Properties

Label 6048.2.a.bk.1.1
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39528.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.87063\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.24049 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-4.24049 q^{5} -1.00000 q^{7} -5.74125 q^{11} +6.06174 q^{13} -2.56098 q^{17} -7.74125 q^{19} -1.67951 q^{23} +12.9817 q^{25} -5.24049 q^{29} -5.24049 q^{31} +4.24049 q^{35} -7.98174 q^{37} -2.24049 q^{41} +5.92000 q^{43} -6.62271 q^{47} +1.00000 q^{49} -2.85826 q^{53} +24.3457 q^{55} -7.48098 q^{59} -12.2222 q^{61} -25.7047 q^{65} -8.16049 q^{67} -2.32049 q^{71} -1.00152 q^{73} +5.74125 q^{77} +7.59951 q^{79} +13.9817 q^{83} +10.8598 q^{85} +11.8030 q^{89} -6.06174 q^{91} +32.8267 q^{95} +3.61777 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - 4q^{7} + 8q^{13} - 2q^{17} - 8q^{19} + 14q^{25} - 6q^{29} - 6q^{31} + 2q^{35} + 6q^{37} + 6q^{41} + 2q^{43} - 2q^{47} + 4q^{49} - 6q^{53} + 12q^{55} + 4q^{61} - 4q^{65} + 4q^{67} - 16q^{71} + 12q^{73} + 2q^{79} + 18q^{83} + 22q^{85} + 8q^{89} - 8q^{91} + 16q^{95} + 24q^{97} + O(q^{100}) \)

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
\(5\) −4.24049 −1.89640 −0.948202 0.317668i \(-0.897100\pi\)
−0.948202 + 0.317668i \(0.897100\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.74125 −1.73105 −0.865526 0.500864i \(-0.833016\pi\)
−0.865526 + 0.500864i \(0.833016\pi\)
\(12\) 0 0
\(13\) 6.06174 1.68122 0.840612 0.541638i \(-0.182196\pi\)
0.840612 + 0.541638i \(0.182196\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.56098 −0.621128 −0.310564 0.950552i \(-0.600518\pi\)
−0.310564 + 0.950552i \(0.600518\pi\)
\(18\) 0 0
\(19\) −7.74125 −1.77596 −0.887982 0.459878i \(-0.847893\pi\)
−0.887982 + 0.459878i \(0.847893\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.67951 −0.350203 −0.175101 0.984550i \(-0.556025\pi\)
−0.175101 + 0.984550i \(0.556025\pi\)
\(24\) 0 0
\(25\) 12.9817 2.59635
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.24049 −0.973134 −0.486567 0.873643i \(-0.661751\pi\)
−0.486567 + 0.873643i \(0.661751\pi\)
\(30\) 0 0
\(31\) −5.24049 −0.941219 −0.470610 0.882341i \(-0.655966\pi\)
−0.470610 + 0.882341i \(0.655966\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.24049 0.716773
\(36\) 0 0
\(37\) −7.98174 −1.31219 −0.656095 0.754678i \(-0.727793\pi\)
−0.656095 + 0.754678i \(0.727793\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.24049 −0.349905 −0.174953 0.984577i \(-0.555977\pi\)
−0.174953 + 0.984577i \(0.555977\pi\)
\(42\) 0 0
\(43\) 5.92000 0.902792 0.451396 0.892324i \(-0.350926\pi\)
0.451396 + 0.892324i \(0.350926\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.62271 −0.966022 −0.483011 0.875614i \(-0.660457\pi\)
−0.483011 + 0.875614i \(0.660457\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.85826 −0.392613 −0.196306 0.980543i \(-0.562895\pi\)
−0.196306 + 0.980543i \(0.562895\pi\)
\(54\) 0 0
\(55\) 24.3457 3.28277
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.48098 −0.973940 −0.486970 0.873419i \(-0.661898\pi\)
−0.486970 + 0.873419i \(0.661898\pi\)
\(60\) 0 0
\(61\) −12.2222 −1.56490 −0.782448 0.622716i \(-0.786029\pi\)
−0.782448 + 0.622716i \(0.786029\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25.7047 −3.18828
\(66\) 0 0
\(67\) −8.16049 −0.996962 −0.498481 0.866901i \(-0.666109\pi\)
−0.498481 + 0.866901i \(0.666109\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.32049 −0.275391 −0.137696 0.990475i \(-0.543970\pi\)
−0.137696 + 0.990475i \(0.543970\pi\)
\(72\) 0 0
\(73\) −1.00152 −0.117220 −0.0586098 0.998281i \(-0.518667\pi\)
−0.0586098 + 0.998281i \(0.518667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.74125 0.654276
\(78\) 0 0
\(79\) 7.59951 0.855012 0.427506 0.904013i \(-0.359392\pi\)
0.427506 + 0.904013i \(0.359392\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.9817 1.53470 0.767348 0.641231i \(-0.221576\pi\)
