Properties

Label 7448.2.a.bn.1.6
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.98211824.1
Defining polynomial: \(x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.04436\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.04436 q^{3} -2.60221 q^{5} +6.26815 q^{9} +O(q^{10})\) \(q+3.04436 q^{3} -2.60221 q^{5} +6.26815 q^{9} +3.68293 q^{11} +6.17208 q^{13} -7.92207 q^{15} -6.92207 q^{17} -1.00000 q^{19} -2.61920 q^{23} +1.77149 q^{25} +9.94945 q^{27} +3.12082 q^{29} +6.97542 q^{31} +11.2122 q^{33} +0.292486 q^{37} +18.7901 q^{39} +5.29265 q^{41} -1.96364 q^{43} -16.3111 q^{45} +7.99528 q^{47} -21.0733 q^{51} +1.79173 q^{53} -9.58376 q^{55} -3.04436 q^{57} +7.12345 q^{59} -2.14820 q^{61} -16.0610 q^{65} -7.63547 q^{67} -7.97378 q^{69} +3.29274 q^{71} +8.82925 q^{73} +5.39308 q^{75} -6.39094 q^{79} +11.4853 q^{81} +7.35997 q^{83} +18.0127 q^{85} +9.50092 q^{87} -13.5002 q^{89} +21.2357 q^{93} +2.60221 q^{95} -8.90409 q^{97} +23.0852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + O(q^{10}) \) \( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99} + O(q^{100}) \)

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
\(3\) 3.04436 1.75766 0.878832 0.477131i \(-0.158323\pi\)
0.878832 + 0.477131i \(0.158323\pi\)
\(4\) 0 0
\(5\) −2.60221 −1.16374 −0.581872 0.813281i \(-0.697679\pi\)
−0.581872 + 0.813281i \(0.697679\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.26815 2.08938
\(10\) 0 0
\(11\) 3.68293 1.11045 0.555223 0.831702i \(-0.312633\pi\)
0.555223 + 0.831702i \(0.312633\pi\)
\(12\) 0 0
\(13\) 6.17208 1.71183 0.855913 0.517119i \(-0.172996\pi\)
0.855913 + 0.517119i \(0.172996\pi\)
\(14\) 0 0
\(15\) −7.92207 −2.04547
\(16\) 0 0
\(17\) −6.92207 −1.67885 −0.839425 0.543476i \(-0.817108\pi\)
−0.839425 + 0.543476i \(0.817108\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.61920 −0.546140 −0.273070 0.961994i \(-0.588039\pi\)
−0.273070 + 0.961994i \(0.588039\pi\)
\(24\) 0 0
\(25\) 1.77149 0.354299
\(26\) 0 0
\(27\) 9.94945 1.91477
\(28\) 0 0
\(29\) 3.12082 0.579522 0.289761 0.957099i \(-0.406424\pi\)
0.289761 + 0.957099i \(0.406424\pi\)
\(30\) 0 0
\(31\) 6.97542 1.25282 0.626411 0.779493i \(-0.284523\pi\)
0.626411 + 0.779493i \(0.284523\pi\)
\(32\) 0 0
\(33\) 11.2122 1.95179
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.292486 0.0480843 0.0240422 0.999711i \(-0.492346\pi\)
0.0240422 + 0.999711i \(0.492346\pi\)
\(38\) 0 0
\(39\) 18.7901 3.00882
\(40\) 0 0
\(41\) 5.29265 0.826573 0.413287 0.910601i \(-0.364381\pi\)
0.413287 + 0.910601i \(0.364381\pi\)
\(42\) 0 0
\(43\) −1.96364 −0.299453 −0.149726 0.988727i \(-0.547839\pi\)
−0.149726 + 0.988727i \(0.547839\pi\)
\(44\) 0 0
\(45\) −16.3111 −2.43151
\(46\) 0 0
\(47\) 7.99528 1.16623 0.583116 0.812389i \(-0.301833\pi\)
0.583116 + 0.812389i \(0.301833\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −21.0733 −2.95085
\(52\) 0 0
\(53\) 1.79173 0.246113 0.123056 0.992400i \(-0.460730\pi\)
0.123056 + 0.992400i \(0.460730\pi\)
\(54\) 0 0
\(55\) −9.58376 −1.29227
\(56\) 0 0
\(57\) −3.04436 −0.403236
\(58\) 0 0
\(59\) 7.12345 0.927395 0.463697 0.885994i \(-0.346523\pi\)
0.463697 + 0.885994i \(0.346523\pi\)
\(60\) 0 0
\(61\) −2.14820 −0.275049 −0.137525 0.990498i \(-0.543915\pi\)
−0.137525 + 0.990498i \(0.543915\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16.0610 −1.99213
\(66\) 0 0
\(67\) −7.63547 −0.932821 −0.466410 0.884568i \(-0.654453\pi\)
−0.466410 + 0.884568i \(0.654453\pi\)
\(68\) 0 0
\(69\) −7.97378 −0.959931
\(70\) 0 0
\(71\) 3.29274 0.390776 0.195388 0.980726i \(-0.437403\pi\)
0.195388 + 0.980726i \(0.437403\pi\)
\(72\) 0 0
\(73\) 8.82925 1.03339 0.516693 0.856171i \(-0.327163\pi\)
0.516693 + 0.856171i \(0.327163\pi\)
\(74\) 0 0
\(75\) 5.39308 0.622739
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.39094 −0.719037 −0.359519 0.933138i \(-0.617059\pi\)
−0.359519 + 0.933138i \(0.617059\pi\)
\(80\) 0 0
\(81\) 11.4853 1.27614
\(82\) 0 0
\(83\) 7.35997 0.807861 0.403931 0.914790i \(-0.367644\pi\)
