Properties

Label 7728.2.a.bk.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.30278 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.30278 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.60555 q^{11} -4.30278 q^{13} -1.30278 q^{15} -3.00000 q^{17} +0.394449 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.30278 q^{25} +1.00000 q^{27} +1.00000 q^{29} -1.00000 q^{31} -3.60555 q^{33} +1.30278 q^{35} +6.21110 q^{37} -4.30278 q^{39} +1.60555 q^{41} -2.90833 q^{43} -1.30278 q^{45} +3.39445 q^{47} +1.00000 q^{49} -3.00000 q^{51} +2.09167 q^{53} +4.69722 q^{55} +0.394449 q^{57} -0.0916731 q^{59} -9.51388 q^{61} -1.00000 q^{63} +5.60555 q^{65} +15.1194 q^{67} +1.00000 q^{69} +1.09167 q^{71} -10.8167 q^{73} -3.30278 q^{75} +3.60555 q^{77} +13.6056 q^{79} +1.00000 q^{81} +7.60555 q^{83} +3.90833 q^{85} +1.00000 q^{87} -0.302776 q^{89} +4.30278 q^{91} -1.00000 q^{93} -0.513878 q^{95} -11.0000 q^{97} -3.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 5 q^{13} + q^{15} - 6 q^{17} + 8 q^{19} - 2 q^{21} + 2 q^{23} - 3 q^{25} + 2 q^{27} + 2 q^{29} - 2 q^{31} - q^{35} - 2 q^{37} - 5 q^{39} - 4 q^{41} + 5 q^{43} + q^{45} + 14 q^{47} + 2 q^{49} - 6 q^{51} + 15 q^{53} + 13 q^{55} + 8 q^{57} - 11 q^{59} - q^{61} - 2 q^{63} + 4 q^{65} + 5 q^{67} + 2 q^{69} + 13 q^{71} - 3 q^{75} + 20 q^{79} + 2 q^{81} + 8 q^{83} - 3 q^{85} + 2 q^{87} + 3 q^{89} + 5 q^{91} - 2 q^{93} + 17 q^{95} - 22 q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.30278 −0.582619 −0.291309 0.956629i \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.60555 −1.08711 −0.543557 0.839372i \(-0.682923\pi\)
−0.543557 + 0.839372i \(0.682923\pi\)
\(12\) 0 0
\(13\) −4.30278 −1.19338 −0.596688 0.802474i \(-0.703517\pi\)
−0.596688 + 0.802474i \(0.703517\pi\)
\(14\) 0 0
\(15\) −1.30278 −0.336375
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 0.394449 0.0904927 0.0452464 0.998976i \(-0.485593\pi\)
0.0452464 + 0.998976i \(0.485593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.30278 −0.660555
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) −3.60555 −0.627646
\(34\) 0 0
\(35\) 1.30278 0.220209
\(36\) 0 0
\(37\) 6.21110 1.02110 0.510549 0.859848i \(-0.329442\pi\)
0.510549 + 0.859848i \(0.329442\pi\)
\(38\) 0 0
\(39\) −4.30278 −0.688996
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) 0 0
\(43\) −2.90833 −0.443516 −0.221758 0.975102i \(-0.571179\pi\)
−0.221758 + 0.975102i \(0.571179\pi\)
\(44\) 0 0
\(45\) −1.30278 −0.194206
\(46\) 0 0
\(47\) 3.39445 0.495131 0.247566 0.968871i \(-0.420369\pi\)
0.247566 + 0.968871i \(0.420369\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 2.09167 0.287313 0.143657 0.989628i \(-0.454114\pi\)
0.143657 + 0.989628i \(0.454114\pi\)
\(54\) 0 0
\(55\) 4.69722 0.633374
\(56\) 0 0
\(57\) 0.394449 0.0522460
\(58\) 0 0
\(59\) −0.0916731 −0.0119348 −0.00596741 0.999982i \(-0.501899\pi\)
−0.00596741 + 0.999982i \(0.501899\pi\)
\(60\) 0 0
\(61\) −9.51388 −1.21813 −0.609064 0.793121i \(-0.708455\pi\)
−0.609064 + 0.793121i \(0.708455\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 5.60555 0.695283
\(66\) 0 0
\(67\) 15.1194 1.84713 0.923566 0.383439i \(-0.125260\pi\)
0.923566 + 0.383439i \(0.125260\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.09167 0.129558 0.0647789 0.997900i \(-0.479366\pi\)
0.0647789 + 0.997900i \(0.479366\pi\)
\(72\) 0 0
\(73\) −10.8167 −1.26599 −0.632997 0.774154i \(-0.718175\pi\)
−0.632997 + 0.774154i \(0.718175\pi\)
\(74\) 0 0
\(75\) −3.30278 −0.381372
\(76\) 0 0
\(77\) 3.60555 0.410891
\(78\) 0 0
\(79\) 13.6056 1.53074 0.765372 0.643588i \(-0.222555\pi\)
0.765372 + 0.643588i \(0.222555\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.60555 0.834818 0.417409 0.908719i \(-0.362938\pi\)
0.417409 + 0.908719i \(0.362938\pi\)
\(84\) 0 0
\(85\) 3.90833 0.423918
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −0.302776 −0.0320942 −0.0160471 0.999871i \(-0.505108\pi\)
−0.0160471 + 0.999871i \(0.505108\pi\)
\(90\) 0 0
