Properties

Label 7728.2.a.ca.1.4
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.256549.1
Defining polynomial: \(x^{4} - 2 x^{3} - 12 x^{2} + 11 x + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.73320\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.20357 q^{11} +0.470368 q^{13} -3.73320 q^{15} -4.20357 q^{17} -4.20357 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.93676 q^{25} -1.00000 q^{27} -0.651155 q^{29} -6.20357 q^{31} -4.20357 q^{33} -3.73320 q^{35} -5.75598 q^{37} -0.470368 q^{39} -3.75598 q^{41} -11.9368 q^{43} +3.73320 q^{45} -6.61168 q^{47} +1.00000 q^{49} +4.20357 q^{51} -3.28561 q^{53} +15.6927 q^{55} +4.20357 q^{57} -8.58792 q^{59} +9.08204 q^{61} -1.00000 q^{63} +1.75598 q^{65} -5.81921 q^{67} +1.00000 q^{69} -9.85075 q^{71} +3.84199 q^{73} -8.93676 q^{75} -4.20357 q^{77} -15.5514 q^{79} +1.00000 q^{81} -10.3677 q^{83} -15.6927 q^{85} +0.651155 q^{87} +12.7915 q^{89} -0.470368 q^{91} +6.20357 q^{93} -15.6927 q^{95} -4.11755 q^{97} +4.20357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} - 2 q^{15} - 2 q^{17} - 2 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} - 4 q^{27} - q^{29} - 10 q^{31} - 2 q^{33} - 2 q^{35} + 5 q^{37} + 13 q^{41} - 20 q^{43} + 2 q^{45} - 17 q^{47} + 4 q^{49} + 2 q^{51} + 13 q^{53} + 7 q^{55} + 2 q^{57} - 5 q^{59} + 25 q^{61} - 4 q^{63} - 21 q^{65} - 23 q^{67} + 4 q^{69} + q^{71} - 8 q^{75} - 2 q^{77} - 14 q^{79} + 4 q^{81} - 4 q^{83} - 7 q^{85} + q^{87} + 7 q^{89} + 10 q^{93} - 7 q^{95} + 11 q^{97} + 2 q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.73320 1.66954 0.834768 0.550601i \(-0.185602\pi\)
0.834768 + 0.550601i \(0.185602\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.20357 1.26742 0.633711 0.773570i \(-0.281531\pi\)
0.633711 + 0.773570i \(0.281531\pi\)
\(12\) 0 0
\(13\) 0.470368 0.130457 0.0652283 0.997870i \(-0.479222\pi\)
0.0652283 + 0.997870i \(0.479222\pi\)
\(14\) 0 0
\(15\) −3.73320 −0.963907
\(16\) 0 0
\(17\) −4.20357 −1.01951 −0.509757 0.860318i \(-0.670265\pi\)
−0.509757 + 0.860318i \(0.670265\pi\)
\(18\) 0 0
\(19\) −4.20357 −0.964364 −0.482182 0.876071i \(-0.660156\pi\)
−0.482182 + 0.876071i \(0.660156\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.93676 1.78735
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.651155 −0.120916 −0.0604582 0.998171i \(-0.519256\pi\)
−0.0604582 + 0.998171i \(0.519256\pi\)
\(30\) 0 0
\(31\) −6.20357 −1.11419 −0.557097 0.830448i \(-0.688085\pi\)
−0.557097 + 0.830448i \(0.688085\pi\)
\(32\) 0 0
\(33\) −4.20357 −0.731747
\(34\) 0 0
\(35\) −3.73320 −0.631026
\(36\) 0 0
\(37\) −5.75598 −0.946277 −0.473138 0.880988i \(-0.656879\pi\)
−0.473138 + 0.880988i \(0.656879\pi\)
\(38\) 0 0
\(39\) −0.470368 −0.0753191
\(40\) 0 0
\(41\) −3.75598 −0.586585 −0.293292 0.956023i \(-0.594751\pi\)
−0.293292 + 0.956023i \(0.594751\pi\)
\(42\) 0 0
\(43\) −11.9368 −1.82034 −0.910170 0.414236i \(-0.864049\pi\)
−0.910170 + 0.414236i \(0.864049\pi\)
\(44\) 0 0
\(45\) 3.73320 0.556512
\(46\) 0 0
\(47\) −6.61168 −0.964412 −0.482206 0.876058i \(-0.660164\pi\)
−0.482206 + 0.876058i \(0.660164\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.20357 0.588617
\(52\) 0 0
\(53\) −3.28561 −0.451313 −0.225656 0.974207i \(-0.572453\pi\)
−0.225656 + 0.974207i \(0.572453\pi\)
\(54\) 0 0
\(55\) 15.6927 2.11601
\(56\) 0 0
\(57\) 4.20357 0.556776
\(58\) 0 0
\(59\) −8.58792 −1.11805 −0.559026 0.829150i \(-0.688825\pi\)
−0.559026 + 0.829150i \(0.688825\pi\)
\(60\) 0 0
\(61\) 9.08204 1.16284 0.581418 0.813605i \(-0.302498\pi\)
0.581418 + 0.813605i \(0.302498\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.75598 0.217802
\(66\) 0 0
\(67\) −5.81921 −0.710930 −0.355465 0.934690i \(-0.615677\pi\)
−0.355465 + 0.934690i \(0.615677\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.85075 −1.16907 −0.584534 0.811369i \(-0.698723\pi\)
−0.584534 + 0.811369i \(0.698723\pi\)
\(72\) 0 0
\(73\) 3.84199 0.449671 0.224836 0.974397i \(-0.427816\pi\)
0.224836 + 0.974397i \(0.427816\pi\)
\(74\) 0 0
\(75\) −8.93676 −1.03193
\(76\) 0 0
\(77\) −4.20357 −0.479041
\(78\) 0 0
