Properties

Label 6048.2.a.bo.1.3
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: 4.4.22896.1
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 1.3
Root \(-0.896343\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.314350 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+0.314350 q^{5} +1.00000 q^{7} -1.47834 q^{11} -1.79269 q^{13} +5.22915 q^{17} +6.22915 q^{19} -6.58683 q^{23} -4.90118 q^{25} -7.43647 q^{29} -4.43647 q^{31} +0.314350 q^{35} -2.79269 q^{37} -8.37952 q^{41} -0.207315 q^{43} +4.06517 q^{47} +1.00000 q^{49} -4.00000 q^{53} -0.464715 q^{55} +4.06517 q^{59} -1.67203 q^{61} -0.563530 q^{65} -13.9230 q^{67} +7.27102 q^{71} +14.6656 q^{73} -1.47834 q^{77} +1.46471 q^{79} -12.2792 q^{83} +1.64378 q^{85} -12.3795 q^{89} -1.79269 q^{91} +1.95813 q^{95} +16.1303 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} + 4q^{7} - 2q^{11} - 8q^{17} - 4q^{19} - 2q^{23} + 8q^{25} - 8q^{29} + 4q^{31} - 2q^{35} - 4q^{37} - 2q^{41} - 8q^{43} - 12q^{47} + 4q^{49} - 16q^{53} + 4q^{55} - 12q^{59} - 8q^{61} - 24q^{65} + 8q^{67} + 18q^{71} + 8q^{73} - 2q^{77} - 8q^{85} - 18q^{89} - 10q^{95} + 8q^{97} + O(q^{100}) \)

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
\(5\) 0.314350 0.140582 0.0702908 0.997527i \(-0.477607\pi\)
0.0702908 + 0.997527i \(0.477607\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.47834 −0.445735 −0.222867 0.974849i \(-0.571542\pi\)
−0.222867 + 0.974849i \(0.571542\pi\)
\(12\) 0 0
\(13\) −1.79269 −0.497201 −0.248601 0.968606i \(-0.579971\pi\)
−0.248601 + 0.968606i \(0.579971\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.22915 1.26826 0.634128 0.773228i \(-0.281359\pi\)
0.634128 + 0.773228i \(0.281359\pi\)
\(18\) 0 0
\(19\) 6.22915 1.42907 0.714533 0.699602i \(-0.246639\pi\)
0.714533 + 0.699602i \(0.246639\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.58683 −1.37345 −0.686725 0.726917i \(-0.740952\pi\)
−0.686725 + 0.726917i \(0.740952\pi\)
\(24\) 0 0
\(25\) −4.90118 −0.980237
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.43647 −1.38092 −0.690459 0.723372i \(-0.742591\pi\)
−0.690459 + 0.723372i \(0.742591\pi\)
\(30\) 0 0
\(31\) −4.43647 −0.796813 −0.398407 0.917209i \(-0.630437\pi\)
−0.398407 + 0.917209i \(0.630437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.314350 0.0531348
\(36\) 0 0
\(37\) −2.79269 −0.459115 −0.229557 0.973295i \(-0.573728\pi\)
−0.229557 + 0.973295i \(0.573728\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.37952 −1.30866 −0.654331 0.756209i \(-0.727049\pi\)
−0.654331 + 0.756209i \(0.727049\pi\)
\(42\) 0 0
\(43\) −0.207315 −0.0316152 −0.0158076 0.999875i \(-0.505032\pi\)
−0.0158076 + 0.999875i \(0.505032\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.06517 0.592966 0.296483 0.955038i \(-0.404186\pi\)
0.296483 + 0.955038i \(0.404186\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −0.464715 −0.0626621
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.06517 0.529240 0.264620 0.964353i \(-0.414753\pi\)
0.264620 + 0.964353i \(0.414753\pi\)
\(60\) 0 0
\(61\) −1.67203 −0.214081 −0.107041 0.994255i \(-0.534138\pi\)
−0.107041 + 0.994255i \(0.534138\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.563530 −0.0698973
\(66\) 0 0
\(67\) −13.9230 −1.70097 −0.850484 0.526001i \(-0.823691\pi\)
−0.850484 + 0.526001i \(0.823691\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.27102 0.862911 0.431456 0.902134i \(-0.358000\pi\)
0.431456 + 0.902134i \(0.358000\pi\)
\(72\) 0 0
\(73\) 14.6656 1.71648 0.858241 0.513247i \(-0.171558\pi\)
0.858241 + 0.513247i \(0.171558\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.47834 −0.168472
\(78\) 0 0
\(79\) 1.46471 0.164793 0.0823966 0.996600i \(-0.473743\pi\)
0.0823966 + 0.996600i \(0.473743\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.2792 −1.34782 −0.673911 0.738813i \(-0.735387\pi\)
−0.673911 + 0.738813i \(0.735387\pi\)
\(84\) 0 0
