Properties

Label 7448.2.a.x.1.1
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +O(q^{10})\) \(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} -3.24264 q^{11} +3.41421 q^{13} -3.41421 q^{15} +6.82843 q^{17} +1.00000 q^{19} +4.41421 q^{23} -4.00000 q^{25} -19.3137 q^{27} -9.41421 q^{29} -4.58579 q^{31} +11.0711 q^{33} -6.58579 q^{37} -11.6569 q^{39} +5.65685 q^{41} -3.24264 q^{43} +8.65685 q^{45} +1.24264 q^{47} -23.3137 q^{51} -5.41421 q^{53} -3.24264 q^{55} -3.41421 q^{57} +0.828427 q^{59} +5.00000 q^{61} +3.41421 q^{65} +3.17157 q^{67} -15.0711 q^{69} -7.75736 q^{71} +7.00000 q^{73} +13.6569 q^{75} +3.89949 q^{79} +39.9706 q^{81} -14.8995 q^{83} +6.82843 q^{85} +32.1421 q^{87} -13.5563 q^{89} +15.6569 q^{93} +1.00000 q^{95} +5.65685 q^{97} -28.0711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.41421 −1.97120 −0.985599 0.169102i \(-0.945913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) −3.24264 −0.977693 −0.488846 0.872370i \(-0.662582\pi\)
−0.488846 + 0.872370i \(0.662582\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.41421 0.920427 0.460214 0.887808i \(-0.347773\pi\)
0.460214 + 0.887808i \(0.347773\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −19.3137 −3.71692
\(28\) 0 0
\(29\) −9.41421 −1.74818 −0.874088 0.485768i \(-0.838540\pi\)
−0.874088 + 0.485768i \(0.838540\pi\)
\(30\) 0 0
\(31\) −4.58579 −0.823632 −0.411816 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 0 0
\(33\) 11.0711 1.92723
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58579 −1.08270 −0.541348 0.840798i \(-0.682086\pi\)
−0.541348 + 0.840798i \(0.682086\pi\)
\(38\) 0 0
\(39\) −11.6569 −1.86659
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) −3.24264 −0.494498 −0.247249 0.968952i \(-0.579527\pi\)
−0.247249 + 0.968952i \(0.579527\pi\)
\(44\) 0 0
\(45\) 8.65685 1.29049
\(46\) 0 0
\(47\) 1.24264 0.181258 0.0906289 0.995885i \(-0.471112\pi\)
0.0906289 + 0.995885i \(0.471112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −23.3137 −3.26457
\(52\) 0 0
\(53\) −5.41421 −0.743699 −0.371850 0.928293i \(-0.621276\pi\)
−0.371850 + 0.928293i \(0.621276\pi\)
\(54\) 0 0
\(55\) −3.24264 −0.437238
\(56\) 0 0
\(57\) −3.41421 −0.452224
\(58\) 0 0
\(59\) 0.828427 0.107852 0.0539260 0.998545i \(-0.482826\pi\)
0.0539260 + 0.998545i \(0.482826\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.41421 0.423481
\(66\) 0 0
\(67\) 3.17157 0.387469 0.193735 0.981054i \(-0.437940\pi\)
0.193735 + 0.981054i \(0.437940\pi\)
\(68\) 0 0
\(69\) −15.0711 −1.81434
\(70\) 0 0
\(71\) −7.75736 −0.920629 −0.460315 0.887756i \(-0.652263\pi\)
−0.460315 + 0.887756i \(0.652263\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 13.6569 1.57696
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.89949 0.438727 0.219364 0.975643i \(-0.429602\pi\)
0.219364 + 0.975643i \(0.429602\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 0 0
\(83\) −14.8995 −1.63543 −0.817716 0.575622i \(-0.804760\pi\)
−0.817716 + 0.575622i \(0.804760\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 0 0
\(87\) 32.1421 3.44600
\(88\) 0 0
\(89\) −13.5563 −1.43697 −0.718485 0.695542i \(-0.755164\pi\)
−0.718485 + 0.695542i \(0.755164\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 15.6569 1.62354
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 5.65685 0.574367 0.287183 0.957876i \(-0.407281\pi\)
0.287183 + 0.957876i \(0.407281\pi\)
\(98\) 0 0
\(99\) −28.0711 −2.82125
\(100\) 0 0
\(101\) −1.48528 −0.147791 −0.0738955 0.997266i \(-0.523543\pi\)
−0.0738955 + 0.997266i \(0.523543\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.0711 1.65032 0.825161 0.564897i \(-0.191084\pi\)
0.825161 + 0.564897i \(0.191084\pi\)
\(108\) 0 0
\(109\) −19.3137 −1.84992 −0.924959 0.380067i \(-0.875901\pi\)
