Properties

Label 768.4.a.e.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} +53.6888 q^{11} +7.15559 q^{13} +55.2666 q^{15} +39.6888 q^{17} +125.689 q^{19} +67.2666 q^{21} -99.1556 q^{23} +214.378 q^{25} -27.0000 q^{27} +205.800 q^{29} -147.489 q^{31} -161.066 q^{33} +413.066 q^{35} +125.689 q^{37} -21.4668 q^{39} -506.444 q^{41} -413.689 q^{43} -165.800 q^{45} +313.911 q^{47} +159.755 q^{49} -119.066 q^{51} -44.3331 q^{53} -989.066 q^{55} -377.066 q^{57} -324.000 q^{59} +324.000 q^{61} -201.800 q^{63} -131.822 q^{65} -464.266 q^{67} +297.467 q^{69} -1052.84 q^{71} +1022.27 q^{73} -643.133 q^{75} -1203.82 q^{77} +602.910 q^{79} +81.0000 q^{81} -15.8217 q^{83} -731.156 q^{85} -617.400 q^{87} +381.378 q^{89} -160.444 q^{91} +442.466 q^{93} -2315.47 q^{95} +659.154 q^{97} +483.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{5} - 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{5} - 16 q^{7} + 18 q^{9} - 8 q^{11} + 72 q^{13} + 24 q^{15} - 36 q^{17} + 136 q^{19} + 48 q^{21} - 256 q^{23} + 198 q^{25} - 54 q^{27} + 152 q^{29} + 80 q^{31} + 24 q^{33} + 480 q^{35} + 136 q^{37} - 216 q^{39} - 436 q^{41} - 712 q^{43} - 72 q^{45} + 224 q^{47} - 142 q^{49} + 108 q^{51} + 344 q^{53} - 1632 q^{55} - 408 q^{57} - 648 q^{59} + 648 q^{61} - 144 q^{63} + 544 q^{65} + 456 q^{67} + 768 q^{69} - 2048 q^{71} + 660 q^{73} - 594 q^{75} - 1600 q^{77} - 496 q^{79} + 162 q^{81} + 776 q^{83} - 1520 q^{85} - 456 q^{87} + 532 q^{89} + 256 q^{91} - 240 q^{93} - 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −18.4222 −1.64773 −0.823866 0.566785i \(-0.808187\pi\)
−0.823866 + 0.566785i \(0.808187\pi\)
\(6\) 0 0
\(7\) −22.4222 −1.21069 −0.605343 0.795965i \(-0.706964\pi\)
−0.605343 + 0.795965i \(0.706964\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 53.6888 1.47162 0.735809 0.677190i \(-0.236802\pi\)
0.735809 + 0.677190i \(0.236802\pi\)
\(12\) 0 0
\(13\) 7.15559 0.152662 0.0763309 0.997083i \(-0.475679\pi\)
0.0763309 + 0.997083i \(0.475679\pi\)
\(14\) 0 0
\(15\) 55.2666 0.951319
\(16\) 0 0
\(17\) 39.6888 0.566233 0.283116 0.959086i \(-0.408632\pi\)
0.283116 + 0.959086i \(0.408632\pi\)
\(18\) 0 0
\(19\) 125.689 1.51763 0.758816 0.651306i \(-0.225778\pi\)
0.758816 + 0.651306i \(0.225778\pi\)
\(20\) 0 0
\(21\) 67.2666 0.698989
\(22\) 0 0
\(23\) −99.1556 −0.898929 −0.449465 0.893298i \(-0.648385\pi\)
−0.449465 + 0.893298i \(0.648385\pi\)
\(24\) 0 0
\(25\) 214.378 1.71502
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 205.800 1.31780 0.658898 0.752232i \(-0.271023\pi\)
0.658898 + 0.752232i \(0.271023\pi\)
\(30\) 0 0
\(31\) −147.489 −0.854508 −0.427254 0.904132i \(-0.640519\pi\)
−0.427254 + 0.904132i \(0.640519\pi\)
\(32\) 0 0
\(33\) −161.066 −0.849639
\(34\) 0 0
\(35\) 413.066 1.99489
\(36\) 0 0
\(37\) 125.689 0.558463 0.279231 0.960224i \(-0.409920\pi\)
0.279231 + 0.960224i \(0.409920\pi\)
\(38\) 0 0
\(39\) −21.4668 −0.0881393
\(40\) 0 0
\(41\) −506.444 −1.92910 −0.964552 0.263892i \(-0.914994\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(42\) 0 0
\(43\) −413.689 −1.46714 −0.733569 0.679615i \(-0.762147\pi\)
−0.733569 + 0.679615i \(0.762147\pi\)
\(44\) 0 0
\(45\) −165.800 −0.549244
\(46\) 0 0
\(47\) 313.911 0.974226 0.487113 0.873339i \(-0.338050\pi\)
0.487113 + 0.873339i \(0.338050\pi\)
\(48\) 0 0
\(49\) 159.755 0.465759
\(50\) 0 0
\(51\) −119.066 −0.326914
\(52\) 0 0
\(53\) −44.3331 −0.114898 −0.0574492 0.998348i \(-0.518297\pi\)
−0.0574492 + 0.998348i \(0.518297\pi\)
\(54\) 0 0
\(55\) −989.066 −2.42483
\(56\) 0 0
\(57\) −377.066 −0.876205
\(58\) 0 0
\(59\) −324.000 −0.714936 −0.357468 0.933925i \(-0.616360\pi\)
−0.357468 + 0.933925i \(0.616360\pi\)
\(60\) 0 0
\(61\) 324.000 0.680065 0.340032 0.940414i \(-0.389562\pi\)
0.340032 + 0.940414i \(0.389562\pi\)
\(62\) 0 0
\(63\) −201.800 −0.403562
\(64\) 0 0
\(65\) −131.822 −0.251546
\(66\) 0 0
\(67\) −464.266 −0.846554 −0.423277 0.906000i \(-0.639120\pi\)
−0.423277 + 0.906000i \(0.639120\pi\)
\(68\) 0 0
\(69\) 297.467 0.518997
\(70\) 0 0
\(71\) −1052.84 −1.75985 −0.879927 0.475109i \(-0.842409\pi\)
−0.879927 + 0.475109i \(0.842409\pi\)
\(72\) 0 0
\(73\) 1022.27 1.63900 0.819501 0.573078i \(-0.194251\pi\)
0.819501 + 0.573078i \(0.194251\pi\)
\(74\) 0 0
\(75\) −643.133 −0.990168
