Properties

Label 6048.2.j.c.5615.13
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.13
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.14

$q$-expansion

\(f(q)\) \(=\) \(q-0.206991 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-0.206991 q^{5} +1.00000i q^{7} +6.42190i q^{11} -3.52336i q^{13} -3.90616i q^{17} +5.48691 q^{19} -0.880512 q^{23} -4.95715 q^{25} +3.20814 q^{29} +0.0631261i q^{31} -0.206991i q^{35} -9.91837i q^{37} +5.94221i q^{41} +1.13960 q^{43} +6.81266 q^{47} -1.00000 q^{49} +12.6324 q^{53} -1.32928i q^{55} -4.23338i q^{59} +12.3006i q^{61} +0.729305i q^{65} +5.74666 q^{67} -10.1795 q^{71} +8.27618 q^{73} -6.42190 q^{77} -4.86426i q^{79} +7.51539i q^{83} +0.808542i q^{85} -4.44682i q^{89} +3.52336 q^{91} -1.13574 q^{95} -17.3929 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.206991 −0.0925694 −0.0462847 0.998928i \(-0.514738\pi\)
−0.0462847 + 0.998928i \(0.514738\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.42190i 1.93628i 0.250418 + 0.968138i \(0.419432\pi\)
−0.250418 + 0.968138i \(0.580568\pi\)
\(12\) 0 0
\(13\) − 3.52336i − 0.977204i −0.872507 0.488602i \(-0.837507\pi\)
0.872507 0.488602i \(-0.162493\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.90616i − 0.947383i −0.880691 0.473692i \(-0.842921\pi\)
0.880691 0.473692i \(-0.157079\pi\)
\(18\) 0 0
\(19\) 5.48691 1.25878 0.629392 0.777088i \(-0.283304\pi\)
0.629392 + 0.777088i \(0.283304\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.880512 −0.183599 −0.0917997 0.995777i \(-0.529262\pi\)
−0.0917997 + 0.995777i \(0.529262\pi\)
\(24\) 0 0
\(25\) −4.95715 −0.991431
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.20814 0.595737 0.297868 0.954607i \(-0.403724\pi\)
0.297868 + 0.954607i \(0.403724\pi\)
\(30\) 0 0
\(31\) 0.0631261i 0.0113378i 0.999984 + 0.00566890i \(0.00180448\pi\)
−0.999984 + 0.00566890i \(0.998196\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 0.206991i − 0.0349879i
\(36\) 0 0
\(37\) − 9.91837i − 1.63057i −0.579060 0.815285i \(-0.696580\pi\)
0.579060 0.815285i \(-0.303420\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.94221i 0.928017i 0.885831 + 0.464009i \(0.153589\pi\)
−0.885831 + 0.464009i \(0.846411\pi\)
\(42\) 0 0
\(43\) 1.13960 0.173788 0.0868938 0.996218i \(-0.472306\pi\)
0.0868938 + 0.996218i \(0.472306\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.81266 0.993729 0.496864 0.867828i \(-0.334485\pi\)
0.496864 + 0.867828i \(0.334485\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.6324 1.73520 0.867598 0.497265i \(-0.165662\pi\)
0.867598 + 0.497265i \(0.165662\pi\)
\(54\) 0 0
\(55\) − 1.32928i − 0.179240i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.23338i − 0.551139i −0.961281 0.275570i \(-0.911134\pi\)
0.961281 0.275570i \(-0.0888664\pi\)
\(60\) 0 0
\(61\) 12.3006i 1.57493i 0.616359 + 0.787465i \(0.288607\pi\)
−0.616359 + 0.787465i \(0.711393\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.729305i 0.0904591i
\(66\) 0 0
\(67\) 5.74666 0.702067 0.351033 0.936363i \(-0.385830\pi\)
0.351033 + 0.936363i \(0.385830\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.1795 −1.20809 −0.604043 0.796952i \(-0.706444\pi\)
−0.604043 + 0.796952i \(0.706444\pi\)
\(72\) 0 0
\(73\) 8.27618 0.968653 0.484327 0.874887i \(-0.339065\pi\)
0.484327 + 0.874887i \(0.339065\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.42190 −0.731843
\(78\) 0 0
\(79\) − 4.86426i − 0.547272i −0.961833 0.273636i \(-0.911774\pi\)
0.961833 0.273636i \(-0.0882264\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.51539i 0.824921i 0.910976 + 0.412460i \(0.135331\pi\)
−0.910976 + 0.412460i \(0.864669\pi\)
\(84\) 0 0
\(85\) 0.808542i 0.0876987i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.44682i − 0.471362i −0.971830 0.235681i \(-0.924268\pi\)
0.971830 0.235681i \(-0.0757320\pi\)
\(90\) 0 0
\(91\) 3.52336 0.369348
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.13574 −0.116525
\(96\) 0 0
