Properties

Label 6048.2.c.d.3025.7
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 16
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: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.7
Root \(-0.987688 + 0.156434i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.d.3025.10

$q$-expansion

\(f(q)\) \(=\) \(q-0.0549306i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-0.0549306i q^{5} +1.00000 q^{7} -2.63506i q^{11} -3.67853i q^{13} -3.16228 q^{17} -3.07768i q^{19} +2.86834 q^{23} +4.99698 q^{25} -10.1898i q^{29} -9.32437 q^{31} -0.0549306i q^{35} +0.774554i q^{37} -6.36183 q^{41} +9.98927i q^{43} -12.3126 q^{47} +1.00000 q^{49} +3.39291i q^{53} -0.144746 q^{55} +6.93263i q^{59} -8.35114i q^{61} -0.202064 q^{65} +8.93584i q^{67} +4.28255 q^{71} +8.38081 q^{73} -2.63506i q^{77} +3.03186 q^{79} +10.3255i q^{83} +0.173706i q^{85} -12.8155 q^{89} -3.67853i q^{91} -0.169059 q^{95} -10.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{7} + O(q^{10}) \) \( 16q + 16q^{7} - 32q^{25} - 40q^{31} + 16q^{49} + 72q^{55} + 24q^{73} - 24q^{79} + 16q^{97} + O(q^{100}) \)

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
\(5\) − 0.0549306i − 0.0245657i −0.999925 0.0122828i \(-0.996090\pi\)
0.999925 0.0122828i \(-0.00390985\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.63506i − 0.794502i −0.917710 0.397251i \(-0.869964\pi\)
0.917710 0.397251i \(-0.130036\pi\)
\(12\) 0 0
\(13\) − 3.67853i − 1.02024i −0.860103 0.510121i \(-0.829601\pi\)
0.860103 0.510121i \(-0.170399\pi\)
\(14\) 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
\(19\) − 3.07768i − 0.706069i −0.935610 0.353035i \(-0.885150\pi\)
0.935610 0.353035i \(-0.114850\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.86834 0.598090 0.299045 0.954239i \(-0.403332\pi\)
0.299045 + 0.954239i \(0.403332\pi\)
\(24\) 0 0
\(25\) 4.99698 0.999397
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 10.1898i − 1.89219i −0.323890 0.946095i \(-0.604991\pi\)
0.323890 0.946095i \(-0.395009\pi\)
\(30\) 0 0
\(31\) −9.32437 −1.67471 −0.837353 0.546662i \(-0.815898\pi\)
−0.837353 + 0.546662i \(0.815898\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 0.0549306i − 0.00928496i
\(36\) 0 0
\(37\) 0.774554i 0.127336i 0.997971 + 0.0636679i \(0.0202798\pi\)
−0.997971 + 0.0636679i \(0.979720\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.36183 −0.993551 −0.496775 0.867879i \(-0.665483\pi\)
−0.496775 + 0.867879i \(0.665483\pi\)
\(42\) 0 0
\(43\) 9.98927i 1.52335i 0.647960 + 0.761674i \(0.275622\pi\)
−0.647960 + 0.761674i \(0.724378\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.3126 −1.79598 −0.897990 0.440015i \(-0.854973\pi\)
−0.897990 + 0.440015i \(0.854973\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.39291i 0.466052i 0.972470 + 0.233026i \(0.0748628\pi\)
−0.972470 + 0.233026i \(0.925137\pi\)
\(54\) 0 0
\(55\) −0.144746 −0.0195175
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.93263i 0.902552i 0.892384 + 0.451276i \(0.149031\pi\)
−0.892384 + 0.451276i \(0.850969\pi\)
\(60\) 0 0
\(61\) − 8.35114i − 1.06925i −0.845088 0.534627i \(-0.820452\pi\)
0.845088 0.534627i \(-0.179548\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.202064 −0.0250629
\(66\) 0 0
\(67\) 8.93584i 1.09169i 0.837887 + 0.545843i \(0.183791\pi\)
−0.837887 + 0.545843i \(0.816209\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.28255 0.508245 0.254123 0.967172i \(-0.418213\pi\)
0.254123 + 0.967172i \(0.418213\pi\)
\(72\) 0 0
\(73\) 8.38081 0.980900 0.490450 0.871469i \(-0.336832\pi\)
0.490450 + 0.871469i \(0.336832\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.63506i − 0.300293i
\(78\) 0 0
\(79\) 3.03186 0.341111 0.170556 0.985348i \(-0.445444\pi\)
0.170556 + 0.985348i \(0.445444\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3255i 1.13338i 0.823932 + 0.566688i \(0.191776\pi\)
−0.823932 + 0.566688i \(0.808224\pi\)
\(84\) 0 0
\(85\) 0.173706i 0.0188410i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.8155 −1.35844 −0.679222 0.733933i \(-0.737683\pi\)
