Properties

Label 6048.2.j.c.5615.24
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
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 5615.24
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.23

$q$-expansion

\(f(q)\) \(=\) \(q+1.17792 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+1.17792 q^{5} -1.00000i q^{7} +2.90595i q^{11} +1.05317i q^{13} -6.48837i q^{17} +3.11597 q^{19} +0.812516 q^{23} -3.61251 q^{25} -4.03607 q^{29} -0.108779i q^{31} -1.17792i q^{35} -9.97034i q^{37} -9.44298i q^{41} -10.3446 q^{43} +4.03126 q^{47} -1.00000 q^{49} -2.97772 q^{53} +3.42296i q^{55} +0.868874i q^{59} +5.33629i q^{61} +1.24054i q^{65} +3.97987 q^{67} -0.844302 q^{71} +3.59513 q^{73} +2.90595 q^{77} -9.48320i q^{79} -2.71280i q^{83} -7.64276i q^{85} +2.22575i q^{89} +1.05317 q^{91} +3.67035 q^{95} -3.93019 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

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
\(5\) 1.17792 0.526780 0.263390 0.964689i \(-0.415159\pi\)
0.263390 + 0.964689i \(0.415159\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.90595i 0.876176i 0.898932 + 0.438088i \(0.144344\pi\)
−0.898932 + 0.438088i \(0.855656\pi\)
\(12\) 0 0
\(13\) 1.05317i 0.292096i 0.989278 + 0.146048i \(0.0466554\pi\)
−0.989278 + 0.146048i \(0.953345\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.48837i − 1.57366i −0.617169 0.786830i \(-0.711721\pi\)
0.617169 0.786830i \(-0.288279\pi\)
\(18\) 0 0
\(19\) 3.11597 0.714853 0.357426 0.933941i \(-0.383654\pi\)
0.357426 + 0.933941i \(0.383654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.812516 0.169421 0.0847107 0.996406i \(-0.473003\pi\)
0.0847107 + 0.996406i \(0.473003\pi\)
\(24\) 0 0
\(25\) −3.61251 −0.722503
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.03607 −0.749480 −0.374740 0.927130i \(-0.622268\pi\)
−0.374740 + 0.927130i \(0.622268\pi\)
\(30\) 0 0
\(31\) − 0.108779i − 0.0195372i −0.999952 0.00976861i \(-0.996891\pi\)
0.999952 0.00976861i \(-0.00310949\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.17792i − 0.199104i
\(36\) 0 0
\(37\) − 9.97034i − 1.63911i −0.572998 0.819557i \(-0.694220\pi\)
0.572998 0.819557i \(-0.305780\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.44298i − 1.47475i −0.675486 0.737373i \(-0.736066\pi\)
0.675486 0.737373i \(-0.263934\pi\)
\(42\) 0 0
\(43\) −10.3446 −1.57754 −0.788770 0.614688i \(-0.789282\pi\)
−0.788770 + 0.614688i \(0.789282\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.03126 0.588019 0.294010 0.955802i \(-0.405010\pi\)
0.294010 + 0.955802i \(0.405010\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.97772 −0.409022 −0.204511 0.978864i \(-0.565560\pi\)
−0.204511 + 0.978864i \(0.565560\pi\)
\(54\) 0 0
\(55\) 3.42296i 0.461552i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.868874i 0.113118i 0.998399 + 0.0565589i \(0.0180129\pi\)
−0.998399 + 0.0565589i \(0.981987\pi\)
\(60\) 0 0
\(61\) 5.33629i 0.683242i 0.939838 + 0.341621i \(0.110976\pi\)
−0.939838 + 0.341621i \(0.889024\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.24054i 0.153870i
\(66\) 0 0
\(67\) 3.97987 0.486218 0.243109 0.969999i \(-0.421833\pi\)
0.243109 + 0.969999i \(0.421833\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.844302 −0.100200 −0.0501001 0.998744i \(-0.515954\pi\)
−0.0501001 + 0.998744i \(0.515954\pi\)
\(72\) 0 0
\(73\) 3.59513 0.420778 0.210389 0.977618i \(-0.432527\pi\)
0.210389 + 0.977618i \(0.432527\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.90595 0.331164
\(78\) 0 0
\(79\) − 9.48320i − 1.06694i −0.845818 0.533472i \(-0.820887\pi\)
0.845818 0.533472i \(-0.179113\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 2.71280i − 0.297769i −0.988855 0.148884i \(-0.952432\pi\)
0.988855 0.148884i \(-0.0475682\pi\)
\(84\) 0 0
\(85\) − 7.64276i − 0.828973i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.22575i 0.235930i 0.993018 + 0.117965i \(0.0376370\pi\)
−0.993018 + 0.117965i \(0.962363\pi\)
\(90\) 0 0
\(91\) 1.05317 0.110402
