Properties

Label 6048.2.a.bn.1.4
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.31526\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.58060 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+2.58060 q^{5} +1.00000 q^{7} -4.04992 q^{11} -0.190188 q^{13} -6.82071 q^{17} -1.34049 q^{19} +0.240108 q^{23} +1.65951 q^{25} +0.190188 q^{29} +2.46932 q^{31} +2.58060 q^{35} +10.4512 q^{37} -2.77079 q^{41} -11.4512 q^{43} +4.29003 q^{47} +1.00000 q^{49} +9.26104 q^{53} -10.4512 q^{55} -12.9710 q^{59} -13.6414 q^{61} -0.490799 q^{65} -0.871175 q^{67} -11.0608 q^{71} +12.5126 q^{73} -4.04992 q^{77} -15.0709 q^{79} -4.19019 q^{83} -17.6015 q^{85} -10.3904 q^{89} -0.190188 q^{91} -3.45928 q^{95} -10.3224 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} + 4q^{7} - 2q^{11} - 4q^{13} - 4q^{17} - 4q^{19} - 10q^{23} + 8q^{25} + 4q^{29} + 8q^{31} - 2q^{35} - 8q^{37} - 2q^{41} + 4q^{43} - 8q^{47} + 4q^{49} - 16q^{53} + 8q^{55} - 24q^{59} - 8q^{61} + 4q^{65} - 4q^{67} - 10q^{71} + 4q^{73} - 2q^{77} - 4q^{79} - 20q^{83} - 16q^{85} - 26q^{89} - 4q^{91} - 34q^{95} + 8q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.58060 1.15408 0.577040 0.816716i \(-0.304208\pi\)
0.577040 + 0.816716i \(0.304208\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.04992 −1.22110 −0.610548 0.791979i \(-0.709051\pi\)
−0.610548 + 0.791979i \(0.709051\pi\)
\(12\) 0 0
\(13\) −0.190188 −0.0527486 −0.0263743 0.999652i \(-0.508396\pi\)
−0.0263743 + 0.999652i \(0.508396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.82071 −1.65427 −0.827133 0.562007i \(-0.810029\pi\)
−0.827133 + 0.562007i \(0.810029\pi\)
\(18\) 0 0
\(19\) −1.34049 −0.307530 −0.153765 0.988107i \(-0.549140\pi\)
−0.153765 + 0.988107i \(0.549140\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.240108 0.0500661 0.0250330 0.999687i \(-0.492031\pi\)
0.0250330 + 0.999687i \(0.492031\pi\)
\(24\) 0 0
\(25\) 1.65951 0.331901
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.190188 0.0353170 0.0176585 0.999844i \(-0.494379\pi\)
0.0176585 + 0.999844i \(0.494379\pi\)
\(30\) 0 0
\(31\) 2.46932 0.443503 0.221751 0.975103i \(-0.428823\pi\)
0.221751 + 0.975103i \(0.428823\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.58060 0.436201
\(36\) 0 0
\(37\) 10.4512 1.71817 0.859086 0.511831i \(-0.171033\pi\)
0.859086 + 0.511831i \(0.171033\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.77079 −0.432725 −0.216362 0.976313i \(-0.569419\pi\)
−0.216362 + 0.976313i \(0.569419\pi\)
\(42\) 0 0
\(43\) −11.4512 −1.74630 −0.873148 0.487455i \(-0.837925\pi\)
−0.873148 + 0.487455i \(0.837925\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.29003 0.625765 0.312883 0.949792i \(-0.398705\pi\)
0.312883 + 0.949792i \(0.398705\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.26104 1.27210 0.636051 0.771647i \(-0.280567\pi\)
0.636051 + 0.771647i \(0.280567\pi\)
\(54\) 0 0
\(55\) −10.4512 −1.40924
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.9710 −1.68868 −0.844341 0.535806i \(-0.820008\pi\)
−0.844341 + 0.535806i \(0.820008\pi\)
\(60\) 0 0
\(61\) −13.6414 −1.74660 −0.873302 0.487178i \(-0.838026\pi\)
−0.873302 + 0.487178i \(0.838026\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.490799 −0.0608761
\(66\) 0 0
\(67\) −0.871175 −0.106431 −0.0532155 0.998583i \(-0.516947\pi\)
−0.0532155 + 0.998583i \(0.516947\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.0608 −1.31268 −0.656339 0.754466i \(-0.727896\pi\)
−0.656339 + 0.754466i \(0.727896\pi\)
\(72\) 0 0
\(73\) 12.5126 1.46449 0.732244 0.681042i \(-0.238473\pi\)
0.732244 + 0.681042i \(0.238473\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.04992 −0.461531
\(78\) 0 0
\(79\) −15.0709 −1.69560 −0.847802 0.530313i \(-0.822074\pi\)
−0.847802 + 0.530313i \(0.822074\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.19019 −0.459933 −0.229966 0.973199i \(-0.573862\pi\)
−0.229966 + 0.973199i \(0.573862\pi\)