−0.679222 + 0.733933i \(0.737683\pi\)
\(90\) 0 0
\(91\) − 3.67853i − 0.385615i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.169059 −0.0173451
\(96\) 0 0
\(97\) −10.1803 −1.03366 −0.516828 0.856089i \(-0.672887\pi\)
−0.516828 + 0.856089i \(0.672887\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.02749i − 0.102239i −0.998693 0.0511194i \(-0.983721\pi\)
0.998693 0.0511194i \(-0.0162789\pi\)
\(102\) 0 0
\(103\) −1.72905 −0.170368 −0.0851842 0.996365i \(-0.527148\pi\)
−0.0851842 + 0.996365i \(0.527148\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.11985i 0.591629i 0.955245 + 0.295814i \(0.0955910\pi\)
−0.955245 + 0.295814i \(0.904409\pi\)
\(108\) 0 0
\(109\) − 8.85597i − 0.848248i −0.905604 0.424124i \(-0.860582\pi\)
0.905604 0.424124i \(-0.139418\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.02383 −0.848891 −0.424445 0.905454i \(-0.639531\pi\)
−0.424445 + 0.905454i \(0.639531\pi\)
\(114\) 0 0
\(115\) − 0.157559i − 0.0146925i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) 4.05644 0.368767
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 0.549140i − 0.0491166i
\(126\) 0 0
\(127\) −18.2656 −1.62081 −0.810406 0.585868i \(-0.800754\pi\)
−0.810406 + 0.585868i \(0.800754\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 17.4470i − 1.52435i −0.647373 0.762174i \(-0.724132\pi\)
0.647373 0.762174i \(-0.275868\pi\)
\(132\) 0 0
\(133\) − 3.07768i − 0.266869i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.73613 −0.660942 −0.330471 0.943816i \(-0.607208\pi\)
−0.330471 + 0.943816i \(0.607208\pi\)
\(138\) 0 0
\(139\) − 0.802034i − 0.0680276i −0.999421 0.0340138i \(-0.989171\pi\)
0.999421 0.0340138i \(-0.0108290\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.69316 −0.810583
\(144\) 0 0
\(145\) −0.559729 −0.0464829
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 15.7165i − 1.28755i −0.765215 0.643775i \(-0.777367\pi\)
0.765215 0.643775i \(-0.222633\pi\)
\(150\) 0 0
\(151\) −2.58129 −0.210062 −0.105031 0.994469i \(-0.533494\pi\)
−0.105031 + 0.994469i \(0.533494\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.512193i 0.0411403i
\(156\) 0 0
\(157\) − 0.680724i − 0.0543277i −0.999631 0.0271638i \(-0.991352\pi\)
0.999631 0.0271638i \(-0.00864758\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.86834 0.226057
\(162\) 0 0
\(163\) 21.8339i 1.71016i 0.518494 + 0.855081i \(0.326493\pi\)
−0.518494 + 0.855081i \(0.673507\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.2203 −1.33255 −0.666274 0.745707i \(-0.732112\pi\)
−0.666274 + 0.745707i \(0.732112\pi\)
\(168\) 0 0
\(169\) −0.531594 −0.0408918
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 2.57218i − 0.195559i −0.995208 0.0977797i \(-0.968826\pi\)
0.995208 0.0977797i \(-0.0311741\pi\)
\(174\) 0 0
\(175\) 4.99698 0.377736
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 5.26703i − 0.393676i −0.980436 0.196838i \(-0.936933\pi\)
0.980436 0.196838i \(-0.0630673\pi\)
\(180\) 0 0
\(181\) 16.0189i 1.19068i 0.803475 + 0.595339i \(0.202982\pi\)
−0.803475 + 0.595339i \(0.797018\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0425467 0.00312809
\(186\) 0 0
\(187\) 8.33280i 0.609355i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.83730 −0.132943 −0.0664713 0.997788i \(-0.521174\pi\)
−0.0664713 + 0.997788i \(0.521174\pi\)
\(192\) 0 0
\(193\) 15.3429 1.10441 0.552204 0.833709i \(-0.313787\pi\)
0.552204 + 0.833709i \(0.313787\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.02447i 0.357979i 0.983851 + 0.178989i \(0.0572828\pi\)
−0.983851 + 0.178989i \(0.942717\pi\)
\(198\) 0 0
\(199\) 22.0333 1.56190 0.780949 0.624595i \(-0.214736\pi\)
0.780949 + 0.624595i \(0.214736\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 10.1898i − 0.715180i
\(204\) 0 0
\(205\) 0.349459i 0.0244073i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.10989 −0.560973
\(210\) 0 0
\(211\) − 8.50651i − 0.585612i −0.956172 0.292806i \(-0.905411\pi\)
