Properties

Label 6048.2.c.f.3025.12
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 24
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: \(24\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.12
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.13

$q$-expansion

\(f(q)\) \(=\) \(q-0.940450i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-0.940450i q^{5} -1.00000 q^{7} +5.98632i q^{11} +6.59017i q^{13} +2.64225 q^{17} -5.83430i q^{19} +2.88452 q^{23} +4.11555 q^{25} +3.09794i q^{29} -3.52412 q^{31} +0.940450i q^{35} +0.213560i q^{37} +1.63291 q^{41} +7.16716i q^{43} -9.32639 q^{47} +1.00000 q^{49} -7.51044i q^{53} +5.62983 q^{55} -11.9280i q^{59} -1.48304i q^{61} +6.19772 q^{65} +13.0555i q^{67} -1.54642 q^{71} -2.96871 q^{73} -5.98632i q^{77} -15.9515 q^{79} +8.74782i q^{83} -2.48490i q^{85} -7.50339 q^{89} -6.59017i q^{91} -5.48686 q^{95} +10.7529 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) − 0.940450i − 0.420582i −0.977639 0.210291i \(-0.932559\pi\)
0.977639 0.210291i \(-0.0674411\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.98632i 1.80494i 0.430749 + 0.902472i \(0.358249\pi\)
−0.430749 + 0.902472i \(0.641751\pi\)
\(12\) 0 0
\(13\) 6.59017i 1.82778i 0.405957 + 0.913892i \(0.366938\pi\)
−0.405957 + 0.913892i \(0.633062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.64225 0.640840 0.320420 0.947276i \(-0.396176\pi\)
0.320420 + 0.947276i \(0.396176\pi\)
\(18\) 0 0
\(19\) − 5.83430i − 1.33848i −0.743047 0.669240i \(-0.766620\pi\)
0.743047 0.669240i \(-0.233380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.88452 0.601464 0.300732 0.953709i \(-0.402769\pi\)
0.300732 + 0.953709i \(0.402769\pi\)
\(24\) 0 0
\(25\) 4.11555 0.823111
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.09794i 0.575274i 0.957740 + 0.287637i \(0.0928696\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(30\) 0 0
\(31\) −3.52412 −0.632950 −0.316475 0.948601i \(-0.602499\pi\)
−0.316475 + 0.948601i \(0.602499\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.940450i 0.158965i
\(36\) 0 0
\(37\) 0.213560i 0.0351090i 0.999846 + 0.0175545i \(0.00558806\pi\)
−0.999846 + 0.0175545i \(0.994412\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.63291 0.255017 0.127509 0.991837i \(-0.459302\pi\)
0.127509 + 0.991837i \(0.459302\pi\)
\(42\) 0 0
\(43\) 7.16716i 1.09298i 0.837465 + 0.546491i \(0.184037\pi\)
−0.837465 + 0.546491i \(0.815963\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.32639 −1.36039 −0.680197 0.733030i \(-0.738106\pi\)
−0.680197 + 0.733030i \(0.738106\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.51044i − 1.03164i −0.856698 0.515819i \(-0.827488\pi\)
0.856698 0.515819i \(-0.172512\pi\)
\(54\) 0 0
\(55\) 5.62983 0.759126
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 11.9280i − 1.55289i −0.630186 0.776444i \(-0.717021\pi\)
0.630186 0.776444i \(-0.282979\pi\)
\(60\) 0 0
\(61\) − 1.48304i − 0.189884i −0.995483 0.0949419i \(-0.969733\pi\)
0.995483 0.0949419i \(-0.0302665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.19772 0.768733
\(66\) 0 0
\(67\) 13.0555i 1.59498i 0.603329 + 0.797492i \(0.293840\pi\)
−0.603329 + 0.797492i \(0.706160\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.54642 −0.183526 −0.0917632 0.995781i \(-0.529250\pi\)
−0.0917632 + 0.995781i \(0.529250\pi\)
\(72\) 0 0
\(73\) −2.96871 −0.347461 −0.173731 0.984793i \(-0.555582\pi\)
−0.173731 + 0.984793i \(0.555582\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.98632i − 0.682204i
\(78\) 0 0
\(79\) −15.9515 −1.79468 −0.897341 0.441338i \(-0.854504\pi\)
−0.897341 + 0.441338i \(0.854504\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.74782i 0.960198i 0.877214 + 0.480099i \(0.159399\pi\)
−0.877214 + 0.480099i \(0.840601\pi\)
\(84\) 0 0
\(85\) − 2.48490i − 0.269526i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.50339 −0.795358 −0.397679 0.917525i \(-0.630184\pi\)
−0.397679 + 0.917525i \(0.630184\pi\)
\(90\) 0 0
