Properties

Label 6048.2.j.d.5615.15
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 48
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: \(48\)
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 5615.15
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.16

$q$-expansion

\(f(q)\) \(=\) \(q-1.27819 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-1.27819 q^{5} -1.00000i q^{7} +2.84853i q^{11} +0.223683i q^{13} -0.397844i q^{17} +4.64862 q^{19} -7.36177 q^{23} -3.36622 q^{25} +10.6317 q^{29} -7.67558i q^{31} +1.27819i q^{35} -4.74489i q^{37} +1.97843i q^{41} -7.62888 q^{43} -11.0724 q^{47} -1.00000 q^{49} -0.295366 q^{53} -3.64098i q^{55} +7.25779i q^{59} +9.45592i q^{61} -0.285910i q^{65} +3.52321 q^{67} +8.37710 q^{71} +10.6704 q^{73} +2.84853 q^{77} -5.22135i q^{79} -9.11868i q^{83} +0.508521i q^{85} +8.94887i q^{89} +0.223683 q^{91} -5.94184 q^{95} +0.228078 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

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
\(5\) −1.27819 −0.571625 −0.285813 0.958285i \(-0.592264\pi\)
−0.285813 + 0.958285i \(0.592264\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.84853i 0.858865i 0.903099 + 0.429433i \(0.141286\pi\)
−0.903099 + 0.429433i \(0.858714\pi\)
\(12\) 0 0
\(13\) 0.223683i 0.0620385i 0.999519 + 0.0310192i \(0.00987531\pi\)
−0.999519 + 0.0310192i \(0.990125\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.397844i − 0.0964913i −0.998836 0.0482456i \(-0.984637\pi\)
0.998836 0.0482456i \(-0.0153630\pi\)
\(18\) 0 0
\(19\) 4.64862 1.06647 0.533234 0.845968i \(-0.320977\pi\)
0.533234 + 0.845968i \(0.320977\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.36177 −1.53503 −0.767517 0.641028i \(-0.778508\pi\)
−0.767517 + 0.641028i \(0.778508\pi\)
\(24\) 0 0
\(25\) −3.36622 −0.673244
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.6317 1.97425 0.987124 0.159957i \(-0.0511356\pi\)
0.987124 + 0.159957i \(0.0511356\pi\)
\(30\) 0 0
\(31\) − 7.67558i − 1.37857i −0.724488 0.689287i \(-0.757924\pi\)
0.724488 0.689287i \(-0.242076\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.27819i 0.216054i
\(36\) 0 0
\(37\) − 4.74489i − 0.780055i −0.920803 0.390027i \(-0.872466\pi\)
0.920803 0.390027i \(-0.127534\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.97843i 0.308979i 0.987994 + 0.154490i \(0.0493733\pi\)
−0.987994 + 0.154490i \(0.950627\pi\)
\(42\) 0 0
\(43\) −7.62888 −1.16339 −0.581697 0.813406i \(-0.697611\pi\)
−0.581697 + 0.813406i \(0.697611\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.0724 −1.61508 −0.807538 0.589815i \(-0.799201\pi\)
−0.807538 + 0.589815i \(0.799201\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.295366 −0.0405717 −0.0202858 0.999794i \(-0.506458\pi\)
−0.0202858 + 0.999794i \(0.506458\pi\)
\(54\) 0 0
\(55\) − 3.64098i − 0.490949i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.25779i 0.944884i 0.881362 + 0.472442i \(0.156627\pi\)
−0.881362 + 0.472442i \(0.843373\pi\)
\(60\) 0 0
\(61\) 9.45592i 1.21071i 0.795957 + 0.605354i \(0.206968\pi\)
−0.795957 + 0.605354i \(0.793032\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.285910i − 0.0354628i
\(66\) 0 0
\(67\) 3.52321 0.430428 0.215214 0.976567i \(-0.430955\pi\)
0.215214 + 0.976567i \(0.430955\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.37710 0.994179 0.497089 0.867699i \(-0.334402\pi\)
0.497089 + 0.867699i \(0.334402\pi\)
\(72\) 0 0
\(73\) 10.6704 1.24887 0.624437 0.781076i \(-0.285329\pi\)
0.624437 + 0.781076i \(0.285329\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.84853 0.324621
\(78\) 0 0
\(79\) − 5.22135i − 0.587448i −0.955890 0.293724i \(-0.905105\pi\)
0.955890 0.293724i \(-0.0948947\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.11868i − 1.00091i −0.865764 0.500453i \(-0.833167\pi\)
0.865764 0.500453i \(-0.166833\pi\)
\(84\) 0 0
\(85\) 0.508521i 0.0551569i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.94887i 0.948578i 0.880369 + 0.474289i \(0.157295\pi\)
