Properties

Label 7728.2.a.cb.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: 4.4.2225.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.74301 q^{11} -5.07020 q^{13} +1.32719 q^{15} +3.38351 q^{17} +3.74301 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.23856 q^{25} +1.00000 q^{27} -8.01012 q^{29} +10.2151 q^{31} -3.74301 q^{33} -1.32719 q^{35} -0.911372 q^{37} -5.07020 q^{39} -7.38351 q^{41} +4.08408 q^{43} +1.32719 q^{45} -0.831637 q^{47} +1.00000 q^{49} +3.38351 q^{51} +10.0664 q^{53} -4.96769 q^{55} +3.74301 q^{57} -6.42594 q^{59} +2.86019 q^{61} -1.00000 q^{63} -6.72913 q^{65} -2.70058 q^{67} +1.00000 q^{69} -0.210796 q^{71} -12.8418 q^{73} -3.23856 q^{75} +3.74301 q^{77} -10.2151 q^{79} +1.00000 q^{81} +1.20126 q^{83} +4.49056 q^{85} -8.01012 q^{87} -14.8929 q^{89} +5.07020 q^{91} +10.2151 q^{93} +4.96769 q^{95} -12.8418 q^{97} -3.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(5\) 1.32719 0.593538 0.296769 0.954949i \(-0.404091\pi\)
0.296769 + 0.954949i \(0.404091\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.74301 −1.12856 −0.564280 0.825583i \(-0.690846\pi\)
−0.564280 + 0.825583i \(0.690846\pi\)
\(12\) 0 0
\(13\) −5.07020 −1.40622 −0.703110 0.711081i \(-0.748206\pi\)
−0.703110 + 0.711081i \(0.748206\pi\)
\(14\) 0 0
\(15\) 1.32719 0.342679
\(16\) 0 0
\(17\) 3.38351 0.820621 0.410311 0.911946i \(-0.365420\pi\)
0.410311 + 0.911946i \(0.365420\pi\)
\(18\) 0 0
\(19\) 3.74301 0.858705 0.429353 0.903137i \(-0.358742\pi\)
0.429353 + 0.903137i \(0.358742\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.23856 −0.647713
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.01012 −1.48744 −0.743721 0.668490i \(-0.766941\pi\)
−0.743721 + 0.668490i \(0.766941\pi\)
\(30\) 0 0
\(31\) 10.2151 1.83469 0.917347 0.398088i \(-0.130326\pi\)
0.917347 + 0.398088i \(0.130326\pi\)
\(32\) 0 0
\(33\) −3.74301 −0.651574
\(34\) 0 0
\(35\) −1.32719 −0.224336
\(36\) 0 0
\(37\) −0.911372 −0.149829 −0.0749144 0.997190i \(-0.523868\pi\)
−0.0749144 + 0.997190i \(0.523868\pi\)
\(38\) 0 0
\(39\) −5.07020 −0.811882
\(40\) 0 0
\(41\) −7.38351 −1.15311 −0.576555 0.817058i \(-0.695603\pi\)
−0.576555 + 0.817058i \(0.695603\pi\)
\(42\) 0 0
\(43\) 4.08408 0.622817 0.311409 0.950276i \(-0.399199\pi\)
0.311409 + 0.950276i \(0.399199\pi\)
\(44\) 0 0
\(45\) 1.32719 0.197846
\(46\) 0 0
\(47\) −0.831637 −0.121307 −0.0606534 0.998159i \(-0.519318\pi\)
−0.0606534 + 0.998159i \(0.519318\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.38351 0.473786
\(52\) 0 0
\(53\) 10.0664 1.38273 0.691366 0.722505i \(-0.257009\pi\)
0.691366 + 0.722505i \(0.257009\pi\)
\(54\) 0 0
\(55\) −4.96769 −0.669843
\(56\) 0 0
\(57\) 3.74301 0.495774
\(58\) 0 0
\(59\) −6.42594 −0.836586 −0.418293 0.908312i \(-0.637372\pi\)
−0.418293 + 0.908312i \(0.637372\pi\)
\(60\) 0 0
\(61\) 2.86019 0.366209 0.183105 0.983093i \(-0.441385\pi\)
0.183105 + 0.983093i \(0.441385\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −6.72913 −0.834645
\(66\) 0 0
\(67\) −2.70058 −0.329928 −0.164964 0.986300i \(-0.552751\pi\)
−0.164964 + 0.986300i \(0.552751\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.210796 −0.0250169 −0.0125084 0.999922i \(-0.503982\pi\)
−0.0125084 + 0.999922i \(0.503982\pi\)
\(72\) 0 0
\(73\) −12.8418 −1.50301 −0.751507 0.659725i \(-0.770673\pi\)
−0.751507 + 0.659725i \(0.770673\pi\)
\(74\) 0 0
\(75\) −3.23856 −0.373957
\(76\) 0 0
\(77\) 3.74301 0.426556
\(78\) 0 0
\(79\) −10.2151 −1.14929 −0.574647 0.818401i \(-0.694861\pi\)
−0.574647 + 0.818401i \(0.694861\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.20126 0.131856 0.0659278 0.997824i \(-0.478999\pi\)
0.0659278 + 0.997824i \(0.478999\pi\)
\(84\) 0 0
\(85\) 4.49056 0.487070
\(86\) 0 0
\(87\) −8.01012 −0.858775
\(88\) 0 0
\(89\) −14.8929 −1.57865 −0.789325 0.613976i \(-0.789569\pi\)
−0.789325 + 0.613976i \(0.789569\pi\)
\(90\) 0 0
\(91\) 5.07020 0.531501
\(92\) 0 0
\(93\) 10.2151 1.05926
\(94\) 0 0
\(95\) 4.96769 0.509674
\(96\) 0 0
\(97\) −12.8418 −1.30388 −0.651942 0.758269i \(-0.726045\pi\)
