Properties

Label 6048.2.a.bk.1.3
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 0
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39528.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48380\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.79835 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.79835 q^{5} -1.00000 q^{7} +2.96759 q^{11} +6.95053 q^{13} -6.11977 q^{17} +0.967592 q^{19} +7.91812 q^{23} -1.76594 q^{25} +0.798349 q^{29} +0.798349 q^{31} -1.79835 q^{35} +6.76594 q^{37} +3.79835 q^{41} -9.71647 q^{43} -11.0703 q^{47} +1.00000 q^{49} +13.6670 q^{53} +5.33677 q^{55} +4.59670 q^{59} +8.56429 q^{61} +12.4995 q^{65} +13.5148 q^{67} -11.9181 q^{71} +4.33849 q^{73} -2.96759 q^{77} -17.6346 q^{79} -0.765941 q^{83} -11.0055 q^{85} +3.98294 q^{89} -6.95053 q^{91} +1.74007 q^{95} -6.86865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - 4q^{7} + 8q^{13} - 2q^{17} - 8q^{19} + 14q^{25} - 6q^{29} - 6q^{31} + 2q^{35} + 6q^{37} + 6q^{41} + 2q^{43} - 2q^{47} + 4q^{49} - 6q^{53} + 12q^{55} + 4q^{61} - 4q^{65} + 4q^{67} - 16q^{71} + 12q^{73} + 2q^{79} + 18q^{83} + 22q^{85} + 8q^{89} - 8q^{91} + 16q^{95} + 24q^{97} + O(q^{100}) \)

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
\(5\) 1.79835 0.804246 0.402123 0.915586i \(-0.368272\pi\)
0.402123 + 0.915586i \(0.368272\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.96759 0.894763 0.447381 0.894343i \(-0.352357\pi\)
0.447381 + 0.894343i \(0.352357\pi\)
\(12\) 0 0
\(13\) 6.95053 1.92773 0.963865 0.266392i \(-0.0858315\pi\)
0.963865 + 0.266392i \(0.0858315\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.11977 −1.48426 −0.742131 0.670254i \(-0.766185\pi\)
−0.742131 + 0.670254i \(0.766185\pi\)
\(18\) 0 0
\(19\) 0.967592 0.221981 0.110990 0.993821i \(-0.464598\pi\)
0.110990 + 0.993821i \(0.464598\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.91812 1.65104 0.825521 0.564371i \(-0.190881\pi\)
0.825521 + 0.564371i \(0.190881\pi\)
\(24\) 0 0
\(25\) −1.76594 −0.353188
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.798349 0.148250 0.0741248 0.997249i \(-0.476384\pi\)
0.0741248 + 0.997249i \(0.476384\pi\)
\(30\) 0 0
\(31\) 0.798349 0.143388 0.0716938 0.997427i \(-0.477160\pi\)
0.0716938 + 0.997427i \(0.477160\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.79835 −0.303976
\(36\) 0 0
\(37\) 6.76594 1.11231 0.556157 0.831077i \(-0.312275\pi\)
0.556157 + 0.831077i \(0.312275\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.79835 0.593202 0.296601 0.955001i \(-0.404147\pi\)
0.296601 + 0.955001i \(0.404147\pi\)
\(42\) 0 0
\(43\) −9.71647 −1.48175 −0.740874 0.671644i \(-0.765588\pi\)
−0.740874 + 0.671644i \(0.765588\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.0703 −1.61477 −0.807385 0.590025i \(-0.799118\pi\)
−0.807385 + 0.590025i \(0.799118\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.6670 1.87731 0.938653 0.344862i \(-0.112074\pi\)
0.938653 + 0.344862i \(0.112074\pi\)
\(54\) 0 0
\(55\) 5.33677 0.719609
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.59670 0.598439 0.299220 0.954184i \(-0.403274\pi\)
0.299220 + 0.954184i \(0.403274\pi\)
\(60\) 0 0
\(61\) 8.56429 1.09654 0.548272 0.836300i \(-0.315286\pi\)
0.548272 + 0.836300i \(0.315286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.4995 1.55037
\(66\) 0 0
\(67\) 13.5148 1.65110 0.825549 0.564331i \(-0.190866\pi\)
0.825549 + 0.564331i \(0.190866\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.9181 −1.41442 −0.707210 0.707003i \(-0.750047\pi\)
−0.707210 + 0.707003i \(0.750047\pi\)
\(72\) 0 0
\(73\) 4.33849 0.507781 0.253891 0.967233i \(-0.418290\pi\)
0.253891 + 0.967233i \(0.418290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.96759 −0.338189
\(78\) 0 0
\(79\) −17.6346 −1.98405 −0.992023 0.126055i \(-0.959768\pi\)
−0.992023 + 0.126055i \(0.959768\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.765941 −0.0840730 −0.0420365 0.999116i \(-0.513385\pi\)