\(91\) 4.30278 0.451053
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −0.513878 −0.0527228
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) −3.60555 −0.362372
\(100\) 0 0
\(101\) 8.51388 0.847163 0.423581 0.905858i \(-0.360773\pi\)
0.423581 + 0.905858i \(0.360773\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 1.30278 0.127138
\(106\) 0 0
\(107\) 5.51388 0.533047 0.266523 0.963828i \(-0.414125\pi\)
0.266523 + 0.963828i \(0.414125\pi\)
\(108\) 0 0
\(109\) −0.0916731 −0.00878069 −0.00439034 0.999990i \(-0.501397\pi\)
−0.00439034 + 0.999990i \(0.501397\pi\)
\(110\) 0 0
\(111\) 6.21110 0.589532
\(112\) 0 0
\(113\) −14.5139 −1.36535 −0.682675 0.730722i \(-0.739184\pi\)
−0.682675 + 0.730722i \(0.739184\pi\)
\(114\) 0 0
\(115\) −1.30278 −0.121484
\(116\) 0 0
\(117\) −4.30278 −0.397792
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 1.60555 0.144768
\(124\) 0 0
\(125\) 10.8167 0.967471
\(126\) 0 0
\(127\) 0.302776 0.0268670 0.0134335 0.999910i \(-0.495724\pi\)
0.0134335 + 0.999910i \(0.495724\pi\)
\(128\) 0 0
\(129\) −2.90833 −0.256064
\(130\) 0 0
\(131\) 16.8167 1.46928 0.734639 0.678458i \(-0.237352\pi\)
0.734639 + 0.678458i \(0.237352\pi\)
\(132\) 0 0
\(133\) −0.394449 −0.0342030
\(134\) 0 0
\(135\) −1.30278 −0.112125
\(136\) 0 0
\(137\) 20.0278 1.71109 0.855543 0.517731i \(-0.173223\pi\)
0.855543 + 0.517731i \(0.173223\pi\)
\(138\) 0 0
\(139\) 6.69722 0.568051 0.284026 0.958817i \(-0.408330\pi\)
0.284026 + 0.958817i \(0.408330\pi\)
\(140\) 0 0
\(141\) 3.39445 0.285864
\(142\) 0 0
\(143\) 15.5139 1.29734
\(144\) 0 0
\(145\) −1.30278 −0.108190
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −1.81665 −0.148826 −0.0744130 0.997228i \(-0.523708\pi\)
−0.0744130 + 0.997228i \(0.523708\pi\)
\(150\) 0 0
\(151\) −0.605551 −0.0492791 −0.0246395 0.999696i \(-0.507844\pi\)
−0.0246395 + 0.999696i \(0.507844\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 1.30278 0.104641
\(156\) 0 0
\(157\) −13.0278 −1.03973 −0.519864 0.854249i \(-0.674017\pi\)
−0.519864 + 0.854249i \(0.674017\pi\)
\(158\) 0 0
\(159\) 2.09167 0.165880
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 5.48612 0.429706 0.214853 0.976646i \(-0.431073\pi\)
0.214853 + 0.976646i \(0.431073\pi\)
\(164\) 0 0
\(165\) 4.69722 0.365678
\(166\) 0 0
\(167\) −4.81665 −0.372724 −0.186362 0.982481i \(-0.559670\pi\)
−0.186362 + 0.982481i \(0.559670\pi\)
\(168\) 0 0
\(169\) 5.51388 0.424144
\(170\) 0 0
\(171\) 0.394449 0.0301642
\(172\) 0 0
\(173\) 18.0278 1.37062 0.685312 0.728249i \(-0.259666\pi\)
0.685312 + 0.728249i \(0.259666\pi\)
\(174\) 0 0
\(175\) 3.30278 0.249666
\(176\) 0 0
\(177\) −0.0916731 −0.00689057
\(178\) 0 0
\(179\) −9.90833 −0.740583 −0.370292 0.928916i \(-0.620742\pi\)
−0.370292 + 0.928916i \(0.620742\pi\)
\(180\) 0 0
\(181\) 19.4222 1.44364 0.721821 0.692080i \(-0.243306\pi\)
0.721821 + 0.692080i \(0.243306\pi\)
\(182\) 0 0
\(183\) −9.51388 −0.703286
\(184\) 0 0
\(185\) −8.09167 −0.594912
\(186\) 0 0
\(187\) 10.8167 0.790992
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 15.0278 1.08737 0.543685 0.839289i \(-0.317029\pi\)
0.543685 + 0.839289i \(0.317029\pi\)
\(192\) 0 0
\(193\) −11.8167 −0.850581 −0.425291 0.905057i \(-0.639828\pi\)
−0.425291 + 0.905057i \(0.639828\pi\)
\(194\) 0 0
\(195\) 5.60555 0.401422
\(196\) 0 0
\(197\) 19.6972 1.40337 0.701685 0.712488i \(-0.252432\pi\)
0.701685 + 0.712488i \(0.252432\pi\)
\(198\) 0 0
\(199\) −23.3028 −1.65189 −0.825945 0.563751i \(-0.809358\pi\)
−0.825945 + 0.563751i \(0.809358\pi\)
\(200\) 0 0
\(201\) 15.1194 1.06644
\(202\) 0 0
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) −2.09167 −0.146089
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −1.42221 −0.0983760
\(210\) 0 0
\(211\) 6.39445 0.440212 0.220106 0.975476i \(-0.429360\pi\)
0.220106 + 0.975476i \(0.429360\pi\)
\(212\) 0 0
\(213\) 1.09167 0.0748002
\(214\) 0 0
\(215\) 3.78890 0.258401
\(216\) 0 0
\(217\) 1.00000 0.0678844
\(218\) 0 0
\(219\) −10.8167 −0.730922