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.67035 0.376570
\(96\) 0 0
\(97\) −3.93019 −0.399050 −0.199525 0.979893i \(-0.563940\pi\)
−0.199525 + 0.979893i \(0.563940\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0700 1.79803 0.899016 0.437915i \(-0.144283\pi\)
0.899016 + 0.437915i \(0.144283\pi\)
\(102\) 0 0
\(103\) 17.8616i 1.75995i 0.475017 + 0.879977i \(0.342442\pi\)
−0.475017 + 0.879977i \(0.657558\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.67720i − 0.742183i −0.928596 0.371091i \(-0.878984\pi\)
0.928596 0.371091i \(-0.121016\pi\)
\(108\) 0 0
\(109\) − 12.6991i − 1.21635i −0.793801 0.608177i \(-0.791901\pi\)
0.793801 0.608177i \(-0.208099\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 15.4017i − 1.44887i −0.689342 0.724437i \(-0.742100\pi\)
0.689342 0.724437i \(-0.257900\pi\)
\(114\) 0 0
\(115\) 0.957076 0.0892478
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.48837 −0.594788
\(120\) 0 0
\(121\) 2.55546 0.232315
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.1448 −0.907380
\(126\) 0 0
\(127\) 17.1514i 1.52194i 0.648787 + 0.760970i \(0.275276\pi\)
−0.648787 + 0.760970i \(0.724724\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 10.5118i − 0.918421i −0.888328 0.459210i \(-0.848132\pi\)
0.888328 0.459210i \(-0.151868\pi\)
\(132\) 0 0
\(133\) − 3.11597i − 0.270189i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 13.6268i − 1.16422i −0.813110 0.582110i \(-0.802227\pi\)
0.813110 0.582110i \(-0.197773\pi\)
\(138\) 0 0
\(139\) 5.37620 0.456003 0.228002 0.973661i \(-0.426781\pi\)
0.228002 + 0.973661i \(0.426781\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.06045 −0.255928
\(144\) 0 0
\(145\) −4.75416 −0.394811
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.3187 −1.41880 −0.709401 0.704805i \(-0.751035\pi\)
−0.709401 + 0.704805i \(0.751035\pi\)
\(150\) 0 0
\(151\) 4.50773i 0.366834i 0.983035 + 0.183417i \(0.0587158\pi\)
−0.983035 + 0.183417i \(0.941284\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.128132i − 0.0102918i
\(156\) 0 0
\(157\) − 8.84696i − 0.706064i −0.935611 0.353032i \(-0.885151\pi\)
0.935611 0.353032i \(-0.114849\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 0.812516i − 0.0640352i
\(162\) 0 0
\(163\) 8.38238 0.656558 0.328279 0.944581i \(-0.393531\pi\)
0.328279 + 0.944581i \(0.393531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.93563 −0.691460 −0.345730 0.938334i \(-0.612369\pi\)
−0.345730 + 0.938334i \(0.612369\pi\)
\(168\) 0 0
\(169\) 11.8908 0.914680
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.1612 −1.07666 −0.538328 0.842736i \(-0.680944\pi\)
−0.538328 + 0.842736i \(0.680944\pi\)
\(174\) 0 0
\(175\) 3.61251i 0.273080i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 20.3695i − 1.52249i −0.648467 0.761243i \(-0.724589\pi\)
0.648467 0.761243i \(-0.275411\pi\)
\(180\) 0 0
\(181\) − 18.2659i − 1.35770i −0.734279 0.678848i \(-0.762480\pi\)
0.734279 0.678848i \(-0.237520\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 11.7442i − 0.863453i
\(186\) 0 0
\(187\) 18.8549 1.37880
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.83849 0.567173 0.283587 0.958947i \(-0.408476\pi\)
0.283587 + 0.958947i \(0.408476\pi\)
\(192\) 0 0
\(193\) −13.0241 −0.937498 −0.468749 0.883331i \(-0.655295\pi\)
−0.468749 + 0.883331i \(0.655295\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.9576 1.06568 0.532842 0.846215i \(-0.321124\pi\)
0.532842 + 0.846215i \(0.321124\pi\)
\(198\) 0 0
\(199\) 16.9147i 1.19905i 0.800356 + 0.599525i \(0.204644\pi\)
−0.800356 + 0.599525i \(0.795356\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.03607i 0.283277i
\(204\) 0 0
\(205\) − 11.1230i − 0.776867i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.05485i 0.626337i
\(210\) 0 0
\(211\) −5.68386 −0.391293 −0.195647 0.980674i \(-0.562681\pi\)
−0.195647 + 0.980674i \(0.562681\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.1851 −0.831017
\(216\) 0 0
\(217\) −0.108779 −0.00738437