\(79\) −15.5514 −1.74967 −0.874836 0.484419i \(-0.839031\pi\)
−0.874836 + 0.484419i \(0.839031\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.3677 −1.13800 −0.568999 0.822338i \(-0.692669\pi\)
−0.568999 + 0.822338i \(0.692669\pi\)
\(84\) 0 0
\(85\) −15.6927 −1.70212
\(86\) 0 0
\(87\) 0.651155 0.0698111
\(88\) 0 0
\(89\) 12.7915 1.35589 0.677947 0.735111i \(-0.262870\pi\)
0.677947 + 0.735111i \(0.262870\pi\)
\(90\) 0 0
\(91\) −0.470368 −0.0493079
\(92\) 0 0
\(93\) 6.20357 0.643280
\(94\) 0 0
\(95\) −15.6927 −1.61004
\(96\) 0 0
\(97\) −4.11755 −0.418074 −0.209037 0.977908i \(-0.567033\pi\)
−0.209037 + 0.977908i \(0.567033\pi\)
\(98\) 0 0
\(99\) 4.20357 0.422474
\(100\) 0 0
\(101\) −13.6927 −1.36248 −0.681239 0.732061i \(-0.738559\pi\)
−0.681239 + 0.732061i \(0.738559\pi\)
\(102\) 0 0
\(103\) 16.3667 1.61266 0.806328 0.591469i \(-0.201452\pi\)
0.806328 + 0.591469i \(0.201452\pi\)
\(104\) 0 0
\(105\) 3.73320 0.364323
\(106\) 0 0
\(107\) −1.27953 −0.123697 −0.0618484 0.998086i \(-0.519700\pi\)
−0.0618484 + 0.998086i \(0.519700\pi\)
\(108\) 0 0
\(109\) 0.266803 0.0255551 0.0127775 0.999918i \(-0.495933\pi\)
0.0127775 + 0.999918i \(0.495933\pi\)
\(110\) 0 0
\(111\) 5.75598 0.546333
\(112\) 0 0
\(113\) −17.1996 −1.61800 −0.809001 0.587808i \(-0.799991\pi\)
−0.809001 + 0.587808i \(0.799991\pi\)
\(114\) 0 0
\(115\) −3.73320 −0.348122
\(116\) 0 0
\(117\) 0.470368 0.0434855
\(118\) 0 0
\(119\) 4.20357 0.385340
\(120\) 0 0
\(121\) 6.66996 0.606360
\(122\) 0 0
\(123\) 3.75598 0.338665
\(124\) 0 0
\(125\) 14.6967 1.31451
\(126\) 0 0
\(127\) 10.3439 0.917872 0.458936 0.888469i \(-0.348231\pi\)
0.458936 + 0.888469i \(0.348231\pi\)
\(128\) 0 0
\(129\) 11.9368 1.05097
\(130\) 0 0
\(131\) 6.20357 0.542008 0.271004 0.962578i \(-0.412644\pi\)
0.271004 + 0.962578i \(0.412644\pi\)
\(132\) 0 0
\(133\) 4.20357 0.364495
\(134\) 0 0
\(135\) −3.73320 −0.321302
\(136\) 0 0
\(137\) 1.23032 0.105113 0.0525565 0.998618i \(-0.483263\pi\)
0.0525565 + 0.998618i \(0.483263\pi\)
\(138\) 0 0
\(139\) 16.5475 1.40354 0.701769 0.712405i \(-0.252394\pi\)
0.701769 + 0.712405i \(0.252394\pi\)
\(140\) 0 0
\(141\) 6.61168 0.556803
\(142\) 0 0
\(143\) 1.97722 0.165344
\(144\) 0 0
\(145\) −2.43089 −0.201874
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 4.04556 0.331425 0.165712 0.986174i \(-0.447008\pi\)
0.165712 + 0.986174i \(0.447008\pi\)
\(150\) 0 0
\(151\) 8.32111 0.677163 0.338581 0.940937i \(-0.390053\pi\)
0.338581 + 0.940937i \(0.390053\pi\)
\(152\) 0 0
\(153\) −4.20357 −0.339838
\(154\) 0 0
\(155\) −23.1591 −1.86019
\(156\) 0 0
\(157\) −14.5712 −1.16291 −0.581455 0.813579i \(-0.697516\pi\)
−0.581455 + 0.813579i \(0.697516\pi\)
\(158\) 0 0
\(159\) 3.28561 0.260566
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −15.3312 −1.20083 −0.600415 0.799689i \(-0.704998\pi\)
−0.600415 + 0.799689i \(0.704998\pi\)
\(164\) 0 0
\(165\) −15.6927 −1.22168
\(166\) 0 0
\(167\) 17.7155 1.37087 0.685434 0.728135i \(-0.259613\pi\)
0.685434 + 0.728135i \(0.259613\pi\)
\(168\) 0 0
\(169\) −12.7788 −0.982981
\(170\) 0 0
\(171\) −4.20357 −0.321455
\(172\) 0 0
\(173\) −2.90126 −0.220578 −0.110289 0.993900i \(-0.535178\pi\)
−0.110289 + 0.993900i \(0.535178\pi\)
\(174\) 0 0
\(175\) −8.93676 −0.675556
\(176\) 0 0
\(177\) 8.58792 0.645507
\(178\) 0 0
\(179\) 10.2589 0.766783 0.383391 0.923586i \(-0.374756\pi\)
0.383391 + 0.923586i \(0.374756\pi\)
\(180\) 0 0
\(181\) −13.0908 −0.973031 −0.486516 0.873672i \(-0.661732\pi\)
−0.486516 + 0.873672i \(0.661732\pi\)
\(182\) 0 0
\(183\) −9.08204 −0.671364
\(184\) 0 0
\(185\) −21.4882 −1.57984
\(186\) 0 0
\(187\) −17.6700 −1.29216
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 8.76870 0.634481 0.317241 0.948345i \(-0.397244\pi\)
0.317241 + 0.948345i \(0.397244\pi\)
\(192\) 0 0
\(193\) 10.3211 0.742930 0.371465 0.928447i \(-0.378856\pi\)
0.371465 + 0.928447i \(0.378856\pi\)
\(194\) 0 0
\(195\) −1.75598 −0.125748
\(196\) 0 0
\(197\) −11.4032 −0.812441 −0.406221 0.913775i \(-0.633154\pi\)
−0.406221 + 0.913775i \(0.633154\pi\)
\(198\) 0 0
\(199\) 14.6660 1.03964 0.519822 0.854275i \(-0.325998\pi\)
0.519822 + 0.854275i \(0.325998\pi\)
