Properties

Label 6048.2.a.bt.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 \(-2.50948\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.29751 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+2.29751 q^{5} +1.00000 q^{7} +5.01897 q^{11} -4.55358 q^{13} +0.762893 q^{17} -3.01897 q^{19} -1.53461 q^{23} +0.278540 q^{25} +3.29751 q^{29} +3.29751 q^{31} +2.29751 q^{35} +4.72146 q^{37} +0.297507 q^{41} -3.83212 q^{43} +5.79069 q^{47} +1.00000 q^{49} +11.3857 q^{53} +11.5311 q^{55} -3.59501 q^{59} +2.42395 q^{61} -10.4619 q^{65} +4.12963 q^{67} -2.46539 q^{71} +16.6329 q^{73} +5.01897 q^{77} -5.36674 q^{79} +1.27854 q^{83} +1.75275 q^{85} +9.57255 q^{89} -4.55358 q^{91} -6.93610 q^{95} +14.0882 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} + 4q^{7} + 8q^{13} + 2q^{17} + 8q^{19} + 14q^{25} + 6q^{29} + 6q^{31} + 2q^{35} + 6q^{37} - 6q^{41} - 2q^{43} - 2q^{47} + 4q^{49} + 6q^{53} - 12q^{55} + 4q^{61} + 4q^{65} - 4q^{67} - 16q^{71} + 12q^{73} - 2q^{79} + 18q^{83} + 22q^{85} - 8q^{89} + 8q^{91} + 16q^{95} + 24q^{97} + O(q^{100}) \)

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
\(5\) 2.29751 1.02748 0.513738 0.857947i \(-0.328260\pi\)
0.513738 + 0.857947i \(0.328260\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.01897 1.51328 0.756638 0.653834i \(-0.226841\pi\)
0.756638 + 0.653834i \(0.226841\pi\)
\(12\) 0 0
\(13\) −4.55358 −1.26294 −0.631468 0.775402i \(-0.717547\pi\)
−0.631468 + 0.775402i \(0.717547\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.762893 0.185029 0.0925144 0.995711i \(-0.470510\pi\)
0.0925144 + 0.995711i \(0.470510\pi\)
\(18\) 0 0
\(19\) −3.01897 −0.692599 −0.346299 0.938124i \(-0.612562\pi\)
−0.346299 + 0.938124i \(0.612562\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.53461 −0.319989 −0.159995 0.987118i \(-0.551148\pi\)
−0.159995 + 0.987118i \(0.551148\pi\)
\(24\) 0 0
\(25\) 0.278540 0.0557081
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.29751 0.612332 0.306166 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(30\) 0 0
\(31\) 3.29751 0.592250 0.296125 0.955149i \(-0.404306\pi\)
0.296125 + 0.955149i \(0.404306\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.29751 0.388350
\(36\) 0 0
\(37\) 4.72146 0.776203 0.388102 0.921617i \(-0.373131\pi\)
0.388102 + 0.921617i \(0.373131\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.297507 0.0464629 0.0232314 0.999730i \(-0.492605\pi\)
0.0232314 + 0.999730i \(0.492605\pi\)
\(42\) 0 0
\(43\) −3.83212 −0.584393 −0.292197 0.956358i \(-0.594386\pi\)
−0.292197 + 0.956358i \(0.594386\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.79069 0.844659 0.422329 0.906442i \(-0.361213\pi\)
0.422329 + 0.906442i \(0.361213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.3857 1.56395 0.781973 0.623312i \(-0.214213\pi\)
0.781973 + 0.623312i \(0.214213\pi\)
\(54\) 0 0
\(55\) 11.5311 1.55486
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.59501 −0.468031 −0.234016 0.972233i \(-0.575187\pi\)
−0.234016 + 0.972233i \(0.575187\pi\)
\(60\) 0 0
\(61\) 2.42395 0.310355 0.155178 0.987887i \(-0.450405\pi\)
0.155178 + 0.987887i \(0.450405\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.4619 −1.29764
\(66\) 0 0
\(67\) 4.12963 0.504514 0.252257 0.967660i \(-0.418827\pi\)
0.252257 + 0.967660i \(0.418827\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.46539 −0.292587 −0.146294 0.989241i \(-0.546734\pi\)
−0.146294 + 0.989241i \(0.546734\pi\)
\(72\) 0 0
\(73\) 16.6329 1.94674 0.973370 0.229241i \(-0.0736244\pi\)
0.973370 + 0.229241i \(0.0736244\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.01897 0.571964
\(78\) 0 0
\(79\) −5.36674 −0.603805 −0.301902 0.953339i \(-0.597622\pi\)
−0.301902 + 0.953339i \(0.597622\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.27854 0.140338 0.0701690 0.997535i \(-0.477646\pi\)
0.0701690 + 0.997535i \(0.477646\pi\)