\(91\) − 6.59017i − 0.690837i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.48686 −0.562940
\(96\) 0 0
\(97\) 10.7529 1.09179 0.545894 0.837854i \(-0.316190\pi\)
0.545894 + 0.837854i \(0.316190\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 3.54920i − 0.353159i −0.984286 0.176579i \(-0.943497\pi\)
0.984286 0.176579i \(-0.0565033\pi\)
\(102\) 0 0
\(103\) −15.3303 −1.51054 −0.755271 0.655413i \(-0.772495\pi\)
−0.755271 + 0.655413i \(0.772495\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.41633i 0.233596i 0.993156 + 0.116798i \(0.0372629\pi\)
−0.993156 + 0.116798i \(0.962737\pi\)
\(108\) 0 0
\(109\) 6.86711i 0.657750i 0.944373 + 0.328875i \(0.106669\pi\)
−0.944373 + 0.328875i \(0.893331\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.4539 1.45378 0.726891 0.686752i \(-0.240964\pi\)
0.726891 + 0.686752i \(0.240964\pi\)
\(114\) 0 0
\(115\) − 2.71274i − 0.252965i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.64225 −0.242215
\(120\) 0 0
\(121\) −24.8360 −2.25782
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 8.57272i − 0.766767i
\(126\) 0 0
\(127\) −7.96760 −0.707010 −0.353505 0.935433i \(-0.615010\pi\)
−0.353505 + 0.935433i \(0.615010\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1.53780i − 0.134358i −0.997741 0.0671790i \(-0.978600\pi\)
0.997741 0.0671790i \(-0.0213999\pi\)
\(132\) 0 0
\(133\) 5.83430i 0.505898i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.34042 −0.541699 −0.270849 0.962622i \(-0.587305\pi\)
−0.270849 + 0.962622i \(0.587305\pi\)
\(138\) 0 0
\(139\) 8.13659i 0.690137i 0.938578 + 0.345068i \(0.112144\pi\)
−0.938578 + 0.345068i \(0.887856\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −39.4509 −3.29905
\(144\) 0 0
\(145\) 2.91346 0.241950
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.2594i 1.25010i 0.780586 + 0.625048i \(0.214921\pi\)
−0.780586 + 0.625048i \(0.785079\pi\)
\(150\) 0 0
\(151\) 0.728441 0.0592797 0.0296398 0.999561i \(-0.490564\pi\)
0.0296398 + 0.999561i \(0.490564\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.31425i 0.266207i
\(156\) 0 0
\(157\) 12.1207i 0.967337i 0.875251 + 0.483669i \(0.160696\pi\)
−0.875251 + 0.483669i \(0.839304\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.88452 −0.227332
\(162\) 0 0
\(163\) − 5.45067i − 0.426929i −0.976951 0.213465i \(-0.931525\pi\)
0.976951 0.213465i \(-0.0684748\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.96762 −0.152259 −0.0761297 0.997098i \(-0.524256\pi\)
−0.0761297 + 0.997098i \(0.524256\pi\)
\(168\) 0 0
\(169\) −30.4303 −2.34079
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 4.48602i − 0.341066i −0.985352 0.170533i \(-0.945451\pi\)
0.985352 0.170533i \(-0.0545489\pi\)
\(174\) 0 0
\(175\) −4.11555 −0.311107
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 24.8086i − 1.85428i −0.374716 0.927140i \(-0.622260\pi\)
0.374716 0.927140i \(-0.377740\pi\)
\(180\) 0 0
\(181\) 22.6114i 1.68070i 0.542048 + 0.840348i \(0.317649\pi\)
−0.542048 + 0.840348i \(0.682351\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.200842 0.0147662
\(186\) 0 0
\(187\) 15.8174i 1.15668i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.3035 1.17968 0.589838 0.807521i \(-0.299191\pi\)
0.589838 + 0.807521i \(0.299191\pi\)
\(192\) 0 0
\(193\) 6.98388 0.502711 0.251355 0.967895i \(-0.419124\pi\)
0.251355 + 0.967895i \(0.419124\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.5891i 0.968182i 0.875018 + 0.484091i \(0.160850\pi\)
−0.875018 + 0.484091i \(0.839150\pi\)
\(198\) 0 0
\(199\) −20.2305 −1.43410 −0.717052 0.697020i \(-0.754509\pi\)
−0.717052 + 0.697020i \(0.754509\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 3.09794i − 0.217433i
\(204\) 0 0
\(205\) − 1.53567i − 0.107256i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 34.9260 2.41588
\(210\) 0 0
\(211\) 19.1090i 1.31552i 0.753230 + 0.657758i \(0.228495\pi\)
−0.753230 + 0.657758i \(0.771505\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.74035 0.459688