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.83333 0.459660
\(222\) 0 0
\(223\) − 3.43963i − 0.230335i −0.993346 0.115167i \(-0.963260\pi\)
0.993346 0.115167i \(-0.0367404\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.86522i − 0.588406i −0.955743 0.294203i \(-0.904946\pi\)
0.955743 0.294203i \(-0.0950541\pi\)
\(228\) 0 0
\(229\) 17.3653i 1.14753i 0.819019 + 0.573767i \(0.194518\pi\)
−0.819019 + 0.573767i \(0.805482\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 9.99149i − 0.654564i −0.944927 0.327282i \(-0.893867\pi\)
0.944927 0.327282i \(-0.106133\pi\)
\(234\) 0 0
\(235\) 4.74848 0.309757
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.5708 1.58935 0.794677 0.607032i \(-0.207640\pi\)
0.794677 + 0.607032i \(0.207640\pi\)
\(240\) 0 0
\(241\) 23.1786 1.49306 0.746532 0.665349i \(-0.231717\pi\)
0.746532 + 0.665349i \(0.231717\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.17792 −0.0752543
\(246\) 0 0
\(247\) 3.28164i 0.208806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 21.3670i − 1.34868i −0.738423 0.674338i \(-0.764429\pi\)
0.738423 0.674338i \(-0.235571\pi\)
\(252\) 0 0
\(253\) 2.36113i 0.148443i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 29.5594i − 1.84386i −0.387352 0.921932i \(-0.626610\pi\)
0.387352 0.921932i \(-0.373390\pi\)
\(258\) 0 0
\(259\) −9.97034 −0.619527
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.8001 −0.974277 −0.487139 0.873325i \(-0.661959\pi\)
−0.487139 + 0.873325i \(0.661959\pi\)
\(264\) 0 0
\(265\) −3.50751 −0.215464
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.41502 0.391131 0.195565 0.980691i \(-0.437346\pi\)
0.195565 + 0.980691i \(0.437346\pi\)
\(270\) 0 0
\(271\) − 0.0927686i − 0.00563529i −0.999996 0.00281765i \(-0.999103\pi\)
0.999996 0.00281765i \(-0.000896886\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 10.4978i − 0.633040i
\(276\) 0 0
\(277\) 19.7900i 1.18907i 0.804071 + 0.594534i \(0.202663\pi\)
−0.804071 + 0.594534i \(0.797337\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.4404i 0.861443i 0.902485 + 0.430721i \(0.141741\pi\)
−0.902485 + 0.430721i \(0.858259\pi\)
\(282\) 0 0
\(283\) −20.3759 −1.21122 −0.605612 0.795760i \(-0.707071\pi\)
−0.605612 + 0.795760i \(0.707071\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.44298 −0.557402
\(288\) 0 0
\(289\) −25.0989 −1.47641
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.2919 −0.601262 −0.300631 0.953741i \(-0.597197\pi\)
−0.300631 + 0.953741i \(0.597197\pi\)
\(294\) 0 0
\(295\) 1.02346i 0.0595882i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.855715i 0.0494873i
\(300\) 0 0
\(301\) 10.3446i 0.596254i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.28571i 0.359919i
\(306\) 0 0
\(307\) −19.8799 −1.13461 −0.567304 0.823508i \(-0.692014\pi\)
−0.567304 + 0.823508i \(0.692014\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.94464 −0.450499 −0.225250 0.974301i \(-0.572320\pi\)
−0.225250 + 0.974301i \(0.572320\pi\)
\(312\) 0 0
\(313\) −28.2451 −1.59651 −0.798254 0.602321i \(-0.794243\pi\)
−0.798254 + 0.602321i \(0.794243\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.8609 −0.778506 −0.389253 0.921131i \(-0.627267\pi\)
−0.389253 + 0.921131i \(0.627267\pi\)
\(318\) 0 0
\(319\) − 11.7286i − 0.656677i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 20.2176i − 1.12494i
\(324\) 0 0
\(325\) − 3.80458i − 0.211040i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 4.03126i − 0.222250i
\(330\) 0 0
\(331\) 14.5450 0.799468 0.399734 0.916631i \(-0.369103\pi\)
0.399734 + 0.916631i \(0.369103\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.68795 0.256130
\(336\) 0 0
\(337\) 5.26170 0.286623 0.143312 0.989678i \(-0.454225\pi\)
0.143312 + 0.989678i \(0.454225\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.316105 0.0171181
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 15.5549i − 0.835029i −0.908670 0.417514i \(-0.862901\pi\)
0.908670 0.417514i \(-0.137099\pi\)