\(97\) −17.3929 −1.76599 −0.882993 0.469387i \(-0.844475\pi\)
−0.882993 + 0.469387i \(0.844475\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.0273 1.89329 0.946643 0.322284i \(-0.104450\pi\)
0.946643 + 0.322284i \(0.104450\pi\)
\(102\) 0 0
\(103\) 10.7903i 1.06320i 0.846995 + 0.531601i \(0.178409\pi\)
−0.846995 + 0.531601i \(0.821591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.97087i 0.480553i 0.970705 + 0.240276i \(0.0772380\pi\)
−0.970705 + 0.240276i \(0.922762\pi\)
\(108\) 0 0
\(109\) − 2.84810i − 0.272798i −0.990654 0.136399i \(-0.956447\pi\)
0.990654 0.136399i \(-0.0435530\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.88561i 0.271455i 0.990746 + 0.135728i \(0.0433372\pi\)
−0.990746 + 0.135728i \(0.956663\pi\)
\(114\) 0 0
\(115\) 0.182258 0.0169957
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.90616 0.358077
\(120\) 0 0
\(121\) −30.2408 −2.74916
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.06105 0.184345
\(126\) 0 0
\(127\) − 7.23082i − 0.641631i −0.947142 0.320816i \(-0.896043\pi\)
0.947142 0.320816i \(-0.103957\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.8026i 1.11857i 0.828975 + 0.559286i \(0.188924\pi\)
−0.828975 + 0.559286i \(0.811076\pi\)
\(132\) 0 0
\(133\) 5.48691i 0.475776i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 0.363873i − 0.0310878i −0.999879 0.0155439i \(-0.995052\pi\)
0.999879 0.0155439i \(-0.00494798\pi\)
\(138\) 0 0
\(139\) 9.32338 0.790799 0.395399 0.918509i \(-0.370606\pi\)
0.395399 + 0.918509i \(0.370606\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 22.6266 1.89214
\(144\) 0 0
\(145\) −0.664057 −0.0551470
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.82519 0.395295 0.197648 0.980273i \(-0.436670\pi\)
0.197648 + 0.980273i \(0.436670\pi\)
\(150\) 0 0
\(151\) 12.7969i 1.04140i 0.853740 + 0.520700i \(0.174329\pi\)
−0.853740 + 0.520700i \(0.825671\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.0130666i − 0.00104953i
\(156\) 0 0
\(157\) 15.5455i 1.24067i 0.784337 + 0.620335i \(0.213003\pi\)
−0.784337 + 0.620335i \(0.786997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 0.880512i − 0.0693940i
\(162\) 0 0
\(163\) −18.4630 −1.44613 −0.723067 0.690778i \(-0.757268\pi\)
−0.723067 + 0.690778i \(0.757268\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.5320 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(168\) 0 0
\(169\) 0.585951 0.0450731
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.49171 −0.417527 −0.208763 0.977966i \(-0.566944\pi\)
−0.208763 + 0.977966i \(0.566944\pi\)
\(174\) 0 0
\(175\) − 4.95715i − 0.374726i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.5830i 1.08998i 0.838441 + 0.544992i \(0.183467\pi\)
−0.838441 + 0.544992i \(0.816533\pi\)
\(180\) 0 0
\(181\) 17.2753i 1.28407i 0.766677 + 0.642033i \(0.221909\pi\)
−0.766677 + 0.642033i \(0.778091\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.05302i 0.150941i
\(186\) 0 0
\(187\) 25.0850 1.83440
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.29828 −0.528085 −0.264042 0.964511i \(-0.585056\pi\)
−0.264042 + 0.964511i \(0.585056\pi\)
\(192\) 0 0
\(193\) 12.1106 0.871742 0.435871 0.900009i \(-0.356440\pi\)
0.435871 + 0.900009i \(0.356440\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.3925 1.23917 0.619584 0.784931i \(-0.287301\pi\)
0.619584 + 0.784931i \(0.287301\pi\)
\(198\) 0 0
\(199\) 3.07583i 0.218040i 0.994040 + 0.109020i \(0.0347712\pi\)
−0.994040 + 0.109020i \(0.965229\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.20814i 0.225167i
\(204\) 0 0
\(205\) − 1.22999i − 0.0859060i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 35.2364i 2.43735i
\(210\) 0 0
\(211\) −9.57154 −0.658932 −0.329466 0.944167i \(-0.606869\pi\)
−0.329466 + 0.944167i \(0.606869\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.235888 −0.0160874
\(216\) 0 0
\(217\) −0.0631261 −0.00428528
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −13.7628 −0.925786
