Properties

Label 966.2.a.p.1.2
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +4.47214 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -4.47214 q^{17} +1.00000 q^{18} -6.47214 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +4.47214 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +6.47214 q^{31} +1.00000 q^{32} -4.47214 q^{34} +2.00000 q^{35} +1.00000 q^{36} -10.9443 q^{37} -6.47214 q^{38} +4.47214 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +12.9443 q^{43} +2.00000 q^{45} -1.00000 q^{46} +6.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.47214 q^{51} +4.47214 q^{52} +6.94427 q^{53} +1.00000 q^{54} +1.00000 q^{56} -6.47214 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +6.47214 q^{62} +1.00000 q^{63} +1.00000 q^{64} +8.94427 q^{65} -12.9443 q^{67} -4.47214 q^{68} -1.00000 q^{69} +2.00000 q^{70} -12.9443 q^{71} +1.00000 q^{72} -2.94427 q^{73} -10.9443 q^{74} -1.00000 q^{75} -6.47214 q^{76} +4.47214 q^{78} +12.9443 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.47214 q^{83} +1.00000 q^{84} -8.94427 q^{85} +12.9443 q^{86} -2.00000 q^{87} -17.4164 q^{89} +2.00000 q^{90} +4.47214 q^{91} -1.00000 q^{92} +6.47214 q^{93} +6.47214 q^{94} -12.9443 q^{95} +1.00000 q^{96} +8.47214 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{27} + 2 q^{28} - 4 q^{29} + 4 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} + 2 q^{42} + 8 q^{43} + 4 q^{45} - 2 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{53} + 2 q^{54} + 2 q^{56} - 4 q^{57} - 4 q^{58} + 8 q^{59} + 4 q^{60} - 12 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{67} - 2 q^{69} + 4 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} + 8 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{83} + 2 q^{84} + 8 q^{86} - 4 q^{87} - 8 q^{89} + 4 q^{90} - 2 q^{92} + 4 q^{93} + 4 q^{94} - 8 q^{95} + 2 q^{96} + 8 q^{97} + 2 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.47214 0.877058
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 6.47214 1.16243 0.581215 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.47214 −0.766965
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) −6.47214 −1.04992
\(39\) 4.47214 0.716115
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.9443 1.97398 0.986991 0.160773i \(-0.0513986\pi\)
0.986991 + 0.160773i \(0.0513986\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −1.00000 −0.147442
\(47\) 6.47214 0.944058 0.472029 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −4.47214 −0.626224
\(52\) 4.47214 0.620174
\(53\) 6.94427 0.953869 0.476935 0.878939i \(-0.341748\pi\)
0.476935 + 0.878939i \(0.341748\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.47214 −0.857255
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 6.47214 0.821962
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 8.94427 1.10940
\(66\) 0 0
\(67\) −12.9443 −1.58139 −0.790697 0.612207i \(-0.790282\pi\)
−0.790697 + 0.612207i \(0.790282\pi\)
\(68\) −4.47214 −0.542326
\(69\) −1.00000 −0.120386
\(70\) 2.00000 0.239046
\(71\) −12.9443 −1.53620 −0.768101 0.640328i \(-0.778798\pi\)
−0.768101 + 0.640328i \(0.778798\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.94427 −0.344601 −0.172300 0.985044i \(-0.555120\pi\)
−0.172300 + 0.985044i \(0.555120\pi\)
\(74\) −10.9443 −1.27225
\(75\) −1.00000 −0.115470
\(76\) −6.47214 −0.742405
\(77\) 0 0
\(78\) 4.47214 0.506370
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −6.47214 −0.710409 −0.355205 0.934789i \(-0.615589\pi\)
−0.355205 + 0.934789i \(0.615589\pi\)
\(84\) 1.00000 0.109109
\(85\) −8.94427 −0.970143
\(86\) 12.9443 1.39582
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −17.4164 −1.84614 −0.923068 0.384637i \(-0.874327\pi\)
−0.923068 + 0.384637i \(0.874327\pi\)
\(90\) 2.00000 0.210819
\(91\) 4.47214 0.468807
\(92\) −1.00000 −0.104257
\(93\) 6.47214 0.671129
\(94\) 6.47214 0.667550
\(95\) −12.9443 −1.32805
\(96\) 1.00000 0.102062
\(97\) 8.47214 0.860215 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 17.4164 1.73300 0.866499 0.499179i \(-0.166365\pi\)
0.866499 + 0.499179i \(0.166365\pi\)
\(102\) −4.47214 −0.442807
\(103\) 12.9443 1.27544 0.637719 0.770270i \(-0.279878\pi\)
0.637719 + 0.770270i \(0.279878\pi\)
\(104\) 4.47214 0.438529
\(105\) 2.00000 0.195180
\(106\) 6.94427 0.674487
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.9443 1.43140 0.715701 0.698407i \(-0.246107\pi\)