\(220\) 0 0
\(221\) 12.9083 0.868308
\(222\) 0 0
\(223\) 9.48612 0.635238 0.317619 0.948218i \(-0.397117\pi\)
0.317619 + 0.948218i \(0.397117\pi\)
\(224\) 0 0
\(225\) −3.30278 −0.220185
\(226\) 0 0
\(227\) −23.3028 −1.54666 −0.773330 0.634004i \(-0.781410\pi\)
−0.773330 + 0.634004i \(0.781410\pi\)
\(228\) 0 0
\(229\) −15.3028 −1.01124 −0.505618 0.862757i \(-0.668735\pi\)
−0.505618 + 0.862757i \(0.668735\pi\)
\(230\) 0 0
\(231\) 3.60555 0.237228
\(232\) 0 0
\(233\) −18.3305 −1.20087 −0.600437 0.799672i \(-0.705006\pi\)
−0.600437 + 0.799672i \(0.705006\pi\)
\(234\) 0 0
\(235\) −4.42221 −0.288473
\(236\) 0 0
\(237\) 13.6056 0.883776
\(238\) 0 0
\(239\) −13.3028 −0.860485 −0.430243 0.902713i \(-0.641572\pi\)
−0.430243 + 0.902713i \(0.641572\pi\)
\(240\) 0 0
\(241\) 20.2111 1.30191 0.650956 0.759116i \(-0.274368\pi\)
0.650956 + 0.759116i \(0.274368\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.30278 −0.0832313
\(246\) 0 0
\(247\) −1.69722 −0.107992
\(248\) 0 0
\(249\) 7.60555 0.481982
\(250\) 0 0
\(251\) 10.6056 0.669416 0.334708 0.942322i \(-0.391362\pi\)
0.334708 + 0.942322i \(0.391362\pi\)
\(252\) 0 0
\(253\) −3.60555 −0.226679
\(254\) 0 0
\(255\) 3.90833 0.244749
\(256\) 0 0
\(257\) −23.0278 −1.43643 −0.718216 0.695820i \(-0.755041\pi\)
−0.718216 + 0.695820i \(0.755041\pi\)
\(258\) 0 0
\(259\) −6.21110 −0.385939
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 24.6333 1.51895 0.759477 0.650534i \(-0.225455\pi\)
0.759477 + 0.650534i \(0.225455\pi\)
\(264\) 0 0
\(265\) −2.72498 −0.167394
\(266\) 0 0
\(267\) −0.302776 −0.0185296
\(268\) 0 0
\(269\) 24.7250 1.50751 0.753754 0.657156i \(-0.228241\pi\)
0.753754 + 0.657156i \(0.228241\pi\)
\(270\) 0 0
\(271\) −10.8167 −0.657065 −0.328532 0.944493i \(-0.606554\pi\)
−0.328532 + 0.944493i \(0.606554\pi\)
\(272\) 0 0
\(273\) 4.30278 0.260416
\(274\) 0 0
\(275\) 11.9083 0.718099
\(276\) 0 0
\(277\) 23.3305 1.40180 0.700898 0.713262i \(-0.252783\pi\)
0.700898 + 0.713262i \(0.252783\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −3.81665 −0.227682 −0.113841 0.993499i \(-0.536316\pi\)
−0.113841 + 0.993499i \(0.536316\pi\)
\(282\) 0 0
\(283\) −9.30278 −0.552993 −0.276496 0.961015i \(-0.589173\pi\)
−0.276496 + 0.961015i \(0.589173\pi\)
\(284\) 0 0
\(285\) −0.513878 −0.0304395
\(286\) 0 0
\(287\) −1.60555 −0.0947727
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) 0 0
\(293\) 1.21110 0.0707534 0.0353767 0.999374i \(-0.488737\pi\)
0.0353767 + 0.999374i \(0.488737\pi\)
\(294\) 0 0
\(295\) 0.119429 0.00695345
\(296\) 0 0
\(297\) −3.60555 −0.209215
\(298\) 0 0
\(299\) −4.30278 −0.248836
\(300\) 0 0
\(301\) 2.90833 0.167633
\(302\) 0 0
\(303\) 8.51388 0.489110
\(304\) 0 0
\(305\) 12.3944 0.709704
\(306\) 0 0
\(307\) 15.4222 0.880192 0.440096 0.897951i \(-0.354944\pi\)
0.440096 + 0.897951i \(0.354944\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 4.90833 0.278326 0.139163 0.990270i \(-0.455559\pi\)
0.139163 + 0.990270i \(0.455559\pi\)
\(312\) 0 0
\(313\) −10.7889 −0.609825 −0.304912 0.952380i \(-0.598627\pi\)
−0.304912 + 0.952380i \(0.598627\pi\)
\(314\) 0 0
\(315\) 1.30278 0.0734031
\(316\) 0 0
\(317\) −1.30278 −0.0731712 −0.0365856 0.999331i \(-0.511648\pi\)
−0.0365856 + 0.999331i \(0.511648\pi\)
\(318\) 0 0
\(319\) −3.60555 −0.201872
\(320\) 0 0
\(321\) 5.51388 0.307755
\(322\) 0 0
\(323\) −1.18335 −0.0658431
\(324\) 0 0
\(325\) 14.2111 0.788290
\(326\) 0 0
\(327\) −0.0916731 −0.00506953
\(328\) 0 0
\(329\) −3.39445 −0.187142
\(330\) 0 0
\(331\) −4.78890 −0.263222 −0.131611 0.991301i \(-0.542015\pi\)
−0.131611 + 0.991301i \(0.542015\pi\)
\(332\) 0 0
\(333\) 6.21110 0.340366
\(334\) 0 0
\(335\) −19.6972 −1.07617
\(336\) 0 0
\(337\) −5.51388 −0.300360 −0.150180 0.988659i \(-0.547985\pi\)
−0.150180 + 0.988659i \(0.547985\pi\)
\(338\) 0 0
\(339\) −14.5139 −0.788286
\(340\) 0 0
\(341\) 3.60555 0.195252
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.30278 −0.0701391
\(346\) 0 0
\(347\) −7.18335 −0.385622 −0.192811 0.981236i \(-0.561760\pi\)