0.403931 + 0.914790i \(0.367644\pi\)
\(84\) 0 0
\(85\) 18.0127 1.95375
\(86\) 0 0
\(87\) 9.50092 1.01861
\(88\) 0 0
\(89\) −13.5002 −1.43102 −0.715509 0.698603i \(-0.753805\pi\)
−0.715509 + 0.698603i \(0.753805\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 21.2357 2.20204
\(94\) 0 0
\(95\) 2.60221 0.266981
\(96\) 0 0
\(97\) −8.90409 −0.904073 −0.452037 0.891999i \(-0.649302\pi\)
−0.452037 + 0.891999i \(0.649302\pi\)
\(98\) 0 0
\(99\) 23.0852 2.32015
\(100\) 0 0
\(101\) 14.8128 1.47392 0.736962 0.675934i \(-0.236259\pi\)
0.736962 + 0.675934i \(0.236259\pi\)
\(102\) 0 0
\(103\) 16.8373 1.65902 0.829512 0.558489i \(-0.188619\pi\)
0.829512 + 0.558489i \(0.188619\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.1650 −1.27271 −0.636353 0.771398i \(-0.719558\pi\)
−0.636353 + 0.771398i \(0.719558\pi\)
\(108\) 0 0
\(109\) 10.0004 0.957866 0.478933 0.877852i \(-0.341024\pi\)
0.478933 + 0.877852i \(0.341024\pi\)
\(110\) 0 0
\(111\) 0.890433 0.0845162
\(112\) 0 0
\(113\) −17.6013 −1.65579 −0.827895 0.560883i \(-0.810462\pi\)
−0.827895 + 0.560883i \(0.810462\pi\)
\(114\) 0 0
\(115\) 6.81569 0.635567
\(116\) 0 0
\(117\) 38.6876 3.57667
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.56398 0.233089
\(122\) 0 0
\(123\) 16.1128 1.45284
\(124\) 0 0
\(125\) 8.40125 0.751430
\(126\) 0 0
\(127\) 13.8533 1.22928 0.614640 0.788807i \(-0.289301\pi\)
0.614640 + 0.788807i \(0.289301\pi\)
\(128\) 0 0
\(129\) −5.97805 −0.526338
\(130\) 0 0
\(131\) 5.67842 0.496126 0.248063 0.968744i \(-0.420206\pi\)
0.248063 + 0.968744i \(0.420206\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −25.8906 −2.22830
\(136\) 0 0
\(137\) 20.8928 1.78499 0.892495 0.451057i \(-0.148953\pi\)
0.892495 + 0.451057i \(0.148953\pi\)
\(138\) 0 0
\(139\) −18.3740 −1.55847 −0.779233 0.626735i \(-0.784391\pi\)
−0.779233 + 0.626735i \(0.784391\pi\)
\(140\) 0 0
\(141\) 24.3406 2.04984
\(142\) 0 0
\(143\) 22.7313 1.90089
\(144\) 0 0
\(145\) −8.12104 −0.674415
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.86236 −0.316417 −0.158208 0.987406i \(-0.550572\pi\)
−0.158208 + 0.987406i \(0.550572\pi\)
\(150\) 0 0
\(151\) 21.4781 1.74786 0.873932 0.486047i \(-0.161562\pi\)
0.873932 + 0.486047i \(0.161562\pi\)
\(152\) 0 0
\(153\) −43.3886 −3.50776
\(154\) 0 0
\(155\) −18.1515 −1.45796
\(156\) 0 0
\(157\) −1.22729 −0.0979485 −0.0489742 0.998800i \(-0.515595\pi\)
−0.0489742 + 0.998800i \(0.515595\pi\)
\(158\) 0 0
\(159\) 5.45467 0.432584
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.6255 −1.06723 −0.533615 0.845727i \(-0.679167\pi\)
−0.533615 + 0.845727i \(0.679167\pi\)
\(164\) 0 0
\(165\) −29.1764 −2.27138
\(166\) 0 0
\(167\) −23.8703 −1.84714 −0.923570 0.383430i \(-0.874743\pi\)
−0.923570 + 0.383430i \(0.874743\pi\)
\(168\) 0 0
\(169\) 25.0946 1.93035
\(170\) 0 0
\(171\) −6.26815 −0.479338
\(172\) 0 0
\(173\) 2.40276 0.182678 0.0913392 0.995820i \(-0.470885\pi\)
0.0913392 + 0.995820i \(0.470885\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 21.6864 1.63005
\(178\) 0 0
\(179\) −4.56753 −0.341393 −0.170697 0.985324i \(-0.554602\pi\)
−0.170697 + 0.985324i \(0.554602\pi\)
\(180\) 0 0
\(181\) 18.6789 1.38839 0.694196 0.719786i \(-0.255760\pi\)
0.694196 + 0.719786i \(0.255760\pi\)
\(182\) 0 0
\(183\) −6.53991 −0.483444
\(184\) 0 0
\(185\) −0.761109 −0.0559578
\(186\) 0 0
\(187\) −25.4935 −1.86427
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.89439 −0.137074 −0.0685368 0.997649i \(-0.521833\pi\)
−0.0685368 + 0.997649i \(0.521833\pi\)
\(192\) 0 0
\(193\) 26.7270 1.92385 0.961927 0.273306i \(-0.0881173\pi\)
0.961927 + 0.273306i \(0.0881173\pi\)
\(194\) 0 0
\(195\) −48.8957 −3.50149
\(196\) 0 0
\(197\) −13.5172 −0.963060 −0.481530 0.876430i \(-0.659919\pi\)
−0.481530 + 0.876430i \(0.659919\pi\)
\(198\) 0 0
\(199\) −10.8090 −0.766228 −0.383114 0.923701i \(-0.625148\pi\)
−0.383114 + 0.923701i \(0.625148\pi\)
\(200\) 0 0
\(201\) −23.2451 −1.63959
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −13.7726 −0.961919
\(206\) 0 0