−0.651942 + 0.758269i \(0.726045\pi\)
\(98\) 0 0
\(99\) −3.74301 −0.376187
\(100\) 0 0
\(101\) 9.10809 0.906289 0.453145 0.891437i \(-0.350302\pi\)
0.453145 + 0.891437i \(0.350302\pi\)
\(102\) 0 0
\(103\) −2.02777 −0.199802 −0.0999009 0.994997i \(-0.531853\pi\)
−0.0999009 + 0.994997i \(0.531853\pi\)
\(104\) 0 0
\(105\) −1.32719 −0.129521
\(106\) 0 0
\(107\) 14.0766 1.36083 0.680416 0.732826i \(-0.261799\pi\)
0.680416 + 0.732826i \(0.261799\pi\)
\(108\) 0 0
\(109\) −8.24370 −0.789603 −0.394801 0.918766i \(-0.629187\pi\)
−0.394801 + 0.918766i \(0.629187\pi\)
\(110\) 0 0
\(111\) −0.911372 −0.0865036
\(112\) 0 0
\(113\) −8.51833 −0.801337 −0.400669 0.916223i \(-0.631222\pi\)
−0.400669 + 0.916223i \(0.631222\pi\)
\(114\) 0 0
\(115\) 1.32719 0.123761
\(116\) 0 0
\(117\) −5.07020 −0.468740
\(118\) 0 0
\(119\) −3.38351 −0.310166
\(120\) 0 0
\(121\) 3.01012 0.273648
\(122\) 0 0
\(123\) −7.38351 −0.665749
\(124\) 0 0
\(125\) −10.9341 −0.977980
\(126\) 0 0
\(127\) −6.44359 −0.571776 −0.285888 0.958263i \(-0.592289\pi\)
−0.285888 + 0.958263i \(0.592289\pi\)
\(128\) 0 0
\(129\) 4.08408 0.359584
\(130\) 0 0
\(131\) 2.10251 0.183697 0.0918486 0.995773i \(-0.470722\pi\)
0.0918486 + 0.995773i \(0.470722\pi\)
\(132\) 0 0
\(133\) −3.74301 −0.324560
\(134\) 0 0
\(135\) 1.32719 0.114226
\(136\) 0 0
\(137\) −14.1874 −1.21211 −0.606055 0.795423i \(-0.707249\pi\)
−0.606055 + 0.795423i \(0.707249\pi\)
\(138\) 0 0
\(139\) −21.3980 −1.81495 −0.907477 0.420103i \(-0.861994\pi\)
−0.907477 + 0.420103i \(0.861994\pi\)
\(140\) 0 0
\(141\) −0.831637 −0.0700365
\(142\) 0 0
\(143\) 18.9778 1.58700
\(144\) 0 0
\(145\) −10.6310 −0.882854
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 15.3518 1.25767 0.628834 0.777540i \(-0.283533\pi\)
0.628834 + 0.777540i \(0.283533\pi\)
\(150\) 0 0
\(151\) −2.85941 −0.232695 −0.116348 0.993209i \(-0.537119\pi\)
−0.116348 + 0.993209i \(0.537119\pi\)
\(152\) 0 0
\(153\) 3.38351 0.273540
\(154\) 0 0
\(155\) 13.5575 1.08896
\(156\) 0 0
\(157\) 4.58477 0.365905 0.182952 0.983122i \(-0.441435\pi\)
0.182952 + 0.983122i \(0.441435\pi\)
\(158\) 0 0
\(159\) 10.0664 0.798321
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −11.0526 −0.865703 −0.432852 0.901465i \(-0.642493\pi\)
−0.432852 + 0.901465i \(0.642493\pi\)
\(164\) 0 0
\(165\) −4.96769 −0.386734
\(166\) 0 0
\(167\) −22.9175 −1.77341 −0.886706 0.462333i \(-0.847012\pi\)
−0.886706 + 0.462333i \(0.847012\pi\)
\(168\) 0 0
\(169\) 12.7069 0.977457
\(170\) 0 0
\(171\) 3.74301 0.286235
\(172\) 0 0
\(173\) −20.4873 −1.55762 −0.778808 0.627262i \(-0.784176\pi\)
−0.778808 + 0.627262i \(0.784176\pi\)
\(174\) 0 0
\(175\) 3.23856 0.244812
\(176\) 0 0
\(177\) −6.42594 −0.483003
\(178\) 0 0
\(179\) −15.5297 −1.16074 −0.580372 0.814352i \(-0.697093\pi\)
−0.580372 + 0.814352i \(0.697093\pi\)
\(180\) 0 0
\(181\) −21.3417 −1.58631 −0.793157 0.609018i \(-0.791564\pi\)
−0.793157 + 0.609018i \(0.791564\pi\)
\(182\) 0 0
\(183\) 2.86019 0.211431
\(184\) 0 0
\(185\) −1.20957 −0.0889290
\(186\) 0 0
\(187\) −12.6645 −0.926120
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 2.85941 0.206899 0.103450 0.994635i \(-0.467012\pi\)
0.103450 + 0.994635i \(0.467012\pi\)
\(192\) 0 0
\(193\) 16.3657 1.17803 0.589013 0.808123i \(-0.299517\pi\)
0.589013 + 0.808123i \(0.299517\pi\)
\(194\) 0 0
\(195\) −6.72913 −0.481883
\(196\) 0 0
\(197\) −5.90184 −0.420489 −0.210244 0.977649i \(-0.567426\pi\)
−0.210244 + 0.977649i \(0.567426\pi\)
\(198\) 0 0
\(199\) −1.08032 −0.0765821 −0.0382911 0.999267i \(-0.512191\pi\)
−0.0382911 + 0.999267i \(0.512191\pi\)
\(200\) 0 0
\(201\) −2.70058 −0.190484
\(202\) 0 0
\(203\) 8.01012 0.562200
\(204\) 0 0
\(205\) −9.79933 −0.684415
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −14.0101 −0.969100
\(210\) 0 0
\(211\) 10.8418 0.746378 0.373189 0.927755i \(-0.378264\pi\)
0.373189 + 0.927755i \(0.378264\pi\)
\(212\) 0 0
\(213\) −0.210796 −0.0144435
\(214\) 0 0
\(215\) 5.42036 0.369666
\(216\) 0 0
\(217\) −10.2151 −0.693449
\(218\) 0 0
