Properties

Label 1792.2.a.q.1.2
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 448)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{3} -2.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q+2.73205 q^{3} -2.73205 q^{5} +1.00000 q^{7} +4.46410 q^{9} -5.46410 q^{11} -6.73205 q^{13} -7.46410 q^{15} +2.00000 q^{17} -1.26795 q^{19} +2.73205 q^{21} -3.46410 q^{23} +2.46410 q^{25} +4.00000 q^{27} +1.46410 q^{29} -4.00000 q^{31} -14.9282 q^{33} -2.73205 q^{35} +1.46410 q^{37} -18.3923 q^{39} +2.00000 q^{41} -5.46410 q^{43} -12.1962 q^{45} -2.92820 q^{47} +1.00000 q^{49} +5.46410 q^{51} -12.0000 q^{53} +14.9282 q^{55} -3.46410 q^{57} +9.66025 q^{59} +11.1244 q^{61} +4.46410 q^{63} +18.3923 q^{65} +8.00000 q^{67} -9.46410 q^{69} +2.92820 q^{71} -12.9282 q^{73} +6.73205 q^{75} -5.46410 q^{77} +10.9282 q^{79} -2.46410 q^{81} -5.66025 q^{83} -5.46410 q^{85} +4.00000 q^{87} -11.8564 q^{89} -6.73205 q^{91} -10.9282 q^{93} +3.46410 q^{95} +8.92820 q^{97} -24.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} - 10q^{13} - 8q^{15} + 4q^{17} - 6q^{19} + 2q^{21} - 2q^{25} + 8q^{27} - 4q^{29} - 8q^{31} - 16q^{33} - 2q^{35} - 4q^{37} - 16q^{39} + 4q^{41} - 4q^{43} - 14q^{45} + 8q^{47} + 2q^{49} + 4q^{51} - 24q^{53} + 16q^{55} + 2q^{59} - 2q^{61} + 2q^{63} + 16q^{65} + 16q^{67} - 12q^{69} - 8q^{71} - 12q^{73} + 10q^{75} - 4q^{77} + 8q^{79} + 2q^{81} + 6q^{83} - 4q^{85} + 8q^{87} + 4q^{89} - 10q^{91} - 8q^{93} + 4q^{97} - 28q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 0 0
\(5\) −2.73205 −1.22181 −0.610905 0.791704i \(-0.709194\pi\)
−0.610905 + 0.791704i \(0.709194\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) −5.46410 −1.64749 −0.823744 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) 0 0
\(13\) −6.73205 −1.86713 −0.933567 0.358402i \(-0.883322\pi\)
−0.933567 + 0.358402i \(0.883322\pi\)
\(14\) 0 0
\(15\) −7.46410 −1.92722
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.26795 −0.290887 −0.145444 0.989367i \(-0.546461\pi\)
−0.145444 + 0.989367i \(0.546461\pi\)
\(20\) 0 0
\(21\) 2.73205 0.596182
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 1.46410 0.271877 0.135938 0.990717i \(-0.456595\pi\)
0.135938 + 0.990717i \(0.456595\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −14.9282 −2.59867
\(34\) 0 0
\(35\) −2.73205 −0.461801
\(36\) 0 0
\(37\) 1.46410 0.240697 0.120348 0.992732i \(-0.461599\pi\)
0.120348 + 0.992732i \(0.461599\pi\)
\(38\) 0 0
\(39\) −18.3923 −2.94513
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −5.46410 −0.833268 −0.416634 0.909074i \(-0.636790\pi\)
−0.416634 + 0.909074i \(0.636790\pi\)
\(44\) 0 0
\(45\) −12.1962 −1.81810
\(46\) 0 0
\(47\) −2.92820 −0.427122 −0.213561 0.976930i \(-0.568506\pi\)
−0.213561 + 0.976930i \(0.568506\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.46410 0.765127
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 14.9282 2.01292
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) 9.66025 1.25766 0.628829 0.777544i \(-0.283535\pi\)
0.628829 + 0.777544i \(0.283535\pi\)
\(60\) 0 0
\(61\) 11.1244 1.42433 0.712164 0.702013i \(-0.247715\pi\)
0.712164 + 0.702013i \(0.247715\pi\)
\(62\) 0 0
\(63\) 4.46410 0.562424
\(64\) 0 0
\(65\) 18.3923 2.28128
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −9.46410 −1.13934
\(70\) 0 0
\(71\) 2.92820 0.347514 0.173757 0.984789i \(-0.444409\pi\)
0.173757 + 0.984789i \(0.444409\pi\)
\(72\) 0 0
\(73\) −12.9282 −1.51313 −0.756566 0.653917i \(-0.773124\pi\)
−0.756566 + 0.653917i \(0.773124\pi\)
\(74\) 0 0
\(75\) 6.73205 0.777350
\(76\) 0 0
\(77\) −5.46410 −0.622692
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) −5.66025 −0.621294 −0.310647 0.950525i \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(84\) 0 0
\(85\) −5.46410 −0.592665
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −11.8564 −1.25678 −0.628388 0.777900i \(-0.716285\pi\)
−0.628388 + 0.777900i \(0.716285\pi\)
\(90\) 0 0
