 1.61089 0.805445 0.592670i \(-0.201926\pi\)
0.805445 + 0.592670i \(0.201926\pi\)
\(228\) −17.9443 −1.18839
\(229\) 11.1246 0.735135 0.367568 0.929997i \(-0.380191\pi\)
0.367568 + 0.929997i \(0.380191\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 3.23607 0.212458
\(233\) −10.9098 −0.714727 −0.357363 0.933965i \(-0.616324\pi\)
−0.357363 + 0.933965i \(0.616324\pi\)
\(234\) 7.70820 0.503901
\(235\) −9.23607 −0.602495
\(236\) −7.61803 −0.495892
\(237\) 22.1803 1.44077
\(238\) 1.61803 0.104882
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.61803 −0.168993
\(241\) 6.27051 0.403919 0.201960 0.979394i \(-0.435269\pi\)
0.201960 + 0.979394i \(0.435269\pi\)
\(242\) 0 0
\(243\) −21.6525 −1.38901
\(244\) 3.52786 0.225848
\(245\) −1.00000 −0.0638877
\(246\) −27.0344 −1.72365
\(247\) 13.7082 0.872232
\(248\) 1.23607 0.0784904
\(249\) −27.0344 −1.71324
\(250\) 1.00000 0.0632456
\(251\) 26.8328 1.69367 0.846836 0.531854i \(-0.178504\pi\)
0.846836 + 0.531854i \(0.178504\pi\)
\(252\) −3.85410 −0.242786
\(253\) 0 0
\(254\) −11.4164 −0.716329
\(255\) −4.23607 −0.265273
\(256\) 1.00000 0.0625000
\(257\) 29.5623 1.84405 0.922023 0.387135i \(-0.126535\pi\)
0.922023 + 0.387135i \(0.126535\pi\)
\(258\) 4.85410 0.302203
\(259\) 6.47214 0.402159
\(260\) 2.00000 0.124035
\(261\) −12.4721 −0.772006
\(262\) −7.79837 −0.481785
\(263\) −20.1803 −1.24437 −0.622187 0.782869i \(-0.713755\pi\)
−0.622187 + 0.782869i \(0.713755\pi\)
\(264\) 0 0
\(265\) −1.23607 −0.0759311
\(266\) −6.85410 −0.420252
\(267\) 34.2705 2.09732
\(268\) 6.09017 0.372016
\(269\) −12.1803 −0.742648 −0.371324 0.928503i \(-0.621096\pi\)
−0.371324 + 0.928503i \(0.621096\pi\)
\(270\) 2.23607 0.136083
\(271\) 9.41641 0.572006 0.286003 0.958229i \(-0.407673\pi\)
0.286003 + 0.958229i \(0.407673\pi\)
\(272\) 1.61803 0.0981077
\(273\) 5.23607 0.316901
\(274\) 1.09017 0.0658596
\(275\) 0 0
\(276\) 15.7082 0.945523
\(277\) −6.29180 −0.378037 −0.189019 0.981973i \(-0.560531\pi\)
−0.189019 + 0.981973i \(0.560531\pi\)
\(278\) 18.4721 1.10789
\(279\) −4.76393 −0.285209
\(280\) −1.00000 −0.0597614
\(281\) 24.0344 1.43377 0.716887 0.697189i \(-0.245566\pi\)
0.716887 + 0.697189i \(0.245566\pi\)
\(282\) −24.1803 −1.43992
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −9.70820 −0.576076
\(285\) 17.9443 1.06293
\(286\) 0 0
\(287\) −10.3262 −0.609539
\(288\) −3.85410 −0.227105
\(289\) −14.3820 −0.845998
\(290\) −3.23607 −0.190028
\(291\) −25.0344 −1.46754
\(292\) −13.8541 −0.810750
\(293\) −16.4721 −0.962312 −0.481156 0.876635i \(-0.659783\pi\)
−0.481156 + 0.876635i \(0.659783\pi\)
\(294\) −2.61803 −0.152687
\(295\) 7.61803 0.443539
\(296\) 6.47214 0.376185
\(297\) 0 0
\(298\) 24.1803 1.40073
\(299\) −12.0000 −0.693978
\(300\) 2.61803 0.151152
\(301\) 1.85410 0.106869
\(302\) 2.29180 0.131878
\(303\) −23.4164 −1.34524
\(304\) −6.85410 −0.393110
\(305\) −3.52786 −0.202005
\(306\) −6.23607 −0.356492
\(307\) 27.5623 1.57306 0.786532 0.617550i \(-0.211875\pi\)
0.786532 + 0.617550i \(0.211875\pi\)
\(308\) 0 0
\(309\) −44.3607 −2.52359
\(310\) −1.23607 −0.0702039
\(311\) −29.1246 −1.65151 −0.825753 0.564032i \(-0.809249\pi\)
−0.825753 + 0.564032i \(0.809249\pi\)
\(312\) 5.23607 0.296434
\(313\) −0.854102 −0.0482767 −0.0241383 0.999709i \(-0.507684\pi\)
−0.0241383 + 0.999709i \(0.507684\pi\)
\(314\) −0.291796 −0.0164670
\(315\) 3.85410 0.217154
\(316\) 8.47214 0.476595
\(317\) −20.7639 −1.16622 −0.583109 0.812394i \(-0.698164\pi\)
−0.583109 + 0.812394i \(0.698164\pi\)
\(318\) −3.23607 −0.181470
