Properties

Label 6048.2.c.f.3025.6
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 3025.6
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.19

$q$-expansion

\(f(q)\) \(=\) \(q-2.99001i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.99001i q^{5} -1.00000 q^{7} +2.13956i q^{11} +0.665739i q^{13} -7.04931 q^{17} +1.39525i q^{19} +0.184371 q^{23} -3.94016 q^{25} -1.27564i q^{29} +7.62308 q^{31} +2.99001i q^{35} -6.69127i q^{37} +0.274106 q^{41} -2.63971i q^{43} -5.77739 q^{47} +1.00000 q^{49} +7.48353i q^{53} +6.39730 q^{55} +9.03787i q^{59} +9.80298i q^{61} +1.99057 q^{65} +10.8191i q^{67} +7.93573 q^{71} -2.80408 q^{73} -2.13956i q^{77} -7.98350 q^{79} -6.76125i q^{83} +21.0775i q^{85} -16.7549 q^{89} -0.665739i q^{91} +4.17180 q^{95} -0.107694 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) − 2.99001i − 1.33717i −0.743634 0.668587i \(-0.766900\pi\)
0.743634 0.668587i \(-0.233100\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.13956i 0.645101i 0.946552 + 0.322550i \(0.104540\pi\)
−0.946552 + 0.322550i \(0.895460\pi\)
\(12\) 0 0
\(13\) 0.665739i 0.184643i 0.995729 + 0.0923214i \(0.0294287\pi\)
−0.995729 + 0.0923214i \(0.970571\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.04931 −1.70971 −0.854855 0.518867i \(-0.826354\pi\)
−0.854855 + 0.518867i \(0.826354\pi\)
\(18\) 0 0
\(19\) 1.39525i 0.320092i 0.987110 + 0.160046i \(0.0511642\pi\)
−0.987110 + 0.160046i \(0.948836\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.184371 0.0384439 0.0192220 0.999815i \(-0.493881\pi\)
0.0192220 + 0.999815i \(0.493881\pi\)
\(24\) 0 0
\(25\) −3.94016 −0.788032
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.27564i − 0.236880i −0.992961 0.118440i \(-0.962211\pi\)
0.992961 0.118440i \(-0.0377893\pi\)
\(30\) 0 0
\(31\) 7.62308 1.36915 0.684573 0.728944i \(-0.259989\pi\)
0.684573 + 0.728944i \(0.259989\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.99001i 0.505404i
\(36\) 0 0
\(37\) − 6.69127i − 1.10004i −0.835152 0.550019i \(-0.814621\pi\)
0.835152 0.550019i \(-0.185379\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.274106 0.0428081 0.0214041 0.999771i \(-0.493186\pi\)
0.0214041 + 0.999771i \(0.493186\pi\)
\(42\) 0 0
\(43\) − 2.63971i − 0.402553i −0.979535 0.201276i \(-0.935491\pi\)
0.979535 0.201276i \(-0.0645089\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.77739 −0.842719 −0.421359 0.906894i \(-0.638447\pi\)
−0.421359 + 0.906894i \(0.638447\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.48353i 1.02794i 0.857808 + 0.513971i \(0.171826\pi\)
−0.857808 + 0.513971i \(0.828174\pi\)
\(54\) 0 0
\(55\) 6.39730 0.862611
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.03787i 1.17663i 0.808632 + 0.588315i \(0.200208\pi\)
−0.808632 + 0.588315i \(0.799792\pi\)
\(60\) 0 0
\(61\) 9.80298i 1.25514i 0.778559 + 0.627571i \(0.215951\pi\)
−0.778559 + 0.627571i \(0.784049\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.99057 0.246900
\(66\) 0 0
\(67\) 10.8191i 1.32177i 0.750488 + 0.660884i \(0.229819\pi\)
−0.750488 + 0.660884i \(0.770181\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.93573 0.941798 0.470899 0.882187i \(-0.343930\pi\)
0.470899 + 0.882187i \(0.343930\pi\)
\(72\) 0 0
\(73\) −2.80408 −0.328193 −0.164097 0.986444i \(-0.552471\pi\)
−0.164097 + 0.986444i \(0.552471\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.13956i − 0.243825i
\(78\) 0 0
\(79\) −7.98350 −0.898214 −0.449107 0.893478i \(-0.648258\pi\)
−0.449107 + 0.893478i \(0.648258\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.76125i − 0.742143i −0.928604 0.371072i \(-0.878990\pi\)
0.928604 0.371072i \(-0.121010\pi\)
\(84\) 0 0
\(85\) 21.0775i 2.28618i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −16.7549 −1.77602 −0.888010 0.459825i \(-0.847912\pi\)
−0.888010 + 0.459825i \(0.847912\pi\)
\(90\) 0 0
\(91\) − 0.665739i − 0.0697884i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.17180 0.428018
\(96\) 0 0
