Properties

Label 4608.2.a.e.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.585786 q^{5} +3.41421 q^{7} +O(q^{10})\) \(q-0.585786 q^{5} +3.41421 q^{7} -2.00000 q^{11} +2.82843 q^{13} -3.65685 q^{17} -5.65685 q^{19} -1.17157 q^{23} -4.65685 q^{25} -0.585786 q^{29} +4.58579 q^{31} -2.00000 q^{35} -9.65685 q^{37} +11.6569 q^{41} +1.65685 q^{43} -12.4853 q^{47} +4.65685 q^{49} -11.8995 q^{53} +1.17157 q^{55} +4.00000 q^{59} -9.65685 q^{61} -1.65685 q^{65} -8.00000 q^{67} -9.17157 q^{71} -1.65685 q^{73} -6.82843 q^{77} +5.75736 q^{79} -9.31371 q^{83} +2.14214 q^{85} -2.00000 q^{89} +9.65685 q^{91} +3.31371 q^{95} +13.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 4 q^{7} - 4 q^{11} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 12 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{41} - 8 q^{43} - 8 q^{47} - 2 q^{49} - 4 q^{53} + 8 q^{55} + 8 q^{59} - 8 q^{61} + 8 q^{65} - 16 q^{67} - 24 q^{71} + 8 q^{73} - 8 q^{77} + 20 q^{79} + 4 q^{83} - 24 q^{85} - 4 q^{89} + 8 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.585786 −0.108778 −0.0543889 0.998520i \(-0.517321\pi\)
−0.0543889 + 0.998520i \(0.517321\pi\)
\(30\) 0 0
\(31\) 4.58579 0.823632 0.411816 0.911267i \(-0.364895\pi\)
0.411816 + 0.911267i \(0.364895\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −9.65685 −1.58758 −0.793789 0.608194i \(-0.791894\pi\)
−0.793789 + 0.608194i \(0.791894\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.6569 1.82049 0.910247 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(42\) 0 0
\(43\) 1.65685 0.252668 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.4853 −1.82117 −0.910583 0.413327i \(-0.864367\pi\)
−0.910583 + 0.413327i \(0.864367\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −11.8995 −1.63452 −0.817261 0.576268i \(-0.804508\pi\)
−0.817261 + 0.576268i \(0.804508\pi\)
\(54\) 0 0
\(55\) 1.17157 0.157975
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −9.65685 −1.23643 −0.618217 0.786008i \(-0.712145\pi\)
−0.618217 + 0.786008i \(0.712145\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.65685 −0.205507
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.17157 −1.08847 −0.544233 0.838934i \(-0.683179\pi\)
−0.544233 + 0.838934i \(0.683179\pi\)
\(72\) 0 0
\(73\) −1.65685 −0.193920 −0.0969601 0.995288i \(-0.530912\pi\)
−0.0969601 + 0.995288i \(0.530912\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.82843 −0.778171
\(78\) 0 0
\(79\) 5.75736 0.647754 0.323877 0.946099i \(-0.395014\pi\)
0.323877 + 0.946099i \(0.395014\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.31371 −1.02231 −0.511156 0.859488i \(-0.670783\pi\)
−0.511156 + 0.859488i \(0.670783\pi\)
\(84\) 0 0
\(85\) 2.14214 0.232347
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.31371 0.339979
\(96\) 0 0
\(97\) 13.3137 1.35180 0.675901 0.736992i \(-0.263755\pi\)
0.675901 + 0.736992i \(0.263755\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.5563 1.74692 0.873461 0.486894i \(-0.161870\pi\)
0.873461 + 0.486894i \(0.161870\pi\)
\(102\) 0 0
\(103\) 9.07107 0.893799 0.446899 0.894584i \(-0.352528\pi\)
0.446899 + 0.894584i \(0.352528\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.3137 1.48043 0.740216 0.672369i \(-0.234723\pi\)
0.740216 + 0.672369i \(0.234723\pi\)
\(108\) 0 0
\(109\) −13.1716 −1.26161 −0.630804 0.775942i \(-0.717275\pi\)
−0.630804 + 0.775942i \(0.717275\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0.686292 0.0639970