0.956172 0.292806i \(-0.0945890\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.548716 0.0374221
\(216\) 0 0
\(217\) −9.32437 −0.632980
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.6325i 0.782489i
\(222\) 0 0
\(223\) 13.5574 0.907872 0.453936 0.891034i \(-0.350019\pi\)
0.453936 + 0.891034i \(0.350019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 16.0721i − 1.06674i −0.845882 0.533370i \(-0.820925\pi\)
0.845882 0.533370i \(-0.179075\pi\)
\(228\) 0 0
\(229\) − 5.50870i − 0.364025i −0.983296 0.182013i \(-0.941739\pi\)
0.983296 0.182013i \(-0.0582612\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −28.6599 −1.87757 −0.938787 0.344498i \(-0.888049\pi\)
−0.938787 + 0.344498i \(0.888049\pi\)
\(234\) 0 0
\(235\) 0.676339i 0.0441195i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.82998 0.247741 0.123870 0.992298i \(-0.460469\pi\)
0.123870 + 0.992298i \(0.460469\pi\)
\(240\) 0 0
\(241\) −1.72834 −0.111332 −0.0556660 0.998449i \(-0.517728\pi\)
−0.0556660 + 0.998449i \(0.517728\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.0549306i − 0.00350938i
\(246\) 0 0
\(247\) −11.3214 −0.720361
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.245657i 0.0155057i 0.999970 + 0.00775286i \(0.00246784\pi\)
−0.999970 + 0.00775286i \(0.997532\pi\)
\(252\) 0 0
\(253\) − 7.55825i − 0.475183i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.2120 1.01128 0.505638 0.862746i \(-0.331257\pi\)
0.505638 + 0.862746i \(0.331257\pi\)
\(258\) 0 0
\(259\) 0.774554i 0.0481284i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −23.3265 −1.43837 −0.719187 0.694817i \(-0.755485\pi\)
−0.719187 + 0.694817i \(0.755485\pi\)
\(264\) 0 0
\(265\) 0.186375 0.0114489
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.6947i 1.44469i 0.691534 + 0.722344i \(0.256935\pi\)
−0.691534 + 0.722344i \(0.743065\pi\)
\(270\) 0 0
\(271\) −3.52786 −0.214302 −0.107151 0.994243i \(-0.534173\pi\)
−0.107151 + 0.994243i \(0.534173\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 13.1674i − 0.794022i
\(276\) 0 0
\(277\) − 28.6014i − 1.71849i −0.511561 0.859247i \(-0.670932\pi\)
0.511561 0.859247i \(-0.329068\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 25.6546 1.53043 0.765214 0.643776i \(-0.222633\pi\)
0.765214 + 0.643776i \(0.222633\pi\)
\(282\) 0 0
\(283\) − 14.8539i − 0.882974i −0.897268 0.441487i \(-0.854451\pi\)
0.897268 0.441487i \(-0.145549\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.36183 −0.375527
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 21.6946i − 1.26741i −0.773574 0.633706i \(-0.781533\pi\)
0.773574 0.633706i \(-0.218467\pi\)
\(294\) 0 0
\(295\) 0.380813 0.0221718
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 10.5513i − 0.610195i
\(300\) 0 0
\(301\) 9.98927i 0.575772i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.458733 −0.0262670
\(306\) 0 0
\(307\) 0.422903i 0.0241363i 0.999927 + 0.0120682i \(0.00384151\pi\)
−0.999927 + 0.0120682i \(0.996158\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.6732 −0.775338 −0.387669 0.921799i \(-0.626720\pi\)
−0.387669 + 0.921799i \(0.626720\pi\)
\(312\) 0 0
\(313\) −7.68665 −0.434475 −0.217237 0.976119i \(-0.569705\pi\)
−0.217237 + 0.976119i \(0.569705\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 24.2068i − 1.35959i −0.733401 0.679796i \(-0.762068\pi\)
0.733401 0.679796i \(-0.237932\pi\)
\(318\) 0 0
\(319\) −26.8506 −1.50335
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.73249i 0.541530i
\(324\) 0 0
\(325\) − 18.3816i − 1.01963i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.3126 −0.678817
\(330\) 0 0
\(331\) − 9.15756i − 0.503345i −0.967812 0.251672i \(-0.919019\pi\)
0.967812 0.251672i \(-0.0809806\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.490851 0.0268180
\(336\) 0 0
\(337\) 33.2441 1.81092 0.905460 0.424432i \(-0.139526\pi\)
0.905460 + 0.424432i \(0.139526\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 24.5703i 1.33056i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 8.03418i − 0.431297i −0.976471 0.215649i \(-0.930813\pi\)