\(84\) 0 0
\(85\) 1.75275 0.190113
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.57255 1.01469 0.507344 0.861744i \(-0.330627\pi\)
0.507344 + 0.861744i \(0.330627\pi\)
\(90\) 0 0
\(91\) −4.55358 −0.477345
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.93610 −0.711629
\(96\) 0 0
\(97\) 14.0882 1.43044 0.715220 0.698900i \(-0.246327\pi\)
0.715220 + 0.698900i \(0.246327\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0379 1.19782 0.598910 0.800817i \(-0.295601\pi\)
0.598910 + 0.800817i \(0.295601\pi\)
\(102\) 0 0
\(103\) 3.80433 0.374852 0.187426 0.982279i \(-0.439986\pi\)
0.187426 + 0.982279i \(0.439986\pi\)
\(104\) 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
\(109\) 10.7215 1.02693 0.513465 0.858111i \(-0.328362\pi\)
0.513465 + 0.858111i \(0.328362\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.01897 −0.848433 −0.424217 0.905561i \(-0.639450\pi\)
−0.424217 + 0.905561i \(0.639450\pi\)
\(114\) 0 0
\(115\) −3.52579 −0.328781
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.762893 0.0699343
\(120\) 0 0
\(121\) 14.1900 1.29000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.8476 −0.970238
\(126\) 0 0
\(127\) −11.7404 −1.04179 −0.520897 0.853619i \(-0.674403\pi\)
−0.520897 + 0.853619i \(0.674403\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −18.9494 −1.65562 −0.827809 0.561009i \(-0.810413\pi\)
−0.827809 + 0.561009i \(0.810413\pi\)
\(132\) 0 0
\(133\) −3.01897 −0.261778
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.981033 −0.0838153 −0.0419077 0.999121i \(-0.513344\pi\)
−0.0419077 + 0.999121i \(0.513344\pi\)
\(138\) 0 0
\(139\) 23.3162 1.97765 0.988825 0.149078i \(-0.0476307\pi\)
0.988825 + 0.149078i \(0.0476307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −22.8543 −1.91117
\(144\) 0 0
\(145\) 7.57605 0.629157
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.3354 −0.928636 −0.464318 0.885669i \(-0.653700\pi\)
−0.464318 + 0.885669i \(0.653700\pi\)
\(150\) 0 0
\(151\) −7.82329 −0.636651 −0.318325 0.947982i \(-0.603120\pi\)
−0.318325 + 0.947982i \(0.603120\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.57605 0.608523
\(156\) 0 0
\(157\) 3.36355 0.268441 0.134220 0.990952i \(-0.457147\pi\)
0.134220 + 0.990952i \(0.457147\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.53461 −0.120945
\(162\) 0 0
\(163\) −14.4651 −1.13299 −0.566496 0.824065i \(-0.691701\pi\)
−0.566496 + 0.824065i \(0.691701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.4236 1.11613 0.558067 0.829796i \(-0.311543\pi\)
0.558067 + 0.829796i \(0.311543\pi\)
\(168\) 0 0
\(169\) 7.73510 0.595008
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.4429 1.02205 0.511023 0.859567i \(-0.329267\pi\)
0.511023 + 0.859567i \(0.329267\pi\)
\(174\) 0 0
\(175\) 0.278540 0.0210557
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.20900 0.464082 0.232041 0.972706i \(-0.425460\pi\)
0.232041 + 0.972706i \(0.425460\pi\)
\(180\) 0 0
\(181\) 4.93960 0.367158 0.183579 0.983005i \(-0.441232\pi\)
0.183579 + 0.983005i \(0.441232\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.8476 0.797531
\(186\) 0 0
\(187\) 3.82894 0.280000
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.36705 0.0989163 0.0494582 0.998776i \(-0.484251\pi\)
0.0494582 + 0.998776i \(0.484251\pi\)
\(192\) 0 0
\(193\) −5.63295 −0.405469 −0.202734 0.979234i \(-0.564983\pi\)
−0.202734 + 0.979234i \(0.564983\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.68321 −0.618653 −0.309327 0.950956i \(-0.600104\pi\)
−0.309327 + 0.950956i \(0.600104\pi\)
\(198\) 0 0
\(199\) −2.75275 −0.195138 −0.0975688 0.995229i \(-0.531107\pi\)
−0.0975688 + 0.995229i \(0.531107\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.29751 0.231440
\(204\) 0 0
\(205\) 0.683526 0.0477395
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −15.1521 −1.04809
\(210\) 0 0
\(211\) −20.0468 −1.38008 −0.690038 0.723773i \(-0.742406\pi\)