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.4853 −1.14452
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 6.72792 0.597007 0.298503 0.954409i \(-0.403513\pi\)
0.298503 + 0.954409i \(0.403513\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −19.3137 −1.67471
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.34315 −0.371060 −0.185530 0.982639i \(-0.559400\pi\)
−0.185530 + 0.982639i \(0.559400\pi\)
\(138\) 0 0
\(139\) 19.3137 1.63817 0.819084 0.573674i \(-0.194482\pi\)
0.819084 + 0.573674i \(0.194482\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 0.343146 0.0284967
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.2426 −1.16680 −0.583401 0.812184i \(-0.698278\pi\)
−0.583401 + 0.812184i \(0.698278\pi\)
\(150\) 0 0
\(151\) 4.58579 0.373186 0.186593 0.982437i \(-0.440255\pi\)
0.186593 + 0.982437i \(0.440255\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.68629 −0.215768
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 6.34315 0.496834 0.248417 0.968653i \(-0.420090\pi\)
0.248417 + 0.968653i \(0.420090\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6569 1.67586 0.837929 0.545779i \(-0.183766\pi\)
0.837929 + 0.545779i \(0.183766\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.7279 −1.42386 −0.711929 0.702252i \(-0.752178\pi\)
−0.711929 + 0.702252i \(0.752178\pi\)
\(174\) 0 0
\(175\) −15.8995 −1.20189
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) 7.31371 0.534831
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −21.6569 −1.55889 −0.779447 0.626468i \(-0.784500\pi\)
−0.779447 + 0.626468i \(0.784500\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.92893 −0.208678 −0.104339 0.994542i \(-0.533273\pi\)
−0.104339 + 0.994542i \(0.533273\pi\)
\(198\) 0 0
\(199\) −13.5563 −0.960984 −0.480492 0.876999i \(-0.659542\pi\)
−0.480492 + 0.876999i \(0.659542\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −6.82843 −0.476918
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 0 0
\(211\) −19.3137 −1.32961 −0.664805 0.747017i \(-0.731485\pi\)
−0.664805 + 0.747017i \(0.731485\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.970563 −0.0661918
\(216\) 0 0
\(217\) 15.6569 1.06286
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.3431 −0.695755
\(222\) 0 0
\(223\) 7.89949 0.528989 0.264495 0.964387i \(-0.414795\pi\)
0.264495 + 0.964387i \(0.414795\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −29.3137 −1.94562 −0.972810 0.231606i \(-0.925602\pi\)
−0.972810 + 0.231606i \(0.925602\pi\)
\(228\) 0 0
\(229\) −16.4853 −1.08938 −0.544689 0.838638i \(-0.683352\pi\)
−0.544689 + 0.838638i \(0.683352\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.31371 −0.610161 −0.305081 0.952327i \(-0.598683\pi\)
−0.305081 + 0.952327i \(0.598683\pi\)
\(234\) 0 0
\(235\) 7.31371 0.477094
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −17.6569 −1.13738 −0.568689 0.822553i \(-0.692549\pi\)
−0.568689 + 0.822553i \(0.692549\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.72792 −0.174281
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.0000 1.38863 0.694314 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(252\) 0 0
\(253\) 2.34315 0.147312
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.31371 0.0819469 0.0409734 0.999160i \(-0.486954\pi\)
0.0409734 + 0.999160i \(0.486954\pi\)
\(258\) 0 0
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.6569 0.842118 0.421059 0.907033i \(-0.361659\pi\)
0.421059 + 0.907033i \(0.361659\pi\)