\(76\) 0 0
\(77\) −1203.82 −1.78167
\(78\) 0 0
\(79\) 602.910 0.858642 0.429321 0.903152i \(-0.358753\pi\)
0.429321 + 0.903152i \(0.358753\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −15.8217 −0.0209236 −0.0104618 0.999945i \(-0.503330\pi\)
−0.0104618 + 0.999945i \(0.503330\pi\)
\(84\) 0 0
\(85\) −731.156 −0.933000
\(86\) 0 0
\(87\) −617.400 −0.760830
\(88\) 0 0
\(89\) 381.378 0.454224 0.227112 0.973869i \(-0.427072\pi\)
0.227112 + 0.973869i \(0.427072\pi\)
\(90\) 0 0
\(91\) −160.444 −0.184825
\(92\) 0 0
\(93\) 442.466 0.493350
\(94\) 0 0
\(95\) −2315.47 −2.50065
\(96\) 0 0
\(97\) 659.154 0.689969 0.344984 0.938608i \(-0.387884\pi\)
0.344984 + 0.938608i \(0.387884\pi\)
\(98\) 0 0
\(99\) 483.199 0.490539
\(100\) 0 0
\(101\) 498.510 0.491125 0.245562 0.969381i \(-0.421027\pi\)
0.245562 + 0.969381i \(0.421027\pi\)
\(102\) 0 0
\(103\) 196.821 0.188285 0.0941425 0.995559i \(-0.469989\pi\)
0.0941425 + 0.995559i \(0.469989\pi\)
\(104\) 0 0
\(105\) −1239.20 −1.15175
\(106\) 0 0
\(107\) −359.378 −0.324695 −0.162347 0.986734i \(-0.551907\pi\)
−0.162347 + 0.986734i \(0.551907\pi\)
\(108\) 0 0
\(109\) 1969.73 1.73088 0.865441 0.501011i \(-0.167038\pi\)
0.865441 + 0.501011i \(0.167038\pi\)
\(110\) 0 0
\(111\) −377.066 −0.322429
\(112\) 0 0
\(113\) −693.643 −0.577456 −0.288728 0.957411i \(-0.593232\pi\)
−0.288728 + 0.957411i \(0.593232\pi\)
\(114\) 0 0
\(115\) 1826.66 1.48119
\(116\) 0 0
\(117\) 64.4003 0.0508873
\(118\) 0 0
\(119\) −889.911 −0.685529
\(120\) 0 0
\(121\) 1551.49 1.16566
\(122\) 0 0
\(123\) 1519.33 1.11377
\(124\) 0 0
\(125\) −1646.53 −1.17816
\(126\) 0 0
\(127\) −2656.78 −1.85631 −0.928153 0.372199i \(-0.878604\pi\)
−0.928153 + 0.372199i \(0.878604\pi\)
\(128\) 0 0
\(129\) 1241.07 0.847053
\(130\) 0 0
\(131\) −615.734 −0.410664 −0.205332 0.978692i \(-0.565827\pi\)
−0.205332 + 0.978692i \(0.565827\pi\)
\(132\) 0 0
\(133\) −2818.22 −1.83737
\(134\) 0 0
\(135\) 497.400 0.317106
\(136\) 0 0
\(137\) −613.290 −0.382459 −0.191230 0.981545i \(-0.561247\pi\)
−0.191230 + 0.981545i \(0.561247\pi\)
\(138\) 0 0
\(139\) 1899.29 1.15896 0.579480 0.814987i \(-0.303256\pi\)
0.579480 + 0.814987i \(0.303256\pi\)
\(140\) 0 0
\(141\) −941.733 −0.562469
\(142\) 0 0
\(143\) 384.175 0.224660
\(144\) 0 0
\(145\) −3791.29 −2.17137
\(146\) 0 0
\(147\) −479.266 −0.268906
\(148\) 0 0
\(149\) −976.377 −0.536832 −0.268416 0.963303i \(-0.586500\pi\)
−0.268416 + 0.963303i \(0.586500\pi\)
\(150\) 0 0
\(151\) −683.132 −0.368162 −0.184081 0.982911i \(-0.558931\pi\)
−0.184081 + 0.982911i \(0.558931\pi\)
\(152\) 0 0
\(153\) 357.199 0.188744
\(154\) 0 0
\(155\) 2717.07 1.40800
\(156\) 0 0
\(157\) −511.109 −0.259815 −0.129907 0.991526i \(-0.541468\pi\)
−0.129907 + 0.991526i \(0.541468\pi\)
\(158\) 0 0
\(159\) 132.999 0.0663366
\(160\) 0 0
\(161\) 2223.29 1.08832
\(162\) 0 0
\(163\) −2425.95 −1.16574 −0.582869 0.812566i \(-0.698070\pi\)
−0.582869 + 0.812566i \(0.698070\pi\)
\(164\) 0 0
\(165\) 2967.20 1.39998
\(166\) 0 0
\(167\) −337.332 −0.156309 −0.0781544 0.996941i \(-0.524903\pi\)
−0.0781544 + 0.996941i \(0.524903\pi\)
\(168\) 0 0
\(169\) −2145.80 −0.976694
\(170\) 0 0
\(171\) 1131.20 0.505877
\(172\) 0 0
\(173\) −2648.29 −1.16385 −0.581924 0.813243i \(-0.697700\pi\)
−0.581924 + 0.813243i \(0.697700\pi\)
\(174\) 0 0
\(175\) −4806.82 −2.07635
\(176\) 0 0
\(177\) 972.000 0.412768
\(178\) 0 0
\(179\) −2907.29 −1.21397 −0.606986 0.794713i \(-0.707621\pi\)
−0.606986 + 0.794713i \(0.707621\pi\)
\(180\) 0 0
\(181\) 3682.80 1.51238 0.756188 0.654354i \(-0.227060\pi\)
0.756188 + 0.654354i \(0.227060\pi\)
\(182\) 0 0
\(183\) −972.000 −0.392636
\(184\) 0 0
\(185\) −2315.47 −0.920197
\(186\) 0 0
\(187\) 2130.85 0.833277
\(188\) 0 0
\(189\) 605.400 0.232996
\(190\) 0 0
\(191\) −1279.56 −0.484740 −0.242370 0.970184i \(-0.577925\pi\)
−0.242370 + 0.970184i \(0.577925\pi\)
\(192\) 0 0
\(193\) −4836.84 −1.80396 −0.901978 0.431783i \(-0.857885\pi\)
−0.901978 + 0.431783i \(0.857885\pi\)
\(194\) 0 0
\(195\) 395.465 0.145230
\(196\) 0 0
\(197\) −2869.31 −1.03771 −0.518857 0.854861i \(-0.673642\pi\)
−0.518857 + 0.854861i \(0.673642\pi\)
\(198\) 0 0
\(199\) −652.242 −0.232343 −0.116171 0.993229i \(-0.537062\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(200\) 0 0