\(85\) 1.64378 0.178293
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.3795 −1.31223 −0.656113 0.754662i \(-0.727801\pi\)
−0.656113 + 0.754662i \(0.727801\pi\)
\(90\) 0 0
\(91\) −1.79269 −0.187924
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.95813 0.200900
\(96\) 0 0
\(97\) 16.1303 1.63779 0.818894 0.573945i \(-0.194588\pi\)
0.818894 + 0.573945i \(0.194588\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.77085 −0.275709 −0.137855 0.990452i \(-0.544021\pi\)
−0.137855 + 0.990452i \(0.544021\pi\)
\(102\) 0 0
\(103\) 18.5668 1.82944 0.914721 0.404086i \(-0.132410\pi\)
0.914721 + 0.404086i \(0.132410\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.1858 1.17805 0.589024 0.808115i \(-0.299512\pi\)
0.589024 + 0.808115i \(0.299512\pi\)
\(108\) 0 0
\(109\) −10.1085 −0.968219 −0.484109 0.875008i \(-0.660856\pi\)
−0.484109 + 0.875008i \(0.660856\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.95667 −0.654429 −0.327214 0.944950i \(-0.606110\pi\)
−0.327214 + 0.944950i \(0.606110\pi\)
\(114\) 0 0
\(115\) −2.07057 −0.193082
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.22915 0.479356
\(120\) 0 0
\(121\) −8.81452 −0.801320
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.11244 −0.278385
\(126\) 0 0
\(127\) −7.05008 −0.625594 −0.312797 0.949820i \(-0.601266\pi\)
−0.312797 + 0.949820i \(0.601266\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.8078 −0.944279 −0.472140 0.881524i \(-0.656518\pi\)
−0.472140 + 0.881524i \(0.656518\pi\)
\(132\) 0 0
\(133\) 6.22915 0.540136
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.6072 −1.58972 −0.794861 0.606792i \(-0.792456\pi\)
−0.794861 + 0.606792i \(0.792456\pi\)
\(138\) 0 0
\(139\) 8.54497 0.724775 0.362387 0.932028i \(-0.381962\pi\)
0.362387 + 0.932028i \(0.381962\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.65019 0.221620
\(144\) 0 0
\(145\) −2.33765 −0.194132
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.08701 0.580590 0.290295 0.956937i \(-0.406246\pi\)
0.290295 + 0.956937i \(0.406246\pi\)
\(150\) 0 0
\(151\) 3.67203 0.298826 0.149413 0.988775i \(-0.452262\pi\)
0.149413 + 0.988775i \(0.452262\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.39460 −0.112017
\(156\) 0 0
\(157\) −13.0803 −1.04392 −0.521959 0.852971i \(-0.674799\pi\)
−0.521959 + 0.852971i \(0.674799\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.58683 −0.519115
\(162\) 0 0
\(163\) −8.25099 −0.646268 −0.323134 0.946353i \(-0.604736\pi\)
−0.323134 + 0.946353i \(0.604736\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.9567 −0.847853 −0.423926 0.905697i \(-0.639348\pi\)
−0.423926 + 0.905697i \(0.639348\pi\)
\(168\) 0 0
\(169\) −9.78628 −0.752791
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.45650 −0.186764 −0.0933819 0.995630i \(-0.529768\pi\)
−0.0933819 + 0.995630i \(0.529768\pi\)
\(174\) 0 0
\(175\) −4.90118 −0.370495
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.14215 0.160111 0.0800557 0.996790i \(-0.474490\pi\)
0.0800557 + 0.996790i \(0.474490\pi\)
\(180\) 0 0
\(181\) −18.7863 −1.39637 −0.698187 0.715916i \(-0.746009\pi\)
−0.698187 + 0.715916i \(0.746009\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.877880 −0.0645430
\(186\) 0 0
\(187\) −7.73044 −0.565306
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.4164 −1.47728 −0.738641 0.674099i \(-0.764532\pi\)
−0.738641 + 0.674099i \(0.764532\pi\)
\(192\) 0 0
\(193\) 21.7157 1.56313 0.781565 0.623823i \(-0.214421\pi\)
0.781565 + 0.623823i \(0.214421\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.7590 −1.76401 −0.882004 0.471241i \(-0.843806\pi\)
−0.882004 + 0.471241i \(0.843806\pi\)
\(198\) 0 0
\(199\) 4.05008 0.287103 0.143551 0.989643i \(-0.454148\pi\)
0.143551 + 0.989643i \(0.454148\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.43647 −0.521938
\(204\) 0 0
\(205\) −2.63410 −0.183974
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.20878 −0.636985
