Properties

Label 8470.2.a.by.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 $1$

Related objects

Downloads

Learn more

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: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} -1.00000 q^{10} +2.23607 q^{12} -1.00000 q^{13} -1.00000 q^{14} -2.23607 q^{15} +1.00000 q^{16} -3.23607 q^{17} +2.00000 q^{18} -2.23607 q^{19} -1.00000 q^{20} -2.23607 q^{21} -8.23607 q^{23} +2.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} -2.23607 q^{27} -1.00000 q^{28} +4.47214 q^{29} -2.23607 q^{30} -0.763932 q^{31} +1.00000 q^{32} -3.23607 q^{34} +1.00000 q^{35} +2.00000 q^{36} -8.94427 q^{37} -2.23607 q^{38} -2.23607 q^{39} -1.00000 q^{40} +2.76393 q^{41} -2.23607 q^{42} +1.23607 q^{43} -2.00000 q^{45} -8.23607 q^{46} -5.70820 q^{47} +2.23607 q^{48} +1.00000 q^{49} +1.00000 q^{50} -7.23607 q^{51} -1.00000 q^{52} +7.70820 q^{53} -2.23607 q^{54} -1.00000 q^{56} -5.00000 q^{57} +4.47214 q^{58} -8.70820 q^{59} -2.23607 q^{60} +12.4721 q^{61} -0.763932 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -2.76393 q^{67} -3.23607 q^{68} -18.4164 q^{69} +1.00000 q^{70} -10.4721 q^{71} +2.00000 q^{72} +2.76393 q^{73} -8.94427 q^{74} +2.23607 q^{75} -2.23607 q^{76} -2.23607 q^{78} -5.76393 q^{79} -1.00000 q^{80} -11.0000 q^{81} +2.76393 q^{82} -4.23607 q^{83} -2.23607 q^{84} +3.23607 q^{85} +1.23607 q^{86} +10.0000 q^{87} -12.7639 q^{89} -2.00000 q^{90} +1.00000 q^{91} -8.23607 q^{92} -1.70820 q^{93} -5.70820 q^{94} +2.23607 q^{95} +2.23607 q^{96} +12.4721 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + 2q^{8} + 4q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + 2q^{8} + 4q^{9} - 2q^{10} - 2q^{13} - 2q^{14} + 2q^{16} - 2q^{17} + 4q^{18} - 2q^{20} - 12q^{23} + 2q^{25} - 2q^{26} - 2q^{28} - 6q^{31} + 2q^{32} - 2q^{34} + 2q^{35} + 4q^{36} - 2q^{40} + 10q^{41} - 2q^{43} - 4q^{45} - 12q^{46} + 2q^{47} + 2q^{49} + 2q^{50} - 10q^{51} - 2q^{52} + 2q^{53} - 2q^{56} - 10q^{57} - 4q^{59} + 16q^{61} - 6q^{62} - 4q^{63} + 2q^{64} + 2q^{65} - 10q^{67} - 2q^{68} - 10q^{69} + 2q^{70} - 12q^{71} + 4q^{72} + 10q^{73} - 16q^{79} - 2q^{80} - 22q^{81} + 10q^{82} - 4q^{83} + 2q^{85} - 2q^{86} + 20q^{87} - 30q^{89} - 4q^{90} + 2q^{91} - 12q^{92} + 10q^{93} + 2q^{94} + 16q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.23607 0.912871
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 2.00000 0.666667
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.23607 0.645497
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.23607 −0.577350
\(16\) 1.00000 0.250000
\(17\) −3.23607 −0.784862 −0.392431 0.919781i \(-0.628366\pi\)
−0.392431 + 0.919781i \(0.628366\pi\)
\(18\) 2.00000 0.471405
\(19\) −2.23607 −0.512989 −0.256495 0.966546i \(-0.582568\pi\)
−0.256495 + 0.966546i \(0.582568\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.23607 −0.487950
\(22\) 0 0
\(23\) −8.23607 −1.71734 −0.858669 0.512530i \(-0.828708\pi\)
−0.858669 + 0.512530i \(0.828708\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) −2.23607 −0.430331
\(28\) −1.00000 −0.188982
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) −2.23607 −0.408248
\(31\) −0.763932 −0.137206 −0.0686031 0.997644i \(-0.521854\pi\)
−0.0686031 + 0.997644i \(0.521854\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.23607 −0.554981
\(35\) 1.00000 0.169031
\(36\) 2.00000 0.333333
\(37\) −8.94427 −1.47043 −0.735215 0.677834i \(-0.762919\pi\)
−0.735215 + 0.677834i \(0.762919\pi\)
\(38\) −2.23607 −0.362738
\(39\) −2.23607 −0.358057
\(40\) −1.00000 −0.158114
\(41\) 2.76393 0.431654 0.215827 0.976432i \(-0.430755\pi\)
0.215827 + 0.976432i \(0.430755\pi\)
\(42\) −2.23607 −0.345033
\(43\) 1.23607 0.188499 0.0942493 0.995549i \(-0.469955\pi\)
0.0942493 + 0.995549i \(0.469955\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −8.23607 −1.21434
\(47\) −5.70820 −0.832627 −0.416314 0.909221i \(-0.636678\pi\)
−0.416314 + 0.909221i \(0.636678\pi\)
\(48\) 2.23607 0.322749
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −7.23607 −1.01325
\(52\) −1.00000 −0.138675
\(53\) 7.70820 1.05880 0.529402 0.848371i \(-0.322416\pi\)
0.529402 + 0.848371i \(0.322416\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −5.00000 −0.662266
\(58\) 4.47214 0.587220
\(59\) −8.70820 −1.13371 −0.566856 0.823817i \(-0.691840\pi\)
−0.566856 + 0.823817i \(0.691840\pi\)
\(60\) −2.23607 −0.288675
\(61\) 12.4721 1.59689 0.798447 0.602066i \(-0.205655\pi\)
0.798447 + 0.602066i \(0.205655\pi\)
\(62\) −0.763932 −0.0970195
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −2.76393 −0.337668 −0.168834 0.985644i \(-0.554000\pi\)
−0.168834 + 0.985644i \(0.554000\pi\)
\(68\) −3.23607 −0.392431
\(69\) −18.4164 −2.21707
\(70\) 1.00000 0.119523
\(71\) −10.4721 −1.24281 −0.621407 0.783488i \(-0.713439\pi\)
−0.621407 + 0.783488i \(0.713439\pi\)
\(72\) 2.00000 0.235702