0.976471 0.215649i \(-0.0691866\pi\)
\(348\) 0 0
\(349\) 5.46493i 0.292531i 0.989245 + 0.146265i \(0.0467254\pi\)
−0.989245 + 0.146265i \(0.953275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.85159 0.417898 0.208949 0.977927i \(-0.432996\pi\)
0.208949 + 0.977927i \(0.432996\pi\)
\(354\) 0 0
\(355\) − 0.235243i − 0.0124854i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.4616 −1.60770 −0.803851 0.594831i \(-0.797219\pi\)
−0.803851 + 0.594831i \(0.797219\pi\)
\(360\) 0 0
\(361\) 9.52786 0.501467
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 0.460363i − 0.0240965i
\(366\) 0 0
\(367\) −8.97241 −0.468356 −0.234178 0.972194i \(-0.575240\pi\)
−0.234178 + 0.972194i \(0.575240\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.39291i 0.176151i
\(372\) 0 0
\(373\) 19.6614i 1.01803i 0.860759 + 0.509013i \(0.169990\pi\)
−0.860759 + 0.509013i \(0.830010\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −37.4833 −1.93049
\(378\) 0 0
\(379\) − 24.9871i − 1.28350i −0.766914 0.641750i \(-0.778209\pi\)
0.766914 0.641750i \(-0.221791\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −20.3323 −1.03893 −0.519465 0.854492i \(-0.673869\pi\)
−0.519465 + 0.854492i \(0.673869\pi\)
\(384\) 0 0
\(385\) −0.144746 −0.00737691
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 2.92753i − 0.148432i −0.997242 0.0742159i \(-0.976355\pi\)
0.997242 0.0742159i \(-0.0236454\pi\)
\(390\) 0 0
\(391\) −9.07048 −0.458714
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 0.166542i − 0.00837964i
\(396\) 0 0
\(397\) 8.20287i 0.411690i 0.978585 + 0.205845i \(0.0659943\pi\)
−0.978585 + 0.205845i \(0.934006\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −36.5590 −1.82567 −0.912834 0.408332i \(-0.866111\pi\)
−0.912834 + 0.408332i \(0.866111\pi\)
\(402\) 0 0
\(403\) 34.3000i 1.70860i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.04100 0.101169
\(408\) 0 0
\(409\) −6.01783 −0.297562 −0.148781 0.988870i \(-0.547535\pi\)
−0.148781 + 0.988870i \(0.547535\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.93263i 0.341133i
\(414\) 0 0
\(415\) 0.567188 0.0278422
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.25247i 0.110040i 0.998485 + 0.0550201i \(0.0175223\pi\)
−0.998485 + 0.0550201i \(0.982478\pi\)
\(420\) 0 0
\(421\) − 2.87431i − 0.140085i −0.997544 0.0700425i \(-0.977686\pi\)
0.997544 0.0700425i \(-0.0223135\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.8018 −0.766502
\(426\) 0 0
\(427\) − 8.35114i − 0.404140i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.7211 −0.757261 −0.378630 0.925548i \(-0.623605\pi\)
−0.378630 + 0.925548i \(0.623605\pi\)
\(432\) 0 0
\(433\) −3.44027 −0.165329 −0.0826644 0.996577i \(-0.526343\pi\)
−0.0826644 + 0.996577i \(0.526343\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 8.82783i − 0.422292i
\(438\) 0 0
\(439\) 12.2955 0.586833 0.293417 0.955985i \(-0.405208\pi\)
0.293417 + 0.955985i \(0.405208\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.8932i 0.755107i 0.925988 + 0.377553i \(0.123235\pi\)
−0.925988 + 0.377553i \(0.876765\pi\)
\(444\) 0 0
\(445\) 0.703964i 0.0333711i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.97450 −0.423533 −0.211766 0.977320i \(-0.567922\pi\)
−0.211766 + 0.977320i \(0.567922\pi\)
\(450\) 0 0
\(451\) 16.7638i 0.789377i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.202064 −0.00947290
\(456\) 0 0
\(457\) −15.0037 −0.701845 −0.350922 0.936405i \(-0.614132\pi\)
−0.350922 + 0.936405i \(0.614132\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.69884i 0.405145i 0.979267 + 0.202573i \(0.0649303\pi\)
−0.979267 + 0.202573i \(0.935070\pi\)
\(462\) 0 0
\(463\) 9.34522 0.434309 0.217155 0.976137i \(-0.430322\pi\)
0.217155 + 0.976137i \(0.430322\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.23326i 0.427265i 0.976914 + 0.213632i \(0.0685294\pi\)
−0.976914 + 0.213632i \(0.931471\pi\)
\(468\) 0 0
\(469\) 8.93584i 0.412619i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26.3223 1.21030