\(348\) 0 0
\(349\) 17.1607i 0.918594i 0.888283 + 0.459297i \(0.151899\pi\)
−0.888283 + 0.459297i \(0.848101\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.5438i 0.827314i 0.910433 + 0.413657i \(0.135749\pi\)
−0.910433 + 0.413657i \(0.864251\pi\)
\(354\) 0 0
\(355\) −0.994517 −0.0527835
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.9471 0.788876 0.394438 0.918923i \(-0.370939\pi\)
0.394438 + 0.918923i \(0.370939\pi\)
\(360\) 0 0
\(361\) −9.29073 −0.488986
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.23476 0.221657
\(366\) 0 0
\(367\) 2.86184i 0.149387i 0.997207 + 0.0746934i \(0.0237978\pi\)
−0.997207 + 0.0746934i \(0.976202\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.97772i 0.154596i
\(372\) 0 0
\(373\) 20.7327i 1.07350i 0.843743 + 0.536748i \(0.180347\pi\)
−0.843743 + 0.536748i \(0.819653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.25066i − 0.218920i
\(378\) 0 0
\(379\) −28.1561 −1.44628 −0.723141 0.690700i \(-0.757302\pi\)
−0.723141 + 0.690700i \(0.757302\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.9733 −1.53156 −0.765781 0.643101i \(-0.777648\pi\)
−0.765781 + 0.643101i \(0.777648\pi\)
\(384\) 0 0
\(385\) 3.42296 0.174450
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.4931 1.49536 0.747679 0.664060i \(-0.231168\pi\)
0.747679 + 0.664060i \(0.231168\pi\)
\(390\) 0 0
\(391\) − 5.27190i − 0.266612i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 11.1704i − 0.562045i
\(396\) 0 0
\(397\) − 22.0845i − 1.10839i −0.832386 0.554196i \(-0.813026\pi\)
0.832386 0.554196i \(-0.186974\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 7.42054i − 0.370564i −0.982685 0.185282i \(-0.940680\pi\)
0.982685 0.185282i \(-0.0593198\pi\)
\(402\) 0 0
\(403\) 0.114562 0.00570674
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.9733 1.43615
\(408\) 0 0
\(409\) 37.3983 1.84923 0.924615 0.380904i \(-0.124387\pi\)
0.924615 + 0.380904i \(0.124387\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.868874 0.0427545
\(414\) 0 0
\(415\) − 3.19546i − 0.156859i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 8.55960i − 0.418164i −0.977898 0.209082i \(-0.932952\pi\)
0.977898 0.209082i \(-0.0670475\pi\)
\(420\) 0 0
\(421\) − 6.56688i − 0.320050i −0.987113 0.160025i \(-0.948842\pi\)
0.987113 0.160025i \(-0.0511575\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 23.4393i 1.13697i
\(426\) 0 0
\(427\) 5.33629 0.258241
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.61014 −0.270231 −0.135116 0.990830i \(-0.543141\pi\)
−0.135116 + 0.990830i \(0.543141\pi\)
\(432\) 0 0
\(433\) 7.96675 0.382857 0.191429 0.981507i \(-0.438688\pi\)
0.191429 + 0.981507i \(0.438688\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.53178 0.121111
\(438\) 0 0
\(439\) − 33.9176i − 1.61880i −0.587258 0.809400i \(-0.699792\pi\)
0.587258 0.809400i \(-0.300208\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 6.31862i − 0.300207i −0.988670 0.150103i \(-0.952039\pi\)
0.988670 0.150103i \(-0.0479606\pi\)
\(444\) 0 0
\(445\) 2.62175i 0.124283i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 11.9562i − 0.564246i −0.959378 0.282123i \(-0.908961\pi\)
0.959378 0.282123i \(-0.0910387\pi\)
\(450\) 0 0
\(451\) 27.4408 1.29214
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.24054 0.0581575
\(456\) 0 0
\(457\) 10.7774 0.504144 0.252072 0.967708i \(-0.418888\pi\)
0.252072 + 0.967708i \(0.418888\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.36887 −0.110329 −0.0551647 0.998477i \(-0.517568\pi\)
−0.0551647 + 0.998477i \(0.517568\pi\)
\(462\) 0 0
\(463\) − 38.4537i − 1.78709i −0.448969 0.893547i \(-0.648209\pi\)
0.448969 0.893547i \(-0.351791\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 5.09381i − 0.235713i −0.993031 0.117857i \(-0.962398\pi\)
0.993031 0.117857i \(-0.0376023\pi\)
\(468\) 0 0
\(469\) − 3.97987i − 0.183773i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 30.0609i − 1.38220i
\(474\) 0 0
\(475\) −11.2565 −0.516483