\(216\) 0 0
\(217\) 3.52412 0.239233
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.4129i 1.17132i
\(222\) 0 0
\(223\) 20.9241 1.40118 0.700592 0.713562i \(-0.252920\pi\)
0.700592 + 0.713562i \(0.252920\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.48329i − 0.164821i −0.996598 0.0824107i \(-0.973738\pi\)
0.996598 0.0824107i \(-0.0262619\pi\)
\(228\) 0 0
\(229\) − 10.3155i − 0.681669i −0.940123 0.340835i \(-0.889290\pi\)
0.940123 0.340835i \(-0.110710\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.5208 1.27885 0.639424 0.768855i \(-0.279173\pi\)
0.639424 + 0.768855i \(0.279173\pi\)
\(234\) 0 0
\(235\) 8.77100i 0.572157i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.8520 −1.15475 −0.577376 0.816478i \(-0.695923\pi\)
−0.577376 + 0.816478i \(0.695923\pi\)
\(240\) 0 0
\(241\) 6.72097 0.432935 0.216468 0.976290i \(-0.430546\pi\)
0.216468 + 0.976290i \(0.430546\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.940450i − 0.0600831i
\(246\) 0 0
\(247\) 38.4490 2.44645
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 28.1618i 1.77756i 0.458336 + 0.888779i \(0.348445\pi\)
−0.458336 + 0.888779i \(0.651555\pi\)
\(252\) 0 0
\(253\) 17.2677i 1.08561i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.9286 −1.55500 −0.777502 0.628880i \(-0.783514\pi\)
−0.777502 + 0.628880i \(0.783514\pi\)
\(258\) 0 0
\(259\) − 0.213560i − 0.0132700i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.0635 1.17550 0.587751 0.809042i \(-0.300013\pi\)
0.587751 + 0.809042i \(0.300013\pi\)
\(264\) 0 0
\(265\) −7.06319 −0.433888
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 10.2410i − 0.624406i −0.950015 0.312203i \(-0.898933\pi\)
0.950015 0.312203i \(-0.101067\pi\)
\(270\) 0 0
\(271\) 4.45674 0.270728 0.135364 0.990796i \(-0.456780\pi\)
0.135364 + 0.990796i \(0.456780\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.6370i 1.48567i
\(276\) 0 0
\(277\) 16.0596i 0.964929i 0.875915 + 0.482465i \(0.160258\pi\)
−0.875915 + 0.482465i \(0.839742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3739 1.87161 0.935806 0.352516i \(-0.114674\pi\)
0.935806 + 0.352516i \(0.114674\pi\)
\(282\) 0 0
\(283\) 16.4627i 0.978605i 0.872114 + 0.489302i \(0.162749\pi\)
−0.872114 + 0.489302i \(0.837251\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.63291 −0.0963874
\(288\) 0 0
\(289\) −10.0185 −0.589324
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.78733i 0.338100i 0.985608 + 0.169050i \(0.0540699\pi\)
−0.985608 + 0.169050i \(0.945930\pi\)
\(294\) 0 0
\(295\) −11.2177 −0.653117
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.0095i 1.09935i
\(300\) 0 0
\(301\) − 7.16716i − 0.413108i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.39472 −0.0798617
\(306\) 0 0
\(307\) 23.0312i 1.31446i 0.753691 + 0.657229i \(0.228272\pi\)
−0.753691 + 0.657229i \(0.771728\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.7123 0.890964 0.445482 0.895291i \(-0.353032\pi\)
0.445482 + 0.895291i \(0.353032\pi\)
\(312\) 0 0
\(313\) −6.53462 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 19.9169i − 1.11865i −0.828949 0.559324i \(-0.811061\pi\)
0.828949 0.559324i \(-0.188939\pi\)
\(318\) 0 0
\(319\) −18.5453 −1.03834
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 15.4157i − 0.857751i
\(324\) 0 0
\(325\) 27.1222i 1.50447i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.32639 0.514180
\(330\) 0 0
\(331\) 21.3692i 1.17456i 0.809384 + 0.587280i \(0.199801\pi\)
−0.809384 + 0.587280i \(0.800199\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.2780 0.670821
\(336\) 0 0
\(337\) −9.56768 −0.521185 −0.260592 0.965449i \(-0.583918\pi\)
−0.260592 + 0.965449i \(0.583918\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 21.0965i − 1.14244i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 5.63471i − 0.302487i −0.988497 0.151243i \(-0.951672\pi\)
0.988497 0.151243i \(-0.0483278\pi\)
\(348\) 0 0
\(349\) − 33.2219i − 1.77833i −0.457591 0.889163i \(-0.651287\pi\)
