Properties

Label 5472.2.a.bf.1.2
Level $5472$
Weight $2$
Character 5472.1
Self dual yes
Analytic conductor $43.694$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5472.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q+2.56155 q^{5} +3.00000 q^{7} -0.561553 q^{11} -1.56155 q^{13} -0.123106 q^{17} +1.00000 q^{19} -5.56155 q^{23} +1.56155 q^{25} -4.68466 q^{29} +9.12311 q^{31} +7.68466 q^{35} +7.12311 q^{37} -4.00000 q^{41} +5.43845 q^{43} +3.68466 q^{47} +2.00000 q^{49} +4.43845 q^{53} -1.43845 q^{55} +4.43845 q^{59} +14.8078 q^{61} -4.00000 q^{65} -0.438447 q^{67} +4.24621 q^{71} -9.24621 q^{73} -1.68466 q^{77} +10.0000 q^{79} +17.3693 q^{83} -0.315342 q^{85} +15.3693 q^{89} -4.68466 q^{91} +2.56155 q^{95} +7.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + 6q^{7} + O(q^{10}) \) \( 2q + q^{5} + 6q^{7} + 3q^{11} + q^{13} + 8q^{17} + 2q^{19} - 7q^{23} - q^{25} + 3q^{29} + 10q^{31} + 3q^{35} + 6q^{37} - 8q^{41} + 15q^{43} - 5q^{47} + 4q^{49} + 13q^{53} - 7q^{55} + 13q^{59} + 9q^{61} - 8q^{65} - 5q^{67} - 8q^{71} - 2q^{73} + 9q^{77} + 20q^{79} + 10q^{83} - 13q^{85} + 6q^{89} + 3q^{91} + q^{95} + 6q^{97} + O(q^{100}) \)

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
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.561553 −0.169315 −0.0846573 0.996410i \(-0.526980\pi\)
−0.0846573 + 0.996410i \(0.526980\pi\)
\(12\) 0 0
\(13\) −1.56155 −0.433097 −0.216548 0.976272i \(-0.569480\pi\)
−0.216548 + 0.976272i \(0.569480\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.123106 −0.0298575 −0.0149287 0.999889i \(-0.504752\pi\)
−0.0149287 + 0.999889i \(0.504752\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.68466 −0.869919 −0.434960 0.900450i \(-0.643237\pi\)
−0.434960 + 0.900450i \(0.643237\pi\)
\(30\) 0 0
\(31\) 9.12311 1.63856 0.819279 0.573395i \(-0.194374\pi\)
0.819279 + 0.573395i \(0.194374\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.68466 1.29894
\(36\) 0 0
\(37\) 7.12311 1.17103 0.585516 0.810661i \(-0.300892\pi\)
0.585516 + 0.810661i \(0.300892\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 5.43845 0.829355 0.414678 0.909968i \(-0.363894\pi\)
0.414678 + 0.909968i \(0.363894\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.68466 0.537463 0.268731 0.963215i \(-0.413396\pi\)
0.268731 + 0.963215i \(0.413396\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) 0 0
\(55\) −1.43845 −0.193960
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.43845 0.577837 0.288918 0.957354i \(-0.406704\pi\)
0.288918 + 0.957354i \(0.406704\pi\)
\(60\) 0 0
\(61\) 14.8078 1.89594 0.947970 0.318360i \(-0.103132\pi\)
0.947970 + 0.318360i \(0.103132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −0.438447 −0.0535648 −0.0267824 0.999641i \(-0.508526\pi\)
−0.0267824 + 0.999641i \(0.508526\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.24621 0.503933 0.251966 0.967736i \(-0.418923\pi\)
0.251966 + 0.967736i \(0.418923\pi\)
\(72\) 0 0
\(73\) −9.24621 −1.08219 −0.541094 0.840962i \(-0.681990\pi\)
−0.541094 + 0.840962i \(0.681990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.68466 −0.191985
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.3693 1.90653 0.953265 0.302135i \(-0.0976994\pi\)
0.953265 + 0.302135i \(0.0976994\pi\)
\(84\) 0 0
\(85\) −0.315342 −0.0342036
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.3693 1.62914 0.814572 0.580062i \(-0.196972\pi\)
0.814572 + 0.580062i \(0.196972\pi\)
\(90\) 0 0
\(91\) −4.68466 −0.491086
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.56155 0.262810
\(96\) 0 0
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) 14.4924 1.42798 0.713990 0.700155i \(-0.246886\pi\)
