Properties

Label 6048.2.c.c.3025.5
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.5
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.5
Root \(-0.178197 + 1.40294i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.c.3025.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.356394i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+0.356394i q^{5} +1.00000 q^{7} +5.96816i q^{11} -2.87298i q^{13} -3.16228 q^{17} +0.127017i q^{19} -2.80588 q^{23} +4.87298 q^{25} -2.44949i q^{29} +7.00000 q^{31} +0.356394i q^{35} +1.87298i q^{37} +6.99208 q^{41} +1.12702i q^{43} -7.34847 q^{47} +1.00000 q^{49} -2.12702 q^{55} +6.63568i q^{59} +13.7460i q^{61} +1.02391 q^{65} -8.61895i q^{67} +5.96816 q^{71} +0.872983 q^{73} +5.96816i q^{77} -12.6190 q^{79} +6.63568i q^{83} -1.12702i q^{85} +6.99208 q^{89} -2.87298i q^{91} -0.0452680 q^{95} -9.74597 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} + 8q^{25} + 56q^{31} + 8q^{49} - 48q^{55} - 24q^{73} - 8q^{79} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) 0.356394i 0.159384i 0.996820 + 0.0796921i \(0.0253937\pi\)
−0.996820 + 0.0796921i \(0.974606\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.96816i 1.79947i 0.436438 + 0.899734i \(0.356240\pi\)
−0.436438 + 0.899734i \(0.643760\pi\)
\(12\) 0 0
\(13\) − 2.87298i − 0.796822i −0.917207 0.398411i \(-0.869562\pi\)
0.917207 0.398411i \(-0.130438\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.16228 −0.766965 −0.383482 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) 0.127017i 0.0291396i 0.999894 + 0.0145698i \(0.00463788\pi\)
−0.999894 + 0.0145698i \(0.995362\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.80588 −0.585067 −0.292534 0.956255i \(-0.594498\pi\)
−0.292534 + 0.956255i \(0.594498\pi\)
\(24\) 0 0
\(25\) 4.87298 0.974597
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.44949i − 0.454859i −0.973795 0.227429i \(-0.926968\pi\)
0.973795 0.227429i \(-0.0730321\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.356394i 0.0602416i
\(36\) 0 0
\(37\) 1.87298i 0.307917i 0.988077 + 0.153958i \(0.0492021\pi\)
−0.988077 + 0.153958i \(0.950798\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.99208 1.09198 0.545989 0.837792i \(-0.316154\pi\)
0.545989 + 0.837792i \(0.316154\pi\)
\(42\) 0 0
\(43\) 1.12702i 0.171868i 0.996301 + 0.0859342i \(0.0273875\pi\)
−0.996301 + 0.0859342i \(0.972613\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.34847 −1.07188 −0.535942 0.844255i \(-0.680044\pi\)
−0.535942 + 0.844255i \(0.680044\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −2.12702 −0.286807
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.63568i 0.863892i 0.901899 + 0.431946i \(0.142173\pi\)
−0.901899 + 0.431946i \(0.857827\pi\)
\(60\) 0 0
\(61\) 13.7460i 1.75999i 0.474982 + 0.879995i \(0.342454\pi\)
−0.474982 + 0.879995i \(0.657546\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.02391 0.127001
\(66\) 0 0
\(67\) − 8.61895i − 1.05297i −0.850184 0.526486i \(-0.823509\pi\)
0.850184 0.526486i \(-0.176491\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.96816 0.708290 0.354145 0.935190i \(-0.384772\pi\)
0.354145 + 0.935190i \(0.384772\pi\)
\(72\) 0 0
\(73\) 0.872983 0.102175 0.0510875 0.998694i \(-0.483731\pi\)
0.0510875 + 0.998694i \(0.483731\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.96816i 0.680135i
\(78\) 0 0
\(79\) −12.6190 −1.41974 −0.709871 0.704331i \(-0.751247\pi\)
−0.709871 + 0.704331i \(0.751247\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.63568i 0.728361i 0.931328 + 0.364180i \(0.118651\pi\)
−0.931328 + 0.364180i \(0.881349\pi\)
\(84\) 0 0
\(85\) − 1.12702i − 0.122242i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.99208 0.741158 0.370579 0.928801i \(-0.379159\pi\)
0.370579 + 0.928801i \(0.379159\pi\)
\(90\) 0 0
\(91\) − 2.87298i − 0.301170i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.0452680 −0.00464440
