Properties

Label 2898.2.a.v.1.2
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
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: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} +4.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.47214 q^{17} -6.47214 q^{19} -2.00000 q^{20} +1.00000 q^{23} -1.00000 q^{25} -4.47214 q^{26} +1.00000 q^{28} +2.00000 q^{29} +6.47214 q^{31} -1.00000 q^{32} -4.47214 q^{34} -2.00000 q^{35} -10.9443 q^{37} +6.47214 q^{38} +2.00000 q^{40} +6.00000 q^{41} +12.9443 q^{43} -1.00000 q^{46} -6.47214 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.47214 q^{52} -6.94427 q^{53} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -6.00000 q^{61} -6.47214 q^{62} +1.00000 q^{64} -8.94427 q^{65} -12.9443 q^{67} +4.47214 q^{68} +2.00000 q^{70} +12.9443 q^{71} -2.94427 q^{73} +10.9443 q^{74} -6.47214 q^{76} +12.9443 q^{79} -2.00000 q^{80} -6.00000 q^{82} +6.47214 q^{83} -8.94427 q^{85} -12.9443 q^{86} +17.4164 q^{89} +4.47214 q^{91} +1.00000 q^{92} +6.47214 q^{94} +12.9443 q^{95} +8.47214 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{20} + 2 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} + 4 q^{31} - 2 q^{32} - 4 q^{35} - 4 q^{37} + 4 q^{38} + 4 q^{40} + 12 q^{41} + 8 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{53} - 2 q^{56} - 4 q^{58} - 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 4 q^{70} + 8 q^{71} + 12 q^{73} + 4 q^{74} - 4 q^{76} + 8 q^{79} - 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{86} + 8 q^{89} + 2 q^{92} + 4 q^{94} + 8 q^{95} + 8 q^{97} - 2 q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −4.47214 −0.877058
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 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
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 6.47214 1.04992
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −6.47214 −0.944058 −0.472029 0.881583i \(-0.656478\pi\)
−0.472029 + 0.881583i \(0.656478\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.47214 0.620174
\(53\) −6.94427 −0.953869 −0.476935 0.878939i \(-0.658252\pi\)
−0.476935 + 0.878939i \(0.658252\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −6.47214 −0.821962
\(63\) 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
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 12.9443 1.53620 0.768101 0.640328i \(-0.221202\pi\)
0.768101 + 0.640328i \(0.221202\pi\)
\(72\) 0 0
\(73\) −2.94427 −0.344601 −0.172300 0.985044i \(-0.555120\pi\)
−0.172300 + 0.985044i \(0.555120\pi\)
\(74\) 10.9443 1.27225
\(75\) 0 0
\(76\) −6.47214 −0.742405
\(77\) 0 0
\(78\) 0 0
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) 0 0
\(89\) 17.4164 1.84614 0.923068 0.384637i \(-0.125673\pi\)
0.923068 + 0.384637i \(0.125673\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 6.47214 0.667550
\(95\) 12.9443 1.32805
\(96\) 0 0
\(97\) 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 \(-0.833635\pi\)
−0.866499 + 0.499179i \(0.833635\pi\)
\(102\) 0 0
\(103\) 12.9443 1.27544 0.637719 0.770270i \(-0.279878\pi\)
0.637719 + 0.770270i \(0.279878\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) 6.94427 0.674487
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 14.9443 1.43140 0.715701 0.698407i \(-0.246107\pi\)
0.715701 + 0.698407i \(0.246107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 1.05573 0.0993145 0.0496573 0.998766i \(-0.484187\pi\)