\(84\) 0 0
\(85\) −17.6015 −1.90915
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.3904 −1.10138 −0.550691 0.834709i \(-0.685636\pi\)
−0.550691 + 0.834709i \(0.685636\pi\)
\(90\) 0 0
\(91\) −0.190188 −0.0199371
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.45928 −0.354915
\(96\) 0 0
\(97\) −10.3224 −1.04808 −0.524041 0.851693i \(-0.675576\pi\)
−0.524041 + 0.851693i \(0.675576\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.50170 0.746447 0.373223 0.927742i \(-0.378252\pi\)
0.373223 + 0.927742i \(0.378252\pi\)
\(102\) 0 0
\(103\) 2.08894 0.205830 0.102915 0.994690i \(-0.467183\pi\)
0.102915 + 0.994690i \(0.467183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.65951 −0.933820 −0.466910 0.884305i \(-0.654633\pi\)
−0.466910 + 0.884305i \(0.654633\pi\)
\(108\) 0 0
\(109\) −2.46932 −0.236518 −0.118259 0.992983i \(-0.537731\pi\)
−0.118259 + 0.992983i \(0.537731\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.83880 0.643340 0.321670 0.946852i \(-0.395756\pi\)
0.321670 + 0.946852i \(0.395756\pi\)
\(114\) 0 0
\(115\) 0.619624 0.0577803
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.82071 −0.625253
\(120\) 0 0
\(121\) 5.40186 0.491078
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.62048 −0.771040
\(126\) 0 0
\(127\) 13.4512 1.19360 0.596802 0.802389i \(-0.296438\pi\)
0.596802 + 0.802389i \(0.296438\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.3514 −1.16652 −0.583258 0.812287i \(-0.698222\pi\)
−0.583258 + 0.812287i \(0.698222\pi\)
\(132\) 0 0
\(133\) −1.34049 −0.116236
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.77025 −0.749293 −0.374646 0.927168i \(-0.622236\pi\)
−0.374646 + 0.927168i \(0.622236\pi\)
\(138\) 0 0
\(139\) 1.06136 0.0900236 0.0450118 0.998986i \(-0.485667\pi\)
0.0450118 + 0.998986i \(0.485667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.770246 0.0644112
\(144\) 0 0
\(145\) 0.490799 0.0407587
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.54158 0.781677 0.390838 0.920459i \(-0.372185\pi\)
0.390838 + 0.920459i \(0.372185\pi\)
\(150\) 0 0
\(151\) 7.64142 0.621850 0.310925 0.950434i \(-0.399361\pi\)
0.310925 + 0.950434i \(0.399361\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.37233 0.511838
\(156\) 0 0
\(157\) 19.7736 1.57811 0.789054 0.614324i \(-0.210571\pi\)
0.789054 + 0.614324i \(0.210571\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.240108 0.0189232
\(162\) 0 0
\(163\) −16.3899 −1.28375 −0.641877 0.766808i \(-0.721844\pi\)
−0.641877 + 0.766808i \(0.721844\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.77743 −0.447071 −0.223536 0.974696i \(-0.571760\pi\)
−0.223536 + 0.974696i \(0.571760\pi\)
\(168\) 0 0
\(169\) −12.9638 −0.997218
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.240108 −0.0182551 −0.00912755 0.999958i \(-0.502905\pi\)
−0.00912755 + 0.999958i \(0.502905\pi\)
\(174\) 0 0
\(175\) 1.65951 0.125447
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.30093 0.545697 0.272848 0.962057i \(-0.412034\pi\)
0.272848 + 0.962057i \(0.412034\pi\)
\(180\) 0 0
\(181\) −18.1997 −1.35277 −0.676386 0.736547i \(-0.736455\pi\)
−0.676386 + 0.736547i \(0.736455\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 26.9705 1.98291
\(186\) 0 0
\(187\) 27.6233 2.02002
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.1821 −1.31561 −0.657807 0.753187i \(-0.728516\pi\)
−0.657807 + 0.753187i \(0.728516\pi\)
\(192\) 0 0
\(193\) 8.93864 0.643417 0.321709 0.946839i \(-0.395743\pi\)
0.321709 + 0.946839i \(0.395743\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.54158 0.394821 0.197411 0.980321i \(-0.436747\pi\)
0.197411 + 0.980321i \(0.436747\pi\)
\(198\) 0 0
\(199\) 18.8316 1.33494 0.667469 0.744638i \(-0.267378\pi\)
0.667469 + 0.744638i \(0.267378\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.190188 0.0133486
\(204\) 0 0
\(205\) −7.15031 −0.499399