0.767348 + 0.641231i \(0.221576\pi\)
\(84\) 0 0
\(85\) 10.8598 1.17791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.8030 1.25111 0.625557 0.780178i \(-0.284872\pi\)
0.625557 + 0.780178i \(0.284872\pi\)
\(90\) 0 0
\(91\) −6.06174 −0.635443
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 32.8267 3.36795
\(96\) 0 0
\(97\) 3.61777 0.367329 0.183665 0.982989i \(-0.441204\pi\)
0.183665 + 0.982989i \(0.441204\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.48250 0.943544 0.471772 0.881721i \(-0.343615\pi\)
0.471772 + 0.881721i \(0.343615\pi\)
\(102\) 0 0
\(103\) −20.1037 −1.98088 −0.990438 0.137961i \(-0.955945\pi\)
−0.990438 + 0.137961i \(0.955945\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −1.98174 −0.189816 −0.0949080 0.995486i \(-0.530256\pi\)
−0.0949080 + 0.995486i \(0.530256\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.74125 −0.163803 −0.0819015 0.996640i \(-0.526099\pi\)
−0.0819015 + 0.996640i \(0.526099\pi\)
\(114\) 0 0
\(115\) 7.12195 0.664125
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.56098 0.234764
\(120\) 0 0
\(121\) 21.9620 1.99654
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −33.8465 −3.02732
\(126\) 0 0
\(127\) −11.7230 −1.04025 −0.520123 0.854091i \(-0.674114\pi\)
−0.520123 + 0.854091i \(0.674114\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.50229 0.655478 0.327739 0.944768i \(-0.393713\pi\)
0.327739 + 0.944768i \(0.393713\pi\)
\(132\) 0 0
\(133\) 7.74125 0.671252
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.7413 1.00312 0.501561 0.865122i \(-0.332759\pi\)
0.501561 + 0.865122i \(0.332759\pi\)
\(138\) 0 0
\(139\) 0.902774 0.0765723 0.0382861 0.999267i \(-0.487810\pi\)
0.0382861 + 0.999267i \(0.487810\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −34.8020 −2.91029
\(144\) 0 0
\(145\) 22.2222 1.84546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.24201 −0.675212 −0.337606 0.941288i \(-0.609617\pi\)
−0.337606 + 0.941288i \(0.609617\pi\)
\(150\) 0 0
\(151\) 13.3624 1.08742 0.543710 0.839273i \(-0.317019\pi\)
0.543710 + 0.839273i \(0.317019\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 22.2222 1.78493
\(156\) 0 0
\(157\) −15.0237 −1.19902 −0.599510 0.800367i \(-0.704638\pi\)
−0.599510 + 0.800367i \(0.704638\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.67951 0.132364
\(162\) 0 0
\(163\) −1.08152 −0.0847115 −0.0423557 0.999103i \(-0.513486\pi\)
−0.0423557 + 0.999103i \(0.513486\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.6242 −1.20904 −0.604520 0.796590i \(-0.706635\pi\)
−0.604520 + 0.796590i \(0.706635\pi\)
\(168\) 0 0
\(169\) 23.7447 1.82651
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.9635 0.909566 0.454783 0.890602i \(-0.349717\pi\)
0.454783 + 0.890602i \(0.349717\pi\)
\(174\) 0 0
\(175\) −12.9817 −0.981327
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.22070 0.240727 0.120363 0.992730i \(-0.461594\pi\)
0.120363 + 0.992730i \(0.461594\pi\)
\(180\) 0 0
\(181\) 1.19854 0.0890865 0.0445433 0.999007i \(-0.485817\pi\)
0.0445433 + 0.999007i \(0.485817\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 33.8465 2.48844
\(186\) 0 0
\(187\) 14.7032 1.07520
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0015 1.37490 0.687451 0.726231i \(-0.258730\pi\)
0.687451 + 0.726231i \(0.258730\pi\)
\(192\) 0 0
\(193\) 12.0015 0.863889 0.431944 0.901900i \(-0.357828\pi\)
0.431944 + 0.901900i \(0.357828\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.09875 0.149530 0.0747649 0.997201i \(-0.476179\pi\)
0.0747649 + 0.997201i \(0.476179\pi\)
\(198\) 0 0
\(199\) 11.8598 0.840718 0.420359 0.907358i \(-0.361904\pi\)
0.420359 + 0.907358i \(0.361904\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.24049 0.367810
\(204\) 0 0
\(205\) 9.50076 0.663562
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 44.4445 3.07429
\(210\) 0 0