\(97\) −0.107694 −0.0109346 −0.00546732 0.999985i \(-0.501740\pi\)
−0.00546732 + 0.999985i \(0.501740\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.8524i 1.47787i 0.673775 + 0.738937i \(0.264672\pi\)
−0.673775 + 0.738937i \(0.735328\pi\)
\(102\) 0 0
\(103\) 16.1461 1.59092 0.795459 0.606007i \(-0.207230\pi\)
0.795459 + 0.606007i \(0.207230\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.6131i 1.50937i 0.656084 + 0.754687i \(0.272211\pi\)
−0.656084 + 0.754687i \(0.727789\pi\)
\(108\) 0 0
\(109\) − 4.15829i − 0.398292i −0.979970 0.199146i \(-0.936183\pi\)
0.979970 0.199146i \(-0.0638167\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.4682 −1.73734 −0.868672 0.495388i \(-0.835026\pi\)
−0.868672 + 0.495388i \(0.835026\pi\)
\(114\) 0 0
\(115\) − 0.551270i − 0.0514062i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.04931 0.646209
\(120\) 0 0
\(121\) 6.42230 0.583845
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 3.16893i − 0.283437i
\(126\) 0 0
\(127\) 4.37444 0.388169 0.194084 0.980985i \(-0.437826\pi\)
0.194084 + 0.980985i \(0.437826\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.76264i 0.503484i 0.967794 + 0.251742i \(0.0810035\pi\)
−0.967794 + 0.251742i \(0.918997\pi\)
\(132\) 0 0
\(133\) − 1.39525i − 0.120983i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.5191 0.898704 0.449352 0.893355i \(-0.351655\pi\)
0.449352 + 0.893355i \(0.351655\pi\)
\(138\) 0 0
\(139\) − 7.27000i − 0.616633i −0.951284 0.308316i \(-0.900234\pi\)
0.951284 0.308316i \(-0.0997656\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.42439 −0.119113
\(144\) 0 0
\(145\) −3.81416 −0.316749
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 3.06625i − 0.251197i −0.992081 0.125599i \(-0.959915\pi\)
0.992081 0.125599i \(-0.0400851\pi\)
\(150\) 0 0
\(151\) 15.8992 1.29386 0.646930 0.762549i \(-0.276053\pi\)
0.646930 + 0.762549i \(0.276053\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 22.7931i − 1.83079i
\(156\) 0 0
\(157\) 18.3211i 1.46219i 0.682278 + 0.731093i \(0.260989\pi\)
−0.682278 + 0.731093i \(0.739011\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.184371 −0.0145304
\(162\) 0 0
\(163\) 22.0479i 1.72692i 0.504415 + 0.863462i \(0.331708\pi\)
−0.504415 + 0.863462i \(0.668292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.2650 0.794332 0.397166 0.917747i \(-0.369994\pi\)
0.397166 + 0.917747i \(0.369994\pi\)
\(168\) 0 0
\(169\) 12.5568 0.965907
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.5234i − 1.18022i −0.807322 0.590111i \(-0.799084\pi\)
0.807322 0.590111i \(-0.200916\pi\)
\(174\) 0 0
\(175\) 3.94016 0.297848
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 17.2028i − 1.28580i −0.765952 0.642898i \(-0.777732\pi\)
0.765952 0.642898i \(-0.222268\pi\)
\(180\) 0 0
\(181\) 10.5145i 0.781538i 0.920489 + 0.390769i \(0.127791\pi\)
−0.920489 + 0.390769i \(0.872209\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −20.0070 −1.47094
\(186\) 0 0
\(187\) − 15.0824i − 1.10293i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.3893 1.04117 0.520587 0.853808i \(-0.325713\pi\)
0.520587 + 0.853808i \(0.325713\pi\)
\(192\) 0 0
\(193\) 11.3579 0.817562 0.408781 0.912632i \(-0.365954\pi\)
0.408781 + 0.912632i \(0.365954\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 6.44465i − 0.459163i −0.973289 0.229581i \(-0.926264\pi\)
0.973289 0.229581i \(-0.0737357\pi\)
\(198\) 0 0
\(199\) 6.48055 0.459394 0.229697 0.973262i \(-0.426226\pi\)
0.229697 + 0.973262i \(0.426226\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.27564i 0.0895321i
\(204\) 0 0
\(205\) − 0.819579i − 0.0572419i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.98521 −0.206491
\(210\) 0 0
\(211\) 10.3150i 0.710112i 0.934845 + 0.355056i \(0.115538\pi\)
−0.934845 + 0.355056i \(0.884462\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.89277 −0.538283
\(216\) 0 0
\(217\) −7.62308 −0.517489
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.69300i − 0.315686i