\(207\) −16.4175 −1.14110
\(208\) 0 0
\(209\) −3.68293 −0.254754
\(210\) 0 0
\(211\) −9.11248 −0.627329 −0.313664 0.949534i \(-0.601557\pi\)
−0.313664 + 0.949534i \(0.601557\pi\)
\(212\) 0 0
\(213\) 10.0243 0.686853
\(214\) 0 0
\(215\) 5.10981 0.348486
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 26.8794 1.81634
\(220\) 0 0
\(221\) −42.7236 −2.87390
\(222\) 0 0
\(223\) 26.2526 1.75801 0.879003 0.476817i \(-0.158210\pi\)
0.879003 + 0.476817i \(0.158210\pi\)
\(224\) 0 0
\(225\) 11.1040 0.740267
\(226\) 0 0
\(227\) −22.4173 −1.48789 −0.743945 0.668241i \(-0.767047\pi\)
−0.743945 + 0.668241i \(0.767047\pi\)
\(228\) 0 0
\(229\) 15.0486 0.994441 0.497221 0.867624i \(-0.334354\pi\)
0.497221 + 0.867624i \(0.334354\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.82983 −0.316413 −0.158206 0.987406i \(-0.550571\pi\)
−0.158206 + 0.987406i \(0.550571\pi\)
\(234\) 0 0
\(235\) −20.8054 −1.35720
\(236\) 0 0
\(237\) −19.4564 −1.26383
\(238\) 0 0
\(239\) −7.09173 −0.458726 −0.229363 0.973341i \(-0.573664\pi\)
−0.229363 + 0.973341i \(0.573664\pi\)
\(240\) 0 0
\(241\) 18.4405 1.18786 0.593930 0.804517i \(-0.297576\pi\)
0.593930 + 0.804517i \(0.297576\pi\)
\(242\) 0 0
\(243\) 5.11707 0.328260
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.17208 −0.392720
\(248\) 0 0
\(249\) 22.4064 1.41995
\(250\) 0 0
\(251\) 14.6808 0.926641 0.463321 0.886191i \(-0.346658\pi\)
0.463321 + 0.886191i \(0.346658\pi\)
\(252\) 0 0
\(253\) −9.64631 −0.606459
\(254\) 0 0
\(255\) 54.8372 3.43404
\(256\) 0 0
\(257\) 10.0101 0.624415 0.312208 0.950014i \(-0.398932\pi\)
0.312208 + 0.950014i \(0.398932\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 19.5618 1.21085
\(262\) 0 0
\(263\) −11.2916 −0.696268 −0.348134 0.937445i \(-0.613185\pi\)
−0.348134 + 0.937445i \(0.613185\pi\)
\(264\) 0 0
\(265\) −4.66245 −0.286412
\(266\) 0 0
\(267\) −41.0995 −2.51525
\(268\) 0 0
\(269\) −25.9497 −1.58218 −0.791089 0.611701i \(-0.790486\pi\)
−0.791089 + 0.611701i \(0.790486\pi\)
\(270\) 0 0
\(271\) −20.2412 −1.22956 −0.614781 0.788698i \(-0.710756\pi\)
−0.614781 + 0.788698i \(0.710756\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.52429 0.393430
\(276\) 0 0
\(277\) 13.0873 0.786340 0.393170 0.919466i \(-0.371378\pi\)
0.393170 + 0.919466i \(0.371378\pi\)
\(278\) 0 0
\(279\) 43.7230 2.61763
\(280\) 0 0
\(281\) 21.1772 1.26333 0.631664 0.775242i \(-0.282372\pi\)
0.631664 + 0.775242i \(0.282372\pi\)
\(282\) 0 0
\(283\) 7.32367 0.435347 0.217673 0.976022i \(-0.430153\pi\)
0.217673 + 0.976022i \(0.430153\pi\)
\(284\) 0 0
\(285\) 7.92207 0.469263
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 30.9151 1.81854
\(290\) 0 0
\(291\) −27.1073 −1.58906
\(292\) 0 0
\(293\) 17.5488 1.02521 0.512606 0.858624i \(-0.328680\pi\)
0.512606 + 0.858624i \(0.328680\pi\)
\(294\) 0 0
\(295\) −18.5367 −1.07925
\(296\) 0 0
\(297\) 36.6431 2.12625
\(298\) 0 0
\(299\) −16.1659 −0.934897
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 45.0954 2.59067
\(304\) 0 0
\(305\) 5.59007 0.320087
\(306\) 0 0
\(307\) 32.5131 1.85562 0.927810 0.373053i \(-0.121689\pi\)
0.927810 + 0.373053i \(0.121689\pi\)
\(308\) 0 0
\(309\) 51.2587 2.91601
\(310\) 0 0
\(311\) −8.72637 −0.494827 −0.247414 0.968910i \(-0.579581\pi\)
−0.247414 + 0.968910i \(0.579581\pi\)
\(312\) 0 0
\(313\) 10.6345 0.601096 0.300548 0.953767i \(-0.402830\pi\)
0.300548 + 0.953767i \(0.402830\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.2316 −0.799326 −0.399663 0.916662i \(-0.630873\pi\)
−0.399663 + 0.916662i \(0.630873\pi\)
\(318\) 0 0
\(319\) 11.4938 0.643528
\(320\) 0 0
\(321\) −40.0790 −2.23699
\(322\) 0 0
\(323\) 6.92207 0.385155
\(324\) 0 0
\(325\) 10.9338 0.606498
\(326\) 0 0
\(327\) 30.4449 1.68361
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 29.8929 1.64306 0.821532 0.570163i \(-0.193120\pi\)
0.821532 + 0.570163i \(0.193120\pi\)
\(332\) 0 0
\(333\) 1.83335 0.100467
\(334\) 0 0
\(335\) 19.8691 1.08556
\(336\) 0 0
\(337\) 17.6020 0.958841 0.479421 0.877585i \(-0.340847\pi\)