−0.0420365 + 0.999116i \(0.513385\pi\)
\(84\) 0 0
\(85\) −11.0055 −1.19371
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.98294 0.422190 0.211095 0.977466i \(-0.432297\pi\)
0.211095 + 0.977466i \(0.432297\pi\)
\(90\) 0 0
\(91\) −6.95053 −0.728613
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.74007 0.178527
\(96\) 0 0
\(97\) −6.86865 −0.697406 −0.348703 0.937233i \(-0.613378\pi\)
−0.348703 + 0.937233i \(0.613378\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.93518 −0.789580 −0.394790 0.918771i \(-0.629183\pi\)
−0.394790 + 0.918771i \(0.629183\pi\)
\(102\) 0 0
\(103\) −12.4736 −1.22906 −0.614530 0.788893i \(-0.710654\pi\)
−0.614530 + 0.788893i \(0.710654\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 12.7659 1.22276 0.611378 0.791339i \(-0.290616\pi\)
0.611378 + 0.791339i \(0.290616\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.96759 0.655456 0.327728 0.944772i \(-0.393717\pi\)
0.327728 + 0.944772i \(0.393717\pi\)
\(114\) 0 0
\(115\) 14.2395 1.32784
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.11977 0.560999
\(120\) 0 0
\(121\) −2.19340 −0.199400
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1675 −1.08830
\(126\) 0 0
\(127\) 11.7335 1.04118 0.520591 0.853806i \(-0.325712\pi\)
0.520591 + 0.853806i \(0.325712\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.507730 −0.0443606 −0.0221803 0.999754i \(-0.507061\pi\)
−0.0221803 + 0.999754i \(0.507061\pi\)
\(132\) 0 0
\(133\) −0.967592 −0.0839009
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.03241 0.259076 0.129538 0.991574i \(-0.458651\pi\)
0.129538 + 0.991574i \(0.458651\pi\)
\(138\) 0 0
\(139\) 18.1269 1.53750 0.768750 0.639549i \(-0.220879\pi\)
0.768750 + 0.639549i \(0.220879\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.6263 1.72486
\(144\) 0 0
\(145\) 1.43571 0.119229
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.13684 0.256980 0.128490 0.991711i \(-0.458987\pi\)
0.128490 + 0.991711i \(0.458987\pi\)
\(150\) 0 0
\(151\) 14.4412 1.17521 0.587604 0.809149i \(-0.300071\pi\)
0.587604 + 0.809149i \(0.300071\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.43571 0.115319
\(156\) 0 0
\(157\) 8.24287 0.657852 0.328926 0.944356i \(-0.393313\pi\)
0.328926 + 0.944356i \(0.393313\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.91812 −0.624035
\(162\) 0 0
\(163\) −11.3780 −0.891192 −0.445596 0.895234i \(-0.647008\pi\)
−0.445596 + 0.895234i \(0.647008\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.7318 −1.13998 −0.569991 0.821651i \(-0.693053\pi\)
−0.569991 + 0.821651i \(0.693053\pi\)
\(168\) 0 0
\(169\) 35.3098 2.71614
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −17.5319 −1.33292 −0.666462 0.745539i \(-0.732192\pi\)
−0.666462 + 0.745539i \(0.732192\pi\)
\(174\) 0 0
\(175\) 1.76594 0.133493
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.2258 −0.913799 −0.456900 0.889518i \(-0.651040\pi\)
−0.456900 + 0.889518i \(0.651040\pi\)
\(180\) 0 0
\(181\) 3.67858 0.273426 0.136713 0.990611i \(-0.456346\pi\)
0.136713 + 0.990611i \(0.456346\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.1675 0.894574
\(186\) 0 0
\(187\) −18.1610 −1.32806
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.6615 0.988512 0.494256 0.869316i \(-0.335440\pi\)
0.494256 + 0.869316i \(0.335440\pi\)
\(192\) 0 0
\(193\) 6.66151 0.479506 0.239753 0.970834i \(-0.422934\pi\)
0.239753 + 0.970834i \(0.422934\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.4653 −1.45810 −0.729048 0.684463i \(-0.760037\pi\)
−0.729048 + 0.684463i \(0.760037\pi\)
\(198\) 0 0
\(199\) −10.0055 −0.709270 −0.354635 0.935005i \(-0.615395\pi\)
−0.354635 + 0.935005i \(0.615395\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.798349 −0.0560331
\(204\) 0 0
\(205\) 6.83076 0.477081
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.87142 0.198620
