Properties

Label 6048.2.h.b.2591.8
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
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.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 \(-0.965926 + 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.b.2591.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.38134i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+4.38134i q^{5} +1.00000i q^{7} +2.31079 q^{11} -4.73205 q^{13} +5.27792i q^{17} +5.00000i q^{19} +0.517638 q^{23} -14.1962 q^{25} -4.24264i q^{29} +3.53590i q^{31} -4.38134 q^{35} -4.46410 q^{37} -6.45189i q^{41} +1.26795i q^{43} -8.10634 q^{47} -1.00000 q^{49} -5.93426i q^{53} +10.1244i q^{55} -6.31319 q^{59} +8.92820 q^{61} -20.7327i q^{65} -9.66025i q^{67} -12.4877 q^{71} +10.1962 q^{73} +2.31079i q^{77} +12.1962i q^{79} +17.6269 q^{83} -23.1244 q^{85} +0.517638i q^{89} -4.73205i q^{91} -21.9067 q^{95} +0.928203 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{13} - 72q^{25} - 8q^{37} - 8q^{49} + 16q^{61} + 40q^{73} - 88q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) 4.38134i 1.95940i 0.200480 + 0.979698i \(0.435750\pi\)
−0.200480 + 0.979698i \(0.564250\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.31079 0.696729 0.348365 0.937359i \(-0.386737\pi\)
0.348365 + 0.937359i \(0.386737\pi\)
\(12\) 0 0
\(13\) −4.73205 −1.31243 −0.656217 0.754572i \(-0.727845\pi\)
−0.656217 + 0.754572i \(0.727845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.27792i 1.28008i 0.768340 + 0.640041i \(0.221083\pi\)
−0.768340 + 0.640041i \(0.778917\pi\)
\(18\) 0 0
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.517638 0.107935 0.0539675 0.998543i \(-0.482813\pi\)
0.0539675 + 0.998543i \(0.482813\pi\)
\(24\) 0 0
\(25\) −14.1962 −2.83923
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.787839i −0.919145 0.393919i \(-0.871119\pi\)
0.919145 0.393919i \(-0.128881\pi\)
\(30\) 0 0
\(31\) 3.53590i 0.635066i 0.948247 + 0.317533i \(0.102854\pi\)
−0.948247 + 0.317533i \(0.897146\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.38134 −0.740582
\(36\) 0 0
\(37\) −4.46410 −0.733894 −0.366947 0.930242i \(-0.619597\pi\)
−0.366947 + 0.930242i \(0.619597\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.45189i − 1.00762i −0.863816 0.503808i \(-0.831932\pi\)
0.863816 0.503808i \(-0.168068\pi\)
\(42\) 0 0
\(43\) 1.26795i 0.193360i 0.995315 + 0.0966802i \(0.0308224\pi\)
−0.995315 + 0.0966802i \(0.969178\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.10634 −1.18243 −0.591216 0.806513i \(-0.701352\pi\)
−0.591216 + 0.806513i \(0.701352\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.93426i − 0.815133i −0.913176 0.407566i \(-0.866378\pi\)
0.913176 0.407566i \(-0.133622\pi\)
\(54\) 0 0
\(55\) 10.1244i 1.36517i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.31319 −0.821908 −0.410954 0.911656i \(-0.634804\pi\)
−0.410954 + 0.911656i \(0.634804\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 20.7327i − 2.57158i
\(66\) 0 0
\(67\) − 9.66025i − 1.18019i −0.807335 0.590094i \(-0.799091\pi\)
0.807335 0.590094i \(-0.200909\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4877 −1.48202 −0.741008 0.671496i \(-0.765652\pi\)
−0.741008 + 0.671496i \(0.765652\pi\)
\(72\) 0 0
\(73\) 10.1962 1.19337 0.596685 0.802476i \(-0.296484\pi\)
0.596685 + 0.802476i \(0.296484\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.31079i 0.263339i
\(78\) 0 0
\(79\) 12.1962i 1.37217i 0.727519 + 0.686087i \(0.240673\pi\)
−0.727519 + 0.686087i \(0.759327\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.6269 1.93480 0.967402 0.253246i \(-0.0814982\pi\)
0.967402 + 0.253246i \(0.0814982\pi\)
\(84\) 0 0
\(85\) −23.1244 −2.50819
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.517638i 0.0548695i 0.999624 + 0.0274348i \(0.00873385\pi\)
−0.999624 + 0.0274348i \(0.991266\pi\)
\(90\) 0 0