\(474\) 0 0
\(475\) − 15.3791i − 0.705643i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.66050 0.395708 0.197854 0.980231i \(-0.436603\pi\)
0.197854 + 0.980231i \(0.436603\pi\)
\(480\) 0 0
\(481\) 2.84922 0.129913
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.559212i 0.0253925i
\(486\) 0 0
\(487\) −11.1424 −0.504912 −0.252456 0.967608i \(-0.581238\pi\)
−0.252456 + 0.967608i \(0.581238\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.4479i − 1.14845i −0.818698 0.574224i \(-0.805304\pi\)
0.818698 0.574224i \(-0.194696\pi\)
\(492\) 0 0
\(493\) 32.2228i 1.45124i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.28255 0.192099
\(498\) 0 0
\(499\) 4.17422i 0.186864i 0.995626 + 0.0934318i \(0.0297837\pi\)
−0.995626 + 0.0934318i \(0.970216\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.6106 −0.473105 −0.236552 0.971619i \(-0.576017\pi\)
−0.236552 + 0.971619i \(0.576017\pi\)
\(504\) 0 0
\(505\) −0.0564404 −0.00251156
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 24.1111i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(510\) 0 0
\(511\) 8.38081 0.370745
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.0949777i 0.00418522i
\(516\) 0 0
\(517\) 32.4445i 1.42691i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.5642 1.42666 0.713331 0.700827i \(-0.247185\pi\)
0.713331 + 0.700827i \(0.247185\pi\)
\(522\) 0 0
\(523\) 18.6166i 0.814046i 0.913418 + 0.407023i \(0.133433\pi\)
−0.913418 + 0.407023i \(0.866567\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.4863 1.28444
\(528\) 0 0
\(529\) −14.7726 −0.642289
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 23.4022i 1.01366i
\(534\) 0 0
\(535\) 0.336167 0.0145338
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.63506i − 0.113500i
\(540\) 0 0
\(541\) − 41.3556i − 1.77802i −0.457891 0.889008i \(-0.651395\pi\)
0.457891 0.889008i \(-0.348605\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.486463 −0.0208378
\(546\) 0 0
\(547\) 18.8135i 0.804408i 0.915550 + 0.402204i \(0.131756\pi\)
−0.915550 + 0.402204i \(0.868244\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −31.3608 −1.33602
\(552\) 0 0
\(553\) 3.03186 0.128928
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.05808i 0.171946i 0.996297 + 0.0859731i \(0.0273999\pi\)
−0.996297 + 0.0859731i \(0.972600\pi\)
\(558\) 0 0
\(559\) 36.7458 1.55418
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 45.2454i − 1.90687i −0.301605 0.953433i \(-0.597522\pi\)
0.301605 0.953433i \(-0.402478\pi\)
\(564\) 0 0
\(565\) 0.495684i 0.0208536i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.6255 −0.571212 −0.285606 0.958347i \(-0.592195\pi\)
−0.285606 + 0.958347i \(0.592195\pi\)
\(570\) 0 0
\(571\) − 30.9574i − 1.29553i −0.761842 0.647763i \(-0.775705\pi\)
0.761842 0.647763i \(-0.224295\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.3330 0.597729
\(576\) 0 0
\(577\) −38.4675 −1.60142 −0.800712 0.599049i \(-0.795545\pi\)
−0.800712 + 0.599049i \(0.795545\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.3255i 0.428376i
\(582\) 0 0
\(583\) 8.94054 0.370279
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 28.6131i − 1.18099i −0.807041 0.590495i \(-0.798932\pi\)
0.807041 0.590495i \(-0.201068\pi\)
\(588\) 0 0
\(589\) 28.6975i 1.18246i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.5554 −0.926239 −0.463119 0.886296i \(-0.653270\pi\)
−0.463119 + 0.886296i \(0.653270\pi\)
\(594\) 0 0
\(595\) 0.173706i 0.00712124i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 42.1743 1.72319 0.861597 0.507593i \(-0.169465\pi\)
0.861597 + 0.507593i \(0.169465\pi\)
\(600\) 0 0
\(601\) −14.0155 −0.571705 −0.285853 0.958274i \(-0.592277\pi\)
−0.285853 + 0.958274i \(0.592277\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 0.222823i − 0.00905902i
\(606\) 0 0
\(607\) 21.5893 0.876282 0.438141 0.898906i \(-0.355637\pi\)
0.438141 + 0.898906i \(0.355637\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 45.2924i 1.83233i