0.0496573 + 0.998766i \(0.484187\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 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
\(133\) −6.47214 −0.561205
\(134\) 12.9443 1.11821
\(135\) 0 0
\(136\) −4.47214 −0.383482
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −12.9443 −1.08626
\(143\) 0 0
\(144\) 0 0
\(145\) −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 \(-0.217252\pi\)
0.775988 + 0.630748i \(0.217252\pi\)
\(150\) 0 0
\(151\) −12.9443 −1.05339 −0.526695 0.850054i \(-0.676569\pi\)
−0.526695 + 0.850054i \(0.676569\pi\)
\(152\) 6.47214 0.524960
\(153\) 0 0
\(154\) 0 0
\(155\) −12.9443 −1.03971
\(156\) 0 0
\(157\) 14.9443 1.19268 0.596341 0.802731i \(-0.296620\pi\)
0.596341 + 0.802731i \(0.296620\pi\)
\(158\) −12.9443 −1.02979
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 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
\(171\) 0 0
\(172\) 12.9443 0.986991
\(173\) 8.47214 0.644125 0.322062 0.946718i \(-0.395624\pi\)
0.322062 + 0.946718i \(0.395624\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −17.4164 −1.30541
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.47214 −0.331497
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 21.8885 1.60928
\(186\) 0 0
\(187\) 0 0
\(188\) −6.47214 −0.472029
\(189\) 0 0
\(190\) −12.9443 −0.939076
\(191\) 20.9443 1.51547 0.757737 0.652560i \(-0.226305\pi\)
0.757737 + 0.652560i \(0.226305\pi\)
\(192\) 0 0
\(193\) −23.8885 −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(194\) −8.47214 −0.608264
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.94427 −0.209771 −0.104885 0.994484i \(-0.533448\pi\)
−0.104885 + 0.994484i \(0.533448\pi\)
\(198\) 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
\(201\) 0 0
\(202\) 17.4164 1.22541
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) −12.9443 −0.901870
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −25.8885 −1.76558
\(216\) 0 0
\(217\) 6.47214 0.439357
\(218\) −14.9443 −1.01215
\(219\) 0 0
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) 9.52786 0.638033 0.319016 0.947749i \(-0.396647\pi\)
0.319016 + 0.947749i \(0.396647\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −1.05573 −0.0702260
\(227\) −14.4721 −0.960549 −0.480275 0.877118i \(-0.659463\pi\)
−0.480275 + 0.877118i \(0.659463\pi\)
\(228\) 0 0
\(229\) −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 \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 12.9443 0.844391
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −4.47214 −0.289886
\(239\) −3.05573 −0.197659 −0.0988293 0.995104i \(-0.531510\pi\)
−0.0988293 + 0.995104i \(0.531510\pi\)
\(240\) 0 0
\(241\) 11.5279 0.742575 0.371288 0.928518i \(-0.378916\pi\)
0.371288 + 0.928518i \(0.378916\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −28.9443 −1.84168
\(248\) −6.47214 −0.410981
\(249\) 0 0
\(250\) −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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.9443 −0.932198 −0.466099 0.884733i \(-0.654341\pi\)
−0.466099 + 0.884733i \(0.654341\pi\)
\(258\) 0 0
\(259\) −10.9443 −0.680044
\(260\) −8.94427 −0.554700
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 13.8885 0.853166
\(266\) 6.47214 0.396832
\(267\) 0 0
\(268\) −12.9443 −0.790697
\(269\) −7.52786 −0.458982 −0.229491 0.973311i \(-0.573706\pi\)
−0.229491 + 0.973311i \(0.573706\pi\)
\(270\) 0 0
\(271\) −11.4164 −0.693497 −0.346749 0.937958i \(-0.612714\pi\)