\(210\) 0 0
\(211\) −4.22248 −0.290687 −0.145344 0.989381i \(-0.546429\pi\)
−0.145344 + 0.989381i \(0.546429\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −17.4736 −1.19169
\(216\) 0 0
\(217\) −0.798349 −0.0541955
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −42.5356 −2.86126
\(222\) 0 0
\(223\) 19.5456 1.30887 0.654436 0.756118i \(-0.272906\pi\)
0.654436 + 0.756118i \(0.272906\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.9593 1.39112 0.695560 0.718468i \(-0.255156\pi\)
0.695560 + 0.718468i \(0.255156\pi\)
\(228\) 0 0
\(229\) 24.2395 1.60179 0.800897 0.598802i \(-0.204356\pi\)
0.800897 + 0.598802i \(0.204356\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.6104 1.15370 0.576849 0.816851i \(-0.304282\pi\)
0.576849 + 0.816851i \(0.304282\pi\)
\(234\) 0 0
\(235\) −19.9083 −1.29867
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −19.7236 −1.27581 −0.637905 0.770115i \(-0.720199\pi\)
−0.637905 + 0.770115i \(0.720199\pi\)
\(240\) 0 0
\(241\) 30.4005 1.95827 0.979135 0.203210i \(-0.0651374\pi\)
0.979135 + 0.203210i \(0.0651374\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.79835 0.114892
\(246\) 0 0
\(247\) 6.72528 0.427919
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.9593 −1.25982 −0.629911 0.776667i \(-0.716909\pi\)
−0.629911 + 0.776667i \(0.716909\pi\)
\(252\) 0 0
\(253\) 23.4978 1.47729
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.49775 −0.342940 −0.171470 0.985189i \(-0.554852\pi\)
−0.171470 + 0.985189i \(0.554852\pi\)
\(258\) 0 0
\(259\) −6.76594 −0.420415
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.6923 0.844304 0.422152 0.906525i \(-0.361275\pi\)
0.422152 + 0.906525i \(0.361275\pi\)
\(264\) 0 0
\(265\) 24.5780 1.50982
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.4599 −0.942604 −0.471302 0.881972i \(-0.656216\pi\)
−0.471302 + 0.881972i \(0.656216\pi\)
\(270\) 0 0
\(271\) 10.0055 0.607790 0.303895 0.952706i \(-0.401713\pi\)
0.303895 + 0.952706i \(0.401713\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.24059 −0.316020
\(276\) 0 0
\(277\) 15.1044 0.907537 0.453769 0.891120i \(-0.350079\pi\)
0.453769 + 0.891120i \(0.350079\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.33849 0.258812 0.129406 0.991592i \(-0.458693\pi\)
0.129406 + 0.991592i \(0.458693\pi\)
\(282\) 0 0
\(283\) 30.4653 1.81098 0.905488 0.424371i \(-0.139505\pi\)
0.905488 + 0.424371i \(0.139505\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.79835 −0.224209
\(288\) 0 0
\(289\) 20.4516 1.20304
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.2774 1.18462 0.592310 0.805710i \(-0.298216\pi\)
0.592310 + 0.805710i \(0.298216\pi\)
\(294\) 0 0
\(295\) 8.26647 0.481292
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 55.0351 3.18276
\(300\) 0 0
\(301\) 9.71647 0.560048
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.4016 0.881892
\(306\) 0 0
\(307\) −5.14509 −0.293646 −0.146823 0.989163i \(-0.546905\pi\)
−0.146823 + 0.989163i \(0.546905\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.0889674 0.00504488 0.00252244 0.999997i \(-0.499197\pi\)
0.00252244 + 0.999997i \(0.499197\pi\)
\(312\) 0 0
\(313\) −27.6077 −1.56048 −0.780239 0.625482i \(-0.784902\pi\)
−0.780239 + 0.625482i \(0.784902\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.6615 −0.654976 −0.327488 0.944855i \(-0.606202\pi\)
−0.327488 + 0.944855i \(0.606202\pi\)
\(318\) 0 0
\(319\) 2.36917 0.132648
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.92144 −0.329478
\(324\) 0 0
\(325\) −12.2742 −0.680851
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0703 0.610325
\(330\) 0 0
\(331\) −15.8154 −0.869294 −0.434647 0.900601i \(-0.643127\pi\)
−0.434647 + 0.900601i \(0.643127\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.3044 1.32789
\(336\) 0 0
\(337\) −9.60872 −0.523420 −0.261710 0.965147i \(-0.584286\pi\)
