Properties

Label 6048.2.h.g.2591.8
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.8
Root \(-0.258819 + 0.965926i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.g.2591.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.24969i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+3.24969i q^{5} +1.00000i q^{7} +6.07812 q^{11} +0.0963763 q^{13} -3.41421i q^{17} +5.89898i q^{19} -7.54222 q^{23} -5.56048 q^{25} +5.81382i q^{29} -1.07055i q^{31} -3.24969 q^{35} +7.82843 q^{37} -0.614014i q^{41} +7.75323i q^{43} -0.585786 q^{47} -1.00000 q^{49} +7.39836i q^{53} +19.7520i q^{55} +9.01461 q^{59} -8.92820 q^{61} +0.313193i q^{65} -10.4887i q^{67} -9.19980 q^{71} +11.8238 q^{73} +6.07812i q^{77} +11.0951i q^{79} -15.6118 q^{83} +11.0951 q^{85} +3.01362i q^{89} +0.0963763i q^{91} -19.1698 q^{95} +18.5851 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} - 8q^{13} + 8q^{23} - 8q^{25} - 8q^{35} + 40q^{37} - 16q^{47} - 8q^{49} + 64q^{59} - 16q^{61} - 72q^{71} + 24q^{73} - 32q^{83} + 8q^{85} - 56q^{95} + 48q^{97} + O(q^{100}) \)

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
\(5\) 3.24969i 1.45331i 0.687005 + 0.726653i \(0.258925\pi\)
−0.687005 + 0.726653i \(0.741075\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.07812 1.83262 0.916310 0.400469i \(-0.131153\pi\)
0.916310 + 0.400469i \(0.131153\pi\)
\(12\) 0 0
\(13\) 0.0963763 0.0267300 0.0133650 0.999911i \(-0.495746\pi\)
0.0133650 + 0.999911i \(0.495746\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.41421i − 0.828068i −0.910261 0.414034i \(-0.864119\pi\)
0.910261 0.414034i \(-0.135881\pi\)
\(18\) 0 0
\(19\) 5.89898i 1.35332i 0.736296 + 0.676659i \(0.236573\pi\)
−0.736296 + 0.676659i \(0.763427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.54222 −1.57266 −0.786331 0.617806i \(-0.788022\pi\)
−0.786331 + 0.617806i \(0.788022\pi\)
\(24\) 0 0
\(25\) −5.56048 −1.11210
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.81382i 1.07960i 0.841794 + 0.539799i \(0.181500\pi\)
−0.841794 + 0.539799i \(0.818500\pi\)
\(30\) 0 0
\(31\) − 1.07055i − 0.192277i −0.995368 0.0961384i \(-0.969351\pi\)
0.995368 0.0961384i \(-0.0306492\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.24969 −0.549298
\(36\) 0 0
\(37\) 7.82843 1.28699 0.643493 0.765452i \(-0.277485\pi\)
0.643493 + 0.765452i \(0.277485\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 0.614014i − 0.0958929i −0.998850 0.0479465i \(-0.984732\pi\)
0.998850 0.0479465i \(-0.0152677\pi\)
\(42\) 0 0
\(43\) 7.75323i 1.18236i 0.806541 + 0.591178i \(0.201337\pi\)
−0.806541 + 0.591178i \(0.798663\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.585786 −0.0854457 −0.0427229 0.999087i \(-0.513603\pi\)
−0.0427229 + 0.999087i \(0.513603\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.39836i 1.01624i 0.861286 + 0.508121i \(0.169660\pi\)
−0.861286 + 0.508121i \(0.830340\pi\)
\(54\) 0 0
\(55\) 19.7520i 2.66336i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.01461 1.17360 0.586801 0.809731i \(-0.300387\pi\)
0.586801 + 0.809731i \(0.300387\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.313193i 0.0388468i
\(66\) 0 0
\(67\) − 10.4887i − 1.28140i −0.767793 0.640698i \(-0.778645\pi\)
0.767793 0.640698i \(-0.221355\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.19980 −1.09182 −0.545908 0.837845i \(-0.683815\pi\)
−0.545908 + 0.837845i \(0.683815\pi\)
\(72\) 0 0
\(73\) 11.8238 1.38387 0.691935 0.721960i \(-0.256759\pi\)
0.691935 + 0.721960i \(0.256759\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.07812i 0.692666i
\(78\) 0 0
\(79\) 11.0951i 1.24830i 0.781305 + 0.624150i \(0.214555\pi\)
−0.781305 + 0.624150i \(0.785445\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.6118 −1.71361 −0.856807 0.515637i \(-0.827555\pi\)
−0.856807 + 0.515637i \(0.827555\pi\)
\(84\) 0 0
\(85\) 11.0951 1.20344
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.01362i 0.319443i 0.987162 + 0.159721i \(0.0510596\pi\)
