Properties

Label 768.4.a.v.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.9792.1
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.06909\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -17.4288 q^{5} -2.99032 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -17.4288 q^{5} -2.99032 q^{7} +9.00000 q^{9} -10.6274 q^{11} +43.3156 q^{13} -52.2865 q^{15} -37.8823 q^{17} +79.8823 q^{19} -8.97095 q^{21} -191.204 q^{23} +178.765 q^{25} +27.0000 q^{27} -138.918 q^{29} -212.136 q^{31} -31.8823 q^{33} +52.1177 q^{35} +270.404 q^{37} +129.947 q^{39} +441.411 q^{41} +64.1177 q^{43} -156.860 q^{45} +436.234 q^{47} -334.058 q^{49} -113.647 q^{51} -278.348 q^{53} +185.224 q^{55} +239.647 q^{57} +830.039 q^{59} +724.580 q^{61} -26.9128 q^{63} -754.940 q^{65} +859.529 q^{67} -573.613 q^{69} +681.264 q^{71} +785.058 q^{73} +536.294 q^{75} +31.7793 q^{77} +1018.82 q^{79} +81.0000 q^{81} +467.334 q^{83} +660.244 q^{85} -416.753 q^{87} -510.706 q^{89} -129.527 q^{91} -636.409 q^{93} -1392.26 q^{95} -234.235 q^{97} -95.6468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 36 q^{9} + 48 q^{11} + 120 q^{17} + 48 q^{19} + 172 q^{25} + 108 q^{27} + 144 q^{33} + 480 q^{35} + 408 q^{41} + 528 q^{43} + 836 q^{49} + 360 q^{51} + 144 q^{57} + 1872 q^{59} - 576 q^{65} + 2352 q^{67} + 968 q^{73} + 516 q^{75} + 324 q^{81} + 3408 q^{83} - 3672 q^{89} + 5184 q^{91} - 1480 q^{97} + 432 q^{99}+O(q^{100}) \) 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
\(5\) −17.4288 −1.55888 −0.779441 0.626475i \(-0.784497\pi\)
−0.779441 + 0.626475i \(0.784497\pi\)
\(6\) 0 0
\(7\) −2.99032 −0.161462 −0.0807310 0.996736i \(-0.525725\pi\)
−0.0807310 + 0.996736i \(0.525725\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −10.6274 −0.291299 −0.145649 0.989336i \(-0.546527\pi\)
−0.145649 + 0.989336i \(0.546527\pi\)
\(12\) 0 0
\(13\) 43.3156 0.924121 0.462061 0.886848i \(-0.347110\pi\)
0.462061 + 0.886848i \(0.347110\pi\)
\(14\) 0 0
\(15\) −52.2865 −0.900021
\(16\) 0 0
\(17\) −37.8823 −0.540459 −0.270229 0.962796i \(-0.587100\pi\)
−0.270229 + 0.962796i \(0.587100\pi\)
\(18\) 0 0
\(19\) 79.8823 0.964539 0.482270 0.876023i \(-0.339813\pi\)
0.482270 + 0.876023i \(0.339813\pi\)
\(20\) 0 0
\(21\) −8.97095 −0.0932201
\(22\) 0 0
\(23\) −191.204 −1.73343 −0.866714 0.498806i \(-0.833772\pi\)
−0.866714 + 0.498806i \(0.833772\pi\)
\(24\) 0 0
\(25\) 178.765 1.43012
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −138.918 −0.889530 −0.444765 0.895647i \(-0.646713\pi\)
−0.444765 + 0.895647i \(0.646713\pi\)
\(30\) 0 0
\(31\) −212.136 −1.22906 −0.614529 0.788894i \(-0.710654\pi\)
−0.614529 + 0.788894i \(0.710654\pi\)
\(32\) 0 0
\(33\) −31.8823 −0.168181
\(34\) 0 0
\(35\) 52.1177 0.251700
\(36\) 0 0
\(37\) 270.404 1.20146 0.600731 0.799451i \(-0.294876\pi\)
0.600731 + 0.799451i \(0.294876\pi\)
\(38\) 0 0
\(39\) 129.947 0.533542
\(40\) 0 0
\(41\) 441.411 1.68139 0.840693 0.541511i \(-0.182148\pi\)
0.840693 + 0.541511i \(0.182148\pi\)
\(42\) 0 0
\(43\) 64.1177 0.227392 0.113696 0.993516i \(-0.463731\pi\)
0.113696 + 0.993516i \(0.463731\pi\)
\(44\) 0 0
\(45\) −156.860 −0.519628
\(46\) 0 0
\(47\) 436.234 1.35386 0.676929 0.736049i \(-0.263311\pi\)
0.676929 + 0.736049i \(0.263311\pi\)
\(48\) 0 0
\(49\) −334.058 −0.973930
\(50\) 0 0
\(51\) −113.647 −0.312034
\(52\) 0 0
\(53\) −278.348 −0.721398 −0.360699 0.932682i \(-0.617462\pi\)
−0.360699 + 0.932682i \(0.617462\pi\)
\(54\) 0 0
\(55\) 185.224 0.454101
\(56\) 0 0
\(57\) 239.647 0.556877
\(58\) 0 0
\(59\) 830.039 1.83156 0.915778 0.401684i \(-0.131575\pi\)
0.915778 + 0.401684i \(0.131575\pi\)
\(60\) 0 0
\(61\) 724.580 1.52087 0.760434 0.649416i \(-0.224986\pi\)
0.760434 + 0.649416i \(0.224986\pi\)
\(62\) 0 0
\(63\) −26.9128 −0.0538206
\(64\) 0 0
\(65\) −754.940 −1.44060
\(66\) 0 0
\(67\) 859.529 1.56729 0.783643 0.621211i \(-0.213359\pi\)
0.783643 + 0.621211i \(0.213359\pi\)
\(68\) 0 0
\(69\) −573.613 −1.00079
\(70\) 0 0
\(71\) 681.264 1.13875 0.569374 0.822078i \(-0.307186\pi\)
0.569374 + 0.822078i \(0.307186\pi\)
\(72\) 0 0
\(73\) 785.058 1.25869 0.629343 0.777128i \(-0.283324\pi\)
0.629343 + 0.777128i \(0.283324\pi\)
\(74\) 0 0
\(75\) 536.294 0.825678
\(76\) 0 0
\(77\) 31.7793 0.0470337
\(78\) 0 0