\(96\) 0 0
\(97\) −9.74597 −0.989553 −0.494776 0.869020i \(-0.664750\pi\)
−0.494776 + 0.869020i \(0.664750\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.5347i 1.14774i 0.818946 + 0.573871i \(0.194559\pi\)
−0.818946 + 0.573871i \(0.805441\pi\)
\(102\) 0 0
\(103\) 10.7460 1.05883 0.529416 0.848363i \(-0.322411\pi\)
0.529416 + 0.848363i \(0.322411\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.3858i 1.39073i 0.718657 + 0.695364i \(0.244757\pi\)
−0.718657 + 0.695364i \(0.755243\pi\)
\(108\) 0 0
\(109\) 12.7460i 1.22084i 0.792077 + 0.610421i \(0.209000\pi\)
−0.792077 + 0.610421i \(0.791000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.8353 −1.58373 −0.791866 0.610695i \(-0.790890\pi\)
−0.791866 + 0.610695i \(0.790890\pi\)
\(114\) 0 0
\(115\) − 1.00000i − 0.0932505i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) −24.6190 −2.23809
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.51867i 0.314720i
\(126\) 0 0
\(127\) −10.8730 −0.964821 −0.482411 0.875945i \(-0.660239\pi\)
−0.482411 + 0.875945i \(0.660239\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.77405i 0.766592i 0.923626 + 0.383296i \(0.125211\pi\)
−0.923626 + 0.383296i \(0.874789\pi\)
\(132\) 0 0
\(133\) 0.127017i 0.0110137i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.58785 0.391967 0.195983 0.980607i \(-0.437210\pi\)
0.195983 + 0.980607i \(0.437210\pi\)
\(138\) 0 0
\(139\) − 14.0000i − 1.18746i −0.804663 0.593732i \(-0.797654\pi\)
0.804663 0.593732i \(-0.202346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 17.1464 1.43386
\(144\) 0 0
\(145\) 0.872983 0.0724973
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.79796i − 0.802680i −0.915929 0.401340i \(-0.868545\pi\)
0.915929 0.401340i \(-0.131455\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.49476i 0.200384i
\(156\) 0 0
\(157\) − 7.12702i − 0.568798i −0.958706 0.284399i \(-0.908206\pi\)
0.958706 0.284399i \(-0.0917940\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.80588 −0.221135
\(162\) 0 0
\(163\) − 22.8730i − 1.79155i −0.444507 0.895775i \(-0.646621\pi\)
0.444507 0.895775i \(-0.353379\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.42558 0.110314 0.0551572 0.998478i \(-0.482434\pi\)
0.0551572 + 0.998478i \(0.482434\pi\)
\(168\) 0 0
\(169\) 4.74597 0.365074
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.2156i 1.38491i 0.721462 + 0.692454i \(0.243470\pi\)
−0.721462 + 0.692454i \(0.756530\pi\)
\(174\) 0 0
\(175\) 4.87298 0.368363
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.77405i 0.655803i 0.944712 + 0.327901i \(0.106341\pi\)
−0.944712 + 0.327901i \(0.893659\pi\)
\(180\) 0 0
\(181\) 21.7460i 1.61636i 0.588932 + 0.808182i \(0.299549\pi\)
−0.588932 + 0.808182i \(0.700451\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.667520 −0.0490770
\(186\) 0 0
\(187\) − 18.8730i − 1.38013i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −21.7795 −1.57591 −0.787956 0.615731i \(-0.788861\pi\)
−0.787956 + 0.615731i \(0.788861\pi\)
\(192\) 0 0
\(193\) −9.49193 −0.683244 −0.341622 0.939837i \(-0.610976\pi\)
−0.341622 + 0.939837i \(0.610976\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.04783i − 0.145902i −0.997336 0.0729508i \(-0.976758\pi\)
0.997336 0.0729508i \(-0.0232416\pi\)
\(198\) 0 0
\(199\) −0.127017 −0.00900397 −0.00450199 0.999990i \(-0.501433\pi\)
−0.00450199 + 0.999990i \(0.501433\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.44949i − 0.171920i
\(204\) 0 0
\(205\) 2.49193i 0.174044i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.758056 −0.0524358
\(210\) 0 0
\(211\) 20.0000i 1.37686i 0.725304 + 0.688428i \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.401662 −0.0273931
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.08517i 0.611135i
\(222\) 0 0
\(223\) −7.61895 −0.510203 −0.255101 0.966914i \(-0.582109\pi\)