0.715701 + 0.698407i \(0.246107\pi\)
\(110\) 0 0
\(111\) −10.9443 −1.03878
\(112\) 1.00000 0.0944911
\(113\) −1.05573 −0.0993145 −0.0496573 0.998766i \(-0.515813\pi\)
−0.0496573 + 0.998766i \(0.515813\pi\)
\(114\) −6.47214 −0.606171
\(115\) −2.00000 −0.186501
\(116\) −2.00000 −0.185695
\(117\) 4.47214 0.413449
\(118\) 4.00000 0.368230
\(119\) −4.47214 −0.409960
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) 6.47214 0.581215
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) 3.05573 0.271152 0.135576 0.990767i \(-0.456712\pi\)
0.135576 + 0.990767i \(0.456712\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.9443 1.13968
\(130\) 8.94427 0.784465
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −6.47214 −0.561205
\(134\) −12.9443 −1.11821
\(135\) 2.00000 0.172133
\(136\) −4.47214 −0.383482
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) 2.00000 0.169031
\(141\) 6.47214 0.545052
\(142\) −12.9443 −1.08626
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −2.94427 −0.243670
\(147\) 1.00000 0.0824786
\(148\) −10.9443 −0.899614
\(149\) −18.9443 −1.55198 −0.775988 0.630748i \(-0.782748\pi\)
−0.775988 + 0.630748i \(0.782748\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.9443 −1.05339 −0.526695 0.850054i \(-0.676569\pi\)
−0.526695 + 0.850054i \(0.676569\pi\)
\(152\) −6.47214 −0.524960
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 12.9443 1.03971
\(156\) 4.47214 0.358057
\(157\) 14.9443 1.19268 0.596341 0.802731i \(-0.296620\pi\)
0.596341 + 0.802731i \(0.296620\pi\)
\(158\) 12.9443 1.02979
\(159\) 6.94427 0.550717
\(160\) 2.00000 0.158114
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 8.94427 0.700569 0.350285 0.936643i \(-0.386085\pi\)
0.350285 + 0.936643i \(0.386085\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −6.47214 −0.502335
\(167\) 11.4164 0.883428 0.441714 0.897156i \(-0.354371\pi\)
0.441714 + 0.897156i \(0.354371\pi\)
\(168\) 1.00000 0.0771517
\(169\) 7.00000 0.538462
\(170\) −8.94427 −0.685994
\(171\) −6.47214 −0.494937
\(172\) 12.9443 0.986991
\(173\) −8.47214 −0.644125 −0.322062 0.946718i \(-0.604376\pi\)
−0.322062 + 0.946718i \(0.604376\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −17.4164 −1.30541
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.47214 0.331497
\(183\) −6.00000 −0.443533
\(184\) −1.00000 −0.0737210
\(185\) −21.8885 −1.60928
\(186\) 6.47214 0.474560
\(187\) 0 0
\(188\) 6.47214 0.472029
\(189\) 1.00000 0.0727393
\(190\) −12.9443 −0.939076
\(191\) −20.9443 −1.51547 −0.757737 0.652560i \(-0.773695\pi\)
−0.757737 + 0.652560i \(0.773695\pi\)
\(192\) 1.00000 0.0721688
\(193\) −23.8885 −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(194\) 8.47214 0.608264
\(195\) 8.94427 0.640513
\(196\) 1.00000 0.0714286
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) 0 0
\(199\) 20.9443 1.48470 0.742350 0.670012i \(-0.233711\pi\)
0.742350 + 0.670012i \(0.233711\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.9443 −0.913019
\(202\) 17.4164 1.22541
\(203\) −2.00000 −0.140372
\(204\) −4.47214 −0.313112
\(205\) −12.0000 −0.838116
\(206\) 12.9443 0.901870
\(207\) −1.00000 −0.0695048
\(208\) 4.47214 0.310087
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) 16.9443 1.16649 0.583246 0.812296i \(-0.301782\pi\)
0.583246 + 0.812296i \(0.301782\pi\)
\(212\) 6.94427 0.476935
\(213\) −12.9443 −0.886927
\(214\) −8.00000 −0.546869
\(215\) 25.8885 1.76558
\(216\) 1.00000 0.0680414
\(217\) 6.47214 0.439357
\(218\) 14.9443 1.01215
\(219\) −2.94427 −0.198955
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) −10.9443 −0.734531
\(223\) 9.52786 0.638033 0.319016 0.947749i \(-0.396647\pi\)
0.319016 + 0.947749i \(0.396647\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −1.05573 −0.0702260
\(227\) 14.4721 0.960549 0.480275 0.877118i \(-0.340537\pi\)
0.480275 + 0.877118i \(0.340537\pi\)
\(228\) −6.47214 −0.428628
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.47214 0.292353
\(235\) 12.9443 0.844391
\(236\) 4.00000 0.260378
\(237\) 12.9443 0.840821
\(238\) −4.47214 −0.289886
\(239\) 3.05573 0.197659 0.0988293 0.995104i \(-0.468490\pi\)
0.0988293 + 0.995104i \(0.468490\pi\)
\(240\) 2.00000 0.129099
\(241\) 11.5279 0.742575 0.371288 0.928518i \(-0.378916\pi\)
0.371288 + 0.928518i \(0.378916\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 2.00000 0.127775