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.42889 0.375524
\(210\) 0 0
\(211\) 12.7028 0.874496 0.437248 0.899341i \(-0.355953\pi\)
0.437248 + 0.899341i \(0.355953\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −29.5511 −2.01537
\(216\) 0 0
\(217\) 2.46932 0.167628
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.29722 0.0872602
\(222\) 0 0
\(223\) 8.45123 0.565936 0.282968 0.959129i \(-0.408681\pi\)
0.282968 + 0.959129i \(0.408681\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.54158 −0.633297 −0.316648 0.948543i \(-0.602558\pi\)
−0.316648 + 0.948543i \(0.602558\pi\)
\(228\) 0 0
\(229\) −2.35858 −0.155859 −0.0779297 0.996959i \(-0.524831\pi\)
−0.0779297 + 0.996959i \(0.524831\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.4189 −1.01012 −0.505061 0.863083i \(-0.668530\pi\)
−0.505061 + 0.863083i \(0.668530\pi\)
\(234\) 0 0
\(235\) 11.0709 0.722183
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.96012 −0.644266 −0.322133 0.946694i \(-0.604400\pi\)
−0.322133 + 0.946694i \(0.604400\pi\)
\(240\) 0 0
\(241\) 21.7736 1.40256 0.701282 0.712884i \(-0.252612\pi\)
0.701282 + 0.712884i \(0.252612\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.58060 0.164869
\(246\) 0 0
\(247\) 0.254946 0.0162218
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0034 −0.947006 −0.473503 0.880792i \(-0.657011\pi\)
−0.473503 + 0.880792i \(0.657011\pi\)
\(252\) 0 0
\(253\) −0.972420 −0.0611355
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.66954 −0.228900 −0.114450 0.993429i \(-0.536511\pi\)
−0.114450 + 0.993429i \(0.536511\pi\)
\(258\) 0 0
\(259\) 10.4512 0.649408
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.6406 −1.27275 −0.636376 0.771379i \(-0.719567\pi\)
−0.636376 + 0.771379i \(0.719567\pi\)
\(264\) 0 0
\(265\) 23.8991 1.46811
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −17.7019 −1.07931 −0.539653 0.841888i \(-0.681444\pi\)
−0.539653 + 0.841888i \(0.681444\pi\)
\(270\) 0 0
\(271\) 4.30061 0.261244 0.130622 0.991432i \(-0.458303\pi\)
0.130622 + 0.991432i \(0.458303\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.72087 −0.405284
\(276\) 0 0
\(277\) −18.1936 −1.09315 −0.546573 0.837411i \(-0.684068\pi\)
−0.546573 + 0.837411i \(0.684068\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.3514 1.03510 0.517549 0.855654i \(-0.326845\pi\)
0.517549 + 0.855654i \(0.326845\pi\)
\(282\) 0 0
\(283\) −4.38038 −0.260386 −0.130193 0.991489i \(-0.541560\pi\)
−0.130193 + 0.991489i \(0.541560\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.77079 −0.163555
\(288\) 0 0
\(289\) 29.5221 1.73659
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.56306 0.266577 0.133288 0.991077i \(-0.457446\pi\)
0.133288 + 0.991077i \(0.457446\pi\)
\(294\) 0 0
\(295\) −33.4730 −1.94888
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.0456657 −0.00264092
\(300\) 0 0
\(301\) −11.4512 −0.660038
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −35.2031 −2.01572
\(306\) 0 0
\(307\) −8.65951 −0.494224 −0.247112 0.968987i \(-0.579482\pi\)
−0.247112 + 0.968987i \(0.579482\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.8350 1.18144 0.590722 0.806875i \(-0.298843\pi\)
0.590722 + 0.806875i \(0.298843\pi\)
\(312\) 0 0
\(313\) 20.1902 1.14122 0.570608 0.821222i \(-0.306707\pi\)
0.570608 + 0.821222i \(0.306707\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.4128 −0.809501 −0.404750 0.914427i \(-0.632642\pi\)
−0.404750 + 0.914427i \(0.632642\pi\)
\(318\) 0 0
\(319\) −0.770246 −0.0431255
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.14312 0.508737
\(324\) 0 0
\(325\) −0.315618 −0.0175073
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.29003 0.236517
\(330\) 0 0
\(331\) 18.9386 1.04096 0.520481 0.853873i \(-0.325753\pi\)
0.520481 + 0.853873i \(0.325753\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.24816 −0.122830
\(336\) 0 0
\(337\) 2.61962 0.142700 0.0713500 0.997451i \(-0.477269\pi\)