\(211\) −4.92494 −0.339047 −0.169523 0.985526i \(-0.554223\pi\)
−0.169523 + 0.985526i \(0.554223\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −25.1037 −1.71206
\(216\) 0 0
\(217\) 5.24049 0.355748
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −15.5240 −1.04425
\(222\) 0 0
\(223\) −1.62082 −0.108538 −0.0542692 0.998526i \(-0.517283\pi\)
−0.0542692 + 0.998526i \(0.517283\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.9437 −1.19096 −0.595482 0.803368i \(-0.703039\pi\)
−0.595482 + 0.803368i \(0.703039\pi\)
\(228\) 0 0
\(229\) 17.1220 1.13145 0.565725 0.824594i \(-0.308596\pi\)
0.565725 + 0.824594i \(0.308596\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.8617 0.908109 0.454054 0.890974i \(-0.349977\pi\)
0.454054 + 0.890974i \(0.349977\pi\)
\(234\) 0 0
\(235\) 28.0835 1.83197
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.57820 0.619562 0.309781 0.950808i \(-0.399744\pi\)
0.309781 + 0.950808i \(0.399744\pi\)
\(240\) 0 0
\(241\) −9.58125 −0.617183 −0.308591 0.951195i \(-0.599858\pi\)
−0.308591 + 0.951195i \(0.599858\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.24049 −0.270915
\(246\) 0 0
\(247\) −46.9254 −2.98579
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.9437 1.19572 0.597858 0.801602i \(-0.296019\pi\)
0.597858 + 0.801602i \(0.296019\pi\)
\(252\) 0 0
\(253\) 9.64250 0.606219
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.35750 0.521327 0.260663 0.965430i \(-0.416059\pi\)
0.260663 + 0.965430i \(0.416059\pi\)
\(258\) 0 0
\(259\) 7.98174 0.495961
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.5412 1.20496 0.602481 0.798133i \(-0.294179\pi\)
0.602481 + 0.798133i \(0.294179\pi\)
\(264\) 0 0
\(265\) 12.1204 0.744552
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.7610 −0.899996 −0.449998 0.893029i \(-0.648575\pi\)
−0.449998 + 0.893029i \(0.648575\pi\)
\(270\) 0 0
\(271\) −11.8598 −0.720431 −0.360215 0.932869i \(-0.617297\pi\)
−0.360215 + 0.932869i \(0.617297\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −74.5314 −4.49441
\(276\) 0 0
\(277\) −4.98326 −0.299415 −0.149708 0.988730i \(-0.547833\pi\)
−0.149708 + 0.988730i \(0.547833\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00152 −0.0597459 −0.0298730 0.999554i \(-0.509510\pi\)
−0.0298730 + 0.999554i \(0.509510\pi\)
\(282\) 0 0
\(283\) 7.90125 0.469681 0.234840 0.972034i \(-0.424543\pi\)
0.234840 + 0.972034i \(0.424543\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.24049 0.132252
\(288\) 0 0
\(289\) −10.4414 −0.614200
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.00341485 0.000199498 0 9.97490e−5 1.00000i \(-0.499968\pi\)
9.97490e−5 1.00000i \(0.499968\pi\)
\(294\) 0 0
\(295\) 31.7230 1.84698
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.1808 −0.588769
\(300\) 0 0
\(301\) −5.92000 −0.341223
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 51.8282 2.96767
\(306\) 0 0
\(307\) −23.9604 −1.36749 −0.683747 0.729719i \(-0.739651\pi\)
−0.683747 + 0.729719i \(0.739651\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.97869 −0.225611 −0.112805 0.993617i \(-0.535984\pi\)
−0.112805 + 0.993617i \(0.535984\pi\)
\(312\) 0 0
\(313\) 28.2005 1.59399 0.796995 0.603986i \(-0.206422\pi\)
0.796995 + 0.603986i \(0.206422\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.0015 −0.954901 −0.477450 0.878659i \(-0.658439\pi\)
−0.477450 + 0.878659i \(0.658439\pi\)
\(318\) 0 0
\(319\) 30.0870 1.68455
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 19.8252 1.10310
\(324\) 0 0
\(325\) 78.6919 4.36504
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.62271 0.365122
\(330\) 0 0
\(331\) −1.95652 −0.107540 −0.0537701 0.998553i \(-0.517124\pi\)
−0.0537701 + 0.998553i \(0.517124\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 34.6045 1.89064
\(336\) 0 0
\(337\) −30.2089 −1.64558 −0.822792 0.568343i \(-0.807585\pi\)