\(246\) −6.00000 −0.382546
\(247\) −28.9443 −1.84168
\(248\) 6.47214 0.410981
\(249\) −6.47214 −0.410155
\(250\) −12.0000 −0.758947
\(251\) −14.4721 −0.913473 −0.456737 0.889602i \(-0.650982\pi\)
−0.456737 + 0.889602i \(0.650982\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 3.05573 0.191733
\(255\) −8.94427 −0.560112
\(256\) 1.00000 0.0625000
\(257\) 14.9443 0.932198 0.466099 0.884733i \(-0.345659\pi\)
0.466099 + 0.884733i \(0.345659\pi\)
\(258\) 12.9443 0.805875
\(259\) −10.9443 −0.680044
\(260\) 8.94427 0.554700
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 13.8885 0.853166
\(266\) −6.47214 −0.396832
\(267\) −17.4164 −1.06587
\(268\) −12.9443 −0.790697
\(269\) 7.52786 0.458982 0.229491 0.973311i \(-0.426294\pi\)
0.229491 + 0.973311i \(0.426294\pi\)
\(270\) 2.00000 0.121716
\(271\) −11.4164 −0.693497 −0.346749 0.937958i \(-0.612714\pi\)
−0.346749 + 0.937958i \(0.612714\pi\)
\(272\) −4.47214 −0.271163
\(273\) 4.47214 0.270666
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 23.8885 1.43532 0.717662 0.696392i \(-0.245212\pi\)
0.717662 + 0.696392i \(0.245212\pi\)
\(278\) −8.94427 −0.536442
\(279\) 6.47214 0.387477
\(280\) 2.00000 0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 6.47214 0.385410
\(283\) 9.52786 0.566373 0.283186 0.959065i \(-0.408609\pi\)
0.283186 + 0.959065i \(0.408609\pi\)
\(284\) −12.9443 −0.768101
\(285\) −12.9443 −0.766752
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 3.00000 0.176471
\(290\) −4.00000 −0.234888
\(291\) 8.47214 0.496645
\(292\) −2.94427 −0.172300
\(293\) −7.88854 −0.460854 −0.230427 0.973090i \(-0.574012\pi\)
−0.230427 + 0.973090i \(0.574012\pi\)
\(294\) 1.00000 0.0583212
\(295\) 8.00000 0.465778
\(296\) −10.9443 −0.636123
\(297\) 0 0
\(298\) −18.9443 −1.09741
\(299\) −4.47214 −0.258630
\(300\) −1.00000 −0.0577350
\(301\) 12.9443 0.746095
\(302\) −12.9443 −0.744859
\(303\) 17.4164 1.00055
\(304\) −6.47214 −0.371202
\(305\) −12.0000 −0.687118
\(306\) −4.47214 −0.255655
\(307\) 7.05573 0.402692 0.201346 0.979520i \(-0.435468\pi\)
0.201346 + 0.979520i \(0.435468\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) 12.9443 0.735185
\(311\) 6.47214 0.367001 0.183501 0.983020i \(-0.441257\pi\)
0.183501 + 0.983020i \(0.441257\pi\)
\(312\) 4.47214 0.253185
\(313\) 13.4164 0.758340 0.379170 0.925327i \(-0.376210\pi\)
0.379170 + 0.925327i \(0.376210\pi\)
\(314\) 14.9443 0.843354
\(315\) 2.00000 0.112687
\(316\) 12.9443 0.728172
\(317\) −30.9443 −1.73800 −0.869002 0.494809i \(-0.835238\pi\)
−0.869002 + 0.494809i \(0.835238\pi\)
\(318\) 6.94427 0.389415
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) −8.00000 −0.446516
\(322\) −1.00000 −0.0557278
\(323\) 28.9443 1.61050
\(324\) 1.00000 0.0555556
\(325\) −4.47214 −0.248069
\(326\) 8.94427 0.495377
\(327\) 14.9443 0.826420
\(328\) −6.00000 −0.331295
\(329\) 6.47214 0.356820
\(330\) 0 0
\(331\) 21.8885 1.20310 0.601552 0.798834i \(-0.294549\pi\)
0.601552 + 0.798834i \(0.294549\pi\)
\(332\) −6.47214 −0.355205
\(333\) −10.9443 −0.599742
\(334\) 11.4164 0.624678
\(335\) −25.8885 −1.41444
\(336\) 1.00000 0.0545545
\(337\) 14.9443 0.814066 0.407033 0.913413i \(-0.366563\pi\)
0.407033 + 0.913413i \(0.366563\pi\)
\(338\) 7.00000 0.380750
\(339\) −1.05573 −0.0573393
\(340\) −8.94427 −0.485071
\(341\) 0 0
\(342\) −6.47214 −0.349973
\(343\) 1.00000 0.0539949
\(344\) 12.9443 0.697908
\(345\) −2.00000 −0.107676
\(346\) −8.47214 −0.455465
\(347\) 16.9443 0.909616 0.454808 0.890589i \(-0.349708\pi\)
0.454808 + 0.890589i \(0.349708\pi\)
\(348\) −2.00000 −0.107211
\(349\) 15.5279 0.831188 0.415594 0.909550i \(-0.363574\pi\)
0.415594 + 0.909550i \(0.363574\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 4.47214 0.238705
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 4.00000 0.212598
\(355\) −25.8885 −1.37402
\(356\) −17.4164 −0.923068
\(357\) −4.47214 −0.236691
\(358\) −20.0000 −1.05703
\(359\) 22.8328 1.20507 0.602535 0.798092i \(-0.294157\pi\)
0.602535 + 0.798092i \(0.294157\pi\)
\(360\) 2.00000 0.105409
\(361\) 22.8885 1.20466
\(362\) 2.00000 0.105118
\(363\) −11.0000 −0.577350
\(364\) 4.47214 0.234404
\(365\) −5.88854 −0.308220
\(366\) −6.00000 −0.313625
\(367\) −22.8328 −1.19186 −0.595932 0.803035i \(-0.703217\pi\)
−0.595932 + 0.803035i \(0.703217\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) −21.8885 −1.13793