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.8166 −0.951137 −0.475568 0.879679i \(-0.657758\pi\)
−0.475568 + 0.879679i \(0.657758\pi\)
\(480\) 0 0
\(481\) 10.5004 0.478778
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.62943 −0.210212
\(486\) 0 0
\(487\) 26.1716i 1.18595i 0.805222 + 0.592973i \(0.202046\pi\)
−0.805222 + 0.592973i \(0.797954\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 27.7686i − 1.25318i −0.779349 0.626590i \(-0.784450\pi\)
0.779349 0.626590i \(-0.215550\pi\)
\(492\) 0 0
\(493\) 26.1875i 1.17943i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.844302i 0.0378721i
\(498\) 0 0
\(499\) 13.7781 0.616791 0.308395 0.951258i \(-0.400208\pi\)
0.308395 + 0.951258i \(0.400208\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.5875 1.45300 0.726502 0.687165i \(-0.241145\pi\)
0.726502 + 0.687165i \(0.241145\pi\)
\(504\) 0 0
\(505\) 21.2850 0.947168
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.1161 0.758657 0.379328 0.925262i \(-0.376155\pi\)
0.379328 + 0.925262i \(0.376155\pi\)
\(510\) 0 0
\(511\) − 3.59513i − 0.159039i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.0394i 0.927109i
\(516\) 0 0
\(517\) 11.7146i 0.515209i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 22.6308i − 0.991471i −0.868473 0.495736i \(-0.834898\pi\)
0.868473 0.495736i \(-0.165102\pi\)
\(522\) 0 0
\(523\) 19.0252 0.831914 0.415957 0.909384i \(-0.363447\pi\)
0.415957 + 0.909384i \(0.363447\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.705796 −0.0307450
\(528\) 0 0
\(529\) −22.3398 −0.971296
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.94503 0.430767
\(534\) 0 0
\(535\) − 9.04310i − 0.390967i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.90595i − 0.125168i
\(540\) 0 0
\(541\) 17.3996i 0.748066i 0.927415 + 0.374033i \(0.122025\pi\)
−0.927415 + 0.374033i \(0.877975\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 14.9585i − 0.640751i
\(546\) 0 0
\(547\) −4.96571 −0.212318 −0.106159 0.994349i \(-0.533855\pi\)
−0.106159 + 0.994349i \(0.533855\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.5763 −0.535768
\(552\) 0 0
\(553\) −9.48320 −0.403267
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.67400 0.240415 0.120208 0.992749i \(-0.461644\pi\)
0.120208 + 0.992749i \(0.461644\pi\)
\(558\) 0 0
\(559\) − 10.8946i − 0.460793i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 25.4752i − 1.07365i −0.843693 0.536826i \(-0.819623\pi\)
0.843693 0.536826i \(-0.180377\pi\)
\(564\) 0 0
\(565\) − 18.1420i − 0.763238i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.5120i 0.817986i 0.912537 + 0.408993i \(0.134120\pi\)
−0.912537 + 0.408993i \(0.865880\pi\)
\(570\) 0 0
\(571\) 38.0268 1.59137 0.795686 0.605709i \(-0.207111\pi\)
0.795686 + 0.605709i \(0.207111\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.93522 −0.122407
\(576\) 0 0
\(577\) 5.32632 0.221738 0.110869 0.993835i \(-0.464637\pi\)
0.110869 + 0.993835i \(0.464637\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.71280 −0.112546
\(582\) 0 0
\(583\) − 8.65311i − 0.358375i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.6946i 1.06053i 0.847832 + 0.530265i \(0.177908\pi\)
−0.847832 + 0.530265i \(0.822092\pi\)
\(588\) 0 0
\(589\) − 0.338951i − 0.0139662i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 35.9249i − 1.47526i −0.675205 0.737630i \(-0.735945\pi\)
0.675205 0.737630i \(-0.264055\pi\)
\(594\) 0 0
\(595\) −7.64276 −0.313322
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.08090 0.330177 0.165088 0.986279i \(-0.447209\pi\)
0.165088 + 0.986279i \(0.447209\pi\)
\(600\) 0 0
\(601\) 39.0492 1.59285 0.796425 0.604738i \(-0.206722\pi\)
0.796425 + 0.604738i \(0.206722\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.01012 0.122379
\(606\) 0 0
\(607\) − 39.0636i − 1.58554i −0.609519 0.792772i \(-0.708637\pi\)
0.609519 0.792772i \(-0.291363\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.24559i 0.171758i
\(612\) 0 0