−0.880369 + 0.474289i \(0.842705\pi\)
\(90\) 0 0
\(91\) 0.223683 0.0234483
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.94184 −0.609620
\(96\) 0 0
\(97\) 0.228078 0.0231578 0.0115789 0.999933i \(-0.496314\pi\)
0.0115789 + 0.999933i \(0.496314\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.36504 0.633346 0.316673 0.948535i \(-0.397434\pi\)
0.316673 + 0.948535i \(0.397434\pi\)
\(102\) 0 0
\(103\) − 11.6557i − 1.14847i −0.818690 0.574236i \(-0.805299\pi\)
0.818690 0.574236i \(-0.194701\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.01403i − 0.871419i −0.900087 0.435709i \(-0.856498\pi\)
0.900087 0.435709i \(-0.143502\pi\)
\(108\) 0 0
\(109\) − 5.45206i − 0.522213i −0.965310 0.261106i \(-0.915913\pi\)
0.965310 0.261106i \(-0.0840873\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 7.00229i − 0.658720i −0.944204 0.329360i \(-0.893167\pi\)
0.944204 0.329360i \(-0.106833\pi\)
\(114\) 0 0
\(115\) 9.40976 0.877465
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.397844 −0.0364703
\(120\) 0 0
\(121\) 2.88586 0.262351
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.6936 0.956469
\(126\) 0 0
\(127\) − 9.48046i − 0.841255i −0.907234 0.420627i \(-0.861810\pi\)
0.907234 0.420627i \(-0.138190\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.34169i 0.554076i 0.960859 + 0.277038i \(0.0893528\pi\)
−0.960859 + 0.277038i \(0.910647\pi\)
\(132\) 0 0
\(133\) − 4.64862i − 0.403087i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.3108i − 0.966349i −0.875524 0.483174i \(-0.839484\pi\)
0.875524 0.483174i \(-0.160516\pi\)
\(138\) 0 0
\(139\) −21.7583 −1.84551 −0.922757 0.385382i \(-0.874070\pi\)
−0.922757 + 0.385382i \(0.874070\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.637168 −0.0532827
\(144\) 0 0
\(145\) −13.5893 −1.12853
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.42505 0.198668 0.0993340 0.995054i \(-0.468329\pi\)
0.0993340 + 0.995054i \(0.468329\pi\)
\(150\) 0 0
\(151\) − 20.0398i − 1.63082i −0.578887 0.815408i \(-0.696513\pi\)
0.578887 0.815408i \(-0.303487\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.81087i 0.788028i
\(156\) 0 0
\(157\) − 15.5990i − 1.24494i −0.782644 0.622470i \(-0.786129\pi\)
0.782644 0.622470i \(-0.213871\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.36177i 0.580189i
\(162\) 0 0
\(163\) −12.8755 −1.00848 −0.504242 0.863562i \(-0.668228\pi\)
−0.504242 + 0.863562i \(0.668228\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.55771 −0.120539 −0.0602697 0.998182i \(-0.519196\pi\)
−0.0602697 + 0.998182i \(0.519196\pi\)
\(168\) 0 0
\(169\) 12.9500 0.996151
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.60553 0.198095 0.0990474 0.995083i \(-0.468420\pi\)
0.0990474 + 0.995083i \(0.468420\pi\)
\(174\) 0 0
\(175\) 3.36622i 0.254462i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.9293i 1.41484i 0.706793 + 0.707420i \(0.250141\pi\)
−0.706793 + 0.707420i \(0.749859\pi\)
\(180\) 0 0
\(181\) − 15.6813i − 1.16558i −0.812621 0.582792i \(-0.801960\pi\)
0.812621 0.582792i \(-0.198040\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.06488i 0.445899i
\(186\) 0 0
\(187\) 1.13327 0.0828730
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.86888 −0.279943 −0.139971 0.990156i \(-0.544701\pi\)
−0.139971 + 0.990156i \(0.544701\pi\)
\(192\) 0 0
\(193\) −12.1061 −0.871419 −0.435710 0.900087i \(-0.643503\pi\)
−0.435710 + 0.900087i \(0.643503\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.20058 −0.370526 −0.185263 0.982689i \(-0.559314\pi\)
−0.185263 + 0.982689i \(0.559314\pi\)
\(198\) 0 0
\(199\) 15.5030i 1.09898i 0.835501 + 0.549489i \(0.185177\pi\)
−0.835501 + 0.549489i \(0.814823\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 10.6317i − 0.746196i
\(204\) 0 0
\(205\) − 2.52882i − 0.176620i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.2418i 0.915952i
\(210\) 0 0
\(211\) 9.04096 0.622406 0.311203 0.950344i \(-0.399268\pi\)
