Properties

Label 384.4.d.f.193.8
Level $384$
Weight $4$
Character 384.193
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1534132224.8
Defining polynomial: \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.8
Root \(-0.0690906i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.4.d.f.193.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} +17.4288i q^{5} +2.99032 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +17.4288i q^{5} +2.99032 q^{7} -9.00000 q^{9} +10.6274i q^{11} +43.3156i q^{13} -52.2865 q^{15} -37.8823 q^{17} +79.8823i q^{19} +8.97095i q^{21} +191.204 q^{23} -178.765 q^{25} -27.0000i q^{27} -138.918i q^{29} -212.136 q^{31} -31.8823 q^{33} +52.1177i q^{35} -270.404i q^{37} -129.947 q^{39} -441.411 q^{41} -64.1177i q^{43} -156.860i q^{45} +436.234 q^{47} -334.058 q^{49} -113.647i q^{51} +278.348i q^{53} -185.224 q^{55} -239.647 q^{57} -830.039i q^{59} +724.580i q^{61} -26.9128 q^{63} -754.940 q^{65} +859.529i q^{67} +573.613i q^{69} -681.264 q^{71} -785.058 q^{73} -536.294i q^{75} +31.7793i q^{77} +1018.82 q^{79} +81.0000 q^{81} +467.334i q^{83} -660.244i q^{85} +416.753 q^{87} +510.706 q^{89} +129.527i q^{91} -636.409i q^{93} -1392.26 q^{95} -234.235 q^{97} -95.6468i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 17.4288i 1.55888i 0.626475 + 0.779441i \(0.284497\pi\)
−0.626475 + 0.779441i \(0.715503\pi\)
\(6\) 0 0
\(7\) 2.99032 0.161462 0.0807310 0.996736i \(-0.474275\pi\)
0.0807310 + 0.996736i \(0.474275\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 10.6274i 0.291299i 0.989336 + 0.145649i \(0.0465271\pi\)
−0.989336 + 0.145649i \(0.953473\pi\)
\(12\) 0 0
\(13\) 43.3156i 0.924121i 0.886848 + 0.462061i \(0.152890\pi\)
−0.886848 + 0.462061i \(0.847110\pi\)
\(14\) 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
\(19\) 79.8823i 0.964539i 0.876023 + 0.482270i \(0.160187\pi\)
−0.876023 + 0.482270i \(0.839813\pi\)
\(20\) 0 0
\(21\) 8.97095i 0.0932201i
\(22\) 0 0
\(23\) 191.204 1.73343 0.866714 0.498806i \(-0.166228\pi\)
0.866714 + 0.498806i \(0.166228\pi\)
\(24\) 0 0
\(25\) −178.765 −1.43012
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) − 138.918i − 0.889530i −0.895647 0.444765i \(-0.853287\pi\)
0.895647 0.444765i \(-0.146713\pi\)
\(30\) 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
\(35\) 52.1177i 0.251700i
\(36\) 0 0
\(37\) − 270.404i − 1.20146i −0.799451 0.600731i \(-0.794876\pi\)
0.799451 0.600731i \(-0.205124\pi\)
\(38\) 0 0
\(39\) −129.947 −0.533542
\(40\) 0 0
\(41\) −441.411 −1.68139 −0.840693 0.541511i \(-0.817852\pi\)
−0.840693 + 0.541511i \(0.817852\pi\)
\(42\) 0 0
\(43\) − 64.1177i − 0.227392i −0.993516 0.113696i \(-0.963731\pi\)
0.993516 0.113696i \(-0.0362690\pi\)
\(44\) 0 0
\(45\) − 156.860i − 0.519628i
\(46\) 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
\(51\) − 113.647i − 0.312034i
\(52\) 0 0
\(53\) 278.348i 0.721398i 0.932682 + 0.360699i \(0.117462\pi\)
−0.932682 + 0.360699i \(0.882538\pi\)
\(54\) 0 0
\(55\) −185.224 −0.454101
\(56\) 0 0
\(57\) −239.647 −0.556877
\(58\) 0 0
\(59\) − 830.039i − 1.83156i −0.401684 0.915778i \(-0.631575\pi\)
0.401684 0.915778i \(-0.368425\pi\)
\(60\) 0 0
\(61\) 724.580i 1.52087i 0.649416 + 0.760434i \(0.275014\pi\)
−0.649416 + 0.760434i \(0.724986\pi\)
\(62\) 0 0
\(63\) −26.9128 −0.0538206
\(64\) 0 0
\(65\) −754.940 −1.44060
\(66\) 0 0
\(67\) 859.529i 1.56729i 0.621211 + 0.783643i \(0.286641\pi\)
−0.621211 + 0.783643i \(0.713359\pi\)
\(68\) 0 0
\(69\) 573.613i 1.00079i
\(70\) 0 0
\(71\) −681.264 −1.13875 −0.569374 0.822078i \(-0.692814\pi\)
−0.569374 + 0.822078i \(0.692814\pi\)
\(72\) 0 0
\(73\) −785.058 −1.25869 −0.629343 0.777128i \(-0.716676\pi\)
−0.629343 + 0.777128i \(0.716676\pi\)
\(74\) 0 0
\(75\) − 536.294i − 0.825678i
\(76\) 0 0
\(77\) 31.7793i 0.0470337i
\(78\) 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