\(91\) −6.73205 −0.705711
\(92\) 0 0
\(93\) −10.9282 −1.13320
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 8.92820 0.906522 0.453261 0.891378i \(-0.350261\pi\)
0.453261 + 0.891378i \(0.350261\pi\)
\(98\) 0 0
\(99\) −24.3923 −2.45152
\(100\) 0 0
\(101\) 1.66025 0.165201 0.0826007 0.996583i \(-0.473677\pi\)
0.0826007 + 0.996583i \(0.473677\pi\)
\(102\) 0 0
\(103\) 9.85641 0.971181 0.485590 0.874187i \(-0.338605\pi\)
0.485590 + 0.874187i \(0.338605\pi\)
\(104\) 0 0
\(105\) −7.46410 −0.728422
\(106\) 0 0
\(107\) −5.07180 −0.490309 −0.245155 0.969484i \(-0.578839\pi\)
−0.245155 + 0.969484i \(0.578839\pi\)
\(108\) 0 0
\(109\) −12.3923 −1.18697 −0.593484 0.804846i \(-0.702248\pi\)
−0.593484 + 0.804846i \(0.702248\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −6.53590 −0.614846 −0.307423 0.951573i \(-0.599467\pi\)
−0.307423 + 0.951573i \(0.599467\pi\)
\(114\) 0 0
\(115\) 9.46410 0.882532
\(116\) 0 0
\(117\) −30.0526 −2.77836
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) 0 0
\(123\) 5.46410 0.492681
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 11.4641 1.01727 0.508637 0.860981i \(-0.330149\pi\)
0.508637 + 0.860981i \(0.330149\pi\)
\(128\) 0 0
\(129\) −14.9282 −1.31436
\(130\) 0 0
\(131\) −10.7321 −0.937664 −0.468832 0.883287i \(-0.655325\pi\)
−0.468832 + 0.883287i \(0.655325\pi\)
\(132\) 0 0
\(133\) −1.26795 −0.109945
\(134\) 0 0
\(135\) −10.9282 −0.940550
\(136\) 0 0
\(137\) −19.8564 −1.69645 −0.848224 0.529638i \(-0.822328\pi\)
−0.848224 + 0.529638i \(0.822328\pi\)
\(138\) 0 0
\(139\) 9.26795 0.786097 0.393049 0.919518i \(-0.371420\pi\)
0.393049 + 0.919518i \(0.371420\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 36.7846 3.07608
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 2.73205 0.225336
\(148\) 0 0
\(149\) −1.07180 −0.0878050 −0.0439025 0.999036i \(-0.513979\pi\)
−0.0439025 + 0.999036i \(0.513979\pi\)
\(150\) 0 0
\(151\) −2.39230 −0.194683 −0.0973415 0.995251i \(-0.531034\pi\)
−0.0973415 + 0.995251i \(0.531034\pi\)
\(152\) 0 0
\(153\) 8.92820 0.721802
\(154\) 0 0
\(155\) 10.9282 0.877774
\(156\) 0 0
\(157\) 12.1962 0.973359 0.486679 0.873581i \(-0.338208\pi\)
0.486679 + 0.873581i \(0.338208\pi\)
\(158\) 0 0
\(159\) −32.7846 −2.59999
\(160\) 0 0
\(161\) −3.46410 −0.273009
\(162\) 0 0
\(163\) −13.4641 −1.05459 −0.527295 0.849682i \(-0.676794\pi\)
−0.527295 + 0.849682i \(0.676794\pi\)
\(164\) 0 0
\(165\) 40.7846 3.17508
\(166\) 0 0
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) 0 0
\(169\) 32.3205 2.48619
\(170\) 0 0
\(171\) −5.66025 −0.432850
\(172\) 0 0
\(173\) −3.80385 −0.289201 −0.144601 0.989490i \(-0.546190\pi\)
−0.144601 + 0.989490i \(0.546190\pi\)
\(174\) 0 0
\(175\) 2.46410 0.186269
\(176\) 0 0
\(177\) 26.3923 1.98377
\(178\) 0 0
\(179\) −2.92820 −0.218864 −0.109432 0.993994i \(-0.534903\pi\)
−0.109432 + 0.993994i \(0.534903\pi\)
\(180\) 0 0
\(181\) 10.7321 0.797707 0.398854 0.917015i \(-0.369408\pi\)
0.398854 + 0.917015i \(0.369408\pi\)
\(182\) 0 0
\(183\) 30.3923 2.24666
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −10.9282 −0.799149
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 6.53590 0.470464 0.235232 0.971939i \(-0.424415\pi\)
0.235232 + 0.971939i \(0.424415\pi\)
\(194\) 0 0
\(195\) 50.2487 3.59838
\(196\) 0 0
\(197\) −25.8564 −1.84219 −0.921096 0.389335i \(-0.872705\pi\)
−0.921096 + 0.389335i \(0.872705\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 21.8564 1.54163
\(202\) 0 0
\(203\) 1.46410 0.102760
\(204\) 0 0
\(205\) −5.46410 −0.381629
\(206\) 0 0
\(207\) −15.4641 −1.07483
\(208\) 0 0
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) 16.7846 1.15550 0.577750 0.816214i \(-0.303931\pi\)
0.577750 + 0.816214i \(0.303931\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 14.9282 1.01810
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) −35.3205 −2.38674
\(220\) 0 0
\(221\) −13.4641 −0.905693