0.311203 + 0.950344i \(0.399268\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.75118 0.665025
\(216\) 0 0
\(217\) −7.67558 −0.521052
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.0889909 0.00598617
\(222\) 0 0
\(223\) − 10.2327i − 0.685231i −0.939476 0.342616i \(-0.888687\pi\)
0.939476 0.342616i \(-0.111313\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 10.9987i − 0.730007i −0.931006 0.365003i \(-0.881068\pi\)
0.931006 0.365003i \(-0.118932\pi\)
\(228\) 0 0
\(229\) − 24.1988i − 1.59910i −0.600600 0.799550i \(-0.705072\pi\)
0.600600 0.799550i \(-0.294928\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 21.5448i − 1.41144i −0.708489 0.705722i \(-0.750623\pi\)
0.708489 0.705722i \(-0.249377\pi\)
\(234\) 0 0
\(235\) 14.1527 0.923219
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.5708 −1.33061 −0.665307 0.746570i \(-0.731699\pi\)
−0.665307 + 0.746570i \(0.731699\pi\)
\(240\) 0 0
\(241\) −11.0840 −0.713984 −0.356992 0.934107i \(-0.616198\pi\)
−0.356992 + 0.934107i \(0.616198\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.27819 0.0816608
\(246\) 0 0
\(247\) 1.03982i 0.0661620i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 20.7016i − 1.30667i −0.757068 0.653336i \(-0.773369\pi\)
0.757068 0.653336i \(-0.226631\pi\)
\(252\) 0 0
\(253\) − 20.9702i − 1.31839i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.6516i − 0.664427i −0.943204 0.332213i \(-0.892205\pi\)
0.943204 0.332213i \(-0.107795\pi\)
\(258\) 0 0
\(259\) −4.74489 −0.294833
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.36313 0.392367 0.196184 0.980567i \(-0.437145\pi\)
0.196184 + 0.980567i \(0.437145\pi\)
\(264\) 0 0
\(265\) 0.377535 0.0231918
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.3840 −0.877007 −0.438503 0.898730i \(-0.644491\pi\)
−0.438503 + 0.898730i \(0.644491\pi\)
\(270\) 0 0
\(271\) − 12.9317i − 0.785548i −0.919635 0.392774i \(-0.871515\pi\)
0.919635 0.392774i \(-0.128485\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 9.58880i − 0.578226i
\(276\) 0 0
\(277\) − 13.5242i − 0.812593i −0.913741 0.406297i \(-0.866820\pi\)
0.913741 0.406297i \(-0.133180\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 20.7882i − 1.24012i −0.784554 0.620061i \(-0.787108\pi\)
0.784554 0.620061i \(-0.212892\pi\)
\(282\) 0 0
\(283\) 17.6119 1.04692 0.523459 0.852051i \(-0.324641\pi\)
0.523459 + 0.852051i \(0.324641\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.97843 0.116783
\(288\) 0 0
\(289\) 16.8417 0.990689
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.42581 −0.141718 −0.0708588 0.997486i \(-0.522574\pi\)
−0.0708588 + 0.997486i \(0.522574\pi\)
\(294\) 0 0
\(295\) − 9.27687i − 0.540120i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.64670i − 0.0952312i
\(300\) 0 0
\(301\) 7.62888i 0.439721i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 12.0865i − 0.692071i
\(306\) 0 0
\(307\) 5.15871 0.294423 0.147212 0.989105i \(-0.452970\pi\)
0.147212 + 0.989105i \(0.452970\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −31.1304 −1.76524 −0.882620 0.470086i \(-0.844223\pi\)
−0.882620 + 0.470086i \(0.844223\pi\)
\(312\) 0 0
\(313\) −21.8080 −1.23266 −0.616330 0.787488i \(-0.711381\pi\)
−0.616330 + 0.787488i \(0.711381\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.309869 −0.0174040 −0.00870199 0.999962i \(-0.502770\pi\)
−0.00870199 + 0.999962i \(0.502770\pi\)
\(318\) 0 0
\(319\) 30.2846i 1.69561i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 1.84943i − 0.102905i
\(324\) 0 0
\(325\) − 0.752966i − 0.0417670i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0724i 0.610441i
\(330\) 0 0
\(331\) −28.5893 −1.57141 −0.785706 0.618600i \(-0.787700\pi\)
−0.785706 + 0.618600i \(0.787700\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.50334 −0.246044
\(336\) 0 0
\(337\) 26.7103 1.45500 0.727502 0.686106i \(-0.240681\pi\)
0.727502 + 0.686106i \(0.240681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.8641 1.18401
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.0302i 0.592132i 0.955168 + 0.296066i \(0.0956748\pi\)