−0.987162 + 0.159721i \(0.948940\pi\)
\(90\) 0 0
\(91\) 0.0963763i 0.0101030i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −19.1698 −1.96678
\(96\) 0 0
\(97\) 18.5851 1.88703 0.943513 0.331335i \(-0.107499\pi\)
0.943513 + 0.331335i \(0.107499\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 0.858380i − 0.0854120i −0.999088 0.0427060i \(-0.986402\pi\)
0.999088 0.0427060i \(-0.0135979\pi\)
\(102\) 0 0
\(103\) − 16.3843i − 1.61439i −0.590285 0.807195i \(-0.700985\pi\)
0.590285 0.807195i \(-0.299015\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.34366 0.516591 0.258296 0.966066i \(-0.416839\pi\)
0.258296 + 0.966066i \(0.416839\pi\)
\(108\) 0 0
\(109\) −1.85425 −0.177605 −0.0888025 0.996049i \(-0.528304\pi\)
−0.0888025 + 0.996049i \(0.528304\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.0413286i 0.00388787i 0.999998 + 0.00194393i \(0.000618774\pi\)
−0.999998 + 0.00194393i \(0.999381\pi\)
\(114\) 0 0
\(115\) − 24.5099i − 2.28556i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.41421 0.312980
\(120\) 0 0
\(121\) 25.9435 2.35850
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.82138i − 0.162909i
\(126\) 0 0
\(127\) 7.63103i 0.677144i 0.940940 + 0.338572i \(0.109944\pi\)
−0.940940 + 0.338572i \(0.890056\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.05594 −0.0922579 −0.0461290 0.998935i \(-0.514689\pi\)
−0.0461290 + 0.998935i \(0.514689\pi\)
\(132\) 0 0
\(133\) −5.89898 −0.511506
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.7420i 1.77211i 0.463580 + 0.886055i \(0.346565\pi\)
−0.463580 + 0.886055i \(0.653435\pi\)
\(138\) 0 0
\(139\) 2.53590i 0.215092i 0.994200 + 0.107546i \(0.0342993\pi\)
−0.994200 + 0.107546i \(0.965701\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.585786 0.0489859
\(144\) 0 0
\(145\) −18.8931 −1.56899
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 12.6848i − 1.03918i −0.854415 0.519591i \(-0.826084\pi\)
0.854415 0.519591i \(-0.173916\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.47896 0.279437
\(156\) 0 0
\(157\) 13.7532 1.09763 0.548814 0.835945i \(-0.315080\pi\)
0.548814 + 0.835945i \(0.315080\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.54222i − 0.594410i
\(162\) 0 0
\(163\) − 5.70158i − 0.446582i −0.974752 0.223291i \(-0.928320\pi\)
0.974752 0.223291i \(-0.0716801\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.95544 −0.692992 −0.346496 0.938051i \(-0.612629\pi\)
−0.346496 + 0.938051i \(0.612629\pi\)
\(168\) 0 0
\(169\) −12.9907 −0.999286
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.82138i 0.138477i 0.997600 + 0.0692384i \(0.0220569\pi\)
−0.997600 + 0.0692384i \(0.977943\pi\)
\(174\) 0 0
\(175\) − 5.56048i − 0.420333i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.11563 0.307617 0.153808 0.988101i \(-0.450846\pi\)
0.153808 + 0.988101i \(0.450846\pi\)
\(180\) 0 0
\(181\) −19.4548 −1.44606 −0.723032 0.690814i \(-0.757252\pi\)
−0.723032 + 0.690814i \(0.757252\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 25.4400i 1.87038i
\(186\) 0 0
\(187\) − 20.7520i − 1.51754i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.80071 0.419725 0.209862 0.977731i \(-0.432698\pi\)
0.209862 + 0.977731i \(0.432698\pi\)
\(192\) 0 0
\(193\) −21.1210 −1.52032 −0.760160 0.649736i \(-0.774879\pi\)
−0.760160 + 0.649736i \(0.774879\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.3971i 1.45323i 0.687043 + 0.726617i \(0.258909\pi\)
−0.687043 + 0.726617i \(0.741091\pi\)
\(198\) 0 0
\(199\) − 15.8956i − 1.12681i −0.826182 0.563404i \(-0.809492\pi\)
0.826182 0.563404i \(-0.190508\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.81382 −0.408050
\(204\) 0 0
\(205\) 1.99536 0.139362
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 35.8547i 2.48012i
\(210\) 0 0
\(211\) 8.19275i 0.564012i 0.959413 + 0.282006i \(0.0909999\pi\)
−0.959413 + 0.282006i \(0.909000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −25.1956 −1.71832