−0.346749 + 0.937958i \(0.612714\pi\)
\(272\) 4.47214 0.271163
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 23.8885 1.43532 0.717662 0.696392i \(-0.245212\pi\)
0.717662 + 0.696392i \(0.245212\pi\)
\(278\) 8.94427 0.536442
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 9.52786 0.566373 0.283186 0.959065i \(-0.408609\pi\)
0.283186 + 0.959065i \(0.408609\pi\)
\(284\) 12.9443 0.768101
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −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
\(300\) 0 0
\(301\) 12.9443 0.746095
\(302\) 12.9443 0.744859
\(303\) 0 0
\(304\) −6.47214 −0.371202
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 7.05573 0.402692 0.201346 0.979520i \(-0.435468\pi\)
0.201346 + 0.979520i \(0.435468\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.9443 0.735185
\(311\) −6.47214 −0.367001 −0.183501 0.983020i \(-0.558743\pi\)
−0.183501 + 0.983020i \(0.558743\pi\)
\(312\) 0 0
\(313\) 13.4164 0.758340 0.379170 0.925327i \(-0.376210\pi\)
0.379170 + 0.925327i \(0.376210\pi\)
\(314\) −14.9443 −0.843354
\(315\) 0 0
\(316\) 12.9443 0.728172
\(317\) 30.9443 1.73800 0.869002 0.494809i \(-0.164762\pi\)
0.869002 + 0.494809i \(0.164762\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −28.9443 −1.61050
\(324\) 0 0
\(325\) −4.47214 −0.248069
\(326\) −8.94427 −0.495377
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) −8.94427 −0.485071
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −12.9443 −0.697908
\(345\) 0 0
\(346\) −8.47214 −0.455465
\(347\) −16.9443 −0.909616 −0.454808 0.890589i \(-0.650292\pi\)
−0.454808 + 0.890589i \(0.650292\pi\)
\(348\) 0 0
\(349\) 15.5279 0.831188 0.415594 0.909550i \(-0.363574\pi\)
0.415594 + 0.909550i \(0.363574\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) −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 \(-0.705843\pi\)
−0.602535 + 0.798092i \(0.705843\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 4.47214 0.234404
\(365\) 5.88854 0.308220
\(366\) 0 0
\(367\) −22.8328 −1.19186 −0.595932 0.803035i \(-0.703217\pi\)
−0.595932 + 0.803035i \(0.703217\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) −21.8885 −1.13793
\(371\) −6.94427 −0.360529
\(372\) 0 0
\(373\) −10.9443 −0.566673 −0.283336 0.959021i \(-0.591441\pi\)
−0.283336 + 0.959021i \(0.591441\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) −20.9443 −1.07160
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.8885 1.21589
\(387\) 0 0
\(388\) 8.47214 0.430108
\(389\) 1.05573 0.0535275 0.0267638 0.999642i \(-0.491480\pi\)
0.0267638 + 0.999642i \(0.491480\pi\)
\(390\) 0 0
\(391\) 4.47214 0.226166
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 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
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 28.9443 1.44182
\(404\) −17.4164 −0.866499
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) 19.8885 0.983425 0.491713 0.870758i \(-0.336371\pi\)
0.491713 + 0.870758i \(0.336371\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) 12.9443 0.637719
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −12.9443 −0.635409
\(416\) −4.47214 −0.219265
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 6.94427 0.337244
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) −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 \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(432\) 0 0