−0.924959 + 0.380067i \(0.875901\pi\)
\(110\) 0 0
\(111\) 22.4853 2.13421
\(112\) 0 0
\(113\) 11.0711 1.04148 0.520739 0.853716i \(-0.325656\pi\)
0.520739 + 0.853716i \(0.325656\pi\)
\(114\) 0 0
\(115\) 4.41421 0.411628
\(116\) 0 0
\(117\) 29.5563 2.73249
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.485281 −0.0441165
\(122\) 0 0
\(123\) −19.3137 −1.74146
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −19.5563 −1.73535 −0.867673 0.497136i \(-0.834385\pi\)
−0.867673 + 0.497136i \(0.834385\pi\)
\(128\) 0 0
\(129\) 11.0711 0.974753
\(130\) 0 0
\(131\) −1.65685 −0.144760 −0.0723800 0.997377i \(-0.523059\pi\)
−0.0723800 + 0.997377i \(0.523059\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −19.3137 −1.66226
\(136\) 0 0
\(137\) 0.656854 0.0561188 0.0280594 0.999606i \(-0.491067\pi\)
0.0280594 + 0.999606i \(0.491067\pi\)
\(138\) 0 0
\(139\) 18.5563 1.57393 0.786964 0.616998i \(-0.211651\pi\)
0.786964 + 0.616998i \(0.211651\pi\)
\(140\) 0 0
\(141\) −4.24264 −0.357295
\(142\) 0 0
\(143\) −11.0711 −0.925809
\(144\) 0 0
\(145\) −9.41421 −0.781808
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.65685 −0.545351 −0.272675 0.962106i \(-0.587908\pi\)
−0.272675 + 0.962106i \(0.587908\pi\)
\(150\) 0 0
\(151\) 5.65685 0.460348 0.230174 0.973149i \(-0.426070\pi\)
0.230174 + 0.973149i \(0.426070\pi\)
\(152\) 0 0
\(153\) 59.1127 4.77898
\(154\) 0 0
\(155\) −4.58579 −0.368339
\(156\) 0 0
\(157\) 1.68629 0.134581 0.0672904 0.997733i \(-0.478565\pi\)
0.0672904 + 0.997733i \(0.478565\pi\)
\(158\) 0 0
\(159\) 18.4853 1.46598
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 14.0711 1.10213 0.551066 0.834462i \(-0.314221\pi\)
0.551066 + 0.834462i \(0.314221\pi\)
\(164\) 0 0
\(165\) 11.0711 0.861881
\(166\) 0 0
\(167\) −18.5858 −1.43821 −0.719106 0.694901i \(-0.755448\pi\)
−0.719106 + 0.694901i \(0.755448\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 8.65685 0.662006
\(172\) 0 0
\(173\) 9.65685 0.734197 0.367099 0.930182i \(-0.380351\pi\)
0.367099 + 0.930182i \(0.380351\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.82843 −0.212598
\(178\) 0 0
\(179\) 6.48528 0.484733 0.242366 0.970185i \(-0.422076\pi\)
0.242366 + 0.970185i \(0.422076\pi\)
\(180\) 0 0
\(181\) 14.7279 1.09472 0.547359 0.836898i \(-0.315633\pi\)
0.547359 + 0.836898i \(0.315633\pi\)
\(182\) 0 0
\(183\) −17.0711 −1.26193
\(184\) 0 0
\(185\) −6.58579 −0.484197
\(186\) 0 0
\(187\) −22.1421 −1.61919
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.24264 0.668774 0.334387 0.942436i \(-0.391471\pi\)
0.334387 + 0.942436i \(0.391471\pi\)
\(192\) 0 0
\(193\) 2.24264 0.161429 0.0807144 0.996737i \(-0.474280\pi\)
0.0807144 + 0.996737i \(0.474280\pi\)
\(194\) 0 0
\(195\) −11.6569 −0.834765
\(196\) 0 0
\(197\) −0.514719 −0.0366722 −0.0183361 0.999832i \(-0.505837\pi\)
−0.0183361 + 0.999832i \(0.505837\pi\)
\(198\) 0 0
\(199\) −2.07107 −0.146814 −0.0734071 0.997302i \(-0.523387\pi\)
−0.0734071 + 0.997302i \(0.523387\pi\)
\(200\) 0 0
\(201\) −10.8284 −0.763778
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.65685 0.395092
\(206\) 0 0
\(207\) 38.2132 2.65600
\(208\) 0 0
\(209\) −3.24264 −0.224298
\(210\) 0 0
\(211\) 12.1421 0.835899 0.417950 0.908470i \(-0.362749\pi\)
0.417950 + 0.908470i \(0.362749\pi\)
\(212\) 0 0
\(213\) 26.4853 1.81474
\(214\) 0 0
\(215\) −3.24264 −0.221146
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −23.8995 −1.61498
\(220\) 0 0
\(221\) 23.3137 1.56825
\(222\) 0 0
\(223\) 16.2426 1.08769 0.543844 0.839186i \(-0.316968\pi\)
0.543844 + 0.839186i \(0.316968\pi\)
\(224\) 0 0
\(225\) −34.6274 −2.30849
\(226\) 0 0
\(227\) −1.41421 −0.0938647 −0.0469323 0.998898i \(-0.514945\pi\)
−0.0469323 + 0.998898i \(0.514945\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.34315 −0.415553 −0.207777 0.978176i \(-0.566623\pi\)