\(216\) 0 0
\(217\) 1.07055 0.0726738
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.329049i − 0.0221343i
\(222\) 0 0
\(223\) 5.61088i 0.375732i 0.982195 + 0.187866i \(0.0601571\pi\)
−0.982195 + 0.187866i \(0.939843\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.9988 −0.862758 −0.431379 0.902171i \(-0.641973\pi\)
−0.431379 + 0.902171i \(0.641973\pi\)
\(228\) 0 0
\(229\) 7.60521 0.502566 0.251283 0.967914i \(-0.419147\pi\)
0.251283 + 0.967914i \(0.419147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.898979i 0.0588941i 0.999566 + 0.0294471i \(0.00937464\pi\)
−0.999566 + 0.0294471i \(0.990625\pi\)
\(234\) 0 0
\(235\) − 1.90362i − 0.124179i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.7712 −0.696734 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(240\) 0 0
\(241\) −15.3463 −0.988544 −0.494272 0.869307i \(-0.664565\pi\)
−0.494272 + 0.869307i \(0.664565\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.24969i − 0.207615i
\(246\) 0 0
\(247\) 0.568522i 0.0361742i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.6982 −0.864622 −0.432311 0.901725i \(-0.642302\pi\)
−0.432311 + 0.901725i \(0.642302\pi\)
\(252\) 0 0
\(253\) −45.8425 −2.88209
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 23.3419i − 1.45603i −0.685561 0.728015i \(-0.740443\pi\)
0.685561 0.728015i \(-0.259557\pi\)
\(258\) 0 0
\(259\) 7.82843i 0.486435i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27.3543 −1.68674 −0.843368 0.537336i \(-0.819431\pi\)
−0.843368 + 0.537336i \(0.819431\pi\)
\(264\) 0 0
\(265\) −24.0424 −1.47691
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.47720i 0.577835i 0.957354 + 0.288918i \(0.0932954\pi\)
−0.957354 + 0.288918i \(0.906705\pi\)
\(270\) 0 0
\(271\) 21.6544i 1.31541i 0.753276 + 0.657705i \(0.228472\pi\)
−0.753276 + 0.657705i \(0.771528\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −33.7972 −2.03805
\(276\) 0 0
\(277\) −5.37519 −0.322964 −0.161482 0.986876i \(-0.551627\pi\)
−0.161482 + 0.986876i \(0.551627\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.0958i 1.25847i 0.777215 + 0.629235i \(0.216632\pi\)
−0.777215 + 0.629235i \(0.783368\pi\)
\(282\) 0 0
\(283\) − 30.2326i − 1.79714i −0.438827 0.898571i \(-0.644606\pi\)
0.438827 0.898571i \(-0.355394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.614014 0.0362441
\(288\) 0 0
\(289\) 5.34315 0.314303
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 24.6009i − 1.43720i −0.695423 0.718600i \(-0.744783\pi\)
0.695423 0.718600i \(-0.255217\pi\)
\(294\) 0 0
\(295\) 29.2947i 1.70560i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.726891 −0.0420372
\(300\) 0 0
\(301\) −7.75323 −0.446889
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 29.0139i − 1.66133i
\(306\) 0 0
\(307\) − 17.6143i − 1.00530i −0.864490 0.502650i \(-0.832358\pi\)
0.864490 0.502650i \(-0.167642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.34242 0.132826 0.0664131 0.997792i \(-0.478844\pi\)
0.0664131 + 0.997792i \(0.478844\pi\)
\(312\) 0 0
\(313\) 9.97418 0.563774 0.281887 0.959448i \(-0.409040\pi\)
0.281887 + 0.959448i \(0.409040\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.0144i 1.57345i 0.617307 + 0.786723i \(0.288224\pi\)
−0.617307 + 0.786723i \(0.711776\pi\)
\(318\) 0 0
\(319\) 35.3370i 1.97849i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.1404 1.12064
\(324\) 0 0
\(325\) −0.535898 −0.0297263
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 0.585786i − 0.0322955i
\(330\) 0 0
\(331\) 28.1810i 1.54897i 0.632594 + 0.774483i \(0.281990\pi\)
−0.632594 + 0.774483i \(0.718010\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 34.0850 1.86226
\(336\) 0 0
\(337\) 4.12096 0.224483 0.112241 0.993681i \(-0.464197\pi\)
0.112241 + 0.993681i \(0.464197\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 6.50694i − 0.352371i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.4619 0.883719 0.441860 0.897084i \(-0.354319\pi\)