\(222\) 0 0
\(223\) −23.8589 −1.59771 −0.798854 0.601526i \(-0.794560\pi\)
−0.798854 + 0.601526i \(0.794560\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.88092i − 0.124841i −0.998050 0.0624206i \(-0.980118\pi\)
0.998050 0.0624206i \(-0.0198820\pi\)
\(228\) 0 0
\(229\) 18.8128i 1.24319i 0.783340 + 0.621594i \(0.213515\pi\)
−0.783340 + 0.621594i \(0.786485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.18759 −0.601899 −0.300949 0.953640i \(-0.597304\pi\)
−0.300949 + 0.953640i \(0.597304\pi\)
\(234\) 0 0
\(235\) 17.2745i 1.12686i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.1200 1.88362 0.941808 0.336152i \(-0.109126\pi\)
0.941808 + 0.336152i \(0.109126\pi\)
\(240\) 0 0
\(241\) 25.4640 1.64028 0.820142 0.572160i \(-0.193894\pi\)
0.820142 + 0.572160i \(0.193894\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.99001i − 0.191025i
\(246\) 0 0
\(247\) −0.928871 −0.0591027
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.72427i 0.171955i 0.996297 + 0.0859773i \(0.0274012\pi\)
−0.996297 + 0.0859773i \(0.972599\pi\)
\(252\) 0 0
\(253\) 0.394471i 0.0248002i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.6181 1.09899 0.549494 0.835498i \(-0.314821\pi\)
0.549494 + 0.835498i \(0.314821\pi\)
\(258\) 0 0
\(259\) 6.69127i 0.415775i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.42576 0.334567 0.167283 0.985909i \(-0.446501\pi\)
0.167283 + 0.985909i \(0.446501\pi\)
\(264\) 0 0
\(265\) 22.3758 1.37454
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.6635i 1.07696i 0.842637 + 0.538482i \(0.181002\pi\)
−0.842637 + 0.538482i \(0.818998\pi\)
\(270\) 0 0
\(271\) −11.5223 −0.699929 −0.349965 0.936763i \(-0.613806\pi\)
−0.349965 + 0.936763i \(0.613806\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 8.43020i − 0.508360i
\(276\) 0 0
\(277\) − 7.84872i − 0.471584i −0.971804 0.235792i \(-0.924232\pi\)
0.971804 0.235792i \(-0.0757684\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.1308 1.85711 0.928555 0.371195i \(-0.121052\pi\)
0.928555 + 0.371195i \(0.121052\pi\)
\(282\) 0 0
\(283\) 24.7863i 1.47339i 0.676224 + 0.736696i \(0.263615\pi\)
−0.676224 + 0.736696i \(0.736385\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.274106 −0.0161800
\(288\) 0 0
\(289\) 32.6928 1.92311
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 28.9898i − 1.69360i −0.531909 0.846802i \(-0.678525\pi\)
0.531909 0.846802i \(-0.321475\pi\)
\(294\) 0 0
\(295\) 27.0233 1.57336
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.122743i 0.00709839i
\(300\) 0 0
\(301\) 2.63971i 0.152151i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 29.3110 1.67834
\(306\) 0 0
\(307\) 18.8831i 1.07772i 0.842397 + 0.538858i \(0.181144\pi\)
−0.842397 + 0.538858i \(0.818856\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.147540 0.00836624 0.00418312 0.999991i \(-0.498668\pi\)
0.00418312 + 0.999991i \(0.498668\pi\)
\(312\) 0 0
\(313\) −23.3154 −1.31786 −0.658932 0.752202i \(-0.728992\pi\)
−0.658932 + 0.752202i \(0.728992\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.43318i 0.0804954i 0.999190 + 0.0402477i \(0.0128147\pi\)
−0.999190 + 0.0402477i \(0.987185\pi\)
\(318\) 0 0
\(319\) 2.72929 0.152811
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 9.83554i − 0.547264i
\(324\) 0 0
\(325\) − 2.62312i − 0.145505i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.77739 0.318518
\(330\) 0 0
\(331\) 23.3196i 1.28176i 0.767640 + 0.640882i \(0.221431\pi\)
−0.767640 + 0.640882i \(0.778569\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 32.3493 1.76743
\(336\) 0 0
\(337\) 0.484629 0.0263994 0.0131997 0.999913i \(-0.495798\pi\)
0.0131997 + 0.999913i \(0.495798\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.3100i 0.883237i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.0395i 0.538947i 0.963008 + 0.269473i \(0.0868496\pi\)
−0.963008 + 0.269473i \(0.913150\pi\)
\(348\) 0 0
\(349\) − 9.55791i − 0.511623i −0.966727 0.255812i \(-0.917657\pi\)