0.713990 + 0.700155i \(0.246886\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.9309 −1.63677 −0.818384 0.574671i \(-0.805130\pi\)
−0.818384 + 0.574671i \(0.805130\pi\)
\(108\) 0 0
\(109\) −3.56155 −0.341135 −0.170567 0.985346i \(-0.554560\pi\)
−0.170567 + 0.985346i \(0.554560\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.87689 −0.458780 −0.229390 0.973335i \(-0.573673\pi\)
−0.229390 + 0.973335i \(0.573673\pi\)
\(114\) 0 0
\(115\) −14.2462 −1.32847
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.369317 −0.0338552
\(120\) 0 0
\(121\) −10.6847 −0.971333
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) 15.1231 1.34196 0.670979 0.741476i \(-0.265874\pi\)
0.670979 + 0.741476i \(0.265874\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.4384 1.17412 0.587061 0.809542i \(-0.300285\pi\)
0.587061 + 0.809542i \(0.300285\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.3693 −1.22765 −0.613827 0.789441i \(-0.710371\pi\)
−0.613827 + 0.789441i \(0.710371\pi\)
\(138\) 0 0
\(139\) 5.43845 0.461283 0.230642 0.973039i \(-0.425918\pi\)
0.230642 + 0.973039i \(0.425918\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.876894 0.0733296
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.5616 −0.865236 −0.432618 0.901577i \(-0.642410\pi\)
−0.432618 + 0.901577i \(0.642410\pi\)
\(150\) 0 0
\(151\) −22.4924 −1.83041 −0.915204 0.402992i \(-0.867970\pi\)
−0.915204 + 0.402992i \(0.867970\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.3693 1.87707
\(156\) 0 0
\(157\) −7.75379 −0.618820 −0.309410 0.950929i \(-0.600131\pi\)
−0.309410 + 0.950929i \(0.600131\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.6847 −1.31494
\(162\) 0 0
\(163\) 16.4924 1.29179 0.645893 0.763428i \(-0.276485\pi\)
0.645893 + 0.763428i \(0.276485\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −10.5616 −0.812427
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) 0 0
\(175\) 4.68466 0.354127
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.24621 −0.167890 −0.0839449 0.996470i \(-0.526752\pi\)
−0.0839449 + 0.996470i \(0.526752\pi\)
\(180\) 0 0
\(181\) −0.246211 −0.0183007 −0.00915037 0.999958i \(-0.502913\pi\)
−0.00915037 + 0.999958i \(0.502913\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.2462 1.34149
\(186\) 0 0
\(187\) 0.0691303 0.00505531
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 0.0723575 0.0361787 0.999345i \(-0.488481\pi\)
0.0361787 + 0.999345i \(0.488481\pi\)
\(192\) 0 0
\(193\) 18.4924 1.33111 0.665557 0.746347i \(-0.268194\pi\)
0.665557 + 0.746347i \(0.268194\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.3693 −0.810030 −0.405015 0.914310i \(-0.632734\pi\)
−0.405015 + 0.914310i \(0.632734\pi\)
\(198\) 0 0
\(199\) −23.7386 −1.68279 −0.841394 0.540423i \(-0.818264\pi\)
−0.841394 + 0.540423i \(0.818264\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.0540 −0.986396
\(204\) 0 0
\(205\) −10.2462 −0.715626
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.561553 −0.0388434
\(210\) 0 0
\(211\) 15.5616 1.07130 0.535651 0.844440i \(-0.320066\pi\)
0.535651 + 0.844440i \(0.320066\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.9309 0.950077
\(216\) 0 0
\(217\) 27.3693 1.85795
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.192236 0.0129312
\(222\) 0 0
\(223\) −1.36932 −0.0916962 −0.0458481 0.998948i \(-0.514599\pi\)
−0.0458481 + 0.998948i \(0.514599\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.6847 −1.10740 −0.553700 0.832716i \(-0.686784\pi\)
−0.553700 + 0.832716i \(0.686784\pi\)
\(228\) 0 0
\(229\) −12.5616 −0.830091 −0.415045 0.909801i \(-0.636234\pi\)