\(83\) 467.334i 0.618031i 0.951057 + 0.309015i \(0.0999995\pi\)
−0.951057 + 0.309015i \(0.900001\pi\)
\(84\) 0 0
\(85\) − 660.244i − 0.842512i
\(86\) 0 0
\(87\) 416.753 0.513570
\(88\) 0 0
\(89\) 510.706 0.608256 0.304128 0.952631i \(-0.401635\pi\)
0.304128 + 0.952631i \(0.401635\pi\)
\(90\) 0 0
\(91\) 129.527i 0.149210i
\(92\) 0 0
\(93\) − 636.409i − 0.709597i
\(94\) 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
\(99\) − 95.6468i − 0.0970996i
\(100\) 0 0
\(101\) − 205.555i − 0.202509i −0.994861 0.101255i \(-0.967714\pi\)
0.994861 0.101255i \(-0.0322857\pi\)
\(102\) 0 0
\(103\) 391.379 0.374405 0.187203 0.982321i \(-0.440058\pi\)
0.187203 + 0.982321i \(0.440058\pi\)
\(104\) 0 0
\(105\) −156.353 −0.145319
\(106\) 0 0
\(107\) − 934.274i − 0.844109i −0.906570 0.422055i \(-0.861309\pi\)
0.906570 0.422055i \(-0.138691\pi\)
\(108\) 0 0
\(109\) − 584.123i − 0.513292i −0.966505 0.256646i \(-0.917383\pi\)
0.966505 0.256646i \(-0.0826174\pi\)
\(110\) 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
\(115\) 3332.47i 2.70221i
\(116\) 0 0
\(117\) − 389.840i − 0.308040i
\(118\) 0 0
\(119\) −113.280 −0.0872635
\(120\) 0 0
\(121\) 1218.06 0.915145
\(122\) 0 0
\(123\) − 1324.23i − 0.970749i
\(124\) 0 0
\(125\) − 937.053i − 0.670501i
\(126\) 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
\(131\) 98.4323i 0.0656494i 0.999461 + 0.0328247i \(0.0104503\pi\)
−0.999461 + 0.0328247i \(0.989550\pi\)
\(132\) 0 0
\(133\) 238.873i 0.155736i
\(134\) 0 0
\(135\) 470.579 0.300007
\(136\) 0 0
\(137\) 2171.06 1.35391 0.676956 0.736024i \(-0.263299\pi\)
0.676956 + 0.736024i \(0.263299\pi\)
\(138\) 0 0
\(139\) 1624.70i 0.991407i 0.868492 + 0.495703i \(0.165090\pi\)
−0.868492 + 0.495703i \(0.834910\pi\)
\(140\) 0 0
\(141\) 1308.70i 0.781650i
\(142\) 0 0
\(143\) −460.333 −0.269195
\(144\) 0 0
\(145\) 2421.17 1.38667
\(146\) 0 0
\(147\) − 1002.17i − 0.562299i
\(148\) 0 0
\(149\) 636.658i 0.350047i 0.984564 + 0.175024i \(0.0560002\pi\)
−0.984564 + 0.175024i \(0.944000\pi\)
\(150\) 0 0
\(151\) 1819.34 0.980503 0.490252 0.871581i \(-0.336905\pi\)
0.490252 + 0.871581i \(0.336905\pi\)
\(152\) 0 0
\(153\) 340.940 0.180153
\(154\) 0 0
\(155\) − 3697.29i − 1.91596i
\(156\) 0 0
\(157\) 1656.50i 0.842059i 0.907047 + 0.421029i \(0.138331\pi\)
−0.907047 + 0.421029i \(0.861669\pi\)
\(158\) 0 0
\(159\) −835.045 −0.416499
\(160\) 0 0
\(161\) 571.761 0.279882
\(162\) 0 0
\(163\) 2228.82i 1.07101i 0.844532 + 0.535505i \(0.179879\pi\)
−0.844532 + 0.535505i \(0.820121\pi\)
\(164\) 0 0
\(165\) − 555.671i − 0.262175i
\(166\) 0 0
\(167\) 1667.01 0.772439 0.386219 0.922407i \(-0.373781\pi\)
0.386219 + 0.922407i \(0.373781\pi\)
\(168\) 0 0
\(169\) 320.761 0.146000
\(170\) 0 0
\(171\) − 718.940i − 0.321513i
\(172\) 0 0
\(173\) 2500.53i 1.09891i 0.835522 + 0.549457i \(0.185165\pi\)
−0.835522 + 0.549457i \(0.814835\pi\)
\(174\) 0 0
\(175\) −534.562 −0.230909
\(176\) 0 0
\(177\) 2490.12 1.05745
\(178\) 0 0
\(179\) − 378.742i − 0.158148i −0.996869 0.0790740i \(-0.974804\pi\)
0.996869 0.0790740i \(-0.0251963\pi\)
\(180\) 0 0
\(181\) 3093.88i 1.27053i 0.772294 + 0.635265i \(0.219109\pi\)
−0.772294 + 0.635265i \(0.780891\pi\)
\(182\) 0 0
\(183\) −2173.74 −0.878073
\(184\) 0 0
\(185\) 4712.82 1.87294
\(186\) 0 0
\(187\) − 402.590i − 0.157435i
\(188\) 0 0
\(189\) − 80.7385i − 0.0310734i
\(190\) 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
\(195\) − 2264.82i − 0.831729i
\(196\) 0 0
\(197\) − 1147.74i − 0.415091i −0.978225 0.207546i \(-0.933452\pi\)
0.978225 0.207546i \(-0.0665475\pi\)
\(198\) 0 0
\(199\) −4842.73 −1.72508 −0.862542 0.505986i \(-0.831129\pi\)
−0.862542 + 0.505986i \(0.831129\pi\)
\(200\) 0 0
\(201\) −2578.59 −0.904873
\(202\) 0 0
\(203\) − 415.408i − 0.143625i
\(204\) 0 0
\(205\) − 7693.29i − 2.62109i
\(206\) 0 0
\(207\) −1720.84 −0.577809
\(208\) 0 0
\(209\) −848.942 −0.280969