0.966727 0.255812i \(-0.0823427\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.2931 −1.50589 −0.752945 0.658083i \(-0.771368\pi\)
−0.752945 + 0.658083i \(0.771368\pi\)
\(354\) 0 0
\(355\) − 23.7279i − 1.25935i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.8316 −1.36334 −0.681671 0.731659i \(-0.738746\pi\)
−0.681671 + 0.731659i \(0.738746\pi\)
\(360\) 0 0
\(361\) 17.0533 0.897541
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.38424i 0.438851i
\(366\) 0 0
\(367\) 1.78123 0.0929793 0.0464896 0.998919i \(-0.485197\pi\)
0.0464896 + 0.998919i \(0.485197\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 7.48353i − 0.388525i
\(372\) 0 0
\(373\) − 22.1074i − 1.14468i −0.820017 0.572340i \(-0.806036\pi\)
0.820017 0.572340i \(-0.193964\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.849241 0.0437381
\(378\) 0 0
\(379\) 35.4109i 1.81893i 0.415776 + 0.909467i \(0.363510\pi\)
−0.415776 + 0.909467i \(0.636490\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.1927 −1.23619 −0.618095 0.786104i \(-0.712095\pi\)
−0.618095 + 0.786104i \(0.712095\pi\)
\(384\) 0 0
\(385\) −6.39730 −0.326036
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 13.0465i − 0.661483i −0.943721 0.330741i \(-0.892701\pi\)
0.943721 0.330741i \(-0.107299\pi\)
\(390\) 0 0
\(391\) −1.29969 −0.0657279
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 23.8707i 1.20107i
\(396\) 0 0
\(397\) − 36.0707i − 1.81034i −0.425053 0.905168i \(-0.639745\pi\)
0.425053 0.905168i \(-0.360255\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.5149 0.774778 0.387389 0.921916i \(-0.373377\pi\)
0.387389 + 0.921916i \(0.373377\pi\)
\(402\) 0 0
\(403\) 5.07499i 0.252803i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.3163 0.709635
\(408\) 0 0
\(409\) −28.6937 −1.41881 −0.709406 0.704800i \(-0.751037\pi\)
−0.709406 + 0.704800i \(0.751037\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 9.03787i − 0.444724i
\(414\) 0 0
\(415\) −20.2162 −0.992374
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.9355i 1.12047i 0.828333 + 0.560236i \(0.189290\pi\)
−0.828333 + 0.560236i \(0.810710\pi\)
\(420\) 0 0
\(421\) 13.1562i 0.641193i 0.947216 + 0.320596i \(0.103883\pi\)
−0.947216 + 0.320596i \(0.896117\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 27.7754 1.34731
\(426\) 0 0
\(427\) − 9.80298i − 0.474399i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −39.6373 −1.90926 −0.954631 0.297792i \(-0.903750\pi\)
−0.954631 + 0.297792i \(0.903750\pi\)
\(432\) 0 0
\(433\) 11.5906 0.557010 0.278505 0.960435i \(-0.410161\pi\)
0.278505 + 0.960435i \(0.410161\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.257243i 0.0123056i
\(438\) 0 0
\(439\) 9.88971 0.472010 0.236005 0.971752i \(-0.424162\pi\)
0.236005 + 0.971752i \(0.424162\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2.14448i − 0.101888i −0.998702 0.0509438i \(-0.983777\pi\)
0.998702 0.0509438i \(-0.0162229\pi\)
\(444\) 0 0
\(445\) 50.0974i 2.37485i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.65352 −0.219613 −0.109807 0.993953i \(-0.535023\pi\)
−0.109807 + 0.993953i \(0.535023\pi\)
\(450\) 0 0
\(451\) 0.586465i 0.0276156i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.99057 −0.0933192
\(456\) 0 0
\(457\) 19.0517 0.891203 0.445601 0.895232i \(-0.352990\pi\)
0.445601 + 0.895232i \(0.352990\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 26.0354i − 1.21259i −0.795240 0.606294i \(-0.792655\pi\)
0.795240 0.606294i \(-0.207345\pi\)
\(462\) 0 0
\(463\) −29.6096 −1.37607 −0.688036 0.725676i \(-0.741527\pi\)
−0.688036 + 0.725676i \(0.741527\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.8983i 1.19843i 0.800588 + 0.599216i \(0.204521\pi\)
−0.800588 + 0.599216i \(0.795479\pi\)
\(468\) 0 0
\(469\) − 10.8191i − 0.499581i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.64782 0.259687
\(474\) 0 0
\(475\) − 5.49750i − 0.252243i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.2498 −0.879544 −0.439772 0.898109i \(-0.644941\pi\)