\(222\) 0 0
\(223\) − 26.5694i − 1.77922i −0.456722 0.889610i \(-0.650976\pi\)
0.456722 0.889610i \(-0.349024\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.7345i 1.37620i 0.725616 + 0.688100i \(0.241555\pi\)
−0.725616 + 0.688100i \(0.758445\pi\)
\(228\) 0 0
\(229\) − 19.2625i − 1.27290i −0.771316 0.636452i \(-0.780401\pi\)
0.771316 0.636452i \(-0.219599\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.0038i 1.63805i 0.573756 + 0.819027i \(0.305486\pi\)
−0.573756 + 0.819027i \(0.694514\pi\)
\(234\) 0 0
\(235\) −1.41016 −0.0919889
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0439 0.973107 0.486553 0.873651i \(-0.338254\pi\)
0.486553 + 0.873651i \(0.338254\pi\)
\(240\) 0 0
\(241\) −3.23274 −0.208239 −0.104120 0.994565i \(-0.533203\pi\)
−0.104120 + 0.994565i \(0.533203\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.206991 0.0132242
\(246\) 0 0
\(247\) − 19.3324i − 1.23009i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.4258i 0.658071i 0.944318 + 0.329036i \(0.106724\pi\)
−0.944318 + 0.329036i \(0.893276\pi\)
\(252\) 0 0
\(253\) − 5.65456i − 0.355499i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.12452i 0.506794i 0.967362 + 0.253397i \(0.0815479\pi\)
−0.967362 + 0.253397i \(0.918452\pi\)
\(258\) 0 0
\(259\) 9.91837 0.616297
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.7639 0.910382 0.455191 0.890394i \(-0.349571\pi\)
0.455191 + 0.890394i \(0.349571\pi\)
\(264\) 0 0
\(265\) −2.61480 −0.160626
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.8961 −1.63989 −0.819943 0.572445i \(-0.805995\pi\)
−0.819943 + 0.572445i \(0.805995\pi\)
\(270\) 0 0
\(271\) 17.5914i 1.06860i 0.845295 + 0.534301i \(0.179425\pi\)
−0.845295 + 0.534301i \(0.820575\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 31.8343i − 1.91968i
\(276\) 0 0
\(277\) − 24.8383i − 1.49239i −0.665729 0.746194i \(-0.731879\pi\)
0.665729 0.746194i \(-0.268121\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.13370i 0.246596i 0.992370 + 0.123298i \(0.0393471\pi\)
−0.992370 + 0.123298i \(0.960653\pi\)
\(282\) 0 0
\(283\) 23.9297 1.42247 0.711236 0.702954i \(-0.248136\pi\)
0.711236 + 0.702954i \(0.248136\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.94221 −0.350758
\(288\) 0 0
\(289\) 1.74190 0.102465
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0225 0.702360 0.351180 0.936308i \(-0.385781\pi\)
0.351180 + 0.936308i \(0.385781\pi\)
\(294\) 0 0
\(295\) 0.876273i 0.0510186i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.10236i 0.179414i
\(300\) 0 0
\(301\) 1.13960i 0.0656855i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.54612i − 0.145790i
\(306\) 0 0
\(307\) 5.82060 0.332199 0.166100 0.986109i \(-0.446883\pi\)
0.166100 + 0.986109i \(0.446883\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.7884 0.725161 0.362580 0.931952i \(-0.381896\pi\)
0.362580 + 0.931952i \(0.381896\pi\)
\(312\) 0 0
\(313\) 1.74881 0.0988483 0.0494242 0.998778i \(-0.484261\pi\)
0.0494242 + 0.998778i \(0.484261\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.5938 1.71832 0.859159 0.511709i \(-0.170987\pi\)
0.859159 + 0.511709i \(0.170987\pi\)
\(318\) 0 0
\(319\) 20.6024i 1.15351i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 21.4328i − 1.19255i
\(324\) 0 0
\(325\) 17.4658i 0.968830i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.81266i 0.375594i
\(330\) 0 0
\(331\) 20.7269 1.13925 0.569627 0.821904i \(-0.307088\pi\)
0.569627 + 0.821904i \(0.307088\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.18951 −0.0649899
\(336\) 0 0
\(337\) 19.9716 1.08792 0.543961 0.839110i \(-0.316924\pi\)
0.543961 + 0.839110i \(0.316924\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.405390 −0.0219531
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.6443i − 0.571414i −0.958317 0.285707i \(-0.907772\pi\)
0.958317 0.285707i \(-0.0922284\pi\)
\(348\) 0 0
\(349\) − 21.1575i − 1.13254i −0.824221 0.566268i \(-0.808387\pi\)
