Properties

Label 2646.2.a.bp.1.1
Level $2646$
Weight $2$
Character 2646.1
Self dual yes
Analytic conductor $21.128$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2646.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.58579 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.58579 q^{5} +1.00000 q^{8} +1.58579 q^{10} +1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} +2.82843 q^{17} +0.171573 q^{19} +1.58579 q^{20} +1.00000 q^{22} +3.24264 q^{23} -2.48528 q^{25} +4.24264 q^{26} -8.24264 q^{29} -1.24264 q^{31} +1.00000 q^{32} +2.82843 q^{34} -3.24264 q^{37} +0.171573 q^{38} +1.58579 q^{40} +11.8284 q^{41} +10.4853 q^{43} +1.00000 q^{44} +3.24264 q^{46} +0.343146 q^{47} -2.48528 q^{50} +4.24264 q^{52} -8.00000 q^{53} +1.58579 q^{55} -8.24264 q^{58} +0.343146 q^{59} -9.17157 q^{61} -1.24264 q^{62} +1.00000 q^{64} +6.72792 q^{65} -10.2426 q^{67} +2.82843 q^{68} -7.24264 q^{71} +8.82843 q^{73} -3.24264 q^{74} +0.171573 q^{76} +10.4853 q^{79} +1.58579 q^{80} +11.8284 q^{82} +15.8995 q^{83} +4.48528 q^{85} +10.4853 q^{86} +1.00000 q^{88} +14.6569 q^{89} +3.24264 q^{92} +0.343146 q^{94} +0.272078 q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 6q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 6q^{5} + 2q^{8} + 6q^{10} + 2q^{11} + 2q^{16} + 6q^{19} + 6q^{20} + 2q^{22} - 2q^{23} + 12q^{25} - 8q^{29} + 6q^{31} + 2q^{32} + 2q^{37} + 6q^{38} + 6q^{40} + 18q^{41} + 4q^{43} + 2q^{44} - 2q^{46} + 12q^{47} + 12q^{50} - 16q^{53} + 6q^{55} - 8q^{58} + 12q^{59} - 24q^{61} + 6q^{62} + 2q^{64} - 12q^{65} - 12q^{67} - 6q^{71} + 12q^{73} + 2q^{74} + 6q^{76} + 4q^{79} + 6q^{80} + 18q^{82} + 12q^{83} - 8q^{85} + 4q^{86} + 2q^{88} + 18q^{89} - 2q^{92} + 12q^{94} + 26q^{95} + O(q^{100}) \)

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
\(5\) 1.58579 0.709185 0.354593 0.935021i \(-0.384620\pi\)
0.354593 + 0.935021i \(0.384620\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.58579 0.501470
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 0.171573 0.0393615 0.0196808 0.999806i \(-0.493735\pi\)
0.0196808 + 0.999806i \(0.493735\pi\)
\(20\) 1.58579 0.354593
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 3.24264 0.676137 0.338069 0.941121i \(-0.390226\pi\)
0.338069 + 0.941121i \(0.390226\pi\)
\(24\) 0 0
\(25\) −2.48528 −0.497056
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) −8.24264 −1.53062 −0.765310 0.643662i \(-0.777414\pi\)
−0.765310 + 0.643662i \(0.777414\pi\)
\(30\) 0 0
\(31\) −1.24264 −0.223185 −0.111592 0.993754i \(-0.535595\pi\)
−0.111592 + 0.993754i \(0.535595\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) −3.24264 −0.533087 −0.266543 0.963823i \(-0.585882\pi\)
−0.266543 + 0.963823i \(0.585882\pi\)
\(38\) 0.171573 0.0278328
\(39\) 0 0
\(40\) 1.58579 0.250735
\(41\) 11.8284 1.84729 0.923645 0.383249i \(-0.125195\pi\)
0.923645 + 0.383249i \(0.125195\pi\)
\(42\) 0 0
\(43\) 10.4853 1.59899 0.799495 0.600672i \(-0.205100\pi\)
0.799495 + 0.600672i \(0.205100\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 3.24264 0.478101
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.48528 −0.351472
\(51\) 0 0
\(52\) 4.24264 0.588348
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 1.58579 0.213827
\(56\) 0 0
\(57\) 0 0
\(58\) −8.24264 −1.08231
\(59\) 0.343146 0.0446738 0.0223369 0.999751i \(-0.492889\pi\)
0.0223369 + 0.999751i \(0.492889\pi\)
\(60\) 0 0
\(61\) −9.17157 −1.17430 −0.587150 0.809478i \(-0.699750\pi\)
−0.587150 + 0.809478i \(0.699750\pi\)
\(62\) −1.24264 −0.157816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.72792 0.834496
\(66\) 0 0
\(67\) −10.2426 −1.25134 −0.625669 0.780089i \(-0.715174\pi\)
−0.625669 + 0.780089i \(0.715174\pi\)
\(68\) 2.82843 0.342997
\(69\) 0 0
\(70\) 0 0
\(71\) −7.24264 −0.859543 −0.429772 0.902938i \(-0.641406\pi\)
−0.429772 + 0.902938i \(0.641406\pi\)
\(72\) 0 0
\(73\) 8.82843 1.03329 0.516645 0.856200i \(-0.327181\pi\)
0.516645 + 0.856200i \(0.327181\pi\)
\(74\) −3.24264 −0.376949
\(75\) 0 0
\(76\) 0.171573 0.0196808
\(77\) 0 0
\(78\) 0 0
\(79\) 10.4853 1.17969 0.589843 0.807518i \(-0.299190\pi\)
0.589843 + 0.807518i \(0.299190\pi\)
\(80\) 1.58579 0.177296
\(81\) 0 0
\(82\) 11.8284 1.30623
\(83\) 15.8995 1.74520 0.872598 0.488439i \(-0.162433\pi\)