\(613\) 27.0877i 1.09406i 0.837112 + 0.547031i \(0.184242\pi\)
−0.837112 + 0.547031i \(0.815758\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.2143i 0.411213i 0.978635 + 0.205607i \(0.0659167\pi\)
−0.978635 + 0.205607i \(0.934083\pi\)
\(618\) 0 0
\(619\) −28.5400 −1.14712 −0.573560 0.819164i \(-0.694438\pi\)
−0.573560 + 0.819164i \(0.694438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.22575 0.0891730
\(624\) 0 0
\(625\) 6.11281 0.244512
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −64.6912 −2.57941
\(630\) 0 0
\(631\) 17.9037i 0.712734i 0.934346 + 0.356367i \(0.115985\pi\)
−0.934346 + 0.356367i \(0.884015\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.2029i 0.801728i
\(636\) 0 0
\(637\) − 1.05317i − 0.0417280i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 36.5576i 1.44394i 0.691925 + 0.721969i \(0.256763\pi\)
−0.691925 + 0.721969i \(0.743237\pi\)
\(642\) 0 0
\(643\) −28.5306 −1.12514 −0.562568 0.826751i \(-0.690187\pi\)
−0.562568 + 0.826751i \(0.690187\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.0326 0.984134 0.492067 0.870557i \(-0.336242\pi\)
0.492067 + 0.870557i \(0.336242\pi\)
\(648\) 0 0
\(649\) −2.52490 −0.0991112
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −43.1196 −1.68740 −0.843699 0.536816i \(-0.819627\pi\)
−0.843699 + 0.536816i \(0.819627\pi\)
\(654\) 0 0
\(655\) − 12.3820i − 0.483806i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.36352i 0.325796i 0.986643 + 0.162898i \(0.0520842\pi\)
−0.986643 + 0.162898i \(0.947916\pi\)
\(660\) 0 0
\(661\) − 8.48754i − 0.330127i −0.986283 0.165064i \(-0.947217\pi\)
0.986283 0.165064i \(-0.0527829\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.67035i − 0.142330i
\(666\) 0 0
\(667\) −3.27937 −0.126978
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.5070 −0.598641
\(672\) 0 0
\(673\) −25.4287 −0.980206 −0.490103 0.871664i \(-0.663041\pi\)
−0.490103 + 0.871664i \(0.663041\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.0820 −0.656515 −0.328258 0.944588i \(-0.606461\pi\)
−0.328258 + 0.944588i \(0.606461\pi\)
\(678\) 0 0
\(679\) 3.93019i 0.150827i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0552i 1.68573i 0.538128 + 0.842863i \(0.319132\pi\)
−0.538128 + 0.842863i \(0.680868\pi\)
\(684\) 0 0
\(685\) − 16.0513i − 0.613288i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 3.13604i − 0.119474i
\(690\) 0 0
\(691\) 18.3483 0.698002 0.349001 0.937122i \(-0.386521\pi\)
0.349001 + 0.937122i \(0.386521\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.33271 0.240213
\(696\) 0 0
\(697\) −61.2695 −2.32075
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.6724 1.27179 0.635894 0.771777i \(-0.280632\pi\)
0.635894 + 0.771777i \(0.280632\pi\)
\(702\) 0 0
\(703\) − 31.0673i − 1.17172i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 18.0700i − 0.679592i
\(708\) 0 0
\(709\) 15.9147i 0.597689i 0.954302 + 0.298845i \(0.0966013\pi\)
−0.954302 + 0.298845i \(0.903399\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 0.0883844i − 0.00331002i
\(714\) 0 0
\(715\) −3.60495 −0.134818
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.6441 −1.29201 −0.646004 0.763334i \(-0.723561\pi\)
−0.646004 + 0.763334i \(0.723561\pi\)
\(720\) 0 0
\(721\) 17.8616 0.665200
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.5804 0.541501
\(726\) 0 0
\(727\) 38.6477i 1.43336i 0.697400 + 0.716682i \(0.254340\pi\)
−0.697400 + 0.716682i \(0.745660\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 67.1197i 2.48251i
\(732\) 0 0
\(733\) − 4.45768i − 0.164648i −0.996606 0.0823240i \(-0.973766\pi\)
0.996606 0.0823240i \(-0.0262342\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.5653i 0.426013i
\(738\) 0 0
\(739\) 32.0619 1.17942 0.589709 0.807616i \(-0.299243\pi\)
0.589709 + 0.807616i \(0.299243\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.9201 1.64796 0.823979 0.566620i \(-0.191749\pi\)
0.823979 + 0.566620i \(0.191749\pi\)
\(744\) 0 0
\(745\) −20.4000 −0.747397