0.0713500 + 0.997451i \(0.477269\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0005 −0.541560
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.8539 −1.17318 −0.586591 0.809884i \(-0.699530\pi\)
−0.586591 + 0.809884i \(0.699530\pi\)
\(348\) 0 0
\(349\) 1.61962 0.0866965 0.0433482 0.999060i \(-0.486197\pi\)
0.0433482 + 0.999060i \(0.486197\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.74899 0.252763 0.126382 0.991982i \(-0.459664\pi\)
0.126382 + 0.991982i \(0.459664\pi\)
\(354\) 0 0
\(355\) −28.5436 −1.51494
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.92055 0.154141 0.0770704 0.997026i \(-0.475443\pi\)
0.0770704 + 0.997026i \(0.475443\pi\)
\(360\) 0 0
\(361\) −17.2031 −0.905425
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 32.2900 1.69014
\(366\) 0 0
\(367\) 7.13222 0.372299 0.186149 0.982521i \(-0.440399\pi\)
0.186149 + 0.982521i \(0.440399\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.26104 0.480809
\(372\) 0 0
\(373\) −13.7883 −0.713933 −0.356966 0.934117i \(-0.616189\pi\)
−0.356966 + 0.934117i \(0.616189\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.0361714 −0.00186292
\(378\) 0 0
\(379\) 7.83161 0.402283 0.201141 0.979562i \(-0.435535\pi\)
0.201141 + 0.979562i \(0.435535\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.780828 0.0398985 0.0199492 0.999801i \(-0.493650\pi\)
0.0199492 + 0.999801i \(0.493650\pi\)
\(384\) 0 0
\(385\) −10.4512 −0.532644
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.8411 1.71581 0.857906 0.513807i \(-0.171765\pi\)
0.857906 + 0.513807i \(0.171765\pi\)
\(390\) 0 0
\(391\) −1.63771 −0.0828226
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −38.8919 −1.95686
\(396\) 0 0
\(397\) 24.6019 1.23473 0.617366 0.786676i \(-0.288200\pi\)
0.617366 + 0.786676i \(0.288200\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.1506 −0.656711 −0.328355 0.944554i \(-0.606494\pi\)
−0.328355 + 0.944554i \(0.606494\pi\)
\(402\) 0 0
\(403\) −0.469634 −0.0233942
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −42.3267 −2.09805
\(408\) 0 0
\(409\) 19.1718 0.947984 0.473992 0.880529i \(-0.342813\pi\)
0.473992 + 0.880529i \(0.342813\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.9710 −0.638262
\(414\) 0 0
\(415\) −10.8132 −0.530799
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.3643 −1.58110 −0.790549 0.612398i \(-0.790205\pi\)
−0.790549 + 0.612398i \(0.790205\pi\)
\(420\) 0 0
\(421\) −29.1322 −1.41982 −0.709909 0.704294i \(-0.751264\pi\)
−0.709909 + 0.704294i \(0.751264\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11.3190 −0.549053
\(426\) 0 0
\(427\) −13.6414 −0.660155
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.240108 −0.0115656 −0.00578281 0.999983i \(-0.501841\pi\)
−0.00578281 + 0.999983i \(0.501841\pi\)
\(432\) 0 0
\(433\) −25.6509 −1.23270 −0.616352 0.787471i \(-0.711390\pi\)
−0.616352 + 0.787471i \(0.711390\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.321864 −0.0153968
\(438\) 0 0
\(439\) 18.5221 0.884011 0.442006 0.897012i \(-0.354267\pi\)
0.442006 + 0.897012i \(0.354267\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.6947 1.17328 0.586641 0.809847i \(-0.300450\pi\)
0.586641 + 0.809847i \(0.300450\pi\)
\(444\) 0 0
\(445\) −26.8135 −1.27108
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.4005 0.585214 0.292607 0.956233i \(-0.405477\pi\)
0.292607 + 0.956233i \(0.405477\pi\)
\(450\) 0 0
\(451\) 11.2215 0.528399
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.490799 −0.0230090
\(456\) 0 0
\(457\) 33.6448 1.57384 0.786919 0.617056i \(-0.211675\pi\)
0.786919 + 0.617056i \(0.211675\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −39.0628 −1.81934 −0.909668 0.415336i \(-0.863664\pi\)
−0.909668 + 0.415336i \(0.863664\pi\)
\(462\) 0 0
\(463\) −5.61962 −0.261166 −0.130583 0.991437i \(-0.541685\pi\)