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −40.2705 −2.24768
\(322\) 6.00000 0.334367
\(323\) −11.0902 −0.617074
\(324\) −5.70820 −0.317122
\(325\) −2.00000 −0.110940
\(326\) −2.85410 −0.158074
\(327\) 1.23607 0.0683547
\(328\) −10.3262 −0.570171
\(329\) −9.23607 −0.509201
\(330\) 0 0
\(331\) 6.03444 0.331683 0.165841 0.986152i \(-0.446966\pi\)
0.165841 + 0.986152i \(0.446966\pi\)
\(332\) −10.3262 −0.566726
\(333\) −24.9443 −1.36694
\(334\) 6.47214 0.354140
\(335\) −6.09017 −0.332742
\(336\) −2.61803 −0.142825
\(337\) 15.6180 0.850769 0.425384 0.905013i \(-0.360139\pi\)
0.425384 + 0.905013i \(0.360139\pi\)
\(338\) 9.00000 0.489535
\(339\) −24.4164 −1.32612
\(340\) −1.61803 −0.0877502
\(341\) 0 0
\(342\) 26.4164 1.42844
\(343\) −1.00000 −0.0539949
\(344\) 1.85410 0.0999665
\(345\) −15.7082 −0.845701
\(346\) 12.0000 0.645124
\(347\) −5.32624 −0.285927 −0.142964 0.989728i \(-0.545663\pi\)
−0.142964 + 0.989728i \(0.545663\pi\)
\(348\) −8.47214 −0.454154
\(349\) −5.52786 −0.295900 −0.147950 0.988995i \(-0.547267\pi\)
−0.147950 + 0.988995i \(0.547267\pi\)
\(350\) 1.00000 0.0534522
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) −28.4508 −1.51429 −0.757143 0.653249i \(-0.773405\pi\)
−0.757143 + 0.653249i \(0.773405\pi\)
\(354\) 19.9443 1.06003
\(355\) 9.70820 0.515258
\(356\) 13.0902 0.693778
\(357\) −4.23607 −0.224196
\(358\) 6.38197 0.337297
\(359\) −18.6525 −0.984440 −0.492220 0.870471i \(-0.663814\pi\)
−0.492220 + 0.870471i \(0.663814\pi\)
\(360\) 3.85410 0.203129
\(361\) 27.9787 1.47256
\(362\) −3.52786 −0.185420
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 13.8541 0.725157
\(366\) −9.23607 −0.482777
\(367\) 28.1803 1.47100 0.735501 0.677524i \(-0.236947\pi\)
0.735501 + 0.677524i \(0.236947\pi\)
\(368\) 6.00000 0.312772
\(369\) 39.7984 2.07182
\(370\) −6.47214 −0.336470
\(371\) −1.23607 −0.0641735
\(372\) −3.23607 −0.167782
\(373\) 24.4721 1.26712 0.633560 0.773694i \(-0.281593\pi\)
0.633560 + 0.773694i \(0.281593\pi\)
\(374\) 0 0
\(375\) −2.61803 −0.135195
\(376\) −9.23607 −0.476314
\(377\) 6.47214 0.333332
\(378\) 2.23607 0.115011
\(379\) −27.8541 −1.43077 −0.715385 0.698731i \(-0.753748\pi\)
−0.715385 + 0.698731i \(0.753748\pi\)
\(380\) 6.85410 0.351608
\(381\) 29.8885 1.53124
\(382\) 1.52786 0.0781723
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −2.61803 −0.133601
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −7.14590 −0.363246
\(388\) −9.56231 −0.485453
\(389\) −32.8328 −1.66469 −0.832345 0.554258i \(-0.813002\pi\)
−0.832345 + 0.554258i \(0.813002\pi\)
\(390\) −5.23607 −0.265139
\(391\) 9.70820 0.490965
\(392\) −1.00000 −0.0505076
\(393\) 20.4164 1.02987
\(394\) 6.00000 0.302276
\(395\) −8.47214 −0.426279
\(396\) 0 0
\(397\) −16.9443 −0.850409 −0.425204 0.905097i \(-0.639798\pi\)
−0.425204 + 0.905097i \(0.639798\pi\)
\(398\) 19.4164 0.973257
\(399\) 17.9443 0.898337
\(400\) 1.00000 0.0500000
\(401\) −26.5066 −1.32368 −0.661838 0.749647i \(-0.730223\pi\)
−0.661838 + 0.749647i \(0.730223\pi\)
\(402\) −15.9443 −0.795228
\(403\) 2.47214 0.123146
\(404\) −8.94427 −0.444994
\(405\) 5.70820 0.283643
\(406\) −3.23607 −0.160603
\(407\) 0 0
\(408\) −4.23607 −0.209717
\(409\) −27.8885 −1.37900 −0.689500 0.724286i \(-0.742170\pi\)
−0.689500 + 0.724286i \(0.742170\pi\)
\(410\) 10.3262 0.509977
\(411\) −2.85410 −0.140782
\(412\) −16.9443 −0.834784
\(413\) 7.61803 0.374859
\(414\) −23.1246 −1.13651
\(415\) 10.3262 0.506895
\(416\) 2.00000 0.0980581
\(417\) −48.3607 −2.36823
\(418\) 0 0
\(419\) 12.3262 0.602176 0.301088 0.953596i \(-0.402650\pi\)