\(264\) 0 0
\(265\) 6.97056 0.428198
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.24264 0.380621 0.190310 0.981724i \(-0.439051\pi\)
0.190310 + 0.981724i \(0.439051\pi\)
\(270\) 0 0
\(271\) −19.4142 −1.17933 −0.589665 0.807648i \(-0.700740\pi\)
−0.589665 + 0.807648i \(0.700740\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.31371 0.561638
\(276\) 0 0
\(277\) −7.51472 −0.451516 −0.225758 0.974183i \(-0.572486\pi\)
−0.225758 + 0.974183i \(0.572486\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 15.3137 0.910305 0.455153 0.890413i \(-0.349585\pi\)
0.455153 + 0.890413i \(0.349585\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 39.7990 2.34926
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −27.8995 −1.62991 −0.814953 0.579527i \(-0.803237\pi\)
−0.814953 + 0.579527i \(0.803237\pi\)
\(294\) 0 0
\(295\) −2.34315 −0.136423
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.31371 −0.191637
\(300\) 0 0
\(301\) 5.65685 0.326056
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.65685 0.323911
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.07107 −0.284820 −0.142410 0.989808i \(-0.545485\pi\)
−0.142410 + 0.989808i \(0.545485\pi\)
\(318\) 0 0
\(319\) 1.17157 0.0655955
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.6863 1.15102
\(324\) 0 0
\(325\) −13.1716 −0.730627
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −42.6274 −2.35013
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.68629 0.256039
\(336\) 0 0
\(337\) 20.9706 1.14234 0.571170 0.820832i \(-0.306490\pi\)
0.571170 + 0.820832i \(0.306490\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −9.17157 −0.496669
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.3137 −1.35891 −0.679456 0.733717i \(-0.737784\pi\)
−0.679456 + 0.733717i \(0.737784\pi\)
\(348\) 0 0
\(349\) −0.686292 −0.0367363 −0.0183682 0.999831i \(-0.505847\pi\)
−0.0183682 + 0.999831i \(0.505847\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.3137 0.708617 0.354309 0.935129i \(-0.384716\pi\)
0.354309 + 0.935129i \(0.384716\pi\)
\(354\) 0 0
\(355\) 5.37258 0.285147
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.7990 0.833839 0.416919 0.908943i \(-0.363110\pi\)
0.416919 + 0.908943i \(0.363110\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.970563 0.0508016
\(366\) 0 0
\(367\) −22.7279 −1.18639 −0.593194 0.805060i \(-0.702133\pi\)
−0.593194 + 0.805060i \(0.702133\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −40.6274 −2.10927
\(372\) 0 0
\(373\) −24.2843 −1.25739 −0.628696 0.777651i \(-0.716411\pi\)
−0.628696 + 0.777651i \(0.716411\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.65685 −0.0853323
\(378\) 0 0
\(379\) −18.3431 −0.942224 −0.471112 0.882073i \(-0.656147\pi\)
−0.471112 + 0.882073i \(0.656147\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.3137 0.986884 0.493442 0.869779i \(-0.335738\pi\)
0.493442 + 0.869779i \(0.335738\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.2426 1.12775 0.563873 0.825861i \(-0.309311\pi\)
0.563873 + 0.825861i \(0.309311\pi\)
\(390\) 0 0
\(391\) 4.28427 0.216665
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.37258 −0.169693
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.6569 1.58087 0.790434 0.612547i \(-0.209855\pi\)
0.790434 + 0.612547i \(0.209855\pi\)
\(402\) 0 0
\(403\) 12.9706 0.646110
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.3137 0.957345
\(408\) 0 0
\(409\) 1.31371 0.0649587 0.0324794 0.999472i \(-0.489660\pi\)
