Properties

Label 4140.2.a.q.1.2
Level $4140$
Weight $2$
Character 4140.1
Self dual yes
Analytic conductor $33.058$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.0580664368\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 4140.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.44949 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.44949 q^{7} -2.44949 q^{11} -0.449490 q^{13} -0.550510 q^{17} -6.44949 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.89898 q^{29} -7.00000 q^{31} +1.44949 q^{35} -8.34847 q^{37} +1.89898 q^{41} +0.898979 q^{43} -2.44949 q^{47} -4.89898 q^{49} -10.3485 q^{53} -2.44949 q^{55} -7.89898 q^{59} +9.34847 q^{61} -0.449490 q^{65} -3.44949 q^{67} -3.00000 q^{71} -5.34847 q^{73} -3.55051 q^{77} -4.00000 q^{79} -4.34847 q^{83} -0.550510 q^{85} +7.10102 q^{89} -0.651531 q^{91} -6.44949 q^{95} -5.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 8 q^{19} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 14 q^{31} - 2 q^{35} - 2 q^{37} - 6 q^{41} - 8 q^{43} - 6 q^{53} - 6 q^{59} + 4 q^{61} + 4 q^{65} - 2 q^{67} - 6 q^{71} + 4 q^{73} - 12 q^{77} - 8 q^{79} + 6 q^{83} - 6 q^{85} + 24 q^{89} - 16 q^{91} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.44949 0.547856 0.273928 0.961750i \(-0.411677\pi\)
0.273928 + 0.961750i \(0.411677\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) −0.449490 −0.124666 −0.0623330 0.998055i \(-0.519854\pi\)
−0.0623330 + 0.998055i \(0.519854\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.550510 −0.133518 −0.0667592 0.997769i \(-0.521266\pi\)
−0.0667592 + 0.997769i \(0.521266\pi\)
\(18\) 0 0
\(19\) −6.44949 −1.47961 −0.739807 0.672819i \(-0.765083\pi\)
−0.739807 + 0.672819i \(0.765083\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.89898 1.46680 0.733402 0.679795i \(-0.237931\pi\)
0.733402 + 0.679795i \(0.237931\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.44949 0.245008
\(36\) 0 0
\(37\) −8.34847 −1.37248 −0.686240 0.727375i \(-0.740740\pi\)
−0.686240 + 0.727375i \(0.740740\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.89898 0.296571 0.148285 0.988945i \(-0.452625\pi\)
0.148285 + 0.988945i \(0.452625\pi\)
\(42\) 0 0
\(43\) 0.898979 0.137093 0.0685465 0.997648i \(-0.478164\pi\)
0.0685465 + 0.997648i \(0.478164\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.44949 −0.357295 −0.178647 0.983913i \(-0.557172\pi\)
−0.178647 + 0.983913i \(0.557172\pi\)
\(48\) 0 0
\(49\) −4.89898 −0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.3485 −1.42147 −0.710736 0.703459i \(-0.751638\pi\)
−0.710736 + 0.703459i \(0.751638\pi\)
\(54\) 0 0
\(55\) −2.44949 −0.330289
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.89898 −1.02836 −0.514180 0.857682i \(-0.671904\pi\)
−0.514180 + 0.857682i \(0.671904\pi\)
\(60\) 0 0
\(61\) 9.34847 1.19695 0.598474 0.801142i \(-0.295774\pi\)
0.598474 + 0.801142i \(0.295774\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.449490 −0.0557523
\(66\) 0 0
\(67\) −3.44949 −0.421422 −0.210711 0.977548i \(-0.567578\pi\)
−0.210711 + 0.977548i \(0.567578\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −5.34847 −0.625991 −0.312995 0.949755i \(-0.601332\pi\)
−0.312995 + 0.949755i \(0.601332\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.55051 −0.404618
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.34847 −0.477307 −0.238653 0.971105i \(-0.576706\pi\)
−0.238653 + 0.971105i \(0.576706\pi\)
\(84\) 0 0
\(85\) −0.550510 −0.0597112
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.10102 0.752707 0.376353 0.926476i \(-0.377178\pi\)