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.67720 −0.280519
\(750\) 0 0
\(751\) 20.4697i 0.746951i 0.927640 + 0.373475i \(0.121834\pi\)
−0.927640 + 0.373475i \(0.878166\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.30973i 0.193241i
\(756\) 0 0
\(757\) − 11.4003i − 0.414352i −0.978304 0.207176i \(-0.933573\pi\)
0.978304 0.207176i \(-0.0664273\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.5142i 1.17864i 0.807901 + 0.589319i \(0.200604\pi\)
−0.807901 + 0.589319i \(0.799396\pi\)
\(762\) 0 0
\(763\) −12.6991 −0.459739
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.915070 −0.0330413
\(768\) 0 0
\(769\) 25.1134 0.905614 0.452807 0.891609i \(-0.350423\pi\)
0.452807 + 0.891609i \(0.350423\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.4890 1.52822 0.764112 0.645083i \(-0.223177\pi\)
0.764112 + 0.645083i \(0.223177\pi\)
\(774\) 0 0
\(775\) 0.392964i 0.0141157i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 29.4240i − 1.05423i
\(780\) 0 0
\(781\) − 2.45350i − 0.0877930i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 10.4210i − 0.371941i
\(786\) 0 0
\(787\) −19.1415 −0.682322 −0.341161 0.940005i \(-0.610820\pi\)
−0.341161 + 0.940005i \(0.610820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −15.4017 −0.547623
\(792\) 0 0
\(793\) −5.62001 −0.199572
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.9802 0.530625 0.265312 0.964162i \(-0.414525\pi\)
0.265312 + 0.964162i \(0.414525\pi\)
\(798\) 0 0
\(799\) − 26.1563i − 0.925343i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.4473i 0.368675i
\(804\) 0 0
\(805\) − 0.957076i − 0.0337325i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 20.4899i − 0.720388i −0.932877 0.360194i \(-0.882711\pi\)
0.932877 0.360194i \(-0.117289\pi\)
\(810\) 0 0
\(811\) −19.0258 −0.668087 −0.334043 0.942558i \(-0.608413\pi\)
−0.334043 + 0.942558i \(0.608413\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.87374 0.345862
\(816\) 0 0
\(817\) −32.2335 −1.12771
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −39.0750 −1.36373 −0.681863 0.731480i \(-0.738830\pi\)
−0.681863 + 0.731480i \(0.738830\pi\)
\(822\) 0 0
\(823\) − 38.1369i − 1.32937i −0.747124 0.664684i \(-0.768566\pi\)
0.747124 0.664684i \(-0.231434\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.8842i 0.482802i 0.970425 + 0.241401i \(0.0776070\pi\)
−0.970425 + 0.241401i \(0.922393\pi\)
\(828\) 0 0
\(829\) 0.705296i 0.0244960i 0.999925 + 0.0122480i \(0.00389875\pi\)
−0.999925 + 0.0122480i \(0.996101\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.48837i 0.224809i
\(834\) 0 0
\(835\) −10.5254 −0.364248
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −50.4627 −1.74217 −0.871083 0.491136i \(-0.836582\pi\)
−0.871083 + 0.491136i \(0.836582\pi\)
\(840\) 0 0
\(841\) −12.7101 −0.438280
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.0064 0.481835
\(846\) 0 0
\(847\) − 2.55546i − 0.0878068i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 8.10106i − 0.277701i
\(852\) 0 0
\(853\) 10.7370i 0.367628i 0.982961 + 0.183814i \(0.0588444\pi\)
−0.982961 + 0.183814i \(0.941156\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 9.75153i − 0.333106i −0.986033 0.166553i \(-0.946736\pi\)
0.986033 0.166553i \(-0.0532637\pi\)
\(858\) 0 0
\(859\) −28.9169 −0.986630 −0.493315 0.869851i \(-0.664215\pi\)
−0.493315 + 0.869851i \(0.664215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16.0259 −0.545528 −0.272764 0.962081i \(-0.587938\pi\)
−0.272764 + 0.962081i \(0.587938\pi\)
\(864\) 0 0
\(865\) −16.6807 −0.567161
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 27.5577 0.934831
\(870\) 0 0
\(871\) 4.19146i 0.142022i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.1448i 0.342958i
\(876\) 0 0
\(877\) 2.24225i 0.0757154i 0.999283 + 0.0378577i \(0.0120534\pi\)
−0.999283 + 0.0378577i \(0.987947\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.97799i 0.336167i 0.985773 + 0.168083i \(0.0537578\pi\)
−0.985773 + 0.168083i \(0.946242\pi\)