0.824221 0.566268i \(-0.191613\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.85498i 0.418079i 0.977907 + 0.209039i \(0.0670337\pi\)
−0.977907 + 0.209039i \(0.932966\pi\)
\(354\) 0 0
\(355\) 2.10707 0.111832
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.0102 −1.16165 −0.580826 0.814028i \(-0.697270\pi\)
−0.580826 + 0.814028i \(0.697270\pi\)
\(360\) 0 0
\(361\) 11.1062 0.584537
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.71310 −0.0896676
\(366\) 0 0
\(367\) − 5.61196i − 0.292942i −0.989215 0.146471i \(-0.953209\pi\)
0.989215 0.146471i \(-0.0467915\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.6324i 0.655843i
\(372\) 0 0
\(373\) − 21.5334i − 1.11496i −0.830192 0.557478i \(-0.811769\pi\)
0.830192 0.557478i \(-0.188231\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.3034i − 0.582156i
\(378\) 0 0
\(379\) −36.2973 −1.86447 −0.932233 0.361858i \(-0.882143\pi\)
−0.932233 + 0.361858i \(0.882143\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.4362 −0.635462 −0.317731 0.948181i \(-0.602921\pi\)
−0.317731 + 0.948181i \(0.602921\pi\)
\(384\) 0 0
\(385\) 1.32928 0.0677463
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.47648 0.277669 0.138834 0.990316i \(-0.455664\pi\)
0.138834 + 0.990316i \(0.455664\pi\)
\(390\) 0 0
\(391\) 3.43942i 0.173939i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00686i 0.0506606i
\(396\) 0 0
\(397\) − 9.58947i − 0.481282i −0.970614 0.240641i \(-0.922642\pi\)
0.970614 0.240641i \(-0.0773576\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.76702i 0.0882408i 0.999026 + 0.0441204i \(0.0140485\pi\)
−0.999026 + 0.0441204i \(0.985951\pi\)
\(402\) 0 0
\(403\) 0.222416 0.0110793
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 63.6948 3.15723
\(408\) 0 0
\(409\) −5.87490 −0.290495 −0.145248 0.989395i \(-0.546398\pi\)
−0.145248 + 0.989395i \(0.546398\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.23338 0.208311
\(414\) 0 0
\(415\) − 1.55562i − 0.0763624i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 18.9459i − 0.925567i −0.886471 0.462783i \(-0.846851\pi\)
0.886471 0.462783i \(-0.153149\pi\)
\(420\) 0 0
\(421\) 25.2598i 1.23109i 0.788103 + 0.615544i \(0.211064\pi\)
−0.788103 + 0.615544i \(0.788936\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 19.3634i 0.939265i
\(426\) 0 0
\(427\) −12.3006 −0.595268
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37.5774 1.81004 0.905020 0.425369i \(-0.139856\pi\)
0.905020 + 0.425369i \(0.139856\pi\)
\(432\) 0 0
\(433\) 21.1040 1.01419 0.507097 0.861889i \(-0.330719\pi\)
0.507097 + 0.861889i \(0.330719\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.83129 −0.231112
\(438\) 0 0
\(439\) 2.88405i 0.137648i 0.997629 + 0.0688240i \(0.0219247\pi\)
−0.997629 + 0.0688240i \(0.978075\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.5978i 1.64379i 0.569640 + 0.821895i \(0.307083\pi\)
−0.569640 + 0.821895i \(0.692917\pi\)
\(444\) 0 0
\(445\) 0.920453i 0.0436337i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 14.8661i − 0.701575i −0.936455 0.350787i \(-0.885914\pi\)
0.936455 0.350787i \(-0.114086\pi\)
\(450\) 0 0
\(451\) −38.1603 −1.79690
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.729305 −0.0341903
\(456\) 0 0
\(457\) 11.0037 0.514730 0.257365 0.966314i \(-0.417146\pi\)
0.257365 + 0.966314i \(0.417146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 32.9687 1.53551 0.767753 0.640746i \(-0.221375\pi\)
0.767753 + 0.640746i \(0.221375\pi\)
\(462\) 0 0
\(463\) − 12.3348i − 0.573245i −0.958044 0.286622i \(-0.907468\pi\)
0.958044 0.286622i \(-0.0925325\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.5264i 1.55142i 0.631091 + 0.775709i \(0.282607\pi\)
−0.631091 + 0.775709i \(0.717393\pi\)
\(468\) 0 0
\(469\) 5.74666i 0.265356i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.31840i 0.336501i
\(474\) 0 0
\(475\) −27.1995 −1.24800
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.65791 0.441281 0.220641 0.975355i \(-0.429185\pi\)