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 3.80385 0.252470 0.126235 0.992000i \(-0.459711\pi\)
0.126235 + 0.992000i \(0.459711\pi\)
\(228\) 0 0
\(229\) 17.2679 1.14110 0.570549 0.821263i \(-0.306730\pi\)
0.570549 + 0.821263i \(0.306730\pi\)
\(230\) 0 0
\(231\) −14.9282 −0.982204
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 29.8564 1.93938
\(238\) 0 0
\(239\) 12.5359 0.810880 0.405440 0.914122i \(-0.367118\pi\)
0.405440 + 0.914122i \(0.367118\pi\)
\(240\) 0 0
\(241\) −18.7846 −1.21002 −0.605012 0.796217i \(-0.706832\pi\)
−0.605012 + 0.796217i \(0.706832\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 0 0
\(245\) −2.73205 −0.174544
\(246\) 0 0
\(247\) 8.53590 0.543126
\(248\) 0 0
\(249\) −15.4641 −0.979998
\(250\) 0 0
\(251\) 15.8038 0.997530 0.498765 0.866737i \(-0.333787\pi\)
0.498765 + 0.866737i \(0.333787\pi\)
\(252\) 0 0
\(253\) 18.9282 1.19001
\(254\) 0 0
\(255\) −14.9282 −0.934840
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 1.46410 0.0909748
\(260\) 0 0
\(261\) 6.53590 0.404562
\(262\) 0 0
\(263\) 10.9282 0.673862 0.336931 0.941529i \(-0.390611\pi\)
0.336931 + 0.941529i \(0.390611\pi\)
\(264\) 0 0
\(265\) 32.7846 2.01394
\(266\) 0 0
\(267\) −32.3923 −1.98238
\(268\) 0 0
\(269\) −12.5885 −0.767532 −0.383766 0.923430i \(-0.625373\pi\)
−0.383766 + 0.923430i \(0.625373\pi\)
\(270\) 0 0
\(271\) −14.9282 −0.906824 −0.453412 0.891301i \(-0.649793\pi\)
−0.453412 + 0.891301i \(0.649793\pi\)
\(272\) 0 0
\(273\) −18.3923 −1.11315
\(274\) 0 0
\(275\) −13.4641 −0.811916
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 0 0
\(279\) −17.8564 −1.06904
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −16.5885 −0.986081 −0.493041 0.870006i \(-0.664115\pi\)
−0.493041 + 0.870006i \(0.664115\pi\)
\(284\) 0 0
\(285\) 9.46410 0.560605
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 24.3923 1.42990
\(292\) 0 0
\(293\) −12.1962 −0.712507 −0.356253 0.934389i \(-0.615946\pi\)
−0.356253 + 0.934389i \(0.615946\pi\)
\(294\) 0 0
\(295\) −26.3923 −1.53662
\(296\) 0 0
\(297\) −21.8564 −1.26824
\(298\) 0 0
\(299\) 23.3205 1.34866
\(300\) 0 0
\(301\) −5.46410 −0.314946
\(302\) 0 0
\(303\) 4.53590 0.260581
\(304\) 0 0
\(305\) −30.3923 −1.74026
\(306\) 0 0
\(307\) −8.58846 −0.490169 −0.245085 0.969502i \(-0.578816\pi\)
−0.245085 + 0.969502i \(0.578816\pi\)
\(308\) 0 0
\(309\) 26.9282 1.53189
\(310\) 0 0
\(311\) −5.85641 −0.332086 −0.166043 0.986118i \(-0.553099\pi\)
−0.166043 + 0.986118i \(0.553099\pi\)
\(312\) 0 0
\(313\) −19.8564 −1.12235 −0.561175 0.827697i \(-0.689651\pi\)
−0.561175 + 0.827697i \(0.689651\pi\)
\(314\) 0 0
\(315\) −12.1962 −0.687175
\(316\) 0 0
\(317\) −28.7846 −1.61670 −0.808352 0.588699i \(-0.799640\pi\)
−0.808352 + 0.588699i \(0.799640\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −13.8564 −0.773389
\(322\) 0 0
\(323\) −2.53590 −0.141101
\(324\) 0 0
\(325\) −16.5885 −0.920162
\(326\) 0 0
\(327\) −33.8564 −1.87226
\(328\) 0 0
\(329\) −2.92820 −0.161437
\(330\) 0 0
\(331\) 16.3923 0.901003 0.450501 0.892776i \(-0.351245\pi\)
0.450501 + 0.892776i \(0.351245\pi\)
\(332\) 0 0
\(333\) 6.53590 0.358165
\(334\) 0 0
\(335\) −21.8564 −1.19414
\(336\) 0 0
\(337\) 16.3923 0.892946 0.446473 0.894797i \(-0.352680\pi\)
0.446473 + 0.894797i \(0.352680\pi\)
\(338\) 0 0
\(339\) −17.8564 −0.969827
\(340\) 0 0
\(341\) 21.8564 1.18359
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 25.8564 1.39206
\(346\) 0 0
\(347\) 35.3205 1.89610 0.948052 0.318115i \(-0.103050\pi\)
0.948052 + 0.318115i \(0.103050\pi\)
\(348\) 0 0
\(349\) −14.0526 −0.752216 −0.376108 0.926576i \(-0.622738\pi\)
−0.376108 + 0.926576i \(0.622738\pi\)
\(350\) 0 0
\(351\) −26.9282 −1.43732
\(352\) 0 0
\(353\) 0.928203 0.0494033 0.0247016 0.999695i \(-0.492136\pi\)
0.0247016 + 0.999695i \(0.492136\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 5.46410 0.289191
\(358\) 0 0
\(359\) −27.4641 −1.44950 −0.724750 0.689012i \(-0.758045\pi\)
−0.724750 + 0.689012i \(0.758045\pi\)
\(360\) 0 0