\(433\) −1.41641 −0.0680682 −0.0340341 0.999421i \(-0.510835\pi\)
−0.0340341 + 0.999421i \(0.510835\pi\)
\(434\) −6.47214 −0.310672
\(435\) 0 0
\(436\) 14.9443 0.715701
\(437\) −6.47214 −0.309604
\(438\) 0 0
\(439\) −16.3607 −0.780853 −0.390426 0.920634i \(-0.627672\pi\)
−0.390426 + 0.920634i \(0.627672\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) 18.8328 0.894774 0.447387 0.894340i \(-0.352355\pi\)
0.447387 + 0.894340i \(0.352355\pi\)
\(444\) 0 0
\(445\) −34.8328 −1.65123
\(446\) −9.52786 −0.451157
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.05573 0.0496573
\(453\) 0 0
\(454\) 14.4721 0.679211
\(455\) −8.94427 −0.419314
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 29.3050 1.35607 0.678036 0.735029i \(-0.262831\pi\)
0.678036 + 0.735029i \(0.262831\pi\)
\(468\) 0 0
\(469\) −12.9443 −0.597711
\(470\) −12.9443 −0.597075
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) 6.47214 0.296962
\(476\) 4.47214 0.204980
\(477\) 0 0
\(478\) 3.05573 0.139766
\(479\) −12.9443 −0.591439 −0.295719 0.955275i \(-0.595559\pi\)
−0.295719 + 0.955275i \(0.595559\pi\)
\(480\) 0 0
\(481\) −48.9443 −2.23167
\(482\) −11.5279 −0.525080
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −16.9443 −0.769400
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −23.0557 −1.04049 −0.520245 0.854017i \(-0.674159\pi\)
−0.520245 + 0.854017i \(0.674159\pi\)
\(492\) 0 0
\(493\) 8.94427 0.402830
\(494\) 28.9443 1.30226
\(495\) 0 0
\(496\) 6.47214 0.290607
\(497\) 12.9443 0.580630
\(498\) 0 0
\(499\) 15.0557 0.673987 0.336993 0.941507i \(-0.390590\pi\)
0.336993 + 0.941507i \(0.390590\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) 3.05573 0.135576
\(509\) −33.4164 −1.48116 −0.740578 0.671970i \(-0.765448\pi\)
−0.740578 + 0.671970i \(0.765448\pi\)
\(510\) 0 0
\(511\) −2.94427 −0.130247
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.9443 0.659164
\(515\) −25.8885 −1.14079
\(516\) 0 0
\(517\) 0 0
\(518\) 10.9443 0.480864
\(519\) 0 0
\(520\) 8.94427 0.392232
\(521\) −6.58359 −0.288432 −0.144216 0.989546i \(-0.546066\pi\)
−0.144216 + 0.989546i \(0.546066\pi\)
\(522\) 0 0
\(523\) 37.3050 1.63123 0.815616 0.578594i \(-0.196398\pi\)
0.815616 + 0.578594i \(0.196398\pi\)
\(524\) −12.0000 −0.524222
\(525\) 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
\(531\) 0 0
\(532\) −6.47214 −0.280603
\(533\) 26.8328 1.16226
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 12.9443 0.559107
\(537\) 0 0
\(538\) 7.52786 0.324549
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 11.4164 0.490377
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 0 0
\(551\) −12.9443 −0.551445
\(552\) 0 0
\(553\) 12.9443 0.550446
\(554\) −23.8885 −1.01493
\(555\) 0 0
\(556\) −8.94427 −0.379322
\(557\) 34.9443 1.48064 0.740318 0.672257i \(-0.234675\pi\)
0.740318 + 0.672257i \(0.234675\pi\)
\(558\) 0 0
\(559\) 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 \(-0.620422\pi\)
−0.369356 + 0.929288i \(0.620422\pi\)
\(564\) 0 0
\(565\) −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 \(-0.698738\pi\)
−0.584574 + 0.811340i \(0.698738\pi\)
\(570\) 0 0
\(571\) 4.94427 0.206911 0.103456 0.994634i \(-0.467010\pi\)
0.103456 + 0.994634i \(0.467010\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 14.9443 0.622138 0.311069 0.950387i \(-0.399313\pi\)
0.311069 + 0.950387i \(0.399313\pi\)
\(578\) −3.00000 −0.124784