−0.207777 + 0.978176i \(0.566623\pi\)
\(234\) 0 0
\(235\) 1.24264 0.0810609
\(236\) 0 0
\(237\) −13.3137 −0.864818
\(238\) 0 0
\(239\) −4.34315 −0.280935 −0.140467 0.990085i \(-0.544861\pi\)
−0.140467 + 0.990085i \(0.544861\pi\)
\(240\) 0 0
\(241\) −8.72792 −0.562215 −0.281107 0.959676i \(-0.590702\pi\)
−0.281107 + 0.959676i \(0.590702\pi\)
\(242\) 0 0
\(243\) −78.5269 −5.03750
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.41421 0.217241
\(248\) 0 0
\(249\) 50.8701 3.22376
\(250\) 0 0
\(251\) 22.2132 1.40208 0.701042 0.713120i \(-0.252718\pi\)
0.701042 + 0.713120i \(0.252718\pi\)
\(252\) 0 0
\(253\) −14.3137 −0.899895
\(254\) 0 0
\(255\) −23.3137 −1.45996
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −81.4975 −5.04457
\(262\) 0 0
\(263\) −21.1716 −1.30550 −0.652748 0.757575i \(-0.726384\pi\)
−0.652748 + 0.757575i \(0.726384\pi\)
\(264\) 0 0
\(265\) −5.41421 −0.332592
\(266\) 0 0
\(267\) 46.2843 2.83255
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −2.27208 −0.138019 −0.0690095 0.997616i \(-0.521984\pi\)
−0.0690095 + 0.997616i \(0.521984\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.9706 0.782154
\(276\) 0 0
\(277\) −27.9706 −1.68059 −0.840294 0.542131i \(-0.817618\pi\)
−0.840294 + 0.542131i \(0.817618\pi\)
\(278\) 0 0
\(279\) −39.6985 −2.37669
\(280\) 0 0
\(281\) −17.8995 −1.06779 −0.533897 0.845549i \(-0.679273\pi\)
−0.533897 + 0.845549i \(0.679273\pi\)
\(282\) 0 0
\(283\) −14.5563 −0.865285 −0.432643 0.901566i \(-0.642419\pi\)
−0.432643 + 0.901566i \(0.642419\pi\)
\(284\) 0 0
\(285\) −3.41421 −0.202241
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) −19.3137 −1.13219
\(292\) 0 0
\(293\) −9.65685 −0.564159 −0.282080 0.959391i \(-0.591024\pi\)
−0.282080 + 0.959391i \(0.591024\pi\)
\(294\) 0 0
\(295\) 0.828427 0.0482329
\(296\) 0 0
\(297\) 62.6274 3.63401
\(298\) 0 0
\(299\) 15.0711 0.871582
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.07107 0.291325
\(304\) 0 0
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) −18.6274 −1.06312 −0.531561 0.847020i \(-0.678395\pi\)
−0.531561 + 0.847020i \(0.678395\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −1.14214 −0.0645573 −0.0322787 0.999479i \(-0.510276\pi\)
−0.0322787 + 0.999479i \(0.510276\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.2426 −0.575284 −0.287642 0.957738i \(-0.592871\pi\)
−0.287642 + 0.957738i \(0.592871\pi\)
\(318\) 0 0
\(319\) 30.5269 1.70918
\(320\) 0 0
\(321\) −58.2843 −3.25311
\(322\) 0 0
\(323\) 6.82843 0.379944
\(324\) 0 0
\(325\) −13.6569 −0.757546
\(326\) 0 0
\(327\) 65.9411 3.64655
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −18.4853 −1.01604 −0.508021 0.861344i \(-0.669623\pi\)
−0.508021 + 0.861344i \(0.669623\pi\)
\(332\) 0 0
\(333\) −57.0122 −3.12425
\(334\) 0 0
\(335\) 3.17157 0.173282
\(336\) 0 0
\(337\) 5.07107 0.276239 0.138119 0.990416i \(-0.455894\pi\)
0.138119 + 0.990416i \(0.455894\pi\)
\(338\) 0 0
\(339\) −37.7990 −2.05296
\(340\) 0 0
\(341\) 14.8701 0.805259
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −15.0711 −0.811399
\(346\) 0 0
\(347\) 12.4142 0.666430 0.333215 0.942851i \(-0.391867\pi\)
0.333215 + 0.942851i \(0.391867\pi\)
\(348\) 0 0
\(349\) −5.65685 −0.302804 −0.151402 0.988472i \(-0.548379\pi\)
−0.151402 + 0.988472i \(0.548379\pi\)
\(350\) 0 0
\(351\) −65.9411 −3.51968
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −7.75736 −0.411718
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 36.6985 1.93687 0.968436 0.249262i \(-0.0801882\pi\)
0.968436 + 0.249262i \(0.0801882\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.65685 0.0869623
\(364\) 0 0
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) −16.6274 −0.867944 −0.433972 0.900926i \(-0.642888\pi\)