\(73\) 2.76393 0.323494 0.161747 0.986832i \(-0.448287\pi\)
0.161747 + 0.986832i \(0.448287\pi\)
\(74\) −8.94427 −1.03975
\(75\) 2.23607 0.258199
\(76\) −2.23607 −0.256495
\(77\) 0 0
\(78\) −2.23607 −0.253185
\(79\) −5.76393 −0.648493 −0.324247 0.945973i \(-0.605111\pi\)
−0.324247 + 0.945973i \(0.605111\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 2.76393 0.305225
\(83\) −4.23607 −0.464969 −0.232484 0.972600i \(-0.574685\pi\)
−0.232484 + 0.972600i \(0.574685\pi\)
\(84\) −2.23607 −0.243975
\(85\) 3.23607 0.351001
\(86\) 1.23607 0.133289
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −12.7639 −1.35297 −0.676487 0.736455i \(-0.736499\pi\)
−0.676487 + 0.736455i \(0.736499\pi\)
\(90\) −2.00000 −0.210819
\(91\) 1.00000 0.104828
\(92\) −8.23607 −0.858669
\(93\) −1.70820 −0.177132
\(94\) −5.70820 −0.588756
\(95\) 2.23607 0.229416
\(96\) 2.23607 0.228218
\(97\) 12.4721 1.26635 0.633177 0.774007i \(-0.281751\pi\)
0.633177 + 0.774007i \(0.281751\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.00000 0.0995037 0.0497519 0.998762i \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) −7.23607 −0.716477
\(103\) −8.18034 −0.806033 −0.403016 0.915193i \(-0.632038\pi\)
−0.403016 + 0.915193i \(0.632038\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.23607 0.218218
\(106\) 7.70820 0.748687
\(107\) −6.47214 −0.625685 −0.312842 0.949805i \(-0.601281\pi\)
−0.312842 + 0.949805i \(0.601281\pi\)
\(108\) −2.23607 −0.215166
\(109\) −9.70820 −0.929877 −0.464939 0.885343i \(-0.653924\pi\)
−0.464939 + 0.885343i \(0.653924\pi\)
\(110\) 0 0
\(111\) −20.0000 −1.89832
\(112\) −1.00000 −0.0944911
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −5.00000 −0.468293
\(115\) 8.23607 0.768017
\(116\) 4.47214 0.415227
\(117\) −2.00000 −0.184900
\(118\) −8.70820 −0.801655
\(119\) 3.23607 0.296650
\(120\) −2.23607 −0.204124
\(121\) 0 0
\(122\) 12.4721 1.12917
\(123\) 6.18034 0.557262
\(124\) −0.763932 −0.0686031
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −1.76393 −0.156524 −0.0782618 0.996933i \(-0.524937\pi\)
−0.0782618 + 0.996933i \(0.524937\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.76393 0.243351
\(130\) 1.00000 0.0877058
\(131\) −19.1803 −1.67579 −0.837897 0.545828i \(-0.816215\pi\)
−0.837897 + 0.545828i \(0.816215\pi\)
\(132\) 0 0
\(133\) 2.23607 0.193892
\(134\) −2.76393 −0.238767
\(135\) 2.23607 0.192450
\(136\) −3.23607 −0.277491
\(137\) 4.52786 0.386842 0.193421 0.981116i \(-0.438042\pi\)
0.193421 + 0.981116i \(0.438042\pi\)
\(138\) −18.4164 −1.56771
\(139\) 0.236068 0.0200230 0.0100115 0.999950i \(-0.496813\pi\)
0.0100115 + 0.999950i \(0.496813\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.7639 −1.07492
\(142\) −10.4721 −0.878802
\(143\) 0 0
\(144\) 2.00000 0.166667
\(145\) −4.47214 −0.371391
\(146\) 2.76393 0.228745
\(147\) 2.23607 0.184428
\(148\) −8.94427 −0.735215
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 2.23607 0.182574
\(151\) −7.76393 −0.631820 −0.315910 0.948789i \(-0.602310\pi\)
−0.315910 + 0.948789i \(0.602310\pi\)
\(152\) −2.23607 −0.181369
\(153\) −6.47214 −0.523241
\(154\) 0 0
\(155\) 0.763932 0.0613605
\(156\) −2.23607 −0.179029
\(157\) 17.4721 1.39443 0.697214 0.716863i \(-0.254423\pi\)
0.697214 + 0.716863i \(0.254423\pi\)
\(158\) −5.76393 −0.458554
\(159\) 17.2361 1.36691
\(160\) −1.00000 −0.0790569
\(161\) 8.23607 0.649093
\(162\) −11.0000 −0.864242
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 2.76393 0.215827
\(165\) 0 0
\(166\) −4.23607 −0.328783
\(167\) 11.7082 0.906008 0.453004 0.891508i \(-0.350352\pi\)
0.453004 + 0.891508i \(0.350352\pi\)
\(168\) −2.23607 −0.172516
\(169\) −12.0000 −0.923077
\(170\) 3.23607 0.248195
\(171\) −4.47214 −0.341993
\(172\) 1.23607 0.0942493
\(173\) 15.8885 1.20798 0.603992 0.796991i \(-0.293576\pi\)
0.603992 + 0.796991i \(0.293576\pi\)
\(174\) 10.0000 0.758098
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −19.4721 −1.46362
\(178\) −12.7639 −0.956697
\(179\) −6.76393 −0.505560 −0.252780 0.967524i \(-0.581345\pi\)
−0.252780 + 0.967524i \(0.581345\pi\)
\(180\) −2.00000 −0.149071
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 1.00000 0.0741249
\(183\) 27.8885 2.06158
\(184\) −8.23607 −0.607171
\(185\) 8.94427 0.657596
\(186\) −1.70820 −0.125252
\(187\) 0 0
\(188\) −5.70820 −0.416314
\(189\) 2.23607 0.162650
\(190\) 2.23607 0.162221
\(191\) 2.81966 0.204023 0.102012 0.994783i \(-0.467472\pi\)
0.102012 + 0.994783i \(0.467472\pi\)
\(192\) 2.23607 0.161374
\(193\) 9.94427 0.715804 0.357902 0.933759i \(-0.383492\pi\)
0.357902 + 0.933759i \(0.383492\pi\)
\(194\) 12.4721 0.895447
\(195\) 2.23607 0.160128
\(196\) 1.00000 0.0714286
\(197\) 16.4721 1.17359 0.586796 0.809735i \(-0.300389\pi\)
0.586796 + 0.809735i \(0.300389\pi\)
\(198\) 0 0
\(199\) 4.47214 0.317021 0.158511 0.987357i \(-0.449331\pi\)