0.457591 0.889163i \(-0.348713\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.20112 −0.330052 −0.165026 0.986289i \(-0.552771\pi\)
−0.165026 + 0.986289i \(0.552771\pi\)
\(354\) 0 0
\(355\) 1.45433i 0.0771879i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.76894 0.0933609 0.0466805 0.998910i \(-0.485136\pi\)
0.0466805 + 0.998910i \(0.485136\pi\)
\(360\) 0 0
\(361\) −15.0390 −0.791526
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.79192i 0.146136i
\(366\) 0 0
\(367\) −9.62872 −0.502615 −0.251308 0.967907i \(-0.580861\pi\)
−0.251308 + 0.967907i \(0.580861\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.51044i 0.389922i
\(372\) 0 0
\(373\) − 19.6983i − 1.01994i −0.860192 0.509970i \(-0.829657\pi\)
0.860192 0.509970i \(-0.170343\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.4160 −1.05148
\(378\) 0 0
\(379\) − 16.3324i − 0.838941i −0.907769 0.419470i \(-0.862216\pi\)
0.907769 0.419470i \(-0.137784\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.7037 0.904618 0.452309 0.891861i \(-0.350600\pi\)
0.452309 + 0.891861i \(0.350600\pi\)
\(384\) 0 0
\(385\) −5.62983 −0.286923
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.4174i 1.23801i 0.785387 + 0.619005i \(0.212464\pi\)
−0.785387 + 0.619005i \(0.787536\pi\)
\(390\) 0 0
\(391\) 7.62162 0.385442
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.0016i 0.754811i
\(396\) 0 0
\(397\) 21.4996i 1.07903i 0.841975 + 0.539517i \(0.181393\pi\)
−0.841975 + 0.539517i \(0.818607\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.0629 −1.45133 −0.725667 0.688046i \(-0.758469\pi\)
−0.725667 + 0.688046i \(0.758469\pi\)
\(402\) 0 0
\(403\) − 23.2245i − 1.15690i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.27844 −0.0633698
\(408\) 0 0
\(409\) −5.96315 −0.294859 −0.147429 0.989073i \(-0.547100\pi\)
−0.147429 + 0.989073i \(0.547100\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.9280i 0.586937i
\(414\) 0 0
\(415\) 8.22689 0.403842
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 28.8317i − 1.40852i −0.709941 0.704261i \(-0.751279\pi\)
0.709941 0.704261i \(-0.248721\pi\)
\(420\) 0 0
\(421\) 30.3324i 1.47831i 0.673534 + 0.739156i \(0.264776\pi\)
−0.673534 + 0.739156i \(0.735224\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.8743 0.527482
\(426\) 0 0
\(427\) 1.48304i 0.0717693i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 32.3775 1.55957 0.779785 0.626048i \(-0.215329\pi\)
0.779785 + 0.626048i \(0.215329\pi\)
\(432\) 0 0
\(433\) −30.9777 −1.48869 −0.744346 0.667794i \(-0.767239\pi\)
−0.744346 + 0.667794i \(0.767239\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 16.8291i − 0.805047i
\(438\) 0 0
\(439\) −8.76381 −0.418274 −0.209137 0.977886i \(-0.567065\pi\)
−0.209137 + 0.977886i \(0.567065\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.53477i 0.357988i 0.983850 + 0.178994i \(0.0572842\pi\)
−0.983850 + 0.178994i \(0.942716\pi\)
\(444\) 0 0
\(445\) 7.05656i 0.334513i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.5584 −1.53653 −0.768263 0.640134i \(-0.778879\pi\)
−0.768263 + 0.640134i \(0.778879\pi\)
\(450\) 0 0
\(451\) 9.77510i 0.460292i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.19772 −0.290554
\(456\) 0 0
\(457\) −3.81122 −0.178281 −0.0891407 0.996019i \(-0.528412\pi\)
−0.0891407 + 0.996019i \(0.528412\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.29754i 0.293306i 0.989188 + 0.146653i \(0.0468500\pi\)
−0.989188 + 0.146653i \(0.953150\pi\)
\(462\) 0 0
\(463\) −12.8418 −0.596811 −0.298405 0.954439i \(-0.596455\pi\)
−0.298405 + 0.954439i \(0.596455\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 34.5588i − 1.59919i −0.600538 0.799596i \(-0.705047\pi\)
0.600538 0.799596i \(-0.294953\pi\)
\(468\) 0 0
\(469\) − 13.0555i − 0.602847i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −42.9049 −1.97277
\(474\) 0 0
\(475\) − 24.0114i − 1.10172i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.5950 −0.758244 −0.379122 0.925347i \(-0.623774\pi\)