0.872598 + 0.488439i \(0.162433\pi\)
\(84\) 0 0
\(85\) 4.48528 0.486497
\(86\) 10.4853 1.13066
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 14.6569 1.55362 0.776812 0.629733i \(-0.216836\pi\)
0.776812 + 0.629733i \(0.216836\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.24264 0.338069
\(93\) 0 0
\(94\) 0.343146 0.0353928
\(95\) 0.272078 0.0279146
\(96\) 0 0
\(97\) −11.3137 −1.14873 −0.574367 0.818598i \(-0.694752\pi\)
−0.574367 + 0.818598i \(0.694752\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.48528 −0.248528
\(101\) 2.48528 0.247295 0.123647 0.992326i \(-0.460541\pi\)
0.123647 + 0.992326i \(0.460541\pi\)
\(102\) 0 0
\(103\) 4.07107 0.401134 0.200567 0.979680i \(-0.435722\pi\)
0.200567 + 0.979680i \(0.435722\pi\)
\(104\) 4.24264 0.416025
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 18.9706 1.83395 0.916977 0.398941i \(-0.130622\pi\)
0.916977 + 0.398941i \(0.130622\pi\)
\(108\) 0 0
\(109\) −6.75736 −0.647238 −0.323619 0.946188i \(-0.604900\pi\)
−0.323619 + 0.946188i \(0.604900\pi\)
\(110\) 1.58579 0.151199
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264 0.399114 0.199557 0.979886i \(-0.436050\pi\)
0.199557 + 0.979886i \(0.436050\pi\)
\(114\) 0 0
\(115\) 5.14214 0.479507
\(116\) −8.24264 −0.765310
\(117\) 0 0
\(118\) 0.343146 0.0315891
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −9.17157 −0.830355
\(123\) 0 0
\(124\) −1.24264 −0.111592
\(125\) −11.8701 −1.06169
\(126\) 0 0
\(127\) −3.75736 −0.333412 −0.166706 0.986007i \(-0.553313\pi\)
−0.166706 + 0.986007i \(0.553313\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 6.72792 0.590078
\(131\) −2.48528 −0.217140 −0.108570 0.994089i \(-0.534627\pi\)
−0.108570 + 0.994089i \(0.534627\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.2426 −0.884829
\(135\) 0 0
\(136\) 2.82843 0.242536
\(137\) −10.2426 −0.875088 −0.437544 0.899197i \(-0.644152\pi\)
−0.437544 + 0.899197i \(0.644152\pi\)
\(138\) 0 0
\(139\) 3.17157 0.269009 0.134505 0.990913i \(-0.457056\pi\)
0.134505 + 0.990913i \(0.457056\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.24264 −0.607789
\(143\) 4.24264 0.354787
\(144\) 0 0
\(145\) −13.0711 −1.08549
\(146\) 8.82843 0.730646
\(147\) 0 0
\(148\) −3.24264 −0.266543
\(149\) 14.2426 1.16680 0.583401 0.812184i \(-0.301722\pi\)
0.583401 + 0.812184i \(0.301722\pi\)
\(150\) 0 0
\(151\) −10.7279 −0.873026 −0.436513 0.899698i \(-0.643787\pi\)
−0.436513 + 0.899698i \(0.643787\pi\)
\(152\) 0.171573 0.0139164
\(153\) 0 0
\(154\) 0 0
\(155\) −1.97056 −0.158279
\(156\) 0 0
\(157\) −19.0711 −1.52204 −0.761018 0.648730i \(-0.775300\pi\)
−0.761018 + 0.648730i \(0.775300\pi\)
\(158\) 10.4853 0.834164
\(159\) 0 0
\(160\) 1.58579 0.125367
\(161\) 0 0
\(162\) 0 0
\(163\) 10.7279 0.840276 0.420138 0.907460i \(-0.361982\pi\)
0.420138 + 0.907460i \(0.361982\pi\)
\(164\) 11.8284 0.923645
\(165\) 0 0
\(166\) 15.8995 1.23404
\(167\) 24.3848 1.88695 0.943475 0.331443i \(-0.107535\pi\)
0.943475 + 0.331443i \(0.107535\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 4.48528 0.344005
\(171\) 0 0
\(172\) 10.4853 0.799495
\(173\) −4.75736 −0.361695 −0.180848 0.983511i \(-0.557884\pi\)
−0.180848 + 0.983511i \(0.557884\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 14.6569 1.09858
\(179\) −22.9706 −1.71690 −0.858450 0.512897i \(-0.828572\pi\)
−0.858450 + 0.512897i \(0.828572\pi\)
\(180\) 0 0
\(181\) −1.75736 −0.130623 −0.0653117 0.997865i \(-0.520804\pi\)
−0.0653117 + 0.997865i \(0.520804\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.24264 0.239051
\(185\) −5.14214 −0.378057
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) 0.343146 0.0250265
\(189\) 0 0
\(190\) 0.272078 0.0197386
\(191\) 9.24264 0.668774 0.334387 0.942436i \(-0.391471\pi\)
0.334387 + 0.942436i \(0.391471\pi\)
\(192\) 0 0
\(193\) −12.4853 −0.898710 −0.449355 0.893353i \(-0.648346\pi\)
−0.449355 + 0.893353i \(0.648346\pi\)
\(194\) −11.3137 −0.812277
\(195\) 0 0
\(196\) 0 0
\(197\) −22.7279 −1.61930 −0.809649 0.586915i \(-0.800342\pi\)
−0.809649 + 0.586915i \(0.800342\pi\)
\(198\) 0 0
\(199\) −13.2426 −0.938746 −0.469373 0.883000i \(-0.655520\pi\)
−0.469373 + 0.883000i \(0.655520\pi\)
\(200\) −2.48528 −0.175736