\(219\) −12.8418 −0.867766
\(220\) 0 0
\(221\) −17.1551 −1.15397
\(222\) 0 0
\(223\) −5.89172 −0.394538 −0.197269 0.980349i \(-0.563207\pi\)
−0.197269 + 0.980349i \(0.563207\pi\)
\(224\) 0 0
\(225\) −3.23856 −0.215904
\(226\) 0 0
\(227\) 17.2626 1.14576 0.572879 0.819640i \(-0.305827\pi\)
0.572879 + 0.819640i \(0.305827\pi\)
\(228\) 0 0
\(229\) 10.8410 0.716392 0.358196 0.933646i \(-0.383392\pi\)
0.358196 + 0.933646i \(0.383392\pi\)
\(230\) 0 0
\(231\) 3.74301 0.246272
\(232\) 0 0
\(233\) −13.9766 −0.915636 −0.457818 0.889046i \(-0.651369\pi\)
−0.457818 + 0.889046i \(0.651369\pi\)
\(234\) 0 0
\(235\) −1.10374 −0.0720002
\(236\) 0 0
\(237\) −10.2151 −0.663545
\(238\) 0 0
\(239\) 5.89172 0.381103 0.190552 0.981677i \(-0.438972\pi\)
0.190552 + 0.981677i \(0.438972\pi\)
\(240\) 0 0
\(241\) −19.0290 −1.22577 −0.612883 0.790174i \(-0.709990\pi\)
−0.612883 + 0.790174i \(0.709990\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.32719 0.0847911
\(246\) 0 0
\(247\) −18.9778 −1.20753
\(248\) 0 0
\(249\) 1.20126 0.0761269
\(250\) 0 0
\(251\) −24.6328 −1.55481 −0.777404 0.629002i \(-0.783464\pi\)
−0.777404 + 0.629002i \(0.783464\pi\)
\(252\) 0 0
\(253\) −3.74301 −0.235321
\(254\) 0 0
\(255\) 4.49056 0.281210
\(256\) 0 0
\(257\) −8.12016 −0.506521 −0.253261 0.967398i \(-0.581503\pi\)
−0.253261 + 0.967398i \(0.581503\pi\)
\(258\) 0 0
\(259\) 0.911372 0.0566299
\(260\) 0 0
\(261\) −8.01012 −0.495814
\(262\) 0 0
\(263\) 12.1505 0.749233 0.374617 0.927180i \(-0.377774\pi\)
0.374617 + 0.927180i \(0.377774\pi\)
\(264\) 0 0
\(265\) 13.3601 0.820704
\(266\) 0 0
\(267\) −14.8929 −0.911433
\(268\) 0 0
\(269\) −24.6689 −1.50409 −0.752043 0.659114i \(-0.770932\pi\)
−0.752043 + 0.659114i \(0.770932\pi\)
\(270\) 0 0
\(271\) 9.74800 0.592149 0.296074 0.955165i \(-0.404322\pi\)
0.296074 + 0.955165i \(0.404322\pi\)
\(272\) 0 0
\(273\) 5.07020 0.306863
\(274\) 0 0
\(275\) 12.1220 0.730983
\(276\) 0 0
\(277\) 11.0980 0.666812 0.333406 0.942783i \(-0.391802\pi\)
0.333406 + 0.942783i \(0.391802\pi\)
\(278\) 0 0
\(279\) 10.2151 0.611565
\(280\) 0 0
\(281\) −7.56062 −0.451029 −0.225514 0.974240i \(-0.572406\pi\)
−0.225514 + 0.974240i \(0.572406\pi\)
\(282\) 0 0
\(283\) 21.6929 1.28951 0.644753 0.764391i \(-0.276960\pi\)
0.644753 + 0.764391i \(0.276960\pi\)
\(284\) 0 0
\(285\) 4.96769 0.294261
\(286\) 0 0
\(287\) 7.38351 0.435835
\(288\) 0 0
\(289\) −5.55187 −0.326581
\(290\) 0 0
\(291\) −12.8418 −0.752797
\(292\) 0 0
\(293\) 25.9859 1.51811 0.759057 0.651024i \(-0.225660\pi\)
0.759057 + 0.651024i \(0.225660\pi\)
\(294\) 0 0
\(295\) −8.52845 −0.496546
\(296\) 0 0
\(297\) −3.74301 −0.217191
\(298\) 0 0
\(299\) −5.07020 −0.293217
\(300\) 0 0
\(301\) −4.08408 −0.235403
\(302\) 0 0
\(303\) 9.10809 0.523246
\(304\) 0 0
\(305\) 3.79601 0.217359
\(306\) 0 0
\(307\) 26.2354 1.49733 0.748666 0.662947i \(-0.230694\pi\)
0.748666 + 0.662947i \(0.230694\pi\)
\(308\) 0 0
\(309\) −2.02777 −0.115356
\(310\) 0 0
\(311\) 18.1462 1.02898 0.514488 0.857498i \(-0.327982\pi\)
0.514488 + 0.857498i \(0.327982\pi\)
\(312\) 0 0
\(313\) 0.597283 0.0337604 0.0168802 0.999858i \(-0.494627\pi\)
0.0168802 + 0.999858i \(0.494627\pi\)
\(314\) 0 0
\(315\) −1.32719 −0.0747788
\(316\) 0 0
\(317\) 3.06008 0.171871 0.0859356 0.996301i \(-0.472612\pi\)
0.0859356 + 0.996301i \(0.472612\pi\)
\(318\) 0 0
\(319\) 29.9820 1.67867
\(320\) 0 0
\(321\) 14.0766 0.785677
\(322\) 0 0
\(323\) 12.6645 0.704672
\(324\) 0 0
\(325\) 16.4202 0.910827
\(326\) 0 0
\(327\) −8.24370 −0.455877
\(328\) 0 0
\(329\) 0.831637 0.0458497
\(330\) 0 0
\(331\) 33.9088 1.86380 0.931898 0.362721i \(-0.118152\pi\)
0.931898 + 0.362721i \(0.118152\pi\)
\(332\) 0 0
\(333\) −0.911372 −0.0499429
\(334\) 0 0
\(335\) −3.58418 −0.195825
\(336\) 0 0
\(337\) 11.6675 0.635568 0.317784 0.948163i \(-0.397061\pi\)
0.317784 + 0.948163i \(0.397061\pi\)
\(338\) 0 0
\(339\) −8.51833 −0.462652
\(340\) 0 0
\(341\) −38.2354 −2.07056
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.32719 0.0714536
\(346\) 0 0