0.376353 + 0.926476i \(0.377178\pi\)
\(90\) 0 0
\(91\) −0.651531 −0.0682990
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.44949 −0.661704
\(96\) 0 0
\(97\) −5.10102 −0.517930 −0.258965 0.965887i \(-0.583381\pi\)
−0.258965 + 0.965887i \(0.583381\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.797959 0.0793999 0.0396999 0.999212i \(-0.487360\pi\)
0.0396999 + 0.999212i \(0.487360\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.1464 1.94763 0.973814 0.227345i \(-0.0730044\pi\)
0.973814 + 0.227345i \(0.0730044\pi\)
\(108\) 0 0
\(109\) 3.34847 0.320725 0.160363 0.987058i \(-0.448734\pi\)
0.160363 + 0.987058i \(0.448734\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.65153 0.155363 0.0776815 0.996978i \(-0.475248\pi\)
0.0776815 + 0.996978i \(0.475248\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.797959 −0.0731488
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.3485 −1.00701 −0.503507 0.863991i \(-0.667957\pi\)
−0.503507 + 0.863991i \(0.667957\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.10102 −0.0961966 −0.0480983 0.998843i \(-0.515316\pi\)
−0.0480983 + 0.998843i \(0.515316\pi\)
\(132\) 0 0
\(133\) −9.34847 −0.810615
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.10102 0.606681 0.303341 0.952882i \(-0.401898\pi\)
0.303341 + 0.952882i \(0.401898\pi\)
\(138\) 0 0
\(139\) 7.69694 0.652846 0.326423 0.945224i \(-0.394157\pi\)
0.326423 + 0.945224i \(0.394157\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.10102 0.0920720
\(144\) 0 0
\(145\) 7.89898 0.655975
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.24745 0.511811 0.255905 0.966702i \(-0.417626\pi\)
0.255905 + 0.966702i \(0.417626\pi\)
\(150\) 0 0
\(151\) −0.202041 −0.0164419 −0.00822093 0.999966i \(-0.502617\pi\)
−0.00822093 + 0.999966i \(0.502617\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 0 0
\(157\) 0.348469 0.0278109 0.0139054 0.999903i \(-0.495574\pi\)
0.0139054 + 0.999903i \(0.495574\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.44949 −0.114236
\(162\) 0 0
\(163\) 4.69694 0.367893 0.183946 0.982936i \(-0.441113\pi\)
0.183946 + 0.982936i \(0.441113\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.2474 1.87632 0.938162 0.346197i \(-0.112527\pi\)
0.938162 + 0.346197i \(0.112527\pi\)
\(168\) 0 0
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.79796 −0.744925 −0.372463 0.928047i \(-0.621486\pi\)
−0.372463 + 0.928047i \(0.621486\pi\)
\(174\) 0 0
\(175\) 1.44949 0.109571
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −8.89898 −0.661456 −0.330728 0.943726i \(-0.607294\pi\)
−0.330728 + 0.943726i \(0.607294\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.34847 −0.613792
\(186\) 0 0
\(187\) 1.34847 0.0986098
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.4495 −1.47967 −0.739837 0.672787i \(-0.765097\pi\)
−0.739837 + 0.672787i \(0.765097\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.79796 −0.270593 −0.135297 0.990805i \(-0.543199\pi\)
−0.135297 + 0.990805i \(0.543199\pi\)
\(198\) 0 0
\(199\) −7.79796 −0.552783 −0.276391 0.961045i \(-0.589139\pi\)
−0.276391 + 0.961045i \(0.589139\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.4495 0.803597
\(204\) 0 0
\(205\) 1.89898 0.132630
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.7980 1.09277
\(210\) 0 0
\(211\) −15.6969 −1.08062 −0.540311 0.841465i \(-0.681693\pi\)
−0.540311 + 0.841465i \(0.681693\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.898979 0.0613099
\(216\) 0 0
\(217\) −10.1464 −0.688784
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.247449 0.0166452