−0.433972 + 0.900926i \(0.642888\pi\)
\(368\) 0 0
\(369\) 48.9706 2.54931
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.34315 −0.328436 −0.164218 0.986424i \(-0.552510\pi\)
−0.164218 + 0.986424i \(0.552510\pi\)
\(374\) 0 0
\(375\) 30.7279 1.58678
\(376\) 0 0
\(377\) −32.1421 −1.65540
\(378\) 0 0
\(379\) 12.5858 0.646488 0.323244 0.946316i \(-0.395226\pi\)
0.323244 + 0.946316i \(0.395226\pi\)
\(380\) 0 0
\(381\) 66.7696 3.42071
\(382\) 0 0
\(383\) 3.75736 0.191992 0.0959960 0.995382i \(-0.469396\pi\)
0.0959960 + 0.995382i \(0.469396\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −28.0711 −1.42693
\(388\) 0 0
\(389\) −15.3137 −0.776436 −0.388218 0.921568i \(-0.626909\pi\)
−0.388218 + 0.921568i \(0.626909\pi\)
\(390\) 0 0
\(391\) 30.1421 1.52435
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) 3.89949 0.196205
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.62742 −0.430833 −0.215416 0.976522i \(-0.569111\pi\)
−0.215416 + 0.976522i \(0.569111\pi\)
\(402\) 0 0
\(403\) −15.6569 −0.779923
\(404\) 0 0
\(405\) 39.9706 1.98615
\(406\) 0 0
\(407\) 21.3553 1.05854
\(408\) 0 0
\(409\) 29.3137 1.44947 0.724735 0.689028i \(-0.241962\pi\)
0.724735 + 0.689028i \(0.241962\pi\)
\(410\) 0 0
\(411\) −2.24264 −0.110621
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −14.8995 −0.731387
\(416\) 0 0
\(417\) −63.3553 −3.10252
\(418\) 0 0
\(419\) 30.6985 1.49972 0.749860 0.661597i \(-0.230121\pi\)
0.749860 + 0.661597i \(0.230121\pi\)
\(420\) 0 0
\(421\) −24.5858 −1.19824 −0.599119 0.800660i \(-0.704482\pi\)
−0.599119 + 0.800660i \(0.704482\pi\)
\(422\) 0 0
\(423\) 10.7574 0.523041
\(424\) 0 0
\(425\) −27.3137 −1.32491
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 37.7990 1.82495
\(430\) 0 0
\(431\) −13.8995 −0.669515 −0.334758 0.942304i \(-0.608654\pi\)
−0.334758 + 0.942304i \(0.608654\pi\)
\(432\) 0 0
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 0 0
\(435\) 32.1421 1.54110
\(436\) 0 0
\(437\) 4.41421 0.211160
\(438\) 0 0
\(439\) −26.4853 −1.26407 −0.632037 0.774938i \(-0.717781\pi\)
−0.632037 + 0.774938i \(0.717781\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.6863 −0.792790 −0.396395 0.918080i \(-0.629739\pi\)
−0.396395 + 0.918080i \(0.629739\pi\)
\(444\) 0 0
\(445\) −13.5563 −0.642633
\(446\) 0 0
\(447\) 22.7279 1.07499
\(448\) 0 0
\(449\) 30.6274 1.44540 0.722699 0.691163i \(-0.242901\pi\)
0.722699 + 0.691163i \(0.242901\pi\)
\(450\) 0 0
\(451\) −18.3431 −0.863745
\(452\) 0 0
\(453\) −19.3137 −0.907437
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.82843 0.272642 0.136321 0.990665i \(-0.456472\pi\)
0.136321 + 0.990665i \(0.456472\pi\)
\(458\) 0 0
\(459\) −131.882 −6.15574
\(460\) 0 0
\(461\) −10.7990 −0.502959 −0.251480 0.967863i \(-0.580917\pi\)
−0.251480 + 0.967863i \(0.580917\pi\)
\(462\) 0 0
\(463\) −27.7279 −1.28863 −0.644313 0.764762i \(-0.722857\pi\)
−0.644313 + 0.764762i \(0.722857\pi\)
\(464\) 0 0
\(465\) 15.6569 0.726069
\(466\) 0 0
\(467\) 7.72792 0.357606 0.178803 0.983885i \(-0.442778\pi\)
0.178803 + 0.983885i \(0.442778\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −5.75736 −0.265285
\(472\) 0 0
\(473\) 10.5147 0.483467
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −46.8701 −2.14603
\(478\) 0 0
\(479\) 9.92893 0.453664 0.226832 0.973934i \(-0.427163\pi\)
0.226832 + 0.973934i \(0.427163\pi\)
\(480\) 0 0
\(481\) −22.4853 −1.02524
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.65685 0.256865
\(486\) 0 0
\(487\) −11.8579 −0.537331 −0.268666 0.963234i \(-0.586583\pi\)
−0.268666 + 0.963234i \(0.586583\pi\)
\(488\) 0 0
\(489\) −48.0416 −2.17252
\(490\) 0 0
\(491\) −18.6985 −0.843851 −0.421925 0.906631i \(-0.638646\pi\)
−0.421925 + 0.906631i \(0.638646\pi\)
\(492\) 0 0