\(210\) 0 0
\(211\) 3222.35i 1.05135i 0.850684 + 0.525677i \(0.176188\pi\)
−0.850684 + 0.525677i \(0.823812\pi\)
\(212\) 0 0
\(213\) − 2043.79i − 0.657457i
\(214\) 0 0
\(215\) 1117.50 0.354478
\(216\) 0 0
\(217\) −634.355 −0.198446
\(218\) 0 0
\(219\) − 2355.17i − 0.726703i
\(220\) 0 0
\(221\) − 1640.89i − 0.499449i
\(222\) 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
\(227\) − 3619.49i − 1.05830i −0.848529 0.529150i \(-0.822511\pi\)
0.848529 0.529150i \(-0.177489\pi\)
\(228\) 0 0
\(229\) − 305.759i − 0.0882320i −0.999026 0.0441160i \(-0.985953\pi\)
0.999026 0.0441160i \(-0.0140471\pi\)
\(230\) 0 0
\(231\) −95.3380 −0.0271549
\(232\) 0 0
\(233\) 639.648 0.179849 0.0899244 0.995949i \(-0.471337\pi\)
0.0899244 + 0.995949i \(0.471337\pi\)
\(234\) 0 0
\(235\) 7603.05i 2.11050i
\(236\) 0 0
\(237\) 3056.45i 0.837713i
\(238\) 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
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 5822.24i − 1.51824i
\(246\) 0 0
\(247\) −3460.15 −0.891351
\(248\) 0 0
\(249\) −1402.00 −0.356820
\(250\) 0 0
\(251\) 1317.45i 0.331301i 0.986185 + 0.165650i \(0.0529723\pi\)
−0.986185 + 0.165650i \(0.947028\pi\)
\(252\) 0 0
\(253\) 2032.01i 0.504945i
\(254\) 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
\(259\) − 808.592i − 0.193990i
\(260\) 0 0
\(261\) 1250.26i 0.296510i
\(262\) 0 0
\(263\) −1655.76 −0.388206 −0.194103 0.980981i \(-0.562180\pi\)
−0.194103 + 0.980981i \(0.562180\pi\)
\(264\) 0 0
\(265\) −4851.29 −1.12458
\(266\) 0 0
\(267\) 1532.12i 0.351177i
\(268\) 0 0
\(269\) 5292.72i 1.19964i 0.800135 + 0.599820i \(0.204761\pi\)
−0.800135 + 0.599820i \(0.795239\pi\)
\(270\) 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
\(275\) − 1899.80i − 0.416591i
\(276\) 0 0
\(277\) 5692.81i 1.23483i 0.786638 + 0.617415i \(0.211820\pi\)
−0.786638 + 0.617415i \(0.788180\pi\)
\(278\) 0 0
\(279\) 1909.23 0.409686
\(280\) 0 0
\(281\) 2024.83 0.429861 0.214931 0.976629i \(-0.431047\pi\)
0.214931 + 0.976629i \(0.431047\pi\)
\(282\) 0 0
\(283\) 247.761i 0.0520419i 0.999661 + 0.0260210i \(0.00828366\pi\)
−0.999661 + 0.0260210i \(0.991716\pi\)
\(284\) 0 0
\(285\) − 4176.77i − 0.868106i
\(286\) 0 0
\(287\) −1319.96 −0.271480
\(288\) 0 0
\(289\) −3477.94 −0.707905
\(290\) 0 0
\(291\) − 702.706i − 0.141558i
\(292\) 0 0
\(293\) 8133.61i 1.62174i 0.585225 + 0.810871i \(0.301006\pi\)
−0.585225 + 0.810871i \(0.698994\pi\)
\(294\) 0 0
\(295\) 14466.6 2.85518
\(296\) 0 0
\(297\) 286.940 0.0560605
\(298\) 0 0
\(299\) 8282.12i 1.60190i
\(300\) 0 0
\(301\) − 191.732i − 0.0367152i
\(302\) 0 0
\(303\) 616.664 0.116919
\(304\) 0 0
\(305\) −12628.6 −2.37085
\(306\) 0 0
\(307\) − 2974.82i − 0.553035i −0.961009 0.276518i \(-0.910820\pi\)
0.961009 0.276518i \(-0.0891805\pi\)
\(308\) 0 0
\(309\) 1174.14i 0.216163i
\(310\) 0 0
\(311\) −4451.52 −0.811648 −0.405824 0.913951i \(-0.633015\pi\)
−0.405824 + 0.913951i \(0.633015\pi\)
\(312\) 0 0
\(313\) −8273.75 −1.49412 −0.747061 0.664755i \(-0.768536\pi\)
−0.747061 + 0.664755i \(0.768536\pi\)
\(314\) 0 0
\(315\) − 469.060i − 0.0839001i
\(316\) 0 0
\(317\) − 429.036i − 0.0760160i −0.999277 0.0380080i \(-0.987899\pi\)
0.999277 0.0380080i \(-0.0121012\pi\)
\(318\) 0 0
\(319\) 1476.34 0.259119
\(320\) 0 0
\(321\) 2802.82 0.487347
\(322\) 0 0
\(323\) − 3026.12i − 0.521293i
\(324\) 0 0
\(325\) − 7743.29i − 1.32160i
\(326\) 0 0
\(327\) 1752.37 0.296349
\(328\) 0 0
\(329\) 1304.48 0.218596
\(330\) 0 0
\(331\) 8196.71i 1.36112i 0.732691 + 0.680562i \(0.238264\pi\)
−0.732691 + 0.680562i \(0.761736\pi\)
\(332\) 0 0
\(333\) 2433.63i 0.400487i
\(334\) 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
\(339\) − 1748.12i − 0.280073i
\(340\) 0 0
\(341\) − 2254.46i − 0.358023i
\(342\) 0 0
\(343\) −2024.62 −0.318715
\(344\) 0 0
\(345\) −9997.40 −1.56012
\(346\) 0 0