\(347\) 4.45948 0.239398 0.119699 0.992810i \(-0.461807\pi\)
0.119699 + 0.992810i \(0.461807\pi\)
\(348\) 0 0
\(349\) 2.96769 0.158857 0.0794284 0.996841i \(-0.474690\pi\)
0.0794284 + 0.996841i \(0.474690\pi\)
\(350\) 0 0
\(351\) −5.07020 −0.270627
\(352\) 0 0
\(353\) −7.15824 −0.380995 −0.190497 0.981688i \(-0.561010\pi\)
−0.190497 + 0.981688i \(0.561010\pi\)
\(354\) 0 0
\(355\) −0.279767 −0.0148485
\(356\) 0 0
\(357\) −3.38351 −0.179074
\(358\) 0 0
\(359\) −21.3424 −1.12641 −0.563206 0.826317i \(-0.690432\pi\)
−0.563206 + 0.826317i \(0.690432\pi\)
\(360\) 0 0
\(361\) −4.98988 −0.262625
\(362\) 0 0
\(363\) 3.01012 0.157990
\(364\) 0 0
\(365\) −17.0435 −0.892096
\(366\) 0 0
\(367\) 15.1257 0.789557 0.394779 0.918776i \(-0.370821\pi\)
0.394779 + 0.918776i \(0.370821\pi\)
\(368\) 0 0
\(369\) −7.38351 −0.384370
\(370\) 0 0
\(371\) −10.0664 −0.522624
\(372\) 0 0
\(373\) 21.6506 1.12103 0.560513 0.828145i \(-0.310604\pi\)
0.560513 + 0.828145i \(0.310604\pi\)
\(374\) 0 0
\(375\) −10.9341 −0.564637
\(376\) 0 0
\(377\) 40.6129 2.09167
\(378\) 0 0
\(379\) −2.41660 −0.124132 −0.0620662 0.998072i \(-0.519769\pi\)
−0.0620662 + 0.998072i \(0.519769\pi\)
\(380\) 0 0
\(381\) −6.44359 −0.330115
\(382\) 0 0
\(383\) −8.92663 −0.456129 −0.228065 0.973646i \(-0.573240\pi\)
−0.228065 + 0.973646i \(0.573240\pi\)
\(384\) 0 0
\(385\) 4.96769 0.253177
\(386\) 0 0
\(387\) 4.08408 0.207606
\(388\) 0 0
\(389\) 8.97717 0.455161 0.227580 0.973759i \(-0.426919\pi\)
0.227580 + 0.973759i \(0.426919\pi\)
\(390\) 0 0
\(391\) 3.38351 0.171111
\(392\) 0 0
\(393\) 2.10251 0.106058
\(394\) 0 0
\(395\) −13.5575 −0.682149
\(396\) 0 0
\(397\) −1.42892 −0.0717155 −0.0358577 0.999357i \(-0.511416\pi\)
−0.0358577 + 0.999357i \(0.511416\pi\)
\(398\) 0 0
\(399\) −3.74301 −0.187385
\(400\) 0 0
\(401\) −22.8695 −1.14205 −0.571025 0.820933i \(-0.693454\pi\)
−0.571025 + 0.820933i \(0.693454\pi\)
\(402\) 0 0
\(403\) −51.7928 −2.57999
\(404\) 0 0
\(405\) 1.32719 0.0659487
\(406\) 0 0
\(407\) 3.41128 0.169091
\(408\) 0 0
\(409\) 17.1492 0.847971 0.423986 0.905669i \(-0.360631\pi\)
0.423986 + 0.905669i \(0.360631\pi\)
\(410\) 0 0
\(411\) −14.1874 −0.699812
\(412\) 0 0
\(413\) 6.42594 0.316200
\(414\) 0 0
\(415\) 1.59430 0.0782613
\(416\) 0 0
\(417\) −21.3980 −1.04786
\(418\) 0 0
\(419\) 16.4412 0.803205 0.401603 0.915814i \(-0.368453\pi\)
0.401603 + 0.915814i \(0.368453\pi\)
\(420\) 0 0
\(421\) 12.1487 0.592092 0.296046 0.955174i \(-0.404332\pi\)
0.296046 + 0.955174i \(0.404332\pi\)
\(422\) 0 0
\(423\) −0.831637 −0.0404356
\(424\) 0 0
\(425\) −10.9577 −0.531527
\(426\) 0 0
\(427\) −2.86019 −0.138414
\(428\) 0 0
\(429\) 18.9778 0.916257
\(430\) 0 0
\(431\) 17.8004 0.857413 0.428707 0.903444i \(-0.358969\pi\)
0.428707 + 0.903444i \(0.358969\pi\)
\(432\) 0 0
\(433\) 30.9277 1.48629 0.743144 0.669131i \(-0.233334\pi\)
0.743144 + 0.669131i \(0.233334\pi\)
\(434\) 0 0
\(435\) −10.6310 −0.509716
\(436\) 0 0
\(437\) 3.74301 0.179052
\(438\) 0 0
\(439\) 28.3571 1.35341 0.676706 0.736254i \(-0.263407\pi\)
0.676706 + 0.736254i \(0.263407\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −35.5618 −1.68959 −0.844796 0.535088i \(-0.820278\pi\)
−0.844796 + 0.535088i \(0.820278\pi\)
\(444\) 0 0
\(445\) −19.7658 −0.936988
\(446\) 0 0
\(447\) 15.3518 0.726115
\(448\) 0 0
\(449\) 9.67761 0.456715 0.228357 0.973577i \(-0.426665\pi\)
0.228357 + 0.973577i \(0.426665\pi\)
\(450\) 0 0
\(451\) 27.6365 1.30135
\(452\) 0 0
\(453\) −2.85941 −0.134347
\(454\) 0 0
\(455\) 6.72913 0.315466
\(456\) 0 0
\(457\) 0.827288 0.0386989 0.0193494 0.999813i \(-0.493840\pi\)
0.0193494 + 0.999813i \(0.493840\pi\)
\(458\) 0 0
\(459\) 3.38351 0.157929
\(460\) 0 0
\(461\) 21.4170 0.997491 0.498746 0.866748i \(-0.333794\pi\)
0.498746 + 0.866748i \(0.333794\pi\)
\(462\) 0 0
\(463\) −37.2427 −1.73082 −0.865408 0.501068i \(-0.832941\pi\)
−0.865408 + 0.501068i \(0.832941\pi\)
\(464\) 0 0
\(465\) 13.5575 0.628712
\(466\) 0 0
\(467\) −2.59372 −0.120023 −0.0600114 0.998198i \(-0.519114\pi\)
−0.0600114 + 0.998198i \(0.519114\pi\)