\(201\) 0 0
\(202\) 2.48528 0.174864
\(203\) 0 0
\(204\) 0 0
\(205\) 18.7574 1.31007
\(206\) 4.07107 0.283645
\(207\) 0 0
\(208\) 4.24264 0.294174
\(209\) 0.171573 0.0118679
\(210\) 0 0
\(211\) −12.7279 −0.876226 −0.438113 0.898920i \(-0.644353\pi\)
−0.438113 + 0.898920i \(0.644353\pi\)
\(212\) −8.00000 −0.549442
\(213\) 0 0
\(214\) 18.9706 1.29680
\(215\) 16.6274 1.13398
\(216\) 0 0
\(217\) 0 0
\(218\) −6.75736 −0.457666
\(219\) 0 0
\(220\) 1.58579 0.106914
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 1.24264 0.0832134 0.0416067 0.999134i \(-0.486752\pi\)
0.0416067 + 0.999134i \(0.486752\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.24264 0.282216
\(227\) −12.7279 −0.844782 −0.422391 0.906414i \(-0.638809\pi\)
−0.422391 + 0.906414i \(0.638809\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 5.14214 0.339062
\(231\) 0 0
\(232\) −8.24264 −0.541156
\(233\) −7.75736 −0.508202 −0.254101 0.967178i \(-0.581779\pi\)
−0.254101 + 0.967178i \(0.581779\pi\)
\(234\) 0 0
\(235\) 0.544156 0.0354968
\(236\) 0.343146 0.0223369
\(237\) 0 0
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) −14.4853 −0.933079 −0.466539 0.884500i \(-0.654499\pi\)
−0.466539 + 0.884500i \(0.654499\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) −9.17157 −0.587150
\(245\) 0 0
\(246\) 0 0
\(247\) 0.727922 0.0463166
\(248\) −1.24264 −0.0789078
\(249\) 0 0
\(250\) −11.8701 −0.750728
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 0 0
\(253\) 3.24264 0.203863
\(254\) −3.75736 −0.235758
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.48528 0.342162 0.171081 0.985257i \(-0.445274\pi\)
0.171081 + 0.985257i \(0.445274\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.72792 0.417248
\(261\) 0 0
\(262\) −2.48528 −0.153541
\(263\) 21.7279 1.33980 0.669901 0.742451i \(-0.266337\pi\)
0.669901 + 0.742451i \(0.266337\pi\)
\(264\) 0 0
\(265\) −12.6863 −0.779313
\(266\) 0 0
\(267\) 0 0
\(268\) −10.2426 −0.625669
\(269\) 21.3848 1.30385 0.651926 0.758282i \(-0.273961\pi\)
0.651926 + 0.758282i \(0.273961\pi\)
\(270\) 0 0
\(271\) −29.3137 −1.78068 −0.890340 0.455295i \(-0.849534\pi\)
−0.890340 + 0.455295i \(0.849534\pi\)
\(272\) 2.82843 0.171499
\(273\) 0 0
\(274\) −10.2426 −0.618781
\(275\) −2.48528 −0.149868
\(276\) 0 0
\(277\) 28.2132 1.69517 0.847584 0.530662i \(-0.178057\pi\)
0.847584 + 0.530662i \(0.178057\pi\)
\(278\) 3.17157 0.190218
\(279\) 0 0
\(280\) 0 0
\(281\) 6.48528 0.386879 0.193440 0.981112i \(-0.438036\pi\)
0.193440 + 0.981112i \(0.438036\pi\)
\(282\) 0 0
\(283\) −20.1421 −1.19733 −0.598663 0.801001i \(-0.704301\pi\)
−0.598663 + 0.801001i \(0.704301\pi\)
\(284\) −7.24264 −0.429772
\(285\) 0 0
\(286\) 4.24264 0.250873
\(287\) 0 0
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) −13.0711 −0.767560
\(291\) 0 0
\(292\) 8.82843 0.516645
\(293\) 11.6569 0.681001 0.340500 0.940244i \(-0.389404\pi\)
0.340500 + 0.940244i \(0.389404\pi\)
\(294\) 0 0
\(295\) 0.544156 0.0316820
\(296\) −3.24264 −0.188475
\(297\) 0 0
\(298\) 14.2426 0.825054
\(299\) 13.7574 0.795609
\(300\) 0 0
\(301\) 0 0
\(302\) −10.7279 −0.617323
\(303\) 0 0
\(304\) 0.171573 0.00984038
\(305\) −14.5442 −0.832796
\(306\) 0 0
\(307\) 16.7990 0.958769 0.479384 0.877605i \(-0.340860\pi\)
0.479384 + 0.877605i \(0.340860\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.97056 −0.111920
\(311\) −15.5563 −0.882120 −0.441060 0.897478i \(-0.645397\pi\)
−0.441060 + 0.897478i \(0.645397\pi\)
\(312\) 0 0
\(313\) 10.2426 0.578948 0.289474 0.957186i \(-0.406520\pi\)
0.289474 + 0.957186i \(0.406520\pi\)
\(314\) −19.0711 −1.07624
\(315\) 0 0
\(316\) 10.4853 0.589843
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −8.24264 −0.461499
\(320\) 1.58579 0.0886482
\(321\) 0 0
\(322\) 0 0
\(323\) 0.485281 0.0270018
\(324\) 0 0
\(325\) −10.5442 −0.584885
\(326\) 10.7279 0.594165
\(327\) 0 0
\(328\) 11.8284 0.653116
\(329\) 0 0
\(330\) 0 0
\(331\) 24.4853 1.34583 0.672916 0.739719i \(-0.265041\pi\)
0.672916 + 0.739719i \(0.265041\pi\)
\(332\) 15.8995 0.872598
\(333\) 0 0
\(334\) 24.3848 1.33428
\(335\) −16.2426 −0.887430
\(336\) 0 0
\(337\) 13.4853 0.734590 0.367295 0.930104i \(-0.380284\pi\)
0.367295 + 0.930104i \(0.380284\pi\)
\(338\) 5.00000 0.271964