\(79\) 1018.82 1.45096 0.725481 0.688243i \(-0.241618\pi\)
0.725481 + 0.688243i \(0.241618\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 467.334 0.618031 0.309015 0.951057i \(-0.400001\pi\)
0.309015 + 0.951057i \(0.400001\pi\)
\(84\) 0 0
\(85\) 660.244 0.842512
\(86\) 0 0
\(87\) −416.753 −0.513570
\(88\) 0 0
\(89\) −510.706 −0.608256 −0.304128 0.952631i \(-0.598365\pi\)
−0.304128 + 0.952631i \(0.598365\pi\)
\(90\) 0 0
\(91\) −129.527 −0.149210
\(92\) 0 0
\(93\) −636.409 −0.709597
\(94\) 0 0
\(95\) −1392.26 −1.50360
\(96\) 0 0
\(97\) −234.235 −0.245186 −0.122593 0.992457i \(-0.539121\pi\)
−0.122593 + 0.992457i \(0.539121\pi\)
\(98\) 0 0
\(99\) −95.6468 −0.0970996
\(100\) 0 0
\(101\) 205.555 0.202509 0.101255 0.994861i \(-0.467714\pi\)
0.101255 + 0.994861i \(0.467714\pi\)
\(102\) 0 0
\(103\) −391.379 −0.374405 −0.187203 0.982321i \(-0.559942\pi\)
−0.187203 + 0.982321i \(0.559942\pi\)
\(104\) 0 0
\(105\) 156.353 0.145319
\(106\) 0 0
\(107\) 934.274 0.844109 0.422055 0.906570i \(-0.361309\pi\)
0.422055 + 0.906570i \(0.361309\pi\)
\(108\) 0 0
\(109\) −584.123 −0.513292 −0.256646 0.966505i \(-0.582617\pi\)
−0.256646 + 0.966505i \(0.582617\pi\)
\(110\) 0 0
\(111\) 811.211 0.693664
\(112\) 0 0
\(113\) −582.706 −0.485101 −0.242551 0.970139i \(-0.577984\pi\)
−0.242551 + 0.970139i \(0.577984\pi\)
\(114\) 0 0
\(115\) 3332.47 2.70221
\(116\) 0 0
\(117\) 389.840 0.308040
\(118\) 0 0
\(119\) 113.280 0.0872635
\(120\) 0 0
\(121\) −1218.06 −0.915145
\(122\) 0 0
\(123\) 1324.23 0.970749
\(124\) 0 0
\(125\) −937.053 −0.670501
\(126\) 0 0
\(127\) −1461.03 −1.02083 −0.510416 0.859928i \(-0.670509\pi\)
−0.510416 + 0.859928i \(0.670509\pi\)
\(128\) 0 0
\(129\) 192.353 0.131285
\(130\) 0 0
\(131\) 98.4323 0.0656494 0.0328247 0.999461i \(-0.489550\pi\)
0.0328247 + 0.999461i \(0.489550\pi\)
\(132\) 0 0
\(133\) −238.873 −0.155736
\(134\) 0 0
\(135\) −470.579 −0.300007
\(136\) 0 0
\(137\) −2171.06 −1.35391 −0.676956 0.736024i \(-0.736701\pi\)
−0.676956 + 0.736024i \(0.736701\pi\)
\(138\) 0 0
\(139\) −1624.70 −0.991407 −0.495703 0.868492i \(-0.665090\pi\)
−0.495703 + 0.868492i \(0.665090\pi\)
\(140\) 0 0
\(141\) 1308.70 0.781650
\(142\) 0 0
\(143\) −460.333 −0.269195
\(144\) 0 0
\(145\) 2421.17 1.38667
\(146\) 0 0
\(147\) −1002.17 −0.562299
\(148\) 0 0
\(149\) −636.658 −0.350047 −0.175024 0.984564i \(-0.556000\pi\)
−0.175024 + 0.984564i \(0.556000\pi\)
\(150\) 0 0
\(151\) −1819.34 −0.980503 −0.490252 0.871581i \(-0.663095\pi\)
−0.490252 + 0.871581i \(0.663095\pi\)
\(152\) 0 0
\(153\) −340.940 −0.180153
\(154\) 0 0
\(155\) 3697.29 1.91596
\(156\) 0 0
\(157\) 1656.50 0.842059 0.421029 0.907047i \(-0.361669\pi\)
0.421029 + 0.907047i \(0.361669\pi\)
\(158\) 0 0
\(159\) −835.045 −0.416499
\(160\) 0 0
\(161\) 571.761 0.279882
\(162\) 0 0
\(163\) 2228.82 1.07101 0.535505 0.844532i \(-0.320121\pi\)
0.535505 + 0.844532i \(0.320121\pi\)
\(164\) 0 0
\(165\) 555.671 0.262175
\(166\) 0 0
\(167\) −1667.01 −0.772439 −0.386219 0.922407i \(-0.626219\pi\)
−0.386219 + 0.922407i \(0.626219\pi\)
\(168\) 0 0
\(169\) −320.761 −0.146000
\(170\) 0 0
\(171\) 718.940 0.321513
\(172\) 0 0
\(173\) 2500.53 1.09891 0.549457 0.835522i \(-0.314835\pi\)
0.549457 + 0.835522i \(0.314835\pi\)
\(174\) 0 0
\(175\) −534.562 −0.230909
\(176\) 0 0
\(177\) 2490.12 1.05745
\(178\) 0 0
\(179\) −378.742 −0.158148 −0.0790740 0.996869i \(-0.525196\pi\)
−0.0790740 + 0.996869i \(0.525196\pi\)
\(180\) 0 0
\(181\) −3093.88 −1.27053 −0.635265 0.772294i \(-0.719109\pi\)
−0.635265 + 0.772294i \(0.719109\pi\)
\(182\) 0 0
\(183\) 2173.74 0.878073
\(184\) 0 0
\(185\) −4712.82 −1.87294
\(186\) 0 0
\(187\) 402.590 0.157435
\(188\) 0 0
\(189\) −80.7385 −0.0310734
\(190\) 0 0
\(191\) 3656.98 1.38539 0.692695 0.721230i \(-0.256423\pi\)
0.692695 + 0.721230i \(0.256423\pi\)
\(192\) 0 0
\(193\) 2788.12 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(194\) 0 0
\(195\) −2264.82 −0.831729
\(196\) 0 0
\(197\) 1147.74 0.415091 0.207546 0.978225i \(-0.433452\pi\)
0.207546 + 0.978225i \(0.433452\pi\)
\(198\) 0 0
\(199\) 4842.73 1.72508 0.862542 0.505986i \(-0.168871\pi\)
0.862542 + 0.505986i \(0.168871\pi\)
\(200\) 0 0
\(201\) 2578.59 0.904873
\(202\) 0 0
\(203\) 415.408 0.143625
\(204\) 0 0
\(205\) −7693.29 −2.62109