−0.130583 + 0.991437i \(0.541685\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.85169 0.455882 0.227941 0.973675i \(-0.426801\pi\)
0.227941 + 0.973675i \(0.426801\pi\)
\(468\) 0 0
\(469\) −0.871175 −0.0402271
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 46.3766 2.13240
\(474\) 0 0
\(475\) −2.22456 −0.102070
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.960434 −0.0438833 −0.0219417 0.999759i \(-0.506985\pi\)
−0.0219417 + 0.999759i \(0.506985\pi\)
\(480\) 0 0
\(481\) −1.98770 −0.0906312
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.6380 −1.20957
\(486\) 0 0
\(487\) 14.5706 0.660255 0.330128 0.943936i \(-0.392908\pi\)
0.330128 + 0.943936i \(0.392908\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.01004 −0.0907118 −0.0453559 0.998971i \(-0.514442\pi\)
−0.0453559 + 0.998971i \(0.514442\pi\)
\(492\) 0 0
\(493\) −1.29722 −0.0584237
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.0608 −0.496146
\(498\) 0 0
\(499\) 34.0743 1.52537 0.762687 0.646768i \(-0.223880\pi\)
0.762687 + 0.646768i \(0.223880\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.7736 1.06001 0.530007 0.847993i \(-0.322189\pi\)
0.530007 + 0.847993i \(0.322189\pi\)
\(504\) 0 0
\(505\) 19.3589 0.861460
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.2045 −0.806899 −0.403450 0.915002i \(-0.632189\pi\)
−0.403450 + 0.915002i \(0.632189\pi\)
\(510\) 0 0
\(511\) 12.5126 0.553525
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.39073 0.237544
\(516\) 0 0
\(517\) −17.3743 −0.764120
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.5938 1.25272 0.626359 0.779535i \(-0.284545\pi\)
0.626359 + 0.779535i \(0.284545\pi\)
\(522\) 0 0
\(523\) −33.2215 −1.45267 −0.726337 0.687339i \(-0.758779\pi\)
−0.726337 + 0.687339i \(0.758779\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.8425 −0.733671
\(528\) 0 0
\(529\) −22.9423 −0.997493
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.526971 0.0228256
\(534\) 0 0
\(535\) −24.9273 −1.07770
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.04992 −0.174442
\(540\) 0 0
\(541\) −19.7702 −0.849989 −0.424995 0.905196i \(-0.639724\pi\)
−0.424995 + 0.905196i \(0.639724\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.37233 −0.272961
\(546\) 0 0
\(547\) 14.2577 0.609613 0.304807 0.952414i \(-0.401408\pi\)
0.304807 + 0.952414i \(0.401408\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.254946 −0.0108610
\(552\) 0 0
\(553\) −15.0709 −0.640878
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.83880 −0.120284 −0.0601418 0.998190i \(-0.519155\pi\)
−0.0601418 + 0.998190i \(0.519155\pi\)
\(558\) 0 0
\(559\) 2.17789 0.0921147
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.7518 −0.579571 −0.289786 0.957092i \(-0.593584\pi\)
−0.289786 + 0.957092i \(0.593584\pi\)
\(564\) 0 0
\(565\) 17.6482 0.742466
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.5054 0.901554 0.450777 0.892637i \(-0.351147\pi\)
0.450777 + 0.892637i \(0.351147\pi\)
\(570\) 0 0
\(571\) −27.9420 −1.16934 −0.584669 0.811272i \(-0.698776\pi\)
−0.584669 + 0.811272i \(0.698776\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.398461 0.0166170
\(576\) 0 0
\(577\) 1.69939 0.0707465 0.0353732 0.999374i \(-0.488738\pi\)
0.0353732 + 0.999374i \(0.488738\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4.19019 −0.173838
\(582\) 0 0
\(583\) −37.5065 −1.55336
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.4836 −1.13437 −0.567185 0.823590i \(-0.691968\pi\)
−0.567185 + 0.823590i \(0.691968\pi\)
\(588\) 0 0
\(589\) −3.31011 −0.136391
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −41.5915 −1.70796 −0.853979 0.520307i \(-0.825817\pi\)
−0.853979 + 0.520307i \(0.825817\pi\)
\(594\) 0 0
\(595\) −17.6015 −0.721593
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.9234 1.30435 0.652177 0.758066i \(-0.273856\pi\)
0.652177 + 0.758066i \(0.273856\pi\)
\(600\) 0 0
\(601\) 25.4512 1.03818 0.519089 0.854720i \(-0.326271\pi\)