\(339\) 0 0
\(340\) 4.48528 0.243249
\(341\) −1.24264 −0.0672928
\(342\) 0 0
\(343\) 0 0
\(344\) 10.4853 0.565328
\(345\) 0 0
\(346\) −4.75736 −0.255757
\(347\) 35.4853 1.90495 0.952475 0.304617i \(-0.0985285\pi\)
0.952475 + 0.304617i \(0.0985285\pi\)
\(348\) 0 0
\(349\) −22.2426 −1.19062 −0.595311 0.803496i \(-0.702971\pi\)
−0.595311 + 0.803496i \(0.702971\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −8.65685 −0.460758 −0.230379 0.973101i \(-0.573997\pi\)
−0.230379 + 0.973101i \(0.573997\pi\)
\(354\) 0 0
\(355\) −11.4853 −0.609575
\(356\) 14.6569 0.776812
\(357\) 0 0
\(358\) −22.9706 −1.21403
\(359\) −33.4558 −1.76573 −0.882866 0.469625i \(-0.844389\pi\)
−0.882866 + 0.469625i \(0.844389\pi\)
\(360\) 0 0
\(361\) −18.9706 −0.998451
\(362\) −1.75736 −0.0923648
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) 11.1005 0.579442 0.289721 0.957111i \(-0.406438\pi\)
0.289721 + 0.957111i \(0.406438\pi\)
\(368\) 3.24264 0.169034
\(369\) 0 0
\(370\) −5.14214 −0.267327
\(371\) 0 0
\(372\) 0 0
\(373\) −6.75736 −0.349883 −0.174941 0.984579i \(-0.555974\pi\)
−0.174941 + 0.984579i \(0.555974\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) 0.343146 0.0176964
\(377\) −34.9706 −1.80108
\(378\) 0 0
\(379\) −30.9706 −1.59085 −0.795425 0.606051i \(-0.792753\pi\)
−0.795425 + 0.606051i \(0.792753\pi\)
\(380\) 0.272078 0.0139573
\(381\) 0 0
\(382\) 9.24264 0.472895
\(383\) −30.0416 −1.53506 −0.767528 0.641016i \(-0.778513\pi\)
−0.767528 + 0.641016i \(0.778513\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12.4853 −0.635484
\(387\) 0 0
\(388\) −11.3137 −0.574367
\(389\) 3.02944 0.153599 0.0767993 0.997047i \(-0.475530\pi\)
0.0767993 + 0.997047i \(0.475530\pi\)
\(390\) 0 0
\(391\) 9.17157 0.463826
\(392\) 0 0
\(393\) 0 0
\(394\) −22.7279 −1.14502
\(395\) 16.6274 0.836616
\(396\) 0 0
\(397\) −16.5858 −0.832417 −0.416208 0.909269i \(-0.636641\pi\)
−0.416208 + 0.909269i \(0.636641\pi\)
\(398\) −13.2426 −0.663794
\(399\) 0 0
\(400\) −2.48528 −0.124264
\(401\) −10.2426 −0.511493 −0.255747 0.966744i \(-0.582321\pi\)
−0.255747 + 0.966744i \(0.582321\pi\)
\(402\) 0 0
\(403\) −5.27208 −0.262621
\(404\) 2.48528 0.123647
\(405\) 0 0
\(406\) 0 0
\(407\) −3.24264 −0.160732
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 18.7574 0.926360
\(411\) 0 0
\(412\) 4.07107 0.200567
\(413\) 0 0
\(414\) 0 0
\(415\) 25.2132 1.23767
\(416\) 4.24264 0.208013
\(417\) 0 0
\(418\) 0.171573 0.00839190
\(419\) 12.0416 0.588272 0.294136 0.955764i \(-0.404968\pi\)
0.294136 + 0.955764i \(0.404968\pi\)
\(420\) 0 0
\(421\) 18.2132 0.887657 0.443829 0.896112i \(-0.353620\pi\)
0.443829 + 0.896112i \(0.353620\pi\)
\(422\) −12.7279 −0.619586
\(423\) 0 0
\(424\) −8.00000 −0.388514
\(425\) −7.02944 −0.340978
\(426\) 0 0
\(427\) 0 0
\(428\) 18.9706 0.916977
\(429\) 0 0
\(430\) 16.6274 0.801845
\(431\) −19.2426 −0.926885 −0.463443 0.886127i \(-0.653386\pi\)
−0.463443 + 0.886127i \(0.653386\pi\)
\(432\) 0 0
\(433\) −30.7279 −1.47669 −0.738345 0.674423i \(-0.764392\pi\)
−0.738345 + 0.674423i \(0.764392\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.75736 −0.323619
\(437\) 0.556349 0.0266138
\(438\) 0 0
\(439\) 28.9706 1.38269 0.691345 0.722525i \(-0.257019\pi\)
0.691345 + 0.722525i \(0.257019\pi\)
\(440\) 1.58579 0.0755994
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −28.9411 −1.37503 −0.687517 0.726168i \(-0.741299\pi\)
−0.687517 + 0.726168i \(0.741299\pi\)
\(444\) 0 0
\(445\) 23.2426 1.10181
\(446\) 1.24264 0.0588407
\(447\) 0 0
\(448\) 0 0
\(449\) −11.2721 −0.531962 −0.265981 0.963978i \(-0.585696\pi\)
−0.265981 + 0.963978i \(0.585696\pi\)
\(450\) 0 0
\(451\) 11.8284 0.556979
\(452\) 4.24264 0.199557
\(453\) 0 0
\(454\) −12.7279 −0.597351
\(455\) 0 0
\(456\) 0 0
\(457\) −13.9706 −0.653515 −0.326758 0.945108i \(-0.605956\pi\)
−0.326758 + 0.945108i \(0.605956\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 5.14214 0.239753
\(461\) 6.89949 0.321342 0.160671 0.987008i \(-0.448634\pi\)
0.160671 + 0.987008i \(0.448634\pi\)
\(462\) 0 0
\(463\) −13.2721 −0.616806 −0.308403 0.951256i \(-0.599794\pi\)
−0.308403 + 0.951256i \(0.599794\pi\)
\(464\) −8.24264 −0.382655
\(465\) 0 0
\(466\) −7.75736 −0.359353
\(467\) −23.6569 −1.09471 −0.547354 0.836901i \(-0.684365\pi\)