−0.690038 + 0.723773i \(0.742406\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.80433 −0.600450
\(216\) 0 0
\(217\) 3.29751 0.223849
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.47390 −0.233680
\(222\) 0 0
\(223\) −23.1777 −1.55209 −0.776047 0.630675i \(-0.782778\pi\)
−0.776047 + 0.630675i \(0.782778\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.53143 0.168017 0.0840084 0.996465i \(-0.473228\pi\)
0.0840084 + 0.996465i \(0.473228\pi\)
\(228\) 0 0
\(229\) 13.5258 0.893809 0.446905 0.894582i \(-0.352526\pi\)
0.446905 + 0.894582i \(0.352526\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.1398 −1.12286 −0.561432 0.827523i \(-0.689749\pi\)
−0.561432 + 0.827523i \(0.689749\pi\)
\(234\) 0 0
\(235\) 13.3041 0.867867
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.9112 1.93479 0.967396 0.253267i \(-0.0815050\pi\)
0.967396 + 0.253267i \(0.0815050\pi\)
\(240\) 0 0
\(241\) 5.35472 0.344928 0.172464 0.985016i \(-0.444827\pi\)
0.172464 + 0.985016i \(0.444827\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.29751 0.146782
\(246\) 0 0
\(247\) 13.7471 0.874708
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.53143 −0.0966630 −0.0483315 0.998831i \(-0.515390\pi\)
−0.0483315 + 0.998831i \(0.515390\pi\)
\(252\) 0 0
\(253\) −7.70218 −0.484232
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −25.7022 −1.60326 −0.801629 0.597822i \(-0.796033\pi\)
−0.801629 + 0.597822i \(0.796033\pi\)
\(258\) 0 0
\(259\) 4.72146 0.293377
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.6744 1.39816 0.699081 0.715042i \(-0.253593\pi\)
0.699081 + 0.715042i \(0.253593\pi\)
\(264\) 0 0
\(265\) 26.1587 1.60692
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.930456 −0.0567309 −0.0283655 0.999598i \(-0.509030\pi\)
−0.0283655 + 0.999598i \(0.509030\pi\)
\(270\) 0 0
\(271\) 2.75275 0.167218 0.0836089 0.996499i \(-0.473355\pi\)
0.0836089 + 0.996499i \(0.473355\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.39799 0.0843017
\(276\) 0 0
\(277\) 25.3544 1.52340 0.761699 0.647931i \(-0.224365\pi\)
0.761699 + 0.647931i \(0.224365\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.6329 −0.992239 −0.496119 0.868254i \(-0.665242\pi\)
−0.496119 + 0.868254i \(0.665242\pi\)
\(282\) 0 0
\(283\) −1.31679 −0.0782751 −0.0391375 0.999234i \(-0.512461\pi\)
−0.0391375 + 0.999234i \(0.512461\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.297507 0.0175613
\(288\) 0 0
\(289\) −16.4180 −0.965764
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.24593 0.306471 0.153235 0.988190i \(-0.451031\pi\)
0.153235 + 0.988190i \(0.451031\pi\)
\(294\) 0 0
\(295\) −8.25957 −0.480891
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.98799 0.404126
\(300\) 0 0
\(301\) −3.83212 −0.220880
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.56905 0.318883
\(306\) 0 0
\(307\) 33.8230 1.93038 0.965190 0.261551i \(-0.0842339\pi\)
0.965190 + 0.261551i \(0.0842339\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −26.5444 −1.50520 −0.752598 0.658480i \(-0.771200\pi\)
−0.752598 + 0.658480i \(0.771200\pi\)
\(312\) 0 0
\(313\) 6.10052 0.344822 0.172411 0.985025i \(-0.444844\pi\)
0.172411 + 0.985025i \(0.444844\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.632949 −0.0355500 −0.0177750 0.999842i \(-0.505658\pi\)
−0.0177750 + 0.999842i \(0.505658\pi\)
\(318\) 0 0
\(319\) 16.5501 0.926627
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.30315 −0.128151
\(324\) 0 0
\(325\) −1.26836 −0.0703558
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.79069 0.319251
\(330\) 0 0
\(331\) 25.2750 1.38924 0.694621 0.719376i \(-0.255572\pi\)
0.694621 + 0.719376i \(0.255572\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.48785 0.518377
\(336\) 0 0
\(337\) 20.0243 1.09079 0.545396 0.838178i \(-0.316379\pi\)
0.545396 + 0.838178i \(0.316379\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.5501 0.896237