\(200\) 0 0
\(201\) 5.81921 0.410456
\(202\) 0 0
\(203\) 0.651155 0.0457021
\(204\) 0 0
\(205\) −14.0218 −0.979325
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −17.6700 −1.22226
\(210\) 0 0
\(211\) 13.7155 0.944215 0.472108 0.881541i \(-0.343493\pi\)
0.472108 + 0.881541i \(0.343493\pi\)
\(212\) 0 0
\(213\) 9.85075 0.674962
\(214\) 0 0
\(215\) −44.5623 −3.03912
\(216\) 0 0
\(217\) 6.20357 0.421125
\(218\) 0 0
\(219\) −3.84199 −0.259618
\(220\) 0 0
\(221\) −1.97722 −0.133002
\(222\) 0 0
\(223\) 26.0464 1.74419 0.872097 0.489333i \(-0.162760\pi\)
0.872097 + 0.489333i \(0.162760\pi\)
\(224\) 0 0
\(225\) 8.93676 0.595784
\(226\) 0 0
\(227\) 16.3044 1.08216 0.541081 0.840971i \(-0.318015\pi\)
0.541081 + 0.840971i \(0.318015\pi\)
\(228\) 0 0
\(229\) 14.9240 0.986208 0.493104 0.869970i \(-0.335862\pi\)
0.493104 + 0.869970i \(0.335862\pi\)
\(230\) 0 0
\(231\) 4.20357 0.276574
\(232\) 0 0
\(233\) 0.172841 0.0113232 0.00566158 0.999984i \(-0.498198\pi\)
0.00566158 + 0.999984i \(0.498198\pi\)
\(234\) 0 0
\(235\) −24.6827 −1.61012
\(236\) 0 0
\(237\) 15.5514 1.01017
\(238\) 0 0
\(239\) −28.0999 −1.81763 −0.908815 0.417200i \(-0.863011\pi\)
−0.908815 + 0.417200i \(0.863011\pi\)
\(240\) 0 0
\(241\) 16.1175 1.03822 0.519111 0.854707i \(-0.326263\pi\)
0.519111 + 0.854707i \(0.326263\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.73320 0.238505
\(246\) 0 0
\(247\) −1.97722 −0.125808
\(248\) 0 0
\(249\) 10.3677 0.657023
\(250\) 0 0
\(251\) 21.4259 1.35239 0.676197 0.736721i \(-0.263627\pi\)
0.676197 + 0.736721i \(0.263627\pi\)
\(252\) 0 0
\(253\) −4.20357 −0.264276
\(254\) 0 0
\(255\) 15.6927 0.982717
\(256\) 0 0
\(257\) −10.3616 −0.646337 −0.323169 0.946341i \(-0.604748\pi\)
−0.323169 + 0.946341i \(0.604748\pi\)
\(258\) 0 0
\(259\) 5.75598 0.357659
\(260\) 0 0
\(261\) −0.651155 −0.0403055
\(262\) 0 0
\(263\) −0.487069 −0.0300340 −0.0150170 0.999887i \(-0.504780\pi\)
−0.0150170 + 0.999887i \(0.504780\pi\)
\(264\) 0 0
\(265\) −12.2658 −0.753484
\(266\) 0 0
\(267\) −12.7915 −0.782826
\(268\) 0 0
\(269\) 10.1869 0.621104 0.310552 0.950556i \(-0.399486\pi\)
0.310552 + 0.950556i \(0.399486\pi\)
\(270\) 0 0
\(271\) −9.26283 −0.562677 −0.281338 0.959609i \(-0.590778\pi\)
−0.281338 + 0.959609i \(0.590778\pi\)
\(272\) 0 0
\(273\) 0.470368 0.0284680
\(274\) 0 0
\(275\) 37.5663 2.26533
\(276\) 0 0
\(277\) 26.4220 1.58754 0.793771 0.608217i \(-0.208115\pi\)
0.793771 + 0.608217i \(0.208115\pi\)
\(278\) 0 0
\(279\) −6.20357 −0.371398
\(280\) 0 0
\(281\) 14.3211 0.854326 0.427163 0.904175i \(-0.359513\pi\)
0.427163 + 0.904175i \(0.359513\pi\)
\(282\) 0 0
\(283\) −6.71439 −0.399129 −0.199565 0.979885i \(-0.563953\pi\)
−0.199565 + 0.979885i \(0.563953\pi\)
\(284\) 0 0
\(285\) 15.6927 0.929558
\(286\) 0 0
\(287\) 3.75598 0.221708
\(288\) 0 0
\(289\) 0.669960 0.0394094
\(290\) 0 0
\(291\) 4.11755 0.241375
\(292\) 0 0
\(293\) −1.22335 −0.0714689 −0.0357344 0.999361i \(-0.511377\pi\)
−0.0357344 + 0.999361i \(0.511377\pi\)
\(294\) 0 0
\(295\) −32.0604 −1.86663
\(296\) 0 0
\(297\) −4.20357 −0.243916
\(298\) 0 0
\(299\) −0.470368 −0.0272021
\(300\) 0 0
\(301\) 11.9368 0.688024
\(302\) 0 0
\(303\) 13.6927 0.786627
\(304\) 0 0
\(305\) 33.9051 1.94140
\(306\) 0 0
\(307\) −26.9774 −1.53968 −0.769840 0.638237i \(-0.779664\pi\)
−0.769840 + 0.638237i \(0.779664\pi\)
\(308\) 0 0
\(309\) −16.3667 −0.931067
\(310\) 0 0
\(311\) 12.9100 0.732060 0.366030 0.930603i \(-0.380717\pi\)
0.366030 + 0.930603i \(0.380717\pi\)
\(312\) 0 0
\(313\) −7.47434 −0.422475 −0.211237 0.977435i \(-0.567749\pi\)
−0.211237 + 0.977435i \(0.567749\pi\)
\(314\) 0 0
\(315\) −3.73320 −0.210342
\(316\) 0 0
\(317\) −25.7788 −1.44788 −0.723940 0.689863i \(-0.757671\pi\)
−0.723940 + 0.689863i \(0.757671\pi\)
\(318\) 0 0
\(319\) −2.73717 −0.153252
\(320\) 0 0
\(321\) 1.27953 0.0714164
\(322\) 0 0
\(323\) 17.6700 0.983183
\(324\) 0 0
\(325\) 4.20357 0.233172
\(326\) 0 0
\(327\) −0.266803 −0.0147542
\(328\) 0 0
\(329\) 6.61168 0.364513
\(330\) 0 0
\(331\) 35.1029 1.92943 0.964714 0.263300i \(-0.0848110\pi\)
0.964714 + 0.263300i \(0.0848110\pi\)
\(332\) 0 0