\(493\) −64.2843 −2.89522
\(494\) 0 0
\(495\) −28.0711 −1.26170
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.757359 −0.0339041 −0.0169520 0.999856i \(-0.505396\pi\)
−0.0169520 + 0.999856i \(0.505396\pi\)
\(500\) 0 0
\(501\) 63.4558 2.83500
\(502\) 0 0
\(503\) −21.7279 −0.968800 −0.484400 0.874847i \(-0.660962\pi\)
−0.484400 + 0.874847i \(0.660962\pi\)
\(504\) 0 0
\(505\) −1.48528 −0.0660942
\(506\) 0 0
\(507\) 4.58579 0.203662
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −19.3137 −0.852721
\(514\) 0 0
\(515\) −2.34315 −0.103251
\(516\) 0 0
\(517\) −4.02944 −0.177214
\(518\) 0 0
\(519\) −32.9706 −1.44725
\(520\) 0 0
\(521\) 21.8995 0.959434 0.479717 0.877423i \(-0.340739\pi\)
0.479717 + 0.877423i \(0.340739\pi\)
\(522\) 0 0
\(523\) −7.31371 −0.319806 −0.159903 0.987133i \(-0.551118\pi\)
−0.159903 + 0.987133i \(0.551118\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −31.3137 −1.36405
\(528\) 0 0
\(529\) −3.51472 −0.152814
\(530\) 0 0
\(531\) 7.17157 0.311220
\(532\) 0 0
\(533\) 19.3137 0.836570
\(534\) 0 0
\(535\) 17.0711 0.738047
\(536\) 0 0
\(537\) −22.1421 −0.955504
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −29.1421 −1.25292 −0.626459 0.779454i \(-0.715496\pi\)
−0.626459 + 0.779454i \(0.715496\pi\)
\(542\) 0 0
\(543\) −50.2843 −2.15790
\(544\) 0 0
\(545\) −19.3137 −0.827308
\(546\) 0 0
\(547\) −21.5563 −0.921683 −0.460841 0.887482i \(-0.652452\pi\)
−0.460841 + 0.887482i \(0.652452\pi\)
\(548\) 0 0
\(549\) 43.2843 1.84733
\(550\) 0 0
\(551\) −9.41421 −0.401059
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 22.4853 0.954447
\(556\) 0 0
\(557\) 35.4853 1.50356 0.751780 0.659414i \(-0.229196\pi\)
0.751780 + 0.659414i \(0.229196\pi\)
\(558\) 0 0
\(559\) −11.0711 −0.468256
\(560\) 0 0
\(561\) 75.5980 3.19175
\(562\) 0 0
\(563\) −42.8701 −1.80676 −0.903379 0.428844i \(-0.858921\pi\)
−0.903379 + 0.428844i \(0.858921\pi\)
\(564\) 0 0
\(565\) 11.0711 0.465763
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.31371 −0.222762 −0.111381 0.993778i \(-0.535527\pi\)
−0.111381 + 0.993778i \(0.535527\pi\)
\(570\) 0 0
\(571\) −42.6985 −1.78688 −0.893438 0.449187i \(-0.851714\pi\)
−0.893438 + 0.449187i \(0.851714\pi\)
\(572\) 0 0
\(573\) −31.5563 −1.31829
\(574\) 0 0
\(575\) −17.6569 −0.736342
\(576\) 0 0
\(577\) −28.1716 −1.17280 −0.586399 0.810022i \(-0.699455\pi\)
−0.586399 + 0.810022i \(0.699455\pi\)
\(578\) 0 0
\(579\) −7.65685 −0.318208
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 17.5563 0.727110
\(584\) 0 0
\(585\) 29.5563 1.22200
\(586\) 0 0
\(587\) 34.0000 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(588\) 0 0
\(589\) −4.58579 −0.188954
\(590\) 0 0
\(591\) 1.75736 0.0722881
\(592\) 0 0
\(593\) −4.31371 −0.177143 −0.0885714 0.996070i \(-0.528230\pi\)
−0.0885714 + 0.996070i \(0.528230\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.07107 0.289400
\(598\) 0 0
\(599\) 21.2132 0.866748 0.433374 0.901214i \(-0.357323\pi\)
0.433374 + 0.901214i \(0.357323\pi\)
\(600\) 0 0
\(601\) 16.5269 0.674147 0.337073 0.941478i \(-0.390563\pi\)
0.337073 + 0.941478i \(0.390563\pi\)
\(602\) 0 0
\(603\) 27.4558 1.11809
\(604\) 0 0
\(605\) −0.485281 −0.0197295
\(606\) 0 0
\(607\) −15.8995 −0.645341 −0.322670 0.946511i \(-0.604581\pi\)
−0.322670 + 0.946511i \(0.604581\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.24264 0.171639
\(612\) 0 0
\(613\) −32.4853 −1.31207 −0.656034 0.754731i \(-0.727767\pi\)
−0.656034 + 0.754731i \(0.727767\pi\)
\(614\) 0 0
\(615\) −19.3137 −0.778804
\(616\) 0 0
\(617\) 27.2843 1.09842 0.549212 0.835683i \(-0.314928\pi\)
0.549212 + 0.835683i \(0.314928\pi\)
\(618\) 0 0
\(619\) −9.10051 −0.365780 −0.182890 0.983133i \(-0.558545\pi\)
−0.182890 + 0.983133i \(0.558545\pi\)
\(620\) 0 0
\(621\) −85.2548 −3.42116