0.301088 + 0.953596i \(0.402650\pi\)
\(420\) 2.61803 0.127747
\(421\) −6.36068 −0.310001 −0.155000 0.987914i \(-0.549538\pi\)
−0.155000 + 0.987914i \(0.549538\pi\)
\(422\) 11.2705 0.548640
\(423\) 35.5967 1.73077
\(424\) −1.23607 −0.0600288
\(425\) 1.61803 0.0784862
\(426\) 25.4164 1.23143
\(427\) −3.52786 −0.170725
\(428\) −15.3820 −0.743515
\(429\) 0 0
\(430\) −1.85410 −0.0894127
\(431\) −31.2361 −1.50459 −0.752294 0.658827i \(-0.771053\pi\)
−0.752294 + 0.658827i \(0.771053\pi\)
\(432\) 2.23607 0.107583
\(433\) 16.4508 0.790577 0.395289 0.918557i \(-0.370645\pi\)
0.395289 + 0.918557i \(0.370645\pi\)
\(434\) −1.23607 −0.0593332
\(435\) 8.47214 0.406208
\(436\) 0.472136 0.0226112
\(437\) −41.1246 −1.96726
\(438\) 36.2705 1.73307
\(439\) 28.6525 1.36751 0.683754 0.729713i \(-0.260346\pi\)
0.683754 + 0.729713i \(0.260346\pi\)
\(440\) 0 0
\(441\) 3.85410 0.183529
\(442\) 3.23607 0.153924
\(443\) 28.0344 1.33196 0.665978 0.745971i \(-0.268014\pi\)
0.665978 + 0.745971i \(0.268014\pi\)
\(444\) −16.9443 −0.804140
\(445\) −13.0902 −0.620534
\(446\) −6.29180 −0.297925
\(447\) −63.3050 −2.99422
\(448\) −1.00000 −0.0472456
\(449\) 24.4508 1.15391 0.576953 0.816777i \(-0.304241\pi\)
0.576953 + 0.816777i \(0.304241\pi\)
\(450\) −3.85410 −0.181684
\(451\) 0 0
\(452\) −9.32624 −0.438669
\(453\) −6.00000 −0.281905
\(454\) −24.2705 −1.13907
\(455\) −2.00000 −0.0937614
\(456\) 17.9443 0.840318
\(457\) 31.2705 1.46277 0.731386 0.681963i \(-0.238874\pi\)
0.731386 + 0.681963i \(0.238874\pi\)
\(458\) −11.1246 −0.519819
\(459\) 3.61803 0.168875
\(460\) −6.00000 −0.279751
\(461\) −6.94427 −0.323427 −0.161713 0.986838i \(-0.551702\pi\)
−0.161713 + 0.986838i \(0.551702\pi\)
\(462\) 0 0
\(463\) −7.41641 −0.344670 −0.172335 0.985038i \(-0.555131\pi\)
−0.172335 + 0.985038i \(0.555131\pi\)
\(464\) −3.23607 −0.150231
\(465\) 3.23607 0.150069
\(466\) 10.9098 0.505388
\(467\) −17.8885 −0.827783 −0.413892 0.910326i \(-0.635831\pi\)
−0.413892 + 0.910326i \(0.635831\pi\)
\(468\) −7.70820 −0.356312
\(469\) −6.09017 −0.281218
\(470\) 9.23607 0.426028
\(471\) 0.763932 0.0352001
\(472\) 7.61803 0.350648
\(473\) 0 0
\(474\) −22.1803 −1.01878
\(475\) −6.85410 −0.314488
\(476\) −1.61803 −0.0741625
\(477\) 4.76393 0.218125
\(478\) 0 0
\(479\) 37.8885 1.73117 0.865586 0.500761i \(-0.166946\pi\)
0.865586 + 0.500761i \(0.166946\pi\)
\(480\) 2.61803 0.119496
\(481\) 12.9443 0.590208
\(482\) −6.27051 −0.285614
\(483\) −15.7082 −0.714748
\(484\) 0 0
\(485\) 9.56231 0.434202
\(486\) 21.6525 0.982176
\(487\) −26.8328 −1.21591 −0.607955 0.793971i \(-0.708010\pi\)
−0.607955 + 0.793971i \(0.708010\pi\)
\(488\) −3.52786 −0.159699
\(489\) 7.47214 0.337902
\(490\) 1.00000 0.0451754
\(491\) 32.5066 1.46700 0.733501 0.679689i \(-0.237885\pi\)
0.733501 + 0.679689i \(0.237885\pi\)
\(492\) 27.0344 1.21881
\(493\) −5.23607 −0.235821
\(494\) −13.7082 −0.616761
\(495\) 0 0
\(496\) −1.23607 −0.0555011
\(497\) 9.70820 0.435472
\(498\) 27.0344 1.21144
\(499\) −15.2016 −0.680518 −0.340259 0.940332i \(-0.610515\pi\)
−0.340259 + 0.940332i \(0.610515\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −16.9443 −0.757014
\(502\) −26.8328 −1.19761
\(503\) 17.1246 0.763549 0.381774 0.924256i \(-0.375313\pi\)
0.381774 + 0.924256i \(0.375313\pi\)
\(504\) 3.85410 0.171675
\(505\) 8.94427 0.398015
\(506\) 0 0
\(507\) −23.5623 −1.04644
\(508\) 11.4164 0.506521
\(509\) −25.5279 −1.13150 −0.565751 0.824576i \(-0.691414\pi\)
−0.565751 + 0.824576i \(0.691414\pi\)
\(510\) 4.23607 0.187576