−0.379122 + 0.925347i \(0.623774\pi\)
\(480\) 0 0
\(481\) −1.40739 −0.0641717
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 10.1125i − 0.459186i
\(486\) 0 0
\(487\) −26.4209 −1.19724 −0.598622 0.801032i \(-0.704285\pi\)
−0.598622 + 0.801032i \(0.704285\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 3.70049i − 0.167001i −0.996508 0.0835003i \(-0.973390\pi\)
0.996508 0.0835003i \(-0.0266100\pi\)
\(492\) 0 0
\(493\) 8.18554i 0.368658i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.54642 0.0693664
\(498\) 0 0
\(499\) 2.12488i 0.0951229i 0.998868 + 0.0475614i \(0.0151450\pi\)
−0.998868 + 0.0475614i \(0.984855\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −43.3846 −1.93443 −0.967213 0.253968i \(-0.918264\pi\)
−0.967213 + 0.253968i \(0.918264\pi\)
\(504\) 0 0
\(505\) −3.33785 −0.148532
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.34482i − 0.103932i −0.998649 0.0519662i \(-0.983451\pi\)
0.998649 0.0519662i \(-0.0165488\pi\)
\(510\) 0 0
\(511\) 2.96871 0.131328
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.4174i 0.635306i
\(516\) 0 0
\(517\) − 55.8307i − 2.45543i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8.53436 −0.373897 −0.186949 0.982370i \(-0.559860\pi\)
−0.186949 + 0.982370i \(0.559860\pi\)
\(522\) 0 0
\(523\) − 30.8607i − 1.34944i −0.738072 0.674722i \(-0.764264\pi\)
0.738072 0.674722i \(-0.235736\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.31160 −0.405620
\(528\) 0 0
\(529\) −14.6796 −0.638241
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.7611i 0.466116i
\(534\) 0 0
\(535\) 2.27244 0.0982461
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.98632i 0.257849i
\(540\) 0 0
\(541\) − 15.2568i − 0.655940i −0.944688 0.327970i \(-0.893636\pi\)
0.944688 0.327970i \(-0.106364\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.45817 0.276638
\(546\) 0 0
\(547\) 35.6416i 1.52393i 0.647621 + 0.761963i \(0.275764\pi\)
−0.647621 + 0.761963i \(0.724236\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 18.0743 0.769992
\(552\) 0 0
\(553\) 15.9515 0.678326
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.3871i − 1.20280i −0.798949 0.601399i \(-0.794610\pi\)
0.798949 0.601399i \(-0.205390\pi\)
\(558\) 0 0
\(559\) −47.2328 −1.99773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 13.0963i − 0.551942i −0.961166 0.275971i \(-0.911001\pi\)
0.961166 0.275971i \(-0.0889993\pi\)
\(564\) 0 0
\(565\) − 14.5336i − 0.611435i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20.0434 −0.840263 −0.420132 0.907463i \(-0.638016\pi\)
−0.420132 + 0.907463i \(0.638016\pi\)
\(570\) 0 0
\(571\) 6.91621i 0.289434i 0.989473 + 0.144717i \(0.0462272\pi\)
−0.989473 + 0.144717i \(0.953773\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.8714 0.495071
\(576\) 0 0
\(577\) 23.2279 0.966992 0.483496 0.875347i \(-0.339367\pi\)
0.483496 + 0.875347i \(0.339367\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 8.74782i − 0.362921i
\(582\) 0 0
\(583\) 44.9599 1.86205
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.76774i − 0.0729624i −0.999334 0.0364812i \(-0.988385\pi\)
0.999334 0.0364812i \(-0.0116149\pi\)
\(588\) 0 0
\(589\) 20.5607i 0.847191i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.8807 −0.611079 −0.305540 0.952179i \(-0.598837\pi\)
−0.305540 + 0.952179i \(0.598837\pi\)
\(594\) 0 0
\(595\) 2.48490i 0.101871i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.47561 0.0602916 0.0301458 0.999546i \(-0.490403\pi\)
0.0301458 + 0.999546i \(0.490403\pi\)
\(600\) 0 0
\(601\) −11.1957 −0.456684 −0.228342 0.973581i \(-0.573330\pi\)
−0.228342 + 0.973581i \(0.573330\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.3570i 0.949598i
\(606\) 0 0
\(607\) 31.4554 1.27674 0.638368 0.769731i \(-0.279610\pi\)
0.638368 + 0.769731i \(0.279610\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 61.4625i − 2.48651i
\(612\) 0 0