\(222\) 0 0
\(223\) −26.8990 −1.80129 −0.900644 0.434557i \(-0.856905\pi\)
−0.900644 + 0.434557i \(0.856905\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 17.7980 1.17612 0.588061 0.808816i \(-0.299891\pi\)
0.588061 + 0.808816i \(0.299891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.4949 1.60471 0.802357 0.596844i \(-0.203579\pi\)
0.802357 + 0.596844i \(0.203579\pi\)
\(234\) 0 0
\(235\) −2.44949 −0.159787
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.1010 −0.653381 −0.326690 0.945131i \(-0.605933\pi\)
−0.326690 + 0.945131i \(0.605933\pi\)
\(240\) 0 0
\(241\) −11.3485 −0.731019 −0.365510 0.930808i \(-0.619105\pi\)
−0.365510 + 0.930808i \(0.619105\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.89898 −0.312984
\(246\) 0 0
\(247\) 2.89898 0.184458
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.8990 0.687937 0.343969 0.938981i \(-0.388229\pi\)
0.343969 + 0.938981i \(0.388229\pi\)
\(252\) 0 0
\(253\) 2.44949 0.153998
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.65153 0.290155 0.145077 0.989420i \(-0.453657\pi\)
0.145077 + 0.989420i \(0.453657\pi\)
\(258\) 0 0
\(259\) −12.1010 −0.751921
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.24745 0.200246 0.100123 0.994975i \(-0.468076\pi\)
0.100123 + 0.994975i \(0.468076\pi\)
\(264\) 0 0
\(265\) −10.3485 −0.635701
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −23.6969 −1.44483 −0.722414 0.691461i \(-0.756967\pi\)
−0.722414 + 0.691461i \(0.756967\pi\)
\(270\) 0 0
\(271\) −3.69694 −0.224573 −0.112287 0.993676i \(-0.535817\pi\)
−0.112287 + 0.993676i \(0.535817\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.44949 −0.147710
\(276\) 0 0
\(277\) −5.10102 −0.306491 −0.153245 0.988188i \(-0.548972\pi\)
−0.153245 + 0.988188i \(0.548972\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.34847 −0.438373 −0.219186 0.975683i \(-0.570340\pi\)
−0.219186 + 0.975683i \(0.570340\pi\)
\(282\) 0 0
\(283\) 13.4495 0.799489 0.399745 0.916627i \(-0.369099\pi\)
0.399745 + 0.916627i \(0.369099\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.75255 0.162478
\(288\) 0 0
\(289\) −16.6969 −0.982173
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.65153 0.447007 0.223504 0.974703i \(-0.428251\pi\)
0.223504 + 0.974703i \(0.428251\pi\)
\(294\) 0 0
\(295\) −7.89898 −0.459896
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.449490 0.0259947
\(300\) 0 0
\(301\) 1.30306 0.0751072
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.34847 0.535292
\(306\) 0 0
\(307\) −7.55051 −0.430930 −0.215465 0.976512i \(-0.569127\pi\)
−0.215465 + 0.976512i \(0.569127\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.6969 −1.17362 −0.586808 0.809726i \(-0.699616\pi\)
−0.586808 + 0.809726i \(0.699616\pi\)
\(312\) 0 0
\(313\) 18.3485 1.03712 0.518558 0.855042i \(-0.326469\pi\)
0.518558 + 0.855042i \(0.326469\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.247449 −0.0138981 −0.00694905 0.999976i \(-0.502212\pi\)
−0.00694905 + 0.999976i \(0.502212\pi\)
\(318\) 0 0
\(319\) −19.3485 −1.08331
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.55051 0.197556
\(324\) 0 0
\(325\) −0.449490 −0.0249332
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.55051 −0.195746
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.44949 −0.188466
\(336\) 0 0
\(337\) 24.8990 1.35633 0.678167 0.734908i \(-0.262775\pi\)
0.678167 + 0.734908i \(0.262775\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.1464 0.928531
\(342\) 0 0
\(343\) −17.2474 −0.931275
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.7980 −1.17018 −0.585088 0.810970i \(-0.698940\pi\)