\(210\) 0 0
\(211\) 15.2009 1.04647 0.523237 0.852187i \(-0.324724\pi\)
0.523237 + 0.852187i \(0.324724\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.0651694 −0.00444452
\(216\) 0 0
\(217\) −4.43647 −0.301167
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.37423 −0.630579
\(222\) 0 0
\(223\) 0.792685 0.0530821 0.0265411 0.999648i \(-0.491551\pi\)
0.0265411 + 0.999648i \(0.491551\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.8296 0.917903 0.458952 0.888461i \(-0.348225\pi\)
0.458952 + 0.888461i \(0.348225\pi\)
\(228\) 0 0
\(229\) 0.842770 0.0556918 0.0278459 0.999612i \(-0.491135\pi\)
0.0278459 + 0.999612i \(0.491135\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.1737 −1.12508 −0.562542 0.826769i \(-0.690177\pi\)
−0.562542 + 0.826769i \(0.690177\pi\)
\(234\) 0 0
\(235\) 1.27789 0.0833601
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.4734 1.25963 0.629815 0.776746i \(-0.283131\pi\)
0.629815 + 0.776746i \(0.283131\pi\)
\(240\) 0 0
\(241\) −0.337654 −0.0217502 −0.0108751 0.999941i \(-0.503462\pi\)
−0.0108751 + 0.999941i \(0.503462\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.314350 0.0200831
\(246\) 0 0
\(247\) −11.1669 −0.710534
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.2876 0.712465 0.356233 0.934397i \(-0.384061\pi\)
0.356233 + 0.934397i \(0.384061\pi\)
\(252\) 0 0
\(253\) 9.73755 0.612194
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.60867 0.100346 0.0501732 0.998741i \(-0.484023\pi\)
0.0501732 + 0.998741i \(0.484023\pi\)
\(258\) 0 0
\(259\) −2.79269 −0.173529
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.1289 0.809561 0.404781 0.914414i \(-0.367348\pi\)
0.404781 + 0.914414i \(0.367348\pi\)
\(264\) 0 0
\(265\) −1.25740 −0.0772415
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.28610 −0.505213 −0.252606 0.967569i \(-0.581288\pi\)
−0.252606 + 0.967569i \(0.581288\pi\)
\(270\) 0 0
\(271\) −22.0437 −1.33906 −0.669529 0.742786i \(-0.733504\pi\)
−0.669529 + 0.742786i \(0.733504\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.24559 0.436926
\(276\) 0 0
\(277\) −5.12066 −0.307670 −0.153835 0.988097i \(-0.549162\pi\)
−0.153835 + 0.988097i \(0.549162\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.5235 1.10502 0.552509 0.833507i \(-0.313670\pi\)
0.552509 + 0.833507i \(0.313670\pi\)
\(282\) 0 0
\(283\) −2.74260 −0.163031 −0.0815153 0.996672i \(-0.525976\pi\)
−0.0815153 + 0.996672i \(0.525976\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.37952 −0.494627
\(288\) 0 0
\(289\) 10.3441 0.608474
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.7440 −1.15345 −0.576727 0.816937i \(-0.695670\pi\)
−0.576727 + 0.816937i \(0.695670\pi\)
\(294\) 0 0
\(295\) 1.27789 0.0744014
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11.8081 0.682881
\(300\) 0 0
\(301\) −0.207315 −0.0119494
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.525602 −0.0300959
\(306\) 0 0
\(307\) 11.5975 0.661906 0.330953 0.943647i \(-0.392630\pi\)
0.330953 + 0.943647i \(0.392630\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.8242 −1.40765 −0.703826 0.710373i \(-0.748526\pi\)
−0.703826 + 0.710373i \(0.748526\pi\)
\(312\) 0 0
\(313\) 24.7960 1.40155 0.700775 0.713382i \(-0.252838\pi\)
0.700775 + 0.713382i \(0.252838\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.8514 −1.84512 −0.922561 0.385852i \(-0.873907\pi\)
−0.922561 + 0.385852i \(0.873907\pi\)
\(318\) 0 0
\(319\) 10.9936 0.615523
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 32.5732 1.81242
\(324\) 0 0
\(325\) 8.78628 0.487375
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.06517 0.224120
\(330\) 0 0
\(331\) 13.3877 0.735857 0.367928 0.929854i \(-0.380067\pi\)
0.367928 + 0.929854i \(0.380067\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.37670 −0.239125
\(336\) 0 0
\(337\) 16.9133 0.921328 0.460664 0.887575i \(-0.347611\pi\)