\(468\) 0 0
\(469\) 2.70058 0.124701
\(470\) 0 0
\(471\) 4.58477 0.211255
\(472\) 0 0
\(473\) −15.2868 −0.702886
\(474\) 0 0
\(475\) −12.1220 −0.556194
\(476\) 0 0
\(477\) 10.0664 0.460911
\(478\) 0 0
\(479\) 1.56829 0.0716568 0.0358284 0.999358i \(-0.488593\pi\)
0.0358284 + 0.999358i \(0.488593\pi\)
\(480\) 0 0
\(481\) 4.62084 0.210692
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −17.0435 −0.773904
\(486\) 0 0
\(487\) −7.26335 −0.329134 −0.164567 0.986366i \(-0.552623\pi\)
−0.164567 + 0.986366i \(0.552623\pi\)
\(488\) 0 0
\(489\) −11.0526 −0.499814
\(490\) 0 0
\(491\) −7.20308 −0.325071 −0.162535 0.986703i \(-0.551967\pi\)
−0.162535 + 0.986703i \(0.551967\pi\)
\(492\) 0 0
\(493\) −27.1023 −1.22063
\(494\) 0 0
\(495\) −4.96769 −0.223281
\(496\) 0 0
\(497\) 0.210796 0.00945549
\(498\) 0 0
\(499\) 9.19296 0.411533 0.205767 0.978601i \(-0.434031\pi\)
0.205767 + 0.978601i \(0.434031\pi\)
\(500\) 0 0
\(501\) −22.9175 −1.02388
\(502\) 0 0
\(503\) 7.73471 0.344874 0.172437 0.985021i \(-0.444836\pi\)
0.172437 + 0.985021i \(0.444836\pi\)
\(504\) 0 0
\(505\) 12.0882 0.537917
\(506\) 0 0
\(507\) 12.7069 0.564335
\(508\) 0 0
\(509\) 9.59865 0.425453 0.212726 0.977112i \(-0.431766\pi\)
0.212726 + 0.977112i \(0.431766\pi\)
\(510\) 0 0
\(511\) 12.8418 0.568086
\(512\) 0 0
\(513\) 3.74301 0.165258
\(514\) 0 0
\(515\) −2.69124 −0.118590
\(516\) 0 0
\(517\) 3.11283 0.136902
\(518\) 0 0
\(519\) −20.4873 −0.899290
\(520\) 0 0
\(521\) −41.1418 −1.80245 −0.901227 0.433348i \(-0.857332\pi\)
−0.901227 + 0.433348i \(0.857332\pi\)
\(522\) 0 0
\(523\) 4.29488 0.187802 0.0939010 0.995582i \(-0.470066\pi\)
0.0939010 + 0.995582i \(0.470066\pi\)
\(524\) 0 0
\(525\) 3.23856 0.141343
\(526\) 0 0
\(527\) 34.5630 1.50559
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.42594 −0.278862
\(532\) 0 0
\(533\) 37.4359 1.62153
\(534\) 0 0
\(535\) 18.6823 0.807706
\(536\) 0 0
\(537\) −15.5297 −0.670155
\(538\) 0 0
\(539\) −3.74301 −0.161223
\(540\) 0 0
\(541\) −41.7402 −1.79455 −0.897276 0.441469i \(-0.854457\pi\)
−0.897276 + 0.441469i \(0.854457\pi\)
\(542\) 0 0
\(543\) −21.3417 −0.915858
\(544\) 0 0
\(545\) −10.9410 −0.468659
\(546\) 0 0
\(547\) −21.1561 −0.904570 −0.452285 0.891874i \(-0.649391\pi\)
−0.452285 + 0.891874i \(0.649391\pi\)
\(548\) 0 0
\(549\) 2.86019 0.122070
\(550\) 0 0
\(551\) −29.9820 −1.27727
\(552\) 0 0
\(553\) 10.2151 0.434392
\(554\) 0 0
\(555\) −1.20957 −0.0513432
\(556\) 0 0
\(557\) −15.0326 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(558\) 0 0
\(559\) −20.7071 −0.875818
\(560\) 0 0
\(561\) −12.6645 −0.534696
\(562\) 0 0
\(563\) 3.93460 0.165824 0.0829118 0.996557i \(-0.473578\pi\)
0.0829118 + 0.996557i \(0.473578\pi\)
\(564\) 0 0
\(565\) −11.3055 −0.475624
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 30.7948 1.29098 0.645492 0.763767i \(-0.276652\pi\)
0.645492 + 0.763767i \(0.276652\pi\)
\(570\) 0 0
\(571\) 38.8848 1.62728 0.813639 0.581371i \(-0.197483\pi\)
0.813639 + 0.581371i \(0.197483\pi\)
\(572\) 0 0
\(573\) 2.85941 0.119453
\(574\) 0 0
\(575\) −3.23856 −0.135057
\(576\) 0 0
\(577\) 18.4505 0.768106 0.384053 0.923311i \(-0.374528\pi\)
0.384053 + 0.923311i \(0.374528\pi\)
\(578\) 0 0
\(579\) 16.3657 0.680134
\(580\) 0 0
\(581\) −1.20126 −0.0498367
\(582\) 0 0
\(583\) −37.6788 −1.56050
\(584\) 0 0
\(585\) −6.72913 −0.278215
\(586\) 0 0
\(587\) −44.5471 −1.83866 −0.919329 0.393491i \(-0.871267\pi\)
−0.919329 + 0.393491i \(0.871267\pi\)
\(588\) 0 0
\(589\) 38.2354 1.57546
\(590\) 0 0
\(591\) −5.90184 −0.242769
\(592\) 0 0
\(593\) 1.69261 0.0695070 0.0347535 0.999396i \(-0.488935\pi\)
0.0347535 + 0.999396i \(0.488935\pi\)
\(594\) 0 0
\(595\) −4.49056 −0.184095
\(596\) 0 0
\(597\) −1.08032 −0.0442147
\(598\) 0 0
\(599\) 34.1180 1.39402 0.697012 0.717059i \(-0.254512\pi\)
0.697012 + 0.717059i \(0.254512\pi\)
\(600\) 0 0
\(601\) 44.0346 1.79621 0.898104 0.439783i \(-0.144945\pi\)
0.898104 + 0.439783i \(0.144945\pi\)
\(602\) 0 0
\(603\) −2.70058 −0.109976
\(604\) 0 0
\(605\) 3.99501 0.162420