−0.192811 + 0.981236i \(0.561760\pi\)
\(348\) 0 0
\(349\) −22.5139 −1.20514 −0.602570 0.798066i \(-0.705857\pi\)
−0.602570 + 0.798066i \(0.705857\pi\)
\(350\) 0 0
\(351\) −4.30278 −0.229665
\(352\) 0 0
\(353\) 4.39445 0.233893 0.116946 0.993138i \(-0.462689\pi\)
0.116946 + 0.993138i \(0.462689\pi\)
\(354\) 0 0
\(355\) −1.42221 −0.0754828
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 21.9083 1.15628 0.578139 0.815939i \(-0.303779\pi\)
0.578139 + 0.815939i \(0.303779\pi\)
\(360\) 0 0
\(361\) −18.8444 −0.991811
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 14.0917 0.737592
\(366\) 0 0
\(367\) −2.72498 −0.142243 −0.0711214 0.997468i \(-0.522658\pi\)
−0.0711214 + 0.997468i \(0.522658\pi\)
\(368\) 0 0
\(369\) 1.60555 0.0835817
\(370\) 0 0
\(371\) −2.09167 −0.108594
\(372\) 0 0
\(373\) 2.57779 0.133473 0.0667366 0.997771i \(-0.478741\pi\)
0.0667366 + 0.997771i \(0.478741\pi\)
\(374\) 0 0
\(375\) 10.8167 0.558570
\(376\) 0 0
\(377\) −4.30278 −0.221604
\(378\) 0 0
\(379\) −9.21110 −0.473143 −0.236571 0.971614i \(-0.576024\pi\)
−0.236571 + 0.971614i \(0.576024\pi\)
\(380\) 0 0
\(381\) 0.302776 0.0155117
\(382\) 0 0
\(383\) −6.02776 −0.308004 −0.154002 0.988071i \(-0.549216\pi\)
−0.154002 + 0.988071i \(0.549216\pi\)
\(384\) 0 0
\(385\) −4.69722 −0.239393
\(386\) 0 0
\(387\) −2.90833 −0.147839
\(388\) 0 0
\(389\) 30.2111 1.53176 0.765882 0.642981i \(-0.222303\pi\)
0.765882 + 0.642981i \(0.222303\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) 16.8167 0.848288
\(394\) 0 0
\(395\) −17.7250 −0.891841
\(396\) 0 0
\(397\) −3.39445 −0.170362 −0.0851812 0.996365i \(-0.527147\pi\)
−0.0851812 + 0.996365i \(0.527147\pi\)
\(398\) 0 0
\(399\) −0.394449 −0.0197471
\(400\) 0 0
\(401\) 13.4222 0.670273 0.335136 0.942170i \(-0.391218\pi\)
0.335136 + 0.942170i \(0.391218\pi\)
\(402\) 0 0
\(403\) 4.30278 0.214337
\(404\) 0 0
\(405\) −1.30278 −0.0647354
\(406\) 0 0
\(407\) −22.3944 −1.11005
\(408\) 0 0
\(409\) 7.42221 0.367004 0.183502 0.983019i \(-0.441257\pi\)
0.183502 + 0.983019i \(0.441257\pi\)
\(410\) 0 0
\(411\) 20.0278 0.987896
\(412\) 0 0
\(413\) 0.0916731 0.00451094
\(414\) 0 0
\(415\) −9.90833 −0.486381
\(416\) 0 0
\(417\) 6.69722 0.327964
\(418\) 0 0
\(419\) 8.51388 0.415930 0.207965 0.978136i \(-0.433316\pi\)
0.207965 + 0.978136i \(0.433316\pi\)
\(420\) 0 0
\(421\) 1.09167 0.0532049 0.0266024 0.999646i \(-0.491531\pi\)
0.0266024 + 0.999646i \(0.491531\pi\)
\(422\) 0 0
\(423\) 3.39445 0.165044
\(424\) 0 0
\(425\) 9.90833 0.480624
\(426\) 0 0
\(427\) 9.51388 0.460409
\(428\) 0 0
\(429\) 15.5139 0.749017
\(430\) 0 0
\(431\) 12.9083 0.621772 0.310886 0.950447i \(-0.399374\pi\)
0.310886 + 0.950447i \(0.399374\pi\)
\(432\) 0 0
\(433\) 1.02776 0.0493908 0.0246954 0.999695i \(-0.492138\pi\)
0.0246954 + 0.999695i \(0.492138\pi\)
\(434\) 0 0
\(435\) −1.30278 −0.0624633
\(436\) 0 0
\(437\) 0.394449 0.0188690
\(438\) 0 0
\(439\) 10.6333 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −19.6333 −0.932807 −0.466403 0.884572i \(-0.654450\pi\)
−0.466403 + 0.884572i \(0.654450\pi\)
\(444\) 0 0
\(445\) 0.394449 0.0186987
\(446\) 0 0
\(447\) −1.81665 −0.0859248
\(448\) 0 0
\(449\) 33.1194 1.56300 0.781501 0.623904i \(-0.214454\pi\)
0.781501 + 0.623904i \(0.214454\pi\)
\(450\) 0 0
\(451\) −5.78890 −0.272589
\(452\) 0 0
\(453\) −0.605551 −0.0284513
\(454\) 0 0
\(455\) −5.60555 −0.262792
\(456\) 0 0
\(457\) 31.7250 1.48403 0.742016 0.670382i \(-0.233870\pi\)
0.742016 + 0.670382i \(0.233870\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −25.5416 −1.18959 −0.594796 0.803876i \(-0.702767\pi\)
−0.594796 + 0.803876i \(0.702767\pi\)
\(462\) 0 0
\(463\) −11.7889 −0.547877 −0.273938 0.961747i \(-0.588326\pi\)
−0.273938 + 0.961747i \(0.588326\pi\)
\(464\) 0 0
\(465\) 1.30278 0.0604148
\(466\) 0 0
\(467\) 12.3944 0.573547 0.286773 0.957998i \(-0.407417\pi\)
0.286773 + 0.957998i \(0.407417\pi\)
\(468\) 0 0
\(469\) −15.1194 −0.698150
\(470\) 0 0
\(471\) −13.0278 −0.600287