0.0324794 + 0.999472i \(0.489660\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.6569 0.672010
\(414\) 0 0
\(415\) 5.45584 0.267817
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.31371 0.0641789 0.0320894 0.999485i \(-0.489784\pi\)
0.0320894 + 0.999485i \(0.489784\pi\)
\(420\) 0 0
\(421\) 22.1421 1.07914 0.539571 0.841940i \(-0.318587\pi\)
0.539571 + 0.841940i \(0.318587\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.0294 0.826049
\(426\) 0 0
\(427\) −32.9706 −1.59556
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −35.1127 −1.69132 −0.845660 0.533723i \(-0.820793\pi\)
−0.845660 + 0.533723i \(0.820793\pi\)
\(432\) 0 0
\(433\) −16.6274 −0.799063 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.62742 0.317032
\(438\) 0 0
\(439\) −15.8995 −0.758841 −0.379421 0.925224i \(-0.623877\pi\)
−0.379421 + 0.925224i \(0.623877\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.9411 1.32752 0.663761 0.747944i \(-0.268959\pi\)
0.663761 + 0.747944i \(0.268959\pi\)
\(444\) 0 0
\(445\) 1.17157 0.0555379
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.65685 0.361349 0.180675 0.983543i \(-0.442172\pi\)
0.180675 + 0.983543i \(0.442172\pi\)
\(450\) 0 0
\(451\) −23.3137 −1.09780
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.65685 −0.265197
\(456\) 0 0
\(457\) −17.3137 −0.809901 −0.404951 0.914339i \(-0.632711\pi\)
−0.404951 + 0.914339i \(0.632711\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.9289 0.509011 0.254506 0.967071i \(-0.418087\pi\)
0.254506 + 0.967071i \(0.418087\pi\)
\(462\) 0 0
\(463\) 2.44365 0.113566 0.0567830 0.998387i \(-0.481916\pi\)
0.0567830 + 0.998387i \(0.481916\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.31371 0.430987 0.215494 0.976505i \(-0.430864\pi\)
0.215494 + 0.976505i \(0.430864\pi\)
\(468\) 0 0
\(469\) −27.3137 −1.26123
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.31371 −0.152364
\(474\) 0 0
\(475\) 26.3431 1.20871
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.48528 0.204938 0.102469 0.994736i \(-0.467326\pi\)
0.102469 + 0.994736i \(0.467326\pi\)
\(480\) 0 0
\(481\) −27.3137 −1.24540
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.79899 −0.354134
\(486\) 0 0
\(487\) −5.55635 −0.251782 −0.125891 0.992044i \(-0.540179\pi\)
−0.125891 + 0.992044i \(0.540179\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.686292 0.0309719 0.0154860 0.999880i \(-0.495070\pi\)
0.0154860 + 0.999880i \(0.495070\pi\)
\(492\) 0 0
\(493\) 2.14214 0.0964769
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −31.3137 −1.40461
\(498\) 0 0
\(499\) 23.3137 1.04366 0.521832 0.853048i \(-0.325249\pi\)
0.521832 + 0.853048i \(0.325249\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.48528 0.199989 0.0999944 0.994988i \(-0.468118\pi\)
0.0999944 + 0.994988i \(0.468118\pi\)
\(504\) 0 0
\(505\) −10.2843 −0.457644
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −23.2132 −1.02891 −0.514454 0.857518i \(-0.672005\pi\)
−0.514454 + 0.857518i \(0.672005\pi\)
\(510\) 0 0
\(511\) −5.65685 −0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.31371 −0.234150
\(516\) 0 0
\(517\) 24.9706 1.09820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.9706 0.480629 0.240315 0.970695i \(-0.422749\pi\)
0.240315 + 0.970695i \(0.422749\pi\)
\(522\) 0 0
\(523\) 26.3431 1.15191 0.575953 0.817483i \(-0.304631\pi\)
0.575953 + 0.817483i \(0.304631\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.7696 −0.730493