−0.415045 + 0.909801i \(0.636234\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.9309 1.30571 0.652857 0.757481i \(-0.273570\pi\)
0.652857 + 0.757481i \(0.273570\pi\)
\(234\) 0 0
\(235\) 9.43845 0.615696
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −14.2462 −0.917679 −0.458840 0.888519i \(-0.651735\pi\)
−0.458840 + 0.888519i \(0.651735\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.12311 0.327303
\(246\) 0 0
\(247\) −1.56155 −0.0993592
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.68466 −0.358812 −0.179406 0.983775i \(-0.557418\pi\)
−0.179406 + 0.983775i \(0.557418\pi\)
\(252\) 0 0
\(253\) 3.12311 0.196348
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −26.2462 −1.63719 −0.818597 0.574369i \(-0.805248\pi\)
−0.818597 + 0.574369i \(0.805248\pi\)
\(258\) 0 0
\(259\) 21.3693 1.32782
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.0540 −0.681617 −0.340809 0.940133i \(-0.610701\pi\)
−0.340809 + 0.940133i \(0.610701\pi\)
\(264\) 0 0
\(265\) 11.3693 0.698412
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.4924 1.12750 0.563751 0.825944i \(-0.309358\pi\)
0.563751 + 0.825944i \(0.309358\pi\)
\(270\) 0 0
\(271\) 10.0540 0.610736 0.305368 0.952234i \(-0.401221\pi\)
0.305368 + 0.952234i \(0.401221\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.876894 −0.0528787
\(276\) 0 0
\(277\) −11.6847 −0.702063 −0.351032 0.936364i \(-0.614169\pi\)
−0.351032 + 0.936364i \(0.614169\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.630683 0.0376234 0.0188117 0.999823i \(-0.494012\pi\)
0.0188117 + 0.999823i \(0.494012\pi\)
\(282\) 0 0
\(283\) 28.5616 1.69781 0.848904 0.528547i \(-0.177263\pi\)
0.848904 + 0.528547i \(0.177263\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −16.9848 −0.999109
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −15.3153 −0.894732 −0.447366 0.894351i \(-0.647638\pi\)
−0.447366 + 0.894351i \(0.647638\pi\)
\(294\) 0 0
\(295\) 11.3693 0.661947
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.68466 0.502247
\(300\) 0 0
\(301\) 16.3153 0.940401
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 37.9309 2.17192
\(306\) 0 0
\(307\) 4.63068 0.264287 0.132144 0.991231i \(-0.457814\pi\)
0.132144 + 0.991231i \(0.457814\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.12311 0.347209 0.173605 0.984815i \(-0.444458\pi\)
0.173605 + 0.984815i \(0.444458\pi\)
\(312\) 0 0
\(313\) −0.930870 −0.0526159 −0.0263079 0.999654i \(-0.508375\pi\)
−0.0263079 + 0.999654i \(0.508375\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.4384 −0.923275 −0.461638 0.887069i \(-0.652738\pi\)
−0.461638 + 0.887069i \(0.652738\pi\)
\(318\) 0 0
\(319\) 2.63068 0.147290
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.123106 −0.00684978
\(324\) 0 0
\(325\) −2.43845 −0.135261
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0540 0.609425
\(330\) 0 0
\(331\) 4.43845 0.243959 0.121980 0.992533i \(-0.461076\pi\)
0.121980 + 0.992533i \(0.461076\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.12311 −0.0613618
\(336\) 0 0
\(337\) −20.4924 −1.11629 −0.558147 0.829742i \(-0.688487\pi\)
−0.558147 + 0.829742i \(0.688487\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.12311 −0.277432
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.56155 0.244877 0.122438 0.992476i \(-0.460929\pi\)
0.122438 + 0.992476i \(0.460929\pi\)
\(348\) 0 0
\(349\) 13.6847 0.732523 0.366261 0.930512i \(-0.380638\pi\)
0.366261 + 0.930512i \(0.380638\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.9309 1.53983 0.769917 0.638144i \(-0.220297\pi\)
0.769917 + 0.638144i \(0.220297\pi\)
\(354\) 0 0
\(355\) 10.8769 0.577286
