Properties

Label 6048.2.j.c.5615.10
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 5615.10
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.9

$q$-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 \(-0.584841\pi\)
−0.263390 + 0.964689i \(0.584841\pi\)
\(6\) 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 \(-0.953345\pi\)
0.989278 0.146048i \(-0.0466554\pi\)
\(14\) 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 \(-0.526997\pi\)
−0.0847107 + 0.996406i \(0.526997\pi\)
\(24\) 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 \(-0.377732\pi\)
0.374740 + 0.927130i \(0.377732\pi\)
\(30\) 0 0
\(31\) 0.108779i 0.0195372i 0.999952 + 0.00976861i \(0.00310949\pi\)
−0.999952 + 0.00976861i \(0.996891\pi\)
\(32\) 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 \(0.305780\pi\)
−0.572998 + 0.819557i \(0.694220\pi\)
\(38\) 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 \(-0.594990\pi\)
−0.294010 + 0.955802i \(0.594990\pi\)
\(48\) 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 \(-0.434440\pi\)
0.204511 + 0.978864i \(0.434440\pi\)
\(54\) 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 \(-0.889024\pi\)
0.939838 0.341621i \(-0.110976\pi\)
\(62\) 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 \(-0.484046\pi\)
0.0501001 + 0.998744i \(0.484046\pi\)
\(72\) 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 \(0.179113\pi\)
−0.845818 + 0.533472i \(0.820887\pi\)
\(80\) 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 \(-0.855717\pi\)
−0.899016 + 0.437915i \(0.855717\pi\)
\(102\) 0 0
\(103\) − 17.8616i − 1.75995i −0.475017 0.879977i \(-0.657558\pi\)
0.475017 0.879977i \(-0.342442\pi\)
\(104\) 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 \(0.208099\pi\)
−0.793801 + 0.608177i \(0.791901\pi\)
\(110\) 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 \(-0.724724\pi\)
0.648787 0.760970i \(-0.275276\pi\)
\(128\) 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 \(-0.248965\pi\)
0.709401 + 0.704805i \(0.248965\pi\)
\(150\) 0 0
\(151\) − 4.50773i − 0.366834i −0.983035 0.183417i \(-0.941284\pi\)
0.983035 0.183417i \(-0.0587158\pi\)
\(152\) 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 \(0.114849\pi\)
−0.935611 + 0.353032i \(0.885151\pi\)
\(158\) 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 \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(168\) 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 \(-0.319056\pi\)
0.538328 + 0.842736i \(0.319056\pi\)
\(174\) 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 \(0.237520\pi\)
−0.734279 + 0.678848i \(0.762480\pi\)
\(182\) 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 \(-0.591524\pi\)
−0.283587 + 0.958947i \(0.591524\pi\)
\(192\) 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 \(-0.678876\pi\)
−0.532842 + 0.846215i \(0.678876\pi\)
\(198\) 0 0
\(199\) − 16.9147i − 1.19905i −0.800356 0.599525i \(-0.795356\pi\)
0.800356 0.599525i \(-0.204644\pi\)
\(200\) 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 \(0.0367404\pi\)
−0.993346 + 0.115167i \(0.963260\pi\)
\(224\) 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 \(-0.805482\pi\)
0.819019 0.573767i \(-0.194518\pi\)
\(230\) 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 \(-0.792360\pi\)
−0.794677 + 0.607032i \(0.792360\pi\)
\(240\) 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 \(-0.338041\pi\)
0.487139 + 0.873325i \(0.338041\pi\)
\(264\) 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 \(-0.562654\pi\)
−0.195565 + 0.980691i \(0.562654\pi\)
\(270\) 0 0
\(271\) 0.0927686i 0.00563529i 0.999996 + 0.00281765i \(0.000896886\pi\)
−0.999996 + 0.00281765i \(0.999103\pi\)
\(272\) 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 \(-0.797337\pi\)
0.804071 0.594534i \(-0.202663\pi\)
\(278\) 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 \(-0.402803\pi\)
0.300631 + 0.953741i \(0.402803\pi\)
\(294\) 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 \(-0.427680\pi\)
0.225250 + 0.974301i \(0.427680\pi\)
\(312\) 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 \(-0.372733\pi\)
0.389253 + 0.921131i \(0.372733\pi\)
\(318\) 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 \(-0.848101\pi\)
0.888283 0.459297i \(-0.151899\pi\)
\(350\) 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 \(-0.629061\pi\)
−0.394438 + 0.918923i \(0.629061\pi\)
\(360\) 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 \(-0.976202\pi\)
0.997207 0.0746934i \(-0.0237978\pi\)
\(368\) 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 \(-0.819653\pi\)
0.843743 0.536748i \(-0.180347\pi\)
\(374\) 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 \(-0.222352\pi\)
0.765781 + 0.643101i \(0.222352\pi\)
\(384\) 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 \(-0.768832\pi\)
−0.747679 + 0.664060i \(0.768832\pi\)
\(390\) 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 \(0.186974\pi\)
−0.832386 + 0.554196i \(0.813026\pi\)
\(398\) 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 \(0.0511575\pi\)
−0.987113 + 0.160025i \(0.948842\pi\)
\(422\) 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 \(-0.456859\pi\)
0.135116 + 0.990830i \(0.456859\pi\)
\(432\) 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 \(0.300208\pi\)
−0.587258 + 0.809400i \(0.699792\pi\)
\(440\) 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 \(-0.482432\pi\)