\(206\) 0 0
\(207\) −1720.84 −0.577809
\(208\) 0 0
\(209\) −848.942 −0.280969
\(210\) 0 0
\(211\) 3222.35 1.05135 0.525677 0.850684i \(-0.323812\pi\)
0.525677 + 0.850684i \(0.323812\pi\)
\(212\) 0 0
\(213\) 2043.79 0.657457
\(214\) 0 0
\(215\) −1117.50 −0.354478
\(216\) 0 0
\(217\) 634.355 0.198446
\(218\) 0 0
\(219\) 2355.17 0.726703
\(220\) 0 0
\(221\) −1640.89 −0.499449
\(222\) 0 0
\(223\) 4932.61 1.48122 0.740610 0.671935i \(-0.234537\pi\)
0.740610 + 0.671935i \(0.234537\pi\)
\(224\) 0 0
\(225\) 1608.88 0.476705
\(226\) 0 0
\(227\) −3619.49 −1.05830 −0.529150 0.848529i \(-0.677489\pi\)
−0.529150 + 0.848529i \(0.677489\pi\)
\(228\) 0 0
\(229\) 305.759 0.0882320 0.0441160 0.999026i \(-0.485953\pi\)
0.0441160 + 0.999026i \(0.485953\pi\)
\(230\) 0 0
\(231\) 95.3380 0.0271549
\(232\) 0 0
\(233\) −639.648 −0.179849 −0.0899244 0.995949i \(-0.528663\pi\)
−0.0899244 + 0.995949i \(0.528663\pi\)
\(234\) 0 0
\(235\) −7603.05 −2.11050
\(236\) 0 0
\(237\) 3056.45 0.837713
\(238\) 0 0
\(239\) 1744.94 0.472262 0.236131 0.971721i \(-0.424121\pi\)
0.236131 + 0.971721i \(0.424121\pi\)
\(240\) 0 0
\(241\) 3357.29 0.897354 0.448677 0.893694i \(-0.351895\pi\)
0.448677 + 0.893694i \(0.351895\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 5822.24 1.51824
\(246\) 0 0
\(247\) 3460.15 0.891351
\(248\) 0 0
\(249\) 1402.00 0.356820
\(250\) 0 0
\(251\) −1317.45 −0.331301 −0.165650 0.986185i \(-0.552972\pi\)
−0.165650 + 0.986185i \(0.552972\pi\)
\(252\) 0 0
\(253\) 2032.01 0.504945
\(254\) 0 0
\(255\) 1980.73 0.486424
\(256\) 0 0
\(257\) −3036.12 −0.736917 −0.368459 0.929644i \(-0.620114\pi\)
−0.368459 + 0.929644i \(0.620114\pi\)
\(258\) 0 0
\(259\) −808.592 −0.193990
\(260\) 0 0
\(261\) −1250.26 −0.296510
\(262\) 0 0
\(263\) 1655.76 0.388206 0.194103 0.980981i \(-0.437820\pi\)
0.194103 + 0.980981i \(0.437820\pi\)
\(264\) 0 0
\(265\) 4851.29 1.12458
\(266\) 0 0
\(267\) −1532.12 −0.351177
\(268\) 0 0
\(269\) 5292.72 1.19964 0.599820 0.800135i \(-0.295239\pi\)
0.599820 + 0.800135i \(0.295239\pi\)
\(270\) 0 0
\(271\) −8010.52 −1.79559 −0.897795 0.440414i \(-0.854832\pi\)
−0.897795 + 0.440414i \(0.854832\pi\)
\(272\) 0 0
\(273\) −388.582 −0.0861467
\(274\) 0 0
\(275\) −1899.80 −0.416591
\(276\) 0 0
\(277\) −5692.81 −1.23483 −0.617415 0.786638i \(-0.711820\pi\)
−0.617415 + 0.786638i \(0.711820\pi\)
\(278\) 0 0
\(279\) −1909.23 −0.409686
\(280\) 0 0
\(281\) −2024.83 −0.429861 −0.214931 0.976629i \(-0.568953\pi\)
−0.214931 + 0.976629i \(0.568953\pi\)
\(282\) 0 0
\(283\) −247.761 −0.0520419 −0.0260210 0.999661i \(-0.508284\pi\)
−0.0260210 + 0.999661i \(0.508284\pi\)
\(284\) 0 0
\(285\) −4176.77 −0.868106
\(286\) 0 0
\(287\) −1319.96 −0.271480
\(288\) 0 0
\(289\) −3477.94 −0.707905
\(290\) 0 0
\(291\) −702.706 −0.141558
\(292\) 0 0
\(293\) −8133.61 −1.62174 −0.810871 0.585225i \(-0.801006\pi\)
−0.810871 + 0.585225i \(0.801006\pi\)
\(294\) 0 0
\(295\) −14466.6 −2.85518
\(296\) 0 0
\(297\) −286.940 −0.0560605
\(298\) 0 0
\(299\) −8282.12 −1.60190
\(300\) 0 0
\(301\) −191.732 −0.0367152
\(302\) 0 0
\(303\) 616.664 0.116919
\(304\) 0 0
\(305\) −12628.6 −2.37085
\(306\) 0 0
\(307\) −2974.82 −0.553035 −0.276518 0.961009i \(-0.589180\pi\)
−0.276518 + 0.961009i \(0.589180\pi\)
\(308\) 0 0
\(309\) −1174.14 −0.216163
\(310\) 0 0
\(311\) 4451.52 0.811648 0.405824 0.913951i \(-0.366985\pi\)
0.405824 + 0.913951i \(0.366985\pi\)
\(312\) 0 0
\(313\) 8273.75 1.49412 0.747061 0.664755i \(-0.231464\pi\)
0.747061 + 0.664755i \(0.231464\pi\)
\(314\) 0 0
\(315\) 469.060 0.0839001
\(316\) 0 0
\(317\) −429.036 −0.0760160 −0.0380080 0.999277i \(-0.512101\pi\)
−0.0380080 + 0.999277i \(0.512101\pi\)
\(318\) 0 0
\(319\) 1476.34 0.259119
\(320\) 0 0
\(321\) 2802.82 0.487347
\(322\) 0 0
\(323\) −3026.12 −0.521293
\(324\) 0 0
\(325\) 7743.29 1.32160
\(326\) 0 0
\(327\) −1752.37 −0.296349
\(328\) 0 0
\(329\) −1304.48 −0.218596
\(330\) 0 0
\(331\) −8196.71 −1.36112 −0.680562 0.732691i \(-0.738264\pi\)
−0.680562 + 0.732691i \(0.738264\pi\)
\(332\) 0 0
\(333\) 2433.63 0.400487
\(334\) 0 0
\(335\) −14980.6 −2.44322
\(336\) 0 0
\(337\) −2000.35 −0.323341 −0.161670 0.986845i \(-0.551688\pi\)
−0.161670 + 0.986845i \(0.551688\pi\)
\(338\) 0 0
\(339\) −1748.12 −0.280073