\(472\) 0 0
\(473\) 10.4861 0.482152
\(474\) 0 0
\(475\) −1.30278 −0.0597754
\(476\) 0 0
\(477\) 2.09167 0.0957711
\(478\) 0 0
\(479\) −13.4222 −0.613276 −0.306638 0.951826i \(-0.599204\pi\)
−0.306638 + 0.951826i \(0.599204\pi\)
\(480\) 0 0
\(481\) −26.7250 −1.21855
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 14.3305 0.650716
\(486\) 0 0
\(487\) 34.2111 1.55025 0.775127 0.631806i \(-0.217686\pi\)
0.775127 + 0.631806i \(0.217686\pi\)
\(488\) 0 0
\(489\) 5.48612 0.248091
\(490\) 0 0
\(491\) −27.3305 −1.23341 −0.616705 0.787194i \(-0.711533\pi\)
−0.616705 + 0.787194i \(0.711533\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) 4.69722 0.211125
\(496\) 0 0
\(497\) −1.09167 −0.0489682
\(498\) 0 0
\(499\) 17.5416 0.785271 0.392636 0.919694i \(-0.371563\pi\)
0.392636 + 0.919694i \(0.371563\pi\)
\(500\) 0 0
\(501\) −4.81665 −0.215192
\(502\) 0 0
\(503\) 12.6972 0.566141 0.283071 0.959099i \(-0.408647\pi\)
0.283071 + 0.959099i \(0.408647\pi\)
\(504\) 0 0
\(505\) −11.0917 −0.493573
\(506\) 0 0
\(507\) 5.51388 0.244880
\(508\) 0 0
\(509\) −8.60555 −0.381434 −0.190717 0.981645i \(-0.561081\pi\)
−0.190717 + 0.981645i \(0.561081\pi\)
\(510\) 0 0
\(511\) 10.8167 0.478501
\(512\) 0 0
\(513\) 0.394449 0.0174153
\(514\) 0 0
\(515\) −20.8444 −0.918514
\(516\) 0 0
\(517\) −12.2389 −0.538264
\(518\) 0 0
\(519\) 18.0278 0.791331
\(520\) 0 0
\(521\) 39.2111 1.71787 0.858935 0.512085i \(-0.171127\pi\)
0.858935 + 0.512085i \(0.171127\pi\)
\(522\) 0 0
\(523\) −2.00000 −0.0874539 −0.0437269 0.999044i \(-0.513923\pi\)
−0.0437269 + 0.999044i \(0.513923\pi\)
\(524\) 0 0
\(525\) 3.30278 0.144145
\(526\) 0 0
\(527\) 3.00000 0.130682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −0.0916731 −0.00397827
\(532\) 0 0
\(533\) −6.90833 −0.299233
\(534\) 0 0
\(535\) −7.18335 −0.310563
\(536\) 0 0
\(537\) −9.90833 −0.427576
\(538\) 0 0
\(539\) −3.60555 −0.155302
\(540\) 0 0
\(541\) −8.60555 −0.369982 −0.184991 0.982740i \(-0.559226\pi\)
−0.184991 + 0.982740i \(0.559226\pi\)
\(542\) 0 0
\(543\) 19.4222 0.833487
\(544\) 0 0
\(545\) 0.119429 0.00511580
\(546\) 0 0
\(547\) −16.5139 −0.706082 −0.353041 0.935608i \(-0.614852\pi\)
−0.353041 + 0.935608i \(0.614852\pi\)
\(548\) 0 0
\(549\) −9.51388 −0.406042
\(550\) 0 0
\(551\) 0.394449 0.0168041
\(552\) 0 0
\(553\) −13.6056 −0.578567
\(554\) 0 0
\(555\) −8.09167 −0.343472
\(556\) 0 0
\(557\) 35.4500 1.50206 0.751032 0.660266i \(-0.229557\pi\)
0.751032 + 0.660266i \(0.229557\pi\)
\(558\) 0 0
\(559\) 12.5139 0.529281
\(560\) 0 0
\(561\) 10.8167 0.456679
\(562\) 0 0
\(563\) 18.1194 0.763643 0.381821 0.924236i \(-0.375297\pi\)
0.381821 + 0.924236i \(0.375297\pi\)
\(564\) 0 0
\(565\) 18.9083 0.795479
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 0.816654 0.0341759 0.0170879 0.999854i \(-0.494560\pi\)
0.0170879 + 0.999854i \(0.494560\pi\)
\(572\) 0 0
\(573\) 15.0278 0.627794
\(574\) 0 0
\(575\) −3.30278 −0.137735
\(576\) 0 0
\(577\) 25.6333 1.06713 0.533564 0.845760i \(-0.320852\pi\)
0.533564 + 0.845760i \(0.320852\pi\)
\(578\) 0 0
\(579\) −11.8167 −0.491083
\(580\) 0 0
\(581\) −7.60555 −0.315531
\(582\) 0 0
\(583\) −7.54163 −0.312343
\(584\) 0 0
\(585\) 5.60555 0.231761
\(586\) 0 0
\(587\) −2.30278 −0.0950457 −0.0475229 0.998870i \(-0.515133\pi\)
−0.0475229 + 0.998870i \(0.515133\pi\)
\(588\) 0 0
\(589\) −0.394449 −0.0162530
\(590\) 0 0
\(591\) 19.6972 0.810236
\(592\) 0 0
\(593\) −25.0278 −1.02777 −0.513883 0.857860i \(-0.671794\pi\)
−0.513883 + 0.857860i \(0.671794\pi\)
\(594\) 0 0
\(595\) −3.90833 −0.160226
\(596\) 0 0
\(597\) −23.3028 −0.953719
\(598\) 0 0
\(599\) 14.7250 0.601646 0.300823 0.953680i \(-0.402739\pi\)
0.300823 + 0.953680i \(0.402739\pi\)
\(600\) 0 0
\(601\) 10.9361 0.446092 0.223046 0.974808i \(-0.428400\pi\)
0.223046 + 0.974808i \(0.428400\pi\)
\(602\) 0 0
\(603\) 15.1194 0.615711
\(604\) 0 0
\(605\) −2.60555 −0.105931
\(606\) 0 0
\(607\) −28.1194 −1.14133 −0.570666 0.821182i \(-0.693315\pi\)