\(91\) − 4.73205i − 0.496054i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.9067 −2.24758
\(96\) 0 0
\(97\) 0.928203 0.0942448 0.0471224 0.998889i \(-0.484995\pi\)
0.0471224 + 0.998889i \(0.484995\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.1769i 1.01264i 0.862346 + 0.506320i \(0.168994\pi\)
−0.862346 + 0.506320i \(0.831006\pi\)
\(102\) 0 0
\(103\) 10.4641i 1.03106i 0.856872 + 0.515529i \(0.172405\pi\)
−0.856872 + 0.515529i \(0.827595\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.7632 1.33054 0.665269 0.746603i \(-0.268317\pi\)
0.665269 + 0.746603i \(0.268317\pi\)
\(108\) 0 0
\(109\) 13.5885 1.30154 0.650769 0.759276i \(-0.274447\pi\)
0.650769 + 0.759276i \(0.274447\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.48528i 0.798228i 0.916901 + 0.399114i \(0.130682\pi\)
−0.916901 + 0.399114i \(0.869318\pi\)
\(114\) 0 0
\(115\) 2.26795i 0.211487i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.27792 −0.483826
\(120\) 0 0
\(121\) −5.66025 −0.514569
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 40.2915i − 3.60378i
\(126\) 0 0
\(127\) − 0.339746i − 0.0301476i −0.999886 0.0150738i \(-0.995202\pi\)
0.999886 0.0150738i \(-0.00479832\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.41421 0.123560 0.0617802 0.998090i \(-0.480322\pi\)
0.0617802 + 0.998090i \(0.480322\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.31319i − 0.539372i −0.962948 0.269686i \(-0.913080\pi\)
0.962948 0.269686i \(-0.0869200\pi\)
\(138\) 0 0
\(139\) − 23.3205i − 1.97802i −0.147850 0.989010i \(-0.547235\pi\)
0.147850 0.989010i \(-0.452765\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.9348 −0.914411
\(144\) 0 0
\(145\) 18.5885 1.54369
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.7990i 1.62200i 0.585049 + 0.810998i \(0.301075\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(150\) 0 0
\(151\) − 1.07180i − 0.0872216i −0.999049 0.0436108i \(-0.986114\pi\)
0.999049 0.0436108i \(-0.0138862\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −15.4920 −1.24435
\(156\) 0 0
\(157\) 3.26795 0.260811 0.130405 0.991461i \(-0.458372\pi\)
0.130405 + 0.991461i \(0.458372\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.517638i 0.0407956i
\(162\) 0 0
\(163\) 20.0526i 1.57064i 0.619092 + 0.785319i \(0.287501\pi\)
−0.619092 + 0.785319i \(0.712499\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.20788 −0.635145 −0.317572 0.948234i \(-0.602868\pi\)
−0.317572 + 0.948234i \(0.602868\pi\)
\(168\) 0 0
\(169\) 9.39230 0.722485
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 11.7298i − 0.891801i −0.895083 0.445900i \(-0.852884\pi\)
0.895083 0.445900i \(-0.147116\pi\)
\(174\) 0 0
\(175\) − 14.1962i − 1.07313i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.17209 −0.162350 −0.0811748 0.996700i \(-0.525867\pi\)
−0.0811748 + 0.996700i \(0.525867\pi\)
\(180\) 0 0
\(181\) −16.7846 −1.24759 −0.623795 0.781588i \(-0.714410\pi\)
−0.623795 + 0.781588i \(0.714410\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 19.5588i − 1.43799i
\(186\) 0 0
\(187\) 12.1962i 0.891871i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.1087 −0.876158 −0.438079 0.898936i \(-0.644341\pi\)
−0.438079 + 0.898936i \(0.644341\pi\)
\(192\) 0 0
\(193\) −25.3205 −1.82261 −0.911305 0.411732i \(-0.864924\pi\)
−0.911305 + 0.411732i \(0.864924\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.37945i 0.383270i 0.981466 + 0.191635i \(0.0613790\pi\)
−0.981466 + 0.191635i \(0.938621\pi\)
\(198\) 0 0
\(199\) 17.7321i 1.25699i 0.777813 + 0.628496i \(0.216329\pi\)
−0.777813 + 0.628496i \(0.783671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.24264 0.297775
\(204\) 0 0
\(205\) 28.2679 1.97432
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.5539i 0.799203i
\(210\) 0 0
\(211\) − 21.4641i − 1.47765i −0.673898 0.738825i \(-0.735381\pi\)
0.673898 0.738825i \(-0.264619\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.55532 −0.378870