−0.955168 + 0.296066i \(0.904325\pi\)
\(348\) 0 0
\(349\) 9.30132i 0.497888i 0.968518 + 0.248944i \(0.0800836\pi\)
−0.968518 + 0.248944i \(0.919916\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.4585i 0.982446i 0.871034 + 0.491223i \(0.163450\pi\)
−0.871034 + 0.491223i \(0.836550\pi\)
\(354\) 0 0
\(355\) −10.7076 −0.568298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.70098 0.406442 0.203221 0.979133i \(-0.434859\pi\)
0.203221 + 0.979133i \(0.434859\pi\)
\(360\) 0 0
\(361\) 2.60971 0.137353
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.6388 −0.713888
\(366\) 0 0
\(367\) − 36.0127i − 1.87985i −0.341386 0.939923i \(-0.610896\pi\)
0.341386 0.939923i \(-0.389104\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.295366i 0.0153346i
\(372\) 0 0
\(373\) − 32.8248i − 1.69960i −0.527103 0.849801i \(-0.676722\pi\)
0.527103 0.849801i \(-0.323278\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.37812i 0.122479i
\(378\) 0 0
\(379\) 27.8200 1.42902 0.714509 0.699626i \(-0.246650\pi\)
0.714509 + 0.699626i \(0.246650\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.9047 −0.608301 −0.304151 0.952624i \(-0.598373\pi\)
−0.304151 + 0.952624i \(0.598373\pi\)
\(384\) 0 0
\(385\) −3.64098 −0.185561
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.8388 1.25938 0.629688 0.776848i \(-0.283183\pi\)
0.629688 + 0.776848i \(0.283183\pi\)
\(390\) 0 0
\(391\) 2.92883i 0.148117i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.67390i 0.335800i
\(396\) 0 0
\(397\) − 4.49543i − 0.225619i −0.993617 0.112810i \(-0.964015\pi\)
0.993617 0.112810i \(-0.0359850\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.8094i 0.739545i 0.929122 + 0.369772i \(0.120564\pi\)
−0.929122 + 0.369772i \(0.879436\pi\)
\(402\) 0 0
\(403\) 1.71689 0.0855246
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.5160 0.669962
\(408\) 0 0
\(409\) −33.4088 −1.65196 −0.825980 0.563699i \(-0.809378\pi\)
−0.825980 + 0.563699i \(0.809378\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.25779 0.357133
\(414\) 0 0
\(415\) 11.6554i 0.572143i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.31558i 0.308536i 0.988029 + 0.154268i \(0.0493020\pi\)
−0.988029 + 0.154268i \(0.950698\pi\)
\(420\) 0 0
\(421\) 10.6716i 0.520104i 0.965595 + 0.260052i \(0.0837397\pi\)
−0.965595 + 0.260052i \(0.916260\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.33923i 0.0649622i
\(426\) 0 0
\(427\) 9.45592 0.457604
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.8686 1.72773 0.863865 0.503723i \(-0.168037\pi\)
0.863865 + 0.503723i \(0.168037\pi\)
\(432\) 0 0
\(433\) 4.43908 0.213328 0.106664 0.994295i \(-0.465983\pi\)
0.106664 + 0.994295i \(0.465983\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −34.2221 −1.63706
\(438\) 0 0
\(439\) − 18.7939i − 0.896985i −0.893787 0.448493i \(-0.851961\pi\)
0.893787 0.448493i \(-0.148039\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.9630i 1.47109i 0.677473 + 0.735547i \(0.263075\pi\)
−0.677473 + 0.735547i \(0.736925\pi\)
\(444\) 0 0
\(445\) − 11.4384i − 0.542232i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.6744i 1.16446i 0.813025 + 0.582229i \(0.197819\pi\)
−0.813025 + 0.582229i \(0.802181\pi\)
\(450\) 0 0
\(451\) −5.63563 −0.265372
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.285910 −0.0134037
\(456\) 0 0
\(457\) 13.3915 0.626429 0.313214 0.949682i \(-0.398594\pi\)
0.313214 + 0.949682i \(0.398594\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.3835 0.576759 0.288379 0.957516i \(-0.406884\pi\)
0.288379 + 0.957516i \(0.406884\pi\)
\(462\) 0 0
\(463\) − 3.34707i − 0.155551i −0.996971 0.0777757i \(-0.975218\pi\)
0.996971 0.0777757i \(-0.0247818\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.05078i − 0.141173i −0.997506 0.0705866i \(-0.977513\pi\)
0.997506 0.0705866i \(-0.0224871\pi\)
\(468\) 0 0
\(469\) − 3.52321i − 0.162687i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 21.7311i − 0.999198i