\(201\) 1392.80 0.488758
\(202\) 0 0
\(203\) −4614.49 −1.59544
\(204\) 0 0
\(205\) 9329.82 3.17865
\(206\) 0 0
\(207\) −892.400 −0.299643
\(208\) 0 0
\(209\) 6748.08 2.23337
\(210\) 0 0
\(211\) −537.511 −0.175373 −0.0876866 0.996148i \(-0.527947\pi\)
−0.0876866 + 0.996148i \(0.527947\pi\)
\(212\) 0 0
\(213\) 3158.53 1.01605
\(214\) 0 0
\(215\) 7621.06 2.41745
\(216\) 0 0
\(217\) 3307.02 1.03454
\(218\) 0 0
\(219\) −3066.80 −0.946278
\(220\) 0 0
\(221\) 283.997 0.0864421
\(222\) 0 0
\(223\) 4041.98 1.21377 0.606885 0.794790i \(-0.292419\pi\)
0.606885 + 0.794790i \(0.292419\pi\)
\(224\) 0 0
\(225\) 1929.40 0.571674
\(226\) 0 0
\(227\) −3070.22 −0.897699 −0.448850 0.893607i \(-0.648166\pi\)
−0.448850 + 0.893607i \(0.648166\pi\)
\(228\) 0 0
\(229\) −205.110 −0.0591881 −0.0295940 0.999562i \(-0.509421\pi\)
−0.0295940 + 0.999562i \(0.509421\pi\)
\(230\) 0 0
\(231\) 3611.47 1.02864
\(232\) 0 0
\(233\) 13.3776 0.00376137 0.00188068 0.999998i \(-0.499401\pi\)
0.00188068 + 0.999998i \(0.499401\pi\)
\(234\) 0 0
\(235\) −5782.93 −1.60526
\(236\) 0 0
\(237\) −1808.73 −0.495737
\(238\) 0 0
\(239\) −4327.99 −1.17136 −0.585679 0.810543i \(-0.699172\pi\)
−0.585679 + 0.810543i \(0.699172\pi\)
\(240\) 0 0
\(241\) −1508.31 −0.403150 −0.201575 0.979473i \(-0.564606\pi\)
−0.201575 + 0.979473i \(0.564606\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −2943.04 −0.767446
\(246\) 0 0
\(247\) 899.378 0.231684
\(248\) 0 0
\(249\) 47.4652 0.0120803
\(250\) 0 0
\(251\) 4871.47 1.22504 0.612518 0.790456i \(-0.290157\pi\)
0.612518 + 0.790456i \(0.290157\pi\)
\(252\) 0 0
\(253\) −5323.55 −1.32288
\(254\) 0 0
\(255\) 2193.47 0.538668
\(256\) 0 0
\(257\) −1665.55 −0.404258 −0.202129 0.979359i \(-0.564786\pi\)
−0.202129 + 0.979359i \(0.564786\pi\)
\(258\) 0 0
\(259\) −2818.22 −0.676122
\(260\) 0 0
\(261\) 1852.20 0.439265
\(262\) 0 0
\(263\) −7167.64 −1.68052 −0.840258 0.542188i \(-0.817596\pi\)
−0.840258 + 0.542188i \(0.817596\pi\)
\(264\) 0 0
\(265\) 816.713 0.189322
\(266\) 0 0
\(267\) −1144.13 −0.262246
\(268\) 0 0
\(269\) −5453.84 −1.23616 −0.618079 0.786116i \(-0.712089\pi\)
−0.618079 + 0.786116i \(0.712089\pi\)
\(270\) 0 0
\(271\) 5416.20 1.21406 0.607031 0.794678i \(-0.292360\pi\)
0.607031 + 0.794678i \(0.292360\pi\)
\(272\) 0 0
\(273\) 481.332 0.106709
\(274\) 0 0
\(275\) 11509.7 2.52385
\(276\) 0 0
\(277\) −2648.75 −0.574542 −0.287271 0.957849i \(-0.592748\pi\)
−0.287271 + 0.957849i \(0.592748\pi\)
\(278\) 0 0
\(279\) −1327.40 −0.284836
\(280\) 0 0
\(281\) 6664.57 1.41486 0.707429 0.706784i \(-0.249855\pi\)
0.707429 + 0.706784i \(0.249855\pi\)
\(282\) 0 0
\(283\) 5630.84 1.18275 0.591376 0.806396i \(-0.298585\pi\)
0.591376 + 0.806396i \(0.298585\pi\)
\(284\) 0 0
\(285\) 6946.40 1.44375
\(286\) 0 0
\(287\) 11355.6 2.33554
\(288\) 0 0
\(289\) −3337.80 −0.679381
\(290\) 0 0
\(291\) −1977.46 −0.398354
\(292\) 0 0
\(293\) 908.374 0.181119 0.0905593 0.995891i \(-0.471135\pi\)
0.0905593 + 0.995891i \(0.471135\pi\)
\(294\) 0 0
\(295\) 5968.79 1.17802
\(296\) 0 0
\(297\) −1449.60 −0.283213
\(298\) 0 0
\(299\) −709.517 −0.137232
\(300\) 0 0
\(301\) 9275.82 1.77624
\(302\) 0 0
\(303\) −1495.53 −0.283551
\(304\) 0 0
\(305\) −5968.79 −1.12056
\(306\) 0 0
\(307\) 414.671 0.0770896 0.0385448 0.999257i \(-0.487728\pi\)
0.0385448 + 0.999257i \(0.487728\pi\)
\(308\) 0 0
\(309\) −590.463 −0.108706
\(310\) 0 0
\(311\) −1615.91 −0.294629 −0.147315 0.989090i \(-0.547063\pi\)
−0.147315 + 0.989090i \(0.547063\pi\)
\(312\) 0 0
\(313\) −8479.33 −1.53125 −0.765623 0.643289i \(-0.777569\pi\)
−0.765623 + 0.643289i \(0.777569\pi\)
\(314\) 0 0
\(315\) 3717.60 0.664962
\(316\) 0 0
\(317\) −6774.73 −1.20034 −0.600169 0.799873i \(-0.704900\pi\)
−0.600169 + 0.799873i \(0.704900\pi\)
\(318\) 0 0
\(319\) 11049.2 1.93929
\(320\) 0 0
\(321\) 1078.13 0.187463
\(322\) 0 0
\(323\) 4988.44 0.859332
\(324\) 0 0
\(325\) 1534.00 0.261818
\(326\) 0 0
\(327\) −5909.20 −0.999325
\(328\) 0 0
\(329\) −7038.57 −1.17948
\(330\) 0 0
\(331\) −9292.36 −1.54306 −0.771532 0.636191i \(-0.780509\pi\)
−0.771532 + 0.636191i \(0.780509\pi\)
\(332\) 0 0
\(333\) 1131.20 0.186154
\(334\) 0 0
\(335\) 8552.80 1.39489
\(336\) 0 0
\(337\) 6563.78 1.06098 0.530492 0.847690i \(-0.322007\pi\)