\(612\) 0 0
\(613\) − 0.788508i − 0.0318475i −0.999873 0.0159238i \(-0.994931\pi\)
0.999873 0.0159238i \(-0.00506891\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.4936 1.38866 0.694331 0.719656i \(-0.255701\pi\)
0.694331 + 0.719656i \(0.255701\pi\)
\(618\) 0 0
\(619\) − 47.8005i − 1.92127i −0.277822 0.960633i \(-0.589612\pi\)
0.277822 0.960633i \(-0.410388\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.8155 −0.513443
\(624\) 0 0
\(625\) 24.9547 0.998190
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 2.44935i − 0.0976621i
\(630\) 0 0
\(631\) 21.7200 0.864659 0.432330 0.901716i \(-0.357692\pi\)
0.432330 + 0.901716i \(0.357692\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00334i 0.0398164i
\(636\) 0 0
\(637\) − 3.67853i − 0.145749i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.0166 −0.672117 −0.336058 0.941841i \(-0.609094\pi\)
−0.336058 + 0.941841i \(0.609094\pi\)
\(642\) 0 0
\(643\) − 38.4170i − 1.51502i −0.652824 0.757509i \(-0.726416\pi\)
0.652824 0.757509i \(-0.273584\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.7497 0.894382 0.447191 0.894439i \(-0.352425\pi\)
0.447191 + 0.894439i \(0.352425\pi\)
\(648\) 0 0
\(649\) 18.2679 0.717079
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 12.9198i − 0.505591i −0.967520 0.252795i \(-0.918650\pi\)
0.967520 0.252795i \(-0.0813499\pi\)
\(654\) 0 0
\(655\) −0.958371 −0.0374466
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.1905i 1.83828i 0.393932 + 0.919140i \(0.371115\pi\)
−0.393932 + 0.919140i \(0.628885\pi\)
\(660\) 0 0
\(661\) 31.5648i 1.22773i 0.789412 + 0.613864i \(0.210386\pi\)
−0.789412 + 0.613864i \(0.789614\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.169059 −0.00655582
\(666\) 0 0
\(667\) − 29.2276i − 1.13170i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −22.0058 −0.849524
\(672\) 0 0
\(673\) −37.1186 −1.43082 −0.715408 0.698707i \(-0.753759\pi\)
−0.715408 + 0.698707i \(0.753759\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 15.8912i − 0.610750i −0.952232 0.305375i \(-0.901218\pi\)
0.952232 0.305375i \(-0.0987818\pi\)
\(678\) 0 0
\(679\) −10.1803 −0.390686
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 27.4319i 1.04965i 0.851210 + 0.524826i \(0.175869\pi\)
−0.851210 + 0.524826i \(0.824131\pi\)
\(684\) 0 0
\(685\) 0.424950i 0.0162365i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.4809 0.475486
\(690\) 0 0
\(691\) − 6.57770i − 0.250227i −0.992142 0.125114i \(-0.960070\pi\)
0.992142 0.125114i \(-0.0399296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.0440562 −0.00167115
\(696\) 0 0
\(697\) 20.1179 0.762018
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.07153i 0.342627i 0.985217 + 0.171314i \(0.0548011\pi\)
−0.985217 + 0.171314i \(0.945199\pi\)
\(702\) 0 0
\(703\) 2.38383 0.0899079
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.02749i − 0.0386426i
\(708\) 0 0
\(709\) 6.16889i 0.231678i 0.993268 + 0.115839i \(0.0369556\pi\)
−0.993268 + 0.115839i \(0.963044\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −26.7454 −1.00162
\(714\) 0 0
\(715\) 0.532451i 0.0199125i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.5440 −1.13910 −0.569550 0.821956i \(-0.692883\pi\)
−0.569550 + 0.821956i \(0.692883\pi\)
\(720\) 0 0
\(721\) −1.72905 −0.0643932
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 50.9180i − 1.89105i
\(726\) 0 0
\(727\) 21.3429 0.791565 0.395782 0.918344i \(-0.370473\pi\)
0.395782 + 0.918344i \(0.370473\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 31.5888i − 1.16836i
\(732\) 0 0
\(733\) − 15.1235i − 0.558599i −0.960204 0.279300i \(-0.909898\pi\)
0.960204 0.279300i \(-0.0901023\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.5465 0.867347
\(738\) 0 0
\(739\) 38.7212i 1.42438i 0.701985 + 0.712192i \(0.252297\pi\)
−0.701985 + 0.712192i \(0.747703\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.7258 1.45740 0.728698 0.684835i \(-0.240126\pi\)
0.728698 + 0.684835i \(0.240126\pi\)