\(216\) 0 0
\(217\) −3.53590 −0.240032
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 24.9754i − 1.68003i
\(222\) 0 0
\(223\) 0.124356i 0.00832747i 0.999991 + 0.00416374i \(0.00132536\pi\)
−0.999991 + 0.00416374i \(0.998675\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.6274 1.50183 0.750917 0.660396i \(-0.229612\pi\)
0.750917 + 0.660396i \(0.229612\pi\)
\(228\) 0 0
\(229\) 0.535898 0.0354132 0.0177066 0.999843i \(-0.494364\pi\)
0.0177066 + 0.999843i \(0.494364\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 0.757875i − 0.0496500i −0.999692 0.0248250i \(-0.992097\pi\)
0.999692 0.0248250i \(-0.00790286\pi\)
\(234\) 0 0
\(235\) − 35.5167i − 2.31685i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.62587 −0.493277 −0.246638 0.969108i \(-0.579326\pi\)
−0.246638 + 0.969108i \(0.579326\pi\)
\(240\) 0 0
\(241\) 28.1962 1.81627 0.908137 0.418673i \(-0.137505\pi\)
0.908137 + 0.418673i \(0.137505\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4.38134i − 0.279914i
\(246\) 0 0
\(247\) − 23.6603i − 1.50547i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.3538 −1.28472 −0.642360 0.766403i \(-0.722045\pi\)
−0.642360 + 0.766403i \(0.722045\pi\)
\(252\) 0 0
\(253\) 1.19615 0.0752015
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.9719i − 0.684411i −0.939625 0.342205i \(-0.888826\pi\)
0.939625 0.342205i \(-0.111174\pi\)
\(258\) 0 0
\(259\) − 4.46410i − 0.277386i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.96764 0.491306 0.245653 0.969358i \(-0.420998\pi\)
0.245653 + 0.969358i \(0.420998\pi\)
\(264\) 0 0
\(265\) 26.0000 1.59717
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 16.6288i − 1.01388i −0.861983 0.506938i \(-0.830777\pi\)
0.861983 0.506938i \(-0.169223\pi\)
\(270\) 0 0
\(271\) − 15.8564i − 0.963208i −0.876389 0.481604i \(-0.840054\pi\)
0.876389 0.481604i \(-0.159946\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −32.8043 −1.97817
\(276\) 0 0
\(277\) 5.92820 0.356191 0.178096 0.984013i \(-0.443006\pi\)
0.178096 + 0.984013i \(0.443006\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 29.6985i − 1.77166i −0.464007 0.885832i \(-0.653589\pi\)
0.464007 0.885832i \(-0.346411\pi\)
\(282\) 0 0
\(283\) 24.5359i 1.45851i 0.684243 + 0.729254i \(0.260133\pi\)
−0.684243 + 0.729254i \(0.739867\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.45189 0.380843
\(288\) 0 0
\(289\) −10.8564 −0.638612
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 21.7680i − 1.27170i −0.771812 0.635850i \(-0.780650\pi\)
0.771812 0.635850i \(-0.219350\pi\)
\(294\) 0 0
\(295\) − 27.6603i − 1.61044i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.44949 −0.141658
\(300\) 0 0
\(301\) −1.26795 −0.0730834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 39.1175i 2.23986i
\(306\) 0 0
\(307\) 9.00000i 0.513657i 0.966457 + 0.256829i \(0.0826776\pi\)
−0.966457 + 0.256829i \(0.917322\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.27792 0.299283 0.149642 0.988740i \(-0.452188\pi\)
0.149642 + 0.988740i \(0.452188\pi\)
\(312\) 0 0
\(313\) −8.05256 −0.455158 −0.227579 0.973760i \(-0.573081\pi\)
−0.227579 + 0.973760i \(0.573081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.72689i 0.153157i 0.997064 + 0.0765787i \(0.0243997\pi\)
−0.997064 + 0.0765787i \(0.975600\pi\)
\(318\) 0 0
\(319\) − 9.80385i − 0.548910i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −26.3896 −1.46836
\(324\) 0 0
\(325\) 67.1769 3.72630
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 8.10634i − 0.446917i
\(330\) 0 0
\(331\) − 16.7846i − 0.922566i −0.887253 0.461283i \(-0.847389\pi\)
0.887253 0.461283i \(-0.152611\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 42.3249 2.31245
\(336\) 0 0
\(337\) −5.53590 −0.301560 −0.150780 0.988567i \(-0.548178\pi\)
−0.150780 + 0.988567i \(0.548178\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.17072i 0.442469i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.5278 1.15567 0.577836 0.816153i \(-0.303897\pi\)