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32.9706 1.42811
\(534\) 0 0
\(535\) −8.97056 −0.387831
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.31371 −0.401170
\(540\) 0 0
\(541\) −5.17157 −0.222343 −0.111172 0.993801i \(-0.535460\pi\)
−0.111172 + 0.993801i \(0.535460\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.71573 0.330506
\(546\) 0 0
\(547\) 28.9706 1.23869 0.619346 0.785118i \(-0.287398\pi\)
0.619346 + 0.785118i \(0.287398\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.31371 0.141169
\(552\) 0 0
\(553\) 19.6569 0.835894
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.0711 0.553839 0.276919 0.960893i \(-0.410686\pi\)
0.276919 + 0.960893i \(0.410686\pi\)
\(558\) 0 0
\(559\) 4.68629 0.198209
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 28.6274 1.20650 0.603251 0.797551i \(-0.293872\pi\)
0.603251 + 0.797551i \(0.293872\pi\)
\(564\) 0 0
\(565\) 3.51472 0.147865
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.9706 −0.795287 −0.397644 0.917540i \(-0.630172\pi\)
−0.397644 + 0.917540i \(0.630172\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.45584 0.227524
\(576\) 0 0
\(577\) 28.2843 1.17749 0.588745 0.808319i \(-0.299622\pi\)
0.588745 + 0.808319i \(0.299622\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −31.7990 −1.31924
\(582\) 0 0
\(583\) 23.7990 0.985653
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.68629 0.358522 0.179261 0.983802i \(-0.442629\pi\)
0.179261 + 0.983802i \(0.442629\pi\)
\(588\) 0 0
\(589\) −25.9411 −1.06889
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 7.31371 0.299833
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −28.4853 −1.16388 −0.581939 0.813233i \(-0.697706\pi\)
−0.581939 + 0.813233i \(0.697706\pi\)
\(600\) 0 0
\(601\) 5.65685 0.230748 0.115374 0.993322i \(-0.463193\pi\)
0.115374 + 0.993322i \(0.463193\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.10051 0.166709
\(606\) 0 0
\(607\) −15.8995 −0.645341 −0.322670 0.946511i \(-0.604581\pi\)
−0.322670 + 0.946511i \(0.604581\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −35.3137 −1.42864
\(612\) 0 0
\(613\) 34.6274 1.39859 0.699294 0.714834i \(-0.253498\pi\)
0.699294 + 0.714834i \(0.253498\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.9411 1.76900 0.884502 0.466537i \(-0.154499\pi\)
0.884502 + 0.466537i \(0.154499\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.82843 −0.273575
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 35.3137 1.40805
\(630\) 0 0
\(631\) 33.0711 1.31654 0.658269 0.752783i \(-0.271289\pi\)
0.658269 + 0.752783i \(0.271289\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.94113 −0.156399
\(636\) 0 0
\(637\) 13.1716 0.521877
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.9706 −1.69724 −0.848618 0.529007i \(-0.822565\pi\)
−0.848618 + 0.529007i \(0.822565\pi\)
\(642\) 0 0
\(643\) 24.2843 0.957678 0.478839 0.877903i \(-0.341058\pi\)
0.478839 + 0.877903i \(0.341058\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.1716 0.675084 0.337542 0.941310i \(-0.390404\pi\)
0.337542 + 0.941310i \(0.390404\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.4142 −0.916269 −0.458134 0.888883i \(-0.651482\pi\)
−0.458134 + 0.888883i \(0.651482\pi\)
\(654\) 0 0
\(655\) −2.34315 −0.0915543
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.6863 0.650006 0.325003 0.945713i \(-0.394635\pi\)
0.325003 + 0.945713i \(0.394635\pi\)
\(660\) 0 0
\(661\) 16.6863 0.649022 0.324511 0.945882i \(-0.394800\pi\)