−0.261710 + 0.965147i \(0.584286\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.36917 0.128298
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.8345 −1.01109 −0.505545 0.862800i \(-0.668709\pi\)
−0.505545 + 0.862800i \(0.668709\pi\)
\(348\) 0 0
\(349\) 4.04775 0.216671 0.108336 0.994114i \(-0.465448\pi\)
0.108336 + 0.994114i \(0.465448\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.52307 0.347188 0.173594 0.984817i \(-0.444462\pi\)
0.173594 + 0.984817i \(0.444462\pi\)
\(354\) 0 0
\(355\) −21.4329 −1.13754
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.6093 1.50994 0.754970 0.655760i \(-0.227652\pi\)
0.754970 + 0.655760i \(0.227652\pi\)
\(360\) 0 0
\(361\) −18.0638 −0.950724
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.80211 0.408381
\(366\) 0 0
\(367\) −1.66700 −0.0870166 −0.0435083 0.999053i \(-0.513853\pi\)
−0.0435083 + 0.999053i \(0.513853\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.6670 −0.709555
\(372\) 0 0
\(373\) 20.1506 1.04336 0.521679 0.853142i \(-0.325306\pi\)
0.521679 + 0.853142i \(0.325306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.54895 0.285785
\(378\) 0 0
\(379\) 26.7797 1.37558 0.687790 0.725910i \(-0.258581\pi\)
0.687790 + 0.725910i \(0.258581\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.7966 −0.653877 −0.326939 0.945046i \(-0.606017\pi\)
−0.326939 + 0.945046i \(0.606017\pi\)
\(384\) 0 0
\(385\) −5.33677 −0.271987
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.0620 −1.21999 −0.609997 0.792404i \(-0.708829\pi\)
−0.609997 + 0.792404i \(0.708829\pi\)
\(390\) 0 0
\(391\) −48.4571 −2.45058
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −31.7131 −1.59566
\(396\) 0 0
\(397\) −7.15927 −0.359313 −0.179657 0.983729i \(-0.557499\pi\)
−0.179657 + 0.983729i \(0.557499\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.7373 −0.586133 −0.293066 0.956092i \(-0.594676\pi\)
−0.293066 + 0.956092i \(0.594676\pi\)
\(402\) 0 0
\(403\) 5.54895 0.276413
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0786 0.995257
\(408\) 0 0
\(409\) −2.57803 −0.127475 −0.0637377 0.997967i \(-0.520302\pi\)
−0.0637377 + 0.997967i \(0.520302\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.59670 −0.226189
\(414\) 0 0
\(415\) −1.37743 −0.0676154
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.7373 0.915377 0.457688 0.889113i \(-0.348678\pi\)
0.457688 + 0.889113i \(0.348678\pi\)
\(420\) 0 0
\(421\) −18.7560 −0.914110 −0.457055 0.889438i \(-0.651096\pi\)
−0.457055 + 0.889438i \(0.651096\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.8072 0.524224
\(426\) 0 0
\(427\) −8.56429 −0.414455
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.9352 −0.863907 −0.431954 0.901896i \(-0.642176\pi\)
−0.431954 + 0.901896i \(0.642176\pi\)
\(432\) 0 0
\(433\) 25.6587 1.23308 0.616540 0.787323i \(-0.288534\pi\)
0.616540 + 0.787323i \(0.288534\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.66151 0.366500
\(438\) 0 0
\(439\) −20.7473 −0.990213 −0.495107 0.868832i \(-0.664871\pi\)
−0.495107 + 0.868832i \(0.664871\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.35543 0.111910 0.0559550 0.998433i \(-0.482180\pi\)
0.0559550 + 0.998433i \(0.482180\pi\)
\(444\) 0 0
\(445\) 7.16271 0.339545
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.5643 1.06487 0.532437 0.846469i \(-0.321276\pi\)
0.532437 + 0.846469i \(0.321276\pi\)
\(450\) 0 0
\(451\) 11.2720 0.530775
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.4995 −0.585984
\(456\) 0 0
\(457\) −17.9714 −0.840665 −0.420332 0.907370i \(-0.638086\pi\)
−0.420332 + 0.907370i \(0.638086\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.29610 0.339813 0.169907 0.985460i \(-0.445653\pi\)
0.169907 + 0.985460i \(0.445653\pi\)
\(462\) 0 0
\(463\) 32.8280 1.52565 0.762823 0.646607i \(-0.223813\pi\)
0.762823 + 0.646607i \(0.223813\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.5439 0.673012 0.336506 0.941681i \(-0.390755\pi\)