0.577836 + 0.816153i \(0.303897\pi\)
\(348\) 0 0
\(349\) −20.5359 −1.09926 −0.549631 0.835408i \(-0.685232\pi\)
−0.549631 + 0.835408i \(0.685232\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 8.52245i − 0.453604i −0.973941 0.226802i \(-0.927173\pi\)
0.973941 0.226802i \(-0.0728270\pi\)
\(354\) 0 0
\(355\) − 54.7128i − 2.90385i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.1112 −0.850314 −0.425157 0.905120i \(-0.639781\pi\)
−0.425157 + 0.905120i \(0.639781\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 44.6728i 2.33828i
\(366\) 0 0
\(367\) 30.5167i 1.59296i 0.604667 + 0.796478i \(0.293306\pi\)
−0.604667 + 0.796478i \(0.706694\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.93426 0.308091
\(372\) 0 0
\(373\) 4.66025 0.241299 0.120649 0.992695i \(-0.461502\pi\)
0.120649 + 0.992695i \(0.461502\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.0764i 1.03399i
\(378\) 0 0
\(379\) − 28.1962i − 1.44834i −0.689622 0.724170i \(-0.742223\pi\)
0.689622 0.724170i \(-0.257777\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.82843 0.144526 0.0722629 0.997386i \(-0.476978\pi\)
0.0722629 + 0.997386i \(0.476978\pi\)
\(384\) 0 0
\(385\) −10.1244 −0.515985
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 24.4206i − 1.23817i −0.785323 0.619086i \(-0.787503\pi\)
0.785323 0.619086i \(-0.212497\pi\)
\(390\) 0 0
\(391\) 2.73205i 0.138166i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −53.4355 −2.68863
\(396\) 0 0
\(397\) 4.92820 0.247339 0.123670 0.992323i \(-0.460534\pi\)
0.123670 + 0.992323i \(0.460534\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 9.41902i − 0.470363i −0.971951 0.235182i \(-0.924431\pi\)
0.971951 0.235182i \(-0.0755685\pi\)
\(402\) 0 0
\(403\) − 16.7321i − 0.833483i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.3156 −0.511325
\(408\) 0 0
\(409\) 3.41154 0.168690 0.0843450 0.996437i \(-0.473120\pi\)
0.0843450 + 0.996437i \(0.473120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 6.31319i − 0.310652i
\(414\) 0 0
\(415\) 77.2295i 3.79105i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −37.8792 −1.85052 −0.925259 0.379336i \(-0.876152\pi\)
−0.925259 + 0.379336i \(0.876152\pi\)
\(420\) 0 0
\(421\) −20.3205 −0.990361 −0.495180 0.868790i \(-0.664898\pi\)
−0.495180 + 0.868790i \(0.664898\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 74.9261i − 3.63445i
\(426\) 0 0
\(427\) 8.92820i 0.432066i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.79126 0.134450 0.0672252 0.997738i \(-0.478585\pi\)
0.0672252 + 0.997738i \(0.478585\pi\)
\(432\) 0 0
\(433\) −28.7321 −1.38077 −0.690387 0.723440i \(-0.742560\pi\)
−0.690387 + 0.723440i \(0.742560\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.58819i 0.123810i
\(438\) 0 0
\(439\) 38.2487i 1.82551i 0.408506 + 0.912756i \(0.366050\pi\)
−0.408506 + 0.912756i \(0.633950\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.0077 −0.808062 −0.404031 0.914745i \(-0.632391\pi\)
−0.404031 + 0.914745i \(0.632391\pi\)
\(444\) 0 0
\(445\) −2.26795 −0.107511
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.31749i 0.439719i 0.975532 + 0.219860i \(0.0705599\pi\)
−0.975532 + 0.219860i \(0.929440\pi\)
\(450\) 0 0
\(451\) − 14.9090i − 0.702036i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 20.7327 0.971965
\(456\) 0 0
\(457\) 36.1769 1.69228 0.846142 0.532957i \(-0.178919\pi\)
0.846142 + 0.532957i \(0.178919\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 17.8671i − 0.832156i −0.909329 0.416078i \(-0.863404\pi\)
0.909329 0.416078i \(-0.136596\pi\)
\(462\) 0 0
\(463\) 36.2487i 1.68462i 0.538993 + 0.842310i \(0.318805\pi\)
−0.538993 + 0.842310i \(0.681195\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0101 0.972233 0.486116 0.873894i \(-0.338413\pi\)
0.486116 + 0.873894i \(0.338413\pi\)