−0.255101 + 0.966914i \(0.582109\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 25.9205i − 1.72040i −0.509955 0.860201i \(-0.670338\pi\)
0.509955 0.860201i \(-0.329662\pi\)
\(228\) 0 0
\(229\) 15.7460i 1.04052i 0.854007 + 0.520261i \(0.174165\pi\)
−0.854007 + 0.520261i \(0.825835\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.61177 0.367639 0.183820 0.982960i \(-0.441154\pi\)
0.183820 + 0.982960i \(0.441154\pi\)
\(234\) 0 0
\(235\) − 2.61895i − 0.170841i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.2474 −0.792222 −0.396111 0.918203i \(-0.629640\pi\)
−0.396111 + 0.918203i \(0.629640\pi\)
\(240\) 0 0
\(241\) 28.6190 1.84351 0.921754 0.387774i \(-0.126756\pi\)
0.921754 + 0.387774i \(0.126756\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.356394i 0.0227692i
\(246\) 0 0
\(247\) 0.364917 0.0232191
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.8353i 1.06263i 0.847173 + 0.531317i \(0.178303\pi\)
−0.847173 + 0.531317i \(0.821697\pi\)
\(252\) 0 0
\(253\) − 16.7460i − 1.05281i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.0428 −1.25024 −0.625119 0.780529i \(-0.714950\pi\)
−0.625119 + 0.780529i \(0.714950\pi\)
\(258\) 0 0
\(259\) 1.87298i 0.116382i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.80588 0.173018 0.0865091 0.996251i \(-0.472429\pi\)
0.0865091 + 0.996251i \(0.472429\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.5560i 0.643612i 0.946806 + 0.321806i \(0.104290\pi\)
−0.946806 + 0.321806i \(0.895710\pi\)
\(270\) 0 0
\(271\) 19.7460 1.19948 0.599741 0.800194i \(-0.295270\pi\)
0.599741 + 0.800194i \(0.295270\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 29.0828i 1.75376i
\(276\) 0 0
\(277\) 5.87298i 0.352873i 0.984312 + 0.176437i \(0.0564571\pi\)
−0.984312 + 0.176437i \(0.943543\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0454 1.31512 0.657559 0.753403i \(-0.271589\pi\)
0.657559 + 0.753403i \(0.271589\pi\)
\(282\) 0 0
\(283\) 11.4919i 0.683125i 0.939859 + 0.341562i \(0.110956\pi\)
−0.939859 + 0.341562i \(0.889044\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.99208 0.412729
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 31.8434i 1.86031i 0.367168 + 0.930155i \(0.380327\pi\)
−0.367168 + 0.930155i \(0.619673\pi\)
\(294\) 0 0
\(295\) −2.36492 −0.137691
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.06126i 0.466195i
\(300\) 0 0
\(301\) 1.12702i 0.0649602i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.89898 −0.280515
\(306\) 0 0
\(307\) 29.6190i 1.69044i 0.534416 + 0.845221i \(0.320531\pi\)
−0.534416 + 0.845221i \(0.679469\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.7104 −1.17438 −0.587189 0.809450i \(-0.699765\pi\)
−0.587189 + 0.809450i \(0.699765\pi\)
\(312\) 0 0
\(313\) 2.87298 0.162391 0.0811953 0.996698i \(-0.474126\pi\)
0.0811953 + 0.996698i \(0.474126\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.6730i 0.767954i 0.923343 + 0.383977i \(0.125446\pi\)
−0.923343 + 0.383977i \(0.874554\pi\)
\(318\) 0 0
\(319\) 14.6190 0.818504
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 0.401662i − 0.0223491i
\(324\) 0 0
\(325\) − 14.0000i − 0.776580i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.34847 −0.405134
\(330\) 0 0
\(331\) − 10.0000i − 0.549650i −0.961494 0.274825i \(-0.911380\pi\)
0.961494 0.274825i \(-0.0886199\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.07174 0.167827
\(336\) 0 0
\(337\) −3.25403 −0.177258 −0.0886292 0.996065i \(-0.528249\pi\)
−0.0886292 + 0.996065i \(0.528249\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 41.7771i 2.26236i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 36.4765i − 1.95816i −0.203475 0.979080i \(-0.565223\pi\)
0.203475 0.979080i \(-0.434777\pi\)
\(348\) 0 0
\(349\) − 23.4919i − 1.25749i −0.777610 0.628747i \(-0.783568\pi\)