0.158511 + 0.987357i \(0.449331\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.18034 −0.435928
\(202\) 1.00000 0.0703598
\(203\) −4.47214 −0.313882
\(204\) −7.23607 −0.506626
\(205\) −2.76393 −0.193041
\(206\) −8.18034 −0.569951
\(207\) −16.4721 −1.14489
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 2.23607 0.154303
\(211\) −12.4721 −0.858617 −0.429309 0.903158i \(-0.641243\pi\)
−0.429309 + 0.903158i \(0.641243\pi\)
\(212\) 7.70820 0.529402
\(213\) −23.4164 −1.60447
\(214\) −6.47214 −0.442426
\(215\) −1.23607 −0.0842991
\(216\) −2.23607 −0.152145
\(217\) 0.763932 0.0518591
\(218\) −9.70820 −0.657523
\(219\) 6.18034 0.417629
\(220\) 0 0
\(221\) 3.23607 0.217681
\(222\) −20.0000 −1.34231
\(223\) 14.1803 0.949586 0.474793 0.880098i \(-0.342523\pi\)
0.474793 + 0.880098i \(0.342523\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.00000 0.133333
\(226\) −1.00000 −0.0665190
\(227\) −19.4164 −1.28871 −0.644356 0.764726i \(-0.722875\pi\)
−0.644356 + 0.764726i \(0.722875\pi\)
\(228\) −5.00000 −0.331133
\(229\) −10.9443 −0.723218 −0.361609 0.932330i \(-0.617772\pi\)
−0.361609 + 0.932330i \(0.617772\pi\)
\(230\) 8.23607 0.543070
\(231\) 0 0
\(232\) 4.47214 0.293610
\(233\) 16.4164 1.07547 0.537737 0.843112i \(-0.319279\pi\)
0.537737 + 0.843112i \(0.319279\pi\)
\(234\) −2.00000 −0.130744
\(235\) 5.70820 0.372362
\(236\) −8.70820 −0.566856
\(237\) −12.8885 −0.837201
\(238\) 3.23607 0.209763
\(239\) 2.70820 0.175179 0.0875896 0.996157i \(-0.472084\pi\)
0.0875896 + 0.996157i \(0.472084\pi\)
\(240\) −2.23607 −0.144338
\(241\) −22.3607 −1.44038 −0.720189 0.693778i \(-0.755945\pi\)
−0.720189 + 0.693778i \(0.755945\pi\)
\(242\) 0 0
\(243\) −17.8885 −1.14755
\(244\) 12.4721 0.798447
\(245\) −1.00000 −0.0638877
\(246\) 6.18034 0.394044
\(247\) 2.23607 0.142278
\(248\) −0.763932 −0.0485097
\(249\) −9.47214 −0.600272
\(250\) −1.00000 −0.0632456
\(251\) 5.88854 0.371682 0.185841 0.982580i \(-0.440499\pi\)
0.185841 + 0.982580i \(0.440499\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) −1.76393 −0.110679
\(255\) 7.23607 0.453140
\(256\) 1.00000 0.0625000
\(257\) −9.23607 −0.576130 −0.288065 0.957611i \(-0.593012\pi\)
−0.288065 + 0.957611i \(0.593012\pi\)
\(258\) 2.76393 0.172075
\(259\) 8.94427 0.555770
\(260\) 1.00000 0.0620174
\(261\) 8.94427 0.553637
\(262\) −19.1803 −1.18497
\(263\) 23.6525 1.45847 0.729237 0.684261i \(-0.239875\pi\)
0.729237 + 0.684261i \(0.239875\pi\)
\(264\) 0 0
\(265\) −7.70820 −0.473511
\(266\) 2.23607 0.137102
\(267\) −28.5410 −1.74668
\(268\) −2.76393 −0.168834
\(269\) 25.8328 1.57505 0.787527 0.616280i \(-0.211361\pi\)
0.787527 + 0.616280i \(0.211361\pi\)
\(270\) 2.23607 0.136083
\(271\) 19.4164 1.17946 0.589731 0.807599i \(-0.299234\pi\)
0.589731 + 0.807599i \(0.299234\pi\)
\(272\) −3.23607 −0.196215
\(273\) 2.23607 0.135333
\(274\) 4.52786 0.273538
\(275\) 0 0
\(276\) −18.4164 −1.10854
\(277\) 25.5967 1.53796 0.768980 0.639273i \(-0.220765\pi\)
0.768980 + 0.639273i \(0.220765\pi\)
\(278\) 0.236068 0.0141584
\(279\) −1.52786 −0.0914708
\(280\) 1.00000 0.0597614
\(281\) 27.8328 1.66037 0.830183 0.557491i \(-0.188236\pi\)
0.830183 + 0.557491i \(0.188236\pi\)
\(282\) −12.7639 −0.760081
\(283\) −21.7639 −1.29373 −0.646866 0.762604i \(-0.723921\pi\)
−0.646866 + 0.762604i \(0.723921\pi\)
\(284\) −10.4721 −0.621407
\(285\) 5.00000 0.296174
\(286\) 0 0
\(287\) −2.76393 −0.163150
\(288\) 2.00000 0.117851
\(289\) −6.52786 −0.383992
\(290\) −4.47214 −0.262613
\(291\) 27.8885 1.63486
\(292\) 2.76393 0.161747
\(293\) 5.00000 0.292103 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(294\) 2.23607 0.130410
\(295\) 8.70820 0.507011
\(296\) −8.94427 −0.519875
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 8.23607 0.476304
\(300\) 2.23607 0.129099
\(301\) −1.23607 −0.0712458
\(302\) −7.76393 −0.446764
\(303\) 2.23607 0.128459
\(304\) −2.23607 −0.128247
\(305\) −12.4721 −0.714152
\(306\) −6.47214 −0.369987
\(307\) 15.4164 0.879861 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(308\) 0 0
\(309\) −18.2918 −1.04058
\(310\) 0.763932 0.0433884
\(311\) 13.1246 0.744228 0.372114 0.928187i \(-0.378633\pi\)
0.372114 + 0.928187i \(0.378633\pi\)
\(312\) −2.23607 −0.126592
\(313\) 28.4721 1.60934 0.804670 0.593722i \(-0.202342\pi\)
0.804670 + 0.593722i \(0.202342\pi\)
\(314\) 17.4721 0.986010
\(315\) 2.00000 0.112687
\(316\) −5.76393 −0.324247
\(317\) −30.6525 −1.72161 −0.860807 0.508931i \(-0.830041\pi\)
−0.860807 + 0.508931i \(0.830041\pi\)
\(318\) 17.2361 0.966551
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −14.4721 −0.807756
\(322\) 8.23607 0.458978
\(323\) 7.23607 0.402626
\(324\) −11.0000 −0.611111
\(325\) −1.00000 −0.0554700
\(326\) −6.00000 −0.332309
\(327\) −21.7082 −1.20047
\(328\) 2.76393 0.152613
\(329\) 5.70820 0.314703
\(330\) 0 0
\(331\) −8.29180 −0.455758 −0.227879 0.973689i \(-0.573179\pi\)
−0.227879 + 0.973689i \(0.573179\pi\)