\(468\) 0 0
\(469\) 9.66025 0.446069
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.92996i 0.134720i
\(474\) 0 0
\(475\) − 70.9808i − 3.25682i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.28221 −0.378424 −0.189212 0.981936i \(-0.560593\pi\)
−0.189212 + 0.981936i \(0.560593\pi\)
\(480\) 0 0
\(481\) 21.1244 0.963188
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.06678i 0.184663i
\(486\) 0 0
\(487\) 14.5885i 0.661066i 0.943794 + 0.330533i \(0.107229\pi\)
−0.943794 + 0.330533i \(0.892771\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −23.3209 −1.05246 −0.526229 0.850343i \(-0.676395\pi\)
−0.526229 + 0.850343i \(0.676395\pi\)
\(492\) 0 0
\(493\) 22.3923 1.00850
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.4877i − 0.560149i
\(498\) 0 0
\(499\) 16.3397i 0.731467i 0.930720 + 0.365734i \(0.119182\pi\)
−0.930720 + 0.365734i \(0.880818\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.5211 −0.647463 −0.323731 0.946149i \(-0.604937\pi\)
−0.323731 + 0.946149i \(0.604937\pi\)
\(504\) 0 0
\(505\) −44.5885 −1.98416
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 31.2886i − 1.38684i −0.720533 0.693421i \(-0.756103\pi\)
0.720533 0.693421i \(-0.243897\pi\)
\(510\) 0 0
\(511\) 10.1962i 0.451051i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −45.8468 −2.02025
\(516\) 0 0
\(517\) −18.7321 −0.823835
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.9329i 0.522789i 0.965232 + 0.261395i \(0.0841824\pi\)
−0.965232 + 0.261395i \(0.915818\pi\)
\(522\) 0 0
\(523\) 11.3397i 0.495852i 0.968779 + 0.247926i \(0.0797491\pi\)
−0.968779 + 0.247926i \(0.920251\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.6622 −0.812937
\(528\) 0 0
\(529\) −22.7321 −0.988350
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.5307i 1.32243i
\(534\) 0 0
\(535\) 60.3013i 2.60705i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.31079 −0.0995327
\(540\) 0 0
\(541\) −43.1051 −1.85323 −0.926617 0.376007i \(-0.877297\pi\)
−0.926617 + 0.376007i \(0.877297\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 59.5357i 2.55023i
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.2132 0.903713
\(552\) 0 0
\(553\) −12.1962 −0.518633
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.23835i − 0.0524705i −0.999656 0.0262352i \(-0.991648\pi\)
0.999656 0.0262352i \(-0.00835190\pi\)
\(558\) 0 0
\(559\) − 6.00000i − 0.253773i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.2934 −1.65602 −0.828009 0.560715i \(-0.810526\pi\)
−0.828009 + 0.560715i \(0.810526\pi\)
\(564\) 0 0
\(565\) −37.1769 −1.56404
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 44.7744i − 1.87704i −0.345226 0.938519i \(-0.612198\pi\)
0.345226 0.938519i \(-0.387802\pi\)
\(570\) 0 0
\(571\) 46.2487i 1.93545i 0.252012 + 0.967724i \(0.418908\pi\)
−0.252012 + 0.967724i \(0.581092\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.34847 −0.306452
\(576\) 0 0
\(577\) −5.60770 −0.233451 −0.116726 0.993164i \(-0.537240\pi\)
−0.116726 + 0.993164i \(0.537240\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.6269i 0.731287i
\(582\) 0 0
\(583\) − 13.7128i − 0.567927i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0096 0.660788 0.330394 0.943843i \(-0.392818\pi\)
0.330394 + 0.943843i \(0.392818\pi\)
\(588\) 0 0
\(589\) −17.6795 −0.728471
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 10.1397i − 0.416389i −0.978087 0.208194i \(-0.933241\pi\)
0.978087 0.208194i \(-0.0667587\pi\)
\(594\) 0 0
\(595\) − 23.1244i − 0.948006i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −23.6999 −0.968350 −0.484175 0.874971i \(-0.660880\pi\)
−0.484175 + 0.874971i \(0.660880\pi\)
\(600\) 0 0
\(601\) −30.3397 −1.23758 −0.618792 0.785555i \(-0.712378\pi\)
−0.618792 + 0.785555i \(0.712378\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 24.7995i − 1.00824i
\(606\) 0 0
\(607\) 39.1769i 1.59014i 0.606516 + 0.795071i \(0.292566\pi\)