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 11.0711 0.442136
\(628\) 0 0
\(629\) −44.9706 −1.79309
\(630\) 0 0
\(631\) −17.3848 −0.692077 −0.346039 0.938220i \(-0.612473\pi\)
−0.346039 + 0.938220i \(0.612473\pi\)
\(632\) 0 0
\(633\) −41.4558 −1.64772
\(634\) 0 0
\(635\) −19.5563 −0.776070
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −67.1543 −2.65658
\(640\) 0 0
\(641\) 10.6863 0.422083 0.211042 0.977477i \(-0.432314\pi\)
0.211042 + 0.977477i \(0.432314\pi\)
\(642\) 0 0
\(643\) −27.5147 −1.08507 −0.542537 0.840032i \(-0.682536\pi\)
−0.542537 + 0.840032i \(0.682536\pi\)
\(644\) 0 0
\(645\) 11.0711 0.435923
\(646\) 0 0
\(647\) −27.9289 −1.09800 −0.549000 0.835822i \(-0.684991\pi\)
−0.549000 + 0.835822i \(0.684991\pi\)
\(648\) 0 0
\(649\) −2.68629 −0.105446
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.1421 −1.17955 −0.589776 0.807567i \(-0.700784\pi\)
−0.589776 + 0.807567i \(0.700784\pi\)
\(654\) 0 0
\(655\) −1.65685 −0.0647387
\(656\) 0 0
\(657\) 60.5980 2.36415
\(658\) 0 0
\(659\) 24.3848 0.949896 0.474948 0.880014i \(-0.342467\pi\)
0.474948 + 0.880014i \(0.342467\pi\)
\(660\) 0 0
\(661\) −8.87006 −0.345005 −0.172503 0.985009i \(-0.555185\pi\)
−0.172503 + 0.985009i \(0.555185\pi\)
\(662\) 0 0
\(663\) −79.5980 −3.09133
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −41.5563 −1.60907
\(668\) 0 0
\(669\) −55.4558 −2.14405
\(670\) 0 0
\(671\) −16.2132 −0.625904
\(672\) 0 0
\(673\) −40.6274 −1.56607 −0.783036 0.621976i \(-0.786330\pi\)
−0.783036 + 0.621976i \(0.786330\pi\)
\(674\) 0 0
\(675\) 77.2548 2.97354
\(676\) 0 0
\(677\) −11.3137 −0.434821 −0.217411 0.976080i \(-0.569761\pi\)
−0.217411 + 0.976080i \(0.569761\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4.82843 0.185026
\(682\) 0 0
\(683\) −24.1421 −0.923773 −0.461887 0.886939i \(-0.652827\pi\)
−0.461887 + 0.886939i \(0.652827\pi\)
\(684\) 0 0
\(685\) 0.656854 0.0250971
\(686\) 0 0
\(687\) 54.6274 2.08417
\(688\) 0 0
\(689\) −18.4853 −0.704233
\(690\) 0 0
\(691\) 16.6274 0.632537 0.316268 0.948670i \(-0.397570\pi\)
0.316268 + 0.948670i \(0.397570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.5563 0.703882
\(696\) 0 0
\(697\) 38.6274 1.46312
\(698\) 0 0
\(699\) 21.6569 0.819137
\(700\) 0 0
\(701\) −20.7990 −0.785567 −0.392784 0.919631i \(-0.628488\pi\)
−0.392784 + 0.919631i \(0.628488\pi\)
\(702\) 0 0
\(703\) −6.58579 −0.248388
\(704\) 0 0
\(705\) −4.24264 −0.159787
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 52.5980 1.97536 0.987679 0.156492i \(-0.0500184\pi\)
0.987679 + 0.156492i \(0.0500184\pi\)
\(710\) 0 0
\(711\) 33.7574 1.26600
\(712\) 0 0
\(713\) −20.2426 −0.758093
\(714\) 0 0
\(715\) −11.0711 −0.414034
\(716\) 0 0
\(717\) 14.8284 0.553778
\(718\) 0 0
\(719\) −3.31371 −0.123580 −0.0617902 0.998089i \(-0.519681\pi\)
−0.0617902 + 0.998089i \(0.519681\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 29.7990 1.10824
\(724\) 0 0
\(725\) 37.6569 1.39854
\(726\) 0 0
\(727\) −3.24264 −0.120263 −0.0601314 0.998190i \(-0.519152\pi\)
−0.0601314 + 0.998190i \(0.519152\pi\)
\(728\) 0 0
\(729\) 148.196 5.48874
\(730\) 0 0
\(731\) −22.1421 −0.818956
\(732\) 0 0
\(733\) 51.3137 1.89532 0.947658 0.319289i \(-0.103444\pi\)
0.947658 + 0.319289i \(0.103444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.2843 −0.378826
\(738\) 0 0
\(739\) 44.6274 1.64165 0.820823 0.571183i \(-0.193515\pi\)
0.820823 + 0.571183i \(0.193515\pi\)
\(740\) 0 0
\(741\) −11.6569 −0.428225
\(742\) 0 0
\(743\) 21.2721 0.780397 0.390198 0.920731i \(-0.372406\pi\)
0.390198 + 0.920731i \(0.372406\pi\)
\(744\) 0 0
\(745\) −6.65685 −0.243888
\(746\) 0 0
\(747\) −128.983 −4.71923
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.51472 0.201235 0.100617 0.994925i \(-0.467918\pi\)