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 6.47214 0.268509
\(582\) 0 0
\(583\) 0 0
\(584\) 2.94427 0.121835
\(585\) 0 0
\(586\) −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
\(591\) 0 0
\(592\) −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
\(595\) −8.94427 −0.366679
\(596\) 18.9443 0.775988
\(597\) 0 0
\(598\) −4.47214 −0.182879
\(599\) 1.88854 0.0771638 0.0385819 0.999255i \(-0.487716\pi\)
0.0385819 + 0.999255i \(0.487716\pi\)
\(600\) 0 0
\(601\) −12.8328 −0.523461 −0.261731 0.965141i \(-0.584293\pi\)
−0.261731 + 0.965141i \(0.584293\pi\)
\(602\) −12.9443 −0.527569
\(603\) 0 0
\(604\) −12.9443 −0.526695
\(605\) 22.0000 0.894427
\(606\) 0 0
\(607\) 16.3607 0.664060 0.332030 0.943269i \(-0.392267\pi\)
0.332030 + 0.943269i \(0.392267\pi\)
\(608\) 6.47214 0.262480
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) −28.9443 −1.17096
\(612\) 0 0
\(613\) −12.8328 −0.518313 −0.259156 0.965835i \(-0.583444\pi\)
−0.259156 + 0.965835i \(0.583444\pi\)
\(614\) −7.05573 −0.284746
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −25.5279 −1.02605 −0.513026 0.858373i \(-0.671475\pi\)
−0.513026 + 0.858373i \(0.671475\pi\)
\(620\) −12.9443 −0.519854
\(621\) 0 0
\(622\) 6.47214 0.259509
\(623\) 17.4164 0.697774
\(624\) 0 0
\(625\) −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
\(633\) 0 0
\(634\) −30.9443 −1.22895
\(635\) −6.11146 −0.242526
\(636\) 0 0
\(637\) 4.47214 0.177192
\(638\) 0 0
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) −1.52786 −0.0602531 −0.0301265 0.999546i \(-0.509591\pi\)
−0.0301265 + 0.999546i \(0.509591\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 28.9443 1.13880
\(647\) 32.3607 1.27223 0.636115 0.771594i \(-0.280540\pi\)
0.636115 + 0.771594i \(0.280540\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.47214 0.175412
\(651\) 0 0
\(652\) 8.94427 0.350285
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 6.47214 0.252310
\(659\) 6.83282 0.266169 0.133084 0.991105i \(-0.457512\pi\)
0.133084 + 0.991105i \(0.457512\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −6.47214 −0.251168
\(665\) 12.9443 0.501957
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) −11.4164 −0.441714
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) 39.8885 1.53304 0.766521 0.642220i \(-0.221986\pi\)
0.766521 + 0.642220i \(0.221986\pi\)
\(678\) 0 0
\(679\) 8.47214 0.325131
\(680\) 8.94427 0.342997
\(681\) 0 0
\(682\) 0 0
\(683\) 0.944272 0.0361316 0.0180658 0.999837i \(-0.494249\pi\)
0.0180658 + 0.999837i \(0.494249\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 12.9443 0.493496
\(689\) −31.0557 −1.18313
\(690\) 0 0
\(691\) −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
\(696\) 0 0
\(697\) 26.8328 1.01637
\(698\) −15.5279 −0.587738
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −32.8328 −1.24008 −0.620039 0.784571i \(-0.712883\pi\)
−0.620039 + 0.784571i \(0.712883\pi\)
\(702\) 0 0
\(703\) 70.8328 2.67151
\(704\) 0 0
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −17.4164 −0.655011
\(708\) 0 0
\(709\) −10.9443 −0.411021 −0.205510 0.978655i \(-0.565885\pi\)
−0.205510 + 0.978655i \(0.565885\pi\)
\(710\) 25.8885 0.971580
\(711\) 0 0
\(712\) −17.4164 −0.652707
\(713\) 6.47214 0.242383
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 22.8328 0.852113
\(719\) 14.4721 0.539720 0.269860 0.962900i \(-0.413023\pi\)
0.269860 + 0.962900i \(0.413023\pi\)
\(720\) 0 0
\(721\) 12.9443 0.482070