−0.606516 + 0.795071i \(0.707434\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 38.3596 1.55186
\(612\) 0 0
\(613\) −29.5885 −1.19507 −0.597533 0.801844i \(-0.703852\pi\)
−0.597533 + 0.801844i \(0.703852\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.9396i 0.762479i 0.924476 + 0.381239i \(0.124503\pi\)
−0.924476 + 0.381239i \(0.875497\pi\)
\(618\) 0 0
\(619\) 28.6603i 1.15195i 0.817466 + 0.575976i \(0.195378\pi\)
−0.817466 + 0.575976i \(0.804622\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.517638 −0.0207387
\(624\) 0 0
\(625\) 105.550 4.22200
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 23.5612i − 0.939445i
\(630\) 0 0
\(631\) − 16.0526i − 0.639042i −0.947579 0.319521i \(-0.896478\pi\)
0.947579 0.319521i \(-0.103522\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.48854 0.0590710
\(636\) 0 0
\(637\) 4.73205 0.187491
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.07055i 0.0817819i 0.999164 + 0.0408910i \(0.0130196\pi\)
−0.999164 + 0.0408910i \(0.986980\pi\)
\(642\) 0 0
\(643\) 22.8564i 0.901369i 0.892683 + 0.450684i \(0.148820\pi\)
−0.892683 + 0.450684i \(0.851180\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.3249 1.66396 0.831981 0.554804i \(-0.187207\pi\)
0.831981 + 0.554804i \(0.187207\pi\)
\(648\) 0 0
\(649\) −14.5885 −0.572647
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.48099i 0.214488i 0.994233 + 0.107244i \(0.0342025\pi\)
−0.994233 + 0.107244i \(0.965797\pi\)
\(654\) 0 0
\(655\) 6.19615i 0.242104i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.7351 0.963544 0.481772 0.876297i \(-0.339993\pi\)
0.481772 + 0.876297i \(0.339993\pi\)
\(660\) 0 0
\(661\) −5.26795 −0.204899 −0.102450 0.994738i \(-0.532668\pi\)
−0.102450 + 0.994738i \(0.532668\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 21.9067i − 0.849506i
\(666\) 0 0
\(667\) − 2.19615i − 0.0850354i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 20.6312 0.796458
\(672\) 0 0
\(673\) −12.9282 −0.498346 −0.249173 0.968459i \(-0.580159\pi\)
−0.249173 + 0.968459i \(0.580159\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 39.1547i 1.50484i 0.658686 + 0.752418i \(0.271113\pi\)
−0.658686 + 0.752418i \(0.728887\pi\)
\(678\) 0 0
\(679\) 0.928203i 0.0356212i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.41662 0.207261 0.103631 0.994616i \(-0.466954\pi\)
0.103631 + 0.994616i \(0.466954\pi\)
\(684\) 0 0
\(685\) 27.6603 1.05684
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28.0812i 1.06981i
\(690\) 0 0
\(691\) 4.67949i 0.178016i 0.996031 + 0.0890081i \(0.0283697\pi\)
−0.996031 + 0.0890081i \(0.971630\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 102.175 3.87572
\(696\) 0 0
\(697\) 34.0526 1.28983
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 12.4505i − 0.470250i −0.971965 0.235125i \(-0.924450\pi\)
0.971965 0.235125i \(-0.0755499\pi\)
\(702\) 0 0
\(703\) − 22.3205i − 0.841834i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.1769 −0.382742
\(708\) 0 0
\(709\) 16.6603 0.625689 0.312844 0.949804i \(-0.398718\pi\)
0.312844 + 0.949804i \(0.398718\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.83032i 0.0685459i
\(714\) 0 0
\(715\) − 47.9090i − 1.79169i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 47.2982 1.76392 0.881962 0.471320i \(-0.156222\pi\)
0.881962 + 0.471320i \(0.156222\pi\)
\(720\) 0 0
\(721\) −10.4641 −0.389704
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 60.2292i 2.23686i
\(726\) 0 0
\(727\) 5.07180i 0.188103i 0.995567 + 0.0940513i \(0.0299818\pi\)
−0.995567 + 0.0940513i \(0.970018\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.69213 −0.247517
\(732\) 0 0
\(733\) −25.8038 −0.953087 −0.476543 0.879151i \(-0.658111\pi\)
−0.476543 + 0.879151i \(0.658111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 22.3228i − 0.822271i
\(738\) 0 0
\(739\) − 14.9282i − 0.549143i −0.961567 0.274571i \(-0.911464\pi\)