0.441860 + 0.897084i \(0.354319\pi\)
\(348\) 0 0
\(349\) 21.4032 1.14569 0.572843 0.819665i \(-0.305841\pi\)
0.572843 + 0.819665i \(0.305841\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.7118i 0.889479i 0.895660 + 0.444740i \(0.146704\pi\)
−0.895660 + 0.444740i \(0.853296\pi\)
\(354\) 0 0
\(355\) − 29.8965i − 1.58674i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −23.9973 −1.26653 −0.633264 0.773936i \(-0.718285\pi\)
−0.633264 + 0.773936i \(0.718285\pi\)
\(360\) 0 0
\(361\) −15.7980 −0.831472
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 38.4236i 2.01118i
\(366\) 0 0
\(367\) − 20.8661i − 1.08920i −0.838695 0.544602i \(-0.816681\pi\)
0.838695 0.544602i \(-0.183319\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.39836 −0.384104
\(372\) 0 0
\(373\) 23.3184 1.20738 0.603689 0.797220i \(-0.293697\pi\)
0.603689 + 0.797220i \(0.293697\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.560314i 0.0288576i
\(378\) 0 0
\(379\) − 11.8117i − 0.606725i −0.952875 0.303363i \(-0.901891\pi\)
0.952875 0.303363i \(-0.0981093\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.69818 −0.291164 −0.145582 0.989346i \(-0.546505\pi\)
−0.145582 + 0.989346i \(0.546505\pi\)
\(384\) 0 0
\(385\) −19.7520 −1.00665
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 3.72940i − 0.189088i −0.995521 0.0945440i \(-0.969861\pi\)
0.995521 0.0945440i \(-0.0301393\pi\)
\(390\) 0 0
\(391\) 25.7507i 1.30227i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −36.0557 −1.81416
\(396\) 0 0
\(397\) 33.4947 1.68105 0.840525 0.541773i \(-0.182247\pi\)
0.840525 + 0.541773i \(0.182247\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.79744i 0.339448i 0.985492 + 0.169724i \(0.0542876\pi\)
−0.985492 + 0.169724i \(0.945712\pi\)
\(402\) 0 0
\(403\) − 0.103176i − 0.00513956i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 47.5821 2.35856
\(408\) 0 0
\(409\) 0.411329 0.0203389 0.0101695 0.999948i \(-0.496763\pi\)
0.0101695 + 0.999948i \(0.496763\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.01461i 0.443580i
\(414\) 0 0
\(415\) − 50.7334i − 2.49041i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −20.1396 −0.983886 −0.491943 0.870627i \(-0.663713\pi\)
−0.491943 + 0.870627i \(0.663713\pi\)
\(420\) 0 0
\(421\) −6.03047 −0.293907 −0.146954 0.989143i \(-0.546947\pi\)
−0.146954 + 0.989143i \(0.546947\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.9847i 0.920891i
\(426\) 0 0
\(427\) − 8.92820i − 0.432066i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.8255 −0.665949 −0.332974 0.942936i \(-0.608052\pi\)
−0.332974 + 0.942936i \(0.608052\pi\)
\(432\) 0 0
\(433\) −6.55923 −0.315217 −0.157608 0.987502i \(-0.550378\pi\)
−0.157608 + 0.987502i \(0.550378\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 44.4914i − 2.12831i
\(438\) 0 0
\(439\) 10.9799i 0.524040i 0.965062 + 0.262020i \(0.0843886\pi\)
−0.965062 + 0.262020i \(0.915611\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.9204 0.851425 0.425713 0.904858i \(-0.360023\pi\)
0.425713 + 0.904858i \(0.360023\pi\)
\(444\) 0 0
\(445\) −9.79331 −0.464248
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 9.14537i − 0.431597i −0.976438 0.215798i \(-0.930765\pi\)
0.976438 0.215798i \(-0.0692354\pi\)
\(450\) 0 0
\(451\) − 3.73205i − 0.175735i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.313193 −0.0146827
\(456\) 0 0
\(457\) 24.3629 1.13965 0.569823 0.821767i \(-0.307012\pi\)
0.569823 + 0.821767i \(0.307012\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 19.4651i − 0.906580i −0.891363 0.453290i \(-0.850250\pi\)
0.891363 0.453290i \(-0.149750\pi\)
\(462\) 0 0
\(463\) − 3.07860i − 0.143075i −0.997438 0.0715373i \(-0.977210\pi\)
0.997438 0.0715373i \(-0.0227905\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −31.8829 −1.47537 −0.737683 0.675148i \(-0.764080\pi\)