−0.570666 + 0.821182i \(0.693315\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −14.6056 −0.590877
\(612\) 0 0
\(613\) 20.4500 0.825966 0.412983 0.910739i \(-0.364487\pi\)
0.412983 + 0.910739i \(0.364487\pi\)
\(614\) 0 0
\(615\) −2.09167 −0.0843444
\(616\) 0 0
\(617\) 18.1194 0.729461 0.364730 0.931113i \(-0.381161\pi\)
0.364730 + 0.931113i \(0.381161\pi\)
\(618\) 0 0
\(619\) 7.33053 0.294639 0.147319 0.989089i \(-0.452935\pi\)
0.147319 + 0.989089i \(0.452935\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 0.302776 0.0121304
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) 0 0
\(627\) −1.42221 −0.0567974
\(628\) 0 0
\(629\) −18.6333 −0.742959
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 6.39445 0.254157
\(634\) 0 0
\(635\) −0.394449 −0.0156532
\(636\) 0 0
\(637\) −4.30278 −0.170482
\(638\) 0 0
\(639\) 1.09167 0.0431859
\(640\) 0 0
\(641\) −39.1472 −1.54622 −0.773110 0.634271i \(-0.781300\pi\)
−0.773110 + 0.634271i \(0.781300\pi\)
\(642\) 0 0
\(643\) 17.2750 0.681260 0.340630 0.940197i \(-0.389360\pi\)
0.340630 + 0.940197i \(0.389360\pi\)
\(644\) 0 0
\(645\) 3.78890 0.149188
\(646\) 0 0
\(647\) 33.9083 1.33307 0.666537 0.745472i \(-0.267776\pi\)
0.666537 + 0.745472i \(0.267776\pi\)
\(648\) 0 0
\(649\) 0.330532 0.0129745
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) 0 0
\(653\) 10.5139 0.411440 0.205720 0.978611i \(-0.434046\pi\)
0.205720 + 0.978611i \(0.434046\pi\)
\(654\) 0 0
\(655\) −21.9083 −0.856029
\(656\) 0 0
\(657\) −10.8167 −0.421998
\(658\) 0 0
\(659\) −27.6056 −1.07536 −0.537680 0.843149i \(-0.680699\pi\)
−0.537680 + 0.843149i \(0.680699\pi\)
\(660\) 0 0
\(661\) 6.21110 0.241584 0.120792 0.992678i \(-0.461457\pi\)
0.120792 + 0.992678i \(0.461457\pi\)
\(662\) 0 0
\(663\) 12.9083 0.501318
\(664\) 0 0
\(665\) 0.513878 0.0199273
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 9.48612 0.366755
\(670\) 0 0
\(671\) 34.3028 1.32424
\(672\) 0 0
\(673\) −33.4222 −1.28833 −0.644166 0.764886i \(-0.722795\pi\)
−0.644166 + 0.764886i \(0.722795\pi\)
\(674\) 0 0
\(675\) −3.30278 −0.127124
\(676\) 0 0
\(677\) 34.3028 1.31836 0.659181 0.751984i \(-0.270903\pi\)
0.659181 + 0.751984i \(0.270903\pi\)
\(678\) 0 0
\(679\) 11.0000 0.422141
\(680\) 0 0
\(681\) −23.3028 −0.892964
\(682\) 0 0
\(683\) 40.3944 1.54565 0.772825 0.634619i \(-0.218843\pi\)
0.772825 + 0.634619i \(0.218843\pi\)
\(684\) 0 0
\(685\) −26.0917 −0.996912
\(686\) 0 0
\(687\) −15.3028 −0.583837
\(688\) 0 0
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −37.9361 −1.44316 −0.721578 0.692333i \(-0.756583\pi\)
−0.721578 + 0.692333i \(0.756583\pi\)
\(692\) 0 0
\(693\) 3.60555 0.136964
\(694\) 0 0
\(695\) −8.72498 −0.330957
\(696\) 0 0
\(697\) −4.81665 −0.182444
\(698\) 0 0
\(699\) −18.3305 −0.693324
\(700\) 0 0
\(701\) 27.1194 1.02429 0.512143 0.858900i \(-0.328852\pi\)
0.512143 + 0.858900i \(0.328852\pi\)
\(702\) 0 0
\(703\) 2.44996 0.0924020
\(704\) 0 0
\(705\) −4.42221 −0.166550
\(706\) 0 0
\(707\) −8.51388 −0.320197
\(708\) 0 0
\(709\) 16.9083 0.635006 0.317503 0.948257i \(-0.397156\pi\)
0.317503 + 0.948257i \(0.397156\pi\)
\(710\) 0 0
\(711\) 13.6056 0.510248
\(712\) 0 0
\(713\) −1.00000 −0.0374503
\(714\) 0 0
\(715\) −20.2111 −0.755852
\(716\) 0 0
\(717\) −13.3028 −0.496801
\(718\) 0 0
\(719\) 25.0000 0.932343 0.466171 0.884694i \(-0.345633\pi\)
0.466171 + 0.884694i \(0.345633\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 20.2111 0.751659
\(724\) 0 0
\(725\) −3.30278 −0.122662
\(726\) 0 0
\(727\) 3.60555 0.133722 0.0668612 0.997762i \(-0.478702\pi\)
0.0668612 + 0.997762i \(0.478702\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.72498 0.322705
\(732\) 0 0
\(733\) −13.0278 −0.481191 −0.240596 0.970625i \(-0.577343\pi\)
−0.240596 + 0.970625i \(0.577343\pi\)
\(734\) 0 0
\(735\) −1.30278 −0.0480536
\(736\) 0 0
\(737\) −54.5139 −2.00804
\(738\) 0 0
\(739\) −13.9722 −0.513977 −0.256989 0.966414i \(-0.582730\pi\)
−0.256989 + 0.966414i \(0.582730\pi\)
\(740\) 0 0
\(741\) −1.69722 −0.0623491