\(613\) 18.9222i 0.764262i 0.924108 + 0.382131i \(0.124810\pi\)
−0.924108 + 0.382131i \(0.875190\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.1549 0.489339 0.244669 0.969607i \(-0.421321\pi\)
0.244669 + 0.969607i \(0.421321\pi\)
\(618\) 0 0
\(619\) 14.9619i 0.601371i 0.953723 + 0.300686i \(0.0972155\pi\)
−0.953723 + 0.300686i \(0.902784\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.50339 0.300617
\(624\) 0 0
\(625\) 12.5156 0.500623
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.564278i 0.0224992i
\(630\) 0 0
\(631\) 8.71232 0.346832 0.173416 0.984849i \(-0.444519\pi\)
0.173416 + 0.984849i \(0.444519\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.49313i 0.297356i
\(636\) 0 0
\(637\) 6.59017i 0.261112i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.31021 −0.288736 −0.144368 0.989524i \(-0.546115\pi\)
−0.144368 + 0.989524i \(0.546115\pi\)
\(642\) 0 0
\(643\) − 11.1766i − 0.440763i −0.975414 0.220381i \(-0.929270\pi\)
0.975414 0.220381i \(-0.0707302\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.13625 −0.280555 −0.140277 0.990112i \(-0.544799\pi\)
−0.140277 + 0.990112i \(0.544799\pi\)
\(648\) 0 0
\(649\) 71.4046 2.80288
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 15.6385i − 0.611982i −0.952034 0.305991i \(-0.901012\pi\)
0.952034 0.305991i \(-0.0989878\pi\)
\(654\) 0 0
\(655\) −1.44622 −0.0565086
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.73739i 0.0676791i 0.999427 + 0.0338395i \(0.0107735\pi\)
−0.999427 + 0.0338395i \(0.989226\pi\)
\(660\) 0 0
\(661\) 22.7968i 0.886691i 0.896351 + 0.443345i \(0.146208\pi\)
−0.896351 + 0.443345i \(0.853792\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.48686 0.212771
\(666\) 0 0
\(667\) 8.93608i 0.346006i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.87795 0.342729
\(672\) 0 0
\(673\) −13.1667 −0.507538 −0.253769 0.967265i \(-0.581670\pi\)
−0.253769 + 0.967265i \(0.581670\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.3183i 1.43426i 0.696941 + 0.717129i \(0.254544\pi\)
−0.696941 + 0.717129i \(0.745456\pi\)
\(678\) 0 0
\(679\) −10.7529 −0.412657
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.72259i 0.257233i 0.991694 + 0.128616i \(0.0410536\pi\)
−0.991694 + 0.128616i \(0.958946\pi\)
\(684\) 0 0
\(685\) 5.96285i 0.227829i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 49.4951 1.88561
\(690\) 0 0
\(691\) 8.18845i 0.311503i 0.987796 + 0.155752i \(0.0497800\pi\)
−0.987796 + 0.155752i \(0.950220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.65205 0.290259
\(696\) 0 0
\(697\) 4.31455 0.163425
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.85226i 0.221037i 0.993874 + 0.110518i \(0.0352511\pi\)
−0.993874 + 0.110518i \(0.964749\pi\)
\(702\) 0 0
\(703\) 1.24597 0.0469927
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.54920i 0.133482i
\(708\) 0 0
\(709\) 45.3444i 1.70294i 0.524399 + 0.851472i \(0.324290\pi\)
−0.524399 + 0.851472i \(0.675710\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.1654 −0.380697
\(714\) 0 0
\(715\) 37.1015i 1.38752i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.1182 1.08592 0.542962 0.839757i \(-0.317303\pi\)
0.542962 + 0.839757i \(0.317303\pi\)
\(720\) 0 0
\(721\) 15.3303 0.570931
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 12.7498i 0.473514i
\(726\) 0 0
\(727\) 42.5099 1.57661 0.788303 0.615287i \(-0.210960\pi\)
0.788303 + 0.615287i \(0.210960\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 18.9374i 0.700426i
\(732\) 0 0
\(733\) 2.79234i 0.103137i 0.998669 + 0.0515687i \(0.0164221\pi\)
−0.998669 + 0.0515687i \(0.983578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −78.1544 −2.87886
\(738\) 0 0
\(739\) − 15.4715i − 0.569129i −0.958657 0.284565i \(-0.908151\pi\)
0.958657 0.284565i \(-0.0918490\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.2897 −0.670984 −0.335492 0.942043i \(-0.608903\pi\)
−0.335492 + 0.942043i \(0.608903\pi\)
\(744\) 0 0
\(745\) 14.3507 0.525768