\(511\) 13.8541 0.612869
\(512\) −1.00000 −0.0441942
\(513\) −15.3262 −0.676670
\(514\) −29.5623 −1.30394
\(515\) 16.9443 0.746654
\(516\) −4.85410 −0.213690
\(517\) 0 0
\(518\) −6.47214 −0.284369
\(519\) −31.4164 −1.37903
\(520\) −2.00000 −0.0877058
\(521\) 35.7984 1.56836 0.784178 0.620536i \(-0.213085\pi\)
0.784178 + 0.620536i \(0.213085\pi\)
\(522\) 12.4721 0.545891
\(523\) −5.97871 −0.261431 −0.130715 0.991420i \(-0.541727\pi\)
−0.130715 + 0.991420i \(0.541727\pi\)
\(524\) 7.79837 0.340674
\(525\) −2.61803 −0.114260
\(526\) 20.1803 0.879905
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 1.23607 0.0536914
\(531\) −29.3607 −1.27414
\(532\) 6.85410 0.297163
\(533\) −20.6525 −0.894558
\(534\) −34.2705 −1.48303
\(535\) 15.3820 0.665020
\(536\) −6.09017 −0.263055
\(537\) −16.7082 −0.721012
\(538\) 12.1803 0.525132
\(539\) 0 0
\(540\) −2.23607 −0.0962250
\(541\) −14.8328 −0.637713 −0.318856 0.947803i \(-0.603299\pi\)
−0.318856 + 0.947803i \(0.603299\pi\)
\(542\) −9.41641 −0.404469
\(543\) 9.23607 0.396358
\(544\) −1.61803 −0.0693726
\(545\) −0.472136 −0.0202241
\(546\) −5.23607 −0.224083
\(547\) 17.9098 0.765769 0.382885 0.923796i \(-0.374931\pi\)
0.382885 + 0.923796i \(0.374931\pi\)
\(548\) −1.09017 −0.0465698
\(549\) 13.5967 0.580295
\(550\) 0 0
\(551\) 22.1803 0.944914
\(552\) −15.7082 −0.668586
\(553\) −8.47214 −0.360272
\(554\) 6.29180 0.267313
\(555\) 16.9443 0.719244
\(556\) −18.4721 −0.783393
\(557\) −38.1803 −1.61775 −0.808876 0.587979i \(-0.799924\pi\)
−0.808876 + 0.587979i \(0.799924\pi\)
\(558\) 4.76393 0.201673
\(559\) 3.70820 0.156840
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −24.0344 −1.01383
\(563\) −3.49342 −0.147230 −0.0736151 0.997287i \(-0.523454\pi\)
−0.0736151 + 0.997287i \(0.523454\pi\)
\(564\) 24.1803 1.01818
\(565\) 9.32624 0.392358
\(566\) −12.0000 −0.504398
\(567\) 5.70820 0.239722
\(568\) 9.70820 0.407347
\(569\) 8.09017 0.339158 0.169579 0.985517i \(-0.445759\pi\)
0.169579 + 0.985517i \(0.445759\pi\)
\(570\) −17.9443 −0.751603
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 10.3262 0.431009
\(575\) 6.00000 0.250217
\(576\) 3.85410 0.160588
\(577\) 25.2148 1.04971 0.524853 0.851193i \(-0.324120\pi\)
0.524853 + 0.851193i \(0.324120\pi\)
\(578\) 14.3820 0.598211
\(579\) 5.23607 0.217604
\(580\) 3.23607 0.134370
\(581\) 10.3262 0.428405
\(582\) 25.0344 1.03771
\(583\) 0 0
\(584\) 13.8541 0.573287
\(585\) 7.70820 0.318695
\(586\) 16.4721 0.680458
\(587\) 2.74265 0.113201 0.0566006 0.998397i \(-0.481974\pi\)
0.0566006 + 0.998397i \(0.481974\pi\)
\(588\) 2.61803 0.107966
\(589\) 8.47214 0.349088
\(590\) −7.61803 −0.313629
\(591\) −15.7082 −0.646149
\(592\) −6.47214 −0.266003
\(593\) −12.0902 −0.496484 −0.248242 0.968698i \(-0.579853\pi\)
−0.248242 + 0.968698i \(0.579853\pi\)
\(594\) 0 0
\(595\) 1.61803 0.0663329
\(596\) −24.1803 −0.990465
\(597\) −50.8328 −2.08045
\(598\) 12.0000 0.490716
\(599\) −25.2361 −1.03112 −0.515559 0.856854i \(-0.672416\pi\)
−0.515559 + 0.856854i \(0.672416\pi\)
\(600\) −2.61803 −0.106881
\(601\) 29.0902 1.18661 0.593306 0.804977i \(-0.297822\pi\)
0.593306 + 0.804977i \(0.297822\pi\)
\(602\) −1.85410 −0.0755676
\(603\) 23.4721 0.955859
\(604\) −2.29180 −0.0932519
\(605\) 0 0
\(606\) 23.4164 0.951227
\(607\) 29.5967 1.20129 0.600647 0.799514i \(-0.294910\pi\)
0.600647 + 0.799514i \(0.294910\pi\)
\(608\) 6.85410 0.277971
\(609\) 8.47214 0.343308
\(610\) 3.52786 0.142839
\(611\) −18.4721 −0.747303
\(612\) 6.23607 0.252078
\(613\) 37.4164 1.51123 0.755617 0.655013i \(-0.227337\pi\)