\(744\) 0 0
\(745\) −0.863319 −0.0316296
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.11985i 0.223615i
\(750\) 0 0
\(751\) 2.96814 0.108309 0.0541544 0.998533i \(-0.482754\pi\)
0.0541544 + 0.998533i \(0.482754\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.141792i 0.00516032i
\(756\) 0 0
\(757\) 47.1588i 1.71401i 0.515305 + 0.857007i \(0.327679\pi\)
−0.515305 + 0.857007i \(0.672321\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.8000 −1.47900 −0.739499 0.673158i \(-0.764937\pi\)
−0.739499 + 0.673158i \(0.764937\pi\)
\(762\) 0 0
\(763\) − 8.85597i − 0.320608i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.5019 0.920821
\(768\) 0 0
\(769\) −52.9121 −1.90806 −0.954029 0.299714i \(-0.903109\pi\)
−0.954029 + 0.299714i \(0.903109\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.0168i 1.40334i 0.712503 + 0.701669i \(0.247561\pi\)
−0.712503 + 0.701669i \(0.752439\pi\)
\(774\) 0 0
\(775\) −46.5937 −1.67370
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.5797i 0.701515i
\(780\) 0 0
\(781\) − 11.2848i − 0.403802i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.0373925 −0.00133460
\(786\) 0 0
\(787\) − 1.01675i − 0.0362431i −0.999836 0.0181215i \(-0.994231\pi\)
0.999836 0.0181215i \(-0.00576858\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.02383 −0.320851
\(792\) 0 0
\(793\) −30.7199 −1.09090
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 50.9667i − 1.80533i −0.430339 0.902667i \(-0.641606\pi\)
0.430339 0.902667i \(-0.358394\pi\)
\(798\) 0 0
\(799\) 38.9359 1.37745
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 22.0840i − 0.779327i
\(804\) 0 0
\(805\) − 0.157559i − 0.00555324i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.3800 0.716522 0.358261 0.933622i \(-0.383370\pi\)
0.358261 + 0.933622i \(0.383370\pi\)
\(810\) 0 0
\(811\) − 40.7638i − 1.43141i −0.698402 0.715706i \(-0.746105\pi\)
0.698402 0.715706i \(-0.253895\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.19935 0.0420113
\(816\) 0 0
\(817\) 30.7438 1.07559
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 35.8394i − 1.25080i −0.780303 0.625402i \(-0.784935\pi\)
0.780303 0.625402i \(-0.215065\pi\)
\(822\) 0 0
\(823\) −34.8745 −1.21565 −0.607824 0.794071i \(-0.707958\pi\)
−0.607824 + 0.794071i \(0.707958\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.4727i 1.47692i 0.674296 + 0.738461i \(0.264447\pi\)
−0.674296 + 0.738461i \(0.735553\pi\)
\(828\) 0 0
\(829\) 33.1210i 1.15034i 0.818034 + 0.575169i \(0.195064\pi\)
−0.818034 + 0.575169i \(0.804936\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.16228 −0.109566
\(834\) 0 0
\(835\) 0.945922i 0.0327350i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 43.5817 1.50461 0.752304 0.658817i \(-0.228943\pi\)
0.752304 + 0.658817i \(0.228943\pi\)
\(840\) 0 0
\(841\) −74.8311 −2.58038
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.0292007i 0.00100454i
\(846\) 0 0
\(847\) 4.05644 0.139381
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.22168i 0.0761582i
\(852\) 0 0
\(853\) − 36.8928i − 1.26319i −0.775300 0.631593i \(-0.782401\pi\)
0.775300 0.631593i \(-0.217599\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.2607 0.657933 0.328967 0.944342i \(-0.393300\pi\)
0.328967 + 0.944342i \(0.393300\pi\)
\(858\) 0 0
\(859\) 19.4667i 0.664195i 0.943245 + 0.332098i \(0.107756\pi\)
−0.943245 + 0.332098i \(0.892244\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.4715 0.730898 0.365449 0.930831i \(-0.380915\pi\)
0.365449 + 0.930831i \(0.380915\pi\)
\(864\) 0 0
\(865\) −0.141291 −0.00480405
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 7.98916i − 0.271014i
\(870\) 0 0
\(871\) 32.8708 1.11378
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 0.549140i − 0.0185643i
\(876\) 0 0
\(877\) − 23.8421i − 0.805091i −0.915400 0.402545i \(-0.868126\pi\)
0.915400 0.402545i \(-0.131874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.81134 0.296862 0.148431 0.988923i \(-0.452578\pi\)
0.148431 + 0.988923i \(0.452578\pi\)