0.777610 0.628747i \(-0.216432\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.3166 −0.708773 −0.354386 0.935099i \(-0.615310\pi\)
−0.354386 + 0.935099i \(0.615310\pi\)
\(354\) 0 0
\(355\) 2.12702i 0.112890i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.1733 1.53971 0.769854 0.638221i \(-0.220329\pi\)
0.769854 + 0.638221i \(0.220329\pi\)
\(360\) 0 0
\(361\) 18.9839 0.999151
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.311126i 0.0162851i
\(366\) 0 0
\(367\) −21.8730 −1.14176 −0.570880 0.821033i \(-0.693398\pi\)
−0.570880 + 0.821033i \(0.693398\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 28.2379i 1.46210i 0.682322 + 0.731052i \(0.260970\pi\)
−0.682322 + 0.731052i \(0.739030\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.03734 −0.362442
\(378\) 0 0
\(379\) − 8.87298i − 0.455775i −0.973688 0.227887i \(-0.926818\pi\)
0.973688 0.227887i \(-0.0731818\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.4576 0.892039 0.446020 0.895023i \(-0.352841\pi\)
0.446020 + 0.895023i \(0.352841\pi\)
\(384\) 0 0
\(385\) −2.12702 −0.108403
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.4097i 0.781304i 0.920538 + 0.390652i \(0.127750\pi\)
−0.920538 + 0.390652i \(0.872250\pi\)
\(390\) 0 0
\(391\) 8.87298 0.448726
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 4.49732i − 0.226285i
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.5242 −0.825178 −0.412589 0.910917i \(-0.635375\pi\)
−0.412589 + 0.910917i \(0.635375\pi\)
\(402\) 0 0
\(403\) − 20.1109i − 1.00179i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.1783 −0.554086
\(408\) 0 0
\(409\) −12.3649 −0.611406 −0.305703 0.952127i \(-0.598891\pi\)
−0.305703 + 0.952127i \(0.598891\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.63568i 0.326521i
\(414\) 0 0
\(415\) −2.36492 −0.116089
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.7716i 1.40559i 0.711394 + 0.702793i \(0.248064\pi\)
−0.711394 + 0.702793i \(0.751936\pi\)
\(420\) 0 0
\(421\) − 19.1109i − 0.931407i −0.884941 0.465704i \(-0.845801\pi\)
0.884941 0.465704i \(-0.154199\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.4097 −0.747482
\(426\) 0 0
\(427\) 13.7460i 0.665214i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.1146 1.11339 0.556695 0.830717i \(-0.312069\pi\)
0.556695 + 0.830717i \(0.312069\pi\)
\(432\) 0 0
\(433\) −4.87298 −0.234181 −0.117090 0.993121i \(-0.537357\pi\)
−0.117090 + 0.993121i \(0.537357\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 0.356394i − 0.0170486i
\(438\) 0 0
\(439\) −25.7460 −1.22879 −0.614394 0.788999i \(-0.710599\pi\)
−0.614394 + 0.788999i \(0.710599\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.7024i 1.31618i 0.752938 + 0.658091i \(0.228636\pi\)
−0.752938 + 0.658091i \(0.771364\pi\)
\(444\) 0 0
\(445\) 2.49193i 0.118129i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 30.9100 1.45873 0.729366 0.684123i \(-0.239815\pi\)
0.729366 + 0.684123i \(0.239815\pi\)
\(450\) 0 0
\(451\) 41.7298i 1.96498i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.02391 0.0480018
\(456\) 0 0
\(457\) 9.25403 0.432885 0.216443 0.976295i \(-0.430555\pi\)
0.216443 + 0.976295i \(0.430555\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.9389i − 0.649198i −0.945852 0.324599i \(-0.894771\pi\)
0.945852 0.324599i \(-0.105229\pi\)
\(462\) 0 0
\(463\) −0.508067 −0.0236119 −0.0118059 0.999930i \(-0.503758\pi\)
−0.0118059 + 0.999930i \(0.503758\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.401662i 0.0185867i 0.999957 + 0.00929335i \(0.00295821\pi\)
−0.999957 + 0.00929335i \(0.997042\pi\)
\(468\) 0 0
\(469\) − 8.61895i − 0.397986i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.72622 −0.309272
\(474\) 0 0
\(475\) 0.618950i 0.0283994i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.712788 −0.0325681 −0.0162841 0.999867i \(-0.505184\pi\)
−0.0162841 + 0.999867i \(0.505184\pi\)
\(480\) 0 0