−0.822792 + 0.568343i \(0.807585\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 30.0870 1.62930
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −23.9882 −1.28775 −0.643877 0.765129i \(-0.722675\pi\)
−0.643877 + 0.765129i \(0.722675\pi\)
\(348\) 0 0
\(349\) 29.2855 1.56762 0.783808 0.621003i \(-0.213275\pi\)
0.783808 + 0.621003i \(0.213275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.0420 0.800602 0.400301 0.916384i \(-0.368906\pi\)
0.400301 + 0.916384i \(0.368906\pi\)
\(354\) 0 0
\(355\) 9.84000 0.522253
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.9990 −1.00273 −0.501363 0.865237i \(-0.667168\pi\)
−0.501363 + 0.865237i \(0.667168\pi\)
\(360\) 0 0
\(361\) 40.9270 2.15405
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.24695 0.222296
\(366\) 0 0
\(367\) 14.8583 0.775595 0.387797 0.921745i \(-0.373236\pi\)
0.387797 + 0.921745i \(0.373236\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.85826 0.148394
\(372\) 0 0
\(373\) 17.1006 0.885438 0.442719 0.896660i \(-0.354014\pi\)
0.442719 + 0.896660i \(0.354014\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −31.7665 −1.63606
\(378\) 0 0
\(379\) 20.3609 1.04587 0.522935 0.852373i \(-0.324837\pi\)
0.522935 + 0.852373i \(0.324837\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.1067 −1.58948 −0.794740 0.606950i \(-0.792393\pi\)
−0.794740 + 0.606950i \(0.792393\pi\)
\(384\) 0 0
\(385\) −24.3457 −1.24077
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.5797 0.536413 0.268207 0.963361i \(-0.413569\pi\)
0.268207 + 0.963361i \(0.413569\pi\)
\(390\) 0 0
\(391\) 4.30119 0.217521
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −32.2256 −1.62145
\(396\) 0 0
\(397\) 1.35598 0.0680545 0.0340272 0.999421i \(-0.489167\pi\)
0.0340272 + 0.999421i \(0.489167\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.23555 0.461201 0.230601 0.973048i \(-0.425931\pi\)
0.230601 + 0.973048i \(0.425931\pi\)
\(402\) 0 0
\(403\) −31.7665 −1.58240
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 45.8252 2.27147
\(408\) 0 0
\(409\) 9.87957 0.488513 0.244257 0.969711i \(-0.421456\pi\)
0.244257 + 0.969711i \(0.421456\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.48098 0.368115
\(414\) 0 0
\(415\) −59.2894 −2.91040
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.23555 −0.109214 −0.0546069 0.998508i \(-0.517391\pi\)
−0.0546069 + 0.998508i \(0.517391\pi\)
\(420\) 0 0
\(421\) 1.83695 0.0895276 0.0447638 0.998998i \(-0.485746\pi\)
0.0447638 + 0.998998i \(0.485746\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −33.2459 −1.61266
\(426\) 0 0
\(427\) 12.2222 0.591475
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.517500 −0.0249271 −0.0124635 0.999922i \(-0.503967\pi\)
−0.0124635 + 0.999922i \(0.503967\pi\)
\(432\) 0 0
\(433\) −21.0607 −1.01211 −0.506056 0.862500i \(-0.668897\pi\)
−0.506056 + 0.862500i \(0.668897\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.0015 0.621947
\(438\) 0 0
\(439\) −5.61967 −0.268212 −0.134106 0.990967i \(-0.542816\pi\)
−0.134106 + 0.990967i \(0.542816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.7443 1.03310 0.516551 0.856256i \(-0.327216\pi\)
0.516551 + 0.856256i \(0.327216\pi\)
\(444\) 0 0
\(445\) −50.0504 −2.37262
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.77777 0.0838983 0.0419492 0.999120i \(-0.486643\pi\)
0.0419492 + 0.999120i \(0.486643\pi\)
\(450\) 0 0
\(451\) 12.8632 0.605704
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 25.7047 1.20506
\(456\) 0 0
\(457\) −11.7462 −0.549464 −0.274732 0.961521i \(-0.588589\pi\)
−0.274732 + 0.961521i \(0.588589\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.5980 −0.586747 −0.293373 0.955998i \(-0.594778\pi\)
−0.293373 + 0.955998i \(0.594778\pi\)
\(462\) 0 0
\(463\) −16.5615 −0.769677 −0.384838 0.922984i \(-0.625743\pi\)
−0.384838 + 0.922984i \(0.625743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.7264 0.820280 0.410140 0.912023i \(-0.365480\pi\)