0.519089 + 0.854720i \(0.326271\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.9400 0.566743
\(606\) 0 0
\(607\) −3.58345 −0.145448 −0.0727239 0.997352i \(-0.523169\pi\)
−0.0727239 + 0.997352i \(0.523169\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.815911 −0.0330082
\(612\) 0 0
\(613\) −40.8135 −1.64844 −0.824221 0.566268i \(-0.808387\pi\)
−0.824221 + 0.566268i \(0.808387\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.9710 1.32736 0.663682 0.748015i \(-0.268993\pi\)
0.663682 + 0.748015i \(0.268993\pi\)
\(618\) 0 0
\(619\) −7.17789 −0.288504 −0.144252 0.989541i \(-0.546078\pi\)
−0.144252 + 0.989541i \(0.546078\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3904 −0.416283
\(624\) 0 0
\(625\) −30.5436 −1.22174
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −71.2848 −2.84231
\(630\) 0 0
\(631\) 17.8098 0.708997 0.354499 0.935057i \(-0.384652\pi\)
0.354499 + 0.935057i \(0.384652\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.7123 1.37751
\(636\) 0 0
\(637\) −0.190188 −0.00753552
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.1830 0.599692 0.299846 0.953988i \(-0.403065\pi\)
0.299846 + 0.953988i \(0.403065\pi\)
\(642\) 0 0
\(643\) 33.1018 1.30541 0.652704 0.757613i \(-0.273634\pi\)
0.652704 + 0.757613i \(0.273634\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.706576 −0.0277784 −0.0138892 0.999904i \(-0.504421\pi\)
−0.0138892 + 0.999904i \(0.504421\pi\)
\(648\) 0 0
\(649\) 52.5316 2.06205
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.4283 1.38642 0.693209 0.720736i \(-0.256196\pi\)
0.693209 + 0.720736i \(0.256196\pi\)
\(654\) 0 0
\(655\) −34.4546 −1.34625
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −28.2496 −1.10045 −0.550224 0.835017i \(-0.685458\pi\)
−0.550224 + 0.835017i \(0.685458\pi\)
\(660\) 0 0
\(661\) −9.48740 −0.369017 −0.184509 0.982831i \(-0.559069\pi\)
−0.184509 + 0.982831i \(0.559069\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.45928 −0.134145
\(666\) 0 0
\(667\) 0.0456657 0.00176818
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 55.2467 2.13277
\(672\) 0 0
\(673\) 26.3442 1.01549 0.507747 0.861506i \(-0.330478\pi\)
0.507747 + 0.861506i \(0.330478\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.5829 0.829499 0.414749 0.909936i \(-0.363869\pi\)
0.414749 + 0.909936i \(0.363869\pi\)
\(678\) 0 0
\(679\) −10.3224 −0.396138
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −51.3167 −1.96358 −0.981789 0.189975i \(-0.939159\pi\)
−0.981789 + 0.189975i \(0.939159\pi\)
\(684\) 0 0
\(685\) −22.6325 −0.864744
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.76134 −0.0671017
\(690\) 0 0
\(691\) 2.80371 0.106658 0.0533291 0.998577i \(-0.483017\pi\)
0.0533291 + 0.998577i \(0.483017\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.73896 0.103894
\(696\) 0 0
\(697\) 18.8988 0.715841
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −39.0565 −1.47514 −0.737571 0.675269i \(-0.764028\pi\)
−0.737571 + 0.675269i \(0.764028\pi\)
\(702\) 0 0
\(703\) −14.0098 −0.528390
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.50170 0.282130
\(708\) 0 0
\(709\) −14.1687 −0.532117 −0.266058 0.963957i \(-0.585721\pi\)
−0.266058 + 0.963957i \(0.585721\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.592904 0.0222044
\(714\) 0 0
\(715\) 1.98770 0.0743357
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.36979 0.312141 0.156070 0.987746i \(-0.450117\pi\)
0.156070 + 0.987746i \(0.450117\pi\)
\(720\) 0 0
\(721\) 2.08894 0.0777963
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.315618 0.0117218
\(726\) 0 0
\(727\) −42.9025 −1.59116 −0.795582 0.605846i \(-0.792835\pi\)
−0.795582 + 0.605846i \(0.792835\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 78.1055 2.88884
\(732\) 0 0
\(733\) −0.211984 −0.00782982 −0.00391491 0.999992i \(-0.501246\pi\)
−0.00391491 + 0.999992i \(0.501246\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.52819 0.129963