−0.737683 + 0.675148i \(0.764080\pi\)
\(468\) 0 0
\(469\) 10.4887 0.484322
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 47.1250i 2.16681i
\(474\) 0 0
\(475\) − 32.8011i − 1.50502i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.5851 1.12332 0.561660 0.827368i \(-0.310163\pi\)
0.561660 + 0.827368i \(0.310163\pi\)
\(480\) 0 0
\(481\) 0.754475 0.0344011
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 60.3957i 2.74243i
\(486\) 0 0
\(487\) 35.5298i 1.61001i 0.593269 + 0.805004i \(0.297837\pi\)
−0.593269 + 0.805004i \(0.702163\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.2772 −1.59204 −0.796019 0.605271i \(-0.793065\pi\)
−0.796019 + 0.605271i \(0.793065\pi\)
\(492\) 0 0
\(493\) 19.8496 0.893981
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 9.19980i − 0.412667i
\(498\) 0 0
\(499\) − 35.4169i − 1.58548i −0.609562 0.792739i \(-0.708654\pi\)
0.609562 0.792739i \(-0.291346\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.90074 0.307689 0.153844 0.988095i \(-0.450835\pi\)
0.153844 + 0.988095i \(0.450835\pi\)
\(504\) 0 0
\(505\) 2.78947 0.124130
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.0285i 1.50829i 0.656710 + 0.754143i \(0.271948\pi\)
−0.656710 + 0.754143i \(0.728052\pi\)
\(510\) 0 0
\(511\) 11.8238i 0.523053i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 53.2438 2.34620
\(516\) 0 0
\(517\) −3.56048 −0.156590
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.4125i 1.11334i 0.830733 + 0.556671i \(0.187921\pi\)
−0.830733 + 0.556671i \(0.812079\pi\)
\(522\) 0 0
\(523\) − 6.46750i − 0.282804i −0.989952 0.141402i \(-0.954839\pi\)
0.989952 0.141402i \(-0.0451610\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.65509 −0.159218
\(528\) 0 0
\(529\) 33.8850 1.47326
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.0591764i − 0.00256322i
\(534\) 0 0
\(535\) 17.3652i 0.750765i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.07812 −0.261803
\(540\) 0 0
\(541\) −29.6823 −1.27614 −0.638072 0.769977i \(-0.720268\pi\)
−0.638072 + 0.769977i \(0.720268\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 6.02574i − 0.258114i
\(546\) 0 0
\(547\) 0.327058i 0.0139840i 0.999976 + 0.00699199i \(0.00222564\pi\)
−0.999976 + 0.00699199i \(0.997774\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −34.2956 −1.46104
\(552\) 0 0
\(553\) −11.0951 −0.471813
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.19632i 0.0506895i 0.999679 + 0.0253448i \(0.00806836\pi\)
−0.999679 + 0.0253448i \(0.991932\pi\)
\(558\) 0 0
\(559\) 0.747228i 0.0316044i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.1156 −0.679193 −0.339596 0.940571i \(-0.610291\pi\)
−0.339596 + 0.940571i \(0.610291\pi\)
\(564\) 0 0
\(565\) −0.134305 −0.00565026
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 32.6098i − 1.36707i −0.729916 0.683537i \(-0.760441\pi\)
0.729916 0.683537i \(-0.239559\pi\)
\(570\) 0 0
\(571\) 36.9189i 1.54501i 0.635010 + 0.772504i \(0.280996\pi\)
−0.635010 + 0.772504i \(0.719004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 41.9383 1.74895
\(576\) 0 0
\(577\) 43.5866 1.81454 0.907268 0.420554i \(-0.138164\pi\)
0.907268 + 0.420554i \(0.138164\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 15.6118i − 0.647686i
\(582\) 0 0
\(583\) 44.9681i 1.86239i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.6120 −1.05712 −0.528560 0.848896i \(-0.677268\pi\)
−0.528560 + 0.848896i \(0.677268\pi\)
\(588\) 0 0
\(589\) 6.31517 0.260212
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 24.4984i − 1.00603i −0.864278 0.503014i \(-0.832224\pi\)
0.864278 0.503014i \(-0.167776\pi\)
\(594\) 0 0
\(595\) 11.0951i 0.454856i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 37.9534 1.55073 0.775367 0.631510i \(-0.217565\pi\)
0.775367 + 0.631510i \(0.217565\pi\)
\(600\) 0 0
\(601\) 16.4181 0.669709 0.334855 0.942270i \(-0.391313\pi\)
0.334855 + 0.942270i \(0.391313\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 84.3083i 3.42762i