0.410140 + 0.912023i \(0.365480\pi\)
\(468\) 0 0
\(469\) 8.16049 0.376816
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.9882 −1.56278
\(474\) 0 0
\(475\) −100.495 −4.61102
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −42.0291 −1.92036 −0.960180 0.279383i \(-0.909870\pi\)
−0.960180 + 0.279383i \(0.909870\pi\)
\(480\) 0 0
\(481\) −48.3832 −2.20608
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.3411 −0.696605
\(486\) 0 0
\(487\) 15.9604 0.723236 0.361618 0.932326i \(-0.382224\pi\)
0.361618 + 0.932326i \(0.382224\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.4311 1.77950 0.889751 0.456446i \(-0.150878\pi\)
0.889751 + 0.456446i \(0.150878\pi\)
\(492\) 0 0
\(493\) 13.4208 0.604441
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.32049 0.104088
\(498\) 0 0
\(499\) 25.3229 1.13361 0.566804 0.823853i \(-0.308180\pi\)
0.566804 + 0.823853i \(0.308180\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.1417 −0.541373 −0.270687 0.962668i \(-0.587251\pi\)
−0.270687 + 0.962668i \(0.587251\pi\)
\(504\) 0 0
\(505\) −40.2104 −1.78934
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −35.4859 −1.57289 −0.786443 0.617663i \(-0.788080\pi\)
−0.786443 + 0.617663i \(0.788080\pi\)
\(510\) 0 0
\(511\) 1.00152 0.0443048
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 85.2495 3.75654
\(516\) 0 0
\(517\) 38.0227 1.67223
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.7699 −1.39186 −0.695932 0.718108i \(-0.745008\pi\)
−0.695932 + 0.718108i \(0.745008\pi\)
\(522\) 0 0
\(523\) −8.97680 −0.392528 −0.196264 0.980551i \(-0.562881\pi\)
−0.196264 + 0.980551i \(0.562881\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.4208 0.584618
\(528\) 0 0
\(529\) −20.1792 −0.877358
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13.5813 −0.588269
\(534\) 0 0
\(535\) 16.9620 0.733329
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.74125 −0.247293
\(540\) 0 0
\(541\) −19.5816 −0.841879 −0.420940 0.907089i \(-0.638300\pi\)
−0.420940 + 0.907089i \(0.638300\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.40354 0.359968
\(546\) 0 0
\(547\) 44.8480 1.91756 0.958781 0.284147i \(-0.0917104\pi\)
0.958781 + 0.284147i \(0.0917104\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 40.5679 1.72825
\(552\) 0 0
\(553\) −7.59951 −0.323164
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.57326 −0.0666613 −0.0333306 0.999444i \(-0.510611\pi\)
−0.0333306 + 0.999444i \(0.510611\pi\)
\(558\) 0 0
\(559\) 35.8855 1.51779
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.4231 −0.987168 −0.493584 0.869698i \(-0.664313\pi\)
−0.493584 + 0.869698i \(0.664313\pi\)
\(564\) 0 0
\(565\) 7.38375 0.310637
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.3255 1.14555 0.572773 0.819714i \(-0.305868\pi\)
0.572773 + 0.819714i \(0.305868\pi\)
\(570\) 0 0
\(571\) −34.1304 −1.42831 −0.714157 0.699986i \(-0.753190\pi\)
−0.714157 + 0.699986i \(0.753190\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −21.8030 −0.909247
\(576\) 0 0
\(577\) 4.35750 0.181405 0.0907026 0.995878i \(-0.471089\pi\)
0.0907026 + 0.995878i \(0.471089\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.9817 −0.580060
\(582\) 0 0
\(583\) 16.4100 0.679633
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.2287 −0.546006 −0.273003 0.962013i \(-0.588017\pi\)
−0.273003 + 0.962013i \(0.588017\pi\)
\(588\) 0 0
\(589\) 40.5679 1.67157
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.2405 0.913307 0.456654 0.889645i \(-0.349048\pi\)
0.456654 + 0.889645i \(0.349048\pi\)
\(594\) 0 0
\(595\) −10.8598 −0.445208
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.70119 −0.314662 −0.157331 0.987546i \(-0.550289\pi\)
−0.157331 + 0.987546i \(0.550289\pi\)
\(600\) 0 0
\(601\) −45.1872 −1.84323 −0.921613 0.388111i \(-0.873128\pi\)