\(738\) 0 0
\(739\) −39.2610 −1.44424 −0.722120 0.691767i \(-0.756832\pi\)
−0.722120 + 0.691767i \(0.756832\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.87782 0.289009 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(744\) 0 0
\(745\) 24.6230 0.902118
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.65951 −0.352951
\(750\) 0 0
\(751\) 54.3175 1.98207 0.991037 0.133585i \(-0.0426489\pi\)
0.991037 + 0.133585i \(0.0426489\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.7195 0.717665
\(756\) 0 0
\(757\) −8.50310 −0.309050 −0.154525 0.987989i \(-0.549385\pi\)
−0.154525 + 0.987989i \(0.549385\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 47.9658 1.73876 0.869380 0.494144i \(-0.164519\pi\)
0.869380 + 0.494144i \(0.164519\pi\)
\(762\) 0 0
\(763\) −2.46932 −0.0893953
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.46693 0.0890757
\(768\) 0 0
\(769\) 33.3715 1.20341 0.601703 0.798720i \(-0.294489\pi\)
0.601703 + 0.798720i \(0.294489\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.9811 0.538831 0.269416 0.963024i \(-0.413169\pi\)
0.269416 + 0.963024i \(0.413169\pi\)
\(774\) 0 0
\(775\) 4.09785 0.147199
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.71423 0.133076
\(780\) 0 0
\(781\) 44.7954 1.60291
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 51.0279 1.82126
\(786\) 0 0
\(787\) −50.4859 −1.79963 −0.899814 0.436273i \(-0.856298\pi\)
−0.899814 + 0.436273i \(0.856298\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.83880 0.243160
\(792\) 0 0
\(793\) 2.59443 0.0921310
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 50.2427 1.77969 0.889844 0.456264i \(-0.150813\pi\)
0.889844 + 0.456264i \(0.150813\pi\)
\(798\) 0 0
\(799\) −29.2610 −1.03518
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −50.6750 −1.78828
\(804\) 0 0
\(805\) 0.619624 0.0218389
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.5344 −0.862583 −0.431292 0.902213i \(-0.641942\pi\)
−0.431292 + 0.902213i \(0.641942\pi\)
\(810\) 0 0
\(811\) 6.01840 0.211335 0.105667 0.994402i \(-0.466302\pi\)
0.105667 + 0.994402i \(0.466302\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −42.2957 −1.48155
\(816\) 0 0
\(817\) 15.3503 0.537039
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.5149 −1.58848 −0.794241 0.607603i \(-0.792131\pi\)
−0.794241 + 0.607603i \(0.792131\pi\)
\(822\) 0 0
\(823\) 12.6380 0.440534 0.220267 0.975440i \(-0.429307\pi\)
0.220267 + 0.975440i \(0.429307\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.1088 1.60336 0.801680 0.597754i \(-0.203940\pi\)
0.801680 + 0.597754i \(0.203940\pi\)
\(828\) 0 0
\(829\) −3.55216 −0.123372 −0.0616858 0.998096i \(-0.519648\pi\)
−0.0616858 + 0.998096i \(0.519648\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.82071 −0.236324
\(834\) 0 0
\(835\) −14.9093 −0.515956
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.80981 0.131529 0.0657647 0.997835i \(-0.479051\pi\)
0.0657647 + 0.997835i \(0.479051\pi\)
\(840\) 0 0
\(841\) −28.9638 −0.998753
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −33.4545 −1.15087
\(846\) 0 0
\(847\) 5.40186 0.185610
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.50943 0.0860221
\(852\) 0 0
\(853\) 0.960434 0.0328846 0.0164423 0.999865i \(-0.494766\pi\)
0.0164423 + 0.999865i \(0.494766\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.41221 −0.287356 −0.143678 0.989625i \(-0.545893\pi\)
−0.143678 + 0.989625i \(0.545893\pi\)
\(858\) 0 0
\(859\) −22.7822 −0.777320 −0.388660 0.921381i \(-0.627062\pi\)
−0.388660 + 0.921381i \(0.627062\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −52.0105 −1.77046 −0.885229 0.465155i \(-0.845998\pi\)
−0.885229 + 0.465155i \(0.845998\pi\)
\(864\) 0 0
\(865\) −0.619624 −0.0210679
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 61.0358 2.07050
\(870\) 0 0
\(871\) 0.165687 0.00561409
