Properties

Label 6762.2.a.bz.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
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: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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 \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 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 \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) −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 \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.47214 0.766965
\(35\) 0 0
\(36\) 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 \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 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 \(-0.656478\pi\)
−0.472029 + 0.881583i \(0.656478\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) −6.47214 −0.857255
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −6.47214 −0.821962
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) −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 \(-0.444880\pi\)
0.172300 + 0.985044i \(0.444880\pi\)
\(74\) −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 \(-0.384411\pi\)
0.355205 + 0.934789i \(0.384411\pi\)
\(84\) 0 0
\(85\) −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 \(-0.125673\pi\)
0.923068 + 0.384637i \(0.125673\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −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 \(-0.641524\pi\)
−0.430108 + 0.902778i \(0.641524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −17.4164 −1.73300 −0.866499 0.499179i \(-0.833635\pi\)
−0.866499 + 0.499179i \(0.833635\pi\)
\(102\) −4.47214 −0.442807
\(103\) −12.9443 −1.27544 −0.637719 0.770270i \(-0.720122\pi\)
−0.637719 + 0.770270i \(0.720122\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) 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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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
\(126\) 0 0
\(127\) 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 \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −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 \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) −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 \(-0.703380\pi\)
−0.596341 + 0.802731i \(0.703380\pi\)
\(158\) 12.9443 1.02979
\(159\) −6.94427 −0.550717
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 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 \(-0.645629\pi\)
−0.441714 + 0.897156i \(0.645629\pi\)
\(168\) 0 0
\(169\) 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 \(-0.395624\pi\)
0.322062 + 0.946718i \(0.395624\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 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 \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) −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
\(196\) 0 0
\(197\) 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 \(-0.766289\pi\)
−0.742350 + 0.670012i \(0.766289\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.9443 0.913019
\(202\) −17.4164 −1.22541
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) 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
\(217\) 0 0
\(218\) 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 \(-0.603353\pi\)
−0.319016 + 0.947749i \(0.603353\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −1.05573 −0.0702260
\(227\) −14.4721 −0.960549 −0.480275 0.877118i \(-0.659463\pi\)
−0.480275 + 0.877118i \(0.659463\pi\)
\(228\) −6.47214 −0.428628
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 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
\(238\) 0 0
\(239\) 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 \(-0.621084\pi\)
−0.371288 + 0.928518i \(0.621084\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −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 \(-0.349018\pi\)
0.456737 + 0.889602i \(0.349018\pi\)
\(252\) 0 0
\(253\) 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 \(-0.654341\pi\)
−0.466099 + 0.884733i \(0.654341\pi\)
\(258\) −12.9443 −0.805875
\(259\) 0 0
\(260\) 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
\(266\) 0 0
\(267\) −17.4164 −1.06587
\(268\) −12.9443 −0.790697
\(269\) −7.52786 −0.458982 −0.229491 0.973311i \(-0.573706\pi\)
−0.229491 + 0.973311i \(0.573706\pi\)
\(270\) 2.00000 0.121716
\(271\) 11.4164 0.693497 0.346749 0.937958i \(-0.387286\pi\)
0.346749 + 0.937958i \(0.387286\pi\)
\(272\) 4.47214 0.271163
\(273\) 0 0
\(274\) −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
\(280\) 0 0
\(281\) 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 \(-0.591391\pi\)
−0.283186 + 0.959065i \(0.591391\pi\)
\(284\) −12.9443 −0.768101
\(285\) 12.9443 0.766752
\(286\) 0 0
\(287\) 0 0
\(288\) 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 \(-0.425988\pi\)
0.230427 + 0.973090i \(0.425988\pi\)
\(294\) 0 0
\(295\) 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
\(301\) 0 0
\(302\) −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 \(-0.564532\pi\)
−0.201346 + 0.979520i \(0.564532\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) 12.9443 0.735185
\(311\) −6.47214 −0.367001 −0.183501 0.983020i \(-0.558743\pi\)
−0.183501 + 0.983020i \(0.558743\pi\)
\(312\) 4.47214 0.253185
\(313\) −13.4164 −0.758340 −0.379170 0.925327i \(-0.623790\pi\)
−0.379170 + 0.925327i \(0.623790\pi\)
\(314\) −14.9443 −0.843354
\(315\) 0 0
\(316\) 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
\(322\) 0 0
\(323\) 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
\(329\) 0 0