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.8769 −1.47129 −0.735643 0.677369i \(-0.763120\pi\)
−0.735643 + 0.677369i \(0.763120\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −23.6847 −1.23971
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.3153 0.691298
\(372\) 0 0
\(373\) −14.9309 −0.773091 −0.386546 0.922270i \(-0.626332\pi\)
−0.386546 + 0.922270i \(0.626332\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.31534 0.376759
\(378\) 0 0
\(379\) 16.0540 0.824637 0.412319 0.911040i \(-0.364719\pi\)
0.412319 + 0.911040i \(0.364719\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.2462 −1.23892 −0.619462 0.785027i \(-0.712649\pi\)
−0.619462 + 0.785027i \(0.712649\pi\)
\(384\) 0 0
\(385\) −4.31534 −0.219930
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.8078 −1.35921 −0.679604 0.733579i \(-0.737848\pi\)
−0.679604 + 0.733579i \(0.737848\pi\)
\(390\) 0 0
\(391\) 0.684658 0.0346247
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.6155 1.28886
\(396\) 0 0
\(397\) −5.43845 −0.272948 −0.136474 0.990644i \(-0.543577\pi\)
−0.136474 + 0.990644i \(0.543577\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −23.3693 −1.16701 −0.583504 0.812110i \(-0.698319\pi\)
−0.583504 + 0.812110i \(0.698319\pi\)
\(402\) 0 0
\(403\) −14.2462 −0.709654
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 18.8769 0.933402 0.466701 0.884415i \(-0.345442\pi\)
0.466701 + 0.884415i \(0.345442\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.3153 0.655205
\(414\) 0 0
\(415\) 44.4924 2.18405
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) 0 0
\(421\) 20.0540 0.977371 0.488685 0.872460i \(-0.337477\pi\)
0.488685 + 0.872460i \(0.337477\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.192236 −0.00932481
\(426\) 0 0
\(427\) 44.4233 2.14979
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.1231 1.49915 0.749574 0.661921i \(-0.230259\pi\)
0.749574 + 0.661921i \(0.230259\pi\)
\(432\) 0 0
\(433\) 39.6155 1.90380 0.951900 0.306408i \(-0.0991271\pi\)
0.951900 + 0.306408i \(0.0991271\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.56155 −0.266045
\(438\) 0 0
\(439\) 1.75379 0.0837038 0.0418519 0.999124i \(-0.486674\pi\)
0.0418519 + 0.999124i \(0.486674\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.68466 −0.0800405 −0.0400203 0.999199i \(-0.512742\pi\)
−0.0400203 + 0.999199i \(0.512742\pi\)
\(444\) 0 0
\(445\) 39.3693 1.86628
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.1231 0.713703 0.356852 0.934161i \(-0.383850\pi\)
0.356852 + 0.934161i \(0.383850\pi\)
\(450\) 0 0
\(451\) 2.24621 0.105770
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −36.1231 −1.68977 −0.844884 0.534950i \(-0.820330\pi\)
−0.844884 + 0.534950i \(0.820330\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 33.9309 1.58032 0.790159 0.612902i \(-0.209998\pi\)
0.790159 + 0.612902i \(0.209998\pi\)
\(462\) 0 0
\(463\) 15.0540 0.699618 0.349809 0.936821i \(-0.386247\pi\)
0.349809 + 0.936821i \(0.386247\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.6847 −0.818348 −0.409174 0.912456i \(-0.634183\pi\)
−0.409174 + 0.912456i \(0.634183\pi\)
\(468\) 0 0
\(469\) −1.31534 −0.0607368
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.05398 −0.140422
\(474\) 0 0
\(475\) 1.56155 0.0716490
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.2462 0.650926 0.325463 0.945555i \(-0.394480\pi\)
0.325463 + 0.945555i \(0.394480\pi\)
\(480\) 0 0
\(481\) −11.1231 −0.507170
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.2462 0.828518
\(486\) 0 0
\(487\) −6.49242 −0.294200 −0.147100 0.989122i \(-0.546994\pi\)