\(722\) −22.8885 −0.851823
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 20.9443 0.776780 0.388390 0.921495i \(-0.373031\pi\)
0.388390 + 0.921495i \(0.373031\pi\)
\(728\) −4.47214 −0.165748
\(729\) 0 0
\(730\) −5.88854 −0.217945
\(731\) 57.8885 2.14109
\(732\) 0 0
\(733\) −33.0557 −1.22094 −0.610471 0.792039i \(-0.709020\pi\)
−0.610471 + 0.792039i \(0.709020\pi\)
\(734\) 22.8328 0.842775
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) 8.94427 0.329020 0.164510 0.986375i \(-0.447396\pi\)
0.164510 + 0.986375i \(0.447396\pi\)
\(740\) 21.8885 0.804639
\(741\) 0 0
\(742\) 6.94427 0.254932
\(743\) −30.8328 −1.13115 −0.565573 0.824698i \(-0.691345\pi\)
−0.565573 + 0.824698i \(0.691345\pi\)
\(744\) 0 0
\(745\) −37.8885 −1.38813
\(746\) 10.9443 0.400698
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −17.8885 −0.652762 −0.326381 0.945238i \(-0.605829\pi\)
−0.326381 + 0.945238i \(0.605829\pi\)
\(752\) −6.47214 −0.236015
\(753\) 0 0
\(754\) −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 \(0.749441\pi\)
\(762\) 0 0
\(763\) 14.9443 0.541019
\(764\) 20.9443 0.757737
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −17.8885 −0.645918
\(768\) 0 0
\(769\) −14.3607 −0.517859 −0.258930 0.965896i \(-0.583370\pi\)
−0.258930 + 0.965896i \(0.583370\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −23.8885 −0.859768
\(773\) −22.9443 −0.825248 −0.412624 0.910901i \(-0.635388\pi\)
−0.412624 + 0.910901i \(0.635388\pi\)
\(774\) 0 0
\(775\) −6.47214 −0.232486
\(776\) −8.47214 −0.304132
\(777\) 0 0
\(778\) −1.05573 −0.0378497
\(779\) −38.8328 −1.39133
\(780\) 0 0
\(781\) 0 0
\(782\) −4.47214 −0.159923
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −29.8885 −1.06677
\(786\) 0 0
\(787\) −27.4164 −0.977289 −0.488645 0.872483i \(-0.662509\pi\)
−0.488645 + 0.872483i \(0.662509\pi\)
\(788\) −2.94427 −0.104885
\(789\) 0 0
\(790\) 25.8885 0.921073
\(791\) 1.05573 0.0375374
\(792\) 0 0
\(793\) −26.8328 −0.952861
\(794\) 27.5279 0.976927
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) −28.9443 −1.01952
\(807\) 0 0
\(808\) 17.4164 0.612707
\(809\) −0.111456 −0.00391859 −0.00195930 0.999998i \(-0.500624\pi\)
−0.00195930 + 0.999998i \(0.500624\pi\)
\(810\) 0 0
\(811\) −2.11146 −0.0741433 −0.0370716 0.999313i \(-0.511803\pi\)
−0.0370716 + 0.999313i \(0.511803\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −17.8885 −0.626608
\(816\) 0 0
\(817\) −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 \(-0.722201\pi\)
−0.642737 + 0.766087i \(0.722201\pi\)
\(822\) 0 0
\(823\) 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 \(-0.489546\pi\)
0.0328356 + 0.999461i \(0.489546\pi\)
\(828\) 0 0
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 12.9443 0.449302
\(831\) 0 0
\(832\) 4.47214 0.155043
\(833\) 4.47214 0.154950
\(834\) 0 0
\(835\) 22.8328 0.790162
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 16.9443 0.583246
\(845\) −14.0000 −0.481615
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) −6.94427 −0.238467
\(849\) 0 0
\(850\) 4.47214 0.153393
\(851\) −10.9443 −0.375165
\(852\) 0 0
\(853\) 38.3607 1.31344 0.656722 0.754132i \(-0.271942\pi\)
0.656722 + 0.754132i \(0.271942\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −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 \(-0.417316\pi\)
0.256847 + 0.966452i \(0.417316\pi\)
\(860\) −25.8885 −0.882792
\(861\) 0 0
\(862\) −17.8885 −0.609286