−0.547354 + 0.836901i \(0.684365\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.544156 0.0251000
\(471\) 0 0
\(472\) 0.343146 0.0157946
\(473\) 10.4853 0.482114
\(474\) 0 0
\(475\) −0.426407 −0.0195649
\(476\) 0 0
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 18.3431 0.838120 0.419060 0.907959i \(-0.362360\pi\)
0.419060 + 0.907959i \(0.362360\pi\)
\(480\) 0 0
\(481\) −13.7574 −0.627282
\(482\) −14.4853 −0.659786
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −17.9411 −0.814665
\(486\) 0 0
\(487\) −21.5147 −0.974925 −0.487462 0.873144i \(-0.662077\pi\)
−0.487462 + 0.873144i \(0.662077\pi\)
\(488\) −9.17157 −0.415178
\(489\) 0 0
\(490\) 0 0
\(491\) −0.514719 −0.0232289 −0.0116145 0.999933i \(-0.503697\pi\)
−0.0116145 + 0.999933i \(0.503697\pi\)
\(492\) 0 0
\(493\) −23.3137 −1.05000
\(494\) 0.727922 0.0327508
\(495\) 0 0
\(496\) −1.24264 −0.0557962
\(497\) 0 0
\(498\) 0 0
\(499\) 18.9706 0.849239 0.424620 0.905372i \(-0.360408\pi\)
0.424620 + 0.905372i \(0.360408\pi\)
\(500\) −11.8701 −0.530845
\(501\) 0 0
\(502\) −11.3137 −0.504956
\(503\) −39.5563 −1.76373 −0.881865 0.471502i \(-0.843712\pi\)
−0.881865 + 0.471502i \(0.843712\pi\)
\(504\) 0 0
\(505\) 3.94113 0.175378
\(506\) 3.24264 0.144153
\(507\) 0 0
\(508\) −3.75736 −0.166706
\(509\) −22.2843 −0.987733 −0.493866 0.869538i \(-0.664417\pi\)
−0.493866 + 0.869538i \(0.664417\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 5.48528 0.241945
\(515\) 6.45584 0.284478
\(516\) 0 0
\(517\) 0.343146 0.0150915
\(518\) 0 0
\(519\) 0 0
\(520\) 6.72792 0.295039
\(521\) −5.14214 −0.225281 −0.112641 0.993636i \(-0.535931\pi\)
−0.112641 + 0.993636i \(0.535931\pi\)
\(522\) 0 0
\(523\) 0.514719 0.0225071 0.0112535 0.999937i \(-0.496418\pi\)
0.0112535 + 0.999937i \(0.496418\pi\)
\(524\) −2.48528 −0.108570
\(525\) 0 0
\(526\) 21.7279 0.947382
\(527\) −3.51472 −0.153104
\(528\) 0 0
\(529\) −12.4853 −0.542838
\(530\) −12.6863 −0.551057
\(531\) 0 0
\(532\) 0 0
\(533\) 50.1838 2.17370
\(534\) 0 0
\(535\) 30.0833 1.30061
\(536\) −10.2426 −0.442415
\(537\) 0 0
\(538\) 21.3848 0.921963
\(539\) 0 0
\(540\) 0 0
\(541\) −33.2426 −1.42921 −0.714606 0.699527i \(-0.753394\pi\)
−0.714606 + 0.699527i \(0.753394\pi\)
\(542\) −29.3137 −1.25913
\(543\) 0 0
\(544\) 2.82843 0.121268
\(545\) −10.7157 −0.459011
\(546\) 0 0
\(547\) 7.51472 0.321306 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(548\) −10.2426 −0.437544
\(549\) 0 0
\(550\) −2.48528 −0.105973
\(551\) −1.41421 −0.0602475
\(552\) 0 0
\(553\) 0 0
\(554\) 28.2132 1.19866
\(555\) 0 0
\(556\) 3.17157 0.134505
\(557\) 6.72792 0.285071 0.142536 0.989790i \(-0.454474\pi\)
0.142536 + 0.989790i \(0.454474\pi\)
\(558\) 0 0
\(559\) 44.4853 1.88153
\(560\) 0 0
\(561\) 0 0
\(562\) 6.48528 0.273565
\(563\) −19.4142 −0.818212 −0.409106 0.912487i \(-0.634159\pi\)
−0.409106 + 0.912487i \(0.634159\pi\)
\(564\) 0 0
\(565\) 6.72792 0.283046
\(566\) −20.1421 −0.846637
\(567\) 0 0
\(568\) −7.24264 −0.303894
\(569\) −36.2426 −1.51937 −0.759685 0.650291i \(-0.774647\pi\)
−0.759685 + 0.650291i \(0.774647\pi\)
\(570\) 0 0
\(571\) −41.2132 −1.72472 −0.862359 0.506297i \(-0.831014\pi\)
−0.862359 + 0.506297i \(0.831014\pi\)
\(572\) 4.24264 0.177394
\(573\) 0 0
\(574\) 0 0
\(575\) −8.05887 −0.336078
\(576\) 0 0
\(577\) 28.2426 1.17576 0.587878 0.808949i \(-0.299963\pi\)
0.587878 + 0.808949i \(0.299963\pi\)
\(578\) −9.00000 −0.374351
\(579\) 0 0
\(580\) −13.0711 −0.542747
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 8.82843 0.365323
\(585\) 0 0
\(586\) 11.6569 0.481540
\(587\) −16.2426 −0.670406 −0.335203 0.942146i \(-0.608805\pi\)
−0.335203 + 0.942146i \(0.608805\pi\)
\(588\) 0 0
\(589\) −0.213203 −0.00878489
\(590\) 0.544156 0.0224025
\(591\) 0 0
\(592\) −3.24264 −0.133272
\(593\) 8.65685 0.355494 0.177747 0.984076i \(-0.443119\pi\)
0.177747 + 0.984076i \(0.443119\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.2426 0.583401
\(597\) 0 0
\(598\) 13.7574 0.562580
\(599\) 5.72792 0.234037 0.117018 0.993130i \(-0.462666\pi\)
0.117018 + 0.993130i \(0.462666\pi\)
\(600\) 0 0
\(601\) −5.27208 −0.215053 −0.107526 0.994202i \(-0.534293\pi\)
−0.107526 + 0.994202i \(0.534293\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.7279 −0.436513