0.530492 + 0.847690i \(0.322007\pi\)
\(338\) 0 0
\(339\) 2080.93 0.333394
\(340\) 0 0
\(341\) −7918.49 −1.25751
\(342\) 0 0
\(343\) 4108.75 0.646798
\(344\) 0 0
\(345\) −5479.99 −0.855168
\(346\) 0 0
\(347\) 3870.93 0.598855 0.299427 0.954119i \(-0.403204\pi\)
0.299427 + 0.954119i \(0.403204\pi\)
\(348\) 0 0
\(349\) 3474.57 0.532922 0.266461 0.963846i \(-0.414146\pi\)
0.266461 + 0.963846i \(0.414146\pi\)
\(350\) 0 0
\(351\) −193.201 −0.0293798
\(352\) 0 0
\(353\) −4308.58 −0.649639 −0.324820 0.945776i \(-0.605304\pi\)
−0.324820 + 0.945776i \(0.605304\pi\)
\(354\) 0 0
\(355\) 19395.7 2.89977
\(356\) 0 0
\(357\) 2669.73 0.395791
\(358\) 0 0
\(359\) 8161.19 1.19981 0.599904 0.800072i \(-0.295205\pi\)
0.599904 + 0.800072i \(0.295205\pi\)
\(360\) 0 0
\(361\) 8938.68 1.30320
\(362\) 0 0
\(363\) −4654.47 −0.672992
\(364\) 0 0
\(365\) −18832.4 −2.70064
\(366\) 0 0
\(367\) −4427.66 −0.629760 −0.314880 0.949132i \(-0.601964\pi\)
−0.314880 + 0.949132i \(0.601964\pi\)
\(368\) 0 0
\(369\) −4558.00 −0.643035
\(370\) 0 0
\(371\) 994.045 0.139106
\(372\) 0 0
\(373\) 11278.5 1.56562 0.782812 0.622258i \(-0.213785\pi\)
0.782812 + 0.622258i \(0.213785\pi\)
\(374\) 0 0
\(375\) 4939.60 0.680213
\(376\) 0 0
\(377\) 1472.62 0.201177
\(378\) 0 0
\(379\) −709.683 −0.0961846 −0.0480923 0.998843i \(-0.515314\pi\)
−0.0480923 + 0.998843i \(0.515314\pi\)
\(380\) 0 0
\(381\) 7970.33 1.07174
\(382\) 0 0
\(383\) 1233.69 0.164592 0.0822962 0.996608i \(-0.473775\pi\)
0.0822962 + 0.996608i \(0.473775\pi\)
\(384\) 0 0
\(385\) 22177.1 2.93571
\(386\) 0 0
\(387\) −3723.20 −0.489046
\(388\) 0 0
\(389\) −6830.06 −0.890226 −0.445113 0.895474i \(-0.646836\pi\)
−0.445113 + 0.895474i \(0.646836\pi\)
\(390\) 0 0
\(391\) −3935.37 −0.509003
\(392\) 0 0
\(393\) 1847.20 0.237097
\(394\) 0 0
\(395\) −11106.9 −1.41481
\(396\) 0 0
\(397\) −11289.7 −1.42724 −0.713618 0.700535i \(-0.752945\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(398\) 0 0
\(399\) 8454.66 1.06081
\(400\) 0 0
\(401\) −3055.59 −0.380521 −0.190261 0.981734i \(-0.560933\pi\)
−0.190261 + 0.981734i \(0.560933\pi\)
\(402\) 0 0
\(403\) −1055.37 −0.130451
\(404\) 0 0
\(405\) −1492.20 −0.183081
\(406\) 0 0
\(407\) 6748.08 0.821843
\(408\) 0 0
\(409\) 4089.01 0.494349 0.247175 0.968971i \(-0.420498\pi\)
0.247175 + 0.968971i \(0.420498\pi\)
\(410\) 0 0
\(411\) 1839.87 0.220813
\(412\) 0 0
\(413\) 7264.79 0.865562
\(414\) 0 0
\(415\) 291.471 0.0344765
\(416\) 0 0
\(417\) −5697.86 −0.669126
\(418\) 0 0
\(419\) −15397.0 −1.79520 −0.897602 0.440806i \(-0.854693\pi\)
−0.897602 + 0.440806i \(0.854693\pi\)
\(420\) 0 0
\(421\) 1034.45 0.119753 0.0598766 0.998206i \(-0.480929\pi\)
0.0598766 + 0.998206i \(0.480929\pi\)
\(422\) 0 0
\(423\) 2825.20 0.324742
\(424\) 0 0
\(425\) 8508.40 0.971101
\(426\) 0 0
\(427\) −7264.79 −0.823344
\(428\) 0 0
\(429\) −1152.53 −0.129707
\(430\) 0 0
\(431\) 4943.86 0.552523 0.276261 0.961083i \(-0.410904\pi\)
0.276261 + 0.961083i \(0.410904\pi\)
\(432\) 0 0
\(433\) 337.202 0.0374247 0.0187124 0.999825i \(-0.494043\pi\)
0.0187124 + 0.999825i \(0.494043\pi\)
\(434\) 0 0
\(435\) 11373.9 1.25364
\(436\) 0 0
\(437\) −12462.7 −1.36424
\(438\) 0 0
\(439\) −4493.93 −0.488573 −0.244286 0.969703i \(-0.578554\pi\)
−0.244286 + 0.969703i \(0.578554\pi\)
\(440\) 0 0
\(441\) 1437.80 0.155253
\(442\) 0 0
\(443\) −4292.26 −0.460341 −0.230171 0.973150i \(-0.573928\pi\)
−0.230171 + 0.973150i \(0.573928\pi\)
\(444\) 0 0
\(445\) −7025.82 −0.748440
\(446\) 0 0
\(447\) 2929.13 0.309940
\(448\) 0 0
\(449\) −4167.96 −0.438081 −0.219040 0.975716i \(-0.570293\pi\)
−0.219040 + 0.975716i \(0.570293\pi\)
\(450\) 0 0
\(451\) −27190.4 −2.83890
\(452\) 0 0
\(453\) 2049.40 0.212559
\(454\) 0 0
\(455\) 2955.73 0.304543
\(456\) 0 0
\(457\) 301.643 0.0308759 0.0154380 0.999881i \(-0.495086\pi\)
0.0154380 + 0.999881i \(0.495086\pi\)
\(458\) 0 0
\(459\) −1071.60 −0.108971
\(460\) 0 0
\(461\) 9611.88 0.971085 0.485542 0.874213i \(-0.338622\pi\)
0.485542 + 0.874213i \(0.338622\pi\)
\(462\) 0 0
\(463\) 13251.0 1.33008 0.665041 0.746807i \(-0.268414\pi\)
0.665041 + 0.746807i \(0.268414\pi\)
\(464\) 0 0
\(465\) −8151.20 −0.812909
\(466\) 0 0
\(467\) −4432.00 −0.439161 −0.219581 0.975594i \(-0.570469\pi\)
−0.219581 + 0.975594i \(0.570469\pi\)