−0.147100 + 0.989122i \(0.546994\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.4924 1.46636 0.733181 0.680033i \(-0.238035\pi\)
0.733181 + 0.680033i \(0.238035\pi\)
\(492\) 0 0
\(493\) 0.576708 0.0259736
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.7386 0.571406
\(498\) 0 0
\(499\) −3.68466 −0.164948 −0.0824740 0.996593i \(-0.526282\pi\)
−0.0824740 + 0.996593i \(0.526282\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.43845 0.108725 0.0543625 0.998521i \(-0.482687\pi\)
0.0543625 + 0.998521i \(0.482687\pi\)
\(504\) 0 0
\(505\) 0.630683 0.0280650
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.6155 0.869443 0.434721 0.900565i \(-0.356847\pi\)
0.434721 + 0.900565i \(0.356847\pi\)
\(510\) 0 0
\(511\) −27.7386 −1.22708
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 37.1231 1.63584
\(516\) 0 0
\(517\) −2.06913 −0.0910002
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.3693 −1.02383 −0.511914 0.859037i \(-0.671063\pi\)
−0.511914 + 0.859037i \(0.671063\pi\)
\(522\) 0 0
\(523\) −32.5464 −1.42315 −0.711577 0.702608i \(-0.752019\pi\)
−0.711577 + 0.702608i \(0.752019\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.12311 −0.0489232
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.24621 0.270553
\(534\) 0 0
\(535\) −43.3693 −1.87502
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.12311 −0.0483756
\(540\) 0 0
\(541\) −35.4384 −1.52362 −0.761809 0.647802i \(-0.775688\pi\)
−0.761809 + 0.647802i \(0.775688\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.12311 −0.390791
\(546\) 0 0
\(547\) 24.4924 1.04722 0.523610 0.851958i \(-0.324585\pi\)
0.523610 + 0.851958i \(0.324585\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.68466 −0.199573
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.9309 0.759755 0.379878 0.925037i \(-0.375966\pi\)
0.379878 + 0.925037i \(0.375966\pi\)
\(558\) 0 0
\(559\) −8.49242 −0.359191
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.2462 1.27473 0.637363 0.770564i \(-0.280025\pi\)
0.637363 + 0.770564i \(0.280025\pi\)
\(564\) 0 0
\(565\) −12.4924 −0.525560
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.7386 1.12094 0.560471 0.828174i \(-0.310620\pi\)
0.560471 + 0.828174i \(0.310620\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.68466 −0.362175
\(576\) 0 0
\(577\) 5.24621 0.218403 0.109201 0.994020i \(-0.465171\pi\)
0.109201 + 0.994020i \(0.465171\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 52.1080 2.16180
\(582\) 0 0
\(583\) −2.49242 −0.103226
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.9309 1.40048 0.700238 0.713909i \(-0.253077\pi\)
0.700238 + 0.713909i \(0.253077\pi\)
\(588\) 0 0
\(589\) 9.12311 0.375911
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −32.7386 −1.34441 −0.672207 0.740363i \(-0.734654\pi\)
−0.672207 + 0.740363i \(0.734654\pi\)
\(594\) 0 0
\(595\) −0.946025 −0.0387832
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −43.8617 −1.79214 −0.896071 0.443911i \(-0.853591\pi\)
−0.896071 + 0.443911i \(0.853591\pi\)
\(600\) 0 0
\(601\) −4.63068 −0.188890 −0.0944448 0.995530i \(-0.530108\pi\)
−0.0944448 + 0.995530i \(0.530108\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.3693 −1.11272
\(606\) 0 0
\(607\) −12.8769 −0.522657 −0.261329 0.965250i \(-0.584161\pi\)
−0.261329 + 0.965250i \(0.584161\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.75379 −0.232773
\(612\) 0 0
\(613\) −41.5464 −1.67804 −0.839022 0.544098i \(-0.816872\pi\)
−0.839022 + 0.544098i \(0.816872\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.6847 −0.872991 −0.436496 0.899706i \(-0.643781\pi\)
−0.436496 + 0.899706i \(0.643781\pi\)
\(618\) 0 0