\(481\) 5.38105 0.245355
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 3.47340i − 0.157719i
\(486\) 0 0
\(487\) −26.1109 −1.18320 −0.591599 0.806233i \(-0.701503\pi\)
−0.591599 + 0.806233i \(0.701503\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 13.4072i − 0.605057i −0.953140 0.302528i \(-0.902169\pi\)
0.953140 0.302528i \(-0.0978307\pi\)
\(492\) 0 0
\(493\) 7.74597i 0.348861i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.96816 0.267709
\(498\) 0 0
\(499\) 34.3649i 1.53838i 0.639017 + 0.769192i \(0.279341\pi\)
−0.639017 + 0.769192i \(0.720659\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.34847 0.327652 0.163826 0.986489i \(-0.447616\pi\)
0.163826 + 0.986489i \(0.447616\pi\)
\(504\) 0 0
\(505\) −4.11088 −0.182932
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.311126i 0.0137904i 0.999976 + 0.00689521i \(0.00219483\pi\)
−0.999976 + 0.00689521i \(0.997805\pi\)
\(510\) 0 0
\(511\) 0.872983 0.0386185
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.82980i 0.168761i
\(516\) 0 0
\(517\) − 43.8569i − 1.92882i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.8408 1.30735 0.653675 0.756776i \(-0.273226\pi\)
0.653675 + 0.756776i \(0.273226\pi\)
\(522\) 0 0
\(523\) − 34.7460i − 1.51934i −0.650312 0.759668i \(-0.725362\pi\)
0.650312 0.759668i \(-0.274638\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22.1359 −0.964257
\(528\) 0 0
\(529\) −15.1270 −0.657696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 20.0881i − 0.870113i
\(534\) 0 0
\(535\) −5.12702 −0.221660
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.96816i 0.257067i
\(540\) 0 0
\(541\) − 30.8569i − 1.32664i −0.748336 0.663320i \(-0.769147\pi\)
0.748336 0.663320i \(-0.230853\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.54259 −0.194583
\(546\) 0 0
\(547\) 39.2379i 1.67769i 0.544369 + 0.838846i \(0.316769\pi\)
−0.544369 + 0.838846i \(0.683231\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.311126 0.0132544
\(552\) 0 0
\(553\) −12.6190 −0.536612
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.17571i 0.388787i 0.980924 + 0.194394i \(0.0622739\pi\)
−0.980924 + 0.194394i \(0.937726\pi\)
\(558\) 0 0
\(559\) 3.23790 0.136949
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 7.34847i − 0.309701i −0.987938 0.154851i \(-0.950510\pi\)
0.987938 0.154851i \(-0.0494896\pi\)
\(564\) 0 0
\(565\) − 6.00000i − 0.252422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.52123 0.231462 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(570\) 0 0
\(571\) − 32.9839i − 1.38033i −0.723651 0.690166i \(-0.757538\pi\)
0.723651 0.690166i \(-0.242462\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.6730 −0.570205
\(576\) 0 0
\(577\) −22.2540 −0.926448 −0.463224 0.886241i \(-0.653307\pi\)
−0.463224 + 0.886241i \(0.653307\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.63568i 0.275294i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.8568i 1.56252i 0.624208 + 0.781259i \(0.285422\pi\)
−0.624208 + 0.781259i \(0.714578\pi\)
\(588\) 0 0
\(589\) 0.889117i 0.0366354i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.8035 0.936426 0.468213 0.883616i \(-0.344898\pi\)
0.468213 + 0.883616i \(0.344898\pi\)
\(594\) 0 0
\(595\) − 1.12702i − 0.0462032i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −41.3755 −1.69056 −0.845278 0.534327i \(-0.820565\pi\)
−0.845278 + 0.534327i \(0.820565\pi\)
\(600\) 0 0
\(601\) 32.1109 1.30983 0.654915 0.755702i \(-0.272704\pi\)
0.654915 + 0.755702i \(0.272704\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 8.77405i − 0.356716i
\(606\) 0 0
\(607\) −15.2379 −0.618487 −0.309244 0.950983i \(-0.600076\pi\)
−0.309244 + 0.950983i \(0.600076\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.1120i 0.854101i
\(612\) 0 0
\(613\) 40.2379i 1.62519i 0.582826 + 0.812597i \(0.301947\pi\)