\(468\) 0 0
\(469\) 10409.9 1.02491
\(470\) 0 0
\(471\) 1533.33 0.150004
\(472\) 0 0
\(473\) −22210.5 −2.15907
\(474\) 0 0
\(475\) 26944.9 2.60277
\(476\) 0 0
\(477\) −398.998 −0.0382995
\(478\) 0 0
\(479\) −9076.49 −0.865794 −0.432897 0.901443i \(-0.642509\pi\)
−0.432897 + 0.901443i \(0.642509\pi\)
\(480\) 0 0
\(481\) 899.378 0.0852559
\(482\) 0 0
\(483\) −6669.86 −0.628342
\(484\) 0 0
\(485\) −12143.1 −1.13688
\(486\) 0 0
\(487\) 3343.89 0.311142 0.155571 0.987825i \(-0.450278\pi\)
0.155571 + 0.987825i \(0.450278\pi\)
\(488\) 0 0
\(489\) 7277.86 0.673040
\(490\) 0 0
\(491\) −2423.73 −0.222773 −0.111386 0.993777i \(-0.535529\pi\)
−0.111386 + 0.993777i \(0.535529\pi\)
\(492\) 0 0
\(493\) 8167.95 0.746179
\(494\) 0 0
\(495\) −8901.60 −0.808277
\(496\) 0 0
\(497\) 23607.1 2.13063
\(498\) 0 0
\(499\) 811.819 0.0728296 0.0364148 0.999337i \(-0.488406\pi\)
0.0364148 + 0.999337i \(0.488406\pi\)
\(500\) 0 0
\(501\) 1012.00 0.0902449
\(502\) 0 0
\(503\) −18192.4 −1.61264 −0.806320 0.591479i \(-0.798544\pi\)
−0.806320 + 0.591479i \(0.798544\pi\)
\(504\) 0 0
\(505\) −9183.65 −0.809242
\(506\) 0 0
\(507\) 6437.39 0.563895
\(508\) 0 0
\(509\) 5645.44 0.491610 0.245805 0.969319i \(-0.420948\pi\)
0.245805 + 0.969319i \(0.420948\pi\)
\(510\) 0 0
\(511\) −22921.5 −1.98432
\(512\) 0 0
\(513\) −3393.60 −0.292068
\(514\) 0 0
\(515\) −3625.88 −0.310243
\(516\) 0 0
\(517\) 16853.5 1.43369
\(518\) 0 0
\(519\) 7944.87 0.671948
\(520\) 0 0
\(521\) 12338.7 1.03756 0.518780 0.854908i \(-0.326386\pi\)
0.518780 + 0.854908i \(0.326386\pi\)
\(522\) 0 0
\(523\) −10609.8 −0.887062 −0.443531 0.896259i \(-0.646274\pi\)
−0.443531 + 0.896259i \(0.646274\pi\)
\(524\) 0 0
\(525\) 14420.5 1.19878
\(526\) 0 0
\(527\) −5853.65 −0.483850
\(528\) 0 0
\(529\) −2335.17 −0.191926
\(530\) 0 0
\(531\) −2916.00 −0.238312
\(532\) 0 0
\(533\) −3623.91 −0.294501
\(534\) 0 0
\(535\) 6620.53 0.535010
\(536\) 0 0
\(537\) 8721.86 0.700887
\(538\) 0 0
\(539\) 8577.07 0.685419
\(540\) 0 0
\(541\) −4035.42 −0.320696 −0.160348 0.987061i \(-0.551262\pi\)
−0.160348 + 0.987061i \(0.551262\pi\)
\(542\) 0 0
\(543\) −11048.4 −0.873171
\(544\) 0 0
\(545\) −36286.8 −2.85203
\(546\) 0 0
\(547\) 7407.45 0.579012 0.289506 0.957176i \(-0.406509\pi\)
0.289506 + 0.957176i \(0.406509\pi\)
\(548\) 0 0
\(549\) 2916.00 0.226688
\(550\) 0 0
\(551\) 25866.7 1.99993
\(552\) 0 0
\(553\) −13518.6 −1.03954
\(554\) 0 0
\(555\) 6946.40 0.531276
\(556\) 0 0
\(557\) 9500.77 0.722730 0.361365 0.932424i \(-0.382311\pi\)
0.361365 + 0.932424i \(0.382311\pi\)
\(558\) 0 0
\(559\) −2960.19 −0.223976
\(560\) 0 0
\(561\) −6392.54 −0.481093
\(562\) 0 0
\(563\) 2700.26 0.202136 0.101068 0.994880i \(-0.467774\pi\)
0.101068 + 0.994880i \(0.467774\pi\)
\(564\) 0 0
\(565\) 12778.4 0.951492
\(566\) 0 0
\(567\) −1816.20 −0.134521
\(568\) 0 0
\(569\) 15904.9 1.17183 0.585913 0.810374i \(-0.300736\pi\)
0.585913 + 0.810374i \(0.300736\pi\)
\(570\) 0 0
\(571\) 18234.0 1.33638 0.668188 0.743992i \(-0.267070\pi\)
0.668188 + 0.743992i \(0.267070\pi\)
\(572\) 0 0
\(573\) 3838.67 0.279865
\(574\) 0 0
\(575\) −21256.7 −1.54168
\(576\) 0 0
\(577\) 4869.57 0.351339 0.175670 0.984449i \(-0.443791\pi\)
0.175670 + 0.984449i \(0.443791\pi\)
\(578\) 0 0
\(579\) 14510.5 1.04151
\(580\) 0 0
\(581\) 354.758 0.0253319
\(582\) 0 0
\(583\) −2380.19 −0.169086
\(584\) 0 0
\(585\) −1186.40 −0.0838486
\(586\) 0 0
\(587\) −1616.99 −0.113697 −0.0568485 0.998383i \(-0.518105\pi\)
−0.0568485 + 0.998383i \(0.518105\pi\)
\(588\) 0 0
\(589\) −18537.7 −1.29683
\(590\) 0 0
\(591\) 8607.93 0.599125
\(592\) 0 0
\(593\) 8117.01 0.562101 0.281050 0.959693i \(-0.409317\pi\)
0.281050 + 0.959693i \(0.409317\pi\)
\(594\) 0 0
\(595\) 16394.1 1.12957
\(596\) 0 0
\(597\) 1956.73 0.134143
\(598\) 0 0
\(599\) −9536.40 −0.650495 −0.325248 0.945629i \(-0.605448\pi\)
−0.325248 + 0.945629i \(0.605448\pi\)
\(600\) 0 0
\(601\) −16247.8 −1.10276 −0.551381 0.834253i \(-0.685899\pi\)
−0.551381 + 0.834253i \(0.685899\pi\)
\(602\) 0 0
\(603\) −4178.39 −0.282185
\(604\) 0 0
\(605\) −28581.9 −1.92069
\(606\) 0 0
\(607\) −27725.7 −1.85396 −0.926980 0.375111i \(-0.877605\pi\)
−0.926980 + 0.375111i \(0.877605\pi\)
\(608\) 0 0
\(609\) 13843.5 0.921125
\(610\) 0 0
\(611\) 2246.22 0.148727