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 2.41633i − 0.0882908i
\(750\) 0 0
\(751\) −11.1167 −0.405655 −0.202828 0.979214i \(-0.565013\pi\)
−0.202828 + 0.979214i \(0.565013\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 0.685062i − 0.0249320i
\(756\) 0 0
\(757\) 11.8173i 0.429507i 0.976668 + 0.214753i \(0.0688948\pi\)
−0.976668 + 0.214753i \(0.931105\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.2039 0.442390 0.221195 0.975230i \(-0.429004\pi\)
0.221195 + 0.975230i \(0.429004\pi\)
\(762\) 0 0
\(763\) − 6.86711i − 0.248606i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 78.6073 2.83835
\(768\) 0 0
\(769\) −7.32578 −0.264175 −0.132087 0.991238i \(-0.542168\pi\)
−0.132087 + 0.991238i \(0.542168\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.26470i 0.261293i 0.991429 + 0.130647i \(0.0417053\pi\)
−0.991429 + 0.130647i \(0.958295\pi\)
\(774\) 0 0
\(775\) −14.5037 −0.520988
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 9.52686i − 0.341335i
\(780\) 0 0
\(781\) − 9.25737i − 0.331255i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.3989 0.406844
\(786\) 0 0
\(787\) − 3.02431i − 0.107805i −0.998546 0.0539025i \(-0.982834\pi\)
0.998546 0.0539025i \(-0.0171660\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −15.4539 −0.549478
\(792\) 0 0
\(793\) 9.77348 0.347067
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 20.6317i − 0.730813i −0.930848 0.365406i \(-0.880930\pi\)
0.930848 0.365406i \(-0.119070\pi\)
\(798\) 0 0
\(799\) −24.6426 −0.871794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 17.7716i − 0.627148i
\(804\) 0 0
\(805\) 2.71274i 0.0956117i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −29.5889 −1.04029 −0.520145 0.854078i \(-0.674122\pi\)
−0.520145 + 0.854078i \(0.674122\pi\)
\(810\) 0 0
\(811\) − 42.6230i − 1.49670i −0.663307 0.748348i \(-0.730847\pi\)
0.663307 0.748348i \(-0.269153\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.12608 −0.179559
\(816\) 0 0
\(817\) 41.8153 1.46293
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 48.6214i 1.69690i 0.529276 + 0.848450i \(0.322464\pi\)
−0.529276 + 0.848450i \(0.677536\pi\)
\(822\) 0 0
\(823\) −23.7228 −0.826924 −0.413462 0.910521i \(-0.635680\pi\)
−0.413462 + 0.910521i \(0.635680\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.3889i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(828\) 0 0
\(829\) − 26.4785i − 0.919635i −0.888013 0.459818i \(-0.847915\pi\)
0.888013 0.459818i \(-0.152085\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.64225 0.0915485
\(834\) 0 0
\(835\) 1.85045i 0.0640375i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.51211 −0.121252 −0.0606258 0.998161i \(-0.519310\pi\)
−0.0606258 + 0.998161i \(0.519310\pi\)
\(840\) 0 0
\(841\) 19.4027 0.669060
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28.6182i 0.984496i
\(846\) 0 0
\(847\) 24.8360 0.853376
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.616017i 0.0211168i
\(852\) 0 0
\(853\) − 28.2378i − 0.966843i −0.875388 0.483422i \(-0.839394\pi\)
0.875388 0.483422i \(-0.160606\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.8126 0.437669 0.218834 0.975762i \(-0.429775\pi\)
0.218834 + 0.975762i \(0.429775\pi\)
\(858\) 0 0
\(859\) − 35.4496i − 1.20952i −0.796406 0.604762i \(-0.793268\pi\)
0.796406 0.604762i \(-0.206732\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.3418 −1.10093 −0.550465 0.834858i \(-0.685549\pi\)
−0.550465 + 0.834858i \(0.685549\pi\)
\(864\) 0 0
\(865\) −4.21887 −0.143446
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 95.4907i − 3.23930i
\(870\) 0 0
\(871\) −86.0380 −2.91529
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.57272i 0.289811i
\(876\) 0 0
\(877\) − 6.53592i − 0.220702i −0.993893 0.110351i \(-0.964802\pi\)
0.993893 0.110351i \(-0.0351976\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 53.0057 1.78581 0.892904 0.450248i \(-0.148664\pi\)
0.892904 + 0.450248i \(0.148664\pi\)
\(882\) 0 0