−0.921613 + 0.388111i \(0.873128\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −93.1294 −3.78625
\(606\) 0 0
\(607\) 16.4494 0.667660 0.333830 0.942633i \(-0.391659\pi\)
0.333830 + 0.942633i \(0.391659\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −40.1452 −1.62410
\(612\) 0 0
\(613\) 27.5634 1.11327 0.556637 0.830756i \(-0.312092\pi\)
0.556637 + 0.830756i \(0.312092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.48402 0.341554 0.170777 0.985310i \(-0.445372\pi\)
0.170777 + 0.985310i \(0.445372\pi\)
\(618\) 0 0
\(619\) −36.5896 −1.47066 −0.735330 0.677709i \(-0.762973\pi\)
−0.735330 + 0.677709i \(0.762973\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −11.8030 −0.472877
\(624\) 0 0
\(625\) 78.6168 3.14467
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.4410 0.815038
\(630\) 0 0
\(631\) 10.7976 0.429844 0.214922 0.976631i \(-0.431050\pi\)
0.214922 + 0.976631i \(0.431050\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 49.7112 1.97273
\(636\) 0 0
\(637\) 6.06174 0.240175
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −38.2696 −1.51156 −0.755779 0.654826i \(-0.772742\pi\)
−0.755779 + 0.654826i \(0.772742\pi\)
\(642\) 0 0
\(643\) −35.8499 −1.41378 −0.706891 0.707323i \(-0.749903\pi\)
−0.706891 + 0.707323i \(0.749903\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26.4810 −1.04107 −0.520537 0.853839i \(-0.674268\pi\)
−0.520537 + 0.853839i \(0.674268\pi\)
\(648\) 0 0
\(649\) 42.9502 1.68594
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.00646 0.0393859 0.0196930 0.999806i \(-0.493731\pi\)
0.0196930 + 0.999806i \(0.493731\pi\)
\(654\) 0 0
\(655\) −31.8134 −1.24305
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.2485 −0.438178 −0.219089 0.975705i \(-0.570309\pi\)
−0.219089 + 0.975705i \(0.570309\pi\)
\(660\) 0 0
\(661\) −14.2192 −0.553062 −0.276531 0.961005i \(-0.589185\pi\)
−0.276531 + 0.961005i \(0.589185\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −32.8267 −1.27296
\(666\) 0 0
\(667\) 8.80146 0.340794
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 70.1709 2.70892
\(672\) 0 0
\(673\) 32.2302 1.24238 0.621192 0.783659i \(-0.286649\pi\)
0.621192 + 0.783659i \(0.286649\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −48.5710 −1.86673 −0.933367 0.358923i \(-0.883144\pi\)
−0.933367 + 0.358923i \(0.883144\pi\)
\(678\) 0 0
\(679\) −3.61777 −0.138837
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.2222 0.391143 0.195571 0.980689i \(-0.437344\pi\)
0.195571 + 0.980689i \(0.437344\pi\)
\(684\) 0 0
\(685\) −49.7886 −1.90233
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.3260 −0.660069
\(690\) 0 0
\(691\) −31.2207 −1.18769 −0.593846 0.804579i \(-0.702391\pi\)
−0.593846 + 0.804579i \(0.702391\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.82820 −0.145212
\(696\) 0 0
\(697\) 5.73784 0.217336
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.3822 −0.694287 −0.347144 0.937812i \(-0.612848\pi\)
−0.347144 + 0.937812i \(0.612848\pi\)
\(702\) 0 0
\(703\) 61.7886 2.33040
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.48250 −0.356626
\(708\) 0 0
\(709\) 46.5466 1.74809 0.874047 0.485841i \(-0.161486\pi\)
0.874047 + 0.485841i \(0.161486\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.80146 0.329617
\(714\) 0 0
\(715\) 147.577 5.51908
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −44.6998 −1.66702 −0.833510 0.552504i \(-0.813672\pi\)
−0.833510 + 0.552504i \(0.813672\pi\)
\(720\) 0 0
\(721\) 20.1037 0.748701
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −68.0306 −2.52659
\(726\) 0 0
\(727\) −22.5285 −0.835537 −0.417769 0.908553i \(-0.637188\pi\)
−0.417769 + 0.908553i \(0.637188\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.1610 −0.560749
\(732\) 0 0
\(733\) −8.92189 −0.329538 −0.164769 0.986332i \(-0.552688\pi\)
−0.164769 + 0.986332i \(0.552688\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 46.8514 1.72579