0.479421 + 0.877585i \(0.340847\pi\)
\(338\) 0 0
\(339\) −53.5847 −2.91032
\(340\) 0 0
\(341\) 25.6900 1.39119
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 20.7495 1.11711
\(346\) 0 0
\(347\) −1.55959 −0.0837234 −0.0418617 0.999123i \(-0.513329\pi\)
−0.0418617 + 0.999123i \(0.513329\pi\)
\(348\) 0 0
\(349\) −27.9765 −1.49755 −0.748773 0.662826i \(-0.769357\pi\)
−0.748773 + 0.662826i \(0.769357\pi\)
\(350\) 0 0
\(351\) 61.4088 3.27776
\(352\) 0 0
\(353\) −4.89276 −0.260415 −0.130208 0.991487i \(-0.541564\pi\)
−0.130208 + 0.991487i \(0.541564\pi\)
\(354\) 0 0
\(355\) −8.56840 −0.454763
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −37.7566 −1.99272 −0.996360 0.0852473i \(-0.972832\pi\)
−0.996360 + 0.0852473i \(0.972832\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 7.80568 0.409692
\(364\) 0 0
\(365\) −22.9756 −1.20260
\(366\) 0 0
\(367\) 14.1666 0.739493 0.369746 0.929133i \(-0.379445\pi\)
0.369746 + 0.929133i \(0.379445\pi\)
\(368\) 0 0
\(369\) 33.1751 1.72703
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.80652 −0.0935380 −0.0467690 0.998906i \(-0.514892\pi\)
−0.0467690 + 0.998906i \(0.514892\pi\)
\(374\) 0 0
\(375\) 25.5765 1.32076
\(376\) 0 0
\(377\) 19.2620 0.992042
\(378\) 0 0
\(379\) −35.3424 −1.81542 −0.907710 0.419599i \(-0.862171\pi\)
−0.907710 + 0.419599i \(0.862171\pi\)
\(380\) 0 0
\(381\) 42.1745 2.16066
\(382\) 0 0
\(383\) 19.8292 1.01323 0.506614 0.862173i \(-0.330897\pi\)
0.506614 + 0.862173i \(0.330897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.3084 −0.625672
\(388\) 0 0
\(389\) −22.2057 −1.12587 −0.562936 0.826501i \(-0.690328\pi\)
−0.562936 + 0.826501i \(0.690328\pi\)
\(390\) 0 0
\(391\) 18.1303 0.916887
\(392\) 0 0
\(393\) 17.2872 0.872023
\(394\) 0 0
\(395\) 16.6306 0.836775
\(396\) 0 0
\(397\) −0.875987 −0.0439645 −0.0219823 0.999758i \(-0.506998\pi\)
−0.0219823 + 0.999758i \(0.506998\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38.6864 −1.93191 −0.965953 0.258717i \(-0.916700\pi\)
−0.965953 + 0.258717i \(0.916700\pi\)
\(402\) 0 0
\(403\) 43.0528 2.14461
\(404\) 0 0
\(405\) −29.8871 −1.48510
\(406\) 0 0
\(407\) 1.07720 0.0533950
\(408\) 0 0
\(409\) −11.6808 −0.577580 −0.288790 0.957392i \(-0.593253\pi\)
−0.288790 + 0.957392i \(0.593253\pi\)
\(410\) 0 0
\(411\) 63.6052 3.13741
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −19.1522 −0.940143
\(416\) 0 0
\(417\) −55.9373 −2.73926
\(418\) 0 0
\(419\) −11.2773 −0.550932 −0.275466 0.961311i \(-0.588832\pi\)
−0.275466 + 0.961311i \(0.588832\pi\)
\(420\) 0 0
\(421\) −5.72033 −0.278792 −0.139396 0.990237i \(-0.544516\pi\)
−0.139396 + 0.990237i \(0.544516\pi\)
\(422\) 0 0
\(423\) 50.1157 2.43671
\(424\) 0 0
\(425\) −12.2624 −0.594815
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 69.2025 3.34113
\(430\) 0 0
\(431\) 15.3845 0.741045 0.370522 0.928824i \(-0.379179\pi\)
0.370522 + 0.928824i \(0.379179\pi\)
\(432\) 0 0
\(433\) 20.1376 0.967749 0.483875 0.875137i \(-0.339229\pi\)
0.483875 + 0.875137i \(0.339229\pi\)
\(434\) 0 0
\(435\) −24.7234 −1.18540
\(436\) 0 0
\(437\) 2.61920 0.125293
\(438\) 0 0
\(439\) 10.7327 0.512245 0.256122 0.966644i \(-0.417555\pi\)
0.256122 + 0.966644i \(0.417555\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.1006 1.66768 0.833840 0.552006i \(-0.186137\pi\)
0.833840 + 0.552006i \(0.186137\pi\)
\(444\) 0 0
\(445\) 35.1304 1.66534
\(446\) 0 0
\(447\) −11.7584 −0.556154
\(448\) 0 0
\(449\) −35.3230 −1.66699 −0.833497 0.552524i \(-0.813665\pi\)
−0.833497 + 0.552524i \(0.813665\pi\)
\(450\) 0 0
\(451\) 19.4925 0.917864
\(452\) 0 0
\(453\) 65.3872 3.07216
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.6444 1.43348 0.716741 0.697339i \(-0.245633\pi\)
0.716741 + 0.697339i \(0.245633\pi\)
\(458\) 0 0
\(459\) −68.8709 −3.21462
\(460\) 0 0
\(461\) 2.47914 0.115465 0.0577325 0.998332i \(-0.481613\pi\)
0.0577325 + 0.998332i \(0.481613\pi\)
\(462\) 0 0
\(463\) −6.46751 −0.300571 −0.150285 0.988643i \(-0.548019\pi\)
−0.150285 + 0.988643i \(0.548019\pi\)