\(371\) 6.94427 0.360529
\(372\) 6.47214 0.335565
\(373\) −10.9443 −0.566673 −0.283336 0.959021i \(-0.591441\pi\)
−0.283336 + 0.959021i \(0.591441\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 6.47214 0.333775
\(377\) −8.94427 −0.460653
\(378\) 1.00000 0.0514344
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) −12.9443 −0.664027
\(381\) 3.05573 0.156550
\(382\) −20.9443 −1.07160
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −23.8885 −1.21589
\(387\) 12.9443 0.657994
\(388\) 8.47214 0.430108
\(389\) −1.05573 −0.0535275 −0.0267638 0.999642i \(-0.508520\pi\)
−0.0267638 + 0.999642i \(0.508520\pi\)
\(390\) 8.94427 0.452911
\(391\) 4.47214 0.226166
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 2.94427 0.148330
\(395\) 25.8885 1.30259
\(396\) 0 0
\(397\) −27.5279 −1.38158 −0.690792 0.723054i \(-0.742738\pi\)
−0.690792 + 0.723054i \(0.742738\pi\)
\(398\) 20.9443 1.04984
\(399\) −6.47214 −0.324012
\(400\) −1.00000 −0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) −12.9443 −0.645602
\(403\) 28.9443 1.44182
\(404\) 17.4164 0.866499
\(405\) 2.00000 0.0993808
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) −4.47214 −0.221404
\(409\) 19.8885 0.983425 0.491713 0.870758i \(-0.336371\pi\)
0.491713 + 0.870758i \(0.336371\pi\)
\(410\) −12.0000 −0.592638
\(411\) −14.0000 −0.690569
\(412\) 12.9443 0.637719
\(413\) 4.00000 0.196827
\(414\) −1.00000 −0.0491473
\(415\) −12.9443 −0.635409
\(416\) 4.47214 0.219265
\(417\) −8.94427 −0.438003
\(418\) 0 0
\(419\) 19.4164 0.948554 0.474277 0.880376i \(-0.342710\pi\)
0.474277 + 0.880376i \(0.342710\pi\)
\(420\) 2.00000 0.0975900
\(421\) −20.8328 −1.01533 −0.507665 0.861555i \(-0.669491\pi\)
−0.507665 + 0.861555i \(0.669491\pi\)
\(422\) 16.9443 0.824834
\(423\) 6.47214 0.314686
\(424\) 6.94427 0.337244
\(425\) 4.47214 0.216930
\(426\) −12.9443 −0.627152
\(427\) −6.00000 −0.290360
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 25.8885 1.24846
\(431\) −17.8885 −0.861661 −0.430830 0.902433i \(-0.641779\pi\)
−0.430830 + 0.902433i \(0.641779\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.41641 −0.0680682 −0.0340341 0.999421i \(-0.510835\pi\)
−0.0340341 + 0.999421i \(0.510835\pi\)
\(434\) 6.47214 0.310672
\(435\) −4.00000 −0.191785
\(436\) 14.9443 0.715701
\(437\) 6.47214 0.309604
\(438\) −2.94427 −0.140683
\(439\) −16.3607 −0.780853 −0.390426 0.920634i \(-0.627672\pi\)
−0.390426 + 0.920634i \(0.627672\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −20.0000 −0.951303
\(443\) −18.8328 −0.894774 −0.447387 0.894340i \(-0.647645\pi\)
−0.447387 + 0.894340i \(0.647645\pi\)
\(444\) −10.9443 −0.519392
\(445\) −34.8328 −1.65123
\(446\) 9.52786 0.451157
\(447\) −18.9443 −0.896033
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −1.05573 −0.0496573
\(453\) −12.9443 −0.608175
\(454\) 14.4721 0.679211
\(455\) 8.94427 0.419314
\(456\) −6.47214 −0.303086
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −14.0000 −0.654177
\(459\) −4.47214 −0.208741
\(460\) −2.00000 −0.0932505
\(461\) 33.4164 1.55636 0.778179 0.628043i \(-0.216144\pi\)
0.778179 + 0.628043i \(0.216144\pi\)
\(462\) 0 0
\(463\) 41.8885 1.94673 0.973363 0.229270i \(-0.0736339\pi\)
0.973363 + 0.229270i \(0.0736339\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 12.9443 0.600276
\(466\) −6.00000 −0.277945
\(467\) −29.3050 −1.35607 −0.678036 0.735029i \(-0.737169\pi\)
−0.678036 + 0.735029i \(0.737169\pi\)
\(468\) 4.47214 0.206725
\(469\) −12.9443 −0.597711
\(470\) 12.9443 0.597075
\(471\) 14.9443 0.688596
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 12.9443 0.594550
\(475\) 6.47214 0.296962
\(476\) −4.47214 −0.204980
\(477\) 6.94427 0.317956
\(478\) 3.05573 0.139766
\(479\) 12.9443 0.591439 0.295719 0.955275i \(-0.404441\pi\)
0.295719 + 0.955275i \(0.404441\pi\)
\(480\) 2.00000 0.0912871
\(481\) −48.9443 −2.23167
\(482\) 11.5279 0.525080
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) 16.9443 0.769400
\(486\) 1.00000 0.0453609
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −6.00000 −0.271607
\(489\) 8.94427 0.404474
\(490\) 2.00000 0.0903508
\(491\) 23.0557 1.04049 0.520245 0.854017i \(-0.325841\pi\)
0.520245 + 0.854017i \(0.325841\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.94427 0.402830
\(494\) −28.9443 −1.30226
\(495\) 0 0