0.220641 + 0.975355i \(0.429185\pi\)
\(480\) 0 0
\(481\) −34.9460 −1.59340
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.60019 0.163476
\(486\) 0 0
\(487\) 25.1008i 1.13742i 0.822537 + 0.568712i \(0.192558\pi\)
−0.822537 + 0.568712i \(0.807442\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 10.5524i − 0.476224i −0.971238 0.238112i \(-0.923471\pi\)
0.971238 0.238112i \(-0.0765285\pi\)
\(492\) 0 0
\(493\) − 12.5315i − 0.564391i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.1795i − 0.456613i
\(498\) 0 0
\(499\) 11.6619 0.522059 0.261029 0.965331i \(-0.415938\pi\)
0.261029 + 0.965331i \(0.415938\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.0202 −1.29394 −0.646972 0.762513i \(-0.723965\pi\)
−0.646972 + 0.762513i \(0.723965\pi\)
\(504\) 0 0
\(505\) −3.93849 −0.175260
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.0894 −0.979096 −0.489548 0.871976i \(-0.662838\pi\)
−0.489548 + 0.871976i \(0.662838\pi\)
\(510\) 0 0
\(511\) 8.27618i 0.366116i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 2.23350i − 0.0984199i
\(516\) 0 0
\(517\) 43.7502i 1.92413i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 14.5460i − 0.637273i −0.947877 0.318636i \(-0.896775\pi\)
0.947877 0.318636i \(-0.103225\pi\)
\(522\) 0 0
\(523\) −31.6350 −1.38330 −0.691651 0.722232i \(-0.743116\pi\)
−0.691651 + 0.722232i \(0.743116\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.246581 0.0107412
\(528\) 0 0
\(529\) −22.2247 −0.966291
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.9365 0.906862
\(534\) 0 0
\(535\) − 1.02893i − 0.0444844i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 6.42190i − 0.276611i
\(540\) 0 0
\(541\) 22.7160i 0.976639i 0.872665 + 0.488319i \(0.162390\pi\)
−0.872665 + 0.488319i \(0.837610\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.589532i 0.0252528i
\(546\) 0 0
\(547\) 22.2601 0.951774 0.475887 0.879506i \(-0.342127\pi\)
0.475887 + 0.879506i \(0.342127\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.6028 0.749904
\(552\) 0 0
\(553\) 4.86426 0.206849
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.99670 0.126974 0.0634872 0.997983i \(-0.479778\pi\)
0.0634872 + 0.997983i \(0.479778\pi\)
\(558\) 0 0
\(559\) − 4.01522i − 0.169826i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13.4913i 0.568589i 0.958737 + 0.284295i \(0.0917594\pi\)
−0.958737 + 0.284295i \(0.908241\pi\)
\(564\) 0 0
\(565\) − 0.597296i − 0.0251284i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 40.7445i 1.70810i 0.520191 + 0.854050i \(0.325861\pi\)
−0.520191 + 0.854050i \(0.674139\pi\)
\(570\) 0 0
\(571\) −18.0110 −0.753739 −0.376869 0.926266i \(-0.622999\pi\)
−0.376869 + 0.926266i \(0.622999\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.36483 0.182026
\(576\) 0 0
\(577\) 29.3495 1.22184 0.610919 0.791693i \(-0.290800\pi\)
0.610919 + 0.791693i \(0.290800\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.51539 −0.311791
\(582\) 0 0
\(583\) 81.1242i 3.35982i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.1732i 0.461168i 0.973052 + 0.230584i \(0.0740637\pi\)
−0.973052 + 0.230584i \(0.925936\pi\)
\(588\) 0 0
\(589\) 0.346368i 0.0142718i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.9792i 1.23110i 0.788099 + 0.615549i \(0.211066\pi\)
−0.788099 + 0.615549i \(0.788934\pi\)
\(594\) 0 0
\(595\) −0.808542 −0.0331470
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.38388 0.260838 0.130419 0.991459i \(-0.458368\pi\)
0.130419 + 0.991459i \(0.458368\pi\)
\(600\) 0 0
\(601\) −23.5445 −0.960402 −0.480201 0.877159i \(-0.659436\pi\)
−0.480201 + 0.877159i \(0.659436\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.25958 0.254488
\(606\) 0 0
\(607\) − 33.9599i − 1.37839i −0.724577 0.689194i \(-0.757965\pi\)
0.724577 0.689194i \(-0.242035\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 24.0034i − 0.971076i
\(612\) 0 0
\(613\) − 20.6042i − 0.832197i −0.909320 0.416098i \(-0.863397\pi\)