\(464\) 0 0
\(465\) −55.2598 −2.56261
\(466\) 0 0
\(467\) 6.61582 0.306144 0.153072 0.988215i \(-0.451083\pi\)
0.153072 + 0.988215i \(0.451083\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.73632 −0.172161
\(472\) 0 0
\(473\) −7.23196 −0.332526
\(474\) 0 0
\(475\) −1.77149 −0.0812817
\(476\) 0 0
\(477\) 11.2308 0.514224
\(478\) 0 0
\(479\) −10.5627 −0.482621 −0.241310 0.970448i \(-0.577577\pi\)
−0.241310 + 0.970448i \(0.577577\pi\)
\(480\) 0 0
\(481\) 1.80524 0.0823121
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 23.1703 1.05211
\(486\) 0 0
\(487\) −25.1801 −1.14102 −0.570510 0.821291i \(-0.693254\pi\)
−0.570510 + 0.821291i \(0.693254\pi\)
\(488\) 0 0
\(489\) −41.4810 −1.87583
\(490\) 0 0
\(491\) −10.3251 −0.465966 −0.232983 0.972481i \(-0.574849\pi\)
−0.232983 + 0.972481i \(0.574849\pi\)
\(492\) 0 0
\(493\) −21.6026 −0.972931
\(494\) 0 0
\(495\) −60.0725 −2.70006
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −23.0441 −1.03160 −0.515798 0.856710i \(-0.672504\pi\)
−0.515798 + 0.856710i \(0.672504\pi\)
\(500\) 0 0
\(501\) −72.6699 −3.24665
\(502\) 0 0
\(503\) 5.09453 0.227154 0.113577 0.993529i \(-0.463769\pi\)
0.113577 + 0.993529i \(0.463769\pi\)
\(504\) 0 0
\(505\) −38.5459 −1.71527
\(506\) 0 0
\(507\) 76.3970 3.39291
\(508\) 0 0
\(509\) 39.2110 1.73800 0.868999 0.494815i \(-0.164764\pi\)
0.868999 + 0.494815i \(0.164764\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −9.94945 −0.439279
\(514\) 0 0
\(515\) −43.8141 −1.93068
\(516\) 0 0
\(517\) 29.4461 1.29504
\(518\) 0 0
\(519\) 7.31487 0.321087
\(520\) 0 0
\(521\) −8.68475 −0.380486 −0.190243 0.981737i \(-0.560928\pi\)
−0.190243 + 0.981737i \(0.560928\pi\)
\(522\) 0 0
\(523\) −33.5727 −1.46803 −0.734017 0.679131i \(-0.762357\pi\)
−0.734017 + 0.679131i \(0.762357\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −48.2843 −2.10330
\(528\) 0 0
\(529\) −16.1398 −0.701731
\(530\) 0 0
\(531\) 44.6509 1.93768
\(532\) 0 0
\(533\) 32.6667 1.41495
\(534\) 0 0
\(535\) 34.2580 1.48110
\(536\) 0 0
\(537\) −13.9052 −0.600055
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.5514 −1.74344 −0.871720 0.490004i \(-0.836995\pi\)
−0.871720 + 0.490004i \(0.836995\pi\)
\(542\) 0 0
\(543\) 56.8654 2.44033
\(544\) 0 0
\(545\) −26.0232 −1.11471
\(546\) 0 0
\(547\) −12.7094 −0.543414 −0.271707 0.962380i \(-0.587588\pi\)
−0.271707 + 0.962380i \(0.587588\pi\)
\(548\) 0 0
\(549\) −13.4653 −0.574683
\(550\) 0 0
\(551\) −3.12082 −0.132952
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.31709 −0.0983551
\(556\) 0 0
\(557\) −16.8513 −0.714011 −0.357005 0.934102i \(-0.616202\pi\)
−0.357005 + 0.934102i \(0.616202\pi\)
\(558\) 0 0
\(559\) −12.1198 −0.512611
\(560\) 0 0
\(561\) −77.6116 −3.27676
\(562\) 0 0
\(563\) 5.30235 0.223467 0.111734 0.993738i \(-0.464360\pi\)
0.111734 + 0.993738i \(0.464360\pi\)
\(564\) 0 0
\(565\) 45.8022 1.92691
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.4621 0.522438 0.261219 0.965280i \(-0.415876\pi\)
0.261219 + 0.965280i \(0.415876\pi\)
\(570\) 0 0
\(571\) 15.5137 0.649230 0.324615 0.945846i \(-0.394765\pi\)
0.324615 + 0.945846i \(0.394765\pi\)
\(572\) 0 0
\(573\) −5.76723 −0.240929
\(574\) 0 0
\(575\) −4.63989 −0.193497
\(576\) 0 0
\(577\) 11.8485 0.493260 0.246630 0.969110i \(-0.420677\pi\)
0.246630 + 0.969110i \(0.420677\pi\)
\(578\) 0 0
\(579\) 81.3668 3.38149
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.59881 0.273295
\(584\) 0 0
\(585\) −100.673 −4.16232
\(586\) 0 0
\(587\) 21.7714 0.898603 0.449302 0.893380i \(-0.351673\pi\)
0.449302 + 0.893380i \(0.351673\pi\)
\(588\) 0 0
\(589\) −6.97542 −0.287417
\(590\) 0 0
\(591\) −41.1513 −1.69274
\(592\) 0 0
\(593\) 28.0822 1.15320 0.576599 0.817028i \(-0.304380\pi\)
0.576599 + 0.817028i \(0.304380\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −32.9065 −1.34677
\(598\) 0 0
\(599\) 29.0002 1.18492 0.592459 0.805601i \(-0.298157\pi\)
0.592459 + 0.805601i \(0.298157\pi\)
\(600\) 0 0
\(601\) −5.11828 −0.208779 −0.104390 0.994536i \(-0.533289\pi\)