\(496\) 6.47214 0.290607
\(497\) −12.9443 −0.580630
\(498\) −6.47214 −0.290023
\(499\) 15.0557 0.673987 0.336993 0.941507i \(-0.390590\pi\)
0.336993 + 0.941507i \(0.390590\pi\)
\(500\) −12.0000 −0.536656
\(501\) 11.4164 0.510047
\(502\) −14.4721 −0.645923
\(503\) 25.8885 1.15431 0.577157 0.816634i \(-0.304162\pi\)
0.577157 + 0.816634i \(0.304162\pi\)
\(504\) 1.00000 0.0445435
\(505\) 34.8328 1.55004
\(506\) 0 0
\(507\) 7.00000 0.310881
\(508\) 3.05573 0.135576
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) −8.94427 −0.396059
\(511\) −2.94427 −0.130247
\(512\) 1.00000 0.0441942
\(513\) −6.47214 −0.285752
\(514\) 14.9443 0.659164
\(515\) 25.8885 1.14079
\(516\) 12.9443 0.569840
\(517\) 0 0
\(518\) −10.9443 −0.480864
\(519\) −8.47214 −0.371885
\(520\) 8.94427 0.392232
\(521\) 6.58359 0.288432 0.144216 0.989546i \(-0.453934\pi\)
0.144216 + 0.989546i \(0.453934\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 37.3050 1.63123 0.815616 0.578594i \(-0.196398\pi\)
0.815616 + 0.578594i \(0.196398\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) −28.9443 −1.26083
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 13.8885 0.603280
\(531\) 4.00000 0.173585
\(532\) −6.47214 −0.280603
\(533\) −26.8328 −1.16226
\(534\) −17.4164 −0.753682
\(535\) −16.0000 −0.691740
\(536\) −12.9443 −0.559107
\(537\) −20.0000 −0.863064
\(538\) 7.52786 0.324549
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −11.4164 −0.490377
\(543\) 2.00000 0.0858282
\(544\) −4.47214 −0.191741
\(545\) 29.8885 1.28028
\(546\) 4.47214 0.191390
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 12.9443 0.551445
\(552\) −1.00000 −0.0425628
\(553\) 12.9443 0.550446
\(554\) 23.8885 1.01493
\(555\) −21.8885 −0.929117
\(556\) −8.94427 −0.379322
\(557\) −34.9443 −1.48064 −0.740318 0.672257i \(-0.765325\pi\)
−0.740318 + 0.672257i \(0.765325\pi\)
\(558\) 6.47214 0.273987
\(559\) 57.8885 2.44842
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 17.5279 0.738711 0.369356 0.929288i \(-0.379578\pi\)
0.369356 + 0.929288i \(0.379578\pi\)
\(564\) 6.47214 0.272526
\(565\) −2.11146 −0.0888296
\(566\) 9.52786 0.400486
\(567\) 1.00000 0.0419961
\(568\) −12.9443 −0.543130
\(569\) 27.8885 1.16915 0.584574 0.811340i \(-0.301262\pi\)
0.584574 + 0.811340i \(0.301262\pi\)
\(570\) −12.9443 −0.542176
\(571\) 4.94427 0.206911 0.103456 0.994634i \(-0.467010\pi\)
0.103456 + 0.994634i \(0.467010\pi\)
\(572\) 0 0
\(573\) −20.9443 −0.874960
\(574\) −6.00000 −0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 14.9443 0.622138 0.311069 0.950387i \(-0.399313\pi\)
0.311069 + 0.950387i \(0.399313\pi\)
\(578\) 3.00000 0.124784
\(579\) −23.8885 −0.992774
\(580\) −4.00000 −0.166091
\(581\) −6.47214 −0.268509
\(582\) 8.47214 0.351181
\(583\) 0 0
\(584\) −2.94427 −0.121835
\(585\) 8.94427 0.369800
\(586\) −7.88854 −0.325873
\(587\) 16.9443 0.699365 0.349682 0.936868i \(-0.386289\pi\)
0.349682 + 0.936868i \(0.386289\pi\)
\(588\) 1.00000 0.0412393
\(589\) −41.8885 −1.72599
\(590\) 8.00000 0.329355
\(591\) 2.94427 0.121111
\(592\) −10.9443 −0.449807
\(593\) 24.8328 1.01976 0.509881 0.860245i \(-0.329690\pi\)
0.509881 + 0.860245i \(0.329690\pi\)
\(594\) 0 0
\(595\) −8.94427 −0.366679
\(596\) −18.9443 −0.775988
\(597\) 20.9443 0.857192
\(598\) −4.47214 −0.182879
\(599\) −1.88854 −0.0771638 −0.0385819 0.999255i \(-0.512284\pi\)
−0.0385819 + 0.999255i \(0.512284\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −12.8328 −0.523461 −0.261731 0.965141i \(-0.584293\pi\)
−0.261731 + 0.965141i \(0.584293\pi\)
\(602\) 12.9443 0.527569
\(603\) −12.9443 −0.527132
\(604\) −12.9443 −0.526695
\(605\) −22.0000 −0.894427
\(606\) 17.4164 0.707493
\(607\) 16.3607 0.664060 0.332030 0.943269i \(-0.392267\pi\)
0.332030 + 0.943269i \(0.392267\pi\)
\(608\) −6.47214 −0.262480
\(609\) −2.00000 −0.0810441
\(610\) −12.0000 −0.485866
\(611\) 28.9443 1.17096
\(612\) −4.47214 −0.180775
\(613\) −12.8328 −0.518313 −0.259156 0.965835i \(-0.583444\pi\)
−0.259156 + 0.965835i \(0.583444\pi\)
\(614\) 7.05573 0.284746
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 12.9443 0.520695
\(619\) −25.5279 −1.02605 −0.513026 0.858373i \(-0.671475\pi\)
−0.513026 + 0.858373i \(0.671475\pi\)
\(620\) 12.9443 0.519854
\(621\) −1.00000 −0.0401286
\(622\) 6.47214 0.259509
\(623\) −17.4164 −0.697774
\(624\) 4.47214 0.179029
\(625\) −19.0000 −0.760000