\(619\) −47.1231 −1.89404 −0.947019 0.321178i \(-0.895921\pi\)
−0.947019 + 0.321178i \(0.895921\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 46.1080 1.84728
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.876894 −0.0349641
\(630\) 0 0
\(631\) −45.3002 −1.80337 −0.901686 0.432391i \(-0.857670\pi\)
−0.901686 + 0.432391i \(0.857670\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 38.7386 1.53730
\(636\) 0 0
\(637\) −3.12311 −0.123742
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.1080 −0.952207 −0.476103 0.879389i \(-0.657951\pi\)
−0.476103 + 0.879389i \(0.657951\pi\)
\(642\) 0 0
\(643\) 27.3002 1.07661 0.538307 0.842749i \(-0.319064\pi\)
0.538307 + 0.842749i \(0.319064\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −47.0000 −1.84776 −0.923880 0.382682i \(-0.875001\pi\)
−0.923880 + 0.382682i \(0.875001\pi\)
\(648\) 0 0
\(649\) −2.49242 −0.0978361
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8.80776 −0.344674 −0.172337 0.985038i \(-0.555132\pi\)
−0.172337 + 0.985038i \(0.555132\pi\)
\(654\) 0 0
\(655\) 34.4233 1.34503
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.5464 1.26783 0.633914 0.773404i \(-0.281447\pi\)
0.633914 + 0.773404i \(0.281447\pi\)
\(660\) 0 0
\(661\) −21.1771 −0.823693 −0.411846 0.911253i \(-0.635116\pi\)
−0.411846 + 0.911253i \(0.635116\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.68466 0.297998
\(666\) 0 0
\(667\) 26.0540 1.00881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.31534 −0.321010
\(672\) 0 0
\(673\) −21.1231 −0.814236 −0.407118 0.913376i \(-0.633466\pi\)
−0.407118 + 0.913376i \(0.633466\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.43845 0.0937171 0.0468586 0.998902i \(-0.485079\pi\)
0.0468586 + 0.998902i \(0.485079\pi\)
\(678\) 0 0
\(679\) 21.3693 0.820079
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.7538 0.679330 0.339665 0.940547i \(-0.389686\pi\)
0.339665 + 0.940547i \(0.389686\pi\)
\(684\) 0 0
\(685\) −36.8078 −1.40635
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.93087 −0.264045
\(690\) 0 0
\(691\) −45.0540 −1.71393 −0.856967 0.515371i \(-0.827654\pi\)
−0.856967 + 0.515371i \(0.827654\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.9309 0.528428
\(696\) 0 0
\(697\) 0.492423 0.0186518
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13.6155 −0.514251 −0.257126 0.966378i \(-0.582775\pi\)
−0.257126 + 0.966378i \(0.582775\pi\)
\(702\) 0 0
\(703\) 7.12311 0.268653
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.738634 0.0277792
\(708\) 0 0
\(709\) 50.4924 1.89628 0.948141 0.317849i \(-0.102960\pi\)
0.948141 + 0.317849i \(0.102960\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −50.7386 −1.90018
\(714\) 0 0
\(715\) 2.24621 0.0840035
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23.4924 −0.876120 −0.438060 0.898946i \(-0.644334\pi\)
−0.438060 + 0.898946i \(0.644334\pi\)
\(720\) 0 0
\(721\) 43.4773 1.61918
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.31534 −0.271685
\(726\) 0 0
\(727\) 22.3693 0.829632 0.414816 0.909905i \(-0.363846\pi\)
0.414816 + 0.909905i \(0.363846\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.669503 −0.0247625
\(732\) 0 0
\(733\) 3.36932 0.124449 0.0622243 0.998062i \(-0.480181\pi\)
0.0622243 + 0.998062i \(0.480181\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.246211 0.00906931
\(738\) 0 0
\(739\) 1.43845 0.0529141 0.0264571 0.999650i \(-0.491577\pi\)
0.0264571 + 0.999650i \(0.491577\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22.8769 −0.839272 −0.419636 0.907693i \(-0.637842\pi\)
−0.419636 + 0.907693i \(0.637842\pi\)
\(744\) 0 0
\(745\) −27.0540 −0.991181