−0.582826 + 0.812597i \(0.698053\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.7209 −0.632898 −0.316449 0.948610i \(-0.602491\pi\)
−0.316449 + 0.948610i \(0.602491\pi\)
\(618\) 0 0
\(619\) 0.491933i 0.0197725i 0.999951 + 0.00988624i \(0.00314694\pi\)
−0.999951 + 0.00988624i \(0.996853\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.99208 0.280132
\(624\) 0 0
\(625\) 23.1109 0.924435
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 5.92289i − 0.236161i
\(630\) 0 0
\(631\) 8.87298 0.353228 0.176614 0.984280i \(-0.443486\pi\)
0.176614 + 0.984280i \(0.443486\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 3.87507i − 0.153777i
\(636\) 0 0
\(637\) − 2.87298i − 0.113832i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.1703 −0.717685 −0.358843 0.933398i \(-0.616829\pi\)
−0.358843 + 0.933398i \(0.616829\pi\)
\(642\) 0 0
\(643\) 8.38105i 0.330516i 0.986250 + 0.165258i \(0.0528457\pi\)
−0.986250 + 0.165258i \(0.947154\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.4232 0.842231 0.421116 0.907007i \(-0.361639\pi\)
0.421116 + 0.907007i \(0.361639\pi\)
\(648\) 0 0
\(649\) −39.6028 −1.55455
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 0.311126i − 0.0121753i −0.999981 0.00608765i \(-0.998062\pi\)
0.999981 0.00608765i \(-0.00193777\pi\)
\(654\) 0 0
\(655\) −3.12702 −0.122183
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 35.3620i − 1.37751i −0.724994 0.688755i \(-0.758158\pi\)
0.724994 0.688755i \(-0.241842\pi\)
\(660\) 0 0
\(661\) − 45.8569i − 1.78362i −0.452405 0.891812i \(-0.649434\pi\)
0.452405 0.891812i \(-0.350566\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.0452680 −0.00175542
\(666\) 0 0
\(667\) 6.87298i 0.266123i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −82.0381 −3.16705
\(672\) 0 0
\(673\) 29.4919 1.13683 0.568415 0.822742i \(-0.307557\pi\)
0.568415 + 0.822742i \(0.307557\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.9045i 0.688125i 0.938947 + 0.344063i \(0.111803\pi\)
−0.938947 + 0.344063i \(0.888197\pi\)
\(678\) 0 0
\(679\) −9.74597 −0.374016
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.44157i − 0.361271i −0.983550 0.180636i \(-0.942184\pi\)
0.983550 0.180636i \(-0.0578155\pi\)
\(684\) 0 0
\(685\) 1.63508i 0.0624733i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000i 0.836919i 0.908235 + 0.418460i \(0.137430\pi\)
−0.908235 + 0.418460i \(0.862570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.98952 0.189263
\(696\) 0 0
\(697\) −22.1109 −0.837509
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 48.7692i − 1.84199i −0.389577 0.920994i \(-0.627379\pi\)
0.389577 0.920994i \(-0.372621\pi\)
\(702\) 0 0
\(703\) −0.237900 −0.00897257
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.5347i 0.433806i
\(708\) 0 0
\(709\) 21.2540i 0.798212i 0.916905 + 0.399106i \(0.130679\pi\)
−0.916905 + 0.399106i \(0.869321\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −19.6412 −0.735568
\(714\) 0 0
\(715\) 6.11088i 0.228534i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.8329 1.37363 0.686817 0.726830i \(-0.259007\pi\)
0.686817 + 0.726830i \(0.259007\pi\)
\(720\) 0 0
\(721\) 10.7460 0.400201
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 11.9363i − 0.443304i
\(726\) 0 0
\(727\) −3.49193 −0.129509 −0.0647543 0.997901i \(-0.520626\pi\)
−0.0647543 + 0.997901i \(0.520626\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 3.56394i − 0.131817i
\(732\) 0 0
\(733\) 8.36492i 0.308965i 0.987996 + 0.154483i \(0.0493711\pi\)
−0.987996 + 0.154483i \(0.950629\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 51.4393 1.89479
\(738\) 0 0
\(739\) 0.254033i 0.00934477i 0.999989 + 0.00467238i \(0.00148727\pi\)
−0.999989 + 0.00467238i \(0.998513\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.3037 1.40523 0.702614 0.711571i \(-0.252016\pi\)
0.702614 + 0.711571i \(0.252016\pi\)
\(744\) 0 0
\(745\) 3.49193 0.127935