\(606\) 0 0
\(607\) −14.9410 −0.606435 −0.303217 0.952921i \(-0.598061\pi\)
−0.303217 + 0.952921i \(0.598061\pi\)
\(608\) 0 0
\(609\) 8.01012 0.324587
\(610\) 0 0
\(611\) 4.21657 0.170584
\(612\) 0 0
\(613\) 6.47200 0.261401 0.130701 0.991422i \(-0.458277\pi\)
0.130701 + 0.991422i \(0.458277\pi\)
\(614\) 0 0
\(615\) −9.79933 −0.395147
\(616\) 0 0
\(617\) −10.0513 −0.404651 −0.202326 0.979318i \(-0.564850\pi\)
−0.202326 + 0.979318i \(0.564850\pi\)
\(618\) 0 0
\(619\) −10.0095 −0.402315 −0.201158 0.979559i \(-0.564470\pi\)
−0.201158 + 0.979559i \(0.564470\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 14.8929 0.596673
\(624\) 0 0
\(625\) 1.68111 0.0672445
\(626\) 0 0
\(627\) −14.0101 −0.559510
\(628\) 0 0
\(629\) −3.08364 −0.122953
\(630\) 0 0
\(631\) −12.9251 −0.514539 −0.257269 0.966340i \(-0.582823\pi\)
−0.257269 + 0.966340i \(0.582823\pi\)
\(632\) 0 0
\(633\) 10.8418 0.430921
\(634\) 0 0
\(635\) −8.55187 −0.339371
\(636\) 0 0
\(637\) −5.07020 −0.200889
\(638\) 0 0
\(639\) −0.210796 −0.00833896
\(640\) 0 0
\(641\) −13.3676 −0.527989 −0.263994 0.964524i \(-0.585040\pi\)
−0.263994 + 0.964524i \(0.585040\pi\)
\(642\) 0 0
\(643\) −25.5994 −1.00954 −0.504772 0.863253i \(-0.668423\pi\)
−0.504772 + 0.863253i \(0.668423\pi\)
\(644\) 0 0
\(645\) 5.42036 0.213427
\(646\) 0 0
\(647\) −15.9953 −0.628839 −0.314419 0.949284i \(-0.601810\pi\)
−0.314419 + 0.949284i \(0.601810\pi\)
\(648\) 0 0
\(649\) 24.0524 0.944138
\(650\) 0 0
\(651\) −10.2151 −0.400363
\(652\) 0 0
\(653\) 17.9397 0.702036 0.351018 0.936369i \(-0.385836\pi\)
0.351018 + 0.936369i \(0.385836\pi\)
\(654\) 0 0
\(655\) 2.79043 0.109031
\(656\) 0 0
\(657\) −12.8418 −0.501005
\(658\) 0 0
\(659\) 25.8366 1.00645 0.503225 0.864155i \(-0.332147\pi\)
0.503225 + 0.864155i \(0.332147\pi\)
\(660\) 0 0
\(661\) 5.27587 0.205207 0.102604 0.994722i \(-0.467283\pi\)
0.102604 + 0.994722i \(0.467283\pi\)
\(662\) 0 0
\(663\) −17.1551 −0.666248
\(664\) 0 0
\(665\) −4.96769 −0.192639
\(666\) 0 0
\(667\) −8.01012 −0.310153
\(668\) 0 0
\(669\) −5.89172 −0.227787
\(670\) 0 0
\(671\) −10.7057 −0.413289
\(672\) 0 0
\(673\) −49.9947 −1.92715 −0.963577 0.267431i \(-0.913825\pi\)
−0.963577 + 0.267431i \(0.913825\pi\)
\(674\) 0 0
\(675\) −3.23856 −0.124652
\(676\) 0 0
\(677\) −24.9637 −0.959434 −0.479717 0.877423i \(-0.659261\pi\)
−0.479717 + 0.877423i \(0.659261\pi\)
\(678\) 0 0
\(679\) 12.8418 0.492822
\(680\) 0 0
\(681\) 17.2626 0.661503
\(682\) 0 0
\(683\) −26.8418 −1.02707 −0.513536 0.858068i \(-0.671665\pi\)
−0.513536 + 0.858068i \(0.671665\pi\)
\(684\) 0 0
\(685\) −18.8294 −0.719433
\(686\) 0 0
\(687\) 10.8410 0.413609
\(688\) 0 0
\(689\) −51.0389 −1.94443
\(690\) 0 0
\(691\) −4.02602 −0.153157 −0.0765785 0.997064i \(-0.524400\pi\)
−0.0765785 + 0.997064i \(0.524400\pi\)
\(692\) 0 0
\(693\) 3.74301 0.142185
\(694\) 0 0
\(695\) −28.3992 −1.07724
\(696\) 0 0
\(697\) −24.9822 −0.946267
\(698\) 0 0
\(699\) −13.9766 −0.528643
\(700\) 0 0
\(701\) −18.0109 −0.680262 −0.340131 0.940378i \(-0.610472\pi\)
−0.340131 + 0.940378i \(0.610472\pi\)
\(702\) 0 0
\(703\) −3.41128 −0.128659
\(704\) 0 0
\(705\) −1.10374 −0.0415693
\(706\) 0 0
\(707\) −9.10809 −0.342545
\(708\) 0 0
\(709\) 46.0104 1.72796 0.863978 0.503530i \(-0.167965\pi\)
0.863978 + 0.503530i \(0.167965\pi\)
\(710\) 0 0
\(711\) −10.2151 −0.383098
\(712\) 0 0
\(713\) 10.2151 0.382560
\(714\) 0 0
\(715\) 25.1872 0.941947
\(716\) 0 0
\(717\) 5.89172 0.220030
\(718\) 0 0
\(719\) 16.1521 0.602371 0.301186 0.953566i \(-0.402618\pi\)
0.301186 + 0.953566i \(0.402618\pi\)
\(720\) 0 0
\(721\) 2.02777 0.0755180
\(722\) 0 0
\(723\) −19.0290 −0.707696
\(724\) 0 0
\(725\) 25.9413 0.963435
\(726\) 0 0
\(727\) −24.7771 −0.918933 −0.459467 0.888195i \(-0.651959\pi\)
−0.459467 + 0.888195i \(0.651959\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.8185 0.511097
\(732\) 0 0
\(733\) 44.5111 1.64405 0.822027 0.569449i \(-0.192843\pi\)
0.822027 + 0.569449i \(0.192843\pi\)
\(734\) 0 0
\(735\) 1.32719 0.0489542
\(736\) 0 0
\(737\) 10.1083 0.372343