\(330\) 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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) 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 \(-0.636426\pi\)
−0.415594 + 0.909550i \(0.636426\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 4.00000 0.212598
\(355\) 25.8885 1.37402
\(356\) 17.4164 0.923068
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) −5.88854 −0.308220
\(366\) −6.00000 −0.313625
\(367\) 22.8328 1.19186 0.595932 0.803035i \(-0.296783\pi\)
0.595932 + 0.803035i \(0.296783\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) 21.8885 1.13793
\(371\) 0 0
\(372\) 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
\(378\) 0 0
\(379\) 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 \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −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
\(392\) 0 0
\(393\) 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 \(-0.257262\pi\)
0.690792 + 0.723054i \(0.257262\pi\)
\(398\) −20.9443 −1.04984
\(399\) 0 0
\(400\) −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
\(406\) 0 0
\(407\) 0 0
\(408\) −4.47214 −0.221404
\(409\) −19.8885 −0.983425 −0.491713 0.870758i \(-0.663629\pi\)
−0.491713 + 0.870758i \(0.663629\pi\)
\(410\) −12.0000 −0.592638
\(411\) 14.0000 0.690569
\(412\) −12.9443 −0.637719
\(413\) 0 0
\(414\) −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 \(-0.657290\pi\)
−0.474277 + 0.880376i \(0.657290\pi\)
\(420\) 0 0
\(421\) −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
\(427\) 0 0
\(428\) −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 \(-0.489165\pi\)
0.0340341 + 0.999421i \(0.489165\pi\)
\(434\) 0 0
\(435\) −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 \(-0.372328\pi\)
0.390426 + 0.920634i \(0.372328\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) −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 \(-0.783856\pi\)
−0.778179 + 0.628043i \(0.783856\pi\)
\(462\) 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 \(-0.262831\pi\)
0.678036 + 0.735029i \(0.262831\pi\)
\(468\) −4.47214 −0.206725
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) 6.94427 0.317956
\(478\) 3.05573 0.139766
\(479\) −12.9443 −0.591439 −0.295719 0.955275i \(-0.595559\pi\)
−0.295719 + 0.955275i \(0.595559\pi\)
\(480\) 2.00000 0.0912871
\(481\) 48.9443 2.23167
\(482\) −11.5279 −0.525080
\(483\) 0 0
\(484\) −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
\(490\) 0 0
\(491\) 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
\(497\) 0 0
\(498\) −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 \(-0.695838\pi\)
−0.577157 + 0.816634i \(0.695838\pi\)
\(504\) 0 0
\(505\) 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 \(-0.765448\pi\)
−0.740578 + 0.671970i \(0.765448\pi\)
\(510\) 8.94427 0.396059
\(511\) 0 0
\(512\) 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
\(518\) 0 0
\(519\) −8.47214 −0.371885
\(520\) 8.94427 0.392232
\(521\) −6.58359 −0.288432 −0.144216 0.989546i \(-0.546066\pi\)
−0.144216 + 0.989546i \(0.546066\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −37.3050 −1.63123 −0.815616 0.578594i \(-0.803602\pi\)
−0.815616 + 0.578594i \(0.803602\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −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
\(532\) 0 0
\(533\) −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
\(546\) 0 0
\(547\) 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
\(553\) 0 0
\(554\) 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
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −17.5279 −0.738711 −0.369356 0.929288i \(-0.620422\pi\)
−0.369356 + 0.929288i \(0.620422\pi\)
\(564\) 6.47214 0.272526
\(565\) 2.11146 0.0888296
\(566\) −9.52786 −0.400486
\(567\) 0 0
\(568\) −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
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −14.9443 −0.622138 −0.311069 0.950387i \(-0.600687\pi\)
−0.311069 + 0.950387i \(0.600687\pi\)
\(578\) 3.00000 0.124784
\(579\) 23.8885 0.992774
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) 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 \(-0.613711\pi\)
−0.349682 + 0.936868i \(0.613711\pi\)
\(588\) 0 0
\(589\) −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 \(-0.670310\pi\)
−0.509881 + 0.860245i \(0.670310\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −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 \(-0.415707\pi\)
0.261731 + 0.965141i \(0.415707\pi\)
\(602\) 0 0
\(603\) −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 \(-0.607733\pi\)
−0.332030 + 0.943269i \(0.607733\pi\)
\(608\) 6.47214 0.262480
\(609\) 0 0
\(610\) −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 \(-0.328525\pi\)
0.513026 + 0.858373i \(0.328525\pi\)
\(620\) 12.9443 0.519854
\(621\) 1.00000 0.0401286
\(622\) −6.47214 −0.259509
\(623\) 0 0
\(624\) 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
\(630\) 0 0
\(631\) −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
\(637\) 0 0
\(638\) 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 \(-0.490409\pi\)
0.0301265 + 0.999546i \(0.490409\pi\)
\(644\) 0 0
\(645\) 25.8885 1.01936
\(646\) 28.9443 1.13880
\(647\) 32.3607 1.27223 0.636115 0.771594i \(-0.280540\pi\)
0.636115 + 0.771594i \(0.280540\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.47214 0.175412
\(651\) 0 0
\(652\) 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
\(658\) 0 0
\(659\) −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 \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 21.8885 0.850722
\(663\) 20.0000 0.776736
\(664\) 6.47214 0.251168