\(605\) −15.8579 −0.644714
\(606\) 0 0
\(607\) 30.0000 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) 0.171573 0.00695820
\(609\) 0 0
\(610\) −14.5442 −0.588876
\(611\) 1.45584 0.0588971
\(612\) 0 0
\(613\) 3.72792 0.150569 0.0752847 0.997162i \(-0.476013\pi\)
0.0752847 + 0.997162i \(0.476013\pi\)
\(614\) 16.7990 0.677952
\(615\) 0 0
\(616\) 0 0
\(617\) −6.97056 −0.280624 −0.140312 0.990107i \(-0.544811\pi\)
−0.140312 + 0.990107i \(0.544811\pi\)
\(618\) 0 0
\(619\) −22.7990 −0.916369 −0.458184 0.888857i \(-0.651500\pi\)
−0.458184 + 0.888857i \(0.651500\pi\)
\(620\) −1.97056 −0.0791397
\(621\) 0 0
\(622\) −15.5563 −0.623753
\(623\) 0 0
\(624\) 0 0
\(625\) −6.39697 −0.255879
\(626\) 10.2426 0.409378
\(627\) 0 0
\(628\) −19.0711 −0.761018
\(629\) −9.17157 −0.365695
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 10.4853 0.417082
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) −5.95837 −0.236451
\(636\) 0 0
\(637\) 0 0
\(638\) −8.24264 −0.326329
\(639\) 0 0
\(640\) 1.58579 0.0626837
\(641\) −1.21320 −0.0479187 −0.0239593 0.999713i \(-0.507627\pi\)
−0.0239593 + 0.999713i \(0.507627\pi\)
\(642\) 0 0
\(643\) 0.171573 0.00676617 0.00338309 0.999994i \(-0.498923\pi\)
0.00338309 + 0.999994i \(0.498923\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.485281 0.0190931
\(647\) −21.5147 −0.845831 −0.422915 0.906169i \(-0.638993\pi\)
−0.422915 + 0.906169i \(0.638993\pi\)
\(648\) 0 0
\(649\) 0.343146 0.0134696
\(650\) −10.5442 −0.413576
\(651\) 0 0
\(652\) 10.7279 0.420138
\(653\) −27.9411 −1.09342 −0.546710 0.837322i \(-0.684120\pi\)
−0.546710 + 0.837322i \(0.684120\pi\)
\(654\) 0 0
\(655\) −3.94113 −0.153993
\(656\) 11.8284 0.461822
\(657\) 0 0
\(658\) 0 0
\(659\) 24.9411 0.971568 0.485784 0.874079i \(-0.338534\pi\)
0.485784 + 0.874079i \(0.338534\pi\)
\(660\) 0 0
\(661\) −0.727922 −0.0283129 −0.0141564 0.999900i \(-0.504506\pi\)
−0.0141564 + 0.999900i \(0.504506\pi\)
\(662\) 24.4853 0.951647
\(663\) 0 0
\(664\) 15.8995 0.617020
\(665\) 0 0
\(666\) 0 0
\(667\) −26.7279 −1.03491
\(668\) 24.3848 0.943475
\(669\) 0 0
\(670\) −16.2426 −0.627508
\(671\) −9.17157 −0.354065
\(672\) 0 0
\(673\) 49.4558 1.90638 0.953191 0.302368i \(-0.0977771\pi\)
0.953191 + 0.302368i \(0.0977771\pi\)
\(674\) 13.4853 0.519434
\(675\) 0 0
\(676\) 5.00000 0.192308
\(677\) −38.6985 −1.48730 −0.743652 0.668567i \(-0.766908\pi\)
−0.743652 + 0.668567i \(0.766908\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.48528 0.172003
\(681\) 0 0
\(682\) −1.24264 −0.0475832
\(683\) −13.9706 −0.534569 −0.267284 0.963618i \(-0.586126\pi\)
−0.267284 + 0.963618i \(0.586126\pi\)
\(684\) 0 0
\(685\) −16.2426 −0.620599
\(686\) 0 0
\(687\) 0 0
\(688\) 10.4853 0.399748
\(689\) −33.9411 −1.29305
\(690\) 0 0
\(691\) 21.9411 0.834680 0.417340 0.908750i \(-0.362962\pi\)
0.417340 + 0.908750i \(0.362962\pi\)
\(692\) −4.75736 −0.180848
\(693\) 0 0
\(694\) 35.4853 1.34700
\(695\) 5.02944 0.190777
\(696\) 0 0
\(697\) 33.4558 1.26723
\(698\) −22.2426 −0.841896
\(699\) 0 0
\(700\) 0 0
\(701\) 19.7574 0.746225 0.373113 0.927786i \(-0.378291\pi\)
0.373113 + 0.927786i \(0.378291\pi\)
\(702\) 0 0
\(703\) −0.556349 −0.0209831
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −8.65685 −0.325805
\(707\) 0 0
\(708\) 0 0
\(709\) −27.2426 −1.02312 −0.511559 0.859248i \(-0.670932\pi\)
−0.511559 + 0.859248i \(0.670932\pi\)
\(710\) −11.4853 −0.431035
\(711\) 0 0
\(712\) 14.6569 0.549289
\(713\) −4.02944 −0.150904
\(714\) 0 0
\(715\) 6.72792 0.251610
\(716\) −22.9706 −0.858450
\(717\) 0 0
\(718\) −33.4558 −1.24856
\(719\) −23.6569 −0.882252 −0.441126 0.897445i \(-0.645421\pi\)
−0.441126 + 0.897445i \(0.645421\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.9706 −0.706011
\(723\) 0 0
\(724\) −1.75736 −0.0653117
\(725\) 20.4853 0.760804
\(726\) 0 0
\(727\) 14.1421 0.524503 0.262251 0.965000i \(-0.415535\pi\)
0.262251 + 0.965000i \(0.415535\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) 29.6569 1.09690
\(732\) 0 0
\(733\) −27.9411 −1.03203 −0.516015 0.856580i \(-0.672585\pi\)
−0.516015 + 0.856580i \(0.672585\pi\)
\(734\) 11.1005 0.409727
\(735\) 0 0
\(736\) 3.24264 0.119525
\(737\) −10.2426 −0.377293
\(738\) 0 0