\(742\) 0 0
\(743\) −25.7250 −0.943758 −0.471879 0.881663i \(-0.656424\pi\)
−0.471879 + 0.881663i \(0.656424\pi\)
\(744\) 0 0
\(745\) 2.36669 0.0867089
\(746\) 0 0
\(747\) 7.60555 0.278273
\(748\) 0 0
\(749\) −5.51388 −0.201473
\(750\) 0 0
\(751\) −17.4861 −0.638078 −0.319039 0.947742i \(-0.603360\pi\)
−0.319039 + 0.947742i \(0.603360\pi\)
\(752\) 0 0
\(753\) 10.6056 0.386488
\(754\) 0 0
\(755\) 0.788897 0.0287109
\(756\) 0 0
\(757\) −11.4222 −0.415147 −0.207574 0.978219i \(-0.566557\pi\)
−0.207574 + 0.978219i \(0.566557\pi\)
\(758\) 0 0
\(759\) −3.60555 −0.130873
\(760\) 0 0
\(761\) 1.21110 0.0439024 0.0219512 0.999759i \(-0.493012\pi\)
0.0219512 + 0.999759i \(0.493012\pi\)
\(762\) 0 0
\(763\) 0.0916731 0.00331879
\(764\) 0 0
\(765\) 3.90833 0.141306
\(766\) 0 0
\(767\) 0.394449 0.0142427
\(768\) 0 0
\(769\) 36.6611 1.32203 0.661016 0.750372i \(-0.270126\pi\)
0.661016 + 0.750372i \(0.270126\pi\)
\(770\) 0 0
\(771\) −23.0278 −0.829324
\(772\) 0 0
\(773\) 12.0278 0.432608 0.216304 0.976326i \(-0.430600\pi\)
0.216304 + 0.976326i \(0.430600\pi\)
\(774\) 0 0
\(775\) 3.30278 0.118639
\(776\) 0 0
\(777\) −6.21110 −0.222822
\(778\) 0 0
\(779\) 0.633308 0.0226906
\(780\) 0 0
\(781\) −3.93608 −0.140844
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 16.9722 0.605765
\(786\) 0 0
\(787\) −41.3583 −1.47426 −0.737132 0.675749i \(-0.763820\pi\)
−0.737132 + 0.675749i \(0.763820\pi\)
\(788\) 0 0
\(789\) 24.6333 0.876969
\(790\) 0 0
\(791\) 14.5139 0.516054
\(792\) 0 0
\(793\) 40.9361 1.45368
\(794\) 0 0
\(795\) −2.72498 −0.0966451
\(796\) 0 0
\(797\) −7.39445 −0.261925 −0.130962 0.991387i \(-0.541807\pi\)
−0.130962 + 0.991387i \(0.541807\pi\)
\(798\) 0 0
\(799\) −10.1833 −0.360261
\(800\) 0 0
\(801\) −0.302776 −0.0106981
\(802\) 0 0
\(803\) 39.0000 1.37628
\(804\) 0 0
\(805\) 1.30278 0.0459168
\(806\) 0 0
\(807\) 24.7250 0.870361
\(808\) 0 0
\(809\) −46.5416 −1.63632 −0.818158 0.574993i \(-0.805005\pi\)
−0.818158 + 0.574993i \(0.805005\pi\)
\(810\) 0 0
\(811\) 25.2389 0.886256 0.443128 0.896458i \(-0.353869\pi\)
0.443128 + 0.896458i \(0.353869\pi\)
\(812\) 0 0
\(813\) −10.8167 −0.379357
\(814\) 0 0
\(815\) −7.14719 −0.250355
\(816\) 0 0
\(817\) −1.14719 −0.0401350
\(818\) 0 0
\(819\) 4.30278 0.150351
\(820\) 0 0
\(821\) −13.8167 −0.482205 −0.241102 0.970500i \(-0.577509\pi\)
−0.241102 + 0.970500i \(0.577509\pi\)
\(822\) 0 0
\(823\) 11.6695 0.406772 0.203386 0.979099i \(-0.434805\pi\)
0.203386 + 0.979099i \(0.434805\pi\)
\(824\) 0 0
\(825\) 11.9083 0.414595
\(826\) 0 0
\(827\) 19.5416 0.679529 0.339765 0.940511i \(-0.389653\pi\)
0.339765 + 0.940511i \(0.389653\pi\)
\(828\) 0 0
\(829\) −37.8444 −1.31439 −0.657195 0.753720i \(-0.728257\pi\)
−0.657195 + 0.753720i \(0.728257\pi\)
\(830\) 0 0
\(831\) 23.3305 0.809327
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 6.27502 0.217156
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −34.3028 −1.18426 −0.592132 0.805841i \(-0.701713\pi\)
−0.592132 + 0.805841i \(0.701713\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −3.81665 −0.131453
\(844\) 0 0
\(845\) −7.18335 −0.247115
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −9.30278 −0.319270
\(850\) 0 0
\(851\) 6.21110 0.212914
\(852\) 0 0
\(853\) 0.183346 0.00627765 0.00313883 0.999995i \(-0.499001\pi\)
0.00313883 + 0.999995i \(0.499001\pi\)
\(854\) 0 0
\(855\) −0.513878 −0.0175743
\(856\) 0 0
\(857\) −4.18335 −0.142900 −0.0714502 0.997444i \(-0.522763\pi\)
−0.0714502 + 0.997444i \(0.522763\pi\)
\(858\) 0 0
\(859\) −4.78890 −0.163395 −0.0816975 0.996657i \(-0.526034\pi\)
−0.0816975 + 0.996657i \(0.526034\pi\)
\(860\) 0 0
\(861\) −1.60555 −0.0547170
\(862\) 0 0
\(863\) 26.6056 0.905663 0.452832 0.891596i \(-0.350414\pi\)
0.452832 + 0.891596i \(0.350414\pi\)
\(864\) 0 0
\(865\) −23.4861 −0.798552
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −49.0555 −1.66409
\(870\) 0 0
\(871\) −65.0555 −2.20432
\(872\) 0 0
\(873\) −11.0000 −0.372294
\(874\) 0 0
\(875\) −10.8167 −0.365670