0.909320 0.416098i \(-0.136603\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 37.4489i − 1.50764i −0.657083 0.753818i \(-0.728210\pi\)
0.657083 0.753818i \(-0.271790\pi\)
\(618\) 0 0
\(619\) −14.2766 −0.573825 −0.286913 0.957957i \(-0.592629\pi\)
−0.286913 + 0.957957i \(0.592629\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.44682 0.178158
\(624\) 0 0
\(625\) 24.3592 0.974366
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −38.7427 −1.54477
\(630\) 0 0
\(631\) − 7.00448i − 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.49672i 0.0593954i
\(636\) 0 0
\(637\) 3.52336i 0.139601i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 9.23092i − 0.364599i −0.983243 0.182300i \(-0.941646\pi\)
0.983243 0.182300i \(-0.0583541\pi\)
\(642\) 0 0
\(643\) −1.48603 −0.0586031 −0.0293016 0.999571i \(-0.509328\pi\)
−0.0293016 + 0.999571i \(0.509328\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.2654 −0.639457 −0.319729 0.947509i \(-0.603592\pi\)
−0.319729 + 0.947509i \(0.603592\pi\)
\(648\) 0 0
\(649\) 27.1863 1.06716
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.92310 0.270922 0.135461 0.990783i \(-0.456748\pi\)
0.135461 + 0.990783i \(0.456748\pi\)
\(654\) 0 0
\(655\) − 2.65003i − 0.103545i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 11.3057i − 0.440407i −0.975454 0.220203i \(-0.929328\pi\)
0.975454 0.220203i \(-0.0706721\pi\)
\(660\) 0 0
\(661\) 18.9877i 0.738536i 0.929323 + 0.369268i \(0.120392\pi\)
−0.929323 + 0.369268i \(0.879608\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.13574i − 0.0440422i
\(666\) 0 0
\(667\) −2.82481 −0.109377
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −78.9932 −3.04950
\(672\) 0 0
\(673\) 17.6164 0.679061 0.339531 0.940595i \(-0.389732\pi\)
0.339531 + 0.940595i \(0.389732\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.42579 −0.285396 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(678\) 0 0
\(679\) − 17.3929i − 0.667480i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 22.8832i − 0.875600i −0.899072 0.437800i \(-0.855758\pi\)
0.899072 0.437800i \(-0.144242\pi\)
\(684\) 0 0
\(685\) 0.0753186i 0.00287778i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 44.5086i − 1.69564i
\(690\) 0 0
\(691\) −33.6439 −1.27987 −0.639936 0.768428i \(-0.721039\pi\)
−0.639936 + 0.768428i \(0.721039\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.92986 −0.0732038
\(696\) 0 0
\(697\) 23.2112 0.879188
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 35.4474 1.33883 0.669414 0.742890i \(-0.266545\pi\)
0.669414 + 0.742890i \(0.266545\pi\)
\(702\) 0 0
\(703\) − 54.4212i − 2.05253i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.0273i 0.715595i
\(708\) 0 0
\(709\) 8.71118i 0.327155i 0.986530 + 0.163578i \(0.0523034\pi\)
−0.986530 + 0.163578i \(0.947697\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 0.0555833i − 0.00208161i
\(714\) 0 0
\(715\) −4.68352 −0.175154
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.821108 0.0306222 0.0153111 0.999883i \(-0.495126\pi\)
0.0153111 + 0.999883i \(0.495126\pi\)
\(720\) 0 0
\(721\) −10.7903 −0.401853
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −15.9032 −0.590632
\(726\) 0 0
\(727\) − 28.0509i − 1.04035i −0.854059 0.520176i \(-0.825866\pi\)
0.854059 0.520176i \(-0.174134\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 4.45147i − 0.164643i
\(732\) 0 0
\(733\) − 21.9677i − 0.811395i −0.914007 0.405697i \(-0.867029\pi\)
0.914007 0.405697i \(-0.132971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.9045i 1.35939i
\(738\) 0 0
\(739\) 18.6044 0.684374 0.342187 0.939632i \(-0.388832\pi\)
0.342187 + 0.939632i \(0.388832\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.0849 −1.06702 −0.533512 0.845793i \(-0.679128\pi\)
−0.533512 + 0.845793i \(0.679128\pi\)
\(744\) 0 0
\(745\) −0.998774 −0.0365922
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.97087 −0.181632
\(750\) 0 0