−0.439772 + 0.898109i \(0.644941\pi\)
\(480\) 0 0
\(481\) 4.45464 0.203114
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.322005i 0.0146215i
\(486\) 0 0
\(487\) −2.32047 −0.105150 −0.0525752 0.998617i \(-0.516743\pi\)
−0.0525752 + 0.998617i \(0.516743\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 0.578109i − 0.0260897i −0.999915 0.0130449i \(-0.995848\pi\)
0.999915 0.0130449i \(-0.00415242\pi\)
\(492\) 0 0
\(493\) 8.99235i 0.404995i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.93573 −0.355966
\(498\) 0 0
\(499\) − 26.3969i − 1.18169i −0.806785 0.590845i \(-0.798795\pi\)
0.806785 0.590845i \(-0.201205\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.40780 0.374885 0.187443 0.982276i \(-0.439980\pi\)
0.187443 + 0.982276i \(0.439980\pi\)
\(504\) 0 0
\(505\) 44.4090 1.97617
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.9679i 1.46128i 0.682764 + 0.730639i \(0.260778\pi\)
−0.682764 + 0.730639i \(0.739222\pi\)
\(510\) 0 0
\(511\) 2.80408 0.124045
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 48.2769i − 2.12733i
\(516\) 0 0
\(517\) − 12.3611i − 0.543638i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.9299 −1.35506 −0.677531 0.735494i \(-0.736950\pi\)
−0.677531 + 0.735494i \(0.736950\pi\)
\(522\) 0 0
\(523\) − 2.73852i − 0.119747i −0.998206 0.0598735i \(-0.980930\pi\)
0.998206 0.0598735i \(-0.0190697\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −53.7375 −2.34084
\(528\) 0 0
\(529\) −22.9660 −0.998522
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.182483i 0.00790421i
\(534\) 0 0
\(535\) 46.6833 2.01830
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.13956i 0.0921572i
\(540\) 0 0
\(541\) 32.3372i 1.39029i 0.718872 + 0.695143i \(0.244659\pi\)
−0.718872 + 0.695143i \(0.755341\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.4333 −0.532585
\(546\) 0 0
\(547\) 20.4045i 0.872434i 0.899842 + 0.436217i \(0.143682\pi\)
−0.899842 + 0.436217i \(0.856318\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.77983 0.0758232
\(552\) 0 0
\(553\) 7.98350 0.339493
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.16560i − 0.176502i −0.996098 0.0882510i \(-0.971872\pi\)
0.996098 0.0882510i \(-0.0281278\pi\)
\(558\) 0 0
\(559\) 1.75736 0.0743285
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 4.37435i − 0.184357i −0.995743 0.0921784i \(-0.970617\pi\)
0.995743 0.0921784i \(-0.0293830\pi\)
\(564\) 0 0
\(565\) 55.2202i 2.32313i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.59520 −0.192641 −0.0963204 0.995350i \(-0.530707\pi\)
−0.0963204 + 0.995350i \(0.530707\pi\)
\(570\) 0 0
\(571\) − 30.6111i − 1.28103i −0.767944 0.640517i \(-0.778720\pi\)
0.767944 0.640517i \(-0.221280\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.726450 −0.0302951
\(576\) 0 0
\(577\) −8.32558 −0.346598 −0.173299 0.984869i \(-0.555443\pi\)
−0.173299 + 0.984869i \(0.555443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.76125i 0.280504i
\(582\) 0 0
\(583\) −16.0114 −0.663126
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.4147i 1.00770i 0.863791 + 0.503851i \(0.168084\pi\)
−0.863791 + 0.503851i \(0.831916\pi\)
\(588\) 0 0
\(589\) 10.6361i 0.438252i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.57185 0.228809 0.114404 0.993434i \(-0.463504\pi\)
0.114404 + 0.993434i \(0.463504\pi\)
\(594\) 0 0
\(595\) − 21.0775i − 0.864094i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −28.6835 −1.17198 −0.585988 0.810319i \(-0.699294\pi\)
−0.585988 + 0.810319i \(0.699294\pi\)
\(600\) 0 0
\(601\) −28.9086 −1.17921 −0.589603 0.807693i \(-0.700716\pi\)
−0.589603 + 0.807693i \(0.700716\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 19.2027i − 0.780702i
\(606\) 0 0
\(607\) 10.3353 0.419498 0.209749 0.977755i \(-0.432735\pi\)
0.209749 + 0.977755i \(0.432735\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 3.84624i − 0.155602i
\(612\) 0 0