0.336506 + 0.941681i \(0.390755\pi\)
\(468\) 0 0
\(469\) −13.5148 −0.624056
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −28.8345 −1.32581
\(474\) 0 0
\(475\) −1.70871 −0.0784010
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.2517 0.879632 0.439816 0.898088i \(-0.355044\pi\)
0.439816 + 0.898088i \(0.355044\pi\)
\(480\) 0 0
\(481\) 47.0269 2.14424
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.3522 −0.560886
\(486\) 0 0
\(487\) −2.85491 −0.129368 −0.0646841 0.997906i \(-0.520604\pi\)
−0.0646841 + 0.997906i \(0.520604\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.95557 −0.0882538 −0.0441269 0.999026i \(-0.514051\pi\)
−0.0441269 + 0.999026i \(0.514051\pi\)
\(492\) 0 0
\(493\) −4.88571 −0.220041
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.9181 0.534601
\(498\) 0 0
\(499\) 7.58629 0.339609 0.169804 0.985478i \(-0.445686\pi\)
0.169804 + 0.985478i \(0.445686\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −28.6670 −1.27820 −0.639099 0.769124i \(-0.720693\pi\)
−0.639099 + 0.769124i \(0.720693\pi\)
\(504\) 0 0
\(505\) −14.2702 −0.635017
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.3423 −1.69949 −0.849745 0.527194i \(-0.823244\pi\)
−0.849745 + 0.527194i \(0.823244\pi\)
\(510\) 0 0
\(511\) −4.33849 −0.191923
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −22.4319 −0.988467
\(516\) 0 0
\(517\) −32.8521 −1.44484
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.7285 −0.645267 −0.322633 0.946524i \(-0.604568\pi\)
−0.322633 + 0.946524i \(0.604568\pi\)
\(522\) 0 0
\(523\) 20.7049 0.905362 0.452681 0.891673i \(-0.350468\pi\)
0.452681 + 0.891673i \(0.350468\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.88571 −0.212825
\(528\) 0 0
\(529\) 39.6966 1.72594
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26.4005 1.14353
\(534\) 0 0
\(535\) −7.19340 −0.310998
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.96759 0.127823
\(540\) 0 0
\(541\) −10.5539 −0.453747 −0.226873 0.973924i \(-0.572850\pi\)
−0.226873 + 0.973924i \(0.572850\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.9576 0.983396
\(546\) 0 0
\(547\) 17.8290 0.762315 0.381157 0.924510i \(-0.375526\pi\)
0.381157 + 0.924510i \(0.375526\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.772476 0.0329086
\(552\) 0 0
\(553\) 17.6346 0.749899
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.6625 1.80767 0.903834 0.427883i \(-0.140740\pi\)
0.903834 + 0.427883i \(0.140740\pi\)
\(558\) 0 0
\(559\) −67.5346 −2.85641
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.2175 0.936358 0.468179 0.883634i \(-0.344910\pi\)
0.468179 + 0.883634i \(0.344910\pi\)
\(564\) 0 0
\(565\) 12.5302 0.527148
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −32.0451 −1.34340 −0.671700 0.740823i \(-0.734436\pi\)
−0.671700 + 0.740823i \(0.734436\pi\)
\(570\) 0 0
\(571\) 7.44624 0.311615 0.155808 0.987787i \(-0.450202\pi\)
0.155808 + 0.987787i \(0.450202\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.9829 −0.583129
\(576\) 0 0
\(577\) −9.49775 −0.395397 −0.197698 0.980263i \(-0.563347\pi\)
−0.197698 + 0.980263i \(0.563347\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.765941 0.0317766
\(582\) 0 0
\(583\) 40.5581 1.67974
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.03617 −0.0840418 −0.0420209 0.999117i \(-0.513380\pi\)
−0.0420209 + 0.999117i \(0.513380\pi\)
\(588\) 0 0
\(589\) 0.772476 0.0318293
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.2017 0.665322 0.332661 0.943046i \(-0.392054\pi\)
0.332661 + 0.943046i \(0.392054\pi\)
\(594\) 0 0
\(595\) 11.0055 0.451181
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −33.1253 −1.35346 −0.676731 0.736231i \(-0.736604\pi\)
−0.676731 + 0.736231i \(0.736604\pi\)
\(600\) 0 0
\(601\) 10.4347 0.425639 0.212819 0.977092i \(-0.431735\pi\)