\(333\) −5.75598 −0.315426
\(334\) 0 0
\(335\) −21.7243 −1.18692
\(336\) 0 0
\(337\) 34.8746 1.89974 0.949872 0.312640i \(-0.101213\pi\)
0.949872 + 0.312640i \(0.101213\pi\)
\(338\) 0 0
\(339\) 17.1996 0.934154
\(340\) 0 0
\(341\) −26.0771 −1.41215
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.73320 0.200989
\(346\) 0 0
\(347\) 21.4575 1.15190 0.575949 0.817486i \(-0.304633\pi\)
0.575949 + 0.817486i \(0.304633\pi\)
\(348\) 0 0
\(349\) −15.0881 −0.807649 −0.403824 0.914837i \(-0.632319\pi\)
−0.403824 + 0.914837i \(0.632319\pi\)
\(350\) 0 0
\(351\) −0.470368 −0.0251064
\(352\) 0 0
\(353\) −0.651155 −0.0346575 −0.0173287 0.999850i \(-0.505516\pi\)
−0.0173287 + 0.999850i \(0.505516\pi\)
\(354\) 0 0
\(355\) −36.7748 −1.95180
\(356\) 0 0
\(357\) −4.20357 −0.222476
\(358\) 0 0
\(359\) −25.5988 −1.35105 −0.675526 0.737336i \(-0.736083\pi\)
−0.675526 + 0.737336i \(0.736083\pi\)
\(360\) 0 0
\(361\) −1.33004 −0.0700021
\(362\) 0 0
\(363\) −6.66996 −0.350082
\(364\) 0 0
\(365\) 14.3429 0.750742
\(366\) 0 0
\(367\) 19.5217 1.01902 0.509512 0.860464i \(-0.329826\pi\)
0.509512 + 0.860464i \(0.329826\pi\)
\(368\) 0 0
\(369\) −3.75598 −0.195528
\(370\) 0 0
\(371\) 3.28561 0.170580
\(372\) 0 0
\(373\) 12.7292 0.659094 0.329547 0.944139i \(-0.393104\pi\)
0.329547 + 0.944139i \(0.393104\pi\)
\(374\) 0 0
\(375\) −14.6967 −0.758935
\(376\) 0 0
\(377\) −0.306282 −0.0157743
\(378\) 0 0
\(379\) −0.250102 −0.0128469 −0.00642343 0.999979i \(-0.502045\pi\)
−0.00642343 + 0.999979i \(0.502045\pi\)
\(380\) 0 0
\(381\) −10.3439 −0.529934
\(382\) 0 0
\(383\) 7.55143 0.385860 0.192930 0.981213i \(-0.438201\pi\)
0.192930 + 0.981213i \(0.438201\pi\)
\(384\) 0 0
\(385\) −15.6927 −0.799776
\(386\) 0 0
\(387\) −11.9368 −0.606780
\(388\) 0 0
\(389\) −36.1937 −1.83509 −0.917546 0.397630i \(-0.869833\pi\)
−0.917546 + 0.397630i \(0.869833\pi\)
\(390\) 0 0
\(391\) 4.20357 0.212583
\(392\) 0 0
\(393\) −6.20357 −0.312928
\(394\) 0 0
\(395\) −58.0566 −2.92114
\(396\) 0 0
\(397\) −37.3855 −1.87632 −0.938162 0.346198i \(-0.887473\pi\)
−0.938162 + 0.346198i \(0.887473\pi\)
\(398\) 0 0
\(399\) −4.20357 −0.210441
\(400\) 0 0
\(401\) 2.19562 0.109644 0.0548220 0.998496i \(-0.482541\pi\)
0.0548220 + 0.998496i \(0.482541\pi\)
\(402\) 0 0
\(403\) −2.91796 −0.145354
\(404\) 0 0
\(405\) 3.73320 0.185504
\(406\) 0 0
\(407\) −24.1956 −1.19933
\(408\) 0 0
\(409\) 7.13635 0.352870 0.176435 0.984312i \(-0.443543\pi\)
0.176435 + 0.984312i \(0.443543\pi\)
\(410\) 0 0
\(411\) −1.23032 −0.0606870
\(412\) 0 0
\(413\) 8.58792 0.422584
\(414\) 0 0
\(415\) −38.7045 −1.89993
\(416\) 0 0
\(417\) −16.5475 −0.810332
\(418\) 0 0
\(419\) −5.81921 −0.284287 −0.142144 0.989846i \(-0.545399\pi\)
−0.142144 + 0.989846i \(0.545399\pi\)
\(420\) 0 0
\(421\) 28.5535 1.39161 0.695807 0.718229i \(-0.255047\pi\)
0.695807 + 0.718229i \(0.255047\pi\)
\(422\) 0 0
\(423\) −6.61168 −0.321471
\(424\) 0 0
\(425\) −37.5663 −1.82223
\(426\) 0 0
\(427\) −9.08204 −0.439511
\(428\) 0 0
\(429\) −1.97722 −0.0954612
\(430\) 0 0
\(431\) −15.6937 −0.755940 −0.377970 0.925818i \(-0.623378\pi\)
−0.377970 + 0.925818i \(0.623378\pi\)
\(432\) 0 0
\(433\) 15.3074 0.735627 0.367814 0.929900i \(-0.380106\pi\)
0.367814 + 0.929900i \(0.380106\pi\)
\(434\) 0 0
\(435\) 2.43089 0.116552
\(436\) 0 0
\(437\) 4.20357 0.201084
\(438\) 0 0
\(439\) −29.8595 −1.42512 −0.712558 0.701613i \(-0.752464\pi\)
−0.712558 + 0.701613i \(0.752464\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −11.4715 −0.545027 −0.272514 0.962152i \(-0.587855\pi\)
−0.272514 + 0.962152i \(0.587855\pi\)
\(444\) 0 0
\(445\) 47.7531 2.26372
\(446\) 0 0
\(447\) −4.04556 −0.191348
\(448\) 0 0
\(449\) −19.9358 −0.940828 −0.470414 0.882446i \(-0.655895\pi\)
−0.470414 + 0.882446i \(0.655895\pi\)
\(450\) 0 0
\(451\) −15.7885 −0.743451
\(452\) 0 0
\(453\) −8.32111 −0.390960
\(454\) 0 0
\(455\) −1.75598 −0.0823214
\(456\) 0 0
\(457\) −34.2263 −1.60104 −0.800520 0.599305i \(-0.795443\pi\)
−0.800520 + 0.599305i \(0.795443\pi\)
\(458\) 0 0
\(459\) 4.20357 0.196206
\(460\) 0 0
\(461\) 7.12152 0.331682 0.165841 0.986152i \(-0.446966\pi\)