0.460664 + 0.887575i \(0.347611\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.55859 0.355168
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −23.8812 −1.28201 −0.641004 0.767538i \(-0.721482\pi\)
−0.641004 + 0.767538i \(0.721482\pi\)
\(348\) 0 0
\(349\) −14.4146 −0.771597 −0.385799 0.922583i \(-0.626074\pi\)
−0.385799 + 0.922583i \(0.626074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −27.0384 −1.43911 −0.719554 0.694436i \(-0.755654\pi\)
−0.719554 + 0.694436i \(0.755654\pi\)
\(354\) 0 0
\(355\) 2.28564 0.121309
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.2758 1.75623 0.878114 0.478452i \(-0.158802\pi\)
0.878114 + 0.478452i \(0.158802\pi\)
\(360\) 0 0
\(361\) 19.8024 1.04223
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.61014 0.241306
\(366\) 0 0
\(367\) 31.7658 1.65816 0.829080 0.559129i \(-0.188864\pi\)
0.829080 + 0.559129i \(0.188864\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −4.49296 −0.232637 −0.116318 0.993212i \(-0.537109\pi\)
−0.116318 + 0.993212i \(0.537109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.3312 0.686594
\(378\) 0 0
\(379\) 8.62194 0.442880 0.221440 0.975174i \(-0.428924\pi\)
0.221440 + 0.975174i \(0.428924\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −34.4311 −1.75935 −0.879673 0.475580i \(-0.842238\pi\)
−0.879673 + 0.475580i \(0.842238\pi\)
\(384\) 0 0
\(385\) −0.464715 −0.0236840
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.61554 0.487527 0.243764 0.969835i \(-0.421618\pi\)
0.243764 + 0.969835i \(0.421618\pi\)
\(390\) 0 0
\(391\) −34.4436 −1.74189
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.460433 0.0231669
\(396\) 0 0
\(397\) −7.36837 −0.369808 −0.184904 0.982757i \(-0.559197\pi\)
−0.184904 + 0.982757i \(0.559197\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.5507 −1.12613 −0.563065 0.826413i \(-0.690378\pi\)
−0.563065 + 0.826413i \(0.690378\pi\)
\(402\) 0 0
\(403\) 7.95319 0.396177
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.12852 0.204643
\(408\) 0 0
\(409\) 4.63546 0.229209 0.114604 0.993411i \(-0.463440\pi\)
0.114604 + 0.993411i \(0.463440\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.06517 0.200034
\(414\) 0 0
\(415\) −3.85998 −0.189479
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.7590 −1.79580 −0.897898 0.440203i \(-0.854906\pi\)
−0.897898 + 0.440203i \(0.854906\pi\)
\(420\) 0 0
\(421\) −36.2543 −1.76693 −0.883463 0.468502i \(-0.844794\pi\)
−0.883463 + 0.468502i \(0.844794\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −25.6290 −1.24319
\(426\) 0 0
\(427\) −1.67203 −0.0809152
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.7139 0.564238 0.282119 0.959379i \(-0.408963\pi\)
0.282119 + 0.959379i \(0.408963\pi\)
\(432\) 0 0
\(433\) 9.24772 0.444417 0.222208 0.974999i \(-0.428673\pi\)
0.222208 + 0.974999i \(0.428673\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −41.0304 −1.96275
\(438\) 0 0
\(439\) −12.9166 −0.616477 −0.308238 0.951309i \(-0.599739\pi\)
−0.308238 + 0.951309i \(0.599739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.2213 −1.38835 −0.694173 0.719809i \(-0.744230\pi\)
−0.694173 + 0.719809i \(0.744230\pi\)
\(444\) 0 0
\(445\) −3.89150 −0.184475
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.45796 −0.446349 −0.223174 0.974779i \(-0.571642\pi\)
−0.223174 + 0.974779i \(0.571642\pi\)
\(450\) 0 0
\(451\) 12.3877 0.583316
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.563530 −0.0264187
\(456\) 0 0
\(457\) 9.84277 0.460425 0.230213 0.973140i \(-0.426058\pi\)
0.230213 + 0.973140i \(0.426058\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.95813 0.463797 0.231898 0.972740i \(-0.425506\pi\)
0.231898 + 0.972740i \(0.425506\pi\)
\(462\) 0 0
\(463\) −1.58537 −0.0736784 −0.0368392 0.999321i \(-0.511729\pi\)
−0.0368392 + 0.999321i \(0.511729\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.1794 −0.887518 −0.443759 0.896146i \(-0.646355\pi\)