\(361\) −17.3923 −0.915384
\(362\) 0 0
\(363\) 51.5167 2.70392
\(364\) 0 0
\(365\) 35.3205 1.84876
\(366\) 0 0
\(367\) −20.7846 −1.08495 −0.542474 0.840073i \(-0.682512\pi\)
−0.542474 + 0.840073i \(0.682512\pi\)
\(368\) 0 0
\(369\) 8.92820 0.464784
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 30.9282 1.60140 0.800701 0.599064i \(-0.204461\pi\)
0.800701 + 0.599064i \(0.204461\pi\)
\(374\) 0 0
\(375\) 18.9282 0.977448
\(376\) 0 0
\(377\) −9.85641 −0.507631
\(378\) 0 0
\(379\) 14.2487 0.731907 0.365954 0.930633i \(-0.380743\pi\)
0.365954 + 0.930633i \(0.380743\pi\)
\(380\) 0 0
\(381\) 31.3205 1.60460
\(382\) 0 0
\(383\) 32.7846 1.67522 0.837608 0.546272i \(-0.183954\pi\)
0.837608 + 0.546272i \(0.183954\pi\)
\(384\) 0 0
\(385\) 14.9282 0.760812
\(386\) 0 0
\(387\) −24.3923 −1.23993
\(388\) 0 0
\(389\) −23.3205 −1.18240 −0.591198 0.806526i \(-0.701345\pi\)
−0.591198 + 0.806526i \(0.701345\pi\)
\(390\) 0 0
\(391\) −6.92820 −0.350374
\(392\) 0 0
\(393\) −29.3205 −1.47902
\(394\) 0 0
\(395\) −29.8564 −1.50224
\(396\) 0 0
\(397\) 4.87564 0.244702 0.122351 0.992487i \(-0.460957\pi\)
0.122351 + 0.992487i \(0.460957\pi\)
\(398\) 0 0
\(399\) −3.46410 −0.173422
\(400\) 0 0
\(401\) 13.4641 0.672365 0.336183 0.941797i \(-0.390864\pi\)
0.336183 + 0.941797i \(0.390864\pi\)
\(402\) 0 0
\(403\) 26.9282 1.34139
\(404\) 0 0
\(405\) 6.73205 0.334518
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 3.07180 0.151891 0.0759453 0.997112i \(-0.475803\pi\)
0.0759453 + 0.997112i \(0.475803\pi\)
\(410\) 0 0
\(411\) −54.2487 −2.67589
\(412\) 0 0
\(413\) 9.66025 0.475350
\(414\) 0 0
\(415\) 15.4641 0.759103
\(416\) 0 0
\(417\) 25.3205 1.23995
\(418\) 0 0
\(419\) 23.8038 1.16289 0.581447 0.813584i \(-0.302487\pi\)
0.581447 + 0.813584i \(0.302487\pi\)
\(420\) 0 0
\(421\) 25.8564 1.26016 0.630082 0.776529i \(-0.283021\pi\)
0.630082 + 0.776529i \(0.283021\pi\)
\(422\) 0 0
\(423\) −13.0718 −0.635573
\(424\) 0 0
\(425\) 4.92820 0.239053
\(426\) 0 0
\(427\) 11.1244 0.538345
\(428\) 0 0
\(429\) 100.497 4.85206
\(430\) 0 0
\(431\) 35.4641 1.70825 0.854123 0.520071i \(-0.174095\pi\)
0.854123 + 0.520071i \(0.174095\pi\)
\(432\) 0 0
\(433\) 15.8564 0.762010 0.381005 0.924573i \(-0.375578\pi\)
0.381005 + 0.924573i \(0.375578\pi\)
\(434\) 0 0
\(435\) −10.9282 −0.523967
\(436\) 0 0
\(437\) 4.39230 0.210112
\(438\) 0 0
\(439\) 30.9282 1.47612 0.738061 0.674734i \(-0.235742\pi\)
0.738061 + 0.674734i \(0.235742\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 32.3923 1.53554
\(446\) 0 0
\(447\) −2.92820 −0.138499
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −10.9282 −0.514589
\(452\) 0 0
\(453\) −6.53590 −0.307083
\(454\) 0 0
\(455\) 18.3923 0.862245
\(456\) 0 0
\(457\) −15.3205 −0.716663 −0.358332 0.933594i \(-0.616654\pi\)
−0.358332 + 0.933594i \(0.616654\pi\)
\(458\) 0 0
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −24.1962 −1.12693 −0.563464 0.826141i \(-0.690531\pi\)
−0.563464 + 0.826141i \(0.690531\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 29.8564 1.38456
\(466\) 0 0
\(467\) 0.875644 0.0405200 0.0202600 0.999795i \(-0.493551\pi\)
0.0202600 + 0.999795i \(0.493551\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 33.3205 1.53533
\(472\) 0 0
\(473\) 29.8564 1.37280
\(474\) 0 0
\(475\) −3.12436 −0.143355
\(476\) 0 0
\(477\) −53.5692 −2.45277
\(478\) 0 0
\(479\) 9.85641 0.450351 0.225175 0.974318i \(-0.427705\pi\)
0.225175 + 0.974318i \(0.427705\pi\)
\(480\) 0 0
\(481\) −9.85641 −0.449413
\(482\) 0 0
\(483\) −9.46410 −0.430632
\(484\) 0 0
\(485\) −24.3923 −1.10760
\(486\) 0 0
\(487\) −21.6077 −0.979138 −0.489569 0.871965i \(-0.662846\pi\)
−0.489569 + 0.871965i \(0.662846\pi\)
\(488\) 0 0
\(489\) −36.7846 −1.66346
\(490\) 0 0
\(491\) −32.7846 −1.47955 −0.739774 0.672855i \(-0.765068\pi\)
−0.739774 + 0.672855i \(0.765068\pi\)
\(492\) 0 0
\(493\) 2.92820 0.131880
\(494\) 0 0
\(495\) 66.6410 2.99529
\(496\) 0 0
\(497\) 2.92820 0.131348