0.165841 + 0.986152i \(0.446966\pi\)
\(462\) 0 0
\(463\) −24.2341 −1.12626 −0.563128 0.826370i \(-0.690402\pi\)
−0.563128 + 0.826370i \(0.690402\pi\)
\(464\) 0 0
\(465\) 23.1591 1.07398
\(466\) 0 0
\(467\) 23.8666 1.10441 0.552206 0.833707i \(-0.313786\pi\)
0.552206 + 0.833707i \(0.313786\pi\)
\(468\) 0 0
\(469\) 5.81921 0.268706
\(470\) 0 0
\(471\) 14.5712 0.671406
\(472\) 0 0
\(473\) −50.1770 −2.30714
\(474\) 0 0
\(475\) −37.5663 −1.72366
\(476\) 0 0
\(477\) −3.28561 −0.150438
\(478\) 0 0
\(479\) 32.8203 1.49960 0.749800 0.661665i \(-0.230150\pi\)
0.749800 + 0.661665i \(0.230150\pi\)
\(480\) 0 0
\(481\) −2.70743 −0.123448
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −15.3716 −0.697989
\(486\) 0 0
\(487\) −25.2224 −1.14293 −0.571467 0.820625i \(-0.693626\pi\)
−0.571467 + 0.820625i \(0.693626\pi\)
\(488\) 0 0
\(489\) 15.3312 0.693299
\(490\) 0 0
\(491\) 17.6612 0.797039 0.398520 0.917160i \(-0.369524\pi\)
0.398520 + 0.917160i \(0.369524\pi\)
\(492\) 0 0
\(493\) 2.73717 0.123276
\(494\) 0 0
\(495\) 15.6927 0.705336
\(496\) 0 0
\(497\) 9.85075 0.441866
\(498\) 0 0
\(499\) 8.50896 0.380913 0.190457 0.981696i \(-0.439003\pi\)
0.190457 + 0.981696i \(0.439003\pi\)
\(500\) 0 0
\(501\) −17.7155 −0.791471
\(502\) 0 0
\(503\) 20.9495 0.934092 0.467046 0.884233i \(-0.345318\pi\)
0.467046 + 0.884233i \(0.345318\pi\)
\(504\) 0 0
\(505\) −51.1177 −2.27471
\(506\) 0 0
\(507\) 12.7788 0.567524
\(508\) 0 0
\(509\) 32.8853 1.45761 0.728807 0.684719i \(-0.240075\pi\)
0.728807 + 0.684719i \(0.240075\pi\)
\(510\) 0 0
\(511\) −3.84199 −0.169960
\(512\) 0 0
\(513\) 4.20357 0.185592
\(514\) 0 0
\(515\) 61.1000 2.69239
\(516\) 0 0
\(517\) −27.7926 −1.22232
\(518\) 0 0
\(519\) 2.90126 0.127351
\(520\) 0 0
\(521\) −5.27685 −0.231183 −0.115592 0.993297i \(-0.536876\pi\)
−0.115592 + 0.993297i \(0.536876\pi\)
\(522\) 0 0
\(523\) −28.9328 −1.26514 −0.632571 0.774502i \(-0.718000\pi\)
−0.632571 + 0.774502i \(0.718000\pi\)
\(524\) 0 0
\(525\) 8.93676 0.390032
\(526\) 0 0
\(527\) 26.0771 1.13594
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.58792 −0.372684
\(532\) 0 0
\(533\) −1.76669 −0.0765238
\(534\) 0 0
\(535\) −4.77674 −0.206516
\(536\) 0 0
\(537\) −10.2589 −0.442702
\(538\) 0 0
\(539\) 4.20357 0.181060
\(540\) 0 0
\(541\) 1.46639 0.0630452 0.0315226 0.999503i \(-0.489964\pi\)
0.0315226 + 0.999503i \(0.489964\pi\)
\(542\) 0 0
\(543\) 13.0908 0.561780
\(544\) 0 0
\(545\) 0.996027 0.0426651
\(546\) 0 0
\(547\) 30.8606 1.31951 0.659753 0.751483i \(-0.270661\pi\)
0.659753 + 0.751483i \(0.270661\pi\)
\(548\) 0 0
\(549\) 9.08204 0.387612
\(550\) 0 0
\(551\) 2.73717 0.116607
\(552\) 0 0
\(553\) 15.5514 0.661314
\(554\) 0 0
\(555\) 21.4882 0.912123
\(556\) 0 0
\(557\) 36.1997 1.53383 0.766916 0.641747i \(-0.221790\pi\)
0.766916 + 0.641747i \(0.221790\pi\)
\(558\) 0 0
\(559\) −5.61467 −0.237475
\(560\) 0 0
\(561\) 17.6700 0.746026
\(562\) 0 0
\(563\) −1.41110 −0.0594709 −0.0297355 0.999558i \(-0.509466\pi\)
−0.0297355 + 0.999558i \(0.509466\pi\)
\(564\) 0 0
\(565\) −64.2095 −2.70131
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −20.9732 −0.879244 −0.439622 0.898183i \(-0.644888\pi\)
−0.439622 + 0.898183i \(0.644888\pi\)
\(570\) 0 0
\(571\) −21.6165 −0.904621 −0.452310 0.891861i \(-0.649400\pi\)
−0.452310 + 0.891861i \(0.649400\pi\)
\(572\) 0 0
\(573\) −8.76870 −0.366318
\(574\) 0 0
\(575\) −8.93676 −0.372689
\(576\) 0 0
\(577\) −43.6285 −1.81628 −0.908140 0.418668i \(-0.862497\pi\)
−0.908140 + 0.418668i \(0.862497\pi\)
\(578\) 0 0
\(579\) −10.3211 −0.428931
\(580\) 0 0
\(581\) 10.3677 0.430123
\(582\) 0 0
\(583\) −13.8113 −0.572004
\(584\) 0 0
\(585\) 1.75598 0.0726007
\(586\) 0 0
\(587\) 17.5701 0.725195 0.362598 0.931946i \(-0.381890\pi\)
0.362598 + 0.931946i \(0.381890\pi\)
\(588\) 0 0
\(589\) 26.0771 1.07449
\(590\) 0 0
\(591\) 11.4032 0.469063
\(592\) 0 0
\(593\) 2.99335 0.122922 0.0614611 0.998109i \(-0.480424\pi\)
0.0614611 + 0.998109i \(0.480424\pi\)
\(594\) 0 0
\(595\) 15.6927 0.643340
\(596\) 0 0
\(597\) −14.6660 −0.600239
\(598\) 0 0
\(599\) 18.1314 0.740829 0.370414 0.928867i \(-0.379216\pi\)