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.2333 1.56933 0.784663 0.619923i \(-0.212836\pi\)
0.784663 + 0.619923i \(0.212836\pi\)
\(348\) 0 0
\(349\) −13.6105 −0.728552 −0.364276 0.931291i \(-0.618684\pi\)
−0.364276 + 0.931291i \(0.618684\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.35791 −0.498071 −0.249036 0.968494i \(-0.580114\pi\)
−0.249036 + 0.968494i \(0.580114\pi\)
\(354\) 0 0
\(355\) −5.66424 −0.300627
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.4268 −1.50031 −0.750155 0.661262i \(-0.770021\pi\)
−0.750155 + 0.661262i \(0.770021\pi\)
\(360\) 0 0
\(361\) −9.88584 −0.520307
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 38.2143 2.00023
\(366\) 0 0
\(367\) −23.3857 −1.22072 −0.610362 0.792123i \(-0.708976\pi\)
−0.610362 + 0.792123i \(0.708976\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3857 0.591116
\(372\) 0 0
\(373\) 36.0702 1.86765 0.933823 0.357736i \(-0.116451\pi\)
0.933823 + 0.357736i \(0.116451\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −15.0155 −0.773336
\(378\) 0 0
\(379\) −32.4562 −1.66717 −0.833583 0.552395i \(-0.813714\pi\)
−0.833583 + 0.552395i \(0.813714\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.4616 1.04554 0.522769 0.852475i \(-0.324899\pi\)
0.522769 + 0.852475i \(0.324899\pi\)
\(384\) 0 0
\(385\) 11.5311 0.587680
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.2782 −0.673233 −0.336616 0.941642i \(-0.609283\pi\)
−0.336616 + 0.941642i \(0.609283\pi\)
\(390\) 0 0
\(391\) −1.17075 −0.0592072
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.3301 −0.620395
\(396\) 0 0
\(397\) 36.3351 1.82361 0.911804 0.410626i \(-0.134690\pi\)
0.911804 + 0.410626i \(0.134690\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.1764 −1.50694 −0.753468 0.657484i \(-0.771621\pi\)
−0.753468 + 0.657484i \(0.771621\pi\)
\(402\) 0 0
\(403\) −15.0155 −0.747974
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.6968 1.17461
\(408\) 0 0
\(409\) −4.15874 −0.205636 −0.102818 0.994700i \(-0.532786\pi\)
−0.102818 + 0.994700i \(0.532786\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.59501 −0.176899
\(414\) 0 0
\(415\) 2.93746 0.144194
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −23.1764 −1.13224 −0.566120 0.824323i \(-0.691556\pi\)
−0.566120 + 0.824323i \(0.691556\pi\)
\(420\) 0 0
\(421\) 32.9301 1.60492 0.802458 0.596708i \(-0.203525\pi\)
0.802458 + 0.596708i \(0.203525\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.212497 0.0103076
\(426\) 0 0
\(427\) 2.42395 0.117303
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −22.0379 −1.06153 −0.530765 0.847519i \(-0.678095\pi\)
−0.530765 + 0.847519i \(0.678095\pi\)
\(432\) 0 0
\(433\) −19.8732 −0.955047 −0.477523 0.878619i \(-0.658465\pi\)
−0.477523 + 0.878619i \(0.658465\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.63295 0.221624
\(438\) 0 0
\(439\) 28.4752 1.35905 0.679524 0.733653i \(-0.262186\pi\)
0.679524 + 0.733653i \(0.262186\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.2849 −1.15381 −0.576904 0.816812i \(-0.695739\pi\)
−0.576904 + 0.816812i \(0.695739\pi\)
\(444\) 0 0
\(445\) 21.9930 1.04257
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.4240 −0.775094 −0.387547 0.921850i \(-0.626678\pi\)
−0.387547 + 0.921850i \(0.626678\pi\)
\(450\) 0 0
\(451\) 1.49318 0.0703111
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.4619 −0.490461
\(456\) 0 0
\(457\) 21.8978 1.02434 0.512169 0.858885i \(-0.328842\pi\)
0.512169 + 0.858885i \(0.328842\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 27.9997 1.30408 0.652038 0.758187i \(-0.273914\pi\)
0.652038 + 0.758187i \(0.273914\pi\)
\(462\) 0 0
\(463\) 6.55677 0.304719 0.152359 0.988325i \(-0.451313\pi\)
0.152359 + 0.988325i \(0.451313\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.9864 −0.508388 −0.254194 0.967153i \(-0.581810\pi\)