0.755617 + 0.655013i \(0.227337\pi\)
\(614\) −27.5623 −1.11232
\(615\) −27.0344 −1.09013
\(616\) 0 0
\(617\) 23.0344 0.927332 0.463666 0.886010i \(-0.346534\pi\)
0.463666 + 0.886010i \(0.346534\pi\)
\(618\) 44.3607 1.78445
\(619\) 18.6738 0.750562 0.375281 0.926911i \(-0.377546\pi\)
0.375281 + 0.926911i \(0.377546\pi\)
\(620\) 1.23607 0.0496417
\(621\) 13.4164 0.538382
\(622\) 29.1246 1.16779
\(623\) −13.0902 −0.524447
\(624\) −5.23607 −0.209610
\(625\) 1.00000 0.0400000
\(626\) 0.854102 0.0341368
\(627\) 0 0
\(628\) 0.291796 0.0116439
\(629\) −10.4721 −0.417551
\(630\) −3.85410 −0.153551
\(631\) −23.7082 −0.943809 −0.471904 0.881650i \(-0.656433\pi\)
−0.471904 + 0.881650i \(0.656433\pi\)
\(632\) −8.47214 −0.337003
\(633\) −29.5066 −1.17278
\(634\) 20.7639 0.824641
\(635\) −11.4164 −0.453046
\(636\) 3.23607 0.128318
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −37.4164 −1.48017
\(640\) 1.00000 0.0395285
\(641\) 40.3262 1.59279 0.796395 0.604776i \(-0.206738\pi\)
0.796395 + 0.604776i \(0.206738\pi\)
\(642\) 40.2705 1.58935
\(643\) −14.9787 −0.590703 −0.295351 0.955389i \(-0.595437\pi\)
−0.295351 + 0.955389i \(0.595437\pi\)
\(644\) −6.00000 −0.236433
\(645\) 4.85410 0.191130
\(646\) 11.0902 0.436337
\(647\) 0.180340 0.00708989 0.00354495 0.999994i \(-0.498872\pi\)
0.00354495 + 0.999994i \(0.498872\pi\)
\(648\) 5.70820 0.224239
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 3.23607 0.126832
\(652\) 2.85410 0.111775
\(653\) 26.8328 1.05005 0.525025 0.851087i \(-0.324056\pi\)
0.525025 + 0.851087i \(0.324056\pi\)
\(654\) −1.23607 −0.0483341
\(655\) −7.79837 −0.304708
\(656\) 10.3262 0.403172
\(657\) −53.3951 −2.08314
\(658\) 9.23607 0.360059
\(659\) −4.09017 −0.159330 −0.0796652 0.996822i \(-0.525385\pi\)
−0.0796652 + 0.996822i \(0.525385\pi\)
\(660\) 0 0
\(661\) −45.8885 −1.78486 −0.892429 0.451188i \(-0.851000\pi\)
−0.892429 + 0.451188i \(0.851000\pi\)
\(662\) −6.03444 −0.234535
\(663\) −8.47214 −0.329030
\(664\) 10.3262 0.400736
\(665\) −6.85410 −0.265791
\(666\) 24.9443 0.966571
\(667\) −19.4164 −0.751806
\(668\) −6.47214 −0.250414
\(669\) 16.4721 0.636850
\(670\) 6.09017 0.235284
\(671\) 0 0
\(672\) 2.61803 0.100993
\(673\) 38.8541 1.49772 0.748858 0.662731i \(-0.230603\pi\)
0.748858 + 0.662731i \(0.230603\pi\)
\(674\) −15.6180 −0.601584
\(675\) 2.23607 0.0860663
\(676\) −9.00000 −0.346154
\(677\) 13.5967 0.522565 0.261283 0.965262i \(-0.415855\pi\)
0.261283 + 0.965262i \(0.415855\pi\)
\(678\) 24.4164 0.937706
\(679\) 9.56231 0.366968
\(680\) 1.61803 0.0620488
\(681\) 63.5410 2.43490
\(682\) 0 0
\(683\) −31.0557 −1.18831 −0.594157 0.804349i \(-0.702514\pi\)
−0.594157 + 0.804349i \(0.702514\pi\)
\(684\) −26.4164 −1.01006
\(685\) 1.09017 0.0416533
\(686\) 1.00000 0.0381802
\(687\) 29.1246 1.11117
\(688\) −1.85410 −0.0706870
\(689\) −2.47214 −0.0941809
\(690\) 15.7082 0.598001
\(691\) 21.1459 0.804428 0.402214 0.915546i \(-0.368241\pi\)
0.402214 + 0.915546i \(0.368241\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 5.32624 0.202181
\(695\) 18.4721 0.700688
\(696\) 8.47214 0.321135
\(697\) 16.7082 0.632868
\(698\) 5.52786 0.209233
\(699\) −28.5623 −1.08033
\(700\) −1.00000 −0.0377964
\(701\) 12.1803 0.460045 0.230023 0.973185i \(-0.426120\pi\)
0.230023 + 0.973185i \(0.426120\pi\)
\(702\) 4.47214 0.168790
\(703\) 44.3607 1.67309
\(704\) 0 0
\(705\) −24.1803 −0.910684
\(706\) 28.4508 1.07076
\(707\) 8.94427 0.336384
\(708\) −19.9443 −0.749552
\(709\) −24.9443 −0.936802 −0.468401 0.883516i \(-0.655170\pi\)