\(751\) 14.4693i 0.527991i 0.964524 + 0.263995i \(0.0850403\pi\)
−0.964524 + 0.263995i \(0.914960\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2.64886i − 0.0964018i
\(756\) 0 0
\(757\) − 13.6988i − 0.497891i −0.968517 0.248946i \(-0.919916\pi\)
0.968517 0.248946i \(-0.0800840\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 46.5579i − 1.68772i −0.536562 0.843861i \(-0.680277\pi\)
0.536562 0.843861i \(-0.319723\pi\)
\(762\) 0 0
\(763\) 2.84810 0.103108
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.9157 −0.538575
\(768\) 0 0
\(769\) 50.6427 1.82622 0.913111 0.407711i \(-0.133673\pi\)
0.913111 + 0.407711i \(0.133673\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.66928 −0.239877 −0.119939 0.992781i \(-0.538270\pi\)
−0.119939 + 0.992781i \(0.538270\pi\)
\(774\) 0 0
\(775\) − 0.312926i − 0.0112406i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.6044i 1.16817i
\(780\) 0 0
\(781\) − 65.3718i − 2.33919i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 3.21779i − 0.114848i
\(786\) 0 0
\(787\) 0.759732 0.0270815 0.0135408 0.999908i \(-0.495690\pi\)
0.0135408 + 0.999908i \(0.495690\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.88561 −0.102600
\(792\) 0 0
\(793\) 43.3394 1.53903
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.1758 −0.927194 −0.463597 0.886046i \(-0.653441\pi\)
−0.463597 + 0.886046i \(0.653441\pi\)
\(798\) 0 0
\(799\) − 26.6114i − 0.941442i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 53.1488i 1.87558i
\(804\) 0 0
\(805\) 0.182258i 0.00642376i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 14.6876i − 0.516388i −0.966093 0.258194i \(-0.916873\pi\)
0.966093 0.258194i \(-0.0831273\pi\)
\(810\) 0 0
\(811\) 4.13912 0.145344 0.0726721 0.997356i \(-0.476847\pi\)
0.0726721 + 0.997356i \(0.476847\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.82168 0.133868
\(816\) 0 0
\(817\) 6.25289 0.218761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30.7760 −1.07409 −0.537045 0.843554i \(-0.680460\pi\)
−0.537045 + 0.843554i \(0.680460\pi\)
\(822\) 0 0
\(823\) 28.3750i 0.989090i 0.869152 + 0.494545i \(0.164665\pi\)
−0.869152 + 0.494545i \(0.835335\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 26.2522i − 0.912880i −0.889754 0.456440i \(-0.849124\pi\)
0.889754 0.456440i \(-0.150876\pi\)
\(828\) 0 0
\(829\) − 8.40833i − 0.292033i −0.989282 0.146017i \(-0.953355\pi\)
0.989282 0.146017i \(-0.0466453\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.90616i 0.135340i
\(834\) 0 0
\(835\) −2.59401 −0.0897695
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −10.3099 −0.355936 −0.177968 0.984036i \(-0.556952\pi\)
−0.177968 + 0.984036i \(0.556952\pi\)
\(840\) 0 0
\(841\) −18.7078 −0.645098
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.121287 −0.00417239
\(846\) 0 0
\(847\) − 30.2408i − 1.03909i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.73324i 0.299372i
\(852\) 0 0
\(853\) 42.6992i 1.46199i 0.682382 + 0.730996i \(0.260944\pi\)
−0.682382 + 0.730996i \(0.739056\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 5.06990i − 0.173185i −0.996244 0.0865923i \(-0.972402\pi\)
0.996244 0.0865923i \(-0.0275977\pi\)
\(858\) 0 0
\(859\) −10.9220 −0.372654 −0.186327 0.982488i \(-0.559658\pi\)
−0.186327 + 0.982488i \(0.559658\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38.1030 −1.29704 −0.648520 0.761197i \(-0.724612\pi\)
−0.648520 + 0.761197i \(0.724612\pi\)
\(864\) 0 0
\(865\) 1.13674 0.0386502
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 31.2378 1.05967
\(870\) 0 0
\(871\) − 20.2476i − 0.686062i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.06105i 0.0696760i
\(876\) 0 0
\(877\) 26.1674i 0.883610i 0.897111 + 0.441805i \(0.145662\pi\)
−0.897111 + 0.441805i \(0.854338\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 29.2903i − 0.986817i −0.869798 0.493408i \(-0.835751\pi\)
0.869798 0.493408i \(-0.164249\pi\)
\(882\) 0 0
\(883\) 49.0887 1.65197 0.825983 0.563696i \(-0.190621\pi\)