−0.254194 + 0.967153i \(0.581810\pi\)
\(468\) 0 0
\(469\) 4.12963 0.190689
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −19.2333 −0.884348
\(474\) 0 0
\(475\) −0.840904 −0.0385833
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.44856 0.340333 0.170167 0.985415i \(-0.445569\pi\)
0.170167 + 0.985415i \(0.445569\pi\)
\(480\) 0 0
\(481\) −21.4995 −0.980295
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 32.3677 1.46974
\(486\) 0 0
\(487\) −25.8230 −1.17015 −0.585075 0.810979i \(-0.698935\pi\)
−0.585075 + 0.810979i \(0.698935\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −25.4482 −1.14846 −0.574232 0.818693i \(-0.694699\pi\)
−0.574232 + 0.818693i \(0.694699\pi\)
\(492\) 0 0
\(493\) 2.51565 0.113299
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.46539 −0.110588
\(498\) 0 0
\(499\) −29.6463 −1.32715 −0.663575 0.748110i \(-0.730962\pi\)
−0.663575 + 0.748110i \(0.730962\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.61430 −0.161154 −0.0805768 0.996748i \(-0.525676\pi\)
−0.0805768 + 0.996748i \(0.525676\pi\)
\(504\) 0 0
\(505\) 27.6572 1.23073
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.71613 0.386336 0.193168 0.981166i \(-0.438124\pi\)
0.193168 + 0.981166i \(0.438124\pi\)
\(510\) 0 0
\(511\) 16.6329 0.735798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.74047 0.385151
\(516\) 0 0
\(517\) 29.0633 1.27820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.2614 −0.887668 −0.443834 0.896109i \(-0.646382\pi\)
−0.443834 + 0.896109i \(0.646382\pi\)
\(522\) 0 0
\(523\) 19.1574 0.837696 0.418848 0.908056i \(-0.362434\pi\)
0.418848 + 0.908056i \(0.362434\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.51565 0.109583
\(528\) 0 0
\(529\) −20.6450 −0.897607
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.35472 −0.0586796
\(534\) 0 0
\(535\) −9.19003 −0.397320
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.01897 0.216182
\(540\) 0 0
\(541\) −34.6652 −1.49038 −0.745188 0.666855i \(-0.767640\pi\)
−0.745188 + 0.666855i \(0.767640\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.6326 1.05515
\(546\) 0 0
\(547\) 17.4805 0.747414 0.373707 0.927547i \(-0.378087\pi\)
0.373707 + 0.927547i \(0.378087\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.95507 −0.424100
\(552\) 0 0
\(553\) −5.36674 −0.228217
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 44.7901 1.89782 0.948908 0.315553i \(-0.102190\pi\)
0.948908 + 0.315553i \(0.102190\pi\)
\(558\) 0 0
\(559\) 17.4499 0.738051
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.6965 −0.703675 −0.351838 0.936061i \(-0.614443\pi\)
−0.351838 + 0.936061i \(0.614443\pi\)
\(564\) 0 0
\(565\) −20.7211 −0.871745
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.6396 1.41024 0.705122 0.709086i \(-0.250892\pi\)
0.705122 + 0.709086i \(0.250892\pi\)
\(570\) 0 0
\(571\) −35.8251 −1.49923 −0.749617 0.661871i \(-0.769762\pi\)
−0.749617 + 0.661871i \(0.769762\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.427452 −0.0178260
\(576\) 0 0
\(577\) 21.7022 0.903473 0.451737 0.892151i \(-0.350805\pi\)
0.451737 + 0.892151i \(0.350805\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.27854 0.0530428
\(582\) 0 0
\(583\) 57.1445 2.36668
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.9358 1.73088 0.865438 0.501017i \(-0.167041\pi\)
0.865438 + 0.501017i \(0.167041\pi\)
\(588\) 0 0
\(589\) −9.95507 −0.410191
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20.2975 −0.833519 −0.416759 0.909017i \(-0.636834\pi\)
−0.416759 + 0.909017i \(0.636834\pi\)
\(594\) 0 0
\(595\) 1.75275 0.0718559
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15.0101 −0.613298 −0.306649 0.951823i \(-0.599208\pi\)
−0.306649 + 0.951823i \(0.599208\pi\)
\(600\) 0 0
\(601\) 12.4998 0.509878 0.254939 0.966957i \(-0.417945\pi\)
0.254939 + 0.966957i \(0.417945\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 32.6017 1.32545