\(347\) − 7707.48i − 1.19239i −0.802840 0.596195i \(-0.796679\pi\)
0.802840 0.596195i \(-0.203321\pi\)
\(348\) 0 0
\(349\) 9681.98i 1.48500i 0.669847 + 0.742499i \(0.266360\pi\)
−0.669847 + 0.742499i \(0.733640\pi\)
\(350\) 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
\(355\) − 11873.6i − 1.77518i
\(356\) 0 0
\(357\) − 339.840i − 0.0503816i
\(358\) 0 0
\(359\) 514.158 0.0755884 0.0377942 0.999286i \(-0.487967\pi\)
0.0377942 + 0.999286i \(0.487967\pi\)
\(360\) 0 0
\(361\) 477.826 0.0696641
\(362\) 0 0
\(363\) 3654.17i 0.528359i
\(364\) 0 0
\(365\) − 13682.7i − 1.96214i
\(366\) 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
\(371\) 832.350i 0.116478i
\(372\) 0 0
\(373\) − 6956.92i − 0.965726i −0.875696 0.482863i \(-0.839597\pi\)
0.875696 0.482863i \(-0.160403\pi\)
\(374\) 0 0
\(375\) 2811.16 0.387114
\(376\) 0 0
\(377\) 6017.30 0.822034
\(378\) 0 0
\(379\) − 10201.3i − 1.38260i −0.722569 0.691299i \(-0.757039\pi\)
0.722569 0.691299i \(-0.242961\pi\)
\(380\) 0 0
\(381\) − 4383.10i − 0.589378i
\(382\) 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
\(387\) 577.060i 0.0757974i
\(388\) 0 0
\(389\) − 546.451i − 0.0712240i −0.999366 0.0356120i \(-0.988662\pi\)
0.999366 0.0356120i \(-0.0113381\pi\)
\(390\) 0 0
\(391\) −7243.25 −0.936846
\(392\) 0 0
\(393\) −295.297 −0.0379027
\(394\) 0 0
\(395\) 17756.8i 2.26188i
\(396\) 0 0
\(397\) − 2084.56i − 0.263529i −0.991281 0.131764i \(-0.957936\pi\)
0.991281 0.131764i \(-0.0420642\pi\)
\(398\) 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
\(403\) − 9188.81i − 1.13580i
\(404\) 0 0
\(405\) 1411.74i 0.173209i
\(406\) 0 0
\(407\) 2873.69 0.349984
\(408\) 0 0
\(409\) −6659.89 −0.805160 −0.402580 0.915385i \(-0.631886\pi\)
−0.402580 + 0.915385i \(0.631886\pi\)
\(410\) 0 0
\(411\) 6513.17i 0.781681i
\(412\) 0 0
\(413\) − 2482.08i − 0.295727i
\(414\) 0 0
\(415\) −8145.09 −0.963438
\(416\) 0 0
\(417\) −4874.11 −0.572389
\(418\) 0 0
\(419\) − 10576.7i − 1.23318i −0.787283 0.616592i \(-0.788513\pi\)
0.787283 0.616592i \(-0.211487\pi\)
\(420\) 0 0
\(421\) − 4871.09i − 0.563901i −0.959429 0.281951i \(-0.909019\pi\)
0.959429 0.281951i \(-0.0909814\pi\)
\(422\) 0 0
\(423\) −3926.11 −0.451286
\(424\) 0 0
\(425\) 6772.00 0.772918
\(426\) 0 0
\(427\) 2166.72i 0.245562i
\(428\) 0 0
\(429\) − 1381.00i − 0.155420i
\(430\) 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
\(435\) 7263.52i 0.800596i
\(436\) 0 0
\(437\) 15273.8i 1.67196i
\(438\) 0 0
\(439\) −1856.28 −0.201812 −0.100906 0.994896i \(-0.532174\pi\)
−0.100906 + 0.994896i \(0.532174\pi\)
\(440\) 0 0
\(441\) 3006.52 0.324643
\(442\) 0 0
\(443\) − 1472.21i − 0.157893i −0.996879 0.0789465i \(-0.974844\pi\)
0.996879 0.0789465i \(-0.0251556\pi\)
\(444\) 0 0
\(445\) 8901.02i 0.948200i
\(446\) 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
\(451\) − 4691.06i − 0.489786i
\(452\) 0 0
\(453\) 5458.03i 0.566094i
\(454\) 0 0
\(455\) −2257.51 −0.232602
\(456\) 0 0
\(457\) 3613.53 0.369877 0.184938 0.982750i \(-0.440791\pi\)
0.184938 + 0.982750i \(0.440791\pi\)
\(458\) 0 0
\(459\) 1022.82i 0.104011i
\(460\) 0 0
\(461\) 17710.7i 1.78931i 0.446759 + 0.894654i \(0.352578\pi\)
−0.446759 + 0.894654i \(0.647422\pi\)
\(462\) 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
\(467\) 15208.3i 1.50697i 0.657466 + 0.753484i \(0.271628\pi\)
−0.657466 + 0.753484i \(0.728372\pi\)
\(468\) 0 0
\(469\) 2570.26i 0.253057i
\(470\) 0 0
\(471\) −4969.51 −0.486163
\(472\) 0 0
\(473\) 681.406 0.0662391
\(474\) 0 0
\(475\) − 14280.1i − 1.37940i
\(476\) 0 0
\(477\) − 2505.14i − 0.240466i
\(478\) 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
\(483\) 1715.28i 0.161590i
\(484\) 0 0
\(485\) − 4082.45i − 0.382216i
\(486\) 0 0
\(487\) −11337.7 −1.05495 −0.527474 0.849571i \(-0.676861\pi\)
−0.527474 + 0.849571i \(0.676861\pi\)
\(488\) 0 0
\(489\) −6686.46 −0.618348
\(490\) 0 0