\(739\) 48.7279 1.79249 0.896243 0.443564i \(-0.146286\pi\)
0.896243 + 0.443564i \(0.146286\pi\)
\(740\) −5.14214 −0.189029
\(741\) 0 0
\(742\) 0 0
\(743\) 14.7574 0.541395 0.270698 0.962664i \(-0.412746\pi\)
0.270698 + 0.962664i \(0.412746\pi\)
\(744\) 0 0
\(745\) 22.5858 0.827479
\(746\) −6.75736 −0.247405
\(747\) 0 0
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) 39.4558 1.43976 0.719882 0.694096i \(-0.244196\pi\)
0.719882 + 0.694096i \(0.244196\pi\)
\(752\) 0.343146 0.0125132
\(753\) 0 0
\(754\) −34.9706 −1.27355
\(755\) −17.0122 −0.619137
\(756\) 0 0
\(757\) −47.4558 −1.72481 −0.862406 0.506217i \(-0.831043\pi\)
−0.862406 + 0.506217i \(0.831043\pi\)
\(758\) −30.9706 −1.12490
\(759\) 0 0
\(760\) 0.272078 0.00986930
\(761\) −29.3137 −1.06262 −0.531311 0.847177i \(-0.678300\pi\)
−0.531311 + 0.847177i \(0.678300\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9.24264 0.334387
\(765\) 0 0
\(766\) −30.0416 −1.08545
\(767\) 1.45584 0.0525675
\(768\) 0 0
\(769\) 44.1838 1.59331 0.796654 0.604436i \(-0.206601\pi\)
0.796654 + 0.604436i \(0.206601\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.4853 −0.449355
\(773\) 39.3848 1.41657 0.708286 0.705926i \(-0.249469\pi\)
0.708286 + 0.705926i \(0.249469\pi\)
\(774\) 0 0
\(775\) 3.08831 0.110935
\(776\) −11.3137 −0.406138
\(777\) 0 0
\(778\) 3.02944 0.108611
\(779\) 2.02944 0.0727121
\(780\) 0 0
\(781\) −7.24264 −0.259162
\(782\) 9.17157 0.327975
\(783\) 0 0
\(784\) 0 0
\(785\) −30.2426 −1.07941
\(786\) 0 0
\(787\) 53.3137 1.90043 0.950214 0.311597i \(-0.100864\pi\)
0.950214 + 0.311597i \(0.100864\pi\)
\(788\) −22.7279 −0.809649
\(789\) 0 0
\(790\) 16.6274 0.591577
\(791\) 0 0
\(792\) 0 0
\(793\) −38.9117 −1.38179
\(794\) −16.5858 −0.588608
\(795\) 0 0
\(796\) −13.2426 −0.469373
\(797\) −25.2426 −0.894140 −0.447070 0.894499i \(-0.647533\pi\)
−0.447070 + 0.894499i \(0.647533\pi\)
\(798\) 0 0
\(799\) 0.970563 0.0343360
\(800\) −2.48528 −0.0878680
\(801\) 0 0
\(802\) −10.2426 −0.361680
\(803\) 8.82843 0.311548
\(804\) 0 0
\(805\) 0 0
\(806\) −5.27208 −0.185701
\(807\) 0 0
\(808\) 2.48528 0.0874319
\(809\) 18.2426 0.641377 0.320689 0.947185i \(-0.396086\pi\)
0.320689 + 0.947185i \(0.396086\pi\)
\(810\) 0 0
\(811\) 45.4264 1.59514 0.797568 0.603228i \(-0.206119\pi\)
0.797568 + 0.603228i \(0.206119\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.24264 −0.113654
\(815\) 17.0122 0.595911
\(816\) 0 0
\(817\) 1.79899 0.0629387
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 18.7574 0.655035
\(821\) 13.0294 0.454730 0.227365 0.973810i \(-0.426989\pi\)
0.227365 + 0.973810i \(0.426989\pi\)
\(822\) 0 0
\(823\) 29.7574 1.03728 0.518638 0.854994i \(-0.326439\pi\)
0.518638 + 0.854994i \(0.326439\pi\)
\(824\) 4.07107 0.141822
\(825\) 0 0
\(826\) 0 0
\(827\) −12.9411 −0.450007 −0.225004 0.974358i \(-0.572239\pi\)
−0.225004 + 0.974358i \(0.572239\pi\)
\(828\) 0 0
\(829\) 15.8579 0.550766 0.275383 0.961335i \(-0.411195\pi\)
0.275383 + 0.961335i \(0.411195\pi\)
\(830\) 25.2132 0.875163
\(831\) 0 0
\(832\) 4.24264 0.147087
\(833\) 0 0
\(834\) 0 0
\(835\) 38.6690 1.33820
\(836\) 0.171573 0.00593397
\(837\) 0 0
\(838\) 12.0416 0.415971
\(839\) 17.6152 0.608145 0.304073 0.952649i \(-0.401653\pi\)
0.304073 + 0.952649i \(0.401653\pi\)
\(840\) 0 0
\(841\) 38.9411 1.34280
\(842\) 18.2132 0.627668
\(843\) 0 0
\(844\) −12.7279 −0.438113
\(845\) 7.92893 0.272764
\(846\) 0 0
\(847\) 0 0
\(848\) −8.00000 −0.274721
\(849\) 0 0
\(850\) −7.02944 −0.241108
\(851\) −10.5147 −0.360440
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.9706 0.648400
\(857\) −33.7696 −1.15355 −0.576773 0.816904i \(-0.695688\pi\)
−0.576773 + 0.816904i \(0.695688\pi\)
\(858\) 0 0
\(859\) 29.8284 1.01773 0.508866 0.860846i \(-0.330065\pi\)
0.508866 + 0.860846i \(0.330065\pi\)
\(860\) 16.6274 0.566990
\(861\) 0 0
\(862\) −19.2426 −0.655407
\(863\) −2.97056 −0.101119 −0.0505596 0.998721i \(-0.516100\pi\)
−0.0505596 + 0.998721i \(0.516100\pi\)
\(864\) 0 0
\(865\) −7.54416 −0.256509
\(866\) −30.7279 −1.04418
\(867\) 0 0
\(868\) 0 0
\(869\) 10.4853 0.355689
\(870\) 0 0
\(871\) −43.4558 −1.47245
\(872\) −6.75736 −0.228833
\(873\) 0 0
\(874\) 0.556349 0.0188188
\(875\) 0 0
\(876\) 0 0