\(606\) 0 0
\(607\) 35.7268 1.45011 0.725053 0.688693i \(-0.241815\pi\)
0.725053 + 0.688693i \(0.241815\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −26.3684 −1.06675
\(612\) 0 0
\(613\) 29.9438 1.20942 0.604709 0.796447i \(-0.293289\pi\)
0.604709 + 0.796447i \(0.293289\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.6709 1.23476 0.617382 0.786664i \(-0.288193\pi\)
0.617382 + 0.786664i \(0.288193\pi\)
\(618\) 0 0
\(619\) −6.47954 −0.260435 −0.130217 0.991485i \(-0.541568\pi\)
−0.130217 + 0.991485i \(0.541568\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.57255 0.383516
\(624\) 0 0
\(625\) −26.3151 −1.05260
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.60197 0.143620
\(630\) 0 0
\(631\) −20.5125 −0.816588 −0.408294 0.912850i \(-0.633876\pi\)
−0.408294 + 0.912850i \(0.633876\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −26.9737 −1.07042
\(636\) 0 0
\(637\) −4.55358 −0.180419
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.1511 −0.519435 −0.259718 0.965685i \(-0.583630\pi\)
−0.259718 + 0.965685i \(0.583630\pi\)
\(642\) 0 0
\(643\) −14.0935 −0.555794 −0.277897 0.960611i \(-0.589637\pi\)
−0.277897 + 0.960611i \(0.589637\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.5950 −0.888302 −0.444151 0.895952i \(-0.646495\pi\)
−0.444151 + 0.895952i \(0.646495\pi\)
\(648\) 0 0
\(649\) −18.0433 −0.708260
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.5118 1.54622 0.773109 0.634274i \(-0.218701\pi\)
0.773109 + 0.634274i \(0.218701\pi\)
\(654\) 0 0
\(655\) −43.5364 −1.70111
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 48.8473 1.90282 0.951410 0.307928i \(-0.0996355\pi\)
0.951410 + 0.307928i \(0.0996355\pi\)
\(660\) 0 0
\(661\) −34.8419 −1.35519 −0.677597 0.735433i \(-0.736979\pi\)
−0.677597 + 0.735433i \(0.736979\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.93610 −0.268970
\(666\) 0 0
\(667\) −5.06040 −0.195940
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.1657 0.469653
\(672\) 0 0
\(673\) −40.5687 −1.56381 −0.781905 0.623398i \(-0.785752\pi\)
−0.781905 + 0.623398i \(0.785752\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −37.2210 −1.43052 −0.715259 0.698860i \(-0.753691\pi\)
−0.715259 + 0.698860i \(0.753691\pi\)
\(678\) 0 0
\(679\) 14.0882 0.540655
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.42395 −0.169278 −0.0846389 0.996412i \(-0.526974\pi\)
−0.0846389 + 0.996412i \(0.526974\pi\)
\(684\) 0 0
\(685\) −2.25393 −0.0861183
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −51.8457 −1.97516
\(690\) 0 0
\(691\) 34.2090 1.30137 0.650686 0.759347i \(-0.274482\pi\)
0.650686 + 0.759347i \(0.274482\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 53.5690 2.03199
\(696\) 0 0
\(697\) 0.226966 0.00859697
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 7.91181 0.298825 0.149412 0.988775i \(-0.452262\pi\)
0.149412 + 0.988775i \(0.452262\pi\)
\(702\) 0 0
\(703\) −14.2539 −0.537597
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0379 0.452733
\(708\) 0 0
\(709\) 18.5894 0.698139 0.349069 0.937097i \(-0.386498\pi\)
0.349069 + 0.937097i \(0.386498\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.06040 −0.189514
\(714\) 0 0
\(715\) −52.5079 −1.96368
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −31.4170 −1.17166 −0.585828 0.810435i \(-0.699231\pi\)
−0.585828 + 0.810435i \(0.699231\pi\)
\(720\) 0 0
\(721\) 3.80433 0.141681
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.918489 0.0341118
\(726\) 0 0
\(727\) −49.3727 −1.83113 −0.915567 0.402166i \(-0.868258\pi\)
−0.915567 + 0.402166i \(0.868258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.92350 −0.108130
\(732\) 0 0
\(733\) −19.2191 −0.709875 −0.354938 0.934890i \(-0.615498\pi\)
−0.354938 + 0.934890i \(0.615498\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.7265 0.763469
\(738\) 0 0