\(340\) 0 0
\(341\) 2254.46 0.358023
\(342\) 0 0
\(343\) 2024.62 0.318715
\(344\) 0 0
\(345\) 9997.40 1.56012
\(346\) 0 0
\(347\) 7707.48 1.19239 0.596195 0.802840i \(-0.296679\pi\)
0.596195 + 0.802840i \(0.296679\pi\)
\(348\) 0 0
\(349\) 9681.98 1.48500 0.742499 0.669847i \(-0.233640\pi\)
0.742499 + 0.669847i \(0.233640\pi\)
\(350\) 0 0
\(351\) 1169.52 0.177847
\(352\) 0 0
\(353\) −10540.3 −1.58925 −0.794626 0.607099i \(-0.792333\pi\)
−0.794626 + 0.607099i \(0.792333\pi\)
\(354\) 0 0
\(355\) −11873.6 −1.77518
\(356\) 0 0
\(357\) 339.840 0.0503816
\(358\) 0 0
\(359\) −514.158 −0.0755884 −0.0377942 0.999286i \(-0.512033\pi\)
−0.0377942 + 0.999286i \(0.512033\pi\)
\(360\) 0 0
\(361\) −477.826 −0.0696641
\(362\) 0 0
\(363\) −3654.17 −0.528359
\(364\) 0 0
\(365\) −13682.7 −1.96214
\(366\) 0 0
\(367\) 11272.4 1.60331 0.801657 0.597785i \(-0.203952\pi\)
0.801657 + 0.597785i \(0.203952\pi\)
\(368\) 0 0
\(369\) 3972.70 0.560462
\(370\) 0 0
\(371\) 832.350 0.116478
\(372\) 0 0
\(373\) 6956.92 0.965726 0.482863 0.875696i \(-0.339597\pi\)
0.482863 + 0.875696i \(0.339597\pi\)
\(374\) 0 0
\(375\) −2811.16 −0.387114
\(376\) 0 0
\(377\) −6017.30 −0.822034
\(378\) 0 0
\(379\) 10201.3 1.38260 0.691299 0.722569i \(-0.257039\pi\)
0.691299 + 0.722569i \(0.257039\pi\)
\(380\) 0 0
\(381\) −4383.10 −0.589378
\(382\) 0 0
\(383\) 2461.56 0.328406 0.164203 0.986427i \(-0.447495\pi\)
0.164203 + 0.986427i \(0.447495\pi\)
\(384\) 0 0
\(385\) −553.877 −0.0733200
\(386\) 0 0
\(387\) 577.060 0.0757974
\(388\) 0 0
\(389\) 546.451 0.0712240 0.0356120 0.999366i \(-0.488662\pi\)
0.0356120 + 0.999366i \(0.488662\pi\)
\(390\) 0 0
\(391\) 7243.25 0.936846
\(392\) 0 0
\(393\) 295.297 0.0379027
\(394\) 0 0
\(395\) −17756.8 −2.26188
\(396\) 0 0
\(397\) −2084.56 −0.263529 −0.131764 0.991281i \(-0.542064\pi\)
−0.131764 + 0.991281i \(0.542064\pi\)
\(398\) 0 0
\(399\) −716.620 −0.0899144
\(400\) 0 0
\(401\) −9710.59 −1.20929 −0.604643 0.796497i \(-0.706684\pi\)
−0.604643 + 0.796497i \(0.706684\pi\)
\(402\) 0 0
\(403\) −9188.81 −1.13580
\(404\) 0 0
\(405\) −1411.74 −0.173209
\(406\) 0 0
\(407\) −2873.69 −0.349984
\(408\) 0 0
\(409\) 6659.89 0.805160 0.402580 0.915385i \(-0.368114\pi\)
0.402580 + 0.915385i \(0.368114\pi\)
\(410\) 0 0
\(411\) −6513.17 −0.781681
\(412\) 0 0
\(413\) −2482.08 −0.295727
\(414\) 0 0
\(415\) −8145.09 −0.963438
\(416\) 0 0
\(417\) −4874.11 −0.572389
\(418\) 0 0
\(419\) −10576.7 −1.23318 −0.616592 0.787283i \(-0.711487\pi\)
−0.616592 + 0.787283i \(0.711487\pi\)
\(420\) 0 0
\(421\) 4871.09 0.563901 0.281951 0.959429i \(-0.409019\pi\)
0.281951 + 0.959429i \(0.409019\pi\)
\(422\) 0 0
\(423\) 3926.11 0.451286
\(424\) 0 0
\(425\) −6772.00 −0.772918
\(426\) 0 0
\(427\) −2166.72 −0.245562
\(428\) 0 0
\(429\) −1381.00 −0.155420
\(430\) 0 0
\(431\) 16916.7 1.89060 0.945302 0.326196i \(-0.105767\pi\)
0.945302 + 0.326196i \(0.105767\pi\)
\(432\) 0 0
\(433\) −1163.88 −0.129174 −0.0645870 0.997912i \(-0.520573\pi\)
−0.0645870 + 0.997912i \(0.520573\pi\)
\(434\) 0 0
\(435\) 7263.52 0.800596
\(436\) 0 0
\(437\) −15273.8 −1.67196
\(438\) 0 0
\(439\) 1856.28 0.201812 0.100906 0.994896i \(-0.467826\pi\)
0.100906 + 0.994896i \(0.467826\pi\)
\(440\) 0 0
\(441\) −3006.52 −0.324643
\(442\) 0 0
\(443\) 1472.21 0.157893 0.0789465 0.996879i \(-0.474844\pi\)
0.0789465 + 0.996879i \(0.474844\pi\)
\(444\) 0 0
\(445\) 8901.02 0.948200
\(446\) 0 0
\(447\) −1909.97 −0.202100
\(448\) 0 0
\(449\) 9620.94 1.01123 0.505613 0.862761i \(-0.331267\pi\)
0.505613 + 0.862761i \(0.331267\pi\)
\(450\) 0 0
\(451\) −4691.06 −0.489786
\(452\) 0 0
\(453\) −5458.03 −0.566094
\(454\) 0 0
\(455\) 2257.51 0.232602
\(456\) 0 0
\(457\) −3613.53 −0.369877 −0.184938 0.982750i \(-0.559209\pi\)
−0.184938 + 0.982750i \(0.559209\pi\)
\(458\) 0 0
\(459\) −1022.82 −0.104011
\(460\) 0 0
\(461\) 17710.7 1.78931 0.894654 0.446759i \(-0.147422\pi\)
0.894654 + 0.446759i \(0.147422\pi\)
\(462\) 0 0
\(463\) −1674.57 −0.168087 −0.0840433 0.996462i \(-0.526783\pi\)
−0.0840433 + 0.996462i \(0.526783\pi\)
\(464\) 0 0
\(465\) 11091.9 1.10618
\(466\) 0 0
\(467\) 15208.3 1.50697 0.753484 0.657466i \(-0.228372\pi\)
0.753484 + 0.657466i \(0.228372\pi\)
\(468\) 0 0
\(469\) −2570.26 −0.253057
\(470\) 0 0
\(471\) 4969.51 0.486163