\(738\) 0 0
\(739\) −33.4846 −1.23175 −0.615875 0.787844i \(-0.711198\pi\)
−0.615875 + 0.787844i \(0.711198\pi\)
\(740\) 0 0
\(741\) −18.9778 −0.697167
\(742\) 0 0
\(743\) 18.5588 0.680857 0.340429 0.940270i \(-0.389428\pi\)
0.340429 + 0.940270i \(0.389428\pi\)
\(744\) 0 0
\(745\) 20.3748 0.746473
\(746\) 0 0
\(747\) 1.20126 0.0439519
\(748\) 0 0
\(749\) −14.0766 −0.514346
\(750\) 0 0
\(751\) −29.4651 −1.07520 −0.537598 0.843201i \(-0.680668\pi\)
−0.537598 + 0.843201i \(0.680668\pi\)
\(752\) 0 0
\(753\) −24.6328 −0.897669
\(754\) 0 0
\(755\) −3.79498 −0.138113
\(756\) 0 0
\(757\) 4.68610 0.170319 0.0851597 0.996367i \(-0.472860\pi\)
0.0851597 + 0.996367i \(0.472860\pi\)
\(758\) 0 0
\(759\) −3.74301 −0.135863
\(760\) 0 0
\(761\) 0.936750 0.0339572 0.0169786 0.999856i \(-0.494595\pi\)
0.0169786 + 0.999856i \(0.494595\pi\)
\(762\) 0 0
\(763\) 8.24370 0.298442
\(764\) 0 0
\(765\) 4.49056 0.162357
\(766\) 0 0
\(767\) 32.5808 1.17643
\(768\) 0 0
\(769\) −18.7024 −0.674426 −0.337213 0.941428i \(-0.609484\pi\)
−0.337213 + 0.941428i \(0.609484\pi\)
\(770\) 0 0
\(771\) −8.12016 −0.292440
\(772\) 0 0
\(773\) −5.35827 −0.192724 −0.0963618 0.995346i \(-0.530721\pi\)
−0.0963618 + 0.995346i \(0.530721\pi\)
\(774\) 0 0
\(775\) −33.0824 −1.18835
\(776\) 0 0
\(777\) 0.911372 0.0326953
\(778\) 0 0
\(779\) −27.6365 −0.990182
\(780\) 0 0
\(781\) 0.789011 0.0282330
\(782\) 0 0
\(783\) −8.01012 −0.286258
\(784\) 0 0
\(785\) 6.08487 0.217178
\(786\) 0 0
\(787\) −26.0308 −0.927897 −0.463948 0.885862i \(-0.653568\pi\)
−0.463948 + 0.885862i \(0.653568\pi\)
\(788\) 0 0
\(789\) 12.1505 0.432570
\(790\) 0 0
\(791\) 8.51833 0.302877
\(792\) 0 0
\(793\) −14.5017 −0.514971
\(794\) 0 0
\(795\) 13.3601 0.473834
\(796\) 0 0
\(797\) 28.7343 1.01782 0.508910 0.860820i \(-0.330049\pi\)
0.508910 + 0.860820i \(0.330049\pi\)
\(798\) 0 0
\(799\) −2.81385 −0.0995469
\(800\) 0 0
\(801\) −14.8929 −0.526216
\(802\) 0 0
\(803\) 48.0668 1.69624
\(804\) 0 0
\(805\) −1.32719 −0.0467773
\(806\) 0 0
\(807\) −24.6689 −0.868385
\(808\) 0 0
\(809\) −31.7817 −1.11738 −0.558692 0.829375i \(-0.688697\pi\)
−0.558692 + 0.829375i \(0.688697\pi\)
\(810\) 0 0
\(811\) 44.6848 1.56909 0.784547 0.620069i \(-0.212896\pi\)
0.784547 + 0.620069i \(0.212896\pi\)
\(812\) 0 0
\(813\) 9.74800 0.341877
\(814\) 0 0
\(815\) −14.6689 −0.513828
\(816\) 0 0
\(817\) 15.2868 0.534816
\(818\) 0 0
\(819\) 5.07020 0.177167
\(820\) 0 0
\(821\) 30.3581 1.05951 0.529753 0.848152i \(-0.322284\pi\)
0.529753 + 0.848152i \(0.322284\pi\)
\(822\) 0 0
\(823\) 14.1004 0.491508 0.245754 0.969332i \(-0.420964\pi\)
0.245754 + 0.969332i \(0.420964\pi\)
\(824\) 0 0
\(825\) 12.1220 0.422033
\(826\) 0 0
\(827\) −51.9559 −1.80668 −0.903342 0.428922i \(-0.858893\pi\)
−0.903342 + 0.428922i \(0.858893\pi\)
\(828\) 0 0
\(829\) 42.8782 1.48922 0.744611 0.667498i \(-0.232635\pi\)
0.744611 + 0.667498i \(0.232635\pi\)
\(830\) 0 0
\(831\) 11.0980 0.384984
\(832\) 0 0
\(833\) 3.38351 0.117232
\(834\) 0 0
\(835\) −30.4160 −1.05259
\(836\) 0 0
\(837\) 10.2151 0.353087
\(838\) 0 0
\(839\) 23.3157 0.804948 0.402474 0.915431i \(-0.368150\pi\)
0.402474 + 0.915431i \(0.368150\pi\)
\(840\) 0 0
\(841\) 35.1621 1.21249
\(842\) 0 0
\(843\) −7.56062 −0.260402
\(844\) 0 0
\(845\) 16.8645 0.580158
\(846\) 0 0
\(847\) −3.01012 −0.103429
\(848\) 0 0
\(849\) 21.6929 0.744497
\(850\) 0 0
\(851\) −0.911372 −0.0312414
\(852\) 0 0
\(853\) −14.7024 −0.503400 −0.251700 0.967805i \(-0.580990\pi\)
−0.251700 + 0.967805i \(0.580990\pi\)
\(854\) 0 0
\(855\) 4.96769 0.169891
\(856\) 0 0
\(857\) 30.3835 1.03788 0.518940 0.854811i \(-0.326327\pi\)
0.518940 + 0.854811i \(0.326327\pi\)
\(858\) 0 0
\(859\) 24.0202 0.819560 0.409780 0.912184i \(-0.365605\pi\)
0.409780 + 0.912184i \(0.365605\pi\)
\(860\) 0 0
\(861\) 7.38351 0.251629
\(862\) 0 0
\(863\) −4.21657 −0.143534 −0.0717668 0.997421i \(-0.522864\pi\)
−0.0717668 + 0.997421i \(0.522864\pi\)
\(864\) 0 0
\(865\) −27.1905 −0.924505
\(866\) 0 0
\(867\) −5.55187 −0.188551
\(868\) 0 0
\(869\) 38.2354 1.29705