\(474\) 0 0
\(475\) −15.6483 −0.717993
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.4934 −0.936368 −0.468184 0.883631i \(-0.655092\pi\)
−0.468184 + 0.883631i \(0.655092\pi\)
\(480\) 0 0
\(481\) 1.06135 0.0483934
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.291528 −0.0132376
\(486\) 0 0
\(487\) − 21.9186i − 0.993226i −0.867972 0.496613i \(-0.834577\pi\)
0.867972 0.496613i \(-0.165423\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 10.2880i − 0.464291i −0.972681 0.232146i \(-0.925425\pi\)
0.972681 0.232146i \(-0.0745746\pi\)
\(492\) 0 0
\(493\) − 4.22974i − 0.190498i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.37710i − 0.375764i
\(498\) 0 0
\(499\) 1.87931 0.0841294 0.0420647 0.999115i \(-0.486606\pi\)
0.0420647 + 0.999115i \(0.486606\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.5489 0.648702 0.324351 0.945937i \(-0.394854\pi\)
0.324351 + 0.945937i \(0.394854\pi\)
\(504\) 0 0
\(505\) −8.13576 −0.362036
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −25.7122 −1.13967 −0.569837 0.821758i \(-0.692994\pi\)
−0.569837 + 0.821758i \(0.692994\pi\)
\(510\) 0 0
\(511\) − 10.6704i − 0.472030i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.8983i 0.656496i
\(516\) 0 0
\(517\) − 31.5401i − 1.38713i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 8.03293i − 0.351929i −0.984397 0.175965i \(-0.943696\pi\)
0.984397 0.175965i \(-0.0563044\pi\)
\(522\) 0 0
\(523\) −17.2938 −0.756206 −0.378103 0.925764i \(-0.623423\pi\)
−0.378103 + 0.925764i \(0.623423\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.05368 −0.133020
\(528\) 0 0
\(529\) 31.1956 1.35633
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.442542 −0.0191686
\(534\) 0 0
\(535\) 11.5217i 0.498125i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.84853i − 0.122695i
\(540\) 0 0
\(541\) − 6.58504i − 0.283113i −0.989930 0.141557i \(-0.954789\pi\)
0.989930 0.141557i \(-0.0452107\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.96879i 0.298510i
\(546\) 0 0
\(547\) 33.4478 1.43012 0.715062 0.699061i \(-0.246398\pi\)
0.715062 + 0.699061i \(0.246398\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 49.4226 2.10547
\(552\) 0 0
\(553\) −5.22135 −0.222035
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.92223 0.250933 0.125467 0.992098i \(-0.459957\pi\)
0.125467 + 0.992098i \(0.459957\pi\)
\(558\) 0 0
\(559\) − 1.70645i − 0.0721751i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 23.1453i − 0.975459i −0.872995 0.487730i \(-0.837825\pi\)
0.872995 0.487730i \(-0.162175\pi\)
\(564\) 0 0
\(565\) 8.95028i 0.376541i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.41139i 0.0591686i 0.999562 + 0.0295843i \(0.00941835\pi\)
−0.999562 + 0.0295843i \(0.990582\pi\)
\(570\) 0 0
\(571\) 6.82194 0.285489 0.142745 0.989760i \(-0.454407\pi\)
0.142745 + 0.989760i \(0.454407\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.7813 1.03345
\(576\) 0 0
\(577\) 6.97587 0.290409 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.11868 −0.378307
\(582\) 0 0
\(583\) − 0.841360i − 0.0348456i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 40.3481i − 1.66534i −0.553766 0.832672i \(-0.686810\pi\)
0.553766 0.832672i \(-0.313190\pi\)
\(588\) 0 0
\(589\) − 35.6809i − 1.47020i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 6.40544i − 0.263040i −0.991314 0.131520i \(-0.958014\pi\)
0.991314 0.131520i \(-0.0419858\pi\)
\(594\) 0 0
\(595\) 0.508521 0.0208473
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.80603 0.114651 0.0573257 0.998356i \(-0.481743\pi\)
0.0573257 + 0.998356i \(0.481743\pi\)
\(600\) 0 0
\(601\) 11.6165 0.473847 0.236924 0.971528i \(-0.423861\pi\)
0.236924 + 0.971528i \(0.423861\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.68868 −0.149966
\(606\) 0 0
\(607\) 38.5388i 1.56424i 0.623126 + 0.782122i \(0.285862\pi\)
−0.623126 + 0.782122i \(0.714138\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.47671i − 0.100197i
\(612\) 0 0