\(863\) −3.05573 −0.104018 −0.0520091 0.998647i \(-0.516562\pi\)
−0.0520091 + 0.998647i \(0.516562\pi\)
\(864\) 0 0
\(865\) −16.9443 −0.576123
\(866\) 1.41641 0.0481315
\(867\) 0 0
\(868\) 6.47214 0.219679
\(869\) 0 0
\(870\) 0 0
\(871\) −57.8885 −1.96148
\(872\) −14.9443 −0.506077
\(873\) 0 0
\(874\) 6.47214 0.218923
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 54.7214 1.84781 0.923905 0.382623i \(-0.124979\pi\)
0.923905 + 0.382623i \(0.124979\pi\)
\(878\) 16.3607 0.552146
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) −18.8328 −0.632701
\(887\) 14.4721 0.485927 0.242963 0.970035i \(-0.421881\pi\)
0.242963 + 0.970035i \(0.421881\pi\)
\(888\) 0 0
\(889\) 3.05573 0.102486
\(890\) 34.8328 1.16760
\(891\) 0 0
\(892\) 9.52786 0.319016
\(893\) 41.8885 1.40175
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 12.9443 0.431716
\(900\) 0 0
\(901\) −31.0557 −1.03462
\(902\) 0 0
\(903\) 0 0
\(904\) −1.05573 −0.0351130
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) −6.11146 −0.202928 −0.101464 0.994839i \(-0.532353\pi\)
−0.101464 + 0.994839i \(0.532353\pi\)
\(908\) −14.4721 −0.480275
\(909\) 0 0
\(910\) 8.94427 0.296500
\(911\) −40.7214 −1.34916 −0.674579 0.738202i \(-0.735675\pi\)
−0.674579 + 0.738202i \(0.735675\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 52.9443 1.74647 0.873235 0.487299i \(-0.162018\pi\)
0.873235 + 0.487299i \(0.162018\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 33.4164 1.10051
\(923\) 57.8885 1.90542
\(924\) 0 0
\(925\) 10.9443 0.359845
\(926\) −41.8885 −1.37654
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −6.47214 −0.212116
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −29.3050 −0.958887
\(935\) 0 0
\(936\) 0 0
\(937\) −40.2492 −1.31488 −0.657442 0.753505i \(-0.728362\pi\)
−0.657442 + 0.753505i \(0.728362\pi\)
\(938\) 12.9443 0.422645
\(939\) 0 0
\(940\) 12.9443 0.422196
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 16.9443 0.550615 0.275307 0.961356i \(-0.411220\pi\)
0.275307 + 0.961356i \(0.411220\pi\)
\(948\) 0 0
\(949\) −13.1672 −0.427425
\(950\) −6.47214 −0.209984
\(951\) 0 0
\(952\) −4.47214 −0.144943
\(953\) 26.9443 0.872811 0.436405 0.899750i \(-0.356251\pi\)
0.436405 + 0.899750i \(0.356251\pi\)
\(954\) 0 0
\(955\) −41.8885 −1.35548
\(956\) −3.05573 −0.0988293
\(957\) 0 0
\(958\) 12.9443 0.418210
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 48.9443 1.57803
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 16.9443 0.544048
\(971\) −3.41641 −0.109638 −0.0548189 0.998496i \(-0.517458\pi\)
−0.0548189 + 0.998496i \(0.517458\pi\)
\(972\) 0 0
\(973\) −8.94427 −0.286740
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 23.0557 0.735738
\(983\) 57.8885 1.84636 0.923179 0.384371i \(-0.125581\pi\)
0.923179 + 0.384371i \(0.125581\pi\)
\(984\) 0 0
\(985\) 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
\(993\) 0 0
\(994\) −12.9443 −0.410567
\(995\) −41.8885 −1.32796
\(996\) 0 0
\(997\) −1.63932 −0.0519178 −0.0259589 0.999663i \(-0.508264\pi\)
−0.0259589 + 0.999663i \(0.508264\pi\)
\(998\) −15.0557 −0.476581
\(999\) 0 0
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 2898.2.a.v.1.2 2
3.2 odd 2 966.2.a.p.1.2 2
12.11 even 2 7728.2.a.bd.1.2 2
21.20 even 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 3.2 odd 2
2898.2.a.v.1.2 2 1.1 even 1 trivial
6762.2.a.bz.1.1 2 21.20 even 2
7728.2.a.bd.1.2 2 12.11 even 2