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −8.62048 −0.291426
\(876\) 0 0
\(877\) 37.9638 1.28195 0.640974 0.767563i \(-0.278531\pi\)
0.640974 + 0.767563i \(0.278531\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −16.1678 −0.544709 −0.272354 0.962197i \(-0.587802\pi\)
−0.272354 + 0.962197i \(0.587802\pi\)
\(882\) 0 0
\(883\) −16.8807 −0.568080 −0.284040 0.958812i \(-0.591675\pi\)
−0.284040 + 0.958812i \(0.591675\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.7341 −1.23341 −0.616705 0.787195i \(-0.711533\pi\)
−0.616705 + 0.787195i \(0.711533\pi\)
\(888\) 0 0
\(889\) 13.4512 0.451140
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5.75076 −0.192442
\(894\) 0 0
\(895\) 18.8408 0.629778
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.469634 0.0156632
\(900\) 0 0
\(901\) −63.1669 −2.10439
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −46.9661 −1.56121
\(906\) 0 0
\(907\) 42.9890 1.42743 0.713713 0.700438i \(-0.247012\pi\)
0.713713 + 0.700438i \(0.247012\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 53.2068 1.76282 0.881410 0.472352i \(-0.156595\pi\)
0.881410 + 0.472352i \(0.156595\pi\)
\(912\) 0 0
\(913\) 16.9699 0.561623
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.3514 −0.440902
\(918\) 0 0
\(919\) 16.8009 0.554211 0.277105 0.960840i \(-0.410625\pi\)
0.277105 + 0.960840i \(0.410625\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.10363 0.0692419
\(924\) 0 0
\(925\) 17.3439 0.570264
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −57.9229 −1.90039 −0.950194 0.311660i \(-0.899115\pi\)
−0.950194 + 0.311660i \(0.899115\pi\)
\(930\) 0 0
\(931\) −1.34049 −0.0439329
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 71.2848 2.33126
\(936\) 0 0
\(937\) −31.5623 −1.03110 −0.515548 0.856861i \(-0.672411\pi\)
−0.515548 + 0.856861i \(0.672411\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.8846 0.974210 0.487105 0.873343i \(-0.338053\pi\)
0.487105 + 0.873343i \(0.338053\pi\)
\(942\) 0 0
\(943\) −0.665290 −0.0216648
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.8695 1.52306 0.761528 0.648132i \(-0.224450\pi\)
0.761528 + 0.648132i \(0.224450\pi\)
\(948\) 0 0
\(949\) −2.37974 −0.0772498
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.4730 1.01951 0.509756 0.860319i \(-0.329736\pi\)
0.509756 + 0.860319i \(0.329736\pi\)
\(954\) 0 0
\(955\) −46.9209 −1.51832
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.77025 −0.283206
\(960\) 0 0
\(961\) −24.9025 −0.803305
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 23.0671 0.742555
\(966\) 0 0
\(967\) 51.7252 1.66337 0.831685 0.555248i \(-0.187377\pi\)
0.831685 + 0.555248i \(0.187377\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 60.2528 1.93360 0.966802 0.255528i \(-0.0822493\pi\)
0.966802 + 0.255528i \(0.0822493\pi\)
\(972\) 0 0
\(973\) 1.06136 0.0340257
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.7423 0.567628 0.283814 0.958879i \(-0.408400\pi\)
0.283814 + 0.958879i \(0.408400\pi\)
\(978\) 0 0
\(979\) 42.0804 1.34489
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 42.9928 1.37126 0.685629 0.727951i \(-0.259527\pi\)
0.685629 + 0.727951i \(0.259527\pi\)
\(984\) 0 0
\(985\) 14.3006 0.455655
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.74954 −0.0874302
\(990\) 0 0
\(991\) 31.3933 0.997240 0.498620 0.866821i \(-0.333840\pi\)
0.498620 + 0.866821i \(0.333840\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 48.5969 1.54062
\(996\) 0 0
\(997\) 7.36197 0.233156 0.116578 0.993182i \(-0.462807\pi\)
0.116578 + 0.993182i \(0.462807\pi\)
\(998\) 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 6048.2.a.bn.1.4 yes 4
3.2 odd 2 6048.2.a.bu.1.1 yes 4
4.3 odd 2 6048.2.a.bl.1.4 4
12.11 even 2 6048.2.a.bq.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bl.1.4 4 4.3 odd 2
6048.2.a.bn.1.4 yes 4 1.1 even 1 trivial
6048.2.a.bq.1.1 yes 4 12.11 even 2
6048.2.a.bu.1.1 yes 4 3.2 odd 2