\(613\) 15.8274i 0.639261i 0.947542 + 0.319631i \(0.103559\pi\)
−0.947542 + 0.319631i \(0.896441\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.9040i 0.962339i 0.876628 + 0.481170i \(0.159788\pi\)
−0.876628 + 0.481170i \(0.840212\pi\)
\(618\) 0 0
\(619\) 31.4193 1.26285 0.631424 0.775437i \(-0.282471\pi\)
0.631424 + 0.775437i \(0.282471\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.94887 0.358529
\(624\) 0 0
\(625\) 3.16256 0.126502
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.88772 −0.0752685
\(630\) 0 0
\(631\) 6.94219i 0.276364i 0.990407 + 0.138182i \(0.0441260\pi\)
−0.990407 + 0.138182i \(0.955874\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.1179i 0.480883i
\(636\) 0 0
\(637\) − 0.223683i − 0.00886264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 31.1823i − 1.23163i −0.787892 0.615814i \(-0.788827\pi\)
0.787892 0.615814i \(-0.211173\pi\)
\(642\) 0 0
\(643\) −22.6881 −0.894733 −0.447366 0.894351i \(-0.647638\pi\)
−0.447366 + 0.894351i \(0.647638\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.7608 1.48453 0.742266 0.670106i \(-0.233751\pi\)
0.742266 + 0.670106i \(0.233751\pi\)
\(648\) 0 0
\(649\) −20.6741 −0.811528
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −47.8848 −1.87388 −0.936939 0.349494i \(-0.886353\pi\)
−0.936939 + 0.349494i \(0.886353\pi\)
\(654\) 0 0
\(655\) − 8.10591i − 0.316724i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 7.00120i − 0.272728i −0.990659 0.136364i \(-0.956458\pi\)
0.990659 0.136364i \(-0.0435417\pi\)
\(660\) 0 0
\(661\) 41.8042i 1.62599i 0.582267 + 0.812997i \(0.302166\pi\)
−0.582267 + 0.812997i \(0.697834\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.94184i 0.230415i
\(666\) 0 0
\(667\) −78.2677 −3.03054
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −26.9355 −1.03983
\(672\) 0 0
\(673\) 22.2519 0.857747 0.428874 0.903364i \(-0.358911\pi\)
0.428874 + 0.903364i \(0.358911\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.2219 −1.12309 −0.561544 0.827447i \(-0.689793\pi\)
−0.561544 + 0.827447i \(0.689793\pi\)
\(678\) 0 0
\(679\) − 0.228078i − 0.00875283i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.0080i 1.33954i 0.742567 + 0.669772i \(0.233608\pi\)
−0.742567 + 0.669772i \(0.766392\pi\)
\(684\) 0 0
\(685\) 14.4574i 0.552390i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 0.0660684i − 0.00251700i
\(690\) 0 0
\(691\) −1.33849 −0.0509185 −0.0254593 0.999676i \(-0.508105\pi\)
−0.0254593 + 0.999676i \(0.508105\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.8113 1.05494
\(696\) 0 0
\(697\) 0.787107 0.0298138
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0621 0.871044 0.435522 0.900178i \(-0.356564\pi\)
0.435522 + 0.900178i \(0.356564\pi\)
\(702\) 0 0
\(703\) − 22.0572i − 0.831903i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.36504i − 0.239382i
\(708\) 0 0
\(709\) 46.8150i 1.75818i 0.476660 + 0.879088i \(0.341847\pi\)
−0.476660 + 0.879088i \(0.658153\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 56.5058i 2.11616i
\(714\) 0 0
\(715\) 0.814424 0.0304577
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.8849 −1.30099 −0.650493 0.759512i \(-0.725438\pi\)
−0.650493 + 0.759512i \(0.725438\pi\)
\(720\) 0 0
\(721\) −11.6557 −0.434082
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −35.7885 −1.32915
\(726\) 0 0
\(727\) 16.4491i 0.610063i 0.952342 + 0.305032i \(0.0986671\pi\)
−0.952342 + 0.305032i \(0.901333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.03510i 0.112257i
\(732\) 0 0
\(733\) 35.9855i 1.32915i 0.747220 + 0.664577i \(0.231388\pi\)
−0.747220 + 0.664577i \(0.768612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0360i 0.369680i
\(738\) 0 0
\(739\) 5.44866 0.200432 0.100216 0.994966i \(-0.468047\pi\)
0.100216 + 0.994966i \(0.468047\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.2115 1.14504 0.572520 0.819891i \(-0.305966\pi\)
0.572520 + 0.819891i \(0.305966\pi\)
\(744\) 0 0
\(745\) −3.09969 −0.113564