\(882\) 0 0
\(883\) − 30.0935i − 1.01273i −0.862320 0.506363i \(-0.830989\pi\)
0.862320 0.506363i \(-0.169011\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.0549 0.438340 0.219170 0.975687i \(-0.429665\pi\)
0.219170 + 0.975687i \(0.429665\pi\)
\(888\) 0 0
\(889\) −18.2656 −0.612609
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 37.8944i 1.26809i
\(894\) 0 0
\(895\) −0.289321 −0.00967092
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 95.0131i 3.16886i
\(900\) 0 0
\(901\) − 10.7293i − 0.357446i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.879929 0.0292498
\(906\) 0 0
\(907\) 35.8987i 1.19200i 0.802985 + 0.595999i \(0.203244\pi\)
−0.802985 + 0.595999i \(0.796756\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 46.3926 1.53705 0.768527 0.639817i \(-0.220990\pi\)
0.768527 + 0.639817i \(0.220990\pi\)
\(912\) 0 0
\(913\) 27.2085 0.900469
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 17.4470i − 0.576149i
\(918\) 0 0
\(919\) 9.87878 0.325871 0.162935 0.986637i \(-0.447904\pi\)
0.162935 + 0.986637i \(0.447904\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 15.7535i − 0.518533i
\(924\) 0 0
\(925\) 3.87043i 0.127259i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.21688 0.105543 0.0527713 0.998607i \(-0.483195\pi\)
0.0527713 + 0.998607i \(0.483195\pi\)
\(930\) 0 0
\(931\) − 3.07768i − 0.100867i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.457725 0.0149692
\(936\) 0 0
\(937\) 54.7874 1.78983 0.894913 0.446241i \(-0.147237\pi\)
0.894913 + 0.446241i \(0.147237\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 8.31449i − 0.271045i −0.990774 0.135522i \(-0.956729\pi\)
0.990774 0.135522i \(-0.0432713\pi\)
\(942\) 0 0
\(943\) −18.2479 −0.594232
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 12.2157i − 0.396958i −0.980105 0.198479i \(-0.936400\pi\)
0.980105 0.198479i \(-0.0636001\pi\)
\(948\) 0 0
\(949\) − 30.8291i − 1.00075i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 50.2955 1.62923 0.814615 0.580002i \(-0.196948\pi\)
0.814615 + 0.580002i \(0.196948\pi\)
\(954\) 0 0
\(955\) 0.100924i 0.00326583i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.73613 −0.249813
\(960\) 0 0
\(961\) 55.9439 1.80464
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 0.842795i − 0.0271305i
\(966\) 0 0
\(967\) −6.17661 −0.198626 −0.0993132 0.995056i \(-0.531665\pi\)
−0.0993132 + 0.995056i \(0.531665\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.74342i 0.120132i 0.998194 + 0.0600660i \(0.0191311\pi\)
−0.998194 + 0.0600660i \(0.980869\pi\)
\(972\) 0 0
\(973\) − 0.802034i − 0.0257120i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.1012 1.37893 0.689464 0.724320i \(-0.257846\pi\)
0.689464 + 0.724320i \(0.257846\pi\)
\(978\) 0 0
\(979\) 33.7697i 1.07929i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 46.8079 1.49294 0.746470 0.665419i \(-0.231747\pi\)
0.746470 + 0.665419i \(0.231747\pi\)
\(984\) 0 0
\(985\) 0.275997 0.00879399
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 28.6526i 0.911099i
\(990\) 0 0
\(991\) 44.6923 1.41970 0.709850 0.704353i \(-0.248763\pi\)
0.709850 + 0.704353i \(0.248763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.21030i − 0.0383691i
\(996\) 0 0
\(997\) 41.1361i 1.30279i 0.758737 + 0.651397i \(0.225817\pi\)
−0.758737 + 0.651397i \(0.774183\pi\)
\(998\) 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.d.3025.7 16
3.2 odd 2 inner 6048.2.c.d.3025.9 16
4.3 odd 2 1512.2.c.d.757.6 16
8.3 odd 2 1512.2.c.d.757.7 yes 16
8.5 even 2 inner 6048.2.c.d.3025.10 16
12.11 even 2 1512.2.c.d.757.11 yes 16
24.5 odd 2 inner 6048.2.c.d.3025.8 16
24.11 even 2 1512.2.c.d.757.10 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.6 16 4.3 odd 2
1512.2.c.d.757.7 yes 16 8.3 odd 2
1512.2.c.d.757.10 yes 16 24.11 even 2
1512.2.c.d.757.11 yes 16 12.11 even 2
6048.2.c.d.3025.7 16 1.1 even 1 trivial
6048.2.c.d.3025.8 16 24.5 odd 2 inner
6048.2.c.d.3025.9 16 3.2 odd 2 inner
6048.2.c.d.3025.10 16 8.5 even 2 inner