\(332\) −4.23607 −0.232484
\(333\) −17.8885 −0.980286
\(334\) 11.7082 0.640644
\(335\) 2.76393 0.151010
\(336\) −2.23607 −0.121988
\(337\) −32.8885 −1.79155 −0.895777 0.444505i \(-0.853380\pi\)
−0.895777 + 0.444505i \(0.853380\pi\)
\(338\) −12.0000 −0.652714
\(339\) −2.23607 −0.121447
\(340\) 3.23607 0.175500
\(341\) 0 0
\(342\) −4.47214 −0.241825
\(343\) −1.00000 −0.0539949
\(344\) 1.23607 0.0666443
\(345\) 18.4164 0.991506
\(346\) 15.8885 0.854173
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 10.0000 0.536056
\(349\) 26.4164 1.41404 0.707019 0.707195i \(-0.250040\pi\)
0.707019 + 0.707195i \(0.250040\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.23607 0.119352
\(352\) 0 0
\(353\) −9.23607 −0.491586 −0.245793 0.969322i \(-0.579048\pi\)
−0.245793 + 0.969322i \(0.579048\pi\)
\(354\) −19.4721 −1.03493
\(355\) 10.4721 0.555803
\(356\) −12.7639 −0.676487
\(357\) 7.23607 0.382973
\(358\) −6.76393 −0.357485
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −2.00000 −0.105409
\(361\) −14.0000 −0.736842
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) −2.76393 −0.144671
\(366\) 27.8885 1.45776
\(367\) −14.1803 −0.740208 −0.370104 0.928990i \(-0.620678\pi\)
−0.370104 + 0.928990i \(0.620678\pi\)
\(368\) −8.23607 −0.429335
\(369\) 5.52786 0.287769
\(370\) 8.94427 0.464991
\(371\) −7.70820 −0.400190
\(372\) −1.70820 −0.0885662
\(373\) 14.4721 0.749339 0.374669 0.927158i \(-0.377756\pi\)
0.374669 + 0.927158i \(0.377756\pi\)
\(374\) 0 0
\(375\) −2.23607 −0.115470
\(376\) −5.70820 −0.294378
\(377\) −4.47214 −0.230327
\(378\) 2.23607 0.115011
\(379\) −15.4164 −0.791888 −0.395944 0.918275i \(-0.629583\pi\)
−0.395944 + 0.918275i \(0.629583\pi\)
\(380\) 2.23607 0.114708
\(381\) −3.94427 −0.202071
\(382\) 2.81966 0.144266
\(383\) −12.7639 −0.652206 −0.326103 0.945334i \(-0.605736\pi\)
−0.326103 + 0.945334i \(0.605736\pi\)
\(384\) 2.23607 0.114109
\(385\) 0 0
\(386\) 9.94427 0.506150
\(387\) 2.47214 0.125666
\(388\) 12.4721 0.633177
\(389\) −31.7082 −1.60767 −0.803835 0.594852i \(-0.797210\pi\)
−0.803835 + 0.594852i \(0.797210\pi\)
\(390\) 2.23607 0.113228
\(391\) 26.6525 1.34787
\(392\) 1.00000 0.0505076
\(393\) −42.8885 −2.16344
\(394\) 16.4721 0.829854
\(395\) 5.76393 0.290015
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 4.47214 0.224168
\(399\) 5.00000 0.250313
\(400\) 1.00000 0.0500000
\(401\) 24.8328 1.24009 0.620046 0.784566i \(-0.287114\pi\)
0.620046 + 0.784566i \(0.287114\pi\)
\(402\) −6.18034 −0.308247
\(403\) 0.763932 0.0380542
\(404\) 1.00000 0.0497519
\(405\) 11.0000 0.546594
\(406\) −4.47214 −0.221948
\(407\) 0 0
\(408\) −7.23607 −0.358239
\(409\) −20.4721 −1.01228 −0.506141 0.862451i \(-0.668928\pi\)
−0.506141 + 0.862451i \(0.668928\pi\)
\(410\) −2.76393 −0.136501
\(411\) 10.1246 0.499410
\(412\) −8.18034 −0.403016
\(413\) 8.70820 0.428503
\(414\) −16.4721 −0.809561
\(415\) 4.23607 0.207940
\(416\) −1.00000 −0.0490290
\(417\) 0.527864 0.0258496
\(418\) 0 0
\(419\) −23.7639 −1.16094 −0.580472 0.814280i \(-0.697132\pi\)
−0.580472 + 0.814280i \(0.697132\pi\)
\(420\) 2.23607 0.109109
\(421\) −20.9443 −1.02076 −0.510381 0.859949i \(-0.670495\pi\)
−0.510381 + 0.859949i \(0.670495\pi\)
\(422\) −12.4721 −0.607134
\(423\) −11.4164 −0.555085
\(424\) 7.70820 0.374343
\(425\) −3.23607 −0.156972
\(426\) −23.4164 −1.13453
\(427\) −12.4721 −0.603569
\(428\) −6.47214 −0.312842
\(429\) 0 0
\(430\) −1.23607 −0.0596085
\(431\) 2.34752 0.113076 0.0565381 0.998400i \(-0.481994\pi\)
0.0565381 + 0.998400i \(0.481994\pi\)
\(432\) −2.23607 −0.107583
\(433\) 18.6525 0.896381 0.448190 0.893938i \(-0.352069\pi\)
0.448190 + 0.893938i \(0.352069\pi\)
\(434\) 0.763932 0.0366699
\(435\) −10.0000 −0.479463
\(436\) −9.70820 −0.464939
\(437\) 18.4164 0.880976
\(438\) 6.18034 0.295308
\(439\) 25.1246 1.19913 0.599566 0.800325i \(-0.295340\pi\)
0.599566 + 0.800325i \(0.295340\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 3.23607 0.153924
\(443\) 20.3607 0.967365 0.483683 0.875244i \(-0.339299\pi\)
0.483683 + 0.875244i \(0.339299\pi\)
\(444\) −20.0000 −0.949158
\(445\) 12.7639 0.605068
\(446\) 14.1803 0.671459
\(447\) −8.94427 −0.423050
\(448\) −1.00000 −0.0472456
\(449\) −29.8328 −1.40790 −0.703949 0.710251i \(-0.748582\pi\)
−0.703949 + 0.710251i \(0.748582\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) −1.00000 −0.0470360
\(453\) −17.3607 −0.815676
\(454\) −19.4164 −0.911257
\(455\) −1.00000 −0.0468807
\(456\) −5.00000 −0.234146
\(457\) 9.58359 0.448302 0.224151 0.974554i \(-0.428039\pi\)
0.224151 + 0.974554i \(0.428039\pi\)
\(458\) −10.9443 −0.511392
\(459\) 7.23607 0.337751
\(460\) 8.23607 0.384009
\(461\) −11.8885 −0.553705 −0.276852 0.960912i \(-0.589291\pi\)
−0.276852 + 0.960912i \(0.589291\pi\)
\(462\) 0 0
\(463\) −2.70820 −0.125861 −0.0629305 0.998018i \(-0.520045\pi\)
−0.0629305 + 0.998018i \(0.520045\pi\)