\(491\) 14946.7i 1.37380i 0.726754 + 0.686898i \(0.241028\pi\)
−0.726754 + 0.686898i \(0.758972\pi\)
\(492\) 0 0
\(493\) 5262.51i 0.480754i
\(494\) 0 0
\(495\) 1667.01 0.151367
\(496\) 0 0
\(497\) −2037.19 −0.183865
\(498\) 0 0
\(499\) 2631.77i 0.236101i 0.993008 + 0.118051i \(0.0376645\pi\)
−0.993008 + 0.118051i \(0.962336\pi\)
\(500\) 0 0
\(501\) 5001.04i 0.445968i
\(502\) 0 0
\(503\) 6907.45 0.612302 0.306151 0.951983i \(-0.400959\pi\)
0.306151 + 0.951983i \(0.400959\pi\)
\(504\) 0 0
\(505\) 3582.58 0.315689
\(506\) 0 0
\(507\) 962.283i 0.0842929i
\(508\) 0 0
\(509\) − 12020.3i − 1.04674i −0.852107 0.523368i \(-0.824675\pi\)
0.852107 0.523368i \(-0.175325\pi\)
\(510\) 0 0
\(511\) −2347.57 −0.203230
\(512\) 0 0
\(513\) 2156.82 0.185626
\(514\) 0 0
\(515\) 6821.29i 0.583654i
\(516\) 0 0
\(517\) 4636.04i 0.394377i
\(518\) 0 0
\(519\) −7501.60 −0.634458
\(520\) 0 0
\(521\) 15846.1 1.33249 0.666247 0.745731i \(-0.267899\pi\)
0.666247 + 0.745731i \(0.267899\pi\)
\(522\) 0 0
\(523\) − 8891.64i − 0.743411i −0.928351 0.371706i \(-0.878773\pi\)
0.928351 0.371706i \(-0.121227\pi\)
\(524\) 0 0
\(525\) − 1603.69i − 0.133316i
\(526\) 0 0
\(527\) 8036.20 0.664255
\(528\) 0 0
\(529\) 24392.0 2.00477
\(530\) 0 0
\(531\) 7470.35i 0.610519i
\(532\) 0 0
\(533\) − 19120.0i − 1.55381i
\(534\) 0 0
\(535\) 16283.3 1.31587
\(536\) 0 0
\(537\) 1136.23 0.0913068
\(538\) 0 0
\(539\) − 3550.17i − 0.283705i
\(540\) 0 0
\(541\) − 12833.5i − 1.01988i −0.860210 0.509940i \(-0.829667\pi\)
0.860210 0.509940i \(-0.170333\pi\)
\(542\) 0 0
\(543\) −9281.63 −0.733541
\(544\) 0 0
\(545\) 10180.6 0.800162
\(546\) 0 0
\(547\) − 16257.0i − 1.27075i −0.772204 0.635375i \(-0.780845\pi\)
0.772204 0.635375i \(-0.219155\pi\)
\(548\) 0 0
\(549\) − 6521.22i − 0.506956i
\(550\) 0 0
\(551\) 11097.1 0.857986
\(552\) 0 0
\(553\) 3046.59 0.234275
\(554\) 0 0
\(555\) 14138.5i 1.08134i
\(556\) 0 0
\(557\) − 1558.32i − 0.118542i −0.998242 0.0592712i \(-0.981122\pi\)
0.998242 0.0592712i \(-0.0188777\pi\)
\(558\) 0 0
\(559\) 2777.30 0.210138
\(560\) 0 0
\(561\) 1207.77 0.0908951
\(562\) 0 0
\(563\) 9782.16i 0.732272i 0.930561 + 0.366136i \(0.119319\pi\)
−0.930561 + 0.366136i \(0.880681\pi\)
\(564\) 0 0
\(565\) − 10155.9i − 0.756216i
\(566\) 0 0
\(567\) 242.216 0.0179402
\(568\) 0 0
\(569\) 7887.05 0.581094 0.290547 0.956861i \(-0.406163\pi\)
0.290547 + 0.956861i \(0.406163\pi\)
\(570\) 0 0
\(571\) − 21819.3i − 1.59914i −0.600573 0.799570i \(-0.705061\pi\)
0.600573 0.799570i \(-0.294939\pi\)
\(572\) 0 0
\(573\) 10970.9i 0.799856i
\(574\) 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
\(579\) 8364.35i 0.600363i
\(580\) 0 0
\(581\) 1397.48i 0.0997884i
\(582\) 0 0
\(583\) −2958.12 −0.210142
\(584\) 0 0
\(585\) 6794.46 0.480199
\(586\) 0 0
\(587\) − 13305.6i − 0.935569i −0.883843 0.467785i \(-0.845052\pi\)
0.883843 0.467785i \(-0.154948\pi\)
\(588\) 0 0
\(589\) − 16945.9i − 1.18548i
\(590\) 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
\(595\) − 1974.34i − 0.136034i
\(596\) 0 0
\(597\) − 14528.2i − 0.995978i
\(598\) 0 0
\(599\) 2185.21 0.149058 0.0745288 0.997219i \(-0.476255\pi\)
0.0745288 + 0.997219i \(0.476255\pi\)
\(600\) 0 0
\(601\) −3542.25 −0.240418 −0.120209 0.992749i \(-0.538357\pi\)
−0.120209 + 0.992749i \(0.538357\pi\)
\(602\) 0 0
\(603\) − 7735.76i − 0.522429i
\(604\) 0 0
\(605\) 21229.3i 1.42660i
\(606\) 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
\(611\) 18895.7i 1.25113i
\(612\) 0 0
\(613\) 22514.2i 1.48343i 0.670717 + 0.741713i \(0.265986\pi\)
−0.670717 + 0.741713i \(0.734014\pi\)
\(614\) 0 0
\(615\) 23079.9 1.51328
\(616\) 0 0
\(617\) 4255.88 0.277691 0.138845 0.990314i \(-0.455661\pi\)
0.138845 + 0.990314i \(0.455661\pi\)
\(618\) 0 0
\(619\) 228.949i 0.0148663i 0.999972 + 0.00743315i \(0.00236607\pi\)
−0.999972 + 0.00743315i \(0.997634\pi\)
\(620\) 0 0