−0.585088 + 0.810970i \(0.698940\pi\)
\(348\) 0 0
\(349\) 25.6969 1.37553 0.687763 0.725935i \(-0.258593\pi\)
0.687763 + 0.725935i \(0.258593\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.55051 −0.508322 −0.254161 0.967162i \(-0.581799\pi\)
−0.254161 + 0.967162i \(0.581799\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.1464 1.22162 0.610811 0.791777i \(-0.290844\pi\)
0.610811 + 0.791777i \(0.290844\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.34847 −0.279952
\(366\) 0 0
\(367\) −19.2474 −1.00471 −0.502354 0.864662i \(-0.667533\pi\)
−0.502354 + 0.864662i \(0.667533\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.55051 −0.182861
\(378\) 0 0
\(379\) −33.3939 −1.71533 −0.857664 0.514210i \(-0.828085\pi\)
−0.857664 + 0.514210i \(0.828085\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.34847 −0.222196 −0.111098 0.993809i \(-0.535437\pi\)
−0.111098 + 0.993809i \(0.535437\pi\)
\(384\) 0 0
\(385\) −3.55051 −0.180951
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.1010 0.664248 0.332124 0.943236i \(-0.392235\pi\)
0.332124 + 0.943236i \(0.392235\pi\)
\(390\) 0 0
\(391\) 0.550510 0.0278405
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 17.7980 0.893254 0.446627 0.894720i \(-0.352625\pi\)
0.446627 + 0.894720i \(0.352625\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.6969 −1.03356 −0.516778 0.856120i \(-0.672869\pi\)
−0.516778 + 0.856120i \(0.672869\pi\)
\(402\) 0 0
\(403\) 3.14643 0.156735
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.4495 1.01364
\(408\) 0 0
\(409\) 18.5959 0.919509 0.459754 0.888046i \(-0.347937\pi\)
0.459754 + 0.888046i \(0.347937\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −11.4495 −0.563393
\(414\) 0 0
\(415\) −4.34847 −0.213458
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −17.1464 −0.837658 −0.418829 0.908065i \(-0.637559\pi\)
−0.418829 + 0.908065i \(0.637559\pi\)
\(420\) 0 0
\(421\) −35.3485 −1.72278 −0.861389 0.507945i \(-0.830405\pi\)
−0.861389 + 0.507945i \(0.830405\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.550510 −0.0267037
\(426\) 0 0
\(427\) 13.5505 0.655755
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.5959 −0.943902 −0.471951 0.881625i \(-0.656450\pi\)
−0.471951 + 0.881625i \(0.656450\pi\)
\(432\) 0 0
\(433\) 36.8434 1.77058 0.885290 0.465040i \(-0.153960\pi\)
0.885290 + 0.465040i \(0.153960\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.44949 0.308521
\(438\) 0 0
\(439\) −2.89898 −0.138361 −0.0691804 0.997604i \(-0.522038\pi\)
−0.0691804 + 0.997604i \(0.522038\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.9444 0.710029 0.355015 0.934861i \(-0.384476\pi\)
0.355015 + 0.934861i \(0.384476\pi\)
\(444\) 0 0
\(445\) 7.10102 0.336621
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.3939 −1.52876 −0.764381 0.644765i \(-0.776955\pi\)
−0.764381 + 0.644765i \(0.776955\pi\)
\(450\) 0 0
\(451\) −4.65153 −0.219032
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.651531 −0.0305442
\(456\) 0 0
\(457\) −11.6515 −0.545036 −0.272518 0.962151i \(-0.587856\pi\)
−0.272518 + 0.962151i \(0.587856\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −38.6969 −1.80230 −0.901148 0.433511i \(-0.857274\pi\)
−0.901148 + 0.433511i \(0.857274\pi\)
\(462\) 0 0
\(463\) −20.0454 −0.931589 −0.465795 0.884893i \(-0.654231\pi\)
−0.465795 + 0.884893i \(0.654231\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 35.9444 1.66331 0.831654 0.555294i \(-0.187394\pi\)
0.831654 + 0.555294i \(0.187394\pi\)
\(468\) 0 0
\(469\) −5.00000 −0.230879