\(472\) 0 0
\(473\) −681.406 −0.0662391
\(474\) 0 0
\(475\) 14280.1 1.37940
\(476\) 0 0
\(477\) −2505.14 −0.240466
\(478\) 0 0
\(479\) 6458.37 0.616056 0.308028 0.951377i \(-0.400331\pi\)
0.308028 + 0.951377i \(0.400331\pi\)
\(480\) 0 0
\(481\) 11712.7 1.11030
\(482\) 0 0
\(483\) 1715.28 0.161590
\(484\) 0 0
\(485\) 4082.45 0.382216
\(486\) 0 0
\(487\) 11337.7 1.05495 0.527474 0.849571i \(-0.323139\pi\)
0.527474 + 0.849571i \(0.323139\pi\)
\(488\) 0 0
\(489\) 6686.46 0.618348
\(490\) 0 0
\(491\) −14946.7 −1.37380 −0.686898 0.726754i \(-0.741028\pi\)
−0.686898 + 0.726754i \(0.741028\pi\)
\(492\) 0 0
\(493\) 5262.51 0.480754
\(494\) 0 0
\(495\) 1667.01 0.151367
\(496\) 0 0
\(497\) −2037.19 −0.183865
\(498\) 0 0
\(499\) 2631.77 0.236101 0.118051 0.993008i \(-0.462336\pi\)
0.118051 + 0.993008i \(0.462336\pi\)
\(500\) 0 0
\(501\) −5001.04 −0.445968
\(502\) 0 0
\(503\) −6907.45 −0.612302 −0.306151 0.951983i \(-0.599041\pi\)
−0.306151 + 0.951983i \(0.599041\pi\)
\(504\) 0 0
\(505\) −3582.58 −0.315689
\(506\) 0 0
\(507\) −962.283 −0.0842929
\(508\) 0 0
\(509\) −12020.3 −1.04674 −0.523368 0.852107i \(-0.675325\pi\)
−0.523368 + 0.852107i \(0.675325\pi\)
\(510\) 0 0
\(511\) −2347.57 −0.203230
\(512\) 0 0
\(513\) 2156.82 0.185626
\(514\) 0 0
\(515\) 6821.29 0.583654
\(516\) 0 0
\(517\) −4636.04 −0.394377
\(518\) 0 0
\(519\) 7501.60 0.634458
\(520\) 0 0
\(521\) −15846.1 −1.33249 −0.666247 0.745731i \(-0.732101\pi\)
−0.666247 + 0.745731i \(0.732101\pi\)
\(522\) 0 0
\(523\) 8891.64 0.743411 0.371706 0.928351i \(-0.378773\pi\)
0.371706 + 0.928351i \(0.378773\pi\)
\(524\) 0 0
\(525\) −1603.69 −0.133316
\(526\) 0 0
\(527\) 8036.20 0.664255
\(528\) 0 0
\(529\) 24392.0 2.00477
\(530\) 0 0
\(531\) 7470.35 0.610519
\(532\) 0 0
\(533\) 19120.0 1.55381
\(534\) 0 0
\(535\) −16283.3 −1.31587
\(536\) 0 0
\(537\) −1136.23 −0.0913068
\(538\) 0 0
\(539\) 3550.17 0.283705
\(540\) 0 0
\(541\) −12833.5 −1.01988 −0.509940 0.860210i \(-0.670333\pi\)
−0.509940 + 0.860210i \(0.670333\pi\)
\(542\) 0 0
\(543\) −9281.63 −0.733541
\(544\) 0 0
\(545\) 10180.6 0.800162
\(546\) 0 0
\(547\) −16257.0 −1.27075 −0.635375 0.772204i \(-0.719155\pi\)
−0.635375 + 0.772204i \(0.719155\pi\)
\(548\) 0 0
\(549\) 6521.22 0.506956
\(550\) 0 0
\(551\) −11097.1 −0.857986
\(552\) 0 0
\(553\) −3046.59 −0.234275
\(554\) 0 0
\(555\) −14138.5 −1.08134
\(556\) 0 0
\(557\) −1558.32 −0.118542 −0.0592712 0.998242i \(-0.518878\pi\)
−0.0592712 + 0.998242i \(0.518878\pi\)
\(558\) 0 0
\(559\) 2777.30 0.210138
\(560\) 0 0
\(561\) 1207.77 0.0908951
\(562\) 0 0
\(563\) 9782.16 0.732272 0.366136 0.930561i \(-0.380681\pi\)
0.366136 + 0.930561i \(0.380681\pi\)
\(564\) 0 0
\(565\) 10155.9 0.756216
\(566\) 0 0
\(567\) −242.216 −0.0179402
\(568\) 0 0
\(569\) −7887.05 −0.581094 −0.290547 0.956861i \(-0.593837\pi\)
−0.290547 + 0.956861i \(0.593837\pi\)
\(570\) 0 0
\(571\) 21819.3 1.59914 0.799570 0.600573i \(-0.205061\pi\)
0.799570 + 0.600573i \(0.205061\pi\)
\(572\) 0 0
\(573\) 10970.9 0.799856
\(574\) 0 0
\(575\) −34180.5 −2.47900
\(576\) 0 0
\(577\) 7190.22 0.518774 0.259387 0.965773i \(-0.416479\pi\)
0.259387 + 0.965773i \(0.416479\pi\)
\(578\) 0 0
\(579\) 8364.35 0.600363
\(580\) 0 0
\(581\) −1397.48 −0.0997884
\(582\) 0 0
\(583\) 2958.12 0.210142
\(584\) 0 0
\(585\) −6794.46 −0.480199
\(586\) 0 0
\(587\) 13305.6 0.935569 0.467785 0.883843i \(-0.345052\pi\)
0.467785 + 0.883843i \(0.345052\pi\)
\(588\) 0 0
\(589\) −16945.9 −1.18548
\(590\) 0 0
\(591\) 3443.21 0.239653
\(592\) 0 0
\(593\) −8062.23 −0.558307 −0.279153 0.960246i \(-0.590054\pi\)
−0.279153 + 0.960246i \(0.590054\pi\)
\(594\) 0 0
\(595\) −1974.34 −0.136034
\(596\) 0 0
\(597\) 14528.2 0.995978
\(598\) 0 0
\(599\) −2185.21 −0.149058 −0.0745288 0.997219i \(-0.523745\pi\)
−0.0745288 + 0.997219i \(0.523745\pi\)
\(600\) 0 0
\(601\) 3542.25 0.240418 0.120209 0.992749i \(-0.461643\pi\)
0.120209 + 0.992749i \(0.461643\pi\)
\(602\) 0 0
\(603\) 7735.76 0.522429
\(604\) 0 0
\(605\) 21229.3 1.42660
\(606\) 0 0
\(607\) −6050.64 −0.404593 −0.202296 0.979324i \(-0.564840\pi\)
−0.202296 + 0.979324i \(0.564840\pi\)
\(608\) 0 0
\(609\) 1246.22 0.0829220
\(610\) 0 0
\(611\) 18895.7 1.25113
\(612\) 0 0
\(613\) −22514.2 −1.48343 −0.741713 0.670717i \(-0.765986\pi\)