\(870\) 0 0
\(871\) 13.6925 0.463952
\(872\) 0 0
\(873\) −12.8418 −0.434628
\(874\) 0 0
\(875\) 10.9341 0.369642
\(876\) 0 0
\(877\) 56.1035 1.89448 0.947241 0.320522i \(-0.103858\pi\)
0.947241 + 0.320522i \(0.103858\pi\)
\(878\) 0 0
\(879\) 25.9859 0.876483
\(880\) 0 0
\(881\) −49.7605 −1.67647 −0.838237 0.545307i \(-0.816413\pi\)
−0.838237 + 0.545307i \(0.816413\pi\)
\(882\) 0 0
\(883\) 37.7548 1.27055 0.635275 0.772286i \(-0.280887\pi\)
0.635275 + 0.772286i \(0.280887\pi\)
\(884\) 0 0
\(885\) −8.52845 −0.286681
\(886\) 0 0
\(887\) 24.9385 0.837353 0.418676 0.908135i \(-0.362494\pi\)
0.418676 + 0.908135i \(0.362494\pi\)
\(888\) 0 0
\(889\) 6.44359 0.216111
\(890\) 0 0
\(891\) −3.74301 −0.125396
\(892\) 0 0
\(893\) −3.11283 −0.104167
\(894\) 0 0
\(895\) −20.6109 −0.688945
\(896\) 0 0
\(897\) −5.07020 −0.169289
\(898\) 0 0
\(899\) −81.8246 −2.72900
\(900\) 0 0
\(901\) 34.0599 1.13470
\(902\) 0 0
\(903\) −4.08408 −0.135910
\(904\) 0 0
\(905\) −28.3245 −0.941537
\(906\) 0 0
\(907\) −52.2735 −1.73571 −0.867857 0.496814i \(-0.834503\pi\)
−0.867857 + 0.496814i \(0.834503\pi\)
\(908\) 0 0
\(909\) 9.10809 0.302096
\(910\) 0 0
\(911\) 10.4494 0.346203 0.173101 0.984904i \(-0.444621\pi\)
0.173101 + 0.984904i \(0.444621\pi\)
\(912\) 0 0
\(913\) −4.49634 −0.148807
\(914\) 0 0
\(915\) 3.79601 0.125492
\(916\) 0 0
\(917\) −2.10251 −0.0694310
\(918\) 0 0
\(919\) 7.14281 0.235620 0.117810 0.993036i \(-0.462413\pi\)
0.117810 + 0.993036i \(0.462413\pi\)
\(920\) 0 0
\(921\) 26.2354 0.864486
\(922\) 0 0
\(923\) 1.06878 0.0351793
\(924\) 0 0
\(925\) 2.95154 0.0970460
\(926\) 0 0
\(927\) −2.02777 −0.0666006
\(928\) 0 0
\(929\) −59.6248 −1.95623 −0.978113 0.208073i \(-0.933281\pi\)
−0.978113 + 0.208073i \(0.933281\pi\)
\(930\) 0 0
\(931\) 3.74301 0.122672
\(932\) 0 0
\(933\) 18.1462 0.594079
\(934\) 0 0
\(935\) −16.8082 −0.549688
\(936\) 0 0
\(937\) 38.6177 1.26158 0.630792 0.775952i \(-0.282730\pi\)
0.630792 + 0.775952i \(0.282730\pi\)
\(938\) 0 0
\(939\) 0.597283 0.0194916
\(940\) 0 0
\(941\) −57.9022 −1.88756 −0.943779 0.330576i \(-0.892757\pi\)
−0.943779 + 0.330576i \(0.892757\pi\)
\(942\) 0 0
\(943\) −7.38351 −0.240440
\(944\) 0 0
\(945\) −1.32719 −0.0431735
\(946\) 0 0
\(947\) −34.9823 −1.13677 −0.568386 0.822762i \(-0.692432\pi\)
−0.568386 + 0.822762i \(0.692432\pi\)
\(948\) 0 0
\(949\) 65.1103 2.11357
\(950\) 0 0
\(951\) 3.06008 0.0992298
\(952\) 0 0
\(953\) 54.8953 1.77823 0.889117 0.457681i \(-0.151320\pi\)
0.889117 + 0.457681i \(0.151320\pi\)
\(954\) 0 0
\(955\) 3.79498 0.122803
\(956\) 0 0
\(957\) 29.9820 0.969179
\(958\) 0 0
\(959\) 14.1874 0.458134
\(960\) 0 0
\(961\) 73.3492 2.36610
\(962\) 0 0
\(963\) 14.0766 0.453611
\(964\) 0 0
\(965\) 21.7204 0.699204
\(966\) 0 0
\(967\) 46.6096 1.49886 0.749432 0.662081i \(-0.230327\pi\)
0.749432 + 0.662081i \(0.230327\pi\)
\(968\) 0 0
\(969\) 12.6645 0.406843
\(970\) 0 0
\(971\) −15.4409 −0.495521 −0.247760 0.968821i \(-0.579695\pi\)
−0.247760 + 0.968821i \(0.579695\pi\)
\(972\) 0 0
\(973\) 21.3980 0.685988
\(974\) 0 0
\(975\) 16.4202 0.525866
\(976\) 0 0
\(977\) 35.6040 1.13907 0.569537 0.821966i \(-0.307122\pi\)
0.569537 + 0.821966i \(0.307122\pi\)
\(978\) 0 0
\(979\) 55.7444 1.78160
\(980\) 0 0
\(981\) −8.24370 −0.263201
\(982\) 0 0
\(983\) −9.59009 −0.305877 −0.152938 0.988236i \(-0.548874\pi\)
−0.152938 + 0.988236i \(0.548874\pi\)
\(984\) 0 0
\(985\) −7.83287 −0.249576
\(986\) 0 0
\(987\) 0.831637 0.0264713
\(988\) 0 0
\(989\) 4.08408 0.129866
\(990\) 0 0
\(991\) −15.9792 −0.507596 −0.253798 0.967257i \(-0.581680\pi\)
−0.253798 + 0.967257i \(0.581680\pi\)
\(992\) 0 0
\(993\) 33.9088 1.07606
\(994\) 0 0
\(995\) −1.43380 −0.0454544
\(996\) 0 0
\(997\) −40.2809 −1.27571 −0.637855 0.770156i \(-0.720178\pi\)
−0.637855 + 0.770156i \(0.720178\pi\)
\(998\) 0 0
\(999\) −0.911372 −0.0288345
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.cb.1.4 4
4.3 odd 2 3864.2.a.r.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.r.1.4 4 4.3 odd 2
7728.2.a.cb.1.4 4 1.1 even 1 trivial