\(626\) 13.4164 0.536228
\(627\) 0 0
\(628\) 14.9443 0.596341
\(629\) 48.9443 1.95154
\(630\) 2.00000 0.0796819
\(631\) −28.9443 −1.15225 −0.576127 0.817360i \(-0.695436\pi\)
−0.576127 + 0.817360i \(0.695436\pi\)
\(632\) 12.9443 0.514895
\(633\) 16.9443 0.673474
\(634\) −30.9443 −1.22895
\(635\) 6.11146 0.242526
\(636\) 6.94427 0.275358
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) −12.9443 −0.512067
\(640\) 2.00000 0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −8.00000 −0.315735
\(643\) −1.52786 −0.0602531 −0.0301265 0.999546i \(-0.509591\pi\)
−0.0301265 + 0.999546i \(0.509591\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 25.8885 1.01936
\(646\) 28.9443 1.13880
\(647\) −32.3607 −1.27223 −0.636115 0.771594i \(-0.719460\pi\)
−0.636115 + 0.771594i \(0.719460\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.47214 −0.175412
\(651\) 6.47214 0.253663
\(652\) 8.94427 0.350285
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 14.9443 0.584367
\(655\) 24.0000 0.937758
\(656\) −6.00000 −0.234261
\(657\) −2.94427 −0.114867
\(658\) 6.47214 0.252310
\(659\) −6.83282 −0.266169 −0.133084 0.991105i \(-0.542488\pi\)
−0.133084 + 0.991105i \(0.542488\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 21.8885 0.850722
\(663\) −20.0000 −0.776736
\(664\) −6.47214 −0.251168
\(665\) −12.9443 −0.501957
\(666\) −10.9443 −0.424082
\(667\) 2.00000 0.0774403
\(668\) 11.4164 0.441714
\(669\) 9.52786 0.368369
\(670\) −25.8885 −1.00016
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 14.9443 0.575632
\(675\) −1.00000 −0.0384900
\(676\) 7.00000 0.269231
\(677\) −39.8885 −1.53304 −0.766521 0.642220i \(-0.778014\pi\)
−0.766521 + 0.642220i \(0.778014\pi\)
\(678\) −1.05573 −0.0405450
\(679\) 8.47214 0.325131
\(680\) −8.94427 −0.342997
\(681\) 14.4721 0.554573
\(682\) 0 0
\(683\) −0.944272 −0.0361316 −0.0180658 0.999837i \(-0.505751\pi\)
−0.0180658 + 0.999837i \(0.505751\pi\)
\(684\) −6.47214 −0.247468
\(685\) −28.0000 −1.06983
\(686\) 1.00000 0.0381802
\(687\) −14.0000 −0.534133
\(688\) 12.9443 0.493496
\(689\) 31.0557 1.18313
\(690\) −2.00000 −0.0761387
\(691\) −32.9443 −1.25326 −0.626630 0.779317i \(-0.715566\pi\)
−0.626630 + 0.779317i \(0.715566\pi\)
\(692\) −8.47214 −0.322062
\(693\) 0 0
\(694\) 16.9443 0.643196
\(695\) −17.8885 −0.678551
\(696\) −2.00000 −0.0758098
\(697\) 26.8328 1.01637
\(698\) 15.5279 0.587738
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) 32.8328 1.24008 0.620039 0.784571i \(-0.287117\pi\)
0.620039 + 0.784571i \(0.287117\pi\)
\(702\) 4.47214 0.168790
\(703\) 70.8328 2.67151
\(704\) 0 0
\(705\) 12.9443 0.487509
\(706\) 2.00000 0.0752710
\(707\) 17.4164 0.655011
\(708\) 4.00000 0.150329
\(709\) −10.9443 −0.411021 −0.205510 0.978655i \(-0.565885\pi\)
−0.205510 + 0.978655i \(0.565885\pi\)
\(710\) −25.8885 −0.971580
\(711\) 12.9443 0.485448
\(712\) −17.4164 −0.652707
\(713\) −6.47214 −0.242383
\(714\) −4.47214 −0.167365
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 3.05573 0.114118
\(718\) 22.8328 0.852113
\(719\) −14.4721 −0.539720 −0.269860 0.962900i \(-0.586977\pi\)
−0.269860 + 0.962900i \(0.586977\pi\)
\(720\) 2.00000 0.0745356
\(721\) 12.9443 0.482070
\(722\) 22.8885 0.851823
\(723\) 11.5279 0.428726
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) −11.0000 −0.408248
\(727\) 20.9443 0.776780 0.388390 0.921495i \(-0.373031\pi\)
0.388390 + 0.921495i \(0.373031\pi\)
\(728\) 4.47214 0.165748
\(729\) 1.00000 0.0370370
\(730\) −5.88854 −0.217945
\(731\) −57.8885 −2.14109
\(732\) −6.00000 −0.221766
\(733\) −33.0557 −1.22094 −0.610471 0.792039i \(-0.709020\pi\)
−0.610471 + 0.792039i \(0.709020\pi\)
\(734\) −22.8328 −0.842775
\(735\) 2.00000 0.0737711
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 8.94427 0.329020 0.164510 0.986375i \(-0.447396\pi\)
0.164510 + 0.986375i \(0.447396\pi\)
\(740\) −21.8885 −0.804639
\(741\) −28.9443 −1.06329
\(742\) 6.94427 0.254932
\(743\) 30.8328 1.13115 0.565573 0.824698i \(-0.308655\pi\)
0.565573 + 0.824698i \(0.308655\pi\)
\(744\) 6.47214 0.237280
\(745\) −37.8885 −1.38813
\(746\) −10.9443 −0.400698
\(747\) −6.47214 −0.236803
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) −12.0000 −0.438178
\(751\) −17.8885 −0.652762 −0.326381 0.945238i \(-0.605829\pi\)
−0.326381 + 0.945238i \(0.605829\pi\)
\(752\) 6.47214 0.236015
\(753\) −14.4721 −0.527394