−0.741713 + 0.670717i \(0.765986\pi\)
\(614\) 0 0
\(615\) −23079.9 −1.51328
\(616\) 0 0
\(617\) −4255.88 −0.277691 −0.138845 0.990314i \(-0.544339\pi\)
−0.138845 + 0.990314i \(0.544339\pi\)
\(618\) 0 0
\(619\) −228.949 −0.0148663 −0.00743315 0.999972i \(-0.502366\pi\)
−0.00743315 + 0.999972i \(0.502366\pi\)
\(620\) 0 0
\(621\) −5162.51 −0.333598
\(622\) 0 0
\(623\) 1527.17 0.0982102
\(624\) 0 0
\(625\) −6013.82 −0.384884
\(626\) 0 0
\(627\) −2546.83 −0.162218
\(628\) 0 0
\(629\) −10243.5 −0.649340
\(630\) 0 0
\(631\) −11429.2 −0.721058 −0.360529 0.932748i \(-0.617404\pi\)
−0.360529 + 0.932748i \(0.617404\pi\)
\(632\) 0 0
\(633\) 9667.04 0.606999
\(634\) 0 0
\(635\) 25464.1 1.59136
\(636\) 0 0
\(637\) −14469.9 −0.900030
\(638\) 0 0
\(639\) 6131.38 0.379583
\(640\) 0 0
\(641\) 29381.4 1.81045 0.905223 0.424938i \(-0.139704\pi\)
0.905223 + 0.424938i \(0.139704\pi\)
\(642\) 0 0
\(643\) 249.316 0.0152909 0.00764546 0.999971i \(-0.497566\pi\)
0.00764546 + 0.999971i \(0.497566\pi\)
\(644\) 0 0
\(645\) −3352.49 −0.204658
\(646\) 0 0
\(647\) 16025.8 0.973785 0.486893 0.873462i \(-0.338130\pi\)
0.486893 + 0.873462i \(0.338130\pi\)
\(648\) 0 0
\(649\) −8821.17 −0.533530
\(650\) 0 0
\(651\) 1903.06 0.114573
\(652\) 0 0
\(653\) −14008.6 −0.839511 −0.419755 0.907637i \(-0.637884\pi\)
−0.419755 + 0.907637i \(0.637884\pi\)
\(654\) 0 0
\(655\) −1715.56 −0.102340
\(656\) 0 0
\(657\) 7065.52 0.419562
\(658\) 0 0
\(659\) −1011.91 −0.0598152 −0.0299076 0.999553i \(-0.509521\pi\)
−0.0299076 + 0.999553i \(0.509521\pi\)
\(660\) 0 0
\(661\) −23619.4 −1.38985 −0.694923 0.719084i \(-0.744561\pi\)
−0.694923 + 0.719084i \(0.744561\pi\)
\(662\) 0 0
\(663\) −4922.67 −0.288357
\(664\) 0 0
\(665\) 4163.28 0.242775
\(666\) 0 0
\(667\) 26561.6 1.54194
\(668\) 0 0
\(669\) 14797.8 0.855183
\(670\) 0 0
\(671\) −7700.41 −0.443027
\(672\) 0 0
\(673\) −25811.9 −1.47842 −0.739208 0.673477i \(-0.764800\pi\)
−0.739208 + 0.673477i \(0.764800\pi\)
\(674\) 0 0
\(675\) 4826.64 0.275226
\(676\) 0 0
\(677\) 18255.2 1.03634 0.518172 0.855277i \(-0.326613\pi\)
0.518172 + 0.855277i \(0.326613\pi\)
\(678\) 0 0
\(679\) 700.438 0.0395881
\(680\) 0 0
\(681\) −10858.5 −0.611009
\(682\) 0 0
\(683\) −20090.4 −1.12553 −0.562765 0.826617i \(-0.690262\pi\)
−0.562765 + 0.826617i \(0.690262\pi\)
\(684\) 0 0
\(685\) 37839.0 2.11059
\(686\) 0 0
\(687\) 917.278 0.0509408
\(688\) 0 0
\(689\) −12056.8 −0.666659
\(690\) 0 0
\(691\) 16521.5 0.909563 0.454782 0.890603i \(-0.349717\pi\)
0.454782 + 0.890603i \(0.349717\pi\)
\(692\) 0 0
\(693\) 286.014 0.0156779
\(694\) 0 0
\(695\) 28316.7 1.54549
\(696\) 0 0
\(697\) −16721.7 −0.908720
\(698\) 0 0
\(699\) −1918.95 −0.103836
\(700\) 0 0
\(701\) 12431.4 0.669795 0.334897 0.942255i \(-0.391298\pi\)
0.334897 + 0.942255i \(0.391298\pi\)
\(702\) 0 0
\(703\) 21600.4 1.15886
\(704\) 0 0
\(705\) −22809.2 −1.21850
\(706\) 0 0
\(707\) −614.674 −0.0326976
\(708\) 0 0
\(709\) −980.957 −0.0519614 −0.0259807 0.999662i \(-0.508271\pi\)
−0.0259807 + 0.999662i \(0.508271\pi\)
\(710\) 0 0
\(711\) 9169.36 0.483654
\(712\) 0 0
\(713\) 40561.4 2.13048
\(714\) 0 0
\(715\) 8023.06 0.419644
\(716\) 0 0
\(717\) 5234.81 0.272660
\(718\) 0 0
\(719\) 4115.73 0.213478 0.106739 0.994287i \(-0.465959\pi\)
0.106739 + 0.994287i \(0.465959\pi\)
\(720\) 0 0
\(721\) 1170.35 0.0604522
\(722\) 0 0
\(723\) 10071.9 0.518088
\(724\) 0 0
\(725\) −24833.5 −1.27213
\(726\) 0 0
\(727\) 20850.1 1.06367 0.531833 0.846849i \(-0.321503\pi\)
0.531833 + 0.846849i \(0.321503\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2428.92 −0.122896
\(732\) 0 0
\(733\) −31517.2 −1.58815 −0.794074 0.607821i \(-0.792044\pi\)
−0.794074 + 0.607821i \(0.792044\pi\)
\(734\) 0 0
\(735\) 17466.7 0.876558
\(736\) 0 0
\(737\) −9134.57 −0.456549
\(738\) 0 0
\(739\) 11415.0 0.568213 0.284106 0.958793i \(-0.408303\pi\)
0.284106 + 0.958793i \(0.408303\pi\)
\(740\) 0 0
\(741\) 10380.4 0.514622
\(742\) 0 0
\(743\) 5732.08 0.283028 0.141514 0.989936i \(-0.454803\pi\)
0.141514 + 0.989936i \(0.454803\pi\)
\(744\) 0 0
\(745\) 11096.2 0.545683
\(746\) 0 0
\(747\) 4206.01 0.206010
\(748\) 0 0
\(749\) −2793.78 −0.136291
\(750\) 0 0
\(751\) 7843.07 0.381089 0.190544 0.981679i \(-0.438975\pi\)