\(464\) 4.47214 0.207614
\(465\) 1.70820 0.0792161
\(466\) 16.4164 0.760475
\(467\) −25.1803 −1.16521 −0.582604 0.812756i \(-0.697966\pi\)
−0.582604 + 0.812756i \(0.697966\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 2.76393 0.127627
\(470\) 5.70820 0.263300
\(471\) 39.0689 1.80020
\(472\) −8.70820 −0.400828
\(473\) 0 0
\(474\) −12.8885 −0.591990
\(475\) −2.23607 −0.102598
\(476\) 3.23607 0.148325
\(477\) 15.4164 0.705869
\(478\) 2.70820 0.123870
\(479\) −21.5967 −0.986781 −0.493390 0.869808i \(-0.664243\pi\)
−0.493390 + 0.869808i \(0.664243\pi\)
\(480\) −2.23607 −0.102062
\(481\) 8.94427 0.407824
\(482\) −22.3607 −1.01850
\(483\) 18.4164 0.837976
\(484\) 0 0
\(485\) −12.4721 −0.566331
\(486\) −17.8885 −0.811441
\(487\) −2.23607 −0.101326 −0.0506630 0.998716i \(-0.516133\pi\)
−0.0506630 + 0.998716i \(0.516133\pi\)
\(488\) 12.4721 0.564587
\(489\) −13.4164 −0.606711
\(490\) −1.00000 −0.0451754
\(491\) −26.7639 −1.20784 −0.603920 0.797045i \(-0.706395\pi\)
−0.603920 + 0.797045i \(0.706395\pi\)
\(492\) 6.18034 0.278631
\(493\) −14.4721 −0.651792
\(494\) 2.23607 0.100605
\(495\) 0 0
\(496\) −0.763932 −0.0343016
\(497\) 10.4721 0.469739
\(498\) −9.47214 −0.424457
\(499\) −26.8328 −1.20120 −0.600601 0.799549i \(-0.705072\pi\)
−0.600601 + 0.799549i \(0.705072\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 26.1803 1.16965
\(502\) 5.88854 0.262819
\(503\) 23.5967 1.05213 0.526063 0.850445i \(-0.323667\pi\)
0.526063 + 0.850445i \(0.323667\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −1.00000 −0.0444994
\(506\) 0 0
\(507\) −26.8328 −1.19169
\(508\) −1.76393 −0.0782618
\(509\) −6.88854 −0.305329 −0.152665 0.988278i \(-0.548785\pi\)
−0.152665 + 0.988278i \(0.548785\pi\)
\(510\) 7.23607 0.320418
\(511\) −2.76393 −0.122269
\(512\) 1.00000 0.0441942
\(513\) 5.00000 0.220755
\(514\) −9.23607 −0.407385
\(515\) 8.18034 0.360469
\(516\) 2.76393 0.121675
\(517\) 0 0
\(518\) 8.94427 0.392989
\(519\) 35.5279 1.55950
\(520\) 1.00000 0.0438529
\(521\) −18.9443 −0.829964 −0.414982 0.909830i \(-0.636212\pi\)
−0.414982 + 0.909830i \(0.636212\pi\)
\(522\) 8.94427 0.391480
\(523\) 4.34752 0.190104 0.0950520 0.995472i \(-0.469698\pi\)
0.0950520 + 0.995472i \(0.469698\pi\)
\(524\) −19.1803 −0.837897
\(525\) −2.23607 −0.0975900
\(526\) 23.6525 1.03130
\(527\) 2.47214 0.107688
\(528\) 0 0
\(529\) 44.8328 1.94925
\(530\) −7.70820 −0.334823
\(531\) −17.4164 −0.755808
\(532\) 2.23607 0.0969458
\(533\) −2.76393 −0.119719
\(534\) −28.5410 −1.23509
\(535\) 6.47214 0.279815
\(536\) −2.76393 −0.119384
\(537\) −15.1246 −0.652675
\(538\) 25.8328 1.11373
\(539\) 0 0
\(540\) 2.23607 0.0962250
\(541\) −27.3050 −1.17393 −0.586966 0.809612i \(-0.699678\pi\)
−0.586966 + 0.809612i \(0.699678\pi\)
\(542\) 19.4164 0.834006
\(543\) 15.6525 0.671712
\(544\) −3.23607 −0.138745
\(545\) 9.70820 0.415854
\(546\) 2.23607 0.0956949
\(547\) 24.6525 1.05406 0.527032 0.849846i \(-0.323305\pi\)
0.527032 + 0.849846i \(0.323305\pi\)
\(548\) 4.52786 0.193421
\(549\) 24.9443 1.06460
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) −18.4164 −0.783854
\(553\) 5.76393 0.245107
\(554\) 25.5967 1.08750
\(555\) 20.0000 0.848953
\(556\) 0.236068 0.0100115
\(557\) −9.12461 −0.386622 −0.193311 0.981138i \(-0.561923\pi\)
−0.193311 + 0.981138i \(0.561923\pi\)
\(558\) −1.52786 −0.0646796
\(559\) −1.23607 −0.0522801
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 27.8328 1.17406
\(563\) −12.5967 −0.530890 −0.265445 0.964126i \(-0.585519\pi\)
−0.265445 + 0.964126i \(0.585519\pi\)
\(564\) −12.7639 −0.537458
\(565\) 1.00000 0.0420703
\(566\) −21.7639 −0.914806
\(567\) 11.0000 0.461957
\(568\) −10.4721 −0.439401
\(569\) 1.47214 0.0617151 0.0308576 0.999524i \(-0.490176\pi\)
0.0308576 + 0.999524i \(0.490176\pi\)
\(570\) 5.00000 0.209427
\(571\) 8.36068 0.349884 0.174942 0.984579i \(-0.444026\pi\)
0.174942 + 0.984579i \(0.444026\pi\)
\(572\) 0 0
\(573\) 6.30495 0.263393
\(574\) −2.76393 −0.115364
\(575\) −8.23607 −0.343468
\(576\) 2.00000 0.0833333
\(577\) 41.0132 1.70740 0.853700 0.520765i \(-0.174353\pi\)
0.853700 + 0.520765i \(0.174353\pi\)
\(578\) −6.52786 −0.271523
\(579\) 22.2361 0.924099
\(580\) −4.47214 −0.185695
\(581\) 4.23607 0.175742
\(582\) 27.8885 1.15602
\(583\) 0 0
\(584\) 2.76393 0.114372
\(585\) 2.00000 0.0826898
\(586\) 5.00000 0.206548
\(587\) 35.0689 1.44745 0.723724 0.690090i \(-0.242429\pi\)
0.723724 + 0.690090i \(0.242429\pi\)
\(588\) 2.23607 0.0922139
\(589\) 1.70820 0.0703853
\(590\) 8.70820 0.358511
\(591\) 36.8328 1.51510
\(592\) −8.94427 −0.367607
\(593\) −19.0557 −0.782525 −0.391262 0.920279i \(-0.627962\pi\)
−0.391262 + 0.920279i \(0.627962\pi\)
\(594\) 0 0
\(595\) −3.23607 −0.132666
\(596\) −4.00000 −0.163846
\(597\) 10.0000 0.409273
\(598\) 8.23607 0.336798
\(599\) −31.0689 −1.26944 −0.634720 0.772742i \(-0.718885\pi\)