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −50.7926 −1.85592
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −57.6155 −2.09684
\(756\) 0 0
\(757\) 21.0540 0.765220 0.382610 0.923910i \(-0.375025\pi\)
0.382610 + 0.923910i \(0.375025\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.8617 1.48124 0.740618 0.671926i \(-0.234533\pi\)
0.740618 + 0.671926i \(0.234533\pi\)
\(762\) 0 0
\(763\) −10.6847 −0.386811
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.93087 −0.250259
\(768\) 0 0
\(769\) 12.3693 0.446049 0.223024 0.974813i \(-0.428407\pi\)
0.223024 + 0.974813i \(0.428407\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32.0540 1.15290 0.576451 0.817132i \(-0.304437\pi\)
0.576451 + 0.817132i \(0.304437\pi\)
\(774\) 0 0
\(775\) 14.2462 0.511739
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −2.38447 −0.0853231
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −19.8617 −0.708896
\(786\) 0 0
\(787\) 22.4384 0.799844 0.399922 0.916549i \(-0.369037\pi\)
0.399922 + 0.916549i \(0.369037\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.6307 −0.520207
\(792\) 0 0
\(793\) −23.1231 −0.821126
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.68466 0.0950955 0.0475477 0.998869i \(-0.484859\pi\)
0.0475477 + 0.998869i \(0.484859\pi\)
\(798\) 0 0
\(799\) −0.453602 −0.0160473
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.19224 0.183230
\(804\) 0 0
\(805\) −42.7386 −1.50634
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.12311 −0.215277 −0.107638 0.994190i \(-0.534329\pi\)
−0.107638 + 0.994190i \(0.534329\pi\)
\(810\) 0 0
\(811\) −11.4233 −0.401126 −0.200563 0.979681i \(-0.564277\pi\)
−0.200563 + 0.979681i \(0.564277\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 42.2462 1.47982
\(816\) 0 0
\(817\) 5.43845 0.190267
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.31534 −0.150606 −0.0753032 0.997161i \(-0.523992\pi\)
−0.0753032 + 0.997161i \(0.523992\pi\)
\(822\) 0 0
\(823\) 20.3693 0.710030 0.355015 0.934861i \(-0.384476\pi\)
0.355015 + 0.934861i \(0.384476\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19.5616 −0.680222 −0.340111 0.940385i \(-0.610465\pi\)
−0.340111 + 0.940385i \(0.610465\pi\)
\(828\) 0 0
\(829\) 18.5464 0.644143 0.322072 0.946715i \(-0.395621\pi\)
0.322072 + 0.946715i \(0.395621\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.246211 −0.00853071
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −29.6155 −1.02244 −0.511221 0.859449i \(-0.670807\pi\)
−0.511221 + 0.859449i \(0.670807\pi\)
\(840\) 0 0
\(841\) −7.05398 −0.243241
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −27.0540 −0.930685
\(846\) 0 0
\(847\) −32.0540 −1.10139
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −39.6155 −1.35800
\(852\) 0 0
\(853\) 32.2462 1.10409 0.552045 0.833815i \(-0.313848\pi\)
0.552045 + 0.833815i \(0.313848\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.6307 −0.431456 −0.215728 0.976454i \(-0.569212\pi\)
−0.215728 + 0.976454i \(0.569212\pi\)
\(858\) 0 0
\(859\) −35.9309 −1.22595 −0.612973 0.790104i \(-0.710026\pi\)
−0.612973 + 0.790104i \(0.710026\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.6307 −0.770357 −0.385179 0.922842i \(-0.625860\pi\)
−0.385179 + 0.922842i \(0.625860\pi\)
\(864\) 0 0
\(865\) 51.8617 1.76335
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.61553 −0.190494
\(870\) 0 0
\(871\) 0.684658 0.0231988
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −26.4233 −0.893270
\(876\) 0 0
\(877\) 2.68466 0.0906545 0.0453272 0.998972i \(-0.485567\pi\)
0.0453272 + 0.998972i \(0.485567\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −49.6847 −1.67392 −0.836959 0.547265i \(-0.815669\pi\)