\(739\) 40.5944 1.49329 0.746644 0.665224i \(-0.231664\pi\)
0.746644 + 0.665224i \(0.231664\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.77490 −0.248547 −0.124274 0.992248i \(-0.539660\pi\)
−0.124274 + 0.992248i \(0.539660\pi\)
\(744\) 0 0
\(745\) −26.0433 −0.954151
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 46.0309 1.67969 0.839846 0.542824i \(-0.182645\pi\)
0.839846 + 0.542824i \(0.182645\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.9741 −0.654144
\(756\) 0 0
\(757\) −33.1571 −1.20512 −0.602558 0.798075i \(-0.705852\pi\)
−0.602558 + 0.798075i \(0.705852\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.22347 0.225600 0.112800 0.993618i \(-0.464018\pi\)
0.112800 + 0.993618i \(0.464018\pi\)
\(762\) 0 0
\(763\) 10.7215 0.388143
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.3702 0.591093
\(768\) 0 0
\(769\) 1.06923 0.0385573 0.0192787 0.999814i \(-0.493863\pi\)
0.0192787 + 0.999814i \(0.493863\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.4875 −0.413178 −0.206589 0.978428i \(-0.566236\pi\)
−0.206589 + 0.978428i \(0.566236\pi\)
\(774\) 0 0
\(775\) 0.918489 0.0329931
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.898165 −0.0321801
\(780\) 0 0
\(781\) −12.3737 −0.442765
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.72778 0.275816
\(786\) 0 0
\(787\) −32.7335 −1.16682 −0.583411 0.812177i \(-0.698282\pi\)
−0.583411 + 0.812177i \(0.698282\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.01897 −0.320678
\(792\) 0 0
\(793\) −11.0377 −0.391959
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.4496 1.22027 0.610133 0.792299i \(-0.291116\pi\)
0.610133 + 0.792299i \(0.291116\pi\)
\(798\) 0 0
\(799\) 4.41768 0.156286
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 83.4802 2.94595
\(804\) 0 0
\(805\) −3.52579 −0.124268
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.6845 −0.762387 −0.381194 0.924495i \(-0.624487\pi\)
−0.381194 + 0.924495i \(0.624487\pi\)
\(810\) 0 0
\(811\) 21.9497 0.770760 0.385380 0.922758i \(-0.374070\pi\)
0.385380 + 0.922758i \(0.374070\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33.2336 −1.16412
\(816\) 0 0
\(817\) 11.5690 0.404750
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.6765 0.756516 0.378258 0.925700i \(-0.376523\pi\)
0.378258 + 0.925700i \(0.376523\pi\)
\(822\) 0 0
\(823\) −30.8540 −1.07550 −0.537751 0.843104i \(-0.680726\pi\)
−0.537751 + 0.843104i \(0.680726\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0503 −0.766763 −0.383381 0.923590i \(-0.625240\pi\)
−0.383381 + 0.923590i \(0.625240\pi\)
\(828\) 0 0
\(829\) 22.1464 0.769177 0.384588 0.923088i \(-0.374343\pi\)
0.384588 + 0.923088i \(0.374343\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.762893 0.0264327
\(834\) 0 0
\(835\) 33.1384 1.14680
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.6450 1.26512 0.632562 0.774510i \(-0.282003\pi\)
0.632562 + 0.774510i \(0.282003\pi\)
\(840\) 0 0
\(841\) −18.1264 −0.625050
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17.7714 0.611356
\(846\) 0 0
\(847\) 14.1900 0.487575
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.24562 −0.248377
\(852\) 0 0
\(853\) −14.1296 −0.483789 −0.241895 0.970303i \(-0.577769\pi\)
−0.241895 + 0.970303i \(0.577769\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.9329 0.851692 0.425846 0.904796i \(-0.359977\pi\)
0.425846 + 0.904796i \(0.359977\pi\)
\(858\) 0 0
\(859\) −23.1654 −0.790392 −0.395196 0.918597i \(-0.629323\pi\)
−0.395196 + 0.918597i \(0.629323\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.3440 1.37332 0.686662 0.726977i \(-0.259075\pi\)
0.686662 + 0.726977i \(0.259075\pi\)
\(864\) 0 0
\(865\) 30.8852 1.05013
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −26.9355 −0.913723
\(870\) 0 0
\(871\) −18.8046 −0.637170
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.8476 −0.366715