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.01403 −0.329365
\(750\) 0 0
\(751\) − 23.1812i − 0.845894i −0.906154 0.422947i \(-0.860996\pi\)
0.906154 0.422947i \(-0.139004\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.6147i 0.932216i
\(756\) 0 0
\(757\) 39.3962i 1.43188i 0.698162 + 0.715940i \(0.254002\pi\)
−0.698162 + 0.715940i \(0.745998\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 17.1356i − 0.621166i −0.950546 0.310583i \(-0.899476\pi\)
0.950546 0.310583i \(-0.100524\pi\)
\(762\) 0 0
\(763\) −5.45206 −0.197378
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.62344 −0.0586192
\(768\) 0 0
\(769\) −44.9084 −1.61944 −0.809718 0.586819i \(-0.800380\pi\)
−0.809718 + 0.586819i \(0.800380\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.0273 0.972106 0.486053 0.873929i \(-0.338436\pi\)
0.486053 + 0.873929i \(0.338436\pi\)
\(774\) 0 0
\(775\) 25.8377i 0.928117i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.19699i 0.329516i
\(780\) 0 0
\(781\) 23.8624i 0.853866i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 19.9386i 0.711639i
\(786\) 0 0
\(787\) 34.1562 1.21754 0.608769 0.793348i \(-0.291664\pi\)
0.608769 + 0.793348i \(0.291664\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.00229 −0.248973
\(792\) 0 0
\(793\) −2.11513 −0.0751104
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −46.2712 −1.63901 −0.819505 0.573072i \(-0.805751\pi\)
−0.819505 + 0.573072i \(0.805751\pi\)
\(798\) 0 0
\(799\) 4.40509i 0.155841i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.3949i 1.07261i
\(804\) 0 0
\(805\) − 9.40976i − 0.331651i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 26.3429i − 0.926167i −0.886315 0.463084i \(-0.846743\pi\)
0.886315 0.463084i \(-0.153257\pi\)
\(810\) 0 0
\(811\) −2.96401 −0.104080 −0.0520402 0.998645i \(-0.516572\pi\)
−0.0520402 + 0.998645i \(0.516572\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.4573 0.576475
\(816\) 0 0
\(817\) −35.4638 −1.24072
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.3996 −0.362949 −0.181475 0.983396i \(-0.558087\pi\)
−0.181475 + 0.983396i \(0.558087\pi\)
\(822\) 0 0
\(823\) − 39.8598i − 1.38942i −0.719288 0.694712i \(-0.755532\pi\)
0.719288 0.694712i \(-0.244468\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 27.5060i − 0.956477i −0.878230 0.478239i \(-0.841275\pi\)
0.878230 0.478239i \(-0.158725\pi\)
\(828\) 0 0
\(829\) − 38.0086i − 1.32009i −0.751225 0.660047i \(-0.770536\pi\)
0.751225 0.660047i \(-0.229464\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.397844i 0.0137845i
\(834\) 0 0
\(835\) 1.99106 0.0689034
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.9037 1.06691 0.533457 0.845827i \(-0.320893\pi\)
0.533457 + 0.845827i \(0.320893\pi\)
\(840\) 0 0
\(841\) 84.0320 2.89765
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −16.5526 −0.569425
\(846\) 0 0
\(847\) − 2.88586i − 0.0991592i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 34.9308i 1.19741i
\(852\) 0 0
\(853\) 25.3302i 0.867290i 0.901084 + 0.433645i \(0.142773\pi\)
−0.901084 + 0.433645i \(0.857227\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.1574i 1.33759i 0.743447 + 0.668795i \(0.233190\pi\)
−0.743447 + 0.668795i \(0.766810\pi\)
\(858\) 0 0
\(859\) −11.2783 −0.384810 −0.192405 0.981316i \(-0.561629\pi\)
−0.192405 + 0.981316i \(0.561629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15.6543 −0.532878 −0.266439 0.963852i \(-0.585847\pi\)
−0.266439 + 0.963852i \(0.585847\pi\)
\(864\) 0 0
\(865\) −3.33037 −0.113236
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.8732 0.504539
\(870\) 0 0
\(871\) 0.788081i 0.0267031i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.6936i − 0.361511i
\(876\) 0 0
\(877\) 14.6357i 0.494211i 0.968989 + 0.247106i \(0.0794795\pi\)
−0.968989 + 0.247106i \(0.920521\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.9054i 0.738013i 0.929427 + 0.369007i \(0.120302\pi\)
−0.929427 + 0.369007i \(0.879698\pi\)
\(882\) 0 0