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.20204 −0.101250
\(474\) 0 0
\(475\) −6.44949 −0.295923
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.3485 1.15820 0.579101 0.815256i \(-0.303404\pi\)
0.579101 + 0.815256i \(0.303404\pi\)
\(480\) 0 0
\(481\) 3.75255 0.171102
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.10102 −0.231625
\(486\) 0 0
\(487\) −33.1464 −1.50201 −0.751004 0.660298i \(-0.770430\pi\)
−0.751004 + 0.660298i \(0.770430\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.3939 0.920363 0.460181 0.887825i \(-0.347784\pi\)
0.460181 + 0.887825i \(0.347784\pi\)
\(492\) 0 0
\(493\) −4.34847 −0.195845
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.34847 −0.195056
\(498\) 0 0
\(499\) 14.7980 0.662448 0.331224 0.943552i \(-0.392538\pi\)
0.331224 + 0.943552i \(0.392538\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.1464 −0.898285 −0.449142 0.893460i \(-0.648270\pi\)
−0.449142 + 0.893460i \(0.648270\pi\)
\(504\) 0 0
\(505\) 0.797959 0.0355087
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 35.3939 1.56881 0.784403 0.620252i \(-0.212969\pi\)
0.784403 + 0.620252i \(0.212969\pi\)
\(510\) 0 0
\(511\) −7.75255 −0.342953
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16.6515 −0.729517 −0.364758 0.931102i \(-0.618848\pi\)
−0.364758 + 0.931102i \(0.618848\pi\)
\(522\) 0 0
\(523\) −31.7980 −1.39043 −0.695214 0.718803i \(-0.744690\pi\)
−0.695214 + 0.718803i \(0.744690\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.85357 0.167864
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.853572 −0.0369723
\(534\) 0 0
\(535\) 20.1464 0.871006
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 24.8990 1.07049 0.535245 0.844697i \(-0.320219\pi\)
0.535245 + 0.844697i \(0.320219\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.34847 0.143433
\(546\) 0 0
\(547\) 25.3939 1.08576 0.542882 0.839809i \(-0.317333\pi\)
0.542882 + 0.839809i \(0.317333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −50.9444 −2.17030
\(552\) 0 0
\(553\) −5.79796 −0.246554
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.4495 1.24781 0.623907 0.781498i \(-0.285544\pi\)
0.623907 + 0.781498i \(0.285544\pi\)
\(558\) 0 0
\(559\) −0.404082 −0.0170909
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.85357 0.162409 0.0812043 0.996697i \(-0.474123\pi\)
0.0812043 + 0.996697i \(0.474123\pi\)
\(564\) 0 0
\(565\) 1.65153 0.0694804
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 47.3939 1.98685 0.993427 0.114465i \(-0.0365152\pi\)
0.993427 + 0.114465i \(0.0365152\pi\)
\(570\) 0 0
\(571\) 12.0454 0.504085 0.252042 0.967716i \(-0.418898\pi\)
0.252042 + 0.967716i \(0.418898\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 42.2929 1.76067 0.880337 0.474348i \(-0.157316\pi\)
0.880337 + 0.474348i \(0.157316\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.30306 −0.261495
\(582\) 0 0
\(583\) 25.3485 1.04983
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.5959 1.55175 0.775875 0.630887i \(-0.217309\pi\)
0.775875 + 0.630887i \(0.217309\pi\)
\(588\) 0 0
\(589\) 45.1464 1.86023
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.85357 0.281442 0.140721 0.990049i \(-0.455058\pi\)
0.140721 + 0.990049i \(0.455058\pi\)
\(594\) 0 0
\(595\) −0.797959 −0.0327131
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −4.79796 −0.195713 −0.0978564 0.995201i \(-0.531199\pi\)
−0.0978564 + 0.995201i \(0.531199\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 39.3485 1.59711 0.798553 0.601925i \(-0.205599\pi\)
0.798553 + 0.601925i \(0.205599\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.10102 0.0445425