−0.443759 + 0.896146i \(0.646355\pi\)
\(468\) 0 0
\(469\) −13.9230 −0.642906
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.306481 0.0140920
\(474\) 0 0
\(475\) −30.5302 −1.40082
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.4281 0.750621 0.375310 0.926899i \(-0.377536\pi\)
0.375310 + 0.926899i \(0.377536\pi\)
\(480\) 0 0
\(481\) 5.00641 0.228272
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.07057 0.230243
\(486\) 0 0
\(487\) 4.44863 0.201586 0.100793 0.994907i \(-0.467862\pi\)
0.100793 + 0.994907i \(0.467862\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.23602 −0.0557808 −0.0278904 0.999611i \(-0.508879\pi\)
−0.0278904 + 0.999611i \(0.508879\pi\)
\(492\) 0 0
\(493\) −38.8864 −1.75136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.27102 0.326150
\(498\) 0 0
\(499\) 18.1207 0.811192 0.405596 0.914052i \(-0.367064\pi\)
0.405596 + 0.914052i \(0.367064\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.4500 0.867232 0.433616 0.901098i \(-0.357237\pi\)
0.433616 + 0.901098i \(0.357237\pi\)
\(504\) 0 0
\(505\) −0.871015 −0.0387597
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.1888 −0.628905 −0.314453 0.949273i \(-0.601821\pi\)
−0.314453 + 0.949273i \(0.601821\pi\)
\(510\) 0 0
\(511\) 14.6656 0.648769
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.83647 0.257186
\(516\) 0 0
\(517\) −6.00968 −0.264306
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.0633554 0.00277565 0.00138782 0.999999i \(-0.499558\pi\)
0.00138782 + 0.999999i \(0.499558\pi\)
\(522\) 0 0
\(523\) −6.20091 −0.271147 −0.135573 0.990767i \(-0.543288\pi\)
−0.135573 + 0.990767i \(0.543288\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −23.1990 −1.01056
\(528\) 0 0
\(529\) 20.3864 0.886365
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.0218 0.650668
\(534\) 0 0
\(535\) 3.83061 0.165612
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.47834 −0.0636764
\(540\) 0 0
\(541\) −23.3377 −1.00336 −0.501682 0.865052i \(-0.667285\pi\)
−0.501682 + 0.865052i \(0.667285\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.17761 −0.136114
\(546\) 0 0
\(547\) −35.8460 −1.53267 −0.766333 0.642443i \(-0.777921\pi\)
−0.766333 + 0.642443i \(0.777921\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −46.3229 −1.97342
\(552\) 0 0
\(553\) 1.46471 0.0622860
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.2302 −0.560580 −0.280290 0.959915i \(-0.590431\pi\)
−0.280290 + 0.959915i \(0.590431\pi\)
\(558\) 0 0
\(559\) 0.371650 0.0157191
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 21.2388 0.895110 0.447555 0.894256i \(-0.352295\pi\)
0.447555 + 0.894256i \(0.352295\pi\)
\(564\) 0 0
\(565\) −2.18683 −0.0920006
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.9599 0.752920 0.376460 0.926433i \(-0.377141\pi\)
0.376460 + 0.926433i \(0.377141\pi\)
\(570\) 0 0
\(571\) −22.3717 −0.936224 −0.468112 0.883669i \(-0.655066\pi\)
−0.468112 + 0.883669i \(0.655066\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 32.2833 1.34631
\(576\) 0 0
\(577\) −4.04368 −0.168341 −0.0841703 0.996451i \(-0.526824\pi\)
−0.0841703 + 0.996451i \(0.526824\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.2792 −0.509429
\(582\) 0 0
\(583\) 5.91334 0.244906
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.12999 −0.129188 −0.0645942 0.997912i \(-0.520575\pi\)
−0.0645942 + 0.997912i \(0.520575\pi\)
\(588\) 0 0
\(589\) −27.6355 −1.13870
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.42185 −0.0583882 −0.0291941 0.999574i \(-0.509294\pi\)
−0.0291941 + 0.999574i \(0.509294\pi\)
\(594\) 0 0
\(595\) 1.64378 0.0673886
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.570396 −0.0233057 −0.0116529 0.999932i \(-0.503709\pi\)
−0.0116529 + 0.999932i \(0.503709\pi\)
\(600\) 0 0
\(601\) 5.79269 0.236289 0.118144 0.992996i \(-0.462305\pi\)