\(498\) 0 0
\(499\) −13.0718 −0.585174 −0.292587 0.956239i \(-0.594516\pi\)
−0.292587 + 0.956239i \(0.594516\pi\)
\(500\) 0 0
\(501\) −13.8564 −0.619059
\(502\) 0 0
\(503\) 30.9282 1.37902 0.689510 0.724276i \(-0.257826\pi\)
0.689510 + 0.724276i \(0.257826\pi\)
\(504\) 0 0
\(505\) −4.53590 −0.201845
\(506\) 0 0
\(507\) 88.3013 3.92160
\(508\) 0 0
\(509\) −3.12436 −0.138485 −0.0692423 0.997600i \(-0.522058\pi\)
−0.0692423 + 0.997600i \(0.522058\pi\)
\(510\) 0 0
\(511\) −12.9282 −0.571910
\(512\) 0 0
\(513\) −5.07180 −0.223925
\(514\) 0 0
\(515\) −26.9282 −1.18660
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) −10.3923 −0.456172
\(520\) 0 0
\(521\) 14.7846 0.647726 0.323863 0.946104i \(-0.395018\pi\)
0.323863 + 0.946104i \(0.395018\pi\)
\(522\) 0 0
\(523\) −3.41154 −0.149176 −0.0745882 0.997214i \(-0.523764\pi\)
−0.0745882 + 0.997214i \(0.523764\pi\)
\(524\) 0 0
\(525\) 6.73205 0.293811
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 43.1244 1.87144
\(532\) 0 0
\(533\) −13.4641 −0.583195
\(534\) 0 0
\(535\) 13.8564 0.599065
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −5.46410 −0.235356
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 29.3205 1.25826
\(544\) 0 0
\(545\) 33.8564 1.45025
\(546\) 0 0
\(547\) −46.2487 −1.97745 −0.988726 0.149736i \(-0.952158\pi\)
−0.988726 + 0.149736i \(0.952158\pi\)
\(548\) 0 0
\(549\) 49.6603 2.11945
\(550\) 0 0
\(551\) −1.85641 −0.0790856
\(552\) 0 0
\(553\) 10.9282 0.464714
\(554\) 0 0
\(555\) −10.9282 −0.463876
\(556\) 0 0
\(557\) 23.7128 1.00474 0.502372 0.864652i \(-0.332461\pi\)
0.502372 + 0.864652i \(0.332461\pi\)
\(558\) 0 0
\(559\) 36.7846 1.55582
\(560\) 0 0
\(561\) −29.8564 −1.26054
\(562\) 0 0
\(563\) 34.0526 1.43514 0.717572 0.696484i \(-0.245253\pi\)
0.717572 + 0.696484i \(0.245253\pi\)
\(564\) 0 0
\(565\) 17.8564 0.751225
\(566\) 0 0
\(567\) −2.46410 −0.103483
\(568\) 0 0
\(569\) 5.46410 0.229067 0.114534 0.993419i \(-0.463463\pi\)
0.114534 + 0.993419i \(0.463463\pi\)
\(570\) 0 0
\(571\) −25.1769 −1.05362 −0.526811 0.849983i \(-0.676612\pi\)
−0.526811 + 0.849983i \(0.676612\pi\)
\(572\) 0 0
\(573\) −43.7128 −1.82613
\(574\) 0 0
\(575\) −8.53590 −0.355972
\(576\) 0 0
\(577\) −31.0718 −1.29354 −0.646768 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(578\) 0 0
\(579\) 17.8564 0.742087
\(580\) 0 0
\(581\) −5.66025 −0.234827
\(582\) 0 0
\(583\) 65.5692 2.71560
\(584\) 0 0
\(585\) 82.1051 3.39463
\(586\) 0 0
\(587\) 33.2679 1.37312 0.686558 0.727075i \(-0.259121\pi\)
0.686558 + 0.727075i \(0.259121\pi\)
\(588\) 0 0
\(589\) 5.07180 0.208980
\(590\) 0 0
\(591\) −70.6410 −2.90578
\(592\) 0 0
\(593\) 12.1436 0.498678 0.249339 0.968416i \(-0.419787\pi\)
0.249339 + 0.968416i \(0.419787\pi\)
\(594\) 0 0
\(595\) −5.46410 −0.224006
\(596\) 0 0
\(597\) −10.9282 −0.447262
\(598\) 0 0
\(599\) 27.7128 1.13231 0.566157 0.824297i \(-0.308429\pi\)
0.566157 + 0.824297i \(0.308429\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 35.7128 1.45434
\(604\) 0 0
\(605\) −51.5167 −2.09445
\(606\) 0 0
\(607\) 27.7128 1.12483 0.562414 0.826856i \(-0.309873\pi\)
0.562414 + 0.826856i \(0.309873\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 19.7128 0.797495
\(612\) 0 0
\(613\) −39.3205 −1.58814 −0.794070 0.607826i \(-0.792042\pi\)
−0.794070 + 0.607826i \(0.792042\pi\)
\(614\) 0 0
\(615\) −14.9282 −0.601963
\(616\) 0 0
\(617\) 24.3923 0.981997 0.490999 0.871160i \(-0.336632\pi\)
0.490999 + 0.871160i \(0.336632\pi\)
\(618\) 0 0
\(619\) 24.1962 0.972525 0.486263 0.873813i \(-0.338360\pi\)
0.486263 + 0.873813i \(0.338360\pi\)
\(620\) 0 0
\(621\) −13.8564 −0.556038
\(622\) 0 0
\(623\) −11.8564 −0.475017
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) 18.9282 0.755920
\(628\) 0 0
\(629\) 2.92820 0.116755
\(630\) 0 0
\(631\) −49.5692 −1.97332 −0.986660 0.162796i \(-0.947949\pi\)
−0.986660 + 0.162796i \(0.947949\pi\)
\(632\) 0 0
\(633\) 45.8564 1.82263
\(634\) 0 0
\(635\) −31.3205 −1.24292