0.100617 + 0.994925i \(0.467918\pi\)
\(752\) 0 0
\(753\) −75.8406 −2.76379
\(754\) 0 0
\(755\) 5.65685 0.205874
\(756\) 0 0
\(757\) −31.6863 −1.15166 −0.575829 0.817570i \(-0.695321\pi\)
−0.575829 + 0.817570i \(0.695321\pi\)
\(758\) 0 0
\(759\) 48.8701 1.77387
\(760\) 0 0
\(761\) 13.1421 0.476402 0.238201 0.971216i \(-0.423442\pi\)
0.238201 + 0.971216i \(0.423442\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 59.1127 2.13722
\(766\) 0 0
\(767\) 2.82843 0.102129
\(768\) 0 0
\(769\) 3.54416 0.127806 0.0639028 0.997956i \(-0.479645\pi\)
0.0639028 + 0.997956i \(0.479645\pi\)
\(770\) 0 0
\(771\) 47.7990 1.72144
\(772\) 0 0
\(773\) 0.928932 0.0334114 0.0167057 0.999860i \(-0.494682\pi\)
0.0167057 + 0.999860i \(0.494682\pi\)
\(774\) 0 0
\(775\) 18.3431 0.658905
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.65685 0.202678
\(780\) 0 0
\(781\) 25.1543 0.900093
\(782\) 0 0
\(783\) 181.823 6.49784
\(784\) 0 0
\(785\) 1.68629 0.0601863
\(786\) 0 0
\(787\) −11.7574 −0.419105 −0.209552 0.977797i \(-0.567201\pi\)
−0.209552 + 0.977797i \(0.567201\pi\)
\(788\) 0 0
\(789\) 72.2843 2.57339
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 17.0711 0.606211
\(794\) 0 0
\(795\) 18.4853 0.655605
\(796\) 0 0
\(797\) 45.4975 1.61160 0.805802 0.592186i \(-0.201735\pi\)
0.805802 + 0.592186i \(0.201735\pi\)
\(798\) 0 0
\(799\) 8.48528 0.300188
\(800\) 0 0
\(801\) −117.355 −4.14655
\(802\) 0 0
\(803\) −22.6985 −0.801012
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −54.6274 −1.92298
\(808\) 0 0
\(809\) −27.9706 −0.983393 −0.491696 0.870767i \(-0.663623\pi\)
−0.491696 + 0.870767i \(0.663623\pi\)
\(810\) 0 0
\(811\) −32.4264 −1.13865 −0.569323 0.822114i \(-0.692794\pi\)
−0.569323 + 0.822114i \(0.692794\pi\)
\(812\) 0 0
\(813\) 7.75736 0.272062
\(814\) 0 0
\(815\) 14.0711 0.492888
\(816\) 0 0
\(817\) −3.24264 −0.113446
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.02944 0.280229 0.140115 0.990135i \(-0.455253\pi\)
0.140115 + 0.990135i \(0.455253\pi\)
\(822\) 0 0
\(823\) 9.72792 0.339094 0.169547 0.985522i \(-0.445770\pi\)
0.169547 + 0.985522i \(0.445770\pi\)
\(824\) 0 0
\(825\) −44.2843 −1.54178
\(826\) 0 0
\(827\) −0.686292 −0.0238647 −0.0119323 0.999929i \(-0.503798\pi\)
−0.0119323 + 0.999929i \(0.503798\pi\)
\(828\) 0 0
\(829\) −39.3137 −1.36542 −0.682711 0.730689i \(-0.739199\pi\)
−0.682711 + 0.730689i \(0.739199\pi\)
\(830\) 0 0
\(831\) 95.4975 3.31277
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −18.5858 −0.643188
\(836\) 0 0
\(837\) 88.5685 3.06138
\(838\) 0 0
\(839\) −30.4853 −1.05247 −0.526234 0.850340i \(-0.676397\pi\)
−0.526234 + 0.850340i \(0.676397\pi\)
\(840\) 0 0
\(841\) 59.6274 2.05612
\(842\) 0 0
\(843\) 61.1127 2.10483
\(844\) 0 0
\(845\) −1.34315 −0.0462056
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 49.6985 1.70565
\(850\) 0 0
\(851\) −29.0711 −0.996543
\(852\) 0 0
\(853\) 52.4558 1.79605 0.898027 0.439940i \(-0.145000\pi\)
0.898027 + 0.439940i \(0.145000\pi\)
\(854\) 0 0
\(855\) 8.65685 0.296058
\(856\) 0 0
\(857\) −7.02944 −0.240121 −0.120061 0.992767i \(-0.538309\pi\)
−0.120061 + 0.992767i \(0.538309\pi\)
\(858\) 0 0
\(859\) 27.1838 0.927498 0.463749 0.885967i \(-0.346504\pi\)
0.463749 + 0.885967i \(0.346504\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.8701 −0.846587 −0.423293 0.905993i \(-0.639126\pi\)
−0.423293 + 0.905993i \(0.639126\pi\)
\(864\) 0 0
\(865\) 9.65685 0.328343
\(866\) 0 0
\(867\) −101.154 −3.43538
\(868\) 0 0
\(869\) −12.6447 −0.428941
\(870\) 0 0
\(871\) 10.8284 0.366907
\(872\) 0 0
\(873\) 48.9706 1.65740
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 36.8701 1.24501 0.622507 0.782614i \(-0.286114\pi\)
0.622507 + 0.782614i \(0.286114\pi\)
\(878\) 0 0
\(879\) 32.9706 1.11207
\(880\) 0 0