\(606\) 0 0
\(607\) − 0.921404i − 0.0373986i −0.999825 0.0186993i \(-0.994047\pi\)
0.999825 0.0186993i \(-0.00595252\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.0564559 −0.00228396
\(612\) 0 0
\(613\) 34.7614 1.40400 0.702000 0.712177i \(-0.252291\pi\)
0.702000 + 0.712177i \(0.252291\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 47.4692i − 1.91104i −0.294929 0.955519i \(-0.595296\pi\)
0.294929 0.955519i \(-0.404704\pi\)
\(618\) 0 0
\(619\) − 26.2138i − 1.05362i −0.849982 0.526811i \(-0.823388\pi\)
0.849982 0.526811i \(-0.176612\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.01362 −0.120738
\(624\) 0 0
\(625\) −21.8835 −0.875339
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 26.7279i − 1.06571i
\(630\) 0 0
\(631\) 5.00568i 0.199273i 0.995024 + 0.0996364i \(0.0317680\pi\)
−0.995024 + 0.0996364i \(0.968232\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.7985 −0.984097
\(636\) 0 0
\(637\) −0.0963763 −0.00381857
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.17011i 0.362198i 0.983465 + 0.181099i \(0.0579654\pi\)
−0.983465 + 0.181099i \(0.942035\pi\)
\(642\) 0 0
\(643\) 38.4655i 1.51693i 0.651714 + 0.758465i \(0.274050\pi\)
−0.651714 + 0.758465i \(0.725950\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.65310 0.143618 0.0718091 0.997418i \(-0.477123\pi\)
0.0718091 + 0.997418i \(0.477123\pi\)
\(648\) 0 0
\(649\) 54.7919 2.15077
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 48.3662i − 1.89272i −0.323121 0.946358i \(-0.604732\pi\)
0.323121 0.946358i \(-0.395268\pi\)
\(654\) 0 0
\(655\) − 3.43148i − 0.134079i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.6738 0.805338 0.402669 0.915346i \(-0.368083\pi\)
0.402669 + 0.915346i \(0.368083\pi\)
\(660\) 0 0
\(661\) 32.9729 1.28250 0.641249 0.767333i \(-0.278416\pi\)
0.641249 + 0.767333i \(0.278416\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 19.1698i − 0.743375i
\(666\) 0 0
\(667\) − 43.8491i − 1.69784i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −54.2667 −2.09494
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 15.7446i − 0.605114i −0.953131 0.302557i \(-0.902160\pi\)
0.953131 0.302557i \(-0.0978402\pi\)
\(678\) 0 0
\(679\) 18.5851i 0.713229i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −34.4490 −1.31816 −0.659078 0.752075i \(-0.729053\pi\)
−0.659078 + 0.752075i \(0.729053\pi\)
\(684\) 0 0
\(685\) −67.4051 −2.57542
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.713026i 0.0271641i
\(690\) 0 0
\(691\) 14.1504i 0.538306i 0.963097 + 0.269153i \(0.0867438\pi\)
−0.963097 + 0.269153i \(0.913256\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.24088 −0.312594
\(696\) 0 0
\(697\) −2.09638 −0.0794059
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.9289i 0.563858i 0.959435 + 0.281929i \(0.0909743\pi\)
−0.959435 + 0.281929i \(0.909026\pi\)
\(702\) 0 0
\(703\) 46.1797i 1.74170i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.858380 0.0322827
\(708\) 0 0
\(709\) −44.6389 −1.67645 −0.838224 0.545326i \(-0.816406\pi\)
−0.838224 + 0.545326i \(0.816406\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.07434i 0.302386i
\(714\) 0 0
\(715\) 1.90362i 0.0711915i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.2268 0.866214 0.433107 0.901343i \(-0.357417\pi\)
0.433107 + 0.901343i \(0.357417\pi\)
\(720\) 0 0
\(721\) 16.3843 0.610182
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 32.3276i − 1.20062i
\(726\) 0 0
\(727\) − 33.7153i − 1.25043i −0.780452 0.625216i \(-0.785011\pi\)
0.780452 0.625216i \(-0.214989\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 26.4712 0.979072
\(732\) 0 0
\(733\) 18.2984 0.675867 0.337934 0.941170i \(-0.390272\pi\)
0.337934 + 0.941170i \(0.390272\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 63.7514i − 2.34831i
\(738\) 0 0
\(739\) 14.3246i 0.526938i 0.964668 + 0.263469i \(0.0848666\pi\)