0.370414 + 0.928867i \(0.379216\pi\)
\(600\) 0 0
\(601\) −22.6335 −0.923239 −0.461619 0.887078i \(-0.652731\pi\)
−0.461619 + 0.887078i \(0.652731\pi\)
\(602\) 0 0
\(603\) −5.81921 −0.236977
\(604\) 0 0
\(605\) 24.9003 1.01234
\(606\) 0 0
\(607\) −40.3430 −1.63747 −0.818736 0.574170i \(-0.805325\pi\)
−0.818736 + 0.574170i \(0.805325\pi\)
\(608\) 0 0
\(609\) −0.651155 −0.0263861
\(610\) 0 0
\(611\) −3.10992 −0.125814
\(612\) 0 0
\(613\) −16.4992 −0.666397 −0.333199 0.942857i \(-0.608128\pi\)
−0.333199 + 0.942857i \(0.608128\pi\)
\(614\) 0 0
\(615\) 14.0218 0.565413
\(616\) 0 0
\(617\) −24.7125 −0.994889 −0.497444 0.867496i \(-0.665728\pi\)
−0.497444 + 0.867496i \(0.665728\pi\)
\(618\) 0 0
\(619\) −4.38435 −0.176222 −0.0881110 0.996111i \(-0.528083\pi\)
−0.0881110 + 0.996111i \(0.528083\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −12.7915 −0.512480
\(624\) 0 0
\(625\) 10.1819 0.407276
\(626\) 0 0
\(627\) 17.6700 0.705670
\(628\) 0 0
\(629\) 24.1956 0.964743
\(630\) 0 0
\(631\) 5.71552 0.227531 0.113766 0.993508i \(-0.463709\pi\)
0.113766 + 0.993508i \(0.463709\pi\)
\(632\) 0 0
\(633\) −13.7155 −0.545143
\(634\) 0 0
\(635\) 38.6158 1.53242
\(636\) 0 0
\(637\) 0.470368 0.0186367
\(638\) 0 0
\(639\) −9.85075 −0.389690
\(640\) 0 0
\(641\) −7.35760 −0.290608 −0.145304 0.989387i \(-0.546416\pi\)
−0.145304 + 0.989387i \(0.546416\pi\)
\(642\) 0 0
\(643\) −13.1075 −0.516909 −0.258455 0.966023i \(-0.583213\pi\)
−0.258455 + 0.966023i \(0.583213\pi\)
\(644\) 0 0
\(645\) 44.5623 1.75464
\(646\) 0 0
\(647\) −40.3369 −1.58581 −0.792904 0.609346i \(-0.791432\pi\)
−0.792904 + 0.609346i \(0.791432\pi\)
\(648\) 0 0
\(649\) −36.0999 −1.41704
\(650\) 0 0
\(651\) −6.20357 −0.243137
\(652\) 0 0
\(653\) −21.1540 −0.827821 −0.413911 0.910317i \(-0.635837\pi\)
−0.413911 + 0.910317i \(0.635837\pi\)
\(654\) 0 0
\(655\) 23.1591 0.904902
\(656\) 0 0
\(657\) 3.84199 0.149890
\(658\) 0 0
\(659\) 25.7410 1.00273 0.501363 0.865237i \(-0.332832\pi\)
0.501363 + 0.865237i \(0.332832\pi\)
\(660\) 0 0
\(661\) 20.5377 0.798825 0.399412 0.916771i \(-0.369214\pi\)
0.399412 + 0.916771i \(0.369214\pi\)
\(662\) 0 0
\(663\) 1.97722 0.0767889
\(664\) 0 0
\(665\) 15.6927 0.608538
\(666\) 0 0
\(667\) 0.651155 0.0252128
\(668\) 0 0
\(669\) −26.0464 −1.00701
\(670\) 0 0
\(671\) 38.1770 1.47381
\(672\) 0 0
\(673\) 40.6808 1.56813 0.784065 0.620679i \(-0.213143\pi\)
0.784065 + 0.620679i \(0.213143\pi\)
\(674\) 0 0
\(675\) −8.93676 −0.343976
\(676\) 0 0
\(677\) −4.50092 −0.172984 −0.0864922 0.996253i \(-0.527566\pi\)
−0.0864922 + 0.996253i \(0.527566\pi\)
\(678\) 0 0
\(679\) 4.11755 0.158017
\(680\) 0 0
\(681\) −16.3044 −0.624786
\(682\) 0 0
\(683\) −14.0771 −0.538645 −0.269322 0.963050i \(-0.586800\pi\)
−0.269322 + 0.963050i \(0.586800\pi\)
\(684\) 0 0
\(685\) 4.59301 0.175490
\(686\) 0 0
\(687\) −14.9240 −0.569387
\(688\) 0 0
\(689\) −1.54544 −0.0588767
\(690\) 0 0
\(691\) −15.5376 −0.591077 −0.295539 0.955331i \(-0.595499\pi\)
−0.295539 + 0.955331i \(0.595499\pi\)
\(692\) 0 0
\(693\) −4.20357 −0.159680
\(694\) 0 0
\(695\) 61.7749 2.34326
\(696\) 0 0
\(697\) 15.7885 0.598032
\(698\) 0 0
\(699\) −0.172841 −0.00653743
\(700\) 0 0
\(701\) −6.92403 −0.261517 −0.130759 0.991414i \(-0.541741\pi\)
−0.130759 + 0.991414i \(0.541741\pi\)
\(702\) 0 0
\(703\) 24.1956 0.912555
\(704\) 0 0
\(705\) 24.6827 0.929604
\(706\) 0 0
\(707\) 13.6927 0.514968
\(708\) 0 0
\(709\) 11.0721 0.415823 0.207911 0.978148i \(-0.433333\pi\)
0.207911 + 0.978148i \(0.433333\pi\)
\(710\) 0 0
\(711\) −15.5514 −0.583224
\(712\) 0 0
\(713\) 6.20357 0.232325
\(714\) 0 0
\(715\) 7.38136 0.276047
\(716\) 0 0
\(717\) 28.0999 1.04941
\(718\) 0 0
\(719\) −17.2204 −0.642213 −0.321106 0.947043i \(-0.604055\pi\)
−0.321106 + 0.947043i \(0.604055\pi\)
\(720\) 0 0
\(721\) −16.3667 −0.609527
\(722\) 0 0
\(723\) −16.1175 −0.599418
\(724\) 0 0
\(725\) −5.81921 −0.216120
\(726\) 0 0
\(727\) −16.7273 −0.620380 −0.310190 0.950675i \(-0.600393\pi\)
−0.310190 + 0.950675i \(0.600393\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.1770 1.85586
\(732\) 0 0
\(733\) −31.3855 −1.15925 −0.579625 0.814884i \(-0.696801\pi\)