−0.468401 + 0.883516i \(0.655170\pi\)
\(710\) −9.70820 −0.364342
\(711\) 32.6525 1.22456
\(712\) −13.0902 −0.490575
\(713\) −7.41641 −0.277747
\(714\) 4.23607 0.158531
\(715\) 0 0
\(716\) −6.38197 −0.238505
\(717\) 0 0
\(718\) 18.6525 0.696104
\(719\) 32.3607 1.20685 0.603425 0.797420i \(-0.293802\pi\)
0.603425 + 0.797420i \(0.293802\pi\)
\(720\) −3.85410 −0.143634
\(721\) 16.9443 0.631038
\(722\) −27.9787 −1.04126
\(723\) 16.4164 0.610533
\(724\) 3.52786 0.131112
\(725\) −3.23607 −0.120185
\(726\) 0 0
\(727\) 38.8328 1.44023 0.720115 0.693855i \(-0.244089\pi\)
0.720115 + 0.693855i \(0.244089\pi\)
\(728\) −2.00000 −0.0741249
\(729\) −39.5623 −1.46527
\(730\) −13.8541 −0.512763
\(731\) −3.00000 −0.110959
\(732\) 9.23607 0.341375
\(733\) 13.2361 0.488885 0.244443 0.969664i \(-0.421395\pi\)
0.244443 + 0.969664i \(0.421395\pi\)
\(734\) −28.1803 −1.04016
\(735\) −2.61803 −0.0965676
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) −39.7984 −1.46500
\(739\) 24.6738 0.907639 0.453820 0.891094i \(-0.350061\pi\)
0.453820 + 0.891094i \(0.350061\pi\)
\(740\) 6.47214 0.237920
\(741\) 35.8885 1.31840
\(742\) 1.23607 0.0453775
\(743\) 12.1115 0.444326 0.222163 0.975010i \(-0.428688\pi\)
0.222163 + 0.975010i \(0.428688\pi\)
\(744\) 3.23607 0.118640
\(745\) 24.1803 0.885899
\(746\) −24.4721 −0.895989
\(747\) −39.7984 −1.45615
\(748\) 0 0
\(749\) 15.3820 0.562045
\(750\) 2.61803 0.0955971
\(751\) −32.6525 −1.19151 −0.595753 0.803168i \(-0.703146\pi\)
−0.595753 + 0.803168i \(0.703146\pi\)
\(752\) 9.23607 0.336805
\(753\) 70.2492 2.56002
\(754\) −6.47214 −0.235701
\(755\) 2.29180 0.0834070
\(756\) −2.23607 −0.0813250
\(757\) 19.8885 0.722861 0.361431 0.932399i \(-0.382288\pi\)
0.361431 + 0.932399i \(0.382288\pi\)
\(758\) 27.8541 1.01171
\(759\) 0 0
\(760\) −6.85410 −0.248624
\(761\) 31.7984 1.15269 0.576345 0.817206i \(-0.304478\pi\)
0.576345 + 0.817206i \(0.304478\pi\)
\(762\) −29.8885 −1.08275
\(763\) −0.472136 −0.0170925
\(764\) −1.52786 −0.0552762
\(765\) −6.23607 −0.225466
\(766\) 24.0000 0.867155
\(767\) 15.2361 0.550143
\(768\) 2.61803 0.0944702
\(769\) 7.88854 0.284468 0.142234 0.989833i \(-0.454571\pi\)
0.142234 + 0.989833i \(0.454571\pi\)
\(770\) 0 0
\(771\) 77.3951 2.78732
\(772\) 2.00000 0.0719816
\(773\) −24.6525 −0.886688 −0.443344 0.896352i \(-0.646208\pi\)
−0.443344 + 0.896352i \(0.646208\pi\)
\(774\) 7.14590 0.256854
\(775\) −1.23607 −0.0444009
\(776\) 9.56231 0.343267
\(777\) 16.9443 0.607872
\(778\) 32.8328 1.17711
\(779\) −70.7771 −2.53585
\(780\) 5.23607 0.187481
\(781\) 0 0
\(782\) −9.70820 −0.347165
\(783\) −7.23607 −0.258596
\(784\) 1.00000 0.0357143
\(785\) −0.291796 −0.0104146
\(786\) −20.4164 −0.728229
\(787\) −51.1033 −1.82164 −0.910818 0.412807i \(-0.864548\pi\)
−0.910818 + 0.412807i \(0.864548\pi\)
\(788\) −6.00000 −0.213741
\(789\) −52.8328 −1.88090
\(790\) 8.47214 0.301425
\(791\) 9.32624 0.331603
\(792\) 0 0
\(793\) −7.05573 −0.250556
\(794\) 16.9443 0.601330
\(795\) −3.23607 −0.114772
\(796\) −19.4164 −0.688196
\(797\) −6.36068 −0.225307 −0.112653 0.993634i \(-0.535935\pi\)
−0.112653 + 0.993634i \(0.535935\pi\)
\(798\) −17.9443 −0.635220
\(799\) 14.9443 0.528690
\(800\) −1.00000 −0.0353553
\(801\) 50.4508 1.78259
\(802\) 26.5066 0.935980
\(803\) 0 0
\(804\) 15.9443 0.562311
\(805\) 6.00000 0.211472
\(806\) −2.47214 −0.0870773
\(807\) −31.8885 −1.12253
\(808\) 8.94427 0.314658
\(809\) 48.8115 1.71612 0.858061 0.513548i \(-0.171669\pi\)
0.858061 + 0.513548i \(0.171669\pi\)
\(810\) −5.70820 −0.200566