\(621\) − 5162.51i − 0.333598i
\(622\) 0 0
\(623\) 1527.17 0.0982102
\(624\) 0 0
\(625\) −6013.82 −0.384884
\(626\) 0 0
\(627\) − 2546.83i − 0.162218i
\(628\) 0 0
\(629\) 10243.5i 0.649340i
\(630\) 0 0
\(631\) 11429.2 0.721058 0.360529 0.932748i \(-0.382596\pi\)
0.360529 + 0.932748i \(0.382596\pi\)
\(632\) 0 0
\(633\) −9667.04 −0.606999
\(634\) 0 0
\(635\) − 25464.1i − 1.59136i
\(636\) 0 0
\(637\) − 14469.9i − 0.900030i
\(638\) 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
\(643\) 249.316i 0.0152909i 0.999971 + 0.00764546i \(0.00243365\pi\)
−0.999971 + 0.00764546i \(0.997566\pi\)
\(644\) 0 0
\(645\) 3352.49i 0.204658i
\(646\) 0 0
\(647\) −16025.8 −0.973785 −0.486893 0.873462i \(-0.661870\pi\)
−0.486893 + 0.873462i \(0.661870\pi\)
\(648\) 0 0
\(649\) 8821.17 0.533530
\(650\) 0 0
\(651\) − 1903.06i − 0.114573i
\(652\) 0 0
\(653\) − 14008.6i − 0.839511i −0.907637 0.419755i \(-0.862116\pi\)
0.907637 0.419755i \(-0.137884\pi\)
\(654\) 0 0
\(655\) −1715.56 −0.102340
\(656\) 0 0
\(657\) 7065.52 0.419562
\(658\) 0 0
\(659\) − 1011.91i − 0.0598152i −0.999553 0.0299076i \(-0.990479\pi\)
0.999553 0.0299076i \(-0.00952131\pi\)
\(660\) 0 0
\(661\) 23619.4i 1.38985i 0.719084 + 0.694923i \(0.244561\pi\)
−0.719084 + 0.694923i \(0.755439\pi\)
\(662\) 0 0
\(663\) 4922.67 0.288357
\(664\) 0 0
\(665\) −4163.28 −0.242775
\(666\) 0 0
\(667\) − 26561.6i − 1.54194i
\(668\) 0 0
\(669\) 14797.8i 0.855183i
\(670\) 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
\(675\) 4826.64i 0.275226i
\(676\) 0 0
\(677\) − 18255.2i − 1.03634i −0.855277 0.518172i \(-0.826613\pi\)
0.855277 0.518172i \(-0.173387\pi\)
\(678\) 0 0
\(679\) −700.438 −0.0395881
\(680\) 0 0
\(681\) 10858.5 0.611009
\(682\) 0 0
\(683\) 20090.4i 1.12553i 0.826617 + 0.562765i \(0.190262\pi\)
−0.826617 + 0.562765i \(0.809738\pi\)
\(684\) 0 0
\(685\) 37839.0i 2.11059i
\(686\) 0 0
\(687\) 917.278 0.0509408
\(688\) 0 0
\(689\) −12056.8 −0.666659
\(690\) 0 0
\(691\) 16521.5i 0.909563i 0.890603 + 0.454782i \(0.150283\pi\)
−0.890603 + 0.454782i \(0.849717\pi\)
\(692\) 0 0
\(693\) − 286.014i − 0.0156779i
\(694\) 0 0
\(695\) −28316.7 −1.54549
\(696\) 0 0
\(697\) 16721.7 0.908720
\(698\) 0 0
\(699\) 1918.95i 0.103836i
\(700\) 0 0
\(701\) 12431.4i 0.669795i 0.942255 + 0.334897i \(0.108702\pi\)
−0.942255 + 0.334897i \(0.891298\pi\)
\(702\) 0 0
\(703\) 21600.4 1.15886
\(704\) 0 0
\(705\) −22809.2 −1.21850
\(706\) 0 0
\(707\) − 614.674i − 0.0326976i
\(708\) 0 0
\(709\) 980.957i 0.0519614i 0.999662 + 0.0259807i \(0.00827084\pi\)
−0.999662 + 0.0259807i \(0.991729\pi\)
\(710\) 0 0
\(711\) −9169.36 −0.483654
\(712\) 0 0
\(713\) −40561.4 −2.13048
\(714\) 0 0
\(715\) − 8023.06i − 0.419644i
\(716\) 0 0
\(717\) 5234.81i 0.272660i
\(718\) 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
\(723\) 10071.9i 0.518088i
\(724\) 0 0
\(725\) 24833.5i 1.27213i
\(726\) 0 0
\(727\) −20850.1 −1.06367 −0.531833 0.846849i \(-0.678497\pi\)
−0.531833 + 0.846849i \(0.678497\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 2428.92i 0.122896i
\(732\) 0 0
\(733\) − 31517.2i − 1.58815i −0.607821 0.794074i \(-0.707956\pi\)
0.607821 0.794074i \(-0.292044\pi\)
\(734\) 0 0
\(735\) 17466.7 0.876558
\(736\) 0 0
\(737\) −9134.57 −0.456549
\(738\) 0 0
\(739\) 11415.0i 0.568213i 0.958793 + 0.284106i \(0.0916969\pi\)
−0.958793 + 0.284106i \(0.908303\pi\)
\(740\) 0 0
\(741\) − 10380.4i − 0.514622i
\(742\) 0 0
\(743\) −5732.08 −0.283028 −0.141514 0.989936i \(-0.545197\pi\)
−0.141514 + 0.989936i \(0.545197\pi\)
\(744\) 0 0
\(745\) −11096.2 −0.545683
\(746\) 0 0
\(747\) − 4206.01i − 0.206010i
\(748\) 0 0
\(749\) − 2793.78i − 0.136291i
\(750\) 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
\(755\) 31709.0i 1.52849i
\(756\) 0 0
\(757\) − 29125.9i − 1.39841i −0.714920 0.699206i \(-0.753537\pi\)
0.714920 0.699206i \(-0.246463\pi\)
\(758\) 0 0