\(876\) 0 0
\(877\) −2.78890 −0.0941744 −0.0470872 0.998891i \(-0.514994\pi\)
−0.0470872 + 0.998891i \(0.514994\pi\)
\(878\) 0 0
\(879\) 1.21110 0.0408495
\(880\) 0 0
\(881\) −32.6611 −1.10038 −0.550190 0.835040i \(-0.685445\pi\)
−0.550190 + 0.835040i \(0.685445\pi\)
\(882\) 0 0
\(883\) 12.1194 0.407851 0.203926 0.978986i \(-0.434630\pi\)
0.203926 + 0.978986i \(0.434630\pi\)
\(884\) 0 0
\(885\) 0.119429 0.00401458
\(886\) 0 0
\(887\) −53.5139 −1.79682 −0.898410 0.439158i \(-0.855277\pi\)
−0.898410 + 0.439158i \(0.855277\pi\)
\(888\) 0 0
\(889\) −0.302776 −0.0101548
\(890\) 0 0
\(891\) −3.60555 −0.120791
\(892\) 0 0
\(893\) 1.33894 0.0448058
\(894\) 0 0
\(895\) 12.9083 0.431478
\(896\) 0 0
\(897\) −4.30278 −0.143665
\(898\) 0 0
\(899\) −1.00000 −0.0333519
\(900\) 0 0
\(901\) −6.27502 −0.209051
\(902\) 0 0
\(903\) 2.90833 0.0967831
\(904\) 0 0
\(905\) −25.3028 −0.841093
\(906\) 0 0
\(907\) 39.7250 1.31905 0.659523 0.751684i \(-0.270758\pi\)
0.659523 + 0.751684i \(0.270758\pi\)
\(908\) 0 0
\(909\) 8.51388 0.282388
\(910\) 0 0
\(911\) 7.39445 0.244989 0.122495 0.992469i \(-0.460911\pi\)
0.122495 + 0.992469i \(0.460911\pi\)
\(912\) 0 0
\(913\) −27.4222 −0.907543
\(914\) 0 0
\(915\) 12.3944 0.409748
\(916\) 0 0
\(917\) −16.8167 −0.555335
\(918\) 0 0
\(919\) 34.2666 1.13035 0.565176 0.824971i \(-0.308808\pi\)
0.565176 + 0.824971i \(0.308808\pi\)
\(920\) 0 0
\(921\) 15.4222 0.508179
\(922\) 0 0
\(923\) −4.69722 −0.154611
\(924\) 0 0
\(925\) −20.5139 −0.674492
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 3.90833 0.128228 0.0641140 0.997943i \(-0.479578\pi\)
0.0641140 + 0.997943i \(0.479578\pi\)
\(930\) 0 0
\(931\) 0.394449 0.0129275
\(932\) 0 0
\(933\) 4.90833 0.160691
\(934\) 0 0
\(935\) −14.0917 −0.460847
\(936\) 0 0
\(937\) −44.8722 −1.46591 −0.732955 0.680277i \(-0.761859\pi\)
−0.732955 + 0.680277i \(0.761859\pi\)
\(938\) 0 0
\(939\) −10.7889 −0.352082
\(940\) 0 0
\(941\) 8.84441 0.288320 0.144160 0.989554i \(-0.453952\pi\)
0.144160 + 0.989554i \(0.453952\pi\)
\(942\) 0 0
\(943\) 1.60555 0.0522839
\(944\) 0 0
\(945\) 1.30278 0.0423793
\(946\) 0 0
\(947\) 19.5778 0.636193 0.318096 0.948058i \(-0.396956\pi\)
0.318096 + 0.948058i \(0.396956\pi\)
\(948\) 0 0
\(949\) 46.5416 1.51081
\(950\) 0 0
\(951\) −1.30278 −0.0422454
\(952\) 0 0
\(953\) 7.33053 0.237459 0.118730 0.992927i \(-0.462118\pi\)
0.118730 + 0.992927i \(0.462118\pi\)
\(954\) 0 0
\(955\) −19.5778 −0.633523
\(956\) 0 0
\(957\) −3.60555 −0.116551
\(958\) 0 0
\(959\) −20.0278 −0.646730
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 5.51388 0.177682
\(964\) 0 0
\(965\) 15.3944 0.495565
\(966\) 0 0
\(967\) −49.6333 −1.59610 −0.798050 0.602592i \(-0.794135\pi\)
−0.798050 + 0.602592i \(0.794135\pi\)
\(968\) 0 0
\(969\) −1.18335 −0.0380146
\(970\) 0 0
\(971\) 54.7250 1.75621 0.878104 0.478470i \(-0.158808\pi\)
0.878104 + 0.478470i \(0.158808\pi\)
\(972\) 0 0
\(973\) −6.69722 −0.214703
\(974\) 0 0
\(975\) 14.2111 0.455120
\(976\) 0 0
\(977\) 28.4861 0.911352 0.455676 0.890146i \(-0.349398\pi\)
0.455676 + 0.890146i \(0.349398\pi\)
\(978\) 0 0
\(979\) 1.09167 0.0348900
\(980\) 0 0
\(981\) −0.0916731 −0.00292690
\(982\) 0 0
\(983\) 49.2389 1.57048 0.785238 0.619194i \(-0.212541\pi\)
0.785238 + 0.619194i \(0.212541\pi\)
\(984\) 0 0
\(985\) −25.6611 −0.817629
\(986\) 0 0
\(987\) −3.39445 −0.108046
\(988\) 0 0
\(989\) −2.90833 −0.0924794
\(990\) 0 0
\(991\) −12.3028 −0.390811 −0.195405 0.980723i \(-0.562602\pi\)
−0.195405 + 0.980723i \(0.562602\pi\)
\(992\) 0 0
\(993\) −4.78890 −0.151971
\(994\) 0 0
\(995\) 30.3583 0.962422
\(996\) 0 0
\(997\) 41.4222 1.31185 0.655927 0.754824i \(-0.272278\pi\)
0.655927 + 0.754824i \(0.272278\pi\)
\(998\) 0 0
\(999\) 6.21110 0.196511
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.bk.1.1 2
4.3 odd 2 1932.2.a.e.1.1 2
12.11 even 2 5796.2.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.e.1.1 2 4.3 odd 2
5796.2.a.k.1.2 2 12.11 even 2
7728.2.a.bk.1.1 2 1.1 even 1 trivial