0.324511 + 0.945882i \(0.394800\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.3137 0.438727
\(666\) 0 0
\(667\) 0.686292 0.0265733
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 19.3137 0.745597
\(672\) 0 0
\(673\) 36.6274 1.41188 0.705942 0.708270i \(-0.250524\pi\)
0.705942 + 0.708270i \(0.250524\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.21320 −0.277226 −0.138613 0.990347i \(-0.544264\pi\)
−0.138613 + 0.990347i \(0.544264\pi\)
\(678\) 0 0
\(679\) 45.4558 1.74444
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.3137 0.968602 0.484301 0.874901i \(-0.339074\pi\)
0.484301 + 0.874901i \(0.339074\pi\)
\(684\) 0 0
\(685\) 2.54416 0.0972072
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −33.6569 −1.28222
\(690\) 0 0
\(691\) −17.6569 −0.671698 −0.335849 0.941916i \(-0.609023\pi\)
−0.335849 + 0.941916i \(0.609023\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.3137 −0.429153
\(696\) 0 0
\(697\) −42.6274 −1.61463
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −7.41421 −0.280031 −0.140015 0.990149i \(-0.544715\pi\)
−0.140015 + 0.990149i \(0.544715\pi\)
\(702\) 0 0
\(703\) 54.6274 2.06031
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 59.9411 2.25432
\(708\) 0 0
\(709\) 31.1127 1.16846 0.584231 0.811587i \(-0.301396\pi\)
0.584231 + 0.811587i \(0.301396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.37258 −0.201205
\(714\) 0 0
\(715\) 3.31371 0.123926
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −25.1716 −0.938741 −0.469371 0.883001i \(-0.655519\pi\)
−0.469371 + 0.883001i \(0.655519\pi\)
\(720\) 0 0
\(721\) 30.9706 1.15340
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.72792 0.101312
\(726\) 0 0
\(727\) −13.7574 −0.510232 −0.255116 0.966910i \(-0.582114\pi\)
−0.255116 + 0.966910i \(0.582114\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.05887 −0.224096
\(732\) 0 0
\(733\) 40.4853 1.49536 0.747679 0.664060i \(-0.231168\pi\)
0.747679 + 0.664060i \(0.231168\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 34.6274 1.27379 0.636895 0.770951i \(-0.280218\pi\)
0.636895 + 0.770951i \(0.280218\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 8.34315 0.305669
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 52.2843 1.91043
\(750\) 0 0
\(751\) 30.7279 1.12128 0.560639 0.828060i \(-0.310556\pi\)
0.560639 + 0.828060i \(0.310556\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.68629 −0.0977642
\(756\) 0 0
\(757\) 15.5147 0.563892 0.281946 0.959430i \(-0.409020\pi\)
0.281946 + 0.959430i \(0.409020\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −24.3431 −0.882438 −0.441219 0.897399i \(-0.645454\pi\)
−0.441219 + 0.897399i \(0.645454\pi\)
\(762\) 0 0
\(763\) −44.9706 −1.62804
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.3137 0.408514
\(768\) 0 0
\(769\) −13.6569 −0.492479 −0.246239 0.969209i \(-0.579195\pi\)
−0.246239 + 0.969209i \(0.579195\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.7279 −0.673597 −0.336798 0.941577i \(-0.609344\pi\)
−0.336798 + 0.941577i \(0.609344\pi\)
\(774\) 0 0
\(775\) −21.3553 −0.767106
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −65.9411 −2.36259
\(780\) 0 0
\(781\) 18.3431 0.656369
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.7157 0.418152
\(786\) 0 0
\(787\) 10.3431 0.368693 0.184347 0.982861i \(-0.440983\pi\)
0.184347 + 0.982861i \(0.440983\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.4853 −0.728373
\(792\) 0 0