\(754\) −8.94427 −0.325731
\(755\) −25.8885 −0.942181
\(756\) 1.00000 0.0363696
\(757\) −25.0557 −0.910666 −0.455333 0.890321i \(-0.650480\pi\)
−0.455333 + 0.890321i \(0.650480\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) −12.9443 −0.469538
\(761\) 38.9443 1.41173 0.705864 0.708347i \(-0.250559\pi\)
0.705864 + 0.708347i \(0.250559\pi\)
\(762\) 3.05573 0.110697
\(763\) 14.9443 0.541019
\(764\) −20.9443 −0.757737
\(765\) −8.94427 −0.323381
\(766\) 16.0000 0.578103
\(767\) 17.8885 0.645918
\(768\) 1.00000 0.0360844
\(769\) −14.3607 −0.517859 −0.258930 0.965896i \(-0.583370\pi\)
−0.258930 + 0.965896i \(0.583370\pi\)
\(770\) 0 0
\(771\) 14.9443 0.538205
\(772\) −23.8885 −0.859768
\(773\) 22.9443 0.825248 0.412624 0.910901i \(-0.364612\pi\)
0.412624 + 0.910901i \(0.364612\pi\)
\(774\) 12.9443 0.465272
\(775\) −6.47214 −0.232486
\(776\) 8.47214 0.304132
\(777\) −10.9443 −0.392624
\(778\) −1.05573 −0.0378497
\(779\) 38.8328 1.39133
\(780\) 8.94427 0.320256
\(781\) 0 0
\(782\) 4.47214 0.159923
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 29.8885 1.06677
\(786\) 12.0000 0.428026
\(787\) −27.4164 −0.977289 −0.488645 0.872483i \(-0.662509\pi\)
−0.488645 + 0.872483i \(0.662509\pi\)
\(788\) 2.94427 0.104885
\(789\) −24.0000 −0.854423
\(790\) 25.8885 0.921073
\(791\) −1.05573 −0.0375374
\(792\) 0 0
\(793\) −26.8328 −0.952861
\(794\) −27.5279 −0.976927
\(795\) 13.8885 0.492576
\(796\) 20.9443 0.742350
\(797\) 24.8328 0.879623 0.439812 0.898090i \(-0.355045\pi\)
0.439812 + 0.898090i \(0.355045\pi\)
\(798\) −6.47214 −0.229111
\(799\) −28.9443 −1.02397
\(800\) −1.00000 −0.0353553
\(801\) −17.4164 −0.615379
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) −12.9443 −0.456509
\(805\) −2.00000 −0.0704907
\(806\) 28.9443 1.01952
\(807\) 7.52786 0.264993
\(808\) 17.4164 0.612707
\(809\) 0.111456 0.00391859 0.00195930 0.999998i \(-0.499376\pi\)
0.00195930 + 0.999998i \(0.499376\pi\)
\(810\) 2.00000 0.0702728
\(811\) −2.11146 −0.0741433 −0.0370716 0.999313i \(-0.511803\pi\)
−0.0370716 + 0.999313i \(0.511803\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −11.4164 −0.400391
\(814\) 0 0
\(815\) 17.8885 0.626608
\(816\) −4.47214 −0.156556
\(817\) −83.7771 −2.93099
\(818\) 19.8885 0.695387
\(819\) 4.47214 0.156269
\(820\) −12.0000 −0.419058
\(821\) 36.8328 1.28547 0.642737 0.766087i \(-0.277799\pi\)
0.642737 + 0.766087i \(0.277799\pi\)
\(822\) −14.0000 −0.488306
\(823\) 3.05573 0.106516 0.0532580 0.998581i \(-0.483039\pi\)
0.0532580 + 0.998581i \(0.483039\pi\)
\(824\) 12.9443 0.450935
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −1.88854 −0.0656711 −0.0328356 0.999461i \(-0.510454\pi\)
−0.0328356 + 0.999461i \(0.510454\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) −12.9443 −0.449302
\(831\) 23.8885 0.828684
\(832\) 4.47214 0.155043
\(833\) −4.47214 −0.154950
\(834\) −8.94427 −0.309715
\(835\) 22.8328 0.790162
\(836\) 0 0
\(837\) 6.47214 0.223710
\(838\) 19.4164 0.670729
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 2.00000 0.0690066
\(841\) −25.0000 −0.862069
\(842\) −20.8328 −0.717946
\(843\) 10.0000 0.344418
\(844\) 16.9443 0.583246
\(845\) 14.0000 0.481615
\(846\) 6.47214 0.222517
\(847\) −11.0000 −0.377964
\(848\) 6.94427 0.238467
\(849\) 9.52786 0.326995
\(850\) 4.47214 0.153393
\(851\) 10.9443 0.375165
\(852\) −12.9443 −0.443463
\(853\) 38.3607 1.31344 0.656722 0.754132i \(-0.271942\pi\)
0.656722 + 0.754132i \(0.271942\pi\)
\(854\) −6.00000 −0.205316
\(855\) −12.9443 −0.442685
\(856\) −8.00000 −0.273434
\(857\) −12.1115 −0.413719 −0.206860 0.978371i \(-0.566324\pi\)
−0.206860 + 0.978371i \(0.566324\pi\)
\(858\) 0 0
\(859\) 15.0557 0.513695 0.256847 0.966452i \(-0.417316\pi\)
0.256847 + 0.966452i \(0.417316\pi\)
\(860\) 25.8885 0.882792
\(861\) −6.00000 −0.204479
\(862\) −17.8885 −0.609286
\(863\) 3.05573 0.104018 0.0520091 0.998647i \(-0.483438\pi\)
0.0520091 + 0.998647i \(0.483438\pi\)
\(864\) 1.00000 0.0340207
\(865\) −16.9443 −0.576123
\(866\) −1.41641 −0.0481315
\(867\) 3.00000 0.101885
\(868\) 6.47214 0.219679
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) −57.8885 −1.96148
\(872\) 14.9443 0.506077
\(873\) 8.47214 0.286738
\(874\) 6.47214 0.218923
\(875\) −12.0000 −0.405674
\(876\) −2.94427 −0.0994777
\(877\) 54.7214 1.84781 0.923905 0.382623i \(-0.124979\pi\)
0.923905 + 0.382623i \(0.124979\pi\)