−0.104390 + 0.994536i \(0.533289\pi\)
\(602\) 0 0
\(603\) −47.8603 −1.94902
\(604\) 0 0
\(605\) −6.67200 −0.271256
\(606\) 0 0
\(607\) −1.89786 −0.0770317 −0.0385159 0.999258i \(-0.512263\pi\)
−0.0385159 + 0.999258i \(0.512263\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 49.3475 1.99639
\(612\) 0 0
\(613\) −30.0880 −1.21524 −0.607621 0.794227i \(-0.707876\pi\)
−0.607621 + 0.794227i \(0.707876\pi\)
\(614\) 0 0
\(615\) −41.9288 −1.69073
\(616\) 0 0
\(617\) −14.3309 −0.576941 −0.288470 0.957489i \(-0.593147\pi\)
−0.288470 + 0.957489i \(0.593147\pi\)
\(618\) 0 0
\(619\) 16.3218 0.656027 0.328013 0.944673i \(-0.393621\pi\)
0.328013 + 0.944673i \(0.393621\pi\)
\(620\) 0 0
\(621\) −26.0596 −1.04573
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −30.7193 −1.22877
\(626\) 0 0
\(627\) −11.2122 −0.447771
\(628\) 0 0
\(629\) −2.02461 −0.0807264
\(630\) 0 0
\(631\) 12.0842 0.481065 0.240533 0.970641i \(-0.422678\pi\)
0.240533 + 0.970641i \(0.422678\pi\)
\(632\) 0 0
\(633\) −27.7417 −1.10263
\(634\) 0 0
\(635\) −36.0492 −1.43057
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 20.6394 0.816482
\(640\) 0 0
\(641\) −8.04908 −0.317919 −0.158960 0.987285i \(-0.550814\pi\)
−0.158960 + 0.987285i \(0.550814\pi\)
\(642\) 0 0
\(643\) 4.82685 0.190352 0.0951762 0.995460i \(-0.469659\pi\)
0.0951762 + 0.995460i \(0.469659\pi\)
\(644\) 0 0
\(645\) 15.5561 0.612522
\(646\) 0 0
\(647\) −26.1071 −1.02637 −0.513187 0.858277i \(-0.671535\pi\)
−0.513187 + 0.858277i \(0.671535\pi\)
\(648\) 0 0
\(649\) 26.2352 1.02982
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −39.5954 −1.54949 −0.774743 0.632276i \(-0.782121\pi\)
−0.774743 + 0.632276i \(0.782121\pi\)
\(654\) 0 0
\(655\) −14.7764 −0.577364
\(656\) 0 0
\(657\) 55.3431 2.15914
\(658\) 0 0
\(659\) 34.5877 1.34735 0.673674 0.739029i \(-0.264715\pi\)
0.673674 + 0.739029i \(0.264715\pi\)
\(660\) 0 0
\(661\) −16.4839 −0.641149 −0.320574 0.947223i \(-0.603876\pi\)
−0.320574 + 0.947223i \(0.603876\pi\)
\(662\) 0 0
\(663\) −130.066 −5.05135
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.17404 −0.316500
\(668\) 0 0
\(669\) 79.9225 3.08998
\(670\) 0 0
\(671\) −7.91168 −0.305427
\(672\) 0 0
\(673\) −23.4114 −0.902442 −0.451221 0.892412i \(-0.649011\pi\)
−0.451221 + 0.892412i \(0.649011\pi\)
\(674\) 0 0
\(675\) 17.6254 0.678402
\(676\) 0 0
\(677\) 35.5410 1.36595 0.682977 0.730440i \(-0.260685\pi\)
0.682977 + 0.730440i \(0.260685\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −68.2465 −2.61521
\(682\) 0 0
\(683\) −16.7886 −0.642397 −0.321198 0.947012i \(-0.604086\pi\)
−0.321198 + 0.947012i \(0.604086\pi\)
\(684\) 0 0
\(685\) −54.3674 −2.07727
\(686\) 0 0
\(687\) 45.8135 1.74789
\(688\) 0 0
\(689\) 11.0587 0.421302
\(690\) 0 0
\(691\) −40.5064 −1.54094 −0.770468 0.637478i \(-0.779978\pi\)
−0.770468 + 0.637478i \(0.779978\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 47.8131 1.81365
\(696\) 0 0
\(697\) −36.6361 −1.38769
\(698\) 0 0
\(699\) −14.7038 −0.556148
\(700\) 0 0
\(701\) −1.84233 −0.0695838 −0.0347919 0.999395i \(-0.511077\pi\)
−0.0347919 + 0.999395i \(0.511077\pi\)
\(702\) 0 0
\(703\) −0.292486 −0.0110313
\(704\) 0 0
\(705\) −63.3392 −2.38549
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.60227 0.135286 0.0676430 0.997710i \(-0.478452\pi\)
0.0676430 + 0.997710i \(0.478452\pi\)
\(710\) 0 0
\(711\) −40.0594 −1.50235
\(712\) 0 0
\(713\) −18.2700 −0.684216
\(714\) 0 0
\(715\) −59.1517 −2.21215
\(716\) 0 0
\(717\) −21.5898 −0.806286
\(718\) 0 0
\(719\) 36.7155 1.36926 0.684629 0.728892i \(-0.259964\pi\)
0.684629 + 0.728892i \(0.259964\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 56.1397 2.08786
\(724\) 0 0
\(725\) 5.52852 0.205324
\(726\) 0 0
\(727\) −26.0217 −0.965090 −0.482545 0.875871i \(-0.660287\pi\)
−0.482545 + 0.875871i \(0.660287\pi\)
\(728\) 0 0
\(729\) −18.8777 −0.699173
\(730\) 0 0
\(731\) 13.5925 0.502736
\(732\) 0 0
\(733\) −16.8036 −0.620656 −0.310328 0.950630i \(-0.600439\pi\)