\(811\) 29.9230 1.05074 0.525369 0.850874i \(-0.323927\pi\)
0.525369 + 0.850874i \(0.323927\pi\)
\(812\) 3.23607 0.113564
\(813\) 24.6525 0.864600
\(814\) 0 0
\(815\) −2.85410 −0.0999748
\(816\) 4.23607 0.148292
\(817\) 12.7082 0.444604
\(818\) 27.8885 0.975100
\(819\) 7.70820 0.269346
\(820\) −10.3262 −0.360608
\(821\) −31.0557 −1.08385 −0.541926 0.840426i \(-0.682305\pi\)
−0.541926 + 0.840426i \(0.682305\pi\)
\(822\) 2.85410 0.0995482
\(823\) −2.83282 −0.0987457 −0.0493729 0.998780i \(-0.515722\pi\)
−0.0493729 + 0.998780i \(0.515722\pi\)
\(824\) 16.9443 0.590282
\(825\) 0 0
\(826\) −7.61803 −0.265065
\(827\) 40.7426 1.41676 0.708380 0.705831i \(-0.249426\pi\)
0.708380 + 0.705831i \(0.249426\pi\)
\(828\) 23.1246 0.803636
\(829\) 34.1803 1.18713 0.593566 0.804785i \(-0.297720\pi\)
0.593566 + 0.804785i \(0.297720\pi\)
\(830\) −10.3262 −0.358429
\(831\) −16.4721 −0.571412
\(832\) −2.00000 −0.0693375
\(833\) 1.61803 0.0560616
\(834\) 48.3607 1.67459
\(835\) 6.47214 0.223978
\(836\) 0 0
\(837\) −2.76393 −0.0955355
\(838\) −12.3262 −0.425803
\(839\) 4.47214 0.154395 0.0771976 0.997016i \(-0.475403\pi\)
0.0771976 + 0.997016i \(0.475403\pi\)
\(840\) −2.61803 −0.0903308
\(841\) −18.5279 −0.638892
\(842\) 6.36068 0.219204
\(843\) 62.9230 2.16718
\(844\) −11.2705 −0.387947
\(845\) 9.00000 0.309609
\(846\) −35.5967 −1.22384
\(847\) 0 0
\(848\) 1.23607 0.0424467
\(849\) 31.4164 1.07821
\(850\) −1.61803 −0.0554981
\(851\) −38.8328 −1.33117
\(852\) −25.4164 −0.870752
\(853\) 4.29180 0.146948 0.0734741 0.997297i \(-0.476591\pi\)
0.0734741 + 0.997297i \(0.476591\pi\)
\(854\) 3.52786 0.120721
\(855\) 26.4164 0.903422
\(856\) 15.3820 0.525745
\(857\) 9.27051 0.316675 0.158337 0.987385i \(-0.449387\pi\)
0.158337 + 0.987385i \(0.449387\pi\)
\(858\) 0 0
\(859\) −7.43769 −0.253771 −0.126885 0.991917i \(-0.540498\pi\)
−0.126885 + 0.991917i \(0.540498\pi\)
\(860\) 1.85410 0.0632244
\(861\) −27.0344 −0.921331
\(862\) 31.2361 1.06390
\(863\) 6.76393 0.230247 0.115123 0.993351i \(-0.463274\pi\)
0.115123 + 0.993351i \(0.463274\pi\)
\(864\) −2.23607 −0.0760726
\(865\) 12.0000 0.408012
\(866\) −16.4508 −0.559023
\(867\) −37.6525 −1.27875
\(868\) 1.23607 0.0419549
\(869\) 0 0
\(870\) −8.47214 −0.287232
\(871\) −12.1803 −0.412715
\(872\) −0.472136 −0.0159885
\(873\) −36.8541 −1.24732
\(874\) 41.1246 1.39106
\(875\) 1.00000 0.0338062
\(876\) −36.2705 −1.22547
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −28.6525 −0.966974
\(879\) −43.1246 −1.45456
\(880\) 0 0
\(881\) 4.14590 0.139679 0.0698394 0.997558i \(-0.477751\pi\)
0.0698394 + 0.997558i \(0.477751\pi\)
\(882\) −3.85410 −0.129774
\(883\) 2.43769 0.0820349 0.0410175 0.999158i \(-0.486940\pi\)
0.0410175 + 0.999158i \(0.486940\pi\)
\(884\) −3.23607 −0.108841
\(885\) 19.9443 0.670419
\(886\) −28.0344 −0.941835
\(887\) 4.36068 0.146417 0.0732086 0.997317i \(-0.476676\pi\)
0.0732086 + 0.997317i \(0.476676\pi\)
\(888\) 16.9443 0.568613
\(889\) −11.4164 −0.382894
\(890\) 13.0902 0.438783
\(891\) 0 0
\(892\) 6.29180 0.210665
\(893\) −63.3050 −2.11842
\(894\) 63.3050 2.11723
\(895\) 6.38197 0.213326
\(896\) 1.00000 0.0334077
\(897\) −31.4164 −1.04896
\(898\) −24.4508 −0.815935
\(899\) 4.00000 0.133407
\(900\) 3.85410 0.128470
\(901\) 2.00000 0.0666297
\(902\) 0 0
\(903\) 4.85410 0.161534
\(904\) 9.32624 0.310186
\(905\) −3.52786 −0.117270
\(906\) 6.00000 0.199337
\(907\) 22.2705 0.739480 0.369740 0.929135i \(-0.379447\pi\)
0.369740 + 0.929135i \(0.379447\pi\)
\(908\) 24.2705 0.805445
\(909\) −34.4721 −1.14337
\(910\) 2.00000 0.0662994