\(878\) −16.3607 −0.552146
\(879\) −7.88854 −0.266074
\(880\) 0 0
\(881\) −22.3607 −0.753350 −0.376675 0.926345i \(-0.622933\pi\)
−0.376675 + 0.926345i \(0.622933\pi\)
\(882\) 1.00000 0.0336718
\(883\) −50.8328 −1.71066 −0.855330 0.518083i \(-0.826646\pi\)
−0.855330 + 0.518083i \(0.826646\pi\)
\(884\) −20.0000 −0.672673
\(885\) 8.00000 0.268917
\(886\) −18.8328 −0.632701
\(887\) −14.4721 −0.485927 −0.242963 0.970035i \(-0.578119\pi\)
−0.242963 + 0.970035i \(0.578119\pi\)
\(888\) −10.9443 −0.367266
\(889\) 3.05573 0.102486
\(890\) −34.8328 −1.16760
\(891\) 0 0
\(892\) 9.52786 0.319016
\(893\) −41.8885 −1.40175
\(894\) −18.9443 −0.633591
\(895\) −40.0000 −1.33705
\(896\) 1.00000 0.0334077
\(897\) −4.47214 −0.149320
\(898\) 18.0000 0.600668
\(899\) −12.9443 −0.431716
\(900\) −1.00000 −0.0333333
\(901\) −31.0557 −1.03462
\(902\) 0 0
\(903\) 12.9443 0.430758
\(904\) −1.05573 −0.0351130
\(905\) 4.00000 0.132964
\(906\) −12.9443 −0.430045
\(907\) −6.11146 −0.202928 −0.101464 0.994839i \(-0.532353\pi\)
−0.101464 + 0.994839i \(0.532353\pi\)
\(908\) 14.4721 0.480275
\(909\) 17.4164 0.577666
\(910\) 8.94427 0.296500
\(911\) 40.7214 1.34916 0.674579 0.738202i \(-0.264325\pi\)
0.674579 + 0.738202i \(0.264325\pi\)
\(912\) −6.47214 −0.214314
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) −12.0000 −0.396708
\(916\) −14.0000 −0.462573
\(917\) 12.0000 0.396275
\(918\) −4.47214 −0.147602
\(919\) 52.9443 1.74647 0.873235 0.487299i \(-0.162018\pi\)
0.873235 + 0.487299i \(0.162018\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 7.05573 0.232494
\(922\) 33.4164 1.10051
\(923\) −57.8885 −1.90542
\(924\) 0 0
\(925\) 10.9443 0.359845
\(926\) 41.8885 1.37654
\(927\) 12.9443 0.425146
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 12.9443 0.424459
\(931\) −6.47214 −0.212116
\(932\) −6.00000 −0.196537
\(933\) 6.47214 0.211888
\(934\) −29.3050 −0.958887
\(935\) 0 0
\(936\) 4.47214 0.146176
\(937\) −40.2492 −1.31488 −0.657442 0.753505i \(-0.728362\pi\)
−0.657442 + 0.753505i \(0.728362\pi\)
\(938\) −12.9443 −0.422645
\(939\) 13.4164 0.437828
\(940\) 12.9443 0.422196
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 14.9443 0.486911
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) −16.9443 −0.550615 −0.275307 0.961356i \(-0.588780\pi\)
−0.275307 + 0.961356i \(0.588780\pi\)
\(948\) 12.9443 0.420410
\(949\) −13.1672 −0.427425
\(950\) 6.47214 0.209984
\(951\) −30.9443 −1.00344
\(952\) −4.47214 −0.144943
\(953\) −26.9443 −0.872811 −0.436405 0.899750i \(-0.643749\pi\)
−0.436405 + 0.899750i \(0.643749\pi\)
\(954\) 6.94427 0.224829
\(955\) −41.8885 −1.35548
\(956\) 3.05573 0.0988293
\(957\) 0 0
\(958\) 12.9443 0.418210
\(959\) −14.0000 −0.452084
\(960\) 2.00000 0.0645497
\(961\) 10.8885 0.351243
\(962\) −48.9443 −1.57803
\(963\) −8.00000 −0.257796
\(964\) 11.5279 0.371288
\(965\) −47.7771 −1.53800
\(966\) −1.00000 −0.0321745
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −11.0000 −0.353553
\(969\) 28.9443 0.929824
\(970\) 16.9443 0.544048
\(971\) 3.41641 0.109638 0.0548189 0.998496i \(-0.482542\pi\)
0.0548189 + 0.998496i \(0.482542\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.94427 −0.286740
\(974\) 24.0000 0.769010
\(975\) −4.47214 −0.143223
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 8.94427 0.286006
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 14.9443 0.477134
\(982\) 23.0557 0.735738
\(983\) −57.8885 −1.84636 −0.923179 0.384371i \(-0.874419\pi\)
−0.923179 + 0.384371i \(0.874419\pi\)
\(984\) −6.00000 −0.191273
\(985\) 5.88854 0.187625
\(986\) 8.94427 0.284844
\(987\) 6.47214 0.206010
\(988\) −28.9443 −0.920840
\(989\) −12.9443 −0.411604
\(990\) 0 0
\(991\) 43.0557 1.36771 0.683855 0.729618i \(-0.260302\pi\)
0.683855 + 0.729618i \(0.260302\pi\)
\(992\) 6.47214 0.205491
\(993\) 21.8885 0.694612
\(994\) −12.9443 −0.410567
\(995\) 41.8885 1.32796
\(996\) −6.47214 −0.205077
\(997\) −1.63932 −0.0519178 −0.0259589 0.999663i \(-0.508264\pi\)
−0.0259589 + 0.999663i \(0.508264\pi\)
\(998\) 15.0557 0.476581
\(999\) −10.9443 −0.346261
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 966.2.a.p.1.2 2
3.2 odd 2 2898.2.a.v.1.2 2
4.3 odd 2 7728.2.a.bd.1.2 2
7.6 odd 2 6762.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.2 2 1.1 even 1 trivial
2898.2.a.v.1.2 2 3.2 odd 2
6762.2.a.bz.1.1 2 7.6 odd 2
7728.2.a.bd.1.2 2 4.3 odd 2