\(636\) 0 0
\(637\) −6.73205 −0.266734
\(638\) 0 0
\(639\) 13.0718 0.517112
\(640\) 0 0
\(641\) −22.2487 −0.878771 −0.439386 0.898299i \(-0.644804\pi\)
−0.439386 + 0.898299i \(0.644804\pi\)
\(642\) 0 0
\(643\) 14.3397 0.565504 0.282752 0.959193i \(-0.408753\pi\)
0.282752 + 0.959193i \(0.408753\pi\)
\(644\) 0 0
\(645\) 40.7846 1.60589
\(646\) 0 0
\(647\) −1.85641 −0.0729829 −0.0364914 0.999334i \(-0.511618\pi\)
−0.0364914 + 0.999334i \(0.511618\pi\)
\(648\) 0 0
\(649\) −52.7846 −2.07198
\(650\) 0 0
\(651\) −10.9282 −0.428310
\(652\) 0 0
\(653\) −38.5359 −1.50803 −0.754013 0.656859i \(-0.771885\pi\)
−0.754013 + 0.656859i \(0.771885\pi\)
\(654\) 0 0
\(655\) 29.3205 1.14565
\(656\) 0 0
\(657\) −57.7128 −2.25159
\(658\) 0 0
\(659\) 3.32051 0.129349 0.0646743 0.997906i \(-0.479399\pi\)
0.0646743 + 0.997906i \(0.479399\pi\)
\(660\) 0 0
\(661\) −28.9808 −1.12722 −0.563611 0.826041i \(-0.690588\pi\)
−0.563611 + 0.826041i \(0.690588\pi\)
\(662\) 0 0
\(663\) −36.7846 −1.42860
\(664\) 0 0
\(665\) 3.46410 0.134332
\(666\) 0 0
\(667\) −5.07180 −0.196381
\(668\) 0 0
\(669\) 18.9282 0.731807
\(670\) 0 0
\(671\) −60.7846 −2.34656
\(672\) 0 0
\(673\) 4.14359 0.159724 0.0798619 0.996806i \(-0.474552\pi\)
0.0798619 + 0.996806i \(0.474552\pi\)
\(674\) 0 0
\(675\) 9.85641 0.379373
\(676\) 0 0
\(677\) 14.4449 0.555161 0.277581 0.960702i \(-0.410467\pi\)
0.277581 + 0.960702i \(0.410467\pi\)
\(678\) 0 0
\(679\) 8.92820 0.342633
\(680\) 0 0
\(681\) 10.3923 0.398234
\(682\) 0 0
\(683\) −30.6410 −1.17245 −0.586223 0.810150i \(-0.699386\pi\)
−0.586223 + 0.810150i \(0.699386\pi\)
\(684\) 0 0
\(685\) 54.2487 2.07274
\(686\) 0 0
\(687\) 47.1769 1.79991
\(688\) 0 0
\(689\) 80.7846 3.07765
\(690\) 0 0
\(691\) 11.1244 0.423190 0.211595 0.977357i \(-0.432134\pi\)
0.211595 + 0.977357i \(0.432134\pi\)
\(692\) 0 0
\(693\) −24.3923 −0.926587
\(694\) 0 0
\(695\) −25.3205 −0.960462
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −16.3923 −0.620014
\(700\) 0 0
\(701\) −19.6077 −0.740572 −0.370286 0.928918i \(-0.620740\pi\)
−0.370286 + 0.928918i \(0.620740\pi\)
\(702\) 0 0
\(703\) −1.85641 −0.0700157
\(704\) 0 0
\(705\) 21.8564 0.823160
\(706\) 0 0
\(707\) 1.66025 0.0624403
\(708\) 0 0
\(709\) −48.1051 −1.80663 −0.903313 0.428982i \(-0.858872\pi\)
−0.903313 + 0.428982i \(0.858872\pi\)
\(710\) 0 0
\(711\) 48.7846 1.82957
\(712\) 0 0
\(713\) 13.8564 0.518927
\(714\) 0 0
\(715\) −100.497 −3.75839
\(716\) 0 0
\(717\) 34.2487 1.27904
\(718\) 0 0
\(719\) −21.0718 −0.785845 −0.392923 0.919572i \(-0.628536\pi\)
−0.392923 + 0.919572i \(0.628536\pi\)
\(720\) 0 0
\(721\) 9.85641 0.367072
\(722\) 0 0
\(723\) −51.3205 −1.90863
\(724\) 0 0
\(725\) 3.60770 0.133986
\(726\) 0 0
\(727\) 10.9282 0.405305 0.202652 0.979251i \(-0.435044\pi\)
0.202652 + 0.979251i \(0.435044\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −10.9282 −0.404194
\(732\) 0 0
\(733\) −13.6603 −0.504553 −0.252276 0.967655i \(-0.581179\pi\)
−0.252276 + 0.967655i \(0.581179\pi\)
\(734\) 0 0
\(735\) −7.46410 −0.275318
\(736\) 0 0
\(737\) −43.7128 −1.61018
\(738\) 0 0
\(739\) −3.32051 −0.122147 −0.0610734 0.998133i \(-0.519452\pi\)
−0.0610734 + 0.998133i \(0.519452\pi\)
\(740\) 0 0
\(741\) 23.3205 0.856700
\(742\) 0 0
\(743\) 32.2487 1.18309 0.591545 0.806272i \(-0.298518\pi\)
0.591545 + 0.806272i \(0.298518\pi\)
\(744\) 0 0
\(745\) 2.92820 0.107281
\(746\) 0 0
\(747\) −25.2679 −0.924506
\(748\) 0 0
\(749\) −5.07180 −0.185319
\(750\) 0 0
\(751\) −24.2487 −0.884848 −0.442424 0.896806i \(-0.645881\pi\)
−0.442424 + 0.896806i \(0.645881\pi\)
\(752\) 0 0
\(753\) 43.1769 1.57345
\(754\) 0 0
\(755\) 6.53590 0.237866
\(756\) 0 0
\(757\) 10.2487 0.372496 0.186248 0.982503i \(-0.440367\pi\)
0.186248 + 0.982503i \(0.440367\pi\)
\(758\) 0 0
\(759\) 51.7128 1.87706
\(760\) 0 0
\(761\) 21.7128 0.787089 0.393544 0.919306i \(-0.371249\pi\)
0.393544 + 0.919306i \(0.371249\pi\)
\(762\) 0 0
\(763\) −12.3923 −0.448632
\(764\) 0 0