−0.579625 + 0.814884i \(0.696801\pi\)
\(734\) 0 0
\(735\) −3.73320 −0.137701
\(736\) 0 0
\(737\) −24.4614 −0.901049
\(738\) 0 0
\(739\) −25.4645 −0.936728 −0.468364 0.883536i \(-0.655156\pi\)
−0.468364 + 0.883536i \(0.655156\pi\)
\(740\) 0 0
\(741\) 1.97722 0.0726351
\(742\) 0 0
\(743\) 33.8568 1.24209 0.621043 0.783776i \(-0.286709\pi\)
0.621043 + 0.783776i \(0.286709\pi\)
\(744\) 0 0
\(745\) 15.1029 0.553326
\(746\) 0 0
\(747\) −10.3677 −0.379333
\(748\) 0 0
\(749\) 1.27953 0.0467530
\(750\) 0 0
\(751\) 8.37033 0.305438 0.152719 0.988270i \(-0.451197\pi\)
0.152719 + 0.988270i \(0.451197\pi\)
\(752\) 0 0
\(753\) −21.4259 −0.780804
\(754\) 0 0
\(755\) 31.0644 1.13055
\(756\) 0 0
\(757\) −6.12461 −0.222603 −0.111301 0.993787i \(-0.535502\pi\)
−0.111301 + 0.993787i \(0.535502\pi\)
\(758\) 0 0
\(759\) 4.20357 0.152580
\(760\) 0 0
\(761\) 29.4310 1.06687 0.533437 0.845840i \(-0.320900\pi\)
0.533437 + 0.845840i \(0.320900\pi\)
\(762\) 0 0
\(763\) −0.266803 −0.00965890
\(764\) 0 0
\(765\) −15.6927 −0.567372
\(766\) 0 0
\(767\) −4.03948 −0.145857
\(768\) 0 0
\(769\) −48.4498 −1.74715 −0.873573 0.486693i \(-0.838203\pi\)
−0.873573 + 0.486693i \(0.838203\pi\)
\(770\) 0 0
\(771\) 10.3616 0.373163
\(772\) 0 0
\(773\) 12.6432 0.454745 0.227372 0.973808i \(-0.426987\pi\)
0.227372 + 0.973808i \(0.426987\pi\)
\(774\) 0 0
\(775\) −55.4398 −1.99146
\(776\) 0 0
\(777\) −5.75598 −0.206494
\(778\) 0 0
\(779\) 15.7885 0.565681
\(780\) 0 0
\(781\) −41.4083 −1.48170
\(782\) 0 0
\(783\) 0.651155 0.0232704
\(784\) 0 0
\(785\) −54.3972 −1.94152
\(786\) 0 0
\(787\) 21.5207 0.767130 0.383565 0.923514i \(-0.374696\pi\)
0.383565 + 0.923514i \(0.374696\pi\)
\(788\) 0 0
\(789\) 0.487069 0.0173401
\(790\) 0 0
\(791\) 17.1996 0.611547
\(792\) 0 0
\(793\) 4.27190 0.151700
\(794\) 0 0
\(795\) 12.2658 0.435024
\(796\) 0 0
\(797\) 12.0071 0.425312 0.212656 0.977127i \(-0.431789\pi\)
0.212656 + 0.977127i \(0.431789\pi\)
\(798\) 0 0
\(799\) 27.7926 0.983232
\(800\) 0 0
\(801\) 12.7915 0.451965
\(802\) 0 0
\(803\) 16.1501 0.569923
\(804\) 0 0
\(805\) 3.73320 0.131578
\(806\) 0 0
\(807\) −10.1869 −0.358595
\(808\) 0 0
\(809\) −4.88456 −0.171732 −0.0858659 0.996307i \(-0.527366\pi\)
−0.0858659 + 0.996307i \(0.527366\pi\)
\(810\) 0 0
\(811\) 8.65910 0.304062 0.152031 0.988376i \(-0.451419\pi\)
0.152031 + 0.988376i \(0.451419\pi\)
\(812\) 0 0
\(813\) 9.26283 0.324862
\(814\) 0 0
\(815\) −57.2343 −2.00483
\(816\) 0 0
\(817\) 50.1770 1.75547
\(818\) 0 0
\(819\) −0.470368 −0.0164360
\(820\) 0 0
\(821\) −27.5199 −0.960451 −0.480226 0.877145i \(-0.659445\pi\)
−0.480226 + 0.877145i \(0.659445\pi\)
\(822\) 0 0
\(823\) −41.3617 −1.44178 −0.720889 0.693050i \(-0.756266\pi\)
−0.720889 + 0.693050i \(0.756266\pi\)
\(824\) 0 0
\(825\) −37.5663 −1.30789
\(826\) 0 0
\(827\) −29.2697 −1.01781 −0.508904 0.860823i \(-0.669949\pi\)
−0.508904 + 0.860823i \(0.669949\pi\)
\(828\) 0 0
\(829\) 53.8340 1.86973 0.934867 0.354999i \(-0.115519\pi\)
0.934867 + 0.354999i \(0.115519\pi\)
\(830\) 0 0
\(831\) −26.4220 −0.916568
\(832\) 0 0
\(833\) −4.20357 −0.145645
\(834\) 0 0
\(835\) 66.1355 2.28871
\(836\) 0 0
\(837\) 6.20357 0.214427
\(838\) 0 0
\(839\) −24.5920 −0.849011 −0.424506 0.905425i \(-0.639552\pi\)
−0.424506 + 0.905425i \(0.639552\pi\)
\(840\) 0 0
\(841\) −28.5760 −0.985379
\(842\) 0 0
\(843\) −14.3211 −0.493245
\(844\) 0 0
\(845\) −47.7056 −1.64112
\(846\) 0 0
\(847\) −6.66996 −0.229183
\(848\) 0 0
\(849\) 6.71439 0.230437
\(850\) 0 0
\(851\) 5.75598 0.197312
\(852\) 0 0
\(853\) 5.83503 0.199787 0.0998937 0.994998i \(-0.468150\pi\)
0.0998937 + 0.994998i \(0.468150\pi\)
\(854\) 0 0
\(855\) −15.6927 −0.536680
\(856\) 0 0
\(857\) 33.3450 1.13904 0.569522 0.821976i \(-0.307128\pi\)
0.569522 + 0.821976i \(0.307128\pi\)
\(858\) 0 0
\(859\) −1.26381 −0.0431206 −0.0215603 0.999768i \(-0.506863\pi\)
−0.0215603 + 0.999768i \(0.506863\pi\)
\(860\) 0 0
\(861\) −3.75598 −0.128003
\(862\) 0 0
\(863\) 41.0179 1.39627 0.698133 0.715968i \(-0.254014\pi\)
0.698133 + 0.715968i \(0.254014\pi\)
\(864\) 0 0
\(865\) −10.8310 −0.368264
\(866\) 0 0