−0.836959 + 0.547265i \(0.815669\pi\)
\(882\) 0 0
\(883\) 0.0691303 0.00232642 0.00116321 0.999999i \(-0.499630\pi\)
0.00116321 + 0.999999i \(0.499630\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.2311 1.25010 0.625048 0.780586i \(-0.285079\pi\)
0.625048 + 0.780586i \(0.285079\pi\)
\(888\) 0 0
\(889\) 45.3693 1.52164
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.68466 0.123302
\(894\) 0 0
\(895\) −5.75379 −0.192328
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −42.7386 −1.42541
\(900\) 0 0
\(901\) −0.546398 −0.0182032
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.630683 −0.0209646
\(906\) 0 0
\(907\) −15.5616 −0.516713 −0.258356 0.966050i \(-0.583181\pi\)
−0.258356 + 0.966050i \(0.583181\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.12311 0.302262 0.151131 0.988514i \(-0.451708\pi\)
0.151131 + 0.988514i \(0.451708\pi\)
\(912\) 0 0
\(913\) −9.75379 −0.322803
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.3153 1.33133
\(918\) 0 0
\(919\) −24.3002 −0.801589 −0.400795 0.916168i \(-0.631266\pi\)
−0.400795 + 0.916168i \(0.631266\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.63068 −0.218252
\(924\) 0 0
\(925\) 11.1231 0.365725
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 55.1771 1.81030 0.905151 0.425091i \(-0.139758\pi\)
0.905151 + 0.425091i \(0.139758\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.177081 0.00579117
\(936\) 0 0
\(937\) −12.7538 −0.416648 −0.208324 0.978060i \(-0.566801\pi\)
−0.208324 + 0.978060i \(0.566801\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.3002 −0.596569 −0.298285 0.954477i \(-0.596414\pi\)
−0.298285 + 0.954477i \(0.596414\pi\)
\(942\) 0 0
\(943\) 22.2462 0.724436
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.9848 −1.20185 −0.600923 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(948\) 0 0
\(949\) 14.4384 0.468692
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.24621 0.267121 0.133560 0.991041i \(-0.457359\pi\)
0.133560 + 0.991041i \(0.457359\pi\)
\(954\) 0 0
\(955\) 2.56155 0.0828899
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −43.1080 −1.39203
\(960\) 0 0
\(961\) 52.2311 1.68487
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 47.3693 1.52487
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 23.5076 0.754394 0.377197 0.926133i \(-0.376888\pi\)
0.377197 + 0.926133i \(0.376888\pi\)
\(972\) 0 0
\(973\) 16.3153 0.523046
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 53.6155 1.71531 0.857656 0.514223i \(-0.171920\pi\)
0.857656 + 0.514223i \(0.171920\pi\)
\(978\) 0 0
\(979\) −8.63068 −0.275838
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −11.6155 −0.370478 −0.185239 0.982694i \(-0.559306\pi\)
−0.185239 + 0.982694i \(0.559306\pi\)
\(984\) 0 0
\(985\) −29.1231 −0.927939
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.2462 −0.961774
\(990\) 0 0
\(991\) −18.2462 −0.579610 −0.289805 0.957086i \(-0.593590\pi\)
−0.289805 + 0.957086i \(0.593590\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −60.8078 −1.92774
\(996\) 0 0
\(997\) −46.9157 −1.48584 −0.742918 0.669383i \(-0.766559\pi\)
−0.742918 + 0.669383i \(0.766559\pi\)
\(998\) 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 5472.2.a.bf.1.2 2
3.2 odd 2 608.2.a.h.1.1 yes 2
4.3 odd 2 5472.2.a.bc.1.2 2
12.11 even 2 608.2.a.g.1.2 2
24.5 odd 2 1216.2.a.s.1.2 2
24.11 even 2 1216.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.g.1.2 2 12.11 even 2
608.2.a.h.1.1 yes 2 3.2 odd 2
1216.2.a.s.1.2 2 24.5 odd 2
1216.2.a.t.1.1 2 24.11 even 2
5472.2.a.bc.1.2 2 4.3 odd 2
5472.2.a.bf.1.2 2 1.1 even 1 trivial