\(613\) 2.77963i 0.112268i 0.998423 + 0.0561341i \(0.0178774\pi\)
−0.998423 + 0.0561341i \(0.982123\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.1145 −0.890296 −0.445148 0.895457i \(-0.646849\pi\)
−0.445148 + 0.895457i \(0.646849\pi\)
\(618\) 0 0
\(619\) 8.20284i 0.329700i 0.986319 + 0.164850i \(0.0527140\pi\)
−0.986319 + 0.164850i \(0.947286\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.7549 0.671272
\(624\) 0 0
\(625\) −29.1759 −1.16704
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 47.1688i 1.88074i
\(630\) 0 0
\(631\) 28.2572 1.12490 0.562450 0.826831i \(-0.309859\pi\)
0.562450 + 0.826831i \(0.309859\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 13.0796i − 0.519049i
\(636\) 0 0
\(637\) 0.665739i 0.0263776i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.7144 1.92410 0.962051 0.272870i \(-0.0879730\pi\)
0.962051 + 0.272870i \(0.0879730\pi\)
\(642\) 0 0
\(643\) − 23.8806i − 0.941757i −0.882198 0.470879i \(-0.843937\pi\)
0.882198 0.470879i \(-0.156063\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.9383 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(648\) 0 0
\(649\) −19.3370 −0.759045
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.4564i 1.38752i 0.720208 + 0.693759i \(0.244047\pi\)
−0.720208 + 0.693759i \(0.755953\pi\)
\(654\) 0 0
\(655\) 17.2304 0.673246
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 4.40093i − 0.171436i −0.996319 0.0857179i \(-0.972682\pi\)
0.996319 0.0857179i \(-0.0273184\pi\)
\(660\) 0 0
\(661\) 13.8183i 0.537470i 0.963214 + 0.268735i \(0.0866056\pi\)
−0.963214 + 0.268735i \(0.913394\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.17180 −0.161776
\(666\) 0 0
\(667\) − 0.235190i − 0.00910658i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20.9740 −0.809693
\(672\) 0 0
\(673\) 28.7125 1.10679 0.553393 0.832921i \(-0.313333\pi\)
0.553393 + 0.832921i \(0.313333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.5789i 1.17524i 0.809136 + 0.587621i \(0.199936\pi\)
−0.809136 + 0.587621i \(0.800064\pi\)
\(678\) 0 0
\(679\) 0.107694 0.00413290
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.0994i 0.386443i 0.981155 + 0.193221i \(0.0618936\pi\)
−0.981155 + 0.193221i \(0.938106\pi\)
\(684\) 0 0
\(685\) − 31.4521i − 1.20172i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.98208 −0.189802
\(690\) 0 0
\(691\) − 1.93187i − 0.0734916i −0.999325 0.0367458i \(-0.988301\pi\)
0.999325 0.0367458i \(-0.0116992\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.7374 −0.824545
\(696\) 0 0
\(697\) −1.93226 −0.0731895
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.6511i 0.893288i 0.894712 + 0.446644i \(0.147381\pi\)
−0.894712 + 0.446644i \(0.852619\pi\)
\(702\) 0 0
\(703\) 9.33598 0.352113
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 14.8524i − 0.558584i
\(708\) 0 0
\(709\) − 20.8417i − 0.782728i −0.920236 0.391364i \(-0.872003\pi\)
0.920236 0.391364i \(-0.127997\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.40547 0.0526354
\(714\) 0 0
\(715\) 4.25893i 0.159275i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.81902 0.105132 0.0525658 0.998617i \(-0.483260\pi\)
0.0525658 + 0.998617i \(0.483260\pi\)
\(720\) 0 0
\(721\) −16.1461 −0.601311
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.02621i 0.186669i
\(726\) 0 0
\(727\) 14.6918 0.544887 0.272444 0.962172i \(-0.412168\pi\)
0.272444 + 0.962172i \(0.412168\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 18.6082i 0.688248i
\(732\) 0 0
\(733\) − 45.7679i − 1.69048i −0.534389 0.845239i \(-0.679458\pi\)
0.534389 0.845239i \(-0.320542\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.1482 −0.852673
\(738\) 0 0
\(739\) − 37.7771i − 1.38965i −0.719178 0.694826i \(-0.755481\pi\)
0.719178 0.694826i \(-0.244519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.38675 −0.234307 −0.117154 0.993114i \(-0.537377\pi\)
−0.117154 + 0.993114i \(0.537377\pi\)
\(744\) 0 0
\(745\) −9.16812 −0.335894