−0.964668 + 0.263469i \(0.915133\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.2374 1.76966 0.884829 0.465916i \(-0.154275\pi\)
0.884829 + 0.465916i \(0.154275\pi\)
\(744\) 0 0
\(745\) 41.2218 1.51025
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.34366i 0.195253i
\(750\) 0 0
\(751\) 39.4689i 1.44024i 0.693850 + 0.720120i \(0.255913\pi\)
−0.693850 + 0.720120i \(0.744087\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 35.3112 1.28341 0.641704 0.766952i \(-0.278228\pi\)
0.641704 + 0.766952i \(0.278228\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 32.5093i − 1.17846i −0.807965 0.589231i \(-0.799431\pi\)
0.807965 0.589231i \(-0.200569\pi\)
\(762\) 0 0
\(763\) − 1.85425i − 0.0671284i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.868795 0.0313704
\(768\) 0 0
\(769\) −39.2362 −1.41489 −0.707447 0.706766i \(-0.750153\pi\)
−0.707447 + 0.706766i \(0.750153\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 7.29301i − 0.262311i −0.991362 0.131156i \(-0.958131\pi\)
0.991362 0.131156i \(-0.0418688\pi\)
\(774\) 0 0
\(775\) 5.95278i 0.213830i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.62206 0.129774
\(780\) 0 0
\(781\) −55.9175 −2.00088
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 44.6937i 1.59519i
\(786\) 0 0
\(787\) 21.3044i 0.759421i 0.925105 + 0.379710i \(0.123976\pi\)
−0.925105 + 0.379710i \(0.876024\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.0413286 −0.00146948
\(792\) 0 0
\(793\) −0.860467 −0.0305561
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.71735i − 0.344206i −0.985079 0.172103i \(-0.944944\pi\)
0.985079 0.172103i \(-0.0550562\pi\)
\(798\) 0 0
\(799\) 2.00000i 0.0707549i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 71.8663 2.53611
\(804\) 0 0
\(805\) 24.5099 0.863859
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.1981i 1.09687i 0.836195 + 0.548433i \(0.184775\pi\)
−0.836195 + 0.548433i \(0.815225\pi\)
\(810\) 0 0
\(811\) 9.29945i 0.326548i 0.986581 + 0.163274i \(0.0522054\pi\)
−0.986581 + 0.163274i \(0.947795\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.5284 0.649020
\(816\) 0 0
\(817\) −45.7361 −1.60011
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 46.4539i 1.62125i 0.585565 + 0.810626i \(0.300873\pi\)
−0.585565 + 0.810626i \(0.699127\pi\)
\(822\) 0 0
\(823\) − 16.9282i − 0.590080i −0.955485 0.295040i \(-0.904667\pi\)
0.955485 0.295040i \(-0.0953330\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −13.0242 −0.452895 −0.226447 0.974023i \(-0.572711\pi\)
−0.226447 + 0.974023i \(0.572711\pi\)
\(828\) 0 0
\(829\) 44.9632 1.56164 0.780819 0.624758i \(-0.214802\pi\)
0.780819 + 0.624758i \(0.214802\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.41421i 0.118295i
\(834\) 0 0
\(835\) − 29.1024i − 1.00713i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.08392 −0.140993 −0.0704963 0.997512i \(-0.522458\pi\)
−0.0704963 + 0.997512i \(0.522458\pi\)
\(840\) 0 0
\(841\) −4.80045 −0.165533
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 42.2158i − 1.45227i
\(846\) 0 0
\(847\) 25.9435i 0.891429i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −59.0437 −2.02399
\(852\) 0 0
\(853\) 43.1515 1.47748 0.738740 0.673990i \(-0.235421\pi\)
0.738740 + 0.673990i \(0.235421\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 0.542431i − 0.0185291i −0.999957 0.00926455i \(-0.997051\pi\)
0.999957 0.00926455i \(-0.00294904\pi\)
\(858\) 0 0
\(859\) 38.6159i 1.31756i 0.752337 + 0.658778i \(0.228926\pi\)
−0.752337 + 0.658778i \(0.771074\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31.6834 −1.07851 −0.539257 0.842141i \(-0.681295\pi\)
−0.539257 + 0.842141i \(0.681295\pi\)
\(864\) 0 0
\(865\) −5.91891 −0.201249
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 67.4375i 2.28766i
\(870\) 0 0
\(871\) − 1.01086i − 0.0342517i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.82138 0.0615738
\(876\) 0 0