\(612\) 0 0
\(613\) −6.69694 −0.270487 −0.135243 0.990812i \(-0.543182\pi\)
−0.135243 + 0.990812i \(0.543182\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.65153 0.0664881 0.0332441 0.999447i \(-0.489416\pi\)
0.0332441 + 0.999447i \(0.489416\pi\)
\(618\) 0 0
\(619\) 17.7980 0.715360 0.357680 0.933844i \(-0.383568\pi\)
0.357680 + 0.933844i \(0.383568\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.2929 0.412375
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.59592 0.183251
\(630\) 0 0
\(631\) 2.24745 0.0894695 0.0447348 0.998999i \(-0.485756\pi\)
0.0447348 + 0.998999i \(0.485756\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.3485 −0.450350
\(636\) 0 0
\(637\) 2.20204 0.0872480
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21.5505 −0.851194 −0.425597 0.904913i \(-0.639936\pi\)
−0.425597 + 0.904913i \(0.639936\pi\)
\(642\) 0 0
\(643\) 9.65153 0.380619 0.190310 0.981724i \(-0.439051\pi\)
0.190310 + 0.981724i \(0.439051\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.5505 −0.611354 −0.305677 0.952135i \(-0.598883\pi\)
−0.305677 + 0.952135i \(0.598883\pi\)
\(648\) 0 0
\(649\) 19.3485 0.759494
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.4495 −1.26985 −0.634923 0.772575i \(-0.718968\pi\)
−0.634923 + 0.772575i \(0.718968\pi\)
\(654\) 0 0
\(655\) −1.10102 −0.0430204
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.55051 −0.372035 −0.186018 0.982546i \(-0.559558\pi\)
−0.186018 + 0.982546i \(0.559558\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9.34847 −0.362518
\(666\) 0 0
\(667\) −7.89898 −0.305850
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −22.8990 −0.884005
\(672\) 0 0
\(673\) −26.0454 −1.00398 −0.501988 0.864874i \(-0.667398\pi\)
−0.501988 + 0.864874i \(0.667398\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.7526 0.566987 0.283493 0.958974i \(-0.408507\pi\)
0.283493 + 0.958974i \(0.408507\pi\)
\(678\) 0 0
\(679\) −7.39388 −0.283751
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.65153 0.177986 0.0889929 0.996032i \(-0.471635\pi\)
0.0889929 + 0.996032i \(0.471635\pi\)
\(684\) 0 0
\(685\) 7.10102 0.271316
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.65153 0.177209
\(690\) 0 0
\(691\) −15.3939 −0.585611 −0.292805 0.956172i \(-0.594589\pi\)
−0.292805 + 0.956172i \(0.594589\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.69694 0.291962
\(696\) 0 0
\(697\) −1.04541 −0.0395976
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.0454 −1.05926 −0.529630 0.848229i \(-0.677669\pi\)
−0.529630 + 0.848229i \(0.677669\pi\)
\(702\) 0 0
\(703\) 53.8434 2.03074
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.15663 0.0434997
\(708\) 0 0
\(709\) −0.944387 −0.0354672 −0.0177336 0.999843i \(-0.505645\pi\)
−0.0177336 + 0.999843i \(0.505645\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) 1.10102 0.0411758
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.2929 −0.719502 −0.359751 0.933048i \(-0.617138\pi\)
−0.359751 + 0.933048i \(0.617138\pi\)
\(720\) 0 0
\(721\) −5.79796 −0.215927
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.89898 0.293361
\(726\) 0 0
\(727\) −32.3485 −1.19974 −0.599869 0.800098i \(-0.704781\pi\)
−0.599869 + 0.800098i \(0.704781\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.494897 −0.0183044
\(732\) 0 0
\(733\) 36.3485 1.34256 0.671281 0.741203i \(-0.265745\pi\)
0.671281 + 0.741203i \(0.265745\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.44949 0.311241
\(738\) 0 0
\(739\) −16.3031 −0.599718 −0.299859 0.953984i \(-0.596940\pi\)