\(738\) 0 0
\(739\) −13.6770 −0.503115 −0.251557 0.967842i \(-0.580943\pi\)
−0.251557 + 0.967842i \(0.580943\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.7417 −0.944373 −0.472186 0.881499i \(-0.656535\pi\)
−0.472186 + 0.881499i \(0.656535\pi\)
\(744\) 0 0
\(745\) 34.9502 1.28047
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 47.5329 1.73450 0.867251 0.497872i \(-0.165885\pi\)
0.867251 + 0.497872i \(0.165885\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −56.6633 −2.06219
\(756\) 0 0
\(757\) −7.57479 −0.275310 −0.137655 0.990480i \(-0.543957\pi\)
−0.137655 + 0.990480i \(0.543957\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.2874 0.880417 0.440208 0.897896i \(-0.354905\pi\)
0.440208 + 0.897896i \(0.354905\pi\)
\(762\) 0 0
\(763\) 1.98174 0.0717437
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −45.3477 −1.63741
\(768\) 0 0
\(769\) 1.35902 0.0490077 0.0245038 0.999700i \(-0.492199\pi\)
0.0245038 + 0.999700i \(0.492199\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.2024 0.762599 0.381299 0.924452i \(-0.375477\pi\)
0.381299 + 0.924452i \(0.375477\pi\)
\(774\) 0 0
\(775\) −68.0306 −2.44373
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.3442 0.621420
\(780\) 0 0
\(781\) 13.3225 0.476717
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 63.7078 2.27383
\(786\) 0 0
\(787\) 37.1990 1.32600 0.663001 0.748618i \(-0.269282\pi\)
0.663001 + 0.748618i \(0.269282\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.74125 0.0619117
\(792\) 0 0
\(793\) −74.0879 −2.63094
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.2875 1.21453 0.607263 0.794501i \(-0.292268\pi\)
0.607263 + 0.794501i \(0.292268\pi\)
\(798\) 0 0
\(799\) 16.9606 0.600023
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.75000 0.202913
\(804\) 0 0
\(805\) −7.12195 −0.251016
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 11.2424 0.395261 0.197630 0.980277i \(-0.436675\pi\)
0.197630 + 0.980277i \(0.436675\pi\)
\(810\) 0 0
\(811\) −10.8997 −0.382741 −0.191371 0.981518i \(-0.561293\pi\)
−0.191371 + 0.981518i \(0.561293\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.58619 0.160647
\(816\) 0 0
\(817\) −45.8282 −1.60333
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.5497 1.45009 0.725047 0.688700i \(-0.241818\pi\)
0.725047 + 0.688700i \(0.241818\pi\)
\(822\) 0 0
\(823\) 27.3999 0.955102 0.477551 0.878604i \(-0.341525\pi\)
0.477551 + 0.878604i \(0.341525\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33.1003 −1.15101 −0.575505 0.817798i \(-0.695194\pi\)
−0.575505 + 0.817798i \(0.695194\pi\)
\(828\) 0 0
\(829\) −24.4623 −0.849612 −0.424806 0.905284i \(-0.639658\pi\)
−0.424806 + 0.905284i \(0.639658\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.56098 −0.0887325
\(834\) 0 0
\(835\) 66.2544 2.29283
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.1792 1.24905 0.624523 0.781006i \(-0.285293\pi\)
0.624523 + 0.781006i \(0.285293\pi\)
\(840\) 0 0
\(841\) −1.53729 −0.0530099
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −100.689 −3.46381
\(846\) 0 0
\(847\) −21.9620 −0.754622
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.4054 0.459532
\(852\) 0 0
\(853\) −18.1605 −0.621803 −0.310902 0.950442i \(-0.600631\pi\)
−0.310902 + 0.950442i \(0.600631\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.4499 1.21094 0.605472 0.795866i \(-0.292984\pi\)
0.605472 + 0.795866i \(0.292984\pi\)
\(858\) 0 0
\(859\) −34.2036 −1.16701 −0.583506 0.812109i \(-0.698319\pi\)
−0.583506 + 0.812109i \(0.698319\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.91354 0.0651376 0.0325688 0.999469i \(-0.489631\pi\)
0.0325688 + 0.999469i \(0.489631\pi\)
\(864\) 0 0
\(865\) −50.7310 −1.72490
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −43.6307 −1.48007
\(870\) 0 0
\(871\) −49.4667 −1.67612
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 33.8465 1.14422