−0.634720 + 0.772742i \(0.718885\pi\)
\(600\) 2.23607 0.0912871
\(601\) 4.65248 0.189778 0.0948892 0.995488i \(-0.469750\pi\)
0.0948892 + 0.995488i \(0.469750\pi\)
\(602\) −1.23607 −0.0503784
\(603\) −5.52786 −0.225112
\(604\) −7.76393 −0.315910
\(605\) 0 0
\(606\) 2.23607 0.0908341
\(607\) −38.9443 −1.58070 −0.790350 0.612656i \(-0.790101\pi\)
−0.790350 + 0.612656i \(0.790101\pi\)
\(608\) −2.23607 −0.0906845
\(609\) −10.0000 −0.405220
\(610\) −12.4721 −0.504982
\(611\) 5.70820 0.230929
\(612\) −6.47214 −0.261621
\(613\) −41.1246 −1.66101 −0.830504 0.557013i \(-0.811947\pi\)
−0.830504 + 0.557013i \(0.811947\pi\)
\(614\) 15.4164 0.622156
\(615\) −6.18034 −0.249215
\(616\) 0 0
\(617\) −8.11146 −0.326555 −0.163277 0.986580i \(-0.552207\pi\)
−0.163277 + 0.986580i \(0.552207\pi\)
\(618\) −18.2918 −0.735804
\(619\) 19.5410 0.785420 0.392710 0.919662i \(-0.371538\pi\)
0.392710 + 0.919662i \(0.371538\pi\)
\(620\) 0.763932 0.0306802
\(621\) 18.4164 0.739025
\(622\) 13.1246 0.526249
\(623\) 12.7639 0.511376
\(624\) −2.23607 −0.0895144
\(625\) 1.00000 0.0400000
\(626\) 28.4721 1.13798
\(627\) 0 0
\(628\) 17.4721 0.697214
\(629\) 28.9443 1.15408
\(630\) 2.00000 0.0796819
\(631\) 28.3607 1.12902 0.564510 0.825426i \(-0.309065\pi\)
0.564510 + 0.825426i \(0.309065\pi\)
\(632\) −5.76393 −0.229277
\(633\) −27.8885 −1.10847
\(634\) −30.6525 −1.21737
\(635\) 1.76393 0.0699995
\(636\) 17.2361 0.683455
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −20.9443 −0.828543
\(640\) −1.00000 −0.0395285
\(641\) 29.8328 1.17833 0.589163 0.808014i \(-0.299458\pi\)
0.589163 + 0.808014i \(0.299458\pi\)
\(642\) −14.4721 −0.571170
\(643\) 41.8885 1.65192 0.825961 0.563727i \(-0.190633\pi\)
0.825961 + 0.563727i \(0.190633\pi\)
\(644\) 8.23607 0.324547
\(645\) −2.76393 −0.108830
\(646\) 7.23607 0.284699
\(647\) −4.76393 −0.187289 −0.0936447 0.995606i \(-0.529852\pi\)
−0.0936447 + 0.995606i \(0.529852\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 1.70820 0.0669498
\(652\) −6.00000 −0.234978
\(653\) 39.7082 1.55390 0.776951 0.629561i \(-0.216765\pi\)
0.776951 + 0.629561i \(0.216765\pi\)
\(654\) −21.7082 −0.848858
\(655\) 19.1803 0.749438
\(656\) 2.76393 0.107913
\(657\) 5.52786 0.215663
\(658\) 5.70820 0.222529
\(659\) 25.2361 0.983058 0.491529 0.870861i \(-0.336438\pi\)
0.491529 + 0.870861i \(0.336438\pi\)
\(660\) 0 0
\(661\) −22.4164 −0.871897 −0.435949 0.899972i \(-0.643587\pi\)
−0.435949 + 0.899972i \(0.643587\pi\)
\(662\) −8.29180 −0.322270
\(663\) 7.23607 0.281026
\(664\) −4.23607 −0.164391
\(665\) −2.23607 −0.0867110
\(666\) −17.8885 −0.693167
\(667\) −36.8328 −1.42617
\(668\) 11.7082 0.453004
\(669\) 31.7082 1.22591
\(670\) 2.76393 0.106780
\(671\) 0 0
\(672\) −2.23607 −0.0862582
\(673\) 27.9443 1.07717 0.538586 0.842570i \(-0.318959\pi\)
0.538586 + 0.842570i \(0.318959\pi\)
\(674\) −32.8885 −1.26682
\(675\) −2.23607 −0.0860663
\(676\) −12.0000 −0.461538
\(677\) −7.83282 −0.301040 −0.150520 0.988607i \(-0.548095\pi\)
−0.150520 + 0.988607i \(0.548095\pi\)
\(678\) −2.23607 −0.0858757
\(679\) −12.4721 −0.478637
\(680\) 3.23607 0.124098
\(681\) −43.4164 −1.66372
\(682\) 0 0
\(683\) 38.3607 1.46783 0.733915 0.679241i \(-0.237691\pi\)
0.733915 + 0.679241i \(0.237691\pi\)
\(684\) −4.47214 −0.170996
\(685\) −4.52786 −0.173001
\(686\) −1.00000 −0.0381802
\(687\) −24.4721 −0.933670
\(688\) 1.23607 0.0471246
\(689\) −7.70820 −0.293659
\(690\) 18.4164 0.701101
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 15.8885 0.603992
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) −0.236068 −0.00895457
\(696\) 10.0000 0.379049
\(697\) −8.94427 −0.338788
\(698\) 26.4164 0.999876
\(699\) 36.7082 1.38843
\(700\) −1.00000 −0.0377964
\(701\) −31.5967 −1.19339 −0.596696 0.802467i \(-0.703520\pi\)
−0.596696 + 0.802467i \(0.703520\pi\)
\(702\) 2.23607 0.0843949
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 12.7639 0.480717
\(706\) −9.23607 −0.347604
\(707\) −1.00000 −0.0376089
\(708\) −19.4721 −0.731808
\(709\) 36.9443 1.38747 0.693736 0.720230i \(-0.255964\pi\)
0.693736 + 0.720230i \(0.255964\pi\)
\(710\) 10.4721 0.393012
\(711\) −11.5279 −0.432329
\(712\) −12.7639 −0.478349
\(713\) 6.29180 0.235630
\(714\) 7.23607 0.270803
\(715\) 0 0
\(716\) −6.76393 −0.252780
\(717\) 6.05573 0.226155
\(718\) −16.0000 −0.597115
\(719\) −46.8328 −1.74657 −0.873285 0.487210i \(-0.838015\pi\)
−0.873285 + 0.487210i \(0.838015\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 8.18034 0.304652
\(722\) −14.0000 −0.521026
\(723\) −50.0000 −1.85952
\(724\) 7.00000 0.260153
\(725\) 4.47214 0.166091
\(726\) 0 0
\(727\) 23.3050 0.864333 0.432166 0.901794i \(-0.357749\pi\)
0.432166 + 0.901794i \(0.357749\pi\)
\(728\) 1.00000 0.0370625
\(729\) −7.00000 −0.259259
\(730\) −2.76393 −0.102298
\(731\) −4.00000 −0.147945
\(732\) 27.8885 1.03079
\(733\) 21.5836 0.797208 0.398604 0.917123i \(-0.369495\pi\)