0.212819 + 0.977092i \(0.431735\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.94449 −0.160366
\(606\) 0 0
\(607\) −10.1896 −0.413584 −0.206792 0.978385i \(-0.566302\pi\)
−0.206792 + 0.978385i \(0.566302\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −76.9444 −3.11284
\(612\) 0 0
\(613\) 3.78794 0.152993 0.0764967 0.997070i \(-0.475627\pi\)
0.0764967 + 0.997070i \(0.475627\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.2737 −0.574636 −0.287318 0.957835i \(-0.592764\pi\)
−0.287318 + 0.957835i \(0.592764\pi\)
\(618\) 0 0
\(619\) −31.8159 −1.27879 −0.639394 0.768880i \(-0.720815\pi\)
−0.639394 + 0.768880i \(0.720815\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.98294 −0.159573
\(624\) 0 0
\(625\) −13.0517 −0.522070
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −41.4060 −1.65097
\(630\) 0 0
\(631\) 40.9917 1.63186 0.815928 0.578154i \(-0.196227\pi\)
0.815928 + 0.578154i \(0.196227\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.1010 0.837367
\(636\) 0 0
\(637\) 6.95053 0.275390
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 29.0500 1.14741 0.573703 0.819063i \(-0.305506\pi\)
0.573703 + 0.819063i \(0.305506\pi\)
\(642\) 0 0
\(643\) −34.4450 −1.35838 −0.679188 0.733964i \(-0.737668\pi\)
−0.679188 + 0.733964i \(0.737668\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.4033 −0.566252 −0.283126 0.959083i \(-0.591372\pi\)
−0.283126 + 0.959083i \(0.591372\pi\)
\(648\) 0 0
\(649\) 13.6411 0.535461
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.6005 0.414828 0.207414 0.978253i \(-0.433495\pi\)
0.207414 + 0.978253i \(0.433495\pi\)
\(654\) 0 0
\(655\) −0.913076 −0.0356768
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.46363 −0.368651 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(660\) 0 0
\(661\) −4.11268 −0.159965 −0.0799824 0.996796i \(-0.525486\pi\)
−0.0799824 + 0.996796i \(0.525486\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.74007 −0.0674770
\(666\) 0 0
\(667\) 6.32142 0.244766
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 25.4153 0.981148
\(672\) 0 0
\(673\) 15.6977 0.605101 0.302551 0.953133i \(-0.402162\pi\)
0.302551 + 0.953133i \(0.402162\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.90450 0.0731958 0.0365979 0.999330i \(-0.488348\pi\)
0.0365979 + 0.999330i \(0.488348\pi\)
\(678\) 0 0
\(679\) 6.86865 0.263595
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −10.5643 −0.404231 −0.202116 0.979362i \(-0.564782\pi\)
−0.202116 + 0.979362i \(0.564782\pi\)
\(684\) 0 0
\(685\) 5.45333 0.208361
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 94.9929 3.61894
\(690\) 0 0
\(691\) −15.7742 −0.600079 −0.300039 0.953927i \(-0.597000\pi\)
−0.300039 + 0.953927i \(0.597000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 32.5984 1.23653
\(696\) 0 0
\(697\) −23.2450 −0.880468
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.8686 −1.09035 −0.545177 0.838321i \(-0.683538\pi\)
−0.545177 + 0.838321i \(0.683538\pi\)
\(702\) 0 0
\(703\) 6.54667 0.246912
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.93518 0.298433
\(708\) 0 0
\(709\) 2.68351 0.100781 0.0503906 0.998730i \(-0.483953\pi\)
0.0503906 + 0.998730i \(0.483953\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.32142 0.236739
\(714\) 0 0
\(715\) 37.0933 1.38721
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.43842 0.314700 0.157350 0.987543i \(-0.449705\pi\)
0.157350 + 0.987543i \(0.449705\pi\)
\(720\) 0 0
\(721\) 12.4736 0.464541
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.40984 −0.0523600
\(726\) 0 0
\(727\) −33.5202 −1.24319 −0.621597 0.783337i \(-0.713516\pi\)
−0.621597 + 0.783337i \(0.713516\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 59.4626 2.19930
\(732\) 0 0
\(733\) −18.8995 −0.698067 −0.349034 0.937110i \(-0.613490\pi\)
−0.349034 + 0.937110i \(0.613490\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.1065 1.47734