\(765\) −24.3923 −0.881906
\(766\) 0 0
\(767\) −65.0333 −2.34822
\(768\) 0 0
\(769\) −10.7846 −0.388903 −0.194451 0.980912i \(-0.562293\pi\)
−0.194451 + 0.980912i \(0.562293\pi\)
\(770\) 0 0
\(771\) −16.3923 −0.590354
\(772\) 0 0
\(773\) −20.8756 −0.750845 −0.375422 0.926854i \(-0.622502\pi\)
−0.375422 + 0.926854i \(0.622502\pi\)
\(774\) 0 0
\(775\) −9.85641 −0.354053
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) −2.53590 −0.0908580
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 5.85641 0.209291
\(784\) 0 0
\(785\) −33.3205 −1.18926
\(786\) 0 0
\(787\) 20.5885 0.733899 0.366950 0.930241i \(-0.380402\pi\)
0.366950 + 0.930241i \(0.380402\pi\)
\(788\) 0 0
\(789\) 29.8564 1.06292
\(790\) 0 0
\(791\) −6.53590 −0.232390
\(792\) 0 0
\(793\) −74.8897 −2.65941
\(794\) 0 0
\(795\) 89.5692 3.17669
\(796\) 0 0
\(797\) 12.9808 0.459802 0.229901 0.973214i \(-0.426160\pi\)
0.229901 + 0.973214i \(0.426160\pi\)
\(798\) 0 0
\(799\) −5.85641 −0.207185
\(800\) 0 0
\(801\) −52.9282 −1.87013
\(802\) 0 0
\(803\) 70.6410 2.49287
\(804\) 0 0
\(805\) 9.46410 0.333566
\(806\) 0 0
\(807\) −34.3923 −1.21067
\(808\) 0 0
\(809\) −23.3205 −0.819905 −0.409953 0.912107i \(-0.634455\pi\)
−0.409953 + 0.912107i \(0.634455\pi\)
\(810\) 0 0
\(811\) 27.1244 0.952465 0.476232 0.879319i \(-0.342002\pi\)
0.476232 + 0.879319i \(0.342002\pi\)
\(812\) 0 0
\(813\) −40.7846 −1.43038
\(814\) 0 0
\(815\) 36.7846 1.28851
\(816\) 0 0
\(817\) 6.92820 0.242387
\(818\) 0 0
\(819\) −30.0526 −1.05012
\(820\) 0 0
\(821\) 6.92820 0.241796 0.120898 0.992665i \(-0.461423\pi\)
0.120898 + 0.992665i \(0.461423\pi\)
\(822\) 0 0
\(823\) −2.92820 −0.102071 −0.0510354 0.998697i \(-0.516252\pi\)
−0.0510354 + 0.998697i \(0.516252\pi\)
\(824\) 0 0
\(825\) −36.7846 −1.28068
\(826\) 0 0
\(827\) 13.8564 0.481834 0.240917 0.970546i \(-0.422552\pi\)
0.240917 + 0.970546i \(0.422552\pi\)
\(828\) 0 0
\(829\) 40.9808 1.42332 0.711660 0.702524i \(-0.247944\pi\)
0.711660 + 0.702524i \(0.247944\pi\)
\(830\) 0 0
\(831\) 10.9282 0.379095
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −39.7128 −1.37104 −0.685519 0.728054i \(-0.740425\pi\)
−0.685519 + 0.728054i \(0.740425\pi\)
\(840\) 0 0
\(841\) −26.8564 −0.926083
\(842\) 0 0
\(843\) −5.46410 −0.188194
\(844\) 0 0
\(845\) −88.3013 −3.03766
\(846\) 0 0
\(847\) 18.8564 0.647914
\(848\) 0 0
\(849\) −45.3205 −1.55540
\(850\) 0 0
\(851\) −5.07180 −0.173859
\(852\) 0 0
\(853\) 4.98076 0.170538 0.0852690 0.996358i \(-0.472825\pi\)
0.0852690 + 0.996358i \(0.472825\pi\)
\(854\) 0 0
\(855\) 15.4641 0.528861
\(856\) 0 0
\(857\) 30.7846 1.05158 0.525791 0.850614i \(-0.323769\pi\)
0.525791 + 0.850614i \(0.323769\pi\)
\(858\) 0 0
\(859\) 30.4449 1.03877 0.519383 0.854542i \(-0.326162\pi\)
0.519383 + 0.854542i \(0.326162\pi\)
\(860\) 0 0
\(861\) 5.46410 0.186216
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 10.3923 0.353349
\(866\) 0 0
\(867\) −35.5167 −1.20621
\(868\) 0 0
\(869\) −59.7128 −2.02562
\(870\) 0 0
\(871\) −53.8564 −1.82485
\(872\) 0 0
\(873\) 39.8564 1.34893
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) 20.3923 0.688599 0.344300 0.938860i \(-0.388116\pi\)
0.344300 + 0.938860i \(0.388116\pi\)
\(878\) 0 0
\(879\) −33.3205 −1.12387
\(880\) 0 0
\(881\) −50.7846 −1.71098 −0.855488 0.517822i \(-0.826743\pi\)
−0.855488 + 0.517822i \(0.826743\pi\)
\(882\) 0 0
\(883\) 5.07180 0.170680 0.0853398 0.996352i \(-0.472802\pi\)
0.0853398 + 0.996352i \(0.472802\pi\)
\(884\) 0 0
\(885\) −72.1051 −2.42379
\(886\) 0 0
\(887\) −46.6410 −1.56605 −0.783026 0.621989i \(-0.786325\pi\)
−0.783026 + 0.621989i \(0.786325\pi\)
\(888\) 0 0
\(889\) 11.4641 0.384494
\(890\) 0 0
\(891\) 13.4641 0.451064
\(892\) 0 0
\(893\) 3.71281 0.124245
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 63.7128 2.12731
\(898\) 0 0
\(899\) −5.85641 −0.195322
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) −14.9282 −0.496779
\(904\) 0 0
\(905\) −29.3205 −0.974647
\(906\) 0 0