0.118144 + 0.992996i \(0.462305\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.77085 −0.112651
\(606\) 0 0
\(607\) −7.71571 −0.313171 −0.156585 0.987664i \(-0.550049\pi\)
−0.156585 + 0.987664i \(0.550049\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.28757 −0.294823
\(612\) 0 0
\(613\) 4.43647 0.179187 0.0895937 0.995978i \(-0.471443\pi\)
0.0895937 + 0.995978i \(0.471443\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.6101 −0.829733 −0.414866 0.909882i \(-0.636172\pi\)
−0.414866 + 0.909882i \(0.636172\pi\)
\(618\) 0 0
\(619\) −36.3877 −1.46255 −0.731273 0.682085i \(-0.761074\pi\)
−0.731273 + 0.682085i \(0.761074\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.3795 −0.495975
\(624\) 0 0
\(625\) 23.5275 0.941101
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.6034 −0.582275
\(630\) 0 0
\(631\) −16.8633 −0.671316 −0.335658 0.941984i \(-0.608959\pi\)
−0.335658 + 0.941984i \(0.608959\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.21619 −0.0879470
\(636\) 0 0
\(637\) −1.79269 −0.0710288
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.7154 0.581222 0.290611 0.956841i \(-0.406141\pi\)
0.290611 + 0.956841i \(0.406141\pi\)
\(642\) 0 0
\(643\) 10.0988 0.398258 0.199129 0.979973i \(-0.436189\pi\)
0.199129 + 0.979973i \(0.436189\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 49.4529 1.94419 0.972097 0.234578i \(-0.0753709\pi\)
0.972097 + 0.234578i \(0.0753709\pi\)
\(648\) 0 0
\(649\) −6.00968 −0.235901
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.8514 0.815980 0.407990 0.912986i \(-0.366230\pi\)
0.407990 + 0.912986i \(0.366230\pi\)
\(654\) 0 0
\(655\) −3.39742 −0.132748
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.1372 0.940252 0.470126 0.882599i \(-0.344208\pi\)
0.470126 + 0.882599i \(0.344208\pi\)
\(660\) 0 0
\(661\) 31.0130 1.20626 0.603132 0.797641i \(-0.293919\pi\)
0.603132 + 0.797641i \(0.293919\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.95813 0.0759332
\(666\) 0 0
\(667\) 48.9828 1.89662
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.47182 0.0954236
\(672\) 0 0
\(673\) 25.7157 0.991268 0.495634 0.868532i \(-0.334936\pi\)
0.495634 + 0.868532i \(0.334936\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.0301 −0.769818 −0.384909 0.922955i \(-0.625767\pi\)
−0.384909 + 0.922955i \(0.625767\pi\)
\(678\) 0 0
\(679\) 16.1303 0.619026
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 45.6087 1.74517 0.872584 0.488465i \(-0.162443\pi\)
0.872584 + 0.488465i \(0.162443\pi\)
\(684\) 0 0
\(685\) −5.84918 −0.223485
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.17074 0.273183
\(690\) 0 0
\(691\) −11.0404 −0.419997 −0.209998 0.977702i \(-0.567346\pi\)
−0.209998 + 0.977702i \(0.567346\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.68611 0.101890
\(696\) 0 0
\(697\) −43.8178 −1.65972
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.9545 1.39575 0.697877 0.716218i \(-0.254128\pi\)
0.697877 + 0.716218i \(0.254128\pi\)
\(702\) 0 0
\(703\) −17.3961 −0.656105
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.77085 −0.104208
\(708\) 0 0
\(709\) 46.7107 1.75426 0.877128 0.480257i \(-0.159457\pi\)
0.877128 + 0.480257i \(0.159457\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29.2223 1.09438
\(714\) 0 0
\(715\) 0.833087 0.0311557
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.4368 0.762165 0.381082 0.924541i \(-0.375551\pi\)
0.381082 + 0.924541i \(0.375551\pi\)
\(720\) 0 0
\(721\) 18.5668 0.691464
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 36.4475 1.35363
\(726\) 0 0
\(727\) 28.4885 1.05658 0.528290 0.849064i \(-0.322834\pi\)
0.528290 + 0.849064i \(0.322834\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.08408 −0.0400962
\(732\) 0 0
\(733\) −18.5353 −0.684616 −0.342308 0.939588i \(-0.611209\pi\)
−0.342308 + 0.939588i \(0.611209\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.5829 0.758181
\(738\) 0 0