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 15.6131i − 0.570490i
\(750\) 0 0
\(751\) −52.5372 −1.91711 −0.958554 0.284910i \(-0.908036\pi\)
−0.958554 + 0.284910i \(0.908036\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 47.5388i − 1.73012i
\(756\) 0 0
\(757\) 3.80676i 0.138359i 0.997604 + 0.0691794i \(0.0220381\pi\)
−0.997604 + 0.0691794i \(0.977962\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.9307 0.939986 0.469993 0.882670i \(-0.344256\pi\)
0.469993 + 0.882670i \(0.344256\pi\)
\(762\) 0 0
\(763\) 4.15829i 0.150540i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.01686 −0.217256
\(768\) 0 0
\(769\) −42.2680 −1.52422 −0.762112 0.647445i \(-0.775838\pi\)
−0.762112 + 0.647445i \(0.775838\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.4481i 0.483696i 0.970314 + 0.241848i \(0.0777536\pi\)
−0.970314 + 0.241848i \(0.922246\pi\)
\(774\) 0 0
\(775\) −30.0362 −1.07893
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.382445i 0.0137025i
\(780\) 0 0
\(781\) 16.9790i 0.607555i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 54.7804 1.95520
\(786\) 0 0
\(787\) 1.40104i 0.0499418i 0.999688 + 0.0249709i \(0.00794932\pi\)
−0.999688 + 0.0249709i \(0.992051\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.4682 0.656654
\(792\) 0 0
\(793\) −6.52623 −0.231753
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.3605i 0.685783i 0.939375 + 0.342891i \(0.111406\pi\)
−0.939375 + 0.342891i \(0.888594\pi\)
\(798\) 0 0
\(799\) 40.7266 1.44080
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 5.99950i − 0.211718i
\(804\) 0 0
\(805\) 0.551270i 0.0194297i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.4386 0.507633 0.253816 0.967252i \(-0.418314\pi\)
0.253816 + 0.967252i \(0.418314\pi\)
\(810\) 0 0
\(811\) 42.1307i 1.47941i 0.672932 + 0.739705i \(0.265035\pi\)
−0.672932 + 0.739705i \(0.734965\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 65.9234 2.30920
\(816\) 0 0
\(817\) 3.68305 0.128854
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.21315i 0.251741i 0.992047 + 0.125870i \(0.0401723\pi\)
−0.992047 + 0.125870i \(0.959828\pi\)
\(822\) 0 0
\(823\) 6.47104 0.225566 0.112783 0.993620i \(-0.464023\pi\)
0.112783 + 0.993620i \(0.464023\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 55.8363i 1.94162i 0.239850 + 0.970810i \(0.422902\pi\)
−0.239850 + 0.970810i \(0.577098\pi\)
\(828\) 0 0
\(829\) 14.1807i 0.492517i 0.969204 + 0.246258i \(0.0792011\pi\)
−0.969204 + 0.246258i \(0.920799\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.04931 −0.244244
\(834\) 0 0
\(835\) − 30.6925i − 1.06216i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.4229 1.08484 0.542420 0.840108i \(-0.317508\pi\)
0.542420 + 0.840108i \(0.317508\pi\)
\(840\) 0 0
\(841\) 27.3728 0.943888
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 37.5449i − 1.29159i
\(846\) 0 0
\(847\) −6.42230 −0.220673
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.23367i − 0.0422898i
\(852\) 0 0
\(853\) − 23.3929i − 0.800957i −0.916306 0.400479i \(-0.868844\pi\)
0.916306 0.400479i \(-0.131156\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0033 0.819936 0.409968 0.912100i \(-0.365540\pi\)
0.409968 + 0.912100i \(0.365540\pi\)
\(858\) 0 0
\(859\) 41.5626i 1.41810i 0.705160 + 0.709048i \(0.250875\pi\)
−0.705160 + 0.709048i \(0.749125\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.43323 0.116868 0.0584342 0.998291i \(-0.481389\pi\)
0.0584342 + 0.998291i \(0.481389\pi\)
\(864\) 0 0
\(865\) −46.4151 −1.57816
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 17.0812i − 0.579438i
\(870\) 0 0
\(871\) −7.20273 −0.244055
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.16893i 0.107129i
\(876\) 0 0
\(877\) − 47.9141i − 1.61794i −0.587847 0.808972i \(-0.700024\pi\)
0.587847 0.808972i \(-0.299976\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −25.7071 −0.866095 −0.433047 0.901371i \(-0.642562\pi\)
−0.433047 + 0.901371i \(0.642562\pi\)
\(882\) 0 0