0.190544 + 0.981679i \(0.438975\pi\)
\(752\) 0 0
\(753\) −3952.34 −0.191277
\(754\) 0 0
\(755\) 31709.0 1.52849
\(756\) 0 0
\(757\) 29125.9 1.39841 0.699206 0.714920i \(-0.253537\pi\)
0.699206 + 0.714920i \(0.253537\pi\)
\(758\) 0 0
\(759\) 6096.02 0.291530
\(760\) 0 0
\(761\) −14228.5 −0.677768 −0.338884 0.940828i \(-0.610049\pi\)
−0.338884 + 0.940828i \(0.610049\pi\)
\(762\) 0 0
\(763\) 1746.71 0.0828771
\(764\) 0 0
\(765\) 5942.19 0.280837
\(766\) 0 0
\(767\) 35953.6 1.69258
\(768\) 0 0
\(769\) −28133.7 −1.31928 −0.659641 0.751581i \(-0.729292\pi\)
−0.659641 + 0.751581i \(0.729292\pi\)
\(770\) 0 0
\(771\) −9108.35 −0.425459
\(772\) 0 0
\(773\) 14686.9 0.683377 0.341689 0.939813i \(-0.389001\pi\)
0.341689 + 0.939813i \(0.389001\pi\)
\(774\) 0 0
\(775\) −37922.5 −1.75770
\(776\) 0 0
\(777\) −2425.78 −0.112000
\(778\) 0 0
\(779\) 35260.9 1.62176
\(780\) 0 0
\(781\) −7240.08 −0.331716
\(782\) 0 0
\(783\) −3750.78 −0.171190
\(784\) 0 0
\(785\) −28870.9 −1.31267
\(786\) 0 0
\(787\) 21001.0 0.951214 0.475607 0.879658i \(-0.342228\pi\)
0.475607 + 0.879658i \(0.342228\pi\)
\(788\) 0 0
\(789\) 4967.27 0.224131
\(790\) 0 0
\(791\) 1742.48 0.0783253
\(792\) 0 0
\(793\) 31385.6 1.40547
\(794\) 0 0
\(795\) 14553.9 0.649274
\(796\) 0 0
\(797\) −13362.7 −0.593892 −0.296946 0.954894i \(-0.595968\pi\)
−0.296946 + 0.954894i \(0.595968\pi\)
\(798\) 0 0
\(799\) −16525.5 −0.731704
\(800\) 0 0
\(801\) −4596.36 −0.202752
\(802\) 0 0
\(803\) −8343.14 −0.366654
\(804\) 0 0
\(805\) −9965.13 −0.436304
\(806\) 0 0
\(807\) 15878.2 0.692612
\(808\) 0 0
\(809\) 39481.6 1.71582 0.857911 0.513798i \(-0.171762\pi\)
0.857911 + 0.513798i \(0.171762\pi\)
\(810\) 0 0
\(811\) −31157.5 −1.34906 −0.674531 0.738247i \(-0.735654\pi\)
−0.674531 + 0.738247i \(0.735654\pi\)
\(812\) 0 0
\(813\) −24031.6 −1.03668
\(814\) 0 0
\(815\) −38845.8 −1.66958
\(816\) 0 0
\(817\) 5121.87 0.219329
\(818\) 0 0
\(819\) −1165.75 −0.0497368
\(820\) 0 0
\(821\) 21229.9 0.902470 0.451235 0.892405i \(-0.350984\pi\)
0.451235 + 0.892405i \(0.350984\pi\)
\(822\) 0 0
\(823\) 24603.8 1.04208 0.521041 0.853532i \(-0.325544\pi\)
0.521041 + 0.853532i \(0.325544\pi\)
\(824\) 0 0
\(825\) −5699.41 −0.240519
\(826\) 0 0
\(827\) −13668.1 −0.574710 −0.287355 0.957824i \(-0.592776\pi\)
−0.287355 + 0.957824i \(0.592776\pi\)
\(828\) 0 0
\(829\) −27518.8 −1.15291 −0.576457 0.817127i \(-0.695565\pi\)
−0.576457 + 0.817127i \(0.695565\pi\)
\(830\) 0 0
\(831\) −17078.4 −0.712929
\(832\) 0 0
\(833\) 12654.9 0.526369
\(834\) 0 0
\(835\) 29054.1 1.20414
\(836\) 0 0
\(837\) −5727.68 −0.236532
\(838\) 0 0
\(839\) −29951.5 −1.23247 −0.616234 0.787563i \(-0.711343\pi\)
−0.616234 + 0.787563i \(0.711343\pi\)
\(840\) 0 0
\(841\) −5090.88 −0.208737
\(842\) 0 0
\(843\) −6074.48 −0.248180
\(844\) 0 0
\(845\) 5590.49 0.227596
\(846\) 0 0
\(847\) 3642.38 0.147761
\(848\) 0 0
\(849\) −743.283 −0.0300464
\(850\) 0 0
\(851\) −51702.3 −2.08265
\(852\) 0 0
\(853\) −5174.61 −0.207708 −0.103854 0.994593i \(-0.533118\pi\)
−0.103854 + 0.994593i \(0.533118\pi\)
\(854\) 0 0
\(855\) −12530.3 −0.501201
\(856\) 0 0
\(857\) −9258.34 −0.369030 −0.184515 0.982830i \(-0.559071\pi\)
−0.184515 + 0.982830i \(0.559071\pi\)
\(858\) 0 0
\(859\) −24353.0 −0.967304 −0.483652 0.875260i \(-0.660690\pi\)
−0.483652 + 0.875260i \(0.660690\pi\)
\(860\) 0 0
\(861\) −3959.88 −0.156739
\(862\) 0 0
\(863\) 42283.4 1.66784 0.833919 0.551887i \(-0.186092\pi\)
0.833919 + 0.551887i \(0.186092\pi\)
\(864\) 0 0
\(865\) −43581.4 −1.71308
\(866\) 0 0
\(867\) −10433.8 −0.408709
\(868\) 0 0
\(869\) −10827.4 −0.422663
\(870\) 0 0
\(871\) 37231.0 1.44836
\(872\) 0 0
\(873\) −2108.12 −0.0817286
\(874\) 0 0
\(875\) 2802.08 0.108260
\(876\) 0 0
\(877\) 49843.1 1.91914 0.959568 0.281476i \(-0.0908239\pi\)
0.959568 + 0.281476i \(0.0908239\pi\)
\(878\) 0 0
\(879\) −24400.8 −0.936314
\(880\) 0 0
\(881\) 8986.94 0.343675 0.171837 0.985125i \(-0.445030\pi\)
0.171837 + 0.985125i \(0.445030\pi\)
\(882\) 0 0
\(883\) −3693.99 −0.140784 −0.0703922 0.997519i \(-0.522425\pi\)
−0.0703922 + 0.997519i \(0.522425\pi\)
\(884\) 0 0
\(885\) −43399.8 −1.64844
\(886\) 0 0
\(887\) −51613.0 −1.95377 −0.976886 0.213763i \(-0.931428\pi\)
−0.976886 + 0.213763i \(0.931428\pi\)
\(888\) 0 0