\(739\) 30.6888 1.12891 0.564453 0.825465i \(-0.309087\pi\)
0.564453 + 0.825465i \(0.309087\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.23097 −0.338651 −0.169326 0.985560i \(-0.554159\pi\)
−0.169326 + 0.985560i \(0.554159\pi\)
\(744\) 0 0
\(745\) 2.22780 0.0816203
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.1858 0.445260
\(750\) 0 0
\(751\) −3.66235 −0.133641 −0.0668205 0.997765i \(-0.521285\pi\)
−0.0668205 + 0.997765i \(0.521285\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.15430 0.0420094
\(756\) 0 0
\(757\) 20.7561 0.754394 0.377197 0.926133i \(-0.376888\pi\)
0.377197 + 0.926133i \(0.376888\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.1425 0.838915 0.419457 0.907775i \(-0.362220\pi\)
0.419457 + 0.907775i \(0.362220\pi\)
\(762\) 0 0
\(763\) −10.1085 −0.365952
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.28757 −0.263139
\(768\) 0 0
\(769\) 9.90951 0.357346 0.178673 0.983908i \(-0.442820\pi\)
0.178673 + 0.983908i \(0.442820\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35.8160 1.28821 0.644106 0.764936i \(-0.277230\pi\)
0.644106 + 0.764936i \(0.277230\pi\)
\(774\) 0 0
\(775\) 21.7440 0.781066
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −52.1973 −1.87016
\(780\) 0 0
\(781\) −10.7490 −0.384630
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.11178 −0.146756
\(786\) 0 0
\(787\) 7.30108 0.260255 0.130128 0.991497i \(-0.458461\pi\)
0.130128 + 0.991497i \(0.458461\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.95667 −0.247351
\(792\) 0 0
\(793\) 2.99742 0.106442
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.7756 0.452535 0.226267 0.974065i \(-0.427348\pi\)
0.226267 + 0.974065i \(0.427348\pi\)
\(798\) 0 0
\(799\) 21.2574 0.752033
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −21.6807 −0.765096
\(804\) 0 0
\(805\) −2.07057 −0.0729780
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.8948 −0.769779 −0.384890 0.922963i \(-0.625761\pi\)
−0.384890 + 0.922963i \(0.625761\pi\)
\(810\) 0 0
\(811\) 21.4146 0.751969 0.375985 0.926626i \(-0.377305\pi\)
0.375985 + 0.926626i \(0.377305\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.59370 −0.0908533
\(816\) 0 0
\(817\) −1.29140 −0.0451802
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.7405 −0.723848 −0.361924 0.932208i \(-0.617880\pi\)
−0.361924 + 0.932208i \(0.617880\pi\)
\(822\) 0 0
\(823\) 19.7459 0.688298 0.344149 0.938915i \(-0.388167\pi\)
0.344149 + 0.938915i \(0.388167\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 53.5532 1.86223 0.931113 0.364730i \(-0.118839\pi\)
0.931113 + 0.364730i \(0.118839\pi\)
\(828\) 0 0
\(829\) −37.5386 −1.30377 −0.651884 0.758319i \(-0.726021\pi\)
−0.651884 + 0.758319i \(0.726021\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.22915 0.181179
\(834\) 0 0
\(835\) −3.44423 −0.119192
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −44.7677 −1.54555 −0.772777 0.634678i \(-0.781133\pi\)
−0.772777 + 0.634678i \(0.781133\pi\)
\(840\) 0 0
\(841\) 26.3011 0.906934
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.07632 −0.105829
\(846\) 0 0
\(847\) −8.81452 −0.302871
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 18.3950 0.630571
\(852\) 0 0
\(853\) 15.3011 0.523899 0.261949 0.965082i \(-0.415635\pi\)
0.261949 + 0.965082i \(0.415635\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19.8346 −0.677535 −0.338768 0.940870i \(-0.610010\pi\)
−0.338768 + 0.940870i \(0.610010\pi\)
\(858\) 0 0
\(859\) −21.2452 −0.724878 −0.362439 0.932007i \(-0.618056\pi\)
−0.362439 + 0.932007i \(0.618056\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.7039 −0.602648 −0.301324 0.953522i \(-0.597429\pi\)
−0.301324 + 0.953522i \(0.597429\pi\)
\(864\) 0 0
\(865\) −0.772199 −0.0262556
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.16534 −0.0734541
\(870\) 0 0
\(871\) 24.9596 0.845724