\(883\) 28.0481i 0.943894i 0.881627 + 0.471947i \(0.156449\pi\)
−0.881627 + 0.471947i \(0.843551\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.58627 0.153992 0.0769959 0.997031i \(-0.475467\pi\)
0.0769959 + 0.997031i \(0.475467\pi\)
\(888\) 0 0
\(889\) 7.96760 0.267225
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 54.4129i 1.82086i
\(894\) 0 0
\(895\) −23.3312 −0.779876
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 10.9175i − 0.364120i
\(900\) 0 0
\(901\) − 19.8445i − 0.661115i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.2649 0.706870
\(906\) 0 0
\(907\) − 31.8338i − 1.05703i −0.848925 0.528513i \(-0.822750\pi\)
0.848925 0.528513i \(-0.177250\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 55.9123 1.85246 0.926228 0.376964i \(-0.123032\pi\)
0.926228 + 0.376964i \(0.123032\pi\)
\(912\) 0 0
\(913\) −52.3673 −1.73310
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.53780i 0.0507826i
\(918\) 0 0
\(919\) 51.6749 1.70460 0.852299 0.523055i \(-0.175208\pi\)
0.852299 + 0.523055i \(0.175208\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 10.1912i − 0.335447i
\(924\) 0 0
\(925\) 0.878917i 0.0288986i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10.6988 −0.351016 −0.175508 0.984478i \(-0.556157\pi\)
−0.175508 + 0.984478i \(0.556157\pi\)
\(930\) 0 0
\(931\) − 5.83430i − 0.191211i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.8754 0.486478
\(936\) 0 0
\(937\) 29.3422 0.958568 0.479284 0.877660i \(-0.340896\pi\)
0.479284 + 0.877660i \(0.340896\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 37.2035i 1.21280i 0.795160 + 0.606399i \(0.207387\pi\)
−0.795160 + 0.606399i \(0.792613\pi\)
\(942\) 0 0
\(943\) 4.71015 0.153384
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 25.7353i − 0.836283i −0.908382 0.418142i \(-0.862682\pi\)
0.908382 0.418142i \(-0.137318\pi\)
\(948\) 0 0
\(949\) − 19.5643i − 0.635084i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 17.6526 0.571824 0.285912 0.958256i \(-0.407703\pi\)
0.285912 + 0.958256i \(0.407703\pi\)
\(954\) 0 0
\(955\) − 15.3326i − 0.496150i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.34042 0.204743
\(960\) 0 0
\(961\) −18.5806 −0.599374
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 6.56799i − 0.211431i
\(966\) 0 0
\(967\) −9.22421 −0.296631 −0.148315 0.988940i \(-0.547385\pi\)
−0.148315 + 0.988940i \(0.547385\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 0.945630i − 0.0303467i −0.999885 0.0151734i \(-0.995170\pi\)
0.999885 0.0151734i \(-0.00483002\pi\)
\(972\) 0 0
\(973\) − 8.13659i − 0.260847i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −34.7944 −1.11317 −0.556585 0.830791i \(-0.687889\pi\)
−0.556585 + 0.830791i \(0.687889\pi\)
\(978\) 0 0
\(979\) − 44.9177i − 1.43558i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.7133 0.756336 0.378168 0.925737i \(-0.376554\pi\)
0.378168 + 0.925737i \(0.376554\pi\)
\(984\) 0 0
\(985\) 12.7799 0.407200
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.6738i 0.657389i
\(990\) 0 0
\(991\) 24.1109 0.765908 0.382954 0.923767i \(-0.374907\pi\)
0.382954 + 0.923767i \(0.374907\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.0258i 0.603158i
\(996\) 0 0
\(997\) − 30.0398i − 0.951371i −0.879615 0.475685i \(-0.842200\pi\)
0.879615 0.475685i \(-0.157800\pi\)
\(998\) 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.f.3025.12 24
3.2 odd 2 inner 6048.2.c.f.3025.14 24
4.3 odd 2 1512.2.c.g.757.6 yes 24
8.3 odd 2 1512.2.c.g.757.5 24
8.5 even 2 inner 6048.2.c.f.3025.13 24
12.11 even 2 1512.2.c.g.757.19 yes 24
24.5 odd 2 inner 6048.2.c.f.3025.11 24
24.11 even 2 1512.2.c.g.757.20 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.5 24 8.3 odd 2
1512.2.c.g.757.6 yes 24 4.3 odd 2
1512.2.c.g.757.19 yes 24 12.11 even 2
1512.2.c.g.757.20 yes 24 24.11 even 2
6048.2.c.f.3025.11 24 24.5 odd 2 inner
6048.2.c.f.3025.12 24 1.1 even 1 trivial
6048.2.c.f.3025.13 24 8.5 even 2 inner
6048.2.c.f.3025.14 24 3.2 odd 2 inner