\(793\) −27.3137 −0.969938
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −32.5858 −1.15425 −0.577124 0.816657i \(-0.695825\pi\)
−0.577124 + 0.816657i \(0.695825\pi\)
\(798\) 0 0
\(799\) 45.6569 1.61522
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.31371 0.116938
\(804\) 0 0
\(805\) 2.34315 0.0825850
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.9706 1.37013 0.685066 0.728481i \(-0.259773\pi\)
0.685066 + 0.728481i \(0.259773\pi\)
\(810\) 0 0
\(811\) −29.6569 −1.04139 −0.520697 0.853742i \(-0.674328\pi\)
−0.520697 + 0.853742i \(0.674328\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.71573 −0.130156
\(816\) 0 0
\(817\) −9.37258 −0.327905
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 44.8701 1.56598 0.782988 0.622037i \(-0.213695\pi\)
0.782988 + 0.622037i \(0.213695\pi\)
\(822\) 0 0
\(823\) −3.21320 −0.112005 −0.0560026 0.998431i \(-0.517836\pi\)
−0.0560026 + 0.998431i \(0.517836\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −37.9411 −1.31934 −0.659671 0.751554i \(-0.729304\pi\)
−0.659671 + 0.751554i \(0.729304\pi\)
\(828\) 0 0
\(829\) 20.7696 0.721356 0.360678 0.932690i \(-0.382545\pi\)
0.360678 + 0.932690i \(0.382545\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −17.0294 −0.590035
\(834\) 0 0
\(835\) −12.6863 −0.439027
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.5147 −0.397532 −0.198766 0.980047i \(-0.563693\pi\)
−0.198766 + 0.980047i \(0.563693\pi\)
\(840\) 0 0
\(841\) −28.6569 −0.988167
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.92893 0.100758
\(846\) 0 0
\(847\) −23.8995 −0.821196
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137 0.387829
\(852\) 0 0
\(853\) −10.6274 −0.363876 −0.181938 0.983310i \(-0.558237\pi\)
−0.181938 + 0.983310i \(0.558237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37.5980 1.28432 0.642161 0.766570i \(-0.278038\pi\)
0.642161 + 0.766570i \(0.278038\pi\)
\(858\) 0 0
\(859\) 11.0294 0.376320 0.188160 0.982138i \(-0.439748\pi\)
0.188160 + 0.982138i \(0.439748\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48.9706 −1.66698 −0.833489 0.552537i \(-0.813660\pi\)
−0.833489 + 0.552537i \(0.813660\pi\)
\(864\) 0 0
\(865\) 10.9706 0.373010
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.5147 −0.390610
\(870\) 0 0
\(871\) −22.6274 −0.766701
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 19.3137 0.652923
\(876\) 0 0
\(877\) −0.686292 −0.0231744 −0.0115872 0.999933i \(-0.503688\pi\)
−0.0115872 + 0.999933i \(0.503688\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −22.6863 −0.764321 −0.382160 0.924096i \(-0.624820\pi\)
−0.382160 + 0.924096i \(0.624820\pi\)
\(882\) 0 0
\(883\) 41.6569 1.40186 0.700932 0.713228i \(-0.252767\pi\)
0.700932 + 0.713228i \(0.252767\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.3431 0.615903 0.307951 0.951402i \(-0.400357\pi\)
0.307951 + 0.951402i \(0.400357\pi\)
\(888\) 0 0
\(889\) 22.9706 0.770408
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 70.6274 2.36346
\(894\) 0 0
\(895\) −2.34315 −0.0783227
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.68629 −0.0895928
\(900\) 0 0
\(901\) 43.5147 1.44969
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.97056 −0.165227
\(906\) 0 0
\(907\) −27.5980 −0.916376 −0.458188 0.888855i \(-0.651501\pi\)
−0.458188 + 0.888855i \(0.651501\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −51.3137 −1.70010 −0.850050 0.526703i \(-0.823428\pi\)
−0.850050 + 0.526703i \(0.823428\pi\)