\(883\) 41.7112i 1.40369i 0.712328 + 0.701847i \(0.247641\pi\)
−0.712328 + 0.701847i \(0.752359\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.6475 −1.09620 −0.548099 0.836414i \(-0.684648\pi\)
−0.548099 + 0.836414i \(0.684648\pi\)
\(888\) 0 0
\(889\) −4.37444 −0.146714
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 8.06089i − 0.269747i
\(894\) 0 0
\(895\) −51.4365 −1.71933
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 9.72428i − 0.324323i
\(900\) 0 0
\(901\) − 52.7537i − 1.75748i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 31.4385 1.04505
\(906\) 0 0
\(907\) 22.4610i 0.745804i 0.927871 + 0.372902i \(0.121637\pi\)
−0.927871 + 0.372902i \(0.878363\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.7360 −0.687015 −0.343507 0.939150i \(-0.611615\pi\)
−0.343507 + 0.939150i \(0.611615\pi\)
\(912\) 0 0
\(913\) 14.4661 0.478757
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5.76264i − 0.190299i
\(918\) 0 0
\(919\) 42.9827 1.41787 0.708935 0.705274i \(-0.249176\pi\)
0.708935 + 0.705274i \(0.249176\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.28313i 0.173896i
\(924\) 0 0
\(925\) 26.3647i 0.866865i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.62200 −0.118834 −0.0594170 0.998233i \(-0.518924\pi\)
−0.0594170 + 0.998233i \(0.518924\pi\)
\(930\) 0 0
\(931\) 1.39525i 0.0457274i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −45.0965 −1.47481
\(936\) 0 0
\(937\) 21.8189 0.712793 0.356396 0.934335i \(-0.384005\pi\)
0.356396 + 0.934335i \(0.384005\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 16.8639i − 0.549747i −0.961480 0.274873i \(-0.911364\pi\)
0.961480 0.274873i \(-0.0886359\pi\)
\(942\) 0 0
\(943\) 0.0505370 0.00164571
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.5568i − 0.603016i −0.953464 0.301508i \(-0.902510\pi\)
0.953464 0.301508i \(-0.0974900\pi\)
\(948\) 0 0
\(949\) − 1.86679i − 0.0605985i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.6643 −1.18767 −0.593836 0.804586i \(-0.702387\pi\)
−0.593836 + 0.804586i \(0.702387\pi\)
\(954\) 0 0
\(955\) − 43.0242i − 1.39223i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.5191 −0.339678
\(960\) 0 0
\(961\) 27.1114 0.874562
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 33.9604i − 1.09322i
\(966\) 0 0
\(967\) 11.1440 0.358367 0.179184 0.983816i \(-0.442654\pi\)
0.179184 + 0.983816i \(0.442654\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.98773i 0.127972i 0.997951 + 0.0639862i \(0.0203814\pi\)
−0.997951 + 0.0639862i \(0.979619\pi\)
\(972\) 0 0
\(973\) 7.27000i 0.233065i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.7959 1.04923 0.524617 0.851338i \(-0.324209\pi\)
0.524617 + 0.851338i \(0.324209\pi\)
\(978\) 0 0
\(979\) − 35.8481i − 1.14571i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 26.0499 0.830864 0.415432 0.909624i \(-0.363630\pi\)
0.415432 + 0.909624i \(0.363630\pi\)
\(984\) 0 0
\(985\) −19.2696 −0.613980
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 0.486686i − 0.0154757i
\(990\) 0 0
\(991\) 17.5053 0.556075 0.278037 0.960570i \(-0.410316\pi\)
0.278037 + 0.960570i \(0.410316\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 19.3769i − 0.614289i
\(996\) 0 0
\(997\) 29.5008i 0.934300i 0.884178 + 0.467150i \(0.154719\pi\)
−0.884178 + 0.467150i \(0.845281\pi\)
\(998\) 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.3025.6 24
3.2 odd 2 inner 6048.2.c.f.3025.20 24
4.3 odd 2 1512.2.c.g.757.7 24
8.3 odd 2 1512.2.c.g.757.8 yes 24
8.5 even 2 inner 6048.2.c.f.3025.19 24
12.11 even 2 1512.2.c.g.757.18 yes 24
24.5 odd 2 inner 6048.2.c.f.3025.5 24
24.11 even 2 1512.2.c.g.757.17 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.7 24 4.3 odd 2
1512.2.c.g.757.8 yes 24 8.3 odd 2
1512.2.c.g.757.17 yes 24 24.11 even 2
1512.2.c.g.757.18 yes 24 12.11 even 2
6048.2.c.f.3025.5 24 24.5 odd 2 inner
6048.2.c.f.3025.6 24 1.1 even 1 trivial
6048.2.c.f.3025.19 24 8.5 even 2 inner
6048.2.c.f.3025.20 24 3.2 odd 2 inner