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 14.3858i 0.525646i
\(750\) 0 0
\(751\) 51.3488 1.87374 0.936872 0.349673i \(-0.113707\pi\)
0.936872 + 0.349673i \(0.113707\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.27673i 0.155646i
\(756\) 0 0
\(757\) − 25.7460i − 0.935753i −0.883794 0.467877i \(-0.845019\pi\)
0.883794 0.467877i \(-0.154981\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34.6945 −1.25768 −0.628838 0.777537i \(-0.716469\pi\)
−0.628838 + 0.777537i \(0.716469\pi\)
\(762\) 0 0
\(763\) 12.7460i 0.461435i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 19.0642 0.688368
\(768\) 0 0
\(769\) 11.6351 0.419572 0.209786 0.977747i \(-0.432723\pi\)
0.209786 + 0.977747i \(0.432723\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 22.0001i − 0.791290i −0.918404 0.395645i \(-0.870521\pi\)
0.918404 0.395645i \(-0.129479\pi\)
\(774\) 0 0
\(775\) 34.1109 1.22530
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.888110i 0.0318198i
\(780\) 0 0
\(781\) 35.6190i 1.27455i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.54003 0.0906574
\(786\) 0 0
\(787\) − 4.25403i − 0.151640i −0.997122 0.0758200i \(-0.975843\pi\)
0.997122 0.0758200i \(-0.0241574\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.8353 −0.598594
\(792\) 0 0
\(793\) 39.4919 1.40240
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 31.2664i − 1.10751i −0.832679 0.553756i \(-0.813194\pi\)
0.832679 0.553756i \(-0.186806\pi\)
\(798\) 0 0
\(799\) 23.2379 0.822098
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.21011i 0.183861i
\(804\) 0 0
\(805\) − 1.00000i − 0.0352454i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.58785 −0.161300 −0.0806502 0.996742i \(-0.525700\pi\)
−0.0806502 + 0.996742i \(0.525700\pi\)
\(810\) 0 0
\(811\) 41.9839i 1.47425i 0.675755 + 0.737126i \(0.263818\pi\)
−0.675755 + 0.737126i \(0.736182\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.15179 0.285545
\(816\) 0 0
\(817\) −0.143150 −0.00500818
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 22.1359i − 0.772550i −0.922384 0.386275i \(-0.873762\pi\)
0.922384 0.386275i \(-0.126238\pi\)
\(822\) 0 0
\(823\) 37.4919 1.30689 0.653443 0.756975i \(-0.273324\pi\)
0.653443 + 0.756975i \(0.273324\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 30.1519i − 1.04849i −0.851569 0.524243i \(-0.824348\pi\)
0.851569 0.524243i \(-0.175652\pi\)
\(828\) 0 0
\(829\) − 2.36492i − 0.0821370i −0.999156 0.0410685i \(-0.986924\pi\)
0.999156 0.0410685i \(-0.0130762\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.16228 −0.109566
\(834\) 0 0
\(835\) 0.508067i 0.0175824i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37.5457 1.29622 0.648110 0.761547i \(-0.275560\pi\)
0.648110 + 0.761547i \(0.275560\pi\)
\(840\) 0 0
\(841\) 23.0000 0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.69143i 0.0581871i
\(846\) 0 0
\(847\) −24.6190 −0.845917
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 5.25537i − 0.180152i
\(852\) 0 0
\(853\) − 5.23790i − 0.179342i −0.995971 0.0896711i \(-0.971418\pi\)
0.995971 0.0896711i \(-0.0285816\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −47.6095 −1.62631 −0.813155 0.582048i \(-0.802252\pi\)
−0.813155 + 0.582048i \(0.802252\pi\)
\(858\) 0 0
\(859\) − 31.8730i − 1.08749i −0.839250 0.543746i \(-0.817005\pi\)
0.839250 0.543746i \(-0.182995\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.3221 −0.896016 −0.448008 0.894030i \(-0.647866\pi\)
−0.448008 + 0.894030i \(0.647866\pi\)
\(864\) 0 0
\(865\) −6.49193 −0.220732
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 75.3119i − 2.55478i
\(870\) 0 0
\(871\) −24.7621 −0.839032
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.51867i 0.118953i
\(876\) 0 0
\(877\) − 13.4919i − 0.455590i −0.973709 0.227795i \(-0.926848\pi\)