−0.310328 + 0.950630i \(0.600439\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −28.1209 −1.03585
\(738\) 0 0
\(739\) 0.614452 0.0226030 0.0113015 0.999936i \(-0.496403\pi\)
0.0113015 + 0.999936i \(0.496403\pi\)
\(740\) 0 0
\(741\) −18.7901 −0.690270
\(742\) 0 0
\(743\) 4.75699 0.174517 0.0872585 0.996186i \(-0.472189\pi\)
0.0872585 + 0.996186i \(0.472189\pi\)
\(744\) 0 0
\(745\) 10.0507 0.368228
\(746\) 0 0
\(747\) 46.1334 1.68793
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.0603 0.914463 0.457232 0.889348i \(-0.348841\pi\)
0.457232 + 0.889348i \(0.348841\pi\)
\(752\) 0 0
\(753\) 44.6936 1.62872
\(754\) 0 0
\(755\) −55.8906 −2.03407
\(756\) 0 0
\(757\) −13.3559 −0.485429 −0.242715 0.970098i \(-0.578038\pi\)
−0.242715 + 0.970098i \(0.578038\pi\)
\(758\) 0 0
\(759\) −29.3669 −1.06595
\(760\) 0 0
\(761\) −45.7947 −1.66005 −0.830027 0.557723i \(-0.811675\pi\)
−0.830027 + 0.557723i \(0.811675\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 112.906 4.08214
\(766\) 0 0
\(767\) 43.9665 1.58754
\(768\) 0 0
\(769\) −25.8371 −0.931708 −0.465854 0.884862i \(-0.654253\pi\)
−0.465854 + 0.884862i \(0.654253\pi\)
\(770\) 0 0
\(771\) 30.4745 1.09751
\(772\) 0 0
\(773\) −15.7479 −0.566412 −0.283206 0.959059i \(-0.591398\pi\)
−0.283206 + 0.959059i \(0.591398\pi\)
\(774\) 0 0
\(775\) 12.3569 0.443873
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.29265 −0.189629
\(780\) 0 0
\(781\) 12.1269 0.433936
\(782\) 0 0
\(783\) 31.0505 1.10965
\(784\) 0 0
\(785\) 3.19367 0.113987
\(786\) 0 0
\(787\) −1.22949 −0.0438267 −0.0219133 0.999760i \(-0.506976\pi\)
−0.0219133 + 0.999760i \(0.506976\pi\)
\(788\) 0 0
\(789\) −34.3756 −1.22380
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −13.2589 −0.470836
\(794\) 0 0
\(795\) −14.1942 −0.503416
\(796\) 0 0
\(797\) −47.3799 −1.67828 −0.839141 0.543913i \(-0.816942\pi\)
−0.839141 + 0.543913i \(0.816942\pi\)
\(798\) 0 0
\(799\) −55.3440 −1.95793
\(800\) 0 0
\(801\) −84.6214 −2.98995
\(802\) 0 0
\(803\) 32.5175 1.14752
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −79.0002 −2.78094
\(808\) 0 0
\(809\) −19.8717 −0.698653 −0.349326 0.937001i \(-0.613590\pi\)
−0.349326 + 0.937001i \(0.613590\pi\)
\(810\) 0 0
\(811\) −23.5779 −0.827931 −0.413966 0.910293i \(-0.635857\pi\)
−0.413966 + 0.910293i \(0.635857\pi\)
\(812\) 0 0
\(813\) −61.6214 −2.16116
\(814\) 0 0
\(815\) 35.4564 1.24198
\(816\) 0 0
\(817\) 1.96364 0.0686992
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −28.4995 −0.994638 −0.497319 0.867568i \(-0.665682\pi\)
−0.497319 + 0.867568i \(0.665682\pi\)
\(822\) 0 0
\(823\) −21.0443 −0.733558 −0.366779 0.930308i \(-0.619540\pi\)
−0.366779 + 0.930308i \(0.619540\pi\)
\(824\) 0 0
\(825\) 19.8623 0.691517
\(826\) 0 0
\(827\) −0.283949 −0.00987387 −0.00493693 0.999988i \(-0.501571\pi\)
−0.00493693 + 0.999988i \(0.501571\pi\)
\(828\) 0 0
\(829\) 47.7341 1.65787 0.828937 0.559342i \(-0.188946\pi\)
0.828937 + 0.559342i \(0.188946\pi\)
\(830\) 0 0
\(831\) 39.8425 1.38212
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 62.1156 2.14960
\(836\) 0 0
\(837\) 69.4016 2.39887
\(838\) 0 0
\(839\) −24.9296 −0.860664 −0.430332 0.902671i \(-0.641603\pi\)
−0.430332 + 0.902671i \(0.641603\pi\)
\(840\) 0 0
\(841\) −19.2605 −0.664154
\(842\) 0 0
\(843\) 64.4712 2.22051
\(844\) 0 0
\(845\) −65.3013 −2.24643
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 22.2959 0.765194
\(850\) 0 0
\(851\) −0.766077 −0.0262608
\(852\) 0 0
\(853\) 30.1668 1.03289 0.516447 0.856319i \(-0.327254\pi\)
0.516447 + 0.856319i \(0.327254\pi\)
\(854\) 0 0
\(855\) 16.3111 0.557826
\(856\) 0 0
\(857\) −30.7276 −1.04963 −0.524817 0.851215i \(-0.675866\pi\)
−0.524817 + 0.851215i \(0.675866\pi\)
\(858\) 0 0
\(859\) −29.1043 −0.993025 −0.496513 0.868029i \(-0.665386\pi\)
−0.496513 + 0.868029i \(0.665386\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.8047 0.878401 0.439201 0.898389i \(-0.355262\pi\)
0.439201 + 0.898389i \(0.355262\pi\)
\(864\) 0 0
\(865\) −6.25248 −0.212591
\(866\) 0 0
\(867\) 94.1169 3.19638