\(907\) 43.7128 1.45146 0.725730 0.687980i \(-0.241502\pi\)
0.725730 + 0.687980i \(0.241502\pi\)
\(908\) 0 0
\(909\) 7.41154 0.245825
\(910\) 0 0
\(911\) 4.53590 0.150281 0.0751405 0.997173i \(-0.476059\pi\)
0.0751405 + 0.997173i \(0.476059\pi\)
\(912\) 0 0
\(913\) 30.9282 1.02357
\(914\) 0 0
\(915\) −83.0333 −2.74500
\(916\) 0 0
\(917\) −10.7321 −0.354404
\(918\) 0 0
\(919\) −7.21539 −0.238014 −0.119007 0.992893i \(-0.537971\pi\)
−0.119007 + 0.992893i \(0.537971\pi\)
\(920\) 0 0
\(921\) −23.4641 −0.773168
\(922\) 0 0
\(923\) −19.7128 −0.648855
\(924\) 0 0
\(925\) 3.60770 0.118620
\(926\) 0 0
\(927\) 44.0000 1.44515
\(928\) 0 0
\(929\) 37.7128 1.23732 0.618659 0.785660i \(-0.287676\pi\)
0.618659 + 0.785660i \(0.287676\pi\)
\(930\) 0 0
\(931\) −1.26795 −0.0415554
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) 29.8564 0.976409
\(936\) 0 0
\(937\) −8.64102 −0.282290 −0.141145 0.989989i \(-0.545078\pi\)
−0.141145 + 0.989989i \(0.545078\pi\)
\(938\) 0 0
\(939\) −54.2487 −1.77034
\(940\) 0 0
\(941\) −44.9808 −1.46633 −0.733165 0.680050i \(-0.761958\pi\)
−0.733165 + 0.680050i \(0.761958\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) 0 0
\(945\) −10.9282 −0.355494
\(946\) 0 0
\(947\) −24.3923 −0.792643 −0.396322 0.918112i \(-0.629714\pi\)
−0.396322 + 0.918112i \(0.629714\pi\)
\(948\) 0 0
\(949\) 87.0333 2.82522
\(950\) 0 0
\(951\) −78.6410 −2.55011
\(952\) 0 0
\(953\) 19.8564 0.643212 0.321606 0.946874i \(-0.395777\pi\)
0.321606 + 0.946874i \(0.395777\pi\)
\(954\) 0 0
\(955\) 43.7128 1.41451
\(956\) 0 0
\(957\) −21.8564 −0.706517
\(958\) 0 0
\(959\) −19.8564 −0.641197
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −22.6410 −0.729597
\(964\) 0 0
\(965\) −17.8564 −0.574818
\(966\) 0 0
\(967\) −15.1769 −0.488057 −0.244028 0.969768i \(-0.578469\pi\)
−0.244028 + 0.969768i \(0.578469\pi\)
\(968\) 0 0
\(969\) −6.92820 −0.222566
\(970\) 0 0
\(971\) 55.5167 1.78161 0.890807 0.454381i \(-0.150140\pi\)
0.890807 + 0.454381i \(0.150140\pi\)
\(972\) 0 0
\(973\) 9.26795 0.297117
\(974\) 0 0
\(975\) −45.3205 −1.45142
\(976\) 0 0
\(977\) −28.1436 −0.900393 −0.450197 0.892929i \(-0.648646\pi\)
−0.450197 + 0.892929i \(0.648646\pi\)
\(978\) 0 0
\(979\) 64.7846 2.07053
\(980\) 0 0
\(981\) −55.3205 −1.76625
\(982\) 0 0
\(983\) −47.7128 −1.52180 −0.760901 0.648868i \(-0.775243\pi\)
−0.760901 + 0.648868i \(0.775243\pi\)
\(984\) 0 0
\(985\) 70.6410 2.25081
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 18.9282 0.601882
\(990\) 0 0
\(991\) −24.7846 −0.787309 −0.393655 0.919258i \(-0.628789\pi\)
−0.393655 + 0.919258i \(0.628789\pi\)
\(992\) 0 0
\(993\) 44.7846 1.42120
\(994\) 0 0
\(995\) 10.9282 0.346447
\(996\) 0 0
\(997\) 5.26795 0.166838 0.0834188 0.996515i \(-0.473416\pi\)
0.0834188 + 0.996515i \(0.473416\pi\)
\(998\) 0 0
\(999\) 5.85641 0.185289
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 1792.2.a.q.1.2 2
4.3 odd 2 1792.2.a.i.1.1 2
8.3 odd 2 1792.2.a.s.1.2 2
8.5 even 2 1792.2.a.k.1.1 2
16.3 odd 4 448.2.b.d.225.1 yes 4
16.5 even 4 448.2.b.c.225.1 4
16.11 odd 4 448.2.b.d.225.4 yes 4
16.13 even 4 448.2.b.c.225.4 yes 4
48.5 odd 4 4032.2.c.k.2017.4 4
48.11 even 4 4032.2.c.n.2017.4 4
48.29 odd 4 4032.2.c.k.2017.1 4
48.35 even 4 4032.2.c.n.2017.1 4
112.13 odd 4 3136.2.b.g.1569.1 4
112.27 even 4 3136.2.b.h.1569.1 4
112.69 odd 4 3136.2.b.g.1569.4 4
112.83 even 4 3136.2.b.h.1569.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
448.2.b.c.225.1 4 16.5 even 4
448.2.b.c.225.4 yes 4 16.13 even 4
448.2.b.d.225.1 yes 4 16.3 odd 4
448.2.b.d.225.4 yes 4 16.11 odd 4
1792.2.a.i.1.1 2 4.3 odd 2
1792.2.a.k.1.1 2 8.5 even 2
1792.2.a.q.1.2 2 1.1 even 1 trivial
1792.2.a.s.1.2 2 8.3 odd 2
3136.2.b.g.1569.1 4 112.13 odd 4
3136.2.b.g.1569.4 4 112.69 odd 4
3136.2.b.h.1569.1 4 112.27 even 4
3136.2.b.h.1569.4 4 112.83 even 4
4032.2.c.k.2017.1 4 48.29 odd 4
4032.2.c.k.2017.4 4 48.5 odd 4
4032.2.c.n.2017.1 4 48.35 even 4
4032.2.c.n.2017.4 4 48.11 even 4