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.11244 −0.105220
\(876\) 0 0
\(877\) −52.9058 −1.78650 −0.893251 0.449558i \(-0.851582\pi\)
−0.893251 + 0.449558i \(0.851582\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −24.8543 −0.837362 −0.418681 0.908133i \(-0.637507\pi\)
−0.418681 + 0.908133i \(0.637507\pi\)
\(882\) 0 0
\(883\) −26.5886 −0.894779 −0.447390 0.894339i \(-0.647646\pi\)
−0.447390 + 0.894339i \(0.647646\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.3465 −0.347403 −0.173701 0.984798i \(-0.555573\pi\)
−0.173701 + 0.984798i \(0.555573\pi\)
\(888\) 0 0
\(889\) −7.05008 −0.236452
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.3226 0.847387
\(894\) 0 0
\(895\) 0.673383 0.0225087
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.9917 1.10033
\(900\) 0 0
\(901\) −20.9166 −0.696834
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.90547 −0.196304
\(906\) 0 0
\(907\) −11.9563 −0.397003 −0.198502 0.980101i \(-0.563607\pi\)
−0.198502 + 0.980101i \(0.563607\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 56.4330 1.86971 0.934854 0.355032i \(-0.115530\pi\)
0.934854 + 0.355032i \(0.115530\pi\)
\(912\) 0 0
\(913\) 18.1528 0.600771
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.8078 −0.356904
\(918\) 0 0
\(919\) −56.3910 −1.86017 −0.930084 0.367347i \(-0.880266\pi\)
−0.930084 + 0.367347i \(0.880266\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13.0347 −0.429041
\(924\) 0 0
\(925\) 13.6875 0.450041
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 33.4898 1.09877 0.549383 0.835571i \(-0.314863\pi\)
0.549383 + 0.835571i \(0.314863\pi\)
\(930\) 0 0
\(931\) 6.22915 0.204152
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.43006 −0.0794716
\(936\) 0 0
\(937\) 4.18683 0.136778 0.0683889 0.997659i \(-0.478214\pi\)
0.0683889 + 0.997659i \(0.478214\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 51.3176 1.67291 0.836453 0.548038i \(-0.184625\pi\)
0.836453 + 0.548038i \(0.184625\pi\)
\(942\) 0 0
\(943\) 55.1945 1.79738
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.5367 −0.634859 −0.317430 0.948282i \(-0.602820\pi\)
−0.317430 + 0.948282i \(0.602820\pi\)
\(948\) 0 0
\(949\) −26.2908 −0.853437
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.4500 −0.630047 −0.315023 0.949084i \(-0.602012\pi\)
−0.315023 + 0.949084i \(0.602012\pi\)
\(954\) 0 0
\(955\) −6.41791 −0.207679
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.6072 −0.600858
\(960\) 0 0
\(961\) −11.3177 −0.365088
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.82633 0.219747
\(966\) 0 0
\(967\) −48.8923 −1.57227 −0.786135 0.618054i \(-0.787921\pi\)
−0.786135 + 0.618054i \(0.787921\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.05201 0.290493 0.145246 0.989396i \(-0.453603\pi\)
0.145246 + 0.989396i \(0.453603\pi\)
\(972\) 0 0
\(973\) 8.54497 0.273939
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.3581 0.395372 0.197686 0.980265i \(-0.436657\pi\)
0.197686 + 0.980265i \(0.436657\pi\)
\(978\) 0 0
\(979\) 18.3011 0.584905
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.7513 0.470493 0.235246 0.971936i \(-0.424410\pi\)
0.235246 + 0.971936i \(0.424410\pi\)
\(984\) 0 0
\(985\) −7.78300 −0.247987
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.36555 0.0434219
\(990\) 0 0
\(991\) −45.9969 −1.46114 −0.730569 0.682838i \(-0.760745\pi\)
−0.730569 + 0.682838i \(0.760745\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.27314 0.0403614
\(996\) 0 0
\(997\) 38.9166 1.23250 0.616251 0.787550i \(-0.288651\pi\)
0.616251 + 0.787550i \(0.288651\pi\)
\(998\) 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.bo.1.3 yes 4
3.2 odd 2 6048.2.a.bv.1.2 yes 4
4.3 odd 2 6048.2.a.bm.1.3 4
12.11 even 2 6048.2.a.br.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bm.1.3 4 4.3 odd 2
6048.2.a.bo.1.3 yes 4 1.1 even 1 trivial
6048.2.a.br.1.2 yes 4 12.11 even 2
6048.2.a.bv.1.2 yes 4 3.2 odd 2