\(868\) 0 0
\(869\) −23.5374 −0.798451
\(870\) 0 0
\(871\) −47.1267 −1.59683
\(872\) 0 0
\(873\) −55.8122 −1.88896
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.4888 −1.02953 −0.514766 0.857331i \(-0.672121\pi\)
−0.514766 + 0.857331i \(0.672121\pi\)
\(878\) 0 0
\(879\) 53.4249 1.80198
\(880\) 0 0
\(881\) −12.0111 −0.404663 −0.202331 0.979317i \(-0.564852\pi\)
−0.202331 + 0.979317i \(0.564852\pi\)
\(882\) 0 0
\(883\) −29.4398 −0.990729 −0.495365 0.868685i \(-0.664966\pi\)
−0.495365 + 0.868685i \(0.664966\pi\)
\(884\) 0 0
\(885\) −56.4325 −1.89696
\(886\) 0 0
\(887\) 14.9000 0.500295 0.250147 0.968208i \(-0.419521\pi\)
0.250147 + 0.968208i \(0.419521\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 42.2995 1.41709
\(892\) 0 0
\(893\) −7.99528 −0.267552
\(894\) 0 0
\(895\) 11.8857 0.397294
\(896\) 0 0
\(897\) −49.2148 −1.64324
\(898\) 0 0
\(899\) 21.7690 0.726038
\(900\) 0 0
\(901\) −12.4025 −0.413186
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −48.6064 −1.61573
\(906\) 0 0
\(907\) 15.9921 0.531009 0.265505 0.964110i \(-0.414461\pi\)
0.265505 + 0.964110i \(0.414461\pi\)
\(908\) 0 0
\(909\) 92.8487 3.07960
\(910\) 0 0
\(911\) 7.54020 0.249818 0.124909 0.992168i \(-0.460136\pi\)
0.124909 + 0.992168i \(0.460136\pi\)
\(912\) 0 0
\(913\) 27.1062 0.897086
\(914\) 0 0
\(915\) 17.0182 0.562605
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4.87076 0.160672 0.0803358 0.996768i \(-0.474401\pi\)
0.0803358 + 0.996768i \(0.474401\pi\)
\(920\) 0 0
\(921\) 98.9817 3.26156
\(922\) 0 0
\(923\) 20.3230 0.668941
\(924\) 0 0
\(925\) 0.518137 0.0170362
\(926\) 0 0
\(927\) 105.539 3.46634
\(928\) 0 0
\(929\) −49.7616 −1.63263 −0.816313 0.577610i \(-0.803986\pi\)
−0.816313 + 0.577610i \(0.803986\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −26.5663 −0.869740
\(934\) 0 0
\(935\) 66.3395 2.16953
\(936\) 0 0
\(937\) 19.8753 0.649299 0.324649 0.945834i \(-0.394754\pi\)
0.324649 + 0.945834i \(0.394754\pi\)
\(938\) 0 0
\(939\) 32.3752 1.05652
\(940\) 0 0
\(941\) 17.1505 0.559090 0.279545 0.960133i \(-0.409816\pi\)
0.279545 + 0.960133i \(0.409816\pi\)
\(942\) 0 0
\(943\) −13.8625 −0.451425
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.8371 −0.449645 −0.224823 0.974400i \(-0.572180\pi\)
−0.224823 + 0.974400i \(0.572180\pi\)
\(948\) 0 0
\(949\) 54.4948 1.76898
\(950\) 0 0
\(951\) −43.3262 −1.40495
\(952\) 0 0
\(953\) 19.7888 0.641023 0.320511 0.947245i \(-0.396145\pi\)
0.320511 + 0.947245i \(0.396145\pi\)
\(954\) 0 0
\(955\) 4.92961 0.159518
\(956\) 0 0
\(957\) 34.9912 1.13111
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 17.6564 0.569562
\(962\) 0 0
\(963\) −82.5201 −2.65917
\(964\) 0 0
\(965\) −69.5493 −2.23887
\(966\) 0 0
\(967\) −13.5184 −0.434721 −0.217361 0.976091i \(-0.569745\pi\)
−0.217361 + 0.976091i \(0.569745\pi\)
\(968\) 0 0
\(969\) 21.0733 0.676972
\(970\) 0 0
\(971\) −26.7127 −0.857250 −0.428625 0.903482i \(-0.641002\pi\)
−0.428625 + 0.903482i \(0.641002\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 33.2865 1.06602
\(976\) 0 0
\(977\) 46.5330 1.48872 0.744361 0.667778i \(-0.232754\pi\)
0.744361 + 0.667778i \(0.232754\pi\)
\(978\) 0 0
\(979\) −49.7203 −1.58907
\(980\) 0 0
\(981\) 62.6841 2.00135
\(982\) 0 0
\(983\) −28.4260 −0.906648 −0.453324 0.891346i \(-0.649762\pi\)
−0.453324 + 0.891346i \(0.649762\pi\)
\(984\) 0 0
\(985\) 35.1746 1.12076
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.14317 0.163543
\(990\) 0 0
\(991\) 7.12413 0.226305 0.113153 0.993578i \(-0.463905\pi\)
0.113153 + 0.993578i \(0.463905\pi\)
\(992\) 0 0
\(993\) 91.0049 2.88795
\(994\) 0 0
\(995\) 28.1272 0.891693
\(996\) 0 0
\(997\) 27.2902 0.864290 0.432145 0.901804i \(-0.357757\pi\)
0.432145 + 0.901804i \(0.357757\pi\)
\(998\) 0 0
\(999\) 2.91007 0.0920706
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 7448.2.a.bn.1.6 yes 6
7.6 odd 2 7448.2.a.bm.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7448.2.a.bm.1.1 6 7.6 odd 2
7448.2.a.bn.1.6 yes 6 1.1 even 1 trivial