\(882\) 0 0
\(883\) −19.8382 −0.667609 −0.333805 0.942642i \(-0.608333\pi\)
−0.333805 + 0.942642i \(0.608333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −34.8500 −1.17015 −0.585074 0.810980i \(-0.698935\pi\)
−0.585074 + 0.810980i \(0.698935\pi\)
\(888\) 0 0
\(889\) 17.1514 0.575239
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.5613 0.420347
\(894\) 0 0
\(895\) − 23.9935i − 0.802016i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.439039i 0.0146428i
\(900\) 0 0
\(901\) 19.3206i 0.643661i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 21.5157i − 0.715207i
\(906\) 0 0
\(907\) 3.69626 0.122732 0.0613662 0.998115i \(-0.480454\pi\)
0.0613662 + 0.998115i \(0.480454\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.1975 1.69625 0.848125 0.529796i \(-0.177732\pi\)
0.848125 + 0.529796i \(0.177732\pi\)
\(912\) 0 0
\(913\) 7.88327 0.260898
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.5118 −0.347130
\(918\) 0 0
\(919\) 50.4632i 1.66463i 0.554306 + 0.832313i \(0.312984\pi\)
−0.554306 + 0.832313i \(0.687016\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 0.889190i − 0.0292681i
\(924\) 0 0
\(925\) 36.0180i 1.18426i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 8.22557i − 0.269872i −0.990854 0.134936i \(-0.956917\pi\)
0.990854 0.134936i \(-0.0430829\pi\)
\(930\) 0 0
\(931\) −3.11597 −0.102122
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 22.2095 0.726327
\(936\) 0 0
\(937\) 0.968784 0.0316488 0.0158244 0.999875i \(-0.494963\pi\)
0.0158244 + 0.999875i \(0.494963\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.2805 −0.367734 −0.183867 0.982951i \(-0.558862\pi\)
−0.183867 + 0.982951i \(0.558862\pi\)
\(942\) 0 0
\(943\) − 7.67257i − 0.249853i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 57.8442i 1.87968i 0.341609 + 0.939842i \(0.389028\pi\)
−0.341609 + 0.939842i \(0.610972\pi\)
\(948\) 0 0
\(949\) 3.78627i 0.122907i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 23.4368i − 0.759193i −0.925152 0.379596i \(-0.876063\pi\)
0.925152 0.379596i \(-0.123937\pi\)
\(954\) 0 0
\(955\) 9.23309 0.298776
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.6268 −0.440034
\(960\) 0 0
\(961\) 30.9882 0.999618
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.3413 −0.493855
\(966\) 0 0
\(967\) 44.8294i 1.44162i 0.693135 + 0.720808i \(0.256229\pi\)
−0.693135 + 0.720808i \(0.743771\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.3436i 0.428216i 0.976810 + 0.214108i \(0.0686844\pi\)
−0.976810 + 0.214108i \(0.931316\pi\)
\(972\) 0 0
\(973\) − 5.37620i − 0.172353i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 11.0654i − 0.354015i −0.984210 0.177007i \(-0.943358\pi\)
0.984210 0.177007i \(-0.0566416\pi\)
\(978\) 0 0
\(979\) −6.46793 −0.206716
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 48.7128 1.55370 0.776849 0.629687i \(-0.216817\pi\)
0.776849 + 0.629687i \(0.216817\pi\)
\(984\) 0 0
\(985\) 17.6188 0.561381
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.40517 −0.267269
\(990\) 0 0
\(991\) − 44.6581i − 1.41861i −0.704901 0.709306i \(-0.749009\pi\)
0.704901 0.709306i \(-0.250991\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.9241i 0.631636i
\(996\) 0 0
\(997\) − 20.5408i − 0.650534i −0.945622 0.325267i \(-0.894546\pi\)
0.945622 0.325267i \(-0.105454\pi\)
\(998\) 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.j.c.5615.24 32
3.2 odd 2 inner 6048.2.j.c.5615.9 32
4.3 odd 2 1512.2.j.c.323.1 32
8.3 odd 2 inner 6048.2.j.c.5615.10 32
8.5 even 2 1512.2.j.c.323.31 yes 32
12.11 even 2 1512.2.j.c.323.32 yes 32
24.5 odd 2 1512.2.j.c.323.2 yes 32
24.11 even 2 inner 6048.2.j.c.5615.23 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.1 32 4.3 odd 2
1512.2.j.c.323.2 yes 32 24.5 odd 2
1512.2.j.c.323.31 yes 32 8.5 even 2
1512.2.j.c.323.32 yes 32 12.11 even 2
6048.2.j.c.5615.9 32 3.2 odd 2 inner
6048.2.j.c.5615.10 32 8.3 odd 2 inner
6048.2.j.c.5615.23 32 24.11 even 2 inner
6048.2.j.c.5615.24 32 1.1 even 1 trivial