−0.299859 + 0.953984i \(0.596940\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.7980 1.23993 0.619963 0.784631i \(-0.287147\pi\)
0.619963 + 0.784631i \(0.287147\pi\)
\(744\) 0 0
\(745\) 6.24745 0.228889
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 29.2020 1.06702
\(750\) 0 0
\(751\) 12.6515 0.461661 0.230830 0.972994i \(-0.425856\pi\)
0.230830 + 0.972994i \(0.425856\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.202041 −0.00735303
\(756\) 0 0
\(757\) 35.2474 1.28109 0.640545 0.767921i \(-0.278708\pi\)
0.640545 + 0.767921i \(0.278708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −45.4949 −1.64919 −0.824594 0.565724i \(-0.808597\pi\)
−0.824594 + 0.565724i \(0.808597\pi\)
\(762\) 0 0
\(763\) 4.85357 0.175711
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.55051 0.128201
\(768\) 0 0
\(769\) −3.14643 −0.113463 −0.0567316 0.998389i \(-0.518068\pi\)
−0.0567316 + 0.998389i \(0.518068\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.69694 0.312807 0.156404 0.987693i \(-0.450010\pi\)
0.156404 + 0.987693i \(0.450010\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.2474 −0.438810
\(780\) 0 0
\(781\) 7.34847 0.262949
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.348469 0.0124374
\(786\) 0 0
\(787\) −48.1464 −1.71623 −0.858117 0.513454i \(-0.828366\pi\)
−0.858117 + 0.513454i \(0.828366\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.39388 0.0851165
\(792\) 0 0
\(793\) −4.20204 −0.149219
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.1464 1.35122 0.675608 0.737261i \(-0.263881\pi\)
0.675608 + 0.737261i \(0.263881\pi\)
\(798\) 0 0
\(799\) 1.34847 0.0477054
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.1010 0.462325
\(804\) 0 0
\(805\) −1.44949 −0.0510878
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.797959 −0.0280547 −0.0140274 0.999902i \(-0.504465\pi\)
−0.0140274 + 0.999902i \(0.504465\pi\)
\(810\) 0 0
\(811\) −16.7980 −0.589856 −0.294928 0.955519i \(-0.595296\pi\)
−0.294928 + 0.955519i \(0.595296\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.69694 0.164527
\(816\) 0 0
\(817\) −5.79796 −0.202845
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.5959 −0.893304 −0.446652 0.894708i \(-0.647384\pi\)
−0.446652 + 0.894708i \(0.647384\pi\)
\(822\) 0 0
\(823\) 3.10102 0.108095 0.0540474 0.998538i \(-0.482788\pi\)
0.0540474 + 0.998538i \(0.482788\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.2474 −1.57341 −0.786704 0.617330i \(-0.788214\pi\)
−0.786704 + 0.617330i \(0.788214\pi\)
\(828\) 0 0
\(829\) 34.3939 1.19455 0.597274 0.802037i \(-0.296250\pi\)
0.597274 + 0.802037i \(0.296250\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.69694 0.0934434
\(834\) 0 0
\(835\) 24.2474 0.839118
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.5959 −0.469383 −0.234692 0.972070i \(-0.575408\pi\)
−0.234692 + 0.972070i \(0.575408\pi\)
\(840\) 0 0
\(841\) 33.3939 1.15151
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.7980 −0.440263
\(846\) 0 0
\(847\) −7.24745 −0.249025
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.34847 0.286182
\(852\) 0 0
\(853\) 4.20204 0.143875 0.0719376 0.997409i \(-0.477082\pi\)
0.0719376 + 0.997409i \(0.477082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.2929 1.78629 0.893145 0.449769i \(-0.148494\pi\)
0.893145 + 0.449769i \(0.148494\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13.5959 0.462810 0.231405 0.972857i \(-0.425668\pi\)
0.231405 + 0.972857i \(0.425668\pi\)
\(864\) 0 0
\(865\) −9.79796 −0.333141
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.79796 0.332373