\(881\) −7.31371 −0.246405 −0.123203 0.992382i \(-0.539316\pi\)
−0.123203 + 0.992382i \(0.539316\pi\)
\(882\) 0 0
\(883\) −25.1716 −0.847091 −0.423545 0.905875i \(-0.639215\pi\)
−0.423545 + 0.905875i \(0.639215\pi\)
\(884\) 0 0
\(885\) −2.82843 −0.0950765
\(886\) 0 0
\(887\) 28.2843 0.949693 0.474846 0.880069i \(-0.342504\pi\)
0.474846 + 0.880069i \(0.342504\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −129.610 −4.34210
\(892\) 0 0
\(893\) 1.24264 0.0415834
\(894\) 0 0
\(895\) 6.48528 0.216779
\(896\) 0 0
\(897\) −51.4558 −1.71806
\(898\) 0 0
\(899\) 43.1716 1.43985
\(900\) 0 0
\(901\) −36.9706 −1.23167
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.7279 0.489573
\(906\) 0 0
\(907\) −49.5563 −1.64549 −0.822746 0.568410i \(-0.807559\pi\)
−0.822746 + 0.568410i \(0.807559\pi\)
\(908\) 0 0
\(909\) −12.8579 −0.426468
\(910\) 0 0
\(911\) −25.5563 −0.846720 −0.423360 0.905962i \(-0.639149\pi\)
−0.423360 + 0.905962i \(0.639149\pi\)
\(912\) 0 0
\(913\) 48.3137 1.59895
\(914\) 0 0
\(915\) −17.0711 −0.564352
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.58579 0.184258 0.0921290 0.995747i \(-0.470633\pi\)
0.0921290 + 0.995747i \(0.470633\pi\)
\(920\) 0 0
\(921\) 63.5980 2.09562
\(922\) 0 0
\(923\) −26.4853 −0.871774
\(924\) 0 0
\(925\) 26.3431 0.866157
\(926\) 0 0
\(927\) −20.2843 −0.666223
\(928\) 0 0
\(929\) 35.6274 1.16890 0.584449 0.811431i \(-0.301311\pi\)
0.584449 + 0.811431i \(0.301311\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 27.3137 0.894211
\(934\) 0 0
\(935\) −22.1421 −0.724125
\(936\) 0 0
\(937\) −30.5147 −0.996872 −0.498436 0.866926i \(-0.666092\pi\)
−0.498436 + 0.866926i \(0.666092\pi\)
\(938\) 0 0
\(939\) 3.89949 0.127255
\(940\) 0 0
\(941\) −27.4142 −0.893678 −0.446839 0.894614i \(-0.647450\pi\)
−0.446839 + 0.894614i \(0.647450\pi\)
\(942\) 0 0
\(943\) 24.9706 0.813153
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.0000 −0.844886 −0.422443 0.906389i \(-0.638827\pi\)
−0.422443 + 0.906389i \(0.638827\pi\)
\(948\) 0 0
\(949\) 23.8995 0.775810
\(950\) 0 0
\(951\) 34.9706 1.13400
\(952\) 0 0
\(953\) −59.1543 −1.91620 −0.958098 0.286439i \(-0.907528\pi\)
−0.958098 + 0.286439i \(0.907528\pi\)
\(954\) 0 0
\(955\) 9.24264 0.299085
\(956\) 0 0
\(957\) −104.225 −3.36913
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) 147.782 4.76220
\(964\) 0 0
\(965\) 2.24264 0.0721932
\(966\) 0 0
\(967\) 16.9706 0.545737 0.272868 0.962051i \(-0.412028\pi\)
0.272868 + 0.962051i \(0.412028\pi\)
\(968\) 0 0
\(969\) −23.3137 −0.748944
\(970\) 0 0
\(971\) −48.8701 −1.56831 −0.784157 0.620562i \(-0.786905\pi\)
−0.784157 + 0.620562i \(0.786905\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 46.6274 1.49327
\(976\) 0 0
\(977\) 17.3137 0.553915 0.276957 0.960882i \(-0.410674\pi\)
0.276957 + 0.960882i \(0.410674\pi\)
\(978\) 0 0
\(979\) 43.9584 1.40492
\(980\) 0 0
\(981\) −167.196 −5.33816
\(982\) 0 0
\(983\) 49.9411 1.59287 0.796437 0.604721i \(-0.206715\pi\)
0.796437 + 0.604721i \(0.206715\pi\)
\(984\) 0 0
\(985\) −0.514719 −0.0164003
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −14.3137 −0.455149
\(990\) 0 0
\(991\) −43.4558 −1.38042 −0.690210 0.723609i \(-0.742482\pi\)
−0.690210 + 0.723609i \(0.742482\pi\)
\(992\) 0 0
\(993\) 63.1127 2.00282
\(994\) 0 0
\(995\) −2.07107 −0.0656573
\(996\) 0 0
\(997\) −19.5980 −0.620674 −0.310337 0.950627i \(-0.600442\pi\)
−0.310337 + 0.950627i \(0.600442\pi\)
\(998\) 0 0
\(999\) 127.196 4.02430
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7448.2.a.x.1.1 2
7.3 odd 6 1064.2.q.k.457.1 yes 4
7.5 odd 6 1064.2.q.k.305.1 4
7.6 odd 2 7448.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.k.305.1 4 7.5 odd 6
1064.2.q.k.457.1 yes 4 7.3 odd 6
7448.2.a.x.1.1 2 1.1 even 1 trivial
7448.2.a.bd.1.2 2 7.6 odd 2