0.961567 0.274571i \(-0.0885360\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.8004 0.506286 0.253143 0.967429i \(-0.418536\pi\)
0.253143 + 0.967429i \(0.418536\pi\)
\(744\) 0 0
\(745\) −86.7461 −3.17813
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.7632i 0.502896i
\(750\) 0 0
\(751\) 13.6603i 0.498470i 0.968443 + 0.249235i \(0.0801791\pi\)
−0.968443 + 0.249235i \(0.919821\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.69591 0.170902
\(756\) 0 0
\(757\) −16.6410 −0.604828 −0.302414 0.953177i \(-0.597793\pi\)
−0.302414 + 0.953177i \(0.597793\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.859411i 0.0311536i 0.999879 + 0.0155768i \(0.00495845\pi\)
−0.999879 + 0.0155768i \(0.995042\pi\)
\(762\) 0 0
\(763\) 13.5885i 0.491935i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.8744 1.07870
\(768\) 0 0
\(769\) 17.6603 0.636845 0.318423 0.947949i \(-0.396847\pi\)
0.318423 + 0.947949i \(0.396847\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23.9773i 0.862402i 0.902256 + 0.431201i \(0.141910\pi\)
−0.902256 + 0.431201i \(0.858090\pi\)
\(774\) 0 0
\(775\) − 50.1962i − 1.80310i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.2595 1.15582
\(780\) 0 0
\(781\) −28.8564 −1.03256
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.3180i 0.511031i
\(786\) 0 0
\(787\) 7.32051i 0.260948i 0.991452 + 0.130474i \(0.0416499\pi\)
−0.991452 + 0.130474i \(0.958350\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.48528 −0.301702
\(792\) 0 0
\(793\) −42.2487 −1.50030
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.20599i − 0.326093i −0.986618 0.163046i \(-0.947868\pi\)
0.986618 0.163046i \(-0.0521320\pi\)
\(798\) 0 0
\(799\) − 42.7846i − 1.51361i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 23.5612 0.831455
\(804\) 0 0
\(805\) −2.26795 −0.0799347
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 28.1827i 0.990852i 0.868650 + 0.495426i \(0.164988\pi\)
−0.868650 + 0.495426i \(0.835012\pi\)
\(810\) 0 0
\(811\) 44.6603i 1.56823i 0.620613 + 0.784117i \(0.286884\pi\)
−0.620613 + 0.784117i \(0.713116\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −87.8571 −3.07750
\(816\) 0 0
\(817\) −6.33975 −0.221800
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 38.8129i − 1.35458i −0.735716 0.677290i \(-0.763154\pi\)
0.735716 0.677290i \(-0.236846\pi\)
\(822\) 0 0
\(823\) 7.07180i 0.246507i 0.992375 + 0.123254i \(0.0393329\pi\)
−0.992375 + 0.123254i \(0.960667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.85752 0.0645924 0.0322962 0.999478i \(-0.489718\pi\)
0.0322962 + 0.999478i \(0.489718\pi\)
\(828\) 0 0
\(829\) 36.1962 1.25714 0.628572 0.777751i \(-0.283640\pi\)
0.628572 + 0.777751i \(0.283640\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.27792i − 0.182869i
\(834\) 0 0
\(835\) − 35.9615i − 1.24450i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −55.5061 −1.91628 −0.958141 0.286297i \(-0.907576\pi\)
−0.958141 + 0.286297i \(0.907576\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 41.1509i 1.41563i
\(846\) 0 0
\(847\) − 5.66025i − 0.194489i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.31079 −0.0792128
\(852\) 0 0
\(853\) 18.0000 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 46.8821i 1.60146i 0.599025 + 0.800731i \(0.295555\pi\)
−0.599025 + 0.800731i \(0.704445\pi\)
\(858\) 0 0
\(859\) 55.1051i 1.88016i 0.340951 + 0.940081i \(0.389251\pi\)
−0.340951 + 0.940081i \(0.610749\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.93855 0.304272 0.152136 0.988360i \(-0.451385\pi\)
0.152136 + 0.988360i \(0.451385\pi\)
\(864\) 0 0
\(865\) 51.3923 1.74739
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 28.1827i 0.956034i
\(870\) 0 0
\(871\) 45.7128i 1.54892i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 40.2915 1.36210
\(876\) 0 0
\(877\) −18.1436 −0.612666 −0.306333 0.951924i \(-0.599102\pi\)