0.398604 + 0.917123i \(0.369495\pi\)
\(734\) −14.1803 −0.523406
\(735\) −2.23607 −0.0824786
\(736\) −8.23607 −0.303585
\(737\) 0 0
\(738\) 5.52786 0.203483
\(739\) 40.0689 1.47396 0.736979 0.675916i \(-0.236252\pi\)
0.736979 + 0.675916i \(0.236252\pi\)
\(740\) 8.94427 0.328798
\(741\) 5.00000 0.183680
\(742\) −7.70820 −0.282977
\(743\) −28.0000 −1.02722 −0.513610 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) −1.70820 −0.0626258
\(745\) 4.00000 0.146549
\(746\) 14.4721 0.529863
\(747\) −8.47214 −0.309979
\(748\) 0 0
\(749\) 6.47214 0.236487
\(750\) −2.23607 −0.0816497
\(751\) 11.1803 0.407976 0.203988 0.978973i \(-0.434610\pi\)
0.203988 + 0.978973i \(0.434610\pi\)
\(752\) −5.70820 −0.208157
\(753\) 13.1672 0.479839
\(754\) −4.47214 −0.162866
\(755\) 7.76393 0.282558
\(756\) 2.23607 0.0813250
\(757\) −11.3475 −0.412433 −0.206216 0.978506i \(-0.566115\pi\)
−0.206216 + 0.978506i \(0.566115\pi\)
\(758\) −15.4164 −0.559949
\(759\) 0 0
\(760\) 2.23607 0.0811107
\(761\) −36.5410 −1.32461 −0.662305 0.749234i \(-0.730422\pi\)
−0.662305 + 0.749234i \(0.730422\pi\)
\(762\) −3.94427 −0.142886
\(763\) 9.70820 0.351461
\(764\) 2.81966 0.102012
\(765\) 6.47214 0.234001
\(766\) −12.7639 −0.461180
\(767\) 8.70820 0.314435
\(768\) 2.23607 0.0806872
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) −20.6525 −0.743781
\(772\) 9.94427 0.357902
\(773\) −10.0557 −0.361679 −0.180840 0.983513i \(-0.557882\pi\)
−0.180840 + 0.983513i \(0.557882\pi\)
\(774\) 2.47214 0.0888591
\(775\) −0.763932 −0.0274412
\(776\) 12.4721 0.447724
\(777\) 20.0000 0.717496
\(778\) −31.7082 −1.13679
\(779\) −6.18034 −0.221434
\(780\) 2.23607 0.0800641
\(781\) 0 0
\(782\) 26.6525 0.953091
\(783\) −10.0000 −0.357371
\(784\) 1.00000 0.0357143
\(785\) −17.4721 −0.623607
\(786\) −42.8885 −1.52978
\(787\) −39.7771 −1.41790 −0.708950 0.705259i \(-0.750831\pi\)
−0.708950 + 0.705259i \(0.750831\pi\)
\(788\) 16.4721 0.586796
\(789\) 52.8885 1.88288
\(790\) 5.76393 0.205071
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) −12.4721 −0.442899
\(794\) 6.00000 0.212932
\(795\) −17.2361 −0.611300
\(796\) 4.47214 0.158511
\(797\) −6.88854 −0.244005 −0.122002 0.992530i \(-0.538932\pi\)
−0.122002 + 0.992530i \(0.538932\pi\)
\(798\) 5.00000 0.176998
\(799\) 18.4721 0.653497
\(800\) 1.00000 0.0353553
\(801\) −25.5279 −0.901983
\(802\) 24.8328 0.876877
\(803\) 0 0
\(804\) −6.18034 −0.217964
\(805\) −8.23607 −0.290283
\(806\) 0.763932 0.0269084
\(807\) 57.7639 2.03339
\(808\) 1.00000 0.0351799
\(809\) 46.3607 1.62995 0.814977 0.579493i \(-0.196749\pi\)
0.814977 + 0.579493i \(0.196749\pi\)
\(810\) 11.0000 0.386501
\(811\) 6.47214 0.227267 0.113634 0.993523i \(-0.463751\pi\)
0.113634 + 0.993523i \(0.463751\pi\)
\(812\) −4.47214 −0.156941
\(813\) 43.4164 1.52268
\(814\) 0 0
\(815\) 6.00000 0.210171
\(816\) −7.23607 −0.253313
\(817\) −2.76393 −0.0966977
\(818\) −20.4721 −0.715791
\(819\) 2.00000 0.0698857
\(820\) −2.76393 −0.0965207
\(821\) −30.9443 −1.07996 −0.539981 0.841677i \(-0.681569\pi\)
−0.539981 + 0.841677i \(0.681569\pi\)
\(822\) 10.1246 0.353136
\(823\) −28.3607 −0.988591 −0.494296 0.869294i \(-0.664574\pi\)
−0.494296 + 0.869294i \(0.664574\pi\)
\(824\) −8.18034 −0.284976
\(825\) 0 0
\(826\) 8.70820 0.302997
\(827\) 10.1803 0.354005 0.177003 0.984210i \(-0.443360\pi\)
0.177003 + 0.984210i \(0.443360\pi\)
\(828\) −16.4721 −0.572446
\(829\) 5.94427 0.206453 0.103227 0.994658i \(-0.467083\pi\)
0.103227 + 0.994658i \(0.467083\pi\)
\(830\) 4.23607 0.147036
\(831\) 57.2361 1.98550
\(832\) −1.00000 −0.0346688
\(833\) −3.23607 −0.112123
\(834\) 0.527864 0.0182784
\(835\) −11.7082 −0.405179
\(836\) 0 0
\(837\) 1.70820 0.0590442
\(838\) −23.7639 −0.820911
\(839\) 35.5967 1.22894 0.614468 0.788942i \(-0.289371\pi\)
0.614468 + 0.788942i \(0.289371\pi\)
\(840\) 2.23607 0.0771517
\(841\) −9.00000 −0.310345
\(842\) −20.9443 −0.721787
\(843\) 62.2361 2.14352
\(844\) −12.4721 −0.429309
\(845\) 12.0000 0.412813
\(846\) −11.4164 −0.392504
\(847\) 0 0
\(848\) 7.70820 0.264701
\(849\) −48.6656 −1.67020
\(850\) −3.23607 −0.110996
\(851\) 73.6656 2.52523
\(852\) −23.4164 −0.802233
\(853\) 5.47214 0.187362 0.0936812 0.995602i \(-0.470137\pi\)
0.0936812 + 0.995602i \(0.470137\pi\)
\(854\) −12.4721 −0.426788
\(855\) 4.47214 0.152944
\(856\) −6.47214 −0.221213
\(857\) 42.8328 1.46314 0.731571 0.681766i \(-0.238788\pi\)
0.731571 + 0.681766i \(0.238788\pi\)
\(858\) 0 0
\(859\) −16.9443 −0.578131 −0.289066 0.957309i \(-0.593345\pi\)
−0.289066 + 0.957309i \(0.593345\pi\)
\(860\) −1.23607 −0.0421496
\(861\) −6.18034 −0.210625
\(862\) 2.34752 0.0799570
\(863\) 32.7214 1.11385 0.556924 0.830563i \(-0.311981\pi\)
0.556924 + 0.830563i \(0.311981\pi\)
\(864\) −2.23607 −0.0760726
\(865\) −15.8885 −0.540227
\(866\) 18.6525 0.633837
\(867\) −14.5967 −0.495732
\(868\) 0.763932 0.0259295
\(869\) 0 0