\(912\) 0 0
\(913\) 18.6274 0.616478
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.6569 0.450989
\(918\) 0 0
\(919\) 53.3553 1.76003 0.880015 0.474946i \(-0.157532\pi\)
0.880015 + 0.474946i \(0.157532\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −25.9411 −0.853863
\(924\) 0 0
\(925\) 44.9706 1.47862
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 49.5980 1.62726 0.813628 0.581385i \(-0.197489\pi\)
0.813628 + 0.581385i \(0.197489\pi\)
\(930\) 0 0
\(931\) −26.3431 −0.863362
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.28427 −0.140111
\(936\) 0 0
\(937\) −8.62742 −0.281845 −0.140923 0.990021i \(-0.545007\pi\)
−0.140923 + 0.990021i \(0.545007\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −16.3848 −0.534128 −0.267064 0.963679i \(-0.586054\pi\)
−0.267064 + 0.963679i \(0.586054\pi\)
\(942\) 0 0
\(943\) −13.6569 −0.444728
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10.6274 −0.345345 −0.172672 0.984979i \(-0.555240\pi\)
−0.172672 + 0.984979i \(0.555240\pi\)
\(948\) 0 0
\(949\) −4.68629 −0.152123
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.97056 0.0962260 0.0481130 0.998842i \(-0.484679\pi\)
0.0481130 + 0.998842i \(0.484679\pi\)
\(954\) 0 0
\(955\) −9.94113 −0.321687
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.8284 −0.478835
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.6863 0.408386
\(966\) 0 0
\(967\) 10.2426 0.329381 0.164691 0.986345i \(-0.447337\pi\)
0.164691 + 0.986345i \(0.447337\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.68629 −0.214573 −0.107287 0.994228i \(-0.534216\pi\)
−0.107287 + 0.994228i \(0.534216\pi\)
\(972\) 0 0
\(973\) 65.9411 2.11398
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26.2843 −0.840908 −0.420454 0.907314i \(-0.638129\pi\)
−0.420454 + 0.907314i \(0.638129\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52.2843 −1.66761 −0.833805 0.552060i \(-0.813842\pi\)
−0.833805 + 0.552060i \(0.813842\pi\)
\(984\) 0 0
\(985\) 1.71573 0.0546677
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.94113 −0.0617242
\(990\) 0 0
\(991\) −7.89949 −0.250936 −0.125468 0.992098i \(-0.540043\pi\)
−0.125468 + 0.992098i \(0.540043\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.94113 0.251751
\(996\) 0 0
\(997\) −20.9706 −0.664144 −0.332072 0.943254i \(-0.607748\pi\)
−0.332072 + 0.943254i \(0.607748\pi\)
\(998\) 0 0
\(999\) 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 4608.2.a.e.1.2 2
3.2 odd 2 1536.2.a.l.1.1 yes 2
4.3 odd 2 4608.2.a.a.1.2 2
8.3 odd 2 4608.2.a.n.1.1 2
8.5 even 2 4608.2.a.r.1.1 2
12.11 even 2 1536.2.a.e.1.1 yes 2
16.3 odd 4 4608.2.d.o.2305.3 4
16.5 even 4 4608.2.d.c.2305.2 4
16.11 odd 4 4608.2.d.o.2305.2 4
16.13 even 4 4608.2.d.c.2305.3 4
24.5 odd 2 1536.2.a.b.1.2 2
24.11 even 2 1536.2.a.g.1.2 yes 2
48.5 odd 4 1536.2.d.a.769.1 4
48.11 even 4 1536.2.d.f.769.3 4
48.29 odd 4 1536.2.d.a.769.4 4
48.35 even 4 1536.2.d.f.769.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.2 2 24.5 odd 2
1536.2.a.e.1.1 yes 2 12.11 even 2
1536.2.a.g.1.2 yes 2 24.11 even 2
1536.2.a.l.1.1 yes 2 3.2 odd 2
1536.2.d.a.769.1 4 48.5 odd 4
1536.2.d.a.769.4 4 48.29 odd 4
1536.2.d.f.769.2 4 48.35 even 4
1536.2.d.f.769.3 4 48.11 even 4
4608.2.a.a.1.2 2 4.3 odd 2
4608.2.a.e.1.2 2 1.1 even 1 trivial
4608.2.a.n.1.1 2 8.3 odd 2
4608.2.a.r.1.1 2 8.5 even 2
4608.2.d.c.2305.2 4 16.5 even 4
4608.2.d.c.2305.3 4 16.13 even 4
4608.2.d.o.2305.2 4 16.11 odd 4
4608.2.d.o.2305.3 4 16.3 odd 4