−0.306333 + 0.951924i \(0.599102\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 8.04197i − 0.270941i −0.990781 0.135470i \(-0.956745\pi\)
0.990781 0.135470i \(-0.0432546\pi\)
\(882\) 0 0
\(883\) − 10.3923i − 0.349729i −0.984593 0.174864i \(-0.944051\pi\)
0.984593 0.174864i \(-0.0559487\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −45.1533 −1.51610 −0.758050 0.652197i \(-0.773848\pi\)
−0.758050 + 0.652197i \(0.773848\pi\)
\(888\) 0 0
\(889\) 0.339746 0.0113947
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 40.5317i − 1.35634i
\(894\) 0 0
\(895\) − 9.51666i − 0.318107i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0015 0.500330
\(900\) 0 0
\(901\) 31.3205 1.04344
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 73.5391i − 2.44452i
\(906\) 0 0
\(907\) − 28.9282i − 0.960545i −0.877119 0.480273i \(-0.840538\pi\)
0.877119 0.480273i \(-0.159462\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.30942 0.275303 0.137652 0.990481i \(-0.456045\pi\)
0.137652 + 0.990481i \(0.456045\pi\)
\(912\) 0 0
\(913\) 40.7321 1.34803
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.41421i 0.0467014i
\(918\) 0 0
\(919\) 29.7128i 0.980135i 0.871684 + 0.490068i \(0.163028\pi\)
−0.871684 + 0.490068i \(0.836972\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 59.0924 1.94505
\(924\) 0 0
\(925\) 63.3731 2.08369
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.93855i 0.293264i 0.989191 + 0.146632i \(0.0468434\pi\)
−0.989191 + 0.146632i \(0.953157\pi\)
\(930\) 0 0
\(931\) − 5.00000i − 0.163868i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −53.4355 −1.74753
\(936\) 0 0
\(937\) −51.5692 −1.68469 −0.842346 0.538936i \(-0.818826\pi\)
−0.842346 + 0.538936i \(0.818826\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.00292i 0.293487i 0.989175 + 0.146743i \(0.0468792\pi\)
−0.989175 + 0.146743i \(0.953121\pi\)
\(942\) 0 0
\(943\) − 3.33975i − 0.108757i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.2209 1.24201 0.621007 0.783805i \(-0.286724\pi\)
0.621007 + 0.783805i \(0.286724\pi\)
\(948\) 0 0
\(949\) −48.2487 −1.56622
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 25.2800i − 0.818899i −0.912333 0.409449i \(-0.865721\pi\)
0.912333 0.409449i \(-0.134279\pi\)
\(954\) 0 0
\(955\) − 53.0526i − 1.71674i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.31319 0.203864
\(960\) 0 0
\(961\) 18.4974 0.596691
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 110.938i − 3.57121i
\(966\) 0 0
\(967\) 24.2487i 0.779786i 0.920860 + 0.389893i \(0.127488\pi\)
−0.920860 + 0.389893i \(0.872512\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −40.5317 −1.30072 −0.650362 0.759624i \(-0.725383\pi\)
−0.650362 + 0.759624i \(0.725383\pi\)
\(972\) 0 0
\(973\) 23.3205 0.747621
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.9192i 1.50108i 0.660825 + 0.750540i \(0.270206\pi\)
−0.660825 + 0.750540i \(0.729794\pi\)
\(978\) 0 0
\(979\) 1.19615i 0.0382292i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.42280 −0.236751 −0.118375 0.992969i \(-0.537769\pi\)
−0.118375 + 0.992969i \(0.537769\pi\)
\(984\) 0 0
\(985\) −23.5692 −0.750978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.656339i 0.0208704i
\(990\) 0 0
\(991\) − 46.8372i − 1.48783i −0.668273 0.743916i \(-0.732966\pi\)
0.668273 0.743916i \(-0.267034\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −77.6902 −2.46294
\(996\) 0 0
\(997\) −28.9282 −0.916165 −0.458083 0.888910i \(-0.651464\pi\)
−0.458083 + 0.888910i \(0.651464\pi\)
\(998\) 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.b.2591.8 yes 8
3.2 odd 2 inner 6048.2.h.b.2591.2 yes 8
4.3 odd 2 inner 6048.2.h.b.2591.7 yes 8
12.11 even 2 inner 6048.2.h.b.2591.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.b.2591.1 8 12.11 even 2 inner
6048.2.h.b.2591.2 yes 8 3.2 odd 2 inner
6048.2.h.b.2591.7 yes 8 4.3 odd 2 inner
6048.2.h.b.2591.8 yes 8 1.1 even 1 trivial