\(870\) −10.0000 −0.339032
\(871\) 2.76393 0.0936523
\(872\) −9.70820 −0.328761
\(873\) 24.9443 0.844236
\(874\) 18.4164 0.622944
\(875\) 1.00000 0.0338062
\(876\) 6.18034 0.208814
\(877\) −41.9574 −1.41680 −0.708401 0.705810i \(-0.750583\pi\)
−0.708401 + 0.705810i \(0.750583\pi\)
\(878\) 25.1246 0.847915
\(879\) 11.1803 0.377104
\(880\) 0 0
\(881\) 12.9443 0.436104 0.218052 0.975937i \(-0.430030\pi\)
0.218052 + 0.975937i \(0.430030\pi\)
\(882\) 2.00000 0.0673435
\(883\) −13.5279 −0.455249 −0.227624 0.973749i \(-0.573096\pi\)
−0.227624 + 0.973749i \(0.573096\pi\)
\(884\) 3.23607 0.108841
\(885\) 19.4721 0.654549
\(886\) 20.3607 0.684030
\(887\) −0.763932 −0.0256503 −0.0128252 0.999918i \(-0.504082\pi\)
−0.0128252 + 0.999918i \(0.504082\pi\)
\(888\) −20.0000 −0.671156
\(889\) 1.76393 0.0591604
\(890\) 12.7639 0.427848
\(891\) 0 0
\(892\) 14.1803 0.474793
\(893\) 12.7639 0.427129
\(894\) −8.94427 −0.299141
\(895\) 6.76393 0.226093
\(896\) −1.00000 −0.0334077
\(897\) 18.4164 0.614906
\(898\) −29.8328 −0.995534
\(899\) −3.41641 −0.113944
\(900\) 2.00000 0.0666667
\(901\) −24.9443 −0.831014
\(902\) 0 0
\(903\) −2.76393 −0.0919779
\(904\) −1.00000 −0.0332595
\(905\) −7.00000 −0.232688
\(906\) −17.3607 −0.576770
\(907\) −4.36068 −0.144794 −0.0723970 0.997376i \(-0.523065\pi\)
−0.0723970 + 0.997376i \(0.523065\pi\)
\(908\) −19.4164 −0.644356
\(909\) 2.00000 0.0663358
\(910\) −1.00000 −0.0331497
\(911\) −12.3475 −0.409092 −0.204546 0.978857i \(-0.565572\pi\)
−0.204546 + 0.978857i \(0.565572\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) 9.58359 0.316997
\(915\) −27.8885 −0.921967
\(916\) −10.9443 −0.361609
\(917\) 19.1803 0.633391
\(918\) 7.23607 0.238826
\(919\) 4.94427 0.163096 0.0815482 0.996669i \(-0.474014\pi\)
0.0815482 + 0.996669i \(0.474014\pi\)
\(920\) 8.23607 0.271535
\(921\) 34.4721 1.13590
\(922\) −11.8885 −0.391528
\(923\) 10.4721 0.344695
\(924\) 0 0
\(925\) −8.94427 −0.294086
\(926\) −2.70820 −0.0889971
\(927\) −16.3607 −0.537355
\(928\) 4.47214 0.146805
\(929\) 10.8328 0.355413 0.177707 0.984084i \(-0.443132\pi\)
0.177707 + 0.984084i \(0.443132\pi\)
\(930\) 1.70820 0.0560142
\(931\) −2.23607 −0.0732842
\(932\) 16.4164 0.537737
\(933\) 29.3475 0.960795
\(934\) −25.1803 −0.823926
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −11.5279 −0.376599 −0.188299 0.982112i \(-0.560298\pi\)
−0.188299 + 0.982112i \(0.560298\pi\)
\(938\) 2.76393 0.0902456
\(939\) 63.6656 2.07765
\(940\) 5.70820 0.186181
\(941\) 20.1115 0.655615 0.327807 0.944745i \(-0.393690\pi\)
0.327807 + 0.944745i \(0.393690\pi\)
\(942\) 39.0689 1.27293
\(943\) −22.7639 −0.741296
\(944\) −8.70820 −0.283428
\(945\) −2.23607 −0.0727393
\(946\) 0 0
\(947\) −48.0689 −1.56203 −0.781014 0.624513i \(-0.785297\pi\)
−0.781014 + 0.624513i \(0.785297\pi\)
\(948\) −12.8885 −0.418600
\(949\) −2.76393 −0.0897210
\(950\) −2.23607 −0.0725476
\(951\) −68.5410 −2.22259
\(952\) 3.23607 0.104882
\(953\) 10.8885 0.352715 0.176357 0.984326i \(-0.443569\pi\)
0.176357 + 0.984326i \(0.443569\pi\)
\(954\) 15.4164 0.499125
\(955\) −2.81966 −0.0912421
\(956\) 2.70820 0.0875896
\(957\) 0 0
\(958\) −21.5967 −0.697759
\(959\) −4.52786 −0.146212
\(960\) −2.23607 −0.0721688
\(961\) −30.4164 −0.981174
\(962\) 8.94427 0.288375
\(963\) −12.9443 −0.417123
\(964\) −22.3607 −0.720189
\(965\) −9.94427 −0.320117
\(966\) 18.4164 0.592538
\(967\) 43.1935 1.38901 0.694505 0.719488i \(-0.255624\pi\)
0.694505 + 0.719488i \(0.255624\pi\)
\(968\) 0 0
\(969\) 16.1803 0.519787
\(970\) −12.4721 −0.400456
\(971\) 24.1246 0.774196 0.387098 0.922039i \(-0.373478\pi\)
0.387098 + 0.922039i \(0.373478\pi\)
\(972\) −17.8885 −0.573775
\(973\) −0.236068 −0.00756799
\(974\) −2.23607 −0.0716482
\(975\) −2.23607 −0.0716115
\(976\) 12.4721 0.399223
\(977\) −28.0557 −0.897582 −0.448791 0.893637i \(-0.648145\pi\)
−0.448791 + 0.893637i \(0.648145\pi\)
\(978\) −13.4164 −0.429009
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −19.4164 −0.619918
\(982\) −26.7639 −0.854071
\(983\) −55.1935 −1.76040 −0.880200 0.474604i \(-0.842591\pi\)
−0.880200 + 0.474604i \(0.842591\pi\)
\(984\) 6.18034 0.197022
\(985\) −16.4721 −0.524846
\(986\) −14.4721 −0.460887
\(987\) 12.7639 0.406280
\(988\) 2.23607 0.0711388
\(989\) −10.1803 −0.323716
\(990\) 0 0
\(991\) 32.7082 1.03901 0.519505 0.854467i \(-0.326116\pi\)
0.519505 + 0.854467i \(0.326116\pi\)
\(992\) −0.763932 −0.0242549
\(993\) −18.5410 −0.588381
\(994\) 10.4721 0.332156
\(995\) −4.47214 −0.141776
\(996\) −9.47214 −0.300136
\(997\) 6.16718 0.195317 0.0976583 0.995220i \(-0.468865\pi\)
0.0976583 + 0.995220i \(0.468865\pi\)
\(998\) −26.8328 −0.849378
\(999\) 20.0000 0.632772
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.by.1.2 yes 2
11.10 odd 2 8470.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bn.1.2 2 11.10 odd 2
8470.2.a.by.1.2 yes 2 1.1 even 1 trivial