\(876\) 0 0
\(877\) 46.6237 1.57437 0.787185 0.616717i \(-0.211538\pi\)
0.787185 + 0.616717i \(0.211538\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.6795 0.663020 0.331510 0.943452i \(-0.392442\pi\)
0.331510 + 0.943452i \(0.392442\pi\)
\(882\) 0 0
\(883\) 1.83305 0.0616870 0.0308435 0.999524i \(-0.490181\pi\)
0.0308435 + 0.999524i \(0.490181\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −47.6648 −1.60043 −0.800213 0.599715i \(-0.795281\pi\)
−0.800213 + 0.599715i \(0.795281\pi\)
\(888\) 0 0
\(889\) 11.7230 0.393176
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 51.2681 1.71562
\(894\) 0 0
\(895\) −13.6573 −0.456515
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 27.4627 0.915933
\(900\) 0 0
\(901\) 7.31994 0.243863
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.08238 −0.168944
\(906\) 0 0
\(907\) −30.0805 −0.998806 −0.499403 0.866370i \(-0.666447\pi\)
−0.499403 + 0.866370i \(0.666447\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 41.5843 1.37775 0.688875 0.724880i \(-0.258105\pi\)
0.688875 + 0.724880i \(0.258105\pi\)
\(912\) 0 0
\(913\) −80.2727 −2.65664
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.50229 −0.247747
\(918\) 0 0
\(919\) −19.0820 −0.629458 −0.314729 0.949182i \(-0.601914\pi\)
−0.314729 + 0.949182i \(0.601914\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −14.0662 −0.462994
\(924\) 0 0
\(925\) −103.617 −3.40690
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.9684 1.54098 0.770492 0.637450i \(-0.220011\pi\)
0.770492 + 0.637450i \(0.220011\pi\)
\(930\) 0 0
\(931\) −7.74125 −0.253709
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −62.3487 −2.03902
\(936\) 0 0
\(937\) −10.7397 −0.350852 −0.175426 0.984493i \(-0.556130\pi\)
−0.175426 + 0.984493i \(0.556130\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 35.2024 1.14757 0.573783 0.819007i \(-0.305475\pi\)
0.573783 + 0.819007i \(0.305475\pi\)
\(942\) 0 0
\(943\) 3.76293 0.122538
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.7196 −0.640800 −0.320400 0.947282i \(-0.603817\pi\)
−0.320400 + 0.947282i \(0.603817\pi\)
\(948\) 0 0
\(949\) −6.07098 −0.197072
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −42.0870 −1.36333 −0.681665 0.731664i \(-0.738744\pi\)
−0.681665 + 0.731664i \(0.738744\pi\)
\(954\) 0 0
\(955\) −80.5757 −2.60737
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.7413 −0.379145
\(960\) 0 0
\(961\) −3.53729 −0.114106
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −50.8923 −1.63828
\(966\) 0 0
\(967\) 6.87500 0.221085 0.110543 0.993871i \(-0.464741\pi\)
0.110543 + 0.993871i \(0.464741\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −35.9718 −1.15439 −0.577195 0.816606i \(-0.695853\pi\)
−0.577195 + 0.816606i \(0.695853\pi\)
\(972\) 0 0
\(973\) −0.902774 −0.0289416
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.6014 0.339169 0.169584 0.985516i \(-0.445757\pi\)
0.169584 + 0.985516i \(0.445757\pi\)
\(978\) 0 0
\(979\) −67.7639 −2.16574
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.3027 1.15788 0.578938 0.815371i \(-0.303467\pi\)
0.578938 + 0.815371i \(0.303467\pi\)
\(984\) 0 0
\(985\) −8.89973 −0.283569
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.94271 −0.316160
\(990\) 0 0
\(991\) −15.0049 −0.476648 −0.238324 0.971186i \(-0.576598\pi\)
−0.238324 + 0.971186i \(0.576598\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −50.2913 −1.59434
\(996\) 0 0
\(997\) −37.6677 −1.19295 −0.596474 0.802632i \(-0.703432\pi\)
−0.596474 + 0.802632i \(0.703432\pi\)
\(998\) 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.a.bk.1.1 4
3.2 odd 2 6048.2.a.bs.1.4 yes 4
4.3 odd 2 6048.2.a.bp.1.1 yes 4
12.11 even 2 6048.2.a.bt.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bk.1.1 4 1.1 even 1 trivial
6048.2.a.bp.1.1 yes 4 4.3 odd 2
6048.2.a.bs.1.4 yes 4 3.2 odd 2
6048.2.a.bt.1.4 yes 4 12.11 even 2