\(612\) 0 0
\(613\) −927.190 −0.0610911 −0.0305456 0.999533i \(-0.509724\pi\)
−0.0305456 + 0.999533i \(0.509724\pi\)
\(614\) 0 0
\(615\) −27989.5 −1.83519
\(616\) 0 0
\(617\) −18727.8 −1.22196 −0.610982 0.791644i \(-0.709225\pi\)
−0.610982 + 0.791644i \(0.709225\pi\)
\(618\) 0 0
\(619\) 3210.22 0.208449 0.104224 0.994554i \(-0.466764\pi\)
0.104224 + 0.994554i \(0.466764\pi\)
\(620\) 0 0
\(621\) 2677.20 0.172999
\(622\) 0 0
\(623\) −8551.33 −0.549922
\(624\) 0 0
\(625\) 3535.57 0.226276
\(626\) 0 0
\(627\) −20244.3 −1.28944
\(628\) 0 0
\(629\) 4988.44 0.316220
\(630\) 0 0
\(631\) 11911.2 0.751468 0.375734 0.926728i \(-0.377391\pi\)
0.375734 + 0.926728i \(0.377391\pi\)
\(632\) 0 0
\(633\) 1612.53 0.101252
\(634\) 0 0
\(635\) 48943.7 3.05869
\(636\) 0 0
\(637\) 1143.14 0.0711036
\(638\) 0 0
\(639\) −9475.60 −0.586618
\(640\) 0 0
\(641\) 17232.6 1.06185 0.530924 0.847419i \(-0.321845\pi\)
0.530924 + 0.847419i \(0.321845\pi\)
\(642\) 0 0
\(643\) −12754.6 −0.782262 −0.391131 0.920335i \(-0.627916\pi\)
−0.391131 + 0.920335i \(0.627916\pi\)
\(644\) 0 0
\(645\) −22863.2 −1.39572
\(646\) 0 0
\(647\) −9441.73 −0.573714 −0.286857 0.957973i \(-0.592610\pi\)
−0.286857 + 0.957973i \(0.592610\pi\)
\(648\) 0 0
\(649\) −17395.2 −1.05211
\(650\) 0 0
\(651\) −9921.06 −0.597292
\(652\) 0 0
\(653\) 5198.99 0.311565 0.155783 0.987791i \(-0.450210\pi\)
0.155783 + 0.987791i \(0.450210\pi\)
\(654\) 0 0
\(655\) 11343.2 0.676664
\(656\) 0 0
\(657\) 9200.39 0.546334
\(658\) 0 0
\(659\) 6508.01 0.384698 0.192349 0.981327i \(-0.438389\pi\)
0.192349 + 0.981327i \(0.438389\pi\)
\(660\) 0 0
\(661\) −25280.2 −1.48757 −0.743785 0.668419i \(-0.766972\pi\)
−0.743785 + 0.668419i \(0.766972\pi\)
\(662\) 0 0
\(663\) −851.991 −0.0499074
\(664\) 0 0
\(665\) 51917.8 3.02750
\(666\) 0 0
\(667\) −20406.2 −1.18460
\(668\) 0 0
\(669\) −12125.9 −0.700770
\(670\) 0 0
\(671\) 17395.2 1.00079
\(672\) 0 0
\(673\) −5525.89 −0.316504 −0.158252 0.987399i \(-0.550586\pi\)
−0.158252 + 0.987399i \(0.550586\pi\)
\(674\) 0 0
\(675\) −5788.20 −0.330056
\(676\) 0 0
\(677\) 6293.21 0.357264 0.178632 0.983916i \(-0.442833\pi\)
0.178632 + 0.983916i \(0.442833\pi\)
\(678\) 0 0
\(679\) −14779.7 −0.835335
\(680\) 0 0
\(681\) 9210.66 0.518287
\(682\) 0 0
\(683\) −5675.91 −0.317984 −0.158992 0.987280i \(-0.550824\pi\)
−0.158992 + 0.987280i \(0.550824\pi\)
\(684\) 0 0
\(685\) 11298.2 0.630190
\(686\) 0 0
\(687\) 615.331 0.0341722
\(688\) 0 0
\(689\) −317.229 −0.0175406
\(690\) 0 0
\(691\) −3617.79 −0.199171 −0.0995854 0.995029i \(-0.531752\pi\)
−0.0995854 + 0.995029i \(0.531752\pi\)
\(692\) 0 0
\(693\) −10834.4 −0.593888
\(694\) 0 0
\(695\) −34989.1 −1.90966
\(696\) 0 0
\(697\) −20100.2 −1.09232
\(698\) 0 0
\(699\) −40.1329 −0.00217163
\(700\) 0 0
\(701\) 7938.43 0.427718 0.213859 0.976865i \(-0.431397\pi\)
0.213859 + 0.976865i \(0.431397\pi\)
\(702\) 0 0
\(703\) 15797.7 0.847540
\(704\) 0 0
\(705\) 17348.8 0.926799
\(706\) 0 0
\(707\) −11177.7 −0.594597
\(708\) 0 0
\(709\) 25691.6 1.36088 0.680442 0.732802i \(-0.261788\pi\)
0.680442 + 0.732802i \(0.261788\pi\)
\(710\) 0 0
\(711\) 5426.19 0.286214
\(712\) 0 0
\(713\) 14624.3 0.768142
\(714\) 0 0
\(715\) −7077.35 −0.370179
\(716\) 0 0
\(717\) 12984.0 0.676284
\(718\) 0 0
\(719\) −36803.3 −1.90895 −0.954473 0.298298i \(-0.903581\pi\)
−0.954473 + 0.298298i \(0.903581\pi\)
\(720\) 0 0
\(721\) −4413.16 −0.227954
\(722\) 0 0
\(723\) 4524.94 0.232759
\(724\) 0 0
\(725\) 44118.9 2.26005
\(726\) 0 0
\(727\) −28333.2 −1.44542 −0.722709 0.691152i \(-0.757103\pi\)
−0.722709 + 0.691152i \(0.757103\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −16418.8 −0.830742
\(732\) 0 0
\(733\) 10767.9 0.542592 0.271296 0.962496i \(-0.412548\pi\)
0.271296 + 0.962496i \(0.412548\pi\)
\(734\) 0 0
\(735\) 8829.13 0.443085
\(736\) 0 0
\(737\) −24925.9 −1.24580
\(738\) 0 0
\(739\) −17301.4 −0.861221 −0.430610 0.902538i \(-0.641702\pi\)
−0.430610 + 0.902538i \(0.641702\pi\)
\(740\) 0 0
\(741\) −2698.13 −0.133763
\(742\) 0 0
\(743\) −24110.8 −1.19050 −0.595248 0.803542i \(-0.702946\pi\)
−0.595248 + 0.803542i \(0.702946\pi\)
\(744\) 0 0
\(745\) 17987.0 0.884555
\(746\) 0 0
\(747\) −142.396 −0.00697455
\(748\) 0 0
\(749\) 8058.04 0.393103
\(750\) 0 0