\(870\) 0 0
\(871\) 1.55051 0.0525370
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.44949 0.0490017
\(876\) 0 0
\(877\) 4.20204 0.141893 0.0709464 0.997480i \(-0.477398\pi\)
0.0709464 + 0.997480i \(0.477398\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.853572 0.0287576 0.0143788 0.999897i \(-0.495423\pi\)
0.0143788 + 0.999897i \(0.495423\pi\)
\(882\) 0 0
\(883\) −42.4495 −1.42854 −0.714270 0.699871i \(-0.753241\pi\)
−0.714270 + 0.699871i \(0.753241\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.59592 −0.0535857 −0.0267928 0.999641i \(-0.508529\pi\)
−0.0267928 + 0.999641i \(0.508529\pi\)
\(888\) 0 0
\(889\) −16.4495 −0.551698
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.7980 0.528659
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −55.2929 −1.84412
\(900\) 0 0
\(901\) 5.69694 0.189793
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.89898 −0.295812
\(906\) 0 0
\(907\) 12.3485 0.410024 0.205012 0.978759i \(-0.434277\pi\)
0.205012 + 0.978759i \(0.434277\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.4949 0.811552 0.405776 0.913973i \(-0.367001\pi\)
0.405776 + 0.913973i \(0.367001\pi\)
\(912\) 0 0
\(913\) 10.6515 0.352514
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.59592 −0.0527019
\(918\) 0 0
\(919\) 27.1010 0.893980 0.446990 0.894539i \(-0.352496\pi\)
0.446990 + 0.894539i \(0.352496\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.34847 0.0443854
\(924\) 0 0
\(925\) −8.34847 −0.274496
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 31.5959 1.03551
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.34847 0.0440997
\(936\) 0 0
\(937\) −0.696938 −0.0227680 −0.0113840 0.999935i \(-0.503624\pi\)
−0.0113840 + 0.999935i \(0.503624\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −27.5505 −0.898121 −0.449060 0.893501i \(-0.648241\pi\)
−0.449060 + 0.893501i \(0.648241\pi\)
\(942\) 0 0
\(943\) −1.89898 −0.0618393
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.4949 0.406030 0.203015 0.979176i \(-0.434926\pi\)
0.203015 + 0.979176i \(0.434926\pi\)
\(948\) 0 0
\(949\) 2.40408 0.0780398
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 51.7980 1.67790 0.838950 0.544208i \(-0.183170\pi\)
0.838950 + 0.544208i \(0.183170\pi\)
\(954\) 0 0
\(955\) −20.4495 −0.661730
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.2929 0.332374
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −23.3485 −0.750836 −0.375418 0.926856i \(-0.622501\pi\)
−0.375418 + 0.926856i \(0.622501\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.4949 −1.17118 −0.585588 0.810608i \(-0.699137\pi\)
−0.585588 + 0.810608i \(0.699137\pi\)
\(972\) 0 0
\(973\) 11.1566 0.357665
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.6515 −0.820665 −0.410333 0.911936i \(-0.634587\pi\)
−0.410333 + 0.911936i \(0.634587\pi\)
\(978\) 0 0
\(979\) −17.3939 −0.555911
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.65153 0.0526757 0.0263378 0.999653i \(-0.491615\pi\)
0.0263378 + 0.999653i \(0.491615\pi\)
\(984\) 0 0
\(985\) −3.79796 −0.121013
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.898979 −0.0285859
\(990\) 0 0
\(991\) 39.8990 1.26743 0.633716 0.773565i \(-0.281529\pi\)
0.633716 + 0.773565i \(0.281529\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.79796 −0.247212
\(996\) 0 0
\(997\) 32.0000 1.01345 0.506725 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4140.2.a.q.1.2 yes 2
3.2 odd 2 4140.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.a.l.1.2 2 3.2 odd 2
4140.2.a.q.1.2 yes 2 1.1 even 1 trivial