0.973709 0.227795i \(-0.0731516\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 43.4233 1.46297 0.731484 0.681859i \(-0.238828\pi\)
0.731484 + 0.681859i \(0.238828\pi\)
\(882\) 0 0
\(883\) − 15.2379i − 0.512796i −0.966571 0.256398i \(-0.917464\pi\)
0.966571 0.256398i \(-0.0825358\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0564 −1.61358 −0.806788 0.590841i \(-0.798796\pi\)
−0.806788 + 0.590841i \(0.798796\pi\)
\(888\) 0 0
\(889\) −10.8730 −0.364668
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 0.933378i − 0.0312343i
\(894\) 0 0
\(895\) −3.12702 −0.104525
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 17.1464i − 0.571865i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.75013 −0.257623
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 33.3595 1.10525 0.552624 0.833430i \(-0.313626\pi\)
0.552624 + 0.833430i \(0.313626\pi\)
\(912\) 0 0
\(913\) −39.6028 −1.31066
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.77405i 0.289744i
\(918\) 0 0
\(919\) 14.5081 0.478577 0.239288 0.970949i \(-0.423086\pi\)
0.239288 + 0.970949i \(0.423086\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 17.1464i − 0.564382i
\(924\) 0 0
\(925\) 9.12702i 0.300094i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23.4710 0.770058 0.385029 0.922904i \(-0.374191\pi\)
0.385029 + 0.922904i \(0.374191\pi\)
\(930\) 0 0
\(931\) 0.127017i 0.00416280i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.72622 0.219971
\(936\) 0 0
\(937\) 41.4919 1.35548 0.677741 0.735301i \(-0.262959\pi\)
0.677741 + 0.735301i \(0.262959\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11.4894i 0.374544i 0.982308 + 0.187272i \(0.0599645\pi\)
−0.982308 + 0.187272i \(0.940036\pi\)
\(942\) 0 0
\(943\) −19.6190 −0.638881
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.00256i 0.0650745i 0.999471 + 0.0325372i \(0.0103587\pi\)
−0.999471 + 0.0325372i \(0.989641\pi\)
\(948\) 0 0
\(949\) − 2.50807i − 0.0814153i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.311126 −0.0100784 −0.00503918 0.999987i \(-0.501604\pi\)
−0.00503918 + 0.999987i \(0.501604\pi\)
\(954\) 0 0
\(955\) − 7.76210i − 0.251176i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.58785 0.148150
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 3.38287i − 0.108898i
\(966\) 0 0
\(967\) −25.2379 −0.811596 −0.405798 0.913963i \(-0.633006\pi\)
−0.405798 + 0.913963i \(0.633006\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.7636i 0.441694i 0.975309 + 0.220847i \(0.0708821\pi\)
−0.975309 + 0.220847i \(0.929118\pi\)
\(972\) 0 0
\(973\) − 14.0000i − 0.448819i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.1599 0.740949 0.370475 0.928843i \(-0.379195\pi\)
0.370475 + 0.928843i \(0.379195\pi\)
\(978\) 0 0
\(979\) 41.7298i 1.33369i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −53.6682 −1.71175 −0.855875 0.517183i \(-0.826981\pi\)
−0.855875 + 0.517183i \(0.826981\pi\)
\(984\) 0 0
\(985\) 0.729833 0.0232544
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 3.16228i − 0.100555i
\(990\) 0 0
\(991\) −17.3810 −0.552127 −0.276064 0.961139i \(-0.589030\pi\)
−0.276064 + 0.961139i \(0.589030\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 0.0452680i − 0.00143509i
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 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.c.c.3025.5 8
3.2 odd 2 inner 6048.2.c.c.3025.3 8
4.3 odd 2 1512.2.c.c.757.7 yes 8
8.3 odd 2 1512.2.c.c.757.8 yes 8
8.5 even 2 inner 6048.2.c.c.3025.4 8
12.11 even 2 1512.2.c.c.757.2 yes 8
24.5 odd 2 inner 6048.2.c.c.3025.6 8
24.11 even 2 1512.2.c.c.757.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.c.757.1 8 24.11 even 2
1512.2.c.c.757.2 yes 8 12.11 even 2
1512.2.c.c.757.7 yes 8 4.3 odd 2
1512.2.c.c.757.8 yes 8 8.3 odd 2
6048.2.c.c.3025.3 8 3.2 odd 2 inner
6048.2.c.c.3025.4 8 8.5 even 2 inner
6048.2.c.c.3025.5 8 1.1 even 1 trivial
6048.2.c.c.3025.6 8 24.5 odd 2 inner