0.0551647 + 0.998477i \(0.482432\pi\)
\(462\) 0 0
\(463\) 38.4537i 1.78709i 0.448969 + 0.893547i \(0.351791\pi\)
−0.448969 + 0.893547i \(0.648209\pi\)
\(464\) 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 \(-0.342242\pi\)
0.475568 + 0.879679i \(0.342242\pi\)
\(480\) 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 \(-0.797954\pi\)
0.805222 0.592973i \(-0.202046\pi\)
\(488\) 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 \(-0.758855\pi\)
−0.726502 + 0.687165i \(0.758855\pi\)
\(504\) 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 \(-0.623845\pi\)
−0.379328 + 0.925262i \(0.623845\pi\)
\(510\) 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 \(-0.877975\pi\)
0.927415 0.374033i \(-0.122025\pi\)
\(542\) 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 \(-0.538356\pi\)
−0.120208 + 0.992749i \(0.538356\pi\)
\(558\) 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 \(-0.552791\pi\)
−0.165088 + 0.986279i \(0.552791\pi\)
\(600\) 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 \(0.291363\pi\)
−0.609519 + 0.792772i \(0.708637\pi\)
\(608\) 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 \(-0.815758\pi\)
0.837112 0.547031i \(-0.184242\pi\)
\(614\) 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 \(-0.884015\pi\)
0.934346 0.356367i \(-0.115985\pi\)
\(632\) 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 \(-0.663758\pi\)
−0.492067 + 0.870557i \(0.663758\pi\)
\(648\) 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 \(-0.180373\pi\)
0.843699 + 0.536816i \(0.180373\pi\)
\(654\) 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 \(0.0527829\pi\)
−0.986283 + 0.165064i \(0.947217\pi\)
\(662\) 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 \(-0.393539\pi\)
0.328258 + 0.944588i \(0.393539\pi\)
\(678\) 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 \(-0.719368\pi\)
−0.635894 + 0.771777i \(0.719368\pi\)
\(702\) 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 \(-0.903399\pi\)
0.954302 0.298845i \(-0.0966013\pi\)
\(710\) 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 \(-0.276439\pi\)
0.646004 + 0.763334i \(0.276439\pi\)
\(720\) 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 \(-0.745660\pi\)
0.697400 0.716682i \(-0.254340\pi\)
\(728\) 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 \(0.0262342\pi\)
−0.996606 + 0.0823240i \(0.973766\pi\)
\(734\) 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 \(-0.808251\pi\)
−0.823979 + 0.566620i \(0.808251\pi\)
\(744\) 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 \(-0.878166\pi\)
0.927640 0.373475i \(-0.121834\pi\)
\(752\) 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 \(0.0664273\pi\)
−0.978304 + 0.207176i \(0.933573\pi\)
\(758\) 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 \(-0.776823\pi\)
−0.764112 + 0.645083i \(0.776823\pi\)
\(774\) 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 \(-0.585475\pi\)
−0.265312 + 0.964162i \(0.585475\pi\)
\(798\) 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 \(-0.261170\pi\)
0.681863 + 0.731480i \(0.261170\pi\)
\(822\) 0 0
\(823\) 38.1369i 1.32937i 0.747124 + 0.664684i \(0.231434\pi\)
−0.747124 + 0.664684i \(0.768566\pi\)
\(824\) 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 \(-0.996101\pi\)
0.999925 0.0122480i \(-0.00389875\pi\)
\(830\) 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 \(-0.163418\pi\)
0.871083 + 0.491136i \(0.163418\pi\)
\(840\) 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 \(-0.941156\pi\)
0.982961 0.183814i \(-0.0588444\pi\)
\(854\) 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 \(-0.412062\pi\)
0.272764 + 0.962081i \(0.412062\pi\)
\(864\) 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 \(-0.987947\pi\)
0.999283 0.0378577i \(-0.0120534\pi\)
\(878\) 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 \(-0.301065\pi\)
0.585074 + 0.810980i \(0.301065\pi\)
\(888\) 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 \(-0.822268\pi\)
−0.848125 + 0.529796i \(0.822268\pi\)
\(912\) 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 \(-0.687016\pi\)
0.554306 0.832313i \(-0.312984\pi\)
\(920\) 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 \(-0.441138\pi\)
0.183867 + 0.982951i \(0.441138\pi\)
\(942\) 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 \(-0.743771\pi\)
0.693135 0.720808i \(-0.256229\pi\)
\(968\) 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 \(-0.783183\pi\)
−0.776849 + 0.629687i \(0.783183\pi\)
\(984\) 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 \(0.250991\pi\)
−0.704901 + 0.709306i \(0.749009\pi\)
\(992\) 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 \(0.105454\pi\)
−0.945622 + 0.325267i \(0.894546\pi\)
\(998\) 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.5615.10 32
3.2 odd 2 inner 6048.2.j.c.5615.23 32
4.3 odd 2 1512.2.j.c.323.31 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.24 32
8.5 even 2 1512.2.j.c.323.1 32
12.11 even 2 1512.2.j.c.323.2 yes 32
24.5 odd 2 1512.2.j.c.323.32 yes 32
24.11 even 2 inner 6048.2.j.c.5615.9 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.1 32 8.5 even 2
1512.2.j.c.323.2 yes 32 12.11 even 2
1512.2.j.c.323.31 yes 32 4.3 odd 2
1512.2.j.c.323.32 yes 32 24.5 odd 2
6048.2.j.c.5615.9 32 24.11 even 2 inner
6048.2.j.c.5615.10 32 1.1 even 1 trivial
6048.2.j.c.5615.23 32 3.2 odd 2 inner
6048.2.j.c.5615.24 32 8.3 odd 2 inner