\(759\) −6096.02 −0.291530
\(760\) 0 0
\(761\) 14228.5 0.677768 0.338884 0.940828i \(-0.389951\pi\)
0.338884 + 0.940828i \(0.389951\pi\)
\(762\) 0 0
\(763\) − 1746.71i − 0.0828771i
\(764\) 0 0
\(765\) 5942.19i 0.280837i
\(766\) 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
\(771\) − 9108.35i − 0.425459i
\(772\) 0 0
\(773\) − 14686.9i − 0.683377i −0.939813 0.341689i \(-0.889001\pi\)
0.939813 0.341689i \(-0.110999\pi\)
\(774\) 0 0
\(775\) 37922.5 1.75770
\(776\) 0 0
\(777\) 2425.78 0.112000
\(778\) 0 0
\(779\) − 35260.9i − 1.62176i
\(780\) 0 0
\(781\) − 7240.08i − 0.331716i
\(782\) 0 0
\(783\) −3750.78 −0.171190
\(784\) 0 0
\(785\) −28870.9 −1.31267
\(786\) 0 0
\(787\) 21001.0i 0.951214i 0.879658 + 0.475607i \(0.157772\pi\)
−0.879658 + 0.475607i \(0.842228\pi\)
\(788\) 0 0
\(789\) − 4967.27i − 0.224131i
\(790\) 0 0
\(791\) −1742.48 −0.0783253
\(792\) 0 0
\(793\) −31385.6 −1.40547
\(794\) 0 0
\(795\) − 14553.9i − 0.649274i
\(796\) 0 0
\(797\) − 13362.7i − 0.593892i −0.954894 0.296946i \(-0.904032\pi\)
0.954894 0.296946i \(-0.0959682\pi\)
\(798\) 0 0
\(799\) −16525.5 −0.731704
\(800\) 0 0
\(801\) −4596.36 −0.202752
\(802\) 0 0
\(803\) − 8343.14i − 0.366654i
\(804\) 0 0
\(805\) 9965.13i 0.436304i
\(806\) 0 0
\(807\) −15878.2 −0.692612
\(808\) 0 0
\(809\) −39481.6 −1.71582 −0.857911 0.513798i \(-0.828238\pi\)
−0.857911 + 0.513798i \(0.828238\pi\)
\(810\) 0 0
\(811\) 31157.5i 1.34906i 0.738247 + 0.674531i \(0.235654\pi\)
−0.738247 + 0.674531i \(0.764346\pi\)
\(812\) 0 0
\(813\) − 24031.6i − 1.03668i
\(814\) 0 0
\(815\) −38845.8 −1.66958
\(816\) 0 0
\(817\) 5121.87 0.219329
\(818\) 0 0
\(819\) − 1165.75i − 0.0497368i
\(820\) 0 0
\(821\) − 21229.9i − 0.902470i −0.892405 0.451235i \(-0.850984\pi\)
0.892405 0.451235i \(-0.149016\pi\)
\(822\) 0 0
\(823\) −24603.8 −1.04208 −0.521041 0.853532i \(-0.674456\pi\)
−0.521041 + 0.853532i \(0.674456\pi\)
\(824\) 0 0
\(825\) 5699.41 0.240519
\(826\) 0 0
\(827\) 13668.1i 0.574710i 0.957824 + 0.287355i \(0.0927759\pi\)
−0.957824 + 0.287355i \(0.907224\pi\)
\(828\) 0 0
\(829\) − 27518.8i − 1.15291i −0.817127 0.576457i \(-0.804435\pi\)
0.817127 0.576457i \(-0.195565\pi\)
\(830\) 0 0
\(831\) −17078.4 −0.712929
\(832\) 0 0
\(833\) 12654.9 0.526369
\(834\) 0 0
\(835\) 29054.1i 1.20414i
\(836\) 0 0
\(837\) 5727.68i 0.236532i
\(838\) 0 0
\(839\) 29951.5 1.23247 0.616234 0.787563i \(-0.288657\pi\)
0.616234 + 0.787563i \(0.288657\pi\)
\(840\) 0 0
\(841\) 5090.88 0.208737
\(842\) 0 0
\(843\) 6074.48i 0.248180i
\(844\) 0 0
\(845\) 5590.49i 0.227596i
\(846\) 0 0
\(847\) 3642.38 0.147761
\(848\) 0 0
\(849\) −743.283 −0.0300464
\(850\) 0 0
\(851\) − 51702.3i − 2.08265i
\(852\) 0 0
\(853\) 5174.61i 0.207708i 0.994593 + 0.103854i \(0.0331175\pi\)
−0.994593 + 0.103854i \(0.966882\pi\)
\(854\) 0 0
\(855\) 12530.3 0.501201
\(856\) 0 0
\(857\) 9258.34 0.369030 0.184515 0.982830i \(-0.440929\pi\)
0.184515 + 0.982830i \(0.440929\pi\)
\(858\) 0 0
\(859\) 24353.0i 0.967304i 0.875260 + 0.483652i \(0.160690\pi\)
−0.875260 + 0.483652i \(0.839310\pi\)
\(860\) 0 0
\(861\) − 3959.88i − 0.156739i
\(862\) 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
\(867\) − 10433.8i − 0.408709i
\(868\) 0 0
\(869\) 10827.4i 0.422663i
\(870\) 0 0
\(871\) −37231.0 −1.44836
\(872\) 0 0
\(873\) 2108.12 0.0817286
\(874\) 0 0
\(875\) − 2802.08i − 0.108260i
\(876\) 0 0
\(877\) 49843.1i 1.91914i 0.281476 + 0.959568i \(0.409176\pi\)
−0.281476 + 0.959568i \(0.590824\pi\)
\(878\) 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
\(883\) − 3693.99i − 0.140784i −0.997519 0.0703922i \(-0.977575\pi\)
0.997519 0.0703922i \(-0.0224251\pi\)
\(884\) 0 0
\(885\) 43399.8i 1.64844i
\(886\) 0 0
\(887\) 51613.0 1.95377 0.976886 0.213763i \(-0.0685720\pi\)
0.976886 + 0.213763i \(0.0685720\pi\)
\(888\) 0 0
\(889\) −4368.95 −0.164825
\(890\) 0 0
\(891\) 860.821i 0.0323665i