\(876\) 0 0
\(877\) 17.7971 0.600964 0.300482 0.953788i \(-0.402852\pi\)
0.300482 + 0.953788i \(0.402852\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −19.5346 −0.658138 −0.329069 0.944306i \(-0.606735\pi\)
−0.329069 + 0.944306i \(0.606735\pi\)
\(882\) 0 0
\(883\) −46.3822 −1.56089 −0.780443 0.625227i \(-0.785006\pi\)
−0.780443 + 0.625227i \(0.785006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.65887 0.290736 0.145368 0.989378i \(-0.453563\pi\)
0.145368 + 0.989378i \(0.453563\pi\)
\(888\) 0 0
\(889\) −11.7404 −0.393761
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −17.4819 −0.585009
\(894\) 0 0
\(895\) 14.2652 0.476834
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.8736 0.362653
\(900\) 0 0
\(901\) 8.68608 0.289375
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.3488 0.377246
\(906\) 0 0
\(907\) 23.9618 0.795637 0.397818 0.917464i \(-0.369767\pi\)
0.397818 + 0.917464i \(0.369767\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.62062 −0.285614 −0.142807 0.989751i \(-0.545613\pi\)
−0.142807 + 0.989751i \(0.545613\pi\)
\(912\) 0 0
\(913\) 6.41695 0.212370
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −18.9494 −0.625765
\(918\) 0 0
\(919\) −4.67120 −0.154089 −0.0770443 0.997028i \(-0.524548\pi\)
−0.0770443 + 0.997028i \(0.524548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11.2263 0.369519
\(924\) 0 0
\(925\) 1.31512 0.0432408
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.32180 0.0433669 0.0216835 0.999765i \(-0.493097\pi\)
0.0216835 + 0.999765i \(0.493097\pi\)
\(930\) 0 0
\(931\) −3.01897 −0.0989426
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.79701 0.287693
\(936\) 0 0
\(937\) −17.6140 −0.575424 −0.287712 0.957717i \(-0.592895\pi\)
−0.287712 + 0.957717i \(0.592895\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25.4875 −0.830870 −0.415435 0.909623i \(-0.636371\pi\)
−0.415435 + 0.909623i \(0.636371\pi\)
\(942\) 0 0
\(943\) −0.456559 −0.0148676
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.50551 −0.0489224 −0.0244612 0.999701i \(-0.507787\pi\)
−0.0244612 + 0.999701i \(0.507787\pi\)
\(948\) 0 0
\(949\) −75.7395 −2.45861
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4.55008 −0.147392 −0.0736958 0.997281i \(-0.523479\pi\)
−0.0736958 + 0.997281i \(0.523479\pi\)
\(954\) 0 0
\(955\) 3.14081 0.101634
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.981033 −0.0316792
\(960\) 0 0
\(961\) −20.1264 −0.649240
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.9417 −0.416609
\(966\) 0 0
\(967\) −45.7401 −1.47090 −0.735451 0.677577i \(-0.763030\pi\)
−0.735451 + 0.677577i \(0.763030\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.5677 0.563775 0.281888 0.959447i \(-0.409039\pi\)
0.281888 + 0.959447i \(0.409039\pi\)
\(972\) 0 0
\(973\) 23.3162 0.747482
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.7538 −0.663971 −0.331986 0.943284i \(-0.607719\pi\)
−0.331986 + 0.943284i \(0.607719\pi\)
\(978\) 0 0
\(979\) 48.0443 1.53550
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 15.5378 0.495579 0.247789 0.968814i \(-0.420296\pi\)
0.247789 + 0.968814i \(0.420296\pi\)
\(984\) 0 0
\(985\) −19.9497 −0.635652
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.88083 0.186999
\(990\) 0 0
\(991\) −7.87888 −0.250281 −0.125140 0.992139i \(-0.539938\pi\)
−0.125140 + 0.992139i \(0.539938\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.32447 −0.200499
\(996\) 0 0
\(997\) 15.6987 0.497182 0.248591 0.968609i \(-0.420032\pi\)
0.248591 + 0.968609i \(0.420032\pi\)
\(998\) 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.bt.1.3 yes 4
3.2 odd 2 6048.2.a.bp.1.2 yes 4
4.3 odd 2 6048.2.a.bs.1.3 yes 4
12.11 even 2 6048.2.a.bk.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bk.1.2 4 12.11 even 2
6048.2.a.bp.1.2 yes 4 3.2 odd 2
6048.2.a.bs.1.3 yes 4 4.3 odd 2
6048.2.a.bt.1.3 yes 4 1.1 even 1 trivial