\(877\) −14.9706 −0.505520 −0.252760 0.967529i \(-0.581338\pi\)
−0.252760 + 0.967529i \(0.581338\pi\)
\(878\) 28.9706 0.977709
\(879\) 0 0
\(880\) 1.58579 0.0534568
\(881\) 54.9411 1.85101 0.925507 0.378731i \(-0.123639\pi\)
0.925507 + 0.378731i \(0.123639\pi\)
\(882\) 0 0
\(883\) 11.5147 0.387501 0.193751 0.981051i \(-0.437935\pi\)
0.193751 + 0.981051i \(0.437935\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −28.9411 −0.972296
\(887\) 47.3553 1.59004 0.795018 0.606585i \(-0.207461\pi\)
0.795018 + 0.606585i \(0.207461\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 23.2426 0.779095
\(891\) 0 0
\(892\) 1.24264 0.0416067
\(893\) 0.0588745 0.00197016
\(894\) 0 0
\(895\) −36.4264 −1.21760
\(896\) 0 0
\(897\) 0 0
\(898\) −11.2721 −0.376154
\(899\) 10.2426 0.341611
\(900\) 0 0
\(901\) −22.6274 −0.753829
\(902\) 11.8284 0.393844
\(903\) 0 0
\(904\) 4.24264 0.141108
\(905\) −2.78680 −0.0926363
\(906\) 0 0
\(907\) −8.72792 −0.289806 −0.144903 0.989446i \(-0.546287\pi\)
−0.144903 + 0.989446i \(0.546287\pi\)
\(908\) −12.7279 −0.422391
\(909\) 0 0
\(910\) 0 0
\(911\) −12.4853 −0.413656 −0.206828 0.978377i \(-0.566314\pi\)
−0.206828 + 0.978377i \(0.566314\pi\)
\(912\) 0 0
\(913\) 15.8995 0.526196
\(914\) −13.9706 −0.462105
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) 12.4853 0.411851 0.205926 0.978568i \(-0.433979\pi\)
0.205926 + 0.978568i \(0.433979\pi\)
\(920\) 5.14214 0.169531
\(921\) 0 0
\(922\) 6.89949 0.227223
\(923\) −30.7279 −1.01142
\(924\) 0 0
\(925\) 8.05887 0.264974
\(926\) −13.2721 −0.436148
\(927\) 0 0
\(928\) −8.24264 −0.270578
\(929\) 16.2843 0.534270 0.267135 0.963659i \(-0.413923\pi\)
0.267135 + 0.963659i \(0.413923\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −7.75736 −0.254101
\(933\) 0 0
\(934\) −23.6569 −0.774076
\(935\) 4.48528 0.146684
\(936\) 0 0
\(937\) 0.686292 0.0224202 0.0112101 0.999937i \(-0.496432\pi\)
0.0112101 + 0.999937i \(0.496432\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.544156 0.0177484
\(941\) 10.0711 0.328307 0.164154 0.986435i \(-0.447511\pi\)
0.164154 + 0.986435i \(0.447511\pi\)
\(942\) 0 0
\(943\) 38.3553 1.24902
\(944\) 0.343146 0.0111684
\(945\) 0 0
\(946\) 10.4853 0.340906
\(947\) −16.0294 −0.520887 −0.260443 0.965489i \(-0.583869\pi\)
−0.260443 + 0.965489i \(0.583869\pi\)
\(948\) 0 0
\(949\) 37.4558 1.21587
\(950\) −0.426407 −0.0138345
\(951\) 0 0
\(952\) 0 0
\(953\) 17.7574 0.575217 0.287609 0.957748i \(-0.407140\pi\)
0.287609 + 0.957748i \(0.407140\pi\)
\(954\) 0 0
\(955\) 14.6569 0.474285
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 18.3431 0.592640
\(959\) 0 0
\(960\) 0 0
\(961\) −29.4558 −0.950189
\(962\) −13.7574 −0.443555
\(963\) 0 0
\(964\) −14.4853 −0.466539
\(965\) −19.7990 −0.637352
\(966\) 0 0
\(967\) −52.6690 −1.69372 −0.846861 0.531814i \(-0.821511\pi\)
−0.846861 + 0.531814i \(0.821511\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) −17.9411 −0.576055
\(971\) 55.4975 1.78100 0.890499 0.454984i \(-0.150355\pi\)
0.890499 + 0.454984i \(0.150355\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −21.5147 −0.689376
\(975\) 0 0
\(976\) −9.17157 −0.293575
\(977\) 16.9706 0.542936 0.271468 0.962447i \(-0.412491\pi\)
0.271468 + 0.962447i \(0.412491\pi\)
\(978\) 0 0
\(979\) 14.6569 0.468435
\(980\) 0 0
\(981\) 0 0
\(982\) −0.514719 −0.0164253
\(983\) −1.79899 −0.0573789 −0.0286894 0.999588i \(-0.509133\pi\)
−0.0286894 + 0.999588i \(0.509133\pi\)
\(984\) 0 0
\(985\) −36.0416 −1.14838
\(986\) −23.3137 −0.742460
\(987\) 0 0
\(988\) 0.727922 0.0231583
\(989\) 34.0000 1.08114
\(990\) 0 0
\(991\) 0.242641 0.00770774 0.00385387 0.999993i \(-0.498773\pi\)
0.00385387 + 0.999993i \(0.498773\pi\)
\(992\) −1.24264 −0.0394539
\(993\) 0 0
\(994\) 0 0
\(995\) −21.0000 −0.665745
\(996\) 0 0
\(997\) −39.8995 −1.26363 −0.631815 0.775119i \(-0.717690\pi\)
−0.631815 + 0.775119i \(0.717690\pi\)
\(998\) 18.9706 0.600503
\(999\) 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 2646.2.a.bp.1.1 yes 2
3.2 odd 2 2646.2.a.be.1.2 2
7.6 odd 2 2646.2.a.bk.1.2 yes 2
21.20 even 2 2646.2.a.bj.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.a.be.1.2 2 3.2 odd 2
2646.2.a.bj.1.1 yes 2 21.20 even 2
2646.2.a.bk.1.2 yes 2 7.6 odd 2
2646.2.a.bp.1.1 yes 2 1.1 even 1 trivial