\(751\) −30052.8 −1.46024 −0.730121 0.683318i \(-0.760536\pi\)
−0.730121 + 0.683318i \(0.760536\pi\)
\(752\) 0 0
\(753\) −14614.4 −0.707275
\(754\) 0 0
\(755\) 12584.8 0.606633
\(756\) 0 0
\(757\) −25599.4 −1.22910 −0.614549 0.788879i \(-0.710662\pi\)
−0.614549 + 0.788879i \(0.710662\pi\)
\(758\) 0 0
\(759\) 15970.6 0.763765
\(760\) 0 0
\(761\) −28904.5 −1.37685 −0.688427 0.725306i \(-0.741698\pi\)
−0.688427 + 0.725306i \(0.741698\pi\)
\(762\) 0 0
\(763\) −44165.7 −2.09555
\(764\) 0 0
\(765\) −6580.40 −0.311000
\(766\) 0 0
\(767\) −2318.41 −0.109143
\(768\) 0 0
\(769\) −8756.13 −0.410603 −0.205302 0.978699i \(-0.565818\pi\)
−0.205302 + 0.978699i \(0.565818\pi\)
\(770\) 0 0
\(771\) 4996.66 0.233399
\(772\) 0 0
\(773\) −28289.8 −1.31632 −0.658159 0.752879i \(-0.728665\pi\)
−0.658159 + 0.752879i \(0.728665\pi\)
\(774\) 0 0
\(775\) −31618.3 −1.46550
\(776\) 0 0
\(777\) 8454.66 0.390359
\(778\) 0 0
\(779\) −63654.4 −2.92767
\(780\) 0 0
\(781\) −56526.0 −2.58983
\(782\) 0 0
\(783\) −5556.60 −0.253610
\(784\) 0 0
\(785\) 9415.75 0.428105
\(786\) 0 0
\(787\) 4859.74 0.220116 0.110058 0.993925i \(-0.464896\pi\)
0.110058 + 0.993925i \(0.464896\pi\)
\(788\) 0 0
\(789\) 21502.9 0.970246
\(790\) 0 0
\(791\) 15553.0 0.699117
\(792\) 0 0
\(793\) 2318.41 0.103820
\(794\) 0 0
\(795\) −2450.14 −0.109305
\(796\) 0 0
\(797\) −17361.7 −0.771623 −0.385811 0.922578i \(-0.626078\pi\)
−0.385811 + 0.922578i \(0.626078\pi\)
\(798\) 0 0
\(799\) 12458.8 0.551638
\(800\) 0 0
\(801\) 3432.40 0.151408
\(802\) 0 0
\(803\) 54884.2 2.41198
\(804\) 0 0
\(805\) −40957.8 −1.79326
\(806\) 0 0
\(807\) 16361.5 0.713696
\(808\) 0 0
\(809\) −24475.3 −1.06367 −0.531833 0.846849i \(-0.678497\pi\)
−0.531833 + 0.846849i \(0.678497\pi\)
\(810\) 0 0
\(811\) 19875.4 0.860566 0.430283 0.902694i \(-0.358414\pi\)
0.430283 + 0.902694i \(0.358414\pi\)
\(812\) 0 0
\(813\) −16248.6 −0.700939
\(814\) 0 0
\(815\) 44691.4 1.92083
\(816\) 0 0
\(817\) −51996.1 −2.22658
\(818\) 0 0
\(819\) −1444.00 −0.0616085
\(820\) 0 0
\(821\) −21682.1 −0.921693 −0.460846 0.887480i \(-0.652454\pi\)
−0.460846 + 0.887480i \(0.652454\pi\)
\(822\) 0 0
\(823\) −6698.17 −0.283698 −0.141849 0.989888i \(-0.545305\pi\)
−0.141849 + 0.989888i \(0.545305\pi\)
\(824\) 0 0
\(825\) −34529.0 −1.45715
\(826\) 0 0
\(827\) −3390.85 −0.142577 −0.0712886 0.997456i \(-0.522711\pi\)
−0.0712886 + 0.997456i \(0.522711\pi\)
\(828\) 0 0
\(829\) 40093.4 1.67974 0.839869 0.542789i \(-0.182632\pi\)
0.839869 + 0.542789i \(0.182632\pi\)
\(830\) 0 0
\(831\) 7946.25 0.331712
\(832\) 0 0
\(833\) 6340.50 0.263728
\(834\) 0 0
\(835\) 6214.40 0.257555
\(836\) 0 0
\(837\) 3982.19 0.164450
\(838\) 0 0
\(839\) 6172.12 0.253975 0.126988 0.991904i \(-0.459469\pi\)
0.126988 + 0.991904i \(0.459469\pi\)
\(840\) 0 0
\(841\) 17964.6 0.736585
\(842\) 0 0
\(843\) −19993.7 −0.816869
\(844\) 0 0
\(845\) 39530.3 1.60933
\(846\) 0 0
\(847\) −34787.8 −1.41124
\(848\) 0 0
\(849\) −16892.5 −0.682862
\(850\) 0 0
\(851\) −12462.7 −0.502018
\(852\) 0 0
\(853\) 276.632 0.0111040 0.00555198 0.999985i \(-0.498233\pi\)
0.00555198 + 0.999985i \(0.498233\pi\)
\(854\) 0 0
\(855\) −20839.2 −0.833550
\(856\) 0 0
\(857\) 3704.41 0.147655 0.0738274 0.997271i \(-0.476479\pi\)
0.0738274 + 0.997271i \(0.476479\pi\)
\(858\) 0 0
\(859\) 26915.5 1.06909 0.534544 0.845141i \(-0.320484\pi\)
0.534544 + 0.845141i \(0.320484\pi\)
\(860\) 0 0
\(861\) −34066.8 −1.34842
\(862\) 0 0
\(863\) 23623.9 0.931828 0.465914 0.884830i \(-0.345726\pi\)
0.465914 + 0.884830i \(0.345726\pi\)
\(864\) 0 0
\(865\) 48787.3 1.91771
\(866\) 0 0
\(867\) 10013.4 0.392241
\(868\) 0 0
\(869\) 32369.5 1.26359
\(870\) 0 0
\(871\) −3322.10 −0.129236
\(872\) 0 0
\(873\) 5932.39 0.229990
\(874\) 0 0
\(875\) 36918.9 1.42638
\(876\) 0 0
\(877\) −14094.0 −0.542667 −0.271333 0.962485i \(-0.587465\pi\)
−0.271333 + 0.962485i \(0.587465\pi\)
\(878\) 0 0
\(879\) −2725.12 −0.104569
\(880\) 0 0
\(881\) 18967.3 0.725341 0.362671 0.931917i \(-0.381865\pi\)
0.362671 + 0.931917i \(0.381865\pi\)
\(882\) 0 0
\(883\) 32886.0 1.25334 0.626672 0.779283i \(-0.284417\pi\)
0.626672 + 0.779283i \(0.284417\pi\)
\(884\) 0 0
\(885\) −17906.4 −0.680132
\(886\) 0 0
\(887\) 27945.6 1.05786 0.528929 0.848666i \(-0.322594\pi\)
0.528929 + 0.848666i \(0.322594\pi\)