\(867\) −0.669960 −0.0227530
\(868\) 0 0
\(869\) −65.3715 −2.21757
\(870\) 0 0
\(871\) −2.73717 −0.0927455
\(872\) 0 0
\(873\) −4.11755 −0.139358
\(874\) 0 0
\(875\) −14.6967 −0.496840
\(876\) 0 0
\(877\) 21.6031 0.729484 0.364742 0.931109i \(-0.381157\pi\)
0.364742 + 0.931109i \(0.381157\pi\)
\(878\) 0 0
\(879\) 1.22335 0.0412626
\(880\) 0 0
\(881\) 24.8872 0.838472 0.419236 0.907877i \(-0.362298\pi\)
0.419236 + 0.907877i \(0.362298\pi\)
\(882\) 0 0
\(883\) −41.7223 −1.40407 −0.702034 0.712144i \(-0.747724\pi\)
−0.702034 + 0.712144i \(0.747724\pi\)
\(884\) 0 0
\(885\) 32.0604 1.07770
\(886\) 0 0
\(887\) −18.8370 −0.632486 −0.316243 0.948678i \(-0.602421\pi\)
−0.316243 + 0.948678i \(0.602421\pi\)
\(888\) 0 0
\(889\) −10.3439 −0.346923
\(890\) 0 0
\(891\) 4.20357 0.140825
\(892\) 0 0
\(893\) 27.7926 0.930044
\(894\) 0 0
\(895\) 38.2983 1.28017
\(896\) 0 0
\(897\) 0.470368 0.0157051
\(898\) 0 0
\(899\) 4.03948 0.134724
\(900\) 0 0
\(901\) 13.8113 0.460120
\(902\) 0 0
\(903\) −11.9368 −0.397231
\(904\) 0 0
\(905\) −48.8705 −1.62451
\(906\) 0 0
\(907\) −6.72145 −0.223182 −0.111591 0.993754i \(-0.535595\pi\)
−0.111591 + 0.993754i \(0.535595\pi\)
\(908\) 0 0
\(909\) −13.6927 −0.454159
\(910\) 0 0
\(911\) −10.3587 −0.343200 −0.171600 0.985167i \(-0.554894\pi\)
−0.171600 + 0.985167i \(0.554894\pi\)
\(912\) 0 0
\(913\) −43.5811 −1.44232
\(914\) 0 0
\(915\) −33.9051 −1.12087
\(916\) 0 0
\(917\) −6.20357 −0.204860
\(918\) 0 0
\(919\) −24.0041 −0.791823 −0.395911 0.918289i \(-0.629571\pi\)
−0.395911 + 0.918289i \(0.629571\pi\)
\(920\) 0 0
\(921\) 26.9774 0.888934
\(922\) 0 0
\(923\) −4.63347 −0.152513
\(924\) 0 0
\(925\) −51.4398 −1.69133
\(926\) 0 0
\(927\) 16.3667 0.537552
\(928\) 0 0
\(929\) 10.0473 0.329640 0.164820 0.986324i \(-0.447296\pi\)
0.164820 + 0.986324i \(0.447296\pi\)
\(930\) 0 0
\(931\) −4.20357 −0.137766
\(932\) 0 0
\(933\) −12.9100 −0.422655
\(934\) 0 0
\(935\) −65.9654 −2.15730
\(936\) 0 0
\(937\) −29.5304 −0.964717 −0.482359 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(938\) 0 0
\(939\) 7.47434 0.243916
\(940\) 0 0
\(941\) 15.6333 0.509632 0.254816 0.966990i \(-0.417985\pi\)
0.254816 + 0.966990i \(0.417985\pi\)
\(942\) 0 0
\(943\) 3.75598 0.122311
\(944\) 0 0
\(945\) 3.73320 0.121441
\(946\) 0 0
\(947\) −9.66898 −0.314200 −0.157100 0.987583i \(-0.550214\pi\)
−0.157100 + 0.987583i \(0.550214\pi\)
\(948\) 0 0
\(949\) 1.80715 0.0586625
\(950\) 0 0
\(951\) 25.7788 0.835933
\(952\) 0 0
\(953\) 15.3627 0.497647 0.248823 0.968549i \(-0.419956\pi\)
0.248823 + 0.968549i \(0.419956\pi\)
\(954\) 0 0
\(955\) 32.7353 1.05929
\(956\) 0 0
\(957\) 2.73717 0.0884802
\(958\) 0 0
\(959\) −1.23032 −0.0397290
\(960\) 0 0
\(961\) 7.48422 0.241426
\(962\) 0 0
\(963\) −1.27953 −0.0412323
\(964\) 0 0
\(965\) 38.5308 1.24035
\(966\) 0 0
\(967\) −11.5929 −0.372802 −0.186401 0.982474i \(-0.559682\pi\)
−0.186401 + 0.982474i \(0.559682\pi\)
\(968\) 0 0
\(969\) −17.6700 −0.567641
\(970\) 0 0
\(971\) −2.79943 −0.0898379 −0.0449190 0.998991i \(-0.514303\pi\)
−0.0449190 + 0.998991i \(0.514303\pi\)
\(972\) 0 0
\(973\) −16.5475 −0.530487
\(974\) 0 0
\(975\) −4.20357 −0.134622
\(976\) 0 0
\(977\) −27.5137 −0.880243 −0.440121 0.897938i \(-0.645065\pi\)
−0.440121 + 0.897938i \(0.645065\pi\)
\(978\) 0 0
\(979\) 53.7698 1.71849
\(980\) 0 0
\(981\) 0.266803 0.00851835
\(982\) 0 0
\(983\) −42.9564 −1.37010 −0.685048 0.728498i \(-0.740219\pi\)
−0.685048 + 0.728498i \(0.740219\pi\)
\(984\) 0 0
\(985\) −42.5702 −1.35640
\(986\) 0 0
\(987\) −6.61168 −0.210452
\(988\) 0 0
\(989\) 11.9368 0.379567
\(990\) 0 0
\(991\) 25.7698 0.818606 0.409303 0.912399i \(-0.365772\pi\)
0.409303 + 0.912399i \(0.365772\pi\)
\(992\) 0 0
\(993\) −35.1029 −1.11396
\(994\) 0 0
\(995\) 54.7510 1.73572
\(996\) 0 0
\(997\) 25.1618 0.796883 0.398441 0.917194i \(-0.369551\pi\)
0.398441 + 0.917194i \(0.369551\pi\)
\(998\) 0 0
\(999\) 5.75598 0.182111
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 7728.2.a.ca.1.4 4
4.3 odd 2 3864.2.a.u.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.u.1.4 4 4.3 odd 2
7728.2.a.ca.1.4 4 1.1 even 1 trivial