\(889\) 4368.95 0.164825
\(890\) 0 0
\(891\) −860.821 −0.0323665
\(892\) 0 0
\(893\) 34847.4 1.30585
\(894\) 0 0
\(895\) 6601.03 0.246534
\(896\) 0 0
\(897\) −24846.4 −0.924856
\(898\) 0 0
\(899\) 29469.5 1.09328
\(900\) 0 0
\(901\) 10544.5 0.389886
\(902\) 0 0
\(903\) −575.197 −0.0211975
\(904\) 0 0
\(905\) 53922.7 1.98061
\(906\) 0 0
\(907\) 17016.8 0.622971 0.311486 0.950251i \(-0.399173\pi\)
0.311486 + 0.950251i \(0.399173\pi\)
\(908\) 0 0
\(909\) 1849.99 0.0675032
\(910\) 0 0
\(911\) −2991.72 −0.108804 −0.0544019 0.998519i \(-0.517325\pi\)
−0.0544019 + 0.998519i \(0.517325\pi\)
\(912\) 0 0
\(913\) −4966.55 −0.180032
\(914\) 0 0
\(915\) −37885.7 −1.36881
\(916\) 0 0
\(917\) −294.344 −0.0105999
\(918\) 0 0
\(919\) −5174.82 −0.185747 −0.0928736 0.995678i \(-0.529605\pi\)
−0.0928736 + 0.995678i \(0.529605\pi\)
\(920\) 0 0
\(921\) −8924.46 −0.319295
\(922\) 0 0
\(923\) 29509.3 1.05234
\(924\) 0 0
\(925\) 48338.6 1.71823
\(926\) 0 0
\(927\) −3522.41 −0.124802
\(928\) 0 0
\(929\) 49256.5 1.73956 0.869780 0.493439i \(-0.164260\pi\)
0.869780 + 0.493439i \(0.164260\pi\)
\(930\) 0 0
\(931\) −26685.3 −0.939394
\(932\) 0 0
\(933\) 13354.6 0.468605
\(934\) 0 0
\(935\) −7016.69 −0.245423
\(936\) 0 0
\(937\) −31566.5 −1.10057 −0.550283 0.834978i \(-0.685480\pi\)
−0.550283 + 0.834978i \(0.685480\pi\)
\(938\) 0 0
\(939\) 24821.3 0.862632
\(940\) 0 0
\(941\) −37575.6 −1.30173 −0.650866 0.759193i \(-0.725594\pi\)
−0.650866 + 0.759193i \(0.725594\pi\)
\(942\) 0 0
\(943\) −84399.7 −2.91456
\(944\) 0 0
\(945\) 1407.18 0.0484397
\(946\) 0 0
\(947\) −10289.3 −0.353070 −0.176535 0.984294i \(-0.556489\pi\)
−0.176535 + 0.984294i \(0.556489\pi\)
\(948\) 0 0
\(949\) 34005.2 1.16318
\(950\) 0 0
\(951\) −1287.11 −0.0438879
\(952\) 0 0
\(953\) 36779.7 1.25017 0.625085 0.780557i \(-0.285064\pi\)
0.625085 + 0.780557i \(0.285064\pi\)
\(954\) 0 0
\(955\) −63736.9 −2.15966
\(956\) 0 0
\(957\) 4429.01 0.149602
\(958\) 0 0
\(959\) 6492.15 0.218605
\(960\) 0 0
\(961\) 15210.9 0.510586
\(962\) 0 0
\(963\) 8408.47 0.281370
\(964\) 0 0
\(965\) −48593.6 −1.62102
\(966\) 0 0
\(967\) −35228.9 −1.17155 −0.585773 0.810475i \(-0.699209\pi\)
−0.585773 + 0.810475i \(0.699209\pi\)
\(968\) 0 0
\(969\) −9078.36 −0.300969
\(970\) 0 0
\(971\) −15314.8 −0.506155 −0.253077 0.967446i \(-0.581443\pi\)
−0.253077 + 0.967446i \(0.581443\pi\)
\(972\) 0 0
\(973\) 4858.38 0.160074
\(974\) 0 0
\(975\) 23229.9 0.763027
\(976\) 0 0
\(977\) 24074.3 0.788337 0.394168 0.919038i \(-0.371033\pi\)
0.394168 + 0.919038i \(0.371033\pi\)
\(978\) 0 0
\(979\) 5427.49 0.177184
\(980\) 0 0
\(981\) −5257.10 −0.171097
\(982\) 0 0
\(983\) 15244.1 0.494620 0.247310 0.968936i \(-0.420453\pi\)
0.247310 + 0.968936i \(0.420453\pi\)
\(984\) 0 0
\(985\) −20003.7 −0.647079
\(986\) 0 0
\(987\) −3913.43 −0.126207
\(988\) 0 0
\(989\) −12259.6 −0.394168
\(990\) 0 0
\(991\) 37518.6 1.20264 0.601321 0.799008i \(-0.294641\pi\)
0.601321 + 0.799008i \(0.294641\pi\)
\(992\) 0 0
\(993\) −24590.1 −0.785845
\(994\) 0 0
\(995\) −84403.1 −2.68920
\(996\) 0 0
\(997\) 1717.01 0.0545420 0.0272710 0.999628i \(-0.491318\pi\)
0.0272710 + 0.999628i \(0.491318\pi\)
\(998\) 0 0
\(999\) 7300.90 0.231221
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.v.1.1 4
3.2 odd 2 2304.4.a.by.1.4 4
4.3 odd 2 768.4.a.u.1.1 4
8.3 odd 2 inner 768.4.a.v.1.4 4
8.5 even 2 768.4.a.u.1.4 4
12.11 even 2 2304.4.a.cb.1.4 4
16.3 odd 4 384.4.d.f.193.4 yes 8
16.5 even 4 384.4.d.f.193.1 8
16.11 odd 4 384.4.d.f.193.5 yes 8
16.13 even 4 384.4.d.f.193.8 yes 8
24.5 odd 2 2304.4.a.cb.1.1 4
24.11 even 2 2304.4.a.by.1.1 4
48.5 odd 4 1152.4.d.p.577.8 8
48.11 even 4 1152.4.d.p.577.7 8
48.29 odd 4 1152.4.d.p.577.2 8
48.35 even 4 1152.4.d.p.577.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.1 8 16.5 even 4
384.4.d.f.193.4 yes 8 16.3 odd 4
384.4.d.f.193.5 yes 8 16.11 odd 4
384.4.d.f.193.8 yes 8 16.13 even 4
768.4.a.u.1.1 4 4.3 odd 2
768.4.a.u.1.4 4 8.5 even 2
768.4.a.v.1.1 4 1.1 even 1 trivial
768.4.a.v.1.4 4 8.3 odd 2 inner
1152.4.d.p.577.1 8 48.35 even 4
1152.4.d.p.577.2 8 48.29 odd 4
1152.4.d.p.577.7 8 48.11 even 4
1152.4.d.p.577.8 8 48.5 odd 4
2304.4.a.by.1.1 4 24.11 even 2
2304.4.a.by.1.4 4 3.2 odd 2
2304.4.a.cb.1.1 4 24.5 odd 2
2304.4.a.cb.1.4 4 12.11 even 2