\(911\) −51.5967 −1.70948 −0.854738 0.519059i \(-0.826282\pi\)
−0.854738 + 0.519059i \(0.826282\pi\)
\(912\) −17.9443 −0.594194
\(913\) 0 0
\(914\) −31.2705 −1.03434
\(915\) −9.23607 −0.305335
\(916\) 11.1246 0.367568
\(917\) −7.79837 −0.257525
\(918\) −3.61803 −0.119413
\(919\) 8.83282 0.291368 0.145684 0.989331i \(-0.453462\pi\)
0.145684 + 0.989331i \(0.453462\pi\)
\(920\) 6.00000 0.197814
\(921\) 72.1591 2.37772
\(922\) 6.94427 0.228697
\(923\) 19.4164 0.639099
\(924\) 0 0
\(925\) −6.47214 −0.212803
\(926\) 7.41641 0.243718
\(927\) −65.3050 −2.14490
\(928\) 3.23607 0.106229
\(929\) −3.25735 −0.106870 −0.0534352 0.998571i \(-0.517017\pi\)
−0.0534352 + 0.998571i \(0.517017\pi\)
\(930\) −3.23607 −0.106115
\(931\) −6.85410 −0.224634
\(932\) −10.9098 −0.357363
\(933\) −76.2492 −2.49629
\(934\) 17.8885 0.585331
\(935\) 0 0
\(936\) 7.70820 0.251951
\(937\) −33.3951 −1.09097 −0.545486 0.838120i \(-0.683655\pi\)
−0.545486 + 0.838120i \(0.683655\pi\)
\(938\) 6.09017 0.198851
\(939\) −2.23607 −0.0729713
\(940\) −9.23607 −0.301247
\(941\) −34.1803 −1.11425 −0.557124 0.830430i \(-0.688095\pi\)
−0.557124 + 0.830430i \(0.688095\pi\)
\(942\) −0.763932 −0.0248903
\(943\) 61.9574 2.01761
\(944\) −7.61803 −0.247946
\(945\) 2.23607 0.0727393
\(946\) 0 0
\(947\) −41.9098 −1.36189 −0.680943 0.732336i \(-0.738430\pi\)
−0.680943 + 0.732336i \(0.738430\pi\)
\(948\) 22.1803 0.720384
\(949\) 27.7082 0.899446
\(950\) 6.85410 0.222376
\(951\) −54.3607 −1.76277
\(952\) 1.61803 0.0524408
\(953\) −29.4508 −0.954007 −0.477003 0.878902i \(-0.658277\pi\)
−0.477003 + 0.878902i \(0.658277\pi\)
\(954\) −4.76393 −0.154238
\(955\) 1.52786 0.0494405
\(956\) 0 0
\(957\) 0 0
\(958\) −37.8885 −1.22412
\(959\) 1.09017 0.0352034
\(960\) −2.61803 −0.0844967
\(961\) −29.4721 −0.950714
\(962\) −12.9443 −0.417340
\(963\) −59.2837 −1.91039
\(964\) 6.27051 0.201960
\(965\) −2.00000 −0.0643823
\(966\) 15.7082 0.505403
\(967\) −27.0132 −0.868685 −0.434342 0.900748i \(-0.643019\pi\)
−0.434342 + 0.900748i \(0.643019\pi\)
\(968\) 0 0
\(969\) −29.0344 −0.932721
\(970\) −9.56231 −0.307027
\(971\) −26.8328 −0.861106 −0.430553 0.902565i \(-0.641681\pi\)
−0.430553 + 0.902565i \(0.641681\pi\)
\(972\) −21.6525 −0.694503
\(973\) 18.4721 0.592189
\(974\) 26.8328 0.859779
\(975\) −5.23607 −0.167688
\(976\) 3.52786 0.112924
\(977\) 43.3050 1.38545 0.692724 0.721203i \(-0.256410\pi\)
0.692724 + 0.721203i \(0.256410\pi\)
\(978\) −7.47214 −0.238933
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 1.81966 0.0580973
\(982\) −32.5066 −1.03733
\(983\) −14.6525 −0.467341 −0.233671 0.972316i \(-0.575074\pi\)
−0.233671 + 0.972316i \(0.575074\pi\)
\(984\) −27.0344 −0.861827
\(985\) 6.00000 0.191176
\(986\) 5.23607 0.166750
\(987\) −24.1803 −0.769669
\(988\) 13.7082 0.436116
\(989\) −11.1246 −0.353742
\(990\) 0 0
\(991\) 20.6525 0.656048 0.328024 0.944669i \(-0.393617\pi\)
0.328024 + 0.944669i \(0.393617\pi\)
\(992\) 1.23607 0.0392452
\(993\) 15.7984 0.501346
\(994\) −9.70820 −0.307926
\(995\) 19.4164 0.615542
\(996\) −27.0344 −0.856619
\(997\) 23.1246 0.732364 0.366182 0.930543i \(-0.380665\pi\)
0.366182 + 0.930543i \(0.380665\pi\)
\(998\) 15.2016 0.481199
\(999\) −14.4721 −0.457878
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.bt.1.2 2
11.3 even 5 770.2.n.b.141.1 yes 4
11.4 even 5 770.2.n.b.71.1 4
11.10 odd 2 8470.2.a.cf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.b.71.1 4 11.4 even 5
770.2.n.b.141.1 yes 4 11.3 even 5
8470.2.a.bt.1.2 2 1.1 even 1 trivial
8470.2.a.cf.1.2 2 11.10 odd 2