0.825983 + 0.563696i \(0.190621\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.0617 −0.673606 −0.336803 0.941575i \(-0.609346\pi\)
−0.336803 + 0.941575i \(0.609346\pi\)
\(888\) 0 0
\(889\) 7.23082 0.242514
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 37.3805 1.25089
\(894\) 0 0
\(895\) − 3.01855i − 0.100899i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.202518i 0.00675434i
\(900\) 0 0
\(901\) − 49.3443i − 1.64390i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 3.57585i − 0.118865i
\(906\) 0 0
\(907\) −17.0142 −0.564948 −0.282474 0.959275i \(-0.591155\pi\)
−0.282474 + 0.959275i \(0.591155\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.54462 −0.283096 −0.141548 0.989931i \(-0.545208\pi\)
−0.141548 + 0.989931i \(0.545208\pi\)
\(912\) 0 0
\(913\) −48.2631 −1.59727
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.8026 −0.422780
\(918\) 0 0
\(919\) 6.49841i 0.214363i 0.994239 + 0.107181i \(0.0341826\pi\)
−0.994239 + 0.107181i \(0.965817\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 35.8660i 1.18055i
\(924\) 0 0
\(925\) 49.1669i 1.61660i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.9216i 0.555181i 0.960699 + 0.277591i \(0.0895358\pi\)
−0.960699 + 0.277591i \(0.910464\pi\)
\(930\) 0 0
\(931\) −5.48691 −0.179826
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.19237 −0.169809
\(936\) 0 0
\(937\) 14.4249 0.471240 0.235620 0.971845i \(-0.424288\pi\)
0.235620 + 0.971845i \(0.424288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 58.5364 1.90823 0.954116 0.299436i \(-0.0967986\pi\)
0.954116 + 0.299436i \(0.0967986\pi\)
\(942\) 0 0
\(943\) − 5.23218i − 0.170383i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 15.3006i − 0.497201i −0.968606 0.248601i \(-0.920029\pi\)
0.968606 0.248601i \(-0.0799707\pi\)
\(948\) 0 0
\(949\) − 29.1599i − 0.946571i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 14.9133i − 0.483089i −0.970390 0.241544i \(-0.922346\pi\)
0.970390 0.241544i \(-0.0776539\pi\)
\(954\) 0 0
\(955\) 1.51068 0.0488845
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.363873 0.0117501
\(960\) 0 0
\(961\) 30.9960 0.999871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.50680 −0.0806966
\(966\) 0 0
\(967\) 27.1896i 0.874358i 0.899375 + 0.437179i \(0.144022\pi\)
−0.899375 + 0.437179i \(0.855978\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 35.0892i − 1.12607i −0.826434 0.563034i \(-0.809634\pi\)
0.826434 0.563034i \(-0.190366\pi\)
\(972\) 0 0
\(973\) 9.32338i 0.298894i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 32.0438i − 1.02517i −0.858636 0.512586i \(-0.828688\pi\)
0.858636 0.512586i \(-0.171312\pi\)
\(978\) 0 0
\(979\) 28.5570 0.912687
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 43.5719 1.38973 0.694864 0.719142i \(-0.255465\pi\)
0.694864 + 0.719142i \(0.255465\pi\)
\(984\) 0 0
\(985\) −3.60011 −0.114709
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.00343 −0.0319073
\(990\) 0 0
\(991\) 15.2929i 0.485794i 0.970052 + 0.242897i \(0.0780978\pi\)
−0.970052 + 0.242897i \(0.921902\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 0.636670i − 0.0201838i
\(996\) 0 0
\(997\) 29.8050i 0.943933i 0.881616 + 0.471967i \(0.156456\pi\)
−0.881616 + 0.471967i \(0.843544\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.j.c.5615.13 32
3.2 odd 2 inner 6048.2.j.c.5615.20 32
4.3 odd 2 1512.2.j.c.323.24 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.19 32
8.5 even 2 1512.2.j.c.323.10 yes 32
12.11 even 2 1512.2.j.c.323.9 32
24.5 odd 2 1512.2.j.c.323.23 yes 32
24.11 even 2 inner 6048.2.j.c.5615.14 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.9 32 12.11 even 2
1512.2.j.c.323.10 yes 32 8.5 even 2
1512.2.j.c.323.23 yes 32 24.5 odd 2
1512.2.j.c.323.24 yes 32 4.3 odd 2
6048.2.j.c.5615.13 32 1.1 even 1 trivial
6048.2.j.c.5615.14 32 24.11 even 2 inner
6048.2.j.c.5615.19 32 8.3 odd 2 inner
6048.2.j.c.5615.20 32 3.2 odd 2 inner