\(738\) 0 0
\(739\) 38.1889 1.40480 0.702401 0.711782i \(-0.252111\pi\)
0.702401 + 0.711782i \(0.252111\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35.9153 1.31761 0.658803 0.752315i \(-0.271063\pi\)
0.658803 + 0.752315i \(0.271063\pi\)
\(744\) 0 0
\(745\) 5.64113 0.206675
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −27.0979 −0.988816 −0.494408 0.869230i \(-0.664615\pi\)
−0.494408 + 0.869230i \(0.664615\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.9703 0.945156
\(756\) 0 0
\(757\) 42.0010 1.52655 0.763276 0.646073i \(-0.223590\pi\)
0.763276 + 0.646073i \(0.223590\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.6637 0.894057 0.447029 0.894520i \(-0.352482\pi\)
0.447029 + 0.894520i \(0.352482\pi\)
\(762\) 0 0
\(763\) −12.7659 −0.462158
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 31.9495 1.15363
\(768\) 0 0
\(769\) −17.8362 −0.643191 −0.321596 0.946877i \(-0.604219\pi\)
−0.321596 + 0.946877i \(0.604219\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.99174 −0.323411 −0.161705 0.986839i \(-0.551699\pi\)
−0.161705 + 0.986839i \(0.551699\pi\)
\(774\) 0 0
\(775\) −1.40984 −0.0506428
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.67525 0.131680
\(780\) 0 0
\(781\) −35.3681 −1.26557
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.8236 0.529075
\(786\) 0 0
\(787\) −13.2692 −0.472995 −0.236498 0.971632i \(-0.576000\pi\)
−0.236498 + 0.971632i \(0.576000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.96759 −0.247739
\(792\) 0 0
\(793\) 59.5263 2.11384
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −49.2385 −1.74412 −0.872058 0.489402i \(-0.837215\pi\)
−0.872058 + 0.489402i \(0.837215\pi\)
\(798\) 0 0
\(799\) 67.7477 2.39674
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.8749 0.454344
\(804\) 0 0
\(805\) −14.2395 −0.501878
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.8176 1.08349 0.541744 0.840543i \(-0.317764\pi\)
0.541744 + 0.840543i \(0.317764\pi\)
\(810\) 0 0
\(811\) −38.8038 −1.36259 −0.681293 0.732010i \(-0.738582\pi\)
−0.681293 + 0.732010i \(0.738582\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.4616 −0.716738
\(816\) 0 0
\(817\) −9.40158 −0.328920
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.9935 −0.453475 −0.226738 0.973956i \(-0.572806\pi\)
−0.226738 + 0.973956i \(0.572806\pi\)
\(822\) 0 0
\(823\) −47.9224 −1.67047 −0.835236 0.549892i \(-0.814669\pi\)
−0.835236 + 0.549892i \(0.814669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.19617 −0.180688 −0.0903442 0.995911i \(-0.528797\pi\)
−0.0903442 + 0.995911i \(0.528797\pi\)
\(828\) 0 0
\(829\) 33.3170 1.15715 0.578574 0.815630i \(-0.303609\pi\)
0.578574 + 0.815630i \(0.303609\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.11977 −0.212038
\(834\) 0 0
\(835\) −26.4929 −0.916826
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.6966 −0.818099 −0.409049 0.912512i \(-0.634140\pi\)
−0.409049 + 0.912512i \(0.634140\pi\)
\(840\) 0 0
\(841\) −28.3626 −0.978022
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 63.4994 2.18445
\(846\) 0 0
\(847\) 2.19340 0.0753660
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 53.5735 1.83648
\(852\) 0 0
\(853\) 3.51482 0.120345 0.0601725 0.998188i \(-0.480835\pi\)
0.0601725 + 0.998188i \(0.480835\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −44.1374 −1.50770 −0.753852 0.657044i \(-0.771807\pi\)
−0.753852 + 0.657044i \(0.771807\pi\)
\(858\) 0 0
\(859\) 32.2846 1.10154 0.550769 0.834658i \(-0.314335\pi\)
0.550769 + 0.834658i \(0.314335\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23.3169 −0.793718 −0.396859 0.917880i \(-0.629900\pi\)
−0.396859 + 0.917880i \(0.629900\pi\)
\(864\) 0 0
\(865\) −31.5284 −1.07200
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −52.3323 −1.77525
\(870\) 0 0
\(871\) 93.9351 3.18287
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.1675 0.411337