\(877\) −26.4999 −0.894838 −0.447419 0.894324i \(-0.647657\pi\)
−0.447419 + 0.894324i \(0.647657\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.8866i 0.838451i 0.907882 + 0.419226i \(0.137698\pi\)
−0.907882 + 0.419226i \(0.862302\pi\)
\(882\) 0 0
\(883\) − 44.8978i − 1.51093i −0.655187 0.755466i \(-0.727410\pi\)
0.655187 0.755466i \(-0.272590\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.7844 −0.630718 −0.315359 0.948972i \(-0.602125\pi\)
−0.315359 + 0.948972i \(0.602125\pi\)
\(888\) 0 0
\(889\) −7.63103 −0.255937
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 3.45554i − 0.115635i
\(894\) 0 0
\(895\) 13.3745i 0.447061i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.22399 0.207582
\(900\) 0 0
\(901\) 25.2596 0.841518
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 63.2221i − 2.10157i
\(906\) 0 0
\(907\) − 54.7660i − 1.81848i −0.416276 0.909238i \(-0.636665\pi\)
0.416276 0.909238i \(-0.363335\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.32405 0.242657 0.121328 0.992612i \(-0.461285\pi\)
0.121328 + 0.992612i \(0.461285\pi\)
\(912\) 0 0
\(913\) −94.8902 −3.14041
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.05594i − 0.0348702i
\(918\) 0 0
\(919\) − 30.4838i − 1.00557i −0.864412 0.502784i \(-0.832309\pi\)
0.864412 0.502784i \(-0.167691\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.886643 −0.0291842
\(924\) 0 0
\(925\) −43.5298 −1.43125
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 20.7385i − 0.680408i −0.940352 0.340204i \(-0.889504\pi\)
0.940352 0.340204i \(-0.110496\pi\)
\(930\) 0 0
\(931\) − 5.89898i − 0.193331i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 67.4375 2.20544
\(936\) 0 0
\(937\) 38.5402 1.25905 0.629527 0.776979i \(-0.283249\pi\)
0.629527 + 0.776979i \(0.283249\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 44.2505i − 1.44253i −0.692662 0.721263i \(-0.743562\pi\)
0.692662 0.721263i \(-0.256438\pi\)
\(942\) 0 0
\(943\) 4.63103i 0.150807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.9046 −0.581820 −0.290910 0.956750i \(-0.593958\pi\)
−0.290910 + 0.956750i \(0.593958\pi\)
\(948\) 0 0
\(949\) 1.13953 0.0369908
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.8668i 0.708334i 0.935182 + 0.354167i \(0.115235\pi\)
−0.935182 + 0.354167i \(0.884765\pi\)
\(954\) 0 0
\(955\) 18.8505i 0.609988i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.7420 −0.669795
\(960\) 0 0
\(961\) 29.8539 0.963030
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 68.6365i − 2.20949i
\(966\) 0 0
\(967\) 42.8761i 1.37880i 0.724379 + 0.689402i \(0.242126\pi\)
−0.724379 + 0.689402i \(0.757874\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.19045 0.230753 0.115376 0.993322i \(-0.463193\pi\)
0.115376 + 0.993322i \(0.463193\pi\)
\(972\) 0 0
\(973\) −2.53590 −0.0812972
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.78834i 0.0572142i 0.999591 + 0.0286071i \(0.00910716\pi\)
−0.999591 + 0.0286071i \(0.990893\pi\)
\(978\) 0 0
\(979\) 18.3171i 0.585417i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32.6201 1.04042 0.520211 0.854038i \(-0.325853\pi\)
0.520211 + 0.854038i \(0.325853\pi\)
\(984\) 0 0
\(985\) −66.2843 −2.11199
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 58.4766i − 1.85945i
\(990\) 0 0
\(991\) 21.3866i 0.679369i 0.940539 + 0.339684i \(0.110320\pi\)
−0.940539 + 0.339684i \(0.889680\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 51.6557 1.63760
\(996\) 0 0
\(997\) 20.0560 0.635179 0.317589 0.948228i \(-0.397127\pi\)
0.317589 + 0.948228i \(0.397127\pi\)
\(998\) 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.h.g.2591.8 yes 8
3.2 odd 2 6048.2.h.a.2591.1 8
4.3 odd 2 6048.2.h.a.2591.8 yes 8
12.11 even 2 inner 6048.2.h.g.2591.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.a.2591.1 8 3.2 odd 2
6048.2.h.a.2591.8 yes 8 4.3 odd 2
6048.2.h.g.2591.1 yes 8 12.11 even 2 inner
6048.2.h.g.2591.8 yes 8 1.1 even 1 trivial