\(892\) 0 0
\(893\) 34847.4i 1.30585i
\(894\) 0 0
\(895\) 6601.03 0.246534
\(896\) 0 0
\(897\) −24846.4 −0.924856
\(898\) 0 0
\(899\) 29469.5i 1.09328i
\(900\) 0 0
\(901\) − 10544.5i − 0.389886i
\(902\) 0 0
\(903\) 575.197 0.0211975
\(904\) 0 0
\(905\) −53922.7 −1.98061
\(906\) 0 0
\(907\) − 17016.8i − 0.622971i −0.950251 0.311486i \(-0.899173\pi\)
0.950251 0.311486i \(-0.100827\pi\)
\(908\) 0 0
\(909\) 1849.99i 0.0675032i
\(910\) 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
\(915\) − 37885.7i − 1.36881i
\(916\) 0 0
\(917\) 294.344i 0.0105999i
\(918\) 0 0
\(919\) 5174.82 0.185747 0.0928736 0.995678i \(-0.470395\pi\)
0.0928736 + 0.995678i \(0.470395\pi\)
\(920\) 0 0
\(921\) 8924.46 0.319295
\(922\) 0 0
\(923\) − 29509.3i − 1.05234i
\(924\) 0 0
\(925\) 48338.6i 1.71823i
\(926\) 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
\(931\) − 26685.3i − 0.939394i
\(932\) 0 0
\(933\) − 13354.6i − 0.468605i
\(934\) 0 0
\(935\) 7016.69 0.245423
\(936\) 0 0
\(937\) 31566.5 1.10057 0.550283 0.834978i \(-0.314520\pi\)
0.550283 + 0.834978i \(0.314520\pi\)
\(938\) 0 0
\(939\) − 24821.3i − 0.862632i
\(940\) 0 0
\(941\) − 37575.6i − 1.30173i −0.759193 0.650866i \(-0.774406\pi\)
0.759193 0.650866i \(-0.225594\pi\)
\(942\) 0 0
\(943\) −84399.7 −2.91456
\(944\) 0 0
\(945\) 1407.18 0.0484397
\(946\) 0 0
\(947\) − 10289.3i − 0.353070i −0.984294 0.176535i \(-0.943511\pi\)
0.984294 0.176535i \(-0.0564889\pi\)
\(948\) 0 0
\(949\) − 34005.2i − 1.16318i
\(950\) 0 0
\(951\) 1287.11 0.0438879
\(952\) 0 0
\(953\) −36779.7 −1.25017 −0.625085 0.780557i \(-0.714936\pi\)
−0.625085 + 0.780557i \(0.714936\pi\)
\(954\) 0 0
\(955\) 63736.9i 2.15966i
\(956\) 0 0
\(957\) 4429.01i 0.149602i
\(958\) 0 0
\(959\) 6492.15 0.218605
\(960\) 0 0
\(961\) 15210.9 0.510586
\(962\) 0 0
\(963\) 8408.47i 0.281370i
\(964\) 0 0
\(965\) 48593.6i 1.62102i
\(966\) 0 0
\(967\) 35228.9 1.17155 0.585773 0.810475i \(-0.300791\pi\)
0.585773 + 0.810475i \(0.300791\pi\)
\(968\) 0 0
\(969\) 9078.36 0.300969
\(970\) 0 0
\(971\) 15314.8i 0.506155i 0.967446 + 0.253077i \(0.0814428\pi\)
−0.967446 + 0.253077i \(0.918557\pi\)
\(972\) 0 0
\(973\) 4858.38i 0.160074i
\(974\) 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
\(979\) 5427.49i 0.177184i
\(980\) 0 0
\(981\) 5257.10i 0.171097i
\(982\) 0 0
\(983\) −15244.1 −0.494620 −0.247310 0.968936i \(-0.579547\pi\)
−0.247310 + 0.968936i \(0.579547\pi\)
\(984\) 0 0
\(985\) 20003.7 0.647079
\(986\) 0 0
\(987\) 3913.43i 0.126207i
\(988\) 0 0
\(989\) − 12259.6i − 0.394168i
\(990\) 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
\(995\) − 84403.1i − 2.68920i
\(996\) 0 0
\(997\) − 1717.01i − 0.0545420i −0.999628 0.0272710i \(-0.991318\pi\)
0.999628 0.0272710i \(-0.00868170\pi\)
\(998\) 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 384.4.d.f.193.8 yes 8
3.2 odd 2 1152.4.d.p.577.2 8
4.3 odd 2 inner 384.4.d.f.193.4 yes 8
8.3 odd 2 inner 384.4.d.f.193.5 yes 8
8.5 even 2 inner 384.4.d.f.193.1 8
12.11 even 2 1152.4.d.p.577.1 8
16.3 odd 4 768.4.a.v.1.4 4
16.5 even 4 768.4.a.v.1.1 4
16.11 odd 4 768.4.a.u.1.1 4
16.13 even 4 768.4.a.u.1.4 4
24.5 odd 2 1152.4.d.p.577.8 8
24.11 even 2 1152.4.d.p.577.7 8
48.5 odd 4 2304.4.a.by.1.4 4
48.11 even 4 2304.4.a.cb.1.4 4
48.29 odd 4 2304.4.a.cb.1.1 4
48.35 even 4 2304.4.a.by.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.1 8 8.5 even 2 inner
384.4.d.f.193.4 yes 8 4.3 odd 2 inner
384.4.d.f.193.5 yes 8 8.3 odd 2 inner
384.4.d.f.193.8 yes 8 1.1 even 1 trivial
768.4.a.u.1.1 4 16.11 odd 4
768.4.a.u.1.4 4 16.13 even 4
768.4.a.v.1.1 4 16.5 even 4
768.4.a.v.1.4 4 16.3 odd 4
1152.4.d.p.577.1 8 12.11 even 2
1152.4.d.p.577.2 8 3.2 odd 2
1152.4.d.p.577.7 8 24.11 even 2
1152.4.d.p.577.8 8 24.5 odd 2
2304.4.a.by.1.1 4 48.35 even 4
2304.4.a.by.1.4 4 48.5 odd 4
2304.4.a.cb.1.1 4 48.29 odd 4
2304.4.a.cb.1.4 4 48.11 even 4