\(876\) 0 0
\(877\) −54.8252 −1.85132 −0.925658 0.378361i \(-0.876488\pi\)
−0.925658 + 0.378361i \(0.876488\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.0819 0.339667 0.169834 0.985473i \(-0.445677\pi\)
0.169834 + 0.985473i \(0.445677\pi\)
\(882\) 0 0
\(883\) 13.9144 0.468255 0.234128 0.972206i \(-0.424777\pi\)
0.234128 + 0.972206i \(0.424777\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 40.3088 1.35344 0.676718 0.736242i \(-0.263401\pi\)
0.676718 + 0.736242i \(0.263401\pi\)
\(888\) 0 0
\(889\) −11.7335 −0.393530
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.7115 −0.358448
\(894\) 0 0
\(895\) −21.9863 −0.734920
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.637361 0.0212572
\(900\) 0 0
\(901\) −83.6389 −2.78642
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.61537 0.219902
\(906\) 0 0
\(907\) 7.23129 0.240111 0.120055 0.992767i \(-0.461693\pi\)
0.120055 + 0.992767i \(0.461693\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.07751 −0.300751 −0.150376 0.988629i \(-0.548048\pi\)
−0.150376 + 0.988629i \(0.548048\pi\)
\(912\) 0 0
\(913\) −2.27300 −0.0752254
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.507730 0.0167667
\(918\) 0 0
\(919\) 23.5698 0.777495 0.388748 0.921344i \(-0.372908\pi\)
0.388748 + 0.921344i \(0.372908\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −82.8372 −2.72662
\(924\) 0 0
\(925\) −11.9483 −0.392856
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 32.4071 1.06324 0.531621 0.846983i \(-0.321583\pi\)
0.531621 + 0.846983i \(0.321583\pi\)
\(930\) 0 0
\(931\) 0.967592 0.0317116
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −32.6598 −1.06809
\(936\) 0 0
\(937\) −7.37089 −0.240797 −0.120398 0.992726i \(-0.538417\pi\)
−0.120398 + 0.992726i \(0.538417\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.00826 0.163264 0.0816322 0.996663i \(-0.473987\pi\)
0.0816322 + 0.996663i \(0.473987\pi\)
\(942\) 0 0
\(943\) 30.0758 0.979402
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0110 0.780252 0.390126 0.920762i \(-0.372432\pi\)
0.390126 + 0.920762i \(0.372432\pi\)
\(948\) 0 0
\(949\) 30.1548 0.978865
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14.3692 −0.465463 −0.232732 0.972541i \(-0.574766\pi\)
−0.232732 + 0.972541i \(0.574766\pi\)
\(954\) 0 0
\(955\) 24.5682 0.795007
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.03241 −0.0979215
\(960\) 0 0
\(961\) −30.3626 −0.979440
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.9797 0.385641
\(966\) 0 0
\(967\) 10.4374 0.335645 0.167823 0.985817i \(-0.446326\pi\)
0.167823 + 0.985817i \(0.446326\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −41.6845 −1.33772 −0.668860 0.743389i \(-0.733217\pi\)
−0.668860 + 0.743389i \(0.733217\pi\)
\(972\) 0 0
\(973\) −18.1269 −0.581120
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.9813 0.351324 0.175662 0.984451i \(-0.443793\pi\)
0.175662 + 0.984451i \(0.443793\pi\)
\(978\) 0 0
\(979\) 11.8197 0.377760
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.7956 −0.695171 −0.347585 0.937648i \(-0.612998\pi\)
−0.347585 + 0.937648i \(0.612998\pi\)
\(984\) 0 0
\(985\) −36.8038 −1.17267
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −76.9362 −2.44643
\(990\) 0 0
\(991\) −29.9389 −0.951042 −0.475521 0.879704i \(-0.657740\pi\)
−0.475521 + 0.879704i \(0.657740\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −17.9934 −0.570428
\(996\) 0 0
\(997\) −22.9164 −0.725770 −0.362885 0.931834i \(-0.618208\pi\)
−0.362885 + 0.931834i \(0.618208\pi\)
\(998\) 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.bk.1.3 4
3.2 odd 2 6048.2.a.bs.1.2 yes 4
4.3 odd 2 6048.2.a.bp.1.3 yes 4
12.11 even 2 6048.2.a.bt.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bk.1.3 4 1.1 even 1 trivial
6048.2.a.bp.1.3 yes 4 4.3 odd 2
6048.2.a.bs.1.2 yes 4 3.2 odd 2
6048.2.a.bt.1.2 yes 4 12.11 even 2