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) −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 \(-0.221986\pi\)
0.766521 + 0.642220i \(0.221986\pi\)
\(678\) 1.05573 0.0405450
\(679\) 0 0
\(680\) −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
\(686\) 0 0
\(687\) −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 \(-0.284434\pi\)
0.626630 + 0.779317i \(0.284434\pi\)
\(692\) 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
\(700\) 0 0
\(701\) 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
\(707\) 0 0
\(708\) 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
\(714\) 0 0
\(715\) 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 \(-0.413023\pi\)
0.269860 + 0.962900i \(0.413023\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 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 \(-0.626969\pi\)
−0.388390 + 0.921495i \(0.626969\pi\)
\(728\) 0 0
\(729\) 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 \(-0.290980\pi\)
0.610471 + 0.792039i \(0.290980\pi\)
\(734\) 22.8328 0.842775
\(735\) 0 0
\(736\) −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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) −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
\(756\) 0 0
\(757\) −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 \(-0.749441\pi\)
−0.705864 + 0.708347i \(0.749441\pi\)
\(762\) −3.05573 −0.110697
\(763\) 0 0
\(764\) −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 \(-0.416630\pi\)
0.258930 + 0.965896i \(0.416630\pi\)
\(770\) 0 0
\(771\) 14.9443 0.538205
\(772\) −23.8885 −0.859768
\(773\) −22.9443 −0.825248 −0.412624 0.910901i \(-0.635388\pi\)
−0.412624 + 0.910901i \(0.635388\pi\)
\(774\) 12.9443 0.465272
\(775\) 6.47214 0.232486
\(776\) −8.47214 −0.304132
\(777\) 0 0
\(778\) −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
\(784\) 0 0
\(785\) 29.8885 1.06677
\(786\) 12.0000 0.428026
\(787\) 27.4164 0.977289 0.488645 0.872483i \(-0.337491\pi\)
0.488645 + 0.872483i \(0.337491\pi\)
\(788\) 2.94427 0.104885
\(789\) 24.0000 0.854423
\(790\) −25.8885 −0.921073
\(791\) 0 0
\(792\) 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 \(-0.644955\pi\)
−0.439812 + 0.898090i \(0.644955\pi\)
\(798\) 0 0
\(799\) −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
\(805\) 0 0
\(806\) 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 \(-0.488197\pi\)
0.0370716 + 0.999313i \(0.488197\pi\)
\(812\) 0 0
\(813\) −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
\(819\) 0 0
\(820\) −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
\(826\) 0 0
\(827\) −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 \(-0.425151\pi\)
0.232986 + 0.972480i \(0.425151\pi\)
\(830\) −12.9443 −0.449302
\(831\) −23.8885 −0.828684
\(832\) −4.47214 −0.155043
\(833\) 0 0
\(834\) −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 \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −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
\(847\) 0 0
\(848\) 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 \(-0.728058\pi\)
−0.656722 + 0.754132i \(0.728058\pi\)
\(854\) 0 0
\(855\) −12.9443 −0.442685
\(856\) −8.00000 −0.273434
\(857\) 12.1115 0.413719 0.206860 0.978371i \(-0.433676\pi\)
0.206860 + 0.978371i \(0.433676\pi\)
\(858\) 0 0
\(859\) −15.0557 −0.513695 −0.256847 0.966452i \(-0.582684\pi\)
−0.256847 + 0.966452i \(0.582684\pi\)
\(860\) −25.8885 −0.882792
\(861\) 0 0
\(862\) −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
\(868\) 0 0
\(869\) 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
\(875\) 0 0
\(876\) −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 \(-0.377067\pi\)
0.376675 + 0.926345i \(0.377067\pi\)
\(882\) 0 0
\(883\) −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 \(-0.421881\pi\)
0.242963 + 0.970035i \(0.421881\pi\)
\(888\) 10.9443 0.367266
\(889\) 0 0
\(890\) −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
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) −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
\(910\) 0 0
\(911\) 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
\(917\) 0 0
\(918\) −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 \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −12.9443 −0.424459
\(931\) 0 0
\(932\) −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 \(-0.271638\pi\)
0.657442 + 0.753505i \(0.271638\pi\)
\(938\) 0 0
\(939\) 13.4164 0.437828
\(940\) 12.9443 0.422196
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 14.9443 0.486911
\(943\) −6.00000 −0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) −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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) −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 \(-0.517458\pi\)
−0.0548189 + 0.998496i \(0.517458\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) 14.9443 0.477134
\(982\) 23.0557 0.735738
\(983\) 57.8885 1.84636 0.923179 0.384371i \(-0.125581\pi\)
0.923179 + 0.384371i \(0.125581\pi\)
\(984\) −6.00000 −0.191273
\(985\) −5.88854 −0.187625
\(986\) −8.94427 −0.284844
\(987\) 0 0
\(988\) −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
\(994\) 0 0
\(995\) 41.8885 1.32796
\(996\) −6.47214 −0.205077
\(997\) 1.63932 0.0519178 0.0259589 0.999663i \(-0.491736\pi\)
0.0259589 + 0.999663i \(0.491736\pi\)
\(998\) 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 6762.2.a.bz.1.1 2
7.6 odd 2 966.2.a.p.1.2 2
21.20 even 2 2898.2.a.v.1.2 2
28.27 even 2 7728.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.2 2 7.6 odd 2
2898.2.a.v.1.2 2 21.20 even 2
6762.2.a.bz.1.1 2 1.1 even 1 trivial
7728.2.a.bd.1.2 2 28.27 even 2