\(883\) 36.3852 1.22446 0.612230 0.790680i \(-0.290273\pi\)
0.612230 + 0.790680i \(0.290273\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.3554 −0.347700 −0.173850 0.984772i \(-0.555621\pi\)
−0.173850 + 0.984772i \(0.555621\pi\)
\(888\) 0 0
\(889\) −9.48046 −0.317964
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −51.4715 −1.72243
\(894\) 0 0
\(895\) − 24.1953i − 0.808759i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 81.6040i − 2.72165i
\(900\) 0 0
\(901\) 0.117510i 0.00391481i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.0438i 0.666278i
\(906\) 0 0
\(907\) −10.5615 −0.350689 −0.175344 0.984507i \(-0.556104\pi\)
−0.175344 + 0.984507i \(0.556104\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.4732 −0.612044 −0.306022 0.952024i \(-0.598998\pi\)
−0.306022 + 0.952024i \(0.598998\pi\)
\(912\) 0 0
\(913\) 25.9749 0.859643
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.34169 0.209421
\(918\) 0 0
\(919\) − 27.7252i − 0.914569i −0.889320 0.457285i \(-0.848822\pi\)
0.889320 0.457285i \(-0.151178\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.87381i 0.0616773i
\(924\) 0 0
\(925\) 15.9723i 0.525167i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 57.6318i − 1.89084i −0.325858 0.945419i \(-0.605653\pi\)
0.325858 0.945419i \(-0.394347\pi\)
\(930\) 0 0
\(931\) −4.64862 −0.152353
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.44854 −0.0473723
\(936\) 0 0
\(937\) −44.9081 −1.46708 −0.733542 0.679644i \(-0.762134\pi\)
−0.733542 + 0.679644i \(0.762134\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 25.0167 0.815521 0.407761 0.913089i \(-0.366310\pi\)
0.407761 + 0.913089i \(0.366310\pi\)
\(942\) 0 0
\(943\) − 14.5648i − 0.474294i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.7909i 1.29303i 0.762901 + 0.646516i \(0.223775\pi\)
−0.762901 + 0.646516i \(0.776225\pi\)
\(948\) 0 0
\(949\) 2.38678i 0.0774782i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.4544i 0.403437i 0.979444 + 0.201718i \(0.0646526\pi\)
−0.979444 + 0.201718i \(0.935347\pi\)
\(954\) 0 0
\(955\) 4.94518 0.160022
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.3108 −0.365246
\(960\) 0 0
\(961\) −27.9145 −0.900466
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.4740 0.498125
\(966\) 0 0
\(967\) 6.97106i 0.224174i 0.993698 + 0.112087i \(0.0357536\pi\)
−0.993698 + 0.112087i \(0.964246\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.18374i 0.294720i 0.989083 + 0.147360i \(0.0470776\pi\)
−0.989083 + 0.147360i \(0.952922\pi\)
\(972\) 0 0
\(973\) 21.7583i 0.697539i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 39.1316i − 1.25193i −0.779851 0.625965i \(-0.784705\pi\)
0.779851 0.625965i \(-0.215295\pi\)
\(978\) 0 0
\(979\) −25.4912 −0.814701
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50.4121 1.60790 0.803948 0.594700i \(-0.202729\pi\)
0.803948 + 0.594700i \(0.202729\pi\)
\(984\) 0 0
\(985\) 6.64735 0.211802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 56.1620 1.78585
\(990\) 0 0
\(991\) 55.9071i 1.77595i 0.459894 + 0.887974i \(0.347887\pi\)
−0.459894 + 0.887974i \(0.652113\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 19.8158i − 0.628204i
\(996\) 0 0
\(997\) 11.6669i 0.369494i 0.982786 + 0.184747i \(0.0591466\pi\)
−0.982786 + 0.184747i \(0.940853\pi\)
\(998\) 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.d.5615.15 48
3.2 odd 2 inner 6048.2.j.d.5615.34 48
4.3 odd 2 1512.2.j.d.323.24 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.33 48
8.5 even 2 1512.2.j.d.323.26 yes 48
12.11 even 2 1512.2.j.d.323.25 yes 48
24.5 odd 2 1512.2.j.d.323.23 48
24.11 even 2 inner 6048.2.j.d.5615.16 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.23 48 24.5 odd 2
1512.2.j.d.323.24 yes 48 4.3 odd 2
1512.2.j.d.323.25 yes 48 12.11 even 2
1512.2.j.d.323.26 yes 48 8.5 even 2
6048.2.j.d.5615.15 48 1.1 even 1 trivial
6048.2.j.d.5615.16 48 24.11 even 2 inner
6048.2.j.d.5615.33 48 8.3 odd 2 inner
6048.2.j.d.5615.34 48 3.2 odd 2 inner