\(888\) 0 0
\(889\) 59570.8 2.24740
\(890\) 0 0
\(891\) 4348.79 0.163513
\(892\) 0 0
\(893\) 39455.1 1.47852
\(894\) 0 0
\(895\) 53558.6 2.00030
\(896\) 0 0
\(897\) 2128.55 0.0792310
\(898\) 0 0
\(899\) −30353.1 −1.12607
\(900\) 0 0
\(901\) −1759.53 −0.0650592
\(902\) 0 0
\(903\) −27827.4 −1.02551
\(904\) 0 0
\(905\) −67845.2 −2.49199
\(906\) 0 0
\(907\) −5544.89 −0.202993 −0.101497 0.994836i \(-0.532363\pi\)
−0.101497 + 0.994836i \(0.532363\pi\)
\(908\) 0 0
\(909\) 4486.59 0.163708
\(910\) 0 0
\(911\) −19638.9 −0.714233 −0.357116 0.934060i \(-0.616240\pi\)
−0.357116 + 0.934060i \(0.616240\pi\)
\(912\) 0 0
\(913\) −849.451 −0.0307916
\(914\) 0 0
\(915\) 17906.4 0.646958
\(916\) 0 0
\(917\) 13806.1 0.497184
\(918\) 0 0
\(919\) −22128.7 −0.794297 −0.397149 0.917754i \(-0.630000\pi\)
−0.397149 + 0.917754i \(0.630000\pi\)
\(920\) 0 0
\(921\) −1244.01 −0.0445077
\(922\) 0 0
\(923\) −7533.72 −0.268663
\(924\) 0 0
\(925\) 26944.9 0.957775
\(926\) 0 0
\(927\) 1771.39 0.0627616
\(928\) 0 0
\(929\) 17599.3 0.621544 0.310772 0.950484i \(-0.399412\pi\)
0.310772 + 0.950484i \(0.399412\pi\)
\(930\) 0 0
\(931\) 20079.5 0.706850
\(932\) 0 0
\(933\) 4847.72 0.170104
\(934\) 0 0
\(935\) −39254.9 −1.37302
\(936\) 0 0
\(937\) 441.299 0.0153859 0.00769297 0.999970i \(-0.497551\pi\)
0.00769297 + 0.999970i \(0.497551\pi\)
\(938\) 0 0
\(939\) 25438.0 0.884065
\(940\) 0 0
\(941\) −5408.01 −0.187350 −0.0936749 0.995603i \(-0.529861\pi\)
−0.0936749 + 0.995603i \(0.529861\pi\)
\(942\) 0 0
\(943\) 50216.8 1.73413
\(944\) 0 0
\(945\) −11152.8 −0.383916
\(946\) 0 0
\(947\) −34237.1 −1.17482 −0.587410 0.809289i \(-0.699852\pi\)
−0.587410 + 0.809289i \(0.699852\pi\)
\(948\) 0 0
\(949\) 7314.92 0.250213
\(950\) 0 0
\(951\) 20324.2 0.693015
\(952\) 0 0
\(953\) 21410.7 0.727766 0.363883 0.931445i \(-0.381451\pi\)
0.363883 + 0.931445i \(0.381451\pi\)
\(954\) 0 0
\(955\) 23572.2 0.798722
\(956\) 0 0
\(957\) −33147.5 −1.11965
\(958\) 0 0
\(959\) 13751.3 0.463038
\(960\) 0 0
\(961\) −8038.09 −0.269816
\(962\) 0 0
\(963\) −3234.40 −0.108232
\(964\) 0 0
\(965\) 89105.3 2.97243
\(966\) 0 0
\(967\) 53874.6 1.79161 0.895806 0.444445i \(-0.146599\pi\)
0.895806 + 0.444445i \(0.146599\pi\)
\(968\) 0 0
\(969\) −14965.3 −0.496136
\(970\) 0 0
\(971\) 42901.5 1.41789 0.708947 0.705262i \(-0.249171\pi\)
0.708947 + 0.705262i \(0.249171\pi\)
\(972\) 0 0
\(973\) −42586.2 −1.40314
\(974\) 0 0
\(975\) −4602.00 −0.151161
\(976\) 0 0
\(977\) 58636.5 1.92011 0.960055 0.279812i \(-0.0902721\pi\)
0.960055 + 0.279812i \(0.0902721\pi\)
\(978\) 0 0
\(979\) 20475.7 0.668444
\(980\) 0 0
\(981\) 17727.6 0.576961
\(982\) 0 0
\(983\) 25296.7 0.820793 0.410396 0.911907i \(-0.365390\pi\)
0.410396 + 0.911907i \(0.365390\pi\)
\(984\) 0 0
\(985\) 52859.0 1.70988
\(986\) 0 0
\(987\) 21115.7 0.680973
\(988\) 0 0
\(989\) 41019.6 1.31885
\(990\) 0 0
\(991\) −10605.2 −0.339944 −0.169972 0.985449i \(-0.554368\pi\)
−0.169972 + 0.985449i \(0.554368\pi\)
\(992\) 0 0
\(993\) 27877.1 0.890888
\(994\) 0 0
\(995\) 12015.7 0.382839
\(996\) 0 0
\(997\) 5770.19 0.183294 0.0916469 0.995792i \(-0.470787\pi\)
0.0916469 + 0.995792i \(0.470787\pi\)
\(998\) 0 0
\(999\) −3393.60 −0.107476
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 768.4.a.e.1.1 2
3.2 odd 2 2304.4.a.bp.1.2 2
4.3 odd 2 768.4.a.k.1.1 2
8.3 odd 2 768.4.a.j.1.2 2
8.5 even 2 768.4.a.p.1.2 2
12.11 even 2 2304.4.a.bq.1.2 2
16.3 odd 4 384.4.d.c.193.4 yes 4
16.5 even 4 384.4.d.e.193.3 yes 4
16.11 odd 4 384.4.d.c.193.1 4
16.13 even 4 384.4.d.e.193.2 yes 4
24.5 odd 2 2304.4.a.s.1.1 2
24.11 even 2 2304.4.a.t.1.1 2
48.5 odd 4 1152.4.d.o.577.4 4
48.11 even 4 1152.4.d.i.577.4 4
48.29 odd 4 1152.4.d.o.577.1 4
48.35 even 4 1152.4.d.i.577.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 16.11 odd 4
384.4.d.c.193.4 yes 4 16.3 odd 4
384.4.d.e.193.2 yes 4 16.13 even 4
384.4.d.e.193.3 yes 4 16.5 even 4
768.4.a.e.1.1 2 1.1 even 1 trivial
768.4.a.j.1.2 2 8.3 odd 2
768.4.a.k.1.1 2 4.3 odd 2
768.4.a.p.1.2 2 8.5 even 2
1152.4.d.i.577.1 4 48.35 even 4
1152.4.d.i.577.4 4 48.11 even 4
1152.4.d.o.577.1 4 48.29 odd 4
1152.4.d.o.577.4 4 48.5 odd 4
2304.4.a.s.1.1 2 24.5 odd 2
2304.4.a.t.1.1 2 24.11 even 2
2304.4.a.bp.1.2 2 3.2 odd 2
2304.4.a.bq.1.2 2 12.11 even 2